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+arXiv:1001.0001v1  [cs.IT]  30 Dec 2009On the structure of non-full-rank perfect codes
+Olof Heden and Denis S. Krotov∗
+Abstract
+The Krotov combining construction of perfect 1-error-corr ecting binary codes
+from 2000 and a theorem of Heden saying that every non-full-r ank perfect 1-error-
+correcting binary code can be constructed by this combining construction is gener-
+alized to the q-ary case. Simply, every non-full-rank perfect code Cis the union of
+a well-defined family of ¯ µ-components K¯µ, where ¯µbelongs to an “outer” perfect
+codeC⋆, and these components are at distance three from each other. Compo-
+nents from distinct codes can thus freely be combined to obta in new perfect codes.
+The Phelps general product construction of perfect binary c ode from 1984 is gen-
+eralized to obtain ¯ µ-components, and new lower bounds on the number of perfect
+1-error-correcting q-ary codes are presented.
+1. Introduction
+LetFqdenote the finite field with qelements. A perfect1-error-correcting q-ary code of
+lengthn, for short here a perfect code , is a subset Cof the direct product Fn
+q, ofncopies of
+Fq, having the property that any element of Fn
+qdiffers in at most one coordinate position
+from a unique element of C.
+The family of all perfect codes is far from classified or enumerated. We will in this
+short note say something about the structure of these codes. W e need the concept of
+rank.
+We consider Fn
+qas a vector space of dimension nover the finite field Fq. Therank
+of aq-ary codeC, here denoted rank( C), is the dimension of the linear span < C >of
+the elements of C. Trivial, and well known, counting arguments give that if there exist s
+a perfect code in Fn
+qthenn= (qm−1)/(q−1), for some integer m, and|C|=qn−m. So,
+for every perfect code C,
+n−m≤rank(C)≤n.
+If rank(C) =nwe will say that Chasfull rank.
+∗This research collaboration was partially supported by a grant from Swedish Institute; the work of
+the second author was partially supported by the Federal Target Program “Scientific and Educational
+PersonnelofInnovation Russia”for 2009-2013(governmentco ntract No. 02.740.11.0429)and the Russian
+Foundation for Basic Research (grant 08-01-00673).
+1We will show thatevery non-full-rankperfect code isa unionofso ca lled ¯µ-components
+K¯µ, and that these components may be enumerated by some other pe rfect codeC⋆, i.e,
+¯µ∈C⋆. Further, the distance between any two such components will be a t least three.
+This implies that we will be completely free to combine ¯ µ-components from different
+perfect codesofsamelength, toobtainotherperfect codes. Ge neralizing aconstruction by
+Phelps of perfect 1-error correcting binary codes [8], we will obtain further ¯µ-components.
+As an application of our results we will be able to slightly improve the lowe r bound on
+the number of perfect codes given in [6].
+Our results generalize corresponding results for the binary case. In [3] it was shown
+that a binary perfect code can be constructed as the union of diffe rent subcodes (¯ µ-
+components) satisfying some generalized parity-check property , each of them being con-
+structed independently or taken from another perfect code. In [2] it was shown that every
+non-full-rank perfect binary code can be obtained by this combining construction.
+2. Every non-full-rank perfect code is the union of ¯µ-
+components
+We start with some notation. Assume we have positive integers n,t,n1, ...,ntsuch that
+n1+...+nt≤n. Anyq-aryword ¯xwill berepresented intheblockform ¯ x= (¯x1|¯x2|...|
+¯xt|¯x0) = (¯x∗|¯x0), where ¯xi= (xi1,xi2,...,x ini),i= 0,1,...,t,n0=n−n1−...−nt,
+¯x∗= (¯x1|¯x2|...|¯xt). For every block ¯ xi,i= 1,2,...,t, we define σi(¯xi) by
+σi(¯xi) =ni/summationdisplay
+j=1xij,
+and, for ¯x,
+¯σ(¯x) = ¯σ(¯x∗) = (σ1(¯x1),σ2(¯x2),...,σ t(¯xt))
+Recall that the Hamming distance d(¯x,¯y) between two words ¯ x, ¯yof the same length
+means the number of positions in which they differ.
+Amonomial transformation is a map of the space Fn
+qthat can be composed by a
+permutation of the set of coordinate positions and the multiplication in each coordinate
+position with some non-zero element of the finite field Fq.
+Aq-ary codeCislinearifCis a subspace of Fn
+q. A linear perfect code is called a
+Hamming code .
+Theorem 1. LetCbe any non-full-rank perfect code Cof lengthn= (qm−1)/(q−1).
+To any integer r<m, satisfying
+1≤r≤n−rank(C),
+there is aq-ary Hamming code C⋆of lengtht= (qr−1)/(q−1), such that for some
+monomial transformation ψ
+ψ(C) =/uniondisplay
+¯µ∈C⋆K¯µ,
+2where
+K¯µ={(¯x1|¯x2|...|¯xt|¯x0) : ¯σ(¯x) = ¯µ,¯x1,¯x2,...,¯xt∈Fqs
+q,¯x0∈C¯µ(¯x∗)}(1)
+for some family of perfect codes C¯µ(¯x), of length 1+q+q2+...+qs−1, wheres=m−r,
+and satisfying, for each ¯µ∈C⋆,
+d(¯x∗,¯x′
+∗)≤2 =⇒C¯µ(¯x∗)∩C¯µ(¯x′
+∗) =∅. (2)
+The codeC⋆will be called an outercode toψ(C). The subcodes K¯µwill be called
+¯µ-components ofψ(C). As the minimum distance of Cis three, the distance between any
+two distinct ¯ µ-components will be at least three.
+Proof. LetDbe any subspace of Fn
+qcontaining<C >, and of dimension n−r. By
+using a monomial transformation ψof space we may achieve that the dual space of ψ(D)
+is the nullspace of a r×n-matrix
+H=
+| | | | | | | |
+¯α11···¯α1n1¯α21···¯α2n2···¯αt1···¯αtnt¯0···¯0
+| | | | | | | |
+
+where ¯αij= ¯αi, fori= 1,2,...,t, the first non-zero coordinate in each vector ¯ αiequals
+1, ¯αi/ne}ationslash= ¯αi′, fori/ne}ationslash=i′, and where the columns of Hare in lexicographic order, according
+to some given ordering of Fq.
+To avoid too much notation we assume that Cwas such that ψ= id.
+LetC⋆be the null space of the matrix
+H⋆=
+| | |
+¯α1¯α2···¯αt
+| | |
+
+Define, for ¯ µ∈C⋆,
+K¯µ={(¯x1|¯x2|...|¯xt|¯x0)∈C: (σ1(¯x1),σ2(¯x2),...,σ(¯xt)) = ¯µ}.
+Then,
+C=/uniondisplay
+¯µ∈C⋆K¯µ.
+Further, since any two columns of H⋆are linearly independent, for any two distinct words
+¯µand ¯µ′ofC⋆
+d(K¯µ,K¯µ′)≥3. (3)
+We will show that K¯µhas the properties given in Equation (1).
+Any word ¯x= (¯x1|¯x2|...|¯xt|¯x0) must be at distance at most one from a word of
+C, and hence, the word ( σ1(¯x1),σ2(¯x2),...,σ t(¯xt)) is at distance at most one from some
+word ofC⋆. It follows that C⋆is a perfect code, and as a consequence, as C⋆is linear, it
+is a Hamming code with parity-check matrix H⋆. As the number of rows of H⋆isr, we
+then get that the number tof columns of H⋆is equal to
+t=qr−1
+q−1= 1+q+q2+...+qr−1.
+3For any word ¯ x∗ofFn1+n2+...+ntq with ¯σ(¯x∗) = ¯µ∈C⋆, we now define the code C¯µ(¯x∗)
+of lengthn0by
+C¯µ(¯x∗) ={¯c∈Fn0
+q: (¯x∗|¯c)∈C}.
+Again, using the fact that Cis a perfect code, we may deduce that for any ¯ x∗such
+that the set C¯µ(¯x∗) is non empty, the set C¯µ(¯x∗) must be a perfect code of length n0=
+(qs−1)/(q−1), for some integer s.
+From the fact that the minimum distance of Cequals three, we get the property in
+Equation (2).
+Let ¯eidenote a word of weight one with the entry 1 in the coordinate positio ni. It
+then follows that the two perfect codes C¯µ(¯x∗) andC¯µ(¯x∗+ ¯e1−¯ei), fori= 2,3,...,n 1,
+must be mutually disjoint. Hence, n1is at most equal to the number of perfect codes in
+a partition of Fn0qinto perfect codes, i.e.,
+n1≤(q−1)n0+1 =qs.
+Similarly,ni≤qs, fori= 2,3,...,t.
+Reversing these arguments, using Equation (3) and the fact that Cis a perfect code,
+we find that ni, for eachi= 1,2,...,t, is at least equal to the number of words in an
+1-ball ofFn0q.
+We conclude that ni=qs, fori= 1,2,...,t, and finally
+n=qs(1+q+q2+...+qr−1)+1+q+q2+...+qs−1= 1+q+q2+...+qr+s−1.
+Givenr, we can then find sfrom the equality
+n= 1+q+q2+...+qm−1.
+△
+3. Combining construction of perfect codes
+In the previous section, it was shown that a perfect code, depend ing on its rank, can
+be divided onto small or large number of so-called ¯ µ-components, which satisfy some
+equation with ¯ σ. The construction described in the following theorem realizes the ide a
+of combining independent ¯ µ-components, differently constructed or taken from different
+perfect codes, in one perfect code.
+A functionf: Σn→Σ, where Σ is some set, is called an n-ary(ormultary)quasigroup
+of order |Σ|if in the equality z0=f(z1,...,z n) knowledge of any nelements of z0,z1,
+...,znuniquely specifies the remaining one.
+Theorem 2. Letmandrbe integers, m>r,qbe a prime power, n= (qm−1)/(q−1)
+andt= (qr−1)/(q−1). Assume that C∗is a perfect code in Ft
+qand for every ¯µ∈C∗
+we have a distance- 3codeK¯µ⊂Fn
+qof cardinality qn−m−(t−r)that satisfies the following
+generalized parity-check law:
+¯σ(¯x) = (σ1(x1,...,x l),...,σ t(xlt−l+1,...,x lt)) = ¯µ
+4for every ¯x= (x1,...,x n)∈K¯µ, wherel=qm−rand¯σ= (σ1,...,σ t)is a collections of
+l-ary quasigroups of order q. Then the union
+C=/uniondisplay
+¯µ∈C∗K¯µ
+is a perfect code in Fn
+q.
+Proof. It is easy to check that Chas the cardinality of a perfect code. The distance
+at least 3 between different words ¯ x, ¯yfromCfollows from the code distances of K¯µ(if
+¯x, ¯ybelong to the same K¯µ) andC∗(if ¯x, ¯ybelong to different K¯µ′,K¯µ′′, ¯µ′,¯µ′′∈C∗).△
+The ¯µ-components K¯µcanbeconstructedindependentlyortakenfromdifferentperfec t
+codes. In the important case when all σiare linear quasigroups (e.g., σi(y1,...,y l) =
+y1+...+yl) the components can be taken from any perfect code of rank at m ostn−r, as
+followsfromtheprevioussection(itshouldbenotedthatif ¯ σislinear, thena ¯ µ-component
+can be obtained from any ¯ µ′-component by adding a vector ¯ zsuch that ¯σ(¯z) = ¯µ−¯µ′).
+In general, the existence of ¯ µ-components that satisfy the generalized parity-check law
+for arbitrary ¯ σis questionable. But for some class of ¯ σsuch components exist, as we will
+see from the following two subsections.
+Remark. It is worth mentioning that ¯ µ-components can exist for arbitrary length tof
+¯µ(for example, in the next two subsections there are no restriction s ont), if we do not
+require the possibility to combine them into a perfect code. This is esp ecially important
+for the study of perfect codes of small ranks (close to the rank o f a linear perfect code):
+once we realize that the code is the union of ¯ µ-components of some special form, we may
+forget about the code length and consider ¯ µ-components for arbitrary length of ¯ µ, which
+allows to use recursive approaches.
+3.1. Mollard-Phelps construction
+Here we describe the way to construct ¯ µ-components derived from the product construc-
+tion discovered independently in [7] and [9]. In terms of ¯ µ-components, the construction
+in [9] is more general; it allows substitution of arbitrary multary quasig roups, and we will
+use this possibility in Section 4.
+Lemma 1. Let¯µ∈Ft
+qand letC#be a perfect code in Fk
+q. Letvandhbe(q−1)-ary
+quasigroups of order qsuch that the code {(¯y|v(¯y)|h(¯y)) : ¯y∈Fq−1
+q}is perfect. Let
+V1, ...,VtandH1, ...,Hkbe respectively (k+1)-ary and (t+1)-ary quasigroups of order
+q. Then the set
+K¯µ=/braceleftBig
+(¯x11|...|¯x1k|y1|¯x21|...|¯x2k|y2|...|¯xt1|...|¯xtk|yt|z1|z2|...|zk) :
+¯xij∈Fq−1
+q,
+(V1(v(¯x11),...,v(¯x1k),y1),...,V t(v(¯xt1),...,v(¯xtk),yt)) = ¯µ,
+(H1(h(¯x11),...,h(¯xt1),z1),...,H k(h(¯x1k),...,h(¯xtk),zk))∈C#/bracerightBig
+is a¯µ-component that satisfies the generalized parity-check law with
+σi(·,...,·,·) =Vi(v(·),...,v(·),·).
+5(The elements of F(q−1)kt+k+t
+q in this construction may be thought of as three-dimensional
+arrays where the elements of ¯xijare z-lined, every underlined block is y-lined, and the
+tuple of blocks is x-lined. Naturally, the multary quasigroups Vimay be named “vertical”
+andHi, “horizontal”.)
+The proof of the code distance is similar to that in [9], and the other pr operties of a
+¯µ-component are straightforward. The existence of admissible ( q−1)-ary quasigroups v
+andhis the only restriction on the q(this concerns the next subsection as well). If Fqis
+a finite field, there are linear examples: v(y1,...,y q−1) =y1+...+yq−1,v(y1,...,y q−1) =
+α1y1+...+αq−1yq−1whereα1, ...,αq−1are all the non-zero elements of Fq. Ifqis not
+a prime power, the existence of a q-ary perfect code of length q+1 is an open problem
+(with the only exception q= 6, when the nonexistence follows from the nonexistence of
+two orthogonal 6 ×6 Latin squares [1, Th.6]).
+3.2. Generalized Phelps construction
+Here we describe another way to construct ¯ µ-components, which generalizes the construc-
+tion of binary perfect codes from [8].
+Lemma 2. Let¯µ∈Ft
+q. Let for every ifrom1tot+1the codesCi,j,j= 0,1,...,qk−k
+form a partition of Fk
+qinto perfect codes and γi:Fk
+q→ {0,1,...,qk−k}be the corre-
+sponding partition function:
+γi(¯y) =j⇐⇒¯y∈Ci,j.
+Letvandhbe(q−1)-ary quasigroups of order qsuch that the code {(¯y|v(¯y)|h(¯y)) :
+¯y∈Fq−1
+q}is perfect. Let V1, ...,Vtbe(k+ 1)-ary quasigroups of order qandQbe a
+t-ary quasigroup of order qk−k+1.
+K¯µ=/braceleftBig
+(¯x11|...|¯x1k|y1|¯x21|...|¯x2k|y2|...|¯xt1|...|¯xtk|yt|z1|z2|...|zk) :
+¯xij∈Fq−1
+q,
+(V1(v(¯x11),...,v(¯x1k),y1),...,V t(v(¯xt1),...,v(¯xtk),yt)) = ¯µ,
+Q(γ1(h(¯x11),...,h(¯x1k)),...,γ t(h(¯xt1),...,h(¯xtk))) =γt+1(z1,...,zk)/bracerightBig
+is a¯µ-component that satisfies the generalized parity-check law with
+σi(·,...,·,·) =Vi(v(·),...,v(·),·).
+The proof consists of trivial verifications.
+4. On the number of perfect codes
+In this section we discuss some observations, which result in the bes t known lower bound
+on the number of q-ary perfect codes, q≥3. The basic facts are already contained in
+other known results: lower bounds on the number of multary quasig roups of order q, the
+6construction [9] of perfect codes from multary quasigroups of or derq, and the possibility
+to choose the quasigroup independently for every vector of the o uter code (this possibility
+was not explicitly mentioned in [9], but used in the previous paper [8]).
+A general lower bound, in terms of the number of multary quasigrou ps, is given by
+Lemma 3. In combination with Lemma 4, it gives explicit numbers.
+Lemma 3. The number of q-ary perfect codes of length nis not less than
+Q/parenleftBiggn−1
+q,q/parenrightBiggRn−1
+q
+whereQ(m,q)is the number of m-ary quasigroups of order qand whereRn′=qn′/(n′q−
+q+1)is the cardinality of a perfect code of length n′.
+Proof. Constructing a perfect code like in Theorem 2 with t=n−1
+q, we combine
+Rn−1
+qdifferent ¯µ-components.
+Constructing every such a component as in Lemma 2, k= 1,t=n−1
+q, we are free
+to choose the t-ary quasigroup Qof orderqinQ(t,q) ways. Clearly, different t-ary
+quasigroups give different components. (Equivalently, we can use L emma 1 and choose
+the (t+1)-ary quasigroup H1, but should note that the value of H1in the construction is
+always fixed when k= 1, because C#consists of only one vertex; so we again have Q(t,q)
+different choices, not Q(t+1,q)). △
+Lemma 4. The number Q(m,q)ofm-ary quasigroups of order qsatisfies:
+(a) [5]Q(m,3) = 3·2m;
+(b) [11]Q(m,4) = 3m+1·22m+1(1+o(1));
+(c) [4]Q(m,5)≥23n/3−0.072;
+(d) [10]Q(m,q)≥2((q2−4q+3)/4)n/2for oddq(the previous bound [4]wasQ(m,q)≥
+2⌊q/3⌋n);
+(e) [4]Q(m,q1q2)≥Q(m,q1)·Q(m,q2)qm
+1.
+For oddq≥5, the number of codes given by Lemmas 3 and 4(c,d) improves the
+constantcin the lower estimation of form eecn(1+o(1))for the number of perfect codes, in
+comparison with the last known lower bound [6]. Informally, this can be explained in the
+following way: the construction in [6] can be described in terms of mu tually independent
+small modifications of the linear multary quasigroup of order q, while the lower bounds
+in Lemma 4(c,d) are based on a specially-constructed nonlinear multa ry quasigroup that
+allows a lager number of independent modifications. For q= 3 andq= 2s, the number
+of codes given by Lemmas 3 and 4(a,b,e) also slightly improves the boun d in [6], but do
+not affect on the constant c.
+7References
+1. S. W. Golomb and E. C. Posner. Rook domains, latin squares, and e rror-distributing
+codes.IEEE Trans. Inf. Theory , 10(3):196–208, 1964.
+2. O. Heden. On the classification of perfect binary 1-error corre cting codes. Preprint
+TRITA-MAT-2002-01, KTH, Stockholm, 2002.
+3. D. S. Krotov. Combining construction of perfect binary codes. Probl. Inf. Transm. ,
+36(4):349–353, 2000. translated from Probl. Peredachi Inf. 36 (4) (2000), 74-79.
+4. D. S. Krotov, V. N. Potapov, and P. V. Sokolova. On reconstru cting reducible n-ary
+quasigroups and switching subquasigroups. Quasigroups Relat. Syst. , 16(1):55–67,
+2008. ArXiv:math/0608269
+5. C. F. Laywine and G. L. Mullen. Discrete Mathematics Using Latin Squares . Wiley,
+New York, 1998.
+6. A. V. Los’. Construction of perfect q-ary codes by switchings o f simple components.
+Probl. Inf. Transm. , 42(1):30–37, 2006. DOI: 10.1134/S0032946006010030 transla ted
+from Probl. Peredachi Inf. 42(1) (2006), 34-42.
+7. M. Mollard. A generalized parity function and its use in the constru ction of perfect
+codes.SIAM J. Algebraic Discrete Methods , 7(1):113–115, 1986.
+8. K. T. Phelps. A general product construction for error corre cting codes. SIAM J.
+Algebraic Discrete Methods , 5(2):224–228, 1984.
+9. K. T. Phelps. A product construction for perfect codes over a rbitrary alphabets.
+IEEE Trans. Inf. Theory , 30(5):769–771, 1984.
+10. V. N. Potapov and D. S. Krotov. On the number of n-ary quasigroups of finite order.
+Submitted. ArXiv:0912.5453
+11. V.N.PotapovandD.S.Krotov. Asymptoticsforthenumbero fn-quasigroupsoforder
+4.Sib. Math. J. , 47(4):720–731, 2006. DOI: 10.1007/s11202-006-0083-9 tran slated
+from Sib. Mat. Zh. 47(4) (2006), 873-887. ArXiv:math/0605104
+O. Heden
+Department of Mathematics, KTH
+S-100 44 Stockholm, Sweden
+email:olohed@math.kth.se
+D. Krotov
+Sobolev Institute of Mathematics
+and
+Mechanics and Mathematics Department, Novosibirsk State Univer sity
+Novosibirsk, Russia
+email:krotov@math.nsc.ru
+8
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+arXiv:1001.0002v2  [hep-th]  9 Mar 2010Gravity duals for logarithmic conformal field theories
+Daniel Grumiller and Niklas Johansson
+Institute for Theoretical Physics, Vienna University of Te chnology
+Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria
+E-mail:grumil@hep.itp.tuwien.ac.at, niklasj@hep.itp.tuwien. ac.at
+Abstract. Logarithmic conformal fieldtheories with vanishingcentra l charge describe systems
+withquencheddisorder, percolation ordiluteself-avoidi ngpolymers. Inthesetheories theenergy
+momentum tensor acquires a logarithmic partner. In this tal k we address the construction of
+possible gravity duals for these logarithmic conformal fiel d theories and present two viable
+candidates for such duals, namely theories of massive gravi ty in three dimensions at a chiral
+point.
+Outline
+Thistalk isorganized asfollows. Insection 1werecall sali ent featuresof2-dimensionalconformal
+field theories. In section 2 we review a specific class of logar ithmic conformal field theories where
+the energy momentum tensor acquires a logarithmic partner. In section 3 we present a wish-list
+for gravity duals to logarithmic conformal field theories. I n section 4 we discuss two examples
+of massive gravity theories that comply with all the items on that list. In section 5 we address
+possible applications of an Anti-deSitter/logarithmic co nformal field theory correspondence in
+condensed matter physics.
+1. Conformal field theory distillate
+Conformal field theories (CFTs) are quantum field theories th at exhibit invariance under angle
+preserving transformations: translations, rotations, bo osts, dilatations and special conformal
+transformations. In two dimensions the conformal algebra i s infinite dimensional, and thus
+two-dimensional CFTs exhibit a particularly rich structur e. They arise in various contexts in
+physics, including string theory, statistical mechanics a nd condensed matter physics, see e.g. [1].
+The main observables in any field theory are correlation func tions between gauge invariant
+operators. There exist powerful tools to calculate these co rrelators in a CFT. The operator
+content of various CFTs may differ, but all CFTs contain at leas t an energy momentum tensor
+Tµν. Conformal invariance requires the energy momentum tensor to be traceless, Tµ
+µ= 0,
+in addition to its conservation, ∂µTµν= 0. In lightcone gauge for the Minkowski metric,
+ds2= 2dzd¯z, these equations take a particularly simple form: Tz¯z= 0,Tzz=Tzz(z) :=OL(z)
+andT¯z¯z=T¯z¯z(¯z) :=OR(¯z). Conformal Ward identities determine essentially unique ly the form
+of 2- and3-point correlators between thefluxcomponents OL/Rof theenergy momentum tensor:∝an}b∇acketle{tOR(¯z)OR(0)∝an}b∇acket∇i}ht=cR
+2¯z4(1a)
+∝an}b∇acketle{tOL(z)OL(0)∝an}b∇acket∇i}ht=cL
+2z4(1b)
+∝an}b∇acketle{tOL(z)OR(0)∝an}b∇acket∇i}ht= 0 (1c)
+∝an}b∇acketle{tOR(¯z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht=cR
+¯z2¯z′2(¯z−¯z′)2(1d)
+∝an}b∇acketle{tOL(z)OL(z′)OL(0)∝an}b∇acket∇i}ht=cL
+z2z′2(z−z′)2(1e)
+∝an}b∇acketle{tOL(z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht= 0 (1f)
+∝an}b∇acketle{tOL(z)OL(z′)OR(0)∝an}b∇acket∇i}ht= 0 (1g)
+The real numbers cL,cRare the left and right central charges, which determine key p roperties of
+the CFT. We have omitted terms that are less divergent in the n ear coincidence limit z,¯z→0 as
+well as contact terms, i.e., contributions that are localiz ed (δ-functions and derivatives thereof).
+If someone provides us with a traceless energy momentum tens or and gives us a prescription
+how to calculate correlators,1but does not reveal whether the underlying field theory is a CF T,
+thenwecanperformthefollowing check. Wecalculate all 2- a nd3-point correlators of theenergy
+momentum tensor with itself, and if at least one of the correl ators does not match precisely with
+the corresponding correlator in (1) then we know that the fiel d theory in question cannot be a
+CFT. On the other hand, if all the correlators match with corr esponding ones in (1) we have
+non-trivial evidence that the field theory in question might be a CFT. Let us keep this stringent
+check in mind for later purposes, but switch gears now and con sider a specific class of CFTs,
+namely logarithmic CFTs (LCFTs).
+2. Logarithmic CFTs with an energetic partner
+LCFTs were introduced in physics by Gurarie [2]. We focus now on some properties of LCFTs
+and postpone a physics discussion until the end of the talk, s ee [3,4] for reviews. There are two
+conceptually different, but mathematically equivalent, way s to define LCFTs. In both versions
+there exists at least one operator that acquires a logarithm ic partner, which we denote by Olog.
+We focus in this talk exclusively on theories where one (or bo th) of the energy momentum
+tensor flux components is the operator acquiring such a partn er, for instance OL. We discuss
+now briefly both ways of defining LCFTs.
+According to the first definition “acquiring a logarithmic pa rtner” means that the
+Hamiltonian Hcannot be diagonalized. For example
+H/parenleftbigg
+Olog
+OL/parenrightbigg
+=/parenleftbigg
+2 1
+0 2/parenrightbigg/parenleftbigg
+Olog
+OL/parenrightbigg
+(2)
+Theangularmomentum operator Jmay ormay not bediagonalizable. Weconsider onlytheories
+whereJis diagonalizable:
+J/parenleftbigg
+Olog
+OL/parenrightbigg
+=/parenleftbigg
+2 0
+0 2/parenrightbigg/parenleftbigg
+Olog
+OL/parenrightbigg
+(3)
+The eigenvalues 2 arise because the energy momentum tensor a nd its logarithmic partner both
+correspond to spin-2 excitations.
+1This is exactly what the AdS/CFT correspondence does: given a gravity dual we can calculate the energy
+momentum tensor and correlators.The second definition makes it more transparent why these CFT s are called “logarithmic”
+in the first place. Suppose that in addition to OL/Rwe have an operator OMwith conformal
+weightsh= 2+ε,¯h=ε, meaning that its 2-point correlator with itself is given by
+∝an}b∇acketle{tOM(z,¯z)OM(0,0)∝an}b∇acket∇i}ht=ˆB
+z4+2ε¯z2ε(4)
+The correlator of OMwithOLvanishes since the latter has conformal weights h= 2,¯h= 0, and
+operators whose conformal weights do not match lead to vanis hing correlators. Suppose now
+that we send the central charge cLand the parameter εto zero, and simultaneously send ˆBto
+infinity, such that the following limits exist:
+bL:= lim
+cL→0−cL
+ε∝ne}ationslash= 0B:= lim
+cL→0/parenleftbigˆB+2
+cL/parenrightbig
+(5)
+Then we can define a new operator Ologthat linearly combines OL/M.
+Olog=bLOL
+cL+bL
+2OM(6)
+Taking the limit cL→0 leads to the following 2-point correlators:
+∝an}b∇acketle{tOL(z)OL(0,0)∝an}b∇acket∇i}ht= 0 (7a)
+∝an}b∇acketle{tOL(z)Olog(0,0)∝an}b∇acket∇i}ht=bL
+2z4(7b)
+∝an}b∇acketle{tOlog(z,¯z)Olog(0,0)∝an}b∇acket∇i}ht=−bLln(m2
+L|z|2)
+z4(7c)
+These 2-point correlators exhibit several remarkable feat ures. The flux component OLof the
+energy momentum tensor becomes a zero norm state (7a). Never theless, the theory does not
+become chiral, because the left-moving sector is not trivia l:OLhas a non-vanishing correlator
+(7b) with its logarithmic partner Olog. The 2-point correlator (7c) between two logarithmic
+operators Ologmakes it clear why such CFTs have the attribute “logarithmic ”. The constant
+bL, sometimes called “new anomaly”, defines crucial propertie s of the LCFT, much like the
+central charges do in ordinary CFTs. The mass scale mLappearing in the last correlator above
+has no significance, and is determined by the value of Bin (5). It can be changed to any finite
+value by the redefinition Olog→ Olog+γOLwith some finite γ. We setmL= 1 for convenience.
+Conformal Ward identities determine again essentially uni quely the form of 2- and 3-point
+correlators in a LCFT. For the specific case where the energy m omentum tensor acquires a
+logarithmic partner the 3-point correlators were calculat ed in [5]. The non-vanishing ones are
+given by
+∝an}b∇acketle{tOL(z,¯z)OL(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=bL
+z2z′2(z−z′)2(8a)
+∝an}b∇acketle{tOL(z,¯z)Olog(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=−2bLln|z′|2+bL
+2
+z2z′2(z−z′)2(8b)
+∝an}b∇acketle{tOlog(z,¯z)Olog(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=lengthy
+z2z′2(z−z′)2(8c)
+If alsoORacquires a logarithmic partner O/tildewiderlogthen the construction above can be repeated,
+changing everywhere L→R,z→¯zetc. In that case we have a LCFT with cL=cR= 0 andbL,bR∝ne}ationslash= 0. Alternatively, it may happen that only OLhas a logarithmic partner Olog. In that
+case we have a LCFT with cL=bR= 0 andbL,cR∝ne}ationslash= 0. This concludes our brief excursion into
+the realm of LCFTs.
+Given that LCFTs are interesting in physics (see section 5) a nd that a powerful way to
+describe strongly coupled CFTs is to exploit the AdS/CFT cor respondence [6] it is natural to
+inquire whether there are any gravity duals to LCFTs.
+3. Wish-list for gravity duals to LCFTs
+In this section we establish necessary properties required for gravity duals to LCFTs. We
+formulate them as a wish-list and explain afterwards each it em on this list.
+(i) We wishfora 3-dimensional action Sthat dependsonthemetric gµνandpossiblyonfurther
+fields that we summarily denote by φ.
+(ii) We wish for the existence of AdS 3vacua with finite AdS radius ℓ.
+(iii) We wish for a finite, conserved and traceless Brown–Yor k stress tensor, given by the first
+variation of the full on-shell action (including boundary t erms) with respect to the metric.
+(iv) We wish that the 2- and 3-point correlators of the Brown– York stress tensor with itself are
+given by (1).
+(v) We wish for central charges (a la Brown–Henneaux [7]) tha t can be tuned to zero, without
+requiring a singular limit of the AdS radius or of Newton’s co nstant. For concreteness we
+assumecL= 0 (in addition cRmay also vanish, but it need not).
+(vi) We wish for a logarithmic partner to the Brown–York stre ss tensor, so that we obtain a
+Jordan-block structure like in (2) and (3).
+(vii) We wish that the 2- and non-vanishing 3-point correlat ors of the Brown–York stress tensor
+with its logarithmic partner are given by (7) and (8) (and the right-handed analog thereof).
+We explain now why each of these items is necessary. (i) is req uired since the AdS/CFT
+correspondence relates a gravity theory in d+1 dimensions to a CFT in ddimensions, and we
+chosed= 2 on the CFT side. (ii) is required since we are not merely loo king for a gauge/gravity
+duality, butreallyforanAdS/CFTcorrespondence,whichre quirestheexistenceofAdSsolutions
+on the gravity side. (iii) is required since we desire consis tency with the AdS dictionary, which
+relates the vacuum expectation value of the renormalized en ergy momentum tensor in the CFT
+∝an}b∇acketle{tTij∝an}b∇acket∇i}htto the Brown–York stress tensor TBY
+ij:
+∝an}b∇acketle{tTij∝an}b∇acket∇i}ht=TBY
+ij=2√−gδS
+δgij/vextendsingle/vextendsingle/vextendsingle
+EOM(9)
+The right hand side of this equation contains the first variat ion of the full on-shell action with
+respect to the metric, which by definition yields the Brown–Y ork stress tensor. (iv) is required
+since the 2- and 3-point correlators of a CFT are fixed by confo rmal Ward identities to take
+the form (1). (v) is required because of the construction pre sented in section 2, where a LCFT
+emerges from taking an appropriate limit of vanishing centr al charge, so we need to be able
+to tune the central charge without generating parametric si ngularities. Actually, there are
+two cases: either left and right central charge vanish and bo th energy momentum tensor flux
+components acquire a logarithmic partner, or only one of the m acquires a logarithmic partner,
+which for sake of specificity we always choose to be left. (vi) is required, since we consider
+exclusively LCFTs where the energy momentum tensor acquire s a logarithmic partner. (vii) is
+required since the 2- and 3-point correlators of a LCFT are fix ed by conformal Ward identities to
+taketheform(7), (8). Ifanyoftheitemsonthewish-listabo veisnotfulfilleditisimpossiblethat
+the gravitational theory under consideration is a gravity d ual to a LCFT of the type discussedin section 2.2On the other hand, if all the wishes are granted by a given grav itational theory
+there are excellent chances that this theory is dual to a LCFT . Until recently no good gravity
+duals for LCFTs were known [8–12].
+Before addressing candidate theories that may comply with a ll wishes we review briefly how
+to calculate correlators on the gravity side [6], since we sh all need such calculations for checking
+several items on the wish-list. The basic identity of the AdS /CFT dictionary is
+∝an}b∇acketle{tO1(z1)O2(z2)...On(zn)∝an}b∇acket∇i}ht=δ(n)S
+δj1(z1)δj2(z2)...δjn(zn)/vextendsingle/vextendsingle/vextendsingle
+ji=0(10)
+The left hand side is the CFT correlator between noperators Oi, whereOiin our case comprise
+theleft-andright-moving fluxcomponentsoftheenergymome ntumtensor andtheirlogarithmic
+partners. The right hand side contains the gravitational ac tionSdifferentiated with respect to
+appropriate sources jifor the corresponding operators. According to the AdS/CFT d ictionary
+“appropriate sources” refers to non-normalizable solutio ns of the linearized equations of motion.
+We shall be more concrete about the operators, actions, sour ces and non-normalizable solutions
+to the linearized equations of motion in the next section. Fo r now we address possible candidate
+theories of gravity duals to LCFTs.
+The simplest candidate, pure 3-dimensional Einstein gravi ty with a cosmological constant
+described by the action
+SEH=−1
+8πGN/integraldisplay
+Md3x√−g/bracketleftig
+R+2
+ℓ2/bracketrightig
+−1
+4πGN/integraldisplay
+∂Md2x√−γ/bracketleftig
+K−1
+ℓ/bracketrightig
+(11)
+does not comply with the whole wish list. Only the first four wi shes are granted: The 3-
+dimensional action (12) depends on the metric. The equation s of motion are solved by AdS 3.
+ds2
+AdS3=gAdS3µνdxµdxν=ℓ2/parenleftbig
+dρ2−1
+4cosh2ρ(du+dv)2+1
+4sinh2ρ(du−dv)2/parenrightbig
+(12)
+The Brown–York stress tensor (9) is finite, conserved and tra celess. The 2- and 3-point
+correlators on the gravity side match precisely with (1). Ho wever, the central charges are given
+by [7]
+cL=cR=3ℓ
+2GN(13)
+and therefore allow no tuning to cL= 0 without taking a singular limit. Moreover, there is no
+candidate for a logarithmic partner to the Brown–York stres s tensor. Thus, pure 3-dimensional
+Einstein gravity cannot be dual to a LCFT.
+Adding matter fields to Einstein gravity does not help neithe r. While this may lead to other
+kinds of LCFTs, it cannot produce a logarithmic partner for t he energy momentum tensor. This
+is so, because the energy momentum tensor corresponds to gra viton (spin-2) excitations in the
+bulk, and the only field producing such excitations is the met ric.
+Therefore, what we need is a way to provide additional degree s of freedom in the gravity
+sector. The most natural way to do this is by considering high er derivative interactions of the
+metric. Thefirstgravity modelofthistypewas constructedb yDeser, Jackiw andTempleton [13]
+who introduced a Chern–Simons term for the Christoffel connec tion.
+SCS=−1
+16πGNµ/integraldisplay
+d3xǫλµνΓρσλ/bracketleftig
+∂µΓσρν+2
+3ΓσκµΓκσν/bracketrightig
+(14)
+2Other types of LCFTs exist, e.g. with non-vanishing central charge or with logarithmic partners to operators
+other than the energy momentum tensor. The gravity duals for such LCFTs need not comply with all the items
+on our wish list.Hereµis a real coupling constant. Adding this action to the Einste in–Hilbert action (11)
+generates massive graviton excitations in the bulk, which i s encouraging for our wish list since
+we need these extra degrees of freedom. The model that arises when summing the actions (11)
+and (14),
+SCTMG=SEH+SCS (15)
+is known as “cosmological topologically massive gravity” ( CTMG) [14]. It was demonstrated by
+KrausandLarsen[15]that thecentral charges inCTMG areshi ftedfromtheir Brown–Henneaux
+values:
+cL=3ℓ
+2GN/parenleftbig
+1−1
+µℓ/parenrightbig
+cR=3ℓ
+2GN/parenleftbig
+1+1
+µℓ/parenrightbig
+(16)
+This is again good news concerning our wish list, since cLcan be made vanishing by a (non-
+singular) tuning of parameters in the action.
+µℓ= 1 (17)
+CTMG (15) with the tuning above (17) is known as “cosmologica l topologically massive gravity
+at the chiral point” (CCTMG). It complies with the first five it ems on our wish list, but we still
+have to prove that also the last two wishes are granted. To thi s end we need to find a suitable
+partner for the graviton.
+4. Keeping logs in massive gravity
+4.1. Login
+In this section we discuss the evidence for the existence of s pecific gravity duals to LCFTs that
+has accumulated over the past two years. We start with the the ory introduced above, CCTMG,
+and we end with a relatively new theory, new massive gravity [ 16].
+4.2. Seeds of logs
+Given that we want a partner for the graviton we consider now g raviton excitations ψaround
+the AdS background (12) in CCTMG.
+gµν=gAdS3µν+ψµν (18)
+Li,SongandStrominger[17]foundanicewaytoconstructthe m,andwefollowtheirconstruction
+here. Imposing transverse gauge ∇µψµν= 0 and defining the mutually commuting first order
+operators
+/parenleftbig
+DM/parenrightbigβ
+µ=δβ
+µ+1
+µεµαβ∇α/parenleftbig
+DL/R/parenrightbigβ
+µ=δβ
+µ±ℓεµαβ∇α (19)
+allows to write the linearized equations of motion around th e AdS background (12) as follows.
+(DMDLDRψ)µν= 0 (20)
+A mode annihilated by DM(DL) [DR]{(DL)2but not by DL}is called massive (left-moving)
+[right-moving] {logarithmic }and is denoted by ψM(ψL) [ψR]{ψlog}. Away from the chiral
+point,µℓ∝ne}ationslash= 1, the general solution to the linearized equations of moti on (20) is obtained from
+linearly combining left, right and massive modes [17]. At th e chiral point DMdegenerates with
+DLand the general solution to the linearized equations of moti on (20) is obtained from linearly
+combining left, right and logarithmic modes [18]. Interest ingly, we discovered in [18] that the
+modesψlogandψLbehave as follows:
+(L0+¯L0)/parenleftbigg
+ψlog
+ψL/parenrightbigg
+=/parenleftbigg
+2 1
+0 2/parenrightbigg/parenleftbigg
+ψlog
+ψL/parenrightbigg
+(21)whereL0=i∂u,¯L0=i∂vand
+(L0−¯L0)/parenleftbigg
+ψlog
+ψL/parenrightbigg
+=/parenleftbigg
+2 0
+0 2/parenrightbigg/parenleftbigg
+ψlog
+ψL/parenrightbigg
+(22)
+If we define naturally the Hamiltonian by H=L0+¯L0and the angular momentum by
+J=L0−¯L0we recover exactly (2) and (3), which suggests that the CFT du al to CCTMG
+(if it exists) is logarithmic, as conjectured in [18]. It was further shown with Jackiw that the
+existence of the logarithmic excitations ψlogis not an artifact of the linearized approach, but
+persists in the full theory [19].
+Thus, also the sixth wish is granted in CCTMG. The rest of this section discusses the last
+wish.
+4.3. Growing logs
+We assume now that there is a standard AdS/CFT dictionary [6] available for LCFTs and check
+if CCTMG indeed leads to the correct 2- and 3-point correlato rs. To this end we have to identify
+the sources jithat appear on the right hand side of the correlator equation (10). Following the
+standard AdS/CFT prescription the sources for the operator sOL(OR) [Olog] are given by left
+(right) [logarithmic] non-normalizablesolutions tothel inearized equations of motion (20). Thus,
+our first task is to find all solutions of the linearized equati ons of motion and to classify them
+into normalizable and non-normalizable ones, where “norma lizable” refers to asymptotic (large
+ρ) behavior that is exponentially suppressed as compared to t he AdS background (12).
+A construction of all normalizable left and right solutions was provided in [17], and the
+normalizable logarithmic solutions were constructed in [1 8].3The non-normalizable solutions
+were constructed in [25]. It turned out to be convenient to wo rk in momentum space
+ψL/R/log
+µν(h,¯h) =e−ih(t+φ)−i¯h(t−φ)FL/R/log
+µν(ρ) (23)
+The momenta h,¯hare called “weights”. All components of the tensor Fµνare determined
+algebraically, except for one that is determined from a seco nd order (hypergeometric) differential
+equation. Ingeneral oneofthelinearcombinations of theso lutionsis singularattheorigin ρ= 0,
+whiletheother isregular there. We keep onlyregular soluti ons. For each given set ofweights h,¯h
+the regular solution is either normalizable or non-normali zable. It turns out that normalizable
+solutions exist for integer weights h≥2,¯h≥0 (orh≤ −2,¯h≤0). All other solutions are
+non-normalizable.
+An example for a normalizable left mode is given by the primar y with weights h= 2,¯h= 0
+ψL
+µν(2,0) =e−2iu
+cosh4ρ
+1
+4sinh2(2ρ) 0i
+2sinh(2ρ)
+0 0 0
+i
+2sinh(2ρ) 0 −1
+
+µν(24)
+Note that all components of this mode behave asymptotically (ρ→ ∞) at most like a constant.
+The corresponding logarithmic mode is given by
+ψlog
+µν(2,0) =−1
+2(i(u+v)+lncosh2ρ)ψL
+µν(2,0) (25)
+Evidently, it behaves asymptotically like its left partner (24), except for overall linear growth in
+ρ. It is also worthwhile emphasizing that the logarithmic mod e (25) depends linearly on time
+3All these modes are compatible with asymptotic AdS behavior [20,21], and they appear in vacuum expectation
+values of 1-point functions. Indeed, the 1-point function /angbracketleftTij/angbracketrightinvolves both ψlogandψR[21–24].t= (u+v)/2. Both features are inherent to all logarithmic modes. All o ther normalizable
+modes can be constructed from the primaries (24), (25) algeb raically.
+An example for a non-normalizable left mode is given by the mo de with weights h= 1,
+¯h=−1
+ψL
+µν(1,−1) =1
+4e−iu+iv
+0 0 0
+0 cosh(2 ρ)−1−2i/radicalig
+cosh(2ρ)−1
+cosh(2ρ)+1
+0−2i/radicalig
+cosh(2ρ)−1
+cosh(2ρ)+1−4
+cosh(2ρ)+1
+
+µν(26)
+Note that all components of this mode behave asymptotically (ρ→ ∞) at most like a constant,
+except for the vv-component, which grows like e2ρ. The corresponding logarithmic mode grows
+again faster than its left partner (26) by a factor of ρand depends again linearly on time.
+Given a non-normalizable solution ψLobviously also αψLis a non-normalizable solution,
+with some constant α. To fix this normalization ambiguity we demand standard coup ling of the
+metric to the stress tensor:
+S(ψuL
+v,Tv
+u) =1
+2/integraldisplay
+dtdφ/radicalig
+−g(0)ψuu
+LTuu=/integraldisplay
+dtdφe−ihu−i¯hvTuu (27)
+HereSis either someCFT action withbackgroundmetric g(0)or adualgravitational action with
+boundary metric g(0). The non-normalizable mode ψLis the source for the energy-momentum
+flux component Tuu. The requirement (27) fixes the normalization. The discussi on above
+focussed on left modes. For the right modes essentially the s ame discussion applies, but with
+the substitutions L↔R,h↔¯handu↔v.
+4.4. Logging correlators
+Generically the 2-point correlators on the gravity side bet ween two modes ψ1(h,¯h) andψ2(h′,¯h′)
+in momentum space are determined by
+∝an}b∇acketle{tψ1(h,¯h)ψ2(h′,¯h′)∝an}b∇acket∇i}ht=1
+2/parenleftbig
+δ(2)SCCTMG(ψ1,ψ2)+δ(2)SCCTMG(ψ2,ψ1)/parenrightbig
+(28)
+where∝an}b∇acketle{tψ1ψ2∝an}b∇acket∇i}htstands for the correlation function of the CFT operators dua l to the graviton
+modesψ1andψ2. On the right hand side one has to plug the non-normalizable m odesψ1
+andψ2into the second variation of the on-shell action and symmetr ize with respect to the two
+modes. The second variation of the on-shell action of CCTMG
+δ(2)SCCTMG=−1
+16πGN/integraldisplay
+d3x√−g/parenleftbig
+DLψ1∗/parenrightbigµνδGµν(ψ2)+boundary terms (29)
+turns out to be very similar to the second variation of the on- shell Einstein–Hilbert action
+δ(2)SEH=−1
+16πGN/integraldisplay
+d3x√−gψ1µν∗δGµν(ψ2)+boundary terms (30)
+Thissimilarity allows ustoexploitresultsfromEinsteing ravity forCCTMG,aswenowexplain.4
+The bulk term in CCTMG (29) has the same form as in Einstein the ory (30) with ψ1replaced
+byDLψ1. Now, consider boundary terms. Possible obstructions to a w ell-defined Dirichlet
+boundary value problem can come only from the variation δGµν(ψ2), sinceDLis a first order
+operator. Thus any boundary terms appearing in (29) contain ing normal derivatives must be
+4Alternatively, one can follow the program of holographic re normalization, as it was done by Skenderis, Taylor
+and van Rees [23]. Their results for 2-point correlators agr ee with the results presented here.identical with those in Einstein gravity upon substituting ψ1→ DLψ1. In addition there can be
+boundary terms which do not contain normal derivatives of th e metric. However, it turns out
+that such terms can at most lead to contact terms in the hologr aphic computation of 2-point
+functions. The upshot of this discussion is that we can reduc e the calculation of all possible 2-
+point functions in CCTMG to the equivalent calculation in Ei nstein gravity with suitable source
+terms. To continue we go on-shell.5
+DLψL= 0 DLψR= 2ψRDLψlog=−2ψL(31)
+These relations together with the comparison between CCTMG (29) and Einstein gravity (30)
+then establish
+∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼2∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htEH (32a)
+∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32b)
+∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32c)
+∝an}b∇acketle{tψR(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32d)
+∝an}b∇acketle{tψL(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼ −2∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htEH (32e)
+Here the sign ∼means equality up to contact terms. Evaluating the right han d sides in Einstein
+gravity yields
+∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htEH=δh,h′δ¯h,¯h′cBH
+24h
+¯h(h2−1)t1/integraldisplay
+t0dt (33)
+and similarly for the right modes, with h↔¯h. The quantity cBHis the Brown–Henneaux
+central charge (13). The calculation of the 2-point correla tor between two logarithmic modes
+cannot be reduced to a correlator known from Einstein gravit y. The result is given by [25]
+∝an}b∇acketle{tψlog(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼ −δh,h′δ¯h,¯h′ℓ
+4GNh
+¯h(h2−1)/parenleftbig
+ψ(h−1)+ψ(−¯h)/parenrightbigt1/integraldisplay
+t0dt(34)
+whereψis the digamma function. An ambiguity in defining ψlog, viz.,ψlog→ψlog+γψL, was
+fixed conveniently in the result (34). This ambiguity corres ponds precisely to the ambiguity of
+the LCFT mass scale mLin (7c) (see also the discussion below that equation).
+To compare the results (32)-(34) with the Euclidean 2-point correlators in the short-
+distance limit (1), (7) we take the limit of large weights h,−¯h→ ∞(e.g. lim h→∞ψ(h) =
+lnh+O(1/h)) and Fourier-transform back to coordinate space (e.g. h3/¯his Fourier-transformed
+into∂4
+z/(∂z∂¯z)δ(2)(z,¯z)∝∂4
+zln|z| ∝1/z4). Straightforward calculation establishes perfect
+agreement with the LCFT correlators (1), (7), provided we us e the values
+cL= 0 cR=3ℓ
+GNbL=−3ℓ
+GN(35)
+These are exactly the values for central charges cL,cR[15] and new anomaly bL[23,25] found
+before. Thus, at the level of 2-point correlators CCTMG is in deed a gravity dual for a LCFT.
+5Above by “on-shell” we meant that the background metric is Ad S3(12) and therefore a solution of the classical
+equations of motion. Here by “on-shell” we mean additionall y that the linearized equations of motion (20) hold.Ψ1
+Ψ3Ψ2
+Figure 1. Witten diagram for three graviton correlator
+We evaluate now the Witten diagram in Fig. 1, which yields the 3-point correlator on the
+gravity side between three modes ψ1(h,¯h),ψ2(h′,¯h′) andψ3(h′′,¯h′′) in momentum space.
+∝an}b∇acketle{tψ1(h,¯h)ψ2(h′,¯h′)ψ3(h′′,¯h′′)∝an}b∇acket∇i}ht=1
+6/parenleftbig
+δ(3)SCCTMG(ψ1,ψ2,ψ3)+5 permutations/parenrightbig
+(36)
+On the right hand side one has to plug the non-normalizable mo desψ1,ψ2andψ3into the third
+variation of the on-shell action and symmetrize with respec t to all three modes.
+δ(3)SCCTMG∼ −1
+16πGN/integraldisplay
+d3x√−g/bracketleftig/parenleftbig
+DLψ1/parenrightbigµνδ(2)Rµν(ψ2,ψ3)+ψ1µν∆µν(ψ2,ψ3)/bracketrightig
+(37)
+The quantity δ(2)Rµν(ψ2,ψ3) denotes the second variation of the Ricci-tensor and the te nsor
+∆µν(ψ2,ψ3) vanishes if evaluated on left- and/or right-moving soluti ons. All boundary terms
+turn out to be contact terms, which is why only bulk terms are p resent in the result (37) for the
+third variation of the on-shell action. We compare again wit h Einstein gravity.
+δ(3)SEH∼ −1
+16πGN/integraldisplay
+d3x√−gψ1µνδ(2)Rµν(ψ2,ψ3) (38)
+Once more we can exploit some results from Einstein gravity f or CCTMG, and we find the
+following results [25] for 3-point correlators without log -insertions:
+∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼2∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htEH (39a)
+∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39b)
+∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39c)
+∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψL(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39d)
+with one log-insertion:
+∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (40a)
+∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (40b)
+∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼ −2∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψL(h′′,¯h′′)∝an}b∇acket∇i}htEH (40c)and with two or more log-insertions:
+lim
+|weights|→∞∝an}b∇acketle{tψR(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (41a)
+lim
+|weights|→∞∝an}b∇acketle{tψL(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼δh′′,−h−h′δ¯h′′,−¯h−¯h′Plog(h,h′,¯h,¯h′)
+¯h¯h′(¯h+¯h′)(41b)
+lim
+|weights|→∞∝an}b∇acketle{tψlog(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼δh′′,−h−h′δ¯h′′,−¯h−¯h′lengthy
+¯h¯h′(¯h+¯h′)(41c)
+Thelast two correlators so far could becalculated qualitat ively only (Plogis a known polynomial
+in the weights and also contains logarithms in the weights, a s expected on general grounds),
+and it would be interesting to calculate them exactly. They a re in qualitative agreement with
+corresponding LCFT correlators. All other correlators hav e been calculated exactly [25], and
+they are in precise agreement with the LCFT correlators (1), (8), provided we use again the
+values (35) for central charges and new anomaly.
+Inconclusion, also theseventh wishisgranted forCCTMG.6Thus, thereareexcellent chances
+that CCTMG is dual to a LCFT with values for central charges an d new anomaly given by (35).
+4.5. Logs don’t grow on trees
+From the discussion above it is clear that possible gravity d uals for LCFTs are sparse in theory
+space: Einstein gravity (11) does not provide a gravity dual for any tuning of parameters and
+CTMG (15) does potentially provide a gravity dual only for a s pecific tuning of parameters (17).
+Any candidate for a novel gravity dual to a LCFT is therefore w elcomed as a rare entity.
+Very recently another plausible candidate for such a gravit ational theory was found [26].
+That theory is known as “new massive gravity” [16].
+SNMG=1
+16πGN/integraldisplay
+d3x√−g/bracketleftig
+σR+1
+m2/parenleftbig
+RµνRµν−3
+8R2/parenrightbig
+−2λm2/bracketrightig
+(42)
+Heremis a mass parameter, λa dimensionless cosmological parameter and σ=±1 the sign of
+the Einstein-Hilbert term. If they are tuned as follows
+λ= 3 ⇒m2=−σ
+2ℓ2(43)
+then essentially the same story unfolds as for CTMG at the chi ral point. The main difference
+to CCTMG is that both central charges vanish in new massive gr avity at the chiral point
+(CNMG) [27,28].
+cL=cR=3ℓ
+2GN/parenleftbigg
+σ+1
+2ℓ2m2/parenrightbigg
+= 0 (44)
+Therefore, both left and right flux component of the energy mo mentum tensor acquire a
+logarithmic partner. It is easy to check that CNMG grants us t he first six wishes from section
+3. The seventh wish requires again the calculation of correl ators. The 3-point correlators have
+not been calculated so far, but at the level of 2-point correl ators again perfect agreement with
+a LCFT was found, provided we use the values [26]
+cL=cR= 0bL=bR=−σ12ℓ
+GN(45)
+6The sole caveat is that two of the ten 3-point correlators wer e calculated only qualitatively. It would be
+particularly interesting to calculate the correlator betw een three logarithmic modes (41c), since it contains an
+additional parameter independent from the central charges and new anomaly that determines LCFT properties.Itislikely thatasimilarstorycanberepeatedforgeneralm assivegravity [16], whichcombines
+new massive gravity (42) with a gravitational Chern–Simons term (14). Thus, even though they
+are sparse in theory space we have found a few good candidates for gravity duals to LCFTs:
+cosmological topologically massive gravity, new massive g ravity and general massive gravity. In
+all cases we have to tune parameters in such a way that a “chira l point” emerges where at least
+one of the central charges vanishes.
+4.6. Chopping logs?
+Sofarwe were exclusively concerned with findinggravitatio nal theories wherelogarithmic modes
+can arise. In this subsection we try to get rid of them. The rat ionale behind the desire to
+eliminate the logarithmic modes is unitarity of quantum gra vity. Gravity in 2+1 dimensions is
+simple and yet relevant, as it contains black holes [29], pos sibly gravity waves [13] and solutions
+that are asymptotically AdS. Thus, it could provide an excel lent arena to study quantum gravity
+in depth provided one is able to come up with a consistent (uni tary) theory of quantum gravity,
+for instance by constructing its dual (unitary) CFT. Indeed , two years ago Witten suggested a
+specific CFT dual to 3-dimensional quantum gravity in AdS [30 ]. This proposal engendered a
+lot of further research (see [31–37] for some early referenc es), including the suggestion by Li,
+Song and Strominger [17] to construct a quantum theory of gra vity that is purely right-moving,
+dubbed“chiral gravity”. To make a long story [18,19,24,38– 81] short, “chiral gravity” is nothing
+but CCTMG with the logarithmic modes truncated in some consi stent way.
+We discuss now two conceptually different possibilities of im plementing such a truncation.
+The first option was proposed in [18]. If one imposes periodic ity in time for all modes, t→t+β,
+then only the left- and right-moving modes are allowed, whil e the logarithmic modes are
+eliminated since they grow linearly in time, see e.g. (25). T he other possibility was pursued
+in [22]. It is based upon the observation that logarithmic mo des grow logarithmically faster in
+e2ρthan their left partners, see e.g. (25). Thus, imposing boun dary conditions that prohibit this
+logarithmic growth eliminates all logarithmic modes.
+Currently it is not known whether chiral gravity has its own d ual CFT or if it exists merely
+as a zero-charge superselection sector of the logarithmic C FT. In the latter case it is unclear
+whether or not the zero-charge superselection sector is a fu lly-fledged CFT. Another alternative
+is that neither the LCFT nor its chiral truncation dual to chi ral gravity exists. In that case
+CTMG is unlikely to exist as a consistent quantum theory on it s own. Rather, it would require
+a UV completion, such as string theory.
+4.7. Logout
+We summarize now the key results reviewed in this section as w ell as some open issues.
+Cosmological topologically massive gravity (15) at the chi ral point (17) is likely to be dual
+to a LCFT with a logarithmic partner for one flux component of t he energy momentum tensor
+since 2- [23] and 3-point correlators [25] match. The values of central charges and new anomaly
+are given by (35). The detailed calculation of the correlato r with three log-insertions (41c)
+still needs to be performed and will determine another param eter of the LCFT. New massive
+gravity (42) at the chiral point (43) is likely to be dual to a L CFT with a logarithmic partner
+for both flux components of the energy momentum tensor since 2 -point correlators match [26].
+The central charges vanish and the new anomalies are given by (45). The calculation of 3-
+point correlators still needs to be performed and will provi de a more stringent test of the
+conjectured duality to a LCFT. A similar story is likely to re peat for general massive gravity
+(the combination of topologically and new massive gravity) at a chiral point, and it could be
+rewardingtoinvestigate thisissue. Finallyweaddressedp ossibilitiestoeliminatethelogarithmic
+modes and their partners, since such an elimination might le ad to a chiral theory of quantum
+gravity [17], called “chiral gravity”. The issue of whether chiral gravity exists still remains open.5. Towards condensed matter applications
+In this final section we review briefly some condensed matter s ystems where LCFTs do arise,
+see [3,4] for more comprehensive reviews. We focus on LCFTs w here the energy-momentum
+tensor acquires a logarithmic partner, i.e., the class of LC FTs for which we have found possible
+gravity duals.7Condensed matter systems described by such LCFTs are for ins tance systems
+at (or near) a critical point with quenched disorder, like sp in glasses [83]/quenched random
+magnets [84,85], dilute self-avoiding polymers or percola tion [86]. “Quenched disorder” arises
+in a condensed matter system with random variables that do no t evolve with time. If the
+amount of disorder is sufficiently large one cannot study the e ffects of disorder by perturbing
+around a critical point without disorder — standard mean fiel d methods break down. The
+system is then driven towards a random critical point, and it is a challenge to understand its
+precise nature. Mathematically, the essence of the problem lies in the infamous denominator
+arising in correlation functions of some operator Oaveraged over disordered configurations (see
+e.g. chapter VI.7 in [87])
+∝an}b∇acketle{tO(z)O(0)∝an}b∇acket∇i}ht=/integraldisplay
+DVP[V]/integraltext
+Dφexp/parenleftbig
+−S[φ]−/integraltext
+d2z′V(z′)O(z′)/parenrightbig
+O(z)O(0)/integraltext
+Dφexp/parenleftbig
+−S[φ]−/integraltext
+d2z′V(z′)O(z′)/parenrightbig (46)
+HereS[φ] is some 2-dimensional8quantum field theory action for some field(s) φandV(z) is a
+random potential with some probability distribution. For w hite noise one takes the Gaussian
+probability distribution P[V]∝exp/parenleftbig
+−/integraltext
+d2zV2(z)/(2g2)/parenrightbig
+, wheregis a coupling constant that
+measuresthestrengthoftheimpurities. Ifit werenot forth edenominatorappearingontheright
+hand side of the averaged correlator (46) we could simply per form the Gaussian integral over
+the impurities encoded in the random potential V(z). This denominator is therefore the source
+of all complications and to deal with it requires suitable me thods, see e.g. [88]. One possibility is
+to eliminate the denominator by introducing ghosts. This so -called “supersymmetric method”
+works well if the original quantum field theory described by t he actionS[φ] is very simple, like a
+free field theory. Another option is the so-called replica tr ick, where one introduces ncopies of
+the original quantum field theory, calculates correlators i n this setup and takes the limit n→0
+in the end, which formally reproduces the denominator in (46 ). Recently, Fujita, Hikida, Ryu
+and Takayanagi combined the replica method with the AdS/CFT correspondence to describe
+disordered systems [89] (see [90,91] for related work), ess entially by taking ncopies of the CFT,
+exploiting AdS/CFT to calculate correlators and taking for mally the limit n→0 in the end.
+Like other replica tricks their approach relies on the exist ence of the limit n→0.
+One of the results obtained by the supersymmetric method or r eplica trick is that correlators
+like the one in (46) develop a logarithmic behavior, exactly as in a LCFT [84]. In fact, in
+then→0 limit prescribed by the replica trick, the conformal dimen sions of certain operators
+degenerate. This produces a Jordan block structure for the H amiltonian in precise parallel to
+theµℓ→1 limit of CTMG. More concretely, LCFTs can be used to compute correlators of
+quenched random systems!
+This suggests yet-another route to describe systems with qu enched disorder, and our present
+results add to this toolbox. Namely, instead of taking ncopies of an ordinary CFT we may
+start directly with a LCFT. If this LCFT is weakly coupled we c an work on the LCFT side
+perturbatively, using the results mentioned above [3,4,84 –86]. On the other hand, if the LCFT
+becomes strongly coupled, perturbative methods fail. To ge t a handle on these situations we
+can exploit the AdS/LCFT correspondence and work on the grav ity side. Of course, to this end
+7A well-studied alternative case is a LCFT with c=−2 [2,82]. There is no obvious way to construct a gravity
+dual for such LCFTs, even when considering CTMG or new massiv e gravity away from the chiral point. We thank
+Ivo Sachs for discussions on this issue.
+8Analog constructions work in higher dimensions, but we focu s here on two dimensions.one needs to construct gravity duals for LCFTs. The models re viewed in this talk are simple
+and natural examples of such constructions.
+Acknowledgments
+We thank Matthias Gaberdiel, Gaston Giribet, Olaf Hohm, Rom an Jackiw, David Lowe, Hong
+Liu, Alex Maloney, John McGreevy, Ivo Sachs, Kostas Skender is, Wei Song, Andy Strominger
+and Marika Taylor for discussions. DG thanks the organizers of the “First Mediterranean
+Conference on Classical and Quantum Gravity” for the kind in vitation and for all their efforts to
+make the meeting very enjoyable. DG and NJ are supported by th e START project Y435-N16
+of the Austrian Science Foundation (FWF). During the final st age NJ has been supported by
+project P21927-N16 of FWF. NJ acknowledges financial suppor t from the Erwin-Schr¨ odinger-
+Institute (ESI) during the workshop “Gravity in three dimen sions”.
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+arXiv:1001.0003v3  [hep-th]  10 May 2010Preprint typeset in JHEP style - HYPER VERSION KUL-TF-09/28
+HD-THEP-09-31
+A landscape of non-supersymmetric AdS vacua on
+coset manifolds
+Paul Koerber∗
+Instituut voor Theoretische Fysica, Katholieke Universit eit Leuven, Celestijnenlaan
+200D, B-3001 Leuven, Belgium
+Email:koerber atitf.fys.kuleuven.be
+Simon K¨ ors
+Institut f¨ ur Theoretische Physik, Universit¨ at Heidelbe rg, Philosophenweg 16-19, D-69120
+Heidelberg, Germany
+Email:s.koers atthphys.uni-heidelberg.de
+Abstract: We construct new families of non-supersymmetric sourceles s type IIA AdS 4
+vacua on those coset manifolds that also admit supersymmetr ic solutions. We investigate
+the spectrum of left-invariant modes and find that most, but n ot all, of the vacua are stable
+under these fluctuations. Generically, there are also no mas sless moduli.
+∗Postdoctoral Fellow FWO – Vlaanderen.Contents
+1. Introduction 1
+2. Ansatz 3
+3. Solutions 6
+4. Stability analysis 11
+5. Conclusions 15
+A. SU(3)-structure 15
+B. Type II supergravity 16
+1. Introduction
+The reasons for studying AdS 4vacua of type IIA supergravity are twofold: first they are
+examples of flux compactifications away from the Calabi-Yau r egime, where all the moduli
+can be stabilized at the classical level. Secondly, they can serve as a gravity dual in the
+AdS4/CFT3-correspondence, which became the focus of attention due to recent progress
+in the understanding of the CFT-side as a Chern-Simons-matt er theory describing the
+world-volume of coinciding M2-branes [1].
+Itismucheasiertofindsupersymmetricsolutionsofsupergr avityasthesupersymmetry
+conditions are simpler than the full equations of motion, wh ile at the same time there
+are general theorems stating that the former – supplemented with the Bianchi identities
+of the form fields – imply the latter [2, 3, 4, 5]. Although spec ial type IIA solutions
+that came from the reduction of supersymmetric M-theory vac ua were already known (see
+e.g. [6, 7, 8]), it was only in [3] that the supersymmetry cond itions for type IIA vacua with
+SU(3)-structure were first worked out in general. It was disc overed that there are natural
+solutions to these equations on the four coset manifolds G/Hthat have a nearly-K¨ ahler
+limit [9, 10, 11, 12, 13, 14] (solutions on other manifolds ca n be found in e.g. [3, 15, 16]).1
+To be precise these are the manifolds SU(2) ×SU(2),G2
+SU(3),Sp(2)
+S(U(2)×U(1))andSU(3)
+U(1)×U(1).2
+These solutions are particularly simple in the sense that bo th the SU(3)-structure, which
+determines the metric, as well as all the form fluxes can be exp anded in terms of forms
+which are left-invariant under the action of the group G. The supersymmetry equations
+1For an early appearance of these coset manifolds in the strin g literature see e.g. [17].
+2See [18] for a review and a proof that these are the only homoge neous manifolds admitting a nearly-
+K¨ ahler geometry.
+– 1 –of [3] then reduce to purely algebraic equations and can be ex plicitly solved. Nevertheless,
+these solutions still have non-trivial geometric fluxes as o pposed to the Calabi-Yau or torus
+orientifolds of [15, 16]. Similarly to those papers it is pos sible to classically stabilize all
+left-invariant moduli [14]. Inspired by the AdS 4/CFT3correspondence more complicated
+type IIA solutions have in the meantime been proposed. The so lutions have a more generic
+form for the supersymmetry generators, called SU(3) ×SU(3)-structure [19], and are not
+left-invariant anymore [20, 21, 22, 23] (see also [24]). Sup ersymmetric AdS 4vacua in type
+IIB with SU(2)-structure have also been studied in [25, 26, 2 7, 28] and in particular it has
+been shown in [28] that also in this setup classical moduli st abilization is possible.
+At some point, however, supersymmetry has to be broken and we have to leave
+the safe haven of the supersymmetry conditions. In this pape r we construct new non-
+supersymmetric AdS 4vacua without source terms. This means that the more complic ated
+equations of motion of supergravity should be tackled direc tly3. In order to simplify the
+equations we use a specific ansatz: we start from a supersymme tric AdS 4solution and scan
+for non-supersymmetric solutions with the samegeometry (and thus SU(3)-structure), but
+withdifferent NSNS- and RR-fluxes. Moreover, we expand these form fields in t erms of the
+SU(3)-structure and its torsion classes. This may seem rest rictive at first, but it works for
+11D supergravity, where solutions like this have been found and are known as Englert-type
+solutions [31, 32, 33] (see [34] for a review). To be specific, for each supersymmetric M-
+theory solution of Freund-Rubin type (which means the M-the ory four-form flux has only
+legs along the external AdS 4space, i.e.F4=fvol4wherefis called the Freund-Rubin
+parameter) it is possible to construct a non-supersymmetri c solution with the same inter-
+nal geometry but with a different four-form flux. The modified fo ur-form of the Englert
+solution has then a non-zero internal part: ˆF4∝η†γm1m2m3m4ηdxm1m2m3m4, whereηis
+the 7D supersymmetry generator, and a different Freund-Rubin parameterfE=−(2/3)f.
+Also the Ricci scalar of the AdS 4space, and thus the effective 4D cosmological constant,
+differs:R4D,E= (5/6)R4D. In type IIA with non-zero Romans mass (so that there is no lif t
+to M-theory) non-supersymmetric solutions of this form hav e been found as well: for the
+nearly-K¨ ahler geometry in [35, 29, 36] and for the K¨ ahler- Einstein geometry in [35, 20, 37].
+In this paper we show that this type of solutions is not restri cted to these limits and sys-
+tematically scan for them. Applying our ansatz to the coset m anifolds with nearly-K¨ ahler
+limit, mentioned above, we find that the most interesting man ifolds areSp(2)
+S(U(2)×U(1))and
+SU(3)
+U(1)×U(1), on which we find several families of non-supersymmetric AdS 4solutions. We
+also find some non-supersymmetric solutions in regimes of th e geometry that do not allow
+for a supersymmetric solution.
+These non-supersymmetric solutions are not necessarily st able. For instance, it is
+known that if there is more than one Killing spinor on the inte rnal manifold (which holds
+in particular for S7, the M-theory lift of CP3=Sp(2)
+S(U(2)×U(1))), the Englert-type solution is
+unstable [38]. We investigate stability of our solutions ag ainst left-invariant fluctuations.
+This means we calculate the spectrum of left-invariant mode s, and check for each mode
+3Anotherroute would be tofindsome alternative first-ordereq uations, which extendthe supersymmetry
+conditions in that they still automatically imply the full e quations of motion in certain non-supersymmetric
+cases, see e.g. [29, 30].
+– 2 –whether the mass-squared is above the Breitenlohner-Freed man bound [39, 40]. This is not
+a complete stability analysis in that there could still be no n-left-invariant modes that are
+unstable. We do believe it provides a good first indication. I n particular, we find for the
+type IIA reduction of the Englert solution on S7that the unstable mode of [38] is among
+our left-invariant fluctuations and we find the exact same mas s-squared.
+These non-supersymmetric AdS 4vacua are interesting, because, provided they are
+stable, they should have a CFT-dual. For instance in [20] the CFT-dual for a non-
+supersymmetric K¨ ahler-Einstein solution on CP3was proposed. Furthermore, for phe-
+nomenologically more realistic vacua, supersymmetry-bre aking is essential. Really, one
+would like to construct classical solutions with a dS 4-factor, which are necessarily non-
+supersymmetric. Because of a series of no-go theorems – from very general to more specific:
+[41, 42, 43, 44, 45] – this is a very non-trivial task. For pape rs nevertheless addressing this
+problemsee[46,47,45,48,49,28]. Inthiscontext thelands capeofthenon-supersymmetric
+AdS4vacua of this paper can be considered as a playground to gain e xperience before try-
+ing to construct dS 4-vacua. In fact, in [48] an ansatz very similar to the one used in this
+paper was proposed in order to construct dS 4-vacua. Applied to the coset manifolds above,
+it did however not yield any solutions, in agreement with the no-go theorem of [45].
+In section 2 we explain our ansatz in full detail, while in sec tion 3 we present the
+explicit solutions we found on the coset manifolds. In secti on 4 we analyse the stability
+against left-invariant fluctuations before ending with som e short conclusions. We provide
+an appendix with some useful formulae involving SU(3)-stru ctures and an appendix on our
+supergravity conventions.
+Thenon-supersymmetricsolutions of this paperappearedbe forein thesecond author’s
+PhD thesis [50].
+2. Ansatz
+In this section we explain the ansatz for our non-supersymme tric solutions. The reader
+interested in the details might want to check out our SU(3)-s tructure conventions in ap-
+pendix A, while towards the end of the section we need the type II supergravity equations
+of motion outlined in appendix B.
+We start with a supersymmetric SU(3)-structure solution of type IIA supergravity.
+The SU(3)-structure is defined by a real two-form Jand a complex decomposable three-
+form Ω satisfying (A.1). Moreover, Jand Ω together determine the metric as in (A.2). In
+order for the solution to preserve at least one supersymmetr y (N= 1) [3] one finds that
+the warp factor Aand the dilaton Φ should be constant, the torsion classes W1,W2purely
+imaginary and all other torsion classes zero (for the definit ion of the torsion classes see
+(A.3)). This implies
+dJ=3
+2W1ReΩ, (2.1a)
+dReΩ = 0, (2.1b)
+dImΩ =W1J∧J+W2∧J, (2.1c)
+– 3 –where we defined W1≡ −iW1andW2≡ −iW2. The fluxes can then be expressed in terms
+of Ω,Jand the torsion classes and are given by
+eΦˆF0=f1, (2.2a)
+eΦˆF2=f2J+f3ˆW2, (2.2b)
+eΦˆF4=f4J∧J+f5ˆW2∧J, (2.2c)
+eΦˆF6=f6vol6, (2.2d)
+H=f7ReΩ, (2.2e)
+where for the supersymmetric solution
+f1=eΦm, f 2=−W1
+4, f3=−w2, f4=3eΦm
+10,
+f5= 0, f 6=9W1
+4, f7=2eΦm
+5.(2.3)
+Using the duality relation f=˜F0=−⋆6ˆF6=−e−Φf6(see (B.6)) we find that f6is
+proportional to the Freund-Rubin parameter f, whilef1is proportional to the Romans
+massm. Furthermore, we introduced here a normalized version of W2, enabling us later
+on to use (2.2) as an ansatz for the fluxes also in the limit W2→0:
+ˆW2=W2
+w2,withw2=±/radicalbig
+(W2)2, (2.4)
+where one can choose a convenient sign in the last expression .
+The Bianchi identity for ˆF2imposes dW2∝ReΩ. Working out the proportionality
+constant [3] we find
+dW2=−1
+4(W2)2ReΩ. (2.5)
+Furthermore, using the values for the fluxes (2.3) it fixes the Romans mass:
+e2Φm2=5
+16/parenleftbig
+3(W1)2−2(W2)2/parenrightbig
+. (2.6)
+We now want to construct non-supersymmetric AdS solutions o n the manifolds men-
+tioned in the introduction with the samegeometry as in the supersymmetric solution, and
+thus the same SU(3)-structure ( J,Ω), but with different fluxes. We make the ansatz that
+the fluxes can still be expanded in terms of J,Ω and the torsion class ˆW2as in (2.2), but
+with different values for the coefficients fi. To this end we plug the ansatz for the geometry
+(J,Ω) — eqs. (2.1) — and the ansatz for the fluxes — eqs. (2.2) — into the equations of
+motion (B.7) and solve for the fi. We will make one more assumption, namely that
+ˆW2∧ˆW2=cJ∧J+pˆW2∧J, (2.7)
+withc,psome parameters. This is an extra constraint only for theSU(3)
+U(1)×U(1)coset and
+we will discuss its relaxation later.4Wedging with Jwe find then immediately c=−1/6.
+4With the ansatz (2.2) the constraint is forced upon us. Indee d, suppose that instead ˆW2∧ˆW2=
+−1/6J∧J+pˆW2∧J+P∧J, wherePis a non-zero simple (1,1)-form independent of ˆW2. We find then
+from the equation of motion for Hand the internal part of the Einstein equation respectively f5f3= 0 and
+(f3)2−(f5)2−(w2)2= 0. So the only possibility is then f5= 0 and f3=±w2, which leads in the end to
+the supersymmetric solution. They way out is to also include Pas an expansion form in (2.2).
+– 4 –Furthermore we need expressions for the Ricci scalar and ten sor, which for a manifold with
+SU(3)-structure can be expressed in terms of the torsion cla sses [51]. Taking into account
+that onlyW1,2are non-zero we find:
+R6D=15(W1)2
+2−(W2)2
+2, (2.8a)
+Rmn=1
+6gmnR6D+W1
+4W2(m·Jn)+1
+2[W2m·W2n]0+1
+2Re/bracketleftbig
+dW2|(2,1)m·¯Ωn/bracketrightbig
+,(2.8b)
+where (P)2andPm·Pnfor a form Pare defined in (B.2) and |0indicates taking the
+traceless part. From eq. (2.5) follows that for our purposes dW2|2,1= 0 so that the last
+term in (2.8b) vanishes. Moreover, using (2.7) [ W2m·W2n]0can be expressed in terms of
+W2(m·Jn).
+Plugging the ansatz for the fluxes (2.2) into the equations of motion (B.7) and using
+eqs. (2.1), (2.5), (A.5), (2.7), (B.5), (B.6) and (2.8b) we fi nd:
+BianchiF2: 0 =3
+2W1f2−1
+4w2f3+f1f7,
+eomF4: 0 = 3W1f4+1
+4w2f5−f6f7,
+eomH: 0 = 6W1f7−3f1f2−12f4f2−6f4f6−f3f5,
+0 =w2f7+f1f3+f2f5−2f3f4−f5f6+pf3f5, (2.9)
+dilaton eom : 0 = R4D+R6D−2f2
+7,
+Einstein ext. : 0 = R4D+(f1)2+3(f2)2+12(f4)2+(f6)2+(f3)2+(f5)2,
+Einstein int. : 0 = R6D−6(f7)2+1
+2/bracketleftbig
+3(f1)2+3(f2)2−12(f4)2−3(f6)2+(f3)2−(f5)2/bracketrightbig
+,
+0 = 4(f2f3+2f4f5)−w2W1−p/bracketleftbig
+(f3)2−(f5)2−(w2)2/bracketrightbig
+.
+In the equation of motion for Hwe get separate conditions from the coefficients of J∧J
+andˆW2∧Jrespectively. In the internal Einstein equation we find like wise a separate
+condition from the trace and the coefficient of W2(m·Jn). In the next section we find
+explicit solutions to these equations for the coset manifol ds with nearly-K¨ ahler limit, the
+stability of which we investigate in section 4.
+Flipping signs
+The Einstein and dilaton equation are quadratic in the form fl uxes and thus insensitive to
+flipping the signs of these fluxes. Taking into account also th e flux equations of motion
+and Bianchi identities, we find that for each solution to the s upergravity equations, we
+automatically obtain new ones by making the following sign fl ips:
+H→ −H,ˆF0→ −ˆF0,ˆF2→ˆF2,ˆF4→ −ˆF4,ˆF6→ˆF6,
+H→ −H,ˆF0→ˆF0,ˆF2→ −ˆF2,ˆF4→ˆF4,ˆF6→ −ˆF6,
+H→H,ˆF0→ −ˆF0,ˆF2→ −ˆF2,ˆF4→ −ˆF4,ˆF6→ −ˆF6.(2.10)
+In particular, these sign flips will transform a supersymmet ric solution into another super-
+symmetric solution (as can be verified using the conditions ( 2.1),(2.3) allowing for suitable
+– 5 –sign flips of J, ReΩ and ImΩ compatible with the metric). If some fluxes are ze ro, more
+sign flips are possible. For instance for ˆF0=ˆF4= 0 we find the following extra sign-flip,
+known as skew-whiffing in the M-theory compactification literature [52] (see also t he review
+[34])
+H→ ±H,ˆF2→ˆF2,ˆF6→ −ˆF6, (2.11)
+which transforms a supersymmetric solution into a non-supersymmetric one. When dis-
+cussing different solutions, we will from now on implicitly co nsider each solution together
+with its signed-flipped counterparts.
+3. Solutions
+Let us now solve the equations obtained in the previous secti on for the coset manifolds that
+admit sourceless supersymmetric solutions, namelyG2
+SU(3), SU(2)×SU(2),Sp(2)
+S(U(2)×U(1))and
+SU(3)
+U(1)×U(1). For the supersymmetricsolutions on these manifolds we wil l use the conventions
+and presentation of [13, 14]. For moredetails, includingin particular ourchoice of structure
+constants for the relevant algebras, we refer to these paper s.
+On a coset manifold G/Hone can define a coframe emthrough the decomposition of
+the Lie-valued one-form L−1dL=emKm+ωaHain terms of the algebras of GandH. Here
+Lis a coset representative, the Haspan the algebra of Hand theKmspan the complement
+of this algebra within the algebra of G. The exterior derivative on the emis then given
+in terms of the structure constants through the Maurer-Cart an relation. Furthermore,
+the forms that are left-invariant under the action of Gare precisely those forms that are
+constant in the basis spanned by emand for which the exterior derivative is also constant
+in this basis. For these forms the exterior derivative can th en be expressed solely in terms
+of the structure constants only involving the Km. We refer to [53, 54] for a review on coset
+technology or to the above papers for a quick explanation.
+G2
+SU(3)and SU(2) ×SU(2)
+We start from the supersymmetric nearly-K¨ ahler solution o nG2
+SU(3). The SU(3)-structure
+is given by
+J=a(e12−e34+e56),
+Ω =a3/2/bracketleftbig
+(e245−e236−e146−e135)+i(e246+e235+e145−e136)/bracketrightbig
+,(3.1)
+whereais the overall scale.
+Since this SU(3)-structure corresponds to a nearly-K¨ ahle r geometry the torsion class
+W2is zero. Furthermore we find
+W1=−2√
+3a−1/2, w2=p= 0. (3.2)
+– 6 –Plugging this into the equations (2.9) we find exactly three s olutions for ( f1,...,f7) (up
+to the sign flips (2.10)):
+a−1/2(√
+5
+2,1
+2√
+3,0,3
+4√
+5,0,−9
+2√
+3,1√
+5),
+a−1/2(/radicalbigg
+5
+3,0,0,0,0,5√
+3,0),
+a−1/2(1,1√
+3,0,−1
+2,0,√
+3,1).(3.3)
+The first is the supersymmetric solution, while the last two a re non-supersymmetric solu-
+tions, which were already found in [35, 29, 36]. Truncating t o the 4D effective theory it
+was shown in [30] that a generalization of this family of solu tions is quite universal as it
+appears in a large class of N= 2 gauged supergravities.
+On the SU(2) ×SU(2) manifold, requiring the same geometry as the supersym metric
+solution and not allowing for source terms will restrict us t o the nearly-K¨ ahler point. The
+analysis is then basically the same as forG2
+SU(3)above.
+Sp(2)
+S(U(2)×U(1))
+The family of supersymmetric solutions on this manifold has , next to the overall scale,
+an extra parameter determining the shape of the solutions. I t is then possible to turn on
+the torsion class W2and venture away from the nearly-K¨ ahler geometry. This mak es this
+class much richer and enables us this time to find new non-supe rsymmetric solutions. The
+SU(3)-structure is given by [12, 13, 14]
+J=a(e12+e34−σe56),
+Ω =a3/2σ1/2/bracketleftbig
+(e245−e236−e146−e135)+i(e246+e235+e145−e136)/bracketrightbig
+,(3.4)
+whereais the overall scale and σis the shape parameter. We find for the torsion classes
+and the parameter p:
+W1= (aσ)−1/22+σ
+3,
+(W2)2= (aσ)−18(1−σ)2
+3⇒w2= (aσ)−1/22√
+2(1−σ)√
+3,
+ˆW2=−1√
+3/parenleftbig
+e12+e34+2σe56/parenrightbig
+,
+p=−/radicalbig
+2/3.(3.5)
+We easily read off that σ= 1 corresponds to the nearly-K¨ ahler geometry. Note that ev en
+thoughW2→0 forσ→1,ˆW2is well-defined and non-zero in this limit so that we can
+still use it as an expansion form for the fluxes. The points σ= 2 andσ= 2/5 are also
+special, since eq. (2.6) then implies that the supersymmetr ic solution has zero Romans
+mass and, in particular, can be lifted to M-theory. Moreover , these are the endpoints of
+the interval where supersymmetric solutions exist (since o utside this interval we would find
+from eq. (2.6) that m2<0). They are indicated as vertical dashed lines in the plots.
+– 7 –Figure 1:Sp(2)
+S(U(2)×U(1))-model: plot of aR4Dfor the supersymmetric solutions (light green) and
+the new non-supersymmetric solutions (other colors) in terms of t he shape parameter σ. Unstable
+solutions are indicated in red.
+Pluggingeqs.(3.5) intothesupergravityequationsofmoti on(2.9)wefindnumericallya
+rich spectrumofsolutions, whicharedisplayed infigures1a nd2. Note that thedependence
+on the overall scale can be easily extracted from all plotted quantities by multiplying by
+ato a suitable power. We plotted the value of the 4D Ricci scala rR4Dof the AdS-space
+against the shape parameter σin figure 1. Note that R4Dis inversely proportional to the
+AdS-radius squared and related to the effective 4D cosmologic al constant and the vev of
+the 4D scalar potential Vas follows
+Λ =∝angb∇acketleftV∝angb∇acket∇ight=R4D/4. (3.6)
+The supersymmetric solutions are plotted in light green, wh ile red is used for the non-
+supersymmetric solutions found to be unstable in section 4. For completeness of the pre-
+sentation of our numeric results, we provide the values of ea ch of the coefficients fiof the
+ansatz (2.2) in figure 2.
+The first point to note is that where the supersymmetric solut ions are restricted to
+the interval σ∈[2/5,2], there exist non-supersymmetric solutions in the somewh at larger
+intervalσ∈[0.39958,2.13327]. Furthermore, there are up to five non-supersymmetri c
+solutions for each supersymmetric solution.
+We remark that the parameters σand the overall scale are not continuous moduli since
+they are determined by the vevs of the fluxes, which in a proper string theory treatment
+shouldbequantized. Indeed, inthenextsection wewill show that generically all moduliare
+stabilized. We leave the analysis of flux quantization, whic h is complicated by the fact that
+– 8 –(a) Plot of a1/2f1(Romans mass)
+ (b) Plot of a1/2f2(J-part of ˆF2)
+(c) Plot of a1/2f3(ˆW2-part of ˆF2)
+ (d) Plot of a1/2f4(J∧J-part of ˆF4)
+(e) Plot of a1/2f5(J∧ˆW2-part of ˆF4)
+ (f) Plot of a1/2f6(Freund-Rubin parameter)
+(g) Plot of a1/2f7(ReΩ part of H)
+Figure 2: Plots of the solutions on the cosetSp(2)
+S(U(2)×U(1)). Different colors indicate different
+solutions. Unstable solutions are indicated in red (see section 4) and the supersymmetric solutions
+in light green. By a suitable rescaling of the coefficients the dependen ce on the overall scale ais
+taken out.
+– 9 –there is non-trivial H-flux (twisting the RR-charges), to further work. The expect ation is
+that the continuous line of supergravity solutions is repla ced by discrete solutions.
+Let us now take a look at some special values of σ. Forσ= 1 we find five solutions
+of which three (including the supersymmetric one) are up to s caling equivalent to the
+solutions (3.3) onG2
+SU(3)of the previous section [35, 29, 36, 30]. They have f3=f5= 0 and
+so the fluxes are completely expressed in terms of J. However, there are also two new non-
+supersymmetric solutions (the dark green and the purple one ) which have f3∝negationslash= 0,f5∝negationslash= 0.
+Next we turn to the case σ= 2. This point is special in that the metric becomes
+the Fubini-Study metric on CP3and the bosonic symmetry of the geometry enhances
+from Sp(2) to SU(4). In fact, since the RR-forms of the supers ymmetric solution can be
+expanded in terms of the closed K¨ ahler form ˜J= (1/3)J+(2a)1/2W2of the Fubini-Study
+metric, the symmetry group of the whole supersymmetric solu tion is SU(4). One can also
+show that the supersymmetry enhances from the generic N= 1 toN= 6 [6]. In [37] it
+was found that there is an infinite continuous family of non-s upersymmetric solutions and
+two discrete separate solutions (see also [35] for an incomp lete early discussion), which all
+have SU(4)-symmetry. They are notdisplayed in the plot since they can not be found by
+taking a continuous limit σ→2. For these solutions H= 0 (f7= 0) and ˆF2andˆF4are
+expanded in terms of ˜J(for more details see [37]).
+Instead, in the plot we find apart from the supersymmetric sol ution (which merges
+with the dark green solution at σ= 2) two more discrete non-supersymmetric solutions,
+which have only Sp(2)-symmetry (since the fluxes cannot be ex pressed in terms of ˜Jonly).
+The blue one is new, while the red one turns out to be the reduct ion of the Englert-type
+solution. Indeed for the Englert-type solution we expect
+f1= 0, no Romans mass ,(3.7a)
+f2=f2,susy, f3=f3,susy, same geometry in M ⇒sameˆF2as susy,(3.7b)
+f7=−2f4=−(1/3)f6,susy, f5= 0, fromˆF4in M-theory ,(3.7c)
+f6= (−2/3)f6,susy, Freund-Rubin parameter changes ,(3.7d)
+R4D= (5/6)R4D,susy, 4D Λ changes ,(3.7e)
+which agrees with the values displayed in the figures for the r ed curve at σ= 2.
+Also forσ= 2/5 we find apart from the supersymmetric solution, the Englert solution
+(the purplecurve) andoneextra non-supersymmetricsoluti on (the darkgreen curve). Note
+that while the supersymmetric curve joins the olive green cu rve atσ= 2/5, the purple
+curve only joins the dark green curve at σ= 0.39958.
+SU(3)
+U(1)×U(1)
+For this manifold the SU(3)-structure is given by [13, 14]:
+J=a(−e12+ρe34−σe56),
+Ω =a3/2(ρσ)1/2/bracketleftbig
+(e245+e135+e146−e236)+i(e235+e136+e246−e145)/bracketrightbig
+,(3.8)
+– 10 –whereρandσare the shape parameters of the model. Furthermore we find for the torsion
+classes:
+W1=−(aρσ)−1/21+ρ+σ
+3,
+W2=−(2/3)a1/2(ρσ)−1/2/bracketleftbig
+(2−ρ−σ)e12+ρ(1−2ρ+σ)e34−σ(1+ρ−2σ)e56/bracketrightbig
+.(3.9)
+It turns out that the ansatz (2.7) is only satisfied for
+ρ= 1, σ= 1 orρ=σ. (3.10)
+In all three of these cases the equations (2.9) forSU(3)
+U(1)×U(1)reduce to exactly the same
+equations as forSp(2)
+S(U(2)×U(1))so that we obtain the same solution space. However, as we
+will see in the next section, the stability analysis will be d ifferent since the model on
+SU(3)
+U(1)×U(1)has two extra left-invariant modes.
+In order to find further non-supersymmetric solutions, we sh ould go beyond the ansatz
+(2.7). Let us put
+ˆW2∧ˆW2= (−1/6)J∧J+p1ˆW2∧J+p2ˆP∧J, (3.11)
+whereˆPis a primitive normalized (1,1)-form (so that it is orthogon al toJandˆP2= 1).
+Furthermore, we also choose it orthogonal to ˆW2i.e.
+ˆW2·ˆP= 0 or equivalently J∧ˆW2∧ˆP= 0. (3.12)
+From the last equation one finds, using (2.1c), that d ˆP∧ImΩ = 0, which implies on
+SU(3)
+U(1)×U(1)that
+dˆP= 0. (3.13)
+One can now allow the RR-fluxes ˆF2andˆF4to have pieces proportional to ˆPandˆP∧
+Jrespectively and adapt the equations (2.9) accordingly to a ccommodate for the new
+contributions. Now it is possible to numerically find non-su persymmetric solutions for ρ
+andσnot satisfying (3.10). In particular, there are Englert-ty pe solutions on the ellipse of
+values for (ρ,σ) where the supersymmetric solution has zero Romans mass. Fr om eq. (2.6)
+we find that this ellipse is described by
+m2=5
+16ρσ/bracketleftbig
+−5(ρ2+σ2)+6(ρ+σ+ρσ)−5/bracketrightbig
+= 0. (3.14)
+We will not go into more detail on these solutions in this pape r.
+4. Stability analysis
+Inthissectionweinvestigate whetherthenewnon-supersym metricsolutionsonSp(2)
+S(U(2)×U(1))
+andSU(3)
+U(1)×U(1)are stable5. To this end we calculate the spectrum of scalar fluctuations . We
+5In [36] it was found that the non-supersymmetric solutions o nG2
+SU(3)and the similar solutions on the
+nearly-K¨ ahler limits of the other two coset manifolds unde r study are stable. We find exactly the same
+spectrum as the authors of that paper, which provides a consi stency check on our approach. We thank
+Davide Cassani for providing us with these numbers, which ar e not explicitly given in their paper. We did
+not investigate the spectrum of the similar solution on SU(2 )×SU(2), which is more complicated as there
+are more modes.
+– 11 –use the well-known result of [39, 40] that in an AdS 4vacuum a tachyonic mode does not yet
+signal an instability. Only a mode with a mass-squared below the Breitenlohner-Freedman
+bound,
+M2<−3|Λ|
+4, (4.1)
+where Λ<0 is the 4D effective cosmological constant, leads to an instab ility. We restrict
+ourselves to left-invariant fluctuations, which implies th at even if we do not find any modes
+below the Breitenlohner-Freedman bound, the vacuum might s till be unstable, since there
+might be fluctuations with sufficiently negative mass-square d that are not left-invariant.
+This analysis can however pinpoint many unstable vacua and w e do believe it gives a
+valuable first indication for the stability of the others.
+Truncatingtotheleft-invariant modesonthecoset manifol dsunderstudyleads toa4D
+N= 2 gauged supergravity6. It has been shown in [36] that this truncation is consistent .
+The spectrum of the scalar fields can then be obtained from the 4D scalar potential. In
+fact, this computation is analogous to the one performed in [ 14] for the supersymmetric
+N= 1 vacua on the coset spaces. As opposed to the models here, th e models in that
+paper included orientifolds, which broke the supersymmetr y of the 4D effective theory
+fromN= 2 toN= 1. However, also in the present case the N= 1 approach is applicable
+and effectively we have used exactly the same procedure, i.e. u sing theN= 1 scalar
+fluctuations and obtaining the scalar potential from the N= 1 superpotential and K¨ ahler
+potential (see [55, 56, 57, 58]).7The reason is the following. The N= 2 scalar fluctuations
+in the vector multiplets are
+Jc=J−iB= (ki−ibi)ωi=tiωi, (4.2)
+whereωispan the left-invariant two-forms of the coset manifold. Th e orientifold projection
+of theN= 1 theory would then project out the scalar fluctuations comi ng from expanding
+oneventwo-forms, which are absent for the N= 1 theory on the coset manifolds under
+study. The scalar fluctuations in the N= 2 vector multiplets are thus exactly the same as
+the scalars in the chiral multiplets of the K¨ ahler moduli se ctor of the N= 1 theory. The
+6It is important to make the distinction between the number of supersymmetries of respectively the
+4D effective theory, the 10D compactifications, and their 4D t runcation (which are the solutions of the
+4D effective theory [36]). In the presence of one left-invari ant internal spinor, the effective theory will be
+N= 2 since this same spinor can be used in the 4+ 6 decomposition of both ten-dimensional Majorana-
+Weyl supersymmetry generators, but multiplied with indepe ndent four-dimensional spinors. On the other
+hand, for a certain compactification to preserve the supersy mmetry, certain differential conditions, which
+follow from putting the variations of the fermions to zero mu st be satisfied. In the presence of RR-fluxes,
+these conditions mix both ten-dimensional Majorana-Weyl s pinors, putting the four-dimensional spinors in
+both decompositions equal. A generic supersymmetric compa ctification therefore only preserves N= 1.
+Theσ= 2 supersymmetric K¨ ahler-Einstein solution on CP3on the other hand is non-generic in that it
+preserves N= 6, of which only one internal spinor is left-invariant unde r the action of Sp(2) and remains
+after truncation to 4D.
+7It is interesting to note that (in N= 1 language) all the D-terms vanish, so that the supersymmet ry
+breaking is purely due to F-terms. Indeed, in [58] it is shown thatD= 0 is equivalent to d H(e2A−ΦReΨ1) =
+0 in the generalized geometry formalism. For SU(3)-structu re this translates to d( e2A−ΦReΩ) = 0 and
+H∧ReΩ = 0, which is satisfied for our ansatz, eq. (2.1) and (2.2).
+– 12 –(a) Spectrum ofSp(2)
+S(U(2)×U(1))
+(b) Two extra modes of theSU(3)
+U(1)×U(1)-model
+Figure 3: Spectrum of left-invariant modes of the solutions onSp(2)
+S(U(2)×U(1))andSU(3)
+U(1)×U(1).
+expansion forms can then be chosen to bethe same as the Y(2−)
+iof [14]. Furthermore, there
+is one tensor multiplet, which contains the dilaton Φ, the tw o-formBµνand two axions ξ
+and˜ξcoming from the expansion of the RR-potential C3:
+C3=ξα+˜ξβ, (4.3)
+where a choice for αandβspanning the left-invariant three-forms would be Y(3−)and
+Y(3+)of [14] respectively. In the presence of Romans mass or ˆF2-flux the two-form Bµν
+becomes massive and cannot be dualized to a scalar. The dilat on and˜ξappear in a chiral
+multiplet of the complex moduli sector of the N= 1 theory, while Bµνandξare projected
+out by the orientifold. By using the N= 1 approach we thus loose the information on just
+one scalarξ. A proper N= 2 analysis would however learn that ξdoes not appear in the
+scalarpotential (seee.g.[36]), implyingthatitismassle ssandthusabovetheBreitenlohner-
+Freedman bound. Moreover, the scalar potential should be th e same whether it is obtained
+directly from reducing the 10D supergravity action (as in [5 9]) or whether it is obtained
+usingN= 2 orN= 1 technology8. Furthermore we note that the massless scalar field ξ
+not appearing in the potential is not a modulus, since it is ch arged [60, 61], and therefore
+eaten by a vector field becoming massive.
+Thespectraof left-invariant modesforSp(2)
+S(U(2)×U(1))andSU(3)
+U(1)×U(1)aredisplayed infigure
+3. The Breitenlohner-Freedman bound is indicated as a horiz ontal dashed line. The Sp(2)-
+model has six scalar fluctuations entering the potential: ki,biwithi= 1,2 from the two
+vector multiplets, and Φ ,˜ξfrom the universal hypermultiplet, while the SU(3)-model h as
+two more fluctuations from the extra vector multiplet. These two extra modes make a big
+difference for the stability analysis since one of them tends t o be below the Breitenlohner-
+Freedman bound for the purple and dark green solution. As a re sult, even though the
+solutions for the Sp(2)- and SU(3)-model take the same form, the SU(3)-model has more
+unstable solutions: compare figure 1 and 4.
+8The only potential difference between the latter two would be the contribution from the orientifold.
+We have checked that this contribution vanishes in the scala r potentials of [14] in the limit of the orientifold
+chargeµ→0.
+– 13 –Figure 4:SU(3)
+U(1)×U(1)-model: plot of aR4Din terms of the shape parameter σ. Unstable solutions
+are indicated in red.
+Inparticular, wenotethatthereductionoftheEnglert-typ esolutionisunstablefor σ=
+2 in the Sp(2)-model, in agreement with [38], since the M-the ory lift of the corresponding
+supersymmetric solution has eight Killing spinors. We inde ed find the same negative mass-
+squaredM2=−(4/5)|Λ|for the unstable mode as in that paper. On the other hand,
+forσ= 2/5 the Englert-type solution is stable against left-invaria nt fluctuations. This is
+still in agreement with [38] which relied on the existence of at least two Killing spinors,
+while the M-theory lift of the N= 1 supersymmetric solution at σ= 2/5 has only one
+Killing-spinor. For the SU(3)-model, all Englert-type sol utions turn out to be unstable
+(including the ones outside the condition (3.10)).
+We also investigated the stability of the additional soluti ons at the special point σ= 2
+found in [37]. We found that for the Sp(2)-model all these sol utions are stable against left-
+invariant fluctuations. For the SU(3)-model on theother han dit turnsout that thediscrete
+solutions ineqs.(3.16) and(3.17) ofthatreferenceareuns table, whilethecontinuous family
+of eq. (3.18) becomes unstable for
+γ2
+β2>5(75∓16√
+21)
+8217, (4.4)
+for the±sign choice in front of the square root in eq. (3.18) of that pa per respectively
+(note that the supersymmetric solution corresponds to the p ointγ2/β2= 0 in this family).
+Finally, we note that generically (i.e. unless an eigenvalu e is crossing zero at a special
+value forσ) all the plotted modes are massive. For a range of values for σone of the
+eigenvalues for the dark green and purple solution takes a sm all, but still non-zero value.
+– 14 –5. Conclusions
+In this paper we presented new families of non-supersymmetr ic AdS 4vacua. In fact,
+extrapolating from our analysis on these specific coset mani folds and under the assumption
+that a proper treatment of flux quantization does not kill muc h more vacua than in the
+supersymmetric case, it would seem that there are more of the se non-supersymmetric
+vacua than supersymmetric ones. This would imply that such v acua cannot be ignored
+in landscape studies. We have moreover shown that many of the m are stable against a
+specific set of fluctuations, namely the ones that can be expan ded in terms of left-invariant
+forms. If these vacua turn out to be stable against all fluctua tions they should also have
+a CFT-dual, which could be studied along the lines of [20], wh ere the three-dimensional
+Chern-Simons-matter theory dual to a particular highly sym metric non-supersymmetric
+vacuum was proposed. Furthermore, the nice property of some IIA vacua that all moduli
+enter the superpotential and thus can be stabilized at a clas sical level [15] also extends to
+our non-supersymmetric vacua.
+A next step would be to relax the constraint that the solution s should have the same
+geometry as the supersymmetric solution. It is also interes ting to investigate whether a
+similar ansatz and techniques can be used to look for tree-le vel dS-vacua [62].
+Acknowledgments
+We thank Davide Cassani for useful email correspondence and proofreading, and further-
+more Claudio Caviezel for active discussions and initial co llaboration. We would further
+like to thank the Max-Planck-Institut f¨ ur Physik in Munich , where both of the authors
+were affiliated during the bulk of the work on this paper. P.K. i s a Postdoctoral Fellow
+of the FWO – Vlaanderen. The work of P.K. is further supported in part by the FWO –
+Vlaanderen project G.0235.05 and in part by the Federal Office for Scientific, Technical and
+Cultural Affairs through the ’Interuniversity Attraction Po les Programme Belgian Science
+Policy’ P6/11-P. S.K. is supported by the SFB – Transregio 33 “The Dark Universe” by
+the DFG.
+A. SU(3)-structure
+A real non-degenerate two-form Jand a complex decomposable three-form Ω define an
+SU(3)-structure on the 6D manifold M6iff:
+Ω∧J= 0, (A.1a)
+Ω∧¯Ω =8i
+3!J∧J∧J∝negationslash= 0, (A.1b)
+and the associated metric is positive-definite. This metric is determined by Jand Ω as
+follows:
+gmn=−JmpIpn, (A.2)
+withIthe complex structure associated (in the way of [63]) to Ω. Th e volume-form is
+given by vol 6=1
+3!J3=−(i/8)Ω∧¯Ω.
+– 15 –Theintrinsictorsionofthemanifold M6decomposesintofivetorsionclasses W1,...,W5.
+Alternatively they correspond to the SU(3)-decomposition of the exterior derivatives of J
+and Ω [64]. Intuitively, they parameterize the failure of th e manifold to be of special
+holonomy, which can also be thought of as the deviation from c losure ofJand Ω. More
+specifically we have:
+dJ=3
+2Im(W1¯Ω)+W4∧J+W3,
+dΩ =W1J∧J+W2∧J+¯W5∧Ω,(A.3)
+whereW1is a scalar, W2is a primitive (1,1)-form, W3is a real primitive (1 ,2)+(2,1)-form,
+W4is a real one-form and W5a complex (1,0)-form. In this paper only the torsion classes
+W1,W2are non-vanishing and they are purely imaginary, so it will b e convenient to define
+W1,2so thatW1,2=iW1,2. A primitive (1,1)-form P(such asW2) transforms under the 8
+of SU(3) and satisfies
+P∧J∧J= 0. (A.4)
+The Hodge dual is given by
+⋆6P=−P∧J. (A.5)
+A primitive (1 ,2)(or (2,1))-formQon the other hand transforms as a 6(or¯6) under SU(3)
+and satisfies
+Q∧J= 0. (A.6)
+B. Type II supergravity
+The bosonic content of type II supergravity consists of a met ricG, a dilaton Φ, an NSNS
+three-form Hand RR-fields Fn. We use the democratic formalism of [65], in which the
+number of RR-fields is doubled, so that nruns over 0 ,2,4,6,8,10 in type IIA and over
+1,3,5,7,9 in IIB. We will often collectively denote the RR-fields with the polyform F=/summationtext
+nFn. We have also doubled the RR-potentials, collectively deno ted byC=/summationtext
+nC(n−1).
+These potentials satisfy F= dHC+me−B= (d +H∧)C+me−B. In type IIB there is
+of course no Romans mass m, so that the second term vanishes. In type IIA we find in
+particularF0=m.
+The bosonic part of the pseudo-action of the democratic form alism then simply reads
+S=1
+2κ2
+10/integraldisplay
+d10X√
+−G/braceleftbigg
+e−2Φ/bracketleftbigg
+R+4(dΦ)2−1
+2H2/bracketrightbigg
+−1
+4F2/bracerightbigg
+, (B.1)
+where we defined F2=/summationtext
+nF2
+nand the square of an l-formPas follows
+P2=P·P=1
+l!Pm1...mlPm1...ml, (B.2a)
+where the indices are raised with the inverse of the metric Gmnor the internal metric gmn
+(defined later on), depending on the context. In the followin g it will also be convenient to
+define:
+Pm·Pn=ιmP·ιnP=1
+(l−1)!Pmm2...mlPnm2...ml. (B.2b)
+– 16 –The extra degrees of freedom for the RR-fields in the democrat ic formalism have to be
+removed by hand by imposing the following duality condition at the level of the equations
+of motion after deriving them from the action (B.1):
+Fn= (−1)(n−1)(n−2)
+2⋆10F10−n. (B.3)
+That is why (B.1) is only a pseudo-action.
+The fermionic content consists of a doublet of gravitinos ψMand a doublet of dilatinos
+λ. The components of the doublets are of different chirality in t ype IIA and of the same
+chirality in type IIB.
+In this paper we look for vacuum solutions that take the form A dS4×M6. In principle
+there could also be a warp factor A, but it will always be constant for the solutions in this
+paper. We can choose it to be zero. The compactification ansat z for the metric then reads
+ds2
+10=GmndXmdXn= ds2
+4+gmndxmdxn, (B.4)
+where ds2
+4is the line-element for AdS 4andgmnis the metric on the internal space M6. For
+the RR-fluxes the ansatz becomes
+F=ˆF+vol4∧˜F, (B.5)
+whereˆFand˜Fonly have internal indices. The duality constraint (B.3) im plies that ˜Fis
+not independent of ˆF, and given by
+˜Fn= (−1)(n−1)(n−2)
+2⋆6ˆF6−n. (B.6)
+What we need in this paper are the type II equations of motion, which can be found
+from the pseudo-action (B.1). We use them as they are written down in [5] (originally they
+were obtained for massive type IIA in [35]), but take some lin ear combinations in order
+to further simplify then. Without source terms (i.e. we put jtotal= 0 in the equations of
+motion of [5]), they then read:
+dHF= 0 (Bianchi RR fields) , (B.7a)
+d−H⋆10F= 0 (eom RR fields) , (B.7b)
+dH= 0 (BianchiH), (B.7c)
+d/parenleftbig
+e−2Φ⋆10H/parenrightbig
+−1
+2/summationdisplay
+n⋆10Fn∧Fn−2= 0 (eom H), (B.7d)
+2R−H2+8/parenleftbig
+∇2Φ−(∂Φ)2/parenrightbig
+= 0 (dilaton eom) , (B.7e)
+2(∂Φ)2−∇2Φ−1
+2H2−e2Φ
+8/summationdisplay
+nnF2
+n= 0 (trace Einstein/dilaton eom) ,(B.7f)
+RMN+2∇M∂NΦ−1
+2HM·HN−e2Φ
+4/summationdisplay
+nFnM·FnN= 0 (B.7g)
+(Einstein eq./dilaton/trace) .
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\ No newline at end of file
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+arXiv:1001.0004v1  [quant-ph]  31 Dec 2009The Lie Algebraic Significance of
+Symmetric Informationally Complete Measurements
+D.M. Appleby, Steven T. Flammia and Christopher A. Fuchs
+Perimeter Institute for Theoretical Physics
+Waterloo, Ontario N2L 2Y5, Canada
+December 30, 2009
+Abstract
+Examplesofsymmetric informationallycomplete positiveoperatorva lued mea-
+sures (SIC-POVMs) have been constructed in every dimension ≤67. However,
+it remains an open question whether they exist in all finite dimensions. A SIC-
+POVM is usually thought of as a highly symmetric structure in quantum state
+space. However, its elements can equally well be regarded as a basis for the Lie
+algebra gl(d,C). In this paper we examine the resulting structure constants,
+which are calculated from the traces of the triple products of the S IC-POVM
+elements and which, it turns out, characterize the SIC-POVM up to unitary
+equivalence. We show that the structure constants have numero us remarkable
+properties. In particular we show that the existence of a SIC-POV M in di-
+mensiondis equivalent to the existence of a certain structure in the adjoint
+representation of gl( d,C). We hope that transforming the problem in this way,
+from a question about quantum state space to a question about Lie algebras,
+may help to make the existence problem tractable.
+Contents
+1. Introduction 1
+2. The Angle Tensors 7
+3. Spectral Decompositions 14
+4. TheQ-QTProperty 18
+5. Lie Algebraic Formulation of the Existence Problem 21
+6. The Algebra sl( d,C) 31
+7. Further Identities 33
+8. Geometrical Considerations 36
+9. TheP-PTProperty 49
+10. Conclusion 52
+11. Acknowledgements 53
+References 531
+1.Introduction
+Symmetric informationally complete positive operator-valued measu res (SIC-
+POVMs) present us with what is, simultaneously, one of the most inte resting, and
+one of the most difficult and tantalizing problems in quantum informatio n [1–46].
+SIC-POVMs are important practically, with applications to quantum t omography
+and cryptography [ 4,8,12,15,20,29], and to classical signal processing [ 24,36].
+However, without in any way wishing to impugn the significance of the a pplications
+which have so far been proposed, it appears to us that the interes t of SIC-POVMs
+stems less from these particular proposed uses than from rather broader, more gen-
+eral considerations: the sense one gets that SICs are telling us so mething deep,
+and hitherto unsuspected about the structure of quantum stat e space. In spite of
+its being the central object about which the rest of quantum mech anics rotates,
+and notwithstanding the efforts of numerous investigators [ 47], the geometry of
+quantum state space continues to be surprisingly ill-understood. T he hope which
+inspires our efforts is that a solution to the SIC problem will prove to b e the key,
+not just to SIC-POVMs narrowly conceived, but to the geometry o f state space in
+general. Such things are, by nature, unpredictable. However, it is not unreasonable
+to speculate that a better theoretical understanding of the geo metry of quantum
+state space might have important practical consequences: not o nly the applica-
+tions listed above, but perhaps other applications which have yet to be conceived.
+On a more foundational level one may hope that it will lead to a much imp roved
+understanding of the conceptual message of quantum mechanics [7,43,45,48].
+Having said why we describe the problem as interesting, let us now exp lain why
+we describe it as tantalizing. The trouble is that, although there is an abundance of
+reasons for suspecting that SIC-POVMs exist in every finite dimens ion (exact and
+high-precision numerical examples [ 1,2,5,11,16,19,28,39,46] having now been
+constructed in every dimension up to 67), and in spite of the intense efforts of many
+people [1–46] extending over a period of more than ten years, a general existe nce
+proof continues to elude us. In their seminal paper on the subject , published in
+2004, Renes et al[5] say “A rigorous proof of existence of SIC-POVMs in all finite
+dimensions seems tantalizingly close, yet remains somehow distant.” T hey could
+have said the same if they were writing today.
+The purposeofthis paperis totryto takeourunderstandingofSI C mathematics
+(as it might be called) a little further forward. The research we repo rt began with
+a chance numerical discovery made while we were working on a differen t problem.
+Pursuing that initial numerical hint we uncovered a rich and interest ing set of
+connections between SIC-POVMs in dimension dand the Lie Algebra gl( d,C). The
+existence of these connections came as a surprise to us. However , in retrospect it
+is, perhaps, not so surprising. Interest in SIC-POVMs has, to dat e, focused on the
+fact that an arbitrary density matrix can be expanded in terms of a SIC-POVM.
+However, a SIC-POVM in dimension ddoes in fact provide a basis, not just for the
+space of density matrices, but for the space of all d×dcomplex matrices— i.e.the
+Liealgebragl( d,C). Boykin et al[49] haverecentlyshownthatthere isaconnection
+betweentheexistenceproblemformaximalsetsofMUBs(mutuallyu nbiasedbases)
+and the theory of Lie algebras. Since SIC-POVMs share with MUBs th e property
+of being highly symmetrical structures in quantum state space it mig ht have been
+anticipated that there are also some interesting connections betw een SIC-POVMs
+and Lie algebras.2
+Our main result (proved in Sections 3,4and5) is that the proposition, that a
+SIC-POVM exists in dimension d, is equivalent to a proposition about the adjoint
+representation of gl( d,C). Our hope is that transforming the problem in this way,
+from a question about quantum state space to a question about Lie algebras, may
+help to make the SIC-existence problem tractable. But even if this h ope fails to
+materialize we feel that this result, along with the many other result s we obtain,
+provides some additional insight into these structures.
+Inddimensional Hilbert space Hda SIC-POVM is a set of d2operatorsE1,
+...,Ed2of the form
+Er=1
+dΠr (1)
+where the Π rare rank-1 projectors with the property
+Tr(ΠrΠs) =/braceleftigg
+1r=s
+1
+d+1r/ne}ationslash=s(2)
+We will refer to the Π ras SIC projectors, and we will say that {Πr:r= 1,...,d2}
+is a SIC set.
+It follows from this definition that the Ersatisfy
+d2/summationdisplay
+r=1Er=I (3)
+(sotheyconstitute aPOVM),andthattheyarelinearlyindependen t (sothePOVM
+is informationally complete).
+It is an open question whether SIC-POVMs exist for all values of d. However,
+examples have been constructed analytically in dimensions 2–15 inclus ive [1,2,11,
+16,19,28,39,46], and in dimensions 19, 24, 35 and 48 [ 16,46]. Moreover, high
+precisionnumerical solutionshave been constructed in dimensions 2 –67inclusive [ 5,
+46]. Thislendssomeplausibilitytothe speculationthat theyexistinalldime nsions.
+For a comprehensive account of the current state of knowledge in this regard, and
+many new results, see the recent study by Scott and Grassl [ 46].
+All known SIC-POVMs have a group covariance property. In other words, there
+exists
+(1) a group Ghavingd2elements
+(2) a projective unitary representation of GonHd:i.e.a mapg→UgfromG
+to the set of unitaries such that Ug1Ug2∼Ug1g2for allg1,g2(where the
+notation “ ∼” means “equals up to a phase”)
+(3) a normalized vector |ψ/an}bracketri}ht(the fiducial vector)
+such that the SIC-projectors are given by
+Πg=Ug|ψ/an}bracketri}ht/an}bracketle{tψ|U†
+g (4)
+(where we label the projector by the group element g, rather than the integer ras
+above).
+Most known SIC-POVMs are covariant under the action of the Weyl- Heisenberg
+group (though not all—see Renes et al[5] and, for an explicit example of a non
+Weyl-Heisenberg SIC-POVM, Grassl [ 19]). Here the group is Zd×Zd, and the
+projective representation is p→Dp, wherep= (p1,p2)∈Zd×ZdandDpis the3
+corresponding Weyl-Heisenberg displacement operator
+Dp=d−1/summationdisplay
+rτ(2r+p1)p2|r+p1/an}bracketri}ht/an}bracketle{tr| (5)
+In this expression τ=eiπ(d+1)
+d, the vectors |0/an}bracketri}ht,...|d−1/an}bracketri}htare an orthonormal basis,
+and the addition in |r+p1/an}bracketri}htis modulod. For more details see, for example, ref. [ 16].
+One should not attach too much weight to the fact that all known SI C-POVMs
+have a group covariance property as this may only reflect the fact that group co-
+variant SIC-POVMs are much easier to construct. So in this paper w e will try to
+prove as much as we can without assuming such a property. One pot ential benefit
+ofthis attitude is that, by accumulatingenough facts about SIC-P OVMsin general,
+we may eventually get to the point where we can answer the question , whether all
+SIC-POVMs actually do have a group covariance property.
+The fact that the d2operatorsΠ rare linearly independent means that they form
+a basis for the complex Lie algebra gl( d,C) (the set of all operators acting on Hd).
+Since the Π rare Hermitian, then iΠrforms a basis also for the real Lie algebra
+u(d) (the set of all anti-Hermitian operators acting on Hd). So for any operator
+A∈gl(d,C) there is a unique set of expansion coefficients arsuch that
+A=d2/summationdisplay
+r=1arΠr (6)
+To find the expansion coefficients we can use the fact that
+d2/summationdisplay
+s=1Tr(ΠrΠs)/parenleftbiggd+1
+dδst−1
+d2/parenrightbigg
+=δrt (7)
+from which it follows
+ar=d+1
+dTr(ΠrA)−1
+dTr(A) (8)
+Specializing to the case A= ΠrΠswe find
+ΠrΠs=d+1
+d
+d2/summationdisplay
+t=1TrstΠt
+−dδrs+1
+d+1I (9)
+where
+Trst= Tr(Π rΠsΠt) (10)
+To a large extent this paper consists in an exploration of the proper ties of these
+important quantities, which we will refer to as the triple products. T hey are inti-
+mately related to the geometric phase, in which context they are us ually referred
+to as 3-vertex Bargmann invariants (see Mukunda et al[50], and references cited
+therein). We have, as an immediate consequence of the definition,
+Trst=Ttrs=Tstr=T∗
+rts=T∗
+tsr=T∗
+srt (11)
+It is convenient to define
+Jrst=d+1
+d(Trst−T∗
+rst) (12)
+Rrst=d+1
+d(Trst+T∗
+rst) (13)4
+SoJrstis imaginary and completely anti-symmetric; Rrstis real and completely
+symmetric. Both these quantities play a significant role in the theory . It follows
+from Eq. ( 9) that
+[Πr,Πs] =d2/summationdisplay
+t=1JrstΠt (14)
+So theJrstare structure constants for the Lie algebra gl( d,C). As an immediate
+consequence of this they satisfy the Jacobi identity:
+d2/summationdisplay
+b=1/parenleftbig
+JrsbJtba+JstbJrba+JtrbJsba/parenrightbig
+= 0 (15)
+for allr,s,t,a. The Jacobi identity holds for any representation of the structu re
+constants. In the following sections we will derive many other identit ies which are
+specific to this particular representation.
+Turning to the quantities Rrst, it follows from Eq. ( 9) that they feature in the
+expression for the anti-commutator
+{Πr,Πs}=/summationdisplay
+tRrstΠt−2(dδrs+1)
+d+1I (16)
+They also play an important role in the description of quantum state s pace. Let
+ρbe any density matrix and let pr=1
+dTr(Πrρ) be the probability of obtaining
+outcomerin the measurement described by the POVM with elements1
+dΠr. Then
+it follows from Eq. ( 8) thatρcan be reconstructed from the probabilities by
+ρ=d2/summationdisplay
+r=1/parenleftbigg
+(d+1)pr−1
+d/parenrightbigg
+Πr (17)
+Suppose, now, that the prareanyset ofd2real numbers. So we do not assume
+that theprare even probabilities, let alone the probabilities coming from a density
+matrix according to the prescription pr=1
+dTr(Πrρ). Then it is shown in ref. [ 34]
+that theprare in fact the probabilities coming from a pure state if and only if they
+satisfy the two conditions
+d2/summationdisplay
+r=1p2
+r=2
+d(d+1)(18)
+d2/summationdisplay
+r,s,t=1Rrstprpspt=2(d+7)
+d(d+1)2(19)
+Let us look at the quantities JrstandRrstin a little more detail. For each r
+choose a unit vector |ψr/an}bracketri}htsuch that Π r=|ψr/an}bracketri}ht/an}bracketle{tψr|. Then the Gram matrix for these
+vectors is of the form
+Grs=/an}bracketle{tψr|ψs/an}bracketri}ht=Krseiθrs(20)
+where the matrix θrsis anti-symmetric and
+Krs=/radicalbigg
+dδrs+1
+d+1(21)
+Note that the SIC-POVM does not determine the angles θrsuniquely since making
+the replacements |ψr/an}bracketri}ht →eiφr|ψr/an}bracketri}htleaves the SIC-POVM unaltered, but changes5
+the angles θrsaccording to the prescription θrs→θrs−φr+φs. This freedom
+to rephase the vectors |ψr/an}bracketri}htis not usually important. However, it sometimes has
+interesting consequences (see Section 9). It can be thought of as a kind of gauge
+freedom.
+The Gram matrix satisfies an important identity. Every SIC-POVM ha s the
+2-design property [ 5,17]
+d2/summationdisplay
+r=1Πr⊗Πr=2d
+d+1Psym (22)
+wherePsymis the projector onto the symmetric subspace of Hd⊗Hd. Expressed
+in terms of the Gram matrix this becomes
+d2/summationdisplay
+r=1Gs1rGs2rGrt1Grt2=d
+d+1/parenleftbig
+Gs1t1Gs2t2+Gs1t2Gs2t1/parenrightbig
+(23)
+Turning to the triple products we have
+Trst=GrsGstGtr=KrsKstKtreiθrst(24)
+where
+θrst=θrs+θst+θtr (25)
+Note that the tensor θrstis completely anti-symmetric. In particular θrst= 0 if any
+two of the indices are the same. Note also that re-phasing the vect ors|ψr/an}bracketri}htleaves
+the tensors Trstandθrstunchanged. They are in that sense gauge invariant.
+Finally, we have the following expressions for JrstandRrst:
+Jrst=2i
+d√
+d+1sinθrst (26)
+Rrst=2(d+1)
+dKrsKstKtrcosθrst (27)
+Like the triple products, JrstandRrstare gauge invariant.
+For later reference let us note that the matrix Jr, with matrix elements
+(Jr)st=Jrst (28)
+is the adjoint representative of Π rin the SIC-projector basis:
+adΠrΠs= [Πr,Πs] =d2/summationdisplay
+t=1JrstΠt (29)
+It can be seen that all the interesting features of the tensor Grs(respectively,
+the tensors Trst,JrstandRrst) are contained in the order-2 angle tensor θrs(re-
+spectively, the order-3 angle tensor θrst). It is also easy to see that, for any unitary
+U, the transformation
+Πr→UΠrU†(30)
+leaves the angle tensors invariant. This suggests that we shift our focus from indi-
+vidual SIC-POVMs to families of unitarily equivalent SIC-POVMs—SIC- families,
+as we will call them for short.
+We begin our investigation in Section 2by giving necessary and sufficient con-
+ditions for an arbitrary tensor θrs(respectively θrst) to be the rank-2 (respectively
+rank-3) angle tensor corresponding to a SIC-family. We also show t hat either angle
+tensor uniquely determines the corresponding SIC-family. Finally we describe a6
+method for reconstructing the SIC-family, starting from a knowle dge of either of
+the two angle tensors.
+In Sections 3,4and5we prove the central result of this paper: namely, that
+the existence of a SIC-POVM in dimension dis equivalent to the existence of a
+certain very special set of matrices in the adjoint representation of gl(d,C). In
+Section3we show that, for any SIC-POVM, the adjoint matrices Jrhave the
+spectral decomposition
+Jr=Qr−QT
+r (31)
+whereQris a rankd−1 projector which has the remarkable property of being
+orthogonal to its own transpose:
+QrQT
+r= 0 (32)
+We refer to this feature of the adjoint matrices as the Q-QTproperty. In Section 3
+we also show that from a knowledge of the Jmatrices it is possible to reconstruct
+the corresponding SIC-family. In Section 4we characterize the general class of
+projectors which have the property of being orthogonal to their own transpose.
+Then, in Section 5, we prove a converse of the result established in Section 3. The
+Q-QTproperty is not completely equivalent to the property of being a SIC set.
+However, it turns out that it is, in a certain sense, very nearly equiv alent. To be
+more specific: let Lrbe any set of d2Hermitian operators which constitute a basis
+for gl(d,C) and letCrbe the adjoint representative of Lrin this basis. Then the
+necessary and sufficient condition for the Crto have the spectral decomposition
+Cr=Qr−QT
+r (33)
+whereQris a rankd−1 projector such that QrQT
+r= 0 is that there exists a
+SIC set Π rsuch thatLr=ǫr(Πr+αI) for some fixed number α∈Rand signs
+ǫr=±1. In particular, the existence of an Hermitian basis for gl( d,C) having the
+Q-QTproperty is both necesary and sufficient for the existence of a SIC -POVM in
+dimensiond.
+In Section 6we digress briefly, and consider sl( d,C) (the Lie algebra consisting
+of all trace-zero d×dcomplex matrices). As we have explained, this paper is
+motivated by the hope that a Lie algebraic perspective will cast light o n the SIC-
+existence problem, rather than by an interest in Lie algebras as suc h. We focus on
+gl(d,C) because that is the casewherethe connection with SIC-POVMsse ems most
+straightforward. However a SIC-POVM also gives rise to an interes ting geometrical
+structure in sl( d,C), as we show in Section 6.
+In Section 7we derive a number of additional identities satisfied by the Jand
+Qmatrices.
+The complex projectors Qr,QT
+rand the real projector Qr+QT
+rdefine three
+families of subspaces. It turns out that there are some interestin g geometrical
+relationships between these subspaces, which we study in Section 8.
+Finally, in Section 9we show that, with the appropriate choice of gauge, the
+Gram matrix corresponding to a Weyl-Heisenberg covariant SIC-fa mily has a fea-
+ture analogous to the Q-QTproperty, which we call the P-PTproperty. It is an
+open question whether this result generalizes to other SIC-families , not covariant
+with respect to the Weyl-Heisenberg group.7
+2.The Angle Tensors
+The purpose of this section is to establish the necessary and sufficie nt conditions
+for an arbitrary tensor θrs(respectively θrst) to be the order-2 (respectively order-
+3) angle tensor for a SIC-family. We will also show that either one of t he angle
+tensors is enough to uniquely determine the SIC-family. Moreover, we will describe
+explicit procedures for reconstructing the family, starting from a knowledge of one
+of the angle tensors.
+We begin by considering the general class of POVMs (not just SIC-P OVMs)
+which consist of d2rank-1 elements. A POVM of this type is thus defined by a set
+ofd2vectors|ξ1/an}bracketri}ht,...,|ξd2/an}bracketri}htwith the property
+d2/summationdisplay
+r=1|ξr/an}bracketri}ht/an}bracketle{tξr|=I (34)
+Note that/summationtextd2
+r=1/vextenddouble/vextenddouble|ξr/an}bracketri}ht/vextenddouble/vextenddouble2=d, so the vectors |ξr/an}bracketri}htcannot all be normalized. In the
+particular case of a SIC-POVM the vectors all have the same norm/vextenddouble/vextenddouble|ξr/an}bracketri}ht/vextenddouble/vextenddouble=1√
+d.
+However in the general case they may have different norms.
+Given a set of such vectors consider the Gram matrix
+Prs=/an}bracketle{tξr|ξs/an}bracketri}ht (35)
+Clearly the Gram matrix cannot determine the POVM uniquely since if Uis any
+unitary operator then the vectors U|ξr/an}bracketri}htwill define another POVM having the same
+Gram matrix. However, the theorem we now prove shows that this is the only free-
+dom. In other words, the Gram matrix fixes the POVM up to unitary e quivalence.
+The theorem also provides us with a criterion for deciding whether an arbitrary
+d2×d2matrixPis the Gram matrix corresponding to a POVM of the specified
+type. As a corollary this will give us a criterion for deciding whether an arbitrary
+tensorθrsis specifically the order-2 angle tensor for a SIC-family.
+Theorem 1. LetPbe anyd2×d2Hermitian matrix. Then the following conditions
+are equivalent:
+(1)Pis a rankdprojector.
+(2)Psatisfies the trace identities
+Tr(P) = Tr(P2) = Tr(P3) = Tr(P4) =d (36)
+(3)Pis the Gram matrix for a set of d2vectors|ξr/an}bracketri}ht(not all normalized) such
+that|ξr/an}bracketri}ht/an}bracketle{tξr|is a POVM:
+/an}bracketle{tξr|ξs/an}bracketri}ht=Prs (37)
+d2/summationdisplay
+r=1|ξr/an}bracketri}ht/an}bracketle{tξr|=I (38)
+SupposePsatisfies these conditions. To construct a POVM correspondi ng toP
+let thedcolumn vectors
+ξ11
+ξ12
+...
+ξ1d2
+,
+ξ21
+ξ22
+...
+ξ2d2
+,...,
+ξd1
+ξd2
+...
+ξdd2
+(39)8
+be any orthonormal basis for the subspace onto which Pprojects. Define
+|ξr/an}bracketri}ht=d/summationdisplay
+a=1ξ∗
+ar|a/an}bracketri}ht (40)
+where the vectors |a/an}bracketri}htare any orthonormal basis for Hd. ThenPis the Gram matrix
+for the vectors |ξ1/an}bracketri}ht,...,|ξd2/an}bracketri}ht. Moreover, the necessary and sufficient condition for
+any other set of vectors |η1/an}bracketri}ht,...,|ηd2/an}bracketri}htto have Gram matrix Pis that there exist a
+unitary operator Usuch that
+|ηr/an}bracketri}ht=U|ξr/an}bracketri}ht (41)
+for allr.
+Proof.We begin by showing that (3) = ⇒(1). Suppose |ξ1/an}bracketri}ht,...|ξd2/an}bracketri}htis any set of
+d2vectors such that |ξr/an}bracketri}ht/an}bracketle{tξr|is a POVM. So
+d2/summationdisplay
+r=1|ξr/an}bracketri}ht/an}bracketle{tξr|=I (42)
+Let
+Prs=/an}bracketle{tξr|ξs/an}bracketri}ht (43)
+be the Gram matrix. Then Pis Hermitian. Moreover, P2=Psince
+d2/summationdisplay
+t=1PrtPts=/an}bracketle{tξr|
+d2/summationdisplay
+t=1|ξt/an}bracketri}ht/an}bracketle{tξs|
+|ξr/an}bracketri}ht
+=/an}bracketle{tξr|ξs/an}bracketri}ht
+=Prs (44)
+Also
+Tr(P) =d2/summationdisplay
+r=1/an}bracketle{tξr|ξr/an}bracketri}ht=d (45)
+(as can be seen by taking the trace on both sides of Eq. ( 42)). SoPis a rank-d
+projector.
+We next show that (1) = ⇒(3). LetPbe a rank-dprojector, and let the d
+column vectors
+ξ11
+ξ12
+...
+ξ1d2
+,
+ξ21
+ξ22
+...
+ξ2d2
+,...,
+ξd1
+ξd2
+...
+ξdd2
+(46)
+be an orthonormal basis for the subspace onto which it projects. So
+d2/summationdisplay
+r=1ξ∗
+arξbr=δab (47)
+for alla,b, and
+d2/summationdisplay
+a=1ξarξ∗
+as=Prs (48)9
+for allr,s. Now let |ξ1/an}bracketri}ht,...|ξd2/an}bracketri}htbe the vectors defined by Eq. ( 40). Then it follows
+from Eq. ( 47) that
+d2/summationdisplay
+r=1|ξr/an}bracketri}ht/an}bracketle{tξr|=d/summationdisplay
+a,b=1
+d2/summationdisplay
+r=1ξ∗
+arξbr
+|a/an}bracketri}ht/an}bracketle{tb|
+=d/summationdisplay
+a=1|a/an}bracketri}ht/an}bracketle{ta|
+=I (49)
+implying that |ξr/an}bracketri}ht/an}bracketle{tξr|is POVM. Also, it follows from Eq. ( 48) that
+/an}bracketle{tξr|ξs/an}bracketri}ht=d/summationdisplay
+a=1ξarξ∗
+as=Prs (50)
+implying that the |ξr/an}bracketri}hthave Gram matrix P.
+We next turn to condition (2). The fact that (1) = ⇒(2) is immediate. To
+prove the reverse implication observe that condition (2) implies
+Tr(P4)−2Tr(P3)+Tr(P2) = 0 (51)
+Letλ1,...,λ d2be the eigenvalues of P. Then Eq. ( 51) implies
+d2/summationdisplay
+r=1λ2
+r(λr−1)2= 0 (52)
+It follows that each eigenvalue is either 0 or 1. Since Tr( P) =dwe must have d
+eigenvalues = 1 and the rest all = 0. So Pis a rank-dprojector.
+It remains to show that the POVM corresponding to a given rank- dprojector
+is unique up to unitary equivalence. To prove this let Pbe a rank-dprojector, let
+|ξr/an}bracketri}htbe the vectors defined by Eq. ( 40), and let |η1/an}bracketri}ht,...,|ηd2/an}bracketri}htbe any other set of
+vectors such that
+/an}bracketle{tηr|ηs/an}bracketri}ht=Prs (53)
+for allr,s. Define
+ηar=/an}bracketle{tηr|a/an}bracketri}ht (54)
+Then
+d2/summationdisplay
+r=1η∗
+arηbr=/an}bracketle{ta|
+d2/summationdisplay
+r=1|ηr/an}bracketri}ht/an}bracketle{tηr|
+|b/an}bracketri}ht=δab (55)
+(because |ηr/an}bracketri}ht/an}bracketle{tηr|is a POVM) and
+d/summationdisplay
+a=1ηarη∗
+as=Prs (56)
+(because the |ηr/an}bracketri}hthave Gram matrix P). So thedcolumn vectors
+
+η11
+η12
+...
+η1d2
+,
+η21
+η22
+...
+η2d2
+,...,
+ηd1
+ηd2
+...
+ηdd2
+(57)10
+are an orthonormal basis for the subspace onto which Pprojects. But the column
+vectors 
+ξ11
+ξ12
+...
+ξ1d2
+,
+ξ21
+ξ22
+...
+ξ2d2
+,...,
+ξd1
+ξd2
+...
+ξdd2
+(58)
+are also an orthonormal basis for this subspace. So there must ex ist ad×dunitary
+matrixUabsuch that
+ηar=d/summationdisplay
+b=1Uabξbr (59)
+for alla,r. Define
+U=d/summationdisplay
+a,b=1U∗
+ab|a/an}bracketri}ht/an}bracketle{tb| (60)
+Then
+|ηr/an}bracketri}ht=U|ξr/an}bracketri}ht (61)
+for allr. /square
+In the case of a SIC-POVM we have
+|ξr/an}bracketri}ht=1√
+d|ψr/an}bracketri}ht (62)
+where the vectors |ψr/an}bracketri}htare normalized, and
+Prs=1
+dGrs=1
+dKrseiθrs(63)
+whereGis the Gram matrix of the vectors |ψr/an}bracketri}htandθrsis the order-2 angle tensor.
+In the sequel we will distinguish these matrices by referring to Gas the Gram
+matrix and Pas the Gram projector.
+We have
+Corollary 2. Letθrsbe a real anti-symmetric tensor. Then the following state-
+ments are equivalent:
+(1)θrsis an order- 2angle tensor corresponding to a SIC-family.
+(2)θrssatisfies
+d2/summationdisplay
+t=1KrtKtsei(θrt+θts)=dKrseiθrs(64)
+for allr,s.
+(3)θrssatisfies
+d2/summationdisplay
+r,s,t=1KrsKstKtrei(θrs+θst+θtr)=d4(65)
+and
+d2/summationdisplay
+r,s,t,u=1KrsKstKtuKurei(θrs+θst+θtu+θur)=d5(66)11
+LetΠr,Π′
+rbe two different SIC-sets, and let θrs,θ′
+rsbe corresponding order- 2
+angle tensors. Then there exists a unitary Usuch that
+Π′
+r=UΠrU†(67)
+for allrif and only if
+θ′
+rs=θrs−φr+φs (68)
+for some arbitrary set of phase angles φr(in other words two SIC-sets are unitarily
+equivalent if and only if their order- 2angle tensors are gauge equivalent).
+A SIC-family can be reconstructed from its order- 2angle tensor θrsby calculating
+an orthonormal basis for the subspace onto which the Gram pro jector
+Prs=1
+dKrseiθrs(69)
+projects, as described in Theorem 1.
+Remark. The sense in which we areusing the term “gaugeequivalence”is explain ed
+in the passage immediately following Eq. ( 21).
+Note that condition (2) imposes d2(d2−1)/2 independent constraints (taking
+account of the anti-symmetry of θrs). Condition (3), by contrast, only imposes 2
+independent constraints. It is to be observed, however, that th e price we pay for
+the reduction in the number of equations is that Eqs. ( 65) and (65) are respectively
+cubic and quartic in the phases, whereas Eq. ( 64) is only quadratic.
+Proof.Letθrsbe an arbitrary anti-symmetric tensor, and define
+Prs=1
+dKrseiθrs(70)
+The anti-symmetry of θrsmeans that Pis automatically Hermitian. So it follows
+from Theorem 1that a necessary and sufficient condition for Prsto be a rank- d
+projector, and for θrsto be the order-2 angle tensor of a SIC-family, is that
+d2/summationdisplay
+t=1KrtKtsei(θrt+θts)=dKrseiθrs(71)
+for allr,s.
+To prove the equivalence of conditions (1) and (3) note that the co nditions
+Tr(P) = Tr(P2) =dare an automatic consequence of Phaving the specified form.
+So it follows from Theorem 1thatθrsis the order-2 angle tensor of a SIC-family if
+and only if Eqs. ( 65) and (66) are satisfied.
+Now let Πr, Π′
+rbe two SIC-sets and let θrs,θ′
+rsbe order-2 angle tensors corre-
+sponding to them. Then there exist normalized vectors |ψr/an}bracketri}ht,|ψ′
+r/an}bracketri}htsuch that
+Πr=|ψr/an}bracketri}ht/an}bracketle{tψr| Π′
+r=|ψ′
+r/an}bracketri}ht/an}bracketle{tψ′
+r| (72)
+for allr, and
+/an}bracketle{tψr|ψs/an}bracketri}ht=Krseiθrs/an}bracketle{tψ′
+r|ψ′
+s/an}bracketri}ht=Krseiθ′
+rs (73)
+for allr,s.
+Suppose, first of all, that there exists a unitary Usuch that
+Π′
+r=UΠrU†(74)12
+Then there exist phase angles φrsuch that
+|ψ′
+r/an}bracketri}ht=eiφrU|ψr/an}bracketri}ht (75)
+for allr, which is easily seen to imply that
+θ′
+rs=θrs−φr+φs (76)
+for allr,s. Soθrs,θ′
+rsare gauge equivalent.
+Conversely, suppose there exist phase angles φrsuch that
+θ′
+rs=θrs−φr+φs (77)
+Define
+|ψ′′
+r/an}bracketri}ht=e−iφr|ψ′
+r/an}bracketri}ht (78)
+Then
+/an}bracketle{tψ′′
+r|ψ′′
+s/an}bracketri}ht=Krseiθrs=/an}bracketle{tψr|ψs/an}bracketri}ht (79)
+for allr,s. So it follows from Theorem 1that there exists a unitary Usuch that
+|ψ′′
+r/an}bracketri}ht=U|ψr/an}bracketri}ht (80)
+for allr. Consequently
+Π′
+r=|ψ′′
+r/an}bracketri}ht/an}bracketle{tψ′′
+r|=UΠrU†(81)
+for allr. So Πrand Π′
+rare unitarily equivalent. /square
+We now turn to the order-3 angle tensors. We have
+Theorem 3. Letθrstbe a real completely anti-symmetric tensor. Then the follow -
+ing conditions are equivalent:
+(1)θrstis the order- 3angle tensor for a SIC-family
+(2)For some fixed aand allr,s,t
+θars+θast+θatr=θrst (82)
+and for all r,s
+d2/summationdisplay
+t=1KrtKtseiθrst=dKrs (83)
+(3)For some fixed aand allr,s,t
+θars+θast+θatr=θrst (84)
+and
+d2/summationdisplay
+r,s,t=1KrsKstKtreiθrst=d4(85)
+d2/summationdisplay
+r,s,t,u=1KrsKstKtuKurei(θrst+θtur)=d5(86)13
+LetΠr,Π′
+rbe two different SIC-sets and let θrst,θ′
+rstbe the corresponding order-
+3angle tensors. Then the necessary and sufficient condition fo r there to exist a
+unitaryUsuch that
+Π′
+r=UΠrU†(87)
+for allris thatθ′
+rst=θrstfor allr,s,t(in other words two SIC-sets are unitarily
+equivalent if and only if their order- 3angle tensors are identical).
+Letθrstbe the order- 3angle tensor corresponding to a SIC-family. Then the
+order-2angle tensor is given by (up to gauge freedom)
+θrs=θars (88)
+for any fixed a, from which the SIC-family can be reconstructed using the me thod
+described in Theorem 1.
+Remark. Unlike the order-2tensor, the order-3angletensoris gaugeinvar iant. This
+means that it provides what is, in many ways, a more useful charact erization of
+the SIC-family. For that reason we will be almost exclusively concern ed with the
+order-3 tensor in the remainder of this paper.
+Proof.The fact that (1) = ⇒(2) is an immediate consequence of the definition of
+theorder-3angletensorandcondition(2)ofCorollary 2. Toprovethat(2) = ⇒(1)
+letθrstbe a completely anti-symmetric tensor such that condition (2) holds . Define
+θrs=θars (89)
+for allr,s. Then Eq. ( 83) implies
+d2/summationdisplay
+t=1KrtKtsei(θrt+θts)=eiθrs
+d2/summationdisplay
+t=1KrtKtseiθrst
+∗
+=dKrseiθrs(90)
+for allr,s. It follows from Corollary 2thatθrsis the order-2 and θrstthe order-3
+angle tensor of a SIC-family.
+The equivalence of conditions (1) and (3) is proved similarly.
+It remains to show that two SIC-sets are unitarily equivalent if and o nly if
+their order-3 angle tensors are identical. To see this let Πr=|ψr/an}bracketri}ht/an}bracketle{tψr|and Π′
+r=
+|ψ′
+r/an}bracketri}ht/an}bracketle{tψ′
+r|be two different SIC-sets having the same order-3 angle tensor θrst. Let
+θrs(respectively θ′
+rs) be the order-2 angle tensor corresponding to the vectors |ψr/an}bracketri}ht
+(respectively |ψ′
+r/an}bracketri}ht). Choose some fixed index a. We have
+θ′
+ar+θ′
+sa+θ′
+rs=θar+θsa+θrs (91)
+for allr,s. Consequently
+θ′
+rs=θrs+φr−φs (92)
+for allr,s, where
+φr=θar−θ′
+ar (93)
+Soθ′
+rsandθrsare gauge equivalent. It follows from Corollary 2that Πrand Π′
+rare
+unitarily equivalent. Conversely, suppose that Πrand Π′
+rare unitarily equivalent,
+and letθrs,θ′
+rsbe order-2 angle tensors corresponding to them. It follows from
+Corollary 2thatθrsandθ′
+rsare gauge equivalent. It is then immediate that the
+order-3 angle tensors are identical. /square14
+Finally, let us note that when expressed in terms of the triple produc ts Eq. (83)
+reads
+d2/summationdisplay
+t=1Trst=dK2
+rs (94)
+while Eq. ( 85) reads
+d2/summationdisplay
+r,s,t=1Trst=d4(95)
+For Eq. ( 86) we have to work a little harder. We have
+d2/summationdisplay
+r,s,t,u=11
+K2
+rtTrstTtur=d5(96)
+from which it follows
+d5=d2/summationdisplay
+r,s,t,u=1/parenleftbig
+−dδrt+d+1/parenrightbig
+TrstTtur
+= (d+1)d2/summationdisplay
+r,s,t,u=1TrstTtur−dd2/summationdisplay
+r,s,u=1K2
+rsK2
+ru
+= (d+1)d2/summationdisplay
+r,s,t,u=1TrstTtur−d5(97)
+Consequently
+d2/summationdisplay
+s,u=1Tr/parenleftbig
+TsTu/parenrightbig
+=d2/summationdisplay
+r,s,t,u=1TrstTtur=2d5
+d+1(98)
+This equation be alternatively written
+d2/summationdisplay
+r,s=1Tr/parenleftbig
+TrTs/parenrightbig
+=2d5
+d+1(99)
+whereTris the matrix with matrix elements ( Tr)uv=Truv.
+When they are written like this, in terms of the triple products, the f act that
+Eq. (94) implies Eqs. ( 95) and (98) becomes almost obvious. The reverse implica-
+tion, by contrast, is rather less obvious.
+3.Spectral Decompositions
+LetTr,Jr,Rrbe thed2×d2matrices whose matrix elements are
+(Tr)st=Trst (Jr)st=Jrst (Rr)st=Rrst(100)
+whereJrst,RrstarethequantitiesdefinedbyEqs.( 12)and(13). SoJristheadjoint
+representation matrix of Π r. In this section we derive the spectral decompositions
+of these matrices. To avoid confusion we will use the notation |ψ/an}bracketri}htto denote a ket in
+ddimensional Hilbert space Hd, and/bardblψ/an}bracketri}ht/an}bracketri}htto denote a ket in d2dimensional Hilbert15
+spaceHd2. In terms of this notation the spectral decompositions will turn ou t to
+be:
+Tr=d
+d+1Qr+2d
+d+1/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (101)
+Jr=Qr−QT
+r (102)
+Rr=Qr+QT
+r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (103)
+In these expressions the vector /bardbler/an}bracketri}ht/an}bracketri}htis normalized, and its components in the stan-
+dard basis are all real. Qris a rankd−1 projector such that
+Qr/bardbler/an}bracketri}ht/an}bracketri}ht=QT
+r/bardbler/an}bracketri}ht/an}bracketri}ht= 0 (104)
+and which has, in addition, the remarkable property of being orthog onal to its own
+transpose (also a rank d−1 projector):
+QrQT
+r= 0 (105)
+Explicit expressions for /bardbler/an}bracketri}ht/an}bracketri}htandQrwill be given below.
+It will be convenient to define the rank 2( d−1) projector
+¯Rr=Qr+QT
+r (106)
+We have
+¯Rr=J2
+r (107)
+and
+Rr=¯Rr+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (108)
+SinceQris Hermitian we have
+QT
+r=Q∗
+r (109)
+whereQ∗
+ris the matrix whose elements are the complex conjugates of the cor re-
+sponding elements of Qr. So¯Rris twice the real part of Qrand−iJris twice the
+imaginary part.
+In Section 5we will show that Eq. ( 102) is essentially definitive of a SIC-POVM.
+To be more specific, let Lrbe any set of d2Hermitian matrices which constitute a
+basis for gl( d,C), and letCrbe the adjoint representative of Lrin that basis. Then
+we will show that Crhas the spectral decomposition
+Cr=Qr−QT
+r (110)
+whereQris a rankd−1 projector which is orthogonal to its own transpose if and
+only if the Lrare a family of SIC projectors up to multiplication by a sign and
+shifting by a multiple of the identity.
+Having stated our results let us now turn to the task of proving the m. We begin
+byderivingthespectraldecompositionof Tr. Multiplyingboth sidesoftheequation
+ΠrΠs=d+1
+dd2/summationdisplay
+t=1TrstΠt−K2
+rsI (111)
+by Πrwe find
+ΠrΠs=d+1
+dd2/summationdisplay
+t=1TrstΠrΠt−K2
+rsΠr16
+=(d+1)2
+d2d2/summationdisplay
+t=1(Tr)2
+stΠt−d+1
+dd2/summationdisplay
+t=1TrstK2
+rtI−K2
+rsΠr(112)
+We have
+d2/summationdisplay
+t=1TrstK2
+rt=1
+d+1d2/summationdisplay
+t=1Trst(dδrt+1)
+=1
+d+1
+dTrsr+d2/summationdisplay
+t=1Trst
+
+=2d
+d+1Tsrr
+=2d
+d+1K2
+rs (113)
+Consequently
+ΠrΠs=d+1
+dd2/summationdisplay
+t=1/parenleftbiggd+1
+d(Tr)2
+st−K2
+rsK2
+rt/parenrightbigg
+Πt−K2
+rsI (114)
+Comparing with Eq. ( 111) we deduce
+(Tr)2
+rs=d
+d+1Trst+d
+d+1K2
+rsK2
+rt (115)
+Now define
+/bardbler/an}bracketri}ht/an}bracketri}ht=/radicalbigg
+d+1
+2dd2/summationdisplay
+s=1K2
+rs/bardbls/an}bracketri}ht/an}bracketri}ht (116)
+where the basis kets /bardbls/an}bracketri}ht/an}bracketri}htare given by (in column vector form)
+/bardbl1/an}bracketri}ht/an}bracketri}ht=
+1
+0
+...
+0
+,/bardbl2/an}bracketri}ht/an}bracketri}ht=
+0
+1
+...
+0
+,...,/bardbld2/an}bracketri}ht/an}bracketri}ht=
+0
+0
+...
+1
+(117)
+It is easily verified that /bardbler/an}bracketri}ht/an}bracketri}htis normalized. Eq. ( 115) then becomes
+T2
+r=d
+d+1Tr+2d2
+(d+1)2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (118)
+Using Eq. ( 113) we find
+/an}bracketle{t/an}bracketle{ts/bardblTr/bardbler/an}bracketri}ht/an}bracketri}ht=/radicalbigg
+d+1
+2dd2/summationdisplay
+t=1TrstK2
+rt
+=/radicalbigg
+2d
+d+1K2
+rs
+=2d
+d+1/an}bracketle{t/an}bracketle{ts/bardbler/an}bracketri}ht/an}bracketri}ht (119)
+So/bardbler/an}bracketri}ht/an}bracketri}htis an eigenvector of Trwith eigenvalue2d
+d+1.17
+Also define
+Qr=d+1
+dTr−2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (120)
+So in terms of the order-3 angle tensor the matrix elements of Qrare
+Qrst=d+1
+dKrsKrt/parenleftbig
+Ksteiθrst−KrsKrt/parenrightbig
+(121)
+Qris Hermitian (because Tris Hermitian). Moreover
+Q2
+r=(d+1)2
+d2T2
+r−4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl=Qr (122)
+SoQris a projection operator. Since
+Tr(Tr) =/summationdisplay
+uTruu=d2/summationdisplay
+u=1K2
+ru=d (123)
+we have
+Tr(Qr) =d−1 (124)
+We have thus proved that the spectral decomposition of Tris
+Tr=d
+d+1Qr+2d
+d+1/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (125)
+whereQris a rankd−1 projector, as claimed.
+We next prove that QT
+r/bardbler/an}bracketri}ht/an}bracketri}ht= 0. The fact that the components of /bardbler/an}bracketri}ht/an}bracketri}htin the
+standard basis are all real means
+/an}bracketle{t/an}bracketle{ts/bardblTT
+r/bardbler/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{ter/bardblTr/bardbls/an}bracketri}ht/an}bracketri}ht=2d
+d+1/an}bracketle{t/an}bracketle{ts/bardbler/an}bracketri}ht/an}bracketri}ht (126)
+So/bardbler/an}bracketri}ht/an}bracketri}htis an eigenvector of TT
+ras well asTr, again with the eigenvalue2d
+d+1. In
+view of Eq. ( 120) it follows that QT
+r/bardbler/an}bracketri}ht/an}bracketri}ht= 0.
+Turning to the problem of showing that Qris orthogonal to its own transpose.
+We have
+QrQT
+r=/parenleftbiggd+1
+dTr−2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightbigg/parenleftbiggd+1
+dTT
+r−2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightbigg
+=(d+1)2
+d2TrTT
+r−4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (127)
+It follows from Eq. ( 24) that
+/an}bracketle{t/an}bracketle{ts/bardblTrTT
+r/bardblt/an}bracketri}ht/an}bracketri}ht=d2/summationdisplay
+u=1TrsuTrtu
+=GrsGrtd2/summationdisplay
+u=1GsuGtuGurGur (128)
+In view of Eq. ( 23) (i.e.the fact that every SIC-POVM is a 2-design) this implies
+/an}bracketle{t/an}bracketle{ts/bardblTrTT
+r/bardblt/an}bracketri}ht/an}bracketri}ht=2d
+d+1|Grs|2|Grt|2
+=2d
+d+1K2
+rsK2
+rt18
+=4d2
+(d+1)2/an}bracketle{t/an}bracketle{ts/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardblt/an}bracketri}ht/an}bracketri}ht (129)
+So
+TrTT
+r=4d2
+(d+1)2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (130)
+and consequently
+QrQT
+r= 0 (131)
+Eqs. (102) and (103) are immediate consequences of the results already proved
+and the definitions of Jr,Rr.
+We definedthe Jmatricestobe theadjointrepresentativesofthe SIC-projecto rs,
+considered as a basis for the Lie algebra gl( d,C), and that is certainly a most
+important fact about them. However, the results of this section s how that, along
+with the vectors /bardbler/an}bracketri}ht/an}bracketri}ht, they actually determine the whole structure. Specifically,
+we have
+Qr=1
+2/parenleftbig
+Jr+J2
+r/parenrightbig
+(132)
+Rr=J2
+r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (133)
+Tr=d
+2(d+1)/parenleftig
+Jr+J2
+r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightig
+(134)
+Moreover, if we know the Tmatrices then we know the order-3 angle tensor, which
+in view of Theorem 3means we can reconstruct the SIC-projectors. Since the
+vectors/bardbler/an}bracketri}ht/an}bracketri}htare given, once and for all, this means that the problem of proving th e
+existenceofa SIC-POVMin dimension dis equivalent to the problem ofprovingthe
+existence of a certain remarkable structure in the adjoint repres entation of gl( d,C)
+(as we will see in more detail in Section 5).
+In the Introduction webegan with the concept ofa SIC-POVM,and then defined
+theJmatrices in terms of it. However, one could, if one wished, go in the op posite
+direction, and take the Lie algebraic structure to be primary, with t he SIC-POVM
+being the secondary, derivative entity.
+4.TheQ-QTProperty
+The next five sections are devoted to a study of the Jmatrices which, as we will
+see, have numerous interesting properties. We begin our investiga tion by trying to
+get some additional insight into what we will call the Q-QTproperty: namely, the
+fact that the Jmatrices have the spectral decomposition
+Jr=Qr−QT
+r (135)
+whereQrisarankd−1projectorwhichisorthogonaltoits owntranspose. We wish
+to characterize the general class of matrices which are of this typ e. The following
+theorem provides one such characterization.
+Theorem 4. LetAbe a Hermitian matrix. Then the following statements are
+equivalent:19
+(1)Ahas the spectral decomposition
+A=P−PT(136)
+wherePis a projector which is orthogonal to its own transpose.
+(2)Ais pure imaginary and A2is a projector.
+Proof.To show that (1) = ⇒(2) observe that the fact that Pis Hermitian means
+PT=P∗(137)
+whereP∗is the matrix whose elements are the complex conjugates of the cor re-
+sponding elements of P. So Eq. ( 136) implies that the components of Aare pure
+imaginary. Since PPT= 0 it also implies that A2is a projector.
+To show that (2) = ⇒(1) observe that the fact that A2is a projector means
+that the eigenvalues of A=±1 or 0. So
+A=P−P′(138)
+whereP,P′are orthogonal projectors. Since Ais pure imaginary we must have
+PT−(P′)T=AT=A∗=−A=P′−P (139)
+PTand (P′)Tare also orthogonal projectors. So if PT|ψ/an}bracketri}ht=|ψ/an}bracketri}ht, and|ψ/an}bracketri}htis nor-
+malized, we must have
+1 =/an}bracketle{tψ|PT|ψ/an}bracketri}ht
+=/angbracketleftbig
+ψ/vextendsingle/vextendsingle/parenleftbig
+PT−(P′)T/parenrightbig/vextendsingle/vextendsingleψ/angbracketrightbig
+=/an}bracketle{tψ|P′|ψ/an}bracketri}ht−/an}bracketle{tψ|P|ψ/an}bracketri}ht (140)
+Since
+0≤ /an}bracketle{tψ|P′|ψ/an}bracketri}ht ≤1 (141)
+0≤ /an}bracketle{tψ|P|ψ/an}bracketri}ht ≤1 (142)
+we must have /an}bracketle{tψ|P′|ψ/an}bracketri}ht= 1, implying P′|ψ/an}bracketri}ht=|ψ/an}bracketri}ht. Similarly P′|ψ/an}bracketri}ht=|ψ/an}bracketri}htimplies
+PT|ψ/an}bracketri}ht=|ψ/an}bracketri}ht. So
+P′=PT(143)
+/square
+We also have the following statement, inspired in part by Ref. [ 51],
+Theorem 5. The necessary and sufficient condition for a matrix Pto be a projector
+which is orthogonal to its own transpose is that
+P=SDST(144)
+whereSis an any real orthogonal matrix and Dhas the block-diagonal form
+D=
+σ ... 0 0...0
+............
+0... σ 0...0
+0...0 0...0
+............
+0...0 0...0
+(145)20
+with
+σ=1
+2/parenleftbigg
+1−i
+i1/parenrightbigg
+(146)
+In other words Dhasncopies ofσon the diagonal, where n= rank(P), and0
+everywhere else.
+Proof.Sufficiency is an immediate consequence of the fact that σis a rank 1 pro-
+jector such that σσT= 0.
+To prove necessity let dbe the dimension of the space and nthe rank of P. It
+will be convenient to define
+|1/an}bracketri}ht=
+1
+0
+...
+0
+,|2/an}bracketri}ht=
+0
+1
+...
+0
+, ... |d/an}bracketri}ht=
+0
+0
+...
+1
+(147)
+In terms of these basis vectors we have
+P=d/summationdisplay
+r,s=1Prs|r/an}bracketri}ht/an}bracketle{ts| (148)
+Now let |a1/an}bracketri}ht,...,|an/an}bracketri}htbe an orthonormal basis for the subspace onto which P
+projects, and let |a∗
+r/an}bracketri}htbe the column vector which is obtained from |ar/an}bracketri}htby tak-
+ing the complex conjugate of each of its components. Taking comple x conjugates
+on each side of the equation
+P|ar/an}bracketri}ht=|ar/an}bracketri}ht (149)
+gives
+P∗|a∗
+r/an}bracketri}ht=|a∗
+r/an}bracketri}ht (150)
+So|a∗
+1/an}bracketri}ht,...,|a∗
+n/an}bracketri}htis an orthonormal basis for the subspace onto which PT=P∗
+projects. Since PTis orthogonal to Pwe conclude that
+/an}bracketle{tar|a∗
+s/an}bracketri}ht= 0 (151)
+for allr,s.
+Next define vectors |b1/an}bracketri}ht,...,|b2n/an}bracketri}htby
+|b2r−1/an}bracketri}ht=1√
+2/parenleftbig
+|a∗
+r/an}bracketri}ht−|ar/an}bracketri}ht/parenrightbig
+(152)
+|b2r/an}bracketri}ht=i√
+2/parenleftbig
+|a∗
+r/an}bracketri}ht+|ar/an}bracketri}ht/parenrightbig
+(153)
+By construction these vectors are orthonormal and real. So we c an extend them
+to an orthonormal basis for the full space by adding a further d−2nvectors
+|b2n+1/an}bracketri}ht,...,|bd/an}bracketri}ht, which can also be chosen to be real. We have
+P=n/summationdisplay
+r=1|ar/an}bracketri}ht/an}bracketle{tar|
+=1
+2n/summationdisplay
+r=1/parenleftig
+|b2r−1/an}bracketri}ht/an}bracketle{tb2r−1|−i|b2r−1/an}bracketri}ht/an}bracketle{tb2r|+i|b2r/an}bracketri}ht/an}bracketle{tb2r−1|+|b2r/an}bracketri}ht/an}bracketle{tb2r|/parenrightig
+(154)21
+So if we define
+S=d/summationdisplay
+r=1|br/an}bracketri}ht/an}bracketle{tr| (155)
+thenSis a real orthogonal matrix such that
+P=SDST(156)
+where
+D=1
+2n/summationdisplay
+r=1/parenleftig
+|2r−1/an}bracketri}ht/an}bracketle{t2r−1|−i|2r−1/an}bracketri}ht/an}bracketle{t2r|+i|2r/an}bracketri}ht/an}bracketle{t2r−1|+|2r/an}bracketri}ht/an}bracketle{t2r|/parenrightig
+(157)
+is the matrix defined by Eq. ( 145). /square
+This result implies the following alternative characterization of the cla ss of ma-
+trices to which the Jmatrices belong
+Corollary 6. LetAbe a Hermitian matrix. Then the following statements are
+equivalent:
+(1)Ahas the spectral decomposition
+A=P−PT(158)
+wherePis a projector which is orthogonal to its own transpose.
+(2)There exists a real orthogonal matrix Ssuch that
+A=SDST(159)
+whereDhas the block diagonal form
+D=
+σy...0 0...0
+............
+0... σ y0...0
+0...0 0...0
+............
+0...0 0...0
+(160)
+σybeing the Pauli matrix
+σy=/parenleftbigg0−i
+i0/parenrightbigg
+(161)
+In other words Dhasncopies ofσyon the diagonal, where n=1
+2rank(A),
+and0everywhere else (note that a matrix of this type must have eve n rank).
+Proof.Immediate consequence of Theorem 5. /square
+5.Lie Algebraic Formulation of the Existence Problem
+This section is the core of the paper. We show that the problem of pr oving the
+existence of a SIC-POVM in dimension dis equivalent to the problem of proving
+the existence of an Hermitian basis for gl( d,C) all of whose elements have the Q-QT
+property. We hope that this new way of thinking will help make the SIC -existence
+problem more amenable to solution.
+The result we prove is the following:22
+Theorem 7. LetLrbe a set ofd2Hermitian matrices forming a basis for gl(d,C).
+LetCrstbe the structure constants relative to this basis, so that
+[Lr,Ls] =d2/summationdisplay
+t=1CrstLt (162)
+and letCrbe the matrix with matrix elements (Cr)st=Crst. Then the following
+statements are equivalent
+(1)EachCrhas the spectral decomposition
+Cr=Pr−PT
+r (163)
+wherePris a rankd−1projector which is orthogonal to its own transpose.
+(2)There exists a SIC-set Πr, a set of signs ǫr=±1and a real constant
+α/ne}ationslash=−1
+dsuch that
+Lr=ǫr(Πr+αI) (164)
+Remark. The restriction to values of α/ne}ationslash=−1
+dis needed to ensure that the matrices
+Lrare linearly independent, and therefore constitute a basis for gl( d,C) (otherwise
+they would all have trace = 0). The Q-QTproperty continues to hold even if α
+does =−1
+d.
+It will be seen that it is not only SIC-sets which have the Q-QTproperty, but
+also any set of operators obtained from a SIC-set by shifting by a c onstant and
+multiplying by an r-dependent sign. Sothe Q-QTpropertyis not strictly equivalent
+to the property of being a SIC-set. However, it could be said that t he properties
+are almost equivalent. In particular, the existence of an Hermitian b asis for gl(d,C)
+having the Q-QTproperty implies the existence of a SIC-POVM in dimension d,
+and conversely.
+Proof that (2) =⇒(1).Taking the trace on both sides of
+[Πr,Πs] =d2/summationdisplay
+t=1JrstΠt (165)
+we deduce that
+d2/summationdisplay
+t=1Jrst= 0 (166)
+Then from the definition of Lrin terms of Π rwe find
+Crst=ǫrǫsǫtJrst (167)
+Consequently
+Cr=Pr−PT
+r (168)
+where
+Pr=ǫrSQrS (169)
+Sbeing the symmetric orthogonal matrix
+S=
+ǫ10...0
+0ǫ2...0
+.........
+0 0... ǫ d2
+(170)
+The claim is now immediate.23
+Proof that (1) =⇒(2).Forthis we need to workharder. Since the proofis rather
+lengthy we will break it into a number of lemmas. We first collect a few ele mentary
+facts which will be needed in the sequel:
+Lemma 8. LetLrbe any Hermitian basis for gl(d,C), and letCrstandCrbe
+the structure constants and adjoint representatives as defi ned in the statement of
+Theorem 7. Letlr= Tr(Lr). Then
+(1)Thelrare not all zero.
+(2)TheCrstare pure imaginary and antisymmetric in the first pair of indi ces.
+(3)TheCrstare completely antisymmetric if and only if the Crare Hermitian.
+(4)In every case
+d2/summationdisplay
+t=1Crstlt= 0 (171)
+for allr,s.
+(5)In the special case that the Crare Hermitian
+d2/summationdisplay
+r=1lrLr=κI (172)
+where
+κ=1
+d
+d2/summationdisplay
+r=1l2
+r
+>0 (173)
+Proof.To prove (1) observe that if the lrwere all zero it would mean that the
+identity was not in the span of the Lr—contrary to the assumption that they form
+a basis.
+To prove(2) observethat taking Hermitian conjugates on both sid es of Eq. ( 162)
+gives
+−[Lr,Ls] =d2/summationdisplay
+t=1C∗
+rstLt (174)
+from which it follows that C∗
+rst=−Crst. The fact that Csrt=−Crstis an imme-
+diate consequence of the definition.
+(3) is now immediate.
+(4) is proved in the same way as Eq. ( 166).
+To prove (5) observe that if the Crare Hermitian it follows from (2) and (3) that
+d2/summationdisplay
+r=1lrCrst= 0 (175)
+for alls,t. Consequently the matrix
+d2/summationdisplay
+r=1lrLr (176)24
+commutes with everything. But the only matrices for which that is tr ue are multi-
+ples of the identity. It follows that
+d2/summationdisplay
+r=1lrLr=κI (177)
+for some real κ. Taking the trace on both sides of this equation we deduce
+d2/summationdisplay
+r=1l2
+r=dκ (178)
+The fact that κ>0 is a consequence of this and statement (1). /square
+We next observe that if the Crhave theQ-QTproperty they must, in particular,
+be Hermitian. It turns out that that is, by itself, already a very str ong constraint.
+Before stating the result it may be helpful if we explain the essential idea on
+which it depends. Although we have not done so before, and will not d o so again, it
+will be convenient to make use of the covariant/contravariantinde x notation which
+is often used to describe the structure constants. Define the me tric tensor
+Mrs= Tr(LrLs) (179)
+and letMrsbe its inverse. So
+d2/summationdisplay
+t=1MrtMts=Mr
+s=/braceleftigg
+1r=s
+0r/ne}ationslash=s(180)
+We can use these tensors to raise and lower indices (we use the Hilber t-Schmidt
+inner product for this purpose because the fact that gl( d,C) is not semi-simple
+means that its Killing form is degenerate [ 52–55]). In particular, the matrices
+Lr=d2/summationdisplay
+t=1MrsLs (181)
+are the basis dual to the Lr:
+Tr(LrLs) =Mr
+s (182)
+Suppose we now define structure constants ˜Crstby
+[Lr,Ls] =d2/summationdisplay
+t=1˜CrstLt(183)
+(so in terms of the Crstwe have ˜Ct
+rs=Crst). It follows from the relation
+˜Crst= Tr/parenleftbig
+[Lr,Ls]Lt/parenrightbig
+= Tr/parenleftbig
+Lr[Ls,Lt]/parenrightbig
+(184)
+that the ˜Crstare completely antisymmetric for any choice of the Lr. If we now
+require that the matrices Crbe Hermitian it means that, not only the ˜Crst, but
+also theCrstmust be completely antisymmetric. Since the two quantities are
+related by
+˜Crst=d2/summationdisplay
+u=1CrsuMut (185)25
+this is a very strong requirement. It means that the Lrmust, in a certain sense,
+be close to orthonormal (relative to the Hilbert-Schmidt inner prod uct). More
+precisely, it means we have the following lemma:
+Lemma 9. LetLr,CrstandCrbe defined as in the statement of Theorem 7, and
+letlr= Tr(Lr). Then the Crare Hermitian if and only if
+Tr(LrLs) =βδrs+γlrls (186)
+whereβ,γare real constants such that β >0andγ <1
+d.
+If this condition is satisfied we also have
+d2/summationdisplay
+r=1lrLr=β
+1−dγI (187)
+d2/summationdisplay
+r=1l2
+r=dβ
+1−dγ(188)
+Proof.To prove sufficiency observe that, in view of Eq. ( 185), the condition means
+˜Crst=βCrst+γltd2/summationdisplay
+u=1Crsulu (189)
+In view of Lemma 8, and the fact that β/ne}ationslash= 0, this implies
+Crst=1
+β˜Crst (190)
+Since the ˜Crstare completely antisymmetric we conclude that the Crstmust be
+also. It follows that the Crare Hermitian.
+To prove necessity let ˜Cr(respectively M) be the matrix whose matrix elements
+are˜Crst(respectively Mst). Then Eq. ( 185) can be written
+˜Cr=CrM (191)
+Taking the transpose (or, equivalently, the Hermitian conjugate) on both sides of
+this equation we find
+˜Cr=MCr (192)
+implying
+[M,Cr] = 0 (193)
+for allr. Since the Lrare a basis for gl( d,C) we deduce
+[M,adA] = 0 (194)
+for allA∈gl(d,C). Eq. (186) is a straightforward consequence of this, the fact
+that gl(d,C) has the direct sum decomposition CI⊕sl(d,C), the fact that sl( d,C)
+is simple, and Schur’s lemma [ 52–55]. However, for the benefit of the reader who is
+not so familiar with the theory of Lie algebras we will give the argument in a little
+more detail.26
+Given arbitrary A=/summationtextd2
+r=1arLr, let/bardblA/an}bracketri}ht/an}bracketri}htdenote the column vector
+/bardblA/an}bracketri}ht/an}bracketri}ht=
+a1
+a2
+...
+ad2
+(195)
+So
+/bardblLr/an}bracketri}ht/an}bracketri}ht=
+1
+0
+...
+0
+/bardblL2/an}bracketri}ht/an}bracketri}ht=
+0
+1
+...
+0
+/bardblLd2/an}bracketri}ht/an}bracketri}ht=
+0
+0
+...
+1
+(196)
+In view of Lemma 8we then have
+/bardblI/an}bracketri}ht/an}bracketri}ht=1
+κd2/summationdisplay
+r=1lr/bardblLr/an}bracketri}ht/an}bracketri}ht (197)
+Since
+Tr(A) =d2/summationdisplay
+r=1arlr=κ/an}bracketle{t/an}bracketle{tI/bardblA/an}bracketri}ht/an}bracketri}ht (198)
+we have that A∈sl(d,C) if and only if /an}bracketle{t/an}bracketle{tI/bardblA/an}bracketri}ht/an}bracketri}ht= 0.
+Now observe that it follows from Lemma 8and the definition of Mthat
+M/bardblI/an}bracketri}ht/an}bracketri}ht=κ/bardblI/an}bracketri}ht/an}bracketri}ht (199)
+IfMis a multiple of the identity we have Mrs=κδrsand the lemma is proved.
+OtherwiseMhas at least one more eigenvalue, βsay. Let Ebe the corresponding
+eigenspace. Since Eis orthogonal to /bardblI/an}bracketri}ht/an}bracketri}htit follows from Eq. ( 198) thatE⊆sl(d,C).
+SinceMcommutes with every adjoint representation matrix we have
+adAE⊆E (200)
+for allA∈sl(d,C). SoEis an ideal of sl( d,C). However sl( d,C) is a simple Lie
+algebra, meaning it has no proper ideals [ 52–55]. So we must have E= sl(d,C). It
+follows that if we define
+˜Lr=Lr−lr
+dI (201)
+then
+M/bardblLr/an}bracketri}ht/an}bracketri}ht=lr
+dM/bardblI/an}bracketri}ht/an}bracketri}ht+M/bardbl˜Lr/an}bracketri}ht/an}bracketri}ht (202)
+=κlr
+d/bardblI/an}bracketri}ht/an}bracketri}ht+β/bardbl˜Lr/an}bracketri}ht/an}bracketri}ht (203)
+=d2/summationdisplay
+s=1(βδrs+γlrls)/bardblLs/an}bracketri}ht/an}bracketri}ht (204)
+whereγ=1
+d/parenleftig
+1−β
+κ/parenrightig
+. Eqs. (186), (187) and (188) are now immediate (in view of
+Lemma8).27
+It remains to establish the bounds on β,γ. LetA=/summationtextd2
+r=1arLrbe any non-zero
+element of sl( d,C). Then/summationtextd2
+r=1arlr= 0, so in view of Eq. ( 186) we have
+0<Tr(A2) =βd2/summationdisplay
+r=1a2
+r (205)
+It follows that β >0. Also, using Lemma 8once more, we find
+lr=1
+κd2/summationdisplay
+s=1lsTr(LrLs)
+=βlr
+κ+γlr
+κd2/summationdisplay
+s=1l2
+s
+=lr/parenleftbiggβ
+κ+dγ/parenrightbigg
+(206)
+Since thelrcannot all be zero this implies
+β
+κ= 1−dγ (207)
+Sinceβ
+κ>0 we deduce that γ <1
+d. /square
+Eq. (186) only depends on the Crbeing Hermitian. If we make the assumption
+that theCrhave theQ-QTproperty we get a stronger statement:
+Corollary 10. LetLr,CrstandCrbe as defined in the statement of Theorem 7.
+Suppose that the Crhave the spectral decomposition
+Cr=Pr−PT
+r (208)
+wherePris a rankd−1projector which is orthogonal to its own transpose. Then
+(1)For allr
+Tr(Lr) =ǫ′
+rl (209)
+(2)For allr,s
+Tr(LrLs) =d
+d+1δrs+ǫ′
+rǫ′
+s
+d/parenleftbigg
+l2−1
+d+1/parenrightbigg
+(210)
+(3)
+d2/summationdisplay
+r=1ǫ′
+rLr=dlI (211)
+for some real constant l>0and signsǫ′
+r=±1.
+Proof.The proof relies on the fact that the Killing form for gl( d,C) is related to
+the Hilbert-Schmidt inner product by [ 55]
+Tr(adAadB) = 2dTr(AB)−2Tr(A)Tr(B) (212)
+Specializing to the case A=B=Lrand making use of the Q-QTproperty we find
+d−1 =dTr(L2
+r)−l2
+r (213)
+Using Lemma 9we deduce
+l2
+r=dβ−d+1
+1−dγ(214)28
+It follows that
+lr=ǫ′
+rl (215)
+for some real constant l≥0 and signs ǫ′
+r=±1. The fact that the Lrare a basis
+for gl(d,C) means the lrcannot all be zero. So we must in fact have l>0. Using
+this result in Eq. ( 188) we find
+β+d2l2γ=dl2(216)
+while Eq. ( 214) implies
+dβ+dl2γ=d−1+l2(217)
+This gives us a pair of simultaneous equations in βandγ. Solving them we obtain
+β=d
+d+1(218)
+γ=1
+dl2/parenleftbigg
+l2−1
+d+1/parenrightbigg
+(219)
+Substituting these expressions into Eqs. ( 186) and (187) we deduce Eqs. ( 210)
+and (211). /square
+The next lemma shows that each Lris a linear combination of a rank-1projector
+and the identity:
+Lemma 11. LetLbe any Hermitian matrix ∈gl(d,C)which is not a multiple of
+the identity. Then
+rank(ad L)≥2(d−1) (220)
+The lower bound is achieved if and only if Lis of the form
+L=ηI+ξP (221)
+wherePis a rank- 1projector and η,ξare any pair of real numbers. The eigenvalues
+ofadLare then ±ξ(each with multiplicity d−1) and0(with multiplicity d2−2d+2).
+Proof.Letλ1≥λ2≥ ··· ≥λdbe the eigenvalues of Larranged in decreasing
+order, and let |b1/an}bracketri}ht,|b2/an}bracketri}ht,...,|bd/an}bracketri}htbe the correspondingeigenvectors. We mayassume,
+without loss of generality, that the |br/an}bracketri}htare orthonormal. We have
+adL/parenleftbig
+|br/an}bracketri}ht/an}bracketle{tbs|/parenrightbig
+=/bracketleftbig
+L,|br/an}bracketri}ht/an}bracketle{tbs|/bracketrightbig
+= (λr−λs)|br/an}bracketri}ht/an}bracketle{tbs| (222)
+So the eigenvalues of ad Lareλr−λs. SinceLis not a multiple of the identity we
+must haveλr/ne}ationslash=λr+1for somerin the range 1 ≤r≤d−1. We then have that
+λs−λt/ne}ationslash= 0 if either s≤r<tort≤r<s. There are 2 r(d−r) such pairs s,t. So
+rank(ad L)≥2r(d−r)≥2(d−1) (223)
+Suppose, now that the lower bound is achieved. Then r(d−r) =d−1, implying
+thatr= 1 ord−1. Also we must have λs=λs+1for alls/ne}ationslash=r. So either
+L=λ2I+(λ1−λ2)|b1/an}bracketri}ht/an}bracketle{tb1| (224)
+or
+L=λd−1I−(λd−1−λd)|bd/an}bracketri}ht/an}bracketle{tbd| (225)
+Either way Land the spectrum of ad Lare as described. /square
+The final ingredient needed to complete the proof is29
+Lemma 12. LetLr,CrstandCrbe as defined in the statement of Theorem 7.
+Suppose that the Crhave the spectral decomposition
+Cr=Pr−PT
+r (226)
+wherePris a rankd−1projector which is orthogonal to its own transpose. Let l,
+ǫ′
+rbe as in the statement of Corollary 10. Then there is a fixed sign ǫ=±1such
+that
+Πr=ǫǫ′
+rLr−ǫl−1
+dI (227)
+is a rank- 1projector for all r.
+Proof.Define
+L′
+r=ǫ′
+rLr−l−1
+dI (228)
+Then it follows from Corollary 10that
+Tr(L′
+r) = 1 (229)
+for allr,
+Tr(L′
+rL′
+s) =dδrs+1
+d+1(230)
+for allr,s, and
+d2/summationdisplay
+r=1L′
+r=dI (231)
+It is also easily seen that if we define C′
+rst=ǫ′
+rǫ′
+sǫ′
+tCrstthen
+[L′
+r,L′
+s] =d2/summationdisplay
+t=1C′
+rstL′
+t (232)
+and
+C′
+r=P′
+r−P′
+rT(233)
+whereP′
+ris a rank-1 projector which is orthogonal to its own transpose (se e the
+first part of the proof of Theorem 7). In particular
+rank/parenleftbig
+adL′r/parenrightbig
+= 2(d−1) (234)
+and the eigenvalues of ad L′rall equal to ±1 or 0. So, taking account of the fact
+that Tr(L′
+r) = 1, we can use Lemma 11to deduce that there is a family of rank-1
+projectors Π′
+rand signsξr=±1 such that
+L′
+r=ξrΠ′
+r+1−ξr
+dI (235)
+Ifξr= +1 (respectively −1) for allrthen Eq. ( 227) holds with Π r= Π′
+randǫ= +1
+(respectively −1). Also, if d= 2 thenL′
+ris a rank-1 projector irrespective of the
+value ofξr, so Eq. ( 227) holds with Π r=L′
+randǫ= +1. The problem therefore
+reduces to showing that if d >2 it cannot happen that ξr= +1 for some values
+ofrand−1 for others. We will do this by assuming the contrary and deducing a
+contradiction.
+Letmbe the number of values of rfor whichξr= +1. We are assuming that
+mis in the range 1 ≤m≤d2−1. We may also assume, without loss of generality,30
+that the labelling is such that ξr= +1 for the first mvalues ofr, and−1 for the
+rest. So
+L′
+r=/braceleftigg
+Π′
+r ifr≤m
+2
+dI−Π′
+rifr>m(236)
+Now define
+˜Trst= Tr/parenleftbig
+L′
+rL′
+sL′
+t/parenrightbig
+(237)
+Eqs. (230) and (231) mean that the same argument which led to Eq. ( 9) can be
+used to deduce
+L′
+rL′
+s=d+1
+d
+d2/summationdisplay
+t=1˜TrstL′
+t
+−K2
+rsI (238)
+SinceL′
+1is a projector it follows that
+L′
+1L′
+s=/parenleftbig
+L′
+1/parenrightbig2L′
+s=d+1
+d
+d2/summationdisplay
+t=1˜T1stL′
+1L′
+t
+−K2
+1sL′
+1 (239)
+By essentially the same argument which led to Eq. ( 118) we can use this to infer
+/parenleftbig˜T′
+1/parenrightbig2=d
+d+1˜T1+2d2
+(d+1)2/bardble1/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{te1/bardbl (240)
+where˜T′
+1is the matrix with matrix elements ˜T′
+1rsand/bardble1/an}bracketri}ht/an}bracketri}htis the vector defined by
+Eq.(116). Asbefore /bardble1/an}bracketri}ht/an}bracketri}htisaneigenvectorof ˜T′
+1witheigenvalue2d
+d+1. Consequently
+the matrix
+˜Q1=d+1
+d˜T′
+1−2/bardble1/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{te1/bardbl (241)
+is a projector. But that means Tr( ˜Q1) must be an integer. We now use this to
+derive a contradiction.
+It follows from Eq. ( 236) that
+(L′
+r)2=/braceleftigg
+L′
+r r≤m
+2(d−2)
+d2I−d−4
+dL′
+rr>m(242)
+Consequently
+˜T1rr=/braceleftigg
+K2
+1r r≤m
+2(d−2)
+d2−d−4
+dK2
+1rr>m(243)
+and so
+Tr(˜Q1) =d+1
+dd2/summationdisplay
+r=1˜T1rr−2
+=d+1−4d2+2m(d−2)
+d3(244)
+So if Tr( ˜Q1) is an integer/parenleftbig
+4d2+2n(d−2)/parenrightbig
+/d3must also be an integer. But the
+fact that 1 ≤m<d2, together with the fact that d>2 means
+4
+d<4d2+2m(d−2)
+d3<2 (245)31
+Ifd= 3 or 4 there are no integers in this interval, which gives us a contrad iction
+straight away. If, on the other hand, d≥5 there is the possibility
+4d2+2m(d−2)
+d3= 1 (246)
+implying
+m=d2(d−4)
+2(d−2)(247)
+This equationhasthe solution d= 6,m= 9(this is in fact the only integersolution,
+ascanbe seenfrom ananalysisofthe possible primefactorizationso fthe numerator
+and denominator on the right hand side). To eliminate this possibility de fine
+L′′
+r=2
+dI−L′
+d2+1−r (248)
+for allr. It is easily verified that
+Tr(L′′
+rL′′
+s) =dδrs+1
+d+1(249)
+d2/summationdisplay
+r=1L′′
+r=dI (250)
+and
+L′′
+r=/braceleftigg
+Πr r≤d2−m
+2
+dI−Πrr>d2−m(251)
+So we can go through the same argument as before to deduce
+d2−m=d2(d−4)
+2(d−2)(252)
+Eqs. (247) and (252) have no joint solutions at all with d/ne}ationslash= 0, integer or otherwise.
+/square
+To complete the proof of Theorem 7observe that Eqs. ( 210) and (227) imply
+Tr(ΠrΠs) =dδrs+1
+d+1(253)
+So the Π rare a SIC-set. Moreover
+Lr=ǫr(Πr+αI) (254)
+whereǫr=ǫǫ′
+randα= (ǫl−1)/d.
+6.The Algebra sl(d,C)
+The motivation for this paper is the hope that a Lie algebraic perspec tive may
+cast some light on the SIC-existence problem, and on the mathemat ics of SIC-
+POVMs generally. We have focused on gl( d,C) as that is the case where the con-
+nection with Lie algebras seems most straightforward. However, it may be worth
+mentioning that a SIC-POVM also gives rise to an interesting geometr ical structure
+in sl(d,C) (the Lie algebra consisting of all trace-zero d×dcomplex matrices).32
+Let Πrbe a SIC-set and define
+Br=/radicaligg
+d+1
+2(d2−1)/parenleftbigg
+Πr−1
+dI/parenrightbigg
+(255)
+SoBr∈sl(d,C). Let
+/an}bracketle{tA,A′/an}bracketri}ht= Tr(ad AadA′) = 2dTr(AA′) (256)
+be the Killing form [ 55] on sl(d,C). Then
+/an}bracketle{tBr,Bs/an}bracketri}ht=/braceleftigg
+1 r=s
+−1
+d2−1r/ne}ationslash=s(257)
+So theBrform a regular simplex in sl( d,C). Since sl( d,C) isd2−1 dimensional
+theBrare an overcomplete set. However, the fact that
+d2/summationdisplay
+r=1Br= 0 (258)
+means that for each A∈sl(d,C) there is a unique set of numbers arsuch that
+A=d2/summationdisplay
+r=1arBr (259)
+and
+d2/summationdisplay
+r=1ar= 0 (260)
+Thearcan be calculated using
+ar=d2−1
+d2/an}bracketle{tA,Br/an}bracketri}ht (261)
+Similarly, given any linear transformation M: sl(d,C)→sl(d,C), there is a unique
+set of numbers Mrssuch that
+MBr=d2/summationdisplay
+s=1MrsBs (262)
+and
+d2/summationdisplay
+s=1Mrs=d2/summationdisplay
+s=1Msr= 0 (263)
+for allr. TheMrscan be calculated using
+Mrs=d2−1
+d2/an}bracketle{tBs,MBr/an}bracketri}ht (264)
+In short, the Brretain many analogous properties of, and can be used in much the
+same way as, a basis. It could be said that they form a simplicial basis.33
+7.Further Identities
+In the preceding pages we have seen that there are five different f amilies of ma-
+trices naturally associated with a SIC-POVM: namely, the projecto rsQrtogether
+with the matrices
+Jr=Qr−QT
+r (265)
+¯Rr=Qr+QT
+r (266)
+Rr=Qr+QT
+r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (267)
+Tr=d
+d+1Qr+2d
+d+1/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (268)
+(see Section 3). As we noted previously, it is possible to define everything in terms
+of the adjoint representation matrices Jrand the rank-1 projectors /bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl:
+Qr=1
+2Jr(Jr+I) (269)
+¯Rr=J2
+r (270)
+Rr=J2
+r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (271)
+Tr=d
+2(d+1)Jr(Jr+I)+2d
+d+1/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (272)
+In that sense the structure constants of the Lie algebra, supple mented with the
+vectors/bardbler/an}bracketri}ht/an}bracketri}ht, determine everything else.
+In the next section we will show that there are some interesting geo metrical
+relationships between the hyperplanes onto which Qr,QT
+rand¯Rrproject. In this
+section, as a preliminary to that investigation, we prove a number of identities
+satisfied by the Q,Jand¯Rmatries. We start by computing their Hilbert-Schmidt
+inner products:
+Theorem 13. For allr,s
+Tr/parenleftbig
+QrQs/parenrightbig
+=d3δrs+d2−d−1
+(d+1)2(273)
+Tr/parenleftbig
+QrQT
+s/parenrightbig
+=d2(1−δrs)
+(d+1)2(274)
+Tr/parenleftbig
+JrJs/parenrightbig
+=2(d2δrs−1)
+d+1(275)
+Tr/parenleftbig¯Rr¯Rs/parenrightbig
+=2(d−1)(d2δrs+2d+1)
+(d+1)2(276)
+Tr/parenleftbig
+Jr¯Rs/parenrightbig
+= 0 (277)
+Proof.Let us first calculate some auxiliary quantities. It follows from the de finition
+ofTr, andthe factthat the matrix P=1
+dGdefined byEq.( 63) isarankdprojector,
+that
+Tr(TrTs) =d2/summationdisplay
+u,v=1TruvTsvu34
+=d2/summationdisplay
+u,v=1K2
+uvGruGusGsvGvr
+=d
+d+1d2/summationdisplay
+u=1K2
+ruK2
+su+d4
+d+1d2/summationdisplay
+u,v=1PruPusPsvPvr
+=d2(dδrs+d+2)
+(d+1)3+d4
+d+1/vextendsingle/vextendsinglePrs/vextendsingle/vextendsingle2
+=d2(dδrs+d+2)
+(d+1)3+d2
+d+1K2
+rs
+=d2/parenleftbig
+d(d+2)δrs+2d+3/parenrightbig
+(d+1)3(278)
+Also
+Tr/parenleftbig
+TrTT
+s/parenrightbig
+=d2/summationdisplay
+u,v=1TruvTsuv
+=d2/summationdisplay
+u=1GruGsu
+d2/summationdisplay
+v=1GuvGuvGvrGvs
+
+=2d
+d+1d2/summationdisplay
+u=1GruGsuGurGus
+=2d2
+(d+1)2/parenleftbig
+1+K2
+rs/parenrightbig
+=2d2(dδrs+d+2)
+(d+1)3(279)
+where we made two applications of Eq. ( 23) (i.e.the fact that every SIC-POVM is
+a 2-design). Finally, it is a straightforward consequence of the defi nitions ofTr,TT
+r
+and/bardbler/an}bracketri}ht/an}bracketri}htthat
+/an}bracketle{t/an}bracketle{ter/bardblTs/bardbler/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{ter/bardblTT
+s/bardbler/an}bracketri}ht/an}bracketri}ht
+=d+1
+2dd2/summationdisplay
+u,v=1TsuvK2
+ruK2
+rv
+=1
+2d(d+1)
+d2Tsrr+dd2/summationdisplay
+v=1Tsrv+dd2/summationdisplay
+u=1Tsur+d2/summationdisplay
+u,v=1Tsuv
+
+=d
+2(d+1)/parenleftbig
+3K2
+rs+1/parenrightbig
+=d(3dδrs+d+4)
+2(d+1)2(280)35
+and
+/an}bracketle{t/an}bracketle{ter/bardbles/an}bracketri}ht/an}bracketri}ht=d+1
+2dd2/summationdisplay
+u=1K2
+ruK2
+su
+=dδrs+d+2
+2(d+1)(281)
+Using these results in the expressions
+Tr/parenleftbig
+QrQs/parenrightbig
+= Tr/parenleftigg/parenleftbiggd+1
+dTr−2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightbigg/parenleftbiggd+1
+dTs−2/bardbles/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tes/bardbl/parenrightbigg/parenrightigg
+(282)
+and
+Tr/parenleftbig
+QrQT
+s/parenrightbig
+= Tr/parenleftigg/parenleftbiggd+1
+dTr−2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightbigg/parenleftbiggd+1
+dTT
+s−2/bardbles/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tes/bardbl/parenrightbigg/parenrightigg
+(283)
+the first two statements follow. The remaining statements are imme diate conse-
+quences of these and the fact that
+Jr=Qr−QT
+r (284)
+¯Rr=Qr+QT
+r (285)
+/square
+Now define
+/bardblv0/an}bracketri}ht/an}bracketri}ht=1
+dd2/summationdisplay
+r=1/bardblr/an}bracketri}ht/an}bracketri}ht (286)
+where/bardblr/an}bracketri}ht/an}bracketri}htis the basis defined in Eq. ( 117). The following result shows (among
+other things) that the subspaces onto which the Qr(respectively QT
+r,Rr) project
+span the orthogonal complement of /bardblv0/an}bracketri}ht/an}bracketri}ht.
+Theorem 14. For allr
+Qr/bardblv0/an}bracketri}ht/an}bracketri}ht=QT
+r/bardblv0/an}bracketri}ht/an}bracketri}ht=Jr/bardblv0/an}bracketri}ht/an}bracketri}ht=Rr/bardblv0/an}bracketri}ht/an}bracketri}ht= 0 (287)
+Moreover
+d2/summationdisplay
+r=1Qr=d2/summationdisplay
+r=1QT
+r=d2
+d+1/parenleftbig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightbig
+(288)
+d2/summationdisplay
+r=1Jr= 0 (289)
+d2/summationdisplay
+r=1¯Rr=2d2
+d+1/parenleftbig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightbig
+(290)
+Proof.Some of this is a straightforward consequence of the fact that Jris the
+adjoint representative of Π r. Since
+d2/summationdisplay
+s=1Πs=dI (291)36
+we must have
+d2/summationdisplay
+s,t=1JrstΠt=d2/summationdisplay
+s=1adΠrΠs= 0 (292)
+In view of the antisymmetry of the Jrstit follows that
+d2/summationdisplay
+r=1Jr= 0 (293)
+and
+Jr/bardblv0/an}bracketri}ht/an}bracketri}ht= 0 (294)
+Using the relations
+Qr=1
+2Jr(Jr+I) (295)
+QT
+r=1
+2Jr(Jr−I) (296)
+¯Rr=J2
+r (297)
+we deduce
+Qr/bardblv0/an}bracketri}ht/an}bracketri}ht=QT
+r/bardblv0/an}bracketri}ht/an}bracketri}ht=¯Rr/bardblv0/an}bracketri}ht/an}bracketri}ht= 0 (298)
+It remains to prove Eqs. ( 288) and (290). It follows from Eq. ( 120) that
+d2/summationdisplay
+r=1Qrst=d+1
+dd2/summationdisplay
+r=1Trst−2d2/summationdisplay
+r=1/an}bracketle{t/an}bracketle{ts/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardblt/an}bracketri}ht/an}bracketri}ht
+= (d+1)K2
+st−d+1
+dd2/summationdisplay
+r=1K2
+rsK2
+rt
+=d2δst−1
+d+1(299)
+from which it follows
+d2/summationdisplay
+r=1Qr=d2/summationdisplay
+r=1QT
+r=d2
+d+1/parenleftbig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightbig
+(300)
+Eq. (290) follows from this and the fact that Rr=Qr+QT
+r.
+/square
+8.Geometrical Considerations
+In this section we show that there are some interesting geometrica l relationships
+between the subspaces onto which the operators Qr,QT
+rand¯Rrproject. The
+original motivation for this work was an observation concerning the subspaces onto
+which the ¯Rrproject. ¯Rris a real matrix, and so it defines a 2( d−2) subspace
+inRd2, which we will denote Rr. We noticed that for each pair of distinct indices
+randsthe intersection Rr∩Rsis a 1-dimensional line. This led us to speculate
+that a set of hyperplanes parallel to the Rrmight be the edges of an interesting
+polytope. We continue to think that this could be the case. Unfortu nately we have
+not been able to prove it. However, it appears to us that the result s we obtained37
+while trying to prove it have an interest which is independent of the tr uth of the
+motivating speculation.
+We will begin with some terminology. Let Pbe any projector (on either RN
+orCN), letPbe the subspace onto which Pprojects, and let |ψ/an}bracketri}htbe any non-zero
+vector. Then we define the angle between |ψ/an}bracketri}htandPin the usual way, to be
+θ= cos−1/parenleftigg/vextenddouble/vextenddoubleP|ψ/an}bracketri}ht/vextenddouble/vextenddouble
+/vextenddouble/vextenddouble|ψ/an}bracketri}ht/vextenddouble/vextenddouble/parenrightigg
+(301)
+(soθis the smallest angle between |ψ/an}bracketri}htand any of the vectors in P).
+Suppose, now, that P′is another projector, and let P′be the subspace onto
+whichP′projects. We will say that P′is uniformly inclined to Pif every vector in
+P′makes the same angle θwithP. Ifθ= 0 this means that P′⊆P, while ifθ=π
+2
+it means P′⊥P. Suppose, on the other hand, that 0 < θ <π
+2. Let|u′
+1/an}bracketri}ht,...,|u′
+n/an}bracketri}ht
+be any orthonormal basis for P′, and define |ur/an}bracketri}ht= secθP|u′
+r/an}bracketri}ht. Then/an}bracketle{tur|ur/an}bracketri}ht= 1
+for allr. Moreover, if P,P′are complex projectors,
+/an}bracketle{tu′
+r+eiφu′
+s|P|u′
+r+eiφu′
+s/an}bracketri}ht= 2cos2θ/parenleftig
+1+Re/parenleftbig
+eiφ/an}bracketle{tur|us/an}bracketri}ht/parenrightbig/parenrightig
+(302)
+for allφandr/ne}ationslash=s. On the other hand it follows from the assumption that P′is
+uniformly inclined to Pthat
+/an}bracketle{tu′
+r+eiφu′
+s|P|u′
+r+eiφu′
+s/an}bracketri}ht= 2cos2θ (303)
+for allφandr/ne}ationslash=s. Consequently
+/an}bracketle{tur|us/an}bracketri}ht=δrs (304)
+for allr,s. It is easily seen that the same is true if P,P′are real projectors.
+Suppose we now make the further assumption that dim( P′) = dim( P) =n. Then
+|u1/an}bracketri}ht,...,|un/an}bracketri}htis an orthonormal basis for P, and we can write
+P=n/summationdisplay
+r=1|ur/an}bracketri}ht/an}bracketle{tur| (305)
+P′=n/summationdisplay
+r=1|u′
+r/an}bracketri}ht/an}bracketle{tu′
+r| (306)
+Observe that
+/an}bracketle{tu′
+r|us/an}bracketri}ht=/an}bracketle{tu′
+r|P|us/an}bracketri}ht= cosθ/an}bracketle{tur|us/an}bracketri}ht= cosθδrs (307)
+for allr,s. Consequently
+P′|ur/an}bracketri}ht= cosθ|ur/an}bracketri}ht (308)
+for allr. It follows that
+/vextenddouble/vextenddoubleP′|ψ/an}bracketri}ht/vextenddouble/vextenddouble=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay
+r=1cosθ/an}bracketle{tur|ψ/an}bracketri}ht|u′
+r/an}bracketri}ht/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble= cosθ/vextenddouble/vextenddouble|ψ/an}bracketri}ht/vextenddouble/vextenddouble (309)
+for all|ψ/an}bracketri}ht ∈P. SoPis uniformly inclined to P′at the same angle θ.
+It follows from Eqs. ( 305) and (306) that
+PP′P= cos2θP (310)
+P′PP′= cos2θP′(311)
+Eq. (310), or equivalently Eq. ( 311), is not only necessary but also sufficient for
+the subspaces to be uniformly inclined. In fact, let P,P′be any two subspaces38
+which have the same dimension n, but which are not assumed at the outset to be
+uniformly inclined, and let P,P′be the corresponding projectors. Suppose
+PP′P= cos2θP (312)
+for someθin the range 0 ≤θ≤π
+2. It is immediate that P=P′ifθ= 0, and
+P⊥P′ifθ=π
+2. Either way, the subspaces are uniformly inclined. Suppose, on
+the other hand, that 0 <θ<π
+2. Let|u′
+1/an}bracketri}ht,...,|u′
+n/an}bracketri}htbe any orthonormal basis for P′,
+and define |ur/an}bracketri}ht= secθP|u′
+r/an}bracketri}ht. Eq. (305) then implies
+P= sec2θn/summationdisplay
+r=1P|u′
+r/an}bracketri}ht/an}bracketle{tu′
+r|P=n/summationdisplay
+r=1|ur/an}bracketri}ht/an}bracketle{tur| (313)
+Given any |ψ/an}bracketri}ht ∈Pwe have
+|ψ/an}bracketri}ht=P|ψ/an}bracketri}ht=n/summationdisplay
+r=1/an}bracketle{tur|ψ/an}bracketri}ht|ur/an}bracketri}ht (314)
+Since dim( P) =nit follows that the |ur/an}bracketri}htare linearly independent. In particular
+|ur/an}bracketri}ht=P|ur/an}bracketri}ht=n/summationdisplay
+s=1/an}bracketle{tus|ur/an}bracketri}ht|us/an}bracketri}ht (315)
+Since the |ur/an}bracketri}htare linearly independent this means
+/an}bracketle{tus|ur/an}bracketri}ht=δrs (316)
+So the|ur/an}bracketri}htare an orthonormal basis for P. It follows, that if |ψ′/an}bracketri}htis any vector in
+P′, then
+/vextenddouble/vextenddoubleP|ψ′/an}bracketri}ht/vextenddouble/vextenddouble=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay
+r=1/an}bracketle{tu′
+r|ψ′/an}bracketri}htP|u′
+r/an}bracketri}ht/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble= cosθ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay
+r=1/an}bracketle{tu′
+r|ψ′/an}bracketri}ht|ur/an}bracketri}ht/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble= cosθ/vextenddouble/vextenddouble|ψ′/an}bracketri}ht/vextenddouble/vextenddouble(317)
+implying that P′is uniformly inclined to Pat angleθ.
+It will be convenient to summarise all this in the form of a lemma:
+Lemma 15. LetP,P′be any two subspaces, real or complex, having the same
+dimensionn. LetP,P′be the corresponding projectors. Then the following state-
+ments are equivalent:
+(a)P′is uniformly inclined to Pat angleθ.
+(b)Pis uniformly inclined to P′at angleθ.
+(c)
+PP′P= cos2θP (318)
+(d)
+P′PP′= cos2θP′(319)
+Suppose these conditions are satisfied for some θin the range 0< θ <π
+2, and
+let|u1/an}bracketri}ht,...|un/an}bracketri}htbe any orthonormal basis for P. Then there exists an orthonormal
+basis|u′
+1/an}bracketri}ht,...,|u′
+n/an}bracketri}htforP′such that
+P′|ur/an}bracketri}ht= cosθ|u′
+r/an}bracketri}ht (320)
+P|u′
+r/an}bracketri}ht= cosθ|ur/an}bracketri}ht (321)
+We are now in a position to state the main results of this section. Let Qr
+(respectively ¯Qr) be the subspace onto which Qr(respectively QT
+r) projects. We
+then have39
+Theorem 16. For each pair of distinct indices r,sthe subspaces Qr,¯Qrhave the
+orthogonal decomposition
+Qr=Q0
+rs⊕Qrs (322)
+¯Qr=¯Q0
+rs⊕¯Qrs (323)
+where
+Q0
+rs⊥Qrs dim(Q0
+rs) = 1 dim( Qrs) =d−2
+¯Q0
+rs⊥¯Qrs dim(¯Q0
+rs) = 1 dim( ¯Qrs) =d−2
+We have
+(a)Relation of the subspaces QrandQs:
+(1)Q0
+rs⊥QsrandQrs⊥Q0
+sr.
+(2)Q0
+rsandQ0
+srare inclined at angle cos−1/parenleftbig1
+d+1/parenrightbig
+.
+(3)QrsandQsrare uniformly inclined at angle cos−1/parenleftig
+1√d+1/parenrightig
+.
+(b)Relation of the subspaces ¯Qrand¯Qs:
+(1)¯Q0
+rs⊥¯Qsrand¯Qrs⊥¯Q0
+sr.
+(2)¯Q0
+rsand¯Q0
+srare inclined at angle cos−1/parenleftbig1
+d+1/parenrightbig
+.
+(3)¯Qrsand¯Qsrare uniformly inclined at angle cos−1/parenleftig
+1√d+1/parenrightig
+.
+(c)Relation of the subspaces Qrand¯Qs:
+(1)Q0
+rs⊥¯Qsr,Qrs⊥¯Q0
+srandQrs⊥¯Qsr.
+(2)Q0
+rsand¯Q0
+srare inclined at angle cos−1/parenleftbigd
+d+1/parenrightbig
+.
+The relations between these subspaces are, perhaps, easier to a ssimilate if pre-
+sented pictorially. In the following diagrams the line joining each pair of subspaces
+is labelled with the cosine of the angle between them. In particular a 0 o n the line
+joining two subspaces indicates that they are orthogonal.
+Q0
+rs Qrs
+Q0
+sr Qsr0
+01
+d+11√d+1
+
+0❅
+❅
+❅
+❅
+❅
+❅❅0¯Q0
+rs¯Qrs
+¯Q0
+sr¯Qsr0
+01
+d+11√d+1
+
+0❅
+❅
+❅
+❅
+❅
+❅❅0
+Q0
+rs Qrs
+¯Q0
+sr¯Qsr0
+0d
+d+10
+
+0❅
+❅
+❅
+❅
+❅
+❅❅040
+Wewillprovethistheorembelow. Beforedoingso,however,letusst atetheother
+mainresult ofthis section. Let Rrbe the subspace ontowhichthe ¯Rrproject. Since
+¯Rris a real matrix we regard Rras a subspace of Rd2. We have
+Theorem 17. For each pair of distinct indices r,sthe subspace Rrhas the decom-
+position
+Rr=R0
+rs⊕R1
+rs⊕Rrs (324)
+whereR0
+rs,R1
+rs,Rrsare pairwise orthogonal and
+dim(R0
+rs) = 1 dim( R1
+rs) = 1 dim( Rrs) = 2d−4 (325)
+We have
+(1)R0
+rs=R0
+sr.
+(2)R1
+rs⊥RsrandRrs⊥R1
+sr.
+(3)R1
+rsandR1
+srare inclined at angle cos−1/parenleftbigd−1
+d+1/parenrightbig
+.
+(4)RrsandRsrare uniformly inclined at angle cos−1/parenleftig/radicalig
+1
+d+1/parenrightig
+In particular, the subspaces ¯Rrand¯Rsintersect in a line.
+In diagrammatic form the relations between these subspaces are
+R0
+rs=R0
+srR1
+rs Rrs
+R1
+sr Rsr0
+0d−1
+d+1/radicalig
+1
+d+1
+
+0❅
+❅
+❅
+❅
+❅
+❅❅0✟✟✟✟✟0
+❍❍❍❍❍00
+0
+where, as before, each line is labelled with the cosine of the angle betw een the two
+subspaces it connects.
+Proof of Theorem 16.Let/bardbl1/an}bracketri}ht/an}bracketri}ht,...,/bardbld2/an}bracketri}ht/an}bracketri}htbethestandardbasisfor Hd2, asdefined
+by Eq. (117). For each pair of distinct indices r,sdefine
+/bardblfrs/an}bracketri}ht/an}bracketri}ht=i√
+d+1Qr/bardbls/an}bracketri}ht/an}bracketri}ht (326)
+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=−i√
+d+1QT
+r/bardbls/an}bracketri}ht/an}bracketri}ht (327)
+The significance of these vectors is that /bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl(respectively /bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl) will
+turn out to be the projectoronto the 1-dimensionalsubspace Q0
+rs(respectively ¯Q0
+rs).41
+Note that the fact that Qris Hermitian means
+QT
+r=Q∗
+r (328)
+(whereQ∗
+ris the matrix whose elements are the complex conjugates of the cor re-
+sponding elements of Qr). Consequently
+/an}bracketle{t/an}bracketle{tt/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=/parenleftig
+/an}bracketle{t/an}bracketle{tt/bardblfrs/an}bracketri}ht/an}bracketri}ht/parenrightig∗
+(329)
+for allr,s,t.
+It is easily seen that /bardblfrs/an}bracketri}ht/an}bracketri}ht,/bardblf∗
+rs/an}bracketri}ht/an}bracketri}htare normalized. In fact, it follows from
+Eqs. (116) and (120) that
+/an}bracketle{t/an}bracketle{tfrs/bardblfrs/an}bracketri}ht/an}bracketri}ht= (d+1)/an}bracketle{t/an}bracketle{ts/bardblQr/bardbls/an}bracketri}ht/an}bracketri}ht
+=(d+1)2
+dTrss−2(d+1)/an}bracketle{t/an}bracketle{ts/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbls/an}bracketri}ht/an}bracketri}ht
+=(d+1)2
+d/parenleftbig
+K2
+rs−K4
+rs/parenrightbig
+= 1 (330)
+for allr/ne}ationslash=s. In view of Eq. ( 329) we then have
+/an}bracketle{t/an}bracketle{tf∗
+rs/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=/parenleftig
+/an}bracketle{t/an}bracketle{tfrs/bardblfrs/an}bracketri}ht/an}bracketri}ht/parenrightig∗
+= 1 (331)
+for allr/ne}ationslash=s. The fact that QrQT
+r= 0 means we also have
+/an}bracketle{t/an}bracketle{tfrs/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht= 0 (332)
+for allr/ne}ationslash=s.
+Note that, although we required that r/ne}ationslash=sin the definitions of /bardblfrs/an}bracketri}ht/an}bracketri}ht,/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht,
+the definitions continue to make sense when r=s. However, the vectors are then
+zero (as can be seen by setting r=sin Eq. (121)).
+The vectors /bardblfrs/an}bracketri}ht/an}bracketri}ht,/bardblf∗
+rs/an}bracketri}ht/an}bracketri}htsatisfy a number of identities, which it will be conve-
+nient to collect in a lemma:
+Lemma 18. For allr/ne}ationslash=s
+/bardblfrs/an}bracketri}ht/an}bracketri}ht=−/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht+i/radicalbigg
+2
+d/parenleftig
+/bardbles/an}bracketri}ht/an}bracketri}ht−/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
+(333)
+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=−/bardblfsr/an}bracketri}ht/an}bracketri}ht−i/radicalbigg
+2
+d/parenleftig
+/bardbles/an}bracketri}ht/an}bracketri}ht−/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
+(334)
+(where/bardbler/an}bracketri}ht/an}bracketri}htis the vector defined by Eq. ( 116))
+Qr/bardblfrs/an}bracketri}ht/an}bracketri}ht=/bardblfrs/an}bracketri}ht/an}bracketri}ht QT
+r/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht (335)
+QT
+r/bardblfrs/an}bracketri}ht/an}bracketri}ht= 0 Qr/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht= 0 (336)
+Qs/bardblfrs/an}bracketri}ht/an}bracketri}ht=−1
+d+1/bardblfsr/an}bracketri}ht/an}bracketri}ht QT
+s/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=−1
+d+1/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht(337)
+QT
+s/bardblfrs/an}bracketri}ht/an}bracketri}ht=−d
+d+1/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht Qs/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=−d
+d+1/bardblfsr/an}bracketri}ht/an}bracketri}ht(338)
+/an}bracketle{t/an}bracketle{tfrs/bardblfsr/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tf∗
+rs/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht=−1
+d+1(339)42
+/an}bracketle{t/an}bracketle{tfrs/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tf∗
+rs/bardblfsr/an}bracketri}ht/an}bracketri}ht=−d
+d+1(340)
+Proof.It follows from Eqs. ( 116) and (120) that
+/an}bracketle{t/an}bracketle{tt/bardblfrs/an}bracketri}ht/an}bracketri}ht+/an}bracketle{t/an}bracketle{tt/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht=i√
+d+1/parenleftbig
+Qrts−Qsrt/parenrightbig
+=i√
+d+1/parenleftbiggd+1
+d/parenleftbig
+Trts−Tsrt/parenrightbig
+−2/parenleftbig
+/an}bracketle{t/an}bracketle{tt/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbls/an}bracketri}ht/an}bracketri}ht−/an}bracketle{t/an}bracketle{tr/bardbles/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tes/bardblt/an}bracketri}ht/an}bracketri}ht/parenrightbigg
+=i/radicalbigg
+2
+d/parenleftbig
+/an}bracketle{t/an}bracketle{tt/bardbles/an}bracketri}ht/an}bracketri}ht−/an}bracketle{t/an}bracketle{tt/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightbig
+(341)
+where we used the fact that Trts=Tsrtin the third step, and the fact that /an}bracketle{t/an}bracketle{tt/bardbles/an}bracketri}ht/an}bracketri}htis
+real in the last. This establishes Eq. ( 333). Eq. (334) is obtained by taking complex
+conjugates on both sides, and using the fact that the vectors /bardbles/an}bracketri}ht/an}bracketri}htare real.
+Eqs. (335) and (336) are immediate consequences of the definitions, and the fact
+thatQrQT
+r= 0. Turning to the proof of Eqs. ( 337) and (338), it follows from
+Eqs. (119) and (120) that
+Qs/bardbles/an}bracketri}ht/an}bracketri}ht= 0 (342)
+Using this and the fact that Qs/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht= 0 in Eq. ( 333) we find
+Qs/bardblfrs/an}bracketri}ht/an}bracketri}ht=−i/radicalbigg
+2
+dQs/bardbler/an}bracketri}ht/an}bracketri}ht (343)
+Since
+/bardbler/an}bracketri}ht/an}bracketri}ht=/radicaligg
+d
+2(d+1)/parenleftig
+/bardblr/an}bracketri}ht/an}bracketri}ht+/bardblv0/an}bracketri}ht/an}bracketri}ht/parenrightig
+(344)
+and taking account of the fact that Qs/bardblv0/an}bracketri}ht/an}bracketri}ht= 0 (see Eq. ( 287)) we deduce
+Qs/bardblfrs/an}bracketri}ht/an}bracketri}ht=−i/radicalbigg
+1
+d+1Qs/bardblr/an}bracketri}ht/an}bracketri}ht=−1
+d+1/bardblfsr/an}bracketri}ht/an}bracketri}ht (345)
+Taking complex conjugates on both sides of this equation we deduce the second
+identity in Eq. ( 337).
+In the same way, acting on both sides of Eq. ( 333) withQT
+swe find
+QT
+s/bardblfrs/an}bracketri}ht/an}bracketri}ht=−/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht−i/radicalbigg
+2
+dQT
+s/bardbler/an}bracketri}ht/an}bracketri}ht
+=−/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht−i/radicalbigg
+1
+d+1QT
+s/bardblr/an}bracketri}ht/an}bracketri}ht
+=−d
+d+1/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht (346)
+Taking complex conjugates on both sides of this equation we deduce the second
+identity in Eq. ( 338).
+Turning to the last group of identities we have
+/an}bracketle{t/an}bracketle{tfrs/bardblfsr/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tfrs/bardblQr/bardblfsr/an}bracketri}ht/an}bracketri}ht=−1
+d+1/an}bracketle{t/an}bracketle{tfrs/bardblfrs/an}bracketri}ht/an}bracketri}ht=−1
+d+1(347)
+and
+/an}bracketle{t/an}bracketle{tfrs/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tfrs/bardblQr/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht=−d
+d+1/an}bracketle{t/an}bracketle{tfrs/bardblfrs/an}bracketri}ht/an}bracketri}ht=−d
+d+1(348)43
+The other two identities are obtained by taking complex conjugates on both sides
+of the two just derived. /square
+This lemma provides a substantial part of what we need to prove the theorem.
+The remaining part is provided by
+Lemma 19. For allr/ne}ationslash=s
+QrQsQr=1
+d+1Qr−d
+(d+1)2/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (349)
+QrQT
+sQr=d2
+(d+1)2/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (350)
+Proof.It follows from Eq. ( 120) that
+QrQsQr=d+1
+dQrTsQr−2Qr/bardbles/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tes/bardblQr (351)
+QrQT
+sQr=d+1
+dQrTT
+sQr−2Qr/bardbles/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tes/bardblQr (352)
+In view of Eqs. ( 344), (287) and the definition of /bardblfrs/an}bracketri}ht/an}bracketri}htwe have
+Qr/bardbles/an}bracketri}ht/an}bracketri}ht=/radicaligg
+d
+2(d+1)Qr/bardbls/an}bracketri}ht/an}bracketri}ht=−i√
+d√
+2(d+1)/bardblfrs/an}bracketri}ht/an}bracketri}ht (353)
+Substituting this expression into Eqs. ( 351) and (352) we obtain
+QrQsQr=d+1
+dQrTsQr−d
+(d+1)2/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (354)
+QrQT
+sQr=d+1
+dQrTT
+sQr−d
+(d+1)2/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (355)
+The problem therefore reduces to showing
+QrTsQr=d
+(d+1)2Qr (356)
+QrTT
+sQr=d2
+(d+1)2/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (357)
+Using Eq. ( 120) we find
+/an}bracketle{t/an}bracketle{ta/bardblQrTsQr/bardblb/an}bracketri}ht/an}bracketri}ht=(d+1)2
+d2/an}bracketle{t/an}bracketle{ta/bardblTrTsTr/bardblb/an}bracketri}ht/an}bracketri}ht
+−1
+2/parenleftbigg2(d+1)
+d/parenrightbigg3
+2/parenleftig
+K2
+ra/an}bracketle{t/an}bracketle{ter/bardblTsTr/bardblb/an}bracketri}ht/an}bracketri}ht+K2
+rb/an}bracketle{t/an}bracketle{ta/bardblTrTs/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
++2(d+1)
+dK2
+raK2
+rb/an}bracketle{t/an}bracketle{ter/bardblTs/bardbler/an}bracketri}ht/an}bracketri}ht (358)
+/an}bracketle{t/an}bracketle{ta/bardblQrTT
+sQr/bardblb/an}bracketri}ht/an}bracketri}ht=(d+1)2
+d2/an}bracketle{t/an}bracketle{ta/bardblTrTT
+sTr/bardblb/an}bracketri}ht/an}bracketri}ht
+−1
+2/parenleftbigg2(d+1)
+d/parenrightbigg3
+2/parenleftig
+K2
+ra/an}bracketle{t/an}bracketle{ter/bardblTT
+sTr/bardblb/an}bracketri}ht/an}bracketri}ht+K2
+rb/an}bracketle{t/an}bracketle{ta/bardblTrTT
+s/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
++2(d+1)
+dK2
+raK2
+rb/an}bracketle{t/an}bracketle{ter/bardblTT
+s/bardbler/an}bracketri}ht/an}bracketri}ht (359)44
+Using the definitions of Tr,/bardbler/an}bracketri}ht/an}bracketri}htand Eq. ( 23) (the 2-design property) we find, after
+some algebra,
+/an}bracketle{t/an}bracketle{ta/bardblTrTsTr/bardblb/an}bracketri}ht/an}bracketri}ht=d2
+(d+1)2/parenleftig
+K2
+raTrsb+K2
+rbTras+K2
+rsTrab+K2
+raK2
+rb/parenrightig
+(360)
+/an}bracketle{t/an}bracketle{ter/bardblTsTr/bardblb/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
+2(d+1)/parenrightbigg3
+2/parenleftig
+2K2
+rsK2
+rb+K2
+rb+Trsb/parenrightig
+(361)
+/an}bracketle{t/an}bracketle{ta/bardblTrTs/bardbler/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
+2(d+1)/parenrightbigg3
+2/parenleftig
+2K2
+rsK2
+ra+K2
+ra+Tras/parenrightig
+(362)
+/an}bracketle{t/an}bracketle{ter/bardblTs/bardbler/an}bracketri}ht/an}bracketri}ht=d
+2(d+1)/parenleftbig
+3K2
+rs+1/parenrightbig
+(363)
+and
+/an}bracketle{t/an}bracketle{ta/bardblTrTT
+sTr/bardblb/an}bracketri}ht/an}bracketri}ht=d2
+(d+1)2/parenleftig
+GraGasGsbGbr
++K2
+raTrsb+K2
+rbTras+K2
+raK2
+rb/parenrightig
+=d2
+(d+1)2/parenleftig
+(d+1)TrasTrsb
++K2
+raTrsb+K2
+rbTras+K2
+raK2
+rb/parenrightig
+(364)
+/an}bracketle{t/an}bracketle{ter/bardblTT
+sTr/bardblb/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
+2(d+1)/parenrightbigg3
+2/parenleftig
+K2
+rsK2
+rb+K2
+rb+2Trsb/parenrightig
+(365)
+/an}bracketle{t/an}bracketle{ta/bardblTrTT
+s/bardbler/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
+2(d+1)/parenrightbigg3
+2/parenleftig
+K2
+rsK2
+ra+K2
+ra+2Tras/parenrightig
+(366)
+/an}bracketle{t/an}bracketle{ter/bardblTT
+s/bardbler/an}bracketri}ht/an}bracketri}ht=d
+2(d+1)/parenleftbig
+3K2
+rs+1/parenrightbig
+(367)
+where in deriving Eq. ( 364) we used the fact that GraGasGsbGbr= (d+1)TrasTrsb
+(in view of the fact that r/ne}ationslash=s). Substituting these expressions into Eqs. ( 358)
+and (359) we deduce Eqs. ( 356) and (357). /square
+Now define the rank d−1 projectors
+Qrs=Qr−/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (368)
+QT
+rs=QT
+r−/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl (369)
+andletQ0
+rs,Qrs,¯Q0
+rsand¯Qrsbe, respectively, the subspacesontowhich /bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl,
+Qrs,/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardblandQ∗
+rsproject. It is immediate that we have the orthogonal
+decompositions
+Qr=Q0
+rs⊕Qrs (370)
+¯Qr=¯Q0
+rs⊕¯Qrs (371)
+Using Lemma 18we find
+Qsr/bardblfrs/an}bracketri}ht/an}bracketri}ht=Qrs/bardblfsr/an}bracketri}ht/an}bracketri}ht= 0 (372)45
+implying that Q0
+rs⊥QsrandQrs⊥Q0
+sr, and
+/vextendsingle/vextendsingle/an}bracketle{t/an}bracketle{tfrs/bardblfsr/an}bracketri}ht/an}bracketri}ht/vextendsingle/vextendsingle=1
+d+1(373)
+implying that Q0
+rsandQ0
+srare inclined at angle cos−1/parenleftbig1
+d+1/parenrightbig
+. Using Lemma 18
+together with Lemma 19we find
+QrsQsrQrs=QrsQsQrs
+=QrQsQr−/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardblQsQr−QrQs/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl
++/an}bracketle{t/an}bracketle{tfrs/bardblQs/bardblfrs/an}bracketri}ht/an}bracketri}ht/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl
+=1
+d+1Qr−1
+d+1/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl
+=1
+d+1Qrs (374)
+which in view of Lemma 15implies that QrsandQsrare uniformly inclined at angle
+cos−1/parenleftbig1√d+1/parenrightbig
+. This proves part (a) of the theorem. Parts (b) and (c) are prov ed
+similarly.
+Proof of Theorem 17.Define
+/bardblgrs/an}bracketri}ht/an}bracketri}ht=1√
+2/parenleftbig
+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht+/bardblfrs/an}bracketri}ht/an}bracketri}ht/parenrightbig
+(375)
+/bardbl¯grs/an}bracketri}ht/an}bracketri}ht=i√
+2/parenleftbig
+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht−/bardblfrs/an}bracketri}ht/an}bracketri}ht/parenrightbig
+(376)
+By construction the components of /bardblgrs/an}bracketri}ht/an}bracketri}ht,/bardbl¯grs/an}bracketri}ht/an}bracketri}htin the standard basis are real, so
+we can regard them as ∈Rd2. They are orthonormal:
+/an}bracketle{t/an}bracketle{tgrs/bardblgrs/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{t¯grs/bardbl¯grs/an}bracketri}ht/an}bracketri}ht= 1 and /an}bracketle{t/an}bracketle{tgrs/bardbl¯grs/an}bracketri}ht/an}bracketri}ht= 0 (377)
+It is also readily verified, using Lemma 18, that
+¯Rr/bardblgrs/an}bracketri}ht/an}bracketri}ht=/bardblgrs/an}bracketri}ht/an}bracketri}ht (378)
+¯Rr/bardbl¯grs/an}bracketri}ht/an}bracketri}ht=/bardbl¯grs/an}bracketri}ht/an}bracketri}ht (379)
+So
+Rrs=¯Rr−/bardblgrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgrs/bardbl−/bardbl¯grs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯grs/bardbl (380)
+is a rank 2 d−4 projector. If we define R0
+rs,R1
+rsandRrsto be, respectively,
+the subspaces onto which /bardblgrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgrs/bardbl,/bardbl¯grs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯grs/bardblandRrsproject we have the
+orthogonal decomposition
+Rr=R0
+rs⊕R1
+rs⊕Rrs (381)
+It follows from Eqs. ( 333) and (334) that
+/bardblgrs/an}bracketri}ht/an}bracketri}ht=−/bardblgsr/an}bracketri}ht/an}bracketri}ht (382)
+implying that R0
+rs=R0
+srfor allr/ne}ationslash=s. It is also easily verified, using Lemma 18,
+that/vextendsingle/vextendsingle/an}bracketle{t/an}bracketle{t¯grs/bardbl¯gsr/an}bracketri}ht/an}bracketri}ht/vextendsingle/vextendsingle=d−1
+d+1(383)
+from which it follows that R1
+rsandR1
+srare inclined at angle cos−1/parenleftbigd−1
+d+1/parenrightbig
+. We next
+observe that
+Rrs=Qrs+QT
+rs (384)46
+Using Lemma 18once again we deduce
+Rrs/bardbl¯gsr/an}bracketri}ht/an}bracketri}ht=Rsr/bardbl¯grs/an}bracketri}ht/an}bracketri}ht= 0 (385)
+from which it follows that R1
+rs⊥RsrandRrs⊥R1
+sr. Finally, we know from
+Theorem 16thatQT
+rsQsr=QrsQT
+sr= 0. Consequently
+RrsRsrRrs=QrsQsrQrs+QT
+rsQT
+srQT
+rs
+=d
+d+1Qrs+d
+d+1QT
+rs
+=1
+d+1Rrs (386)
+In view of Lemma 15it follows that RrsandRsrare uniformly inclined at angle
+cos−1/parenleftbig1√d+1/parenrightbig
+.
+Further Identities. We conclude this section with another set of identities in-
+volving the vectors /bardblfrs/an}bracketri}ht/an}bracketri}ht,/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht,/bardblgrs/an}bracketri}ht/an}bracketri}htand/bardbl¯grs/an}bracketri}ht/an}bracketri}ht.
+Define
+/bardbl¯er/an}bracketri}ht/an}bracketri}ht=/radicalbigg
+2d
+d−1/bardbler/an}bracketri}ht/an}bracketri}ht−/radicalbigg
+d+1
+d−1/bardblv0/an}bracketri}ht/an}bracketri}ht (387)
+where/bardblv0/an}bracketri}ht/an}bracketri}htis the vector defined by Eq. ( 286). It is readily verified that
+/an}bracketle{t/an}bracketle{t¯er/bardbl¯er/an}bracketri}ht/an}bracketri}ht= 0 and /an}bracketle{t/an}bracketle{t¯er/bardblv0/an}bracketri}ht/an}bracketri}ht= 0 (388)
+So/bardbl¯er/an}bracketri}ht/an}bracketri}ht,/bardblv0/an}bracketri}ht/an}bracketri}htis an orthonormal basis for the 2-dimensional subspace spanned b y
+/bardbler/an}bracketri}ht/an}bracketri}ht,/bardblv0/an}bracketri}ht/an}bracketri}ht. Note that
+Qr/bardbl¯er/an}bracketri}ht/an}bracketri}ht=QT
+r/bardbl¯er/an}bracketri}ht/an}bracketri}ht=¯Rr/bardbl¯er/an}bracketri}ht/an}bracketri}ht= 0 (389)
+We then have
+Theorem 20. For allr
+1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl=Qr (390)
+1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl=QT
+r (391)
+2
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblgrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgrs/bardbl=¯Rr (392)
+2
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardbl¯grs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯grs/bardbl=¯Rr (393)
+and
+1
+d−1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl=QT
+r+/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl+1
+d2−1/parenleftig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig
+(394)47
+1
+d−1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl=Qr+/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl+1
+d2−1/parenleftig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig
+(395)
+2
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblgsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgsr/bardbl=¯Rr (396)
+2
+d−3d2/summationdisplay
+s=1
+(s/negationslash=r)/bardbl¯gsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯gsr/bardbl=¯Rr+4(d−1)
+d−3/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl+4
+(d+1)(d−3)/parenleftig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig
+(397)
+Proof.It follows from the definition of /bardblfrs/an}bracketri}ht/an}bracketri}htthat
+1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl=d2/summationdisplay
+s=1
+(s/negationslash=r)Qr/bardbls/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ts/bardblQr
+=Qr
+d2/summationdisplay
+s=1/bardbls/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ts/bardbl
+Qr
+=Qr (398)
+where in the second step we used the fact that Qr/bardblr/an}bracketri}ht/an}bracketri}ht= 0 (as can be seen by setting
+r=sin Eq. (121)). Eq. (391) is obtained by taking the complex conjugate on both
+sides.
+We also have
+1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl=−d2/summationdisplay
+s=1
+(s/negationslash=r)Qr/bardbls/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ts/bardblQT
+r
+=−Qr
+d2/summationdisplay
+s=1/bardbls/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ts/bardbl
+QT
+r
+=−QrQT
+r
+= 0 (399)
+Taking the complex conjugate on both sides we find
+1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl= 0 (400)
+Consequently
+2
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblgrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgrs/bardbl=1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/parenleftig
+/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl
++/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl/parenrightig
+=¯Rr (401)48
+Eq. (393) is proved similarly.
+To prove the second group of identities we have to work a little harde r. Using
+Eqs. (116) and (120) we find
+1
+d−1d2/summationdisplay
+s=1
+(s/negationslash=r)/an}bracketle{t/an}bracketle{ta/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardblb/an}bracketri}ht/an}bracketri}ht=d+1
+d−1d2/summationdisplay
+s=1/an}bracketle{t/an}bracketle{ta/bardblQs/bardblr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tr/bardblQs/bardblb/an}bracketri}ht/an}bracketri}ht
+=(d+1)3
+d2(d−1)d2/summationdisplay
+s=1/parenleftig
+TsarTsrb−K2
+saK2
+srTsrb
+−K2
+srK2
+sbTsar+K2
+saK4
+srK2
+sb/parenrightig
+(402)
+(where we used the fact that Qs/bardbls/an}bracketri}ht/an}bracketri}ht= 0 in the first step). After some algebra we
+find
+d2/summationdisplay
+s=1TsarTsrb=d
+d+1/parenleftigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg
++Trba/parenrightigg
+(403)
+d2/summationdisplay
+s=1K2
+saK2
+srTsrb=d
+d+1/parenleftigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+2d+1
+d(d+1)/parenrightigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg
++1
+d+1Trba/parenrightigg
+(404)
+d2/summationdisplay
+s=1K2
+srK2
+sbTsar=d
+d+1/parenleftigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+2d+1
+d(d+1)/parenrightigg
++1
+d+1Trba/parenrightigg
+(405)
+d2/summationdisplay
+s=1K2
+saK4
+srK2
+sb=d
+(d+1)/parenleftigg
+d+2
+d+1/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg
++d
+(d+1)3δab+d+2
+(d+1)3/parenrightigg
+(406)
+wherewe usedEq. ( 23) to derivethe firstexpression. Substituting these expressions
+into Eq. ( 402) and using
+/an}bracketle{t/an}bracketle{ta/bardblQT
+r/bardblb/an}bracketri}ht/an}bracketri}ht=d+1
+d/parenleftigg
+Trba−/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg/parenrightigg
+(407)
+we deduce Eq. ( 394). Taking complex conjugates on both sides we obtain Eq. ( 395).
+Eq. (396) is an immediate consequence of Eq. ( 392) and the fact that /bardblgsr/an}bracketri}ht/an}bracketri}ht=
+−/bardblgrs/an}bracketri}ht/an}bracketri}htfor allr,s.49
+To prove Eq. ( 397) observe that it follows from Eqs. ( 394)–(396) that
+d2/summationdisplay
+s=1
+(s/negationslash=r)/parenleftig
+/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl+/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl/parenrightig
+=d2/summationdisplay
+s=1
+(s/negationslash=r)/parenleftig
+2/bardblgsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgsr/bardbl−/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl−/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl/parenrightig
+= 2/parenleftig
+¯Rr−(d−1)/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl
+−1
+d+1/parenleftig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig/parenrightbigg
+(408)
+Hence
+2
+d−3d2/summationdisplay
+s=1
+(s/negationslash=r)/bardbl¯gsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯gsr/bardbl=1
+d−3d2/summationdisplay
+s=1
+(s/negationslash=r)/parenleftig
+/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl+/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl
+−/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl−/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl/parenrightig
+=¯Rr+4(d−1)
+d−3/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl+4
+(d+1)(d−3)/parenleftig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig
+(409)
+/square
+9.TheP-PTProperty
+In the preceding sections the Q-QTproperty has played a prominent role. In
+this section we show that in the particular case ofa Weyl-Heisenberg covariantSIC-
+POVM, and with the appropriate choice of gauge, the Gram project or (defined in
+Eq. (63)) has an analogous property, which we call the P-PTproperty. Specifically
+one has
+PPT=PTP=/bardblh/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{th/bardbl (410)
+where/bardblh/an}bracketri}ht/an}bracketri}htis a normalized vector whose components in the standard basis are a ll
+real. In odd dimensions the components of /bardblh/an}bracketri}ht/an}bracketri}htin the standard basis can be simply
+expressed in terms of the Wigner function of the fiducial vector. I t could be said
+thattheprojectors PandPTarealmostorthogonal(bycontrastwiththeprojectors
+QrandQT
+rwhich are completely orthogonal). More precisely Phas the spectral
+decomposition
+P=¯P+/bardblh/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{th/bardbl (411)
+where¯Pis a rank (d−1) projector with the property
+¯P¯PT= 0 (412)
+This means that the matrix
+JP=P−PT(413)
+is a pure imaginary Hermitian matrix with the property that J2
+Pis a real rank
+2d−2 projector ( c.f.the discussion in Section 4).
+Although we are mainly interested in the P-PTproperty as it applies to SIC-
+POVMs, itshould benoted that itactuallyholdsforanyWeyl-Heisenbe rgcovariant
+POVM (with the appropriate choice of gauge). So we will prove the ab ove propo-
+sitions for this more general case.50
+Let us begin by fixing notation. Let |0/an}bracketri}ht,...,|d−1/an}bracketri}htbe an orthonormal basis for
+d-dimensional Hilbert space and let XandZbe the operators whose action on the
+|r/an}bracketri}htis
+X|a/an}bracketri}ht=|a+1/an}bracketri}ht (414)
+Z|a/an}bracketri}ht=ωa|a/an}bracketri}ht (415)
+whereω=e2πi
+dand the addition of indices in the first equation is modd. We then
+define the Weyl-Heisenberg displacement operators by (adopting t he convention
+used in, for example, ref. [ 16])
+Dp=τp1p2Xp1Zp2(416)
+wherepis the vector ( p1,p2) (p1,p2being integers) and τ=e(d+1)πi
+d. Generally
+speaking the decision to insert the phase τp1p2is a matter of convention, and many
+authors define it differently, or else omit altogether. However, for the purposes of
+this section it is essential, as a different choice of phase at this stage would lead to a
+different gauge in the class of POVMs to be defined below, and the Gra m projector
+would then typically not have the P-PTproperty.
+Note thatτ2=τd2=ωin every dimension. If the dimension is odd we can write
+τ=ωd+1
+2. Soτis adthroot of unity. However, if the dimension is even τd=−1.
+This has the consequence that
+Dp+du= (−1)u1p2+u2p1Dp (417)
+Soin even dimension p=q(modd) does notnecessarilyimply Dp=Dq(although
+the operators are, of course, equal if p=q(mod 2d))
+In every dimension (even or odd) we have
+D†
+p=D−p (418)
+for allp
+(Dp)n=Dnp (419)
+for allp,nand
+DpDq=τ/angbracketleftp,q/angbracketrightDp+q (420)
+for allp,q. In the last expression /an}bracketle{tp,q/an}bracketri}htis the symplectic form
+/an}bracketle{tp,q/an}bracketri}ht=p2q1−p1q2 (421)
+Now let|ψ/an}bracketri}htbe any normalized vector (not necessarily a SIC-fiducial vector), and
+define
+|ψp/an}bracketri}ht=Dp|ψ/an}bracketri}ht (422)
+Let
+L=/summationdisplay
+p∈Z2
+d|ψp/an}bracketri}ht/an}bracketle{tψp| (423)
+It is easily seen that/bracketleftbig
+Dp,L/bracketrightbig
+= 0 (424)
+for allp.51
+We now appeal to the fact that there is no non-trivial subspace of Hdwhich
+the displacement operators leave invariant. To see this assume the contrary. Then
+there would exist non-zero vectors |φ/an}bracketri}ht,|χ/an}bracketri}htsuch that
+/an}bracketle{tφ|Dp|χ/an}bracketri}ht= 0 (425)
+for allp. Writing the left-hand side out in full this gives
+d−1/summationdisplay
+a=0ωp2a/an}bracketle{tφ|a+p1/an}bracketri}ht/an}bracketle{ta|χ/an}bracketri}ht= 0 (426)
+for allp1,p2. Taking the discrete Fourier transform with respect to p2, we have
+/an}bracketle{tφ|a+p1/an}bracketri}ht/an}bracketle{ta|χ/an}bracketri}ht= 0 (427)
+for alla,p1, implying that either |φ/an}bracketri}ht= 0 or|χ/an}bracketri}ht= 0—contrary to assumption. We
+can therefore use Schur’s lemma [ 55] to deduce that
+L=kI (428)
+for some constant k. Taking the trace on both sides of this equation we infer
+thatk=d. We conclude that1
+d|ψp/an}bracketri}ht/an}bracketle{tψp|is a POVM. We refer to POVMs of this
+general class as Weyl-Heisenberg covariant POVMs. We refer to th e vector |ψ/an}bracketri}ht
+which generates the POVM as the fiducial vector (with no implication t hat it is
+necessarily a SIC-fiducial).
+Now consider the Gram projector
+P=/summationdisplay
+p,q∈Z2
+dPp,q/bardblp/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tq/bardbl (429)
+where
+Pp,q=1
+d/an}bracketle{tψp|ψq/an}bracketri}ht (430)
+and where we label the matrix elements of Pand the standard basis kets with the
+vectorsp,qrather than with the single integer indices r,sas in the rest of this
+paper. We know from Theorem 1thatPis a rankdprojector.
+In view of Eqs. ( 418) and (420) we have
+/an}bracketle{t/an}bracketle{tp/bardblP/bardblq/an}bracketri}ht/an}bracketri}ht=Pp,q
+=1
+dτ−/angbracketleftp,q/angbracketright/an}bracketle{tψ|Dq−p|ψ/an}bracketri}ht
+=1
+dd−1/summationdisplay
+a=0τp1p2+q1q2ωaq2−(q1+a)p2/an}bracketle{tψ|a+q1−p1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht(431)
+Hence
+/an}bracketle{t/an}bracketle{tp/bardblPPT/bardblq/an}bracketri}ht/an}bracketri}ht=/summationdisplay
+u∈Zd/an}bracketle{t/an}bracketle{tp/bardblP/bardblu/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tq/bardblP/bardblu/an}bracketri}ht/an}bracketri}ht
+=1
+d2d−1/summationdisplay
+a,b,u1,u2=0τp1p2+q1q2ωu2(u1+a+b)−(u1+a)p2−(u1+b)q2
+×/an}bracketle{tψ|a+u1−p1/an}bracketri}ht/an}bracketle{tψ|b+u1−q1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht/an}bracketle{tb|ψ/an}bracketri}ht52
+=1
+dd−1/summationdisplay
+a,b=0τp1p2+q1q2ωp2b+q2a/an}bracketle{tψ|−b−p1/an}bracketri}ht/an}bracketle{tb|ψ/an}bracketri}ht/an}bracketle{tψ|−a−q1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht
+=/an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{th/bardblq/an}bracketri}ht/an}bracketri}ht (432)
+where/bardblh/an}bracketri}ht/an}bracketri}htis the vector with components
+/an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}ht=1√
+dd−1/summationdisplay
+a=0τp1p2ωp2a/an}bracketle{tψ|−a−p1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht (433)
+It is easily verified that /bardblh/an}bracketri}ht/an}bracketri}htis normalized, and that /an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}htis real.
+Finally, suppose that the dimension is odd. Then the Wigner function o f the
+state|ψ/an}bracketri}htis [56,57]
+W(p) =1
+d/an}bracketle{tψ|DpUPD†
+p|ψ/an}bracketri}ht=1
+d/an}bracketle{tψ|D2pUP|ψ/an}bracketri}ht (434)
+whereUPistheparityoperator,whoseactiononthestandardbasisis UP|a/an}bracketri}ht=|−a/an}bracketri}ht.
+It is straightforward to show
+/an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}ht=√
+dW(−2−1p) (435)
+where 2−1= (d+1)/2 is the multiplicative inverse of 2 considered as an element of
+Zd:i.e.the unique integer 0 ≤m<dsuch that 2 m= 1 (modd).
+10.Conclusion
+A curious fact about SIC-POVMs is that, although they are charac terized by
+their being highly symmetric structures, they do not wear this prop erty on their
+sleeve (so to speak). If one casually inspects the components of a SIC-fiducial,
+without knowing in advance that that is what they are, there does n ot seem to be
+anything special about them at all. Indeed, so far from there being any obvious
+pattern to the components, they seem, to a casual inspection, lik e a completely
+random collection of numbers. Moreover, this is just as true of an e xact fiducial as
+it is of a numerical one (see, for instance, the tabulations in Scott a nd Grassl [ 46]).
+It is only when one looks at them through the right pair of spectacles , and takes the
+trouble to calculate the overlaps Tr(Π rΠs), that the symmetry becomes apparent.
+The situation is a little reminiscent of a hologram, which only takes on th e aspect of
+a meaningful image when it is viewed in the right way. If one wanted to s ummarize
+the content of this paper in a nutshell it could be said that we have ex hibited some
+other pairs of spectacles—other ways of looking at a SIC—which cau se its inner
+secrets (or at any rate some of its inner secrets) to become manif est.
+Rather than focusing on the SIC-vectors |ψr/an}bracketri}ht, as is usually done, we havefocused
+on the angle tensors θrsandθrst, and on the T,JandRmatrices defined in
+terms of them. This is an important change of emphasis because, ra ther than
+being tied to any particular SIC, these quantities characterize ent ire families of
+unitarily equivalent SICs. Like the components of a SIC-fiducial, the angle tensors
+appear, to a casual inspection, like a random collection of numbers. However, if
+one examines the spectra of the T,JandRmatrices one realizes that, underlying
+the appearance of randomness, there is a high degree of order. I f one then goes on
+to examine the geometrical relationships between the subspaces o nto which the Q,
+QTand¯Rmatrices project, as we did in Section 8, one finds yet more instances
+of structure and order. To our minds what is particularly interestin g about all of53
+this is that none of it is obviously suggested by the defining property of a SIC, that
+Tr(ΠrΠs) = 1/(d+1) forr/ne}ationslash=s.
+In the course of this paper we have several times expressed the h ope that the
+Lie algebraic perspective on a SIC will lead to a solution to the existenc e problem.
+Of course, that is only a hope, and it may not be fulfilled. However, we feel on
+rather safer ground when we suggest that the solution is likely to co me, if not from
+this investigation, then from one which is like it to the extent that it fo cuses on a
+feature of a SIC which is not immediately apparent to the eye.
+Specializing to the case of a Weyl-Heisenberg covariant SIC, a fiducia l vector|ψ/an}bracketri}ht
+is a solution to the equations
+/vextendsingle/vextendsingle/an}bracketle{tψ|Dp|ψ/an}bracketri}ht/vextendsingle/vextendsingle2=dδp,0+1
+d+1(436)
+Allowing for the arbitrariness of the overall phase of |ψ/an}bracketri}ht, and taking |ψ/an}bracketri}htto be nor-
+malized, this gives us d2−1 conditions on only 2 d−2 independent real parameters.
+The equations are thus over-determined, and very highly over-de termined when d
+is large. Nevertheless, they have turned out to be soluble in every c ase which has
+been investigated to date. It seems likely that progress will depend on finding the
+structural feature which is responsible for this remarkable fact. The motivation for
+this paper is the belief that it may be structural features of the Lie algebra gl(d,C)
+which are responsible. That suggestion may or may not be correct. But if it turns
+out to be incorrect, the amount of effort which has been expended on this problem
+over a period of more than ten years, so far without fruit, sugges ts to us that the
+solution will depend on finding some other structural feature of a S IC, which is not
+obvious, and which has hitherto escaped attention.
+11.Acknowledgements
+The authors thank I. Bengtsson for discussions. Two authors, D MA and CAF,
+were supported in part by the U. S. Office of Naval Research (Gran t No. N00014-
+09-1-0247). Research at Perimeter Institute is supported by th e Government of
+Canada through Industry Canada and by the Province of Ontario t hrough the
+Ministry of Research & Innovation.
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\ No newline at end of file
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+arXiv:1001.0005v1  [astro-ph.CO]  30 Dec 2009Astronomy& Astrophysics manuscriptno.akari˙RXJ1716˙v5 c∝circlecopyrtESO 2018
+October30,2018
+Environmentaldependenceof 8 µmluminosityfunctionsof
+galaxiesatz ∼0.8
+Comparison between RXJ1716.4 +6708 andthe AKARI NEP deep field.⋆,⋆⋆
+Tomotsugu Goto1,2,⋆⋆⋆, Yusei Koyama3,T.Wada4,C.Pearson5,6,7,H.Matsuhara4,T.Takagi4, H.Shim8, M.Im8,
+M.G.Lee8, H.Inami4,9,10,M.Malkan11, S.Okamura3,T.T.Takeuchi12, S.Serjeant7, T.Kodama2, T.Nakagawa4,
+S.Oyabu4,Y.Ohyama13, H.M.Lee8, N.Hwang2, H.Hanami14, K.Imai15,and T.Ishigaki16
+1Institute for Astronomy, University of Hawaii,2680 Woodla wnDrive, Honolulu, HI,96822, USA
+e-mail:tomo@ifa.hawaii.edu
+2National Astronomical Observatory, 2-21-1 Osawa,Mitaka, Tokyo, 181-8588,Japan
+3Department of Astronomy, School of Science,The University of Tokyo, Tokyo113-0033, Japan
+4Institute of Space and Astronautical Science, JapanAerosp ace Exploration Agency, Sagamihara,Kanagawa 229-8510
+5Rutherford Appleton Laboratory, Chilton, Didcot,Oxfords hire OX110QX, UK
+6Department of Physics,Universityof Lethbridge, 4401 Univ ersity Drive,Lethbridge, AlbertaT1J 1B1, Canada
+7Astrophysics Group, Department of Physics, The OpenUniver sity, MiltonKeynes, MK76AA, UK
+8Department of Physics& Astronomy, FPRD,Seoul National Uni versity, Shillim-Dong,Kwanak-Gu, Seoul 151-742, Korea
+9Spitzer Science Center,California Institute ofTechnolog y, Pasadena, CA91125
+10Department of Astronomical Science,The Graduate Universi tyfor Advanced Studies
+11Department of Physicsand Astronomy, UCLA,Los Angeles, CA, 90095-1547 USA
+12Institute for Advanced Research, Nagoya University, Furo- cho, Chikusa-ku, Nagoya 464-8601
+13Academia Sinica,Institute of Astronomyand Astrophysics, Taiwan
+14Physics Section,Facultyof Humanities and SocialSciences , Iwate University, Morioka, 020-8550
+15TOMER&D Inc. Kawasaki, Kanagawa 2130012, Japan
+16Asahikawa National College of Technology, 2-1-6 2-joShunk ohdai, Asahikawa-shi, Hokkaido 071-8142
+Received September 15, 2009; accepted December 16, 2009
+ABSTRACT
+Aims.Weaim to reveal environmental dependence of infraredlumin osity functions (IR LFs)of galaxies at z ∼0.8 using the AKARI
+satellite. AKARI’s wide field of view and unique mid-IR filter s help us to construct restframe 8 µm LFs directly without relying on
+SEDmodels.
+Methods. We construct restframe 8 µm IR LFs in the cluster region RXJ1716.4 +6708 at z=0.81, and compare them with a blank
+field using the AKARI North Ecliptic Pole deep field data at the same redshift. AKARI’s wide field of view (10’ ×10’) is suitable to
+investigate wide range of galaxy environments. AKARI’s 15 µm filter is advantageous here since it directly probes restfr ame 8µm at
+z∼0.8, without relyingona large extrapolation based ona SEDfi t,which was the largestuncertainty inprevious work.
+Results. We have found that cluster IR LFsat restframe 8 µm have a factor of 2.4smaller L∗and a steeper faint-end slope than that
+of the field. Confirming this trend, we also found that faint-e nd slopes of the cluster LFs becomes flatter and flatter with de creasing
+local galaxy density. These changes in LFs cannot be explain ed by a simple infall of field galaxy population into a cluster . Physics
+that canpreferentiallysuppress IR luminous galaxies inhi gh density regions is requiredtoexplain the observed resul ts.
+Keywords. galaxies: evolution, galaxies:interactions, galaxies:s tarburst, galaxies:peculiar, galaxies:formation
+1. Introduction
+It hasbeenobservedthat galaxypropertieschangeas a funct ion
+of galaxyenvironment;the morphology-densityrelation re ports
+that fractionof elliptical galaxiesis largerat highergal axyden-
+sity(Gotoetal.,2003);thestarformationrate(SFR)ishig herin
+lower galaxy density (G´ omezet al., 2003; Tanakaet al., 200 4)
+. However, despite accumulating observational evidence, w e
+⋆This research is based on the observations with AKARI, a JAXA
+project withthe participationof ESA.
+⋆⋆Based on data collected at Subaru Telescope, which is operat ed by
+the National Astronomical Observatory ofJapan.
+⋆⋆⋆JSPSSPDfellowstill do not fully understand the underlying physics govern ing
+environmental-dependentevolutionofgalaxies.
+Infrared (IR) emission of galaxies is an important
+probe of galaxy activity since at higher redshift, a sig-
+nificant fraction of star formation is obscured by dust
+(Takeuchi,Buat,&Burgarella, 2005; Gotoetal., 2010).
+Although there exist low-z cluster studies (Baiet al., 2006 ;
+Shimet al., 2010; Tranetal., 2010), not much attention has
+been paid to the infrared properties of high redshift cluste r
+galaxies, mainly due to the lack of sensitivity in previous I R
+satellites such as ISO and IRAS. Superb sensitivity of recen tly
+launched Spitzer and AKARI satellites can revolutionize th e
+infraredviewofenvironmentaldependenceofgalaxyevolut ion.2 Gotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8
+Fig.1.Restframe 8 µm LFs of cluster RXJ1716.4 +6708 at
+z=0.81 in the squares, and those of the AKARI NEP deep
+field in the triangles. For RXJ1716.4 +6708, only photometric
+and spectroscopic cluster member galaxies are used. For the
+NEP deep field, galaxies with photo-z/specz in the range of
+0.65< z <0.9are used. The dot-dashed lines are 8 µm LFs
+of RXJ1716.4 +6708, but scaled down for easier comparison.
+Thethindottedlinesarethebest-fitdoublepowerlaws.Vert ical
+arrows show the 5 σflux limits of deep/shallow regions of the
+cluster (red) and the NEP deep field (blue) in terms of L8µmat
+z=0.81.
+In this work, we comparerestframe8 µm LFs between clus-
+ter and field regions at z=0.8 using data from the AKARI.
+Monochromaticrestframe 8 µm luminosity ( L8µm) is important
+since it is known to correlate well with the total IR luminosi ty
+(Babbedgeet al., 2006; Huanget al., 2007), andhence,with t he
+SFR of galaxies (Kennicutt, 1998). This is especially true f or
+star-forminggalaxiesbecausethe rest-frame8 µm fluxare dom-
+inatedbyprominentPAHfeaturessuchasat6.2,7.7and8.6 µm
+(Desert,Boulanger,&Puget, 1990).
+ImportantadvantagesbroughtbytheAKARIareasfollows:
+(i) At z=0.8, AKARI’s 15 µm filter (L15) covers the redshifted
+restframe 8 µm, thus we can estimate 8 µm LFs without using
+a large extrapolation based on SED models, which were the
+largest uncertainty in previous work. (ii) Large field of vie w of
+the AKARI’smid-IRcamera(IRC, 10’ ×10’)allowsustostudy
+wider area including cluster outskirts, where important ev olu-
+tionary mechanisms are suggested to be at work (Gotoet al.,
+2004; Kodamaet al., 2004). For example, passive spiral gala x-
+ies have been observed in such an environment (Gotoet al.,
+2003). Unless otherwise stated, we adopt a cosmology with
+(h,Ωm,ΩΛ) = (0.7,0.3,0.7)(Komatsuet al., 2009).
+2. Data & Analysis
+2.1. LFs ofclusterRXJ1716.4 +6708
+The AKARI is a Japanese infrared satellite (Murakamiet al.,
+2007), which has continuous filter coverage in the mid
+IR wavelengths ( N2,N3,N4,S7,S9W,S11,L15,L18Wand
+L24). The AKARI has observed a massive galaxy cluster,Fig.2.Restframe 8 µm LFs of cluster RXJ1716.4 +6708 at
+z=0.81, divided according to the local galaxy density ( Σ5th).
+Thestars,circlesandsquaresareforgalaxieswith logΣ5th≥2,
+1.6≤logΣ5th<2,andlogΣ5th<1.6,respectively.
+RXJ1716.4 +6708, in N3,S7andL15(Koyamaetal., 2008).
+RXJ1716.4 +6708 is at z=0.81 and has σ= 1522+215
+−150km s−1,
+LXbol= 13.86±1.04×1044ergs−1,kT= 6.8+1.0
+−0.6keV.Mass
+estimate from weak lensing and X-ray are 3.7 ±1.3×1014M⊙
+and 4.35 ±0.83×1014M⊙, respectively (see Koyamaet al.,
+2007, forreferences).
+An important advantage of the AKARI observation is L15
+filter, which corresponds to the restframe 8 µm at z=0.81. With
+15 (3) pointings, L15reaches 66.5 (96.5) µJy in deep (shal-
+low) regions at 5 σ. Here flux is measured in 11” aperture,
+and coverted to total flux using AKARI’s IRC correction table
+(2009.5.1)1.ClusterstudieswiththeSpitzerareoftenperformed
+in 24µm and thus needed a large extrapolation to estimate ei-
+therL8µmor total infrared luminosity ( LTIR,8−1000µm).
+Note that we do not claim the L8µmis a better indicator of
+thetotalIRluminositythanotherindicators(Brandlet al. ,2006;
+Calzetti et al., 2007; Riekeet al., 2009), but it is importan t that
+theAKARIcanmeausureredshifted 8µmfluxdirectlyinoneof
+thefilters.
+Thanks to the AKARI’s wide field of view (10’ ×10’), the
+total area coverage around the cluster is 200 arcmin2, which
+cover larger area than previous cluster studies with the Spi tzer,
+allowingustostudyIRsourcesintheoutskirts,whereimpor tant
+galaxyevolutiontakesplace(e.g.,Gotoet al.,2003).Prev iously,
+Koyamaet al. (2008) reporteda high fractionof L15sourcesin
+the intermediatedensity regionin the cluster,suggesting a pres-
+enceofenvironmentaleffectintheintermediatedensityen viron-
+ment.
+Thissameregionwasimagedwith Suprime-Camin VRi′z′
+and has a good photometric redshift estimate (Koyamaet al.,
+2007).Usedinthisworkare54 L15-detectedgalaxieswhichare
+well identifiedwithopticalsourceswith 0.76≤zphoto≤0.83.
+With the L15filter covering the restframe 8 µm, we simply
+convert the observed flux to 8 µm monochromatic luminosity
+1http://www.ir.isas.jaxa.jp/ASTRO-
+F/Observation/DataReduction/IRC/ApertureCorrection 090501.htmlGotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8 3
+Table 1.Best doublepower-lawfit parametersforLFs
+Sample L∗
+8µm(L⊙)φ∗(Mpc−3dex−1)α β
+NEPDeepfield 6.1 ±0.5×10100.0010±0.0003 1.1 ±0.3 5.7 ±1.2
+RXJ1716.4 +67082.5±0.1×10100.74±0.04 2.6 ±0.1 5.5 ±0.4
+(L8µm) using a standard cosmology. Completeness was mea-
+sured by distributing artificial point sources with varying flux
+withinthe field andby examiningwhat fractionofthem wasre-
+coveredasafunctionofinputflux.Sincewehavedeepercover -
+age at the center of the cluster, the completeness was measur ed
+separately in the central deep region and the outer regions o f
+the field. More detail of the method is described in Wada et al.
+(2008).
+Oncethefluxisconvertedtoluminosityandcompletenessis
+takenintoaccount,it is straightforwardto construct L8µmLFs,
+which we show in the squares in Fig.1. Errors of the LFs are
+assumedtofollowPoissondistribution.Here,wetakeanang ular
+distance of the most distant source from the cluster center a s
+a cluster radius ( Rmax= 6.2Mpc). We assumed4
+3πR3
+maxas
+the volume of the cluster to obtain galaxy density ( φ). This is
+only one of many ways to define a cluster volume, and thus, a
+cautionmustbetakentocompare absolute valuesofourLFsto
+other work such as Shimet al. (2010). This cluster is elongat ed
+inangulardirection(Koyamaet al.,2007),andthus,thevol ume
+mightnotbespherical.Yet,comparisonofthe shapeoftheLFs
+isvalid.
+2.2. LFs inthe AKARI NEP Deepfield
+Our field LFs are based on the AKARI NEP Deep field
+data. The AKARI performed deep imaging in the North
+Ecliptic Pole Region (NEP) from 2-24 µm, with 4 pointings
+in each field over 0.4 deg2(Matsuharaet al., 2006, 2007;
+Wada etal., 2008). The 5 σsensitivity in the AKARI IR filters
+(N2,N3,N4,S7,S9W,S11,L15,L18WandL24) are 14.2,
+11.0, 8.0, 48, 58, 71, 117, 121 and 275 µJy (Wada etal., 2008).
+Flux is measured in 3 pix radius aperture (=7”), then correct ed
+tototal flux.
+AsubregionoftheNEP-Deepfield(0.25deg2)hasancillary
+datafromSubaru BVRi′z′(Imaiet al.,2007;Wada etal.,2008),
+CFHTu′(Serjeant et al. in prep.), KPNO2m/FLAMINGOs J
+andKs(Imaietal., 2007), GALEX FUVandNUV(Malkan
+et al. in prep.). For the optical identification of MIR source s,
+we adopt the likelihood ratio method (Sutherland&Saunders ,
+1992).Usingthesedata,weestimatephotometricredshifto fL15
+detectedsourcesintheregionwiththe LePhare (Ilbertet al.,
+2006; Arnoutset al., 2007).Themeasurederrorsonthephoto -z
+against 293 spec-z galaxies from Keck/DEIMOS (Takagi et al.
+in prep.) are∆z
+1+z=0.036 at z≤0.8. We have excluded those
+sourcesbetterfit with QSO templatesfromtheLFs.
+To construct field LFs, we have selected L15sources at
+0.65< zphotoz<0.9. There remained 289 IR galaxies with
+a median redshift of 0.76. L15flux is converted to L8µmus-
+ing the photometric redshift of each galaxy. LFs are com-
+puted using the 1/ Vmaxmethod. We used the SED templates
+(Lagache,Dole,&Puget, 2003) for k-corrections to obtain the
+maximumobservableredshiftfromthefluxlimit.Completene ss
+of theL15detection is corrected using Pearsonet al. (2009b).
+Thiscorrectionis25%atmaximum,sincewe onlyusethesam-
+plewherethecompletenessisgreaterthan80%.
+The resulting field LFs are shown in the dotted line and tri-
+angles in Fig.1. Errors of the LFs are computed using a 1000Monte Caro simulation with varying zand flux within their er-
+rors. These estimated errors are added to the Poisson errors in
+eachLFbinin quadrature.
+We performed a detaild comparison of restframe 8 µm
+LFs to those in the literature in Gotoetal. (2010). Briefly,
+there is an oder of difference between Caputiet al. (2007) an d
+Babbedgeetal. (2006), reflecting difficulty in estimating L8µm
+dominatedbyPAHemissionsusingSpitzer24 µmflux.Ourfield
+8µm LF lies between Caputi etal. (2007) and Babbedgeet al.
+(2006). Compared with these work, we have directly observed
+restframe 8 µm using the AKARI L15filter, eliminating the un-
+certaintlyinfluxconversionbasedonSEDmodels.Moredetai ls
+andevolutionoffieldIRLFsaredescribedinGotoet al.(2010 ).
+3. Results& Discussion
+3.1. 8µmIRLFs
+In Fig.1, we show restframe 8 µm LFs of cluster
+RXJ1716.4 +6708 in the squares, and LFs of the field re-
+gion in the triangles. First of all, cluster LFs have by a fact or
+of∼700 higher density than the field LFs, reflecting the fact
+the galaxy clusters is indeed high density regions in terms o f
+infraredsources.
+Next, to compare the shape of the LFs, we normalized the
+cluster LF to match the field LFs at the faintest end, and show
+in the dash-dottedline. In contrast to the field LFs, which sh ow
+flattening of the slope at log L8µm<10.8L⊙, the cluster LF
+maintainsthesteepslopeintherangeof 10.0L⊙<logL8µm<
+10.6L⊙.Thedifferenceissignificant,consideringthesizeofer-
+rorsoneachLF.
+Wefitadouble-powerlawtobothclusterandfieldLFsusing
+thefollowingformulae.
+Φ(L)dL/L∗= Φ∗/parenleftbiggL
+L∗/parenrightbigg1−α
+dL/L∗,(L < L∗) (1)
+Φ(L)dL/L∗= Φ∗/parenleftbiggL
+L∗/parenrightbigg1−β
+dL/L∗,(L > L∗) (2)
+Free parameters are: L∗(characteristic luminosity, L⊙),φ∗
+(normalization, Mpc−3),αandβ(faint and bright end slopes),
+respectively.ThebestfitvaluesforfieldandclusterLFsare sum-
+marisedinTable1andshowninthedottedlinesinFig.1.
+The bright-end slopes are not very different, but L∗of the
+cluster LF is smaller than the field by a factor of 2.4, and the
+faint-endtailofclusterLF issteeperthanthatoffieldLF.
+To further examine the difference at the faint end of the
+LFs, we divide the cluster LF using the local galaxy density
+(Σ5th) measuredbyKoyamaet al. (2008). Thisdensityis based
+on the distance to the 5th nearest neighbor in the transverse di-
+rection using all the optical photo-z members, and thus, is a
+surface galaxy density. We separate LFs using similar crite ria,
+logΣ5th≥2(dense),1.6≤logΣ5th<2(intermediate), and
+logΣ5th<1.6(sparse), then plot LFs of each region in the4 Gotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8
+stars, circles, and squares in Fig.2. A fraction of the total vol-
+umeofthe clusteris assignedto eachdensitygroupin invers ely
+proportionaltothe sumof Σ3/2
+5thofeachgroup.
+Interestingly, the faint-end slope becomes flatter and flatt er
+with decreasing local galaxy density. This result is consis tent
+with our comparison with the field in Fig.1. In fact, the lowes t
+densityLF(squares)hasaflatfaint-endtailsimilartothat ofthe
+fieldLF.SincetheseLFsarebasedonthesamedata,changesin
+the faint-end slope are not likely due to the errors in comple te-
+ness correction nor calibration problems. The completenes s of
+the deep and shallow regions of the cluster are measured sep-
+arately. The changes in the slope is much larger than the maxi -
+mumcompletenesscorrectionof25%.Wealsocheckedtheclus -
+ter LFs as a function of cluster centric radius, to find no sign ifi-
+cantdifference,perhapsduetotheelongatedmorphologyof this
+cluster. At the same time, assuming the same cluster volume,
+Fig.2 shows that a possible contamination from the field gala x-
+ies to cluster LFs is only ∼0.1% in the dense region and ∼1%
+eveninthe sparseregion.
+It is interesting that not just the change in the scale of the
+LFs, but there is a change in the L∗and the faint-end slope ( α)
+of the LFs, resulting in the deficit in the 10.2L⊙<logL8µm<
+10.8L⊙for cluster LFs. One might imaginea change just in L∗
+might explain the difference in Fig.1. However, in Fig.2, th ere
+clearlyisachangein theslopeasafunctionof Σ5th.
+However,interpretationis rathercomplicated;a shapeofL F
+would not change if field galaxies infall into cluster unifor mly
+withoutchangingtheirstar-formationactivity.Although inclus-
+ter environment,a fractionof MIR luminousgalaxiesis smal ler
+than field (Koyamaet al., 2008), uniformand instant quenchi ng
+of star-formation activity of field galaxies can only shift a LF,
+butcannotaccountforachangein L∗andαoftheLFs.
+Two important findings in this work are; (i) L∗is smaller
+in the cluster. (ii) the faint-end slopes become steeper tow ard
+higher-density regions. To explain these changes in LFs, IR -
+luminousgalaxiesneedtobepreferentiallyreduced,witha rela-
+tive increase of IR-faint galaxies. However, an environmen tal-
+driven physical process such as the ram-pressure stripping or
+galaxy-merging would quench star-formation not only in mas -
+sivegalaxiesbutinlessmassivegalaxiesaswell,andthusi snot
+abletoexplaintheobservedchangesinLFs.
+Ontheotherhand,ithasbeenfrequentlyobservedthatmore
+massive galaxies formed earlier in the Universe. This downs iz-
+ing scenario also depends on the environment,in the sense th at
+galaxieswith same mass are moreevolvedin higherdensityen -
+vironmentsthangalaxisin less denseenvironments(Gotoet al.,
+2005; Tanakaet al., 2005, 2008). Statistically, a good corr ela-
+tionhasbeenfoundbetween LTIRandstellarmass(Elbazet al.,
+2007). Our finding of the relative lack of IR-luminous galaxi es
+in the cluster environmentmay be consistent with the downsi z-
+ing scenario, where higher density regions have more evolve d
+galaxies and lacks massive star-forming galaxies. In contr ast,
+in lower density regions more massive galaxies are still sta r-
+forming. However, since the data we have shown is in IR lumi-
+nosity, to conclude on this, we need good stellar mass estima te
+basedondeepernear-IRdata.
+Although a specific mechanism is unclear, the steep faint-
+end could also result from the enhanced star-formation in le ss
+massive galaxies. In the above scenario, massive galaxies h ave
+already ceased their star-formation in the cluster, but les s mas-
+sive galaxiesare still formingstars. These less massive ga laxies
+may stop star-formation soon to join the faint-end of the red -
+sequence(Koyamaet al., 2007).Fig.3.TotalinfraredLFsofclusterRXJ1716.4 +6708atz=0.81
+in the solid line, and those of the AKARI NEP deep field in the
+dashed line. Overplottedare the LFs of MS1054 from Bai et al.
+(2007).
+3.2. Total IRLFs
+To compare the L8µmLF in Fig.1 to those in the literature, we
+needtoconvert L8µmtoLTIR.Weusethethefollowingrelation
+byCaputiet al.(2007);
+LTIR= 1.91×(νLν8µm)1.06(±55%) (3)
+Thisis better tunedfor a similar luminosityrange used here
+than the originalrelationby Bavouzetetal. (2008). The con ver-
+sion, however, has been the largest source of errors in estim at-
+ingLTIRfromL8µm.Caputi etal.(2007)report55%ofdisper-
+sion around the relation. It should be kept in mind that the re st-
+frame8µm is sensitive to the star-formation activity, but at the
+same time, it is where the SED models have strongest discrep-
+anciesduetothecomplicatedPAHemissionlines(seeFig.12 of
+Caputiet al.,2007; Gotoetal., 2010).
+Theestimated LTIRcanbe,then,convertedtoSFRusingthe
+followingrelationfor a Salpeter IMF, φ(m)∝m−2.35between
+0.1−100M⊙(Kennicutt, 1998).
+SFR(M⊙yr−1) = 1.72×10−10LTIR(L⊙) (4)
+In Fig.3, we show the LTIRLFs. Symbols are the same as
+Fig.1. Inthe topaxis,we showcorrespondingSFR. Overplott ed
+asterisks are cluster LF of MS1054 at z=0.83 with ×2 larger
+mass by Bai et al. (2007), which show good agreement with
+ourLFsofRXJ1716.4 +6708in thesquares.Notethat Bai et al.
+(2007) covered only the central region of MS1054 due to the
+smaller field of view of the Spitzer. The shape of their LF look s
+more similar to our LFs in the highest density bin in Fig.2. A
+shift in scale is perhaps due to difference in esimating clus ter
+volumes.
+Amajordifferenceofourworktothat ofBai etal. (2007)is
+that they were not able to compare in detail on the shape of the
+LFs between field and cluster regions, due mainly to a smaller
+fieldcoverageandlargererrorsonLFs.Theyhadtofixthefain t-
+end slope with a local value. The largest source of errors is i nGotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8 5
+converting Spitzer 24 µm flux into 8 µm. Both cluster and field
+LFs of this work use L15filter, which measures restframe 8 µm
+fluxdirectly,eliminatingthelargestsourceoferrors.Ina ddition,
+bothclusterandfiledLFsaremeasuredwithanessentiallysa me
+methodology,allowingusa faircomparisonofLFs.
+4. Summary
+We constructed restframe 8 µm LFs of a massive galaxy cluster
+(RXJ1716.4 +6708) and a rarefied field region (the NEP deep
+field)at z ∼0.8usingessentially thesame methodanddata from
+the AKARI telescope. AKARI’s 15 µm filter nicely covers rest-
+frame8µm at z∼0.8,and thuswe do not needa large interpola-
+tion based on SED models. AKARI’s wide field of view allows
+ustoinvestigatevarietyofclusterenvironmentswith2ord ersof
+differenceinlocal galaxydensity.
+We found that L∗of the cluster 8 µm LF is smaller than the
+field by a factor of 2.4, and the faint-end tail of cluster IR LF s
+becomesteeperandsteeperwithincreasinglocalgalaxyden sity.
+This difference cannot be explained by a simple infall of fiel d
+galaxies into a cluster. Physics that preferentially supre sses IR
+luminous galaxes in higer density regions is needed to expla in
+theobservedresults.
+Acknowledgments
+Wethanktheanonymousrefereeformanyinsightfulcomments ,
+which significantly improved the paper. We are greateful to
+MasayukiTanakaforusefuldiscussion.WethankL.Baiforpr o-
+vidingdataforcomparison.
+T.G.,Y.K. and H.I. acknowledgesfinancial support from the
+JapanSocietyforthePromotionofScience(JSPS)throughJS PS
+ResearchFellowshipsforYoungScientists.MIwassupporte dby
+the Korea Science and Engineering Foundation(KOSEF) grant
+No. 2009-0063616, funded by the Korea government(MEST)”
+HML acknowledgesthe supportfrom KASI throughits cooper-
+ativefundin2008.
+This research is based on the observations with AKARI, a
+JAXA projectwiththe participationofESA.
+Theauthorswishtorecognizeandacknowledgetheverysig-
+nificant cultural role and reverence that the summit of Mauna
+Kea has always had within the indigenous Hawaiian commu-
+nity. We are most fortunate to have the opportunity to conduc t
+observationsfromthissacredmountain.
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+arXiv:1001.0006v2  [astro-ph.CO]  4 May 2010Draft version November 2, 2018
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+COMPARISON OF HECTOSPEC VIRIAL MASSES WITH SZE MEASUREMENT S
+Kenneth Rines1,2, Margaret J. Geller2, and Antonaldo Diaferio3,4
+Draft version November 2, 2018
+ABSTRACT
+We present the first comparison of virial masses of galaxy clusters with their Sunyaev-Zel’dovich
+Effect (SZE) signals. We study 15 clusters from the Hectospec Clus ter Survey (HeCS) with
+MMT/Hectospec spectroscopy and published SZE signals. We measu re virial masses of these clusters
+from an average of 90 member redshifts inside the radius r100. The virial masses of the clusters are
+strongly correlated with their SZE signals (at the 99% confidence lev el using a Spearman rank-sum
+test). This correlation suggests that YSZcan be used as a measure of virial mass. Simulations predict
+a powerlaw scaling of YSZ∝Mα
+200withα≈1.6. Observationally, we find α=1.11±0.16, significantly
+shallower (given the formal uncertainty) than the theoretical pr ediction. However, the selection func-
+tion of our sample is unknown and a bias against less massive clusters c annot be excluded (such a
+selection bias could artificially flatten the slope). Moreover, our sam ple indicates that the relation
+between velocity dispersion (or virial mass estimate) and SZE signal has significant intrinsic scatter,
+comparable to the range of our current sample. More detailed stud ies of scaling relations are therefore
+needed to derive a robust determination of the relation between clu ster mass and SZE.
+Subject headings: galaxies: clusters: individual — galaxies: kinematics and dynamics — co smology:
+observations
+1.INTRODUCTION
+Clusters of galaxies are the most massive virialized
+systems in the universe. The normalization and evo-
+lution of the cluster mass function is therefore a sen-
+sitive probe of the growth of structure and thus cos-
+mology (e.g., Rines et al. 2007, 2008; Vikhlinin et al.
+2009; Henry et al. 2009; Mantz et al. 2008; Rozo et al.
+2008, and references therein). Many methods exist
+to estimate cluster masses, including dynamical masses
+from either galaxies (Zwicky 1937) or intracluster gas
+(e.g., Fabricant et al. 1980), gravitational lensing (e.g.,
+Smith et al.2005;Richard et al.2010), andthe Sunyaev-
+Zel’dovich effect (SZE Sunyaev & Zeldovich 1972). In
+practice, these estimates are often made using simple
+observables, such as velocity dispersion for galaxy dy-
+namics or X-ray temperature for the intracluster gas.
+If one of these observable properties of clusters has a
+well-defined relation to the cluster mass, a large survey
+can yield tight constraints on cosmological parameters
+(e.g., Majumdar & Mohr 2004). There is thus much
+interest in identifying cluster observables that exhibit
+tight scaling relations with mass (Kravtsov et al. 2006;
+Rozo et al. 2008). Numerical simulations indicate that
+X-ray gas observables (Nagai et al. 2007) and SZE sig-
+nals (Motl et al. 2005) are both candidates for tight scal-
+ing relations. Both methods are beginning to gain ob-
+servational support (e.g., Henry et al. 2009; Lopes et al.
+2009; Mantz et al. 2009; Locutus Huang et al. 2009).
+Dynamical masses from galaxy velocities are unbiased
+kenneth.rines@wwu.edu
+1Department of Physics & Astronomy, Western Washington
+University, Bellingham, WA 98225; kenneth.rines@wwu.edu
+2Smithsonian Astrophysical Observatory, 60 Garden St, Cam-
+bridge, MA 02138
+3Universit` a degli Studi di Torino, Dipartimento di Fisica G en-
+erale “Amedeo Avogadro”, Torino, Italy
+4Istituto Nazionale di Fisica Nucleare (INFN), Sezione di
+Torino, Torino, Italyin numerical simulations (Diaferio 1999; Evrard et al.
+2008), and recent results from hydrodynamical simula-
+tions indicate that virial masses may have scatter as
+small as ∼5% (Lau et al. 2010).
+Previous studies have compared SZE signals to hydro-
+staticX-raymasses(Bonamente et al.2008;Plagge et al.
+2010) and gravitational lensing masses (Marrone et al.
+2009, hereafter M09). Here, we make the first compar-
+ison between virial masses of galaxy clusters and their
+SZE signals. We use SZE measurements from the lit-
+erature and newly-measured virial masses of 15 clus-
+ters from extensive MMT/Hectospec spectroscopy. This
+comparison tests the robustness of the SZE as a proxy
+for cluster mass and the physical relationship between
+the SZE signal and cluster mass. Large SZ cluster sur-
+veys are underway and are beginning to yield cosmologi-
+calconstraints(Carlstrom et al.2010;Hincks et al.2010;
+Staniszewski et al. 2009).
+We assume a cosmology of Ω m=0.3, Ω Λ=0.7, and
+H0=70 km s−1Mpc−1for all calculations.
+2.OBSERVATIONS
+2.1.Optical Photometry and Spectroscopy
+We are completing the Hectospec Cluster Survey
+(HeCS), a study of an X-ray flux-limited sample of 53
+galaxy clusters at moderate redshift with extensive spec-
+troscopy from MMT/Hectospec. HeCS includes all clus-
+ters with ROSAT X-ray fluxes of fX>5×10−12erg
+s−1at [0.5-2.0]keVfrom the Bright Cluster Survey (BCS
+Ebeling et al.1998)orREFLEXsurvey(B¨ ohringer et al.
+2004) with optical imaging in the Sixth Data Release
+(DR6) of SDSS (Adelman-McCarthy et al. 2008). We
+use DR6 photometry to select Hectospec targets. The
+HeCS targets are all brighter than r=20.8 (SDSS cata-
+logs are 95% complete for point sources to r≈22.2). Out
+of the HeCS sample, 15 clusters have published SZ mea-
+surements.2 Rines, Geller, & Diaferio
+2.1.1.Spectroscopy: MMT/Hectospec and SDSS
+HeCS is a spectroscopic survey of clusters in the red-
+shift range 0.10 ≤z≤0.30. We measure spectra with
+the Hectospec instrument (Fabricant et al. 2005) on the
+MMT 6.5m telescope. Hectospec provides simultaneous
+spectroscopy of up to 300 objects across a diameter of
+1◦. This telescope and instrument combination is ideal
+for studying the virial regions and outskirts of clusters
+at these redshifts. We use the red sequence to preselect
+likely cluster members as primary targets, and we fill
+fibers with bluer targets (Rines et al. in prep. describes
+the details of target selection). We eliminate all targets
+withexistingSDSSspectroscopyfromourtargetlistsbut
+include these in our final redshift catalogs.
+Ofthe15clustersstudiedhere,onewasobservedwitha
+single Hectospec pointing and the remaining 14 were ob-
+served with two pointings. Using multiple pointings and
+incorporatingSDSS redshifts of brighterobjectsmitigate
+fiber collision issues. Because the galaxy targets are rel-
+atively bright ( r≤20.8), the spectra were obtained with
+relativelyshortexposuretimes of3x600sto 4x900sunder
+a variety of observing conditions.
+Figure 1 shows the redshifts of galaxies versus their
+projected clustrocentric radii for the 15 clusters stud-
+ied here. The infall patterns are clearly present in all
+clusters. We use the caustic technique (Diaferio 1999)
+to determine cluster membership. Briefly, the caustic
+technique uses a redshift-radius diagram to isolate clus-
+ter members in phase space by using an adaptive ker-
+nel estimator to smooth out the galaxies in phase space,
+and then determining the edges of this distribution (see
+Diaferio 2009, for a recent review). This technique has
+been successfully applied to optical studies of X-ray clus-
+ters, and yields cluster mass estimates in agreement
+with estimatesfromX-rayobservationsandgravitational
+lensing (e.g., Rines et al. 2003; Biviano & Girardi 2003;
+Diaferio et al. 2005; Rines & Diaferio 2006; Rines et al.
+2007, and references therein).
+We apply the prescription of Danese et al. (1980) to
+determine the mean redshift cz⊙and projected velocity
+dispersion σpof each cluster from all galaxies within the
+caustics. We calculate σpusing only the cluster members
+projected within r100estimated from the caustic mass
+profile.
+2.2.SZE Measurements
+The SZE detections are primarily from
+Bonamente et al. (2008, hereafter B08), supplemented
+by three measurements from Marrone et al. (2009,
+hereafter M09). Most of the SZ data were obtained with
+the OVRO/BIMA arrays; the additional clusters from
+M09 were observed with the Sunyaev-Zel’dovich Array
+(SZA; e.g., Muchovej et al. 2007).
+Numerical simulations indicate that the integrated
+Compton y-parameter YSZhas smaller scatter than the
+peak y-decrement ypeak(Motl et al. 2005), so B08 and
+M09 report only YSZ. Although ypeakshould be nearly
+independent of redshift, YSZdepends on the angular size
+of the cluster. The quantity YSZD2
+Aremoves this depen-
+dence. Thus, we compare our dynamical mass estimates
+to this quantity rather than ypeakorYSZ. Table 1 sum-
+marizes the SZ data and optical spectroscopy.
+It is also critical to determine the radius within whichYSZis determined. B08 use r2500, the radius that en-
+closes an average density of 2500 times the critical den-
+sity at the cluster’s redshift; r2500has physical values of
+300-700kpc forthe massiveclustersstudied by B08(470-
+670kpcforthesubsamplestudiedhere). M09useaphys-
+ical radius of 350 kpc because this radius best matches
+their lensing data.
+To use both sets of data, we must estimate the con-
+version between YSZ(r2500) measured within r2500and
+YSZ(r= 350 kpc) measured within the smaller radius
+r=350 kpc. There are 8 clusters analyzed in both B08
+and M09 (5 of which are in HeCS). We perform a least-
+squaresfit to YSZ(r2500)−YSZ(r= 350kpc) to determine
+an approximate aperture correction for the M09 clusters.
+We list both quantities in Table 1.
+3.RESULTS
+We examine two issues: (1) the strength of the corre-
+lation between SZE signal and the dynamical mass and
+(2) the slope of the relationship between them. Figure 2
+shows the YSZ−σprelation. Here, we compute σpfor all
+galaxies inside both the caustics and the radius r100,cde-
+fined by the caustic mass profile [ rδis the radius within
+which the enclosed density is δtimes the critical density
+ρc(z)].
+Because we make the first comparison of dynami-
+cal properties and SZE signals, we first confirm that
+these two variables are well correlated. A nonparametric
+Spearman rank-sum test (one-tailed) rejects the hypoth-
+esis of uncorrelated data at the 98.4% confidence level.
+The strong correlation in the data suggests that both σp
+andYSZD2
+Aincrease with increasing cluster mass.
+Hydrodynamic numerical simulations indicate that
+YSZ(integrated to r500) scales with cluster mass as
+YSZ∝Mα
+500, whereα=1.60 with radiative cooling and
+star formation, and 1.61 for simulations with radiative
+cooling, star formation, and AGN feedback ( α=1.70 for
+non-radiative simulations, Motl et al. 2005). Combin-
+ing this result with the virial scaling relation of dark
+matter particles, σp∝M0.336±0.003
+200 (Evrard et al. 2008),
+the expected scaling is YSZ∝σ4.76(we assume that
+M100∝M500). The right panels of Figure 2 shows this
+predicted slope (dashed lines).
+The bisector of the least-squares fits to the data has
+a slope of 2 .94±0.74, significantly shallower than the
+predicted slope of 4.8.
+We recompute the velocity dispersions σp,Afor all
+galaxies within one Abell radius (2.14 Mpc) and in-
+side the caustics. Surprisingly, the correlation is slightly
+stronger (99.4% confidence level). This result supports
+the idea that velocity dispersions computed within a
+fixedphysicalradiusretainstrongcorrelationswith other
+cluster observables, even though we measure the velocity
+dispersion inside different fractions of the virial radius
+for clusters of different masses. Because cluster veloc-
+ity dispersions decline with radius (e.g. Rines et al. 2003;
+Rines & Diaferio 2006), σp,Amay be smaller than σp,100
+(measured within r100,c) for low-mass clusters, perhaps
+exaggerating the difference in measured velocity disper-
+sionsrelativeto the differences in virialmass(i.e., σp,Aof
+a low-mass cluster may be measured within 2 r100while
+σp,Aof a high-mass cluster may be measured within r100;
+the ratio σp,Aof these clusters would be exaggerated rel-Hectospec Virial Masses and SZE 3
+Fig. 1.— Redshift versus projected clustrocentric radius for the 15 HeCS clusters studied here. Clusters are ordered left-to-r ight and
+top-to-bottom by decreasing values of YSZD2
+A(r2500). The solid lines show the locations of the caustics, which w e use to identify cluster
+members. The Hectospec data extend out to ∼8 Mpc; the figure shows only the inner 4 Mpc to focus on the viria l regions.
+ative to the ratio σp,100). Future cluster surveys with
+enough redshifts to estimate velocity dispersions but too
+few to perform a caustic analysis should still be sufficient
+for analyzing scaling relations.
+Because of random errors in the mass estimation, the
+virial mass and the caustic mass within a given radius
+do not necessarily coincide. Therefore, the radius r100
+depends on the mass estimator used. Figure 2 shows
+the scaling relationsfor two estimated masses M100,cand
+M100,v;M100,cis the mass estimated within r100,c(where
+bothquantitiesaredefinedfromthecausticmassprofile),
+andM100,vis the mass estimated within r100,v(both
+quantities are estimated with the virial theorem, e.g.,
+Rines & Diaferio 2006). including galaxies projected in-
+sider100,v. Similar to σp, there is a clear correlation
+between M100,vandYSZD2
+A(99.0% confidence with a
+Spearman test). The strong correlation of dynamical
+mass with SZE also holds for M100,cestimated directlyfrom the caustic technique (99.8% confidence).
+The bisector of the least-squares fits has a slope of
+1.11±0.16, again significantly shallower than the pre-
+dicted slope of 1.6. This discrepancy has two distinct
+origins. By looking at the distribution of the SZE sig-
+nals in Figure 2, we see that, at a given velocity disper-
+sion or mass, the SZE signals have a scatter which is a
+factor of ∼2. Alternatively, at fixed SZE signal, there
+is a scatter of a factor of ∼2 in estimated virial mass.
+Unless the observational uncertainties are significantly
+underestimated, the data show substantial intrinsic scat-
+ter. Moreover, this scatter is comparable to the range of
+our sample and, therefore, the error on the slope derived
+from our least-squares fit to the data is likely to be un-
+derestimated (see Andreon & Hurn 2010, for a detailed
+discussionofaBayesianapproachtofittingrelationswith
+measurement uncertainties and intrinsic scatter in both
+quantities).4 Rines, Geller, & Diaferio
+TABLE 1
+HeCS Dynamical Masses and SZE Signals
+Cluster z σ p M100,vM100,c YSZD2
+AYSZD2
+ASZE
+(350 kpc) ( r2500)
+km s−11014M⊙1014M⊙10−5Mpc−210−4Mpc2Ref.
+A267 0.2288 743+81
+−616.86±0.82 4.26 ±0.14 3.08 ±0.34 0.42 ±0.06 1
+A697 0.2812 784+77
+−596.11±0.69 5.96 ±3.51 – 1.29 ±0.15 1
+A773 0.2174 1066+77
+−6318.4±1.7 16.3 ±0.7 5.40 ±0.57 0.90 ±0.10 1
+Zw2701 0.2160 564+63
+−473.47±0.42 2.69 ±0.30 1.46 ±0.016 0.17 ±0.02a2
+Zw3146 0.2895 752+92
+−676.87±0.89 4.96 ±0.91 – 0.71 ±0.09 1
+A1413 0.1419 674+81
+−606.60±0.85 3.49 ±0.15 3.47 ±0.24 0.81 ±0.12 1
+A1689 0.1844 886+63
+−5215.3±1.4 9.44 ±5.66 7.51 ±0.60 1.50 ±0.14 1
+A1763 0.2315 1042+79
+−6416.9±1.6 12.6 ±1.5 3.10 ±0.32 0.46 ±0.05a2
+A1835 0.2507 1046+66
+−5519.6±1.6 20.6 ±0.3 6.82 ±0.48 1.37 ±0.11 1
+A1914 0.1659 698+46
+−386.70±0.57 6.21 ±0.21 – 1.08 ±0.09 1
+A2111 0.2290 661+57
+−454.01±0.41 4.77 ±1.23 – 0.55 ±0.12 1
+A2219 0.2256 915+53
+−4512.8±1.0 12.0 ±4.7 6.27 ±0.26 1.19 ±0.05a2
+A2259 0.1606 735+67
+−535.59±0.60 4.90 ±1.69 – 0.27 ±0.10 1
+A2261 0.2249 725+75
+−577.13±0.83 5.10 ±2.07 – 0.71 ±0.09 1
+RXJ2129 0.2338 684+88
+−644.31±0.57 2.94 ±0.13 – 0.40 ±0.07 1
+Note. —aExtrapolated to r2500using the best-fit relation between YSZD2
+A(350kpc) and YSZD2
+A(r2500) for eight clusters in common
+between B08 and M09.
+Note. — Redshift zand velocity dispersion σpare computed for galaxies defined as members using the causti cs. Masses M100,vand
+M100,care evaluated using the virial mass profile and caustic mass p rofile respectively.
+Note. — REFERENCES: SZE data are from (1) Bonamente et al. 2008 and (2) Marrone et al. 2009.
+Our shallow slopes may also arise in part from the fact
+that our sample, which has been assembled from the lit-
+erature and whose selection function is difficult to deter-
+mine, is likely to be biased against clusters with small
+mass and low SZE signal. Larger samples should deter-
+mine whether unknown observational biases or issues in
+the physical understanding of the relation account for
+this discrepancy.
+4.DISCUSSION
+Thestrongcorrelationbetweenmassesfromgalaxydy-
+namics and SZE signals indicates that the SZE is a rea-
+sonableproxyforcluster mass. B08compareSZEsignals
+toX-rayobservables,inparticularthetemperature TXof
+the intracluster medium and YX=MgasTX, whereMgas
+isthemassoftheICM(seealsoPlagge et al.2010). Both
+of these quantities are measured within r500, a signifi-
+cantly smaller radius than r100where we measure virial
+mass. M09 compare SZE signals to masses estimated
+from gravitational lensing measurements. The lensing
+masses are measured within a radius of 350 kpc. For the
+clusters studied here, this radius is smaller than r2500
+and much smaller than r100. Numerical simulations indi-
+cate that the scatter in masses measured within an over-
+densityδdecreases as δdecreases (White 2002), largely
+because variations in cluster cores are averaged out at
+larger radii. Thus, the dynamical measurement reaching
+to larger radius may provide a more robust indication
+of the relationship between the SZE measurements and
+cluster mass.
+TheYSZD2
+A−Mlensdata presented in M09 show a
+weakercorrelationthanouropticaldynamicalproperties.
+A Spearman test rejects the hypothesis of uncorrelated
+data for the M09 data at only the 94.8% confidence level,
+compared to the 98.4-99.8% confidence levels for our op-
+tical dynamical properties. One possibility is that Mlensis more strongly affected by substructure in cluster cores
+and by line-of-sight structures than are the virial masses
+and velocity dispersions we derive.
+Few measurements of SZE at large radii ( > r500) are
+currently available. Hopefully, future SZ data will allow
+a comparisonbetween virialmass and YSZwithin similar
+apertures.
+5.CONCLUSIONS
+Our first direct comparison of virial masses, velocity
+dispersions, and SZ measurements for a sizable clus-
+ter sample demonstrates a strong correlation between
+these observables (98.4-99.8% confidence). The SZE sig-
+nal increases with cluster mass. However, the slopes of
+both the YSZ−σrelation ( YSZ∝σ2.94±0.74
+p) and the
+YSZ−M100relation ( YSZ∝M1.11±0.16
+100) are significantly
+shallower(giventheformaluncertainties)thantheslopes
+predictedbynumericalsimulations(4.76and1.60respec-
+tively).
+This result may be partly explained by a bias against
+less massive clusters that could artificially flatten our
+measured slopes. Unfortunately, the selection function
+of our sample is unknown and we are unable to quan-
+tify the size of this effect. More importantly, our sample
+indicates that the relation between SZE and virial mass
+estimates (or velocity dispersion) has a non-negligible in-
+trinsicscatter. Acomplete, representativeclustersample
+is required to robustly determine the size of this scatter,
+its origin, and its possible effect on the SZE as a mass
+proxy.
+Curiously, YSZis more strongly correlated with both
+σpandM100than with Mlens(M09). Comparison of
+lensingmassesandclustervelocitydispersions(andvirial
+masses)forlarger,complete, objectivelyselected samples
+of clusters may resolve these differences.
+Thefull HeCS sampleof53clusterswill providealargeHectospec Virial Masses and SZE 5
+Fig. 2.— Integrated S-Z Compton parameter YSZD2
+Aversus dynamical properties for 15 clusters from HeCS. Left panels: SZE data
+versus virial mass M100estimated from the virial mass profile (top) and the caustic m ass profile (bottom). Solid and open points indicate
+SZ measurements from B08 and M09 respectively. The dashed li ne shows the slope of the scaling predicted from numerical si mulations:
+YSZ∝M1.6(Motl et al. 2005), while the solid line shows the ordinary le ast-squares bisector. Arrows show the aperture correction s to
+the SZE measurements (see text). Right panels: SZE data versus projected velocity dispersions measured fo r galaxies inside the caustics
+and (top) inside r100,cestimated from the caustic mass profile and (bottom) inside t he Abell radius 2.14 Mpc. The dashed line shows the
+scaling predicted from simulations: YSZ∝M1.6(Motl et al. 2005) and σ∝M0.33(Evrard et al. 2008). The solid line shows the ordinary
+least-squares bisector. Data points and arrows are defined a s in the left panels.
+sample of clusters with robustly measured velocity dis-
+persions and virial masses as a partial foundation for
+these comparisons.
+We thank Stefano Andreon for fruitful discussions
+about fitting scaling relations with measurement errorsand intrinsic scatter in both quantities. AD gratefully
+acknowledges partial support from INFN grant PD51.
+We thank Susan Tokarz for reducing the spectroscopic
+data and Perry Berlind and Mike Calkins for assisting
+with the observations.
+Facilities: MMT (Hectospec)
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+arXiv:1001.0007v1  [astro-ph.CO]  30 Dec 2009Cosmicstarformation history
+revealedby the AKARI
+& Spatially-resolvedspectroscopyofan E+A(Post-starbur st)system
+Tomotsugu GOTO∗, the AKARINEPDteam†,M.Yagi∗∗andC.Yamauchi†
+∗InstituteforAstronomy,Universityof Hawaii,2680Woodla wnDrive, Honolulu,HI,96822,USA
+†JapanAerospaceExplorationAgency,Sagamihara,Kanagawa 229-8510,Japan
+∗∗NationalAstronomicalObservatory,2-21-1Osawa,Mitaka, Tokyo,181-8588,Japan
+Abstract. We reveal cosmic star-formation history obscured by dust us ing deep infrared observa-
+tionwiththeAKARI.Acontinuousfiltercoverageinthemid-I Rwavelength(2.4,3.2,4.1,7,9,11,
+15, 18, and 24 µm) by the AKARI satellite allows us to estimate restframe 8 µm and 12 µm lumi-
+nositieswithoutusingalargeextrapolationbasedonaSEDfi t,whichwasthelargestuncertaintyin
+previouswork. We found that restframe 8 µm (0.38<z<2.2), 12µm (0.15<z<1.16), and total
+infrared (TIR) luminosity functions (LFs) (0 .2<z<1.6) constructed from the AKARI NEP deep
+data, show a continuous and strong evolution toward higher r edshift. In terms of cosmic infrared
+luminosity density ( ΩIR), which was obtained by integrating analytic fits to the LFs, we found a
+goodagreementwithpreviousworkat z<1.2,withΩIR∝(1+z)4.4±1.0.Whenweseparatecontri-
+butionsto ΩIRby LIRGs and ULIRGs, we foundmore IR luminoussourcesare inc reasinglymore
+importantathigherredshift.WefoundthattheULIRG(LIRG) contributionincreasesbyafactorof
+10(1.8)from z=0.35toz=1.4.
+Keywords: galaxies:evolution,galaxies:starburst
+PACS:98.70.Lt
+Introduction .Revealingthecosmicstarformationhistoryisoneofthemaj orgoals
+of the observational astronomy. However, UV/optical estim ation only provides us with
+alowerlimitofthestarformationrate(SFR) duetotheobscu rationbydust.Astraight-
+forward way to overcome this problem is to observe in infrare d, which can capture the
+starformation activityinvisiblein the UV. The superb sens itivitiesofrecently launched
+SpitzerandAKARI satellitescan revolutionizethefield.
+However,most of theSpitzer work relied on a large extrapola tionfrom 24 µm flux to
+estimate the 8, 12 µm or total infrared (TIR) luminosity, due to the limited numb er of
+mid-IR filters. AKARI has continuous filter coverage across t he mid-IR wavelengths,
+thus, allows us to estimate mid-IR luminosity without using a largek-correction based
+on the SED models, eliminating the largest uncertainty in pr evious work. By taking
+advantage of this, we present the restframe 8, 12 µm TIR LFs, and thereby the cosmic
+starformationhistoryderivedfrom theseusingtheAKARINE P-Deep data.
+Data&Analysis .TheAKARIhasobservedtheNEPdeepfield(0.4deg2)in9filters
+(N2,N3,N4,S7,S9W,S11,L15,L18WandL24) to the depths of 14.2, 11.0, 8.0, 48, 58,
+71, 117, 121 and 275 µJy (5σ)[14]. This region is also observed in BVRi′z′(Subaru),
+u′(CFHT), FUV,NUV(GALEX), and J,Ks(KPNO2m), with which we computed
+photo-zwithΔz
+1+z=0.043. Objects which are better fit with a QSO template are re movedFIGURE 1. (left) Restframe 8 µm LFs. The blue diamonds, purple triangles, red squares, and orange
+crosses show the 8 µm LFs at 0 .38<z<0.58,0.65<z<0.90,1.1<z<1.4, and 1.8<z<2.2,
+respectively. The dotted lines show analytical fits with a do uble-power law. Vertical arrows show the
+8µm luminosity corresponding to the flux limit at the central re dshift in each redshift bin. Overplotted
+are Babbedge et al. [1] in the pink dash-dotted lines, Caputi et al. [2] in the cyan dash-dotted lines,
+and Huang et al. [6] in the green dash-dotted lines. AGNs are e xcluded from the sample. (middle)
+Restframe 12 µm LFs. The blue diamonds, purple triangles, and red squares s how the 12 µm LFs at
+0.15<z<0.35,0.38<z<0.62, and 0 .84<z<1.16, respectively. Overplotted are Pérez-González
+et al. [11] at z=0.3,0.5and 0.9 in the cyan dash-dottedlines, and Rush, Mal kan, & Spinoglio [12] at z=0
+inthegreendash-dottedlines. (right)TIRLFs.
+from the analysis. We compute LFs using the 1/ Vmaxmethod. Data are used to 5 σwith
+completeness correction. Errors of the LFs are from 1000 rea lization of Monte Carlo
+simulation.
+8µm LF.Monochromatic 8 µm luminosity ( L8µm) is known to correlate well with
+the TIR luminosity [1, 6], especially for star-forming gala xies because the rest-frame
+8µmfluxaredominatedbyprominentPAHfeaturessuchasat6.2,7 .7and8.6 µm.The
+leftpanelofFig.1showsastrongevoltuionof8 µmLFs.Overplottedpreviousworkhad
+torelyonSEDmodelstoestimate L8µmfromtheSpitzer S24µmintheMIRwavelengths
+whereSEDmodelingisdifficultduetothecomplicatedPAHemi ssions.Here,AKARI’s
+mid-IR bands are advantageous in directly observing redshi fted restframe 8 µm flux in
+one of the AKARI’s filters, leading to more reliable measurem ent of 8µm LFs without
+uncertaintyfromtheSED modeling.
+12µm LF.12µm luminosity ( L12µm) represents mid-IR continuum, and known to
+correlate closely with TIR luminosity [11]. The middle pane l of Fig.1 shows a strong
+evoltuion of 12 µm LFs. Here the agreement with previous work is better becaus e (i)
+12µm continuum is easier to be modeled, and (ii) the Spitzer also captures restframe
+12µm inS24µmat z=1.
+TIRLF.Lastly,weshowtheTIRLFsintherightpanelofFig.1.Weused Lagache,
+Dole, & Puget [8]’s SED templates to fit the photometry using t he AKARI bands
+at>6µm (S7,S9W,S11,L15,L18WandL24). The TIR LFs show a strong evolution
+comparedto localLFs. At 0 .25<z<1.3,L∗
+TIRevolvesas ∝(1+z)4.1±0.4.FIGURE2. Evolutionof TIRluminositydensitybasedon TIRLFs (redcir cles),8µmLFs (stars), and
+12µm LFs (filled triangles). The blue open squares and orange fill ed squares are for LIRG and ULIRGs
+only,alsobasedonour LTIRLFs.Overplotteddot-dashedlinesareestimatesfromtheli terature:LeFloc’h
+et al. [9], Magnelli et al. [10] , Pérez-González et al. [11], Caputi et al. [2], and Babbedge et al. [1] are
+in cyan, yellow, green, navy, and pink, respectively. The pu rple dash-dotted line shows UV estimate by
+Schiminovichet al.[13].Thepinkdashedlineshowsthe tota lestimateofIR(TIRLF)andUV [13].
+Cosmic star formation history .We fit LFs in Fig.1 with a double-power law, then
+integrate to estimate total infrared luminosity density at various z. The restframe 8
+and 12µm LFs are converted to LTIRusing [11, 2] before integration. The resulting
+evolution of the TIR density is shown in Fig.2. The right axis shows the star formation
+densityassumingKennicutt[7].We obtain ΩIR(z)∝(1+z)4.4±1.0. Comparisonto ΩUV
+[13] suggests that ΩTIRexplains 70% of Ωtotalatz=0.25, and that by z=1.3, 90% of
+the cosmic SFD is explained by the infrared. This implies tha tΩTIRprovides good
+approximationofthe Ωtotalatz>1.
+In Fig.2, we also show the contributions to ΩTIRfrom LIRGs and ULIRGs. From
+z=0.35 to z=1.4,ΩIRby LIRGs increases by a factor of ∼1.6, andΩIRby ULIRGs
+increases byafactorof ∼10. Moredetailsarein Gotoet al. [3].
+Spatially-Resolved Spectroscopy of an E+A (post-starburs t) System .We per-
+formed a spatially-resolved medium resolution long-slit s pectroscopy of a nearby E+A
+(post-starburst) galaxy system with FOCAS/Subaru [4]. Thi s E+A galaxy has an obvi-
+ous companion galaxy 14kpc in front (Fig.3, left) with the ve locity difference of 61.8
+km/s.
+WefoundthatH δequivalentwidth(EW)oftheE+Agalaxyisgreaterthan7Å gal axy
+wide (8.5 kpc) with no significant spatial variation. We dete cted a rotational velocity in
+the companion galaxy of >175km/s. The progenitor of the companion may have beenFIGURE 3. (left) The SDSS g,r,i-composite image of the J1613+5103. The long-slit position s are
+overlayed.The E+A galaxy is to the right (west), with bluer c olour. The companion galaxy is to the left
+(east). (right) H δEW is plotted against D4000. The diamonds and triangles are f or the E+A core/north
+spectra, respectively. The squares and crosses are for the c ompanion galaxy’s core/north spectra. Gray
+lines are population synthesis models with 5-100% delta bur st population added to the 10G-year-old
+exponentially-decaying( τ=1Gyr)underlyingstellarpopulation.SalpeterIMFandmet allicityof Z=0.008
+areassumed.Onthe models,burstagesof0.1,0.25,0.5and2 G yraremarkedwiththefilled circles.
+a rotationally-supported, but yet passive S0 galaxy. The ag e of the E+A galaxy after
+quenching the star formation is estimated to be 100-500Myr, with its centre having
+slightly younger stellar population. The companion galaxy is estimated to have older
+stellarpopulationof >2 Gyrs ofagewithnosignificantspatialvariation(Fig.3, ri ght).
+Thesefindingsareinconsistentwithasimplepicturewheret hedynamicalinteraction
+createsinfallofthegasreservoirthatcausesthecentrals tarburst/post-starburst.Instead,
+ourresultspresentanimportantexamplewherethegalaxy-g alaxyinteractioncantrigger
+agalaxy-widepost-starburstphenomena.
+REFERENCES
+1. BabbedgeT.S.R., et al.,2006,MNRAS, 370,1159
+2. CaputiK.I.,et al.,2007,ApJ,660,97
+3. GotoT.,et al. 2010,A&AAKARI specialissue
+4. GotoT.,YagiM.,YamauchiC., 2008,MNRAS, 391,700
+5. HopkinsA.M.,ConnollyA. J.,HaarsmaD. B.,CramL. E.,200 1,AJ, 122,288
+6. HuangJ.-S.,et al.,2007,ApJ, 664,840
+7. KennicuttR. C.,Jr., 1998,ARA&A,36,189
+8. LagacheG., DoleH.,PugetJ.-L.,2003,MNRAS, 338,555
+9. LeFloc’hE.,etal., 2005,ApJ,632,169
+10. MagnelliB., et al.2009,A&A,496,57
+11. Pérez-GonzálezP. G.,etal., 2005,ApJ,630,82
+12. RushB., MalkanM. A.,SpinoglioL.,1993,ApJS,89,1
+13. SchiminovichD.,et al.,2005,ApJ, 619,L47
+14. Wada T.,et al.,2008,PASJ, 60,517
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@@ -0,0 +1,322 @@
+arXiv:1001.0008v2  [hep-th]  6 Jan 2010Multi-Stream Inflation: Bifurcations and Recombinations i n the Multiverse
+Yi Wang∗
+Physics Department, McGill University, Montreal, H3A2T8, Canada
+In this Letter, we briefly review the multi-stream inflation s cenario, and discuss its implications in
+the string theory landscape and the inflationary multiverse . In multi-stream inflation, the inflation
+trajectory encounters bifurcations. If these bifurcation s are in the observable stage of inflation, then
+interesting observational effects can take place, such as do main fences, non-Gaussianities, features
+and asymmetries in the CMB. On the other hand, if the bifurcat ion takes place in the eternal stage
+of inflation, it provides an alternative creation mechanism of bubbles universes in eternal inflation,
+as well as a mechanism to locally terminate eternal inflation , which reduces the measure of eternal
+inflation.
+I. INTRODUCTION
+Inflation [1] has become the leading paradigm for the
+very early universe. However, the detailed mechanism
+for inflation still remains unknown. Inspired by the pic-
+ture of string theory landscape [2], one could expect that
+the inflationary potential has very complicated structure
+[3]. Inflation in the string theory landscape has impor-
+tantimplicationsinbothobservablestageofinflationand
+eternal inflation.
+The complicated inflationary potentials in the string
+theory landscape open up a great number of interest-
+ing observational effects during observable inflation. Re-
+searchesinvestigatingthecomplicatedstructureofthein-
+flationary potential include multi-stream inflation [4, 5],
+quasi-single field inflation [6], meandering inflation [7],
+old curvaton [8], etc.
+Thestringtheorylandscapealsoprovidesaplayground
+for eternal inflation. Eternal inflation is an very early
+stage of inflation, during which the universe reproduces
+itself, so that inflation becomes eternal to the future.
+Eternal inflation, if indeed happened (for counter ar-
+guments see, for example [9]), can populate the string
+theory landscape, providing an explanation for the cos-
+mological constant problem in our bubble universe by
+anthropic arguments.
+In this Letter, we shall focus on the multi-stream infla-
+tion scenario. Multi-stream inflation is proposed in [4].
+And in [5], it is pointed out that the bifurcations can
+lead to multiverse. Multi-stream inflation assumes that
+during inflation there exist bifurcation(s) in the inflation
+trajectory. For example, the bifurcations take place nat-
+urally in a random potential, as illustrated in Fig. 1. We
+briefly review multi-stream inflation in Section II. The
+details of some contents in Section II can be found in
+[4]. We discuss some new implications of multi-stream
+inflation for the inflationary multiverse in Section III.
+∗wangyi@hep.physics.mcgill.ca
+FIG. 1. In this figure, we use a tilted random potential to
+mimic a inflationary potential in the string theory landscap e.
+One can expect that in such a random potential, bifurcation
+effects happens generically, as illustrated in the trajecto ries
+in the figure.
+FIG. 2. One sample bifurcation in multi-stream inflation.
+The inflation trajectory bifurcates into AandBwhen the
+comoving scale k1exits the horizon, and recombines when
+the comoving scale k2exits the horizon.
+II. OBSERVABLE BIFURCATIONS
+In this section, we discuss the possibility that the bi-
+furcation of multi-stream inflation happens during the
+observable stage of inflation. We review the production
+of non-Gaussianities, features and asymmetries [4] in the2
+FIG. 3. In multi-stream inflation, the universe breaks up
+into patches with comoving scale k1. Each patch experienced
+inflation either along trajectories AorB. These different
+patches can be responsible for the asymmetries in the CMB.
+CMB, and investigate some other possible observational
+effects.
+To be explicit, we focus on one single bifurcation, as
+illustrated in Fig. 2. We denote the initial (before bifur-
+cation) inflationary direction by ϕ, and the initial isocur-
+vature direction by χ. For simplicity, we let χ= 0 before
+bifurcation. When comoving wave number k1exits the
+horizon, the inflation trajectory bifurcates into Aand
+B. When comoving wave number k2exits the horizon,
+the trajectories recombines into a single trajectory. The
+universe breaks into of order k1/k0patches (where k0de-
+notes the comoving scale of the current observable uni-
+verse), each patch experienced inflation either along tra-
+jectories AorB. The choice of the trajectories is made
+by the isocurvature perturbation δχat scale k1. This
+picture is illustrated in Fig. 3.
+We shall classify the bifurcation into three cases:
+Symmetric bifurcation . If the bifurcation is symmetric,
+in other words, V(ϕ,χ) =V(ϕ,−χ), then there are two
+potentially observable effects, namely, quasi-single field
+inflation, and a effect from a domain-wall-like objects,
+which we call domain fences.
+As discussed in [4], the discussion of the bifurcation
+effect becomes simpler when the isocurvature direction
+has mass of order the Hubble parameter. In this case,
+except for the bifurcation and recombination points, tra-
+jectoryAand trajectory Bexperience quasi-single field
+inflation respectively. As there are turnings of these tra-
+jectories, the analysis in [6] can be applied here. The
+perturbations, especially non-Gaussianities in the isocur-
+vature directions are projected onto the curvature direc-
+tion, resultingin a correctionto the powerspectrum, and
+potentially large non-Gaussianities. As shown in [6], the
+amount of non-Gaussianity is of order
+fNL∼P−1/2
+ζ/parenleftbigg1
+H∂3V
+∂χ3/parenrightbigg/parenleftBigg˙θ
+H/parenrightBigg3
+, (1)
+whereθdenotes the angle between the true inflation di-
+rection and the ϕdirection.
+As shown in Fig. 3, the universe is broken into patches
+during multi-stream inflation. There arewall-likebound-
+aries between these patches. During inflation, theseboundaries are initially domain walls. However, after
+the recombination of the trajectories, the tensions of
+these domain walls vanish. We call these objects domain
+fences. As is well known, domain wall causes disasters
+in cosmology because of its tension. However, without
+tension, domain fence does not necessarily cause such
+disasters. It is interesting to investigate whether there
+are observational sequences of these domain fences.
+Nearly symmetric bifurcation If the bifurcation is
+nearly symmetric, in other words, V(ϕ,χ)≃V(ϕ,−χ),
+but not equal exactly, which can be achieved by a spon-
+taneous breaking and restoring of an approximate sym-
+metry, then besides the quasi-single field effect and the
+domain fence effect, there will be four more potentially
+observable effects in multi-stream inflation, namely, the
+features and asymmetries in CMB, non-Gaussianity at
+scalek1and squeezed non-Gaussianity correlating scale
+k1and scale kwithk1< k < k 2.
+The CMB power asymmetries are produced because,
+as in Fig. 3, patches coming from trajectory AorBcan
+have different power spectra PA
+ζandPB
+ζ, which are de-
+termined by their local potentials. If the scale k1is near
+to the scale of the observational universe k0, then multi-
+stream inflation provides an explanation of the hemi-
+spherical asymmetry problem [10].
+The features in the CMB (here feature denotes extra
+large perturbation at a single scale k1) are produced as
+a result of the e-folding number difference δNbetween
+two trajectories. From the δNformalism, the curvature
+perturbation in the uniform density slice at scale k1has
+an additional contribution
+δζk1∼δN≡ |NA−NB|. (2)
+These features in the CMB are potentially observable
+in the future precise CMB measurements. As the addi-
+tional fluctuation δζk1does not obey Gaussian distribu-
+tion, there will be non-Gaussianity at scale k1.
+Finally, there are also correlations between scale k1
+and scale kwithk1< k < k 2. This is because the ad-
+ditional fluctuation δζk1and the asymmetry at scale k
+are both controlled by the isocurvature perturbation at
+scalek1. Thus the fluctuations at these two scales are
+correlated. As estimated in [4], this correlation results in
+a non-Gaussianity of order
+fNL∼δζk1
+ζk1PA
+ζ−PB
+ζ
+PA
+ζP−1/2
+ζ. (3)
+Non-symmetric bifurcation If the bifurcation is not
+symmetric at all, especially with large e-folding number
+differences (of order O(1) or greater) along different tra-
+jectories, the anisotropy in the CMB and the large scale
+structure becomes too large at scale k1. However, in
+this case, regions with smaller e-folding number will have
+exponentially small volume compared with regions with
+larger e-folding number. Thus the anisotropy can behave
+in the form of great voids. We shall address this issue in
+more detail in [11]. Trajectories with e-folding number3
+difference from O(10−5) toO(1) in the observable stage
+of inflation are ruled out by the large scale isotropy of
+the observable universe.
+At the remainderof this section, we would like to make
+several additional comments for multi-stream inflation:
+The possibility that the bifurcated trajectories never re-
+combine. In this case, one needs to worry about the do-
+main walls, which do not become domain fence during
+inflation. These domain walls may eventually become
+domain fence after reheating anyway. Another prob-
+lem is that the e-folding numbers along different tra-
+jectories may differ too much, which produce too much
+anisotropies in the CMB and the large scale structure.
+However, similar to the discussion in the case of non-
+symmetric bifurcation, in this case, the observable effect
+could become great voids due to a large e-folding number
+difference. The case without recombination of trajectory
+also has applications in eternal inflation, as we shall dis-
+cuss in the next section.
+Probabilities for different trajectories . In [4], we con-
+sidered the simple example that during the bifurcation,
+the inflaton will run into trajectories AandBwith equal
+probabilities. Actually, this assumption does not need to
+be satisfied for more general cases. The probability to
+run into different trajectories can be of the same order
+of magnitude, or different exponentially. In the latter
+case, there is a potential barrier in front of one trajec-
+tory, which can be leaped over by a large fluctuation of
+theisocurvaturefield. Alargefluctuationoftheisocurva-
+ture field is exponentially rare, resulting in exponentially
+different probabilities for different trajectories. The bi-
+furcation of this kind is typically non-symmetric.
+Bifurcation point itself does not result in eternal infla-
+tion. As is well known, in single field inflation, if the
+inflaton releases at a local maxima on a “top of the hill”,
+a stage of eternal inflation is usually obtained. However,
+at the bifurcation point, it is not the case. Because al-
+though the χdirection releases at a local maxima, the ϕ
+direction keeps on rolling at the same time. The infla-
+tiondirectionisacombinationofthesetwodirections. So
+multi-stream inflation can coexist with eternal inflation,
+but itself is not necessarily eternal.
+III. ETERNAL BIFURCATIONS
+In multi-stream inflation, the bifurcation effect may ei-
+ther take place at an eternal stage of inflation. In this
+case, it provides interesting ingredients to eternal infla-
+tion. These ingredients include alternative mechanism to
+producedifferentbubble universesandlocalterminations
+for eternal inflation, as we shall discuss separately.
+Multi-stream bubble universes . The most discussed
+mechanisms to produce bubble universes are tunneling
+processes, such as Coleman de Luccia instantons [12] and
+Hawking Moss instantons [13]. In these processes, the
+tunneling events, which are usually exponentially sup-
+pressed, create new bubble universes, while most parts
+FIG. 4. Cascade creation of bubble universes. In this figure,
+we assume trajectory Ais the eternal inflation trajectory, and
+trajectory Bis the non-eternal inflation trajectory.
+of the spatial volume remain in the old bubble universe
+at the instant of tunneling.
+If bifurcations of multi-stream inflation happen dur-
+ing eternal inflation, two kinds of new bubble universes
+can be created with similar probabilities. In this case,
+at the instant of bifurcation, both kinds of bubble uni-
+verseshavenearlyequalspatialvolume. Withachangeof
+probabilities, the measures for eternal inflation should be
+reconsideredformulti-streamtype bubble creationmech-
+anism.
+If the inflation trajectories recombine after a period of
+inflation, the different bubble universes will eventually
+have the same physical laws and constants of nature. On
+the other hand, if the different inflation trajectories do
+not recombine, then the different bubble universes cre-
+ated by the bifurcation will have different vacuum ex-
+pectation values of the scalar fields, resulting to different
+physical laws or constants of nature. It is interesting
+to investigate whether the bifurcation effect is more ef-
+fective than the tunneling effect to populate the string
+theory landscape.
+Note that in multi-stream inflation, it is still possi-
+ble that different trajectorieshaveexponentiallydifferent
+probabilities, as discussed in the previous section. In this
+case, multi-stream inflation behaves similar to Hawking
+Moss instantons during eternal inflation.
+Local terminations for eternal inflation . It is possible
+that during multi-stream inflation, a inflation trajectory
+bifurcates in to one eternal inflation trajectory and one
+non-eternal inflation trajectory with similar probability.
+Inthiscase,theinflatonintheeternalinflationtrajectory
+frequently jumps back to the bifurcation point, resulting
+in a cascade creation of bubble universes, as illustrated
+in Fig. 4. This cascade creation of bubble universes, if4
+realized, is more efficient in producing reheating bubbles
+than tunneling effects. Thus it reduces the measure for
+eternal inflation.
+There are some other interesting issues for bifurcation
+in the multiverse. For example, the bubble walls may
+be observable in the present observable universe, and the
+bifurcations can lead to multiverse without eternal infla-
+tion. These possibilities are discussed in [5].
+IV. CONCLUSION AND DISCUSSION
+To conclude, webriefly reviewedmulti-stream inflation
+during observable inflation. Some new issues such as do-main fences and connection with quasi-single field infla-
+tion are discussed. We also discussed multi-stream infla-
+tion in the context of eternal inflation. The bifurcation
+effect in multi-stream inflation provides an alternative
+mechanism for creating bubble universes and populating
+the string theory landscape. The bifurcation effect also
+provides a very efficient mechanism to locally terminate
+eternal inflation.
+ACKNOWLEDGMENT
+We thank Yifu Cai for discussion. This work was sup-
+ported by NSERC and an IPP postdoctoral fellowship.
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\ No newline at end of file
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+arXiv:1001.0009v1  [q-bio.BM]  30 Dec 2009Jamming proteins with slipknots and their free energy lands cape
+Joanna I. Su/suppress lkowska1, Piotr Su/suppress lkowski2,3,4and Jos´ e N. Onuchic1
+1Center for Theoretical Biological Physics,
+University of California San Diego,
+Gilman Drive 9500, La Jolla 92037,
+2Physikalisches Institute and Bethe Center for Theoretical Physics,
+Universit¨ at Bonn, Nussallee 12, 53115 Bonn, Germany
+3California Institute of Technology, Pasadena, CA 92215,
+4Institute for Nuclear Studies,
+Ho˙ za 69, 00-681 Warsaw, Poland
+Theoretical studies of stretching proteins with slipknots reveal a surprising growth of their un-
+folding times when the stretching force crosses an intermed iate threshold. This behavior arises as
+a consequence of the existence of alternative unfolding rou tes that are dominant at different force
+ranges. Responsible for longer unfolding times at higher fo rces is the existence of an intermediate,
+metastable configuration where the slipknot is jammed. Simu lations are performed with a coarsed
+grained model with further quantification using a refined des cription of the geometry of the slip-
+knots. The simulation data is used to determine the free ener gy landscape (FEL) of the protein,
+which supports recent analytical predictions.
+PACS numbers: 87.15.ap, 87.14.E-, 87.15.La, 82.37.Gk, 87. 10.+e
+The large increase in determining new protein struc-
+tures has led to the discovery of several proteins with
+complicated topology. This new fact has arised the ques-
+tion if their energy landscape and the folding mechanism
+is similar to typical proteins. One class of such proteins
+includes knotted proteins which comprise around 1% of
+all structures deposited in the PDB database [1, 2]. A
+related class of proteins contains more subtle geometric
+configurations called slipknots [3, 4]. Recent theoretical
+studies using structure-based models (where native con-
+tacts are dominant) suggest that slipknot-like conforma-
+tions act like intermediates during the folding of knotted
+proteins [5]. This entire new mechanism is consistent
+with energy landscape theory (FEL) and the funnel con-
+cept [7, 8]. It was shown that the slipknot formation
+reduces the topological barrier. Complementing regular
+folding studies, additional information about the land-
+scape was obtained by mechanical manipulation of the
+knotted protein with atomic force microscopy [9] both
+experimentally in [10, 11] and theoretically in [12, 13, 14].
+For example, [12] it has been showen that unfolding pro-
+ceeds via a series of jumps between various metastable
+conformations, a mechanism opposite to the smooth un-
+folding in knotted homopolymers.
+Motivated by these early results, we now propose a uni-
+fied picture for the mechanical unfolding of proteins with
+slipknots. In this Letter this question is addressed by
+explaining the role of topological barriers along their me-
+chanical unfolding pathways. Supported by our previous
+results that knotted proteins can still have a minimally
+frustrated funnel-like energy landscape, structure-based
+theoretical coarse-grained models are used [15] to ana-
+lyze the behavior of a slipknot protein under stretching.
+Studies are performed for the α/β class protein thymi-dine kinase (PDB code: 1e2i [17]).
+2 3 4 5F/LBracket1Ε/Slash1/Angstrom/RBracket17.27.57.88.1logΤ
+FIG. 1: Dependence of the unfolding times τon the stretch-
+ing force Ffor 1e2i (solid line, in red). In this Letter we
+describe this mechanism as a superposition of two unfolding
+pathways: I for small forces (dashed (lower) line, in blue),
+and II for intermidiate and large forces (dashed-dotted (up -
+per) line, green).
+Most of our analysis is based on stretching simulations
+under constant force [16]. The crucial signature for this
+process is the overall unfolding time from the beginning
+of the stretching until the protein fully unfolds. Normally
+one expects that the transition between the native and
+the unfolded basins to be limited by overcoming the free
+energy barrier, which gets effectively reduced upon an
+application of a stretching force. The rate by which this
+barrier is reduced depends on the distance between the
+unfolded basin and the top of the barrier measured along
+the stretching coordinate x. This idea was first devel-
+oped in the phenomenological model of Bell [18], which
+states that the unfolding time τdecreases exponentially
+with applied stretching force Fasτ(F) =τ0e−Fx
+kBT. A2
+refined analysis performed in ref. [19] revealed that this
+dependence is more complicated but still monotonically
+decreasing.
+The unfolding times for 1e2i measured in our simula-
+tions are shown as the red curve in Fig. 1. In contrast to
+the above expectations, increasing the force in the range
+3-3.5ǫ/˚A surprisingly results in a larger stability of the
+protein. ǫis the typical effective energy of tertiary na-
+tive contacts that is consistent with the value ǫ/˚A≃71
+pN derived in [15]. A solution for this paradox is accom-
+plished by realizing that unfolding is dominated by two
+distinct, alternative routes that are dominant at different
+force regimes. A routing switch occurs when threshold is
+crossed between weak and intermediate forces. At higher
+forces, mechanical unfolding is dominated by a route that
+involves a jammed slipknot. This jamming gives rise to
+the unexpected dependence of unfolding time on applied
+force. Characterizing this mechanism is the central goal
+of this Letter.
+FIG. 2: A slipknot (left) consists of a threaded loop (k1−k2,
+in red) which is partialy threaded through a knotting loop
+(k2−k3, in blue). An example of a protein configuration with
+a tightened slipknot is shown in the right panel.
+To describe the evolution of a slipknot quantitatively
+requires a refined description. A slipknot is character-
+ized by the three points shown in Fig. 2. The first
+pointk1is determined by eliminating amino acids con-
+secutively from one terminus until the knot configura-
+tion is reached (which can be detected e.g. by applying
+the KMT algorithm [20]). The two additional points,
+k2andk3, correspond to the ends of this knot. In the
+native state the protein 1e2i contains a slipknot with
+k1= 10,k2= 128,k3= 298. These three points divide
+the slipknot into two loops, which are called the knotting
+loop and thethreaded loop . The former one is the loop of
+the trefoil knot and the latter one is threaded through the
+knotting loop. Unfolding of the slipknot upon stretch-
+ing depends on the relative shrinking velocity of these
+two loops (see Fig. 3). When the threaded loop shrinks
+faster than the knotting loop, the slipknot unties. In the
+opposite case the slipknot gets (temporarily) tightened
+or jammed, resulting in a metastable state associated
+to a local minimum in the protein’s FEL. Upon further
+stretching, this configuration eventually also unties. The
+evolution of both loops of the slipknot is encoded in thetime dependence of the points k1,k2,k3, see Fig. 3.
+pathway I pathway II
+catch−bonds slip−bondspathway II
+catch−bonds slip−bondspathway I
+FIG. 3: The behavior of the slipknot during stretching (top)
+is determined by the relative behavior of its two loops, en-
+coded in the time dependence of k1,k2andk3(bottom). If the
+threaded loop shrinks faster than the knotting loop, k1merges
+withk2(bottom left) and the slipknot untightens (pathway I,
+top left). If the knotting loop shrinks faster, k2approaches k3
+(bottom right, ≃14000τ) and the slipknot gets temporarily
+tightened (pathway II, top right). This is a metastable stat e
+which can eventually untie further stretching , with k1finally
+merging with k2(bottom right, ≃19000τ). Kinetic stud-
+ies were performed slightly above folding temperature usin g
+overdamped Langevin dynamics with typical folding times of
+10000τ.
+Before discussing the stretching of 1e2i, we explain why
+a slipknot formed by a uniformly elastic polymer should
+smoothly unfold under stretching. To simplify the discus-
+sion we approximate the threaded and knotting loops by
+circles of size RtandRk. These two loops shrink during
+stretching and, when the threaded one eventually van-
+ishes, the slipknot gets untied. If both loops have similar
+sizes, the slipknot is very unstable and unties immedi-
+ately. When the threaded loop is much larger than the
+knotting one, Rt>> R k, untightening can be explained
+as follows. The elastic energy associated to local bend-
+ing is proportional to the square of the curvature. If the
+loop is approximated by a circle of radius R, then its local
+curvature is constant and equals R−1. The total elastic
+energy is/contintegraltext
+dsR−2∼R−1[21]. From the assumption
+Rt>> R kwe conclude that upon stretching it is ener-
+getically favorable to decrease Rtrather than Rk. This
+happens until both radii become equal and then, just as
+above, the slipknot gets very unstable and untightens. In
+this discussion we have not yet taken into account that
+when a slipknot is stretched some parts of a chain slide
+along each other. This effect could be incorporated by in-
+cluding the friction generated by the sliding [22]. But in
+the slipknot the sliding region associated with the knot-
+ting loop is much longer than the region associated to3
+the threaded loop. Thus this effect results in a faster
+tightening of the threaded rather than the knotting loop,
+facilitating even more the untightening of the slipknot.
+The above argument should apply to slipknots in
+biomolecules because they are characterized by a per-
+sistence length that in principle is simply related to their
+elasticity [23]. For DNA this effect is described by worm-
+like-chain models (WLC) [24] and it has been confirmed
+experimentally. Although WLC models are too simple
+to describe the protein general behavior, they are use-
+ful in some limited applications. Thus at first sight one
+might expect that slipknots in proteins should smoothly
+untie upon stretching. Proteins, however, are much more
+complicated than DNA or uniformly elastic polymers.
+The presence of stabilizing native tertiary contacts leads
+to a jumping character during stretching [12]. In addi-
+tion their bending energy is not uniform along the chain
+due to the heterogeneity of the amino-acid sequence. As
+a consequence it turns out that the intuition obtained
+through the above analysis of polymers or WLC models
+is misleading.
+2 3 4 5F/LBracket1Ε/Slash1/Angstrom/RBracket10.51Prob/LParen1pathway I/RParen1
+FIG. 4: Dependence on the applied stretching force of the
+probability of choosing pathway I rather than II (see Fig. 3) .
+This varying probability leads to the complicated dependen ce
+of the total unfolding time on the stretching force observed in
+Fig. 1.
+Our analysis of the evolution of the endpoints k1,k2,k3
+(Fig. 3, bottom) reveals that for various stretching forces
+unfolding proceeds along two distinct pathways (Fig. 3,
+top). In pathway I the slipknot smoothly unties, which
+is observed for relatively weak forces. At intermediate
+forces pathway II starts to dominate and the knotting
+loop can shrink tightly before the threaded one vanishes.
+In this regime the protein gets temporarily jammed (Fig.
+3, right), leading to much longer unfolding times (catch
+pathway). The probability of choosing pathway I at dif-
+ferent forces is shown in Fig. 4. This pathway competi-
+tion explains the nontrivial total unfolding time depen-
+dence observed in Fig. 1.
+The two different pathways I and II arise from com-
+pletely different unfolding mechanisms. Pathway I starts
+and continues mostly from the C-terminal side, along
+16α, 15β, 14α, 13β, 12(helices bundle), 11 α(here the
+number denotes a consecutive secondary structure ascounted from N-terminal, and αorβspecifies whether
+this is a helix or a β-sheet; for more details about the
+structure of 1e2i see the PDB). This is followed by unfold-
+ing of helices 11 α, 10αthat allows breaking of the con-
+tacts inside the β-sheet created by the N-terminal, with
+unfolding proceeding also from the N-terminal. Pathway
+II also starts from the C-terminal but rapidly (as soon
+as helix 15 is unfolded) switches to the N-terminal. In
+this case, differently from pathway I, the β-sheet from
+the N-terminal unfolds even before 13 β. These scenarios
+indicate that the pathway I should be dominant at weak
+forces since they are not sufficient to break the β-sheet
+during first steps of unfolding. The jammed pathway is
+typical only if stretching forces are sufficiently strong for
+unfolding to proceed from the two terminals of the pro-
+tein.
+A similar phenomenon was firstly proposed in ref. [25]
+and referred to as catch-bonds. Experimental evidence
+suggesting this mechanism was first observed for adhe-
+sion complexes [26, 27]. Using AFM, at large forces the
+ligand-receptor pair becomes entangled and therefore ex-
+pands the unfolding time. A theoretical description of
+this mechanism was given in ref. [28, 29, 30].
+The kinetic data can also be used to determine the as-
+sociated free energy landscape (FEL) [7]. In an initial
+simplification we associate the barriers along the stretch-
+ing coordinate as the the kinetic bottlenecks during the
+mechanical unfolding event. Generalizing Bell’s model,
+a recent description of two-state mechanical unfolding in
+the presence of a single transition barrier has been devel-
+oped in [19], with the rate equation
+τ(F) =τ0/parenleftBig
+1−νFx†
+∆G/parenrightBig1−1/ν
+e−∆G
+kBT/parenleftbig
+1−(1−νFx†/∆G)1/ν/parenrightbig
+,
+(1)
+whereνencodes the shape of the barrier. Here x†denotes
+the distance between the barrier and the unfolded basin
+(in a first approximation it can be regarded as Findepen-
+dent) and lies on the reaction coordinate along the AFM
+pulling direction. It can be experimentally determined
+by measuring how the stretching force modulates the un-
+folding times τ. The height of the barrier is denoted by
+∆G. Fig. 1 (unfolding times are given by solid red line)
+shows that this single barrier theory is not sufficient for
+the full range of forces. As described before, in the higher
+force regime, additional basins have to be included in the
+energy landscape. Models with several metastable basins
+have been called multi-state FEL models [31]. Evidence
+supporting the need of multi-states FEL was confirmed
+by AFM experiments in different systems [32, 33].
+To construct a multi-state FEL that incorporates two
+unfolding pathways I and II we use a linear combina-
+tion of eq. (1)-like expressions with different shapes and
+barrier heights. Each one of them essentially accounts
+for the distinct barrier along a relevant unfolding route.
+Fitting the stretching data to eq. (1) with a cusp-like4
+2.5 33.5 4F/LBracket1Ε/Slash1/Angstrom/RBracket16.67.6logΤ
+N U I
+FIG. 5: Pathway II with two barriers. Left: dependence of the
+unfolding time on the applied force with the data and the fit
+to the formula (1) for the first maximum (lower, in green) and
+for the second maximum (upper, in blue). Right: schematic
+free energy landscape for this pathway, with jammed slipkno t
+in a minimum between two barriers.
+ν= 1/2 approximation (another possibility ν= 2/3 for
+the cubic potential in general leads to similar results [19])
+determines accurately the location and the height of the
+potential barriers. Pathway II involves two barriers: first
+until the moment of creation of the intermediate which
+is followed the untieing event. They are characterized by
+(x1,∆G1) and (x2,∆G2) arising respectively from the
+lower and upper fits in Fig. 5 (left). The superposition
+of these two fits gives the overall mean unfolding time for
+pathway II (dotted-dashed curve in green in Fig. 1). For
+the ordinary slipknot unfolding (pathway I), the results
+xIandGIarise from the dashed blue curve in Fig. 1.
+This analysis leads to the results
+x1= 2.3kBT˚A
+ǫ, x2= 0.7kBT˚A
+ǫ, xI= 1.4kBT˚A
+ǫ,
+∆G1= 8.0kBT, ∆G2= 4.2kBT, ∆GI= 4.7kBT.
+We conclude that the free energy landscape consists of
+two “valleys”. The force-dependent probability of choos-
+ing one of the valleys during stretching depends on the
+details of the protein structure. It is determined from our
+simulations as shown in Fig. 4. Using these probability
+values and the parameters above for xand ∆G, we can
+accurately represent the simulation data using a linear
+combination of equations of the form (1). This agreement
+supports our analytical analysis and generalizes eq. (1)
+for the full of range forces. In addition it demonstrates
+that structure-based models sufficiently capture the ma-
+jor geometrical properties of a slipknotted protein. A
+schematic representation of the free energy landscape for
+pathway II is shown in Fig. 5 (right).
+Summarizing, we have analyzed the process of tighten-
+ing of the slipknot in protein 1e2i and determined the cor-
+responding free energy landscape. Its main feature is the
+presence of a metastable configuration with a tightened
+slipknot, which is observed for sufficiently large pulling
+forces. This phenomenon does not exist for uniformly
+elastic polymers. In this Letter we concentrated on pro-
+tein 1e2i but similar behavior has also been observed forother proteins with slipknots, e.g. 1p6x. Our results
+provide testable predictions that can now be verified by
+AFM stretching experiments.
+We appreciate useful comments of O. Dudko. The
+work of J.S. was supported by the Center for Theo-
+retical Biological Physics sponsored by the NSF (Grant
+PHY-0822283) with additional support from NSF-MCB-
+0543906. P.S. acknowledges the support of Hum-
+boldt Fellowship, DOE grant DE-FG03-92ER40701FG-
+02, Marie-Curie IOF Fellowship, and Foundation for Pol-
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+arXiv:1001.0011v2  [cond-mat.mes-hall]  16 Apr 2010Guided plasmons in graphene p-njunctions
+E. G. Mishchenko,1A. V. Shytov∗,1and P. G. Silvestrov2
+1Department of Physics and Astronomy, University of Utah, Sa lt Lake City, Utah 84112, USA
+2Theoretische Physik III, Ruhr-Universit¨ at Bochum, 44780 Bochum, Germany
+Spatial separation of electrons and holes in graphene gives rise to existence of plasmon waves
+confined to the boundary region. Theory of such guided plasmo n modes within hydrodynamics of
+electron-hole liquid is developed. For plasmon wavelength s smaller than the size of charged domains
+plasmon dispersion is found to be ω∝q1/4. Frequency, velocity and direction of propagation of
+guided plasmon modes can be easily controlled by external el ectric field. In the presence of magnetic
+field spectrum of additional gapless magnetoplasmon excita tions is obtained. Our findings indicate
+that graphene is a promising material for nanoplasmonics.
+PACS numbers: 73.23.-b, 72.30.+q
+Introduction . Breakthrough progress in synthesis and
+characterization has made graphene [2] a promising ob-
+ject for nanoelectronics. Operation of graphene-based
+transistors [3] and other components would rely on the
+propertiesofits single-particle excitations–electronsand
+holes. However, one can also envisage a completely dif-
+ferent set of applications which employ collective excita-
+tions, such as plasmons. Currently, plasmon excitations
+in metallic structures are a subject of nanoplasmonics, a
+new field which has emerged at the confluence of optics
+and condensed matter physics with one of the aims be-
+ing the developing of plasmon-enhanced high resolution
+near-field imaging methods [4, 5]. Another objective is
+possible utilization of plasmons in integrated optical cir-
+cuits. However, perspectives of graphene for nanoplas-
+monics are largely unexplored since plasmon modes of
+graphene flakes have not been addressed so far. As our
+results indicate a great amount of control over graphene
+plasmon properties makes it a very promising material
+for applications.
+Fundamentally, the spectrum of collective chargeoscil-
+lations reflects the long-rangenature of Coulomb interac-
+tion. In conventional two dimensional systems, such as
+those created in semiconducting heterostructures, plas-
+mons are gapless, ω2(q) = 2πe2nq/m∗, withnandm∗
+being electron density and effective mass, respectively
+[6]. Such oscillations can be treated hydrodynamically.
+In clean graphene at zero temperature the plasmon fre-
+quency,ω2∝ |EF|, vanishes with decreasing the doping
+levelEF. It has been argued [7] that the interaction be-
+tweenelectronsandholesinthefinalstatecanmodifythe
+response functions of Dirac fermions and open up a pos-
+sibility for the propagation of charge oscillations at low
+frequencies ω < qv, wherevis electron velocity. Still, hy-
+drodynamic( ω > qv)analogofconventionalplasmonsre-
+mains absent unless either temperature is non-zero [8] or
+graphene is driven away from the charge neutrality point
+by doping or gating [9]. Expectedly, in both cases plas-
+mon spectrum has the conventional form, ω(q)∝q1/2.
+In the present paper we investigate spectra of hydro-
+dynamic plasmons in spatially inhomogeneous grapheneflakes. Realistic graphene samples are typically subject
+to disorder potential and mechanical strain [10] that lead
+totheformationofchargedelectronandholepuddles[11]
+with boundaries between nandpregions being the lines
+ofzerochemicalpotential. Moreover,controlled p-njunc-
+tions can be made with the help of metallic gates [12].
+Alsop-njunctions can be created by applying electric
+field within the plane of a graphene flake, see Fig. 1a.
+The field separates electrons and holes spatially in a way
+that allows control of both the amount of induced charge
+(and thus plasmon frequency) and spatial orientation of
+the junction (the direction of plasmon propagation).
+b)2d 2d
+Ea)
+0n n
+p p
+FIG.1: Twotypesofgraphene p-njunctions: a)field-induced,
+b) gate-induced. Dot-dashed line indicates boundary betwe en
+electron and hole regions and, correspondingly, the direct ion
+of plasmon propagation. In case of field-induced junction it
+is controlled by the direction of external electric field E0.
+Below, we demonstrate that such p-njunctions can
+guide plasmons. We show the existence of charge oscil-
+lations which are localized at the junction and have the
+amplitude decaying with the distance to the junction.
+For wavelengths shorter than the width of the charged
+domains, we find the plasmon spectrum of the form,
+ω2
+n(q) =αne2v
+¯h/radicalbigg
+q|ρ′
+0|
+e, (1)
+whereρ′
+0is the gradient of equilibrium charge density
+at the junction, vis electron velocity, and n= 0,1,2,...2
+enumerates the solutions. The lowest mode has α0=
+4√
+2πΓ(3/4)/Γ(1/4)≈3.39.
+Below we derive this result and discuss plasmon prop-
+erties for the two types of p-njunctions: electric field
+controlled and gate controlled, as shown in Fig. 1.
+Hydrodynamics of charge density oscillations. We uti-
+lize the hydrodynamic approach to describe the motion
+of charged Dirac fermions. The rate of change of electric
+current density Jdue to dynamic electric field Efollows
+from the usual intra-band Drude conductivity with the
+corresponding density of states [13],
+˙J(r,t) =e2
+π¯h2|µ(r)|E(r,t), (2)
+determined by the local value of chemical potential µ(r)
+as measured from the Dirac point (positive for electrons
+and negative for holes). Electric current is related to the
+variation of charge density δρby means of the continuity
+equation,
+δ˙ρ(r,t)+∇·J(r,t) = 0. (3)
+Finally, the variation of charge density produces electric
+field according to the Coulomb law [14],
+E(r,t) =−∇/integraldisplay
+d2r′δρ(r′,t)
+|r−r′|. (4)
+Equations (2)-(4) give a closed system for plasmon exci-
+tations in graphene flakes. We apply it to a p-njunction
+created in a strip infinite along the y-axis (direction of
+plasmon propagation). Using the Fourier representation,
+δρ(r,t) =δρ(x)exp(iqy−iωt), and eliminating Eand
+Jwe arrive at the equation for the oscillating part of
+electron density,
+ω2δρ(x)+2e2v√π¯h/braceleftBigg
+d
+dx/radicalbigg
+|ρ0(x)|
+ed
+dx−q2/radicalbigg
+|ρ0(x)|
+e/bracerightBigg
+×/integraldisplayd
+−ddx′δρ(x′)K0(|q||x−x′|) = 0,(5)
+HereK0is the modified Bessel function and 2 dis
+the width of graphene flake. Within the Thomas-
+Fermi approximation equilibrium charge density ρ0(x)
+is related to the chemical potential via ρ0(x) =
+−sgn(µ)eµ2(x)/π¯h2v2(electron charge is taken to be
+−e). This follows from the condition that the electro-
+chemical potential µ(x)−eφ(x) is constant throughout
+the system. The solutions of Eq. (5) will now be consid-
+ered for large and small plasmon momenta separately.
+Short wavelength, q≫1/d. In this case the decay
+of plasmon density δρ(x) occurs over a distance much
+smaller than the width of the system and the limits
+of integration in Eq. (5) can be extended to infinity.
+Assuming (cf. Eq. (11) below) the linear dependence,
+ρ0(x) =ρ′
+0x, we observe that the integro-differentialequation (5) acquires obvious scaling property. Intro-
+ducing the variable ξ=qxwe arrive at the plasmon
+spectrum in the form (1), with dimensionless constants
+αndetermined from the eigenvalue problem:
+−2√π/parenleftbiggd
+dξ/radicalbig
+|ξ|d
+dξ−/radicalbig
+|ξ|/parenrightbigg
+×/integraldisplay∞
+−∞dξ′δρ(n)(ξ′)K0(|ξ−ξ′|) =αnδρ(n)(ξ).(6)
+Interestingly, this integro-differential equation allows a
+complete analytic solution, though the detailed analysis
+is beyond the scope of this paper. Our main findings
+are as follows. Solutions are enumerated by n= 0,1,2...
+with even/odd numbers corresponding to even/odd den-
+sity profile, δρ(n)(−ξ) = (−1)nδρ(n)(ξ). Surprisingly,
+eigenvalues are doubly-degenerate and given by
+α2n=α02n+1
+4n+1·3·7··(4n−1)
+1·5··(4n−3), α2n+1=α2n.(7)
+At large distances all modes have exponential depen-
+dence,δρ(n)(ξ)∼e−|ξ|, while at |ξ| ≪1 even and
+odd solutions exhibit different behavior, δρ(even)∼1−
+const/radicalbig
+|ξ|andδρ(odd)∼sign(ξ)//radicalbig
+|ξ|. The first pair
+of solutions (belonging to the lowest eigenvalue α0) in
+the Fourier representation δρ(n)(k) =/integraltext
+dξδρ(n)(ξ)eikξ
+acquires a simple form:
+δρ(0)(k)∝1
+(1+k2)3/4, δρ(1)(k)∝k
+(1+k2)3/4.(8)
+Long wavelength, q≪1/d. In contrast to the above
+result (1) plasmon spectrum at small qis sensitive to a
+specific realization of the p-njunction. We address the
+long-wavelength behavior of plasmons in field controlled
+junctions. We expect this case to be of more interest,
+in addition it allows a more complete description. Be-
+fore analyzing plasmons in this structure, we discuss the
+equilibrium density profile. As shown in Fig. 1a the flake
+of width 2 dis placed in external electric field E0applied
+along the x-direction. The equilibrium density distribu-
+tionρ(x) is found from,
+E0x+sgn(x)¯hv
+e/radicalbiggπ
+e|ρ0(x)|+2/integraldisplayd
+0dx′ρ0(x′)lnx+x′
+|x−x′|= 0,
+(9)
+where it is used that ρ0(x) =−ρ0(−x). Prior to solv-
+ing Eq. (9) it is instructive to analyze validity of the
+semiclassical approach. The first condition implies that
+the change of the electron wavelength is smooth on the
+scale of itself, d/dx(¯hv/µ)≪1. Estimating µ(x)∼eE0x
+we obtain that the distance to the p-njunction line
+(x= 0) should exceed the characteristic electric field
+lengthlE=/radicalbig
+e/E0≪x. The second condition requires
+that the electron wavelength is small compared with the
+width of the system, d≫¯hv/µ. Noting that in graphene3
+¯hv∼e2we can rewrite this second condition simply as
+lE≪d. Thus, the Thomas-Fermi equation (9) for the
+equilibrium charge density and the hydrodynamic equa-
+tion (5) for its variation are applicable as long as
+lE≪d, q≪1/lE. (10)
+However, the ratio of qand 1/dcan be arbitrary. For a
+moderate external electric field ∼104V/m the value of
+electric length lE∼0.4µm and the first of the conditions
+(10) is satisfied easily for micron-sized samples.
+AnalyticsolutionofEq.(9)ispossiblewhenthe second
+term is small, in which case the charge density is [15]
+ρ0(x) =E0x√
+d2−x2. (11)
+Substituting this expression back into Eq. (9) we ob-
+serve that the second term is indeed negligible as long
+asx≫l2
+E/d. This is assured whenever the condi-
+tions (10) are satisfied. It is also worth pointing out
+that Eq. (11) justifies the linear approximation for the
+charge density used in deriving Eq. (1) for q≫1/d, with
+ρ′
+0/e= 1/(l2
+Ed).
+We now turn to the analysis of plasma oscillations
+propagating on top of the density distribution, Eq. (11).
+For small plasmon momenta, q≪1/d, electric field ex-
+tends beyond the width of the flake and the equation (5)
+needs to be supplemented with the boundary condition,
+which ensures that electric field (and thus the current)
+vanishes at the edges, x=±d:
+P/integraldisplayd
+−ddxδρ(x)
+x±d= 0. (12)
+The spectrum of the lowest symmetric mode can be most
+easily found by integrating Eq. (5) across the width of
+the flake. The first term in the brackets will then van-
+ish exactly due to the boundary condition (12). The
+remaining integral can now be calculated to the log-
+arithmic accuracy with the help of the approximation
+K0(q|x−x′|) =−lnq|x−x′|:
+/integraldisplayd
+−ddx/radicalbigg
+|ρ0(x)|
+eln(q|x−x′|)≈2dΓ2(3/4)
+lE√πln(qd).
+(13)
+Eqs. (5) and (13) combine to give the equation, [ ω2−
+ω2
+0(q)]/integraltextd
+−ddxδρ(x) = 0, that yields the dispersion of the
+gapless symmetric plasmon,
+ω2
+0(q) = Γ2(3/4)4e2vd
+π¯hlEq2ln(1/qd),(14)
+reminiscent of the plasmon spectrum in quasi-one-
+dimensional wires, The remaining modes, n≥1, are
+gapped. For these modes/integraltextd
+−ddxδρ(x) = 0 and simple
+procedure of integrating Eq. (5) over the width of theflake is not useful. Instead, the equation for the n-th fre-
+quency gap can be obtained by setting q= 0 in Eq. (5).
+We observe that
+ω2
+n(0) =βne2v
+¯hlEd, (15)
+whereβnare the eigenvalues of the equation,
+2√πd
+dξ/radicalbig
+|ξ|
+(1−ξ2)1/4/integraldisplay1
+−1dξ′δρ(n)(ξ′)
+ξ−ξ′=βnδρ(n)(ξ).(16)
+The zeroth mode β0= 0, see Eq. (14), is found ana-
+lytically: δρ(0)∝1//radicalbig
+1−ξ2. It describes charge dis-
+tribution in the strip in response to a (uniform along
+xdirection and smooth along y-direction) change of its
+chemical potential [16]. Other solutions of Eq. (16) are
+found numerically:
+β1= 1.41, β2= 6.49, β3= 6.75,... (17)
+With increasing nthe eigenmodes of integro-differential
+equation (16) oscillate faster, but in generaldo not follow
+the oscillation theorem familiar from quantum mechan-
+ics. In particular, the solutions with n= 0 andn= 3 are
+even while n= 1,n= 2 are odd [17].
+Finally, we mention the case of a gate-controlled p-n
+junction, Fig.1b. Theequilibriumdensityprofileislinear
+nearx= 0 and saturates for large |x|[18]. Eq. (1) is still
+applicable for q >1/d. In the limit q <1/done should
+take into account the screening of long-range Coulomb
+interaction by metallic gates. In this case the logarithm
+in the spectrum of the gapless plasmon disappears, and
+the lowest mode Eq. (14) becomes sound-like.
+Magnetoplasmons. If external magnetic field Bis ap-
+plied perpendicularly to the plane of graphene the plas-
+mon spectra acquire new modes. The equation of motion
+(2) should now be modified to include the Lorentz force,
+˙J(r,t) =e2
+π¯h2|µ(x)|E(r,t)−ev2
+cµ(x)J×B.(18)
+The relative coefficient between electric and magnetic
+terms in this equation follows from the expression for
+the Lorentz force acting on a single particle. The last
+term has opposite sign for electrons and holes. Note that
+the frequency of cyclotron motion ωB(x) =ev2B/cµ(x)
+in graphene p-njunctions is position-dependent. The
+remaining equations (3)-(4) are intact in the presence of
+magnetic field. The boundary condition requires now the
+vanishing of the normal component of electric current at
+the boundary, rather than simply vanishing of the elec-
+tric field, as in Eq. (12). Eliminating JandEwe arrive
+at the generalization of equation (5),
+δρ(x)+2e2
+π/braceleftbigg
+q2Z −q
+ω(ωBZ)′−d
+dxZd
+dx/bracerightbigg
+×/integraldisplayd
+−ddx′δρ(x′)K0(|q||x−x′|) = 0,(19)4
+whereZ(x) =|µ(x)|/(ω2
+B(x)−ω2).
+The most interesting effect described by Eq. (19) is
+the appearance of a set of new modes, chiral magne-
+toplasmons, similar to those considered in Ref. [19] for
+conventional 2D electron systems with smooth bound-
+aries. To find their dispersion in strong magnetic fields,
+whenω≪ωB(x) (the exact condition is given below),
+one should retain only the second term in Eq. (19).
+Noticing that ( ωBZ)′=πl2
+Bρ′
+0(x)/e=πl2
+B/(l2
+Ed), where
+lB=/radicalbig
+¯hc/eBis the magnetic length, we arrive at the
+integral equation
+−2c
+Bq
+ωdρ0(x)
+dx/integraldisplayd
+−ddx′δρ(x′)K0(|q||x−x′|) =δρ(x).(20)
+SinceK0is positive, propagation of magnetoplasmons
+withq >0is quenched, indicative oftheir chiral property
+[20]. As seen from Eq. (20), the plasmon density δρ(x) is
+concentratedwhere ρ′
+0(x) isthestrongest. Thederivative
+of the charge density in field-induced junctions (11) fea-
+tures strong singularitynearthe edges of the flake. Thus,
+low-frequency magnetoplasmon spectrum is strongly de-
+pendent on the microscopic regularization of this singu-
+lar behavior and is, therefore, beyond the scope of the
+Thomas-Fermi approximation used throughout this pa-
+per.
+Thegate-induced junctions, however, allow a rather
+simple analytical description of these modes if we ap-
+proximate that ρ′
+0(x) =e/l2
+Edfor|x| ≤dandρ′
+0(x) = 0
+for|x|> d. The oscillating density δρ(x) then vanishes
+for|x|> d. The solution inside the strip, |x| ≤d, can
+be easily found for q≫1/d, where one can assume the
+range of integration in Eq. (20) to be infinite. The eigen-
+functions of Eq. (20) are simply given by sin[ q⊥(x+d)],
+with the values of q⊥=πn/2ddetermined from the con-
+dition,δρq(±d) = 0. The spectrum of magnetoplasmons
+is then found to be,
+ωn(q) =−2πe2l2
+B
+¯hl2
+Edq/radicalbig
+q2+π2n2/4d2, n= 1,2...(21)
+The magnetoplasmon spectrum (21) is derived under
+the assumption that magnetic field is strong, ωB(d)≫ω,
+which implies that lB≪lE. In order to neglect the first
+and third terms in the brackets in Eq. (19) one has to
+ensure that q≪(lE/lB)4/d. This condition might turn
+out to be more orless restrictivethan the hydrodynamics
+condition q≪1/lE, depending on the particular value of
+the ratio lB/lE. Note that the smallness of this ratio is
+not in contradiction to the non-quantized description of
+electron motion in magnetic filed. The latter is valid as
+long as the filling factor is large, eEd≫ωB(d), which
+means that lB≫l2
+E/d. For magnetic field ∼1T, and
+lB∼25nm, using the estimate below Eq. (10) that lE∼
+400nm we conclude that the width of the flake should
+exceedd >10µm. The magnetoplasmon modes (21) are
+∼(lB/lE)2slowerthan electrons. Note that these modesare undamped since single-particle excitations cannot be
+induced at frequencies below cyclotron frequency ωB.
+Conclusions . Graphene p-njunctions are among the
+most simple and promising applications of this material.
+Single-electron properties of p-njunctions have been ex-
+tensively studied. In the present paper we investigated
+their collective excitations both with and without mag-
+netic field. We anticipate that plasmon modes will be
+crucial for the optical response of graphene nanostruc-
+tures and realistic samples containing electron-hole pud-
+dles. High degree of experimental control should make
+them of special interest to nanoplasmonics and electron-
+ics. Among the most promising applications of plasmons
+inp-njunctions we envisage a possibility of a “plasmon
+transistor” [4]. In particular, by simply switching the
+direction of electric field from across the flake to along
+it (and back) the propagation of plasmons can be facil-
+itated (or prevented). In addition, as follows from the
+above Eqs. (1), (11), the plasmon velocity can be con-
+trolled with simple change in the magnitude of electric
+field. This is in a sharp contrast to plasmons in metal-
+lic nanostructures, whose spectra are typically fixed once
+devices are fabricated.
+Acknowledgments. Useful discussions with M. Raikh
+and O. Starykh are gratefully acknowledged. This
+work was supported by DOE, Grant No. DE-FG02-
+06ER46313. P.G.S. was supported by the SFB TR 12.
+[*] Present address: School of Physics, University of Exete r,
+EX4 4QL, U.K.
+[2] M. Wilson, Phys. Today 59, No. 1, 21 (2006).
+[3] A.K. Geim and K.S. Novoselov, Nature Mater. 6, 183
+(2007).
+[4] H.A. Atwater, Sci. Am. 296, 56 (2007).
+[5] S.A. Maier, Plasmonics: Fundamentals and Applications
+(Springer, New York, 2007).
+[6] F. Stern, Phys. Rev. Lett. 18, 546 (1967).
+[7] S. Gangadharaiah, A.M. Farid, and E.G. Mishchenko,
+Phys. Rev. Lett. 100, 166802 (2008).
+[8] O. Vafek, Phys. Rev. Lett. 97, 266406 (2006).
+[9] E.H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418
+(2007).
+[10] A.H. Castro Neto et al.,Rev. Mod. Phys. 81, 109 (2009).
+[11] J. Martin et al.,Nature Physics 4, 144 (2008).
+[12] J.R. Williams, L. DiCarlo, and C.M. Marcus, Science
+317, 638 (2007).
+[13] Rigorous derivation of Eq. (2) is based on the “rela-
+tivistic” stress energy-momentum tensor, see L.D. Lan-
+dau and E. M. Lifshitz, Fluid Mechanics , Butterworth-
+Heinemann, Oxford (1987), Ch. 15; M. Mueller, L. Fritz,
+S. Sachdev, and J. Schmalian, arXiv:0810.3657.
+[14] In the case of gate controlled junctions the image charg es
+induced at the gates should be included into Eq. (4).
+[15] T.A. Sedrakyan, E.G. Mishchenko, and M.E. Raikh,
+Phys. Rev. B 74, 235423 (2006).
+[16] P.G. SilvestrovandK.B. Efetov, Phys.Rev.B 77, 155436
+(2008).5
+[17] In addition even and odd solutions with n >0 have dif-
+ferent singular behavior at |ξ| ≪1:δρ(even)∼/radicalbig
+|ξ|,
+δρ(odd)∼sign(ξ)//radicalbig
+|ξ|. Atξ→ ±1 all solutions diverge
+asδρ∼1//radicalbig
+1−ξ2.
+[18] L.M. Zhang and M.M. Fogler, Phys. Rev. Lett. 100,116804 (2008).
+[19] I.L. Aleiner and L.I. Glazman, Phys. Rev. Lett. 72, 2935
+(1994).
+[20] V.A. Volkov and S. A. Mikhailov, JETP Lett. 42, 556
+(1985).
\ No newline at end of file
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@@ -0,0 +1,1189 @@
+arXiv:1001.0012v2  [astro-ph.EP]  20 Dec 2010Draft version May 20, 2018
+Preprint typeset using L ATEX style emulateapj v. 8/13/10
+THE STATISTICS OF ALBEDO AND HEAT RECIRCULATION ON HOT EXOPL ANETS
+Nicolas B. Cowan1,2, Eric Agol2,
+Draft version May 20, 2018
+ABSTRACT
+If both the day-side and night-side effective temperatures of a pla net can be measured, it is possible
+to estimate its Bond albedo, 0 < AB<1, as well as its day–night heat redistribution efficiency,
+0< ε <1. We attempt a statistical analysis of the albedo and redistribution efficiency for 24
+transiting exoplanets that have at least one published secondary e clipse. For each planet, we show
+how to calculate a sub-stellar equilibrium temperature, T0, and associated uncertainty. We then use
+a simple model-independent technique to estimate a planet’s effective temperature from planet/star
+flux ratios. We use thermal secondary eclipse measurements —tho se obtained at λ >0.8 micron—
+to estimate day-side effective temperatures, Td, and thermal phase variations —when available— to
+estimatenight-sideeffectivetemperature. Westronglyruleoutth e“nullhypothesis”ofasingle ABand
+εforall 24planets. If wealloweachplanet to havedifferent paramete rs,we find that lowBond albedos
+are favored ( AB<0.35 at 1σconfidence), which is an independent confirmation of the low albedos
+inferred from non-detection of reflected light. Our sample exhibits a wide variety of redistribution
+efficiencies. When normalized by T0, the day-side effective temperatures of the 24 planets describe
+a uni-modal distribution. The two biggest outliers are GJ 436b (abno rmally hot) and HD 80606b
+(abnormally cool), and these are the only eccentric planets in our sa mple. The dimensionless quantity
+Td/T0exhibits no trend with the presence or absence of stratospheric in versions. There is also no
+clear trend between Td/T0andT0. That said, the 6 planets with the greatest sub-stellar equilibrium
+temperatures ( T >2400 K) have low ε, as opposed to the 18 cooler planets, which show a variety
+of recirculation efficiencies. This hints that the very hottest trans iting giant planets are qualitatively
+different from the merely hot Jupiters. We propose an explanation o f this trend based on how a
+planet’s radiative and advective times scale with temperature: both timescales are expected to be
+shorter for hotter planets, but the temperature-dependance of the radiative timescale is stronger,
+leading to decreased heat recirculation efficiency.
+Subject headings: methods: data analysis — (stars:) planetary systems —
+1.INTRODUCTION
+Short-period exoplanets are expected to have atmo-
+spheric compositions and dynamics that differ signifi-
+cantly from Solar System giant planets3. These planets
+orbit∼100×closer to their host stars than Jupiter does
+from the Sun. As a result, they receive ∼104×more flux
+andexperiencetidalforces ∼106×strongerthanJupiter.
+In contrast to Jupiter, which releases roughly as much
+power in its interior as it receives from the Sun, short-
+period exoplanets have power budgets dictated by the
+flux they receive from their host stars. Roughly speak-
+ing, the stellar flux incident on a planet does one of two
+things: it is reflected back into space, or advected else-
+where on the planet and re-radiated at different wave-
+lengths. The physical parameters that describe these
+processes are the planet’s Bond albedo and redistribu-
+tion efficiency.
+1.1.Albedo
+1CIERA Fellow, Northwestern University, 2131 Tech Dr,
+Evanston, IL 60208
+email: n-cowan@northwestern.edu
+2Astronomy Department, University of Washington, Box
+351580, Seattle, WA 98195
+3For our purposes a “short period” exoplanet is one where the
+periastron distance is less than 0 .1 AU, regardless of its actual
+period, and regardless of mass, which may range from Neptune -
+sized to Brown Dwarf. They are all Class IV and V extrasolar
+giant planets in the scheme of Sudarsky et al. (2003).Giant planets in the Solar System have albedos greater
+than 50%because ofthe presenceofcondensedmolecules
+(H2O, CH 4, NH3, etc.) in their atmospheres. Planets
+with effective temperatures exceeding ∼400 K should be
+cloud free, leading to albedos of 0.05–0.4 (Marley et al.
+1999). If pressure-broadenedNa and K opacity is impor-
+tant at optical wavelengths (as it is for brown dwarfs,
+Burrows et al. 2000), then the Bond albedos of hot
+Jupiters may be less than 10% (Sudarsky et al. 2000).
+But the very hottest planets, the so-called class V extra-
+solar giant planets ( Teff>1500 K), might have very high
+albedosdue to a high silicate cloud layer(Sudarsky et al.
+2000). For a planet whose albedo is dominated by
+clouds (as opposed to Rayleigh scattering) the albedo
+depends on the composition and size of cloud particles
+(Seager et al. 2000).
+Earlyattempts to observe reflected light from exoplan-
+ets (Charbonneau et al. 1999; Collier Cameron et al.
+2002a; Leigh et al. 2003a,b; Rodler et al. 2008, 2010) in-
+dicated that they might not be as reflective as Solar Sys-
+tem gas giants (for a review, see Langford et al. 2010).
+Measurements of HD 209458b taken with the Cana-
+dian MOST satellite revealed a very low albedo ( <8%,
+Rowe et al.2008), andit hassincebeentakenforgranted
+that all short-period planets have albedos on par with
+that of charcoal.
+From the standpoint of the planet’s climate, the im-
+portant factor is not the albedo at any one wavelength,2 Cowan & Agol
+Aλ, but rather the integrated albedo, weighted by the in-
+cident stellar spectrum, known as the Bond albedo and
+denoted in this paper as AB. The relation between Aλ
+and the planet’s Bond albedo is not trivial. If the albedo
+is dominated by gray clouds, then the albedo at a sin-
+gle wavelength can indeed be extrapolated to obtain AB.
+For non-grayreflectance spectra, however, it is critical to
+measureAλat the peak emitting wavelength of the host
+startoobtainagoodestimateofthe planet’senergybud-
+get. For example, as pointed out in Marley et al. (1999),
+planets with identical albedo spectra, Aλ, mayhaveradi-
+cally different ABdepending on the spectraltype oftheir
+host stars.
+1.2.Redistribution Efficiency
+The first few measurements of hot Jupiter phase vari-
+ations showed signs that these planets are not all cut
+from the same cloth. Harrington et al. (2006) and
+Knutson et al. (2007a) quoted very different phase func-
+tion amplitudes for the υAndromeda and HD 189733
+systems. It was not clear whether the differences were
+intrinsic to the planets, however, because the data
+were taken with different instruments, at different wave-
+lengths, and with very different observation schemes (in
+any case, subsequent re-analysis of the original data and
+newly aquired Spitzerobservations of υAndromeda b
+paint a completely different picture of that system:
+Crossfield et al. 2010).
+The uniform study presented in Cowan et al. (2007),
+on the other hand, showed that HD 179949b and
+HD209458bexhibit significantlydifferentdegreesofheat
+recirculation, confirming suspicions. But it was not clear
+whether hot exoplanets were uni-modal or bi-modal in
+redistribution: are HD 179949b and HD 209458b end-
+members of a single distribution, or prototypes for two
+fundamentally different sorts of exoplanets?
+The presence or lack of a stratospheric tempera-
+ture inversion (Hubeny et al. 2003; Fortney et al. 2006;
+Burrows et al. 2007, 2008; Zahnle et al. 2009) on the
+day-sides of exoplanets has been invoked to explain
+a purported bi-modality in recirculation efficiency on
+hot Jupiters (Fortney et al. 2008). The argument, sim-
+ply put, is that optical absorbers high in the atmo-
+sphere of extremely hot Jupiters (equilibrium temper-
+atures greater than ∼1700 K) would absorb incident
+photons where the radiative timescales are short, mak-
+ingit difficult forthese planets torecirculateenergy. The
+most robust detection of this temperature inversionis for
+HD 209458b (Knutson et al. 2008), but this planet does
+not exhibit a large day-night brightness contrast at 8 µm
+(Cowan et al. 2007). So while temperature inversions
+seem to exist in the majority of hot Jupiter atmospheres
+(Knutson et al. 2010), their connection to circulation ef-
+ficiency —if any— is not clear.
+1.3.Outline of Paper
+It has been suggested (e.g., Harrington et al. 2006;
+Cowan et al. 2007) that observations of secondary
+eclipses and phase variations each constrain a combina-
+tion of a planet’s Bond albedo and circulation efficiency.
+But observations —even phase variations— at a single
+waveband do little to constrain a planet’s energy bud-
+get. In this work we show how observations in differentwavebands and for different planets can be meaningfully
+combined to estimate these planetary parameters.
+In§2 we introduce a simple model to quantify the
+day-side and night-side energy budget of a short-period
+planet, and show how a planet’s Bond albedo, AB, and
+redistribution efficiency, ε, can be constrained by ob-
+servations. In §3 we use published observations of
+24 transiting planets to estimate day-side and —where
+appropriate—night-sideeffective temperatures. We con-
+struct a two-dimensionaldistribution function in ABand
+εin§4. We state our conclusions in §5.
+2.PARAMETERIZING THE ENERGY BUDGET
+2.1.Incident Flux
+Short-period planets have a power budget entirely dic-
+tated by the flux they receive from their host star,
+which dwarfs tidal heating or remnant heat of forma-
+tion. Following Hansen (2008), we define the equi-
+librium temperature at the planet’s sub-stellar point:
+T0(t) =Teff(R∗/r(t))1/2, whereTeffandR∗are the star’s
+effective temperature and radius, and r(t) is the planet–
+star distance (for a circular orbit ris simply equal to the
+semi-major axis, a). For shorthand, we define the geo-
+metrical factor a∗=a/R∗, which is directly constrained
+by transit lightcurves (Seager & Mall´ en-Ornelas 2003).
+The incident flux on the planet is given by Finc=
+1
+2σBT4
+0, and it is significant that this quantity has some
+associated uncertainty. For a planet on a circular orbit,
+the uncertainty in T0=Teff/√a∗is related —to first
+order— to the uncertainties in the host star’s effective
+temperature, and the geometrical factor:
+σ2
+T0
+T2
+0=σ2
+Teff
+T2
+eff+σ2
+a∗
+4a2∗. (1)
+For a planet with non-zero eccentricity, T0varies with
+time, but we are only concerned with its value at su-
+perior conjunction: secondary eclipse occurs at superior
+conjunction, when we are seeing the planet’s day-side.
+At that point in the orbit, the planet–star distance is
+rsc=a(1−e2)/(1−esinω), whereeandωare the
+planet’s orbital eccentricity and argument of periastron,
+respectively.
+For planets with non-zero eccentricity, the uncertainty
+inT0is given by
+σ2
+T0
+T2
+0=σ2
+Teff
+T2
+eff+σ2
+a∗
+4a2∗+/parenleftBig
+e2cos2ω
+1−e2/parenrightBig
+σ2
+ecosω
++/parenleftBig
+esinω
+1−e2−1
+2(1−esinω)/parenrightBig
+σ2
+esinω,(2)
+whereσecosωandσesinωarethe observationaluncertain-
+ties in the two components of the planet’s eccentricity4.
+2.2.Emergent Flux
+At secondary eclipse, and in the absence of albedo or
+energy circulation, the equilibrium temperature of a re-
+gion on the planet depends on the normalized projected
+4This formulation is preferable to an error estimate based on σe
+andσω, because the eccentricity and argument of periastron are
+highlycorrelated inorbitalfits. Thatsaid, the uncertaint iesσecosω
+andσesinωare often not included in the literature, in which case
+we use a slightly different —and more conservative— formulat ion
+of the error budget using σeandσω.Albedo and Heat Recirculation on Hot Exoplanets 3
+distance,γ, from the center of the planetary disc as
+T(γ) =T0(1−γ2)1/8. The thermal secondary eclipse
+depth in this limit is given by:
+Fday
+F∗=/parenleftbiggRp
+R∗/parenrightbigg2/parenleftbigghc
+λkT0/parenrightbigg8/parenleftBig
+ehc/λkT∗
+b−1/parenrightBig
+×/integraldisplay(λkT0/hc)8
+0dx
+exp(x−1/8)−1, (3)
+whereT∗
+bis the brightness temperature of the star at
+wavelength λ.
+In the no-circulation limit, then, the day-side emer-
+gent spectrum is not exactly that of a blackbody, even
+if each annulus has a blackbody spectrum. But these
+differences are not important for the present study, since
+we are concerned with bolometric flux. By integrating
+Equation 3 over λ, one obtains the effective tempera-
+tureoftheday-sideintheno-albedo,no-circulationlimit:
+Tε=0= (2/3)1/4T0(see also Burrows et al. 2008; Hansen
+2008). Indeed, treatingtheplanet’sday-sideasauniform
+hemisphere emitting at this temperature gives nearly the
+same wavelength dependence as the more complex Equa-
+tion 3. The Tε=0temperatures for our sample of 24 tran-
+siting planets are shown in Table 1. These set the max-
+imum possible day-side effective temperature we should
+expect to measure.
+The integrated day-side flux in the general —non-zero
+circulation— case is more subtle: heat may be trans-
+ported to the planet’s night-side, and/or to its poles. In
+this paper we neglect the E-W asymetry in the planet’s
+temperature map due to zonal flows and hence phase
+offsets in the thermal phase variations. Under this as-
+sumption, the day-night temperature contrast can more
+directly be extracted from the observed thermal phase
+variations.
+In practice, manystudies haveadopted asingle param-
+eter to represent bothzonal and meridional transport. It
+is instructive to consider the apparent day-side effective
+temperatures in variouslimits: uniform day-sidetemper-
+ature andT= 0 on the night-side (this is often referred
+to as the planet’s “equilibrium temperature”): Tequ=
+(1/2)1/4T0; in the case of perfect longitudinal transport
+but no latitudinal transport: Tlong= (8/(3π2))1/4T0;
+and in the limit of a uniform temperature everywhere
+on the planet: Tuni= (1/4)1/4T0.
+Comparing the apparent day-side temperatures in the
+three limits of circulation above leads to the following
+simple parametrization of the day-side effective temper-
+ature in terms of the planetary albedo, AB, and circula-
+tion efficiency, ε:
+Td=T0(1−AB)1/4/parenleftbigg2
+3−5
+12ε/parenrightbigg1/4
+,(4)
+where 0< ε <1. Note that εis related to —but dif-
+ferent from— the ǫused in (Cowan & Agol 2010). The
+former is merely a parametrization of the observed disk-
+integrated effective temperature, while the latter, which
+can take values from 0 to ∞, is a precisely defined ratio
+of radiative and advective timescales. The ǫ= 0 case is
+precisely equal to the ε= 0 case, while the ǫ→ ∞limit
+is equivalent to ǫ≈0.95.
+Our definition of εis similar to the Burrows et al.(2006) definition of Pnandyieldsthe sameno-circulation
+limit. But our ε= 1 limit produces a lower day-side
+brightness than the Pn= 0.5 limit, because we as-
+sume that the planet’s day-side has a uniform tempera-
+ture distribution in that limit (for a discussion of differ-
+ent redistribution parameterizations, see the appendix of
+Spiegel & Burrows 2010).
+In reality, efficient longitudinal transport (read: fast
+zonalwinds) mayleadtomorebandingandthereforeless
+efficient latitudinal transport. So one could argue that
+in the limit of perfect day-night temperature homoge-
+nization, both the day and night apparent temperatures
+should beTd= (8/(3π2))1/4T0, in between the Burrows
+et al. value of Td= (1/3)1/4T0and that suggested by
+our parameterization, Td= (1/4)1/4T0. At moderate
+day-night recirculation efficiencies, however, there is a
+good deal of latitudinal transport (I. Dobbs-Dixon, priv.
+comm.), so implicitly assuming a constant T∝cos1/4
+latitudinal dependence —as done by Burrows et al.— is
+not founded, either. The bottom line is that any single-
+parameter implementation of advection is incapable of
+capturing the real complexities involved, but longitudi-
+nal transport is the dominant factor in determining day
+and night effective temperatures.
+Not withstanding the subtleties discussed above and
+noting that cooling tends to latitudinaly homogenize
+night-side temperatures (Cowan & Agol 2010), we get a
+night-side temperature of:
+Tn=T0(1−AB)1/4/parenleftBigε
+4/parenrightBig1/4
+. (5)
+Note thatTdandTnare the equator-weighted tempera-
+tures of their respective hemispheres (ie, as seen by an
+edge-on viewer). As such, they will tend to be slightly
+higher than the hemisphere-averaged temperature, ex-
+cept in the ε= 1 limit. This is also why the quantity
+T4
+d+T4
+nis still a weak function of ε.
+Fig. 1.— Different kinds of idealized observations constrain the
+Bond albedo, ABand circulation efficiency, ε, differently. A mea-
+surement of the secondary eclipse depth at optical waveleng ths is
+a measure of albedo (solid line). A secondary eclipse depth a t
+thermal wavelengths gives a joint constraint on albedo and r ecir-
+culation (dotted line). A measurement of the night-side effe ctive
+temperature from thermal phase variations yields a constra int (the
+dashed line) nearly orthogonal to the day-side measurement .
+In Figure 1 we show how different kinds of observa-4 Cowan & Agol
+tions constrain ABandε. For this example, we chose
+constraints consistent with AB= 0.2 andε= 0.3. The
+solid line is a locus of constant AB; the dotted line is
+the locus of constant Td/T0; the dashed line is a lo-
+cus of constant Tn/T0. From this figure it is clear that
+the measurements complement each other: measuring
+two of the three quantities (Bond albedo, effective day-
+side or night-side temperatures) uniquely determines the
+planet’s albedo and circulation efficiency. When obser-
+vations have some associated uncertainty, they define a
+swath through the AB–εplane.
+3.ANALYSIS
+3.1.Planetary & Stellar Data
+We begin by considering all the photometric obser-
+vations of short-period exoplanets published through
+November 2010, summarized in Table 1. We have dis-
+carded photometric observations of non-transiting plan-
+ets because of their unknown radius and orbital inclina-
+tion5. This leaves us with 24 transiting exoplanets for
+which there are observations in at least one waveband
+at superior conjunction, and in some cases in multiple
+wavebands and at multiple planetary phases.
+Stellar and planetary data are taken from the Ex-
+oplanet Encyclopedia (exoplanet.eu), and references
+therein. We repeated parts of the analysis with the
+Exoplanet Data Explorer database (exoplanets.org) and
+found identical results, within the uncertainties. When
+the stellar data are not available, we have assumed typi-
+cal parameters for the appropriate spectral class, and so-
+lar metallicity. Insofar as we are only concerned with the
+broadband brightnesses of the stars, our results should
+not depend sensitively on the input stellar parameters.
+Knowing the stars’ Teff, loggand [Fe/H], we
+use the PHOENIX/NextGen stellar spectrum grids
+(Hauschildt et al. 1999) to determine their brightness
+temperatures at the observed bandpasses. At each wave-
+band for which eclipse or phase observations have been
+obtained, we determine the ratio of the stellar flux to the
+blackbodyfluxatthatgridstar’s Teff. Wethenapplythis
+factor to the Teffof the observed star.
+It is worth noting that the choice of stellar model leads
+to systematic uncertainties in the planetary brightness
+that are of order the photometric uncertainties. For
+example, Christiansen et al. (2010) use stellar models
+for HAT-P-7 from Kurucz (2005), while we use those
+of Hauschildt et al. (1999). The resulting 8 µm bright-
+ness temperatures for HAT-P-7b differ by as much as
+600 K, or slightly more than 1 σ. Our uniform use
+of Hauschildt et al. (1999) models should alleviate this
+problem, however.
+3.2.From Flux Ratios to Effective Temperature
+The planet’s albedo and recirculation efficiency gov-
+ern its effective day-side and night-side temperatures, Td
+andTn, respectively. Observationally, we can only mea-
+sure the brightness temperature, ideally at a number of
+different wavelengths: Tb(λ). If one knew, a priori, the
+5For completeness, these are: τ-Bootis b, υ-Andromeda b,
+51 Peg b, Gl 876d, HD 75289b, HD 179949b and HD 46375b
+(Charbonneau et al. 1999; Collier Cameron et al. 2002b;
+Leigh et al. 2003a,b; Harrington et al. 2006; Cowan et al. 200 7;
+Seager & Deming 2009; Crossfield et al. 2010; Gaulme et al. 201 0)emergent spectrum of a planet, one could trivially con-
+vert a single brightness temperature to an effective tem-
+perature. Alternatively, if observations were obtained at
+a number of wavelengths bracketing the planet’s black-
+body peak, it would be possible to estimate the planet’s
+bolometric flux and hence its effective temperature in a
+model-independent way (e.g., Barman 2008).
+We adopt the latter empirical approach of converting
+observed flux ratios into brightness temperatures, then
+using these to estimate the planet’s effective tempera-
+ture. The secondary eclipse depth in some waveband di-
+vided by the transit depth is a direct measureofthe ratio
+of the planet’s day-side intensity to the star’s intensity
+at that wavelength, ψ(λ). Knowing the star’s brightness
+temperature at a given wavelength, it is possible to com-
+pute the apparent brightness temperature of the planet’s
+day side:
+Tb(λ) =hc
+λk/bracketleftbigg
+log/parenleftbigg
+1+ehc/λkT∗
+b(λ)−1
+ψ(λ)/parenrightbigg/bracketrightbigg−1
+.(6)
+On the Rayleigh-Jeans tail, the fractional uncertainty
+in the brightness temperature is roughly equal to the
+fractional uncertainty in the eclipse depth; on the Wien
+tail, the fractional error on brightness temperature can
+be smaller because the flux is very sensitive to tempera-
+ture.
+By the same token, a secondary eclipse depth and
+phase variation amplitude at a given wavelength can be
+combined to obtain a measure of the planet’s night-side
+brightness temperature at that waveband.
+Since the albedo and recirculation efficiency of the
+planet are not known ahead of time, it is not immedi-
+atelyobviouswhich wavelengthsaresensitiveto reflected
+light and which are dominated by thermal emission. For
+each planet, we compute the expected blackbody peak if
+the planet has no albedo and no recirculation of energy,
+λε=0= 2898/Tε=0µm. Insofar as real planets will have
+non-zero albedo and non-zero recirculation, the day side
+should never reach Tε=0, and the actual spectral energy
+distributionwillpeakatslightlylongerwavelengths. The
+coolest planet in our sample, Gl 436b, would exhibit a
+blackbody peak at λε=0= 3.1µm, while the hottest
+planet we consider, WASP-12b, has λε=0= 0.9µm.
+In practice this means that ground-based near-IR and
+space-based mid-IR (e.g., Spitzer) observations are as-
+sumed to measure thermal emission, while space-based
+optical observations (MOST, CoRoT, Kepler) may be
+contaminated by reflected starlight.
+In Figure2, wedemonstratetwo alternativetechniques
+to convert an array of brightness temperatures, Tb(λ),
+into an estimate of a planet’s effective temperature, Teff.
+The solid black line shows a model spectrum of ther-
+mal emission from Fortney et al. (2008), with an ef-
+fective temperature of Teff= 1941 K shown with the
+black dashed line. The expected blackbody peak of
+the planet is marked with a vertical dotted line. The
+red points are the expected brightness temperatures in
+the J, H, and K sbands (crosses), as well as the IRAC
+(asterisks) and MIPS (diamond) instruments on Spitzer
+(Fazio et al. 2004; Rieke et al. 2004; Werner et al. 2004).
+Since the majority of the observations of exoplanets have
+been obtained with SpitzerIRAC, we focus on estimat-
+ingTeffbasedonlyon brightness temperatures in thoseAlbedo and Heat Recirculation on Hot Exoplanets 5
+Fig. 2.— The solid black line shows a model spectrum from
+Fortney et al. (2008) including only thermal emission (ie: n o re-
+flected light). The planet’s effective temperature is shown w ith the
+black dashed line, while the expected wavelength of the blac kbody
+peak of the planet is marked with a black dotted line. The red
+points show the expected brightness temperatures in the J, H , and
+Ksbands (crosses), as well as the IRAC (asterisks) and MIPS (di a-
+mond) instruments on Spitzer. The linear interpolation technique
+described in the text is shown with the red line.
+four bandpasses.
+Wien Displacement: The first approach is to simply
+adopt the brightness temperature of the bandpass clos-
+est to the planet’s blackbody peak (the black dotted
+line). If only the four IRAC channels are available, the
+best one can do is the 3.6 µm measurement, yielding
+Teff= 1925 K. There is —however— some subtlety in
+estimating the peak wavelength, as this is dependent on
+knowing the planet’s temperature (and hence ABandε)
+a priori.
+Linear Interpolation: The linear interpolation tech-
+nique, shown with the red line in Figure 2, obviates the
+need for an estimate of the planet’s temperature. The
+brightness temperature is assumed to be constant short-
+ward of the shortest- λobservation, and longward of the
+longest-λobservation. Between bandpasses, the bright-
+ness temperature changes linearly with λ. As long as
+the various brightness temperatures do not differ grossly
+from one another, this technique implicitly gives more
+weight to observations near the hypothetical blackbody
+peak. The bolometric flux of this “model” spectrum is
+then computed, and admits a single effective tempera-
+ture, which is Teff= 1927 K for the current example.
+Since we hope to apply our routine to planets with well
+sampled blackbody peaks, we adopt the linear interpola-
+tion technique, as it can make use of multiple brightness
+temperature estimates near the peak.
+Thetwotechniquesdescribedaboveproducesimilaref-
+fective temperatures, though —unsurprisingly— neither
+gives precisely the correct answer. But these system-
+atic errors are comparable or smaller than the photo-
+metric uncertainty in observations of individual bright-
+ness temperatures (see Table 1). The best IR observa-
+tions for the nearest, brightest planetary systems (e.g.,
+HD 189733b and HD 209458b) lead to observational un-
+certainties of approximately 50 K in brightness temper-
+ature. For many planets, the uncertainty is 100–200 K.
+By that metric, either the Wien displacement or the lin-
+ear interpolation routines give adequate estimates of the
+effective temperature, with errors of 16 K and 14 K, re-spectively.
+Wemakeamorequantitativeanalysisofthesystematic
+uncertainties involved in the Linear Interpolation tem-
+perature estimates as follows. We produce 8800 mock
+data sets: 100 realizations for 11 models and data in
+up to 8 wavebands (J, H, K, IRAC, MIPS; Since this nu-
+mericalexperiment choosesrandom bands from the eight
+available, the results should not be very different if ad-
+ditional wavebands are considered). We run our Linear
+Interpolation technique on each of these and plot in Fig-
+ure 3 the estimated day-side temperature normalized by
+the actual model effective temperature versus the num-
+ber of wavebands used in the estimate. The temperature
+estimates cluster near Test/Teff= 1, indicating that the
+technique is not significantly biased. The scatter in es-
+timates decreases as more wavebands are used, from a
+standard deviation of 7.6% if only a single brightness
+temperature is used, down to 2.4% if photometry is ac-
+quired in eight bands. We incorporate this systematic
+error into our analysis by adding it in quadrature to
+the observational uncertainties described in the follow-
+ing paragraph. This has the desirable effect that planets
+with fewer observations have a larger systematic uncer-
+tainty on their effective temperature.
+Fig. 3.— The Linear Interpolation technique for estimating day-
+side effective as tested on a suite of eleven hot Jupiter spect ral
+models provided by J.J. Fortney. The y-axis shows the estima ted
+day-side effective temperature normalized by the actual mod el ef-
+fective temperature. The x-axis represents the number of br ight-
+ness temperatures used in the estimate. Each color correspo nds to
+one of the eleven models used in the comparison. The black err or
+bars represent the standard deviation in the normalized tem pera-
+ture estimates.
+Inpractice,wewouldliketopropagatethephotometric
+uncertainties to the estimate of Teff. For the Wien Dis-
+placement technique, this uncertainty propagates triv-
+ially to the effective temperature. For the linear inter-
+polation technique, a Monte Carlo can be used to esti-
+mate the uncertainty in Teff: the input eclipse depths
+are randomly shifted 1000 times in a manner consistent
+with their photometric uncertainties —assuming Gaus-
+sianerrors—andtheeffectivetemperatureisrecomputed
+repeatedly. Thescatterintheresultingvaluesof Teffpro-
+vides an estimate of the observational uncertainty in the
+parameter, to which we add in quadrature the estimate
+ofsystematicerrordescribedabove. The resultinguncer-
+tainties are listed in Table 1. These uncertainties should6 Cowan & Agol
+be compared to the uncertainties in Tε=0(also listed in
+Table 1), which are computed using the uncertainty in
+the star’s properties and the planet’s orbit.
+There are two practical issues with the linear interpo-
+lation temperature estimation technique. In some cases,
+onlyupperlimitshavebeenobtained, thereforeonecould
+setψ= 0, with the appropriate1-sigmauncertainty. But
+this approach leads to huge uncertainties in Tefffor plan-
+ets with a secondary eclipse upper-limit near their black-
+body peak. Instead of “punishing” these planets, we opt
+to not use upper-limits (though for completeness we in-
+clude them in Table 1). Secondly, when multiple mea-
+surements of an eclipse depth have been published for
+a given waveband, we use the most recent observation,
+indicated with a superscript “ e” in Table 1. In all cases
+these observations either explicitly agree with their older
+counterpart, or agree with the re-analyzed older data.
+4.RESULTS
+4.1.Looking for Reflected Light
+For each planet, we use thermal observations (essen-
+tially those in the J, H, K s, andSpitzerbands) to es-
+timate the planet’s effective day-side temperature, Td,
+and —when phase variations are available— Tn. These
+values are listed in Table 1. In five cases (CoRoT-
+1b, CoRoT-2b, HAT-P-7b, HD 209458b, TrES-2b), sec-
+ondary eclipses and/or phase variations have been ob-
+tained at optical wavelengths. Such observations have
+the potential to directly constrain the albedo of these
+planets. One approach is to adopt the Tdfrom thermal
+observations and calculate the expected contrast ratio at
+optical wavelengths, under the assumption of blackbody
+emission (see also Kipping & Bakos 2010). Insofar as
+the observed eclipse depths are deeper than this calcu-
+lated depth, one can invoke the contribution of reflected
+light and compute a geometric albedo, Ag. If one treats
+the planet as a uniform Lambert sphere, the geometric
+albedo is related to the spherical albedo at that wave-
+length byAλ=3
+2Ag. These values are listed in Table 1.
+But reflected light is not the only explanation for an
+unexpectedly deep optical eclipse. Alternatively, the
+emissivity of the planets may simply be greater at op-
+tical wavelengths than at mid-IR wavelengths, in agree-
+mentwith realisticspectralmodelsofhotJupiters, which
+predict brightness temperatures greater than Teffon the
+Wien tail (see, for example, the Fortney et al. model
+showninFigure2, whichdoesnotincludereflectedlight).
+Note that this increasein emissivityshould occurregard-
+less of whether or not the planet has a stratosphere: by
+definition, the depth at which the optical thermal emis-
+sion is emitted is the depth at which incident starlight
+is absorbed, which will necessarily be a hot layer —
+assuming the incident stellar spectrum peaks in the op-
+tical.
+Determining the albedo directly (ie: by observing re-
+flected light) can be difficult for short period planets,
+because there is no way to distinguish between reflected
+and re-radiated photons. The blackbody peaks of the
+star and planet often differ by less than a micron. There-
+fore, unlike Solar System planets, these worlds do not
+exhibit a minimum in their spectral energy distribution
+between the reflected and thermal peaks. The hottest
+—and therefore most ambiguous case— of the five tran-siting planets with optical constraints is HAT-P-7b. If
+one takes the mid-IR eclipse depths at face value, the
+planet has a day-side effective temperature of ∼2000 K.
+When combined with the Kepler observations, one com-
+putesanalbedoofgreaterthan50%. Thelargeday-night
+amplitude seen in the Kepler bandpass is then simply
+due to the fact that the planet’s night-side reflects no
+starlight, and the cool day-side can be attributed to high
+ABand/orε. If, on the other hand, one takes the op-
+tical flux to be entirely thermal in origin ( Aλ= 0), the
+day-side effective temperature is ∼2800 K. This is very
+close to that planet’s Tε=0, leaving very little power left
+for the night-side, again explaining the large day-night
+contrast observed by Kepler. The truth probably lies
+somewhere between these two extremes, but in any case
+this degeneracy will be neatly broken with Warm Spitzer
+observations: the two scenarios outlined above will lead
+to small and large thermal phase variations, respectively.
+It is telling that the only optical measurement in Table 1
+that is unanimously considered to constrain albedo —
+and not thermal emission— is the MOST observations
+of HD 209458b (Rowe et al. 2008), the coolest of the five
+transiting planets with optical photometric constraints.
+The bottom line is that extracting a constraint on re-
+flected light from optical measurements of hot Jupiters is
+best done with a detailed spectral model. But even when
+reflectedlightcanbedirectlyconstrained,convertingthis
+constraint on Aλinto a constraint on ABalso requires
+detailedknowledgeofboththestarandtheplanet’sspec-
+tral energy distributions, making for a model-dependent
+exercise.
+4.2.Populating the AB-εPlane
+Setting aside optical eclipses and direct measurements
+of albedo, we may use the rich near- and mid-IR data to
+constrain the Bond albedo and redistribution efficiency
+of short-period giant planets. We define a 20 ×20 grid in
+ABandεand use Equations 4 & 5 to calculate the nor-
+malized day-side and night-side effective temperatures,
+Td/T0andTn/T0, at each grid point, ( i,j). For each
+planet, we have an observational estimate of the day-side
+effective temperature, and in three cases we also have an
+estimate of the night-side effective temperature (as well
+as associated uncertainties).
+We first verifywhether ornot the observationsarecon-
+sistent with a single ABandε. To evaluate this “null
+hypothesis”, we compute the usual χ2=/summationtext24
+i=1(model−
+data)2/error2at each grid point. We use only the esti-
+mates of day-side and (when available) night-side effec-
+tive temperatures to calculate the χ2, giving us 27-2=25
+degreesoffreedom. The“best-fit”has χ2= 132(reduced
+χ2= 5.3), so the current observations strongly rule out
+a single Bond albedo and redistribution efficiency for all
+24 planets.
+For 21 of the 24 planets considered here, we construct
+a two-dimensional distribution function for each planet
+as follows:
+PDF(i,j) =1/radicalbig
+2πσ2
+de−(Td−Td(i,j))2/(2σd)2.(7)
+This defines a swath through parameter space with the
+same shape as the dotted line in Figure 1.
+For the three remaining planets (HD 149026b,Albedo and Heat Recirculation on Hot Exoplanets 7
+HD 189733b, HD 209458b), phase variation measure-
+ments help break the degeneracy:
+PDF(i,j) =1√
+2πσ2
+de−(Td−Td(i,j))2/(2σd)2
+×1√
+2πσ2ne−(Tn−Tn(i,j))2/(2σn)2.(8)
+Fig. 4.— The global distribution function for short-period exo-
+planets in the AB–εplane. The gray-scale shows the sum of the
+normalized probability distribution function for the 24 pl anets in
+our sample. The data mostly consist of infrared day-side flux es,
+leading to the dominant degeneracy (see first the dotted line in
+Figure 1).
+We create a two-dimensional normalized probability
+distribution function (PDF) for each planet, then add
+these together to create the global PDF shown in Fig-
+ure 4. This is a democratic way of representing the data,
+since each planet’s distribution contributes equally.
+In Figures 5 and 6 we show the distribution functions
+for the albedo and circulation of the 24 planets in our
+sample,obtainedbymarginalizingtheglobalPDFofFig-
+ure 4 over either ABorε.
+Fig. 5.— The solid black line shows the projection of the 2-
+dimensional probability distribution function (the gray- scale of
+Figure 4) projected onto the ε-axis. The dashed line shows the
+ε-distribution if one requires that all planets have Bond alb edos
+less than 0.1; under this assumption, we see hints of a bimoda l
+distribution in heat circulation efficiency.Fig. 6.— The solid black line shows the projection of the 2-
+dimensionalprobabilitydistributionfunction (the gray- scale ofFig-
+ure 4) projected onto the AB-axis. The cumulative distribution
+function (not shown) yields a 1 σupper limit of AB<0.35.
+The solid line in Figure 5 shows no evidence of bi-
+modality in heat redistribution efficiency, although there
+is a wide range of behaviors. The dashed line in Figure 5
+shows theε-distribution if one requires the albedo to be
+low,AB<0.1. There are then many high-recirculation
+planets, since advection is the only way to depress the
+day-side temperature in the absence of albedo. Inter-
+estingly, the dashed line doesshow tentative evidence of
+two separate peaks in ε: if short-period giant planets
+have uniformly low albedos, then there appear to be two
+modes of heat recirculation efficiency. We revisit this
+idea below.
+Figure 6 shows that planets in this sample are consis-
+tent with a low Bond albedo. Note that this constraint
+is based entirely on near- and mid-infrared observations,
+and is thus independent from the claims of low albedo
+based on searches for reflected light (Rowe et al. 2008,
+and references therein). Furthermore, this is a constraint
+on the Bond albedo, rather than the albedo in any lim-
+ited wavelength range.
+In Figure 7 we plot the dimensionless day-side effec-
+tive temperature, Td/T0, against the maximum expected
+day-side temperature, Tε=0. The cyan asterisks in Fig-
+ure 7 show the four hot Jupiters without temperature
+inversions, while most of the remaining planets have in-
+versions (Knutson et al. 2010). The presence or absence
+of an inversion does not appear to affect the efficiency of
+day–night heat recirculation.
+Planets should lie below the solid red line in Figure 7,
+which denotes Tε=0= (2/3)1/4T0. Of the 24 planets in
+our sample, only one (Gl 436b) has a day-side effective
+temperature significantly above the Tε=0limit6. This
+planet is by far the coolest in our sample, it is on an ec-
+centric orbit, and observations indicate that it may have
+a non-equilibrium atmosphere (Stevenson et al. 2010).
+There is no reason, on the other hand, that planets
+shouldn’t lie below the red dotted line in Figure 7:
+all it would take is non-zero Bond albedo. That said,
+only 3 of the 24 planets we consider are in this region,
+6This is driven by the abnormally high 3.6 micron brightness
+temperature; including the 4.5 micron eclipse upper limit d oes not
+significantly change our estimate of this planet’s effective temper-
+ature.8 Cowan & Agol
+Fig. 7.— The dimensionless day-side effective temperature,
+Td/T0, plotted against the maximum expected day-side temper-
+ature,Tε=0. The red lines correspond to three fiducial limits of
+recirculation, assuming AB= 0: no recirculation (solid), uniform
+day-hemisphere (dashed), and uniform planet (dotted). The gray
+points indicate the default values (using only observation s with
+λ >0.8 micron) for the four planets whose optical eclipse depths
+may be probing thermal emission rather than just reflected li ght
+(from left to right: TrES-2b, CoRoT-2b, CoRoT-1b, HAT-P-7b ).
+For these planets we have here elected to include optical mea sure-
+ments in our estimate of the day-side bolometric flux and effec tive
+temperature, shown in black. The cyan asterisks denote thos e hot
+Jupiters known notto have a stratospheric inversion according
+to (Knutson et al. 2010). They are, from left to right: TrES-1 b,
+HD 189733b, TrES-3b, WASP-4b. The two red x’s denote the ec-
+centric planets in our sample, which are also the two worst ou tliers.
+with the greatest outlier being HD 80606b, a planet on
+an extremely eccentric orbit with superior conjunction
+nearly coinciding with periastron. As such, it is likely
+that much of the energy absorbed by the planet at that
+point in its orbit performs mechanical work (speeding up
+winds, puffingupthe planet, etc. SeealsoCowan & Agol
+2010) rather than merely warming the gas. Gl 436b and
+HD 80606b are denoted by red x’s in Figure 7.
+The gray points in Figure 7 indicate the default val-
+ues (using only observationswith λ>0.8 micron) for the
+four planets whose optical eclipse depths may be probing
+thermal emission rather than just reflected light (from
+left to right: TrES-2b, CoRoT-2b, CoRoT-1b, HAT-
+P-7b). For these planets we have here elected to use
+all available flux ratios (including optical observations
+potentially contaminated by reflected light) to estimate
+the day-side bolometric flux and effective temperature,
+shown as black points in Figure 7.
+If one takes these day-side effective temperature es-
+timates at face value, it appears that the planets with
+Tε=0<2400 K exhibit a wide-variety of redistribution
+efficiencies and/or Bond albedos, but are consistent with
+AB= 0. It is worth noting that many of the best char-
+acterized planets in this region have Td/T0≈0.75, and
+this accounts for the sharp peak in the dotted line of Fig-
+ure 5 atε= 0.75. The hottest 6 planets, on the other
+hand, have uniformly high Td/T0, indicating that they
+have both low Bond albedo andlow redistribution effi-
+ciency. These planets must not have the high-altitude,
+reflective silicate clouds hypothesized in Sudarsky et al.
+(2000). But this conclusion is dependent on how one
+interprets the Keplerobservations of HAT-P-7b: if the
+large optical flux ratio is due to reflected light, then this
+planet is cooler than we think, and even the hottest tran-siting planets exhibit a variety of behaviors.
+5.SUMMARY & CONCLUSIONS
+We have described how to estimate a planet’s incident
+power budget ( T0), where the uncertainties are driven by
+the uncertainties in the host star’s effective temperature
+and size, as well as the planet’s orbit. We then described
+a model-independent technique to estimate the effective
+temperature of a planet based on planet/star flux ra-
+tiosobtained at variouswavelengths. When the observed
+day-side and night-side effective temperatures are com-
+pared, one can constrain a combination of the planet’s
+Bond albedo, AB, and its recirculation efficiency, ε. We
+applied this analysis on 24 known transiting planets with
+measured infrared eclipse depths.
+Our principal results are:
+1. Essentially all of the planets are consistent with low
+Bond albedo.
+2. We firmly rule out the “null hypothesis”, whereby all
+transiting planets can be fit by a single ABandε. It
+is not immediately clear whether this stems from differ-
+ences in Bond albedo, recirculation efficiency, or both.
+3. In the few cases where it is possible to unambiguously
+infer an albedo based on optical eclipse depths, they are
+extremely low, implying correspondingly low Bond albe-
+dos (<10%). If one adopts such low albedos for all
+the planets in our sample, the discrepancies in day-side
+effective temperature must be due to differences in recir-
+culation efficiency.
+4. These differences in recirculation efficiency do not
+appear to be correlated with the presence or absence of
+a stratospheric inversion.
+5. Planets cooler than Tε=0= 2400 K exhibit a wide va-
+riety of circulation efficiencies that do not appear to be
+correlated with equilibrium temperature. Alternatively,
+theseplanetsmayhavedifferent (but generallylow)albe-
+dos. Planets hotter than Tε=0= 2400 K have uniformly
+low redistribution efficiencies and albedos.
+The apparent decrease in advective efficiency with
+increasing planetary temperature remains unexplained.
+One hypothesis, mentioned earlier, is that TiO and VO
+would provide additional optical opacity in atmospheres
+hotter than T∼1700 K, leading to temperature in-
+versions and reduced heat recirculation on these plan-
+ets (Fortney et al. 2008). But if our sample shows any
+sharp change it behavior it occurs near 2400 K, rather
+than 1700K. One couldinvokeanotheroptical absorber,
+but in any case the lack of correlation —pointed out in
+thisworkandelsewhere—betweenthepresenceofatem-
+perature inversionand the efficiency of heat recirculation
+makes this explanation suspect. Another possible expla-
+nation for the observed trend is that the hottest planets
+have the most ionized atmospheres and may suffer the
+most severe magnetic drag (Perna et al. 2010).
+The simplest explanation for this trend is simply that
+the radiative time is a steeper function of temperature
+than the advective time: advective efficiency is given
+roughly by the ratio of the radiative and advective times
+(eg: Cowan & Agol 2010). In the limit of Newtonian
+cooling, the radiative time scales as τrad∝T−3. If one
+assumes the wind speed to be of order the local sound
+speed, then the advective time scales as τadv∝T−0.5.
+One might therefore naively expect the advective effi-
+ciency to scale as T−2.5. Such an explanation would notAlbedo and Heat Recirculation on Hot Exoplanets 9
+explain the apparent sharp transition seen at 2400 K,
+however.
+The combination of optical observations of secondary
+eclipses and thermal observations of phase variations is
+the best way to constrain planetary albedo and circu-
+lation. The optical observations should be taken near
+the star’s blackbody peak, both to maximize signal-to-
+noise, and to avoidcontaminationfrom the planet’s ther-
+mal emission, but this separationmay not be possible for
+the hottest transiting planets. The thermal observations,
+likewise, should be near the planet’s blackbody peak to
+better constrain its bolometric flux. Note that this wave-
+length is shortwardof the ideal contrastratio, which typ-
+ically falls on the planet’s Rayleigh-Jeans tail. Further-
+more, the thermal phase observations should span a full
+planetaryorbit: thelightcurveminimumisthemostsen-
+sitive measure of ε, and should occur nearly half an orbit
+apart from the light curve maximum, despite skewed di-
+urnal heatingpatterns (Cowan & Agol 2008, 2010). This
+means that observing campaigns that only cover a little
+more than half an orbit (transit →eclipse) are probably
+underestimating the real peak-trough phase amplitude.A possible improvement to this study would be to per-
+form a uniform data reduction for all the Spitzerexo-
+planet observations of hot Jupiters. These data make up
+the majority of the constraints presented in our study
+and most are publicly available. And while the pub-
+lished observations were analyzed in disparate ways, a
+consensus approach to correcting detector systematics is
+beginning to emerge.
+N.B.C. acknowledges useful discussions of aspects of
+this work with T. Robinson, M.S. Marley, J.J. Fort-
+ney, T.S. Barman and D.S. Spiegel. Thanks to our
+referee B.M.S. Hansen for insightful feedback, and to
+E.D. Feigelson for suggestions about statistical methods.
+N.B.C. was supported by the Natural Sciences and Engi-
+neering Research Council of Canada. E.A. is supported
+by a National Science Foundation Career Grant. Sup-
+port for this work was provided by NASA through an
+award issued by JPL/Caltech. This research has made
+use of the Exoplanet Orbit Database and the Exoplanet
+Data Explorer at exoplanets.org.
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+Todorov, K., Deming, D., Harrington, J., Stevenson, K. B.,
+Bowman, W. C., Nymeyer, S., Fortney, J. J., & Bakos, G. A.
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+Werner, M. W. et al. 2004, ApJS, 154, 1
+Wheatley, P. J., et al. 2010, arXiv:1004.0836
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+Fortney, J. J. 2009, ApJ, 701, L20Albedo and Heat Recirculation on Hot Exoplanets 11
+TABLE 1
+Secondary Eclipses & Phase Variations of Exoplanets
+Planet Tε=0[K]aλ[µm]bEclipse DepthcTbright[K] Phase AmplitudecDerived Quantitiesd
+CoRoT-1b12424(84) 0.60(0.42) 1 .6(6)×10−42726(141) Td=2674(144) K
+0.71(0.25) 1 .26(33)×10−42409(75) 1 .0(3)×10−4Aλ<0.1
+2.10(0.02) 2 .8(5)×10−32741(125) Td(A= 0)=2515(84) K
+2.15(0.32) 3 .36(42)×10−32490(157)
+3.6(0.75) 4 .15(42)×10−32098(116)
+4.5(1.0) 4 .82(42)×10−32084(106)
+CoRoT-2b21964(42) 0.60(0.42) 6(2) ×10−52315(85) Td=1864(233) K
+0.71(0.25) 1 .02(20)×10−42215(49) Aλ= 0.16(7)
+1.65(0.25) <1.7×10−3(3σ) Td(A= 0)=2010(144) K
+2.15(0.32) 1 .6(9)×10−31914(292)
+3.6(0.75) 3 .55(20)×10−31798(40)
+4.5(1.0)e4.75(19)×10−31791(33)
+4.5(1.0) 5 .10(42)×10−3
+8.0(2.9) 4 .1(1.1)×10−3
+8.0(2.9)e4.09(80)×10−31318(143)
+Gl 436b3934(41) 3.6(0.75) 4 .1(3)×10−41145(23) Td=1082(38) K
+4.5(1.0) <1.0×10−4(3σ)
+5.8(1.4) 3 .3(1.4)×10−4797(106)
+8.0(2.9)e4.52(27)×10−4737(17)
+8.0(2.9) 5 .7(8)×10−4
+8.0(2.9) 5 .4(7)×10−4
+16(5) 1 .40(27)×10−3963(126)
+24(9) 1 .75(41)×10−31016(182)
+HAT-P-1b41666(38) 3.6(0.75) 8 .0(8)×10−41420(47) Td=1439(59) K
+4.5(1.0) 1 .35(22)×10−31507(100)
+5.8(1.4) 2 .03(31)×10−31626(128)
+8.0(2.9) 2 .38(40)×10−31564(151)
+HAT-P-7b52943(95) 0.65(0.4) 1 .30(11)×10−43037(35) 1 .22(16)×10−4Td=2086(156) K
+3.6(0.75) 9 .8(1.7)×10−42063(152) Aλ= 0.58(5)
+4.5(1.0) 1 .59(22)×10−32378(179) Td(A= 0)=2830(86) K
+5.8(1.4) 2 .45(31)×10−32851(235)
+8.0(2.9) 2 .25(52)×10−32512(403)
+HD 80606b61799(50) 8.0(2.9) 1 .36(18)×10−31137(73) Td=1137(113) K
+HD 149026b71871(17) 8.0(2.9)e3.7(0.8)×10−4976(276) 2 .3(7)×10−4Td=1571(231) K
+8.0(2.9) 8 .4(1.1)×10−4Tn=976(286) K
+HD 189733b81537(16) 2.15(32) <4.0×10−4(1σ) Td=1605(52) K
+3.6(0.75) 2 .56(14)×10−31639(34) Tn=1107(132) K
+4.5(1.0) 2 .14(20)×10−31318(45)
+5.8(1.4) 3 .10(34)×10−31368(69)
+8.0(2.9) 3 .381(55)×10−3
+8.0(2.9) 3 .91(22)×10−31.2(2)×10−3
+8.0(2.9)e3.440(36)×10−31259(7) 1 .2(4)×10−3
+16(5) 5 .51(30)×10−31338(52)
+24(9) 5 .98(38)×10−3
+24(9)e5.36(27)×10−31202(46) 1 .3(3)×10−3
+HD 209458b91754(15) 0.5(0.3) 7(9) ×10−62368(156) Td=1486(53) K
+2.15(0.32) <3×10−4(1σ) Aλ= 0.09(7)
+3.6(0.75) 9 .4(9)×10−41446(45) Td(A= 0)=2031(128) K
+4.5(1.0) 2 .13(15)×10−31757(57) Tn=1476(304) K
+5.8(1.4) 3 .01(43)×10−31890(149)
+8.0(2.9) 2 .40(26)×10−31480(94) <1.5×10−3(2σ)
+24(9) 2 .60(44)×10−31131(143)
+OGLE-TR-56b102874(84) 0.90(0.15) 3 .63(91)×10−42696(116) Td=2696(236) K
+OGLE-TR-113b111716(33) 2.15(0.32) 1 .7(5)×10−31918(164) Td=1918(219) K
+TrES-1b121464(16) 3.6(0.75) <1.5×10−3(1σ) Td=998(67) K
+4.5(1.0) 6 .6(1.3)×10−4972(56)
+8.0(2.9) 2 .25(36)×10−31152(94)
+TrES-2b131917(21) 0.65(0.4) 1 .14(78)×10−52020(132) Td=1623(76) K
+2.15(0.32) 6 .2(1.2)×10−41655(80) Aλ= 0.06(3)
+3.6(0.75) 1 .27(21)×10−31490(84) Td(A= 0) = 1751(80) K
+4.5(1.0) 2 .30(24)×10−31652(74)
+5.8(1.4) 1 .99(54)×10−31373(177)
+8.0(2.9) 3 .59(60)×10−31659(163)
+TrES-3b142093(32) 0.7(0.3) <6.2×10−4(1σ) Td=1761(66) K
+1.25(0.16) <5.1×10−4(3σ)
+2.15(0.32) 2 .41(43)×10−3
+2.15(0.32)e1.33(17)×10−31770(58)
+3.6(0.75) 3 .46(35)×10−31818(73)12 Cowan & Agol
+TABLE 1
+Secondary Eclipses & Phase Variations of Exoplanets
+4.5(1.0) 3 .72(54)×10−31649(107)
+5.8(1.4) 4 .49(97)×10−31621(173)
+8.0(2.9) 4 .75(46)×10−31480(82)
+TrES-4b152250(37) 3.6(0.75) 1 .37(11)×10−31889(63) Td=1891(81) K
+4.5(1.0) 1 .48(16)×10−31727(83)
+5.8(1.4) 2 .61(59)×10−32112(283)
+8.0(2.9) 3 .18(44)×10−32168(197)
+WASP-1b162347(35) 3.6(0.75) 1 .17(16)×10−31678(87) Td=1719(89) K
+4.5(1.0) 2 .12(21)×10−31923(91)
+5.8(1.4) 2 .82(60)×10−32042(253)
+8.0(2.9) 4 .70(46)×10−32587(176)
+WASP-2b171661(69) 3.6(0.75) 8 .3(3.5)×10−41264(164) Td=1280(121) K
+4.5(1.0) 1 .69(17)×10−31380(53)
+5.8(1.4) 1 .92(77)×10−31299(232)
+8.0(2.9) 2 .85(59)×10−31372(154)
+WASP-4b182163(60) 3.6(0.75) 3 .19(31)×10−32156(97) Td=2146(140) K
+4.5(1.0) 3 .43(27)×10−31971(75)
+WASP-12b193213(119) 0.9(0.15) 8 .2(1.5)×10−43002(104) Td=2939(98) K
+1.25(0.16) 1 .31(28)×10−32894(149)
+1.65(0.25) 1 .76(18)×10−32823(88)
+2.15(0.32) 3 .09(13)×10−33018(51)
+3.6(0.75) 3 .79(13)×10−32704(49)
+4.5(1.0) 3 .82(19)×10−32486(68)
+5.8(1.4) 6 .29(52)×10−33167(179)
+8.0(2.9) 6 .36(67)×10−32996(229)
+WASP-18b203070(50) 3.6(0.75) 3 .1(2)×10−33000(107) Td=2998(138) K
+4.5(1.0) 3 .8(3)×10−33128(150)
+5.8(1.4) 4 .1(2)×10−33095(103)
+8.0(2.9) 4 .3(3)×10−32991(153)
+WASP-19b212581(49) 1.65(0.25) 2 .59(45)×10−32677(135) Td=2677(244) K
+XO-1b221526(24) 3.6(0.75) 8 .6(7)×10−41300(32) Td=1306(47) K
+4.5(1.0) 1 .22(9)×10−31265(34)
+5.8(1.4) 2 .61(31)×10−31546(89)
+8.0(2.9) 2 .10(29)×10−31211(87)
+XO-2231685(33) 3.6(0.75) 8 .1(1.7)×10−41447(102) Td=1431(98) K
+4.5(1.0) 9 .8(2.0)×10−41341(105)
+5.8(1.4) 1 .67(36)×10−31497(155)
+8.0(2.9) 1 .33(49)×10−31179(219)
+XO-3241982(82) 3.6(0.75) 1 .01(4)×10−31875(30) Td=1871(63) K
+4.5(1.0) 1 .43(6)×10−31965(40)
+5.8(1.4) 1 .34(49)×10−31716(330)
+8.0(2.9) 1 .50(36)×10−31625(236)
+aThe planet’s expected day-side effective temperature in the absence of reflection or recirculation ( AB= 0,ε= 0). The 1 σuncertainty is shown
+in parenthese.
+bThe bandwidth is shown in parenthese.
+cEclipse depths and phase amplitudes are unitless, since the y are measured relative to stellar flux.
+dTdandTndenote the day-side and night-side effective temperatures o f the planet, as estimated from thermal secondary eclipse de pths and
+thermal phase variations, respectively. The estimated 1 σuncertainties are shown in parentheses. The default day-si de temperature is computed
+using only observations at λ >0.8µm. Eclipse measurements at shorter wavelengths may then be u sed to estimate the planet’s albedo at those
+wavelengths, Aλ. Note that this is a spherical albedo; the geometric albedo i s given by Ag=2
+3Aλ. If —on the other hand— AB= 0 is assumed,
+then all the day-side flux is thermal, regardless of waveband , yielding the second Tdestimate.
+eWhen multiple measurements of an eclipse depth have been pub lished in a given waveband, we use the most recent observatio n. In all cases
+these observations are either explicitly agree with their o lder counterpart, or agree with the re-analyzed older data.
+1Snellen et al. (2009); Alonso et al. (2009b); Gillon et al. (2 009); Rogers et al. (2009); Deming et al. (2010),2Alonso et al. (2009a); Snellen et al.
+(2010); Gillon et al. (2010); Alonso et al. (2010); Deming et al. (2010),3Deming et al. (2007); Demory et al. (2007); Stevenson et al. ( 2010);
+Knutson et al. in prep.,4Todorov et al. (2010),5Borucki et al. (2009); Christiansen et al. (2010),6Laughlin et al. (2009),7Knutson et al.
+(2009b),8Deming et al. (2006); Knutson et al. (2007a); Barnes et al. (2 007); Charbonneau et al. (2008); Knutson et al. (2009c); Ago l et al.
+(2010),9Richardson et al. (2003); Deming et al. (2005); Cowan et al. ( 2007); Rowe et al. (2008); Knutson et al. (2008),10Sing & L´ opez-Morales
+(2009),11Snellen & Covino (2007),12Charbonneau et al. (2005); Knutson et al. (2007b),13O’Donovan et al. (2010); Croll et al. (2010a);
+Kipping & Bakos (2010b),14Fressin et al. (2010); Croll et al. (2010b); Christiansen et al. (2010b),15Knutson et al. (2009a),16,17Wheatley et al.
+(2010),18Beerer et al. (2010),19L´ opez-Morales et al. (2010); Campo et al. (2010); Croll et a l. (2010c),20Nymeyer et al. (2010),21Anderson et al.
+(2010),22Machalek et al. (2008),23Machalek et al. (2009),24Machalek et al. (2010)
\ No newline at end of file
diff --git a/1001.0013.txt b/1001.0013.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d06769decd311651d25f19d15ae7cda437585a94
--- /dev/null
+++ b/1001.0013.txt
@@ -0,0 +1,1084 @@
+arXiv:1001.0013v2  [astro-ph.CO]  8 Jan 2010Astronomy& Astrophysics manuscriptno.akari˙LF˙aa˙v7 c∝circlecopyrtESO 2018
+October30,2018
+EvolutionofInfraredLuminosityfunctionsofGalaxiesint he
+AKARINEP-Deepfield
+Revealing thecosmic star formationhistory hidden by dust⋆,⋆⋆
+Tomotsugu Goto1,2,⋆⋆⋆,T.Takagi3,H.Matsuhara3,T.T.Takeuchi4,C.Pearson5,6,7, T.Wada3,T.Nakagawa3,O.Ilbert8,
+E.LeFloc’h9,S.Oyabu3, Y.Ohyama10,M.Malkan11, H.M.Lee12, M.G.Lee12,H.Inami3,13,14, N.Hwang2, H.Hanami15,
+M.Im12, K.Imai16,T.Ishigaki17,S.Serjeant7,and H.Shim12
+1Institute for Astronomy, University of Hawaii,2680 Woodla wnDrive, Honolulu, HI,96822, USA
+e-mail:tomo@ifa.hawaii.edu
+2National Astronomical Observatory, 2-21-1 Osawa,Mitaka, Tokyo, 181-8588,Japan
+3Institute of Space and Astronautical Science, JapanAerosp ace Exploration Agency, Sagamihara,Kanagawa 229-8510
+4Institute for Advanced Research, Nagoya University, Furo- cho, Chikusa-ku, Nagoya 464-8601
+5Rutherford Appleton Laboratory, Chilton, Didcot,Oxfords hire OX110QX, UK
+6Department of Physics,Universityof Lethbridge, 4401 Univ ersity Drive,Lethbridge, AlbertaT1J 1B1, Canada
+7Astrophysics Group, Department of Physics, The OpenUniver sity, MiltonKeynes, MK76AA, UK
+8Laboratoire d’Astrophysique de Marseille, BP8,Traverse d u Siphon, 13376 Marseille Cedex 12, France
+9CEA-Saclay,Service d’Astrophysique, France
+10Academia Sinica,Institute of Astronomyand Astrophysics, Taiwan
+11Department of Physicsand Astronomy, UCLA,Los Angeles, CA, 90095-1547 USA
+12Department of Physics& Astronomy, FPRD,Seoul National Uni versity, Shillim-Dong,Kwanak-Gu, Seoul 151-742, Korea
+13Spitzer Science Center,California Institute ofTechnolog y, Pasadena, CA91125
+14Department of Astronomical Science,The Graduate Universi tyfor Advanced Studies
+15Physics Section,Facultyof Humanities and SocialSciences , Iwate University, Morioka, 020-8550
+16TOMER&D Inc. Kawasaki, Kanagawa 2130012, Japan
+17Asahikawa National College of Technology, 2-1-6 2-joShunk ohdai, Asahikawa-shi, Hokkaido 071-8142
+Received September 15, 2009; accepted December 16, 2009
+ABSTRACT
+Aims.Dust-obscured star-formation becomes much more important with increasing intensity, and increasing redshift. We aim to
+reveal cosmic star-formationhistoryobscured bydust usin g deep infraredobservation withthe AKARI.
+Methods. We construct restframe 8 µm, 12µm, and total infrared (TIR) luminosity functions (LFs) at 0.15< z <2.2using 4128
+infraredsources intheAKARINEP-Deepfield.Acontinuous fil tercoverage inthemid-IRwavelength(2.4,3.2,4.1,7,9,11 , 15,18,
+and 24µm) by the AKARI satellite allows us to estimate restframe 8 µm and 12 µm luminosities without using a large extrapolation
+based ona SEDfit,which was the largestuncertainty inprevio us work.
+Results. Wehavefoundthatall8 µm(0.38< z <2.2),12µm(0.15< z <1.16),andTIRLFs( 0.2< z <1.6),showacontinuous
+andstrongevolutiontowardhigher redshift.Intermsofcos micinfraredluminositydensity( ΩIR),whichwasobtainedbyintegrating
+analytic fits to the LFs,we found a good agreement withprevio us work at z <1.2. We found the ΩIRevolves as ∝(1+z)4.4±1.0.
+Whenweseparatecontributionsto ΩIRbyLIRGsandULIRGs,wefoundmoreIRluminoussourcesareinc reasinglymoreimportant
+at higher redshift. Wefound that the ULIRG(LIRG)contribut ionincreases bya factor of 10(1.8) from z=0.35 toz=1.4.
+Keywords. galaxies: evolution, galaxies:interactions, galaxies:s tarburst, galaxies:peculiar, galaxies:formation
+1. Introduction
+Studies of the extragalactic background suggest at least ha lf
+the luminous energy generated by stars has been reprocessed
+into the infrared(IR) by dust (Lagacheetal., 1999; Pugetet al.,
+1996; Franceschini,Rodighiero,&Vaccari, 2008), suggest ing
+that dust-obscured star formation was much more important a t
+higherredshiftsthantoday.
+⋆This research is based on the observations with AKARI, a JAXA
+project withthe participationof ESA.
+⋆⋆Based on data collected at Subaru Telescope, which is operat ed by
+the National Astronomical Observatory ofJapan.
+⋆⋆⋆JSPSSPDfellowBell etal. (2005) estimate that IR luminosity density is 7
+times higher than the UV luminosity density at z ∼0.7 than lo-
+cally. Takeuchi,Buat, &Burgarella (2005) reported that UV -to-
+IRluminositydensityratio, ρL(UV)/ρL(dust),evolvesfrom3.75
+(z=0) to 15.1 by z=1.0 with a careful treatment of the sample
+selection effect, and that 70% of star formation activity is ob-
+scured by dust at 0.5 < z <1.2. Both works highlight the im-
+portance of probing cosmic star formation activity at high r ed-
+shift in the infrared bands. Several works found that most ex -
+tremestar-forming(SF) galaxies,whichareincreasinglyi mpor-
+tant at higher redshifts, are also more heavily obscured by d ust
+(Hopkinsetal., 2001; Sullivanet al., 2001; Buatet al.,200 7).2 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+Despite the value of infrared observations, studies of
+infrared galaxies by the IRAS and the ISO were re-
+stricted to bright sources due to the limited sensitiv-
+ities (Saundersetal., 1990; Rowan-Robinsonet al., 1997;
+Floreset al., 1999; Serjeantet al., 2004; Takeuchiet al., 2 006;
+Takeuchi,Yoshikawa,&Ishii, 2003), until the recent launc h of
+theSpitzer andtheAKARI satellites. Theirenormousimprov ed
+sensitivitieshaverevolutionizedthefield.Forexample:
+Le Floc’het al. (2005) analyzed the evolution of the total
+and 15µm IR luminosity functions (LFs) at 0< z <1based
+on the the Spitzer MIPS 24 µm data (>83µJy andR <24) in
+the CDF-S, and found a positive evolution in both luminosity
+and density, suggesting increasing importance of the LIRG a nd
+ULIRGpopulationsathigherredshifts.
+P´ erez-Gonz´ alezetal. (2005) used MIPS 24 µm observations
+oftheCDF-SandHDF-N( >83µJy)tofindthatthat L∗steadily
+increasesbyanorderofmagnitudeto z∼2,suggestingthatthe
+luminosity evolution is stronger than the density evolutio n. The
+ΩTIRscalesas(1+z)4.0±0.2fromz=0to0.8.
+Babbedgeet al. (2006) constructed LFs at 3.6, 4.5, 5.8, 8
+and 24µm over0< z < 2using the data from the Spitzer
+Wide-areaInfraredExtragalactic(SWIRE)Surveyin a 6.5de g2
+(S24µm>230µJy). They found a clear luminosity evolu-
+tion in all the bands, but the evolution is more pronounced at
+longer wavelength; extrapolatingfrom 24 µm, they inferred that
+ΩTIR∝(1+z)4.5. They constructed separate LFs for three dif-
+ferentgalaxySED (spectral energydistribution)typesand Type
+1 AGN, finding that starburst and late-type galaxies showed
+strongerevolution.Comparisonof3.6and4.5 µmLFswithsemi-
+analytic and spectrophotometricmodelssuggested that the IMF
+is skewed towards higher mass star formation in more intense
+starbursts.
+Caputi etal.(2007)estimatedrestframe8 µmLFsofgalaxies
+over 0.08deg2in the GOODS fields based on Spitzer 24 µm (>
+80µJy) atz=1 and 2. They found a continuousand strong posi-
+tiveluminosityevolutionfrom z=0toz=1,andto z=2.However,
+theyalsofoundthatthenumberdensityofstar-forminggala xies
+withνL8µm
+ν>1010.5L⊙(AGNs are excluded.) increases by a
+factor of 20 from z=0 to 1, but decreases by half from z=1 to 2
+mainlyduetothe decreaseofLIRGs.
+Magnelliet al. (2009) investigated restframe 15 µm, 35µm
+and total infrared (TIR) LFs using deep 70 µm observations
+(∼300µJy) in the Spitzer GOODS and FIDEL (Far Infrared
+Deep Extragalactic Legacy Survey) fields (0.22 deg2in total)
+atz <1.3. They stacked 70 µm flux at the positions of 24 µm
+sources when sources are not detected in 70 µm. They found no
+changeintheshapeoftheLFs,butfoundapureluminosityevo -
+lutionproportionalto(1+z)3.6±0.5,andthatLIRGsandULIRGs
+have increased by a factor of 40 and 100 in number density by
+z∼1.
+Also, see Daiet al. (2009) for 3.6-8.0 µm LFs based on the
+IRACphotometryintheNOAODeepWide-FieldSurveyBootes
+field.
+However, most of the Spitzer work relied on a large
+extrapolation from 24 µm flux to estimate the 8, 12 µm or
+TIR luminosity. Consequently, Spitzer results heavily de-
+pended on the assumed IR SED library (Dale&Helou, 2002;
+Lagache,Dole,&Puget, 2003; Chary& Elbaz, 2001). Indeed
+many authors pointed out that the largest uncertainty in the se
+previous IR LFs came from SED models, especially when one
+computesTIRluminositysolelyfromobserved24 µmflux(e.g.,
+see Fig.5ofCaputiet al.,2007).
+AKARI, the first Japanese IR dedicated satellite, has con-
+tinuous filter coverage across the mid-IR wavelengths, thus , al-Fig.1. Photometric redshift estimates with LePhare
+(Ilbertet al., 2006; Arnoutset al., 2007; Ilbertet al., 200 9)
+for spectroscopically observed galaxies with Keck/DEIMOS
+(Takagi et al. in prep.). Red squares show objects where AGN
+templates were better fit. Errors of the photoz is∆z
+1+z=0.036 for
+z≤0.8, but becomes worse at z >0.8to be∆z
+1+z=0.10 due
+mainlyto therelativelyshallownear-IRdata.
+lows us to estimate MIR (mid-infrared)-luminositywithout us-
+ing a large k-correction based on the SED models, eliminating
+thelargestuncertaintyinpreviouswork.Bytakingadvanta geof
+this, we present the restframe 8, 12 µm and TIR LFs using the
+AKARI NEP-Deepdatainthiswork.
+Restframe 8 µm luminosity in particular is of primary rele-
+vance for star-forming galaxies, as it includes polycyclic aro-
+matic hydrocarbon (PAH) emission. PAH molecules charac-
+terize star-forming regions (Desert,Boulanger,&Puget, 1 990),
+and the associated emission lines between 3.3 and 17 µm dom-
+inate the SED of star-forming galaxies with a main bump lo-
+cated around 7.7 µm. Restframe 8 µm luminosities have been
+confirmed to be good indicators of knots of star formation
+(Calzetti etal., 2005) and of the overall star formation act ivity
+of star forming galaxies (Wuet al., 2005). At z=0.375, 0.875,
+1.25 and 2, the restframe 8 µm is covered by the AKARI S11,
+L15,L18WandL24filters. We present the restframe 8 µm LFs
+at theseredshiftsatSection3.1.
+Restframe 12 µm luminosity functions have also been
+studied extensively (Rush,Malkan,& Spinoglio, 1993;
+P´ erez-Gonz´ alezet al., 2005). At z=0.25, 0.5 and 1, the
+restframe12 µmiscoveredbytheAKARI L15,L18WandL24
+filters. We present the restframe 12 µm LFs at these redshifts in
+Section3.3.
+We also estimate TIR LFs through the SED fit using all
+the mid-IR bands of the AKARI. The results are presented in
+Section3.5.
+Unless otherwise stated, we adopt a cosmology with
+(h,Ωm,ΩΛ) = (0.7,0.3,0.7)(Komatsuet al., 2008).
+2. Data & Analysis
+2.1. Multi-wavelength data inthe AKARI NEP Deepfield
+AKARI, the Japanese infraredsatellite (Murakamiet al., 20 07),
+performed deep imaging in the North Ecliptic Region (NEP)
+from 2-24 µm, with 14 pointings in each field over 0.4
+deg2(Matsuharaet al., 2006, 2007; Wada et al., 2008). DueGotoet al.:InfraredLuminosityfunctions withthe AKARI 3
+Fig.2.Photometricredshiftdistribution.
+Fig.3.8µmluminositydistributionsofsamplesusedtocompute
+restframe 8 µm LFs. From low redshift, 533, 466, 236 and 59
+galaxiesarein eachredshiftbin.
+to the solar synchronous orbit of the AKARI, the NEP
+is the only AKARI field with very deep imaging at these
+wavelengths. The 5 σsensitivity in the AKARI IR filters
+(N2,N3,N4,S7,S9W,S11,L15,L18WandL24) are 14.2,
+11.0, 8.0, 48, 58, 71, 117, 121 and 275 µJy (Wada etal., 2008).
+These filters provide us with a unique continuous wavelength
+coverage at 2-24 µm, where there is a gap between the Spitzer
+IRAC and MIPS, and the ISO LW2andLW3. Please consult
+Wada etal. (2007, 2008); Pearsonet al. (2009a,b) for data ve ri-
+ficationandcompletenessestimateatthesefluxes.ThePSFsi zes
+are 4.4, 5.1, and 5.4” in 2−4,7−11,15−24µm bands. The
+depths of near-IR bands are limited by source confusion, but
+thoseofmid-IRbandsarebyskynoise.In analyzingthese observations,we first combinedthe three
+images of the MIR channels, i.e. MIR-S( S7,S9W, andS11)
+and MIR-L( L15,L18WandL24), in order to obtain two high-
+quality images. In the resulting MIR-S and MIR-L images, the
+residual sky has been reduced significantly, which helps to o b-
+tain more reliable source catalogues. For both the MIR-S and
+MIR-Lchannels,we use SExtractorforthecombinedimagesto
+determineinitialsourcepositions.
+We follow Takagietal. (2007) procedures for photometry
+and band-merging of IRC sources. But this time, to maximize
+the number of MIR sources, we made two IRC band-merged
+catalogues based on the combined MIR-S and MIR-L images,
+andthenconcatenatedthese catalogues,eliminatingdupli cates.
+Intheband-mergingprocess,thesourcecentroidineachIRC
+image has beendetermined,starting fromthe sourcepositio n in
+the combined images as the initial guess. If the centroid det er-
+mined in this way is shifted from the original position by >3′′,
+we reject such a source as the counterpart. We note that this
+band-mergingmethodisusedonlyforIRCbands.
+We comparedraw numbercountswith previouswork based
+on the same data but with different source extraction method s
+(Wadaet al., 2008; Pearsonet al., 2009a,b) and found a good
+agreement.
+A subregion of the NEP-Deep field was observed in the
+BVRi′z′-bands with the Subaru telescope (Imaiet al., 2007;
+Wada etal., 2008), reaching limiting magnitudes of zAB=26
+in one field of view of the Suprime-Cam.We restrict our analy-
+sis to the data in this Suprime-Cam field (0.25 deg2), where we
+have enough UV-opical-NIR coverage to estimate good photo-
+metricredshifts.The u′-bandphotometryinthisareaisprovided
+by the CFHT (Serjeant et al. in prep.). The same field was also
+observed with the KPNO2m/FLAMINGOs in JandKsto the
+depth ofKsVega<20(Imaiet al., 2007). GALEX coveredthe
+entirefieldtodepthsof FUV <25andNUV < 25(Malkanet
+al.in prep.).
+In the Suprime-Cam field of the AKARI NEP-Deep field,
+there are a total of 4128 infrared sources down to ∼100µJy in
+theL18Wfilter. All magnitudesare given in AB system in this
+paper.
+For the optical identification of MIR sources, we adopt the
+likelihood ratio (LR) method (Sutherland&Saunders, 1992) .
+For the probability distribution functions of magnitude an d an-
+gular separation based on correct optical counterparts (an d for
+this purpose only), we use a subset of IRC sources, which are
+detected in all IRC bands. For this subset of 1100 all-band–
+detected sources, the optical counterparts are all visuall y in-
+spected and ambiguous cases are excluded. There are multipl e
+opticalcounterpartsfor35%ofMIRsourceswithin <3′′. Ifwe
+adoptedthenearestneighborapproachfortheopticalident ifica-
+tion,theopticalcounterpartsdiffersfromthat oftheLRme thod
+for20%ofthesourceswith multipleopticalcounterparts.T hus,
+in total we estimate that less than 15% of MIR sources suffer
+fromseriousproblemsofopticalidentification.
+2.2. Photometric redshift estimation
+For these infrared sources, we have computed photomet-
+ric redshift using a publicly available code, LePhare1
+(Ilbertet al., 2006; Arnoutsetal., 2007; Ilbertet al., 200 9).
+The input magnitudes are FUV,NUV (GALEX), u(CFHT),
+B,V,R,i′,z′(Subaru), J,andK(KPNO2m).Wesummarizethe
+filtersusedinTable1.
+1http://www.cfht.hawaii.edu/∼arnouts/lephare.html4 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+Table 1.Summaryoffiltersused.
+Estimate Redshift Filter
+Photoz0.15<z<2.2FUV,NUV ,u,B,V,R,i′,z,J, andK
+8µm LF 0.38 <z<0.58 S11(11 µm)
+8µm LF 0.65 <z<0.90 L15(15 µm)
+8µm LF 1.1 <z<1.4 L18W (18 µm)
+8µm LF 1.8 <z<2.2 L24(24 µm)
+12µm LF 0.15 <z<0.35 L15(15 µm)
+12µm LF 0.38 <z<0.62 L18W (18 µm)
+12µm LF 0.84 <z<1.16 L24(24 µm)
+TIRLF 0.2 <z<0.5S7,S9W,S11,L15,L18WandL24
+TIRLF 0.5 <z<0.8S7,S9W,S11,L15,L18WandL24
+TIRLF 0.8 <z<1.2S7,S9W,S11,L15,L18WandL24
+TIRLF 1.2 <z<1.6S7,S9W,S11,L15,L18WandL24
+Among various templates and fitting parameters we tried,
+we found the best results were obtained with the following: w e
+used modified CWW (Coleman,Wu,& Weedman, 1980) and
+QSO templates.TheseCWW templatesareinterpolatedandad-
+justed to better match VVDS spectra (Arnoutsetal., 2007). W e
+included strong emission lines in computing colors. We used
+the Calzetti extinction law. More details in training LePhare
+isgiveninIlbertet al.(2006).
+The resulting photometric redshift estimates agree reason -
+ably well with 293 galaxies ( R <24) with spectroscopic red-
+shifts taken with Keck/DEIMOS in the NEP field (Takagi et al.
+inprep.).Themeasurederrorsonthephoto- zare∆z
+1+z=0.036for
+z≤0.8and∆z
+1+z=0.10 for z >0.8. The∆z
+1+zbecomes signifi-
+cantly larger at z >0.8, where we suffer from relative shallow-
+ness of our near-IR data. The rate of catastrophic failures i s 4%
+(∆z
+1+z>0.2)amongthespectroscopicsample.
+In Fig.1, we compare spectroscopic redshifts from
+Keck/DEIMOS (Takagi et al.) and our photometric red-
+shift estimation. Those SEDs which are better fit with a QSO
+template are shown as red triangles. We remove those red
+triangle objects ( ∼2% of the sample) from the LFs presented
+below. We caution that this can only remove extreme type-1
+AGNs, and thus, fainter, type-2 AGN that could be removedby
+X-raysoropticalspectroscopystill remainin thesample.
+Fig.2showsthedistributionofphotometricredshift.Thed is-
+tributionhasseveralpeaks,whichcorrespondstogalaxycl usters
+in the field (Gotoetal., 2008). We have 12% of sources that do
+nothaveagoodSEDfit toobtainareliablephotometricredshi ft
+estimation.Weapplythisphoto- zcompletenesscorrectiontothe
+LFs we obtain.Readers are referredto Negrelloet atal. (200 9),
+who estimated photometricredshifts using only the AKARI fil -
+terstoobtain10%accuracy.
+2.3. The1/ Vmaxmethod
+WecomputeLFsusingthe1/ Vmaxmethod(Schmidt,1968).The
+advantage of the 1/ Vmaxmethod is that it allows us to compute
+a LF directly from data, with no parameter dependence or an
+assumed model. A drawback is that it assumes a homogeneous
+galaxy distribution, and is thus vulnerable to local over-/ under-
+densities(Takeuchi,Yoshikawa,&Ishii,2000).
+A comoving volume associated with any source of a given
+luminosity is defined as Vmax=Vzmax−Vzmin, wherezmin
+is the lower limit of the redshift bin and zmaxis the maximum
+redshiftat whichthe objectcouldbe seen giventhe fluxlimit of
+the survey, with a maximum value corresponding to the upperredshiftoftheredshiftbin.Moreprecisely,
+zmax= min(z maxof the bin ,zmaxfromthe flux limit) (1)
+We usedtheSED templates(Lagache,Dole,&Puget, 2003) for
+k-corrections to obtain the maximum observable redshift fro m
+thefluxlimit.
+Foreachluminositybinthen,theLFisderivedas
+φ=1
+∆L/summationdisplay
+i1
+Vmax,iwi, (2)
+whereVmaxis a comoving volume over which the ith galaxy
+couldbeobserved, ∆Listhesizeoftheluminositybin(0.2dex),
+andwiis the completeness correction factor of the ith galaxy.
+WeusecompletenesscorrectionmeasuredbyWadaet al.(2008 )
+for11and24 µmandPearsonet al.(2009a,b)for15and18 µm.
+Thiscorrectionis25%atmaximum,sincewe onlyusethesam-
+plewherethecompletenessisgreaterthan80%.
+2.4. Monte Carlo simulation
+Uncertainties of the LF values stem from various factors suc h
+as fluctuations in the numberof sources in each luminosity bi n,
+the photometric redshift uncertainties, the k-correction uncer-
+tainties, and the flux errors. To compute these errors we per-
+formedMonteCarlosimulationsbycreating1000simulatedc at-
+alogs,whereeach catalogcontainsthesame numberof source s,
+but we assign each source a new redshift following a Gaussian
+distribution centered at the photometric redshift with the mea-
+sured dispersion of ∆z/(1 +z) =0.036 for z≤0.8and
+∆z/(1+z) =0.10forz >0.8(Fig.1). The flux of each source
+is also allowed to vary accordingto the measuredflux error fo l-
+lowingaGaussiandistribution.For8 µmand12µmLFs,wecan
+ignore the errors due to the k-correction thanks to the AKARI
+MIR filter coverage. For TIR LFs, we have added 0.05 dex of
+error for uncertaintyin the SED fitting following the discus sion
+in Magnelliet al. (2009). We did not consider the uncertaint y
+on the cosmic variance here since the AKARI NEP field cov-
+ers a large volume and has comparable number counts to other
+generalfields(Imaiet al.,2007,2008).Eachredshiftbinwe use
+covers∼106Mpc3of volume. See Matsuharaetal. (2006) for
+morediscussion on the cosmic variancein the NEP field. These
+estimated errors are added to the Poisson errors in each LF bi n
+inquadrature.
+3. Results
+3.1. 8µm LF
+Monochromatic 8 µm luminosity ( L8µm) is known to cor-
+relate well with the TIR luminosity (Babbedgeet al., 2006;
+Huanget al.,2007),especiallyforstar-forminggalaxiesb ecause
+the rest-frame 8 µm flux are dominated by prominent PAH fea-
+turessuchasat 6.2,7.7and8.6 µm.
+Since the AKARI has continuous coverage in the mid-IR
+wavelengthrange,therestframe8 µmluminositycanbeobtained
+without a large uncertainty in k-correction at a corresponding
+redshift and filter. For example, at z=0.375, restframe 8 µm is
+redshiftedinto S11filter. Similarly, L15,L18WandL24cover
+restframe 8 µm atz=0.875, 1.25 and 2. This continuous filter
+coverageisanadvantagetoAKARIdata.OftenSEDmodelsare
+used to extrapolate from Spitzer 24 µm flux in previous work,Gotoet al.:InfraredLuminosityfunctions withthe AKARI 5
+producingasourceofthe largestuncertainty.We summarise fil-
+tersusedinTable1.
+To obtain restframe 8 µm LF, we applied a flux limit
+of F(S11) <70.9, F(L15) <117, F(L18W) <121.4, and
+F(L24)<275.8µJy atz=0.38-0.58, z=0.65-0.90, z=1.1-1.4
+andz=1.8-2.2,respectively.Thesearethe5 σlimitsmeasuredin
+Wada etal. (2008). We exclude those galaxies whose SEDs are
+betterfit withQSO templates( §2).
+We use the completeness curve presented in Wada et al.
+(2008) and Pearsonet al. (2009a,b) to correct for the incom-
+pleteness of the detection. However, this correction is 25% at
+maximumsincethesampleis80%completeatthe5 σlimit.Our
+mainconclusionsarenotaffectedbythisincompletenessco rrec-
+tion. To compensatefor the increasing uncertaintyin incre asing
+z, we use redshift binsize of 0.38 < z <0.58, 0.65 < z <0.90,
+1.1< z <1.4,and 1.8 < z <2.2.We show the L8µmdistribution
+in each redshift rangein Fig.3. Within each redshift bin, we use
+1/Vmaxmethodto compensateforthefluxlimit ineachfilter.
+We show the computed restframe 8 µm LF in Fig.4. Arrows
+show the 8 µm luminosity correspondingto the flux limit at the
+central redshift in each redshift bin. Errorbarson each poi nt are
+basedontheMonteCarlosimulation( §2.3).
+For a comparison, as the green dot-dashed line, we also
+show the 8 µm LF of star-forming galaxies at 0< z < 0.3
+by Huanget al. (2007), using the 1/ Vmaxmethod applied to the
+IRAC 8µm GTO data. Compared to the local LF, our 8 µm LFs
+showstrongevolutionin luminosity.Intherangeof 0.48< z <
+2,L∗
+8µmevolvesas ∝(1+z)1.6±0.2. Detailedcomparisonwith
+theliteraturewill bepresentedin §4.
+3.2. Bolometric IR luminosity density basedonthe 8 µm
+LF
+Constraining the star formation history of galaxies as a fun c-
+tion of redshift is a key to understanding galaxy formation i n
+the Universe. One of the primary purposes in computing IR
+LFs is to estimate the IR luminosity density, which in turn is a
+goodestimatorof thedust hiddencosmic star formationdens ity
+(Kennicutt, 1998). Since dust obscurationis more importan t for
+more actively star forming galaxies at higher redshift, and such
+star formationcannotbeobservedinUV light,it is importan tto
+obtainIR-basedestimateinordertofullyunderstandtheco smic
+star formationhistoryoftheUniverse.
+Weestimatethetotalinfraredluminositydensitybyintegr at-
+ingtheLFweightedbytheluminosity.First, weneedtoconve rt
+L8µmto the bolometric infrared luminosity. The bolometric IR
+luminosity of a galaxy is produced by the thermal emission of
+its interstellarmatter. Instar-forminggalaxies,the UV r adiation
+producedbyyoungstarsheatstheinterstellardust,andthe repro-
+cessed lightisemittedin theIR. Forthisreason,in star-fo rming
+galaxies,thebolometricIRluminosityisagoodestimatoro fthe
+current SFR (star formation rate) of the galaxy. Bavouzetet al.
+(2008) showed a strong correlation between L8µmand total in-
+frared luminosity ( LTIR) for 372 local star-forming galaxies.
+TheconversiongivenbyBavouzetet al.(2008)is:
+LTIR= 377.9×(νLν)0.83
+rest8µm(±37%) (3)
+Caputi etal. (2007) further constrained the sample to lumi-
+nous, high S/N galaxies ( νL8µm
+ν>1010L⊙and S/N>3in all
+MIPS bands) in order to better match their sample, and derive d
+thefollowingequation.Fig.4.Restframe 8 µm LFs based on the AKARI NEP-Deep
+field. The blue diamons, purple triangles, red squares, and o r-
+ange crosses show the 8 µm LFs at 0.38< z <0.58,0.65<
+z <0.90,1.1< z <1.4, and1.8< z <2.2, respectively.
+AKARI’s MIR filters can observe restframe 8 µm at these red-
+shifts in a corresponding filter. Errorbars are from the Mont e
+Caro simulations ( §2.4). The dotted lines show analytical fits
+with a double-power law. Vertical arrows show the 8 µm lumi-
+nosity corresponding to the flux limit at the central redshif t in
+each redshift bin. Overplotted are Babbedgeet al. (2006) in the
+pink dash-dotted lines, Caputiet al. (2007) in the cyan dash -
+dotted lines, and Huanget al. (2007) in the green dash-dotte d
+lines.AGNsareexcludedfromthe sample( §2.2).
+LTIR= 1.91×(νLν)1.06
+rest8µm(±55%) (4)
+Since ours is also a sample of bright galaxies, we use this
+equation to convert L8µmtoLTIR. Because the conversion is
+based on local star-forming galaxies, it is a concern if it ho lds
+at higher redshift or not. Bavouzetet al. (2008) checked thi s by
+stacking 24 µm sources at 1.3< z <2.3in the GOODS fields
+to find the stacked sources are consistent with the local rela -
+tion. They concluded that equation (3) is valid to link L8µm
+andLTIRat1.3< z <2.3. Takagiet al. (2010) also show
+that local L7.7µmvsLTIRrelation holds true for IR galaxies
+at z∼1 (see their Fig.10). Popeetal. (2008) showed that z∼2
+sub-millimeter galaxies lie on the relation between LTIRand
+LPAH,7.7that has been established for local starburst galaxies.
+S70/S24ratios of 70 µm sources in Papovichet al. (2007) are
+also consistent with local SED templates. These results sug gest
+it isreasonabletouse equation(4) foroursample.
+The conversion, however, has been the largest source of er-
+rorinestimating LTIRfromL8µm.Bavouzetet al.(2008)them-
+selvesquote37%ofuncertainty,andthatCaputietal.(2007 )re-
+port 55% of dispersion around the relation. It should be kept in
+mind that the restframe 8µm is sensitive to the star-formation
+activity, but at the same time, it is where the SED models have
+strongest discrepancies due to the complicated PAH emissio n
+lines. A detailed comparison of different conversions is pr e-
+sented in Fig.12 of Caputiet al. (2007), who reported factor of
+∼5ofdifferencesamongvariousmodels.6 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+Then the 8 µm LF is weighted by the LTIRand integrated
+to obtain TIR density. For integration, we first fit an ana-
+lytical function to the LFs. In the literature, IR LFs were
+fit better by a double-power law (Babbedgeet al., 2006) or
+a double-exponential (Saunderset al., 1990; Pozziet al., 2 004;
+Takeuchiet al., 2006; Le Floc’het al., 2005) than a Schechte r
+function, which declines too suddenlly at the high luminosi ty,
+underestimating the number of bright galaxies. In this work ,
+we fit the 8 µm LFs using a double-powerlaw (Babbedgeet al.,
+2006)asshownbelow.
+Φ(L)dL/L∗= Φ∗/parenleftbiggL
+L∗/parenrightbigg1−α
+dL/L∗,(L < L∗) (5)
+Φ(L)dL/L∗= Φ∗/parenleftbiggL
+L∗/parenrightbigg1−β
+dL/L∗,(L > L∗) (6)
+First, the double-powerlaw is fitted to the lowest redshift L F at
+0.38< z <0.58 to determine the normalization( Φ∗) and slopes
+(α,β).Forhigherredshiftswedonothaveenoughstatisticstosi -
+multaneouslyfit 4parameters( Φ∗,L∗,α,andβ).Therefore,we
+fixedtheslopesandnormalizationat the localvaluesandvar ied
+onlyL∗atforthehigher-redshiftLFs.Fixingthefaint-endslope
+isacommonprocedurewiththedepthofcurrentIRsatellites ur-
+veys (Babbedgeet al., 2006; Caputi etal., 2007). The strong er
+evolution in luminosity than in density found by previous wo rk
+(P´ erez-Gonz´ alezet al., 2005; LeFloc’het al., 2005) also justi-
+fies this parametrization. Best fit parameters are presented in
+Table2.Oncethebest-fitparametersarefound,weintegrate the
+doublepowerlawoutsidetheluminosityrangeinwhichwehav e
+data to obtain estimate of the total infrared luminosity den sity,
+ΩTIR.
+The resulting total luminosity density ( ΩIR) is shown in
+Fig.5 as a function of redshift. Errors are estimated by vary ing
+thefit within1 σofuncertaintyin LFs, thenerrorsin conversion
+fromL8µmtoLTIRare added. The latter is by far the larger
+source of uncertainty. Simply switching from equation (3) ( or-
+ange dashed line) to (4) (red solid line) produces a ∼50% dif-
+ference. Cyan dashed lines show results from LeFloc’het al.
+(2005) for a comparision. The lowest redshift point was cor-
+rectedfollowingMagnellietal. (2009).
+We also show the evolution of monochromatic 8 µm lumi-
+nosity (L8µm), which is obtained by integrating the fits, but
+without converting to LTIRin Fig.6. The Ω8µmevolves as
+∝(1+z)1.9±0.7.
+The SFR and LTIRare related by the following equation
+for a Salpeter IMF, φ(m)∝m−2.35between0.1−100M⊙
+(Kennicutt,1998).
+SFR(M⊙yr−1) = 1.72×10−10LTIR(L⊙) (7)
+The right ticks of Fig.5 shows the star formation density
+scale,convertedfrom ΩIRusingtheaboveequation.
+In Fig.5, ΩIRmonotonically increases toward higher z.
+Comparedwith z=0,ΩIRis∼10timeslargerat z=1.Theevolu-
+tionbetween z=0.5andz=1.2isalittleflatter,butthisisperhaps
+duetoamoreirregularshapeofLFsat0.65 < z <0.90,andthus,
+wedonotconsideritsignificant.Theresultsobtainedherea gree
+with previous work (e.g., Le Floc’het al., 2005) within the e r-
+rors. We compare the results with previous work in more detai l
+in§4.Fig.5.Evolution of TIR luminosity density computed by inte-
+grating the 8 µm LFs in Fig.4.The red solid lines use the con-
+version in equation (4). The orange dashed lines use equatio n
+(3).ResultsfromLeFloc’hetal.(2005)areshownwiththecy an
+dottedlines.
+Fig.6.Evolution of 8 µm IR luminosity density computed by
+integrating the 8 µm LFs in Fig.4. The lowest redshift point is
+fromHuanget al.(2007).
+3.3. 12µm LF
+In this subsection we estimate restframe 12 µm LFs based
+on the AKARI NEP-Deep data. 12 µm luminosity ( L12µm)
+has been well studied through ISO and IRAS, and known to
+correlate closely with TIR luminosity (Spinoglioetal., 19 95;
+P´ erez-Gonz´ alezet al.,2005).
+As was the case for the 8 µm LF, it is advantageous that
+AKARI’s continuous filters in the mid-IR allow us to estimate
+restframe 12 µm luminosity without much extrapolation based
+onSEDmodels.Gotoet al.:InfraredLuminosityfunctions withthe AKARI 7
+Table 2.Best fit parametersfor8,12 µmandTIRLFs
+Redshift λ L∗(L⊙)Φ∗(Mpc−3dex−1)α β
+0.38<z<0.58 8 µm (2.2+0.3
+−0.1)×1010(2.1+0.3
+−0.4)×10−31.75+0.01
+−0.013.5+0.2
+−0.4
+0.65<z<0.90 8 µm (2.8+0.1
+−0.1)×10102.1×10−31.75 3.5
+1.1<z<1.4 8 µm (3.3+0.2
+−0.2)×10102.1×10−31.75 3.5
+1.8<z<2.2 8 µm (8.2+1.2
+−1.8)×10102.1×10−31.75 3.5
+0.15<z<0.35 12 µm (6.8+0.1
+−0.1)×109(4.2+0.7
+−0.6)×10−31.20+0.01
+−0.022.9+0.4
+−0.2
+0.38<z<0.62 12 µm (11.7+0.3
+−0.5)×1094.2×10−31.20 2.9
+0.84<z<1.16 12 µm (14+2
+−3)×1094.2×10−31.20 2.9
+0.2<z<0.5 Total (1.2+0.1
+−0.2)×1011(5.6+1.5
+−0.2)×10−41.8+0.1
+−0.43.0+1.0
+−1.0
+0.5<z<0.8 Total (2.4+1.8
+−1.6)×10115.6×10−41.8 3.0
+0.8<z<1.2 Total (3.9+2.3
+−2.2)×10115.6×10−41.8 3.0
+1.2<z<1.6 Total (14+1
+−2)×10115.6×10−41.8 3.0
+Fig.7.12µm luminosity distributions of samples used to com-
+puterestframe12 µmLFs. Fromlowredshift,335,573,and213
+galaxiesarein eachredshiftbin.
+Targeted redshifts are z=0.25, 0.5 and 1 where L15,L18W
+andL24filterscovertherestframe12 µm,respectively.Wesum-
+marise the filters used in Table 1. Methodology is the same as
+for the 8µm LF; we used the sample to the 5 σlimit, corrected
+for the completeness, then used the 1/ Vmaxmethod to com-
+pute LF in each redshift bin. The histogram of L12µmdistri-
+bution is presented in Fig.7. The resulting 12 µm LF is shown
+in Fig.8. Compared with Rush,Malkan,& Spinoglio (1993)’s
+z=0 LF based on IRAS Faint Source Catalog, the 12 µm LFs
+show steady evolution with increasing redshift. In the rang e of
+0.25< z <1,L∗
+12µmevolvesas ∝(1+z)1.5±0.4.
+3.4. Bolometric IR luminosity density basedonthe 12 µm
+LF
+12µm is one of the most frequentlyused monochromaticfluxes
+to estimate LTIR. The total infrared luminosity is computed
+from theL12µmusing the conversionin Chary& Elbaz (2001);
+P´ erez-Gonz´ alezet al.(2005).
+logLTIR= log(0.89+0.38
+−0.27)+1.094logL12µm (8)Fig.8.Restframe 12 µm LFs based on the AKARI NEP-Deep
+field.Thebluediamonds,purpletriangles,andredsquaress how
+the 12µm LFs at 0.15< z <0.35,0.38< z <0.62, and
+0.84< z <1.16, respectively. Vertical arrows show the 12 µm
+luminosity corresponding to the flux limit at the central red -
+shift in each redshift bin. Overplotted are P´ erez-Gonz´ al ezet al.
+(2005) at z=0.3,0.5 and 0.9 in the cyan dash-dotted lines, and
+Rush,Malkan,& Spinoglio (1993) at z=0 in the green dash-
+dottedlines. AGNsareexcludedfromthesample( §2.2).
+Takeuchietal. (2005) independently estimated the relatio n
+tobe
+logLTIR= 1.02+0.972logL12µm, (9)
+which we also use to check our conversion. As both au-
+thors state, these conversions contain an error of factor of 2-3.
+Therefore, we should avoid conclusions that could be affect ed
+bysucherrors.
+Then the 12 µm LF is weighted by the LTIRand integrated
+to obtain TIR density. Errors are estimated by varying the fit
+within 1σof uncertainty in LFs, and errors in converting from
+L12µmtoLTIRareadded.Thelatter isbyfarthe largestsource
+of uncertainty. Best fit parameters are presented in Table 2. In
+Fig.10,we showtotal luminositydensitybasedonthe12 µmLF8 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+Fig.9.Evolution of 12 µm IR luminosity density computed by
+integratingthe12 µmLFsinFig.8.
+Fig.10. TIR luminosity density computed by integrating the
+12µmLFsin Fig.8.
+presented in Fig.8. The results show a rapid increase of ΩIR,
+agreeing with previous work (LeFloc’hetal., 2005) within t he
+errors.
+We also integrate monochromatic L12µmover the LFs
+(without converting to LTIR) to derive the evolution of to-
+tal12µmmonochromatic luminosity density, Ω12µm. The re-
+sults are shown in Fig.9, which shows a strong evolution of
+Ω12µm∝(1 +z)1.4±1.4. It is interesting to compare this to
+Ω8µm∝(1 +z)1.9±0.7obtained in §3.2. Although errors are
+significantonbothestimates, Ω12µmandΩ8µmshowa possibly
+differentevolution,suggestingthatthecosmicinfrareds pectrum
+changesits SED shape.Whetherthisisdueto evolutionindus t,
+or dusty AGN contribution is an interesting subject for futu re
+work.Fig.11.An example of the SED fit. The red dashed line shows
+thebest-fitSEDfortheUV-optical-NIRSED,mainlytoestima te
+photometricredshift.Thebluesolidlineshowsthebest-fit model
+fortheIRSEDat λ >6µm,toestimate LTIR.
+3.5. TIRLF
+AKARI’scontinuousmid-IRcoverageisalsosuperiorforSED -
+fitting to estimate LTIR, since for star-forming galaxies, the
+mid-IR part of the IR SED is dominated by the PAH emissions
+whichreflectthe SFR ofgalaxies,andthus,correlateswell w ith
+LTIR, which is also a good indicator of the galaxy SFR. The
+AKARI’scontinuousMIRcoveragehelpsustoestimate LTIR.
+After photometric redshifts are estimated using the UV-
+optical-NIRphotometry,we fix the redshift at the photo- z,then
+use the same LePhare code to fit the infrared part of the SED
+to estimate TIR luminosity. We used Lagache,Dole,&Puget
+(2003)’s SED templates to fit the photometryusing the AKARI
+bands at >6µm (S7,S9W,S11,L15,L18WandL24). We
+showanexampleoftheSEDfitinFig.11,wherethereddashed
+and blue solid lines show the best-fit SEDs for the UV-optical -
+NIR and IR SED at λ >6µm, respectively. The obtained total
+infraredluminosity( LTIR) is shown as a functionofredshift in
+Fig.12,withspectroscopicgalaxiesinlargetriangles.Th efigure
+shows that the AKARI can detect LIRGs ( LTIR>1011L⊙)
+up toz=1 and ULIRGs ( LTIR>1012L⊙) toz=2. We also
+checkedthatusingdifferentSEDmodels(Chary& Elbaz,2001 ;
+Dale& Helou,2002) doesnotchangeouressentialresults.
+Galaxies in the targeted redshift range are best sampled in
+the 18µm band due to the wide bandpass of the L18Wfilter
+(Matsuharaet al., 2006). In fact, in a single-band detectio n, the
+18µm image returns the largest number of sources. Therefore,
+we applied the 1/ Vmaxmethod using the detection limit at
+L18W. We also checked that using the L15flux limit does
+not change our main results. The same Lagache,Dole,&Puget
+(2003)’s models are also used for k-corrections necessary to
+compute VmaxandVmin. The redshift bins used are 0.2 <
+z <0.5,0.5< z <0.8,0.8< z <1.2,and 1.2 < z <1.6. A distri-
+butionof LTIRineachredshiftbinis showninFig.13.
+Theobtained LTIRLFsareshowninFig.14.Theuncertain-
+ties are esimated through the Monte Carlo simulations ( §2.4).
+For a local benchmark, we overplot Sanderset al. (2003) who
+derived LFs from the analytical fit to the IRAS Revised Bright
+Galaxy Sample, i.e., φ∝L−0.6forL < L∗andφ∝L−2.2for
+L > L∗withL∗= 1010.5L⊙. The TIR LFs show a strong evo-
+lutioncomparedtolocalLFs.At 0.25< z <1.3,L∗
+TIRevolvesGotoet al.:InfraredLuminosityfunctions withthe AKARI 9
+Fig.12.TIR luminosity is shown as a function of photometric
+redshift. The photo- zis estimated using UV-optical-NIR pho-
+tometry.LTIRisobtainedthroughSED fit in7-24 µm.
+Fig.13.AhistogramofTIRluminosity.Fromlow-redshift,144,
+192, 394, and 222 galaxies are in 0.2 < z <0.5, 0.5< z <0.8,
+0.8< z <1.2,and1.2 < z <1.6,respectively.
+as∝(1 +z)4.1±0.4. We further compare LFs to the previous
+workin§4.
+3.6. Bolometric IR luminosity density basedonthe TIRLF
+Using the same methodology as in previous sections, we inte-
+grateLTIRLFs in Fig.14 through a double-power law fit (eq.
+5 and 6). The resulting evolution of the TIR density is shown
+with red diamonds in Fig.15, which in in good agreement with
+LeFloc’hetal.(2005)withintheerrors.Errorsareestimat edby
+varying the fit within 1 σof uncertainty in LFs. For uncertainty
+intheSEDfit,weadded0.15dexoferror.Bestfitparametersar e
+presented in Table 2. In Fig.15, we also show the contributio ns
+toΩTIRfromLIRGsandULIRGswiththebluesquaresandor-
+ange triangles, respectively. We further discuss the evolu tion of
+ΩTIRin§4.Fig.14.TIRLFs.Verticallinesshowtheluminositycorrespond-
+ing to the flux limit at the central redshift in each redshift b in.
+AGNsareexcludedfromthesample( §2.2).
+Fig.15. TIR luminosity density (red diamonds) computed by
+integrating the total LF in Fig.14. The blue squares and oran ge
+trianglesareforLIRG andULIRGsonly.
+4. Discussion
+4.1. Comparison with previouswork
+In this section, we compare our results to previous work, esp e-
+ciallythosebasedontheSpitzerdata.Comparisonsarebest done
+inthesamewavelengths,sincetheconversionfromeither L8µm
+orL12µmtoLTIRinvolves the largest uncertainty. Hubble pa-
+rametersinthepreviousworkareconvertedto h= 0.7forcom-
+parison.10 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+4.1.1. 8µm LFs
+Recently, using the Spitzer space telescope, restframe 8 µm LFs
+ofz∼1 galaxies have been computed in detail by Caputiet al.
+(2007) in the GOODS fields and by Babbedgeetal. (2006) in
+theSWIREfield.Inthissection,wecompareourrestframe8 µm
+LFs(Fig.4)tothese anddiscusspossibledifferences.
+In Fig.4, we overplot Caputi etal. (2007)’s LFs at z=1 and
+z=2inthecyandash-dottedlines.Their z=2LFisingoodagree-
+ment with our LF at 1.8 < z <2.2. However, their z=1 LF is
+larger than ours by a factor of 3-5 at logL >11.2. Note that
+the brightest ends( logL∼11.4)are consistent with each other
+to within 1 σ. They have excluded AGN using optical-to-X-ray
+flux ratio, and we also have excluded AGN through the optical
+SED fit. Therefore, especially at the faint-end, the contami na-
+tionfromAGN isnot likelyto be the maincauseof differences .
+Since Caputiet al. (2007) uses GOODS fields, cosmic variance
+may play a role here. The exact reason for the difference is un -
+known, but we point out that their ΩIRestimate at z=1 is also
+higherthanotherestimatesbyafactorofafew(seetheirFig .15).
+Once converted into LTIR, Magnelliet al. (2009) also reported
+Caputiet al.(2007)’s z=1LF ishigherthantheirestimatebased
+on 70µm by a factor of several (see their Fig.12). They con-
+cluded the difference is from different SED models used, sin ce
+their LF matched with that of Caputi etal. (2007)’s once the
+same SED models were used. We will compare our total LFs
+tothosein theliteraturebelow.
+Babbedgeet al. (2006) also computed restframe 8 µm LFs
+using the Spitzer/SWIRE data. We overplot their results at
+0.25< z <0.5and0.5< z <1in Fig.4 with the pink dot-
+dashedlines.Inbothredshiftranges,goodagreementisfou ndat
+higherluminositybins( L8µm>1010.5L⊙).However,atallred-
+shift ranges including the ones not shown here, Babbedgeet a l.
+(2006) tends to show a flatter faint-end tail than ours, and a
+smallerφby a factor of ∼3. Although the exact reason is un-
+known, the deviation starts toward the fainter end, where bo th
+works approach the flux limits of the surveys. Therefore,pos si-
+blyincompletesamplingmaybeoneofthereasons.Itisalsor e-
+portedthat thefaint-endof IRLFsdependson theenvironmen t,
+in the sense that higher-density environment has steeper fa int-
+end tail (Gotoet al., 2010). Note that at z=1, Babbedgeet al.
+(2006)’s LF (pink) deviates from that by Caputiet al. (2007)
+(cyan) by almost a magnitude. Our 8 µm LFs are between these
+works.
+These comparisons suggest that even with the current gen-
+eration of satellites and state-of-the-art SED models, fac tor-of-
+several uncertainties still remain in estimating the 8 µm LFs
+at z∼1. More accurate determination has to await a larger
+and deeper survey by the next generation IR satellites such a s
+HerschelandWISE.
+To summarise, our 8 µm LFs are between those by
+Babbedgeetal.(2006)andCaputiet al.(2007),anddiscrepa ncy
+is by a factor of several at most. We note that both of the previ -
+ous works had to rely on SED models to estimate L8µmfrom
+the Spitzer S24µmin the MIR wavelengths where SED model-
+ing is difficult. Here, AKARI’s mid-IR bands are advantageou s
+indirectlyobservingredshiftedrestframe8 µmfluxinoneofthe
+AKARI’s filters, leading to more reliable measurement of 8 µm
+LFswithoutuncertaintyfromtheSED modeling.
+4.1.2. 12 µm LFs
+P´ erez-Gonz´ alezet al. (2005) investigated the evolution of rest-
+frame12µmLFsusingthe SpitzerCDF-S andHDF-N data.Weoverplot their results in similar redshift ranges as the cya n dot-
+dashed lines in Fig.8. Consideringboth LFs have significant er-
+ror bars, these LFs are in good agreement with our LFs, and
+show significant evolution in the 12 µm LFs compared with the
+z=012µmLFbyRush,Malkan,&Spinoglio(1993).Theagree-
+ment is in a stark contrast to the comparison in 8 µm LFs in
+§4.1.1, wherewe sufferedfromdifferncesof a factor of sever al.
+Apossiblereasonforthisisthat12 µmissufficientlyredderthan
+8µm, that it is easier to be extrapolated from S24µmin case of
+the Spitzer work. In fact, at z=1, both the Spitzer 24 µm band
+and AKARI L24observe the restframe 12 µm directly. In addi-
+ton, mid-IR SEDs around 12 µm are flatter than at 8 µm, where
+PAH emissions are prominent.Therefore,SED modelscan pre-
+dict the flux more accurately. In fact, this is part of the rea-
+sonwhyP´ erez-Gonz´ alezet al.(2005)chosetoinvestigate 12µm
+LFs. P´ erez-Gonz´ alezetal. (2005) used Chary&Elbaz (2001 )’s
+SEDtoextrapolate S24µm,andyet,theyagreewellwithAKARI
+results, which are derived from filters that cover the restfr ame
+12µm. However, in other words, the discrepancy in 8 µm LFs
+highlights the fact that the SED models are perhaps still imp er-
+fect in the 8 µm wavelengthrange, and thus, MIR-spectroscopic
+data that covers wider luminosity and redshift ranges will b e
+necessary to refine SED models in the mid-IR. AKARI’s mid-
+IR slitless spectroscopy survey (Wada, 2008) may help in thi s
+regard.
+4.1.3. TIRLFs
+Lastly,we compareourTIRLFs(Fig.14) withthoseinthelite r-
+ature.AlthoughtheTIRLFs canalso be obtainedbyconvertin g
+8µmLFsor12 µmLFs,wealreadycomparedourresultsinthese
+wavelengths in the last subsections. Here, we compare our TI R
+LFstoLe Floc’het al.(2005)andMagnellietal. (2009).
+LeFloc’het al. (2005) obtained TIR LFs using the Spitzer
+CDF-S data. They have used the best-fit SED among various
+templatestoestimate LTIR.WeoverplottheirtotalLFsinFig.14
+with the cyan dash-dotted lines. Only LFs that overlapwith o ur
+redshit ranges are shown. The agreement at 0.3< z <0.45
+and0.6< z <0.8is reasonable, considering the error bars on
+bothsides.However,inallthreeredshiftranges,LeFloc’h et al.
+(2005)’sLFsare higherthanours,especiallyfor 1.0< z <1.2.
+We also overplot TIR LFs by Magnellietal. (2009), who
+used Spitzer 70 µm flux and Chary& Elbaz (2001)’s model to
+estimateLTIR.Inthetwobins(centeredon z=0.55and z=0.85;
+pink dash-dotted lines) which closely overlap with our reds hift
+bins, excellent agreement is found. We also plot Huynhet al.
+(2007)’s LF at 0.6< z <0.9in the navy dash-dotted lines,
+whichis computedfromSpitzer 70µmimagingin the GOODS-
+N, and this also shows very good agreement with ours. These
+LFs are on top of each other within the error bars, despite the
+fact that these measurements are from different data sets us ing
+differentanalyses.
+This means that LeFloc’hetal. (2005)’s LFs is also higher
+thanthatofMagnelliet al.(2009),inadditiontoours.Apos sible
+reasonis that both Magnelliet al. (2009) and we removedAGN
+(at least bright ones), whereas Le Floc’het al. (2005) inclu ded
+them. This also is consistent with the fact that the differen ce
+is larger at 1.0< z <1.2where both surveys are only sen-
+sitive to luminous IR galaxies, which are dominated by AGN.
+Another possible source of uncertainty is that Magnelliet a l.
+(2009) and we used a single SED library, while LeFloc’het al.
+(2005)pickedthebestSEDtemplateamongseverallibraries for
+eachgalaxy.Gotoet al.:InfraredLuminosityfunctions withthe AKARI 11
+Fig.16.EvolutionofTIRluminositydensitybasedonTIRLFs(redcir cles),8µmLFs(stars),and12 µmLFs(filledtriangles).The
+blue open squaresand orangefilled squaresare for LIRG and UL IRGs only, also based on our LTIRLFs. Overplotteddot-dashed
+lines are estimates from the literature: LeFloc’het al. (20 05), Magnelliet al. (2009) , P´ erez-Gonz´ alezet al. (2005) , Caputiet al.
+(2007), and Babbedgeet al. (2006) are in cyan, yellow, green , navy,and pink, respectively.The purple dash-dottedline shows UV
+estimatebySchiminovichetal. (2005).Thepinkdashedline showsthetotalestimateofIR(TIRLF)andUV (Schiminoviche t al.,
+2005).
+4.2. Evolution of ΩIR
+In this section, we compare the evolution of ΩIRas a function
+ofredshift.InFig.16, weplot ΩIRestimatedfromTIRLFs(red
+circles), 8 µm LFs (brown stars), and 12 µm LFs (pink filled tri-
+angles),as a functionof redshift.Estimatesbased on12 µmLFs
+and TIR LFs agree each other very well, while those from 8 µm
+LFs show a slightly higher value by a factor of a few than oth-
+ers. This perhaps reflects the fact that 8 µm is a more difficult
+part of the SED to be modeled, as we had a poorer agreement
+amongpapersintheliteraturein8 µmLFs.Thebright-endslope
+of the double-power law was 3.5+0.2
+−0.4in Table 2. This is flat-
+ter than a Schechter fit by Babbedgeet al. (2006) and a double-
+exponential fit by Caputiet al. (2007). This is perhaps why we
+obtainedhigher ΩIRin8µm.
+We overplot estimates from various papers in the litera-
+ture(LeFloc’hetal.,2005; Babbedgeet al.,2006;Caputiet al.,
+2007; P´ erez-Gonz´ alezet al., 2005; Magnelliet al., 2009) in the
+dash-dottedlines. Our ΩIRhasverygoodagreementwith these
+at0< z <1.2,withalmostallthedash-dottedlineslyingwithin
+ourerrorbarsof ΩIRfromLTIRand12µmLFs.Thisisperhaps
+because an estimate of an integrated value such as ΩIRis more
+reliablethanthat ofLFs.
+Atz >1.2, ourΩIRshows a hint of continuous increase,
+while Caputiet al. (2007) and Babbedgeetal. (2006) observe da slight decline at z >1. However,as both authorsalso pointed
+out, at this high-redshift range, both the AKARI and Spitzer
+satellites are sensitive to onlyLIRGs and ULIRGs, and thust he
+extrapolationto fainterluminositiesassumesthefaint-e ndslope
+of the LFs, which couldbe uncertain.In addition,this work h as
+a poorerphoto-zestimate at z >0.8(∆z
+1+z=0.10)due to the rel-
+atively shallow near-IR data. Several authors tried to over come
+thisproblembystackingundetectedsources.However,ifan un-
+detectedsourceisalsonotdetectedatshorterwavelengths where
+positions for stacking are obtained, it would not be include d in
+the stacking either. Next generation satellite such as Hers chel,
+WISE, and SPICA (Nakagawa, 2008) will determine the faint-
+endslopeat z >1moreprecisely.
+We parameterize the evolution of ΩIRusing a following
+function.
+ΩIR(z)∝(1+z)γ(10)
+By fitting this to the ΩIRfrom TIR LFs, we obtained γ=
+4.4±1.0. This is consistent with most previous works.
+For example, LeFloc’hetal. (2005) obtained γ= 3.9±
+0.4, P´ erez-Gonz´ alezet al. (2005) obtained γ= 4.0±0.2,
+Babbedgeetal. (2006) obtained γ= 4.5+0.7
+−0.6, Magnelliet al.
+(2009) obtained γ= 3.6±0.4. The agreement was expected
+fromFig.16,butconfirmsastrongevolutionof ΩIR.12 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
+Fig.17. Contribution of ΩTIRtoΩtotal= ΩUV+ ΩTIRis
+shownasa functionofredshift.
+4.3. Differential evolution among ULIRG,LIRG,normal
+galaxies
+In Fig. 15, we also plot the contributions to ΩIRfrom LIRGs
+and ULIRGs (measured from TIR LFs) with the blue open
+squares and orange filled squares, respectively. Both LIRGs
+and ULIRGs show strong evolution, as has been seen for to-
+talΩIRin the red filled circles. Normal galaxies ( LTIR<
+1011L⊙) are still dominant, but decrease their contribution to-
+ward higher redshifts. In contrast, ULIRGs continueto incr ease
+their contribution. From z=0.35 to z=1.4,ΩIRby LIRGs in-
+creases by a factor of ∼1.6, andΩIRby ULIRGs increases by
+a factor of ∼10. The physical origin of ULIRGs in the local
+Universe is often merger/interaction(Sanders& Mirabel, 1 996;
+Taniguchi&Shioya, 1998; Goto, 2005). It would be interesti ng
+to investigate whether the merger rate also increases in pro por-
+tion to the ULIRG fraction, or if different mechanisms can al so
+produceULIRGsathigherredshift.
+4.4. Comparison tothe UVestimate
+We have been emphasizing the importance of IR probes of the
+total SFRD of the Universe. However, the IR estimates do not
+take into account the contribution of the unabsorbed UV ligh t
+produced by the young stars. Therefore, it is important to es ti-
+matehowsignificantthisUV contributionis.
+Schiminovichet al. (2005) found that the energy density
+measured at 1500 ˚A evolves as ∝(1+z)2.5±0.7at0< z <1
+and∝(1 +z)0.5±0.4atz >1. using the GALEX data sup-
+plemented by the VVDS spectroscopic redshifts. We overplot
+their UV estimate of ρSFRwith the purple dot-dashed line in
+Fig.16. The UV estimate is almost a factor of 10 smaller than
+the IR estimate at most of the redshifts, confirming the impor -
+tanceofIRprobeswheninvestingtheevolutionofthetotalc os-
+mic star formation density. In Fig.16 we also plot total SFD ( or
+Ωtotal)byadding ΩUVandΩTIR,withthemagentadashedline.
+In Fig.17, we show the ratio of the IR contribution to the to-
+tal SFRD of the Universe ( ΩTIR/ΩTIR+ ΩUV) as a function
+of redshift. Although the errors are large, Fig.17 agrees wi thTakeuchi,Buat,& Burgarella (2005), and suggests that ΩTIR
+explains 70% of Ωtotalatz=0.25, and that by z=1.3, 90% of
+the cosmic SFD is explained by the infrared. This implies tha t
+ΩTIRprovidesgoodapproximationofthe Ωtotalatz >1.
+5. Summary
+We have estimated restframe 8 µm, 12µm, and total infrared lu-
+minosity functions using the AKARI NEP-Deep data. Our ad-
+vantage over previous work is AKARI’s continuous filter cov-
+erage in the mid-IR wavelengths (2.4, 3.2, 4.1, 7, 9, 11, 15, 1 8,
+and24µm),whichallowustoestimate mid-IRluminositywith-
+out a large extrapolationbased on SED models, which were the
+largest uncertainty in previous work. Even for LTIR, the SED
+fitting is much more reliable due to this continuouscoverage of
+mid-IRfilters.
+Ourfindingsareasfollows:
+–8µm LFs show a strong and continuous evolution from
+z=0.35 to z=2.2. Our LFs are larger than Babbedgeet al.
+(2006), but smaller than Caputi etal. (2007). The differenc e
+perhaps stems from the different SED models, highlighting
+a difficulty in SED modeling at wavelengths crowded by
+strong PAH emissions. L∗
+8µmshows a continuous evolution
+asL∗
+8µm∝(1+z)1.6±0.2in therangeof 0.48< z <2.
+–12µm LFs show a strong and continuous evolution from
+z=0.15toz=1.16with L∗
+12µm∝(1+z)1.5±0.4. Thisagrees
+well with P´ erez-Gonz´ alezet al. (2005), including a flatte r
+faint-endslope. A better agreementthan with 8 µm LFs was
+obtained, perhaps because of smaller uncertainty in model-
+ing the 12 µm SED, and less extrapolationneededin Spitzer
+24µmobservations.
+–The TIR LFs show good agreement with Magnelliet al.
+(2009), but are smaller than Le Floc’het al. (2005). At
+0.25< z <1.3,L∗
+TIRevolvesas ∝(1+z)4.1±0.4.Possible
+causes of the disagreement include different treatment of
+SEDmodelsinestimating LTIR,andAGNcontamination.
+–TIR densities estimated from 12 µm and TIR LFs show a
+strong evolution as a function of redshift, with ΩIR∝
+(1 +z)4.4±1.0.ΩIR(z)also show an excellent agreement
+withpreviousworkat z <1.2.
+–We investigated the differential contribution to ΩIRby
+ULIRGsandLIRGs.WefoundthattheULIRG(LIRG)con-
+tribution increases by a factor of 10 (1.8) from z=0.35 to
+z=1.4, suggesting IR galaxies are more dominant source of
+ΩIRathigherredshift.
+–We estimated that ΩIRcaptures80% of the cosmic star for-
+mationatredshiftslessthan1,andvirtuallyallofitathig her
+redshift.Thusaddingtheunobscuredstarformationdetect ed
+at UV wavelengths would not change SFRD estimates sig-
+nificantly.
+Acknowledgments
+We are grateful to S.Arnouts for providing the LePhare code,
+and kindly helping us in using the code. We thank the anony-
+mousrefereeformanyinsightfulcomments,whichsignifican tly
+improvedthe paper.
+T.G. and H.I. acknowledgefinancial supportfrom the Japan
+Society for the Promotion of Science (JSPS) through JSPS
+Research Fellowships for Young Scientists. HML acknowl-
+edges the support from KASI through its cooperative fund in
+2008. TTT has been supported by Program for Improvement
+of Research Environment for Young Researchers from SpecialGotoet al.:InfraredLuminosityfunctions withthe AKARI 13
+CoordinationFundsforPromotingScienceandTechnology,a nd
+the Grant-in-Aid for the Scientific Research Fund (20740105 )
+commissioned by the Ministry of Education, Culture, Sports ,
+Science and Technology (MEXT) of Japan. TTT has been also
+partially supported from the Grand-in-Aid for the Global CO E
+Program “Quest for Fundamental Principles in the Universe:
+from Particles to the Solar System and the Cosmos” from the
+MEXT.
+This research is based on the observations with AKARI, a
+JAXA projectwiththe participationofESA.
+Theauthorswishtorecognizeandacknowledgetheverysig-
+nificant cultural role and reverence that the summit of Mauna
+Kea has always had within the indigenous Hawaiian commu-
+nity. We are most fortunate to have the opportunity to conduc t
+observationsfromthissacredmountain.
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+arXiv:1001.0015v2  [astro-ph.CO]  10 May 2010DRAFT VERSION MAY12, 2010
+Preprint typesetusingL ATEX styleemulateapjv. 11/10/09
+ACOMPREHENSIVE ANALYSISOFUNCERTAINTIESAFFECTING THE
+STELLARMASS –HALO MASS RELATIONFOR0 <z<4
+PETERS. BEHROOZI1, CHARLIECONROY2, RISAH. WECHSLER1
+Draftversion May12, 2010
+ABSTRACT
+We conductacomprehensiveanalysisoftherelationshipbet weencentralgalaxiesandtheirhostdarkmatter
+halos, as characterized by the stellar mass – halo mass (SM–H M) relation, with rigorous consideration of
+uncertainties. Our analysis focuses on results from the abu ndance matching technique, which assumes that
+every dark matter halo or subhalo above a specific mass thresh old hosts one galaxy. We provide a robust
+estimate of the SM–HMrelationfor 0 <z<1 anddiscussthe quantitativeeffectsof uncertaintiesin o bserved
+galaxystellarmassfunctions(GSMFs)(includingstellarm assestimatesandcountinguncertainties),halomass
+functions (including cosmology and uncertainties from sub structure), and the abundance matching technique
+used to link galaxies to halos (including scatter in this con nection). Our analysis results in a robust estimate
+of the SM–HM relation and its evolution from z=0 to z=4. The sh ape and evolution are well constrained for
+z<1. The largest uncertainties at these redshifts are due to st ellar mass estimates (0.25 dex uncertainty in
+normalization); however, failure to account for scatter in stellar masses at fixed halo mass can lead to errors
+of similar magnitude in the SM–HM relation for central galax ies in massive halos. We also investigate the
+SM–HM relation to z= 4, although the shape of the relation at higher redshifts re mains fairly unconstrained
+whenuncertaintiesaretakenintoaccount. Wefindthatthein tegratedstarformationatagivenhalomasspeaks
+at 10-20%ofavailable baryonsforall redshiftsfrom0 to 4. T hispeak occursat a halomass of7 ×1011M⊙at
+z=0andthismassincreasesbyafactorof5to z=4. Atlowerandhighermasses,starformationissubstantia lly
+lessefficient,withstellarmassscalingas M∗∼M2.3
+hatlowmassesand M∗∼M0.29
+hathighmasses. Thetypical
+stellarmassforhaloswithmasslessthan1012M⊙hasincreasedby0 .3−0.45dexforhalossince z∼1. These
+resultswill providea powerfultoolto informgalaxyevolut ionmodels.
+Subject headings: dark matter — galaxies: abundances — galaxies: evolution — g alaxies: stellar content —
+methods: N-bodysimulations
+1.INTRODUCTION
+A variety of physical processes are thought to be respon-
+sible for the observed distribution of galaxy properties, a nd
+distinguishing among them is one of the principal goals of
+modern galaxy formation theory. Among the relevant mech-
+anisms are those responsible for galaxy growth, such as star
+formation and galaxy mergers, as well as those responsible
+forregulatinggrowth,includingenergeticfeedbackbysup er-
+novae, active galactic nuclei, cosmic ray pressure, and lon g
+gascoolingtimes.
+A fruitful approach to separating the influence of different
+mechanisms is to constrain the redshift–dependent relatio n
+between physical characteristics of galaxies, such as stel lar
+mass,andthemassoftheirdarkmatterhalos. Thisispossibl e
+because it is expected that many of these physical processes
+depend primarily on the mass of a galaxy’sdark matter halo.
+By connecting galaxies to their parent halos, one is able to
+moreclearlyidentifyandconstrainthephysicalprocesses re-
+sponsibleforgalaxygrowth.
+The galaxystellar mass – halo massrelation hasadditional
+utility because many properties of both galaxies and halos
+are tightly correlated with halo mass. In addition, the stel -
+lar mass – halo mass relation providesa mechanism for con-
+necting predictions for the halo mass function and the mass-
+1Kavli Institute for Particle Astrophysics and Cosmology; P hysics De-
+partment, Stanford University, and SLAC National Accelera tor Labora-
+tory, Stanford, CA 94305
+2Department of Astrophysical Sciences, Princeton Universi ty, Prince-
+ton, NJ 08544dependent spatial clustering of halos to the abundances and
+clustering of galaxies. If a model for galaxy evolution is
+able to reproduce the intrinsic galaxy mass – halo mass re-
+lation in the correct cosmological model, then such a model
+will match both the observed stellar mass function and the
+stellar mass dependent clustering of galaxies. Simultane-
+ously matching these two observational quantities and thei r
+evolution has been difficult with either hydrodynamicalsim -
+ulations or semi-analytic models of galaxy formation (e.g. ,
+Weinbergetal. 2004; Liet al.2007).
+There are several ways to constrain the galaxy mass –
+halo mass relation. The first type of approach attempts
+to directly measure the mass of galactic halos. Tech-
+niques include weak lensing (e.g., Guzik&Seljak 2002;
+Sheldonet al. 2004; Mandelbaumetal. 2006) and the use of
+satellite galaxy or stellar velocities as tracers of the hal o po-
+tentialwell(e.g.,Ashmanetal.1993;Zaritsky& White1994 ;
+Pradaet al. 2003; vandenBoschet al. 2004; Conroyet al.
+2007). While such methods are a relatively direct probe of
+halo mass, they are limited in dynamic range; current obser-
+vationsprobehalo massesfromroughly1012–1014M⊙at the
+present epoch, and a smaller range at higher redshift. A sec-
+ond approach is to identify groups and clusters of galaxies
+either through optical or X-ray selected cluster catalogs, and
+then directly measure their galaxy content (e.g., Lin&Mohr
+2004; Hansenet al. 2009; Yanget al. 2007). This is limited
+to relatively massive halos (although for optically identi fied
+groupsit could extend to lower masses as new surveysprobe
+dimmer galaxies in large enough volumes), and it also re-
+quires accurate knowledge of the mass–observable relation2 BEHROOZI,CONROY& WECHSLER
+(Yangetal. 2007; Hansenetal. 2009).
+An alternative approach is to assume that the properties
+of the halo population are known, for example from cos-
+mological simulations, and then find a functional form re-
+lating galaxies to halos which achieves agreement with a
+variety of observations. This approach is less direct but
+can be applied over a much larger dynamic range. Halo
+occupation (e.g., Berlind&Weinberg 2002; Bullocketal.
+2002; Cooray& Sheth 2002; Tinkeret al. 2005; Zhengetal.
+2007) and conditional luminosity function modeling (e.g.,
+Yanget al.2003; Cooray2006) fallintothiscategory.
+In the past decade, a number of studies have found that
+this latter approach can be greatly simplified using a tech-
+nique called abundance matching. In its most basic form,
+the technique assigns the most massive (or the most lumi-
+nous) galaxiesto the most massive halos monotonically. The
+techniquethusrequiresasinputonlytheobservedabundanc e
+of galaxies as a function of mass, namely the galaxy stel-
+lar mass function (alternatively the galaxy luminosity fun c-
+tion) and the abundance of dark matter halos as a func-
+tion of mass, namely the halo mass function. This tech-
+nique has been shown to accurately reproduce a variety
+of observational results including various measures of the
+redshift– and scale–dependent spatial clustering of galax ies
+(Colínetal. 1999; Kravtsov& Klypin 1999; Neyrincketal.
+2004; Kravtsovet al. 2004; Vale&Ostriker 2004, 2006;
+Tasitsiomi etal. 2004; Conroyetal. 2006; Shankaretal.
+2006; Berrieret al. 2006; Marínet al. 2008; Guoet al. 2009;
+Mosteretal. 2009). In the context of this technique, not
+only central halos but also subhalos (halos contained withi n
+the virial radii of larger halos) are included in the matchin g
+process, meaning that satellite galaxies can be accounted f or
+withoutanyadditionalparameters.
+Applications of the abundance matching technique have
+typically focused on using the default modeling assumption s
+to derive statistical information about the galaxy – halo co n-
+nection. Uncertaintiesinthederivedgalaxymass–halomas s
+relationhavereceivedlittlesystematicattentioninthec ontext
+of this technique (though see Mosteret al. 2009, for a recent
+treatment of the attendant uncertainties). An accurate ass ess-
+ment of the uncertainties is necessary to make strong state-
+ments regarding the underlying physical processes respons i-
+bleforthederivedgalaxy–halorelation. Inthepresentwor k
+we undertake an exhaustive exploration of the uncertaintie s
+relevantinconstructingthegalaxystellarmass–halomass re-
+lationfromtheabundancematchingtechnique. Weconsidera
+rangeofuncertaintiesrelatedtotheobservationalstella rmass
+function,the theoretical halo mass function,and the under ly-
+ing technique of abundance matching. The resulting galaxy
+stellar mass – halo mass relation and associated uncertain-
+tieswill provideabenchmarkagainstwhichgalaxyevolutio n
+modelsmaybefruitfullytested.
+This paper is divided into several sections. In §2 we detail
+known sources of uncertainty which may affect our results.
+Our methodologyfor modeling the effects of uncertainties i s
+discussed in §3, providing simple conversions where possi-
+ble to allow for different modeling choices. We present our
+resulting estimates of the galaxy stellar mass – halo mass re -
+lation for z<1 in §4 and describe the contribution of each
+of the uncertainties to the overall error budget. Estimates for
+theevolutionoftherelationoutto z∼4,forwhichtheuncer-
+tainties are significantly less well-understood, are prese nted
+in §5. Finally, we discuss the implications of this work in §6
+andsummarizeourconclusionsin§7.Stellar masses throughout are quoted assuming a Chabrier
+(2003) initial mass function (IMF), the stellar population
+synthesis models of Bruzual& Charlot (2003), and the age
+and dust models in Blanton& Roweis (2007). We consider
+multiple cosmologies in this paper, but the main results as-
+sume a WMAP5+SN+BAO concordance ΛCDM cosmology
+(Komatsuet al. 2009) with ΩM=0.27,ΩΛ=1−ΩM,h=0.7,
+σ8=0.8,andns=0.96.
+2.UNCERTAINTIESAFFECTINGTHESTELLARMASS
+TOHALO MASS RELATION
+Uncertainties in the abundance matching technique for as-
+signinggalaxiestodarkmatterhaloscanbeconceptuallyse p-
+arated into three classes. The first is uncertainty in the abu n-
+dance of galaxies as a function of stellar mass. This class
+includes both uncertainties in counting galaxies due to sho t
+noiseandsamplevariance,aswellasuncertaintiesinthest el-
+larmassestimatesthemselves. Thesecondclassconcernsth e
+darkmatter halos, andincludesuncertaintiesin cosmologi cal
+parameters,theimpactofbaryoncondensation,andsubstru c-
+ture. Finally, there are uncertainties in the process of mat ch-
+inggalaxiesto halosarising primarilyfromthe intrinsics cat-
+ter between galaxystellar mass and halo mass. Each of these
+sources of uncertainty are described in detail below. The de -
+tailedmodelingoftheseuncertaintiesis describedin§3.
+2.1.UncertaintiesintheStellarMassFunction
+Galaxystellar massesare notmeasureddirectly,but are in-
+stead inferred from photometry and/or spectra. In particul ar,
+as the observed stellar light is a function of many physical
+processes (e.g., stellar evolution, star formation histor y and
+metal–enrichmenthistory,andwavelength–dependentdust at-
+tenuation),stellarmassesareestimatedviacomplicatedm od-
+els to find the best fit to galaxy observations in a very large
+parameter space. Assumptions and simplifications in these
+models, along with the fact that the best fit may be only one
+ofanumberoflikelypossibilities,meanthattherecanbesu b-
+stantialuncertaintiesin calculatedstellar masses.
+Different types of observations can yield different uncer-
+taintiesinthesecalculations. Spectroscopicsurveysgen erally
+recover more spectral features per galaxy and more accurate
+redshifts than photometric surveys. However, spectroscop y
+requires substantially more telescope time than photometr y.
+Therefore, spectroscopic samples tend to be limited both in
+area and depth, which translates into limitations in both vo l-
+ume and the minimum stellar mass probed. An additional
+problem for spectroscopic surveys such as the Sloan Digital
+SkySurvey(SDSS,Yorket al.2000) isthatthespectraprobe
+only the central regionsof galaxies (the SDSS spectra gathe r
+onaverage1 /3 of the total flux fromgalaxiesat z=0.1). For
+galaxies containing both a bulge and a disk, or galaxies with
+radial gradients, the spectra will therefore not provide a f air
+sampleoftheentiregalaxy(e.g.,Kewleyet al.2005);furth er-
+more, this bias will be a function of redshift. For these rea-
+sons,especiallyforrobustcomparisonsofevolution,we co n-
+fine thispaperto photometric–basedstellar masses; howeve r,
+samples with spectroscopic redshifts are used where avail-
+able.
+2.1.1.Principal Uncertainties inStellar Mass Functions
+In this paper, we analyze seven main sources of uncertain-
+ties in stellar mass functions applicable to photometric su r-
+veys,listedbelowinroughorderofimportance.UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 3
+1.Choice of stellar initial mass function (IMF): The lu-
+minosity of stars scales as their mass to a large power
+(dlnL/dlnM∼3.5),whiletheIMF,definedasthenum-
+berofstarsformedperunitmass,scalesapproximately
+asξ∝M−2.3(Salpeter 1955). Thus, the light from a
+galaxy is dominated by the most massive stars, while
+the total stellar mass is dominated by the lowest mass
+stars. This fact, which has been known for decades
+(e.g., Tinsley&Gunn 1976), implies that the assumed
+form of the IMF at M/lessorsimilar0.5M⊙will have no effect on
+the integratedlight from galaxies, but will have a large
+effectonthetotalstellarmass. Nonetheless,constraints
+onthe total dynamicalmassofspheroidalsystemspro-
+vide a valuable independent check on the form of the
+IMF at low masses. Cappellariet al. (2006) find, for
+example, that the IMF proposed by Chabrier (2003)
+for the Solar neighborhood is consistent with dynam-
+ical constraintsonmassesofnearbyellipticals.
+We do not marginalize over IMFs in our error calcula-
+tionsforthereasonthatitisrelativelysimpletoconvert
+fromourchoiceofIMF(Chabrier2003)tootherIMFs.
+For our purposes the IMF becomes a serious source
+of systematic uncertainty only if the IMF varies with
+galaxy properties or if it evolves. It should be noted,
+however,thattheIMF canalsointroducemorecompli-
+catedsystematiceffectsassociatedwiththeinferredstar
+formationrate,whichmayinturnimpactstellarmasses
+in non–trivialways(e.g.Conroyetal. 2010).
+2.Choice of Stellar Population Synthesis (SPS) model:
+SPS modeling efforts have grown substantially in so-
+phistication in the past decade (e.g. Leithereretal.
+1999; Bruzual&Charlot 2003; LeBorgneet al. 2004;
+Maraston 2005). Yet, significant uncertainties re-
+main (e.g., Charlotetal. 1996; Charlot 1996; Yi 2003;
+Lee etal. 2007; Conroyetal. 2009). Treatments of
+convection vary, leading to different main sequence
+turn off times for intermediate mass stars. Advanced
+stages of stellar evolution, including blue stragglers,
+thermally–pulsating AGB stars (TP–AGB), and hori-
+zontal branch stars, are poorly understood, both obser-
+vationally and theoretically. The theoretical spectral
+libraries contain known deficiencies, especially for M
+giants, where the effective temperatures are low and
+where hydrodynamic effects become important. Em-
+pirical stellar libraries to some extent circumventthese
+issues, althoughthey are plaguedbyincompletecover-
+age in the HR diagram and difficulties associated with
+deriving stellar parameters. See Conroyet al. (2009)
+andPercival&Salaris(2009) forrecentreviews.
+Differentstellarpopulationsynthesismodelstreatthese
+issues differently, which can result in large system-
+atic differences in the derived stellar mass. For
+instance, the model of Maraston (2005) compared
+to Bruzual&Charlot (2003) has systematic differ-
+ences of 0.1dex in stellar mass (Salimbeniet al. 2009;
+Pérez-Gonzálezetal. 2008). However,Salimbenietal.
+(2009) reports that the model in Bruzual (2007) (with
+a revisedtreatmentofTP-AGBstars)yieldssystematic
+differencesinstellarmassrelativetoBruzual&Charlot
+(2003), which ranges from 0.05dex for 1011M⊙galax-
+iesto0.3dexfor109.5M⊙galaxies. Conroyet al.(2009)
+show thatuncertaintyin theluminosityofthe TP-AGBphasecanshiftstellar massesbyasmuchas ±0.2dex.
+3.Parameterization of star formation histories: In or-
+der to estimate stellar masses, model libraries are con-
+structed with a large range in star formation histories
+(SFHs),dustattenuation(seebelow),and,oftenbutnot
+always,metallicity. Theadoptedfunctionalformofthe
+SFHisanothersourceofsystematicuncertainty,astyp-
+ically very simple functional forms are assumed. Sev-
+eral authors have investigated various aspects of this
+problem. When attempting to model observed pho-
+tometry, Pérez-Gonzálezet al. (2008) found that a sin-
+gle stellar population model (in particular, star forma-
+tion proportional to e−t/τ, which is a commonly-used
+parameterization) systematically underpredicts stellar
+mass by 0.18 dex comparedto a double stellar popula-
+tion model (exponentially decaying star formation fol-
+lowed by a later starburst). The particular parameteri-
+zation of SFHs may also lead to systematic differences
+asafunctionofstellarmass. Leeet al.(2009)analyzed
+a sample of mock Lyman–break galaxies at z∼4−5
+and found that simple SFHs produced best–fit stellar
+masses that were under or overestimated by ∼ ±50%
+dependingontherest-framegalaxycolor. Thisbiaswas
+attributedto thechaoticSFHsofthemockgalaxies.
+4.Choice of dust attenuation model: Because dust red-
+dens starlight, it is difficult to separate the effects of
+dustfromstellarpopulationeffects,especiallywhenfit-
+ting optical photometry. The effects of dust are also
+knowntochangedependingongalaxyinclination(e.g.,
+Driveret al. 2007). Hence, the choice of dust attenu-
+ation law has a nontrivial effect on the inferred stel-
+lar population ages and, consequently, star formation
+histories derived from photometric and even spectro-
+scopic surveys (Panteret al. 2007). In terms of the
+effect on stellar masses, Pérez-Gonzálezetal. (2008)
+compared the dust models of Calzetti et al. (2000) and
+Charlot&Fall (2000), finding a systematic difference
+of 0.10dex. Panteret al. (2007) found a similar differ-
+ence between Calzetti et al. and models based on ex-
+tinction curves from the Small and Large Magellanic
+Clouds. The effectsof varyingdust attenuationmodels
+have also been explored recently by Marchesiniet al.
+(2009) and Muzzinet al. (2009), with similar results.
+We have used the stellar population fitting proce-
+dure described in Conroyet al. (2009) to compare the
+Calzetti et al. dust attenuation law to the dust model
+used in the kcorrect package (Blanton&Roweis
+2007). We find a median offset of 0.02dex but also a
+systematic trend such that two galaxies whose stellar
+massesareestimatedwithCalzettidustattenuationand
+are separated by 1.0dex will have kcorrect masses
+separatedby(onaverage)only0.92dex.
+5.Statistical errors in individual stellar mass estimates:
+Stellar mass estimates for each galaxy are subject to
+statistical errors due to uncertainty in photometry as
+well as uncertainty in the SPS parameters for a given
+set of model assumptions. We treat this here as a ran-
+dom statistical error. While it may seem that random
+scatter in individual stellar masses should on average
+have no systematic effect, it in fact introduces a sys-
+tematic error analogous to Eddington bias (Eddington4 BEHROOZI,CONROY& WECHSLER
+1940) observed in luminosity functions. As the stellar
+mass function drops off steeply beyond a certain char-
+acteristic stellar mass, there are many more low stellar
+mass galaxies that can be up-scattered than there are
+high stellar mass galaxies that can be down-scattered
+by errors in stellar mass estimates. This asymmetric
+scatter implies that the drop-off in number density at
+highmassesbecomesshallowerinthepresenceofscat-
+ter. We discuss this effect in detail in §3.1.2 (see also
+the AppendixinCattaneoet al.2008).
+6.Sample variance: Surveysof finite regionsof the Uni-
+verse are susceptible to large–scale fluctuations in the
+number density of galaxies. This is no longer a dom-
+inant source of uncertainty for the volumes probed at
+lowredshiftbytheSDSS,butitisanimportantconsid-
+eration for higher–redshift surveys, which cover much
+smaller comoving volumes. Most authors who con-
+sidersamplevarianceattempttominimizeitbyaverag-
+ingoverseveralfields(e.g.,Pérez-Gonzálezetal.2008;
+Marchesinietal. 2009). Very few surveys at z>0 at-
+tempt to estimate the magnitude of the error except by
+computing the field–to–field variance, which is often
+an underestimate when insufficient volume is probed
+(Crocceet al.2009). Wedetailamoreaccuratemethod
+based on simulations to model the error arising from
+samplevariancein §3.2.4.
+7.Redshift errors: Photometric redshift errors blur the
+distinction between GSMFs at different redshifts.
+While a galaxy may be scattered either up or down in
+redshift space, volume-limited survey lightcones will
+contain larger numbers of galaxies at higher redshifts,
+meaning that the GSMF as reported at lower redshifts
+willbeartificiallyinflated. Moreover,asgalaxiesatear-
+liertimeshavelowerstellarmasses,surveyswilltendto
+report artificially larger faint-end slopes in the GSMF.
+However, as these errors are well known, it is easy to
+correct for their effects on the stellar mass function,
+as has been done for the data in Pérez-Gonzálezetal.
+(2008) (seetheappendixofPérez-Gonzálezet al.2005
+fordetailsonthisprocess).
+For completeness, we remark that galaxy-galaxy lensing
+will also result in systematic errorsin the GSMF at high red-
+shifts because galaxy magnification will result in higher ob -
+served luminosities. However, from ray-tracing studies of
+the Millennium simulation (Hilbertet al. 2007), the expect ed
+scatter in galaxystellar masses fromlensingis minimal (e. g.,
+0.04 dex at z= 1) compared to the other sources of scatter
+above (e.g., 0.25 dex from different model choices). For tha t
+reason,we donot modelgalaxy-galaxylensing effectsin thi s
+paper.
+2.1.2.Additional Systematics atz >1
+Recently, it has become clear that current estimates of
+the evolution in the cosmic SFR density are not consistent
+with estimates of the evolution of the stellar mass density
+atz>1 (Nagamineet al. 2006; Hopkins&Beacom 2006;
+Pérez-Gonzálezet al. 2008; Wilkinset al. 2008a). The ori-
+gin of this discrepancy is currently a matter of debate. One
+solution involves allowing for an evolving IMF with red-
+shift (Davé 2008; Wilkinsetal. 2008a). While such a so-
+lution is controversial, a number of independent lines ofevidence suggest that the IMF was different at high red-
+shift(Lucatelloetal. 2005;Tumlinson2007a,b; vanDokkum
+2008). Reddy&Steidel (2009) offer a more mundane ex-
+planation for the discrepancy. They appeal to luminosity–
+dependentreddeningcorrectionsin the ultraviolet lumino sity
+functionsat highredshift,anddemonstratethat the purpor ted
+discrepancythenlargelyvanishes.
+In contrast to results at z>1, there does seem to be
+an accord that for z<1 both the integrated SFR and the
+total stellar mass are in good agreement if one assumes
+(as we have) a Chabrier (2003) IMF (see Wilkinset al.
+2008b; Pérez-Gonzálezetal. 2008; Hopkins& Beacom
+2006;Nagamineet al.2006; Conroy&Wechsler 2009).
+Because of the discrepancy between reported SFRs and
+stellar massesin the literature,it is clearthat estimates ofun-
+certaintiesin galaxystellar mass functionsandSFRs at z>1
+tend to underestimate the true uncertainties; for this reas on,
+we separately analyze results for z<1 in §4 and z>1 in §5
+ofthispaper.
+2.2.Uncertaintiesin theHaloMassFunction
+Darkmatterhalopropertiesoverthemassrange1010−1015
+M⊙have been extensively analyzed in simulations (e.g.,
+Jenkinset al. 2001; Warrenet al. 2006; Tinkeret al. 2008),
+and the overall cosmology has been constrained by probes
+such as WMAP (Spergeletal. 2003; Komatsuetal. 2009).
+As such, uncertainties in the halo mass function have on the
+wholemuch less impact thanuncertaintiesin the stellar mas s
+function. We present our primary results for a fixed cosmol-
+ogy (WMAP5), but we also calculate the impact of uncer-
+tain cosmological parameters on our error bars. We do not
+marginalize over the mass function uncertainties for a give n
+cosmology,astherelevantuncertaintiesareconstraineda tthe
+5% level (when baryonic effects are neglected, see below;
+Tinkeret al. 2008). Additionally, in Appendix A, a simple
+method is described to convert our results to a different cos -
+mology using an arbitrary mass function. For completeness,
+wementionthethreemostsignificantuncertaintieshere:
+1.Cosmologicalmodel: Thestellarmass–halomassrela-
+tionhasdependenceoncosmologicalparametersdueto
+the resulting differences in halo number densities. We
+investigate this both by calculating the relation for two
+specific cosmological modes (WMAP1 and WMAP5
+parameters)andthenbycalculatingtheuncertaintiesin
+the relation over the full range of cosmologiesallowed
+by WMAP5 data. We findthat in all casesthese uncer-
+tainties are small compared to the uncertainties inher-
+entinstellarmassmodeling(§2.1.1),althoughtheyare
+larger than the statistical errors for typical halo masses
+at lowredshift.
+2.Uncertainties in substructure identification: Different
+simulations have different methods of identifying and
+assigning masses to substructure. Our matching meth-
+ods make use only of the subhalo mass at the epoch
+of accretion ( Macc) as this results in a better match to
+clustering and pair–count results (Conroyetal. 2006;
+Berrieret al. 2006), so we are largely immune to the
+problem of different methods for calculating subhalo
+masses. Ofgreaterconcernistheabilitytoreliablyfol-
+lowsubhalosinsimulationsastheyaretidallystripped.
+Two related issues apply here. The first is that it is not
+clear how to account for subhaloswhich fall below theUncertaintiesAffectingtheStellar Mass–HaloMassRelati on 5
+resolution limit of the simulation. The second is that
+theformationofgalaxieswilldramaticallyincreasethe
+binding energy of the central regions of subhalos, po-
+tentiallymakingthemmoreresilienttotidaldisruption.
+Hydrodynamicsimulationssuggestthatthislattereffect
+issmallexceptforsubhalosthatorbitnearthecentersof
+themostmassiveclusters(Weinbergetal.2008). How-
+ever, while these details are important for accurately
+predicting the clustering strength on small scales ( /lessorsimilar1
+Mpc), they are not a substantial source of uncertainty
+fortheglobalhalomass—stellarmassrelationbecause
+satellites are always sub-dominant ( /lessorsimilar20%) by num-
+ber. We discuss the analytic method we use to model
+the satellite contribution to the halo mass function in
+§3.2.2.
+3.Baryonic physics: Recent work by Staneket al. (2009)
+suggests that gas physics can affect halo masses rela-
+tive to dark matter-onlysimulations by -16% to +17%,
+leading to number density shifts of up to 30% in the
+halo massfunctionat 1014M⊙. Withoutevidencefora
+clear bias in one direction or the other—the models of
+gasphysicsstillremaintoouncertain—wedonotapply
+a correction for this effect in our mass functions. Un-
+certainties of this magnitude are larger than the statis-
+tical errors in individual stellar masses at low redshift,
+but are still small in comparisonto systematic errorsin
+calculatingstellar masses.
+For completeness, we note that the effects of sample vari-
+ance on halo mass functions estimated from simulations are
+small. Current simulations readily probe volumes of 1000
+(h−1Mpc)3(Tinkeretal. 2008), and so the effects of sample
+varianceonthe halomassfunctionaredwarfedbythe effects
+of sample variance on the stellar mass function; we therefor e
+donotanalyzethemseparatelyinthispaper.
+We also remark on the issue of mass definitions. Al-
+though abundance matching implies matching the most mas-
+sive galaxiesto the most massivehalos, thereis little cons en-
+susonwhichhalomassdefinitiontouse,withpopularchoices
+beingMvir(mass within the virial radius), M200(mass within
+a sphere with mean density 200 ρcrit), andMfof(mass deter-
+minedby a friends-of-friendsparticle linkingalgorithm) . We
+chooseMvirfor this paper and note that the largest effect of
+choosinganothermassalgorithmwill beapurelydefinitiona l
+shift in halo masses. We expect that scatter between any two
+of these mass definitions is degenerate with and smaller than
+the amountofscatter in stellar massesat fixedhalo mass(the
+lattereffectisdiscussedin§2.3).
+2.3.Uncertaintiesin AbundanceMatching
+Finally, there are two primary uncertainties concerningth e
+abundancematchingtechniqueitself:
+1.Nonzero scatter in assigning galaxies to halos: While
+host halo mass is strongly correlated with stellar mass,
+the correlation is not perfect. At a given halo mass,
+the halomergerhistory,angularmomentumproperties,
+and cooling and feedback processes can induce scatter
+between halo mass and galaxystellar mass. This is ex-
+pectedtoresultinscatterinstellarof ∼0.1–0.2dexata
+given halo mass, see §3.3.1 for discussion. The scatter
+between halo mass and stellar mass will have system-
+atic effects on the mean relation for reasons analogousto those mentioned for statistical error in stellar mass
+measurements. At the high mass end where both the
+halo and stellar mass functions are exponential, scat-
+ter in stellar mass at fixed halo mass (or vice versa)
+will alter the average relation because there are more
+low mass galaxies that are upscattered than high mass
+galaxiesthataredownscattered.
+2.Uncertainty in Assigning Galaxies to Satellite Halos:
+It is not clear that the halo mass — stellar mass rela-
+tion should be the same for satellite and central galax-
+ies. Once a halo is accreted onto a larger halo, it starts
+to lose halo mass because of dynamicaleffects such as
+tidal stripping. While stripping of the halo appears to
+be a relatively dramatic process (e.g., Kravtsovet al.
+2004), the stripping of the stellar component proba-
+bly does not occur unless the satellite passes very near
+to the central object because the stellar component is
+muchmoretightlyboundthanthehalo. Itisclearfrom
+the observed color–density relation (Dressler 1980;
+Postman&Geller 1984; Hansenet al. 2009) that star
+formation in satellite galaxies must eventually cease
+with respect to galaxiesin the field. It is less clear how
+quicklystar formationceases, andwhetherornot there
+is a burst ofstar formationuponaccretion. All ofthese
+issues can potentially alter the relation between halo
+andstellarmassforsatellites(althoughthemodelingre-
+sults ofWang etal. 2006suggestthat the halo–satellite
+relation is indistinguishable from the overall galaxy–
+halorelation).
+3.METHODOLOGY
+Ourprimarygoalistoprovidearobustestimateofthestel-
+lar mass – halo mass relation over a significant fraction of
+cosmic time via the abundance matching technique. We aim
+to constructthis relation by taking into account all of the r el-
+evant sources of uncertainty. This section describes in de-
+tail a number of aspects of our methodology, including our
+approach for incorporating uncertainties in the stellar ma ss
+function ( §3.1), a summary of the adopted halo mass func-
+tionsand associateduncertainties( §3.2), the uncertaintiesas-
+sociatedwithabundancematching(§3.3),ourchoiceoffunc -
+tionalformforthestellarmass–halomassrelation,includ ing
+adiscussionofwhycertainfunctionsshouldbepreferredov er
+others (§3.4), and the Markov Chain Monte Carlo parameter
+estimationtechnique( §3.5). Forreadersinterestedinthegen-
+eral outline of our process but not the details, we conclude
+witha briefsummaryofourmethodology(§3.6).
+3.1.ModelingStellarMassFunctionUncertainties
+Asdiscussedin§2,thereareseveralclassesofuncertainti es
+affectingthewaythestellarmassfunctionisusedintheabu n-
+dance matching process. In this section, we discuss system-
+aticshiftsinstellarmassestimatesandtheeffectsofstat istical
+errorsonthestellar massfunction.
+3.1.1.Modeling Systematic ShiftsinStellar Mass Estimates
+Most studies on the GSMF report Schechter function fits
+as well as individual data points; many also provide statist i-
+calerrors. However,evenwhensystematicerrorsarereport ed
+(either in Schechter parameters or at individual data point s),
+the systematic error estimates are of limited value unless o ne
+is also able to model shifts in the GSMF caused by such er-
+rors.6 BEHROOZI,CONROY& WECHSLER
+Fortunately, based on the discussion in §2.1.1, there seem
+to be two main classes of systematic errors causing shifts in
+theGSMF:
+1. Over/underestimationofallstellarmassesbyaconstant
+factorµ. This appears to cover the majority of errors,
+includingmostdifferencesinSPSmodeling,dustatten-
+uationassumptions,andstellar populationagemodels.
+2. Over/underestimation of stellar masses by a factor
+which depends linearly on the logarithm of the stel-
+lar mass (i.e., depends on a power of the stellar mass).
+Thiscoversthemajorityoftheremainingdiscrepancies
+between different SPS models and different stellar age
+models.
+Bothformsoferrorare modeledwith theequation
+log10/parenleftbiggM∗,meas
+M∗,true/parenrightbigg
+=µ+κlog10/parenleftbiggM∗,true
+M0/parenrightbigg
+.(1)
+Without loss of generality, we may take M0= 1011.3M⊙(the
+fixed point of the variation between the Bruzual 2007 and
+Bruzual& Charlot 2003 models found by Salimbenietal.
+2009), allowing the prior on M0to be absorbed into the prior
+onµ.
+For the prior on µ, we consider four contributing sources
+of uncertainty. We adopt estimates of the uncertainty from
+the SPS model( ≈0.1dex),the dust model( ≈0.1dex),and as-
+sumptions about the star formation history ( ≈0.2dex) from
+Pérez-Gonzálezet al. (2008) as detailed in §2.1.1. Additio n-
+ally, we have the variation in κlog10(M0) (at most 0.1dex, as
+|κ|/lessorsimilar0.15 — see below). Assuming that these are statisti-
+cally independent, they combine to give a total uncertainty
+of 0.25dex, which is consistent with the accepted range for
+systematicuncertaintiesinstellarmass(Pérez-González etal.
+2008; Kannappan& Gawiser 2007; vanderWel et al. 2006;
+Marchesiniet al. 2009). For lack of adequate information
+(i.e., different models) to infer a more complicated distri bu-
+tion, we assume that µhas a Gaussian prior. As more stud-
+ies ofthe overallsystematic shift µbecomeavailable,ouras-
+sumptions for the prior on µand the probability distribution
+will likely need corrections. We remark, however, that our
+results can easily be converted to a different assumption fo r
+µ, asµsimply imparts a uniform shift in the intrinsic stellar
+massesrelativeto theobservedstellar masses.
+For the prior on κ, the result of Salimbeniet al. (2009)
+would suggest |κ|/lessorsimilar0.15. As mentionedin §2.1.1, we found
+that|κ| ≈0.08 between the Blanton& Roweis (2007) and
+Calzetti et al.(2000)modelsfordustattenuation. Li &Whit e
+(2009) finds |κ|/lessorsimilar0.10 between Blanton& Roweis (2007)
+and Bell etal. (2003) stellar masses. Without a large num-
+ber of other comparisons, it is difficult to robustly determi ne
+the priordistributionfor κ; however,motivatedby the results
+just mentioned, we assume that the prior on κis a Gaussian
+ofwidth0.10centeredat0.0.
+We remark that some authors have considered much more
+complicated parameterizations of the systematic error. Fo r
+example, Li &White (2009) considers a four-parameter hy-
+perbolic tangent fit to differences in the GSMF caused by
+different SPS models, as well as a five-parameter quartic fit.
+However,wedonotconsiderhigher-ordermodelsforsystem-
+atic errors for several reasons. First, given that second- a nd
+higher-ordercorrectionswill resultonlyinverysmall cor rec-
+tions to the stellar masses in comparison to the zeroth-orde rcorrection ( µ≈0.25dex), the corrections will not substan-
+tiallyeffectthesystematicerrorbars. Second,wedonotkn ow
+ofanystudieswhichwouldallowustoconstructpriorsonthe
+higher-order corrections. Finally, with higher-order mod els,
+there is the serious danger of over-fitting—that is, with ver y
+loose priors on systematic errors, the best-fit parameters f or
+the systematic errors will be influenced by bumps and wig-
+gles in the stellar mass function due to statistical and samp le
+variance errors. Hence, the interpretive value of the syste m-
+aticerrorsbecomesincreasinglydubiouswitheachadditio nal
+parameter.
+3.1.2.Modeling Statistical ErrorsinIndividual Stellar Mass
+Measurements
+In addition to the systematic effectsdiscussed in the previ -
+oussection,measurementofstellarmassesissubjecttosta tis-
+ticalerrors. Evenforafixedsetofassumptionsaboutthedus t
+model, SPS model, and the parameterization of star forma-
+tion histories, stellar masses will carry uncertainties be cause
+the mapping between observables and stellar masses is not
+one-to-one. This additional source of uncertainty has uniq ue
+effects on the GSMF. Observers will see an GSMF ( φmeas)
+which is the true or “intrinsic” GSMF ( φtrue) convolved with
+theprobabilitydistributionfunctionofthemeasurements cat-
+ter. Forinstance,ifthescatterisuniformacrossstellarm asses
+and has the shape of a certain probability distribution P, we
+have:
+φmeas(M)=/integraldisplay∞
+−∞φtrue(10y)P/parenleftbig
+y−log10(M)/parenrightbig
+dy,(2)
+whereyis the integrationvariable,in units of log10mass. As
+derived in Appendix B, the approximate effect of the convo-
+lutionis
+log10/parenleftbiggφmeas(M)
+φtrue(M)/parenrightbigg
+≈σ2
+2ln(10)/parenleftbiggdlogφtrue(M)
+dlogM/parenrightbigg2
+,(3)
+whereσis the standard deviation of P. That is to say, the
+effectof the convolutiondependsstronglyon the logarithm ic
+slope ofφtrue. Where the slope is small (i.e., for low-mass
+galaxies), there is almost no effect. Above 1011M⊙, where
+the GSMF becomes exponential, there can be a dramatic ef-
+fect, with the result that φtrueis more than an order of mag-
+nitudelessthan φmeasbecauseit becomesfar morelikelythat
+stellar mass calculation errors produce a galaxy of very hig h
+perceived stellar mass than it is for there to be such a galaxy
+inreality(seeforexampleCattaneoet al. 2008).
+For the observed z∼0 GSMF, we take the probabil-
+ity distribution Pto be log-normal with 1 σwidth 0.07dex
+fromtheanalysisofthephotometryoflow–redshiftluminou s
+red galaxies (LRGs) (Conroyet al. 2009). Kauffmannet al.
+(2003) found similar results regarding the width of P. This
+function only accounts for the statistical uncertainties m en-
+tioned above and does not include additional systematic un-
+certainties. In light of Equation 3, we use LRGs to esti-
+matePbecause LRGs occupy the high stellar mass regime
+where measurementerrors are most likely to affect the shape
+of the observed GSMF. However, the single most important
+attribute of the distribution Pis its width; the main results do
+not change substantially if an alternate distribution with non-
+Gaussiantailsbeyondthe1 σlimitsofPisused.
+For higher redshifts, we scale the width of the probabil-
+ity distributionto accountfor the fact that mass estimates be-
+come less certain at higher redshift (e.g., Conroyet al. 200 9;UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 7
+Kajisawaet al. 2009):
+P(∆log10M∗,z)=σ0
+σ(z)P0/parenleftbiggσ0
+σ(z)∆log10M∗/parenrightbigg
+,(4)
+whereP0is the probability distribution at z=0 (as discussed
+above),σ0is the standard deviation of P0, andσ(z) gives the
+evolutionofthe standarddeviationasa functionofredshif t.
+Conroyet al. (2009) did not give a functional form for
+σ(z), but they calculate fora handfulof massive galaxiesthat
+σ(z= 2) is≈0.18dex, as compared to σ(z= 0)≈0.07dex.
+Kajisawaet al. (2009) performeda similar calculation (alb eit
+with a differentSPS model)ofthe distributioninseveralre d-
+shift bins; their resultsshow gradualevolutionfor σ(z) out to
+z=3.5 for high stellar mass galaxies consistent with a linear
+fit:
+σ(z)=σ0+σzz. (5)
+The results of Kajisawaet al. (2009) suggest that σz=0.03-
+0.06dexforLRGs. Asthisisconsistentwiththevalueof σz=
+0.05dexwhichwouldcorrespondtoConroyet al.(2009),we
+adopt the linear scaling of Equation 5 with a Gaussian prior
+ofσz=0.05±0.015dex.
+Note that the effect of this statistical error on the stellar
+mass functionis minimalbelow 1011M⊙, andthereforedoes
+notaffectthestellarmass–halomassrelationforhalosbel ow
+∼1013M⊙,asdiscussedin §4.2. Whilethisscatterdoeshave
+an effect on the shape of the stellar mass function for high-
+mass galaxies, the qualitative predictions we make from thi s
+analysisaregenericto alltypesofrandomscatter.
+3.2.HaloMassFunctions
+The halo mass function specifies the abundance of halos
+as a function of mass and redshift. A number of analytic
+modelsandsimulation–basedfittingfunctionshavebeenpre -
+sented for computing mass functions given an input cos-
+mology (e.g., Press& Schechter 1974; Jenkinset al. 2001;
+Warrenet al. 2006; Tinkeret al. 2008). For most of our re-
+sultswewilladopttheuniversalmassfunctionofTinkereta l.
+(2008), as described below. Analytic mass functions are
+preferableasthey1)allowmassfunctionstobecomputedfor
+arangeofcosmologiesand2)donotsuffersignificantlyfrom
+sample variance uncertainties, because the analytic relat ions
+are typically calibrated with very large or multiple N−body
+simulations.
+For some purposes it will be useful to also consider full
+halo merger trees derived directly from N−body simulations
+that have sufficient resolution to follow halo substructure s.
+The simulations used herein will be described below, in ad-
+ditiontoourmethodsformodelinguncertaintiesintheunde r-
+lyingmassfunction,includingcosmologyuncertainties,s am-
+plevarianceinthegalaxysurveys,andourmodelsforsatell ite
+treatment.
+3.2.1.Simulations
+For the principal simulation in this study (“L80G”), we
+used a pure dark matter N-body simulation based on Adap-
+tive Refinement Tree (ART) code (Kravtsovet al. 1997;
+Kravtsov&Klypin 1999). The simulation assumed flat, con-
+cordance ΛCDM (ΩM=0.3,ΩΛ=0.7,h=0.7, andσ8=0.9)
+and included 5123particles in a cubic box with periodic
+boundary conditions and comoving side length 80 h−1Mpc.
+These parameters correspondto a particle mass resolution o f≈3.2×108h−1M⊙. For this simulation, the ART code be-
+gins with a spatial grid size of 5123; it refines the grid up to
+eight times in locally dense regions, leading to an adaptive
+distance resolution of ≈1.2h−1kpc (comoving units) in the
+densest parts and ≈0.31h−1Mpc in the sparsest parts of the
+simulation.
+In this simulation, halos and subhalos were identified
+using a variant of the Bound Density Maxima algorithm
+(Klypinetal. 1999). Halo centers are located at peaks in the
+density field smoothed over a 24-particle SPH kernel (for a
+minimumresolvable halomass of 7 .7×109h−1M⊙). Nearby
+particles are classified as bound or unbound in an iterative
+process;onceall thelocallyboundparticleshavebeenfoun d,
+halo parameters such as the virial mass Mvirand maximum
+circularvelocity Vmaxmaybe calculated. (See Kravtsovet al.
+2004 for complete details on the algorithm). The simulation
+is complete down to Vmax≈100 km s−1, corresponding to a
+galaxystellar massof108.75M⊙atz=0.
+The ability of L80G to track satellites with high mass and
+forceresolutiongivesitseveraluses. MergertreesfromL8 0G
+informourprescriptionforconvertinganalytical central -only
+halo mass functions to mass functions which include satel-
+lite halos (see §3.2.2). Additionally, the merger trees all ow
+forevaluationofdifferentmodelsofsatellitestellar evo lution
+with full consistency (see §3.3.2). Finally, the knowledge of
+which satellite halos are associated with which central hal os
+allowsforestimatesofthetotalstellarmass(inthecentra land
+allsatellite galaxies)— halomassrelation(see §4.3.6).
+We also make use of a secondary simulation from the
+Large Suite of Dark Matter Simulations (LasDamas Project,
+http://lss.phy.vanderbilt.edu/lasdamas/) in our sample vari-
+ancecalculations. TheL80Gsimulationistoosmallforusei n
+calculatingthesamplevariancebetweenmultipleindepend ent
+mocksurveys,butthelargersizeoftheLasDamassimulation
+(420h−1Mpc,14003particles)makesitidealforthispurpose.
+However, the LasDamas simulation has poorer mass resolu-
+tion (a minimum particle size of 1 .9×109M⊙) and force
+resolution (8 h−1kpc), making it unable to resolve subhalos
+(particularlyafteraccretion)aswell asL80G.TheLasDama s
+simulation assumes a flat, ΛCDM cosmology ( ΩM= 0.25,
+ΩΛ= 0.75,h= 0.7, andσ8= 0.8) which is very close to the
+WMAP5best-fitcosmology(Komatsuetal.2009). Collision-
+less gravitational evolution was provided by the GADGET-2
+code (Springel 2005). Halos are identified using friends of
+friendswith a linkinglengthof 0.164. The subfind algorithm
+Springel(2005) isusedtoidentifysubstructure.
+Asmentioned,theprimaryuseoftheLasDamassimulation
+is in sampling the halo mass functions in mock surveys to
+model the effects of sample variance on high-redshiftpenci l-
+beam galaxy surveys. The mock surveys are constructed so
+as to mimic the observationsin Pérez-Gonzálezet al. (2008) .
+Ineachmocksurvey,threepencil-beamlightcones(matchin g
+the angular sizes of the three fields in Pérez-Gonzálezet al.
+2008) with random orientations are sampled from a random
+originin the simulationvolumeoutto z=1.3. Thus,bycom-
+paring the halo mass functionsin individualmock surveysto
+themassfunctionoftheensemble,theeffectsofsamplevari -
+ancemaybecalculatedwithfullconsiderationofthe correl a-
+tionsbetweenhalocountsat differentmasses.
+3.2.2.AnalyticMass Functions
+TheanalyticmassfunctionsofTinkeret al.(2008)areused
+to calculate the abundance of halos in several cosmological8 BEHROOZI,CONROY& WECHSLER
+0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
+Scale Factor-1-0.9-0.8-0.7-0.6Δlog10φ0 L80G
+ Fit
+0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
+Scale Factor-0.16-0.12-0.08-0.0400.04Δlog10M0 L80G
+ Fit
+0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
+Scale Factor-0.16-0.0800.080.16Δlog10α
+ L80G
+ Fit
+Figure1. Differences between the fitted Schechter function paramete rs for the satellite halo mass (at accretion) function and th e central halo mass function, as
+a function of scale factor; e.g., ∆log10φ0corresponds to log10(φ0,sats/φ0,centrals). The black lines are calculated from a simulation using WMA P1 cosmology
+(L80G),and the red lines represent the fits to the simulation results in Equation 7.
+models. We calculate mass functions defined by Mvir, using
+the overdensity specified by Bryan&Norman (1998)3This
+results in an overdensity (compared to the mean background
+density)∆virwhichrangesfrom337at z=0to203at z=1and
+smoothly approaches 180 at very high redshifts. Following
+Tinkeret al. (2008), we use spline interpolation to calcula te
+mass functions for overdensities between the discrete inte r-
+valspresentedintheirpaper.
+ThemassfunctionsinTinkeret al.(2008)onlyincludecen-
+tral halos. We model the small ( ≈20% atz=0) correctionto
+the mass function introduced by subhalos to first order only,
+as the overall uncertaintyin the central halo mass function is
+alreadyoforder5%(Tinkeret al.2008). Inparticular,weca l-
+culate satellite (massat accretion)and centralmass funct ions
+in our simulation (L80G) and fit Schechter functionsto both,
+excluding halos below our completeness limit (1010.3M⊙).
+Then, we plot the difference between the Schechter param-
+eters (the difference in characteristic mass, ∆log10M∗; the
+difference in characteristic density, ∆log10φ0; and the dif-
+ference in faint-end slopes, ∆α) as a function of scale factor
+(a). This gives the satellite mass function ( φs) as a function
+of the central mass function ( φc), which allows us to use this
+(first-order)correction for central mass functionsof diff erent
+cosmologies:
+φs(M)=10∆log10φ0/parenleftbiggM
+M0·10∆log10M0/parenrightbigg−∆α
+φc(M/10∆log10M0).
+(6)
+Fromoursimulation,we findfitsasshownin Figure1:4
+∆log10φ0(a)=−0.736−0.213a,
+∆log10M0(a)=0.134−0.306a, (7)
+∆α(a)=−0.306+1.08a−0.570a2.
+Themassfunctionused heremaybe beeasily replacedby an
+arbitrarymassfunction,asdetailedinAppendixA.
+3.2.3.Modeling Uncertainties inCosmological Parameters
+Our fiducial results are calculated assuming WMAP5 cos-
+mologicalparameters. In orderto modeluncertaintiesin co s-
+mological parameters, we have sampled an additional 100
+setsofcosmologicalparametersfromtheWMAP5+BAO+SN
+3∆vir=(18π2+82x−39x2)/(1+x);x=(1+ρΛ(z)/ρM(z))−1−1
+4Comparing these fits to satellite mass functions from a more r ecent sim-
+ulation (Klypin etal. 2010, the “Bolshoi” simulation), we h ave verified that
+applying these fits to mass functions for the WMAP5 cosmology introduces
+errorsonly onthelevel of5%inoverall number density, simi lar totheuncer-
+tainty with which the mass function isknown.MCMC chains (from the models in Komatsuet al. 2009)
+and generated mass functions for each one according to the
+methodinthe previoussection. Hence,todeterminethevari -
+anceinthederivedstellarmass–halomassrelationcausedb y
+cosmology uncertainties, we recalculate the relation for e ach
+sampled mass function according to the method described in
+AppendixA.
+3.2.4.EstimatingSample Variance Effectsforthe Stellar Mass
+Function
+Large–scalemodesinthematterpowerspectrumimplythat
+finitesurveyswillobtainabiasedestimateofthenumberden -
+sities of galaxies and halos as compared to the full universe .
+That is to say, matching observed GSMFs measured from a
+finitesurveytothehalomassfunctionestimatedfromamuch
+largervolumewill introducesystematic errorsintothe res ult-
+ingSM–HMrelation. Theseerrorscannotbecorrectedunless
+one has knowledge of the halo mass function for the specific
+surveyin question,whichisin generalnotpossible.
+However,wecanstillcalculatetheuncertaintiesintroduc ed
+by the limited sample size. While we cannot determine the
+true halo mass function for the survey, we can calculate the
+probabilitydistribution of halo mass functionsfor identi cally
+shaped surveys via sampling lightcones from simulations. I f
+we rematch galaxy abundances from the observed GSMF to
+the abundances of halos in each of the sampled lightcones,
+thenthe uncertaintyintroducedbysample varianceis exact ly
+capturedin thevarianceoftheresultingSM–HMrelations.
+In detail, we create our distribution of halo mass func-
+tions by sampling one thousand mock surveys from the Las-
+Damassimulation(see§3.2.1)correspondingtotheexactsu r-
+vey parameters used in Pérez-Gonzálezet al. (2008). We fit
+Schechter functions to the halo mass functionsof each mock
+survey (over all redshifts), and we calculate the change in
+Schechter parameters ( ∆log10φ0,∆log10M0, and∆α) as
+compared to a Schechter fit to the ensemble average of the
+mass functions. Using the distribution of the changes in
+Schechter parameters, we may mimic to first order the ex-
+pecteddistributionofhalomassfunctionsforanycosmolog y.
+In particular, we use an equation exactly analogous to Equa-
+tion6to convertthe massfunctionforthe fulluniverse( φfull)
+and the distribution of ∆log10φ0,∆log10M0, and∆αinto a
+distributionofpossiblesurveymassfunctions( φobs):
+φobs(M)=10∆log10φ0/parenleftbiggM
+M0·10∆log10M0/parenrightbigg−∆α
+φfull(M/10∆log10M0).
+(8)
+Hence,toobtainthevarianceinthestellarmass–halomassUncertaintiesAffectingtheStellar Mass–HaloMassRelati on 9
+relation caused by finite survey size, we recalculate the rel a-
+tionforeachoneofthesurveymassfunctionsthuscomputed
+accordingtothemethoddescribedin AppendixA.
+3.3.Uncertaintiesin AbundanceMatching
+3.3.1.Scatter inStellar Mass at FixedHalo Mass
+An important uncertainty in the abundance matching pro-
+cedure is introduced by intrinsic scatter in stellar mass at a
+given halo mass. Suppose that M∗(Mh) is the average (true)
+galaxy stellar mass as a function of host halo mass. For a
+perfect monotonic correlation between stellar mass and hal o
+mass, i.e., without scatter between stellar and halo mass, i t
+is straightforwardto relate the true or “intrinsic” stella r mass
+function( φtrue)to thehalomassfunction( φh) via
+dN
+dlog10M∗=dN
+dlog10MhdlogMh
+dlogM∗, (9)
+whereNisthenumberdensityofgalaxies,sothat
+φtrue(M∗(Mh))=φh(Mh)/parenleftbiggdlogM∗(Mh)
+dlogMh/parenrightbigg−1
+.(10)
+Intuitively,asthehalosofmass Mhgetassignedstellarmasses
+ofM∗(Mh),thenumberdensityofgalaxieswithmass M∗(Mh)
+willbeproportionaltothenumberdensityofhaloswithmass
+Mh. Theaboveequationsaresimplyamathematicalrepresen-
+tationofthetraditionalabundancematchingtechnique.
+Equation10remainsusefulinthepresenceofscatter. Ifwe
+knowthe expectedscatter aboutthe meanstellar mass, sayin
+the formof a probabilitydensity function Ps(∆log10M∗|Mh),
+then we may still relate φtruetoφhvia an integral similar to a
+convolution:
+φtrue(x)=/integraltext∞
+0φh(Mh(M∗))dlogMh(M∗)
+dlogM∗×
+×Ps(log10x
+M∗|Mh(M∗))dlog10M∗,(11)
+whereMh(M∗)istheinversefunctionof M∗(Mh).
+This similarity to a convolution is no coincidence—
+mathematically,it isanalogoustohowwemodelrandomsta-
+tisticalerrorsinstellarmassmeasurementsin§3.1.2. Nam ely,
+ifwedefine φdirecttoequaltheright-handsideofEquation10,
+φdirect(M∗)≡φh(Mh(M∗))dlogMh
+dlogM∗, (12)
+and if we assume a probability density distribution indepen -
+dent of halo mass (i.e., scatter in stellar mass at fixed halo
+mass is independent of halo mass), then φtrueis exactly re-
+latedtoφdirectbyaconvolution:
+φtrue(M∗)=/integraldisplay∞
+−∞φdirect(10y)Ps(y−log10M∗)dy,(13)
+whichis mathematicallyidenticaltoEquation2in§3.1.2.
+Then, if one calculates φdirectfromφtrue, one may find
+Mh(M∗) via direct abundance matching. Namely, integrating
+equation12,we have:
+/integraldisplay∞
+Mh(M∗)φh(M)dlog10M=/integraldisplay∞
+M∗φdirect(M∗)dlog10M∗.(14)
+Equivalently, letting Φh(Mh)≡/integraltext∞
+Mhφh(M)dlog10Mbe the
+cumulative halo mass function, and letting Φdirect(M∗)≡/integraltext∞
+M∗φdirect(M∗)dlog10M∗be the cumulative “direct” stellar
+massfunction,wehave
+Mh(M∗)=Φ−1
+h(Φdirect(M∗)), (15)
+andonemaysimilarlyfind M∗(Mh)byinvertingthisrelation.
+Our approach in all equations except for Equation 13 al-
+lows a halo mass-dependentscatter in the stellar mass, but t o
+date the data appears to be consistent with a constant scatte r
+value. For example, using the kinematics of satellite galax -
+ies, Moreet al. (2009) finds that the scatter in galaxy lumi-
+nosity at a given halo mass is 0 .16±0.04 dex, independent
+of halo mass. Using a catalog of galaxy groups, Yanget al.
+(2009b) find a value of 0 .17 dex for the scatter in the stel-
+lar massat a givenhalomass, also independentof halomass.
+Here, we thus assume a fixed value for the scatter in stellar
+mass at fixed halo mass, ξ, to specify the standard deviation
+ofPs(∆log10M∗). As the Yangetal. (2009b) value is consis-
+tent with the Moreet al. (2009) value, we set the prior using
+the Moreetal. (2009) value and error bounds on ξ, We as-
+sume a Gaussian prior on the probability distribution for ξ,
+andwe assumethatthescatter itself islog-normal.
+3.3.2.The Treatment of Satellites
+Whenagalaxyisaccretedintoalargersystem,itwilllikely
+bestrippedofdarkmattermuchmorerapidlythanstellarmas s
+because the stars are much more tightly bound than the halo.
+It has been demonstratedthat variousgalaxyclusteringpro p-
+erties compare favorably to samples of halos where satellit e
+halos— i.e., subhalos— are selected accordingto their halo
+mass at the epoch of accretion, Macc, rather than their cur-
+rent mass (e.g., Nagai&Kravtsov 2005; Conroyet al. 2006;
+Vale&Ostriker 2006; Berrieretal. 2006). Theseresultssup -
+port the idea that satellite systems lose dark matter more
+rapidlythanstellar mass.
+As commonly implemented (e.g. Conroyetal. 2006), the
+abundancematchingtechniquematchesthestellarmassfunc -
+tionataparticularepochtothehalomassfunctionatthesam e
+epoch, using Maccrather than the present mass for subhalos.
+AsMaccremainsfixedaslongasthesatelliteisresolvable,the
+standard technique implies that the satellite galaxy’s ste llar
+mass will continue to evolve in the same way as for centrals
+ofthat halomass. Therefore,a subtle implicationof thesta n-
+dardtechniqueisthatsatellitesmaycontinuetogrowinste llar
+mass, even though Maccremainsthe same. A differentmodel
+forsatellitestellarevolution(e.g.,inwhichstellarmas swhich
+does not evolve after accretion) would therefore involve di f-
+ferentchoicesinthesatellite matchingprocess.
+The fiducial results presented here use the standard model
+where satellites are assigned stellar masses based on the cu r-
+rent stellar mass function and their accretion–epoch masse s.
+However, we also present results for comparison in which
+satellite masses are assigned utilizing the stellar mass fu nc-
+tion at the epoch of accretion, correspondingto a situation in
+which satellite stellar masses do not change after the epoch
+ofaccretion. In orderto maintainself-consistencyforthe lat-
+ter method, we use full merger trees (from L80G, the simu-
+lation described in §3.2.1) to keep track of satellites and t o
+assure that, e.g.,mergersbetween satellites beforethey r each
+thecentralhalopreservestellar mass.
+Finally, we note that any specific halo–finding algorithm
+may introduce artifacts in the halo mass function in terms
+of when a satellite halo is considered absorbed/destroyed.
+This can have a small effect on satellite clustering as well a s10 BEHROOZI,CONROY & WECHSLER
+number density counts. Wetzel & White (2009) suggest an
+approach that avoids some of the problems associated with
+resolving satellites after accretion. Namely, they sugges t a
+model where satellites remain in orbit for a duration that is a
+function of the satellite mass, the host mass, and the Hubble
+time, after which time they dissolve or merge with the cen-
+tralobject. Althoughwehavenotmodeledthisexplicitly,o ur
+satellite counts are consistent with their recommendedcut off
+—theysuggestconsideringasatellitehaloabsorbedwhenit s
+presentmassislessthan0.03timesitsinfallmass;inoursi m-
+ulation,only0.1%ofall satellitesfall belowthisthresho ld.
+3.4.FunctionalFormsfortheStellarMass–HaloMass
+Relation
+Inordertodeterminetheprobabilitydistributionofourun -
+derlying model parameters, we must first define an allowed
+parameterspaceforthestellarmass–halomassrelation. Id e-
+ally, one would like a simple, accurate, physically intuiti ve,
+andorthogonalparameterization;inpractice,weseektheb est
+compromise with these four goals in mind. We consider one
+of the most popular methods for choosing a functional form
+(indirect parameterization via the stellar mass function) be-
+fore discussing the method we use in this paper (parameteri-
+zationvia deconvolutionofthe stellarmassfunction).
+3.4.1.Parameterizing the Stellar Mass Function
+In abundance matching, knowledge of the halo mass func-
+tion and the stellar mass function uniquely determines the
+stellar mass – halo mass relation. Hence, parameterizing
+the stellar mass function yields an indirect parameterizat ion
+for the stellar mass – halo mass relation as well. Numer-
+ouspapers(e.g.Cole et al.2001;Bell etal.2003;Pantereta l.
+2004; Pérez-Gonzálezet al.2008) havefoundthat theGSMF
+iswell-approximatedbyaSchechterfunction:
+φ(M∗,z)=φ⋆(z)/parenleftbiggM∗
+M(z)/parenrightbigg−α(z)
+exp/parenleftbigg
+−M∗
+M(z)/parenrightbigg
+,(16)
+where the Schechter parameters φ⋆(z),M(z), andα(z) evolve
+as functions of the redshift z. In many previous works
+on abundance matching (e.g. Conroyetal. 2009), it is the
+Schechter function for the stellar mass function that sets t he
+formoftheSM–HMrelation.
+More recently, however, several authors have noted that
+the GSMF cannot be matched by a single Schechter function
+forz<0.2 to within statistical errors (e.g. Li &White 2009;
+Baldryetal. 2008), in part because of an upturn in the slope
+of the GSMF for galaxies below 109M⊙in stellar mass. It
+is possible that a conspiracy of systematic errors causes th e
+observeddeviations,butthereisnofundamentalreasontoe x-
+pecttheintrinsicGSMF tobefitexactlybyaSchechterfunc-
+tion (see discussion in AppendixC). In any case, our full pa-
+rameterization —either the stellar mass function or the err or
+parameterization— mustbe able to capture all the subtleties
+of the observedstellar massfunction. Hence, we are incline d
+toadopta moreflexiblemodelthanthe Schechterfunctionof
+equation16. Otherauthors,wrestlingwiththesameproblem ,
+have chosen to adopt multiple Schechter functions, includ-
+ing the eleven-parameter triple piecewise Schechter-func tion
+fit used by Li& White (2009). While accurate, these models
+oftenaddcomplicationwithoutincreasingintuition.
+3.4.2.Deconvolving the Observed Stellar Mass Function11 12 13 14 15
+log10(Mh) [MO•]8.89.29.61010.410.811.211.6log10(M*) [MO•]
+Direct Deconvolution
+Functional Fit
+Figure2. Relation between halo massandstellar massinthelocalUniv erse,
+obtained via direct deconvolution of the stellar mass funct ion in Li&White
+(2009) matched to halos in a WMAP5 cosmology. The deconvolut ion in-
+cludes the most likely value of scatter in stellar mass at a gi ven halo mass as
+wellasstatisticalerrorsinindividualstellarmasses. Th edirectdeconvolution
+(solid line) is compared to thebest fitto Eq. 21 ( red dashed line ).
+Rather than attempting to parameterize the stellar mass
+function, we could use abundance matching directly to de-
+rive the stellar mass – halo mass relation for the maximal-
+likelihoodstellar mass function,and thenfind a fit which can
+parameterize the uncertainties in the shape of the relation .
+This process is complicated by the various errors which we
+musttakeintoaccount. Recall fromEquations2and13that
+φmeas(M∗)=φdirect(M∗)◦Ps(∆log10M∗)◦P(∆log10M∗),
+(17)
+(where “◦” denotes the convolutionoperation, Psis the prob-
+ability distribution for the scatter in stellar mass at fixed halo
+mass, and Pis the probability distribution for errors in ob-
+served stellar mass at fixed true stellar mass). However,
+if we obtain φdirectby deconvolution of the observed stellar
+mass function φmeas, we may use direct abundance matching
+(Equation 15) to determine the maximum likelihood form of
+Mh(M∗).
+Figure 2 shows the result of calculating Mh(M∗) atz∼0.1
+via deconvolution and direct matching of the stellar mass
+function as described in the previous section. We choose
+themaximum-likelihoodvalueforthedistributionfunctio nPs
+(namely, 0.16 dex log-normal scatter), and we use WMAP5
+cosmologyforthehalomassfunction φhinthederivation.
+While deconvolutionplusdirectabundancematchinggives
+anunbiasedcalculationoftherelation,thereareseveralp rob-
+lems which prevent it from being used directly to calculate
+uncertainties:
+1. Deconvolutionwilltendtoamplifystatisticalvariatio ns
+in the stellar mass function—that is, shallow bumps
+in the GSMF will be interpreted as convolutions of a
+sharperfeature.
+2. Deconvolutionwill give different results depending on
+the boundary conditions imposed on the stellar mass
+function (i.e., how the GSMF is extrapolated beyond
+the reporteddata points)—the effects of which may be
+seenat theedgesofthedeconvolutioninFigure2.
+3. Deconvolution becomes substantially more problem-UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 11
+atic when the convolutionfunctionvaries over the red-
+shift range, as it does for our higher-redshift data ( z>
+0.2).
+4. Deconvolution cannot extract the relation at a single
+redshift—instead, it will only return the relation aver-
+aged over the redshift range of galaxiesin the reported
+GSMF.
+For these reasons, we choose to find a fitting formula in-
+stead. In the discussion that follows, we fit Mh(M∗) (the halo
+massforwhichtheaveragestellarmassis M∗)ratherthanthe
+moreintuitive M∗(Mh)(theaveragestellarmassatahalomass
+Mh) primarily for reasons of computational efficiency. From
+Equations12and17,thecalculationofwhatobserverswould
+see(φmeas)foratrialstellarmass–halomassrelationrequires
+many evaluations of Mh(M∗) and no evaluations of M∗(Mh).
+Ifwehadinsteadparameterized M∗(Mh),andtheninvertedas
+necessary in the calculation of φmeas, our calculations would
+havetakenanorderofmagnitudemorecomputertime.
+3.4.3.Fittingthe Deconvolved Relation
+It is well-known from comparing the GSMF (or the lu-
+minosity function) to the halo mass function that high-mass
+(M∗/greaterorsimilar1010.5M⊙) galaxieshave a significantly differentstel-
+lar mass-halo mass scaling than low-mass galaxies, which
+is usually attributed to different feedback mechanisms dom -
+inating in high-mass vs. low-mass galaxies. The transition
+point between low-mass and high-mass galaxies—seen as a
+turnoverintheplotof Mh(M∗)aroundM∗=1010.6M⊙inFig-
+ure 2—defines a characteristic stellar mass ( M∗,0) and an as-
+sociated characteristic halo mass ( M1). Hence, we consider
+functionalformswhichrespectthisgeneralstructureofa l ow
+stellarmassregimeandahighstellarmassregimewithachar -
+acteristictransitionpoint:
+log10(Mh(M∗))= log10(M1) [CharacteristicHaloMass]
++flow(M∗/M∗,0) [Low-massfunctionalform]
++fhigh(M∗/M∗,0) [High-massfunctionalform]
+whereflowandfhighare dimensionless functions dominating
+belowandabove M∗,0, respectively.
+Forlow-massgalaxies( M∗<1010.5M⊙),wefindthestellar
+mass–halomassrelationtobeconsistentwithapower–law:
+Mh(M∗)
+M1≈/parenleftbiggM∗
+M∗,0/parenrightbiggβ
+,or
+log(Mh(M∗))≈log(M1)+βlog/parenleftbiggM∗
+M∗,0/parenrightbigg
+.(18)
+Forhigh-massgalaxies,we findthestellar mass–halomass
+relation to be inconsistent with a power–law. In particu-
+lar, the logarithmic slope of Mh(M∗) changes with M∗, with
+dlogMh/dlogM∗always increasing as M∗increases. This
+may seem like a small detail; after all, by eye, it appears tha t
+a power law could be a reasonable fit for high-mass galax-
+ies in Figure 2. In addition, because previous authors (e.g. ,
+Mosteretal. 2009; Yanget al. 2009a) have used power laws,
+it maynotseem necessarytouse a differentfunctionalform.
+In order to explore this issue, we tried a general dou-
+ble power–law functional form for Mh(M∗) which parame-
+terized a superset of the fits used in Mosteretal. (2009) and
+Yanget al. (2009a) (in particular, the same form as in Equa-
+tionC2inAppendixC). Wefoundthatthisapproachhadtwo
+majorproblemscommonto anysuchpower–lawform:1. As the logarithmic slope of Mh(M∗) increases with
+increasing M∗, the best-fit power–law for high-mass
+galaxies will depend on the upper limit of M∗in the
+available data for the GSMF. Thus, the best-fit power–
+law will depend on the number density limit of the
+observational survey used—rather than on any funda-
+mental physics. Moreover, for studies such as this one
+whichconsiderredshiftevolution,thedifferentnumber
+densities probed at different redshifts result in a com-
+pletelyartificial“evolution”ofthebest-fitpower–law.
+2. The best–fit power–law will not depend on the high-
+est mass galaxies alone; instead, it will be something
+of an average overall the high-massgalaxies. Because
+thelogarithmicslopeisincreasingwith M∗,thismeans
+thatthebest-fitpowerlawfor Mh(M∗)willincreasingly
+underestimate the true Mh(M∗) at high M∗. Namely,
+the fit will underestimate the halo mass correspond-
+ing to a given stellar mass, and therefore (as lower-
+masshaloshavehighernumberdensities)resultinstel-
+larmassfunctions systematically biasedaboveobserva-
+tional values. However, a systematic bias in our func-
+tional form will influencethe best-fit valuesof the sys-
+tematic error parameters. The systematic bias caused
+byassumingapower–lawformturnsouttobemostde-
+generate with the scatter in stellar mass at fixed halo
+mass (ξ). As a result, for the MCMC chains which as-
+sumed a double power–law form for Mh(M∗), the pos-
+terior distribution of ξwas 0.09±0.02 dex, which just
+barelylieswithin2 σoftheconstraintsfromMoreet al.
+(2009).
+These problemsare not as significant if one only considers
+thestellarmassfunctionatasingleredshift,orifonedoes not
+allowforthesystematicerrorswhichchangetheoverallsha pe
+ofthestellarmassfunction( κ,ξ,andσ(z)). However,wefind
+that the issues listed above exclude the use of a power–law
+for our purposes. Instead, we find that Mh(M∗) asymptotes
+toasub-exponential functionforhigh M∗, namely,afunction
+which climbsmore rapidly than any power–lawfunction,but
+lessrapidlythananyexponentialfunction. Wefindthathigh –
+massgalaxies( M∗>1010.5M⊙)arewell fit bytherelation
+Mh(M∗)∼∝10/parenleftBig
+M∗
+M∗,0/parenrightBigδ
+,or
+log10(Mh(M∗))→log10(M1)+/parenleftbiggM∗
+M∗,0/parenrightbiggδ
+(19)
+whereδsets how rapidly the function climbs; δ→0 would
+correspond to a power–law, and δ= 1 would correspond to a
+pureexponential. Typicalvaluesof δatz=0rangefrom0 .5−
+0.6. It is not obvious what physical meaning can be directly
+inferredfromthechoiceofa sub-exponentialfunction—aft er
+all, the stellar mass of a galaxyis a complicatedintegralov er
+the merger and evolution history of the galaxy—but it could
+suggest that the physics drivingthe Mh(M∗) relation at high–
+massis notscale–free.
+Although this form now matches the asymptotic behavior
+for the highest and lowest stellar mass galaxies, one addi-
+tional parameteris necessary to match the functionalform o f
+the deconvolution. That is to say, galaxiesin between the ex -
+tremes in stellar mass will lie in a transition region, as the y
+may have been substantially affected by multiple feedback
+mechanisms. The width of this transition region will depend
+on many things—e.g., how long galaxies take to gain stellar12 BEHROOZI,CONROY & WECHSLER
+mass,howmuchofthestellar masspresentcamefromquies-
+cent star formation as opposed to mergers, and the degree of
+interaction between multiple feedback mechanisms. Hence,
+instead of having Mh(M∗) become suddenly sub-exponential
+forgalaxieslargerthan M∗,0,weallowforaslow“turn-on”of
+the morerapid growth. The behaviorof Mh(M∗) is best fit by
+modifyingthepreviousequationto
+log10(Mh(M∗))→log10(M1)+/parenleftBig
+M∗
+M∗,0/parenrightBigδ
+1+/parenleftBig
+M∗
+M∗,0/parenrightBig−γ(20)
+The denominator,1 +(M∗/M∗,0)−γ, is largefor M∗<M∗,0,
+anditfallstounityfor M∗>M∗,0ataratecontrolledby γ. A
+larger value of γimplies a more rapid transition between the
+power–law and sub-exponential behavior (typical values fo r
+(γ)atz=0are1.3-1.7). Asthenon-constantpieceof Mh(M∗)
+inEquation20is1
+2forM∗=M∗,0, weadda finalfactorof −1
+2tocompensatesothat Mh(M∗,0)=M1.
+To summarize, our resulting best–fit functional form has
+fiveparameters:
+log10(Mh(M∗))=
+log10(M1)+βlog10/parenleftbiggM∗
+M∗,0/parenrightbigg
++/parenleftBig
+M∗
+M∗,0/parenrightBigδ
+1+/parenleftBig
+M∗
+M∗,0/parenrightBig−γ−1
+2.(21)
+WhereM1isacharacteristichalomass, M∗,0isacharacteristic
+stellar mass, βis the faint-end slope, and δandγcontrol the
+massive-end slope. The best fit using this functional form is
+shown in Figure 2, and it achieves excellent agreement over
+theentirerangeofstellar masses.
+Deconvolving the GSMF at higher redshifts does not sug-
+gest that anything more than linear evolution in the parame-
+tersisnecessary,at least outto z=1. While the characteristic
+mass of the GSMF and the characteristic mass of the halo
+mass function certainly evolve, the change in the shapesof
+thetwofunctionsisrelativelyslight. Aswewishforthefun c-
+tionalformtohaveanaturalextensiontohigherredshifts, we
+parameterizethe evolutionin termsofthescale factor( a):
+log10(M1(a))=M1,0+M1,a(a−1),
+log10(M∗,0(a))=M∗,0,0+M∗,0,a(a−1),
+β(a)=β0+βa(a−1), (22)
+δ(a)=δ0+δa(a−1),
+γ(a)=γ0+γa(a−1),
+wherea=1isthescale factortoday.
+3.5.CalculatingModelLikelihoods
+We make use of a Markov Chain Monte Carlo (MCMC)
+method to generate a probability distribution in our com-
+plete parameter space of stellar mass function parame-
+ters (M1,0,M1,a,M∗,0,0,M∗,0,a,β0,βa,δ0,δa,γ0,γa), systematic
+modeling errors ( κ,µ,σz), and the scatter in stellar mass at
+fixedhalomass( ξ). Abriefsummaryofeachoftheseparam-
+eters appears in Table 1 along with a reference to the section
+inwhichitwasfirstdescribed. Usingthisfullmodel,wemay
+calculate the stellar mass functions expected to be seen by
+observers ( φexpect) for a large number of points in parameter
+space, and compare them to observed GSMFs (Li&White2009; Pérez-Gonzálezet al. 2008). Note that, as the observa -
+tionaldataalwayscoversarangeofredshifts,wemustmimic
+thisin ourcalculationof φexpect:
+φexpect=/integraltextz2
+z1φfit(z)dVC(z)/integraltextz2
+z1dVC(z), (23)
+wheredVC(z) is the comoving volume element per unit solid
+angle as a functionof redshift. Then, we can write the likeli -
+hoodasL=exp/parenleftbig
+−χ2/2/parenrightbig
+, where
+χ2=/integraldisplay/bracketleftbigglog10[φexpect(M∗)/φmeas(M∗)]
+σobs(M∗)/bracketrightbigg2
+dlog10(M∗),(24)
+andwhere σobs(M∗)isthereportedstatistical errorin φmeasas
+afunctionofstellarmass.
+Note that, as defined above, the equation for χ2contains
+the assumption that there is only one independent observa-
+tion point for the GSMF per decade in stellar mass (from the
+weightof dlog10(M)). Wemaytunethisassumptionintroduc-
+inganotherparameter n—thenumberofnon-correlatedobser-
+vations per decade in stellar mass—which would change the
+likelihood function to L= exp/parenleftbig
+−nχ2/2/parenrightbig
+. Here, we assume
+that each of the data points reported by Li &White (2009)
+and Pérez-Gonzálezet al. (2008) are independent—suchthat
+n=10fortheformerpaperand n=5forthelatterpaper.
+The MCMC chains each contain 222≈4×106points.
+We verify convergence according to the algorithm in
+Dunkleyet al. (2005); in all cases, the ratio of the sample
+mean variance to the distribution variance (the “convergen ce
+ratio”)isbelow0.005.
+3.6.MethodologySummary
+Our procedureto calculate the stellar mass – halo mass re-
+lation, taking into account all mentioned uncertainties, m ay
+besummarizedin sevensteps:
+1. We select a trial point in the parameter space of SM–
+HM relations as well as a trial point in our parameter
+space of systematics ( µ,κ,σz,ξ). A complete list of
+parametersanddescriptionsisgiveninTable1.
+2. The trial SM–HM relation gives a one-to-onemapping
+between halo masses and stellar masses, giving a di-
+rect conversion from the halo mass function to a trial
+galaxystellarmassfunction(correspondingto φdirectin
+§3.3.1).
+3. This trial GSMF is convolvedwith the probability dis-
+tributions for scatter in stellar mass at fixed halo mass
+(controlledby ξ,see§3.3.1)andforscatterinobserver-
+determined stellar mass at fixed true stellar mass (par-
+tially controlledby σz,see §3.1.2).
+4. The resulting GSMF is shifted by a uniform offset in
+stellar masses (controlled by µ) to account for uni-
+formsystematicdifferencesbetweenouradoptedstellar
+masses and the true underlyingmasses. Also, its shape
+is stretched or compressed to account for stellar mass–
+dependentoffsets between our masses and the true un-
+derlyingmasses(controlledby κ, see §3.1.1).
+5. We repeat steps 2-4 for all redshifts in the range cov-
+ered by the observed data set. We may then calculateUncertaintiesAffectingtheStellar Mass–HaloMassRelati on 13
+Table 1
+Summaryof Model Parameters
+Symbol Description PrioraSection
+Mh(M∗) Thehalo massfor which the average stellar massis M∗ N/A 3.4.3
+M1 Characteristic Halo Mass Flat (Log) 3.4.3
+M∗,0 Characteristic Stellar Mass Flat (Log) 3.4.3
+β Faint-end power law ( Mh∼Mβ
+∗) Flat (Linear) 3.4.3
+δ Massive-end sub-exponential (log10(Mh)∼Mδ
+∗) Flat (Linear) 3.4.3
+γ Transition width between faint- and massive-end relations Flat (Linear) 3.4.3
+(x)0Value of thevariable ( x) atthe present epoch, where ( x) is oneof ( M1,M∗,0,β,δ,γ) (see above) 3.4.3
+(x)a Evolution of the variable ( x) with scale factor (same as for ( x)0) 3.4.3
+µ Systematic offset in M∗calculations G(0,0.25) (Log) 3.1.1
+κ Systematic mass-dependent offset in M∗calculations G(0,0.10) (Linear) 3.1.1
+σz Redshift scaling of statistical errors in M∗calculations G(0.05,0.015) (Log) 3.1.2
+ξ Scatter in M∗at fixedMh G(0.16,0.04) (Log) 3.3.1
+aSee Equations 1, 5, 21-23. G(x,s) denotes a Gaussian prior centered at xwith standard deviation s, in either linear or logarithmic
+units. ‘Flat’ denotes auniform prior in either linear or log arithmic units.
+the expected GSMF in each redshift bin for which ob-
+servers have reported data. The likelihood of the ex-
+pectedGSMFsgiventhemeasuredGSMFsisthenused
+to determinethe nextstep intheMCMCchain.
+6. To account for sample variance in the observed stel-
+lar mass functions above z∼0.2, we recalculate each
+SM–HMrelationinthechainforanalternatehalomass
+function taken from a randomly sampled mock survey
+(see §3.2.4) and re-fit our functional form to the red-
+shift evolutionof the relation. Similarly, for the results
+which include cosmology uncertainty, we recalculate
+each SM–HM relation for an alternate halo mass func-
+tion randomly selected from the MCMC chain used to
+determinetheWMAP5 cosmologyuncertainties.
+7. We repeat steps 1-6 to build a joint probability distri-
+butionfortheSM–HMrelationandthesystematicspa-
+rameter space. The steps are repeated until the joint
+probabilitydistributionhasconvergedtotheunderlying
+posteriordistribution.
+4.RESULTS FOR0 <z<1
+We now present the results of this approach to determine
+theSM–HMrelationandrelatedquantities. In§4.1, wecom-
+pareGSMFsgeneratedfromourbestfitstoobserveddataand
+comment on the effects of systematic observational biases.
+We present our best-fitting results for the SM–HM relation
+with full error bars in §4.2. We evaluate the relative impor-
+tance of each of the contributing types of error in §4.3 and
+summarize the most relevant contributionsin §4.3.7. Final ly,
+our derived SM–HM relation is compared to other published
+resultsin §4.4.
+4.1.GalaxyStellarMassFunctions
+To demonstrate that our functional form for Mh(M∗) is ca-
+pable of reproducingobserved galaxy stellar mass function s,
+we show a comparison between our best–fit models and the
+observed data in Figs. 3 and 4 at several redshifts. For our
+best-fit models, both φtrue(the true or “intrinsic” stellar mass
+function)and φmeas(theGSMFthatobserverswouldmeasure)
+are shown. Recall that φmeasincorporates the effects of the
+systematic observational biases; namely, the overall shif t in
+stellarmasscalculations, µ,thelinearlymass-dependentshift,
+κ, and the statistical errorsin stellar mass calculationsfo r in-
+dividual galaxies, σ(z). The fact that the best-values of thesystematic parameters ( µ,κ,ξ,σz) are very close to the cen-
+ters of their prior distributionsprovidesconfirmation tha t the
+functional form for the SM–HM relation outlined in §3.4.3
+doesnotbiasourbest-fit results.
+As our best-fit values for µandκare close to zero (see
+Table 2), the differencebetween φtrueandφmeasis almost ex-
+clusively due to the scatter σ(z) in calculated stellar masses.
+The differencebetween φtrueandφmeasonly becomesevident
+for galaxies above 1011M⊙, where the falling slope of the
+GSMF becomes severe enough for the scatter σ(z) to signifi-
+cantly raise number counts in the observed GSMF. At z∼0,
+thesystematiceffectof σ(z)putstheintrinsicGSMF wellbe-
+lowthesmall statistical errorbars.
+At higher redshifts, although the effect of σ(z) is larger,
+current surveys at z>0.2 do not yet cover sufficient volume
+to constrain the shape of the GSMF well at the massive end.
+Nonetheless, for future wide-field surveys at z>0.2, correc-
+tion to the GSMF for scatter in calculated stellar masses wil l
+beanimportantconsideration.
+4.2.TheBest-Fit StellarMass–HaloMassRelations
+We plotthe averagestellar massas a functionofhalo mass
+forz= 0−1 in Figure 5 to show the evolution of the stellar
+mass – halo mass relation. Note that as the stellar mass at a
+givenhalomasshasalog-normalscatter(see §2.3),weusege-
+ometricaveragesforstellarmassesratherthanlinearones . To
+highlighttheeffectsofhalomassonstarformationefficien cy,
+we also present the SM–HM relation in terms of the average
+stellar mass fraction (stellar mass / halo mass) for z= 0−1
+as a function of halo mass in the same figure. We focus on
+this quantity for the remainder of the paper. The best-fit pa-
+rameters for the function Mh(M∗) are given in Table 2, and
+thenumericalvaluesforthestellarmassfractionsarelist edin
+AppendixD.
+The stellar mass fractions for central galaxies consistent ly
+show a maximum for halo masses near 1012M⊙. While the
+location of this maximum evolves with time, it clearly il-
+lustrates that star-formation efficiency must fall off for b oth
+higher and lower-mass halos. The slopes of the SM–HM re-
+lation above and below this characteristic halo mass are in-
+dicative of at least two processes limiting star-formation effi-
+ciency,althoughmergerscomplicate direct analysis for hi gh-
+masshalos. Atthelow-massend,theSM–HMrelationscales
+asM∗∼M2.3
+hatz= 0 and as M∗∼M2.9
+hatz= 1. However,
+giventhe lack of informationabout low stellar-mass galaxi es
+atz>0.5,thestatisticalsignificanceofthisevolutionisweak;14 BEHROOZI,CONROY & WECHSLER
+-8-7-6-5-4-3-2log10(φ) [Mpc-3 (log M)-1]
+z = 0.1, φtrue 
+z = 0.1, φmeas 
+z = 0.1, Li & White (2009)
+9 10 11 12
+log10(M*) [MO•]-0.300.3log10(φ/φmeas)
+Figure3. Comparison of the best fit φtrue(the true or “intrinsic” GSMF)
+in our model to the resulting φmeas(what an observer would report for the
+GSMF, which includes the effects of the systematic biases µ,κ, andσ) at
+z=0. Sincethebest–fitvaluesof µandκareveryclosetozero,thedifference
+betweenφmeasandφtruealmost exclusively comes from the uncertainty in
+measuring stellar masses ( σ).
+Table 2
+Bestfits for the redshift evolution of Mh(M∗)
+Parameter Free ( µ,κ)µ=κ=0 Free ( µ,κ)
+0<z<1 0<z<1 0.8<z<4
+M∗,0,010.72+0.22
+−0.2910.72+0.02
+−0.1211.09+0.54
+−0.31
+M∗,0,a0.55+0.18
+−0.790.59+0.15
+−0.850.56+0.89
+−0.44
+M∗,0,a2 N/A N /A6.99+2.69
+−3.51
+M1,012.35+0.07
+−0.1612.35+0.02
+−0.1512.27+0.59
+−0.27
+M1,a0.28+0.19
+−0.970.30+0.14
+−1.02−0.84+0.87
+−0.58
+β00.44+0.04
+−0.060.43+0.01
+−0.050.65+0.26
+−0.20
+βa0.18+0.08
+−0.340.18+0.06
+−0.340.31+0.38
+−0.47
+δ00.57+0.15
+−0.060.56+0.14
+−0.050.56+1.33
+−0.29
+δa0.17+0.42
+−0.410.18+0.41
+−0.42−0.12+0.76
+−0.50
+γ01.56+0.12
+−0.381.54+0.03
+−0.401.12+7.47
+−0.36
+γa2.51+0.15
+−1.832.52+0.03
+−1.89−0.53+7.87
+−2.50
+µ0.00+0.24
+−0.25N/A0.00+0.25
+−0.25
+κ0.02+0.11
+−0.07N/A0.00+0.14
+−0.04
+ξ0.15+0.04
+−0.020.15+0.04
+−0.010.16+0.07
+−0.01
+σz0.05+0.02
+−0.010.05+0.02
+−0.010.05+0.02
+−0.01
+Note. —See Table1 and Equations 1,5, 21-23,25.
+noevolutioninthelow-massslopeoftherelationisconsist ent
+within our one-sigma errors. Several studies (most recentl y,
+Baldryetal. 2008; Droryetal. 2009) have reported that the
+GSMF has an upturn in slope for very low stellar masses,
+particularly below 108.5M⊙; this would imply that our best
+fits may overestimate the scaling relation for galaxies belo w
+108.5M⊙. At the high mass end, our best fitting function re-
+sults in a progressively shallower relation for the growth o f
+stellar mass with halo mass, so that no single power law can
+describe the scaling. However, for halos close to 1014M⊙,
+the best-fit relation scales locally as M∗∼M0.28
+hatz=0 and9 10 11 12
+log10(M*) [MO•]-8-7-6-5-4-3-2log10(φ) [Mpc-3 (log M)-1]
+z = 0.5, φtrue 
+z = 0.5, φmeas 
+z = 0.5, Pérez-González et. al. (2008)
+9 10 11 12
+log10(M*) [MO•]-8-7-6-5-4-3-2log10(φ) [Mpc-3 (log M)-1]
+z = 1.15, φtrue 
+z = 1.15, φmeas 
+z = 1.15, Pérez-González et. al. (2008)
+Figure4. Comparison of the best fit φtrue(the true or “intrinsic” GSMF) to
+theresulting φmeas(as in Figure 3),for z=0.5 andz=1.15. Statistical errors
+inindividual stellar masseshavealarger effect athigher r edshift, resulting in
+asteeper intrinsic bright end than measured.
+M∗∼M0.34
+hatz=1,inaccordwithpreviousstudies(see§4.4).
+The results for high mass halos are also consistent with no
+evolutionin theslopeoftheSM–HMrelation.
+Figure 6 shows the stellar mass fraction for 0 <z<1 ex-
+cluding the effects of systematic shifts in stellar mass cal cu-
+lations (i.e., assuming µ=κ=0). Under the assumption that
+systematic errorsin stellar mass calculations result in si milar
+biasesin stellar masses at z=0 as they do at higherredshifts,
+thisallowsustoconsidertheevolutioninnormalizationof the
+SM–HM relation. Low-mass halos (below 1012M⊙) display
+clearlyhigherstellar massfractionsat lateredshiftstha nthey
+doatearlyredshifts. Bycontrast,theevolutioninstellar mass
+fractionsfor high mass halos (above 1013.5M⊙) is not statis-
+ticallysignificant,anditisconstrainedtobesubstantial lyless
+than for low-mass halos. In the time since z= 1, this means
+thatthe star formationrates forhigh-masshalostypically fall
+relative to their dark matter accretion rates, whereas the o p-
+posite is true for low-mass halos (Conroy&Wechsler 2009).
+The best-fitting parameters for the SM–HM relation assum-
+ingµ=κ=0 appear in Table 2, and the data pointsin Figure
+6appearinAppendixD.
+4.3.ImpactofUncertaintiesUncertaintiesAffectingtheStellar Mass–HaloMassRelati on 15
+11 12 13 14 15
+log10(Mh) [MO•]89101112log10(M*) [MO•]
+z = 0.1
+z = 0.5
+z = 1.0
+11 12 13 14 15
+log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
+z = 0.1
+z = 0.5
+z = 1.0
+Figure5. Top panel : Stellar mass – halo mass relation as a function of red-
+shift for our preferred model. Bottom panel : Evolution of the derived stellar
+mass fractions ( M∗/Mh). In each case, the lines show the mean values for
+central galaxies. These relations also characterize the sa tellite galaxy pop-
+ulation if the horizontal axis is interpreted as the halo mas s at the time of
+accretion. Errors bars include both systematic and statist ical uncertainties,
+calculated for afixed cosmological model (with WMAP5parame ters).
+4.3.1.Systematic ShiftsinStellar Mass Calculations
+ByfarthelargestcontributortotheerrorbudgetoftheSM–
+HM relation is the systematic error parameter µ. As the ef-
+fect ofµis to multiply all stellar masses by a constant factor,
+and as the width of the error bars in Figure 5 correspondsal-
+mostexactlytotheprioron µ,wemayconcludethatreducing
+the error on the systematic shifts in stellar mass calculati ons
+wouldrepresent the single largest improvementin ourunder -
+standingoftheshapeoftheSM–HMrelation. Figure6shows
+thesubstantiallysmallererrorbarsthatresultifsystema ticer-
+rors(µandκ)inthe stellarmasscalculationsareneglected.
+4.3.2.Scatter inStellar Mass at FixedHalo Mass
+The effect of ignoring scatter in stellar mass at fixed halo
+mass(i.e.,setting ξ=0)isshownat tworedshiftsinFigure7.
+We find that the changeis insignificant below halo masses of
+1012M⊙, and is within statistical error bars below 1013M⊙
+forz=1. Thisisaresultofthefactthattheslopeofthestellar
+mass function below 1010.5M⊙in stellar mass (correspond-
+ing to 1012M⊙in halo mass) is not steep enough for scat-11 12 13 14 15
+log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
+z = 0.1
+z = 0.5
+z = 1.0
+Figure6. Evolution of the derived stellar mass fractions ( M∗/Mh) in the
+absence of systematic errors. This result is analogous to Fi gure 5, bottom
+panel, calculated under theassumption thatthetrue values ofthesystematics
+µandκin thestellar mass function are zero at all redshifts.
+tertohavesignificantimpact(seealsoTasitsiomi et al.200 4).
+Becauseξ >0 results in high stellar–mass galaxies being as-
+signedtolower-masshalosthantheywouldbeotherwise(due
+to the higher numberdensity of lower-mass halos), the effec t
+is that higher-masshalos contain fewer stars on average tha n
+they would for ξ=0. The effect of setting ξ=0 exceedssys-
+tematicerrorbarsonlyfortheveryhighestmasshalos,abov e
+1014.5M⊙.
+We note that our posterior distribution constrains ξto be
+less than 0.22 dex at the 98% confidence level. Higher val-
+ues forξwould result in GSMFs inconsistent with the steep
+falloff of the Li &White (2009) GSMF (see also discussion
+inGuoetal. 2009).
+4.3.3.Statistical ErrorsinStellar Mass Calculations
+The significance of includingor excludingrandomstatisti-
+calerrorsinstellarmasscalculations, σ(z),isalsoshownFig-
+ure7. TheeffectofthistypeofscatterontheSM–HMrelation
+is mathematically identical to the effect of scatter in stel lar
+mass at fixed halo mass. As σ(z= 0) (∼0.07 dex) is much
+smaller than the expected value of ξ(∼0.16 dex), the con-
+volution of the two effects is only marginally different fro m
+including ξaloneatz=0;thisresultsinonlyaminoreffecton
+the SM–HM relation. The effect becomes more pronounced
+atz=1forthereasonthat σ(z=1)(∼0.12dex)becomesmore
+comparableto ξ—andsoincludingtheeffectsofstatisticaler-
+rorsin stellar massbecomesas importantasmodelingscatte r
+instellar massat fixedhalomass.
+4.3.4.Cosmology Uncertainties
+InFigure8,weshowacomparisonofbestfitsforthestellar
+mass fraction using abundance matching with three differen t
+halo mass functions: analytic prescriptions for WMAP5 and
+WMAP1 (see §3.2.2) as well as the mass function taken di-
+rectly from the L80G simulation(see §3.2.1). The differenc e
+betweentheL80GsimulationandtheanalyticWMAP1mass
+functionisslight,astheL80GsimulationusesWMAP1initia l
+conditions( h=0.7,Ωm=0.3,ΩΛ=0.7,σ8=0.9,ns=1); the
+differenceisconsistentwithsamplevariancefortherelat ively
+small (80 h−1Mpc)size ofthe simulation. Thedifferencebe-
+tween SM–HM relations using WMAP1 and WMAP5 cos-16 BEHROOZI,CONROY & WECHSLER
+11 12 13 14 15
+log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
+z = 0.1 (incl. σ(z), ξ=0.16dex)
+z = 0.1 (excl. σ(z), ξ=0.16dex)
+z = 0.1 (incl. σ(z), ξ=0.0dex)
+11 12 13 14 15
+log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
+z = 1.0 (incl. σ(z), ξ=0.16dex)
+z = 1.0 (excl. σ(z), ξ=0.16dex)
+z = 1.0 (incl. σ(z), ξ=0.0dex)
+Figure7. Comparison between SM–HM relations derived in the preferre d model (including the effects of the statistical errors σ(z) and taking the scatter in
+stellar mass at a given halo mass to be ξ= 0.16dex) to those excluding the effects of σ(z) or taking ξ= 0, atz= 0 (left panel ) andz= 1 (right panel ). Light
+shaded regions denote 1- σerrors including both systematic and statistical errors; d ark shaded regions denote the 1- σerrors if the systematic offsets in stellar
+masscalculations ( µandκ)are fixed to 0.
+mologies is within the systematic errors at all masses. When
+systematic errors are neglected, the two cosmologies yield
+SM–HM relations that are noticeably different only at low
+halomasses( M<1012M⊙).
+Figure 9 show the results of including uncertainties in the
+WMAP5cosmologicalparameters. Asdescribedin§3.1,this
+is doneusinghalo mass functionscalculated with parameter s
+resampled from the cosmological parameter chains provided
+by the WMAP team. Only at z∼0 are the changes in error
+bars significant enough to justify mention. Here, the uncer-
+tainty in cosmology begins to exceed other sources of statis -
+tical error for halos below 1012M⊙due to the small errors
+on the GSMF at the stellar masses associated with such ha-
+los(Li &White2009). However,thecosmologyuncertainties
+arestill well withinthesystematicerrorbars.
+4.3.5.Sample Variance
+Because of the large volumeof the SDSS, sample variance
+contributesinsignificantlytotheerrorbudgetfortheSM–H M
+relationbelow z=0.2. Abovethatredshift,thecomparatively
+limitedsurveyvolumeofPérez-Gonzálezetal.(2008)resul ts
+in sample variance becoming an important contributor to the
+statistical error for halos below 1012M⊙(Poisson noise dom-
+inatesforlargerhalos). Iftheeffectsofsamplevariancew ere
+ignored, the statistical error spreads for our derived SM–H M
+relations at z=1 would shrink from 0.12 dex to 0.09 dex for
+1011M⊙halos, and from 0.05 dex to 0.04 dex for 1012.25M⊙
+halos. As with other types of errors, these considerationsa re
+well belowthelimitsofthesystematic errorbars.
+We caution that our error bars including sample variance
+atz>0 have a very specific meaning. Namely, they include
+the standard deviation in our fitting form which might be ex-
+pected if the surveyin Pérez-Gonzálezet al. (2008) had been
+conductedonalternatepatchesofthesky. Samplevariancea t
+redshiftsz>0 impacts only the linear evolution of the SM–
+HM relations we derive, as the large volume probed by the
+SDSSconstrainstheSM–HMrelationverywell at z∼0. Be-
+cause our fit is matched to the ensemble of reported data be-
+tween 0<z<1, it is less vulnerableto the effects of sample
+varianceinindividualredshiftbins. Instead,itisaffect edmost
+by overall shifts in the number densities reported for the en -tire high-redshift survey. While this means that our fit give s
+a more robust SM–HM relation at all redshifts, some caution
+must be used when comparing our relation to results derived
+from the GSMF in a single redshift bin (e.g., 0 .2<z<0.4).
+Thesewillhavemuchlargeruncertaintiesduetosamplevari -
+ancethan SM–HM relationsderived(like ours)fromGSMFs
+alongalightconeprobinga largeredshiftrange.
+Todemonstratethecredibilityofourapproachforcalculat -
+ing the appropriate error bars including sample variance, w e
+repeated our analysis of the SM–HM relation using GSMFs
+from Droryetal. (2009) (which appeared as we were com-
+pleting this work) instead of Pérez-Gonzálezetal. (2008)
+for 0.2<z<1 and retaining the GSMF from Li &White
+(2009) for z<0.2. Although the COSMOS survey in
+Droryetal. (2009) covers a much larger area ( ∼9x the area
+in Pérez-Gonzálezet al. 2008), the fact that it is a single
+fieldmeansthatthe expectedsamplevarianceis onlyslightl y
+smaller than for the combined fields in Pérez-Gonzálezet al.
+(2008). Asmightbeexpected,theSM–HMrelationat z=0.1
+using the Droryet al. (2009) GSMF is identical to the result
+in Figure 6 because of the strong constraining power of the
+SDSS data sample. At z=1, the SM–HM relation generated
+by using the Droryet al. (2009) GSMF is within our quoted
+statistical and sample varianceerrors, as shown in Figure 1 2.
+This may not be surprising unless one considers that clus-
+tering results suggest an overdensity at the 2-3 σlevel in the
+COSMOS redshift bin z=1 (Meneuxetal. 2009). However,
+becauseourmethodfitstheDroryetal.(2009)GSMFsacross
+the entire redshift range, the excess at z=1 is partially offset
+by an underdensity at z= 0.5. This demonstrates the robust-
+ness of our fitting method to the effects of sample variance
+exceptonthescaleofthe entiresurvey.
+4.3.6.Satellite Treatment
+Finally, we consider the changes in both the stellar mass
+fraction and total stellar mass fraction (total stellar mas s in
+central galaxy and all satellites / total halo mass) induced by
+different satellite evolution models (see §3.3.2 for detai ls on
+thetwomodels). AsshowninFigure10,fixingsatellitestell ar
+mass at the redshift of accretion (lines labeled as “SMF acc”)
+hasvirtuallynoeffectoneitherfractionascomparedtoall ow-UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 17
+11 12 13 14 15
+log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
+z = 0.1 (WMAP5)
+z = 0.1 (L80G)
+z = 0.1 (WMAP1)
+Figure8. Comparison between stellar mass fractions in different cos molo-
+gies. Lightshaded regionsdenotesystematicerrorspreads whiledarkshaded
+regions denote error spreads assuming µ=κ= 0, both about the WMAP5
+model. The dot-dashed blue line shows the fiducial relation f or a WMAP1
+cosmological model (using our analytic model) The dashed re d line shows
+therelation forasimulation oftheWMAP1cosmology. Differ ences between
+this and the analytic modelare within the expected sampleva riance errors.
+ing satellite stellar mass to evolve the same way as centrals
+with the same mass (labeled as SMF now). Because compar-
+ison of different satellite evolution models requires trac king
+satellites through merger trees, Figure 10 shows results on ly
+forsatellites intheL80Gsimulation.
+Thetreatmentofsatellitesmayhaveasomewhatlargerim-
+pact on the total stellar mass fraction, including the stell ar
+massofbothcentralandsatellite galaxieswithin a halo. Th is
+is shown for both models in Figure 10. Because of the steep
+fall-off in stellar mass for low mass galaxies, the total ste llar
+mass fraction has only minimal contribution from satellite s
+forlowmasshalos,anddeviatessignificantlyfromthestell ar
+massfractionforcentralsonlyathalomasses Mh>1012.5M⊙.
+At cluster-scale masses ( Mh∼1014M⊙), accreted satellites
+haveonaveragea higherratio ofstarsto darkmatter thanthe
+centralgalaxy,andthetotalstellarmassfractioncanbema ny
+times the central stellar mass fraction. However, the impac t
+ofthetwomodelsforsatellitetreatmentonthisratioissma ll.
+Profilesofsatellitegalaxiesinclustersshouldbeabletob etter
+distinguishbetweensuchmodels.
+4.3.7.Summary of Most Important Uncertainties
+Systematic stellar mass offsets resulting from modeling
+choices result in the single largest source of uncertaintie s
+(∼0.25 dex at all redshifts). The contribution from all other
+sourcesof error is much smaller, rangingfrom 0.02-0.12dex
+atz= 0 and from 0.07-0.16 dex at z= 1. On the other hand,
+this statement is only true when all contributing sources of
+scatter in stellar masses are considered. Models that do not
+accountforscatterinstellarmassatfixedhalomasswillove r-
+predict stellar masses in 1014.25M⊙halos by 0.13-0.19 dex,
+depending on the redshift. Models that do not account for
+scatterincalculatedstellarmassatfixedtruestellarmass will
+overpredictstellar masses in 1014.25M⊙halos by 0.12 dex at
+z= 1. Hence, it is important to take both these effects into
+account when considering the SM–HM connection either at
+highmassesorat highredshifts.11 12 13 14 15
+log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
+z = 0.1
+Figure9. Effect of cosmological uncertainties on the stellar mass fr action
+atz= 0.1. The error bars show the spread in stellar mass fractions in clud-
+ing both statistical errors and cosmology uncertainties (f rom WMAP5 con-
+straints, Komatsu etal. 2009). For comparison, the light sh aded region in-
+cludesstatistical andsystematicerrors,whilethedarksh adedregionincludes
+only statistical errors.
+11 12 13 14 15
+log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
+z = 0.0 (L80G) [SMFnow]
+z = 0.0 (L80G) [SMFacc]
+z = 0.0 (L80G) [SMFnow] (Total M*/Mh)
+z = 0.0 (L80G) [SMFacc] (Total M*/Mh)
+Figure10. Comparison between stellar massfractions and total stella r mass
+fractions(labeled as“TotalM ∗/Mh”)derived byassumingdifferentmatching
+epochs for satellite galaxies. The L80G simulation was used here in order
+to follow the accretion histories of the subhalos. The relat ions terminate at
+highmasseswherethehalo statistics becomeunreliable due tofinite–volume
+effects.
+4.4.Comparisonwith otherwork
+Acomparisonofourresultswithseveralresultsintheliter -
+atureatz∼0.1isshowninFigure11. Suchcomparisonisnot
+always straightforward, as other papers have often made dif -
+ferentassumptionsforthecosmologicalmodel,thedefiniti on
+of halo mass, or the measurement of stellar mass. In addi-
+tion, some papers report the average stellar mass at a given
+halomass(aswedo),andothersreporttheaveragehalomass
+at a given stellar mass. Given the scatter in stellar mass at
+fixedhalomass,theaveragingmethodcanaffecttheresultin g
+stellarmassfractions,particularlyforgroup-andcluste r-scale
+halo masses. To facilitate comparison with both approaches ,
+we plot our main results (labeled as “ ∝angbracketleftM∗/Mh|Mh∝angbracketright”) along
+withresultsforwhichthestellarmassfractionshavebeena v-
+eraged at a given stellar mass (labeled as “ ∝angbracketleftM∗/Mh|M∗∝angbracketright”).18 BEHROOZI,CONROY & WECHSLER
+11 12 13 14 15
+log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)This work, < M*/Mh | Mh >
+This work, M* / < Mh | M* >
+Moster et al. 2009 (AM)
+Guo et al. 2009 (AM)
+Wang & Jing 2009 (AM+CC)
+Zheng et al. 2007 (HOD)
+Mandelbaum et al. 2006 (WL)
+Klypin et al. in prep. (SD)
+Gavazzi et al. 2007 (SL)
+Yang et al. 2009a (CL)
+Hansen et al. 2009 (CL)
+Lin & Mohr 2004 (CL)
+Figure11. Comparison of our best-fit model at z= 0.1 to previously published results. Results shown include ot her results from abundance matching
+(Moster etal. 2009 and Guo et al. 2009); abundance matching p lus clustering constraints (Wang &Jing 2009); HOD modeling (Zheng etal. 2007); direct mea-
+surements from weak lensing (Mandelbaum etal. 2006), state llite dynamics (Klypin et al. 2009) and strong lensing (Gava zzi etal. 2007); and clusters selected
+from SDSS spectroscopic data (Yang etal. 2009a), SDSS photo metric data (the maxBCG sample Hansen et al. 2009), and X-ray selected clusters (Lin &Mohr
+2004). Dark grey shading indicates statistical and sample v ariance errors; light grey shading includes systematic err ors. Thered line shows our results averaged
+over stellar mass instead of halo mass;scatter affects thes e relations differently athigh masses. Theresults of Mande lbaum et al. (2006)and Klypin etal. (2009)
+are determined by stacking galaxies in bins of stellar mass, and so aremoreappropriately compared to this red line.
+In the comparisons below, we have not adjusted the assump-
+tions used to derive stellar masses, because such adjustmen ts
+can be complex and difficult to apply using simple conver-
+sions. Additionally,we haveonlycorrectedfordifference sin
+the underlyingcosmology for those papers using a variant of
+abundance matching method (Mosteret al. 2009; Guoetal.
+2009; Wang &Jing 2009; Conroyetal. 2009) using the pro-
+cess described in Appendix A, as alternate methods require
+corrections which are much more complicated. We have,
+however,adjustedtheIMFofall quotedstellarmassesto tha t
+of Chabrier (2003), and we have converted all quoted halo
+massestovirialmassesasdefinedin §3.2.2.
+Theclosestcomparisonwithourwork,usingaverysimilar
+method, is the result from Mosteretal. (2009). This result i s
+in excellent agreementwith oursat the high mass end, and is
+within our systematic errorsfor all masses considered. How -
+ever, their less flexible choice of functional form, and thei r
+use of a different stellar mass function(estimated from spe c-
+troscopy using the results of Panteret al. 2007) results in a
+differentvalueforthehalomass Mpeakwithpeakstellarmass
+fractionandashallowerscalingofstellarmasswithhaloma ssat the low mass end. Their error estimates only account for
+statisticalvariationsingalaxynumbercounts,andtheydo not
+include sample variance or variations in modeling assump-
+tions. Guoet al.(2009)useasimilarapproachtoMosteret al .
+(2009),usingstellarmassesfromLi& White(2009),butthey
+do not account for scatter in stellar mass at fixed halo mass.
+Consequently, their results match ours for 1012M⊙and less
+massive halos, but overpredict the stellar mass for larger h a-
+los.
+Wang&Jing (2009) use a parameterization for the SM–
+HM relation for both satellites and centrals, and they attem pt
+to simultaneously fit both the stellar mass function and clus -
+tering constraints, including the effects of scatter in ste llar
+massatfixedhalomass. At z∼0.1,theirdatasourcematches
+ours (Li&White 2009), but their approach finds a best-fit
+scatter in stellar mass at fixed halo mass of ξ= 0.2 dex, es-
+sentially the highest value allowed by the stellar mass func -
+tion(Guoet al.2009). Asthisishigherthanourbest-fitvalu e
+forξ, their SM–HM relation falls below ours for high-mass
+galaxies. Possiblybecauseofthelimitedflexibilityofthe irfit-
+tingform(theyuseonlyafour-parameterdoublepower-law) ,UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 19
+their SM–HM relation is in excess of ours for halo masses
+near1012M⊙.
+Zhengetal. (2007) used the galaxy clustering for
+luminosity-selectedsamplesintheSDSStoconstraintheha lo
+occupation distribution. This gives a direct constraint on the
+r−band luminosity of central galaxies as a function of halo
+mass. Stellar masses for this sample were determined us-
+ing theg−rcolor and the r-band luminosity as given by
+theBell etal. (2003) relation,anda WMAP1 cosmologywas
+assumed. This method allows for scatter in the luminosity
+at fixed halo mass to be constrained as a parameter in the
+model; results for this scatter are consistent with Moreeta l.
+(2009), although they are less well constrained. According
+toLi &White (2009), stellar massesfortheBell et al. (2003)
+relation are systematically larger than those calculated u sing
+Blanton&Roweis(2007) by0.1–0.3dex. However,as Ωmin
+WMAP1 is larger than in WMAP5, halo masses in WMAP1
+will be higher at a given number density than in WMAP5,
+somewhatcompensatingforthehigherstellarmasses.
+We next compare to constraints from direct measure-
+mentsofhalomassesfromdynamicsorgravitationallensing .
+Mandelbaumetal. (2006) have used weak lensing to mea-
+sure the galaxy–mass correlation function for SDSS galax-
+ies and derive a mean halo mass as a function of stellar
+mass. Mandelbaumet al. (2006) assume a WMAP1 cos-
+mology and uses spectroscopic stellar masses, calculated p er
+Kauffmannetal. (2003). Klypin et al (in preparation) have
+derived the mean halo mass as a function of stellar mass us-
+ing satellite dynamicsof SDSS galaxies(see also Pradaetal .
+2003; vandenBoschet al. 2004; Conroyet al. 2007). Their
+results are generally within our systematic errors but lowe r
+than others at the lowest masses and with a somewhat dif-
+ferent shape. This may be due to selection effects, as their
+work uses only isolated galaxies, which may have somewhat
+loweraveragestellarmasses. Gavazziet al.(2007)useaset of
+stronglensesfromthe SLACS surveyalong with a modelfor
+simultaneouslyfitting the stellar anddarkmatter componen ts
+ofthestackedlensprofiles. Thisresult,atonemassscale,i sa
+bithigherthanourerrorrangebutwithin1.5 σ. Theselection
+effects relevant to strong lenses are beyond the scope of thi s
+paper; however, within the effective radius, the stellar ma ss
+can easily contribute more to the lensing effect than the dar k
+matter. Thus,atanygivenhalomass,thehaloswithlessmas-
+sivegalaxiesaremuchlesslikelytobestronglenses,resul ting
+inabiastowardshigherstellarmassfractionsinstronglen ses
+ascomparedtohalosselectedat random.
+Atthehighmassend,onecandirectlyidentifyclustersand
+groups corresponding to dark matter halos, and measure the
+stellar masses of their central galaxies. Yanget al. (2009a )
+useagroupcatalogmatchedtohalostodeterminehalomasses
+(viaaniteratively-computedgroupluminosity–massrelat ion).
+StellarmassesinthisworkaredeterminedusingtheBell eta l.
+(2003)relationbetween g−rcolorand M/L; a WMAP3 cos-
+mologywasassumed. Theirresultsagreeverywell withours
+for low-masshalos, but they beginto differ at highermasses .
+This may be partially due to scatter between their calculate d
+halo masses (based on total stellar mass in the groups) and
+the true halo masses, resulting in additional scatter in the ir
+stellar masses at fixed halo mass. It could also be due to dif-
+ferences in stellar modeling; their results remain at all ti mes
+within oursystematic errors. We also compareto directmea-
+surements of massive clusters by Hansenet al. (2009) and
+Lin&Mohr (2004). In order to convert luminosities to stel-
+lar masses, we assume M/Li0.25= 3.3M⊙/L⊙,i0.25andM/LK= 0.83M⊙/L⊙,Kbased on the population synthesis code of
+Conroyetal.(2009). Thesemeasurementsarebothsomewhat
+higherthanourresultsformassiveclusters,theone-sigma er-
+ror estimates overlap. The discrepancies may be due to is-
+sues with cluster selection and with modeling scatter in the
+mass-observable relation; in each case the cluster mass is a n
+average mass for the given observable (X-ray luminosity or
+cluster richness), and can result in a bias if central galaxi es
+are correlated with this observable. More detailed modelin g
+of the scatter and correlations will be required to determin e
+whetherthisis canaccountfortheoffsets.
+A comparison of our results to others at z∼1 is shown in
+Figure 12. As may be expected, it is much harder to directly
+measurethe SM–HM relationat higherredshifts, resultingi n
+relatively fewer published results with which we may com-
+pare. We first note that we have compared the impact of
+two independent measurements of the GSMF from different
+surveys. As discussed in 4.3.5, because we simultaneously
+fit our model with linear evolution to the GSMF at redshifts
+0<z<1, our results are less sensitive to sample variance.
+In contrast to the conclusion of Droryet al. (2009) which fit
+theirresultstospecificredshiftbins,Figure12showsthat the
+Droryetal. (2009) and Pérez-Gonzálezetal. (2008) results
+are in agreement within statistical errors at z∼1 when fit-
+tingthefullredshiftrange. TheresultsinMosteret al.(20 09)
+are also very similar to ours at z∼1. However, Mosteret al.
+(2009)donotgivefitsfortheSM–HMrelationwhichinclude
+the effects of scatter in stellar mass at fixed halo mass excep t
+atz∼0anddonotingeneralincludescatterinmeasuredstel-
+lar mass with respect to the true stellar mass. Hence, we in-
+clude for comparison an SM–HM relation derived using our
+analysis but excluding both of these effects. The remaining
+deviance most likely stems from sample variance due to the
+muchsmallersurveyvolumeonwhichtheirfit isbased.
+Atz∼1, Wang& Jing (2009) make use of clustering
+data and stellar mass functions from the VVDS survey
+(Meneuxet al. 2008; Pozzettiet al. 2007) at z∼0.8.At the
+sametime,thehigh-massandlow-massslopesoftheirpower-
+law relation are not re-fit for the higher redshift, leading t o
+similar deviations from our results as at z∼0.1. The results
+in Zhenget al. (2007) lie slightly below our results at z∼1.
+This is partially due to their use of a WMAP1 cosmology.
+However,it isdifficultto say exactlyhowmuch,as theirstel -
+lar masses at z∼1 are a hybrid of Ks-derived masses from
+Bundyetal.(2006)andcolor–basedmassesderivedinaman-
+ner analogousto Bell et al. (2003). Nonetheless, they remai n
+wellwithinoursystematicerrorbars.
+We also include a comparison to the z= 1 SM–HM rela-
+tion presented in Conroy& Wechsler (2009), who also use
+an abundance matching technique to assign galaxies to ha-
+los. Unique to the work of Conroy& Wechsler (2009) is
+their attempt to jointly fit both the redshift–dependent ste l-
+lar mass functions and the redshift–dependentstar formati on
+rate – stellar mass relations. In their model, halo growth is
+tracked through time using results derived from halo merger
+trees, allowing galaxies to be identified across epochs. The
+evolution of halos in conjunction with standard abundance
+matchingprovidesmodelpredictionsforstarformationrat es.
+The SM–HM relation from Conroy&Wechsler (2009) lies
+slightly above our best–fit relation, and it is within statis tical
+errorbars exceptat the veryhighest halo masses. Thisis due
+tothedifferentassumptionsmadeabouttheGSMFinthetwo
+worksandalso to the absenceofcorrectionsfor thescatter i n
+stellarmassat fixedhalomassinConroy&Wechsler (2009).20 BEHROOZI,CONROY & WECHSLER
+Finally, the strong lensing survey of Gavazzietal. (2007)
+has been extended by (Lagattutaetal. 2009) out to z∼0.9
+using lenses observed in the CASTLES program, as well as
+in COSMOS and in the EGS. While the same caveats about
+selection effects apply as for lower redshifts, Lagattutae tal.
+(2009) find that the evolution in the stellar mass fraction fo r
+Mh∼1013.5M⊙halosiswithin0-0.3dexgreaterat z∼0.9as
+compared to z∼0, consistent with our limits of 0-0.15 dex
+forthe allowedevolutionoverthatredshiftinterval.
+5.RESULTS BEYOND z=1
+5.1.MethodologyandDataLimitations
+As discussed in §2.1.2, published results for the galaxy
+stellar mass function beyond z= 1 suffer from the important
+caveat that integrated SFRs are inconsistent with galaxy st el-
+lar mass functions when both sets of observations are taken
+at face value with a constant IMF. Nonetheless, one may use
+similar methodology as in §3 to derive stellar mass – halo
+mass relations at higher redshifts under the assumption tha t
+the observed stellar mass functions are correct. Here we as-
+sume the GSMFs of Marchesiniet al. (2009), which cover a
+redshiftrangeof1 .3<z<4.
+AsthereisnoguaranteethattheevolutionoftheSM–HM
+relation at high redshifts will have the same form as its evo-
+lutionatlowredshifts,were-examinetheassumptionsaffe ct-
+ing our evolution parameterization in Equation 23. As with
+all current high-redshift data, the results in Marchesinie tal.
+(2009) are limited to luminous (massive) galaxies, so littl e
+information about the value of β(the faint-end slope of the
+galaxy-halo mass relation) is available. Hence, we continu e
+to assume a linear functional form for its evolution; as the
+valueofβevolveslinearlywith scalefactor( a)inourfit, this
+means that it is largely constrained to be consistent with th e
+evolutionat lowerredshifts(1 <z<2). Naturally,if theevo-
+lution ofβat high redshifts is significantly different than for
+1<z<2,thenourerrorbarsfor βmayunderestimatethefull
+uncertaintiesintheparameter.
+Additionally, the systematics affecting high-mass galaxi es
+at high redshifts are much more severe than for z<1. Not
+onlyaretheerrorsinstellarmasscalculationssignificant (due
+to larger photometry errors, limited templates, etc.), but any
+miscalibration in correcting photometric redshift errors will
+result in low-redshift galaxies masquerading as very brigh t
+high-redshift galaxies. These combined uncertainties res ult
+in poor constraints on high-mass galaxies. For that reason,
+we do not attempt to assume a more complicated functional
+formforthe evolutionof δandγ, whichmeansasbeforethat
+theirratesofevolutionarelargelyconstrainedtobeconsi stent
+withlowerredshifts.
+However, we find that individual fits at each redshift do
+suggestapossibleevolutionforthecharacteristicstella rmass
+which is nonlinear in the scale factor. Hence, we expand the
+form of the evolution of M∗,0to include a quadratic depen-
+denceonscale factor:
+M∗,0(a)=M∗,0,0+M∗,0,a(a−1)+M∗,0,a2(a−0.5)2,(25)
+where (a−0.5)2is used instead of ( a−1)2to minimize the
+degeneracy between M∗,0,aandM∗,0,a2. We parameterize the
+evolutionof all other parametersas in Equation23. All othe r
+methodologyremainsthe same as for lower redshifts, as out-
+linedin§3.6.
+5.2.Results11 12 13 14 15
+log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
+This work
+This work, Drory et al. 2009 GSMF
+This work, ( σ(z) = 0, ξ=0)
+Moster et al. 2009 (AM)
+Wang & Jing 2009 (AM+CC)
+Zheng et al. 2007 (HOD)
+Conroy & Wechsler 2009 (AM)
+Figure12. Comparison of our best-fit model at z= 1.0 for different model
+assumptionsandtopreviously published results. Darkgrey shading indicates
+statistical and sample variance errors; light grey shading includes system-
+atic errors. The error bars for the red line, calculated usin g the Drory etal.
+(2009)GSMFincludestatistical errorsonly—i.e.,theydon otincludesample
+variance. Theresults ofMoster etal.(2009)(green line) do notinclude mod-
+eling of scatter or statistical errors in stellar masses, so for comparison, we
+present ourresults excluding the effects of σ(z) andξ(blue line). Theresults
+of Conroy &Wechsler (2009) made slightly different assumpt ions about the
+stellar massfunction evolution.
+To maintain some overlapwith the z<1 results, we evalu-
+atethelikelihoodfunctionforeachSM–HMrelationagainst
+GSMFsfor0 .8<z<4. Thisdatarangeincludestworedshift
+bins from Pérez-Gonzálezet al. (2008) (0.8-1.0, 1.0-1.3)a nd
+three redshift bins from Marchesiniet al. (2009) (1.3-2, 2- 3,
+3-4). IncludingmoreredshiftbinsfromPérez-Gonzálezet a l.
+(2008) would improve the continuity of the fits to the low-
+redshift results; however, doing so would require a more
+complicated redshift parameterization than what we have as -
+sumed. The evolution of the best-fit stellar mass fraction fo r
+0<z<4isshowninFigure13. AlldatapointsforFigure13
+arelistedin AppendixD.
+Asmaybeexpected,uncertaintiesathighredshiftsaresub-
+stantially larger than at lower redshifts. The contributio n of
+systematic errors in stellar masses to the error budget (0.2 5
+dex) is still important, but it is no longer the only dominant
+factor. Statistical errorsdue to the comparativelysmall n um-
+ber of galaxy observationsat highredshifts can contribute an
+equaluncertainty(up to 0.25dex)to the derivedSM–HM re-
+lation. The statistical errors are large not only for massiv e
+galaxies with low number counts, but also for halos below
+1012M⊙, where magnitude limits on surveys make observa-
+tionsofthecorrespondinggalaxiesdifficult.
+The contribution from other sources of uncertainties (e.g. ,
+sample variance, cosmology uncertainties) is substantial ly
+smaller than the current statistical errors. The effectsof sam-
+ple variance on uncertainties at high redshift is less than f or
+low redshifts because the volume probed in the high redshift
+sample is five times larger than for the low redshift sample
+(≈5×106Mpc3vs. 106Mpc3, respectively). While cos-
+mology uncertainties are somewhat larger at high redshifts ,
+theircontributiontotheoveralllevelsofuncertaintyare again
+much smaller than the statistical errors, as was true even by
+z=1forthe low-redshiftsample.
+The statistical uncertainties at high redshifts mean that i t
+is difficult to draw strong conclusions about the evolutionUncertaintiesAffectingtheStellar Mass–HaloMassRelati on 21
+11 12 13 14 15
+log10(Mh) [MO•]-4-3.5-3-2.5-2-1.5log10(M* / Mh)
+z = 0.1
+z = 1.0
+z = 2.0
+z = 3.0
+z = 4.0
+Figure13. Evolution of the derived stellar mass fractions for central galax-
+ies, from z=4 to the present. The best–fit relations are shown only over t he
+mass range where constraining data are available. At higher redshifts, cer-
+tainty about the shape of the curves drops precipitously owi ng to a lack of
+constraining data beyond the knee of the stellar mass functi on. Combined
+systematic and statistical error bars areshown for three re dshift bins only.
+of the SM–HM relation. However, the indication is that the
+mass corresponding to the peak efficiency for star formation
+evolves slowly, and is roughly a factor of five larger at z=4.
+At all redshifts, the integrated star formation peaks at ∼10-
+20 per cent of the universal baryon fraction; the current dat a
+indicatesthat this value may start high at veryhigh redshif ts,
+shrinkashalosgrowfasterthantheyformstars,andthensta rt
+growing again after z= 2. However, with current uncertain-
+ties these results are tenuous. The single most effective wa y
+to reduce current uncertainties on both the SM–HM relation
+at individualredshiftsandonitsevolutionis to conductmo re
+high-redshiftgalaxysurveys,bothtoprobefaintergalaxi esto
+determinthe shapeofthe GSMF andto get betterstatistics at
+thehighmassend.
+6.DISCUSSION AND IMPLICATIONS
+Atz∼0, the majority of published results are in accord
+within our full systematic error bars, regardless of the tec h-
+nique used. All reported results appear to be consistent wit h
+the principlesnecessary for abundancematching over a wide
+rangeofhalomasses(1011−1015M⊙)—thateachdarkmatter
+halo and subhaloabovethe masses we have consideredhosts
+a galaxy with a reasonably tight relationship between their
+masses, and that average stellar mass — halo mass relation
+increasesmonotonicallywith halomass.
+Because of the available statistics of halo and galaxy stel-
+lar mass functions,especially at z=0, the techniqueof abun-
+dancematchingoffersthetightestconstraintsontheSM–HM
+relationcurrentlyavailable,anditisinagreementwithre sults
+from a broad variety of additional techniques. Under the as-
+sumptionthatsystematicerrorsinstellarmasscalculatio nsdo
+not change substantially with redshift, abundance matchin g
+offers tight constraints on the evolution of the SM–HM rela-
+tion from z=1 to the present. These in turn will serve as im-
+portant new tests for star formation prescriptionsand reci pes
+in both hydrodynamical simulations and semi-analytic mod-
+els, as they will applyon the level of individualhalosinste ad
+ofonthesimulatedvolumeasa whole.
+At the same time, abundance matching offers these con-straints with a minimal number of parameters. The Halo
+OccupationDistribution (HOD) technique requiresmodelin g
+P(N|Mh), the probabilitydistributionof the numberof galax-
+iesperhaloasafunctionofhalomass,inseveraldifferentl u-
+minosity bins. In the model proposed in Zhenget al. (2007),
+this results in 45 fitted parametersjust to models the occupa -
+tionatz=0(fiveparametersfornineluminositybins). Condi-
+tional Luminosity Function (CLF) modeling requires param-
+eterizing a form for φ(L,Mh), the number density of galaxies
+as a functionof luminosityand host halo mass, which results
+in approximately a dozen parameters to model occupation at
+z=0(Cooray2006). Becauseoftheadditionalconstraintsim-
+posedbyassumingthateachhalohostsagalaxy,ourapproach
+uses fewer parameters. Abundancematching, as discussed in
+this paper, results in a model with only six independent pa-
+rameters(fiveto empiricallyfit the derivedSM–HMrelation,
+andonetomodelthescatterinobservedstellarmassesatfixe d
+halomass) todescribethe populationofgalaxiesinhalos.
+Theabundancematchingapproachto the SM–HM relation
+requires so few parameters in comparison to other methods
+becauseofthefairlysmallscatter( ≈0.16dex)betweenstellar
+mass and halo mass at high masses (the scatter has a negligi-
+bleimpactonthe averageSM–HMrelationat lowermasses),
+and the requirement that satellite galaxies live in satelli te ha-
+los(subhalos). Itmaywellbe thatamorecomplicatedmodel
+must be adopted for satellites to quantitatively match the
+small-scale clustering observations (e.g. Wang etal. 2006 ).
+However, such changes will affect the clustering much more
+than the derived SM–HM relation, as suggested by the min-
+imal changes in Figures 7 and 10 for mass scales ( /lessorsimilar1012.5
+M⊙) wheresatellites are a non-negligiblefractionofthe tota l
+halopopulation.
+ThelargestuncertaintiesintheSM—HMrelationat z<1
+comefromassumptionsinconvertinggalaxyluminositiesin to
+stellar masses, which amount to uncertaintieson the order o f
+0.25 dex in the normalization of the relation. However, the
+systematicbiasesintroducedbythecombinedsourcesofsca t-
+terbetweencalculatedstellarmassesandhalomassescanri se
+toequivalentsignificanceforhalosabove1014.5M⊙. Because
+the GSMF is monotonicallydecreasing, results which do not
+account for all sources of scatter in stellar mass will over-
+predictthe average stellar mass in halos by 0.17-0.25dex fo r
+thesemassivehalos.
+Using abundance matching to find confidence intervals for
+the SM–HM relation is an even more involved process, as
+each of the ways in which the systematics might vary must
+also be taken into account. While future work on constrain-
+ing stellar masses will be the most valuable in terms of re-
+ducing uncertainties for the lowest redshift data, wider an d
+deeper surveys and some resolution to the discrepancy be-
+tween high-redshift cosmic star formationdensity and stel lar
+massfunctionsmust occurin orderto improveconstraintson
+therelationat highredshifts.
+Asmentionedintheintroduction,abundancematchingmay
+be used equallywell to assign galaxyluminositiesand color s
+to halos. In this case, the galaxy luminosity — halo mass
+relation may be derived using identical methodology to that
+presentedin§3,withtheexceptionthatthesystematics µand
+κmaybeneglected,leavingonly σ(z)(effectively,theredshift
+scalingof photometryerrors)and ξ(effectively,the scatter in
+luminosityatfixedhalomass). Asthesesystematicsaremuch
+betterconstrainedthantheirstellarmasscounterparts(s imply
+as luminosities may be measured directly),this approachca n22 BEHROOZI,CONROY & WECHSLER
+yield very powerful constraints on the normalization as wel l
+as the evolution of the luminosity–mass relation. The highe r
+accuracy possible compared to the stellar mass–mass rela-
+tion will generally notremove uncertainties in comparing to
+galaxyformationmodels. Galaxyformationcodeswhichcal-
+culate luminosities must include modeling for all the effec ts
+in §2.1.1, meaning that constraintson the underlyingphysi cs
+are subject to the same uncertainties. However, tighter con -
+straints on the luminosity–mass relation will be nonethele ss
+helpful for applicationswhich are concernedwith cosmolog -
+icalconstraintsfromlargeluminosity-selectedsurveys.
+7.CONCLUSIONS
+We have performed an extensive exploration of the uncer-
+taintiesrelevantto determiningthe relationshipbetween dark
+matter halos and galaxy stellar masses from the halo abun-
+dancematchingtechnique. Errorsrelatedtotheobservedst el-
+lar mass function, the theoretical halo mass function, and t he
+underlying technique of abundance matching are all consid-
+ered. We focus on the mean stellar mass to halo mass ratio
+forcentralgalaxiesasafunctionofhalomass,andpresentr e-
+sultsforthisrelationshipatthepresentepochandextendi ngto
+z∼4. Weaccountseparatelyforstatistical errorsandforsys-
+tematic errors resulting from uncertainties in stellar mas s es-
+timation, and also investigatethe relative contributiono f var-
+ious sources of error including cosmological uncertainty a nd
+the scatter between stellar mass and halo mass. An analytic
+modelhasbeendevelopedwhichcanbeusedtoconstrainthis
+connectionintheabsenceofhighresolutionsimulations.
+Ourprimaryconclusionsareasfollows:
+1. The peak integrated star formation efficiency occurs at
+a halo mass near 1012M⊙, with a relatively low frac-
+tion — 20% at z= 0 — of baryons currently locked
+up in stars. This peak value declines to z∼2 but re-
+mains between10–20%for all redshiftsbetween z=0–
+4. Thisimpliesthat30 −40%ofbaryonswereconverted
+intostarsoverthelifetimeofagalaxywithcurrenthalo
+massof1012M⊙.
+2. At low masses, the stellar mass – halo mass relation at
+z=0scalesas M∗∼M2.3
+h. Athighmasses,around1014
+M⊙, stellar mass scales as M∗∼M0.29
+h. However, the
+highmassscalingmaynotbeapowerlaw,asourmodel
+indicates that this slope decreases with increasing halo
+mass.
+3. Within statistical uncertainties,the stellar mass cont ent
+of halos has increased by 0 .3−0.45dex for halos with
+mass less than 1012M⊙sincez∼1. Systematic biases
+in stellar mass calculations between different redshifts
+could broaden the uncertaintiesin this number, but the
+conclusion that significant evolution has occurred for
+low-mass halos would remain robust. For halos with
+mass greater than 1013M⊙, our best-fit results indicate
+more growth in halo masses than stellar masses since
+z∼1,butareconsistentwithnoevolutioninthestellar
+massfractionsoverthistime.
+4. Systematic, uniform offsets in the galaxy stellar mass
+functionand its evolutionare the dominantuncertainty
+in the stellar mass – halo mass relation at low redshift.
+Statistical errors in the estimation of individual stellar
+masses impact the high mass end of the GSMF, and athigher redshifts may result in an observed GSMF that
+deviatesfroma Schechterfunction.
+5. Current uncertainties in the underlying cosmological
+model are sub-dominant to the systematic errors, but
+are larger than other sources of statistical error for ha-
+losbelow Mh∼1012M⊙forlowredshifts( z<0.2).
+6. Given current constraints from other methods, uncer-
+tainty in the value of scatter between stellar mass and
+halo mass is important in the mean relation for masses
+aboveMh∼1012.5M⊙, although it is subdominant to
+systematicerrorsforallmassesbelow Mh∼1014.5M⊙.
+7. Other uncertainties in the galaxy–halo assignment, in-
+cluding different assumptions about the treatment of
+satellite galaxies, are subdominant when considering
+themeanrelationforcentralgalaxies.
+8. At higher redshifts (1 <z<4), systematic uncertain-
+tiesremainimportant,butstatisticaluncertaintiesreac h
+equal significance. The shape of the relation is fairly
+unconstrained at z>2, where improved statistics and
+constraintsonthe GSMFbelow M∗areneeded.
+Wehavepresentedabest–fitgalaxystellarmass–halomass
+relation including an estimate of the total statistical and sys-
+tematic errors using available data from z= 0−4, although
+caution should be used at redshifts higher than z∼1. We
+also presentan algorithmto generalizethis relationforan ar-
+bitrary cosmological model or halo mass function. The fact
+that assignment errors are sub-dominant and scatter can be
+well–constrained by other means gives increased confidence
+inusingthesimpleabundancematchingapproachtoconstrai n
+this relation. These results provide a powerful constraint on
+modelsofgalaxyformationandevolution.
+PSB andRHW receivedsupportfromthe U.S. Department
+of Energy under contract number DE-AC02-76SF00515 and
+froma TermanFellowship at StanfordUniversity. CC is sup-
+ported by the Porter Ogden Jacobus Fellowship at Princeton
+University. We thank Michael Blanton, Niv Drory, Raphael
+Gavazzi, Qi Guo, Sarah Hansen, Anatoly Klypin, Cheng
+Li, Yen-TingLin, Pablo Pérez-González, Danilo Marchesini ,
+Benjamin Moster, Lan Wang, Xiaohu Yang, Zheng Zheng,
+as well as their co-authors for the use of electronic ver-
+sions of their data. We appreciate many helpful discus-
+sionsandcommentsfromIvanBaldry,MichaelBusha,Simon
+Driver,NivDrory,AnatolyKlypin,AriMaller,DaniloMarch -
+esini, Phil Marshall, Pablo Pérez-González, Paolo Salucci ,
+Jeremy Tinker, Frank van den Bosch, the Santa Cruz Galaxy
+Workshop, and the anonymous referee for this paper. The
+ART simulation (L80G) used here was run by Anatoly
+Klypin, and we thank him for allowing us access to these
+data. We are grateful to Michael Busha for providing the
+halo catalogs we used to estimate sample variance errors.
+These simulations were produced by the LasDamas project (
+http://lss.phy.vanderbilt.edu/lasdamas/ );
+we thank the LasDamas collaboration for providing us with
+thisdata.
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+APPENDIX
+CONVERTING RESULTS TO OTHER HALO MASS FUNCTIONS
+FromEquation14,itispossibletosimplyconvertfromourha lomassfunction φhtoanyhalomassfunctionofchoice( φh,r). In
+particular,the function Mh(M∗) is defined by the fact that the numberdensity of halos with ma ss aboveMh(M∗) must match the
+numberdensityofgalaxieswithstellarmassabove M∗(withtheappropriatedeconvolutionstepsapplied). Recal lfromEquation
+14that
+/integraldisplay∞
+Mh(M∗)φh(M)dlog10M=/integraldisplay∞
+M∗φdirect(M∗)dlog10M∗. (A1)
+Naturally,thecorrectmass-stellarmassrelationforthea lternatehalomassfunction φh,r(whichwewilllabelas Mh,r(M∗))must
+satisfythissameequation,withtheresult that:
+/integraldisplay∞
+Mh(M∗)φh(M)dlog10M=/integraldisplay∞
+Mh,r(M∗)φh,r(M)dlog10M. (A2)
+Tomakethecalculationevenmoreexplicit,let Φh(M)=/integraltext∞
+Mφhdlog10Mbeourcumulativehalomassfunction,andlet Φh,r(M)
+bethecorrespondingcumulativehalomassfunctionfor φh,r. Then,we find:
+Mh,r(M∗)=Φ−1
+h,r(Φh(Mh(M∗))). (A3)
+Massfunctionsfromdifferentcosmologiesthanthose assum edin thispaperwill alsorequireconvertingstellar masses if their
+choicesof hdifferfromtheWMAP5 best-fitvalue.
+EFFECTS OF SCATTER ON THE STELLAR MASS FUNCTION
+Thissectionisintendedtoprovidebasicintuitionforthee ffectsofboth ξandσ(z),whichmaybemodeledasconvolutions. The
+classic examplein this case is convolutionof the GSMF with a log-normaldistributionof some width ω. While the convolution
+(evenofa Schechterfunction)with a Gaussian hasno knownan alyticalsolution, we may approximatethe result byconside ring
+the case where the logarithmic slope of the GSMF changes very little over the width of the Gaussian. Then, locally, the ste llar
+mass function is proportional to a power function, say φ(M∗)∝(M∗)α. Then, if we let x= log10M∗(so thatφ(10x)∝10αx),
+findingtheconvolutionisequivalenttocalculatingthefol lowingintegral:
+φconv(10x)∝/integraldisplay∞
+−∞10αb
+√
+2πω2exp/parenleftbigg
+−(x−b)2
+2ω2/parenrightbigg
+db
+= 10αx101
+2α2ω2ln(10). (B1)
+That is to say, the stellar mass function is shifted upward by approximately1 .15(αω)2dex. Hence, for parts of the stellar mass
+functionwith shallow slopes, the shift is completely insig nificant, as it is proportionalto α2. However,it matters much more in
+thesteeperpartofthestellarmassfunction,tothepointth atforgalaxiesofmass1012M⊙,theobservedstellarmassfunctioncan
+beseveralordersofmagnitudeabovethe intrinsicGSMF.
+A SAMPLE CALCULATION OF THE FUNCTIONAL FORM OF THE STELLAR MA SS FUNCTION
+Galaxy formation models typically assume at least two feedb ack mechanisms to limit star formation for low-mass galaxie s
+and for high-mass galaxies. Thus, one of the simplest fiducia l star formation rate (SFR) as a function of halo mass ( Mh) would
+assumea doublepower-lawform:
+SFR(Mh)∝/parenleftbiggMh
+M0/parenrightbigga
++/parenleftbiggMh
+M0/parenrightbiggb
+. (C1)
+We mightexpectthe total stellar massas a functionofhaloma ssto take a similar form,exceptperhapswith a wider regiono f
+transitionbetween galaxieswhose historiesare predomina ntlylow mass, and those with historieswhich are predominan tlyhigh
+mass, for the reason that some galaxies’ accretion historie s may have caused them to be affected comparably by both feedb ack
+mechanisms.
+Hence, assuming that the relation between halo mass and stel lar mass follows a double power-law form, we adopt a simple
+functionalformto convertfromthestellar massofagalaxyt othehalomass:
+Mh(M∗)=M1/bracketleftBigg/parenleftbiggM∗
+M∗,0/parenrightbiggβ/γ
++/parenleftbiggM∗
+M∗,0/parenrightbiggδ/γ/bracketrightBiggγ
+. (C2)
+Here,βmaybethoughtofasthefaint-endslope, δasthemassive-endslope(although βandδarefunctionallyinterchangeable),
+andγasthetransitionwidth(larger γmeansa slowertransitionbetweenthe massiveandfaint-end slopes).
+We first calculatedlog(Mh)
+dlog(M∗):UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 25
+log(Mh)=log(M1)+γlog/bracketleftBigg/parenleftbiggM∗
+M∗,0/parenrightbiggβ/γ
++/parenleftbiggM∗
+M∗,0/parenrightbiggδ/γ/bracketrightBigg
+, (C3)
+dlog(Mh)
+dlog(M∗)=dlog(Mh)
+dM∗dM∗
+dlog(M∗)(C4)
+=M∗ln(10)dlog(Mh)
+dM∗(C5)
+=β/parenleftBig
+M∗
+M∗,0/parenrightBigβ/γ
++δ/parenleftBig
+M∗
+M∗,0/parenrightBigδ/γ
+/parenleftBig
+M∗
+M∗,0/parenrightBigβ/γ
++/parenleftBig
+M∗
+M∗,0/parenrightBigδ/γ(C6)
+=β+(δ−β)/parenleftBigg
+1+/parenleftbiggM∗
+M∗,0/parenrightbiggβ−δ
+γ/parenrightBigg−1
+. (C7)
+This justifies the earlier intuition that the functional for m forMh(M∗) transitions between slopes of βandδwith a width that
+increases with γ. Note thatdlog(Mh)
+dlog(M∗)is always of order one, as the stellar mass is always assumed t o increase with the halo mass
+andvice versa(namely, β >0andδ >0).
+Next,we approachdN
+dlogMh. Fromanalyticalresults, weexpectaSchechterfunctionfo rthehalomassfunction,namely:
+dN
+dlog(Mh)=φ0ln(10)/parenleftbiggMh
+M0/parenrightbigg1−α
+exp/parenleftbigg
+−Mh
+M0/parenrightbigg
+. (C8)
+Substitutinginthe equationfor Mh(M∗),we have
+dN
+dlog(Mh)=φ0ln(10)/bracketleftBigg/parenleftbiggM∗
+M∗,0/parenrightbiggβ/γ
++/parenleftbiggM∗
+M∗,0/parenrightbiggδ/γ/bracketrightBiggγ(1−α)
+×/parenleftbiggMh
+M0/parenrightbigg1−α
+exp/parenleftBigg
+−M1
+M0/bracketleftBigg/parenleftbiggM∗
+M∗,0/parenrightbiggβ/γ
++/parenleftbiggM∗
+M∗,0/parenrightbiggδ/γ/bracketrightBiggγ/parenrightBigg
+. (C9)
+Already evident is the generic result that there will be sepa rate faint-end and massive-end slopes in the stellar mass fu nction,
+and that the falloff is not generically specified by an expone ntial. We may make one simplification in this model—namely, t o
+note that Mh(M∗,0) correspondsto the halo mass at which the slope of Mh(M∗) begins to transition from βtoδ. We expect this
+transition to correspondto the transition between superno vafeedbackand AGN feedbackin semi-analyticmodels—namel y,for
+a halo mass which is too large to be affectedmuch by supernova feedback,but which is yet too small to host a large AGN. This
+implies that Mh(M∗,0) is expected to be around 1012M⊙or less, meaning that Mh/M0is small until stellar masses well beyond
+M∗,0, meaning that we may neglect the faint-end slope of the Mh(M∗) relation in the exponential portion of the stellar mass
+function:
+dN
+dlog(Mh)=φ0ln(10)/parenleftbiggM1
+M0/parenrightbigg1−α/bracketleftBigg/parenleftbiggM∗
+M∗,0/parenrightbiggβ/γ
++/parenleftbiggM∗
+M∗,0/parenrightbiggδ/γ/bracketrightBiggγ(1−α)
+×exp/parenleftBigg
+−M1
+M0/parenleftbiggM∗
+M∗,0/parenrightbiggδ/parenrightBigg
+. (C10)
+Hence,we maycombinethese twoequationstoobtaintheexpre ssionforthestellar massfunction:
+dN
+dlog(M∗)=φ0ln(10)/parenleftbiggM1
+M0/parenrightbigg1−α/bracketleftBigg/parenleftbiggM∗
+M∗,0/parenrightbiggβ/γ
++/parenleftbiggM∗
+M∗,0/parenrightbiggδ/γ/bracketrightBiggγ(1−α)
+×exp/parenleftBigg
+−M1
+M0/parenleftbiggM∗
+M∗,0/parenrightbiggδ/parenrightBigg
+×
+β+(δ−β)/parenleftBigg
+1+/parenleftbiggM∗
+M∗,0/parenrightbiggβ−δ
+γ/parenrightBigg−1
+. (C11)26 BEHROOZI,CONROY & WECHSLER
+Whilethisseemscomplicated,it maybeintuitivelydeconst ructedas:
+dN
+dlog(M∗)=[constant]/bracketleftbig
+doublepowerlaw/bracketrightbig
+×/bracketleftbig
+exponentialdropoff/bracketrightbig
+O(1). (C12)
+As mentioned previously, this functional form is equivalen t toφdirect. To convert to the true stellar mass function φtrueor the
+observed stellar mass function φmeas, it must be convolved with the scatter in stellar mass at fixed halo mass and (for φmeas)
+the scatter in calculated stellar mass at fixed true stellar m ass. As such, it should be clear that—while the final form may b e
+Schechter– like—there is certainly much more flexibility in the final shape of the GSMF than a Schechter function alone would
+allow,asevidencedbythefiveparametersrequiredtofullys pecifyequationC11.
+DATA TABLES
+WereproduceherelistingsofthedatapointsinFigures5,6, and13inTables3,4,and5,respectively. Seesections4.2an d5.2
+fordetailsonthedatapointsineachtable.
+Table3
+Stellar Mass Fractions For0 <z<1 Including Full Systematics
+z=0.1 z=0.5 z=1.0
+log10(Mh) log10(M∗/Mh) log10(M∗/Mh) log10(M∗/Mh)
+11.00 −2.30+0.26
+−0.23
+11.25 −1.96+0.25
+−0.23−2.11+0.22
+−0.26
+11.50 −1.67+0.24
+−0.24−1.84+0.22
+−0.26−2.01+0.25
+−0.24
+11.75 −1.53+0.23
+−0.24−1.70+0.24
+−0.24−1.85+0.26
+−0.23
+12.00 −1.54+0.24
+−0.24−1.68+0.26
+−0.23−1.77+0.26
+−0.23
+12.25 −1.62+0.24
+−0.24−1.72+0.26
+−0.23−1.77+0.25
+−0.23
+12.50 −1.74+0.24
+−0.24−1.81+0.25
+−0.23−1.81+0.24
+−0.24
+12.75 −1.87+0.23
+−0.25−1.92+0.25
+−0.23−1.89+0.23
+−0.26
+13.00 −2.02+0.23
+−0.25−2.05+0.24
+−0.24−1.99+0.22
+−0.27
+13.25 −2.18+0.22
+−0.26−2.19+0.24
+−0.25−2.11+0.21
+−0.28
+13.50 −2.35+0.22
+−0.26−2.34+0.23
+−0.25−2.25+0.21
+−0.29
+13.75 −2.52+0.22
+−0.27−2.51+0.23
+−0.26−2.39+0.21
+−0.30
+14.00 −2.70+0.21
+−0.28−2.68+0.23
+−0.26−2.55+0.22
+−0.30
+14.25 −2.88+0.21
+−0.28−2.86+0.23
+−0.26
+14.50 −3.07+0.20
+−0.29−3.04+0.23
+−0.27
+14.75 −3.26+0.20
+−0.30
+15.00 −3.45+0.20
+−0.30
+Note. — Halo massesare in units of M⊙. Constraints are quoted over themass range probed by the obs erved GSMF.UncertaintiesAffectingtheStellar Mass–HaloMassRelati on 27
+Table4
+Stellar Mass Fractions without Systematic Errors ( µ=κ=0)
+z=0.1 z=0.5 z=1.0
+log10(Mh) log10(M∗/Mh) log10(M∗/Mh) log10(M∗/Mh)
+11.00 −2.30+0.03
+−0.02
+11.25 −1.96+0.04
+−0.01−2.10+0.04
+−0.08
+11.50 −1.67+0.03
+−0.01−1.83+0.05
+−0.06−2.02+0.10
+−0.06
+11.75 −1.53+0.01
+−0.01−1.71+0.06
+−0.03−1.85+0.10
+−0.04
+12.00 −1.54+0.01
+−0.02−1.68+0.06
+−0.02−1.78+0.09
+−0.03
+12.25 −1.62+0.00
+−0.02−1.72+0.06
+−0.01−1.77+0.08
+−0.03
+12.50 −1.74+0.01
+−0.02−1.81+0.05
+−0.02−1.81+0.06
+−0.04
+12.75 −1.87+0.01
+−0.03−1.92+0.05
+−0.02−1.88+0.04
+−0.06
+13.00 −2.02+0.01
+−0.03−2.05+0.04
+−0.03−1.98+0.03
+−0.08
+13.25 −2.18+0.02
+−0.04−2.19+0.04
+−0.04−2.10+0.04
+−0.10
+13.50 −2.35+0.02
+−0.05−2.34+0.04
+−0.05−2.24+0.04
+−0.13
+13.75 −2.52+0.03
+−0.06−2.51+0.05
+−0.06−2.38+0.05
+−0.15
+14.00 −2.70+0.03
+−0.07−2.68+0.05
+−0.07−2.54+0.06
+−0.17
+14.25 −2.88+0.04
+−0.08−2.85+0.06
+−0.08
+14.50 −3.07+0.04
+−0.09−3.04+0.06
+−0.10
+14.75 −3.25+0.05
+−0.10
+15.00 −3.45+0.05
+−0.11
+Note. — Halo massesare in units of M⊙.
+Table5
+Stellar Mass Fractions For 0 .8<z<4 Including Full Systematics
+z=1.0 z=2.0 z=3.0 z=4.0
+log10(Mh) log10(M∗/Mh) log10(M∗/Mh) log10(M∗/Mh) log10(M∗/Mh)
+11.50 −2.01+0.25
+−0.24
+11.75 −1.85+0.26
+−0.23
+12.00 −1.77+0.26
+−0.23−1.89+0.22
+−0.27−1.89+0.25
+−0.27
+12.25 −1.77+0.25
+−0.23−1.76+0.24
+−0.25−1.67+0.21
+−0.29−1.58+0.22
+−0.32
+12.50 −1.81+0.24
+−0.24−1.78+0.23
+−0.26−1.63+0.21
+−0.30−1.50+0.20
+−0.35
+12.75 −1.89+0.23
+−0.26−1.87+0.20
+−0.30−1.71+0.19
+−0.35−1.56+0.19
+−0.40
+13.00 −1.99+0.22
+−0.27−2.00+0.17
+−0.35−1.83+0.17
+−0.39−1.68+0.19
+−0.44
+13.25 −2.11+0.21
+−0.28−2.14+0.15
+−0.39−1.97+0.16
+−0.44−1.82+0.19
+−0.49
+13.50 −2.25+0.21
+−0.29−2.30+0.15
+−0.42−2.13+0.17
+−0.47−1.98+0.19
+−0.52
+13.75 −2.39+0.21
+−0.30−2.47+0.17
+−0.45−2.30+0.19
+−0.49
+14.00 −2.55+0.22
+−0.30−2.64+0.20
+−0.47
+Note. — Halo massesare in units of M⊙.
\ No newline at end of file
diff --git a/1001.0016.txt b/1001.0016.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7f1654d3d9f0714d43fb7c75fafbafd0ccc3df6c
--- /dev/null
+++ b/1001.0016.txt
@@ -0,0 +1,2042 @@
+arXiv:1001.0016v4  [hep-th]  1 Feb 2011ExactResultsandHolographyof WilsonLoops
+in
+N=2Superconformal(Quiver)GaugeTheories
+Soo-JongReya,b, Takao Suyamaa
+aSchool ofPhysicsand Astronomy&Center forTheoreticalPhy sics
+Seoul NationalUniversity,Seoul 141-747 KOREA
+bSchool ofNaturalSciences, InstituteforAdvancedStudy,P rinceton NJ 08540 USA
+sjrey@snu.ac.kr suyama@phya.snu.ac.kr
+ABSTRACT
+Using localization, matrix model and saddle-point techniq ues, we determine exact behavior of
+circularWilsonloopin N=2superconformal(quiver)gaugetheoriesinthelargenumbe rlimit
+of colors. Focusing at planar and large ‘t Hooft couling limi ts, we compare its asymptotic be-
+havior with well-known exponential growth of Wilson loop in N=4 super Yang-Mills theory
+with respect to ‘t Hooft coupling. For theory with gauge grou p SU(N)coupled to 2 Nfunda-
+mental hypermultiplets,we find that Wilson loop exhibits non-exponential growth – at most, it
+can grow as a power of ‘t Hooft coupling. For theory with gauge group SU( N)×SU(N)and
+bifundamental hypermultiplets, there are two Wilson loops associated with two gauge groups.
+We find Wilson loop in untwisted sector grows exponentially l arge as in N=4 super Yang-
+Mills theory. We then find Wilson loop in twisted sector exhib itsnon-analytic behavior with
+respecttodifferenceofthetwo‘tHooftcouplingconstants . Bylettingonegaugecouplingcon-
+stanthierarchically larger/smallerthan theother, wesho wthatWilsonloops inthesecond type
+theory interpolate to Wilson loops in the first type theory. W e infer implications of these find-
+ings from holographic dual description in terms of minimal s urface of dual string worldsheet.
+We suggest intuitive interpretation that in both classes of theory holographic dual background
+must involve string scale geometry even at planar and large ‘ t Hooft coupling limit and that
+new resultsfound in thegaugetheorysideare attributablet o worldsheet instantonsand infinite
+resummation therein. Our interpretation also indicates th at holographic dual of these gauge
+theoriesis providedby certain non-critical stringtheories.1 Introduction
+AdS/CFTcorrespondence[1]between N=4superYang-MillstheoryandTypeIIBstringthe-
+ory onAdS5×S5has been studied extensively during the last decade. One rem arkable result
+obtained from thestudy is exact computationforexpectatio n valueofWilson loopoperators at
+strongcoupling[2][3]. Forahalf-BPS circularWilsonloop ,based on perturbativecalculations
+at weak ‘t Hooft coupling [4], exact form of the expectation v alue was conjectured in [5], pre-
+ciselyreproducingtheresultexpectedfromthestringtheo rycomputation[2],[3]andconformal
+anomalytherein. Theirconjecturewas confirmed laterin[6] usingalocalizationtechnique.
+Inthispaper,westudyaspectsofhalf-BPScircularWilsonl oopsin N=2supersymmetric
+gaugetheories. Wefocusonaclassof N=2superconformalgaugetheories—the A1(quiver)
+gaugetheory of gaugegroup SU (N)and 2Nfundamentalhypermultipletsand ˆA1quivergauge
+theory of gauge group SU( N)×SU(N)and bifundamental hypermultiplets— and compute the
+Wilson loop expectation value by adapting the localization technique of [6]. We then compare
+the results with the N=4 super Yang-Mills theory, which is a special limit of the ˆA0quiver
+gauge theory of gauge group SU( N) and an adjoint hypermultiplet. Their quiver diagrams are
+depictedin Fig. 1.
+(a)                                                     (b)                                                           (c)
+Figure 1: Quiver diagram of N=2superconformal gauge theories under study: (a) ˆA0theory with G
+= SU(N)and one adjoint hypermultiplet, (b) A1theory with G=SU(N) and2Nfundamental hypermul-
+tiplets, (c) ˆA1theory with G=SU(N)×SU(N) and2Nbifundamental hypermultiplets. The A1theory
+is obtainable from ˆA1theory by tuning ratio of coupling constants to 0 or ∞. See sections 3 and 4 for
+explanations.
+We show that, on general grounds, path integral of these N=2 superconformal gauge
+theories on S4is reducible to a finite-dimensional matrix integral. The re sulting matrix model
+turns out very complicated mainly because the one-loop dete rminant around the localization
+fixed point is non-trivial. This is in shartp contrast to the N=4 super Yang-Mills theory,
+where the one-loop determinant is absent and further evalua tionof Wilson loops or correlation
+1functionsisstraightforwardmanipulationinGaussianmat rixintegral.
+Nevertheless, in the N→∞planar limit, we show that expectation value of the half-BPS
+circular Wilson loop is determinable provided the ’t Hooft coupling λis large. In the large λ
+limit, the one-loop determinant evaluated by the zeta-func tion regularization admits a suitable
+asymptotic expansion. Using this expansion, we can solve th e saddle-point equation of the
+matrixmodelandobtainlarge λbehavioroftheWilsonloopexpectationvalue. In N=4super
+Yang-Mills theory, it is known that the Wilson loop grows exp onentially large ∼exp(√
+2λ)as
+λbecomesinfinitelystrong.
+InˆA0gauge theory, we find that the Wilson loop expectation value g rows exponentially,
+exactly the same as the N=4 super Yang-Mills theory. The result for A1gauge theory is
+surprising. We find that the Wilson loop is finite at large λ. This means that the Wilson loop
+exhibitsnon-exponential growth. The ˆA1quiver gauge theory is also interesting. There are
+two Wilsonloops associated witheach gaugegroups, equival ently,onein untwistedsector and
+anotherin twistedsector. Wefind that theWilsonloopin untw istedsector scales exponentially
+large, coincidingwith the behavior of the Wilson loop N=4 super Yang-Millstheory and the
+ˆA0gauge theory. On the other hand, the Wilson loop in twisted se ctor exhibits non-analytic
+behavior with respect to difference of two ‘t Hooft coupling constants. We also find that we
+can interpolate the two surprising results in A1andˆA1gauge theories by tuning the two ‘t
+Hooft couplings in ˆA1theory hierarchically different. In all these, we ignored p ossible non-
+perturbative corrections to the Wilson loops. This is becau se, recalling the fishnet picture for
+the stringy interpretation of Wilson loops, the perturbati ve contributions would be the most
+relevantpart forexploringtheAdS/CFT correspondenceand theholographytherein.
+We also studied how holographic dual descriptions may expla in the exact results. Expec-
+tation value of the Wilson loop is described by worldsheet pa th integral of Type IIB string in
+dual geometry and that, in case the dual geometry is macrosco pically large such as AdS 5×S5,
+itisevaluatedbysaddle-pointsofthepathintegral–world sheetconfigurationsofextremalarea
+surface. We first suggest that non-exponential growth of the A1Wilson loop arise from deli-
+catecancelationamongmultiple—possiblyinfinitelymany— saddle-points. Thisimpliesthat
+holographicdualgeometryofthe N=2A1gaugetheoryoughttobe(AdS 5×M2)×Mwhere
+the internal space M= [S1×S2]necessarily involves a geometry of string scale. The string
+worldsheet sweeps on average an extremal area surface insid e AdS5, but many nearby saddle-
+point configurations whose worldsheet sweep two cycles over Mcancel among the leading,
+exponential contributions of each. We next suggest that ˆA1Wilson loop in untwisted sector is
+givenbyamacroscopicstringinAdS 5×S5/Z2andhencegrowsexponentiallywithaverageof
+thetwo‘tHooftcouplingconstants. Intwistedsector,howe ver,itisnegligiblysmallandscales
+withdifferenceofthetwo‘tHooftcouplingconstants. This isagainduetodelicatecancelation
+2among multiple worldsheet instantons that sweep around col lapsed two cycles at the Z2orb-
+ifold fixed point. We also demonstrate that Wilson loop expec tation values are interpolatable
+between ˆA1andA1behaviors(orviceversa)bytuningNS-NS2-formpotentialo nthecollapsed
+twocyclefrom 1 /2to0,1 orviceversa.
+This paperis organized as follows. In section 2, we showthat evaluationof theexpectation
+value of the half-BPS circular Wilson loop in a generic N=2 superconformal gauge theory
+reduces to a related problem in a one-matrix model. The reduc tion procedure is based on lo-
+calization technique and is parallel to [6]. Compared to [6] , our derivations are more direct
+and elementary and hence makes foregoing analysis in thepla nar limitfar clearer physicswise.
+In section 3, we evaluate the Wilson loop at large ‘t Hooft cou pling limit. Based on general
+analysis for one-matrix model (subsection 3.1), we evaluat e the matrix model action which is
+induced by the one-loop determinant (subsection 3.2). As a r esult, we obtain a saddle-point
+equationwhosesolutionprovidesthelarge‘tHooftcouplin gbehavioroftheWilsonloop(sub-
+section 3.3). In section 5, we discuss interpretation of the se results in holographic dual string
+theory. For both A1andˆA1types, we argue contribution of worldsheet instanton effec ts can
+explain non-analytic behavior of the exact gauge theory res ults. Section 7 is devoted to dis-
+cussion, including a possible implication of the present re sults to our previous work [7] (see
+also [8][9]) on ABJM theory [10]. We relegated several techn ical points in the appendices. In
+appendixA,wesummarizeKillingspinorson S4. InappendixB, weworkoutoff-shellclosure
+ofsupersymmetryalgebra. InappendixC,wepresentasympto ticexpansionoftheWilsonloop.
+In appendix D, we present detailed computation of c1that arise in the evaluation of one-loop
+determinant.
+Results of this work were previously reported at KEK worksho p and at Strings 2009 con-
+ference. Foronlineproceedings,see[11]and [12], respect ively.
+2 ReductiontoOne-MatrixModel
+The work [6] provided a proof for the conjecture [4, 5] that th e evaluation of the half-BPS
+Wilson loop in N=4 super Yang-Mills theory [2, 3] is reduced to a related probl em in a
+Gaussian Hermitian one-matrix model. In this section, we sh ow that the similarreduction also
+works for N=2 superconformal gauge theories of general quiver type. The resulting matrix
+model is, however, not Gaussian but includes non-trivial ve rtices due to nontrivial one-loop
+determinant.
+32.1 From N=4toN=2
+A shortcut route to an N=2 gauge theory of general quiver type — with matters in variou s
+different representations and coupling constants in diffe rent values — is to start with N=4
+super Yang-Mills theory. In this section, for completeness of our treatment, we elaborate on
+this route. Let Gbe the gauge group. The latter theory consists of a gauge field Amwith
+m=1,2,3,4, scalar fields A0,A5,···,A9and anSO(9,1)Majorana-Weyl spinor Ψ, all in the
+adjointrepresentationof G. Theaction can bewrittencompactlyas
+SN=4=/integraldisplay
+R4d4xTr/parenleftBig
+−1
+4FMNFMN−i
+2ΨΓMDMΨ/parenrightBig
+, (2.1)
+whereM,N=0,···,9and
+FMN=∂MAN−∂NAM−ig[AM,AN], (2.2)
+DMΨ=∂MΨ−ig[AM,Ψ], (2.3)
+ΓΨ= +Ψ. (2.4)
+Note that the metric of the base manifold R4is taken in the Euclidean signature, while the
+ten-dimensional’metric’ ηMNis taken Lorentzian with η00=−1. As usual in thedimensional
+reduction,thederivativesotherthan ∂mare setto zero.
+Theaction (2.1)is invariantunderthesupersymmetrytrans formations
+δAM=−iξΓMΨ, (2.5)
+δΨ=1
+2FMNΓMNξ, (2.6)
+whereξis a constant SO(9,1)Majorana-Weyl spinor-valued supersymmetry parameter sat is-
+fying the chirality condition Γξ=+ξ. In what follows, we rewrite the action (2.1) so that the
+resulting action provides a useful guide to deduce the actio n of an N=2 gauge theory with
+hypermultipletfields ofarbitrary representations.
+We first choose which half of the supercharges of the N=4 supersymmetry is to be pre-
+served. This choice corresponds to the choice of embedding t he SU(2) R-symmetry of N=2
+theory intotheSU(4)R-symmetryofthe N=4theory. Consideronesuchembeddingdefined
+by thematrix
+M:=
+x6+ix7−(x8−ix9)
+x8+ix9x6−ix7
+. (2.7)
+Its determinantis
+detM=(x6)2+(x7)2+(x8)2+(x9)2, (2.8)
+4soit isobviousthatany transformationoftheform
+M→gLMgR,gL∈SU(2)L,gR∈SU(2)R (2.9)
+belongs to the SO(4) transformation acting on (x6,···,x9)∈R4. Note that this transformation
+preserves the embedding (2.7). In the ten-dimensional lang uage, SU(4) R-symmetry of the
+N=4theoryisrealizedastherotationalsymmetrySO(6)of R6. Therefore,oneembeddingof
+SU(2) R-symmetry into SU(4) is chosen by selecting SU (2)Lor SU(2)R. We choose the latter
+as theR-symmetryofthe N=2 theories.
+There is a U(1) subgroup of SU (2)Lgenerated by σ3. LetR(θ)be an element of this U(1).
+This isθ-rotation in 67-plane and (−θ)-rotation in 89-plane. In the following, we require
+that the supercharges preserved in N=2 theory should be invariant under the R(θ). For an
+infinitesimal θ,R(θ)acts onthesupersymmetrytransformationparameter ξas
+δθξ=−1
+2θ(Γ6Γ7−Γ8Γ9)ξ. (2.10)
+Therefore, ξshouldsatisfy
+Γ6789ξ=−ξ, (2.11)
+selectingeightcomponentsoutoftheoriginalsixteenones .
+The scalar fields Aswiths=6,7,8,9 can be combined into the doublet qα(α=1,2) of
+SU(2)Ras
+q1:=1√
+2(A6−iA7),q2:=−1√
+2(A8+iA9), (2.12)
+and their conjugates qα=(qα)†. Gamma matrices γα,γαare defined similarly in terms of Γs.
+Theysatisfy
+{γα,γβ}=2δα
+β,{γα,γβ}=0={γα,γβ}. (2.13)
+Notethat,forarbitrary vectors VsandWs, onehas
+VsWs=VαWα+VαWα. (2.14)
+The Majorana-Weyl spinor Ψis split into the chirality eigenstates with respect to Γ6789as
+follows:
+λ:=1
+2(1−Γ6789)Ψ,η:=1
+2(1+Γ6789)Ψ. (2.15)
+Both fermionsareMajorana-Weyl. We furthersplit ηintoη±, which areeigenstatesof
+γ:=1
+2[γα,γα]=i
+2(Γ6Γ7−Γ8Γ9). (2.16)
+Notethat γisthegeneratorfor R(θ)and hencesatisfies
+γ2=1
+2(1+Γ6789),[γ,γα]=+γα,[γ,γα]=−γα. (2.17)
+5Now,η±arenotMajorana-Weyl. Infact, theyarerelated by chargeconjugation
+(ηA
+±)∗=CηA
+∓, (2.18)
+whereAistheindexfortheadjointrepresentationof GandCisthecomplexconjugationmatrix.
+So, weshalldenote η−byψ. Then,moduloa phasefactor, η+isψ†.
+In termsof Aµ(µ=0,···,5),qα,qα,λandψ, theaction (2.1)can bewritten as
+SN=4=/integraldisplay
+R4d4xTr/parenleftBig
+−1
+4FµνFµν−DµqαDµqα−i
+2λΓµDµλ−iψΓµDµψ
+−gλγα[qα,ψ]−gψγα[qα,λ]−g2[qα,qβ][qβ,qα]+1
+2g2[qα,qα][qβ,qβ]/parenrightBig
+,(2.19)
+with the understanding that the dimensional reduction sets ∂µ=0 forµ=0,5. The supersym-
+metrytransformations(2.5),(2.6)can bewrittenas
+δAµ=−iξΓµλ, (2.20)
+δqα=−iξγαψ, (2.21)
+δqα=−iψγαξ (2.22)
+δλ= +1
+2FµνΓµνξ−ig[qα,qβ]γα
+βξ, (2.23)
+δψ= +DµqαΓµγαξ. (2.24)
+Again,if ξobeystheprojectioncondition(2.11), theaction (2.19)ha sN=2 supersymmetry.
+At this stage, we shall be explicit of representation conten ts of(qα,ψ)fields and their con-
+jugates. Let (TA)B
+C=−ifAB
+Cbe the generators of Lie (G)in the adjoint representation. We also
+impose on ξthe projection condition (2.11). In terms of them, the actio n (2.19) can be written
+as
+SN=2=/integraldisplay
+R4d4x/parenleftBig
+−1
+4tr(FµνFµν)−i
+2tr(λΓµDµλ)−DµqαDµqα−iψΓµDµψ
++gλAγαqαTAψ+gψγαTAqαλA−g2(qαTAqβ)2+1
+2g2(qαTAqα)2/parenrightBig
+,(2.25)
+wherethegaugecovariantderivativesare
+Dµqα=∂µqα−iAA
+µTAqα, (2.26)
+Dµqα=∂µqα+iqαTAAA
+µ, (2.27)
+Dµψ=∂µψ−iAA
+µTAψ. (2.28)
+6TheN=2 supersymmetrytransformationrules are
+δAµ=−iξΓµλ, (2.29)
+δλA= +1
+2FA
+µνΓµνξ+iqαTAqβγα
+βξ, (2.30)
+δqα=−iξγαψ, (2.31)
+δqα=−iψγαξ (2.32)
+δψ= +DµqαΓµγαξ. (2.33)
+Theaboveaction(2.25)isequivalenttotheoriginalaction (2.1): wehavejustrewrittentheorig-
+inal action in terms of renamed component fields. The supersy mmetry transformations (2.29)-
+(2.33)are also equivalentto (2.5) -(2.6) in so far as ξis projected to N=2 supersymmetryas
+(2.11).
+It turns out that the action (2.25) is invariant under N=2 supersymmetry transformations
+(2.29)-(2.33) even for TAin a generic representation Rof the gauge group G, which can also
+bereducible. Therefore, (2.25) defines an N=2 gaugetheory withmatterfields (qα,ψ)in the
+representation Rand theirconjugates.
+It is also possible to treat ˆAk−1quiver gauge theories on the same footing. We embed the
+orbifold action Zkinto SU(2)L. In thispaper, we shall focus on ˆA1quivergaugetheory. In this
+case, weshouldsubstitute
+Aµ=
+Aµ(1)
+Aµ(2)
+,λ=
+λ(1)
+λ(2)
+,
+qα=
+q(1)α
+q(2)α
+,ψ=
+ψ(1)
+ψ(2)
+. (2.34)
+into(2.19). Notethatthe N=2supersymmetry(2.29)-(2.33)ispreservedevenwhenthega uge
+couplingconstant gis replaced withthematrix-valuedone:
+g=
+g1I
+g2I
+. (2.35)
+Ingeneral, g1/ne}ationslash=g2andcanbeextendedtocomplexdomain. Extensionto ˆAk(k≥2)isstraight-
+forward.
+72.2 Superconformal symmetryon S4
+Following [6], we now define the N=2 superconformal gauge theory on S4of radius r. For
+definiteness, we consider the round-sphere with the metric hmninduced through the standard
+stereographicprojection. Details aresummarizedinAppen dixA.
+For this purpose, it also turns out convenient to start with N=4 super Yang-Mills theory
+defined on S4. To maintain conformal invariance, the scalars ought to hav e the conformal
+couplingtothecurvaturescalarof S4. Theactionthusreads
+SN=4=/integraldisplay
+S4d4x√
+hTr/parenleftBig
+−1
+4FMNFMN−1
+r2ASAS−i
+2ΨΓMDMΨ/parenrightBig
+, (2.36)
+whereS=0,5,6,···,9. Theactionisinvariantunderthe N=4supersymmetrytransformations
+δAM=−iξΓMΨ, (2.37)
+δΨ= +1
+2FMNΓMNξ−2ΓSAS/tildewideξ, (2.38)
+providedthat ξand/tildewideξsatisfytheconformal Killingequations:
+∇mξ=Γm/tildewideξ,∇m/tildewideξ=−1
+4r2Γmξ. (2.39)
+Explicitform ofthesolutiontotheseequationsare givenin AppendixA.
+The action of an N=2 gauge theory on S4with a hypermultiplet of representation Rcan
+bededuced easilyas intheprevioussubsection. Oneobtains
+SN=2=/integraldisplay
+S4d4x√
+h/parenleftBig
+−1
+4Tr(FµνFµν)−i
+2Tr(λΓµDµλ)−1
+r2Tr(AaAa)
+−DµqαDµqα−iψΓµDµψ−2
+r2qαqα
++gλAγαqαTAψ+gψγαTAqαλA−g2(qαTAqβ)2+1
+2g2(qαTAqα)2/parenrightBig
+,(2.40)
+wherea=0,5. Theactionisinvariantunderthe N=2 superconformalsymmetry
+δAµ=−iξΓµλ,
+δλA= +1
+2FA
+µνΓµνξ+igqαTAqβγα
+βξ−2ΓaAA
+a/tildewideξ,
+δqα=−iξγαψ,
+δqα=−iψγαξ
+δψ= +DµqαΓµγαξ−2γαqα/tildewideξ,
+8whereξsatisfies the conformal Killing equations (2.39) in additio n to the projection condition
+(2.11). We emphasize that this is the transformation of the N=2 superconformal symmetry,
+not just the Poincar´ e part of it. This can be checked explici tly, for example, by examining the
+commutatoroftwo transformationsonthefields.
+We find it convenient to define a fermionic transformation Qcorresponding to the above
+superconformal transformation δ. It is obtained easily by the replacement δ→θQandξ→θξ
+withθareal Grassmannparameter. Theresultingtransformationi s
+QAµ=−iξΓµλ,
+QλA= +1
+2FA
+µνΓµνξ+igqαTAqβγα
+βξ−2ΓaAA
+a/tildewideξ,
+Qqα=−iξγαψ,
+Qqα=−iψγαξ,
+Qψ= +DµqαΓµγαξ−2γαqα/tildewideξ, (2.41)
+where now ξand/tildewideξarebosonicSO(9,1) Majorana-Weyl spinors satisfying N=2 projection
+(2.11)andconformal Killingequation(2.39).
+2.3 Localization
+By extending the localization technique of [6], we now show t hat computation of Wilson loop
+expectation value in N=2 superconformal gauge theory of quiver type can be reduced t o
+computationofaone-matrixintegral.
+LetQbe a fermionic transformation. Suppose that an action Sunder consideration is in-
+variantunder Q. Then, thefollowingmodification
+S(t):=S+t/integraldisplay
+d4x√
+hQV(x) (2.42)
+does notchangethepartitionfunctionprovidedthat
+/integraldisplay
+d4x√
+hQ2V(x)=0. (2.43)
+Likewise,correlationfunctionsremainunchangedifopera torsunderconsiderationare Q-invariant.
+We shall choose V(x)such that the bosonic part of QV(x)is positive semi-definite. For this
+choice, since tcan be chosen to be an arbitrary value, we can take the limit t→+∞so that
+9the path-integral is localized to configurations where the b osonic part of QV(x)vanishes. It
+willturn out laterthat thevanishinglocusof QV(x)is parametrized by a constantmatrix. This
+is why the evaluation of the expectation value of a Q-invariant operator reduces to a matrix
+integral. Theaction oftheresultingmatrix modelis thesum ofSevaluatedat thevanishinglo-
+cus and the one-loop determinant obtained from the quadrati c terms of QV(x)when expanded
+around thevanishinglocus.
+One might think that the fermionic transformation Qdefined in the previous section can be
+used asQabove. In fact, Q2is asumofbosonictransformations,and therefore, (2.43)a ppears
+toholdaslongas V(x)isinvariantunderthetransformations. Theproblemofthis choiceisthat
+Q2is such a sum only on-shell. According to [13],[14] and [15], Qhas to be modified so that
+theresulting Qclosestoasumofbosonictransformationsfor off-shell.
+To this end, we introduce auxiliary fields K˙m(˙m=ˆ2,ˆ3,ˆ4),KαandKα. They transform in
+the adjoint, RandRrepresentations of the gauge group G, respectively. Utilizing them, we
+modifytheaction (2.40)in atrivialmanner:
+SN=2=/integraldisplay
+S4d4x/parenleftBig
+−1
+4Tr(FµνFµν)−i
+2Tr(λΓµDµλ)−1
+r2Tr(AaAa)
+−DµqαDµqα−iψΓµDµψ−2
+r2qαqα
++gλAγαqαTAψ+gψγαTAqαλA−g2(qαTAqβ)2+1
+2g2(qαTAqα)2
++1
+2K˙mK˙m+KαKα/parenrightBig
+. (2.44)
+Evidently,this action is physicallyequivalentto the orig inalone. Themodified action (2.44) is
+nowinvariantunderthefollowing Qtransformations:
+QAµ=−iξΓµλ,
+QλA= +1
+2FA
+µνΓµνξ+igqαTAqβγα
+βξ−2ΓaAA
+a/tildewideξ+K˙mAν˙m,
+Qqα=−iξγαψ,
+Qqα=−iψγαξ,
+Qψ= +DµqαΓµγαξ−2γαqα/tildewideξ+Kανα,
+Qψ= +DµqαξγαΓµ+2/tildewideξγαqα+Kανα,
+QK˙mA=−ν˙m/parenleftBig
+−iΓµDµλA+gγαqαTAψ−gγαψ∗TAqα/parenrightBig
+,
+QKα=−να/parenleftBig
+−iΓµDµψ+γβTAqβgλA/parenrightBig
+,
+QKα=−/parenleftBig
+−iDµψΓµ−gλAγβqβTA/parenrightBig
+να. (2.45)
+10To makeQ2close to a sum of bosonic transformations off-shell, the spi norsν˙m,να,ναshould
+be chosen appropriately out of ξ,/tildewideξ. Details on them are summarized in Appendix B. With the
+correct choice, Q2closes,forexample,on λas follows:
+−iQ2λ=/parenleftbigg
+vm∇mλ−1
+2(ξΓmn/tildewideξ)Γmnλ−ig[vµAµ,λ]/parenrightbigg
++1
+2(ξΓst/tildewideξ)Γstλ.(2.46)
+Thisshowsthat Q2isasumofadiffeomorphismon S4,aGgaugetransformationand aglobal
+SU(2)Rtransformation. In particular, notice that ξΓst/tildewideξturns out to be independent of xm. The
+actionof Q2on theauxiliaryfields isslightlydifferent. Forexample,o nK˙m, oneobtains
+−iQ2K˙m=vk∇kK˙m−ig[vµAµ,K˙m]+ν˙mΓk∇kν˙nK˙n. (2.47)
+Here, the index ˙ mdoes not transform as a part of the four-vector on S4. This is not a problem
+sinceK˙mis contracted with ν˙minVdefined below, and not with some other four-vectors. The
+Qdefined aboveis therighttransformationavailableforthel ocalizationprocedure.
+We areat thepositiontochoose V. Wetake
+V:=Tr(Vλλ)+Vψψ+ψVψ, (2.48)
+where
+Vλ=1
+2FµνξΓ0Γµν+igqαTAqβtAξΓ0γα
+β+2/tildewideξΓ0ΓaAa+K˙mν˙mΓ0,(2.49)
+Vψ=DµqαξΓ0Γµγα+2/tildewideξΓ0γαqα+KαναΓ0, (2.50)
+Vψ=DµqαγαΓµΓ0ξ−2γαqαΓ0/tildewideξ+KαΓ0να. (2.51)
+Notethat Visascalarwithrespecttoaparticularcombinationofthedi ffeomorphismon S4,the
+GgaugetransformationandtheglobalSU (2)Rtransformation. Thisfollowsfromtheidentities
+forthespinors,forexample,
+vm∇mξ−1
+2(ξΓmn/tildewideξ)Γmnξ+1
+2(ξΓst/tildewideξ)Γstξ=0, (2.52)
+and similarones for/tildewideξandνIwhichare summarizedin AppendixA and B. Therefore, (2.43)i s
+satisfiedwith thischoice, as required.
+Afterstraightforward buttediousalgebra, oneobtainsthe bosonicpart of QVexpressedas
+Tr(VλQλ)+VψQψ+QψVψ/vextendsingle/vextendsingle/vextendsingle
+bosonic
+=Tr/bracketleftBig
+cos2θ
+2(F+
+mn+w+
+mnA5)2+sin2θ
+2(F−
+mn+w−
+mnA5)2−(K˙m−2A0ν˙m/tildewideξ)2
++DmAaDmAa−1
+2g2[Aa,Ab]2+g2tAtB(2qαTAqβqβTBqα−qαTAqαqβTBqβ)/bracketrightBig
++2D0qαD0qα+2|D˙µqα+ξΓ0˙µγα
+β/tildewideξqβ|2+3
+2r2qαqα−2KαKα, (2.53)
+11whereθisthepolarangleon S4, ˙µ=1,2,···,5and
+w+
+mn:=1
+cos2θ
+2ξΓ05Γmn1−Γˆ1ˆ2ˆ3ˆ4
+2/tildewideξ, (2.54)
+w−
+mn:=1
+sin2θ
+2ξΓ05Γmn1+Γˆ1ˆ2ˆ3ˆ4
+2/tildewideξ. (2.55)
+Here, thehatted indicesaretheLorentzones. Theaboveexpr essionshowsthat,aftera suitable
+Wick rotation for A0and the auxiliary fields, the bosonic part of QVis positive semi-definite.
+Therefore, by taking the limit t→+∞, the path-integral is localized at the vanishing locus of
+QV. Itturns outthat,as in[6], non-zero fields at thevanishing locusare
+A0=−i
+grΦ,Kˆ2=−i
+gr2Φ, (2.56)
+whereΦis aconstantHermitianmatrix. Thecoefficients are chosenforlaterconv enience.
+Now, the path-integralis reduced to an integraloverthe Her mitian matrix Φ. The action of
+the corresponding matrix model is a sum of the action (2.44) e valuated at the vanishing locus
+and the one-loop determinant for the quadratic terms in QV. Note that higher-loop contribu-
+tions vanish in the large tlimit since t−1plays the role of the loop-counting parameter. At the
+vanishinglocus,theaction(2.44)takesthevalue
+S=−/integraldisplay
+S4d4x√
+hTr/parenleftBig1
+r2(A0)2+1
+2(Kˆ2)2/parenrightBig
+=4π2
+g2TrΦ2. (2.57)
+An importantdifference from the N=4 superYang-Millstheory isthat theone-loop determi-
+nant around the vanishinglocus does not cancel and has a comp licated functional structure. In
+the next section, we show that the presence of the non-trivia l one-loop determinant is crucial
+fordeterminingthelarge‘t Hooftcouplingbehavioroftheh alf-BPS Wilsonloop.
+Thehalf-BPS Wilsonloopof N=2 gaugetheory hasthefollowingform:
+W[C]:=TrPsexp/bracketleftBig
+ig/integraldisplay2π
+0ds/parenleftBig
+˙xmAm(x)+θaAa(x)/parenrightBig/bracketrightBig
+. (2.58)
+The functions xm(s),θa(s)are chosen appropriately to preserve a half of the N=2 supercon-
+formal symmetry. We shall choose Cto be the great circle at the equator of S4(i.e.θ=π
+2)
+specified by
+(x1,x2,x3,x4)=(2rcoss,2rsins,0,0), (2.59)
+andθaas
+θ0=r,θ5=0. (2.60)
+12Forthischoice,onecan showthat
+˙xmAm(x)+θaAa(x)=−rvµAµ(x), (2.61)
+wherevµ=ξΓµξ. See Appendix A for theexplicit expressionsof vµ. This implies that W[C]is
+invariantunder Qdueto theidentity
+ξΓµξξΓµλ=0. (2.62)
+Thus, we have shown that /an}bracketle{tW[C]/an}bracketri}htis calculable by a finite-dimensional matrix integral. The
+operatorwhoseexpectationvaluein thematrixmodelis equa lto/an}bracketle{tW[C]/an}bracketri}htis
+Trexp/parenleftBig
+2πΦ/parenrightBig
+. (2.63)
+Noticethatitissolelygovernedbytheconstant-valued,He rmitianmatrix Φ. Thisenablesusto
+compute the Wilson loops in terms of a matrix integral. This o bservation will also play a role
+inidentifyingholographicdual geometrylater.
+3 Wilson loopsatLarge‘t HooftCoupling
+We have shown that evaluation of the Wilson loop /an}bracketle{tW[C]/an}bracketri}htis reduced to a related problem in
+a one-Hermitian matrix model. Still, the matrix model is too complicated to solve exactly.
+In the following, we focus our attention to either the N=2 superconformal gauge theory
+ofA1type with G=U(N)coupled to 2 Nfundamental hypermultiplets and of ˆA1type with
+G=U(N)×U(N), both at large Nlimit. For these theories, we show that the large ‘t Hooft
+couplingbehaviorisdeterminablebyafewquantitiesextra ctedfromtheone-loopdeterminant.
+This allows us to exactly evaluate the Wilson loop /an}bracketle{tW[C]/an}bracketri}htin the large Nand large ’t Hooft
+couplinglimit.
+3.1 General resultsin one matrixmodel
+Consider a matrix model for an N×NHermitian matrix X. In the large Nlimit, expectation
+valueofanyoperatorinthismodelisdeterminableintermso feigenvaluedensityfunction ρ(x)
+ofthematrix X. By definition, ρ(x)isnormalizedby
+/integraldisplay
+dxρ(x)=1. (3.1)
+13LetDdenotethesupportof ρ(x). Weassumethat1
+min{D}=:b<0<a:=max{D}. (3.2)
+Expectationvalueoftheoperator1
+NTr(ecX) (c>0)isgivenintermsof ρ(x)as
+W:=/angbracketleftbigg1
+NTr(ecX)/angbracketrightbigg
+=/integraldisplay
+dxρ(x)ecx. (3.3)
+By theassumptiononthesupport D,thevalueof Wis bounded:
+ecb≤W≤eca. (3.4)
+b                                               a                        x βα(a - x)
+Figure2: Typical distribution of the eigenvalue density ρ.
+Weareinterestedinthebehaviorof Winthelimit a→+∞. Introducingtherescaleddensity
+function/tildewideρ(x)=aρ(ax),Wis writtenas
+W=eca/integraldisplay1−b
+a
+0du/tildewideρ(1−u)e−cauwhere x=a(1−u). (3.5)
+At therightedgeofthesupport D,weexpect thatthedensitycutsoffwithapower-lawtail:
+/tildewideρ(1−u)=βuα+χ(u)where |χ(u)|≤Kuα+ε,u∈(0,δ) (3.6)
+for a positive K,ε,δ. See figure 2. Here, α>0 signifies the leading powerof the fall-off at the
+rightedge: χrefers tothesub-leadingremainder. Then,fora largeposit ivea, (3.6)leads to the
+followingasymptoticbehavior:
+W∼βΓ(α+1)(ca)−α−1eca, (3.7)
+1IfXis traceless, the assumption is always valid since/integraltextdxρ(x)x=0 must hold. In the large Nlimit, the
+contributionfromthetracepartisnegligible.
+14Detailsofthederivationof(3.7)are relegatedtoAppendix C.
+Wehavefoundthatthelarge abehaviorof Wisdeterminedbythefunctionalformof ρ(x)in
+thevicinityoftherightedgeofits support. In particular, we foundthat theleadingexponential
+part isdeterminedsolelyby thelocationoftherightedgeof theeigenvaluedistribution.
+For comparison, let us recall the exact form of the Wilson loo p inN=4 super Yang-Mills
+theory [4], which is a special case of the ˆA0gauge theory. In this case, the eigenvalue density
+functionisgivenby
+ρ(x)=4π
+λ/radicalbigg
+λ
+2π2−x2, (3.8)
+whichis thesolutionofthesaddle-pointequation
+4π2
+λφ=/integraldisplay
+−dφ′ρ(φ′)
+φ−φ′. (3.9)
+TheWilsonloopisevaluatedas follows:
+/an}bracketle{tW[C]/an}bracketri}ht=4π
+λ/integraldisplay+√
+λ/π
+−√
+λ/πdxe2πx/radicalbigg
+λ
+2π2−x2
+=2√
+2λI1(√
+2λ)
+∼/radicalbigg
+2
+π(2λ)−3
+4e√
+2λ. (3.10)
+Weseethat thisasymptoticbehavioris reproduced exactlyb y(3.7)with α=1
+2of(3.8)2.
+3.2 One-loop determinant and zetafunction regularization
+Let us return to the evaluation of /an}bracketle{tW[C]/an}bracketri}ht. To determine the eigenvalue density function ρof
+the Hermitian matrix Φ, it is necessary to know the explicit functional form of the o ne-loop
+determinant. However,thisisaformidabletask forageneri cN=2gaugetheory. Fortunately,
+as shown in the previous subsection, the leading behavior of /an}bracketle{tW[C]/an}bracketri}htis governed by a small
+numberofdataif a=max(D)islarge.
+So, we shall assume that the limit λ→+∞induces indefinite growth of a. This is a rea-
+sonable assumption since otherwise /an}bracketle{tW[C]/an}bracketri}htdoes not grow exponentially in the limit λ→+∞,
+implying that any N=2 gauge theory with such a behavior of the Wilson loop cannot h ave
+an AdS dual in the usual sense. In other words, we assume that t he rescaled density function
+2Here,thedefinitionofthegaugecouplingconstant gisdifferentbythe factor2fromthatin[4]
+15λγρ(λγx)has a reasonable large λlimit for a positiveγ. Under this assumption, we now show
+that the large λbehavior of the Wilson loop is determined by the behavior of t he one-loop de-
+terminant in the region where the eigenvalues of Φare large. The asymptoticbehavior in such
+a limit is most transparently derivable from the heat-kerne l expansion for a certain differential
+operatorinthezeta-functionregularizationoftheone-lo opdeterminant.
+•A1gaugetheory :
+Consider first the A1gauge theory. There are contributions to the one-loop effec tive action
+both from the hypermultiplet and the vector multiplet. We fir st focus on the hypermultiplet
+contribution. If QVis expanded around the vanishing locus (2.56), quadratic te rms of the
+hypermultipletscalars become:
+−qα(Δ)α
+βqβ+1
+r2ΦAΦBqαTATBqα, (3.11)
+where
+(Δ)α
+β= (∇mδα
+γ+Vmαγ)(∇mδγ
+β+Vmγ
+β)−1
+4r2(3+cos2θ)δα
+β, (3.12)
+Vmα
+β=ξΓ0mγα
+β/tildewideξ. (3.13)
+IfΦis diagonalizedas Φ=diag(φ1,···,φN), thenthesecond termin (3.11)can bewrittenas
+2N
+r2N
+∑
+i=1(φi)2qiαqα
+i. (3.14)
+Nowthequadratictermsaredecomposedintothesumoftermsf orcomponents qα
+i. So,theone-
+loop determinant of the hypermultiplet scalars is the produ ct of determinants for each compo-
+nents. Let FB
+h(Φ)denoteapartofthematrixmodelactioninducedbytheone-lo opdeterminant
+forthehypermultipletscalars qα. Itscontributionto theeffectiveaction can bewrittenas
+FB
+h(Φ)=2NN
+∑
+i=1FB
+h(φi), (3.15)
+whereFB
+h(m)is formallygivenas
+FB
+h(m):=logDet/parenleftBig
+−Δ+m2
+r2/parenrightBig
+. (3.16)
+Noticethat the eigenvalues φienteras masses of qα
+i. Therefore, what we need to analyze is the
+largembehaviorof FB
+h(m).
+We now evaluate the function FB
+h(m)in the limit m→∞. In terms of Feynman diagram-
+matics, this amounts to expanding the one-loop determinant in the background of scalar field
+16(m/r)2. LetD(m)=Det(−Δ+m2/r2). The relation (3.16) is afflicted by ultraviolet infinities,
+so it should be regularized appropriately. The determinant is formally defined over the space
+spanned by the normalizable eigenfunctions of −Δ. Letλk(k=0,1,2,···)be eigenvalues of
+−Δ:
+−Δψk=λkψk. (3.17)
+Then,D(m)can beformallywrittenas
+D(m)=∞
+∏
+k=0/parenleftBig
+λk+m2
+r2/parenrightBig
+. (3.18)
+To makethisexpressionwell-defined, letus definearegulari zed function
+ζ(s,m):=r−2s∞
+∑
+k=01
+(λk+m2/r2)s, (3.19)
+wheresisacomplexvariable. Thissummationmaybewell-definedfor swithsufficientlylarge
+Re(s). Onecan formallydifferentiate ζ(s,m)withrespect to stoobtain
+∂sζ(s,m)/vextendsingle/vextendsingle/vextendsingle
+s=0=−∞
+∑
+k=0log(r2λk+m2)=−log[r2D(m)]. (3.20)
+Since the left-hand side makes sense via a suitable analytic continuation of (3.19), it can be
+regarded that the right-hand side is defined by the left-hand side. Therefore, we define the
+functionFB
+h(m)viathezeta-function regularization:
+FB
+h(m):=−∂sζ(s,m)/vextendsingle/vextendsingle/vextendsingle
+s=0. (3.21)
+The large mbehavior of FB
+h(m)is determined as follows. For a suitable range of s,ζ(s,m)
+can bewrittenas
+ζ(s,m)=r−2s
+Γ(s)/integraldisplay∞
+0dtts−1e−m2t/r2K(t), (3.22)
+where
+K(t):=∞
+∑
+k=0e−λkt=Tr(etΔ) (3.23)
+is the heat-kernel of Δ. The convergence of this sum is assumed. The asymptoticexpa nsion of
+K(t)is knownas theheat-kernel expansion. Forareviewon thissu bject, seee.g. [16]. Since Δ
+isadifferential operatoron S4, theheat-kernel expansionhastheform
+K(t)∼∞
+∑
+i=0ti−2a2i(Δ) (3.24)
+In theexpansion, a2i(Δ)are knownas theheat-kernel coefficients for Δ.
+17Theexpression(3.22)of ζ(s,m)isonlyvalidforarangeof s,butζ(s,m)canbeanalytically
+continued to theentire complex plane provided that the asym ptoticexpansion (3.24) is known.
+In particular, there exists a formulafor the asymptoticexp ansion of ζ(s,m)in the large mlimit
+[17]
+ζ(s,m)∼∞
+∑
+i=0a2i(Δ)r2i−4Γ(s+i−2)
+Γ(s)m−2s−2i+4, (3.25)
+valid in the entire complex s-plane. Note that a2i(Δ)r2i−4are dimensionless combinations.
+Differentiatingwith respect to sandsetting s=0, oneobtains
+FB
+h(m) =/parenleftBig1
+2m4logm2−3
+4m4/parenrightBig
+a0(Δ)r−4−/parenleftBig
+m2logm2−m2/parenrightBig
+a2(Δ)r−2
++logm2a4(Δ)+O(m−2logm). (3.26)
+The evaluation of the one-loop determinant for the hypermul tiplet fermions can be done
+similarly. Thequadratictermsofthefermionsaregivenby
+iψΓm∇mψ−i
+rψΓ0ΦATAψ+i
+2(ξΓµν/tildewideξ)ψΓ0Γµνψ. (3.27)
+Weneed to evaluate −logDet(iD/)where
+iD/:=iΓm∇m−m
+riΓ0+κ
+2(ξΓµν/tildewideξ)Γ0Γµν(3.28)
+withκ=i. Inthefollowing,wewillevaluate −1
+2logDet(iD/)2withareal κ,forwhich (iD/)2is
+non-negativeand its heat-kernel is well-defined, and then s ubstituteκ=iinto the final expres-
+sion. Thevalidityofthisprocedure isjustifiedbyconverge nceoftheresult.
+Theexplicitform of (iD/)2isgivenby
+(iD/)2=−(∇m+Vm)(∇m+Vm)−1
+2Γmn[∇m,∇n]−3κ2
+4r2sin2θ
+−κ2
+4(ξΓµν/tildewideξ)(ξΓρσ/tildewideξ)ΓµνΓρσ+iκm
+r(ξΓµν/tildewideξ)Γµν+m2
+r2
+:=−ΔF+m2
+r2. (3.29)
+where
+Vm=iκ(ξΓmµ/tildewideξ)Γ0Γµ. (3.30)
+The fermion case is slightly different from the scalar case s ince there is a term linear in m
+in−ΔF. However,theasymptoticexpansionofthezeta-function-r egularizedone-loopdetermi-
+nant can be made in the fermion case as well. The part FF
+h(Φ)of the matrix model action due
+toψhasa similarform with FB
+h(Φ),withdifferentcoefficients.
+18The total one-loop contribution of hypermultiplet to the ef fective action is Fh=FB
+h+FF
+h.
+Because ofunderlyingsupersymmetry,thetermsoforder m4andm4logm2cancel between FB
+h
+andFF
+h. Theresultingexpressionfor Fhis
+Fh=2NN
+∑
+i=1F(φi), (3.31)
+F(m) =c1m2logm2+c2m2+c3logm2+O(m−2logm). (3.32)
+The fact that c1is positive will turn out to be important later, while the exa ct values of the
+coefficients are irrelevant for the large ‘t Hooft coupling b ehavior of the Wilson loop. We
+presented details of computation of c1in Appendix D. Notice that, at least up to this order,
+F(m)is an evenfunctionof m.
+Obviously, Fhdepends on field contents. The expression for FhwhenRis the adjoint rep-
+resentation can be found easily by noticing that, for exampl e, the ’mass’ term of qαcan be put
+to
+1
+r2∑
+i/ne}ationslash=j(φi−φj)2qijαqα
+ji. (3.33)
+In thiscase, Fhis writtenas
+Fh/vextendsingle/vextendsingle/vextendsingle
+adj.=∑
+i/ne}ationslash=jF(φi−φj). (3.34)
+Notethat F(m)here isthesamefunctionas (3.32).
+Direct evaluation of the contribution from the vector multi plet, which we denote as Fv,
+appears morecomplicatedsincetherearemixingtermsbetwe enAmandAa. Fortunately,itwas
+shown in [6] that FvandFhcancel each other in N=4 super Yang-Mills theory. This implies
+from (3.34)that
+Fv=−∑
+i/ne}ationslash=jF(φi−φj). (3.35)
+•ˆA1gaugetheory :
+We next consider the ˆA1quiver gauge theory. In this case, qαandψconsist of bi-fundamental
+fields. The Φis ablock-diagonalmatrix:
+Φ=
+Φ(1)
+Φ(2)
+, (3.36)
+in which Φ(1)=diag(φ(1)
+1,···,φ(1)
+N)andΦ(2)=diag(φ(2)
+1,···,φ(2)
+N), respectively. By repeating
+thesimilarcomputations,onecan easilyshowthat Fhhastheform
+Fh=2N
+∑
+i,j=1F(φ(1)
+i−φ(2)
+j), (3.37)
+19andFvhas theform
+Fv=−∑
+i/ne}ationslash=jF(φ(1)
+i−φ(1)
+j)−∑
+i/ne}ationslash=jF(φ(2)
+i−φ(2)
+j). (3.38)
+Thetotalone-loopcontributionisthesum F=Fh+Fv.
+As a consistency check of the above result, consider taking t he two nodes identical. This
+reduces the number of nodes from two to one, and hence must map theˆA1gauge theory to ˆA0
+one. The reduction puts Φ(1)andΦ(2)equal. Then, up to an irrelevant constant, Fvis precisely
+minus of Fh. We thus see that Fvanishes identically, reproducing the known result of the ˆA0
+gaugetheory.
+3.3 Saddle-point equations
+We can now extract the saddle-point equations for the matrix model and determine the large ‘t
+HooftcouplingbehavioroftheWilsonloopfromthem.
+•A1gaugetheory :
+In thistheory,thesaddle-pointequationreads
+8π2
+λφk+2F′(φk)−2
+N∑
+i/ne}ationslash=kF′(φk−φi)=2
+N∑
+i/ne}ationslash=k1
+φk−φi. (3.39)
+Asexplainedbefore,weassumethat λγρ(λγφ)forapositiveγhasasensiblelarge λasymptote.
+By rescaling φk→λγφk, oneobtains
+8π2φk+2λ1−γF′(λγφk)−2
+N∑
+i/ne}ationslash=kλ1−γF′(λγ(φk−φi))=2
+Nλ1−2γ∑
+i/ne}ationslash=k1
+φk−φi.(3.40)
+Recall that F(x)∼c1x2logx2for largex. This shows that the leading-order equation for large
+λisgivenby
+4c1φklogφk+2(c1+c2)φk−2
+N∑
+i/ne}ationslash=k/bracketleftBig
+2c1(φk−φi)log(φk−φi)+(c1+c2)(φk−φi)/bracketrightBig
+=0.(3.41)
+Differentiatingtwicewithrespect to φk, oneobtains
+1
+φk=1
+N∑
+i/ne}ationslash=k1
+φk−φi. (3.42)
+Notice that c1andc2dropped out. Now, this equation has no sensible solution. Th erefore, we
+conclude that the scaling assumption we started with is inva lid, implying that the Wilson loop
+inthistheory cannotgrowexponentiallyin thelarge‘t Hoof t couplinglimit.
+20There is another way to check the finiteness of the Wilson loop . Let us rewrite the saddle-
+pointequationas follows:
+8π2
+λφk+2F′(φk)=2
+N∑
+i/ne}ationslash=kF′(φk−φi)+2
+N∑
+i/ne}ationslash=k1
+φk−φi. (3.43)
+The left-hand side represents the external force acting on t he eigenvalues, whilethe right-hand
+side represents the interactions among the eigenvalues. Fo r a large φk, the external force is
+dominated by 2 F′(φk), which is nonzero. This implies that the large λlimit must be smooth,
+and the Wilson loop expectation value approaches a finite val ue. Recall that in the case of
+N=4 super Yang-Mill theory, the large λlimit renders the external force to vanish, resulting
+in an indefinite spread of the eigenvalues. This is reflected i n the exponential growth of the
+Wilsonloopexpectationvalue.
+Implicationsofthissurprisingconclusionarefarreachin g: the N=2supersymmetricgauge
+theorycoupledto2 Nfundamentalhypermultiplets,althoughsuperconformal,m usthaveaholo-
+graphic dual whose geometry does not belong to the more famil iar cases such as N=4 super
+Yang-Mills theory. Central to this phenomenon is that there are two ‘t Hooft coupling param-
+eters whose ratio can be tuned hierarchically large or small . In particular, we can tune one of
+them to be smaller than O(1), which also renders two widely separated length scales (in u nits
+of string scale) in the putative gravity dual background. In the next section, we shall discuss
+how nonstandard the dual geometry ought to be by using the non -exponential behavior of the
+Wilsonloopas aprobe.
+•ˆA1gaugetheory :
+In this theory, there are two saddle-point equations corres ponding to two matrices Φ(1)and
+Φ(2):
+8π2
+λ1φ(1)
+k+2
+NN
+∑
+i=1F′(φ(1)
+k−φ(2)
+i)−2
+N∑
+i/ne}ationslash=kF′(φ(1)
+k−φ(1)
+i)=2
+N∑
+i/ne}ationslash=k1
+φ(1)
+k−φ(1)
+i,(3.44)
+8π2
+λ2φ(2)
+k+2
+NN
+∑
+i=1F′(φ(2)
+k−φ(1)
+i)−2
+N∑
+i/ne}ationslash=kF′(φ(2)
+k−φ(2)
+i)=2
+N∑
+i/ne}ationslash=k1
+φ(2)
+k−φ(2)
+i,(3.45)
+whereλ1=g2
+1Nandλ2=g2
+2Nare the‘t Hooftcouplingconstantsofeach gaugegroups.
+Denoteρ(1)(φ),ρ(2)(φ)the eigenvalue distribution functions for the Φ(1),Φ(2)matrices,
+respectively. Itis convenientto define
+ρ(φ):=1
+2(ρ(1)(φ)+ρ(2)(φ)), (3.46)
+δρ(φ):=1
+2(ρ(1)(φ)−ρ(2)(φ)). (3.47)
+21In termsofthem,theabovesaddle-pointequationsaresimpl ifiedas follows:
+4π2
+λφ=/integraldisplay
+−dφ′ρ(φ′)
+φ−φ′, (3.48)
+2π2/bracketleftBig1
+λ1−1
+λ2/bracketrightBig
+φ−2/integraldisplay
+−dφ′δρ(φ′)F′(φ−φ′) =/integraldisplay
+−dφ′δρ(φ′)
+φ−φ′, (3.49)
+where
+1
+λ:=1
+|Γ|/parenleftbigg1
+λ1+1
+λ2/parenrightbigg
+and|Γ|=2. (3.50)
+For obvious reasons, we refer these two as untwisted and twis ted saddle-point equations. By
+the scaling argument, one can show that δρ(φ)is negligible compared to ρ(φ)in the large λ
+limit. In particular,when λ1=λ2,itfollowsthat δρ=0is asolution,consistentwith Z2parity
+exchangingthetwonodes. Therefore, thelarge λbehavioroftheWilsonloopisdeterminedby
+(6.7), which is exactly the same as (3.9). Indeed, λdefined by (3.50) is exactly what is related
+togsN[18].
+The two Wilson loops are then obtainablefrom the one-matrix model with eigenvalueden-
+sityρ±δρ:
+W1=/integraldisplay
+Ddxeaxρ(1)(x) =/integraldisplay
+Ddxeax[ρ(x)+δρ(x)]
+W2=/integraldisplay
+Ddxeaxρ(2)(x) =/integraldisplay
+Ddxeax[ρ(x)−δρ(x)]. (3.51)
+Weseethat theuntwistedandthetwistedWilsonloopsare giv enby
+W(0):=1
+2(W1+W2)=/integraldisplay
+Ddxeaxρ(x)
+W(1):=1
+2(W1−W2)=/integraldisplay
+Ddxeaxδρ(x). (3.52)
+Inferring from the saddle-point equations (3.48, 3.49), we see that these Wilson loops are di-
+rectly related to the average and difference of the two gauge coupling constants. It also shows
+thatthetwistedWilsonloopwillhavenonzero expectationv alueoncethetwogaugecouplings
+are set different. In the next section, we shall see that they descend from moduli parameters of
+six-dimensionaltwistedsectors at theorbifoldsingulari tyin theholographicdual description.
+We have found the following result for the Wilson loop in ˆA1quiver gauge theory. The
+two Wilson loops, corresponding to the two quiver gauge grou ps, have exponentially growing
+behavior at large ‘t Hooft coupling limit. Its functional fo rm is exactly the same as the one
+exhibitedby theWilsonloopin N=4superYang-Millstheory.
+223.4 Interpolationamongthe quivers
+Withthesaddle-pointequationsathand,wenowdiscussvari ousinterpolationsamong ˆA0,A1,ˆA1
+theories and learn about the gauge dynamics. Our starting po int is the ˆA1theory, whose quiver
+diagramhas twonodes. Seefigure 1.
+•Considerthesymmetricquiverforwhichthetwo‘tHooftcoup lingconstantstaketheratio
+λ1/λ2=1. Then the twisted saddle-point equation (3.49) asserts th atδρ=0 is the solution. It
+follows that /an}bracketle{tW1/an}bracketri}ht−/an}bracketle{tW2/an}bracketri}ht=0, viz. the Wilson loop in the twisted sector vanishes identi cally.
+Intuitively,the two gauge interactions are of equal streng th, so the two Wilson loops are indis-
+tinguishable. Moreover,fromtheuntwistedsaddle-pointe quation(3.48),weseethattheWilson
+loopintheuntwistedsectorbehavesexactlythesameastheo neinˆA0theoryand,inparticular,
+N=4 superYang-Millstheory:
+W(0)=1
+2/parenleftBig
+/an}bracketle{tW1/an}bracketri}ht+/an}bracketle{tW2/an}bracketri}ht/parenrightBig
+=1√
+2λI1(√
+2λ). (3.53)
+It follows that the Wilson loop grows exponentially at large ‘t Hooft coupling limit, much the
+sameway asthe ˆA0theory does.
+•Considertheasymmetricquiverwherethe ratio λ1/λ2/ne}ationslash=1 but finite. Thetwisted saddle-
+pointequation(3.49)can berecast as
+1
+λ/parenleftbigg
+B−1
+2/parenrightbigg/integraldisplay
+−dφ′ρ(φ′)
+φ−φ′=/integraldisplay
+−dφ′δρ(φ′)/bracketleftbigg1
+21
+φ−φ′+F′(φ−φ′)/bracketrightbigg
+. (3.54)
+Here, weparametrized thedifferenceoftwoinverse‘tHooft couplingsas
+/parenleftbigg
+B−1
+2/parenrightbigg
+:=1
+2/parenleftbigg1
+λ1−1
+λ2/parenrightbigg/slashBig/parenleftbigg1
+λ1+1
+λ2/parenrightbigg
+. (3.55)
+Obviously, taking into account the Z2exchange symmetry between the two quiver nodes, B
+ranges overtheinterval [0,+1]. Thesymmetricquiverconsidered abovecorresponds to B=1
+2.
+Solvingfirst ρfrom(3.48)andsubstitutingthesolutionto(3.54),onesol vesδρasafunctionof
+B. Weseefrom(3.54)that δρoughttobea linearfunctionof Bthroughouttheinterval [0,+1].
+Equivalently, extending the range of Bto(−∞,+∞), we see that δρis a sawtooth function,
+piecewiselinearovereach unitintervalof B. Inparticular,itisdiscontinuousacross B=0(and
+across all other nonzero integer values). This is depicted i n figure 3. Therefore, we conclude
+that the Wilson loops W1,W2at strong ‘t Hooft coupling limit are nonanalytic not only in λ
+but also in B. In fact, as we shall recall in the next section, B=0 is a special point where
+thespacetimegaugesymmetryisenhancedandtheworldsheet conformalfieldtheorybecomes
+23singular. Nevertheless,the Wilsonloopin theuntwistedse ctorbehaves exactlythesameas the
+symmetric quiver, viz. (3.53). We conclude that the untwist ed Wilson loop is independent of
+strengthofthegaugeinteractions.
+-1               -1/2                 0                +1/2               +1   B  tW
+Figure 3: Dependence of twisted sector Wilson loops on the parameter B. It shows discontinuity at
+B=0,resulting in non-analytic behavior of the Wilson loops tob oth gauge couplings.
+•Consider an extreme limit of the asymmetric quiver where the ratioλ1/λ2→0, equiva-
+lently,λ2/λ1→∞,viz. thetwo‘tHooftcouplingsarehierarchicallyseparat ed. Inthiscase,one
+gauge group is infinitely stronger than the other gauge group and theˆA1quiver gauge theory
+ought to become the A1gauge theory . This can be seen as follows. In the ˆA1saddle-point
+equations (3.45), we see that φ(1)→0 solves the first equation. Plugging this into the second
+equation, we see it is reduced to the A1saddle-point equation (3.43). This reduction poses a
+very interesting physics since from the above consideratio ns the Wilson expectation value in-
+terpolates from the exponential growth of the ˆA1quiver gauge theory to the non-exponential
+behavior of the A1gauge theory. In the next section, we shall argue that this is a clear demon-
+stration (as probed by the Wilson loops) that holographic du al of theA1gauge theory ought to
+haveinternalgeometryof stringscale size.
+Wecanalsounderstandtheinterpolationdirectlyintermso ftheWilsonloop. Consider,for
+example, λ2/λ1→∞. From the ˆA1Wilson loops, using the fact that ρ(1)(x),ρ(2)(x)are strictly
+positive-definite,wehave
+/an}bracketle{tW2/an}bracketri}ht=/integraldisplay
+dλρ(2)(λ)eλ
+24≤2/integraldisplay
+dλ1
+2[ρ(1)(λ)+ρ(2)(λ)]eλ
+=4√
+2λI1(√
+2λ). (3.56)
+Sinceλ∼λ1→0, the Wilson loop is bounded from above by a constant. Note th at the limit
+λ1→0 can besafely taken: thesaddle-pointequation(3.48)isin fact exact in λ.
+•Considerthelimit λ1,λ2→0. In thislimit,
+λ=2λ1λ2
+λ1+λ2→0,κ:=λ2
+λ1=fixed (3.57)
+and theexact result(3.53)isexpandablein powerseries of λandκ:
+W(0)/vextendsingle/vextendsingle/vextendsingle
+exact=1
+2/parenleftBig
+/an}bracketle{tW1/an}bracketri}ht+/an}bracketle{tW2/an}bracketri}ht/parenrightBig
+=1
+2∞
+∑
+ℓ=0∞
+∑
+m=0(−)m(ℓ+m−1)!
+(ℓ−1)!ℓ!(ℓ+1)!λℓ
+1κℓ+m. (3.58)
+Here, the exact result (3.53) is symmetric under λ1↔λ2, so we assumed in (3.58) that κ<1.
+Ontheotherhand,fromstandpointofthequivergaugetheory ,theWilsonloopinthefixed-order
+perturbationtheoryis givenby powerseries in λ1orλ2:
+W(0)/vextendsingle/vextendsingle/vextendsingle
+pert=∞
+∑
+ℓ=0∞
+∑
+m=0Wℓ,mλℓ
+1λm
+2=∞
+∑
+ℓ=1∞
+∑
+m=1Wℓ,mλℓ+m
+1κm. (3.59)
+Weseethattheexactresult(3.58)andtheperturbativeresu lt(3.59)donotagreeeachother.
+Recallthatbothresultsareobtainedatplanarlimit N→andoughttobeabsolutelyconvergentin
+(λ,B)andin(λ1,λ2),respectively. Thereasonmaybethatthetwosetsofcouplin gconstantsare
+notanalyticin C2complexplane. Infact,from(3.57),weseethat λ(λ1,λ2)hasacodimension-1
+singularityat λ1+λ2=0. Anexceptionalsituationiswhen λ1=λ2. Inthiscase,thesingularity
+disappearsand,withthesamepowerseriesexpansion,weexp ecttheexactresult(3.58)andthe
+perturbativeresult (3.59)are thesame.
+We should note that the change of variables is well-defined at strong coupling regime. In
+thisregime,powerseriesexpansionsin1 /λ1and1/λ2isrelatedunambiguouslytopowerseries
+expansionsin 1 /λandB. In fact, thechangeofvariables
+/parenleftBig1
+λ1,1
+λ2/parenrightBig
+−→/parenleftBig1
+λ,B/parenrightBig
+(3.60)
+isanalyticanddoesnotintroduceanysingularityaround λ1,λ2=∞. Infact,aswewillrecapit-
+ulate,theseare thevariablesnaturallyintroducedin theg ravitydualdescription.
+WeremarkthattheanalyticstructureoftheWilsonloopsinq uivergaugetheoriesissimilar
+totheIsingmodelinamagneticfieldonaplanarrandomlattic e[20]. Thelatterisdefined bya
+25matrix modelinvolvingtwo interactingHermitian matrices and involvestwo couplingparame-
+ters: average‘tHooftcouplingandmagneticfield. Hereagai n,byturningonthemagneticfield,
+one can scale two independent ‘t Hooft coupling parameters d ifferently. In light of our results,
+it would be extremely interesting to study this system in the limit the magnetic field is sent to
+infinity.
+4 IntuitiveUnderstandingofNon-Analyticity
+In the last section, the distinguishingfeature of the A1theory from the ˆA0,ˆA1theories was that
+growth of the Wilson loop expectation value was less than exp onential. Yet, these theories are
+connected one another by continuously deforming gauge coup ling parameters. How can then
+suchanon-analyticbehaviorcomeabout?3In thissection,weofferanintuitiveunderstanding
+ofthis in termsof competitionbetween screening and over-s creeningof colorcharges and also
+draw analogytotheKondoeffect ofmagneticimpurityinamet al.
+•screeningversusanti-screening :
+Consider first the weak coupling regime. The representation contents of these N=2 quiver
+gaugetheoriesaresuchthatthe ˆA0theorycontainsfieldcontentsinadjointrepresentationso nly,
+while the ˆA1and theA1theories contain additional field contents in bi-fundament al or funda-
+mental representations, respectively. The A1theory contains additional massless multiplets in
+fundamental representation, so we see immediately that the theory is capable of screening an
+external color charge sourced by the Wilson loop for any repr esentations. Since the theory is
+conformal, the screening length ought to be infinite (zero is also compatible with conformal
+symmetry, but it just means there is no screening) and impedi ng creation of an excitation en-
+ergyabovethegroundstate. Evenmoreso,‘tension’oftheco lorfluxtubewouldgotozero. In
+other words, once a static color charge is introduced to the t heory, massless hypermultipletsin
+fundamental representation will immediately screen out th e charge to arbitrary long distances.
+Though this intuitive picture is based on weak coupling dyna mics, due to conformal symme-
+try, it fits well with the non-exponential growth of the Wilso n loop in the A1theory, which we
+derivedintheprevioussectionin theplanarlimit.
+We stress that the screening has nothingto do with supersymm etrybut is a consequence of
+elementary consideration of gauge dynamics with massless m atter in complex representations.
+Thisisclearlyillustratedbythewellknowntwo-dimension alSchwingermodel. Generalization
+of this Schwinger mechanism to nonabelian gauge theories sh owed that massless fermions in
+arbitrarycomplexrepresentationscreenstheheavyprobechargeinth efundamentalrepresenta-
+3ThisquestionwasraisedtousbyJuanMaldacena.
+26tion[21]. The screening and consequentstring breakingby t hedynamical masslessmatterwas
+observedconvincinglyinbothtwo-dimensionalQED[22]and three-dimensionalQCD[23]. In
+four-dimensional lattice QCD, the static quark potential V(R)awas computed ( adenotes the
+lattice spacing) for fermions in both quenched and dynamica l simulations [24]. For quenched
+simulation,thepotentialscaledlinearlywith R/a,indicatingconfinementbehavior. Fordynam-
+ical simulation, the potential exhibited flattening over a w ide range of the separation distance
+R/a.
+             (a)                                                                              (b)
+Figure4: Responseofgaugetheoriestoexternalcolorchargesource. (a)ForA1theory,anexternalcolor
+charge infundamental representation ofthegaugegroupiss creened bythe Nf=2Ncflavorsofmassless
+matter fields, which are in fundamental representation (blu e arrow). (b) For ˆA1theory, an external color
+charge in fundamental representation of the first gauge grou p is screened by the massless matter fields.
+As the matter fields are in bi-fundamental representations ( black and white arrows), color charge in the
+secondgaugegroupisregenerated andanti-screened. Thepr ocessrepeatsbetweenthetwogaugegroups
+and leads thetheory to exhibit Coulomb behavior.
+The case of ˆA1theory is more interesting. Having two gauge groups associa ted with each
+nodes,considerintroducingastaticcolorcharge oftherep resentation Rfor, say,thefirst gauge
+groupinSU (N)×SU(N). Thehypermultipletstransformingin (N,N)and(N,N)areindefining
+representations with respect to the first gauge group, so the y will rearrange their ground-state
+configuration to screen out the color charge. But then, as the se hypermultipletsare in defining
+representationwithrespecttothesecondgaugegroupaswel l,acompletescreeningwithrespect
+to the first gauge group will reassemble the resulting configu ration to be in the representation
+27Rof the second gauge group in SU (N)×SU(N). This configuration is essentially the same as
+thestartingconfigurationexceptthatthetwogaugegroupsa reinterchanged(alongwithcharge
+conjugation). The hypermultiplets may opt to rearrange the ir ground-state configuration to
+screenoutthecolorchargeofthesecondgaugegroup,butthe ntheprocesswillrepeatitselfand
+returns back to the original static color charge of the first g auge group — in ˆA1theory, perfect
+screeningofthefirstgaugegroupisaccompaniedbyperfecta nti-screeningofthesecondgauge
+group and vice versa. This is depicted in figure 4. Consequent ly, a complete screening never
+takes place for bothgauge groups simultaneously. Instead, the external color c harge excites
+the ground-state to a conformally invariant configuration w ith the Coulomb energy. Again, we
+formulated this intuitive picture from weak coupling regim e, but the picture fits well with the
+exponentialgrowthoftheWilsonloopexpectationvalueof ˆA1theorywederivedintheprevious
+sectionat planarlimit.
+•AnalogytoKondoeffect :
+It is interesting to observe that the screening vs. anti-scr eening process described above is
+reminiscentofthemulti-channelKondoeffectinametal[25 ]. There,astaticmagneticimpurity
+carrying aspin Sinteracts withconductionelectronsand profoundlyaffect s electrical transport
+propertyatlongdistances. Supposeinametalthereare Nfflavorsofconductionbandelectrons.
+Thus,thereare Nfchannels and theyare mutuallynon-interacting. Theantife rromagneticspin-
+spin interaction between the impurity and the conduction el ectrons leads at weak coupling to
+screening of the impurity spin StoSren= (S−Nf/2). We see that the system with Nf<2S
+is under-screened, leading to an asymptotic screening of th e impurity spin and that the system
+withNf>2Sisover-screened,leadingtoanasymptoticanti-screening oftheimpurityspin. The
+marginallyscreenedcase, Nf=2S,isattheborderbetweenthescreeningandtheanti-screeni ng:
+thespinSof themagneticimpurityis intact underrenormalization by the conductionelectrons
+(modulo overall flip of the spin orientation, which is a symme try of the system). We thus
+observe that the Coulomb behavior of the external color sour ce inˆA1theory is tantalizingly
+parallel tothemarginallyscreened caseofthemulti-chann elKondoeffect.
+•Interpretationviabraneconfigurations :
+We can also understand the screening-Coulomb transition fr om the brane configurations de-
+scribingˆA1andA1theories4. Consider Type IIA string theory on R8,1×S1, where the circle
+direction is along x9and have circumference L. We set up thebrane configuration by introduc-
+ing two NS5-branes stretched along (012345)directions and Nstack of D4-branes stretched
+along(01239)directions on intervals between the two NS5-branes. Generi cally, the two NS5-
+4Fora comprehensivereviewofbraneconfigurations,see [26] .
+28branes are located at separate position on S1and this corresponds to the ˆA1theory. The gauge
+couplings 1 /g2
+1and 1/g2
+2of the two quiver gauge groups are proportional to the length of the
+twox9-intervalsoftheD4-branes. WhenthetwoNS5-branesareloc atedatdiagonallyopposite
+points,say,at x9=0,L/2, thetwogaugecouplingsofthe ˆA1theoryare equal. Thisis depicted
+in figure 5(a). By approaching one NS5-brane to another, say, atx9=0, we can obtain the
+configurationin figure5(b). Thiscorrespondsto A1theory sincethegaugecouplingoftheD4-
+branes encircling the S1becomes arbitrarily weak compared to that of the D4-branes s tretched
+infinitesimallybetweenthetwooverlappingNS5-branes.
+NS5                                                NS5                                                                         NS5-NS5
+(a)                                                                                          (b) F1                                F1                                              F1                                    F1
+Figure5: SemiclassicalWilsonloopinbraneconfigurationof N=2superconformal gaugetheoriesun-
+der study: (a) ˆA1theory with G=SU(N)×SU(N) and2Nbifundamental hypermultiplets. ND4-branes
+stretch between twowidely separated NS5-branes on acircle . TheF1(fundamental string) ending on or
+emanating from D4-brane represent static charges. On D4-br anes, having finite gauge coupling, conser-
+vation of the F1 flux is manifestly. (b) A1theory with G=SU(N) and2Nfundamental hypermultiplets.
+TheA1theory is obtained from ˆA1in (a) by approaching the two NS5-branes. The flux is leaked in to
+the coincident NS5-branes and run along their worldvolumes . On D4-branes, having vanishing gauge
+coupling, conservation of the F1fluxisnot manifest.
+We now introduce external color charge to the D4-branes and e xamine fate of the color
+fluxes. Theexternalcolorsourcesareprovidedbyamacrosco picIIAfundamentalstringending
+on the stacked D4-branes. Consider first the configuration of theˆA1theory. The color charge
+is an endpoint of the fundamental string on one stack of the D4 -branes, viz. one of the two
+quiver gauge groups. Along the D4-branes, the endpoint sour ces color Coulomb field. The
+color field will sink at another external color charge locate d at a finite distance from the first
+external charge. See figure 2(a). We see that the color flux is c onserved on the first stack of
+D4-branes. Wealsoseethat,atweakcouplingregime,effect softheNS5-branesarenegligible.
+Considernexttheconfigurationofthe A1theory. Basedontheconsiderationsoftheprevious
+section,weconsideranexternalcolorchargetothestackof D4-branesencirclingthe S1. Inthis
+29configuration, the two NS5-branes are coincident and this op ens up a new possible color flux
+configuration. To understand this, we recall the situation o f stack of D1-D5 branes, which is
+related to the macroscopic IIA string and stack of NS5-brane s. In the D1-D5 system, it is well
+known that there are threshold bound states of D1-branes on D 5-branesprovided two or more
+D5-branes are stacked. For a single D5-brane, the D1-brane b ound-state does not exist. This
+suggestsinthebraneconfigurationofthe A1theorythatthecolorfluxmaynowbepulledtoand
+smear out along the two coincident NS5-branes. From theview pointof stack of the D4-branes
+encircling S1, thecolorflux appears not conserved.
+5 HolographicDual
+Theexactresultsofthe N=2Wilsonloopsatstrong‘tHooftcouplinglimitweobtainedi nthe
+previoussectionrevealedmanyintriguingaspects. Inpart icular,comparedtothemorefamiliar,
+exponentialgrowthbehaviorofthe N=4Wilsonloops,wefoundthefollowingdistinguishing
+features and consequences:
+•InA1gauge theory, the Wilson loop /an}bracketle{tW/an}bracketri}htdoesnotexhibit the exponential growth. Re-
+placing 2 Nfundamental representation hypermultiplets by single adj oint representation
+hypermultipletrestores the exponential growth, since the latter is nothing but the N=4
+counterpart. This suggests that /an}bracketle{tW/an}bracketri}htinˆA1gauge theory has (possibly infinitely) many
+saddle points and potential leading exponential growth is c anceled upon summing over
+the saddle points. We stress that, in this case, the ratio of t wo ‘t Hooft coupling goes
+to zero, equivalently, infinite. The limit decouples dynami cs of the two quiver gauge
+groupsandrendertheglobalgaugesymmetryasanewlyemerge ntflavorsymmetry. The
+non-exponential behavior of the Wilson loop originates fro m the decoupling, as can be
+understoodintuitivelyfrom thescreening phenomenon.
+•InˆA1quivergaugetheory,thetwoWilsonloops /an}bracketle{tW1/an}bracketri}ht,/an}bracketle{tW2/an}bracketri}htassociatedwiththetwoquiver
+nodes exhibit the same exponential growth as the N=4 counterpart. The exponents
+depend not only on the largest edge of the eigenvalue distrib ution but also on the two ‘t
+Hooftcouplingconstants, λ1,λ2, equivalently, λ,B.
+•InˆA1quiver gauge theory, in case the two ‘t Hooft couplings are th e same, so are the
+two Wilson loops. If the two ‘t Hooft couplings differ butremain finite, the two Wilson
+loops will also differ. As such, /an}bracketle{tW1/an}bracketri}ht−/an}bracketle{tW2/an}bracketri}htis an order parameter of the Z2parity ex-
+changing the two quiver nodes. It scales linearly with Band shows non-analyticity over
+thefundamentaldomain [−1
+2,+1
+2].
+30In this section, we pose these features from holographic dua l viewpoint and extract several
+new perspectives. Much of success of the AdS/CFT correspond ence was based on the obser-
+vation that holographic dual geometry is macroscopically l arge compared to the string scale.
+In this limit, string scale effects are suppressed and physi cal observables and correlators are
+computable in saddle-point, supergravity approximation. For example, the AdS 5×S5dual to
+theN=4superYang-Millstheory has thesize R2=O(√
+λ):
+ds2=R2ds2(AdS5)+R2dΩ2
+5(S5), (5.1)
+growing arbitrarily large at strong ‘t Hooft coupling. Many other examples of the AdS/CFT
+correspondence share essentially the same behavior. In suc h a background, expectation value
+oftheWilsonloop /an}bracketle{tW/an}bracketri}htisevaluatedbythePolyakovpathintegralofafundamentals tringinthe
+holographicdualbackground:
+/an}bracketle{tW/an}bracketri}ht:=/integraldisplay
+C[DXDh]⊥exp(iSws[X∗g]) (5.2)
+withaprescribedboundaryconditionalongthecontour CoftheWilsonloopattimelikeinfinity.
+The worldsheet coupling parameter is set by the pull-back of the spacetime metric, and hence
+byR2. AsRgrows large at strong ‘t Hooft coupling, the path integral is dominatedby a saddle
+point and /an}bracketle{tW/an}bracketri}htexhibits exponential growth whose Euclidean geometry is th e minimal surface
+Acl:
+/an}bracketle{tW/an}bracketri}ht ≃eAclwhere Acl≃O(R2). (5.3)
+NotethattheminimalsurfaceoftheWilsonloopsweepsoutan AdS3foliationinsidetheAdS 5.
+Thisexplainsthe R2growthoftheareaoftheminimalsurface atstrong‘t Hooftco upling.
+Central to our discussionswill consist of re-examination o n global geometry of the gravity
+dualto N=2superconformalgaugetheoriesincomparisonto N=4superYang-Millstheory.
+5.1 Holographic dualof A1gauge theory
+At present, gravity dual to the A1gauge theory is not known. Still, it is not difficult to guess
+whatthedualtheorywouldbe. Ingeneral, N=2gaugetheoryisdefinedinperturbationtheory
+by threecouplingparameters:
+λ,g2
+c:=1
+N2,go:=Nf
+N, (5.4)
+associated ‘t Hooftcoupling,closedsurface couplingasso ciatedwith adjointvectorand hyper-
+multiplets, and open puncture coupling associated with fun damental hypermultiplets. For A1
+31gauge theory, go=2∼O(1)and it indicates that dual string theory is described by the w orld-
+sheet with proliferating open boundaries. Moreover, as we s tudied in earlier sections, the A1
+gaugetheoryis related to the ˆA1quivergaugetheory as thelimitwhere oneofthetwo ‘t Hooft
+coupling constants is sent to zero while the other is held fini te. Equivalently, in the large N
+limit,oneofthetwo‘tHooftcouplingconstantsisdialedin finitelystrongerthantheother. This
+hierarchical scaling limit of the two ‘t Hooft coupling cons tants, along with the PSU (2,2|2)
+superconformal symmetry and the SU(2) ×U(1) R-symmetry imply that the gravity dual is a
+noncritical superstring theory involving AdS 5andS2×S1space. One thus expects that the
+gravitydual of A1gaugetheory hasthelocalgeometry oftheform:
+(AdS5×M2)×[S1×S2]. (5.5)
+By local geometry, we mean that the internal space consists o fS1andS2, possibly fibered or
+warped over an appropriate 2-dimensionalbase-space M25. The curvature scales of AdS 5and
+ofM2are equal and are set by R∼λ1/4, much as in the N=4 super Yang-Mills theory. The
+remaining internal geometry [S1×S2]involves geometry of string scale, and is describable in
+termsofa(singular)superconformalfieldtheory. Inpartic ular,theinternalspace [S1×S2]may
+havecollapsed2-cycles. Therefore, theten-dimensionalg eometryis schematicallygivenby
+ds2=R2(ds2(AdS5)+ds2(M2))+r2ds2([S1×S2]) (5.6)
+whereR,rare the curvature radii that are hierarchically different, r≪R(measured in string
+scale). Inparticular, rcanbecomesmallerthan O(1)intheregimethatthetwo‘tHooftcoupling
+constantsaretaken hierarchically disparate.
+Consider now evaluatingthe Wilson loop /an}bracketle{tW[C]/an}bracketri}htin thegravity dual (5.5). As well-known,
+the Wilson loop is holographically computed by free energy o f a macroscopic string whose
+endpoint sweeps the contour C. From the viewpointof evaluatingit in terms ofa minimalare a
+worldsheet, since the internal space has nontrivial 2-cycl es, there will not be just one saddle-
+point but infinitely many. These saddle-point configuration s are approximately a combination
+of minimal surface of area Aswinside the AdS 5and surfaces of area a(i)
+swwrapping 2-cycles
+insidetheinternalspacemultipletimes. Notethat Aswhastheareaoforder O(r2)≫1instring
+unit anda(i)
+swhas the area of order O(1)since the 2-cycles are collapsed. Therefore, all these
+configurations have nearly degenerate total worldsheet are a and correspond to infinitely many,
+5The expected gravity dual (5.5) may be anticipated from the A rgyres-Seiberg S-duality [19]. At finite N, S-
+duality maps an infinite coupling N=2 superconformal gauge theory to a weak coupling N=2 gauge theory
+combined with strongly interacting, isolated conformal fie ld theory. The presence of the strongly interacting,
+isolated conformal field theory suggests that putative holo graphic dual ought to involve a string geometry whose
+size istypicallyoforder O(1)instringunit.
+32nearbysaddlepoints. Ineffect,thesurfacesofarea a(i)
+swwrappingthecollapsed2-cyclemultiple
+timesproducesizableworldsheetinstantoneffects. Wethu shave
+/an}bracketle{tW/an}bracketri}ht=∑
+i=saddlescaexp/parenleftBig
+Asw+a(i)
+sw+···/parenrightBig
+≃/bracketleftBig
+∑
+i=saddlescaexp(a(i)
+sw)/bracketrightBig
+·exp(Asw), (5.7)
+wherecadenotes calculablecoefficients of each saddle-point,incl udingone-loop stringworld-
+sheet determinants and integrals over moduli parameters, i f present. This is depicted in figure
+6. Since we do not have exact worldsheet result for each saddl e point configurations available,
+we can only guess what must happen in order for the final result to yield the exact result we
+derived from the gauge theory side. In the last expression of (5.7), even though contribution
+of individual saddle point is same order, summing up infinite ly many of them could produce
+an exponentially small effect of order O(exp(−Asw)). What then happens is that summing
+up infinitely many worldsheet instantons over the internal s pace cancels against the leading
+O(exp(Asw))contributionfromtheworldsheetinsidetheAdS 5. Afterthecancelation,thelead-
+ing nonzero contribution is of the same order as the pre-expo nential contribution. It scales as
+Rνforsome finitevalueoftheexponent νat strong‘t Hooftcoupling.
+<W>    =                             +                                  +                                  +                                 +  ....
+Figure 6: Schematic view of holographic computation of Wilson loop ex pectation value in instanton
+expansion. Each hemisphere represents minimal surface of s emiclassical string in AdS spacetime. In-
+stantons are string worldsheets P1’s stretched into the internal space X5. Their sizes are of string scale,
+and hence of order O(1)for any number of instantons. The gauge theory computations indicate that
+these worldsheet instantons ought to proliferate and lead t o delicate cancelations of the leading-order
+result (the first term) upon resummation.
+At the orbifold fixed point, there are in general torsion comp onents of the NS-NS 2-form
+potential B2, whoseintegralovera2-cycleisdenotedby B:
+Ba:=/contintegraldisplay
+CaB2
+2π,Ba=[0,1) (5.8)
+33TheA1theory has the global flavor symmetry Gf=U(Nf)=U(2N). For a well-defined con-
+formal field theory of the internal geometry, Bamust take the value 1 /2. But then, the string
+worldsheetwrappingthe2-cycle Canatimespicksup thephasefactor
+∞
+∏
+a=1exp(2πiBana)=∞
+∏
+a=1(−)na, (5.9)
+givingriseto ±relativesignsamongvariousworldsheetinstantoncontrib utionstotheminimal
+surface dualtotheWilsonloop.
+5.2 Holographic dualof ˆA1quiver gauge theory
+Considernextholographicdescriptionof the ˆA1quivergaugetheory. It is knownthattheholo-
+graphic dual is provided by the AdS 5×S5/Z2orbifold, where the Z2acts onC2⊂C3of the
+coveringspaceof S5. Locally,thespacetimegeometryisexactly thesameas AdS 5×S5:
+ds2=R2ds2(AdS5)+R2dΩ2
+5(S5). (5.10)
+Thesizeof boththe AdS5and theS5/Z2isR, which growsas (λ)1/4at large ‘t Hooft coupling
+limit.
+Located at the orbifold fixed point is a twisted sector. The ma ssless fields of the twisted
+sector consists of a tensor multiplet of (5+1)-dimensional (2,0) chiral supersymmetry. The
+multiplet contains five massless scalars. Three of them are a ssociated with S2replacing the
+orbifoldfixed point,and theothertwo areassociated with
+B=/contintegraldisplay
+S2B2
+2πandC=/contintegraldisplay
+S2C2
+2π, (5.11)
+whereB2,C2are NS-NS and R-R 2-form potentials. Both of them are periodi c, ranging over
+B,C=[0,1)6. Thesetwomasslessmoduliarewell-definedeveninthelimit thattheotherthree
+modulivanish, viz. S2shrinks back to theorbifold singularity. Along withthe typ eIIB dilaton
+andaxionoftheuntwistedsector, thesetwotwistedscalarfi elds arerelatedtothegaugetheory
+parameters. In particular,wehave
+1
+gs=1
+g2
+1+1
+g2
+2;1
+gs(B−1
+2)=1
+g2
+1−1
+g2
+2. (5.12)
+The other moduli field Cis related to the theta angles. This can be seen by uplifting t he brane
+configuration to M-theory. There, the theta angle is nothing but the M-theory circle. It would
+varyifweturn onC-potentialon twocycles.
+6The periodicitycan be seen from the T-dual, brane configurat ionas well. Consider the moduli B. The quiver
+gauge theories are mapped to D4 branes connecting adjacent N S5 branes on a circle in two different directions.
+Thesumovergaugecouplingsisthenrelatedtocirclesize,w hilethedifferencebetweenadjacentgaugecouplings
+isgivenbythelengthofeachinterval. Evidently,theinter valcannotbelongerthanthe circumference.
+34Consider now computation of the Wilson loop expectation val ue from the Polyakov path
+integral(5.2). Again,asthecontour CoftheWilsonloopliesattheboundaryofAdS 3foliation
+insideAdS 5, theTypeIIB stringworldsheetwouldsweep a minimalsurfac ein AdS 3. Thearea
+isoforder O(R2). Ontheotherhand,theTypeIIBstringmaysweepoverthevani shingS2atthe
+orbifold fixed point. As the area of the cycle vanishes, the co rresponding worldsheet instanton
+effect is of order O(1)and unsuppressed. Thus, the situation is similar to the A1case. In the
+ˆA1case, however, we have a new direction of turning on the twist ed moduli associated with B.
+From (5.12), we see that this amounts to turning on the two gau ge couplings asymmetrically.
+Now, for the worldsheet instanton configuration, the Type II B string worldsheet couples to the
+B2field. Therefore, theWilsonloopwillget contributionsofe xp(±2πiB)oncethemoduli Bis
+turnedon.
+There is another reason why infinitely many worldsheet insta ntons needs to be resummed.
+We proved that the twisted sector Wilson loop is proportiona l to|B|. AsBranges over the in-
+terval[−1
+2,+1
+2],weseethattheWilsonloophasnonanalyticbehaviorat B=0. Ingravitydual,
+we argued that the Wilson loop depends on Bthrough the string worldsheet sweeping vanish-
+ing two-cycle at the orbifold fixed point. The ninstanton effect is proportional to exp (2πinB)
+forn=±1,±2,···. It shows that Bhas the periodicity over [−1
+2,+1
+2]and effect of individual
+instantonis analytic overthe period. Obviously,in order t o exhibitnon-analyticity such as |B|,
+infinitelymanyinstantoneffects needsto beresummed.
+5.3 CommentsonWilsonloopsin Higgsphase
+Startingfrom the ˆA1quivergaugetheory,wehaveanotherlimitwecan take. Consi dernowthe
+D3-branesdisplacedawayfromtheorbifoldsingularity. If allthebranesaremovedtoasmooth
+point,thenthequivergaugesymmetry Gisbroken tothediagonalsubgroup GD:
+G=U(N)×U(N)→GD=UD(N) (5.13)
+modulo center-of-mass U(1) group. Of the two bifundamental hypermultiplets, one of them is
+Higgsed away and the other forms a hypermultiplet transform ing in adjoint representation of
+the diagonal subgroup. This theory flows in the infrared belo w the Higgs scale to the N=4
+superconformal Yang-Mills theory, as expected since the ND3-branes are stacked now at a
+smoothpoint.
+We should be able to understand the two Wilson loops of the ˆA1quiver gauge theory in
+this limit. Obviously, the two Wilson loops W1,W2are independent and distinguishable at an
+energy above the Higgs scale, while they are reduced to one an d the same Wilson loop at an
+energy below the Higgs scale. Noting that Higgs scale is set b y the location of the D3-branes
+from the orbifold singularity, we therefore see that the min imal surface of the macroscopic
+35string worldsheet must exhibita crossover. How this crosso vertakes place is a very interesting
+problemleft forthefuture.
+Theaboveconsiderationisalsogeneralizableto variouspa rtialbreaking patternssuchas
+SU(2N)×SU(2N)→SU(N)×SU(N)×SUD(N). (5.14)
+Now,thereareseveraltypesofstrings. Therearestringsco rrespondingtoWilsonloopsofthree
+SU(N)’s. There are also W-bosons that connect diagonal SU( N) to either of the two SU( N)’s.
+The fields now transform as (N,N;1),(N,N;1)and(1,1,N2−1). As the theory is Higgsed,
+localization method we relied on is no longer valid. Still, N evertheless, taking holographic
+geometry of the conformal points of quiver gauge theories as the starting point, the gravity
+dual is expected to be a certain class of multi-centered defo rmations. We expect that one can
+stilllearn a lot of (quiver)gaugetheory dynamics by taking suitableapproximategravity duals
+and then computing Wilson loop expectation values and compa ring them with weak ‘t Hooft
+couplingperturbativeresults.
+6 Generalizationto ˆAk−1QuiverGaugeTheories
+So far, we were mainly concerned with A1andˆA1ofN=2 (quiver) gauge theories. These
+are the simplest two within a series of ˆAk−1type. These quiver gauge theories are obtainable
+fromD3-branessittingattheorbifoldsingularity C×(C2/Zk). Thereare (k−1)orbifoldfixed
+pointswhoseblow-upconsistsof S2
+i(i=1,···,k−1). ThetwistedsectoroftheTypeIIBstring
+theory includes (k−1)tensor multiplets of (5+1)-dimensional (2,0) chiral supersymmetry.
+Two setsof (k−1)scalarfields areassociated with
+Bi=/contintegraldisplay
+S2
+iB2
+2πandCi=/contintegraldisplay
+S2
+iC2
+2π(i=1,···,k−1). (6.1)
+Again,afterT-dualitytoTypeIIA stringtheory,weobtaint heˆAk−1braneconfiguration. Asfor
+k=2, we first partially compactify the orbifold to S1of a fixed asymptotic radius and resolve
+theˆAk−1singularities. This results in a hyperk¨ ahler space where t heS1is fibered over the
+base space R3. The manifold is known as k-centered Taub-NUT space. There are 3 (k−1)
+geometricmoduliassociatedwith (k−1)degenerationcenters(wherethe S1fiberdegenerates)
+which, along with the 2 (k−1)moduli in (6.1), constitute5 scalar fields of the aforementi oned
+(k−1)tensor multiplets. Now, T-dualizing along the S1fiber, we obtain Type IIA background
+involving kNS5-branes,whichsourcenontrivialdilatonandNS-NS H3fieldstrength,sittingat
+36the degeneration centers on the base space R3and at various positions on the T-dual circle /tildewideS1
+set bythe Bi’sin(6.1).
+In the Type IIA brane configuration, there are various limits where global symmetries are
+enhanced. Atgenericdistributionof kNS5-branesonthedualcircle /hatwideS1,theglobalsymmetryis
+givenbySU (2)×U(1)associatedwiththebasespace R3andthedualcircle /hatwideS1. When(fraction
+of)NS5-branesallcoalescetogether,thespacetransverse totheNS5-branesapproaches C2very
+close to them and the U (1)symmetry is enhanced to SU(2). In this limit, (a subset of) ga uge
+couplings of D4-branes become zero and we have global symmet ry enhancement. It is well
+known that k-stack of NS5-branes, which source the dilation and the NS-N SH3field strength,
+generate the near-horizon geometry of linear dilaton [27]. In string frame, the geometry is the
+exact conformalfield theory[28]
+R5,1×/parenleftBig
+Rφ,Q×SU(2)k/parenrightBig
+where Q=/radicalbigg
+2
+k. (6.2)
+Modulo the center of mass part, the worldvolume dynamics on D 4-branes stretched between
+various NS5-branes can be described in terms of various boun dary states [29], representing
+localized andextendedstates inthebulk.
+Thestringtheoryinthisbackgroundbreaksdownatthelocat ionofNS5-branes,asthestring
+couplingbecomesinfinitelystrong. Toregularizethegeome tryand definethestringtheory,we
+maytake Cinsidetheaforementionednear-horizon C2,splitthecoincident kNS5-branesatthe
+centerandarraythemonaconcentriccircleofanonzeroradi us. Thestringcouplingisthencut
+off at a value set by the radius. The resulting worldsheet the ory is the N=2 supersymmetric
+Liouvilletheory.
+In the regime we are interested in, ktakes values larger than 2, k=3,4,···. In this regime,
+theN=2 Liouville theory (6.2) is strongly coupled. By the supersy mmetric extension of the
+Fateev-Zamolodchikov-Zamolodchikov(FZZ) duality, we ca n turn the N=2 supersymmetric
+Liouville theory to Kazama-Suzuki coset theory. To do so, we T-dualize along the angular
+direction of the arrayed NS5-branes. Conserved winding mod es around the angular direction
+is mapped to conserved momentum modes and the resulting Type IIB background is given by
+anotherexactconformal field theory
+R5,1×/parenleftBigSL(2;R)k
+U(1)×SU(2)k
+U(1)/parenrightBig
+(6.3)
+moduloZkorbifolding. Forlarge k,theconformalfieldtheoryisweaklycoupledanddescribes
+thewell-knowncigargeometry[30].
+In the large (finite or infinite) k, what do we expect for the Wilson loop expectation value
+and,fromtheexpectationvalues,whatinformationcanweex tractfortheholographicgeometry
+37of gravity dual? Here, we shall remark several essential poi nts that are extendible straightfor-
+wardly from the results of ˆA1and relegate further aspects in a separate work. For ˆAk−1quiver
+gaugetheories,thereare knodesofgaugegroupsU( N). Associatedwiththemare kindependent
+Wilsonloops:
+W(i)[C]:=Tr(i)Psexp/bracketleftBig
+ig/integraldisplay
+Cd/parenleftBig
+˙xmA(i)
+m(x)+θaA(i)
+a(x)/parenrightBig/bracketrightBig
+(i=1,···,k).(6.4)
+From these,wecan constructtheWilsonloopin untwistedand twistedsectors. Explicitly,they
+are
+W0=1
+k/parenleftBig
+W(1)+W(2)+···+W(k−1)+W(k)/parenrightBig
+(6.5)
+fortheuntwistedsectorWilsonloopand
+W1=W(1)+ωW(2)+···+ωk−1W(k)
+W2=W(1)+ω2W(2)+···+ω2(k−1)W(k)
+···
+Wk−1=W(1)+ωk−1W(2)+···+ω(k−1)2W(k)(6.6)
+for the(k−1)independent twisted sector Wilson loops. They are simply knormal modes
+of Wilson loops constructed from {ωn|n=0,···,k−1}Fourier series of Zkover thekquiver
+nodes. Considernowtheplanarlimit N→∞. TheWilsonloops W(i)areallsame. Equivalently,
+all the twisted Wilson loops vanish. Furthermore, as in ˆA1quiver gauge theory, the untwisted
+Wilsonloopwillshowexponentialgrowthat large‘t Hooft co upling.
+It isnot difficult to extendthegaugetheory results to ˆAk−1case. Aftertaking large Nlimit,
+thesaddlepointequationsnowread
+4π2
+λφ=/integraldisplay
+−dφ′ρ(φ′)
+φ−φ′, (6.7)
+2π2
+λaφ−(1−ω)/integraldisplay
+−dφ′δaρ(φ′)F′(φ−φ′) =/integraldisplay
+−dφ′δaρ(φ′)
+φ−φ′,(a=1,···,k−1)
+(6.8)
+where
+ρ:=1
+k/parenleftBig
+ρ(1)+···+ρ(k)/parenrightBig
+δaρ:=1
+kk
+∑
+i=1ωi−1ρ(i)(a=1,2,···,k−1), (6.9)
+38and
+1
+λ:=1
+k/parenleftBig1
+λ(1)+···+1
+λ(k)/parenrightBig
+1
+λa:=1
+kk
+∑
+i=1ωi−11
+λ(i)(a=1,2,···,k−1). (6.10)
+It isevidentthat δaρisproportionalto 1 /λalinearly,and henceexhibits non-analytic behavior.
+BytheAdS/CFTcorrespondence,theWilsonloopsaremappedt omacroscopicfundamental
+TypeIIBstringinthegeometryAdS 5×S5/Zk. Thereare (k−1)2-cyclesofvanishingvolume.
+As in the ˆA1case,nworldsheet instanton picks up a phase factor exp (2πiBn). Again, since
+B=1/2 for the exact conformal field theory, the phase factor is giv en by(−)n. As (fraction
+of)thegaugecouplingsaretunedtozero,weagainseefrom(6 .8)thattwistedWilsonloopsare
+suppressedbytheworldsheetinstantoneffects. Thisisthe effect ofthescreening weexplained
+intheprevioussection,butnowextendedtothe ˆAk−1quivertheories. Thesuppression,however,
+is less significant as kbecomes large since the one-loop contribution in (6.8) is hi erarchically
+small compared to the classical contribution. We see this as a manifestation of the fact we
+recalled abovethat,at k→∞, theworldsheet conformalfield theory isweakly coupled in T ype
+IIB setupand theholographicdual geometry,thecigargeome try,becomes weaklycurved.
+It is also illuminating to understand the above Wilson loops from the viewpoint of the
+brane configuration. For the brane configuration, we start fr om the Type IIA theory on a
+compact spatial circle of circumference L. We place kNS5-branes on the circle on intervals
+La,(a=1,2,···,k)such that L1+L2+···+Lk=Land then stretch ND4-branes on each in-
+terval. The low-energy dynamics of these D4-branes is then d escribed by N=2 quivergauge
+theory of ˆAk−1type. In this setup, the W(a)Wilson loop is represented by a semi-infinite,
+macroscopic string emanating from a-th D4-brane to infinity. Since there are kdifferent states
+for identical macroscopic strings, we can also form linear c ombinations of them. There are k
+different normal modes: the untwisted Wilson loop W0is the lowest normal mode obtained by
+algebraic average of the kstrings,W1is the next lowest normal mode obtained by discrete lat-
+ticetranslation ωforadjacentstrings, ···,andtheWk−1isthehighestnormalmodeobtainedby
+discretelatticetranslation ωk−1(whichis thesameas theconfigurationwithlatticemomentum
+ωby theUnklappprocess)foradjacent strings.
+If the intervals are all equal, L1=L2=···=Lk=(L/k), then the brane configuration has
+cyclicpermutationsymmetry. Thissymmetrythenensuresth atalltwistedWilsonloopsvanish.
+If the intervals are different, (someof) the twisted Wilson loops are non-vanishing. If (fraction
+of) NS5-branes become coalescing, the geometry and the worl dvolume global symmetries get
+enhanced. We see that fundamental strings ending on the weak ly coupled D4-branes will be
+pulled to the coalescing NS5-branes. The difference from th eA1theory is that, effect of other
+39NS5-branes away from the coalescing ones becomes larger as kgets larger. This is the brane
+configuration counterpart of the suppression of twisted Wil son loop expectation value which
+wereattributedearlier totheweak curvatureofthehologra phicgeometry(6.3)inthislimit.
+7 Discussion
+In this paper, we investigated aspects of four-dimensional N=2 superconformal gauge theo-
+ries. Utilizingthe localization technique, we showed that thepath integralof these theories are
+reducedtoafinite-dimensionalmatrixintegral,muchasfor theN=4superYang-Millstheory.
+The resulting matrix model is, however, non-Gaussian. Expe ctation value of half-BPS Wilson
+loops in these theories can also be evaluated using the matri x model techniques. We studied
+two theories in detail: A1gauge theory with gauge group U (N)and 2Nfundamental hyper-
+multiplets and ˆA1quivergauge theory with gauge group U (N)×U(N)and two bi-fundamental
+hypermultiplets.
+In the planar limit, N→∞, we determined exactly the leading asymptotes of the circul ar
+Wilson loops as the ‘t Hooft coupling becomes strong, λ→∞and then compared it to the
+exponentialgrowth ∼exp(√
+λ)seeninthe N=4superYang-Millstheory. Inthe A1theory,we
+found the Wilson loop exhibits non-exponential growth: it is bounded from above in the large
+λlimit. In the ˆA1theory, there are two Wilson loops, corresponding to the two U(N)gauge
+groups. WefoundthattheuntwistedWilsonloopexhibitsexp onentialgrowth,exactlythesame
+leading behavior as the Wilson loop in N=4 super Yang-Millstheory, but the twisted Wilson
+loopexhibitsanew non-analytic behaviorindifference ofthetwogaugecouplingconstants.
+Wealsostudiedholographicdualofthese N=2theoriesandmacroscopicstringconfigura-
+tionsrepresentingtheWilsonloops. Wearguedthatboththe non-exponential behaviorofthe A1
+Wilsonloop and the non-analytic behaviorofthe ˆA1Wilson loopsare indicativeofstringscale
+geometriesofthegravitydual. Forgravitydualof A1theory,thereareinfinitelymanyvanishing
+2-cyclesaroundwhichthemacroscopicstringwrapsarounda ndproduceworldsheetinstantons.
+These different saddle-points interfere among themselves , canceling out the would-be leading
+exponentialgrowth. What remains thereafter thenyields an on-exponentialbehavior, matching
+with the exact gauge theory results. For gravity dual of ˆA1theory, there is again a vanishing
+2-cycle at the Z2orbifold singularity. On the 2-cycle, NS-NS 2-form potenti al can be turned
+on and it is set by asymmetry between the two gauge coupling co nstants. The macroscopic
+string wraps around and each worldsheet instanton is weight ed by exp (2πiB). Again, since the
+2-cycle has a vanishing area, infinite number of worldsheet i nstantons needs to be resummed.
+The resummation can then yield a non-analytic dependence on B, and this fits well with the
+40exact gaugetheoryresult.
+A key lesson drawn from the present work is that holographic d ual of these N=2 super-
+conformal gauge theories must involvegeometry of string sc ale. ForA1theory, suppression of
+exponential growth of Wilson loop expectation value hints t hat the holographic duals must be
+a noncritical string theory. In the brane construction view point, this arose because the two co-
+inciding NS5-branes generates the well-known linear dilat on background near the horizon and
+macroscopicstring is pulled to theNS5-branes. In theholog raphicdual gravity viewpoint,this
+arosebecauseworldsheetofmacroscopicstringrepresenti ngtheWilsonloopisnotpeakedtoa
+semiclassicalsaddle-pointbutisaffectedbyproliferati ngworldsheetinstantons. Wearguedthat
+delicate cancelation among the instanton sums lead to non-e xponential behavior of the Wilson
+loop.
+It should be possible to extend the analysis in this paper to g eneral N=2 superconformal
+gauge theories. Recently, various quiver constructions we re put forward [31] and some of its
+gravity duals were studied [32]. Main focus of this line of re search were on quivergeneraliza-
+tion of the Argyres-Seiberg S-duality, which does not commu te with the large Nlimit. Aim of
+the present work was to characterize behavior of the Wilson l oop in large Nlimit in terms of
+representationcontentsofmatterfieldsand,fromtheresul ts,infertheholographicgeometryof
+gravityduals. Wealsoremarkedthatourapproachiscomplem entarytotheresearchesbasedon
+variousworldsheetformulations[33][34][35][36].
+Recently, localization in the N=6 superconformal Chern-Simons theory was obtained
+and Wilson loops therein was studied in detail [37]. It shoul d also be possible to extend the
+analysis to the superconformal (quiver) Chern-Simons theo ries. In particular, given that these
+twotypesoftheoriesarerelatedroughlyspeakingbypartia llycompactifyingon S1andflowing
+intoinfrared,understandingsimilaritiesanddifference sbetweenquivergaugetheoriesin(3+1)
+dimensionsandin(2+1)dimensionswouldbeextremelyusefu lforelucidatingfurtherrelations
+ingaugeandstringdynamics.
+Finally, it should be possible to extend the analysis in this work to N=1 superconformal
+quiver gauge theories and study implications to the Seiberg duality. Candidate non-critical
+stringdualsofthesegaugetheorieswere proposedby[38].
+Wearecurrentlyinvestigatingtheseissuesbutwillrelega tereportingourfindingstofollow-
+up publications.
+41Acknowledgments
+WearegratefultoZoltanBajnok,DongsuBak,DavidGrossand JuanMaldacenaforusefuldis-
+cussionsontopicsrelatedtothisworkandcomments. SJRtha nksKavliInstituteforTheoretical
+Physics for hospitality during this work. TS thanks KEK Theo ry Group, Institute for Physics
+andMathematicsoftheUniverseandAsia-PacificCenterforT heoreticalPhysicsforhospitality
+duringthiswork. ThisworkwassupportedinpartbytheNatio nalScienceFoundationofKorea
+Grants 2005-084-C00003, 2009-008-0372, 2010-220-C00003 , EU-FP Marie Curie Research
+& Training Networks HPRN-CT-2006-035863 (2009-06318) and U.S. Department of Energy
+Grant DE-FG02-90ER40542.
+A Killingspinoron S4
+TheKillingspinorson S4aredefinedasfollows. Let ya(a=1,···,5)becoordinatesof R5. We
+embedS4intoR5bythehypersurface
+(ya+za)2=r2,za=(0,···,0,r). (A.1)
+Eachpointon S4canbemappedtoapointonafour-dimensionalhyperplane R4,y5=0,tangent
+totheNorthPolethrough
+ya=−2za+eΩ(xa+2za),eΩ=/parenleftbigg
+1+x2
+4r2/parenrightbigg−1
+, (A.2)
+wherexa=(xm,x5=0). Thisdescribes aprojectionon R4from theSouthPoleof S4. Accord-
+ingly,theinducedmetricon S4isgivenby
+ds2=hmndxmdxn
+=e2Ωδmndxmdxn. (A.3)
+Letθbe the polar angle measured from the North Pole, viz. the orig in of theR4. Then, for a
+fixedθ,thecoordinates xmsatisfy
+4
+∑
+m=1(xm)2=4r2tan2θ
+2. (A.4)
+Wealso denoteorthonormalframecoordinatesas xˆm,(ˆm=ˆ1,···,ˆ4)withvierbein eˆm
+m=δˆm
+meΩ.
+42It isstraightforward toshowthatthespinors
+ξ=e1
+2Ω(ξs+xˆmΓˆmξc), (A.5)
+/tildewideξ=e1
+2Ω(ξc−1
+4r2xˆmΓˆmξs), (A.6)
+whereξsandξcare arbitrary constant Majorana-Weyl spinors, satisfy the conformal Killing
+spinorequations
+∇mξ=Γm/tildewideξ,∇m/tildewideξ=−1
+4r2Γmξ. (A.7)
+We furtherimposeanti-chiralitycondition:
+Γˆ1ˆ2ˆ3ˆ4ξs=−ξs,ξc=1
+2rΓ0ˆ1ˆ2ξs. (A.8)
+Theseequationsimply
+ξ/tildewideξ=0,ξΓ05/tildewideξ=0. (A.9)
+Onecan showthatthecomponentsof vM=ξΓMξhavethefollowingexplicitforms:
+v1=x2
+r,v2=−x1
+r, (A.10)
+v3=x4
+r,v4=−x3
+r, (A.11)
+v0=−1,v5=cosθ, (A.12)
+v6,7,8,9=0, (A.13)
+wherewenormalized ξssuchthat ξsΓ0ξs=−1.
+Theexpression(A.5) can berewrittenas follows:
+ξ=e1
+2Ωξs+1
+2e−1
+2ΩvˆmΓˆmΓ5ξs. (A.14)
+Wedefine
+nˆm:=vˆm
+sinθ(A.15)
+sothat
+(nˆmΓˆmΓ5)2=−1. (A.16)
+Then, itiseasy to showthat theconformal Killingspinorise xpressibleas
+ξ(x) =/parenleftbigg
+cosθ
+2+sinθ
+2nˆm(x)ΓˆmΓ5/parenrightbigg
+ξs
+=exp/parenleftbiggθ
+2nˆm(x)ΓˆmΓ5/parenrightbigg
+ξs. (A.17)
+43Theconformal Killingspinors ξand/tildewideξsatisfythefollowingidentities:
+vm∇mξ−1
+2(ξΓmn/tildewideξ)Γmnξ+1
+2(ξΓst/tildewideξ)Γstξ=0, (A.18)
+vm∇m/tildewideξ−1
+2(ξΓmn/tildewideξ)Γmn/tildewideξ+1
+2(ξΓst/tildewideξ)Γst/tildewideξ=0. (A.19)
+B Spinorsfor off-shellclosure
+Wedefine
+ν˙m
+0:=Γ˙mΓˆ1ξs,νs
+0:=ΓsΓˆ1ξs, (B.1)
+where ˙m=ˆ2,ˆ3,ˆ4. LetI=(˙m,s). Itcan beshownthat
+ξsΓMνI
+0=0, (B.2)
+νI
+0ΓMνJ
+0=δIJξsΓMξs, (B.3)
+1
+2vM
+sΓM=ξsξs+νI
+0ν0I (B.4)
+hold,where vM
+s=ξsΓMξs. Sinceξisobtainedfrom ξsthrougharotation,ifwe define
+νI:=exp/parenleftBigθ
+2nˆmΓ5Γˆm/parenrightBig
+νI
+0, (B.5)
+thenthefollowingrelationsfollow:
+ξΓMνI=0, (B.6)
+νIΓMνJ=δIJξΓMξ, (B.7)
+1
+2vMΓM=ξξ+νIνI (B.8)
+Ifthelastequationis projectedontothespaceof λ,onefinds
+1
+2vMΓM=ξξ+ν˙mν˙m, (B.9)
+whilein thespace of ψ, itbecomes
+1
+2vMΓM=νανα. (B.10)
+44Thespinorssatisfythefollowingidentities:
+vm∇mν˙k−1
+2(ξΓmn/tildewideξ)Γmnν˙k+1
+2(ξΓst/tildewideξ)Γstν˙k+(ν˙kΓm∇mν˙n)ν˙n=0,(B.11)
+vm∇mνα−1
+2(ξΓmn/tildewideξ)Γmnνα−νβνβΓm∇mνα=0.(B.12)
+Dueto theabovechoiceofspinors, Q2closes onfields as follows:
+−iQ2Am=vn∇nAm+∇mvnAn−ig[vµAµ,Am]−∇m(vµAµ), (B.13)
+−iQ2Aa=vm∇mAa−ig[vµAµ,Aa], (B.14)
+−iQ2qα=vm∇mqα−ig(vµAA
+µ)TAqα+2ξγα
+β/tildewideξqβ, (B.15)
+−iQ2qα=vm∇mqα+ig(vµAA
+µ)qαTA−2qβξγβα/tildewideξ, (B.16)
+−iQ2λ=vm∇mλ−1
+2(ξΓmn/tildewideξ)Γmnλ−ig[vµAµ,λ]+1
+2(ξΓst/tildewideξ)Γstλ,(B.17)
+−iQ2ψ=vm∇mψ−1
+2(ξΓmn/tildewideξ)Γmnψ−ig(vµAA
+µ)TAψ, (B.18)
+−iQ2ψ=vm∇mψ+1
+2(ξΓmn/tildewideξ)ψΓmn+ig(vµAA
+µ)ψTA, (B.19)
+−iQ2K˙m=vk∇kK˙m−ig[vµAµ,K˙m]+ν˙mΓk∇kν˙nK˙n, (B.20)
+−iQ2Kα=vm∇mKα−ig(vµAA
+µ)TAKα+ναΓm∇mνβKβ, (B.21)
+−iQ2Kα=vm∇mKα+ig(vµAA
+µ)KαTA−KβνβΓm∇mνα. (B.22)
+C AsymptoticexpansionofWilson loop
+Inthisappendix,weprovidedetailsoftheasymptoticexpan sionoftheWilsonloopinthelarge
+alimit.
+We firstestimatethefollowingintegral:
+I(α,a):=/integraldisplay∞
+δdu uαe−au, (C.1)
+wherea,α,δ>0. Thissatisfiestherelation
+I(α,a)=δα
+ae−δa+α
+aI(α−1,a). (C.2)
+45There exists an integer Kfor which α−K+1>0 andα−K<0. Then, repeating integration
+by parts,I(α,a)can bewrittenas
+I(α,a)=K−1
+∑
+n=0δα−n
+an+1Γ(α+1)
+Γ(α+1−n)e−δa+1
+aKΓ(α+1)
+Γ(α+1−K)I(α−K,a).(C.3)
+I(α−K,a)isestimatedas follows:
+I(α−K,a)≤δα−K/integraldisplay∞
+δdue−au=δα−K
+ae−δa. (C.4)
+Therefore, forlarge a,I(α,a)is estimatedto be
+I(α,a)=O(a−1e−δa). (C.5)
+With the above result, we now estimate W. With the assumed behavior of rescaled density
+function/tildewideρinsection3, onecan write e−caWas
+/integraldisplay1−a
+b
+0du/tildewideρ(1−u)e−cau=β/integraldisplayδ
+0duuαe−cau+/integraldisplayδ
+0duχ(u)e−cau+/integraldisplay1−a
+b
+δdu/tildewideρ(1−u)e−cau.(C.6)
+Thefirst termoftheright-handsideis
+β/integraldisplayδ
+0duuαe−cau=β/integraldisplay∞
+0du uαe−cau−βI(α,ca)
+=βΓ(α+1)(ca)−α−1+O((ca)−1e−δca). (C.7)
+The second term can be evaluated similarly, and it turns out t o be negligible compared to the
+first term. Thethirdterm is
+/integraldisplay1−a
+b
+δdu/tildewideρ(1−u)e−cau≤e−δca/integraldisplay1−a
+b
+δdu/tildewideρ(u−1)≤e−δca. (C.8)
+Thiscompletestheproofoftheproclaimedestimate(3.7)in thelargealimit.
+D Coefficient c1
+In this appendix, we elaborate detailed calculation of the c oefficient c1of the leading term in
+theone-loopdeterminant. Theheat-kernel coefficient a2(Δ)is
+a2(Δ)=1
+(4π)2/integraldisplay
+S4d4x√
+htrB/bracketleftBig
+−1
+4r2(3+cos2θ)+1
+6R/bracketrightBig
+, (D.1)
+46where tr Bis the trace over the indices α,β. The second term is canceled by the fermionic
+contribution. Thefirst termyields −5
+12r2.
+Thecoefficient a2(ΔF)forthefermionsis
+a2(ΔF)=1
+(4π)2/integraldisplay
+S4d4x√
+htrF/bracketleftBig3κ2
+r3+κ2
+4(ξΓµν/tildewideξ)(ξΓρσ/tildewideξ)ΓµνΓρσ+1
+6R/bracketrightBig
+,(D.2)
+where tr Fis the trace over the subspace of the spinor corresponding to ψ. One can show that
+thefirst twotermscancel each other.
+As the−ΔFhas the term linear in m,a4(ΔF)also contribute to c1. The relevant part of the
+coefficient a4(ΔF)is
+1
+(4π)2/integraldisplay
+S4d4x√
+htrF/bracketleftBig1
+2/parenleftBig
+iκm
+r(ξΓµν/tildewideξ)Γµν/parenrightBig2/bracketrightBig
+=−2
+3m2. (D.3)
+As aresult,it followsthat
+c1=−/parenleftBig
+−5
+12/parenrightBig
+−1
+2/parenleftBig
+−2
+3/parenrightBig
+=3
+4. (D.4)
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+arXiv:1001.0018v2  [quant-ph]  28 Jan 2010Nonadaptive quantum query complexity
+Ashley Montanaro∗
+October 1, 2018
+Abstract
+We studythe powerofnonadaptivequantum queryalgorithms,whic h arealgorithms
+whose queries to the input do not depend on the result of previous q ueries. First, we
+show that any bounded-error nonadaptive quantum query algorit hm that computes
+some total boolean function depending on nvariables must make Ω( n) queries to the
+input in total. Second, we show that, if there exists a quantum algor ithm that uses k
+nonadaptive oracle queries to learn which one of a set of mboolean functions it has
+been given, there exists a nonadaptive classical algorithm using O(klogm) queries to
+solve the same problem. Thus, in the nonadaptive setting, quantum algorithms can
+achieve at most a very limited speed-up over classical query algorith ms.
+1 Introduction
+Many of the best-known results showing that quantum compute rs outperform their classical
+counterparts are proven in the query complexity model. This model studies the number of
+queries to the input xwhich are required to compute some function f(x). In this work, we
+study two broad classes of problem that fit into this model.
+In the first class of problems, computational problems, one wishes to compute some
+boolean function f(x1,...,x n) using a small number of queries to the bits of the input
+x∈ {0,1}n. The query complexity of fis the minimum number of queries required for any
+algorithm to compute f, with some requirement on the success probability. The dete rmin-
+istic query complexity of f,D(f), is the minimum number of queries that a deterministic
+classical algorithm requires to compute fwith certainty. D(f) is also known as the decision
+tree complexity of f. Similarly, the randomised query complexity R2(f) is the minimum
+number of queries required for a randomised classical algor ithm to compute fwith success
+probability at least 2 /3. The choice of 2 /3 is arbitrary; any constant strictly between 1 /2
+and 1 would give the same complexity, up to constant factors.
+There is a natural generalisation of the query complexity mo del to quantum computa-
+tion, which gives rise to the exact and bounded-error quantu m query complexities QE(f),
+Q2(f) (respectively). In this generalisation, the quantum algo rithm is given access to the
+∗Department of Computer Science, University of Bristol, Woo dland Road, Bristol, BS8 1UB, UK;
+montanar@cs.bris.ac.uk .
+1inputxthrough a unitary oracle operator Ox. Many of the best-known quantum speed-ups
+can be understood in the query complexity model. Indeed, it i s known that, for certain
+partial functions f(i.e. functions where there is a promise on the input), Q2(f) may be ex-
+ponentially smaller than R2(f)[14]. However, if fis atotal function, D(f) =O(Q2(f)6) [4].
+See [6, 10] for good reviews of quantum and classical query co mplexity.
+In the second class of problems, learning problems, one is given as an oracle an unknown
+functionf?(x1,...,x n), which is picked from a known set Cofmboolean functions f:
+{0,1}n→ {0,1}. These functions can be identified with n-bit strings or subsets of [ n], the
+integers between 1 and n. The goal is to determine which of the functions in Cthe oraclef?
+is, with some requirement on the success probability, using the minimum number of queries
+tof?. Note that the success probability required should be stric tly greater than 1 /2 for
+this model to make sense.
+Borrowing terminology fromthe machinelearning literatur e, each function in Cis known
+as aconcept, andCis known as a concept class [13]. We say that an algorithm that can
+identify any f∈ Cwith worst-case success probability plearnsCwith success probability
+p. This problem is known classically as exact learning from me mbership queries [3, 13],
+and also in the literature on quantum computation as the orac le identification problem [2].
+Many interesting results in quantum algorithmics fit into th is framework, a straightforward
+example being Grover’s quantum search algorithm [9]. It has been shown by Servedio and
+Gortler that the speed-up that may be obtained by quantum que ry algorithms in this model
+is at most polynomial [13].
+1.1 Nonadaptive query algorithms
+This paper considers query algorithms of a highly restricti ve form, where oracle queries are
+not allowed to depend on previous queries. In other words, th e queries must all be made
+at the start of the algorithm. We call such algorithms nonadaptive , but one could also call
+themparallel, in contrast to the usual serial model of query complexity, w here one query
+follows another. It is easy to see that, classically, a deter ministic nonadaptive algorithm
+that computes a function f:{0,1}n→ {0,1}which depends on all ninput variables must
+query allnvariables (x1,...,x n). Indeed, for any 1 ≤i≤n, consider an input xfor which
+f(x) = 0, butf(x⊕ei) = 1, where eiis the bit string which has a 1 at position i, and is 0
+elsewhere. Then, if the i’th variable were not queried, changing the input from xtox⊕ei
+would change the output of the function, but the algorithm wo uld not notice.
+In the case of learning, the exact number of queries required by a nonadaptive determin-
+istic classical algorithm to learn any concept class Ccan also be calculated. Identify each
+concept in Cwith ann-bit string, and imagine an algorithm Athat queries some subset
+S⊆[n] of the input bits. If there are two or more concepts in Cthat do not differ on any of
+the bits inS, thenAcannot distinguish between these two concepts, and so canno t succeed
+with certainty. On the other hand, if every concept x∈ Cis unique when restricted to S,
+thenxcan be identified exactly by A. Thus the number of queries required is the minimum
+size of a subset S⊆[n] such that every pair of concepts in Cdiffers on at least one bit in S.
+We will be concerned with the speed-up over classical query a lgorithms that can be
+2achieved by nonadaptive quantum query algorithms. Interes tingly, it is known that speed-
+ups can indeed be found in this model. In the case of computing partial functions, the
+speed-up can be dramatic; Simon’s algorithm for the hidden s ubgroup problem over Zn
+2, for
+example, is nonadaptive and gives an exponential speed-up o ver the best possible classical
+algorithm [14]. Thereare also known speed-upsfor computin g total functions. For example,
+the parity of nbits can be computed exactly using only ⌈n/2⌉nonadaptive quantum queries
+[8]. More generally, anyfunction of nbits can be computed with bounded error using only
+n/2+O(√n)nonadaptivequeries, byaremarkablealgorithmofvanDam[ 7]. Thisalgorithm
+in fact retrieves allthe bits of the input xsuccessfully with constant probability, so can also
+be seen as an algorithm that learns the concept class consist ing of all boolean functions on
+nbits usingn/2+O(√n) nonadaptive queries.
+Finally, one of the earliest results in quantum computation can be understood as a
+nonadaptive learning algorithm. The quantum algorithm sol ving the Bernstein-Vazirani
+parity problem [5] uses one query to learn a concept class of s ize 2n, for which any classical
+learning algorithm requires nqueries, showing that there can be an asymptotic quantum-
+classical separation for learning problems.
+1.2 New results
+We show here that these results are essentially the best poss ible. First, any nonadap-
+tive quantum query algorithm that computes a total boolean f unction with a constant
+probability of success greater than 1 /2 can only obtain a constant factor reduction in the
+number of queries used. In particular, if we restrict to nona daptive query algorithms, then
+Q2(f) = Θ(D(f)). In the case of exact nonadaptive algorithms, we show that the factor of
+2 speed-up obtained for computing parity is tight. More form ally, our result is the following
+theorem.
+Theorem 1. Letf:{0,1}n→ {0,1}be a total function that depends on all nvariables,
+and letAbe a nonadaptive quantum query algorithm that uses kqueries to the input to
+computef, and succeeds with probability at least 1−ǫon every input. Then
+k≥n
+2/parenleftBig
+1−2/radicalbig
+ǫ(1−ǫ)/parenrightBig
+.
+In the case of learning, we show that the speed-up obtained by the Bernstein-Vazirani
+algorithm [5] is asymptotically tight. That is, the query co mplexities of quantum and
+classical nonadaptive learning are equivalent, up to a loga rithmic term. This is formalised
+as the following theorem.
+Theorem 2. LetCbe a concept class containing mconcepts, and let Abe a nonadaptive
+quantum query algorithm that uses kqueries to the input to learn C, and succeeds with
+probability at least 1−ǫon every input, for some ǫ <1/2. Then there exists a classical
+nonadaptive query algorithm that learns Cwith certainty using at most
+4klog2m
+1−2/radicalbig
+ǫ(1−ǫ)
+queries to the input.
+31.3 Related work
+We note that the question of putting lower bounds on nonadapt ive quantum query algo-
+rithms has been studied previously. First, Zalka has obtain ed a tight lower bound on the
+nonadaptive quantum query complexity of the unordered sear ch problem, which is a par-
+ticular learning problem [15]. Second, in [12], Nishimura a nd Yamakami give lower bounds
+on the nonadaptive quantum query complexity of a multiple-b lock variant of the ordered
+search problem. Finally, Koiran et al [11] develop the weigh ted adversary argument of Am-
+bainis [1] to obtain lower bounds that are specific to the nona daptive setting. Unlike the
+situation considered here, their bounds also apply to quant um algorithms for computing
+partial functions.
+We now turn to proving the new results: nonadaptive computat ion in Section 2, and
+nonadaptive learning in Section 3.
+2 Nonadaptive quantum query complexity of computation
+LetAbe a nonadaptive quantum query algorithm. We will use what is essentially the
+standard model of quantum query complexity [10]. Ais given access to the input x=
+x1...xnvia an oracle Oxwhich acts on an n+1 dimensional space indexed by basis states
+|0/an}brack⌉tri}ht,...,|n/an}brack⌉tri}ht, and performs the operation Ox|i/an}brack⌉tri}ht= (−1)xi|i/an}brack⌉tri}ht. We define Ox|0/an}brack⌉tri}ht=|0/an}brack⌉tri}htfor
+technical reasons (otherwise, Acould not distinguish between xand ¯x). Assume that A
+makeskqueries toOx. As the queries are nonadaptive, we may assume they are made i n
+parallel. Therefore, the existence of a nonadaptive quantu m query algorithm that computes
+fand fails with probability ǫis equivalent to the existence of an input state |ψ/an}brack⌉tri}htand a
+measurement specified by positive operators {M0,I−M0}, such that /an}brack⌉tl⌉{tψ|O⊗k
+xM0O⊗k
+x|ψ/an}brack⌉tri}ht ≥
+1−ǫfor all inputs xwheref(x) = 0, and /an}brack⌉tl⌉{tψ|O⊗k
+xM0O⊗k
+x|ψ/an}brack⌉tri}ht ≤ǫfor all inputs xwhere
+f(x) = 1.
+The intuition behind the proof of Theorem 1 is much the same as that behind “adver-
+sary” arguments lower bounding quantum query complexity [1 0]. As in Section 1.1, let ej
+denote the n-bit string which contains a single 1, at position j. In order to distinguish two
+inputsx,x⊕ejwheref(x)/n⌉}ationslash=f(x⊕ej), the algorithm must invest amplitude of |ψ/an}brack⌉tri}htin
+components where the oracle gives information about j. But, unless kis large, it is not
+possible to invest in many variables simultaneously.
+We will use the following well-known fact from [5].
+Fact 3(Bernstein and Vazirani [5]) .Imagine there exists a positive operator M≤Iand
+states|ψ1/an}brack⌉tri}ht,|ψ2/an}brack⌉tri}htsuch that /an}brack⌉tl⌉{tψ1|M|ψ1/an}brack⌉tri}ht ≤ǫ, but/an}brack⌉tl⌉{tψ2|M|ψ2/an}brack⌉tri}ht ≥1−ǫ. Then |/an}brack⌉tl⌉{tψ1|ψ2/an}brack⌉tri}ht|2≤
+4ǫ(1−ǫ).
+We now turn to the proof itself. Write the input state |ψ/an}brack⌉tri}htas
+|ψ/an}brack⌉tri}ht=/summationdisplay
+i1,...,ikαi1,...,ik|i1,...,ik/an}brack⌉tri}ht,
+4where, for each m, 0≤im≤n. It is straightforward to compute that
+O⊗k
+x|i1,...,ik/an}brack⌉tri}ht= (−1)xi1+···+xik|i1,...,ik/an}brack⌉tri}ht.
+Asfdepends on all ninputs, for any j, there exists a bit string xjsuch thatf(xj)/n⌉}ationslash=
+f(xj⊕ej). Then
+(OxjOxj⊕ej)⊗k|i1,...,ik/an}brack⌉tri}ht= (−1)|{m:im=j}||i1,...,ik/an}brack⌉tri}ht;
+in other words ( OxjOxj⊕ej)⊗knegates those basis states that correspond to bit strings
+i1,...,ikwherejoccurs an odd number of times in the string. Therefore, we hav e
+|/an}brack⌉tl⌉{tψ|(OxjOxj⊕ej)⊗k|ψ/an}brack⌉tri}ht|2=
+/summationdisplay
+i1,...,ik|αi1,...,ik|2(−1)|{m:im=j}|
+2
+=
+1−2/summationdisplay
+i1,...,ik|αi1,...,ik|2[|{m:im=j}|odd]
+2
+=: (1−2Wj)2.
+Now, by Fact 3, (1 −2Wj)2≤4ǫ(1−ǫ) for allj, so
+Wj≥1
+2/parenleftBig
+1−2/radicalbig
+ǫ(1−ǫ)/parenrightBig
+.
+On the other hand,
+n/summationdisplay
+j=1Wj=n/summationdisplay
+j=1/summationdisplay
+i1,...,ik|αi1,...,ik|2[|{m:im=j}|odd]
+=/summationdisplay
+i1,...,ik|αi1,...,ik|2n/summationdisplay
+j=1[|{m:im=j}|odd]
+≤/summationdisplay
+i1,...,ik|αi1,...,ik|2k=k.
+Combining these two inequalities, we have
+k≥n
+2/parenleftBig
+1−2/radicalbig
+ǫ(1−ǫ)/parenrightBig
+.
+3 Nonadaptive quantum query complexity of learning
+In the case of learning, we use a very similar model to the prev ious section. Let Abe a
+nonadaptivequantumqueryalgorithm. Aisgiven access toanoracle Ox, whichcorresponds
+toabit-string xpickedfromaconcept class C.Oxactsonann+1dimensionalspaceindexed
+by basis states |0/an}brack⌉tri}ht,...,|n/an}brack⌉tri}ht, and performs the operation Ox|i/an}brack⌉tri}ht= (−1)xi|i/an}brack⌉tri}ht, withOx|0/an}brack⌉tri}ht=|0/an}brack⌉tri}ht.
+5Assume that Amakeskqueries toOxand outputs xwith probability strictly greater than
+1/2 for allx∈ C.
+We will prove limitations on nonadaptive quantum algorithm s in this model as follows.
+First, we show that a nonadaptive quantum query algorithm th at useskqueries to learn C
+is equivalent to an algorithm using one query to learn a relat ed concept class C′. We then
+show that existence of a quantum algorithm using one query th at learns C′with constant
+success probability greater than 1 /2 implies existence of a deterministic classical algorithm
+usingO(log|C′|) queries. Combining these two results gives Theorem 2.
+Lemma 4. LetCbe a concept class over n-bit strings, and let C⊗kbe the concept class
+defined by
+C⊗k={x⊗k:x∈ C},
+wherex⊗kdenotes the (n+ 1)k-bit string indexed by 0≤i1,...,ik≤n, withx⊗k
+i1,...,ik=
+xi1⊕ ··· ⊕xik, and we define x0= 0. Then, if there exists a classical nonadaptive query
+algorithm that learns C⊗kwith success probability pand usesqqueries, there exists a classical
+nonadaptive query algorithm that learns Cwith success probability pand uses at most kq
+queries.
+Proof.Given access to x, an algorithm Acan simulate a query of index ( x1,...,x k) ofx⊗k
+by using at most kqueries to compute x1⊕··· ⊕xk. Hence, by simulating the algorithm
+for learning C⊗k,Acan learn C⊗kwith success probability pusing at most kqnonadaptive
+queries. Learning C⊗ksuffices to learn C, because each concept in C⊗kuniquely corresponds
+to a concept in C(to see this, note that the first nbits ofx⊗kare equal to x).
+Lemma 5. LetCbe a concept class containing mconcepts. Assume that Ccan be learned
+using one quantum query by an algorithm that fails with proba bility at most ǫ, for some
+ǫ<1/2. Then there exists a classical algorithm that uses at most (4log2m)/(1−2/radicalbig
+ǫ(1−ǫ))
+queries and learns Cwith certainty.
+Proof.Associate each concept with an n-bit string, for some n, and suppose there exists a
+quantum algorithm that uses one query to learn Cand fails with probability ǫ<1/2. Then
+by Fact 3 there exists an input state |ψ/an}brack⌉tri}ht=/summationtextn
+i=0αi|i/an}brack⌉tri}htsuch that, for all x/n⌉}ationslash=y∈ C,
+|/an}brack⌉tl⌉{tψ|OxOy|ψ/an}brack⌉tri}ht|2≤4ǫ(1−ǫ),
+or in other words/parenleftBiggn/summationdisplay
+i=0|αi|2(−1)xi+yi/parenrightBigg2
+≤4ǫ(1−ǫ). (1)
+We now show that, if this constraint holds, there must exist a subset of the inputs S⊆[n]
+such that every pair of concepts in Cdiffers on at least one input in S, and|S|=O(logm).
+By the argument of Section 1.1, this implies that there is a no nadaptive classical algorithm
+that learns Mwith certainty using O(logm) queries.
+We will use the probabilistic method to show the existence of S. For anyk, form a
+subsetSof at most kinputs between 1 and nby a process of krandom, independent
+6choices of input, where at each stage input iis picked to add to Swith probability |αi|2.
+Now consider an arbitrary pair of concepts x/n⌉}ationslash=y, and letS+,S−be the set of inputs on
+which the concepts are equal and differ, respectively. By the c onstraint (1), we have
+4ǫ(1−ǫ)≥/parenleftBiggn/summationdisplay
+i=0|αi|2(−1)xi+yi/parenrightBigg2
+=
+/summationdisplay
+i∈S+|αi|2−/summationdisplay
+i∈S−|αi|2
+2
+=
+1−2/summationdisplay
+i∈S−|αi|2
+2
+,
+so/summationdisplay
+i∈S−|αi|2≥1
+2−/radicalbig
+ǫ(1−ǫ).
+Therefore, at each stage of adding an input to S, the probability that an input in S−is
+added is at least1
+2−/radicalbig
+ǫ(1−ǫ). So, after kstages of doing so, the probability that none
+of these inputs has been added is at most/parenleftBig
+1
+2+/radicalbig
+ǫ(1−ǫ)/parenrightBigk
+. As there are/parenleftbigm
+2/parenrightbig
+pairs of
+conceptsx/n⌉}ationslash=y, by a union bound the probability that none of the pairs of con cepts differs
+on any of the inputs in Sis upper bounded by
+/parenleftbiggm
+2/parenrightbigg/parenleftbigg1
+2+/radicalbig
+ǫ(1−ǫ)/parenrightbiggk
+≤m2/parenleftbigg1
+2+/radicalbig
+ǫ(1−ǫ)/parenrightbiggk
+.
+For anykgreater than
+2log2m
+log22/(1+2/radicalbig
+ǫ(1−ǫ))<4log2m
+1−2/radicalbig
+ǫ(1−ǫ)
+this probability is strictly less than 1, implying that ther e exists some choice of S⊆[n]
+with|S| ≤ksuch that every pair of concepts differs on at least one of the in puts inS. This
+completes the proof.
+We are finally ready to prove Theorem 2, which we restate for cl arity.
+Theorem. LetCbe a concept class containing mconcepts, and let Abe a nonadaptive
+quantum query algorithm that uses kqueries to the input to learn C, and succeeds with
+probability at least 1−ǫon every input, for some ǫ <1/2. Then there exists a classical
+nonadaptive query algorithm that learns Cwith certainty using at most
+4klog2m
+1−2/radicalbig
+ǫ(1−ǫ)
+queries to the input.
+Proof.LetOxbe the oracle operator corresponding to the concept x. Then a nonadaptive
+quantum algorithm Athat learns xusingkqueries to Oxis equivalent to a quantum
+algorithm that uses one query to O⊗k
+xto learnx. It is easy to see that this is equivalent to
+Ain fact using one query to learn the concept class C⊗k. By Lemma 5, this implies that
+there exists a classical algorithm that uses at most (4 klog2m)/(1−2/radicalbig
+ǫ(1−ǫ)) queries
+to learn C⊗kwith certainty. Finally, by Lemma 4, this implies in turn tha t there exists a
+classical algorithm that uses the same number of queries and learnsCwith certainty.
+7Acknowledgements
+I would like to thank Aram Harrow and Dan Shepherdfor helpful discussions and comments
+on a previous version. This work was supported by the EC-FP6- STREP network QICS and
+an EPSRC Postdoctoral Research Fellowship.
+References
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+9
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+arXiv:1001.0019v1  [gr-qc]  30 Dec 2009On the instability of Reissner-Nordstr¨ om black holes in de Sitter backgrounds
+Vitor Cardoso∗
+CENTRA, Departamento de F´ ısica, Instituto Superior T´ ecn ico,
+Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal &
+Department of Physics and Astronomy, The University of Miss issippi, University, MS 38677-1848, USA
+Madalena Lemos†and Miguel Marques‡
+CENTRA, Departamento de F´ ısica, Instituto Superior T´ ecn ico,
+Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
+(Dated: November 3, 2018)
+Recent numerical investigations have uncovered a surprisi ng result: Reissner-Nordstr¨ om-de Sitter
+black holes are unstable for spacetime dimensions larger th an 6. Here we prove the existence of
+such instability analytically, and we compute the timescal e in the near-extremal limit. We find very
+good agreement with the previous numerical results. Our res ults may me helpful in shedding some
+light on the nature of the instability.
+PACS numbers: 04.50.Gh,04.70.-s
+I. INTRODUCTION
+In physics, stability of a given configuration (solution
+of some set of equations), is a useful criterium for rele-
+vance of that solution. Unstable configurations are likely
+not tobe realizablein practice, and representaninterme-
+diate stage in the evolution of the system. Nevertheless,
+the instability itself is of great interest, since an under-
+standing of the mechanism behind it may help one to
+better grasp the physics involved. In particular, it is of
+interest to be able to predict which other systems display
+similar instabilities, or even have a deeper understanding
+of the physics behind the instability (why is the system
+unstable? is there some fundamental principle behind
+the instability?).
+In General Relativity, the Kerr family exhausts the
+blackhole solutionsto the electro-vacEinstein equations.
+Kerr black holes are stable, and can therefore describe
+astrophysicalobjects. However,there aremanyinstances
+of instabilities afflicting objects with an event horizon,
+such as the Gregory-Laflamme [1], the ultra-spinning [2]
+or superradiant instabilities [3] and other instabilities of
+higher-dimensional black holes in alternative theories [4,
+5](for a review see Ref. [6]).
+Konoplya and Zhidenko (hereafter KZ) recently stud-
+ied small perturbations in the vicinity of a charged black
+hole in de Sitter background, a Reissner-Nordstr¨ om de
+Sitter black hole (RNdS) [7]. Their (numerical) results
+show that when the spacetime dimensionality D >6, the
+spacetime is unstable, provided the charge is larger than
+agiventhreshold, determined byKZforeach D. Because
+∗Electronic address: vitor.cardoso@ist.utl.pt
+†Electronic address: madalena.dal@gmail.com
+‡Electronic address: miguel.e.marques@gmail.comthe results are so surprising (the mechanism behind it is
+not yet understood), we set out to to investigate this in-
+stability and hopefully understand it better. Our results
+can be summarized as follows: (i) we can prove analyti-
+cally the existence of unstable modes for charge Qhigher
+thanacertainthreshold. (ii)inthenear-extremalregime,
+we are able to find an explicit solution for the unstable
+modes, determining the instability timescale analytically.
+We hope that our incursion in this topic helps to better
+understand the physics at work.
+II. EQUATIONS
+This work focuses on the higher dimensional RNdS ge-
+ometry, described by the line element
+ds2=−f dt2+f−1dr2+r2dΩ2
+n, (1)
+wheredΩ2
+nis the line element of the nsphere and
+f= 1−λr2−2M
+rn−1+Q2
+r2n−2. (2)
+the background electric field is E0=q/rn, withqthe
+electric charge. The quantities MandQare related to
+the physical mass M and charge qof the black hole [8],
+andλto the cosmological constant. The spacetime di-
+mensionality is D=n+2.
+The above geometry possesses three horizons: the
+black-holeCauchyhorizonat r=ra, the black hole event
+horizon is at r=rband the cosmological horizon is at
+r=rc, whererc> rb> ra, the only real, positive zeroes
+off. For convenience, we set rb= 1, i.e., we measure all
+quantities in terms of the event horizon rb. We thus get
+2M= 1+Q2−λ, (3)2
+Furthermore, we can also write
+λ=r−4−n
+c(rn+2
+c−r3
+c)(rn+2
+c−Q2r3
+c)
+rn+2c−rc.(4)
+For a fixed rcand spacetime dimension D, the existence
+ofaregulareventhorizonimposesthatthecharge Qmust
+be smaller than a certain value Qext. With our units this
+maximum charge is
+Q2
+ext=rn
+c/parenleftbig
+−2rc+(n+1)rn
+c−(n−1)rn+2
+c/parenrightbig
+−rc/parenleftbig
+rc(n+1)−2nrnc+(n−1)r2n+1c/parenrightbig.(5)
+Gravitational perturbations of this spacetime couple to
+the electromagnetic field, and were completely character-
+ized by Kodama and Ishibashi [8]. They can be reduced
+to a set of two second order ordinary differential equa-
+tions of the form,
+d2
+dr2∗Φ±+/parenleftbig
+ω2−VS±/parenrightbig
+Φ±= 0, (6)where the tortoise coordinate r∗and the potentials VS±
+are defined through
+r∗≡/integraldisplay
+f−1dr, V S±=fU±
+64r2H2
+±.(7)
+We have
+H+= 1−n(n+1)
+2δx, (8)
+H−=m+n(n+1)
+2(1+mδ)x, (9)
+and the quantities U±are given by
+U+=/bracketleftbig
+−4n3(n+2)(n+1)2δ2x2−48n2(n+1)(n−2)δx
+−16(n−2)(n−4)]y−δ3n3(3n−2)(n+1)4(1+mδ)x4
++4δ2n2(n+1)2/braceleftbig
+(n+1)(3n−2)mδ+4n2+n−2/bracerightbig
+x3
++4δ(n+1)/braceleftbig
+(n−2)(n−4)(n+1)(m+n2K)δ−7n3+7n2−14n+8/bracerightbig
+x2
++/braceleftbig
+16(n+1)/parenleftbig
+−4m+3n2(n−2)K/parenrightbig
+δ−16(3n−2)(n−2)/bracerightbig
+x
++64m+16n(n+2)K, (10)
+U−=/bracketleftbig
+−4n3(n+2)(n+1)2(1+mδ)2x2+48n2(n+1)(n−2)m(1+mδ)x
+−16(n−2)(n−4)m2/bracketrightbig
+y−n3(3n−2)(n+1)4δ(1+mδ)3x4
+−4n2(n+1)2(1+mδ)2/braceleftbig
+(n+1)(3n−2)mδ−n2/bracerightbig
+x3
++4(n+1)(1+ mδ)/braceleftbig
+m(n−2)(n−4)(n+1)(m+n2K)δ
++4n(2n2−3n+4)m+n2(n−2)(n−4)(n+1)K/bracerightbig
+x2
+−16m/braceleftbig
+(n+1)m/parenleftbig
+−4m+3n2(n−2)K/parenrightbig
+δ
++3n(n−4)m+3n2(n+1)(n−2)K/bracerightbig
+x
++64m3+16n(n+2)m2K. (11)
+The variables x,yand parameters µ,mare defined
+through
+x≡2M
+rn−1, y≡λr2, (12)
+µ2≡M2+4mQ2
+(n+1)2, m≡k2−nK,(13)
+andthe quantity δis implicitly givenby µ= (1+2mδ)M.
+Note that the following relations holds Q2= (n+
+1)2M2δ(1+mδ).
+Note also that for the spacetime considered in this pa-
+perK= 1, whichmeansthatthe eigenvalues k2aregivenbyk2=l(l+n−1), where lis the angular quantum
+number, that gives the multipolarity of the field. The
+behavior of the potentials varies considerably over the
+range of parameters. In Fig. 1 we show V−forD= 8,
+rc= 1/0.95,l= 2andthreedifferentvaluesofthecharge,
+Q= 0.2,0.35,0.44.
+III. A CRITERIUM FOR INSTABILITY
+A sufficient (but not necessary) condition for the exis-
+tence of an unstable mode has been proven by Buell and3
+/s48/s44/s48/s48 /s48/s44/s48/s49 /s48/s44/s48/s50 /s48/s44/s48/s51 /s48/s44/s48/s52 /s48/s44/s48/s53/s45/s50/s48/s50/s52/s54
+/s49/s48/s52
+/s32/s86
+/s45/s49/s48/s52
+/s32/s86
+/s45
+/s32/s32/s86
+/s45
+/s114/s45/s49/s32/s81/s61/s48/s46/s50/s48
+/s32/s81/s61/s48/s46/s51/s53
+/s32/s81/s61/s48/s46/s52/s52/s49/s48/s51
+/s32/s86
+/s45
+FIG. 1: Behavior of V−for different parameters, for D= 8.
+Here we fix the event horizon at rb= 1, and the cosmological
+horizon at rc= 1/0.95. We consider l= 2 modes and three
+different charges, Q= 0.2,0.35,0.44.
+Shadwick [9] and is the following,
+/integraldisplayrc
+rbV
+fdr <0. (14)
+The instability region is depicted in figure 2 for several
+/s48/s44/s48 /s48/s44/s50 /s48/s44/s52 /s48/s44/s54 /s48/s44/s56 /s49/s44/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s32
+/s32/s81/s47/s81
+/s101/s120/s116
+/s114
+/s98/s47/s114
+/s99
+FIG. 2: The parametric region of instability in Q/Qext−rb/rc
+coordinates, according to criterim (14), for l= 2. Top to
+bottom, D= 7,8,9,10,11.
+spacetime-dimension D, which can be compared with the
+numerical results by KZ, their figure 4. It is apparent
+that condition (14) very accurately describes the numer-
+ical results for rb/rc∼1, a regime we explore below in
+Section IV. As one moves away from extremality cri-
+terium (14) is just too restrictive. An improved analysis
+and refined criterium would be necessary to describe the
+whole rangeofthe numericalresults. Nevertheless, figure2 is very clear: higher-dimensional ( D >6) RNdS black
+holes are unstable for a wide range of parameters.
+IV. AN EXACT SOLUTION IN THE NEAR
+EXTREMAL RNDS BLACK HOLE
+Let us now specialize to the near extremal RNdS black
+hole, which we define as the spacetime for which the cos-
+mological horizon rcis very close (in the rcoordinate)
+to the black hole horizon rb, i.e.rc−rb
+rb≪1. The wave
+equationin this spacetime can be solvedexactly, in terms
+of hypergeometric functions [10]. The key point is that
+the physical region of interest (where the boundary con-
+ditions are imposed), lies between rbandrc. Thus,
+f∼2κb(r−rb)(rc−r)
+rc−rb, (15)
+where we have introduced the surface gravity κbassoci-
+ated with the event horizon at r=rb, as defined by the
+relationκb=1
+2df/drr=rb. For near-extremal black holes,
+it is approximately
+κb∼(rc−rb)(n−1)
+2r2
+b/parenleftbig
+1−nQ2/parenrightbig
+.(16)
+In this limit, one can invert the relation r∗(r) of (7) to
+get
+r=rce2κbr∗+rb
+1+e2κbr∗. (17)
+Substituting this on the expression (15) for fwe find
+f=(rc−rb)κb
+2cosh(κbr∗)2. (18)
+As such, and taking into account the functional form of
+the potentials for wave propagation, we see that for the
+near extremal RNdS black hole the wave equation (6) is
+of the form
+d2Φ(ω,r)
+dr2∗+/bracketleftBigg
+ω2−V0
+cosh(κbr∗)2/bracketrightBigg
+Φ(ω,r) = 0,(19)
+with
+V0=(rc−rb)κb
+2VS±(rb)
+f(20)
+The potential in (19) is the well known P¨ oshl-Teller po-
+tential [11]. The solutions to (19) were studied and they
+are of the hypergeometric type, (for details see Refs.
+[12, 13]). Itshouldbesolvedunderappropriateboundary
+conditions:
+Φ∼e−iωr∗, r∗→ −∞ (21)
+Φ∼eiωr∗, r∗→ ∞. (22)4
+These boundary conditions impose a non-trivial condi-
+tion onω[12, 13], and those that satisfy both simultane-
+ously are called quasinormal frequencies. For the P¨ oshl-
+Teller potential one can show [12, 13] that they are given
+by
+ω=κb/bracketleftBigg
+−/parenleftbigg
+j+1
+2/parenrightbigg
+i+/radicalBigg
+V0
+κ2
+b−1
+4/bracketrightBigg
+, j= 0,1,....
+(23)
+We conclude therefore that an instability is present
+TABLE I: The threshold of instability for near-extremal
+RNdS black holes (i.e., black holes for which the cosmologic al
+and event horizon almost coincide) for l= 2 modes. We show
+the prediction from the exact, analytic expression obtaine d
+in the near extremal limit (24), which we label Q/QN
+extand
+the one from criterium (14) which we label as Q/QV
+ext. Both
+these results are compared to the numerical results by KZ.
+D
+7 8 9 10 11 D→ ∞
+Q/QN
+ext0.913 0.774 0.683 0.617 0.567p
+2/D
+Q/QV
+ext0.913 0.775 0.684 0.618 0.568p
+2/D
+Q/QNum
+ext0.94 0.78 0.68 0.61 0.55 —
+whenever V0is negative. The threshold of stability in
+the near-extremal regime is therefore given by
+VS±(rb)
+f= 0, (24)
+The expression for VS±(rb)/fis lengthy, and we won’t
+presentit here. Thevaluesofthe charge Q/Qextthat sat-
+isfy the condition above are given in Table I (for l= 2),
+and compared to the prediction from the analysis in Sec-
+tion III, criterium (14). The agreement is excellent. Fur-thermore, we compare these predictions against the nu-
+merical results by KZ, extrapolated to the extremal limit
+(ρ= 1 in KZ notation). The agreement is remarkable.
+V. CONCLUSIONS
+We have shown analytically that charged black holes
+in de Sitter backgrounds are unstable for a wide range of
+charge and mass of the black hole, confirming previous
+numerical studies [7]. The stability properties of the ex-
+tremalD= 6 black hole remain unknown. Our methods
+and results and inconclusive at this precise point, further
+dedicated investigations would be necessary.
+Ouranalyticalresultinthenear-extremalregimecould
+be used to investigate further the nature of this instabil-
+ity, something we have not attempted to do here. A
+possible refinement concerns the large- Dlimit of the in-
+stability, where it couldbe possible to find an analytical
+expression throughout all range of parameters. We have
+inmind resultsandtechniquessimilartothoseofKoland
+Sorkin [14]. It would also be interesting to investigate
+the stability properties, using this or other techniques, of
+near-extremal Kerr-dS black holes, which have recently
+been conjectured to have an holographic description [15].
+Acknowledgements
+We warmly thank Roman Konoplya and Alexander
+Zhidenko for useful correspondence and for sharing their
+numerical results with us. This work was partially
+funded by Funda¸ c˜ ao para a Ciˆ encia e Tecnologia (FCT)-
+Portugal through projects PTDC/FIS/64175/2006,
+PTDC/ FIS/098025/2008,PTDC/FIS/098032/2008and
+CERN/FP/109290/2009.
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\ No newline at end of file
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+arXiv:1001.0020v2  [nlin.SI]  3 Mar 2010Classification of integrable hydrodynamic chains
+A.V. Odesskii1,2, V.V. Sokolov1
+1L.D. Landau Institute for Theoretical Physics (Russia)
+2Brock University (Canada)
+Abstract Using the method of hydrodynamic reductions, we find all inte-
+grable infinite (1+1)-dimensional hydrodynamic-type chains of shif t one. A
+class of integrable infinite (2+1)-dimensional hydrodynamic-type c hains is
+constructed.
+MSC numbers: 17B80, 17B63, 32L81, 14H70
+Address : L.D. Landau Institute for Theoretical Physics of Russian Academ y of Sciences,
+Kosygina 2, 119334, Moscow, Russia
+E-mail: aodesski@brocku.ca, sokolov@itp.ac.ru
+1Contents
+1 Introduction 3
+2 Integrable chains and hydrodynamic reductions 4
+3 GT-systems 5
+4 Canonical forms of GT-systems associated
+with integrable chains 7
+5 Generic case 12
+6 Trivial GT-system and 2+1-dimensional integrable hydrodynamic chains 14
+7 Infinitesimal symmetries of triangular GT-systems 17
+21 Introduction
+We consider integrable infinite quasilinear chains of the form
+uα,t=φα,1u1,x+···+φα,α+1uα+1,x, α= 1,2,..., φ α,α+1/negationslash= 0, (1.1)
+whereφα,j=φα,j(u1,...,uα+1).Two chains are called equivalent if they are related by a trans-
+formation of the form
+uα→Ψα(u1,...,uα),∂Ψα
+∂uα/negationslash= 0, α= 1,2,... (1.2)
+By integrability we mean the existence of an infinite set of hydrodyna mic reductions [1, 2,
+3, 4, 5, 6].
+Example 1. The Benney equations [7, 8, 9]
+u1,t=u2,x, u 2,t=u1u1,x+u3,x,... u αt= (α−1)uα−1u1,x+uα+1,x,... (1.3)
+provide the most known example of integrable chain (1.1). The hydro dynamic reductions for
+the Benney chain were investigated in [10]. /square
+In [4, 5, 6] integrable divergent chains of the form
+u1t=F1(u1,u2)x, u2t=F2(u1,u2,u3)x,···, uit=Fi(u1,u2,...,ui+1)x,··· (1.4)
+were considered. In [6] some necessary integrability conditions we re obtained. Namely, a non-
+linear overdetermined system of PDEs for functions F1,F2was presented. The general solution
+of the system was not found. Another open problem was to prove t hat the conditions are
+sufficient. In other words, for any solution F1,F2of the system one should find functions
+Fi,i>2 such that the resulting chain is integrable.
+Probably any integrable chain (1.1) is equivalent to a divergent chain. However, the diver-
+gent coordinates are not suitable for explicit formulas. Our main obs ervation is that a conve-
+nient coordinates are those, in which the so-called Gibbons-Tsarev type system (GT-system)
+related to integrable chain is in a canonical form.
+Using our version (see [11, 12]) of the hydrodynamic reduction meth od, we describe all
+integrable chains (1.1). We establish an one-to-one corresponden ce between integrable chains
+(1.1) and infinite triangular GT-systems of the form
+∂ipj=P(pi,pj)
+pi−pj∂iu1, i/negationslash=j, (1.5)
+∂i∂ju1=Q(pi,pj)
+(pi−pj)2∂iu1∂ju1, i/negationslash=j, (1.6)
+∂ium= (gm,0+gm,1pi+···+gm,m−1pm−1
+i)∂iu1, g m,j=gm,j(u1,...,um), gm,m−1/negationslash= 0,
+3wherem= 2,3,...andi,j= 1,2,3.The functions P,Qare polynomials quadratic in each of
+variablespiandpj,with coefficients being functions of u1,u2.The functions p1,p2,p3,u1,u2,...
+in (3.11) depend on r1,r2,r3,and∂i=∂
+∂ri.
+Example 1-1 (continuation of Example 1.) The system (1.5),(1.6) corresponding t o the
+Benney chain has the following form
+∂ipj=∂iu1
+pi−pj, ∂ i∂ju1=2∂iu1∂ju1
+(pi−pj)2, (1.7)
+∂ium= (−(m−2)um−2−···−2u2pm−2
+i−u1pm−3
+i+pm−1
+i)∂iu1. (1.8)
+Equations (1.7) were firstly obtained in [10]. /square
+Given GT-system (1.5), (1.6) the coefficients of (1.1) are uniquely de fined by the following
+relations
+pi∂ium=φm,1∂iu1+···+φm,m+1∂ium+1, m= 2,3,... (1.9)
+Namely, equating the coefficients at different powers of piin (1.9), we get a triangular system
+of linear algebraic equations for φi,j. Thus, the classification problem for chains (1.1) is reduced
+to a description of all GT-systems (1.5), (1.6) . The latter problem is solved in Section 4-6.
+The paper is organized as follows. Following [11, 12], we recall main defin itions in Section
+2 (see [1, 2, 3, 11] for details). We consider only 3-component hyd rodynamic reductions since
+the existence of reductions with N >3 gives nothing new [1]. In Section 3 we formulate
+our previous results that are needed in the paper. Section 4 is devo ted to a classification of
+admissible polynomials PandQin (1.5), (1.6). In Sections 5,6 we construct integrable chains
+for the generic case and for some degenerations. Section 6 also co ntains examples of (2+1)-
+dimensional infinite hydrodynamic-type chains integrable from the v iewpoint of the method
+of hydrodynamic reductions. Infinitesimal symmetries of GT-syst ems are studied in Section 7.
+These symmetries seem to be important basic objects in the hydrod ynamic reduction approach.
+Acknowledgments. Authors thank M.V. Pavlov for fruitful discussions. V.S. is gratefu l to
+Brock University for hospitality. He was partially supported by the R FBR grants 08-01-464,
+09-01-22442-KE, and NS 3472.2008.2.
+2 Integrable chains and hydrodynamic reductions
+According to [1, 2, 3, 4, 5, 6] a chain (1.1) is called integrable if it admits sufficiently many
+so-called hydrodynamic reductions.
+Definition. A hydrodynamic (1+1)-dimensional N-component reduction of a chain (1.1)
+is a semi-Hamiltonian (see formula (3.18) ) system of the form
+ri
+t=pi(r1,...,rN)ri
+x, i= 1,..,N (2.10)
+4and functions uj(r1,...,rN), j= 1,2,...such that for each solution of (2.10) functions uj=
+uj(r1,...,rN), i= 1,...satisfy (1.1).
+Substituting ui=ui(r1,...,rN), i= 1,...into (1.1), calculating tandx-derivatives by virtue
+of (2.10) and equating coefficients at rs
+xto zero, we obtain
+∂suαps=φα,1∂su1+···+φα,α+1∂suα+1, α= 1,2,...
+It is clear from this system that
+∂suk=gk(ps,u1,...,uk)∂su1, k= 2,3,...
+wheregk(p,u1,...,uk) is a polynomial of degree k−1 inpfor eachk= 2,3,...Compatibility
+conditions∂i∂juk=∂j∂iukgive us a system of linear equations for ∂ipj, ∂jpi, ∂i∂ju1, i/negationslash=j.
+This system should have a solution (otherwise we would not have suffic iently many reductions).
+Moreover, expressions for ∂suk, k= 2,3,..., ∂jpi, ∂i∂ju1, i/negationslash=jshould be compatible and form
+a so-called GT-system.
+Remark. In the sequel we assume N= 3 because the case N >3 gives nothing new [1].
+3 GT-systems
+Definition. A compatible system of PDEs of the form
+∂ipj=f(pi,pj,u1,...,un), ∂iu1j/negationslash=i,
+∂i∂ju1=h(pi,pj,u1,...,un)∂iu1∂ju1, j/negationslash=i, (3.11)
+∂iuk=gk(pi,u1,...,un)∂iu1, k= 1,...,n−1,
+wherei,j= 1,2,3 is called n-fields GT-system . Herep1,p2,p3,u1,...,unare functions of
+r1,r2,r3and∂i=∂
+∂ri.
+Definition. Two GT-systems are called equivalent if they are related by a transformation
+of the form
+pi→λ(pi,u1,...,un), (3.12)
+uk→µk(u1,...,un), k= 1,...,n. (3.13)
+Example 2 [13]. Leta0,a1,a2be arbitrary constants, R(x) =a2x2+a1x+a0. Then the
+system
+∂ipj=a2p2
+j+a1pj+a0
+pi−pj∂iu1, ∂ i∂ju1=2a2pipj+a1(pi+pj)+2a0
+(pi−pj)2∂iu1∂ju1(3.14)
+is an one-field GT-system. The original Gibbons-Tsarev system (1.7 ) corresponds to a2=a1=
+0,a0= 1.The polynomial R(x) can be reduced to one of the following canonical forms: R= 1,
+5R=x,R=x2, orR=x(x−1) by a linear transformation (3.12). A wide class of integrable
+3D-systems of hydrodynamic type related to (3.14) is described in [1 3]. An elliptic version of
+this GT-system and the corresponding integrable 3D-systems wer e constructed in [15]. /square
+Definition. An additional system
+∂iuk=gk(pi,u1,...,un+m)∂iun, k=n+1,...,n+m (3.15)
+suchthat(3.11)and(3.15)arecompatibleiscalled an extension of(3.11)byfields un+1,...,un+m.
+It turns our that
+∂iun+1=f(pi,un+1,u1,...,un)∂iu1
+is an extension for GT-system (3.11). Stress that here fis the same function as in (3.11). We
+call this extension the regular extension byun+1.
+Example 2-1. The generic case of Example 2 corresponds to R=x(x−1). The regular
+extension by u2is given by
+∂iu2=u2(u2−1)
+pi−u2∂iu1.
+If we express u1from this formula and substitute it to (3.14), we get the following one -field
+GT-system
+∂ipj=pj(pj−1)(pi−u1)
+u1(u1−1)(pi−pj)∂iu1,
+∂i∂ju1=pipj(pi+pj)−p2
+i−p2
+j+(p2
+i+p2
+j−4pipj+pi+pj)u1
+u1(u1−1)(pi−pj)2∂iu1∂ju1./square(3.16)
+The second basic notion of the hydrodynamic reduction method is so -called GT-family of
+(1+1)-dimensional hydrodynamic-type systems.
+Definition. An (1+1)-dimensional 3-component hydrodynamic-type system o f the form
+ri
+t=vi(r1,...,rN)ri
+x, i= 1,2,3, (3.17)
+is called semi-Hamiltonian if the following relation holds
+∂j∂ivk
+vi−vk=∂i∂jvk
+vj−vk, i/negationslash=j/negationslash=k. (3.18)
+Definition. A Gibbons-Tsarev family associated with the Gibbons-Tsarev type s ystem
+(4.25) is a (1+1)-dimensional hydrodynamic-type system of the fo rm
+ri
+t=F(pi,u1,...,um)ri
+x, i= 1,2,3, (3.19)
+semi-Hamiltonian by virtue of (3.11).
+6Example 2-2 [13]. Applying the regular extension to the generic GT-system (3.14) two
+times, we get the following GT-system:
+∂ipj=pj(pj−1)
+pi−pj∂iw, ∂ ijw=2pipj−pi−pj
+(pi−pj)2∂iw∂jw, i/negationslash=j, (3.20)
+∂iuj=uj(uj−1)∂iw
+pi−uj, j= 1,2. (3.21)
+Consider the generalized hypergeometric [14] linear system of the f orm
+∂2h
+∂uj∂uk=sj
+uj−uk·∂h
+∂uk+sk
+uk−uj·∂h
+∂uj, j/negationslash=k, (3.22)
+∂2h
+∂uj∂uj=−/parenleftBigg
+1+n+2/summationdisplay
+k=1sk/parenrightBigg
+sj
+uj(uj−1)·h+sj
+uj(uj−1)n/summationdisplay
+k/negationslash=juk(uk−1)
+uk−uj·∂h
+∂uk+
+/parenleftBiggn/summationdisplay
+k/negationslash=jsk
+uj−uk+sj+sn+1
+uj+sj+sn+2
+uj−1/parenrightBigg
+·∂h
+∂uj.(3.23)
+Herei,j= 1,2 ands1,...,s4are arbitrary parameters. It easy to verify that this system is in
+involution and therefore the solution space is 3-dimensional. Let h1,h2,h3be a basis of this
+space. For any hwe put
+S(p,h) =u1(u1−1)(p−u2)hh1,u1−hu1h1
+h1+u2(u2−1)(p−u1)hh1,u2−hu2h1
+h1.
+Then the formula
+F=S(p,h3)
+S(p,h2)(3.24)
+defines the generic linear fractional GT-family for (3.20). /square
+4 Canonical forms of GT-systems associated
+with integrable chains
+For integrable chains the corresponding GT-systems involve infinite number of fields ui, i=
+1,2,...(see Example 1-1). In this Section we show that these GT-systems are equivalent to
+infinite triangular extensions of one-field GT-systems from Example s 2,3.
+A compatible system of PDEs of the form
+∂ipj=f(pi,pj,u1,...,un)∂iu1, i/negationslash=j,
+∂iuk=gk(pi,u1,...,uk)∂iu1, k= 1,2,...,, (4.25)
+7∂i∂ju1=h(pi,pj,u1,...,un)∂iu1∂ju1, i/negationslash=j,
+wherei,j= 1,2,3 is called triangular GT-system . Herep1,p2,p3,u1,u2,...are functions of
+r1,r2,r3,and∂i=∂
+∂ri.
+Definition. A chain (1.1) is called integrable if there exists a Gibbons-Tsarev type system
+of the form (4.25) and a Gibbons-Tsarev family
+ri
+t=F(pi,u1,...,um)ri
+x, i= 1,2,3, (4.26)
+such that (1.1) holds by virtue of (4.25), (4.26).
+Due to the equivalence transformations (3.12) we can assume witho ut loss of generality that
+F(p,u1,...,um) =p. (4.27)
+Under this assumption we have
+uj,t=/summationdisplay
+s∂sujrs
+t=/summationdisplay
+s∂sujpsrs
+x.
+and similar
+uj,x=/summationdisplay
+s∂sujrs
+x.
+Substituting these expressions into (1.1) and equating coefficients atrs
+xto zero, we obtain
+∂suαps=φα,1∂su1+···+φα,α+1∂suα+1, α= 1,2,...
+Using (4.25) and replacing psbyp, we get
+p=φ1,1+φ1,2g2, pg2=φ2,1+φ2,2g2+φ2,3g3, pg3=φ3,1+φ3,2g2+φ3,3g3+φ3,4g4,...
+Solving this system with respect to g2, g3,..., we obtain
+gi(p) =ψi,0+ψi,1p+...+ψi,i−1pi−1.
+Hereψi,jare functions of u1,...,ui. For example,
+g2=−p
+φ1,2−φ1,1
+φ1,2. (4.28)
+Remark. Since we assume that φi,i−1/negationslash= 0,we haveψi,i−1/negationslash= 0 for all i. Therefore g1=
+1,g2,...is a basis in the linear space of all polynomials in p. The coefficients φi,jof our chain
+are just entries of the matrix of multiplication by pin this basis. More generally, if we don’t
+normalizeF=p, then the coefficients φi,jcan be found from the equations
+F(p) =φ1,1+φ1,2g2, F(p)g2=φ2,1+φ2,2g2+φ2,3g3,
+F(p)g3=φ3,1+φ3,2g2+φ3,3g3+φ3,4g4,...(4.29)
+8Compatibility conditions ∂i∂juα=∂j∂iuα, α= 2,3,4 give a system of linear equations for
+∂ipj, ∂jpi, ∂i∂ju1. Solving this system, we obtain formulas (1.5),(1.6), where in principa lP, Q
+coulddependon u1,u2,u3,u4. However, itfollowsfromcompatibility conditions ∂i∂jpk=∂j∂ipk
+thatP, Qdepend onu1, u2only.
+Written (1.5) in the form
+∂ipj=/parenleftbiggR(pj)
+pi−pj+(z4p2
+j+z5pj+z6)pi+z4p3
+j+z3p2
+j+z7pj+z8/parenrightbigg
+∂iu1, (4.30)
+whereR(x) =z4x4+z3x3+z2x2+z1x+z0,one can derive from the compatibility conditions
+∂i∂jpk=∂j∂ipk,∂i∂ju1=∂j∂iu1that the equation (1.6) has the following form
+∂i∂ju1=/parenleftbigg2z4p2
+ip2
+j+z3pipj(pi+pj)+z2(p2
+i+p2
+j)+z1(pi+pj)+2z0
+(pi−pj)2+z9/parenrightbigg
+∂iu1∂ju1.(4.31)
+It is easy to verify that we can normalize z9=z6−z7, g2=pby a transformation (1.2).
+Then the coefficients zi(x,y),i= 0,...,8 satisfy the following pair of compatible dynamical
+systems with respect to yandx:
+z0,y= 2z0z5−z1z6, z 1,y= 4z0z4+z1z5−2z2z6, z 2,y= 3z1z4−3z3z6,
+z3,y= 2z2z4−z3z5−4z4z6, z 4,y=z3z4−2z4z5, z 5,y=z4z7−z4z6−z2
+5,
+z6,y=z4z8−z5z6, z 7,y= 2z1z4−2z3z6−z5z6+z4z8, z 8,y= 2z0z4−z2
+6−z6z7+z5z8,
+and
+z0,x=−z0z2−z0z6+3z0z7−z1z8, z 1,x=−z1z2+3z0z3−z1z6+2z1z7−2z2z8,
+z2,x=−z2
+2+2z1z3+4z0z4−z2z6+z2z7−3z3z8, z 3,x= 3z1z4−z3z6−4z4z8,
+z4,x=z2z4−z4z6−z4z7, z 5,x=z1z4−z5z6−z4z8, z 6,x=z0z4−z2
+6,
+z7,x=z1z3+3z0z4+z1z5−z2z6−z2z7+z2
+7−z3z8−2z5z8,
+z8,x=z0z3+z0z5−z2z8−2z6z8+z7z8.
+These is a complete description of the GT-systems related to integr able chains (1.1).
+To solve the dynamical systems we bring the polynomial Rto a canonical form sacrificing
+to the normalization (4.27).
+It is obvious that linear transformations pi→api+b, wherea,bare functions of u1,u2,
+preserve the form of GT-system (4.30),(4.31). Moreover, there exist transformations of the
+form
+pi=a¯pi+b
+¯pi−ψ, i= 1,2,3 (4.32)
+9preserving the form of GT-system (4.30),(4.31). Such admissible tr ansformations are described
+by the following conditions:
+au2=z4(b+aψ), b u2=z4bψ+z5b−z6a, ψ u2=z4ψ2+z5ψ+z6.
+Under transformations (4.32) the polynomial Ris transformed by the following simple way:
+R(pi)→(pi−ψ)4R/parenleftBigapi+b
+pi−ψ/parenrightBig
+.
+Suppose that Rhas distinct roots. It is possible to verify that by an admissible trans formation
+(4.32) we can move three of the four roots to 0 ,1 and∞. It follows from compatibility
+conditionsfortheGT-system thatthenthefourthroot λ(u1,u2)doesnotdependon u2. Making
+transformation of the form u1→q(u1) we arrive at the canonical forms λ=u1orλ=const. It
+is straightforwardly verified that in the first case equations (4.30) , (4.31) coincides with (3.16).
+In the second case the GT-system does not exist.
+In the case of multiple roots the polynomial R(x) can be reduced to one of the following
+forms:R= 0,R= 1,R=x,R=x2, orR=x(x−1).In all these cases equations (4.30),
+(4.31) coincides with the corresponding equations from Example 2.
+Thus, the following statement is valid:
+Proposition 1. There are 6 non-equivalent cases of GT-systems (4.30), (4.31). T he canon-
+ical forms are:
+Case 1: (3.16) (generic case);
+Case 2: (3.14) with R(x) =x(x−1);
+Case 3: (3.14) with R(x) =x2;
+Case 4: (3.14) with R(x) =x;
+Case 5: (3.14) with R(x) = 1.
+Case 6: (3.14) with R(x) = 0./square
+Remark. Cases 2-6 can be obtained from Case 1 by appropriate limit procedur es. For
+example, Case 2 corresponds to the limit u1→u1
+ε, ε→0.
+It follows from (4.27), (4.28) that for any canonical form the func tionsFandg2have the
+following structure:
+g2(pi) =k1pi+k2
+k3pi+k4, F(pi) =f1pi+f2
+k3pi+k4, (4.33)
+where the coefficients are functions of u1,u2.
+Lemma 1. For theCase 1 any function g2can bereduced by anappropriatetransformation
+10¯u2=σ(u1,u2) to one of the following canonical forms:
+a1:g2(p) =u2(u2−1)(p−u1)
+u1(u1−1)(p−u2)(regular extension);
+b1:g2(p) =1
+p−u1;
+c1:g2(p) =u−λ
+1(u1−1)λ−1
+p−λλ= 1,0;
+d1:g2(p) =u1−u2
+u1(u1−1)p+u2−1
+u1−1./square
+The GT-system from the Case 1 possesses a discrete automorphis m groupS4interchanging
+the points 0 ,1,∞,u1. The group is defined by generators
+σ1:u1→1−u1, pi→1−pi, σ 2:u1→u1
+u1−1, pi→pi
+pi−1,
+and
+σ3:u1→1−u1, pi→(1−u1)pi
+pi−u1.
+Up to this group the cases b1,c1,d1are equivalent and one can take say the case d1for further
+consideration. The case a1is invariant with respect to the group.
+Remark. The casesb1, c1, d1are degenerations of the case a1. Namely, they can be
+obtained as appropriate limit u2→u1,u2→λ, u2→ ∞correspondingly.
+All possible functions g2for Cases 2-5 are described in the following
+Lemma 2. For the GT-system (3.14) (excluding Case 6) any function g2can be reduced
+by an appropriate transformation ¯ u2=σ(u1,u2) to one of the following canonical forms:
+a2:g2(p) =R(u2)
+p−u2(regular extension);
+b2:g2(p) =1
+p−λ,whereR(λ) = 0;
+c2:g2(p) =p−a2u2.
+The discrete automorphism of the GT-system interchanges the ro ots ofRin the case b2./square
+Lemma 3. For the GT-system (3.14) with R(x) = 0 (Case 6) any function g2can be
+reduced to g2(p) =pby an appropriate transformation ¯ u2=σ(u1,u2). Furthermore, the
+corresponding triangular GT-system has the form
+∂ipj= 0, ∂ i∂ju1= 0, ∂ iuk=pk−1
+iu1, k= 2,3,.../square (4.34)
+115 Generic case
+The next step in the classification is to find all functions Fof the form (4.28) for each pair
+consisting of a GT-system from Proposition 1 and the correspondin gg2from Lemmas 1-3.
+The semi-Hamiltonian condition (3.18) yields a non-linear system of PDE s for the functions
+f1(u1,u2),f2(u1,u2).For each case this system can be reduced to the linear generalized h yper-
+geometric system (3.22), (3.23) with a special set of parameters s1,s2,s3,s4or to a degeneration
+of this system.
+The general linear fractional GT-family for the generic case 1, a1is given by (3.24). Ac-
+cording to (4.33), the additional restriction is that the root of the denominator has to be equal
+u2.It is easy to verify that this is equivalent to s2= 0,h1,u2=h2,u2= 0. The latter means that
+h1(u1),h2(u1) are linear independent solutions of the standard hypergeometric equation
+u(u−1)h(u)′′+[s1+s3−(s3+s4+2s1)u]h(u)′+s1(s1+s3+s4+1)h(u) = 0.(5.35)
+The function h3(u1,u2) is arbitrary solution of (3.22), (3.23) with s2= 0 linearly independent
+ofh1(u1),h2(u1). Without loss of generality we can choose
+h3(u1,u2) =/integraldisplayu2
+0(t−u1)s1ts3(t−1)s4dt.
+Formula (3.24) gives
+F(p,u1,u2) =f1(u1,u2)p−f2(u1,u2)
+p−u2, (5.36)
+where
+f1=u2(u2−1)h1h3,u2+u1(u1−1)(h1h3,u1−h3h′
+1)
+u1(u1−1)(h1h′
+2−h2h′
+1),
+f2=u1u2(u2−1)h1h3,u2+u2u1(u1−1)(h1h3,u1−h3h′
+1)
+u1(u1−1)(h1h′
+2−h2h′
+1).
+Notice that h1h′
+2−h2h′
+1=const(u1−1)s1+s4us1+s3
+1.
+For integer values of s1,s3,s4the hypergeometric system can be solved explicitly. For
+example, if s1=s3=s4= 0, the above formulas give rise to F=g2.Ifs4=−2−s1−s3then
+F=(u2−u1)s1+1us3+1
+2(u2−1)−1−s1−s3
+p−u2;
+ifs4= 0,then
+F=(p−1)(u2−u1)s1+1us3+1
+2(u1−1)−1−s1
+p−u2.
+Nowwearetofindthefunctions g3,g4,...in(4.25). Thesefunctionsaredefineuptoarbitrary
+transformation (1.2), where α= 3,4,.... In practice, one can look for functions g3,g4,...linear
+inui,i>2 (cf. (1.8)). An extension linear in ui,i>2 is given by
+g3(p) =−(u1−u2)(u2−1)p
+u1(u1−1)(p−u2)2,
+12gi(p) =(i−3)(u1−u2)(u2−1)pui
+u1(u1−1)(p−u2)2−(u1−u2)i−3(u2−1)2p(p−u1)(p−1)i−4
+u1(u1−1)i−2(p−u2)i−1−
+i−4/summationdisplay
+s=1(i−s−2)(u1−u2)s(u2−1)2p(p−u1)(p−1)s−1ui−s
+u1(u1−1)s+1(p−u2)s+2.
+The coefficients of the chain (1.1) corresponding to Case 1, a1are determined from (4.29),
+whereFis given by (5.36). Relations (4.29) are equivalent to a triangular syst em of linear
+algebraic equations. Solving this system, we find that for i>4 coefficients of the chain read:
+φi,i+1=(u1−1)(f1u2−f2)
+(u2−1)(u1−u2)def=Q1, φ i,i=f2−f1
+u2−1def=Q2,
+φi,4=−uiQ1, φ i,3=−/parenleftBig
+(u4+i−3)ui+(2−i)ui+1/parenrightBig
+Q1def=Ai,
+andφi,j= 0 for all remaining i,j.Fori≤4 we have
+φ1,1=f1u1−f2
+u1−u2, φ 1,2=−u1
+u2Q1,
+φ2,1=(u2−1)(f1u2−f2)
+(u1−1)(u1−u2), φ 2,2=f2u1−f1u2
+2
+u2(u1−u2), φ 2,3=f1u2−f2,
+φ3,1=φ3,2= 0, φ 3,3=Q2−(u4−1)Q1, φ 3,4=−Q1,
+φ4,1=φ4,2= 0, φ 4,3=A4, φ 4,4=Q2−u4Q1, φ 4,5=Q1.(5.37)
+The explicit formulas for other cases of Proposition 1 can be obtaine d by limits from the
+above formulas. We outline the limit procedures for the case 1, d1. In this case the limit is
+given byu2→u1+εu2, ε→0.It is easy to check that under this limit the extension a1
+turns tod1. The limit of the system (3.22), (3.23) with s2= 0 can be easily found. The general
+solution of the system thus obtained is given by h=c1(u2−u1)1+s1+s3+s4+h1,whereh1is the
+general solution of (5.35). Let h1,h2be solutions of (5.35), and h3= (u2−u1)1+s1+s3+s4. Then
+the limit procedure in (5.36) gives rise to
+F(p,u1,u2) =Q×/parenleftBig
+(1+s1+s3+s4)h1(p−u1)+u1(u1−1)h′
+1/parenrightBig
+,
+where
+Q= (u2−u1)1+s1+s3+s4(u1−1)−1−s1−s4u−1−s1−s3
+1.
+As usual, the most degenerate cases in classification of integrable P DEs could be interesting
+for applications. In our classification they are Case 5, c2and Case 6. The Benney chain
+(see Examples 1 and 1-1) belongs to Case 5, case c2(i.eg2=p). Any GT-family has the form
+F=f1(u1,u2)p+f2(u1,u2). Iff1= 1 thenF=p+k2u2+k1u1.The Benney case corresponds to
+13k1=k2= 0. For arbitrary kiwe get the Kupershmidt chain [16]. In the case f1=A(u1),A′/negationslash= 0
+we obtain:
+f1=k2exp(λu1)+k1, f 2=k2k3exp(λu1)+λk1(k3u1−u2).
+In the generic case
+F= exp(λu2)(S1(u1)p+S2(u1)),
+where the functions Sican be expressed in terms of the Airy functions.
+6 Trivial GT-system and 2+1-dimensional integrable hy-
+drodynamic chains
+It was observed in [11] that (2+1)-dimensional systems of hydro dynamic type with the trivial
+GT-system usually admit some integrable multi-dimensional generaliza tions. For the chains
+such GT-system is defined by (4.34). That is why the Case 6 is of a gre at importance in our
+classification. The automorphisms of (4.34) are given by
+pj→pj, j= 1,...,N, u i→νui+γi, i= 1,2,...; (6.38)
+pj→apj+b, j= 1,...,N, u i→ai−1ui+(i−1)ai−2bui−2+...+bi−1u1, i= 1,2,...
+The corresponding GT-families are of the form F(p) =A(u1,u2)p+B(u1,u2), where
+A(x,y),B(x,y) satisfies the following system of PDEs:
+AByy=AyBy, AB xy=AyBx, AB xx=AxBx,
+AAyy=A2
+y, AA xy=AxAy, AA xx=A2
+x+AxBy−AyBx.(6.39)
+This system can be easily solved in elementary functions. For each so lution formula (4.29)
+defines the corresponding integrable chain (1.1).
+It follows from (6.39) that there are two types of u2-dependence:
+1(generic case). F(p) = exp(λu2)/parenleftBig
+a(u1)p+b(u1)/parenrightBig
+,
+2. F(p) =a(u1)p+λu2+b(u1).
+In the first case there are two subcases: b′/negationslash= 0 andb′= 0.The first subcase gives rise to
+a=σ′, b=k1σ σ(x) =c1exp(µ1x)+c2exp(µ2x),wherec1c2(λk1−µ1µ2) = 0.
+The second subcase leads to
+b=c1, a(x) =c2exp(µx)+c3,wherec2(c1λ−c3µ) = 0.
+The same subcases for the case 2 yield
+a=σ′, b=k1σ σ(x) =c1+c2x+c3exp(µx),wherec3(λ−c2µ) = 0,
+14and
+b=c1, a(x) =c2exp(µx)+c3,wherec2(λ−c3µ) = 0.
+It is easy to verify that in the generic case the function Fcan be reduced by (6.38) to the
+form
+F(p) =eu2+u1(p−1)+eu2−u1(p+1).
+In this case the corresponding chain reads as
+uk,t= (eu2+u1+eu2−u1)uk+1,x+(eu2−u1−eu2+u1)uk,x, k= 1,2,3,... (6.40)
+Asusual, thischainisthefirstmember ofaninfinitehierarchy. These condflowofthishierarchy
+is given by
+uk,τ= (eu2+u1+eu2−u1)uk+2,x+(u3−u1)(eu2+u1+eu2−u1)uk+1,x+
+(eu2+u1(u1−u3−1)+eu2−u1(u3−u1−1))uk,x, k= 1,2,3,...
+In the case 2 with c3=λ= 0,k1= 1 we get the chain
+uk,t=uk+1,x+u1uk,x, k= 1,2,3,... (6.41)
+This chain is equivalent to the chain of the so-called universal hierarc hy [17]. The chain (6.41)
+is a degeneration of the chain
+uk,t=uk+1,x+u2uk,x, k= 1,2,3,... (6.42)
+Following the line of [3, 11] it is not difficult to find (2+1)-dimensional inte grable generaliza-
+tions for all (1+1)-dimensional integrable chains constructed abo ve. Some families of functions
+Fdescribed above linearly depend on two parameters. Denote these parameters by γ1,γ2.The
+corresponding integrable chain
+uk,t=γ1(φk,1u1,x+···+φk,k+1uk+1,x)+γ2(ψk,1u1,x+···+ψk,k+1uk+1,x)
+is also linear in γ1,γ2.We claim that the following (2+1)-dimensional chain
+uk,t= (φk,1u1,x+···+φk,k+1uk+1,x)+(ψk,1u1,y+···+ψk,k+1uk+1,y) (6.43)
+is integrable from the viewpoint of the method of hydrodynamic redu ctions. For each case the
+reductions can be easily described.
+For example, in the generic case
+F(p) =γ1eu2+u1(p−1)+γ2eu2−u1(p+1)
+formula (6.43) yields (2+1)-dimensional chain
+uk,t=eu2+u1(uk+1,x−uk,x)+eu2−u1(uk+1,y+uk,y), k= 1,2,3,... (6.44)
+15After a change of variables of the form
+x→ −1
+2x, y→1
+2y, u 1→1
+2u0, u2→u1+1
+2u0, u3→ −2u2+1
+2u0,...
+(6.44) can be written as
+u0,t=eu1u0,y+eu1(u1,y−eu0u1,x), u i,t=eu0+u1ui,x+eu1(eu0ui+1,x−ui+1,y),(6.45)
+wherei= 1,2,.... Probably (6.45) is a first example of a (2+1)-dimensional chain integ rable
+from the viewpoint of the hydrodynamic reduction approach.
+TriangularGT-systemsrelatedtointegrable(2+1)-dimensionalch ainswithfields u0,u1,u2,...
+have the form
+∂ipj=f1(pi,qi,pj,qj,u0,...,un)∂iu0, ∂ iqj=f2(pi,qi,pj,qj,u0,...,un)∂iu0,
+∂i∂ju0=h(pi,qi,pj,qj,u0,...,un)∂iu0∂ju0, (6.46)
+∂iuk=gk(pi,qi,u0,...,uk+1)∂iu0, k= 0,1,2,...
+Herei/negationslash=j, i,j= 1,...,3,p1,...,p3, q1,...,q3,u0,u1,u2,...,arefunctionsof r1,r2,r3.Inparticular,
+the GT-system associated with (6.45) has the form:
+∂ipj=∂i∂ju0= 0, ∂ iqj=/parenleftBigpiqi−pjqj
+pi−pj−qiqj/parenrightBig
+∂iu0, ∂ iuk=−pi
+(pi−1)k∂iu0.
+Thehydrodynamicreductionsof(6.45)isgivenbythepairofsemi-ha miltonian(1+1)-dimensional
+systems
+ri
+y=eu0/parenleftBig
+1−1
+qi/parenrightBig
+ri
+x, ri
+t=eu0+u1/parenleftBig1
+(pi−1)qi+1/parenrightBig
+ri
+x.
+Chain (6.45) is the first member of an infinite hierarchy of pairwise com muting flows where
+the corresponding ”times” are t1=t, t2, t3,.... These flows and their hydrodynamic reductions
+can be described in terms of the generating function U(z) =u1+u2z+u3z2+...The hierarchy
+is given by
+D(z)u0=eU(z)/parenleftBig
+u0,y+U(z)y−eu0U(z)x/parenrightBig
+,
+D(z1)U(z2) =eu0+U(z1)U(z2)x+(1+z1)eU(z1)/parenleftBig
+eu0U(z1)x−U(z2)x
+z1−z2−U(z1)y−U(z2)y
+z1−z2/parenrightBig
+,
+whereD(z) =∂
+∂t1+z∂
+∂t2+z2∂
+∂t3+...The reductions can be written as
+D(z)ri=eu0+U(z)/parenleftBig
+1+1+z
+(pi−1−z)qi/parenrightBig
+ri
+x.
+Other (2+1)-dimensional integrable chains related to 2-dimensiona l vector spaces of solu-
+tions for system (6.39) are degenarations of (6.45). In particular F=γ1eu1p+γ2(p+u2) leads
+to the following (2+1)-dimensional integrable generalization of (6.44 ):
+uk,t=eu1uk+1,x+uk+1,y+u2uk,y, k= 1,2,3,....
+16Conjecture. Any chain of the form (6.43) integrable by the hydrodynamic reduct ion
+method is a degeneration of (6.45).
+We are planning to consider the problem of classification of integrable chains (6.43) in a
+separate paper.
+7 Infinitesimal symmetries of triangular GT-systems
+A scientific way to construct the functions g3,g4,...for different cases from Proposition 1 is
+related to infinitesimal symmetries of the corresponding GT-syste m1. The whole Lie algebra
+of symmetries is one the most important algebraic structures relat ed to any triangular GT-
+system (4.25). In particular, this algebra acts on the hierarchy of the commuting flows for the
+corresponding chain (1.1).
+A vector field
+S=N/summationdisplay
+j=1X(pj,u1,...,us)∂
+∂pj+∞/summationdisplay
+m=1Ym(u1,...,ukm)∂
+∂um,∂Ym
+∂ukm/negationslash= 0 (7.47)
+is called a symmetry of the triangular GT-system (4.25) if it commutes with all ∂i.Notice that
+it follows from the definition that
+S(∂iu1) =∂i(Y1).
+We call (7.47) a symmetry of shift difkm=m+dform>>0.LetMbe the minimal integer
+such thatkm=m+d,m>M. If the functions gi,i= 1,...,M+dfrom (4.25) are known, then
+the functions X,Y1,...YMcan be found from the compatibility conditions
+S(∂ipj) =∂iS(pj), S(∂iuk) =∂iS(uk), k= 1,...,M.
+The functions YM+1,YM+2,...can be chosen arbitrarily. After that gM+d+1,gM+d+2,...are
+uniquely defined by the remaining compatibility conditions.
+The generic case 1, a 1. Looking for symmetries of shift one, we find X=Y1= 0 and
+M= 1. Hence without loss of generality we can take
+S=∞/summationdisplay
+m=2um+1∂
+∂um
+for the symmetry. This fact gives us a way to construct all functio nsgi,i >3 in the infinite
+triangular extension for the case 1, a1.Indeed, it follows from the commutativity conditions
+S(∂iuk) =∂iS(uk) thatgk+1=S(gk),wherek= 2,3,.... In particular,
+g3=(pj−u1)(2pju2−pj−u2
+2)u3
+u1(u1−1)(pj−u2)2.
+1Note that these functions are not unique because of the triangula r group of symmetries (1.2) acting on the
+fieldsu3,u4,...
+17The functions githus constructed are not linear in u3.The corresponding chain (1.1) is equiv-
+alent to the chain constructed in Section 5 but not so simple.
+It would be interesting to describe the Lie algebra of all symmetries in this case. Here we
+present the essential part for symmetry of shift 2:
+X=pj(pj−1)u2
+3
+(pj−u2)u2(u2−1), Y 1=u1(u1−1)u2
+3
+(u1−u2)u2(u2−1),
+Y2=−3
+2u4+(2u1−1)u2
+3
+u2(u2−1)+u3./square
+The case 1, d 1. One can add fields u3,...in such a way that the whole triangular GT-
+system admits the following symmetry of shift 1:
+S=u2
+u1(u1−1)N/summationdisplay
+i=1pi(pi−1)∂
+∂pi+∞/summationdisplay
+i=1ui+1∂
+∂ui.
+As in the previous example, one can easily recover the whole GT-syst em. For example,
+∂iu3=/parenleftbiggu3(pi+u1−1)
+u1(u1−1)+2u2
+2pi(pi−1)
+u2
+1(u1−1)2/parenrightbigg
+∂iu1./square
+Below we describe the symmetry algebra for the case 5, c2(in particular, for the Benney
+chain).
+The case 5, c 2. For the triangular GT-system (1.7), (1.8) there exists an infinite L ie
+algebra of symmetries Si,i∈Z,whereSiis a symmetry of shift i. The simplest symmetries
+are the following:
+S−2=∂
+∂u1+∞/summationdisplay
+i=3/parenleftBig
+−ui−2+/summationdisplay
+k+m=i−3ukum−/summationdisplay
+k+m+l=i−4ukumul+···/parenrightBig∂
+∂ui,
+S−1=N/summationdisplay
+j=1∂
+∂pj+∞/summationdisplay
+i=1(i−1)ui−1∂
+∂ui,
+S0=N/summationdisplay
+j=1pj∂
+∂pj+∞/summationdisplay
+i=1(i+1)ui∂
+∂ui,
+S1=N/summationdisplay
+j=1(p2
+j+3u1)∂
+∂pj+∞/summationdisplay
+i=1(i+3)ui+1∂
+∂ui+∞/summationdisplay
+i=2/summationdisplay
+k+m=iukum∂
+∂ui+∞/summationdisplay
+i=23(i−1)u1ui−1∂
+∂ui,
+S2=N/summationdisplay
+j=1(p3
+j+4u1pj+5u2)∂
+∂pj+∞/summationdisplay
+i=1(i+5)ui+2∂
+∂ui+∞/summationdisplay
+i=14iu1ui∂
+∂ui+∞/summationdisplay
+i=25(i−1)u2ui−1∂
+∂ui+
+18∞/summationdisplay
+i=1/summationdisplay
+k+m=i+13ukum∂
+∂ui+∞/summationdisplay
+i=3/summationdisplay
+k+m+l=iukumul∂
+∂ui.
+The whole algebra is generated by S1,S2,S−1,S−2.It is isomorphic to the Virasoro algebra with
+zero central charge.
+LetDtibe the vector fields corresponding to commuting flows for the Benn ey chain. Here
+Dt1=Dx, Dt2=Dt. Then the commutator relations
+[S1,Dti] = (i+1)Dti+1
+hold. Thus the vector field S1plays the role of a master-symmetry for the Benney hierarchy.
+/square
+The case 6 . In this case there exist infinitesimal symmetries of form
+Ti=ui+1∂
+∂u1+ui+2∂
+∂u2+..., i= 0,1,2,...
+Si=N/summationdisplay
+j=1pi+1
+j∂
+∂pj+ui+2∂
+∂u2+2ui+3∂
+∂u3+3ui+4∂
+∂u4+..., i=−1,0,1,2,...
+Note that [Si,Sj] = (j−i)Si+j,[Ti,Tj] = 0,[Si,Tj] =jTi+j./square
+References
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+[16]B.A. Kupershmidt , Deformations of integrable systems, Proc. Roy. Irish Acad. Sec t. A,
+83(1) (1983) 45-74.
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+20
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+arXiv:1001.0021v3  [cond-mat.quant-gas]  8 Oct 2010Strong-coupling expansionforthe two-species Bose-Hubba rd model
+M. Iskin
+Department of Physics, Koc ¸ University, Rumelifeneri Yolu , 34450 Sariyer, Istanbul, Turkey
+(Dated: August 28, 2018)
+Toanalyze the ground-state phase diagram ofBose-Bose mixt ures loadedinto d-dimensional hypercubic op-
+tical lattices, we perform a strong-coupling power-series expansion in the kinetic energy term (plus a scaling
+analysis) for the two-species Bose-Hubbard model with onsi te boson-boson interactions. We consider both
+repulsive and attractive interspecies interaction, and ob tain an analytical expression for the phase boundary be-
+tweentheincompressibleMottinsulatorandthecompressib lesuperfluidphaseuptothirdorderinthehoppings.
+In particular, we find a re-entrant quantum phase transition from paired superfluid (superfluidity of composite
+bosons, i.e. Bose-Bose pairs) to Mott insulator and again to a paired superfluid in all one, two and three di-
+mensions, whentheinterspecies interactionissufficientl ylargeandattractive. Wehope thatsome ofourresults
+couldbe testedwithultracoldatomic systems.
+PACS numbers: 03.75.-b, 37.10.Jk,67.85.-d
+I. INTRODUCTION
+Single-species Bose-Hubbard (BH) model is the bosonic
+generalization of the Hubbard model, and was introduced
+originallytodescribe4Heinporousmediaordisorderedgran-
+ular superconductors [1]. For hypercubic lattices in all di -
+mensions d, there are only two phases in this model: an in-
+compressible Mott insulator at commensurate (integer) fill -
+ings and a compressible superfluid phase otherwise. The su-
+perfluid phase is well described by weak-coupling theories,
+buttheinsulatingphaseisastrong-couplingphenomenonth at
+only appearswhen the system is on a lattice. Transition from
+the Mott insulator to the superfluid phase occurs as the hop-
+ping, particle-particleinteraction,or the chemical pote ntial is
+varied[1].
+It is the recent observation of this transition in effective ly
+three- [2], one- [3], and two-dimensional [4, 5] optical lat -
+tices, which has been considered one of the most remarkable
+achievements in the field of ultracold atomic gases, since it
+paved the way for studying other strongly correlated phases
+in similar setups. Such lattices are created by the intersec tion
+of laser fields, and they are nondissipative periodic potent ial
+energy surfaces for the atoms. Motivated by this success in
+experimentally simulating the single-species BH model wit h
+ultracoldatomic Bose gasesloaded into optical lattices, t here
+has been recently an intense theoretical activity in analyz ing
+BH aswell asFermi-Hubbardtypemodels[6].
+For instance, in addition to the Mott insulator and single-
+species superfluid phases, it has been predicted that the two -
+species BH model has at least two additional phases: an in-
+compressible super-counter flow and a compressible paired
+superfluidphase[7–16]. Ourmaininteresthereisinthelatt er
+phase,wherea directtransitionfromtheMott insulatorto t he
+paired superfluid phase (superfluidity of composite bosons,
+i.e. Bose-Bose pairs) has been predicted, when both species
+have integer fillings and the interspecies interaction is su ffi-
+ciently large and attractive. Given that the interspecies i nter-
+actions can be fine tuned in ongoing experiments, e.g. with
+41K-87Rb with mixtures [17, 18], via using Feshbach reso-
+nances,we hopethat someof ourresults couldbe tested with
+ultracoldatomicsystems.Inthispaper,weexaminetheground-statephasediagramof
+the two-species BH model with on-site boson-boson interac-
+tionsind-dimensionalhypercubiclattices, includingboth the
+repulsive and attractive interspecies interaction, via a s trong-
+coupling perturbation theory in the hopping. We carry the
+expansion out to third-order in the hopping, and perform a
+scaling analysis using the known critical behavior at the ti p
+of the insulating lobes, which allows us to accurately predi ct
+the critical point, and the shape of the insulating lobes in t he
+plane of the chemical potential and the hopping. This tech-
+niquewaspreviouslyusedtodiscussthephasediagramofthe
+single-species BH model [19–23], extended BH model [24],
+and of the hardcore BH model with a superlattice [25], and
+its results showed an excellent agreement with Monte Carlo
+simulations [23, 25]. Motivated by the success of this tech-
+nique with these models, here we apply it to the two-species
+BH model, hoping to develop an analytical approach which
+couldbeasaccurateasthenumericalones.
+The remaining paper is organized as follows. After in-
+troducing the model Hamiltonian in Sec. II, we develop the
+strong-coupling expansion in Sec. III, where we derive an
+analytical expression for the phase boundary between the in -
+compressible Mott insulator and the compressible superflui d
+phase. Then, in Sec. IV, we proposea chemical-potentialex-
+trapolation technique based on scaling theory to extrapola te
+ourthird-orderpower-seriesexpansioninto a functionalf orm
+thatisappropriatefortheMottlobes,anduse ittoobtainty p-
+ical ground-state phase diagrams. A brief summary of our
+conclusionsisgiveninSec.V.
+II. TWO-SPECIESBOSE-HUBBARDMODEL
+TodescribeBose-Bosemixturesloadedintoopticallattice s,
+weconsiderthe followingtwo-speciesBH Hamiltonian,
+H=−/summationdisplay
+i,j,σtij,σb†
+i,σbj,σ+/summationdisplay
+i,σUσσ
+2/hatwideni,σ(/hatwideni,σ−1)
++U↑↓/summationdisplay
+i/hatwideni,↑/hatwideni,↓−/summationdisplay
+i,σµσ/hatwideni,σ, (1)2
+where the pseudo-spin σ≡ {↑,↓}labels the trapped hyper-
+fine states of a given species of bosons, or labels different
+types of bosons in a two-species mixture, tij,σis the tun-
+neling (or hopping) matrix between sites iandj,b†
+i,σ(bi,σ)
+is the boson creation (annihilation) and /hatwideni,σ=b†
+i,σbi,σis
+the boson number operator at site i,Uσσ′is the strength of
+the onsite boson-bosoninteraction between σandσ′compo-
+nents, and µσis the chemical potential. In this manuscript,
+we considera d-dimensionalhypercubiclattice with Msites,
+forwhich we assume tij,σis a real symmetricmatrix with el-
+ementstij,σ=tσ≥0foriandjnearest neighbors and 0
+otherwise. Thelattice coordinationnumber(orthe numbero f
+nearestneighbors)forsuchlatticesis z= 2d.
+We take the intraspecies interactions to be repulsive
+({U↑↑,U↓↓}>0), but discuss both repulsive and attractive
+interspecies interaction U↑↓as long as U↑↑U↓↓> U2
+↑↓. This
+guarantees the stability of the mixture against collapse wh en
+U↑↓≪0,andagainstphaseseparationwhen U↑↓≫0. How-
+ever,whentheinterspeciesinteractionissufficientlylar geand
+attractive, we note that instead of a direct transition from the
+Mottinsulatortoasingleparticlesuperfluidphase,itispo ssi-
+bletohaveatransitionfromtheMottinsulatortoa pairedsu -
+perfluid phase (superfluidity of composite bosons, i.e. Bose -
+Bose pairs) [7–16]. Therefore, one needs to consider both
+possibilities,asdiscussednext.
+III. STRONG-COUPLINGEXPANSION
+We use the many-body version of Rayleigh-Schr¨ odinger
+perturbation theory in the kinetic energy term to perform th e
+expansion (in powers of t↑andt↓) for the different energies
+needed to carryout our analysis. The strong-couplingexpan -
+sion technique was previously used to discuss the phase di-
+agram of the single-species BH model [19–21, 23], extended
+BHmodel[24],andofthehardcoreBHmodelwithasuperlat-
+tice [25], and its results showed an excellent agreement wit h
+Monte Carlo simulations [23, 25]. Motivated by the success
+of this technique with these models, here we apply it to the
+two-speciesBH model.
+To determine the phase boundary separating the incom-
+pressible Mott phase from the compressible superfluid phase
+within the strong-coupling expansion method, one needs the
+energyoftheMottphaseandofits‘defect’states(thosesta tes
+whichhaveexactlyoneextraelementaryparticleorholeabo ut
+the ground state) as a function of t↑andt↓. At the point
+where the energy of the incompressible state becomes equal
+to its defect state, the system becomes compressible, assum -
+ing that the compressibility approaches zero continuously at
+the phaseboundary. Here,we remarkthat thistechniquecan-
+notbeusedtocalculatethephaseboundarybetweentwocom-
+pressiblephases.A. Ground-StateWave Functions
+The perturbation theory is performed with respect to the
+ground state of the system when t↑=t↓= 0, and therefore
+we first need zeroth order wave functions of the Mott phase
+and of its defect states. To zerothorderin t↑andt↓, the Mott
+insulatorwavefunctioncanbewrittenas,
+|Ψins(0)
+Mott/an}bracketri}ht=1/radicalbig
+n↑!n↓!/productdisplay
+i(b†
+i,↑)n↑(b†
+i,↓)n↓|0/an}bracketri}ht,(2)
+where/an}bracketle{t/hatwideni,σ/an}bracketri}ht=nσis anintegernumbercorrespondingto the
+ground-stateoccupancyofthe pseudo-spin σbosons,/an}bracketle{t···/an}bracketri}htis
+thethermalaverage,and |0/an}bracketri}htisthevacuumstate. Ontheother
+hand, the wave functions of the defect states are determined
+by degenerate perturbation theory. The reason for that lies
+in the fact that when exactly one extra elementary particle o r
+hole is added to the Mott phase, it could go to any of the M
+lattice sites, since all of those states share the same energ y
+whent↑=t↓= 0. Therefore, the initial degeneracy of the
+defectstates isoforder M.
+Whentheelementaryexcitationsinvolveasingle- σ-particle
+(exactly one extra pseudo-spin σboson) or a single- σ-hole
+(exactly one less pseudo-spin σboson), this degeneracy is
+lifted at first order in t↑andt↓. The treatment for this case is
+very similar to the single-species BH model [19, 24], and the
+wave functions(to zerothorderin t↑andt↓) forthe single- σ-
+particleandsingle- σ-holedefectstates turnouttobe
+|Ψsσp(0)
+def/an}bracketri}ht=1√nσ+1/summationdisplay
+ifsσp
+ib†
+i,σ|Ψins(0)
+Mott/an}bracketri}ht,(3)
+|Ψsσh(0)
+def/an}bracketri}ht=1√nσ/summationdisplay
+ifsσh
+ibi,σ|Ψins(0)
+Mott/an}bracketri}ht, (4)
+wherefsσp
+i=fsσh
+iis the eigenvector of the hopping matrix
+tij,σwith the highest eigenvalue (which is ztσwithz= 2d)
+such that/summationtext
+jtij,σfsσp
+j=ztσfsσp
+i.The normalizationcondi-
+tion requires that/summationtext
+i|fsσp
+i|2= 1. Notice that we choose the
+highest eigenvalue of tij,σbecause the hoppingmatrix enters
+theHamiltonianas −tij,σ,andweultimatelywantthelowest-
+energystates.
+However,whentheelementaryexcitationsinvolvetwopar-
+ticles (exactly one extra boson of each species) or two holes
+(exactly one less boson of each species), the degeneracy is
+lifted at second order in t↑andt↓. Such elementary excita-
+tions occur when U↑↓is sufficiently large and attractive [26],
+and the wave functions (to zeroth order in t↑andt↓) for the
+two-particleandtwo-holedefectstatescanbewrittenas
+|Ψtp(0)
+def/an}bracketri}ht=1/radicalbig
+(n↑+1)(n↓+1)/summationdisplay
+iftp
+ib†
+i,↑b†
+i,↓|Ψins(0)
+Mott/an}bracketri}ht,(5)
+|Ψth(0)
+def/an}bracketri}ht=1√n↑n↓/summationdisplay
+ifth
+ibi,↑bi,↓|Ψins(0)
+Mott/an}bracketri}ht, (6)
+whereftp
+i=fth
+iturns out to be the eigenvector of the
+tij,↑tij,↓matrix with the highest eigenvalue (which is zt↑t↓
+withz= 2d)suchthat/summationtext
+jtij,↑tij,↓ftp
+j=zt↑t↓ftp
+i.Sincethe
+elementaryexcitationsinvolvetwo particlesor two holes, the3
+degeneratedefectstatescannotbeconnectedbyonehopping ,
+but rather require two hoppings to be connected. Therefore,
+oneexpectsthedegeneracytobeliftedatleastatsecondord er
+int↑andt↓, asdiscussednext.
+B. Ground-StateEnergies
+Next, we employ the many-body version of Rayleigh-
+Schr¨ odinger perturbation theory in t↑andt↓with respect to
+the ground state of the system when t↑=t↓= 0, and cal-
+culate the energy of the Mott phase and of its defect states.
+The energy of the Mott state is obtained via nondegenerate
+perturbation theory, and to third order in t↑andt↓it is given
+by
+Eins
+Mott
+M=/summationdisplay
+σUσσ
+2nσ(nσ−1)+U↑↓n↑n↓−/summationdisplay
+σµσnσ
+−/summationdisplay
+σnσ(nσ+1)zt2
+σ
+Uσσ+O(t4). (7)Thisis anextensivequantity,i.e. Eins
+Mottis proportionalto the
+number of lattice sites M. The odd-order terms in t↑andt↓
+vanishforthe d-dimensionalhypercubiclatticesconsideredin
+thismanuscript,whichissimplybecausetheMott state give n
+in Eq. (2) cannot be connected to itself by only one hopping,
+but ratherrequirestwo hoppingsto be connected. Notice tha t
+Eq. (7) recovers the known result for the single-species BH
+modelwhenoneofthepseudo-spincomponentshavevanish-
+ingfilling,e.g. n↓= 0[19,24].
+Thecalculationofthedefect-stateenergiesismoreinvolv ed
+since it requires using degenerate perturbation theory. As
+mentioned above, when the elementary excitations involve a
+single-σ-particleorasingle- σ-hole,thedegeneracyisliftedat
+firstorderin t↑andt↓. Alengthybutstraightforwardcalcula-
+tionleadstotheenergyofthesingle- σ-particledefectstateup
+tothirdorderin t↑andt↓as
+Esσp
+def=Eins
+Mott+U↑↓n−σ+Uσσnσ−µσ−(nσ+1)ztσ
+−nσ/bracketleftbiggnσ+2
+2+(nσ+1)(z−3)/bracketrightbiggzt2
+σ
+Uσσ−2n−σ(n−σ+1)U2
+↑↓
+U2
+−σ−σ−U2
+↑↓zt2
+−σ
+U−σ−σ
+−nσ(nσ+1)/bracketleftbig
+nσ(z−1)2+(nσ+1)(z−1)(z−4)+(nσ+2)(3z/4−1)/bracketrightbigzt3
+σ
+U2σσ
+−4(nσ+1)n−σ(n−σ+1)U2
+↑↓
+U2
+−σ−σ−U2
+↑↓/parenleftBigg
+z−1−U2
+−σ−σ
+U2
+−σ−σ−U2
+↑↓/parenrightBigg
+ztσt2
+−σ
+U2
+−σ−σ+O(t4), (8)
+where(− ↑)≡↓and vice versa. Here, we assume Uσσ≫tσand{U−σ−σ,|U−σ−σ±U↑↓|} ≫t−σ. Equation(8) is valid for
+alld-dimensionalhypercubiclattices,andit recoversthe know nresult forthesinglespeciesBH modelwhen n−σ= 0[19, 24].
+Note that this expression also recovers the known result for the single species BH model when U↑↓= 0, which provides an
+independentcheckofthe algebra. To thirdorderin t↑andt↓, we obtaina similarexpressionfortheenergyofthe single- σ-hole
+defectstate givenby
+Esσh
+def=Eins
+Mott−U↑↓n−σ−Uσσ(nσ−1)+µσ−nσztσ
+−(nσ+1)/bracketleftbiggnσ−1
+2+nσ(z−3)/bracketrightbiggzt2
+σ
+Uσσ−2n−σ(n−σ+1)U2
+↑↓
+U2
+−σ−σ−U2
+↑↓zt2
+−σ
+U−σ−σ
+−nσ(nσ+1)/bracketleftbig
+(nσ+1)(z−1)2+nσ(z−1)(z−4)+(nσ−1)(3z/4−1)/bracketrightbigzt3
+σ
+U2σσ
+−4nσn−σ(n−σ+1)U2
+↑↓
+U2
+−σ−σ−U2
+↑↓/parenleftBigg
+z−1−U2
+−σ−σ
+U2
+−σ−σ−U2
+↑↓/parenrightBigg
+ztσt2
+−σ
+U2
+−σ−σ+O(t4), (9)
+which is also valid for all d-dimensional hypercubic lattices, and it also recovers the known result for the single-species BH
+modelwhen n−σ= 0orU↑↓= 0[19, 24]. Here, we againassume Uσσ≫tσand{U−σ−σ,|U−σ−σ±U↑↓|} ≫t−σ. We also
+checkedtheaccuracyofthesecond-ordertermsinEqs.(8)an d(9)viaexactsmall-cluster(two-site)calculationswith oneσand
+two−σparticles.
+We note that the mean-field phase boundarybetween the Mott ph ase and its single- σ-particle and single- σ-holedefect states
+canbecalculatedas
+µpar,hol
+σ=Uσσ(nσ−1/2)+U↑↓n−σ−ztσ/2±/radicalbig
+U2σσ/4−Uσσ(nσ+1/2)ztσ+z2t2σ/4. (10)4
+This expression is exact for infinite-dimensionalhypercub iclattices, and it recoversthe knownresult for the single s pecies BH
+model when n−σ= 0orU↑↓= 0[1]. In the d→ ∞limit (while keeping dtσconstant), we checked that our strong-coupling
+perturbationresultsgiveninEqs.(8)and(9)agreewiththi sexactsolutionwhenthelatterisexpandedouttothirdorde rint↑and
+t↓,providinganindependentcheckofthealgebra. Equation(1 0)alsoshowsthat,forinfinite-dimensionallattices,theM ottlobes
+are separatedby U↑↓n−σ, but theirshapesandcritical points(thelatter are obtain edbysetting µpar
+σ=µhol
+σ) are independentof
+U↑↓. This is not the case for finite-dimensional lattices as can b e clearly seen from our results. It is also important to menti on
+herethat boththe shapesandcritical pointsare independen tofthe sign of U↑↓in finite dimensions(at the third-orderpresented
+here)ascanbeseen inEqs.(8) and(9).
+However, when the elementary excitations involve two parti cles or two holes (which occurs when U↑↓is sufficiently large
+and attractive [26]), the degeneracyis lifted at second ord erint↑andt↓. A lengthybut straightforwardcalculationleads to the
+energyofthetwo-particledefectstate uptothirdorderin t↑andt↓as
+Etp
+def=Eins
+Mott+U↑↓(n↑+n↓+1)+/summationdisplay
+σ(Uσσnσ−µσ)+2(n↑+1)(n↓+1)
+U↑↓zt↑t↓
++/summationdisplay
+σ/bracketleftbigg(nσ+1)2
+U↑↓−nσ(nσ+2)
+2Uσσ+U↑↓+2nσ(nσ+1)
+Uσσ/bracketrightbigg
+zt2
+σ+O(t4). (11)
+Here, we assume {Uσσ,|U↑↓|,2Uσσ+U↑↓} ≫tσ. Equation (11) is valid for all d-dimensional hypercubiclattices, where the
+odd-ordertermsin t↑andt↓vanish[27]. Tothirdorderin t↑andt↓,weobtainasimilarexpressionfortheenergyofthetwo-hol e
+defectstate givenby
+Eth
+def=Eins
+Mott−U↑↓(n↑+n↓−1)−/summationdisplay
+σ[Uσσ(nσ−1)−µσ]+2n↑n↓
+U↑↓zt↑t↓
++/summationdisplay
+σ/bracketleftbiggn2
+σ
+U↑↓−(n2
+σ−1)
+2Uσσ+U↑↓+2nσ(nσ+1)
+Uσσ/bracketrightbigg
+zt2
+σ+O(t4), (12)
+which is also valid for all d-dimensional hypercubic lattices,
+where the odd-order terms in t↑andt↓vanish [27]. Here,
+we again assume {Uσσ,|U↑↓|,2Uσσ+U↑↓} ≫tσ. Since
+the single- σ-particleandsingle- σ-holedefectstateshavecor-
+rections to first order in the hopping, while the two-particl e
+and two-hole defect states have corrections to second order
+in the hopping, the slopes of the Mott lobes are finite as
+{t↑,t↓} →0in the former case, but they vanish in the lat-
+tercase. Hence,theshapeoftheinsulatinglobesareexpect ed
+to be very different for two-particle or two-hole excitatio ns.
+In addition, the chemical-potential widths ( µσ) of all Mott
+lobes are Uσσin the former case, but they [ (µ↑+µ↓)/2] are
+U↑↓+(U↑↑+U↓↓)/2inthelatter.
+We note that in the limit when t↑=t↓=t,U↑↑=U↓↓=
+U0,U↑↓=U′,n↑=n↓=n0,µ↑=µ↓=µ, andz= 2
+(ord= 1), Eq. (12) is in complete agreementwith Eq. (3) of
+Ref. [11], providing an independent check of the algebra. In
+addition, in the limit when t↑=t↓=J,U↑↑=U↓↓=U,
+U↑↓=W≈ −U,n↑=n↓=m, andµ↑=µ↓=µ,
+Eqs. (11) and (12) reduce to those given in Ref. [12] (after
+settingUNN= 0there). However, the terms that are propor-
+tional tot↑t↓are not included in their definitions of the two-
+particle and two-hole excitation gaps. We also checked the
+accuracy of Eqs. (11) and (12) via exact small-cluster (two-
+site) calculationswithoneparticleofeachspecies.
+Wewouldalsoliketoremarkinpassingthattheenergydif-
+ferencebetweentheMottphaseanditsdefectstatesdetermi ne
+the phase boundaryof the particle and hole branches. This is
+because at the point where the energy of the incompressiblestate becomes equal to its defect state, the system becomes
+compressible, assuming that the compressibility approach es
+zero continuously at the phase boundary. While Eins
+Mottand
+its defects Esσp
+def,Esσh
+def,Etp
+defandEth
+defdepend on the lattice
+sizeM, their difference do not. Therefore, the chemical po-
+tentialsthatdeterminetheparticleandholebranchesarei nde-
+pendentof Mat thephaseboundaries. Thisindicatesthat the
+numerical Monte Carlo simulations should not have a strong
+dependenceon M.
+It is known that the third-order strong-coupling expansion
+isnotveryaccuratenearthetipoftheMottlobes,as t↑andt↓
+arenotverysmallthere[19,24]. Forthisreason,anextrapo la-
+tion technique is highly desirable to determine more accura te
+phase diagrams. Therefore, having discussed the third-ord er
+strong-coupling expansion for a general two-species Bose-
+Bose mixtures with arbitary hoppings tσ, interactions Uσσ′,
+densities nσ, and chemical potentials µσ, next we show how
+todevelopa scalingtheory.
+IV. EXTRAPOLATIONTECHNIQUE
+In this section, we propose a chemical potential extrapo-
+lation technique based on scaling theory to extrapolate our
+third-orderpower-seriesexpansionintoafunctionalform that
+is appropriate for the entire Mott lobes. It is known that the
+criticalpointatthetipofthelobeshasthescalingbehavio rof
+a(d+1)-dimensional XYmodel,andthereforethelobeshave5
+Kosterlitz-Thouless shapes for d= 1and power-law shapes
+ford >1. For illustration purposes, here we analyze only
+the latter case, but this techniquecan be easily adapted to t he
+d= 1case [19].
+A. ScalingAnsatz
+Fromnowonwe considera two-speciesmixturewith t↑=
+t↓=t,U↑↑=U↓↓=U,U↑↓=V,n↑=n↓=n, and
+µ↑=µ↓=µ. Whend >1, we proposethe followingansatz
+which includes the known power-law critical behavior of the
+tipofthe lobes
+µ±
+U=A(x)±B(x)(xc−x)zν, (13)
+whereA(x) =a+bx+cx2+dx3+···andB(x) =α+βx+
+γx2+δx3+···areregularfunctionsof x= 2dt/U,xcisthe
+critical point which determines the location of the lobes, a nd
+zνis the critical exponent for the ( d+ 1)-dimensional XY
+model which determines the shape of the lobes near xc=
+2dtc/U. In Eq. (13), the plus sign correspondsto the particle
+branch, and the minus sign corresponds to the hole branch.
+Theformoftheansatzistakentobethesameforbothsingle-
+and two-partice (or single- and two-hole) excitations, but the
+parametersareverydifferent.
+The parameters a,b,candddepend on U,Vandn, and
+they are determined by matching them with the coefficients
+givenbyourthird-orderexpansionsuchthat A(x) = (µpar+
+µhol)/(2U).Here,µparandµholare our strong-couplingex-
+pansion results determined from Eqs. (8) and (9) for the
+single-particle and single-hole excitations, or from Eqs. (11)
+and(12)forthetwo-particleandtwo-holeexcitations,res pec-
+tively. Writing our strong-coupling expansion results for the
+particleandhole branchesin the form µpar=U/summationtext3
+n=0e+
+nxn
+andµhol=U/summationtext3
+n=0e−
+nxn, leads to a= (e+
+0+e−
+0)/2,
+b= (e+
+1+e−
+1)/2,c= (e+
+2+e−
+2)/2, andd= (e+
+3+e−
+3)/2.
+To determine the U,Vandndependence of the parameters
+α,β,γ,δ,xcandzν, we first expand the left hand side of
+B(x)(xc−x)zν= (µpar−µhol)/(2U)in powers of x, and
+matchthecoefficientswiththecoefficientsgivenbyourthir d-
+orderexpansion,leadingto
+α=e+
+0−e−
+0
+2xzνc, (14)
+β
+α=zν
+xc+e+
+1−e−
+1
+e+
+0−e−
+0, (15)
+γ
+α=zν(zν+1)
+2x2c+zν
+xce+
+1−e−
+1
+e+
+0−e−
+0+e+
+2−e−
+2
+e+
+0−e−
+0,(16)
+δ
+α=zν(zν+1)(zν+2)
+6x3c+zν(zν+1)
+2x2ce+
+1−e−
+1
+e+
+0−e−
+0
++zν
+xce+
+2−e−
+2
+e+
+0−e−
+0+e+
+3−e−
+3
+e+
+0−e−
+0. (17)
+We fixzνat its well-known values such that zν≈2/3for
+d= 2andzν= 1/2ford >2. If the exact value of xcis known via other means, e.g. numerical simulations, α,β,
+γandδcan be calculated accordingly, for which the extrap-
+olation technique gives very accurate results [23, 25]. If t he
+exact value of xcis not known, then we set δ= 0, and solve
+Eqs. (14), (15), (16) and the δ= 0equation to determine
+α,β,γandxcself-consistently, which also leads to accurate
+results [19, 24]. Next we present typical ground-state phas e
+diagrams for (d= 2)- and (d= 3)-dimensional hypercubic
+latticesobtainedfromthisextrapolationtechnique.
+B. Numerical Results
+In Figs. 1 and 2, the results of the third-order strong-
+couplingexpansion(dottedlines)arecomparedtothoseoft he
+extrapolationtechnique(hollowpink-squaresandsolidbl ack-
+circles) when V= 0.5UandV=−0.85U, respectively, in
+two (d= 2orz= 4) andthree ( d= 3orz= 6) dimensions.
+We recall here that t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=V,
+n↑=n↓=n, andµ↑=µ↓=µ.
+In Fig. 1, we show the chemical potential µ(in units of U)
+versusx= 2dt/Uphasediagramfor(a)two-dimensionaland
+(b) three-dimensional hypercubic lattices, where we choos e
+the interspecies interaction to be repulsive V= 0.5U. Com-
+paring Eqs. (8) and (9) with Eqs. (11) and (12), we expect
+that the excited state of the system to be the usual superfluid
+for allV >0for allt. The dotted lines correspond to phase
+boundary for the Mott insulator to superfluid state as deter-
+mined from the third-order strong-coupling expansion, and
+the hollow pink-squares correspond to the extrapolation fit s
+forthesingle-particleandsingle-holeexcitationsdiscu ssedin
+the text. We recall here that an incompressible super-count er
+flow phase [7–9, 13] also exists outside of the Mott insulator
+lobes, but our current formalism cannot be used to locate its
+phaseboundary.
+TABLE I. List of the critical points (location of the tips) xc=
+2dtc/Ufor the first two Mott insulator lobes that are found from
+the chemical potential extrapolation technique described in the text.
+Here,t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=V,n↑=n↓=n, and
+µ↑=µ↓=µ. These critical points for the single-particle or single-
+hole excitations are determined from Eqs. (8) and (9), and th ey tend
+tomove inas Vincreases, andare independent of the signof V.
+d= 2 d= 3
+V/Un= 1n= 2n= 1n= 2
+0.00.234 0.138 0.196 0.116
+0.10.234 0.138 0.196 0.115
+0.20.233 0.137 0.195 0.115
+0.30.230 0.136 0.194 0.114
+0.40.227 0.134 0.193 0.113
+0.50.223 0.131 0.190 0.112
+0.60.217 0.128 0.187 0.110
+0.70.208 0.123 0.182 0.107
+0.80.197 0.116 0.174 0.102
+0.90.193 0.113 0.163 0.0956
+ 0 1.5 3 4.5
+ 0  0.09  0.18  0.27µ/U
+x = 2dt/U(a) Two dimensions (V=0.5U)
+n=1n=2n=3sp/sh ext
+third order
+ 0 1.5 3 4.5
+ 0  0.09  0.18  0.27µ/U
+x = 2dt/U(a) Two dimensions (V=0.5U)
+sp/sh ext
+third order
+ 0 1.5 3 4.5
+ 0  0.09  0.18  0.27µ/U
+x = 2dt/U(b) Three dimensions (V=0.5U)
+n=1n=2n=3sp/sh ext
+third order
+ 0 1.5 3 4.5
+ 0  0.09  0.18  0.27µ/U
+x = 2dt/U(b) Three dimensions (V=0.5U)
+sp/sh ext
+third order
+FIG. 1. (Color online) Chemical potential µ(in units of U) versus
+x= 2dt/Uphase diagram for (a) two- and (b) three-dimensional
+hypercubic lattices with t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=
+V= 0.5U,n↑=n↓=n, andµ↑=µ↓=µ. The dotted lines
+correspond to phase boundary for the Mott insulator to super fluid
+state as determined from the third-order strong-coupling e xpansion,
+and the hollow pink-squares to the extrapolation fit for the s ingle-
+particle or single-hole excitations discussed in the text. Recall that
+anincompressiblesuper-counterflowphasealsoexistsouts ideofthe
+Mott insulator lobes.
+Att= 0, the chemical potential width of all Mott lobes
+areU(similar to the single-species BH model), but they are
+separated from each other by Vas a function of µ. Astin-
+creasesfromzero,therangeof µaboutwhichthegroundstate
+is a Mott insulator decreases, and the Mott insulator phasedisappears at a critical value of t, beyond which the system
+becomes a superfluid. In addition, similar to what was found
+forthesingle-speciesBH model[19,24],thestrong-coupli ng
+expansionoverestimatesthe phase boundaries,and it leads to
+unphysical pointed tips for all Mott lobes, which is expecte d
+since a finite-order expansion cannot describe the physics o f
+thecriticalpointcorrectly. Ashortlistof V/Uversusthecrit-
+ical points xc= 2dtc/Uis presented for the first two Mott
+insulator lobes in Table I, where it is shown that the criti-
+cal points tend to move in as Vincreases. This is because
+presence of a second species (say −σones) screens the on-
+site intraspeciesrepulsion Uσσbetweenσ-species, and hence
+increasesthesuperfluidregion.
+In Fig. 2, we show the chemical potential µ(in units of
+U)versusx= 2dt/Uphasediagramfor(a) two-dimensional
+and (b) three-dimensionalhypercubiclattices, where in th ese
+figures we choose the interspecies interaction to be attract ive
+V=−0.85U. Comparing Eqs. (8) and (9) with Eqs. (11)
+and (12), we expect that the excited state of the system to
+be a paired superfluid for all V <0whent→0. This is
+clearlyseen inthefigurewherethedottedlinescorrespondt o
+phaseboundaryfortheMottinsulatortosuperfluidstateasd e-
+termined from the third-orderstrong-couplingexpansion, the
+hollow pink-squares correspond to the extrapolation fits fo r
+thesingle-particleandsingle-holeexcitations(shownon lyfor
+illustration purposes), and the solid black-circles corre spond
+to the extrapolation fits for the two-particle and two-hole e x-
+citations(thisisthe expectedtransition)discussedin th etext.
+Att= 0, the chemical potential width of all Mott lobes
+areV+U= 0.15U, which is in contrast with the single-
+species BH model. As tincreases from zero, the range of µ
+aboutwhichthegroundstateisaMottinsulatordecreaseshe re
+as well, and the Mott insulator phase disappears at a critica l
+value oft, beyondwhich the system becomesa paired super-
+fluid. The strong-couplingexpansionagain overestimatest he
+phaseboundaries,anditagainleadstounphysicalpointedt ips
+for all Mott lobes. In addition, a short list of V/Uversus the
+critical points xc= 2dtc/Uare presented for the first two
+MottinsulatorlobesinTableI. Ourresultsareconsistentw ith
+the expectation that, for small V, the locations of the tips in-
+crease as a function of V, because the presence of a nonzero
+Viswhatallowedthesestatestoforminthefirstplace. How-
+ever, when Vis largerthan some critical value ( ∼0.6U), the
+locationsofthetipsdecrease,andtheyeventuallyvanishw hen
+V=−U. Thismay indicatean instabilitytowardsa collapse
+sinceat thispoint U↑↑U↓↓is exactlyequalto U2
+↑↓.
+Compared to the V >0case shown in Fig. 1, note that
+shape of the Mott insulator to paired superfluidphase bound-
+ary is very different, showing a re-entrant behavior in all d i-
+mensions from paired superfluid to Mott insulator and again
+to a paired superfluid phase, as a function of t. Our results
+are consistent with an early numerical time-evolving block
+decimation (TEBD) calculation [11], where such a re-entran t
+quantumphasetransitionin onedimensionwaspredicted.
+The re-entrant quantum phase transition occurs when co-
+efficient of the hopping term in Eq. (12) is negative [so7
+-0.45-0.3-0.15 0
+ 0 0.1 0.2 0.3 0.4µ/U
+x = 2dt/U(a) Two dimensions (V=-0.85U)
+n=1n=2n=3tp/th ext
+sp/sh ext
+third order
+-0.45-0.3-0.15 0
+ 0 0.1 0.2 0.3 0.4µ/U
+x = 2dt/U(a) Two dimensions (V=-0.85U)
+n=1n=2n=3tp/th ext
+sp/sh ext
+third order
+-0.45-0.3-0.15 0
+ 0 0.1 0.2 0.3 0.4µ/U
+x = 2dt/U(b) Three dimensions (V=-0.85U)
+n=1n=2n=3tp/th ext
+sp/sh ext
+third order
+-0.45-0.3-0.15 0
+ 0 0.1 0.2 0.3 0.4µ/U
+x = 2dt/U(b) Three dimensions (V=-0.85U)
+n=1n=2n=3tp/th ext
+sp/sh ext
+third order
+FIG. 2. (Color online) Chemical potential µ(in units of U) versus
+x= 2dt/Uphase diagram for (a) two- and (b) three-dimensional
+hypercubic lattices with t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=
+V=−0.85U,n↑=n↓=n, andµ↑=µ↓=µ. The dotted lines
+correspond to phase boundary for the Mott insulator to super fluid
+statedeterminedfromthethird-order strong-coupling exp ansion, the
+hollow pink-squares to the extrapolation fit for the single- particle or
+single-hole excitations (shown only for illustration purp oses), and
+the solid black-circles to the extrapolation fit for the two- particle or
+two-hole excitations (the expected transition) discussed inthe text.
+that the two-hole excitation branch has a negative slope in
+(µ↑+µ↓)/2versustσphase diagram when tσ→0], i.e.
+−(2n↑n↓/U↑↓)zt↑t↓−/summationtext
+σ[n2
+σ/U↑↓−(n2
+σ−1)/(2Uσσ+
+U↑↓)+2nσ(nσ+1)/Uσσ]zt2
+σterm,whichoccursforthefirst
+few Mott lobes beyond a critical U↑↓. When this coefficient
+is negative, its value is most negative for the first Mott lobe ,TABLE II. List of the critical points (location of the tips) xc=
+2dtc/Uthat are found from the chemical potential extrapolation
+techniquedescribedinthetext. Here, t↑=t↓=t,U↑↑=U↓↓=U,
+U↑↓=V,n↑=n↓=n, andµ↑=µ↓=µ. These critical
+points for the two-particle or two-hole excitations are det ermined
+from Eqs. (11) and (12) when V <0. Note that, for small V,xc’s
+tend to increase as a function of V, since the presence of a nonzero
+Vis what allowed these states to form in the first place. Howeve r,
+xc’s decrease beyond a critical V, and they eventually vanish when
+V=−U,which mayindicate an instabilitytowards a collapse.
+d= 2 d= 3
+V/Un= 1n= 2n= 1n= 2
+-0.010.0543 0.0337 0.0611 0.0379
+-0.030.0937 0.0582 0.105 0.0655
+-0.050.121 0.0749 0.136 0.0843
+-0.070.142 0.0883 0.160 0.0994
+-0.10.169 0.105 0.190 0.118
+-0.20.233 0.145 0.262 0.164
+-0.30.277 0.173 0.311 0.195
+-0.40.307 0.193 0.345 0.217
+-0.50.325 0.205 0.366 0.230
+-0.60.331 0.209 0.372 0.235
+-0.70.321 0.203 0.362 0.228
+-0.80.291 0.183 0.327 0.206
+-0.90.225 0.141 0.253 0.159
+-0.930.193 0.121 0.217 0.136
+-0.950.166 0.103 0.187 0.116
+-0.970.1304 0.0812 0.147 0.0913
+-0.990.0764 0.0474 0.0860 0.0534
+and thereforethe effect is strongest there. However,the co ef-
+ficientincreasesandeventuallybecomespositiveasafunct ion
+offilling,andthusthere-entrantbehaviorbecomesweakera s
+fillingincreases,anditeventuallydisappearsbeyondacri tical
+filling. For the parametersused in Fig. 2, this occursonlyfo r
+the first lobe, as can be seen in the figures. We also note that
+the sign of this coefficientis independentof the dimensiona l-
+ity of the lattice, since z= 2dentersinto the coefficient only
+asanoverallfactor.
+What happenswhen t↑/ne}ationslash=t↓and/orU↑↑/ne}ationslash=U↓↓? We donot
+expectany qualitativechangefor attractiveinterspecies inter-
+actions. However, for repulsive interspecies interaction s, this
+lifts the degeneracyof the single-particle or single-hole exci-
+tation energies. While the transition is from a double Mott
+insulator to a double superfluid of both species in the degen-
+erate case, it is from a double-Mott insulator of both specie s
+toaMottinsulatorofonespeciesandasuperfluidoftheother
+inthenondegeneratecase.
+V. CONCLUSIONS
+We analyzed the zero temperature phase diagram of the
+two-species Bose-Hubbard (BH) model with on-site boson-
+boson interactions in d-dimensional hypercubic lattices, in-8
+cluding both the repulsive and attractive interspecies in-
+teraction. We used the many-body version of Rayleigh-
+Schr¨ odinger perturbation theory in the kinetic energy ter m
+with respect to the ground state of the system when the ki-
+netic energy term is absent, and calculate ground state ener -
+gies needed to carry out our analysis. This technique was
+previously used to discuss the phase diagram of the single-
+speciesBH model[19–21, 23], extendedBH model[24],and
+of the hardcore BH model with a superlattice [25], and its
+resultsshowedanexcellentagreementwithMonteCarlosim-
+ulations [23, 25]. Motivated by the success of this techniqu e
+with these models, here we generalized it to the two-species
+BH model, hoping to develop an analytical approach which
+couldbeasaccurateasthe numericalones.
+We derived analytical expressions for the phase boundary
+betweentheincompressibleMottinsulatorandthecompress -
+iblesuperfluidphaseuptothirdorderinthehoppings. Weals o
+proposed a chemical potential extrapolation technique bas ed
+on the scaling theory to extrapolateour third-orderpower s e-
+riesexpansionintoafunctionalformthatisappropriatefo rthe
+Mott lobes. In particular, when the interspecies interacti on is
+sufficiently large and attractive, we found a re-entrant qua n-
+tum phase transition from paired superfluid (superfluidity o f
+compositebosons,i.e. Bose-Bosepairs)toMottinsulatora nd
+again to a paired superfluid in all one, two and three dimen-sions. SincetheavailableMonteCarlocalculations[9,10] do
+not provide the Mott insulator to superfluid transition phas e
+boundary in the experimentally more relevant chemical po-
+tentialversushoppingplane,wecouldnotcompareourresul ts
+with them. This comparison is highly desirable to judge the
+accuracyofourstrong-couplingexpansionresults.
+A possible direction to extend this work is to consider the
+limit where hopping of one-species is much larger than the
+other. In this limit, the two-species BH model reduces to
+theBose-BoseversionoftheFalicov-Kimballmodel[28],th e
+Fermi-Fermi version of which has been widely discussed in
+the condensed-matter literature and the Fermi-Bose versio n
+has just been studied [29]. It is known for such models that
+thereisa tendencytowardsbothphaseseparationanddensit y
+wave order [30], which requires a new calculation partially
+similar to that of Ref. [24]. One can also examine how the
+momentumdistributionchangeswiththehoppingintheinsu-
+latingphases[23, 31], whichhasdirect relevanceto ultrac old
+atomicexperiments.
+VI. ACKNOWLEDGMENTS
+The author thanks Anzi Hu, L. Mathey and J. K. Freer-
+icksfordiscussions,andTheScientificandTechnologicalR e-
+searchCouncilofTurkey(T ¨UB˙ITAK)forfinancialsupport.
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+↑↓cannot be greaterthanorequalto U↑↑U↓↓,oth-
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+tations where dtσis a constant when d→ ∞, in the case of
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\ No newline at end of file
diff --git a/1001.0022.txt b/1001.0022.txt
new file mode 100644
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+arXiv:1001.0022v2  [hep-ph]  17 Mar 2010Preprint typeset in JHEP style - HYPER VERSION MADPH–09-1552
+µτProduction at Hadron Colliders
+Tao Han∗, Ian Lewis†
+Department of Physics, University of Wisconsin, Madison, W I 53706, U.S.A.
+Marc Sher‡
+Particle Theory Group, College of William and Mary, William sburg, Virginia 23187
+Abstract: Motivated by large νµ−ντflavor mixing, we consider µτproduction at hadron
+colliders via dimension-6 effective operators, which can be a ttributed to new physics in the
+flavor sector at a higher scale Λ. Current bounds on many of the se operators from low energy
+experiments are very weak or nonexistent, and they may lead t o cleanµ+τ−andµ−τ+signals
+at hadron colliders. At the Tevatron with 8 fb−1, one can exceed current bounds for most
+operators, with most 2 σsensitivities being in the 6 −24 TeV range. We find that at the LHC
+with 1 fb−1(100 fb−1) integrated luminosity, one can reach a2 σsensitivity for Λ ∼3−10 TeV
+(Λ∼6−21 TeV), depending on the Lorentz structure of the operator. For some operators,
+an improvement of several orders of magnitude in sensitivit y can be obtained with only a few
+tens of pb−1at the LHC.
+Keywords: Lepton flavor physics; Hadron collider phenomenology..
+∗than@hep.wisc.edu
+†ilewis@wisc.edu
+‡mtsher@wm.eduContents
+1. Introduction 1
+2.µτProduction at Hadron Colliders 3
+3. Signal Identification and Backgrounds 4
+3.1τDecay to Electrons 5
+3.1.1 Signal Reconstruction 5
+3.1.2 Backgrounds and their Suppression 6
+3.2τDecay to Hadrons 9
+3.3 Sensitivity Reach at the Tevatron 10
+3.4 Sensitivity Reach at the LHC 10
+4. Discussions and Conclusions 13
+A. New Physics Bounds 14
+B. Partial Wave Unitarity Bounds 14
+1. Introduction
+The most important discovery in particle physics in the past decade has only deepened the
+mystery of “flavor” of quarks and leptons. The fact that the mi xing angles in the leptonic
+sector are large [1, 2] stands in sharp contrast with the obse rved small mixing angles in the
+quarksector. Inparticular, mixingbetweenthesecondandt hirdgeneration neutrinosappears
+to be maximal. Of course, this large mixing could occur from d iagonalizing the neutrino mass
+matrix, the charged lepton mass matrix, or both. At present, the source of this large mixing
+is a mystery.
+In view of this, it is tempting to explore other interactions which change lepton flavor
+between the second and third generations. Several years ago , two of us (TH, MS), along with
+Black and He (BHHS) [3], performed a comprehensive analysis of constraints on these inter-
+actions based on low energy meson physics. BHHS chose an effect ive field theory approach,
+in which all dimension-6 operators of the form
+(¯µΓτ)(¯qαΓqβ), (1.1)
+– 1 –were studied, where Γ contains possible Dirac γ-matrices. With six flavors of quarks, there
+were 12 possible combinations of qaandqb(assuming Hermiticity), six diagonal and six off-
+diagonal, and four choices S,P,V,A of the gamma matrices were considered. All of these
+operators were considered, and most were bounded by conside ringτ,K,Bandtdecays.
+In particular, BHHS considered operators of the form
+∆L= ∆L(6)
+τµ=/summationdisplay
+j,α,βCj
+αβ
+Λ2(µΓjτ)/parenleftBig
+qαΓjqβ/parenrightBig
++ H.c., (1.2)
+where Γ j∈(1, γ5, γσ, γσγ5) denotes relevant Dirac matrices, specifying scalar, pseu doscalar,
+vector and axial vector couplings, respectively. They did n ot consider tensor operators since
+the hadronic matrix elements were not known and the bounds we re expected to be weak in
+any event. They chose a value of
+Cj
+αβ= 4πO(1) (default) , (1.3)
+which corresponds to an underlying theory with a strong gaug e coupling of αS=O(1).
+Arguments can be made for multiplying or dividing this by 4 π, for naive dimensional analysis
+or for weakly coupled theories, respectively. A discussion is found in BHHS; we simply choose
+the above definition of Λ and other choices can be made by simpl e rescaling.
+Besides the four fermion operators in Eq. (1.2), there may be other induced operators
+involving the SM gauge bosons, such as the electroweak trans ition operator
+∆L=κv
+Λ2¯µσµντFµν, (1.4)
+wherevis the vacuum expectation value of the Standard Model Higgs fi eld andFµνis the
+electroweak field tensor. However, when these operators are compared to the underlying
+new strong dynamics of the four fermion interaction in Eq. (1 .2), it is found that they are
+suppressed by O(MW/Λ), where MWis the mass of the electroweak gauge boson. For new
+physics scales of order 1 TeV or greater, this is at least an or der of magnitude suppression.
+Thus, we ignore these operators.
+BHHS found that operators involving the three lightest quar ks were strongly bounded,
+with bounds ranging from 3 to 13 TeV on the related value of Λ. T hese bounds can be found
+in Appendix A. Not surprisingly, operators involving the to p quark were either unbounded or
+very weakly bounded, with only the tuoperator for vector and axial vector couplings being
+bounded by Λ <650 GeV (the bound arises through a loop in B→µτdecay). Operators
+involving the b-quark and a light quark also have bounds on Λ which were gener ally in
+the several TeV range. However, there were some surprises. T he scalar and pseudoscalar
+operators involving cuandccwere completely unbounded, and the bboperator was essentially
+unbounded for all S,P,V,A operators. And, as noted above, noneof the tensor operators
+were considered at all, for all quark combinations.
+In this note, we point out that the operators in Eq. (1.1) (wit hout involving top quarks)
+will contribute to µ−τproduction at hadron colliders. Given that many of the possi ble
+– 2 –operators, as noted above, are completely unbounded or weak ly bounded from the current
+low energy data, study of pp→µτat the LHC or pp→µτat the Tevatron will probe
+unexplored territory.
+There have been some previous discussions of µ−τproduction at hadron colliders. Han
+and Marfatia [4] looked at the lepton-violating decay h→µτat hadron colliders, and a very
+detailed analysis of signals and backgrounds was carried ou t by Assamagan et al. [5] after-
+wards. Other work looking at Higgs decays focused on mirror f ermions [6], supersymmetric
+models [7], seesaw neutrino models [8], and Randall-Sundru m models [9]. In addition to
+Higgs decays, others have considered lepton-flavor violati on in the decays of supersymmetric
+particles [10] and in horizontal gauge boson models [11]. Th ese analyses, however, were done
+in the context of very specific models (often relying on the as sumption that the µandτare
+emitted in the decay of a single particle). Here, we will use a much more general effective
+field theory approach.
+This paper is organized as follows. In the next section, we di scuss the cross sections
+forµτproduction via the various operators. A detailed analysis o f the signal identification
+and background subtraction is in Section 3, and Section 4 con tains some discussions and our
+conclusions. Appendix A reiterates the bounds from BHHS for comparison, and Appendix B
+outlines the calculation of partial-wave unitarity bounds .
+2.µτProduction at Hadron Colliders
+Dueto the absenceof appreciable µτproductionin theSM, their production can beestimated
+via the effective operators in Eq. (1.1). On dimensional groun ds, the cross section for ¯ qiqj→
+µτgrows with center of mass energy, i.e.,
+σ(¯qiqj→µτ)∝s
+Λ4, (2.1)
+where√sis the center of mass energy for the partonic system. This gro wth of cross section
+with energy will eventually violate unitarity bounds. Expa nding the scattering amplitudes in
+partial waves, we find the unitarity bounds to be (see Appendi x B)
+s≤/braceleftBigg
+2Λ2for scalar ,pseudoscalar ,and tensor;
+3Λ2vector and axial vector case .(2.2)
+The total cross sections for µτproduction at the hadronic level after convoluting with
+the parton distribution functions (pdfs) are
+σScalar=π
+3S
+Λ4/integraldisplayτmax
+τ0dτ(q⊗q)(τ)/parenleftbigg
+1−τ0
+τ/parenrightbigg2
+τ (2.3)
+σVector=4π
+9S
+Λ4/integraldisplayτmax
+τ0dτ(q⊗q)(τ)/parenleftbigg
+1−τ0
+τ/parenrightbigg2/parenleftbigg
+1+τ0
+2τ/parenrightbigg
+τ (2.4)
+σTensor=8π
+9S
+Λ4/integraldisplayτmax
+τ0dτ(q⊗q)(τ)/parenleftbigg
+1−τ0
+τ/parenrightbigg2/parenleftbigg
+1+2τ0
+τ/parenrightbigg
+τ, (2.5)
+– 3 –whereτ=s/S,τ0=m2
+τ/S,mτis the tau mass, and√
+Sis the center of mass energy in the
+lab frame. The pseudoscalar cross section is of the same form as the scalar cross section, and
+the axial vector cross section is of the same form as the vecto r cross section. Our perturbative
+calculation will become invalid at the unitarity bound, hen ce there is a maximum on the τ
+integration. It is given by τmax= 2Λ2/Sfor the scalar, pseudoscalar, and tensor cases, and
+τmax= 3Λ2/Sfor the vector and axial-vector cases. Also, q(x) is the quark distribution
+function with flavor sum suppressed, and ⊗denotes the convolution defined as
+(g1⊗g2)(y) =/integraldisplay1
+0dx1/integraldisplay1
+0dx2g1(x1)g2(x2)δ(x1x2−y). (2.6)
+The CTEQ6L parton distribution function set is used for all o f the results [12].
+Results for the cross sections for the scalar, pseudoscalar , vector, axial vector, and tensor
+structures at the Tevatron, LHC at 10 TeV and 14 TeV are given i n Table 1. Thecross section
+for the pseudoscalar (axial vector) current is the same as fo r the scalar (vector) current. For
+all cases, Λ is set equal to 2 TeV and the unitarity bounds are t aken into consideration. At
+this rather high scale, the production rates are dominated b y the valence quark contributions.
+The cross sections at the LHC are larger than those at the Teva tron by roughly an order of
+magnitude, reaching about 100 pb.
+For some cases the bounds from BHHS are greater than 2 TeV, hen ce the cross section
+needs to be scaled to determine a realistic cross section at h adron colliders. The partonic
+cross sections scale at Λ−4, but at the hadronic level a complication arises since the un itarity
+bounds introduce a dependence on the new physics scale in the integration over pdfs. If
+the unitarity bounds are ignored ( τmax= 1), one finds that with Λ = 2 TeV neglecting the
+unitarity bounds has at most a 10% effect on the cross sections a t the LHC for both 10 TeV
+and 14 TeV and no effect at the Tevatron since the unitarity boun ds are greater than the lab
+frame energy. Hence, if Λ is increased from 2 TeV, at the LHC it is a good approximation to
+assume the cross section scales as Λ−4and at the Tevatron the cross section scales exactly as
+Λ−4. For example, the lower bound on Λ for the vector u¯ucoupling from BHHS is 12 TeV, so
+the maximum cross section at the 14 TeV LHC from this operator would be approximately
+160×(2/12)4pb = 120 fb. On the other hand, there is no bound whatsoever for the vector
+u¯ccoupling, and thus a cross section limit of 110 pb would yield a new limit of 2 TeV on the
+scale of this operator. This would constitute an improvemen t of many orders of magnitude.
+3. Signal Identification and Backgrounds
+Upon production at hadron colliders, τ’s will promptly decay and are detected via their decay
+products. About 35% of the time the τdecays to two neutrinos and an electron or muon, the
+other 65% of the time the τdecays to a few hadrons plus a neutrino. We will consider the τ
+decay to an electron as well as hadronic decays in this work. T he decay to a muon will result
+in aµ+µ−final state that has a large Drell-Yan background. We will stu dy the signal reach
+at the Tevatron and at the 14 TeV LHC.
+– 4 –Table 1: Cross sections for all the scalar, pseudoscalar, vector, axial ve ctor, and tensor structures at
+the Tevatron at 2 TeV, the LHC at 10 TeV, and the LHC at 14 TeV. Th e pseudoscalar (axial vector)
+cross section is the same as the scalar (vector) cross section. All cross sections were evaluated with
+the new physics scale Λ = 2 TeV and the unitarity bounds are taken int o consideration.
+Tevatron 2 TeV ( p¯p)LHC 10 TeV ( pp) LHC 14 TeV ( pp)
+σ(pb)1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
+u¯u8.4 11 22 63 85 170 120 160 310
+d¯d2.5 3.3 6.7 38 51 100 72 98 190
+s¯s0.18 0.24 0.49 5.5 7.4 15 11 15 30
+d¯s1.3 1.7 3.4 34 45 91 66 89 180
+d¯b0.50 0.67 1.3 17 22 45 34 46 90
+s¯b0.13 0.17 0.34 5.0 6.7 13 11 14 28
+u¯c1.5 2.0 3.9 41 55 110 80 110 210
+c¯c0.070 0.094 0.19 2.6 3.5 7.0 5.5 7.3 15
+b¯b0.021 0.028 0.056 1.1 1.5 2.9 2.4 3.2 6.4
+3.1τDecay to Electrons
+3.1.1 Signal Reconstruction
+Theτdecays to an electron plus two neutrinos about 18% of the time . We thus search for a
+final state of an electron and muon
+e+µ. (3.1)
+The electromagnetic calorimeter resolution is simulated b y smearing the electron energies
+according to a Gaussian distribution with a resolution para meterized by
+σ(E)
+E=a/radicalbig
+E/GeV⊕b, (3.2)
+where the constants are a= 10% and b= 0% at the Tevatron [13], a= 5% and b= 0.55% at
+the LHC [14], and ⊕indicates addition in quadrature. For simplicity, we have u sed the same
+form of smearing for the muons.
+The decay of the τleaves us with some missing energy and we need to consider how to
+effectively reconstructthe τmomentum. Forourprocessallthemissingtransversemoment um
+is coming from the τ, hence
+pτ
+T=pe
+T+pmiss
+T. (3.3)
+At hadron colliders, we have no information on the longitudi nal component of the missing
+momentum on an event-by-event basis. However, the τwill be highly boosted and its decay
+– 5 –products will be collimated. Hence, the missing momentum sh ould be aligned with the
+electron momentum and the ratio pe
+z/pmiss
+zshould be the same as the ratio of the magnitudes
+of the transverse momenta, pe
+T/pmiss
+T. Therefore, the longitudinal component of the τcan be
+reconstructed as [4]
+pτ
+z=pe
+z/parenleftbigg
+1+pmiss
+T
+pe
+T/parenrightbigg
+. (3.4)
+Once the three-momentum is reconstructed, we can solve for t heτenergy,E2
+τ=p2
+τ+m2
+τ.
+Figure 1 illustrates the effectiveness of this method at the Te vatron. Figure 1(a) (Figure 1(b))
+shows the transverse momentum (longitudinal momentum) dis tribution for the theoretically
+generated (solid) and kinematically reconstructed (dashe d)τmomenta. As can be seen, the
+τmomentum is reconstructed effectively.
+We first apply some basic cuts on the transverse momentum and t he pseudo rapidity
+to simulate the detector acceptance and triggering, as well as to isolate the signal from the
+background,
+pµ
+T>20 GeV,|ηµ|<2.5,
+pe
+T>20 GeV,|ηe|<2.5. (3.5)
+Since the signal does not contain any jets, we also require a j et veto such that there are no
+jets with pT>50 GeV and |η|<2.5.
+There are several distinctive kinematic features of our sig nal. The decay products of the
+τwill be highly collimated, and the electron transverse mome ntum will be traveling in the
+same direction as the missing transverse momentum. Also, in the transverse plane the muon
+and tau should be back to back. Since the electron will mostly be in the direction of the τ,
+it will also be nearly back to back with the muon. Finally, the τandµhave equal transverse
+momenta; hence, the decay products of the τhave less transverse momentum than the µ. We
+can measure this discrepancy using the momentum imbalance
+∆pT=pµ
+T−pe
+T. (3.6)
+For the signal, this observable should be positive. Based on the kinematics of our signal, we
+apply the further cuts [5]
+δφ(pµ
+T,pe
+T)>2.75 rad, δφ(pmiss
+T,pe
+T)<0.6 rad, (3.7)
+∆pT>0.
+3.1.2 Backgrounds and their Suppression
+Theleadingbackgrounds are W+W−pair production, Z0/γ⋆→τ+τ−, andt¯tpair production
+[5]. The total rates for these backgrounds at the Tevatron an d the LHC are given in Table 2
+with consecutive cuts. We consider both of the final states wi thµ+andµ−.
+– 6 –0 100 200 300
+pτ
+T (GeV)10-410-310-210-1dσ/dpT (pb/GeV)Generated
+Reconstructed
+(a)-300 -200 -100 0 100 200 300
+pτ
+z (GeV)10-310-210-1dσ/dpz (pb/GeV)Generated
+Reconstructed
+(b)
+Figure 1: Distributions of the theoretically generated (solid line) and kinematic ally reconstructed
+(dashed line) τmomentum at the Tevatron at 2 TeV with a u¯cinitial state, scalar coupling, and new
+physics scale of 1 TeV. Fig. (a) is the τtransverse momentum distribution, and Fig. (b) is the τ
+longitidunal momentum distribution.
+Table 2: Leading backgrounds to the τ’s electronic decay before and after consecutive kinematic and
+invariant mass cuts for (a) the Tevatron at 2 TeV and (b) the LHC a t 14 TeV.
+Backgrounds (pb) No Cuts Cuts Eq. (3.5) + Eq. (3.7) + Eq. (3.8)
+(a) Tevatron 2 TeV
+W+W−→µ±νµτ∓ντ0.032 0.0046 0.0012 2.6×10−4
+W+W−→µ±νµe∓νe0.18 0.13 0.0060 9.8×10−4
+Z0/γ⋆→τ+τ−→µ±νµτ∓610 0.21 0.091 1.4×10−4
+t¯t→µ±νµbτ∓ντ¯b 0.020 6.5×10−47.4×10−54.4×10−5
+t¯t→µ±νµbe∓νe¯b 0.11 0.0099 7.3×10−42.7×10−4
+(b) LHC 14 TeV
+W+W−→µ±νµτ∓ντ0.34 0.030 0.0088 0.0031
+W+W−→µ±νµe∓νe 1.9 0.99 0.051 0.014
+Z0/γ⋆→τ+τ−→µ±νµτ∓2300 1.1 0.49 0.0014
+t¯t→µ±νµbτ∓ντ¯b 1.9 0.070 0.010 0.0077
+t¯t→µ±νµbe∓νe¯b 11 1.5 0.10 0.050
+The partonic cross section of our signal increases with ener gy while the cross sections
+of the backgrounds will decrease with energy. Hence, the inv ariant mass distribution of our
+signal does not fall off as quickly as the backgrounds.
+Figure 2(a) shows the invariant mass distributions of backg rounds and our signal at the
+Tevatron with initial states c¯candu¯cwith various couplings and a new physics scale of
+1 TeV after applying the cuts in Eqs. (3.5) and (3.7). The cros s section for the pseudoscalar
+– 7 –0 200 400 600800 1000
+Mµτ (GeV)10-510-410-310-2dσ/dMµτ (pb/GeV)eµνν
+τ+τ−
+Tensor
+Vector
+Scalaru c-bar
+c c-barTevatron
+(a)0 200 400 600800 1000
+Mµτ (GeV)10-510-410-310-2dσ/dMµτ (pb/GeV)eµνν
+τ+τ−1 TeV
+2 TeV
+3 TeVTevatron
+(b)
+Figure 2: The invariant mass distributions of the reconstructed τ−µsystem at the Tevatron at 2
+TeV. Fig. (a) shows the distributions of the leading backgrounds (d otted and dot-dot-dash) and of
+our signal for the u¯candc¯cinitial states with coupling of various Lorentz structures and a new physics
+scale of 1 TeV. Fig. (b) shows the distributions of the leading backgr ounds (dotted and dashed) and
+of our signal (solid) for the u¯cinitial state with scalar coupling and various new physics scales. The
+cuts in Eqs. (3.5) and (3.7) have been applied.
+(axial-vector) couplings are the same as those for the scala r (vector) couplings. The decline
+in the signal rates is due to a suppression of the pdfs at large x. Although the signal rates
+steeply decline with invariant mass the background falls off faster. The u¯csignal is still clearly
+above background due to a valence quark in the initial state, but thec¯csignal distribution is
+much closer to the background distribution due to the steep f all with invariant mass and a
+lack of an initial state valence quark. Figure 2(b) shows the invariant mass distributions of
+backgroundsandoursignalattheTevatronwithinitial stat eu¯candscalarcouplingforvarious
+new physics scales. The 3 TeV new physics scale invariant mas s distribution is approaching
+the background distribution. A higher cutoff on the invarian t mass will be needed to separate
+the weak signal from the backgrounds. Based on Fig. 2, we prop ose a selection cut on
+Mµτ>250 GeV . (3.8)
+Table 2 shows the effects of the invariant mass cut on the backgr ounds in the last column.
+Similar analyses can be carried out for the LHC. Figure 3(a) s hows the invariant mass
+distribution for our signal with the u¯candc¯cinitial states and various Lorentz structures, as
+well as the backgroundsafter thecuts in Eqs. (3.5) and(3.7) . Thenewphysics scale was set to
+1 TeV and the unitarity bound is imposed. Figure 3(b) shows th e invariant mass distribution
+of theu¯cinitial state with various new physics scales. The cutoff on t he invariant mass
+corresponds to the unitarity bound, the scale at which the pe rturbative calculation becomes
+untrustworthy. In the lack of the knowledge for the new physi cs to show up at the scale Λ,
+we simply impose a sharp cutoff at the unitarity bound. As comp ared with the Tevatron,
+the LHC signal rates fall off much less quickly with invariant mass since the Tevatron’s lower
+– 8 –0 500 1000 1500
+Mµτ (GeV)10-410-310-210-1dσ/dMµτ (pb/GeV)Tensor
+Vector
+Scalar
+bbeµνντ+τ−u c-bar
+c c-barLHC
+(a)0 1 2 3 4
+Mµτ (TeV)10-510-410-310-210-1dσ/dMµτ (pb/GeV)bbeµνν
+τ+τ−1 TeV
+2 TeV
+3 TeV
+4 TeV
+5 TeVLHC
+(b)
+Figure 3: The invariant mass distributions of the reconstructed τ−µsystem at the LHC at 14 TeV.
+Fig. (a) shows the distributions of the leading backgrounds (dotte d and dot-dot-dash) and of our
+signal for the u¯candc¯cinitial states with coupling of various Lorentz structures and a new physics
+scale of 1 TeV. Fig. (b) shows the distributions of the leading backgr ounds (dotted and dashed) and
+of our signal (solid) for the u¯cinitial state with scalar coupling and various new physics scales. The
+cut offs in the distributions at high invariant mass are due to the unita rity bounds. The cuts in Eqs.
+(3.5) and (3.7) have been applied.
+energy leads to a suppression from the pdfs at large x. As can be seen, as the new physics
+scale increases the cross section decreases and the backgro und becomes more problematic at
+lower invariant mass. Also, as the new physics scale increas es the unitarity bound becomes
+less strict. Hence, although the backgrounds at the LHC are c onsiderably larger than at the
+Tevatron, for large new physics scales the LHC has an enhance ment in the signal cross section
+from the large invariant mass region.
+3.2τDecay to Hadrons
+Although with significantly larger backgrounds, the signal fromτhadronic decays can be
+very distinctive as well. We limit the hadronic τdecays to 1-prong decays to pions, i.e.,
+τ±→π±ντ,τ±→π±π0ντ, andτ±→π±2π0ντ. Theτ’s have 1-prong decays to these final
+states about 50% of the time. We thus search for a final state of aτjet and a muon
+jτ+µ. (3.9)
+To simulate detector resolution effects, the energy is smeare d according to Eq. (3.2) with
+a= 80% and b= 0% for the jet at the Tevatron [13] and a= 100% and b= 5% at the LHC
+[14]. As in the electronic decay, the τis highly boosted and its decay products are collimated.
+Hence, all the missing energy in the event should be aligned w ith theτ. The signal is then
+reconstructed as described in Eqs. (3.3) and (3.4) with the e lectron momentum replaced by
+the momentum of the τ-jet.
+– 9 –The hadronic decay of the τalso has the backgrounds W+W−pair production, Z0/γ⋆→
+τ+τ−, andt¯tpair production plus an additional background of W+jet, where the jet is
+misidentified as a τ-jet. At the Tevatron, we assume a τ-jet tagging efficiency of 67% and
+that a light jet is mistagged as a τ-jet 1.1% of the time [15] and at the LHC we assume a τ-jet
+tagging efficiency of 40% and a light jet misidentification rat e of 1% [14]. Even with a low
+rate of misidentification, the W+jet background is large. To suppress this background, we
+note that for hadronic decays most of the τtransverse momentum will be carried by the jet.
+Hence the τ-jet should be traveling in the same direction as the reconst ructedτmomentum.
+Motivated by this observation, we apply the same cuts as Eqs. (3.5), (3.7), and (3.8) with the
+electron momentum replaced by the τ-jet momentum and the additional cuts
+pτ−jet
+T
+pτ
+T>0.6 ∆ R(pτ−jet
+T,pτ
+T)<0.2 rad. (3.10)
+3.3 Sensitivity Reach at the Tevatron
+One can determine the sensitivity of the Tevatron to the new p hysics scale with 8 fb−1of
+data. Table 3 shows the sensitivity of the Tevatron for (a) el ectronic and (b) hadronic τ
+decays. The tables list the maximum new physics scale sensit ivity at 2 σand 5σlevel at the
+Tevatron. The reaches for scalar (vector) and pseudoscalar (axial-vector) are the same at the
+Tevatron, although the previous bounds from BHHS for the sca lar (vector) and pseudoscalar
+(axial-vector) couplings may not be the same. The bounds fro m BHHS can be found in
+Appendix A. If only one of the bounds for scalar (vector) or ps eudoscalar (axial-vector)
+coupling from BHHS is greater than the Tevatron reach one sta r is placed next to the new
+physics scale, if both bounds are greater than the Tevatron r each two stars are placed next
+to the new physics scale. Due to the larger backgrounds from W+jet, the Tevatron is much
+less sensitive to the τhadronic decays than the τelectronic decays.
+There were no bounds from BHHS for the tensor couplings, so th e Tevatron will be
+able to exlude some of the parameter space. Since the tensor c ross sections are generally at
+least twice as large as the scalar cross sections, the Tevatr on is more sensitive to the tensor
+couplings than it is to scalar couplings. Also, in general, t he Tevatron is more sensitive to
+processes with initial state valence quarks than those with out initial state quarks. With 8
+fb−1of data most of the bounds can be increased, some quite string ently.
+Somewhat similar leptonic final states have been searched fo r in a model-independent
+way at the Tevatron [16], although these included substanti al missing energy and possible
+jets. We encourage the Tevatron experimenters to carry out t he analyses as suggested in this
+article.
+3.4 Sensitivity Reach at the LHC
+The LHC is also sensitive to flavor changing operators. For th e signal and background anal-
+ysis, we used the same kinematical cuts as we used at the Tevat ron, see Eqs. (3.5), (3.7),
+and (3.8). Table 4 shows the sensitivity of the LHC to all poss ible initial states and the
+– 10 –Table 3: Maximum new physics scales the Tevatron is sensitive to with 8 fb−1of data at the 2 σ
+and 5σlevels. The sensitivities are presented for both (a) electronic and ( b) hadronic τdecays with
+various initial states. One star indicates that the Tevatron reach is less than only one of the scalar
+(vector) or pseudoscalar (axial-vector) bounds from BHHS, and two stars indicates that the Tevatron
+reach is less than both bounds from BHHS. BHHS does not contain bo unds on the tensor coupling.
+(a)τ→e
+ΛNP(TeV) 2σsensitivity 5σdiscovery
+Coupling 1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
+u¯u20 21 24 14 15 17
+d¯d17 18 21 12 13 15
+s¯s9.9 10 12 7.2* 7.7** 8.7
+d¯s15 16 18 10 11* 13
+d¯b13 14 16 9.8 10* 11
+s¯b9.5 10 11 6.9 7.3 8.3
+u¯c17 18 20 12 13 14
+c¯c7.9 8.3 9.5 5.7 6.0 6.9
+b¯b6.4 6.8 7.7 4.6 4.9 5.6
+(b)τ→h±
+ΛNP(TeV) 2σsensitivity 5σdiscovery
+Coupling 1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
+u¯u8.6** 9.2** 10 6.5** 6.9** 8.1
+d¯d5.7** 6.1** 7.1 4.3** 4.6** 5.4
+s¯s1.8* 1.9** 2.3 1.4** 1.4** 1.7
+d¯s3.7 4.0* 4.6 2.8* 3.0** 3.5
+d¯b2.7* 2.9* 3.4 2.0** 2.2 2.5
+s¯b1.5** 1.6** 1.9 1.1** 1.2** 1.4
+u¯c3.9 4.1 4.8 2.9 3.1 3.6
+c¯c1.1 1.2 1.4 0.89 0.95** 1.1
+b¯b0.91 0.97 1.1 0.68 0.73 0.86
+couplings under consideration with 100 fb−1of data. The table contains the maximum new
+physics scales the LHC is sensitive to at the 2 σand 5σlevels. As with the Tevatron, the
+LHC reach for scalar (vector) couplings is the same as that fo r pseudoscalar (axial-vector)
+couplings, although the bounds from BHHS may be different. If o nly one of the bounds for
+scalar (vector) or pseudoscalar (axial-vector) coupling f rom BHHS is greater than the LHC
+reach one star is placed next to the new physics scale, if both bounds are greater than the
+LHC reach two stars are placed next to the new physics scale. D espite the larger backrounds
+for the hadronic τdecays, at the LHC the reaches for the hadronic and electroni cτdecays are
+– 11 –Table 4: Maximum new physics scales the LHC is sensitive to at 14 TeV with 100 fb−1of data at the
+2σand 5σlevels. The sensitivities are presented for both (a) electronic and ( b) hadronic τdecays with
+various initial states. One star indicates that the LHC reach is less t han only one of the scalar (vector)
+or pseudoscalar (axial-vector) bounds from BHHS, and two stars indicates that the LHC reach is less
+than both bounds from BHHS. BHHS does not contain bounds on the tensor coupling.
+(a)τ→e
+ΛNP(TeV) 2σsensitivity 5σdiscovery
+Coupling 1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
+u¯u18 19 21 14 15 17
+d¯d16 17 19 12 13 15
+s¯s9.0* 9.6* 11 7.1* 7.6** 8.6
+d¯s13 14 16 10 11* 13
+d¯b12 13 14 9.7 10 11
+s¯b8.7 9.2 10 6.8 7.3 8.2
+u¯c15 16 18 12 13 14
+c¯c7.2 7.6 8.6 5.7 6.0 6.8
+b¯b5.8 6.2 7.0 4.6 4.9 5.5
+(b)τ→h±
+ΛNP(TeV) 2σsensitivity 5σdiscovery
+u¯u15 16 18 12 13 14
+d¯d13 14 16 10* 11* 13
+s¯s7.9* 8.4** 9.7 6.2* 6.7** 7.7
+d¯s11 12* 14 9.3 9.9* 11
+d¯b10 11 13 8.4* 8.9 10
+s¯b7.6 8.1 9.3 6.0 6.4 7.4
+u¯c13 14 16 10 11 12
+c¯c6.3 6.7 7.8 5.0 5.3 6.2
+b¯b5.1 5.5 6.3 4.1 4.3 5.0
+much more similar than at the Tevatron since the LHC cross sec tion receives an enhancement
+from the large invariant mass region. For electronic (hadro nic)τdecays the LHC with 100
+fb−1of data is less (more) sensitive than the Tevatron with 8 fb−1of data.
+Figure 4 shows the integrated luminosities needed for 2 σand 5σobservation at the LHC
+with various initial states and τdecay to electrons as a function of the new physics scale.
+For some initial states and Lorentz structures BHHS had a bou nd on the new physics scale
+larger than 1 TeV. In those cases the distribution does not be gin until the BHHS bound on
+the new physics scale. The sensitivity for the pseudoscalar (axial-vector) is the same as the
+scalar (vector) state, although the bounds from BHHS are diffe rent. Note that extraordinary
+– 12 –1 2 3 4 5 6 78 9 10
+ΛNP (TeV)10-310-210-1100101102103L (fb-1)Scalar
+Vector
+Tensor
+2σu c-bar Initial State
+5σLHC 14 TeV
+(a)1 2 3 4 5 6 78 9 10
+ΛNP (TeV)10-310-210-1100101102103L (fb-1)
+Scalar
+Vector
+Tensor
+2σc c-bar Initial State
+5σLHC 14 TeV
+(b)
+1 2 3 4 5 6 78 9 10
+ΛNP (TeV)10-310-210-1100101102103L (fb-1)Scalar
+Vector
+Tensor
+2σd b-bar Initial State
+5σLHC 14 TeV
+(c)1 2 3 4 5 6 78 9 10
+ΛNP (TeV)10-310-210-1100101102103L (fb-1)
+Scalar
+Vector
+Tensor
+2σs b-bar Initial State
+5σLHC 14 TeV
+(d)
+Figure 4: The luminosity at the 14 TeV LHC needed for 2 σand 5σobservation as a function of the
+new physics scales with couplings of various Lorentz structures an d electronic τdecay. The sensitivity
+for theu¯cinitial state is shown in (a), for the c¯cinitial state in (b), for the d¯binitial state in (c), and
+for thes¯binitial state in (d). The lower bounds on the new physics scale were ta ken from BHHS.
+improvementintheboundscouldbefound(oradiscoverymade )withrelatively lowintegrated
+luminosity. Consider, for example, the u¯cinitial state. There is currently no bound at all;
+in principle, Λ could be tens of GeV. The figure shows that a tot al integrated luminosity of
+an inverse picobarn would give a 5 σsensitivity for a Λ of 1 TeV. An integrated luminosity of
+an inverse femtobarn would give substantial improvements f or all of the operators shown in
+Fig. 4.
+4. Discussions and Conclusions
+In a previous article, motivated by discovery of large νµ−ντmixing in charged current
+interactions, bounds on the analogous mixing in neutral cur rent interactions were explored.
+A general formalism for dimension-6 fermionic effective oper ators involving τ−µmixing with
+– 13 –typical Lorentz structure ( µΓτ)(qαΓqβ) was presented, and the low-energy constraints on
+the new physics scale associated with each operator were der ived, mostly from experimental
+bounds on rare decays of τ, hadrons or heavy quarks. Tensor operators were not conside red,
+and some of the operators, such as cuµτ, were completely unbounded.
+Inthis article, weconsider µτproductionat hadroncolliders viatheseoperators. Tables 3
+and4 list thenewphysics scales that are accessible at the Te vatron and theLHC, respectively.
+Duetomuchsmallerbackgrounds, boththeLHCandTevatronar emoresensitivetoelectronic
+τdecays than hadronic τdecays. For hadronic τdecays, the LHC receives an enhancement
+from the large invariant mass region and is more sensitive th an the Tevatron. Since the
+backgrounds to electronic τdecays at the Tevatron are much smaller than those at the LHC,
+the Tevatron is more sensitive than the LHC to electronic τdecays. We found that at the
+Tevatron with 8 fb−1, one can exceed current bounds for most operators, with most 2σ
+sensitivities being in the 6 −24 TeV range. We find that at the LHC with 1 fb−1(100 fb−1)
+integrated luminosity, one can reach a 2 σsensitivity for Λ ∼3−10 TeV (Λ ∼6−21 TeV),
+depending on the Lorentz structure of the operator.
+Acknowledgments
+We would like to thank Vernon Barger and Xerxes Tata for discu ssions. MS would like to
+thank the Wisconsin Phenomenology Institute, in particula r Linda Dolan, for hospitality
+during his visit. The work of TH and IL was supported by the US D OE under contract
+No. DE-FG02-95ER40896, and that of MS was supported in part b y the National Science
+Foundation PHY-0755262.
+A. New Physics Bounds
+The bounds from BHHS in units of TeV are presented in Table 5. T he *s indicate there are
+no bounds on the new physics scale. Also, there are no bounds f rom BHHS for the tensor
+coupling.
+B. Partial Wave Unitarity Bounds
+Since the cross section from our higher-dimensional operat ors increases as s, it is necessary
+to determine the unitarity bound for q¯q→µτ. The partial wave expansion for a+b→1+2
+can be written as
+M(s,t) = 16π∞/summationdisplay
+J=M(2J+1)aJ(s)dJ
+µµ′(cosθ)
+where
+aJ(s) =1
+32π/integraldisplay1
+−1M(s,t)dJ
+µµ′(cosθ)dcosθ,
+µ=sa−sb,µ′=s1−s2andJ≤max(|µ|,|µ′|). The condition for unitarity is |ℜ(aJ)| ≤1/2.
+– 14 –Coupling type 1 γ5 γµ γµγ5
+u¯u 2.6 12 12 11
+d¯d 2.6 12 12 11
+s¯s 1.5 9.9 14 9.5
+d¯s 2.3 3.7 13 3.6
+d¯b 2.2 9.3 2.2 8.2
+s¯b 2.6 2.8 2.6 2.5
+u¯c * * 0.55 0.55
+c¯c * * 1.1 1.1
+b¯b * * 0.18 *
+Table 5: Bounds on the new physics scales from BHHS in units of TeV for variou s operators and the
+scalar, pseudoscalar, vector, and axial-vector couplings. The *s indicate there were no bounds.
+It is straightforward to calculate the coefficients for the S, V,T operators. For example,
+for the scalar operator
+M=4π
+Λ2¯vλ1(p1)uλ2(p2)¯uλ3(p3)vλ4(p4)
+one can just plug in the explicit expressions:
+uλ(p)≡/parenleftBigg/radicalbig
+E−λ|p|χλ(ˆp)/radicalbig
+E+λ|p|χλ(ˆp)/parenrightBigg
+vλ(p)≡/parenleftBigg
+−/radicalbig
+E+λ|p|χ−λ(ˆp)/radicalbig
+E−λ|p|χ−λ(ˆp)/parenrightBigg
+whereχ+(ˆz) =/parenleftbig1
+0/parenrightbig
+,χ−(ˆz) =/parenleftbig0
+1/parenrightbig
+. In the massless limit, this simply gives a0=s/(4Λ2) and
+so the unitarity bound gives s≤2Λ2. For the vector case, a0= 0 and a1=s/(6Λ2) giving
+the unitarity bound s≤3Λ2. The tensor case gets contributions from both a0anda1, and
+the stronger bound then applies.
+References
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+5656 (2001); 82, 1810 (1999); 81, 1562 (1998); 81, 1158 (1998); and T. Toshito,
+[Super-Kamiokande Collaboration], hep-ex/0105023 .
+[2] Q. R. Ahmad, et al.,[SNO collaboration], Phys. Rev. Lett. 87, 071301 (2001). Q. R. Ahmad, et
+al.,[SNO Collaboration], Phys. Rev. Lett. (2002), nucl-ex/0204008 andnucl-ex/0204009 .
+[3] D. Black, T. Han, H. J. He and M. Sher, Phys. Rev. D 66, 053002 (2002)
+[arXiv:hep-ph/0206056].
+– 15 –[4] T. Han and D. Marfatia, Phys. Rev. Lett. 86, 1442 (2001) [arXiv:hep-ph/0008141].
+[5] K. A. Assamagan, A. Deandrea and P. A. Delsart, Phys. Rev. D 67, 035001 (2003)
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+[11] H. U. Bengtsson, W. S. Hou, A. Soni and D. H. Stork, Phys. Re v. Lett.55, 2762 (1985).
+[12] D. Stump, J. Huston, J. Pumplin, W. K. Tung, H. L. Lai, S. Kuhlma nn and J. F. Owens, JHEP
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+– 16 –
\ No newline at end of file
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+arXiv:1001.0023v7  [math.AG]  1 Nov 2016Algebraic Geometry over C∞-rings
+Dominic Joyce
+Abstract
+IfXis a manifold then the R-algebra C∞(X) of smooth functions
+c:X→Ris aC∞-ring. That is, for each smooth function f:Rn→R
+there is an n-fold operation Φ f:C∞(X)n→C∞(X) acting by Φ f:
+(c1,...,c n)/mapsto→f(c1,...,c n), and these operations Φ fsatisfy many natural
+identities. Thus, C∞(X) actually has a far richer structure than the
+obviousR-algebra structure.
+We explain the foundations of a version of algebraic geometr y in which
+rings or algebras are replaced by C∞-rings. As schemes are the basic
+objects in algebraic geometry, the new basic objects are C∞-schemes, a
+category of geometric objects which generalize manifolds, and whose mor-
+phisms generalize smooth maps. We also study quasicoherent sheaves on
+C∞-schemes, and C∞-stacks, in particular Deligne–Mumford C∞-stacks,
+a 2-category of geometric objects generalizing orbifolds.
+Many of these ideas are not new: C∞-rings and C∞-schemes have long
+been part of synthetic differential geometry. But we develop them in new
+directions. In [36–38], the author uses these tools to define d-manifolds
+andd-orbifolds , ‘derived’ versions of manifolds and orbifolds related to
+Spivak’s ‘derived manifolds’ [64].
+Contents
+1 Introduction 3
+2C∞-rings 5
+2.1 Two definitions of C∞-ring . . . . . . . . . . . . . . . . . . . . . 6
+2.2C∞-rings as commutative R-algebras, and ideals . . . . . . . . . 7
+2.3 Local C∞-rings, and localization . . . . . . . . . . . . . . . . . . 9
+2.4 FairC∞-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
+2.5 Pushouts of C∞-rings . . . . . . . . . . . . . . . . . . . . . . . . 15
+2.6 Flat ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
+3 TheC∞-ringC∞(X)of a manifold X 17
+4C∞-ringed spaces and C∞-schemes 20
+4.1 Some basic topology . . . . . . . . . . . . . . . . . . . . . . . . . 20
+4.2 Sheaves on topological spaces . . . . . . . . . . . . . . . . . . . . 21
+4.3C∞-ringed spaces and local C∞-ringed spaces . . . . . . . . . . . 24
+14.4 The spectrum functor . . . . . . . . . . . . . . . . . . . . . . . . 26
+4.5 Affine C∞-schemes and C∞-schemes . . . . . . . . . . . . . . . . 31
+4.6 Complete C∞-rings . . . . . . . . . . . . . . . . . . . . . . . . . . 34
+4.7 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . 37
+4.8 A criterion for affine C∞-schemes . . . . . . . . . . . . . . . . . . 39
+4.9 Quotients of C∞-schemes by finite groups . . . . . . . . . . . . . 42
+5 Modules over C∞-rings and C∞-schemes 44
+5.1 Modules over C∞-rings . . . . . . . . . . . . . . . . . . . . . . . 44
+5.2 Cotangent modules of C∞-rings . . . . . . . . . . . . . . . . . . . 45
+5.3 Sheaves ofOX-modules on a C∞-ringed space ( X,OX) . . . . . 50
+5.4 Sheaves on affine C∞-schemes, MSpec and Γ . . . . . . . . . . . 51
+5.5 Complete modules over C∞-rings . . . . . . . . . . . . . . . . . . 56
+5.6 Cotangent sheaves of C∞-schemes . . . . . . . . . . . . . . . . . 58
+6C∞-stacks 61
+6.1C∞-stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
+6.2 Properties of 1-morphisms of C∞-stacks . . . . . . . . . . . . . . 64
+6.3 Open C∞-substacks and open covers . . . . . . . . . . . . . . . . 66
+6.4 The underlying topological space of a C∞-stack . . . . . . . . . . 67
+6.5 Gluing C∞-stacks by equivalences . . . . . . . . . . . . . . . . . 70
+7 Deligne–Mumford C∞-stacks 71
+7.1 Quotient C∞-stacks, 1-morphisms, and 2-morphisms . . . . . . . 71
+7.2 Deligne–Mumford C∞-stacks . . . . . . . . . . . . . . . . . . . . 73
+7.3 Characterizing Deligne–Mumford C∞-stacks . . . . . . . . . . . . 76
+7.4 Quotient C∞-stacks, 1- and 2-morphisms as local models for
+objects, 1- and 2-morphisms in DMC∞Sta. . . . . . . . . . . . 80
+7.5 Effective Deligne–Mumford C∞-stacks . . . . . . . . . . . . . . . 86
+7.6 Orbifolds as Deligne–Mumford C∞-stacks . . . . . . . . . . . . . 87
+8 Sheaves on Deligne–Mumford C∞-stacks 89
+8.1 Quasicoherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . 89
+8.2 Writing sheaves in terms of a groupoid presentation . . . . . . . 92
+8.3 Pullback of sheaves as a weak 2-functor . . . . . . . . . . . . . . 93
+8.4 Cotangent sheaves of Deligne–Mumford C∞-stacks . . . . . . . . 96
+9 Orbifold strata of C∞-stacks 99
+9.1 The definition of orbifold strata XΓ,...,ˆXΓ
+◦. . . . . . . . . . . . 100
+9.2 Lifting 1- and 2-morphisms to orbifold strata . . . . . . . . . . . 108
+9.3 Orbifold strata of quotient C∞-stacks [X/G] . . . . . . . . . . . 109
+9.4 Sheaves on orbifold strata . . . . . . . . . . . . . . . . . . . . . . 111
+9.5 Sheaves on orbifold strata of quotients [ X/G] . . . . . . . . . . . 114
+9.6 Cotangent sheaves of orbifold strata . . . . . . . . . . . . . . . . 116
+2A Background material on stacks 118
+A.1 Introduction to 2-categories . . . . . . . . . . . . . . . . . . . . . 118
+A.2 Grothendieck topologies, sites, prestacks, and stacks . . . . . . . 122
+A.3 Descent theory on a site . . . . . . . . . . . . . . . . . . . . . . . 125
+A.4 Properties of 1-morphisms . . . . . . . . . . . . . . . . . . . . . . 126
+A.5 Geometric stacks, and stacks associated to groupoids . . . . . . . 128
+References 132
+Glossary of Notation 137
+Index 140
+1 Introduction
+LetXbe a smooth manifold, and write C∞(X) for the set of smooth functions
+c:X→R. ThenC∞(X) is a commutative R-algebra, with operations of
+addition, multiplication, and scalar multiplication defined pointwise. How ever,
+C∞(X) has much more structure than this. For example, if c:X→Ris
+smooth then exp( c) :X→Ris smooth, and this defines an operation exp :
+C∞(X)→C∞(X) which cannot be expressed algebraically in terms of the R-
+algebra structure. More generally, if n/greaterorequalslant0 andf:Rn→Ris smooth, define
+ann-fold operation Φ f:C∞(X)n→C∞(X) by
+/parenleftbig
+Φf(c1,...,cn)/parenrightbig
+(x) =f/parenleftbig
+c1(x),...,cn(x)/parenrightbig
+,
+for allc1,...,cn∈C∞(X) andx∈X. These operations satisfy many identities:
+supposem,n/greaterorequalslant0, andfi:Rn→Rfori= 1,...,mandg:Rm→Rare smooth
+functions. Define a smooth function h:Rn→Rby
+h(x1,...,xn) =g/parenleftbig
+f1(x1,...,xn),...,fm(x1...,xn)/parenrightbig
+,
+for all (x1,...,xn)∈Rn. Then for all c1,...,cn∈C∞(X) we have
+Φh(c1,...,cn) = Φg/parenleftbig
+Φf1(c1,...,cn),...,Φfm(c1,...,cn)/parenrightbig
+.(1.1)
+AC∞-ring/parenleftbig
+C,(Φf)f:Rn→RC∞/parenrightbig
+is a setCwith operations Φ f:Cn→Cfor
+allf:Rn→Rsmooth satisfying identities (1.1), and one other condition. For
+exampleC∞(X) is aC∞-ring for any manifold X, but there are also many C∞-
+rings which do not come from manifolds, and can be thought of as rep resenting
+geometric objects which generalize manifolds.
+The most basic objects in conventional algebraic geometry are com mutative
+ringsR, or commutative K-algebrasRfor some field K. The ‘spectrum’ Spec R
+ofRis an affine scheme, and Ris interpreted as an algebra of functions on
+SpecR. More general kinds of spaces in algebraic geometry — schemes and
+stacks — are locally modelled on affine schemes Spec R. This book lays down
+the foundations of Algebraic Geometry over C∞-rings, in which we replace
+3commutative rings in algebraic geometry by C∞-rings. It includes the study of
+C∞-schemes andDeligne–Mumford C∞-stacks, two classes of geometric spaces
+generalizing manifolds and orbifolds, respectively.
+This is not a new idea, but was studied years ago as part of synthetic dif-
+ferential geometry , which grew out of ideas of Lawvere in the 1960s; see for
+instance Dubuc [23] on C∞-schemes, and the books by Moerdijk and Reyes [54]
+and Kock [44]. However, we have new things to say, as we are motivat ed by
+different problems (see below), and so are asking different question s.
+Following Dubuc’s discussion of ‘models of synthetic differential geome try’
+[21] and oversimplifying a bit, synthetic differential geometers are in terested
+inC∞-schemes as they provide a category C∞Schof geometric objects which
+includes smooth manifolds and certain ‘infinitesimal’ objects, and all fi bre prod-
+ucts exist in C∞Sch, andC∞Schhas some other nice properties to do with
+open covers, and exponentials of infinitesimals.
+Synthetic differential geometry concerns proving theorems abou t manifolds
+usingsyntheticreasoninginvolving‘infinitesimals’. But oneneedstoch eckthese
+methods of synthetic reasoning are valid. To do this you need a ‘mode l’, some
+category of geometric spaces including manifolds and infinitesimals, in which
+you can think of your synthetic arguments as happening. Once you know there
+exists at least one model with the properties you want, then as far as synthetic
+differential geometry is concerned the job is done. For this reason C∞-schemes
+have not been developed very far in synthetic differential geometr y.
+Recently,C∞-rings andC∞-ringed spaces appeared in a very different con-
+text, in the theory of derived differential geometry , the differential-geometric
+analogue of the derived algebraic geometry of Lurie [48] and To¨ en– Vezzosi
+[66,67], which studies derived smooth manifolds andderived smooth orbifolds .
+This began with a short section in Lurie [48, §4.5], where he sketched how to
+define an∞-category of derivedC∞-schemes , including derived manifolds.
+Lurie’s student David Spivak [64] worked out the details of this, defi ning
+an∞-category of derived manifolds. Simplifications and extensions of Sp ivak’s
+theory were given by Borisov and Noel [9,10] and the author [36–38 ]. An al-
+ternative approach to the foundations of derived differential geo metry involving
+differential graded C∞-rings is proposed by Carchedi and Roytenberg [12,13].
+The author’s notion of derived manifolds [36–38] are called d-manifolds , and
+are built using our theory of C∞-schemes and quasicoherent sheaves upon them
+below. They form a 2-category. We also study orbifold versions, d-orbifolds ,
+which are built using our theory of Deligne–Mumford C∞-stacks and their qua-
+sicoherent sheaves below.
+Many areas of symplectic geometry involve studying moduli spaces o fJ-
+holomorphic curves in a symplectic manifold, which are made into Kuranishi
+spacesin the framework of Fukaya, Oh, Ohta and Ono [26,27]. The author
+argues that Kuranishi spaces are really derived orbifolds , and has given a new
+definition [39,41] of a 2-category of Kuranishi spaces Kurwhich is equivalent
+to the 2-category of d-orbifolds dOrbfrom [36–38]. Because of this, derived
+differential geometry will have important applications in symplectic ge ometry.
+To set up our theory of d-manifolds and d-orbifolds requires a lot of pre-
+4liminary work on C∞-schemes and C∞-stacks, and quasicoherent sheaves upon
+them. That is the purpose of this book. We have tried to present a c om-
+plete, self-contained account which should be understandable to r eaders with
+a reasonable background in algebraic geometry, and we assume no f amiliarity
+with synthetic differential geometry. We expect this material may h ave other
+applications quite different to those the author has in mind in [36–38].
+Section 2 explains C∞-rings. The archetypal examples of C∞-rings,C∞(X)
+for manifolds X, are discussed in §3. Section 4 studies C∞-schemes, and§5
+modules over C∞-rings and sheaves of modules over C∞-schemes.
+Sections 6–9 discuss C∞-stacks. Section 6 defines the 2-category C∞Sta
+ofC∞-stacks, analogues of Artin stacks in algebraic geometry, and §7 the 2-
+subcategory DMC∞StaofDeligne–Mumford C∞-stacks, which are C∞-stacks
+locallymodelled on[ U/G]forUanaffineC∞-schemeand Gafinitegroupacting
+onU, and are analogues of Deligne–Mumford stacks in algebraic geometr y. We
+show that orbifolds Orbmay be regarded as a 2-subcategory of DMC∞Sta.
+Section 8 studies quasicoherent sheaves on Deligne–Mumford C∞-stacks, gen-
+eralizing§5, and§9 orbifold strata of Deligne–Mumford C∞-stacks.
+Appendix Asummarizesbackgroundonstacksfrom[3,4,29,46,49 ,55], foruse
+in§6–§9. Stacks are a very technical area, and §A is too terse to help a beginner
+learn the subject, it is intended only to establish notation and definit ions for
+those already familiar with stacks. Readers with no experience of st acks are
+advised to first consult an introductory text such as Vistoli [68], G omez [29],
+Laumon and Moret-Bailly [46], or the online ‘Stacks Project’ [34].
+Much of§2–§4 is already understood in synthetic differential geometry, such
+as in the work of Dubuc [23] and Moerdijk and Reyes [54]. But we believ e it is
+worthwhile giving a detailed and self-contained exposition, from our o wn point
+of view. Sections 5–9 are new, so far as the author knows, though §5–§8 are
+based on well known material in algebraic geometry.
+Acknowledgements. I would like to thank Omar Antolin, Eduardo Dubuc, Kelli
+Francis-Staite, Jacob Gross, Jacob Lurie, and Ieke Moerdijk for helpful conver-
+sations, and a referee for many useful comments. This research was supported
+by EPSRC grants EP/H035303/1 and EP/J016950/1.
+2C∞-rings
+We begin by explaining the basic objects out of which our theories are built,
+C∞-rings, orsmooth rings . The archetypal example of a C∞-ring is the vector
+spaceC∞(X) of smooth functions c:X→Rfor a manifold X. Everything in
+thissectionis knowntoexpertsin syntheticdifferentialgeometry, andmuchofit
+canbe found in Moerdijkand Reyes[54, Ch. I], Dubuc [21–24] orKock [44,§III].
+We introducesomenew notationfor brevity, in particular, our fairC∞-ringsare
+known in the literature as ‘finitely generated and germ determined C∞-rings’.
+52.1 Two definitions of C∞-ring
+We first define C∞-rings in the style of classical algebra.
+Definition 2.1. AC∞-ringis a setCtogether with operations
+Φf:Cn=/rightanglenwncopies/rightanglene
+C×···×C−→C
+for alln/greaterorequalslant0 and smooth maps f:Rn→R, where by convention when n= 0 we
+defineC0to be the single point {∅}. These operations must satisfy the following
+relations: suppose m,n/greaterorequalslant0, andfi:Rn→Rfori= 1,...,mandg:Rm→R
+are smooth functions. Define a smooth function h:Rn→Rby
+h(x1,...,xn) =g/parenleftbig
+f1(x1,...,xn),...,fm(x1...,xn)/parenrightbig
+,
+for all (x1,...,xn)∈Rn. Then for all ( c1,...,cn)∈Cnwe have
+Φh(c1,...,cn) = Φg/parenleftbig
+Φf1(c1,...,cn),...,Φfm(c1,...,cn)/parenrightbig
+.
+We also require that for all 1 /lessorequalslantj/lessorequalslantn, definingπj:Rn→Rbyπj:
+(x1,...,xn)/ma√sto→xj, we have Φ πj(c1,...,cn) =cjfor all (c1,...,cn)∈Cn.
+Usually we refer to Cas theC∞-ring, leaving the operations Φ fimplicit.
+Amorphism betweenC∞-rings/parenleftbig
+C,(Φf)f:Rn→RC∞/parenrightbig
+,/parenleftbig
+D,(Ψf)f:Rn→RC∞/parenrightbig
+is a mapφ:C→Dsuch that Ψ f/parenleftbig
+φ(c1),...,φ(cn)/parenrightbig
+=φ◦Φf(c1,...,cn) for
+all smooth f:Rn→Randc1,...,cn∈C. We will write C∞Ringsfor the
+category of C∞-rings.
+Here is the motivating example, which we will study at greater length in §3:
+Example 2.2. LetXbe a manifold, which may be without boundary, or with
+boundary, or with corners. Write C∞(X) for the set of smooth functions c:
+X→R. Forn/greaterorequalslant0 andf:Rn→Rsmooth, define Φ f:C∞(X)n→C∞(X) by
+/parenleftbig
+Φf(c1,...,cn)/parenrightbig
+(x) =f/parenleftbig
+c1(x),...,cn(x)/parenrightbig
+, (2.1)
+for allc1,...,cn∈C∞(X) andx∈X. It is easy to see that C∞(X) and the
+operations Φ fform aC∞-ring.
+Example 2.3. TakeXto be the point∗in Example 2.2. Then C∞(∗) =R,
+with operations Φ f:Rn→Rgiven by Φ f(x1,...,xn) =f(x1,...,xn). This
+makesRinto the simplest nonzero example of a C∞-ring, the initial object
+inC∞Rings.
+Note thatC∞-rings are far more general than those coming from manifolds.
+For example, if Xis any topological space we could define a C∞-ringC0(X) to
+be the set of continuous c:X→Rwith operations Φ fdefined as in (2.1). For
+Xa manifold with dim X >0, theC∞-ringsC∞(X) andC0(X) are different.
+There is a more succinct definition of C∞-rings using category theory:
+6Definition 2.4. WriteManfor the category of manifolds, and Eucfor the full
+subcategory of Manwith objects the Euclidean spaces Rn. That is, the objects
+ofEucareRnforn= 0,1,2,...,and the morphisms in Eucare smooth maps
+f:Rm→Rn. Write Setsfor the category of sets. In both EucandSets
+we have notions of (finite) products of objects (that is, Rm+n=Rm×Rn, and
+productsS×Tof setsS,T), and products of morphisms.
+Define a ( category-theoretic )C∞-ringto be a product-preserving functor
+F:Euc→Sets. HereFshould also preserve the empty product, that is, it
+mapsR0inEucto the terminal object in Sets, the point∗.
+C∞-rings in this sense are an example of an algebraic theory in the sense of
+Ad´ amek, Rosick´ y and Vitale [1], and many of the basic categorical p roperties
+ofC∞-rings follow from this.
+Here is how this relates to Definition 2.1. Suppose F:Euc→Setsis a
+product-preserving functor. Define C=F(R). Then Cis an object in Sets,
+that is, a set. Suppose n/greaterorequalslant0 andf:Rn→Ris smooth. Then fis a morphism
+inEuc, soF(f) :F(Rn)→F(R) =Cis a morphism in Sets. SinceFpreserves
+productsF(Rn) =F(R)×···×F(R) =Cn, soF(f) mapsCn→C. We define
+Φf:Cn→Cby Φf=F(f). The fact that Fis a functor implies that the Φ f
+satisfy the relations in Definition 2.1, so/parenleftbig
+C,(Φf)f:Rn→RC∞/parenrightbig
+is aC∞ring.
+Conversely, if/parenleftbig
+C,(Φf)f:Rn→RC∞/parenrightbig
+is aC∞-ring then we define F:Euc→
+SetsbyF(Rn) =Cn, and iff:Rn→Rmis smooth then f= (f1,...,fm) for
+fi:Rn→Rsmooth, and we define F(f) :Cn→CmbyF(f) : (c1,...,cn)/ma√sto→/parenleftbig
+Φf1(c1,...,cn),...,Φfm(c1,...,cn)/parenrightbig
+. ThenFis a product-preserving functor.
+This defines a 1-1 correspondence between C∞-rings in the sense of Definition
+2.1, and category-theoretic C∞-rings in the sense of Definition 2.4.
+As in Moerdijk and Reyes [54, p. 21–22] we have:
+Proposition 2.5. In the category C∞RingsofC∞-rings, all limits and all
+filtered colimits exist, and regarding C∞-rings as functors F:Euc→Sets
+as in Definition 2.4,they may be computed objectwise in Eucby taking the
+corresponding limits/filtered colimits in Sets.
+Also, all small colimits exist, though in general they are no t computed ob-
+jectwise in Eucby taking colimits in Sets. In particular, pushouts and all finite
+colimits exist in C∞Rings.
+We will write D∐φ,C,ψEorD∐CEfor the pushout of morphisms φ:C→D,
+ψ:C→EinC∞Rings. WhenC=R, the initial object in C∞Rings, pushouts
+D∐REare called coproducts and are usually written D⊗∞E. ForR-algebras
+A,Bthe coproduct is the tensor product A⊗B. But the coproduct D⊗∞Eof
+C∞-ringsD,Eis generally different from their coproduct D⊗EasR-algebras.
+For example we have C∞(Rm)⊗∞C∞(Rn)∼=C∞(Rm+n), which contains but
+is much larger than the tensor product C∞(Rm)⊗C∞(Rn) form,n>0.
+2.2C∞-rings as commutative R-algebras, and ideals
+EveryC∞-ringChas an underlying commutative R-algebra:
+7Definition 2.6. LetCbe aC∞-ring. Then we may give Cthe structure of
+acommutative R-algebra. Define addition ‘+’ on Cbyc+c′= Φf(c,c′) for
+c,c′∈C, wheref:R2→Risf(x,y) =x+y. Define multiplication ‘ ·’ onCby
+c·c′= Φg(c,c′), whereg:R2→Risf(x,y) =xy. Define scalar multiplication
+byλ∈Rbyλc= Φλ′(c), whereλ′:R→Risλ′(x) =λx. Define elements 0
+and 1 in Cby 0 = Φ 0′(∅) and 1 = Φ 1′(∅), where 0′:R0→Rand 1′:R0→R
+are the maps 0′:∅/ma√sto→0 and 1′:∅/ma√sto→1. The relations on the Φ fimply that
+all the axioms of a commutative R-algebra are satisfied. In Example 2.2, this
+yields the obvious R-algebra structure on the smooth functions c:X→R.
+Here is another way to say this. In an R-algebraA, then-fold ‘operations’
+Φ :An→A, that is, all the maps An→Awe can construct using only addition,
+multiplication, scalar multiplication, and the elements 0 ,1∈A, correspond ex-
+actly to polynomials p:Rn→R. Since polynomials are smooth, the operations
+of anR-algebra are a subset of those of a C∞-ring, and we can truncate from
+C∞-rings to R-algebras. As there are many more smooth functions f:Rn→R
+than there are polynomials, a C∞-ring has far more structure and operations
+than a commutative R-algebra.
+Definition 2.7. AnidealIinCis an idealI⊂CinCregarded as a commu-
+tativeR-algebra. Then we make the quotient C/Iinto aC∞-ring as follows. If
+f:Rn→Ris smooth, define ΦI
+f: (C/I)n→C/Iby
+ΦI
+f(c1+I,...,cn+I) = Φf(c1,...,cn)+I.
+To show this is well-defined, we must show it is independent of the choic e of
+representatives c1,...,cninCforc1+I,...,cn+IinC/I. By Hadamard’s
+Lemma there exist smooth functions gi:R2n→Rfori= 1,...,nwith
+f(y1,...,yn)−f(x1,...,xn) =/summationtextn
+i=1(yi−xi)gi(x1,...,xn,y1,...,yn)
+for allx1,...,xn,y1,...,yn∈R. Ifc′
+1,...,c′
+nare alternative choices for c1,...,
+cn, so thatc′
+i+I=ci+Ifori= 1,...,nandc′
+i−ci∈I, we have
+Φf(c′
+1,...,c′
+n)−Φf(c1,...,cn) =/summationtextn
+i=1(c′
+i−ci)Φgi(c′
+1,...,c′
+n,c1,...,cn).
+The second line lies in Iasc′
+i−ci∈IandIis an ideal, so ΦI
+fis well-defined,
+and clearly/parenleftbig
+C/I,(ΦI
+f)f:Rn→RC∞/parenrightbig
+is aC∞-ring.
+IfCis aC∞-ring, we will use the notation ( fa:a∈A) to denote the
+ideal inCgenerated by a collection of elements fa,a∈AinC, in the sense of
+commutative R-algebras. That is,
+(fa:a∈A) =/braceleftbig/summationtextn
+i=1fai·ci:n/greaterorequalslant0,a1,...,an∈A,c1,...,cn∈C/bracerightbig
+.
+Definition 2.8. AC∞-ringCis calledfinitely generated if there exist c1,...,cn
+inCwhich generate Cover allC∞-operations. That is, for each c∈Cthere
+exists a smooth map f:Rn→Rwithc= Φf(c1,...,cn). (This is a much
+weaker condition than Cbeing finitely generated as a commutative R-algebra.)
+8By Kock [44, Prop. III.5.1], C∞(Rn) is the free C∞-ring withngenerators.
+Given such C,c1,...,cn, defineφ:C∞(Rn)→Cbyφ(f) = Φf(c1,...,cn) for
+smoothf:Rn→R, whereC∞(Rn) is as in Example 2.2 with X=Rn. Then
+φis a surjective morphism of C∞-rings, soI= Kerφis an ideal in C∞(Rn),
+andC∼=C∞(Rn)/Ias aC∞-ring. Thus, Cis finitely generated if and only if
+C∼=C∞(Rn)/Ifor somen/greaterorequalslant0 and ideal IinC∞(Rn).
+AnidealIinaC∞-ringCiscalledfinitelygenerated ifIisafinitelygenerated
+ideal of the underlying commutative R-algebra of Cin Definition 2.6, that is,
+I= (i1,...,ik)forsomei1,...,ik∈C. AC∞-ringCiscalledfinitelypresented if
+C∼=C∞(Rn)/Ifor somen/greaterorequalslant0, whereIis a finitely generated ideal in C∞(Rn).
+A difference with conventional algebraic geometry is that C∞(Rn) is not
+noetherian, so ideals in C∞(Rn) may not be finitely generated, and Cfinitely
+generated does not imply Cfinitely presented.
+WriteC∞RingsfgandC∞Ringsfpfor the full subcategories of finitely
+generated and finitely presented C∞-rings in C∞Rings.
+Example 2.9. AWeil algebra [21, Def. 1.4]is afinite-dimensional commutative
+R-algebraWwhich has a maximal ideal mwithW/m∼=Randmn= 0 for some
+n >0. Then by Dubuc [21, Prop. 1.5] or Kock [44, Th. III.5.3], there is a
+unique way to make Winto aC∞-ring compatible with the given underlying
+commutative R-algebra. This C∞-ring is finitely presented [44, Prop. III.5.11].
+C∞-rings from Weil algebras are important in synthetic differential geo metry,
+in arguments involving infinitesimals. See [11, §2] for a detailed study of this.
+2.3 Local C∞-rings, and localization
+Definition 2.10. AC∞-ringCis called localif regarded as an R-algebra, as
+in Definition 2.6, Cis a local R-algebra with residue field R. That is, Chas a
+unique maximal ideal mCwithC/mC∼=R.
+IfC,Dare localC∞-rings with maximal ideals mC,mD, andφ:C→Dis
+a morphism of C∞rings, then using the fact that C/mC∼=R∼=D/mDwe see
+thatφ−1(mD) =mC, that is,φis alocalmorphism of local C∞-rings. Thus,
+there is no difference between morphisms and local morphisms.
+Remark 2.11. We use the term ‘local C∞-ring’ following Dubuc [23, Def. 4].
+They are also called C∞-local rings in Dubuc [22, Def. 2.13], pointed local C∞-
+ringsin [54,§I.3] andArchimedean local C∞-ringsin [52,§3].
+Moerdijk and Reyes [52–54] use the term ‘local C∞-ring’ to mean a C∞-ring
+which is a local R-algebra, but which need not have residue field R.
+The next example is taken from Moerdijk and Reyes [54, §I.3].
+Example 2.12. WriteC∞(N) for theR-algebraofallfunctions f:N→R. It is
+a finitely generated C∞-ring isomorphic to C∞(R)/{f∈C∞(R) :f|N= 0}. Let
+Fbe anon-principal ultrafilter onN, in the sense of Comfort and Negrepontis
+[16], and let I⊂Cbe the prime ideal of f:N→Rsuch that{n∈N:f(n) = 0}
+lies inF. ThenC=C∞(N)/Iis a finitely generated C∞-ring which is a local
+9R-algebra by [54, Ex. I.3.2], that is, it has a unique maximal ideal mC, but its
+residue field is not Rby [54, Cor. I.3.4]. Hence Cis a localC∞-ring in the sense
+of [52–54], but not in our sense.
+Localizations ofC∞-rings are studied in [22,23,52,53], see [54, p. 23].
+Definition 2.13. LetCbe aC∞-ring andSa subset of C. Alocalization
+C[s−1:s∈S] ofCatSis aC∞-ringD=C[s−1:s∈S] and a morphism
+π:C→Dsuch thatπ(s) is invertible in Dfor alls∈S, with the universal
+property that if Eis aC∞-ring andφ:C→Ea morphism with φ(s) invertible
+inEfor alls∈S, then there is a unique morphism ψ:D→Ewithφ=ψ◦π.
+A localization C[s−1:s∈S] always exists — it can be constructed by
+adjoining an extra generator s−1and an extra relation s·s−1−1 = 0 for each
+s∈S— and is unique up to unique isomorphism. When S={c}we have
+an exact sequence 0 →I→C⊗∞C∞(R)π−→C[c−1]→0, where C⊗∞C∞(R)
+is the coproduct of C,C∞(R) as in§2.1, with pushout morphisms ι1:C→
+C⊗∞C∞(R),ι2:C∞(R)→C⊗∞C∞(R), andIis the ideal in C⊗∞C∞(R)
+generated by ι1(c)·ι2(x)−1, wherexis the generator of C∞(R).
+AnR-pointxof aC∞-ringCis aC∞-ring morphism x:C→R, where
+Ris regarded as a C∞-ring as in Example 2.3. By [54, Prop. I.3.6], a map
+x:C→Ris a morphism of C∞-rings if and only if it is a morphism of the
+underlying R-algebras, as in Definition 2.6. Define Cxto be the localization
+Cx=C[s−1:s∈C,x(s)/\e}atio\slash= 0], with projection πx:C→Cx. ThenCxis a
+localC∞-ring by [53, Lem. 1.1]. The R-points ofC∞(Rn) are just evaluation
+at pointsx∈Rn. This also holds for C∞(X) for any manifold X.
+In a new result, we can describe these local C∞-ringsCxexplicitly. Note
+that the surjectivity of πx:C→Cxin the next proposition is surprising. It
+doesnot hold forgenerallocalizationsof C∞-rings— forinstance, π:C∞(R)→
+C∞(R)[x−1] is injective but not surjective, as x−1/∈Imπ— or for localizations
+πx:A→Axof rings or K-algebras in conventional algebraic geometry.
+Proposition 2.14. LetCbe aC∞-ring,x:C→RanR-point of C,andCx
+the localization, with projection πx:C→Cx. Thenπxis surjective with kernel
+an idealI⊂C,so thatCx∼=C/I,where
+I=/braceleftbig
+c∈C:there exists d∈Cwithx(d)/\e}atio\slash= 0inRandc·d= 0inC/bracerightbig
+.(2.2)
+Proof.ClearlyIin (2.2) is closed under multiplication by elements of C. Let
+c1,c2∈I, so there exist d1,d2∈Cwithx(d1)/\e}atio\slash= 0/\e}atio\slash=x(d2) andc1d1= 0 =c2d2.
+Thend1d2∈Cwithx(d1d2) =x(d1)x(d2)/\e}atio\slash= 0, and(c1+c2)(d1·d2) =d2(c1d1)+
+d1(c2d2) = 0, soc1+c2∈I. HenceIis an ideal, and C/IaC∞-ring.
+Supposec∈I, so there exists d∈Cwithx(d)/\e}atio\slash= 0 andcd= 0. Then πx(d)
+is invertible in Cxby definition. Thus
+πx(c) =πx(c)πx(d)πx(d)−1=πx(cd)πx(d)−1=πx(0)πx(d)−1= 0.
+ThereforeI⊆Kerπx. Soπx:C→Cxfactorizes uniquely as πx=ı◦π, where
+π:C→C/Iis the projection and ı:C/I→Cxis aC∞-ring morphism.
+10Supposec∈Cwithx(c)/\e}atio\slash= 0, and write ǫ=1
+2|x(c)|. Choose smooth
+functionsη:R→R\{0}, so thatη−1:R→R\{0}is also smooth, such that
+η(t) =tfor allt∈(x(c)−ǫ,x(c)+ǫ), andζ:R→Rsuch thatζ(t) = 0 for all
+t∈R\(x(c)−ǫ,x(c)+ǫ), so that (η−idR)·ζ= 0, andζ(x(c)) = 1.
+Setc1= Φη(c),c2= Φη−1(c) andd= Φζ(c) inC, using the C∞-ring
+operations from η,η−1,ζ. Thenc1c2= 1 inC, asη·η−1= 1, andx(d) =
+x(Φζ(c)) =ζ(x(c)) = 1, asx:C→Ris aC∞-ring morphism. Also
+(c1−c)·d=/parenleftbig
+Φη(c)−ΦidR(c)/parenrightbig
+Φζ(c) = Φ(η−idR)ζ(c) = Φ0(c) = 0.
+Hencec1−c∈Iasx(d)/\e}atio\slash= 0, soc+I=c1+I. But then ( c+I)(c2+I) =
+(c1+I)(c2+I) =c1c2+I= 1+IinC/I, soπ(c) =c+Iis invertible in C/I.
+As this holds for all c∈Cwithx(c)/\e}atio\slash= 0, by the universal property of Cx
+there exists a unique C∞-ring morphism :Cx→C/Iwithπ=◦πx. Since
+πx,πare surjective, πx=ı◦πandπ=◦πximply that ı:C/I→Cxand
+:Cx→C/Iare inverse, so both are isomorphisms.
+Example 2.15. Forn/greaterorequalslant0 andp∈Rn, defineC∞
+p(Rn) to be the set of germs
+of smooth functions c:Rn→Ratp∈Rn, made into a C∞-ring in the obvious
+way. Then C∞
+p(Rn) is a localC∞-ring in the sense of Definition 2.10. Here are
+three different ways to define C∞
+p(Rn), which yield isomorphic C∞-rings:
+(a) Defining C∞
+p(Rn) asthe germsoffunctions ofsmoothfunctionsat pmeans
+that points of C∞
+p(Rn) are∼-equivalence classes [( U,c)] of pairs ( U,c),
+whereU⊆Rnis open with p∈Uandc:U→Ris smooth, and
+(U,c)∼(U′,c′) if there exists p∈V⊆U∩U′open withc|V≡c′|V.
+(b) As the localization ( C∞(Rn))p=C∞(Rn)[g∈C∞(Rn) :g(p)/\e}atio\slash= 0]. Then
+points of (C∞(Rn))pare equivalence classes [ f/g] of fractions f/gfor
+f,g∈C∞(Rn) withg(p)/\e}atio\slash= 0, and fractions f/g,f′/g′are equivalent if
+there exists h∈C∞(Rn) withh(p)/\e}atio\slash= 0 andh(fg′−f′g)≡0.
+(c) As the quotient C∞(Rn)/I, whereIis the ideal of f∈C∞(Rn) with
+f≡0 nearp∈Rn.
+One can show (a)–(c) are isomorphic using the fact that if Uis any open neigh-
+bourhood of pinRnthen there exists smooth η:Rn→[0,1] such that η≡0 on
+an open neighbourhood of Rn\UinRnandη≡1 on an open neighbourhood
+ofpinU. By Moerdijk and Reyes [54, Prop. I.3.9], any finitely generated local
+C∞-ring is a quotient of some C∞
+p(Rn).
+2.4 Fair C∞-rings
+We now discuss an important class of C∞-rings, which we call fairC∞-rings,
+for brevity. Although our term ‘fair’ is new, we stress that the idea is already
+well-known, being originally introduced by Dubuc [22], [23, Def. 11], who first
+recognized their significance, under the name ‘ C∞-rings of finite type presented
+by an ideal of local character’, and in more recent works would be re ferred to
+as ‘finitely generated and germ-determined C∞-rings’.
+11Definition 2.16. An idealIinC∞(Rn) is called fairif for eachf∈C∞(Rn),
+flies inIif and only if πp(f) lies inπp(I)⊆C∞
+p(Rn) for allp∈Rn, where
+C∞
+p(Rn) is as in Example 2.15 and πp:C∞(Rn)→C∞
+p(Rn) is the natural
+projectionπp:c/ma√sto→[(Rn,c)]. AC∞-ringCis called fairif it is isomorphic
+toC∞(Rn)/I, whereIis a fair ideal. Equivalently, Cis fair if it is finitely
+generated and whenever c∈Cwithπp(c) = 0 in Cpfor allR-pointsp:C→R
+thenc= 0, using the notation of Definition 2.13.
+Dubuc [22], [23, Def. 11] calls fair ideals ideals of local character , and Mo-
+erdijk and Reyes [54, I.4] call them germ determined , which has now become the
+accepted term. Fair C∞-rings are also sometimes called germ determined C∞-
+rings, a more descriptive term than ‘fair’, but the definition of germ deter mined
+C∞-ringsCin [54, Def. I.4.1] does not require Cfinitely generated, so does not
+equate exactly to our fair C∞-rings. By Dubuc [22, Prop. 1.8], [23, Prop. 12]
+any finitely generated ideal Iis fair, so Cfinitely presented implies Cfair. We
+writeC∞Ringsfafor the full subcategory of fair C∞-rings in C∞Rings.
+Proposition 2.17. SupposeI⊂C∞(Rm)andJ⊂C∞(Rn)are ideals with
+C∞(Rm)/I∼=C∞(Rn)/JasC∞-rings. Then Iis finitely generated, or fair, if
+and only if Jis finitely generated, or fair, respectively.
+Proof.Writeφ:C∞(Rm)/I→C∞(Rn)/Jfor the isomorphism, and x1,...,xm
+for the generators of C∞(Rm), andy1,...,ynfor the generators of C∞(Rn).
+Sinceφis an isomorphism we can choose f1,...,fm∈C∞(Rn) withφ(xi+I) =
+fi+Jfori= 1,...,mandg1,...,gn∈C∞(Rm) withφ(gi+I) =yi+Jfor
+i= 1,...,n. It is now easy to show that
+I=/parenleftbig
+xi−fi/parenleftbig
+g1(x1,...,xm),...,gn(x1,...,xm)/parenrightbig
+, i= 1,...,m,
+andh/parenleftbig
+g1(x1,...,xm),...,gn(x1,...,xm)/parenrightbig
+, h∈J/parenrightbig
+.
+Hence, ifJisgeneratedby h1,...,hkthenIisgeneratedby xi−fi(g1,...,gn)
+fori= 1,...,mandhj(g1,...,gn) forj= 1,...,k, soJfinitely generated
+impliesIfinitelygenerated. Applyingthesameargumentto φ−1:C∞(Rn)/J→
+C∞(Rm)/I, we see that Iis finitely generated if and only if Jis.
+SupposeIis fair, and let f∈C∞(Rn) withπq(f)∈πq(J)⊆C∞
+q(Rn) for
+allq∈Rn. We will show that f∈J, so thatJis fair. Consider the function
+f′=f(g1,...,gn)∈C∞(Rm). Ifp= (p1,...,pm) inRmandq= (q1,...,qn) =/parenleftbig
+g1(p1,...,pm),...,gn(p1,...,pm)/parenrightbig
+thenφ:C∞(Rm)/I→C∞(Rn)/Jlocalizes
+to an isomorphism φp:C∞
+p(Rm)/πp(I)→C∞
+q(Rn)/πq(J) which maps φp:
+πp(f′)+πp(I)/ma√sto→πq(f)+πq(J). Sinceπq(f)∈πq(J), this gives πp(f′)∈πp(I)
+for allp∈Rm, sof′∈IasIis fair. But φ(f′+I) =f+J, sof′∈Iimplies
+f∈J. Therefore Jis fair. Conversely, Jis fair implies Iis fair.
+Example 2.18. The localC∞-ringC∞
+p(Rn) of Example 2.15 is the quotient of
+C∞(Rn) by the ideal Iof functions fwithf≡0 nearp∈Rn. Forn>0 thisI
+is fair, but not finitely generated. So C∞
+p(Rn) is fair, but not finitely presented,
+by Proposition 2.17.
+12The following example taken from Dubuc [24, Ex. 7.2] shows that localiz a-
+tions of fair C∞-rings need not be fair:
+Example 2.19. LetCbe the local C∞-ringC∞
+0(R), as in Example 2.15. Then
+C∼=C∞(R)/I, whereIis the ideal of all f∈C∞(R) withf≡0 near 0 in R.
+ThisIis fair, so Cis fair. Let c= [(x,R)]∈C. Then the localization C[c−1]
+is theC∞-ring of germs at 0 in Rof smooth functions R\{0}→R. Taking
+y=x−1as a generator of C[c−1], we see that C[c−1]∼=C∞(R)/J, whereJis
+the ideal of compactly supported functions in C∞(R). ThisJis not fair, so by
+Proposition 2.17, C[c−1] is not fair.
+Recall from category theory that if Cis a subcategory of a category D, a
+reflectionR:D→Cisaleft adjointtothe inclusion C֒→D. Thatis,R:D→C
+is a functor with natural isomorphisms Hom C(R(D),C)∼=HomD(D,C) for all
+C∈CandD∈D. We will define a reflection for C∞Ringsfa⊂C∞Ringsfg,
+following Moerdijk and Reyes [54, p. 48–49] (see also Dubuc [23, Th. 13]).
+Definition 2.20. LetCbe a finitely generated C∞-ring. LetICbe the ideal
+of allc∈Csuch thatπp(c) = 0 in Cpfor allR-pointsp:C→R. ThenC/IC
+is a finitely generated C∞-ring, with projection π:C→C/IC. It has the same
+R-points as C, that is, morphisms p:C/IC→Rare in 1-1 correspondence
+with morphisms p′:C→Rbyp′=p◦π, and the local rings ( C/IC)pandCp′
+are naturally isomorphic. It follows that C/ICis fair. Define a functor Rfa
+fg:
+C∞Ringsfg→C∞RingsfabyRfa
+fg(C) =C/ICon objects, and if φ:C→D
+is a morphism then φ(IC)⊆ID, soφinduces a morphism φ∗:C/IC→D/ID,
+and we set Rfa
+fg(φ) =φ∗. It is easy to see Rfa
+fgis a reflection.
+IfIis an ideal in C∞(Rn), write¯Ifor the set of f∈C∞(Rn) withπp(f)∈
+πp(I) for allp∈Rn. Then¯Iis the smallest fair ideal in C∞(Rn) containing I,
+thegerm-determined closure ofI, andRfa
+fg/parenleftbig
+C∞(Rn)/I/parenrightbig∼=C∞(Rn)/¯I.
+Example 2.21. Letη:R→[0,∞) be smooth with η(x)>0 forx∈(0,1) and
+η(x) = 0 forx /∈(0,1). DefineI⊆C∞(R) by
+I=/braceleftbig/summationtext
+a∈Aga(x)η(x−a) :A⊂Zis finite,ga∈C∞(R),a∈A/bracerightbig
+.
+ThenIis an ideal in C∞(R), soC=C∞(R)/Iis aC∞-ring. The set of
+f∈C∞(R) such that πp(f) lies inπp(I)⊆C∞
+p(R) for allp∈Ris
+¯I=/braceleftbig/summationtext
+a∈Zga(x)η(x−a) :ga∈C∞(R), a∈Z/bracerightbig
+,
+where the sum/summationtext
+a∈Zga(x)η(x−a) makes sense as at most one term is nonzero
+at any point x∈R. Since¯I/\e}atio\slash=I, we see that Iisnot fair, soC=C∞(R)/Iis
+not a fairC∞-ring. In fact ¯Iis the smallest fair ideal containing I. We have
+IC∞(R)/I=¯I/I, andRfa
+fg/parenleftbig
+C∞(R)/I) =C∞(R)/¯I.
+Proposition 2.22. LetCbe aC∞-ring, andGa finite group acting on Cby
+automorphisms. Then the fixed subset CGofGinChas the structure of a C∞-
+ring in a unique way, such that the inclusion CG֒→Cis aC∞-ring morphism.
+IfCis fair, or finitely presented, then CGis also fair, or finitely presented.
+13Proof.For the first part, let f:Rn→Rbe smooth, and c1,...,cn∈CG. Then
+γ·Φf(c1,...,cn) = Φf(γ·c1,...,γ·cn) = Φf(c1,...,cn) for eachγ∈G, so
+Φf(c1,...,cn)∈CG. Define ΦG
+f: (CG)n→CGby ΦG
+f= Φf|(CG)n. It is now
+trivial to check that the operations ΦG
+ffor smooth f:Rn→RmakeCGinto a
+C∞-ring, uniquely such that CG֒→Cis aC∞-ring morphism.
+Suppose now that Cis finitely generated. Choose a finite set of generators
+forC, and by adding the images of these generators under G, extend to a set
+of (not necessarily distinct) generators x1,...,xnforC, on whichGacts freely
+by permutation. This gives an exact sequence 0 ֒→I→C∞(Rn)→C→0,
+whereC∞(Rn) is freely generated by x1,...,xn. HereRnis a direct sum of
+copies of the regular representation of G, andC∞(Rn)→CisG-equivariant.
+HenceIis aG-invariant ideal in C∞(Rn), which is fair, or finitely generated,
+respectively. Taking G-invariant parts gives an exact sequence 0 ֒→IG→
+C∞(Rn)Gπ−→CG→0, whereC∞(Rn)G,CGare clearlyC∞-rings.
+AsGacts linearly on Rnit acts by automorphisms on the polynomial ring
+R[x1,...,xn]. By a classical theorem of Hilbert [70, p. 274], R[x1,...,xn]G
+is a finitely presented R-algebra, so we can choose generators p1,...,plfor
+R[x1,...,xn]G, which induce a surjective R-algebra morphism R[p1,...,pl]→
+R[x1,...,xn]Gwith kernel generated by q1,...,qm∈R[p1,...,pl].
+By results of Bierstone [6] for Ga finite group and Schwarz [63] for Ga
+compact Lie group, any G-invariant smooth function on Rnmay be written
+as a smooth function of the generators p1,...,plofR[x1,...,xn]G, giving a
+surjective morphism C∞(Rl)→C∞(Rn)G, whose kernel is the ideal in C∞(Rl)
+generated by q1,...,qm. ThusC∞(Rn)Gis finitely presented.
+AlsoCGis generated by π(p1),...,π(pl), soCGis finitely generated, and we
+have an exact sequence 0 ֒→J→C∞(Rl)π−→CG→0, whereJis the ideal in
+C∞(Rl) generated by q1,...,qmand the lifts to C∞(Rl) of a generating set for
+the idealIGinC∞(Rn)G∼=C∞(Rl)/(q1,...,qm).
+Suppose now that Iis fair. Then for f∈C∞(Rn)G,flies inIGif and only
+ifπp(f)∈πp(I)⊆C∞
+p(Rn) for allp∈Rn. IfHis the subgroup of Gfixing
+pthenHacts onC∞
+p(Rn), andπp(f) isH-invariant as fisG-invariant, and
+πp(I)H=πp(IG). Thus we may rewrite the condition as flies inIGif and only
+ifπp(f)∈πp(IG)⊆C∞
+p(Rn) for allp∈Rn. Projecting from RntoRn/G, this
+says thatflies inIGif and only if πp(f) lies inπp(IG)⊆/parenleftbig
+C∞(Rn)G/parenrightbig
+pfor all
+p∈Rn/G. SinceC∞(Rn)Gis finitely presented, it follows as in [54, Cor. I.4.9]
+thatJis fair, so CGis fair.
+SupposeIis finitely generated in C∞(Rn), with generators f1,...,fk. As
+Rnis a sum of copies of the regular representation of G, so that every irre-
+ducible representation of Goccurs as a summand of Rn, one can show that IG
+is generated as an ideal in C∞(Rn/G) by then(k+1) elements fG
+iand (fixj)G
+fori= 1,...,kandj= 1,...,n, wherefG=1
+|G|/summationtext
+γ∈Gf◦γis theG-invariant
+part off∈C∞(Rn). Therefore Jis finitely generated by q1,...,qmand lifts of
+fG
+i,(fixj)G. Hence if Cis finitely presented then CGis finitely presented.
+142.5 Pushouts of C∞-rings
+Proposition 2.5 shows that pushouts of C∞-rings exist. For finitely generated
+C∞-rings, we can describe these pushouts explicitly.
+Example 2.23. Suppose the following is a pushout diagram of C∞-rings:
+Cβ/d47/d47
+α/d15/d15E
+δ/d15/d15
+Dγ/d47/d47F,
+so thatF=D∐CE, withC,D,Efinitely generated. Then we have exact
+sequences
+0→I ֒→C∞(Rl)φ−→C→0,0→J ֒→C∞(Rm)ψ−→D→0,
+and 0→K ֒→C∞(Rn)χ−→E→0,(2.3)
+whereφ,ψ,χare morphisms of C∞-rings, and I,J,Kare ideals in C∞(Rl),
+C∞(Rm),C∞(Rn). Writex1,...,xlandy1,...,ymandz1,...,znfor the gen-
+erators ofC∞(Rl),C∞(Rm),C∞(Rn) respectively. Then φ(x1),...,φ(xl) gen-
+erateC, andα◦φ(x1),...,α◦φ(xl) lie inD, so we may write α◦φ(xi) =ψ(fi)
+fori= 1,...,lasψis surjective, where fi:Rm→Ris smooth. Similarly
+β◦φ(x1),...,β◦φ(xl) lie inE, so we may write β◦φ(xi) =χ(gi) fori= 1,...,l,
+wheregi:Rn→Ris smooth.
+Then from the explicit construction of pushouts of C∞-rings we obtain an
+exact sequence with ξa morphism of C∞-rings
+0 /d47/d47L /d47/d47C∞(Rm+n)ξ/d47/d47F /d47/d470, (2.4)
+where we write the generators of C∞(Rm+n) asy1,...,ym,z1,...,zn, and then
+Lis the ideal in C∞(Rm+n) generated by the elements d(y1,...,ym) ford∈
+J⊆C∞(Rm), ande(z1,...,zn) fore∈K⊆C∞(Rn), andfi(y1,...,ym)−
+gi(z1,...,zn) fori= 1,...,l.
+For the case of coproducts D⊗∞E, withC=R,l= 0 andI={0}, we have
+/parenleftbig
+C∞(Rm)/J/parenrightbig
+⊗∞/parenleftbig
+C∞(Rn)/K/parenrightbig∼=C∞(Rm+n)/(J,K).
+Proposition 2.24. The subcategories C∞RingsfgandC∞Ringsfpare closed
+under pushouts and all finite colimits in C∞Rings.
+Proof.Firstweshow C∞Ringsfg,C∞Ringsfpareclosedunderpushouts. Sup-
+poseC,D,Eare finitely generated, and use the notation of Example 2.23. Then
+Fis finitely generated with generators y1,...,ym,z1,...,zn, soC∞Ringsfg
+is closed under pushouts. If C,D,Eare finitely presented then we can take
+J= (d1,...,dj) andK= (e1,...,ek), and then Example 2.23 gives
+L=/parenleftbig
+dp(y1,...,ym), p= 1,...,j, ep(z1,...,zn), p= 1,...,k,
+fp(y1,...,ym)−gp(z1,...,zn), p= 1,...,l/parenrightbig
+.(2.5)
+15SoLis finitely generated, and F∼=C∞(Rm+n)/Lis finitely presented. Thus
+C∞Ringsfpis closed under pushouts.
+NowRis an initial object in C∞Ringsfg,C∞Ringsfp⊂C∞Rings, and
+all finite colimits may be constructed by repeated pushouts involving the initial
+object. Hence C∞Ringsfg,C∞Ringsfpare closed under finite colimits.
+Here is an example from Dubuc [24, Ex. 7.1], Moerdijk and Reyes [54, p . 49].
+Example 2.25. Consider the coproduct C∞(R)⊗∞C∞
+0(R), whereC∞
+0(R) is
+theC∞-ring of germs of smooth functions at 0 in Ras in Example 2.15. Then
+C∞(R),C∞
+0(R) are fairC∞-rings, but C∞
+0(R) is not finitely presented. By
+Example 2.23, C∞(R)⊗∞C∞
+0(R) =C∞(R)∐RC∞
+0(R)∼=C∞(R2)/L, whereL
+is the ideal in C∞(R2) generated by functions f(x,y) =g(y) forg∈C∞(R)
+withg≡0 near 0∈R. This ideal Lis not fair, since for example one can
+findf∈C∞(R2) withf(x,y) = 0 if and only if |xy|/lessorequalslant1, and then f /∈Lbut
+πp(f)∈πp(L)⊆C∞
+p(R2) for allp∈R2. HenceC∞(R)⊗∞C∞
+0(R) is not a fair
+C∞-ring, by Proposition 2.17, and pushouts of fair C∞-rings need not be fair.
+Our next result is referred to in the last part of Dubuc [23, Th. 13].
+Proposition 2.26. C∞Ringsfais not closed under pushouts in C∞Rings.
+Nonetheless, pushouts and all finite colimits exist in C∞Ringsfa,although they
+may not coincide with pushouts and finite colimits in C∞Rings.
+Proof.Example 2.25 shows that C∞Ringsfais not closed under pushouts in
+C∞Rings. To construct finite colimits in C∞Ringsfa, we first take the colimit
+inC∞Ringsfg, which exists by Propositions 2.5 and 2.24, and then apply the
+reflection functor Rfa
+fg. By the universal properties of colimits and reflection
+functors, the result is a colimit in C∞Ringsfa.
+2.6 Flat ideals
+The following class of ideals in C∞(Rn) is defined by Moerdijk and Reyes [54,
+p. 47, p. 49] (see also Dubuc [22, §1.7(a)]), who call them flat ideals :
+Definition 2.27. LetXbe a closed subset of Rn. Define m∞
+Xto be the ideal
+of all functions g∈C∞(Rn) such that ∂kg|X≡0 for allk/greaterorequalslant0, that is,gand
+all its derivatives vanish at each x∈X. If the interior X◦ofXinRnis dense
+inX, that is (X◦) =X, then∂kg|X≡0 for allk/greaterorequalslant0 if and only if g|X≡0. In
+this caseC∞(Rn)/m∞
+X∼=C∞(X) :=/braceleftbig
+f|X:f∈C∞(Rn)/bracerightbig
+.
+Flat ideals are always fair. Here is an example from [54, Th. I.1.3].
+Example 2.28. TakeXtobethepoint{0}. Iff,f′∈C∞(Rn)thenf−f′liesin
+m∞
+{0}if and only if f,f′have the same Taylor series at 0. Thus C∞(Rn)/m∞
+{0}is
+theC∞-ring of Taylor series at 0 of f∈C∞(Rn). Since any formal power series
+inx1,...,xnis the Taylorseries of some f∈C∞(Rn), we haveC∞(Rn)/m∞
+{0}∼=
+R[[x1,...,xn]]. Thus the R-algebra of formal power series R[[x1,...,xn]] can
+be made into a C∞-ring.
+16The following nontrivial result is proved by Reyes and van Quˆ e [60, Th . 1],
+generalizing an unpublished result of A.P. Calder´ on in the case X=Y={0}.
+It can also be found in Moerdijk and Reyes [54, Cor. I.4.12].
+Proposition 2.29. LetX⊆RmandY⊆Rnbe closed. Then as ideals in
+C∞(Rm+n)we have (m∞
+X,m∞
+Y) =m∞
+X×Y.
+Moerdijk and Reyes [54, Cor. I.4.19] prove:
+Proposition 2.30. LetX⊆Rnbe closed with X/\e}atio\slash=∅,Rn. Then the ideal m∞
+X
+inC∞(Rn)is not countably generated.
+We can use these to study C∞-rings of manifolds with corners.
+Example 2.31. Let 0<k/lessorequalslantn, and consider the closed subset Rn
+k= [0,∞)k×
+Rn−kinRn, the local model for manifolds with corners. Write C∞(Rn
+k) for the
+C∞-ring/braceleftbig
+f|Rn
+k:f∈C∞(Rn)/bracerightbig
+. Since the interior ( Rn
+k)◦= (0,∞)k×Rn−kof
+Rn
+kis dense in Rn
+k, as in Definition 2.27 we have C∞(Rn
+k) =C∞(Rn)/m∞
+Rn
+k. As
+m∞
+Rn
+kis not countably generated by Proposition 2.30, it is not finitely gener ated,
+and thusC∞(Rn
+k) is not a finitely presented C∞-ring, by Proposition 2.17.
+Consider the coproduct C∞(Rm
+k)⊗∞C∞(Rn
+l) inC∞Rings, that is, the
+pushoutC∞(Rm
+k)∐RC∞(Rn
+l) over the trivial C∞-ringR. By Example 2.23
+and Proposition 2.29 we have
+C∞(Rm
+k)⊗∞C∞(Rn
+l)∼=C∞(Rm+n)/(m∞
+Rm
+k,m∞
+Rn
+l) =C∞(Rm+n)/m∞
+Rm
+k×Rn
+l
+=C∞(Rm
+k×Rn
+l)∼=C∞(Rm+n
+k+l).
+This is an example of Theorem 3.5 below, with X=Rm
+k,Y=Rn
+landZ=∗.
+3 The C∞-ringC∞(X)of a manifold X
+We nowstudy the C∞-ringsC∞(X) of manifolds Xdefined in Example 2.2. We
+are interested in manifolds without boundary (locally modelled on Rn), and in
+manifolds with boundary (locally modelled on [0 ,∞)×Rn−1), and in manifolds
+with corners (locally modelled on [0 ,∞)k×Rn−k). Manifolds with corners were
+considered by the author [35,40], and we use the conventions of th ose papers.
+TheC∞-rings of manifolds with boundary are discussed by Reyes [59] and
+Kock [44,§III.9], but Kock appears to have been unaware of Proposition 2.29,
+which makes C∞-rings of manifolds with boundary easier to understand.
+IfX,Yare manifolds with cornersof dimensions m,n, then [40,§2.1] defined
+f:X→Yto beweakly smooth iffis continuous and whenever ( U,φ),(V,ψ)
+are charts on X,Ythenψ−1◦f◦φ: (f◦φ)−1(ψ(V))→Vis a smooth map
+from (f◦φ)−1(ψ(V))⊂RmtoV⊂Rn. Asmooth map is a weakly smooth
+mapfsatisfying some extra conditions over ∂kX,∂lYin [40,§2.1]. If∂Y=∅
+these conditionsarevacuous, sofor manifoldswithout boundary, weaklysmooth
+mapsandsmoothmapscoincide. Write Man,Manb,Mancforthecategoriesof
+manifolds without boundary, and with boundary, and with corners, respectively,
+with morphisms smooth maps.
+17Proposition 3.1. (a) IfXis a manifold without boundary then the C∞-ring
+C∞(X)of Example 2.2is finitely presented.
+(b)IfXis a manifold with boundary, or with corners, and ∂X/\e}atio\slash=∅,then the
+C∞-ringC∞(X)of Example 2.2is fair, but is not finitely presented.
+Proof.Part (a) is proved in Dubuc [23, p. 687] and Moerdijk and Reyes [54,
+Th. I.2.3] following an observation of Lawvere, that if Xis a manifold without
+boundary then we can choose a closed embedding i:X ֒→RNforN≫0, and
+thenXis a retract of an open neighbourhood Uofi(X), so we have an exact
+sequence 0→I→C∞(RN)i∗
+−→C∞(X)→0 in which the ideal Iis finitely
+generated, and thus the C∞-ringC∞(X) is finitely presented.
+For (b), ifXis ann-manifold with boundary, or with corners, then roughly
+by gluing on a ‘collar’ ∂X×(−ǫ,0] toXalong∂Xfor smallǫ >0, we can
+embedXasaclosedsubsetinan n-manifoldX′withoutboundary,suchthatthe
+inclusionX ֒→X′is locally modelled on the inclusion of Rn
+k= [0,∞)k×Rn−kin
+(−ǫ,∞)k×Rn−kfork/lessorequalslantn. Choose a closed embedding i:X′֒→RNforN≫0
+as above, giving 0 →I′→C∞(RN)i∗
+−→C∞(X′)→0 withI′generated by
+f1,...,fk∈C∞(RN). Leti(X′)⊂T⊂RNbe an open tubular neighbourhood
+ofi(X′) inRN, with projection π:T→i(X′). SetU=π−1(i(X◦))⊂T⊂RN,
+whereX◦is the interior of X. ThenUis open in RNwithi(X◦) =U∩i(X′),
+and the closure ¯UofUinRNhasi(X) =¯U∩i(X′).
+LetIbe the ideal ( f1,...,fk,m∞¯U) inC∞(RN). ThenIis fair, as (f1,...,fk)
+andm∞¯Uare fair. Since Uis open in RNand dense in ¯U, as in Definition
+2.27 we have g∈m∞
+¯Uif and only if g|¯U≡0. Therefore the isomorphism
+(i∗)∗:C∞(RN)/I′→C∞(X′) identifies the ideal I/I′inC∞(X′) with the
+ideal off∈C∞(X′) such that f|X≡0, sinceX=i−1(¯U). Hence
+C∞(RN)/I∼=C∞(X′)//braceleftbig
+f∈C∞(X′) :f|X≡0/bracerightbig∼=/braceleftbig
+f|X:f∈C∞(X′)/bracerightbig∼=C∞(X).
+AsIis a fair ideal, this implies that C∞(X) is a fairC∞-ring. If∂X/\e}atio\slash=∅then
+using Proposition 2.30 we can show Iis not countably generated, so C∞(X) is
+not finitely presented by Proposition 2.17.
+Next we consider the transformation X/ma√sto→C∞(X) as a functor.
+Definition 3.2. WriteC∞Ringsop, (C∞Ringsfp)op, (C∞Ringsfa)opfor the
+opposite categories of C∞Rings,C∞Ringsfp,C∞Ringsfa(i.e. directions of
+morphisms are reversed). Define functors
+FC∞Rings
+Man:Man−→(C∞Ringsfp)op⊂C∞Ringsop,
+FC∞Rings
+Manb:Manb−→(C∞Ringsfa)op⊂C∞Ringsop,
+FC∞Rings
+Manc:Manc−→(C∞Ringsfa)op⊂C∞Ringsop
+asfollows. On objectsthe functors FC∞Rings
+Man∗mapX/ma√sto→C∞(X), whereC∞(X)
+is aC∞-ring as in Example 2.2. On morphisms, if f:X→Yis a smooth map
+of manifolds then f∗:C∞(Y)→C∞(X) mapping c/ma√sto→c◦fis a morphism
+18ofC∞-rings, so that f∗:C∞(Y)→C∞(X) is a morphism in C∞Rings,
+andf∗:C∞(X)→C∞(Y) a morphism in C∞Ringsop, andFC∞Rings
+Man∗maps
+f/ma√sto→f∗. ClearlyFC∞Rings
+Man,FC∞Rings
+Manb,FC∞Rings
+Mancare functors.
+Iff:X→Yisonlyweakly smooth thenf∗:C∞(Y)→C∞(X)inDefinition
+3.2 is still a morphism of C∞-rings. From [54, Prop. I.1.5] we deduce:
+Proposition 3.3. LetX,Ybe manifolds with corners. Then the map f/ma√sto→f∗
+from weakly smooth maps f:X→Yto morphisms of C∞-ringsφ:C∞(Y)→
+C∞(X)is a1-1correspondence.
+In the category of manifolds Man, the morphisms are weakly smooth maps.
+SoFC∞Rings
+Man is both injective on morphisms (faithful), and surjective on mor-
+phisms (full), as in Moerdijk and Reyes [54, Th. I.2.8]. But in Manb,Manc
+the morphisms are smooth maps, a proper subset of weakly smooth maps, so
+the functors are injective but not surjective on morphisms. That is:
+Corollary 3.4. The functor FC∞Rings
+Man:Man→(C∞Ringsfp)opis full and
+faithful. However, the functors FC∞Rings
+Manb:Manb→(C∞Ringsfa)opand
+FC∞Rings
+Manc:Manc→(C∞Ringsfa)opare faithful, but not full.
+Of course, if we defined Manb,Mancto have morphisms weakly smooth
+maps, then FC∞Rings
+Manb,FC∞Rings
+Mancwould be full and faithful.
+LetX,Y,Zbe manifolds and f:X→Z,g:Y→Zbe smooth maps. If
+X,Y,Zare without boundary then f,gare called transverse if whenever x∈X
+andy∈Ywithf(x) =g(y) =z∈Zwe haveTzZ= df(TxX)+dg(TyY). If
+f,gare transverse then a fibre product X×ZYexists in Man.
+For manifolds with boundary, or with corners, the situation is more c ompli-
+cated, as explained in [35, §6], [40,§4.3]. In the definition of smoothf:X→Y
+we impose extra conditions over ∂jX,∂kY, and in the definition of transverse
+f,gwe impose extra conditions over ∂jX,∂kY,∂lZ. With these more restrictive
+definitions of smooth and transverse maps, transverse fibre pro ducts exist in
+Mancby [35, Th. 6.3] (see also [40, Th. 4.27]). The na¨ ıve definition of tran sver-
+sality is not a sufficient condition for fibre products to exist. Note to o that a
+fibre product of manifolds with boundary may be a manifold with corne rs, so
+fibre products work best in ManorMancrather than Manb.
+Our next theorem is given in [23, Th. 16] and [54, Prop. I.2.6] for manif olds
+without boundary, and the special case of products Man×Manb→Manb
+follows from Reyes [59, Th. 2.5], see also Kock [44, §III.9]. It can be proved
+by combining the usual proof in the without boundary case, the pro of of [35,
+Th. 6.3], and Proposition 2.29.
+Theorem 3.5. The functors FC∞Rings
+Man,FC∞Rings
+Mancpreserve transverse fibre
+products in Man,Manc,in the sense of [35,§6]. That is, if the following is a
+Cartesian square of manifolds with g,htransverse
+Wf/d47/d47
+e/d15/d15Y
+h/d15/d15
+Xg/d47/d47Z,(3.1)
+19so thatW=X×g,Z,hY,then we have a pushout square of C∞-rings
+C∞(Z)
+h∗/d47/d47
+g∗/d15/d15C∞(Y)
+f∗/d15/d15
+C∞(X)e∗/d47/d47C∞(W),(3.2)
+so thatC∞(W) =C∞(X)∐g∗,C∞(Z),h∗C∞(Y).
+4C∞-ringed spaces and C∞-schemes
+Inalgebraicgeometry,if Aisanaffineschemeand Rtheringofregularfunctions
+onA, then we can recover Aas the spectrum of the ring R,A∼=SpecR. One of
+the ideas of synthetic differential geometry, as in [54, §I], is to regard a manifold
+Xas the ‘spectrum’ of the C∞-ringC∞(X) in Example 2.2. So we can try to
+develop analogues of the tools of scheme theory for smooth manifo lds, replacing
+rings byC∞-rings throughout. This was done by Dubuc [22,23]. The analogues
+of the algebraic geometry notions [31, §II.2] of ringed spaces, locally ringed
+spaces, and schemes, are called C∞-ringed spaces, local C∞-ringed spaces and
+C∞-schemes. The material of §4.6–§4.9 is new.
+4.1 Some basic topology
+Later we will use several properties of topological spaces, e.g. se cond countable,
+metrizable, Lindel¨ of, ..., so we now recall their definitions and som e relation-
+ships between them. Let Xbe a topological space, with topology T. Then:
+•AbasisforTis a familyB⊆Tsuch that every open set in Xis a union
+of sets inB. We callXsecond countable ifThas a countable basis.
+•An open cover{Ui:i∈I}ofXislocally finite if everyx∈Xhas an
+open neighbourhood WwithW∩Ui/\e}atio\slash=∅for only finitely many i∈I.
+An open cover{Vj:j∈J}ofXis arefinement of another open cover
+{Ui:i∈I}if for allj∈Jthere exists i∈IwithVj⊆Ui⊆X.
+We callXparacompact if every open cover {Ui:i∈I}ofXadmits a
+locally finite refinement {Vj:j∈J}.
+•We callXHausdorff if for allx,y∈Xwithx/\e}atio\slash=ythere exist open
+U,V⊆Xwithx∈U,y∈VandU∩V=∅.
+•We callXmetrizable if there exists a metric on Xinducing topology T.
+•We callXregularif for every closed subset C⊆Xand eachx∈X\C
+there exist disjoint open sets U,V⊆XwithC⊆Uandx∈V.
+•We callXcompletely regular if for every closed C⊆Xandx∈X\C
+there exists a continuous f:X→[0,1] withf|C= 0 andf(x) = 1.
+•We callXseparable if it has a countable dense subset S⊆X.
+20•We callXlocally compact if for allx∈Xthere exist x∈U⊆C⊆X
+withUopen andCcompact.
+•We callXLindel¨ of if every open cover of Xhas a countable subcover.
+By well known results in topology, including Urysohn’s metrization the orem,
+the following are equivalent:
+(i)Xis Hausdorff, second countable and regular.
+(ii)Xis second countable and metrizable.
+(iii)Xis separable and metrizable.
+Here are some useful implications:
+•XHausdorff and locally compact imply Xis regular.
+•Xmetrizable implies Xis Hausdorff, paracompact, and regular.
+•Xsecond countable implies Xis Lindel¨ of.
+•XLindel¨ of and regular imply Xis paracompact.
+4.2 Sheaves on topological spaces
+Sheaves are a fundamental concept in algebraic geometry. They a re necessary
+even to define schemes, since a scheme is a topological space Xequipped with
+a sheaf of ringsOX. In this book, sheaves of C∞-rings, and sheaves of modules
+over a sheaf of C∞-rings, play a fundamental rˆ ole.
+We now summarize some basics of sheaf theory, following Hartshorn e [31,
+§II.1]. A more detailed reference is Godement [28]. We concentrate on sheaves
+of abelian groups; to define sheaves of C∞-rings, etc., one replaces abelian
+groups with C∞-rings, etc., throughout. This is justified since limits in all these
+categories (including abelian groups) are computed at the level of u nderlying
+sets, because they are all algebras for algebraic theories.
+Definition 4.1. LetXbe a topological space. A presheaf of abelian groups E
+onXconsists of the data of an abelian group E(U) for every open set U⊆X,
+and a morphism of abelian groups ρUV:E(U)→E(V) called the restriction
+mapfor every inclusion V⊆U⊆Xof open sets, satisfying the conditions that
+(i)E(∅) = 0;
+(ii)ρUU= idE(U):E(U)→E(U) for all open U⊆X; and
+(iii)ρUW=ρVW◦ρUV:E(U)→E(W) for all open W⊆V⊆U⊆X.
+That is, a presheaf is a functor E:Open(X)op→AbGp, whereOpen(X) is
+the category of open subsets of Xwith morphisms inclusions, and AbGpis the
+category of abelian groups.
+A presheaf of abelian groups EonXis called a sheafif it also satisfies
+(iv) IfU⊆Xis open,{Vi:i∈I}is an open cover of U, ands∈E(U) has
+ρUVi(s) = 0 inE(Vi) for alli∈I, thens= 0 inE(U); and
+21(v) IfU⊆Xis open,{Vi:i∈I}is an open cover of U, and we are given
+elementssi∈E(Vi) for alli∈Isuch thatρVi(Vi∩Vj)(si) =ρVj(Vi∩Vj)(sj)
+inE(Vi∩Vj) for alli,j∈I, then there exists s∈E(U) withρUVi(s) =si
+for alli∈I. Thissis unique by (iv).
+SupposeE,Fare presheavesor sheavesof abelian groups on X. Amorphism
+φ:E→Fconsists of a morphism of abelian groups φ(U) :E(U)→F(U) for all
+openU⊆X, suchthatthefollowingdiagramcommutesforallopen V⊆U⊆X
+E(U)
+φ(U)/d47/d47
+ρUV/d15/d15F(U)
+ρ′
+UV/d15/d15
+E(V)φ(V)/d47/d47F(V),
+whereρUVis the restriction map for E, andρ′
+UVthe restriction map for F.
+Definition 4.2. LetEbe a presheaf of abelian groups on X. For eachx∈X,
+thestalkExis the direct limit of the groups E(U) for allx∈U⊆X, via the
+restriction maps ρUV. It is an abelian group. A morphism φ:E→Finduces
+morphisms φx:Ex→Fxfor allx∈X. IfE,Fare sheaves then φis an
+isomorphism if and only if φxis an isomorphism for all x∈X.
+Sheaves of abelian groups on Xform an abelian category Sh(X). Thus we
+have (category-theoretic) notions of when a morphism φ:E→Fin Sh(X) is
+injective orsurjective (epimorphic ), and when a sequence E→F→G in Sh(X)
+isexact. It turns out that φ:E→Fis injective if and only if φ(U) :E(U)→
+F(U) is injective for all open U⊆X. Howeverφ:E→Fsurjective does not
+imply that φ(U) :E(U)→F(U) is surjective for all open U⊆X. Instead,φis
+surjective if and only if φx:Ex→Fxis surjective for all x∈X.
+Definition 4.3. LetEbe a presheaf of abelian groups on X. Asheafification
+ofEis a sheaf of abelian groups ˆEonXand a morphism π:E→ˆE, such that
+wheneverFis a sheaf of abelian groups on Xandφ:E→Fis a morphism,
+there is a unique morphism ˆφ:ˆE→Fwithφ=ˆφ◦π. As in [31, Prop. II.1.2],
+a sheafification always exists, and is unique up to canonical isomorph ism; one
+can be constructed explicitly using the stalks ExofE.
+Next we discuss pushforwards andpullbacks of sheaves by continuous maps.
+Definition 4.4. Letf:X→Ybe a continuous map of topological spaces, and
+Ea sheaf of abelian groups on X. Define the pushforward (direct image ) sheaf
+f∗(E) onYby/parenleftbig
+f∗(E)/parenrightbig
+(U) =E/parenleftbig
+f−1(U)/parenrightbig
+for all open U⊆V, with restriction
+mapsρ′
+UV=ρf−1(U)f−1(V):/parenleftbig
+f∗(E)/parenrightbig
+(U)→/parenleftbig
+f∗(E)/parenrightbig
+(V) for all open V⊆U⊆Y.
+Thenf∗(E) is a sheaf of abelian groups on Y.
+Ifφ:E→Fis a morphism in Sh( X) we define f∗(φ) :f∗(E)→f∗(F) by/parenleftbig
+f∗(φ)/parenrightbig
+(u) =φ/parenleftbig
+f−1(U)/parenrightbig
+for all open U⊆Y. Thenf∗(φ) is a morphism in
+Sh(Y), andf∗is a functor Sh( X)→Sh(Y). It is a left exact functor between
+abelian categories, but in general is not exact. For continuous map sf:X→Y,
+g:Y→Zwe have (g◦f)∗=g∗◦f∗.
+22Definition 4.5. Letf:X→Ybe a continuous map of topological spaces,
+andEa sheaf of abelian groups on Y. Define a presheaf Pf−1(E) onXby/parenleftbig
+Pf−1(E)/parenrightbig
+(U) = limA⊇f(U)E(A) for open A⊆X, where the direct limit is
+taken over all open A⊆Ycontaining f(U), using the restriction maps ρAB
+inE. For open V⊆U⊆X, defineρ′
+UV:/parenleftbig
+Pf−1(E)/parenrightbig
+(U)→/parenleftbig
+Pf−1(E)/parenrightbig
+(V) as
+the direct limit of the morphisms ρABinEforB⊆A⊆Ywithf(U)⊆A
+andf(V)⊆B. Then we define the pullback (inverse image )f−1(E) to be the
+sheafification of the presheaf Pf−1(E).
+Pullbacksf−1(E) are only unique up to canonical isomorphism, rather than
+unique. By convention we choose once and for all a pullback f−1(E) for all
+X,Y,f,E, using the Axiom of Choice if necessary. If φ:E→Fis a morphism
+in Sh(Y), one can define a pullback morphism f−1(φ) :f−1(E)→f−1(F).
+Thenf−1: Sh(Y)→Sh(X) is an exact functor between abelian categories.
+We compare pushforwards and pullbacks:
+Remark 4.6. (a) There are two kinds of pullback, with slightly different no-
+tation. The first kind, written f−1(E) as in Definition 4.5, is used for sheaves
+of abelian groups or C∞-rings. The second kind, written f∗(E) orf∗(E) and
+discussed in§5.3 and§8.3, is used for sheaves of OY-modulesE.
+(b)The definition of pushforward sheaves f∗(E) is wholly elementary. In con-
+trast, the definition of pullbacks f−1(E) is complex, involving a direct limit
+followed by a sheafification, and includes arbitrary choices.
+Pushforwards f∗are strictly functorial in the continuous map f:X→Y,
+thatis, forcontinuous f:X→Y,g:Y→Zwehave(g◦f)∗=g∗◦f∗: Sh(X)→
+Sh(Z). However, pullbacks f−1are only weakly functorial in f: ifE∈Sh(Z)
+then we need not have ( g◦f)−1(E) =f−1(g−1(E)). This is because pullbacks
+are only natural up to canonical isomorphism, and we make an arbitr ary choice
+for each pullback. So although f−1(g−1(E)) is a possible pullback for Ebyg◦f,
+it may not be the one we chose.
+Thus, thereisacanonicalisomorphism( g◦f)−1(E)∼=f−1(g−1(E)), whichwe
+will write as If,g(E) : (g◦f)−1(E)→f−1(g−1(E)). TheIf,g(E) for allE∈Sh(Z)
+comprise a natural isomorphism of functors If,g: (g◦f)−1⇒f−1◦g−1. Sim-
+ilarly, forE ∈Sh(X) we may not have id−1
+X(E) =E, but instead there are
+canonical isomorphisms δX(E) : id−1
+X(E)→E, which make up a natural iso-
+morphismδX: id−1
+X⇒idSh(X). Many authors ignore the natural isomorphisms
+If,g,δXentirely.
+(c)Letf:X→Ybe a continuous map of topological spaces. Then we have
+functorsf∗: Sh(X)→Sh(Y), andf−1: Sh(Y)→Sh(X). Asin[31,Ex.II.1.18],
+f∗is right adjoint to f−1. That is, there is a natural bijection
+HomX/parenleftbig
+f−1(E),F/parenrightbig∼=HomY/parenleftbig
+E,f∗(F)/parenrightbig
+(4.1)
+for allE∈Sh(Y) andF∈Sh(X), with functorial properties.
+We define finesheaves, as in Godement [28, §II.3.7] or Voisin [69, Def. 4.35].
+They will be important in §4.7 and§5.3.
+23Definition 4.7. LetXbe a topological space (usually paracompact), and Ea
+sheaf of abelian groups on X, or more generally a sheaf of rings, or C∞-rings,
+orOX-modules, or any other objects which are also abelian groups. We ca llE
+fineif for any open cover {Ui:i∈I}ofX, a subordinate locally finite partition
+of unity{ζi:i∈I}exists in the sheaf Hom(E,E).
+Hereζi:E→Eis a morphism of sheaves of abelian groups (or rings, C∞-
+rings, ...) for each i∈I. For{ζi:i∈I}to besubordinate to{Ui:i∈I}
+means that ζiis supported in Uifor eachi∈I, that is, there exists open Vi⊆X
+withζi|Vi= 0 andUi∪Vi=X. For{ζi:i∈I}to belocally finite means that
+eachx∈Xhas an open neighbourhood Wwithζi|W/\e}atio\slash= 0 for only finitely many
+i∈I. For{ζi:i∈I}to be apartition of unity means that/summationtext
+i∈Iζi= idE,
+where the sum makes sense as {ζi:i∈I}is locally finite.
+IfE=OXis a sheaf of commutative rings or C∞-rings, then writing ηi=
+ζi(1) inOX(X), we see that ζi=ηi·is multiplication by ηi. So we can regard
+the partition of unity as living in OX(X) rather thanHom(OX,OX).
+4.3C∞-ringed spaces and local C∞-ringed spaces
+Definition 4.8. AC∞-ringed space X= (X,OX) is a topological space X
+with a sheafOXofC∞-rings onX. That is, for each open set U⊆Xwe are
+given aC∞ringOX(U), and for each inclusion of open sets V⊆U⊆Xwe are
+given a morphism of C∞-ringsρUV:OX(U)→OX(V), called the restriction
+maps, and all this data satisfies the sheaf axioms in Definition 4.1.
+Equivalently,OXis a presheaf of C∞-rings onX, that is, a functor
+OX:Open(X)op−→C∞Rings,
+whose underlying presheaf of abelian groups, or of sets, is a sheaf . The sheaf
+axioms Definition 4.1(iv),(v) do not use the C∞-ring structure.
+Amorphismf= (f,f♯) : (X,OX)→(Y,OY) ofC∞ringed spaces is a
+continuous map f:X→Yand a morphism f♯:f−1(OY)→OXof sheaves of
+C∞-rings onX, forf−1(OY) as in Definition 4.5. Since f∗is right adjoint to
+f−1, as in (4.1) there is a natural bijection
+HomX/parenleftbig
+f−1(OY),OX/parenrightbig∼=HomY/parenleftbig
+OY,f∗(OX)/parenrightbig
+. (4.2)
+Writef♯:OY→f∗(OX) for the morphism of sheaves of C∞-rings onYcorre-
+sponding to f♯under (4.2), so that
+f♯:f−1(OY)−→OX/squiggleleftrightf♯:OY−→f∗(OX). (4.3)
+Iff:X→Yandg:Y→ZareC∞-scheme morphisms, the composition is
+g◦f=/parenleftbig
+g◦f,(g◦f)♯/parenrightbig
+=/parenleftbig
+g◦f,f♯◦f−1(g♯)◦If,g(OZ)/parenrightbig
+,
+whereIf,g(OZ) : (g◦f)−1(OZ)→f−1(g−1(OZ)) is the canonical isomorphism
+from Remark 4.6(b). In terms of f♯:OY→f∗(OX), composition is
+(g◦f)♯=g∗(f♯)◦g♯:OZ−→(g◦f)∗(OX) =g∗◦f∗(OX).
+24AlocalC∞-ringed space X= (X,OX) is aC∞-ringed space for which the
+stalksOX,xofOXatxare localC∞-rings for all x∈X. As in Definition
+2.10, since morphisms of local C∞-rings are automatically local morphisms,
+morphisms of local C∞-ringed spaces ( X,OX),(Y,OY) are just morphisms of
+C∞-ringedspaces,without anyadditionallocalitycondition. Moerdijk, vanQuˆ e
+and Reyes [52,§3] call our local C∞-ringed spaces Archimedean C∞-spaces.
+WriteC∞RSfor the category of C∞-ringed spaces, and LC∞RSfor the
+full subcategory of local C∞-ringed spaces.
+For brevity, we will use the notation that underlined upper case lett ers
+X,Y,Z,...representC∞-ringed spaces ( X,OX),(Y,OY),(Z,OZ),...,and un-
+derlined lower case letters f,g,...represent morphisms of C∞-ringed spaces
+(f,f♯),(g,g♯),....When we write ‘ x∈X’ we mean that X= (X,OX) and
+x∈X. When we write ‘ Uis open in X’ we mean that U= (U,OU) and
+X= (X,OX) withU⊆Xan open set andOU=OX|U.
+Remark 4.9. As above, there are two equivalent ways to write morphisms
+ofC∞-ringed spaces ( X,OX)→(Y,OY), either using pullbacks as ( f,f♯) for
+f♯:f−1(OY)→OX, or using pushforwards as ( f,f♯) forf♯:OY→f∗(OX).
+Each definition has advantages and disadvantages. We choose to r egardf♯:
+f−1(OY)→OXas the primary object, and so define morphisms of C∞-ringed
+spaces as (f,f♯) rather than ( f,f♯), although we will use f♯in a few places. We
+can always switch between the two points of view using (4.3).
+Example 4.10. LetXbe a manifold, which may have boundary or corners.
+Define aC∞-ringed space X= (X,OX) to have topological space Xand
+OX(U) =C∞(U) for each open subset U⊆X, whereC∞(U) is theC∞-
+ring of smooth maps c:U→R, and ifV⊆U⊆Xare open we define
+ρUV:C∞(U)→C∞(V) byρUV:c/ma√sto→c|V.
+It is easyto verify that OXis a sheaf of C∞-ringsonX(not just a presheaf),
+soX= (X,OX) is aC∞-ringed space. For each x∈X, the stalkOX,xis the
+localC∞-ring of germs [( c,U)] of smooth functions c:X→Ratx∈X, as in
+Example 2.15, with unique maximal ideal mX,x=/braceleftbig
+[(c,U)]∈OX,x:c(x) = 0/bracerightbig
+andOX,x/mX,x∼=R. HenceXis a localC∞-ringed space.
+LetX,Ybe manifolds and f:X→Ya weakly smooth map. Define
+(X,OX),(Y,OY) as above. For all open U⊆Ydefinef♯(U) :OY(U) =
+C∞(U)→OX(f−1(U)) =C∞(f−1(U)) byf♯(U) :c/ma√sto→c◦ffor allc∈C∞(U).
+Thenf♯(U) is a morphism of C∞-rings, and f♯:OY→f∗(OX) is a morphism
+of sheaves of C∞-rings onY. Letf♯:f−1(OY)→OXcorrespond to f♯un-
+der (4.3). Then f= (f,f♯) : (X,OX)→(Y,OY) is a morphism of (local)
+C∞-ringed spaces.
+As the category Topof topological spaces has all finite limits, and the con-
+structionof C∞RSinvolvesTopinacovariantwayandthecategory C∞Rings
+in a contravariant way, using Proposition 2.5 one may prove:
+Proposition 4.11. All finite limits exist in the category C∞RS.
+Dubuc [23, Prop. 7] proves:
+25Proposition 4.12. The full subcategory LC∞RSof localC∞-ringed spaces in
+C∞RSis closed under finite limits in C∞RS.
+4.4 The spectrum functor
+We now define a spectrum functor Spec :C∞Ringsop→LC∞RS. It is
+equivalent to those constructed by Dubuc [22,23] and Moerdijk, v an Quˆ e and
+Reyes [52,§3], but our presentation is closer to that of Hartshorne [31, p. 70].
+Definition 4.13. LetCbe aC∞-ring, and use the notation of Definition 2.13.
+WriteXCfor the set of all R-pointsxofC. LetTCbe the topology on XC
+generated by the basis of open sets Uc=/braceleftbig
+x∈XC:x(c)/\e}atio\slash= 0/bracerightbig
+for allc∈C.
+For eachc∈Cdefinec∗:XC→Rto mapc∗:x/ma√sto→x(c).
+Example 4.14. Suppose Cis a finitely generated C∞-ring, with exact sequence
+0→I ֒→C∞(Rn)φ−→C→0. Define a map φ∗:XC→Rnbyφ∗:x/ma√sto→/parenleftbig
+x◦φ(x1),...,x◦φ(xn)/parenrightbig
+, wherex1,...,xnare the generators of C∞(Rn). Then
+φ∗gives a homeomorphism
+φ∗:XC∼=−→Xφ
+C=/braceleftbig
+(x1,...,xn)∈Rn:f(x1,...,xn) = 0 for all f∈I/bracerightbig
+,(4.4)
+where the right hand side is a closed subset of Rn. So the topological spaces
+(XC,TC) for finitely generated Care homeomorphic to closed subsets of Rn.
+Recall that a topological space Xisregularif whenever S⊆Xis closed and
+x∈X\Sthen there exist open U,V⊆Xwithx∈U,S⊆VandU∩V=∅.
+Lemma 4.15. In Definition 4.13,the topologyTCis also generated by the basis
+of open sets c−1
+∗(V)for allc∈Cand openV⊆R. That is,TCis the weakest
+topology on XCsuch thatc∗:XC→Ris continuous for all c∈C. Also
+(XC,TC)is a Hausdorff, regular topological space.
+Proof.Supposec∈CandV⊆Ris open. Then there exists smooth f:R→R
+withV={x∈R:f(x)/\e}atio\slash= 0}. Setc′= Φf(c), using the C∞-ring operation
+Φf:C→C. Thenc′
+∗=f◦c∗asc:C→Ris aC∞-ring morphism, so
+Uc′= (c′
+∗)−1(R\{0}) = (f◦c∗)−1(R\{0}) =c−1
+∗[f−1(0)] =c−1
+∗(V).
+Soc−1
+∗(V) is of the form Uc′. Conversely Uc=c−1
+∗(V) forV=R\{0}⊆R. So
+the two given bases for TCare the same, proving the first part.
+Letx,ybe distinct points of XC. Then there exists c∈Cwithx(c)/\e}atio\slash=y(c),
+asx/\e}atio\slash=y. Setǫ=1
+2|x(c)−y(c)|>0 andU=c−1
+∗/parenleftbig
+(x(c)−ǫ,x(c) +ǫ)/parenrightbig
+,
+V=c−1
+∗/parenleftbig
+(y(c)−ǫ,y(c) +ǫ)/parenrightbig
+. ThenU,V⊆XCare disjoint open sets with
+x∈U,y∈V, so (XC,TC) is Hausdorff.
+SupposeS⊆XCis closed, and x∈X\S. Then there exists c∈Cwithx∈
+Uc⊆XC\S, sinceXC\Sis open inXCand theUcare a basis forTC. Therefore
+c∗(x)/\e}atio\slash= 0 andc∗|S= 0. Setǫ=1
+2|c∗(x)|>0,U=c−1
+∗/parenleftbig
+(c∗(x)−ǫ,c∗(x) +ǫ)/parenrightbig
+andV=c−1
+∗/parenleftbig
+(−ǫ,ǫ)/parenrightbig
+. ThenU,V⊆XCare disjoint open sets with x∈U,
+S⊆V, so (XC,TC) is regular.
+26Definition 4.16. LetCbe aC∞-ring, and XCthe topological space from
+Definition4.13. Foreachopen U⊆XC, defineOXC(U) tobe thesetoffunctions
+s:U→/coproducttext
+x∈UCxwiths(x)∈Cxfor allx∈U, and such that Umay be covered
+by open sets W⊆U⊆XCfor which there exist c∈Cwiths(x) =πx(c)∈Cx
+for allx∈W. Define operations Φ fonOXC(U) pointwise in x∈Uusing the
+operations Φ fonCx. This makesOXC(U) into aC∞-ring. IfV⊆U⊆XCare
+open, the restriction map ρUV:OXC(U)→OXC(V) mappingρUV:s/ma√sto→s|Vis
+a morphism of C∞-rings.
+ClearlyOXCis a sheaf of C∞-rings onXC. Lemma 4.18 shows that the stalk
+OXC,xatx∈XCisCx, which is a local C∞-ring. Hence ( XC,OXC) is a local
+C∞-ringed space, which we call the spectrum ofC, and write as Spec C.
+Now letφ:C→Dbe a morphism of C∞-rings. Define fφ:XD→
+XCbyfφ(x) =x◦φ. Thenfφis continuous. For U⊆XCopen define
+(fφ)♯(U) :OXC(U)→OXD(f−1
+φ(U)) by (fφ)♯(U)s:x/ma√sto→φx(s(fφ(x))), where
+φx:Cfφ(x)→Dxis the induced morphism of local C∞-rings. Then ( fφ)♯:
+OXC→(fφ)∗(OXD) is a morphism of sheaves of C∞-rings onXC. Letf♯
+φ:
+f−1
+φ(OXC)→OXDbe the corresponding morphism of sheaves of C∞-rings on
+XDunder (4.3). Then fφ= (fφ,f♯
+φ) : (XD,OXD)→(XC,OXC) is a morphism
+of localC∞-ringed spaces. Define Spec φ: SpecD→SpecCby Specφ=fφ.
+Then Spec is a functor C∞Ringsop→LC∞RS, thespectrum functor .
+Example 4.17. LetXbe a manifold. Then it followsfrom Theorem 4.41below
+that the local C∞-ringed space Xconstructed in Example 4.10 is naturally
+isomorphic to Spec C∞(X).
+Lemma 4.18. In Definition 4.16,the stalkOXC,xofOXCatx∈XCis nat-
+urally isomorphic to Cx.
+Proof.Elements ofOXC,xare∼-equivalence classes [ U,s] of pairs (U,s), where
+Uis an open neighbourhood of xinXCands∈OXC(U), and (U,s)∼(U′,s′) if
+there exists open x∈V⊆U∩U′withs|V=s′|V. Define aC∞-ring morphism
+Π :OXC,x→Cxby Π : [U,s]/ma√sto→s(x).
+Supposecx∈Cx. Thencx=πx(c) for some c∈Cby Proposition 2.14.
+The maps:XC→/coproducttext
+x′∈XCCx′mappings:x′/ma√sto→πx′(c) lies inOXC(XC), and
+Π : [XC,s]/ma√sto→πx(c) =cx. Hence Π :OXC,x→Cxis surjective.
+Suppose [U,s]∈OXC,xwith Π([U,s]) = 0∈Cx. Ass∈OXC(U), there exist
+openx∈V⊆Uandc∈Cwiths(x′) =πx′(c)∈Cx′for allx′∈V. Then
+πx(c) =s(x) = Π([U,s]) = 0, soclies in the ideal Iin (2.2) by Proposition
+2.14. Thus there exists d∈Cwithx(d)/\e}atio\slash= 0 inRandcd= 0 inC. Set
+W={x′∈V:x′(d)/\e}atio\slash= 0}, so thatWis an open neighbourhood of xinU. If
+x′∈Wthenx′(d)/\e}atio\slash= 0, soπx′(d) is invertible in Cx′. Thus
+s(x′) =πx′(c) =πx′(c)πx′(d)πx′(d)−1=πx′(cd)πx′(d)−1=πx′(0)πx′(d)−1= 0.
+Hence [U,s] = [W,s|W] = [W,0] = 0 inOXC,x, so Π :OXC,x→Cxis injective.
+Thus Π :OXC,x→Cxis an isomorphism.
+27Definition 4.19. Theglobal sections functor Γ :LC∞RS→C∞Ringsop
+acts on objects ( X,OX) by Γ : (X,OX)/ma√sto→OX(X) and on morphisms ( f,f♯) :
+(X,OX)→(Y,OY) by Γ : (f,f♯)/ma√sto→f♯(Y), forf♯:OY→f∗(OX) as in (4.3).
+Then Γ◦Spec is a functor C∞Ringsop→C∞Ringsop, or equivalently a
+functorC∞Rings→C∞Rings. For eachC∞-ringCandc∈C, define Ψ C(c) :
+XC→/coproducttext
+x∈XCCxby ΨC(c) :x/ma√sto→πx(c)∈Cx. Then Ψ C(c)∈OXC(XC) =
+Γ◦SpecCby Definition 4.16, soΨ C:C→Γ◦SpecCis amap. Since πx:C→Cx
+is aC∞-ring morphism and the C∞-ring operations on OXC(XC) are defined
+pointwise in the Cx, this Ψ Cis aC∞-ring morphism. It is functorial in C, so
+that the Ψ Cfor allCdefine a natural transformation Ψ : id C∞Rings⇒Γ◦Spec
+of functors id C∞Rings,Γ◦Spec :C∞Rings→C∞Rings.
+Theorem 4.20. The functor Spec :C∞Ringsop→LC∞RSisright adjoint
+toΓ :LC∞RS→C∞Ringsop. That is, for all C∈C∞RingsandX∈
+LC∞RSthere are inverse bijections
+HomC∞Rings(C,Γ(X))LC,X/d47/d47HomLC∞RS(X,SpecC),
+RC,X/d111/d111 (4.5)
+which are functorial in the sense that if λ:C→Dis a morphism in C∞Rings
+ande:X→Ya morphism in LC∞RSthen the following commutes:
+HomC∞Rings(D,Γ(Y))LD,Y/d47/d47
+φ/mapsto→Γ(e)◦φ◦λ/d15/d15HomLC∞RS(Y,SpecD)
+RD,Y/d111/d111
+f/mapsto→Specλ◦f◦e/d15/d15
+HomC∞Rings(C,Γ(X))LC,X/d47/d47HomLC∞RS(X,SpecC).
+RC,X/d111/d111(4.6)
+WhenX= SpecCwe have ΨC=RC,X(idX),so thatΨCis the unit of the
+adjunction between ΓandSpec.
+Proof.LetC∈C∞RingsandX∈LC∞RS. WriteY= (Y,OY) = Spec C.
+DefineRC,Xin (4.5) by, for each morphism f:X→YinLC∞RS, taking
+RC,X(f) :C→Γ(X) to be the composition
+CΨC/d47/d47Γ◦SpecC= Γ(Y)Γ(f)/d47/d47Γ(X). (4.7)
+For the last part, if X= SpecCthen Ψ C=RC,X(idX) as Γ(idX) = idΓ(X).
+Letφ:C→Γ(X) be a morphism in C∞Rings. We will construct a
+morphismg= (g,g♯) :X→YinLC∞RS, and setLC,X(φ) =g. For any
+x∈Xwe have an R-algebra morphism x∗: Γ(X)→Rby composing the stalk
+mapσx: Γ(X)→OX,xwith the unique morphism π:OX,x→R, asOX,xis a
+localC∞-ring. Then x∗◦φ:C→RisanR-algebramorphism, andhenceapoint
+ofY. Defineg:X→Ybyg(x) =x∗◦φ. Ifc∈CthenUc={y∈Y:y(c)/\e}atio\slash= 0}
+is open inY, andg−1(Uc) ={x∈X:x∗(φ(c))/\e}atio\slash= 0}is open inX, asx/ma√sto→x∗(d)
+is a continuous map X→Rfor anyd∈Γ(X). Since the Ucforc∈Care a
+basis for the topology of Yby Definition 4.13, g:X→Yis continuous.
+28Letx∈Xwithg(x) =y∈Y. Consider the diagram of C∞-rings
+C
+πy/d15/d15φ/d47/d47Γ(X)
+σx/d15/d15
+Cy∼=OY,yφx/d47/d47OX,x.(4.8)
+HereCy∼=OY,yby Lemma 4.18. If c∈Cwithy(c)/\e}atio\slash= 0 thenσx◦φ(c)∈OX,x
+withπ[σx◦φ(c)]/\e}atio\slash= 0, soσx◦φ(c) is invertible inOX,xasOX,xis a localC∞-
+ring. Thus by the universal property of πy:C→Cythere is a unique morphism
+φx:OY,y→OX,xmaking (4.8) commute.
+For each open V⊆YwithU=g−1(V)⊆X, defineg♯(V) :OY(V)→
+g∗(OX)(V) =OX(U) byg♯(V)s:x/ma√sto→φx(s(g(x))) fors∈OY(V) andx∈U⊆
+X, so thatg(x)∈V, ands(g(x))∈OY,g(x), andφx(s(g(x)))∈OX,x. Here as
+OXisasheafwemayidentify elementsof OX(U)withmaps t:U→/coproducttext
+x∈UOX,x
+witht(x)∈OX,xforx∈U, such that tsatisfies certain local conditions in U.
+Ifs∈OY(V) andx∈U⊆Xwithg(x) =y∈V⊆Y, then by Definition 4.16
+there is an open neighbourhood WyofyinVandc∈Cwiths(y′) =πy′(c)∈
+Cy′∼=OY,y′for ally′∈Wy. Therefore g♯(V)smapsx′/ma√sto→σx′(φ(c)) for allx′in
+the open neighbourhood g−1(Wy) ofxinU, by (4.8). Since the open subsets
+g−1(Wy) coverU,g♯(V)sis a section ofOX|U, andg♯(V) is well defined.
+As theφxareC∞-ring morphisms, this defines a morphism g♯:OY→
+g∗(OX) of sheaves of C∞-rings onY. Letg♯:g−1(OY)→OXbe the corre-
+sponding morphism of sheaves on Xunder (4.3). The stalk g♯
+x:OY,y→OX,x
+ofg♯atx∈Xwithg(x) =y∈Yisg♯
+x=φx. Theng= (g,g♯) is a morphism
+inLC∞RS. SetLC,X(φ) =g. This defines LC,Xin (4.5).
+Forφ,gas above,c∈C, andx∈Xwithg(x) =y=x∗◦φ∈Y, we have
+σx/bracketleftbig/parenleftbig
+RC,X◦LC,X(φ)/parenrightbig
+(c)/bracketrightbig
+=σx/bracketleftbig
+Γ(g)◦ΨC(c)/bracketrightbig
+=g♯
+x◦σy[ΨC(c)]
+=φx◦σy[ΨC(c)] =φx◦πy(c) =σx◦φ(c),
+usingLC,X(φ) =gand the definition (4.7) of RC,X(g) in the first step, σx◦
+Γ(g) =g♯
+x◦σy: Γ(Y)→OX,xin the second, g♯
+x=φxin the third, σy◦ΨC=πy
+as maps C→OY,y∼=Cyin the fourth, and (4.8) in the fifth. As/producttext
+x∈Xσx:
+Γ(X)→/producttext
+x∈XOX,xis injective, this implies that/parenleftbig
+RC,X◦LC,X(φ)/parenrightbig
+(c) =φ(c)
+for allc∈C, soRC,X◦LC,X(φ) =φ, andRC,X◦LC,X= id.
+Supposef:X→Yis a morphism in LC∞RS, and setφ=RC,X(f) and
+g=LC,X(φ). Letx∈Xwithf(x) =y∈Y. Then we have a commutative
+diagram in C∞Rings
+C
+y
+/d38/d38◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆φ
+/d46/d46
+πy/d15/d15ΨC/d47/d47Γ◦SpecC= Γ(Y)
+σy/d15/d15Γ(f)/d47/d47Γ(X)
+σx/d15/d15x∗
+/d119/d119♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦
+Cy
+π/d43/d43❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲∼=/d47/d47OY,y
+π
+/d15/d15f♯
+x/d47/d47OX,x
+π
+/d115/d115❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣
+R,(4.9)
+29where the isomorphism Cy∼=OY,ycomes from Lemma 4.18. Since g(x) =
+x∗◦φ:C→R, this proves that g(x) =y=f(x), sof=g. Also by definition
+the stalkg♯
+x:OY,y→OX,xisφxin (4.8), so comparing (4.8) and (4.9) and
+usingπy:C→Cysurjective by Proposition 2.14 shows that f♯
+x=g♯
+x. As
+this holds for all x∈Xwe havef♯=g♯, sof= (f,f♯) = (g,g♯) =g. Thus
+LC,X◦RC,X(f) =ffor allf:X→Y, soLC,X◦RC,X= id. Therefore
+LC,X,RC,Xin (4.5) are inverse bijections.
+It is easy to see that the rectangle in (4.6) involving RD,Y,RC,Xcommutes
+using (4.7) and functoriality of the Ψ Cand Γ. Then the rectangle involving
+LD,Y,LC,Xcommutes as LD,Y=R−1
+D,YandLC,X=R−1
+C,X. So (4.6) commutes.
+This completes the proof.
+Remark 4.21. (a) The fact in Theorem 4.20 that Spec : C∞Ringsop→
+LC∞RSis right adjoint to Γ : LC∞RS→C∞Ringsopdetermines Spec
+uniquely up to natural isomorphism, by properties of adjoint funct ors.
+Dubuc [23] and Moerdijk, van Quˆ e and Reyes [52, §3] both prove the ex-
+istence of a right adjoint to Γ : LC∞RS→C∞Ringsop, which is therefore
+naturally isomorphic to our functor Spec in Definition 4.16. But they s how Spec
+exists by category theory, without constructing it explicitly as we d o.
+Moerdijk et al. [52, §3] call our functor Spec the Archimedean spectrum .
+They also give a nonequivalent definition [52, §1] of the spectrum Spec C, in
+which the points are not R-points, but ‘ C∞-radical prime ideals’.
+(b)Since Spec is a right adjoint functor, it preserves limits, as in [23, p. 687].
+Equivalently, Spec takes colimits in C∞Ringsto limits in LC∞RS. So, for
+example, a pushout C=D∐FEof morphisms φ:F→D,ψ:F→Ein
+C∞Ringsis mapped to a fibre product Spec C∼=SpecD×SpecFSpecEof
+morphisms Spec φ: SpecD→SpecF, Specψ: SpecE→SpecFinLC∞RS.
+Here are some properties of finitely generated and fair C∞-rings, due to
+Dubuc [23, Th. 13]. The reflection functor Rfa
+fgis as in Definition 2.20.
+Theorem 4.22. (a) IfCis a finitely generated C∞-ring, there is a natural
+isomorphism Γ◦SpecC∼=Rfa
+fg(C),which identifies ΨC:C→Γ(SpecC)with the
+natural surjective projection C→Rfa
+fg(C).
+These isomorphisms for all Cform a natural isomorphism Rfa
+fg∼=Γ◦Spec
+of functors Rfa
+fg,Γ◦Spec :C∞Ringsfg→C∞Ringsfa.
+Hence, if Cis fair then ΨC:C→Γ(SpecC)∼=Rfa
+fg(C)is an isomorphism.
+(b)IfCis finitely generated then SpecΨ C: SpecC→SpecΓ(Spec C)∼=
+SpecRfa
+fg(C)is an isomorphism in LC∞RS.
+(c)The functor Spec|···: (C∞Ringsfa)op→LC∞RSis full and faithful, and
+takes finite limits in (C∞Ringsfa)opto finite limits in LC∞RS.
+To see that Spec is full and faithful on ( C∞Ringsfa)opin (c), let C,Dbe
+fairC∞-rings. Then putting X= SpecDin (4.5) and using D∼=Γ◦SpecDby
+(a) shows that the following is a bijection.
+Spec : Hom C∞Rings(C,D)−→HomLC∞RS(SpecD,SpecC).
+30Note that Spec is neither full nor faithful on ( C∞Ringsfg)oporC∞Ringsop.
+This is a contrast to conventional algebraic geometry, where Γ(Sp ecR)∼=Rfor
+arbitrary rings R, as in [31, Prop. II.2.2], so that Spec is full and faithful. In
+§4.6 we will generalize Theorem 4.22 to non-finitely-generated C∞-rings.
+4.5 Affine C∞-schemes and C∞-schemes
+As for the usual definitions of affine schemes and schemes, we defin e:
+Definition 4.23. A localC∞-ringed space Xis called an affineC∞-scheme
+if it is isomorphic in LC∞RSto SpecCfor someC∞-ringC. We callXa
+finitely presented , orfair, affineC∞-scheme ifX∼=SpecCforCthat kind of
+C∞-ring. Write AC∞Sch,AC∞Schfp,AC∞Schfafor the full subcategories
+of affineC∞-schemes and of finitely presented, and fair, affine C∞-schemes in
+LC∞RSrespectively.
+We do not define finitely generated affineC∞-schemes, because Theorem
+4.22(b) implies that they coincide with fair affine C∞-schemes.
+LetX= (X,OX) be a local C∞-ringed space. We call XaC∞-schemeif
+Xcan be covered by open sets U⊆Xsuch that ( U,OX|U) is an affine C∞-
+scheme. We call a C∞-schemeXlocally fair , orlocally finitely presented , ifX
+can be covered by open U⊆Xwith (U,OX|U) a fair, or finitely presented,
+affineC∞-scheme, respectively.
+We call aC∞-schemeXHausdorff ,second countable ,Lindel¨ of,compact,
+locally compact ,paracompact ,metrizable ,regular, orseparable , if the topological
+spaceXis. AffineC∞-schemes are Hausdorff and regular by Lemma 4.15.
+WriteC∞Schlf,C∞Schlfp,C∞Schfor the full subcategories in LC∞RS
+of locally fair C∞-schemes, locally finitely presented C∞-schemes, and all C∞-
+schemes, respectively.
+Remark 4.24. Ordinary schemes are a much larger class than ordinary affine
+schemes, and central examples such as CPnare not affine schemes. However,
+affineC∞-schemes are already general enough for many purposes. For ex ample,
+all second countable, metrizable C∞-schemes are affine, as in §4.8, including
+manifolds and manifolds with corners. Affine C∞-schemes are Hausdorff and
+regular, so any non-Hausdorff or non-regular C∞-scheme is not affine.
+For the next theorem, part (a) follows from Propositions 2.5, 2.24 a nd
+2.26, Remark 4.21(b), and Theorem 4.22(c). Part (b) holds as finite lim-
+its inC∞Schlfp,C∞Schlf,C∞Schare locally modelled on finite limits in
+AC∞Schfp,AC∞SchfaandAC∞Sch.
+Theorem 4.25. (a) The full subcategories AC∞Schfp,AC∞Schfa,AC∞Sch
+are closed under all finite limits in LC∞RS. Hence, fibre products and all finite
+limits exist in each of these subcategories.
+(b)The full subcategories C∞Schlfp,C∞SchlfandC∞Schare closed under
+all finite limits in LC∞RS. Hence, fibre products and all finite limits exist in
+each of these subcategories.
+31Definition 4.26. Define functors
+FC∞Sch
+Man:Man−→AC∞Schfp⊂AC∞Sch,
+FC∞Sch
+Manb:Manb−→AC∞Schfa⊂AC∞Sch,
+FC∞Sch
+Manc:Manc−→AC∞Schfa⊂AC∞Sch,
+byFC∞Sch
+Man∗= Spec◦FC∞Rings
+Man∗, in the notation of Definitions 3.2 and 4.16.
+By Example 4.17, if Xis a manifold with corners then FC∞Sch
+Manc(X) is nat-
+urally isomorphic to the local C∞-ringed space Xin Example 4.10.
+IfX,Y,... are manifolds, or f,g,...are (weakly) smooth maps, we may use
+X,Y,...,f,g,...to denote the images of X,Y,...,f,g,... underFC∞Sch
+Manc. So
+for instance we will write Rnand [0,∞)forFC∞Sch
+Man(Rn) andFC∞Sch
+Manb/parenleftbig
+[0,∞)/parenrightbig
+.
+Our categories of spaces so far are related as follows:
+Man
+FC∞Sch
+Man/d15/d15⊂/d47/d47Manb
+FC∞Sch
+Manb/d15/d15⊂/d47/d47Manc
+FC∞Sch
+Manc/d118/d118♥♥♥♥♥♥♥♥♥♥
+AC∞Schfp
+⊂/d47/d47
+⊂/d15/d15AC∞Schfa
+⊂/d47/d47
+⊂/d15/d15AC∞Sch
+⊂/d15/d15⊂
+/d39/d39❖❖❖❖❖❖❖❖❖❖
+C∞Schlfp⊂/d47/d47C∞Schlf⊂/d47/d47C∞Sch⊂/d47/d47LC∞RS⊂/d47/d47C∞RS.
+By Corollary 3.4 and Theorems 3.5 and 4.22(c), we find as in [23, Th. 16]:
+Corollary 4.27. FC∞Sch
+Man:Man֒→AC∞Schfp⊂AC∞Schis a full and
+faithful functor, and FC∞Sch
+Manb:Manb→AC∞Schfa⊂AC∞Sch, FC∞Sch
+Manc:
+Manc→AC∞Schfa⊂AC∞Schare both faithful functors, but are not full.
+Also these functors take transverse fibre products in Man,Mancto fibre prod-
+ucts inAC∞Schfp,AC∞Schfa.
+We study open subspaces of C∞-schemes. The definition of Spec Cimplies:
+Lemma 4.28. LetCbe aC∞-ring, andc∈C. WriteSpecC= (X,OX)and
+Uc={x∈X:x(c)/\e}atio\slash= 0}. ThenUc⊆Xis open with (Uc,OX|Uc)∼=SpecC[c−1].
+Corollary 4.29. LetX= (X,OX)be aC∞-scheme and V⊆Xbe open.
+ThenV= (V,OX|V)is also aC∞-scheme.
+Proof.Letx∈V. Then there exists an open x∈Y⊆XwithY∼=SpecCfor
+someC∞-ringC, asXas aC∞-scheme. Identify Ywith Spec C. AsV∩Yis
+open inY=XC, and the topology on XCis generated by subsets Uc={˜x∈
+XC: ˜x(c)/\e}atio\slash= 0}forc∈C, there exists c∈Csuch thatx∈Uc⊆V∩Y. Then
+(Uc,OX|Uc)∼=SpecC[c−1] by Lemma 4.28. So every x∈Vhas an affine open
+neighbourhood, and Vis aC∞-scheme.
+Lemma 4.30. LetCbe a finitely generated C∞-ring and (X,OX) = Spec C.
+SupposeV⊆Xis open. Then there exists c∈CwithV={x∈X:x(c)/\e}atio\slash= 0}.
+We callcacharacteristic function forV.
+32Proof.AsCis a finitely generated C∞-ring it fits into an exact sequence 0 →
+I ֒→C∞(Rn)φ−→C→0. Example 4.14 gives a homeomorphism φ∗:X→Xφ
+C
+with a closed subset Xφ
+CinRngiven in (4.4). Then φ∗(V) is open in Xφ
+C, so
+there exists an open U⊆RnwithU∩Xφ
+C=φ∗(V). By [54, Lem. I.1.4] there
+existsf∈C∞(Rn) withU=/braceleftbig
+x∈Rn:f(x)/\e}atio\slash= 0/bracerightbig
+. Thenc=φ(f)∈Cis a
+characteristic function for V.
+Example 4.31. LetIbe an infinite set, and write C∞(RI) for the free C∞-
+ring with generators xifori∈I. ThenX= SpecC∞(RI) has topological space
+X=RIwith points ( xi)i∈Iforxi∈R. Elements of C∞(RI) are functions
+c:RI→Rdepending only on xjforjin afinitesubsetJ⊆I, and which are
+smooth functions of these xj,j∈J.
+LetV=RI\{0}. ThenVis open inX. But no characteristic function c
+exists forVinC∞(RI), sincecwould depend only on xjforjin a finite subset
+J⊆I, butVdepends on xifor alli∈I. Thus, infinitely generated C∞-rings
+need not admit characteristic functions, in contrast to Lemma 4.30 .
+IfCis a finitely generated (or finitely presented) C∞-ring andc∈Cthen
+C[c−1] is also finitely generated (or finitely presented), since C[c−1]∼=C[x]/(c·
+x−1) is the result of adding one extra generator and one extra relatio n toC.
+Thus from Lemmas 4.28 and 4.30 we deduce:
+Corollary 4.32. (a) Let(X,OX)be a fair (or finitely presented) affine C∞-
+scheme, and U⊆Xbe an open subset. Then (U,OX|U)is also a fair (or
+finitely presented) affine C∞-scheme.
+(b)Let(X,OX)be a locally fair (or locally finitely presented) C∞-scheme, and
+U⊆Xbe an open subset. Then (U,OX|U)is also a locally fair (or locally
+finitely presented) C∞-scheme.
+Our next result describes the sheaf of C∞-ringsOXin SpecCforCa finitely
+generatedC∞-ring. It is a version of [31, Prop. I.2.2(b)] in algebraic geometry,
+and reduces to Moerdijk and Reyes [54, Prop. I.1.6] when C=C∞(Rn).
+Proposition 4.33. LetCbe a finitely generated C∞-ring, write (X,OX) =
+SpecC,and letU⊆Xbe open. By Lemma 4.30we may choose a character-
+istic function c∈CforU. Then there is a canonical isomorphism OX(U)∼=
+Rfa
+fg(C[c−1]),in the notation of Definitions 2.13and2.20. IfCis finitely pre-
+sented thenOX(U)∼=C[c−1].
+Proof.We have morphisms of C∞-ringsc∗:C∞(R)→Candi∗:C∞(R)→
+C∞(R\{0}),andC∞(R),C∞(R\{0})arefinitelypresented C∞-ringsbyPropo-
+sition 3.1(a). So as Spec preserves limits in ( C∞Ringsfg)opwe have
+Spec/parenleftbig
+C∐c∗,C∞(R),i∗C∞(R\{0})/parenrightbig∼=SpecC×f,R,iR\{0}∼=(U,OX|U).
+ButC∐C∞(R)C∞(R\{0})∼=C[c−1] for formal reasons. Thus Theorem 4.22(a)
+givesOX(U)∼=Γ/parenleftbig
+(U,OX|U)/parenrightbig∼=Rfa
+fg(C[c−1]). IfCis finitely presented then
+C[c−1] is too, as in Corollary 4.32, so C[c−1] is fair and Rfa
+fg/parenleftbig
+C[c−1]/parenrightbig
+=C[c−1],
+and thereforeOX(U)∼=C[c−1].
+334.6 Complete C∞-rings
+The material of this section appears to be new.
+Proposition 4.34. LetCbe aC∞-ring, and ΨCbe as in Definition 4.19. Then
+SpecΨ C: Spec◦Γ◦SpecC→SpecCis an isomorphism in LC∞RS.
+Proof.WriteD= Γ◦SpecC,X= SpecC,Y= SpecD, andf= SpecΨ C:Y→
+X. Letx∈X, and define y=π◦Πx:D→Rto be the composition of the
+projection Π x:D→Cx, noting that D⊆/producttext
+˜x∈XC˜xby Definition 4.19, and the
+unique morphism π:Cx→R, asCxis a localC∞-ring. Then f(y) =π◦πx=
+x:C→Rforπx:C→Cx, sof:Y→Xis surjective.
+Suppose now that y∈Ywithf(y) =x, so thaty:D→Ris anR-algebra
+morphism. We will prove that y=π◦Πxas above. Let d∈D. By definition of
+D=OXC(XC) there exist an open neighbourhood WofxinXandc1∈Csuch
+thatd(˜x) =π˜x(c1) inC˜xfor all ˜x∈W. By definition of the topology TC, there
+existsc2∈Csuch thatUc2={˜x∈X: ˜x(c2)/\e}atio\slash= 0}is an open neighbourhood of
+xinW⊆X. Hencex(c2)/\e}atio\slash= 0 and ˜x(c2) = 0 for all ˜ x∈X\W.
+Choose smooth functions g,h:R→Rwithg(x(c2)) = 1 and g= 0 in an
+open neighbourhood ( −ǫ,ǫ) of 0 in R, andh(0)/\e}atio\slash= 0 andh= 0 outside (−ǫ,ǫ),
+so thatg·h= 0. Setc3= Φg(c2) andc4= Φh(c2), with Φ g,Φh:C→Cthe
+C∞-ring operations. Then x(c3) = 1, andπ˜x(c3) = 0 inC˜xfor all ˜x∈X\W, as
+π˜x(c3)·π˜x(c4) =π˜x/parenleftbig
+Φg(c2)·Φh(c2)/parenrightbig
+=π˜x◦Φgh(c2) =π˜x◦Φ0(c2) = 0,
+butπ˜x(c4) is invertible in C˜xas ˜x(c4) =h(˜x(c2)) =h(0)/\e}atio\slash= 0. Thus we have
+d·ΨC(c3) = ΨC(c1)·ΨC(c3) = ΨC(c1·c3) inD, asd(˜x) = ΨC(c1)˜xfor all ˜x∈W,
+and Ψ C(c3)˜x= 0 for all ˜x∈X\W. Therefore
+y(d) =y(d)·1 =y(d)·x(c3) =y(d)·y(ΨC(c3)) =y/parenleftbig
+d·ΨC(c3)/parenrightbig
+=y/parenleftbig
+ΨC(c1·c3)/parenrightbig
+=x(c1·c3)=x(c1)·x(c3)=/parenleftbig
+π◦Πx(d)/parenrightbig
+·1=π◦Πx(d).
+As this holds for all d∈D, we see that y∈Ywithf(y) =ximplies that
+y=π◦Πx. Hencef:Y→Xis injective, and so bijective.
+From above f:Y→Xis continuous. To show f−1:X→Yis continuous,
+note that the topology on Yis generated by the basis of open sets Vd={y∈
+Y:y(d)/\e}atio\slash= 0}for alld∈D. So it is enough to show that f(Vd) ={x∈X:
+π◦Πx(d) = 0}is open inXfor alld. For fixed d, by definition we may cover
+Xby openW⊆Xfor which there exist c∈Cwithd(x) =πx(c)∈Cxfor all
+x∈W. But then W∩f(Vd) =W∩Uc, whereUc={x∈X:x(c)/\e}atio\slash= 0}is open
+inX. So we can cover Xby openW⊆XwithW∩f(Vd) open, and f(Vd) is
+open. Therefore f−1is continuous, and f:Y→Xis a homeomorphism.
+Lety∈Ywithf(y) =x. Taking stalks of f♯:f−1(OX)→OYatygives a
+morphismf♯
+y:OX,x→OY,y, whereOX,x∼=CxandOY,y∼=Dyby Lemma 4.18,
+and we have a commutative diagram
+C
+πx/d15/d15ΨC/d47/d47D
+πy/d15/d15Πx/d114/d114❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞
+Cx∼=OX,xΨC,x∼=f♯
+y/d47/d47OY,y∼=Dy.(4.10)
+34Here the outer rectangle and top left triangle obviously commute. T o see that
+the bottom right triangle commutes, we use that any d∈D=OXC(XC) has
+d(˜x) = Ψ C(c)˜xfor somec∈Cand all ˜xin an open neighbourhood Wofxin
+X. As in the first part of the proof, we can find c3∈Cwithx(c3) = 1 and
+π˜x(c3) = 0 inC˜xfor all ˜x∈X\W. Then evaluating at ˜ x∈Wand ˜x∈X\W
+we see that Ψ C(c)·ΨC(c3) =d·ΨC(c3), which forces πy(d) =πy(ΨC(c)), since
+πy◦ΨC(c3) is invertible in Dyasπ◦πy◦ΨC(c3) =x(c3) = 1>0. Thus
+πy(d) =πy◦ΨC(c) =f♯
+y◦πx(c) =f♯
+y◦Πx◦ΨC(c) =f♯
+y◦Πx(d).
+Sinceπy:D→Dyis surjective by Proposition 2.14, the bottom right
+trianglein(4.10)impliesthat f♯
+y:OX,x→OY,yissurjective. Suppose cx∈OX,x
+withf♯
+y(cx) = 0 inOY,y. Asπxis surjective by Proposition 2.14 we may
+writecx=πx(c) forc∈C. Thenπy◦ΨC(c) =f♯
+y◦πx(c) =f♯
+y(cx) = 0, so
+ΨC(c)∈Kerπy. WriteI⊂CandJ⊂Dfor the ideals in (2.2) for x,y. Then
+J= Kerπy, so ΨC(c)∈J, and thus there exists d∈Dwithy(d) =π◦Πx(d)/\e}atio\slash= 0
+inRand Ψ C(c)·d= 0 inD. Applying Π xgives
+cx·Πx(d) =πx(c)·Πx(d) = Πx(ΨC(c))·Πx(d) = Πx(ΨC(c)·d) = Πx(0) = 0.
+But Πx(d) is invertible in Cxasπ◦Πx(d)/\e}atio\slash= 0 inR, socx= 0. Thusf♯
+y:OX,x→
+OY,yis injective, and so an isomorphism.
+We have shown that f:Y→Xis a homeomorphism, and f♯
+y:OX,f(y)→
+OY,yis an isomorphism on stalks at all y∈Y. Hence SpecΨ C= (f,f♯) is an
+isomorphism in LC∞RS, as we have to prove.
+Definition 4.35. We call aC∞-ringCcomplete if the morphism Ψ C:C→
+Γ◦SpecCin Definition 4.19 is an isomorphism. Write C∞Ringscofor the full
+subcategory of complete C∞-ringsCinC∞Rings.
+IfCis anyC∞-ring, applying Γ to SpecΨ Cin Proposition 4.34 shows that
+Γ◦SpecΨ C= ΨΓ◦SpecC: Γ◦SpecC−→Γ◦Spec(Γ◦SpecC)
+is an isomorphism in C∞Rings, where we check that Γ ◦SpecΨ C= ΨΓ◦SpecC
+from Definitions 4.16 and 4.19. Hence Γ ◦SpecCis a complete C∞-ring. Define
+a functorRco
+all:C∞Rings→C∞RingscobyRco
+all= Γ◦Spec.
+The next result extends Definition 2.20 and Theorem 4.22 from C∞Ringsfa
+⊂C∞RingsfgtoC∞Ringsco⊂C∞Rings.
+Theorem 4.36. (a) LetXbe an affine C∞-scheme. Then X∼=SpecOX(X),
+whereOX(X)is a complete C∞-ring.
+(b)Spec|(C∞Ringsco)op: (C∞Ringsco)op→LC∞RSis full and faithful, and
+an equivalence of categories Spec|···: (C∞Ringsco)op→AC∞Sch.
+(c)Rco
+all:C∞Rings→C∞Ringscois left adjoint to the inclusion functor
+inc :C∞Ringsco֒→C∞Rings. That is,Rco
+allis areflection functor .
+(d)All small colimits exist in C∞Ringsco,although they may not coincide with
+the corresponding small colimits in C∞Rings.
+35(e)Spec|(C∞Ringsco)op= Spec◦inc : (C∞Ringsco)op→LC∞RSis right
+adjoint toRco
+all◦Γ :LC∞RS→(C∞Ringsco)op. ThusSpec|···takes limits in
+(C∞Ringsco)op(equivalently, colimits in C∞Ringsco) to limits in LC∞RS.
+Proof.For (a), ifXis an affine C∞-scheme then X∼=SpecCfor someC∞-ring
+C, soOX(X)∼=Γ◦SpecC, and thus X∼=SpecOX(X) by Proposition 4.34.
+Also, applying Γ to SpecΨ Cin Proposition 4.34 shows that
+Γ◦SpecΨ C= ΨΓ◦SpecC: Γ◦SpecC−→Γ◦Spec(Γ◦SpecC)
+is an isomorphism in C∞Rings, where Γ◦SpecΨ C= ΨΓ◦SpecCfollows from
+the definitions. Hence Γ ◦SpecC∼=OX(X) is complete, proving (a).
+For (b), if C,Dare complete C∞-ringsthen putting X= SpecDin Theorem
+4.20 and using Γ ◦SpecD∼=D, equation (4.5) shows that
+Spec =LC,X: Hom C∞Rings(C,D)−→HomLC∞RS(SpecD,SpecC)
+is a bijection, where the definition of LC,Xagrees with the definition of Spec on
+morphisms in this case. Thus Spec is full and faithful on complete C∞-rings.
+Therefore Spec|···: (C∞Ringsco)op→LC∞RSis an equivalence of categories
+from (C∞Ringsco)opto its essential image in LC∞RS, which is AC∞Sch.
+For (c), let C,DbeC∞-rings with Dcomplete. Then we have bijections
+HomC∞Ringsco/parenleftbig
+Rco
+all(C),D/parenrightbig∼=HomC∞Rings/parenleftbig
+Γ◦SpecC,Γ◦SpecD/parenrightbig
+∼=HomLC∞RS/parenleftbig
+SpecD,Spec◦Γ◦SpecC/parenrightbig∼=HomLC∞RS/parenleftbig
+SpecD,SpecC/parenrightbig
+∼=HomC∞Rings/parenleftbig
+C,Γ◦SpecD/parenrightbig∼=HomC∞Rings/parenleftbig
+C,D/parenrightbig
+= Hom C∞Rings/parenleftbig
+C,inc(D)/parenrightbig
+, (4.11)
+usingD∼=Γ◦SpecDasDis complete in the first and fifth steps, Theorem
+4.20 in the second and fourth, and Proposition 4.34 in the third. The b ijections
+(4.11) are functorial in C,Das each step is. Hence Rco
+allis left adjoint to inc.
+For (d), note that Rco
+all:C∞Rings→C∞Ringscotakes colimits to colim-
+its, asit isaleft adjointfunctor by(a). Sogivenafunctor F:J→C∞Ringsco
+forJa small category, we may take the colimit C= colim JFinC∞Rings,
+which exists by Proposition 2.5, and then D=Rco
+all(C) is the colimit of Rco
+all◦F
+inC∞Ringsco. ButRco
+all◦F∼=FasRco
+all|C∞Ringsco∼=id. Hence D= colim JF
+inC∞Ringsco, and all small colimits exist in C∞Ringsco. In Example 2.25,
+the colimits in C∞RingscoandC∞Ringsare different.
+The first part of (e) holds by composing (c) and Theorem 4.20, and t he
+second part follows as right adjoint functors preserve limits. This c ompletes the
+proof of Theorem 4.36.
+Remark 4.37. LetCbe aC∞-ring, so that Ψ C:C→Rco
+all(C) is a morphism of
+C∞-rings. If Cis finitely generated then Theorem 4.22(a) gives an isomorphism
+Rco
+all(C)∼=Rfa
+fg(C) identifying Ψ Cwith the surjective projection π:C→Rfa
+fg(C),
+forRfa
+fgas in Definition 2.20. Thus Ψ C:C→Rco
+all(C) is surjective in this case,
+andRco
+all,Rfa
+fgagree on finitely generated C∞-rings up to natural isomorphism.
+36ForCinfinitely generated, Ψ C:C→Rco
+all(C) need not be surjective, and
+Rco
+all(C) can be much larger than C. For example, if Iis an infinite set and
+C=C∞(RI) is as in Example 4.31, then elements of Care functions c:RI→R
+which depend smoothly only on xjforjin a finite subset J⊆I, but elements
+ofRco
+all(C) are functions c:RI→Rwhichlocally in RIdepend smoothly only
+onxjforjin a finite subset J⊆I, but globally may depend on xifor infinitely
+manyi∈I. So Ψ C:C→Rco
+all(C) is injective but not surjective.
+4.7 Partitions of unity
+We now study the existence of smooth partitions on unity on C∞-schemes and
+localC∞-ringed spaces. We will need the next definition.
+Definition 4.38. LetX= (X,OX) be a local C∞-ringed space. Then each
+c∈OX(X) defines a continuous map c∗:X→Rmappingx/ma√sto→π◦πx(c), for
+πx:OX(X)→OX,xandπ:OX,x→Rthe natural C∞-ring morphisms. Thus
+Uc={x∈X:c∗(x)/\e}atio\slash= 0}is open inX. We say that the topology on Xis
+smoothly generated if{Uc:c∈OX(X)}is a basis for the topology on X.
+This implies Xis a regular (and completely regular) topological space.
+Example 4.39. (a) LetXbeacompletelyregulartopologicalspace, anddefine
+a sheaf ofC∞-ringsOXonXby takingOX(U) =C0(U) to be the C∞-ring of
+continuous functions c:U→Rfor all open U⊆X. ThenX= (X,OX) is a
+localC∞-ringed space, and the topology on Xis smoothly generated.
+(b)LetXbe an affine C∞-scheme. Then X∼=SpecOX(X) by Theorem
+4.36(a). So the definition of the topology on Xin Definition 4.13 implies that
+the topology on Xis smoothly generated.
+(c)SupposeXis a regular C∞-scheme, and let T⊆Xbe open and x∈T.
+Thenxhas an affine open neighbourhood YinX. SinceXis regular, there
+exist disjoint open neighbourhoods VofxandWofX\YinX.
+Thenx∈T∩V⊆Y, and the topology on Yis smoothly generated by (b),
+so there exists a∈OY(Y) withx∈UY
+a⊆T∩V. Nowa∗(x)/\e}atio\slash= 0 anda∗(y) = 0
+for ally∈Y\UY
+a, but this does not imply that ais supported in UY
+a, as we
+could have πy(a)/\e}atio\slash= 0 inOY,yeven though π◦πy(a) = 0 in R. Choose smooth
+f:R→Rwithf(a∗(x))/\e}atio\slash= 0 andf(t) = 0 fortin an open neighbourhood of 0
+inR. Setb= Φf(a), for Φf:OY(Y)→OY(Y) theC∞-ring operation.
+Thenb∗(x)/\e}atio\slash= 0, andUY
+b⊆UY
+a⊆T, andbis supported in UY
+a⊆V⊆Y.
+SinceWis open inXwithX\Y⊆W⊆Y\V, there exists a unique c∈OX(X)
+withc|Y=bandc|W= 0. We have x∈UX
+c=UY
+b⊆T. Thus, for each open
+T⊆Xandx∈Twe can find c∈OX(X) withx∈UX
+c⊆T. So the topology
+onXis smoothly generated.
+(d)LetXbe an infinite-dimensional Banach space or Banach manifold, and
+makeXinto a local C∞-ringed space X= (X,OX) as in Example 4.10. The
+question of when the topology of Xis smoothly generated (framed in terms
+of the existence of ‘smooth bump functions’ on X) is very well understood, as
+in Bonic and Frampton [10] and Deville, Godefroy and Zizler [18, §V]. For
+37example, if Xis a Hilbert manifold, or modelled on Lq(Y) orℓqfor evenq/greaterorequalslant2,
+then the topology on Xis smoothly generated, but if Xis modelled on Lq(Y)
+orℓqforq∈[1,∞] not even, the topology on Xis not smoothly generated.
+For the next theorem, §4.1 defined Lindel¨ of spaces, and explained their rela-
+tion to other topological assumptions. Second countable implies Lind el¨ of, and
+Lindel¨ of and regular imply paracompact (note that Xis regular as its topology
+is smoothly generated). It is easy to see that OXfine implies that the topology
+onXis smoothly generated.
+The proof of Theorem 4.40 is based on the proof of the existence of smooth
+partitions on unity on suitable separableBanachmanifolds in Bonic and Framp-
+ton [10, Th. 1] (see also Lang [45, §II.3] and Deville et al. [18, §VIII.3]).
+Theorem 4.40 applies to a very large class of C∞-schemes, showing that
+partitions of unity exist on most interesting examples of C∞-schemes.
+Theorem 4.40. LetX= (X,OX)be a Lindel¨ of local C∞-ringed space, and
+suppose the topology on Xis smoothly generated. Then OXisfine, as in
+Definition 4.7. That is, for every open cover {Vi:i∈I}ofXthere exists a
+subordinate locally finite partition of unity {ηi:i∈I}inOX(X).
+Proof.Forc∈OX(X) andx∈Xwe haveπx(c)∈OX,xandc∗(x) =π◦
+πx(c)∈R, whereπx:OX(X)→OX,xandπ:OX,x→Rare the natural C∞-
+morphisms. Then c∗:X→Ris continuous. Write Uc={x∈X:c∗(x)/\e}atio\slash= 0},
+so thatUcis open inX. Thesupportofcis suppc={x∈X:πx(c)/\e}atio\slash= 0}.
+Then suppcis closed in XwithUc⊆suppc, but suppcmay be larger than
+the closure of Uc. Note that an infinite sum/summationtext
+j∈JcjinOX(X) is defined, as
+a section of the sheaf OX, if{suppcj:j∈J}is locally finite (that is, each
+x∈Xhas an open neighbourhood Wxintersecting supp cjfor only finitely
+manyj∈J), but may not make sense if only {Ucj:j∈J}is locally finite.
+Because of this, we are careful to keep track of both Ucjand suppcjin the
+following proof.
+Let{Vi:i∈I}be an open cover of X. Supposei∈Iandx∈Vi. As the
+topology on Xis smoothly generated there exists c∈OX(X) withx∈Uc⊆Vi.
+Soc∗(x)/\e}atio\slash= 0 andc∗|X\Vi= 0. We do not know that supp c⊆Vi, but we can
+correct this as follows. Choose smooth f:R→Rsuch thatf(c∗(x))/\e}atio\slash= 0 and
+f= 0 in a neighbourhood of 0 in R. Setc′= Φf(c), where Φ f:OX(X)→
+OX(X) is theC∞-ring operation. Then x∈Uc′⊆suppc′⊆Uc⊆Vi⊆X.
+Thus, we can choose a family {cj:j∈J}such thatcj∈OX(X), and
+Ucj⊆suppcj⊆Vij⊆Xfor eachj∈Jand someij∈I, and{Ucj:j∈J}
+is an open cover of X. SinceXis Lindel¨ of we can take Jto be countable, and
+chooseJ=N.
+Replacingcjbyc2
+jwe have (cj)∗/greaterorequalslant0 onX. For eachj∈N, choose smooth
+fj:Rj+1→Rsuch thatfj(t0,t1,...,tj)>0 ifti<1/jfori= 0,1,...,j−1
+andtj>0, andfj(t0,t1,...,tj) = 0 otherwise. Define dj= Φfj(c0,c1,...,cj),
+38with Φfj:OX(X)j+1→OX(X) theC∞-ring operation. Then
+Udj=/braceleftbig
+x∈X: (dj)∗(x)/\e}atio\slash= 0/bracerightbig
+=/braceleftbig
+x∈X: (ci)∗(x)<1/j, i= 0,...,j−1,(cj)∗(x)/\e}atio\slash= 0/bracerightbig
+⊆Vij,
+suppdj⊆/braceleftbig
+x∈X: (ci)∗(x)/lessorequalslant1/j, i= 1,...,j−1/bracerightbig
+∩suppcj⊆Vij.(4.12)
+Fixx∈X. Thenx∈Ucjfor somej∈Nas{Ucj:j∈J}coversX.
+Letj∈Nbe least with x∈Ucj. Then (cj)∗(x)>0 and (ci)∗(x) = 0 for
+i= 0,1,...,j−1. Thusx∈Udj, so{Udj:j∈N}is an open cover of X. Define
+Tx={y∈X: (cj)∗(y)>1
+2(cj)∗(x)}. ThenTxis an open neighbourhood of x
+inX, andTx∩Udk=∅=Tx∩suppdkprovidedk >max/parenleftbig
+j,2(cj)∗(x)−1/parenrightbig
+by
+(4.12). Thus, both {Udj:j∈N}and{suppdj:j∈N}are locally finite.
+For eachi∈I, defineei=/summationtext
+j∈N:ij=idjinOX(X). This is well defined
+as{suppdj:j∈N}is locally finite. We have Uei⊆suppei⊆Vi, since
+Udj⊆suppdj⊆Vifor eachj∈Nwithij=i. Both{Uei:i∈I}and
+{suppei:i∈I}are locally finite, as {Udj:j∈N}and{suppdj:j∈N}are.
+Thuse=/summationtext
+i∈Ieiis well defined in OX(X). Ifx∈Xthen
+e∗(x) =/summationtext
+i∈I(ei)∗(x) =/summationtext
+i∈I/summationtext
+j∈N:ij=i(dj)∗(x) =/summationtext
+j∈N(dj)∗(x)>0,
+where each sum has only finitely many nonzero terms, and/summationtext
+j∈N(dj)∗(x)>0 as
+{Udj:j∈N}coversXwith (dj)∗>0 onUdjand (dj)∗= 0 onX\Udj. Since
+e∗is positive on X,eis invertible inOX(X). Setηi=e−1·eifori∈I. Then
+suppηi⊆Vi, assuppei⊆Vi, and{ηi:i∈I}islocallyfinite, as {suppei:i∈I}
+is, and/summationtext
+i∈Iηi=/summationtext
+i∈Ie−1·ei=e−1·e= 1. Hence{ηi:i∈I}is a locally
+finite partition of unity subordinate to {Vi:i∈I}, soOXis fine.
+4.8 A criterion for affine C∞-schemes
+Here are sufficient conditions for a local C∞-ringed space Xto be an affine C∞-
+scheme. Note that affine C∞-schemes are Hausdorff with smoothly generated
+topology by Lemma 4.15 and Example 4.39(b), so Lindel¨ of is the only co ndition
+in the theorem which is not also necessary.
+Theorem 4.41. LetX= (X,OX)be a Hausdorff, Lindel¨ of, local C∞-ringed
+space, with smoothly generated topology. Then Xis an affine C∞-scheme.
+Proof.LetXbe as in the theorem. Note that Theorem 4.40 shows that OX
+is fine. Write C=OX(X) = Γ(X), andY= SpecC. Define a morphism
+f:X→Ybyf=LC,X(idC), using the notation of Theorem 4.20. We will
+showfis an isomorphism, so that X∼=SpecCis an affine C∞-scheme.
+Pointsx∈XinduceC∞-ring morphisms π◦πx:C=OX(X)→R, where
+πx:OX(X)→OX,xandπ:OX,x→Rare the natural projections. Points
+y∈YareC∞-ring morphisms y:C→R, andf:X→Yisf(x) =π◦πx.
+Supposex,x′∈Xwithx/\e}atio\slash=x′, and setf(x) =yandf(x′) =y′. SinceXis
+Hausdorff there exists open U⊆Xwithx∈Uandx′/∈U. As the topology on
+Xissmoothlygeneratedthereexists c∈OX(X)withc∗(x)/\e}atio\slash= 0andc∗|X\U= 0,
+39so thatc∗(x′) = 0. Then y(c) =c∗(x)/\e}atio\slash= 0 andy′(c) =c∗(x′) = 0, soy/\e}atio\slash=y′.
+Hencef:X→Yis injective.
+Suppose for a contradiction that y∈Y, butf(x)/\e}atio\slash=yfor allx∈X. Then
+for eachx∈X, there exists a∈Cwithy(a)/\e}atio\slash=π◦πx(a). Choose smooth
+g:R→Rwithg(y(a)) = 0 andg= 1 in an open neighbourhood of π◦πx(a) in
+R. Setb= Φg(a), where Φ g:C→Cis theC∞-ring operation. Then y(b) = 0
+andπ◦π˜x(b) = 1 for ˜xin an open neighbourhood VofxinX.
+Thus we may choose a family of pairs {(Vj,bj) :j∈J}such that for each
+j∈Jwe haveVj⊆Xopen andbj∈Cwithy(bj) = 0 andπ◦πx(bj) = 1 for
+x∈Vj, and{Vj:j∈J}is an open cover of X. AsXis Lindel¨ of we can suppose
+Jis countable, and so take J=N. By Theorem 4.40 there exists a locally finite
+partition of unity {ηj:j∈N}inCsubordinate to{Vj:j∈N}.
+Setc=/summationtext
+j∈Nj·ηj·bjinC=OX(X), which makes sense in global sections
+ofOXas{ηj:j∈N}is locally finite. Choose n∈Nwithn>y(c), and define
+d=c−y(c)·1X+/summationtextn−1
+j=0(n−j)·ηj·bjinC, where 1X∈Cis the identity. Then
+y(d) =y(c)−y(c)·y(1X)+/summationtextn−1
+j=0(n−j)·y(ηj)·y(bj) = 0,
+asy(1X) = 1 andy(bj) = 0. And if x∈Xthen
+π◦πx(d) =π◦πx/bracketleftbig/summationtext
+j∈Nj·ηj·bj−y(c)·/summationtext
+j∈Nηj+/summationtextn−1
+j=0(n−j)·ηj·bj/bracketrightbig
+=/summationtext
+j∈N/parenleftbig
+max(j,n)−y(c)/parenrightbig
+π◦πx(ηj)>0,
+whereeachsumhasonlyfinitelymanynonzeroterms,andweuse/summationtext
+j∈Nηj= 1X,
+π◦πx(bj) = 1, and max( j,n)−y(c)>0,π◦πx(ηj)/greaterorequalslant0 forj∈N.
+Sinceπ◦πx(d)>0 for allx∈X, we see that dis invertible in C=OX(X),
+but this contradicts y(d) = 0. Hence each y∈Yhasy=f(x) for somex∈X,
+andfis surjective, so f:X→Yis a bijection. By definition of Y= SpecC,
+the topology on Yis generated by the open sets Uc={y∈Y:y(c)/\e}atio\slash= 0}for
+allc∈C. As the topology on Xis smoothly generated, it is generated by the
+open setsf−1(Uc) ={x∈X:c∗(x)/\e}atio\slash= 0}forc∈C. Therefore f:X→Yis a
+bijection identifying bases for the topologies of X,Y, sofis a homeomorphism.
+Letx∈Xwithf(x) =y∈Y. Taking stalks of f♯:f−1(OY)→OXatx
+gives a morphism f♯
+x:OY,y→OX,x. By the definition of f=LC,X(idC) in the
+proof of Theorem 4.20, f♯
+xagrees with φxin (4.8), and is the unique morphism
+making the following commute, where Cy∼=OY,yby Lemma 4.18:
+C
+πy/d15/d15y
+/d34/d34OX(X)
+πx/d15/d15
+Cy∼=OY,y
+π/d42/d42❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱f♯
+x/d47/d47OX,x
+π/d116/d116✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐
+R.(4.13)
+Supposeay∈OY,ywithf♯
+x(ay) = 0. Then ay=πy(a) for some a∈C=
+OX(X), asπyis surjective by Proposition 2.14, and then πx(a) = 0 inOX,x, as
+40(4.13) commutes. Hence there exists an open neighbourhood UofxinXwith
+a|U= 0 inOX(U). As the topology on Xis smoothly generated, there exists
+b∈OX(X) withb∗(x)/\e}atio\slash= 0 andb∗|X\U= 0. Choose smooth g:R→Rwith
+g(b∗(x))/\e}atio\slash= 0 andg= 0 near 0 in R, and setc= Φg(b), where Φ g:OX(X)→
+OX(X) is theC∞-ring operation. Then y(c) =c∗(x)/\e}atio\slash= 0, andcis supported in
+U. Asa|U= 0 we see that a·c= 0 inOX(X). Thusalies in the ideal Iin (2.2)
+which is the kernel of πy:C→Cy, by Proposition 2.14, and so ay=πy(a) = 0.
+Thereforef♯
+x:OY,y→OX,xis injective.
+Supposeax∈OX,x. Thenby definition of OX,xthereexists open x∈U⊆X
+anda∈OX(U) withπx(a) =ax. As the topology on Xis smoothly generated
+there exists b∈ OX(X) withb∗(x)/\e}atio\slash= 0 andb∗|X\U= 0. Choose smooth
+g:R→Rwithg= 1 nearb∗(x) inRandg= 0 near 0 in R. Setc= Φg(b),
+where Φg:OX(X)→OX(X) is theC∞-ring operation. Then cis supported
+inU, and there exists an open neighbourhood VofxinUwithc|V= 1. Since
+cis supported in U, the section c|U·a∈OX(U) can be extended by zero over
+X\Uto give a unique d∈OX(X) supported in Uwithd|U=c|U·a.
+Thend|V=c|V·a|V= 1·a|V=a|V. Hencef♯
+x◦πy(d) =πx(d) =ax,
+sof♯
+x:OY,y→ OX,xis surjective, and an isomorphism. This proves that
+f♯:f−1(OY)→OXis an isomorphism on stalks at every x∈X, sof♯is
+an isomorphism. As fis a homeomorphism, f= (f,f♯) :X→SpecCis an
+isomorphism. This completes the proof of Theorem 4.41.
+Corollary 4.42. LetX= (X,OX)be a localC∞-ringed space. Then the
+following are equivalent:
+(i)Xis Hausdorff and second countable, with smoothly generated t opology.
+(ii)Xis separable and metrizable, with smoothly generated topol ogy.
+(iii)Xis a Hausdorff, second countable, regular C∞-scheme.
+(iv)Xis a separable, metrizable C∞-scheme.
+(v)Xis a second countable, affine C∞-scheme.
+When these hold, Xis regular, normal, and paracompact, and OXis fine.
+Proof.Section 4.1 implies that (i),(ii) are equivalent (as Xsmoothly generated
+topology implies Xregular), and (iii),(iv) are equivalent. Also (v) implies (iii)
+by Lemma 4.15, and (iii) implies (i) by Example 4.39(b), and (i) implies (v)
+by Theorem 4.41 (as second countable implies Lindel¨ of). Hence (i)–( v) are
+equivalent. The last part follows from §4.1 and Theorem 4.40.
+In comparison to Theorem 4.41, we have strengthened the Lindel¨ o f assump-
+tion to second countable. The category of C∞-schemes in Corollary 4.42 is very
+large, and convenient to work in. They are closed under products, fibre prod-
+ucts, and arbitrary subspaces (Lindel¨ of spaces are none of the se). They have
+partitions of unity, and as they are affine we can argue globally using C∞-rings.
+41Example 4.43. LetX= (X,OX) be a second countable, affine C∞-scheme,
+and letY⊆Xbeanysubset, not necessarily open or closed. Then Y=
+(Y,OX|Y) is also a second countable, affine C∞-scheme by Corollary 4.42, as
+being Hausdorff, second countable, and of smoothly generated to pology, are all
+preserved under passing to subspaces, so Ysatisfies Corollary4.42(i) as Xdoes.
+Example 4.44. LetXbe a separable Banach manifold modelled locally on
+separableBanach spaces Bwhich admit ‘smooth bump functions’ (that is, there
+exists a nonzero smooth function f:B→Rwith bounded support in B). See
+Deville et al. [18, §V] for results on when a Banach space Bhas a smooth bump
+function, for example, every Hilbert space does.
+MakeXinto a local C∞-ringed space X= (X,OX) as in Example 4.10.
+Then the topology on Xis smoothly generated as in Example 4.39(d), so Xis
+an affineC∞-scheme by Corollary 4.42(ii),(v).
+4.9 Quotients of C∞-schemes by finite groups
+Finally we discuss quotients of C∞-schemes by finite groups.
+Definition 4.45. LetX= (X,OX) be a local C∞-ringed space, Ga finite
+group, and r:G→Aut(X) an action of GonX. We will define a local
+C∞-ringed space Y=X/G.
+SetY=X/r(G) to be the quotient topological space. Open sets V⊆Yare
+of the form U/GforU⊆Xopen andG-invariant. Then γ/ma√sto→r♯(γ)(U) gives an
+action ofGon theC∞-ringOX(U), so as in Proposition2.22we havea C∞-ring
+OX(U)G, theG-invariant subspace in OX(U). DefineOY(V) =OX(U)G.
+IfV2⊆V1⊆Yare open then V1=U1/G,V2=U2/GforU2⊆U1⊆X
+open andG-invariant. The restriction morphism ρU1U2:OX(U1)→OX(U2) in
+OXisG-equivariant, and so restricts to ρU1U2|OX(U1)G:OX(U1)G→OX(U2)G.
+SetρV1V2=ρU1U2|OX(U1)G:OY(V1)→OY(V2). It is now easy to check that
+OYis a sheaf of C∞-rings onY, soY= (Y,OY) is aC∞-ringed space.
+Ifx∈Xandy=xG∈Y, the stalkOY,yofOYatyis (OX,x)H, where
+OX,xis a localC∞-ring, andH=/braceleftbig
+γ∈G:γ(x) =x/bracerightbig
+is the stabilizer group
+ofxinG, which acts onOX,xin the obvious way. As OX,xis local there is
+anR-algebra morphism π:OX,x→R, such that c∈OX,xis invertible if and
+only ifπ(c)/\e}atio\slash= 0. Thusπ|(OX,x)H: (OX,x)H→Ris anR-algebra morphism, and
+c∈(OX,x)His invertible inOX,xif and only if π(c)/\e}atio\slash= 0. But if c∈(OX,x)H
+is invertible inOX,xthenc−1isH-invariant, so cis invertible in (OX,x)H.
+ThereforeOY,y∼=(OX,x)His a localC∞-ring, and Yis a localC∞-ringed
+space. Write X/G=Y.
+Defineπ:X→X/Gto be the natural projection. Define a morphism
+π♯:OY→π∗(OX) of sheaves of C∞-rings onY=X/Gby
+π♯(V) = inc :OY(V) =OX(U)G−→OX(U) =π∗(OX)(V)
+for all open V=U/G⊆Y=X/G, where inc :OX(U)G֒→OX(U) is the
+inclusion. Let π♯:π−1(OY)→OXbe the morphism of sheaves of C∞-rings on
+42Xcorrespondingto π♯under (4.3). Then π= (π,π♯) :X→X/Gis a morphism
+of localC∞-ringed spaces.
+It is easy to see that X/G,πhave the universal property that if f:X→Z
+is a morphism in LC∞RSwithf◦r(γ) =ffor allγ∈Gthenf=g◦πfor a
+unique morphism g:X/G→ZinLC∞RS.
+Proposition 4.46. LetX= (X,OX)be an affine C∞-scheme,Ga finite
+group, and r:G→Aut(X)an action of GonX. SupposeXis Lindel¨ of.
+ThenX= SpecCforC=OX(X)a complete C∞-ring, andr= Specsfor
+s:G→Aut(C)a unique action of GonC. Form the G-invariantC∞-ring
+CG⊆Cas in Proposition 2.22. ThenCGis complete, and there is a canonical
+isomorphism X/G∼=SpecCGinLC∞RS.
+Proof.Theorem 4.36(a) shows that X∼=SpecC, whereC=OX(X) is a com-
+pleteC∞-ring. As Spec is full and faithful on complete C∞-rings by Theorem
+4.36(b), Spec : Aut( C)→Aut(X) is an isomorphism, so there is a unique action
+s:G→Aut(C) withr= Specs.
+LetY=X/Gbe as in Definition 4.45. Then Y=X/Gis Hausdorff, as X
+is Hausdorff and Gis finite. Suppose {Vi:i∈I}is an open cover of Y. Then
+Vi=Ui/Gfor{Ui:i∈I}an open cover of X. AsXis Lindel¨ of there exists a
+subcover{Ui:i∈S}for countable S⊆I, and then{Vi:i∈S}is a countable
+subcover of{Vi:i∈I}. HenceYis Lindel¨ of.
+SupposeV⊆Yis open and y∈V. ThenV=U/Gandy=xGforG-
+invariantopen U⊆Xwithx∈U. As the topologyon Xis smoothly generated,
+there exists c∈Cwithc∗(x)/\e}atio\slash= 0 andc∗(x′) = 0 for all x′∈X\U. Define
+d=/summationtext
+γ∈Gγ∗(c2) inC. ThendisG-invariant with d∗(x)>0 andd∗(x′) = 0
+for allx′∈X\U. Henced∈OY(Y) =OX(X)G=CG, withd∗(y)>0 and
+d∗(y′) = 0 for all y′∈Y\V. Thus the topology of Yis smoothly generated.
+Theorem 4.41 now implies that Y=X/Gis an affine C∞-scheme, and
+Theorem4.36(a)givesacanonicalisomorphism X/G∼=SpecOY(Y) = Spec CG,
+whereCGis complete.
+Proposition 4.47. SupposeXis a Hausdorff, second countable C∞-scheme,
+Ga finite group, and r:G→Aut(X)an action of GonX. Then the quotient
+X/Gis also a Hausdorff, second countable C∞-scheme. If Xis locally fair, or
+locally finitely presented, then so is X/G.
+Proof.Letx∈X, and write H=/braceleftbig
+γ∈G:γ(x) =x/bracerightbig
+. Then the G-orbitxG
+is|G|/|H|points. Since Xis Hausdorff and Gis finite, we can find an open
+neighbourhood RofxinXsuch thatRisH-invariant and R∩γ·R=∅for all
+γ∈G\H. AsXis aC∞-scheme, there is an open neighbourhood SofxinR
+with (S,OX|S) an affineC∞-scheme. Then T=/intersectiontext
+γ∈Hγ·Sis anH-invariant
+open neighbourhood of xinS. Choose an open neighbourhood UofxinTwith
+(U,OX|U) an affineC∞-scheme.
+DefineV=/intersectiontext
+γ∈Hγ·U. ThenVis anH-invariant open neighbourhood
+ofxinU⊆T⊆S⊆R⊆X. It is the intersection of the |H|affineC∞-
+subschemes ( γ·U,OX|γ·U) forγ∈Hinside the affine C∞-scheme (S,OX|S).
+43Finite intersections of affine C∞-subschemes in an affine C∞-scheme are affine,
+as such intersections are fibre products and Spec : C∞Ringsop→LC∞RS
+preserves limits by Remark 4.21(b). Thus ( V,OX|V) is an affine C∞-scheme.
+SetW=/uniontext
+γH∈G/Hγ·V. ThenWis aG-invariant open neighbourhood
+ofxinX, and (W,OX|W) is the disjoint union of |G|/|H|affineC∞-schemes
+isomorphic to ( V,OX|V), so it is affine. We have shown that every x∈Xhas
+aG-invariant open neighbourhood W⊆XwithW= (W,OX|W) affine. Then
+W/Gis an open neighbourhood of xGinX/G. AsXis second countable, W
+is second countable and so Lindel¨ of. Thus W/Gis an affine C∞-scheme by
+Proposition 4.46. As we can cover X/Gby such open W/G, it is aC∞-scheme.
+IfXis locally fair, or locally finitely presented, we can do the argument
+above with S,U,V,W,W/Gfair, or finitely presented, using Proposition 2.22
+forW/G, soX/Gis also locally fair, or locally finitely presented.
+5 Modules over C∞-rings and C∞-schemes
+Nextwediscussmodulesover C∞-rings,andsheavesofmoduleson C∞-schemes.
+The author knows of no previous work on these, so all this section m ay be new,
+although much of it is a straightforward generalization of well known facts.
+5.1 Modules over C∞-rings
+Definition 5.1. LetCbe aC∞-ring. A moduleMoverC, orC-module, is a
+module over Cregarded as a commutative R-algebra as in Definition 2.6, and
+morphisms of C-modules are morphisms of R-algebra modules. We will write
+µM:C×M→Mfor the multiplication map, and also write µM(c,m) =c·m
+forc∈Candm∈M. ThenC-modules form an abelian category, which we
+write as C-mod.
+The action of Con itself by multiplication makes Cinto aC-module, and
+moregenerally C⊗RVisaC-moduleforany R-vectorspace V. AC-moduleMis
+finitely generated ifitfitsintoanexactsequence C⊗Rn→M→0inC-mod, and
+finitely presented if it fits into an exact sequence C⊗Rm→C⊗Rn→M→0.
+BecauseC∞-rings such as C∞(Rn) are not noetherian, finitely generated
+C-modules generally need not be finitely presented.
+Now letφ:C→Dbe a morphism of C∞-rings. IfMis aC-module then
+φ∗(M) =M⊗CDis aD-module, and this induces a functor φ∗:C-mod→
+D-mod. Also,any D-moduleNmayberegardedasa C-moduleφ∗(N) =Nwith
+C-actionµφ∗(N)(c,n) =µN(φ(c),n), and this defines a functor φ∗:D-mod→
+C-mod. Note that φ∗:C-mod→D-mod takes finitely generated (or finitely
+presented) C-modules to finitely generated (or finitely presented) D-modules,
+butφ∗:D-mod→C-mod generally does not.
+Vector bundles Eover manifolds Xgive examples of modules over C∞(X).
+Example 5.2. LetXbe a manifold and E→Xbe a vector bundle, and write
+Γ∞(E) for the vector space of smooth sections eofE. This is a module over
+44theC∞-ringC∞(X), multiplying functions on Xby sections of E.
+LetE,F→Xbe vector bundles over Xandλ:E→Fa morphism of
+vector bundles. Then λ∗: Γ∞(E)→Γ∞(F) defined by λ∗:e/ma√sto→λ◦eis a
+morphism of C∞(X)-modules.
+Now letX,Ybe manifolds and f:X→Ya (weakly) smooth map. Then
+f∗:C∞(Y)→C∞(X) is a morphism of C∞-rings. IfE→Yis a vector
+bundle over Y, thenf∗(E) is a vector bundle over X. Under the functor ( f∗)∗:
+C∞(Y)-mod→C∞(X)-mod of Definition 5.1, we see that ( f∗)∗/parenleftbig
+Γ∞(E)/parenrightbig
+=
+Γ∞(E)⊗C∞(Y)C∞(X) is isomorphic as a C∞(X)-module to Γ∞/parenleftbig
+f∗(E)/parenrightbig
+.
+IfE→Xis any vector bundle over a manifold Xthen by choosing sections
+e1,...,en∈Γ∞(E) forn≫0 such that e1|x,...,en|xspanE|xfor allx∈X
+we obtain a surjective morphism of vector bundles ψ:X×Rn→E, whose
+kernel is another vector bundle F. By choosing another surjective morphism
+φ:X×Rm→Fwe obtain an exact sequence of vector bundles
+X×Rmφ/d47/d47X×Rnψ/d47/d47E /d47/d470,
+which induces an exact sequence of C∞(X)-modules
+C∞(X)⊗RRmφ∗/d47/d47C∞(X)⊗RRnψ∗/d47/d47Γ∞(E) /d47/d470.
+Thus Γ∞(E) is a finitely presented C∞(X)-module.
+5.2 Cotangent modules of C∞-rings
+Given aC∞-ringC, we will define the cotangent module ΩCofC. Although
+our definition of C-module only used the commutative R-algebra underlying the
+C∞-ringC, our definition of the particular C-module Ω Cdoes use the C∞-ring
+structure in a nontrivial way. It is a C∞-ring version of the module of relative
+differential forms orK¨ ahler differentials in Hartshorne [31, p. 172], and is an
+example of a construction for Fermat theories by Dubuc and Kock [ 25].
+Definition 5.3. Suppose Cis aC∞-ring, andMaC-module. A C∞-derivation
+is anR-linear map d : C→Msuch that whenever f:Rn→Ris a smooth map
+andc1,...,cn∈C, we have
+dΦf(c1,...,cn) =n/summationtext
+i=1Φ∂f
+∂xi(c1,...,cn)·dci. (5.1)
+Note that d is nota morphism of C-modules. We call such a pair M,d acotan-
+gent module forCif it has the universal property that for any C∞-derivation
+d′:C→M′, there exists a unique morphism of C-modulesλ:M→M′
+with d′=λ◦d.
+There is a natural construction for a cotangent module: we take Mto
+be the quotient of the free C-module with basis of symbols d cforc∈C
+by theC-submodule spanned by all expressions of the form dΦ f(c1,...,cn)−/summationtextn
+i=1Φ∂f
+∂xi(c1,...,cn)·dciforf:Rn→Rsmooth and c1,...,cn∈C. Thus
+45cotangent modules exist, and are unique up to unique isomorphism. W hen we
+speak of ‘the’ cotangent module, we mean that constructed abov e. We write
+dC:C→ΩCfor the cotangent module of C.
+LetC,DbeC∞-rings with cotangent modules Ω C,dC, ΩD,dD, andφ:C→
+Dbe a morphism of C∞-rings. Then we may regard Ω D=φ∗(ΩD) as aC-
+module, and d D◦φ:C→ΩDas aC∞-derivation. Thus by the universal
+property of Ω C, there exists a unique morphism of C-modules Ω φ: ΩC→ΩD
+with d D◦φ= Ωφ◦dC. This then induces a morphism of D-modules (Ω φ)∗:
+ΩC⊗CD→ΩD. Ifφ:C→D,ψ:D→Eare morphisms of C∞-rings
+then Ωψ◦φ= Ωψ◦Ωφ: ΩC→ΩE.
+Example 5.4. LetXbeamanifold. Thenthecotangentbundle T∗Xisavector
+bundle over X, so as in Example 5.2 it yields a C∞(X)-module Γ∞(T∗X). The
+exterior derivative d : C∞(X)→Γ∞(T∗X), d :c/ma√sto→dcis then aC∞-derivation,
+since equation (5.1) follows from
+d/parenleftbig
+f(c1,...,cn)/parenrightbig
+=/summationtextn
+i=1∂f
+∂xi(c1,...,cn)dcn
+forf:Rn→Rsmooth and c1,...,cn∈C∞(X), which holds by the chain rule.
+It is easy to show that Γ∞(T∗X),d have the universal property in Definition
+5.3, and so form a cotangent module for C∞(X).
+Now letX,Ybe manifolds, and f:X→Ya smooth map. Then f∗(T∗Y),
+T∗Xare vector bundles over X, and the derivative of fgives a vector bundle
+morphism d f:f∗(T∗Y)→T∗X. This induces a morphism of C∞(X)-modules
+(df)∗: Γ∞(f∗(T∗Y))→Γ∞(T∗X). This (df)∗is identified with (Ω f∗)∗under
+the natural isomorphism Γ∞(f∗(T∗Y))∼=Γ∞(T∗Y)⊗C∞(Y)C∞(X), where we
+identifyC∞(Y),C∞(X),f∗withC,D,φin Definition 5.3.
+The importance of Definition 5.3 is that it abstracts the notion of cot angent
+bundle of a manifold in a way that makes sense for any C∞-ring.
+Remark 5.5. There is a second way to define a cotangent-type module for a
+C∞-ringC, namely the module Kd CofK¨ ahler differentials of the underlying
+R-algebra of C. This is defined as for Ω C, but requiring (5.1) to hold only when
+f:Rn→Ris a polynomial. Since we impose many fewer relations, Kd Cis
+generally much larger than Ω C, so that Kd C∞(Rn)is not a finitely generated
+C∞(Rn)-module for n>0, for instance.
+Proposition 5.6. IfCis a finitely generated C∞-ring then ΩCis a finitely
+generated C-module. If Cis finitely presented, then ΩCis finitely presented.
+Proof.IfCis finitely generated we have an exact sequence
+0 /d47/d47I /d47/d47C∞(Rn)φ/d47/d47C /d47/d470. (5.2)
+Writex1,...,xnfor the generators of C∞(Rn). Then any c∈Cmay be written
+asφ(f) for somef∈C∞(Rn), and (5.1) implies that
+dc= dΦf/parenleftbig
+φ(x1),...,φ(xn)/parenrightbig
+=/summationtextn
+i=1Φ∂f
+∂xi(φ(x1),...,φ(xn))·d◦φ(xi).
+46Hence the generators d cof ΩCforc∈CareC-linear combinations of d ◦φ(xi),
+i= 1,...,n, so ΩCis spanned by the d ◦φ(xi), and is finitely generated.
+Suppose Cis finitely presented. Then we have an exact sequence (5.2) with
+idealI= (f1,...,fm). We will define an exact sequence of C-modules
+C⊗RRmα/d47/d47C⊗RRnβ/d47/d47ΩC/d47/d470. (5.3)
+Write (a1,...,am), (b1,...,bn) for bases of Rm,Rn. AsC⊗RRm,C⊗RRnare
+freeC-modules, the C-module morphisms α,βare specified uniquely by giving
+α(ai) fori= 1,...,mandβ(bj) forj= 1,...,n, which we define to be
+α:ai/ma√sto−→/summationtextn
+j=1Φ∂fi
+∂xj/parenleftbig
+φ(x1),...,φ(xn)/parenrightbig
+·bjandβ:bj/ma√sto−→dC/parenleftbig
+φ(xj)/parenrightbig
+.
+Then fori= 1,...,mwe have
+β◦α(ai) =/summationtextn
+j=1Φ∂fi
+∂xj/parenleftbig
+φ(x1),...,φ(xn)/parenrightbig
+·dC/parenleftbig
+φ(xj)/parenrightbig
+= dC/parenleftbig
+Φfi/parenleftbig
+φ(x1),...,φ(xn)/parenrightbig/parenrightbig
+= dC◦φ/parenleftbig
+Φfi(x1,...,xn)/parenrightbig
+= dC◦φ/parenleftbig
+fi(x1,...,xn)) = d C(0) = 0,
+using (5.1) in the second step, φa morphism of C∞-rings in the third, the
+definition of C∞(Rn) asaC∞-ringin the fourth, and fi(x1,...,xn)∈I= Kerφ
+in the fifth. Hence β◦α= 0, and (5.3) is a complex.
+Thusβinducesβ∗: (C⊗RRn)/α(C⊗RRm)→ΩC. We will show β∗is an
+isomorphism, so that (5.3) is exact. Define d : C→(C⊗RRn)/α(C⊗RRm) by
+d/parenleftbig
+φ(h)/parenrightbig
+=/summationtextn
+j=1Φ∂h
+∂xj/parenleftbig
+φ(x1),...,φ(xn)/parenrightbig
+·bj+α(C⊗RRm).(5.4)
+Hereeveryc∈Cmaybe written as φ(h) forsomeh∈C∞(Rn) asφis surjective.
+To show (5.4) is well-defined we must show the right hand side is indepen dent
+of the choice of hwithφ(h) =c, that is, we must show that the right hand side
+is zero ifh∈I. It is enough to check this when his a generator f1,...,fmof
+I, and this holds by definition of α. Hence d in (5.4) is well-defined.
+It is easy to see that d is a C∞-derivation, and that β∗◦d = d C. So by
+the universal property of Ω C, there is a unique C-module morphism ψ: ΩC→
+(C⊗RRn)/α(C⊗RRm)withd =ψ◦dC. Thusβ∗◦ψ◦dC=β∗◦d = dC= idΩC◦dC,
+so as Imd Cgenerates Ω Cas anC-module we see that β∗◦ψ= idΩC. Similarly
+ψ◦β∗is the identity, so ψ,β∗are inverse, and β∗is an isomorphism. Therefore
+(5.3) is exact, and Ω Cis finitely presented.
+Cotangent modules behave well under localization.
+Proposition 5.7. LetCbe aC∞-ring,S⊆C,andD=C[s−1:s∈S]be the
+localization of CatSwith projection π:C→D,as in Definition 2.13. Then
+(Ωπ)∗: ΩC⊗CD→ΩDis an isomorphism of D-modules.
+47Proof.Let ΩC,ΩDbe constructed as in Definition 5.3. As D=C[s−1:s∈S]
+isCtogether with an extra generator s−1and an extra relation s·s−1= 1 for
+eachs∈S, we see that the D-module Ω Dmay be constructed from Ω C⊗CD
+by adding an extra generator d( s−1) and an extra relation d( s·s−1−1) = 0 for
+eachs∈S. But using (5.1) and s·s−1= 1 inD, we see that this extra relation
+is equivalent to d( s−1) =−(s−1)2ds. Thus the extra relations exactly cancel
+the effect of adding the extra generators, so (Ω π)∗is an isomorphism.
+Here is a useful exactness property of cotangent modules.
+Theorem 5.8. Suppose we are given a pushout diagram of C∞-rings:
+Cβ/d47/d47
+α/d15/d15E
+δ/d15/d15
+Dγ/d47/d47F,(5.5)
+so thatF=D∐CE. Then the following sequence of F-modules is exact:
+ΩC⊗C,γ◦αF(Ωα)∗⊕−(Ωβ)∗/d47/d47ΩD⊗D,γF⊕
+ΩE⊗E,δF(Ωγ)∗⊕(Ωδ)∗/d47/d47ΩF/d47/d470.(5.6)
+Here(Ωα)∗: ΩC⊗C,γ◦αF→ΩD⊗D,γFis induced by Ωα: ΩC→ΩD,and so
+on. Note the sign of −(Ωβ)∗in(5.6).
+Proof.By Ωψ◦φ= Ωψ◦Ωφin Definition 5.3 and commutativity of (5.5) we have
+Ωγ◦Ωα= Ωγ◦α= Ωδ◦β= Ωδ◦Ωβ: ΩC→ΩF. Tensoring with Fthen gives
+(Ωγ)∗◦(Ωα)∗= (Ωδ)∗◦(Ωβ)∗: ΩC⊗CF→ΩF. Asthe compositionofmorphisms
+in (5.6) is (Ω γ)∗◦(Ωα)∗−(Ωδ)∗◦(Ωβ)∗, this implies (5.6) is a complex.
+For simplicity, first suppose C,D,E,Fare finitely presented. Use the nota-
+tion of Example 2.23 and the proof of Proposition 2.24, with exact seq uences
+(2.3) and (2.4), where I= (h1,...,hi)⊂C∞(Rl),J= (d1,...,dj)⊂C∞(Rm)
+andK= (e1,...,ek)⊂C∞(Rn). ThenLis given by (2.5). Applying the proof
+of Proposition 5.6 to (2.3)–(2.4) yields exact sequences of F-modules
+F⊗RRiǫ1/d47/d47F⊗RRlζ1/d47/d47ΩC⊗CF /d47/d470,(5.7)
+F⊗RRjǫ2/d47/d47F⊗RRmζ2/d47/d47ΩD⊗DF /d47/d470,(5.8)
+F⊗RRkǫ3/d47/d47F⊗RRnζ3/d47/d47ΩE⊗EF /d47/d470,(5.9)
+F⊗RRj+k+lǫ4/d47/d47F⊗RRm+n=F⊗RRm⊕F⊗RRnζ4/d47/d47ΩF/d47/d470,(5.10)
+where for (5.7)–(5.9) we have tensored (5.3) for C,D,EwithF.
+DefineF-module morphisms θ1:F⊗RRl→F⊗RRm,θ2:F⊗RRl→F⊗RRn
+byθ1(a1,...,al) = (b1,...,bm),θ2(a1,...,al) = (c1,...,cn) with
+bq=l/summationdisplay
+p=1Φ∂fp
+∂yq(ξ(y1),...,ξ(ym))·ap, cr=l/summationdisplay
+p=1Φ∂gp
+∂yr(ξ(z1),...,ξ(zn))·ap,
+48forap,bq,cr∈F. Now consider the diagram
+F⊗RRj⊕
+F⊗RRk⊕
+F⊗RRl ǫ 4=/parenleftBigǫ20θ1
+0ǫ3−θ2/parenrightBig/d47/d47
+(0 0ζ1)
+/d15/d15F⊗RRm⊕
+F⊗RRnζ4/d47/d47
+/parenleftBigζ20
+0ζ3/parenrightBig
+/d15/d15ΩF/d47/d47
+idΩF0
+ΩC⊗CF/parenleftbigg
+(Ωα)∗
+−(Ωβ)∗/parenrightbigg
+/d47/d47ΩD⊗DF⊕
+ΩE⊗EF((Ωγ)∗(Ωδ)∗)/d47/d47ΩF/d47/d470,(5.11)
+using matrix notation. The top line is the exact sequence (5.10), whe re the sign
+in−θ2comes fromthe sign of gpin the generators fp(y1,...,ym)−gp(z1,...,zn)
+ofLin (2.5). The bottom line is the complex (5.6).
+The left hand square commutes as ζ2◦ǫ2=ζ3◦ǫ3= 0 by exactness of (5.8)–
+(5.9)andζ2◦θ1= (Ωα)∗◦ζ1followsfrom α◦φ(xp) =ψ(fp), andζ3◦θ2= (Ωβ)∗◦ζ1
+follows from β◦φ(xp) =χ(gp). The right hand square commutes as ζ4and
+(Ωγ)∗◦ζ2act onF⊗RRmby (a1,...,am)/ma√sto→/summationtextm
+q=1aqdF◦ξ(yq), andζ4and
+(Ωδ)∗◦ζ3act onF⊗RRnby (b1,...,bn)/ma√sto→/summationtextn
+r=1brdF◦ξ(zr). Hence (5.11)
+is commutative. The columns are surjective since ζ1,ζ2,ζ3are surjective as
+(5.7)–(5.9) are exact and identities are surjective.
+The bottom right morphism/parenleftbig
+(Ωγ)∗(Ωδ)∗/parenrightbig
+in (5.11) is surjective as ζ4is
+and the right hand square commutes. Also surjectivity of the middle column
+implies that it maps Ker ζ4surjectively onto Ker/parenleftbig
+(Ωγ)∗(Ωδ)∗/parenrightbig
+. But Kerζ4=
+Imǫ4as the top row is exact, so as the left hand square commutes we see that/parenleftbig
+(Ωα)∗−(Ωβ)∗/parenrightbigTsurjects onto Ker/parenleftbig
+(Ωγ)∗(Ωδ)∗/parenrightbig
+, and the bottom row of (5.11)
+is exact. This proves the theorem for C,D,E,Ffinitely presented. For the
+general case we can use the same proof, but allowing i,j,k,l,m,n infinite.
+Here is an example of the situation of Theorem 5.8 for manifolds.
+Example 5.9. LetW,X,Y,Z,e,f,g,h be as in Theorem 3.5, so that (3.1) is
+a Cartesian square of manifolds and (3.2) a pushout square of C∞-rings. We
+have the following sequence of morphisms of vector bundles on W:
+0 /d47/d47(g◦e)∗(T∗Z)e∗(dg∗)⊕−f∗(dh∗)/d47/d47e∗(T∗X)⊕f∗(T∗Y)de∗⊕df∗
+/d47/d47T∗W /d47/d470.(5.12)
+Here dg:TX→g∗(TZ) is a morphism of vector bundles over X, and dg∗:
+g∗(T∗Z)→T∗Xis the dual morphism, and e∗(dg∗) : (g◦e)∗(T∗Z)→e∗(T∗X)
+is the pullback of this dual morphism to W.
+Sinceg◦e=h◦f, we have de∗◦e∗(dg∗) = df∗◦f∗(dh∗), and so (5.12) is a
+complex. As g,haretransverseand(3.1)isCartesian,(5.12)isexact. Sopassing
+to smooth sections in (5.12) we get an exact sequence of C∞(W)-modules:
+0 /d47/d47Γ∞/parenleftbig
+(g◦e)∗(T∗Z)/parenrightbig(e∗(dg∗)⊕
+−f∗(dh∗))∗/d47/d47Γ∞/parenleftbig
+e∗(T∗X)
+⊕f∗(T∗Y)/parenrightbig(de∗⊕
+df∗)∗/d47/d47Γ∞(T∗W) /d47/d470.
+The final four terms are the exact sequence (5.6) for the pushou t diagram (3.2).
+495.3 Sheaves of OX-modules on a C∞-ringed space (X,OX)
+We define sheaves of OX-modules on a C∞-ringed space, following [31, §II.5].
+Definition 5.10. Let (X,OX) be aC∞-ringed space. A sheaf ofOX-modules ,
+or simply anOX-module,EonXassigns a module E(U) over the C∞-ring
+OX(U) for each open set U⊆X, and a linear map EUV:E(U)→E(V) for
+each inclusion of open sets V⊆U⊆X, such that the following commutes
+OX(U)×E(U)
+ρUV×EUV/d15/d15µE(U)/d47/d47E(U)
+EUV/d15/d15
+OX(V)×E(V)µE(V)/d47/d47E(V),(5.13)
+and all this data E(U),EUVsatisfies the sheaf axioms in Definition 4.1.
+Amorphism of sheaves of OX-modulesφ:E→Fassigns a morphism of
+OX(U)-modulesφ(U) :E(U)→F(U) for each open set U⊆X, such that
+φ(V)◦EUV=FUV◦φ(U) for each inclusion of open sets V⊆U⊆X. Then
+OX-modules form an abelian category, which we write as OX-mod.
+AnOX-moduleEis called a vector bundle of rank nif we may cover Xby
+openU⊆XwithE|U∼=OX|U⊗RRn.
+In Definition 4.7 we defined finesheavesEon a topological space X. In§4.7
+we gave sufficient conditions for when a C∞-ringed space X= (X,OX) hasOX
+fine, which hold if Xis an affine C∞-scheme with XLindel¨ of. Now if OXis
+fine, then anyOX-moduleEis also fine, since partitions of unity in OXinduce
+partitions of unity in Hom(E,E).
+As in Voisin [69, Prop.4.36], a fundamental propertyof fine sheaves Eis that
+their cohomology groups Hi(E) are zero for all i >0. This means that H0is
+an exact functor on fine sheaves, rather than just left exact, s inceH1measures
+the failure of H0to be right exact. If Xis second countable then ( U,OX|U) is
+a Lindel¨ of affine C∞-scheme for all open U⊆X. Thus we deduce:
+Proposition 5.11. Let(X,OX)be an affine C∞-scheme with XLindel¨ of, and
+··· /d47/d47Eiφi/d47/d47Ei+1φi+1/d47/d47Ei+2 /d47/d47···
+be an exact sequence in OX-mod. Then
+··· /d47/d47Ei(X)φi(X)/d47/d47Ei+1(X)φi+1(X)/d47/d47Ei+2(X) /d47/d47···
+is an exact sequence of OX(X)-modules. If Xis also second countable then the
+following is an exact sequence of OX(U)-modules for all open U⊆X:
+··· /d47/d47Ei(U)φi(U)/d47/d47Ei+1(U)φi+1(U)/d47/d47Ei+2(U) /d47/d47···.
+Remark 5.12. Recall that a C∞-ringChas an underlying commutative R-
+algebra, and a module over Cis a module over this R-algebra, by Definitions 2.6
+and 5.1. Thus, by truncating the C∞-ringsOX(U) to commutative R-algebras,
+50regarded as rings, a C∞-ringed space ( X,OX) has an underlying ringed space
+in the usual sense of algebraic geometry [31, p. 72], [30, §0.4]. Our definition
+ofOX-modules are simply OX-modules on this underlying ringed space [31,
+§II.5], [30,§0.4.1]. Thus we can apply results from algebraic geometry without
+change, for instance that OX-mod is an abelian category, as in [31, p. 202].
+Definition 5.13. Letf= (f,f♯) : (X,OX)→(Y,OY) be a morphism of
+C∞-ringed spaces, and Ebe anOY-module. Define the pullbackf∗(E) by
+f∗(E) =f−1(E)⊗f−1(OY)OX, wheref−1(E) is as in Definition 4.5, a sheaf of
+modules over the sheaf of C∞-ringsf−1(OY) onX, and the tensor product uses
+the morphism f♯:f−1(OY)→OX. Ifφ:E→Fis a morphism of OY-modules
+we have a morphism of OX-modulesf∗(φ) =f−1(φ)⊗idOX:f∗(E)→f∗(F).
+Remark 5.14. Pullbacksf∗(E) are a kind of fibre product, and may be char-
+acterized by a universal property in OX-mod. So they should be regarded as
+beingunique up to canonical isomorphism , rather than unique. One can give
+an explicit construction for pullbacks, or use the Axiom of Choice to c hoose
+f∗(E) for allf,E, and so speak of ‘the’ pullback f∗(E). However, it may not be
+possible to make these choices strictly functorial in f.
+That is, if f:X→Y,g:Y→Zare morphisms and E∈OZ-mod then
+(g◦f)∗(E),f∗(g∗(E)) are canonically isomorphic in OX-mod, but may not be
+equal. We will write If,g(E) : (g◦f)∗(E)→f∗(g∗(E)) for these canonical
+isomorphisms, as in Remark 4.6(b). Then If,g: (g◦f)∗⇒f∗◦g∗is a natural
+isomorphism of functors. It is common to ignore this point and identif y (g◦f)∗
+withf∗◦g∗. Vistoli [68] makes careful use of natural isomorphisms ( g◦f)∗⇒
+f∗◦g∗in his treatment of descent theory.
+Whenfis the identity id X:X→XandE∈OX-mod we do not require
+id∗
+X(E) =E, but asEis a possible pullback for id∗
+X(E) there is a canonical
+isomorphism δX(E) : id∗
+X(E)→E, and then δX: id∗
+X⇒idOX-modis a natural
+isomorphism of functors.
+By Grothendieck [30, §0.4.3.1] we have:
+Proposition 5.15. LetX,YbeC∞-ringed spaces and f:X→Ya morphism.
+Then pullback f∗:OY-mod→OX-modis aright exact functor between
+abelian categories. That is, if Eφ−→Fψ−→G → 0is exact inOY-modthen
+f∗(E)f∗(φ)−→f∗(F)f∗(ψ)−→f∗(G)→0is exact inOX-mod.
+In generalf∗is not exact, or left exact, unless f:X→Yis flat.
+5.4 Sheaves on affine C∞-schemes, MSpecandΓ
+In§4.4 we defined Spec : C∞Ringsop→LC∞RS. In a similar way, if Cis a
+C∞-ring and (X,OX) = Spec Cwe can define MSpec : C-mod→OX-mod, a
+spectrum functor for modules.
+Definition 5.16. Let (X,OX) = Spec Cfor someC∞-ringCandMbe aC-
+module. We will define an OX-moduleE= MSpecM. For each open U⊆X,
+51defineE(U) to be the R-vector space of functions e:U→/coproducttext
+x∈U(M⊗CCx) with
+e(x)∈M⊗CCxfor allx∈U, and such that Umay be covered by open sets
+W⊆U⊆Xfor which there exist m∈Mwithe(x) =m⊗1∈M⊗CCxfor all
+x∈W. Here the Cx-moduleM⊗CCxis defined using the C-module structure
+onMand the projection πx:C→Cx.
+Definition 4.16 defines OX(U) as a set of functions U→/coproducttext
+x∈UCx. Define
+anOX(U)-module structure µE(U):OX(U)×E(U)→E(U) onE(U) by
+µE(U)(s,e) :x/ma√sto−→s(x)·e(x),
+for alls∈ OX(U),e∈ E(U) andx∈U. For open V⊆U⊆X, define
+EUV:E(U)→E(V) byEUV:e/ma√sto→e|V. It is now easy to check that Eis a sheaf
+ofOX-modules on X. Define MSpec M=EinOX-mod.
+An equivalent way to define MSpec Mis as the sheafification of the presheaf
+U/ma√sto→M⊗COX(U). The definition above performs the sheafification explicitly.
+Now letα:M→Nbe a morphism in C-mod, and setE= MSpecMand
+F= MSpecN. For each open U⊆X, defineλ(U) :E(U)→F(U) by
+λ(U)(e) :x/ma√sto→(α⊗id)(e(x)) forx∈U,
+whereα⊗id mapsM⊗CCx→N⊗CCx. It is easy to check that λ(U) is an
+OX(U)-module morphism and λ(V)◦EUV=FUV◦λ(U) :E(U)→F(V) for
+all openV⊆U⊆X. Henceλ:E→Fis a morphism in OX-mod. Define
+MSpecα=λ, so that MSpec α: MSpecM→MSpecN. This defines a functor
+MSpec : C-mod→OX-mod. It is an exact functor of abelian categories, since
+M/ma√sto→M⊗CCxis an exact functor C-mod→Cx-mod for each x∈X, as the
+localization πx:C→Cxis a flat morphism of R-algebras.
+Definition 5.17. LetCbe aC∞-ring, and ( X,OX) = Spec C. IfEis anOX-
+module thenE(X) is a module over OX(X), so using Ψ C:C→Γ(SpecC) =
+OX(X) we may regard E(X) as aC-module. Define Γ( E) to be the C-module
+E(X). Ifα:E → F is a morphism of OX-modules then Γ( α) :=α(X) :
+E(X)→F(X) is a morphism Γ( α) : Γ(E)→Γ(F) inC-mod. This defines the
+global sections functor Γ :OX-mod→C-mod.
+In general Γ is a left exact functor of abelian categories, but may n ot be
+right exact. However, if Xis Lindel¨ of (for example, if Cis finitely or countably
+generated) then Proposition 5.11 shows that Γ is an exact functor .
+Now Γ◦MSpec is a functor C-mod→C-mod. For each C-moduleMand
+m∈M, define Ψ M(m) :X→/coproducttext
+x∈XM⊗CCxby ΨM(m) :x/ma√sto→m⊗1Cx∈
+M⊗CCx. Then Ψ M(m)∈MSpecM(X) = Γ◦MSpecMby Definition 5.16, so
+ΨM:M→Γ◦MSpecMis a linear map, and in fact a C-module morphism.
+Itisfunctorialin M,sothattheΨ MforallMdefineanaturaltransformation
+Ψ : idC-mod⇒Γ◦MSpec of functors id C-mod,Γ◦MSpec : C-mod→C-mod.
+Here are the analogues of Lemma 4.18 and Theorem 4.20:
+Lemma 5.18. In Definition 5.16,the stalk (MSpecM)x=ExofMSpecMat
+x∈Xis naturally isomorphic to M⊗CCx,as modules over Cx∼=OX,x.
+52Proof.Elements ofExare∼-equivalence classes [ U,e] of pairs (U,e), whereU
+is an open neighbourhood of xinXande∈E(U), and (U,e)∼(U′,e′) if there
+exists open x∈V⊆U∩U′withe|V=e′|V. Define a Cx-module morphism
+Π :Ex→M⊗CCxby Π : [U,e]/ma√sto→e(x).
+Proposition 2.14 shows that Cx∼=C/IforIthe ideal in (2.2). Hence M⊗C
+Cx∼=M/(I·M), and thus every element of M⊗CCxis of the form m⊗1Cx
+for somem∈M. But ΨM(m)∈E(X), so that [ X,ΨM(m)]∈Ex, with Π :
+[X,ΨM(m)]/ma√sto→m⊗1Cx. Hence Π :Ex→M⊗CCxis surjective.
+Suppose [U,e]∈Exwith Π([U,e]) = 0∈M⊗CCx. Ase∈E(U), there exist
+openx∈V⊆Uandm∈Mwithe(x′) =m⊗1Cx′∈M⊗CCx′for allx′∈V.
+Thenm⊗1Cx=e(x) = Π([U,e]) = 0 inM⊗CCx, som∈I·M⊆M, and we
+may writem=/summationtextk
+a=1ia·maforia∈Iandma∈M. By (2.2) we may choose
+d1,...,dk∈Cwithx(da)/\e}atio\slash= 0 andia·da= 0 inCfora= 1,...,k.
+SetW={x′∈V:x′(da)/\e}atio\slash= 0, a= 1,...,k}, so thatWis an open
+neighbourhood of xinU. Ifx′∈Wthenx′(da)/\e}atio\slash= 0, soπx′(da) is invertible in
+Cx′. Butia·da= 0, soπx′(ia) = 0 inCx′fora= 1,...,k. Asm=/summationtextk
+a=1ia·ma
+it follows that e(x′) =m⊗1Cx′= 0 inM⊗CCx′for allx′∈W. Thuse|W= 0 in
+E(W), so [U,e] = [W,e|W] = 0 inEx. Therefore Π :Ex→M⊗CCxis injective,
+and so an isomorphism.
+Theorem 5.19. LetCbe aC∞-ring, and (X,OX) = Spec C. Then Γ :
+OX-mod→C-modisright adjoint toMSpec : C-mod→OX-mod. That
+is, for allM∈C-modandE∈OX-modthere are inverse bijections
+HomC-mod(M,Γ(E))LM,E/d47/d47HomOX-mod(MSpecM,E),
+RM,E/d111/d111 (5.14)
+which are functorial in M,E. WhenE= MSpecMwe have ΨM=RM,E(idE),
+so thatΨMis the unit of the adjunction between ΓandMSpec.
+Proof.LetM∈C-mod andE∈OX-mod, and setD= MSpecM. DefineRM,E
+in (5.14) by, for each morphism α:D→EinOX-mod, taking RM,E(α) :M→
+Γ(E) to be the composition
+MΨM/d47/d47Γ◦MSpecM= Γ(D)Γ(α)/d47/d47Γ(E).
+For the last part, if E= MSpecMthen ΨM=RM,E(idE) as Γ(id E) = idΓ(E).
+Letβ:M→Γ(E) be a morphism in C-mod. We will construct a morphism
+λ:D→EinOX-mod, and set LM,E(β) =λ. Letx∈X. Consider the diagram
+M⊗CC=M
+id⊗πx/d15/d15β/d47/d47Γ(E)
+σx/d15/d15
+M⊗CCx∼=Dxβx/d47/d47Ex(5.15)
+inC-mod, where the isomorphism M⊗CCx∼=Dxcomes from Lemma 5.18.
+HereExis the stalk ofEatx, andσx: Γ(E) =E(X)→Extakes stalks at
+53x. TheC-action on Γ(E) factors via CΨC−→OX(X), and the C-action onEx
+factors via CΨC−→OX(X)π−→OX,x, andβ,σxare both C-module morphisms.
+ButOX,x∼=Cxby Lemma 4.18, so σx◦β:M→Exis aC-module morphism,
+where the C-action onExfactors via Cπx−→Cx. Hence there is a unique OX,x-
+module morphism βx:Dx→Exmaking (5.15) commute.
+For each open U⊆X, defineλ(U) :D(U)→E(U) byλ(U)d:x/ma√sto→βx(d(x))
+ford∈D(U) andx∈U⊆X, andd(x)∈Dx, andβx(d(x))∈Ex. Here asE
+is a sheaf we may identify elements of E(U) with maps e:U→/coproducttext
+x∈UExwith
+e(x)∈Exforx∈U, such that esatisfies certain local conditions in U.
+Ifd∈D(U) = MSpec M(U) andx∈Uthen by Definition 5.16 we may
+coverUby openW⊆Ufor which there exist m∈Mwithd(x) =m⊗1Cxin
+M⊗CCxfor allx∈W. Therefore λ(U)dmapsx/ma√sto→σx(β(m)) for allx∈Wby
+(5.15), soλ(U)dis a section β(m)|WofEonW. Henceλ(U)dis a section of
+E|U, as suchWcoverU, andλ(U) :D(U)→E(U) is well defined.
+Asβxis anOX,x-module morphism for all x∈U,λ(U) :D(U)→E(U) is
+anOX(U)-module morphism. The definition of λ(U) is clearly compatible with
+restriction to open V⊆U⊆X. Thus the λ(U) for all open U⊆Xdefine a
+sheaf morphism λ:D→EinOX-mod. SetLM,E(β) =λ. This defines LM,Ein
+(5.14). A very similar proof to that of Theorem 4.20 shows that LM,E,RM,Eare
+inverse maps, so they are bijections, and that they are functoria l inM,E.
+We show that Γ is a right inverse for MSpec:
+Proposition 5.20. LetCbe aC∞-ring, and (X,OX) = Spec C,andEbe
+anOX-module. Set M= Γ(E)inC-mod,and write ΨE=LM,E(idM). Then
+ΨE: MSpec◦Γ(E)→Eis an isomorphism in OX-mod,for anyE.
+These isomorphisms ΨEare functorial inE,and so define a natural isomor-
+phismΨ : MSpec◦Γ⇒idOX-modof functorsOX-mod→OX-mod.
+Proof.SetD= MSpecM= MSpec◦Γ(E), and letx∈X. Then by definition
+of ΨE=LM,E(idM) :D→Ein the proof of Theorem 5.19, as in (5.15) the
+stalk map Ψ E,x:Dx→Exis the unique morphism of modules over Cx∼=OX,x
+making the following diagram of C-modules commute:
+M⊗CC=M
+id⊗πx/d15/d15idM/d47/d47M= Γ(E)
+σx/d15/d15
+M⊗CCx∼=DxΨE,x/d47/d47Ex.(5.16)
+Let [U,e]∈Ex, so thatx∈U⊆Xis open and e∈E(U). By Definition
+4.13 there exists c∈Csuch thatx(c)/\e}atio\slash= 0 andy(c) = 0 for all y∈X\U.
+Choose smooth f:R→Rsuch thatf= 0 near 0 in Randf= 1 nearx(c)
+inR. Setc′= Φf(c), where Φ f:C→Cis theC∞-ring operation. Then
+η= ΨC(c′)∈OX(X), and there exist open neighbourhoods VofX\UandW
+ofxinXwithη|V= 0 andη|W= 1. Clearly V∩W=∅, sox∈W⊆U. We
+haveη|U·e∈E(U), with (η|U·e)|U∩V= 0 and (η|U·e)|W=e|W.
+54Since{U,V}is an open cover of Xand (η|U·e)|U∩V= 0 = 0|U∩V, by the
+sheaf property of Ethere is a unique e′∈E(X) withe′|U=η|U·eande′|V= 0.
+Thene′|W= (η|U·e)|W=e|W. Thus
+σx(e′) = [X,e′] = [W,e′|W] = [W,e|W] = [U,e]
+inEx. Henceσx: Γ(E)→Exis surjective, so Ψ E,x:Dx→Exis surjective by
+(5.16), asπx:C→Cxis surjective by Proposition 2.14.
+Supposed∈Dxwith Ψ E,x(d) = 0. We may write m⊗1Cx∼=dunder
+the isomorphism M⊗CCx∼=Dxfor somem∈M, and then (5.16) gives
+σx(m) = ΨE,x(d) = 0. Hence there exists open x∈U⊆Xwithm|U= 0. As
+above we may construct η∈OX(X) and open V,W⊆XwithX\U⊆V,
+x∈W⊆U,η|V= 0 andη|W= 1. Then η·m= 0 inMasm|U= 0,η|V= 0
+withU∪V=X, andπx(η) = 1CxinCxasη= 1 nearxinX. Hence
+m⊗1Cx=1Cx·(m⊗1Cx)=πx(η)·(m⊗1Cx)=(η·m)⊗1Cx=0⊗1Cx=0
+inM⊗CCx. Therefore d= 0 inDx, and Ψ E,x:Dx→Exis injective, and so
+an isomorphism. As this holds for all x∈X, ΨE:D→Eis an isomorphism,
+proving the first part of the proposition. The second part follows f romLM,E
+functorial in M,Ein Theorem 5.19.
+As for quasicoherent sheaves in conventional algebraic geometry , we define:
+Definition 5.21. LetX= (X,OX) be aC∞-scheme, andEbe anOX-module.
+We callEquasicoherent if we may cover Xwith open U⊆Xsuch that
+(U,OX|U)∼=SpecCandE|U∼=MSpecMfor someC∞-ringCandC-moduleM.
+We write qcoh( X) for the category of quasicoherent sheaves on X.
+If (X,OX) is aC∞-scheme andEanOX-module, we can cover Xby open
+U⊆Xwith (U,OX|U)∼=SpecCaffine, and then Proposition 5.20 shows that
+E|U∼=MSpecMforM=E(U). Thus we have:
+Corollary 5.22. LetX= (X,OX)be aC∞-scheme. Then every OX-module
+Eis quasicoherent, so that qcoh(X) =OX-mod.
+Remark 5.23. (a) In conventional algebraic geometry, as in Hartshorne [31,
+§II.5], ifRis a ring and ( X,OX) = SpecRthe corresponding affine scheme, we
+also have functors MSpec : R-mod→OX-mod and Γ :OX-mod→R-mod. In
+C∞-algebraic geometry, as in Proposition 5.20, Γ is a right inverse for MS pec,
+but may not be a left inverse. But in algebraic geometry the opposite happens,
+as Γ is a left inverse for MSpec [31, Cor. II.5.5], but may not be a right in verse.
+The fact that Γ is a right inverseforMSpec in C∞-algebraicgeometrymeans
+that allOX-modules on a C∞-scheme (X,OX) are quasicoherent, so quasico-
+herence is not a very useful idea. But in algebraic geometry, as Γ is n ot a right
+inverse for MSpec, this is false: there are many examples of schemes ( X,OX)
+andOX-modulesEwhich are not quasicoherent. For instance, we may take
+X=A1andE(U) = 0 if 0∈U,E(U) =OX(U) if 0/∈Ufor all open U⊆X.
+55In§5.5 we will define a module Mover aC∞ringCto becomplete if
+M∼=Γ◦MSpecM. Then Γ is a left inverse for MSpec on the subcategory
+C-modco⊂C-mod of complete C-modules. In general C-modules need not be
+complete. But in conventional algebraic geometry, as Γ is a left inver se for
+MSpec allR-modules are complete, so completeness is not a useful idea.
+(b)Inconventionalalgebraicgeometryonedefines coherent sheaves [31,§II.5]to
+be quasicoherent sheaves Elocally modelled on MSpec MforMa finitely gener-
+atedC-module. However, coherent sheaves are only well behaved on noetherian
+schemes, and most interesting C∞-rings, such as C∞(Rn) forn >0, are not
+noetherian R-algebras. Because of this, coherent sheaves do not seem to be a
+useful idea in C∞-algebraic geometry (for instance, coh( X) is not closed under
+kernelsin qcoh( X), and is not an abelian category), and we do not discussthem.
+We can understand the pullback functor f∗in Definition 5.13 explicitly in
+terms of modules over the corresponding C∞-rings:
+Proposition 5.24. LetC,DbeC∞-rings,φ:D→Ca morphism, M,NbeD-
+modules, and α:M→Na morphism of D-modules. Write X= SpecC, Y=
+SpecD, f= Specφ:X→Y,andE= MSpecM,F= MSpecNinqcoh(Y).
+Then there are natural isomorphisms f∗(E)∼=MSpec(M⊗DC)andf∗(F)∼=
+MSpec(N⊗DC)inqcoh(X). These identify MSpec(α⊗idC) : MSpec(M⊗D
+C)→MSpec(N⊗DC)withf∗(MSpecα) :f∗(E)→f∗(F).
+Proof.WriteX= (X,OX),Y= (Y,OY) andf= (f,f♯). ThenEis the
+sheafificationofthepresheaf V/ma√sto→M⊗DOY(V), andf−1(E) isthe sheafification
+of the presheaf U/ma√sto→limV⊇f(U)E(V), andf−1(OY) is the sheafification of the
+presheafU/ma√sto→limV⊇f(U)OY(V). Inf∗(E) =f−1(E)⊗f−1(OY)OX, these three
+sheafifications combine into one, so f∗(E) is the sheafification of the presheaf
+U/ma√sto→limV⊇f(U)(M⊗DOY(V))⊗OY(V)OX(U). But
+(M⊗DOY(V))⊗OY(V)OX(U)∼=M⊗DOX(U)∼=(M⊗DC)⊗COX(U),
+sothis iscanonicallyisomorphictothepresheaf U/ma√sto→(M⊗DC)⊗COX(U) whose
+sheafification is MSpec( M⊗DC). This gives a natural isomorphism f∗(E)∼=
+MSpec(M⊗DC). The same holds for N. The identification of MSpec( α⊗idC)
+andf∗(MSpecα)followsbypassingfrommorphismsofpresheavestomorphisms
+of the associated sheaves.
+5.5 Complete modules over C∞-rings
+Here are the module analogues of Definition 4.35 and Theorem 4.36(b) ,(c).
+Definition 5.25. LetCbe aC∞-ring, andMaC-module. We call Mcomplete
+if ΨM:M→Γ◦MSpecMin Definition 5.17 is an isomorphism.
+WriteC-modcofor the full subcategory of complete C-modules in C-mod.
+IfMis aC-module then applying Γ to Proposition 5.20 shows that
+Γ(ΨMSpecM) : Γ◦MSpec(Γ◦MSpecM)−→Γ◦MSpecM
+56is an isomorphism. From the definitions we can show that Ψ Γ◦MSpecM=
+Γ(ΨMSpecM)−1. Thus Γ◦MSpecMis complete, for any C-moduleM. De-
+fine a functor Rco
+all= Γ◦MSpec : C-mod→C-modco.
+Theorem 5.26. LetCbe aC∞-ring, andX= (X,OX) = Spec C. Then
+(a)MSpec|C-modco:C-modco→qcoh(X)is an equivalence of categories.
+(b)Rco
+all:C-mod→C-modcois left adjoint to the inclusion functor inc :
+C-modco֒→C-mod. That is,Rco
+allis areflection functor .
+Proof.For (a), ifM,Nare complete C-modules then putting E= MSpecNin
+Theorem 5.19 and using Γ ◦MSpecN∼=N, equation (5.14) shows that
+MSpec =LM,E: Hom C-modco(M,N)−→HomOX-mod(MSpecM,MSpecN)
+isabijection, wherethedefinitionof LM,EagreeswiththedefinitionofMSpecon
+morphisms in this case. Thus MSpec is full and faithful on complete C-modules.
+IfE ∈OX-mod = qcoh( X) thenE∼=MSpec◦Γ(E) by Proposition 5.20.
+Thus Γ(E)∼=Γ◦MSpec◦Γ(E), so Γ(E) is complete by Definition 5.25. Hence
+E∼=MSpec|C-modco[Γ(E)], and the essential image of MSpec |C-modcois qcoh(X).
+Therefore MSpec |C-modcois an equivalence of categories.
+For (b), let M,NbeC-modules with Ncomplete. Then we have bijections
+HomC-modco/parenleftbig
+Rco
+all(M),N/parenrightbig∼=HomC-mod/parenleftbig
+Γ◦MSpecM,Γ◦MSpecN/parenrightbig
+∼=HomOX-mod/parenleftbig
+MSpec◦Γ◦MSpecM,MSpecN/parenrightbig
+∼=HomOX-mod/parenleftbig
+MSpecM,MSpecN/parenrightbig (5.17)
+∼=HomC-mod/parenleftbig
+M,Γ◦MSpecN/parenrightbig∼=HomC-mod/parenleftbig
+M,N/parenrightbig
+=Hom C-mod/parenleftbig
+M,inc(N)/parenrightbig
+,
+usingN∼=Γ◦MSpecNasNis complete in the first and fifth steps, Theorem
+5.19 in the second and fourth, and Proposition 5.20 in the third. The b ijections
+(5.17) arefunctorial in M,Naseach step is. Hence Rco
+allis left adjoint to inc.
+Proposition 5.27. LetCbe aC∞-ring and (X,OX) = Spec C,and suppose
+Xis Lindel¨ of. Then C-modcois closed under kernels, cokernels and extensions
+inC-mod,that is,C-modcois an abelian subcategory of C-mod.
+Proof.As in§5.4, MSpec : C-mod→OX-mod is an exact functor, and as Xis
+Lindel¨ of Γ :OX-mod→C-mod is also exact by Proposition 5.11. Hence Rco
+all=
+Γ◦MSpec : C-mod→C-mod is an exact functor. Let 0 →M1→M2→M3be
+exact in C-mod withM2,M3complete. Then we have a commutative diagram
+0 /d47/d47M1
+ΨM1/d15/d15/d47/d47M2
+ΨM2∼=/d15/d15/d47/d47M3
+ΨM3∼=/d15/d15
+0 /d47/d47Rco
+all(M1) /d47/d47Rco
+all(M2) /d47/d47Rco
+all(M3)
+inC-mod, where both rows are exact as Rco
+allis an exact functor, and the second
+and third columns are isomorphisms. Hence the first column is also an is omor-
+phism, and M1is complete, so C-modcois closed under kernels in C-mod. It is
+closed under cokernels and extensions by very similar arguments.
+57Example 5.28. LetCbe aC∞-ring with ( X,OX) = Spec C. Then:
+(a)Considering Cas aC-module, we have Γ ◦MSpecC= Γ◦SpecC=OX(X),
+and Ψ C:C→OX(X) in Definitions 4.19 and 5.17 coincide. Hence Cis
+complete as a C-module if and only if it is complete as a C∞-ring, in the
+sense of§4.6. So, if Cis a finitely generated but not fair C∞-ring, as in
+Examples 2.19 and 2.21, then Cis a non-complete C-module.
+(b)Suppose Cis complete and Xis Lindel¨ of. Let Mbe a finitely presented
+C-module, so we have an exact sequence C⊗Rm→C⊗Rn→M→0
+inC-mod. Here C⊗Rm,C⊗Rnare complete as Cis by(a), soMis
+complete by Proposition 5.27 as C-mod is closed under cokernels.
+(c)Suppose Cis complete, Xis Lindel¨ of, and I⊆Cis a finitely generated
+ideal. Choose generators i1,...,inforI. Then we have an exact sequence
+C⊗Rn→C→C/I→0 inC-mod with C⊗Rn,Ccomplete, so C/Iis a
+complete C-module by Proposition 5.27. Also we have an exact sequence
+0→I→C→C/IwithC,C/Icomplete, so Iis a complete C-module.
+(d)LetCbe complete and Vbe an infinite-dimensional R-vector space. One
+can show that C⊗RVis a complete C-module if and only if Xis compact.
+5.6 Cotangent sheaves of C∞-schemes
+We nowdefine cotangent sheaves , the sheafversionofcotangentmodules in §5.2.
+Definition 5.29. LetX= (X,OX) be aC∞-ringed space. Define PT∗Xto
+associate to each open U⊆Xthe cotangent module Ω OX(U)of Definition 5.3,
+regarded as a module over the C∞-ringOX(U), and to each inclusion of open
+setsV⊆U⊆Xthe morphism of OX(U)-modules Ω ρUV: ΩOX(U)→ΩOX(V)
+associated to the morphism of C∞-ringsρUV:OX(U)→OX(V). Then as we
+want for (5.13) the following commutes:
+OX(U)×ΩOX(U)
+ρUV×ΩρUV/d15/d15µOX(U)/d47/d47ΩOX(U)
+ΩρUV/d15/d15
+OX(V)×ΩOX(V)µOX(V)/d47/d47ΩOX(V).
+Using this and functoriality of cotangent modules Ω ψ◦φ= Ωψ◦Ωφin Definition
+5.3, we see thatPT∗Xis a presheaf ofOX-modules on X. Define the cotangent
+sheafT∗XofXto be the sheaf of OX-modules associated to PT∗X.
+IfU⊆Xis open then we have an equality of sheaves of OX|U-modules
+T∗(U,OX|U) =T∗X|U.
+As in Example 5.4, if f:X→Yis a smooth map of manifolds we have a
+morphism d f:f∗(T∗Y)→T∗Xof vector bundles over X. Here is an analogue
+forC∞-ringed spaces. Let f:X→Ybe a morphism of C∞-ringed spaces.
+Then by Definition 5.13, f∗(T∗Y) =f−1(T∗Y)⊗f−1(OY)OX,whereT∗Yis the
+58sheafification of the presheaf V/ma√sto→ΩOY(V), andf−1(T∗Y) the sheafification of
+the presheaf U/ma√sto→limV⊇f(U)(T∗Y)(V), andf−1(OY) the sheafification of the
+presheafU/ma√sto→limV⊇f(U)OY(V). These three sheafifications combine into one,
+so thatf∗(T∗Y) is the sheafification of the presheaf P(f∗(T∗Y)) acting by
+U/ma√sto−→P(f∗(T∗Y))(U) = limV⊇f(U)ΩOY(V)⊗OY(V)OX(U).
+Define a morphism of presheaves PΩf:P(f∗(T∗Y))→PT∗XonXby
+(PΩf)(U) = limV⊇f(U)(Ωρf−1(V)U◦f♯(V))∗,
+where (Ω ρf−1(V)U◦f♯(V))∗: ΩOY(V)⊗OY(V)OX(U)→ΩOX(U)= (PT∗X)(U) is
+constructed as in Definition 5.3 from the C∞-ring morphisms f♯(V) :OY(V)→
+OX(f−1(V)) fromf♯:OY→f∗(OX) corresponding to f♯infas in (4.3), and
+ρf−1(V)U:OX(f−1(V))→OX(U) inOX. Define Ω f:f∗(T∗Y)→T∗Xto be
+the induced morphism of the associated sheaves.
+Remark 5.30. There is an alternative definition of the cotangent sheaf T∗X
+following Hartshorne [31, p. 175]. We can form the product X×XinC∞RS,
+and there is a natural diagonal morphism ∆ X:X→X×X. WriteIXfor
+the sheaf of ideals in OX×Xvanishing on the closed C∞-ringed subspace ∆ X.
+ThenT∗X∼=∆∗
+X(IX/I2
+X). This can be proved using the equivalence of two
+definitions of cotangent module in [31, Prop. II.8.1A]. An affine version of this
+also appears in Dubuc and Kock [25].
+Proposition 5.31. LetCbe aC∞-ring andX= SpecC. Then there is a
+canonical isomorphism T∗X∼=MSpecΩ C.
+Proof.By Definitions 5.16 and 5.29, MSpecΩ CandT∗Xare sheafifications of
+presheavesPMSpecΩ C,PT∗X, where for open U⊆Xwe have
+PMSpecΩ C(U) = ΩC⊗COX(U) andPT∗X(U) = ΩOX(U).
+We haveC∞-ringmorphismsΨ C:C→OX(X) fromDefinition 4.19andrestric-
+tionρXU:OX(X)→OX(U) fromOX, and so as in Definition 5.3 a morphism
+ofOX(U)-modulesPρ(U) := (ρXU◦ΨC)∗: ΩC⊗COX(U)→ΩOX(U). This de-
+fines a morphism of presheaves Pρ:PMSpecΩ C→PT∗X, and so sheafifying
+induces a morphism ρ: MSpecΩ C→T∗X.
+The induced morphism on stalks at x∈Xisρx= (πx)∗: ΩC⊗CCx→ΩCx,
+whereπx:C→Cxisprojectiontothelocal C∞-ringCx, notingthatOX,x∼=Cx.
+ButCxis the localization C[c−1:c∈C,c(x)/\e}atio\slash= 0], so Proposition 5.7 implies
+that (πx)∗: ΩC⊗CCx→ΩCxis an isomorphism. Hence ρ: MSpecΩ C→T∗X
+is a sheaf morphism which induces isomorphisms on stalks at all x∈X, soρis
+an isomorphism.
+Here are some properties of the morphisms Ω fin Definition 5.29. Equation
+(5.20) is an analogue of (5.6) and (5.12).
+59Theorem 5.32. (a) Letf:X→Yandg:Y→Zbe morphisms of C∞-
+schemes. Then
+Ωg◦f= Ωf◦f∗(Ωg)◦If,g(T∗Z) (5.18)
+as morphisms (g◦f)∗(T∗Z)→T∗Xinqcoh(X). HereΩg:g∗(T∗Z)→T∗Yis a
+morphism in qcoh(Y),so applying f∗givesf∗(Ωg) :f∗(g∗(T∗Z))→f∗(T∗Y)in
+qcoh(X),andIf,g(T∗Z) : (g◦f)∗(T∗Z)→f∗(g∗(T∗Z))is as in Remark 5.14.
+(b)Suppose we are given a Cartesian square in C∞Sch
+Wf/d47/d47
+e/d15/d15Y
+h/d15/d15
+Xg/d47/d47Z,(5.19)
+so thatW=X×ZY. Then the following is exact in qcoh(W):
+(g◦e)∗(T∗Z)e∗(Ωg)◦Ie,g(T∗Z)⊕
+−f∗(Ωh)◦If,h(T∗Z)/d47/d47e∗(T∗X)
+⊕f∗(T∗Y)Ωe⊕Ωf/d47/d47T∗W /d47/d470.(5.20)
+Proof.Combining several sheafifications into one as in the proof of Propos ition
+5.24, we see that the sheaves T∗X,f∗(T∗Y),f∗(g∗(T∗Z)) and (g◦f)∗(T∗Z) on
+Xare isomorphic to the sheafifications of the following presheaves:
+T∗X/squigglerightU/ma√sto−→ΩOX(U), (5.21)
+f∗(T∗Y)/squigglerightU/ma√sto−→lim
+V⊇f(U)ΩOY(V)⊗OY(V)OX(U), (5.22)
+f∗(g∗(T∗Z))/squigglerightU/ma√sto−→lim
+V⊇f(U)lim
+W⊇g(V)/parenleftbig
+ΩOZ(W)⊗OZ(W)OY(V)/parenrightbig
+⊗OY(V)OX(U),(5.23)
+(g◦f)∗(T∗Z)/squigglerightU/ma√sto−→lim
+W⊇g◦f(U)ΩOZ(W)⊗OZ(W)OX(U). (5.24)
+Then Ωf,Ωg◦f,f∗(Ωg),If,g(T∗Z) are the morphisms of sheaves associated
+to the following morphisms of the presheaves in (5.21)–(5.24):
+Ωf/squigglerightU/ma√sto−→lim
+V⊇f(U)(Ωρf−1(V)U◦f♯(V))∗, (5.25)
+Ωg◦f/squigglerightU/ma√sto−→lim
+W⊇g◦f(U)(Ωρ(g◦f)−1(W)U◦(g◦f)♯(W))∗,(5.26)
+f∗(Ωg)/squigglerightU/ma√sto−→lim
+V⊇f(U)lim
+W⊇g(V)(Ωρg−1(W)V◦g♯(W))∗,(5.27)
+If,g(T∗Z)/squigglerightU/ma√sto−→lim
+V⊇f(U)lim
+W⊇g(V)IUVW, (5.28)
+by Definition 5.29, where IUVW: ΩOZ(W)⊗OZ(W)OX(U)→/parenleftbig
+ΩOZ(W)⊗OZ(W)
+OY(V)/parenrightbig
+⊗OY(V)OX(U) is the natural isomorphism.
+Now ifU⊆X,V⊆Y,W⊆Zare open with V⊇f(U),W⊇g(V) then
+ρ(g◦f)−1(W)U◦(g◦f)♯(W) =/bracketleftbig
+ρf−1(V)U◦f♯(V)/bracketrightbig
+◦/bracketleftbig
+ρg−1(W)V◦g♯(W)/bracketrightbig
+60as morphismsOZ(W)→OX(U), so Ωφ◦ψ= Ωφ◦Ωψin Definition 5.3 implies
+(Ωρ(g◦f)−1(W)U◦(g◦f)♯(W))∗= (Ωρf−1(V)U◦f♯(V))∗◦(Ωρg−1(W)V◦g♯(W))∗◦IUVW.
+Taking limits lim V⊇f(U)limW⊇g(V)implies that the morphisms of presheaves in
+(5.25)–(5.28) satisfy the analogue of (5.18), so passing to sheave s proves (a).
+For (b), first observe that as (5.19) is commutative, by (a) we hav e
+Ωe◦e∗(Ωg)◦Ie,g(T∗Z) = Ωg◦e= Ωh◦f= Ωf◦f∗(Ωh)◦If,h(T∗Z),
+so Ωe◦/parenleftbig
+e∗(Ωg)◦Ie,g(T∗Z)/parenrightbig
+−Ωf◦/parenleftbig
+f∗(Ωh)◦If,h(T∗Z)/parenrightbig
+= 0,
+and (5.20) is a complex. To show it is exact, note that as in the first pa rt
+of the proof, (5.20) is the sheafification of a complex of presheave s, and the
+presheaves are defined as direct limits. Let S⊆Wbe open. Then the complex
+ofpresheavescorrespondingto (5.20) evaluatedat S⊆Wis the directlimit over
+all openT⊆X,U⊆Y,V⊆Zwithe(S)⊆T,f(S)⊆U,g(T)⊆V,h(U)⊆V
+of equation (5.6) with OZ(V),OX(T),OY(U),OW(S) in place of C,D,E,F.
+Since (5.6) is exact by Theorem 5.8 and direct limits are exact, the com plex
+ofpresheaveswhose sheafificationis (5.20) is exact when evaluate d on each open
+S⊆W, so it is exact. As sheafification is an exact functor, this implies that
+equation (5.20) is exact. This completes the proof.
+6C∞-stacks
+We now discuss C∞-stacks, that is, geometric stacks over the site ( C∞Sch,J)
+ofC∞-schemes with the open cover topology. The author knows of no pr evious
+work on these. For the rest of the book, we will assume the reader has some
+familiarity with stacks in algebraicgeometry. Appendix A summarizes t he main
+definitions and results on stacks that we will use, but it is too brief to help
+someone learn about stacks for the first time. Readers with little ex perience
+of stacks are advised to first consult an introductory text such a s Vistoli [68],
+Gomez [29], Laumon and Moret-Bailly [46], or the online ‘Stacks Project ’ [34].
+The author found Metzler [49] and Noohi [55] useful in writing this section.
+6.1C∞-stacks
+We use the material of §A.2–§A.5.
+Definition 6.1. Define a Grothendieck pretopology PJon the category of
+C∞-schemes C∞Schto have coverings {ia:Ua→U}a∈AwhereVa=ia(Ua)
+is open inUwithia:Ua→(Va,OU|Va) and isomorphism for all a∈A, and
+U=/uniontext
+a∈AVa. Using Corollary 4.29 we see that up to isomorphisms of the
+Ua, the coverings{ia:Ua→U}a∈AofUcorrespond exactly to open covers
+{Va:a∈A}ofU. WriteJfor the associated Grothendieck topology.
+It is a straightforward exercise in sheaf theory to prove:
+61Proposition6.2. The site(C∞Sch,J)has descent for objects and morphisms,
+in the sense of§A.3. Thus it is subcanonical.
+The point here is that since coverings of UinJare just open covers of the
+underlying topological space U, rather than something more complicated like
+´ etale covers in algebraic geometry, proving descent is easy: for o bjects, we glue
+the topological spaces XaofXatogether in the usual way to get a topological
+spaceX, then we glue the OXatogether to get a presheaf of C∞-rings˜OXon
+Xisomorphic toOXaonXa⊆Xfor alla∈A, and finally we sheafify ˜OXto a
+sheaf ofC∞-ringsOXonX, which is still isomorphic to OXaonXa⊆X.
+Definition 6.3. AC∞-stackXis a geometric stack on the site ( C∞Sch,J).
+WriteC∞Stafor the 2-category of C∞-stacks,C∞Sta=GSta(C∞Sch,J).
+As in Definition A.13, we will very often use the notation that if Xis a
+C∞-scheme then ¯Xis the associated C∞-stack, and if f:X→Yis a mor-
+phism ofC∞-schemes then ¯f:¯X→¯Yis the associated 1-morphism of C∞-
+stacks. Write ¯C∞Schlfp,¯C∞Schlf,¯C∞Schfor the full 2-subcategories of C∞-
+stacksXinC∞Stawhich are equivalent to ¯XforXinC∞Schlfp,C∞Schlf
+orC∞Sch, respectively. When we say that a C∞-stackXis aC∞-scheme, we
+mean thatX∈¯C∞Sch.
+Since(C∞Sch,J)isasubcanonicalsite, theembedding C∞Sch→C∞Sta
+takingX/ma√sto→¯X,f/ma√sto→¯fis fully faithful. We write this as a full and faithful
+functorFC∞Sta
+C∞Sch:C∞Sch→C∞StamappingFC∞Sta
+C∞Sch:X/ma√sto→¯Xon objects
+andFC∞Sta
+C∞Sch:f/ma√sto→¯fon (1-)morphisms. Hence ¯C∞Schlfp,¯C∞Schlf,¯C∞Sch
+areequivalentto C∞Schlfp,C∞Schlf,C∞Sch, consideredas 2-categorieswith
+only identity 2-morphisms. In practice one often does not distinguis h between
+schemes and stacks which are equivalent to schemes, that is, one id entifies
+C∞Schlfp,...,C∞Schand¯C∞Schlfp,...,¯C∞Sch.
+Remark 6.4. Behrend and Xu [5, Def. 2.15] use ‘ C∞-stack’to mean something
+different, a stack Xover the site ( Man,JMan) of manifolds with Grothendieck
+topologyJManassociated to the Grothendieck pretopology PJMangiven by
+opencovers,suchthatthereexistsasurjectiverepresentable submersion π:¯U→
+Xfrom some manifold U. These arealsocalled ‘smooth stacks’or‘differentiable
+stacks’in[5,32,49,55]. Thequotient[ V/G]ofamanifold VbyaLiegroup Gisan
+example of a differentiable stack. By Zung’s linearization theorem [71 , Th. 2.3],
+a differentiable stack Xwith proper diagonal is Zariski locally equivalent to
+such a quotient [ V/G] withGcompact. Our C∞-stacks are a far larger class of
+more singular objects than the differentiable stacks of [5,32,49, 55].
+Theorems 4.25(b) and A.23, Corollary A.26 and Proposition 6.2 imply:
+Theorem 6.5. LetXbe aC∞-stack. ThenXis equivalent to the stack
+[V⇒U]associated to a groupoid (U,V,s,t,u,i,m)inC∞Sch. Conversely,
+any groupoid in C∞Schdefines aC∞-stack[V⇒U]. All fibre products exist
+in the2-category C∞Sta.
+QuotientC∞-stacks[X/G] are a special class of C∞-stacks.
+62Definition 6.6. AC∞-groupGis a group object in C∞Sch, that is, a C∞-
+schemeG= (G,OG) equipped with an identity element 1 ∈Gand multiplica-
+tion and inverse morphisms m:G×G→G,i:G→GinC∞Schsuch that
+(∗,G,π,π,1,i,m) is a groupoid in C∞Sch. Here∗= SpecRis a point, and
+π:G→∗is the projection, and we regard 1 ∈Gas a morphism 1 : ∗→G.
+LetGbe aC∞-group, and XaC∞-scheme. A ( left)actionofGonXis a
+morphismµ:G×X→Xsuch that
+/parenleftbig
+X,G×X,πX,µ,1×idX,(i◦πG)×µ,(m◦((πG◦π1)×(πG◦π2)))×(πX◦π2)/parenrightbig
+(6.1)
+is a groupoid object in C∞Sch, where in the final morphism π1,π2are the
+projections from ( G×X)×πX,X,µ(G×X) to the first and second factors
+G×X. Then define the quotientC∞-stack[X/G] to be the stack [ G×X⇒X]
+associated to the groupoid (6.1). It is a C∞-stack.
+IfG= (G,OG) is aC∞-group then the underlying space Gis a topological
+group, and is in particular a group, and if G= (G,OG) acts onX= (X,OX)
+thenGacts continuously on X.
+IfGis a Lie group then G=FC∞Sch
+Man(G) is aC∞-group in a natural way, by
+applyingFC∞Sch
+Mantothesmoothmultiplicationandinversemaps m:G×G→G
+andi:G→G. If a Lie group Gacts smoothly on a manifold Xwith action
+µ:G×X→Xthen theC∞-groupG=FC∞Sch
+Man(G) acts on the C∞-scheme
+X=FC∞Sch
+Man(X) with action µ=FC∞Sch
+Man(µ) :G×X→X, so we can form
+the quotient C∞-stack [X/G].
+Example 6.7. LetGbe aC∞-group, and X=∗be the point in C∞Sch, with
+trivialG-action. The quotient C∞-stack [∗/G] is known as BG, the classifying
+stack for principal G-bundles on C∞-schemes.
+IfSis aC∞-scheme, a principalG-bundle(P,π,µ) overSis aC∞-scheme
+P, a morphism π:P→S, and aG-actionµ:G×P→PofGonP, such
+thatπisG-invariant, and Smay be covered by open C∞-subschemes U⊆S
+such that there exists an isomorphism π−1(U)∼=G×Uwhich identifies the
+G-action onπ−1(U)⊆Pwith the product of the left G-action onGand the
+trivialG-action onU, and identifies π|···:π−1(U)→UwithπU:G×U→U.
+Often we write Pas the principal bundle, leaving π,µimplicit.
+One well known way to write BGexplicitly as a category fibred in groupoids
+pX:X→C∞Sch, as in§A.2, is to defineXto be the category with objects
+pairs (S,P) of aC∞-schemeSandPa principal G-bundle over S, and mor-
+phisms (f,u) : (S,P)→(T,Q) consisting of C∞-scheme morphisms f:S→T
+andu:P→Q, such that uisG-equivariant and
+P
+π/d15/d15u/d47/d47Q
+π/d15/d15
+Sf/d47/d47T(6.2)
+is a Cartesian square in C∞Sch, which implies that Pis canonicallyisomorphic
+to the pullback principal G-bundlef∗(Q). Composition of morphisms is ( g,v)◦
+63(f,u) = (g◦f,v◦u), and identity morphisms are id (S,P)= (idS,idP). The
+functorpX:X→C∞SchmapspX: (S,P)/ma√sto→Sonobjectsand pX: (f,u)/ma√sto→f
+on morphisms.
+In§7.1 we will give a more detailed treatment of quotient C∞-stacks [X/G]
+of aC∞-schemeXby a finite group G.
+6.2 Properties of 1-morphisms of C∞-stacks
+We use the material of §A.4. We define some classes of C∞-scheme morphisms.
+Definition 6.8. Letf= (f,f♯) :X= (X,OX)→Y= (Y,OY) be a morphism
+inC∞Sch. Then:
+•We callfanopen embedding ifV=f(X) is an open subset in Yand
+(f,f♯) : (X,OX)→(V,OY|V) is an isomorphism.
+•We callfaclosed embedding iff:X→Yis a homeomorphism with
+a closed subset of Y, andf♯:f−1(OY)→OXis a surjective morphism
+of sheaves of C∞-rings. Equivalently, fis an isomorphism with a closed
+C∞-subscheme of Y. Over affine open subsets U∼=SpecCinY,fis
+modelled on the natural morphism Spec( C/I)֒→SpecCfor some ideal I
+inC.
+•We callfanembedding if we may write f=g◦hwherehis an open
+embedding and gis a closed embedding.
+•We callf´ etaleif eachx∈Xhas an open neighbourhood UinXsuch
+thatV=f(U) is open in Yand (f|U,f♯|U) : (U,OX|U)→(V,OY|V) is
+an isomorphism. That is, fis a local isomorphism.
+•We callfproperiff:X→Yis a proper map of topological spaces, that
+is, ifS⊆Yis compact then f−1(S)⊆Xis compact.
+•We say that fhas finite fibres iff:X→Yis a finite map, that is, f−1(y)
+is a finite subset of Xfor ally∈Y.
+•We callfseparated iff:X→Yis a separated map of topological spaces,
+that is, ∆ X=/braceleftbig
+(x,x) :x∈X/bracerightbig
+is a closed subset of the topological fibre
+productX×f,Y,fX=/braceleftbig
+(x,x′)∈X×X:f(x) =f(x′)/bracerightbig
+.
+•We callfclosediff:X→Yis a closed map of topological spaces, that
+is,S⊆Xclosed implies f(S)⊆Yclosed.
+•We callfuniversally closed if whenever g:W→Yis a morphism then
+πW:X×f,Y,gW→Wis closed.
+•We callfasubmersion if for allx∈Xwithf(x) =y, there exists an open
+neighbourhood UofyinYand a morphism g= (g,g♯) : (U,OY|U)→
+(X,OX) withg(y) =xandf◦g= id(U,OY|U).
+64•We callflocally fair , orlocally finitely presented , if whenever Uis a locally
+fair, or locally finitely presented C∞-scheme, respectively, and g:U→Y
+is a morphism then X×f,Y,gUis locally fair, or locally finitely presented,
+respectively.
+Remark 6.9. These are mostly analogues of standard concepts in algebraic
+geometry, as in Hartshorne [31] for instance. But because the to pology onC∞-
+schemes is finer than the Zariski topology in algebraic geometry — fo r example,
+affineC∞-schemes are Hausdorff — our definitions of ´ etale and proper are s im-
+pler than in algebraic geometry. (Open or closed) embeddings corre spond to
+(open or closed) immersions in algebraic geometry, but we prefer th e word ‘em-
+bedding’, as immersion has a different meaning in differential geometry . Closed
+morphisms are not invariant under base change, which is why we defin e univer-
+sally closed. If X,Yare manifolds and X,Y=FC∞Sch
+Man(X,Y), thenf:X→Y
+is a submersion of C∞-schemes if and only if f=FC∞Sch
+Man(f) forf:X→Ya
+submersion of manifolds.
+Definition 6.10. LetPbe a property of morphisms in C∞Sch. We say that
+Pis stable under open embedding if whenever f:U→VisPandi:V→W
+is an open embedding, then i◦f:U→WisP.
+The next proposition is elementary. See Laumon and Bailly [46, §3.10] and
+Noohi [55, Ex. 4.6] for similar lists for the ´ etale and topological sites .
+Proposition 6.11. The following properties of morphisms in C∞Schare in-
+variant under base change and local in the target in the site (C∞Sch,J),in
+the sense of§A.4:open embedding, closed embedding, embedding, ´ etale, prop er,
+has finite fibres, separated, universally closed, submersio n, locally fair, locally
+finitely presented. The following properties are also stabl e under open embed-
+ding, in the sense of Definition 6.10:open embedding, embedding, ´ etale, has
+finite fibres, separated, submersion, locally fair, locally finitely presented.
+As in§A.4, this implies that these properties are also defined for repre-
+sentable 1-morphisms in C∞Sta. In particular, if Xis aC∞-stack then ∆ X:
+X →X×X is representable, and if Π : ¯U→Xis an atlas then Π is repre-
+sentable, so we can require that ∆ Xor Π has some of these properties.
+Definition 6.12. LetXbe aC∞-stack. Following [46, Def. 7.6], we say that X
+isseparated if the diagonal 1-morphism ∆ X:X→X×X is universally closed.
+IfX=¯Xfor someC∞-schemeX= (X,OX) thenXis separated if and only if
+∆X:X→X×Xis closed, that is, if and only if Xis Hausdorff.
+Proposition 6.13. LetW=X×f,Z,gYbe a fibre product of C∞-stacks with
+X,Yseparated. ThenWis separated.
+65Proof.We have a 2-commutative diagram with both squares 2-Cartesian:
+W∆W/d47/d47
+/d117/d117❦❦❦❦❦❦❦❦❦❦❦π1/d41/d41❙❙❙❙❙❙❙❙❙ W×W
+/d41/d41❙❙❙❙❙❙❙
+Z
+Z /d41/d41❙❙❙❙❙❙❙❙❙❙❙ X×f◦∆Z,Z×Z,g◦ZY
+/d117/d117❦❦❦❦❦❦❦❦
+/d41/d41❙❙❙❙❙❙π2/d53/d53❦❦❦❦❦❦❦
+X×X×Y×Y .
+IZ X×Y∆X×∆Y/d53/d53❦❦❦❦❦❦❦(6.3)
+Let [V⇒U] be a groupoid presentation of Z, and consider the fourth 2-
+Cartesian diagram of (A.12), with surjective rows. The left hand mo rphism
+¯uׯidUhas a left inverse πU, and so is automatically universally closed. Hence
+Zis universally closed by Propositions A.18(c) and 6.11, so π1in (6.3) is uni-
+versally closed by Propositions A.18(a) and 6.11. Also ∆ X,∆Yare universally
+closed asX,Yare separated, so ∆ X×∆Yin (6.3) is universally closed, and
+π2is universally closed. Thus ∆ W∼=π2◦π1is universally closed, and Wis
+separated.
+6.3 Open C∞-substacks and open covers
+Definition 6.14. LetXbe aC∞-stack. AC∞-substackYinXis a substack
+ofX, in the sense of Definition A.7, which is also a C∞-stack. It has a natural
+inclusion 1-morphism iY:Y֒→X. We callYanopenC∞-substack ofXif
+iYis a representable open embedding, a closedC∞-substack ofXifiYis a
+representable closed embedding, and a locally closed C∞-substack ofXifiYis
+a representable embedding.
+Anopen cover{Ua:a∈A}ofXis a family of open C∞-substacksUainX
+with/coproducttext
+a∈AiUa:/coproducttext
+a∈AUa→Xsurjective. We write U⊆XwhenUis an open
+C∞-substack ofX, and/uniontext
+a∈AU=Xto mean that/coproducttext
+a∈AiUais surjective.
+Somepropertiesof∆ X,ιX,XandatlasesforXcanbetestedontheelements
+of an open cover. The proof is elementary.
+Proposition 6.15. LetXbe aC∞-stack, and{Ua:a∈A}an open cover
+ofX. Suppose PandQare properties of morphisms in C∞Schwhich are
+invariant under base change and local in the target in (C∞Sch,J),and that P
+is stable under open embedding. Then:
+(a)LetΠa:¯Ua→Uabe an atlas forUafora∈A. SetU=/coproducttext
+a∈AUaand
+Π =/coproducttext
+a∈AiUa◦Πa:¯U→X. ThenΠis an atlas forX,andΠisPif
+and only if ΠaisPfor alla∈A.
+(b)∆X:X→X×X isPif and only if ∆Ua:Ua→Ua×UaisPfor alla∈A.
+(c)ιX:IX→XisQif and only if ιUa:IUa→UaisQfor alla∈A.
+(d)X:X→IXisQif and only if Ua:Ua→IUaisQfor alla∈A.
+IfX=¯Ufor someC∞-schemeU= (U,OU), then the open C∞-substacks
+ofXare precisely those subsheaves of the form (V,OU|V) for all open V⊆U,
+that is, they are the images in C∞Staof the open C∞-subschemes of U. We
+can also describe the open substacks of stacks [ V⇒U] associated to groupoids:
+66Proposition 6.16. Let(U,V,s,t,u,i,m)be a groupoid in C∞SchandX=
+[V⇒U]the associated C∞-stack, and write U= (U,OU),and so on. Then open
+C∞-substacksX′ofXare naturally in 1-1correspondence with open subsets
+U′⊆Uwiths−1(U′) =t−1(U′),whereX′= [V′⇒U′]forU′= (U′,OU|U′)
+andV′= (s−1(U′),OV|s−1(U′)). If(U,V,s,t,u,i,m)is as in(6.1),so thatX
+is a quotient C∞-stack[U/G],then openC∞-substacksX′ofXcorrespond to
+G-invariant open subsets U′⊆U.
+Proof.From Theorem A.23, as X= [V⇒U] we have a natural surjective,
+representable 1-morphism Π : ¯U→X. IfX′is an openC∞-substack ofXthen
+¯U×Π,X,iX′X′is an open C∞-substack of ¯U, and so is of the form (U′,OU|U′)
+for some open U′⊆U. We have natural equivalences
+(s−1(U′),OV|s−1(U′))≃¯U′×i¯U′,¯U,¯s¯V≃X′×X(¯U×id¯U,¯U,¯s¯V)≃X′×i′
+X,X,πX¯V
+≃X′×X(¯U×id¯U,¯U,¯t¯V)≃¯U′×i¯U′,¯U,¯t¯V≃(t−1(U′),OV|t−1(U′)),
+by associativity properties of fibre products in 2-categories, whic h implies that
+s−1(U′) =t−1(U′). Conversely, if s−1(U′) =t−1(U′) then defining U′,V′as in
+the proposition, we get a C∞-stackX′= [V′⇒U′] which is naturally an open
+C∞-substack ofX. WhenX= [U/G], we see that s−1(U′) =t−1(U′) if and
+only ifU′isG-invariant.
+6.4 The underlying topological space of a C∞-stack
+Following Noohi [55, §4.3,§11] in the case of topological stacks, we associate a
+topological space Xtopto aC∞-stackX. In§7.4, ifXis a Deligne–Mumford
+C∞-stack, we will also give Xtopthe structure of a C∞-scheme.
+Definition 6.17. LetXbe aC∞-stack. Write∗for the point Spec Rin
+C∞Sch, and¯∗for the associated point in C∞Sta. DefineXtopto be the
+set of 2-isomorphism classes [ x] of 1-morphisms x:¯∗→X.
+SupposeU⊆Xis an openC∞-substack. SinceUis a subcategory of X, any
+1-morphism u:¯∗→U, regardedasafunctorfromthecategory ¯∗tothe category
+U, is also a 1-morphism u:¯∗→X. Also, asUis a strictly full subcategory of
+X, ifx:¯∗→Xis a 1-morphism and η:u⇒xa 2-morphism of 1-morphisms
+¯∗→X, thenxis also a 1-morphism u:¯∗→U, andηis also a 2-morphism of
+1-morphisms ¯∗→U. This implies that Utopis a subset ofXtop.
+DefineTXtop=/braceleftbig
+Utop:U⊆XisanopenC∞-substackinX/bracerightbig
+, asetofsubsets
+ofXtop. We claimthatTXtopisatopologyonXtop. Toseethis, notethattaking
+Uto beXor the empty C∞-substack givesXtop,∅∈TXtop. IfU,V⊆Xare
+openC∞-substacks ofXthen the intersection of subcategories W=U∩Vis
+an openC∞-substack ofXequivalent to the fibre product U×iU,X,iVV, with
+Wtop=Utop∩Vtop, soTXtopis closed under finite intersections.
+If{Ua:a∈A}is a family of open C∞-substacks inX, defineVto be the
+unique smallest strictly full subcategory of Xwhich containsUafor eacha∈A
+and is closed under the stack axiom (A.9) in Definition A.6. Then Vis an open
+67C∞-substack ofX, which we write as V=/uniontext
+a∈AUa, andVtop=/uniontext
+a∈AUatop.
+SoTXtopis closed under arbitrary unions.
+Thus (Xtop,TXtop) is a topological space, which we call the underlying topo-
+logical space ofX, and usually write as Xtop. It has the following properties.
+Iff:X →Y is a 1-morphism of C∞-stacks then there is a natural continu-
+ous mapftop:Xtop→Ytopdefined by ftop([x]) = [f◦x]. Iff,g:X →Y
+are 1-morphisms and η:f⇒gis a 2-isomorphism then ftop=gtop. Map-
+pingX /ma√sto→X top,f/ma√sto→ftopand 2-morphisms to identities defines a 2-functor
+FTop
+C∞Sta:C∞Sta→Top, where the category of topological spaces Topis
+regarded as a 2-category with only identity 2-morphisms.
+IfX= (X,OX) is aC∞-scheme, so that ¯Xis aC∞-stack, then ¯Xtopis
+naturallyhomeomorphicto X, and we will identify ¯XtopwithX. Iff= (f,f♯) :
+X= (X,OX)→Y= (Y,OY) is a morphism of C∞-schemes, so that ¯f:¯X→¯Y
+is a 1-morphism of C∞-stacks, then ¯ftop:¯Xtop→¯Ytopisf:X→Y.
+For aC∞-stackX, we can characterize Xtopby the following universal
+property. We are given a topological space Xtopand for every 1-morphism
+f:¯U→Xfor aC∞-schemeU= (U,OU) we are given a continuous map
+ftop:U→Xtop, such that if fis 2-isomorphic to h◦¯gfor some morphism
+g= (g,g♯) :U→Vand 1-morphism h:V→Xthenftop=htop◦g. IfX′
+top,
+f′
+topare alternative choices of data with these properties then there is a unique
+continuous map j:Xtop→X′
+topwithf′
+top=j◦ftopfor allf.
+We can also make Xtopinto aC∞-ringed spaceXtop:
+Definition 6.18. LetXbe aC∞-stack. Define a sheaf of C∞-ringsOXtop
+onXtopas follows: each open set in XtopisUtopfor some unique open C∞-
+substackU ⊆X. DefineOXtop(Utop) to be the set of 2-isomorphism classes
+[c] of 1-morphisms c:U →¯R. Iff:Rn→Ris smooth and [ c1],...,[cn]∈
+OXtop(Utop), define Φ f/parenleftbig
+[c1],...,[cn]/parenrightbig
+=/bracketleftbig¯f◦(c1×···×cn)/bracketrightbig
+, using the compo-
+sitionUc1×···×cn−→¯R×···×¯R¯f−→¯R. ThenOXtop(Utop) is aC∞-ring.
+IfVtop⊆Utop⊆Xtopare open, so that V ⊆U ⊆X , define aC∞-ring
+morphismρUV:OXtop(Utop)→OXtop(Vtop) byρUV: [c]/ma√sto→[c|V]. It is now
+easy to check that OXtopis a presheaf of C∞-rings onXtop, but it is less
+obvious that it is a sheaf. To see this, note that by general proper ties of stacks,
+U /ma√sto→Hom(U,¯R) is a 2-sheaf (stack) of groupoids on the topological space
+Xtop, whereHom(U,¯R) is the groupoid of 1- and 2-morphisms U →¯R, and
+OXtop(Utop) is its set of isomorphism classes.
+Starting with a 2-sheaf and taking sets of isomorphism classes gene rally
+yields only a presheaf of sets, not a sheaf. But as ¯Ris aC∞-scheme the
+groupoids Hom(U,¯R) are discrete (have no nontrivial automorphisms), so tak-
+ing isomorphism classes loses no information, and the 2-sheaf prope rty implies
+thatOXtopis a sheaf of sets, and so of C∞-rings. ThusXtop= (Xtop,OXtop) is
+aC∞-ringed space, the underlying C∞-ringed space ofX.
+For generalXthisXtopneed not be a C∞-scheme. If it is, we call Xtopthe
+coarse moduli C∞-scheme ofX. Coarse moduli C∞-schemes have the following
+universal property: there is a 1-morphism π:X →¯Xtopcalled the structural
+68morphism , such that if f:X→¯Yis a 1-morphism for any C∞-schemeYthen
+fis 2-isomorphic to ¯ g◦πfor some unique C∞-scheme morphism g:Xtop→Y.
+We can think of a C∞-stackXas being a topological space Xtopequipped
+with some complicated extra geometrical structure, just as manif olds and orb-
+ifolds are usually thought of as topological spaces equipped with ext ra structure
+coming from an atlas of charts. As in Noohi [55, Ex. 4.13], it is easy to d escribe
+Xtopusing a groupoid presentation [ V⇒U] ofX:
+Proposition 6.19. LetXbe equivalent to the C∞-stack[V⇒U]associated to
+a groupoid (U,V,s,t,u,i,m)inC∞Sch,whereU= (U,OU),s= (s,s♯),and so
+on. Define∼onUbyp∼p′if there exists q∈Vwiths(q) =pandt(q) =p′.
+Then∼is an equivalence relation on U,so we can form the quotient U/∼,with
+the quotient topology. There is a natural homeomorphism Xtop∼=U/∼.
+For a quotient C∞-stackX≃[U/G]we haveXtop∼=U/G.
+Using this we can deduce properties of Xtopfrom properties of Xexpressed
+interms ofV⇒U. Forinstance, ifXis separatedthen s×t:V→U×Uis (uni-
+versally) closed, and we can take UHausdorff. But the quotient of a Hausdorff
+topological space by a closed equivalence relation is Hausdorff, yieldin g:
+Lemma 6.20. LetXbe a separated C∞-stack. Then the underlying topological
+spaceXtopis Hausdorff.
+Next we discuss isotropy groups ofC∞-stacks.
+Definition 6.21. LetXbe aC∞-stack, and [ x]∈Xtop. Pick a representative
+xfor [x], so thatx:¯∗→Xis a 1-morphism. Then there exists a C∞-scheme
+G= (G,OG), unique up to isomorphism, with ¯G=¯∗×x,X,x¯∗. Applying the
+construction of the groupoid in Definition A.21 with Π : U→Xreplaced by
+x:¯∗→X, we giveGthe structure of a C∞-group. The underlying group Gis
+canonically isomorphic to the group of 2-morphisms η:x⇒x.
+With [x]fixed, this C∞-groupGisindependent ofchoicesuptononcanonical
+isomorphism; roughly, Gis canonical up to conjugation in G. We define the
+isotropy group (ororbifold group , orstabilizer group )IsoX([x])orIso([x])of[x]to
+be thisC∞-groupG, regarded as a C∞-group up to noncanonical isomorphism.
+IfX= [V⇒U] is associatedto a groupoid( U,V,s,t,u,i,m) thenx:¯∗→X
+factors through ¯ w:¯∗→¯Uup to 2-isomorphism for some point w∈U, and then
+Gis isomorphic to the C∞-subscheme G′=s−1(w)∩t−1(w) inV, with identity
+u|w:∗→G′, inversei|G′:G′→G′, and multiplication m|G′×G′:G′×G′→G′.
+Iff:X→Yis a 1-morphism of C∞-stacks and [ x]∈Xtopwithftop([x]) =
+[y]∈Ytop, fory=f◦x, then at the level of sets we define f∗: IsoX([x])→
+IsoY([y]) byf∗(η) = idf∗η. This is a group morphism, by compatibility of
+horizontal and vertical composition in 2-categories. We can exten df∗naturally
+to a morphism f∗: IsoX([x])→IsoY([y]) ofC∞-groups, such that
+¯f∗:IsoX([x]) =¯∗×x,X,x¯∗−→¯∗×f◦x,Y,f◦x¯∗=IsoY([y])
+is induced from f:X→Yby the universal property of fibre products. Then
+f∗,f∗are independent of the choice of x∈[x] up to conjugation in Iso Y([y]).
+696.5 Gluing C∞-stacks by equivalences
+Here are two propositions on gluing C∞-stacks by equivalences. They are exer-
+cises in stack theory, with no special C∞issues, and also hold for other classes
+of stacks. See Rydh [61, Th. C] for stronger results for algebraic stacks.
+Proposition 6.22. SupposeX,YareC∞-stacks,U ⊆X,V ⊆Yare open
+C∞-substacks, and f:U→Vis an equivalence in C∞Sta. Then there exist
+aC∞-stackZ,openC∞-substacks ˆX,ˆYinZwithZ=ˆX∪ˆY,equivalences
+g:X →ˆXandh:Y→ˆYsuch thatg|Uandh|Vare both equivalences with
+ˆX∩ˆY,and a2-morphism η:g|U⇒h◦f:U→ˆX∩ˆYinC∞Sta. Furthermore,
+Zis independent of choices up to equivalence.
+Proposition 6.23. SupposeX,YareC∞-stacks,U,V ⊆ X are openC∞-
+substacks withX=U∪V, f:U → Y andg:V → Y are1-morphisms,
+andη:f|U∩V⇒g|U∩Vis a2-morphism in C∞Sta. Then there exists a 1-
+morphismh:X →Y and2-morphisms ζ:h|U⇒f, θ:h|V⇒gsuch that
+θ|U∩V=η⊙ζ|U∩V:h|U∩V⇒g|U∩V. Thishis unique up to 2-isomorphism.
+In general, hisnotindependent up to 2-isomorphism of the choice of η.
+Here is an example in which his not independent of ηup to 2-isomorphism
+in the last part of Proposition 6.23.
+Example 6.24. LetXbe theC∞-stack associated to the circle X=/braceleftbig
+(x,y)∈
+R2:x2+y2= 1/bracerightbig
+, andU,V⊆Xthe substacks associated to the open sets U=/braceleftbig
+(x,y)∈X:x >−1
+2/bracerightbig
+andV=/braceleftbig
+(x,y)∈X:x <1
+2/bracerightbig
+. LetYbe the quotient
+C∞-stack [∗/Z2]. Then 1-morphisms f:X →Y correspond to principal Z2-
+bundlesPf→X, and for 1-morphisms f,g:X→Ywith principal Z2-bundles
+Pf,Pg→X,a2-morphism η:f⇒gcorrespondstoanisomorphismofprincipal
+Z2-bundlesPf∼=Pg. The same holds for 1-morphisms U,V,U∪V→Y and
+their 2-morphisms.
+Letf:U → Y andg:V → Y be the 1-morphisms corresponding to
+the trivial Z2-bundlesPf=Z2×U→U,Pg=Z2×V→V. Then 2-
+morphisms η:f|U∩V⇒g|U∩Vcorrespond to automorphisms of the trivial
+Z2-bundleZ2×(U∩V)→U∩V, that is, to continuous maps U∩V→Z2.
+Note thatU∩Vhas two connected components/braceleftbig
+(x,y)∈X:−1
+2< x <1
+2,
+y>0/bracerightbig
+and/braceleftbig
+(x,y)∈X:−1
+2<x<1
+2,y<0/bracerightbig
+.
+Define 2-morphisms η1,η2:f|U∩V⇒g|U∩Vsuch thatη1corresponds to
+the map 1 : ( U∩V)→Z2={±1}, andη1corresponds to the map sign( y) :
+(U∩V)→Z2={±1}. Then Proposition 6.23 gives 1-morphisms h1,h2:
+X→Yfromη1,η2. The associated principal Z2-bundlesPh1,Ph2overXcome
+from gluing Pf,PgoverU,Vusing the transition functions 1 ,sign(y). Therefore
+Ph1is the trivial Z2-bundle over X=S1, andPh2the nontrivial Z2-bundle.
+HencePh1,Ph2are not isomorphic as principal Z2-bundles, and h1,h2are not
+2-isomorphic. Hence in this example, his not independent up to 2-isomorphism
+of the choice of η.
+707 Deligne–Mumford C∞-stacks
+We now introduce Deligne–Mumford C∞-stacks, which are C∞-stacks locally
+modelled on quotients [ U/G] forUan affineC∞-scheme and Ga finite group.
+As we explain in §7.6, orbifolds may be defined as a 2-subcategory of Deligne–
+MumfordC∞-stacks.
+7.1 Quotient C∞-stacks, 1-morphisms, and 2-morphisms
+When aC∞-groupGacts on aC∞-schemeX, Definition 6.6 gives the quotient
+C∞-stack [X/G]. It is a stack associated to a groupoid [ G×X⇒X] from
+Definition A.22, which is the stackification of a certain prestack. By P roposi-
+tion A.9, stackifications always exist, and are unique up to equivalenc e. Thus,
+Definition 6.6 actually only specifies [ X/G] up to equivalence in C∞Sta.
+When afinite group GactsonaC∞-schemeX, wewillnowdefineanexplicit
+C∞-stack [X/G], which is in the equivalence class of [ X/G] in Definition 6.6 for
+G=FC∞Sch
+Man(G). These quotient C∞-stacks [X/G] (forXHausdorff) will be
+our local models for defining Deligne–Mumford C∞-stacks in§7.2.
+We will also define quotient 1-morphisms [f,ρ] : [X/G]→[Y/H] of quotient
+C∞-stacks [X/G],[Y/H] whenρ:G→His a group morphism and f:X→Y
+aρ-equivariant C∞-morphism, and quotient 2-morphisms [δ] : [f,ρ]⇒[g,σ]
+for quotient 1-morphisms [ f,ρ],[g,σ] : [X/G]→[Y/H], whenδ∈Hwith
+σ(γ) =δρ(γ)δ−1for allγ∈G, andg=δ·f. We will see in§7.4 that all 1- and
+2-morphisms of Deligne–Mumford C∞-stacks are locally modelled on quotient
+1- and 2-morphisms.
+Definition 7.1. LetXbe aC∞-scheme,Ga finite group, and r:G→Aut(X)
+an action of GonXby isomorphisms. We will define the quotientC∞-stack
+X= [X/G], generalizing the description of [ ∗/G] in Example 6.24. It is a well
+known construction, as in Behrend et al. [4, Ex. 2.6] and Noohi [55, E x. 12.4].
+Define acategory Xtohaveobjectstriples( S,P,p) whereSisaC∞-scheme,
+andPis a principal G-bundle over Sin the sense of Example 6.7 (or ( P,π,µ)
+rather than P, but we leave π:P→Sand theG-actionµimplicit), and
+p:P→Xis aG-equivariant morphism. Define morphisms ( m,u) : (S,P,p)→
+(T,Q,q) inXto beC∞-scheme morphisms m:S→Tandu:P→Q, such
+thatuisG-equivariant, and (6.2) is a Cartesian square in C∞Sch, andp=
+q◦u:P→X. Composition of morphisms is ( n,v)◦(m,u) = (n◦m,v◦u), and
+identity morphisms are id (S,P,p)= (idS,idP). The functor pX:X →C∞Sch
+mapspX: (S,P,p)/ma√sto→Son objects and pX: (m,u)/ma√sto→mon morphisms.
+ThenXis aC∞-stack, which we write as [ X/G]. It is equivalent in C∞Sta
+to the quotient C∞-stack [X/G] in Definition 6.6 for G=FC∞Sch
+Man(G). To
+see this, note that by Definition A.22 [ X/G] is the stackification of a prestack
+pX′:X′→C∞Sch, whereX′may be written as the category whose objects
+are pairs (S,p′) of aC∞-schemeSand a morphism p′:S→X, and whose
+morphisms ( m,u′) : (S,p′)→(T,q′) consist of morphisms m:S→Tand
+71u′:S→Gwithp′=q′◦m,withcomposition( n,v′)◦(m,u′) = (n◦m,u′·(v′◦m)),
+andpX′mapspX′: (S,p′)/ma√sto→SandpX′: (m,u′)/ma√sto→u′.
+We may identifyX′with the full subcategory of Xwith objects ( S,G×S,p)
+in whichPis the trivial principal G-bundleG×S→S, where (S,p′) inX′is
+identified with ( S,G×S,p) inXforp′=p|S×{1}:S∼=S×{1}→X, and
+(m,u′) : (S,p′)→(T,q′) inX′is identified with ( m,u) : (S,G×S,p)→(T,G×
+T,q) inX, whereu:G×S→G×Tmaps (γ,s)/ma√sto→(γ·u′(s),m(s)). Stackifying
+X′enlarges from trivial principal G-bundles to all principal G-bundles.
+Define a functor π[X/G]:¯X→[X/G] byπ[X/G]: (S,p′)/ma√sto→(S,G×S,p)
+on objects, where p:G×S→Xis the unique G-equivariant morphism with
+p′=p|S×{1}:S∼=S×{1}→X, andπ[X/G]:m/ma√sto→(m,idG×m) on morphisms.
+Thenπ[X/G]:¯X→[X/G] is a representable 1-morphism, and makes ¯Xinto a
+principalG-bundle over [ X/G].
+Definition 7.2. LetX,YbeC∞-schemes acted on by finite groups G,Hwith
+actionsr:G→Aut(X),s:H→Aut(Y), so that we have quotient C∞-stacks
+X= [X/G] andY= [Y/H] as in Definition 7.1. Suppose we have morphisms
+f:X→YofC∞-schemes and ρ:G→Hof groups, with f◦r(γ) =s(ρ(γ))◦f
+for allγ∈G. We will define a quotient 1-morphism [f,ρ] :X→Y.
+Define a functor [ f,ρ] :X →Y by [f,ρ] : (S,P,p)/ma√sto→(S,˜P,˜p) on objects
+(S,P,p) inX, where ˜P= (H×P)/ρGis the principal H-bundle on Scon-
+structedfrom Pandρ:G→H,and˜p:˜P→YistheH-equivariant C∞-scheme
+morphism induced from the ρ-equivariant morphism f◦p:P→Y, which acts
+on points by ˜ p: (h,p)G/ma√sto→h·f◦p(p). Define [f,ρ] : (m,u)/ma√sto→(m,˜u) on mor-
+phisms (m,u) : (S,P,p)→(T,Q,q) inX, where ˜u: (H×P)/ρG→(H×Q)/ρG
+is induced by id H×u:H×P→H×Q. Then [f,ρ] :X →Y is a functor,
+withpX=pY◦[f,ρ], so [f,ρ] is a 1-morphism of C∞-stacks, which we write
+as [f,ρ] : [X/G]→[Y/H].
+It is easy to check that [ f,ρ]◦π[X/G]∼=π[Y/H]◦¯f, and if [f,ρ] : [X/G]→
+[Y/H], [g,σ] : [Y/H]→[Z/I] are quotient 1-morphisms then there is a canon-
+ical 2-isomorphism [ g,σ]◦[f,ρ]∼=[g◦f,σ◦ρ] coming from the canonical C∞-
+scheme isomorphisms/parenleftbig
+I×((H×P)/ρG)/parenrightbig
+/σH∼=(I×P)/σ◦ρG.
+Definition 7.3. Let [f,ρ] : [X/G]→[Y/H] and [g,σ] : [X/G]→[Y/H] be
+quotient 1-morphisms, so that f,g:X→Yandρ,σ:G→Hare morphisms.
+Supposeδ∈Hsatisfiesσ(γ) =δρ(γ)δ−1for allγ∈G, andg=s(δ)◦f. We will
+define a 2-morphism [ δ] : [f,ρ]⇒[g,σ], which we call a quotient 2-morphism .
+Here [δ] must be a natural isomorphism of functors [ f,ρ]⇒[g,σ]. Let
+(S,P,p) be an object in [ X/G]. Define an isomorphism in [ Y/H]:
+[δ]/parenleftbig
+(S,P,p)/parenrightbig
+=/parenleftbig
+idS,(rδ−1×idP)∗/parenrightbig
+: [f,ρ]/parenleftbig
+(S,P,p)/parenrightbig
+=
+(S,(H×P)/ρG,˜p)−→[g,σ]/parenleftbig
+(S,P,p)/parenrightbig
+= (S,(H×P)/σG,˜p),
+whererδ−1:H→Hmapsǫ/ma√sto→ǫδ−1, andrδ−1×idP:H×P→H×Pis
+equivariantundertheactionsof GonH×Pinducedby ρonthedomainand σon
+72the target, so that it descends to an isomorphism ( rδ−1×idP)∗: (H×P)/ρG→
+(H×P)/σG. It is now easy to check that [ δ]/parenleftbig
+(S,P,p)/parenrightbig
+is an isomorphism in
+[Y/H], and [δ] is a natural isomorphism of functors, and a 2-morphism [ δ] :
+[f,ρ]⇒[g,σ] inC∞Sta. Quotient 2-morphisms have functorial properties
+under horizontal and vertical composition. For instance, if [ f,ρ],[g,σ],[h,τ] :
+[X/G]→[Y/H] are quotient 1-morphisms and [ δ] : [f,ρ]⇒[g,σ], [ǫ] : [g,σ]⇒
+[h,τ] are quotient 2-morphisms then [ ǫ]⊙[δ] = [ǫδ] : [f,ρ]⇒[h,τ].
+Remark 7.4. StudyingC∞-stacks [X/G] and their 1- and 2-morphisms is
+a good way to develop geometric intuition about Deligne–Mumford C∞-stacks
+(includingorbifolds)andtheir1-and2-morphisms. If[ X/G],[Y/H]arequotient
+C∞-stacks, then general 1-morphisms f: [X/G]→[Y/H] inC∞Staneed not
+be quotient 1-morphisms [ f,ρ], or even 2-isomorphic to [ f,ρ]. But Theorem
+7.18(b) saysthat f∼=[f,ρ]locally in [ X/G]. If[f,ρ],[g,σ] : [X/G]→[Y/H]are
+quotient 1-morphisms, and [ X/G] is connected, then Proposition 7.19 says that
+all 2-morphisms η: [f,ρ]⇒[g,σ] are quotient 2-morphisms [ δ] : [f,ρ]⇒[g,σ].
+7.2 Deligne–Mumford C∞-stacks
+Deligne–Mumford stacks in algebraic geometry were introduced in [17] to study
+moduli spaces of algebraic curves. As in [46, Th. 6.2], Deligne–Mumfo rd stacks
+are locally modelled (in the ´ etale topology, at least, but with isomorph isms of
+isotropy groups) on quotient C∞-stacks [X/G] forXan affine scheme and Ga
+finite group. This motivates:
+Definition 7.5. ADeligne–Mumford C∞-stackis aC∞-stackXwhich ad-
+mits a (Zariski) open cover {Ua:a∈A}, as in Definition 6.14, with each Ua
+equivalent to a quotient C∞-stack [Ua/Ga] in Definition 7.1 for Uaan affine
+C∞-scheme and Gaa finite group. We call Xlocally fair , orlocally finitely
+presented , orlocally Lindel¨ of , if it admits such an open cover with each Uaa
+fair, or finitely presented, or Lindel¨ of, affine C∞-scheme, respectively. We call
+Xsecond countable if the underlying topologicalspace Xtopis second countable.
+WriteDMC∞Stalf,DMC∞StalfpandDMC∞Stafor the full 2-subcat-
+egories of locally fair, locally finitely presented, and all, Deligne–Mumfo rdC∞-
+stacks in C∞Sta, respectively.
+The functor FC∞Sta
+C∞Sch:C∞Sch→C∞Stain Definition 6.3 maps into
+DMC∞Sta⊂C∞Sta, so the 2-categories ¯C∞Schlf,¯C∞Schlfp,¯C∞Schare
+2-subcategories of DMC∞Stalf,DMC∞Stalfp,DMC∞Sta, respectively. If
+aC∞-stackXis aC∞-scheme, then it is a Deligne–Mumford C∞-stack.
+Proposition 7.6. DMC∞Stalf,DMC∞Stalfp,DMC∞Staare closed under
+taking open C∞-substacks in C∞Sta.
+Proof.LetXlie in one of these 2-categories, and X′be an open C∞-substack
+ofX. ThenXadmits an open cover {Ua:a∈A}withUa≃[Ua/Ga] with
+Uaaffine and Gafinite, and{U′
+a:a∈A}is an open cover of X′, where
+U′
+a=Ua×XX′is an openC∞-substack ofUa. ThusU′
+a≃[U′
+a/Ga] by Propo-
+sition 6.16, where U′
+ais aGa-invariant open C∞-subscheme of Ua. If theUa
+73are fair, or finitely presented then the U′
+aare too by Corollary 4.32(a). Thus
+DMC∞Stalf,DMC∞Stalfpare closed under open subsets.
+ForDMC∞Sta, as open subsets of affine C∞-schemes need not be affine,
+theU′
+aneed not be affine. We will show that we can cover U′
+abyGa-invariant
+open affine C∞-subschemes U′
+au. WriteU′
+a= (U′
+a,OU′a) andGa= (Ga,OGa).
+Then the finite group Gaacts continuously on U′
+a. Letu∈U′
+a, andHu={γ∈
+Ga:γu=u}be the stabilizer of uinGa. Then the orbit {γu:γ∈G}∼=
+Ga/Huofuis a finite set, so as U′
+ais Hausdorff we can choose affine open
+neighbourhoods Vγuofγufor each point in the orbit such that Vγu∩Vγ′u=∅
+ifγu/\e}atio\slash=γ′u. DefineWu=/intersectiontext
+γ∈Gγ−1Vγu. ThenWuis anHu-invariant open
+neighbourhood of uinU′
+a, and ifγ∈Ga\HuthenγWu∩Wu=∅.
+As in Corollary 4.29 we can choose an affine open neighbourhood W′
+uof
+uinWu. DefineW′′
+u=/intersectiontext
+γ∈HuW′
+u, anHu-invariant open neighbourhood of
+uinWu. This a finite intersection of affine open C∞-subschemes W′
+uin the
+affineC∞-schemeVu, and so is affine, since intersection is a kind of fibre prod-
+uct, and AC∞Schis closed under fibre products by Theorem 4.25(a). Define
+U′
+au=/uniontext
+γ∈GaW′′
+u. ThenU′
+auis aGa-invariant open neighbourhood of uin
+U′
+a. SinceW′′
+uisHu-invariant and γW′′
+u∩W′′
+u=∅ifγ∈Ga\Hu, we see
+thatU′
+auis isomorphic to the disjoint union of |Ga|/|Hu|copies ofW′′
+u. Hence
+U′
+au= (U′
+au,OU′a|U′au) is a finite disjoint union of affine C∞-schemes, and is an
+affineC∞-scheme. Therefore we may cover U′
+abyGa-invariant open affine C∞-
+subschemes U′
+au. Using these we obtain an open cover/braceleftbig
+U′
+au:a∈A,u∈Ua/bracerightbig
+ofX′withU′
+au≃[U′
+au/Ga], soX′is Deligne–Mumford.
+The proof of Proposition 7.6 only uses Ua= (Ua,OUa) aC∞-scheme and
+UaHausdorff, it does not need Uato be affine. So the same proof yields:
+Proposition 7.7. AnyC∞-stack of the form [X/G]in§7.1withXa Hausdorff
+C∞-scheme and Gfinite is a separated Deligne–Mumford C∞-stack.
+However, if Xis not Hausdorff then [ X/G] need not be Deligne–Mumford:
+Example 7.8. LetXbe the non-Hausdorff C∞-scheme ( R∐R)/∼, where∼
+is the equivalence relation which identifies the two copies of Ron (0,∞). Let
+G=Z2act onXby exchanging the two copies of R. LetXbe the quotient
+C∞-stack [X/G]. We can think of Xas a like copy of R, where the stabilizer
+group ofx∈Ris{1}ifx∈(−∞,0] andZ2ifx∈(0,∞). Using the obvious
+atlas Π : ¯R→X, the third diagram of (A.12) yields a 2-Cartesian square
+¯R∐(0,∞) /d47/d47
+/d15/d15✘✘✘✘/d72/d80IX
+ιX/d15/d15¯RΠ/d47/d47X.
+As the left hand column is not proper, ιXis not proper, so X= [X/G] is not
+Deligne–Mumford by Corollary 7.14 below.
+We show that the 2-subcategory of quotient C∞-stacks [X/G] inC∞Stais
+closed under fibre products:
+74Proposition 7.9. Supposeg: [X/F]→[Z/H], h: [Y/G]→[Z/H]are1-
+morphisms of quotient C∞-stacks, where X,Y,ZareC∞-schemes and F,G,H
+are finite groups. Then we have a 2-Cartesian square
+[W/(F×G)]f/d47/d47
+e/d15/d15 ✙✙✙✙/d72/d80[Y/G]
+h/d15/d15
+[X/F]g/d47/d47[Z/H],(7.1)
+whereΠX:¯X→[X/F],ΠY:¯Y→[Y/G]are the natural atlases and ¯W=
+¯X×g◦ΠX,[Z/H],h◦ΠY¯Y. IfX,Y,Zare Hausdorff, or locally fair, or locally
+finitely presented, then Wis too.
+Proof.WriteW= [X/F]×[Z/H][Y/G]. ThenfromtheatlasesΠ X,ΠY,Example
+A.25 constructs an atlas Π W:¯W→WforW. Since [X/F]≃[F×X⇒X]
+and [Y/G]≃[G×Y⇒Y] it follows from (A.14) that Wis equivalent to the
+stack associated to the groupoid [( F×G)×W⇒W] for a natural action of
+F×GonW. This proves (7.1).
+IfX,Y,Zare Hausdorff then [ Z/H] is Deligne–Mumford by Proposition
+7.7, so ∆ [Z/H]is separated by Corollary 7.14 below, and thus Wis Hausdorff
+asX,Yare and ¯W∼=(¯XׯY)×[Z/H]×[Z/H],∆[Z/H][Z/H]. Form the diagram
+¯W′
+πW
+/d115/d115❤❤❤❤❤❤❤❤❤❤❤
+/d38/d38◆◆◆◆◆◆◆◆◆◆◆◆/d47/d47¯Y′
+πY/d115/d115❤❤❤❤❤❤❤❤❤❤❤❤
+/d38/d38◆◆◆◆◆◆◆◆◆◆◆◆◆
+¯W
+/d38/d38◆◆◆◆◆◆◆◆◆◆◆◆◆/d47/d47¯Yh◦ΠY
+/d38/d38◆◆◆◆◆◆◆◆◆◆◆
+¯X′/d47/d47
+πX
+/d115/d115❤❤❤❤❤❤❤❤❤❤❤❤¯Z
+ΠZ /d115/d115❤❤❤❤❤❤❤❤❤❤
+Xg◦ΠX/d47/d47[Z/H]
+with squares 2-Cartesian, where W′,X′,Y′areC∞-schemes. Then πW,πX,πY
+are ´ etale and surjective, as Π Zis. IfX,Y,Zare locally fair, then X′,Y′are
+locally fair as X,Yare andπX,πYare ´ etale, so W′∼=X′×ZY′is locally fair
+by Theorem 4.25(b), and thus Wis locally fair as πW:W′→Wis ´ etale and
+surjective. The proof for locally finitely presented is the same.
+Using this we prove:
+Theorem 7.10. DMC∞Sta,DMC∞StalfandDMC∞Stalfpare closed un-
+der fibre products in C∞Sta.
+Proof.LetW=X×ZYbe a fibre product in C∞Staof Deligne–Mumford
+C∞-stacksX,Y,Z. We must showWis Deligne–Mumford. Now Zadmits an
+open cover{Zc:c∈C}withZc≃[Zc/Hc] forZcan affineC∞-scheme and
+Hcfinite. Forc∈CdefineXc=X×ZZcandYc=Y×ZZc, which are open
+C∞-substacks ofX,Y, and so are Deligne–Mumford by Proposition 7.6. Then
+{Xc×ZcYc:c∈C}is an open cover of W, so it is enough to prove Xc×ZcYc
+is Deligne–Mumford. That is, we may replace ZbyZc≃[Zc/Hc].
+75Similarly, by choosing open covers of Xc,Ycby substacks equivalent to
+[X/F],[Y/G], wereducetheproblemtoshowing[ X/F]×[Z/H][Y/G]isDeligne–
+Mumford, for X,Y,ZaffineC∞-schemesand F,G,Hfinite groups. Thisfollows
+from Propositions 7.7 and 7.9, noting that X,Y,Zare Hausdorff as they are
+affine, soWis Hausdorff in Proposition 7.9. This shows DMC∞Stais closed
+under fibre products. For DMC∞Stalf,DMC∞Stalfpwe use the same argu-
+ment withZc,Z,X,Y,Wlocally fair, or locally finitely presented.
+Under weak conditions Deligne–Mumford C∞-stacks have coarse moduli
+C∞-schemes, in the sense of §6.4.
+Theorem 7.11. LetXbe a locally Lindel¨ of Deligne–Mumford C∞-stack. Then
+theC∞-ringed spaceXtopin Definition 6.18is aC∞-scheme. IfXis locally
+fair, or locally finitely presented, then Xtopis too.
+Proof.By definitionXcan be covered by open C∞-substacksUequivalent to
+[Y/G] forYa Lindel¨ of affine C∞-scheme. Then the C∞-ringed spaceUtopis
+isomorphic to Y/Gin Definition 4.45, so Proposition 4.46 shows that Utopis an
+affineC∞-scheme. HenceXtopcan be covered by open affine Utop⊆Xtop, so
+Xtopis aC∞-scheme. IfXis locally fair (or locally finitely presented) the same
+argument works with Yfair (or finitely presented), and then Utop∼=Y/Gis fair
+(or finitely presented), and Xtopis locally fair (or locally finitely presented).
+Remark 7.12. In§4.7 we discussed partitions of unity onC∞-schemes. We
+can use Theorem 7.11 to extend these ideas to Deligne–Mumford C∞-stacks.
+LetXbe a Deligne–Mumford C∞-stack, and suppose Xtopis regular and
+Lindel¨ of. ThenXtopisaC∞-schemebyTheorem7.11, andthetopologyon Xtop
+is smoothly generated by Example 4.39(c) as Xtopis regular. Hence Theorem
+4.40 shows thatOXtopis fine. Suppose{Ua:a∈A}is a (Zariski) open cover of
+X. Then/braceleftbig
+Ua,top:a∈A/bracerightbig
+is an open cover of Xtop, so there exists a partition
+of unity{ηa:a∈A}onXtopsubordinate to/braceleftbig
+Ua,top:a∈A/bracerightbig
+. Therefore
+{π∗(ηa) :a∈A}is (in a suitable sense) a partition of unity on Xsubordinate
+to{Ua:a∈A}, whereπ:X→¯Xtopis the structural morphism.
+7.3 Characterizing Deligne–Mumford C∞-stacks
+We now explore ways to characterize when a C∞-stackXis Deligne–Mumford.
+Proposition 7.13. LetXbe a quotient C∞-stack[U/G]forUaffine and
+Gfinite. Then the natural 1-morphism Π :¯U→ Xis an ´ etale atlas, and
+∆X:X→X×X , ιX:IX→Xare universally closed, proper, and separated,
+with finite fibres, and X:X→IXis an open and closed embedding.
+76Proof.As in (A.12) we have 2-Cartesian diagrams with surjective rows:
+¯GׯU¯µ/d47/d47
+¯πU/d15/d15¯U
+Π/d15/d15¯GׯUΠ◦¯πU/d47/d47
+¯πUׯµ/d15/d15X
+∆X/d15/d15 ✚✚✚✚/d73/d81
+✚✚✚✚/d73/d81
+¯UΠ/d47/d47X,¯UׯUΠ×Π/d47/d47X×X,
+(¯GׯU)ׯUׯU¯U¯µ/d47/d47
+¯πU/d15/d15IX
+ιX/d15/d15¯U
+(1×idU)×idU/d15/d15Π/d47/d47X
+X/d15/d15✚✚✚✚/d73/d81
+✚✚✚✚/d73/d81
+¯UΠ/d47/d47X,(¯GׯU)ׯUׯU¯U¯µ/d47/d47IX.
+The left column ¯ πUin the first diagram is ´ etale. The left columns in the second
+and third diagrams are both universally closed, proper, and separa ted, with
+finite fibres, since Gis finite with the discrete topology, and Uis Hausdorff as U
+is affine. This left column in the fourth is an open and closed embedding. The
+result now follows from Propositions 6.11 and A.18(c).
+Propositions 6.11, 6.15 and 7.13 now imply:
+Corollary 7.14. LetXbe a Deligne–Mumford C∞-stack. ThenXhas an ´ etale
+atlasΠ :¯U→X,the diagonal ∆X:X→X×X is separated with finite fibres, and
+the inertia morphism ιX:IX→Xis universally closed, proper, and separated,
+with finite fibres, and X:X→IXis an open and closed embedding. If Xis
+separated then ∆Xis also universally closed and proper.
+The last part holds as then ∆ Xis universally closed with finite fibres, which
+implies ∆ Xis proper. Note that for Xnot separated we cannot conclude from
+Proposition 7.13 that ∆ Xis universally closed or proper, since these properties
+are not stable under open embedding. Some of the conclusions of Co rollary7.14
+are sufficient forXto be separated and Deligne–Mumford.
+Theorem 7.15. LetXbe aC∞-stack, and suppose Xhas an ´ etale atlas Π :
+¯U→X,and the diagonal ∆X:X→X×X is universally closed and separated.
+ThenXis a separated Deligne–Mumford C∞-stack.
+Proof.Let (U,V,s,t,u,i,m) be the groupoid in C∞Schconstructed from Π :
+¯U→ Xas in§A.5, so thatX ≃[V⇒U]. Then (A.12) gives 2-Cartesian
+diagrams with surjective rows. From the first and Propositions A.18 (a) and
+6.11 we see that s,tare ´ etale, since Π is. From the second s×t:V→U×Uis
+universally closed and separated, as ∆ Xis. Letp∈U. Define
+H=/braceleftbig
+q∈V:s(q) =t(q) =p/bracerightbig
+⊆s−1({p}).
+It has the discrete topology, as s,tare ´ etale.
+Suppose for a contradiction that His infinite. Define a C∞-ring
+C=/braceleftbig
+c:H∐{∞}→ R:c(q) =c(∞) for all but finitely many q∈H/bracerightbig
+,
+withC∞operations defined pointwise in H∐{∞}. Then Spec Chas underly-
+ing topological space the one point compactification H∐{∞}of the discrete
+77topological space H, sinceC=C0(H∐{∞}) is the set of continuous maps
+H∐{∞}→ R, with the natural C∞-ring structure. Define g: SpecC→U×U
+to project Spec Cto the point ( p,p). Then the morphism
+πSpecC:V×s×t,U×U,gSpecC−→SpecC (7.2)
+is the projection H×(H∐{∞})→H∐{∞}. The diagonal in His closed in
+H×(H∐{∞}), but its image is H, which is not closed in H∐{∞}. Hence
+(7.2) is not a closed morphism, contradicting s×tuniversally closed. So His
+finite.
+As (U,V,s,t,u,i,m) is a groupoid, His afinite group , with identity u(p),
+inverse map i|H, and multiplication mH=m|H×H. Sinces,tare ´ etale, we
+can choose small open neighbourhoods ZqofqinVfor allq∈Hsuch that
+s|Zq,t|Zqare isomorphisms with open subsets of U. Ass×tis separated,/braceleftbig
+(v,v) :v∈V/bracerightbig
+is closed in/braceleftbig
+(v,v′)∈V×V:s(v) =s(v′),t(v) =t(v′)/bracerightbig
+,
+which has the subspace topology from V×V. Ifq/\e}atio\slash=q′∈Hthen (q,q′) lies in/braceleftbig
+(v,v′)∈V×V:s(v) =s(v′),t(v) =t(v′)/bracerightbig
+butnotin/braceleftbig
+(v,v) :v∈V/bracerightbig
+, so(q,q′)
+has an open neighbourhood in V×Vwhich does not intersect/braceleftbig
+(v,v) :v∈V/bracerightbig
+.
+MakingZq,Zq′smaller if necessary, we can take this open neighbourhood to be
+Zq×Zq′, andthenZq∩Zq′=∅. Thus, wecan choosethese openneighbourhoods
+Zqforq∈Hto bedisjoint.
+DefineY=/intersectiontext
+q∈Hs(Zq) andY= (Y,OU|Y). ThenYis a small open neigh-
+bourhoodof pinU. MakingYsmallerifnecessarywecansupposeitiscontained
+in an affine open neighbourhood of pinU, and so is Hausdorff. Replace Zqby
+Zq∩s−1(Y) for allq∈H. Thens|Zq: (Zq,OV|Zq)→Yis an isomorphism
+forq∈H. SetZ=/uniontext
+q∈HZq, noting the union is disjoint, and Z= (Z,OV|Z).
+Then we have an isomorphism φ= (φ,φ♯) :H×Y→Z, such that s◦φ= idY
+andφ(q×Y) =Zqforq∈H.
+NowZis open inV, soZ×s,U,tZis open inV×s,U,tV, and we can restrict
+the morphism m:V×s,U,tV→Vtom|Z×UZ:Z×s,U,tZ→V. But
+Z×s,U,tZ∼=(H×Y)×iY◦πY,U,tZ
+∼=H×(Z∩t−1(Y),OV|Z∩t−1(Y))⊆H×Z∼=H×H×Y,
+usingφan isomorphism and s◦φ= idY. Write Φ :Z×s,U,tZ֒→H×H×Yfor
+the induced open embedding. Define a second morphism m′:Z×s,U,tZ→V
+bym′=φ◦(mH×idY)◦Φ, wheremH:H×H→His the group multiplication
+mH:H×H→H, regarded as a morphism of C∞-schemes.
+Following the definitions we find that s◦(m|Z×UZ) =s◦m′:Z×s,U,tZ→
+Y⊂U. AlsoH⊂Z, and the definition of mHfrommimplies that m|Z×UZand
+m′coincide on the finite set H×UHinZ×UZ. Sincesis ´ etale, this implies
+thatm|Z×UZandm′must coincide near the finite set H×UHinZ×UZ.
+Therefore by making the open neighbourhood YofpinUsmaller, and hence
+makingWq,W,Zsmaller too, we can assume that m|Z×UZ=m′.
+Let us summarize what we have done so far. We have constructed a finite
+groupH, a Hausdorff open neighbourhood YofpinU, an open and closed
+78subsetZofs−1(Y) inZwhich contains s−1(p)∩t−1(p), and an isomorphism φ:
+H×Y→Zwiths◦φ=πYwhich identifies the groupoidmultiplication m|Z×UZ
+with the restriction to Z×UZof the morphism mH×idY:H×H×Y→Y
+from multiplication in the finite group H.
+Consider the morphism t◦φ:H×Y→U⊃Y. Roughly speaking, t◦φ
+is anH-action onY. More accurately, there should an H-action on some open
+subset ofUcontainingY, butYmay not be H-invariant, so that t◦φneed not
+mapH×YtoY. ReplaceYbyY′=/intersectiontext
+q∈Ht(Zq), which is an open subset of
+Ysince when qis the identity u(p) inHwe havet(Zu(p)) =s(Zu(p)) =Y, and
+p∈Y′asp=t(q)∈t(Zq) forq∈H. ReplaceZqbyZ′
+q=Zq∩s−1(Y) andZby
+Z′=/uniontext
+q∈HZ′
+q. Then using m|Z×UZ=m′we can show that s(Z′
+q) =t(Z′
+q) =Y′
+for allq∈H, soY′is anH-invariant open set, and t◦φmapsH×Y′→Y′.
+Restricting the groupoid axioms shows that t◦φgives an action of HonY′.
+Now consider the morphism
+s×t|s−1(Y′)∩t−1(Y′):/parenleftbig
+s−1(Y′)∩t−1(Y′),OV|s−1(Y′)∩t−1(Y′)/parenrightbig
+−→Y′×Y′.
+This is closed, as s×tis universally closed. Since Z′is open and closed in
+s−1(Y′)∩t−1(Y′), its complement is closed, so its image/braceleftbig
+(s(v),t(v))∈Y′×
+Y′:v∈V\Z′/bracerightbig
+is closed in Y′. But (p,p) does not lie in this image, since
+s−1(p)∩t−1(p)⊆Z′. Thus, by making the H-invariant open neighbourhood Y′
+ofpinUsmaller if necessary, we can suppose that s−1(Y′)∩t−1(Y′) =Z′.
+Thequotient C∞-stack[Y′/H]isDeligne–MumfordbyProposition7.7, since
+Y′is Hausdorff. Thus there exists an open embedding Yp֒→[Y′/H] with
+Yp≃[Up/Gp] forUpaffine andGpfinite, which includes pin its image. The
+inclusion morphisms Y′֒→U,Z′֒→Vinduce a 1-morphism [ Z′⇒Y′]֒→
+[V⇒U], which is an open embedding as Y′is open in U,Z′is open in V
+ands−1(Y′)∩t−1(Y′) =Z′inV. LetiYp:Yp→ Xbe the composition
+Yp֒→[Y′/H]≃[Z′⇒Y′]֒→[V⇒U]≃X. TheniYpis an open embedding,
+as it is a composition of open embeddings and equivalences. This works for all
+p∈U, and{Yp:p∈U}is an open cover of XwithYp≃[Up/Gp] forUp
+affine andGpfinite. HenceXis Deligne–Mumford. It is separated as ∆ Xis
+universally closed, by assumption.
+Supposef:X→Yis a separated morphism of C∞-schemes with finite
+fibres. Then funiversally closed implies fproper. Conversely, if X,Yare com-
+pactly generated topological spaces then fproper implies funiversally closed.
+IfX,Yare locally fair then X,Yare compactly generated, as they are locally
+homeomorphic to closed subsets of Rn. Thus, in Theorem 7.15, if U,Vare
+locally fair then we can replace ∆ Xuniversally closed by ∆ Xproper, yielding:
+Theorem 7.16. LetXbe aC∞-stack, and suppose Xhas an ´ etale atlas Π :
+¯U→XwithUlocally fair, and the diagonal ∆X:X→X×X is proper and
+separated. ThenXis a separated, locally fair Deligne–Mumford C∞-stack.
+The same holds with locally finitely presented in place of locally fair. If
+X≃[V⇒U] withUa Hausdorff C∞-scheme then Vis Hausdorff if and only
+79if ∆Xis separated. We can always choose UHausdorff, by replacing Uby the
+disjoint union of an open cover of Uby affine open subsets. Thus we can replace
+the condition that ∆ Xis separated by U,VHausdorff. Combining this and the
+results above proves:
+Theorem 7.17. (a) AC∞-stackXis separated and Deligne–Mumford if and
+only if it is equivalent to the stack associated to a groupoid [V⇒U]whereU,V
+are Hausdorff C∞-schemes,s:V→Uis ´ etale, and s×t:V→U×Uis
+universally closed.
+(b)AC∞-stackXis separated, Deligne–Mumford and locally fair (or locally
+finitely presented) if and only if it is equivalent to some [V⇒U]withU,V
+Hausdorff, locally fair (or locally finitely presented) C∞-schemes,s:V→U
+´ etale, ands×t:V→U×Uproper.
+7.4 Quotient C∞-stacks, 1- and 2-morphisms as local
+models for objects, 1- and 2-morphisms in DMC∞Sta
+In our next theorem, we prove that Deligne–Mumford C∞-stacks and their 1-
+and 2-morphisms are (Zariski) locally modelled on quotient C∞-stacks [X/G],
+quotient 1-morphisms [ f,ρ] : [X/G]→[Y/H], and quotient 2-morphisms [ δ] :
+[f,ρ]⇒[g,σ] from§7.1.
+Theorem 7.18. (a) LetXbe a Deligne–Mumford C∞-stack and [x]∈Xtop,
+and writeG= IsoX([x]). Then there exists a quotient C∞-stack[U/G]forU
+an affineC∞-scheme, and a 1-morphism i: [U/G]→Xwhich is an equivalence
+with an open C∞-substackUinX,such thatitop: [u]/ma√sto→[x]∈Utop⊆Xtopfor
+some fixed point uofGinU.
+(b)Letf:X → Y be a1-morphism of Deligne–Mumford C∞-stacks, and
+[x]∈Xtopwithftop: [x]/ma√sto→[y]∈Ytop,and writeG= IsoX([x])andH=
+IsoY([y]). Part(a)gives1-morphisms i: [U/G]→X,j: [V/H]→Ywhich are
+equivalences with open U⊆X,V⊆Y,such thatitop: [u]/ma√sto→[x]∈Utop⊆Xtop,
+jtop: [v]/ma√sto→[y]∈Vtop⊆Ytopforu,vfixed points of G,HinU,V.
+Then there exists a G-invariant open neighbourhood U′ofuinUand a
+quotient 1-morphism [f,ρ] : [U′/G]→[V/H]such thatf(u) =v,andρ:G→
+Hisf∗: IsoX([x])→IsoY([y]),fitting into a 2-commutative diagram:
+[U′/G]
+[f,ρ]/d47/d47
+i|[U′/G]/d15/d15 ✚✚✚✚/d73/d81
+ζ[V/H]
+j/d15/d15
+Xf/d47/d47Y.(7.3)
+(c)Letf,g:X → Y be1-morphisms of Deligne–Mumford C∞-stacks and
+η:f⇒ga2-morphism, let [x]∈Xtopwithftop: [x]/ma√sto→[y]∈Ytop,and write
+G= IsoX([x])andH= IsoY([y]). Part(a)givesi: [U/G]→X,j: [V/H]→Y
+which are equivalences with open U ⊆X,V ⊆Yand mapitop: [u]/ma√sto→[x],
+jtop: [v]/ma√sto→[y]foru,vfixed points of G,H.
+80By making U′smaller, we can take the same U′in(b)for bothf,g. Thus
+part(b)gives aG-invariant open U′⊆U,quotient morphisms [f,ρ] : [U′/G]→
+[V/H]and[g,σ] : [U′/G]→[V/H]withf(u) =g(u) =vandρ=f∗:
+IsoX([x])→IsoY([y]), σ=g∗: IsoX([x])→IsoY([y]),and2-morphisms ζ:
+f◦i|[U′/G]⇒j◦[f,ρ], θ:g◦i|[U′/G]⇒j◦[g,σ].
+Then there exists a G-invariant open neighbourhood U′′ofuinU′and
+δ∈Hsuch thatσ(γ) =δρ(γ)◦δ−1for allγ∈Gandg|U′′=s(δ)◦f|U′′,
+so that[δ] : [f|U′′,ρ]⇒[g|U′′,σ]is a quotient 2-morphism, and the following
+diagram of 2-morphisms in C∞Stacommutes:
+f◦i|[U′′/G]η∗idi|[U′′/G]/d43/d51
+ζ|[U′′/G]/d11/d19g◦i|[U′′/G]
+θ|[U′′/G]/d11/d19
+j◦[f|U′′,ρ]idj∗[δ]/d43/d51j◦[g|U′′,σ].(7.4)
+Proof.In this proof we will use the theory of 2-categories from §A.1, including
+vertical and horizontal composition of 2-morphisms ‘ ∗’, ‘⊙’, and the definition
+of fibre products and 2-Cartesian squares in Definition A.3.
+For (a), asXis Deligne–Mumford it is covered by open C∞-substacksV
+equivalent to [ V/H] forVaffine andHfinite, so we can choose such Vwith
+[x]∈Vtop. ThenVhas an ´ etale atlas Π : ¯V→Vand ∆ Vis universally closed
+and separated by Proposition 7.13, so we can apply the proof of The orem 7.15
+toVfor a point p∈Vwith Π ∗(p) = [x]. This constructs an open C∞-substack
+UinVequivalent to [ U/G], whereUis affine and G= IsoX([x]), as we want.
+For(b), write π[U/G]:¯U→[U/G]andπ[V/H]:¯V→[V/H]forthe projection
+1-morphisms in C∞Sta. They are proper and representable. Let r:G→
+Aut(U) ands:H→Aut(V) be theG- andH-actions on U,V. Then ¯r(γ) :
+¯U→¯Uforγ∈Gand ¯s(δ) :¯V→¯Vforδ∈Hare the corresponding C∞-stack
+1-morphisms, and there are natural 2-morphisms λγ:π[U/G]◦¯r(γ)⇒π[U/G]
+andµδ:π[V/H]◦¯s(δ)⇒π[V/H].
+Consider the C∞-stack fibre product ¯U×f◦i◦π[U/G],Y,j◦π[V/H]¯V. Asπ[V/H]is
+representable and jis an equivalence with an open C∞-substack,j◦π[V/H]is
+representable, and ¯Uis aC∞-stack, so this fibre product is a C∞-scheme. So
+changing the fibre product up to equivalence, we can take ¯U×Y¯V=¯Wfor some
+C∞-schemeWunique up to isomorphism. The fibre product projections are
+1-morphisms ¯W→¯Uand¯W→¯V,so they are 2-isomorphic to ¯ a,¯bfor unique
+morphisms a:W→U,b:W→V. Hence we have a 2-Cartesian square in
+C∞Sta, for some 2-morphism ω:
+¯W¯b/d47/d47
+¯a/d15/d15 ✙✙✙✙/d72/d80ω¯V
+j◦π[V/H]/d15/d15¯Uf◦i◦π[U/G]/d47/d47Y.(7.5)
+We will show that the data r(γ),λγforγ∈Ginduces an action of GonW.
+Letγ∈G, and apply the universal property of the 2-Cartesian square (7.5 ) in
+81DefinitionA.3tothe1-morphisms ¯ r(γ)◦¯a:¯W→¯U,¯b:¯W→¯Vand2-morphism
+ω⊙(idf◦i∗λγ∗id¯a) : (f◦i◦π[U/G])◦(¯r(γ)◦¯a)⇒(j◦π[V/H])◦¯b. This gives
+a 1-morphism cγ:¯W→¯W, unique up to 2-isomorphism, and 2-morphisms
+ζγ: ¯a◦cγ⇒¯r(γ)◦¯a,θγ:¯b◦cγ⇒¯bsuch that (A.6) commutes.
+Nowcγis 2-isomorphic to ¯ cγfor some unique cγ:W→W, so we may
+replacecγby ¯cγ. Thenζγ: ¯a◦¯cγ⇒¯r(γ)◦¯a, so we must have a◦cγ=r(γ)◦a
+andζγ= id¯r(γ)◦¯a. Similarly b◦cγ=bandθγ= id¯b. Therefore (A.6) reduces
+toω∗id¯cγ=ω⊙(idf◦i∗λγ∗id¯a). Usingr(γ)r(γ′) =r(γγ′) and a natural
+compatibility between λγ,λ′
+γ,λγγ′we find that cγ◦cγ′=cγγ′forγ,γ′∈G, and
+asr(1) = idUandλ1= idπ[U/G]we havec1= idW. Henceγ/ma√sto→cγis an action
+ofGonW, anda◦cγ=r(γ)◦ameans that a:W→UisG-equivariant.
+In the same way, we obtain unique isomorphisms dδ:W→Wforδ∈H
+witha◦dδ=a,b◦dδ=s(δ)◦bandω∗id¯dδ= (idj∗µδ∗id¯b)⊙ω, andδ/ma√sto→dδ
+is an action of HonW, andb:W→VisH-equivariant. Using associativity
+of⊙in (idj∗µδ∗id¯b)⊙ω⊙(idf◦i∗λγ∗id¯a), we see that cγanddδcommute.
+Hence (γ,δ)/ma√sto→cγ◦dδis an action of G×HonW.
+Sinceπ[V/H]:¯V→[V/H] is a principal H-bundle, and j: [V/H]→Yis
+an equivalence with V⊆Y, and (7.5) is 2-Cartesian, it follows that a:W→U
+is a principal H-bundle over the open C∞-subscheme ˜UofUmapped toVby
+f◦i◦π[U/G], where the H-action for the principal H-bundle isδ/ma√sto→dδ. As
+u∈˜U, this implies that we can choose a G-invariant open neighbourhood U′
+ofuin˜U⊆Uwith an isomorphism W′=a−1(U′)∼=U′×H, that identifies
+dδ|W′:W′→W′with the product of id U′onU′andǫ/ma√sto→δǫonH.
+Thenγ/ma√sto→cγ|W′is an action of GonW′∼=U′×H, and the projection
+U′×H→U′isG-equivariant. Since u∈U′is a fixed point of G, this implies
+thatcγfixes the finite subset {(u,δ) :δ∈H}inU′×H. Defineρ:G→H
+bycγ(u,1) = (u,ρ(γ)−1) forγ∈G. Sincedδacts by (u,ǫ)/ma√sto→(u,δǫ) andcγ,dδ
+commute, it follows that cγ(u,δ) = (u,δρ(γ)−1) forγ∈G,δ∈H. Hence
+(u,ρ(γγ′)−1)=cγγ′(u,1)=cγ◦cγ′(u,1)=cγ(u,ρ(γ′)−1)=(u,ρ(γ′)−1ρ(γ)−1),
+soρ(γγ′)−1=ρ(γ′)−1ρ(γ)−1, andρ(γγ′) =ρ(γ)ρ(γ′) forγ,γ′∈G. Thus
+ρ:G→His a group morphism.
+UsingW′∼=U′×H,a◦cγ=r(γ)◦a, andcγ(u,δ) = (u,δρ(γ)−1), we see that
+close to{u}×H,cγ|W′:U′×H→U′×Hacts asr(γ) onU′andδ/ma√sto→δρ(γ)−1
+onH. MakingU′smaller if necessary, we can suppose this happens on all of
+U′. Writek:U′֒→Wfor the inclusion of U′as an open C∞-subscheme in
+Wvia the identifications U′∼=U′×{1}⊆U′×H∼=W′⊆W, and define
+f=b◦k:U′→V.
+Letγ∈G. Sincecγ|W′acts asr(γ) onU′andδ/ma√sto→δρ(γ)−1onH, anddρ(γ)
+acts asδ/ma√sto→ρ(γ)δonH, we see that dρ(γ)◦cγacts asr(γ)×id1onU′×{1}.
+Hencek◦r(γ)|U′=dρ(γ)◦cγ◦k. Composing with bgives
+f◦r(γ)|U′=b◦k◦r(γ)|U′=b◦dρ(γ)◦cγ◦k
+=s(ρ(γ))◦b◦cγ◦k=s(ρ(γ))◦b◦k=s(ρ(γ))◦f,
+82usingb◦dδ=s(δ)◦bandb◦cγ=b. We have now constructed a C∞-scheme
+morphismf:U′→Vand a group morphism ρ:G→Hwithf◦r(γ)|U′=
+s(ρ(γ))◦ffor allγ∈G. Thus Definition 7.2 defines [ f,ρ] : [U′/G]→[V/H].
+Consider the diagram of 2-morphisms:
+f◦i|[U′/G]◦π[U′/G]ν/d43/d51j◦[f,ρ]◦π[U′/G] j◦π[V/H]◦¯f
+f◦i◦π[U/G]◦¯a◦¯kω∗id¯k/d43/d51j◦π[V/H]◦¯b◦¯k.(7.6)
+Hereωis as in (7.5), and we have used f=b◦k, so that ¯f=¯b◦¯k, andπ[U′/G]=
+π[U/G]◦¯a◦¯ksincea◦kis the inclusion U′֒→U, and [f,ρ]◦π[U′/G]=π[V/H]◦¯f.
+Thus there is a unique 2-morphism ν=ω∗id¯kmaking (7.6) commute.
+Usingω∗id¯cγ=ω⊙(idf◦i∗λγ∗id¯a) forγ∈Gwe can show that νisG-
+invariant in a suitable sense, and so pushes down from ¯U′to [U′/G]. That is,
+there exists a unique 2-morphism ζ:f◦i|[U′/G]⇒j◦[f,ρ] withν=ζ∗idπ[U′/G].
+So (7.3) 2-commutes, completing part (b).
+For (c), let W,a,b,ω,cγ,dδ,W′,k,f,ρbe the data constructed in (b) above
+forf:X →Y, and let ˆW,ˆa,ˆb,ˆω,ˆcγ,ˆdδ,ˆW′,ˆk,g,σbe the corresponding data
+constructed in (b) for g:X→Y. Then combining η:f⇒gwith the analogue
+of (7.5) for g, we have a 2-morphism
+(η∗idi◦π[U/G]◦¯ˆa)⊙ˆω: (f◦i◦π[U/G])◦¯ˆa=⇒(j◦π[V/H])◦¯ˆb.
+Arguing as in the construction of cγabove, by the 2-Cartesian property of
+(7.5), there exists a 1-morphism e:¯ˆW→¯W, unique up to 2-isomorphism, and
+2-morphisms ˆζ: ¯a◦e⇒¯ˆa,ˆθ:¯b◦e⇒¯ˆbsatisfying (A.6). Then e∼=¯efor a unique
+e:ˆW→W. Replacing eby ¯e, we havea◦e= ˆa,b◦e=ˆb,ˆζ= idˆaandˆθ= idˆb,
+and (A.6) reduces to ω∗id¯e= (η∗idi◦π[U/G]◦¯ˆa)⊙ˆω.
+By repeating this for η−1:g⇒f, we can easily show that e:ˆW→W
+is an isomorphism, and identifies a,b,ω,cγ,dδ,W′withˆW,ˆa,ˆb,ˆω,ˆcγ,ˆdδ,ˆW′,
+respectively. However,theisomorphisms W′∼=U′×HandˆW′∼=U′×Hinvolved
+arbitrary choices of local trivializations of the principal H-bundlesa:W→U
+and ˆa:ˆW→U, soeneed not identify these isomorphisms.
+Abuse notation by identifying W′=U′×HandˆW′=U′×H. Since
+a◦e(u,1) = ˆa(u,1) =uwe see that e′(u,1) = (u,δ) for some unique δ∈H. As
+eidentifiesdǫandˆdǫforǫ∈Hwe have
+e(u,ǫ) =e◦ˆdǫ(u,1) =dǫ◦e(u,1) =dǫ(u,δ) = (u,ǫδ). (7.7)
+Similarly, as eidentifiescǫand ˆcγforγ∈G, andcγ,ˆcγact on{u}×Hby right
+multiplication by ρ(γ)−1,σ(γ)−1inH, we have
+e(u,σ(γ)−1) =e◦ˆcγ(u,1) =cγ◦e(u,1) =cγ(u,δ) = (u,δρ(γ)−1).(7.8)
+Comparing (7.8) and (7.7) with ǫ=σ(γ)−1, we see that σ(γ)−1δ=δρ(γ)−1, so
+σ(γ) =δρ(γ)δ−1for allγ∈Γ.
+83Sincea◦e= ˆa, regarding e|W′as a morphism U′×H→U′×H, we have
+πU′◦e|W′=πU′. So by (7.7), e|W′is near{u}×Hthe product of id U′onU′
+andǫ/ma√sto→ǫδonH. Choose a G-invariant open neighbourhood U′′ofuinU′such
+thate|U′′×His the product of id U′′andǫ/ma√sto→ǫδ. Then
+g|U′′=ˆb◦ˆk|U′′=b◦e◦ˆk|U′′=b◦dδ◦k|U′′=s(δ)◦b◦k|U′′=s(δ)◦f|U′′.
+Henceσ(γ) =δρ(γ)δ−1for allγ∈Gandg|U′′=s(δ)◦f|U′′. Thus byDefinition
+7.3 we have a quotient 2-morphism [ δ] : [f|U′′,ρ]⇒[g|U′′,σ]. An argument
+similar to the last part of (b) then shows that (7.4) commutes.
+Using the method of Theorem 7.18(c), we can also prove:
+Proposition 7.19. Let[f,ρ],[g,σ] : [X/G]→[Y/H]be quotient 1-morphisms
+of quotient C∞-stacks in the sense of §7.1,and suppose [X/G]is connected,
+that is,X/Gis connected as a topological space. Then every 2-morphism η:
+[f,ρ]⇒[g,σ]inC∞Stais a quotient 2-morphism [δ] : [f,ρ]⇒[g,σ]from
+Definition 7.3,for some unique δ∈H.
+Proof.Letη: [f,ρ]⇒[g,σ] be a 2-morphism. The proof of Theorem 7.18(c)
+shows that for each [ x]∈[X/G]top∼=X/G, there exists a unique δ[x]∈Hand
+an open neighbourhood [ U[x]/G] of [x] in [X/G], whereU[x]⊆XisG-invariant
+and open, such that η|[U[x]/G]= [δ[x]]|[U[x]/G]: [f,ρ]|[U[x]/G]⇒[g,σ]|[U[x]/G].
+The mapX/G→Htaking [x]/ma√sto→δ[x]is locally constant, as it is constant
+on each such open [ U[x]/G], so it is globally constant as X/Gis connected,
+andδ[x]=δ∈Hfor all [x]∈X/G. Thus, [X/G] may be covered by open
+[U[x]/G]⊆[X/G] withη|[U[x]/G]= [δ]|[U[x]/G]. As 2-morphisms in C∞Staform
+a sheaf, this proves that η= [δ].
+IfX=¯Xfor someC∞-schemeXthen Iso X([x])∼={1}for all [x]∈Xtop.
+Conversely, a Deligne–Mumford C∞-stack with trivial isotropy groups is a C∞-
+scheme. Notethatinconventionalalgebraicgeometry,aDeligne–M umfordstack
+with trivial stabilizers is an algebraic space , but need not be a scheme.
+Theorem 7.20. SupposeXis a Deligne–Mumford C∞-stack with IsoX([x])∼=
+{1}for all[x]∈Xtop. ThenXis equivalent to ¯Xfor someC∞-schemeX.
+Proof.As Iso X([x])∼={1}for all [x]∈Xtop, by Theorem 7.18(a) there is an
+open cover{Xa:a∈A}ofXwithXa≃[Xa/{1}]≃¯Xafor affineC∞-schemes
+Xa,a∈A. Writeia:¯Xa→Xfor the corresponding open embedding. As ∆ X
+is representable, for a,b∈Athe fibre product ¯Xa×ia,X,ib¯Xbis represented by a
+C∞-schemeXab=Xbawith open embeddings iab:Xab→Xa,iba:Xba→Xb
+identifying Xabwith openC∞-subschemes of Xa,Xb.
+The idea now is that the C∞-stackXis made by gluing the C∞-schemesXa
+fora∈Atogether on the overlaps Xab, that is, we identify Xa⊃iab(Xab)∼=
+Xab=Xba∼=iab(Xba)⊂Xb. This is similar to the notion ofdescent for objects
+in§A.3, and it is easy to check that the natural 1-isomorphisms
+¯Xab×X¯Xc∼=¯Xbc×X¯Xa∼=¯Xca×X¯Xb∼=¯Xa×X¯Xb×X¯Xc
+84imply the obvious compatibility conditions of the gluing morphisms iabon triple
+overlaps, and that Xaa∼=Xa. So by a minor modification of the proof in
+Proposition 6.2 that ( C∞Sch,J) has descent for objects, we construct a C∞-
+schemeXwith open embeddings ja:Xa֒→Xsuch that{Xa:a∈A}is an
+open cover of X, andXa×ja,X,jbXbis identified with Xabfora,b∈A. Then
+by descent for morphisms in ( C∞Sch,J), there exists a 1-morphism i:¯X→X
+withia2-isomorphic to i◦¯jafor alla∈A. Thisiis an equivalence, so X≃¯X,
+as we have to prove.
+In fact in Theorem 7.20 we can take X=Xtop, forXtopas in Definition
+6.18. Recall from Definition A.14 that a 1-morphism of C∞-stacksf:X→Yis
+representable if whenever Uis aC∞-scheme and g:¯U→Ya 1-morphism then
+the fibre product W=X×f,Y,g¯UinC∞Stais equivalent to a C∞-scheme ¯V.
+Corollary 7.21. Letf:X →Ybe a1-morphism of Deligne–Mumford C∞-
+stacks. Then fis representable if and only if f∗: IsoX([x])→IsoY([y])in
+Definition 6.21is injective for all [x]∈Xtopwithftop([x]) = [y]∈Ytop.
+Proof.Supposefis representable, and let [ x]∈Xtopwithftop([x]) = [y]∈
+Ytop. Theny:¯∗→Y, andX×f,Y,y¯∗≃[∗/H], whereH= Ker/parenleftbig
+f∗: IsoX([x])→
+IsoY([y])/parenrightbig
+. Asfis representable, [ ∗/H] is equivalent to a C∞-scheme, so H=
+{1}, andf∗is injective. This proves the ‘only if’ part.
+Now suppose f∗is injective for all [ x]∈ Xtop. LetUbe aC∞-scheme
+andg:¯U→Ya 1-morphism, and define W=X×f,Y,g¯U, with projections
+d:W →X ande:W →¯U. ThenWis a Deligne–Mumford C∞-stack by
+Theorem 7.10, as X,Y,¯Uare. Let [w]∈ Wtop, and set [x] =dtop([w]) in
+Xtop, [u] =etop([w]) in¯Utop, and [y] =ftop([x]) =gtop([u]) inYtop. Then by
+properties of fibre products of C∞-stacks we have a Cartesian square of groups
+IsoW([w])e∗/d47/d47
+d∗/d15/d15Iso¯U([u])
+g∗/d15/d15
+IsoX([x])f∗/d47/d47IsoY([y]).
+But Iso ¯U([u]) ={1}as¯Uis aC∞-scheme, and f∗is injective by assumption,
+so IsoW([w]) ={1}, for all [w]∈Wtop. Thus Theorem 7.20 shows Wis a
+C∞-scheme, and fis representable, proving the ‘if’ part.
+We show thatXbeing Deligne–Mumford is essential in Theorem 7.20:
+Example 7.22. Let the group Z2act onRby (a,b) :x/ma√sto→x+a+b√
+2 for
+a,b∈Zandx∈R. As√
+2 is irrational, this is a free action. It defines
+a groupoid Z2×R⇒RinManwhich is ´ etale, but not proper. Applying
+FC∞Sch
+Mangives a groupoid Z2×R⇒RinC∞Sch, and an associated C∞-stack
+X= [R/Z2] = [Z2×R⇒R]. The underlying topological space XtopisR/Z2.
+Since each orbit of Z2inRis dense in R,Xtophas the indiscrete topology,
+that is, the only open sets are ∅andXtop. ThusXtopis not homeomorphic
+toXfor anyC∞-schemeX= (X,OX), as each point of Xhas an affine and
+85hence Hausdorff open neighbourhood. Therefore Xis not equivalent to ¯Xfor
+anyC∞-schemeX. SoXis not Deligne–Mumford by Theorem 7.20. Hence,
+C∞-stacks with finite isotropy groups need not be Deligne–Mumford.
+7.5 Effective Deligne–Mumford C∞-stacks
+Definition 7.23. A Deligne–Mumford C∞-stackXis called effective if when-
+ever [x]∈XtopandXnear [x] is locally modelled near [ x] on a quotient C∞-
+stack[U/G]near[u],whereG= IsoX([x])andu∈UisfixedbyG, asinTheorem
+7.18(a), then Gacts effectively on Unearu. That is, for each 1 /\e}atio\slash=γ∈G, we
+haver(γ)/\e}atio\slash≡idUnearuinU, wherer:G→Aut(U) is theG-action.
+Here theC∞-schemeUin Theorem 7.18(a) is determined by X,[x] up to
+G-equivariant isomorphism locally near u. Hence to test whether Xis effective,
+it is enough to consider one choice of [ U/G] for each [x]∈Xtop.
+A quotient C∞-stack [X/G] is effective if and only if the action r:G→
+Aut(X) ofGonXislocally effective , that is, if for each 1 /\e}atio\slash=γ∈Gwe have
+r(γ)|U/\e}atio\slash≡idUfor every nonempty open C∞-subscheme U⊆X. If a Deligne–
+MumfordC∞-stackXis aC∞-scheme, it is automatically effective. Quotients
+[∗/G] forG/\e}atio\slash={1}are not effective.
+Here is a uniqueness property of 2-morphisms of effective Deligne–M umford
+C∞-stacks. Embeddings and submersions of C∞-stacks are defined in §6.2.
+Proposition 7.24. Letf,g:X →Y be1-morphisms of Deligne–Mumford
+C∞-stacks. Suppose any one of the following conditions hold:
+(i)Xis effective and fis an embedding of C∞-stacks (this implies f∗:
+IsoX([x])→IsoY(ftop([x]))is an isomorphism for each [x]∈Xtop);
+(ii)Yis effective and fis a submersion; or
+(iii)Yis aC∞-scheme.
+Then there exists at most one 2-morphism η:f⇒g. That is, the groupoid of
+such1-morphisms is equivalent to a set.
+Proof.Supposeη,˜η:f⇒gare 2-morphisms. Let [ x]∈Xtopwithftop([x]) =
+[y]∈ Ytop. Apply Theorem 7.18(c) to η,˜η. This first applies (a) to X,Y
+at [x],[y], givingi: [U/G]∼−→U ⊆ X identifying u∈Uwith [x] andj:
+[V/H]∼−→V⊆Y identifying v∈Uwith [y], and then applies (b) to f,ggiv-
+ingu∈U′⊆Uand 1-morphisms [ f,ρ],[g,σ] : [U/G]→[V/H]. Then (c)
+forηand ˆηgivesG-invariant open u∈U′′,˜U′′⊆U′and elements δ,˜δ∈H
+with 2-morphisms [ δ] : [f|U′′,ρ]⇒[g|U′′,σ], [˜δ] : [f|˜U′′,ρ]⇒[g|˜U′′,σ] such that
+(7.4) and its analogue for ˜ η,˜δ,˜U′′commutes. Making U′′,˜U′′smaller, we can
+takeU′′=˜U′′.
+The 2-morphisms [ δ],[˜δ] : [f|U′′,ρ]⇒[g|U′′,σ] imply that
+s(δ)◦f|U′′=g|U′′=s(ˆδ)◦f|U′′. (7.9)
+86We will show that (7.9) and each of conditions (i)–(iii) force δ=ˆδ. In case (i),
+asfis an embedding, ρ:G→His an isomorphism, and f:U→Vis an
+embedding of C∞-schemes. Hence δ=ρ(γ),ˆδ=ρ(ˆγ) forγ,ˆγ∈G, and
+f◦r(γ)|U′′=s(δ)◦f|U′′=s(ˆδ)◦f|U′′=f◦r(ˆγ)|U′′
+by (7.9). As fis an embedding this implies that r(γ)|U′′=r(ˆγ)|U′′, soγ= ˆγas
+Gacts effectively on UnearusinceXis effective, and thus δ=ˆδ.
+In case (ii), as fis a submersion, f:U→Vis surjective near f(u) =v∈V.
+Hence (7.9) implies that s(δ)|V′′=s(ˆδ)|V′′for some open neighbourhood V′′of
+vinV. ButHacts effectively on VnearvasYis effective, so δ=ˆδ. In case
+(iii)H= IsoY([y]) ={1}asYis aC∞-scheme, so δ=ˆδ= 1. Therefore δ=ˆδin
+each case. Equation (7.4) for η,ˆηnow implies that η∗idi|[U′′/G]= ˆη∗idi|[U′′/G].
+LetU′′⊆U⊆X be the open C∞-substack identified with [ U′′/G]. Then
+i|[U′′/G]: [U′′/G]→U′′is an equivalence, so η∗idi|[U′′/G]= ˆη∗idi|[U′′/G]implies
+thatη|U′′= ˆη|U′′. Thus, each [ x]∈Xtophas an open neighbourhood U′′inX
+withη|U′′= ˆη|U′′. As 2-morphisms form a sheaf on restriction to Zariski open
+C∞-substacks, this implies that η= ˆη, soη:f⇒gis unique.
+Similar arguments show that if f,g:X→Yare arbitrary 1-morphisms of
+Deligne–Mumford C∞-stacks withXconnected, then there are at most finitely
+many 2-morphisms η:f⇒g.
+7.6 Orbifolds as Deligne–Mumford C∞-stacks
+Orbifolds are geometric spaces locally modelled on Rn/GforGa finite group
+acting linearly on Rn, just as manifolds are geometric spaces locally modelled
+onRn. Much has been written about orbifolds, and there are severalde finitions,
+as either categories or 2-categories. See Lerman [47] for a good o verview.
+There are three main definitions of ordinary categories of orbifolds :
+(a) Satake [62] and Thurston [65] defined an orbifold Xto be a Hausdorff
+topological space Xwith an atlas/braceleftbig
+(Vi,Γi,ψi) :i∈I/bracerightbig
+of orbifold charts
+(Vi,Γi,ψi), whereViis a manifold, Γ ia finite group acting on Vi, and
+ψi:Vi/Γi→Xa homeomorphism with an open set in X. Smooth maps
+f:X→Ybetween orbifolds are continuous maps f:X→Ywhich lift
+locally to smooth maps on the charts, giving a category OrbST.
+(b) Chen and Ruan [15, §4] defined orbifolds Xin a similar way to [62,65], but
+using germs of orbifold charts ( Vp,Γp,ψp) forp∈X. Their morphisms
+f:X→Yare called good maps , giving a category OrbCR.
+(c) Moerdijk and Pronk [50,51] defined a category of orbifolds OrbMPas
+proper ´ etale Lie groupoids inMan. Their smooth maps f:X→Y, called
+strong maps [51,§5], are equivalence classes of diagrams Xφ←−X′ψ−→Y,
+whereX′is a third orbifold, and φ,ψare morphisms of groupoids with φ
+an equivalence.
+87A book on orbifolds in the sense of [15,50,51] is Adem, Leida and Rua n [2].
+There are four main definitions of 2-categories of orbifolds:
+(i) Pronk [58] defines a strict 2-category LieGpd of Lie groupoids in Manas
+in (c), with the obvious 1-morphisms of groupoids, and localizes by a c lass
+ofweakequivalences Wtoget aweak2-category OrbPr=LieGpd [W−1].
+(ii) Lerman [47, §3.3] defines a weak 2-category OrbLeof Lie groupoids in
+Manas in (c), with a non-obvious notion of 1-morphism called ‘Hilsum–
+Skandalis morphisms’ involving ‘bibundles’, and does not need to localize .
+Henriques and Metzler [33] also use Hilsum–Skandalis morphisms.
+(iii) Behrend and Xu [5, §2], Lerman [47,§4] and Metzler [49, §3.5] define a
+strict 2-category of orbifolds OrbManStaas a class of Deligne–Mumford
+stacks on the site ( Man,JMan) of manifolds with Grothendieck topology
+JMancoming from open covers.
+(iv) The author [39, §4.5] defines a weak 2-category of orbifolds OrbKuras
+special examples of Kuranishi spaces.
+As in Behrend and Xu [5, §2.6], Lerman [47], Pronk [58], and the author
+[39, Rem. 4.51(a)], approaches (i)–(iv) give equivalent weak 2-cate goriesOrbPr,
+OrbLe,OrbManSta,OrbKur. Properties of localization imply that OrbMP≃
+Ho(OrbPr). Thus, all of (c) and (i)–(iv) are equivalent at the level of weak
+2-categories or homotopy categories.
+Here is yet another definition of a strict 2-category of orbifolds OrbC∞Sta,
+which is similar to (iii), but defining orbifolds as a class of C∞-stacks, that is,
+as stacks on the site ( C∞Sch,J) rather than on ( Man,JMan).
+Definition 7.25. AC∞-stackXis called an orbifoldif it is equivalent to the
+C∞-stack [V⇒U] associated to a groupoid ( U,V,s,t,u,i,m) inC∞Schwhich
+is the image under FC∞Sch
+Manof a groupoid ( U,V,s,t,u,i,m ) inMan, where
+s:V→Uis an ´ etale smooth map, and s×t:V→U×Uis a proper smooth
+map. That is,Xis theC∞-stack associated to a proper ´ etale Lie groupoid in
+Man. WriteOrbC∞Stafor the full 2-subcategory of orbifolds in C∞Sta.
+As in§4.4,U,Vare finitely presented affine C∞-schemes, and thus Xis a
+separated, locally finitely presented Deligne–Mumford C∞-stack by Theorem
+7.17(b). Hence OrbC∞Sta⊂DMC∞Stalfp.
+The next theorem follows from the proofs in [5,39, 47, 58] that (i) –(iv)
+above are equivalent 2-categories (in particular, that orbifolds in ( iii) as stacks
+on (Man,JMan) associated to proper ´ etale Lie groupoids are equivalent to
+(i),(ii),(iv)), and the fact that the inclusion Man֒→C∞Schis full and faith-
+ful, with open covers JinC∞Schrestricting to open covers JManinMan.
+Theorem 7.26. This2-category of orbifolds OrbC∞Stais equivalent to the
+2-categories of orbifolds OrbPr,OrbLe,OrbManSta,OrbKurin[5,39,47,49,58]
+described in (i)–(iv)above. Also the homotopy category Ho(OrbC∞Sta)is equiv-
+alent to the category of orbifolds OrbMPin[50,51]described in (c)above.
+88By Corollary 4.27 FC∞Sch
+Mantakes transverse fibre products in Manto fibre
+products in C∞Sch. As fibre products of orbifolds are locally modelled on fibre
+products of manifolds, and fibre products of Deligne–Mumford C∞-stacks are
+locally modelled on fibre products of C∞-schemes, we deduce:
+Corollary 7.27. Transverse fibre products in OrbC∞Staagree with the corre-
+sponding fibre products in C∞Sta.
+8 Sheaves on Deligne–Mumford C∞-stacks
+Next we discuss quasicoherent sheaves on Deligne–Mumford C∞-stacksX, gen-
+eralizing§5 forC∞-schemes. Some references on sheaves on orbifolds or stacks
+areBehrendand Xu [5, §3.1], Deligne and Mumford [17, Def. 4.10], Heinloth [32,
+§4], Laumon and Moret-Bailly [46, §13], and Moerdijk and Pronk [51, §2]. Our
+definitions are closest to [32,51]. Almost everything in this section is a n exercise
+in stack theory, not special to C∞-stacks, and would also work for sheaves (with
+´ etale descent) on other kinds of Deligne–Mumford stacks.
+8.1 Quasicoherent sheaves
+We build our notions of sheaves on Deligne–Mumford C∞-stacksXfrom those
+of sheaves on C∞-schemesUin§5, by lifting to ´ etale covers ¯U→X. Since
+allOU-modules on a C∞-schemeUare quasicoherent by Corollary 5.22, we do
+not distinguish between OX-modules and quasicoherent sheaves on a Deligne–
+MumfordC∞-stackX, and we will just call them quasicoherent sheaves.
+Definition 8.1. LetXbe a Deligne–Mumford C∞-stack. Define a category
+CXto have objects pairs ( U,u) whereUis aC∞-scheme and u:¯U→Xis an
+´ etale morphism, and morphisms ( f,η) : (U,u)→(V,v) wheref:U→Vis an
+´ etale morphism of C∞-schemes, and η:u⇒v◦¯fis a 2-isomorphism. (Here
+f´ etale is implied by u,v´ etale andu∼=v◦¯f.) If (f,η) : (U,u)→(V,v) and
+(g,ζ) : (V,v)→(W,w) are morphisms in CXthen we define the composition
+(g,ζ)◦(f,η) to be (g◦f,θ) : (U,u)→(W,w), whereθis the composition of
+2-morphisms across the diagram:
+¯U
+¯f
+/d38/d38◆◆◆◆◆◆◆◆◆u
+/d41/d41
+g◦f
+/d15/d15❴❴❴❴ /d107/d115
+id¯Vv/d47/d47
+¯g/d120/d120♣♣♣♣♣♣♣♣☞☞☞☞ /d2/d10η
+X.
+¯Ww/d53/d53☞☞☞☞ /d2/d10ζ
+Define a quasicoherent sheaf EonXto assign a quasicoherent sheaf E(U,u)
+onUfor all objects ( U,u) inCX, and an isomorphism E(f,η):f∗(E(V,v))→
+E(U,u) in qcoh(U) for all morphisms ( f,η) : (U,u)→(V,v) inCX, such that
+89for all (f,η),(g,ζ),(g◦f,θ) as above the following diagram of isomorphisms in
+qcoh(U) commutes:
+(g◦f)∗/parenleftbig
+E(W,w)/parenrightbig
+E(g◦f,θ)/d47/d47
+If,g(E(W,w)) /d42/d42❚❚❚❚❚❚❚E(U,u),
+f∗/parenleftbig
+g∗(E(W,w)/parenrightbigf∗(E(g,ζ))/d47/d47f∗/parenleftbig
+E(V,v)/parenrightbigE(f,η)/d55/d55♣♣♣♣♣♣(8.1)
+forIf,g(E) as in Remark 5.14.
+Amorphism of quasicoherent sheaves φ:E → F assigns a morphism
+φ(U,u) :E(U,u)→ F(U,u) in qcoh(U) for each object ( U,u) inCX, such
+that for all morphisms ( f,η) : (U,u)→(V,v) inCXthe following commutes:
+f∗/parenleftbig
+E(V,v)/parenrightbig
+f∗(φ(V,v))/d15/d15E(f,η)/d47/d47E(U,u)
+φ(U,u)/d15/d15
+f∗/parenleftbig
+F(V,v)/parenrightbig F(f,η)/d47/d47F(U,u).
+We callEavector bundle of rank nifE(U,u) is a vector bundle of rank nfor
+all (U,u)∈CX. Write qcoh(X) for the category of quasicoherent sheaves on X.
+Remark 8.2. (a) Here is a second way to define quasicoherent sheaves, closer
+to [5,§3.1], [17, Def. 4.10]. Define a Grothendieck pretopology PJXonCX
+to have coverings/braceleftbig
+(ia,ηa) : (Ua,ua)→(U,u)/bracerightbig
+a∈Awhereia:Ua→Uis an
+open embedding for all a∈AandU=/uniontext
+a∈Aia(Ua). LetJXbe the associated
+Grothendieck topology. Then ( CX,JX) is a site.
+We can now use the standard notion of sheaves on a site , as in Artin [3] or
+Metzler [49,§2.1]. For all ( U,u) inCX, define aC∞-ringOX(U,u) =OU(U),
+whereU= (U,OU). For all morphisms ( f,η) : (V,v)→(U,u), define a mor-
+phism ofC∞-ringsρ(U,u)(V,v):OX(U,u)→OX(V,v) byρ(U,u)(V,v)=f♯(U) :
+OU(U)→OV(V). ThenOXis asheaf ofC∞-rings on the site (CX,JX).
+Define a quasicoherent sheaf E′to be a sheaf ofOX-modules on (CX,JX).
+That is,E′assignsanOX(U,u)-moduleE′(U,u) for all (U,u) inCX, and a linear
+mapE′
+(f,η):E(U,u)→E(V,v) for all (f,η) : (V,v)→(U,u) inCX, such that
+the analogue of (5.13) commutes, and the axioms for sheaves on a s ite hold.
+IfEis as in Definition 8.1 then defining E′(U,u) = Γ/parenleftbig
+E(U,u)/parenrightbig
+gives a qua-
+sicoherent sheaf in the sense of this second definition. Conversely , any quasico-
+herent sheaf in this second sense extends to one in the first sense uniquely up
+to canonical isomorphism. Thus the two definitions yield equivalent ca tegories.
+(b)As quasicoherent sheaves are a kind of sheaves of sets on a site, n ot sheaves
+of categories on a site as stacks are, qcoh( X) is a category not a 2-category.
+(c)In Definition 8.1 we require the 1-morphisms u,v,wand morphisms f,gto
+be´ etale. This is important in several places below: for instance, if f:U→Vis
+´ etalethenf∗: qcoh(V)→qcoh(U) isexact, notjust rightexact, whichisneeded
+in Proposition 8.3 to show qcoh( X) is abelian, and also Ω f:f∗(T∗V)→T∗U
+is an isomorphism, which is needed to define the cotangent sheaf T∗X. We
+90restricted to Deligne–Mumford C∞-stacksXin order to be able to use ´ etale
+(1-)morphisms in this way. For C∞-stacksXwhich do not admit an ´ etale atlas,
+the approach above is inadequate and would need to be modified.
+(d)Our notion of vector bundles EoverXcorrespond to orbifold vector bundles
+whenXis an orbifold. That is, the isotropy groups Iso X([x]) ofXare allowed
+to act nontrivially on the vector space fibres E|xofE.
+(e)We can also use the method of Definition 8.1 (or the approach of (a))
+to define other kinds of sheaves on a Deligne–Mumford C∞-stackX, such as
+sheaves of sets, abelian groups, C∞-rings, ..., in the obvious way: we just take
+theE(U,u) to be a sheaf of sets, ... on Uinstead of a quasicoherent sheaf.
+Proposition 8.3. LetXbe a Deligne–Mumford C∞-stack. Then qcoh(X)is
+an abelian category.
+Proof.We define a complex in qcoh( X)
+0 /d47/d47Eφ/d47/d47Fψ/d47/d47G /d47/d470
+to beexactif and only if
+0 /d47/d47E(U,u)φ(U,u)/d47/d47F(U,u)ψ(U,u)/d47/d47G(U,u) /d47/d470
+is exact in qcoh( U) for all (U,u) inCX. Since each qcoh( U) in Definition 8.1 is
+abelian, and the functors f∗in Definition 8.1 are exact (not just right exact) as
+fis ´ etale, it is easy to show this makes qcoh( X) into an abelian category.
+Example 8.4. LetXbe aC∞-scheme. ThenX=¯Xis a Deligne–Mumford
+C∞-stack. Wewilldefineanequivalenceofcategories IX: qcoh(X)→qcoh(X).
+LetEbe an object in qcoh( X). If (U,u) is an object in CXthenu:¯U→
+X=¯Xis a 1-morphism, so as C∞Sch,¯C∞Schare equivalent (2-)categories u
+is 2-isomorphic to ¯ u:¯U→¯Xfor some unique morphism u:U→X. Define
+E′(U,u) =u∗(E). If (f,η) : (U,u)→(V,v) is a morphism in CXandu,vare
+associated to u,vas above, so that u=v◦f, then define
+E′
+(f,η)=If,v(E)−1:f∗(E′(V,v)) =f∗/parenleftbig
+v∗(E)/parenrightbig
+→(v◦f)∗(E) =E′(U,u).
+Then (8.1) commutes for all ( f,η),(g,ζ), soE′is a quasicoherent sheaf on X.
+Ifφ:E→Fis a morphism in qcoh( X) define a morphism φ′:E′→F′
+in qcoh(X) byφ′(U,u) =u∗(φ) foruassociated to uas above. Then defining
+IX:E /ma√sto→E′,IX:φ/ma√sto→φ′gives a functor qcoh( X)→qcoh(X). There is a
+natural inverse construction: if ˜Eis an object in qcoh( X) then˜E(X,¯idX) is an
+object in qcoh( X), and˜Eis canonically isomorphic to IX/parenleftbig˜E(X,¯idX)/parenrightbig
+. Using
+this we can show IXis an equivalence of categories.
+918.2 Writing sheaves in terms of a groupoid presentation
+LetXbe a Deligne–Mumford C∞-stack. ThenXadmits an ´ etale atlas Π :
+¯U→X, and as in§A.5 from Π we can construct a groupoid ( U,V,s,t,u,i,m)
+inC∞Sch, withs,t:V→U´ etale, such thatXis equivalent to the associated
+C∞-stack [V⇒U], and we have a 2-Cartesian diagram
+¯V¯t/d47/d47
+¯s/d15/d15 ✗✗✗✗/d71/d79η¯U
+Π/d15/d15¯UΠ/d47/d47X.
+We can now consider the objects ( U,Π) and (V,Π◦s) inCX, and the two
+morphisms ( s,idΠ◦s) : (V,Π◦s)→(U,Π) and (t,η) : (V,Π◦s)→(U,Π).
+Now letEbe an object in qcoh( X). Then we have quasicoherent sheaves
+E=E(U,Π) onUandE′=E(V,Π◦s) onV, and isomorphisms E(s,idΠ◦s):
+s∗(E)→E′andE(t,η):t∗(E)→E′in qcoh(V). Hence Φ =E−1
+(t,η)◦E(s,idΠ◦s)is
+an isomorphism of Φ : s∗(E)→t∗(E) in qcoh(V).
+We also have a 2-commutative diagram with all squares 2-Cartesian:
+¯W¯π1
+/d116/d116✐✐✐✐✐✐✐✐✐✐✐
+¯π2
+/d37/d37❏❏❏❏❏❏❏❏❏❏❏❏ ¯m/d47/d47¯V
+¯t /d116/d116✐✐✐✐✐✐✐✐✐✐✐
+¯s
+/d37/d37❏❏❏❏❏❏❏❏❏❏❏❏❏
+¯V
+¯s
+/d37/d37❏❏❏❏❏❏❏❏❏❏❏❏❏¯t/d47/d47¯U
+Π
+/d37/d37❏❏❏❏❏❏❏❏❏❏❏❏
+¯V¯s/d47/d47¯t
+/d116/d116✐✐✐✐✐✐✐✐✐✐✐¯U
+Π /d116/d116✐✐✐✐✐✐✐✐✐✐✐
+¯UΠ/d47/d47X,
+omitting 2-morphisms, where W=Vׯs,¯U,t¯V, andπ1,π2:W→Vare projec-
+tions to the first and second factors in the fibre product. So we ha ve an object
+(W,Π◦¯s◦¯π1) inCX, and we can define E′′=E(W,Π◦¯s◦¯π1). Then we have
+a commutative diagram of isomorphisms in qcoh( W):
+E′′m∗(E′)E(m,θ3)/d111/d111
+π∗
+1(E′)E(π1,θ1)/d57/d57rrrrrrrrrr(t◦π1)∗(E) =
+(t◦m)∗(E) π∗
+1(E(t,η))
+◦Iπ1,t(E)/d111/d111m∗(E(t,η))
+◦Im,t(E)/d57/d57rrrrrrr
+π∗
+2(E′)E(π2,θ2)/d92/d92✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿(s◦π2)∗(E) =
+(s◦m)∗(E)m∗(E(s,idΠ◦s))
+◦Im,s(E)/d92/d92✿✿✿✿✿✿✿✿✿✿✿✿✿✿
+π∗
+2(E(s,idΠ◦s))
+◦Iπ2,s(E)/d111/d111
+β/d112/d112❦❣❡❞❞❝❝❜❜❜α/d106/d106❯❯❯❯❯❯
+(s◦π1)∗(E) = (t◦π2)∗(E)π∗
+1(E(s,idΠ◦s))
+◦Iπ1,s(E)/d92/d92✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿π∗
+2(E(t,η))
+◦Iπ2,t(E)/d57/d57rrrrrrrrrγ/d66/d66☎
+☎
+☎
+☎
+☎
+☎
+☎(8.2)
+Here the morphisms ‘ /axisshort/axisshort/arrowaxisright’ are given by α=Im,t(E)−1◦m∗(Φ)◦Im,s(E),β=
+Iπ2,t(E)−1◦π∗
+2(Φ)◦Iπ2,s(E) andγ=Iπ1,t(E)−1◦π∗
+1(Φ)◦Iπ1,s(E), and as (8.2)
+commutes we have α=γ◦β. This motivates:
+92Definition 8.5. Let (U,V,s,t,u,i,m) be a groupoid in C∞Sch, withs,t:
+V→U´ etale, which we write as V⇒Ufor short. Define a quasicoherent sheaf
+onV⇒Uto be a pair ( E,Φ) whereEis a quasicoherent sheaf on Uand
+Φ :s∗(E)→t∗(E) is an isomorphism in qcoh( V), such that
+Im,t(E)−1◦m∗(Φ)◦Im,s(E) =/parenleftbig
+Iπ1,t(E)−1◦π∗
+1(Φ)◦Iπ1,s(E)/parenrightbig
+◦
+/parenleftbig
+Iπ2,t(E)−1◦π∗
+2(Φ)◦Iπ2,s(E)/parenrightbig
+in morphisms ( s◦m)∗(E)→(t◦m)∗(E) in qcoh(W). Define a morphism
+φ: (E,Φ)→(F,Ψ) of such sheaves to be a morphism φ:E→Fin qcoh(U)
+such that Ψ◦s∗(φ) =t∗(φ)◦Φ :s∗(E)→t∗(F) in qcoh(V). Then quasicoherent
+sheaves on V⇒Uform an abelian category qcoh( V⇒U).
+IfXis a Deligne–Mumford C∞-stack equivalent to [ V⇒U] with atlas
+Π :¯U→Xthen we have a functor FΠ: qcoh(X)→qcoh(V⇒U) defined by
+FΠ:E/ma√sto→/parenleftbig
+E(U,Π),E−1
+(t,η)◦E(s,idΠ◦s)/parenrightbig
+andFΠ:φ/ma√sto→φ(U,Π).
+The nexttheoremis provedasin LaumonandMoret-Bailly[46, Prop.1 2.4.5]
+or Olsson [56, Prop. 4.4].
+Theorem 8.6. The functor FΠ: qcoh(X)→qcoh(V⇒U)above is an equiva-
+lence of categories.
+For quotient C∞-stacks [U/G] withGa finite group, so that V=G×U, a
+quasicoherent sheaf ( E,Φ) onV⇒Uis a quasicoherent sheaf EonUwith a lift
+Φ of theG-action onUup toE. That is, (E,Φ) is aG-equivariant quasicoherent
+sheaf onU. Hence, ifaDeligne–Mumford C∞-stackXisequivalenttoaquotient
+[U/G] withGfinite, then qcoh( X) is equivalent to the category qcohG(U) of
+G-equivariant quasicoherent sheaves on U.
+8.3 Pullback of sheaves as a weak 2-functor
+In Definition 5.13, for a morphism of C∞-schemesf:X→Ywe defined a
+right exact functor f∗: qcoh(Y)→qcoh(X). As in Remarks 4.6(b) and 5.14,
+pullbacks cannot always be made strictly functorial in f, that is, we do not have
+f∗(g∗(E)) = (g◦f)∗(E) for allf:X→Y,g:Y→ZandE∈qcoh(Z), but
+instead we have canonical isomorphisms If,g(E) : (g◦f)∗(E)→f∗(g∗(E)).
+We now generalize this to pullback for sheaves on Deligne–Mumford C∞-
+stacks. The new factor to consider is that we have not only 1-morp hismsf:
+X→Y, but also 2-morphisms η:f⇒gfor 1-morphisms f,g:X→Y, and we
+must interpret pullback for 2-morphisms as well as 1-morphisms.
+Definition 8.7. Letf:X →Y be a 1-morphism of Deligne–Mumford C∞-
+stacks, andF∈qcoh(Y). Apullback ofFtoXisE∈qcoh(X), together with
+the following data: if U,VareC∞-schemes and u:¯U→Xandv:¯V→Yare
+´ etale 1-morphisms, then there is a C∞-schemeWand morphisms πU:W→U,
+93πV:W→Vgiving a 2-Cartesian diagram:
+¯W¯πV/d47/d47
+¯πU/d15/d15 ✖✖✖✖/d71/d79ζ¯V
+v/d15/d15¯Uf◦u/d47/d47Y.(8.3)
+Then an isomorphism i(F,f,u,v,ζ ) :π∗
+U/parenleftbig
+E(U,u)/parenrightbig
+→π∗
+V/parenleftbig
+F(V,v)/parenrightbig
+in qcoh(W)
+should be given, which is functorial in ( U,u) inCXand (V,v) inCYand the
+2-isomorphism ζin (8.3). We usually write pullbacks Easf∗(F).
+By a similar proof to Theorem 8.6, but using descent for objects and mor-
+phisms for quasicoherent sheaves on C∞-schemesYin the ´ etale topology rather
+than the open cover topology on Y, we can prove:
+Proposition 8.8. Letf:X→Ybe a1-morphism of Deligne–Mumford C∞-
+stacks, andFbe a quasicoherent sheaf on Y. Then a pullback f∗(F)exists in
+qcoh(X),and is unique up to canonical isomorphism.
+From now on we will assume that we have chosena pullback f∗(F) for all
+suchf:X→YandF. This could be done either by some explicit construction
+of pullbacks, as in the C∞-scheme case in§5.3, or by using the Axiom of Choice.
+As in Remark 5.14 we cannot necessarily make these choices functor ial inf.
+Definition 8.9. Choose pullbacks f∗(F) for all 1-morphisms f:X →Y of
+Deligne–Mumford C∞-stacks and allF∈qcoh(Y), as above.
+Letf:X →Y be such a 1-morphism, and φ:E →F be a morphism
+in qcoh(Y). Thenf∗(E),f∗(F)∈qcoh(X). Define the pullback morphism
+f∗(φ) :f∗(E)→f∗(F) to be the morphism in qcoh( X) characterized as follows.
+Letu:¯U→X,v:¯V→Y,W,πU,πVbe as in Definition 8.7, with (8.3)
+2-Cartesian. Then the following diagram of morphisms in qcoh( W) commutes:
+π∗
+U/parenleftbig
+f∗(E)(U,u)/parenrightbig
+i(E,f,u,v,ζ )/d47/d47
+πU∗(f∗(φ)(U,u))/d15/d15π∗
+V/parenleftbig
+E(V,v)/parenrightbig
+πV∗(φ(V,v))/d15/d15
+π∗
+U/parenleftbig
+f∗(F)(U,u)/parenrightbigi(F,f,u,v,ζ )/d47/d47π∗
+V/parenleftbig
+F(V,v)/parenrightbig
+.
+Usingdescentformorphismsinqcoh( Y)onC∞-schemesYinthe´ etaletopology,
+one can show that there is a unique morphism f∗(φ) with this property. This
+defines a functor f∗: qcoh(Y)→qcoh(X).
+Letf:X →Yandg:Y→Zbe 1-morphisms of Deligne–Mumford C∞-
+stacks, andE∈qcoh(Z). Then (g◦f)∗(E) andf∗(g∗(E)) both lie in qcoh( X).
+One can show that f∗(g∗(E)) is a possible pullback of Ebyg◦f. Thus as in
+Remark5.14, wehaveacanonicalisomorphism If,g(E) : (g◦f)∗(E)→f∗(g∗(E)).
+This defines a natural isomorphism of functors If,g: (g◦f)∗⇒f∗◦g∗.
+Letf,g:X→Ybe 1-morphisms of Deligne–Mumford C∞-stacks,η:f⇒g
+a 2-morphism, and E∈qcoh(Y). Then we have f∗(E),g∗(E)∈qcoh(X). Let
+94u:¯U→X,v:¯V→Y,W,πU,πVbe as in Definition 8.7. Then as in (8.3) we
+have 2-Cartesian diagrams
+¯W¯πV/d47/d47
+¯πU/d15/d15 ✓✓✓✓/d69/d77ζ⊙(η∗idu◦¯πU)¯V
+v/d15/d15¯W¯πV/d47/d47
+¯πU/d15/d15✆✆✆✆/d62/d70ζ¯V
+v/d15/d15¯Uf◦u/d47/d47Y, ¯Ug◦u/d47/d47Y,
+whereinζ⊙(η∗idu◦¯πU)‘∗’ishorizontalcompositionand‘ ⊙’verticalcomposition
+of 2-morphisms. Thus we have isomorphisms in qcoh( W):
+π∗
+U/parenleftbig
+f∗(E)(U,u)/parenrightbig
+i(E,f,u,v,ζ ⊙(η∗idu◦¯πU))
+/d45/d45❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬
+/d15/d15π∗
+V/parenleftbig
+E(V,v)/parenrightbig
+.
+π∗
+U/parenleftbig
+g∗(E)(U,u)/parenrightbigi(E,g,u,v,ζ )/d49/d49❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝
+There is a unique isomorphism ‘ /axisshort/axisshort/arrowaxisright’ making this diagram commute. Taken over
+all (V,v), using descent for morphisms we can show these isomorphisms are
+pullbacks of a unique isomorphism f∗(E)(U,u)→g∗(E)(U,u), and taken over
+all (U,u) these give an isomorphism η∗(E) :f∗(E)→g∗(E) in qcoh(X). Over
+allE∈qcoh(Y), this defines a natural isomorphism η∗:f∗⇒g∗.
+IfXisaDeligne–Mumford C∞-stackwithidentity1-morphismid X:X→X
+then for eachE ∈qcoh(X),Eis a possible pullback id∗
+X(E), so we have a
+canonical isomorphism δX(E) : id∗
+X(E)→E. These define a natural isomor-
+phismδX: id∗
+X⇒idqcoh(X).
+The proof of the next theorem is long but straightforward. Weak 2 -functors
+are defined in Definition A.2.
+Theorem 8.10. MappingXtoqcoh(X)for objectsXinDMC∞Sta,and
+mapping 1-morphisms f:X →Y tof∗: qcoh(Y)→qcoh(X),and mapping
+2-morphisms η:f⇒gtoη∗:f∗⇒g∗for1-morphisms f,g:X→Y,and the
+natural isomorphisms If,g: (g◦f)∗⇒f∗◦g∗for all1-morphisms f:X→Y
+andg:Y →Z inDMC∞Sta,andδXfor allX ∈DMC∞Sta,together
+make up a weak 2-functor (DMC∞Sta)op→AbCat,whereAbCat is the
+2-category of abelian categories. That is, they satisfy the c onditions:
+(a)Iff:W→X, g:X→Y, h:Y→Zare1-morphisms in DMC∞Sta
+andE∈qcoh(Z)then the following diagram commutes in qcoh(X) :
+(h◦g◦f)∗(E)
+If,h◦g(E)/d47/d47
+Ig◦f,h(E)/d15/d15f∗/parenleftbig
+(h◦g)∗(E)/parenrightbig
+f∗(Ig,h(E))/d15/d15
+(g◦f)∗/parenleftbig
+h∗(E)/parenrightbigIf,g(h∗(E))/d47/d47f∗/parenleftbig
+g∗(h∗(E))/parenrightbig
+.
+(b)Iff:X→Yis a1-morphism in DMC∞StaandE∈qcoh(Y)then the
+95following pairs of morphisms in qcoh(X)are inverse:
+f∗(E) =
+(f◦idX)∗(E)IidX,f(E)/d45/d45id∗
+X(f∗(E)),
+δX(f∗(E))/d110/d110f∗(E) =
+(idY◦f)∗(E)If,idY(E)/d45/d45f∗(id∗
+Y(E)).
+f∗(δY(E))/d109/d109
+Also(idf)∗(idE) = idf∗(E):f∗(E)→f∗(E).
+(c)Iff,g,h:X → Y are1-morphisms and η:f⇒g, ζ:g⇒hare
+2-morphisms in DMC∞Sta,so thatζ⊙η:f⇒his the vertical compo-
+sition, andE∈qcoh(Y),then
+ζ∗(F)◦η∗(E) = (ζ⊙η)∗(E) :f∗(E)−→h∗(E)inqcoh(X).
+(d)Iff,˜f:X→Y,g,˜g:Y→Zare1-morphisms and η:f⇒f′, ζ:g⇒g′
+2-morphisms in DMC∞Sta,so thatζ∗η:g◦f⇒˜g◦˜fis the horizontal
+composition, and E∈qcoh(Z),then the following commutes in qcoh(X) :
+(g◦f)∗(E)
+(ζ∗η)∗(E)/d47/d47
+If,g(E)/d15/d15(˜g◦˜f)∗(E)
+I˜f,˜g(E)/d15/d15
+f∗(g∗(E))η∗(g∗(E))/d47/d47˜f∗(g∗(E))˜f∗(ζ∗(E))/d47/d47˜f∗(˜g∗(E)).
+Using Proposition 5.15 we may prove:
+Proposition 8.11. Letf:X → Y be a1-morphism of Deligne–Mumford
+C∞-stacks. Then pullback f∗: qcoh(Y)→qcoh(X)is a right exact functor.
+8.4 Cotangent sheaves of Deligne–Mumford C∞-stacks
+We now develop the analogue of the ideas of §5.6.
+Definition 8.12. LetXbe a Deligne–Mumford C∞-stack. Define a quasico-
+herent sheaf T∗XonXcalled the cotangent sheaf ofXby (T∗X)(U,u) =T∗U
+for all objects ( U,u) inCXand (T∗X)(f,η)= Ωf:f∗(T∗V)→T∗Ufor all mor-
+phisms (f,η) : (U,u)→(V,v) inCX, whereT∗Uand Ωfare as in§5.6. Here
+asf:U→Vis ´ etale Ω fis an isomorphism, so ( T∗X)(f,η)is an isomorphism
+in qcoh(U) as required. Also Theorem 5.32(a) shows that (8.1) commutes for
+E=T∗Xfor all such ( f,η),(g,ζ). HenceT∗Xis a quasicoherent sheaf.
+Letf:X →Y be a 1-morphism of Deligne–Mumford C∞-stacks. Define
+Ωf:f∗(T∗Y)→T∗Xto be the unique morphism in qcoh( X) characterized as
+follows. Let u:¯U→X,v:¯V→Y,W,πU,πVbe as in Definition 8.7, with
+(8.3) Cartesian. Then the following diagram in qcoh( W) commutes:
+π∗
+U/parenleftbig
+f∗(T∗Y)(U,u)/parenrightbig
+πU∗(Ωf(U,u))/d15/d15i(T∗Y,f,u,v,ζ )/d47/d47π∗
+V/parenleftbig
+(T∗Y)(V,v)/parenrightbigπ∗
+V(T∗V)
+ΩπV/d15/d15
+π∗
+U/parenleftbig
+(T∗X)(U,u)/parenrightbig(T∗X)(πU,idu◦πU)/d47/d47(T∗X)(W,u◦πU)T∗W.
+96This determines πU∗(Ωf(U,u)) uniquely. Over all ( V,v), using descent for mor-
+phisms in qcoh( U) onC∞-schemesUin the ´ etale topology, this determines the
+morphisms Ω f(U,u), and over all ( U,u) these determine Ω f.
+IfXis an orbifold of dimension nthenT∗Xis a vector bundle of rank n.
+Here is the analogue of Theorem 5.32:
+Theorem 8.13. (a) Letf:X → Y andg:Y → Z be1-morphisms of
+Deligne–Mumford C∞-stacks. Then
+Ωg◦f= Ωf◦f∗(Ωg)◦If,g(T∗Z) (8.4)
+as morphisms (g◦f)∗(T∗Z)→T∗Xinqcoh(X).
+(b)Letf,g:X →Y be1-morphisms of Deligne–Mumford C∞-stacks and
+η:f⇒ga2-morphism. Then Ωf= Ωg◦η∗(T∗Y) :f∗(T∗Y)→T∗X.
+(c)LetW,X,Y,Zbe Deligne–Mumford C∞-stacks in a 2-Cartesian square
+Wf/d47/d47
+e/d15/d15 ✗✗✗✗/d71/d79ηY
+h/d15/d15
+Xg/d47/d47Z
+inDMC∞Sta,so thatW=X×ZY. Then the following is exact in qcoh(W):
+(g◦e)∗(T∗Z)e∗(Ωg)◦Ie,g(T∗Z)⊕
+−f∗(Ωh)◦If,h(T∗Z)◦η∗(T∗Z)/d47/d47e∗(T∗X)⊕
+f∗(T∗Y)Ωe⊕Ωf/d47/d47T∗W /d47/d470.(8.5)
+Proof.For (a), let u:¯U→X,v:¯V→Yandw:¯W→Zbe ´ etale. Then
+there is aC∞-schemeV′with¯V′=¯V×g◦v,Z,w¯W, and fibre product projections
+πV:V′→V,πW:V′→W. Definev′=v◦¯πV:¯V′→Y. Thenv′is´ etale,asvis
+andwis soπVis. Similarly, there is a C∞-schemeU′with¯U′=¯U×f◦u,Y,v′¯V′,
+and fibre product projections πU:U′→U,πV′:U′→V′. Define an ´ etale
+1-morphism u′=u◦¯πU:¯U′→X. Then we have a 2-commutative diagram
+Xf/d47/d47Yg/d47/d47Z
+¯Uu/d107/d107❳❳❳❳❳❳❳❳❳❳❳❳/d51/d51❢❢❢❢❢❢❢❢❢❢❢❢
+❴❴❴❴ /d43/d51η¯Vv/d107/d107❳❳❳❳❳❳❳❳❳❳❳❳❳/d51/d51❢❢❢❢❢❢❢❢❢❢❢❢❢✬✬✬✬/d15/d23ζ
+¯Ww/d79/d79
+¯V′v′/d100/d100■■■■■■■■■■■■■■¯πV/d79/d79
+¯πW/d51/d51❣❣❣❣❣❣❣❣❣❣❣❣
+¯U′u′/d103/d103PPPPPPPPPPPPPPPPPPPPPPPPPPPP¯πU/d99/d99❍❍❍❍❍❍❍❍❍❍❍❍❍❍
+¯πV′/d51/d51❣❣❣❣❣❣❣❣❣❣❣❣
+97with 2-Cartesian squares. On U′andV′we have commutative diagrams:
+π∗
+U/parenleftbig
+f∗(T∗Y)(U,u)/parenrightbigi(T∗Y,f,u,v′,η)
+∼=/d47/d47
+∼=(f∗(T∗Y))(πU,idu′)
+/d15/d15π∗
+V′/parenleftbig
+(T∗Y)(V′,v′)/parenrightbigπ∗
+V′(T∗V′)
+ΩπV′/d15/d15
+(f∗(T∗Y))(U′,u′)Ωf(U′,u′)/d47/d47(T∗X)(U′,u′) T∗U′,(8.6)
+π∗
+V/parenleftbig
+g∗(T∗Z)(V,v)/parenrightbigi(T∗Z,g,v,w,ζ )
+∼=/d47/d47
+∼=(g∗(T∗Z))(πV,idv′)
+/d15/d15πW∗/parenleftbig
+T∗Z(W,w)/parenrightbigπW∗(T∗W)
+ΩπW/d15/d15
+(g∗(T∗Z))(V′,v′)Ωg(V′,v′)/d47/d47(T∗Y)(V′,v′) T∗V′.(8.7)
+Applyingπ∗
+V′to (8.7) we make another commutative diagram on U′:
+π∗
+V′/parenleftbig
+π∗
+V(g∗(T∗Z)(V,v))/parenrightbigπ∗
+V′(i(T∗Z,g,v,w,ζ ))
+∼=/d47/d47
+∼=π∗
+V′((g∗(T∗Z))(πV,idv′))/d15/d15π∗
+V′/parenleftbig
+πW∗(T∗W)/parenrightbig
+π∗
+V′(ΩπW)/d15/d15
+π∗
+V′/parenleftbig
+(g∗(T∗Z))(V′,v′)/parenrightbigπ∗
+V′(Ωg(V′,v′))/d47/d47
+∼=(f∗(g∗(T∗Z)))(πU,idu′)/d15/d15π∗
+V′(T∗V′)
+∼= (f∗(T∗U))(πU,idu′)/d15/d15/parenleftbig
+f∗(g∗(T∗Z))/parenrightbig
+(U′,u′)(f∗(Ωg))(U′,u′)/d47/d47/parenleftbig
+f∗(T∗Y)/parenrightbig
+(U′,u′).(8.8)
+By Theorem 5.32(a) the following commutes:
+(πW◦πV′)∗(T∗W)
+ΩπW◦πV′/d47/d47
+IπV′,πW(T∗W) ∼=/d15/d15T∗U′
+π∗
+V′/parenleftbig
+πW∗(T∗W)/parenrightbigπ∗
+V′(ΩπW)/d47/d47π∗
+V′(T∗V′).ΩπV′/d79/d79
+(8.9)
+Using all this we obtain a commutative diagram on U′:
+/parenleftbig
+(g◦f)∗(T∗Z)/parenrightbig
+(U′,u′)
+Ωg◦f(U′,u′)/d47/d47
+∼=(If,g(T∗Z))(U′,u′)
+/d15/d15/d104/d104∼=
+/d40/d40PPPPPPPPP(T∗X)(U′,u′)
+(πW◦πV′)∗(T∗W) /d47/d47
+∼=/d15/d15T∗U′♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥
+π∗
+V′/parenleftbig
+πW∗(T∗W)/parenrightbig/d47/d47π∗
+V′(T∗V′)/d79/d79
+/parenleftbig
+f∗(g∗(T∗Z))/parenrightbig
+(U′,u′)/d118/d118∼=/d54/d54♥♥♥♥♥♥♥♥(f∗(Ωg))(U′,u′)/d47/d47/parenleftbig
+f∗(T∗Y)/parenrightbig
+(U′,u′)./d40/d40∼=/d104/d104PPPPPPPPΩf(U′,u′)/d79/d79
+(8.10)
+Here the right hand quadrilateralof (8.10) comes from (8.6), the b ottom quadri-
+lateral from (8.8), the central square is (8.9), and the remaining t wo quadrilat-
+erals are similar. Thus, the outer square of (8.10) commutes. But t his is just
+(8.4) evaluated at ( U′,u′). Ifu:¯U→X,v:¯V→Yandw:¯W→Zare ´ etale
+98atlases then u′:¯U′→Xis also an ´ etale atlas, and (8.4) evaluated on an atlas
+implies it in general. This proves part (a).
+Part (b) is immediate from the definitions. For (c), let u:¯U→X,v:
+¯V→Yandw:¯W→Zbe ´ etale. There are C∞-schemesU′,V′,with¯U′=
+¯U×g◦u,Z,w¯W,¯V′=¯V×h◦v,Z,w¯W, and fibre product projections πU:U′→U,
+πW:U′→W,πV:V′→V,πW:V′→W. ThenπU,πVare ´ etale as wis.
+Define aC∞-schemeT=U′×πW,W,πWV′. The 1-morphisms u′◦¯πU′:¯T→X
+andv′◦¯πV′:¯T→Yhave a natural 2-isomorphism g◦(u′◦¯πU′)⇒h◦(v′◦¯πV′)
+constructed from the 2-isomorphisms in the 2-Cartesian squares definingU′,V′.
+Thus asW=X×ZYthere is a 1-morphism t:¯T→W, unique up to 2-
+isomorphism, such that u′◦¯πU′∼=e◦tandv′◦¯πV′∼=f◦t. Alsotis ´ etale. This
+gives a 2-commutative diagram
+¯T¯πU′
+/d117/d117❥❥❥❥❥❥❥❥❥❥
+¯πV′
+/d36/d36■■■■■■■■■■■t/d47/d47W
+e/d117/d117❥❥❥❥❥❥❥❥❥❥
+f
+/d36/d36■■■■■■■■■■■■
+¯U′
+¯πW/d36/d36■■■■■■■■■■■u′/d47/d47X
+g
+/d36/d36■■■■■■■■■■■
+¯V′ v′/d47/d47 ¯πW
+/d117/d117❥❥❥❥❥❥❥❥❥❥ Y
+h /d117/d117❥❥❥❥❥❥❥❥❥❥
+¯Ww/d47/d47Z,
+in which the leftmost and rightmost squares are 2-Cartesian.
+Applying Theorem 5.32(b) to the Cartesian square defining Tgives an exact
+sequence in qcoh( T):
+(πW◦πU′)∗
+(T∗W)π∗
+U′(ΩπW)◦IπU′,πW(T∗W)⊕
+−π∗
+V′(ΩπW)◦IπV′,πW(T∗W)/d47/d47π∗
+U′(T∗U′)
+⊕π∗
+V′(T∗V′)ΩπU′⊕
+ΩπV′/d47/d47T∗T /d47/d470.(8.11)
+By a similar argument to (a), we can use (8.11) to deduce that (8.5) e valuated
+at (T,t) holds. If u:¯U→X,v:¯V→Yandw:¯W→Zare atlases then
+t:¯T→Wis an atlas, so this implies (8.5), and proves (c).
+9 Orbifold strata of C∞-stacks
+LetXbe a Deligne–Mumford C∞-stack, with topological space Xtop. Then
+each point [ x]∈Xtophas an isotropy group Iso X([x]), a finite group defined
+up to isomorphism. For each finite group Γ we write ˜XΓ
+◦,top=/braceleftbig
+[x]∈Xtop:
+IsoX([x])∼=Γ/bracerightbig
+. This is a locally closed subset of Xtop, coming from a locally
+closedC∞-substack ˜XΓ
+◦ofXwith inclusion ˜OΓ
+◦(X) :˜XΓ
+◦→X, with
+Xtop=/coproductdisplay
+isomorphism classes
+of finite groups Γ˜XΓ
+◦,top. (9.1)
+One can show that for each Γ, the closure ˜XΓ
+◦,topof˜XΓ
+◦,topinXtopsatisfies
+˜XΓ
+◦,top⊆/coproductdisplay
+isomorphism classes of finite groups ∆:
+Γ is isomorphic to a subgroup of ∆˜X∆
+◦,top.
+99Thus (9.1) is a stratification of Xtop. The˜XΓ
+◦are called orbifold strata ofX.
+WhenXis an orbifold, as in §7.6, the orbifold strata are manifolds (actually,
+at the level of C∞-stacks, the alternative versions ˆXΓ
+◦below are manifolds), and
+are well studied. Orbifold strata of orbifolds come up in areas such a s the
+Atiyah–Singer Index Theorem for orbifolds as in Kawasaki [42], cobo rdism of
+orbifolds as in Druschel [20], String Theory of orbifolds as in Dixon et a l. [19],
+and (quantum) cohomology of orbifolds as in Chen and Ruan [14].
+However, very little appears to have been done in considering orbifo ld strata
+from the point of view of category theory or stacks, or about orb ifold strata
+of other kinds of Deligne–Mumford stacks. We now define and study orbifold
+strata of Deligne–Mumford C∞-stacks. Actually, almost all of §9 is an exercise
+in stack theory, not specific to C∞-stacks. But the author has been unable to
+find any references on it.
+We will define six variations on ˜XΓ
+◦outlined above, Deligne–Mumford C∞-
+stacks writtenXΓ,˜XΓ,ˆXΓ, and open C∞-substacksXΓ
+◦⊆ XΓ,˜XΓ
+◦⊆˜XΓ,
+ˆXΓ
+◦⊆ˆXΓ. The points and isotropy groups of XΓ,...,ˆXΓ
+◦are given by:
+(i) Points ofXΓare isomorphism classes [ x,ρ], where [x]∈Xtopandρ: Γ→
+IsoX([x]) is an injective morphism, and Iso XΓ([x,ρ]) is the centralizer of
+ρ(Γ) in Iso X([x]). Points ofXΓ
+◦⊆XΓare [x,ρ] withρan isomorphism,
+and Iso XΓ
+◦([x,ρ])∼=C(Γ), the centre of Γ.
+(ii) Points of ˜XΓare pairs [x,∆], where [ x]∈Xtopand ∆⊆IsoX([x]) is
+isomorphic to Γ, and Iso ˜XΓ([x,∆]) is the normalizer of ∆ in Iso X([x]).
+Points of ˜XΓ
+◦⊆˜XΓare [x,∆] with ∆ = Iso X([x]), and Iso ˜XΓ
+◦([x,∆])∼=Γ.
+(iii) Points [ x,∆] ofˆXΓ,ˆXΓ
+◦are the same as for ˜XΓ,˜XΓ
+◦, but with isotropy
+groups Iso ˆXΓ([x,∆])∼=Iso˜XΓ([x,∆])/∆ and Iso ˆXΓ
+◦([x,∆])∼={1}.
+There are 1-morphisms OΓ(X),...,ˆΠΓ
+◦(X) forming a 2-commutative diagram,
+where the columns are inclusions of open C∞-substacks:
+XΓ
+◦˜ΠΓ
+◦(X)/d47/d47
+OΓ
+◦(X)/d42/d42❯❯❯❯❯❯❯❯❯❯❯❯
+⊂
+/d15/d15Aut(Γ)/d44/d44 ˜XΓ
+◦ˆΠΓ
+◦(X)/d47/d47
+˜OΓ
+◦(X)/d116/d116✐✐✐✐✐✐✐✐✐✐✐✐
+⊂
+/d15/d15ˆXΓ
+◦≃¯ˆXΓ
+◦
+⊂
+/d15/d15X
+XΓ
+˜ΠΓ(X)/d47/d47OΓ(X)/d52/d52❤❤❤❤❤❤❤❤❤❤❤❤Aut(Γ) /d50/d50 ˜XΓ
+ˆΠΓ(X)/d47/d47˜OΓ(X)/d106/d106❱❱❱❱❱❱❱❱❱❱❱❱ˆXΓ.(9.2)
+Also Aut(Γ) acts on XΓ,XΓ
+◦, with˜XΓ≃[XΓ/Aut(Γ)], ˜XΓ
+◦≃[XΓ
+◦/Aut(Γ)].
+Note that there are in general no natural 1-morphisms fromˆXΓ,ˆXΓ
+◦to any of
+X,XΓ,XΓ
+◦,˜XΓ,˜XΓ
+◦.
+9.1 The definition of orbifold strata XΓ,...,ˆXΓ
+◦
+We now define the orbifold strata XΓ,...,ˆXΓ
+◦and study their properties.
+Definition 9.1. LetXbe a Deligne–Mumford C∞-stack, and Γ a finite group.
+We will explicitly define another Deligne–Mumford C∞-stackXΓ. SinceXis a
+100stackon the site ( C∞Sch,J),Xis a categorywith a functor pX:X→C∞Sch
+satisfying many conditions. To define XΓwe must define another category XΓ
+and a functor pXΓ:XΓ→C∞Sch.
+Define objects of the category XΓto be pairs ( A,ρ) satisfying:
+(a)Ais an object inX, withpX(A) =Ufor some object U∈C∞Sch;
+(b)ρ: Γ→Aut(A) is a group morphism, where Aut( A) is the group of
+isomorphisms a:A→AinX, andpX◦ρ(γ) = idUfor allγ∈Γ; and
+(c) Letube a point in U, andu:∗→Uthe corresponding morphism in
+C∞Sch. SincepX:X →C∞Schis a category fibred in groupoids,
+as in Definition A.5, there exists a morphism au:Au→AinXwith
+pX(Au) =∗andpX(au) =u, whereAuis unique up to isomorphism in X.
+Havingfixed Au,au, Definition A.5alsoimplies thatforeach γ∈Γthereis
+a unique isomorphism ρu(γ) :Au→Ausuch thatau◦ρu(γ) =ρ(γ)◦au:
+Au→A, andpX(ρu(γ)) = id ∗. Thenρu: Γ→Aut(Au) is a group
+morphism. We require that ρu: Γ→Aut(Au) should be injective for all
+u∈U. This condition is independent of the choice of Au,au.
+Define morphisms c: (A,ρ)→(B,σ) of the category XΓto be morphisms
+c:A→BinXsatisfyingσ(γ)◦c=c◦ρ(γ) :A→BinXfor allγ∈Γ. Given
+morphisms c: (A,ρ)→(B,σ),d: (B,σ)→(C,τ) inXΓ, define composition
+d◦c: (A,ρ)→(C,τ) inXΓto be the composition d◦c:A→CinX. For
+each object ( A,ρ) inXΓ, define the identity morphism id (A,ρ): (A,ρ)→(A,ρ)
+inXΓto be idA:A→AinX. Define a functor pXΓ:XΓ→C∞Schby
+pXΓ: (A,ρ)/ma√sto→U=pX(A) on objects and pXΓ:c/ma√sto→pX(c) on morphisms.
+DefineXΓ
+◦to be the full subcategory of objects ( A,ρ) inXΓsuch that
+ρu: Γ→Aut(Au) in (c) above is an isomorphism for all u∈U. Define a
+functorpXΓ
+◦=pX|XΓ
+◦:XΓ
+◦→C∞Sch. By Theorem 9.5(a) below, XΓis a
+Deligne–Mumford C∞-stack, andXΓ
+◦is an openC∞-substack inXΓ.
+Definition 9.2. LetXbe a Deligne–Mumford C∞-stack, and Γ a finite group.
+Define a category P˜XΓto have objects pairs ( A,∆) satisfying:
+(a)Ais an object inX, withpX(A) =Ufor some object U∈C∞Sch;
+(b) ∆⊆Aut(A) is a subgroup isomorphic to Γ, where Aut( A) is the group of
+isomorphisms a:A→AinX, andpX(δ) = idUfor allδ∈∆; and
+(c) Letube a point in U, andu:∗→Uthe corresponding morphism in
+C∞Sch. SincepX:X→C∞Schis a category fibred in groupoids, there
+exists a morphism au:Au→AinXwithpX(Au) =∗andpX(au) =u,
+whereAuis unique up to isomorphism in X. For each δ∈∆ there is a
+unique isomorphism δu:Au→Ausuch thatau◦δu=δ◦au:Au→A,
+andpX(δu) = id ∗. Then{δu:δ∈∆}is a subgroup of Aut( Au), and
+δ/ma√sto→δuis a group morphism. We require that the map δ/ma√sto→δushould be
+injective for allu∈U.
+101Define morphisms ( A,∆)→(A′,∆′) ofP˜XΓto be pairs ( c,ι), wherec:
+A→A′is a morphism in Xandι: ∆→∆′is a group isomorphism, satisfying
+ι(δ)◦c=c◦δ:A→A′forallδ∈∆. Givenmorphisms( c,ι) : (A,∆)→(A′,∆′),
+(c′,ι′) : (A′,∆′)→(A′′,∆′′) inPXΓ, define composition ( c′,ι′)◦(c,ι) = (c′◦
+c,ι′◦ι). Define identities id (A,∆)= (idA,id∆) : (A,∆)→(A,∆).
+Define a functor pP˜XΓ:P˜XΓ→C∞SchbypP˜XΓ: (A,∆)/ma√sto→U=pX(A)
+on objects and pP˜XΓ: (c,ι)/ma√sto→pX(c) on morphisms. Define P˜XΓ
+◦to be the
+full subcategory of objects ( A,∆) inP˜XΓwith{δu:δ∈∆}= Aut(Au) in (c)
+above for all u∈U. Define a functor pP˜XΓ
+◦=pP˜XΓ|P˜XΓ
+◦:P˜XΓ
+◦→C∞Sch.
+AlthoughP˜XΓ,P˜XΓ
+◦arein generalnot C∞-stacks, they areprestackson the
+site(C∞Sch,J)inthesenseofDefinitionA.6(thatis,morphismsin P˜XΓ,P˜XΓ
+◦
+satisfy a sheaf-like condition over ( C∞Sch,J), but objects may not). Thus,
+P˜XΓ,P˜XΓ
+◦have stackifications ˜XΓ,˜XΓ
+◦, defined up to equivalence, which are
+stacks on the site ( C∞Sch,J). By Theorem 9.5(a) below, ˜XΓis a Deligne–
+MumfordC∞-stack, and ˜XΓ
+◦is an openC∞-substack in ˜XΓ.
+Let (A,∆),(A′,∆′) be objects inP˜XΓ. Define a right action of ∆ on
+morphisms ( c,ι) : (A,∆)→(A′,∆′) inP˜XΓby (c,ι)·δ= (c◦δ,ιδ), where
+ιδ: ∆→∆′mapsιδ:ǫ/ma√sto→ι(δ◦ǫ◦δ−1). If (c′,ι′) : (A′,∆′)→(A′′,∆′′) is
+another morphism and δ′∈∆′, it is easy to show that
+/parenleftbig
+(c′,ι′)·δ′/parenrightbig
+◦/parenleftbig
+(c,ι)·δ/parenrightbig
+=/parenleftbig
+(c′,ι′)◦(c,ι)/parenrightbig
+·(ι−1(δ′)◦δ).(9.3)
+Define a category PˆXΓto have objects ( A,∆) as inP˜XΓ, and to have
+morphisms ( c,ι)∆ : (A,∆)→(A′,∆′) for morphisms ( c,ι) : (A,∆)→(A′,∆′)
+inP˜XΓ, where (c,ι)∆ ={(c,ι)·δ:δ∈∆}is the ∆-orbit of ( c,ι). Define
+composition of morphisms in PˆXΓby/parenleftbig
+(c′,ι′)∆′/parenrightbig
+◦/parenleftbig
+(c,ι)∆/parenrightbig
+=/parenleftbig
+(c′,ι′)◦(c,ι)/parenrightbig
+∆,
+where (c′,ι′)◦(c,ι) is composition of morphisms in P˜XΓ. Equation (9.3) shows
+thisiswell-defined. Defineidentitymorphismsid (A,∆)= (idA,id∆)∆ : (A,∆)→
+(A,∆) inPˆXΓ. Define a functor pPˆXΓ:PˆXΓ→C∞Schto map (A,∆)/ma√sto→
+pX(A) on objects and ( c,ι)∆/ma√sto→pX(c) on morphisms.
+DefinePˆXΓ
+◦to be the full subcategory of PˆXΓwhose objects are objects
+ofP˜XΓ
+◦, and define pPˆXΓ
+◦=pPˆXΓ|PˆXΓ
+◦:PˆXΓ
+◦→C∞Sch. Then as for
+PˆXΓ,PˆXΓ
+◦are prestacks on ( C∞Sch,J), and by Theorem 9.5(a) their stack-
+ifications ˆXΓ,ˆXΓ
+◦are Deligne–Mumford C∞-stacks. Furthermore, by Theorem
+9.5(g) below ˆXΓ
+◦has trivial isotropy groups, so by Theorem 7.20 there is a
+C∞-scheme ˆXΓ
+◦, unique up to isomorphism, such that ˆXΓ
+◦≃¯ˆXΓ
+◦.
+Next, we define all the 1-morphisms in (9.2).
+Definition 9.3. In Definitions 9.1 and 9.2, for Λ ∈Aut(Γ) define functors
+LΓ(Λ,X) :XΓ−→XΓ, OΓ(X) :XΓ−→X,P˜OΓ(X) :P˜XΓ−→X,
+P˜ΠΓ(X) :XΓ−→P˜XΓandPˆΠΓ(X) :P˜XΓ−→PˆXΓ
+on objects by
+LΓ(Λ,X) : (A,ρ)/ma√sto→(A,ρ◦Λ−1), OΓ(X) : (A,ρ)/ma√sto→A,P˜OΓ(X) : (A,∆)/ma√sto→A,
+P˜ΠΓ(X) : (A,ρ)/ma√sto−→/parenleftbig
+A,ρ(Γ)/parenrightbig
+andPˆΠΓ(X) : (A,∆)/ma√sto−→(A,∆),
+102and on morphisms by
+LΓ(Λ,X) :c/ma√sto−→c, OΓ(X) :c/ma√sto−→c,P˜OΓ(X) : (c,ι)/ma√sto−→c,
+P˜ΠΓ(X) :c/ma√sto→(c,σ◦ρ−1) onc: (A,ρ)→(B,σ), and
+PˆΠΓ(X) : (c,ι)/ma√sto→(c,ι)∆ on (c,ι) : (A,∆)→(A′,∆′).
+It istrivialtocheckthat theseareall functors, andcommute with theprojec-
+tionspX,pXΓ,p˜XΓ,pˆXΓtoC∞Sch. HenceLΓ(Λ,X),OΓ(X) are 1-morphismsof
+C∞-stacks. Note that LΓ(Λ,X)◦LΓ(Λ′,X) =LΓ(Λ◦Λ′,X) andLΓ(Λ−1,X) =
+LΓ(Λ,X)−1for Λ,Λ′∈Aut(Γ), so LΓ(−,X) is an action of Aut(Γ) on XΓby
+1-isomorphisms.
+NowP˜OΓ(X),P˜ΠΓ(X),PˆΠΓ(X) are 1-morphisms of prestacks, so stackify-
+ing gives 1-morphisms of C∞-stacks˜OΓ(X) :˜XΓ→X,˜ΠΓ(X) :XΓ→˜XΓ,
+ˆΠΓ(X) :˜XΓ→ˆXΓ. Define 1-morphisms of C∞-stacks
+LΓ
+◦(Λ,X) :XΓ
+◦−→XΓ
+◦, OΓ
+◦(X) :XΓ
+◦−→X,˜OΓ
+◦(X) :˜XΓ
+◦−→X,
+˜ΠΓ
+◦(X) :XΓ
+◦−→˜XΓ
+◦andˆΠΓ
+◦(X) :˜XΓ
+◦−→ˆXΓ
+◦,
+tobe therestrictionsof LΓ(Λ,X),...,ˆΠΓ(X)totheopen C∞-substacksXΓ
+◦,˜XΓ
+◦.
+ThenLΓ
+◦(−,X) is an action of Aut(Γ) on XΓ
+◦by 1-isomorphisms.
+It is easy to see that the analogue of (9.2) with prestacks P˜XΓ,...,PˆXΓ
+◦
+and prestack 1-morphisms P˜OΓ(X),...,PˆΠΓ
+◦(X) is strictly commutative, i.e.
+2-commutativewith identity 2-morphisms. Thus on stackifying, (9.2 ) commutes
+up to canonical 2-isomorphisms.
+Definition 9.4. Let the 1-morphisms OΓ(X) :XΓ→ X,OΓ
+◦(X) :XΓ
+◦→
+Xbe as in Definition 9.3. We will define actions of Γ on OΓ(X),OΓ
+◦(X)
+by 2-morphisms. For each γ∈Γ and (A,ρ)∈ XΓ, define an isomorphism
+EΓ(γ,X)(A,ρ) :OΓ(X)(A,ρ)→OΓ(X)(A,ρ) inXbyEΓ(γ,X) =ρ(γ) :A→
+A. Ifc: (A,ρ)→(B,σ) is a morphism in XΓthen
+OΓ(X)(c)◦EΓ(γ,X)(A,ρ)=c◦ρ(γ)=σ(γ)◦ρ=EΓ(γ,X)(B,σ)◦OΓ(X)(c).
+HenceEΓ(γ,X) :OΓ(X)⇒OΓ(X) is a natural isomorphism of functors. Since
+pX(EΓ(γ,X)(A,ρ)) =pX(ρ(γ)) = idpX(A)forall(A,ρ), wehavepX∗EΓ(γ,X) =
+pXΓ, soEΓ(γ,X) :OΓ(X)⇒OΓ(X) is a 2-morphism of C∞-stacks. Clearly
+EΓ(1,X) = idOΓ(X)andEΓ(γ,X)⊙EΓ(δ,X) =EΓ(γδ,X) for allγ,δ∈Γ, so
+EΓ(−,X) : Γ→Aut/parenleftbig
+OΓ(X)/parenrightbig
+is a group morphism. We define 2-morphisms
+EΓ
+◦(γ,X) :OΓ
+◦(X)⇒OΓ
+◦(X) forγ∈Γ in the same way.
+Here are some basic properties of these definitions.
+Theorem 9.5. (a) XΓ,˜XΓ,ˆXΓare Deligne–Mumford C∞-stacks, andXΓ
+◦⊆
+XΓ,˜XΓ
+◦⊆˜XΓ,ˆXΓ
+◦⊆ˆXΓare openC∞-substacks. Also ˜XΓ≃[XΓ/Aut(Γ)]and
+˜XΓ
+◦≃[XΓ
+◦/Aut(Γ)],where the Aut(Γ)-actions are LΓ(−,X)andLΓ
+◦(−,X).
+(b)IfXis separated, locally fair, locally finitely presented, or s econd count-
+able, thenXΓ,XΓ
+◦,˜XΓ,˜XΓ
+◦,ˆXΓ,ˆXΓ
+◦are separated, locally fair, locally finitely
+presented, or second countable, respectively.
+103IfXis compact thenXΓ,˜XΓ,ˆXΓare compact.
+(c)Points ofXΓ
+topare equivalence classes [x,ρ]of pairs(x,ρ),wherex:¯∗→X
+is a1-morphism and ρ: Γ→Aut(x)is an injective group morphism into
+the group Aut(x)of2-isomorphisms η:x⇒x,and pairs (x,ρ),(x′,ρ′)are
+equivalent if there exists ζ:x⇒x′withζ⊙ρ(γ) =ρ′(γ)⊙ζ:x⇒x′for all
+γ∈Γ. They have isotropy groups
+IsoXΓ([x,ρ]) =/braceleftbig
+η∈Aut(x) :ρ(γ) =ηρ(γ)η−1∀γ∈Γ/bracerightbig
+.
+Points ofXΓ
+◦,topare[x,ρ]withρ: Γ→Aut(x)an isomorphism, and have
+canonical isomorphisms IsoXΓ
+◦([x,ρ])∼=C(Γ),whereC(Γ)is the centre of Γ.
+(d)Points of ˜XΓ
+topare equivalence classes [x,∆]of pairs(x,∆),wherex:¯∗→
+Xis a1-morphism and ∆⊆Aut(x)is a subgroup isomorphic to Γ,and pairs
+(x,∆),(x′,∆′)are equivalent if there exists a 2-isomorphism ζ:x⇒x′with
+∆′=ζ⊙∆⊙ζ−1. They have isotropy groups
+Iso˜XΓ([x,∆])∼=/braceleftbig
+η∈Aut(x) : ∆ =η∆η−1/bracerightbig
+.
+Points of ˜XΓ
+◦,topare[x,∆]with∆ = Aut(x),and have non-canonical isomor-
+phismsIso˜XΓ
+◦([x,∆])∼=Γ.
+(e)As topological spaces ˆXΓ
+top=˜XΓ
+topandˆXΓ
+◦,top=˜XΓ
+◦,top,andˆΠΓ(X)top,
+ˆΠΓ
+◦(X)topare the identity maps. For [x,∆]∈ˆXΓ
+topwe have
+IsoˆXΓ([x,∆])∼=/braceleftbig
+η∈Aut(x) : ∆ =η∆η−1/bracerightbig
+/∆.
+AlsoIsoˆXΓ
+◦([x,∆]) ={1}for all[x,∆]∈ˆXΓ
+◦,top,soˆXΓ
+◦is aC∞-scheme.
+(f)LΓ(Λ,X),LΓ
+◦(Λ,X),OΓ(X),OΓ
+◦(X),˜OΓ(X),˜OΓ
+◦(X),˜ΠΓ(X),˜ΠΓ
+◦(X)are all
+representable, but ˆΠΓ(X),ˆΠΓ
+◦(X)in general are not representable.
+(g)LΓ(Λ,X),LΓ
+◦(Λ,X),OΓ(X),˜OΓ(X),˜ΠΓ(X),˜ΠΓ
+◦(X),ˆΠΓ(X),ˆΠΓ
+◦(X)are all
+proper, but OΓ
+◦(X),˜OΓ
+◦(X)in general are not.
+(h)OΓ
+◦(X)top:XΓ
+◦,top→Xtoptakes|Aut(Γ)|·|C(Γ)|/|Γ|points[x,ρ]ofXΓ
+◦,top
+to each point [x]∈XtopwithIsoX([x])∼=Γ. Also˜OΓ
+◦(X)top:˜XΓ
+◦,top→Xtopis
+a bijection with the subset of [x]∈XtopwithIsoX([x])∼=Γ.
+Proof.For (a), we first prove that XΓis a Deligne–Mumford C∞-stack. The
+inertia stack ofXisthe fibreproduct IX=X×∆X,X×X,∆XX, where∆ X:X→
+X×Xis the diagonal 1-morphism. There is a canonical construction of fib re
+products of stacks. Taking IXto be given by this construction, by definition
+objects of the category IXare triples ( A,B,c) whereA,Bare objects inXwith
+pX(A) =pX(B) =UinC∞Sch, andc: ∆X(A)→∆X(B) is a morphism in
+X×XwithpX×X(c) = idU. But ∆ X(A) = (A,A) and ∆ X(B) = (B,B), so
+c= (c1,c2) forc1,c2:A→Bmorphisms inXwithpX(ci) = idU.
+Thus we may write objects of IXas quadruples ( A,B,c,d ), whereA,Bare
+objects inXwithpX(A) =pX(B) =U, andc,d:A→Bare isomorphisms
+inXwithpX(c) =pX(d) = idU. Morphisms ( A,B,c,d )→(A′,B′,c′,d′) in
+104IXare pairs (a,b) witha:A→A′andb:B→B′morphisms inXsuch
+thatb◦c=c′◦aandb◦d=d′◦a. This forces pX(a) =pX(b). The functor
+pIX:IX→C∞Schacts bypIX: (A,B,c,d )/ma√sto→pX(A) =pX(B) on objects
+andpIX: (a,b)/ma√sto→pX(a) =pX(b) on morphisms.
+WriteiX:X →I Xfor the 1-morphism mapping A/ma√sto→(A,A,idA,idA) on
+objects and a/ma√sto→(a,a) on morphisms. Since Xis Deligne–Mumford, iXis an
+equivalence with an open and closed C∞-substackiX(X) inIX. HereiX(X)
+is the subcategory of objects in IXisomorphic to some ( A,A,idA,idA). Thus
+iX(X) is the full subcategory of objects ( A,B,c,d ) inIXwithc=d.
+SinceiX(X) is open and closed in IX, its complement JX=IX\iX(X) as
+aC∞-stack is also an open and closed C∞-substack inIX. As a subcategory,
+JXis not simply the complement of the subcategory iX(X). Instead,JXis the
+full subcategory of objects ( A,B,c,d ) inIXsatisfying the following condition
+(∗) analogous to Definition 9.1(c):
+(∗) WriteU=pX(A) =pX(B), and letu∈U, andu:∗→Uthe corre-
+sponding morphism in C∞Sch. SincepX:X →C∞Schis a category
+fibred in groupoids, there exist au:Au→A,bu:Bu→BinXwith
+pX(Au) =pX(Bu) =∗andpX(au) =pX(bu) =u, and unique isomor-
+phismscu,du:Au→Busuch thatau◦cu=c◦auandau◦du=d◦au,
+andpX(cu) =pX(du) = id∗. We require that cu/\e}atio\slash=dufor allu∈U.
+Now form the product/producttext
+γ∈ΓXof|Γ|copies ofX, and write ∆Γ
+X:X →/producttext
+γ∈ΓXfor the diagonal 1-morphism. Consider the C∞-stack fibre product
+Y=X×∆Γ
+X,/producttext
+γ∈ΓX,∆Γ
+XX.
+It is a Deligne–Mumford C∞-stack by Theorem 7.10. As for IX, we can take
+objects ofYto be (|Γ|+2)-tuples ( A,B,cγ:γ∈Γ), whereA,Bare objects in
+XwithpX(A) =pY(B) =U, andcγ:A→Bforγ∈Γ are isomorphisms in
+XwithpX(cγ) = idU. Morphisms ( A,B,cγ:γ∈Γ)→(A′,B′,c′
+γ:γ∈Γ) in
+Yare pairs (a,b) witha:A→A′andb:B→B′morphisms inXsuch that
+b◦cγ=c′
+γ◦a:A→B′for allγ∈Γ. The functor pY:Y→C∞Schacts by
+pY: (A,B,cγ:γ∈Γ)/ma√sto→pX(A) =pX(B) on objects and pY: (a,b)/ma√sto→pX(a) =
+pX(b) on morphisms.
+Forδ,ǫ∈Γ defineKδ,ǫ:Y→I Xto map (A,B,cγ:γ∈Γ)/ma√sto→(A,B,cδǫ,cδ◦
+c−1
+1◦cǫ) on objects and ( a,b)/ma√sto→(a,b) on morphisms. It is easy to show that
+Kδ,ǫis a functor, with pIX◦Kδ,ǫ=pY. HenceKδ,ǫ:Y→I Xis a 1-morphism
+of Deligne–Mumford C∞-stacks. Thus K−1
+δ,ǫ/parenleftbig
+iX(X)/parenrightbig
+is an open and closed C∞-
+substack inY, sinceiX(X) is open and closed in IX.
+Similarly, for δ/\e}atio\slash=ǫ∈Γ, defineLδ,ǫ:Y→I Xto map (A,B,cγ:γ∈Γ)/ma√sto→
+(A,B,cδ,cǫ) on objects and ( a,b)/ma√sto→(a,b) on morphisms. Then Lδ,ǫ:Y→I X
+is a 1-morphism, so L−1
+δ,ǫ(JX) is an open and closed C∞-substack inY, since
+JXis open and closed in IX. Define
+Y′=/intersectiondisplay
+δ,ǫ∈ΓK−1
+δ,ǫ/parenleftbig
+iX(X)/parenrightbig
+∩/intersectiondisplay
+δ/\e}atio\slash=ǫ∈ΓL−1
+δ,ǫ/parenleftbig
+JX/parenrightbig
+.
+105ThenY′is an open and closed C∞-substack inY, as it is a finite intersection
+of open and closed C∞-substacks inY.
+Define a functor M:XΓ→Y′to mapM: (A,ρ)/ma√sto→(A,A,ρ(γ) :γ∈Γ)
+on objects and M:a/ma√sto→(a,a) on morphisms. The nontrivial claim here is that
+if (A,ρ) is an object inXΓthenM/parenleftbig
+(A,ρ)/parenrightbig
+= (A,A,ρ(γ) :γ∈Γ) is an object
+inY′. The reason for this is that as ρ: Γ→Aut(A) is a group morphism,
+for eachδ,ǫ∈Γ we have ρ(δǫ) =ρ(δ)ρ(ǫ) =ρ(δ)ρ(1)−1ρ(ǫ), so (A,A,ρ(γ) :
+γ∈Γ) lies inK−1
+δ,ǫ/parenleftbig
+iX(X)/parenrightbig
+. Also, in Definition 9.1(c) ρu: Γ→Aut(Au) is
+injective, so ρu(δ)/\e}atio\slash=ρu(ǫ) forδ/\e}atio\slash=ǫ∈Γ. This is equivalent to condition ( ∗) for
+Lδ,ǫ/parenleftbig
+(A,A,ρ(γ) :γ∈Γ)/parenrightbig
+, so (A,A,ρ(γ) :γ∈Γ) lies inL−1
+δ,ǫ/parenleftbig
+JX/parenrightbig
+.
+Similarly, define a functor N:Y′→XΓto mapN: (A,B,cγ:γ∈Γ)/ma√sto→
+(A,ρ) on objects, where we define ρ(γ) =c−1
+1◦cγforγ∈Γ, and to map
+N: (a,b)/ma√sto→aon morphisms. The nontrivial claim is that if ( A,B,cγ:γ∈Γ)
+is an object inY′thenN/parenleftbig
+(A,B,cγ:γ∈Γ)/parenrightbig
+= (A,ρ) is an object inXΓ. This
+holds because ( A,B,cγ:γ∈Γ)∈K−1
+δ,ǫ(iX(X)) forcesρ(δǫ) =ρ(δ)ρ(ǫ) for all
+δ,ǫ, soρ: Γ→Aut(A) is a group morphism, and ( A,B,cγ:γ∈Γ)∈L−1
+δ,ǫ(JX)
+forδ/\e}atio\slash=ǫforcesρu(δ)/\e}atio\slash=ρu(ǫ) in Definition 9.1(c), so ρuis injective.
+NowN◦M= idXΓ, and there is a natural transformation η:M◦N⇒idY′
+acting byη: (A,B,cγ:γ∈Γ)/ma√sto→(idA,c1). SoXΓ,Y′are equivalent categories.
+AlsopY′◦M=pXΓandpXΓ◦N=pY′. Therefore M,Ndefine equivalences
+ofC∞-stacks, so asY′is a Deligne–Mumford C∞-stack,XΓis also a Deligne–
+MumfordC∞-stack equivalent to Y′. This proves the first part of (a).
+To see thatXΓ
+◦is an openC∞-substack ofXΓ, note that the map Xtop→N
+mapping [x]/ma√sto→/vextendsingle/vextendsingleIsoX([x])/vextendsingle/vextendsingleis upper semicontinuous, so the subset of points [ x]
+inXtopwith/vextendsingle/vextendsingleIsoX([x])/vextendsingle/vextendsingle/lessorequalslant|Γ|is open, and correspondsto an open C∞-substack
+X/lessorequalslant|Γ|inX. But thenXΓ
+◦≃XΓ×OΓ(X),X,incX/lessorequalslant|Γ|, soXΓ
+◦is the open C∞-
+substack inXΓcorresponding to X/lessorequalslant|Γ|inX, as we have to prove.
+NowLΓ(−,X) defines an action of the finite group Aut(Γ) on the Deligne–
+MumfordC∞-stackXΓby 1-isomorphisms, so we may form the quotient C∞-
+stack [XΓ/Aut(Γ)], which is also a Deligne–Mumford C∞-stack. To define
+[XΓ/Aut(Γ)] we first define a prestack XΓ/Aut(Γ) which is the quotient of the
+categoryXΓby Aut(Γ), and then [ XΓ/Aut(Γ)] is its stackification. Since P˜XΓ
+was defined to be equivalent to XΓ/Aut(Γ), its stackification ˜XΓis equivalent
+to [XΓ/Aut(Γ)]. This proves that ˜XΓis a Deligne–Mumford C∞-stack and
+˜XΓ≃[XΓ/Aut(Γ)], as in (a). Similarly ˜XΓ
+◦⊆˜XΓis an open C∞-substack,
+and˜XΓ
+◦≃[XΓ
+◦/Aut(Γ)].
+To show ˆXΓis Deligne–Mumford, we first observethat PˆXΓis a prestack, so
+ˆXΓis a stack on ( C∞Sch,J), and then either note that ˆΠΓ(X) :˜XΓ→ˆXΓhas
+fibre [∗/Γ] and˜XΓis Deligne–Mumford, or use the local models for ˆXΓgiven
+by Theorem 9.10. Then ˆXΓ
+◦⊆ˆXΓis open as forXΓ
+◦,˜XΓ
+◦. This completes (a).
+For (b), ifXis separated, locally fair, locally finitely presented, second
+countable, or compact, then Y=X×/producttext
+γXXis separated, ..., compact, so
+XΓare separated, ..., compact as it is equivalent to an open and close dC∞-
+substackY′ofY, andXΓ
+◦is separated, locally fair, locally finitely presented,
+or second countable (but not necessarily compact) as it is open in XΓ. The
+106result for ˜XΓ,˜XΓ
+◦,ˆXΓ,ˆXΓ
+◦follows as ˜XΓ≃[XΓ/Aut(Γ)], ˜XΓ
+◦≃[XΓ
+◦/Aut(Γ)],
+andˆXΓ,ˆXΓ
+◦fibre over ˜XΓ,˜XΓ
+◦with fibre [ ¯∗/Γ].
+For (c), there is a 1-1 correspondence between 1-morphisms x:¯∗→Xand
+objectsAxinXwithpX(Ax) =∗, and ifx,y:¯∗→Xcorrespond to Ax,Ayin
+Xthere is a 1-1 correspondence between 2-morphisms η:x⇒yand morphisms
+aη:Ax→AyinXwithpX(aη) = id ∗. The same correspondences hold for
+XΓ. Thus, each 1-morphism y:¯∗→XΓcorresponds uniquely to some ( B,σ)
+inXΓwithpX(B) =∗, soB=Axfor some unique 1-morphism x:¯∗→X, and
+eachσ(γ) :Ax→Axisaρ(γ)for some unique 2-morphism ρ(γ) :x⇒x, and
+ρ: Γ→Aut(x) is a group morphism. Definition 9.1 implies that ρis injective.
+This establishes a 1-1 correspondence between 1-morphisms y:¯∗→XΓand
+pairs (x,ρ), wherex:¯∗→Xis a 1-morphism and ρ: Γ→Aut(x) an injective
+group morphism. Similarly, if y,y′:¯∗→XΓcorrespond to ( x,ρ),(x′,ρ′) then
+2-morphisms θ:y⇒y′correspond to 2-morphisms ζ:x⇒x′withζ⊙ρ(γ) =
+ρ′(γ)⊙ζ:x⇒x′for allγ∈Γ. Also 1-morphisms y:¯∗→XΓ
+◦correspond to
+pairs (x,ρ) withρ: Γ→Aut(x) an isomorphism. Part (c) then follows. Parts
+(d),(e) come from the definitions of P˜XΓ,...,PˆXΓ
+◦in the same way, noting that
+stackifying does not change 1-morphisms ¯∗→P˜XΓor their 2-morphisms.
+For (f),LΓ(Λ,X) is representable as it is a 1-isomorphism. Suppose ( A,ρ)
+is an object inXΓwithpXΓ(A,ρ) =U, so thatOΓ(X) : (A,ρ)/ma√sto→A, and
+a:A→A′is an isomorphism in XwithpX(a) = idU. Thena: (A,ρ)→
+(A′,a◦ρ◦a−1) is the unique isomorphism in XΓwithOΓ(X) :a/ma√sto→a, soOΓ(X)
+is representable. The action P˜OΓ(X) : (c,ι)/ma√sto→cofP˜OΓ(X) on 1-morphisms
+is injective, as cdetermines ιbyι(δ)◦c=c◦δforδ∈∆. This implies
+that the stackification ˜OΓ(X) is representable. Then ˜ΠΓ(X) representable fol-
+lows from ˜OΓ(X)◦˜ΠΓ(X)∼=OΓ(X) withOΓ(X),˜OΓ(X) representable. Also
+LΓ
+◦(Λ,X),OΓ
+◦(X),˜OΓ
+◦(X),˜ΠΓ
+◦(X) are representable, as they are restrictions of
+LΓ(Λ,X),...,˜ΠΓ(X) to openC∞-substacks. The actions of ˆΠΓ(X),ˆΠΓ
+◦(X) on
+isotropy groups have kernels isomorphic to Γ. So if Γ /\e}atio\slash={1}these actions are
+not injective, and ˆΠΓ(X),ˆΠΓ
+◦(X) are not representable.
+For (g),LΓ(Λ,X), LΓ
+◦(Λ,X) are 1-isomorphisms, ˜ΠΓ(X),˜ΠΓ
+◦(X) project to
+quotients by Aut(Γ), and ˆΠΓ(X),ˆΠΓ
+◦(X) are fibrations with fibre [ ¯∗/Γ], so these
+are all proper. We can see that OΓ(X),˜OΓ(X) are proper, but OΓ
+◦(X),˜OΓ
+◦(X) in
+general are not, using Theorem 9.10 and the fact that every Delign e–Mumford
+C∞-stack is locally of the form [ X/G].
+For (h), if [ x]∈Xtopwith Iso X([x])∼=Γ, then by (c) points [ x,ρ]∈XΓ
+◦,top
+withOΓ
+◦(X)top: [x,ρ]/ma√sto→[x] are given by isomorphisms ρ: Γ→IsoX([x]).
+There are|Aut(Γ)|suchρ. Ifρ,ρ′are two such isomorphisms, then (c) shows
+[x,ρ] = [x,ρ′] if and only if ρ′=ραfor someα∈Γ, whereρα:γ/ma√sto→αγα−1.
+Forα1,α2∈Γ, we see that ρα1=ρα2if and only if ( α−1
+2α1)γ=γ(α−1
+2α1) for
+allγ∈Γ, that is, if α−1
+2α1∈C(Γ). Hence, the ραforα∈Γ realize|Γ|/|C(Γ)|
+distinct isomorphisms ρ′: Γ→IsoX([x]). So the|Aut(Γ)|isomorphisms ρ:
+Γ→IsoX([x]) areidentified in groupsof |Γ|/|C(Γ)|to make|Aut(Γ)|·|C(Γ)|/|Γ|
+points [x,ρ] inXΓ
+◦,top. The statement for ˜OΓ
+◦(X)topis immediate as if [ x,∆]∈
+˜XΓ
+◦,topthen ∆ = Aut( x), so[x,∆]/ma√sto→[x] isa1-1correspondence. Thiscompletes
+107the proof of Theorem 9.5.
+Example 9.6. LetXbe a Deligne–Mumford C∞-stack, andIXtheinertia
+stackofX, as in the proof of Theorem 9.5. Then there is an equivalence
+IX=X×∆X,X×X,∆XX≃/coproducttext
+k/greaterorequalslant1XZk.
+Toseethis, notethatpointsof IXareequivalenceclasses[ x,η], where[x]∈Xtop
+andη∈IsoX([x]). SinceXis Deligne–Mumford, Iso X([x]) is a finite group, so
+eachη∈IsoX([x]) has some finite order k/greaterorequalslant1, and generates an injective
+morphismρ:Zk→IsoX([x]) mapping ρ:a/ma√sto→ηa. We may identify XZkwith
+the open and closed C∞-substack of [ x,η] inIXfor whichηhas orderk.
+9.2 Lifting 1- and 2-morphisms to orbifold strata
+The construction of XΓ,˜XΓ,ˆXΓextends functorially to 1- and 2-morphisms.
+Definition 9.7. LetX,Ybe Deligne–Mumford C∞-stacks, Γ a finite group,
+andf:X →Y a representable 1-morphism, so that f:X →Y is a functor
+withpY◦f=pX. We will define a representable 1-morphism fΓ:XΓ→YΓ.
+On objects ( A,ρ) inXΓ, definefΓ(A,ρ) = (f(A),f◦ρ). We must check that
+fΓ(A,ρ) satisfies Definition 9.1(a)–(c). Parts (a),(b) hold as fis a functor with
+pY◦f=pX. For (c), if u∈Uthen (c) for ( A,ρ) shows that ρu: Γ→Aut(Au)
+is injective, so f◦ρu: Γ→Aut(f(Au)) is injective as fis representable, and
+this gives (c) for/parenleftbig
+f(A),f◦ρ/parenrightbig
+. On morphisms c: (A,ρ)→(B,σ) inXΓ, define
+fΓ(c) :fΓ(A,ρ)→fΓ(B,σ) byfΓ(c) =f(c) :f(A)→f(B).
+ThenfΓ:XΓ→YΓisafunctor,and pY◦f=pXimpliesthat pYΓ◦fΓ=pXΓ.
+HencefΓ:XΓ→YΓis a 1-morphism of C∞-stacks. It is the unique such 1-
+morphism with OΓ(Y)◦fΓ=f◦OΓ(X) :XΓ→Y. Also,fΓis injective on
+morphisms, as fis, sofΓis representable.
+Now letf,g:X →Y be representable, and η:f⇒gbe a 2-morphism.
+Thenf,g:X →Y are functors, and η:f⇒gis a natural isomorphism.
+DefineηΓ:fΓ⇒gΓby taking the isomorphism ηΓ(A,ρ) :fΓ(A,ρ)→gΓ(A,ρ)
+inYΓfor each object ( A,ρ) inXΓto be the isomorphism ηΓ(A,ρ) =η(A) :
+f(A)→g(A) inY. ThenηΓ:fΓ⇒gΓis a natural isomorphism of functors,
+and hence a 2-morphism in DMC∞Sta. It is the unique such 2-morphism
+with idOΓ(Y)∗ηΓ=η∗idOΓ(X).
+WriteDMC∞Starefor the 2-subcategory of DMC∞Stawith only repre-
+sentable 1-morphisms. Define FΓ:DMC∞Stare→DMC∞StarebyFΓ:
+X /ma√sto→FΓ(X) =XΓon objects, FΓ:f/ma√sto→FΓ(f) =fΓon representable 1-
+morphisms, and FΓ:η/ma√sto→FΓ(η) =ηΓon 2-morphisms. Then FΓis a strict
+2-functor, in the sense of §A.1.
+Definition9.8. LetX,YbeDeligne–Mumford C∞-stacks,Γafinitegroup,and
+f:X →Ya representable 1-morphism. Define functors P˜fΓ:P˜XΓ→P˜YΓ
+mapping (A,∆)/ma√sto→(f(A),f(∆)) on objects and ( c,ι)/ma√sto→(f(c),f◦ι◦f|−1
+∆)
+on morphisms, and PˆfΓ:PˆXΓ→PˆYΓmapping (A,∆)/ma√sto→(f(A),f(∆)) and
+108(c,ι)∆/ma√sto→(f(c),f◦ι◦f|−1
+∆)f(∆). ThenP˜fΓ,PˆfΓare 1-morphisms of prestacks,
+so stackifying gives 1-morphisms ˜fΓ:˜XΓ→˜YΓandˆfΓ:ˆXΓ→ˆYΓ.
+Iff,g:X→Yare representable, and η:f⇒gis a 2-morphism, we define
+P˜ηΓ:P˜fΓ⇒P˜gΓandPˆηΓ:PˆfΓ⇒PˆgΓbyP˜ηΓ: (A,∆)/ma√sto→(η(A),ιη), where
+ιη:f(∆)→g(∆) mapsιη:f(δ)/ma√sto→g(δ) =η(A)◦f(δ)◦η(A)−1forδ∈∆, and
+PˆηΓ: (A,∆)/ma√sto→(η(A),ιη)f(∆). ThenP˜ηΓ,PˆηΓare 2-morphisms of prestacks,
+so stackifying gives 2-morphisms ˜ ηΓ:˜fΓ⇒˜gΓand ˆηΓ:ˆfΓ⇒ˆgΓ.
+As in Definition 9.7, we would like to define 2-functors
+˜FΓ,ˆFΓ:DMC∞Stare−→DMC∞Stare(9.4)
+by˜FΓ(X) =˜XΓon objects, ˜FΓ(f) =˜fΓon 1-morphisms, ˜FΓ(η) = ˜ηΓon 2-
+morphisms, and so on. But there is a difference. Stackifications of 1 -morphisms
+of prestacks involve arbitrary choices, and are unique only up to 2- isomorphism.
+Therefore strict equalities of 1-morphisms of prestacks translat e, on stackifica-
+tion, to 2-isomorphisms of their stackifications, rather than stric t equalities.
+For representable 1-morphisms f:X → Y,g:Y → Z, in prestack 1-
+morphisms we have P˜gΓ◦P˜fΓ=P/tildewidest(g◦f)Γ. Thus, stackification gives a 2-
+isomorphism ˜FΓ
+g,f: ˜gΓ◦˜fΓ⇒/tildewidest(g◦f)Γ, whichneednot bethe identity. Similarly,
+P(/tildewideridX)Γ= idP˜XΓ:P˜XΓ→P˜XΓ, but on stackification we get a 2-isomorphism
+˜FΓ
+X: (/tildewideridX)Γ⇒id˜XΓ, which need not be the identity. Because of this, ˜FΓ,ˆFΓ
+in (9.4) are weak 2-functors rather than strict 2-functors, in th e sense of§A.1.
+The 1-morphisms in (9.2) are compatible with ˜fΓ,ˆfΓby 2-isomorphisms
+˜OΓ(Y)◦˜fΓ∼=f◦˜OΓ(X),˜ΠΓ(Y)◦fΓ∼=˜fΓ◦˜ΠΓ(X),ˆΠΓ(Y)◦˜fΓ∼=ˆfΓ◦ˆΠΓ(X),
+which follow by stackifying equalities of 1-morphisms of prestacks.
+Remark 9.9. Forf:X→Yand Γ as above, the restriction fΓ|XΓ
+◦need not
+mapXΓ
+◦→YΓ
+◦, but onlyXΓ
+◦→YΓ, unlessfinduces isomorphisms on isotropy
+groups. Thus we do not define a 1-morphism fΓ
+◦:XΓ
+◦→YΓ
+◦, or a 2-functor
+FΓ
+◦:DMC∞Stare→DMC∞Stare. The same applies for the actions of fon
+orbifold strata ˜XΓ
+◦,ˆXΓ
+◦.
+9.3 Orbifold strata of quotient C∞-stacks[X/G]
+The next theorem describes XΓ,...,ˆXΓ
+◦explicitly whenXis a quotient C∞-
+stack [X/G], as in§7.1. We can prove it by showing the explicit constructions
+of Definition 7.1 and Definitions 9.1–9.2 commute up to equivalence.
+Theorem 9.10. LetXbe a Hausdorff C∞-scheme and Ga finite group acting
+onXby isomorphisms, and write X= [X/G]for the quotient C∞-stack, which
+is a Deligne–Mumford C∞-stack. Let Γbe a finite group. Then there are
+109equivalences of C∞-stacks
+XΓ≃/bracketleftbig/parenleftbig/coproducttext
+injective group morphisms ρ: Γ→GXρ(Γ)/parenrightbig
+/G/bracketrightbig
+, (9.5)
+XΓ
+◦≃/bracketleftbig/parenleftbig/coproducttext
+injective group morphisms ρ: Γ→GXρ(Γ)
+◦/parenrightbig
+/G/bracketrightbig
+, (9.6)
+˜XΓ≃/bracketleftbig/parenleftbig/coproducttext
+subgroups ∆⊆G:∆∼=ΓX∆/parenrightbig
+/G/bracketrightbig
+, (9.7)
+˜XΓ
+◦≃/bracketleftbig/parenleftbig/coproducttext
+subgroups ∆⊆G:∆∼=ΓX∆
+◦/parenrightbig
+/G/bracketrightbig
+, (9.8)
+where for each subgroup ∆⊆G,we writeX∆for the closed C∞-subscheme in
+Xfixed by∆inG,andX∆
+◦for the open C∞-subscheme in X∆of points in X
+whose stabilizer group in Gis exactly ∆.
+Here the action of Gon/coproducttext
+ρXρ(Γ)in(9.5)is defined as follows. Let g∈G
+andρ: Γ→Gbe an injective morphism. Define another injective morphism
+ρg: Γ→Gbyρg:γ/ma√sto→gρ(γ)g−1. Theng(Xρ(Γ)) =Xρg(Γ),asC∞-subschemes
+ofX,and the action of gon/coproducttext
+ρXρ(Γ)mapsXρ(Γ)→Xρg(Γ)by the restriction
+ofg:X→XtoXρ(Γ). TheG-actions for (9.6)–(9.8)are similar.
+We can also rewrite equations (9.5)–(9.8)as
+XΓ≃/coproductdisplay
+conjugacy classes [ρ]of injective
+group morphisms ρ: Γ→G/bracketleftbig
+Xρ(Γ)//braceleftbig
+g∈G:gρ(γ) =ρ(γ)g∀γ∈Γ/bracerightbig/bracketrightbig
+,(9.9)
+XΓ
+◦≃/coproductdisplay
+conjugacy classes [ρ]of injective
+group morphisms ρ: Γ→G/bracketleftbig
+Xρ(Γ)
+◦//braceleftbig
+g∈G:gρ(γ) =ρ(γ)g∀γ∈Γ/bracerightbig/bracketrightbig
+,(9.10)
+˜XΓ≃/coproductdisplay
+conjugacy classes [∆]of subgroups ∆⊆Gwith∆∼=Γ/bracketleftbig
+X∆//braceleftbig
+g∈G: ∆ =g∆g−1/bracerightbig/bracketrightbig
+, (9.11)
+˜XΓ
+◦≃/coproductdisplay
+conjugacy classes [∆]of subgroups ∆⊆Gwith∆∼=Γ/bracketleftbig
+X∆
+◦//braceleftbig
+g∈G: ∆ =g∆g−1/bracerightbig/bracketrightbig
+. (9.12)
+Here morphisms ρ,ρ′: Γ→Gare conjugate if ρ′=ρgfor someg∈G,and
+subgroups ∆,∆′⊆Gare conjugate if ∆ =g∆′g−1for someg∈G. In(9.9)–
+(9.12)we sum over one representative ρor∆for each conjugacy class.
+In the notation of (9.11)–(9.12), there are equivalences of C∞-stacks
+ˆXΓ≃/coproductdisplay
+conjugacy classes [∆]of subgroups ∆⊆Gwith∆∼=Γ/bracketleftbig
+X∆/slashbig/parenleftbig
+{g∈G: ∆ =g∆g−1}/∆/parenrightbig/bracketrightbig
+,(9.13)
+ˆXΓ
+◦≃/coproductdisplay
+conjugacy classes [∆]of subgroups ∆⊆Gwith∆∼=Γ/bracketleftbig
+X∆
+◦/slashbig/parenleftbig
+{g∈G: ∆ =g∆g−1}/∆/parenrightbig/bracketrightbig
+.(9.14)
+Under the equivalences (9.5)–(9.14), the1-morphisms in (9.2)are identified
+up to2-isomorphism with 1-morphisms between quotient C∞-stacks induced by
+naturalC∞-scheme morphisms between/coproducttext
+ρXρ(Γ),X,....For example, the dis-
+joint union over ρof the inclusion Xρ(Γ)֒→Xis aG-equivariant morphism/coproducttext
+ρXρ(Γ)→X,inducing a 1-morphism [/coproducttext
+ρXρ(Γ)/G]→[X/G]. This is identi-
+fied withOΓ(X) :XΓ→Xby(9.5). Similarly, ˜ΠΓ(X) :XΓ→˜XΓis identified
+110by(9.5), (9.7) with the1-morphism [/coproducttext
+ρXρ(Γ)/G]→[/coproducttext
+∆X∆/G]induced by the
+C∞-scheme morphism/coproducttext
+ρXρ(Γ)→/coproducttext
+∆X∆mapping morphisms ρto subgroups
+∆ =ρ(Γ),and acting by idX∆:Xρ(Γ)→X∆for∆ =ρ(Γ).
+9.4 Sheaves on orbifold strata
+LetXbe a Deligne–Mumford C∞-stack, Γ a finite group, and E∈qcoh(X),
+so thatEΓ:=OΓ(X)∗(E)∈qcoh(XΓ). We will show that there is a natural
+representation of Γ on EΓ, and also the action of Aut(Γ) on XΓlifts toEΓ, so
+that Aut(Γ) ⋉Γ acts equivariantly on EΓ.
+Definition 9.11. LetXbe aDeligne–Mumford C∞-stack, and Γ a finite group,
+so that§9.1 defines the orbifold stratum XΓ, a 1-morphism OΓ(X) :XΓ→X,
+an action of Aut(Γ) on OΓ(X) by 2-isomorphisms EΓ(γ,X) :OΓ(X)⇒OΓ(X),
+and an action of Aut(Γ) on XΓby 1-isomorphisms LΓ(Λ,X) :XΓ→XΓ.
+SupposeEis a quasicoherent sheaf on X,and writeEΓfor the pullback
+sheafOΓ(X)∗(E) in qcoh(XΓ). Using the notation of Definition 8.7, for each
+γ∈Γ and Λ∈Aut(Γ) define morphisms RΓ(γ,E) :EΓ→EΓandSΓ(Λ,E) :
+LΓ(Λ,X)∗(EΓ)→EΓin qcoh(XΓ) by
+RΓ(γ,E) =EΓ(γ,X)∗(E) :OΓ(X)∗(E)−→OΓ(X)∗(E) and
+SΓ(Λ,E) =ILΓ(Λ,X),OΓ(X)(E)−1:LΓ(Λ,X)∗◦OΓ(X)∗(E)−→OΓ(X)∗(E),
+where the definition of SΓ(Λ,E) usesOΓ(X)◦LΓ(Λ,X) =OΓ(X).
+SinceEΓ(1,X) = idOΓ(X)andEΓ(γ,X)⊙EΓ(δ,X) =EΓ(γδ,X) forγ,δ∈Γ
+as in Definition 9.4, we have
+RΓ(1,E) = idEΓandRΓ(γ,E)◦RΓ(δ,E) =RΓ(γδ,E) for allγ,δ∈Γ.
+HenceRΓ(−,E) is an action of Γ on EΓby isomorphisms.
+AsLΓ(idΓ,X) = id XΓandLΓ(Λ,X)◦LΓ(Λ,X) =LΓ(ΛΛ′,X) for Λ,Λ′∈
+Aut(Λ), by properties of morphisms I∗,∗(∗) we find that
+SΓ(idΓ,E) =δXΓ(EΓ) : id∗
+XΓ(EΓ)−→EΓ,and
+SΓ(ΛΛ′,E) =SΓ(Λ′,E)◦LΓ(Λ′,X)∗(SΓ(Λ,E))◦ILΓ(Λ′,X),LΓ(Λ,X)(EΓ).
+This means that the SΓ(Λ,E) define a lift of the action of Aut(Γ) on XΓtoEΓ,
+that is,EΓis an Aut(Γ)-equivariant sheaf on XΓ.
+Ifγ∈Γ and Λ∈Aut(Γ) then noting that OΓ(X)◦LΓ(Λ,X) =OΓ(X), one
+can show from Definitions 9.3 and 9.4 that
+EΓ(Λ(γ),X)∗idLΓ(Λ,X)=EΓ(γ,X) :OΓ(X) =⇒OΓ(X).
+Pulling backEby this equation and using properties of the I∗,∗(∗) we find that
+RΓ(γ,E)◦SΓ(Λ,E) =SΓ(Λ,E)◦LΓ(Λ,X)∗(RΓ(Λ(γ),E)).(9.15)
+111This is a compatibility between the actions of Γ and Aut(Γ) on EΓ. It says that
+the action of Aut(Γ) on XΓlifts to an action of Aut(Γ) ⋉Γ onEΓ.
+Letα:E1→E2be a morphism in qcoh( X). ThenαΓ:=OΓ(X)∗(α) :
+EΓ
+1→EΓ
+2is a morphism in qcoh( XΓ). SinceEΓ(γ,X)∗:OΓ(X)∗⇒OΓ(X)∗is
+a natural isomorphism of functors, we see that
+αΓ◦RΓ(γ,E1) =RΓ(γ,E2)◦αΓforγ∈Γ.
+Similarly we find that
+αΓ◦SΓ(Λ,E1) =SΓ(Λ,E2)◦LΓ(Λ,X)∗(αΓ) for Λ∈Aut(Γ).
+These imply that R(γ,−) andS(Λ,−) are natural isomorphisms of functors.
+Now letf:X →Y be a representable 1-morphism of C∞-stacks, so that
+as in§9.2 we have fΓ:XΓ→YΓ. LetF ∈qcoh(Y). Then we may form
+f∗(F)∈qcoh(X) and hence f∗(F)Γ=OΓ(X)∗(f∗(F))∈qcoh(XΓ), or we may
+formFΓ=OΓ(Y)∗(F)∈qcoh(YΓ) and hence ( fΓ)∗(FΓ)∈qcoh(XΓ). Since
+OΓ(Y)◦fΓ=f◦OΓ(X), these are related by the canonical isomorphism
+TΓ(f,F) :=IfΓ,OΓ(Y)(F)◦IOΓ(X),f(F)−1:f∗(F)Γ−→(fΓ)∗(FΓ).(9.16)
+Using properties of I∗,∗(∗), it is easy to show that
+(fΓ)∗(RΓ(γ,F))◦TΓ(f,F) =TΓ(f,F)◦RΓ(γ,f∗(F)) forγ∈Γ, (9.17)
+and noting that fΓ◦LΓ(Λ,X) =LΓ(Λ,Y)◦fΓ, we also find that
+TΓ(f,F)◦SΓ(Λ,f∗(F)) =(fΓ)∗(SΓ(Λ,F))◦IfΓ,LΓ(Λ,Y)(FΓ)◦
+ILΓ(Λ,X),fΓ(FΓ)−1◦LΓ(Λ,X)∗(TΓ(f,F)).
+This shows that the isomorphisms TΓ(f,F) identify the (Aut(Γ) ⋉Γ)-actions
+onf∗(F)Γand (fΓ)∗(FΓ).
+Now letX,Γ,XΓ,EandEΓbe as above, and write R0,...,Rkfor the irre-
+ducible representations of Γ over R(that is, we choose one representative Riin
+each isomorphism class of irreducible representations), with R0=Rthe trivial
+representation. Then since RΓ(−,E) is an action of Γ on EΓby isomorphisms,
+by elementary representation theory we have a canonical decomp osition
+EΓ∼=/circleplustextk
+i=0EΓ
+i⊗RiforEΓ
+0,...,EΓ
+k∈qcoh(XΓ). (9.18)
+We will be interested in splitting EΓintotrivialandnontrivial representations
+of Γ, denoted by subscripts ‘tr’ and ‘nt’. So we write
+EΓ=EΓ
+tr⊕EΓ
+nt, (9.19)
+whereEΓ
+tr,EΓ
+ntare the subsheavesof EΓcorrespondingto the factors EΓ
+0⊗R0and/circleplustextk
+i=1EΓ
+i⊗Rirespectively. Equivalently, consider1
+|Γ|/summationtext
+γ∈ΓRΓ(γ,E) :EΓ→EΓ.
+It is a projection (its square is itself), with image EΓ
+trand kernelEΓ
+nt.
+112If Γ acts on Ribyρi: Γ→Aut(Ri), and Λ∈Aut(Γ), then ρi◦Λ−1: Γ→
+Aut(Ri) is also an irreducible representation of Γ, and so is isomorphic to RΛ(i)
+for some unique Λ( i) = 0,...,k. This defines an action of Aut(Γ) on {0,...,k}
+by permutations. One can show using (9.15) that SΓ(Λ,E) acts on the splitting
+(9.18) by mapping LΓ(Λ,X)∗(EΓ
+i⊗Ri)→EΓ
+Λ−1(i)⊗RΛ−1(i). Since Λ(0) = 0,
+it follows that SΓ(Λ,E) mapsLΓ(Λ,X)∗(EΓ
+tr)→EΓ
+trandLΓ(Λ,X)∗(EΓ
+nt)→EΓ
+nt,
+that is,SΓ(Λ,E) preserves the splitting (9.19).
+Equation (9.17) implies that TΓ(f,F) canonically maps f∗(F)Γ
+i⊗Ri→
+(fΓ)∗(FΓ
+i⊗Ri) in (9.18) for f∗(F)Γ,FΓ, and so maps f∗(F)Γ
+tr→(fΓ)∗(FΓ
+tr)
+andf∗(F)Γ
+nt→(fΓ)∗(FΓ
+nt) in (9.19).
+The next two definitions explain to what extent this generalizes to ˜XΓ,ˆXΓ.
+Definition 9.12. LetXbe aDeligne–Mumford C∞-stack, and Γ a finite group,
+so that§9.1 defines the orbifold strata XΓ,˜XΓwith˜XΓ≃[XΓ/Aut(Γ)], and
+1-morphisms OΓ(X) :XΓ→X,˜OΓ(X) :˜XΓ→Xand˜ΠΓ(X) :XΓ→˜XΓ
+with˜OΓ(X)◦˜ΠΓ(X) =OΓ(X).
+Let us ask: how much of the structure on EΓin Definition 9.11 descends
+to˜EΓ? It turns out that ˜EΓdoes not have natural representations of Γ or
+Aut(Γ), since we do not have actions of Γ on ˜OΓ(X) by 2-isomorphisms or of
+Aut(Γ) on ˜XΓby 1-isomorphisms. In effect, taking the quotient by Aut(Γ) in
+˜XΓ≃[XΓ/Aut(Γ)] destroys both these actions.
+However, at least part of the natural decompositions (9.18)–(9.1 9) descends
+to˜EΓ. As in Definition 9.11, write R0,...,Rkfor the irreducible representations
+of Γ, so that Aut(Γ) acts on the indexing set {0,...,k}. Form the quotient set
+{0,...,k}/Aut(Γ), so that points of {0,...,k}/Aut(Γ) are orbits Oof Aut(Γ)
+in{0,...,k}. Then we may rewrite (9.18) as
+EΓ∼=/circleplustext
+O∈{0,...,k}/Aut(Γ)/bracketleftbig/circleplustext
+i∈OEΓ
+i⊗Ri/bracketrightbig
+.
+SinceSΓ(Λ,E) mapsLΓ(Λ,X)∗(EΓ
+i⊗Ri)→EΓ
+Λ−1(i)⊗RΛ−1(i), we see that
+SΓ(Λ,E) :LΓ(Λ,X)∗/parenleftbig/circleplustext
+i∈OEΓ
+i⊗Ri/parenrightbig
+−→/circleplustext
+i∈OEΓ
+i⊗Ri
+for eachO∈{0,...,k}/Aut(Γ). Now the SΓ(Λ,E) lift the action of Aut(Γ)
+onXΓtoEΓ, and˜EΓis essentially the quotient of EΓby this lifted action of
+Aut(Γ)undertheequivalence ˜XΓ≃[XΓ/Aut(Γ)]. Thereforeanydecomposition
+ofEΓwhich is invariant under SΓ(Λ,E) for all Λ∈Aut(Γ) corresponds to a
+decomposition of ˜EΓ. Hence there is a canonical splitting
+˜EΓ=/circleplustext
+O∈{0,...,k}/Aut(Γ)˜EΓ
+O,where
+I˜ΠΓ(X),˜OΓ(X)(E)−1/bracketleftbig˜ΠΓ(X)∗(˜EΓ
+O)/bracketrightbig∼=/circleplustext
+i∈OEΓ
+i⊗Riunder (9.18).(9.20)
+As for (9.19) we define the trivialandnontrivial parts of ˜EΓby˜EΓ
+tr=˜EΓ
+{0}and
+˜EΓ
+nt=/circleplustext
+O∈{1,...,k}/Aut(Γ)˜EΓ
+O. Then
+˜EΓ=˜EΓ
+tr⊕˜EΓ
+nt,whereI˜ΠΓ(X),˜OΓ(X)(E)−1/bracketleftbig˜ΠΓ(X)∗(˜EΓ
+tr)/bracketrightbig
+=EΓ
+tr
+andI˜ΠΓ(X),˜OΓ(X)(E)−1/bracketleftbig˜ΠΓ(X)∗(˜EΓ
+nt)/bracketrightbig
+=EΓ
+nt.(9.21)
+113Each point [ x,∆] of˜XΓ
+tophas isotropy group Iso ˜XΓ([x,∆]) with a distin-
+guished subgroup ∆ with a noncanonical isomorphism ∆ ∼=Γ. The fibre of ˜EΓ
+at [x,∆] is a representation of Iso ˜XΓ([x,∆]), and hence a representation of ∆.
+Equation (9.21) corresponds to splitting the fibre of ˜EΓat [x,∆] into trivial and
+nontrivial representations of ∆. Equation (9.20) corresponds to decomposing
+the fibre of ˜EΓat [x,∆] into families of irreducible representations of ∆ ∼=Γ
+that are independent of the choice of isomorphism ∆ ∼=Γ.
+Now letf:X→Ybe a representable 1-morphism of C∞-stacks, so that as
+in§9.2 we have a representable 1-morphism ˜fΓ:˜XΓ→˜YΓwithf◦˜OΓ(X) =
+˜OΓ(Y)◦˜fΓ. LetF∈qcoh(Y), so that ˜FΓ∈qcoh(˜YΓ),f∗(F)∈qcoh(X), and
+/tildewidestf∗(F)Γ∈qcoh(˜XΓ). As for (9.16), we have a canonical isomorphism
+˜TΓ(f,F) :=I˜fΓ,˜OΓ(Y)(F)◦I˜OΓ(X),f(F)−1:/tildewidestf∗(F)Γ−→(˜fΓ)∗(˜FΓ).
+As forTΓ(f,F) in Definition 9.11, ˜TΓ(f,F) maps/tildewidestf∗(F)Γ
+O→(˜fΓ)∗(FΓ
+O) in
+(9.20) for/tildewidestf∗(F)Γ,˜FΓ, and so maps/tildewidestf∗(F)Γ
+tr→(˜fΓ)∗(˜FΓ
+tr) and/tildewidestf∗(F)Γ
+nt→
+(˜fΓ)∗(˜FΓ
+nt) in (9.21).
+Definition 9.13. LetXbe aDeligne–Mumford C∞-stack, and Γ a finite group,
+so that§9.1 defines the orbifold strata ˜XΓ,ˆXΓand 1-morphisms ˜OΓ(X) :˜XΓ→
+XandˆΠΓ:˜XΓ→ˆXΓ, whereˆΠΓis non-representable, with fibre [ ¯∗/Γ].
+SupposeEis a quasicoherent sheaf on X. Since we have no 1-morphism
+ˆXΓ→X, we cannot pull Eback to ˆXΓto define ˆEΓin qcoh( ˆXΓ). But we do
+have˜EΓ=˜OΓ(X)∗(E) in qcoh( ˜XΓ), with splitting ˜EΓ=˜EΓ
+tr⊕˜EΓ
+ntas in (9.21),
+so we can form the pushforward ˆΠΓ
+∗(˜EΓ) in qcoh( ˆXΓ). Now pushforwards take
+global sections of a sheaf on the fibres of the 1-morphism. The fibr es ofˆΠΓare
+[¯∗/Γ]. Quasicoherent sheaves on [ ¯∗/Γ] correspond to Γ-representations, and the
+global sections correspond to the trivial (Γ-invariant) part.
+As the Γ-invariant part of ˜EΓis˜EΓ
+tr, we see that ˆΠΓ
+∗(˜EΓ
+nt) = 0, that is,EΓ
+nt
+and˜EΓ
+ntdo not descend to ˆXΓ. Define ˆEΓ
+tr=ˆΠΓ
+∗(˜EΓ
+tr) in qcoh( ˆXΓ). This is the
+natural analogue of EΓ
+tr,˜EΓ
+tronˆXΓ, and has a canonical isomorphism
+(ˆΠΓ)∗(ˆEΓ
+tr)∼=˜EΓ
+tr. (9.22)
+Now letf:X →Y be a representable 1-morphism of C∞-stacks, so that
+as in§9.2 we have a representable 1-morphism ˜fΓ:˜XΓ→˜YΓ. Then there is a
+canonical isomorphism
+ˆTΓ
+tr(f,F) :/hatwidestf∗(F)Γ
+tr−→(ˆfΓ)∗(ˆFΓ
+tr),
+the composition of the natural isomorphism ˆΠΓ
+∗◦(˜fΓ)∗(˜FΓ
+tr)→(ˆfΓ)∗◦ˆΠΓ
+∗(˜FΓ
+tr)
+withˆΠΓ
+∗(˜TΓ(f,F)|/tildewidestf∗(F)Γ
+tr/parenrightbig
+.
+9.5 Sheaves on orbifold strata of quotients [X/G]
+In the next theorem we take X= [X/G], and use the explicit description of
+XΓin Theorem 9.10 to give an alternative formula for the action RΓ(−,E) of
+114Γ onEΓin Definition 9.11. This then allows us to understand the splittings
+(9.18)–(9.22) in terms of sheaves on X. The proof is a long but straightforward
+consequence of the definitions, and we leave it as an exercise.
+Theorem 9.14. LetXbe a Hausdorff C∞-scheme,Ga finite group, r:G→
+Aut(X)an action of GonX,andX= [X/G]the quotient Deligne–Mumford
+C∞-stack. Then (9.5)gives an equivalence XΓ≃[/coproducttext
+injectiveρ: Γ→GXρ(Γ)/G].
+WriteqcohG(X)for the abelian category of G-equivariant quasicoherent
+sheaves onX,with objects pairs (E,Φ)forE∈qcoh(X)andΦ(g) :r(g)∗(E)→E
+is an isomorphism in qcoh(X)for allg∈Gsatisfying Φ(1) =δX(E)and
+Φ(gh) = Φ(h)◦r(h)∗(Φ(g))◦Ir(h),r(g)(E)for allg,h∈G,
+and morphisms α: (E,Φ)→(F,Ψ)inqcohG(X)are morphisms α:E→Fin
+qcoh(X)withα◦Φ(g) = Ψ(g)◦r(g)∗(α)for allg∈G.
+ThenqcohG(X)is isomorphic to qcoh(G×X⇒X)in Definition 8.5,so
+Theorem 8.6gives an equivalence of categories FΠ: qcoh(X)→qcohG(X).
+Using(9.5)we also get an equivalence FΓ
+Π: qcoh(XΓ)→qcohG(/coproducttext
+ρXρ(Γ)).
+These categories and functors fit into a 2-commutative diagram:
+qcoh(X)FΠ/d47/d47
+OΓ(X)∗/d15/d15 ✚✚✚✚/d73/d81
+NΓ(X)qcohG(X)
+i∗
+X /d15/d15
+qcoh(XΓ)FΓ
+Π/d47/d47qcohG(/coproducttext
+ρXρ(Γ)),(9.23)
+whereiX:/coproducttext
+ρXρ(Γ)→Xis the union over ρof the inclusion morphisms
+Xρ(Γ)→X,which isG-equivariant and so induces a pullback functor i∗
+Xas
+shown, and NΓ(X)is a natural isomorphism of functors.
+Let(E,Φ)∈qcohG(X),so thati∗
+X(E,Φ)∈qcohG(/coproducttext
+ρXρ(Γ)). Define
+¯RΓ/parenleftbig
+γ,(E,Φ)/parenrightbig
+:i∗
+X(E,Φ)→i∗
+X(E,Φ)inqcohG(/coproducttext
+ρXρ(Γ))forγ∈Γsuch that
+¯RΓ/parenleftbig
+γ,(E,Φ)/parenrightbig
+|Xρ(Γ):iX|∗
+Xρ(Γ)(E)−→iX|∗
+Xρ(Γ)(E)is given by
+¯RΓ/parenleftbig
+γ,(E,Φ)/parenrightbig
+|Xρ(Γ)=iX|∗
+Xρ(Γ)(Φ(ρ(γ−1)))◦IiX|Xρ(Γ),r(ρ(γ−1))(E)
+for eachρ,noting that r(ρ(γ−1))◦iX|Xρ(Γ)=iX|Xρ(Γ). Then¯RΓ/parenleftbig
+−,(E,Φ)/parenrightbig
+is
+an action of ΓoniX|∗
+Xρ(Γ)(E)by isomorphisms. Furthermore, for each Ein
+qcoh(X)andγinΓ,the following diagram in qcohG(/coproducttext
+ρXρ(Γ))commutes:
+FΓ
+Π(EΓ)
+FΓ
+Π(RΓ(γ,E))/d47/d47
+NΓ(X)(E)/d15/d15FΓ
+Π(EΓ)
+NΓ(X)(E)/d15/d15
+i∗
+X◦FΠ(E)¯RΓ(γ,FΠ(E))/d47/d47i∗
+X◦FΠ(E).
+That is, the equivalences of categories FΠ,FΓ
+Πin(9.23)identify the Γ-actions
+RΓ(−,−)onOΓ(X)∗and¯RΓ(−,−)oni∗
+Xby natural isomorphisms.
+1159.6 Cotangent sheaves of orbifold strata
+Finally we apply these ideas to write the cotangent sheaves of XΓ,˜XΓ,ˆXΓin
+terms of the pullbacks of T∗X. The theorem illustrates the principle that when
+passing to orbifold strata, it is often natural to restrict to the tr ivial parts
+EΓ
+tr,˜EΓ
+tr,ˆEΓ
+trof the pullbacks of E. The nontrivial parts ( T∗X)Γ
+nt,/tildewider(T∗X)Γ
+ntshould
+be interpreted as the conormal sheaves ofXΓ,˜XΓinX.
+Theorem 9.15. LetXbe a Deligne–Mumford C∞-stack and Γa finite group,
+so that§9.1definesOΓ(X) :XΓ→X. As in Definition 8.12we have cotan-
+gent sheaves T∗X,T∗(XΓ)and a morphism ΩOΓ(X):OΓ(X)∗(T∗X)→T∗(XΓ)
+inqcoh(XΓ). ButOΓ(X)∗(T∗X) = (T∗X)Γ,so by(9.19)we have a splitting
+(T∗X)Γ= (T∗X)Γ
+tr⊕(T∗X)Γ
+nt. ThenΩOΓ(X)|(T∗X)Γ
+tr: (T∗X)Γ
+tr→T∗(XΓ)is an
+isomorphism, and ΩOΓ(X)|(T∗X)Γ
+nt= 0.
+Similarly, using the 1-morphism ˜OΓ(X) :˜XΓ→Xand the splitting (9.21)
+for/tildewider(T∗X)Γwe findthat Ω˜OΓ(X)|/tildewider(T∗X)Γ
+tr:/tildewider(T∗X)Γ
+tr→T∗(˜XΓ)is an isomorphism,
+andΩ˜OΓ(X)|/tildewider(T∗X)Γ
+nt= 0.
+Also, there is a natural isomorphism/hatwider(T∗X)Γ
+tr∼=T∗(ˆXΓ)inqcoh(ˆXΓ).
+Proof.All ofthe claims are localstatements on XΓ,˜XΓ,ˆXΓ, that is, it is enough
+to prove them on open covers of XΓ,˜XΓ,ˆXΓ. AsXis Deligne–Mumford it
+is covered by open C∞-substacksUequivalent to [ U/G] forUan affineC∞-
+schemeand Gafinitegroup. Then XΓ,˜XΓ,ˆXΓarecoveredbythecorresponding
+UΓ,˜UΓ,ˆUΓ. Thus it is sufficient to prove the theorem when X≃[X/G] forX
+an affineC∞-scheme and Ga finite group acting on X. As the theorem is
+independent ofXup to equivalence, we may take X= [X/G].
+ThuswecanapplyTheorems9.10and9.14totranslateeachpartoft he theo-
+rem into statements about X,Xρ(Γ),....For the first part, using the notation of
+Theorem9.14,wefindthat FΠ(T∗X) = (T∗X,Φ),whereΦ( g) = Ωr(g)forg∈G.
+SimilarlyFΓ
+Π(T∗XΓ) =/parenleftbig
+T∗(/coproducttext
+ρXρ(Γ)),ΦΓ/parenrightbig
+, where ΦΓ(g) =/coproducttext
+ρΩr(g)|Xρ(Γ), and
+r(g)|Xρ(Γ)mapsXρ(Γ)→Xρg(Γ).
+Fix an injective morphism ρ: Γ→G, and write iρ
+X:Xρ(Γ)→Xfor the
+inclusion of Xρ(Γ)as aC∞-subscheme. Then ( iρ
+X)∗(T∗X) =i∗
+X(T∗X)|Xρ(Γ)in
+qcoh(Xρ(Γ)), and Ωiρ
+X= ΩiX|Xρ(Γ). Theorem 9.14 and Φ( g) = Ωr(g)show that
+the Γ-action ¯RΓ/parenleftbig
+γ,(T∗X,Φ)/parenrightbig
+on/parenleftbig
+i∗
+X(T∗X),i∗
+X(Φ)/parenrightbig
+acts on (iρ
+X)∗(T∗X) by
+¯RΓ/parenleftbig
+γ,(T∗X,Φ)/parenrightbig
+|(iρ
+X)∗(T∗X)= (iρ
+X)∗(Ωr(ρ(γ−1)))◦Iiρ
+X,r(ρ(γ−1)).
+Let (iρ
+X)∗(T∗X) = (iρ
+X)∗(T∗X)tr⊕(iρ
+X)∗(T∗X)ntbe the decomposition of
+(iρ
+X)∗(T∗X) into trivial and nontrivial Γ-representations under the action of
+¯RΓ(−,(T∗X,Φ)). Since Theorem 9.14 shows that the Γ-actions RΓ(−,T∗X)
+and¯RΓ(−,(T∗X,Φ)) are intertwined by FΓ
+Π, the splitting into trivial and non-
+trivial parts corresponds. As FΓ
+Πis an equivalence of categoriesby Theorem 8.6,
+the first part of the theorem is thus equivalent to showing that
+Ωiρ
+X: (iρ
+X)∗(T∗X) = (iρ
+X)∗(T∗X)tr⊕(iρ
+X)∗(T∗X)nt−→T∗Xρ(Γ)
+116is an isomorphism ( iρ
+X)∗(T∗X)tr→T∗Xρ(Γ), and is zero on ( iρ
+X)∗(T∗X)nt.
+To see this, let x∈Xρ(Γ)⊆X, and write Cxfor the local C∞-ringOX,x,
+the stalk ofOXatx. Then the action ρof Γ onXfixingxinduces an action
+φ: Γ→Aut(Cx) ofΓ on Cx. Sinceeach φ(γ) :Cx→CxactsonCxasaC∞-ring
+isomorphism, itis an R-linearmap, sowemaysplit Cx=Cx,tr⊕Cx,ntinto trivial
+and nontrivial Γ-representations. Write ( Cx,nt) for the ideal in Cxgenerated by
+Cx,nt, andDx=Cx/(Cx,nt) for the quotient C∞-ring, with projection πx:Cx→
+Dx. ThenOXρ(Γ),x∼=Dxandiρ
+X,x∼=πx:Cx→Dx.
+We have cotangent modules Ω Cx,ΩDxwith morphisms Ω πx: ΩCx→ΩDx
+and (Ωπx)∗: ΩCx⊗CxDx→ΩDx. In stalks at x∈Xρ(Γ)⊆Xwe have
+[T∗X]x∼=ΩCx, [T∗Xρ(Γ)]x∼=ΩDx, [(iρ
+X)∗(T∗X)]x∼=ΩCx⊗CxDxand [Ωiρ
+X]x∼=
+(Ωπx)∗: ΩCx⊗CxDx→ΩDx. TheΓ-actionon CxinducesoneonΩ Cx, andhence
+one on Ω Cx⊗CxDx. Thus we split into trivial and nontrivial Γ-representations,
+ΩCx⊗CxDx= (ΩCx⊗CxDx)tr⊕(ΩCx⊗CxDx)nt. This Γ-action is identified
+with that on the stalk [( iρ
+X)∗(T∗X)]x. Hence [(iρ
+X)∗(T∗X)tr]x∼=(ΩCx⊗CxDx)tr
+and [(iρ
+X)∗(T∗X)nt]x∼=(ΩCx⊗CxDx)nt.
+We have a linear map d Cx:Cx→ΩCx, whose image generates Ω Cxas a
+Cx-module. It induces a linear map d Cx⊗πx:Cx→ΩCx⊗CxDx, whose
+image generates Ω Cx⊗CxDxas aDx-module. As d Cx⊗πxis Γ-equivariant,
+it maps Cx,trandCx,ntto (ΩCx⊗CxDx)trand (Ω Cx⊗CxDx)nt, respectively.
+Hence (Ω Cx⊗CxDx)trand (Ω Cx⊗CxDx)ntare generated as Dx-modules by
+(dCx⊗πx)(Cx,tr) and (d Cx⊗πx)(Cx,nt).
+SinceDx=Cx/(Cx,nt), we see that (Ω πx)∗: ΩCx⊗CxDx→ΩDxis surjec-
+tive, with kernel generated by (d Cx⊗πx)/parenleftbig
+(Cx,nt)/parenrightbig
+. It is enough to use not the
+whole ideal ( Cx,nt), but only the generating subspace Cx,nt. TheDx-submodule
+generated by (d Cx⊗πx)(Cx,nt) is (Ω Cx⊗CxDx)nt. Thus, (Ω πx)∗is surjective
+with kernel (Ω Cx⊗CxDx)nt, so (Ωπx)∗|(ΩCx⊗CxDx)tr: (ΩCx⊗CxDx)tr→ΩDxis
+an isomorphism, and (Ω πx)∗|(ΩCx⊗CxDx)nt= 0. Therefore [Ω iρ
+X|(iρ
+X)∗(T∗X)tr]x:
+[(iρ
+X)∗(T∗X)tr]x→[T∗Xρ(Γ)]xis an isomorphism, and [Ω iρ
+X|(iρ
+X)∗(T∗X)nt]x= 0.
+As this holds for all x∈Xρ(Γ)⊆X, the first part follows.
+For the second part, Theorem 8.13(a) and ˜OΓ(X)◦˜ΠΓ(X) =OΓ(X) give a
+commutative diagram in qcoh( XΓ):
+˜ΠΓ(X)∗(˜OΓ(X)∗(T∗X)) =
+˜ΠΓ(X)∗(/tildewider(T∗X)Γ
+tr)⊕˜ΠΓ(X)∗(/tildewider(T∗X)Γ
+nt)˜ΠΓ(X)∗(Ω˜OΓ(X))/d47/d47˜ΠΓ(X)∗(T∗(˜XΓ))
+Ω˜ΠΓ(X)
+/d15/d15OΓ(X)∗(T∗X) =
+(T∗X)Γ
+tr⊕(T∗X)Γ
+ntI˜ΠΓ(X),˜OΓ(X)(E)/d79/d79
+ΩOΓ(X)/d47/d47T∗(XΓ).(9.24)
+As˜ΠΓ(X) is the projection XΓ→[XΓ/Aut(Γ)], it is ´ etale, so Ω ˜ΠΓ(X)is an
+isomorphism. Also I˜ΠΓ(X),˜OΓ(X)(E) identifies ‘tr’,‘nt’ with ‘tr’,‘nt’ components.
+Thus (9.24) and the first part show ˜ΠΓ(X)∗(Ω˜OΓ(X)|/tildewider(T∗X)Γ
+tr):˜ΠΓ(X)∗(/tildewider(T∗X)Γ
+tr)
+→˜ΠΓ(X)∗(T∗(˜XΓ)) is an isomorphism, and ˜ΠΓ(X)∗(Ω˜OΓ(X)|/tildewider(T∗X)Γ
+nt) = 0. As
+117˜ΠΓ(X) is´ etale and surjective, the second part of the theorem follows. The third
+part is proved by a similar argument involving ˆΠΓ.
+A Background material on stacks
+Finally we recall some background material on stacks needed in §6–§9. Readers
+unfamiliarwith stacksareadvised tolookat anintroductorytext su chasOlsson
+[57], Vistoli [68], Gomez [29], or Laumon and Moret-Bailly [46] before re ading
+this section.
+Stacksofanykindformastrict2-category C, withobjectsX,Y, 1-morphisms
+f,g:X→Y, and 2-morphisms η:f⇒g. So we begin in §A.1 with an intro-
+duction to 2-categories. Sections A.2–A.5 cover Grothendieck (pr e)topologies,
+sites, prestacks and stacks, descent theory, properties of 1- morphisms of stacks,
+geometric stacks, and stacks associated to groupoids.
+Our principal references were Artin [3], Behrend et al. [4], Gomez [29], Lau-
+mon and Moret-Bailly [46], Metzler [49], Noohi [55], and Olsson [57]. The
+topological and smooth stacks discussed by Metzler and Noohi are closer to
+our situation than the stacks in algebraic geometry of [4,29,46], so we often fol-
+low [49,55], particularly in §A.5 which is based on Metzler [49, §3]. Heinloth [32]
+and Behrend and Xu [5] also discuss smooth stacks.
+A.1 Introduction to 2-categories
+A good reference on 2-categories for our purposes is Behrend et al. [4, App. B],
+and Borceux [8,§7] and Kelly and Street [43] are also helpful.
+Definition A.1. Astrict2-category Cconsists of a proper class of objects
+Obj(C), for allX,Yin Obj(C) a small category Hom( X,Y), for allXin Obj(C)
+an object id Xin Hom(X,X) called the identity1-morphism , and for all X,Y,Z
+in Obj(C) a functor
+µX,Y,Z: Hom(X,Y)×Hom(Y,Z)−→Hom(X,Z).
+These must satisfy the identity property , that
+µX,X,Y(idX,−) =µX,Y,Y(−,idY) = idHom(X,Y) (A.1)
+as functors Hom( X,Y)→Hom(X,Y), and the associativity property , that
+µW,Y,Z◦(µW,X,Y×idHom(Y,Z)) =µW,X,Z◦(idHom(W,X)×µX,Y,Z) (A.2)
+for allW,X,Y,Z , as functors
+Hom(W,X)×Hom(X,Y)×Hom(Y,Z)−→Hom(W,X).
+Objectsfof Hom(X,Y) are called 1- morphisms , writtenf:X→Y. For
+1-morphisms f,g:X→Y, morphisms η∈HomHom(X,Y)(f,g) are called 2-
+morphisms , writtenη:f⇒g. Thus, a 2-categoryhas objects X, and two kinds
+118of morphisms, 1-morphisms f:X→Ybetween objects, and 2-morphisms
+η:f⇒gbetween 1-morphisms.
+Aweak2-category , orbicategory , is like a strict 2-category, except that the
+equations of functors (A.1), (A.2) are required to hold only up to sp ecified natu-
+ral isomorphisms, which should themselves satisfy identities. Strict 2-categories
+are examples of weak 2-categoriesin which these specified natural isomorphisms
+are identities. We will not give much detail on weak 2-categories, sinc e the 2-
+categories of stacks we are interested in are strict.
+Inmanyexamples, all2-morphismsare2-isomorphisms(i.e.haveanin verse),
+so that the categories Hom( X,Y) are groupoids. Such 2-categories are called
+(2,1)-categories .
+This is quite a complicated structure. There are three kinds of comp osition
+in a 2-category, satisfying various associativity relations. If f:X→Yandg:
+Y→Zare 1-morphisms then µX,Y,Z(f,g) is thecomposition of 1-morphisms ,
+writteng◦f:X→Z. Iff,g,h:X→Yare 1-morphisms and η:f⇒g,
+ζ:g⇒hare 2-morphisms then composition of η,ζin the category Hom( X,Y)
+gives the vertical composition of 2-morphisms ofη,ζ, writtenζ⊙η:f⇒h, as
+a diagram
+Xf
+/d34/d34✤✤✤✤/d11/d19η
+/d60/d60
+h✤✤✤✤/d11/d19ζg/d47/d47Y /d47/d47 /d47/d111 /d47/d111 /d47/d111 Xf
+/d41/d41
+h/d53/d53✤✤✤✤/d11/d19ζ⊙ηY.
+And iff,˜f:X→Yandg,˜g:Y→Zare 1-morphisms and η:f⇒˜f,
+ζ:g⇒˜gare 2-morphisms then µX,Y,Z(η,ζ) is thehorizontal composition of
+2-morphisms , writtenζ∗η:g◦f⇒˜g◦˜f, as a diagram
+Xf
+/d40/d40
+˜f/d54/d54✤✤✤✤/d11/d19ηYg
+/d40/d40
+˜g/d54/d54✤✤✤✤/d11/d19ζZ /d47/d47 /d47/d111 /d47/d111Xg◦f
+/d40/d40
+˜g◦˜f/d54/d54✤✤✤✤/d11/d19ζ∗ηZ.
+There are also two kinds of identity: identity 1-morphisms idX:X→Xand
+identity2-morphisms idf:f⇒f.
+In a strict 2-category C, composition of 1-morphisms is strictly associative,
+(g◦f)◦e=g◦(f◦e), and horizontal composition of 2-morphisms is strictly
+associative, ( ζ∗η)∗ǫ=ζ∗(η∗ǫ). In a weak 2-category C, composition of
+1-morphisms is associative up to specified 2-isomorphisms.
+A basic example is the 2 -category of categories Cat, with objects small cat-
+egoriesC, 1-morphisms functors F:C→D, and 2-morphisms natural trans-
+formations η:F⇒Gfor functors F,G:C→D. Orbifolds naturally form a
+2-category, as do Deligne–Mumford and Artin stacks in algebraic ge ometry.
+In a 2-category C, there are three notions of when objects X,YinCare
+‘the same’: equalityX=Y, andisomorphism , that is we have 1-morphisms
+119f:X→Y,g:Y→Xwithg◦f= idXandf◦g= idY, andequivalence ,
+that is we have 1-morphisms f:X→Y,g:Y→Xand 2-isomorphisms
+η:g◦f⇒idXandζ:f◦g⇒idY. Usually equivalence is the most useful.
+For example, isomorphisms are not preserved by equivalences of 2- categories,
+whereas equivalences are.
+LetCbe a 2-category. The homotopy category Ho(C) ofCis the category
+whose objects are objects of C, and whose morphisms [ f] :X→Yare 2-
+isomorphism classes [ f] of 1-morphisms f:X→YinC. Then equivalences
+inCbecome isomorphisms in Ho( C), 2-commutative diagrams in Cbecome
+commutative diagrams in Ho( C), and so on.
+Commutative diagrams in 2-categories should in general only commute up
+to (specified) 2-isomorphisms , rather than strictly. Then we say the diagram
+2-commutes . A simple example of a commutative diagram in a 2-category Cis
+Yg
+/d42/d42❚❚❚❚❚❚❚❚❚❚❚
+η
+/d11/d19Xf/d52/d52❥❥❥❥❥❥❥❥❥❥❥
+h/d47/d47Z,
+which means that X,Y,Zare objects of C,f:X→Y,g:Y→Zand
+h:X→Zare 1-morphisms in C, andη:g◦f⇒his a 2-isomorphism.
+Next we discuss 2-functors between 2-categories, following Borc eux [8,§7.2,
+§7.5] and Behrend et al. [4, §B.4].
+Definition A.2. LetC,Dbe strict 2-categories. A strict2-functorF:C→D
+assigns an object F(X) inDfor each object XinC, a 1-morphism F(f) :
+F(X)→F(Y) inDfor each 1-morphism f:X→YinC, and a 2-morphism
+F(η) :F(f)⇒F(g) inDfor each 2-morphism η:f⇒ginC, such that F
+preserves all the structures on C,D, that is,
+F(g◦f) =F(g)◦F(f), F(idX) = idF(X), F(ζ∗η) =F(ζ)∗F(η),(A.3)
+F(ζ⊙η) =F(ζ)⊙F(η), F(idf) = idF(f). (A.4)
+Now let C,Dbe weak 2-categories. Then strict 2-functors F:C→Dare
+not well-behaved. To fix this, we need to relax (A.3) to hold only up to s pecified
+2-isomorphisms. A weak2-functor (orpseudofunctor )F:C→Dassigns an
+objectF(X) inDfor each object XinC, a 1-morphism F(f) :F(X)→F(Y)
+inDfor each 1-morphism f:X→YinC, a 2-morphism F(η) :F(f)⇒F(g)
+inDfor each 2-morphism η:f⇒ginC, a 2-isomorphism Fg,f:F(g)◦F(f)⇒
+F(g◦f) inDfor all 1-morphisms f:X→Y,g:Y→ZinC, and a 2-
+isomorphism FX:F(idX)⇒idF(X)inDfor all objects XinCsuch that (A.4)
+holds, and for all e:W→X,f:X→Y,g:Y→ZinCthe following diagram
+of 2-isomorphisms commutes in D:
+(F(g)◦F(f))◦F(e)
+αF(g),F(f),F(e)/d11/d19Fg,f∗idF(e)/d43/d51F(g◦f)◦F(e)Fg◦f,e/d43/d51F((g◦f)◦e)
+F(αg,f,e)/d11/d19
+F(g)◦(F(f)◦F(e))idF(g)∗Ff,e/d43/d51F(g)◦F(f◦e)Fg,f◦e/d43/d51F(g◦(f◦e)),
+120and for all 1-morphisms f:X→YinC, the following commute in D:
+F(f)◦F(idX)Ff,idX/d43/d51
+idF(f)∗FX/d11/d19F(f◦idX)
+F(βf)/d11/d19F(idY)◦F(f)FidY,f/d43/d51
+FY∗idF(f)/d11/d19F(idY◦f)
+F(γf)/d11/d19
+F(f)◦idF(X)βF(f)/d43/d51F(f),idF(Y)◦F(f)γF(f)/d43/d51F(f),
+and iff,˙f:X→Yandg,˙g:Y→Zare 1-morphisms and η:f⇒˙f,ζ:g⇒˙g
+are 2-morphisms in Cthen the following commutes in D:
+F(g)◦F(f)
+F(ζ)∗F(η)/d11/d19Fg,f/d43/d51F(g◦f)
+F(ζ∗η)/d11/d19
+F(˙g)◦F(˙f)F˙g,˙f/d43/d51F(˙g◦˙f).
+There are obvious notions of composition G◦Fof strict and weak 2-functors
+F:C→D,G:D→E,identity2-functors idC, and so on.
+IfC,Dare strict 2-categories, then a strict 2-functor F:C→Dcan be
+made into a weak 2-functor by taking all Fg,f,FXto be identity 2-morphisms.
+Here are some well-known facts about 2-categories and 2-functo rs:
+(i) Every weak 2-category Cis equivalent as a weak 2-category to a strict
+2-category C′, that is, weak 2-categories can always be strictified.
+(ii) IfC,Dare strict 2-categories, and F:C→Dis a weak 2-functor, it
+may not be true that Fis 2-naturally isomorphic to a strict 2-functor
+F′:C→D. That is, weak 2-functors cannot necessarily be strictified.
+Even if one is working with strict 2-categories, weak 2-functors ar e often
+the correct notion of functor between them.
+We define fibre products in 2-categories, following [4, Def. B.13].
+Definition A.3. LetCbe a 2-category and g:X→Z,h:Y→Zbe
+1-morphisms in C. Afibre product X×ZYinCconsists of an object W, 1-
+morphisms e:W→Xandf:W→Yand a 2-isomorphism η:g◦e⇒h◦f
+inC, so that we have a 2-commutative diagram
+Wf/d47/d47
+e/d15/d15 ✘✘✘✘/d72/d80ηY
+h/d15/d15
+Xg/d47/d47Z(A.5)
+with the following universal property: suppose e′:W′→Xandf′:W′→Y
+are 1-morphisms and η′:g◦e′⇒h◦f′is a 2-isomorphism in C. Then there
+should exist a 1-morphism b:W′→Wand 2-isomorphisms ζ:e◦b⇒e′,
+θ:f◦b⇒f′such that the following diagram of 2-isomorphisms commutes:
+g◦e◦bη∗idb/d43/d51
+idg∗ζ/d11/d19h◦f◦b
+idh∗θ/d11/d19
+g◦e′η′/d43/d51h◦f′.(A.6)
+121Furthermore, if ˜b,˜ζ,˜θare alternative choices of b,ζ,θthen there should exist a
+unique 2-isomorphism ǫ:˜b⇒bwith
+˜ζ=ζ⊙(ide∗ǫ) and ˜θ=θ⊙(idf∗ǫ).
+We call such a fibre product diagram (A.5) a 2- Cartesian square . If a fibre
+productX×ZYinCexists then it is unique up to equivalence in C.
+Orbifolds,andstacksinalgebraicgeometry,form2-categories,a ndDefinition
+A.3istherightwaytodefinefibreproductsoforbifoldsorstacks,a sin[4]. Given
+a 2-commutative diagram in a 2-category
+Uf/d47/d47
+e/d15/d15 ✕✕✕✕/d70/d78ηWi/d47/d47
+h/d15/d15 ✕✕✕✕/d70/d78ζY
+k/d15/d15
+Vg/d47/d47Xj/d47/d47Z,
+if the two small rectangles are 2-Cartesian, then the outer recta ngle is too.
+A.2 Grothendieck topologies, sites, prestacks, and stacks
+Some references for this section are Olsson [57], Artin [3], Behrend et al. [4],
+and Laumon and Moret-Bailly [46].
+Definition A.4. LetCbe a category, and U∈C. AsieveSonUis a collection
+of morphisms φ:V→UinCclosed under precomposition, that is, if φ:V→U
+lies inSandψ:W→Vis a morphism in Cthenφ◦ψ:W→Ulies inS.
+AGrothendieck topology onCis a collection of distinguished sieves for each
+objectU∈Ccalledcovering sieves , satisfying some axioms we will not give. A
+site(C,J) is a categoryCwith a Grothendieck topology J.
+It is often convenient to define Grothendieck topologies using Grot hendieck
+pretopologies. A Grothendieck pretopology PJonCis a collection of families
+{ϕa:Ua→U}a∈Aof morphisms inCcalledcoverings , satisfying:
+(i) Ifϕ:V→Uis an isomorphism in C, then{ϕ:V→U}is a covering;
+(ii) If{ϕa:Ua→U}a∈Ais a covering, and {ψab:Vab→Ua}b∈Bais a covering
+for alla∈A, then{ϕa◦ψab:Vab→U}a∈A,b∈Bais a covering.
+(iii) If{ϕa:Ua→U}a∈Ais a covering and ψ:V→Uis a morphism in C
+then{πV:Ua×ϕa,U,ψV→V}a∈Ais a covering, where the fibre product
+Ua×UVexists inCfor alla∈A.
+Each Grothendieck pretopology PJhas an associated Grothendieck topology
+J, in which a sieve SonU∈Cis a covering sieve in Jif and only if it contains
+a covering{ϕa:Ua→U}a∈AinPJ.
+A Grothendieck pretopology PJgives a notion of open cover of objects
+inC. For example, if Cis the category of topological spaces Top, we could
+definePJto be the collection of families {ϕa:Ua→U}a∈AinTopsuch that
+ϕa:Ua→Uis a homeomorphism with an open subset ϕa(Ua)⊆Ufora∈A,
+withU=/uniontext
+a∈Aϕa(Ua), so that{ϕa:Ua→U}a∈Ais an open cover of U.
+122Definition A.5. LetCbe a category. A category fibred in groupoids over Cis
+a functorpX:X →C, whereXis a category, such that given any morphism
+g:C1→C2inCandX2∈XwithpX(X2) =C2, there exists a morphism
+f:X1→X2inXwithpX(f) =g, and given commutative diagrams (on the
+left) inX, in whichgis to be determined, and (on the right) in C:
+X1 g/d47/d47
+f /d38/d38▼▼▼▼▼▼ X2
+h /d120/d120qqqqqq
+X3pX/squigglerightpX(X1)
+g′/d47/d47
+pX(f)/d40/d40❘❘❘pX(X2)
+pX(h)/d118/d118❧❧❧
+pX(X3),(A.7)
+then there exists a unique morphism gas shown with pX(g) =g′andf=h◦g.
+Often we refer to Xas the category fibred in groupoids (or prestack, or stack,
+etc.), leaving pXimplicit.
+IfpX:X→Cis a category fibred in groupoids and Cis an object inC, the
+fibreXCis the subcategory of Xwith objects those X∈XwithpX(X) =C,
+and morphisms those f:X1→X2withpX(f) = idC:C→C. ThenXCis a
+groupoid (i.e. a category with all morphisms isomorphisms).
+Definition A.6. Let (C,J) be a site, and pX:X →Cbe a category fibred
+in groupoids over C. We callXaprestack if whenever{ϕa:Ua→U}a∈Ais
+a covering family in Jand we are given commutative diagrams in X,Cfor all
+a,b∈A, in whichfis to be determined:
+Xab
+/d124/d124①①①
+/d25/d25✸✸✸✸✸✸✸/d47/d47Yab
+/d125/d125③③③
+/d24/d24✶✶✶✶✶✶✶
+Xa
+xa
+/d25/d25✹✹✹✹✹✹✹/d47/d47Ya
+ya
+/d25/d25✷✷✷✷✷✷✷
+Xb/d47/d47xb
+/d123/d123①①①Yb
+yb /d125/d125③③③
+Xf/d47/d47YpX/squigglerightUa×UUbπUa/d120/d120qqqqπUb
+/d29/d29❀❀❀❀❀❀❀Ua×UUb
+πUa /d120/d120qqqq
+πUb
+/d29/d29✿✿✿✿✿✿✿
+Ua
+ϕa/d29/d29❀❀❀❀❀❀❀❀Ua
+ϕa
+/d29/d29❀❀❀❀❀❀❀❀
+Ub
+ϕb/d120/d120rrrrrUb
+ϕb /d120/d120rrrrr
+U U,(A.8)
+then there exists a unique f:X→YinXwithpX(f) = idUmaking (A.8)
+commute for all a∈A.
+LetpX:X→Cbe a prestack. We call Xastackif whenever{ϕa:Ua→
+U}a∈Ais a covering family in Jand we are given commutative diagrams in
+X,Cfor alla,b,c∈A, withXab=Xba,Xabc=Xbac=Xacb, etc., in which the
+objectXand morphisms xaare be determined:
+Xabc
+/d123/d123✈✈✈
+/d26/d26✺✺✺✺✺✺✺/d47/d47Xac
+xac /d123/d123①①xca
+/d25/d25✷✷✷✷✷✷
+Xab
+xba/d26/d26✺✺✺✺✺✺✺xab/d47/d47Xa
+xa
+/d25/d25Xbcxcb /d47/d47xbc
+/d123/d123✈✈✈Xc
+xc /d124/d124
+Xbxb/d47/d47XpX/squigglerightUa×UUb×UUc/d119/d119♦♦♦
+/d31/d31❃❃❃❃❃❃❃/d47/d47Ua×UUc
+/d119/d119♦♦♦♦
+/d31/d31❃❃❃❃❃❃❃
+Ua×UUb
+/d31/d31❃❃❃❃❃❃❃/d47/d47Ua
+ϕa
+/d31/d31❃❃❃❃❃❃❃
+Ub×UUc/d47/d47
+/d119/d119♦♦♦♦Uc
+ϕc /d119/d119♦♦♦♦♦
+Ubϕb/d47/d47U,(A.9)
+then there exists X∈Xand morphisms xa:Xa→XwithpX(xa) =ϕafor all
+a∈A, making (A.9) commute for all a,b,c∈A.
+Thus, in a prestack we have a sheaf-like condition allowing us to glue mo r-
+phisms inXuniquely over covers in C; in a stack we also have a sheaf-like
+condition allowing us to glue objects in Xover covers inC.
+123Definition A.7. Let (C,J) be a site. A 1- morphism between (pre)stacks X,Y
+on (C,J) is a functor F:X→YwithpY◦F=pX:X→C. IfF,G:X→Y
+are 1-morphisms, a 2- morphismη:F⇒Gis an isomorphism of functors with
+idpY∗η= idpX:pY◦F⇒pY◦G. That is, for all X∈ Xwe are given
+an isomorphism η(X) :F(X)→G(X) inYwithpY(η(X)) = idpX(X), such
+that iff:X1→X2is a morphism in Xthenη(X2)◦F(f) =G(f)◦η(X1) :
+F(X1)→G(X2)inY. Withthesedefinitions, thestacksandprestackson( C,J)
+form (strict) 2-categories , which we write as Sta(C,J)andPresta (C,J). All 2-
+morphisms in Sta(C,J),Presta (C,J)are invertible, that is, are 2-isomorphisms,
+soSta(C,J),Presta (C,J)are (2,1)-categories.
+AsubstackYof a stackXis a strictly full subcategory YinXsuch that
+pY:=pX|Y:Y→Cis a stack. The inclusion functor iY:Y֒→Xis then a
+1-morphism of stacks.
+Definition A.8. Let (C,J) be a site, and Xa prestack on (C,J), so that
+Sta(C,J)andPresta (C,J)are 2-categories. A stack associated to X, orstack-
+ification ofX, is a stack ˆXwith a 1-morphism of prestacks i:X →ˆX, such
+that for every stack Y, composition with iyields an equivalence of categories
+Hom(ˆX,Y)i∗
+−→Hom(X,Y).
+As in [46, Lem. 3.2], every prestack has an associated stack, just a s every
+presheaf has an associated sheaf.
+Proposition A.9. For every prestack Xon(C,J)there exists an associated
+stacki:X→ˆX,which is unique up to equivalence in Sta(C,J).
+There is a natural construction of fibre products in the 2-catego rySta(C,J):
+Definition A.10. Let (C,J) be a site,X,Y,Zbe stacks on (C,J), andF:
+X → Z ,G:Y → Z be 1-morphisms. Define a category Wto have ob-
+jects (X,Y,α), whereX∈X,Y∈Yandα:F(X)→G(Y) is an isomor-
+phism inZwithpX(X) =pY(Y) =UandpX(α) = idUinC, and for objects
+(X1,Y1,α1),(X2,Y2,α2) inWa morphism ( f,g) : (X1,Y1,α1)→(X2,Y2,α2)
+inWis a pair of morphisms f:X1→X2inXandg:Y1→Y2inYwith
+pX(f) =pY(g) =ϕ:U→VinCandα2◦F(f) =G(g)◦α1:F(X1)→G(Y2)
+inZ. ThenWis a stack over (C,J).
+Define 1-morphisms pW:W →C bypW: (X,Y,α)/ma√sto→pX(X) andpW:
+(f,g)/ma√sto→pX(f), andπX:W→X byπX: (X,Y,α)/ma√sto→XandπX: (f,g)/ma√sto→f,
+andπY:W →Y byπY: (X,Y,α)/ma√sto→YandπY: (f,g)/ma√sto→g. Define a 2-
+morphismη:F◦πX⇒G◦πYbyη(X,Y,α) =α. ThenW,πX,πY,ηis a fibre
+productX×ZYinSta(C,J), in the sense of Definition A.3.
+The functor id C:C→Cis aterminal object inSta(C,J), andmay be thought
+of as a point∗.ProductsX×YinSta(C,J)are fibre products over ∗. IfXis a
+stack, the diagonal 1-morphism is the natural 1-morphism ∆ X:X→X×X .
+Theinertia stack IXofXis the fibre product X×∆X,X×X,∆XX, with natural
+inertia1-morphism ιX:IX→Xfrom projection to the first factor of X. Then
+124we have a 2-Cartesian diagram in Sta(C,J):
+IX/d47/d47
+ιX/d15/d15✗✗✗✗/d71/d79X
+∆X/d15/d15
+X∆X/d47/d47X×X.
+Thereisalsoanatural1-morphism X:X→IXinduced bythe 1-morphismid X
+fromXto the two factors XinIX=X×X×XXand the identity 2-morphism
+on ∆X◦idX:X→X×X .
+A.3 Descent theory on a site
+The theory of descent in algebraic geometry, due to Grothendieck , says that
+objects and morphisms over a scheme Ucan be described locally on an open
+cover{Ui:i∈I}ofU. It is described by Behrend et al. [4, App. A] and
+Olsson [57,§4], and at length by Vistoli [68]. We shall express descent as
+conditions on a general site ( C,J).
+Definition A.11. Let (C,J) be a site. We say that ( C,J)has descent for
+objectsif whenever{ϕa:Ua→U}a∈Ais a covering inJand we are given mor-
+phismsfa:Xa→UainCfor alla∈Aand isomorphisms gab:Xa×ϕa◦fa,U,ϕb
+Ub→Xb×ϕb◦fb,U,ϕaUainCfor alla,b∈Awithgab=g−1
+basuch that for all
+a,b,c∈Athe following diagram commutes:
+(Xa×ϕa◦fa,U,ϕbUb)×πU,U,ϕcUc∼=
+(Xa×ϕa◦fa,U,ϕcUc)×πU,U,ϕbUbgab×idUc/d47/d47
+gac×idUb
+/d37/d37▲▲▲▲▲▲▲▲▲▲▲▲(Xb×ϕb◦fb,U,ϕcUc)×πU,U,ϕaUa∼=
+(Xb×ϕb◦fb,U,ϕaUa)×πU,U,ϕcUc
+gbc×idUa
+/d121/d121ssssssssssssgba×idUc/d111/d111
+(Xc×ϕc◦fc,U,ϕaUa)×πU,U,ϕbUb∼=
+(Xc×ϕc◦fc,U,ϕbUb)×πU,U,ϕaUa,gca×idUb/d101/d101▲▲▲▲▲▲▲▲▲▲▲▲gcb×idUa/d57/d57ssssssssssss
+then there exist a morphism f:X→UinCand isomorphisms ga:Xa→
+X×f,U,ϕ aUafor alla∈Asuch thatfa=πUa◦gaand the diagram below
+commutes for all a,b∈A:
+Xa×ϕa◦fa,U,ϕbUbga×idUb/d47/d47
+gab
+/d15/d15(X×f,U,ϕ aUa)×ϕa◦πUa,U,ϕbUb
+∼=/d15/d15
+X×f,U,π U(Ua×ϕa,U,ϕbUb)
+∼=/d15/d15
+Xb×ϕb◦fb,U,ϕaUa (X×f,U,ϕ bUb)×ϕb◦πUb,U,ϕaUa.g−1
+b×idUa/d111/d111
+Furthermore X,fshould be unique up to canonical isomorphism. Note that all
+the fibre products used above exist in Cby Definition A.4(iii).
+125Definition A.12. Let (C,J) be a site. We say that ( C,J)has descent for
+morphisms if whenever{ϕa:Ua→U}a∈Ais a covering inJandf:X→U,
+g:Y→Uandha:X×f,U,ϕ aUa→Y×g,U,ϕ aUafor alla∈Aare morphisms
+inCwithπUa◦ha=πUaand for alla,b∈Athe following diagram commutes:
+(X×f,U,ϕ aUa)×ϕa◦πUa,U,ϕbUb
+∼=/d15/d15ha×idUb/d47/d47(Y×g,U,ϕ aUa)×ϕa◦πUa,U,ϕbUb
+∼=/d15/d15
+X×f,U,π U(Ua×ϕa,U,ϕbUb)
+∼=/d15/d15Y×g,U,π U(Ua×ϕa,U,ϕbUb)
+∼=/d15/d15
+(X×f,U,ϕ bUb)×ϕb◦πUb,U,ϕaUahb×idUa/d47/d47(Y×g,U,ϕ bUb)×ϕb◦πUb,U,ϕaUa,
+then there exists a unique h:X→YinCwithha=h×idUafor alla∈A.
+Then [4, Prop.s A.12, A.13 & §A.6] show that descent holds for objects and
+morphisms for affine schemes with the fppf topology, but for arbitr ary schemes
+with the fppf topology, descent holds for morphisms and fails for ob jects.
+A.4 Properties of 1-morphisms
+ObjectsVinCyield stacks ¯Von (C,J).
+Definition A.13. Let (C,J) be a site, and Van object ofC. Define a category
+¯Vto have objects ( U,θ) whereU∈Candθ:U→Vis a morphism in C, and
+to have morphisms ψ: (U1,θ1)→(U2,θ2) whereψ:U1→U2is a morphism in
+Cwithθ2◦ψ=θ1:U1→V. Define a functor p¯V:¯V→Cbyp¯V: (U,θ)/ma√sto→U
+andp¯V:ψ/ma√sto→ψ. Note that p¯Visinjective on morphisms . It is then automatic
+thatp¯V:¯V→Cis a category fibred in groupoids, since in (A.7) we can take
+g=g′. It is also automatic that p¯V:¯V→Cis a prestack, since in (A.8) we
+must haveXa=Ya= (Ua,θa),xa=ya=ϕa,X=Y= (U,θ), etc., and the
+unique solution for fisf= idU.
+The site (C,J) is called subcanonical if¯Vis a stack for all objects V∈C.
+If descent for morphisms holds for ( C,J) then (C,J) is subcanonical. Most
+sites used in practice are subcanonical. Suppose ( C,J) is a subcanonical site.
+Iff:V→Wis a morphism in C, define a 1-morphism ¯f:¯V→¯WinSta(C,J)
+by¯f: (U,θ)/ma√sto→(U,f◦θ) and¯f:ψ/ma√sto→ψ. Then the (2-)functor V/ma√sto→¯V,f/ma√sto→¯f
+embedsCas a full discrete 2-subcategory of Sta(C,J).
+Definition A.14. Let (C,J) be a subcanonical site. A stack Xover (C,J) is
+calledrepresentable if it is equivalent in Sta(C,J)to a stack of the form ¯Vfor
+someV∈C. A 1-morphism F:X →Y inSta(C,J)is called representable if
+for allV∈Cand all 1-morphisms G:¯V→Y, the fibre product X×F,Y,G¯Vin
+Sta(C,J)is a representable stack.
+Remark A.15. For stacks in algebraic geometry, one often takes a different
+definition of representable objects and 1-morphisms: ( C,J) is a category of
+schemes with the ´ etale topology, but stacks are called represent able if they are
+126equivalent to an algebraic space rather than a scheme. This is because schemes
+are not general enough for some purposes, e.g. the quotient of a scheme by an
+´ etale equivalence relation may be an algebraic space but not a schem e.
+In our situation, we will have no need to enlarge C∞-schemes to some cate-
+gory of ‘C∞-algebraic spaces’, as C∞-schemes are already general enough, e.g.
+the quotient of a locally fair C∞-scheme by an ´ etale equivalence relation is a
+locally fair C∞-scheme. This is because the natural topology on C∞-schemes
+is much finer than the Zariski or ´ etale topology on schemes, for ins tance, affine
+C∞-schemes are always Hausdorff.
+Definition A.16. Let (C,J) be a subcanonical site. Let Pbe a property of
+morphisms inC. (For instance, if Cis the category Topof topological spaces,
+thenPcould be ‘proper’, ‘open’, ‘surjective’, ‘covering map’, ...). We say th at
+Pisinvariant under base change if for all Cartesian squares in C
+Wf/d47/d47
+e/d15/d15Y
+h/d15/d15
+Xg/d47/d47Z,
+ifgisP, thenfisP. We say that Pislocal on the target if whenever f:
+U→Vis a morphism in Cand{ϕa:Va→V}a∈Ais a covering inJsuch that
+πVa:U×f,V,ϕ aVa→VaisPfor alla∈A, thenfisP.
+LetPbe invariant under base change and local in the target, and let F:
+X→Ybe a representable 1-morphism in Sta(C,J). IfW∈CandG:¯W→Y
+is a 1-morphism then X×F,Y,G¯Wis equivalent to ¯Vfor someV∈C, and
+under this equivalence the 1-morphism π¯W:X×F,Y,G¯W→¯Wis 2-isomorphic
+to¯f:¯V→¯Wfor some unique morphism f:V→WinC. We say that F
+has property Pif for allW∈Cand 1-morphisms G:¯W→Y, the morphism
+f:V→WinCcorresponding to π¯W:X×F,Y,G¯W→¯Whas property P.
+We define surjective 1-morphisms without requiring them representable.
+Definition A.17. Let (C,J) be a site, and F:X → Y be a 1-morphism
+inSta(C,J). We callFsurjective if whenever Y∈YwithpY(Y) =U∈C,
+there exists a covering {ϕa:Ua→U}a∈AinJsuch that for all a∈Athere
+existsXa∈XwithpX(Xa) =Uaand a morphism ga:F(Xa)→YinY
+withpY(ga) =ϕa.
+Following [46, Prop. 3.8.1, Lem. 4.3.3& Rem. 4.14.1],[55, §6], we may prove:
+Proposition A.18. Let(C,J)be a subcanonical site, and
+Wf/d47/d47
+e/d15/d15✖✖✖✖/d71/d79ηY
+h/d15/d15
+Xg/d47/d47Z
+be a2-Cartesian square in Sta(C,J). LetPbe a property of morphisms in C
+which is invariant under base change and local in the target. Then:
+127(a)Ifhis representable, then eis representable. If also hisP,theneisP.
+(b)Ifgis surjective, then fis surjective.
+Now suppose also that (C,J)has descent for objects and morphisms, and that
+g(and hence f) is surjective. Then:
+(c)Ifeis surjective then his surjective, and if eis representable, then his
+representable, and if also eisP,thenhisP.
+A.5 Geometric stacks, and stacks associated to groupoids
+The 2-category Sta(C,J)of all stacks over a site ( C,J) is usually too general
+to do geometry with. To obtain a smaller 2-category whose objects have better
+properties, we impose extra conditions on a stack X:
+Definition A.19. Let (C,J) be a site. We call a stack Xon (C,J)geometric
+if the diagonal 1-morphism ∆ X:X→X×X is representable, and there exists
+U∈Cand a surjective 1-morphism Π : ¯U→X, which we call an atlasfor
+X. WriteGSta(C,J)for the full 2-subcategory of geometric stacks in Sta(C,J).
+Here ∆ Xrepresentable implies Π is representable.
+To obtain nice classes of stacks, one usually requires further prop ertiesPof
+∆Xand Π. For example, in algebraic geometry with ( C,J) schemes with the
+´ etale topology, we assume ∆ Xis quasicompact and separated, and Π is ´ etale
+for Deligne–Mumford stacks X, and Π is smooth for Artin stacks X.
+The following material is based on Metzler [49, §3.1 &§3.3], Laumon and
+Moret-Bailly [46, §§2.4.3, 3.4.3, 3.8, 4.3], and Lerman [47, §4.4].
+We can characterize geometric stacks Xup to equivalence solely in terms of
+objects and morphisms in C, using the idea of groupoid objects inC.
+Definition A.20. Agroupoid object (U,V,s,t,u,i,m )inacategoryC, orsimply
+groupoid inC, consists of objects U,VinCand morphisms s,t:V→U,
+u:U→V,i:V→Vandm:V×s,U,tV→Vsatisfying the identities
+s◦u=t◦u= idU, s◦i=t, t◦i=s, s◦m=s◦π2, t◦m=t◦π1,
+m◦(i×idV) =u◦s, m◦(idV×i) =u◦t,
+m◦(m×idV) =m◦(idV×m) :V×UV×UV−→V,(A.10)
+m◦(idV×u) =m◦(u×idV) :V=V×UU−→V,
+where we suppose all the fibre products exist.
+Groupoids inCare so called because a groupoid in Setsis a groupoid in
+the usual sense, that is, a category with invertible morphisms, whe reUis the
+set ofobjects,Vthe set of morphisms ,s:V→Uthesourceof a morphism,
+t:V→Uthetargetof a morphism, u:U→VtheunittakingX/ma√sto→idX,ithe
+inversetakingf/ma√sto→f−1, andmthemultiplication taking (f,g)/ma√sto→f◦gwhen
+s(f) =t(g). Then (A.10) reduces to the usual axioms for a groupoid.
+128From a geometric stack with an atlas, we can construct a groupoid in C.
+Definition A.21. Let (C,J) be a subcanonical site, and suppose Xis a geo-
+metric stack on ( C,J) with atlas Π : ¯U→X. Then¯U×Π,X,Π¯Uis equivalent
+to¯Vfor someV∈Cas Π is representable. Hence we can take ¯Vto be the fibre
+product, and we have a 2-Cartesian square
+¯V¯t/d47/d47
+¯s/d15/d15 ✖✖✖✖/d71/d79η¯U
+Π/d15/d15¯UΠ/d47/d47X(A.11)
+inSta(C,J). Here as (C,J) is subcanonical, any 1-morphism ¯V→¯UinSta(C,J)
+is 2-isomorphic to ¯ffor some unique morphism f:V→UinC. Thus we may
+write the projections in (A.11) as ¯ s,¯tfor some unique s,t:V→UinC.
+By the universal property of fibre products there exists a 1-mor phismH:
+¯U→¯V, unique up to 2-isomorphism, with ¯ s◦H∼=id¯U∼=¯t◦H. ThisHis
+2-isomorphic to ¯ u:¯U→¯Vfor some unique morphism u:U→VinC, and
+thens◦u=t◦u= idU. Similarly, exchanging the two factors of Uin the fibre
+product we obtain a unique morphism i:V→VinCwiths◦i=tandt◦i=s.
+InSta(C,J)we have equivalences
+V×s,U,tV≃¯Vׯs,¯U,¯t¯V≃(¯U×X¯U)ׯU(¯U×X¯U)≃¯U×X¯U×X¯U.
+Letm:V×s,U,tV→Vbetheuniquemorphismin Csuchthat ¯mis2-isomorphic
+to the projection V×s,U,tV→¯V=¯U×X¯Ucorresponding to projection to the
+first and third factors of ¯Uin the final fibre product. It is now not difficult to
+verify that ( U,V,s,t,u,i,m ) is a groupoid in C.
+Conversely, given a groupoid in Cwe can construct a stack X.
+Definition A.22. Let (C,J) be a site with descent for morphisms, and ( U,
+V,s,t,u,i,m ) be a groupoid in C. Define a prestack X′on (C,J) as follows:
+letX′be the category whose objects are pairs ( T,f) wheref:T→Uis a
+morphism inC, and morphisms are ( p,q) : (T1,f1)→(T2,f2) wherep:T1→T2
+andq:T1→Vare morphisms in Cwithf1=s◦qandf2◦p=t◦q.
+Given morphisms ( p1,q1) : (T1,f1)→(T2,f2) and (p2,q2) : (T2,f2)→(T3,f3)
+the composition is ( p2,q2)◦(p1,q1) =/parenleftbig
+p2◦p1,m◦(q1×(q2◦p2))/parenrightbig
+, where
+q1×(q2◦p2) :T1→V×t,U,sVis induced by the morphisms q1:T1→Vand
+q2◦p2:T1→V, which satisfy t◦q1=f2◦p1=s◦(q2◦p2).
+Define a functor pX′:X′→CbypX′: (T,f)/ma√sto→TandpX′: (p,q)/ma√sto→p.
+Using the groupoid axioms (A.10) we can show that pX′:X′→Cis a category
+fibred in groupoids. Since ( C,J) has descent for morphisms, we can also show
+X′is a prestack. But in general it is not a stack. Let Xbe the associated
+stack from Proposition A.9. We call Xthestack associated to the groupoid
+(U,V,s,t,u,i,m ). It fits into a natural 2-commutative diagram (A.11).
+Groupoids inCare often written V⇒U, to emphasize s,t:V→U, leaving
+u,i,mimplicit. The associated stack is then written as [ V⇒U].
+129Our next theorem is proved by Metzler [49, Prop. 70] when ( C,J) is the site
+of topological spaces with open covers, but examining the proof sh ows that all
+he uses about (C,J) is that fibre products exist in Cand (C,J) has descent for
+objects and morphisms. See also Lerman [47, Prop. 4.31]. If fibre pr oducts may
+not exist inCthen one must also require the morphisms s,tin (U,V,s,t,u,i,m )
+to berepresentable inC, that is, for all f:T→UinCthe fibre products
+Tf,U,sVandTf,U,tVexist inC.
+Theorem A.23. Let(C,J)be a site, and suppose that all fibre products exist
+inC,and that descent for objects and morphisms holds in (C,J). Then the con-
+structions of Definitions A.21, A.22 are inverse. That is, if (U,V,s,t,u,i,m )is
+a groupoid inCandXis the associated stack, then Xis a geometric stack, and
+the2-commutative diagram (A.11)is2-Cartesian, and Πin(A.11)is surjective
+and so an atlas for X,and(U,V,s,t,u,i,m )is canonically isomorphic to the
+groupoid constructed in Definition A.21from the atlas Π :¯U→X. Conversely,
+ifXis a geometric stack with atlas Π :¯U→X,and(U,V,s,t,u,i,m )is the
+groupoid inCconstructed from Πin Definition A.21,and˜Xis the stack as-
+sociated to (U,V,s,t,u,i,m )in Definition A.22,thenXis equivalent to ˜Xin
+Sta(C,J). Thus every geometric stack is associated to a groupoid.
+In the situation of Theorem A.23 we have 2-Cartesian diagrams
+¯V¯t/d47/d47
+¯s/d15/d15¯U
+Π/d15/d15¯VΠ◦¯s/d47/d47
+¯sׯt/d15/d15X
+∆X/d15/d15 ✚✚✚✚/d73/d81
+✚✚✚✚/d73/d81
+¯UΠ/d47/d47X,¯UׯUΠ×Π/d47/d47X×X,
+¯Vׯsׯt,¯UׯU,∆¯U¯U
+Π◦¯s×Π◦¯t/d47/d47
+π¯U/d15/d15IX
+ιX/d15/d15¯U
+¯uׯidU/d15/d15Π/d47/d47X
+X/d15/d15 ✚✚✚✚/d73/d81
+✚✚✚✚/d73/d81
+¯UΠ/d47/d47X,¯Vׯsׯt,¯UׯU,∆¯U¯UΠ◦¯s×Π◦¯t/d47/d47IX,(A.12)
+with surjective rows. So from Proposition A.18 we deduce:
+Corollary A.24. In the situation of Theorem A.23,letPbe a property of
+morphisms inCwhich is invariant under base change and local in the target.
+Then¯Π :¯U→XisPif and only if s:V→UisP,and∆X:X→X×X
+isPif and only if s×t:V→U×UisP,andιX:IX→XisPif and
+only ifπU:V×s×t,U×U,∆UU→UisP,andX:X→IXisPif and only if
+u×idU:U→V×s×t,U×U,∆UUisP.
+We can describe atlases for fibre products of geometric stacks.
+Example A.25. Suppose (C,J) is a subcanonical site, and all fibre products
+exist inC. Let
+Wf/d47/d47
+e/d15/d15 ✗✗✗✗/d71/d79ηY
+h/d15/d15
+Xg/d47/d47Z
+130be a 2-Cartesian diagram in Sta(C,J), whereX,Y,Zare geometric stacks. Let
+ΠX:¯UX→Xand Π Y:¯UY→Ybe atlases. As ∆ Zis representable the fibre
+product ¯UX×g◦ΠX,Z,h◦ΠY¯UYis represented by an object UWofC. Then we
+have a 2-commutative diagram, where we omit 2-morphisms:
+¯UW:=¯UX×Z¯UYΠW
+/d24/d24π1/d47/d47
+/d116/d116❤❤❤❤❤❤❤❤❤
+/d38/d38▲▲▲▲▲▲▲▲▲▲▲▲▲▲X×Z¯UY
+/d117/d117❦❦❦❦❦❦❦
+/d123/d123①①①①①①①①①①① π2
+/d41/d41❘❘❘❘❘❘❘
+¯UX
+/d38/d38▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ΠX/d47/d47X
+g
+/d120/d120qqqqqqqqqqqqqqqWe/d111/d111
+f
+/d123/d123①①①①①①①①①①①①
+¯UY
+/d115/d115❤❤❤❤❤❤❤❤❤❤❤❤❤ΠY
+/d41/d41❙❙❙❙❙❙❙❙❙❙
+Z Yh/d111/d111(A.13)
+Herethe fivesquaresin (A.13) are2-Cartesian. Define Π W=π2◦π1:¯UW→W,
+whereπ1,π2are as in (A.13). Proposition A.18(a),(b) imply that π1,π2are
+representable and surjective, since Π X,ΠYare. Hence Π W=π2◦π1is also
+representable and surjective, so Wis a geometric stack, and Π Wis an atlas for
+W. In the same way, if Pis a property of morphisms in Cwhich is invariant
+under base change and local in the target and closed under compos itions, and
+ΠX,ΠYareP, then Π WisP.
+Now let ¯VW=¯UW×W¯UWand complete to a groupoid ( UW,VW,sW,tW,
+uW,iW,mW) inCas above, withW≃[VW⇒UW], and do the same for X,Y.
+Then by a diagram chase similar to (A.13) we can show that
+¯VW∼=¯VX×Z¯VYandVW∼=(UW×UXVX)×UYVY. (A.14)
+Corollary A.26. Suppose (C,J)is a subcanonical site, and all fibre products
+exist inC. Then the 2-subcategory GSta(C,J)of geometric stacks is closed
+under fibre products in Sta(C,J).
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+Sci. U.S.A. 42 (1956), 359–363.
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+Lie group , Topology 14 (1975), 63–68.
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+(2010), 55–128. arXiv:0810.5174.
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+lecture notes, Princeton, 1980. Available at
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+ences 1 (2014), 153–240. arXiv:1401.1044.
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+Stacks and Applications , Mem. Amer. Math. Soc. 193 (2008), no. 902.
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+pages 1–104 in B. Fantechi et al., editors, Fundamental algebraic geome-
+try, Math. Surveys and Monographs 123, A.M.S., Providence, RI, 2005 .
+math.AG/0412512.
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+University Press, Cambridge, 2002.
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+and convexity , Ann. Sci. ´Ec. Norm. Sup. 39 (2006), 841–869.
+136Glossary of Notation
+AC∞Schcategory of affine C∞-schemes, 31
+AC∞Schfacategory of fair affine C∞-schemes, 31
+AC∞Schfpcategory of finitely presented affine C∞-schemes, 31
+C,D,E,... C∞-rings, 6
+C∐DEpushout of C∞-ringsC,D,E, 7
+C⊗∞Dcoproduct of C∞-ringsC,D, 7
+CGC∞-subring fixed by finite group Gacting onC∞-ringC, 13
+C-mod abelian category of modules over a C∞-ringC, 44
+C∞Ringscategory of C∞-rings, 6
+C∞Ringsfacategory of fair C∞-rings, 12
+C∞Ringsfgcategory of finitely generated C∞-rings, 9
+C∞Ringsfpcategory of finitely presented C∞-rings, 9
+C∞RScategory of C∞-ringed spaces, 25
+C∞Schcategory of C∞-schemes, 31
+C∞Schlfcategory of locally fair C∞-schemes, 31
+C∞Schlfpcategory of locally finitely presented C∞-schemes, 31
+¯C∞Sch2-subcategory of XinC∞Staequivalent to a C∞-scheme ¯X, 62
+¯C∞Schlf2-subcategory ofXinC∞Staequivalent to ¯XforXlocally fair, 62
+¯C∞Schlfp2-subcategory ofXinC∞Staequivalent to ¯XforXlocally finitely
+presented, 62
+C∞Sta2-category of C∞-stacks, 62
+[δ] : [f,ρ]⇒[g,σ] quotient 2-morphism of quotient 1-morphisms, 72
+δX(E) : id−1
+X(E)→Ecanonical isomorphism of pullback sheaves, 23
+δX(E) : id∗
+X(E)→Ecanonical isomorphism of pullbacks in OX-mod, 51
+DMC∞Sta2-category of Deligne–Mumford C∞-stacks, 73
+DMC∞Stalf2-category of locally fair Deligne–Mumford C∞-stacks, 73
+DMC∞Stalfp2-category of locally finitely presented Deligne–Mumford C∞-
+stacks, 73
+137DMC∞Stare2-category of Deligne–Mumford C∞-stacks with representable
+1-morphisms, 108
+Euc category of Euclidean spaces Rnand smooth maps, 7
+FC∞Sta
+C∞Sch:C∞Sch→C∞Stainclusion from C∞-schemes to C∞-stacks, 62
+f∗(E) pushforward (direct image) sheaf, 22
+f−1(E) pullback (inverse image) sheaf, 23
+f∗(E) pullback of sheaf of OY-modules under f:X→Y, 51
+f∗(E) pullback of quasicoherent sheaf Eunderf:X→Y, 93
+[f,ρ] : [X/G]→[Y/H] quotient 1-morphism of quotient C∞-stacks, 72
+f♯:f−1(OY)→OXmorphism of sheaves of C∞-rings inf:X→Y, 24
+f♯:OY→f∗(OX) morphism of sheaves of C∞-rings inf:X→Y, 24
+f:X→Ymorphism of C∞-ringed spaces or C∞-schemes, 24
+¯f:¯X→¯YC∞-stack 1-morphism from a C∞-scheme morphism f:X→Y,
+62
+Γ :LC∞RS→C∞Ringsopglobal sections functor on C∞-ringed spaces, 28
+Γ :OX-mod→C-mod global sections functor on OX-modules, 52
+GSta(C,J)2-category of geometric stacks on a site ( C,J), 128
+Ho(Orb) homotopy category of the 2-category of orbifolds Orb, 88
+If,g(E) : (g◦f)−1(E)→f−1(g−1(E)) isomorphism of pullback sheaves, 23
+If,g(E) : (g◦f)∗(E)→f−1(g−1(E)) isomorphism of pullbacks in OX-mod, 51
+IX: qcoh(X)→qcoh(X) inclusion functor from sheaves on a C∞-schemeXto
+sheaves on the associated Deligne–Mumford C∞-stackX=¯X, 91
+LC∞RScategory of local C∞-ringed spaces, 25
+Man category of manifolds, 7
+Manbcategory of manifolds with boundary, 17
+Manccategory of manifolds with corners, 17
+MSpec : C-mod→OX-mod spectrum functor for modules over a C∞-ringC,
+52
+m∞
+Xflat ideal of f∈C∞(Rn) vanishing to all orders on X⊆Rn, 16
+138OΓ(X),˜OΓ(X),OΓ
+◦(X),˜OΓ
+◦(X) 1-morphisms of orbifold strata XΓ,...,ˆXΓ
+◦of a
+Deligne–Mumford C∞-stackX, 100
+OX-mod abelian category of OX-modules on C∞-schemeX, 50
+Φf:Cn→Coperations on C∞-ringC, for smooth f:Rn→R, 6
+˜ΠΓ(X),ˆΠΓ(X),˜ΠΓ
+◦(X),ˆΠΓ
+◦(X) 1-morphisms of orbifold strata XΓ,...,ˆXΓ
+◦of a
+Deligne–Mumford C∞-stackX, 100
+Presta (C,J)2-category of prestacks on a site ( C,J), 124
+ΨC:C→Γ◦SpecCcanonical morphism for a C∞-ringC, 28
+ΨM:M→Γ◦MSpecMcanonical morphism for a C-moduleM, 52
+qcoh(X) abelian category of quasicoherent sheaves on C∞-schemeX, 55
+qcoh(X) abelian category of quasicoherent sheaves on Deligne–Mumford C∞-
+stackX, 90
+qcohG(X) abelian category of G-equivariant quasicoherent sheaves on a C∞-
+schemeXacted on by a finite group G, 93
+qcoh(V⇒U) category of quasicoherent sheaves on a groupoid V⇒U, 93
+Rco
+all:C∞Rings→C∞Ringscoreflection functor, 35
+Rco
+all:C-mod→C-modcoreflection functor, 57
+Rfa
+fg:C∞Ringsfg→C∞Ringsfareflection functor, 13
+Sets category of sets, 7
+Sh(X) category of sheaves of abelian groups on topological space X, 22
+Spec :C∞Ringsop→LC∞RSspectrum functor on C∞-rings, 27
+Sta(C,J)2-category of stacks on a site ( C,J), 124
+W,X,Y,Z,... C∞-schemes, 31
+W,X,Y,Z,... C∞-stacks, 62
+¯XC∞-stack associated to a C∞-schemeX, 62
+[X/G] quotient C∞-stack, 71
+XΓ,˜XΓ,ˆXΓ,XΓ
+◦,˜XΓ
+◦,ˆXΓ
+◦orbifold strata of a Deligne–Mumford C∞-stackX,
+100
+Xtopunderlying topological space of a C∞-stackX, 67
+Xtopunderlying C∞-ringed space or C∞-scheme of a C∞-stackX, 68
+139Index
+2-category, 118–122, 124
+1-morphism, 118
+composition, 119
+2-Cartesian square, 66, 74, 77, 81,
+92–94, 121, 124, 128, 130
+2-commutative diagram, 120
+2-morphism, 118
+horizontal composition, 81, 94,
+96, 119
+vertical composition, 70, 73, 81,
+94, 95, 103, 104, 111, 119
+equivalence in, 119
+fibre products in, 121–122, 124
+strict, 118
+weak, 118
+abelian category, 22, 44, 50, 51, 57, 91,
+93, 95, 115
+adjoint functor, 13, 23, 24, 28–30, 35,
+53, 57
+algebraic space, 84, 126
+atlas, 65, 66, 74, 75, 93, 128, 130
+´ etale, 76, 77, 79, 91
+Axiom of Choice, 23, 51, 94
+Banach manifold, 42
+C∞-group, 63, 69, 71
+C∞-ring, 3, 5–20
+ascommutative R-algebra,7–9,44,
+50
+C∞-derivation, 45, 46
+colimit, 7, 15, 16
+complete, 35–37
+coproduct, 7, 10, 15–17
+cotangent module Ω C, 45–49
+definition, 6–7
+fair, 5, 11–14, 16, 18
+finitely generated, 5, 8–9, 11, 13–
+15, 26, 32, 33, 46
+finitely presented, 9, 12, 13, 15,
+17–18, 33, 46, 48
+germ determined, 5, 12ideal,seeideal inC∞-ring
+local, 9–13, 25, 27
+localization, 10–11, 13, 47
+module, seemodule over C∞-ring
+module of K¨ ahler differentials, 46
+not noetherian, 9
+of a manifold X, 17–20
+pushout, 15–16, 48, 49
+R-point, 10
+spectrum functor Spec, 3, 27, 51
+C∞-ringed space, 24–33
+cotangent sheaf, 58
+local, 25–27, 31
+morphism, 24
+sheaves ofOX-modules on, 50–51
+pullback, 51
+C∞-scheme, 31–44
+affine
+sheaves ofOX-modules on, 51–
+56
+C∞-group, 63, 69, 71
+closed embedding, 64, 65
+closed morphism, 64
+compact, 31
+cotangent sheaf, 58–61
+definition, 31
+embedding, 64, 65
+´ etale morphism, 64, 65
+fair affine, 31
+fibre products, 31, 32
+finitely presented affine, 31, 88
+Hausdorff, 31, 71, 109, 114
+Lindel¨ of, 31
+locally compact, 31
+locally fair, 31, 65, 73, 79, 126
+locally fair morphism, 65
+locally finitely presented, 31, 65,
+73, 75
+locallyfinitelypresentedmorphism,
+65
+metrizable, 31
+morphism, 24, 64–65
+morphism with finite fibres, 64, 65
+140open embedding, 64, 65
+paracompact, 31
+proper morphism, 64, 65
+regular, 31
+second countable, 31
+separable, 31
+separated morphism, 64, 65
+sheaves ofOX-modules on, 50–61
+pullback, 56
+submersion, 64, 65
+universally closed morphism, 64,
+65
+C∞-stack, 61–89
+1-morphism with finite fibres, 65,
+76, 77
+associated to a groupoid, 62, 66,
+69, 71, 80, 88, 91–93
+atlas, 65, 66, 74, 75, 93
+´ etale, 76, 77, 79, 91
+C∞-substack, 66
+closed, 66
+locally closed, 66
+open, 66–67, 73, 75, 76,80, 100–
+103, 106
+closed embedding, 65, 76, 77
+coarsemoduli C∞-schemeXtop, 68,
+76
+definition, 62
+Deligne–Mumford, seeDeligne–
+MumfordC∞-stack
+embedding, 65
+´ etale 1-morphism, 65
+fibre products, 62, 65, 67, 74, 75,
+81–84, 89, 92, 104, 105
+gluing by equivalences, 70
+is aC∞-scheme, 62, 73, 84, 86,
+104
+isotropy group Iso X([x]), 69, 80,
+84, 86, 91, 99, 104, 113
+open cover, 66
+open embedding, 65, 76, 77
+orbifold group, seeisotropy group
+proper1-morphism,65, 76, 77, 79,
+104
+quotients [X/G], 62–64, 70–86
+definition, 71orbifoldstrata,109–110,114–115
+quotient1-morphism,72–73, 80–
+86
+quotient2-morphism,72–73, 80–
+86
+representable 1-morphism, 65–67,
+72, 104, 108–109
+separated, 65, 69, 74, 77, 79, 80,
+88, 103
+separated 1-morphism, 65, 76, 77,
+79
+stabilizergroup, seeisotropygroup
+submersion, 65
+underlying C∞-ringed spaceXtop,
+68–69, 76, 85
+underlyingtopologicalspace Xtop,
+67–69, 73
+universallyclosed1-morphism,65,
+76, 77
+C∞-substack, 66
+closed, 66
+locally closed, 66
+open, 66–67, 73, 75, 76, 80, 100–
+103, 106
+Cartesian square, 19, 49, 60, 127
+category
+colimit, 7, 15, 16
+fibre product, 19, 31, 32, 49, 51,
+122
+fibred in groupoids, 122
+groupoid object in, 62, 69, 77–79,
+85, 88, 91–93, 128–131
+limit, 25, 31
+pushout, 7, 15–17, 20, 48, 49
+colimit, 7, 15, 16
+coproduct, 7, 10, 15–17
+d-manifold, 4–5
+d-orbifold, 4–5
+Deligne–Mumford C∞-stack, 71–89
+coarsemoduli C∞-schemeXtop,76
+cotangent sheaf, 96–99
+definition, 73
+effective, 86–87
+locally fair, 73, 76, 79, 103
+141locally finitely presented, 73, 76,
+79, 88, 103
+locally Lindel¨ of, 73, 76
+orbifold, 87–89
+orbifold strata, 99–117
+cotangent sheaves, 115–117
+lifting 1- and 2-morphisms to,
+108–109
+ofquotientC∞-stacks,109–110,
+114–115
+sheaves on, 110–117
+partition of unity on, 76
+quasicoherent sheaves on, 89–99,
+110–117
+morphism, 90
+pullback, 93–96
+second countable, 73, 103
+vector bundles on, 90
+descent theory, 125–126
+´ etale topology, 126
+fibre product, 19, 31, 32, 49, 51, 122
+fine sheaf, 24
+functor
+adjoint, 13, 23, 24, 28–30, 35, 53,
+57
+exact, 22, 23, 50, 51, 90, 91
+faithful, 19
+full, 19, 30
+left exact, 22, 50, 51
+reflection, 13, 16, 33, 35, 57
+right exact, 50, 51, 90, 91, 93, 96
+global sections functor Γ, 28–30
+Grothendieck pretopology, 61, 62, 90,
+122
+Grothendieck topology, 61, 62, 90
+Grothendieck topology, 122
+groupoid object, 62, 69, 77–79, 85, 88,
+91–93, 128–131
+Hadamard’s Lemma, 8
+homotopy category, 88, 120
+ideal inC∞-ring, 7–9
+fair, 12–13, 16finitely generated, 9
+flat, 16–17
+germ-determined, 12
+of local character, 12
+inertia stack, 104, 108, 124
+locally effective group action, 86
+manifold, 17–20
+C∞-ring of, 6
+C∞-scheme of, 25, 27
+cotangent bundle, 46
+transversefibreproduct,19–20,32,
+49, 89
+vector bundles on, 44
+with boundary, seemanifold with
+corners
+withcorners, seemanifoldwithcor-
+ners
+manifold with corners, 17–20
+C∞-ring of, 6, 17
+C∞-scheme of, 25, 27
+smooth map, 17, 19
+transversefibreproduct,19–20,32,
+49
+weakly smooth map, 17, 19, 25,
+32, 45
+module over C∞-ring, 44–49
+C∞-derivation, 45, 46
+complete, 56–58
+cotangent module Ω C, 45–49
+finitely generated, 46
+finitely presented, 46
+module of K¨ ahler differentials, 46
+orbifold, 4, 5, 69, 71, 73, 87–89, 99
+transverse fibre product, 89, 122
+vector bundle, 91, 96
+partition of unity, 24, 76
+presheaf, 21, 58
+sheafification, 22, 23, 58, 60
+prestack, 106, 123
+1-morphism, 123
+2-morphism, 123
+stackification, 71, 102–109, 124
+142pushout, 7, 15–17, 20, 48, 49
+quotientC∞-stack, 62–64, 70–86
+1-morphism, 72–73, 80–86
+2-morphism, 72–73, 80–86
+definition, 71
+orbifold strata, 109–110
+sheaves on, 114–115
+reflection functor, 13, 16, 33, 35, 57
+sheaf, 21–23
+coherent, 56
+definition, 21–22
+direct image, 22
+fine, 24, 38–39, 41
+inverse image, 23
+of abelian groups, 21
+ofC∞-rings, 24–26
+on topological space, 21–23
+presheaf, 21
+pullback, 23
+pushforward, 22
+quasicoherent, 55
+stalk, 22, 25, 27
+site, 62, 65, 90, 100, 102, 122–131
+has descent for morphisms, 125
+has descent for objects, 125
+subcanonical, 62, 126–127
+stack, 119, 122–131
+1-morphism, 123
+representable, 126–130
+surjective, 127
+2-morphism, 123
+associated to a groupoid, 62, 71,
+88, 91–93, 129–131
+atlas, 128
+definition, 123
+fibre product, 124
+geometric, 62, 128–131
+representable, 126
+substack, 124
+symplectic geometry, 4
+synthetic differential geometry, 4–5, 9,
+20
+topological spacecompletely regular, 20
+Hausdorff, 20, 31, 41, 69
+Lindel¨ of, 21, 31
+locally compact, 21, 31
+metrizable, 20, 31, 41
+normal, 41
+paracompact, 20, 31, 41
+regular, 20, 41
+second countable, 20, 31, 41
+separable, 20, 41
+Weil algebra, 9
+Zariski topology, 126
+143
\ No newline at end of file
diff --git a/1001.0024.txt b/1001.0024.txt
new file mode 100644
index 0000000000000000000000000000000000000000..253d2acb84d2b5a1d6c42487063dfc377c986201
--- /dev/null
+++ b/1001.0024.txt
@@ -0,0 +1,659 @@
+arXiv:1001.0024v1  [q-fin.CP]  30 Dec 2009November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Journal of Circuits, Systems, and Computers
+c/circlecopyrtWorld Scientific Publishing Company
+BAYESIAN INFERENCE OF STOCHASTIC VOLATILITY MODEL
+BY HYBRID MONTE CARLO
+Tetsuya Takaishi†
+Hiroshima University of Economics,
+Hiroshima 731-0192 JAPAN
+†takaishi@hiroshima-u.ac.jp
+Received (Day Month Year)
+Revised (Day Month Year)
+Accepted (Day Month Year)
+The hybrid Monte Carlo (HMC) algorithm is applied for the Bay esian inference of the
+stochastic volatility (SV) model. We use the HMC algorithm f or the Markov chain Monte
+Carloupdates of volatility variables of the SV model. First we compute parameters of the
+SV model by using the artificial financial data and compare the results from the HMC
+algorithm with those from the Metropolis algorithm. We find t hat the HMC algorithm
+decorrelates the volatility variables faster than the Metr opolis algorithm. Second we
+make an empirical study for the time series of the Nikkei 225 s tock index by the HMC
+algorithm. We find the similar correlation behavior for the s ampled data to the results
+from the artificial financial data and obtain a φvalue close to one ( φ≈0.977), which
+means that the time series has the strong persistency of the v olatility shock.
+Keywords : Hybrid Monte Carlo Algorithm, Stochastic Volatility Mode l, Markov Chain
+Monte Carlo, Bayesian Inference, Financial Data Analysis
+1. Introduction
+Many empirical studies of financial prices such as stock indexes, ex change rates
+have confirmed that financial time series of price returns shows va rious interesting
+properties which can not be derived from a simple assumption that th e price re-
+turns follow the geometric Brownian motion. Those properties are n ow classified
+as stylized facts1,2. Some examples of the stylized facts are (i) fat-tailed distribu-
+tion of return (ii) volatility clustering (iii) slow decay of the autocorre lation time
+of the absolute returns. The true dynamics behind the stylized fac ts is not fully
+understood. In order to imitate the real financial markets and to understand the
+origins of the stylized facts, a variety of models have been propose d and examined.
+Actually many models are able to capture some of the stylized facts3-14.
+In empirical finance the volatilityis an important value to measurethe risk. One
+of the stylized facts of the volatility is that the volatility of price retu rns changes
+in time and shows clustering, so called ”volatility clustering”. Then the histogram
+of the resulting price returns shows a fat-tailed distribution which in dicates that
+1November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+2Authors’ Names
+the probability of having a large price change is higher than that of th e Gaussian
+distribution. In order to mimic these empirical properties of the vola tility and to
+forecast the future volatility values, Engle advocated the autore gressive conditional
+hetroskedasticity (ARCH) model15where the volatility variable changes determin-
+istically depending on the past squared value of the return. Later t he ARCH model
+is generalized by adding also the past volatility dependence to the vola tility change.
+This model is known asthe generalizedARCH(GARCH) model16. The parameters
+of the GARCH model applied to financial time series are conventionally determined
+by the maximum likelihood method. There are many extended versions of GARCH
+models, such as EGARCH17, GJR18, QGARCH19,20models etc., which are de-
+signed to increase the ability to forecast the volatility value.
+The stochastic volatility (SV) model21,22is another model which captures the
+propertiesofthevolatility.IncontrasttotheGARCHmodel,thevo latilityoftheSV
+model changes stochastically in time. As a result the likelihood functio n of the SV
+model is given as a multiple integral of the volatility variables. Such an in tegral in
+general is not analytically calculable and thus the determination of th e parameters
+of the SV model by the maximum likelihood method becomes difficult. To o vercome
+this difficulty in the maximum likelihood method the Markov Chain Monte Ca rlo
+(MCMC) method based on the Bayesian approach is proposed and de veloped21. In
+the MCMC of the SV model one has to update not only the parameter variables
+but also the volatility ones from a joint probability distribution of the p arameters
+and the volatility variables. The number of the volatility variables to be updated
+increases with the data size of time series. The first proposed upda te scheme of
+the volatility variables is based on the local update such as the Metro polis-type
+algorithm21. It is however known that when the local update scheme is used for
+the volatility variables having interactions to their neighbor variables in time, the
+autocorrelationtime ofsampledvolatilityvariablesbecomeslargeand thusthe local
+update scheme becomes ineffective23. In order to improve the efficiency of the local
+update method the blocked scheme which updates several variable s at once is also
+proposed23,24. A recent survey on the MCMC studies of the SV model is seen in
+Ref.25.
+In our study we use the HMC algorithm26which had not been considered
+seriously for the MCMC simulation of the SV model. In finance there ex ists an
+application of the HMC algorithm to the GARCH model27where three GARCH
+parameters are updated by the HMC scheme. It is more interesting to apply the
+HMC for updates of the volatility variables because the HMC algorithm is a global
+update scheme which can update all variables at once. This feature of the HMC
+algorithm can be used for the global update of the volatility variables which can not
+be achieved by the standard Metropolis algorithm. A preliminary stud y28shows
+that the HMC algorithmsamplesthe volatilityvariableseffectively.In t his paperwe
+give a detailed description of the HMC algorithm and examine the HMC alg orithm
+with artificial financial data up to the data size of T=5000. We also ma ke an
+empirical analysis of the Nikkei 225 stock index by the HMC algorithm.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 3
+2. Stochastic Volatility Model
+The standard version of the SV model21,22is given by
+yt=σtǫt= exp(ht/2)ǫt, (1)
+ht=µ+φ(ht−1−µ)+ηt, (2)
+whereyt= (y1,y2,...,yn) represents the time series data, htis defined by ht= lnσ2
+t
+andσtiscalledvolatility.Wealsocall htvolatilityvariable.Theerrorterms ǫtandηt
+are taken from independent normal distributions N(0,1) andN(0,σ2
+η) respectively.
+We assume that |φ|<1. When φis close to one, the model exhibits the strong
+persistency of the volatility shock.
+For this model the parameters to be determined are µ,φandσ2
+η. Let us use θ
+asθ= (µ,φ,σ2
+η). Then the likelihood function L(θ) for the SV model is written as
+L(θ) =/integraldisplayn/productdisplay
+t=1f(ǫt|σ2
+t)f(ht|θ)dh1dh2...dhn, (3)
+where
+f(ǫt|σ2
+t) =/parenleftbig
+2πσ2
+t/parenrightbig−1
+2exp/parenleftbigg
+−y2
+t
+2σ2
+t/parenrightbigg
+, (4)
+f(h1|θ) =/parenleftBigg
+2πσ2
+η
+1−φ2/parenrightBigg−1
+2
+exp/parenleftbigg
+−[h1−µ]2
+2σ2η/(1−φ2)/parenrightbigg
+, (5)
+f(ht|θ) =/parenleftbig
+2πσ2
+η/parenrightbig−1
+2exp/parenleftbigg
+−[ht−µ−φ(ht−1−µ)]2
+2σ2η/parenrightbigg
+. (6)
+As seen in Eq.(3), L(θ) is constructed as a multiple integral of the volatility vari-
+ables. For such an integral it is difficult to apply the maximum likelihood me thod
+which estimates values of θby maximizing the likelihood function. Instead of using
+the maximum likelihood method we perform the MCMC simulations based o n the
+Bayesian inference as explained in the next section.
+3. Bayesian inference for the SV model
+From the Bayes’ rule, the probability distribution of the parameter sθis given by
+f(θ|y) =1
+ZL(θ)π(θ), (7)
+whereZis the normalization constant Z=/integraltext
+L(θ)π(θ)dθandπ(θ) is a prior disti-
+bution of θfor which we make a certian assumption. The values of the paramete rs
+are inferred as the expectation values of θgiven by
+/an}bracketle{tθ/an}bracketri}ht=/integraldisplay
+θf(θ|y)dθ. (8)
+In general this integral can not be performed analytically. For tha t case, one can
+use the MCMC method to estimate the expectation values numerically .November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+4Authors’ Names
+In the MCMC method, we first generate a series of θwith a probability of
+P(θ) =f(θ|y). Letθ(i)= (θ(1),θ(2),...,θ(k)) be values of θgenerated by the MCMC
+sampling. Then using these kvalues the expectation value of θis estimated by an
+average as
+/an}bracketle{tθ/an}bracketri}ht=1
+kk/summationdisplay
+i=1θ(i). (9)
+The statistical error for kindependent samples is proportional to1√
+k. When the
+sampled data are correlated the statistical error will be proportio nal to/radicalbigg
+2τ
+kwhere
+τis the autocorrelation time between the sampled data. The value of τdepends
+on the MCMC sampling scheme we take. In order to reduce the statis tical error
+within limited sampled data it is better to choose an MCMC method which is able
+to generate data with a small τ.
+3.1.MCMC Sampling of θ
+For the SV model, in addition to θ, volatility variables htalso have to be updated
+sincetheyshouldbeintegratedoutasinEq.(3).Let P(θ,ht)be thejointprobability
+distribution of θandht. ThenP(θ,ht) is given by
+P(θ,ht)∼¯L(θ,ht)π(θ), (10)
+where
+¯L(θ,ht) =n/productdisplay
+t=1f(ǫt|ht)f(ht|θ). (11)
+For the prior π(θ) we assume that π(σ2
+η)∼(σ2
+η)−1and for others π(µ) =π(φ) =
+constant.
+The MCMC sampling methods for θare given in the following21,22. The prob-
+ability distribution for each parameter can be derived from Eq.(10) b y extracting
+the part including the corresponding parameter.
+•σ2
+ηupdate scheme.
+The probability distribution of σ2
+ηis given by
+P(σ2
+η)∼(σ2
+η)−n
+2−1exp/parenleftbigg
+−A
+σ2η/parenrightbigg
+, (12)
+where
+A=1
+2{(1−φ2)(h1−µ)2+n/summationdisplay
+t=2[ht−µ−φ(ht−1−µ)]2}.(13)
+Since Eq.(12) is an inverse gamma distribution we can easily draw a value
+ofσ2
+ηby using an appropriate statistical library in the computer.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 5
+•µupdate scheme.
+The probability distribution of µis given by
+P(µ)∼exp/braceleftbigg
+−B
+2σ2η(µ−C
+B)2/bracerightbigg
+, (14)
+where
+B= (1−φ2)+(n−1)(1−φ)2, (15)
+and
+C= (1−φ2)h1+(1−φ)n/summationdisplay
+t=2(ht−φht−1). (16)
+µis drawn from a Gaussian distribution of Eq.(14).
+•φupdate scheme.
+The probability distribution of φis given by
+P(φ)∼(1−φ2)1/2exp{−D
+2σ2η(φ−E
+D)2}, (17)
+where
+D=−(h1−µ)2+n/summationdisplay
+t=2(ht−1−µ)2, andE=/summationtextn
+t=1(ht−µ)(ht−1−µ).(18)
+In order to update φwith Eq.(17), we use the Metropolis-Hastings
+algorithm30,31. Let us write Eq.(17) as P(φ)∼P1(φ)P2(φ) where
+P1(φ) = (1−φ2)1/2, (19)
+P2(φ)∼exp{−D
+2σ2η(φ−E
+D)2}. (20)
+SinceP2(φ) is a Gaussian distribution we can easily draw φfrom Eq.(20).
+Letφnewbe a candidate given from Eq.(20). Then in order to obtain the
+correct distribution, φnewis accepted with the following probability PMH.
+PMH= min/braceleftbiggP(φnew)P2(φ)
+P(φ)P2(φnew),1/bracerightbigg
+= min/braceleftBigg/radicalBigg
+(1−φ2new)
+(1−φ2),1/bracerightBigg
+.(21)
+In addition to the abovestep we restrict φwithin [−1,1]to avoida negative
+value in the calculation of square root.
+3.2.Probability distribution for ht
+The probability distribution of the volatility variables htis given by
+P(ht)≡P(h1,h2,...,hn)∼ (22)
+exp/parenleftBig
+−/summationtextn
+i=1{ht
+2+ǫ2
+t
+2e−ht}−[h1−µ]2
+2σ2
+η/(1−φ2)−/summationtextn
+i=2[ht−µ−φ(ht−1−µ)]2
+2σ2
+η/parenrightBig
+.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+6Authors’ Names
+Thisprobabilitydistributionisnotasimplefunction todrawvaluesof ht.Aconven-
+tional method is the Metropolis method30,31which updates the variables locally.
+There are several methods21,22,23,24developed to update htfrom Eq.(22). Here
+we use the HMC algorithm to update htglobally. The HMC algorithm is described
+in the next section.
+4. Hybrid Monte Carlo Algorithm
+Originallythe HMCalgorithmis developedforthe MCMCsimulationsofthe lattice
+QuantumChromoDynamics(QCD) calculations26. Amajordifficultyofthe lattice
+QCDcalculationsistheinclusionofdynamicalfermions.Theeffectoft hedynamical
+fermions is incorporated by the determinant of the fermion matrix. The computa-
+tional work of the determinant calculation requires O(V3) arithmetic operations29,
+whereVis the volume of a 4-dimensional lattice. A typical size of the volume is
+V >104. The standard Metropolis algorithm which locally updates variables do es
+not work since each local update requires O(V3) arithmetic operations for a deter-
+minant calculation,which results in unacceptable computational cos t in total. Since
+the HMC algorithm is a global update method, the computational cos t remains in
+the acceptable region.
+The basic idea of the HMC algorithm is a combination of molecular dynamic s
+(MD) simulation and Metropolis accept/reject step. Let us conside r to evaluate the
+following expectation value /an}bracketle{tO(x)/an}bracketri}htby the HMC algorithm.
+/an}bracketle{tO(x)/an}bracketri}ht=/integraldisplay
+O(x)f(x)dx=/integraldisplay
+O(x)elnf(x)dx, (23)
+wherex= (x1,x2,...,xn),f(x) is a probability density and O(x) stands for an
+function of x. First we introduce momentum variables p= (p1,p2,...,pn) conjugate
+to the variables xand then rewrite Eq.(23) as
+/an}bracketle{tO(x)/an}bracketri}ht=1
+Z/integraldisplay
+O(x)e−1
+2p2+lnf(x)dxdp=1
+Z/integraldisplay
+O(x)e−H(p,x)dxdp. (24)
+whereZis a normalization constant given by
+Z=/integraldisplay
+exp/parenleftbigg
+−1
+2p2/parenrightbigg
+dp, (25)
+andp2stands for/summationtextn
+i=1p2
+i.H(p,x) is the Hamiltonian defined by
+H(p,x) =1
+2p2−lnf(x). (26)
+Note that the introduction of pdoes not change the value of /an}bracketle{tO(x)/an}bracketri}ht.
+In the HMC algorithm, new candidates of the variables ( p,x) are drawn by
+integrating the Hamilton’s equations of motion,
+dxi
+dt=∂H
+∂pi, (27)
+dpi
+dt=−∂H
+∂xi. (28)November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 7
+In general the Hamilton’s equations of motion arenot solved analytic ally. Therefore
+wesolvethemnumericallybydoingthe MDsimulation.Let TMD(∆t) beanelemen-
+tary MD step with a step size ∆ t, which evolves ( p(t),x(t)) to (p(t+∆t),x(t+∆t)):
+TMD(∆t) : (p(t),x(t))→(p(t+∆t),x(t+∆t)). (29)
+Any integrator can be used for the MD simulation provided that the f ollowing
+conditions are satisfied26
+•area preserving
+dp(t)dx(t)dx=dp(t+∆t)dx(t+∆t). (30)
+•time reversibility
+TMD(−∆t) : (p(t+∆t),x(t+∆t))→(p(t),x(t)). (31)
+The simplest and often used integrator satisfying the above two co nditions is
+the 2nd order leapfrog integrator given by
+xi(t+∆t/2) =xi(t)+∆t
+2pi(t)
+pi(t+∆t) =p(t)i−∆t∂H
+∂xi
+xi(t+∆t) =xi(t+∆t/2)+∆t
+2pi(t+∆t). (32)
+In this study we use this integrator.The numericalintegration is pe rformedNsteps
+repeatedly by Eq.(32) and in this case the total trajectory length λof the MD is
+λ=N×∆t.
+At the end of the trajectory we obtain new candidates ( p′,x′). These candidates
+are accepted with the Metropolis test, i.e. ( p′,x′) are globally accepted with the
+following probability,
+P= min{1,exp(−H(p′,x′))
+exp(−H(p,x))}= min{1,exp(−∆H)}, (33)
+where∆Histhe energydifferencegivenby∆ H=H(p′,x′)−H(p,x). Sinceweinte-
+grate the Hamilton’s equations of motion approximately by an integra tor, the total
+Hamiltonianisnotconserved,i.e.∆ H/ne}ationslash= 0.Theacceptanceorthe magnitudeof∆ H
+is tuned by the step size ∆ tto obtain a reasonable acceptance. Actually there ex-
+ists the optimal acceptance which is about 60 −70%for 2nd order integrators32,33.
+Surprisingly the optimal acceptance is not dependent of the model we consider. For
+the n-th order integrator the optimal acceptance is expected to be32∼exp/parenleftbigg
+−1
+n/parenrightbigg
+.
+We could also use higher order integrators which give us a smaller ener gy dif-
+ference ∆ H. However the higher order integrators are not always effective sin ce
+they need more arithmetic operations than the lower order integra tors32,33. The
+efficiency of the higher order integrators depends on the model we consider. ThereNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+8Authors’ Names
+also exist improved integrators which have less arithmetic operation s than the con-
+ventional integrators34.
+For the volatility variables ht, from Eq.(22), the Hamiltonian can be defined by
+H(pt,ht) =n/summationdisplay
+i=11
+2p2
+i+n/summationdisplay
+i=1{hi
+2+ǫ2
+i
+2e−hi}+[h1−µ]2
+2σ2η/(1−φ2)+n/summationdisplay
+i=2[hi−µ−φ(hi−1−µ)]2
+2σ2η,(34)
+wherepiis defined as a conjugate momentum to hi. Using this Hamiltonian we
+perform the HMC algorithm for updates of ht.
+5. Numerical Studies
+In order to test the HMC algorithm we use artificial financial time ser ies data
+generatedbythe SVmodel with a setofknownparametersand per formthe MCMC
+simulations to the artificial financial data by the HMC algorithm. We als o perform
+the MCMC simulations by the Metropolis algorithm to the same artificial data and
+compare the results with those from the HMC algorithm.
+Using Eq.(1) with φ= 0.97,σ2
+η= 0.05 andµ=−1 we have generated 5000
+time series data. The time series generated by Eq.(1) is shown in Fig.1. From those
+data we prepared 3 data sets: (1)T=1000 data (the first 1000 of the time series),
+(2)T=2000data (the first 2000ofthe time series)and (3) T=5000 (the whole data).
+To these data sets we made the Bayesian inference by the HMC and M etropolis
+algorithms.Preciselyspeakingboth algorithmsareusedonlyfor the MCMC update
+of the volatility variables. For the update of the SV parameters we u sed the update
+schemes in Sec.3.1.
+For the volatility update in the Metropolis algorithm, we draw a new can didate
+of the volatility variables randomly, i.e. a new volatility hnew
+tis given from the
+previous value hold
+tby
+hnew
+t=hold
+t+δ(r−0.5), (35)
+whereris a uniform random number in [0 ,1) andδis a parameter to tune the
+acceptance. The new volatility hnew
+tis accepted with the acceptance Pmetro
+Pmetro= min/braceleftbigg
+1,P(hnew
+t)
+P(hold
+t)/bracerightbigg
+, (36)
+whereP(ht) is given by Eq.(22).
+The initial parameters for the MCMC simulations are set to φ= 0.5,σ2
+η= 1.0
+andµ= 0. The first 10000 samples are discarded as thermalization or burn -in
+process. Then 200000samples are recorded for analysis. The tot al trajectory length
+λof the HMC algorithm is set to λ= 1 and the step size ∆ tis tuned so that the
+acceptance of the volatility variables becomes more than 50%.
+First we analyze the sampled volatility variables. Fig.2 shows the Mont e Carlo
+(MC) history of the volatility variable h100fromT= 2000 data set. We take h100
+as the representative one of the volatility variables since we have ob served theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 9
+0 1000 2000 3000 4000 5000t-6-4-20246yt
+Fig. 1. The artificial SV time series used for this study.
+50000 55000 60000
+Monte Carlo history-2-10123h100HMC
+50000 55000 60000
+Monte Carlo history-2-10123h100Metropolis
+Fig. 2. Monte Carlo histories of h100generated by HMC (left) and Metropolis (right) with
+T= 2000 data set. The Monte Carlo histories in the window from 5 0000 to 60000 are shown.
+similar behavior for other volatility variables. See also Fig.3 for the sim ilarity of the
+autocorrelation functions of the volatility variables.
+AcomparisonofthevolatilityhistoriesinFig.2clearlyindicatesthatth ecorrela-
+tion of the volatility variable sampled from the HMC algorithm is smaller th an that
+from the Metropolis algorithm. To quantify this we calculate the auto correlation
+function (ACF) of the volatility variable. The ACF is defined as
+ACF(t) =1
+N/summationtextN
+j=1(x(j)−/an}bracketle{tx/an}bracketri}ht)(x(j+t)−/an}bracketle{tx/an}bracketri}ht)
+σ2x, (37)
+where/an}bracketle{tx/an}bracketri}htandσ2
+xare the average value and the variance of xrespectively.
+Fig.3 shows the ACF for three volatility variables, h10,h20andh100sampled
+by the HMC. It is seen that those volatility variables have the similar co rrelation
+behavior. Other volatility variables also show the similar behavior. Thu s hereafter
+we only focus on the volatility variable h100as the representative one.
+Fig.4 compares the ACF of h100by the HMC and Metropolis algorithms. It
+is obvious that the ACF by the HMC decreases more rapidly than that by theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+10Authors’ Names
+0 20 40 60 80t0.010.11ACFh10
+h20
+h100
+Fig. 3. Autocorrelation functions of three volatility vari ablesh10,h20andh100sampled by the
+HMC algorithm for T= 2000 data set. These autocorrelation functions show the si milar behavior.
+0 100 200 300 400 500t0.010.11ACFHMC
+Metropolis
+Fig. 4. Autocorrelation function of the volatility variabl eh100by the HMC and Metropolis
+algorithms for T= 2000 data set.
+Metropolis algorithm. We also calculate the autocorrelation time τintdefined by
+τint=1
+2+∞/summationdisplay
+t=1ACF(t). (38)
+The results of τintof the volatility variables are given in Table 1. The values in
+the parentheses represent the statistical errors estimated by the jackknife method.
+We find that the HMC algorithm gives a smaller autocorrelation time tha n the
+Metropolis algorithm, which means that the HMC algorithm samples the volatility
+variables more effectively than the Metropolis algorithm.
+Next we analyze the sampled SV parameters. Fig.5 shows MC histories of the
+φparameter sampled by the HMC and Metropolis algorithms. It seems t hat both
+algorithms have the similar correlationfor φ. This similarity is also seen in the ACF
+in Fig.6(left), i.e. both autocorrelation functions decrease in the sim ilar rate with
+timet. The autocorrelation times of φare very large as seen in Table 1. We also
+find the similar behavior for σ2
+η, i.e. both autocorrelation times of σ2
+ηare large.
+On the other hand we see small autocorrelations for µas seen in Fig.6(right).
+Furthermore we observe that the HMC algorithm gives a smaller τintforµthanNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 11
+φ µ σ2
+η h100
+true 0.97 -1 0.05
+T=1000 HMC 0.973 -1.13 0.053
+SD 0.010 0.51 0.017
+SE 0.0004 0.003 0.001
+2τint 360(80) 3.1(5) 820(200) 12(1)
+Metropolis 0.973 -1.14 0.053
+SD 0.011 0.40 0.017
+SE 0.0005 0.003 0.0013
+2τint 320(60) 10.1(8) 720(160) 190(20)
+T=2000 HMC 0.978 -0.92 0.053
+SD 0.007 0.26 0.012
+SE 0.0003 0.001 0.0009
+2τint 540(60) 3(1) 1200(150) 18(1)
+Metropolis 0.978 -0.92 0.052
+SD 0.007 0.26 0.011
+SE 0.0003 0.003 0.0009
+2τint 400(100) 13(2) 1000(270) 210(50)
+T=5000 HMC 0.969 -1.00 0.056
+SD 0.005 0.11 0.009
+SE 0.0003 0.0004 0.0007
+2τint 670(100) 4.2(7) 1250(170) 10(1)
+Metropolis 0.970 -1.00 0.054
+SD 0.005 0.12 0.008
+SE 0.00023 0.0011 0.0005
+2τint 510(90) 30(10) 960(180) 230(28)
+Table 1. Results estimated by the HMC and Metropolis algorit hms.SDstands for Standard
+Deviation and SEstands for Statistical Error. The statistical errors are es timated by the jackknife
+method. We observe no significant differences on the autocorr elation times among three data sets.
+that of the Metropolis algorithm, which means that HMC algorithm sam plesµ
+more effectively than the Metropolis algorithm although the values of τintforµ
+take already very small even for the Metropolis algorithm.
+The values of the SV parameters estimated by the HMC and the Metr opolis
+algorithms are listed in Table 1. The results from both algorithms well r eproduce
+the true values used for the generation of the artificial financial d ata. Furthermore
+for each parameter and each data set, the estimated parameter s by the HMC and
+the Metropolis algorithms agree well. And their standard deviations a lso agree
+well. This is not surprising because the same artificial financial data, thus the same
+likelihood function is usedfor both MCMC simulationsby the HMC and Met ropolis
+algorithms. Therefore they should agree each other.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+12Authors’ Names
+40000 45000 50000
+MC history0.940.950.960.970.980.991φ
+HMC
+40000 45000 50000
+MC history0.940.950.960.970.980.991φ
+Metropolis
+Fig. 5. Monte Carlo histories of φgenerated by HMC (left) and Metropolis (right) for T= 2000
+data set.
+0 1000t0.010.11ACFHMC
+Metropolis
+0 100 200 300t0.0010.010.1 ACFHMC
+Metropolis
+Fig. 6. Autocorrelation functions of φ(left) and µ(right) by the HMC and Metropolis algorithm
+forT= 2000 data set.
+6. Empirical Analysis
+In this section we make an empirical study of the SV model by the HMC algorithm.
+The empirical study is based on daily data of the Nikkei 225 stock inde x. The
+sampling period is 4 January 1995 to 30 December 2005 and the numbe r of the
+observations is 2706. Fig.7(left) shows the time series of the data. Letpibe the
+Nikkei 225 index at time i. The Nikkei 225 index piare transformed to returns as
+ri= 100ln( pi/pi−1−¯s), (39)
+where ¯sis the average value of ln( pi/pi−1). Fig.7(right) shows the time series of
+returns calculated by Eq.(39). We perform the same MCMC sampling b y the HMC
+algorithm as in the previous section. The first 10000 MC samples are d iscarded and
+then 20000 samples are recorded for the analysis. The ACF of samp ledh100and
+sampled parameters are shown in Fig.8. Qualitatively the results of t he ACF are
+similar to those from the artificial financial data, i.e. the ACF of the v olatility and
+µdecrease quickly although the ACF of φandσ2
+ηdecrease slowly. The estimated
+values of the parameters are summarized in Table 2. The value of φis estimated to
+beφ≈0.977. This value is very close to one, which means the time series has th e
+strong persistency of the volatility shock. The similar values are also seen in theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 13
+HMC φ µ σ2
+η h100
+0.977 0.52 0.020
+SD 0.006 0.13 0.005
+SE 0.001 0.0016 0.001
+2τint560(190) 4(1) 1120(360) 21(5)
+Table 2. Results estimated by the HMC for the Nikkei 225 index data.
+050010001500200025003000t10000150002000025000
+Nikkei 225 Index
+050010001500200025003000t-505rt
+Fig. 7. Nikkei 225 stock index from 4 January 1995 to 30 Decemb er 2005(left) and returns(right).
+0 20 40 60t0.010.11ACFh100
+0 200 400 600800 1000t0.010.11ACFφ
+ση2
+µ
+Fig. 8. Autocorrelation functions of the volatility variab leh100(left) and the sampled parameters
+(right).
+previous studies21,22.
+7. Conclusions
+We applied the HMC algorithm to the Bayesian inference of the SV mode l and
+examined the property of the HMC algorithm in terms of the autocor relation times
+of the sampled data. We observed that the autocorrelation times o f the volatility
+variables and µparameter are small. On the other hand large autocorrelation times
+are observed for the sampled data of φandσ2
+ηparameters. The similar behavior
+for the autocorrelation times are also seen in the literature22.
+From comparison of the HMC and Metropolis algorithms we find that th e HMCNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
+14Authors’ Names
+algorithmsamplesthevolatilityvariablesand µmoreeffectivelythantheMetropolis
+algorithm. However there is no significant difference for φandσ2
+ηsampling. Since
+the autocorrelation times of µfor both algorithms are estimated to be rather small
+the improvement of sampling µby the HMC algorithm is limited. Therefore the
+overall efficiency is considered to be similar to that of the Metropolis a lgorithm.
+By using the artificial financial data we confirmed that the HMC algor ithm cor-
+rectly reproduces the true parameter values used to generate t he artificial financial
+data. Thus it is concluded that the HMC algorithm can be used as an alt ernative
+algorithm for the Bayesian inference of the SV model.
+If we are only interested in parameter estimations of the SV model, t he HMC
+algorithm may not be a superior algorithm. However the HMC algorithm samples
+thevolatilityvariableseffectively.ThustheHMC algorithmmayservea sanefficient
+algorithm for calculating a certain quantity including the volatility varia bles.
+Acknowledgments.
+The numerical calculations were carried out on SX8 at the Yukawa In stitute for
+Theoretical Physics in Kyoto University and on Altix at the Institute of Statistical
+Mathematics.
+Note added in proof. After this work was completed the author noticed a sim-
+ilar approach by Liu35. The author is grateful to M.A. Girolami for drawing his
+attention to this.
+References
+1. R.Mantegna and H.E.Stanley, Introduction to Econophysics (Cambride University
+Press, 1999).
+2. R. Cont, Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues,
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+3. D. Stauffer and T.J.P. Penna, Crossover in the Cont-Boucha ud percolation model for
+market fluctuations, Physics A 256(1998) 284–290.
+4. T. Lux and M. Marchesi, Scaling and Criticality in a Stocha stic Multi-Agent Model
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+5. G. Iori, Avalanche Dynamics and Trading Friction Effects o n Stock Market Returns,
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+6. L.R. da Silva and D. Stauffer, Ising-correlated clusters i n the Cont-Bouchaud stock
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+7. D. Challet, A. Chessa, M. Marsili and Y-C. Zhang, From Mino rity Games to real
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+8. M. Raberto, S. Cincotti, S.M. Focardi and M. Marchesi, Age nt-based Simulation of a
+Financial Market, Physics A 299(2001) 319–327.
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+Frustration across Scales. Int. J. Mod. Phys. C 12(2001) 667–674.
+10. K. Sznajd-Weron and R. Weron, A Simple Model of Price Form ation.Int. J. Mod.
+Phys. C 13(2002) 115–123.
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+arXiv:1001.0025v1  [cs.CR]  30 Dec 2009GNSS-based Positioning: Attacks and Countermeasures
+Panos Papadimitratos and Aleksandar Jovanovic
+EPFL
+Switzerland
+Email: firstname.lastname@epfl.ch
+Abstract
+Increasing numbers of mobile computing devices, user-
+portable, or embedded in vehicles, cargo containers, or the
+physical space, need to be aware of their location in order
+to provide a wide range of commercial services. Most often,
+mobile devices obtain their own location with the help of
+Global Navigation Satellite Systems (GNSS), integrating,
+for example, a Global Positioning System (GPS) receiver.
+Nonetheless, an adversary can compromise location-aware
+applications by attacking the GNSS-based positioning: It
+can forge navigation messages and mislead the receiver into
+calculating a fake location. In this paper, we analyze this
+vulnerability and propose and evaluate the effectiveness of
+countermeasures. First, we consider replay attacks, which
+can be effective even in the presence of future cryptographic
+GNSS protection mechanisms. Then, we propose and an-
+alyze methods that allow GNSS receivers to detect the re-
+ception of signals generated by an adversary, and then re-
+ject fake locations calculated because of the attack. We
+consider three diverse defense mechanisms, all based on
+knowledge, in particular, own location ,time, andDoppler
+shift, receivers can obtain prior to the onset of an attack.
+We find that inertial mechanisms that estimate location
+can be defeated relatively easy. This is equally true for the
+mechanism that relies on clock readings from off-the-shelf
+devices; as a result, highly stable clocks could be needed.
+On the other hand, our Doppler Shift Test can be effective
+without any specialized hardware, and it can be applied to
+existing devices.
+1 Introduction
+As wireless communications enable an ever-broadening
+spectrum of mobile computing applications, location or
+position information becomes increasingly important for
+those systems. Devices need to determine their own posi-
+tion,1to enable location-based or location-aware function-
+ality and services. Examples of such systems include: sen-
+sors reporting environmental measurements; cellular tele -
+phones or portable digital assistants (PDAs) and comput-
+ers offering users information and services related to their
+1In this paper, we are not concerned with the related but ortho g-
+onal localization problem of allowing a specific entity to de termine
+and ascertain the location of other devices.surroundings; mobile embedded units, such as those for
+Vehicular Communication (VC) systems seeking to pro-
+vide transportation safety and efficiency; or, merchandize
+(container) and fleet (truck) management systems.
+Global navigation satellite systems (GNSS), such as the
+Global Positioning System (GPS), its Russian counter-
+part (GLONAS), and the upcoming European GALILEO
+system, are the most widely used positioning technology.
+GNSS transmit signals bearing reference information from
+a constellation of satellites; computing platforms nodes),
+equipped with the appropriate receiver, can decode them
+and determine their own location.
+However, commercial instantiations of GNSS systems,
+which are within the scope of this paper, are open to
+abuse: An adversary can influence the location informa-
+tion,loc(V), a node Vcalculates, and compromise the node
+operation. For example, in the case of a fleet management
+system, an adversary can target a specific truck. First, the
+adversary can use a transmitter of forged GNSS signals
+that overwrite the legitimate GNSS signals to be received
+by the victim node (truck) V. This would cause a false
+loc(V) to be calculated and then reported to the fleet cen-
+ter, essentially concealing the actual location of Vfrom the
+fleet management system. Once this is achieved, physical
+compromise of the truck (e.g., breaking into the cargo or
+hijacking the vehicle) is possible, as the fleet management
+system would have limited or no ability to protect its as-
+sets.
+This is an important problem, given the consequences
+such attacks can have. In this paper, we are concerned
+with methods to mitigate such a vulnerability. In partic-
+ular, we propose mechanisms to detect and reject forged
+GNSS messages, and thus avoid manipulation of GNSS-
+based positioning. Our investigation is complementary
+to cryptographic protection, which commercial GNSS sys-
+tems do not currently provide but are expected to do so
+in the future (e.g., authentication services by the upcom-
+ing GALILEO system [5]). Our approach is motivated by
+the fundamental vulnerability of GNSS-based positioning
+toreplay attacks [9], which can be mounted even against
+cryptographically protected GNSS.
+The contribution of this paper consists of three mecha-
+nisms that allow receivers to detect forged GNSS messages
+and fake GNSS signals. Our countermeasures rely on in-
+formation the receiver obtained before the onset of an at-
+1tack, or more precisely, before the suspected onset of an
+attack. We investigate mechanisms that rely on own (i)
+location information, calculated by GNSS navigation mes-
+sages, (ii) clock readings, without any re-synchronization
+with the help of the GNSS or any other system, and (iii)
+received GNSS signal Doppler shift measurements. Based
+on those different types of information, our mechanisms
+can detect if the received GNSS signals and messages orig-
+inate from adversarial devices. If so, location informatio n
+induced by the attack can be rejected and manipulation
+of the location-aware functionality be avoided. We clarify
+that the reaction to the detection of an attack, and mecha-
+nisms that mitigate unavailability of legitimate GNSS sig-
+nals is out of the scope of this paper.
+We briefly introduce the GNSS operation and related
+work in Sec. 2. We discuss the adversary model and specific
+attack methods in Sec. 3.2. We then present and analyze
+the three defensive mechanisms in Sec. 4. Our findings
+support that highly accurate clocks can be very effective
+at the expense of appropriate clock hardware; but they
+can otherwise be susceptible, when off-the-shelf hardware
+is used. Location-based mechanisms can also be defeated
+relatively easily. On the contrary, our Doppler Shift Test
+(DST) provides accurate detection of attacks, even against
+a sophisticated adversary.
+2 GNSS Overview
+2.1 Basic Operation
+Each GNSS-equipped node Vcan receive simultaneously
+a set of navigation messages NAV ifrom each satellite Si
+in the visible constellation . Satellite transmitters utilize a
+spread-spectrum technique and each satellite is assigned a
+unique spreading code Ci. These codes are a priori pub-
+licly known. Navigation messages allow Vto determine its
+position, loc(V) = (XV, YV, ZV), in a Cartesian system, as
+well global time, by obtaining a clock correction or time
+offset,tV, also called the synchronization error . At least
+four satellites should be visible in order for a receiver to
+compute position and exact time, the so-called PVT (Po-
+sition, Velocity and Time) or navigation solution [6]. This
+computation relies on the pseudo-range measurements per-
+formed by V, one pseudo-range per visible satellite, that is,
+estimating the satellite-receiver distance based on the es ti-
+mated signal propagation delay, ρi. For each pseudo-range
+ρiestimated at V, the following equation is formed:
+ρi=|si−loc(V)|+c·tV (1)
+The satellite Siposition is si, the receiver position is
+loc(V),cis the speed of light, and tVis the synchronization
+error for V.2.2 Future Cryptographic GNSS Protec-
+tion
+Cryptographic protection ensures the authenticity and in-
+tegrity of GNSS messages, i.e., ensures that NAV messages
+generated solely by GNSS entities, with no modification,
+are accepted and used by nodes. Currently, cryptography is
+used in military systems, but it is not available for commer-
+cial systems to provide authenticity and integrity. Public
+or asymmetric key cryptography is a flexible and scalable
+approach that does not require tamper-resistant receivers .2
+Independently of the number of receivers present in the sys-
+tem (possibly, millions or eventually hundreds of millions ),
+a pair of private/public keys ki, Kican be assigned to each
+satellite Si, with the public key bound to the satellite iden-
+tity via a certificate provided by a Certification Authority.
+Each receiver obtains the certified public keys of all satel-
+lites in order to be able to validate NAV messages digitally
+signed with the corresponding ki.Navigation Message Au-
+thentication (NMA) [5] will be available as a GALILEO
+service.
+To further enhance protection, a different public-key
+NMA approach was proposed in [7]. Each Sichooses a
+secret spreading code for each NAV message but discloses
+this, along with a hidden timing marker , in a delayed and
+authenticated manner to the receiving nodes. If nodes can
+maintain accurate clocks by means other than the GNSS
+system alone, they can then safely detect messages that are
+forged or replayed between the time of their creation and
+the code disclosure. A similar idea using Secret Spreading
+Codes (SSC) was presented in [11].
+3 Attacking GNSS
+3.1 Adversary model
+The location (position) GNSS-equipped nodes obtain can
+be manipulated by an external adversary, without any ad-
+versarial control on the GNSS entities (the system ground
+stations, the satellites, the ground-to-satellite commun ica-
+tion, and the receiver). If any cryptographic protection
+is present, we assume that cryptographic primitives are
+not breakable and that the private keys of satellites can-
+not be compromised. The adversary can receive signals
+from all available satellites (depending on the locations o f
+the adversary-controlled receivers). It is also fully awar e
+of the GNSS implementation specifics and thus can pro-
+duce fully compliant signals, i.e., with the same modula-
+tion, transmission frequency equal to the nominal one, ft,
+or any frequency in the range of received ones, fr; similarly,
+transmitted and received signal powers, as well as message
+preambles and body format (header, content).
+We classify adversaries based on their ability to re-
+produce GNSS messages and signals, considering ones
+equipped with:
+2To prevent the compromise of a single, system-wide symmetri c
+key, shared among the GNSS and all nodes.
+21. Single or multiple radios, each transmitting at the
+same constant power, Pc
+t, and frequency fc
+t.
+2. Single or multiple radios, each being ability to adapt
+its transmission frequency, fj
+t, over time; jis an index
+of adversarial radios.
+3. Multiple radios with adaptive transmission capabili-
+ties as above, and additionally the ability to estab-
+lish fast communication among any of the adversarial
+nodes equipped with those radios.
+Adversarial radios in all above cases can record GNSS
+signals and navigation messages for long periods. For all
+adversaries above, we consider a nominal range R, within
+which adversarial transmissions can be received, with this
+value varying for different adversarial radios. We denote
+this as the area under attack . Clearly, the more powerful
+and the more numerous radios an adversary has, the higher
+its potential impact can be. In the sense, it can influence a
+larger system area and potentially mislead more receivers.
+We assume that the area under attack does not coin-
+cide with the wireless system area. In other words, the
+adversary has limited physical presence and communica-
+tion capabilities. This implies that nodes can lock on ac-
+tual GNSS signals for a period of time before entering an
+area under attack. We do not dwell on how frequently and
+under what circumstances nodes are under attack. Rather,
+we investigate the strength of different defense mechanisms
+given that a node is under attack. We abstract the phys-
+ical properties of the adversarial equipment and consider
+the periods of time it can cause unavailability and maintain
+the receiver locked on the spoofed signal.
+We emphasize that our attack model is notthe worst
+case; this would be a receiver under attack during its cold
+start, that is, the first time it is turned on and searches for
+GNSS signals to lock on. However, our adversary model
+corresponds to a broad range of realistic cases and it is a
+powerful one. For example, returning to the cargo example
+of the introduction: It will be hard for an adversary to
+control a receiver from its installation, e.g., on a contain er,
+and then throughout a trip. But it would be rather easy to
+select a location and time to mount its attack. Regarding
+the strength of the attacker, it is noteworthy that attacks
+are possible without any physical access to and without
+tampering with the victim node(s) software and hardware.
+3.2 Mounting Attacks against GNSS Re-
+ceivers
+The adversary can construct a transmitter that emits sig-
+nals identical to those sent by a satellite, and mislead the
+receiver that signals originate from a visible satellite. H ow-
+ever, the attacker has to first force the receiver to lose
+its “lock” on the satellite signals. This can be achieved
+byjamming legitimate GNSS signals, by transmitting a
+sufficiently powerful signal that interferes with and ob-
+scures the GNSS signals [12]. Jammers are simple to con-
+struct with low cost and very effective: for example, withReceived GNSS signal delayed 
+Transmit after  treplay NAV message buffering 
+Preamble 
+detection 
+Victim receiver 
+V
+Total 
+delay treplay Adversary 
+Figure 1: Illustration of the replay attack: the adversary
+captures and replays the signal after some time treplay =
+tmin
+replay +τ, with the τ≥0 chosen by the adversary, and
+tmin
+replay >0 imposed by the specifics of the attack configu-
+ration and the adversary capabilities.
+1 Watt of transmission power, the reception of GNSS sig-
+nals is stopped within a radius of approximately 35 km
+radius [6,12].
+Then, the adversary can spoof GNSS signals, i.e., forge
+and transmit signals at the same frequency and with power
+thatexceeds that of the legitimate GNSS signal at the re-
+ceiver’s antenna. Satellite simulators are capable of broa d-
+casting simultaneously signals carrying counterfeit navi ga-
+tion data from ten satellites.3The spoofed signal can also
+be generated by manipulating and rebroadcasting actual
+signals ( meaconing ). As long as the lock of the victim re-
+ceiver Von the spoofed signal persists, loc(V) is under the
+influence or full control of the adversary.
+Apart from jamming, the adversary could take advan-
+tage of gaps in coverage , i.e., areas and periods of time for
+which Vcannot lock on to more than three satellite sig-
+nals. Clearly, this can be often possible in urban areas or
+because of the terrain, such as tunnels or obstructions from
+high-rise buildings. We do not consider further this case,
+as such loss of satellite signals is not under the control of
+the attacker. Nonetheless, the tests we propose here are ef-
+fective independently of what causes receivers to loose loc k
+on GNSS signals.
+3.3 Replay attack
+Thereplay attack can be viewed as a part of a more general
+class of relay attacks : the attacker receives at one location
+legitimate GNSS signals, relays those to another location
+3The adversary can deceive the receiver after down-conversi on
+of the satellite signal, with one component in-phase and one in-
+quadrature:
+I(t) =aiCa(t)M(t)cos(ft) (2)
+Q(t) =aqCa(t)M(t)sin(ft) (3)
+Cais the C/A (Course/Aquisition) code, M(t) is the NAV message,
+and coefficients aiandaqrepresent the signal attenuation. The at-
+tacker could pick the amplifying coefficients aiandaqsuch that the
+received signal power exceeds the nominal power od a GPS sign al [13].
+3where it retransmits them without any modification. This
+way the adversary can avoid detection if cryptography is
+employed, while it can “present” a victim with GNSS sig-
+nals that are not normally visible at the victim’s location.
+In this paper, we abstract away the placement of adversar-
+ial nodes, and we characterize the replay attack by two fea-
+tures: (i) the adversarial node capability to receive, reco rd
+and replay GNSS signals, and (ii) the delay treplay between
+reception and re-transmission of a signal.
+The GNSS signal reception and replay can be done
+at the message or symbol level, or it can be done by
+recording the entire frequency band and replaying it with-
+out de-spreading signals. The latter, more involved and
+thus costly, would enable the attacker to mount an at-
+tack against the delayed-disclosure secret spreading code
+approach, as pointed out in [7], not only for long replay-
+ing delays but also for very short ones. Clearly, such an
+instantiation of the replaying attack implies a more sophis -
+ticated adversary than one replaying symbols or messages.
+For example, the adversary would need to infer, possibly by
+possessing a legitimate receiver, the start of NAV messages
+to replay signals accordingly
+Thetreplay delay between reception and re-transmission
+depends on the attack configuration (e.g., the distance be-
+tween the receiving and re-transmitting adversarial radio s,
+the physics of the signal propagation, and, when applica-
+ble, the delay for the adversary to decode the GNSS signal).
+We capture such factors by considering tmin
+replay >0, a min-
+imum delay that the adversary cannot avoid. Beyond this,
+the attacker can choose some additional delay τ≥0, such
+that it replays the signal after treplay =tmin
+replay +τ. We
+illustrate a replay attack in Fig. 1: The recording of the
+NAV message starts after its beginning is detected, due to
+the preamble 10001011, with length of eight chips, and the
+decoding of the NAV message first bit. This corresponds
+totmin
+replay = 20ms: the transmission rate of 50 bit/s implies
+that 20ms are needed for the first bit to be received by an
+adversarial radio.
+The adversary can choose different treplay values for sig-
+nals from different satellites, even though “blind” replayi ng
+of all NAV signals with the same delay can be effective. The
+selection of which signals (from which satellites) to relay of-
+fer flexibility. But even the “blind” replaying of all NAV
+signals (the entire band) can be effective: treplay controls
+the “shift” in the PVT solution. Essentially, treplay con-
+trols the “shift” in the PVT solution the adversary induces
+to the victim node(s).
+Fig. 2 shows the impact of a replay attack as a function
+of the spoofing stage of the attack: (i) the location offset
+or error, i.e., the distance between the attack-induced and
+the actual victim receiver position, and (ii) the time offset
+or error, that is, the time difference between the attack-
+induced clock value and the actual time. We consider for
+this example trelay= 20ms, as the first bit decoding de-
+lay dwarfs the preamble detection and propagation delays.
+This is indeed a very subtle attack we refer to [9] for a range
+oftreplay values, which shows that the larger the treplay, as0 50 100 150 200 250 300010002000300040005000600070008000900010000
+Attack duration  [s]Distance offset  [m]
+(a)
+0 50 100 150 200 250 300050100150200250300350
+Attack duration [s]Time offset   [ms]
+  
+(b)
+Figure 2: Impact of the replay attack, as a function of
+thespoofing attack duration. (a) Location offset or er-
+ror: Distance between the attack-induced and the actual
+victim receiver position. (b) Time offset or error: Time
+difference between the attack-induced clock value and the
+actual time.
+the adversary tunes its τvalue, the higher the location and
+time offsets.
+Even for a very low treplay, while the mobile node re-
+ceiver is still locked on the attacker-transmitted signals , the
+location error increases, with the victim receiver “dragge d”
+away from its actual position. Each millisecond of trelay
+translates approximately into 300m of location offset for
+each pseudorange (as the speed of light, c, is taken into
+account), with the actual “displacement” of the victim de-
+pending on the geometry (e.g., position of the satellite
+whose signals were replayed).
+As for the time offset, which can be viewed as a side-
+effect of the attack: it is in the order of less than one mil-
+lisecond per second, and it can very well go easily unnoticed
+by the user. With a given trelay, every time the victim re-
+ceiver re-synchronizes, typically at the end of a NAV mes-
+sage that lasts 30 sec, treplay will emerge as tVfrom the
+PVT solution and thus will be accumulated as part of the
+time offset shown in Fig. 2.
+4 Defense mechanisms
+We investigate three defense mechanisms that rely on a
+common underlying three-step idea. First, the receiver col -
+lects data for a given parameter during periods of time it
+deems it is not under attack; we term this the normal mode .
+4Second, based on the normal mode data, the receiver pre-
+dicts the value of the parameter in the future. When it
+suspects it is under attack, it enters what we term alert
+mode. In this mode, the receiver compares the predicted
+values with the ones it obtains from the GNSS functional-
+ity. If the GNSS-obtained values differ, beyond a protocol-
+selectable threshold, from the predicted ones, the receive r
+deems it is under attack . In that case, all PVT solutions
+obtained in alert mode are discarded. Otherwise, the sus-
+pected PVT solutions are accepted and the receiver reverts
+to the normal mode.
+In this work, we consider three parameters: location ,
+time, andDoppler Shift , and we present the corresponding
+detection mechanisms, Location Inertial Test ,Clock Offset
+Test, andDoppler Shift Test . We emphasize again that all
+three mechanisms rely on the availability of prior informa-
+tion collected in normal mode. But they are irrelevant if
+the receiver starts its operation without any such informa-
+tion (i.e., a cold start ).
+To evaluate the proposed schemes, we use GPS traces
+collected by an ASHTECH Z-XII3T receiver that out-
+puts observation and navigation (.obs and .nav) data into
+RINEX ( Receiver Independent Exchange Format ) [8]. We
+implement the PVT solution functionality in Matlab, ac-
+cording to the receiver interface specification [8]. Our im-
+plementation operates on the RINEX data, which include
+pseudoranges and Doppler frequency shift and phase mea-
+surements. We simulate the movement of receivers over a
+period of T= 300 s, with their position updated at steps of
+Tstep= 1sec.
+4.1 Location Inertial Test
+At the transition to alert mode, the node utilizes own lo-
+cation information obtained from the PVT solution, to
+predict positions while in attack mode. If those positions
+match the suspected as fraudulent PVT ones, the receiver
+returns to normal mode. We consider two approaches for
+the location prediction: (i) inertial sensors and (ii) Kalm an
+filtering.
+Inertial sensors , i.e., altimeters, speedometers, odome-
+ters, can calculate the node (receiver) location indepen-
+dently of the GNSS functionality.4However, the accuracy
+of such (electro-mechanical) sensors degrades with time.
+One example is the low-cost inertial MEMS Crista IMU-15
+sensor (Inertial Measurement Unit).
+Fig. 3 shows the position error as a function of time [4],
+which is in our context corresponds to the period the re-
+ceiver is in the alert mode. As the inertial sensor inaccurac y
+increases, the node has to accept as normal attack-induced
+locations. Fig. 4 shows a two-dimensional projection of
+two trajectories, the actual one and the estimated and er-
+roneously accepted one. We see that over a short period
+4They have already been used to provide continuous navigatio n
+between the update periods for GNSS receivers, which essent ially are
+discrete-time position/time sensors with sampling interv al of approx-
+imately one second0102030405060708090100050100150200250300
+GNSS unavailability period [s]Inertial navigation error [m]
+  
+Figure 3: Location error of Crista IMU-15 inertial sensor,
+as a function of the GNSS unavailability period.
+3.456 3.458 3.46 3.462 3.464 3.466 3.468
+x 1065.295.35.315.325.335.345.355.365.375.38x 105
+ 
+X coordinate [m] Y coordinate [m]
+Attacker−induced trajectory
+Actual trajectory
+Figure 4: Illustration of location error using inertial sen -
+sors: Actual vs. estimated when under attack trajectory.
+of time, a significant difference is created because of the
+attack.
+A more effective approach is to rely on Kalman filtering
+of location information obtained during normal mode. Pre-
+dicted locations can be obtained by the following system
+model:
+Sk+1= Φ kSk+Wk (4)
+withSkbeing the system state, i.e., location ( Xk, Yk, Zk)
+and velocity ( V xk, V yk, V zk) vectors, Φ kthe transition
+matrix, and Wkthe noise. Fig. 5 illustrates the location
+offset for a set of various trajectories. Unlike the case that
+only inertial sensors are used, with measurements of iner-
+tial sensors (with the error characteristics of Fig. 3 used
+as data when GNSS signals are unavailable, filtering pro-
+vides a linearly increasing error with the period of GNSS
+unavailability.
+Overall, for short unavailability periods, inertial mech-
+anisms can be effective. As long as the error (Y axes of
+Figs. 4, 5) does not grow significantly, the replay attack
+can be detected. But for sufficiently high errors, the re-
+play attack impact can remain undetected. We remind the
+reader that the x-axes in Fig. 2 provide the duration of the
+spoofing attack - the transmission (replay) of GNSS signals
+- and they are not to be confused with the duration of the
+GNSS period of unavailability in the x-axis of Figs. 4, 5.
+50 50 100 150 200 250 300020040060080010001200
+Time [s]Distance offset [m]
+Figure 5: Distance error of inertial mechanisms with
+Kalman filtering, as a function of the GNSS unavailabil-
+ity period.
+0 5 10 15 20 25 30−9−8.5−8−7.5−7−6.5−6x 10−3
+Time  [30s step]Time offset  [s]
+  
+Figure 6: Clock offset for the ASHTECH Z-XII3T receiver,
+during a 900 sec period with no re-synchronization.
+4.2 Clock Offset Test
+Each receiver has a clock that is in general imprecise, due
+to the drift errors of the quartz crystal. If the reception
+of GNSS signals is disrupted, the oscillator switches from
+normal to holdover mode. Then, the time accuracy de-
+pends only on the stability of the local oscillator [2,6]. Th e
+quartz crystals of different clocks run at slightly different
+frequencies, causing the clock values to gradually diverge
+from each other (skew error).
+A simulation based study [2] of quartz clocks claims that
+coarse time synchronization can be maintained at microsec-
+ond accuracy without GPS reception for 350 sec in 95%
+cases. This means that quartz oscillators can maintain
+millisecond synchronization for few hours, including ran-
+dom errors and temperature change inaccuracies. Indeed,
+in such a case, the adversary would need to cause GNSS
+availability for long periods of time, for example, tens of
+hours, before being able to mount a relay attack that causes
+a time offset in the order of tens of milliseconds.
+However, without highly stable clocks, mounting attacks
+against the Clock Offset Test can be significantly easier.
+This can be the case for a ASHTECH receiver, for which
+time offset values are shown at successive points in time,
+each 30 seconds apart, in Fig. 6. We clarify this is notto be perceived as criticism for a given receiver or to be
+the basis for the suitability of the Clock Offset Test. As
+explained above, the stability of the receiver clock deter-
+mines the strength of this test. But the data in Fig. 6,
+over a period of 900 seconds, exactly demonstrates that
+for commodity receivers significant instability is observe d;
+time offset values are in the order of ten milliseconds (or
+slightly less). Consequently, the adversary would need to
+jam for roughly a couple of minutes, force the receiver to
+consider as acceptable a time offset of 20 to 32 millisec-
+onds, and thus be mislead by a replay attack as detailed in
+Sec. 3.
+Finally, we note that we do not consider here the case
+of synchronization by means external to the GNSS system.
+For example, if the receiver could connect to the Internet
+and run NTP, it could obtain accurate time. But this would
+be an infrequent operation (in the order of magnitude of
+days), thus useful only if highly stable clock hardware were
+available.
+4.3 Doppler Shift Test (DST)
+Based on the received GNSS signal Doppler shift, with
+respect to the nominal transmitter frequency ( ft=
+1.575GHz), the receiver can predict future Doppler Shift
+values. Once lock to GNSS signals is obtained again, pre-
+dicted Doppler shift values are compared to the ones cal-
+culated due to the received GNSS signal. If the latter are
+different than the predicted ones beyond a threshold, the
+GNSS signal is deemed adversarial and rejected. What
+makes this approach attractive is the smooth changes of
+Doppler shift and the ability to predict it with low, es-
+sentially constant errors over long periods of time. This
+in dire in contrast to the inertial test based on location,
+whose error grows exponentially with time.
+The Doppler shift is produced due to the relative motion
+of the satellite with respect to the receiver. The satellite
+velocity is computed using ephemeris information and an
+orbital model available at the receiver. The received fre-
+quency, fr, increases as the satellite approaches and de-
+creases as it recedes from the receiver; it can be approxi-
+mated by the classical Doppler equation:
+fr=ft·(1−vr·a
+c) (5)
+where ftis nominal (transmitted) frequency, frreceived
+frequency, vris the satellite-to-user relative velocity vector
+andcspeed of radio signal propagation. The product vr·
+arepresents the radial component of the relative velocity
+vector along the line-of-sight to the satellite.
+If the frequency shift differs from the predicted shift for
+each visible satellite Siin the area depending on the data
+obtained from the almanac (in the case when the naviga-
+tion history is available), for more than defined thresholds
+(∆fmin,∆fmax) or estimated Doppler shift from naviga-
+tion history differs for more than the estimated shift, know-
+ing the rate ( r), the receiver can deem the received signal
+as product of attack.
+650 100 150 200 250 3002300235024002450250025502600265027002750
+Time [s]Frequency offset [Hz]
+  
+Measured Doppler shift [Hz ] 
+Linear approximation
+Prediction bounds
+Figure 7: Measured and approximated Doppler frequency
+shift.
+TheAlmanac contains approximate position of the satel-
+lites, ( Xsi, Y si, Zsi), time and the week number ( WN, t ),
+and the corrections, such that the receiver is aware of the
+expected satellites, their position, and the Doppler offset .
+Because of the high carrier frequencies and large satel-
+lite velocities, large Doppler shifts are produced ( ±5kHz),
+and vary rapidly (1 Hz/s). The oscillator of the receiver
+has frequency shift of ±3KHz, thus the resultant frequency
+shift goes therefore up to ±9KHz. Without the knowledge
+of the shift, the receiver has to perform a search in this
+range of frequencies in order to acquire the signal. The
+rate of Doppler shift receiving frequency caused by the rel-
+ative movement between GPS satellite and vehicles approx-
+imately 40 Hz per minute to the maximum. These varia-
+tions are linear for every satellite. If the receiver is mobi le,
+the Doppler shift variation can be estimated knowing the
+velocity of the receiver( [3]).
+In our simulations, Doppler shift is analyzed for each
+available satellite (number of available satellites varie s). To
+be consistent with results shown for other mechanisms, we
+present results for DST for the 300sec period.
+We observe in Fig. 7 the Doppler shift variation based
+on data collected by an ASHTECH receiver: the maximum
+change in rate is within + /−20Hz around a linear curve
+fitted to the data. This clues that with sufficient samples,
+the future Doppler Shift rate, and thus the shift per se,
+values can be predicted. In practice, we observe that 50
+sec of samples, with one sample per second, appear to be
+sufficient.
+More precisely, the rate of change of the frequency shift,
+Di(t), is computed for each satellite, Si, as:
+ri=dDi(t)
+dt(6)
+which can be approximated by numerical methods. Based
+on prior samples for each Di, available for some time win-
+dow the frequency shift can be predicted based those sam-
+ples and the estimate rate of change of the Doppler shift.
+Based on prior measured statistics of the signal at the re-
+ceiver, the variance σ2of a random component, assumed
+to beN(0, σ2), can be estimated. This random component0 50 100 150 200 250 300−10000100020003000
+Time [s]Frequency offset [Hz]SV−1
+0 50 100 150 200 250 300−10000−50000
+Time [s]Frequency offset [Hz]SV−4
+  
+0 50 100 150 200 250 3000200040006000
+Time [s]Frequency offset [Hz]SV−7
+0 50 100 150 200 250 3000100020003000
+Time [s]Frequency offset [Hz]SV−13
+0 50 100 150 200 250 300−4000−20000
+Time [s]Frequency offset [Hz]SV−20
+0 50 100 150 200 250 300−10000100020003000
+Time [s]Frequency offset [Hz] SV−24
+0 50 100 150 200 250 300−4000−20000
+Time [s]Frequency offset [Hz] SV−25
+Figure 8: Doppler shift attack; unsophisticated adversary .
+The dotted line represents the predicted and the solid line
+the measured frequency offset.
+is due to signal variation (including receiver mobility, RF
+multipath, scattering). Its estimation can serve to deter-
+mine an acceptable interval around the predicted values.
+The adversary is mostly at the ground and static or mov-
+ing with speed that is much smaller than the satellite ve-
+locity, which is in a range around 3km/s. Thus, the adver-
+sary will not be able to produce the same Doppler shift as
+the satellites, unless it changes its transmission frequen cy
+to match the one receivers would obtain from GNSS sig-
+nals due to the Doppler shift. An unsophisticated attacker
+would then be easily detected. This is illustrated in Fig. 8:
+After a “gap” corresponding to jamming, there is a striking
+difference, between 100 and 150 seconds, when comparing
+the Doppler shift due to the attack to the predicted one.
+The case of A sophisticated adversary that controls its
+transmission frequency (the attack starts at 160 s)is shown
+in the Fig. 9. The adversary has multiple adaptive ra-
+dios and it operates according to the following principle: i t
+predicts the Doppler frequency shift at the location of the
+receiver, and it then changes its transmission frequency
+accordingly. If the attacker is not precisely aware of the
+actual location and motion dynamics of the victim node
+(receiver), there is still a significant difference between t he
+predicted and the adversary-caused Doppler shift. This
+is shown, with a magnitude of approximately 300 Hz, in
+Fig. 9; a difference that allows detection of the attack.
+5 Conclusion
+Existing GNSS receivers are vulnerable to a number of
+attacks that manipulate the location and time the re-
+ceivers compute. We qualitatively and quantitatively ana-
+lyze those in this paper, and identify memory-based mech-
+anisms that can help in securing GNNS signals. In particu-
+lar, we realize that location-based inertial mechanisms an d
+a clock offset test can be relatively easily defeated, with th e
+adversary causing (through jamming) a sufficiently long
+period of unavailability. In the latter case, only special-
+ized highly stable clock hardware could enable detection of
+fraudulent GNSS signals. Our Doppler Shift Test provides
+70 50 100 150 200 250 300020004000
+Time [s]Frequency offset [Hz]SV−1
+  
+0 50 100 150 200 250 300−10000−50000
+Time [s]Frequency offset [Hz]SV−21
+  
+0 50 100 150 200 250 3000500010000
+Time [s]Frequency offset [Hz]SV−7
+  
+0 50 100 150 200 250 300020004000
+Time [s]Frequency offset [Hz]SV−25
+  
+0 50 100 150 200 250 300−4000−20000
+Time [s]Frequency offset [Hz]SV−9
+  
+0 50 100 150 200 250 3000100020003000
+Time [s]Frequency offset [Hz]SV−29
+  
+0 50 100 150 200 250 300−4000−20000
+Time [s]Frequency offset [Hz]SV−13
+  
+Figure 9: Doppler shift attack; sophisticated adversary.
+The dotted line represents the predicted and the solid line
+the measured frequency offset.
+resilience to long unavailability periods without special ized
+equipment.
+Our results are the first, to the best of our knowledge,
+to provide tangible demonstration of effective mechanisms
+to secure mobile systems from location information manip-
+ulation via attacks against the GNSS systems.
+As part of on-going and future work, we intent to further
+refine and generalize the simulation framework we utilized
+here, to consider precisely the effect of counter-measures
+that only partially limit the attack impact. Moreover, we
+will consider more closely the cost of mounting attacks of
+differing sophistication levels, especially through proof -of-
+concept implementations.
+References
+[1] N. Bertelsen, K. Borre, The GPS Code Software Re-
+ceiver , Aalborg University, Birkhauser, 2007
+[2] W. Franz and H. Hartenstein, Inter-Vehicle Communi-
+cations, FleetNet project , University Karlruhe, 2005
+[3]http://www.freepatentsonline.com/5036329.html
+[4] S. Godha, Performance Evaluation of Low Cost
+MEMS-Based IMU Integrated with GPS for Land Ve-
+hicle Navigation Appplication , University of Calgary,
+2006
+[5] G.W. Hein and F. Kneissl, Authenticating GNSS Proofs
+Against Spoofs , InsideGNSS, September/October 2007
+[6] E.D. Kaplan, Understanding GPS - Principles and Ap-
+plications , Artech House, 2006
+[7] M. Kuhn, An asymetric Security Mechanism for Nav-
+igation Signals , Sixth Information Hiding Workshop,
+Toronto, Canada, 2004
+[8] NAVSTAR GPS Joint Program Office, NAVSTAR
+Global Positioning System - Interface Specification IS-
+GPS 200 Space Segment/Navigation User Interfaces ,
+SMC/GP, CA, USA, 2004[9] P. Papadimitratos and A. Jovanovic, Protection and
+Fundamental Vulnerability of GNSS , IWSSC, Toulouse,
+2008
+[10] A.D. Rabbany, Introduction to GPS , Artech House,
+2002
+[11] L. Scott, Anti-Spoofing and Authenticated Signal Ar-
+chitectures for Civil Navigation Signals , ION-GNNS,
+Portand, Oregon, 2003
+[12] J.A. Volpe, Vulnearability Assesment of the Trans-
+portation Infrastructure Relying on GPS , NTSC, NAV-
+CEN draft report, 2001
+[13] H. Wen, P. Huang, and J. Fagan, Countermeasures for
+GPS signal spoofing , The University of Oklahoma, 2004
+[14] J. Zogg, GPS Basics - Introduction to the System , U-
+blox AG, 2002
+8
\ No newline at end of file
diff --git a/1001.0026.txt b/1001.0026.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7deedb87766825662cfd54e97e86bb2dc7664bf4
--- /dev/null
+++ b/1001.0026.txt
@@ -0,0 +1,318 @@
+arXiv:1001.0026v1  [astro-ph.SR]  30 Dec 2009Detectionof solar-likeoscillations from Keplerphotometry ofthe open
+cluster NGC 6819
+DennisStello,1Sarbani Basu,2HansBruntt,3Benoˆ ıt Mosser,3Ian R. Stevens,4
+TimothyM.Brown,5Jørgen Christensen-Dalsgaard,6Ronald L. Gilliland,7Hans Kjeldsen,6
+Torben Arentoft,6J´ erˆ omeBallot,8CarolineBarban,3TimothyR. Bedding,1WilliamJ. Chaplin,4
+YvonneP. Elsworth,4Rafael A.Garc´ ıa,9Marie-Jo Goupil,3SaskiaHekker,4Daniel Huber,1
+SavitaMathur,10Søren Meibom,11Reza Samadi,3VinothiniSangaralingam,4
+Charles S. Baldner,2KevinBelkacem,12KatiaBiazzo,13Karsten Brogaard,6
+Juan Carlos Su´ arez,14Francesca D’Antona,15Pierre Demarque,2LisaEsch,2NingGai,2,16
+Frank Grundahl,6YvelineLebreton,17Biwei Jiang,16NadaJevtic,18ChristofferKaroff,4
+AndreaMiglio,12JoannaMolenda- ˙Zakowicz,19JosefinaMontalb´ an,12ArletteNoels,12
+Teodoro RocaCort´ es,20,21Ian W. Roxburgh,22AldoM. Serenelli,23VictorSilvaAguirre,23
+ChristiaanSterken,24Peter Stine,18Robert Szab´ o,25AchimWeiss,23WilliamJ. Borucki,26
+DavidKoch,26JonM. Jenkins27– 2 –
+1SydneyInstituteforAstronomy(SIfA),SchoolofPhysics,U niversityofSydney,NSW2006,Australia
+2DepartmentofAstronomy,YaleUniversity,P.O.Box 208101, New Haven,CT 06520-8101
+3LESIA,CNRS,Universit´ ePierreetMarieCurie,Universit´ eDenisDiderot,ObservatoiredeParis,92195Meudon,
+France
+4SchoolofPhysicsandAstronomy,UniversityofBirmingham, Edgbaston,BirminghamB152TT,UK
+5LasCumbresObservatoryGlobalTelescope,Goleta,CA 93117 ,USA
+6DepartmentofPhysicsandAstronomy,AarhusUniversity,80 00AarhusC,Denmark
+7SpaceTelescopeScienceInstitute,3700San MartinDrive,B altimore,Maryland21218,USA
+8Laboratoired’AstrophysiquedeToulouse-Tarbes,Univers it´ edeToulouse,CNRS,14avE.Belin,31400Toulouse,
+France
+9Laboratoire AIM, CEA/DSM-CNRS, Universit´ e Paris 7 Didero t, IRFU/SAp, Centre de Saclay, 91191, Gif-sur-
+Yvette,France
+10IndianInstituteofAstrophysics,Koramangala,Bangalore 560034,India
+11Harvard-SmithsonianCenterforAstrophysics,60GardenSt reet,Cambridge,MA,02138,USA
+12Institutd’AstrophysiqueetdeG´ eophysiquedel’Universi t´ edeLi` ege,17All´ eedu6Aoˆ ut,B-4000Li` ege,Belgium
+13ArcetriAstrophysicalObservatory,LargoE.Fermi5,50125 ,Firenze,Italy
+14InstitutodeAstrof´ ısicadeAndaluc´ ıa(CSIC),Dept. Stel larPhysics,C.P. 3004,Granada,Spain
+15INAF -Osservatoriodi Roma,via diFrascati 33,I-00040,Mon teporzio,Italy
+16DepartmentofAstronomy,BeijingNormalUniversity,Beiji ng100875,China
+17GEPI,ObservatoiredeParis,CNRS, Universit´ eParisDider ot,5Place JulesJanssen,92195Meudon,France
+18Departmentof Physics& EngineeringTechnology,Bloomsbur gUniversity,400East SecondSt, BloomsburgPA
+17815,USA
+19AstronomicalInstitute,UniversityofWrocław,ul.Kopern ika11,51-622Wrocław,Poland
+20DepartmentodeAstrof´ ıca,Universidadde LaLaguna,38207 LaLaguna,Tenerife,Spain
+21InstitutodeAstrof´ ıcadeCanarias,38205La Laguna,Tener ife,Spain
+22QueenMaryUniversityofLondon,Mile EndRoad,LondonE14NS ,UK
+23MaxPlanckInstituteforAstrophysics,KarlSchwarzschild Str. 1,GarchingbeiM¨ unchen,D-85741,Germany
+24Vrije UniversiteitBrussel, Pleinlaan2,B-1050Brussels, Belgium
+25KonkolyObservatory,H-1525Budapest,P.O. Box67,Hungary
+26NASA AmesResearchCenter,MS 244-30,MoffatField,CA 94035 ,USA
+27SETIInstitute/NASA AmesResearchCenter,MS244-30,Moffa tField, CA 94035,USA– 3 –
+ABSTRACT
+Asteroseismology of stars in clusters has been a long-sough t goal because the as-
+sumption of a common age, distance and initial chemical comp osition allows strong
+tests of the theory of stellar evolution. We report results f rom the first 34 days of sci-
+encedatafromthe KeplerMission fortheopenclusterNGC6819—oneoffourclus-
+ters in the field of view. We obtain the first clear detections o f solar-like oscillations
+in the cluster red giants and are able to measure the large fre quency separation, ∆ν,
+andthefrequencyofmaximumoscillationpower, νmax. Wefindthattheasteroseismic
+parameters allow us to test cluster-membership of the stars , and even with the limited
+seismicdatainhand,wecan alreadyidentifyfourpossiblen on-membersdespitetheir
+havinga betterthan 80% membershipprobabilityfrom radial velocitymeasurements.
+We are also able to determine the oscillation amplitudes for stars that span about two
+orders of magnitude in luminosity and find good agreement wit h the prediction that
+oscillation amplitudesscale as the luminosityto the power of 0.7. These early results
+demonstrate the unique potential of asteroseismology of th e stellar clusters observed
+byKepler.
+Subjectheadings: stars: fundamentalparameters—stars: oscillations—star s: interi-
+ors—techniques: photometric—openclustersandassociati ons: individual(NGC6819)
+1. Introduction
+Openclustersprovideuniqueopportunitiesinastrophysic s. Starsinopenclustersarebelieved
+to be formed from the same cloud of gas at roughly the same time . The fewer free parameters
+available to model cluster stars make them interesting targ ets to analyze as a uniform ensemble,
+especiallyforasteroseismicstudies.
+Asteroseismology is an elegant tool based on the simple prin ciple that the frequency of a
+standing acoustic wave inside a star depends on the sound spe ed, which in turn depends on
+the physical properties of the interior. This technique app lied to the Sun (helioseismology) has
+provided extremely detailed knowledge about the physics th at governs the solar interior, (e.g.,
+Christensen-Dalsgaard2002). Allcoolstarsareexpectedt oexhibitsolar-likeoscillationsofstand-
+ing acoustic waves – called p modes – that are stochastically driven by surface convection. Using
+asteroseismology to probe the interiors of cool stars in clu sters, therefore, holds promise of re-
+warding scientific return (Gough& Novotny 1993; Brown& Gill iland 1994). This potential has
+resulted in several attempts to detect solar-like oscillat ions in clusters using time-series photome-
+try. These attempts were often aimed at red giants, since the iroscillation amplitudesare expected– 4 –
+tobelargerthanthoseofmain-sequenceorsubgiantstarsdu etomorevigoroussurfaceconvection.
+Despite these attempts, only marginal detections have been attained so far, limited either by the
+lengthofthetimeseriesusuallyachievablethroughobserv ationswiththe HubbleSpaceTelescope
+(Edmonds& Gilliland 1996; Stello&Gilliland 2009) or by the difficulty in attaining high preci-
+sion from ground-based campaigns (e.g., Gillilandetal. 19 93; Stelloet al. 2007; Frandsen et al.
+2007).
+InthisLetterwereportcleardetectionsofsolar-likeosci llationsinred-giantstarsintheopen
+cluster NGC 6819 using photometry from NASA’s Kepler Mission (Borucki et al. 2009). This
+cluster,oneoffourinthe Keplerfield, isabout2.5Gyrold. Itisatadistanceof2.3kpc, andha sa
+metallicityof[Fe/H] ∼ −0.05(see Holeet al. 2009, and references herein).
+2. Observations anddata reduction
+The data were obtained between 2009 May 12 and June 14, i.e., t he first 34 days of con-
+tinuous science observations by Kepler(Q1 phase). The spacecraft’s long-cadence mode ( ∆t≃
+30minutes) used in this investigation provided a total of 1639 data points in the time series of
+each observed star. For this Letter we selected 47 stars in th e field of the open cluster NGC 6819
+with membership probability PRV>80% from radial velocity measurements (Holeet al. 2009).
+Figure1showsthecolor-magnitudediagram(CMD)oftheclus terwiththeselectedstarsindicated
+by green symbols. The eleven annotated stars form a represen tative subset, which we will use to
+illustrate our analyses in Sections 3 and 4. We selected the s tars in this subset to cover the same
+brightnessrangeasourfullsample,whilegivinghighweigh ttostarsthatappeartobephotometric
+non-members (i.e., stars located far from the isochrone in t he CMD). Data for each target were
+checked carefully to ensure that the time-series photometr y was not contaminated significantly
+by other stars in the field, which could otherwise complicate the interpretation of the oscillation
+signal.
+Fourteen data points affected by the momentum dumping of the spacecraft were removed
+from the time series of each star. In addition, we removed poi nts that showed a point-to-point
+deviation greater than 4σ, whereσis the local rms of the point-to-point scatter within a 24 hou r
+window. This process removed on average one data-point per t ime series. Finally, we removed a
+linear trend from each time series and then calculated the di screte Fourier transform. The Fourier
+spectraathighfrequencyhavemeanlevelsbelow5partsperm illion(ppm)inamplitude,allowing
+usto search forlow-amplitudesolar-likeoscillations.– 5 –
+3. Extractionofasteroseismicparameters
+Figure 2 shows the Fourier spectra (in power) of 9 stars from o ur subset. These range from
+thelowerred-giant branch to thetip ofthe branch (see Figur e1). The stars are sorted by apparent
+magnitude, which for a cluster is indicative of luminosity, with brightest at the top. Note that the
+redgiantsinNGC6819aresignificantlyfainter( 12/lessorsimilarV/lessorsimilar14)thanthesampleof Keplerfieldred
+giants (8/lessorsimilarV/lessorsimilar12) studied by Beddinget al. (2010). Nevertheless, it is clear from Figure 2 that
+we can detect oscillations for stars that span about two orde rs of magnitude in luminosity along
+theclustersequence.
+Weusedfourdifferentpipelines(Hekkeret al.2009a;Huber et al.2009a;Mathuret al.2009;
+Mosser& Appourchaux 2009) to extract the average frequency separation between modes of the
+same degree (the so-called large frequency separation, ∆ν). We have also obtained the frequency
+of maximum oscillation power, νmax, and the oscillation amplitude. The measured values of ∆ν
+are indicated by vertical dotted lines in Figure 2 centered o n the highest oscillation peaks near
+νmax. While the stars in Figure 2, particularly in the lower panel s, show the regular series of
+peaks expected for solar-likeoscillations,the limitedle ngth of the time-series datadoes not allow
+such structureto be clearly resolved for the mostluminouss tars in our sample— thosewith νmax
+/lessorsimilar20µHz. We do, however, see humps of excess power in the Fourier sp ectra (see Figure 2 star
+no. 2 and 8) with νmaxand amplitude in mutual agreement with oscillations. With l onger time
+series weexpectmorefirm resultsforthesehigh-luminosity giants.
+4. Cluster membership from asteroseismology
+It isimmediatelyclear fromFigure2thatnotallstars follo wtheexpected trendofincreasing
+νmaxwith decreasing apparent magnitude, suggesting that some o f the stars might be intrinsically
+brighterorfainterthanexpected. Sinceoscillationsinas taronlydependonthephysicalproperties
+of the star, we can use asteroseismology to judge whether or n ot a star is likely to be a cluster
+member independentlyof its distanceand of interstellarab sorption and reddening. For cool stars,
+νmaxscaleswiththeacousticcut-offfrequency,anditiswelles tablishedthatwecanestimate νmax
+by scalingfromthesolarvalue(Brownet al. 1991; Kjeldsen& Bedding 1995):
+νmax
+νmax,⊙=M/M⊙(Teff/Teff,⊙)3.5
+L/L⊙, (1)
+whereνmax,⊙= 3100µHz. The accuracy of such estimates is good to within 5% (Stell oet al.
+2009)assumingwehavegoodestimatesofthestellarparamet ersM,L, andTeff.
+In thefollowingweassumetheidealisticscenario whereall clustermembersfollowstandard
+stellar evolutiondescribed by the isochrone. Stellar mass along the red giant branch of thecluster– 6 –
+isochrone varies by less than 1%. The variation is less than 5 % even if we also consider the
+asymptoticgiant branch. For simplicity,we therefore adop t a mass of 1.55M⊙for all stars, which
+is representativefortheisochronefrom Marigoet al. (2008 )(Figure 1) and a similarisochroneby
+VandenBerg etal. (2006). Neglectingbinarity (see Table 1) , we derivethe luminosityof each star
+in our subset from its V-band apparent magnitude, adopting reddening and distance modulus of
+E(B−V) = 0.1and(M−m)V= 12.3,respectively(obtainedfromsimpleisochronefitting,see
+Holeetal.2009). WeusedthecalibrationofFlower(1996)to convertthestellar (B−V)0colorto
+Teff. BolometriccorrectionswerealsotakenfromFlower(1996) . Thederivedquantitieswerethen
+used toestimate νmaxfor each star(Eq.1), and compared withtheobservedvalue(s eeFigure3).
+Figure 3 shows four obvious outliers (no. 1, 3, 8 and 11), thre e of which are also outliers in
+theCMD (no. 1, 3, and11). Fortherest ofthestars weseegood a greement between theexpected
+andobservedvalue,indicatingthattheuncertaintyonthe νmaxestimatesarerelativelysmall. Since
+thevariationsinmassandeffectivetemperatureamongthec lustergiantstarsaresmall,deviations
+fromthedottedlinemustbecausedbyanincorrectestimateo ftheluminosity. Thisimpliesthatthe
+luminositiesofstarsfallingsignificantlyaboveorbelowt helinehavebeenover-orunderestimated,
+respectively. The simplest interpretation is that these ou tliers are fore- or background stars, and
+hence not members of the cluster. To explain the differences between the observed and expected
+value ofνmaxwould require the deviant stars to have Verrors of more than 1 magnitude, and in
+some cases B−Verrors of about 0.2 magnitude if they were cluster members. B inarity may
+explain deviations above the dotted line, but only by up to a f actor of two in L(and hence, in the
+ratio of the observed to expected νmax). The deviation of only one star (no.1) could potentially
+be explained this way. However, that would be in disagreemen t with its single-star classification
+from multi-epoch radial velocity measurements, assuming i t is not a binary viewed pole-on (see
+Table 1). Hence, under the assumptionof a standard stellar e volution, the most likely explanation
+forallfouroutliersinFigure3isthereforethatthesestar sarenotclustermembers. Thisconclusion
+is, however,in disagreementwith theirhighmembershippro babilityfrom measurementsofradial
+velocity (Holeet al. 2009) and proper motion (Sanders 1972) (see Table 1). Another interesting
+possibility is that the anomalous pulsation properties mig ht be explained by more exotic stellar
+evolutionscenariosthan isgenerally anticipatedforopen -clusterstars.
+5. Asteroseismic“color-magnitude diagrams”
+ItisclearfromFigure2thattheamplitudesoftheoscillati onsincreasewithluminosityforthe
+seismicallydeterminedclustermembers. Basedoncalculat ionsbyChristensen-Dalsgaard& Frandsen
+(1983), Kjeldsen& Bedding (1995) have suggested that the ph otometric oscillation amplitude of
+p modes scale as (L/M)sTeff−2, withs= 1(the velocity amplitudes, meanwhile, would scale as– 7 –
+(L/M)s). This was revised by Samadi etal. (2007) to s= 0.7based on models of main sequence
+stars. Takingadvantageofthefewerfreeparameterswithin thisensembleofstars,ourobservations
+allow us to make some progress towards extrapolating this sc aling to red giants and determining
+thevalueof s.
+In Figure4 weintroduceanewtypeofdiagramthatissimilart oaCMD, butwithmagnitude
+replaced by an asteroseismicparameter – in thiscase, theme asured oscillationamplitude. Ampli-
+tudeswereestimatedforallstarsinoursample(exceptfort hefouroutliers)usingmethodssimilar
+tothatofKjeldsenet al.(2008)(seealsoMichelet al.2008) ,whichassumethattherelativepower
+betweenradialandnon-radialmodesisthesameasintheSun. Thisdiagramconfirmstherelation-
+ship between amplitude and luminosity. Despite a large scat ter, which is not surprising from this
+relatively short timeseries, we see that s= 0.7provides a much better match than s= 1.0. Once
+verifiedwithmoredata,thisrelationwillallowtheuseofth emeasuredamplitudeasanadditional
+asteroseismic diagnostic for testing cluster membership a nd for isochrone fitting in general. We
+notethat theother clusters observed by Keplerhave different metallicitiesthan NGC 6819, which
+willallowfutureinvestigationon themetallicitydepende nce oftheoscillationamplitudes.
+We expect to obtain less scatter in the asteroseismic measur ements when longer time series
+become available. That will enable us to expand classical is ochrone fitting techniques to include
+diagramslikethis,whereamplitudecouldalsobereplacedb yνmaxor∆ν. Inparticular,weshould
+beabletodeterminetheabsoluteradiiaidedby ∆νoftheredgiantbranchstars,whichwouldbean
+importantcalibratorfor theoretical isochrones. Additio nally,thedistributionsoftheasteroseismic
+parameters – such as νmax– can potentially be used to test stellar population synthes is models
+(Hekkeret al.2009b;Miglioet al.2009b). Applyingthisapp roachtoclusterscouldleadtofurther
+progress in understanding of physical processes such as mas s loss during the red-giant phase (see
+e.g.,Miglioet al.2009a). Notethatafewclearoutliersare indicativeofnon-membershiporexotic
+stellarevolution,asaresultoffactorssuchasstellarcol lisionsorheavymassloss,whileageneral
+deviationfromthetheoreticalpredictionsbyalargegroup ofstarswouldsuggestthatthestandard
+theorymay need revision.
+Finally, we note that NGC 6819 and another Keplercluster, NGC 6791, contain detached
+eclipsingbinaries(Talamantes& Sandquist2009;Street et al.2005;deMarchi et al.2007;Mochejskaetal.
+2005). For these stars masses and radii can be determined ind ependently (Grundahl et al. 2008),
+whichwillfurtherstrengthenresultsofasteroseismicana lyses.– 8 –
+6. Discussion& Conclusions
+PhotometricdataofredgiantsinNGC6819obtainedbyNASA’s KeplerMission haveenabled
+ustomakethefirst cleardetectionofsolar-likeoscillatio nsin clusterstars. Thegeneral properties
+of the oscillations ( ∆ν,νmax, and amplitudes) agree well with results of field red giants m ade by
+Kepler(Bedding etal.2010)andCoRoT(deRidderet al.2009;Hekker et al.2009b). Wefindthat
+the oscillation amplitudes of the observed stars scale as (L/M)0.7Teff−2, suggesting that previous
+attemptstodetect oscillationsinclustersfrom groundwer eat thelimitofdetection.
+We find that the oscillation properties provide additional t ests for cluster membership, al-
+lowing us to identify four stars that are either non-members or exotic stars. All four stars have
+membership probability higher than 80% from radial-veloci ty measurements, but three of them
+appear to be photometric non-members. We further point out t hat deviations from the theoretical
+predictionsoftheasteroseismicparametersamongalarges ampleofclusterstarshavethepotential
+ofbeingusedasadditionalconstraintsintheisochronefitt ingprocess,whichcanleadtoimproved
+stellarmodels.
+Our results, based on limited data of about one month, highli ght the unique potential of as-
+teroseismologyon the brighteststars in thestellarcluste rs observed by Kepler. With longerseries
+sampled at the spacecraft’s short cadence ( ≃1 minute), we expect to detect oscillations in the
+subgiantsand turn-offstars, as wellas inthebluestraggle rsinthiscluster.
+FundingforthisDiscoverymissionisprovidedbyNASA’sSci enceMissionDirectorate. The
+authorswouldliketothanktheentire Keplerteamwithoutwhomthisinvestigationwouldnothave
+been possible. The authors also thank all funding councils a nd agencies that have supported the
+activitiesofWorkingGroup 2ofthe KeplerAsteroseismicScience Consortium(KASC).
+Facilities: Kepler.
+REFERENCES
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+Kjeldsen,H., etal. 2008,ApJ, 682,1370
+Latham,D. W.,Brown, T.M.,Monet,D. G., Everett,M.,Esquer do,G. A.,& Hergenrother, C. W.
+2005,inBulletinoftheAmerican AstronomicalSociety,Vol . 37,1340
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+A&A,482,883
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+Michel,E., etal. 2008,Science, 322,558
+Miglio, A., Montalb´ an, J., Eggenberger, P., Hekker, S., & N oels, A. 2009a, in American Institute
+ofPhysicsConference Series, Vol.1170,AmericanInstitut eofPhysicsConference Series,
+ed. J.A. Guzik& P. A. Bradley,132– 10 –
+Miglio,A., et al.2009b,A&A,503, L21
+Mochejska,B. J., et al.2005,AJ, 129,2856
+Mosser,B., & Appourchaux,T.2009,A&A,508, 877
+Samadi, R., Georgobiani, D., Trampedach, R., Goupil, M. J., Stein, R. F., & Nordlund, ˚A. 2007,
+A&A,463,297
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+Stello,D., &Gilliland,R. L.2009,ApJ, 700,949
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+Talamantes, A., & Sandquist, E. L. 2009, in Bulletin of the Am erican Astronomical Society,
+Vol.41,320
+VandenBerg, D. A., Bergbusch, P. A., &Dowler,P. D. 2006,ApJ S, 162,375
+ThispreprintwaspreparedwiththeAAS L ATEXmacrosv5.2.– 11 –
+Table1:Cross identificationsandmembership.
+ID ID WOCS ID ID Mem.ship Mem.ship Mem.ship
+Thiswork KICaHoleet al. Sanders Holeet al.bSanderscThiswork
+1 5024272 003003 SM95% no
+2 5024750 001004 141 SM93% 83% yes
+3 5023889 004014 42 SM95% 90% no
+4 5023732 005014 27 SM94% 90% yes
+5 5112950 003005 148 SM95% 92% yes
+6 5112387 003007 73 SM95% 88% yes
+7 5024512 003001 116 SM93% 90% yes
+8 4936335 007021 9 SM95% 68% no
+9 5024405 004001 100 SM93% 91% yes
+10 5112072 009010 39 SM95% 91% yes
+11 4937257 009015 144 SM88% 80% no
+aIDfromthe KeplerInputCatalogue (Lathamet al. 2005).
+bClassification (SM:singlemember)andmembershipprobabil ityfromradialvelocity(Holeetal. 2009).
+cMembershipprobabilityfrompropermotion(Sanders1972).– 12 –
+Fig. 1.— Color-magnitude diagram of NGC 6819. Plotted stars have membership probability
+PRV>80% as determined by Holeet al. (2009). Photometric indices ar e from the same source.
+Theisochroneis from Marigoet al. (2008)(Age=2.4 Gyr, Z=0. 019,modified for theadopted red-
+dening of 0.1mag). Color-coded stars have been analyzed, an d the annotated numbers refer to the
+legend in panels of Figure 2 and star numbers in Figure 3 (see a lso Table 1). Insets show light
+curves in parts per thousand of two red giants oscillating on different timescales. The variations
+ofthelightcurves inPanelA and Baredominatedby thestella roscillationswithperiodsofafew
+days andofaboutsix hours,respectively.– 13 –
+Fig. 2.— Fourierspectraofa representativeset ofred giant salongtheclustersequence sortedby
+apparent magnitude. Annotated numbers in each panel refer t o the star identification (see Fig. 1
+and Table 1). ‘AM’ indicates that the star is an asteroseismi c member. Red solid curves show the
+smoothed spectrum for stars with νmax<20µHz. To guide the eye, we have plotted dotted lines
+toindicatethemeasuredaveragelargefrequencyseparatio n. Thecentraldottedlineiscenteredon
+thehighestoscillationpeaksnear νmax. Notethatsince ∆νisgenerallyfrequencydependent,only
+thecentraldottedlineisexpectedtolineupwithapeakinth eoscillationspectrum. Theredarrows
+indicate the position of the expected νmax(see Eq. 1) for stars where the observed value does not
+agree withtheexpectationsforthiscluster(seeSection 4) .– 14 –
+Fig. 3.— Ratioofobservedandexpected νmax. 1-σerrorbarsindicatetheuncertaintyon νmax(obs).
+Stars clearly above or below the dotted line are either not cl uster members or members whose
+evolutionhavenot followedthestandardscenario.– 15 –
+Fig. 4.— Amplitude color diagram of red giant stars in NGC 681 9 with the Marigoet al. (2008)
+isochrone overlaid with three values of sin the amplitude scaling relation: (L/M)sTeff−2. The
+solarvalueusedin thisscalingis 4.7ppm(Kjeldsen &Bedding 1995).
\ No newline at end of file
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+arXiv:1001.0027v1  [astro-ph.GA]  30 Dec 2009New candidate Planetary Nebulae in the IPHAS survey: the cas e of
+PNe with ISM interaction.
+Laurence SabinA, Albert A. ZijlstraA, Christopher WareingB, Romano L.M.
+CorradiC, Antonio MampasoC, Kerttu ViironenC, Nicholas J. WrightDand
+Quentin A. ParkerE
+AJodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester,
+Manchester M13 9PL, UK
+BDepartment of Applied Mathematics, University of Leeds, Le eds, LS2 9JT, UK
+CInstituto de Astrofisica de Canarias, Tenerife, Spain
+DHarvard-Smithsonian Center for Astrophysics, 60 Garden St reet, Cambridge, MA, 02138, USA
+EMacquarie University/Anglo-Australian Observatory, Dep artment of Physics, North Ryde, Sydney
+NSW 2190, AUSTRALIA
+AEmail: laurence.sabin@manchester.ac.uk
+Abstract: We present the results of the search for candidate Planetary Nebulae interacting with
+the interstellar medium (PN-ISM) in the framework of the INT Photometric H αSurvey (IPHAS)
+and located in the right ascension range 18h-20h. The detect ion capability of this new Northern
+survey, in terms of depth and imaging resolution, has allowe d us to overcome the detection problem
+generally associated to the low surface brightness inheren t to PNe-ISM. We discuss the detection of
+21 IPHAS PN-ISM candidates. Thus, different stages of intera ction were observed, implying various
+morphologies i.e. from the unaffected to totally disrupted s hapes. The majority of the sources belong
+to the so-called WZO2 stage which main characteristic is a br ightening of the nebula’s shell in the
+direction of motion. The new findings are encouraging as they would be a first step into the reduction
+of the scarcity of observational data and they would provide new insights into the physical processes
+occurring in the rather evolved PNe.
+Keywords: Planetary nebulae, ISM interaction, survey.
+1 Introduction
+Large Hαsurveys have so far allowed the detection of
+∼3000 planetary nebulae (PNe) in the Galaxy. The
+data can be principally found in the Strasbourg-ESO
+Catalogue (Acker et al.1992)andtherecentMacquarie-
+AAO-StrasbourgH αPlanetaryNebulaCatalogues: MASH
+IandII(Parker et al.(2006)andMiszalski et al(2008)).
+Unfortunatelyalimitation inour understandingofthis
+short and rather complex phase of stellar evolution lies
+either in the deepness of the detections realised or the
+type of PNe investigated. Indeed, although enormous
+progress has been made over the years in terms of ob-
+servations, the well-studied PNe are generally bright
+and often young. This hampers the study of:
+•PNe hidden by the interstellar medium, partic-
+ularly those located at low galactic height.
+•PNe with (very)low surface brightness where we
+find the group of old PNe.
+•Very distant PNe which appear as unresolved
+and not recognisable as nebulae.•PNe located in crowded areas such as the galac-
+tic plane.
+Moreover, excluding these objects from global studies
+(morphology, abundances,luminosityfunction...etc)may
+bias our understanding of planetary nebulae. As an il-
+lustration, few PNe are described in the literature as
+“PNe with ISM interaction”, which is the step before
+the complete dilution of the nebulae in the interstel-
+lar medium (Borkowski et al. (1990), Ali et al. (2000),
+Xilouris et al. (1996) and Tweedy et al. (1996)). The
+study of the interaction process would give new in-
+sights intoseveral aspects of the PNevolution. Indeed,
+the density difference between ISM and PNe will affect
+their shape. This is expected to be observable in old
+objects where the nebular density declines sufficiently
+to be overcome by the ISM density. Other phenom-
+ena like the flux and brightness enhancement following
+the compression of the external shell, the increase of
+the recombination rate in the PN Rauch et al. (2000),
+the occurrence of turbulent Rayleigh-Taylor instabili-
+ties and the implication of magnetic fields Dgani et al.
+(1998) are among the physical processes which need
+to be addressed not only from a theoretical but also
+observational point of view.
+12 Publications of the Astronomical Society of Australia
+The low surface brightness generally associated to
+PNe-ISM has for a long time prevented any deeper ob-
+servation and good statistical study of these interac-
+tions, where only the interacting rim is well seen. New
+generations of H αsurveys have overcome this prob-
+lem. A perfect example is the discovery of PFP 1 by
+Pierce et al. (2004)intheframeworkoftheAAO/UKST
+SuperCOSMOS H αsurvey (SHS) (Parker et al. 2005).
+This PN, starting to interact with the ISM at the
+rim, is very large (radius = 1.5 ±0.6 pc) and very
+faint (logarithm of the H αsurface brightness equal
+to -6.05 ergcm−2.s−1.sr−1). In order to unveil and
+study this “missing PN population” in the Northern
+hemisphere we need surveys providing the necessary
+observing depth: the Isaac Newton Telescope (INT)
+Photometric H αSurvey (IPHAS) is one of them and
+will complete the work done in the South by the SHS.
+2 IPHAS contribution
+IPHAS is a new fully photometric CCD survey of the
+Northern Galactic Plane, started in 2003 (Drew et al.
+(2005), Gonzalez-Solares et al (2008)) and which has
+now been completed1. Using the 2.5m Isaac Newton
+Telescope (INT)in LaPalma (Canary Islands, SPAIN)
+and the Wide Field Camera (WFC) offering a field of
+view of 34.2 ×34.2 arcmin2, IPHAS targets the Galac-
+tic plane in the Northern hemisphere, at a latitude
+range of -5◦<b<5◦and covers 1800 deg2. This
+international survey is conducted not only in H αbut
+also makes use of two continuum filters, respectively
+the Sloan r’ and i’. IPHAS is viewed as an enhance-
+ment to former narrow-band surveys, first due to the
+use of CCD and the particularly small pixel scale al-
+lowed bytheWFCwith0.33 arcsec pix−1butalso (and
+mainly) due to the depth reached for point sources de-
+tection. Thus sources with a r’ magnitude between 13
+and 19.5-20 could be detected with a very good pho-
+tometric accuracy. The most interesting characteristic
+for our purpose is the ability to detect resolved ex-
+tended emissions with an H αsurface brightness down
+to 2×10−17erg cm−2s−1arcsec−2.
+In this paper we will focus on extended (candi-
+date) PNe (i.e. objects with a size greater than 5 arc-
+sec). They were searched for via a visual inspection of
+2 deg2Hα-r(continuum removal) mosaics made from
+the different IPHAS observations. And in order to al-
+low the detection of objects of multiple size and bright-
+ness level, the mosaics were binned at respectively 15
+pixels×15 pixels (5 arcsec) and 5 pixels ×5 pixels (1.7
+arcsec). The first binning level, which is of particu-
+lar interest to us, helps to detect resolved, low surface
+brightness objects (down to the IPHAS limit) and to
+accentuate the contours/shape of the nebulae (this is
+particularly useful to see, for example, the full extent
+of an outflow or a tail). The second set, is used to de-
+tect intermediate size nebulae i.e. smaller than ∼15-20
+arcsec in diameter.
+1http://www.iphas.orgThe first area that has been fully investigated is the re-
+gion between RA=18h and RA=20h. We detected 233
+candidate PNe among which other nebulosities may be
+found e.g. small HII regions (Sabin, PhD thesis, to be
+published). Around 20% of this sample have been so
+far spectroscopically confirmed as PNe (Sabin et al.,
+in preparation). If we look at the particularities of the
+PNe and candidate PNe uncovered, we observe that
+from thepointofviewofthesize, large objects (greater
+than 20 arcsec) constitute the main new group (Fig.
+1). As large objects are generally considered as more
+evolved, we are confident in finding in this group new
+old PNe and byextension new cases of PNe interacting
+with the surrounding ISM (PNe/ISM).
+Figure 1: Galactic distribution of the IPHAS neb-
+ulae according to their size.
+3 Candidate PNewith ISMin-
+teraction
+Fundamental in PN development, the interaction with
+the ISM does not only concern old PNe, as may be
+commonly thought. Indeed, the PN-ISM interaction
+has mainly been detected in a rather small number of
+nebulae, which are generally bright objects (“young”
+and“mid-age”PNe). Rauch et al.(2000)andWareing et al.
+(2007) showed that different stages of interaction are
+exhibited during the PNe life. The low surface bright-
+ness, generally associated with nebulae mixing with
+the ISM and “old” PNe, has for a long time prevented
+any deeper observation and good statistical study of
+these interactions. Although faint objects will still re-
+main difficult to detect, the IPHAS survey provides
+a noticeable improvement. Nevertheless, a caveat is
+the difficulty to visually separate PNe-ISM from other
+faint and extended structures like old HII regions, Su-
+pernovae (SNRs) or diffuse H αstructures. As an ex-
+ample, faint bow shocks generally characteristics ofwww.publish.csiro.au/journals/pasa 3
+PNemixingwiththeISMcanalsobefilamentarystruc-
+tures from old SNRs. A spectroscopic analysis is the
+only way to have a clear identification.
+The work presented here is based on the classification
+from Wareing et al. (2007) (WZO 1-4 called after the
+authors’ names) and will allow us to establish the de-
+gree of interaction for each nebula. Their classifica-
+tion is the result of the first extensive investigation
+of the applicable parameter space, varying stellar pa-
+rameters, relative velocities through the ISM and ISM
+densities.
+The depth reached by the IPHAS survey combined
+with the binning detection method allowed us to iden-
+tify 21 cases of interacting candidate PNe.
+3.1 WZO1 type
+The first group of PNe/ISM concerns those where the
+main PN is still unaffected and which may display a
+distant bow shock. In our area of study (18h-20h), the
+majority of candidates answer the first condition, but
+none show the outer bow shock. Outside this area, the
+nicknamed“EarNebula”orIPHASXJ205013.7+465518
+with a 6 arcmin size may be coincident with a WZO1
+description as this object is a confirmed bipolar PN
+(Fig. 3) surrounded by a shell which may be an AGB
+remnant shell or would indicate a multiple shell nebula
+(Fig. 2).
+3.2 WZO2 type
+This category concerns PNe showing a bright rim in
+the direction of motion. This is the most common fea-
+ture found in our sample and 17 objects out of 21 fall
+under this classification. Fig. 4 presents 3 examples
+with different angular sizes, although they all display
+a diameter on the order of a few arcmin (we consid-
+ered the assumed full extent of the round nebulae).
+We point out in Fig. 4-Top the difficulty to determine
+the true direction of motion regarding the CS position
+and off-axis bow shock. Such a geometry could be ex-
+plained by an ISM gradient from high on the left to
+low on the right. We also notice a particularly low
+observed surface brightness (SB) which may explain
+previous non detections.
+3.3 WZO3 type
+This type is exemplified by PNe whose geometric cen-
+tres are shifted away from the central star (CS): both
+are no longer coincident. An example, is the ancient
+PN Sh 2-188 around which IPHAS has uncovered an
+extended structure (Wareing et al. 2006). We identi-
+fied 3 candidate PNe coincident with this description.
+The most probing WZO3 type in our sample is pre-
+sented in figure 5 and corresponds, according to hy-
+drodynamical models, to a PN with a CS velocity of
+about 100 km/s.3.4 WZO4 type
+The WZO4 corresponds to the most difficult types of
+PN to be detected: the CS has left the vicinity of
+the now totally disrupted PN, leaving an amorphous
+structure. The challenge does not lie in the detection
+ability (it enters in the IPHAS range of detection) but
+more intheselection oftheobjects as possible PNedue
+to the total lack of symmetry or axi-symmetry. This
+type of interaction is also discussed in more detail by
+Wareing et al. in these proceedings.
+We identified 1 candidate PN which could fit the
+given description. Fig. 6 presents the selected can-
+didate in the top panel. We suggest the the nebular
+material has been moved from the front to the rear
+leaving a remnant “wall of material”. We also notice
+that some features may be linked to turbulence effects.
+The comparison with the hydrodynamical model (bot-
+tom panel) seems to support this hypothesis. Nev-
+ertheless a spectroscopic confirmation of the nebula’s
+nature will be needed. The model implies a velocity
+relative to the ISM of 100 Km/s and an evolution in
+the post-AGB phase of 10 000 years.
+3.5 Distribution of the candidates
+Fig. 7-top shows that the majority of the WZO2 nebu-
+lae typesare locatedinzones ofrelatively lowISMden-
+sity (compared to the Galactic Centre). The low stress
+exerted on the nebulae may explain why they still keep
+their quasi circular shape. The ISM is more dense in
+the Galactic Plane than in the zone towards the anti-
+centreor thezoneaboveaheightof100pc(from obser-
+vation of neutral hydrogen gas, Dickey et al. (1990)).
+We therefore expected a greater influence of the in-
+teraction process in this area. Indeed, we observed
+that the most advanced stages of interaction, namely
+WZO3 and WZO4, are detected in areas of high ISM
+density, where PN are more likely to be affected by
+such densities.
+Thesizedistribution, Fig. 7-bottom, indicatesthat
+althoughmostofthedetectedcandidatePNearelarge2,
+i.e with a size greater than 100 arcsec, or of medium
+size i.e. between 20 and 100 arcsec, small nebulae
+also show signs of interaction. This confirms that the
+ISM interaction process does not “a priori” only im-
+ply “old” nebulae. We also observe that large objects
+mainly lie at higher latitudes than smaller nebulae but
+it is also interesting to notice that we detect large ob-
+jects inzones ofhighextinction; large PNeseemtosur-
+vive at relatively low latitudes. They would undergo
+strong alteration by the ISM and would display more
+advanced stages of interaction. Those disruptions tend
+to affect them more than smaller size nebulae at the
+same latitude range.
+4 Conclusion and Perspectives
+In the first fully analysed area of the Galactic plane,
+RA=18h to RA=20h, the new H αphotometric survey
+2The sizes here are defined in terms of angular sizes, so
+the physical correspondence will depend on the distance.4 Publications of the Astronomical Society of Australia
+IPHASappearstobeanexcellenttooltostudyPNein-
+teracting with the ISM. Indeed the survey contributes
+to the detection of nebulae so far hidden mainly due to
+their faintness. Thus, 21 objects have been identified
+aspossible planetarynebulaeinteractingwiththeISM.
+They show diverse sizes (although the majority display
+a diameter greater than 100 arcsec) and morphologies
+corresponding to the four different cases of interaction
+commonly defined going from the unaffected to the to-
+tally disrupted nebula. The most common stage is the
+WZO2correspondingtonebulaeshowingabrightening
+of their rim in the direction of motion. This is coinci-
+dent with the observations made by Wareing and al (in
+these proceedings) crossing different H αsurveys. We
+were also able to reach those targets at low latitudes
+and found that some could survive in those environ-
+ments although they would be strongly affected by the
+ISM. The total lack of PNe/ISM at the highest point
+of ISM density (b= ±0.5 deg and 30 deg <l<50 deg)
+can either be due to the limitation of IPHAS or be-
+cause they have been totally destroyed by the effects
+of ISM interaction.
+The next logical step is the spectroscopic identification
+of these sources, their central star study and physical
+size determination. The low surface brightness implies
+the use of particular means such as integral field spec-
+troscopy to be able to retrieve the maximum infor-
+mation. Therefore a new programme of IPHAS PN
+candidate follow-up spectroscopy led by Q. Parker, A.
+Zijlstra and R. Corradi is now underway.
+References
+Acker, A., Marcout, J., Ochsenbein, F., Stenholm, B.
+, Tylenda, R.,1992, Garching: European Southern
+Observatory,Strasbourg - ESO catalogue of galactic
+planetary nebulae. Part 1; Part 2.
+Ali, A., El-Nawawy, M. S. and Pfleiderer, J., 2000,
+APSS, 271, 245.
+Borkowski, K. J., Sarazin, C. L. and Soker, N., 1990,
+Apj, 360,173.
+Dgani, R. and Soker, N., 1998, 495, 337.
+Dickey, J. M. and Lockman, F. J.,1990,ARAA,28,215
+Drew, J. E., Greimel, R., Irwin, M. J., Aungwero-
+jwit, A., Barlow, M. J., Corradi, R. L. M., Drake,
+J. J.., G¨ ansicke, B. T., Groot, P. and 26 co-
+authors,2005,MNRAS,362,753.
+Gonzalez-Solares, E. A., Walton, N. A., Greimel, R.
+and Drew, J. E., Irwin, M. J., Sale, S. E., Andrews,
+K., Aungwerojwit, A., Barlow, M. J. and 41 co-
+authors, 2008,MNRAS,388,89.
+Miszalski, B., Parker, Q. A., Acker, A.,
+Birkby, J. L., Frew, D. J. and Kovacevic,
+A.,2008,MNRAS,384,525.Parker, Q. A., Phillipps, S., Pierce, M. J., Hartley, M.,
+Hambly, N. C., Read, M. A., MacGillivray, H. T.,
+Tritton, S. B., Cass, C. P., Cannon, R. D., Cohen,
+M., Drew, J. E., Frew, D. J., Hopewell, E., Mader,
+S., Malin, D. F., Masheder, M. R. W., Morgan,
+D. H., Morris, R. A. H., Russeil, D., Russell, K. S.
+and Walker, R. N. F., 2005, MNRAS, 362, 689.
+Parker,Q. A., Acker, A., Frew, D. J., Hartley,
+M., Peyaud, A. E. J., Ochsenbein, F., Phillipps,
+S., Russeil, D., Beaulieu, S. F., Cohen, M.,
+K¨ oppen, J., Miszalski, B., Morgan, D. H., Mor-
+ris, R. A. H., Pierce, M. J. and Vaughan,
+A. E.,2006,MNRAS,373,79.
+Pierce, M. J., Frew, D. J., Parker, Q. A. and K¨ oppen,
+J., 2004, PASP, 21, 334.
+Rauch,T., Furlan, E., Kerber, F. and Roth,
+M.,2000,ASP Conf. Ser:Asymmetrical Planetary
+Nebulae II,199,341.
+Riesgo, H. and L´ opez, J. A.,2006,Revista Mexicana de
+Astronomia y Astrofisica,42,47-51
+Tweedy, R. W. and Kwitter, K. B., 1996, ApJS, 107,
+255.
+Wareing, C. J., O’Brien, T. J., Zijlstra, A. A., Kwitter,
+K. B., Irwin, J., Wright, N., Greimel, R. and Drew
+, J. E.,2006,MNRAS,366,387.
+Wareing, C. J., Zijlstra, A. A. and O’Brien,
+T. J.,2007,MNRAS,382,1233.
+Xilouris, K. M., Papamastorakis, J., Paleologou, E.
+and Terzian, Y., 1996, AAP, 310, 603.www.publish.csiro.au/journals/pasa 5
+Direction of motionThick outer
+Filamentsshell: rim
++ Bipolar outflowBright edges of the bipolar
+Sharp structures
+Faint opposite edge
+Figure 2: An example of WZO1 type: The “Ear Nebula” IPHAS PN. Nort h on the top and East on the
+left.
+/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1erg/cm2/s/A erg/cm2/s/A
+Figure 3: WHT spectra of the “EarNebula” using the R300Band R158 R gratings. This nebula, for which
+we show some of the “strongest” emission lines useful for an identifi cation, presents a clear [NII] over-
+intensity and it has been confirmed as true PN using the revised diagn ostic diagram from Riesgo et al.
+(2006) (particularly the log [H α/[SII]]vslog [Hα/[NII]] diagram).6 Publications of the Astronomical Society of Australia
+Interacting rimDirection of motion
+CS candidate
+Direction of motion
+Geometric centerBow shock CS candidates
+Direction of motionCS candidateBright rim
+Figure 4: Examples of WZO2 types. Top: Size=1.2 arcmin and SB=3.4e−17erg cm−2s−1arcsec−2.
+Middle: Size= 8.5 arcmin and SB=1.1e−16erg cm−2s−1arcsec−2, Bottom: 4.3 arcmin and SB=2.7e−16
+erg cm−2s−1arcsec−2. North on the top and East on the left.
+ Direction of motion
+Most probable CS
+Bright rim
+Figure 5: WZO3 type of ISM interaction in a IPHAS candidate PNe. Nor th on the top and East on the
+left.www.publish.csiro.au/journals/pasa 7
+ 
+Direction of motion
+Dense "wall" of nebular material gas and dust
+or
+turbulences Traces ofFaint frontal bow shock
+Figure 6: A possible example of WZO4 ISM interaction in one IPHAS PN ca ndidate (top: North on the
+top and East on the left) with the corresponding hydrodynamical m odel (bottom) [reproduction of figure
+5(d) from Wareing et al. (2007)].8 Publications of the Astronomical Society of Australia
+Figure 7: Galactic distribution of the candidate PNe/ISM according t o their stage of interaction and
+their size.
+Figure 8: Example of candidates with sizes greater than 100 arcsec (respectively 7.7 and 2.9 arcmin) and
+located at b= ±1 deg. These objects present a WZO2 stage of interaction and only their (very) faint
+interacting rim are seen.
\ No newline at end of file
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+arXiv:1001.0028v2  [math.CO]  28 Feb 2012CYCLIC SIEVING FOR GENERALISED NON-CROSSING
+PARTITIONS ASSOCIATED WITH COMPLEX REFLECTION
+GROUPS OF EXCEPTIONAL TYPE
+Christian Krattenthaler†andThomas W. M ¨uller‡
+†Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien,
+Nordbergstraße 15, A-1090 Vienna, Austria.
+WWW:http://www.mat.univie.ac.at/ ~kratt
+‡School of Mathematical Sciences,
+Queen Mary & Westfield College, University of London,
+Mile End Road, London E1 4NS, United Kingdom.
+WWW:http://www.maths.qmw.ac.uk/ ~twm/
+Dedicated to the memory of Herb Wilf
+Abstract. We prove that the generalised non-crossing partitions associated with
+well-generated complex reflection groups of exceptional type obe y two different cyclic
+sieving phenomena, as conjectured by Armstrong, and by Bessis a nd Reiner. The
+computational details are provided in the manuscript “Cyclic sieving for generalised
+non-crossing partitions associated with complex reflectio n groups of exceptional type
+— the details” [arχiv:1001.0030 ].
+1.Introduction
+In his memoir [2], Armstrong introduced generalised non-crossing partitions asso-
+ciated with finite (real) reflection groups, thereby embedding Krew eras’ non-crossing
+partitions [22], Edelman’s m-divisible non-crossing partitions [12], thenon-crossing par-
+titions associated with reflection groups due to Bessis [6] and Brady and Watt [10] into
+one uniform framework. Bessis and Reiner [9] observed that Arms trong’s definition can
+be straightforwardly extended to well-generated complex reflection groups (see Section 2
+for the precise definition). These generalised non-crossing partit ions possess a wealth
+of beautiful properties, and they display deep and surprising relat ions to other combi-
+natorial objects defined for reflection groups (such as the gene ralised cluster complex
+2000Mathematics Subject Classification. Primary 05E15; Secondary 05A10 05A15 05A18 06A07
+20F55.
+Key words and phrases. complex reflection groups, unitary reflection groups, m-divisible non-
+crossing partitions, generalised non-crossing partitions, Fuß–Ca talan numbers, cyclic sieving.
+†Research partially supported by the Austrian Science Foundation F WF, grants Z130-N13 and
+S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics
+and Probabilistic Number Theory.”
+‡Research supported by the Austrian Science Foundation FWF, Lise Meitner grant M1201-N13.
+12 C. KRATTENTHALER AND T. W. M ¨ULLER
+of Fomin and Reading [13], or the extended Shi arrangement and the geometric multi-
+chains of filters of Athanasiadis [4, 5]); see Armstrong’s memoir [2] and the references
+given therein.
+Ontheotherhand, cyclic sieving isaphenomenonbroughttolightbyReiner, Stanton
+and White [30]. It extends the so-called “( −1)-phenomenon” of Stembridge [34, 35].
+Cyclic sieving can be defined in three equivalent ways (cf. [30, Prop. 2.1]). The one
+which gives the name can be described as follows: given a set Sof combinatorial
+objects, an action on Sof a cyclic group G=/an}bracketle{tg/an}bracketri}htwith generator gof ordern, and
+a polynomial P(q) inqwith non-negative integer coefficients, we say that the triple
+(S,P,G)exhibits the cyclic sieving phenomenon , if the number of elements of Sfixed
+bygkequalsP(e2πik/n). In [30] it is shown that this phenomenon occurs in surprisingly
+many contexts, and several further instances have been discov ered since then.
+In [2, Conj. 5.4.7] (also appearing in [9, Conj. 6.4]) and [9, Conj. 6.5], Ar mstrong,
+respectively Bessis and Reiner, conjecture that generalised non- crossing partitions for
+irreducible well-generated complex reflection groups exhibit two diffe rent cyclic sieving
+phenomena (see Sections 3 and 7 for the precise statements).
+According to the classification of these groups due to Shephard an d Todd [32], there
+are two infinite families of irreducible well-generated complex reflectio n groups, namely
+the groups G(d,1,n) andG(e,e,n), wheren,d,eare positive integers, and there are 26
+exceptional groups. For the infinite families of types G(d,1,n) andG(e,e,n), the two
+cyclic sieving conjectures follow from the results in [19].
+Thepurposeofthepresent articleistopresent aproofofthecyc licsieving conjectures
+of Armstrong, and of Bessis and Reiner, for the 26 exceptional ty pes, thus completing
+the proof of these conjectures. Since the generalised non-cros sing partitions feature a
+parameterm, from the outset this is nota finite problem. Consequently, we first need
+several auxiliary results to reduce the conjectures for each of t he 26 exceptional types
+to afiniteproblem. Subsequently, we use Stembridge’s Maplepackagecoxeter [36]
+and theGAPpackageCHEVIE[14, 28] to carry out the remaining finitecomputations.
+The details of these computations are provided in [21]. In the presen t paper, we con-
+tent ourselves with exemplifying the necessary computations by go ing through some
+representative cases. It is interesting to observe that, for the verification of the type
+E8case, it is essential to use the decomposition numbers in the sense o f [17, 18, 20] be-
+cause, otherwise, the necessary computations would not be feas ible in reasonable time
+with the currently available computer facilities. We point out that, fo r the special case
+where the aforementioned parameter mis equal to 1, the first cyclic sieving conjecture
+has been proved in a uniform fashion by Bessis and Reiner in [9]. (See [3 ] for a —
+non-uniform — proof of cyclic sieving for non-crossing partitions as sociated with real
+reflection groups under the action of the so-called Kreweras map, a special case of the
+second cyclic sieving phenomenon discussed in the present paper.) T he crucial result on
+which the proof of Bessis and Reiner is based is (5.5) below, and it plays an important
+rolein our reduction of the conjectures forthe 26 exceptional gr oupsto a finite problem.
+Our paper is organised as follows. In the next section, we recall the definition of
+generalised non-crossing partitions for well-generated complex re flection groups and of
+decomposition numbers in the sense of [17, 18, 20], and we review so me basic facts.
+The first cyclic sieving conjecture is subsequently stated in Section 3. In Section 4, we
+outline an elementary proof that the q-Fuß–Catalan number, which is the polynomial
+Pin the cyclic sieving phenomena concerning the generalised non-cros sing partitionsCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 3
+for well-generated complex reflection groups, is always a polynomial with non-negative
+integer coefficients, as required by the definition of cyclic sieving. (F ull details can be
+found in [21, Sec. 4]. The reader is referred to the first paragraph of Section 4 for
+comments on other approaches for establishing polynomiality with no n-negative coeffi-
+cients.) Section 5 contains the announced auxiliary results which, fo r the 26 exceptional
+types, allow a reduction of the conjecture to a finite problem. In Se ction 6, we discuss
+a few cases which, in a representative manner, demonstrate how t o perform the re-
+maining case-by-case verification of the conjecture. For full det ails, we refer the reader
+to [21, Sec. 6]. The second cyclic sieving conjecture is stated in Sect ion 7. Section 8
+contains the auxiliary results which, for the 26 exceptional types, allow a reduction of
+the conjecture to a finite problem, while in Section 9 we discuss some r epresentative
+cases of the remaining case-by-case verification of the conjectu re. Again, for full details
+we refer the reader to [21, Sec. 9].
+2.Preliminaries
+Acomplex reflection group isa groupgeneratedby(complex) reflections in Cn. (Here,
+a reflection is a non-trivial element of GLn(C) which fixes a hyperplane pointwise and
+which hasfiniteorder.) Wereferto[24]foranin-depthexpositionof thetheorycomplex
+reflection groups.
+Shephard and Todd provided a complete classification of all finitecomplex reflection
+groups in [32] (see also [24, Ch. 8]). According to this classification, a n arbitrary
+complex reflection group Wdecomposes into a direct product of irreducible complex
+reflection groups, acting on mutually orthogonal subspaces of th e complex vector space
+onwhichWisacting. Moreover, thelistofirreduciblecomplexreflectiongroups consists
+of the infinite family of groups G(m,p,n), wherem,p,nare positive integers, and 34
+exceptional groups, denoted G4,G5,...,G 37by Shephard and Todd.
+In this paper, we are only interested in finite complex reflection grou ps which are
+well-generated . A complex reflection group of rank nis called well-generated if it is
+generated by nreflections.1Well-generation can be equivalently characterised by a
+duality property due to Orlik and Solomon [29]. Namely, a complex reflec tion group of
+ranknhastwo sets ofdistinguished integers d1≤d2≤ ··· ≤dnandd∗
+1≥d∗
+2≥ ··· ≥d∗
+n,
+called its degreesandcodegrees , respectively (see [24, p. 51 and Def. 10.27]). Orlik and
+Solomon observed, using case-by-case checking, that an irreduc ible complex reflection
+groupWof ranknis well-generated if and only if its degrees and codegrees satisfy
+di+d∗
+i=dn
+for alli= 1,2,...,n. The reader is referred to [24, App. D.2] for a table of the degree s
+and codegrees of all irreducible complex reflection groups. Togeth er with the classi-
+fication of Shephard and Todd [32], this constitutes a classification o f well-generated
+complex reflection groups: the irreducible well-generated complex r eflection groups are
+— the two infinite families G(d,1,n) andG(e,e,n), whered,e,nare positive inte-
+gers,
+— the exceptional groups G4,G5,G6,G8,G9,G10,G14,G16,G17,G18,G20,G21of
+rank 2,
+1We refer to [24, Def. 1.29] for the precise definition of “rank.” Roug hly speaking, the rank of a
+complex reflection group Wis the minimal nsuch that Wcan be realized as reflection group on Cn.4 C. KRATTENTHALER AND T. W. M ¨ULLER
+— the exceptional groups G23=H3,G24,G25,G26,G27of rank 3,
+— the exceptional groups G28=F4,G29,G30=H4,G32of rank 4,
+— the exceptional group G33of rank 5,
+— the exceptional groups G34,G35=E6of rank 6,
+— the exceptional group G36=E7of rank 7,
+— and the exceptional group G37=E8of rank 8.
+In this list, we have made visible the groups H3,F4,H4,E6,E7,E8which appear as
+exceptional groups in the classification of all irreducible realreflection groups (cf. [16]).
+LetWbe a well-generated complex reflection group of rank n, and letT⊆Wdenote
+theset of all(complex) reflections inthegroup. Let ℓT:W→Zdenotethewordlength
+in terms of the generators T. This word length is called absolute length orreflection
+length. Furthermore, we define a partial order ≤TonWby
+u≤Twif and only if ℓT(w) =ℓT(u)+ℓT(u−1w). (2.1)
+This partial order is called absolute order orreflection order . As is well-known and
+easy to see, the equation in (2.1) is equivalent to the statement tha t every shortest
+representation of uby reflections occurs as an initial segment in some shortest produc t
+representation of wby reflections.
+Now fix a (generalised) Coxeter element2c∈Wand a positive integer m. The
+m-divisible non-crossing partitions NCm(W) are defined as the set
+NCm(W) =/braceleftbig
+(w0;w1,...,w m) :w0w1···wm=cand
+ℓT(w0)+ℓT(w1)+···+ℓT(wm) =ℓT(c)/bracerightbig
+.
+A partial order is defined on this set by
+(w0;w1,...,w m)≤(u0;u1,...,u m) if and only if ui≤Twifor 1≤i≤m.
+We have suppressed the dependence on c, since we understand this definition up to
+isomorphism of posets. To be more precise, it can be shown that any two Coxeter
+elements are related to each other by conjugation and (possibly) a n automorphism on
+the field of complex numbers (see [33, Theorem 4.2] or [24, Cor. 11.2 5]), and hence the
+resulting posets NCm(W) are isomorphic to each other. If m= 1, thenNC1(W) can
+be identified with the set NC(W) of non-crossing partitions for the (complex) reflection
+groupWasdefined byBessis andCorran(cf.[8]and[7, Sec.13]; theirdefinit ionextends
+the earlier definition by Bessis [6] and Brady and Watt [10] for real r eflection groups).
+The following result has been proved by a collaborative effort of seve ral authors (see
+[7, Prop. 13.1]).
+2An element of an irreducible well-generated complex reflection group Wof ranknis called a
+Coxeter element if it isregularin the sense of Springer [33] (see also [24, Def. 11.21]) and of order dn.
+An element of Wis called regular if it has an eigenvector which lies in no reflecting hyperp lane of a
+reflection of W. It follows from an observation of Lehrer and Springer, proved un iformly by Lehrer
+and Michel [23] (see [24, Theorem 11.28]), that there is always a regu lar element of order dnin an
+irreducible well-generated complex reflection group Wof rankn. More generally, if a well-generated
+complex reflection group Wdecomposes as W∼=W1×W2×···×Wk, where the Wi’s are irreducible,
+then a Coxeter element of Wis an element of the form c=c1c2···ck, whereciis a Coxeter element of
+Wi,i= 1,2,...,k. IfWis arealreflection group, that is, if all generators in Thave order 2, then the
+notion of generalised Coxeter element given above reduces to that of a Coxeter element in the classical
+sense (cf. [16, Sec. 3.16]).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 5
+Theorem 1. LetWbe an irreducible well-generated complex reflection group, and let
+d1≤d2≤ ··· ≤dnbe its degrees and h:=dnits Coxeter number. Then
+|NCm(W)|=n/productdisplay
+i=1mh+di
+di. (2.2)
+Remark1.(1) The number in (2.2) is called the Fuß–Catalan number for the reflection
+groupW.
+(2) Ifcis a Coxeter element of a well-generated complex reflection group Wof rank
+n, thenℓT(c) =n. (This follows from [7, Sec. 7].)
+We conclude this section by recalling the definition of decomposition nu mbers from
+[17, 18, 20]. Although we need them here only for (very small) real re flection groups,
+and although, strictly speaking, they have been only defined for re al reflection groups in
+[17, 18, 20], this definition can be extended to well-generated comple x reflection groups
+without any extra effort, which we do now.
+Given a well-generated complex reflection group Wof rankn, typesT1,T2,...,T d(in
+the sense of the classification of well-generated complex reflection groups) such that the
+sumoftheranksofthe Ti’sequalsn, andaCoxeter element c, thedecompositionnumber
+NW(T1,T2,...,T d) is defined as the number of “minimal” factorisations c=c1c2···cd,
+“minimal” meaning that ℓT(c1) +ℓT(c2) +···+ℓT(cd) =ℓT(c) =n, such that, for
+i= 1,2,...,d, the type of cias a parabolic Coxeter element is Ti. (Here, the term
+“parabolic Coxeter element” means a Coxeter element in some parab olic subgroup. It
+follows from [31, Prop.6.3] that any element ciis indeed a Coxeter element in a unique
+parabolic subgroup of W.3By definition, the type of ciis the type of this parabolic
+subgroup.) Since any two Coxeter elements are related to each oth er by conjugation
+plus field automorphism, the decomposition numbers are independen t of the choice of
+the Coxeter element c.
+The decomposition numbers for real reflection groups have been c omputed in [17,
+18, 20]. To compute the decomposition numbers for well-generated complex reflection
+groups is a task that remains to be done.
+3.Cyclic sieving I
+In this section we present the first cyclic sieving conjecture due to Armstrong [2,
+Conj. 5.4.7], and to Bessis and Reiner [9, Conj. 6.4].
+Letφ:NCm(W)→NCm(W) be the map defined by
+(w0;w1,...,w m)/mapsto→/parenleftbig
+(cwmc−1)w0(cwmc−1)−1;cwmc−1,w1,w2,...,w m−1/parenrightbig
+.(3.1)
+It is indeed not difficult to see that, if the ( m+ 1)-tuple on the left-hand side is an
+element ofNCm(W), then so is the ( m+1)-tuple on the right-hand side. For m= 1,
+this action reduces to conjugation by the Coxeter element c(applied to w1). Cyclic
+sieving arising from conjugation by chas been the subject of [9].
+3The uniqueness can be argued as follows: suppose that ciwere a Coxeter element in two parabolic
+subgroups of W, sayU1andU2. Then it must also be a Coxeter element in the intersection U1∩U2.
+On the other hand, the absolute length of a Coxeter element of a co mplex reflection group Uis always
+equal to rk( U), the rank of U. (This follows from the fact that, for each element uofU, we have
+ℓT(u) = codim/parenleftbig
+ker(u−id)/parenrightbig
+, with id denoting the identity element in U; see e.g. [31, Prop. 1.3]). We
+conclude that ℓT(ci) = rk(U1) = rk(U2) = rk(U1∩U2), This implies that U1=U2.6 C. KRATTENTHALER AND T. W. M ¨ULLER
+It is easy to see that φmhacts as the identity, where his the Coxeter number of W
+(see (5.1) and Lemma 29 below). By slight abuse of notation, let C1be the cyclic group
+of ordermhgenerated by φ. (The slight abuse consists in the fact that we insist on C1
+to be a cyclic group of order mh, while it may happen that the order of the action of
+φgiven in (3.1) is actually a proper divisor of mh.)
+Given these definitions, we are now in the position to state the first c yclic sieving
+conjecture of Armstrong, respectively of Bessis and Reiner. By t he results of [19] and
+of this paper, it becomes the following theorem.
+Theorem 2. For an irreducible well-generated complex reflection group Wand any
+m≥1, the triple (NCm(W),Catm(W;q),C1), whereCatm(W;q)is theq-analogue of
+the Fuß–Catalan number defined by
+Catm(W;q) :=n/productdisplay
+i=1[mh+di]q
+[di]q, (3.2)
+exhibits the cyclic sieving phenomenon in the sense of Reine r, Stanton and White [30].
+Here,nis the rank of W,d1,d2,...,d nare the degrees of W,his the Coxeter number
+ofW, and[α]q:= (1−qα)/(1−q).
+Remark2.We write Catm(W) for Catm(W;1).
+By definition of the cyclic sieving phenomenon, we have to prove that Catm(W;q) is
+a polynomial in qwith non-negative integer coefficients, and that
+|FixNCm(W)(φp)|= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh, (3.3)
+for allpin the range 0 ≤p<mh. The first fact is established in the next section, while
+the proof of the second is achieved by making use of several auxiliar y results, given
+in Section 5, to reduce the proof to a finite problem, and a subseque nt case-by-case
+analysis. Alldetails ofthisanalysiscanbefoundin[21, Sec. 6]. Inthe present paper, we
+content ourselves with discussing the cases where W=G24and whereW=G37=E8,
+since these suffice to convey the flavour of the necessary comput ations.
+4.Theq-Fusz–Catalan numbers Catm(W;q)
+The purpose of this section is to provide an elementary, self-conta ined proof of the
+fact that, for all irreducible complex reflection groups W, theq-Fuß–Catalan number
+Catm(W;q) is a polynomial in qwith non-negative integer coefficients. For most of
+the groups, this is a known property. However, aside from the fac t that, for many of
+the known cases, the proof is very indirect and uses deep algebraic results on rational
+Cherednik algebras, there still remained some cases where this pro perty had not been
+formally established. The reader is referred to the “Theorem” in Se ction 1.6 of [15],
+whichsaysthat, undertheassumptionofacertainrankcondition( [15, Hypothesis2.4]),
+theq-Fuß–Catalan number Catm(W;q) is a Hilbert series of a finite-dimensional quo-
+tient of the ring of invariants of Wand also the graded character of a finite-dimensional
+irreducible representation of a spherical rational Cherednik algeb ra associated with
+W. At present, this rank condition has been proven for all irreducible well-generated
+complex reflection groups apart from G17,G18,G29,G33,G34; see [26, Tables 8 and 9,
+column “rank”], and the recent paper [27], which establishes the res ult in the case of
+G32.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 7
+In the sequel, aside from the standard notation [ α]q= (1−qα)/(1−q) forq-integers,
+we shall also use the q-binomial coefficient, which is defined by
+/bracketleftbigg
+n
+k/bracketrightbigg
+q:=/braceleftBigg
+1, ifk= 0,
+[n]q[n−1]q···[n−k+1]q
+[k]q[k−1]q···[1]q,ifk>0.
+We begin with several auxiliary results.
+Proposition 3. For all non-negative integers nandk, theq-binomial coefficient [n
+k]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.This is a well-known fact, which can be derived either from the recurr ence rela-
+tion(s) satisfied by the q-binomial coefficients (generalising Pascal’s recurrence relation
+for binomial coefficients; cf. [1, eqs. (3.3.3) and (3.3.4)]), or from th e fact that the q-
+binomial coefficient [n
+k]qis the generating function for (integer) partitions with at most
+kparts all of which are at most n−k(cf. [1, Theorem 3.1]). /square
+Proposition 4. For all non-negative integers mandn, theq-Fuß–Catalan number of
+typeAn,
+1
+[(m+1)n+1]q/bracketleftbigg
+(m+1)n+1
+n/bracketrightbigg
+q,
+is a polynomial in qwith non-negative integer coefficients.
+Proof.In [25, Sec. 3.3], Loehr proves that
+1
+[(m+1)n+1]q/bracketleftbigg
+(m+1)n+1
+n/bracketrightbigg
+q
+=/summationdisplay
+v∈V(m)
+nqm(n
+2)+/summationtext
+i≥0(m(vi
+2)−ivi)/productdisplay
+i≥1qvi/summationtextm
+j=1(m−j)vi−j/bracketleftbigg
+vi+vi−1+···+vi−m−1
+vi/bracketrightbigg
+q,(4.1)
+whereV(m)
+ndenotes the set of all sequences v= (v0,v1,...,v s) (for some s) of non-
+negative integers with v0>0,vs>0, andv0+v1+···+vs=n, and such that there
+is never a string of mor more consecutive zeroes in v. By convention, vi= 0 for all
+negativei. His proof works by showing that the expressions on both sides of ( 4.1)
+satisfy the same recurrence relation and initial conditions, using cla ssicalq-binomial
+identities. We refer the reader to [25] for details. By Proposition 3, the expression on
+the right-hand side of (4.1) is manifestly a polynomial in qwith non-negative integer
+coefficients. /square
+Lemma 5. Ifaandbare coprime positive integers, then
+[ab]q
+[a]q[b]q(4.2)
+is a polynomial in qof degree (a−1)(b−1), all of whose coefficients are in {0,1,−1}.
+Moreover, if one disregards the coefficients which are 0, then+1’s and(−1)’s alternate,
+and the constant coefficient as well as the leading coefficient o f the polynomial equal +1.
+Proof.LetΦn(q)denotethe n-thcyclotomicpolynomialin q. Usingtheclassicalformula
+1−qn=/productdisplay
+d|nΦd(q),8 C. KRATTENTHALER AND T. W. M ¨ULLER
+we see that
+(1−q)(1−qab)
+(1−qa)(1−qb)=/productdisplay
+d1|a,d1/ne}ationslash=1
+d2|a,d2/ne}ationslash=1Φd1d2(q),
+so that, manifestly, the expression in (4.2) is a polynomial in q. The claim concerning
+the degree of this polynomial is obvious.
+In order to establish the claim on the coefficients, we start with a sub -expression of
+(4.2),
+(1−qab)
+(1−qa)(1−qb)=/parenleftbiggb−1/summationdisplay
+i=0qia/parenrightbigg/parenleftbigg∞/summationdisplay
+j=0qjb/parenrightbigg
+=∞/summationdisplay
+k=0Ckqk, (4.3)
+say. The assumption that aandbare coprime implies that 0 ≤Ck≤1 fork≤
+(a−1)(b−1). Multiplying both sides of (4.3) by 1 −q, we obtain the equation
+[ab]q
+[a]q[b]q= (1−q)(a−1)(b−1)/summationdisplay
+k=0Ckqk+(1−q)∞/summationdisplay
+k=(a−1)(b−1)+1Ckqk. (4.4)
+By our previous observation on the coefficients Ckwithk≤(a−1)(b−1), it is obvious
+that the coefficients of the first expression on the right-hand side of (4.4) are alternately
++1 and−1, when 0’s are disregarded. Since we already know that the left-ha nd side is
+a polynomial in qof degree (a−1)(b−1), we may ignore the second expression.
+The proof is concluded by observing that the claims on the constant and leading
+coefficients are obvious. /square
+Corollary 6. Letaandbbe coprime positive integers, and let γbe an integer with
+γ≥(a−1)(b−1). Then the expression
+[γ]q[ab]q
+[a]q[b]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.Let
+[ab]q
+[a]q[b]q=(a−1)(b−1)/summationdisplay
+k=0Dkqk.
+We then have
+[γ]q[ab]q
+[a]q[b]q=(a−1)(b−1)+γ−1/summationdisplay
+N=0qNN/summationdisplay
+k=max{0,N−γ+1}Dk. (4.5)
+IfN≤γ−1, then, by Lemma 5, the sum over kon the right-hand side of (4.5) equals
+1−1+1−1+···, which is manifestly non-negative. On the other hand, if N >γ−1,
+then we may rewrite the sum over kon the right-hand side of (4.5) as
+N/summationdisplay
+k=max{0,N−γ+1}Dk=(a−1)(b−1)/summationdisplay
+k=N−γ+1Dk=(a−1)(b−1)+γ−1−N/summationdisplay
+k=0D(a−1)(b−1)−k.
+Again, by Lemma 5, this sum equals 1 −1 + 1−1 +···, which is manifestly non-
+negative. /squareCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 9
+The next lemmas all have a very similar flavour, and so do their proofs . In order to
+avoid repetition, proof details are only provided for Lemmas 7 and 16 ; the proofs of
+Lemmas 9–15, 22–24 follow the pattern exhibited in the proof of Lem ma 7, while the
+proofs of Lemmas 17–21 follow that of the proof of Lemma 15. Full d etails are found
+in [21, Sec. 4].
+Lemma 7. Letαandβbe positive integers with α≥6andβ≥8. Then the expression
+[α]q3[β]q4[72]q[3]q[4]q
+[8]q[9]q[12]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[72]q[3]q[4]q
+[8]q[9]q[12]q
+= (1−q3+q9−q15+q18)(1−q4+q8−q12+q16−q20+q24−q28+q32).
+It should be observed that both factors on the right-hand side ha ve the property that
+coefficients are in {0,1,−1}and that (+1)’s and ( −1)’s alternate, if one disregards the
+coefficients which are 0. If we now apply the same idea as in the proof o f Corollary 6,
+then we see that [ α]q3times the first factor is a polynomial in qwith non-negative
+integer coefficients, as is [ β]q4times the second factor. Taken together, this establishes
+the claim. /square
+Lemma 8. Letαandβbe positive integers with α≥26andβ≥8. Then the expression
+[α]q[β]q4[15]q
+[3]q[5]q[72]q[3]q[4]q
+[8]q[9]q[12]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 9. Letαandβbe positive integers with α≥18andβ≥3. Then the expression
+[α]q3[β]q4[90]q[3]q[4]q
+[5]q[6]q[9]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 10. Letαandβbe positive integers with α≥20andβ≥18. Then the
+expression
+[α]q[β]q3[90]q[3]q
+[5]q[6]q[9]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 11. Letαbe a positive integer with α≥26. Then the expression
+[α]q[15]q
+[3]q[5]q[12]q3
+[3]q3[4]q3
+is a polynomial in qwith non-negative integer coefficients.10 C. KRATTENTHALER AND T. W. M ¨ULLER
+Lemma 12. Letαbe a positive integer with α≥14. Then the expression
+[α]q[15]q
+[3]q[5]q[6]q3
+[2]q3[3]q3
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 13. Letαandβbe positive integers with α≥30andβ≥20. Then the
+expression
+[α]q[β]q2[84]q[2]q
+[4]q[6]q[7]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 14. Letαandβbe positive integers with α≥24andβ≥68. Then the
+expression
+[α]q[β]q[105]q
+[3]q[5]q[7]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 15. Letαandβbe positive integers with α≥24andβ≥34. Then the
+expression
+[α]q[β]q[70]q
+[2]q[5]q[7]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 16. Letαandβbe positive integers with α≥4andβ≥2. Then the expression
+[α]q2[β]q5[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q= 1+q−q3−q4−q5+q7+q8.
+If we multiply this expression by [ α]q2, then, forα= 4 we obtain
+1+q+q2−q5−q9+q12+q13+q14,
+forα= 5 we obtain
+1+q+q2−q5+q8−q11+q14+q15+q16,
+and, forα≥6, we obtain
+1+q+q2−q5+q8+q10+p1(q)+q2α−4+q2α−2−q2α+1+q2α+4+q2α+5+q2α+6,
+wherep1(q) is a polynomial in qwith non-negative coefficients of order at least 11 and
+degree at most 2 α−5. In all cases it is obvious that the product of the result and [ β]q5,
+withβ≥2, is a polynomial in qwith non-negative coefficients. /squareCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 11
+Lemma 17. Letαandβbe positive integers with α≥14andβ≥2. Then the
+expression
+[α]q[β]q5[14]q
+[2]q[7]q[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 18. Letαandβbe positive integers with α≥32andβ≥12. Then the
+expression
+[α]q[β]q2[35]q
+[5]q[7]q[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 19. Letαandβbe positive integers with α≥16andβ≥2. Then the
+expression
+[α]q2[β]q5[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 20. Letαandβbe positive integers with α≥56andβ≥4. Then the
+expression
+[α]q[β]q2[35]q
+[5]q[7]q[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 21. Letαandβbe positive integers with α≥38andβ≥2. Then the
+expression
+[α]q[β]q5[14]q
+[2]q[7]q[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 22. Letαandβbe positive integers with α≥30andβ≥26. Then the
+expression
+[α]q[β]q3[126]q[3]q
+[6]q[7]q[9]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 23. Letαandβbe positive integers with α≥66andβ≥54. Then the
+expression
+[α]q[β]q3[252]q[3]q
+[7]q[9]q[12]q
+is a polynomial in qwith non-negative integer coefficients.
+Lemma 24. Letαandβbe positive integers with α≥54andβ≥34. Then the
+expression
+[α]q[β]q2[140]q[2]q
+[4]q[7]q[10]q
+is a polynomial in qwith non-negative integer coefficients.12 C. KRATTENTHALER AND T. W. M ¨ULLER
+We are now ready for the proof of the main result of this section.
+Theorem 25. For all irreducible well-generated complex reflection grou ps and posi-
+tive integers m, theq-Fuß–Catalan number Catm(W;q)is a polynomial in qwith non-
+negative integer coefficients.
+Proof.First, letW=An. In this case, the degrees are 2 ,3,...,n+1, and hence
+Catm(An;q) =1
+[(m+1)n+1]q/bracketleftbigg
+(m+1)n+1
+n/bracketrightbigg
+q,
+which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.
+Next, letW=G(d,1,n). In this case, the degrees are d,2d,...,nd , and hence
+Catm(G(d,1,n);q) =/bracketleftbigg
+(m+1)n
+n/bracketrightbigg
+qd,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+Now, letW=G(e,e,n). In this case, the degrees are e,2e,...,(n−1)e,n, and hence
+Catm(G(e,e,n);q) =[m(n−1)e+n]q
+[n]qn−1/productdisplay
+i=1[m(n−1)e+ie]q
+[ie]q
+=/bracketleftbigg
+(m+1)(n−1)
+n−1/bracketrightbigg
+qe+qn[e]qn/bracketleftbigg
+(m+1)(n−1)
+n/bracketrightbigg
+qe,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+It remains to verify the claim for the exceptional groups.
+For the groups W=G6,G9,G14,G17,G21,and partially for the groups W=G20,G23,
+G28,G30,G33,G35,G36,G37(depending on congruence properties of the parameter m),
+polynomiality and non-negativity of coefficients of the correspondin gq-Fuß–Catalan
+number can be directly read off by a proper rearrangement of the t erms in the defining
+expression; for example, for W=G21(with degrees given by 12 ,60) we have
+Catm(G21;q) =[60m+12]q[60m+60]q
+[12]q[60]q= [5m+1]q12[m+1]q60,
+which is manifestly a polynomial in qwith non-negative integer coefficients.
+For the groups G5,G10,G18,G26,G27,G29,G34, the terms in the defining expres-
+sion of the corresponding q-Fuß–Catalan number can be arranged in a manner so
+that aq-binomial coefficient appears; polynomiality and non-negativity of co efficients
+then follow from Proposition 3. For example, for W=G34(with degrees given by
+6,12,18,24,30,42) we have
+Catm(G34;q) =[42m+6]q[42m+12]q[42m+18]q[42m+24]q[42m+30]q[42m+42]q
+[6]q[12]q[18]q[24]q[30]q[42]q
+= [m+1]q42/bracketleftbigg
+7m+5
+5/bracketrightbigg
+q6,
+which, written in this form, is obviously a polynomial in qwith non-negative integer
+coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13
+On the other hand, for the groups G4,G8,G16,G25,G32, the terms in the defining
+expression of the corresponding q-Fuß–Catalan number can be arranged in a manner so
+that aq-Fuß–Catalannumber of type Aappears andProposition 4 applies; for example,
+forW=G32(with degrees given by 12 ,18,24,30) we have
+Catm(G32;q) =[30m+12]q[30m+18]q[30m+24]q[30m+30]q
+[12]q[18]q[24]q[30]q
+=1
+[5m+6]q6/bracketleftbigg
+5m+6
+5/bracketrightbigg
+q6,
+which indeed fits into the framework of Proposition 4 and, hence, is a polynomial in q
+with non-negative integer coefficients.
+In the other cases, the more “specialised” auxiliary results given in C orollary 6 and
+Lemmas7–24havetobeapplied. Forthesakeofillustration, weexhib it oneexample for
+each of them below, with full details being provided in [21, Sec. 4]. In ge neral, the idea
+is that, given a rational expression consisting of cyclotomic factor s, as in the definition
+oftheq-Fuß–Catalannumbers, onetriestoplacedenominator factorsbe lowappropriate
+numerator factors so that one can divide out the denominator fac tor completely. For
+example, if we were to encounter the expression
+[30m+12]q·(other terms)
+[12]q·(other terms)
+and know that mis even, then we would try to simplify this to
+/bracketleftbig5m+2
+2/bracketrightbig
+q12·(other terms)
+(other terms),
+where [5m+2
+2]q12is manifestly a polynomial in qwith non-negative integer coefficients.
+On the other hand, in a situation where twodenominator factors “want” to divide a
+singlenumerator factor, we “extract” as much as we can from the nume rator factor and
+compensate by additional “fudge” factors. To be more concrete , if we encounter the
+expression
+[14m+14]q·(other terms)
+[6]q[14]q·(other terms)
+and we know that m≡0 (mod 3), then we would try the rewriting
+/bracketleftbigm+1
+3/bracketrightbig
+q42[21]q2
+[3]q2[7]q2[2]q·(other terms)
+(other terms),
+with the idea that we might find somewhere else a term [2 α]q, which could be combined
+with the term[2] qin the denominator into [2 α]q/[2]q= [α]q2, andthen apply Corollary6
+to see that
+[α]q2[21]q2
+[3]q2[7]q2
+is a polynomial in qwith non-negative integer coefficients (provided αis at least 12),
+with/bracketleftbigm+1
+3/bracketrightbig
+q42being such a polynomial in any case.
+In situations where threedenominator factors “want” to divide a singlenumerator
+factor, one has to perform more complicated rearrangements, in order to be able to
+apply one of the Lemmas 7–24.14 C. KRATTENTHALER AND T. W. M ¨ULLER
+For example, for W=G24, the degrees are 4 ,6,14, and hence
+Catm(G24;q) =[14m+4]q[14m+6]q[14m+14]q
+[4]q[6]q[14]q.
+We have
+Catm(G24;q) =
+
+/bracketleftbig7m
+2+1/bracketrightbig
+q4/bracketleftbig14m
+6+1/bracketrightbig
+q6[m+1]q14,ifm≡0 (mod 6),/bracketleftbig7m+2
+3/bracketrightbig
+q6/bracketleftbig7m+3
+2/bracketrightbig
+q4[m+1]q14, ifm≡1 (mod 6),
+/bracketleftbig7m
+2+1/bracketrightbig
+q4[7m+3]q2/bracketleftbigm+1
+3/bracketrightbig
+q42[21]q2
+[3]q2[7]q2,ifm≡2 (mod 6),
+[7m+2]q2/bracketleftbig7m
+3+1/bracketrightbig
+q6/bracketleftbigm+1
+2/bracketrightbig
+q28[14]q2
+[2]q2[7]q2,ifm≡3 (mod 6),
+/bracketleftbig7m+2
+6/bracketrightbig
+q12[6]q2
+[2]q2[3]q2[7m+3]q2[m+1]q14,ifm≡4 (mod 6),
+[7m+2]q2/bracketleftbig7m+3
+2/bracketrightbig
+q4/bracketleftbigm+1
+3/bracketrightbig
+q42[21]q2
+[3]q2[7]q2,ifm≡5 (mod 6),
+which, by Corollary 6, are polynomials in qwith non-negative integer coefficients in all
+cases.
+ForW=G30=H4, the degrees are 2 ,12,20,30, and hence
+Catm(H4;q) =[30m+2]q[30m+12]q[30m+20]q[30m+30]q
+[2]q[12]q[20]q[30]q.
+Ifmis odd, then we may write
+Catm(H4;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[5m+2]q6[3m+2]q10/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q6[10]q2[15]q2,
+which, by Lemma 16, is a polynomial in qwith non-negative integer coefficients.
+ForW=G35=E6, the degrees are 2 ,5,6,8,9,12, and hence
+Catm(E6;q) =[12m+2]q[12m+5]q[12m+6]q[12m+8]q[12m+9]q[12m+12]q
+[2]q[5]q[6]q[8]q[9]q[12]q.
+Ifm≡5 (mod 30),then we have
+Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5
+5/bracketrightbig
+q5[2m+1]q6
+×[3m+2]q4[4m+3]q3/bracketleftbigm+1
+6/bracketrightbig
+q72[72]q[3]q[4]q
+[8]q[9]q[12]q,
+which, by Lemma 7, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡7 (mod 30),then we have
+Catm(E6;q) =/bracketleftbig6m+1
+2/bracketrightbig
+q4[12m+5]q/bracketleftbig2m+1
+15/bracketrightbig
+q90
+×[90]q[3]q[4]q
+[5]q[6]q[9]q[3m+2]q4[4m+3]q3/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Corollary 6 and Lemma 9, is a polynomial in qwith non-negative integer
+coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15
+Ifm≡8 (mod 30),then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6/bracketleftbig3m+2
+2/bracketrightbig
+q8
+×/bracketleftbig4m+3
+5/bracketrightbig
+q15[15]q
+[3]q[5]q/bracketleftbigm+1
+3/bracketrightbig
+q36[12]q3
+[3]q3[4]q3,
+which, by Lemma 11, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡13 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+3/bracketrightbig
+q18[6]q3
+[2]q3[3]q3
+×[3m+2]q4/bracketleftbig4m+3
+5/bracketrightbig
+q15[15]q
+[3]q[5]q/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Lemma 12, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡22 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+15/bracketrightbig
+q90[90]q[3]q
+[5]q[6]q[9]q
+×/bracketleftbig3m+2
+2/bracketrightbig
+q8[4m+3]q3[m+1]q12,
+which, by Lemma 10, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡23 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6
+×[3m+2]q4/bracketleftbig4m+3
+5/bracketrightbig
+q15[15]q
+[3]q[5]q/bracketleftbigm+1
+6/bracketrightbig
+q72[72]q[3]q[4]q
+[8]q[9]q[12]q,
+which, by Lemma 8, is a polynomial in qwith non-negative integer coefficients.
+ForW=G36=E7, the degrees are 2 ,6,8,10,12,14,18, and hence
+Catm(E7;q) =[18m+2]q[18m+6]q[18m+8]q[18m+10]q
+[2]q[6]q[8]q[10]q
+×[18m+12]q[18m+14]q[18m+18]q
+[12]q[14]q[18]q.
+Ifm≡18 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2/bracketleftbig3m+2
+28/bracketrightbig
+q168[84]q2[2]q2
+[4]q2[6]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Corollary 6 and Lemma 13, is a polynomial in qwith non-negative integer
+coefficients.16 C. KRATTENTHALER AND T. W. M ¨ULLER
+Ifm≡23 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+35/bracketrightbig
+q210[105]q2
+[3]q2[5]q2[7]q2[9m+4]q2[9m+5]q2
+×[3m+2]q6[9m+7]q2/bracketleftbigm+1
+2/bracketrightbig
+q36[6]q6
+[2]q6[3]q6,
+which, by Corollary 6 and Lemma 14, is a polynomial in qwith non-negative integer
+coefficients.
+Ifm≡54 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+70/bracketrightbig
+q140[70]q2
+[2]q2[5]q2[7]q2[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18.
+Ifonedecomposes[9 m+7]q2as[9m
+2+4]q4+q2[9m
+2+3]q4, thenoneseesthat, byCorollary6
+and Lemma 15, this is a polynomial in qwith non-negative integer coefficients.
+ForW=G37=E8, the degrees are 2 ,8,12,14,18,20,24,30, and hence
+Catm(E7;q) =[30m+2]q[30m+8]q[30m+12]q[30m+14]q
+[2]q[8]q[12]q[14]q
+×[30m+18]q[30m+20]q[30m+24]q[30m+30]q
+[18]q[20]q[24]q[30]q.
+Ifm≡3 (mod 84),then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4/bracketleftbig15m+4
+7/bracketrightbig
+q14[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients.
+Ifm≡8 (mod 84),then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+42/bracketrightbig
+q252[126]q2[3]q2
+[6]q2[7]q2[9]q2[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,
+which, by Lemma 22, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡11 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×/bracketleftbig3m+2
+7/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 17
+which, by Corollary 6 and Lemma 20, is a polynomial in qwith non-negative integer
+coefficients.
+Ifm≡16 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+84/bracketrightbig
+q504[252]q2[3]q2
+[7]q2[9]q2[12]q2[m+1]q30,
+which, by Lemma 23, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡18 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2/bracketleftbig5m+3
+3/bracketrightbig
+q18
+/bracketleftbig3m+2
+28/bracketrightbig
+q280[140]q2[2]q2
+[4]q2[7]q2[10]q2/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30,
+which, by Lemma 24, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡21 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2/bracketleftbig5m+3
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 17, is a polynomial in qwith non-negative integer
+coefficients.
+Ifm≡25 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×/bracketleftbig3m+2
+7/bracketrightbig
+q70[35]q2
+[5]q2[7]q2/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Lemma 18, is a polynomial in qwith non-negative integer coefficients.
+Ifm≡27 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 21, is a polynomial in qwith non-negative integer
+coefficients.
+All other cases are disposed of in a similar fashion. /square
+5.Auxiliary results I
+This section collects several auxiliary results which allow us to reduce the problem
+of proving Theorem 2, or the equivalent statement (3.3), for the 2 6 exceptional groups
+listed in Section 2 to a finite problem. While Lemmas 27 and 28 cover spec ial choices
+of the parameters, Lemmas 26 and 30 afford an inductive procedur e. More precisely,18 C. KRATTENTHALER AND T. W. M ¨ULLER
+if we assume that we have already verified Theorem 2 for all groups o f smaller rank,
+then Lemmas 26 and 30, together with Lemmas 27 and 31, reduce th e verification of
+Theorem 2 for the group that we are currently considering to a finit e problem; see
+Remark 3. The final lemma of this section, Lemma 32, disposes of com plex reflection
+groups with a special property satisfied by their degrees.
+Letp=am+b, 0≤b<m. We have
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;ca+1wm−b+1c−a−1,ca+1wm−b+2c−a−1,...,ca+1wmc−a−1,
+caw1c−a,...,cawm−bc−a/parenrightbig
+,(5.1)
+where∗stands for the element of Wwhich is needed to complete the product of the
+components to c.
+Lemma 26. It suffices to check (3.3)forpa divisor of mh. More precisely, let pbe
+a divisor of mh, and letkbe another positive integer with gcd(k,mh/p) = 1, then we
+have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πikp/mh (5.2)
+and
+|FixNCm(W)(φp)|=|FixNCm(W)(φkp)|. (5.3)
+Proof.For (5.2), this follows immediately from
+lim
+q→ζ[α]q
+[β]q=/braceleftBigg
+α
+βifα≡β≡0 (modd),
+1 otherwise ,(5.4)
+whereζis ad-th root of unity and α,βare non-negative integers such that α≡β
+(modd).
+In order to establish (5.3), suppose that x∈FixNCm(W)(φp), that is,x∈NCm(W)
+andφp(x) =x. It obviously follows that φkp(x) =x, so thatx∈FixNCm(W)(φkp).
+To establish the converse, note that, if gcd( k,mh/p) = 1, then there exists k′with
+k′k≡1 (modmh
+p). It follows that, if x∈FixNCm(W)(φkp), that is, if x∈NCm(W) and
+φkp(x) =x, thenx=φk′kp(x) =φp(x), whencex∈FixNCm(W)(φp). /square
+Lemma 27. Letpbe a divisor of mh. Ifpis divisible by m, then(3.3)is true.
+Proof.According to (5.1), the action of φponNCm(W) is described by
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;cp/mw1c−p/m,...,cp/mwmc−p/m/parenrightbig
+.
+Hence, if (w0;w1,...,w m) is fixed by φp, then each individual wimust be fixed under
+conjugation by cp/m.
+Using the notation W′= Cent W(cp/m), theprevious observationmeans that wi∈W′,
+i= 1,2,...,m. Springer [33, Theorem 4.2] (see also [24, Theorem 11.24(iii)]) prove d
+thatW′is a well-generated complex reflection group whose degrees coincide with those
+degrees ofWthat are divisible by mh/p. It was furthermore shown in [9, Lemma 3.3]
+that
+NC(W)∩W′=NC(W′). (5.5)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 19
+Hence, the tuples ( w0;w1,...,w m) fixed byφpare in fact identical with the elements of
+NCm(W′), which implies that
+|FixNCm(W)(φp)|=|NCm(W′)|. (5.6)
+Application of Theorem 1 with Wreplaced by W′and of the “limit rule” (5.4) then
+yields that
+|NCm(W′)|=/productdisplay
+1≤i≤n
+mh
+p|dimh+di
+di= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh. (5.7)
+Combining (5.6) and (5.7), we obtain (3.3). This finishes the proof of t he lemma. /square
+Lemma 28. Equation (3.3)holds for all divisors pofm.
+Proof.Using (5.4) and the fact that the degrees of irreducible well-genera ted complex
+reflection groups satisfy di<hfor alli<n, we see that
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh=/braceleftBigg
+m+1 ifm=p,
+1 ifm/ne}ationslash=p.
+On the other hand, if ( w0;w1,...,w m) is fixed by φp, then, because of the action (5.1),
+we must have w1=wp+1=···=wm−p+1andw1=cwm−p+1c−1. In particular,
+w1∈CentW(c). By the theorem of Springer cited in the proof of Lemma 27, the
+subgroup Cent W(c) is itself a complex reflection group whose degrees are those degre es
+ofWthat are divisible by h. The only such degree is hitself, hence Cent W(c) is the
+cyclic group generated by c. Moreover, by (5.5), we obtain that w1=ε, the identity
+element of W, orw1=c. Therefore, for m=pthe set Fix NCm(W)(φp) consists of the
+m+1 elements ( w0;w1,...,w m) obtained by choosing wi=cfor a particular ibetween
+0 andm, all otherwj’s being equal to ε, while, for m/ne}ationslash=p, we have
+FixNCm(W)(φp) =/braceleftbig
+(c;ε,...,ε)/bracerightbig
+,
+whence the result. /square
+Lemma 29. LetWbe an irreducible well-generated complex reflection group a ll of
+whose degrees are divisible by d. Then each element of Wis fixed under conjugation by
+ch/d.
+Proof.By the theorem of Springer cited in the proof of Lemma 27, the subg roupW′=
+CentW(ch/d) is itself a complex reflection group whose degrees are those degre es ofW
+that are divisible by d. Thus, by our assumption, the degrees of W′coincide with the
+degrees ofW, and hence W′must be equal to W. Phrased differently, each element of
+Wis fixed under conjugation by ch/d, as claimed. /square
+Lemma 30. LetWbe an irreducible well-generated complex reflection group o f rankn,
+and letp=m1h1be a divisor of mh, wherem=m1m2andh=h1h2. Without loss of
+generality, we assume that gcd(h1,m2) = 1. Suppose that Theorem 2has already been
+verified for all irreducible well-generated complex reflect ion groups with rank <n. Ifh2
+does not divide all degrees di, then Equation (3.3)is satisfied.20 C. KRATTENTHALER AND T. W. M ¨ULLER
+Proof.Let us write h1=am2+b, with 0 ≤b < m 2. The condition gcd( h1,m2) = 1
+translates into gcd( b,m2) = 1. From (5.1), we infer that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;ca+1wm−m1b+1c−a−1,ca+1wm−m1b+2c−a−1,...,ca+1wmc−a−1,
+caw1c−a,...,cawm−m1bc−a/parenrightbig
+.(5.8)
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=ca+1wi+m−m1bc−a−1, i= 1,2,...,m 1b,
+wi=cawi−m1bc−a, i=m1b+1,m1b+2,...,m,
+which, after iteration, implies in particular that
+wi=cb(a+1)+(m2−b)awic−b(a+1)−(m2−b)a=ch1wic−h1, i= 1,2,...,m.
+It is at this point where we need gcd( b,m2) = 1. The last equation shows that each wi,
+i= 1,2,...,m, and thus also w0, lies in Cent W(ch1). By the theorem of Springer cited
+in the proof of Lemma 27, this centraliser subgroup is itself a complex reflection group,
+W′say, whose degrees are those degrees of Wthat are divisible by h/h1=h2. Since,
+by assumption, h2does not divide alldegrees,W′has rank strictly less than n. Again
+by assumption, we know that Theorem 2 is true for W′, so that in particular,
+|FixNCm(W′)(φp)|= Catm(W′;q)/vextendsingle/vextendsingle
+q=e2πip/mh.
+The arguments above together with (5.5) show that Fix NCm(W)(φp) = Fix NCm(W′)(φp).
+On the other hand, using (5.4) it is straightforward to see that
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh= Catm(W′;q)/vextendsingle/vextendsingle
+q=e2πip/mh.
+This proves (3.3) for our particular p, as required. /square
+Lemma 31. LetWbe an irreducible well-generated complex reflection group o f rank
+n, and letp=m1h1be a divisor of mh, wherem=m1m2andh=h1h2. We assume
+thatgcd(h1,m2) = 1. Ifm2>nthen
+FixNCm(W)(φp) =/braceleftbig
+(c;ε,...,ε)/bracerightbig
+.
+Proof.Let us suppose that ( w0;w1,...,w m)∈FixNCm(W)(φp) and that there exists a
+j≥1 such that wj/ne}ationslash=ε. By (5.8), it then follows for such a jthat alsowk/ne}ationslash=εfor
+allk≡j−lm1b(modm), where, as before, bis defined as the unique integer with
+h1=am2+band 0≤b < m 2. Since, by assumption, gcd( b,m2) = 1, there are
+exactlym2suchk’s which are distinct mod m. However, this implies that the sum of
+the absolute lengths of the wi’s, 0≤i≤m, is at least m2> n, a contradiction to
+Remark 1.(2). /square
+Remark 3.(1) If we put ourselves in the situation of the assumptions of Lemma 30,
+then we may conclude that equation (3.3) only needs to be checked f or pairs (m2,h2)
+subject to the following restrictions:
+m2≥2,gcd(h1,m2) = 1,andh2divides all degrees of W. (5.9)
+Indeed, Lemmas 27 and 30 together imply that equation (3.3) is alway s satisfied in all
+other cases.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 21
+(2) Still putting ourselves in the situation of Lemma 30, if m2>nandm2h2does not
+divide any of the degrees of W, then equation (3.3) is satisfied. Indeed, Lemma 31 says
+thatinthiscasetheleft-handsideof (3.3)equals1,whileastraightf orwardcomputation
+using (5.4) shows that in this case the right-hand side of (3.3) equals 1 as well.
+(3)It shouldbeobserved that thisleaves afinitenumber of choices form2to consider,
+whence a finite number of choices for ( m1,m2,h1,h2). Altogether, there remains a finite
+number of choices for p=h1m1to be checked.
+Lemma 32. LetWbe an irreducible well-generated complex reflection group o f rankn
+with the property that di|hfori= 1,2,...,n. Then Theorem 2is true for this group
+W.
+Proof.By Lemma 26, we may restrict ourselves to divisors pofmh.
+Suppose that e2πip/mhis adi-th rootof unity for some i. In other words, mh/pdivides
+di. Sincediis a divisor of hby assumption, the integer mh/palso divides h. But this
+is equivalent to saying that mdividesp, and equation (3.3) holds by Lemma 27.
+Now assume that mh/pdoes not divide any of the di’s. Then, by (5.4), the right-
+hand side of (3.3) equals 1. On the other hand, ( c;ε,...,ε) is always an element of
+FixNCm(W)(φp). To see that there are no others, we make appeal to the classific a-
+tion of all irreducible well-generated complex reflection groups, whic h we recalled in
+Section 2. Inspection reveals that all groups satisfying the hypot heses of the lemma
+have rank n≤2. Except for the groups contained in the infinite series G(d,1,n)
+andG(e,e,n) for which Theorem 2 has been established in [19], these are the grou ps
+G5,G6,G9,G10,G14,G17,G18,G21. We now discuss these groups case by case, keeping
+the notation of Lemma 30. In order to simplify the argument, we not e that Lemma 31
+implies that equation (3.3) holds if m2>2, so that in the following arguments we
+always may assume that m2= 2.
+CaseG5. The degrees are 6 ,12, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.
+CaseG6. The degrees are 4 ,12, and therefore, according to Remark 3.(1), we need
+only consider the casewhere h2= 4andm2= 2, that is, p= 3m/2. Then (5.8) becomes
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+2+1c−2,c2wm
+2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+2c−1/parenrightbig
+.
+(5.10)
+If (w0;w1,...,w m) isfixed by φpandnot equal to ( c;ε,...,ε), there must exist an iwith
+1≤i≤m
+2such thatℓT(wi) =ℓT(wm
+2+i) = 1,wm
+2+i=cwic−1,wiwm
+2+i=wicwic−1=c,
+and allwj, withj/ne}ationslash=i,m
+2+i, equalε. However, with the help of the GAPpackage
+CHEVIE[14, 28], one verifies that there is no wiinG6such that
+ℓT(wi) = 1 and wicwic−1=c
+are simultaneously satisfied. Hence, the left-hand side of (3.3) is eq ual to 1, as required.
+CaseG9. The degrees are 8 ,24, and therefore, according to Remark 3.(1), we need
+only consider the case where h2= 8 andm2= 2, that is, p= 3m/2. This is the same p
+as forG6. Again, CHEVIEfinds no solution. Hence, the left-hand side of (3.3) is equal
+to 1, as required.
+CaseG10. The degrees are 12 ,24, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.22 C. KRATTENTHALER AND T. W. M ¨ULLER
+CaseG14. The degrees are 6 ,24, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.
+CaseG17. The degrees are 20 ,60, and therefore, according to Remark 3.(1), we need
+only consider the cases where h2= 20 orh2= 4. In the first case, p= 3m/2, which is
+the samepas forG6. Again,CHEVIEfinds no solution. In the second case, p= 15m/2.
+Then (5.8) becomes
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c8wm
+2+1c−8,c8wm
+2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
+2c−7/parenrightbig
+.(5.11)
+By Lemma 29, every element of NC(W) is fixed under conjugation by c3, and, thus, on
+elements fixed by φp, the above action of φpreduces to the one in (5.10). This action
+was already discussed in the first case. Hence, in both cases, the le ft-hand side of (3.3)
+is equal to 1, as required.
+CaseG18. The degrees are 30 ,60, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.
+CaseG21. The degrees are 12 ,60, and therefore, according to Remark 3.(1), we need
+only consider the cases where h2= 12 orh2= 4. In the first case, p= 5m/2, so that
+(5.8) becomes
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3wm
+2+1c−3,c3wm
+2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm
+2c−2/parenrightbig
+.(5.12)
+If (w0;w1,...,w m) is fixed by φpand not equal to ( c;ε,...,ε), there must exist an i
+with 1≤i≤m
+2such thatℓT(wi) = 1 andwic2wic−2=c. However, with the help of
+theGAPpackageCHEVIE[14, 28], one verifies that there is no such solution to this
+equation. In the second case, p= 15m/2. Then (5.8) becomes the action in (5.11).
+By Lemma 29, every element of NC(W) is fixed under conjugation by c5, and, thus,
+on elements fixed by φp, the action of φpin (5.11) reduces to the one in the first case.
+Hence, in both cases, the left-hand side of (3.3) is equal to 1, as re quired.
+This completes the proof of the lemma. /square
+6.Exemplification of case-by-case verification of Theorem 2
+It remains to verify Theorem 2 for the groups G4,G8,G16,G20,G23=H3,G24,G25,
+G26,G27,G28=F4,G29,G30=H4,G32,G33,G34,G35=E6,G36=E7,G37=E8. All
+details can be found in [21, Sec. 6]. We content ourselves with illustra ting the type of
+computation that is needed here by going through the case of the g roupG24, and by
+discussing some of the arguments needed for the group G37=E8.
+In the sequel we write ζdfor a primitive d-th root of unity.
+CaseG24.The degrees are 4 ,6,14, and hence we have
+Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q
+[14]q[6]q[4]q.
+Letζbe a 14m-th root of unity. In what follows, we abbreviate the assertion tha t “ζis
+a primitive d-th root of unity” as “ ζ=ζd.” The following cases on the right-hand sideCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 23
+of (3.3) occur:
+lim
+q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (6.1a)
+lim
+q→ζCatm(G24;q) =7m+3
+3,ifζ=ζ6,ζ3,3|m, (6.1b)
+lim
+q→ζCatm(G24;q) =7m+2
+2,ifζ=ζ4,2|m, (6.1c)
+lim
+q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (6.1d)
+lim
+q→ζCatm(G24;q) = 1,otherwise. (6.1e)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.1). The only cases not covered by Lemma 27 are the on es in (6.1b),
+(6.1c), and (6.1e). (In both (6.1a) and (6.1d) we have d|h.)
+We first consider (6.1b). By Lemma 26, we are free to choose p= 7m/3 ifζ=ζ6,
+respectively p= 14m/3 ifζ=ζ3. In both cases, mmust be divisible by 3.
+We start with the case that p= 7m/3. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3w2m
+3+1c−3,c3w2m
+3+2c−3,...,c3wmc−3,c2w1c−2,...,c2w2m
+3c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3w2m
+3+ic−3, i= 1,2,...,m
+3, (6.2a)
+wi=c2wi−m
+3c−2, i=m
+3+1,m
+3+2,...,m. (6.2b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1.
+Writingt1,t2,t3forwi,wi+m
+3,wi+2m
+3, respectively, the equations (6.2) reduce to
+t1=c3t3c−3, (6.3a)
+t2=c2t1c−2, (6.3b)
+t3=c2t2c−2. (6.3c)
+One of these equations is in fact superfluous: if we substitute (6.3b ) and (6.3c) in
+(6.3a), then we obtain t1=c7t1c−7which is automatically satisfied due to Lemma 29
+withd= 2.
+Since (w0;w1,...,w m)∈NCm(G24), we must have t1t2t3=c. Combining this with
+(6.3), we infer that
+t1(c2t1c−2)(c4t1c−4) =c. (6.4)
+With the help of CHEVIE, one obtains 7 solutions for t1in this equation, each of them
+giving rise to m/3 elements of Fix NCm(G24)(φp) sincei(inwi) ranges from 1 to m/3.
+In total, we obtain 1 + 7m
+3=7m+3
+3elements in Fix NCm(G24)(φp), which agrees with
+the limit in (6.1b).
+The case where p= 14m/3 can be treated in a similar fashion. In the end, it
+turns out that we have to solve the same enumeration problem as fo rp= 7m/3, and,24 C. KRATTENTHALER AND T. W. M ¨ULLER
+consequently, the number of elements of Fix NCm(G24)(φp) is the same, namely7m+3
+3, as
+required.
+Our next case is (6.1c). Proceeding in a similar manner as before, we s ee that there is
+againthe trivial possibility ( c;ε,...,ε), and otherwise we have to find t1withℓT(t1) = 1
+satisfying the inequality
+t1(c3t1c−3)≤Tc. (6.5)
+With the help of CHEVIE, one obtains 7 solutions for t1in this relation, each of them
+giving rise to m/2 elements of Fix NCm(G24)(φp) sincei(inwi) ranges from 1 to m/2.
+In total, we obtain 1 + 7m
+2=7m+2
+2elements in Fix NCm(G24)(φp), which agrees with
+the limit in (6.1c).
+Finally, we turn to (6.1e). By Remark 3, the only choices for h2andm2to be consid-
+ered areh2= 1 andm2= 3,h2=m2= 2, andh2= 2 andm2= 3. These correspond
+to the choices p= 14m/3,p= 7m/2, respectively p= 7m/3, all of which have already
+been discussed as they do not belong to (6.1e). Hence, (3.3) must n ecessarily hold, as
+required.
+CaseG37=E8.The degrees are 2 ,8,12,14,18,20,24,30, and hence we have
+Catm(E8;q) =[30m+30]q[30m+24]q[30m+20]q[30m+18]q
+[30]q[24]q[20]q[18]q
+×[30m+14]q[30m+12]q[30m+8]q[30m+2]q
+[14]q[12]q[8]q[2]q.
+Letζbe a 30m-th root of unity. The cases occurring on the right-hand side of (3 .3)
+not covered by Lemma 27 are:
+lim
+q→ζCatm(E8;q) =5m+4
+4,ifζ=ζ24,4|m, (6.6a)
+lim
+q→ζCatm(E8;q) =3m+2
+2,ifζ=ζ20,2|m, (6.6b)
+lim
+q→ζCatm(E8;q) =5m+3
+3,ifζ=ζ18,ζ9,3|m, (6.6c)
+lim
+q→ζCatm(E8;q) =15m+7
+7,ifζ=ζ14,ζ7,7|m, (6.6d)
+lim
+q→ζCatm(E8;q) =(5m+4)(5m+2)
+8,ifζ=ζ12,2|m, (6.6e)
+lim
+q→ζCatm(E8;q) =(5m+4)(15m+4)
+16,ifζ=ζ8,4|m, (6.6f)
+lim
+q→ζCatm(E8;q) =(5m+4)(3m+2)(5m+2)(15m+4)
+64,ifζ=ζ4,2|m,(6.6g)
+lim
+q→ζCatm(E8;q) = Catm(E8),ifζ=−1 orζ= 1, (6.6h)
+lim
+q→ζCatm(E8;q) = 1,otherwise. (6.6i)
+We now have to prove that the left-hand side of (3.3) in each case ag rees with the
+values exhibited in (6.6). Since the corresponding computations in th e various cases are
+very similar, we concentrate here only on the cases (6.6f) and (6.6g ), these two being
+representative of the types of arguments arising. As before, we refer the reader to [21,
+Sec. 6] for full details.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 25
+Letusconsiderthecasein(6.6f)first. ByLemma26, wearefreeto choosep= 15m/4.
+In particular, mmust be divisible by 4. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4wm
+4+1c−4,c4wm
+4+2c−4,...,c4wmc−4,c3w1c−3,...,c3wm
+4c−3/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4wm
+4+ic−4, i= 1,2,...,3m
+4, (6.7a)
+wi=c3wi−3m
+4c−3, i=3m
+4+1,3m
+4+2,...,m. (6.7b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m, which we
+summarise as follows:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+4such that
+1≤ℓT(wi) =ℓT(wi+m
+4) =ℓT(wi+2m
+4) =ℓT(wi+3m
+4)≤2, (6.8a)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iii) there are i1andi2with 1≤i1<i2≤m
+4such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+4) =ℓT(wi2+m
+4)
+=ℓT(wi1+2m
+4) =ℓT(wi2+2m
+4) =ℓT(wi1+3m
+4) =ℓT(wi2+3m
+4) = 1,(6.8b)
+and all other wjare equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have
+wiwi+m
+4wi+2m
+4wi+3m
+4≤Tc,
+or
+wi1wi2wi1+m
+4wi2+m
+4wi1+2m
+4wi2+2m
+4wi1+3m
+4wi2+3m
+4=c.
+Together with equations (6.7)–(6.8), this implies that
+wi=c15wic−15andwi(c11wic−11)(c7wic−7)(c3wic−3)≤Tc, (6.9)
+or that
+wi1=c15wi1c−15, wi1=c15wi2c−15,
+andwi1wi2(c11wi1c−11)(c11wi2c−11)(c7wi1c−7)(c7wi2c−7)(c3wi1c−3)(c3wi2c−3) =c.
+(6.10)
+Here, the first equation in (6.9) and the first two equations in (6.10) are automatically
+satisfied due to Lemma 29 with d= 2.
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 30 solutions
+forwiin (6.9) with ℓT(wi) = 1, 45 solutions for wiwithℓT(wi) = 2 and wiof type
+A2
+1(as a parabolic Coxeter element; see the end of Section 2), and 20 s olutions for
+wiwithℓT(wi) = 2 and wiof typeA2. Each of them gives rise to m/4 elements of
+FixNCm(E8)(φp) sinceiranges from 1 to m/4.
+The number of solutions in Case (iii) can be computed from our knowled ge of the
+solutions in Case (ii) according to type, using some elementary count ing arguments.
+Namely, the number of solutions of (6.10) is equal to
+45·2+20·3 = 150,26 C. KRATTENTHALER AND T. W. M ¨ULLER
+since an element of type A2
+1can be decomposed in two ways into a product of two
+elements of absolute length 1, while for an element of type A2this can be done in 3
+ways.
+In total, we obtain 1 + (30 + 45 + 20)m
+4+ 150/parenleftbigm/4
+2/parenrightbig
+=(5m+4)(15m+4)
+16elements in
+FixNCm(E8)(φp), which agrees with the limit in (6.6f).
+Next, we discuss the case in (6.6g). By Lemma 26, we are free to cho osep= 15m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c8wm
+2+1c−8,c8wm
+2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
+2c−7/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c8wm
+2+ic−8, i= 1,2,...,m
+2, (6.11a)
+wi=c7wi−m
+2c−7, i=m
+2+1,m
+2+2,...,m. (6.11b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+1≤ℓT(wi) =ℓT(wi+m
+2)≤4, (6.12a)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iii) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓ1:=ℓT(wi1) =ℓT(wi1+m
+2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
+2)≥1,andℓ1+ℓ2≤4,
+(6.12b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1,i2,i3with 1≤i1<i2<i3≤m
+2such that
+ℓ1:=ℓT(wi1) =ℓT(wi1+m
+2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
+2)≥1,
+ℓ3:=ℓT(wi3) =ℓT(wi3+m
+2)≥1,andℓ1+ℓ2+ℓ3≤4,(6.12c)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(v) there are i1,i2,i3,i4with 1≤i1<i2<i3<i4≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi3) =ℓT(wi4)
+=ℓT(wi1+m
+2) =ℓT(wi2+m
+2) =ℓT(wi3+m
+2) =ℓT(wi4+m
+2) = 1,(6.12d)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2≤Tc, respectively
+wi1wi2wi3wi1+m
+2wi2+m
+2wi3+m
+2≤Tc,
+respectively
+wi1wi2wi3wi4wi1+m
+2wi2+m
+2wi3+m
+2wi4+m
+2=c.
+Together with equations (6.11)–(6.12), this implies that
+wi=c15wic−15andwi(c7wic−7)≤Tc, (6.13)
+respectively that
+wi1=c15wi1c−15, wi2=c15wi2c−15,andwi1wi2(c7wi1c−7)(c7wi2c−7)≤Tc,(6.14)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 27
+respectively that
+wi1=c15wi1c−15, wi2=c15wi2c−15, wi3=c15wi3c−15,
+andwi1wi2wi3(c7wi1c−7)(c7wi2c−7)(c7wi3c−7)≤Tc,(6.15)
+respectively that
+wi1=c15wi1c−15, wi2=c15wi2c−15, wi3=c15wi3c−15, wi4=c15wi4c−15,
+andwi1wi2wi3wi4(c7wi1c−7)(c7wi2c−7)(c7wi3c−7)(c7wi4c−7) =c.(6.16)
+Here, the first equation in (6.13), the first two in (6.14), the first t hree in (6.15), and
+the first four in (6.16), are all automatically satisfied due to Lemma 2 9 withd= 2.
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains
+— 45 solutions for wiin (6.13) with ℓT(wi) = 1,
+— 150 solutions for wiin (6.13) with ℓT(wi) = 2 andwiof typeA2
+1,
+— 100 solutions for wiin (6.13) with ℓT(wi) = 2 andwiof typeA2,
+— 75 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA3
+1,
+— 165 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA1∗A2,
+— 90 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA3,
+— 15 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA2
+1∗A2,
+— 45 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA1∗A3;
+— 5 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA2
+2,
+— 18 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA4,
+— 5 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeD4.
+Each of them gives rise to m/2 elements of Fix NCm(E8)(φp) sinceiranges from 1 to m/2.
+There are no solutions for wiin (6.13) with wiof typeA4
+1.
+Letting the computer find all solutions in cases (iii)–(v) would take ye ars. However,
+the number of these solutions can be computed from our knowledge of the solutions
+in Case (ii) according to type, if this information is combined with the de composition
+numbers in the sense of [17, 18, 20] (see the end of Section 2) and some elementary
+(multiset) permutation counting. The decomposition numbers for A2,A3,A4, andD4
+of which we make use can be found in the appendix of [18].
+To begin with, the number of solutions of (6.14) with ℓ1=ℓ2= 1 is equal to
+n1,1:= 150·2+100·NA2(A1,A1) = 600,
+since an element of type A2
+1can be decomposed in two ways into a product of two
+elements of absolute length 1, while for an element of type A2this can be done in
+NA2(A1,A1) = 3 ways. Similarly, the number of solutions of (6.14) with ℓ1= 2 and
+ℓ2= 1 is equal to
+n2,1:= 75·3+165·(1+NA2(A1,A1))+90·NA3(A2,A1) = 1425,
+the number of solutions of (6.14) with ℓ1= 3 andℓ2= 1 is equal to
+n3,1:= 15·(2+NA2(A1,A1))+45·(1+NA3(A2,A1))+5·(2NA2(A1,A1))
++18·(NA4(A3,A1)+NA4(A1∗A2,A1))+5·(ND4(A3,A1)+ND4(A3
+1,A1)) = 660,28 C. KRATTENTHALER AND T. W. M ¨ULLER
+the number of solutions of (6.14) with ℓ1=ℓ2= 2 is equal to
+n2,2:= 15·(2+2NA2(A1,A1))+45·(2NA3(A2,A1))+5·(2+NA2(A1,A1)2)
++18·(NA4(A2,A2)+NA4(A2
+1,A2
+1)+2NA4(A2,A2
+1))
++5·(ND4(A2,A2)+2ND4(A2,A2
+1)) = 1195,
+the number of solutions of (6.15) with ℓ1=ℓ2=ℓ3= 1 is equal to
+n1,1,1:= 75·3!+165·(3NA2(A1,A1))+90NA3(A1,A1,A1) = 3375,
+the number of solutions of (6.15) with ℓ1= 2 andℓ2=ℓ3= 1 is equal to
+n2,1,1:= 15·(2+NA2(A1,A1)+2·2·NA2(A1,A1))+45·(2NA3(A2,A1)+NA3(A1,A1,A1))
++5·(2NA2(A1,A1)+2NA2(A1,A1)2)+18·(NA4(A2,A1,A1)+NA4(A2
+1,A1,A1))
++5·(ND4(A2,A1,A1)+ND4(A2
+1,A1,A1)) = 2850,
+and the number of solutions of (6.16) is equal to
+n1,1,1,1:= 15·(12NA2(A1,A1))+45·(4NA3(A1,A1,A1))+5·(6NA2(A1,A1)2)
++18·NA4(A1,A1,A1,A1)+5·ND4(A1,A1,A1,A1) = 6750.
+In total, we obtain
+1+(45+150+100+75+165+90+15+45+5+18+5)m
+2+(n1,1+2n2,1+2n3,1+n2,2)/parenleftbiggm/2
+2/parenrightbigg
++(n1,1,1+3n2,1,1)/parenleftbiggm/2
+3/parenrightbigg
++n1,1,1,1/parenleftbiggm/2
+4/parenrightbigg
+=(5m+4)(3m+2)(5m+2)(15m+4)
+64
+elements in Fix NCm(E8)(φp), which agrees with the limit in (6.6g).
+7.Cyclic sieving II
+In this section we present the second cyclic sieving conjecture due to Bessis and
+Reiner [9, Conj. 6.5].
+Letψ:NCm(W)→NCm(W) be the map defined by
+(w0;w1,...,w m)/mapsto→/parenleftbig
+cwmc−1;w0,w1,...,w m−1/parenrightbig
+. (7.1)
+Form= 1, we have w0=cw−1
+1, so that this action reduces to the inverse of the
+Kreweras complement Kc
+idas defined by Armstrong [2, Def. 2.5.3].
+It is easy to see that ψ(m+1)hacts as the identity, where his the Coxeter number of
+W(see (8.1) below). By slight abuse of notation as before, let C2be the cyclic group
+of order (m+1)hgenerated by ψ.
+Given these definitions, we are now in the position to state the secon d cyclic sieving
+conjecture of Bessis and Reiner. By the results of [19] and of this p aper, it becomes the
+following theorem.
+Theorem 33. For an irreducible well-generated complex reflection group Wand any
+m≥1, the triple (NCm(W),Catm(W;q),C2), whereCatm(W;q)is theq-analogue of
+the Fuß–Catalan number defined in (3.2), exhibits the cyclic sieving phenomenon.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 29
+By definition of the cyclic sieving phenomenon, we have to prove that
+|FixNCm(W)(ψp)|= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h, (7.2)
+for allpin the range 0 ≤p<(m+1)h.
+8.Auxiliary results II
+This section collects several auxiliary results which allow us to reduce the problem of
+proving Theorem 33, respectively the equivalent statement (7.2), for the 26 exceptional
+groups listed in Section 2 to a finite problem. The corresponding lemma s, Lemmas 34–
+39, are analogues of Lemmas 26–28 and 30–32 in Section 5.
+Letp=a(m+1)+b, 0≤b<m+1. We have
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (ca+1wm−b+1c−a−1;ca+1wm−b+2c−a−1,...,ca+1wmc−a−1,
+caw0c−a,...,cawm−bc−a/parenrightbig
+.(8.1)
+Lemma 34. It suffices to check (7.2)forpa divisor of (m+1)h. More precisely, let pbe
+a divisor of (m+1)h, and letkbe another positive integer with gcd(k,(m+1)h/p) = 1,
+then we have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πikp/(m+1)h (8.2)
+and
+|FixNCm(W)(ψp)|=|FixNCm(W)(ψkp)|. (8.3)
+Proof.For (8.3), this follows in the same way as (5.3) in Lemma 26.
+For (8.2), we must argue differently than in Lemma 26. Let us write ζ=e2πip/(m+1)h.
+For a given group W, we writeS1(W) for the set of all indices isuch thatζdi−h= 1,
+and we write S2(W) for the set of all indices isuch thatζdi= 1. By the rule of de
+l’Hospital, we have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h=
+
+0 if |S1(W)|>|S2(W)|,/producttext
+i∈S1(W)(mh+di)/producttext
+i∈S2(W)di/producttext
+i/∈S1(W)(1−ζdi−h)
+/producttext
+i/∈S2(W)(1−ζdi),if|S1(W)|=|S2(W)|.
+(8.4)
+Since, by Theorem 25, Catm(W;q) is a polynomial in q, the case |S1(W)|<|S2(W)|
+cannot occur.
+We claim that, for the case where |S1(W)|=|S2(W)|, the factors in the quotient of
+products/producttext
+i/∈S1(W)(1−ζdi−h)/producttext
+i/∈S2(W)(1−ζdi)
+cancel pairwise. If we assume the correctness of the claim, it is obv ious that we get
+the same result if we replace ζbyζk, where gcd( k,(m+1)h/p) = 1, hence establishing
+(8.2).
+In order to see that our claim is indeed valid, we proceed in a case-by- case fash-
+ion, making appeal to the classification of irreducible well-generated complex reflection
+groups, which werecalled inSection2. Firstofall, since dn=h, thesetS1(W)isalways
+non-empty as it contains the element n. Hence, if we want to have |S1(W)|=|S2(W)|,30 C. KRATTENTHALER AND T. W. M ¨ULLER
+the setS2(W) must be non-empty as well. In other words, the integer ( m+ 1)h/p
+must divide at least one of the degrees d1,d2,...,d n. In particular, this implies that,
+for each fixed reflection group Wof exceptional type, only a finite number of values of
+(m+1)h/phas to be checked. Writing Mfor (m+1)h/p, what needs to be checked is
+whether the multisets (that is, multiplicities of elements must be taken into account)
+{(di−h) modM:i /∈S1(W)}and{dimodM:i /∈S2(W)}
+are the same. Since, for a fixed irreducible well-generated complex r eflection group,
+thereisonlyafinitenumber ofpossibilities for M, thisamountstoaroutineverification.
+/square
+Lemma 35. Letpbe a divisor of (m+ 1)h. Ifpis divisible by m+ 1, then(7.2)is
+true.
+We leave the proof to the reader as it is completely analogous to the p roof of
+Lemma 27.
+Lemma 36. Equation (7.2)holds for all divisors pofm+1.
+Proof.We have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h=/braceleftBigg
+0 ifp<m+1,
+m+1 ifp=m+1.
+Here, the first case follows from (8.4) and the fact that we have S1(W)⊇ {n}and
+S2(W) =∅ifp|(m+1) andp<m+1.
+Ontheother hand, if ( w0;w1,...,w m) is fixed by ψp, then onecanapply anargument
+similar to that in Lemma 28 with any witaking the role of w1, 0≤i≤m. It follows
+that ifp=m+1, the set Fix NCm(W)(ψp) consists of the m+1 elements ( w0;w1,...,w m)
+obtained by choosing wi=cfor a particular ibetween 0 and m, all otherwj’s being
+equal toε. Ifp<m+1, then there is no element in Fix NCm(W)(ψp). /square
+Lemma 37. LetWbe an irreducible well-generated complex reflection group o f rank
+n, and letp=m1h1be a divisor of (m+1)h, wherem+1 =m1m2andh=h1h2. We
+assume that gcd(h1,m2) = 1. Suppose that Theorem 33has already been verified for
+all irreducible well-generated complex reflection groups w ith rank< n. Ifh2does not
+divide all degrees di, then equation (7.2)is satisfied.
+We leave the proof to the reader as it is completely analogous to the p roof of
+Lemma 30.
+Lemma 38. LetWbe an irreducible well-generated complex reflection group o f rank
+n, and letp=m1h1be a divisor of (m+1)h, wherem+1 =m1m2andh=h1h2. We
+assume that gcd(h1,m2) = 1. Ifm2>nthen
+FixNCm(W)(ψp) =∅.
+We leave the proof to the reader as it is analogous to the proof of Le mma 31.
+Remark 4.By applying the same reasoning as in Remark 3 with Lemmas 30 and 31
+replaced by Lemmas 37 and 38, respectively, it follows that we only ne ed to check (7.2)
+for pairs (m2,h2) satisfying (5.9) and m2≤n. This reduces the problem to a finite
+number of choices.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 31
+Lemma 39. LetWbe an irreducible well-generated complex reflection group o f rankn
+with the property that di|hfori= 1,2,...,n. Then Theorem 33is true for this group
+W.
+Proof.Proceeding in a fashion analogous to the beginning of the proof of Le mma 32, we
+mayrestricttothecasewhere p|(m+1)hand(m+1)h/pdoesnotdivideanyofthe di’s.
+Inthiscase, itfollowsfrom(8.4)andthefactthatwehave S1(W)⊇ {n}andS2(W) =∅
+that the right-hand side of (7.2) equals 0. Inspection of the classifi cation of all irre-
+ducible well-generated complex reflection groups, which we recalled in Section 2, reveals
+that all groups satisfying the hypotheses of the lemma have rank n≤2. Except for the
+groups contained in the infinite series G(d,1,n) andG(e,e,n) for which Theorem 2 has
+been established in [19], these are the groups G5,G6,G9,G10,G14,G17,G18,G21. The
+verification of (7.2) can be done in a similar fashion as in the proof of Le mma 32. We
+illustrate this by going through the case of the group G6. In analogy with the earlier
+situation, we note that Lemma 38 implies that equation (7.2) holds if m2>2, so that
+in the following arguments we may assume that m2= 2.
+CaseG6. The degrees are 4 ,12, and therefore, according to Remark 4, we need only
+consider the case where h2= 4 andm2= 2, that is, p= 3(m+1)/2. Then the action
+ofψpis given by
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (c2wm+1
+2c−2;c2wm+3
+2c−2,...,c2wmc−2,cw0c−1,...,cw m−1
+2c−1/parenrightbig
+.
+(8.5)
+If (w0;w1,...,w m) is fixed by ψp, there must exist an iwith 0≤i≤m−1
+2such that
+ℓT(wi) = 1,wicwic−1=c, and allwj,j/ne}ationslash=i,m+1
+2+i, equalε. However, with the help of
+CHEVIE, one verifies that there is no such solution to this equation. Hence, the left-hand
+side of (7.2) is equal to 0, as required.
+This completes the proof of the lemma. /square
+9.Exemplification of case-by-case verification of Theorem 3 3
+It remains to verify Theorem 33 for the groups G4,G8,G16,G20,G23=H3,G24,G25,
+G26,G27,G28=F4,G29,G30=H4,G32,G33,G34,G35=E6,G36=E7,G37=E8. All
+details can be found in [21, Sec. 9]. We content ourselves with discuss ing the case of
+the groupG24, as this suffices to convey the flavour of the necessary computat ions.
+In order to simplify our considerations, it should be observed that t he action of ψ
+(given in(7.1)) is exactly the same as the actionof φ(given in (3.1)) with mreplaced by
+m+1on the components w1,w2,...,w m+1, that is, if we disregard the 0-th component
+of the elements of the generalised non-crossing partitions involved . The only difference
+which arises is that, while the ( m+ 1)-tuples ( w0;w1,...,w m) in (7.1) must satisfy
+w0w1···wm=c, forw1,w2,...,w m+1in (3.1) we only must have w1w2···wm+1≤Tc.
+Consequently, we may use the counting results from Section 6, exc ept that we have to
+restrict our attention to those elements ( w0;w1,...,w m,wm+1)∈NCm+1(W) for which
+w1w2···wm+1=c, or, equivalently, w0=ε.
+CaseG24.The degrees are 4 ,6,14, and hence we have
+Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q
+[14]q[6]q[4]q.32 C. KRATTENTHALER AND T. W. M ¨ULLER
+Letζbe a 14(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (9.1a)
+lim
+q→ζCatm(G24;q) =7m+7
+3,ifζ=ζ6,ζ3,3|(m+1), (9.1b)
+lim
+q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (9.1c)
+lim
+q→ζCatm(G24;q) = 0,otherwise. (9.1d)
+We must now prove that the left-handside of (7.2) in each case agre es with the values
+exhibited in (9.1). The only cases not covered by Lemma 35 are the on es in (9.1b) and
+(9.1d). On the other hand, the only cases left to consider accordin g to Remark 4 are
+the cases where h2= 1 andm2= 3,h2= 2 andm2= 3, andh2=m2= 2. These
+correspond to the choices p= 14(m+1)/3,p= 7(m+1)/3, respectively p= 7(m+1)/2.
+The first two cases belong to (9.1b), while p= 7(m+1)/2 belongs to (9.1d).
+In the case that p= 7(m+1)/3, the action of ψpis given by
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (c3w2m+2
+3c−3;c3w2m+5
+3c−3,...,c3wmc−3,c2w0c−2,...,c2w2m−1
+3c−2/parenrightbig
+.
+Hence, for an iwith 0≤i≤m−2
+3, we must find an element wi=t1, wheret1satisfies
+(6.4), so that we can set wi+m+1
+3=c2t1c−2,wi+2m+2
+3=c4t1c−4, and all other wj’s equal
+toε. We have found seven solutions to the counting problem (6.4), and e ach of them
+gives rise to ( m+1)/3 elements in Fix NCm(G24)(ψp) since the index iranges from 0 to
+(m−2)/3.
+On the other hand, if p= 14(m+1)/3, then the action of ψpis given by
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (c5wm+1
+3c−5;c5wm+4
+3c−5,...,c5wmc−5,c4w0c−4,...,c4wm−2
+3c−4/parenrightbig
+.
+By Lemma 29, every element of NC(W) is fixed under conjugation by c7, and, thus, the
+equations for t1in this case are the same as in the previous one where p= 7(m+1)/3.
+Hence, in either case, we obtain 7m+1
+3=7m+7
+3elements in Fix NCm(G24)(ψp), which
+agrees with the limit in (9.1b).
+Ifp= 7(m+ 1)/2, the relevant counting problem is (6.5). However, no element
+(w0;w1,...,w m)∈FixNCm(G24)(ψp) can be produced in this way since the counting
+problem imposes the restriction that ℓT(w0) +ℓT(w1) +···+ℓT(wm) be even, which
+contradicts the fact that ℓT(c) =n= 3. This is in agreement with the limit in (9.1d).
+Acknowledgements
+The authors thank an anonymous referee for a very careful rea ding of the original
+manuscript, and for numerous pertinent suggestions which have h elped to considerably
+improve the original manuscript.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 33
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\ No newline at end of file
diff --git a/1001.0029.txt b/1001.0029.txt
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index 0000000000000000000000000000000000000000..3668ccbf0664fa08eb6c6dadc2c24dd60a5f7c04
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@@ -0,0 +1,2967 @@
+arXiv:1001.0029v2  [hep-th]  3 Aug 2010Gravity assisted solution
+of the mass gap problem
+for pureYang-Mills fields
+Arkady L.Kholodenko
+375 H.L.Hunter Laboratories, Clemson University, Clemson,
+SC 29634-0973, USA. e-mail: string@clemson.edu
+In 1979 Louis Witten demonstrated that stationary axially symmetr ic Ein-
+stein field equationsand those for static axiallysymmetric self-dual SU(2) gauge
+fields can both be reduced to the same (Ernst) equation. In this pa per we
+use this result as point of departure to prove the existence of the mass gap
+for quantum source-free Yang-Mills (Y-M) fields. The proof is facilit ated by
+results of our recently published paper, JGP 59 (2009) 600-619. S ince both
+pure gravity, the Einstein-Maxwell and pure Y-M fields are describe d for axi-
+ally symmetric configurations by the Ernst equation classically, their quantum
+descriptions are likely to be interrelated. Correctness of this conj ecture is suc-
+cessfully checked by reproducing (by different methods) results o f Korotkin and
+Nicolai, Nucl.Phys.B475 (1996) 397-439, on dimensionally reduced qua ntum
+gravity. Consequently, numerous new results supporting the Fad deev-Skyrme
+(F-S) -type models are obtained. We found that the F-S-like model is best
+suited for description of electroweak interactions while strong inte ractions re-
+quire extension of Witten’s results to the SU(3) gauge group. Such an extension
+is nontrivial. It is linked with the symmetry group SU(3) ×SU(2)×U(1) of the
+Standard Model. This result is quite rigid and should be taken into acco unt in
+development of all grand unified theories. Also, the alternative (to the F-S-like)
+model emerges as by-product of such an extension. Both models a re related to
+each other via known symmetry transformation. Both models poss ess gap in
+their excitation spectrum and are capable of producing knotted/lin ked config-
+urations of gauge/gravity fields. In addition, the paper discusses relevance of
+the obtained results to heterotic strings and to scattering proce sses involving
+topology change. It ends with discussion about usefulness of this in formation
+for searches of Higgs boson.
+Keywords : ExtendedRicciflow; Bose-Einsteincondensation; Ernst,Landa u-
+Lifshitz, Gross-Pitaevski, Richardson-Gaudin equations; Einstein ’s vacuum and
+electrovacuumequations; Floer’stheory; instantons,monopoles ,calorons;knots,
+links and hyperbolic 3-manifolds; Standard Model; Higgs boson.
+Mathematics Subject Classifications 2010 . Primary: 83E99, 53Z05, 53C21,
+83E30,81T13,82B27;Secondary :82B23
+11 Introduction
+1.1 General remarks
+History of physics is full of situations when experimental observat ions lead to
+deep mathematical results. Discoveryof Yang-Mills (Y-M) fields in 19 54[1] falls
+out of this trend. Furthermore, if one believes that theory of the se fields makes
+sense, they should never be directly observed. To make sure that these fields
+do exist, it is necessary to resort to all kinds of indirect methods to probe them.
+Physically, the rationale for the Y-M fields is explained already in the or iginal
+Yang and Mills paper [1]. Mathematically, such a field is easy to understa nd.
+It is a non Abelian extension of Maxwell’s theory of electromagnetism. In
+1956 Utiyama [2] demonstrated that gravity, Y-M and electromagn etism can
+be obtained from general principle of local gauge invariance of the u nderlying
+Lagrangian. The explicit form of the Lagrangian is fixed then by assu mptions
+about its symmetry. For instance, by requiring invariance of such a Lagrangian
+with respect to the Abelian U(1) group, the functional for the Max well field
+is obtained, while doing the same operations but using the Lorentz gr oup the
+Einstein-Hilbert functional for gravitational field is recovered. By employing
+the SU(2) non Abelian gauge group the original Y-M result [1] is recov ered.
+Only Maxwell’s electromagnetic field is reasonably well understood bot h at
+the classical and quantum level. Due to their nonlinearity, the Y-M fie lds are
+much harder to study even at the semi/classical level. In particular , no classical
+solutions e.g. solitons (or lumps) with finite action are known in Minkows ki
+space-time. This result was proven by many authors, e.g. see [3-4] and refr-
+erences therein. The situation changes dramatically in Euclidean spa ce where
+the self-duality constraint allows to obtain meaningful classical solu tions [5,6].
+These are helpful for development of the theory of quantum Y-M fi elds. Such
+solutions are useful in the fields other than quantum chromodynam ics (QCD)
+since the self-duality equations are believed to be at the heart of all exactly
+integrable systems [7]. Although the self-duality equations originate from study
+of the Y-M functional, not all solutions [6] of these equations are re levant to
+QCD. In this paper we discuss the rationale behind the selection proc edure.
+In QCD solutions of self-duality equations, known as instantons , are describ-
+ing tunneling between different QCD vacua [8]. It should be noted thou gh
+that treatment of instantons in mathematics [9-11] and physics lite rature [8]
+is different. This fact is important. It is important since one of the ma jor
+tasks of nonperturbative QCD lies in developing mathematically corre ct and
+physically meanigful description of these vacua. According to a poin t of view
+existing in physics literature the QCD has a countable infinity of topolo gically
+different vacua. Supposedly, the Faddeev-Skyrme (F-S) model is designed for
+description of these vacua. If this model can be used for this purp ose, then
+2each vacuum state is expected to be associated with a particular kn ot (or link)
+configuration. Under these conditions the instantons are believed to be well
+localized objects interpolating between different knotted/linked va cuum config-
+urations [12-16]. These configurations upon quantization are expe cted to posses
+a tower of excited states. Whether or not such a tower has a gap in its spec-
+trum or the spectrum is gaplessis the essence ofthe millennium prize p roblem1.
+Originally, the above results were obtained and discussed only for SU (2) gauge
+fields [17]. They were extended to SU(N) case, N≥2,only quite recently [18]2.
+Although such a description of QCD vacua is in accord with general pr inciples
+of instanton calculations [8], it is in formaldisagreement with results known in
+mathematics [9-11]. Indeed, it is well known that complement of a par ticular
+knot inS3is 3-manifold. Since instantons ”live” in R4(or any Riemannian
+4-manifold allowing an anti self -dual decomposition of the Y-M field (e .g. see
+Ref.[9], pages 38-393), this means that all knots in R4(orS4) are trivial and
+one should talk about knotted spheres instead of knotted rings [20 ]. This known
+topological fact is in apparent contradiction with results of [13-15]. In this work
+we shall provide evidence that such a contradiction is only apparent and that,
+indeed, knotted configurations in S3are consistent with the notion of instan-
+tons as formulated in mathematics. This is achieved by using results b y Floer
+[21]. It should be noted, though, that known to us ”proofs” [22-24 ] of the
+existence of the mass gap in pure Y-M theory done at the physical le vel of rigor
+ignore instanton effects altogether. Among these papers only Ref .[22] uses the
+F-S SU(2) model for such mass gap calculations. It also should be no ted that
+results of such calculation sensitively depend upon the way the F-S m odel is
+quantized. For instance, in the work by Faddeev and Niemi [25], done for the
+SU(2) gauge group, the results of quantization produce gaplesss pectrum. To fix
+the problem the same authors suggested to extend the original mo del in ad hock
+fashion. Other authors, e.g. see Ref.[26], proposed different ad ho c solution of
+the same problem.
+The above results are formally destroyed by the effects of gravity . Indeed, in
+1988 Bartnik and McKinnon numerically demonstrated [27] that the c ombined
+Y-M and gravity fields lead to a stable particle-like (solitonic) solutions while
+neither source-freegravitynor pure Y-M fields are capable of pro ducing such so-
+lutions4. Such situation has interesting cosmological ramifications5[28] causing
+disappearance of singularities in spacetime as shown by Smoller et al [2 9]. In
+this work we do not discuss implications of these remarkable results. Instead, in
+the spirit of Floer’s ideas [21], we argue that even without taking thes e results
+into account, the effects of gravity on processes of high energy p hysics are quite
+substantial.
+1E.g. see
+http://www.claymath.org/millennium/Yang-Mills Theory/
+2In this work, in accord with experimental evidence, we demon strate that N≤3.
+3In physics literature, both anti and self dual instantons ar e allowed to exist, e.g. see
+Ref.[19], page 481.
+4More accurately, neither pure Y-M fields nor pure gravity hav e nontrivial static globally
+regular (i.e.nonsingular, asymptotically flat) solitons.
+5E.g. Einstein-Y-M hairy black holes
+31.2 Statements of the problems to be solved
+Inthispaperseveralproblemsareposedandsolved. Inparticular ,wewouldlike
+to investigate the physics and mathematics behind gravity-Y-M cor respondence
+discovered by Louis Witten [30] for SU(2) gauge fields. Is this corre spondence
+accidental? If it is not accidental, how it should be related to commonly shared
+opinion that the Standard Model (SM) of particle physics does not a ccount
+for gravity? Can this correspondence be extended to other gaug e fields, e.g.
+SU(N), N>2 ? If the answer is ”yes”, will such correspondence be valid for all
+N’s or just for few? In the last case, what such a restriction means physically?
+Howthe noticed correspondenceis helping to solvethe gapproblem? What role
+the F-S model is playing in this solution? Is this model instrumental in s olving
+the gap problem or are there other aspects of this problem which th e F-S model
+is unable to account? How this correspondence affects known strin g-theoretic
+and loop quantum gravity (LQG) results ? What place the topology-c hanging
+(scattering) processes occupy in this correspondence? Is ther e any relevance of
+the results of this work to searches for Higgs boson?
+1.3 Organization of the rest of the paper and summary of
+obtained results
+Sections 2,3 and 6, and Appendix A are devoted to detailed investigat ion of
+gravity-Y-Mcorrespondence. Section4isdevotedtothephysics -styleexposition
+ofworksbyAndreasFloer[11, 21]on Y-Mtheorywith purposeofco nnectinghis
+mathematical formalism for Y-M fields with the F-S model. In the same section
+we also consider the Y-M fields monopole and instanton solutions and t heir
+meaning and place within Floer’s theory. Our exposition is based on res ults of
+Sections 2 and 3. Section 5 is entirely devoted to solution of the gap p roblem for
+pureY-M fields. Although the solution depends onresultsofpreviou ssections,
+numerous additional facts from statistical mechanics and nuclear physics are
+being used. In Section 6 we discuss various implications/corollaries of the
+obtained results, especially for the SM of particle physics. In Sectio n 7 we
+discuss possible directions for further research based on the res ults presented
+in this paper. These include (but not limited to): connections with the LQG,
+the role and place of the Higgs boson, relationship between real spa ce-time
+scattering processes of high energy physics and processes of to pology change
+associated with such scattering. Based on the results of this pape r, we argue
+that this task can be accomplished with help of the formalism develope d by G.
+Perelman for his proof of the Poincare′and geometrization conjectures.
+The major new results of this paper are summarized as follows.
+1. In subsection 5.4.4, while solving the gap problem, we reproduced b y
+employing entirely different methods, the main results of the paper b y Korotkin
+and Nicolai[31]on quantizingdimensionally reducedgravity. From the se results
+it follows that for gravity and Y-M fields possessing the same symmet ry the
+nonperrturbative quantization proceeds essentially in the same wa y.
+2. In subsection 6.3 we demonstrated that gravity-Y-M correspo ndence dis-
+4covered by L.Witten for gauge group SU(2) can be extended onlyto the SU(3)
+gauge group. This group contains SU(2) ×U(1) group as a subgroup. This fact
+allowedustocomeupwiththe anticipated(but neverproven!) conclu sionabout
+symmetry of the SM. It is given by SU(3) ×SU(2)×U(1). The obtained result is
+very rigid. It is deeply rooted into not widely known/appreciated (dis cussed in
+Appendix A) properties of the gravitational field. It is these prope rties which
+ultimately determine the conditions of gravity-Y-M correspondenc e.
+3. The latest papers Refs.[32-34] are aimed at reproduction of the classifi-
+cation scheme of particles and fields in the SM within the framework of LQG
+formalism. These results match perfectly with the results of our pa per be-
+cause of the noticed and developed gravity-Y-M correspondence . In view of this
+correspondence, the results of Refs.[32-34] can be reproduced with help of min-
+imalgravity model described in subsections 3.2, 3.4, and 7.2 . This minimal
+model has differential-geometric /topological meaning in terms of th e dynamics
+of the extended Ricci flow [35,36]. Such a flow is the minimal extension o f the
+Ricci flow now famous because of its relevance in proving the Poincar e′and
+geometrization conjectures.
+4. The formalism developed in this paper explains why using pure gravit y
+onecantalk aboutthe particle/fieldcontentofthe SM. Not onlyit isc ompatible
+with just mentioned LQG results but also with those, coming from non commu-
+tative geometry [37], where it is demonstrated that use of pure gra vity (that is
+”minimal model”) combined with 0- dimensional internal space is sufficie nt for
+description of the SM.
+2 Emergence of the Ernst equation in pure grav-
+ity and Y-M fields
+2.1 Some facts about the Ernst equation
+Study of static vacuum Einstein fields was initiated by Weyl in 1917. Co n-
+siderable progress made in later years is documented in Ref.[38]. To de velop
+formalism of this paper we need to discuss some facts about these s tatic fields.
+Following Wald [39], a spacetime is considered to be stationary if there is a one-
+parameter group of isometries σtwhose orbits are time-like curves e.g. see [40].
+With such group of isometries is associated a time-like Killing vector ξi.Fur-
+thermore, a spacetime is axisymmetric if there exists a one-parameter group of
+isometriesχφwhose orbits are closed spacelikecurves. Thus, a spacelike Killing
+vector field ψihas integral curves which are closed. The spacetime is station-
+ary and axisymmetric if it possesses both of these symmetries, provided that
+σt◦χφ=χφ◦σt.Ifξ= (∂
+∂t) andψ= (∂
+∂φ) so that [ξ,ψ] = 0,one can choose
+coordinates as follows: x0=t,x1=φ,x2=ρ,x3=z.Under such identification,
+the metric tensor gµνbecomes a function of only x2andx3.Explicitly,
+ds2=−V(dt−wdφ)2+V−1[ρ2dφ2+e2γ(dρ2+dz2)],(2.1)
+5where functions V,wandγdepend on ρandzonly. In the case when V=
+1,w=γ= 0,the metric can be presented as ds2=−(dt)2+(d˜s)2, where
+(d˜s)2=ρ2dφ2+dρ2+dz2(2.2)
+is the standard flat 3 dimensional metric written in cylindrical coordin ates. The
+four-dimensional set of vacuum Einstein equations Rij= 0 with help of metric
+given by Eq.(2.2) acquires the following form
+∇·{V−1∇V+ρ−2V2w∇w}= 0 (2.3a)
+and
+∇·{ρ−2V2∇w}= 0. (2.3b)
+Intheseequations ∇·and∇arethree-dimensionalflat(thatiswithmetricgiven
+byEq.(2.2)) divergenceand gradientoperatorsrespectively. In a ddition to these
+two equations, there are another two needed for determination o f factorγin the
+metric, Eq.(2.1). They require knowledge of Vandwas an input. Solutions
+of Eq.s(2.3) is described in great detail in the paper by Reina and Trev ers [41]
+with final result:
+(Reǫ)∇2ǫ=∇ǫ·∇ǫ. (2.4)
+This equation is known in literature as the Ernst equation. The comple x po-
+tentialǫis defined in by ǫ=V+iωwithVdefined as above and ωbeing an
+auxiliary potential whose explicit form we do not need in this work. As it was
+recognized by Ernst [42,43] such an equation can be also used for de scription of
+the combined Einstein-Maxwell fields. We shall exploit this fact in Sect ion 6.
+In Appendix A and in Section 6 we provide proofs that knowledge of st atic vac-
+uum solutions of the Ernst equation is necessary and sufficient for restoration of
+static Einstein-Maxwell fields.6Fields other than Y-M should be also restorable
+7. To proceed, we need to list several properties of the Ernst equa tion to be
+used below. First, following [41] and using prolate spheroidal coordin ates, the
+Ernst equation reproduces the Schwarzschild metric, and with ano ther choice
+of coordinates it reproduces the Kerr and Taub-NUT metric. Thus , the Ernst
+equation is the most general equation describing physically interest ing vac-
+uum spacetimes compatible with the Cauchy formulation of general r elativity
+[39,40,44,45]. Such a formulation is convenient staring point for quant ization
+of gravitational field via superspace formalism [39] leading to the Whe eler- De
+Witt equation, etc. Since in this work we advocate different approach to quan-
+tization of gravity, this topic is not being discussed further. Second, following
+Ref.[38], page 283, a stationary solution of Einstein’s field equations is called
+staticif the timelike Killing vector is orthogonal to the Cauchy surface. In s uch
+a case from the Table 18.1. of the same reference it follows that the Ernst
+6Surprisingly, upon changes of variables in these static sol utions, the exact results for
+propagating gravitational waves can be obtained as well.
+7This is so because each of these fields is a source of gravitati onal field which, in turn, can
+be eliminated locally. See Appendix A.
+6potentialǫis real. This observation allows us to simplify Eq.(2.4) considerably.
+For the sake of notational comparison with Ref.[38] we redefine the potential
+ǫ=V+iΦ.In the static case we have ǫ≡ −F≡ −e2u8.Using this result in
+Eq.(2.4) produces
+∆ρ,zu= 0, (2.5)
+where ∆ρ,zis flat Laplacian written in cylindrical coordinates defined by the
+metric, Eq.(2.2).
+2.2 Isomorphism between the SU(2) self-dual gauge and
+vacuum Einstein field equations
+This isomorphism was discovered by Louis Witten in 1979 [30]. His work wa s
+inspired by earlier works of Ernst [42] and Yang [46]. To our knowledge , since
+time when Ref.[30] was published such an isomorphism was left undevelo ped.
+In this paper we correct this omission in order to demonstrate that when both
+fields are mathematically indistinguishable, their quantization should p roceed
+in the same way. The result analogous to that discovered by Witten w as ob-
+tained using different arguments a year later by Forgacs, Horvath and Palla [47]
+and, in a simpler form, by Singleton [48]. These authors used essentia lly the
+paper by Manton, Ref.[49], in which it was cleverly demonstrated that the ’t
+Hooft-Polyakov monopole can be obtained without actual use of the auxiliary
+Higgs field. Both Refs.[47,48] and the original paper by Witten [30] use the
+axial symmetry of either gravitational or Y-M fields essentially. Only in this
+case it can be shown that the axisymmetric version of the self-dualit y equations
+obtained by Manton can be rewritten in the form of the Ernst equat ion. In
+the light of above information, following Ref.[5 ],we shall discuss briefly con-
+tributions of Yang and Witten. For this purpose, we need to conside r first the
+following auxiliary system of linearequations
+Ψx=XΨ;Ψt=TΨ. (2.6)
+HereΨx=∂
+∂xΨandΨt=∂
+∂tΨ.In this system XandTare square matrices
+of the same dimension and such that
+Xt−Tx+[X,T] = 0 (2.7)
+This result easily follows from the compatibility condition: Ψxt=Ψtx. The
+matrices XandTcan be realized as
+X=/parenleftbigg−iζ q(x,t)
+r(x,t)iζ/parenrightbigg
+,T=/parenleftbiggA B
+C−A/parenrightbigg
+(2.8)
+withζbeing a spectral parameter and, A,BandCbeing some Laurent poly-
+nomials in ζ.The above system can be extended to four variables x1,x2,t1,t2
+8The minus sign in front of Fis written in accord with conventions of Chapter 30.2 of the
+1st edition of Ref.[38].
+7in a simple minded fashion as follows
+(∂
+∂x1+ζ∂
+∂x2)Ψ= (X1+iX2)Ψ, (2.9a)
+(∂
+∂t1+ζ∂
+∂t2)Ψ= (T1+iT2)Ψ. (2.9b)
+In the most general case, the matrices X1,X2,T1,T2are made of functions
+which ”live” in C4.They are representatives of the Lie algebra sl(n,C) ofn×n
+trace-free matrices. The compatibility conditions for this case are equivalent to
+the self-duality condition for the Y-M fields associated with algebra sl(n,C).It
+is instructive to illustrate these general statements explicitly.
+InR4the (anti)self-duality condition for the Y-M curvature reads: ∗F=
+(−1)Fso that for the self-dual case we obtain:
+F01=F23,F02=F31,F03=F12. (2.10)
+In the ”light cone” coordinates σ=1√
+2(x1+ix2),τ=1√
+2(x0+ix3) the Y-M
+field one-form can be written as Aµdxµ=Aσdσ+Aτdτ+A¯σd¯σ+A¯τd¯τwith the
+overbarlabeling the complex conjugation. In such notations A0=1√
+2(Aτ+A¯τ),
+A1=1√
+2(Aσ+A¯σ),A2=1√
+2(Aσ−A¯σ),A3=1√
+2(Aτ−A¯τ).In these notations
+Eq.s (2.9) acquire the following form
+Fστ= 0, F¯σ¯τ= 0 andFσ¯σ+Fτ¯τ= 0. (2.11)
+They can be obtained as compatibility condition for the isospectral lin ear prob-
+lem
+(∂σ+ζ∂¯τ)Ψ= (Aσ+ζA¯τ)Ψand (∂τ−ζ∂¯σ)Ψ= (Aτ−ζA¯σ)Ψ,(2.12)
+where the spectral parameter is ζandΨis the local section of the Y-M fiber
+bundle. The compatibility condition reads: ( ∂σ−ζ∂¯τ)(∂σ+ζ∂¯τ)Ψ= (∂σ+
+ζ∂¯τ)(∂σ−ζ∂¯τ)Ψ,thus leading to
+[Fστ−ζ(Fσ¯σ+Fτ¯τ)+ζ2F¯σ¯τ]Ψ= 0. (2.13)
+This equation allows us to recover Eq.s(2.11). The first two equation s of
+Eq.s(2.11) can be used in order to represent the A-fields as follows: Aσ=
+(∂σC)C−1, Aτ=(∂τC)C−1, A¯σ=(∂¯σD)D−1andA¯τ= (∂τD)D−1,where
+bothCandDare some matrices in the Lie group G, e.g.G=SU(2). By
+introducing the matrix M=C−1D∈Gthe last of equations in Eq.(2.11)
+becomes
+∂¯σ(M−1∂σM)+∂¯τ(M−1∂τM) = 0. (2.14a)
+Thus, the self-duality conditions for the Y-M fields are equivalent to Eq.(2.14a).
+For the future use, following Yang [46], we notice that in such formalis m the
+gauge transformations for Y-M fields are expressible through D→DEand
+8C→CEso thatFσ¯σ→E−1Fσ¯σEandFτ¯τ→E−1Fτ¯τEwith the matrix
+E=E(σ,¯σ,τ,¯τ)∈SU(2) leaving self-duality Eq.s(2.10) (or (2.13)) unchanged.
+To connect Eq.(2.14a) with the Ernst equation, following L.Witten [30] it
+is sufficient to assume that the matrix Mis a function of ρ=/radicalbig
+x2
+1+x2
+2and
+z=x3.In such a case it is useful to remember that ρ2= 2σ¯σandz=i√
+2(τ−¯τ).
+With help of these facts Eq.(2.14a) can be rewritten as
+∂ρ(ρM−1∂ρM)+ρ∂z(M−1∂zM) = 0. (2.14b)
+By assuming that the matrix Mis representable by the SL(2,R)-type matrix,
+and writing it in the form
+M=1
+V/parenleftbigg1 Φ
+Φ Φ2+V2/parenrightbigg
+, (2.15)
+Eq.(2.14b) is reduced to the pair of equations
+V∇2V=∇V·∇V−∇Φ·∇Φ andV∇2Φ = 2∇V·∇Φ.
+With help of the Ernst potential ǫ=V+iΦ these two equations can be brought
+to the canonical form of the Ernst equation, Eq.(2.4). Below, we sh all provide
+sufficientevidencethatsuchareductionoftheErnstequationisco mpatiblewith
+analogous reduction in instanton/monopole calculations for the Y-M fields.
+3 From analysis to synthesis
+3.1 General remarks
+The results of previous section demonstrate that for axially symme tric fields
+both pure gravity and pure self-dual Y-M fields are described by th e same
+(Ernst) equation. In this section we reformulate these results in t erms of the
+nonlinear sigma model with purpose of using such a reformulation late r in the
+text. To do so we need to recallsome results from ourrecent work s, Ref.s[50,51].
+In particular, we notice that under conformal transformations ˆ g=e2ugind-
+dimensions the curvature scalar R(g) changes as follows:
+ˆR(ˆg) =e−2u{R(g)−2(d−1)∆gu−(d−1)(d−2)|▽gu|2}.(3,1)
+Since this equation is Eq.(2.11) of our Ref.[50] we shall be interested o nly in
+transformations for which ˆR(ˆg) is a constant. This is possible only if the total
+volume of the system is conserved. Under this constraint we need t o consider
+Eq.(3.1) for d= 3 in more detail. Without loss of generality we can assume that
+initiallyR(g) = 0. For this case we shall write g=g0so that Eq.(3.1) acquires
+the form
+ˆR(ˆg) =−2e−2u[2∆g0u+|▽g0u|2] (3.2)
+in which ∆ g0is the flat space Laplacian. Now we can formally identify it
+with that in Eq.(2.5). Accordingly, we shall be interested in such conf ormal
+9transformations for which ∆ g0u= 0 in Eq.(3.2). If they exist, Eq.(3.2) can be
+rewritten as
+e2uˆR(ˆg) =−2/parenleftBig
+/vector▽g0u/parenrightBig
+·/parenleftBig
+/vector▽g0u/parenrightBig
+. (3.3)
+This allows us to interpret Eq.(3.3) and
+∆g0u= 0 (3.4)
+as interdependent equations: solutions of Eq.(3.4) determine the s calar cur-
+vatureˆR(ˆg) in Eq.(3.3) .Clearly, under conditions at which these results are
+obtained only those solutions of Eq.(3.4) should be used which yield the con-
+stant scalar curvature ˆR(ˆg).Eq.(3.3) contains information about the Ricci
+tensor. To recover this information we notice that ˆ gij=−e2uδij. Therefore we
+obtain:
+ˆRij(ˆg) = 2∇iu∇ju, (3.5)
+inaccordwithEq.(18.55)ofRef.[38]wherethisresultwasobtainedby employing
+entirely different arguments. From the same reference we find tha t Eq.(3.4)
+comes as result of use of the contracted Bianci identities applied to ˆRij(ˆg)9.
+It is instructive to place the obtained results into broader context . This is
+accomplished in the next subsection.
+3.2 Connection with the nonlinear sigma model
+Some time ago Neugebauer and Kramer (N-K), Ref.[38], obtained Eq.s (3.4) and
+(3.5) usingvariationalprinciple. In less generalform this principle wa sused pre-
+viously by Ernst [42] resulting in now famous Ernst equation. Neugeb auer and
+Kramer proposed the Lagrangian and the associated with it action f unctional
+SN−Kproducing upon minimization both Eq.s(3.4)and (3.5). Todescribe the se
+results, we also use some results by Gal’tsov [52].
+The functional SN−Kis given by
+SN−K=1
+2/integraldisplay
+M/radicalbig
+ˆg[ˆR(ˆg)−ˆgijGAB(ϕ)∂iϕA∂jϕB]d3x, (3.6)
+easily recognizable as three-dimensional nonlinear sigma model coup led to 3-d
+Euclidean gravity. The number of components for the auxiliary field ϕas well
+as the metric tensor GAB(ϕ) of the target space is determined by the problem
+in question. In our case upon variation of SN−Kwith respect to ϕA
+iand ˆgijwe
+should be able to re obtain Eq.s(3.4) and (3.5). To do so, following Ref.[5 3], we
+introduce the current
+Ji=M−1∂iM. (3.7)
+In view of results of subsection 2.2, we have to identify the matrix Mwith that
+defined by Eq.(2.15) and, taking into account Eq.(2.14a), the index ishould
+9Ref.[38], page 283, bottom
+10take two values: σandτ.With such definitions we can replace the functional
+SN−Kby
+S=1
+2/integraldisplay
+M/radicalbig
+ˆg[ˆR(ˆg)−ˆgik1
+4tr(JiJk)]d3x. (3.8)
+The actual calculations with such type of functionals can be made us ing
+results of Ref.[53]. Thus, using this reference we obtain,
+ˆRij(ˆg) =1
+4tr(JiJj) (3.9)
+and
+∂iJi= 0. (3.10)
+Evidently, by construction Eq.(3.10) coincides with Eq.(2.14a) and, u ltimately,
+with Eq.(3.4). It is also easy also to check that Eq.(3.9) does coincide w ith
+Eq.(3.5). For this purpose it is sufficient to notice that
+tr(JiJj) =−tr(∂iM∂jM−1). (3.11)
+To check correctness of our calculations the entries of the matrix M, Eq.(2.15),
+can be restricted to V(that is we can put Φ = 0) .SinceV=−F≡ −e2u(e.g.
+see discussion prior to Eq.(2.5)), a simple calculation indeed brings Eq.( 3.9)
+back to Eq.(3.5) as required.
+It is interesting and important to observe at this point that the equa-
+tion of motion, Eq.(3.10), formally is not affected by effects of gravit y. This
+conclusion requires some explanation. From subsection 2.2, especia lly from
+Eq.s(2.14),(2.15), it should be clear that Eq.(3.10) is the Ernst equat ion deter-
+mining gravitational field. Hence, it is physically wrong to expect that it is
+going to be affected by the effects of gravity. Eq.s(3.9) and (3.10) a re the same
+as Eq.s(3.5) and (3.4) whose meaning was explained in the previous sub section.
+Clearly, the functional, Eq.(3.8), can be used for coupling of otherfields to
+gravity. This is indeed demonstrated in Ref.[52]. This is done with purpo se
+of connecting results for the nonlinear sigma models with those for h eterotic
+strings. We would like to discuss this connection now since it will be used later
+in the text.
+3.3 Connection with heterotic string models
+The functional S, Eq.(3.8), is related to that for the heterotic string model,
+e.g. see Ref.[54]. For such a model the sigma model-like functional is ob tainable
+from 10 dimensional supersymmetric string model by means of comp actifica-
+tion scheme (ideologically similar to that used in the Kaluza-Klein theory of
+gravity and electromagnetism) aimed at bringing the effective dimens ionality
+to physically acceptable values (e.g. 2, 3 or 4). For dimensionality D<10 such
+11compactified/reduced action functional reads (e.g. see Ref.[54], E q.(9.1.8)):
+Sheterotic
+D =/integraldisplay
+dDx√
+−detGe−2φ[R+4∂µφ∂µφ−1
+12ˆHµνρˆHµνρ
+−1
+4(M−1)ijFi
+µνFjµν+1
+8tr(∂µM∂µM−1)]. (3.12)
+The compactification procedure is by no means unique. There are ma ny ways
+to make a compactified action to look exactly like that given by Eq.(3.8) (e.g.
+see [55]). Evidently, there should be a way to relate such actions to each ot her
+since they all arehavingthe sameorigin- 10 dimensionalheterotic st ring action.
+Because of this, we would like to make some comments on action given b y
+Eq.(3.12) by specializing to D= 3 for reasons explained in Refs[ 68,69] and
+to be clarified below, in Section 6. Under such conditions if we require t he
+dilatonφ, the antisymmetric H-field (associated with string orientation) and
+the electromagnetic field Fto vanish,the remaining action will coincide with
+that given by Eq.(3.8). Because of this, the following steps can be ma de.
+First, asexplainedinourwork,Ref.[50],forclosed3-manifoldswecan /should
+dropthedilatonfield φ. Second,byproperlyselectingstringmodelwecanignore
+the antisymmetric field H. Third, by taking into account results of Appendix A
+we can also drop the electromagnetic field since it can be always resto red from
+pure gravity. Thus, we end up with the action functional S, Eq.(3.8), which
+we shall call ” minimal”. In Section 6 we shall provide evidence that its mini-
+mality is deeply rooted into gravity-Y-M correspondence which does not leave
+much room for ”improvements” abundant in physics literature. We s hall begin
+explaining this fact immediately below and will end our arguments in Sect ion
+6.
+3.4 The extended Ricci flow
+Thus far use of the variational principle apparently had not brough t us any
+new results (at least at the classical level). Situation changes in the light of
+recent work by List [35]. Following Ref.s[35,36 ], it is convenient to introduce
+Perelman-like entropy functional F(ˆgij,u,f)
+F(ˆgij,u,f) =/integraldisplay
+M(ˆR(ˆg)−2|∇ˆgu|2+|∇ˆgf|2)e−fdv (3.13)
+coinciding with Eq.(7.22b) of our work, Ref.[50], when u= 0.10. Because of
+this observation, if formally we make a replacement R(ˆg;u) =ˆR(ˆg)−2|∇u|2
+in Eq.(3.13), we are able to identify Eq.(3.13) with Perelman’s entropy f unc-
+tional enabling us to follow the same steps as were made in Perelman’s p apers
+aimed at proofof the geometrizationand Poincare′conjectures. Such a program
+10It should be noted that there is an obvious typographical err or in Eq.(7.22b): the term
+|∇hf|2is typed as |∇hf|.
+12was indeed completed in the PhD thesis by List [36]. Minimization of entro py
+functional F(ˆgij,f) produces the following set of equations
+∂
+∂tgij=−2(ˆRij+∇i∇jf) +4∇iu∇ju, (3.14a)
+∂
+∂tu= ∆ˆgu−(∇u)·(∇f), (3.14b)
+and∂
+∂tf=−ˆR−∆ˆgf+2|∇u|2, (3.14c)
+coinciding with Eq.s(7.28a), (7.28b) of our work, Ref.[50], when u= 0.In these
+equations |∇ˆgu|2= ˆgij∇iu∇ju, etc. From the next section and results below it
+follows that physically we should be interested in closed 3 manifolds. Fo r such
+manifolds onecan use Lemma 2.13, provenby List [36], which canbe for mulated
+as follows:
+Let ˆg,u,fbe a solution of Eq.s(3.14) for t∈[0,T) on a closed manifold M.
+Then the evolution of the entropy is given by
+∂tF(ˆgij,u,f) = 2/integraldisplay
+M[|Rij(ˆg;u)+∇i∇jf|2+2(∆ ˆgu−(∇u)·(∇f))2]e−fdv≥0.
+(3.15)
+Thus, the entropy is non decreasing with equality taking place if and o nly if the
+solution of Eq.(3.14) is a gradient soliton. This happens when the follow ing two
+conditions hold
+Rij(ˆg;u)+∇i∇jf= 0 and ∆ ˆgu−(∇u)·(∇f) = 0.(3.16)
+Foru= 0 the result of Perelman, Eq.(7.30) of Ref [50], for steady gradient
+soliton is reobtained, as required. Since for closed compact manifold sf=const
+Eq.s(3.16) coincide with Eq.s(3.4) and (3.5) as anticipated. Thus, existence
+of steady gradient solitons in the present context is equivalent to e xistence of
+solutions of static Einstein’s equations for pure gravity. This fact alone could
+be mathematically interesting but requires some reinforcement to b e of interest
+physically. We initiate this reinforcement process in the following subs ection.
+3.5 Relationship between the F-S and the Ernst function-
+als
+The F-S functional was mentioned in the Itroduction. In this subse ction we
+would like to initiate study of its connection with the Ernst functional. We
+begin with the following observation. In steps leading to Eq.(2.14b) (o r (3.10))
+the Euclidean time coordinate x0was eventually dropped implying that solu-
+tions of selfduality for Y-M equations, when substituted back into Y -M action
+functional, will produce physically meaningless (divergent) results. While in
+subsection 4.4 we discuss a variety of means for removing of such ap parent
+13divergence, in this subsection we notice that already Ernst [42] sug gested the
+action functional whoseminimization produces the Ernst equation. He gavetwo
+equivalent forms for such a functional, now bearing his name. These are either
+SE1[ǫ] =/integraldisplay
+Mdv∇ǫ·∇ǫ∗
+(Reǫ)2(3.17)
+or
+SE2[ξ] =/integraldisplay
+Mdv∇ξ·∇ξ∗
+(ξξ∗−1)2. (3.18)
+Minimization of SE1[ǫ] leads to Eq.(2.4) while functional, Eq.(3.18), is obtained
+fromSE1[ǫ] by means of substitution: ǫ= (ξ−1)/(ξ+1).In both functionals
+dvis 3-dimensional Euclidean volume element so that apparently the man ifold
+Mis justE3(or, with one point compactification, it is S3).Evidently,both
+SE1[ǫ] andSE2[ξ] are functionals for the nonlinear sigma model. If we drop
+the curvature term in Eq.(3.6) such truncated functional can be id entified, for
+example, with SE2[ξ]. This explains why Eq.(3.10) is formally unaffected by
+gravity. In mathematics literature the nonlinear sigma models are kn own as
+harmonic maps. Since Reina [56] demonstrated that the functional SE2[ξ]
+describes the harmonic map from S3toH2,it is not too difficult to write
+analogous functional SE3[ξ] describing the mapping from S3toS2.It is given
+by
+SE3[ξ] =/integraldisplay
+Mdv∇ξ·∇ξ∗
+(ξξ∗+1)2(3.19)
+and is part of the F-S model. If needed, both SE2[ξ] andSE3[ξ] can be supple-
+mented by additional (topological) terms which in the simplest case ar e wind-
+ing numbers. Thus, we shall be dealing either with the truncated F-S model,
+Eq.(3.19), or with its hyperbolic analog, Eq.(3.18). The choice betwee n these
+models is nontrivial and will discussed in detail in Section 6 .To facilitate this
+discussion, we need to observe the following. In the static case, we argued, e.g.
+see Eq.(2.5), that ǫ=−F=−e2u.Substitution of this result back into SE1[ǫ]
+produces (up to a constant) the following result:
+˜SE1[ǫ] =/integraldisplay
+Mdv∇u·∇u (3.20)
+leading to Eq.(2.5) as anticipated. At the same time, consider the follo wing H-E
+action functional
+SH−E[ˆg] =/integraldisplay
+Mdv/radicalbig
+ˆgˆR(ˆg), (3.21)
+and takeinto accountEq.(3.3)and the fact that ˆ gij=−e2uδij. Straightforward
+calculation leads us then to the result (up to a constant):
+SH−E[ˆg] =−/integraldisplay
+Mdv∇u·∇u. (3.22)
+14The minus sign in front of the integral is important and will be explained be-
+low. Before doing so, we notice that the Ernst functional (in whate ver form)
+is essentially equivalent to the H-E functional! Since in the original pap er by
+ErnstMisE3(orS3),apparently, such a functional should be zero. This is
+surely not the case in general but the explanation is nontrivial. Supp ose that
+minimization of the Ernst functional leads to some knotted/linked st ructures11.
+If such knots/links are hyperbolic then, by construction, complem ents of these
+knots/links in S3areH3modulo some discrete group. This conclusion is in
+accord with properties of the Ernst equation discovered by Reina a nd Trevers
+[41]. Following this reference, we introduce the complex space C×C=C2so
+that∀z= (u,v)∈C2the scalar product z∗
+αzαcan be made with the metric
+παβ=diag{1,−1}. Furthermore, the Ernst Eq.(2.4) can be rewritten with help
+of substitution ǫ= (u−v)/(u+v) as the set of two equations
+zαz∗
+α∇2zβ= 2z∗
+α∇zα·∇zβ. (3.23)
+Such a system of equations is invariant with respect to transforma tions from
+unimodular group SU(1,1) which is equivalent to SL(2, C). But SL(2, C) is the
+group of isometries of hyperbolic space H3as was discussed extensively in our
+work,Ref.[57]. Thus, minimization ofboth the F-S andErnstfunction alsshould
+account for knotted/linked structures. This conclusion is streng thened in the
+next subsection.
+3.6 Relationship between the Ernst and Chern-Simons
+functionals
+Even though we need to find this relationship anticipating results of t he next
+section, by doing so, some unexpected connections with previous s ubsection
+are also going to be revealed. For this purpose, we notice that for u= 0 the
+functional F(ˆgij,u,f) introduced earlier is just Perelman’s entropy functional.
+As such, it was discussed in our work, Ref.[50]. Evidently, both Fand Perel-
+man’s functional can be used for study of topology of 3-manifolds. We believe,
+though, that use of Perelman’s functional is more advantageous a s we would
+like to explain now. For this purpose, it is convenient to introduce the Raleigh
+quotientλgvia
+λg= inf
+ϕ/integraltext
+MdV(4|∇ϕ|2+R(g)ϕ2)
+/integraltext
+MdVϕ2, (3.24)
+e.g. see Eq.(7.24)of [50], to be compared against the Yamabe quotien t (p=2d
+d−2
+andα= 4d−1
+d−2) .
+Yg=/integraltextddx√ˆgˆR(ˆg)
+/parenleftbig/integraltext
+ddx√ˆg/parenrightbig2
+p=/parenleftbigg1/integraltext
+Mddx√gϕp/parenrightbigg2
+p/integraldisplay
+Mddx√g{α(∇gϕ)2+R(g)ϕ2} ≡E[ϕ]
+/ba∇dblϕ/ba∇dbl2
+p
+11We shall postpone detailed discussion of this topic till Sec tion 6.
+15also discussed in [50]. Because of similarity of these two quotients the question
+arises: Can they be equal to each other? The affirmative answerto this question
+is obtained in Ref.[58]. It can be formulated as
+Theorem [58]. Suppose that γis a conformal class on Mwhichdoes not
+contain metric of positive scalar curvature. Then
+Yγ= sup
+g∈γλgV(g)2
+d≡¯λ(M), (3.25a)
+where¯λ(M) is Perelman’s ¯λinvariant. Equivalently,
+λgV(g)2
+d≤Yγ, (3.25b)
+whereV(g) =/integraltext
+ddx√ˆgis the volume.
+The equality happens when gis the Yamabe minimizer. It is metric of
+unit volume for manifold Mof constant scalar curvature (which, according to
+theorem above, should be negative so that Mis hyperbolic 3-manifold). Only
+for hyperbolic 3-manifolds whose Yamabe invariant Y−(M) = supγYγthe
+gravitational Cauchy problem for source-free gravitational field is well posed
+[45,46]. For gwhich is Yamabe minimizer we have SH−E[ˆg]≤Yγ.This result
+can be further extended by noticing that NSH−E[ˆg] =CS(A),whereNis some
+constant whose value depends upon the explicit form of the gauge fi eldA,and
+CS(A) is the Chern-Simons invariant to be described in the next section.
+To demonstrate that NSH−E[ˆg] =CS(A) it is sufficient to use some re-
+sults from works by Chern and Simons [59] and by Chern [60]. In [59] it w as
+proven that: a) the Chern-Simons (C-S) functional CS(A) (to be defined in
+next section) is a conformal invariant of M(Theorem 6.3. of [59]) and, b) that
+the critical points of CS(A) correspond to 3-manifolds which are (at least lo-
+cally) conformally flat (Corollary 6.14 of [59]). Subsequently, these r esults were
+reobtained by Chern, Ref.[60], in much simpler and more physically sugg es-
+tive way. In view of these facts, at least for Yamabe minimizers we ob tain,
+CS(A) =NSY[ϕ],whereNis some constant (different for different gauge
+groups). That this is the case should come as not too big of a surpris e since
+for Lorentzian 2+1 gravity Witten, Ref.[61], demonstrated the equ ivalence of
+the Hilbert-Einstein and C-S functionals without reference to resu lts of Chern
+and Simons just cited. At the same time, the Euclidean/Hyperbolic 3d gravity
+was discussed only much more recently, for instance, in the paper b y Gukov,
+Ref.[62]. To avoid duplications we refer our readers to these papers for further
+details.
+4 Floer-style nonperturbative treatment of Y-
+M fields
+4.1 Physical content of the Floer’s theory
+Strikingresemblancebetweenresultsof nonperturbativetreatm entof4-dimensional
+Y-M fields and two dimensional nonlinear sigma model at the classical le vel is
+16well documented in Ref.[63 ]. Zero curvature equations, e.g. Eq.(2.7), can be
+obtained either by using the two- dimensional nonlinear sigma model o r three-
+dimensional C-S functional. As discussed in previous section, the se lf-duality
+condition for Y-M fields also leads (upon reduction) to zero curvatu re condition.
+Since the Ernst equation describing static gravitational (and elect rovacuum)
+fields is obtainable both from conditions of self-duality for the Y-M fie ld and
+from minimization of 3-dimensional nonlinear sigma model, it follows that 3-d
+gravitational nonlinear sigma model, Eq.(3.8), contains nonperturb ative infor-
+mation about Y-M fields. Furthermore, in view of results of Appendix A, it
+also should contain information about the static electromagnetic fie lds, for the
+combined gravitational and electromagnetic waves and, with minor m odifica-
+tions, for the combined gravitational, electromagnetic and neutrin o fields. The
+nonperturbative treatment of Y-M fields is usually associated eithe r with the in-
+stanton ormonopole calculations. This observation leads to the con clusion that,
+at least in some cases, zero curvature equation should carry all no nperurbative
+information about Y-M fields. This point of view is advocatedand deve loped by
+Floer [11,21 ]. Below,we shall discuss Floer’s point of view now in the language
+used in physics literature. For the sake of illustration, it is convenien t to present
+our arguments for Abelian Y-M (that is electromagnetic) fields first .
+The action functional Sin this case is given by12
+S=1
+2t/integraldisplay
+0dt/integraldisplay
+Mdv[E2−B2], (4.1)
+whereB=∇×AandE=−∇ϕ−∂
+∂tA,ϕ≡A0.It is known that, at least
+for electromagnetic waves, it is sufficient to put A0= 0 (temporal gauge). In
+such a case the above action can be rewritten as
+S[A] =1
+2t/integraldisplay
+0dt/integraldisplay
+Mdv[˙A2−(∇×A)2], (4.2)
+where˙A=∂
+∂tA.From the condition δS/δA= 0 we obtain∂E
+∂t=∇×B. The
+definition of Bguarantees the validity of the condition ∇·B= 0 while from the
+definition of Ewe get another Maxwell equation∂B
+∂t=−∇×E. The question
+arises: will these results imply the remaining Maxwell’s equation ∇·E= 0
+essential for correct formulation of the Cauchy problem? If such a constraint
+satisfied at t= 0,naturally, it will be satisfied for t>0. Unfortunately, for t= 0
+the existence of such a constraint does not follow from the above e quations and
+should be introduced as independent. This causes decomposition of the field
+AasA=A/bardbl+A⊥.Taking into account that E=−∂
+∂tA,we obtain as well
+∇·(E/bardbl+E⊥).Then, by design ∇·E⊥= 0,while∇·E/bardblremains to be defined
+by the initial and boundary data. Because of this, it is alwayspossible to choose
+A/bardbl= 0 and to use only A⊥for description of the field propagation [64]. Hence,
+12Up to an unimportant scale factor.
+17the action functional Scan be finally rewritten as
+S[A⊥] =1
+2t/integraldisplay
+0dt/integraldisplay
+Mdv[˙A2
+⊥−(∇×A⊥)2]. (4.3)
+In such a form it can be used as action in the path integrals, e.g. see R ef.[64],
+page 152, describing free electromagnetic field. Such path integra l can be eval-
+uated both in Minkowski and Eucldean spaces by the saddle point met hod.
+There is, however, a closely related method more suitable for our pu rposes. It
+is described in the monograph by Donaldson, Ref.[11]. Following this ref erence,
+we replace time variable tby−iτin the functional S[A⊥] .Consider now this
+replacement in some detail. We have13
+1
+2T/integraldisplay
+0dτ/integraldisplay
+Mdv[˙A2
+⊥+(∇×A⊥)2]
+=1
+2T/integraldisplay
+0dτ(/integraldisplay
+Mdv[[˙A⊥+(∇×A⊥)]2−∂
+∂τ(A⊥·∇×A⊥)]).(4.4)
+Since variation of A⊥is fixed at the ends of τintegral, the last term can be
+dropped so that we are left with the condition
+∂
+∂τA⊥=−B⊥ (4.5)
+extremizing the Euclidean action SE[A⊥].The above results are transferable
+to the non Abelian Y-M field by continuity and complementarity. Since in
+the Abelian case fields EandBare dual to each other, by applying the curl
+operator to both sides of Eq.(4.5) (and removing the subscript ⊥) we obtain
+the equivalent form of self-duality equations in accord with those on page 33 of
+Ref.[6]. This calculation provides an independent check of Donaldson’s method
+of computation. Since the (anti)self-duality condition in the Abelian c ase can
+be written as B=∓E[9].and since E=−∂
+∂τA, we conclude that Eq.(4.5)
+is the self-duality equation. This conclusion is immediately transferab le to the
+non Abelian Y-M case where the analog of Eq. (4.5) is
+∂
+∂τA=∗F(A(τ)), (4.6)
+in accord with Floer. The symbol * denotes the Hodge star operatio n in 3 di-
+mensions. Following Donaldson [11] this result should be understood a s follows.
+Introduce a connection matrix A=A0dτ+3/summationtext
+i=1Aidxisuch that both A0andAi
+depend upon all four variables τ,x1,x2andx3.In the temporal gauge A0should
+13We shall assume (without loss of generality) that ˙A⊥is collinear with A⊥.
+18be discarded so that τbecomes a parameter in the remaining A′
+is.Evidently,
+it can be associated with the spectral parameter (e.g. see previou s section).
+The Hodge star operator in Eq.(4.6) is needed to make this equation a s an
+equation for one-forms The obtained results fit nicely into Cauchy f ormulation
+of dynamics of both Y-M and gravity. Indeed, under conditions ana logous to
+that discussed in [45,46]the space-time (4-manifold) is decomposab le into direct
+productM×R(a trivial fiber bundle) in such a way that all differential op-
+erations acting on 4-manifold are been projected down to 3-manifo ldM. This
+is essential part of Floer’s theory. Furthermore, since δCS(A)/δA=F(A) the
+above Eq.(4.6) can be equivalently rewritten as
+∂
+∂τA=∗[δCS(A)/δA] (4.7)
+so that the Chern-Simons functional is playing a role of a ”Hamiltonian ” in
+Eq.s(4.7). From the theory of dynamical systems it follows then tha t the dy-
+namics is taking place between the points of equilibria defined by zero c urvature
+condition F(A) = 0.At the same time, using our work, Ref.[50], it is easily rec-
+ognize Eq.(4.7) as an equation for the gradient flow, e.g. see Eq.s(3.1 4). For the
+sake of space we shall not discuss this topic any further. Interes ted readers are
+encouraged to consult Ref.[65]. For supersymmetric Y-M fields part icipating
+in Seiberg-Witten theory the gradient flow equations are discussed in detail in
+Ref.[66]
+The mechanical system described by Eq.(4.7) should be eventually qu an-
+tized. Since the quantization procedure is outlined in Ref.[67], to avoid du-
+plications, we shall concentrate attention of our readers on aspe cts of Floer’s
+theory not covered in [67] but still relevant to this paper. To do so, we follow
+Donaldson [11]. This is accomplished in several steps.
+First, in the previous section we noticed that the axially symmetric self-du al
+solution for Y-M fields does not depend on x0(orτ) variable. Therefore, if such
+solution is substituted back into Y-M functional, it produces diverge nt result.
+Although the cure for this issue is discussed in subsection 4.4, in this s ubsection
+we provide needed background. For this purpose, following Ref.[68] we consider
+the Y-M action S[F] for the pure Y-M field14
+S[F] =−1
+8/integraldisplay
+R4d4xtr(FµνFµν). (4.8)
+The duality condition15∗Fµν=1
+2εµναβFαβallows us then to rewrite this action
+as follows
+S[F] =−1
+16/integraldisplay
+R4d4x[tr((Fµν∓∗Fµν)(Fµν∓∗Fµν))±2tr(Fµν∗Fµν)] (4.9)
+14Strictly following notations of Ref.[68] we do not indicate that in general the integration
+should be made over some 4-manifold M. In physics literature, and inEq.s(2.11), itisassumed
+that we are dealing with R4(orS4upon compactification). In Floer’s theory it is essential
+that the 4-manifold is decomposable as M×R. This decomposition should be treated with
+care as described in the Donaldson’s book [11]
+15With the convention that ε1234=−1.
+19sincetr(FµνFµν) =tr(∗Fµν∗Fµν).The winding number Nfor SU(2) gauge
+field is defined as16
+N=−1
+8π2/integraldisplay
+R4d4xtr(Fµν∗Fµν)≡ −1
+8π2/integraldisplay
+R4tr(Fµν∧Fµν) (4.10)
+so that use of this definition in Eq.s(4.8),(4.9) produces
+S[F]≥π2|N| (4.11)
+with the equality taking place when the (anti) self-duality condition (e .g. see
+Eq.(2.10) ) holds. In such a case the saddle point action is becoming ju st a
+winding number (up to a constant).
+Second, if our space-time 4-manifold Mcan be decomposed as M×[0,1],
+the following identity can be used [11]
+/integraldisplay
+M×[0,1]tr(Fµν∧Fµν) =/integraldisplay
+Mtr(A∧dA+2
+3A∧A∧A)/equalsdotsCS(A).(4.12)
+Here the symbol /equalsdotsmeans ”up to a constant”. The decomposition M×[0,1]
+reflects the fact that the C-S functional is defined up to mod Z. This ambiguity
+can be removed if we agree to consider C-S functional as a quotient R/Z. Ac-
+cordingly, this allows us to replace M×RbyM×[0,1].Details can be found
+in Ref.[11]. Thus, one way or another the winding number Nin Eq.(4.10) can
+be replaced by the Chern-Simons functional.
+Third, since the equation of motion for the C-S functional is zero curvat ure
+condition F= 0, i.e.
+dA+A∧A= 0, (4.13)
+implying that the connection Ais flat, we can use this result in Eq.(4.12) in
+order to rewrite it as (e.g. for SU(2))
+1
+8π2/integraldisplay
+Mtr(A∧dA+2
+3A∧A∧A) =−1
+24π2/integraldisplay
+Mtr(A∧A∧A).(4.14)
+For other groups the prefactor and the domain of integration will b e different
+in general.
+Fourth, zero curvature Eq.(4.13) involves connections which are function s of
+three arguments and a spectral/time parameter. In such setting minimization
+of Y-M functional is not divergent in view of Eq.(4.11).
+Fifth, the obtained result, Eq.(4.14), coincides with that known for the
+winding number for SU(2) instantons in physics literature[8,19] wher e it was
+obtained with help of entirely different arguments. It should be note d though
+that in spite of apparent simplicity of these results, actual calculat ions of C-S
+functionals (invariants) for different 3-manifolds are, in fact, ver y sophisticated
+[69,70]. In accord with Floer and Ref.[67], we conclude that nonpertur batively
+16We follows notations of Ref. [68] in which R4is actually standing for S4∈SU(2)
+20the 4-dimensional pure Y-M quantum field theory is a topological field theory
+of C-S type.
+Sixth, the isomorphism noticed by Louis Witten acquires now natural expla -
+nation. It becomes possible in view of results just presented, on on e hand, and
+the fact that NSH−E[ˆg] =CS(A) (previous section), on another. For fields
+with axial symmetry, equations of motion, Eq.(4.13), for gravity an d Y-M fields
+coincide.
+Seventh, the instantons in Floer’s theory are notthe same as considered
+in physics literature [8,19]. To understand this, we must take into acc ount
+that in Floer’s theory manifolds under consideration are 4-manifolds Mwith
+tubular ends. Such manifolds are complete Riemannian manifolds with fi nite
+number of tubular ends made of half tubes (0 ,∞) so that locally each such
+manifold lookslike Ui=Li×(0,∞) withLibeing a compact 3-manifold(called
+a”crossectionofatube”)and inumberingthetubes. Theclosureof M\/uniontextn
+i=1Ui
+is a compact manifold with boundary. If the crossection is S3,thenUis
+conformallyequivalentto apunctured ball B4\{0}.Thisimplies that amanifold
+Mwith tubular ends is conformally equivalent to a punctured manifold ˜M \
+{p1,...,pn}where˜Mis compact. The instanton moduli problem for Mis
+equivalent to that for the punctured manifold [11]. Recall that the m oduli space
+of instantons is defined as set of solutions of anti self-dual equat ions modulo
+gauge equivalence.
+Being armed with these definitions and taking into account that the ( anti)
+self-duality Eq.(4.7) we can interpret the instanton as a path conne cting one
+flat connection F= 0 at ”time” τ=−∞with another flat connection at ”time”
+τ=∞[11]. It is permissible for the path to begin at one flat connection, to
+wind around a tube (modulo gauge equivalence) and to end up at the s ame flat
+connection, Ref.[11], page 22. Evidently, this caseinvolves4-manifo lds with just
+one tubular end. Physically, each flat connection F= 0 represents the vacuum
+state so that the instantons discussed in the Introduction should be connecting
+different vacua. In this sense there is a difference between the inte rpretation of
+instantons in mathematics and physics literature. As in the case of s tandard
+quantum mechanics, only imposition of some additional physical cons traints
+permitsustoselectbetweenallpossiblesolutionsonlythosewhichar ephysically
+relevant. In the present context it is known that all exactly integr able systems
+are described by the zero curvature equation F= 0 [5,6 ]. It is also known
+that differences between these equations are caused in part by diff erences in a
+way the spectral parameter enters into these equations. Since f or the Floer’s
+instantons F/\e}atio\slash= 0,it means that the curvature Fshould be parametrized in such
+a way that the ”time” parameter should become a spectral parame ter when
+F= 0.In this work we do not investigate this problem17. Instead, we shall
+focus our attention on different vacua, that is on different (knot- like) solutions
+of zero curvature equation F= 0.18
+17See Ref.[7] for introduction into this topic.
+18A complement of each knot in S3is 3-manifold. Floer’s instantons are in fact connecting
+various three-manifolds. These 3- manifolds (with tubular ends) should be glued together to
+formM.The gluing procedure is extremely delicate mathematical op eration [11]. It is above
+214.2 The Faddeev-Skyrme model and vacuum states of the
+Y-M functional
+In the light of results just presented, we would like to argue that th e F-S model
+is indeed capable of representing the vacuum states of pure Y-M fie lds. For
+this purpose it is sufficient to recall the key results of the paper by A uckly and
+Kapitansky [71]. These authors were able to rewrite the Faddeev fu nctional
+E[n] =/integraldisplay
+S3dv{|dn|2+|dn∧dn|2} (4.15)
+in the equivalent form given by
+Eϕ[a] =/integraldisplay
+S3dv{|Daϕ|2+|Daϕ∧Daϕ|2}. (4.16)
+In this expression, the covariant derivative Daϕ=dϕ+[a,ϕ]. Evidently, Eϕ[a]
+acquires its minimum when ϕ=aand the connection becomes flat (that is
+covariant derivative becomes zero). Since this result is compatible w ith those
+discussed in previous subsection, it implies that, indeed, Faddeev’s m odel can
+be used for description of vacuum states for pure Y-M fields. The o nly question
+remains: Is this model the onlymodel describingQCDvacuum? In view ofEq.s
+(3.18),(3.19) it should be clear that this is not the case. The full expla nation
+is given below, in Sections 5,6. In addition, the disadvantage of the F- S
+model as such (that is without modifications) lies in the absence of ga p upon
+its quantization as was recognized already by Faddeev and Niemi in Re f.[25].
+In Sections 5,6 we shall eliminate this deficiency in a way different from t hat
+described in the Introduction (e.g. in Ref.s[25,26]). In the meantime, we would
+like to find the place for monopoles in our calculations.
+4.3 Monopoles and the Ernst equation
+4.3.1 Monopoles versus instantons
+To introduce notations and for the sake of uninterrupted reading , we need to
+describebrieflythealternativepointofviewattheresultsofprevio ussubsection.
+For this purpose, following Manton [49], we need to make a comparison between
+the Lagrangiansfor SU(2) Y-M and the Y-M-Higgs fields described r espectively
+by
+LY−M=−1
+4tr(FµνFµν) (4.17)
+and
+LY−M−H=−1
+4tr(FµνFµν)−1
+2tr(DµΦ·DµΦ)−λ
+2(1−Φ·Φ)2(4.18)
+the level of rigor of this paper. To imagine the connected sum of knots [20] is much easier
+than the connected sum of 3-manifolds. This sum has physical meaning discussed in Section
+6.
+22with covariant derivative for the Higgs field defined as DµΦ=∂µΦ+[Aµ,Φ]
+and with connection Aµusedtodefine the Y-M curvaturetensor Fµν=∂µAν−
+∂νAµ+[Aµ,Aν],provided that Φ=Φata,Aµ=Aa
+µta,and [ta,tb] =−2εabctc.
+Now the self-duality condition F=∗Fcan be equivalently rewritten as Fij=
+−εijkFk0with indices i,j,krunning over 1,2,3. Incidentally, in the temporal
+gaugethisresultisequivalenttoFloer’sEq.(4.6)Considernowthelimit λ→0in
+Eq.(4.18). In the Minkowski spacetime the field equations originating from the
+Y-M-Higgs Lagrangian can be solved by using the Bogomolny ansatz e quations
+Fij=−εijkDkΦin which A0= 0 (temporal gauge). Instead of imposing the
+temporal gauge condition, we can identify the Higgs field ΦwithA0so that
+the Bogomolny equations read now as follows:
+Fij=−εijkDkA0. (4.19)
+Bogomolny demonstrated that the Prasad-Sommerfield monopole s olution can
+be obtained using Eq.(4.19). Thus, any static (that is time-independ ent) so-
+lution of self-duality equations is leading to Bogomolny-Prasad-Somm erfield
+(BPS) monopole solution of the Y-M fields, provided that we interpre t the
+component A0as the Higgs field. Suppose now that there is an axial symme-
+try. Forgacs, Horvath and Palla [72] (FHP) demonstrated equivale nce of the
+set of axially symmetric Bogomolny Eq.s (4.19) to the Ernst equation. The
+static monopole solution is time-independent self-dual gauge field. B ecause of
+this time independence, its four-dimensional action is infinite(because of the
+time translational invariance) while that for instantons is finite. Furthermore,
+the boundary conditions for monopoles and instantons are differen t. The infin-
+ity problem for monopoles can be cured somehow by considering the m onopole
+dynamics [68] but this topic at this moment ”is more art than science” , e.g.
+read [68], page 309. For the same reason we avoid in this section talkin g about
+dyons (pseudo particles having both electric and magnetic charge) . Hence, we
+would like to conclude our discussion with description of more mathema tically
+rigorous treatments. By doing so we shall establish connections wit h results
+presented in previous sections
+4.4 Calorons .
+Calorons are instantons on R3×S1.From this definition it follows that, phys-
+ically, these are just instantons at finite temperature19. Calorons are related
+to both instantons on R4(orS4) and monopoles on R3(orS3).Heuristically,
+the large period calorons are instantons while the small period caloro ns are
+monopoles [73,74]. These results do not account yet for the fact th at both the
+Y-M action and the self-duality equations are conformally invariant. Atiyah
+[75]. noticed that the Euclidean metric can be represented either as
+ds2
+E=/parenleftbig
+dx1/parenrightbig2+/parenleftbig
+dx3/parenrightbig2+/parenleftbig
+dx3/parenrightbig2+/parenleftbig
+dx4/parenrightbig2(4.20a)
+19This explains the word ”caloron”.
+23or as
+ds2=r2
+R2[R2(/parenleftbig
+dx1/parenrightbig2+/parenleftbig
+dx2/parenrightbig2+(dr)2
+r2)+R2dϕ2] (4.20b)
+withRbeing some constant. The above representation involves polar r,ϕ
+coordinates in the ( x3,x4) plane thus implying some kind of axial symmetry.
+Since self-duality equations are conformally invariant, the prefact orr2
+R2can be
+dropped so that the Euclidean space R4becomes conformally equivalent to
+the product H3×S1.For such manifold the constant scalar curvature of the
+hyperbolic 3-space H3is−1/R2.Furthermore, the remaining term represents
+the metric on a circle of radius R. Following Ref.[73], let ( x1,x2,x3) be coor-
+dinates for the hyperbolic ball model of H3so that the radial coordinate be
+R=/radicalBig
+(x1)2+(x2)2+(x3)2. Let 0≤ R ≤R.Letτbe a coordinate on S1
+with period β,then the metric on H3×S1can be represented as
+ds2
+H=dτ2+Λ2(dR2+R2dΩ2), (4.21a)
+where Λ = (1 − R/R)−1anddΩ2is the metric on 2-dimensional sphere. If
+we introduce an auxiliary coordinate µ= (R/2)arctanh( R/R),and complex
+coordinate z=µ+iτ,the above hyperbolic metric can be rewritten as
+ds2
+H=dτ2+dµ2+Ξ2dΩ2(4.21b)
+with Ξ = ( R/2)sinh(2µ/R).By analogy with transition from Eq.(4.20a) to
+(4.20b)wecanproceedasfollows. Let r=/radicalBig
+(y1)2+(y2)2+(y3)2with(y1,y2,y3,y0)
+being coordinates on R4.By lettingt=y0the Euclidean metric can be written
+as usual, i.e.
+ds2
+E=dt2+dr2+r2dΩ2(4.22a)
+so that
+ds2
+H=ξ2ds2
+E, (4.22b)
+withξ= (R/2)[cosh(2µ/R)+cos(2τ/R)].This correspondencebetween R4/integerdivideR2
+andH3×S1is made with help of the mapping w=tanh(z/R) (withw=r+it
+andβ=πR).LetM=H3×S1(orH3×R)then, inviewofconformalinvariance,
+we can rewrite Eq.(4.8) as
+S[F] =−1
+8/integraldisplay
+Mtr(FµνFµν)Ξ2dτdµdΩ. (4.23)
+We have to rewrite the winding number, Eq.(4.10), accordingly. Since it is
+a topological invariant, this means that the self-duality equations m ust be ad-
+justed accordingly. For instance, for the hyperbolic calorons onH3×S1the
+self-duality equation reads
+F0i=1
+2ΛεijkFjk. (4.24)
+The action S[F] nowis finite with tr(FµνFµν)→0whenµ→ ∞.For hyperbolic
+instantons we have finite action with tr(FµνFµν)→0 whenµ2+τ2→ ∞.
+24The results just described match nicely with the results by Witten [76 ] on
+Euclidean SU(2) instantons invariant under the action of SO(3) Lie g roup. His
+results will be discussed in detail in the next section. Notice, that Eu clidean
+metric, Eq.(4.22a), becomes that for H2×S2if we rewrite it as
+ds2
+E=r2(dt2+dr2
+r2+dΩ2) (4.25a)
+and, as before, we drop the conformal factor r2so that it becomes
+ds2
+H=dt2+dr2
+r2+dΩ2. (4.25b)
+Interestingly enough, that results by Witten initially developed for H2×S2
+can be also used without change for H3×S1andH3×Rsince the action of
+SO(3)pullsbacktothesemanifolds[73]. Thisfactisofimportancesincesuchan
+extensionmakeshis resultscompatible with bothFloer’s method ofca lculations
+for Y-M fields and with results of Section 3. Omitting all details, the action,
+Eq.(4.23), is reduced to that known for two dimensional Abelian Ginzb urg-
+Landau (G-L) model ”living” on the hyperbolic 2 manifold Xcoordinatized by
+µandτwith the metric
+ds2
+H=dµ2+dτ2
+Ξ2. (4.26)
+Explicitly, such G-L action functional SG−Lis given by [73]
+SG−L=π
+2/integraldisplay
+Xdτdµ[Ξ2(∇×A)2+2|(∇+iA)φ|2+Ξ−2(1−|φ|2)] (4.27)
+withAandφbeing respectively the Abelian gauge and the Higgs fields, φ=
+φ1+iφ2so that|φ|2=φ2
+1+φ2
+2.This functional is obtained upon substitution of
+solution of the self-duality equations into the Y-M action functional, Eq.(4.23).
+We refer our readers to the original paper, Ref.[73] for details. In the limit
+β→ ∞the above functional coincides with that obtained by Witten [76]. The
+self-dualityequationsobtainedbyWittendescribeinstantonswhich liealongthe
+fixed axis while Fairlie, Corrigan, ’t Hooft, and Wilczek [77]developed an ansatz
+(CFtHWansatz)fortheself-dualityequationsproducinginstanto nsatarbitrary
+locations. Manton[78]demonstratedthatWitten’sandCFtHWmulti- instanton
+solutions are gauge equivalent while Harland [73,74 ]demonstrated how these
+instantons and monopoles can be obtained from calorons in various lim its. The
+obtained results provide needed background information for solut ion of the gap
+problem. This solution is discussed in the next section.
+5 Solution of the gap problem
+5.1 Idea of the proof
+By cleverly using symmetry of the problem Witten [76] reduced the no n Abelian
+Y-M action functional to that for the Abelian G-L model ”living” in the hyper-
+bolic plane. This is one of examples of the Abelian reduction of QCD discu ssed
+25in our paper, Ref.[79]. Vortices existing in the G-L model could be visua lized
+as made of some two-dimensional surfaces (closed strings) living in t he ambi-
+ent space-time. These are known as Nambu-Gotto strings. Their t reatment
+by Polyakov [80] made them to exist in spaces of higher dimensionality. In or-
+der for them to be useful for QCD, Polyakov suggested to modify s tring action
+by adding an extra (rigidity) term into string action functional. By do ing so
+the problem was created of reproducing Polyakov rigid string model from QCD
+action functional. The latest proposal by Polyakov [81] involves con sideration
+of spin chain models while that by Kondo [82] involves the F-S model der ived
+directly from QCD action functional. As explained in [79], in the case of scatter-
+ing processes of high energy physics one is confronted essentially w ith the same
+combinatorialproblems as were encountered at the birth ofquant um mechanics.
+In Ref.[83] we explained in detail why Heisenberg’s (combinatorial) met hod of
+developing quantum mechanical formalism is superior to that by Schr ¨ odinger.
+In Ref.[74] using these general results we demonstrated how the c ombinatorial
+analysis of scattering data leads to spin chain models as microscopic m odels
+describing excitation spectrum of QCD. Thus, the mass gap problem can be
+considered as already solved in principle. Nevertheless, in [ 94] such a solution
+is obtained ”externally”, just based on the rules of combinatorics. As with
+quantum mechanics, where atomic model is used to test Heisenberg ’s ideas,
+there is a need to reproduce this combinatorial result ”internally” b y using mi-
+croscopic model of QCD. For this purpose, we shall use the G-L fun ctional,
+Eq.(4.27). By analogy with the flat case, we expect that it can be rew ritten in
+terms of interacting vortices. In the present case, in view of Eq.(4 .26), vortices
+”live” not in the Euclidean plane but in 3+1 Minkowski space-time. This is
+easy to understand if we recall the SO(3) ⇄SU(2) correspondence and take into
+account the analogous correspondence between SU(1,1) and SO( 2,1).
+Within such a picture it is sufficient to look at evolution dynamics of the
+individual vortex. Typically, it is well described by the dynamics of the continu-
+ousHeisenbergspin chainmodel [84,85]in Euclidean space. In the pre sent case,
+this formalism should be extended to the Minkowski space and, even tually, to
+hyperbolic space (that is to the case of Abelian model discovered by Witten).
+Details of such an extension are summarized in Appendix B. After tha t, the
+next task lies in connecting these results with the Ernst equation. I n the next
+subsection we initiate this study.
+5.2 Heisenberg spin chain model and the Ernst equation
+For the sake of space, this subsection is written under assumption that our read-
+ers are familiar with the book ”Hamiltonian methods in the theory of so litons”
+[86] (or its equivalent) where all needed details can be found. The co ntin-
+uous XXX Heisenberg spin chain is described with help of the spin vecto r20
+20In compliance with [86] we suppress the time-dependence.
+26/vectorS(x) = (S1(x),S2(x),S3(x)) restricted to live on the unit sphere S2:
+/vectorS2(x) =3/summationdisplay
+i=1S2
+i(x) = 1 (5.1)
+while obeying the equation of motion
+∂
+∂t/vectorS=/vectorS×∂2
+∂x2/vectorS (5.2)
+known as the Landau-Lifshitz (L-L) equation. By introducing matr icesU(λ)
+andV(λ) via
+U(λ) =λ
+2iS,V(λ) =iλ2
+2S+λ
+2S∂
+∂xS,S=/vectorS·/vector σ (5.3)
+so thatσiis one of Pauli’s spin matrices and λis the spectral parameter and
+requiring that S2=I,whereIis the unit matrix, the zero curvature condition
+∂
+∂tU−∂
+∂xV+[U,V] = 0 (5.4)
+is obtained. With account of the constraint S2=Iit can be converted into
+equation
+∂
+∂tS=1
+2i[S,∂2
+∂x2S] (5.5)
+equivalent to Eq.(5.2). The correspondence between Eq.s(5.2) and (5.5) can be
+made forS(x,t) matrices of arbitrary dimension.
+Having in mind Witten’s result [76], we want now to extend these Euclidea n
+results to the case of noncompact Heisenberg spin chain model ”livin g” either
+in Minkowski or hyperbolic space. In doing so we follow, in part, Ref.[ 56] and
+Appendix B. For this purpose we need to remind our readers some fa cts about
+the Lie group SU(1,1). Since this group is related to SO(2,1), very mu ch like
+SU(2) is related to SO(3), we can proceed by employing the noticed a nalogy.
+In particular, since S=/vectorS·/vector σ,we can preserve this relation by writing now
+S=/vectorS·/vector τ.Using this result we obtain,
+S=/parenleftbiggSziS−
+iS+−iSz/parenrightbigg
+∈su(1,1), S±=Sx±iSy, (5.6)
+where the form of matrices generating su(1,1) Lie algebra is similar to that
+for Pauli matrices. This time, however, det S=−1 even though S2=I.
+Explicitly, ( Sz)2−(Sx)2−(Sy)2= 1,that is the motion is taking place on the
+unit pseudosphere S1,1.Matricesτigeneratingsu(1,1) are fully characterized
+by the following two properties
+tr(τατβ) = 2gαβ, [τα,τβ] = 2ifαβγτγ;gαβ=diag(−1,−1,1);α,β,γ= 1,2,3
+(5.7)
+27withfαβγbeing structure constants for su(1,1) algebra. An analog of the
+equation of motion, Eq.(5.5), now reads
+∂
+∂tSα=/summationdisplay
+β,γfαβγSβ∂2
+∂x2Sγ. (5.8)
+If we defines the Poisson brackets as {Sα(x),Sβ(y)}=−fαβγSγ(x)δ(x−y),
+then the above equation of motion can be rewritten in the Hamiltonian form
+∂
+∂tSα={H,Sα}, (5.9)
+provided that the Hamiltonian His given by
+H=1
+2∞/integraldisplay
+−∞dx(∇xSα)gαβ/parenleftbig
+∇xSβ/parenrightbig
+≡1
+4tr∞/integraldisplay
+−∞dx(∇xS)2. (5.10)
+Since now the motion takes place on pseudosphere ˇS2, it is convenient to intro-
+duce the pseudospherical coordinates by analogy with spherical, e .g.
+Sx(x,t) = sinhθ(x,t)cosϕ(x,t),Sy(x,t) = sinhθ(x,t)sinϕ(x,t),Sz(x,t) = coshθ(x,t).
+(5.11)
+Also, by analogy with spherical case we can use the stereographic p rojection
+: from pseudosphere to hyperbolic plane. Recall [ 102], that in the case of a
+sphereS2the inverse stereographic projection: from complex plane CtoS2is
+given by
+S+=2z
+1+|z|2,S−=2z∗
+1+|z|2,Sz=1−|z|2
+1+|z|2. (5.12)
+The mapping from CtoH2is obtained with help of Eq.(5.12) in a straightfor-
+ward way as
+S+=2ξ
+1−|ξ|2,S−=2ξ∗
+1−|ξ|2,Sz=1+|ξ|2
+1−|ξ|2. (5.13)
+Using this correspondence the equations of motion, Eq.(5.10), rew ritten in
+terms ofξandξ∗variables (while keeping in mind that they are parametrized
+byxandt) are given by
+i∂
+∂tξ+∂2
+∂x2ξ+2ξ∗
+1−|ξ|2/parenleftbigg∂
+∂xξ/parenrightbigg2
+= 0. (5.14)
+In the static ( t−independent) case the above equation is reduced to
+(|ξ|2−1)∇2
+xξ= 2ξ∗(∇xξ)2(5.15)
+easily recognizable as the Ernst equation. In his paper, Ref. [42], Er nst used
+variational principle applied to the functional Eq.(3.18). From Appen dix B we
+28know that both the L-L equation and its hyperbolic version describe the motion
+of (could be knotted) vortex filament. Because ofthis, the funct ional, Eq.(3.18),
+should undergo the same reduction as was made in going from Eq.(B1.a ) to
+(B1.b). Explicitly, this means that the functional, Eq.(3.18), should b e reduced
+in such a way that the Hamiltonian, Eq.(5.10), should be replaced by
+H=−2∞/integraldisplay
+−∞dx|∇xξ|2
+(1−|ξ|2)2, (5.16)
+where the sign in front is chosen in accord with Ref.[87] and our Eq.(3.2 2).
+The Hamiltonian equation of motion, Eq.(5.9), reproducing Eq.(5.14) c an be
+obtained if the Poisson bracket is defined as by {ξ(x),ξ∗(y)}= (1−|ξ|2)2δ(x−
+y).The obtained results set up the stage for quantization. It will be dis cussed in
+subsection 5.4. In the meantime, we need to connect results of Witt en’s work,
+Ref.[76], with those we just obtained.
+5.3 From Abelian Higgs to Heisenberg spin chain model
+5.3.1 The Abelian Higgs model
+The work by Witten [76] had been further analyzed in the paper by Fo rgacs
+and Manton [88]. The major outcome of their work lies in demonstratio n
+of uniqueness of the self-duality ansatz proposed by Witten. The s elf-duality
+equations obtained in Witten’s work are reduced to the system of th ree coupled
+equations describing interaction between the Abelian Y-M and Higgs fi elds
+∂0ϕ1+A0ϕ2=∂1ϕ2−A1ϕ1, (5.17a)
+∂1ϕ1+A1ϕ2=−(∂0ϕ2−A0ϕ1), (5.17b)
+r2(∂0A1−∂1A0) = 1−ϕ2
+1−ϕ2
+2. (5.17c)
+To analyze these equations, we recall that the original self-duality equations for
+theY-M fieldsareconformallyinvariant. We cantakeadvantageoft hisfact now
+by temporarily dropping the conformal factor r2in Eq.(5.17c). Then, the above
+equations become the Bogomolny equations for the flat space Abelia n Higgs
+model, e.g. for the model described by the action functional, Eq.(4.2 4), with the
+conformal factor Ξ = 1 [89]. Such obtained equations contain all info rmation
+about the Abelian Higgs model and, hence, they are equivalent to th is model.
+It is of importance for us to demonstrate this explicitly for both Euc lidean
+and hyperbolic spaces. For this purpose we introduce a covariant d erivative
+Dµ=∂µ−iAµ,µ= 0,1,and the complex field φ=φ1+iφ2. Consider the
+Bogomolny equation following [68]:
+D0φ+iD1φ= 0. (5.18)
+Usingtheabovedefinitionsstraightforwardcomputationreprodu cesEq.s(5.17a,b).
+These equations can be used to obtain
+r2(D0−iD1)(D0+iD1)φ= 0 (5.19)
+29implying
+r2(D0D0+D1D1)φ=−ir2[D0,D1]φ=−r2(∂0A1−∂1A0)φ=−(1−ϕ2
+1−ϕ2
+2)φ,
+(5.20)
+where the last equality was obtained with help of Eq.(5.17c). Evidently , the
+equation
+(D0D0+D1D1)φ+1
+r2(1−ϕ2
+1−ϕ2
+2)φ= 0 (5.21)
+isoneoftheequationsof”motion”fortheG-Lmodelon H2, e.g. seeRef.[89](Eq.(11.3)
+page 98). The second is the Ampere’s equation
+εµν∂µ(r2B) =i(φ¯Dν¯φ−¯φDνφ) (5.22)
+with the ”magnetic field” B=∂0A1−∂1A0.Details of derivation are given in
+Ref.[68], pages 198-199. Eq.(5.22) also coincides with that given in the book by
+Taubs and Jaffe, Ref.[ 105] (Eq.(11.3) page 98).
+Corollary 1 .Since both equations can be obtained by mimization of the
+functional, Eq. (4.27), they are equivalent to the Abelian Higgs model which, in
+turn, is the reduced form of the Y-M functional for pure gauge fie lds.
+We continue with the discussion of Witten’s treatment of Eq.s(5.17) s ince
+we shall need his results later on. First, he selects physically conven ient gauge
+condition via ∂µAµ= 0.This leads to the choice: Aµ=εµν∂νψ(for some scalar
+ψ). With such a choice for Aµthe first two of Eq.s(5.17) can be rewritten as
+(∂0−∂0ψ)ϕ1= (∂1−∂1ψ)ϕ2, (5.23a)
+(∂1−∂1ψ)ϕ1=−(∂0−∂0ψ)ϕ2. (5.23b)
+Let nowϕ1=eψχ1andϕ2=eψχ2.Then the above equations are reduced to
+the Cauchy-Riemann-type equations: ∂0χ1=∂1χ2and∂1χ1=∂0χ2.Introduce
+the function f=χ1−iχ2. Then, the last of Eq.s(5.17) acquires the form
+−r2∇2ψ= 1−ff∗e2ψ. (5.24)
+Notice that −r2∇2=−r2(∂2
+∂t2+∂2
+∂r2) is the hyperbolic Laplacian [90]. Eq.(5.24)
+is still gauge invariant in the sense that by changing f→fhandψ→ψ−
+1
+2ln(hh∗) in this equation we observe that it preserves its original form. This
+is so because ∇2ln(hh∗) = 0 for any analytic function which does not have
+zeros. Ifhdoes have zeros for r >0, then substitution of ψ→ψ−1
+2ln(hh∗)
+intoEq.(5.24)producesisolatedsingularitiesatthesezeros. After theseremarks,
+Eq.(5.24)canbe simplified further. Forthispurpose, let ψ= lnr−1
+2ln(ff∗)+ρ,
+provided that ∇2ln(ff∗) = 0 for any analytic function fwhich does not have
+zeros21. Under such conditions we end up with the Liouville equation
+∇2ρ=e2ρ. (5.25)
+It is of major importance for what follows.
+21In the case if it does, the treatment is also possible as expla ined by Witten. Following
+his work, we shall temporarily ignore this option.
+305.3.2 The Heisenberg spin chain model
+The results of Appendix B imply that the L-L Eq.(5.2) (or their hyperb olic
+equivalent, Eq.(5.8)) could be interpreted in terms of equations for the Serret-
+Frenet moving triad. Treatment along these lines suitable for immedia te appli-
+cations is given in papers by Lee and Pashaev [91] and Pashaev [92]. Be low we
+superimpose their results with those of our work, Ref.[84], to achiev e our goals.
+We begin with definitions. A collection of smooth vector fields nµ(x,t),
+µ= 0−2, forming an orthogonal basis is called the ”moving frame”. If x∈ S
+whereSis some two dimensional surface, then let n1(x,t) and n2(x,t) form
+basis for the tangent plane to S ∀x∈ S. Then, the Gauss map (that is the
+map from Sto two dimensional sphere S2or pseudosphere S1,1) is given by
+n2(x,t)≡s. By design, it should obey Eq.(5.1). This observation provides
+needed link between the spin and the moving frame vectors. Details a re given
+in [91,92] and Appendix B .It should be clear that since one can draw curves
+on surfaces both formalisms should involve the same elements. The r estriction
+for the curve to lie at the surface causes additional complications in general but
+nonessential in the present case.
+Next, we introduce the combinations n±=n0±in1possessing the following
+properties
+(n+,n+) = (n−,n−) = 0 , (n+,n−) = 2/κ2, (5.26)
+whereκ2= 1 forS2andκ2=−1 forS1,1andH2.Furthermore, ( ..,..) defines
+the scalar product (in Euclidean or pseudo-Euclidean spaces). Also ,
+n+×s=in+,n−×s=−in−,n−×n+=2iκ2s. (5.27)
+In addition, we shall use the vectors
+qµ=κ2
+2(∂µs,n+) and ¯qµ=κ2
+2(∂µs,n−) (5.28)
+in terms of which the equations of motion for the moving frame vecto rs look as
+follows:
+Dµn+=−2κ2qµs, (5.29)
+∂µs=qµn−+ ¯qµn+, (5.30)
+with covariant derivative Dµ=∂µ−i
+2VµandVµ=−2κ2(n1,∂µn0).Consider
+now Eq.(5.30) for µ= 1.Apply to it the operator ∂1and use the equations of
+motion and the definitions just introduced in order to obtain
+∂2
+1s=(D1q1)n−+/parenleftbig¯D1¯q1/parenrightbig
+n+−4
+κ2|q1|2s. (5.31)
+It can be shown that q0=iD1q1.In view of this, Eq.(5.30) for µ= 0 acquires
+the following form:
+∂0s=iD1q1n−−i¯D1¯q1n+. (5.32)
+This equation happens to be of major importance because of the fo llowing.
+Multiply (from the left) Eq.(5.31) by s×and use Eq.s(5.27). Then (depending
+31on signature of κ2) the obtained result is equivalent to the L-L Eq.(5.2) or its
+pseudoeuclidean version, Eq.(5.8). Furthermore, for this to happ en the fields Vµ
+andqµmust be subject to the following constraint equations obtainable dir ectly
+from Eq.s (5.29)
+Dµqν=Dνqµ, (5.33a)
+[Dµ,Dν] =−2κ2(¯qµqν−¯qνqµ). (5.33b)
+Wearegoingtodemonstratenowthattheseequationsareequivale nttoEq.s(5.17)
+obtained by Witten.
+We begin with the following observation. Let indices µandνbe respectively
+1 and 0. Then, taking into account that q0=iD1q1we can rewrite Eq.(5.33b)
+as
+F10=B1=−2κ2i(¯q1D1q1−q1¯D1¯q1). (5.34)
+Surely, by symmetry we could use as well: q1=−iD0q0. This would give us
+an equation similar to Eq.(5.34). Take now the case κ2=−1 (that is consider
+S1,1) in these equations and compare them with the Ampere’s law, Eq.(5.22 ).
+We notice that these equations are not the same. However, since t he G-L model
+was originally designed for phenomenological (thermodynamical) des cription of
+superconductivity(asexplainedindetailinourwork,Ref.[84]),wekn owthatthe
+underlying equations (obtainable from the G-L functional) contain t he London
+equation which reads22
+∇×B=CB (5.35)
+withCbeing someconstant(determined byphysicalconsiderations). Ev idently,
+in view ofthe London(5.35), Eq.s(5.22)and (5.34) become equivalent . Consider
+now Eq.(5.33a). To understand better this equation, it is useful to rewrite
+Eq.(5.18) as follows
+D0φ=−iD1φorD0φ1=D1φ0, (5.36)
+whereφ1=φandφ0=−iφ.Take into account now that φ=a+iband
+identifyφ1withq1andφ0withq0.Then, Eq.(5.33b) acquires the following
+form (κ2=−1) :
+(∂0V1−∂1V0) =−i4(¯φ0φ1−φ0¯φ1) =−4(a2+b2). (5.37)
+Looking at Eq.(5.17c) we can make the following identifications: V1=A1,V0=
+A0,±2a=ϕ1,±2b=ϕ2.Then, comparison between Eq.s(5.17c) and (5.37)
+indicates that we are still missing a factor of r2in the l.h.s. and 1 in the r.h.s.
+Looking at Witten’s derivation of the Liouville Eq.(5.25), we realize that these
+two factors are interdependent. By clever choice of the function ψthey can
+be made to disappear. This makes physical sense since locally the und erlying
+surface is almost flat. This observation makes Eq.s(5.37) and (5.24) (or 5.17c)
+equivalent.
+22This is not the form of the London equation one can find in textb ooks. But in our work,
+Ref.[84], we demonstrated that Eq.(5.35) is equivalent to t he London equation.
+32Corollary 2 .The L-L and 2 dimensional G-L models are essentially equiv-
+alent in the sense just described both in Euclidean and in Minkowski sp aces.
+Corollary 3 .The ”hyperbolc” L-L Eq. (5.14)or its Euclidean analog should
+be identified with Floer’s Eq. (4.6).
+These results play an important role in the rest of this work and, in pa rtic-
+ular, in the next subsection.
+5.4 The proof (implementation)
+5.4.1 General remarks
+In Ref.[79], we demonstrated how treatment of combinatorial data associated
+with real scattering experiments leads to restoration of the unde rlying quantum
+mechanical model reproducing the meson spectrum. It was estab lished that
+the underlying microscopic model is the Richardson-Gaudin (R-G) XX X spin
+chain model originally designed for description of spectrum of excita tions in the
+Bardeen-Cooper-Schriefer (BCS) model of superconductivity. Subsequently,
+the same model was used for description of spectra of the atomic n uclei. Since
+the energy spectrum of the BCS model has the famous gap betwee n the ground
+and the first excited state, the problem emerges :
+Can spectral properties of nonperturbative quantum Y-M fie ld
+theory be described by the R-G model ?
+To answer this question affirmatively the ”equivalence principle” disco vered
+by L.Witten is very helpful. Using it, we can proceed with quantization o f pure
+Y-M fields by using results by Korotkin and Nicolai, Ref.[31], for gravity . By
+comparing the main results of our paper, Ref.[79], done for QCD, with those of
+Ref.[31], done for gravity, we found a complete agreement. In part icular, the
+Knizhnik-Zamolodchikov Eq.s(4.14),(4.15) and the R-G Eq.(4.29) of Re f.[79]
+coincide respectively with Eq.s(4.27),(4.26) and (4.50) of Ref.[31] eve n though
+methods of deriving of these equations are entirely different! Both Ref.s [79]
+and [31] do not reveal the underlying physics sufficiently deeply thou gh. In the
+remainder of this section we shall explain why this is indeed so and demo nstrate
+ways this deficiency can be corrected. Experimentally the challenge lies in de-
+signing scattering experiments providing clean information about th e spectrum
+of glueballs. Thus far this task was accomplished only in lattice calculat ions
+done for unphysically large number of colors, e.g. Nc→ ∞.[23].When it comes
+to interpreting realexperiments (always having only three colors to consider23),
+the situation is even less clear, e.g. see Ref.[93]. Hence, the gap prob lem is full
+ofchallengesforboth theoryand experiment. Fortunately, at lea st theoretically,
+the problem does admit physically meaningful solution as we explained a lready.
+We continue with ramifications in the next subsection.
+23E.g. read Section 6 .
+335.4.2 From Landau-Lifshitz to Gross-Pitaevskiiequation v ia Hashimoto
+map
+Since the F-S model is believed to be capable of describing QCD vacua a nd
+is also capable of describing knotted/linked structures [17], two que stions arise:
+a) Is this the only model capable of describing QCD vacua? b) To what extent
+it matters that the F-S model is also capable of describing knots and links?
+The negative answer to the first question follows from Corollary 3 imp lying
+that, in principle, both Euclidean and hyperbolic versions of the L-L e quation
+are capable of describing QCD vacua: different vacua correspond t o different
+steady-state solutions of the L-L equations. The negative answe r to the second
+question can be found in a review, Ref.[85], by Annalisa Calini. From this
+reference it follows that, besides the F-S model, knotted/linked st ructures can
+be also obtained by using standard (that is Euclidean) L-L equation, e.g. see
+Eq.(B.4) of Appendix B. This fact still does not explain why knots/links are of
+importance to QCD. We address the above issues in more detail in Sec tion 6. In
+view of what is said above, wether or not the hyperbolic version of L- L equation
+is capable of describing knotted structures is not immediately import ant for
+us. Far more important is the connection between the hyperbolic L- L and the
+Ernst equation. Only with this connection it is possible to reproduce r esults by
+Korotkin and Nicolai [31].
+Eq.(3.19)is just the F-S functionalwithout winding numberterm. Wh en the
+commutation relations for su(1,1) introduced in subsection 5.2 are r eplaced by
+those for su(2) this leads to the standard L-L equation (instead o f Eq.(5.14)).
+This replacement causes us to abandon the connection with Ernst e quation
+and, ultimately, with the results of Ref.[31]. In such a case the gap pr oblem
+should be investigated from scratch. In Ref.[25] Faddeev and Niemi indicated
+that, unless some amendments to the F-S model are made, it is gaple ss. At the
+same time from Appendix B it is known that the L-L equation associate d with
+the F-S model can be transformed into the NLSE with help of the Has himoto
+map. Recently, Ding [94] and Ding and Inoguchi [95] were able to find a nalogs
+of the Hashimoto map for the vortex filaments in hyperbolic, de Sitte r and
+anti de Sitter spaces. It is helpful to describe their findings using t erminology
+familiar from physics literature [96].This leads us to the discussion of pr operties
+of the Gross-Pitaevskii equation known in mathematics as the NLSE . In the
+system of units in which ℏ= 1 andm= 1/2 this equation can be written as
+[86]
+iψt=−ψxx+2κ/parenleftBig
+|ψ|2−c2/parenrightBig
+ψ= 0. (5.38)
+Zakharov and Shabat [97,98] performed detailed investigation of th is equation
+for both positive and negative values of the coupling constant κ.Forκ <0
+the above equation is used for description of knots/links [85]. The st andard
+Hashimoto map brings the L-L equation associated with the truncat ed F-S
+model to the NLSE with κ <0 [94, 95]. From the same references it can be
+found that the Hashimoto-like map brings the (hyperbolic) L-L-like e quation to
+the NLSE for which κ >0.Zakharov and Shabat studied in detail differences
+34in treatments of the NLSE for both negative and positive coupling co nstants.
+This difference is caused by difference in underlying physics which in bot h cases
+can be explained in terms of the properties of non ideal Bose gas [99,1 00]. The
+attentive reader might have noticed at this point that Eq.(5.38) app arently
+contains no information about the number of particles in such a gas. This
+parameter, in fact, is hidden in the constant c(the chemical potential) or it can
+be obtained selfconsistently with help of Eq.(5.38)(from which cis removed in a
+way described in Appendix B) as explained in Ref.[100]. With this informat ion
+at our disposal we are ready to make the next step.
+5.4.3 From non ideal Bose gas to Richardson-Gaudin equation s
+Even though statistical mechanics of 1-d interacting Bose gas was considered in
+detailbyLiebandLinger[101],solidstatephysicsliteratureisfullofr efinements
+of their results up to moment of this writing. These refinements hav e been
+inspired by experimental and theoretical advancements in the the ory of Bose
+condensation [96]. Among this literature we selected Ref.s[102,103] a s the most
+relevant to our needs.
+Following [102], the Hamiltonian for Ninteracting bosons moving on the
+circle of length Lis given by
+H=−N/summationdisplay
+i=1∂2
+∂x2
+i+2ˇc/summationdisplay
+1≤i<j≤Nδ(xi−xj) (5.39)
+with constant 2ˇ ccoinciding with 2 κin the system of units ℏ= 1 andm= 1/2.
+The case ˇc >0 (repulsive Bose gas) corresponding to the L-L equation in the
+hyperbolicplane/spacehappens tobe ofimmediate relevance. Onlyforthis case
+it is possible to establish the connection with workby Korotkinand Nico lai[31]!
+We begin by noticing that in the standard BCS theory of supercondu ctivity
+electrons are paired into singlets (Cooper pairs) with zero centre o f mass mo-
+mentum. The pairing interaction term in this theory accounts only fo r pairs
+of attractive electrons with opposite spin and momenta so that the degener-
+acy for each energy state is a doublet, with level degeneracy Ω = 224. In the
+interacting repulsive Bose gas model byRichardson [104] to mimic this pairing
+he coupled two bosons with opposite momenta ±kjinto one (pseudo) Cooper
+pair. An assembly of such formed pairs forms repulsive Bose gas which in the
+simplest case is described by the Hamiltonian, Eq.(5.39). Hence, the fermionic
+BCS-type model with strong attractive pairing interaction can be m apped into
+bosonic repulsive model proposed by Richardson. Although the idea of such
+mapping looks very convincing, its actual implementation in Ref.[102] h as some
+flaws. Because of this, we shall use results of this reference selec tively. For this
+purpose, fist of all we need to make an explicit connection between t he repulsive
+Bose gas model described by Eq.(5.39) and the model proposed by R ichardson.
+In the weak coupling limit ˇ cL≪1 the Bethe ansatz equations for the repulsive
+24We use here the same notations as in our work, Ref.[ 94].
+35Bose gas model described by the Hamiltonian, Eq.(5.39), acquire the following
+form:
+ki=2πdi
+L+2ˇc
+LN/summationdisplay
+j=1
+(j/negationslash=i)1
+kj−ki,i= 1,...,N. (5.40)
+Heredi= 0,±1,±2,...denote the excited states for fixed N. To link this result
+with Richardson’s (repulsive boson) model, consider the case of eve n number of
+bosons and make N= 2M. Next, consider the ground state of this model first.
+To the first order in ˇ c, it is clear that we can write ki=±√Ei. Specifically,
+letk1,2=±√E1,k3,4=±√E3,...,k2M−1,2M=±√EM.Using these results in
+Eq.(5.40), with the accuracy just stated, the Bethe ansatz equa tions after some
+algebra are converted into the following form:
+L
+2ˇc+˜M/summationdisplay
+j=1
+(j/negationslash=i)2
+Ej−Ei=1
+2Ei,i= 1,...,M;˜M≤M. (5.41)
+To analyze these equations, we expect that our readers are familia r with works
+of both Richardson-Sherman, Ref.[105], and Richardson, Ref.[104]. In [105]
+diagonalization of the pairing force Hamiltonian describing the BCS-ty pe su-
+perconductivity was made. Such a Hamiltonian is given by
+H=/summationdisplay
+f2εfˆNf−g/summationdisplay′
+f/summationdisplay′
+f′b†
+fbf′, (5.42)
+whereˆNf=1
+2(a†
+f+af−+a†
+f−af−),bf=af−af+, witha†
+fσandafσbeingfermion
+creation and annihilation operators obeying usual anticommutation relations
+[afσ,a†
+f′σ′]+=δσσ′δff′, whereσ=±denotes states conjugate under time
+reversal. The above Hamiltonian is diagonalized along with the seniority oper-
+ators (taking care of the number of unpaired fermions at each leve lf) defined
+by
+ˆνf=a†
+f+af−−a†
+f−af−. (5.43)
+By construction, [ H,ˆNf] = [H,ˆνf] = 0.The classification of the energy levels
+is done in such a way that the eigenvalues νfof the operator ˆ νf(0 andσ) are
+appropriatefor the case when g= 0.This observation allowsus to subdivide the
+Hamiltonian into two parts: H1,i.e.that which does not contain Cooper pairs
+(for which νf=σ) andH2,i.e.that which may contain such pairs (for which
+νf= 0).The matrix elements of H2are calculated with help of the bosonic-type
+commutation relations
+[bf′,ˆNf′] =δff′bfand [bf,b†
+f′] =δff′(1−2ˆNf′). (5.44)
+These commutators are bosonic but nontraditional. In the traditio nal case we
+have [bf,b†
+f′] =δff′.We refer our readers to Ref.[105] for details of how this
+commutator difficulty is resolved. In the light of this resolution, in Ref .[104]
+36Richardson proposed to deal with the interacting bosons model fr om the be-
+ginning. Supposedly, such bosonic model can be designed to reproduce res ults
+of the fermionic pairing model of Ref.[105]. An attempt to do just this was
+made in Ref.[102]. In the repulsive boson model by Richardson the ”pa iring”
+Hamiltonian is given by25
+H=/summationdisplay
+l2εlˆnl+g
+2/summationdisplay′
+f/summationdisplay′
+f′A†
+fAf′. (5.45)
+in which ˆnlandAf′are bosonic analogs of ˆNfandbf.It is essential that
+the sign of the coupling constant gis nonnegative (repulsive bosons). Upon
+diagonalization, the total energy Eis given by
+E=n/summationdisplay
+l=1εlνl+m/summationdisplay
+j=1Ej (5.46)
+so that summation in the first sum takes place over the unpaired bos ons while
+in the second- over the paired bosons whose energies Ejare determined from
+the Richardson’s equation (Eq.(2.29) of Ref. [104])26
+1
+2g+n/summationdisplay
+l=1dl
+2εl−Ek+m/summationdisplay
+i=1
+i/negationslash=k2
+Ei−Ek= 0,k= 1,...,m (5.47)
+in whichnis the total number of single particle (unpaired) levels, mis the total
+number of pairs, dl=1
+2(2νl+ Ωl).From [104] it follows that for the bosonic
+model to mimic the BCS-type pairing model the degeneracy factor Ω l= 1 and
+νl= 0.It should be noted though that such an identification is not of much
+help in comparing the repulsive bosonic model with the attractive BCS -type
+fermionic model (contrary to claims made in Ref.[102]). This can be eas ily
+seen by comparison between Eq.(5.47) (that is Eq.(2.29) of Ref.[104]) with such
+chosen Ω landνlwith Eq.(3.24) of Ref.[105]. By replacing gin Eq.(5.47) by
+−gwe still will not obtain the analog of the key Eq.(3.24) of Ref.[105]! This
+fact has group-theoretic origin to be explained in the next subsect ion. In the
+meantime, Eq.(5.47) still can be used to connect it with Eq.(5.41) origin ating
+from different bosonic model described by the Hamiltonian Eq.(5.39). To do so
+we follow the path different from that suggested in Ref.[102]. Instea d, following
+the original Richardson’s paper [104], we let n= 1 in Eq.(5.47) then, without
+loosing generality, we can put ε1= 0 so that Eq.(5.47) acquires the following
+form
+1
+Ek=1
+2g+M/summationdisplay
+i=1
+i/negationslash=k2
+Ei−Ek, k= 1,...,M. (5.48)
+25To avoid ambiguities, our coupling constantg
+2is chosen exactly the same as in [104].
+26Since Gaudin’s equation is obtained in the limit |g| → ∞.from Eq.(5.47) the spin -like
+model described by this equation is known as the Richardson- Gaudin (R-G) model.
+37The rationale for replacing mbyMis given on page 1334 of [104]. Evidently,
+Eq.s (5.41) and (5.48) are identical. This observation allows us to use t he
+Richardson model instead of that described by Eq.(5.39). At first s ight such
+an identification looks a bit artificial. To convince our readers that it d oes
+make sense, we would like to use the work by Dhar and Shastry [106,10 7] on
+excitation spectrum of the ferromagnetic Heisenberg spin chain. B y analogy
+with Eq.(5.41) these authors derived a similar equation obtained by re ducing
+the Bethe ansatz equations for Heisenberg ferromagnetic chain. It reads27
+1
+El=πd−d
+n/summationdisplay
+i=1
+i/negationslash=l2
+Ei−El. (5.49)
+Even though Eq.s(5.48) and (5.49) look almost the same, they are no t the same!
+The crucial difference lies in the signs in front of the second term in th e r.h.s. of
+theseequations. BecauseofthisdifferenceHeisenberg’sferroma gneticspinchain
+model is mapped onto Bose gas model with attractive interaction in complete
+accord with what was said immediately after Eq.(5.38). Regrettably, this result
+is still not the same as for the BCS-type model investigated in Richar dson-
+Sherman’s paper, Ref.[105]. This fact was recognized and discussed in some
+detail already by Richardson [104]. For completeness, we mention th at the
+problem of BCS-Bose-Einstein condensation (BEC) crossover whic h follows
+exactly the qualitative picture just described was made quantitativ e only
+very recently in Ref.[108]. Fortunately, it is possible to by-pass this r esult as
+explained in the next subsection.
+5.4.4 From Richardson-Gaudin to Korotkin-Nicolai equatio ns
+In Ref.[109] bosonic and fermionic formalism for pairing models discuss ed in
+the previous subsection was developed. This formalism happens to b e the most
+helpful for investigation of the gap problem. Indeed, define three operators
+ˆnl=/summationtext
+ma†
+lmalm,A†
+l= (Al)†=/summationtext
+ma†
+lma†
+l¯m. Theycanbe used forconstruction
+of operators K0
+l=1
+2ˆnl±1
+4ΩlandK+
+l=1
+2A†
+l=/parenleftbig
+K−
+l/parenrightbig†such that they obey
+the following commutator algebra
+[K0
+l,K+
+l] =δllK+
+l,[K+
+l,K−
+l] =∓2δllK0
+l. (5.50)
+Inthisalgebraaswellasintheprecedingexpressions,theuppersig ncorresponds
+to bosons and the lower to fermions. In Ref.[79], we discussed such a n algebra
+for the fermionic case only, e.g. see Eq.s (4.31) of [79]. These results can
+be extended now for the bosonic case. In fact, such an extension is already
+developed in Ref.[109]. Unlike [79], where we used sl(2,C) Lie algebra, only
+its compact version, that is su(2), was used in [109] for representing fermions.
+For bosonic case the commutation relations, Eq.(5.50), are those f orsu(1,1) Lie
+algebra. Incidentally, in the paper by Korotkinand Nicolai, Ref.[31], ex actly the
+27The physical meaning of constants entering this equation is not important for us. It is
+given in Ref.[106]..
+38same Lie algebra was used. Furthermore, in the same paper it was ar gued that
+it is permissible to replace su(1,1) bysl(2,R) Lie algebrawhile constructing the
+K-Z-type equations, e.g. read p.428 of this reference. Since in [79] thesl(2,C)
+Lie algebra was used, that is a complexified version of sl(2,R),this allows us to
+use many results from our work. Thus, in this subsection we shall dis cuss only
+those results of [109] which are absent in our Ref.[79]. In particular, following
+this reference the set of Gaudin-like commuting Hamiltonians written in terms
+of operators K0
+l,K+
+landK−
+lis given by
+Hl=K0
+l+2g{/summationdisplay
+l′(/negationslash=l)Xll′
+2(K+
+lK−
+l′+K−
+lK+
+l′)∓Yll′K0
+lK0
+l′}.(5.51)
+HereXll′=Yll′= (εl−εl′)−1.Forg→ ∞the first term can be ignored and the
+remainder can be used in the K-Z-type equations. The semiclassical treatment
+of these equations discussed in detail in [79] is resulting in the following set of
+Bethe (or R-G) ansatz equations
+n/summationdisplay
+l=1dl
+2εl−Ek±m/summationdisplay
+i=1
+i/negationslash=k2
+Ei−Ek= 0, k= 1,...,m (5.52)
+to be compared with Eq.(5.47). Unlike Eq.(5.47), in the present case dl=
+1
+2(2νl±Ωl).The bosonic version of Eq.(5.52) corresponding to su(1,1) Lie alge-
+bra coincides with Eq.(4.50) of Korotkin and Nicolai paper, Ref.[31], pr ovided
+that the following identifications are made: dl⇄sl, 2εl⇄γj. Unlike Ref.[31],
+where Eq.(5.52) was obtained by standard mathematical protocol, in this work
+it is obtained based on the underlying physics. Because of this, it is ap propriate
+to extend our physics-stype analysis by considering the case of fin iteg′s.Then,
+Eq.(5.52) should be replaced by
+1
+2g±n/summationdisplay
+l=1dl
+2εl−Ek±m/summationdisplay
+i=1
+i/negationslash=k2
+Ei−Ek= 0,k= 1,...,m. (5.53)
+In Ref.[31] the gap problem was discussed in detail for the fermionic c ase when
+the coupling constant gis negative (BCS pairing Hamiltonian), e.g. see Eq.s
+(4.43)-(4.45) of Ref.[31]. In the present case we are dealing with the bosonic
+case for which the coupling constant is positive. Hence the gap prob lem should
+be re analyzed. For this purpose, it is convenient to consider both p ositive and
+negative coupling constants in parallel for reasons which will become apparent
+upon reading.
+5.4.5 Emergence of the gap and the gap dilemma
+Eq.s(5.53) cannot be solved without some physical input. Initially, su ch an
+input was coming from nuclear physics (e.g. read [110-112]for gene ral informa-
+tion on nuclear physics). Indeed, Richardson’s papers [104,105] we re written
+39having applications to nuclear physics in mind. Given this, the question arises
+about the place of the R-G model among other models describing nuc lear spec-
+tra and nuclear properties. We need an answer to this question to fi nish proof
+of the gap’s existence in QCD.
+Looking at the Gaudin-like Hamiltonian, Eq.(5.51), and comparing it with
+the Hamiltonian, Eq.(6), in Ref.[113]28it is easy to notice that they are almost
+the same! Because of this, it becomes possible to transfer the met hodology of
+Ref.[113]tothepresentcase. Thus, itmakessensetorecallbriefl ycircumstances
+at which the gap emerges in nuclear physics.
+As is well known, the nuclei are made of protons and neutrons. One can
+talk about the number Nof nucleons, the number Z of protons and the number
+N of neutrons in a given nucleus. Nuclear and atomic properties happ en to
+be interrelated. For instance, in analogy with atomic physics one can think of
+some effective nuclear potential in which nucleons can move ”indepen dently”.
+This assumption leads to the shell model of nuclei. Use of Pauli principle guides
+fillings of shells the same way as it guides these fillings in atomic physics. T his
+leads to emergence of magic numbers 2, 8, 20, 28, 50, 82 and 126 fo r either
+protons or neutrons for the totally filled shells. Accordingly, the mo st stable
+are the doubly magic (for both protons and neutrons) nuclei. It is o f interest to
+know what kinds of excitations are possible in such shell models? The s implest
+of these is when some nucleon is moving from the closed shell to the em pty shell
+thus forming a hole. When the numberof nucleonsincreases, the question about
+thevalidity ofthe shellmodel emerges,againin analogywith atomicph ysics. As
+in atomic physics, one can think about the Hartree-Fock(H-F) and other many-
+body computational schemes, including that developed by Richards on-Sherman
+and Gaudin. For our purposes, it is sufficient to use only the Tamm-Da nkoff
+(T-D) approximation to the H-F equations described, for example, in Ref.[112].
+The essence of this approximation lies in restricting the particle-hole interac-
+tions to nucleons lying in the same shell. The T-D approximation is obtainable
+from the isR-G Eq.s (5.53)when the last term (effectively taking care of Pauli
+principle) in these equations is dropped . The T-D approximation was success-
+fully applied for description of the giant nuclear dipole resonance [110 -112]. At
+the classical level the physics of this resonance was explained in the paper by
+Goldhaber and Teller [114]. The resonance is caused by two nuclear vib rational
+modes: one, when protons and neutrons move in the opposite direc tions and
+another- when they move in the same direction. Upon quantization o f such
+classical model and taking into account the isotopic spin of nucleons , the trun-
+cated Eq.s(5.53) are obtained in which both signs for the coupling con stant are
+allowed since the nucleon system is expected to be in two isospin state s :T= 1
+andT= 029. Details of these calculations are given in Ref.[112], page 221. So-
+lutions of the T-D equations can be obtained graphically in complete an alogy
+with that described in our work, Ref.[79]. These graphical solutions r eflect the
+particle-hole duality built into the T-D approximation. Because of this duality,
+28Published in 1961!
+29This can be easily understood based on the fact that isospin f or both particles and holes
+is equal to 1/2 [110-112].
+40the magnitude of the gap in both cases should be the same. To demon strate
+this, the seniority scheme described in [110-112] is helpful. The seniority opera-
+tor was defined by Eq.(5.43). It determines the number of unpaired particles in
+the nuclear system. Since it commutes with the Hamiltonian, the many -body
+states can be classified with help of its eigenvalues νf.Suppose at first that all
+single particle energies εfare the same (that is εf=ε) so that all seniority
+eigenvalues νfareν.Let then Nbe the total number of nucleons. Thus, the
+state for which ν= 0 contains only pairs, analogously, the state ν= 1 contains
+just one unpaired nucleon, ν= 2 has 2 unpaired nucleons and Nshould be
+even and so on. So, states ν= 0,ν= 2,ν= 4,...can exist only in even nuclei.
+For such nuclei the gap is nonzero. To see this, we follow Refs.[110-1 12] which
+we would like now to superimpose with the results of the Richardson-S herman
+paper, Ref.[105]. Specifically, on page 231 of this reference one can find the
+following result for the ground state ( ν= 0) energy
+Eν=0(N) = 2Nε−gN(Ω−N+1) (5.54)
+whereNis the number of pairs. To connect this result with that in Refs.[110-
+112], letN=N/2 and consider the difference
+Eν=0(N) =Eν=0(N/2)−Nε=−g
+4N(2Ω−N+2).(5.55)
+TheobtainedresultcoincideswithEq.(11.14)ofRef.[112]asrequired . Toobtain
+states of seniority ν= 2nwe use Eq.(3.2) of Ref.[105]. It reads
+Eν=2n(N) = 2Nε−g(N−n)(Ω−N−n+1), n= 0,...,N. (5.56)
+Repeating the same steps as in ν= 0 case we obtain,
+Eν(N) =−g
+4(N-ν)(2Ω−N-ν+2). (5.57)
+Finally, consider the difference
+Eν(N)−Eν=0(N)=g
+4ν(2Ω−ν+2). (5.58)
+This result is in accord with Eq.(11.22) of Ref.[112]. Since the obtained d if-
+ference is N-independent it can be used both ways: a) for calculations in the
+thermodynamic limit N → ∞ and b) for making accurate calculations in the
+opposite limit of very small number of nucleons. In the simplest case w e should
+consider only one shell and the first excited state of seniority 2 for this shell.
+Initially (the ground state) we have just one pair while finally (the firs t excited
+state) we have two independent particles occupying single particle le vels.
+Looking at Eq.s(5.53) and letting there m= 1(one pair) we recognize that
+the second sum in this set of equations disappears. Thus, by design , we are left
+with the T-D approximation. Using Eq.(5.58) for ν= 2 we obtain the following
+value of the gap ∆ :
+∆ =E2(N)−E0(N) =gΩ. (5.59a)
+41Notice, that since Ω is the degeneracy, there could be no more than N= Ω
+particles at the single particle level. Thus, in general we should have N ≤Ω.
+Because ofthe particle-holeduality, it is permissible to look alsoat the situation
+for which N ≥Ω.This is equivalent to changing the sign in front of the coupling
+constant. Repeating again all steps leads to the final result for th e gap
+∆ =E2(N)−E0(N)=|g|Ω. (5.59b)
+It is demonstrated in Ref.s [110-112]that in the limit N → ∞,when the contin-
+uum approximation (replacing summation by integration) can be used leading
+to a more familiar BCS-type equation for the gap, the result just ob tained sur-
+vives. Indeed, in Ref.[103] the BCS-type result is obtained in the con tinuum
+approximation for the attractive Bose gas. In view of the results just obtained,
+it should be clear that such a result should hold for both attractive a nd repul-
+sive Bose gases. This conclusion is in accord with accurate recent Be the ansatz
+calculations done in Ref.[115] for systems of finite size. Thus, we jus t arrived
+at the issue which we shall call the gap dilemma . While the results obtained
+above strongly favor use of the repulsive Bose gas model (not linked with the
+F-S model ),the results obtained in this subsection indicate that, after all, the
+F-S model (linked with the attractive Bose gasmodel )can also be used for de-
+scription of the ground and excited states of pure Y-M fields .The essence of
+the dilemma lies in deciding which of these results should ac tually
+be used .
+While the answer is provided in the next section, we are not yet done w ith
+the gap discussion. This is so because the seniority model is applicable only
+to the case when all single-particle levels have the same energy. This is too
+simplistic. We would like now to discuss more realistic case
+Before doing so, few comments are appropriate. In particular, wit h all suc-
+cesses of nuclear physics models, these models are much less convin cing than
+those in atomic physics. Indeed, all nuclei are made of hadrons whic h are made
+of quarks and gluons. Thus the excitations in nuclei are in fact the e xcitations
+of quark-gluon plasma. This observation qualitatively explains why th e R-G
+equations work well both in nuclear and particle physics. Some attem pts to
+look at the processes in nuclear physics from the standpoint of had ron physics
+can be found in Refs.[116,117].
+Now we can return to the discussion of the T-D equations. Fortuna tely, de-
+tailed analytical study of these equations was recently made in Ref.[1 18]. The
+same authors extended these results to the case of two pairs in [11 9]. Since
+the results obtained in [119] are in qualitative agreement with those o btained
+in Ref.[118], we shall focus attention of our readers only on results o f Ref.[118].
+Thus, we need to find some kind of analytic solution of the following T-D equa-
+tion
+L/summationdisplay
+i=1Ωi
+2εi−E=1
+g. (5.60)
+For different ε′
+isnormally it should have Leigenvalues Eµ(1≤µ≤L).Since
+we are interested in finding the gap, the above equation is written fo r just one
+42nucleon pair. Thus the seniority ν= 0.It is of interest to check first what
+happens when all ε′
+iscoalesce. In such a case we obtain,
+Ω
+2¯ε−E=1
+g, (5.61)
+where Ω =/summationtext
+iΩiandεi= ¯ε∀i= 1,...,L.Eq.(5.61) can be equivalently rewrit-
+ten as
+E0= 2¯ε−Ωg. (5.62)
+This result for the ground state is in agreement with Eq.(5.54) for N= 1. The
+first excited state is made of one broken pair so that the pairing disa ppears
+and the energy Eν=2= 2¯ε. From here, the value of the gap is obtained as
+Eν=2−E0=gΩ in agreement with Eq.(5.59). If now we make all energy
+levels different, then one can see that solutions to Eq.(5.60) are sub divided
+into those lying in between the single particle levels ( trapped solutions ) and
+those which lie outside these levels ( collectivized solutions ). For|g|sufficiently
+large the solution, Eq.(5.61), is the leading term (in the sense describ ed below)
+representing the collectivized solution. Since the trapped solutions represent
+corrections to energies of single particle states, they do not cont ribute directly
+to the value of the gap. They do contribute to this value indirectly. I ndeed,
+following Ref.[118] we rewrite Eq.(5.60) as
+L/summationdisplay
+i=1Ωi
+2εi−E=1
+2¯ε−E/summationdisplay
+iΩi
+1+2εi−¯ε
+2¯ε−E=1
+g(5.63)
+and expand the denominator of Eq.(5.63) in a power series. As result , the
+following expansion
+E−2¯ε
+gΩ=−1−α2+γα3+O(α4) (5.64)
+is obtained in which ¯ ε=1
+Ω/summationtext
+iΩiεi,α=2σ
+gΩ,σ=/radicalbigg
+1
+Ω/summationtext
+iΩi(εi−¯ε)2andγis
+related to the higher order moments ( details are in Ref.s[118,119]). U sing these
+results, the gap is obtained in the same way as before.
+The quality of computations in Ref.[118] was tested for 3-dimensiona l har-
+monic oscillator (by adjusting dimensionality of this oscillator it can be t hought
+of as ”closed string model” representing both the shell model for a tomic nu-
+cleus and the gluonic ring for the Y-M fields) for which εi= (i+ 3/2) (in
+the system of units in which /planckover2pi1ω= 1) and Ω i= (i+ 1)(i+ 2)/2.For this 3-
+dimensional oscillator correctionsto the collectivized energy, Eq.(5 .64), become
+negligible already for |g| ≥0.2,provided that L≥8.Obtained results allow us
+to close this section at this point. These results are of no help in solvin g the
+gap dilemma though. This task is accomplished in the next section.
+436 Resolution of the gap dilemma
+6.1 Motivation
+In the previous section we provided evidence linking the gap problem f or Y-M
+fields with the problem about the excitation spectrum of the repulsiv e Bose gas.
+The gap equation, Eq.(5.59), is also used in nuclear physics where it is k nown
+to produce the same value for the gap for both signs of the coupling constantg.
+Since both options are realizable in Nature in the case of nuclear phys ics, the
+question arises about such possibility in the present case. In the ca se of nuclear
+physicsexperimentalrealization(giantnucleardipoleresonance)o fboth options
+for the coupling constant is experimentally testable. Thus, in the pr esent case
+we have to find some alternative physical evidence. If, indeed, suc h evidence
+could be found, this would allow us to bring back into play the well studie d F-S
+model which microscopically is essentially equivalent to the XXX 1d Heise nberg
+ferromagnetas results of Appendix B and subsections 3.5 and 5.2 ind icate. The
+next subsection supplies us with the alternative physical evidence.
+6.2 Some facts about harmonic maps and their uses in
+general relativity
+Suppose we are interested in a map from m−dimensional Riemannian manifold
+Mwith coordinates xaand metric γab(x) ton-dimensional Riemannian man-
+ifoldNwith coordinates ϕAand metric GAB(ϕ). A map M → N is called
+harmonic ifϕA(xa) satisfies the Euler-Lagrange (E-L) equations originating
+from minimization of the following Lagrangian
+L=√γGAB(ϕ)γab(x)ϕA
+,aϕB
+,b (6.1)
+in whichγ= det(γab).Since such defined Lagrangian is part of the La-
+grangian given by Eq.(3.6), the E-L equations for Eq.(6.1), in fact, c oincide
+with Eq.s(3.10). In the most general form they can be written as [38 ]30
+ϕA;a
+,a+ΓA
+BCϕB
+,aϕC,a= 0. (6.2)
+In such a form we can look at transformations ϕA′=ϕA′(ϕB) keeping Lform-
+invariant. To find such transformations, following Neugebauer and Kramer [38],
+we introduce the auxiliary Riemannian space defined by the metric
+dS2=GAB(ϕ)dϕAdϕB. (6.3)
+Use of the above metric allows us to investigate the invariance of Lwith help of
+standardmethods of Riemannian geometry. In the present case, this means that
+one should study Killing’s equations in spaces with metric GAB.Specifically, let
+us consider the Lagrangian for source-free Einstein-Maxwell field s admitting at
+30We use the 1st edition of Ref. [38] for writing this equation. This means that we have to
+define ΓA
+BCas ΓA
+BC=1
+2Gad{∂
+∂ϕcGbd+∂
+∂ϕbGcd−∂
+∂ϕdGbc}.
+44least one non-null Killing vector ξ.To design such a Lagrangian we begin with
+the Ernst equation, Eq.(2.4), for pure gravity and replace the Ern st potential
+ǫ=−F+iω31by two complex potentials Eand Φ. Then, by symmetry, the
+equations for stationary Einstein-Maxwell fields can be written as f ollows [38]
+FE;a
+,a+γabE,a(E,b+2Φ,b¯Φ) = 0,FΦ;a
+,a+γabΦ,a(E,b+2Φ,b¯Φ) = 0.(6.4)
+These equations are obtained by minimization of the Lagrangian
+L=√γ[ˆRab+2F−1γabΦ,aΦ,b+1
+2F−2γab(E,a+2¯ΦΦ,a)(E,b+2¯ΦΦ,b)],(6.5)
+i.e. from equationsδL
+δγab= 0,δL
+δΦ= 0 andδL
+δE= 0.Taking these results into
+account, the auxiliary metric, Eq.(6.3), can now be written as
+dS2= 2F−1dΦd¯Φ+1
+2F−2/vextendsingle/vextendsingledE+2¯ΦdΦ/vextendsingle/vextendsingle2. (6.6)
+The analysis done by Neugebauer and Kramer [38] shows that there are eight
+independent Killing vectors leading to the following finite transformat ions :
+E′=α¯αE, Φ′=αΦ;
+E′=E+ib, Φ′= Φ;
+E′=E(1+icE)−1, Φ′= (1+icE)−1;
+E′=E −2¯βΦ−β¯β, Φ′= Φ+β;
+E′=E(1−2¯γΦ−γ¯γE)−1,Φ′= (Φ+γE)(1−2¯γΦ−γ¯γE)−1.(6.7)
+Complex parameters α,β,γas well as real parameters bandcare connected
+with these eight symmetries. Evidently, solutions E′,Φ′are also solutions of
+Eq.s(6.4), provided that γabstays the same. Therefore if, say, we choose some
+vacuumsolutionasa”seed”,wewouldobtain, say,theelectrovacu umsolutionin
+accord with Appendix A. Incidentally, the electrovacuum solutions o btained by
+Bonnor (Appendix A) cannot be obtained with help of transformatio ns given by
+Eq.s(6.7). They areconsideredseparatelybelow. These observat ionsallowus to
+reduce the Lagrangian Lto the absolute minimum without loss of information.
+In 1973 Kinnersley [38] found that the group of symmetry transfo rmations for
+theEinstein-Maxwellequationswithnonnull Killingvectoristhe group SU(2,1)
+which has eight independent generators. In view of the above ment ioned reduc-
+tion ofLit is sufficient to replace the metric in Eq.(6.6) by a collection of much
+simpler metric related to each other by transformations Eq.(6.7). A ll the possi-
+bilities are described in the Table 34.1 of Ref.[38]. For our needs we focu s only
+on three of these (much simpler/reduced) metric listed in this table. These are
+dS2=2dξd¯ξ
+(1−ξ¯ξ)2,E=1−ξ
+1+ξ, (6.8)
+dS2=2dΦd¯Φ
+(1−Φ¯Φ)2, (6.9)
+31Recall, that −F=Vaccording to notations introduced in connection with Eq.(2 .4).
+45and
+dS2=−2dΦd¯Φ
+(1+Φ¯Φ)2. (6.10)
+The first and the second of these metric correspond to the vacuu m state,
+respectively, with Φ = 0 and E=−1, of pure gravity associated with the
+subgroup SU(1,1) of SU(2,1). The third metric, Eq.(6.10), corresp onds to a
+subgroup SU(2). It is related to the electrostatic fields ( E= 1) such that the
+space-time becomes asymptotically flat for E→0.It is important that the metric,
+Eq.(6.10), is related to the vacuum metric, Eq.s(6.8),(6.9), via trans formations
+either listed in Eq.(6.7) or related to these transformations. In par ticular, the
+related transformations can be obtained as follows. Using Ref.[38], it is conve-
+nient to make the parameters bandcin Eq.s(6.7) complex and to consider all
+eight complex parameters as independent of their complex conjuga tes. Under
+suchconditionsthemetricgivenbyEq.(6.10) isrelatedtothatgivenb yEq.(6.8)
+by the simplest complex transformation: Φ′=iξand¯Φ′=i¯ξ. These transfor-
+mations indicate that, starting with real vacuum solution for pure g ravity as a
+seed, the above transformations are capable of reproducing som e electrovacuum
+solutions. Additional details are discussed below.
+These results can be interpreted as follows. While the Ernst functio nal,
+Eq.(3.18), is representing pure axially symmetric gravity, the F-S-t ype func-
+tional, Eq.(3.19), should describe some special case of electrovacu um (Maxwell-
+Einstein) gravity. In view of results of Appendix C, it is possible to use these
+transformations in reverse (see below), that is to obtain the resu lts for pure
+gravity from those for electrovacuum. This peculiar ”duality” prop erty of grav-
+itational fields provides physically motivated resolution of the gap dile mma and,
+in addition, it allows us to obtain many new results.
+6.3 Resolutionof thegap dilemma and SU(3) ×SU(2)×U(1)
+symmetry of the Standard Model
+The original F-S-type model thus far is limited only to SU(2) gauge th eory.
+SU(2) gauge theory is known to be used for description of electrow eak interac-
+tions where, in fact, one has to use the gauge group SU(2) ×SU(1) [19 ]. The
+hadron physics (that is QCD) requires us to use the gauge group SU (3). This
+is caused by the fact that quark model of hadrons uses flavors (e .g. u,d,s,c,b, t)
+labeling quarks of different masses. Each of these quarks can be in t hree differ-
+ent colors (r,g,b) standing for ”red”, ”green” and ”blue”. Presen ce of different
+colors leads to fractional charges for quarks. Far from the targ et the scattering
+products are always colorless. The gauge group SU(3) is used for d escription of
+these colors. Although theoretically the number of colors can be gr eater than
+three, this number is strictly three experimentally [19]. The results o f this work
+allow us to reproduce this number of colors. For this purpose we hav e to be
+able to provide the answer to the following fundamental question :
+46Can equivalence between gravity and Y-M fields (for SU(2)
+gauge group) discovered by Louis Witten be extended to the gr oup
+SU(3)?
+Very fortunately, this can be done! For the sake of space, we sha ll be brief
+whenever details can be found in literature, e.g. see Refs.[120-122].
+To proceed, first, we have to go back to Eq.s(2.14),(2.15) and to mo dify
+these equations in such a way that instead of the Ernst Eq.(2.4) for the vacuum
+(gravity) field we should be able to obtain Eq.s (6.4) for electrovacuu m. In
+the limit Φ = 0 the obtained set of equations should be reducible to Eq.(2 .4).
+As it was noticed by G¨ urses and Xanthopoulos [120], in general, this t ask can-
+not be accomplished. Indeed, these authors demonstrated that the self-duality
+Eq.s(2.14) for SU(2) and for SU(3) Lie groups look exactly the same for axi-
+ally symmetric fields. Nevertheless, in the last case, upon explicit com putation
+instead of the vacuum Ernst Eq.(2.4) one gets an electovacuum equ ations (e.g.
+see Eq.s(6.4)) which, following Ernst [43], can be explicitly written as
+/parenleftBig
+ReE+|Φ|2/parenrightBig
+∇2E= (∇E+2¯Φ∇Φ)·∇E, (6.11a)
+/parenleftBig
+ReE+|Φ|2/parenrightBig
+∇2Φ = (∇E+2¯Φ∇Φ)·∇Φ. (6.11b)
+These equations are obtained if, instead of the matrix Mgiven by Eq.(2.15),
+one uses
+M=f−1
+1√
+2Φ −i
+2(E −¯E −2Φ¯Φ)√
+2¯Φ −i
+2(E+¯E −2Φ¯Φ) −i√
+2¯ΦE
+i
+2(¯E −E −2Φ¯Φ)i√
+2¯EΦ E¯E
+(6.12)
+in which, instead of the one complex potential ǫ=−F+iωused for solution of
+the vacuum Ernst Eq.(2.4), two complex potentials Eand Φ are being used. In
+this expression the overbars denote the complex conjugation and f=−1
+2(ǫ+
+¯ǫ+ 2Φ¯Φ).Since the Einstein-Maxwell Eq.s(6.4) (or (6.11)) are invariant with
+respect to transformations given by Eq.s(6.7), there should be a m atrixAwith
+constant coefficients such that the M′=AMA†will have primed potentials E
+and Φ takenfrom those listed in the set Eq.(6.7). Authors of [120]fo und explicit
+form of such A-matrices. However, when instead of matrix Mwe substitute the
+matrixM′into self-duality Eq.s(2.14), the combination M′−1∂M′looses this
+information. As result, we are left with the following situation: while on the
+gravity side the matrix M′=AMA†does allow us to obtain new and physically
+meaningful solutions from the old ones, on the Y-M side all this inform ation
+is lost. Thus, the one-to-one correspondence discovered by L.Wit ten for SU(2)
+isapparently lost for SU(3). Very fortunately, this happens only apparently!
+This is so because the Neugebauer- Kramer (N-K) transformation s described
+by Eq.s(6.7) do not exhaust all possible transformations which can b e applied
+to the matrix M, Eq.(6.12). Among those which are not accounted by N-
+K transformations are those by Bonnor [38 ,123] whose work is mentioned in
+Appendix A. These are given by
+E=ǫ¯ǫ;Φ =1
+2(ǫ−¯ǫ) =iω, (6.13)
+47whereǫ=−F+iωis solution of the Ernst Eq.(2.4). In view of the results of
+Appendix A one can be sure that the potentials Eand Φ satisfy Eq.s(6.11).
+This means that one can use these (Bonnor’s) potentials in the matr ixMto
+reproduceEq.s(6.11). Thistime, thereisone-toonecorresponde ncebetweenthe
+self-duality Y-M and the Einstein-Maxwell equations. Even though t his is true,
+the question immediately arises about relevance of such solutions to the solution
+of the gap problem discussed in Section 5. In Section 5 the Ernst Eq.( 2.4) was
+used essentially for this purpose while Eq.s(6.11) are seemingly differe nt from
+Eq.(2.4). Again, fortunately, the difference is only apparent.
+From the definition of Bonnor transformations, Eq.(6.13), it follows that
+the potential Eis real. Also, from the same definition it follows that |Φ|2=
+ω2.Introduce now new potential Z=E+ω2. For it, we obtain
+∇Z=∇(E+ω2) =∇E+2ω∇ω=∇E+2¯Φ∇Φ. (6.14)
+Using this result, Eq.s(6.11) can be rewritten as follows
+/parenleftbig
+Z∇2−∇Z·∇/parenrightbig/parenleftbigg
+E
+ω/parenrightbigg
+= 0. (6.15)
+Furthermore, consider the related equation
+/parenleftbig
+Z∇2−∇Z·∇/parenrightbig
+ω2= 0. (6.16)
+Evidently, if it can be solved, then equation/parenleftbig
+Z∇2−∇Z·∇/parenrightbig
+ω= 0 can be
+solved as well. This being the case, the system of Eq.s(6.15) will be solv ed if
+the Ernst-type vacuum equation
+Z∇2=∇Z·∇Z (6.17)
+of the same type as Eq.(2.4) is solved. The obtained result is opposite to that
+derived by Bonnor, described in Appendix A (see also works Hauser a nd Ernst
+[124] and by Ivanov [125]). This means that, at least in some cases (h aving
+physical significance) the self-dual Y-M fields for both SU(2) and S U(3) gauge
+groups are obtainable as solutions of the Ernst Eq.(2.4). This means that all
+results of Section 5 obtained for SU(2) go through for the gauge g roup SU(3).
+With these results at our disposal we would like to discuss their applica tions
+to the Standard Model [19 ,126]. From Ref.[120] it is known that the matrix
+M∈SU(3) has subgroups which belong to SU(2). In particular, one of s uch
+subgroups is obtained if we let Φ = 0 in Eq.(6.12). Then, in view of Eq.(6.17 ),
+it is permissible to replace Ebyǫof Eq.(2.4). Thus, the obtained matrix Mis
+decomposable as M=M1+M2,where the matrix M1is given by
+M1=f−1
+1 0ω
+0 0 0
+ω0ǫ¯ǫ
+. (6.18)
+48in agreement with the matrix Mdefined by Eq.(2.15) since in this case f=
+−1
+2(ǫ+¯ǫ) =F.At the same time, the matrix M2is given by
+M2=
+0 0 0
+0 1 0
+0 0 0
+. (6.19)
+Using elementary operations with matrices we can represent matrix Min the
+form
+˜M=
+0 0 1
+a b0
+b c0
+ (6.20)
+wherea= 1/F,b=ω/Fandc= (F2+ω2)/F.Such a form of the matrix
+˜Mis typical for the semidirect product of groups (when group elemen ts are
+represented by matrices). In general case one should replace ˜Mby
+˜M=
+0 0 1
+a b α 1
+b c α 2
+
+Since the 2 ×2 submatrix belongs to SU(2) (because its determinant is 1) nor-
+mally describing a rotation in 3d space (in view of SU(2) ⇄SO(3) correspon-
+dence), the parameters α1andα2are responsible for translation. In this, more
+general case, the matrix Mdescribes the Galilean transformations, that is a
+combination of translations and rotations. If the translational mo tion is one
+dimensional it can be compactified to a circle in which case we obtain the cen-
+tralizerofSU(3) as SU(2) ×U(1). At the level ofLie algebrasu(3) this result was
+obtained in Ref.[127], pages 232 and 267. Its physical interpretatio n discussed
+in this reference is essentially the same as ours. The obtained centr alizer is
+the symmetry group of the Weinberg-Salam model (part of the sta ndard model
+describing electroweak interactions).
+All these arguments were meant only to demonstrate that the F-S -type
+model, Eq.(3.19), should be used for description of electroweak inte ractions.
+For description of strong interactions, in accord with Ref.[120], we c laim that
+the matrix Mgiven by Eq.(6.12) in which Eand Φ are taken from Bonnor’s
+Eq.s(6.13) is intrinsically of SU(3) type. That is, it cannot be obtained from
+the matrix M(in which Φ = 0) by applications of the N-K transformations,
+i.e. there are no transformations of the type M′(Φ) =AM(Φ = 0)A†.There-
+fore, this type of SU(3) matrix should be associated with QCD part o f the SM.
+Hence we have to use the Ernst functional, Eq.(3.18), instead of th e F-S-type,
+Eq.(3.19). These results provide resolution of the gap dilemma. Evidently, this
+resolution is equivalent to the statement that the symmetry group of the SM is
+SU(3)×SU(2)×U(1). This result should be taken into account in designing all
+possible grand unified theories (GUT). In the next subsection we sh all discuss
+the rigidity of this result.
+496.3.1 Remarkable rigidity of symmetries of the Standard Mod el and
+the extended Ricci flow
+In addition to Bonnor’s transformations there are many other tra nsformations
+from vacuum to electrovacuum. In particular, in Appendix A we ment ioned
+transformations discovered by Herlt. By looking at Eq.s(A.5)-(A.7) describing
+these transformations and comparing them with those by Bonnor, Eq.(6.13),
+it is an easy exercise to check that all arguments leading from Eq.s(6 .11) to
+(6.17) go through unchanged. By using superposition of N-K trans formations
+and those either by Bonnor or by Herlt it is possible to generate a cou ntable
+infinity of vacuum-to electrovacuum transformations such that t hey could be
+brought back to the vacuum Ernst solution, Eq.(6.17), using result s of previous
+subsection. This property of Einstein and Einstein -Maxwell equatio ns we shall
+call ”rigidity”. In view of results of previous subsection, this rigidity explains
+the remarkableempirical rigidity of symmetries of the SM. Indeed, s uppose that
+the color subgroup SU(3) can be replaced by SU(N), N >3. In such a case it
+is appropriate again to pose a question : Can self-dual Y-M fields-gr avity cor-
+respondence discovered by L.Witten for SU(2) be extended for SU (N), N>3?
+In Ref.[128] G¨ urses demonstrated that, indeed, this is possible bu t under non-
+physical conditions. Indeed, this correspondence requires for S U(n+1) self-dual
+Y-M fields to be in correspondence with the set of n-1 Einstein-Maxw ell fields.
+Sincen= 1 andn= 2 cases have been already described, we need only to
+worry about n >2. In such a case we shall have many-to-one correspondence
+between the replicas of electrovacuum and vacuum Einstein fields wh ich, while
+permissible mathematically, is not permissible physically since the Bonno r-type
+transformations require one-to-one correspondence between the vacuum and
+electrovacuum fields. Herrera-Aguillar and Kechkin, Ref.[129], foun d a way of
+transforming the compactified fields of heterotic string (e.g. see E q.(3.12)) into
+Einstein–multi-Maxwellfields of exactly the same type as discussed in the paper
+by G¨ urses [128]. While in the paper by G¨ urses these replicas of Maxw ell’s fields
+needed to be postulated, in [129] their stringy origin was found explic itly. From
+here, it follows that results obtained in this subsection make the minim al func-
+tional, Eq.(3.8), and the associated with it Perelman-like functional, E q.(3.13),
+universal. The universality of the associated with it Ricci flow, Eq.s (3 .14),
+has physical significance to be discussed below.
+7 Discussion
+7.1 Connections with loop quantum gravity
+A large portion of this paper was spent on justification, extension a nd exploita-
+tion of the remarkable correspondence between gravity and self- dual Y-M fields
+noticed by Louis Witten. Such correspondence is achievable only non per-
+turbatively. In a different form it was emphasized in the paper by Mas on and
+Newman [130] inspired by work by Ashtekar, Jacobson and Smolin [131 ]. It is
+not too difficult to notice that, in fact, papers [130,131] are compat ible with
+50Witten’s result since reobtaining of Nahm’s equations in the context o f grav-
+ity is the main result of Ref.[131]. In this context the Nahm equations a re
+just equations for moving triad on some 3-manifold. Since the conne ction of
+Nam’s equations with monopoles can be found in Ref.[68] and with instan tons
+in Ref.[132] the link with Witten’s results can be established, in principle. Since
+the authors of [131] are the main proponents of loop quantum grav ity (LQG)
+such refinements might be helpful for developments in the field of LQ G. We
+shall continue our discussion of LQG in the next subsection.
+7.2 Topology changing processes, the extended Ricci flow
+and the Higgs boson
+According to the existing opinion the SM does not account for effect s of grav-
+ity. At the same time, in the Introduction we mentioned that in recen t works
+by Smolin and collaborators [32-34] it was shown that ”topological fe atures of
+certain quantum gravity theories32can be interpreted as particles, matching
+known fermions and bosons of the first generation in the Standard Model”.
+Similar results were also independently obtained in works by Finkelstein , e.g.
+see Ref.[133] and references therein. In particular, Finkelstein re cognized that
+all quantum numbers describing basic building blocks(=particles) oft he SM can
+be neatly organized with help of numbers used for description of kno ts. More
+precisely, with projections of these knots onto some plane. It hap pens, that for
+description of all particles of the electroweak portion of the SM the numbers
+describing trefoil knot are sufficient. The task of topological/knot ty description
+of the entire SM was accomplished to some extent in Ref.[33]. This refe rence
+as well as Ref.s[32-34] in addition are capable of describing particle dy nam-
+ics/transformations. All these works share one common feature : calculations
+do notrequire Higgs boson. This fact is consistent with results discussed in
+subsection 4.3.1.
+The question arises: Is this feature a serious deficiency of these t opological
+methods or are these methods so superior to other, that the Higg s boson should
+be looked upon as an artifact of the previously existing perturbativ e methods
+used in SM calculations? To answer this self-imposed question require s several
+steps.
+First, we recall that according to the existing opinion the SM does no t
+account for effects of gravity. In such a case all the above result s should have
+nothing in common with the SM which is not true.
+Second, the results obtained in this paper indicate that knots/links /braids
+mentioned above have not only virtual (combinatorial/topological) b ut also
+differential-geometric description (Appendix B). Because of this, t opological
+description should be looked upon as complementary to that obtaina ble with
+help of the F-S-type models.
+Third, it is known that knot/link- describing Faddeev model can be co n-
+verted into Skyrme model [134]. It is also known that the Skyrme-ty pe models
+32That is LQG.
+51do not account for quarks explicitly , Ref.[68], page 349. This is not a serious
+drawback as we shall explain momentarily.
+Fourth, much more important for us is the fact that the Skyrme mo del can
+be used both in nuclear [135] and high energy [136 ]physics where it is used for
+description of both QCD ( nicely describing the entire known hadron spectra )
+and electroweak interactions.
+To account for quarks one has to go back to the Faddeev-type mo dels capa-
+ble of describing knots/links and to make a connection between thes ephysical
+knots/links and topological/combinatorial knots/ links discussed in Refs[32-
+34,133]. This is still insufficient! It is insufficient because Floer’s Eq.(4.7) co n-
+nects different vacua each is being described by the zero curvatur e condition
+Eq.(4.13). It is always possible to look at such a condition as describing some
+knot/link differential geometrically. With each knot, say in S3,some 3-manifold
+is associated. Furthermore such a manifold should be hyperbolic (su bsection
+3.6), that is either associated with hyperbolic-type knot/links [20,13 7] in S3
+or with knots/links ”living” in hyperboloid embedded in the Minkowski sp ace-
+time. Such a restriction is absent in Ref.s[32-34,133]. At the same time the Y-M
+functional, Eq.(4.12), is defined for a particular 3-manifold whose co nstruction
+is quite sophisticated. Eq.(4.7) describes processes of topology ch ange by con-
+necting different vacua. Such changes formally are not compatible w ith the fact
+that we are dealing with one and the same 3-manifold M ×[0,1]. From the math-
+ematical standpoint [11] no harm is made if one considers just this 3- manifold,
+e.g. read Ref.[11], page 22, bottom. Since particle dynamics is encode d in
+dynamics of transformations between knots/links, it causes us to consider tran-
+sitions between different 3-manifolds. These 3-manifolds should be c arefully
+glued together as described in Ref.[11]. In this picture particle dynam ics involv-
+ing particle scattering/transformation is synonymous with proces ses involving
+topology change. These are carried out naturally by instantons. S uch processes
+can be equivalently and more physically described in terms of the prop erties of
+the (extended) Ricci flow (subsection 3.4) following ideas of Perelma n’s proof of
+the Poincare′conjecture. Indeed, experimentally there is only finite number of
+stable particles. Without an exception, the end products of all sca ttering pro-
+cesses involve only stable particles. This observation matches perf ectly with the
+irreversibility of Ricci flow processes involving changes in topology: f rom more
+complex-to less complex 3-manifolds. Such Ricci flow model upon dev elopment
+could provide mathematical justification to otherwise rather vagu e statements
+by Finkelstein that ”more complicated knots ( particles) can theref ore dynami-
+cally decay to trefoils (stable particles)”, Ref.[133], page 10, botto m.
+7.3 Elementary particles as black holes
+In the paper [138] by Reina and Treves and also in [139] by Ernst it was found
+that for asymptotically flat Einstein-Maxwell fields generated from the vacuum
+fields by means of transformations of the type described above, in Section 6,
+the gyromagnetic factor g= 2. For the sake of space, we refer our readers to a
+recent review by Pfister and King [140] for definitions of gand many historical
+52facts and developments. In [140] it was noticed that such value of gis typical
+for most of stable particles of the SM. In view of the quantum gravit y-Y-M
+correspondence promoted in this paper, the interpretation of ele mentary par-
+ticles as black holes makes sense, especially in view of the following exce rpt
+from Ref.[38], page 526, ”There is one-to-one correspondence be tween station-
+ary vacuum fields with sources characterized by masses and angula r momenta
+and stationary Einstein-Maxwell fields with purely electromagnetic s ources, i.e.
+charges and currents.”
+Appendix A
+Peculiar interrelationship between gravitational, elect romagnetic
+and other fields
+Unification of gravity and electromagnetism was initiated by Nordstr ¨ om in
+1913- before general relativity was formulated by Einstein. Almost immediately
+after Einstein’s formulation, Kaluza, in 1921, and Klein, in 1926, prop osed uni-
+fication of electromagnetism and gravityby embedding Einstein’s 4-d imensional
+theory into 5 dimensional space in which 5th dimension is a circle. These results
+and their generalizations (up to 1987) can be found in the collection o f papers
+compiled by Applequist, Chodos and Freund [141]. Regrettably, this c ollection
+does not contain alternative theories of unification. Since such alte rnative theo-
+ries are much less known/popular to/with string and gravity theore ticians, here
+we provide a brief representative sketch of these alternative the ories.
+The 1st unified Einstein-Maxwell theory in 4-dimenssional space-tim e was
+proposed and solved by Rainich in 1925. It was discussed in great det ail by
+Misner and Wheeler [142]. After Rainich there appeared many other w orks on
+exact solutions of Einstein-Maxwell fields [38]. The most striking outc ome of
+these, more recent, works is the fact that multitude of exact solu tions of the
+combined Einstein-Maxwell equations can be obtained from solutions of the
+vacuumEinstein equations.
+In 1961 Bonnor [123]obtained the following remarkable result (e.g. re ad his
+Theorem 1). Suppose solutions of the vacuum Einstein equations ar e known.
+Using these solutions, it is possible to obtain a certain class of solution s of
+Einstein-Maxwell equations.
+In Section 6 we obtained the reverse result: Einstein’s solutions for pure
+gravity were obtained from solutions of the Einstein-Maxwell equat ions. With-
+out doing extra work, the electrovacuum solution obtained by Bonn or can be
+converted into that describing propagation of the combined cylindr ical gravita-
+tional and electromagnetic waves. With some additional efforts one can use the
+obtained results as an input for results describing the combined gra vitational,
+electromagnetic and neutrino wave propagation [143-144 ].
+The results by Bonnor comprise only a small portion of results conne cting
+static gravity fields with electromagnetic and neutrino fields. The ne xt example
+belongs to Herlt [38 ,145]. It provides a flavor of how this could be achieved.
+53We begin with Eq.(2.5). When written explicitly, this equation reads
+/parenleftbigg
+∂2
+ρ+1
+ρ∂ρ+∂2
+z/parenrightbigg
+u= 0. (A.1)
+Thistypeofsolutionistheresultofuseofthematrix M,Eq.(2.15),inEq.(2.14b).
+Nakamura [146] demonstrated that there is another matrix Qgiven by
+Q=/parenleftbiggf fω
+fω f2ω2−ρ2f−1/parenrightbigg
+(A.2)
+and the associated with it analog of Eq.(2.14b)
+∂ρ(ρ∂ρQ·Q−1)+∂z(ρ∂zQ·Q−1) = 0 (A.3)
+leading to the equation analogous to Eq.(A.1), that is
+/parenleftbigg
+∂2
+ρ−1
+ρ∂ρ+∂2
+z/parenrightbigg
+˜u= 0. (A.4)
+Nakamura demonstrated that the solution ˜ uis obtainable from solution of
+Eq.(A.1). and vice versa. Thus, instead of the Ernst Eq.(2.4) we can use
+Eq.(A.4). This fact plays crucial role in Hertl’s work. In it, he uses Eq.( A.4)
+to obtainuin Eq.(A.1) as follows
+exp(2u) =/parenleftbig
+˜u−1+G/parenrightbig2(A.5)
+withGgiven by
+G= ˜u,ρ[ρ(u2
+,ρ+u2
+,z)−˜u˜u,ρ]−1. (A.6)
+These results allow him to introduce a potential χvia
+χ= ˜u−1−G. (A.7)
+Using the original work of Ernst [43] as well as Ref.[38], we find that so lution
+of the static axially symmetric coupled Einstein-Maxwell equations is g iven in
+terms of complex potentials ǫand Φ.In particular, in purely electrostatic case
+one hasǫ=¯ǫ=e2u−χand Φ = ¯Φ =χwhile the magnetostatic case is obtained
+from the electrostatic by requiring -Φ = ¯Φ =ψandǫ=¯ǫ=e2u−ψ. In this
+caseψis just relabeled χ.Ref.[38] contains many other examples of the cou-
+pledEinstein-Maxwellequationsobtainedfromthevacuum solutions ofEinstein
+equations.
+The aboveresultsshouldbe lookedupon fromthe standpoint offun damental
+problemoftheenergy-momentumconservationingeneralrelativit yrequiringin-
+troduction(inthesimplestcase)oftheLandau-Lifshitz(L-L)ene rgy-momentum
+pseudotensor. The description of more complicated pseudotenso rs (incorporat-
+ing that by L-L) can be found in the monograph by Ortin [147]. To this o ne
+should add the problem about the positivity of mass in general relativ ity. The
+difficulties withthese conceptsstem fromtheverybasicobservatio n, lyingatthe
+54heart of general relativity, that at any given point of space-time g ravity field
+can be eliminated by moving in the appropriately chosen accelerating f rame
+(the equivalence principle). This fact leaves unexplained the origin of the tidal
+forces requiring observation of motion of at least two test particle s separated
+by some nonzero distance. The explanation of this phenomenon with in general
+relativity framework is nontrivial.It can be found in [148]. In turn, it lea ds to
+speculations about the limiting procedure leading to elimination of grav ity at a
+given point33. Apparently, this problem is still not solved rigorously[147]. An
+outstandig collection of rigorous results on general relativity can b e found in
+the recent monograph by Choquet-Bruhat [149] while [150] discuss es peculiar
+relationship between the Newtonian and Einsteinian gravities at the s cale of
+our Solar system.
+Conversely, one can think of other fields at the point/domain where gravity
+is absent as subtle manifestations of gravity. Interestingly enoug h, such an idea
+was originally put forward by Rainich already in 1925 ! Recent status o f these
+ideas is given in paper by Ivanov [125]. From such a standpoint, the fu nctional
+given by Eq.(3.13) (that is the Perelman-like entropy functional) is su fficient for
+description of all fields with integer spin. With minor modifications (e.g. in-
+volving either the Newman-Penrose formalism [143,144] or supersym meric for-
+malism used in calculation of Seiberg-Witten invariants [66]), it can be us ed for
+description of all known fields in nature.
+Appendix B
+Some facts about integrable dynamics of knotted vortex filam ents
+B.1Connection with the Landau-Lifshitz equation
+Following Ref.[85], we discuss motion of a vortex filament in the incompre ss-
+ible fluid. Some historical facts relating this problem to string theory are given
+in our recent work, Ref.[84]. Let ube a velocity field in the fluid such that
+divu= 0. Therefore, we can write u=∇×A. Next, we define the vorticity
+w=∇×uso that eventually,
+u=−1
+4π/integraldisplay
+d3x(x−x′)×w(x′)
+/ba∇dblx−x′/ba∇dbl3. (B.1a)
+This expression can be simplified by assuming that there is a linevortex which
+is modelled by a tube with a cross-sectionalarea dAand such that the vorticity
+wis everywhere tangent to the line vortex and has a constant magnit ude w.
+Let then Γ =/integraltext
+wdAso that
+u=−Γ
+4π/contintegraldisplay(x−x′)×dγ
+/ba∇dblx−x′/ba∇dbl3(B.1b)
+withdγbeing an infinitesimal line segment along the vortex. Such a model
+of a vortex resembles very much model used for description of dyn amics of
+ring polymers [84]. Because of this, it is convenient to make the followin g
+33The abundance of available energy-momentum pseudotensors is result of these specula-
+tions.
+55identification : u(γ(s,t)) =∂γ
+∂t(s,t), withsbeing a position along the vortex
+contour and t-time. This allows us to write
+∂γ
+∂t(s′,t) =−Γ
+4π/contintegraldisplay(γ(s′,t)−γ(s,t))
+/ba∇dblγ(s′,t)−γ(s,t)/ba∇dbl3×∂γ
+∂sds (B.1c)
+and to make a Taylor series expansion in order to rewrite Eq.(B1c) as
+∂γ
+∂t=Γ
+4π[∂γ
+∂s′×∂2γ
+∂s′2/integraldisplayds
+|s−s′|+...]. (B.1d)
+In this expression only the leading order result is written explicitly. By intro-
+ducing a cut off εsuch that |s−s′| ≥εand by rescaling time: t→Γ
+4πtln(ε−1)
+one finally arrives at the basic vortex filament equation
+∂γ
+∂t=∂γ
+∂s′×∂2γ
+∂s′2. (B.2)
+Introduce now the Serret-Frenet frame made of vectors B,TandNso that
+B=T×N,ˇκN=dT
+ds,T=∂γ
+∂s,where ˇκis a curvature of γ. Then, Eq.(B.2) can
+be equivalently rewritten as
+∂γ
+∂t= ˇκB (B.3)
+or, as
+∂T
+∂t=T×Txx. (B.4)
+In the last equation the replacement s⇌xwas made so that the obtained
+equationcoincides with the Landau-Lifshitz (L-L) equationdescrib ing dynamics
+of 1d Heisenberg ferromagnets [86].
+B.2Hashimoto map and the Gross-Pitaevskii equation
+Hashimoto [85] found ingenious way to transform the L-L equation in to the
+nonlinear Scr¨ odinger equation (NLSE) which is also widely known in con densed
+matter physics literature as the Gross-Pitaevskii (G-P) equation [96]. Because
+of its is uses in nonlinear optics and in condensed matter physics for d escrip-
+tion of the Bose-Einstein condensation (BEC) theory of this equat ion is well
+developed. Some facts from this theory are discussed in the main te xt. Here we
+provide a sketch of how Hashimoto arrived at his result.
+LetT,UandVbe another triad such that
+U= cos(x/integraldisplay
+τds)N−sin(x/integraldisplay
+τds)B,V= sin(x/integraldisplay
+τds)N+cos(x/integraldisplay
+τds)B(B.5)
+in whichτis the torsion of the curve. Introduce new curvatures κ1andκ2in
+such a way that
+κ1= ˇκcos(x/integraldisplay
+τds) andκ2= ˇκsin(x/integraldisplay
+τds),
+56then, it can be shown that
+∂γ
+∂t=−κ2U+κ1V (B.6a)
+and
+∂2γ
+∂x2=κ1U+κ2V. (B.6b)
+Using these equations and taking into account that UtV=−UVtafter some
+algebra one obtains the following equation
+iψt+ψxx+[1
+2|ψ|2−A(t)]ψ= 0 (B.7)
+in whichψ=κ1+iκ2andA(t) is some arbitrary x-independent function. By
+replacingψwithψexp(−it/integraltextdt′A(t′)) in this equation we arrive at the canonical
+form of the NLSE which is also known as focussing cubic NLSE.
+iψt+ψxx+1
+2|ψ|2ψ= 0 (B.8)
+Itcanbeshownthatitssolutionallowsustorestoretheshapeofth ecurve/filament
+γ(s,t).The G-P equation can be identified with Eq.(B.7) if we make A(t) time-
+independent. In its canonical form it is written as (in the system of u nits in
+whichℏ= 1,m= 1/2) [86]
+iψt=−ψxx+2κ/parenleftBig
+|ψ|2−c2/parenrightBig
+ψ= 0. (B.9)
+Ingeneral,the signofthecouplingconstant κcanbebothpositiveandnegative.
+In view ofEq.(B.8), when motion of the vortexfilament takes place in E uclidean
+space, the sign of κis negative. This is important if one is interested in dynamic
+ofknottedvortexfilaments[85]. Forpurposesofthisworkitis alsoofinterestt o
+study motion of vortex filaments in the Minkowski and related (hype rbolic, de
+Sitter ) spaces. This should be done with some caution since the tran sition from
+Eq.(B.1a) to (B.2) is specific for Euclidean space. Thus, study can be made at
+the level of Eq.s (B.3) and (B.4). Fortunately, such study was perf ormed quite
+recently [94,95]. The summary of results obtained in these papers ca n be made
+with help of the following definitions. Introduce a vector n={n1,n2,n3}so
+that the unit sphere S2is defined by
+S2:n2
+1+n2
+2+n2
+3= 1. (B.10)
+Respectively, the de Sitter space S1,1(or unit pseudo sphere in R2,1) is defined
+by
+S1,1:n2
+1+n2
+2−n2
+3= 1, (B.11)
+while the hyperbolic space H2(or hyperboloid embedded in R2,1) is defined by
+H2:n2
+1+n2
+2−n2
+3=−1,n3>0. (B.12)
+57Using these definitions, it was proven in [94,95] that: a) for both de S itter and
+H2spaces there are analogs of the L-L equation ( e.g. those discusse d in the
+main text, in subsection 5.2.); b) the Hasimoto map can be extended f or these
+spacesso that the respective L-L equationsare transformed int o the same NLSE
+(or G-P) equation in which κispositive.
+58References
+[1] C.N Yang, R.Mills, Conservation of isotopic spin and isotopic gauge
+invariance, Phys.Rev. 96 (1954) 191-195.
+[2] R.Utiyama, Invariant theoretical interpretation of interaction ,
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\ No newline at end of file
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+arXiv:1001.0030v2  [math.CO]  28 Feb 2012CYCLIC SIEVING FOR GENERALISED NON-CROSSING
+PARTITIONS ASSOCIATED WITH COMPLEX REFLECTION
+GROUPS OF EXCEPTIONAL TYPE — THE DETAILS
+Christian Krattenthaler†andThomas W. M ¨uller‡
+†Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien,
+Nordbergstraße 15, A-1090 Vienna, Austria.
+WWW:http://www.mat.univie.ac.at/ ~kratt
+‡School of Mathematical Sciences,
+Queen Mary & Westfield College, University of London,
+Mile End Road, London E1 4NS, United Kingdom.
+WWW:http://www.maths.qmw.ac.uk/ ~twm/
+Dedicated to the memory of Herb Wilf
+Abstract. We prove that the generalised non-crossing partitions associated with
+well-generated complex reflection groups of exceptional type obe y two different cyclic
+sieving phenomena, as conjectured by Armstrong, and by Bessis a nd Reiner. This
+manuscript accompanies the paper “Cyclic sieving for generalised non-crossing parti-
+tions associated with complex reflection groups of exceptio nal type” [arχiv:1001.0028 ],
+for which it provides the computational details.
+1.Introduction
+In his memoir [2], Armstrong introduced generalised non-crossing partitions asso-
+ciated with finite (real) reflection groups, thereby embedding Krew eras’ non-crossing
+partitions [22], Edelman’s m-divisible non-crossing partitions [12], thenon-crossing par-
+titions associated with reflection groups due to Bessis [6] and Brady and Watt [10] into
+one uniform framework. Bessis and Reiner [9] observed that Arms trong’s definition can
+be straightforwardly extended to well-generated complex reflection groups (see Section 2
+for the precise definition). These generalised non-crossing partit ions possess a wealth
+of beautiful properties, and they display deep and surprising relat ions to other combi-
+natorial objects defined for reflection groups (such as the gene ralised cluster complex
+2000Mathematics Subject Classification. Primary 05E15; Secondary 05A10 05A15 05A18 06A07
+20F55.
+Key words and phrases. complex reflection groups, unitary reflection groups, m-divisible non-
+crossing partitions, generalised non-crossing partitions, Fuß–Ca talan numbers, cyclic sieving.
+†Research partially supported by the Austrian Science Foundation F WF, grants Z130-N13 and
+S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics
+and Probabilistic Number Theory.”
+‡Research supported by the Austrian Science Foundation FWF, Lise Meitner grant M1201-N13.
+12 C. KRATTENTHALER AND T. W. M ¨ULLER
+of Fomin and Reading [13], or the extended Shi arrangement and the geometric multi-
+chains of filters of Athanasiadis [4, 5]); see Armstrong’s memoir [2] and the references
+given therein.
+Ontheotherhand, cyclic sieving isaphenomenonbroughttolightbyReiner, Stanton
+and White [30]. It extends the so-called “( −1)-phenomenon” of Stembridge [34, 35].
+Cyclic sieving can be defined in three equivalent ways (cf. [30, Prop. 2.1]). The one
+which gives the name can be described as follows: given a set Sof combinatorial
+objects, an action on Sof a cyclic group G=/an}bracketle{tg/an}bracketri}htwith generator gof ordern, and
+a polynomial P(q) inqwith non-negative integer coefficients, we say that the triple
+(S,P,G)exhibits the cyclic sieving phenomenon , if the number of elements of Sfixed
+bygkequalsP(e2πik/n). In [30] it is shown that this phenomenon occurs in surprisingly
+many contexts, and several further instances have been discov ered since then.
+In [2, Conj. 5.4.7] (also appearing in [9, Conj. 6.4]) and [9, Conj. 6.5], Ar mstrong,
+respectively Bessis and Reiner, conjecture that generalised non- crossing partitions for
+irreducible well-generated complex reflection groups exhibit two diffe rent cyclic sieving
+phenomena (see Sections 3 and 7 for the precise statements).
+According to the classification of these groups due to Shephard an d Todd [32], there
+are two infinite families of irreducible well-generated complex reflectio n groups, namely
+the groups G(d,1,n) andG(e,e,n), wheren,d,eare positive integers, and there are 26
+exceptional groups. For the infinite families of types G(d,1,n) andG(e,e,n), the two
+cyclic sieving conjectures follow from the results in [19].
+The purpose of the present article is to prove the cyclic sieving conj ectures of Arm-
+strong, and of Bessis and Reiner, for the 26 exceptional types, t hus completing the
+proof of these conjectures. Since the generalised non-crossing partitions feature a pa-
+rameterm, from the outset this is nota finite problem. Consequently, we first need
+several auxiliary results to reduce the conjectures for each of t he 26 exceptional types
+to afiniteproblem. Subsequently, we use Stembridge’s Maplepackagecoxeter [36]
+and theGAPpackageCHEVIE[14, 28] to carry out the remaining finitecomputations.
+It is interesting to observe that, for the verification of the type E8case, it is essential
+to use the decomposition numbers in the sense of [17, 18, 20] becau se, otherwise, the
+necessary computations would not be feasible in reasonable time with the currently
+available computer facilities. We point out that, for the special case where the afore-
+mentioned parameter mis equal to 1, the first cyclic sieving conjecture has been proved
+in a uniform fashion by Bessis and Reiner in [9]. (See [3] for a — non-unifo rm — proof
+of cyclic sieving for non-crossing partitions associated with realreflection groups under
+the action of the so-called Kreweras map, a special case of the sec ond cyclic sieving
+phenomenon discussed in the present paper.) The crucial result on which the proof of
+Bessis andReiner is based is(5.5) below, andit plays animportant role in our reduction
+of the conjectures for the 26 exceptional groups to a finite prob lem.
+Our paper is organised as follows. In the next section, we recall the definition of
+generalised non-crossing partitions for well-generated complex re flection groups and of
+decomposition numbers in the sense of [17, 18, 20], and we review so me basic facts.
+The first cyclic sieving conjecture is subsequently stated in Section 3. In Section 4, we
+outline an elementary proof that the q-Fuß–Catalan number, which is the polynomial
+Pin the cyclic sieving phenomena concerning the generalised non-cros sing partitions
+for well-generated complex reflection groups, is always a polynomial with non-negative
+integer coefficients, asrequired bythedefinitionofcyclic sieving. (T he readerisreferredCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 3
+to the first paragraph of Section 4 for comments on other approa ches for establishing
+polynomiality with non-negative coefficients.) Section 5 contains the a nnounced auxil-
+iary results which, for the 26 exceptional types, allow a reduction o f the conjecture to a
+finite problem. The remaining case-by-case verification of the conj ecture is then carried
+out in Section 6. The second cyclic sieving conjecture is stated in Sec tion 7. Section 8
+contains the auxiliary results which, for the 26 exceptional types, allow a reduction of
+the conjecture to a finite problem, while Section 9 contains the rema ining case-by-case
+verification of the conjecture.
+2.Preliminaries
+Acomplex reflection group isa groupgeneratedby(complex) reflections in Cn. (Here,
+a reflection is a non-trivial element of GLn(C) which fixes a hyperplane pointwise and
+which hasfiniteorder.) Wereferto[24]foranin-depthexpositionof thetheorycomplex
+reflection groups.
+Shephard and Todd provided a complete classification of all finitecomplex reflection
+groups in [32] (see also [24, Ch. 8]). According to this classification, a n arbitrary
+complex reflection group Wdecomposes into a direct product of irreducible complex
+reflection groups, acting on mutually orthogonal subspaces of th e complex vector space
+onwhichWisacting. Moreover, thelistofirreduciblecomplexreflectiongroups consists
+of the infinite family of groups G(m,p,n), wherem,p,nare positive integers, and 34
+exceptional groups, denoted G4,G5,...,G 37by Shephard and Todd.
+In this paper, we are only interested in finite complex reflection grou ps which are
+well-generated . A complex reflection group of rank nis called well-generated if it is
+generated by nreflections.1Well-generation can be equivalently characterised by a
+duality property due to Orlik and Solomon [29]. Namely, a complex reflec tion group of
+ranknhastwo sets ofdistinguished integers d1≤d2≤ ··· ≤dnandd∗
+1≥d∗
+2≥ ··· ≥d∗
+n,
+called its degreesandcodegrees , respectively (see [24, p. 51 and Def. 10.27]). Orlik and
+Solomon observed, using case-by-case checking, that an irreduc ible complex reflection
+groupWof ranknis well-generated if and only if its degrees and codegrees satisfy
+di+d∗
+i=dn
+for alli= 1,2,...,n. The reader is referred to [24, App. D.2] for a table of the degree s
+and codegrees of all irreducible complex reflection groups. Togeth er with the classi-
+fication of Shephard and Todd [32], this constitutes a classification o f well-generated
+complex reflection groups: the irreducible well-generated complex r eflection groups are
+— the two infinite families G(d,1,n) andG(e,e,n), whered,e,nare positive inte-
+gers,
+— the exceptional groups G4,G5,G6,G8,G9,G10,G14,G16,G17,G18,G20,G21of
+rank 2,
+— the exceptional groups G23=H3,G24,G25,G26,G27of rank 3,
+— the exceptional groups G28=F4,G29,G30=H4,G32of rank 4,
+— the exceptional group G33of rank 5,
+— the exceptional groups G34,G35=E6of rank 6,
+— the exceptional group G36=E7of rank 7,
+1We refer to [24, Def. 1.29] for the precise definition of “rank.” Roug hly speaking, the rank of a
+complex reflection group Wis the minimal nsuch that Wcan be realized as reflection group on Cn.4 C. KRATTENTHALER AND T. W. M ¨ULLER
+— and the exceptional group G37=E8of rank 8.
+In this list, we have made visible the groups H3,F4,H4,E6,E7,E8which appear as
+exceptional groups in the classification of all irreducible realreflection groups (cf. [16]).
+LetWbe a well-generated complex reflection group of rank n, and letT⊆Wdenote
+theset of all(complex) reflections inthegroup. Let ℓT:W→Zdenotethewordlength
+in terms of the generators T. This word length is called absolute length orreflection
+length. Furthermore, we define a partial order ≤TonWby
+u≤Twif and only if ℓT(w) =ℓT(u)+ℓT(u−1w). (2.1)
+This partial order is called absolute order orreflection order . As is well-known and
+easy to see, the equation in (2.1) is equivalent to the statement tha t every shortest
+representation of uby reflections occurs as an initial segment in some shortest produc t
+representation of wby reflections.
+Now fix a (generalised) Coxeter element2c∈Wand a positive integer m. The
+m-divisible non-crossing partitions NCm(W) are defined as the set
+NCm(W) =/braceleftbig
+(w0;w1,...,w m) :w0w1···wm=cand
+ℓT(w0)+ℓT(w1)+···+ℓT(wm) =ℓT(c)/bracerightbig
+.
+A partial order is defined on this set by
+(w0;w1,...,w m)≤(u0;u1,...,u m) if and only if ui≤Twifor 1≤i≤m.
+We have suppressed the dependence on c, since we understand this definition up to
+isomorphism of posets. To be more precise, it can be shown that any two Coxeter
+elements are related to each other by conjugation and (possibly) a n automorphism on
+the field of complex numbers (see [33, Theorem 4.2] or [24, Cor. 11.2 5]), and hence the
+resulting posets NCm(W) are isomorphic to each other. If m= 1, thenNC1(W) can
+be identified with the set NC(W) of non-crossing partitions for the (complex) reflection
+groupWasdefined byBessis andCorran(cf.[8]and[7, Sec.13]; theirdefinit ionextends
+the earlier definition by Bessis [6] and Brady and Watt [10] for real r eflection groups).
+The following result has been proved by a collaborative effort of seve ral authors (see
+[7, Prop. 13.1]).
+Theorem 1. LetWbe an irreducible well-generated complex reflection group, and let
+d1≤d2≤ ··· ≤dnbe its degrees and h:=dnits Coxeter number. Then
+|NCm(W)|=n/productdisplay
+i=1mh+di
+di. (2.2)
+2An element of an irreducible well-generated complex reflection group Wof ranknis called a
+Coxeter element if it isregularin the sense of Springer [33] (see also [24, Def. 11.21]) and of order dn.
+An element of Wis called regular if it has an eigenvector which lies in no reflecting hyperp lane of a
+reflection of W. It follows from an observation of Lehrer and Springer, proved un iformly by Lehrer
+and Michel [23] (see [24, Theorem 11.28]), that there is always a regu lar element of order dnin an
+irreducible well-generated complex reflection group Wof rankn. More generally, if a well-generated
+complex reflection group Wdecomposes as W∼=W1×W2×···×Wk, where the Wi’s are irreducible,
+then a Coxeter element of Wis an element of the form c=c1c2···ck, whereciis a Coxeter element of
+Wi,i= 1,2,...,k. IfWis arealreflection group, that is, if all generators in Thave order 2, then the
+notion of generalised Coxeter element given above reduces to that of a Coxeter element in the classical
+sense (cf. [16, Sec. 3.16]).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 5
+Remark1.(1) The number in (2.2) is called the Fuß–Catalan number for the reflection
+groupW.
+(2) Ifcis a Coxeter element of a well-generated complex reflection group Wof rank
+n, thenℓT(c) =n. (This follows from [7, Sec. 7].)
+We conclude this section by recalling the definition of decomposition nu mbers from
+[17, 18, 20]. Although we need them here only for (very small) real re flection groups,
+and although, strictly speaking, they have been only defined for re al reflection groups in
+[17, 18, 20], this definition can be extended to well-generated comple x reflection groups
+without any extra effort, which we do now.
+Given a well-generated complex reflection group Wof rankn, typesT1,T2,...,T d(in
+the sense of the classification of well-generated complex reflection groups) such that the
+sumoftheranksofthe Ti’sequalsn, andaCoxeter element c, thedecompositionnumber
+NW(T1,T2,...,T d) is defined as the number of “minimal” factorisations c=c1c2···cd,
+“minimal” meaning that ℓT(c1) +ℓT(c2) +···+ℓT(cd) =ℓT(c) =n, such that, for
+i= 1,2,...,d, the type of cias a parabolic Coxeter element is Ti. (Here, the term
+“parabolic Coxeter element” means a Coxeter element in some parab olic subgroup. It
+follows from [31, Prop.6.3] that any element ciis indeed a Coxeter element in a unique
+parabolic subgroup of W.3By definition, the type of ciis the type of this parabolic
+subgroup.) Since any two Coxeter elements are related to each oth er by conjugation
+plus field automorphism, the decomposition numbers are independen t of the choice of
+the Coxeter element c.
+The decomposition numbers for real reflection groups have been c omputed in [17,
+18, 20]. To compute the decomposition numbers for well-generated complex reflection
+groups is a task that remains to be done.
+3.Cyclic sieving I
+In this section we present the first cyclic sieving conjecture due to Armstrong [2,
+Conj. 5.4.7], and to Bessis and Reiner [9, Conj. 6.4].
+Letφ:NCm(W)→NCm(W) be the map defined by
+(w0;w1,...,w m)/mapsto→/parenleftbig
+(cwmc−1)w0(cwmc−1)−1;cwmc−1,w1,w2,...,w m−1/parenrightbig
+.(3.1)
+It is indeed not difficult to see that, if the ( m+ 1)-tuple on the left-hand side is an
+element ofNCm(W), then so is the ( m+1)-tuple on the right-hand side. For m= 1,
+this action reduces to conjugation by the Coxeter element c(applied to w1). Cyclic
+sieving arising from conjugation by chas been the subject of [9].
+It is easy to see that φmhacts as the identity, where his the Coxeter number of W
+(see (5.1) and Lemma 29 below). By slight abuse of notation, let C1be the cyclic group
+of ordermhgenerated by φ. (The slight abuse consists in the fact that we insist on C1
+to be a cyclic group of order mh, while it may happen that the order of the action of
+φgiven in (3.1) is actually a proper divisor of mh.)
+3The uniqueness can be argued as follows: suppose that ciwere a Coxeter element in two parabolic
+subgroups of W, sayU1andU2. Then it must also be a Coxeter element in the intersection U1∩U2.
+On the other hand, the absolute length of a Coxeter element of a co mplex reflection group Uis always
+equal to rk( U), the rank of U. (This follows from the fact that, for each element uofU, we have
+ℓT(u) = codim/parenleftbig
+ker(u−id)/parenrightbig
+, with id denoting the identity element in U; see e.g. [31, Prop. 1.3]). We
+conclude that ℓT(ci) = rk(U1) = rk(U2) = rk(U1∩U2), This implies that U1=U2.6 C. KRATTENTHALER AND T. W. M ¨ULLER
+Given these definitions, we are now in the position to state the first c yclic sieving
+conjecture of Armstrong, respectively of Bessis and Reiner. By t he results of [19] and
+of this paper, it becomes the following theorem.
+Theorem 2. For an irreducible well-generated complex reflection group Wand any
+m≥1, the triple (NCm(W),Catm(W;q),C1), whereCatm(W;q)is theq-analogue of
+the Fuß–Catalan number defined by
+Catm(W;q) :=n/productdisplay
+i=1[mh+di]q
+[di]q, (3.2)
+exhibits the cyclic sieving phenomenon in the sense of Reine r, Stanton and White [30].
+Here,nis the rank of W,d1,d2,...,d nare the degrees of W,his the Coxeter number
+ofW, and[α]q:= (1−qα)/(1−q).
+Remark2.We write Catm(W) for Catm(W;1).
+By definition of the cyclic sieving phenomenon, we have to prove that Catm(W;q) is
+a polynomial in qwith non-negative integer coefficients, and that
+|FixNCm(W)(φp)|= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh, (3.3)
+for allpin the range 0 ≤p<mh. The first fact is established in the next section, while
+the proof of the second is achieved by making use of several auxiliar y results, given
+in Section 5, to reduce the proof to a finite problem, and a subseque nt case-by-case
+analysis, which occupies Section 6.
+4.Theq-Fusz–Catalan numbers Catm(W;q)
+The purpose of this section is to provide an elementary, self-conta ined proof of the
+fact that, for all irreducible complex reflection groups W, theq-Fuß–Catalan number
+Catm(W;q) is a polynomial in qwith non-negative integer coefficients. For most of
+the groups, this is a known property. However, aside from the fac t that, for many of
+the known cases, the proof is very indirect and uses deep algebraic results on rational
+Cherednik algebras, there still remained some cases where this pro perty had not been
+formally established. The reader is referred to the “Theorem” in Se ction 1.6 of [15],
+whichsaysthat, undertheassumptionofacertainrankcondition( [15, Hypothesis2.4]),
+theq-Fuß–Catalan number Catm(W;q) is a Hilbert series of a finite-dimensional quo-
+tient of the ring of invariants of Wand also the graded character of a finite-dimensional
+irreducible representation of a spherical rational Cherednik algeb ra associated with
+W. At present, this rank condition has been proven for all irreducible well-generated
+complex reflection groups apart from G17,G18,G29,G33,G34; see [26, Tables 8 and 9,
+column “rank”], and the recent paper [27], which establishes the res ult in the case of
+G32.
+In the sequel, aside from the standard notation [ α]q= (1−qα)/(1−q) forq-integers,
+we shall also use the q-binomial coefficient, which is defined by
+/bracketleftbigg
+n
+k/bracketrightbigg
+q:=/braceleftBigg
+1, ifk= 0,
+[n]q[n−1]q···[n−k+1]q
+[k]q[k−1]q···[1]q,ifk>0.
+We begin with several auxiliary results.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 7
+Proposition 3. For all non-negative integers nandk, theq-binomial coefficient [n
+k]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.This is a well-known fact, which can be derived either from the recurr ence rela-
+tion(s) satisfied by the q-binomial coefficients (generalising Pascal’s recurrence relation
+for binomial coefficients; cf. [1, eqs. (3.3.3) and (3.3.4)]), or from th e fact that the q-
+binomial coefficient [n
+k]qis the generating function for (integer) partitions with at most
+kparts all of which are at most n−k(cf. [1, Theorem 3.1]). /square
+Proposition 4. For all non-negative integers mandn, theq-Fuß–Catalan number of
+typeAn,
+1
+[(m+1)n+1]q/bracketleftbigg
+(m+1)n+1
+n/bracketrightbigg
+q,
+is a polynomial in qwith non-negative integer coefficients.
+Proof.In [25, Sec. 3.3], Loehr proves that
+1
+[(m+1)n+1]q/bracketleftbigg
+(m+1)n+1
+n/bracketrightbigg
+q
+=/summationdisplay
+v∈V(m)
+nqm(n
+2)+/summationtext
+i≥0(m(vi
+2)−ivi)/productdisplay
+i≥1qvi/summationtextm
+j=1(m−j)vi−j/bracketleftbigg
+vi+vi−1+···+vi−m−1
+vi/bracketrightbigg
+q,(4.1)
+whereV(m)
+ndenotes the set of all sequences v= (v0,v1,...,v s) (for some s) of non-
+negative integers with v0>0,vs>0, andv0+v1+···+vs=n, and such that there
+is never a string of mor more consecutive zeroes in v. By convention, vi= 0 for all
+negativei. His proof works by showing that the expressions on both sides of ( 4.1)
+satisfy the same recurrence relation and initial conditions, using cla ssicalq-binomial
+identities. We refer the reader to [25] for details. By Proposition 3, the expression on
+the right-hand side of (4.1) is manifestly a polynomial in qwith non-negative integer
+coefficients. /square
+Lemma 5. Ifaandbare coprime positive integers, then
+[ab]q
+[a]q[b]q(4.2)
+is a polynomial in qof degree (a−1)(b−1), all of whose coefficients are in {0,1,−1}.
+Moreover, if one disregards the coefficients which are 0, then+1’s and(−1)’s alternate,
+and the constant coefficient as well as the leading coefficient o f the polynomial equal +1.
+Proof.LetΦn(q)denotethe n-thcyclotomicpolynomialin q. Usingtheclassicalformula
+1−qn=/productdisplay
+d|nΦd(q),
+we see that
+(1−q)(1−qab)
+(1−qa)(1−qb)=/productdisplay
+d1|a,d1/ne}ationslash=1
+d2|a,d2/ne}ationslash=1Φd1d2(q),
+so that, manifestly, the expression in (4.2) is a polynomial in q. The claim concerning
+the degree of this polynomial is obvious.8 C. KRATTENTHALER AND T. W. M ¨ULLER
+In order to establish the claim on the coefficients, we start with a sub -expression of
+(4.2),
+(1−qab)
+(1−qa)(1−qb)=/parenleftbiggb−1/summationdisplay
+i=0qia/parenrightbigg/parenleftbigg∞/summationdisplay
+j=0qjb/parenrightbigg
+=∞/summationdisplay
+k=0Ckqk, (4.3)
+say. The assumption that aandbare coprime implies that 0 ≤Ck≤1 fork≤
+(a−1)(b−1). Multiplying both sides of (4.3) by 1 −q, we obtain the equation
+[ab]q
+[a]q[b]q= (1−q)(a−1)(b−1)/summationdisplay
+k=0Ckqk+(1−q)∞/summationdisplay
+k=(a−1)(b−1)+1Ckqk. (4.4)
+By our previous observation on the coefficients Ckwithk≤(a−1)(b−1), it is obvious
+that the coefficients of the first expression on the right-hand side of (4.4) are alternately
++1 and−1, when 0’s are disregarded. Since we already know that the left-ha nd side is
+a polynomial in qof degree (a−1)(b−1), we may ignore the second expression.
+The proof is concluded by observing that the claims on the constant and leading
+coefficients are obvious. /square
+Corollary 6. Letaandbbe coprime positive integers, and let γbe an integer with
+γ≥(a−1)(b−1). Then the expression
+[γ]q[ab]q
+[a]q[b]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.Let
+[ab]q
+[a]q[b]q=(a−1)(b−1)/summationdisplay
+k=0Dkqk.
+We then have
+[γ]q[ab]q
+[a]q[b]q=(a−1)(b−1)+γ−1/summationdisplay
+N=0qNN/summationdisplay
+k=max{0,N−γ+1}Dk. (4.5)
+IfN≤γ−1, then, by Lemma 5, the sum over kon the right-hand side of (4.5) equals
+1−1+1−1+···, which is manifestly non-negative. On the other hand, if N >γ−1,
+then we may rewrite the sum over kon the right-hand side of (4.5) as
+N/summationdisplay
+k=max{0,N−γ+1}Dk=(a−1)(b−1)/summationdisplay
+k=N−γ+1Dk=(a−1)(b−1)+γ−1−N/summationdisplay
+k=0D(a−1)(b−1)−k.
+Again, by Lemma 5, this sum equals 1 −1 + 1−1 +···, which is manifestly non-
+negative. /square
+Lemma 7. Letαandβbe positive integers with α≥6andβ≥8. Then the expression
+[α]q3[β]q4[72]q[3]q[4]q
+[8]q[9]q[12]q
+is a polynomial in qwith non-negative integer coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 9
+Proof.We have
+[72]q[3]q[4]q
+[8]q[9]q[12]q
+= (1−q3+q9−q15+q18)(1−q4+q8−q12+q16−q20+q24−q28+q32).
+It should be observed that both factors on the right-hand side ha ve the property that
+coefficients are in {0,1,−1}and that (+1)’s and ( −1)’s alternate, if one disregards the
+coefficients which are 0. If we now apply the same idea as in the proof o f Corollary 6,
+then we see that [ α]q3times the first factor is a polynomial in qwith non-negative
+integer coefficients, as is [ β]q4times the second factor. Taken together, this establishes
+the claim. /square
+Lemma 8. Letαandβbe positive integers with α≥26andβ≥8. Then the expression
+[α]q[β]q4[15]q
+[3]q[5]q[72]q[3]q[4]q
+[8]q[9]q[12]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[15]q[72]q[4]q
+[5]q[8]q[9]q[12]q
+= (1−q+q5−q6+q9−q11+q12−q13+q14−q15+q17−q20+q21−q25+q26)
+×(1−q4+q8−q12+q16−q20+q24−q28+q32).
+Again, if we apply the same idea as in the proof of Corollary 6, then we s ee that [α]q
+times the first factor on the the right-hand side of the above equa tion is a polynomial
+inqwith non-negative integer coefficients, as is [ β]q4times the second factor. Taken
+together, this establishes the claim. /square
+Lemma 9. Letαandβbe positive integers with α≥18andβ≥3. Then the expression
+[α]q3[β]q4[90]q[3]q[4]q
+[5]q[6]q[9]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[90]q[3]q[4]q
+[5]q[6]q[9]q
+= (1−q3+q9−q12+q18−q21+q27−q33+q36−q42+q45−q51+q54)
+×(1−q4+q5+q6−q9+q11+q12−q14+q17+q18−q19+q23)
+= (1−q3+q9−q12+q18−q21+q27−q33+q36−q42+q45−q51+q54)
+×/parenleftbig
+1−q4+q12+q5(1−q4+q12)+q6(1−q8+q12)+q11(1−q8+q12)/parenrightbig
+.
+Again, if we apply the same idea as in the proof of Corollary 6, then we s ee that [α]q3
+times the first factor on the the right-hand side of the above equa tion is a polynomial
+inqwith non-negative integer coefficients, as is [ β]q4times the second factor. Taken
+together, this establishes the claim. /square10 C. KRATTENTHALER AND T. W. M ¨ULLER
+Lemma 10. Letαandβbe positive integers with α≥20andβ≥18. Then the
+expression
+[α]q[β]q3[90]q[3]q
+[5]q[6]q[9]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[90]q[3]q
+[5]q[6]q[9]q
+= (1−q3+q9−q12+q18−q21+q27−q33+q36−q42+q45−q51+q54)
+×(1−q+q5−q7+q10−q13+q15−q19+q20).
+Again, if we apply the same idea as in the proof of Corollary 6, then we s ee that
+[α]q3times the second factor on the the right-hand side of the above eq uation is a
+polynomial in qwith non-negative integer coefficients, as is [ β]q3times the first factor.
+Taken together, this establishes the claim. /square
+Lemma 11. Letαbe a positive integer with α≥26. Then the expression
+[α]q[15]q
+[3]q[5]q[12]q3
+[3]q3[4]q3
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[15]q
+[3]q[5]q[12]q3
+[3]q3[4]q3
+= 1−q+q5−q6+q9−q11+q12−q13+q14−q15+q17−q20+q21−q25+q26.
+Once again, the coefficients of the polynomial on the right-hand side are in{0,1,−1}
+and (+1)’s and ( −1)’s alternate, if one disregards the coefficients which are 0. The
+argument from the proof of Corollary 6 then implies that this is a polyn omial inqwith
+non-negative integer coefficients. /square
+Lemma 12. Letαbe a positive integer with α≥14. Then the expression
+[α]q[15]q
+[3]q[5]q[6]q3
+[2]q3[3]q3
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[15]q
+[3]q[5]q[6]q3
+[2]q3[3]q3= 1−q+q5−q7+q9−q13+q14.
+Repeating the arguments of the previous proof, this establishes t he claim. /squareCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 11
+Lemma 13. Letαandβbe positive integers with α≥30andβ≥20. Then the
+expression
+[α]q[β]q2[84]q[2]q
+[4]q[6]q[7]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[84]q[2]q
+[4]q[6]q[7]q
+= (1−q+q6−q8+q12−q15+q18−q22+q24−q29+q30)
+×(1−q2+q4−q6+q8−q10+q12−q14+q16−q18+q20−q22
++q24−q26+q28−q30+q32−q34+q36−q38+q40).
+Onceagain, ifweapplythesameideaasintheproofofCorollary6, the nweseethat[ α]q
+times the first factor on the the right-hand side of the above equa tion is a polynomial
+inqwith non-negative integer coefficients, as is [ β]q2times the second factor. Taken
+together, this establishes the claim. /square
+Lemma 14. Letαandβbe positive integers with α≥24andβ≥68. Then the
+expression
+[α]q[β]q[105]q
+[3]q[5]q[7]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[105]q
+[3]q[5]q[7]q
+= (1−q+q5−q6+q7−q8+q10−q11+q12−q13+q14−q16
++q17−q18+q19−q23+q24)
+×(1−q+q3−q4+q6−q7+q9−q10+q12−q13+q15−q16+q18−q19
++q21−q22+q24−q25+q27−q28+q30−q31+q33−q34+q35−q37
++q38−q40+q41−q43+q44−q46+q47−q49+q50−q52+q53−q55
++q56−q58+q59−q61+q62−q64+q65−q67+q68).
+Once again, if we apply the same idea as in the proof of Corollary 6, the n we see
+that [α]qtimes the first factor on the the right-hand side of the above equa tion is a
+polynomial in qwith non-negative integer coefficients, as is [ β]qtimes the second factor.
+Taken together, this establishes the claim. /square
+Lemma 15. Letαandβbe positive integers with α≥24andβ≥34. Then the
+expression
+[α]q[β]q[70]q
+[2]q[5]q[7]q
+is a polynomial in qwith non-negative integer coefficients.12 C. KRATTENTHALER AND T. W. M ¨ULLER
+Proof.We have
+[70]q
+[2]q[5]q[7]q
+= (1−q+q5−q6+q7−q8+q10−q11+q12−q13
++q14−q16+q17−q18+q19−q23+q24)
+×(1−q+q2−q3+q4−q5+q6−q7+q8−q9+q10−q11+q12−q13
++q14−q15+q16−q17+q18−q19+q20−q21+q22−q23+q24−q25
++q26−q27+q28−q29+q30−q31+q32−q33+q34).
+Also here, if we apply the same idea as in the proof of Corollary 6, then we see that [ α]q
+times the first factor on the the right-hand side of the above equa tion is a polynomial
+inqwith non-negative integer coefficients, as is [ β]qtimes the second factor. Taken
+together, this establishes the claim. /square
+Lemma 16. Letαandβbe positive integers with α≥4andβ≥2. Then the expression
+[α]q2[β]q5[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q= 1+q−q3−q4−q5+q7+q8.
+If we multiply this expression by [ α]q2, then, forα= 4 we obtain
+1+q+q2−q5−q9+q12+q13+q14,
+forα= 5 we obtain
+1+q+q2−q5+q8−q11+q14+q15+q16,
+and, forα≥6, we obtain
+1+q+q2−q5+q8+q10+p1(q)+q2α−4+q2α−2−q2α+1+q2α+4+q2α+5+q2α+6,
+wherep1(q) is a polynomial in qwith non-negative coefficients of order at least 11 and
+degree at most 2 α−5. In all cases it is obvious that the product of the result and [ β]q5,
+withβ≥2, is a polynomial in qwith non-negative coefficients. /square
+Lemma 17. Letαandβbe positive integers with α≥14andβ≥2. Then the
+expression
+[α]q[β]q5[14]q
+[2]q[7]q[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[14]q
+[2]q[7]q[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q= 1−q3−q5+q6+q7+q8−q9−q11+q14.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13
+If we multiply this expression by [ α]q, then, forα≥14, we obtain
+1+q+q2−q5+q7+2q8+q9+q10+p2(q)
++qα+3+qα+4+2qα+5+qα+6−qα+8+qα+11+qα+12+qα+13,
+wherep2(q) is a polynomial in qwith non-negative coefficients of order at least 11 and
+degree at most α+2. It is obvious that the product of the result and [ β]q5, withβ≥2,
+is a polynomial in qwith non-negative coefficients. /square
+Lemma 18. Letαandβbe positive integers with α≥32andβ≥12. Then the
+expression
+[α]q[β]q2[35]q
+[5]q[7]q[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[35]q
+[5]q[7]q[30]q[2]q[3]q[5]q
+[6]q[10]q[15]q= 1−q2−q3+q5+q6+q7−q8−2q9+q11+q12−q16+q20+q21
+−2q23−q24+q25+q26+q27−q29−q30+q32.
+If we multiply this expression by [ α]q, then, forα≥32, we obtain
+1+q−q3−q4+q6+2q7+q8−q9−q10+q12+q13+q14+q15+q20+2q21+2q22
+−q24+q26+2q27+2q28+q29+p3(q)+qα+2+2qα+4+2qα+5+qα+6−qα+8
++2qα+9+2qα+10+qα+11+qα+16+qα+17+qα+18+qα+19−qα+21−qα+22+qα+23
++2qα+24+qα+25−qα+27−qα+28+qα+30+qα+31,
+wherep3(q) is a polynomial in qwith non-negative coefficients of order at least 30 and
+degree at most α+1. It is obvious that the product of the result and [ β]q2, withβ≥12,
+is a polynomial in qwith non-negative coefficients. /square
+Lemma 19. Letαandβbe positive integers with α≥16andβ≥2. Then the
+expression
+[α]q2[β]q5[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q= 1+q−q3−q4−q5−q6+q8+q9+q10+q11+q12−q14−q15−q16
+−q17−q18+q20+q21+q22+q23+q24−q26−q27−q28−q29+q31+q32.
+If we multiply this expression by [ α]q2, then, forα≥16, we obtain
+1+q+q2−q5−q6−q7+q10+q11+2q12+q13+q14
+−q17−q18−q19+q22+q23+2q24+p4(q)+s4(q),14 C. KRATTENTHALER AND T. W. M ¨ULLER
+wherep4(q) is a polynomial in qwith non-negative coefficients of order at least 25
+and degree at most 2 α+ 5 ands4(q) completes the above expression to a symmetric
+polynomial. It is obvious that the product of the result and [ β]q5, withβ≥2, is a
+polynomial in qwith non-negative coefficients. /square
+Lemma 20. Letαandβbe positive integers with α≥56andβ≥4. Then the
+expression
+[α]q[β]q2[35]q
+[5]q[7]q[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[35]q
+[5]q[7]q[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q= 1−q2−q3+q5+q7−q9+q12−q13+q15−q16−q18
++q19+q20+q22−2q23+q27−q28+q29−2q33+q34+q36+q37−q38
+−q40+q41−q43+q44−q47+q49+q51−q53−q54+q56.
+If we multiply this expression by [ α]q, then, forα≥56, we obtain
+1+q−q3−q4+q7+q8+q12+q15−q18+q20+q21+2q22+q27
++q29+q30+q31+q32−q33+q36+2q37+q38+p5(q)+s5(q),
+wherep5(q) is a polynomial in qwith non-negative coefficients of order at least 39
+and degree at most α+ 16 ands5(q) completes the above expression to a symmetric
+polynomial. It is obvious that the product of the result and [ β]q2, withβ≥4, is a
+polynomial in qwith non-negative coefficients. /square
+Lemma 21. Letαandβbe positive integers with α≥38andβ≥2. Then the
+expression
+[α]q[β]q5[14]q
+[2]q[7]q[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[14]q
+[2]q[7]q[60]q[2]q[3]q[5]q
+[10]q[12]q[15]q= 1−q3−q5+q7+q8−q13+q19−q25+q30+q31−q33−q35+q38.
+If we multiply this expression by [ α]q, then, forα≥38, we obtain
+1+q+q2−q5−q6+q8+q9+q10+q11+p6(q)+s6(q),
+wherep6(q) is a polynomial in qwith non-negative coefficients of order at least 12
+and degree at most α+ 25 ands6(q) completes the above expression to a symmetric
+polynomial. It is obvious that the product of the result and [ β]q2, withβ≥2, is a
+polynomial in qwith non-negative coefficients. /square
+Lemma 22. Letαandβbe positive integers with α≥30andβ≥26. Then the
+expression
+[α]q[β]q3[126]q[3]q
+[6]q[7]q[9]qCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[126]q[3]q
+[6]q[7]q[9]q= (1−q+q6−q8+q12−q15+q18−q22+q24−q29+q30)
+×(1−q3+q9−q12+q18−q21+q27−q30+q36−q39
++q42−q48+q51−q57+q60−q66+q69−q75+q78).
+If we apply the same idea as in the proof of Corollary 6, then we see th at [α]qtimes the
+first factor on the the right-hand side of the above equation is a po lynomial in qwith
+non-negative integer coefficients, as is [ β]q3times the second factor. Taken together,
+this establishes the claim. /square
+Lemma 23. Letαandβbe positive integers with α≥66andβ≥54. Then the
+expression
+[α]q[β]q3[252]q[3]q
+[7]q[9]q[12]q
+is a polynomial in qwith non-negative integer coefficients.
+Proof.We have
+[252]q[3]q
+[7]q[9]q[12]q
+= (1−q+q7−q8+q12−q13+q14−q15+q19−q20+q21−q22+q24−q25
++q26−q27+q28−q29+q31−q32+q33−q34+q35−q37+q38
+−q39+q40−q41+q42−q44+q45−q46+q47−q51+q52−q53
++q54−q58+q59−q65+q66)
+×(1−q3+q9−q12+q18−q21+q27−q30+q36−q39+q45−q48
++q54−q57+q63−q66+q72−q75+q81−q87+q90−q96+q99
+−q105+q108−q114+q117−q123+q126−q132+q135−q141
++q144−q150+q153−q159+q162).
+If we apply the same idea as in the proof of Corollary 6, then we see th at [α]qtimes the
+first factor on the the right-hand side of the above equation is a po lynomial in qwith
+non-negative integer coefficients, as is [ β]q3times the second factor. Taken together,
+this establishes the claim. /square
+Lemma 24. Letαandβbe positive integers with α≥54andβ≥34. Then the
+expression
+[α]q[β]q2[140]q[2]q
+[4]q[7]q[10]q
+is a polynomial in qwith non-negative integer coefficients.16 C. KRATTENTHALER AND T. W. M ¨ULLER
+Proof.We have
+[140]q[2]q
+[4]q[7]q[10]q
+= (1−q+q7−q8+q10−q11+q14−q15+q17−q18+q20−q22+q24−q25
++q27−q29+q30−q32+q34−q36+q37−q39+q40−q43+q44−q46
++q47−q53+q54)
+×(1−q2+q4−q6+q8−q10+q12−q14+q16−q18+q20−q22+q24
+−q26+q28−q30+q32−q34+q36−q38+q40−q42+q44−q46
++q48−q50+q52−q54+q56−q58+q60−q62+q64−q66+q68),
+If we apply the same idea as in the proof of Corollary 6, then we see th at [α]qtimes the
+first factor on the the right-hand side of the above equation is a po lynomial in qwith
+non-negative integer coefficients, as is [ β]q2times the second factor. Taken together,
+this establishes the claim. /square
+We are now ready for the proof of the main result of this section.
+Theorem 25. For all irreducible well-generated complex reflection grou ps and posi-
+tive integers m, theq-Fuß–Catalan number Catm(W;q)is a polynomial in qwith non-
+negative integer coefficients.
+Proof.First, letW=An. In this case, the degrees are 2 ,3,...,n+1, and hence
+Catm(An;q) =1
+[(m+1)n+1]q/bracketleftbigg
+(m+1)n+1
+n/bracketrightbigg
+q,
+which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.
+Next, letW=G(d,1,n). In this case, the degrees are d,2d,...,nd , and hence
+Catm(G(d,1,n);q) =/bracketleftbigg
+(m+1)n
+n/bracketrightbigg
+qd,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+Now, letW=G(e,e,n). In this case, the degrees are e,2e,...,(n−1)e,n, and hence
+Catm(G(e,e,n);q) =[m(n−1)e+n]q
+[n]qn−1/productdisplay
+i=1[m(n−1)e+ie]q
+[ie]q
+=/bracketleftbigg
+(m+1)(n−1)
+n−1/bracketrightbigg
+qe+qn[e]qn/bracketleftbigg
+(m+1)(n−1)
+n/bracketrightbigg
+qe,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+It remains to verify the claim for the exceptional groups.
+ForW=G4, the degrees are 4 ,6, and hence
+Catm(G4;q) =[6m+4]q[6m+6]q
+[4]q[6]q=1
+[3m+4]q2/bracketleftbigg
+3m+4
+3/bracketrightbigg
+q2,
+which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 17
+ForW=G5, the degrees are 6 ,12, and hence
+Catm(G5;q) =[12m+6]q[12m+12]q
+[6]q[12]q=/bracketleftbigg
+2m+2
+2/bracketrightbigg
+q6,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+ForW=G6, the degrees are 4 ,12, and hence
+Catm(G6;q) =[12m+4]q[12m+12]q
+[4]q[12]q= [3m+1]q4[m+1]q12,
+which is manifestly a polynomial in qwith non-negative integer coefficients.
+ForW=G8, the degrees are 8 ,12, and hence
+Catm(G8;q) =[12m+8]q[12m+12]q
+[8]q[12]q=1
+[3m+4]q4/bracketleftbigg
+3m+4
+3/bracketrightbigg
+q4,
+which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.
+ForW=G9, the degrees are 8 ,24, and hence
+Catm(G9;q) =[24m+8]q[24m+24]q
+[8]q[24]q= [3m+1]q8[m+1]q24,
+which is manifestly a polynomial in qwith non-negative integer coefficients.
+ForW=G10, the degrees are 12 ,24, and hence
+Catm(G10;q) =[24m+12]q[24m+24]q
+[12]q[24]q=/bracketleftbigg
+2m+2
+2/bracketrightbigg
+q12,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+ForW=G14, the degrees are 6 ,24, and hence
+Catm(G14;q) =[24m+6]q[24m+24]q
+[6]q[24]q= [4m+1]q6[m+1]q24,
+which is manifestly a polynomial in qwith non-negative integer coefficients.
+ForW=G16, the degrees are 20 ,30, and hence
+Catm(G16;q) =[30m+20]q[30m+30]q
+[20]q[30]q=1
+[3m+4]q10/bracketleftbigg
+3m+4
+3/bracketrightbigg
+q10,
+which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.
+ForW=G17, the degrees are 20 ,60, and hence
+Catm(G17;q) =[60m+20]q[60m+60]q
+[20]q[60]q= [3m+1]q20[m+1]q60,
+which is manifestly a polynomial in qwith non-negative integer coefficients.
+ForW=G18, the degrees are 30 ,60, and hence
+Catm(G18;q) =[60m+30]q[60m+60]q
+[30]q[60]q=/bracketleftbigg
+2m+2
+2/bracketrightbigg
+q30,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.18 C. KRATTENTHALER AND T. W. M ¨ULLER
+ForW=G20, the degrees are 12 ,30, and hence
+Catm(G20;q) =[30m+12]q[30m+30]q
+[12]q[30]q
+=/braceleftBigg/bracketleftbig5m+2
+2/bracketrightbig
+q12[m+1]q30, ifmis even,
+[5m+2]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[10]q6
+[2]q6[5]q6,ifmis odd,
+which, by Corollary 6, are polynomials in qwith non-negative integer coefficients in
+both cases.
+ForW=G21, the degrees are 12 ,60, and hence
+Catm(G21;q) =[60m+12]q[60m+60]q
+[12]q[60]q= [5m+1]q12[m+1]q60,
+which is manifestly a polynomial in qwith non-negative integer coefficients.
+ForW=G23=H3, the degrees are 2 ,6,10, and hence
+Catm(H3;q) =[10m+2]q[10m+6]q[10m+10]q
+[2]q[6]q[10]q
+=
+
+[5m+2]q2/bracketleftbig5m+3
+3/bracketrightbig
+q6[m+1]q10, ifm≡0 (mod 3),/bracketleftbig5m+1
+3/bracketrightbig
+q6[5m+3]q2[m+1]q10, ifm≡1 (mod 3),
+[5m+2]q2[5m+3]q2/bracketleftbigm+1
+3/bracketrightbig
+q30[15]q2
+[3]q2[5]q2,ifm≡2 (mod 3),
+which, by Corollary 6, are polynomials in qwith non-negative integer coefficients in all
+cases.
+ForW=G24, the degrees are 4 ,6,14, and hence
+Catm(G24;q) =[14m+4]q[14m+6]q[14m+14]q
+[4]q[6]q[14]q.
+We have
+Catm(G24;q) =
+
+/bracketleftbig7m
+2+1/bracketrightbig
+q4/bracketleftbig14m
+6+1/bracketrightbig
+q6[m+1]q14,ifm≡0 (mod 6),/bracketleftbig7m+2
+3/bracketrightbig
+q6/bracketleftbig7m+3
+2/bracketrightbig
+q4[m+1]q14, ifm≡1 (mod 6),
+/bracketleftbig7m
+2+1/bracketrightbig
+q4[7m+3]q2/bracketleftbigm+1
+3/bracketrightbig
+q42[21]q2
+[3]q2[7]q2,ifm≡2 (mod 6),
+[7m+2]q2/bracketleftbig7m
+3+1/bracketrightbig
+q6/bracketleftbigm+1
+2/bracketrightbig
+q28[14]q2
+[2]q2[7]q2,ifm≡3 (mod 6),
+/bracketleftbig7m+2
+6/bracketrightbig
+q12[6]q2
+[2]q2[3]q2[7m+3]q2[m+1]q14,ifm≡4 (mod 6),
+[7m+2]q2/bracketleftbig7m+3
+2/bracketrightbig
+q4/bracketleftbigm+1
+3/bracketrightbig
+q42[21]q2
+[3]q2[7]q2,ifm≡5 (mod 6),
+which, by Corollary 6, are polynomials in qwith non-negative integer coefficients in all
+cases.
+ForW=G25, the degrees are 6 ,9,12, and hence
+Catm(G25;q) =[12m+6]q[12m+9]q[12m+12]q
+[6]q[9]q[12]q=1
+[4m+5]q3/bracketleftbigg
+4m+5
+4/bracketrightbigg
+q3,
+which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 19
+ForW=G26, the degrees are 6 ,12,18, and hence
+Catm(G26;q) =[18m+6]q[18m+12]q[18m+18]q
+[6]q[12]q[18]q=/bracketleftbigg
+3m+3
+3/bracketrightbigg
+q6,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+ForW=G27, the degrees are 6 ,12,30, and hence
+Catm(G27;q) =[30m+6]q[30m+12]q[30m+30]q
+[6]q[12]q[30]q= [m+1]q30/bracketleftbigg
+5m+2
+2/bracketrightbigg
+q6,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+ForW=G28=F4, the degrees are 2 ,6,8,12, and hence
+Catm(F4;q) =[12m+2]q[12m+6]q[12m+8]q[12m+12]q
+[2]q[6]q[8]q[12]q
+=
+
+[6m+1]q2[2m+1]q6/bracketleftbig3
+2m+1/bracketrightbig
+q8[m+1]q12, ifmis even,
+[6m+1]q2[2m+1]q6/bracketleftbigm+1
+2/bracketrightbig
+q24[3m+2]q4[6]q4
+[2]q4[3]q4,ifmis odd,
+which in both cases is a polynomial in qwith non-negative integer coefficients; in the
+second case this is due to Corollary 6.
+ForW=G29, the degrees are 4 ,8,12,20, and hence
+Catm(G29;q) =[20m+4]q[20m+8]q[20m+12]q[20m+20]q
+[4]q[8]q[12]q[20]q
+= [m+1]q20/bracketleftbigg
+5m+3
+3/bracketrightbigg
+q4,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+ForW=G30=H4, the degrees are 2 ,12,20,30, and hence
+Catm(H4;q) =[30m+2]q[30m+12]q[30m+20]q[30m+30]q
+[2]q[12]q[20]q[30]q.
+Ifmis even, then we have
+Catm(H4;q) = [15m+1]q2/bracketleftbig5
+2m+1/bracketrightbig
+q12/bracketleftbig3
+2m+1/bracketrightbig
+q20[m+1]q30,
+which is manifestly a polynomial in qwith non-negative integer coefficients. On the
+other hand, if mis odd, then we may write
+Catm(H4;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[5m+2]q6[3m+2]q10/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q6[10]q2[15]q2,
+which, by Lemma 16, is a polynomial in qwith non-negative integer coefficients.
+ForW=G32, the degrees are 12 ,18,24,30, and hence
+Catm(G32;q) =[30m+12]q[30m+18]q[30m+24]q[30m+30]q
+[12]q[18]q[24]q[30]q
+=1
+[5m+6]q6/bracketleftbigg
+5m+6
+5/bracketrightbigg
+q6,
+which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.20 C. KRATTENTHALER AND T. W. M ¨ULLER
+ForW=G33, the degrees are 4 ,6,10,12,18, and hence
+Catm(G33;q) =[18m+4]q[18m+6]q[18m+10]q[18m+12]q[18m+18]q
+[4]q[6]q[10]q[12]q[18]q.
+Ifm≡0 (mod 10), then we have
+Catm(G33;q) =/bracketleftbig9
+2m+1/bracketrightbig
+q4[3m+1]q6/bracketleftbig9
+5m+1/bracketrightbig
+q10/bracketleftbig3
+2m+1/bracketrightbig
+q12[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+1 (mod 10), then we have
+Catm(G33;q) =/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+5
+2/bracketrightbig
+q4/bracketleftbig3m+2
+5/bracketrightbig
+q30[m+1]q18[9m+2]q2[15]q2
+[3]q2[5]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡2 (mod 10), then we have
+Catm(G33;q) =/bracketleftbig9m+2
+10/bracketrightbig
+q20[3m+1]q6/bracketleftbig3
+2m+1/bracketrightbig
+q12[m+1]q18[9m+5]q2[10]q2
+[2]q2[5]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡3 (mod 10), then we have
+Catm(G33;q) =/bracketleftbig3m+1
+10/bracketrightbig
+q60/bracketleftbig9m+5
+2/bracketrightbig
+q4[3m+2]q6[m+1]q18[9m+2]q2[30]q2
+[5]q2[6]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡4 (mod 10), then we have
+Catm(G33;q) =/bracketleftbig9
+2m+1/bracketrightbig
+q4[3m+1]q6/bracketleftbig3
+2m+1/bracketrightbig
+q12/bracketleftbigm+1
+5/bracketrightbig
+q90[9m+5]q2[45]q2
+[5]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡5 (mod 10), then we have
+Catm(G33;q) =/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+5
+10/bracketrightbig
+q20[3m+2]q6[m+1]q18[9m+2]q2[10]q2
+[2]q2[5]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡6 (mod 10), then we have
+Catm(G33;q) =/bracketleftbig9
+2m+1/bracketrightbig
+q4[3m+1]q6/bracketleftbig3m+2
+10/bracketrightbig
+q60[m+1]q18[9m+5]q2[30]q2
+[5]q2[6]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡7 (mod 10), then we have
+Catm(G33;q) =/bracketleftbig9m+2
+5/bracketrightbig
+q10/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+5
+2/bracketrightbig
+q4[3m+2]q6[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+8 (mod 10), then we have
+Catm(G33;q) =/bracketleftbig9
+2m+1/bracketrightbig
+q4/bracketleftbig3m+1
+5/bracketrightbig
+q30/bracketleftbig3
+2m+1/bracketrightbig
+q12[m+1]q18[9m+5]q2[15]q2
+[3]q2[5]q2,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 21
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. Fi-
+nally, ifm≡9 (mod 10), then we have
+Catm(G33;q) =/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+5
+2/bracketrightbig
+q4[3m+2]q6/bracketleftbigm+1
+5/bracketrightbig
+q90[9m+2]q2[45]q2
+[5]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients.
+ForW=G34, the degrees are 6 ,12,18,24,30,42, and hence
+Catm(G34;q) =[42m+6]q[42m+12]q[42m+18]q[42m+24]q[42m+30]q[42m+42]q
+[6]q[12]q[18]q[24]q[30]q[42]q
+= [m+1]q42/bracketleftbigg
+7m+5
+5/bracketrightbigg
+q6,
+which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
+ForW=G35=E6, the degrees are 2 ,5,6,8,9,12, and hence
+Catm(E6;q) =[12m+2]q[12m+5]q[12m+6]q[12m+8]q[12m+9]q[12m+12]q
+[2]q[5]q[6]q[8]q[9]q[12]q.
+Ifm≡0 (mod 30),then we have
+Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5
+5/bracketrightbig
+q5[2m+1]q6/bracketleftbig3m+2
+2/bracketrightbig
+q8/bracketleftbig4m+3
+3/bracketrightbig
+q9[m+1]q12,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+1 (mod 30),then we have
+Catm(E6;q) = [6m+1]q2/bracketleftbig2m+1
+3/bracketrightbig
+q18[4m+3]q3[6]q3
+[2]q3[3]q3
+×/bracketleftbig3m+2
+5/bracketrightbig
+q20[12m+5]q[20]q
+[4]q[5]q/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4.
+If one decomposes [6 m+1]q2as [3m+1]q4+q2[3m]q4, then one sees that, by Corollary 6,
+the above expression is a polynomial in qwith non-negative integer coefficients. If
+m≡2 (mod 30),then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+5/bracketrightbig
+q30[30]q
+[5]q[6]q
+×/bracketleftbig3m+2
+2/bracketrightbig
+q8[4m+3]q3/bracketleftbigm+1
+3/bracketrightbig
+q36[12]q3
+[3]q3[4]q3,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡3 (mod 30),then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6[3m+2]q4
+×/bracketleftbig4m+3
+15/bracketrightbig
+q45[45]q
+[5]q[9]q/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,22 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡4 (mod 30),then we have
+Catm(E6;q) =/bracketleftbig12m+2
+5/bracketrightbig
+q5/bracketleftbig12m+5
+2/bracketrightbig
+q2/bracketleftbig2m+1
+3/bracketrightbig
+q18[6]q3
+[2]q3[3]q3
+×/bracketleftbig3m+2
+2/bracketrightbig
+q8[4m+3]q3[m+1]q12,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡5 (mod 30),then we have
+Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5
+5/bracketrightbig
+q5[2m+1]q6
+×[3m+2]q4[4m+3]q3/bracketleftbigm+1
+6/bracketrightbig
+q72[72]q[3]q[4]q
+[8]q[9]q[12]q,
+which, by Lemma 7, is a polynomial in qwith non-negative integer coefficients. If
+m≡6 (mod 30),then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6/bracketleftbig3m+2
+10/bracketrightbig
+q40[40]q
+[5]q[8]q/bracketleftbig4m+3
+3/bracketrightbig
+q9[m+1]q12,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡7 (mod 30),then we have
+Catm(E6;q) =/bracketleftbig6m+1
+2/bracketrightbig
+q4[12m+5]q/bracketleftbig2m+1
+15/bracketrightbig
+q90
+×[90]q[3]q[4]q
+[5]q[6]q[9]q[3m+2]q4[4m+3]q3/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Corollary 6 and Lemma 9, is a polynomial in qwith non-negative integer
+coefficients. If m≡8 (mod 30),then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6/bracketleftbig3m+2
+2/bracketrightbig
+q8
+×/bracketleftbig4m+3
+5/bracketrightbig
+q15[15]q
+[3]q[5]q/bracketleftbigm+1
+3/bracketrightbig
+q36[12]q3
+[3]q3[4]q3,
+which, by Lemma 11, is a polynomial in qwith non-negative integer coefficients. If
+m≡9 (mod 30),then we have
+Catm(E6;q) =/bracketleftbig12m+2
+5/bracketrightbig
+q5/bracketleftbig12m+5
+2/bracketrightbig
+q2[2m+1]q6[3m+2]q4/bracketleftbig4m+3
+3/bracketrightbig
+q9/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡10 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5
+5/bracketrightbig
+q5/bracketleftbig2m+1
+3/bracketrightbig
+q18[6]q3
+[2]q3[3]q3/bracketleftbig3m+2
+2/bracketrightbig
+q8[4m+3]q3[m+1]q12,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 23
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡11 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6
+×/bracketleftbig3m+2
+5/bracketrightbig
+q20[20]q
+[4]q[5]q[4m+3]q3/bracketleftbigm+1
+6/bracketrightbig
+q72[72]q[3]q[4]q
+[8]q[9]q[12]q.
+If one decomposes [6 m+1]q2as [3m+1]q4+q2[3m]q4, then one sees that, by Corollary 6
+and Lemma 7, this is a polynomial in qwith non-negative integer coefficients. If m≡
+12 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+5/bracketrightbig
+q30[30]q
+[5]q[6]q/bracketleftbig3m+2
+2/bracketrightbig
+q8/bracketleftbig4m+3
+3/bracketrightbig
+q9[m+1]q12,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡13 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+3/bracketrightbig
+q18[6]q3
+[2]q3[3]q3
+×[3m+2]q4/bracketleftbig4m+3
+5/bracketrightbig
+q15[15]q
+[3]q[5]q/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Lemma 12, is a polynomial in qwith non-negative integer coefficients. If
+m≡14 (mod 30) ,then we have
+Catm(E6;q) =/bracketleftbig12m+2
+5/bracketrightbig
+q5/bracketleftbig12m+5
+2/bracketrightbig
+q2[2m+1]q6/bracketleftbig3m+2
+2/bracketrightbig
+q8[4m+3]q3/bracketleftbigm+1
+3/bracketrightbig
+q36[12]q3
+[3]q3[4]q3,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡15 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5
+5/bracketrightbig
+q5[2m+1]q6[3m+2]q4/bracketleftbig4m+3
+3/bracketrightbig
+q9/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡16 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+3/bracketrightbig
+q18[6]q3
+[2]q3[3]q3
+×/bracketleftbig3m+2
+10/bracketrightbig
+q40[40]q
+[5]q[8]q[4m+3]q3[m+1]q12,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡17 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+5/bracketrightbig
+q30[30]q
+[5]q[6]q
+×[3m+2]q4[4m+3]q3/bracketleftbigm+1
+6/bracketrightbig
+q72[72]q[3]q[4]q
+[8]q[9]q[12]q,24 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6 and Lemma 7, is a polynomial in qwith non-negative integer
+coefficients. If m≡18 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6/bracketleftbig3m+2
+2/bracketrightbig
+q8/bracketleftbig4m+3
+15/bracketrightbig
+q45[45]q
+[5]q[9]q[m+1]q12,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡19 (mod 30) ,then we have
+Catm(E6;q) =/bracketleftbig12m+2
+5/bracketrightbig
+q5/bracketleftbig12m+5
+2/bracketrightbig
+q2/bracketleftbig2m+1
+3/bracketrightbig
+q18[6]q3
+[2]q3[3]q3
+×[3m+2]q4[4m+3]q3/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡20 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5
+5/bracketrightbig
+q5[2m+1]q6/bracketleftbig3m+2
+2/bracketrightbig
+q8[4m+3]q3/bracketleftbigm+1
+3/bracketrightbig
+q36[12]q3
+[3]q3[4]q3,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡21 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6
+×/bracketleftbig3m+2
+5/bracketrightbig
+q20[20]q
+[4]q[5]q/bracketleftbig4m+3
+3/bracketrightbig
+q9/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡22 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+15/bracketrightbig
+q90[90]q[3]q
+[5]q[6]q[9]q
+×/bracketleftbig3m+2
+2/bracketrightbig
+q8[4m+3]q3[m+1]q12,
+which, by Lemma 10, is a polynomial in qwith non-negative integer coefficients. If
+m≡23 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6
+×[3m+2]q4/bracketleftbig4m+3
+5/bracketrightbig
+q15[15]q
+[3]q[5]q/bracketleftbigm+1
+6/bracketrightbig
+q72[72]q[3]q[4]q
+[8]q[9]q[12]q,
+which, by Lemma 8, is a polynomial in qwith non-negative integer coefficients. If
+m≡24 (mod 30) ,then we have
+Catm(E6;q) =/bracketleftbig12m+2
+5/bracketrightbig
+q5/bracketleftbig12m+5
+2/bracketrightbig
+q2[2m+1]q6/bracketleftbig3m+2
+2/bracketrightbig
+q8/bracketleftbig4m+3
+3/bracketrightbig
+q9[m+1]q12,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 25
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+25 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5
+5/bracketrightbig
+q5/bracketleftbig2m+1
+3/bracketrightbig
+q18[6]q3
+[2]q3[3]q3
+×[3m+2]q4[4m+3]q3/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡26 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6
+×/bracketleftbig3m+2
+10/bracketrightbig
+q40[40]q
+[5]q[8]q[4m+3]q3/bracketleftbigm+1
+3/bracketrightbig
+q36[12]q3
+[3]q3[4]q3,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡27 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+5/bracketrightbig
+q30[30]q
+[5]q[6]q
+×[3m+2]q4/bracketleftbig4m+3
+3/bracketrightbig
+q9/bracketleftbigm+1
+2/bracketrightbig
+q24[6]q4
+[2]q4[3]q4,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡28 (mod 30) ,then we have
+Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
+3/bracketrightbig
+q18[6]q3
+[2]q3[3]q3
+×/bracketleftbig3m+2
+2/bracketrightbig
+q8/bracketleftbig4m+3
+5/bracketrightbig
+q15[15]q
+[3]q[5]q[m+1]q12,
+which, by Lemma 12, is a polynomial in qwith non-negative integer coefficients. If
+m≡29 (mod 30) ,then we have
+Catm(E6;q) =/bracketleftbig6m+1
+5/bracketrightbig
+q10[10]q
+[2]q[5]q[12m+5]q[2m+1]q6
+×[3m+2]q4[4m+3]q3/bracketleftbigm+1
+6/bracketrightbig
+q72[72]q[3]q[4]q
+[8]q[9]q[12]q,
+which, by Corollary 6 and Lemma 7, is a polynomial in qwith non-negative integer
+coefficients.
+ForW=G36=E7, the degrees are 2 ,6,8,10,12,14,18, and hence
+Catm(E7;q) =[18m+2]q[18m+6]q[18m+8]q[18m+10]q
+[2]q[6]q[8]q[10]q
+×[18m+12]q[18m+14]q[18m+18]q
+[12]q[14]q[18]q.26 C. KRATTENTHALER AND T. W. M ¨ULLER
+Ifm≡0 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8/bracketleftbig9m+5
+5/bracketrightbig
+q10/bracketleftbig3m+2
+2/bracketrightbig
+q12/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+1 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2/bracketleftbig9m+5
+7/bracketrightbig
+q14[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+2 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡3 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2[9m+4]q2
+×/bracketleftbig9m+5
+4/bracketrightbig
+q8[3m+2]q6[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡4 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+20/bracketrightbig
+q40[20]q2
+[4]q2[5]q2[9m+5]q2
+×/bracketleftbig3m+2
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡5 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+7/bracketrightbig
+q14/bracketleftbig9m+5
+5/bracketrightbig
+q10[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+6 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10[3m+1]q6/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡7 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 27
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡8 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2/bracketleftbig9m+4
+4/bracketrightbig
+q8
+×/bracketleftbig9m+5
+7/bracketrightbig
+q14/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡9 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2/bracketleftbig9m+4
+5/bracketrightbig
+q10
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡10 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14[3m+1]q6/bracketleftbig9m+4
+2/bracketrightbig
+q4/bracketleftbig9m+5
+5/bracketrightbig
+q10
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡11 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2/bracketleftbig9m+5
+4/bracketrightbig
+q8
+×/bracketleftbig3m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡12 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡13 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,28 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡14 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+10/bracketrightbig
+q20[10]q2
+[2]q2[5]q2
+×[9m+5]q2/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18.
+Ifonedecomposes[9 m+5]q2as[9m
+2+3]q4+q2[9m
+2+2]q4, thenoneseesthat,byCorollary6,
+this is a polynomial in qwith non-negative integer coefficients. If m≡15 (mod 140) ,
+then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2
+×/bracketleftbig9m+5
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[3m+2]q6[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡16 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+4
+4/bracketrightbig
+q8[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡17 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14/bracketleftbig3m+1
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+4]q2/bracketleftbig9m+5
+2/bracketrightbig
+q4
+×[3m+2]q6/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡18 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2/bracketleftbig3m+2
+28/bracketrightbig
+q168[84]q2[2]q2
+[4]q2[6]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Corollary 6 and Lemma 13, is a polynomial in qwith non-negative integer
+coefficients. If m≡19 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[9m+5]q2
+×[3m+2]q6[9m+7]q2[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 29
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡20 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8/bracketleftbig9m+5
+5/bracketrightbig
+q10/bracketleftbig3m+2
+2/bracketrightbig
+q12
+×[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡21 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+4]q2/bracketleftbig9m+5
+2/bracketrightbig
+q4
+×[3m+2]q6/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡22 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+2/bracketrightbig
+q4/bracketleftbig9m+5
+7/bracketrightbig
+q14/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4
+×/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡23 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+35/bracketrightbig
+q210[105]q2
+[3]q2[5]q2[7]q2[9m+4]q2[9m+5]q2
+×[3m+2]q6[9m+7]q2/bracketleftbigm+1
+2/bracketrightbig
+q36[6]q6
+[2]q6[3]q6,
+which, by Corollary 6 and Lemma 14, is a polynomial in qwith non-negative integer
+coefficients. If m≡24 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14[3m+1]q6/bracketleftbig9m+4
+20/bracketrightbig
+q40[20]q2
+[4]q2[5]q2[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡25 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12
+×[9m+4]q2/bracketleftbig9m+5
+5/bracketrightbig
+q10/bracketleftbig3m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,30 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡26 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10[3m+1]q6/bracketleftbig9m+4
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18.
+Ifonedecomposes[9 m+7]q2as[9m
+2+4]q4+q2[9m
+2+3]q4, thenoneseesthat,byCorollary6,
+this is a polynomial in qwith non-negative integer coefficients. If m≡27 (mod 140) ,
+then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+5/bracketrightbig
+q10/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡28 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2/bracketleftbig9m+4
+4/bracketrightbig
+q8[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡29 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+5/bracketrightbig
+q10/bracketleftbig9m+5
+7/bracketrightbig
+q14[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+30 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+4
+2/bracketrightbig
+q4/bracketleftbig9m+5
+5/bracketrightbig
+q10
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡31 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2
+×/bracketleftbig9m+5
+4/bracketrightbig
+q8[3m+2]q6[9m+7]q2[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 31
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡32 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8
+×[9m+5]q2/bracketleftbig3m+2
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡33 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2/bracketleftbig9m+4
+7/bracketrightbig
+q14
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡34 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+10/bracketrightbig
+q20[10]q2
+[2]q2[5]q2[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2.
+Ifonedecomposes[9 m+7]q2as[9m
+2+4]q4+q2[9m
+2+3]q4, thenoneseesthat,byCorollary6,
+this is a polynomial in qwith non-negative integer coefficients. If m≡35 (mod 140) ,
+then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2/bracketleftbig9m+5
+5/bracketrightbig
+q10[3m+2]q6/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+36 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8/bracketleftbig9m+5
+7/bracketrightbig
+q14/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+37 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+20/bracketrightbig
+q40[20]q2
+[4]q2[5]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡38 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18,32 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡39 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+5/bracketrightbig
+q10[9m+5]q2
+×/bracketleftbig3m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡40 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2
+×/bracketleftbig9m+5
+5/bracketrightbig
+q10/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡41 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡42 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4/bracketleftbig9m+7
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡43 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2[9m+4]q2
+×/bracketleftbig9m+5
+7/bracketrightbig
+q14[3m+2]q6[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡44 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+4
+20/bracketrightbig
+q40[20]q2
+[4]q2[5]q2[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡45 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2/bracketleftbig9m+5
+5/bracketrightbig
+q10[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 33
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+46 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10[3m+1]q6
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2/bracketleftbig3m+2
+28/bracketrightbig
+q168[84]q2[2]q2
+[4]q2[6]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Lemma 13, is a polynomial in qwith non-negative integer coefficients. If
+m≡47 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+7/bracketrightbig
+q14[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+48 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2/bracketleftbig9m+4
+4/bracketrightbig
+q8[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡49 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4/bracketleftbig9m+4
+5/bracketrightbig
+q10/bracketleftbig9m+5
+2/bracketrightbig
+q4
+×[3m+2]q6/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡50 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+2/bracketrightbig
+q4
+×/bracketleftbig9m+5
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡51 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[9m+4]q2
+×/bracketleftbig9m+5
+4/bracketrightbig
+q8[3m+2]q6[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡52 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8[9m+5]q2/bracketleftbig3m+2
+2/bracketrightbig
+q12/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,34 C. KRATTENTHALER AND T. W. M ¨ULLER
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+53 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2[9m+4]q2
+×[9m+5]q2/bracketleftbig3m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡54 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+70/bracketrightbig
+q140[70]q2
+[2]q2[5]q2[7]q2[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18.
+Ifonedecomposes[9 m+7]q2as[9m
+2+4]q4+q2[9m
+2+3]q4, thenoneseesthat, byCorollary6
+and Lemma 15, this is a polynomial in qwith non-negative integer coefficients. If
+m≡55 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2/bracketleftbig9m+5
+5/bracketrightbig
+q10[3m+2]q6
+×[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡56 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8[9m+5]q2/bracketleftbig3m+2
+2/bracketrightbig
+q12/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+57 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+2/bracketrightbig
+q4/bracketleftbig3m+1
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4
+×[9m+4]q2/bracketleftbig9m+5
+7/bracketrightbig
+q14[3m+2]q6/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡58 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+35/bracketrightbig
+q210[105]q2
+[3]q2[5]q2[7]q2/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18,
+which, by Corollary 6 and Lemma 14, is a polynomial in qwith non-negative integer
+coefficients. If m≡59 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+5/bracketrightbig
+q10/bracketleftbig9m+5
+4/bracketrightbig
+q8[3m+2]q6[9m+7]q2[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 35
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+60 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8/bracketleftbig9m+5
+5/bracketrightbig
+q10
+×/bracketleftbig3m+2
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡61 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+7/bracketrightbig
+q14[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+62 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4/bracketleftbig9m+7
+5/bracketrightbig
+q10/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡63 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡64 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+20/bracketrightbig
+q40[20]q2
+[4]q2[5]q2
+×/bracketleftbig9m+5
+7/bracketrightbig
+q14/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡65 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[9m+4]q2
+×/bracketleftbig9m+5
+5/bracketrightbig
+q10[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡66 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[3m+1]q6/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18,36 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡67 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2
+×[9m+5]q2/bracketleftbig3m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡68 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2
+×/bracketleftbig9m+4
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2[9m+5]q2/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡69 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+5/bracketrightbig
+q10
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡70 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+2/bracketrightbig
+q4/bracketleftbig9m+5
+5/bracketrightbig
+q10/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4
+×/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡71 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+20/bracketrightbig
+q40[20]q2
+[4]q2[5]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2
+×/bracketleftbig9m+5
+7/bracketrightbig
+q14[3m+2]q6[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡72 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+4
+4/bracketrightbig
+q8
+×[9m+5]q2/bracketleftbig3m+2
+2/bracketrightbig
+q12/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 37
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡73 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡74 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+10/bracketrightbig
+q20[10]q2
+[2]q2[5]q2
+×[9m+5]q2/bracketleftbig3m+2
+28/bracketrightbig
+q168[84]q2[2]q2
+[4]q2[6]q2[7]q2[9m+7]q2[m+1]q18.
+Ifonedecomposes[9 m+7]q2as[9m
+2+4]q4+q2[9m
+2+3]q4, thenoneseesthat, byCorollary6
+and Lemma 13, this is a polynomial in qwith non-negative integer coefficients. If
+m≡75 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+7/bracketrightbig
+q14/bracketleftbig9m+5
+5/bracketrightbig
+q10[3m+2]q6[9m+7]q2[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+76 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡77 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+4]q2
+×/bracketleftbig9m+5
+2/bracketrightbig
+q4[3m+2]q6/bracketleftbig9m+7
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡78 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4/bracketleftbig9m+5
+7/bracketrightbig
+q14/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18,38 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡79 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2/bracketleftbig9m+4
+5/bracketrightbig
+q10[9m+5]q2
+×[3m+2]q6[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡80 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8/bracketleftbig9m+5
+5/bracketrightbig
+q10/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+81 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2
+×[9m+5]q2/bracketleftbig3m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡82 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2
+×[9m+5]q2/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18.
+Ifonedecomposes[9 m+5]q2as[9m
+2+4]q4+q2[9m
+2+2]q4, thenoneseesthat,byCorollary6,
+this is a polynomial in qwith non-negative integer coefficients. If m≡83 (mod 140) ,
+then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2[9m+4]q2[9m+5]q
+×[3m+2]q6[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡84 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+20/bracketrightbig
+q40[20]q2
+[4]q2[5]q2
+×[9m+5]q2/bracketleftbig3m+2
+2/bracketrightbig
+q12/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡85 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12
+×[9m+4]q2/bracketleftbig9m+5
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 39
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡86 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡87 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2/bracketleftbig9m+5
+4/bracketrightbig
+q8[3m+2]q6/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+88 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2/bracketleftbig9m+4
+4/bracketrightbig
+q8[9m+5]q2
+×/bracketleftbig3m+2
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡89 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡90 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+2/bracketrightbig
+q4/bracketleftbig9m+5
+5/bracketrightbig
+q10/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4
+×[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡91 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2/bracketleftbig9m+5
+4/bracketrightbig
+q8[3m+2]q6/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+92 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8/bracketleftbig9m+5
+7/bracketrightbig
+q14/bracketleftbig3m+2
+2/bracketrightbig
+q12/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,40 C. KRATTENTHALER AND T. W. M ¨ULLER
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+93 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+35/bracketrightbig
+q210[105]q2
+[3]q2[5]q2[7]q2[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8/bracketleftbigm+1
+2/bracketrightbig
+q36[6]q6
+[2]q6[3]q6,
+which, by Corollary 6 and Lemma 14, is a polynomial in qwith non-negative integer
+coefficients. If m≡94 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14[3m+1]q6/bracketleftbig9m+4
+10/bracketrightbig
+q20[10]q2
+[2]q2[5]q2[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18.
+Ifonedecomposes[9 m+7]q2as[9m
+2+4]q4+q2[9m
+2+3]q4, thenoneseesthat,byCorollary6,
+this is a polynomial in qwith non-negative integer coefficients. If m≡95 (mod 140) ,
+then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2/bracketleftbig9m+5
+5/bracketrightbig
+q10
+×/bracketleftbig3m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡96 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10[3m+1]q6/bracketleftbig9m+4
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡97 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+4]q2/bracketleftbig9m+5
+2/bracketrightbig
+q4
+×[3m+2]q6/bracketleftbig9m+7
+5/bracketrightbig
+q10/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡98 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 41
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡99 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+5/bracketrightbig
+q10/bracketleftbig9m+5
+7/bracketrightbig
+q14[3m+2]q6[9m+7]q2[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+100 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+4
+4/bracketrightbig
+q8
+×/bracketleftbig9m+5
+5/bracketrightbig
+q10/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡101 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡102 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2/bracketleftbig3m+2
+28/bracketrightbig
+q168[84]q2[2]q2
+[4]q2[6]q2[7]q2/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which, by Lemma 13, is a polynomial in qwith non-negative integer coefficients. If
+m≡103 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2/bracketleftbig9m+4
+7/bracketrightbig
+q14[9m+5]q2
+×[3m+2]q6[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡104 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+20/bracketrightbig
+q40[20]q2
+[4]q2[5]q2[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡105 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4
+×[9m+4]q2/bracketleftbig9m+5
+10/bracketrightbig
+q20[10]q2
+[2]q2[5]q2[3m+2]q6/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18.42 C. KRATTENTHALER AND T. W. M ¨ULLER
+If one decomposes [9 m+1]q2as [9m+1
+2]q4+q2[9m+1
+2]q4, then one sees that, by Corollary 6,
+this is a polynomial in qwith non-negative integer coefficients. If m≡106 (mod 140) ,
+then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10[3m+1]q6/bracketleftbig9m+4
+2/bracketrightbig
+q4/bracketleftbig9m+5
+7/bracketrightbig
+q14/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4
+×[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡107 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡108 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2/bracketleftbig9m+4
+4/bracketrightbig
+q8[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡109 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+5/bracketrightbig
+q10
+×[9m+5]q2/bracketleftbig3m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡110 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2
+×/bracketleftbig9m+5
+5/bracketrightbig
+q10/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18.
+Ifonedecomposes[9 m+7]q2as[9m
+2+4]q4+q2[9m
+2+3]q4, thenoneseesthat,byCorollary6,
+this is a polynomial in qwith non-negative integer coefficients. If m≡111 (mod 140) ,
+then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2/bracketleftbig9m+5
+4/bracketrightbig
+q8[3m+2]q6
+×[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 43
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡112 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8
+×[9m+5]q2/bracketleftbig3m+2
+2/bracketrightbig
+q12/bracketleftbig9m+7
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡113 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2[9m+4]q2
+×/bracketleftbig9m+5
+7/bracketrightbig
+q14[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡114 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+4
+10/bracketrightbig
+q20[10]q2
+[2]q2[5]q2[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18.
+Ifonedecomposes[9 m+7]q2as[9m
+2+4]q4+q2[9m
+2+3]q4, thenoneseesthat,byCorollary6,
+this is a polynomial in qwith non-negative integer coefficients. If m≡115 (mod 140) ,
+then we have
+Catm(E7;q) =/bracketleftbig9m+1
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2
+×/bracketleftbig9m+5
+5/bracketrightbig
+q10[3m+2]q6[9m+7]q[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡116 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8[9m+5]q2
+×/bracketleftbig3m+2
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡117 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4/bracketleftbig9m+4
+7/bracketrightbig
+q14/bracketleftbig9m+5
+2/bracketrightbig
+q4
+×[3m+2]q6/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,44 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡118 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡119 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+5/bracketrightbig
+q10[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+120 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8
+×/bracketleftbig9m+5
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡121 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡122 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14[3m+1]q6
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡123 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2[9m+4]q2[9m+5]q2
+×/bracketleftbig3m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2[9m+7]q2[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 45
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡124 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2/bracketleftbigm+1
+5/bracketrightbig
+q90[45]q2
+[5]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡125 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+2/bracketrightbig
+q12
+×[9m+4]q2/bracketleftbig9m+5
+5/bracketrightbig
+q10[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡126 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10[3m+1]q6
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4[9m+5]q2/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4/bracketleftbig9m+7
+7/bracketrightbig
+q14[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡127 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12[9m+4]q2/bracketleftbig9m+5
+7/bracketrightbig
+q14[3m+2]q6/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+128 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+35/bracketrightbig
+q210[105]q2
+[3]q2[5]q2[7]q2/bracketleftbig9m+4
+4/bracketrightbig
+q8[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Lemma 14, is a polynomial in qwith non-negative integer coefficients. If
+m≡129 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+7/bracketrightbig
+q14/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+5/bracketrightbig
+q10[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+4/bracketrightbig
+q8[m+1]q18,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+130 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6
+×/bracketleftbig9m+4
+2/bracketrightbig
+q4/bracketleftbig9m+5
+5/bracketrightbig
+q10/bracketleftbig3m+2
+28/bracketrightbig
+q168[84]q2[2]q2
+[4]q2[6]q2[7]q2[9m+7]q2[m+1]q18,
+which, by Lemma 13, is a polynomial in qwith non-negative integer coefficients. If
+m≡131 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+5/bracketrightbig
+q10/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+7/bracketrightbig
+q14/bracketleftbig9m+5
+4/bracketrightbig
+q8[3m+2]q6[9m+7]q2[m+1]q18,46 C. KRATTENTHALER AND T. W. M ¨ULLER
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+132 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8
+×[9m+5]q2/bracketleftbig3m+2
+2/bracketrightbig
+q12/bracketleftbig9m+7
+5/bracketrightbig
+q10/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡133 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+10/bracketrightbig
+q60[30]q2
+[5]q2[6]q2[9m+4]q2
+×[9m+5]q2[3m+2]q6/bracketleftbig9m+7
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡134 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
+10/bracketrightbig
+q20[10]q2
+[2]q2[5]q2
+×/bracketleftbig9m+5
+7/bracketrightbig
+q14/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18.
+Ifonedecomposes[9 m+7]q2as[9m
+2+4]q4+q2[9m
+2+3]q4, thenoneseesthat,byCorollary6,
+this is a polynomial in qwith non-negative integer coefficients. If m≡135 (mod 140) ,
+then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[9m+4]q2
+×/bracketleftbig9m+5
+5/bracketrightbig
+q10[3m+2]q6[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡136 (mod 140) ,then we have
+Catm(E7;q) =/bracketleftbig9m+1
+35/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[3m+1]q6/bracketleftbig9m+4
+4/bracketrightbig
+q8[9m+5]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q12[9m+7]q2[m+1]q18,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡137 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+4]q2/bracketleftbig9m+5
+2/bracketrightbig
+q4
+×/bracketleftbig3m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig9m+7
+5/bracketrightbig
+q10[m+1]q18,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 47
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡138 (mod 140) ,then we have
+Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
+5/bracketrightbig
+q30[15]q2
+[3]q2[5]q2/bracketleftbig9m+4
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2[9m+5]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q24[6]q4
+[2]q4[3]q4[9m+7]q2[m+1]q18.
+Ifonedecomposes[9 m+7]q2as[9m
+2+4]q4+q2[9m
+2+3]q4, thenoneseesthat,byCorollary6,
+this is a polynomial in qwith non-negative integer coefficients. If m≡139 (mod 140) ,
+then we have
+Catm(E7;q) =/bracketleftbig9m+1
+4/bracketrightbig
+q8/bracketleftbig3m+1
+2/bracketrightbig
+q12/bracketleftbig9m+4
+5/bracketrightbig
+q10[9m+5]q2
+×[3m+2]q6[9m+7]q2/bracketleftbigm+1
+7/bracketrightbig
+q126[63]q2
+[7]q2[9]q2,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients.
+ForW=G37=E8, the degrees are 2 ,8,12,14,18,20,24,30, and hence
+Catm(E7;q) =[30m+2]q[30m+8]q[30m+12]q[30m+14]q
+[2]q[8]q[12]q[14]q
+×[30m+18]q[30m+20]q[30m+24]q[30m+30]q
+[18]q[20]q[24]q[30]q.
+Ifm≡0 (mod 84),then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12/bracketleftbig15m+7
+7/bracketrightbig
+q14
+×/bracketleftbig5m+3
+3/bracketrightbig
+q18/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+1 (mod 84),then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2/bracketleftbig5m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×[3m+2]q10/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡2 (mod 84),then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[m+1]q30,48 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡3 (mod 84),then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4/bracketleftbig15m+4
+7/bracketrightbig
+q14[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡4 (mod 84),then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12
+×[15m+7]q2[5m+3]q6/bracketleftbig3m+2
+14/bracketrightbig
+q140[70]q2
+[7]q2[10]q2/bracketleftbig5m+4
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡5 (mod 84),then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+28/bracketrightbig
+q168[84]q2
+[7]q2[12]q2
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡6 (mod 84),then we have
+Catm(E8;q) =/bracketleftbig15m+1
+7/bracketrightbig
+q14/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2
+×/bracketleftbig5m+3
+3/bracketrightbig
+q18/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡7 (mod 84),then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×[3m+2]q10/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡8 (mod 84),then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+42/bracketrightbig
+q252[126]q2[3]q2
+[6]q2[7]q2[9]q2[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 49
+which, by Lemma 22, is a polynomial in qwith non-negative integer coefficients. If
+m≡9 (mod 84),then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6
+×[3m+2]q10/bracketleftbig5m+4
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡10 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2/bracketleftbig5m+2
+4/bracketrightbig
+q24
+×[15m+7]q2[5m+3]q6/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6[m+1]q30.
+If one decomposes [15 m+ 7]q2as [15m
+2+ 4]q4+q2[15m
+2+ 3]q4, then one sees that, by
+Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If m≡
+11 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×/bracketleftbig3m+2
+7/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 20, is a polynomial in qwith non-negative integer
+coefficients. If m≡12 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12
+×[15m+7]q2/bracketleftbig5m+3
+21/bracketrightbig
+q126[63]q2
+[7]q2[9]q2/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡13 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×[3m+2]q10/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡14 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6/bracketleftbig15m+7
+7/bracketrightbig
+q14
+×[5m+3]q6/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30,50 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡15 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2/bracketleftbig5m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡16 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+84/bracketrightbig
+q504[252]q2[3]q2
+[7]q2[9]q2[12]q2[m+1]q30,
+which, by Lemma 23, is a polynomial in qwith non-negative integer coefficients. If
+m≡17 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8/bracketleftbig15m+4
+7/bracketrightbig
+q14/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Lemma 16, is a polynomial in qwith non-negative integer coefficients. If
+m≡18 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2/bracketleftbig5m+3
+3/bracketrightbig
+q18
+/bracketleftbig3m+2
+28/bracketrightbig
+q280[140]q2[2]q2
+[4]q2[7]q2[10]q2/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30,
+which, by Lemma 24, is a polynomial in qwith non-negative integer coefficients. If
+m≡19 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2
+×[3m+2]q10/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡20 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+7/bracketrightbig
+q14/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 51
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡21 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2/bracketleftbig5m+3
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 17, is a polynomial in qwith non-negative integer
+coefficients. If m≡22 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+28/bracketrightbig
+q168[84]q2
+[7]q2[12]q2[15m+7]q2
+×[5m+3]q6/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡23 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×[3m+2]q10/bracketleftbig5m+4
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡24 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2/bracketleftbig5m+2
+2/bracketrightbig
+q12[15m+7]q2
+×/bracketleftbig5m+3
+3/bracketrightbig
+q18/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡25 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×/bracketleftbig3m+2
+7/bracketrightbig
+q70[35]q2
+[5]q2[7]q2/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Lemma 18, is a polynomial in qwith non-negative integer coefficients. If
+m≡26 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6[15m+7]q2/bracketleftbig5m+3
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2
+×/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30.
+If one decomposes [15 m+1]q2as [5m+1]q6+q2[5m]q6+q4[5m]q6, then one sees that,
+by Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If52 C. KRATTENTHALER AND T. W. M ¨ULLER
+m≡27 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 21, is a polynomial in qwith non-negative integer
+coefficients. If m≡28 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12/bracketleftbig15m+7
+7/bracketrightbig
+q14
+×[5m+3]q6/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡29 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2/bracketleftbig5m+2
+21/bracketrightbig
+q126[63]q2
+[7]q2[9]q2/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡30 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2/bracketleftbig5m+3
+3/bracketrightbig
+q18
+/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡31 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4/bracketleftbig15m+4
+7/bracketrightbig
+q14[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×[3m+2]q10/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Lemma 19, is a polynomial in qwith non-negative integer coefficients. If
+m≡32 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+14/bracketrightbig
+q140[70]q2
+[7]q2[10]q2/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 53
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡33 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+84/bracketrightbig
+q504[252]q2[3]q2
+[7]q2[9]q2[12]q2
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Lemmas 16and23, is apolynomial in qwithnon-negative integer coefficients.
+Ifm≡34 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+7/bracketrightbig
+q14/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2
+[5m+3]q6/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡35 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡36 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[15m+7]q2/bracketleftbig5m+3
+3/bracketrightbig
+q18
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡37 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×[3m+2]q10/bracketleftbigg5m+4
+21/bracketrightbigg
+q126[63]q2
+[7]q2[9]q2/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡38 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2/bracketleftbig5m+2
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6
+×[15m+7]q2[5m+3]q6/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30.
+If one decomposes [15 m+ 7]q2as [15m
+2+ 4]q4+q2[15m
+2+ 3]q4, then one sees that, by
+Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If m≡54 C. KRATTENTHALER AND T. W. M ¨ULLER
+39 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6
+×/bracketleftbig3m+2
+7/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 20, is a polynomial in qwith non-negative integer
+coefficients. If m≡40 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12[15m+7]q2/bracketleftbig5m+3
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6[m+1]q30.
+If one decomposes [15 m+7]q2as [5m+1]q6+q2[5m]q6+q4[5m]q6, then one sees that,
+by Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If
+m≡41 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡42 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24/bracketleftbig15m+7
+7/bracketrightbig
+q14
+×/bracketleftbig5m+3
+3/bracketrightbig
+q18/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡43 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2/bracketleftbig5m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×[3m+2]q10/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡44 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+28/bracketrightbig
+q168[84]q2
+[7]q2[12]q2[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 55
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡45 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8/bracketleftbig15m+4
+7/bracketrightbig
+q14[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡46 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2
+[5m+3]q6/bracketleftbig3m+2
+28/bracketrightbig
+q280[140]q2[2]q2
+[4]q2[7]q2[10]q2/bracketleftbig5m+4
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6[m+1]q30,
+which, by Corollary 6 and Lemma 24, is a polynomial in qwith non-negative integer
+coefficients. If m≡47 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡48 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+7/bracketrightbig
+q14/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12[15m+7]q2
+/bracketleftbig5m+3
+3/bracketrightbig
+q18/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,
+which is manifestly a polynomial in qwith non-negative integer coefficients. If m≡
+49 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×[3m+2]q10/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡50 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+28/bracketrightbig
+q168[84]q2
+[7]q2[12]q2[15m+7]q2
+×[5m+3]q6/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+2/bracketrightbig
+q12/bracketleftbigm+1
+3/bracketrightbig
+q90[15]q6
+[3]q6[5]q6,56 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡51 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6
+×[3m+2]q10/bracketleftbig5m+4
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡52 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2/bracketleftbig5m+2
+2/bracketrightbig
+q12[15m+7]q2
+×[5m+3]q6/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡53 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×/bracketleftbig3m+2
+7/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Lemma 18, is a polynomial in qwith non-negative integer coefficients. If
+m≡54 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2
+/bracketleftbig5m+3
+21/bracketrightbig
+q126[63]q2
+[7]q2[9]q2/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡55 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×[3m+2]q10/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Lemma 21, is a polynomial in qwith non-negative integer coefficients. If
+m≡56 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6/bracketleftbig15m+7
+7/bracketrightbig
+q14[5m+3]q6
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 57
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡57 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2/bracketleftbig5m+2
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡58 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbigm+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2[5m+3]q6
+/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+42/bracketrightbig
+q252[126]q2[3]q2
+[6]q2[7]q2[9]q2[m+1]q30,
+which, by Lemma 22, is a polynomial in qwith non-negative integer coefficients. If
+m≡59 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4/bracketleftbig15m+4
+7/bracketrightbig
+q14/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Lemma 19, is a polynomial in qwith non-negative integer coefficients. If
+m≡60 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12[15m+7]q2/bracketleftbig5m+3
+3/bracketrightbig
+q18
+×/bracketleftbig3m+2
+14/bracketrightbig
+q140[70]q2
+[7]q2[10]q2/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡61 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+28/bracketrightbig
+q168[84]q2
+[7]q2[12]q2
+×[3m+2]q10/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡62 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+7/bracketrightbig
+q14/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30,58 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡63 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2/bracketleftbig5m+3
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡64 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+14/bracketrightbig
+q84[42]q2
+[6]q2[7]q2[15m+7]q2
+×[5m+3]q6/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡65 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×[3m+2]q10/bracketleftbig5m+4
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡66 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2
+×/bracketleftbig5m+3
+3/bracketrightbig
+q18/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡67 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×/bracketleftbig3m+2
+7/bracketrightbig
+q70[35]q2
+[5]q2[7]q2/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Lemma 20, is a polynomial in qwith non-negative integer coefficients. If
+m≡68 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6
+×[15m+7]q2/bracketleftbig5m+3
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 59
+If one decomposes [15 m+1]q2as [5m+1]q6+q2[5m]q6+q4[5m]q6, then one sees that,
+by Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If
+m≡69 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡70 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24/bracketleftbig15m+7
+7/bracketrightbig
+q14
+×[5m+3]q6/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡71 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2/bracketleftbig5m+2
+21/bracketrightbig
+q126[63]q2
+[7]q2[9]q2/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡72 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12[15m+7]q2/bracketleftbig5m+3
+3/bracketrightbig
+q18/bracketleftbig3m+2
+2/bracketrightbig
+q20
+/bracketleftbig5m+4
+28/bracketrightbig
+q168[84]q2
+[7]q2[12]q2[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡73 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8/bracketleftbig15m+4
+7/bracketrightbig
+q14[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×[3m+2]q10/bracketleftbig5m+4
+3/bracketrightbig
+q18/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Lemma 16, is a polynomial in qwith non-negative integer coefficients. If
+m≡74 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+28/bracketrightbig
+q280[140]q2[2]q2
+[4]q2[7]q2[10]q2/bracketleftbig5m+4
+2/bracketrightbig
+q12/bracketleftbigm+1
+3/bracketrightbig
+q90[15]q6
+[3]q6[5]q6,60 C. KRATTENTHALER AND T. W. M ¨ULLER
+which, by Corollary 6 and Lemma 24, is a polynomial in qwith non-negative integer
+coefficients. If m≡75 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+42/bracketrightbig
+q252[126]q2[3]q2
+[6]q2[7]q2[9]q2
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Lemmas 19and22, is apolynomial in qwithnon-negative integer coefficients.
+Ifm≡76 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+7/bracketrightbig
+q14/bracketleftbig15m+4
+4/bracketrightbig
+q8/bracketleftbig5m+2
+2/bracketrightbig
+q12[15m+7]q2
+×[5m+3]q6/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡77 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2/bracketleftbig5m+3
+4/bracketrightbig
+q24
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 16, is a polynomial in qwith non-negative integer
+coefficients. If m≡78 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+28/bracketrightbig
+q168[84]q2
+[7]q2[12]q2[15m+7]q2/bracketleftbig5m+3
+3/bracketrightbig
+q18
+×/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+2/bracketrightbig
+q12[m+1]q30,
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡79 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+2/bracketrightbig
+q4[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×[3m+2]q10/bracketleftbig5m+4
+21/bracketrightbig
+q126[63]q2
+[7]q2[9]q2/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
+coefficients. If m≡80 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+28/bracketrightbig
+q56[28]q2
+[4]q2[7]q2/bracketleftbig5m+2
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6[15m+7]q2[5m+3]q6
+×/bracketleftbig3m+2
+2/bracketrightbig
+q20/bracketleftbig5m+4
+4/bracketrightbig
+q24[m+1]q30,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 61
+which, by Corollary 6, is a polynomial in qwith non-negative integer coefficients. If
+m≡81 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+4/bracketrightbig
+q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
+2/bracketrightbig
+q4/bracketleftbig5m+3
+12/bracketrightbig
+q72[12]q6
+[3]q6[4]q6
+×/bracketleftbig3m+2
+7/bracketrightbig
+q70[35]q2
+[5]q2[7]q2[5m+4]q6/bracketleftbigm+1
+2/bracketrightbig
+q60[30]q2[2]q2[3]q2[5]q2
+[6]q2[10]q2[15]q2,
+which, by Corollary 6 and Lemma 18, is a polynomial in qwith non-negative integer
+coefficients. If m≡82 (mod 84) ,then we have
+Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
+2/bracketrightbig
+q4/bracketleftbig5m+2
+4/bracketrightbig
+q24[15m+7]q2/bracketleftbig5m+3
+7/bracketrightbig
+q42[21]q2
+[3]q2[7]q2
+/bracketleftbig3m+2
+4/bracketrightbig
+q40[10]q4
+[2]q4[5]q4/bracketleftbig5m+4
+6/bracketrightbig
+q36[6]q6
+[2]q6[3]q6[m+1]q30.
+If one decomposes [15 m+1]q2as [5m+1]q6+q2[5m]q6+q4[5m]q6, then one sees that,
+by Corollary 6, this is a polynomial in qwith non-negative integer coefficients. If
+m≡83 (mod 84) ,then we have
+Catm(E8;q) =/bracketleftbig15m+1
+14/bracketrightbig
+q28[14]q2
+[2]q2[7]q2[15m+4]q2/bracketleftbig5m+2
+3/bracketrightbig
+q18/bracketleftbig15m+7
+4/bracketrightbig
+q8/bracketleftbig5m+3
+2/bracketrightbig
+q12
+×[3m+2]q10[5m+4]q6/bracketleftbigm+1
+4/bracketrightbig
+q120[60]q2[2]q2[3]q2[5]q2
+[10]q2[12]q2[15]q2,
+which, by Corollary 6 and Lemma 21, is a polynomial in qwith non-negative integer
+coefficients.
+/square
+5.Auxiliary results I
+This section collects several auxiliary results which allow us to reduce the problem
+of proving Theorem 2, or the equivalent statement (3.3), for the 2 6 exceptional groups
+listed in Section 2 to a finite problem. While Lemmas 27 and 28 cover spec ial choices
+of the parameters, Lemmas 26 and 30 afford an inductive procedur e. More precisely,
+if we assume that we have already verified Theorem 2 for all groups o f smaller rank,
+then Lemmas 26 and 30, together with Lemmas 27 and 31, reduce th e verification of
+Theorem 2 for the group that we are currently considering to a finit e problem; see
+Remark 3. The final lemma of this section, Lemma 32, disposes of com plex reflection
+groups with a special property satisfied by their degrees.
+Letp=am+b, 0≤b<m. We have
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;ca+1wm−b+1c−a−1,ca+1wm−b+2c−a−1,...,ca+1wmc−a−1,
+caw1c−a,...,cawm−bc−a/parenrightbig
+,(5.1)
+where∗stands for the element of Wwhich is needed to complete the product of the
+components to c.62 C. KRATTENTHALER AND T. W. M ¨ULLER
+Lemma 26. It suffices to check (3.3)forpa divisor of mh. More precisely, let pbe
+a divisor of mh, and letkbe another positive integer with gcd(k,mh/p) = 1, then we
+have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πikp/mh (5.2)
+and
+|FixNCm(W)(φp)|=|FixNCm(W)(φkp)|. (5.3)
+Proof.For (5.2), this follows immediately from
+lim
+q→ζ[α]q
+[β]q=/braceleftBigg
+α
+βifα≡β≡0 (modd),
+1 otherwise ,(5.4)
+whereζis ad-th root of unity and α,βare non-negative integers such that α≡β
+(modd).
+In order to establish (5.3), suppose that x∈FixNCm(W)(φp), that is,x∈NCm(W)
+andφp(x) =x. It obviously follows that φkp(x) =x, so thatx∈FixNCm(W)(φkp).
+To establish the converse, note that, if gcd( k,mh/p) = 1, then there exists k′with
+k′k≡1 (modmh
+p). It follows that, if x∈FixNCm(W)(φkp), that is, if x∈NCm(W) and
+φkp(x) =x, thenx=φk′kp(x) =φp(x), whencex∈FixNCm(W)(φp). /square
+Lemma 27. Letpbe a divisor of mh. Ifpis divisible by m, then(3.3)is true.
+Proof.According to (5.1), the action of φponNCm(W) is described by
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;cp/mw1c−p/m,...,cp/mwmc−p/m/parenrightbig
+.
+Hence, if (w0;w1,...,w m) is fixed by φp, then each individual wimust be fixed under
+conjugation by cp/m.
+Using the notation W′= Cent W(cp/m), theprevious observationmeans that wi∈W′,
+i= 1,2,...,m. Springer [33, Theorem 4.2] (see also [24, Theorem 11.24(iii)]) prove d
+thatW′is a well-generated complex reflection group whose degrees coincide with those
+degrees ofWthat are divisible by mh/p. It was furthermore shown in [9, Lemma 3.3]
+that
+NC(W)∩W′=NC(W′). (5.5)
+Hence, the tuples ( w0;w1,...,w m) fixed byφpare in fact identical with the elements of
+NCm(W′), which implies that
+|FixNCm(W)(φp)|=|NCm(W′)|. (5.6)
+Application of Theorem 1 with Wreplaced by W′and of the “limit rule” (5.4) then
+yields that
+|NCm(W′)|=/productdisplay
+1≤i≤n
+mh
+p|dimh+di
+di= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh. (5.7)
+Combining (5.6) and (5.7), we obtain (3.3). This finishes the proof of t he lemma. /square
+Lemma 28. Equation (3.3)holds for all divisors pofm.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 63
+Proof.Using (5.4) and the fact that the degrees of irreducible well-genera ted complex
+reflection groups satisfy di<hfor alli<n, we see that
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh=/braceleftBigg
+m+1 ifm=p,
+1 ifm/ne}ationslash=p.
+On the other hand, if ( w0;w1,...,w m) is fixed by φp, then, because of the action (5.1),
+we must have w1=wp+1=···=wm−p+1andw1=cwm−p+1c−1. In particular,
+w1∈CentW(c). By the theorem of Springer cited in the proof of Lemma 27, the
+subgroup Cent W(c) is itself a complex reflection group whose degrees are those degre es
+ofWthat are divisible by h. The only such degree is hitself, hence Cent W(c) is the
+cyclic group generated by c. Moreover, by (5.5), we obtain that w1=ε, the identity
+element of W, orw1=c. Therefore, for m=pthe set Fix NCm(W)(φp) consists of the
+m+1 elements ( w0;w1,...,w m) obtained by choosing wi=cfor a particular ibetween
+0 andm, all otherwj’s being equal to ε, while, for m/ne}ationslash=p, we have
+FixNCm(W)(φp) =/braceleftbig
+(c;ε,...,ε)/bracerightbig
+,
+whence the result. /square
+Lemma 29. LetWbe an irreducible well-generated complex reflection group a ll of
+whose degrees are divisible by d. Then each element of Wis fixed under conjugation by
+ch/d.
+Proof.By the theorem of Springer cited in the proof of Lemma 27, the subg roupW′=
+CentW(ch/d) is itself a complex reflection group whose degrees are those degre es ofW
+that are divisible by d. Thus, by our assumption, the degrees of W′coincide with the
+degrees ofW, and hence W′must be equal to W. Phrased differently, each element of
+Wis fixed under conjugation by ch/d, as claimed. /square
+Lemma 30. LetWbe an irreducible well-generated complex reflection group o f rankn,
+and letp=m1h1be a divisor of mh, wherem=m1m2andh=h1h2. Without loss of
+generality, we assume that gcd(h1,m2) = 1. Suppose that Theorem 2has already been
+verified for all irreducible well-generated complex reflect ion groups with rank <n. Ifh2
+does not divide all degrees di, then Equation (3.3)is satisfied.
+Proof.Let us write h1=am2+b, with 0 ≤b < m 2. The condition gcd( h1,m2) = 1
+translates into gcd( b,m2) = 1. From (5.1), we infer that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;ca+1wm−m1b+1c−a−1,ca+1wm−m1b+2c−a−1,...,ca+1wmc−a−1,
+caw1c−a,...,cawm−m1bc−a/parenrightbig
+.(5.8)
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=ca+1wi+m−m1bc−a−1, i= 1,2,...,m 1b,
+wi=cawi−m1bc−a, i=m1b+1,m1b+2,...,m,
+which, after iteration, implies in particular that
+wi=cb(a+1)+(m2−b)awic−b(a+1)−(m2−b)a=ch1wic−h1, i= 1,2,...,m.64 C. KRATTENTHALER AND T. W. M ¨ULLER
+It is at this point where we need gcd( b,m2) = 1. The last equation shows that each wi,
+i= 1,2,...,m, and thus also w0, lies in Cent W(ch1). By the theorem of Springer cited
+in the proof of Lemma 27, this centraliser subgroup is itself a complex reflection group,
+W′say, whose degrees are those degrees of Wthat are divisible by h/h1=h2. Since,
+by assumption, h2does not divide alldegrees,W′has rank strictly less than n. Again
+by assumption, we know that Theorem 2 is true for W′, so that in particular,
+|FixNCm(W′)(φp)|= Catm(W′;q)/vextendsingle/vextendsingle
+q=e2πip/mh.
+The arguments above together with (5.5) show that Fix NCm(W)(φp) = Fix NCm(W′)(φp).
+On the other hand, using (5.4) it is straightforward to see that
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/mh= Catm(W′;q)/vextendsingle/vextendsingle
+q=e2πip/mh.
+This proves (3.3) for our particular p, as required. /square
+Lemma 31. LetWbe an irreducible well-generated complex reflection group o f rank
+n, and letp=m1h1be a divisor of mh, wherem=m1m2andh=h1h2. We assume
+thatgcd(h1,m2) = 1. Ifm2>nthen
+FixNCm(W)(φp) =/braceleftbig
+(c;ε,...,ε)/bracerightbig
+.
+Proof.Let us suppose that ( w0;w1,...,w m)∈FixNCm(W)(φp) and that there exists a
+j≥1 such that wj/ne}ationslash=ε. By (5.8), it then follows for such a jthat alsowk/ne}ationslash=εfor
+allk≡j−lm1b(modm), where, as before, bis defined as the unique integer with
+h1=am2+band 0≤b < m 2. Since, by assumption, gcd( b,m2) = 1, there are
+exactlym2suchk’s which are distinct mod m. However, this implies that the sum of
+the absolute lengths of the wi’s, 0≤i≤m, is at least m2> n, a contradiction to
+Remark 1.(2). /square
+Remark 3.(1) If we put ourselves in the situation of the assumptions of Lemma 30,
+then we may conclude that equation (3.3) only needs to be checked f or pairs (m2,h2)
+subject to the following restrictions:
+m2≥2,gcd(h1,m2) = 1,andh2divides all degrees of W. (5.9)
+Indeed, Lemmas 27 and 30 together imply that equation (3.3) is alway s satisfied in all
+other cases.
+(2) Still putting ourselves in the situation of Lemma 30, if m2>nandm2h2does not
+divide any of the degrees of W, then equation (3.3) is satisfied. Indeed, Lemma 31 says
+thatinthiscasetheleft-handsideof (3.3)equals1,whileastraightf orwardcomputation
+using (5.4) shows that in this case the right-hand side of (3.3) equals 1 as well.
+(3)It shouldbeobserved that thisleaves afinitenumber of choices form2to consider,
+whence a finite number of choices for ( m1,m2,h1,h2). Altogether, there remains a finite
+number of choices for p=h1m1to be checked.
+Lemma 32. LetWbe an irreducible well-generated complex reflection group o f rankn
+with the property that di|hfori= 1,2,...,n. Then Theorem 2is true for this group
+W.
+Proof.By Lemma 26, we may restrict ourselves to divisors pofmh.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 65
+Suppose that e2πip/mhis adi-th rootof unity for some i. In other words, mh/pdivides
+di. Sincediis a divisor of hby assumption, the integer mh/palso divides h. But this
+is equivalent to saying that mdividesp, and equation (3.3) holds by Lemma 27.
+Now assume that mh/pdoes not divide any of the di’s. Then, by (5.4), the right-
+hand side of (3.3) equals 1. On the other hand, ( c;ε,...,ε) is always an element of
+FixNCm(W)(φp). To see that there are no others, we make appeal to the classific a-
+tion of all irreducible well-generated complex reflection groups, whic h we recalled in
+Section 2. Inspection reveals that all groups satisfying the hypot heses of the lemma
+have rank n≤2. Except for the groups contained in the infinite series G(d,1,n)
+andG(e,e,n) for which Theorem 2 has been established in [19], these are the grou ps
+G5,G6,G9,G10,G14,G17,G18,G21. We now discuss these groups case by case, keeping
+the notation of Lemma 30. In order to simplify the argument, we not e that Lemma 31
+implies that equation (3.3) holds if m2>2, so that in the following arguments we
+always may assume that m2= 2.
+CaseG5. The degrees are 6 ,12, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.
+CaseG6. The degrees are 4 ,12, and therefore, according to Remark 3.(1), we need
+only consider the casewhere h2= 4andm2= 2, that is, p= 3m/2. Then (5.8) becomes
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+2+1c−2,c2wm
+2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+2c−1/parenrightbig
+.
+(5.10)
+If (w0;w1,...,w m) isfixed by φpandnot equal to ( c;ε,...,ε), there must exist an iwith
+1≤i≤m
+2such thatℓT(wi) =ℓT(wm
+2+i) = 1,wm
+2+i=cwic−1,wiwm
+2+i=wicwic−1=c,
+and allwj, withj/ne}ationslash=i,m
+2+i, equalε. However, with the help of the GAPpackage
+CHEVIE[14, 28], one verifies that there is no wiinG6such that
+ℓT(wi) = 1 and wicwic−1=c
+are simultaneously satisfied. Hence, the left-hand side of (3.3) is eq ual to 1, as required.
+CaseG9. The degrees are 8 ,24, and therefore, according to Remark 3.(1), we need
+only consider the case where h2= 8 andm2= 2, that is, p= 3m/2. This is the same p
+as forG6. Again, CHEVIEfinds no solution. Hence, the left-hand side of (3.3) is equal
+to 1, as required.
+CaseG10. The degrees are 12 ,24, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.
+CaseG14. The degrees are 6 ,24, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.
+CaseG17. The degrees are 20 ,60, and therefore, according to Remark 3.(1), we need
+only consider the cases where h2= 20 orh2= 4. In the first case, p= 3m/2, which is
+the samepas forG6. Again,CHEVIEfinds no solution. In the second case, p= 15m/2.
+Then (5.8) becomes
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c8wm
+2+1c−8,c8wm
+2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
+2c−7/parenrightbig
+.(5.11)
+By Lemma 29, every element of NC(W) is fixed under conjugation by c3, and, thus, on
+elements fixed by φp, the above action of φpreduces to the one in (5.10). This action66 C. KRATTENTHALER AND T. W. M ¨ULLER
+was already discussed in the first case. Hence, in both cases, the le ft-hand side of (3.3)
+is equal to 1, as required.
+CaseG18. The degrees are 30 ,60, and therefore Remark 3.(1) implies that equa-
+tion (3.3) is always satisfied.
+CaseG21. The degrees are 12 ,60, and therefore, according to Remark 3.(1), we need
+only consider the cases where h2= 12 orh2= 4. In the first case, p= 5m/2, so that
+(5.8) becomes
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3wm
+2+1c−3,c3wm
+2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm
+2c−2/parenrightbig
+.(5.12)
+If (w0;w1,...,w m) is fixed by φpand not equal to ( c;ε,...,ε), there must exist an i
+with 1≤i≤m
+2such thatℓT(wi) = 1 andwic2wic−2=c. However, with the help of
+theGAPpackageCHEVIE[14, 28], one verifies that there is no such solution to this
+equation. In the second case, p= 15m/2. Then (5.8) becomes the action in (5.11).
+By Lemma 29, every element of NC(W) is fixed under conjugation by c5, and, thus,
+on elements fixed by φp, the action of φpin (5.11) reduces to the one in the first case.
+Hence, in both cases, the left-hand side of (3.3) is equal to 1, as re quired.
+This completes the proof of the lemma. /square
+6.Case-by-case verification of Theorem 2
+In the sequel we write ζdfor a primitive d-th root of unity.
+CaseG4.The degrees are 4 ,6, and hence we have
+Catm(G4;q) =[6m+6]q[6m+4]q
+[6]q[4]q.
+Letζbe a 6m-th root of unity. In what follows, we abbreviate the assertion tha t “ζis
+a primitive d-th root of unity” as “ ζ=ζd.” The following cases on the right-hand side
+of (3.3) occur:
+lim
+q→ζCatm(G4;q) =m+1,ifζ=ζ6,ζ3, (6.1a)
+lim
+q→ζCatm(G4;q) =3m+2
+2,ifζ=ζ4,2|m, (6.1b)
+lim
+q→ζCatm(G4;q) = Catm(G4),ifζ=−1 orζ= 1, (6.1c)
+lim
+q→ζCatm(G4;q) = 1,otherwise. (6.1d)
+We must now prove that the left-hand side of (3.3) in each case agre es with the
+values exhibited in (6.1). The only cases not covered by Lemmas 27 an d 28 are the
+ones in (6.1b) and (6.1d). On the other hand, the only case left to co nsider according
+to Remark 3 is the case where h2=m2= 2, that is the case (6.1b) where p= 3m/2. In
+particular, mmust be divisible by 2. The action of φpis the same as the one in (5.10).
+With the help of CHEVIE, one finds that each of the 3 (complex) reflections in G4which
+are less than the (chosen) Coxeter element is a valid choice for wi, and each of these
+choices gives rise to m/2 elements in NCm(G4) since the index iranges from 1 to m/2.
+Hence, in total, we obtain 1+3m
+2=3m+2
+2elements in Fix NCm(G4)(φp), which agrees
+with the limit in (6.1b).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 67
+CaseG8.The degrees are 8 ,12, and hence we have
+Catm(G8;q) =[12m+12]q[12m+8]q
+[12]q[8]q.
+Letζbe a 12m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(G8;q) =m+1,ifζ=ζ12,ζ6,ζ3, (6.2a)
+lim
+q→ζCatm(G8;q) =3m+2
+2,ifζ=ζ8,2|m, (6.2b)
+lim
+q→ζCatm(G8;q) = Catm(G8),ifζ=ζ4,−1,1, (6.2c)
+lim
+q→ζCatm(G8;q) = 1,otherwise. (6.2d)
+We must now prove that the left-hand side of (3.3) in each case agre es with the
+values exhibited in (6.2). The only cases not covered by Lemmas 27 an d 28 are the
+ones in (6.2b) and (6.2d). On the other hand, the only case left to co nsider according to
+Remark 3 is the case where h2= 4 andm2= 2, that is the case (6.2b) where p= 3m/2.
+In particular, mmust be divisible by 2. The action of φpis the same as the one in
+(5.10). With the help of CHEVIE, one finds that each of the 3 (complex) reflections in
+G8which are less than the (chosen) Coxeter element is a valid choice for wi, and each
+of these choices gives rise to m/2 elements in NCm(G8) since the index iranges from
+1 tom/2.
+Hence, in total, we obtain 1+3m
+2=3m+2
+2elements in Fix NCm(G8)(φp), which agrees
+with the limit in (6.2b).
+CaseG16.The degrees are 20 ,30, and hence we have
+Catm(G16;q) =[30m+30]q[30m+20]q
+[30]q[20]q.
+Letζbe a 30m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(G16;q) =m+1,ifζ=ζ30,ζ15,ζ6,ζ3, (6.3a)
+lim
+q→ζCatm(G16;q) =3m+2
+2,ifζ=ζ20,ζ4,2|m, (6.3b)
+lim
+q→ζCatm(G16;q) = Catm(G16),ifζ=ζ10,ζ5,−1,1, (6.3c)
+lim
+q→ζCatm(G16;q) = 1,otherwise. (6.3d)
+We must now prove that the left-hand side of (3.3) in each case agre es with the
+values exhibited in (6.3). The only cases not covered by Lemmas 27 an d 28 are the
+ones in (6.3b) and (6.3d). On the other hand, the only cases left to c onsider according
+to Remark 3 are the cases where h2= 10 andm2= 2, respectively h2=m2= 2. Both
+cases belong to (6.3b). In the first case, we have p= 3m/2, while in the second case we
+havep= 15m/2. In particular, mmust be divisible by 2. In the first case, the action
+ofφpis the same as the one in (5.10). With the help of CHEVIE, one finds that each of
+the 3 (complex) reflections in G16which are less than the (chosen) Coxeter element is
+a valid choice for wi, and each of these choices gives rise to m/2 elements in NCm(G16)68 C. KRATTENTHALER AND T. W. M ¨ULLER
+since the index iranges from 1 to m/2. On the other hand, if p= 15m/2, then the
+action ofφpis the same as the one in (5.11). By Lemma 29, every element of NC(W)
+is fixed under conjugation by c3, and, thus, on elements fixed by φp, the action of φp
+reduces to the one in the first case.
+Hence, in total, we obtain 1+3m
+2=3m+2
+2elements in Fix NCm(G16)(φp), which agrees
+with the limit in (6.3b).
+CaseG20.The degrees are 12 ,30, and hence we have
+Catm(G20;q) =[30m+30]q[30m+12]q
+[30]q[12]q.
+Letζbe a 30m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(G20;q) =m+1,ifζ=ζ30,ζ15,ζ10,ζ5, (6.4a)
+lim
+q→ζCatm(G20;q) =5m+2
+2,ifζ=ζ12,ζ4,2|m, (6.4b)
+lim
+q→ζCatm(G20;q) = Catm(G20),ifζ=ζ6,ζ3,−1,1, (6.4c)
+lim
+q→ζCatm(G20;q) = 1,otherwise. (6.4d)
+We must now prove that the left-hand side of (3.3) in each case agre es with the
+values exhibited in (6.4). The only cases not covered by Lemmas 27 an d 28 are the
+ones in (6.4b) and (6.4d). On the other hand, the only cases left to c onsider according
+to Remark 3 are the cases where h2= 6 andm2= 2, respectively h2=m2= 2. Both
+cases belong to (6.4b). In the first case, we have p= 5m/2, while in the second case we
+havep= 15m/2. In particular, mmust be divisible by 2. In the first case, the action
+ofφpis the same as the one in (5.12). With the help of CHEVIE, one finds that each of
+the 5 (complex) reflections in G20which are less than the (chosen) Coxeter element is
+a valid choice for wi, and each of these choices gives rise to m/2 elements in NCm(G20)
+since the index iranges from 1 to m/2. On the other hand, if p= 15m/2, then the
+action ofφpis the same as the one in (5.11). By Lemma 29, every element of NC(W)
+is fixed under conjugation by c5, and, thus, on elements fixed by φp, the action of φp
+reduces to the one in the first case.
+Hence, in total, we obtain 1+5m
+2=5m+2
+2elements in Fix NCm(G20)(φp), which agrees
+with the limit in (6.4b).
+CaseG23=H3.The degrees are 2 ,6,10, and hence we have
+Catm(H3;q) =[10m+10]q[10m+6]q[10m+2]q
+[10]q[6]q[2]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 69
+Letζbe a 10m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(H3;q) =m+1,ifζ=ζ10,ζ5, (6.5a)
+lim
+q→ζCatm(H3;q) =10m+6
+6,ifζ=ζ6,ζ3,3|m, (6.5b)
+lim
+q→ζCatm(H3;q) = Catm(H3),ifζ=−1 orζ= 1, (6.5c)
+lim
+q→ζCatm(H3;q) = 1,otherwise. (6.5d)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.5). The only cases not covered by Lemmas 27 and 28 ar e the ones in
+(6.5b) and (6.5d). By Lemma 26, we are free to choose p= 5m/3 ifζ=ζ6, respectively
+p= 10m/3 ifζ=ζ3. In both cases, mmust be divisible by 3.
+We start with the case that p= 5m/3. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+3+1c−2,c2wm
+3+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+3c−1/parenrightbig
+.
+(6.6)
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+3+ic−2, i= 1,2,...,2m
+3, (6.7a)
+wi=cwi−2m
+3c−1, i=2m
+3+1,2m
+3+2,...,m. (6.7b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1.
+Writingt1,t2,t3forwi,wi+m
+3,wi+2m
+3, respectively, the equations (6.7) reduce to
+t1=c2t2c−2, (6.8a)
+t2=c2t3c−2, (6.8b)
+t3=ct1c−1. (6.8c)
+One of these equations is in fact superfluous: if we substitute (6.8b ) and (6.8c) in
+(6.8a), then we obtain t1=c5t1c−5which is automatically satisfied due to Lemma 29
+withd= 2.
+Since (w0;w1,...,w m)∈NCm(H3), we must have t1t2t3=c. Combining this with
+(6.8), we infer that
+t1(c−2t1c2)(ct1c−1) =c. (6.9)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains five solutions
+fort1in this equation:
+t1∈/braceleftbig
+[2],[3],[2,1,2],[1,2,3,2,1],[1,3,2,1,2,1,3]/bracerightbig
+. (6.10)
+Here we have used the short notation of coxeter: if{s1,s2,s3}is a simple system of
+generators of H3, corresponding to the Dynkin diagram displayed in Figure 1, then
+[j1,j2,...,j k] stands for the element sj1sj2...sjk.
+We claim that each of the above five solutions gives rise to m/3 elements of
+FixNCm(H3)(φp). Indeed, given t1, the elements t2andt3can be computed by (6.8a)
+and (6.8c), and there are m/3 possibilities to choose the index iforwi.70 C. KRATTENTHALER AND T. W. M ¨ULLER
+• • •1 2 35
+Figure 1. The Dynkin diagram for H3
+In total, we obtain 1 + 5m
+3=10m+6
+6elements in Fix NCm(H3)(φp), which agrees with
+the limit in (6.5b).
+In the case that p= 10m/3, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4w2m
+3+1c−4,c4w2m
+3+2c−4,...,c4wmc−4,c3w1c−3,...,c3w2m
+3c−3/parenrightbig
+.(6.11)
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4w2m
+3+ic−4, i= 1,2,...,m
+3, (6.12a)
+wi=c3wi−m
+3c−3, i=m
+3+1,m
+3+2,...,m. (6.12b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1.
+Writingt1,t2,t3forwi,wi+m
+3,wi+2m
+3, respectively, the equations (6.12) reduce to
+t1=c4t3c−4, (6.13a)
+t2=c3t1c−3, (6.13b)
+t3=c3t2c−3. (6.13c)
+One of these equations is in fact superfluous: if we substitute (6.13 b) and (6.13c) in
+(6.13a), then we obtain t1=c10t1c−10which is automatically satisfied since c10=ε.
+Since (w0;w1,...,w m)∈NCm(H3), we must have t1t2t3=c. Combining this with
+(6.13), we infer that
+t1(c3t1c−3)(c−4t1c4) =c. (6.14)
+Using that c5t1c−5=t1, due to Lemma 29 with d= 2, we see that this equation is
+equivalent with (6.9). Therefore, we are facing exactly the same en umeration problem
+here as forp= 5m/3, and, consequently, the number of solutions to (6.14) is the same ,
+namely5m+3
+3, as required.
+Finally, we turn to (6.5d). By Remark 3, the only choices for h2andm2to be
+considered are h2= 1 andm2= 3,h2=m2= 2, respectively h2= 2 andm2= 3.
+These correspond to the choices p= 10m/3,p= 5m/2, respectively p= 5m/3, out of
+which only p= 5m/2 has not yet been discussed and belongs to the current case. The
+corresponding action of φpis given by (5.12). A computation with Stembridge’s Maple
+packagecoxeter [36] finds no solution. Hence, the left-hand side of (3.3) is equal to 1 ,
+as required.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 71
+CaseG24.The degrees are 4 ,6,14, and hence we have
+Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q
+[14]q[6]q[4]q.
+Letζbe a 14m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (6.15a)
+lim
+q→ζCatm(G24;q) =7m+3
+3,ifζ=ζ6,ζ3,3|m, (6.15b)
+lim
+q→ζCatm(G24;q) =7m+2
+2,ifζ=ζ4,2|m, (6.15c)
+lim
+q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (6.15d)
+lim
+q→ζCatm(G24;q) = 1,otherwise. (6.15e)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.15). The only cases not covered by Lemmas 27 and 28 a re the ones in
+(6.15b), (6.15c), and (6.15e).
+We first consider (6.15b). By Lemma 26, we are free to choose p= 7m/3 ifζ=ζ6,
+respectively p= 14m/3 ifζ=ζ3. In both cases, mmust be divisible by 3.
+We start with the case that p= 7m/3. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3w2m
+3+1c−3,c3w2m
+3+2c−3,...,c3wmc−3,c2w1c−2,...,c2w2m
+3c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3w2m
+3+ic−3, i= 1,2,...,m
+3, (6.16a)
+wi=c2wi−m
+3c−2, i=m
+3+1,m
+3+2,...,m. (6.16b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1.
+Writingt1,t2,t3forwi,wi+m
+3,wi+2m
+3, respectively, the equations (6.16) reduce to
+t1=c3t3c−3, (6.17a)
+t2=c2t1c−2, (6.17b)
+t3=c2t2c−2. (6.17c)
+One of these equations is in fact superfluous: if we substitute (6.17 b) and (6.17c) in
+(6.17a), then we obtain t1=c7t1c−7which is automatically satisfied due to Lemma 29
+withd= 2.
+Since (w0;w1,...,w m)∈NCm(G24), we must have t1t2t3=c. Combining this with
+(6.17), we infer that
+t1(c2t1c−2)(c4t1c−4) =c. (6.18)
+With the help of CHEVIE, one obtains 7 solutions for t1in this equation:
+t1∈/braceleftbig
+[1],[2],[3],[15],[16],[19],[21]/bracerightbig
+, (6.19)72 C. KRATTENTHALER AND T. W. M ¨ULLER
+each of them giving rise to m/3 elements of Fix NCm(G24)(φp) sinceiranges from 1 to
+m/3. Here we have used the short notation of CHEVIE: [j1,j2,...,j k] stands for the
+elementrj1rj2...rjk, whereriis thei-th (complex) reflection corresponding to the i-th
+root in the internal ordering of the roots of G24inCHEVIE.
+In total, we obtain 1 + 7m
+3=7m+3
+3elements in Fix NCm(G24)(φp), which agrees with
+the limit in (6.15b).
+In the case that p= 14m/3, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c5wm
+3+1c−5,c5wm
+3+2c−5,...,c5wmc−5,c4w1c−4,...,c4wm
+3c−4/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c5wm
+3+ic−5, i= 1,2,...,2m
+3, (6.20a)
+wi=c4wi−2m
+3c−4, i=2m
+3+1,2m
+3+2,...,m. (6.20b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1.
+Writingt1,t2,t3forwi,wi+m
+3,wi+2m
+3, respectively, the equations (6.20) reduce to
+t1=c5t2c−5, (6.21a)
+t2=c5t3c−5, (6.21b)
+t3=c4t1c−4. (6.21c)
+One of these equations is in fact superfluous: if we substitute (6.21 b) and (6.21c) in
+(6.21a), then we obtain t1=c14t1c−14which is automatically satisfied since c14=ε.
+Since (w0;w1,...,w m)∈NCm(G24), we must have t1t2t3=c. Combining this with
+(6.21), we infer that
+t1(c9t1c−9)(c−4t1c4) =c. (6.22)
+Using that c7t1c−7=t1, due to Lemma 29 with d= 2, we see that this equation is
+equivalent with (6.18). Therefore, we are facing exactly the same e numeration problem
+here as forp= 7m/3, and, consequently, the number of solutions to (6.22) is the same ,
+namely7m+3
+3, as required.
+Ournextcaseis(6.15c). ByLemma26, wearefreetochoose p= 7m/2. Inparticular,
+mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4wm
+2+1c−4,c4wm
+2+2c−4,...,c4wmc−4,c3w1c−3,...,c3wm
+2c−3/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4wm
+2+ic−4, i= 1,2,...,m
+2, (6.23a)
+wi=c3wi−m
+2c−3, i=m
+2+1,m
+2+2,...,m. (6.23b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 73
+Writingt1,t2forwi,wi+m
+2, respectively, the equations (6.23) reduce to
+t1=c4t2c−4, (6.24a)
+t2=c3t1c−3. (6.24b)
+One of these equations is in fact superfluous: if we substitute (6.24 b) in (6.24a), then
+we obtaint1=c7t1c−7which is automatically satisfied due to Lemma 29 with d= 2.
+Since (w0;w1,...,w m)∈NCm(G24), we must have t1t2≤Tc, where≤Tis the partial
+order defined in (2.1). Combining this with (6.24), we infer that
+t1(c3t1c−3)≤Tc. (6.25)
+With the help of CHEVIE, one obtains 7 solutions for t1in this relation:
+t1∈/braceleftbig
+[5],[6],[7],[9],[12],[29],[32]/bracerightbig
+, (6.26)
+each of them giving rise to m/2 elements of Fix NCm(G24)(φp) sinceiranges from 1 to
+m/2. Here we have used again the short notation of CHEVIEreferring to the internal
+ordering of the roots of G24inCHEVIE.
+In total, we obtain 1 + 7m
+2=7m+2
+2elements in Fix NCm(G24)(φp), which agrees with
+the limit in (6.15c).
+Finally, we turn to (6.15e). By Remark 3, the only choices for h2andm2to be con-
+sidered are h2= 1 andm2= 3,h2=m2= 2, andh2= 2 andm2= 3. These correspond
+to the choices p= 14m/3,p= 7m/2, respectively p= 7m/3, all of which have already
+been discussed as they do not belong to (6.15e). Hence, (3.3) must necessarily hold, as
+required.
+CaseG25.The degrees are 6 ,9,12, and hence we have
+Catm(G25;q) =[12m+12]q[12m+9]q[12m+6]q
+[12]q[9]q[6]q.
+Letζbe a 12m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(G25;q) =m+1,ifζ=ζ12,ζ4, (6.27a)
+lim
+q→ζCatm(G25;q) =4m+3
+3,ifζ=ζ9,3|m, (6.27b)
+lim
+q→ζCatm(G25;q) = (m+1)(2m+1),ifζ=ζ6,−1 (6.27c)
+lim
+q→ζCatm(G25;q) = Catm(G25),ifζ=ζ3,1, (6.27d)
+lim
+q→ζCatm(G25;q) = 1,otherwise. (6.27e)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.27). The only cases not covered by Lemmas 27 and 28 a re the ones in
+(6.27b) and (6.27e).
+We first consider (6.27b). By Lemma 26, we are free to choose p= 4m/3. In
+particular, mmust be divisible by 3. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2w2m
+3+1c−2,c2w2m
+3+2c−2,...,c2wmc−2,cw1c−1,...,cw 2m
+3c−1/parenrightbig
+.74 C. KRATTENTHALER AND T. W. M ¨ULLER
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2w2m
+3+ic−2, i= 1,2,...,m
+3, (6.28a)
+wi=cwi−m
+3c−1, i=m
+3+1,m
+3+2,...,m. (6.28b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1.
+Writingt1,t2,t3forwi,wi+m
+3,wi+2m
+3, respectively, the equations (6.28) reduce to
+t1=c2t3c−2, (6.29a)
+t2=ct1c−1, (6.29b)
+t3=ct2c−1. (6.29c)
+One of these equations is in fact superfluous: if we substitute (6.29 b) and (6.29c) in
+(6.29a), then we obtain t1=c4t1c−4which is automatically satisfied due to Lemma 29
+withd= 3.
+Since (w0;w1,...,w m)∈NCm(G25), we must have t1t2t3=c. Combining this with
+(6.29), we infer that
+t1(ct1c−1)(c2t1c−2) =c. (6.30)
+With the help of CHEVIE, one obtains four solutions for t1in this equation:
+t1∈/braceleftbig
+[1],[2],[3],[14]/bracerightbig
+, (6.31)
+each of them giving rise to m/3 elements of Fix NCm(G25)(φp) sinceiranges from 1 to
+m/3. Here we have used again the short notation of CHEVIEreferring to the internal
+ordering of the roots of G25inCHEVIE.
+In total, we obtain 1 + 4m
+3=4m+3
+3elements in Fix NCm(G25)(φp), which agrees with
+the limit in (6.27b).
+Finally, we turn to (6.27e). By Remark 3, the only choice for h2andm2to be
+considered are h2=m2= 3. This corresponds to the choice p= 4m/3, which has
+already been discussed as they do not belong to (6.27e). Hence, (3 .3) must necessarily
+hold, as required.
+CaseG26.The degrees are 6 ,12,18, and hence we have
+Catm(G26;q) =[18m+18]q[18m+12]q[18m+6]q
+[18]q[12]q[6]q.
+Letζbe a 14m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(G26;q) =m+1,ifζ=ζ18,ζ9, (6.32a)
+lim
+q→ζCatm(G26;q) =3m+2
+2,ifζ=ζ12,ζ4,2|m, (6.32b)
+lim
+q→ζCatm(G26;q) = Catm(G26),ifζ=ζ6,ζ3,−1,1, (6.32c)
+lim
+q→ζCatm(G26;q) = 1,otherwise. (6.32d)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 75
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.32). The only cases not covered by Lemmas 27 and 28 a re the ones in
+(6.32b) and (6.32d).
+We first consider (6.32b). By Lemma 26, we are free to choose p= 3m/2 ifζ=ζ12,
+respectively p= 9m/2 ifζ=ζ4. In both cases, mmust be divisible by 2.
+We start with the case that p= 3m/2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+2+1c−2,c2wm
+2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+2c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+2+ic−2, i= 1,2,...,m
+2, (6.33a)
+wi=cwi−m
+2c−1, i=m
+2+1,m
+2+2,...,m. (6.33b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1.
+Writingt1,t2forwi,wi+m
+2, respectively, the equations (6.33) reduce to
+t1=c2t2c−2, (6.34a)
+t2=c1t1c−1. (6.34b)
+One of these equations is in fact superfluous: if we substitute (6.34 b) in (6.34a), then
+we obtaint1=c3t1c−3which is automatically satisfied due to Lemma 29 with d= 6.
+Since (w0;w1,...,w m)∈NCm(G26), we must have t1t2≤Tc. Combining this with
+(6.34), we infer that
+t1(ct1c−1)≤Tc. (6.35)
+With the help of CHEVIE, one obtains three solutions for t1in this equation:
+t1∈/braceleftbig
+[2],[3],[12]/bracerightbig
+,
+each of them giving rise to m/2 elements of Fix NCm(G26)(φp) sinceiranges from 1 to
+m/2. Here we have again used the short notation of CHEVIEreferring to the internal
+ordering of the roots of G26inCHEVIE.
+In total, we obtain 1 + 3m
+2=3m+2
+2elements in Fix NCm(G26)(φp), which agrees with
+the limit in (6.32b).
+In the case that p= 9m/2, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c5wm
+2+1c−5,c5wm
+2+2c−5,...,c5wmc−5,c4w1c−4,...,c4wm
+2c−4/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c5wm
+2+ic−5, i= 1,2,...,m
+2, (6.36a)
+wi=c4wi−m
+2c−4, i=m
+2+1,m
+2+2,...,m. (6.36b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+2) = 1.76 C. KRATTENTHALER AND T. W. M ¨ULLER
+Writingt1,t2forwi,wi+m
+2, respectively, the equations (6.36) reduce to
+t1=c5t2c−5, (6.37a)
+t2=c4t1c−4. (6.37b)
+One of these equations is in fact superfluous: if we substitute (6.37 b) in (6.37a), then
+we obtaint1=c9t1c−9which is automatically satisfied due to Lemma 29 with d= 2.
+Since (w0;w1,...,w m)∈NCm(G26), we must have t1t2≤Tc. Combining this with
+(6.37), we infer that
+t1(c4t1c−4)≤Tc. (6.38)
+Using that c3t1c−3=t1, due to Lemma 29 with d= 6, we see that this equation is
+equivalent with (6.35). Therefore, we are facing exactly the same e numeration problem
+here as forp= 3m/2, and, consequently, the number of solutions to (6.38) is the same ,
+namely3m+2
+2, as required.
+Finally, we turn to (6.32d). By Remark 3, the only choices for h2andm2to be
+considered are h2= 6 andm2= 2, respectively h2=m2= 2. These correspond to the
+choicesp= 3m/2, respectively p= 9m/2, all of which have already been discussed as
+they do not belong to (6.32d). Hence, (3.3) must necessarily hold, a s required.
+CaseG27.The degrees are 6 ,12,30, and hence we have
+Catm(G27;q) =[30m+30]q[30m+12]q[30m+6]q
+[30]q[12]q[6]q.
+Letζbe a 14m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(G27;q) =m+1,ifζ=ζ30,ζ15,ζ10,ζ5, (6.39a)
+lim
+q→ζCatm(G27;q) =5m+2
+2,ifζ=ζ12,ζ4,2|m, (6.39b)
+lim
+q→ζCatm(G27;q) = Catm(G27),ifζ=ζ6,ζ3,−1,1, (6.39c)
+lim
+q→ζCatm(G27;q) = 1,otherwise. (6.39d)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.39). The only cases not covered by Lemmas 27 and 28 a re the ones in
+(6.39b) and (6.39d).
+We first consider (6.39b). By Lemma 26, we are free to choose p= 5m/2 ifζ=ζ12,
+respectively p= 15m/2 ifζ=ζ4. In both cases, mmust be divisible by 2.
+We start with the case that p= 5m/2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3wm
+2+1c−3,c3wm
+2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm
+2c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3wm
+2+ic−3, i= 1,2,...,m
+2, (6.40a)
+wi=c2wi−m
+2c−2, i=m
+2+1,m
+2+2,...,m. (6.40b)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 77
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1.
+Writingt1,t2forwi,wi+m
+2, respectively, the equations (6.40) reduce to
+t1=c3t2c−3, (6.41a)
+t2=c2t1c−2. (6.41b)
+One of these equations is in fact superfluous: if we substitute (6.41 b) in (6.41a), then
+we obtaint1=c5t1c−5which is automatically satisfied due to Lemma 29 with d= 6.
+Since (w0;w1,...,w m)∈NCm(G27), we must have t1t2≤Tc. Combining this with
+(6.41), we infer that
+t1(c2t1c−2)≤Tc. (6.42)
+With the help of CHEVIE, one obtains five solutions for t1in this equation:
+t1∈/braceleftbig
+[1],[2],[15],[16],[28]/bracerightbig
+,
+each of them giving rise to m/2 elements of Fix NCm(G27)(φp) sinceiranges from 1 to
+m/2. Here we have used the short notation of CHEVIEreferring to the internal ordering
+of the roots of G27inCHEVIE.
+In total, we obtain 1 + 5m
+2=5m+2
+2elements in Fix NCm(G27)(φp), which agrees with
+the limit in (6.39b).
+In the case that p= 15m/2, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c8wm
+2+1c−8,c8wm
+2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
+2c−7/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c8wm
+2+ic−8, i= 1,2,...,m
+2, (6.43a)
+wi=c7wi−m
+2c−7, i=m
+2+1,m
+2+2,...,m. (6.43b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1.
+Writingt1,t2forwi,wi+m
+2, respectively, the equations (6.43) reduce to
+t1=c8t2c−8, (6.44a)
+t2=c7t1c−7. (6.44b)
+One of these equations is in fact superfluous: if we substitute (6.44 b) in (6.44a), then
+we obtaint1=c15t1c−15which is automatically satisfied due to Lemma 29 with d= 2.
+Since (w0;w1,...,w m)∈NCm(G27), we must have t1t2≤Tc. Combining this with
+(6.44), we infer that
+t1(c7t1c−7)≤Tc. (6.45)
+Using that c5t1c−5=t1, due to Lemma 29 with d= 6, we see that this equation is
+equivalent with (6.42). Therefore, we are facing exactly the same e numeration problem
+here as forp= 5m/2, and, consequently, the number of solutions to (6.45) is the same ,
+namely5m+2
+2, as required.78 C. KRATTENTHALER AND T. W. M ¨ULLER
+Finally, we turn to (6.39d). By Remark 3, the only choices for h2andm2to be
+considered are h2= 6 andm2= 3,h2= 6 andm2= 2,h2=m2= 3, respectively
+h2=m2= 2. These correspond to the choices p= 5m/3, 5m/2, 10m/3, respectively
+15m/2, out of which only p= 5m/3 andp= 10m/3 have not yet been discussed and
+belong tothecurrent case. If p= 5m/3, thecorresponding actionof φpis givenby (6.6),
+so that we have to solve for t1withℓT(t1) = 1 in the equation (6.9). A computation
+with the help of CHEVIEfinds no solution. If p= 10m/3, the corresponding action of
+φpis given by (6.11), so that we have to solve for t1withℓT(t1) in the equation (6.14).
+Using that c5t1c−5=t1, due to Lemma 29 with d= 6, we see that this equation is
+equivalent with the one in (6.9). Hence, in both cases, the left-hand side of (3.3) is
+equal to 1, as required.
+CaseG28=F4.The degrees are 2 ,6,8,12, and hence we have
+Catm(F4;q) =[12m+12]q[12m+8]q[12m+6]q[12m+2]q
+[12]q[8]q[6]q[2]q.
+Letζbe a 12m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(F4;q) =m+1,ifζ=ζ12, (6.46a)
+lim
+q→ζCatm(F4;q) =3m+2
+2,ifζ=ζ8,2|m, (6.46b)
+lim
+q→ζCatm(F4;q) = (m+1)(2m+1),ifζ=ζ6,ζ3, (6.46c)
+lim
+q→ζCatm(F4;q) =(m+1)(3m+2)
+2,ifζ=ζ4, (6.46d)
+lim
+q→ζCatm(F4;q) = Catm(F4),ifζ=−1 orζ= 1, (6.46e)
+lim
+q→ζCatm(F4;q) = 1,otherwise. (6.46f)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.46). The only cases not covered by Lemmas 27 and 28 a re the ones in
+(6.46b) and (6.46f). By Lemma 26, we are free to choose p= 3m/2. In particular, m
+must be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+2+1c−2,c2wm
+2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+2c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+2+ic−2, i= 1,2,...,m
+2, (6.47a)
+wi=cwi−m
+2c−1, i=m
+2+1,m
+2+2,...,m. (6.47b)
+There are four distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 2, (6.48a)
+and all other wj’s are equal to ε,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 79
+(iii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1, (6.48b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+2) =ℓT(wi2+m
+2) = 1, (6.48c)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(F4), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2=c. Together with equations (6.47)–(6.48), this implies that
+wi=c3wic−3andwi(cwic−1) =c, (6.49)
+respectively that
+wi=c3wic−3, wi(cwic−1)≤Tc,andℓT(wi) = 1, (6.50)
+respectively that
+wi1=c3wi1c−3, wi1(cwi1c−1)≤Tc,andℓT(wi1) = 1. (6.51)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains three solutions
+forwiin (6.49):
+wi∈/braceleftbig
+[1,2,3,4,3,2],[2,3],[1,3,2,1,3,4]/bracerightbig
+,
+where we have again used the short notation of coxeter,{s1,s2,s3,s4}being a simple
+system of generators of F4, corresponding to the Dynkin diagram displayed in Figure 2.
+Each of the above solutions for wigives rise to m/2 elements of Fix NCm(F4)(φp) sincei
+ranges from 1 to m/2.
+• • • •1 2 3 44
+Figure 2. The Dynkin diagram for F4
+There are no solutions for wiin (6.50) and for wi1in (6.51).
+In total, we obtain 1+3m
+2=3m+2
+2elements in Fix NCm(F4)(φp), which agrees with the
+limit in (6.46b).
+Finally, we turn to (6.46f). By Remark 3, the are no choices for h2andm2to be
+considered. Hence, the left-hand side of (3.3) is equal to 1, as req uired.
+CaseG29.The degrees are 4 ,8,12,20, and hence we have
+Catm(G29;q) =[20m+20]q[20m+12]q[20m+8]q[20m+4]q
+[20]q[12]q[8]q[4]q.80 C. KRATTENTHALER AND T. W. M ¨ULLER
+Letζbe a 20m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(G29;q) =m+1,ifζ=ζ20,ζ10,ζ5, (6.52a)
+lim
+q→ζCatm(G29;q) =5m+3
+3,ifζ=ζ12,ζ6,ζ3,3|m, (6.52b)
+lim
+q→ζCatm(G29;q) =5m+2
+2,ifζ=ζ8,2|m, (6.52c)
+lim
+q→ζCatm(G29;q) = Catm(G29),ifζ=ζ4,−1,1, (6.52d)
+lim
+q→ζCatm(G29;q) = 1,otherwise. (6.52e)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.52). The only cases not covered by Lemmas 27 and 28 a re the ones in
+(6.52b), (6.52c), and (6.52e).
+We begin with the case in (6.52b). By Lemma 26, we are free to choose p= 5m/3 if
+ζ=ζ12, we are free to choose p= 10m/3 ifζ=ζ6, we are free to choose p= 20m/3 if
+ζ=ζ3. In particular, in all three cases, mmust be divisible by 3.
+We start with the case that p= 5m/3. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+3+1c−2,c2wm
+3+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+3c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+3+ic−2, i= 1,2,...,2m
+3, (6.53a)
+wi=cwi−2m
+3c−1, i=2m
+3+1,2m
+3+2,...,m. (6.53b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1, (6.54a)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G29), we must have wiwi+m
+3wi+2m
+3≤Tc.
+Together with equations (6.53)–(6.54), this implies that
+wi=c5wic−5andwi(c3wic−3)(cwic−1) =c. (6.55)
+With the help of CHEVIE, one obtains five solutions for wiin (6.55):
+wi∈/braceleftbig
+[1],[2],[8],[25],[31]/bracerightbig
+,
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+of the roots of G29inCHEVIE. Each of the above solutions for wigives rise to m/3
+elements of Fix NCm(G29)(φp) sinceiranges from 1 to m/3.
+In total, we obtain 1 + 5m
+3=5m+3
+3elements in Fix NCm(G29)(φp), which agrees with
+the limit in (6.52b).
+In the case that p= 10m/3, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4w2m
+3+1c−4,c4w2m
+3+2c−4,...,c4wmc−4,c3w1c−3,...,c3w2m
+3c−3/parenrightbig
+.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 81
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4w2m
+3+ic−4, i= 1,2,...,m
+3, (6.56a)
+wi=c3wi−m
+3c−3, i=m
+3+1,m
+3+2,...,m. (6.56b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1.
+Writingt1,t2,t3forwi,wi+m
+3,wi+2m
+3, respectively, the equations (6.56) reduce to
+t1=c4t3c−4, (6.57a)
+t2=c3t1c−3, (6.57b)
+t3=c3t2c−3. (6.57c)
+One of these equations is in fact superfluous: if we substitute (6.57 b) and (6.57c) in
+(6.57a), then we obtain t1=c10t1c−10which is automatically satisfied due to Lemma 29
+withd= 2.
+Since (w0;w1,...,w m)∈NCm(G29), we must have t1t2t3≤Tc. Combining this with
+(6.57), we infer that
+t1(c3t1c−3)(c6t1c−6)≤Tc. (6.58)
+Using that c5t1c−5=t1, due to Lemma 29 with d= 4, we see that this equation is
+equivalent with (6.55). Therefore, we are facing exactly the same e numeration problem
+here as forp= 5m/3, and, consequently, the number of solutions to (6.58) is the same ,
+namely5m+3
+3, as required.
+In the case that p= 20m/3, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c7wm
+3+1c−7,c7wm
+3+2c−7,...,c7wmc−7,c6w1c−6,...,c6wm
+3c−6/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c7wm
+3+ic−7, i= 1,2,...,2m
+3, (6.59a)
+wi=c6wi−2m
+3c−6, i=2m
+3+1,2m
+3+2,...,m. (6.59b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1.
+Writingt1,t2,t3forwi,wi+m
+3,wi+2m
+3, respectively, the equations (6.59) reduce to
+t1=c7t2c−7, (6.60a)
+t2=c7t3c−7, (6.60b)
+t3=c6t1c−6. (6.60c)
+One of these equations is in fact superfluous: if we substitute (6.60 b) and (6.60c) in
+(6.60a), then we obtain t1=c20t1c−20which is automatically satisfied since c20=ε.82 C. KRATTENTHALER AND T. W. M ¨ULLER
+Since (w0;w1,...,w m)∈NCm(G29), we must have t1t2t3≤Tc. Combining this with
+(6.60), we infer that
+t1(c13t1c−13)(c6t1c−6)≤Tc. (6.61)
+Using that c5t1c−5=t1, due to Lemma 29 with d= 4, we see that this equation is
+equivalent with (6.55). Therefore, we are facing exactly the same e numeration problem
+here as forp= 5m/3, and, consequently, the number of solutions to (6.61) is the same ,
+namely5m+3
+3, as required.
+Next we discuss the case in (6.52c). By Lemma 26, we are free to cho osep= 5m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3wm
+2+1c−3,c3wm
+2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm
+2c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3wm
+2+ic−3, i= 1,2,...,m
+2, (6.62a)
+wi=c2wi−m
+2c−2, i=m
+2+1,m
+2+2,...,m. (6.62b)
+There are four distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 2, (6.63a)
+and all other wj’s are equal to ε,
+(iii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1, (6.63b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+2) =ℓT(wi2+m
+2) = 1, (6.63c)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G29), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2=c. Together with equations (6.62)–(6.63), this implies that
+wi=c3wic−3andwi(c2wic−2) =c, (6.64)
+respectively that
+wi=c3wic−3, wi(c2wic−2)≤Tc,andℓT(wi) = 1, (6.65)
+respectively that
+wi1=c3wi1c−3, wi2=c3wi2c−3,
+wi1wi2(c2wi1c−2)(c2wi2c−2) =c,andℓT(wi1) =ℓT(wi2) = 1.(6.66)
+With the help of CHEVIE, one obtains five solutions for wiin (6.65):
+wi∈/braceleftbig
+[4],[9],[14],[27],[32]/bracerightbig
+,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 83
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+of the roots of G29inCHEVIE. Each of these solutions for wigives rise to m/2 elements
+of Fix NCm(G29)(φp) sinceiranges from 1 to m/2.
+There are no solutions for wiin (6.64) and for ( wi1,wi2) in (6.66).
+In total, we obtain 1 + 5m
+2=5m+2
+2elements in Fix NCm(G29)(φp), which agrees with
+the limit in (6.52c).
+Finally, we turn to (6.52e). By Remark 3, the only choices for h2andm2to be
+considered are h2= 1 andm2= 3,h2= 2 andm2= 3,h2= 4 andm2= 2,h2= 4
+andm2= 3, respectively h2=m2= 4. These correspond to the choices p= 20m/3,
+p= 10m/3,p= 5m/2,p= 5m/3, respectively p= 5m/4, out of which only p= 5m/4
+has not yet been discussed and belongs to the current case. The c orresponding action
+ofφpis given by
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2w3m
+4+1c−2,c2w3m
+4+2c−2,...,c2wmc−2,cw1c−1,...,cw 3m
+4c−1/parenrightbig
+,
+so that we have to solve
+t1(ct1c−1)(c2t1c−2)(c3t1c−3) =c
+fort1withℓT(t1). A computation with the help of CHEVIEfinds no solution. Hence,
+the left-hand side of (3.3) is equal to 1, as required.
+CaseG30=H4.The degrees are 2 ,12,20,30, and hence we have
+Catm(H4;q) =[30m+30]q[30m+20]q[30m+12]q[30m+2]q
+[30]q[20]q[12]q[2]q.
+Letζbe a 30m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(H4;q) =m+1,ifζ=ζ30,ζ15, (6.67a)
+lim
+q→ζCatm(H4;q) =3m+2
+2,ifζ=ζ20,2|m, (6.67b)
+lim
+q→ζCatm(H4;q) =5m+2
+2,ifζ=ζ12,2|m, (6.67c)
+lim
+q→ζCatm(H4;q) =(m+1)(3m+2)
+2,ifζ=ζ10,ζ5, (6.67d)
+lim
+q→ζCatm(H4;q) =(m+1)(5m+2)
+2,ifζ=ζ6,ζ3, (6.67e)
+lim
+q→ζCatm(H4;q) =(3m+2)(5m+2)
+4,ifζ=ζ4,2|m, (6.67f)
+lim
+q→ζCatm(H4;q) = Catm(H4),ifζ=−1 orζ= 1, (6.67g)
+lim
+q→ζCatm(H4;q) = 1,otherwise. (6.67h)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.67). The only cases not covered by Lemmas 27 and 28 a re the ones in
+(6.67b), (6.67c), (6.67f), and (6.67h).84 C. KRATTENTHALER AND T. W. M ¨ULLER
+We begin with the case in (6.67b). By Lemma 26, we are free to choose p= 3m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+2+1c−2,c2wm
+2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+2c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+2+ic−2, i= 1,2,...,m
+2, (6.68a)
+wi=cwi−m
+2c−1, i=m
+2+1,m
+2+2,...,m. (6.68b)
+There are four distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 2, (6.69a)
+and all other wj’s are equal to ε,
+(iii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1, (6.69b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+2) =ℓT(wi2+m
+2) = 1, (6.69c)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(H4), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2=c. Together with equations (6.68)–(6.69), this implies that
+wi=c3wic−3andwi(cwic−1) =c, (6.70)
+respectively that
+wi=c3wic−3, wi(cwic−1)≤Tc,andℓT(wi) = 1, (6.71)
+respectively that
+wi1=c3wi1c−3, wi1(cwi1c−1)≤Tc,andℓT(wi1) = 1. (6.72)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains three solutions
+forwiin (6.70):
+wi∈/braceleftbig
+[1,2,3,4,3,2,1,2],[2,1,2,3],[1,3,2,1,2,1,3,4]/bracerightbig
+,
+where we have again used the short notation of coxeter,{s1,s2,s3,s4}being a simple
+system of generators of H4, corresponding to the Dynkin diagram displayed in Figure 3.
+Each of the above solutions for wigives rise to m/2 elements of Fix NCm(H4)(φp) sincei
+ranges from 1 to m/2.
+• • • •1 2 3 45
+Figure 3. The Dynkin diagram for H4
+There are no solutions for wiin (6.71) and for wi1in (6.72).
+In total, we obtain 1+3m
+2=3m+2
+2elements in Fix NCm(H4)(φp), which agrees with the
+limit in (6.67b).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 85
+Next we discuss the case in (6.67c). By Lemma 26, we are free to cho osep= 5m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3wm
+2+1c−3,c3wm
+2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm
+2c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3wm
+2+ic−3, i= 1,2,...,m
+2, (6.73a)
+wi=c2wi−m
+2c−2, i=m
+2+1,m
+2+2,...,m. (6.73b)
+There are four distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 2, (6.74a)
+and all other wj’s are equal to ε,
+(iii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1, (6.74b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+2) =ℓT(wi2+m
+2) = 1, (6.74c)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(H4), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2=c. Together with equations (6.73)–(6.74), this implies that
+wi=c3wic−3andwi(c2wic−2) =c, (6.75)
+respectively that
+wi=c3wic−3, wi(c2wic−2)≤Tc,andℓT(wi) = 1, (6.76)
+respectively that
+wi1=c3wi1c−3, wi1(c2wi1c−2)≤Tc,andℓT(wi1) = 1. (6.77)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains five solutions
+forwiin (6.75):
+wi∈/braceleftbig
+[1,3,2,1,2,1,3,2],[1,2,1,4,3,2,1,2,1,3,2,1,4,3],
+[2,1,2,3,2,1,2,4],[2,1,2,1,4,3,2,1,2,1,3,4],[1,2,3,2,1,4,3,2,1,2,1,3]/bracerightbig
+,
+where we used again coxeter’s short notation, {s1,s2,s3,s4}being a simple system of
+generators of H4, corresponding to the Dynkin diagram displayed in Figure 3. Each of
+these solutions for wigives rise to m/2 elements of Fix NCm(H4)(φp) sinceiranges from
+1 tom/2.
+There are no solutions for wiin (6.76) and for wi1in (6.77).
+In total, we obtain 1+5m
+2=5m+2
+2elements in Fix NCm(H4)(φp), which agrees with the
+limit in (6.67c).86 C. KRATTENTHALER AND T. W. M ¨ULLER
+Finallywediscuss thecasein(6.67f). ByLemma 26, wearefreetocho osep= 15m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c8wm
+2+1c−8,c8wm
+2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
+2c−7/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c8wm
+2+ic−8, i= 1,2,...,m
+2, (6.78a)
+wi=c7wi−m
+2c−7, i=m
+2+1,m
+2+2,...,m. (6.78b)
+There are four distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 2, (6.79a)
+and all other wj’s are equal to ε,
+(iii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1, (6.79b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+2) =ℓT(wi2+m
+2) = 1, (6.79c)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(H4), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2=c. Together with equations (6.78)–(6.79), this implies that
+wi=c15wic−15andwi(c7wic−7) =c, (6.80)
+respectively that
+wi=c15wic−15, wi(c7wic−7) =c,andℓT(wi) = 1, (6.81)
+respectively that
+wi1=c15wi1c−15, wi2=c15wi2c−15, wi1wi2(c7wi1c−7)(c7wi2c−7) =c.(6.82)
+Here, the first equations in both (6.80) and (6.81), and the first tw o equations in (6.82)
+are automatically satisfied due to Lemma 29 with d= 2.
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains eight solu-
+tions forwiin (6.80):
+wi∈/braceleftbig
+[1,3,2,1,2,1,3,2],[1,2,3,4,3,2,1,2],[2,1,2,3],
+[1,2,1,4,3,2,1,2,1,3,2,1,4,3],[1,3,2,1,2,1,3,4],[2,1,2,3,2,1,2,4],
+[2,1,2,1,4,3,2,1,2,1,3,4],[1,2,3,2,1,4,3,2,1,2,1,3]/bracerightbig
+,(6.83)
+where{s1,s2,s3,s4}is a simple system of generators of H4, corresponding to the
+Dynkin diagram displayed in Figure 3, and each of them gives rise to m/2 elements ofCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 87
+FixNCm(H4)(φp) sinceiranges from 1 to m/2. Furthermore, one obtains 15 solutions for
+wiin (6.81):
+wi∈/braceleftbig
+[2],[3],[4],[2,1,2],[3,2,1,2,3],[1,3,2,1,2,1,3],[2,1,2,3,2,1,2],[1,2,3,4,3,2,1],
+[2,1,3,2,1,2,1,3,2],[1,4,3,2,1,2,1,3,4],[2,1,2,3,4,3,2,1,2],
+[1,2,1,4,3,2,1,2,1,3,2,1,4],[1,3,2,1,2,3,4,3,2,1,2,1,3],
+[2,1,2,1,4,3,2,1,2,1,3,2,1,2,4],[1,3,2,1,4,3,2,1,2,1,3,2,1,4,3]/bracerightbig
+,
+each of them giving rise to m/2 elements of Fix NCm(H4)(φp) sinceiranges from 1 to
+m/2, and one obtains 30 pairs ( wi1,wi2) of solutions in (6.82):
+(wi1,wi2)∈/braceleftbig
+([2],[2,1,3,2,1,2,1,3,2]),([2],[2,1,2,3,4,3,2,1,2]),([3],[3,2,1,2,3]),
+([3],[1,3,2,1,4,3,2,1,2,1,3,2,1,4,3]),([4],[1,4,3,2,1,2,1,3,4]),
+([4],[2,1,2,3,4,3,2,1,2]),([2,1,2],[3]),([2,1,2],[1,4,3,2,1,2,1,3,4]),
+([3,2,1,2,3],[2,1,3,2,1,2,1,3,2]),([3,2,1,2,3],[1,3,2,1,2,3,4,3,2,1,2,1,3]),
+([1,3,2,1,2,1,3],[2]),([1,3,2,1,2,1,3],[4]),([2,1,2,3,2,1,2],[4]),
+([2,1,2,3,2,1,2],[2,1,2]),([1,2,3,4,3,2,1],[2]),([1,2,3,4,3,2,1],[3,2,1,2,3]),
+([2,1,3,2,1,2,1,3,2],[1,3,2,1,2,1,3]),([2,1,3,2,1,2,1,3,2],[2,1,2,3,2,1,2]),
+([1,4,3,2,1,2,1,3,4],[2,1,2,1,4,3,2,1,2,1,3,2,1,2,4]),
+([1,4,3,2,1,2,1,3,4],[1,3,2,1,4,3,2,1,2,1,3,2,1,4,3]),
+([2,1,2,3,4,3,2,1,2],[2,1,2,3,2,1,2]),
+([2,1,2,3,4,3,2,1,2],[2,1,2,1,4,3,2,1,2,1,3,2,1,2,4]),
+([1,2,1,4,3,2,1,2,1,3,2,1,4],[3]),
+([1,2,1,4,3,2,1,2,1,3,2,1,4],[1,2,3,4,3,2,1]),
+([1,3,2,1,2,3,4,3,2,1,2,1,3],[1,3,2,1,2,1,3]),
+([1,3,2,1,2,3,4,3,2,1,2,1,3],[1,2,3,4,3,2,1]),
+([2,1,2,1,4,3,2,1,2,1,3,2,1,2,4],[2,1,2]),
+([2,1,2,1,4,3,2,1,2,1,3,2,1,2,4],[1,2,1,4,3,2,1,2,1,3,2,1,4]),
+([1,3,2,1,4,3,2,1,2,1,3,2,1,4,3],[1,2,1,4,3,2,1,2,1,3,2,1,4]),
+([1,3,2,1,4,3,2,1,2,1,3,2,1,4,3],[1,3,2,1,2,3,4,3,2,1,2,1,3])/bracerightbig
+,(6.84)
+each of them giving rise to/parenleftbigm/2
+2/parenrightbig
+elements of Fix NCm(H4)(φp) since 1 ≤i1<i2≤m
+2.
+In total, we obtain1+(15+8)m
+2+30/parenleftbigm/2
+2/parenrightbig
+=(3m+2)(5m+2)
+4elements in Fix NCm(H4)(φp),
+which agrees with the limit in (6.67f).
+Finally, we turn to (6.67h). By Remark 3, the only choices for h2andm2to be
+considered are h2= 2 andm2= 4, respectively h2=m2= 2. These correspond to the
+choicesp= 15m/2, respectively p= 15m/4, out of which only p= 15m/4 has not yet
+been discussed and belongs to the current case. The correspond ing action of φpis given
+by
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4wm
+4+1c−4,c4wm
+4+2c−4,...,c4wmc−4,c3w1c−3,...,c3wm
+4c−3/parenrightbig
+,88 C. KRATTENTHALER AND T. W. M ¨ULLER
+so that we have to solve
+t1(c11t1c−11)(c7t1c−7)(c3t1c−3) =c
+fort1withℓT(t1). A computation with Stembridge’s Maplepackagecoxeter [36] finds
+no solution. Hence, the left-hand side of (3.3) is equal to 1, as requ ired.
+CaseG32.The degrees are 12 ,18,24,30, and hence we have
+Catm(G32;q) =[30m+30]q[30m+24]q[30m+18]q[30m+12]q
+[30]q[24]q[18]q[12]q.
+Letζbe a 30m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(G32;q) =m+1,ifζ=ζ30,ζ15,ζ10,ζ5, (6.85a)
+lim
+q→ζCatm(G32;q) =5m+4
+4,ifζ=ζ24,ζ8,4|m, (6.85b)
+lim
+q→ζCatm(G32;q) =5m+3
+3,ifζ=ζ18,ζ9,3|m, (6.85c)
+lim
+q→ζCatm(G32;q) =(5m+4)(5m+2)
+8,ifζ=ζ12,ζ4,2|m, (6.85d)
+lim
+q→ζCatm(G32;q) = Catm(G32),ifζ=ζ6,ζ3,−1,1, (6.85e)
+lim
+q→ζCatm(G32;q) = 1,otherwise. (6.85f)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.85). The only cases not covered by Lemmas 27 and 28 a re the ones in
+(6.85b), (6.85c), (6.85d), and (6.85f).
+We begin with the case in (6.85b). By Lemma 26, we are free to choose p= 5m/4 if
+ζ=ζ24, and we are free to choose p= 15m/4 ifζ=ζ8. In particular, in all both cases,
+mmust be divisible by 4.
+We start with the case that p= 5m/4. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2w3m
+4+1c−2,c2w3m
+4+2c−2,...,c2wmc−2,cw1c−1,...,cw 3m
+4c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2w3m
+4+ic−2, i= 1,2,...,m
+4, (6.86a)
+wi=cwi−m
+4c−1, i=m
+4+1,m
+4+2,...,m. (6.86b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+4such that
+ℓT(wi) =ℓT(wi+m
+4) =ℓT(wi+2m
+4) =ℓT(wi+3m
+4) = 1, (6.87a)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G32), we must have
+wiwi+m
+4wi+2m
+4wi+3m
+4=c.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 89
+Together with equations (6.86)–(6.87), this implies that
+wi=c5wic−5andwi(cwic−1)(c2wic−2)(c3wic−3) =c. (6.88)
+With the help of CHEVIE, one obtains five solutions for wiin (6.88):
+wi∈/braceleftbig
+[1],[2],[3],[4],[27]/bracerightbig
+, (6.89)
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+of the roots of G32inCHEVIE. Each of the above solutions for wigives rise to m/4
+elements of Fix NCm(G32)(φp) sinceiranges from 1 to m/4.
+In total, we obtain 1 + 5m
+4=5m+4
+4elements in Fix NCm(G32)(φp), which agrees with
+the limit in (6.85b).
+In the case that p= 15m/4, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4wm
+4+1c−4,c4wm
+4+2c−4,...,c4wmc−4,c3w1c−3,...,c3wm
+4c−3/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4wm
+4+ic−4, i= 1,2,...,3m
+4, (6.90a)
+wi=c3wi−3m
+4c−3, i=3m
+4+1,3m
+4+2,...,m. (6.90b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+4such that
+ℓT(wi) =ℓT(wi+m
+4) =ℓT(wi+2m
+4) =ℓT(wi+3m
+4) = 1.
+Writingt1,t2,t3,t4forwi,wi+m
+4,wi+2m
+4,wi+3m
+4, respectively, the equations (6.90) reduce
+to
+t1=c4t2c−4, (6.91a)
+t2=c4t3c−4, (6.91b)
+t3=c4t4c−4, (6.91c)
+t4=c3t1c−3. (6.91d)
+One of these equations is infact superfluous: if we substitute (6.91 b)–(6.91d) in (6.91a),
+then we obtain t1=c15t1c−15which is automatically satisfied due to Lemma 29 with
+d= 2.
+Since (w0;w1,...,w m)∈NCm(G32), we must have t1t2t3t4=c. Combining this with
+(6.91), we infer that
+t1(c11t1c−11)(c7t1c−7)(c3t1c−3) =c. (6.92)
+Using that c5t1c−5=t1, due to Lemma 29 with d= 6, we see that this equation is
+equivalent with (6.88). Therefore, we are facing exactly the same e numeration problem
+here as forp= 5m/4, and, consequently, the number of solutions to (6.92) is the same ,
+namely5m+4
+4, as required.
+Next we consider the case in (6.85c). By Lemma 26, we are free to ch oosep= 5m/3
+ifζ=ζ18, and we are free to choose p= 10m/3 ifζ=ζ9. In particular, in both cases,
+mmust be divisible by 3.
+We start with the case that p= 5m/3. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+3+1c−2,c2wm
+3+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+3c−1/parenrightbig
+.90 C. KRATTENTHALER AND T. W. M ¨ULLER
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+3+ic−2, i= 1,2,...,2m
+3, (6.93a)
+wi=cwi−2m
+3c−1, i=2m
+3+1,2m
+3+2,...,m. (6.93b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1, (6.94a)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G32), we must have wiwi+m
+3wi+2m
+3≤Tc.
+Together with equations (6.93)–(6.94), this implies that
+wi=c5wic−5andwi(c3wic−3)(cwic−1)≤Tc. (6.95)
+With the help of CHEVIE, one obtains three solutions for wiin (6.95):
+wi∈/braceleftbig
+[1],[2],[3],[4],[27]/bracerightbig
+,
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+of the roots of G32inCHEVIE. Each of the above solutions for wigives rise to m/3
+elements of Fix NCm(G32)(φp) sinceiranges from 1 to m/3.
+In total, we obtain 1 + 5m
+3=5m+3
+3elements in Fix NCm(G32)(φp), which agrees with
+the limit in (6.85c).
+In the case that p= 10m/3, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4w2m
+3+1c−4,c4w2m
+3+2c−4,...,c4wmc−4,c3w1c−3,...,c3w2m
+3c−3/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4w2m
+3+ic−4, i= 1,2,...,m
+3, (6.96a)
+wi=c3wi−m
+3c−3, i=m
+3+1,m
+3+2,...,m. (6.96b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1.
+Writingt1,t2,t3forwi,wi+m
+3,wi+2m
+3, respectively, the equations (6.96) reduce to
+t1=c4t3c−4, (6.97a)
+t2=c3t1c−3, (6.97b)
+t3=c3t2c−3. (6.97c)
+One of these equations is in fact superfluous: if we substitute (6.97 b) and (6.97c) in
+(6.97a), then we obtain t1=c10t1c−10which is automatically satisfied due to Lemma 29
+withd= 2.
+Since (w0;w1,...,w m)∈NCm(G32), we must have t1t2t3≤Tc. Combining this with
+(6.97), we infer that
+t1(c3t1c−3)(c6t1c−6)≤Tc. (6.98)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 91
+Using that c5t1c−5=t1, due to Lemma 29 with d= 4, we see that this equation is
+equivalent with (6.95). Therefore, we are facing exactly the same e numeration problem
+here as forp= 5m/3, and, consequently, the number of solutions to (6.98) is the same ,
+namely5m+3
+3, as required.
+Next we discuss the case in (6.85d). By Lemma 26, we are free to cho osep= 5m/2
+ifζ=ζ12, and we are free to choose p= 15m/2 ifζ=ζ4. In particular, mmust be
+divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3wm
+2+1c−3,c3wm
+2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm
+2c−2/parenrightbig
+.(6.99)
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3wm
+2+ic−3, i= 1,2,...,m
+2, (6.100a)
+wi=c2wi−m
+2c−2, i=m
+2+1,m
+2+2,...,m. (6.100b)
+There are four distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 2, (6.101a)
+and all other wj’s are equal to ε,
+(iii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1, (6.101b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+2) =ℓT(wi2+m
+2) = 1, (6.101c)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G32), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2=c. Together with equations (6.100)–(6.101), this implies
+that
+wi=c3wic−3andwi(c2wic−2) =c, (6.102)
+respectively that
+wi=c3wic−3, wi(c2wic−2)≤Tc,andℓT(wi) = 1, (6.103)
+respectively that
+wi1=c3wi1c−3, wi2=c3wi2c−3,
+wi1wi2(c2wi1c−2)(c2wi2c−2) =c,andℓT(wi1) =ℓT(wi2) = 1.(6.104)
+With the help of CHEVIE, one obtains ten solutions for wiin (6.102):
+wi∈/braceleftbig
+[3,4],[1,20],[4,7],[1,2],[2,3],[4,27],[3,23],[1,23],[2,16],[12,27]/bracerightbig
+,(6.105)
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+of the roots of G32inCHEVIE, one obtains solutions for wiin (6.103):
+wi∈/braceleftbig
+[4] [3] [20] [1] [7] [2] [16] [12] [27] [23]/bracerightbig
+,92 C. KRATTENTHALER AND T. W. M ¨ULLER
+each of them giving rise to m/2 elements of Fix NCm(G32)(φp) sinceiranges from 1 to
+m/2, and one obtains 25 pairs ( wi1,wi2) satisfying (6.104):
+(wi1,wi2)∈/braceleftbig
+([4],[20]),([4],[7]),([4],[27]),([3],[4]),([3],[12]),([3],[23]),([20],[3]),
+([20],[1]),([1],[20]),([1],[2]),([1],[23]),([7],[4]),([7],[1]),([2],[3]),([2],[7]),
+([2],[16]),([16],[4]),([16],[2]),([12],[2]),([12],[27]),([27],[1]),([27],[16]),
+([27],[12]),([23],[3]),([23],[27])/bracerightbig
+,(6.106)
+each of them giving rise to/parenleftbigm/2
+2/parenrightbig
+elements of Fix NCm(G32)(φp) since 1 ≤i1<i2≤m
+2.
+In total, we obtain 1 + 20m
+2+ 25/parenleftbigm/2
+2/parenrightbig
+=(5m+4)(5m+2)
+8elements in Fix NCm(G32)(φp),
+which agrees with the limit in (6.85d).
+Ifp= 15m/2, then, from (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c8wm
+2+1c−8,c8wm
+2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
+2c−7/parenrightbig
+.
+Using that c5wc−5=wfor allw∈NC(G32, due to Lemma 29 with d= 6, we see that
+this action is identical with the one in (6.99). Therefore, we are facin g exactly the same
+enumeration problem here as for p= 5m/2, and, consequently, the number of elements
+in Fix NCm(G32)(φp) is the same, namely(5m+4)(5m+2)
+8, as required.
+Finally, we turn to (6.85f). By Remark 3, the only choices for h2andm2to be
+considered are h2= 2 andm2= 4,h2=m2= 2,h2=m2= 3,h2= 6 andm2= 4,
+h2= 6 andm2= 3, respectively h2= 6 andm2= 2. These correspond to the choices
+p= 15m/4,p= 15m/2,p= 10m/3,p= 5m/4,p= 5m/3, respectively p= 5m/2, all
+of which have already been discussed as they do not belong to (6.85f ). Hence, (3.3)
+must necessarily hold, as required.
+CaseG33.The degrees are 4 ,6,10,12,18, and hence we have
+Catm(G33;q) =[18m+18]q[18m+12]q[18m+10]q[18m+6]q[18m+4]q
+[18]q[12]q[10]q[6]q[4]q.
+Letζbe a 18m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(G33;q) =m+1,ifζ=ζ18,ζ9, (6.107a)
+lim
+q→ζCatm(G33;q) =3m+2
+2,ifζ=ζ12,2|m, (6.107b)
+lim
+q→ζCatm(G33;q) =9m+5
+5,ifζ=ζ10,ζ5,5|m, (6.107c)
+lim
+q→ζCatm(G33;q) =(m+1)(3m+2)(3m+1)
+2,ifζ=ζ6,ζ3, (6.107d)
+lim
+q→ζCatm(G33;q) =(3m+2)(9m+2)
+4,ifζ=ζ4,2|m, (6.107e)
+lim
+q→ζCatm(G33;q) = Catm(G33),ifζ=−1 orζ= 1, (6.107f)
+lim
+q→ζCatm(G33;q) = 1,otherwise. (6.107g)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 93
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.107). The only cases not covered by Lemmas 27 and 28 are the ones in
+(6.107b), (6.107c), (6.107e), and (6.107g).
+We begin with the case in (6.107b). By Lemma 26, we are free to choos ep= 3m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+2+1c−2,c2wm
+2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+2c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+2+ic−2, i= 1,2,...,m
+2, (6.108a)
+wi=cwi−m
+2c−1, i=m
+2+1,m
+2+2,...,m. (6.108b)
+There are four distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 2, (6.109a)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1, (6.109b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+2) =ℓT(wi2+m
+2) = 1, (6.109c)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G33), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2≤Tc. Together with equations (6.108)–(6.109), this implies
+that
+wi=c3wic−3andwi(cwic−1)≤Tc,andℓT(wi) = 2,(6.110)
+respectively that
+wi=c3wic−3, wi(cwic−1)≤Tc,andℓT(wi) = 1, (6.111)
+respectively that
+wi1=c3wi1c−3, wi1(cwi1c−1)≤Tc,andℓT(wi1) = 1.(6.112)
+With the help of CHEVIE, one obtains three solutions for wiin (6.110):
+wi∈/braceleftbig
+[1,44],[2,4],[5,42]/bracerightbig
+,
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+oftherootsof G33inCHEVIE. Eachofthemgivesriseto m/2elementsofFix NCm(G33)(φp)
+sinceiranges from 1 to m/2.
+There are no solutions to (6.111) and to (6.112).
+In total, we obtain 1 + 3m
+2=3m+2
+2elements in Fix NCm(G33)(φp), which agrees with
+the limit in (6.107b).
+Next we turn to the case in (6.107c). By Lemma 26, we are free to ch oosep= 9m/5
+ifζ=ζ10, and we are free to choose p= 18m/5 ifζ=ζ5. In particular, in all both
+cases,mmust be divisible by 5.94 C. KRATTENTHALER AND T. W. M ¨ULLER
+We start with the case that p= 9m/5. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+5+1c−2,c2wm
+5+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+5c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+5+ic−2, i= 1,2,...,4m
+5, (6.113a)
+wi=cwi−4m
+5c−1, i=4m
+5+1,4m
+5+2,...,m. (6.113b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+5such that
+ℓT(wi) =ℓT(wi+m
+5) =ℓT(wi+2m
+5) =ℓT(wi+3m
+5) =ℓT(wi+4m
+5) = 1,(6.114a)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G33), we must have
+wiwi+m
+5wi+2m
+5wi+3m
+5wi+4m
+5=c.
+Together with equations (6.113)–(6.114), this implies that
+wi=c9wic−9andwi(c7wic−7)(c5wic−5)(c3wic−3)(cwic−1) =c. (6.115)
+With the help of CHEVIE, one obtains nine solutions for wiin (6.115):
+wi∈/braceleftbig
+[5],[4],[1],[2],[10],[27],[42],[44],[52]/bracerightbig
+, (6.116)
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+of the roots of G33inCHEVIE. Each of the above solutions for wigives rise to m/5
+elements of Fix NCm(G33)(φp) sinceiranges from 1 to m/5.
+In total, we obtain 1 + 9m
+5=9m+5
+5elements in Fix NCm(G33)(φp), which agrees with
+the limit in (6.107c).
+In the case that p= 18m/5, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4w2m
+5+1c−4,c4w2m
+5+2c−4,...,c4wmc−4,c3w1c−3,...,c3w2m
+5c−3/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4w2m
+5+ic−4, i= 1,2,...,3m
+5, (6.117a)
+wi=c3wi−3m
+5c−3, i=3m
+5+1,3m
+5+2,...,m. (6.117b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+5such that
+ℓT(wi) =ℓT(wi+m
+5) =ℓT(wi+2m
+5) =ℓT(wi+3m
+5) =ℓT(wi+4m
+5) = 1.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 95
+Writingt1,t2,t3,t4,t5forwi,wi+m
+5,wi+2m
+5,wi+3m
+5,wi+4m
+5, respectively, the equations
+(6.117) reduce to
+t1=c4t3c−4, (6.118a)
+t2=c4t4c−4, (6.118b)
+t3=c4t5c−4, (6.118c)
+t4=c3t1c−3, (6.118d)
+t5=c3t2c−3. (6.118e)
+One of these equations is in fact superfluous: if we substitute (6.11 8b)–(6.118e) in
+(6.118a), then we obtain t1=c18t1c−18which is automatically satisfied since c18=ε.
+Since (w0;w1,...,w m)∈NCm(G33), we must have t1t2t3t4t5=c. Combining this
+with (6.118), we infer that
+t1(c7t1c−7)(c14t1c−14)(c3t1c−3)(c10t1c−10) =c. (6.119)
+Using that c9t1c−9=t1, due to Lemma 29 with d= 2, we see that this equation
+is equivalent with (6.115). Therefore, we are facing exactly the sam e enumeration
+problem here as for p= 9m/5, and, consequently, the number of solutions to (6.119) is
+the same, namely9m+5
+5, as required.
+Next we consider the case in (6.107e). By Lemma 26, we are free to c hoosep= 9m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c5wm
+2+1c−5,c5wm
+2+2c−5,...,c5wmc−5,c4w1c−4,...,c4wm
+2c−4/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c5wm
+2+ic−5, i= 1,2,...,m
+2, (6.120a)
+wi=c4wi−m
+2c−4, i=m
+2+1,m
+2+2,...,m. (6.120b)
+There are four distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 2, (6.121a)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1, (6.121b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+2) =ℓT(wi2+m
+2) = 1, (6.121c)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G33), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2≤Tc. Together with equations (6.120)–(6.121), this implies
+that
+wi=c9wic−9andwi(c4wic−4)≤Tc,andℓT(wi) = 2,(6.122)96 C. KRATTENTHALER AND T. W. M ¨ULLER
+respectively that
+wi=c9wic−9, wi(c4wic−4)≤Tc,andℓT(wi) = 1, (6.123)
+respectively that
+wi1=c9wi1c−9, wi2=c9wi2c−9,
+wi1wi2(c4wi1c−4)(c4wi2c−4)≤Tc,andℓT(wi1) =ℓT(wi2) = 1.(6.124)
+With the help of CHEVIE, one obtains 21 solutions for wiin (6.122):
+wi∈/braceleftbig
+[4,5],[1,20],[5,7],[1,2],[2,4],[10,259],[27,208],[27,44],[4,39],[5,42],[1,39],
+[5,52],[10,208],[1,44],[42,113],[4,113],[2,27],[10,42],[2,49],[52,61],[44,123]/bracerightbig
+,
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+of the roots of G33inCHEVIE, one obtains 18 solutions for wiin (6.123):
+wi∈/braceleftbig
+[5],[4],[20],[208],[259],[1],[2],[7],[10],[27],
+[39],[42],[49],[44],[52],[113],[61],[123]/bracerightbig
+,
+each of them giving rise to m/2 elements of Fix NCm(G33)(φp) sinceiranges from 1 to
+m/2, and one obtains 54 pairs ( wi1,wi2) satisfying (6.124):
+(wi1,wi2)∈/braceleftbig
+([5],[20]),([5],[7]),([5],[42]),([5],[52]),([4],[5]),([4],[10]),([4],[39]),([4],[113]),([20],[4]),
+([20],[1]),([208],[27]),([208],[52]),([259],[10]),([259],[27]),([1],[20]),([1],[2]),([1],[39]),([1],[44]),([2],[4]),
+([2],[7]),([2],[27]),([2],[49]),([7],[5]),([7],[1]),([10],[208]),([10],[259]),([10],[2]),([10],[42]),([27],[5]),
+([27],[208]),([27],[44]),([27],[61]),([39],[4]),([39],[42]),([42],[1]),([42],[27]),([42],[113]),([42],[123]),
+([49],[5]),([49],[2]),([44],[4]),([44],[259]),([44],[52]),([44],[123]),([52],[1]),([52],[10]),([52],[49]),
+([52],[61]),([113],[42]),([113],[44]),([61],[2]),([61],[52]),([123],[10]),([123],[44])/bracerightbig
+,
+each of them giving rise to/parenleftbigm/2
+2/parenrightbig
+elements of Fix NCm(G33)(φp) since 1 ≤i1<i2≤m.
+Intotal,weobtain1+(21+18)m
+2+54/parenleftbigm/2
+2/parenrightbig
+=(3m+2)(9m+2)
+4elementsinFix NCm(G33)(φp),
+which agrees with the limit in (6.107e).
+Finally, we turn to (6.107g). By Remark 3, the only choices for h2andm2to be
+considered are h2= 1 andm2= 5,h2= 2 andm2= 5,h2= 2 andm2= 4, respectively
+h2=m2= 2. These correspond to the choices p= 18m/5,p= 9m/5,p= 9m/4,
+respectively p= 9m/2, out of which only p= 9m/4 has not yet been discussed and
+belongs to the current case. The corresponding action of φpis given by
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3w3m
+4+1c−3,c3w3m
+4+2c−3,...,c3wmc−3,c2w1c−2,...,c2w3m
+4c−2/parenrightbig
+,
+so that we have to solve
+t1(c2t1c−2)(c4t1c−4)(c6t1c−6)≤Tc
+fort1withℓT(t1). A computation with the help of CHEVIEfinds no solution. Hence,
+the left-hand side of (3.3) is equal to 1, as required.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 97
+CaseG34.The degrees are 6 ,12,18,24,30,42, and hence we have
+Catm(G34;q) =[42m+42]q[42m+30]q[42m+24]q
+[42]q[30]q[24]q
+×[42m+18]q[42m+12]q[42m+6]q
+[18]q[12]q[6]q.
+Letζbe a 42m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(G34;q) =m+1,ifζ=ζ42,ζ21,ζ14,ζ7, (6.125a)
+lim
+q→ζCatm(G34;q) =7m+5
+5,ifζ=ζ30,ζ15,ζ10,ζ5,5|m, (6.125b)
+lim
+q→ζCatm(G34;q) =7m+4
+4,ifζ=ζ24,ζ8,4|m, (6.125c)
+lim
+q→ζCatm(G34;q) =7m+3
+3,ifζ=ζ18,ζ9,3|m, (6.125d)
+lim
+q→ζCatm(G34;q) =(7m+4)(7m+2)
+8,ifζ=ζ12,ζ4,2|m, (6.125e)
+lim
+q→ζCatm(G34;q) = Catm(G34),ifζ=ζ6,ζ3,−1,1, (6.125f)
+lim
+q→ζCatm(G34;q) = 1,otherwise. (6.125g)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.125). The only cases not covered by Lemmas 27 and 28 are the ones in
+(6.125b), (6.125c), (6.125d), (6.125e), and (6.125g).
+We begin with the case in (6.125b). By Lemma 26, we are free to choos ep= 7m/5
+ifζ=ζ30, we are free to choose p= 14m/5 ifζ=ζ15, we are free to choose p= 21m/5
+ifζ=ζ10, and we are free to choose p= 42m/5 ifζ=ζ5. In particular, in all cases, m
+must be divisible by 5.
+We start with the case that p= 7m/5. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2w3m
+5+1c−2,c2w3m
+5+2c−2,...,c2wmc−2,cw1c−1,...,cw 3m
+5c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2w3m
+5+ic−2, i= 1,2,...,2m
+5, (6.126a)
+wi=cwi−2m
+5c−1, i=2m
+5+1,2m
+5+2,...,m. (6.126b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+5such that
+ℓT(wi) =ℓT(wi+m
+5) =ℓT(wi+2m
+5) =ℓT(wi+3m
+5) =ℓT(wi+4m
+5) = 1,(6.127a)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G34), we must have
+wiwi+m
+5wi+2m
+5wi+3m
+5wi+4m
+5≤Tc.98 C. KRATTENTHALER AND T. W. M ¨ULLER
+Together with equations (6.126)–(6.127), this implies that
+wi=c7wic−7andwi(c4wic−4)(cwic−1)(c5wic−5)(c2wic−2)≤Tc.(6.128)
+With the help of CHEVIE, one obtains seven solutions for wiin (6.128):
+wi∈/braceleftbig
+[4],[5],[6],[2],[11],[44],[63],[74]/bracerightbig
+,
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+of the roots of G34inCHEVIE. Each of the above solutions for wigives rise to m/5
+elements of Fix NCm(G34)(φp) sinceiranges from 1 to m/5.
+In total, we obtain 1 + 7m
+5=7m+5
+5elements in Fix NCm(G34)(φp), which agrees with
+the limit in (6.125b).
+In the case that p= 14m/5, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3wm
+5+1c−3,c3wm
+5+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm
+5c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3wm
+5+ic−3, i= 1,2,...,4m
+5, (6.129a)
+wi=c2wi−4m
+5c−2, i=4m
+5+1,4m
+5+2,...,m. (6.129b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+5such that
+ℓT(wi) =ℓT(wi+m
+5) =ℓT(wi+2m
+5) =ℓT(wi+3m
+5) =ℓT(wi+4m
+5) = 1.
+Writingt1,t2,t3,t4,t5forwi,wi+m
+5,wi+2m
+5,wi+3m
+5,wi+4m
+5, respectively, the equations
+(6.129) reduce to
+t1=c3t2c−3, (6.130a)
+t2=c3t3c−3, (6.130b)
+t3=c3t4c−3, (6.130c)
+t4=c3t5c−3, (6.130d)
+t5=c2t1c−2. (6.130e)
+One of these equations is in fact superfluous: if we substitute (6.13 0b)–(6.130e) in
+(6.130a), thenweobtain t1=c14t1c−14whichisautomaticallysatisfiedduetoLemma29
+withd= 3.
+Since (w0;w1,...,w m)∈NCm(G34), we must have t1t2t3t4t5≤Tc. Combining this
+with (6.130), we infer that
+t1(c11t1c−11)(c8t1c−8)(c5t1c−5)(c2t1c−2)≤Tc. (6.131)
+Using that c7t1c−7=t1, due to Lemma 29 with d= 6, we see that this equation
+is equivalent with (6.128). Therefore, we are facing exactly the sam e enumeration
+problem here as for p= 7m/5, and, consequently, the number of solutions to (6.131) is
+the same, namely7m+5
+5, as required.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 99
+In the case that p= 21m/5, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c5w4m
+5+1c−5,c5w4m
+5+2c−5,...,c5wmc−5,c4w1c−4,...,c4w4m
+5c−4/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c5w4m
+5+ic−5, i= 1,2,...,m
+5, (6.132a)
+wi=c4wi−m
+5c−4, i=m
+5+1,m
+5+2,...,m. (6.132b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+5such that
+ℓT(wi) =ℓT(wi+m
+5) =ℓT(wi+2m
+5) =ℓT(wi+3m
+5) =ℓT(wi+4m
+5) = 1.
+Writingt1,t2,t3,t4,t5forwi,wi+m
+5,wi+2m
+5,wi+3m
+5,wi+4m
+5, respectively, the equations
+(6.132) reduce to
+t1=c5t5c−5, (6.133a)
+t2=c4t1c−4, (6.133b)
+t3=c4t2c−4, (6.133c)
+t4=c4t3c−4, (6.133d)
+t5=c4t4c−4. (6.133e)
+One of these equations is in fact superfluous: if we substitute (6.13 3b)–(6.133e) in
+(6.133a), thenweobtain t1=c21t1c−21whichisautomaticallysatisfiedduetoLemma29
+withd= 6.
+Since (w0;w1,...,w m)∈NCm(G34), we must have t1t2t3t4t5≤Tc. Combining this
+with (6.133), we infer that
+t1(c4t1c−4)(c8t1c−8)(c12t1c−12)(c16t1c−16)≤Tc. (6.134)
+Using that c7t1c−7=t1, due to Lemma 29 with d= 6, we see that this equation
+is equivalent with (6.128). Therefore, we are facing exactly the sam e enumeration
+problem here as for p= 7m/5, and, consequently, the number of solutions to (6.134) is
+the same, namely7m+5
+5, as required.
+In the case that p= 42m/5, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c9w3m
+5+1c−9,c9w3m
+5+2c−9,...,c9wmc−9,c8w1c−8,...,c8w3m
+5c−8/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c9w3m
+5+ic−9, i= 1,2,...,2m
+5, (6.135a)
+wi=c8wi−2m
+5c−8, i=2m
+5+1,2m
+5+2,...,m. (6.135b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
+are equal to ε, or there is an iwith 1≤i≤m
+5such that
+ℓT(wi) =ℓT(wi+m
+5) =ℓT(wi+2m
+5) =ℓT(wi+3m
+5) =ℓT(wi+4m
+5) = 1.100 C. KRATTENTHALER AND T. W. M ¨ULLER
+Writingt1,t2,t3,t4,t5forwi,wi+m
+5,wi+2m
+5,wi+3m
+5,wi+4m
+5, respectively, the equations
+(6.135) reduce to
+t1=c9t4c−9, (6.136a)
+t2=c9t5c−9, (6.136b)
+t3=c8t1c−8, (6.136c)
+t4=c8t2c−8, (6.136d)
+t5=c8t3c−8. (6.136e)
+One of these equations is in fact superfluous: if we substitute (6.13 6b)–(6.136e) in
+(6.136a), then we obtain t1=c42t1c−42which is automatically satisfied since c42=ε.
+Since (w0;w1,...,w m)∈NCm(G34), we must have t1t2t3t4t5≤Tc. Combining this
+with (6.136), we infer that
+t1(c25t1c−25)(c8t1c−8)(c33t1c−33)(c16t1c−16)≤Tc. (6.137)
+Using that c7t1c−7=t1, due to Lemma 29 with d= 6, we see that this equation
+is equivalent with (6.128). Therefore, we are facing exactly the sam e enumeration
+problem here as for p= 7m/5, and, consequently, the number of solutions to (6.137) is
+the same, namely7m+5
+5, as required.
+Next we consider the case in (6.125c). By Lemma 26, we are free to c hoosep= 7m/4
+ifζ=ζ24, and we are free to choose p= 21m/4 ifζ=ζ8. In both cases, mmust be
+divisible by 4.
+We start with the case that p= 7m/4. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+4+1c−2,c2wm
+4+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+4c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+4+ic−2, i= 1,2,...,3m
+4, (6.138a)
+wi=cwi−3m
+4c−1, i=3m
+4+1,3m
+4+2,...,m. (6.138b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+4) =ℓT(wi+2m
+4) =ℓT(wi+3m
+4) = 1, (6.139a)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G34), we must have
+wiwi+m
+4wi+2m
+4wi+3m
+4≤Tc.
+Together with equations (6.138)–(6.139), this implies that
+wi=c7wic−7andwi(c5wic−5)(c3wic−3)(cwic−1)≤Tc. (6.140)
+With the help of CHEVIE, one obtains seven solutions for wiin (6.140):
+wi∈/braceleftbig
+[1],[2],[28],[34],[61],[46],[168]/bracerightbig
+,
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+oftherootsof G34inCHEVIE. Eachofthemgivesriseto m/4elementsofFix NCm(G34)(φp)
+sinceiranges from 1 to m/4.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 10 1
+In total, we obtain 1 + 7m
+4=7m+4
+4elements in Fix NCm(G34)(φp), which agrees with
+the limit in (6.125c).
+Ifp= 21m/4, then, from (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c6w3m
+4+1c−6,c6w3m
+4+2c−6,...,c6wmc−6,c5w1c−5,...,c5w3m
+4c−5/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c6w3m
+4+ic−6, i= 1,2,...,m
+4, (6.141a)
+wi=c5wi−3m
+4c−5, i=m
+4+1,m
+4+2,...,m. (6.141b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+4) =ℓT(wi+2m
+4) =ℓT(wi+3m
+4) = 1, (6.142a)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G34), we must have
+wiwi+m
+4wi+2m
+4wi+3m
+4≤Tc.
+Together with equations (6.141)–(6.142), this implies that
+wi=c21wic−21andwi(c5wic−5)(c10wic−10)(c15wic−15)≤Tc. (6.143)
+Using that c7t1c−7=t1, due to Lemma 29 with d= 6, we see that this equation
+is equivalent with (6.140). Therefore, we are facing exactly the sam e enumeration
+problem here as for p= 7m/4, and, consequently, the number of solutions to (6.137) is
+the same, namely7m+4
+4, as required.
+Our next case is the case in (6.125d). By Lemma 26, we are free to ch oosep= 7m/3
+ifζ=ζ18, and we are free to choose p= 14m/3 ifζ=ζ9. In both cases, mmust be
+divisible by 3.
+We start with the case that p= 7m/3. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3w2m
+3+1c−3,c3w2m
+3+2c−3,...,c3wmc−3,c2w1c−2,...,c2w2m
+3c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3w2m
+3+ic−3, i= 1,2,...,m
+3, (6.144a)
+wi=c2wi−m
+3c−2, i=m
+3+1,m
+3+2,...,m. (6.144b)
+There are four distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 2, (6.145a)
+and all other wj’s are equal to ε,102 C. KRATTENTHALER AND T. W. M ¨ULLER
+(iii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1, (6.145b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+3) =ℓT(wi2+m
+3) =ℓT(wi1+2m
+3) =ℓT(wi2+2m
+3) = 1,
+(6.145c)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G34), we must have wiwi+m
+3wi+2m
+3≤Tc,
+respectively wi1wi2wi1+m
+3wi2+m
+3wi1+2m
+3wi2+2m
+3=c. Together with equations (6.144)–
+(6.145), this implies that
+wi=c7wic−7andwi(c2wic−2)(c4wic−4) =c, (6.146)
+respectively that
+wi=c7wic−7, wi(c2wic−2)(c4wic−4)≤Tc,andℓT(wi) = 1, (6.147)
+respectively that
+wi1=c7wi1c−7, wi2=c7wi2c−7,
+andwi1wi2(c2wi1c−2)(c2wi2c−2)(c4wi1c−4)(c4wi2c−4) =c.(6.148)
+With the help of CHEVIE, one obtains 21 solutions for wiin (6.147):
+wi∈/braceleftbig
+[4] [5] [6] [11] [44] [63] [74]/bracerightbig
+,
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+oftherootsof G34inCHEVIE. Eachofthemgivesriseto m/3elementsofFix NCm(G34)(φp)
+sinceiranges from 1 to m/3.
+There are no solutions wiin (6.146) and for ( wi1,wi2) in (6.148).
+In total, we obtain 1 + 7m
+3=7m+3
+3elements in Fix NCm(G34)(φp), which agrees with
+the limit in (6.125d).
+In the case that p= 14m/3, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c5wm
+3+1c−5,c5wm
+3+2c−5,...,c5wmc−5,c4w1c−4,...,c4wm
+3c−4/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c5wm
+3+ic−5, i= 1,2,...,2m
+3, (6.149a)
+wi=c4wi−2m
+3c−4, i=2m
+3+1,2m
+3+2,...,m. (6.149b)
+There are four distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 2, (6.150a)
+and all other wj’s are equal to ε,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 10 3
+(iii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1, (6.150b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+3) =ℓT(wi2+m
+3) =ℓT(wi1+2m
+3) =ℓT(wi2+2m
+3) = 1,
+(6.150c)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G34), we must have wiwi+m
+3wi+2m
+3≤Tc,
+respectively wi1wi2wi1+m
+3wi2+m
+3wi1+2m
+3wi2+2m
+3=c. Together with equations (6.149)–
+(6.150), this implies that
+wi=c14wic−14andwi(c9wic−9)(c4wic−4) =c, (6.151)
+respectively that
+wi=c14wic−14, wi(c9wic−9)(c4wic−4)≤Tc,andℓT(wi) = 1,(6.152)
+respectively that
+wi1=c14wi1c−14, wi2=c14wi2c−14,
+andwi1wi2(c9wi1c−9)(c9wi2c−9)(c4wi1c−4)(c4wi2c−4) =c.(6.153)
+Using that c7wc−7=wfor allw∈NC(G34, due to Lemma 29 with d= 6, we see
+that this equation is equivalent with (6.146). Therefore, we are fac ing exactly the same
+enumeration problem here as for p= 7m/3, and, consequently, the number of solutions
+to (6.151) is the same, namely7m+3
+3, as required.
+Next we consider the case in (6.125e). By Lemma 26, we are free to c hoosep= 7m/2
+ifζ=ζ12, and we are free to choose p= 21m/2 ifζ=ζ4. In both cases, mmust be
+divisible by 2.
+We begin with the case that p= 7m/2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4wm
+2+1c−4,c4wm
+2+2c−4,...,c4wmc−4,c3w1c−3,...,c3wm
+2c−3/parenrightbig
+.(6.154)
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4wm
+2+ic−4, i= 1,2,...,m
+2, (6.155a)
+wi=c3wi−m
+2c−3, i=m
+2+1,m
+2+2,...,m. (6.155b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m, which we
+summarise as follows:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+1≤ℓT(wi) =ℓT(wi+m
+2)≤3, (6.156a)
+and the other wj’s, 1≤j≤m, are equal to ε,104 C. KRATTENTHALER AND T. W. M ¨ULLER
+(iii) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓ1=ℓT(wi1) =ℓT(wi1+m
+2)≥1, ℓ2=ℓT(wi2) ==ℓT(wi2+m
+2)≥1, ℓ1+ℓ2≤3,
+(6.156b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1,i2,i3with 1≤i1<i2<i3≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi3) =ℓT(wi1+m
+2) =ℓT(wi2+m
+2) =ℓT(wi3+m
+2) = 1,(6.156c)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(G34), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2≤Tc, respectively
+wi1wi2wi3wi1+m
+2wi2+m
+2wi3+m
+2=c
+. Together with equations (6.155)–(6.156), this implies that
+wi=c7wic−7andwi(c4wic−4)≤Tc, (6.157)
+respectively that
+wi1=c7wi1c−7, wi2=c7wi2c−7,andwi1wi2(c4wi1c−4)(c4wi2c−4)≤Tc,(6.158)
+respectively that
+wi1=c7wi1c−7, wi2=c7wi2c−7, wi3=c7wi3c−7,
+andwi1wi2wi3(c4wi1c−4)(c4wi2c−4)(c4wi3c−4) =c.(6.159)
+With the help of CHEVIE, one obtains 14 solutions for wiin (6.157) with ℓT(wi) = 1:
+wi∈/braceleftbig
+[14],[35],[1],[2],[10],[24],[28],[34],[39],[56],[61],[46],[168],[105]/bracerightbig
+,
+where we have againused the short notationof CHEVIEreferring to the internal ordering
+of the roots of G34inCHEVIE, one obtains 21 solutions for wiin (6.157) with ℓT(wi) = 2:
+wi∈/braceleftbig
+[1,14],[1,35],[1,24],[1,34],[14,61],[2,39],[10,34],[2,56],[2,35],[14,46],[35,168],
+[34,56],[2,61],[28,56],[10,28],[10,61],[34,105],[28,105],[24,61],[39,168],[24,46]/bracerightbig
+,
+each of them giving rise to m/2 elements of Fix NCm(G34)(φp) sinceiranges from 1 to
+m/2, and one obtains 49 pairs ( wi1,wi2) satisfying (6.158):
+(wi1,wi2)∈/braceleftbig
+([14],[1]),([14],[61]),([14],[46]),([35],[1]),([35],[46]),([35],[168]),([1],[14]),([1],[35]),([1],[24]),
+([1],[34]),([2],[35]),([2],[39]),([2],[56]),([2],[61]),([10],[28]),([10],[34]),([10],[61]),([24],[28]),([24],[61]),
+([24],[46]),([28],[1]),([28],[10]),([28],[56]),([28],[105]),([34],[39]),([34],[56]),([34],[46]),([34],[105]),
+([39],[1]),([39],[2]),([39],[168]),([56],[2]),([56],[34]),([56],[168]),([61],[10]),([61],[24]),([61],[168]),
+([61],[105]),([46],[14]),([46],[2]),([46],[10]),([46],[24]),([168],[14]),([168],[35]),([168],[28]),
+([168],[39]),([105],[2]),([105],[28]),([105],[34])/bracerightbig
+,
+each of them giving rise to/parenleftbigm/2
+2/parenrightbig
+elements of Fix NCm(G34)(φp) since 1 ≤i1<i2≤m.
+There are no solutions for wiwithℓT(wi) = 3 in (6.157), and hence no solutions for
+(wi1,wi2) withℓT(wi1) +ℓT(wi2) = 3 in (6.158), and no solutions for ( wi1,wi2,wi3) in
+(6.159).
+Intotal,weobtain1+(14+21)m
+2+49/parenleftbigm/2
+2/parenrightbig
+=(7m+2)(7m+4)
+8elementsinFix NCm(G34)(φp),
+which agrees with the limit in (6.125e).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 10 5
+Ifp= 21m/2, from (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c11wm
+2+1c−11,c11wm
+2+2c−11,...,c11wmc−11,c10w1c−10,...,c10wm
+2c−10/parenrightbig
+.
+Using that c7wc−7=wfor allw∈NC(G34, due to Lemma 29 with d= 6, we see
+that this action is identical with the one in (6.154). Therefore, we ar e facing exactly
+the same enumeration problem here as for p= 7m/2, and, consequently, the number of
+elements in Fix NCm(G34)(φp) is the same, namely(7m+2)(7m+4)
+8, as required.
+Finally, we turn to (6.125g). By Remark 3, the only choices for h2andm2to be
+considered are h2= 1 andm2= 5,h2= 2 andm2= 5,h2= 2 andm2= 4,
+h2=m2= 2,h2= 3 andm2= 5,h2=m2= 3,h2= 6 andm2= 6,h2= 6 and
+m2= 5,h2= 6 andm2= 4,h2= 6 andm2= 3, respectively h2= 6 andm2= 2,
+. These correspond to the choices p= 42m/5,p= 21m/5,p= 21m/4,p= 21m/2,
+p= 14m/3,p= 14m/5,p= 7m/6,p= 7m/5,p= 7m/4,p= 7m/3, respectively
+p= 7m/2, out of which only p= 7m/6 has not yet been discussed and belongs to the
+current case. The corresponding action of φpis given by
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2w5m
+6+1c−2,c2w5m
+6+2c−2,...,c2wmc−2,cw1c−1,...,cw 5m
+6c−1/parenrightbig
+,
+so that we have to solve
+t1(ct1c−1)(c2t1c−2)(c3t1c−3)(c4t1c−4)(c5t1c−5) =c.
+A computation with the help of CHEVIEfinds no solution. Hence, the left-hand side of
+(3.3) is equal to 1, as required.
+CaseG35=E6.The degrees are 2 ,5,6,8,9,12, and hence we have
+Catm(E6;q) =[12m+12]q[12m+9]q[12m+8]q[12m+6]q[12m+5]q[12m+2]q
+[12]q[9]q[8]q[6]q[5]q[2]q.106 C. KRATTENTHALER AND T. W. M ¨ULLER
+Letζbe a 12m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(E6;q) =m+1,ifζ=ζ12, (6.160a)
+lim
+q→ζCatm(E6;q) =4m+3
+3,ifζ=ζ9,3|m, (6.160b)
+lim
+q→ζCatm(E6;q) =3m+2
+2,ifζ=ζ8,2|m, (6.160c)
+lim
+q→ζCatm(E6;q) = (m+1)(2m+1),ifζ=ζ6, (6.160d)
+lim
+q→ζCatm(E6;q) =12m+5
+5,ifζ=ζ5,5|m, (6.160e)
+lim
+q→ζCatm(E6;q) =(m+1)(3m+2)
+2,ifζ=ζ4, (6.160f)
+lim
+q→ζCatm(E6;q) =(m+1)(4m+3)(2m+1)
+3,ifζ=ζ3, (6.160g)
+lim
+q→ζCatm(E6;q) =(m+1)(3m+2)(2m+1)(6m+1)
+2,ifζ=−1, (6.160h)
+lim
+q→ζCatm(E6;q) = Catm(E6),ifζ= 1, (6.160i)
+lim
+q→ζCatm(E6;q) = 1,otherwise. (6.160j)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.160). The only cases not covered by Lemmas 27 and 28 are the ones in
+(6.160b), (6.160c), (6.160e), and (6.160j).
+We begin with the case in (6.160b). By Lemma 26, we are free to choos ep= 4m/3.
+In particular, mmust be divisible by 3. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2w2m
+3+1c−2,c2w2m
+3+2c−2,...,c2wmc−2,cw1c−1,...,cw 2m
+3c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2w2m
+3+ic−2, i= 1,2,...,m
+3, (6.161a)
+wi=cwi−m
+3c−1, i=m
+3+1,m
+3+2,...,m. (6.161b)
+There are four distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 2, (6.162a)
+and all other wj’s are equal to ε,
+(iii) there is an iwith 1≤i≤m
+3such that
+ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3) = 1, (6.162b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1andi2with 1≤i1<i2≤m
+3such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+3) =ℓT(wi2+m
+3) =ℓT(wi1+2m
+3) =ℓT(wi2+2m
+3) = 1,
+(6.162c)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 10 7
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E6), we must have wiwi+m
+3wi+2m
+3≤Tc,
+respectively wi1wi2wi1+m
+3wi2+m
+3wi1+2m
+3wi2+2m
+3=c. Together with equations (6.161)–
+(6.162), this implies that
+wi=c4wic−4andwi(cwic−1)(c2wic−2) =c, (6.163)
+respectively that
+wi=c4wic−4, wi(cwic−1)(c2wic−2)≤Tc,andℓT(wi) = 1, (6.164)
+respectively that
+wi1=c4wi1c−4, wi1(cwi1c−1)(c2wi1c−2)≤Tc,andℓT(wi1) = 1.(6.165)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains four solutions
+forwiin (6.163):
+wi∈/braceleftbig
+[1,2,3,4,5,6,5,4,2,3],[3,4],[2,4,2,5],[1,3,4,5,4,3,1,6]/bracerightbig
+,
+where we have again used the short notation of coxeter,{s1,s2,s3,s4,s5,s6}being a
+simple system of generators of E6, corresponding to the Dynkin diagram displayed in
+Figure4. Eachoftheabovesolutionsfor wigivesriseto m/3elements ofFix NCm(E6)(φp)
+sinceiranges from 1 to m/3.
+• • • • ••
+1 3 4 5 62
+Figure 4. The Dynkin diagram for E6
+There are no solutions for wiin (6.164) and for wi1in (6.165).
+In total, we obtain 1+4m
+3=4m+3
+3elements in Fix NCm(E6)(φp), which agrees with the
+limit in (6.160b).
+Next we discuss the case in (6.160c). By Lemma 26, we are free to ch oosep= 3m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+2+1c−2,c2wm
+2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+2c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+2+ic−2, i= 1,2,...,m
+2, (6.166a)
+wi=cwi−m
+2c−1, i=m
+2+1,m
+2+2,...,m. (6.166b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m, which we
+summarise as follows:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 3, (6.167a)
+and all other wj’s are equal to ε,
+(iii) there is a jwith 1≤j≤m
+2such that
+1≤ℓT(wj) =ℓT(wj+m
+2)≤2. (6.167b)108 C. KRATTENTHALER AND T. W. M ¨ULLER
+Moreover, since ( w0;w1,...,w m)∈NCm(E6), we must have wiwi+m
+2=c, respec-
+tivelywjwj+m
+2≤Tc. Together with equations (6.166)–(6.167), this implies that
+wi=c3wic−3andwi(cwic−1) =c, (6.168)
+respectively that
+wj=c3wjc−3, wj(cwjc−1)≤Tc,and 1≤ℓT(wj)≤2.(6.169)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains three solutions
+forwiin (6.168):
+wi∈/braceleftbig
+[1,3,4,3,5],[1,2,3,4,3,5,6,5,4,3,1],[2,3,4,5,4,2,6]/bracerightbig
+,
+where we used again coxeter’s short notation, {s1,s2,s3,s4,s5,s6}being a simple sys-
+tem of generators of E6, corresponding to the Dynkin diagram displayed in Figure 4.
+Each ofthese solutionsfor wigives rise to m/2 elements of Fix NCm(E6)(φp) sinceiranges
+from 1 tom/2.
+There are no solutions for wjin (6.169).
+In total, we obtain 1+3m
+2=3m+2
+2elements in Fix NCm(E6)(φp), which agrees with the
+limit in (6.160c).
+Finally we discuss the case in (6.160e). By Lemma 26, we are free to ch oosep=
+12m/5. In particular, mmust be divisible by 5. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3w3m
+5+1c−3,c3w3m
+5+2c−3,...,c3wmc−3,c2w1c−2,...,c2w3m
+5c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3w3m
+5+ic−3, i= 1,2,...,2m
+5, (6.170a)
+wi=c2wi−2m
+5c−2, i=2m
+5+1,2m
+5+2,...,m. (6.170b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+5such that
+ℓT(wi) =ℓT(wi+m
+5) =ℓT(wi+2m
+5) =ℓT(wi+3m
+5) =ℓT(wi+4m
+5) = 1, (6.171)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E6), we must have
+wiwi+m
+5wi+2m
+5wi+3m
+5wi+4m
+5≤Tc.
+Together with equations (6.170)–(6.171), this implies that
+wi=c12wic−12andwi(c7wic−7)(c2wic−2)(c9wic−9)(c4wic−4)≤Tc.(6.172)
+Here, the first equation is automatically satisfied since c12=ε.
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 12 solutions
+forwiin (6.172):
+wi∈/braceleftbig
+[1],[3],[4],[5],[6],[2,4,2],[3,4,3],[2,4,5,4,2],[1,3,4,5,4,3,1],
+[1,3,4,5,6,5,4,3,1],[2,3,4,5,6,5,4,2,3],[1,2,3,4,5,6,5,4,2,3,1]/bracerightbig
+,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 10 9
+where{s1,s2,s3,s4,s5,s6}is a simple system of generators of E6, corresponding to the
+Dynkin diagram displayed in Figure 4, and each of them gives rise to m/5 elements of
+FixNCm(E6)(φp) sinceiranges from 1 to m/5.
+In total, we obtain 1+12m
+5=12m+5
+5elements in Fix NCm(E6)(φp), which agrees with
+the limit in (6.160e).
+Finally, we turn to (6.160j). By Remark 3, the only choices for h2andm2to be
+considered are h2= 1 andm2= 5 andh2= 2 andm2= 5. These correspond to the
+choicesp= 12m/5, respectively p= 6m/5, out which only p= 6m/5 has not yet been
+discussed and belongs to the current case. The corresponding ac tion ofφpis given by
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2w4m
+5+1c−2,c2w4m
+5+2c−2,...,c2wmc−2,cw1c−1,...,cw 4m
+5c−1/parenrightbig
+,
+so that we have to solve
+t1(ct1c−1)(c2t1c−2)(c3t1c−3)(c4t1c−4)≤Tc
+fort1withℓT(t1) = 1. A computation with the help of Stembridge’s Maplepackage
+coxeter [36] finds no solution. Hence, the left-hand side of (3.3) is equal to 1 , as
+required.
+CaseG36=E7.The degrees are 2 ,6,8,10,12,14,18, and hence we have
+Catm(E7;q) =[18m+18]q[18m+14]q[18m+12]q
+[18]q[14]q[12]q
+×[18m+10]q[18m+8]q[18m+6]q[18m+2]q
+[10]q[8]q[6]q[2]q.
+Letζbe a 18m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(E7;q) =m+1,ifζ=ζ18,ζ9, (6.173a)
+lim
+q→ζCatm(E7;q) =9m+7
+7,ifζ=ζ14,ζ7,7|m, (6.173b)
+lim
+q→ζCatm(E7;q) =3m+2
+2,ifζ=ζ12,2|m, (6.173c)
+lim
+q→ζCatm(E7;q) =9m+5
+5,ifζ=ζ10,ζ5,5|m, (6.173d)
+lim
+q→ζCatm(E7;q) =9m+4
+4,ifζ=ζ8,4|m, (6.173e)
+lim
+q→ζCatm(E7;q) =(m+1)(3m+2)(3m+1)
+2,ifζ=ζ6,ζ3, (6.173f)
+lim
+q→ζCatm(E7;q) =(3m+2)(9m+4)
+8,ifζ=ζ4,2|m, (6.173g)
+lim
+q→ζCatm(E7;q) = Catm(E7),ifζ=−1 orζ= 1, (6.173h)
+lim
+q→ζCatm(E7;q) = 1,otherwise. (6.173i)110 C. KRATTENTHALER AND T. W. M ¨ULLER
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.173). The only cases not covered by Lemmas 27 and 28 are the ones in
+(6.173b), (6.173c), (6.173d), (6.173e), (6.173g), and (6.173i).
+We begin with the case in (6.173b). By Lemma 26, we are free to choos ep= 9m/7
+ifζ=ζ14, respectively p= 18m/7 ifζ=ζ7. In both cases, mmust be divisible by 7.
+We start with the case that p= 9m/7. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2w5m
+7+1c−2,c2w5m
+7+2c−2,...,c2wmc−2,cw1c−1,...,cw 5m
+7c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2w5m
+7+ic−2, i= 1,2,...,2m
+7, (6.174a)
+wi=cwi−2m
+7c−1, i=2m
+7+1,2m
+7+2,...,m. (6.174b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+7such that
+ℓT(wi) =ℓT(wi+m
+7) =ℓT(wi+2m
+7) =ℓT(wi+3m
+7)
+=ℓT(wi+4m
+7) =ℓT(wi+5m
+7) =ℓT(wi+6m
+7) = 1,(6.175)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E7), we must have
+wiwi+m
+7wi+2m
+7wi+3m
+7wi+4m
+7wi+5m
+7wi+6m
+7=c.
+Together with equations (6.174)–(6.175), this implies that
+wi=c9wic−9andwi(c5wic−5)(cwic−1)(c6wic−6)(c2wic−2)(c7wic−7)(c3wic−3) =c.
+(6.176)
+Here, the first equation is automatically satisfied due to Lemma 29 wit hd= 2.
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 9 solutions
+forwiin (6.176):
+wi∈/braceleftbig
+[4],[5],[6],[7],[3,4,3],[2,4,5,4,2],[1,3,4,5,6,5,4,3,1],
+[2,3,4,5,6,7,6,5,4,2,3],[1,2,3,4,5,6,7,6,5,4,2,3,1]/bracerightbig
+,(6.177)
+where we have again used the short notation of coxeter,{s1,s2,s3,s4,s5,s6,s7}being
+a simple system of generators of E7, corresponding to the Dynkin diagram displayed in
+Figure5. Eachoftheabovesolutionsfor wigivesriseto m/7elements ofFix NCm(E7)(φp)
+sinceiranges from 1 to m/7.
+• • • • • ••
+1 3 4 5 6 72
+Figure 5. The Dynkin diagram for E7
+In total, we obtain 1+9m
+7=9m+7
+7elements in Fix NCm(E7)(φp), which agrees with the
+limit in (6.173b).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 11 1
+In the case that p= 18m/7, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3w3m
+7+1c−3,c3w3m
+7+2c−3,...,c3wmc−3,c2w1c−2,...,c2w3m
+7c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3w3m
+7+ic−3, i= 1,2,...,4m
+7, (6.178a)
+wi=c2wi−4m
+7c−2, i=4m
+7+1,4m
+7+2,...,m. (6.178b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+7such that
+ℓT(wi) =ℓT(wi+m
+7) =ℓT(wi+2m
+7) =ℓT(wi+3m
+7)
+=ℓT(wi+4m
+7) =ℓT(wi+5m
+7) =ℓT(wi+6m
+7) = 1,(6.179)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E7), we must have
+wiwi+m
+7wi+2m
+7wi+3m
+7wi+4m
+7wi+5m
+7wi+6m
+7=c.
+Together with equations (6.178)–(6.179), this implies that
+wi=c18wic−18
+andwi(c5wic−5)(c10wic−10)(c15wic−15)(c2wic−2)(c7wic−7)(c12wic−12) =c.(6.180)
+Here, the first equation is automatically satisfied since c18=ε. Due to Lemma 29 with
+d= 2, wehave c9wic−9=wi, hencealso c10wic−10=cwic−1, etc., sothat(6.180)reduces
+to (6.176). Therefore, we are facing exactly the same enumeratio n problem here as for
+p= 9m/7, and, consequently, the number of solutions to (6.180) is the sam e, namely
+9m+7
+7, as required.
+Next we discuss the case in (6.173c). By Lemma 26, we are free to ch oosep= 3m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+2+1c−2,c2wm
+2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+2c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+2+ic−2, i= 1,2,...,m
+2, (6.181a)
+wi=cwi−m
+2c−1, i=m
+2+1,m
+2+2,...,m. (6.181b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m, which we
+summarise as follows:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+1≤ℓT(wi) =ℓT(wi+m
+2)≤3. (6.182)112 C. KRATTENTHALER AND T. W. M ¨ULLER
+Moreover, since ( w0;w1,...,w m)∈NCm(E7), we must have wiwi+m
+2≤Tc. Together
+with equations (6.181)–(6.182), this implies that
+wi=c3wic−3, wi(cwic−1)≤Tc,and 1≤ℓT(wi)≤3.(6.183)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains three solutions
+forwiin (6.183) with ℓT(wi) = 2:
+wi∈/braceleftbig
+[1,3,4,5,4,3],[2,3,4,5,6,5,4,2],[1,2,4,2,3,4,5,6,7,6,5,4,3,1]/bracerightbig
+,
+where we used again coxeter’s short notation, {s1,s2,s3,s4,s5,s6,s7}being a simple
+system of generators of E7, corresponding to the Dynkin diagram displayed in Figure 5,
+and none if ℓT(wi) = 1 orℓT(wi) = 3. Each of the solutions for wigives rise to m/2
+elements of Fix NCm(E7)(φp) sinceiranges from 1 to m/2.
+In total, we obtain 1+3m
+2=3m+2
+2elements in Fix NCm(E7)(φp), which agrees with the
+limit in (6.173c).
+Next we consider the case in (6.173d). By Lemma 26, we are free to c hoosep= 9m/5
+ifζ=ζ10, respectively p= 18m/5 ifζ=ζ5. In both cases, mmust be divisible by 5.
+We start with the case that p= 9m/5. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+5+1c−2,c2wm
+5+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+5c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+5+ic−2, i= 1,2,...,4m
+5, (6.184a)
+wi=cwi−4m
+5c−1, i=4m
+5+1,4m
+5+2,...,m. (6.184b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+5such that
+ℓT(wi) =ℓT(wi+m
+5) =ℓT(wi+2m
+5) =ℓT(wi+3m
+5) =ℓT(wi+4m
+5) = 1,(6.185a)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E7), we must have
+wiwi+m
+5wi+2m
+5wi+3m
+5wi+4m
+5≤Tc.
+Together with equations (6.184)–(6.185), this implies that
+wi=c9wic−9andwi(c7wic−7)(c5wic−5)(c3wic−3)(cwic−1)≤Tc.(6.186)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 9 solutions for
+wiin (6.186):
+wi∈/braceleftbig
+[4],[5],[6],[7],[3,4,3],[2,4,5,4,2],[1,3,4,5,6,5,4,3,1],
+[2,3,4,5,6,7,6,5,4,2,3],[1,2,3,4,5,6,7,6,5,4,2,3,1]/bracerightbig
+,
+where{s1,s2,s3,s4,s5,s6,s7}is a simple system of generators of E7, corresponding to
+the Dynkin diagram displayed in Figure 5, and each of them gives rise to m/5 elements
+of Fix NCm(E7)(φp) sinceiranges from 1 to m/5.4
+In total, we obtain 1+9m
+5=9m+5
+5elements in Fix NCm(E7)(φp), which agrees with the
+limit in (6.173d).
+4Miraculously, these are exactly the same solutions as in the case of ( 6.173b). We have no explana-
+tion for this phenomenon.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 11 3
+In the case that p= 18m/5, we infer from (5.1) that
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4w2m
+5+1c−4,c4w2m
+5+2c−4,...,c4wmc−4,c3w1c−3,...,c3w2m
+5c−3/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4w2m
+5+ic−4, i= 1,2,...,3m
+5, (6.187a)
+wi=c3wi−3m
+5c−3, i=3m
+5+1,3m
+5+2,...,m. (6.187b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+5such that
+ℓT(wi) =ℓT(wi+m
+5) =ℓT(wi+2m
+5) =ℓT(wi+3m
+5) =ℓT(wi+4m
+5) = 1,(6.188a)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E7), we must have
+wiwi+m
+5wi+2m
+5wi+3m
+5wi+4m
+5≤Tc.
+Together with equations (6.187)–(6.188), this implies that
+wi=c18wic−18andwi(c7wic−7)(c14wic−14)(c3wic−3)(c10wic−10)≤Tc.(6.189)
+Here, the first equation is automatically satisfied since c18=ε. Due to Lemma 29
+withd= 2, we have c9wic−9=wi, hence also c14wic−14=c5wic−5, etc., so that (6.189)
+reduces to (6.186). Therefore, we are facing exactly the same en umeration problem here
+as forp= 9m/5, and, consequently, the number of solutions to (6.189) is the sam e,
+namely9m+5
+5, as required.
+Our next case is the case in (6.173e). By Lemma 26, we are free to ch oosep= 9m/4.
+In particular, mmust be divisible by 4. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3w3m
+4+1c−3,c3w3m
+4+2c−3,...,c3wmc−3,c2w1c−2,...,c2w3m
+4c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3w3m
+4+ic−3, i= 1,2,...,m
+4, (6.190a)
+wi=c2wi−m
+4c−2, i=m
+4+1,m
+4+2,...,m. (6.190b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m, which we
+summarise as follows:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+4such that
+ℓT(wi) =ℓT(wi+m
+4) =ℓT(wi+2m
+4) =ℓT(wi+3m
+4) = 1, (6.191)
+and the other wj’s, 1≤j≤m, are equal to ε,114 C. KRATTENTHALER AND T. W. M ¨ULLER
+Moreover, since ( w0;w1,...,w m)∈NCm(E7), we must have wiwi+m
+4wi+2m
+4wi+3m
+4≤T
+c. Together with equations (6.190)–(6.191), this implies that
+wi=c9wic−9andwi(c2wic−2)(c4wic−4)(c6wic−6)≤Tc. (6.192)
+Here, the first equation in (6.192) is automatically satisfied due to Le mma 29 with
+d= 2.
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 9 solutions
+forwiin (6.192) with ℓT(wi) = 1:
+wi∈/braceleftbig
+[1],[3],[2,4,2],[3,4,5,4,3],[1,3,4,5,4,3,1],[2,4,5,6,5,4,2],[2,3,4,5,6,5,4,2,3],
+[1,3,4,5,6,7,6,5,4,3,1],[1,4,2,3,4,5,6,7,6,5,4,2,3,1,4]/bracerightbig
+,
+where{s1,s2,s3,s4,s5,s6,s7}is a simple system of generators of E7, corresponding to
+the Dynkin diagram displayed in Figure 5, and each of them gives rise to m/4 elements
+of Fix NCm(E7)(φp) sinceiranges from 1 to m/4. Hence, we obtain 1 + 9m
+4=9m+4
+4
+elements in Fix NCm(E7)(φp), which agrees with the limit in (6.173e).
+Finallywediscussthecasein(6.173g). ByLemma26,wearefreetoch oosep= 9m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c5wm
+2+1c−5,c5wm
+2+2c−5,...,c5wmc−5,c4w1c−4,...,c4wm
+2c−4/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c5wm
+2+ic−5, i= 1,2,...,m
+2, (6.193a)
+wi=c4wi−m
+2c−4, i=m
+2+1,m
+2+2,...,m. (6.193b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 3, (6.194a)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iii) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 2, (6.194b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there is an iwith 1≤i≤m
+2such that
+ℓT(wi) =ℓT(wi+m
+2) = 1, (6.194c)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(v) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+2) =ℓT(wi2+m
+2) = 1, (6.194d)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(vi) there are i1andi2with 1≤i1,i2≤m
+2such that
+ℓT(wi1) =ℓT(wi1+m
+2) = 2 and ℓT(wi2) =ℓT(wi2+m
+2) = 1, (6.194e)
+and the other wj’s, 1≤j≤m, are equal to ε,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 11 5
+(vii) there are i1,i2,i3with 1≤i1<i2<i3≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi3) =ℓT(wi1+m
+2) =ℓT(wi2+m
+2) =ℓT(wi3+m
+2) = 1,(6.194f)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E7), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2≤Tc, respectively
+wi1wi2wi3wi1+m
+2wi2+m
+2wi3+m
+2≤Tc.
+Together with equations (6.193)–(6.194), this implies that
+wi=c9wic−9andwi(c4wic−4)≤Tc, (6.195)
+respectively that
+wi1=c9wi1c−9, wi2=c9wi2c−9,andwi1wi2(c4wi1c−4)(c4wi2c−4)≤Tc,(6.196)
+respectively that
+wi1=c9wi1c−9, wi2=c9wi2c−9, wi3=c9wi3c−9,
+andwi1wi2wi3(c4wi1c−4)(c4wi2c−4)(c4wi3c−4)≤Tc.(6.197)
+Here, the first equation in (6.195), the first two in (6.196), and the first three in (6.197),
+are all automatically satisfied due to Lemma 29 with d= 2.
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 9 solutions
+forwiin (6.195) with ℓT(wi) = 1:
+wi∈/braceleftbig
+[1],[3],[2,4,2],[3,4,5,4,3],[1,3,4,5,4,3,1],[2,4,5,6,5,4,2],[2,3,4,5,6,5,4,2,3],
+[1,3,4,5,6,7,6,5,4,3,1],[1,4,2,3,4,5,6,7,6,5,4,2,3,1,4]/bracerightbig
+,
+where{s1,s2,s3,s4,s5,s6,s7}is a simple system of generators of E7, corresponding to
+the Dynkin diagram displayed in Figure 5, and each of them gives rise to m/2 elements
+of Fix NCm(E7)(φp) sinceiranges from 1 to m/2. Furthermore, one obtains 12 solutions
+forwiin (6.195) with ℓT(wi) = 2:
+wi∈/braceleftbig
+[1,2,4,2],[1,3,4,5,4,3],[1,2,4,5,6,5,4,2],[1,3,1,4,5,4,3,1],[2,3,4,5,6,5,4,2],
+[1,3,1,4,5,6,7,6,5,4,3,1],[2,3,4,3,5,6,5,4,2,3],[1,2,4,2,3,4,5,6,7,6,5,4,3,1],
+[2,3,4,2,5,4,2,6,5,4,2,3],[1,3,1,4,3,5,4,3,1,6,7,6,5,4,3,1],
+[1,3,4,2,3,4,5,4,2,6,7,6,5,4,2,3,1,4],[1,2,4,2,3,4,5,4,6,5,4,3,7,6,5,4,2,3,1,4]/bracerightbig
+,116 C. KRATTENTHALER AND T. W. M ¨ULLER
+eachof themgiving riseto m/2 elements of Fix NCm(E7)(φp) sinceirangesfrom1to m/2,
+and one obtains 27 pairs ( wi1,wi2) of solutions in (6.196) with ℓT(wi1) =ℓT(wi2) = 1:
+wi∈/braceleftbig
+([1],[2,4,2]),([1],[3,4,5,4,3]),([1],[2,4,5,6,5,4,2]),([3],[1,3,4,5,4,3,1]),
+([3],[2,4,5,6,5,4,2]),([3],[1,3,4,5,6,7,6,5,4,3,1]),([2,4,2],[1]),
+([2,4,2],[2,3,4,5,6,5,4,2,3]),([2,4,2],[1,3,4,5,6,7,6,5,4,3,1]),
+([3,4,5,4,3],[1,3,4,5,4,3,1]),([3,4,5,4,3],[2,3,4,5,6,5,4,2,3]),
+([3,4,5,4,3],[1,3,4,5,6,7,6,5,4,3,1]),([1,3,4,5,4,3,1],[1]),
+([1,3,4,5,4,3,1],[3]),([1,3,4,5,4,3,1],[1,4,2,3,4,5,6,7,6,5,4,2,3,1,4]),
+([2,4,5,6,5,4,2],[1]),([2,4,5,6,5,4,2],[2,3,4,5,6,5,4,2,3]),
+([2,4,5,6,5,4,2],[1,4,2,3,4,5,6,7,6,5,4,2,3,1,4]),([2,3,4,5,6,5,4,2,3],[3]),
+([2,3,4,5,6,5,4,2,3],[2,4,2]),([2,3,4,5,6,5,4,2,3],[3,4,5,4,3]),
+([1,3,4,5,6,7,6,5,4,3,1],[3]),([1,3,4,5,6,7,6,5,4,3,1],[3,4,5,4,3]),
+([1,3,4,5,6,7,6,5,4,3,1],[1,4,2,3,4,5,6,7,6,5,4,2,3,1,4]),
+([1,4,2,3,4,5,6,7,6,5,4,2,3,1,4],[2,4,2]),
+([1,4,2,3,4,5,6,7,6,5,4,2,3,1,4],[1,3,4,5,4,3,1]),
+([1,4,2,3,4,5,6,7,6,5,4,2,3,1,4],[2,4,5,6,5,4,2])/bracerightbig
+,
+each of them giving rise to/parenleftbigm/2
+2/parenrightbig
+elements of Fix NCm(E7)(φp) since 1 ≤i1<i2≤m
+2.
+There are no solutions for wiin (6.195) with ℓT(wi) = 3, and hence there are no
+solutions for wi1,wi2in (6.196) if we are in case (vi), or for wi1,wi2,wi3in (6.197).
+In total, we obtain 1+(9+12)m
+2+27/parenleftbigm/2
+2/parenrightbig
+=(3m+2)(9m+4)
+8elements in Fix NCm(E7)(φp),
+which agrees with the limit in (6.173g).
+Finally, we turn to (6.173i). By Remark 3, the only choices for h2andm2to be
+considered are h2= 1 andm2= 7,h2= 1 andm2= 5,h2= 2 andm2= 7,h2= 2 and
+m2= 5,h2= 2 andm2= 4, respectively h2=m2= 2. These correspond to the choices
+p= 18m/7,p= 18m/5,p= 9m/7,p= 9m/5,p= 9m/4, respectively p= 9m/2, all
+of which have already been discussed as they do not belong to (6.173 i). Hence, (3.3)
+must necessarily hold, as required.
+CaseG37=E8.The degrees are 2 ,8,12,14,18,20,24,30, and hence we have
+Catm(E8;q) =[30m+30]q[30m+24]q[30m+20]q[30m+18]q
+[30]q[24]q[20]q[18]q
+×[30m+14]q[30m+12]q[30m+8]q[30m+2]q
+[14]q[12]q[8]q[2]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 11 7
+Letζbe a 30m-th root of unity. The following cases on the right-hand side of (3.3)
+occur:
+lim
+q→ζCatm(E8;q) =m+1,ifζ=ζ30,ζ15, (6.198a)
+lim
+q→ζCatm(E8;q) =5m+4
+4,ifζ=ζ24,4|m, (6.198b)
+lim
+q→ζCatm(E8;q) =3m+2
+2,ifζ=ζ20,2|m, (6.198c)
+lim
+q→ζCatm(E8;q) =5m+3
+3,ifζ=ζ18,ζ9,3|m, (6.198d)
+lim
+q→ζCatm(E8;q) =15m+7
+7,ifζ=ζ14,ζ7,7|m, (6.198e)
+lim
+q→ζCatm(E8;q) =(5m+4)(5m+2)
+8,ifζ=ζ12,2|m, (6.198f)
+lim
+q→ζCatm(E8;q) =(m+1)(3m+2)
+2,ifζ=ζ10,ζ5, (6.198g)
+lim
+q→ζCatm(E8;q) =(5m+4)(15m+4)
+16,ifζ=ζ8,4|m, (6.198h)
+lim
+q→ζCatm(E8;q) =(m+1)(5m+4)(5m+3)(5m+2)
+24,ifζ=ζ6,ζ3, (6.198i)
+lim
+q→ζCatm(E8;q) =(5m+4)(3m+2)(5m+2)(15m+4)
+64,ifζ=ζ4,2|m,(6.198j)
+lim
+q→ζCatm(E8;q) = Catm(E8),ifζ=−1 orζ= 1, (6.198k)
+lim
+q→ζCatm(E8;q) = 1,otherwise. (6.198l)
+We must now prove that the left-handside of (3.3) in each case agre es with the values
+exhibited in (6.198). The only cases not covered by Lemmas 27 and 28 are the ones in
+(6.198b), (6.198c), (6.198d), (6.198e), (6.198f), (6.198h), (6.1 98j), and (6.198l).
+We begin with the case in (6.198b). By Lemma 26, we are free to choos ep= 5m/4.
+In particular, mmust be divisible by 4. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2w3m
+4+1c−2,c2w3m
+4+2c−2,...,c2wmc−2,cw1c−1,...,cw 3m
+4c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2w3m
+4+ic−2, i= 1,2,...,m
+4, (6.199a)
+wi=cwi−m
+4c−1, i=m
+4+1,m
+4+2,...,m. (6.199b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m, which we
+summarise as follows:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+4such that
+1≤ℓT(wi) =ℓT(wi+m
+4) =ℓT(wi+2m
+4) =ℓT(wi+3m
+4)≤2. (6.200)
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have
+wiwi+m
+4wi+2m
+4wi+3m
+4≤Tc.118 C. KRATTENTHALER AND T. W. M ¨ULLER
+Together with equations (6.199)–(6.200), this implies that
+wi=c5wic−5andwi(cwic−1)(c2wic−2)(c3wic−3)≤Tc. (6.201)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 5 solutions
+forwiin (6.201) with ℓT(wi) = 2:
+wi∈/braceleftbig
+[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4],[3,4,3,5],[2,4,5,4,2,6],
+[1,3,4,5,6,5,4,3,1,7],[2,3,4,5,6,7,6,5,4,2,3,8]/bracerightbig
+,
+where we have again used the short notation of coxeter,{s1,s2,s3,s4,s5,s6,s7,s8}
+being a simple system of generators of E8, corresponding to the Dynkin diagram dis-
+played in Figure 6. Each of the above solutions for wigives rise to m/4 elements of
+FixNCm(E8)(φp) sinceiranges from 1 to m/4.
+• • • • • • ••
+1 3 4 5 6 7 82
+Figure 6. The Dynkin diagram for E8
+There are no solutions for wiin (6.201) with ℓT(wi) = 1.
+In total, we obtain 1+5m
+4=5m+4
+4elements in Fix NCm(E8)(φp), which agrees with the
+limit in (6.198b).
+Next we discuss the case in (6.198c). By Lemma 26, we are free to ch oosep= 3m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+2+1c−2,c2wm
+2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+2c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+2+ic−2, i= 1,2,...,m
+2, (6.202a)
+wi=cwi−m
+2c−1, i=m
+2+1,m
+2+2,...,m. (6.202b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m, which we
+summarise as follows:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+1≤ℓT(wi) =ℓT(wi+m
+2)≤4. (6.203)
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have wiwi+m
+2≤Tc. Together
+with equations (6.202)–(6.203), this implies that
+wi=c3wic−3, wi(cwic−1)≤Tc,and 1≤ℓT(wi)≤4.(6.204)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains three solutions
+forwiin (6.204) with ℓT(wi) = 4:
+wi∈/braceleftbig
+[1,2,3,4,2,5,4,2,6,5,7,8,7,6,5,4,2,3],[2,3,4,2,5,6,5,4,2,7],
+[1,2,4,2,3,4,5,4,6,7,6,5,4,3,1,8]/bracerightbig
+,
+whereweusedagain coxeter’s short notation, {s1,s2,s3,s4,s5,s6,s7,s8}beingasimple
+system of generators of E8, corresponding to the Dynkin diagram displayed in Figure 6.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 11 9
+Each ofthese solutionsfor wigives rise to m/2 elements of Fix NCm(E8)(φp) sinceiranges
+from 1 tom/2. There are no solutions for wiin (6.204) with 1 ≤ℓT(wi)≤3.
+In total, we obtain 1+3m
+2=3m+2
+2elements in Fix NCm(E8)(φp), which agrees with the
+limit in (6.198c).
+Next we consider the case in (6.198d). If ζ=ζ18, then, by Lemma 26, we are free to
+choosep= 5m/3, whereas, for ζ=ζ9, we can choose p= 10m/3. In particular, in both
+casesmmust be divisible by 3.
+First, letp= 5m/3. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+3+1c−2,c2wm
+3+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+3c−1/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c2wm
+3+ic−2, i= 1,2,...,2m
+3, (6.205a)
+wi=cwi−2m
+3c−1, i=2m
+3+1,2m
+3+2,...,m. (6.205b)
+Thereseveral distinctpossibilities forchoosingthe wi’s, 1≤i≤m, whichwesummarise
+as follows:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+3such that
+1≤ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3)≤2. (6.206)
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have
+wiwi+m
+3wi+2m
+3≤Tc.
+Together with equations (6.205)–(6.206), this implies that
+wi=c5wic−5andwi(c3wic−3)(cwic−1)≤Tc. (6.207)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains five solutions
+forwiin (6.207) with ℓT(wi) = 2:
+wi∈/braceleftbig
+[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4],[3,4,3,5],[2,4,5,4,2,6],
+[1,3,4,5,6,5,4,3,1,7],[2,3,4,5,6,7,6,5,4,2,3,8]/bracerightbig
+,
+where{s1,s2,s3,s4,s5,s6,s7,s8}isasimplesystemofgeneratorsof E8,correspondingto
+the Dynkin diagram displayed in Figure 6, and each of them gives rise to m/3 elements
+of FixNCm(E8)(φp) sinceiranges from 1 to m/3. There are no solutions for wiin (6.207)
+withℓT(wi) = 1.
+In total, we obtain 1+5m
+3=5m+3
+3elements in Fix NCm(E8)(φp), which agrees with the
+limit in (6.198d).
+Now letp= 10m/3. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4w2m
+3+1c−4,c4w2m
+3+2c−4,...,c4wmc−4,c3w1c−3,...,c3w2m
+3c−3/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4w2m
+3+ic−4, i= 1,2,...,m
+3, (6.208a)
+wi=c3wi−m
+3c−3, i=m
+3+1,m
+3+2,...,m. (6.208b)120 C. KRATTENTHALER AND T. W. M ¨ULLER
+Thereseveral distinctpossibilities forchoosingthe wi’s, 1≤i≤m, whichwesummarise
+as follows:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+3such that
+1≤ℓT(wi) =ℓT(wi+m
+3) =ℓT(wi+2m
+3)≤2. (6.209)
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have
+wiwi+m
+3wi+2m
+3≤Tc.
+Together with equations (6.208)–(6.209), this implies that
+wi=c10wic−10andwi(c3wic−3)(c6wic−6)≤Tc. (6.210)
+Due to Lemma 29 with d= 2, we have c15wic−15=wi, hence also c5wic−5=wi, so
+that (6.210) reduces to (6.207). Therefore, we are facing exact ly the same enumeration
+problem here as for p= 5m/3, and, consequently, the number of solutions to (6.210) is
+the same, namely5m+3
+3, as required.
+Our next case is the case in (6.198e). If ζ=ζ14, then, by Lemma 26, we are free to
+choosep= 15m/7, whereas, for ζ=ζ7, we can choose p= 30m/7. In particular, in
+both casesmmust be divisible by 7.
+First, letp= 15m/7. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3w6m
+7+1c−3,c3w6m
+7+2c−3,...,c3wmc−3,c2w1c−2,...,c2w6m
+7c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3w6m
+7+ic−3, i= 1,2,...,m
+7, (6.211a)
+wi=c2wi−m
+7c−2, i=m
+7+1,m
+7+2,...,m. (6.211b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+7such that
+ℓT(wi) =ℓT(wi+m
+7) =ℓT(wi+2m
+7) =ℓT(wi+3m
+7)
+=ℓT(wi+4m
+7) =ℓT(wi+5m
+7) =ℓT(wi+6m
+7) = 1,(6.212)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have
+wiwi+m
+7wi+2m
+7wi+3m
+7wi+4m
+7wi+5m
+7wi+6m
+7≤Tc.
+Together with the equations (6.211)–(6.212), this implies that
+wi=c15wic−15
+andwi(c2wic−2)(c4wic−4)(c6wic−6)(c8wic−8)(c10wic−10)(c12wic−12)≤Tc.(6.213)
+Here, the first equation is automatically satisfied due to Lemma 29 wit hd= 2.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 12 1
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 15 solutions
+forwiin (6.213):
+wi∈/braceleftbig
+[4],[5],[6],[7],[8],[3,4,3],[2,4,5,4,2],[3,4,5,4,3],[2,4,5,6,5,4,2],
+[1,3,4,5,6,5,4,3,1],[1,3,4,5,6,7,6,5,4,3,1],
+[2,3,4,5,6,7,6,5,4,2,3],[2,3,4,5,6,7,8,7,6,5,4,2,3],
+[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1],[1,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4]/bracerightbig
+,
+where we have again used the short notation of coxeter,{s1,s2,s3,s4,s5,s6,s7,s8}
+being a simple system of generators of E8, corresponding to the Dynkin diagram dis-
+played in Figure 6, and each of them gives rise to m/7 elements of Fix NCm(E8)(φp) since
+iranges from 1 to m/7.
+In total, we obtain 1+15m
+7=15m+7
+7elements in Fix NCm(E8)(φp), which agrees with
+the limit in (6.198e).
+Now letp= 30m/7. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c5w5m
+7+1c−5,c5w5m
+7+2c−5,...,c5wmc−5,c4w1c−4,...,c4w5m
+7c−4/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c5w5m
+7+ic−5, i= 1,2,...,2m
+7, (6.214a)
+wi=c4wi−2m
+7c−4, i=2m
+7+1,2m
+7+2,...,m. (6.214b)
+There are two distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+7such that
+ℓT(wi) =ℓT(wi+m
+7) =ℓT(wi+2m
+7) =ℓT(wi+3m
+7)
+=ℓT(wi+4m
+7) =ℓT(wi+5m
+7) =ℓT(wi+6m
+7) = 1,(6.215)
+and the other wj’s, 1≤j≤m, are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have
+wiwi+m
+7wi+2m
+7wi+3m
+7wi+4m
+7wi+5m
+7wi+6m
+7≤Tc.
+Together with the equations (6.214)–(6.215), this implies that
+wi=c30wic−30
+andwi(c17wic−17)(c4wic−4)(c21wic−21)(c8wic−8)(c25wic−25)(c12wic−12)≤Tc.(6.216)
+Here, the first equation is automatically satisfied since c30=ε. Moreover, due to
+Lemma 29 with d= 2, we have c17wic−17=c2wic−2, etc., so that (6.216) reduces to
+(6.213). Therefore, we are facing exactly the same enumeration p roblem here as for
+p= 15m/7, and, consequently, the number of solutions to (6.216) is the sam e, namely
+15m+7
+7, as required.122 C. KRATTENTHALER AND T. W. M ¨ULLER
+We now turn to the case in (6.198f). By Lemma 26, we are free to cho osep= 5m/2.
+In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c3wm
+2+1c−3,c3wm
+2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm
+2c−2/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c3wm
+2+ic−3, i= 1,2,...,m
+2, (6.217a)
+wi=c2wi−m
+2c−2, i=m
+2+1,m
+2+2,...,m. (6.217b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+1≤ℓT(wi) =ℓT(wi+m
+2)≤4, (6.218a)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iii) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓ1:=ℓT(wi1) =ℓT(wi1+m
+2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
+2)≥1,andℓ1+ℓ2≤4,
+(6.218b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1,i2,i3with 1≤i1<i2<i3≤m
+2such that
+ℓ1:=ℓT(wi1) =ℓT(wi1+m
+2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
+2)≥1,
+ℓ3:=ℓT(wi3) =ℓT(wi3+m
+2)≥1,andℓ1+ℓ2+ℓ3≤4,(6.218c)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(v) there are i1,i2,i3,i4with 1≤i1<i2<i3<i4≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi3) =ℓT(wi4)
+=ℓT(wi1+m
+2) =ℓT(wi2+m
+2) =ℓT(wi3+m
+2) =ℓT(wi4+m
+2) = 1,(6.218d)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2≤Tc, respectively
+wi1wi2wi3wi1+m
+2wi2+m
+2wi3+m
+2≤Tc,
+respectively
+wi1wi2wi3wi4wi1+m
+2wi2+m
+2wi3+m
+2wi4+m
+2=c.
+Together with the equations (6.217)–(6.218), this implies that
+wi=c5wic−5andwi(c2wic−2)≤Tc, (6.219)
+respectively that
+wi1=c5wi1c−5, wi2=c5wi2c−5,andwi1wi2(c2wi1c−2)(c2wi2c−2)≤Tc,(6.220)
+respectively that
+wi1=c5wi1c−5, wi2=c5wi2c−5, wi3=c5wi3c−5,
+andwi1wi2wi3(c2wi1c−2)(c2wi2c−2)(c2wi3c−2)≤Tc,(6.221)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 12 3
+respectively that
+wi1=c5wi1c−5, wi2=c5wi2c−5, wi3=c5wi3c−5, wi4=c5wi4c−5,
+andwi1wi2wi3wi4(c2wi1c−2)(c2wi2c−2)(c2wi3c−2)(c2wi4c−2) =c.(6.222)
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 10 solutions
+forwiin (6.219) with ℓT(wi) = 2:
+wi∈/braceleftbig
+[1,4,2,3,4,5,6,7,6,5,4,2,3,4],[3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5],
+[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4],[3,4,3,5],[2,4,5,4,2,6],[1,3,4,5,6,5,4,3,1,7],
+[2,3,4,5,6,7,6,5,4,2,3,8],[2,4,2,3,4,5,6,5,4,3],[1,3,4,5,4,2,3,1,4,5,6,7,6,5,4,2],
+[2,3,1,4,5,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1]/bracerightbig
+,(6.223)
+and one obtains 10 solutions for wiin (6.219) with ℓT(wi) = 4:
+wi∈/braceleftbig
+[1,2,3,4,5,6,7,6,5,4,2,3,4,8],[1,2,4,2,3,4,5,4,3,6,5,7,6,5,4,2,3,4],
+[1,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1,4,5],
+[1,3,1,4,5,4,2,3,4,5,6,5,4,2,7,6,8,7,6,5,4,2,3,1,4,5],
+[1,2,3,4,2,5,4,2,3,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4],[4,2,3,4,5,6],
+[2,3,1,4,2,5,4,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],[1,5,4,2,3,1,4,5,6,7],
+[3,1,6,5,4,2,3,1,4,3,5,6,7,8],[2,3,4,2,3,5,4,6,5,4,2,7,6,5,4,2,3,8]/bracerightbig
+,(6.224)
+where we have again used the short notation of coxeter,{s1,s2,s3,s4,s5,s6,s7,s8}
+being a simple system of generators of E8, corresponding to the Dynkin diagram dis-
+played in Figure 6, and each of them gives rise to m/2 elements of Fix NCm(E8)(φp) since
+iranges from 1 to m/2. There are no solutions for wiin (6.219) with ℓT(wi) = 1 or
+ℓT(wi) = 3.
+Consequently, there are no solutions in Cases (iv) and (v), and the only possible
+solutions occurring in Case (iii) are pairs ( wi1,wi2) of elements of (6.223) whose product
+is in (6.224). Another computation using Stembridge’s Maplepackagecoxeter finds
+the following 25 pairs meeting that description:
+wi∈/braceleftbig
+([1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4],[1,4,2,3,4,5,6,7,6,5,4,2,3,4]),
+([2,4,5,4,2,6],[1,4,2,3,4,5,6,7,6,5,4,2,3,4]),
+([3,4,3,5],[3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5]),
+([1,3,4,5,6,5,4,3,1,7],[3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5]),
+([3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5],[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4]),
+([2,3,4,5,6,7,6,5,4,2,3,8],[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4]),
+([1,3,4,5,4,2,3,1,4,5,6,7,6,5,4,2],[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4]),
+([1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4],[3,4,3,5]),([2,4,2,3,4,5,6,5,4,3],[3,4,3,5]),
+([2,3,1,4,5,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],[3,4,3,5]),
+([1,4,2,3,4,5,6,7,6,5,4,2,3,4],[2,4,5,4,2,6]),([3,4,3,5],[2,4,5,4,2,6]),
+([1,3,4,5,4,2,3,1,4,5,6,7,6,5,4,2],[2,4,5,4,2,6]),
+([3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5],[1,3,4,5,6,5,4,3,1,7]),
+([2,4,5,4,2,6],[1,3,4,5,6,5,4,3,1,7]),
+([2,3,1,4,5,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],[1,3,4,5,6,5,4,3,1,7]),124 C. KRATTENTHALER AND T. W. M ¨ULLER
+([1,4,2,3,4,5,6,7,6,5,4,2,3,4],[2,3,4,5,6,7,6,5,4,2,3,8]),
+([1,3,4,5,6,5,4,3,1,7],[2,3,4,5,6,7,6,5,4,2,3,8]),
+([2,4,2,3,4,5,6,5,4,3],[2,3,4,5,6,7,6,5,4,2,3,8]),([2,4,5,4,2,6],[2,4,2,3,4,5,6,5,4,3]),
+([2,3,4,5,6,7,6,5,4,2,3,8],[2,4,2,3,4,5,6,5,4,3]),
+([1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4],[1,3,4,5,4,2,3,1,4,5,6,7,6,5,4,2]),
+([1,3,4,5,6,5,4,3,1,7],[1,3,4,5,4,2,3,1,4,5,6,7,6,5,4,2]),
+([3,4,3,5],[2,3,1,4,5,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1]),
+([2,3,4,5,6,7,6,5,4,2,3,8],[2,3,1,4,5,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1])/bracerightbig
+,
+where{s1,s2,s3,s4,s5,s6,s7,s8}is a simple system of generators of E8, corresponding
+to the Dynkin diagram displayed in Figure 6, and each of them gives rise to/parenleftbigm/2
+2/parenrightbig
+elements of Fix NCm(E8)(φp) since 1 ≤i1<i2≤m
+2.
+In total, we obtain
+1+20m
+2+25/parenleftbiggm/2
+2/parenrightbigg
+=(5m+4)(5m+2)
+8
+elements in Fix NCm(E8)(φp), which agrees with the limit in (6.198f).
+Next we consider the case in (6.198h). By Lemma 26, we are free to c hoosep=
+15m/4. In particular, mmust be divisible by 4. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c4wm
+4+1c−4,c4wm
+4+2c−4,...,c4wmc−4,c3w1c−3,...,c3wm
+4c−3/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c4wm
+4+ic−4, i= 1,2,...,3m
+4, (6.225a)
+wi=c3wi−3m
+4c−3, i=3m
+4+1,3m
+4+2,...,m. (6.225b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m, which we
+summarise as follows:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+4such that
+1≤ℓT(wi) =ℓT(wi+m
+4) =ℓT(wi+2m
+4) =ℓT(wi+3m
+4)≤2, (6.226a)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iii) there are i1andi2with 1≤i1<i2≤m
+4such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
+4) =ℓT(wi2+m
+4)
+=ℓT(wi1+2m
+4) =ℓT(wi2+2m
+4) =ℓT(wi1+3m
+4) =ℓT(wi2+3m
+4) = 1,(6.226b)
+and all other wjare equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have
+wiwi+m
+4wi+2m
+4wi+3m
+4≤Tc,
+or
+wi1wi2wi1+m
+4wi2+m
+4wi1+2m
+4wi2+2m
+4wi1+3m
+4wi2+3m
+4=c.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 12 5
+Together with equations (6.225)–(6.226), this implies that
+wi=c15wic−15andwi(c11wic−11)(c7wic−7)(c3wic−3)≤Tc, (6.227)
+or that
+wi1=c15wi1c−15, wi1=c15wi2c−15,
+andwi1wi2(c11wi1c−11)(c11wi2c−11)(c7wi1c−7)(c7wi2c−7)(c3wi1c−3)(c3wi2c−3) =c.
+(6.228)
+Here, the first equation in (6.227) and the first two equations in (6.2 28) are automati-
+cally satisfied due to Lemma 29 with d= 2.
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 30 solutions
+forwiin (6.227) with ℓT(wi) = 1:
+wi∈/braceleftbig
+[4],[5],[6],[7],[8],[3,4,3],[4,5,4],[5,6,5],[6,7,6],[7,8,7],[2,4,5,4,2],[3,4,5,4,3],
+[2,4,5,6,5,4,2],[1,3,4,5,6,5,4,3,1],[4,2,3,4,5,4,2,3,4],[1,3,4,5,6,7,6,5,4,3,1],
+[2,3,4,5,6,7,6,5,4,2,3],[1,2,3,4,5,6,7,6,5,4,2,3,1],[5,4,2,3,4,5,6,5,4,2,3,4,5],
+[2,3,4,5,6,7,8,7,6,5,4,2,3],[1,5,4,2,3,4,5,6,5,4,2,3,1,4,5],
+[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1],[4,2,3,4,5,6,7,8,7,6,5,4,2,3,4],
+[1,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4],[3,1,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3],
+[1,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5,6],[3,1,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,3,5,6],
+[4,3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4],
+[3,1,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],
+[2,4,3,1,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7]/bracerightbig
+,
+one obtains 45 solutions for wiin (6.227) with ℓT(wi) = 2 and wiof typeA2
+1(as a
+parabolic Coxeter element; see the end of Section 2):
+wi∈/braceleftbig
+[1,2,3,1,4,5,6,5,4,2,3,4,5,7,6,5,4,2,3,1,4,5,8,7,6,5,4,2,3,1],
+[1,2,3,4,2,3,1,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],
+[1,2,3,4,2,5,4,6,5,4,2,7,6,5,4,2,3,1],[1,2,3,4,3,5,4,6,5,4,3,7,6,5,4,2,3,1],
+[1,2,3,4,5,4,6,5,4,7,8,7,6,5,4,2,3,1],[1,2,3,4,5,4,6,7,6,5,4,2,3,1],
+[1,2,3,4,5,6,5,4,2,3,4,5,7,6,5,4,2,3,4,5,8,7,6,5,4,2,3,1],[1,2,6,5,4,2,3,1,7,6,5,4,2,3,1,4,3,5,4,2,6,5,7,6],
+[1,3,1,4,2,3,4,5,4,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4,3],
+[1,3,1,4,2,7,6,5,4,2,3,1,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,7,6,8,7],
+[1,3,4,3,5,4,3,6,5,4,3,1],[1,3,4,5,4,6,5,4,7,6,5,4,3,1],[1,4,2,3,1,4,3,5,4,3,1,6,7,8,7,6,5,4,2,3,1,4],
+[1,4,2,3,4,5,6,5,7,6,5,8,7,6,5,4,2,3,1,4],[1,4,5,4,2,3,4,5,6,5,4,2,3,1,4,5],
+[1,5,4,2,3,1,4,5,6,5,4,2,3,1,4,5],[1,5,4,2,3,4,5,6,5,4,2,3,1,4,5,8],
+[1,5,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5,6],[2,3,1,4,2,3,4,5,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4,3],
+[2,3,1,6,5,4,2,3,1,4,3,5,6,7,6,5,4,2,3,1,4,3,5,6],
+[2,3,4,2,3,1,4,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],
+[2,3,4,2,3,5,4,2,3,6,7,8,7,6,5,4,2,3],[2,3,4,5,4,6,5,4,7,6,5,4,2,3],[2,3,4,5,6,5,7,6,5,8,7,6,5,4,2,3],
+[2,4,2,3,4,5,4,3,6,7,8,7,6,5,4,2,3,4],[2,4,2,5,4,2,6,5,4,2],
+[2,4,3,1,6,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],[2,4,5,4,2,7,8,7],
+[3,1,4,2,3,4,5,6,5,7,8,7,6,5,4,2,3,1,4,3],[3,1,4,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],
+[3,1,5,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,3,5,6],[3,1,6,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],
+[3,4,2,3,1,5,4,2,3,1,4,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,7,8],[3,4,2,3,4,5,4,2,6,7,8,7,6,5,4,2,3,4],126 C. KRATTENTHALER AND T. W. M ¨ULLER
+[3,4,3,6,7,6],[3,4,5,4,3,7,8,7],[4,2,3,1,5,4,2,3,1,4,3,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4],
+[4,2,3,4,5,4,2,3,4,7],[4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,4],[4,3,1,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,3,5,4],
+[4,5,4,2,3,4,5,6,5,4,2,3,4,5],[4,5,4,8],[4,7,8,7],
+[5,4,2,3,4,5,6,5,4,2,3,4,5,8],[5,4,3,1,6,5,4,2,3,1,4,3,5,4,6,5]/bracerightbig,(6.229)
+and one obtains 20 solutions for wiin (6.227) with ℓT(wi) = 2 andwiof typeA2:
+wi∈/braceleftbig
+[1,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1,4,3],[1,2,3,4,5,6,7,6,5,4,2,3,1,8],
+[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4],[1,2,4,5,4,2,3,4,5,6,5,4,3,1],
+[1,2,4,5,6,5,4,2,3,4,5,6,7,6,5,4,3,1],[1,3,1,4,5,6,5,4,2,3,1,4,5,6,7,6,5,4,2,3],
+[1,3,1,4,5,6,7,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3],[1,3,4,5,6,5,4,3,1,7],
+[1,4,2,3,1,4,5,4,6,7,8,7,6,5,4,2,3,1,4,3,5,4],
+[2,3,4,3,1,5,6,7,6,5,4,2,3,1,4,3,5,6,7,8,7,6,5,4,2,3,1,4],[2,3,4,5,6,7,6,5,4,2,3,8],
+[2,3,4,5,6,7,8,7,6,5,4,2,3,4],[2,4,5,4,2,6],[3,4,2,3,4,5,4,2],[3,4,3,5],
+[3,4,5,4,2,3,4,5,6,5,4,2],[4,5],[5,6],[6,7],[7,8]/bracerightbig
+,(6.230)
+where{s1,s2,s3,s4,s5,s6,s7,s8}isasimplesystemofgeneratorsof E8,correspondingto
+the Dynkin diagram displayed in Figure 6, and each of them gives rise to m/4 elements
+of Fix NCm(E8)(φp) sinceiranges from 1 to m/4.
+The number of solutions in Case (iii) can be computed from our knowled ge of the
+solutions in Case (ii) according to type, using some elementary count ing arguments.
+Namely, the number of solutions of (6.228) is equal to
+45·2+20·3 = 150,
+since an element of type A2
+1can be decomposed in two ways into a product of two
+elements of absolute length 1, while for an element of type A2this can be done in 3
+ways.
+In total, we obtain 1 + (30 + 45 + 20)m
+4+ 150/parenleftbigm/4
+2/parenrightbig
+=(5m+4)(15m+4)
+16elements in
+FixNCm(E8)(φp), which agrees with the limit in (6.198h).
+Finally we discuss the case in (6.198j). By Lemma 26, we are free to ch oosep=
+15m/2. In particular, mmust be divisible by 2. From (5.1), we infer
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c8wm
+2+1c−8,c8wm
+2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
+2c−7/parenrightbig
+.
+Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
+wi=c8wm
+2+ic−8, i= 1,2,...,m
+2, (6.231a)
+wi=c7wi−m
+2c−7, i=m
+2+1,m
+2+2,...,m. (6.231b)
+There are several distinct possibilities for choosing the wi’s, 1≤i≤m:
+(i) all thewi’s are equal to ε(andw0=c),
+(ii) there is an iwith 1≤i≤m
+2such that
+1≤ℓT(wi) =ℓT(wi+m
+2)≤4, (6.232a)
+and the other wj’s, 1≤j≤m, are equal to ε,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 12 7
+(iii) there are i1andi2with 1≤i1<i2≤m
+2such that
+ℓ1:=ℓT(wi1) =ℓT(wi1+m
+2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
+2)≥1,andℓ1+ℓ2≤4,
+(6.232b)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(iv) there are i1,i2,i3with 1≤i1<i2<i3≤m
+2such that
+ℓ1:=ℓT(wi1) =ℓT(wi1+m
+2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
+2)≥1,
+ℓ3:=ℓT(wi3) =ℓT(wi3+m
+2)≥1,andℓ1+ℓ2+ℓ3≤4,(6.232c)
+and the other wj’s, 1≤j≤m, are equal to ε,
+(v) there are i1,i2,i3,i4with 1≤i1<i2<i3<i4≤m
+2such that
+ℓT(wi1) =ℓT(wi2) =ℓT(wi3) =ℓT(wi4)
+=ℓT(wi1+m
+2) =ℓT(wi2+m
+2) =ℓT(wi3+m
+2) =ℓT(wi4+m
+2) = 1,(6.232d)
+and all other wj’s are equal to ε.
+Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have wiwi+m
+2≤Tc, respec-
+tivelywi1wi2wi1+m
+2wi2+m
+2≤Tc, respectively
+wi1wi2wi3wi1+m
+2wi2+m
+2wi3+m
+2≤Tc,
+respectively
+wi1wi2wi3wi4wi1+m
+2wi2+m
+2wi3+m
+2wi4+m
+2=c.
+Together with equations (6.231)–(6.232), this implies that
+wi=c15wic−15andwi(c7wic−7)≤Tc, (6.233)
+respectively that
+wi1=c15wi1c−15, wi2=c15wi2c−15,andwi1wi2(c7wi1c−7)(c7wi2c−7)≤Tc,(6.234)
+respectively that
+wi1=c15wi1c−15, wi2=c15wi2c−15, wi3=c15wi3c−15,
+andwi1wi2wi3(c7wi1c−7)(c7wi2c−7)(c7wi3c−7)≤Tc,(6.235)
+respectively that
+wi1=c15wi1c−15, wi2=c15wi2c−15, wi3=c15wi3c−15, wi4=c15wi4c−15,
+andwi1wi2wi3wi4(c7wi1c−7)(c7wi2c−7)(c7wi3c−7)(c7wi4c−7) =c.(6.236)
+Here, the first equation in (6.233), the first two in (6.234), the firs t three in (6.235), and
+the first four in (6.236), are all automatically satisfied due to Lemma 29 withd= 2.128 C. KRATTENTHALER AND T. W. M ¨ULLER
+With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 45 solutions
+forwiin (6.233) with ℓT(wi) = 1:
+wi∈/braceleftbig
+[1],[3],[4],[5],[6],[7],[8],[2,4,2],[3,4,3],[4,5,4],[5,6,5],[6,7,6],[7,8,7],
+[2,4,5,4,2],[3,4,5,4,3],[1,3,4,5,4,3,1],[2,4,5,6,5,4,2],[3,4,5,6,5,4,3],
+[1,3,4,5,6,5,4,3,1],[4,2,3,4,5,4,2,3,4],[2,3,4,5,6,5,4,2,3],[2,4,5,6,7,6,5,4,2],
+[1,3,4,5,6,7,6,5,4,3,1],[4,2,3,4,5,6,5,4,2,3,4],[2,3,4,5,6,7,6,5,4,2,3],
+[1,2,3,4,5,6,7,6,5,4,2,3,1],[1,3,4,5,6,7,8,7,6,5,4,3,1],[5,4,2,3,4,5,6,5,4,2,3,4,5],
+[4,2,3,4,5,6,7,6,5,4,2,3,4],[2,3,4,5,6,7,8,7,6,5,4,2,3],[1,5,4,2,3,4,5,6,5,4,2,3,1,4,5],
+[1,4,2,3,4,5,6,7,6,5,4,2,3,1,4],[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1],
+[4,2,3,4,5,6,7,8,7,6,5,4,2,3,4],[1,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5],
+[1,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4],[3,1,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3],
+[1,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5,6],[1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5],
+[3,1,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,3,5,6],
+[3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5],
+[4,3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4],
+[3,1,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6],
+[3,1,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],
+[2,4,3,1,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7]/bracerightbig
+,
+one obtains 150 solutions for wiin (6.233) with ℓT(wi) = 2 andwiof typeA2
+1:
+wi∈/braceleftbig
+[1,2,3,1,4,3,5,4,6,5,7,6,5,4,3,1,8,7,6,5,4,2,3,1],
+[1,2,3,1,4,5,6,5,4,2,3,4,5,7,6,5,4,2,3,1,4,5,8,7,6,5,4,2,3,1],[1,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1],
+[1,2,3,4,2,3,1,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],
+[1,2,3,4,2,3,5,4,6,5,7,6,5,4,2,3,8,7,6,5,4,2,3,1],[1,2,3,4,2,5,4,2,6,7,8,7,6,5,4,2,3,1],
+[1,2,3,4,2,5,4,6,5,4,2,7,6,5,4,2,3,1],[1,2,3,4,2,5,4,6,5,4,2,7,8,7,6,5,4,2,3,1],
+[1,2,3,4,3,5,4,6,5,4,3,7,6,5,4,2,3,1],[1,2,3,4,3,5,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1],
+[1,2,3,4,5,4,2,3,4,6,5,7,6,5,4,2,3,4,8,7,6,5,4,2,3,1],[1,2,3,4,5,4,6,5,4,7,8,7,6,5,4,2,3,1],
+[1,2,3,4,5,4,6,7,6,5,4,2,3,1],[1,2,3,4,5,4,6,7,8,7,6,5,4,2,3,1],
+[1,2,3,4,5,6,5,4,2,3,4,5,7,6,5,4,2,3,4,5,8,7,6,5,4,2,3,1],[1,2,3,4,5,6,5,7,8,7,6,5,4,2,3,1],
+[1,2,4,2,3,1,7,6,5,4,2,3,1,4,3,5,4,2,6,7],[1,2,4,2,3,1,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8],
+[1,2,4,2,3,4,5,4,3,6,7,6,5,4,2,3,1,4],[1,2,4,2,3,4,5,4,3,6,7,8,7,6,5,4,2,3,1,4],
+[1,2,4,2,3,4,5,4,6,5,4,3,7,6,5,4,2,3,1,4],[1,2,4,2,3,4,5,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4],
+[1,2,4,5,4,2,3,4,5,6,5,4,3,7,6,5,4,2,3,1,4,5],[1,2,4,5,4,2],[1,2,4,5,6,5,4,2],
+[1,2,6,5,4,2,3,1,7,6,5,4,2,3,1,4,3,5,4,2,6,5,7,6],[1,3,1,4,2,3,4,5,4,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4,3],
+[1,3,1,4,2,5,4,2,3,1,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,7,8],
+[1,3,1,4,2,7,6,5,4,2,3,1,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,7,6,8,7],
+[1,3,1,4,3,1,5,6,7,8,7,6,5,4,3,1],[1,3,1,4,3,5,4,3,1,6,5,4,3,1],[1,3,1,4,3,5,4,3,1,6,7,8,7,6,5,4,3,1],
+[1,3,1,4,5,4,2,3,4,5,6,5,4,2,7,8,7,6,5,4,2,3,1,4,3,5],
+[1,3,1,4,5,4,2,3,4,5,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4,3,5],
+[1,3,1,4,5,6,5,4,2,3,4,5,6,7,6,5,4,2,8,7,6,5,4,2,3,1,4,3,5,6],
+[1,3,1,4,5,6,5,4,3,1],[1,3,1,4,5,6,7,6,5,4,3,1],[1,3,4,2,3,4,5,4,2,6,7,8,7,6,5,4,2,3,1,4],
+[1,3,4,2,3,4,5,4,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4],[1,3,4,3,5,4,3,1],
+[1,3,4,3,5,4,3,6,5,4,3,1],[1,3,4,5,4,2,3,4,5,6,5,4,2,7,6,5,4,2,3,1,4,5],CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 12 9
+[1,3,4,5,4,2,3,4,5,6,5,4,2,7,8,7,6,5,4,2,3,1,4,5],
+[1,3,4,5,4,2,3,4,5,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4,5],[1,3,4,5,4,3,1,7],
+[1,3,4,5,4,6,5,4,7,6,5,4,3,1],[1,3,4,5,4,6,7,6,5,4,3,1],[1,3,4,5,4,6,7,8,7,6,5,4,3,1],[1,3,4,5,6,5,4,3,1,8],
+[1,3,4,5,6,5,7,6,5,4,3,1],[1,3,4,5,6,7,6,8,7,6,5,4,3,1],[1,4,2,3,1,4,3,5,4,3,1,6,7,8,7,6,5,4,2,3,1,4],
+[1,4,2,3,1,4,3,5,4,6,5,7,6,5,4,3,1,8,7,6,5,4,2,3,1,4],[1,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1,4],
+[1,4,2,3,4,5,4,2,3,4,6,5,7,6,5,4,2,3,4,8,7,6,5,4,2,3,1,4],[1,4,2,3,4,5,6,5,7,6,5,4,2,3,1,4],
+[1,4,2,3,4,5,6,5,7,6,5,8,7,6,5,4,2,3,1,4],[1,4,5,4,2,3,4,5,6,5,4,2,3,1,4,5],
+[1,4,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5],[1,4,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5],
+[1,4],[1,5,4,2,3,1,4,5,6,5,4,2,3,1,4,5],[1,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1,4,5],
+[1,5,4,2,3,1,4,5],[1,5,4,2,3,4,5,6,5,4,2,3,1,4,5,8],[1,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,5],
+[1,5,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5,6],[1,6,5,4,2,3,1,4,5,6],[1,6],[1,7,6,5,4,2,3,1,4,5,6,7],[1,8],
+[2,3,1,4,2,3,4,5,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4,3],
+[2,3,1,4,5,6,5,4,2,3,4,5,6,7,6,5,4,3,8,7,6,5,4,2,3,1,4,3,5,6],[2,3,1,6,5,4,2,3,1,4,3,5,6,7,6,5,4,2,3,1,4,3,5,6],
+[2,3,4,2,3,1,4,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],
+[2,3,4,2,3,5,4,2,3,6,7,8,7,6,5,4,2,3],[2,3,4,2,3,5,4,6,5,4,2,3,7,6,5,4,2,3],
+[2,3,4,2,3,5,4,6,5,4,2,3,7,8,7,6,5,4,2,3],[2,3,4,2,3,5,4,6,5,7,6,5,4,2,3,8,7,6,5,4,2,3],
+[2,3,4,2,5,4,2,6,5,4,2,3],[2,3,4,2,5,4,2,6,7,8,7,6,5,4,2,3],[2,3,4,2,5,4,6,5,4,2,7,6,5,4,2,3],
+[2,3,4,2,5,4,6,5,4,2,7,8,7,6,5,4,2,3],[2,3,4,2,5,6,5,4,2,3],[2,3,4,2,5,6,7,6,5,4,2,3],
+[2,3,4,3,5,4,6,5,4,3,7,6,5,4,2,3],[2,3,4,3,5,6,7,6,5,4,2,3],[2,3,4,3,5,6,7,8,7,6,5,4,2,3],
+[2,3,4,5,4,6,5,4,2,3],[2,3,4,5,4,6,5,4,7,6,5,4,2,3],[2,3,4,5,6,5,4,2,3,8],[2,3,4,5,6,5,7,6,5,8,7,6,5,4,2,3],
+[2,3,4,5,6,5,7,8,7,6,5,4,2,3],[2,3,4,5,6,7,6,8,7,6,5,4,2,3],[2,4,2,3,4,5,4,3,6,5,4,2,3,4],
+[2,4,2,3,4,5,4,3,6,7,6,5,4,2,3,4],[2,4,2,3,4,5,4,3,6,7,8,7,6,5,4,2,3,4],[2,4,2,3,4,5,4,6,5,4,3,7,6,5,4,2,3,4],
+[2,4,2,5,4,2,6,5,4,2],[2,4,2,5,6,5,4,2],[2,4,2,5,6,7,6,5,4,2],[2,4,2,6],[2,4,2,8],
+[2,4,3,1,6,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],[2,4,5,4,2,7,8,7],
+[2,4,5,4,2,7],[2,4,5,4,2,8],[2,4,5,4,6,5,4,2],[2,4,5,6,5,7,6,5,4,2],[3,1,4,2,3,4,5,6,5,7,8,7,6,5,4,2,3,1,4,3],
+[3,1,4,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5],[3,1,4,5,4,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6],
+[3,1,4,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6],[3,1,4,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],
+[3,1,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,3,5],[3,1,5,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,3,5,6],
+[3,1,5,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6],[3,1,6,5,4,2,3,1,4,3,5,6],
+[3,1,6,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],[3,1,7,6,5,4,2,3,1,4,3,5,6,7],
+[3,1,8,7,6,5,4,2,3,1,4,3,5,6,7,8],[3,4,2,3,1,5,4,2,3,1,4,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,7,8],
+[3,4,2,3,4,5,4,2,6,5,4,2,3,4],[3,4,2,3,4,5,4,2,6,7,8,7,6,5,4,2,3,4],[3,4,3,5,4,3],[3,4,3,6,7,6],
+[3,4,3,6],[3,4,3,7],[3,4,5,4,3,7,8,7],[3,4,5,4,6,5,4,3],[3,4,5,6,5,4,3,8],[3,5],[3,7],
+[4,2,3,1,5,4,2,3,1,4,3,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4],[4,2,3,4,5,4,2,3,4,7],[4,2,3,4,5,6,5,4,2,3,4,8],
+[4,2,3,4,5,6,5,7,6,5,4,2,3,4],[4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,4],[4,2,3,4],
+[4,3,1,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,3,5,4],[4,5,4,2,3,4,5,6,5,4,2,3,4,5],[4,5,4,8],[4,7,8,7],
+[4,7],[4,8],[5,4,2,3,4,5,6,5,4,2,3,4,5,8],[5,4,2,3,4,5],
+[5,4,3,1,6,5,4,2,3,1,4,3,5,4,6,5],[5,8],[6,5,4,2,3,4,5,6]/bracerightbig,
+one obtains 100 solutions for wiin (6.233) with ℓT(wi) = 2 andwiof typeA2:
+wi∈/braceleftbig
+[1,2,3,1,4,5,4,6,7,8,7,6,5,4,2,3,1,4,3,5],[1,2,3,1,4,5,6,5,4,7,6,5,4,2,3,1,4,3,5,6],
+[1,2,3,1,4,5,6,5,4,7,8,7,6,5,4,2,3,1,4,3,5,6],[1,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1,4,3],
+[1,2,3,4,3,1,5,6,7,6,5,4,3,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],[1,2,3,4,5,4,6,7,6,5,4,2,3,1,4,5],
+[1,2,3,4,5,4,6,7,8,7,6,5,4,2,3,1,4,5],[1,2,3,4,5,6,7,6,5,4,2,3,1,4],
+[1,2,3,4,5,6,7,6,5,4,2,3,1,8],[1,2,3,4,5,6,7,6,5,4,2,3],[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4],
+[1,2,3,4,5,6,7,8,7,6,5,4,2,3],[1,2,4,2,3,1,4,5,6,7,6,5,4,2,3,1,4,3,5,6,7,8,7,6,5,4,2,3],
+[1,2,4,2,3,4,5,6,7,6,5,4,3,1],[1,2,4,2,3,4,5,6,7,8,7,6,5,4,3,1],[1,2,4,5,4,2,3,4,5,6,5,4,3,1],130 C. KRATTENTHALER AND T. W. M ¨ULLER
+[1,2,4,5,4,2,3,4,5,6,7,6,5,4,3,1],[1,2,4,5,4,2,3,4,5,6,7,8,7,6,5,4,3,1],[1,2,4,5,6,5,4,2,3,4,5,6,7,6,5,4,3,1],
+[1,3,1,4,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3],[1,3,1,4,5,6,5,4,2,3,1,4,5,6,7,6,5,4,2,3],
+[1,3,1,4,5,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3],[1,3,1,4,5,6,7,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3],
+[1,3,4,3,1,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3,4],[1,3,4,5,4,2,3,1,4,5,6,5,4,2],
+[1,3,4,5,4,2,3,1,4,5,6,7,6,5,4,2],[1,3,4,5,4,3,1,6,7,6],[1,3,4,5,4,3,1,6],[1,3,4,5,4,3],
+[1,3,4,5,6,5,4,2,3,1,4,5,6,7,6,5,4,2],[1,3,4,5,6,5,4,3,1,7,8,7],[1,3,4,5,6,5,4,3,1,7],
+[1,3,4,5,6,5,4,3],[1,3,4,5,6,7,6,5,4,3,1,8],[1,4,2,3,1,4,5,4,6,7,8,7,6,5,4,2,3,1,4,3,5,4],
+[1,4,2,3,4,5,6,5,7,6,5,4,2,3,1,4,5,6],[1,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5],[1,4,2,3,4,5,6,7,6,5,4,2,3,1,4,8],
+[1,4,2,3,4,5,6,7,6,5,4,2,3,4],[1,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5],[1,4,2,3,4,5,6,7,8,7,6,5,4,2,3,4],
+[1,5,4,2,3,4,5,6,5,4,2,3,1,4,5,7,8,7],[1,5,4,2,3,4,5,6,5,4,2,3,1,4,5,7],[1,5,4,2,3,4,5,6,5,4,2,3,4,5],
+[1,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5,6],[1,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5,8],
+[2,3,1,4,2,3,1,4,5,6,7,6,5,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],[2,3,1,4,5,6,5,4,2,3,1,4,5,6,7,6,5,4,3,1],
+[2,3,1,4,5,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],[2,3,1,4,5,6,7,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],
+[2,3,4,3,1,5,6,7,6,5,4,2,3,1,4,3,5,6,7,8,7,6,5,4,2,3,1,4],[2,3,4,5,4,6,5,4,2,3,4,5],
+[2,3,4,5,6,5,4,2,3,4],[2,3,4,5,6,5,4,2,3,7,8,7],[2,3,4,5,6,5,4,2,3,7],
+[2,3,4,5,6,5,4,2],[2,3,4,5,6,7,6,5,4,2,3,4],[2,3,4,5,6,7,6,5,4,2,3,8],[2,3,4,5,6,7,6,5,4,2],
+[2,3,4,5,6,7,8,7,6,5,4,2,3,4],[2,4,2,3,1,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],
+[2,4,2,3,4,5,4,3],[2,4,2,3,4,5,6,5,4,3],[2,4,2,5,6,5],[2,4,2,5],[2,4,5,4,2,3,4,5,6,5,4,3],
+[2,4,5,4,2,6,7,6],[2,4,5,4,2,6],[2,4,5,6,5,4,2,7],[3,1,4,2,3,4,5,6,5,7,8,7,6,5,4,2,3,1,4,3,5,6],
+[3,1,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5],[3,1,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4],
+[3,1,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,3,5,6,7],[3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4],
+[3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6],[3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5],
+[3,1,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,3,5,6,8],[3,1,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5,6],
+[3,1,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],[3,4,2,3,4,5,4,2],[3,4,2,3,4,5,6,5,4,2],
+[3,4,2,3,4,5,6,7,6,5,4,2],[3,4,3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5],[3,4,3,5,6,5],
+[3,4,3,5],[3,4,5,4,2,3,4,5,6,5,4,2],[3,4,5,4,3,6],[3,4,5,4],[3,4],
+[4,2,3,1,4,5,4,2,3,1,4,3,5,6,7,8,7,6,5,4,3,1],[4,2,3,4,5,4,2,3,4,6,7,6],[4,2,3,4,5,4,2,3,4,6],
+[4,2,3,4,5,6,5,4,2,3,4,5],[4,2,3,4,5,6,5,4,2,3,4,7,8,7],[4,2,3,4,5,6,5,4,2,3,4,7],
+[4,2,3,4,5,6,7,6,5,4,2,3,4,8],[4,5],[5,6],[6,7],[7,8]/bracerightbig,
+one obtains 75 solutions for wiin (6.233) with ℓT(wi) = 3 andwiof typeA3
+1:
+wi∈/braceleftbig
+[1,2,3,1,4,2,5,4,6,5,4,2,7,8,7,6,5,4,2,3,1],[1,2,3,1,4,3,5,4,3,6,5,4,3,7,6,5,4,3,1,8,7,6,5,4,2,3,1],
+[1,2,3,1,4,5,4,6,5,4,2,3,4,5,7,6,5,4,2,3,1,4,5,8,7,6,5,4,2,3,1],[1,2,3,1,4,5,4,6,7,8,7,6,5,4,2,3,1],
+[1,2,3,4,2,3,1,6,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],
+[1,2,3,4,2,3,5,4,6,5,4,7,6,5,4,2,3,8,7,6,5,4,2,3,1],
+[1,2,3,4,2,5,4,2,3,4,6,5,4,3,7,6,5,4,2,3,4,8,7,6,5,4,2,3,1],
+[1,2,3,4,3,1,5,6,7,6,5,4,2,3,1,4,3,5,6,8,7,6,5,4,2,3,1],
+[1,2,3,4,3,5,4,3,6,5,4,3,7,6,5,4,2,3,1],[1,2,3,4,3,5,4,6,5,4,7,6,5,4,3,8,7,6,5,4,2,3,1],
+[1,2,3,4,5,4,6,5,4,2,3,4,5,7,6,5,4,2,3,4,5,8,7,6,5,4,2,3,1],[1,2,3,4,5,6,5,4,2,3,4,5,7,6,5,4,2,3,1],
+[1,2,3,4,5,6,5,4,3,1,7,6,5,4,2,3,1,4,3,5,4,6,5,8,7,6,5,4,2,3,1],
+[1,2,4,2,3,1,4,3,5,4,3,1,6,5,7,6,5,4,3,1,8,7,6,5,4,2,3,1,4],
+[1,2,4,2,3,1,5,6,5,7,6,5,4,2,3,1,4,3,5,4,2,6,7],[1,2,4,2,3,1,6,7,6,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8],
+[1,2,4,2,5,4,2,3,4,5,6,5,4,3,7,6,5,4,2,3,1,4,5],[1,2,4,2,5,6,5,4,2],[1,2,4,5,4,2,8],
+[1,3,1,4,2,3,4,5,4,6,5,4,7,6,5,4,2,8,7,6,5,4,2,3,1,4,3],
+[1,3,1,4,2,3,5,4,2,3,1,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,7,8],
+[1,3,1,4,2,3,7,6,5,4,2,3,1,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,7,6,8,7],
+[1,3,1,4,2,5,4,2,3,1,6,7,6,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,7,8],CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13 1
+[1,3,1,4,3,1,5,6,7,6,8,7,6,5,4,3,1],[1,3,1,4,5,4,6,5,4,2,3,4,5,6,7,6,5,4,2,8,7,6,5,4,2,3,1,4,3,5,6],
+[1,3,1,4,5,4,6,7,6,5,4,3,1],[1,3,4,3,5,4,3,1,7],[1,3,4,3,5,4,3,6,5,4,3,1,8],
+[1,3,4,5,4,2,3,1,4,5,6,5,4,2,7,8,7,6,5,4,2,3,1,4,5],[1,4,2,3,1,4,5,6,5,7,6,5,8,7,6,5,4,2,3,1,4],
+[1,4,2,3,4,5,4,2,3,4,6,7,8,7,6,5,4,2,3,1,4],[1,4,2,3,4,5,6,7,6,5,4,2,3,4,5,6,8,7,6,5,4,2,3,1,4],
+[1,4,5,4,2,3,1,4,5,6,5,4,2,3,1,4,5],[1,4,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1,4,5],
+[1,4,5,4,2,3,4,5,6,5,4,2,3,1,4,5,8],[1,4,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,5],
+[1,4,5,4,6,5,4,2,3,1,4,5,6],[1,4,8],[1,5,4,2,3,1,4,5,7],
+[1,5,4,2,3,4,5,6,5,4,2,3,4,5,7,6,5,4,2,3,1,4,5],[1,6,7,6,5,4,2,3,1,4,5,6,7],
+[2,3,1,4,2,5,6,5,4,2,3,4,5,6,7,6,5,4,3,8,7,6,5,4,2,3,1,4,3,5,6],
+[2,3,1,5,6,5,4,2,3,1,4,3,5,6,7,6,5,4,2,3,1,4,3,5,6],
+[2,3,4,2,3,1,4,6,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],[2,3,4,2,3,5,4,2,3,6,7,6,8,7,6,5,4,2,3],
+[2,3,4,2,3,5,4,3,6,5,4,2,3,7,8,7,6,5,4,2,3],[2,3,4,2,3,5,4,6,5,4,7,6,5,4,2,3,8,7,6,5,4,2,3],
+[2,3,4,2,3,5,6,7,6,5,4,2,3],[2,3,4,3,5,6,5,7,8,7,6,5,4,2,3],[2,3,4,5,4,6,5,4,2,3,8],
+[2,3,4,5,6,5,4,2,3,4,5,7,6,5,4,2,3],[2,4,2,3,1,4,5,4,3,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8],
+[2,4,2,3,1,4,5,6,7,6,5,4,3,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8],[2,4,2,3,4,5,4,3,6,5,4,2,3,4,8],
+[2,4,2,3,4,5,4,3,6,7,6,8,7,6,5,4,2,3,4],[3,1,4,2,3,4,5,6,7,6,5,4,2,3,4,5,6,8,7,6,5,4,2,3,1,4,3],
+[3,1,4,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,3,5],[3,1,4,5,4,6,5,4,2,3,1,4,3,5,6],
+[3,1,4,5,4,8,7,6,5,4,2,3,1,4,3,5,6,7,8],[3,1,4,6,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],
+[3,1,5,6,5,7,6,5,4,2,3,1,4,3,5,6,7],[3,1,6,5,4,2,3,1,4,3,5,6,8],
+[3,1,6,5,4,2,3,4,5,6,7,6,5,4,2,3,4,5,6,8,7,6,5,4,2,3,1,4,3,5,6],[3,1,7,8,7,6,5,4,2,3,1,4,3,5,6,7,8],
+[3,4,2,3,1,5,4,2,3,1,4,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,7,8],[3,4,3,5,4,3,7,8,7],
+[3,4,5,4,6,5,4,3,8],[4,2,3,1,5,4,2,3,1,4,3,5,6,7,6,8,7,6,5,4,2,3,1,4,3,5,4],
+[4,2,3,4,5,4,2,3,4,6,5,4,2,3,4],[4,2,3,4,5,4,2,3,4,6,7,8,7,6,5,4,2,3,4],[4,2,3,4,6],
+[4,5,4,2,3,4,5,6,5,4,2,3,4,5,8],[5,4,2,3,4,5,7,8,7],[5,4,3,1,6,5,4,2,3,1,4,3,5,4,6,5,8],[5,6,5,4,2,3,4,5,6]/bracerightbig
+,
+one obtains 165 solutions for wiin (6.233) with ℓT(wi) = 3 andwiof typeA1∗A2:
+wi∈/braceleftbig[1,3,1,4,5,4,2,3,4,5,6,5,4,2,7,8,7,6,5,4,2,3,1,4,5],
+[1,2,3,1,4,2,5,4,2,6,5,4,2,7,8,7,6,5,4,2,3,1,4,3,5],
+[1,2,3,1,4,2,5,4,6,5,4,2,3,4,5,7,6,5,4,3,1,8,7,6,5,4,2,3,1],
+[1,2,3,1,4,3,5,4,6,5,4,3,7,6,5,4,3,8,7,6,5,4,2,3,1,4,3,5,6],
+[1,2,3,1,4,3,5,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4,3],[1,2,3,1,4,3,5,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1],
+[1,2,3,1,4,5,4,6,5,4,7,6,5,4,2,3,1,4,3,5,6],[1,2,3,1,4,5,4,6,5,4,7,8,7,6,5,4,2,3,1,4,3,5,6],
+[1,2,3,1,4,5,6,5,4,2,3,4,5,7,6,5,4,2,3,4,5,8,7,6,5,4,2,3,1],[1,2,3,1,4,5,6,5,7,8,7,6,5,4,2,3,1,4,3],
+[1,2,3,4,2,3,5,4,6,5,4,7,6,5,4,2,3,4,5,8,7,6,5,4,2,3,1],[1,2,3,4,2,3,5,4,6,5,7,6,5,4,2,3,4,8,7,6,5,4,2,3,1],
+[1,2,3,4,2,5,4,2,3,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1],[1,2,3,4,2,5,4,2,6,5,4,2,7,6,5,4,2,3,1,4,5],
+[1,2,3,4,2,5,4,2,6,5,4,2,7,8,7,6,5,4,2,3,1,4,5],[1,2,3,4,2,5,4,2,6,5,7,8,7,6,5,4,2,3,1],
+[1,2,3,4,2,5,4,2,6,7,8,7,6,5,4,2,3,1,4],[1,2,3,4,2,5,4,6,5,4,2,3,4,5,7,6,5,4,3,8,7,6,5,4,2,3,1],
+[1,2,3,4,2,5,4,6,5,4,2,7,6,5,4,2,3,1,8],[1,2,3,4,3,1,5,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],
+[1,2,3,4,3,5,4,3,6,5,4,3,7,6,5,4,2,3,1,4,5],[1,2,3,4,3,5,4,6,5,4,3,7,6,5,4,2,3,1,4],
+[1,2,3,4,3,5,4,6,5,4,3,7,6,5,4,2,3],[1,2,3,4,3,5,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4],
+[1,2,3,4,5,4,2,3,4,6,5,7,6,5,4,2,3,4,5,8,7,6,5,4,2,3,1],
+[1,2,3,4,5,4,2,3,4,6,5,7,6,5,4,2,3,4,8,7,6,5,4,2,3,1,4],
+[1,2,3,4,5,4,6,5,7,8,7,6,5,4,2,3,1],[1,2,3,4,5,4,6,7,6,5,4,2,3,1,8],
+[1,2,3,4,5,6,5,4,3,1,7,6,5,4,2,3,1,4,3,5,6],[1,2,3,4,5,6,5,7,8,7,6,5,4,2,3],
+[1,2,4,2,3,1,4,3,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,7,8],
+[1,2,4,2,3,1,4,5,6,5,7,6,5,4,2,3,1,4,3,5,6,7,8,7,6,5,4,2,3],
+[1,2,4,2,3,1,6,7,6,5,4,2,3,1,4,3,5,4,2,6,5,7,6],[1,2,4,2,3,4,5,4,2,3,4,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4],132 C. KRATTENTHALER AND T. W. M ¨ULLER
+[1,2,4,2,3,4,5,4,3,6,5,7,6,5,4,2,3,1,4],[1,2,4,2,3,4,5,4,3,6,5,7,6,5,8,7,6,5,4,2,3,1,4],
+[1,2,4,2,3,4,5,4,3,6,7,6,5,4,2,3,1,4,8],[1,2,4,2,3,4,5,4,3,6,7,6,5,4,2,3,4],
+[1,2,4,2,3,4,5,4,3,6,7,8,7,6,5,4,2,3,4],[1,2,4,2,3,4,5,4,6,5,4,3,7,6,5,4,2,3,1,4,5],
+[1,2,4,2,3,4,5,4,6,5,4,3,7,6,5,4,2,3,4],[1,2,4,2,3,4,5,6,5,7,6,5,4,3,1],[1,2,4,5,4,2,3,4,5,6,5,4,3,1,8],
+[1,2,4,5,4,2,3,4,5,6,7,6,8,7,6,5,4,3,1],[1,2,4,5,4,2,6],[1,2,4,5,4,6,5,4,2,3,4,5,6,7,6,5,4,3,1],
+[1,3,1,4,2,3,4,5,4,6,5,4,7,6,5,4,2,8,7,6,5,4,2,3,1,4,3,5,6],
+[1,3,1,4,2,3,4,5,4,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4,3,5],
+[1,3,1,4,2,3,4,5,4,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4],
+[1,3,1,4,2,3,4,5,6,7,6,5,4,2,3,1,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],
+[1,3,1,4,2,5,4,2,3,1,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,7,6,8,7],[1,3,1,4,3,1,5,4,6,7,8,7,6,5,4,3,1],
+[1,3,1,4,3,5,4,3,1,6,5,4,3,1,7,8,7],[1,3,1,4,3,5,4,3,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3],
+[1,3,1,4,3,5,4,3,6,5,4,3,1],[1,3,1,4,5,4,2,3,1,4,5,6,7,6,8,7,6,5,4,2,3],
+[1,3,1,4,5,4,2,3,4,5,6,5,4,2,7,6,8,7,6,5,4,2,3,1,4,3,5],
+[1,3,1,4,5,4,2,3,4,5,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4,3,5,6],
+[1,3,1,4,5,4,2,3,4,5,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4,5],[1,3,1,4,5,6,5,4,3,1,7],
+[1,3,1,6,5,4,2,3,1,4,3,5,4,6,5,7,6,5,4,2,3],[1,3,4,2,3,4,5,4,2,3,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4],
+[1,3,4,2,3,4,5,4,2,6,5,7,6,5,8,7,6,5,4,2,3,1,4],[1,3,4,2,3,4,5,4,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4,5],
+[1,3,4,2,3,5,4,2,3,1,4,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8],
+[1,3,4,2,3,5,6,7,6,5,4,2,3,1,4,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8],
+[1,3,4,3,1,5,4,2,3,1,4,5,6,7,6,8,7,6,5,4,2,3,4],[1,3,4,3,5,4,2,3,1,4,5,6,5,4,2],
+[1,3,4,3,5,4,2,3,1,4,5,6,7,6,5,4,2],[1,3,4,3,5,4,3],[1,3,4,5,4,2,3,4,5,6,5,4,2,7,6,5,4,2,3,1,4,5,8],
+[1,3,4,5,4,2,3,4,5,6,5,4,2,7,6,8,7,6,5,4,2,3,1,4,5],[1,3,4,5,4,6,5,7,6,5,4,3,1],
+[1,3,4,5,4,6,7,6,5,4,3,1,8],[1,3,4,5,6,5,4,3,8],[1,4,2,3,1,4,3,5,4,3,1,6,5,7,6,5,8,7,6,5,4,2,3,1,4],
+[1,4,2,3,1,4,3,5,4,3,6,7,8,7,6,5,4,2,3,1,4],[1,4,2,3,1,4,3,5,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4],
+[1,4,2,3,1,4,3,5,6,7,8,7,6,5,4,3,1],[1,4,2,3,4,5,4,3,1,6,7,8,7,6,5,4,2,3,1,4,3,5,4],
+[1,4,2,3,4,5,6,5,7,6,5,4,2,3,4],[1,4,3,1,5,4,2,3,1,4,3,5,4,6,7,8,7,6,5,4,2,3,4],
+[1,4,5,4,2,3,4,5,6,5,4,2,3,1,4,5,7,8,7],[1,4,5,4,2,3,4,5,6,5,4,2,3,1,4,5,7],
+[1,4,5,4,2,3,4,5,6,5,4,2,3,4,5],[1,4,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5,8],
+[1,5,4,2,3,1,4,3,5,4,6,5,4,3,1],[1,5,4,2,3,1,4,3,5,4,6,7,8,7,6,5,4,3,1],[1,5,4,2,3,1,4,5,6,5,4,2,3,1,4,5,7,8,7],
+[1,5,4,2,3,4,5,6,5,4,2,3,4,5,8],[1,5,4,2,3,4,5],[1,6,5,4,2,3,4,5,6],
+[2,3,1,4,2,3,1,4,5,6,5,7,6,5,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],
+[2,3,1,4,2,3,4,5,4,6,5,4,7,6,5,4,3,8,7,6,5,4,2,3,1,4,3,5,6],
+[2,3,1,4,2,5,4,2,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],[2,3,1,4,2,5,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],
+[2,3,1,4,2,5,6,7,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],
+[2,3,1,4,3,1,7,6,5,4,2,3,1,4,3,5,4,2,6,5,7,6,8,7,6,5,4,2,3,1,4],
+[2,3,1,4,5,4,6,5,4,2,3,1,4,5,6,7,6,5,4,3,1],[2,3,1,4,5,4,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],
+[2,3,4,2,3,1,5,6,7,6,5,4,2,3,1,4,3,5,6,7,8,7,6,5,4,2,3,1,4],[2,3,4,2,3,5,4,2,3,6,5,7,6,5,8,7,6,5,4,2,3],
+[2,3,4,2,3,5,4,2,3,6,5,7,8,7,6,5,4,2,3],[2,3,4,2,3,5,4,2,6,7,8,7,6,5,4,2,3],
+[2,3,4,2,3,5,4,6,5,4,2,3,7,6,5,4,2,3,8],[2,3,4,2,3,5,4,6,5,4,2,3,7,6,8,7,6,5,4,2,3],
+[2,3,4,2,3,5,4,6,5,4,2,7,6,5,4,2,3],[2,3,4,2,3,5,4,6,5,4,2,7,8,7,6,5,4,2,3],[2,3,4,2,3,5,4,6,5,4,3,7,6,5,4,2,3],
+[2,3,4,2,5,4,2,6,5,4,2,3,4],[2,3,4,2,5,4,2,6,5,4,2,3,7,8,7],[2,3,4,2,5,4,2,6,5,7,8,7,6,5,4,2,3],
+[2,3,4,2,5,4,2,6,7,8,7,6,5,4,2,3,4],[2,3,4,2,5,4,6,5,4,2,3],[2,3,4,2,5,4,6,5,4,2,7,6,5,4,2,3,8],
+[2,3,4,2,5,4,6,5,4,7,6,5,4,2,3],[2,3,4,2,5,6,5,4,2,3,7],[2,3,4,3,5,4,6,5,4,3,7,6,5,4,2,3,4],
+[2,3,4,3,5,4,6,5,4,7,6,5,4,2,3],[2,3,4,3,5,6,7,6,5,4,2,3,8],[2,3,4,5,4,6,5,4,2,3,4,5,8],
+[2,3,4,5,4,6,5,4,2],[2,3,4,5,6,5,4,2,3,4,8],[2,3,4,5,6,5,7,6,8,7,6,5,4,2,3],
+[2,3,4,5,6,7,6,8,7,6,5,4,2,3,4],[2,4,2,3,1,4,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],
+[2,4,2,3,1,6,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],[2,4,2,3,1,7,6,5,4,2,3,1,4,3,5,6,7],CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13 3
+[2,4,2,3,1,8,7,6,5,4,2,3,1,4,3,5,6,7,8],[2,4,2,3,4,5,4,3,6,5,4,2,3,4,7,8,7],[2,4,2,3,4,5,4,3,6,5,4,2,3,4,7],
+[2,4,2,3,4,5,4,3,6,5,7,6,5,4,2,3,4],[2,4,2,3,4,5,4,3,6,7,6,5,4,2,3,4,8],[2,4,2,3,4,5,6,5,4,3,8],
+[2,4,2,5,4,6,5,4,2],[2,4,2,5,6,5,4,2,7],[2,4,2,5,8],[2,4,5,4,2,3,4,5,6,5,4,3,8],
+[2,4,5,4,2,7,8],[2,4,5,6,5,4,2,3,1,7,6,5,4,2,3,1,4,3,5,6,7],[3,1,4,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,3,5,6,7],
+[3,1,4,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6],[3,1,4,5,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6],
+[3,1,4,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],[3,1,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,3,5,4],
+[3,1,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,5],[3,1,5,6,5,4,2,3,1,4,3,5,4,6,5],
+[3,1,5,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,3,5,6,8],[3,1,5,6,5,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5,6],
+[3,1,6,5,4,2,3,1,4,5,6],[3,1,7,6,5,4,2,3,1,4,5,6,7],
+[3,4,2,3,4,5,4,2,3,1,4,8,7,6,5,4,2,3,1,4,3,5,6,7,8],[3,4,2,3,4,5,4,2,6,5,4,2,3,4,7,8,7],
+[3,4,2,3,4,5,4,2,7],[3,4,2,3,4,5,6,5,7,6,5,4,2],[3,4,3,1,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,5],
+[3,4,3,5,4,2,3,4,5,6,5,4,2],[3,4,3,6,7],[3,4,5,4,3,1,6,5,4,2,3,1,4,5,6],[3,4,7],
+[4,2,3,1,4,5,4,2,3,1,4,3,5,6,7,6,8,7,6,5,4,3,1],[4,2,3,4,5,6,5,4,2,3,4,5,8],[4,5,8],[4,7,8],
+[1,3,1,4,5,6,5,7,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3]/bracerightbig
+,
+one obtains 90 solutions for wiin (6.233) with ℓT(wi) = 3 andwiof typeA3:
+wi∈/braceleftbig[1,2,3,1,4,3,1,5,6,7,6,5,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],
+[1,2,3,1,4,3,5,4,6,5,7,6,5,4,3,1,8,7,6,5,4,2,3,1,4],[1,2,3,1,4,5,4,6,7,8,7,6,5,4,2,3,1,4,3,5,4],
+[1,2,3,1,4,5,4,6,7,8,7,6,5,4,2,3,1,4,3,5,6],[1,2,3,1,4,5,4,6,7,8,7,6,5,4,2,3,1,4,5],
+[1,2,3,1,4,5,6,5,4,7,6,5,4,2,3,1,4,3,5,6,8],[1,2,3,1,4,5,6,5,7,8,7,6,5,4,2,3,1,4,3,5,6],
+[1,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5],[1,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1,4],
+[1,2,3,4,2,3,5,4,6,5,7,6,5,4,2,3,8,7,6,5,4,2,3],[1,2,3,4,2,5,4,2,6,7,8,7,6,5,4,2,3],
+[1,2,3,4,2,5,4,6,5,4,2,7,6,5,4,2,3],[1,2,3,4,2,5,4,6,5,4,2,7,8,7,6,5,4,2,3],[1,2,3,4,5,4,6,7,6,5,4,2,3,1,4,5,8],
+[1,2,3,4,5,6,7,6,5,4,2,3,1,4,5],[1,2,3,4,5,6,7,6,5,4,2,3,1,4,8],[1,2,3,4,5,6,7,6,5,4,2,3,4],
+[1,2,3,4,5,6,7,6,5,4,2,3,8],[1,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5],
+[1,2,3,4,5,6,7,8,7,6,5,4,2,3,4],[1,2,4,2,3,1,4,3,5,4,3,1,6,7,8,7,6,5,4,3,1],
+[1,2,4,2,3,1,4,5,6,7,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3],[1,2,4,2,3,1,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8],
+[1,2,4,2,3,4,5,4,6,5,4,7,6,5,4,3,1],[1,2,4,2,3,4,5,4,6,7,6,5,4,3,1],[1,2,4,2,3,4,5,4,6,7,8,7,6,5,4,3,1],
+[1,2,4,2,3,4,5,6,7,6,5,4,3,1,8],[1,2,4,5,4,2,3,4,5,6,5,4,3,1,7,8,7],[1,2,4,5,4,2,3,4,5,6,5,4,3,1,7],
+[1,2,4,5,4,2,3,4,5,6,5,4,3],[1,2,4,5,4,2,3,4,5,6,5,7,6,5,4,3,1],[1,2,4,5,4,2,3,4,5,6,7,6,5,4,3,1,8],
+[1,3,1,4,5,4,2,3,1,4,5,6,5,7,6,5,8,7,6,5,4,2,3],[1,3,1,4,5,4,2,3,1,4,5,6,5,7,8,7,6,5,4,2,3],
+[1,3,1,4,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3,4],[1,3,1,4,5,6,5,4,2,3,1,4,5,6,7,6,5,4,2,3,8],
+[1,3,1,4,5,6,5,4,2,3,1,4,5,6,7,6,5,4,2],[1,3,1,4,5,6,5,4,2,3,1,4,5,6,7,6,8,7,6,5,4,2,3],
+[1,3,4,2,3,4,5,4,2,6,7,8,7,6,5,4,2,3,4],[1,3,4,5,4,2,3,1,4,5,6,5,4,2,7],
+[1,3,4,5,4,2,3,1,4,5,6,5,7,6,5,4,2],[1,3,4,5,4,2,3,4,5,6,5,4,2],
+[1,3,4,5,4,3,1,6,7],[1,3,4,5,4,3,6],[1,3,4,5,6,5,4,3,1,7,8],[1,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1,4,5],
+[1,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5,6],[1,4,2,3,4,5,6,7,6,5,4,2,3,1,4,5,8],[1,4,2,3,4,5,6,7,6,5,4,2,3,4,8],
+[1,5,4,2,3,1,4,5,6,7,6],[1,5,4,2,3,1,4,5,6],[1,5,4,2,3,4,5,6,5,4,2,3,1,4,5,7,8],[1,6,5,4,2,3,1,4,5,6,7],
+[2,3,1,4,2,3,4,5,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4],[2,3,1,4,5,6,5,4,2,3,1,4,5,6,7,6,5,4,3,1,8],
+[2,3,1,4,5,6,5,4,2,3,1,4,5,6,7,6,8,7,6,5,4,3,1],[2,3,1,4,5,6,5,4,2,3,4,5,6,7,6,5,4,3,1],[2,3,4,2,5,4,2,6,5,4,2],
+[2,3,4,2,5,6,5,4,2],[2,3,4,2,5,6,7,6,5,4,2],[2,3,4,3,1,5,6,7,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],
+[2,3,4,5,6,5,4,2,3,4,5],[2,3,4,5,6,5,4,2,3,4,7,8,7],[2,3,4,5,6,5,4,2,3,4,7],
+[2,3,4,5,6,5,4,2,3,7,8],[2,3,4,5,6,5,4,2,7],[2,3,4,5,6,7,6,5,4,2,3,4,8],[2,4,2,3,4,5,4,3,6],
+[2,4,2,3,4,5,4,6,5,4,3],[2,4,2,5,6],[2,4,5,4,2,6,7],[3,1,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6],
+[3,1,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5],[3,1,4,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,5],
+[3,1,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],[3,1,6,5,4,2,3,1,4,3,5,6,7,8,7],
+[3,1,6,5,4,2,3,1,4,3,5,6,7],[3,1,7,6,5,4,2,3,1,4,3,5,6,7,8],134 C. KRATTENTHALER AND T. W. M ¨ULLER
+[3,4,2,3,1,4,5,4,2,3,4,5,6,7,8,7,6,5,4,3,1],[3,4,2,3,4,5,4,2,6,7,6],[3,4,2,3,4,5,4,2,6],[3,4,2,3,4,5,4,6,5,4,2],
+[3,4,2,3,4,5,6,5,4,2,7],[3,4,3,5,6],[3,4,5],[4,2,3,4,5,4,2,3,4,6,7],
+[4,2,3,4,5,6,5,4,2,3,4,7,8],[4,2,3,4,5,6,5],[4,2,3,4,5],[5,4,2,3,4,5,6]/bracerightbig,
+one obtains 15 solutions for wiin (6.233) with ℓT(wi) = 4 andwiof typeA2
+1∗A2:
+wi∈/braceleftbig
+[1,2,3,1,4,5,4,6,5,4,2,3,4,5,7,6,5,4,2,3,4,5,8,7,6,5,4,2,3,1],
+[1,2,3,4,3,1,5,6,5,7,6,5,4,2,3,1,4,3,5,4,6,5,8,7,6,5,4,2,3,1],
+[1,2,3,4,5,4,6,5,4,2,3,4,5,7,6,5,4,2,3,1,4,5],
+[1,2,3,4,5,4,6,5,4,3,1,7,6,5,4,2,3,1,4,3,5,6],
+[1,3,1,4,2,3,4,5,6,5,7,6,5,4,2,3,1,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],
+[1,3,1,4,2,3,5,4,2,3,1,7,8,7,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1,7,6,8,7],
+[1,4,3,1,5,4,2,3,1,4,3,5,4,6,7,6,8,7,6,5,4,2,3,4],
+[1,4,5,4,2,3,1,4,5,6,5,4,2,3,1,4,5,7,8,7],
+[1,4,5,4,2,3,4,5,6,5,4,2,3,4,5,8],
+[2,4,2,3,1,4,5,4,3,6,7,6,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8],
+[2,4,2,3,1,4,6,7,6,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],
+[3,1,4,2,3,4,5,6,5,7,6,5,4,2,3,4,5,6,8,7,6,5,4,2,3,1,4,3,5,6],
+[3,1,5,6,5,4,2,3,1,4,3,5,4,6,5,8],
+[3,4,2,3,4,5,4,2,3,1,4,7,8,7,6,5,4,2,3,1,4,3,5,6,7,8],
+[4,2,3,4,5,4,2,3,4,6,5,4,2,3,4,7,8,7]/bracerightbig
+,(6.237)
+one obtains 45 solutions for wiin (6.233) with ℓT(wi) = 4 andwiof typeA1∗A3:
+wi∈/braceleftbig[1,2,3,1,4,2,5,4,2,6,5,4,2,7,8,7,6,5,4,2,3,1,4,5],
+[1,2,3,1,4,2,5,4,6,5,4,2,3,4,5,7,6,5,4,3,8,7,6,5,4,2,3,1],
+[1,2,3,1,4,3,1,5,6,5,7,6,5,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7],
+[1,2,3,1,4,3,5,4,6,5,4,7,6,5,4,3,8,7,6,5,4,2,3,1,4,3,5,6],
+[1,2,3,1,4,5,4,6,5,4,7,6,5,4,2,3,1,4,3,5,6,8],[1,2,3,4,2,3,5,4,6,5,4,7,6,5,4,2,3,8,7,6,5,4,2,3],
+[1,2,3,4,2,3,5,4,6,5,7,6,5,4,2,3,4,5,8,7,6,5,4,2,3,1],[1,2,3,4,2,5,4,2,3,4,6,5,4,7,6,5,4,3,8,7,6,5,4,2,3,1],
+[1,2,3,4,2,5,4,2,6,5,4,2,7,6,5,4,2,3,1,4,5,8],[1,2,3,4,3,5,4,6,5,4,3,7,6,5,4,2,3,1,4,5],
+[1,2,3,4,3,5,4,6,5,4,3,7,6,5,4,2,3,4],[1,2,3,4,5,6,5,4,2,3,4,5,7,6,5,4,2,3],
+[1,2,4,2,3,1,4,5,6,5,7,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,2,3],[1,2,4,2,3,4,5,4,3,6,7,6,5,4,2,3,4,8],
+[1,2,4,5,4,2,3,4,5,6,5,4,3,8],[1,3,1,4,2,3,4,5,4,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4,3,5,6],
+[1,3,1,4,2,3,4,5,4,6,5,7,6,5,4,2,8,7,6,5,4,2,3,1,4,5],[1,3,1,4,5,4,2,3,1,4,5,6,7,6,8,7,6,5,4,2,3,4],
+[1,3,4,2,3,5,4,2,3,1,4,6,7,6,8,7,6,5,4,2,3,1,4,3,5,4,2,6,7,8],[1,3,4,3,5,4,2,3,1,4,5,6,5,4,2,7],
+[1,3,4,3,5,4,2,3,4,5,6,5,4,2],[1,4,2,3,1,4,3,5,4,3,6,5,7,6,5,8,7,6,5,4,2,3,1,4],
+[1,4,2,3,4,5,4,2,3,4,6,5,7,6,5,8,7,6,5,4,2,3,1,4],[1,4,2,3,4,5,4,2,3,4,6,7,8,7,6,5,4,2,3,4],
+[1,4,5,4,2,3,4,5,6,5,4,2,3,1,4,5,7,8],[1,5,4,2,3,1,4,3,5,4,6,5,4,3,1,7,8,7],
+[2,3,1,4,2,5,6,5,4,2,3,1,4,5,6,7,6,8,7,6,5,4,3,1],[2,3,1,4,5,4,6,5,4,2,3,1,4,5,6,7,6,5,4,3,1,8],
+[2,3,1,4,5,4,6,5,4,2,3,4,5,6,7,6,5,4,3,1],[2,3,4,2,3,1,5,6,7,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],
+[2,3,4,2,3,5,4,2,3,6,5,7,6,8,7,6,5,4,2,3],[2,3,4,2,3,5,4,2,6,5,7,8,7,6,5,4,2,3],
+[2,3,4,2,3,5,4,6,5,4,7,6,5,4,2,3],[2,3,4,2,5,4,2,6,5,4,2,3,4,7,8,7],
+[2,3,4,5,6,5,4,2,3,4,5,8],[2,4,2,3,1,4,5,4,8,7,6,5,4,2,3,1,4,3,5,6,7,8],
+[2,4,2,3,1,5,6,5,7,6,5,4,2,3,1,4,3,5,6,7],[2,4,2,3,4,5,4,3,6,5,4,2,3,4,7,8],[2,4,2,3,4,5,4,6,5,4,3,8],
+[3,1,4,2,3,4,5,6,7,6,5,4,2,3,4,5,6,8,7,6,5,4,2,3,1,4],[3,1,4,5,4,2,3,4,5,6,7,6,8,7,6,5,4,2,3,1,4,5],CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13 5
+[3,1,4,5,4,2,3,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6,7],[3,1,4,5,4,6,5,4,2,3,1,4,3,5,6,7,8,7],
+[3,1,4,5,4,6,5,4,2,3,1,4,5,6],[3,4,2,3,1,4,5,4,2,3,4,5,6,7,6,8,7,6,5,4,3,1]/bracerightbig
+,(6.238)
+one obtains 5 solutions for wiin (6.233) with ℓT(wi) = 4 andwiof typeA2
+2:
+wi∈/braceleftbig
+[1,2,3,4,2,5,4,2,3,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4],
+[1,2,4,2,3,4,5,4,3,6,5,7,6,5,4,2,3,4],
+[1,3,1,4,5,4,2,3,4,5,6,5,4,2,7,6,8,7,6,5,4,2,3,1,4,5],
+[2,3,1,4,2,5,4,6,5,4,2,3,1,4,5,6,7,8,7,6,5,4,3,1],
+[2,3,4,2,3,5,4,6,5,4,2,7,6,5,4,2,3,8]/bracerightbig
+,(6.239)
+one obtains 18 solutions for wiin (6.233) with ℓT(wi) = 4 andwiof typeA4:
+wi∈/braceleftbig[1,2,3,1,4,3,5,4,6,5,7,6,5,4,3,8,7,6,5,4,2,3,1,4],[1,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1,4,3,5,6],
+[1,2,3,4,2,5,4,2,6,5,7,8,7,6,5,4,2,3],[1,2,3,4,2,5,4,2,6,7,8,7,6,5,4,2,3,4],
+[1,2,3,4,2,5,4,6,5,4,2,7,6,5,4,2,3,8],[1,2,3,4,5,6,7,6,5,4,2,3,1,4,5,8],[1,2,4,2,3,4,5,4,6,5,7,6,5,4,3,1],
+[1,2,4,2,3,4,5,4,6,7,6,5,4,3,1,8],[1,2,4,5,4,2,3,4,5,6,5,4,3,1,7,8],
+[1,3,1,4,5,4,2,3,1,4,5,6,5,7,6,8,7,6,5,4,2,3],[1,4,2,3,1,4,3,5,4,6,7,8,7,6,5,4,3,1],
+[1,5,4,2,3,4,5,6],[2,3,4,2,5,4,6,5,4,2],[2,3,4,2,5,6,5,4,2,7],[2,3,4,5,6,5,4,2,3,4,7,8],
+[2,4,2,3,1,7,6,5,4,2,3,1,4,3,5,6,7,8],[3,1,6,5,4,2,3,1,4,5,6,7],[3,4,2,3,4,5,4,2,6,7]/bracerightbig,(6.240)
+and one obtains 5 solutions for wiin (6.233) with ℓT(wi) = 4 andwiof typeD4:
+wi∈/braceleftbig
+[1,2,3,1,4,5,6,7,8,7,6,5,4,2,3,1,4,5],[1,2,3,4,5,6,7,6,5,4,2,3,4,8],
+[1,5,4,2,3,1,4,5,6,7],[3,1,6,5,4,2,3,1,4,3,5,6,7,8],[4,2,3,4,5,6]/bracerightbig
+,(6.241)
+where{s1,s2,s3,s4,s5,s6,s7,s8}isasimplesystemofgeneratorsof E8,correspondingto
+the Dynkin diagram displayed in Figure 6, and each of them gives rise to m/2 elements
+of FixNCm(E8)(φp) sinceiranges from 1 to m/2. There are no solutions for wiin (6.233)
+withwiof typeA4
+1.
+Letting the computer find all solutions in cases (iii)–(v) would take ye ars. However,
+the number of these solutions can be computed from our knowledge of the solutions
+in Case (ii) according to type, if this information is combined with the de composition
+numbers in the sense of [17, 18, 20] (see the end of Section 2) and some elementary
+(multiset) permutation counting. The decomposition numbers for A2,A3,A4, andD4
+of which we make use can be found in the appendix of [18].
+To begin with, the number of solutions of (6.234) with ℓ1=ℓ2= 1 is equal to
+n1,1:= 150·2+100·NA2(A1,A1) = 600,
+since an element of type A2
+1can be decomposed in two ways into a product of two
+elements of absolute length 1, while for an element of type A2this can be done in
+NA2(A1,A1) = 3 ways. Similarly, the number of solutions of (6.234) with ℓ1= 2 and
+ℓ2= 1 is equal to
+n2,1:= 75·3+165·(1+NA2(A1,A1))+90·NA3(A2,A1) = 1425,
+the number of solutions of (6.234) with ℓ1= 3 andℓ2= 1 is equal to
+n3,1:= 15·(2+NA2(A1,A1))+45·(1+NA3(A2,A1))+5·(2NA2(A1,A1))
++18·(NA4(A3,A1)+NA4(A1∗A2,A1))+5·(ND4(A3,A1)+ND4(A3
+1,A1)) = 660,136 C. KRATTENTHALER AND T. W. M ¨ULLER
+the number of solutions of (6.234) with ℓ1=ℓ2= 2 is equal to
+n2,2:= 15·(2+2NA2(A1,A1))+45·(2NA3(A2,A1))+5·(2+NA2(A1,A1)2)
++18·(NA4(A2,A2)+NA4(A2
+1,A2
+1)+2NA4(A2,A2
+1))
++5·(ND4(A2,A2)+2ND4(A2,A2
+1)) = 1195,
+the number of solutions of (6.235) with ℓ1=ℓ2=ℓ3= 1 is equal to
+n1,1,1:= 75·3!+165·(3NA2(A1,A1))+90NA3(A1,A1,A1) = 3375,
+the number of solutions of (6.235) with ℓ1= 2 andℓ2=ℓ3= 1 is equal to
+n2,1,1:= 15·(2+NA2(A1,A1)+2·2·NA2(A1,A1))+45·(2NA3(A2,A1)+NA3(A1,A1,A1))
++5·(2NA2(A1,A1)+2NA2(A1,A1)2)+18·(NA4(A2,A1,A1)+NA4(A2
+1,A1,A1))
++5·(ND4(A2,A1,A1)+ND4(A2
+1,A1,A1)) = 2850,
+and the number of solutions of (6.236) is equal to
+n1,1,1,1:= 15·(12NA2(A1,A1))+45·(4NA3(A1,A1,A1))+5·(6NA2(A1,A1)2)
++18·NA4(A1,A1,A1,A1)+5·ND4(A1,A1,A1,A1) = 6750.
+In total, we obtain
+1+(45+150+100+75+165+90+15+45+5+18+5)m
+2+(n1,1+2n2,1+2n3,1+n2,2)/parenleftbiggm/2
+2/parenrightbigg
++(n1,1,1+3n2,1,1)/parenleftbiggm/2
+3/parenrightbigg
++n1,1,1,1/parenleftbiggm/2
+4/parenrightbigg
+=(5m+4)(3m+2)(5m+2)(15m+4)
+64
+elements in Fix NCm(E8)(φp), which agrees with the limit in (6.198j).
+Finally, we turn to (6.198l). By Remark 3, the only choices for h2andm2to be
+considered are h2= 1 andm2= 7,h2= 2 andm2= 8,h2= 2 andm2= 7,h2= 2
+andm2= 4, respectively h2=m2= 2. These correspond to the choices p= 30m/7,
+p= 15m/8,p= 15m/7,p= 15m/4, respectively 15 m/2, out of which only p= 15m/8
+has not yet been discussed and belongs to the current case. The c orresponding action
+ofφpis given by
+φp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (∗;c2wm
+8+1c−2,c2wm
+8+2c−2,...,c2wmc−2,cw1c−1,...,cw m
+8c−1/parenrightbig
+,
+so that we have to solve
+t1(c13t1c−13)(c11t1c−11)(c9t1c−9)(c7t1c−7)(c5t1c−5)(c3t1c−3)(ct1c−1) =c(6.242)
+fort1withℓT(t1) = 1. A computation with the help of Stembridge’s Maplepackage
+coxeter [36] finds no solution. Hence, the left-hand side of (3.3) is equal to 1 , as
+required.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13 7
+7.Cyclic sieving II
+In this section we present the second cyclic sieving conjecture due to Bessis and
+Reiner [9, Conj. 6.5].
+Letψ:NCm(W)→NCm(W) be the map defined by
+(w0;w1,...,w m)/mapsto→/parenleftbig
+cwmc−1;w0,w1,...,w m−1/parenrightbig
+. (7.1)
+Form= 1, we have w0=cw−1
+1, so that this action reduces to the inverse of the
+Kreweras complement Kc
+idas defined by Armstrong [2, Def. 2.5.3].
+It is easy to see that ψ(m+1)hacts as the identity, where his the Coxeter number of
+W(see (8.1) below). By slight abuse of notation as before, let C2be the cyclic group
+of order (m+1)hgenerated by ψ.
+Given these definitions, we are now in the position to state the secon d cyclic sieving
+conjecture of Bessis and Reiner. By the results of [19] and of this p aper, it becomes the
+following theorem.
+Theorem 33. For an irreducible well-generated complex reflection group Wand any
+m≥1, the triple (NCm(W),Catm(W;q),C2), whereCatm(W;q)is theq-analogue of
+the Fuß–Catalan number defined in (3.2), exhibits the cyclic sieving phenomenon.
+By definition of the cyclic sieving phenomenon, we have to prove that
+|FixNCm(W)(ψp)|= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h, (7.2)
+for allpin the range 0 ≤p<(m+1)h.
+8.Auxiliary results II
+This section collects several auxiliary results which allow us to reduce the problem of
+proving Theorem 33, respectively the equivalent statement (7.2), for the 26 exceptional
+groups listed in Section 2 to a finite problem. While Lemmas 35 and 36 cov er special
+choices of the parameters, Lemmas 34 and 37 afford an inductive pr ocedure. More
+precisely, if we assume that we have already verified Theorem 33 for all groups of
+smaller rank, then Lemmas 34 and 37, together with Lemmas 35 and 3 8, reduce the
+verification of Theorem 33 for the group that we are currently con sidering to a finite
+problem; see Remark4. The final lemma ofthis section, Lemma 39, dis poses of complex
+reflection groups with a special property satisfied by their degree s.
+Letp=a(m+1)+b, 0≤b<m+1. We have
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (ca+1wm−b+1c−a−1;ca+1wm−b+2c−a−1,...,ca+1wmc−a−1,
+caw0c−a,...,cawm−bc−a/parenrightbig
+.(8.1)
+Lemma 34. It suffices to check (7.2)forpa divisor of (m+1)h. More precisely, let pbe
+a divisor of (m+1)h, and letkbe another positive integer with gcd(k,(m+1)h/p) = 1,
+then we have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πikp/(m+1)h (8.2)
+and
+|FixNCm(W)(ψp)|=|FixNCm(W)(ψkp)|. (8.3)138 C. KRATTENTHALER AND T. W. M ¨ULLER
+Proof.For (8.3), this follows in the same way as (5.3) in Lemma 26.
+For (8.2), we must argue differently than in Lemma 26. Let us write ζ=e2πip/(m+1)h.
+For a given group W, we writeS1(W) for the set of all indices isuch thatζdi−h= 1,
+and we write S2(W) for the set of all indices isuch thatζdi= 1. By the rule of de
+l’Hospital, we have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h=
+
+0 if |S1(W)|>|S2(W)|,/producttext
+i∈S1(W)(mh+di)/producttext
+i∈S2(W)di/producttext
+i/∈S1(W)(1−ζdi−h)
+/producttext
+i/∈S2(W)(1−ζdi),if|S1(W)|=|S2(W)|.
+(8.4)
+Since, by Theorem 25, Catm(W;q) is a polynomial in q, the case |S1(W)|<|S2(W)|
+cannot occur.
+We claim that, for the case where |S1(W)|=|S2(W)|, the factors in the quotient of
+products/producttext
+i/∈S1(W)(1−ζdi−h)/producttext
+i/∈S2(W)(1−ζdi)
+cancel pairwise. If we assume the correctness of the claim, it is obv ious that we get
+the same result if we replace ζbyζk, where gcd( k,(m+1)h/p) = 1, hence establishing
+(8.2).
+In order to see that our claim is indeed valid, we proceed in a case-by- case fash-
+ion, making appeal to the classification of irreducible well-generated complex reflection
+groups, which werecalled inSection2. Firstofall, since dn=h, thesetS1(W)isalways
+non-empty as it contains the element n. Hence, if we want to have |S1(W)|=|S2(W)|,
+the setS2(W) must be non-empty as well. In other words, the integer ( m+ 1)h/p
+must divide at least one of the degrees d1,d2,...,d n. In particular, this implies that,
+for each fixed reflection group Wof exceptional type, only a finite number of values of
+(m+1)h/phas to be checked. Writing Mfor (m+1)h/p, what needs to be checked is
+whether the multisets (that is, multiplicities of elements must be taken into account)
+{(di−h) modM:i /∈S1(W)}and{dimodM:i /∈S2(W)}
+are the same. Since, for a fixed irreducible well-generated complex r eflection group,
+thereisonlyafinitenumber ofpossibilities for M, thisamountstoaroutineverification.
+/square
+Lemma 35. Letpbe a divisor of (m+ 1)h. Ifpis divisible by m+ 1, then(7.2)is
+true.
+Proof.According to (8.1), the action of ψponNCm(W) is described by
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (cp/(m+1)w0c−p/(m+1);cp/(m+1)w1c−p/(m+1),...,cp/(m+1)wmc−p/(m+1)/parenrightbig
+.
+Hence, if (w0;w1,...,w m) is fixed by ψp, then each individual wimust be fixed under
+conjugation by cp/(m+1).
+Using the notation W′= Cent W(cp/(m+1)), the previous observation means that wi∈
+W′,i= 1,2,...,m. By the theorem of Springer cited in the proof of Lemma 27 and by
+(5.5), the tuples ( w0;w1,...,w m) fixed byψpare in fact identical with the elements ofCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13 9
+NCm(W′), which implies that
+|FixNCm(W)(ψp)|=|NCm(W′)|. (8.5)
+Application of Theorem 1 with Wreplaced by W′and of the “limit rule” (5.4) then
+yields that
+|NCm(W′)|=/productdisplay
+1≤i≤n
+(m+1)h
+p|dimh+di
+di= Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h. (8.6)
+Combining (8.5) and (8.6), we obtain (7.2). This finishes the proof of t he lemma. /square
+Lemma 36. Equation (7.2)holds for all divisors pofm+1.
+Proof.We have
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h=/braceleftBigg
+0 ifp<m+1,
+m+1 ifp=m+1.
+Here, the first case follows from (8.4) and the fact that we have S1(W)⊇ {n}and
+S2(W) =∅ifp|(m+1) andp<m+1.
+On the other hand, if ( w0;w1,...,w m) is fixed by ψp, then, because of the action
+(8.1), we must have w0=wp=···=wm−p+1andw0=cwm−p+1c−1. In particular,
+w0∈CentW(c). By the theorem of Springer cited in the proof of Lemma 27, the
+subgroup Cent W(c) is itself a complex reflection group whose degrees are those degre es
+ofWthat are divisible by h. The only such degree is hitself, hence Cent W(c) is the
+cyclic group generated by c. Moreover, by (5.5), we obtain that w0=εorw0=c.
+Ifp=m+ 1, the set Fix NCm(W)(ψp) consists of the m+ 1 elements ( w0;w1,...,w m)
+obtained by choosing wi=cfor a particular ibetween 0 and m, all otherwj’s being
+equal toε. Ifp<m+1, then there is no element in Fix NCm(W)(ψp). /square
+Lemma 37. LetWbe an irreducible well-generated complex reflection group o f rank
+n, and letp=m1h1be a divisor of (m+1)h, wherem+1 =m1m2andh=h1h2. We
+assume that gcd(h1,m2) = 1. Suppose that Theorem 33 has already been verified for
+all irreducible well-generated complex reflection groups w ith rank< n. Ifh2does not
+divide all degrees di, then equation (7.2)is satisfied.
+Proof.Let us write h1=am2+b, with 0 ≤b < m 2. The condition gcd( h1,m2) = 1
+translates into gcd( b,m2) = 1. From (8.1), we infer that
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (ca+1wm−m1b+1c−a−1;ca+1wm−m1b+2c−a−1,...,ca+1wmc−a−1,
+caw0c−a,...,cawm−m1bc−a/parenrightbig
+.(8.7)
+Supposing that ( w0;w1,...,w m) is fixed by ψp, we obtain the system of equations
+wi=ca+1wi+m−m1b+1c−a−1, i= 0,1,...,m 1b−1,
+wi=cawi−m1bc−a, i=m1b,m1b+1,...,m,
+which, after iteration, implies in particular that
+wi=cb(a+1)+(m2−b)awic−b(a+1)−(m2−b)a=ch1wic−h1, i= 0,1,...,m.140 C. KRATTENTHALER AND T. W. M ¨ULLER
+It is at this point where we need gcd( b,m2) = 1. The last equation shows that each
+wi,i= 0,1,...,m, lies in Cent W(ch1). By the theorem of Springer cited in the proof of
+Lemma 27, this centraliser subgroup is itself a complex reflection gro up,W′say, whose
+degrees are those degrees of Wthat are divisible by h/h1=h2. Since, by assumption,
+h2does not divide alldegrees,W′has rank strictly less than n. Again by assumption,
+we know that Theorem 33 is true for W′, so that in particular,
+|FixNCm(W′)(ψp)|= Catm(W′;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h.
+The arguments above together with (5.5) show that Fix NCm(W)(ψp) = Fix NCm(W′)(ψp).
+On the other hand, it is straightforward to see that
+Catm(W;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h= Catm(W′;q)/vextendsingle/vextendsingle
+q=e2πip/(m+1)h.
+This proves (7.2) for our particular p, as required. /square
+Lemma 38. LetWbe an irreducible well-generated complex reflection group o f rank
+n, and letp=m1h1be a divisor of (m+1)h, wherem+1 =m1m2andh=h1h2. We
+assume that gcd(h1,m2) = 1. Ifm2>nthen
+FixNCm(W)(ψp) =∅.
+Proof.Let us suppose that ( w0;w1,...,w m)∈FixNCm(W)(ψp) and that there exists a
+j≥1 such that wj/ne}ationslash=ε. By (8.7), it then follows for such a jthat alsowk/ne}ationslash=εfor all
+k≡j−lm1b(modm+ 1), where, as before, bis defined as the unique integer with
+h1=am2+band 0≤b<m 2. Since, by assumption, gcd( b,m2) = 1, there are exactly
+m2suchk’s which are distinct mod m+1. However, this implies that the sum of the
+absolute lengths of the wi’s, 0≤i≤m, is at least m2> n, a contradiction. This
+leaves as only possibility ( w0;w1,...,w m) = (c;ε,...,ε). However, this is clearly not
+an element of Fix NCm(W)(ψp) unlesspis divisible by m+1. This is impossible since
+p
+m+1=m1h1
+m1m2=h1
+m2
+is not an integer by our hypotheses. /square
+Remark 4.(1) If we put ourselves in the situation of the assumptions of Lemma 37,
+then we may conclude that equation (7.2) only needs to be checked f or pairs (m2,h2)
+subject to the following restrictions:
+m2≥2,gcd(h1,m2) = 1,andh2divides all degrees of W. (8.8)
+Indeed, Lemmas 35 and 37 together imply that equation (7.2) is alway s satisfied except
+ifm2≥2,h2divides all degrees of W, and gcd(h1,m2) = 1.
+(2) Still putting ourselves in the situation of Lemma 37, if m2> nandm2h2does
+not divide any of the degrees of W, then equation (7.2) is satisfied. Indeed, Lemma 38
+says that in this case the left-hand side of (7.2) equals 0, while it is obv ious that in this
+case the right-hand side of (7.2) equals 0 as well.
+(3)It shouldbeobserved that thisleaves afinitenumber of choices form2to consider,
+whence a finite number of choices for ( m1,m2,h1,h2). Altogether, there remains a finite
+number of choices for p=h1m1to be checked.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 14 1
+Lemma 39. LetWbe an irreducible well-generated complex reflection group o f rankn
+with the property that di|hfori= 1,2,...,n. Then Theorem 33is true for this group
+W.
+Proof.By Lemma 34, we may restrict ourselves to divisors pof (m+1)h.
+Supposethat e2πip/(m+1)hisadi-throotofunityforsome i. Inotherwords, ( m+1)h/p
+dividesdi. Sincediis a divisor of hby assumption, the integer ( m+1)h/palso divides
+h. But this is equivalent to saying that m+ 1 divides p, and equation (7.2) holds by
+Lemma 35.
+Now assume that ( m+1)h/pdoes not divide any of the di’s. In this case, it follows
+from (8.4) and the fact that we have S1(W)⊇ {n}andS2(W) =∅that the right-
+hand side of (7.2) equals 0. Inspection of the classification of all irre ducible well-
+generated complex reflection groups, which we recalled in Section 2, reveals that all
+groups satisfying the hypotheses of the lemma have rank n≤2. Except for the groups
+contained in the infinite series G(d,1,n) andG(e,e,n) for which Theorem 2 has been
+established in [19], these are the groups G5,G6,G9,G10,G14,G17,G18,G21. We now
+discuss these groups case by case, keeping the notation of Lemma 37. In order to
+simplify the argument, we note that Lemma 38 implies that equation (7 .2) holds if
+m2>2, so that in the following arguments we always may assume that m2= 2.
+CaseG5. The degrees are 6 ,12, and therefore Remark 4.(1) implies that equa-
+tion (7.2) is always satisfied.
+CaseG6. The degrees are 4 ,12, and therefore, according to Remark 4.(1), we need
+only consider the case where h2= 4 andm2= 2, that is, p= 3(m+1)/2. Then (8.7)
+becomes
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (c2wm+1
+2c−2;c2wm+3
+2c−2,...,c2wmc−2,cw0c−1,...,cw m−1
+2c−1/parenrightbig
+.
+(8.9)
+If (w0;w1,...,w m) is fixed by ψp, there must exist an iwith 0≤i≤m−1
+2such that
+ℓT(wi) = 1,wicwic−1=c, and allwj,j/ne}ationslash=i,m+1
+2+i, equalε. However, with the help of
+CHEVIE, one verifies that there is no such solution to this equation. Hence, the left-hand
+side of (7.2) is equal to 0, as required.
+CaseG9. The degrees are 8 ,24, and therefore, according to Remark 4.(1), we need
+only consider the case where h2= 8 andm2= 2, that is, p= 3(m+1)/2. This is the
+samepas forG6. Again,CHEVIEfinds no solution. Hence, the left-hand side of (7.2) is
+equal to 0, as required.
+CaseG10. The degrees are 12 ,24, and therefore, according to Remark 4.(1), we need
+only consider the case where h2= 12 andm2= 2, that is, p= 3(m+1)/2. This is the
+samepas forG6. Again,CHEVIEfinds no solution. Hence, the left-hand side of (7.2) is
+equal to 0, as required.
+CaseG14. The degrees are 6 ,24, and therefore Remark 4.(1) implies that equa-
+tion (7.2) is always satisfied.
+CaseG17. The degrees are 20 ,60, and therefore, according to Remark 4.(1), we
+need only consider the cases where h2= 20 andm2= 2, respectively that h2= 4 and
+m2= 2. In the first case, p= 3(m+1)/2, which is the same pas forG6. Again,CHEVIE142 C. KRATTENTHALER AND T. W. M ¨ULLER
+finds no solution. In the second case, p= 15(m+1)/2. Then (8.7) becomes
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (c8wm+1
+2c−8;c8wm+3
+2c−8,...,c8wmc−8,c7w0c−7,...,c7wm−1
+2c−7/parenrightbig
+.(8.10)
+By Lemma 29, every element of NC(W) is fixed under conjugation by c3, and, thus,
+on elements fixed by ψp, the above action of ψpreduces to the one in (8.9). This action
+was already discussed in the first case. Hence, in both cases, the le ft-hand side of (7.2)
+is equal to 0, as required.
+CaseG18. The degrees are 30 ,60, and therefore Remark 4.(1) implies that equa-
+tion (7.2) is always satisfied.
+CaseG21. The degrees are 12 ,60, and therefore, according to Remark 4.(1), we
+need only consider the cases where h2= 5 andm2= 2, respectively that h2= 15 and
+m2= 2. In the first case, p= 5(m+1)/2, so that (8.7) becomes
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (c3wm+1
+2c−3;c3wm+3
+2c−3,...,c3wmc−3,c2w0c−2,...,c2wm−1
+2c−2/parenrightbig
+.(8.11)
+If (w0;w1,...,w m) is fixed by ψp, there must exist an iwith 0≤i≤m−1
+2such that
+ℓT(wi) = 1 and wic2wic−2=c. However, with the help of CHEVIE, one verifies that
+there is no such solution to this equation. In the second case, p= 15(m+1)/2. Then
+(8.7) becomes the action in (8.10). By Lemma 29, every element of NC(W) is fixed
+under conjugation by c5, and, thus, on elements fixed by ψp, the action of ψpin (8.10)
+reduces to the one in the first case. Hence, in both cases, the left -hand side of (7.2) is
+equal to 0, as required.
+This completes the proof of the lemma. /square
+9.Case-by-case verification of Theorem 33
+We now perform a case-by-case verification of Theorem 33. It sho uld be observed
+that theactionof ψ(givenin (7.1)) isexactly thesame astheactionof φ(given in(3.1))
+withmreplaced by m+1on the components w1,w2,...,w m+1, that is, if we disregard
+the 0-th component of the elements of the generalised non-cross ing partitions involved.
+The only difference which arises is that, while the ( m+ 1)-tuples ( w0;w1,...,w m) in
+(7.1) must satisfy w0w1···wm=c, forw1,w2,...,w m+1in (3.1) we only must have
+w1w2···wm+1≤Tc. The condition for ( w0;w1,...,w m) of being in Fix NCm(W)(ψp)
+is therefore exactly the same as the condition on w1,w2,...,w m+1for the element
+(ε;w1,...,w m,wm+1)beinginFix NCm+1(W)(φp). Consequently, wemayusethecounting
+results from Section 6, except that we have to restrict our atten tion to those elements
+(w0;w1,...,w m,wm+1)∈NCm+1(W) for which w1w2···wm+1=c, or, equivalently,
+w0=ε.
+As before, we write ζdfor a primitive d-th root of unity.
+CaseG4.The degrees are 4 ,6, and hence we have
+Catm(G4;q) =[6m+6]q[6m+4]q
+[6]q[4]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 14 3
+Letζbe a 6(m+ 1)-th root of unity. As before, in what follows we abbreviate the
+assertion that “ ζis a primitive d-th root of unity” as “ ζ=ζd.” The following cases on
+the right-hand side of (7.2) occur:
+lim
+q→ζCatm(G4;q) =m+1,ifζ=ζ6,ζ3, (9.1a)
+lim
+q→ζCatm(G4;q) =3m+3
+2,ifζ=ζ4,2|(m+1), (9.1b)
+lim
+q→ζCatm(G4;q) = Catm(G4),ifζ=−1 orζ= 1, (9.1c)
+lim
+q→ζCatm(G4;q) = 0,otherwise. (9.1d)
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.1). The only cases not covered by Lemmas 35 an d 36 are the
+ones in (9.1b) and (9.1d). On the other hand, the only case left to co nsider according to
+Remark 4 is the case where h2=m2= 2, that is the case (9.1b) where p= 3(m+1)/2.
+In particular, m+ 1 must be divisible by 2. The action of ψpis the same as the one
+in (8.9). Hence, the counting problem is the same as there, except t hat the underlying
+group now is G4. With the help of CHEVIE, one finds that each of the 3 (complex)
+reflections in G4which are less than the (chosen) Coxeter element is a valid choice for
+wi, and each of these choices gives rise to ( m+1)/2 elements in Fix NCm(G4)(ψp) since
+the indexiranges from 0 to ( m−1)/2.
+Hence, in total, we obtain 3m+1
+2=3m+3
+2elements in Fix NCm(G4)(ψp), which agrees
+with the limit in (9.1b).
+CaseG8.The degrees are 8 ,12, and hence we have
+Catm(G8;q) =[12m+12]q[12m+8]q
+[12]q[8]q.
+Letζbe a 12(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G8;q) =m+1,ifζ=ζ12,ζ6,ζ3, (9.2a)
+lim
+q→ζCatm(G8;q) =3m+3
+2,ifζ=ζ8,2|(m+1), (9.2b)
+lim
+q→ζCatm(G8;q) = Catm(G8),ifζ=ζ4,−1,1, (9.2c)
+lim
+q→ζCatm(G8;q) = 0,otherwise. (9.2d)
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.2). The only cases not covered by Lemmas 35 an d 36 are the
+ones in (9.2b) and (9.2d). On the other hand, the only case left to co nsider according
+to Remark 4 is the case where h2= 4 andm2= 2, that is the case (9.2b) where
+p= 3(m+ 1)/2. In particular, m+1 must be divisible by 2. The action of ψpis the
+same as the one in (8.9). Hence, the counting problem is the same as t here, except
+that the underlying group now is G8. With the help of CHEVIE, one finds that each
+of the 3 (complex) reflections in G8which are less than the (chosen) Coxeter element
+is a valid choice for wi, and each of these choices gives rise to ( m+ 1)/2 elements in
+FixNCm(G8)(ψp) since the index iranges from 0 to ( m−1)/2.144 C. KRATTENTHALER AND T. W. M ¨ULLER
+Hence, in total, we obtain 3m+1
+2=3m+3
+2elements in Fix NCm(G8)(ψp), which agrees
+with the limit in (9.2b).
+CaseG16.The degrees are 20 ,30, and hence we have
+Catm(G16;q) =[30m+30]q[30m+20]q
+[30]q[20]q.
+Letζbe a 30(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G16;q) =m+1,ifζ=ζ30,ζ15,ζ6,ζ3, (9.3a)
+lim
+q→ζCatm(G16;q) =3m+3
+2,ifζ=ζ20,ζ4,2|(m+1), (9.3b)
+lim
+q→ζCatm(G16;q) = Catm(G16),ifζ=ζ10,ζ5,−1,1, (9.3c)
+lim
+q→ζCatm(G16;q) = 0,otherwise. (9.3d)
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.3). The only cases not covered by Lemmas 35 an d 36 are the
+ones in (9.3b) and (9.3d). On the other hand, the only cases left to c onsider according
+to Remark 4 are the cases where h2= 10 andm2= 2, respectively h2=m2= 2. Both
+cases belong to (9.3b). In the first case, we have p= 3(m+1)/2, while in the second
+case we have p= 15(m+ 1)/2. In particular, m+ 1 must be divisible by 2. In the
+first case, the action of ψpis the same as the one in (8.9). Hence, the counting problem
+is the same as there, except that the underlying group now is G16. With the help of
+CHEVIE, one finds that each of the 3 (complex) reflections in G16which are less than the
+(chosen) Coxeter element is a valid choice for wi, and each of these choices gives rise to
+(m+ 1)/2 elements in Fix NCm(G16)(ψp) since the index iranges from 0 to ( m−1)/2.
+On the other hand, if p= 15(m+1)/2, then the action of ψpis the same as the one in
+(8.10). By Lemma 29, every element of NC(G16) is fixed under conjugation by c3, and,
+thus, on elements fixed by ψp, the action of ψpreduces to the one in the first case.
+Hence, in total, we obtain 3m+1
+2=3m+3
+2elements in Fix NCm(G16)(ψp), which agrees
+with the limit in (9.3b).
+CaseG20.The degrees are 12 ,30, and hence we have
+Catm(G20;q) =[30m+30]q[30m+12]q
+[30]q[12]q.
+Letζbe a 30(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G20;q) =m+1,ifζ=ζ30,ζ15,ζ10,ζ5, (9.4a)
+lim
+q→ζCatm(G20;q) =5m+5
+2,ifζ=ζ12,ζ4,2|(m+1), (9.4b)
+lim
+q→ζCatm(G20;q) = Catm(G20),ifζ=ζ6,ζ3,−1,1, (9.4c)
+lim
+q→ζCatm(G20;q) = 0,otherwise. (9.4d)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 14 5
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.4). The only cases not covered by Lemmas 35 an d 36 are the
+ones in (9.4b) and (9.4d). On the other hand, the only cases left to c onsider according
+to Remark 4 are the cases where h2= 6 andm2= 2, respectively h2=m2= 2. Both
+cases belong to (9.4b). In the first case, we have p= 5(m+1)/2, while in the second
+case we have p= 15(m+1)/2. In particular, m+1 must be divisible by 2. In the first
+case, the action of ψpis the same as the one in (8.11). Hence, the counting problem
+is the same as there, except that the underlying group now is G20. With the help of
+CHEVIE, one finds that each of the 5 (complex) reflections in G20which are less than the
+(chosen) Coxeter element is a valid choice for wi, and each of these choices gives rise to
+(m+ 1)/2 elements in Fix NCm(G20)(ψp) since the index iranges from 0 to ( m−1)/2.
+On the other hand, if p= 15(m+1)/2, then the action of ψpis the same as the one in
+(8.10). By Lemma 29, every element of NC(G20) is fixed under conjugation by c5, and,
+thus, on elements fixed by ψp, the action of ψpreduces to the one in the first case.
+Hence, in total, we obtain 5m+1
+2=5m+5
+2elements in Fix NCm(G20)(ψp), which agrees
+with the limit in (9.4b).
+CaseG23=H3.The degrees are 2 ,6,10, and hence we have
+Catm(H3;q) =[10m+10]q[10m+6]q[10m+2]q
+[10]q[6]q[2]q.
+Letζbe a 10(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(H3;q) =m+1,ifζ=ζ10,ζ5, (9.5a)
+lim
+q→ζCatm(H3;q) =5m+5
+3,ifζ=ζ6,ζ3,3|(m+1), (9.5b)
+lim
+q→ζCatm(H3;q) = Catm(H3),ifζ=−1 orζ= 1, (9.5c)
+lim
+q→ζCatm(H3;q) = 0,otherwise. (9.5d)
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.5). The only cases not covered by Lemmas 35 an d 36 are the
+ones in (9.5b) and (9.5d). On the other hand, the only cases left to c onsider according
+to Remark 4 are the cases where h2= 1 andm2= 3,h2= 2 andm2= 3, and
+h2=m2= 2. These correspond to the choices p= 10(m+ 1)/3,p= 5(m+ 1)/3,
+respectively p= 5(m+1)/2. The first two cases belong to (9.5b), while p= 5(m+1)/2
+belongs to (9.5d).
+In the case that p= 5(m+1)/3, the action of ψpis given by
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (c2wm+1
+3c−2;c2wm+4
+3c−2,...,c2wmc−2,cw0c−1,...,cw m−2
+3c−1/parenrightbig
+.
+Hence, for an iwith 0≤i≤m−2
+3, we must find an element wi=t1, wheret1satisfies
+(6.9), andall other wj,j /∈/braceleftbig
+i,i+m+1
+3,i+2(m+1)
+3/bracerightbig
+, are set equal to ε. We have found five
+solutions to the counting problem (6.9) in (6.10). Each of them gives r ise to (m+1)/3
+elements in Fix NCm(H3)(ψp) since the index iranges from 0 to ( m−2)/3. On the other146 C. KRATTENTHALER AND T. W. M ¨ULLER
+hand, ifp= 10(m+1)/3, then the action of ψpis given by
+ψp/parenleftbig
+(w0;w1,...,w m)/parenrightbig
+= (c4w2m+2
+3c−4;c4w2m+5
+3c−4,...,c4wmc−4,c3w0c−3,...,c3w2m−1
+3c−3/parenrightbig
+.
+By Lemma 29, every element of NC(H3) is fixed under conjugation by c5, and, thus,
+on elements fixed by ψp, the action of ψpreduces to the one in the first case.
+Hence, in total, we obtain 5m+1
+3=5m+5
+3elements in Fix NCm(H3)(ψp), which agrees
+with the limit in (9.5b).
+Ifp= 5(m+ 1)/2, then the action of ψpis the same as the one in (8.11). The
+computation at the end of Case H3in Section 6 did not find any solutions, which is in
+agreement with (9.5d).
+CaseG24.The degrees are 4 ,6,14, and hence we have
+Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q
+[14]q[6]q[4]q.
+Letζbe a 14(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (9.6a)
+lim
+q→ζCatm(G24;q) =7m+7
+3,ifζ=ζ6,ζ3,3|(m+1), (9.6b)
+lim
+q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (9.6c)
+lim
+q→ζCatm(G24;q) = 0,otherwise. (9.6d)
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.6). The only cases not covered by Lemmas 35 an d 36 are the
+ones in (9.6b) and (9.6d). On the other hand, the only cases left to c onsider according
+to Remark 4 are the cases where h2= 1 andm2= 3,h2= 2 andm2= 3, and
+h2=m2= 2. These correspond to the choices p= 14(m+ 1)/3,p= 7(m+ 1)/3,
+respectively p= 7(m+1)/2. The first two cases belong to (9.6b), while p= 7(m+1)/2
+belongs to (9.6d).
+In the case that p= 14(m+1)/3 orp= 7(m+1)/3, we have found seven solutions to
+the counting problem (6.18) in (6.19), and each of them gives rise to ( m+1)/3 elements
+in Fix NCm(G24)(ψp) (in the style as discussed in Case H3). Hence, in total, we obtain
+7m+1
+3=7m+7
+3elements in Fix NCm(G24)(ψp), which agrees with the limit in (9.6b).
+Ifp= 7(m+ 1)/2, the relevant counting problem is (6.25). However, no element
+(w0;w1,...,w m)∈FixNCm(G24)(ψp) can be produced in this way since the counting
+problem imposes the restriction that ℓT(w0) +ℓT(w1) +···+ℓT(wm) be even, which
+contradicts the fact that ℓT(c) =n= 3. This is in agreement with the limit in (9.6d).
+CaseG25.The degrees are 6 ,9,12, and hence we have
+Catm(G25;q) =[12m+12]q[12m+9]q[12m+6]q
+[12]q[9]q[6]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 14 7
+Letζbe a 12(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G25;q) =m+1,ifζ=ζ12,ζ4, (9.7a)
+lim
+q→ζCatm(G25;q) =4m+4
+3,ifζ=ζ9,3|(m+1), (9.7b)
+lim
+q→ζCatm(G25;q) = (m+1)(2m+1),ifζ=ζ6,−1 (9.7c)
+lim
+q→ζCatm(G25;q) = Catm(G25),ifζ=ζ3,1, (9.7d)
+lim
+q→ζCatm(G25;q) = 0,otherwise. (9.7e)
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.7). The only cases not covered by Lemmas 35 an d 36 are the
+ones in (9.7b) and (9.7e). On the other hand, the only case left to co nsider according to
+Remark4isthecasewhere h2=m2= 3. Thiscorrespondstothechoice p= 4(m+1)/3,
+which belongs to (9.7b). We have found four solutions to the countin g problem (6.30)
+in (6.31), and each of them gives rise to ( m+1)/3 elements in Fix NCm(G25)(ψp) (in the
+style as discussed in Case H3). Hence, in total, we obtain 4m+1
+3=4m+4
+3elements in
+FixNCm(G25)(ψp), which agrees with the limit in (9.7b).
+CaseG26.The degrees are 6 ,12,18, and hence we have
+Catm(G26;q) =[18m+18]q[18m+12]q[18m+6]q
+[18]q[12]q[6]q.
+Letζbe a 14(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G26;q) =m+1,ifζ=ζ18,ζ9, (9.8a)
+lim
+q→ζCatm(G26;q) = Catm(G26),ifζ=ζ6,ζ3,−1,1, (9.8b)
+lim
+q→ζCatm(G26;q) = 0,otherwise. (9.8c)
+We must now prove that the left-handside of (7.2) in each case agre es with the values
+exhibited in (9.8). The only case not covered by Lemmas 35 and 36 is th e one in (9.8c).
+On the other hand, the only cases left to consider according to Rem ark 4 are the cases
+whereh2= 6 andm2= 2, respectively h2=m2= 2. These correspond to the choices
+p= 3(m+1)/2,respectively p= 9(m+1)/2,bothofwhichbelongto(9.8c). Therelevant
+counting problem is (6.35). However, no element ( w0;w1,...,w m)∈FixNCm(G26)(ψp)
+can be produced in this way since the counting problem imposes the re striction that
+ℓT(w0)+ℓT(w1)+···+ℓT(wm) be even, which is absurd. This is in agreement with the
+limit in (9.8c).
+CaseG27.The degrees are 6 ,12,30, and hence we have
+Catm(G27;q) =[30m+30]q[30m+12]q[30m+6]q
+[30]q[12]q[6]q.148 C. KRATTENTHALER AND T. W. M ¨ULLER
+Letζbe a 14(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G27;q) =m+1,ifζ=ζ30,ζ15,ζ10,ζ5, (9.9a)
+lim
+q→ζCatm(G27;q) = Catm(G27),ifζ=ζ6,ζ3,−1,1, (9.9b)
+lim
+q→ζCatm(G27;q) = 0,otherwise. (9.9c)
+We must now prove that the left-handside of (7.2) in each case agre es with the values
+exhibited in (9.9). The only case not covered by Lemmas 35 and 36 is th e one in (9.9c).
+On the other hand, the only cases left to consider according to Rem ark 4 are the cases
+whereh2= 6 andm2= 3,h2=m2= 3,h2= 6 andm2= 2, respectively h2=m2= 2.
+These correspond to the choices p= 5(m+1)/3, 10(m+1)/3, 5(m+1)/2, respectively
+15(m+1)/2, all of which belong to (9.9c).
+Ifp= 5(m+ 1)/3 orp= 10(m+ 1)/3, the computation with the help of CHEVIE
+at the end of Case G27in Section 6 did not find any solutions for the corresponding
+counting problem. This is in agreement with the limit in (9.9c).
+In the case that 5( m+1)/2 or 15(m+1)/2, the relevant counting problem is (6.42).
+However, no element ( w0;w1,...,w m)∈FixNCm(G27)(ψp) can be produced in this way
+since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+···+ℓT(wm)
+be even, which is absurd. This is again in agreement with the limit in (9.9c) .
+CaseG28=F4.The degrees are 2 ,6,8,12, and hence we have
+Catm(F4;q) =[12m+12]q[12m+8]q[12m+6]q[12m+2]q
+[12]q[8]q[6]q[2]q.
+Letζbe a 12(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(F4;q) =m+1,ifζ=ζ12, (9.10a)
+lim
+q→ζCatm(F4;q) =3m+3
+2,ifζ=ζ8,2|(m+1), (9.10b)
+lim
+q→ζCatm(F4;q) = (m+1)(2m+1),ifζ=ζ6,ζ3, (9.10c)
+lim
+q→ζCatm(F4;q) =(m+1)(3m+2)
+2,ifζ=ζ4, (9.10d)
+lim
+q→ζCatm(F4;q) = Catm(F4),ifζ=−1 orζ= 1, (9.10e)
+lim
+q→ζCatm(F4;q) = 0,otherwise. (9.10f)
+We must now prove that the left-handside of (7.2) in each case agre es with the values
+exhibited in (9.10). The only cases not covered by Lemmas 35 and 36 a re the ones in
+(9.10b) and (9.10f). On the other hand, according to Remark 4, th e are no choices for
+h2andm2left to be considered.
+CaseG29.The degrees are 4 ,8,12,20, and hence we have
+Catm(G29;q) =[20m+20]q[20m+12]q[20m+8]q[20m+4]q
+[20]q[12]q[8]q[4]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 14 9
+Letζbe a 20(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G29;q) =m+1,ifζ=ζ20,ζ10,ζ5, (9.11a)
+lim
+q→ζCatm(G29;q) = Catm(G29),ifζ=ζ4,−1,1, (9.11b)
+lim
+q→ζCatm(G29;q) = 0,otherwise. (9.11c)
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.11). The only case not covered by Lemmas 35 an d 36 is the one
+in(9.11c). Ontheother hand, theonly cases left to consider accor ding to Remark4, the
+only choices for h2andm2to be considered are h2= 1 andm2= 3,h2= 2 andm2= 3,
+h2= 4 andm2= 3,h2= 4 andm2= 2, respectively h2=m2= 4. These correspond
+to the choices p= 20(m+ 1)/3,p= 10(m+ 1)/3,p= 5(m+ 1)/3,p= 5(m+ 1)/2,
+respectively p= 5(m+1)/4, all of which belong to (9.11c).
+In the case that p= 20(m+1)/3,p= 10(m+1)/3, orp= 5(m+1)/3, the relevant
+counting problem is (6.55). However, no element ( w0;w1,...,w m)∈FixNCm(G27)(ψp)
+can be produced in this way since the counting problem imposes the re striction that
+ℓT(w0)+ℓT(w1)+···+ℓT(wm) be divisible by 3, which is absurd. This is in agreement
+with the limit in (9.11c).
+In the case that p= 5(m+1)/2, the relevant counting problem is (6.64), for which
+we did not find any solutions. This is again in agreement with the limit in (9.1 1c).
+In the case that p= 5(m+1)/4, the computation at the end of Case G29in Section 6
+did not find any solutions, which is as well in agreement with the limit in (9.1 1c).
+CaseG30=H4.The degrees are 2 ,12,20,30, and hence we have
+Catm(H4;q) =[30m+30]q[30m+20]q[30m+12]q[30m+2]q
+[30]q[20]q[12]q[2]q.
+Letζbe a 30(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(H4;q) =m+1,ifζ=ζ30,ζ15, (9.12a)
+lim
+q→ζCatm(H4;q) =3m+3
+2,ifζ=ζ20,2|(m+1), (9.12b)
+lim
+q→ζCatm(H4;q) =5m+5
+2,ifζ=ζ12,2|(m+1), (9.12c)
+lim
+q→ζCatm(H4;q) =(m+1)(3m+2)
+2,ifζ=ζ10,ζ5, (9.12d)
+lim
+q→ζCatm(H4;q) =(m+1)(5m+2)
+2,ifζ=ζ6,ζ3, (9.12e)
+lim
+q→ζCatm(H4;q) =(m+1)(15m+1)
+4,ifζ=ζ4,2|(m+1), (9.12f)
+lim
+q→ζCatm(H4;q) = Catm(H4),ifζ=−1 orζ= 1, (9.12g)
+lim
+q→ζCatm(H4;q) = 0,otherwise. (9.12h)
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.12). The only cases not covered by Lemmas 35 a nd 36 are150 C. KRATTENTHALER AND T. W. M ¨ULLER
+the ones in (9.12b), (9.12c), (9.12f), and (9.12h). On the other ha nd, the only cases
+left to consider according to Remark 4 are the cases where h2= 2 andm2= 4,
+respectively h2=m2= 2. Thesecorrespondtothechoices p= 15(m+1)/2,respectively
+p= 15(m+1)/4, out of which the first belongs to (9.12f), while the second belongs to
+(9.12h).
+In the case that p= 15(m+1)/2, the action of ψpis the same as the one in (8.10).
+We have found eight solutions to the counting problem (6.80) in (6.83) , each of them
+giving rise to ( m+1)/2 elements in Fix NCm(H4)(ψp) since the index i(in (6.80)) ranges
+from 0 to ( m−1)/2, and we have found 30 solutions to the counting problem (6.82)
+in (6.84), each of them giving rise to/parenleftbig(m+1)/2
+2/parenrightbig
+elements in Fix NCm(H4)(ψp) since 0 ≤
+i1< i2≤(m−1)/2 (in (6.82)). Hence, we obtain 8m+1
+2+ 30/parenleftbig(m+1)/2
+2/parenrightbig
+=(m+1)(15m+1)
+4
+elements in Fix NCm(H4)(ψp), which agrees with the limit in (9.12f).
+Ifp= 15(m+1)/4, the computation at the end of Case H4in Section 6 did not find
+any solutions, which is in agreement with the limit in (9.12h).
+CaseG32.The degrees are 12 ,18,24,30, and hence we have
+Catm(G32;q) =[30m+30]q[30m+24]q[30m+18]q[30m+12]q
+[30]q[24]q[18]q[12]q.
+Letζbe a 30(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G32;q) =m+1,ifζ=ζ30,ζ15,ζ10,ζ5, (9.13a)
+lim
+q→ζCatm(G32;q) =5m+5
+4,ifζ=ζ24,ζ8,4|(m+1), (9.13b)
+lim
+q→ζCatm(G32;q) =(5m+5)(5m+3)
+8,ifζ=ζ12,ζ4,2|(m+1), (9.13c)
+lim
+q→ζCatm(G32;q) = Catm(G32),ifζ=ζ6,ζ3,−1,1, (9.13d)
+lim
+q→ζCatm(G32;q) = 0,otherwise. (9.13e)
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.13). The only cases not covered by Lemmas 35 a nd 36 are the
+ones in (9.13b), (9.13c), and (9.13e). On the other hand, the only c ases left to consider
+according to Remark 4 are the cases where h2= 2 andm2= 4,h2= 6 andm2= 4,
+h2=m2= 3,h2= 6 andm2= 3,h2=m2= 2, respectively h2= 6 andm2= 2.
+These correspond to the choices p= 15(m+1)/4,p= 5(m+1)/4,p= 10(m+ 1)/3,
+p= 5(m+1)/3,p= 15(m+1)/2, respectively p= 5(m+1)/2, out of which the first two
+belong to (9.13b), the next two belong to (9.13e), and the last two b elong to (9.13c).
+In the case that p= 15(m+1)/4 orp= 5(m+1)/4, we have found five solutions to
+the counting problem (6.88) in (6.89), each of them giving rise to ( m+1)/4 elements
+in Fix NCm(G32)(ψp). Hence, we obtain 5m+1
+4=5m+5
+4elements in Fix NCm(G32)(ψp), which
+agrees with the limit in (9.13b).
+In the case that p= 10(m+1)/3 orp= 5(m+1)/3, the relevant counting problem
+is (6.95). However, no element ( w0;w1,...,w m)∈FixNCm(G32)(ψp) can be produced
+in this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+
+···+ℓT(wm) be divisible by 3, which is absurd. This is in agreement with the limit in
+(9.13e).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15 1
+In the case that p= 15(m+1)/2 orp= 5(m+1)/2, we have found ten solutions to
+the counting problem (6.102) in (6.105), each of them giving rise to ( m+1)/2 elements
+in Fix NCm(G32)(ψp), and we have found 25 solutions to the counting problem (6.104) in
+(6.106), each of them giving rise to/parenleftbig(m+1)/2
+2/parenrightbig
+elements in Fix NCm(G32)(ψp). Hence, we
+obtain 10m+1
+2+ 25/parenleftbig(m+1)/2
+2/parenrightbig
+=(5m+5)(5m+3)
+8elements in Fix NCm(G32)(ψp), which agrees
+with the limit in (9.13c).
+CaseG33.The degrees are 4 ,6,10,12,18, and hence we have
+Catm(G33;q) =[18m+18]q[18m+12]q[18m+10]q[18m+6]q[18m+4]q
+[18]q[12]q[10]q[6]q[4]q.
+Letζbe a 18(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G33;q) =m+1,ifζ=ζ18,ζ9, (9.14a)
+lim
+q→ζCatm(G33;q) =9m+9
+5,ifζ=ζ10,ζ5,5|(m+1), (9.14b)
+lim
+q→ζCatm(G33;q) =(m+1)(3m+2)(3m+1)
+2,ifζ=ζ6,ζ3, (9.14c)
+lim
+q→ζCatm(G33;q) = Catm(G33),ifζ=−1 orζ= 1, (9.14d)
+lim
+q→ζCatm(G33;q) = 0,otherwise. (9.14e)
+We must now prove that the left-handside of (7.2) in each case agre es with the values
+exhibited in (9.14). The only cases not covered by Lemmas 35 and 36 a re the ones in
+(9.14b) and (9.14e). On the other hand, the only cases left to cons ider according to
+Remark 4 are the cases where h2= 1 andm2= 5,h2= 2 andm2= 5,h2= 2 and
+m2= 4, respectively h2=m2= 2. These correspond to the choices p= 18(m+1)/5,
+p= 9(m+ 1)/5,p= 9(m+ 1)/4, respectively p= 9(m+ 1)/2, out of which the first
+two belong to (9.14b), while the others belong to (9.14e).
+In the case that p= 18(m+1)/5 orp= 9(m+1)/5, we have found nine solutions
+to the counting problem (6.115) in (6.116). Hence, we obtain 9m+1
+5=9m+9
+5elements in
+FixNCm(G33)(ψp), which agrees with the limit in (9.14b).
+Ifp= 9(m+1)/4, the computation at the end of Case G33in Section 6 did not find
+any solutions, which is again in agreement with the limit in (9.13e).
+In the case that p= 9(m+ 1)/2, the relevant counting problems are (6.122) and
+(6.124). However, no element ( w0;w1,...,w m)∈FixNCm(G33)(ψp) can be produced in
+this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+···+
+ℓT(wm) be even, which is absurd. This is in agreement with the limit in (9.14e).
+CaseG34.The degrees are 6 ,12,18,24,30,42, and hence we have
+Catm(G34;q) =[42m+42]q[42m+30]q[42m+24]q
+[42]q[30]q[24]q
+×[42m+18]q[42m+12]q[42m+6]q
+[18]q[12]q[6]q.152 C. KRATTENTHALER AND T. W. M ¨ULLER
+Letζbe a 42(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(G34;q) =m+1,ifζ=ζ42,ζ21,ζ14,ζ7, (9.15a)
+lim
+q→ζCatm(G34;q) = Catm(G34),ifζ=ζ6,ζ3,−1,1, (9.15b)
+lim
+q→ζCatm(G34;q) = 0,otherwise. (9.15c)
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.15). The only case not covered by Lemmas 35 an d 36 is the one
+in (9.15c). On the other hand, the only cases left to consider accor ding to Remark 4
+are the cases where h2= 1 andm2= 5,h2= 2 andm2= 5,h2= 3 andm2= 5,h2= 6
+andm2= 5,h2= 2 andm2= 4,h2= 6 andm2= 4,h2=m2= 3,h2= 6 andm2= 3,
+h2=m2= 2,h2= 6 andm2= 2, respectively h2= 6 andm2= 6. These correspond
+to the choices p= 42(m+1)/5,p= 21(m+1)/5,p= 14(m+1)/5,p= 7(m+ 1)/5,
+p= 21(m+1)/4,p= 7(m+1)/4,p= 14(m+1)/3,p= 7(m+1)/3,p= 21(m+1)/2,
+p= 7(m+1)/2, respectively p= 7(m+1)/6, all of which belong to (9.15c).
+Inthecasethat p= 42(m+1)/5,p= 21(m+1)/5,p= 14(m+1)/5,orp= 7(m+1)/5,
+the relevant counting problem is (6.128). However, no element ( w0;w1,...,w m)∈
+FixNCm(G34)(ψp) can be produced in this way since the counting problem imposes the
+restriction that ℓT(w0)+ℓT(w1)+···+ℓT(wm) be divisible by 5, which is absurd. This
+is in agreement with the limit in (9.15c).
+In the case that p= 21(m+1)/4 orp= 7(m+1)/4, the relevant counting problem
+is (6.140). However, no element ( w0;w1,...,w m)∈FixNCm(G34)(ψp) can be produced
+in this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+
+···+ℓT(wm) be divisible by 4, which is absurd. This is in agreement with the limit in
+(9.15c).
+In the case that p= 14(m+1)/3 orp= 7(m+1)/3, the relevant counting problems
+are (6.146) and (6.148). However, the computations with the help o fCHEVIEperformed
+in CaseG34in Section 6 did not find any solutions for (6.146) or (6.148). This is in
+agreement with the limit in (9.15c).
+In the case that p= 21(m+1)/2, the relevant counting problems are (6.157), (6.158),
+and (6.159). However, the computations with the help of CHEVIEperformed in Case G34
+in Section 6 found no wiwithℓT(wi) = 3 in (6.157), and hence no solutions for( wi1,wi2)
+withℓT(wi1)+ℓT(wi2) = 3 in (6.158), and no solutions for ( wi1,wi2,wi3) in (6.159). This
+is in agreement with the limit in (9.15c).
+Ifp= 7(m+1)/6, the computation at the end of Case G34in Section 6 did not find
+any solutions, which is also in agreement with the limit in (9.15c).
+CaseG35=E6.The degrees are 2 ,5,6,8,9,12, and hence we have
+Catm(E6;q) =[12m+12]q[12m+9]q[12m+8]q[12m+6]q[12m+5]q[12m+2]q
+[12]q[9]q[8]q[6]q[5]q[2]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15 3
+Letζbe a 12(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(E6;q) =m+1,ifζ=ζ12, (9.16a)
+lim
+q→ζCatm(E6;q) =4m+4
+3,ifζ=ζ9,3|(m+1), (9.16b)
+lim
+q→ζCatm(E6;q) =3m+3
+2,ifζ=ζ8,2|(m+1), (9.16c)
+lim
+q→ζCatm(E6;q) = (m+1)(2m+1),ifζ=ζ6, (9.16d)
+lim
+q→ζCatm(E6;q) =(m+1)(3m+2)
+2,ifζ=ζ4, (9.16e)
+lim
+q→ζCatm(E6;q) =(m+1)(4m+3)(2m+1)
+3,ifζ=ζ3, (9.16f)
+lim
+q→ζCatm(E6;q) =(m+1)(3m+2)(2m+1)(6m+1)
+2,ifζ=−1, (9.16g)
+lim
+q→ζCatm(E6;q) = Catm(E6),ifζ= 1, (9.16h)
+lim
+q→ζCatm(E6;q) = 0,otherwise. (9.16i)
+We must now prove that the left-hand side of (7.2) in each case agre es with the
+values exhibited in (9.16). The only cases not covered by Lemmas 35 a nd 36 are the
+ones in (9.16b), (9.16c), and (9.16i). On the other hand, the only c ases left to consider
+according to Remark 4 are the cases where h2= 1 andm2= 5, respectively h2= 2 and
+m2= 5. These correspond to the choices p= 12(m+1)/5, respectively p= 6(m+1)/5,
+both of which belong to (9.16i).
+In the case that p= 12(m+1)/5, the relevant counting problem is (6.172). However,
+no element ( w0;w1,...,w m)∈FixNCm(E6)(ψp) can be produced in this way since the
+counting problemimposes therestriction that ℓT(w0)+ℓT(w1)+···+ℓT(wm)bedivisible
+by 5, which is absurd. This is in agreement with the limit in (9.16i).
+Ifp= 6(m+1)/5, the computation at the end of Case E6in Section 6 did not find
+any solutions, which is also in agreement with the limit in (9.16i).
+CaseG36=E7.The degrees are 2 ,6,8,10,12,14,18, and hence we have
+Catm(E7;q) =[18m+18]q[18m+14]q[18m+12]q
+[18]q[14]q[12]q
+×[18m+10]q[18m+8]q[18m+6]q[18m+2]q
+[10]q[8]q[6]q[2]q.154 C. KRATTENTHALER AND T. W. M ¨ULLER
+Letζbe a 18(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(E7;q) =m+1,ifζ=ζ18,ζ9, (9.17a)
+lim
+q→ζCatm(E7;q) =9m+9
+7,ifζ=ζ14,ζ7,7|(m+1), (9.17b)
+lim
+q→ζCatm(E7;q) =(m+1)(3m+2)(3m+1)
+2,ifζ=ζ6,ζ3, (9.17c)
+lim
+q→ζCatm(E7;q) = Catm(E7),ifζ=−1 orζ= 1, (9.17d)
+lim
+q→ζCatm(E7;q) = 0,otherwise. (9.17e)
+We must now prove that the left-handside of (7.2) in each case agre es with the values
+exhibited in (9.17). The only cases not covered by Lemmas 35 and 36 a re the ones in
+(9.17b) and (9.17e). On the other hand, the only cases left to cons ider according to
+Remark 4 are the cases where h2= 1 andm2= 7,h2= 2 andm2= 7,h2= 1
+andm2= 5,h2= 2 andm2= 5,h2= 2 andm2= 4, respectively h2=m2= 2.
+These correspond to the choices p= 18(m+1)/7,p= 9(m+1)/7,p= 18(m+ 1)/5,
+p= 9(m+ 1)/5,p= 9(m+ 1)/4, respectively p= 9(m+ 1)/2, out of which the first
+two belong to (9.16b), and all others belong to (9.16i).
+In the case that p= 18(m+1)/7 orp= 9(m+1)/7, we have found nine solutions
+to the counting problem (6.176) in (6.177). Hence, we obtain 9m+1
+7=9m+9
+7elements in
+FixNCm(E7)(ψp), which agrees with the limit in (9.17b).
+In the case that p= 18(m+1)/5 orp= 9(m+1)/5, the relevant counting problem
+is (6.186). However, no element ( w0;w1,...,w m)∈FixNCm(E7)(ψp) can be produced
+in this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+
+···+ℓT(wm) be divisible by 5, which is absurd. This is in agreement with the limit in
+(9.17e).
+In the case that p= 9(m+1)/4, the relevant counting problem is (6.192). However,
+no element ( w0;w1,...,w m)∈FixNCm(E7)(ψp) can be produced in this way since the
+counting problemimposes therestriction that ℓT(w0)+ℓT(w1)+···+ℓT(wm)bedivisible
+by 4, which is absurd. This is again in agreement with the limit in (9.17e).
+In the case that p= 9(m+1)/2, the relevant counting problems are (6.195), (6.196),
+and (6.197). However, no element ( w0;w1,...,w m)∈FixNCm(E7)(ψp) can be produced
+in this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+
+···+ℓT(wm) be even, which is absurd. This is also in agreement with the limit in
+(9.17e).
+CaseG37=E8.The degrees are 2 ,8,12,14,18,20,24,30, and hence we have
+Catm(E8;q) =[30m+30]q[30m+24]q[30m+20]q[30m+18]q
+[30]q[24]q[20]q[18]q
+×[30m+14]q[30m+12]q[30m+8]q[30m+2]q
+[14]q[12]q[8]q[2]q.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15 5
+Letζbe a 30(m+ 1)-th root of unity. The following cases on the right-hand side of
+(7.2) occur:
+lim
+q→ζCatm(E8;q) =m+1,ifζ=ζ30,ζ15, (9.18a)
+lim
+q→ζCatm(E8;q) =5m+5
+4,ifζ=ζ24,4|(m+1), (9.18b)
+lim
+q→ζCatm(E8;q) =3m+3
+2,ifζ=ζ20,2|(m+1), (9.18c)
+lim
+q→ζCatm(E8;q) =(5m+5)(5m+3)
+8,ifζ=ζ12,2|(m+1), (9.18d)
+lim
+q→ζCatm(E8;q) =(m+1)(3m+2)
+2,ifζ=ζ10,ζ5, (9.18e)
+lim
+q→ζCatm(E8;q) =(5m+5)(15m+7)
+16,ifζ=ζ8,4|(m+1), (9.18f)
+lim
+q→ζCatm(E8;q) =(m+1)(5m+4)(5m+3)(5m+2)
+24,ifζ=ζ6,ζ3,(9.18g)
+lim
+q→ζCatm(E8;q) =(m+1)(5m+3)(15m+7)(15m+1)
+64,ifζ=ζ4,2|(m+1),
+(9.18h)
+lim
+q→ζCatm(E8;q) = Catm(E8),ifζ=−1 orζ= 1, (9.18i)
+lim
+q→ζCatm(E8;q) = 0,otherwise. (9.18j)
+We must now prove that the left-handside of (7.2) in each case agre es with the values
+exhibited in (9.18). The only cases not covered by Lemmas 35 and 36 a re the ones in
+(9.18b), (9.18c), (9.18d), (9.18f), (9.18h), and (9.18j). On the o ther hand, the only
+cases left to consider according to Remark 4 are the cases where h2= 2 andm2= 8,
+h2= 1 andm2= 7,h2= 2 andm2= 7,h2= 2 andm2= 4, respectively h2=m2= 2.
+These correspond to the choices p= 15(m+1)/8,p= 30(m+1)/7,p= 15(m+1)/7,
+p= 15(m+1)/4, respectively 15( m+1)/2, out of which the first three belong to (9.18j),
+the fourth belongs to (9.18f), and the last belongs to (9.18h).
+Ifp= 15(m+1)/8, the relevant counting problem is (6.242). However, the computa -
+tion at the end of Case E8in Section 6 did not find any solutions, which is in agreement
+with the limit in (9.18j). Hence, the left-hand side of (7.2) is equal to 0 , as required.
+In the case that p= 30(m+1)/7 orp= 15(m+1)/7, the relevant counting problem
+is (6.213). However, no element ( w0;w1,...,w m)∈FixNCm(E8)(ψp) can be produced
+in this way since the counting problem imposes the restriction that ℓT(w0)+ℓT(w1)+
+···+ℓT(wm) be divisible by 7, which is absurd. This is also in agreement with the limit
+in (9.18j).
+In the case that p= 15(m+ 1)/4, the relevant counting problems are (6.227) and
+(6.228). We have found 45 solutions wito (6.227) of type A2
+1in (6.229), and we have
+found 20 solutions wito (6.227) of type A2in (6.230), which implied 150 solutions for
+(wi1,wi2) to (6.228). The first two give rise to to (45 + 20)m+1
+4= 65m+1
+4elements in
+FixNCm(E8)(ψp), while the third give rise to 150/parenleftbig(m+1)/4
+2/parenrightbig
+elements in Fix NCm(E8)(ψp).
+Hence, we obtain 65m+1
+4+ 150/parenleftbig(m+1)/4
+2/parenrightbig
+=(5m+5)(15m+7)
+16elements in Fix NCm(E8)(ψp),
+which agrees with the limit in (9.18f).156 C. KRATTENTHALER AND T. W. M ¨ULLER
+In the case that p= 15(m+1)/2, the relevant counting problems are (6.233), (6.234),
+(6.235), and(6.236). Wehave found15 solutions wito(6.233) of type A2
+1∗A2in(6.237),
+we have found 45 solutions wito (6.233) of type A1∗A3in (6.238), we have found 5
+solutionswito (6.233) of type A2
+2in (6.239), we have found 18 solutions wito (6.233)
+of typeA4in (6.240), we have found 5 solutions wito (6.233) of type D4in (6.241),
+each giving rise to ( m+ 1)/2 elements in Fix NCm(E8)(ψp). Using the notation from
+there, these imply 2 n3,1+n2,2= 2·660+1195 = 2515 solutions for ( wi1,wi2) to (6.234)
+withℓT(wi1) +ℓT(wi2) = 4, each giving rise to/parenleftbig(m+1)/2
+2/parenrightbig
+elements in Fix NCm(E8)(ψp).
+They also imply 3 n2,1,1= 3·2850 = 8550 solutions for ( wi1,wi2,wi3) to (6.235) with
+ℓT(wi1)+ℓT(wi2)+ℓT(wi3) = 4, each giving rise to/parenleftbig(m+1)/2
+3/parenrightbig
+elements in Fix NCm(E8)(ψp).
+Finally, they implyaswell n1,1,1,1= 6750solutionsfor( wi1,wi2,wi3,wi4)to(6.236), each
+giving rise to/parenleftbig(m+1)/2
+4/parenrightbig
+elements in Fix NCm(E8)(ψp).
+In total, we obtain
+(15+45+5+18+5)m+1
+2+2515/parenleftbigg(m+1)/2
+2/parenrightbigg
++8550/parenleftbigg(m+1)/2
+3/parenrightbigg
++6750/parenleftbigg(m+1)/2
+4/parenrightbigg
+=(m+1)(5m+3)(15m+7)(15m+1)
+64
+elements in Fix NCm(E8)(ψp), which agrees with the limit in (9.18h).
+Acknowledgements
+The authors thank an anonymous referee for a very careful rea ding of the paper [21],
+and for the many pertinent suggestions which helped to improve tha t paper and also
+this manuscript considerably.
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+arXiv:1001.0032v1  [astro-ph.SR]  30 Dec 2009Draft version November 15, 2018
+Preprint typeset using L ATEX style emulateapj v. 08/22/09
+ASTEROSEISMIC INVESTIGATION OF KNOWN PLANET HOSTS IN THE KEPLER FIELD
+J. Christensen-Dalsgaard1,2, H. Kjeldsen1,2, T. M. Brown3, R. L. Gilliland4, T. Arentoft1,2, S. Frandsen1,2,
+P.-O. Quirion1,2,5, W. J. Borucki6, D. Koch6, and J. M. Jenkins7
+Draft version November 15, 2018
+ABSTRACT
+In addition to its great potential for characterizing extra-solar p lanetary systems the Kepler mis-
+sionis providing unique data on stellar oscillations. A key aspect of Keplerasteroseismology is the
+application to solar-like oscillations of main-sequence stars. As an ex ample we here consider an ini-
+tial analysis of data for three stars in the Keplerfield for which planetary transits were known from
+ground-based observations. For one of these, HAT-P-7, we obt ain a detailed frequency spectrum and
+hence strong constraints on the stellar properties. The remaining two stars show definite evidence for
+solar-like oscillations, yielding a preliminary estimate of their mean dens ities.
+Subject headings: stars: fundamental parameters — stars: oscillations — planetary systems
+1.INTRODUCTION
+The main goal of the Kepler mission is to character-
+ize extra-solar planetary systems, particularly Earth-like
+planets in the habitable zone (e.g., Borucki et al. 2009).
+The mission detects the presence of planets through the
+minute reduction of the light from a star as a planet
+crosses the line of sight. Several observations of such
+reductions at fixed time intervals for a given star, and
+extensive follow-up observations, are used to verify that
+the effect results from planet transits and to characterize
+the planet. To ensure a reasonable chance of detection
+Keplerobserves more than 100,000 stars simultaneously,
+in a fixed field in the Cygnus-Lyra region. Most stars
+are observed at a cadence of 29.4 min, but a subset of
+up to 512 stars can be observed at a short cadence (SC)
+of 58.85s. Keplerwas launched on 6 March 2009 and
+data from the commissioning period and the first month
+of regular observations are now available.
+The very high photometric accuracy required to detect
+planet transits (Borucki et al. 2010; Koch et al. 2010)
+also makes the Keplerobservations of great interest for
+asteroseismic studies of stellar interiors. In particular,
+the SC data allow investigations of solar-like oscillations
+in main-sequence stars. Apart from the great astrophys-
+ical interest of such investigations they also provide pow-
+erful tools to characterize stars that host planetary sys-
+tems (Kjeldsen et al. 2009).
+In stars with effective temperature Teff<∼7000K we
+expect to see oscillations similar to those observed in the
+Sun (e.g., Christensen-Dalsgaard 2002), excited stochas-
+ticallybythe near-surfaceconvection. Theseareacoustic
+modes of high radial order; in main-sequence stars such
+1Department of Physics and Astronomy, Aarhus University,
+DK-8000 Aarhus C, Denmark: e-mail jcd@phys.au.dk
+2Danish AsteroSeismology Centre
+3Las Cumbres Observatory Global Telescope, Goleta, CA 93117
+4Space Telescope Science Institute, 3700 San Martin Drive, B al-
+timore, MD 21218
+5Canadian Space Agency, 6767 Route de l’A´ eroport, Saint-
+Hubert, QC, J3Y 8Y9 Canada (present address)
+6NASA Ames Research Center, MS 244-30, Moffett Field, CA
+94035, USA
+7SETI Institute/NASA Ames Research Center, MS244-30, Mof-
+fett Field, CA 94035, USAmodes approximately satisfy the asymptotic relation
+νnl≃∆ν0(n+l/2+ǫ)−l(l+1)D0 (1)
+(Vandakurov 1967; Tassoul 1980). Here νnlis the cyclic
+frequency, nis the radial order of the mode and lis
+the degree, l= 0 corresponding to radial (i.e., spher-
+ically symmetric) oscillations. Also, ∆ ν0is essentially
+the inverse sound travel time across the stellar diameter;
+this is closely related to the mean stellar density ∝angbracketleftρ∗∝angbracketright:
+∆ν0∝ ∝angbracketleftρ∗∝angbracketright1/2.D0depends sensitively on conditions
+near the center of the star; for stars during the central
+hydrogenburningphasethisprovidesameasureofstellar
+age. Finally, ǫis determined by conditions near the stel-
+lar surface. This regular form of the frequency spectrum
+simplifies the analysis of the observations, and the close
+relation between the stellar properties and the param-
+eters characterizing the frequencies make them efficient
+diagnostics of the properties of the star. This has been
+demonstrated in the last few years through observations
+of solar-like oscillations from the ground and from space
+(for reviews, see Bedding & Kjeldsen 2008; Aerts et al.
+2009; Gilliland et al. 2010a).
+Even observations allowing a determination of ∆ ν0
+provide useful constraints on ∝angbracketleftρ∗∝angbracketright. With a reliable de-
+termination of individual frequencies ∝angbracketleftρ∗∝angbracketrightis tightly con-
+strained and an estimate of the stellar age can be ob-
+tained. This can greatly aid the interpretation of obser-
+vations of planetary transits (e.g., Gilliland et al. 2010b;
+Nutzman et al. 2010). We note that photometric obser-
+vations such as those carried out by Keplerare predom-
+inantly sensitive to modes of degree l= 0−2. As indi-
+catedbyEq.(1)thesearesufficienttoobtaininformation
+about the core properties of the star.
+Ground-based transit observations have identified
+three planetary systems in the Keplerfield: TrES-2
+(O’Donovan et al. 2006; Sozzetti et al. 2007), HAT-P-7
+(P´ al et al. 2008), and HAT-P-11 (Dittmann et al. 2009;
+Bakos et al. 2010). These systems have been observed
+byKeplerin SC mode. Their properties (cf. Table 1)
+indicate that they should display solar-like oscillations
+at observable amplitudes, and hence they are obvious
+targets for Keplerasteroseismology. Here we report the
+results of a preliminary asteroseismic characterization of2 Christensen-Dalsgaard et al.
+TABLE 1
+Properties of transiting systems.
+Name KIC No Teff(K) [Fe/H] L/L⊙log(g) (cgs) vsiniSource
+(kms−1)
+HAT-P-7 10666592 6350 ±80 0.26±0.08 4 .9±1.1 4.07±0.06 3.8±0.5 (a)
+6525±61 0.31±0.07 4 .09±0.08 (b)
+HAT-P-11 10748390 4780 ±50 0.31±0.05 0.26±0.02 4.59±0.03 1.5±1.5 (c)
+TrES-2 11446443 5850 ±50−0.15±0.10 1.17±0.10 4.4±0.1 2 ±1 (d)
+5795±73 0.06±0.08 4 .30±0.13 (b)
+Note. — Sources: (a): P´ al et al. (2008); (b): Ammler-von Eif et al . (2009); (c): Bakos et al. (2010); (d): Sozzetti et al. (2007 ). In some
+cases asymmetric error bars have been symmetrized.
+the central stars in the systems, based on the early Ke-
+plerdata.
+2.OBSERVATIONS AND DATA ANALYSIS
+We have analyzed data from Kepler for three
+planet-hosting stars using a pipeline developed for
+fast and robust analysis of all Keplerp-mode data
+(Christensen-Dalsgaard et al. 2008; Huber et al. 2009).
+Each time series contains 63324 data points. SC data
+characteristics and minor post-pipeline processing are
+discussed in Gilliland et al. (2010c). In addition a limb-
+darkened transit light curve model fit has been removed
+and 5-σclipping applied to remove outlying data points
+from each of the time series. The frequency analysis con-
+tains four main steps:
+1. We calculate an oversampled (factor of four) ver-
+sion ofthe power spectrum by using a least-squares
+fitting. We smoothed the spectrum to 3 µHz reso-
+lution to remove the fine structure caused by the
+finite mode lifetime.
+2. We correlated the smoothed power spectrum with
+an equally spaced comb of delta functions, sepa-
+ratedby∆ ν0/2,andconfinedtoaGaussian-shaped
+band with a full width at half maximum of 5∆ ν0.
+We adopted the maximum of this convolution over
+lags between 0 and 0.5 ∆ ν0as the filter output for
+each ∆ν0.
+3. After identifying the peak correlation for the best
+matched model filter and extracting the large sep-
+aration corresponding to this peak we calculate the
+folded spectrum (see Fig. 1b), i.e., the sum of the
+power as a function of frequency modulo the opti-
+mumlargeseparation(theonecorrespondingtothe
+peak correlation). The summed power is used to
+locate the p-mode structure and identify the ridges
+corresponding to the different mode degrees (based
+on the asymptotic relation).
+4. From the asymptoticrelationandthe identification
+of mode degrees we finally identify the position of
+the individual p-mode frequencies in the smoothed
+version of the power spectrum; when more than
+one mode is seen near the expected frequency we
+use the power-weighted average of the two peaks.
+Those extracted frequencies and the mode identifi-
+cations are used in the modeling.
+For observations with low signal-to-noise ratio it may
+not be possible to identify the individual frequencies. In       0        1        2
+Fig. 1.— (a)PowerspectrumofHAT-P-7forfrequencies between
+300 and 3000 µHz. The spectrum is smoothed with a gaussian filter
+with a FWHM of 3 µHz. The noise level at high frequencies corre-
+sponds to 1.1 ppm in amplitude. The white curve is a smoothed
+power spectrum with a gaussian filter (150 µHz FWHM). A fit to
+the background (dashed white curve) is also shown. The exces s
+power and the individual p-modes are evident. (b) Folded pow er
+spectrum, between 750 and 1500 µHz, for HAT-P-7 for a large sep-
+aration of 59 .22µHz. Indicated are the positions corresponding to
+radial modes ( l= 0) and non-radial modes with l= 1 and 2. The
+measured positions are used to identify the individual osci llation
+modes in panel (a). (c) ´Echelle diagram (see text) for frequencies
+of degree l= 0, 1, and 2 in HAT-P-7; a frequency separation of
+59.36µHz and a starting frequency of 10 .8µHz were used. The
+filled symbols, coded for degree as indicated, show the obser ved
+frequencies, while the open symbols are for Model 3 in Table 2 ,
+minimizing χ2
+ν.3
+such cases the analysis is carried through step 2, to de-
+termine the maximum response and hence an estimate of
+the large separation.
+Results on the three individual cases are presented in
+§4.
+3.MODEL FITTING
+Stellar evolution models and adiabatic oscillation
+frequencies were computed using the Aarhus codes
+(Christensen-Dalsgaard 2008a,b), with the OPAL
+equation of state (Rogers et al. 1996) and opacity
+(Iglesias & Rogers 1996) and the NACRE nuclear reac-
+tion parameters (Angulo et al. 1999). In some cases (see
+below) diffusion and settling of helium were included,
+using the simplified formulation of Michaud & Proffitt
+(1993). Convection was treated with the B¨ ohm-Vitense
+(1958) mixing-length formulation, with a mixing length
+αML= 2.00 in units of the pressure scale height roughly
+corresponding to a solar calibration. In some models
+with convective cores, overshoot was included over a dis-
+tance of αovpressure scale heights. Evolution started
+from chemically homogeneous zero-age models. The ini-
+tial abundances by mass X0andZ0of hydrogen and
+heavy elements were characterized by the assumed value
+of [Fe/H], using as reference a present solar surface com-
+position with Zs/Xs= 0.0245 (Grevesse & Noels 1993)
+and assuming, from galactic chemical evolution, that
+X0= 0.7679−3Z0.
+From the observed ∆ ν0, effective temperature and
+composition an initial estimate of the stellar parame-
+ters was obtained using the grid-based SEEK pipeline
+(Quirion et al., in preparation). Smaller grids were then
+computed in the vicinity of these initial parameters, to
+obtaintighterconstraintsonstellarproperties. ForHAT-
+P-7 the analysis of the observations yielded frequencies
+of individually identified modes; here the analysis was
+based on
+χ2
+ν=1
+N−1/summationdisplay
+nl/parenleftBigg
+ν(obs)
+nl−ν(mod)
+nl
+σν/parenrightBigg2
+,(2)
+whereν(obs)
+nlandν(mod)
+nlare the observed and model fre-
+quencies, σνis the standard error in the observed fre-
+quencies (assumed to be constant) and Nis the num-
+ber of observed frequencies. In addition, we considered
+χ2=χ2
+ν+χ2
+T, whereχ2
+Tis the corresponding normalized
+square difference between the observed and model effec-
+tive temperature. When χ2
+νwas available we minimized
+it along each evolution track and considered the result-
+ing minimum values, and the corresponding value of χ2,
+as a function of the parameters characterizing the mod-
+els (see Gilliland et al. 2010b, for details). When only
+the large separation ∆ ν0could be determined from the
+observations, we identified the model along each track
+which matched ∆ ν0and considered the resulting χ2
+Tas
+a function of the model parameters.
+4.RESULTS
+4.1.HAT-P-7
+The observed power spectrum for HAT-P-7 is shown
+in Fig. 1a. The presence of solar-like p-mode peaks, with
+a maximum power around 1.1mHz, is evident. At high
+frequency the noise level in the amplitude spectrum is1.1 parts per million (ppm), with some increase at lower
+frequency, likely due to the effects of stellar granulation.
+Carrying out the correlation analysis described in §2
+we determined the large separation as ∆ ν0= 59.22µHz.
+Figure 1b shows the resulting folded spectrum. This
+clearly shows two closely spaced peaks, identified as cor-
+responding to modes of degree l= 0 and 2, and single
+peak separated from these two by approximately ∆ ν0/2,
+corresponding to l= 1. On this basis we finally deter-
+mined the individual frequencies, identifying the modes
+from the asymptotic relation; the final set includes 33
+p-mode frequencies, determined with a standard error
+σν= 1.4µHz. These frequencies, corresponding to ra-
+dial orders between 11 and 24, are illustrated in Fig. 1c
+in an ´ echelle diagram (see below).
+A grid of models was computed for masses between
+1.41 and 1 .61M⊙, [Fe/H] between 0.17 and 0.38, and
+αov= 0,0.1 and 0.2, extending well beyond the end of
+central hydrogen burning. The modeling did not include
+diffusion and settling. At the mass of this star the outer
+convection zone is quite thin, and as a result the set-
+tling timescale is much shorter than the age of the star.
+Including settling, without compensating effects such as
+partial mixing in the radiative region or mass loss, leads
+to a rapid change in the surface composition which is
+inconsistent with the observed [Fe/H]; for simplicity we
+therefore neglected these effects for HAT-P-7.8
+The computed frequencies were corrected according to
+the procedure of Kjeldsen et al. (2008) for errors in the
+modeling of the near-surface layers, by adding a(ν/ν0)b
+wherea= 0.1158µHz,ν0= 1000µHz andb= 4.9. As
+discussed in §3, for each evolution track, characterized
+by a set of model parameters, we minimized the depar-
+tureχ2
+νof the model frequencies from the observations,
+defining the best model for this set.
+We first consider χ2
+νas a function of the effective
+temperature of the models (Fig. 2a). It is evident
+that there is a clear minimum in χ2
+ν; this is consistent
+with the determination of Teffby P´ al et al. (2008) but
+not with the somewhat higher temperature obtained by
+Ammler-von Eif et al. (2009) (see also Table 1). Thus
+in the following we use the observed quantities from
+P´ al et al. (2008).
+Since the frequencies to leading order are determined
+by the mean stellar density ∝angbracketleftρ∗∝angbracketright, Fig. 2b,c show χ2
+νand
+χ2as functions of ∝angbracketleftρ∗∝angbracketright. It is evident that the best-fitting
+modelsoccupyanarrowrangeof ∝angbracketleftρ∗∝angbracketright, withawell-defined
+minimum. Fittingaparabolato χ2inpanel(c)weobtain
+the estimate ∝angbracketleftρ∗∝angbracketright= 0.2712±0.0032gcm−1. In Fig. 2d
+χ2is shown against model age. Here the variation with
+model parameters is substantially stronger, resulting in
+a greater spread in the inferred age; in particular, it is
+evident, not surprisingly, that the results depend on the
+extent of convective overshoot. From the figure we esti-
+matethattheageofHAT-P-7isbetween1.4and2.3Gyr.
+Examples of evolution tracks are shown in Fig. 3; pa-
+rameters for these models are provided in Table 2. They
+were chosen to give the smallest χ2
+νfor each of the three
+values of αovconsidered. Also shown are the locations
+8Artificially suppressing settling in the outer layers, whil e in-
+cluding diffusion and settling in the core, leads to results t hat are
+very similar to those presented here.4 Christensen-Dalsgaard et al.
+TABLE 2
+Stellar evolution models fitting the observed frequencies for HAT-P-7.
+No M ∗/M⊙Age Z0X0αovR∗/R⊙/angbracketleftρ∗/angbracketrightTeffL∗/L⊙χ2
+νχ2
+(Gyr) (gcm−3) (K)
+1 1.53 1.758 0.0270 0.6870 0.0 1.994 0.2718 6379 5.91 1.08 1.2 1
+2 1.52 1.875 0.0290 0.6809 0.1 1.992 0.2708 6355 5.81 1.04 1.0 4
+3 1.50 2.009 0.0270 0.6870 0.2 1.981 0.2718 6389 5.87 1.00 1.2 4
+Note. — Models minimizing χ2
+ν(cf. Eq. 2) along the evolution tracks, illustrated in Fig. 3 . The models have been selected as providing
+the smallest χ2
+νfor each of the three values of the overshoot parameter αov. The smallest value of χ2
+νis obtained for Model 3.
+Fig. 2.— Results of fitting the observed frequencies to a grid of
+stellar models (see text for details). Plusses, stars and di amonds
+correspond to modelswith αov= 0 (no overshoot), 0.1, and 0.2. (a)
+Minimum mean square deviation χ2
+νof the frequencies (cf. Eq. 2)
+along each evolution track, against the effective temperatu reTeff
+of the corresponding models. The vertical dashed and dotted lines
+indicate the effective temperatures found by P´ al et al. (200 8) and
+Ammler-von Eif et al. (2009). (b) Minimum mean square devia-
+tionχ2
+νagainst the mean density /angbracketleftρ∗/angbracketrightof the corresponding models.
+(c) As (b), but showing the combined χ2. (d)χ2against the age
+for the models that minimize χ2
+ν; the different ridges correspond to
+the different masses in the grid, the more massive models resu lting
+in a lower estimate of the age.Fig. 3.— Theoretical HR diagram with selected evolutionary
+tracks, corresponding to the models defined in Table 2. The ’+ ’ in-
+dicate the models along the full set of evolutionary sequenc es mini-
+mizing the difference between the computed and observed freq uen-
+cies. The box is centered on the LandTeffas given by P´ al et al.
+(2008), with a size matching the errors on these quantities.
+of the models minimizing χ2
+νalong each of the computed
+tracks; these evidently fall close to a line in the HR di-
+agram, corresponding to the small range in ∝angbracketleftρ∗∝angbracketright. The
+range of luminosities, from P´ al et al. (2008), is based on
+modeling and hence has not been used in our fit; even
+so, it is gratifying that the present models are essentially
+consistentwiththesevalues. Also,asindicatedbyFig.2a
+andTable 2, the best-fitting models areclose to the value
+ofTeffobtained by P´ al et al. (2008).
+The match of the best-fitting model (Model 3 of Ta-
+ble 2) to the observed frequencies is illustrated in a so-
+called´ echelle diagram (Grec et al. 1983) in Fig. 1c. In
+accordance with Eq. (1) the frequency spectrum is di-
+vided into slices of length ∆ ν, starting at a frequency of
+10.8µHz; the figure shows the location of the observed
+(filled symbols) and computed (open symbols) frequen-
+cies within each slice, against the starting frequency of
+the slice; the model results extend to the acoustical cut-
+off frequency, 1930 µHz, of the model. There is clearly a
+very good overall agreement between model and obser-
+vations, including the detailed variation with frequency
+which reflects the frequency dependence of the large sep-
+aration, as a possible diagnostics of the outer layers of
+the star (e.g., Houdek & Gough 2007).
+We have finally made a fit of the inferred ∝angbracketleftρ∗∝angbracketright, as
+well asTeffand [Fe/H] from P´ al et al. (2008), to com-
+puted evolutionary tracks from the Yonsei-Yale compi-
+lation (Yi et al. 2001). This was based on a Markov
+Chain Monte Carlo analysis to obtain the statistical
+properties of the inferred quantities (see Brown 2010,
+for details). This resulted in M= 1.520±0.036M⊙,
+R= 1.991±0.018R⊙and an age of 2 .14±0.26Gyr.
+We note that the age estimate reflects the specific as-5
+sumptions in the Yonsei-Yale evolution calculations; as
+indicated by Fig. 2d the true uncertainty in the age de-
+termination is likely somewhat larger.
+4.2.HAT-P-11
+For HAT-P-11 the oscillation amplitudes were much
+smaller than in HAT-P-7, as expected from the general
+scaling of amplitudes with stellar mass and luminosity
+(e.g., Kjeldsen & Bedding 1995). Thus with the present
+short run of data it has only been possible to determine
+thelargeseparation∆ ν0= 180.1µHzfromthemaximum
+in the correlation analysis. We have matched this to a
+grid of models, including diffusion and settling of helium,
+with masses between 0.7 and 0 .9M⊙and [Fe/H] between
+0.21 and 0.41. These models provide a good fit to the
+observed TeffandL/L⊙; note that in the presentcase the
+luminosity is based on a reasonably well-determined par-
+allax. We havedetermined an estimateof ∝angbracketleftρ∗∝angbracketrightbyaverag-
+ing the results of those models which match the observed
+∆ν0and lie within 2 standard deviations ( ±100K) from
+the value of Teffprovided by Bakos et al. (2010); the re-
+sult is∝angbracketleftρ∗∝angbracketright= 2.5127±0.0009gcm−3. Although the for-
+mal error is extremely small, owing to a tight relation
+between the large separation and the mean density for
+stars in this region in the HR diagram, the true error is
+undoubtedly substantially larger. In particular, we ne-
+glected the error in the determination of ∆ ν0and these
+data have not allowed a correction for the systematic
+errors in the modeling of the near-surface layers of the
+star.
+4.3.TrES-2
+Here also we were unable to determine individual fre-
+quencies from the present set of data. The expected am-
+plitudes are smaller than for HAT-P-7, and the noise
+level higher due to the fainter magnitude of TrES-2.
+The correlation analysis yielded two possible values of
+∆ν0: 97.7µHz and 130 .7µHz. For this star ∝angbracketleftρ∗∝angbracketrighthas
+been determined from the analysis of the transit light
+curve. Sozzetti et al. (2007) obtained ∝angbracketleftρ∗∝angbracketright= 1.375±
+0.065gcm−3, while Southworth (2009) found ∝angbracketleftρ∗∝angbracketright=
+1.42±0.13gcm−3. From the scaling with ∝angbracketleftρ∗∝angbracketright1/2thesmaller of the two possible values of ∆ ν0is clearly incon-
+sistent with these values of ∝angbracketleftρ∗∝angbracketright, while ∆ ν0= 130.7µHz
+yields models that are consistent with the observed Teff
+and log(g) of Sozzetti et al. (2007) as well as with these
+values of the mean density. Here we considered a grid
+of models with helium diffusion and settling, masses be-
+tween 0.85 and 1 .1M⊙and [Fe/H] between −0.25 and
+−0.05. Determining again the mean value of ∝angbracketleftρ∗∝angbracketrightfor
+those models that matched ∆ ν0and had Teffwithin two
+standard deviations of the value of Sozzetti et al. (2007)
+we obtained ∝angbracketleftρ∗∝angbracketright= 1.3233±0.0027gcm−3. As in the
+case of HAT-P-11 the true error is likely substantially
+higher.
+5.DISCUSSION AND CONCLUSION
+The present preliminary analysis provides a striking
+demonstration of the potential of Keplerasteroseismol-
+ogyanditssupportingroleintheanalysisofplanethosts.
+Thesestarswill undoubtedly be observedthroughout the
+mission and hence the quality of the data will increase
+substantially. For HAT-P-7 the detected frequencies are
+already close to what will be required for a detailed anal-
+ysis of the stellar interior, beyond the determination of
+the basic parameters of the star. Thus here we can look
+forward to a test of the assumptions of the stellar mod-
+eling; the resulting improvements will further constrain
+the overall properties of the star, in particular its age.
+Also, given the observed vsiniwe expect a rotational
+splitting comparable to that observed in the Sun, and
+hence likely detectable with a few months of observa-
+tions. For the other two stars there is strong evidence
+for the presence of solar-like oscillations; thus continued
+observations will very likely result in the determination
+of individual frequencies and hence further constraints
+on the properties of the stars.
+Funding for this Discovery mission is provided by
+NASA’s Science Mission Directorate. We are very grate-
+ful to the entire Keplerteam, whose efforts have led to
+this exceptional mission. The present work was sup-
+ported by the Danish Natural Science Research Council.
+Facilities: The Kepler Mission
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+arXiv:1001.0033v1  [astro-ph.SR]  30 Dec 2009WIYN OPEN CLUSTER STUDY. XXXVIII. STELLAR RADIAL VELOCITIE S IN THE YOUNG OPEN
+CLUSTER M35 (NGC 2168)
+Aaron M. Geller∗, Robert D. Mathieu∗, Ella K. Braden∗, Søren Meibom∗,†
+Department of Astronomy, University of Wisconsin - Madison , WI 53706, USA
+and
+Imants Platais
+Department of Physics and Astronomy, The Johns Hopkins Univ ersity, Baltimore, MD 21218, USA
+and
+Christopher J. Dolan∗
+Department of Astronomy, University of Wisconsin - Madison , WI 53706, USA
+ABSTRACT
+We present 5201 radial-velocity measurements of 1144 stars, as p art of an ongoing study of the
+young (150 Myr) open cluster M35 (NGC 2168). We have observed M 35 since 1997, using the Hydra
+Multi-Object Spectrograph on the WIYN 3.5m telescope. Our stellar sample covers main-sequence
+stars over a magnitude range of 13.0 ≤V≤16.5 (1.6 - 0.8 M ⊙) and extends spatially to a radius of
+30 arcminutes (7 pc in projection at a distance of 805 pc or ∼4 core radii). Due to its youth, M35
+provides a sample of late-type stars with a range of rotation period s. Therefore, we analyze the radial-
+velocity measurement precision as a function of the projected rot ational velocity. For narrow-lined
+stars (vsini≤10 km s−1), the radial velocities have a precision of 0.5 km s−1, which degrades to 1.0
+km s−1for stars with vsini= 50 km s−1. The radial-velocitydistribution shows a well-defined cluster
+peak with a central velocity of -8.16 ±0.05 km s−1, permitting a clean separation of the cluster and
+field stars. For stars with ≥3 measurements, we derive radial-velocity membership probabilities a nd
+identify radial-velocity variables, finding 360 cluster members, 55 of which show significant radial-
+velocity variability. Using these cluster members, we construct a co lor-magnitude diagram for our
+stellar sample cleaned of field star contamination. We also compare th e spatial distribution of the
+single and binary cluster members, finding no evidence for mass segr egation in our stellar sample.
+Accounting for measurement precision, we place an upper limit on the radial-velocity dispersion of
+the cluster of 0 .81±0.08 km s−1. After correction for undetected binaries, we derive a true radia l-
+velocity dispersion of 0 .65±0.10 km s−1.
+(galaxy:) open clusters and associations: individual (NGC 2168) - (s tars:) binaries: spectroscopic
+1.INTRODUCTION
+Young open clusters are laboratories for the direct
+study of the near-primordial characteristics of stellar
+populations. Their properties, and particularly those of
+the binary systems, offer unique insights into how stars
+arebornand provideessentialguidancefor N-bodystud-
+ies of star clusters. Indeed, with sophisticated N-body
+simulations now able to model real open clusters (e.g.,
+Hurley et al. 2005), knowledge of the correct initial con-
+ditions are all the more important. In particular, the ini-
+tial binary population has a vast impact on the dynami-
+cal evolution of the cluster, and the characteristics of the
+initial binary population will affect the overallfrequency,
+formation rate and formation mechanisms of anomalous
+stars, like blue stragglers, as interactions with binaries
+are thought to be catalysts for the formation of these ex-
+otic objects (Hurley et al. 2005; Knigge et al. 2009). As
+a rich open cluster with an age of ∼150 Myr, M35 is a
+prime cluster to define these hitherto poorly known ini-
+tial conditions for the binary population required for any
+∗Visiting Astronomer, Kitt Peak National Observatory, Nati onal
+Optical Astronomy Observatory, which is operated by the Ass o-
+ciation of Universities for Research in Astronomy (AURA) un der
+cooperative agreement with the National Science Foundatio n.
+†Current address: Harvard-Smithsonian Center for Astrophy sics,
+60 Garden Street, Cambridge, MA 02138, USAopen cluster simulation.
+M35 is a fundamental cluster in the WIYN Open Clus-
+ter Study (WOCS; Mathieu 2000), and as such has a
+strong base of astrometric and photometric observations
+fromboththeWOCScollaborationandothers. Theclus-
+ter is centered at α= 6h09m07.s5 andδ= +24◦20′28′′
+(J2000),towardstheGalacticanticenter. Numerouspho-
+tometric studies have identified the rich main-sequence
+population (e.g., Kalirai et al. 2003; von Hippel et al.
+2002; Sung & Bessell 1999). WOCS CCD photometry
+places the cluster at a distance of 805 ±40 pc, with an
+ageof150 ±25Myr, ametallicityof[Fe/H]=-0.18 ±0.05
+and a reddening of E(B−V)=0.20±0.01 (C. Deliyan-
+nis, private communication). The most recent published
+parameters, from Kalirai et al. (2003), place the cluster
+at a distance of 912+70
+−65pc ((m−M)0= 9.80±0.16)
+with an age of 180 Myr, adopting a E(B−V)=0.20 and
+[Fe/H] = -0.21. (See Kalirai et al. (2003) for a thorough
+review of previous photometry references and their de-
+rived cluster parameters). We note that these two recent
+studies used different isochrone families.
+There have been multiple proper-motion studies
+of the cluster (Ebbighausen 1942; Cudworth 1971;
+McNamara & Sekiguchi 1986a), although none deter-
+mine clustermembership for individual starsfainter than2 Geller et al.
+V≈15.0. Using proper motions, Leonard & Merritt
+(1989) derive a cluster mass from 1600-3200 M ⊙within
+3.75 pc. Detailed observations have also been made
+in M35 to study tidal evolution in binary stars
+(Meibom & Mathieu 2005; Meibom et al. 2006, 2007),
+lithium abundances (Steinhauer & Deliyannis 2004;
+Barrado y Navascu´ es et al. 2001), and white dwarfs
+(Reimers & Koester 1988; Williams et al. 2004, 2006,
+2009).
+This is the first paper in a series studying the dy-
+namical state of M35 through the use of radial-velocity
+(RV) measurements. The data and results presented
+in this series will form the largest database of spec-
+troscopic cluster membership and variability in M35 to
+date. In this paper, we present results from our ongo-
+ing radial-velocity study of the cluster, which we began
+in September 1997. Our stellar sample includes solar-
+type main-sequence stars within the magnitude range of
+13.0≤V≤16.5, which corresponds to a mass range1of 1.6
+- 0.8 M ⊙. The main-sequence turnoff is at V∼9.5,∼4
+M⊙. In Section 2, we describe this stellar sample, ob-
+servations and data reduction in detail. We thoroughly
+investigate our RV measurement precision and the effect
+of stellar rotation in Section 3. Then in Section 4 we de-
+rive RV membership probabilities, and use our study of
+the RV precision to identify RV variables, which we as-
+sume to be binaries or higher-order systems. Within this
+mass range, we identify 360 solar-type main-sequence
+members; 305 are single2(non-RV-variable) stars while
+55 show significant RV variability. We then use these re-
+sults to plot a color-magnitude diagram (CMD) cleaned
+offield starcontamination, tosearchfor evidenceofmass
+segregation and to study the cluster RV dispersion (Sec-
+tion5). Finally, inSection6, weprovideabriefsummary.
+In future papers, we will study the binary population of
+M35in detail, providingobservationsthat will be used to
+directly constrain the initial binary population of open
+cluster simulations.
+2.OBSERVATIONS AND DATA REDUCTION
+In the following section, we define our stellar sample,
+provide a detailed description of our observations and
+data reduction process, and discuss the completeness of
+our spectroscopic observations.
+2.1.Photometric Target Selection
+Initially, we created our M35 target list from the stars
+in three wide-field CCD images centered on M35, taken
+by T. von Hippel with the Kitt Peak National Observa-
+tory (KPNO) Burrell Schmidt telescope on November 18
+and 19, 1993. These images have Vexposures of 4 s, 20 s
+and 180 s and Bexposures of 4 s, 25 s and 240s covering
+a 70′×70′field. We obtained BandVphotometry with
+a limiting magnitude of V= 17, denoted as source 1 in
+1This mass range is derived from a 180 Myr Padova isochrone
+(Marigo et al. 2008) using the distance, reddening and metal licity
+from Kalirai et al. (2003).
+2In the following, we use the term “single” to identify stars
+with no significant RV variation. Certainly, many of these st ars
+are also binaries, although generally with longer periods a nd/or
+lower mass ratios ( q=m2/m1) than the binaries identified in this
+study. When applicable, we have attempted to reduce this bin ary
+contamination amongst the single star sample by photometri cally
+identifying objects as binaries that lie well above the sing le-star
+main-sequence (see Section 4.2).Fig. 1.— Color-magnitude diagram for stars in the field
+of M35 highlighting the selected region used in this survey.
+We plot all stars in the field with the gray points to show
+the location of our selected sample relative to the full clus -
+ter. Our stellar sample is bounded by the solid black lines.
+Within this region, we plot observed stars in the solid black
+points. Additionally, for reference we plot a 180 Myr Padova
+isochrone (Marigo et al. 2008) using the distance, reddenin g
+and metallicity from Kalirai et al. (2003)
+Table 3. Additionally, we derive astrometry from these
+plates, tied to the Tycho catalogue.
+More recently, we added to our database the BVpho-
+tometry of Deliyannis (private communication), taken
+on the WIYN30.9m telescope with the S2KB 2K by
+2K CCD. This photometry derives from a mosaic of five
+fields. Each field has a 20′×20′field-of-view, with one
+central field and four tiled around the center, for a total
+field-of-view of of 40′×40′. This photometry is denoted
+as source 2 in Table 3, and covers 74% of the objects
+we have observed in this study. We note that this pho-
+tometry is more precise than that of source 1. The star-
+by-star difference in Vmagnitudes for the two sources is
+roughly Gaussian with σ= 0.06 mag. However there is
+a tail that extends beyond three times this sigma value.
+Therefore we caution the reader when using magnitudes
+from source 1.
+We selected stars for the RV master list based on three
+constraints. The faintest sources that can be observed
+efficiently at echelle resolution using the Hydra Multi-
+Object Spectrograph (MOS) on the WIYN 3.5m have
+V=16.5; this therefore sets our faint limit for observa-
+tions. Stars bluer than ( B−V)∼0.6 ((B−V)0∼0.4)
+do not provide precise RV measurementsdue to rapid ro-
+tationandpaucityofspectrallines; thisthereforesetsour
+blue limit for observations. Finally, we perform a photo-
+metric selection of cluster member candidates, shown as
+the outlined region4in Figure 1. This region includes a
+wide swath above the main sequence so as to not select
+against binary stars (e.g. Dabrowski & Beardsley 1977),
+yet also removes stars that are very likely cluster non-
+members. This photometric selection allows for an effi-
+cient survey of the cluster. Our sample extends radially
+to 30 arcminutes from the cluster center. At a distance
+of 805 pc, this corresponds to the inner ∼7 pc of the
+3The WIYN Observatory is a joint facility of the University of
+Wisconsin-Madison, Indiana University, Yale University, and the
+National Optical Astronomy Observatories.
+4Specifically, we select stars with 0 .6<(B−V)<1.5, 13.0<
+V <16.5 and between the lines defined by 5 .7(B−V)+8.6< V <
+5.7(B−V)+11.0.WOCS. RV Measurements in M35 3
+Fig. 2.— Completeness of our observations as a function of Vmagnitude (left) and projected radius (right). We plot the
+completeness in stars observed ≥3 times with the dashed line, and stars observed ≥1 time with the solid line.
+cluster in projection. Given the core radius derived by
+Mathieu (1983) of1.9 ±0.1pc, oursample isdrawn from
+the inner ∼4 core radii.
+Note that we lack BVphotometry for ∼11% of the
+point sources found within 30 arcminutes from the clus-
+tercenterin2MASS.Formostoftheseobjects, itislikely
+that there is a nearby or overlapping additional object
+which has prevented accurate BVphotometric measure-
+ments from either of our sources. These objects, by de-
+fault, are not included in our stellar sample. In total, our
+stellar sample contains 1344 stars.
+2.2.Spectroscopic Observations
+Since September 1997, we have collected 5201 spec-
+tra of 1144 stars within this stellar sample as part of
+an ongoing observing program using the WIYN Hydra
+MOS. For the majority of these observations, we use Hy-
+dra’s blue-sensitive 300 µm fibers, which project to a 3.1′
+aperture on the sky. We use the 316 lines mm−1echelle
+grating, isolating the 11th order with the X14 filter. The
+resulting spectra span a wavelength range of ∼25 nm,
+with a dispersion of 0.015 nm pixel−1, centered on 512.5
+nm. We have also occasionally centered our observation
+on 637.5 nm using a very similar setup. In this region,
+we use the same grating, but isolate the 9th order with
+the X18 filter. These observations span a slightly larger
+wavelength range of ∼30 nm, and have a dispersion of
+0.017 nm pixel−1. Due to a broken filter, observations
+taken after the spring of 2008 use different observing se-
+tups than discussed above; most are centered on 560 nm
+and all use the echelle grating. We have not noticed any
+decrease in performance from the new wavelength range,
+but we caution the reader that we lack sufficient obser-
+vations in these setups to reliably determine our RV pre-
+cision for these measurements. During this same period
+certain upgrades were made to the spectrograph collima-
+tor5. All observed regions are rich in metal lines. The
+typical velocity resolution is 15 km s−1. In a two-hour
+integration, the spectra have signal-to-noise (S/N) ra-
+tios ranging from ∼18 per resolution element for V=16.5
+stars to∼100 per resolution element for V=13 stars.
+We create fiber configurations ( pointings ) for our ob-
+servations using a similar method as Geller et al. (2008).
+Monte Carlo simulations show that we require at least
+5http://www.astro.wisc.edu/ ∼mab/research/bench upgrade/threeobservationsoverthecourseofayearinordertoen-
+sure 90% confidence that a star is either constant or vari-
+able in RV out to binary periods of 1000 days (Mathieu
+1983, Geller & Mathieu, in preparation). Given three
+observations with consistent RV measurements over a
+timespan of at least a year and typically longer, we clas-
+sify a given star as single (strictly, non-RV variable) and
+finished, andmoveittothelowestpriority. Ifagivenstar
+has three RV measurements with a standard deviation
+>2.0 km s−1(four times our precision for narrow-lined
+stars; see Section 4.2), we classify the star as RV vari-
+able and give it the highest priority for observation on a
+schedule appropriate to its timescale of variability. This
+prioritization allows us to most efficiently derive orbital
+solutions for our detected binaries.
+We place our shortest-period binaries at the highest
+priority for observations each night, followed by longer-
+period binaries to obtain 1-2 observations per run. Be-
+low the confirmed binaries we place, in the following or-
+der, “candidate binaries” (once-observedstars with a RV
+measurement outside the cluster RV distribution or stars
+with a few measurements that span only 1.5 - 2.5 km
+s−1), once observed and then twice observed non-RV-
+variable likely members, twice observed non-RV-variable
+likely non-members, unobserved stars, and finally, “fin-
+ished” stars. Within each group, we prioritize by dis-
+tance from the cluster center, giving those stars nearest
+to the center the highest priority. A typical pointing
+will contain ∼70 fibers placed on individual stars in our
+sample and ∼10 sky fibers.
+For a given pointing we obtain three consecutive ex-
+posures, each of 40 minutes. In poor transparency or
+with a particularly bright sky, we restrict the targets
+toV <15.0 and shorten the integration time, gener-
+ally to 20 minute exposures. We obtain Thorium-Argon
+(ThAr), oroccasionallyCopper-Argon(CuAr), emission-
+lamp comparison spectra (300 s integrations) before and
+after each set of science integrations for wavelength cal-
+ibration and to check for wavelength shifts during the
+observing sequence. For each set of integrations we also
+obtain one flat-field image (200 s) of a white spot on
+the dome illuminated by incandescent lights. Associat-
+ing the flat-field images with the science integrations is
+particularly critical for calibrating throughput variations
+between the fibers in order to apply sky subtractions. In
+total, we have observed 106 distinct pointings in M354 Geller et al.
+over the roughly 11 years since our survey began.
+2.3.Data Reduction
+For a thorough description of our data reduction pro-
+cess, seeGeller et al.(2008). Inshort, weperformastan-
+dard bias and flat-field correction to the images using
+the overscan strip and the flat-field images, respectively.
+The flat-field spectra are used to trace each aperture in
+a given pointing and thereby extract the science spec-
+tra. Wavelength solutions derived from the emission-
+lamp spectra are applied, followed by sky-subtraction
+using sky fibers from each pointing. The three sets of
+spectra (one set from each integration in a given con-
+figuration) are then combined via a median filter to re-
+movecosmicraysignalsandimproveS/N.These reduced
+spectra are cross-correlated with a high S/N solar spec-
+trum, obtained using a dusk sky exposure taken on the
+WIYN 3.5m with the same instrument setup as the given
+pointing. A Gaussian fit to the cross-correlation func-
+tion (CCF) yields a RV and a full width at half maxi-
+mum (FWHM, in km s−1) for each stellar observation.
+The mean UT time is used to find and correct each RV
+measurement for the Earth’s heliocentric velocity. Fi-
+nally we apply the unique fiber-to-fiber RV offsets de-
+rived by Geller et al. (2008) for the WIYN-Hydra data
+to these RVs. As in Geller et al. (2008), to ensure a
+sufficient quality of measurement, we incorporate into
+our database only those spectra with a CCF peak height
+higher than 0.4. Additionally, we examine the distri-
+bution of RVs for each individual star and visually in-
+spect any measurements that are outliers in the distri-
+bution. Occasionally we remove a measurement whose
+CCF, though having a peak height above 0.4, clearly
+provides a spurious measurement (e.g., inadequate sky
+subtraction).
+2.4.Completeness of Spectroscopic Observations
+We have at least one observation for 1144 of the
+1344 stars in our stellar sample, for a completeness of
+85% across our entire sample. 60% of the stars in our
+stellar sample have sufficient observations for their RVs
+to be considered final (813/1344). For these stars, we ei-
+ther have ≥3 RV measurements that show no variation,
+or, if we do see RV variability, we have found a binary
+orbital solution. (These 813 stars comprise the SM, SN,
+BM and BN classes; see Section 4.1). Of those stars not
+finalized, 231 have only one or two observations, and an-
+other 100stars arevariable but do not yet havedefinitive
+orbital solutions.
+In Figure 2, we show the completeness of our observa-
+tions as functions of Vmagnitude (left) and projected
+radius (right). We plot the completeness in stars ob-
+served≥3 times with the dashed line and stars observed
+≥1 time with the solid line. Our prioritization of stars
+by distance from the cluster center is evident by our de-
+creasing completeness with cluster radius. The decreas-
+ing completeness towards fainter stars reflects the need
+for dark skies with minimal sky contamination in order
+to obtain sufficient S/N in our spectra to derive reliable
+RVsforfaint stars. Thereare37starswith V <15in our
+stellar sample that do not have RV measurements, one of
+which is a proper-motion member. 15 were observed but
+did not yield reliable RVs, mostly due to rapid rotation.22 were not observed, 15 of which are farther than 20
+arcminutes in radius from the cluster center.
+Thedifferenceincompletenessbetweenbrightstarsob-
+served≥1 and≥3 times is also a result of an increas-
+ing population of rapidly rotating stars towards bluer
+(B−V) color. For many of these stars, we have multi-
+ple observations of which only a few, and sometimes one,
+exceed this cutoff value of CCF peak height >0.4 and
+therefore are included in our database. For purposes of
+future research we also include seven rapid rotators in
+Table 3 for which we have been unable to derive RVs
+from our spectra.
+3.EFFECTS OF STELLAR ROTATION ON
+MEASUREMENT PRECISION
+3.1.Observed Rotation
+Because of its youth, M35 provides a sample of
+late-type stars with a range of rotational periods
+(Meibom et al. 2009); some of these stars have projected
+rotational velocities that exceed our spectral resolution.
+As such, the cluster presents an opportunity to explore
+empiricallythedependence ofourmeasurementprecision
+on increasing vsini, whereiis the inclination angle of
+the stellar rotation axis to our line of sight.
+Fig. 3.— FWHM as a function of vsinifor observations
+in the 512.5 nm region. FWHM values are measured from
+the CCF peaks derived from a series of artificially broadened
+templates, of known vsini, correlated against the original
+narrow-lined spectrum. We also show a polynomial fit to the
+data, which we then use to derive vsinivalues for observed
+stars in M35. Additionally we plot a dashed line at vsini
+= 10 km s−1, below which the curve flattens out due to our
+spectral resolution. We impose a floor in vsiniat this value
+as we are unable to reliably measure slower rotation.
+The measured FWHM of the CCF for a given star is
+directly related to the vsini(Rhode et al. 2001). Thus
+in order to derive a vsinivalue, we first measure the
+FWHM of the CCF peak. To do so, we fit a Gaussian
+function to the peak, forcing the baseline of the Gaus-
+sian to start at the background level of the CCF. Specif-
+ically, we subtract from the CCF a polynomial fit to this
+backgroundlevel, and then fit the Gaussian to the subse-
+quent “continuum subtracted” CCF. We only use spec-
+tra from the 512.5 nm region to measure the FWHM, as
+the FWHM is dependent on the setup (i.e., the disper-
+sion, etc.), and most of our observations were taken in
+the 512.5 nm region. We then use a similar technique as
+Rhode et al. (2001), to convert this FWHM to a vsini.WOCS. RV Measurements in M35 5
+Fig. 4.— Histogram of vsinimeasurements (left) and vsinias a function of ( B−V)0(right) for the cluster members of M35.
+We have removed double-lined binaries and any binaries with known periods less than 10.2 days, the circularization peri od in
+M35 (Meibom & Mathieu 2005). We only show stars with mean vsinivalues derived from ≥3 observations within the 512.5
+nm region. Notice that the stars with the largest rotation ar e generally also the bluest stars in our sample.
+We create a series of artificially broadened templates by
+convolving our standard solar template with a series of
+theoretical rotation profiles of specific vsinivalues. We
+then cross correlate this series of broadened templates
+with the original narrow-lined template and measure the
+FWHM of the CCF peak as described above. In Fig-
+ure 3 we show the results of this analysis along with a
+polynomial fit to the data. We use this curve to derive
+vsinivalues for all observations of stars in M35 in the
+512.5 nm region. We then take the mean vsinifor each
+star, using only our highest quality (CCF peak height
+>0.4) spectra, and provide these values in Table 3. We
+are unable to reliably measure vsinivalues below 10 km
+s−1, due to the spectral resolution; we therefore impose
+a floor to the vsiniat this value.
+The median FWHM value that we observe is 46.1 km
+s−1which corresponds to vsini= 10.3 km s−1. Exclud-
+ing stars rotating slower than 10 km s−1, we find a preci-
+sion of 1.4 km s−1for individual vsinivalues of ≤25 km
+s−1, which increasesto 1.6km s−1forvsini >25km s−1.
+These precision values were derived in the same manner
+as for our RV precision, with a fit to a χ2function; see
+Section 3.2 and Geller et al. (2008). Where possible, we
+derive a mean vsinifor a given star from multiple, gen-
+erally≥3, observations within the 512.5 nm region. We
+have compared our vsinimeasurements to the rotation
+periods from Meibom et al. (2009) for stars observed in
+both studies, and find the vsiniand rotation periods to
+be consistent.
+In the left panel of Figure 4, we plot a histogram of
+the mean vsinimeasurements for M35 cluster members.
+(See Section 4.1 for our membership criteria.) In this
+and the other panel, we have excluded any binaries with
+periods known to be less than the circularization period
+in M35 of 10.2 days (Meibom & Mathieu 2005), as the
+rotation of the stars in these binaries have likely been af-
+fectedbytidalprocesses. Wehavealsoremovedanystars
+that appearto be indouble-lined binaries, asthe spectral
+lines in many of these observations are broadened due to
+the secondary spectrum at similar, though slightly off-
+set, RV. In the right panel of Figure 4, we plot the mean
+vsinias a function of ( B−V)0for M35 cluster mem-
+bers. We see a clear trend of increasing rotation towards
+bluer stars, as has also been observed in other young
+open clusters and the field (e.g., field, Hyades, Pleiades,Kraft 1967; Pleiades, Soderblom et al. 1993; Blanco 1,
+Mermilliod et al. 2008; IC 2391, Platais et al. 2007).
+3.2.Radial-Velocity Precision
+We determine the RV measurement precision following
+Geller et al. (2008), where a χ2distribution is fit to the
+distribution of the standard deviations of the first three
+RV measurements for each star in an ensemble of stars.
+Here we do this operation on samples of stars with dif-
+feringvsini. Specifically, we consider stars with vsini
+of≤10 km s−1, 10 - 20 km s−1and 20 - 80 km s−1.
+The bin sizes were chosen arbitrarily in order to pro-
+vide sufficiently large samples. The first bin contains all
+narrow-lined stars for which we have imposed a floor to
+thevsini(see Section 3.1); these stars have line widths
+characteristicof the auto-correlationof our spectralreso-
+lution. The remainingbins containstarswith line widths
+increased by stellar rotation.
+A detailed study of the RV measurement precision of
+our observation and data-reduction pipeline has been
+done by Geller et al. (2008) for late-type stars in the
+old open cluster NGC 188. For the narrow-lined stars
+in NGC 188 they find a single-measurement precision of
+0.4 km s−1. This precision is also a function of the S/N
+of the spectrum, as shown in Geller et al. (2008) by the
+degrading precision with increasing Vmagnitude as well
+as decreasing CCF peak height. The largest S/N effect
+seen for narrow-lined stars in NGC 188 is to degrade the
+precision by 0.25 km s−1. The effect of rotation is larger
+than this amount. Here, we derive a relationship be-
+tween the measurement precision and vsiniand use this
+relationship in our analysis throughout this paper.
+In Figure 5 we show the RV precision as a function
+ofvsiniin M35 for observations taken in the 512.5 nm
+region. The narrow-lined stars have a RV precision of
+0.5 km s−1, similar to that found for the narrow-lined
+stars in NGC 188 observed with this same setup. As ex-
+pected, the value of the measurement precision increases
+with increasing line width. For the most rapidly rotating
+stars (vsini >50 km s−1), the measurement precision
+degrades to ∼1.0 km s−1. We fit a linear relationship to
+the points in Figure 5, shown as the dashed line:
+σi= 0.38+0.012(vsini) km s−1,(1)
+whereσiis our precision. We use this equation with
+the mean measured vsinifor a given star to calculate6 Geller et al.
+the single-measurement RV precision for that star. We
+adopt a floor to our precision at 0.5 km s−1, as found for
+our narrow-lined stars, and shown by the break in the
+dashed line in Figure 5.
+Fig. 5.— RV measurement precision as a function of the
+averagevsini(in km s−1) for single lined stars with ≥3 ob-
+servations. The bins are vsiniof≤10 km s−1, 10 - 20 km
+s−1and 20 - 80 km s−1, chosen to provide sufficiently large
+samples. The gray horizontal bars indicate the bin sizes for
+each point. The black vertical error bars show the one sigma
+errors on the precision fit values. The dotted line shows the
+fit to these data, and provided in Equation 1; we impose a
+floor to our precision at 0.5 km s−1.
+We lack sufficient observations to perform this same
+analysis using observations in the 637.5 nm region or
+for observations taken after the spring of 2008 (see Sec-
+tion 2.2). Therefore, for the 129 stars that do not have
+anyobservationsinthe512.5nmregion( ∼11%ofourob-
+served stars), we visually inspect the spectra and CCFs.
+For narrow-lined stars, we set the precision to 0.5 km
+s−1, and for rotating stars we set the precision to 1.0 km
+s−1. We can then use this RV precision value for a given
+star to determine whether our observations for this star
+are constant or variable in velocity (see Section 4.2). We
+note that only 13 of these stars have sufficient observa-
+tions for their RVs to be considered final, and only 2 are
+probable members.
+4.RESULTS
+The full M35 database is available with the electronic
+version of this paper; here we show a sample of our re-
+sults in Table 3. The first column in Table 3 contains
+the WOCS identification number ( IDW). These num-
+bers are defined in the same manner as in Hole et al.
+(2009), with the cluster center set at α= 6h9m7.s5 and
+δ= +24◦20′28′′(J2000). Nextwe givethe corresponding
+IDs from Meibom et al. (2009), McNamara & Sekiguchi
+(1986a) and Cudworth (1971) ( IDM,IDMcandIDC).
+The next few columns provide the right ascension ( RA),
+declination ( DEC), theBVphotometry and the source
+number ( S) for this photometry (see Section 2.1). Next,
+we show the number of RV measurements ( N) and the
+mean and standard error of the RV measurements. For
+stars with only one RV measurements, we show the
+single-measurementRV precision instead of the standard
+error. Next we provide this single-measurement RV pre-
+cision (σi, derived using equation 1), the mean and stan-TABLE 1
+Gaussian Fit Parameters For Cluster
+and Field RV Distributions
+Cluster Field
+Ampl. (Number) 69.0 ±2.0 2.4 ±0.4
+RV(km s−1) -8.17 ±0.05 13 ±4
+σ(km s−1) 0.92 ±0.08 34 ±4
+dard error of the vsinimeasurements6, thee/ivalue
+(see Section 4.2), the calculated RV membership proba-
+bility(P RV, seeSection4.1), theproper-motionmember-
+ship probability from McNamara & Sekiguchi (1986a)
+(PPM1) and Cudworth (1971) ( PPM2), where available,
+andthen, theclassificationoftheobject(seeSection4.3).
+For RV-variable stars with orbital solutions, we present
+the center-of-mass ( γ) RV with the derived error in place
+of the mean RV and its standard error, and add the com-
+ment SB1 or SB2 for single- and double-lined binaries,
+respectively. Additionally, for binaries without orbital
+solutionsthatappeartobedouble-lined, weaddthecom-
+ment of SB2. Finally, for purposes of future research we
+include seven rapid rotators for which we have been un-
+able to derive RVs from our spectra, and label them with
+the comment RR.
+4.1.Membership
+The RV distribution of M35 is clearly distinguished
+from that of the field when we plot a histogram of the
+mean RVs for the observedstars in our stellar sample. In
+Figure 6, we show a histogram of the mean RVs for stars
+with≥3 RV measurements whose standard deviations
+are<2 km s−1, as well as the γ-RVs for binary stars
+with orbitalsolutions, thus excludingfrom the fit anyRV
+variables whose γ-RVs are unknown. The cluster shows
+a well-defined peak rising above the broad distribution
+of the field stars. We simultaneously fit one-dimensional
+Gaussian functions, Fc(v) andFf(v), to represent the
+cluster and field RV distributions, respectively, and then
+use these fits to calculate RV membership probabilities
+for each individual star. We compute the membership
+probability PRV(v) with the usual formula:
+PRV(v) =Fc(v)
+Ff(v)+Fc(v)(2)
+(Vasilevskis et al. 1958). We plot these Gaussian fits in
+Figure 6 with the dashed lines, and show the fit param-
+eters in Table 1.
+For a given single star, we use the mean RV to com-
+pute the RV membership probability. For a given binary
+star with an orbital solution, we compute the RV mem-
+bership probability from the γ-RV. For RV-variable stars
+without orbital solutions, the γ-RVs are not known, and
+therefore we cannot calculate RV membership probabil-
+ities. For these stars, we provide a preliminary member-
+ship classification, described in Section 4.3.
+6For double-lined binaries and stars with no observation in t he
+512.5 nmregion, wedo notderive a vsinivalue. For starswith only
+one measurement in the 512.5 nm region, we convert the 1-sigm a
+error on the FWHM (derived from the Gaussian fit to the CCF
+peak) to an error on the vsiniusing the fit shown in Figure 3. As
+this relationship is not linear, we provide the mean of the de rived
+upper and lower errors on vsini.WOCS. RV Measurements in M35 7
+Fig. 6.— RV histogram for stars in the field of M35. We
+include the mean RVs for stars observed ≥3 times with RV
+standarddeviations <2kms−1andtheγ-RVsfor binarystars
+with orbital solutions, excluding RV variables whose γ-RVs
+are unknown. The bin sizes are 0.5 km s−1, equal to our
+RV precision for narrow-lined stars, as found in Section 3.
+The dashed lines show the simultaneous Gaussian fits to the
+cluster and field RV distributions.
+Fig. 7.— Histogram of membership probabilities, P RV, for
+stars observed ≥3 times with RV standard deviations <2 km
+s−1and for binaries whose γ-RVs are known. For the single
+stars, we compute P RVusing the mean observed RV; for bi-
+naries with orbital solutions, P RVis based on the γ-RV. We
+show our membership cutoff of P RV=50% with the dashed
+line, above which we classify a star as a cluster member. Note
+that we do not show the full height of the bin at lowest mem-
+bership probability for clarity.
+In Figure 7, we show the distribution of RV member-
+ship probabilities, displaying a clean separation between
+the cluster members and field stars. In the following
+analysis, we use a probability cutoff of P RV≥50 % to
+define our cluster member sample. Using the 344 single
+clustermembersandbinaryclustermemberswithorbital
+solutions, we find a mean cluster RV of -8.16 ±0.05 km
+s−1. From the area under the fit to the cluster and field
+distributions, we estimate a field contamination of 6%
+within our cluster member sample (P RV≥50%). Though
+this estimate is derived excluding the RV variables that
+do not have orbital solutions, the percent contamination
+should be valid for the cluster as a whole.
+Our RV membership probabilities agree well with the
+proper-motion memberships of Cudworth (1971) and
+McNamara & Sekiguchi (1986a). We note that our stel-
+lar sample covers only the faintest portion of either
+proper-motion study. There are 24 Cudworth (1971)proper-motionmembers within our observed stellar sam-
+ple, of which we find 14 (58%) to also have ≥50% RV
+membership probabilities. Cudworth (1971) note that
+forV >13 they begin to find significant errors in
+their photometry and expect many field stars to con-
+taminate their proper-motion member sample; this can
+likely explain the 10 discrepant stars. There are 70
+McNamara & Sekiguchi(1986a)proper-motionmembers
+within our observed stellar sample, of which we find 64
+(91%) to also have ≥50% RV membership probabilities.
+McNamara & Sekiguchi (1986a) expects up to 15 field
+stars contaminating their cluster member sample from
+13< V <15, which can easily account for the 6 dis-
+crepant stars.
+We also note that NGC 2158 is only ∼28 arcminutes
+away from the center of M35, at α= 6h07m25sandδ=
++24◦05′48′′(J2000), and thus is within the spatialregion
+that we have surveyed. Scott et al. (1995) find a mean
+RV for NGC 2158 of 28 ±4 km s−1. There are five stars
+within our sample that lie within the cluster radius of 2.5
+arcminutes Carraro et al. (2002) from the center of NGC
+2158 and have RVs within three times the standard error
+(12 km s−1) of the mean RV : 125044, 39017, 111050,
+57037, 54048. Two of these stars (125044 and 57037)
+have less than three observations; the remaining three
+have≥3observationsandappeartobenon-RV-variables.
+4.2.Radial-Velocity Variability
+RV-variable stars are distinguishable by the larger
+standard deviations of their RV measurements. Here,
+we assume that such velocity variability is the result of
+a binary companion, or perhaps multiple companions.
+Specifically, we consider a star to be a RV variable if the
+ratio of the standard deviation of its RV measurements
+to the single-measurement RV precision7(e/i) for that
+star is greater than four (Geller et al. 2008). We provide
+thee/ivalue for each single-lined star in Table 3; we
+label double-lined systems as RV variables directly, and
+include the comment of SB2 in Table 3.
+MonteCarloanalysishasshownthat, forsimilarobser-
+vations ofsolar-typestars in NGC 188, Geller & Mathieu
+(in preparation) can detect the majority of binaries with
+periods less than 104days and a negligible fraction of
+longer-period binaries. Though the slightly poorer preci-
+sionforthe M35datawill effect the specific completeness
+numbers, we can assume a similarly high completeness in
+detected binaries with periods less than 104days and a
+corresponding drop in completeness for longer-period bi-
+naries. Some of the undetected systems are evident from
+their separation from the main sequence (see Figure 8).
+We have currently identified 55 RV-variable members
+of M35, and have derived orbital solutions for 71%
+(39/55) of this sample. In following papers we will pro-
+vide the orbital solutions for these systems, including
+all derived parameters. We will then perform a detailed
+analysis of the distributions of these orbital parameters
+as well as the binary frequency of the cluster.
+7We use the same nomenclature of “ e/i” as in Geller et al.
+(2008), though in other sections, for clarity, we have label ed the
+precision as σi, so as not to confuse the precision with an inclina-
+tion angle.8 Geller et al.
+TABLE 2
+Number of Stars
+or Star Systems
+Within Each
+Membership
+Class
+Class Number
+SM 305
+SN 452
+BM 39
+BN 17
+BLM 16
+BU 16
+BLN 68
+U 231
+4.3.Membership Classification of Radial-Velocity
+Variable Stars
+We follow the same classification system as
+Geller et al. (2008) and Hole et al. (2009) in order
+to provide a qualitative guide to a given star’s mem-
+bership and variability, in addition to the calculated
+RV memberships and e/ivalues. We provide these
+classifications for all observed stars, while the member-
+ships and e/ivalues are only provided for a subset of
+appropriate stars.
+For stars with e/i<4, we classify those with P RV≥50%
+as single members (SM), and those with P RV<50% as
+singlenon-members(SN). Ifa starhas e/i≥4and enough
+measurements from which we are able to derive an or-
+bital solution, we use the γ-RV to compute a secure
+RV membership. For these binaries, we classify those
+with P RV≥50% as binary members (BM) and those with
+PRV<50% as binary non-members (BN). For RV vari-
+ables without orbital solutions, we split our classifica-
+tions into three categories. If the mean RV results in
+PRV≥50%,weclassifythesystemasabinarylikelymem-
+ber (BLM). If the mean RV results in P RV<50% but the
+range of measured RVs includes the cluster mean RV, we
+classify the system as a binary with unknown member-
+ship (BU). Finally, if the RV measurements for a given
+star all lie either at a lower or higher RV than the clus-
+ter distribution, we classify the system as a binary likely
+non-member (BLN), since it is unlikely that any orbital
+solution could place the binary within the cluster distri-
+bution. We classify stars with <3 RV measurements as
+unknown (U), as these stars do not meet our minimum
+criterion for deriving RV memberships or e/imeasure-
+ments. In the following analysis, we include the SM,
+BM and BLM stars as cluster members. Including these
+stars, we find 360 total cluster members in our sample.
+We list the number of stars within each class in Table 2.
+5.DISCUSSION
+In the following section, we present a CMD for M35
+cleaned of field star contamination (Section 5.1), com-
+pare the spatial distribution of the single and binary
+members (Section 5.2), and analyze the RV dispersion
+of the cluster (Section 5.3).
+5.1.Color-Magnitude Diagram
+In Figure 8, we show the CMD for all RV cluster mem-
+bers in M35 from this study for which we have pho-
+tometry from the WIYN 0.9m, as this set of photom-Fig. 8.— Color-magnitude diagram of M35 including only
+cluster members (P RV≥50%) with photometry from WIYN
+0.9m (source 2). We plot the RV variables with orbital so-
+lutions with circles and without orbital solutions with dia -
+monds. We show the 180 Myr Padova isochrone as the black
+line. The solid gray line shows where binaries with mass ra-
+tios of 1.0 lie on the CMD, and the dashed gray line shows the
+deviation from the isochrone of twice the photometric error .
+etry is of higher precision than that taken on the Bur-
+rell Schmidt (see Section 2.1). We also plot a 180 Myr
+Padova isochrone using the cluster parameters derived
+by Kalirai et al. (2003) in the black curve. Binaries with
+orbital solutions are circled and RV variables without or-
+bital solutions are marked by diamonds.
+Additionally, we use the Padova isochrone to plot the
+location on the CMD of binaries with mass ratios q= 1,
+shown as the gray line. We note that there are a number
+ofstarsobservedbrighterandtothe redofthis line, some
+that we have not identified as RV variables. In this loca-
+tion on the CMD one would expect to find either higher-
+order systems or field stars. There are 33 RV members
+that lie above the q= 1 line; 22 are single and 11 show
+RV variability. We expect a 6% field star contamination
+within the cluster members sample (Section 4.1). If we
+include only the 309 cluster members that have photom-
+etry from the WIYN 0.9m (and are therefore shown in
+Figure 8), this results in 19 possible field stars; including
+our entire cluster member sample results in 22 possible
+field stars. Therefore field star contamination cannot ac-
+count for all of these sources, suggesting that a subset
+of these stars are indeed higher-order systems. We also
+notethatthereareanadditional12clustermemberswith
+photometry from the Burrell Schmidt that lie above the
+q= 1 line, but recall that this source of photometry is of
+poorer precision.
+Finally, for use in Sections 5.2, we follow a similar pro-
+cedure as Montgomery et al. (1993) to attempt to photo-
+metricallyidentify binariesthatlie farfromtheisochrone
+ontheCMD. Wederivethedistanceofeachstarfromthe
+main-sequenceisochroneand fit a Gaussianfunction rep-
+resenting the photometric error distribution to the dis-
+tribution of these distances. We notice a clear excess in
+the observed distribution from the Gaussian fit at 2 σ,
+shown as the dashed gray line in Figure 8. We attribute
+this excess to photometric binaries. A 1 M ⊙star in M35
+with the additional light from a companion of mass-ratio
+q= 0.78 would lie on this line. Therefore sources ob-
+served above this line are likely binaries with larger mass
+ratios (q >0.78), or very infrequently, field stars. We
+observe 42 cluster members above this line that showWOCS. RV Measurements in M35 9
+no significant RV variation (and therefore fall into the
+SM class). Many of these are likely long-period binaries
+that are outside of our detection limits, as the hard-soft
+boundary for solar-type stars in M35 is ∼105−106days,
+and we only detect binaries with P/lessorsimilar104days (Geller
+& Mathieu, in preparation).
+5.2.Spatial Distribution and Mass Segregation
+In Figure 9 we compare the cumulative projected ra-
+dial distributions of the single and binary members of
+M35. We have attempted to reduce the contamination
+from undetected binaries within our single-star sample
+by only including stars with no detectable RV variation
+(SM) that arefainter and bluer than the dashed grayline
+in Figure 8. This conservative cut removes large- q(i.e.,
+high total mass) binaries that have periods longer than
+our detection limit. We have not applied any correc-
+tion for the spatial bias found in our observations (Sec-
+tion 2.4), because this bias will be present in both the
+single- and binary-star samples and should therefore not
+effect this analysis. A Kolmogorov-Smirnov test shows
+no significant difference between these two populations
+with a value of 60%. We therefore conclude that the
+solar-type main-sequence binaries in M35 show no evi-
+dence for central concentration as compared to the single
+stars.
+Mathieu (1983) finds a half-mass relaxation time for
+the cluster of 150 Myr, comparable to the cluster age.
+This study of the radial spatial distributions for proper-
+motion-selected member stars in the 8 .0< V < 14.5
+(∼4.4 - 1.2 M ⊙) range revealed mass segregation only
+amongstarsmoremassivethan 2M ⊙. The degreeofseg-
+regationlessenswith decreasingmass, and is largelynon-
+existent among solar-like stars. McNamara & Sekiguchi
+(1986b) found similar results in their proper-motion se-
+lected sample, which covered stars down to V= 14.5 (∼
+1.2 M⊙). We only include primary stars with masses of
+1.6 - 0.8 M ⊙, and therefore most of our binaries have
+total masses that are lower than the higher-mass stars
+that have been shown to be mass segregated. Mathieu
+(1983) found that M35 is fit well with a multi-mass King
+model. In such models the reduction in mass segregation
+for lower-mass systems derives from more severe tidal
+truncation of higher-dispersion velocity distributions in
+a cluster potential dominated by the solar-like stars.
+5.3.Cluster Radial-Velocity Dispersion
+To determine the true RV dispersion of the cluster, we
+followtheprocedureofGeller et al.(2008). Wefirstlimit
+oursampletoonlyincludeSMstarsthathave vsini≤10
+km s−1. We limit the vsinivalue to ensure that we only
+use the highest precision RV measurements for this anal-
+ysis. These narrow-lined stars have a precision σi= 0.5
+km s−1. We will discuss the effect of undetected binaries
+that likely remain within this sample in Section 5.3.1.
+Usingthissampleof67SMstars,wefirstderivetheob-
+serveddispersion σobsbytakingthestandarddeviationof
+themeanRVsforeachstar,andwefind σobs= 0.86±0.07
+km s−1. This observed dispersion is a function of our
+measurement precision and is also inflated by undetected
+binaries. Therefore, in order to derive the true RV dis-
+persion, we must first account for the precision on theseFig. 9.— Cumulative projected radial spatial distributions
+of the M35 single and binary cluster members. We have ex-
+cluded any stars from the single-star sample that are bright er
+and redder than the dashed grey line in Figure 8, as these
+stars are likely long-period binaries that are outside of ou r de-
+tection limits. We plot the single stars with the black point s
+andtheRVvariables with theopen diamonds. We findnosig-
+nificant evidence for central concentration of the RV-varia ble
+population.
+RV measurements. We derive8the “combined RV dis-
+persion” σcbfrom :
+σ2
+cb=σ2
+obs−1
+nn/summationdisplay
+i=1ξ2
+i. (3)
+Here,n= 67 is the number of stars used in this analysis,
+andξiis the mean errorof the RV for the ith star defined
+as,
+ξi=
+m/summationdisplay
+j=1/parenleftbig
+RVj−RVi/parenrightbig2
+m(m−1)
+1/2
+(4)
+whereRVjis one of the mnumber of RV measurements
+for a given star i, andRViis the mean RV for that star.
+WenotethatthesecondterminEquation3isverynearly
+equal to σ2
+i/3, as we have 3 RVs for most of the stars in
+this sample, and these narrow-lined stars all have the
+same precision of σi= 0.5 km s−1. Following this proce-
+dure, wederiveacombineddispersionof σcb= 0.81±0.08
+km s−1. The error on this combined dispersion is almost
+entirely due to the statistical error on σobs.
+We find no significant difference in the combined RV
+dispersion of the SM or BM stars. For the BM stars,
+we use the γ-RVs in place of the mean RVs, and substi-
+tute the measurement precision for the standard devia-
+tion portion in Equation 4. There is also no significant
+variation in the combined RV dispersion as a function
+of radius, although due to the small sample sizes our
+binned RV dispersion values have large uncertainties (of
+0.1 - 0.15 km s−1for bins of 10 arcmin).
+This combined RV dispersion is inflated by undetected
+binaries. In the following section, we quantify this effect
+and apply the correctionto derivethe true RV dispersion
+ofM35, an improvementon the procedureofGeller et al.
+(2008).
+8The use of Equations 3 and 4 is an improvement over the pro-
+cedure of Geller et al. (2008) adapted from McNamara & Sander s
+(1977). The uncertainty on σcbalso follows McNamara & Sanders
+(1977).10 Geller et al.
+5.3.1.Contribution from Undetected Binaries
+The combined RV dispersion defined in Equation 3 is
+also described by,
+σcb=σc+β (5)
+whereσcis the true RV dispersion of the cluster and
+βrepresents the contribution from undetected binaries
+within our sample. Therefore, in order to derive the true
+RV dispersion of the cluster we have performed a Monte
+Carlo analysis to determine this contribution from unde-
+tected binaries.
+We first create a set of simulated binaries with or-
+bital parameters distributed according to the Galactic
+field solar-type binaries studied by Duquennoy & Mayor
+(1991). Specifically, these binaries have a log-normal pe-
+riod distribution centered on log( P[days] ) = 4.8 with
+σ= 2.3, and a Gaussian eccentricity distribution cen-
+tered on e= 0.3. For binaries with periods below the
+circularization period of 10.2 days (Meibom & Mathieu
+2005), we set the eccentricity to zero. We use only solar-
+massprimary stars, and a distribution in secondarymass
+between0.08-1M ⊙describedby aGaussiancenteredon
+M2= 0.23 M⊙withσ= 0.42 M⊙(Kroupa et al. 1990).
+Duquennoy & Mayor (1991) found this Gaussian to be
+the best fit to their solar-type field binaries, and this dis-
+tribution is also consistent with that of Goldberg et al.
+(2003) for their field binaries with primary masses >0.67
+M⊙. The orbital inclinations and phases of the binaries
+are chosen randomly. We then generate three RVs for
+these simulated binaries distributed in time according to
+the actual distribution of our first three observations for
+starsin M35. The majorityofthe SM starsin oursample
+haveonlythreeobservations. TotheseRVs, wealsoadda
+randomerrorgeneratedfromaGaussiancenteredonzero
+and with σ= 0.5 km s−1, the RV precision for narrow-
+lined stars in M35. We also add a random velocity offset
+generated from a Gaussian centered on zero with a stan-
+dard deviation equal to an adopted one-dimensional RV
+dispersion.
+To this sample, we add a number of simulated single
+stars to produce a desired binary frequency. We generate
+three RVs for each single star from a Gaussian described
+byourprecision. To themean RVforeverysinglestarwe
+also add a random offset described by the assumed RV
+dispersion in the same manner as for the simulated bi-
+naries. We then keep only those simulated binaries (and
+single stars) whose first three RVs result in an e/i <4,
+and whose mean RVs are within three standard devia-
+tions of the mean RV from a Gaussian fit to the simu-
+lated RV distribution. This cutoff in standard deviation
+reflects our membership criterion of P RV≥50% for the
+M35 observations. These binaries would be undetected
+within the SM sample.
+We then follow the equations given above to derive β
+fora rangeofbinaryfrequenciesand velocitydispersions,
+σc. In Figure 10, we plot the true cluster RV dispersion
+(σc) as a function of the combined RV dispersion ( σcb)
+for a range of total binary frequencies, where each line
+corresponds to a different binary frequency between 0%
+(far left) to 100% (far right) in steps of 10%. We can
+then use the results shown in Figure 10 to derive the
+true RV dispersion for M35. Furthermore, the results
+shown in this figure are also applicable to RV dispersionFig. 10.— The true cluster RV dispersion ( σc) plotted
+against the combined RV dispersion ( σcb) for a range of total
+binaryfrequencies. Eachlinecorrespondstoadifferentbin ary
+frequency in steps of 10%, with 0% at the far left and 100%
+at the far right. With the vertical gray rectangle, we plot th e
+region included in the combined RV dispersion for M35 of
+0.81±0.08 km s−1. The diagonal gray region covers the pos-
+sible lines within our extrapolated true binary frequency i n
+M35 of 66% ±8% (derived assuming the M35 binaries follow
+a Duquennoy & Mayor (1991) period distribution). Finally,
+we plot the resulting true RV dispersion in M35 of 0.65 ±
+0.10 km s−1with the black point at the intersection of these
+two shaded regions.
+analyses for other star clusters, provided that the binary
+population is consistent with the Duquennoy & Mayor
+(1991) field binaries.
+5.3.2.True Radial-Velocity Dispersion
+To date, we have detected 55 binaries in M35 out
+of 360 cluster members. If we assume a similar com-
+pleteness as in Geller & Mathieu (in preparation), for
+NGC 188, then we can assume that we have detected
+63% of the binaries with periods less than 104days
+(and a negligible fraction of binaries with longer peri-
+ods). This correction results in a binary frequency of
+24%±3% forP <104days. This binary frequency
+is consistent with that of solar-type stars in the Galac-
+tic field Duquennoy & Mayor (1991) out to the same
+period limit. If we assume the M35 binaries follow
+a Duquennoy & Mayor (1991) period distribution, then
+our binary frequency for P <104days implies a total bi-
+nary frequency of 66% ±8%, with the inclusion of wider
+binaries currently beyond our detection limits. We then
+take this value for the total binary frequency and correct
+our combined RV dispersion for undetected binaries.
+In the filled gray areas in Figure 10 we show the re-
+gions defined by our M35 combined RV dispersion and
+the total binary frequency. At the intersection, we plot
+the derivedtrue RV dispersionin M35 of σc= 0.65±0.10
+km s−1. Using a flat distribution in secondary mass (and
+mass ratio), as has been suggested by some studies (e.g.,
+Mazeh et al. 1992, 2003), has a negligible effect on the
+derived true RV dispersion. This true RV dispersion is
+consistent with the projected velocity dispersion of 1.0
+±0.15 km s−1, derived by Leonard & Merritt (1989) us-
+ingtheproper-motiondatafromMcNamara & Sekiguchi
+(1986a).
+6.SUMMARY
+This is the first paper in a series studying the dynam-
+ical state of the young ( ∼150 Myr) open cluster M35WOCS. RV Measurements in M35 11
+(NGC 2168). In this first paper, we present our RV ob-
+servations and provide initial results from this survey.
+Our stellar sample extends to 30 arcminutes in radius
+from the cluster center (7 pc in projection at a distance
+of 805 pc or ∼4 core radii), and we have selected a region
+from aV, (B−V) CMD (Figure 1) which covers a mass
+range of 1.6 - 0.8 M ⊙. We have used the WIYN 3.5m
+telescope with the Hydra MOS to obtain 5201 spectra of
+1144 stars within this stellar sample. From these spec-
+tra, we derive RV measurements with a precision of 0.5
+km s−1for narrow-lined stars. The vast majority of the
+observed stars have multiple measurements, allowing de-
+termination of cluster membership and identification of
+spectroscopic binary stars. We detect 360 cluster mem-
+bers, 55 of which show significant variability in their RV
+measurements. Binary orbital solutions have been ob-
+tained for 39 of these RV variables, which we will present
+in detail in the next paper in this series. Observations
+of the rest of the RV variables and the remainder of our
+stellar sample are ongoing. Table 3 provides the first RV
+membership database for M35 and extends ∼1.5 magni-
+tudes deeper than any previous membership catalogue.
+Using the RV cluster members, we study the spa-
+tial distribution and velocity dispersion of the single
+and binary stars. We find their spatial distributions
+to be indistinguishable. This lack of central concentra-
+tion for the binaries is consistent with earlier observa-
+tional studies of stars in M35 as well as with a fully re-
+laxed dynamical model for the cluster (Mathieu 1983;
+McNamara & Sekiguchi 1986b). In these studies, mass
+segregationisseeninhigher-massstars,butdiminishesto
+being undetectable for stars in our observed mass range.After correcting for measurement precision, but not for
+binaries, we place an upper limit on the RV dispersion of
+the cluster of 0 .81±0.08 km s−1. When we also correct
+for undetected binaries, we derive a true RV dispersion
+of 0.65±0.10 km s−1.
+The WOCS group will continue our survey of M35 in
+ordertoderiveRVmembershipsforallstarsin ourstellar
+sample and obtain orbital solutions for all binaries with
+periods less than a few thousand days, as well as some
+with longer periods. In future papers, we will study the
+binary population of M35 in detail, providing all orbital
+solutions and analyzing the binary frequency and dis-
+tributions of orbital parameters. These data will form
+essential constraints on the hitherto poorly known initial
+binary populations used in sophisticated N-body models
+of open clusters.
+The authors would like to express their gratitude to
+the staff of the WIYN Observatory for their skillful and
+dedicated work that have allowed us to obtain these ex-
+cellent spectra. We thank Ata Sarajedini and Ted von
+Hippel for the acquisition of the Schmidt images, Vera
+Platais for work on the astrometry and photometry as
+well as John Bjorkman for early photometry work. We
+also thank the many undergraduate and graduate stu-
+dents who have contributed late nights to obtain the
+spectra for this project. This work was supported by
+NSF grant AST 0406615and the Wisconsin Space Grant
+Consortium.
+Facilities: WIYN 3.5m
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+Radial-Velocity Data Table
+IDWIDMIDMcIDCRA DEC V (B−V)S N RV RV eσivsini(vsini)ePRVPPM1PPM2e/i Class Comment
+96041 410 ··· ··· 6:10:28.65 24:11:52.0 16.416 0.960 1 4 51.89 1.55 0.52 11.8 1 .1 ··· ·· · · ·· 5.93 BLN ···
+36042 209 ··· ··· 6:10:34.30 24:14:07.8 14.835 0.859 1 3 -8.74 0.55 0.64 21.6 0 .8 96 ·· · · ·· 1.49 SM ···
+36045 209 ··· ··· 6:10:43.69 24:16:08.9 14.497 0.824 1 17 -9.58 0.19 0.50 10.3 0.2 91 ·· · · ·· 77.46 BM SB1
+138057 366 ··· ··· 6:10:50.20 24:04:50.7 16.368 1.165 1 1 -25.13 0.50 0.50 ··· ··· ··· ·· · · ·· · ·· U ···
+64052 312 ··· ··· 6:10:43.70 24:07:00.8 15.884 1.023 1 4 -8.85 0.64 0.55 14.2 4 .2 96 ·· · · ·· 2.34 SM ···
+15036 180 ··· 731 6:10:15.70 24:11:31.7 13.450 0.690 2 1 57.98 0.50 0.50 ··· ··· ··· ·· · 0 · ·· U ···
+49051 227 ··· ··· 6:10:51.32 24:11:10.6 15.086 0.890 1 4 88.83 0.34 0.50 10.0 ··· 0 ·· · · ·· 1.35 SN ···
+29047 87 ··· ··· 6:10:44.59 24:13:44.3 14.948 0.895 1 4 56.95 0.28 0.50 10.0 ··· 0 ·· · · ·· 1.10 SN ···
+40032 ··· ··· ··· 6:10:11.15 24:14:01.5 15.280 0.820 2 3 -7.72 0.32 0.55 14.3 0 .8 96 ·· · · ·· 1.02 SM ···
+20037 193 ··· 758 6:10:22.30 24:14:39.3 14.270 0.650 2 1 3.88 0.50 0.50 ··· ··· ··· ·· · 7 · ·· U ···
+The contents of each column are defined in Section 4.
\ No newline at end of file
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+arXiv:1001.0034v1  [math.NT]  4 Jan 2010NEW IDENTITIES INVOLVING q-EULER
+POLYNOMIALS OF HIGHER ORDER
+T. Kim AND Y. H. Kim
+Abstract. In this paper, we present new generating functions which are relat ed to
+q-Euler numbers and polynomials of higher order. From these genera ting functions, we
+give new identities involving q-Euler numbers and polynomials of higher order.
+§1. Introduction/ Preliminaries
+LetCbe the complex number field. We assume that q∈Cwith|q|<1 and
+theq-number is defined by [ x]q=1−qx
+1−qin this paper. The q-factorial is given by
+[n]q! = [n]q[n−1]q···[2]q[1]qand theq-binomial formulae are known that
+(x:q)n=n/productdisplay
+i=1(1−xqi−1) =n/summationdisplay
+i=0/parenleftbiggn
+i/parenrightbigg
+qq(i
+2)(−x)i,(see [3, 14, 15]) ,
+and
+1
+(x:q)n=n/productdisplay
+i=1/parenleftbigg1
+1−xqi−1/parenrightbigg
+=∞/summationdisplay
+i=0/parenleftbiggn+i−1
+i/parenrightbigg
+qxi,(see [3, 5, 14, 15]) ,
+where/parenleftbign
+i/parenrightbig
+q=[n]q!
+[n−i]q![i]q!=[n]q[n−1]q···[n−i+1]q
+[i]q!.
+The Euler polynomials are defined by2
+et+1ext=/summationtext∞
+n=0En(x)tn
+n!, for|t|< π. In the
+special case x= 0,En(=En(0)) are called the n-th Euler numbers. In this paper, we
+consider the q-extensions of Euler numbers and polynomials of higher orde r. Barnes’
+multiple Bernoulli polynomials are also defined by
+(1)
+tr
+/producttextr
+j=1(eajt−1)ext=∞/summationdisplay
+n=0Bn(x,r|a1,···,ar)tn
+n!,where|t|<max
+1≤i≤r2π
+|ai|, (see [1, 14]).
+Key words and phrases. : multiple q-zeta function, q-Euler numbers and polynomials, higher
+order q-Euler numbers, Laurent series, Cauchy integral.
+2000 AMS Subject Classification: 11B68, 11S80
+The present Research has been conducted by the research Grant of Kw angwoon University in 2010
+Typeset by AMS-TEX
+1In one of an impressive series of papers (see [1, 6, 14]), Barn es developed the so-called
+multiple zeta and multiple gamma function. Let a1,···,aNbe positive parameters.
+Then Barnes’ multiple zeta function is defined by
+ζN(s,w|a1,···,aN) =/summationdisplay
+m1,···,mN=0(w+m1a1+···+mNaN)−s,(see [1]),
+whereℜ(s)> N,ℜ(w)>0. Form∈Z+, we have
+ζN(−m,w|a1,···,aN) =(−1)mm!
+(N+m)!BN+m(w,N|a1,···,aN).
+In this paper, we consider Barnes’ type multiple q-Euler numbers and polynomials.
+The purpose of this paper is to present new generating functi ons which are related
+toq-Euler numbers and polynomials of higher order. From the Mel lin transformation
+of these generating functions, we derive the q-extensions o f Barnes’ type multiple
+zeta functions, which interpolate the q-Euler polynomials of higher order at negative
+integer. Finally, we give new identities involving q-Euler numbers and polynomials of
+higher order.
+§2.q-Euler numbers and polynomials of higher order
+In this section, we assume that q∈Cwith|q|<1. Letx,a1,... ,a rbe complex
+numbers with positive real parts. Barnes’ type multiple Eul er polynomialsare defined
+by
+(2)2r
+/producttextr
+j=1(eajt+1)ext=∞/summationdisplay
+n=0E(r)
+n(x|a1,... ,a r)tn
+n!,for|t|<max
+1≤i≤rπ
+|wi|,(see [6]),
+andE(r)
+n(a1,... ,a r)(=E(r)
+n(0|a1,... ,a r)) are called the n-th Barnes’ type multiple
+Euler numbers. First, we consider the q-extension of Euler polynomials. The q-Euler
+polynomials are defined by
+(3)Fq(t,x) =∞/summationdisplay
+n=0En,q(x)tn
+n!= [2]q∞/summationdisplay
+m=0(−q)me[m+x]qt,(see [8, 11, 13, 14, 15]) .
+From (3), we have
+En,q(x) =[2]q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−1)lqlx
+(1+ql+1).
+In the special case x= 0,En,q(=En,q(0)) are called the n-thq-Euler numbers. From
+(3), we can easily derive the following relation.
+E0,q= 1,andq(qE+1)n+En,q= 0 ifn≥1,(see [8, 16, 17]) ,
+2where we use the standard convention about replacing EkbyEk,q.It is easy to show
+that
+lim
+q→1Fq(t,x) =2
+et+1ext=∞/summationdisplay
+n=0En(x)tn
+n!,(see [2, 3, 19-23]) ,
+whereEn(x) are the n-th Euler polynomials. For r∈N, the Euler polynomials of
+orderris defined by
+(4)/parenleftbigg2
+et+1/parenrightbiggr
+ext=∞/summationdisplay
+n=0E(r)
+n(x)tn
+n!,for|t|< π.
+Now we consider the q-extension of (4).
+(5)F(r)
+q(t,x) = [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mre[m1+···+mr+x]qt=∞/summationdisplay
+n=0E(r)
+n,q(x)tn
+n!,
+whereE(r)
+n,q(x) are called the n-thq-Euler polynomials of order r(see [10-15]). From
+(5), we can derive
+(6) E(r)
+n,q(x) =[2]r
+q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−1)lqlx
+(1+ql+1)r.
+By (5) and (6), we see that
+(7) F(r)
+q(t,x) = [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+(−q)me[m+x]qt.
+Thus, we note that lim q→1F(r)
+q(t,x) =/parenleftBig
+2
+et+1/parenrightBigr
+ext=/summationtext∞
+n=0E(r)
+n(x)tn
+n!.In the special
+casex= 0,E(r)
+n,q(=E(r)
+n,q(0)) are called the n-thq-Euler numbers of order r. By (5),
+(6) and (7), we obtain the following proposition.
+Proposition 1. Forr∈N, let
+F(r)
+q(t,x) = [2]r
+q/summationdisplay
+m1,...,m r=0(−q)m1+···+mre[m1+···+mr+x]qt=∞/summationdisplay
+n=0E(r)
+n,q(x)tn
+n!.
+Then we have
+E(r)
+n,q(x) =[2]r
+q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−1)lqlx
+(1+ql+1)r= [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+(−q)m[m+x]n
+q.
+3From the Mellin transformation of F(r)
+q(t,x), we can derive the following equation.
+1
+Γ(s)/integraldisplay∞
+0F(r)
+q(−t,x)ts−1dt= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr
+[m1+···+mr+x]sq
+= [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+(−q)m1
+[m+x]sq, (8)
+wheres∈C,x/negationslash= 0,−1,−2,.... By (8), we can define the multiple q-zeta function
+related to q-Euler polynomials.
+Definition 2. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the multiple q-zeta
+function related to q-Euler polynomials as
+ζq,r(s,x) = [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr
+[m1+···+mr+x]sq.
+Note that ζq,r(s,x) is a meromorphic function in whole complex s-plane. From (8),
+we also note that
+ζq,r(s,x) = [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+(−q)m1
+[m+x]sq.
+By Laurent series and the Cauchy residue theorem in (5) and (8 ), we see that
+ζq(−n,x) =E(n)
+n,q(x),forn∈Z+.
+Therefore, we obtain the following theorem.
+Theorem 3. Forr∈N,n∈Z+, andx∈Rwithx/negationslash= 0,−1,−2,..., we have
+ζq(−n,x) =E(r)
+n,q(x).
+Letχbe the Dirichlet’s character with conductor f∈Nwithf≡1 (mod 2). Then
+the generalized q-Euler polynomial attached to χare considered by
+Fq,χ(x) =∞/summationdisplay
+n=0En,χ,q(x)tn
+n!= [2]q∞/summationdisplay
+m=0(−q)mχ(m)e[m+x]qt.
+From (3) and (9), we have
+En,χ,q(x) =[2]q
+[2]qff−1/summationdisplay
+a=0(−q)aχ(a)En,qf(x+a
+f).
+4In the special case x= 0,En,χ,q=En,χ,q(0) are called the n-th generated q-Euler
+number attached to χ.
+It is known that the generalized Euler polynomials of order rare defined by
+(10) (2/summationtextf−1
+a=0(−1)aχ(a)eat
+eft+1)rext=∞/summationdisplay
+n=0E(r)
+n,χ(x)tn
+n!,
+for|t|<π
+f.
+We consider the q-extension of (10). The generalized q-Euler polynomials of order
+rattached to χare defined by
+F(r)
+q,χ(t,x) = [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr(r/productdisplay
+i=1χ(mi))e[m1+···+mr+x]qt
+=∞/summationdisplay
+n=0E(r)
+n,χ,q(x)tn
+n!,(see [14, 15]) . (11)
+Note that
+lim
+q→1F(r)
+q,χ(t,x) = (2/summationtextf−1
+a=0(−1)aχ(a)eat
+eft+1)r.
+By (11), we easily see that
+E(r)
+n,χ,q(x) =[2]r
+q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg
+(−qx)lf−1/summationdisplay
+a1,...,ar=0(r/productdisplay
+j=1χ(aj))(−ql+1)/summationtextr
+i=1ai
+(1+q(l+1)f)r
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr(r/productdisplay
+i=1χ(mi))[m1+···+mr+x]n
+q.
+Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we have
+1
+Γ(s)/integraldisplay∞
+0F(r)
+q,χ(−t,x)ts−1dt
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr(/producttextr
+i=1χ(mi))
+[m1+···+mr+x]sq,(see [15]) . (12)
+From (12), we can consider the Dirichlet’s type multiple q-l-function as follows :
+Definition 4. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the Dirichlet’s
+type multiple q-l-function as
+lq(s,x|χ) = [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mr(/producttextr
+i=1χ(mi))
+[m1+···+mr+x]sq,(see [15]) .
+By Laurent series and the Cauchy residue theorem in (11) and ( 12), we obtain the
+following theorem.
+5Theorem 5. Forn∈Z+, we have
+lq(−n,x|χ) =E(r)
+n,χ,q(x).
+Forh∈Zandr∈N, we consider the extended r-pleq-Euler polynomials.
+F(h,r)
+q(t,x) = [2]r
+q∞/summationdisplay
+m1,...,m r=0q/summationtextr
+j=1(h−j+1)mj(−1)/summationtextr
+j=1mje[m1+···+mr+x]qt
+=∞/summationdisplay
+n=0E(h,r)
+n,q(x)tn
+n!. (13)
+Note that
+lim
+q→1F(h,r)
+q(t,x) = (2
+et+1)rext=∞/summationdisplay
+n=0E(r)
+n(x)tn
+n!.
+From (13), we note that
+E(h,r)
+n,q(x) =[2]r
+q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−qx)l
+(−qh−r+l+1:q)r
+= [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+q(−qh−r+1)m[m+x]n
+q. (14)
+By (14), we easily see that
+(15)F(h,r)
+q(t,x) = [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+q(−qh−r+1)me[m+x]qt,(see [11, 13, 14]) .
+Using the Mellin transform for F(h,r)
+q(t,x), we have
+1
+Γ(s)/integraldisplay∞
+0F(r)
+q(−t,x)ts−1dt
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−1)m1+···+mrq/summationtextr
+j=1(h−j+1)mj
+[m1+···+mr+x]sq,(see [13, 14, 15]) ,(16)
+fors∈C,x∈Rwithx/negationslash= 0,−1,−2,.... Now we can define the extended q-zeta
+function associated with E(h,r)
+n,q(x).
+6Definition 6. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the (h, q)-zeta
+function as
+ζ(h)
+q,r(s,x) = [2]r
+q∞/summationdisplay
+m1,...,m r=0(−1)m1+···+mrq/summationtextr
+j=1(h−j+1)mj
+[m1+···+mr+x]sq.
+Notethat ζ(h)
+q,r(s,x)isalsoa meromorphic function inwholecomplex s-plane. From
+(16) and (15), we note that
+(17) ζ(h)
+q,r(s,x) = [2]r
+q∞/summationdisplay
+m=0/parenleftbiggm+r−1
+m/parenrightbigg
+q(−qh−j+1)m1
+[m+x]sq.
+Using the Cauchy residue theorem and Laurent series in (16), we obtain the following
+theorem.
+Theorem 7. Forn∈Z+, we have
+ζ(h)
+q,r(−n,x) =E(h,r)
+n,q(x).
+We consider the extended r-ple generalized q-Euler polynomials as follows :
+F(h,r)
+q,χ(t,x)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0q/summationtextr
+j=1(h−j+1)mj(−1)/summationtextr
+j=1mj(r/productdisplay
+j=1χ(mj))e[m1+···+mr+x]qt(18)
+=∞/summationdisplay
+n=0E(h,r)
+n,χ,q(x)tn
+n!.
+By (18), we see that
+E(h,r)
+n,χ,q(x) =[2]r
+q
+(1−q)nf−1/summationdisplay
+a1,...,ar=0(−1)/summationtextr
+j=1aj(r/productdisplay
+j=1χ(aj))n/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−1)lqlxq(h−j+l+1)aj
+(−q(h−r+l+1)f:qf)r
+=[2]r
+q
+[2]r
+qf[f]n
+qf−1/summationdisplay
+a1,...,ar=0(−1)/summationtextr
+j=1aj(r/productdisplay
+j=1χ(aj))q/summationtextr
+j=1(h−j+1)ajζ(h)
+qf,r(−n,x+/summationtextr
+j=1aj
+f).(19)
+Therefore, we obtain the following theorem.
+7Theorem 8. Forn∈Z+, we have
+E(h,r)
+n,χ,q(x)
+=[2]r
+q
+[2]r
+qf[f]n
+qf−1/summationdisplay
+a1,...,ar=0(−1)/summationtextr
+j=1aj(r/productdisplay
+j=1χ(aj))q/summationtextr
+j=1(h−j+1)ajζ(h)
+qf,r(−n,x+/summationtextr
+j=1aj
+f).
+From (18), we note that
+1
+Γ(s)/integraldisplay∞
+0F(h,r)
+q,χ(−t,x)ts−1dt
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0q/summationtextr
+j=1(h−j+1)mj(/producttextr
+j=1χ(mj))(−1)m1+···+mr
+[m1+···+mr+x]sq, (20)
+wheres∈C,x∈Rwithx/negationslash= 0,−1,−2,....
+From (20), we define the Dirichlet’s type multiple ( h,q)-l-function associated with
+the generalized multiple q-Euler polynomials attached to χ.
+Definition 9. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the Dirichlet’s
+type multiple q-l-function as follows :
+l(h)
+q(s,x|χ) = [2]r
+q∞/summationdisplay
+m1,...,m r=0q/summationtextr
+j=1(h−j+1)mj(/producttextr
+i=1χ(mi))(−1)m1+···+mr
+[m1+···+mr+x]sq.
+Note that l(h)
+q(s,x|χ) is a meromorphic function in whole complex plane. It is easy
+to show that
+l(h)
+q(s,x|χ)
+=[2]r
+q
+[2]r
+qf1
+[f]sqf−1/summationdisplay
+a1,...,ar=0(−1)/summationtextr
+j=1aj(r/productdisplay
+j=1χ(aj))q/summationtextr
+j=1(h−j+1)ajζ(h)
+qf,r(s,x+/summationtextr
+j=1aj
+f).
+By (19) and (20), we obtain the following theorem.
+Theorem 10. Forn∈Z+, we have
+l(h)
+q(−n,x|χ) =E(h,r)
+n,χ,q(x).
+Finally, we give the q-extension of Barnes’ type multiple Euler polynomials in (2 ).
+Forx,a1,... ,a r∈Cwith positive real part, let us define the Barnes’ type mutipl e
+8q-Euler polynomials in Cas follows :
+F(r)
+q(t,x|a1,... ,a r;b1,... ,b r)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−1)m1+···+mrq(b1+1)m1+···+(br+1)mre[a1m1+···+armr+x]t(21)
+=∞/summationdisplay
+n=0E(r)
+n,q(x|a1,... ,a r;b1,... ,b r)tn
+n!,
+whereb1,... ,b r∈Z. By (21), we see that
+E(r)
+n,q(x|a1,... ,a r;b1,... ,b r)
+=[2]r
+q
+(1−q)nn/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg(−1)lqlx
+(1+qla1+b1+1)···(1+qlar+br+1)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−1)m1+···+mrq(b1+1)m1+···+(br+1)mr[a1m1+···+armr+x]n
+q.
+From (21), we note that
+1
+Γ(s)/integraldisplay∞
+0F(r)
+q(−t,x|a1,... ,a r;b1,... ,b r)ts−1dt
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr
+[a1m1+···+armr+x]sq. (22)
+By (22), we define the Barnes’ type multiple q-zeta function as follows :
+ζq,r(s,x|a1,... ,a r;b1,... ,b r)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr
+[a1m1+···+armr+x]sq,
+wheres∈C,x∈Rwithx/negationslash= 0,−1,−2,.... By (21), (22) and (23), we obtain the
+following theorem.
+Theorem 11. Forn∈Z+, we have
+ζq,r(s,x|a1,... ,a r;b1,... ,b r) =E(r)
+n,q(x|a1,... ,a r;b1,... ,b r).
+Letχbe the Dirichlet’s character with conductor f∈Nwithf≡1 (mod 2). Then
+the generalized Barnes’ type multiple q-Euler polynomials attached to χare defined
+9by
+F(r)
+q,χ(t,x|a1,... ,a r;b1,... ,b r)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(r/productdisplay
+i=1χ(mi))e[a1m1+···+armr+x]qt(24)
+=∞/summationdisplay
+n=0E(r)
+n,χ,q(x|a1,... ,a r;b1,... ,b r)tn
+n!,
+From (24), we note that
+1
+Γ(s)/integraldisplay∞
+0F(r)
+q,χ(−t,x|a1,... ,a r;b1,... ,b r)ts−1dt
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(/producttextr
+i=1χ(mi))
+[a1m1+···+armr+x]sq. (25)
+By (25), we can define Barnes’ type multiple q-l-function in C. Fors∈C,x∈Rwith
+x/negationslash= 0,−1,−2,..., let us define the Barnes’ type multiple q-l-function as follows :
+l(r)
+q(s,x|a1,... ,a r;b1,... ,b r)
+= [2]r
+q∞/summationdisplay
+m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(/producttextr
+i=1χ(mi))
+[a1m1+···+armr+x]sq. (26)
+Note that l(r)
+q(s,x|a1,... ,a r;b1,... ,b r) is a meromorphic function in whole complex
+s-plane. By (24), (25) and (26), we easily see that
+l(r)
+q(−n,x|a1,... ,a r;b1,... ,b r) =E(r)
+n,χ,q(x|a1,... ,a r;b1,... ,b r)
+forn∈Z+, (see [1-18]).
+References
+[1] E. W. Barnes, On the theory of multiple gamma function , Trans. Camb. Ohilos. Soc. A
+196(1904), 374-425.
+[2] I. N. Cangul,V. Kurt, H. Ozden, Y. Simsek, On the higher-order w-q-Genocchi numbers ,
+Adv. Stud. Contemp. Math. 19(2009), 39–57.
+[3] N. K.Govil, V. Gupta, Convergence of q-Meyer-Konig-Zeller-Durrmeyer operators , Adv.
+Stud. Contemp. Math. 19(2009), 97–108.
+[4] T. Kim, On aq-analogue of the p-adic log gamma functions and related integrals , J.Number
+Theory76(1999), 320–329.
+[5] T. Kim, q-Volkenborn integration , Russ. J. Math. Phys. 9(2002), 288–299.
+[6] T. Kim, On Euler-Barnes multiple zeta functions , Russ. J. Math. Phys. 10(2003), 261–267.
+10[7] T. Kim, Analytic continuation of multiple q-zeta functions and their values at negative
+integers, Russ. J. Math. Phys. 11(2004), 71–76.
+[8] T. Kim, The modified q-Euler numbers and polynomials , Adv. Stud. Contemp. Math. 16
+(2008), 161–170.
+[9] T. Kim, Note on the q-Euler numbers of higher order , Adv. Stud. Contemp. Math. 19
+(2009), 25–29.
+[10] T. Kim, Note on Dedekind type DC sums , Adv. Stud. Contemp. Math. 18(2009), 249–260.
+[11] T. Kim, Note on the Euler q-zeta functions , J. Number Theory 129(2009), 1798–1804.
+[12] T. Kim, A note on the generalized q-Euler numbers , Proc. Jangjeon Math. Soc. 12(2009),
+45–50.
+[13] T. Kim, Some identities on the q-Euler polynomials of higher order a nd q-stirling numbers
+by the fermionic p-adic integral on Zp, Russ. J. Math. Phys. 16(2009), 1061-9208.
+[14] T. Kim, Barnes type multiple q-zeta functions and q-Euler polynomials , arXiv:0912.5119v1.
+[15] T. Kim, Note on multiple q-zeta functions , to be appeared in Russ. J. Math. Phys.,
+arXiv:0912.5477v1.
+[16] T. Kim, On theq-extension of Euler and Genocchi numbers , J. Math. Anal. Appl. 326,
+1458–1465.
+[17] T. Kim, Onp-adicq-l-functions and sums of powers , J. Math. Anal. Appl. 329, 1472–1481.
+[18] T. Kim, Y. Simsek, Analytic continuation of the multiple Daehee q-l-functions associated
+with Daehee numbers , Russ. J. Math. Phys. 15(2008), 58–65.
+[19] Y. H. Kim, W. Kim, C. S. Ryoo, On the twisted q-Euler zeta function associated with
+twistedq-Euler numbers , Proc. Jangjeon Math. Soc. 12(2009), 93-100.
+[20] H.Ozden, I.N.Cangul, Y.Simsek, Remarks on q-Bernoulli numbers associated with Daehee
+numbers , Adv. Stud. Contemp. Math. 18(2009), 41-48.
+[21] K. Shiratani, S. Yamamoto, On ap-adic interpolation function for the Euler numbers and
+its derivatives , Mem. Fac. Sci., Kyushu University Ser. A 39(1985), 113-125.
+[22] Y. Simsek, Theorems on twisted L-function and twisted Bernoulli numbers , Advan. Stud.
+Contemp. Math. 11(2005), 205–218.
+[23] Z. Zhang, Y. Zhang, Summation formulas of q-series by modified Abel’s lemma , Adv. Stud.
+Contemp. Math. 17(2008), 119–129.
+Taekyun Kim
+Division of General Education-Mathematics, Kwangwoon Uni versity,
+Seoul 139-701, S. Korea e-mail: tkkim@kw.ac.kr
+Young-Hee Kim
+Division of General Education-Mathematics,
+Kwangwoon University,
+Seoul 139-701, S. Korea e-mail: yhkim@kw.ac.kr
+11
\ No newline at end of file
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+arXiv:1001.0035v2  [hep-th]  25 Mar 2010Reconstruction of Baxter Q-operator fromSklyanin SOV
+for cyclic representations ofintegrable quantum models
+G. Niccoli
+DESY,Notkestr. 85, 22603 Hamburg, GermanyDESY 09-227
+Abstract
+In [1], the spectrum (eigenvaluesand eigenstates) of a latt ice regularizationsof the Sine-
+Gordon model has been completely characterized in terms of p olynomial solutions with
+certain propertiesof the Baxter equation. This characteri zation for cyclic representations
+hasbeenderivedbytheuseofthe SeparationofVariables(SO V)methodofSklyaninand
+bythedirectconstructionoftheBaxter Q-operatorfamily. Here,wereconstructtheBaxter
+Q-operatorandthesamecharacterizationofthespectrumbyo nlyusingtheSOVmethod.
+This analysis allows us to deduce the main features required for the extension to cyclic
+representationsofotherintegrablequantummodelsofthis kindofspectrumcharacteriza-
+tion.
+Keywords: Integrable Quantum Systems;Separation of Variables;Baxt erQ-operator; PACScode 02.30.IK2
+1. Introduction
+Theintegrabilityofa quantummodelisbydefinitionrelated to theexistenceofamutuallycommu-
+tativefamily Qofself-adjointoperators Tsuchthat
+(A) [T,T′] = 0,
+(B) [T,U] = 0,
+(C) if [ T,O] = 0,∀T,T′∈ Q,
+∀T∈ Q,
+∀T∈ Q,thenO=O(Q),(1.1)
+whereUis the unitary operatordefining the time-evolutionin the mo del; note that the property(C)
+stays for the completeness of the family Q. In the framework of the quantum inverse scattering
+method [2, 3, 4] the Lax operator L(λ)is the mathematical tool which allows to define the transfer
+matrix:
+T(λ) = trC2M(λ),M(λ)≡/parenleftbiggA(λ)B(λ)
+C(λ)D(λ)/parenrightbigg
+≡LN(λ)...L1(λ),(1.2)
+aoneparameterfamilyofmutualcommutativeself-adjointo perators. Theintegrabilityofthemodel
+follows from T(λ)if the properties (B) and (C) of definition (1.1) can be proven for it. In some
+quantummodeltheintegrabilityisderivedbyprovingtheex istenceofafurtherone-parameterfamily
+ofself-adjointoperatorsthe Q-operatorwhichbydefinitionsatisfiesthefollowingproper ties:
+[Q(λ),Q(µ)] = 0,[T(λ),Q(µ)] = 0,∀λ,µ∈C, (1.3)
+plusthe Baxterequationwith thetransfermatrix:
+T(λ)Q(λ) =a(λ)Q(q−1λ)+d(λ)Q(qλ). (1.4)
+This is in particular the case for those models (like Sine-Go rdon [1]) for which the time-evolution
+operatorUis expressedin terms of Q. A naturalquestionarises: Is the integrablestructure of t hese
+quantummodelscompletelycharacterizedbythetransferma trixT(λ)?
+Note that a standard procedure1to prove the existence of Q(λ)is by a direct construction of an
+operatorsolutionoftheBaxterequation(1.4). Moreover,t hecoefficients a(λ)andd(λ)aswellasthe
+analytic and asymptotics properties of Q(λ)are some model dependent features which are derived
+bythe construction. Let usrecall thatthe generalstrategy [11, 12,13,14, 15] ofthisconstructionis
+to findagaugetransformation2such that the action of each gaugetransformedLax matrixon Q(λ)
+becomesupper-triangular. Thenthe Q-operatorassumesa factorized localformandthe problemof
+its existence in such a form is reduced to the problem of the ex istence of some model dependent
+specialfunction3.
+1Itis worth recalling that there are also others constructio ns of theQ-operator. An interesting example is presented in the
+series of works [5,6,7] by V.V. Bazhanov, S.L.Lukyanov and A .B.Zamolodchikov on the integrable structure of conformal
+field theories. In [6,7] the Q-operator is obtained asatransfer-matrix byatrace proced ure ofafundamental L-operator with
+q-oscillator representation for the auxiliary space (see al so [8, 9]). This construction can be extended to massive inte grable
+quantum field theories as itwas argued by thesame authors in [ 10].
+2Itleaves unchanged thetransfer matrix while modifies the mo nodromy matrix M(λ)defined in (1.2) .
+3Thequantumdilogarithm functions [16,17,18,19,20,21,22,23,24,25]forexample a ppear intheSinh-Gordon model
+[26],in their non-compact form,and in the Sine-Gordon model[1],in their cyclicform.3
+It is worth pointing out that on the one hand the construction of these special functionsfor general
+modelscanrepresenta concretetechnicalproblem4andthat ontheotherhandtheexistenceofsuch
+functionsis onlya sufficientcriterionforthe existence of Q(λ). It is then a relevantquestionif it is
+possibletobypassthiskindofconstructionprovidingadif ferentproofofthe existenceof Q(λ).
+Given an integrable quantum model the first fundamental task to solve is the exact solution of its
+spectral problem , i.e. the determination of the eigenvalues and the simultan eous eigenstates of the
+operator family Q, defined in (1.1). There are several methods to analyze this s pectral problem as
+thecoordinate Bethe ansatz [27, 28, 29], the TQmethod [28], the algebraic Bethe ansatz (ABA)
+[2, 3, 4], the analyticBethe ansatz [30] and the separation of variables (SOV) meth od of Sklyanin
+[31, 32, 33]; this last one seems to be more promising. Indeed , on the one hand it resolves the
+problems related to the reduced applicability of other meth ods (like ABA) and on the other hand
+it directly implies the completeness of the characterizati on of the spectrum which instead for other
+methodshastobeproven.
+For cyclic representations [34] of integrable quantum mode ls the SOV method should lead to the
+characterizationoftheeigenvaluesandthesimultaneouse igenstatesofthetransfermatrix T(λ)bya
+finite5systemofBaxter-likeequations. However,itisworthpoint ingoutthatsuchacharacterization
+of the spectrum is not the most efficient; this is in particula r true in view of the analysis of the
+continuum limit. Here the main question reads: Is it possibl e to define a set of conditions under
+whichtheSOVcharacterizationofthespectrumcanbereform ulatedintermsofafunctionalBaxter
+equation? In fact, this is equivalent to ask if we can reconst ruct theQ-operator from the finite
+system of Baxter-like equations. In this case the solution o f the spectral problem is reduced to the
+classification of the solutions of the Baxter equation which satisfy some analytic and asymptotic
+propertiesfixedbythe operators TandQ.
+The lattice Sine-Gordonmodelis used asa concreteexamplew herethese questionsaboutquantum
+integrability find a complete and affirmative answer. Indeed , in section 3, we show that the SOV
+characterization of the transfer matrix spectrum is exactl y equivalent to a functional equation of
+the form detD(Λ) = 0, whereD(λ)(see (3.21)) is a one-parameter family of quasi-tridiagonal
+matrices. In section 4, we show that this functional equatio n is indeed equivalent to the Baxter
+functional equation and, in section 5, we use these results t o reconstruct the Baxter Q-operator
+with the same level of accuracy obtained by the direct constr uction presented in [1]. It is worth
+pointingoutthat these resultsallowusto provethat thetra nsfermatrix T(λ)(plustheΘ-chargefor
+even chain) describes the family Qof complete commuting self-adjoint charges which implies t he
+quantum integrabilityof the model accordingto definition ( 1.1). So that in the Sine-Gordonmodel
+theBaxter Q-operatorplaysonlytheroleofa usefulauxiliaryobject.
+Let us point out that one of the main advantages of the spectru m characterization derived for the
+Sine-Gordonmodelisthe possibilityto proveanexactrefor mulationin termsof non-linearintegral
+4TheSine-Gordon model at irrational values of the coupling β2is asimple case where this kind of problem emerges.
+5Thenumber of equations in thesystem is finite and related to t hedimension of thecyclic representation.4
+equations6(NLIE).Thiswill bethe subjectof a futurepublicationwher ethe NLIEcharacterization
+will lead us by the implementation of the continuum limit to t he description of the Sine-Gordon
+spectrum in all the interesting regimes. These results will be shown to be consistent with those
+obtained previously in the literature7[37, 38, 39, 40, 41, 42] (see [43, 44] for reviews). Note that
+the methodbasedon thereformulationofthe spectralproble min termsofNLIEhasbeenalso used
+recently [49] to derive the Sinh-Gordonspectrum in finite vo lume and to characterize the spectrum
+in theinfraredandultravioletlimits.
+The analysis of the Sine-Gordon model allows us to infer the m ain features required to extend this
+kindofspectrumcharacterizationtocyclicrepresentatio nsofotherintegrablequantummodels. This
+is particularly relevant for those models for which a direct construction of the Baxter Q-operator
+encounterstechnicaldifficulties.
+Acknowledgments. I would like to thank J. Teschner for stimulating discussion s and suggestions on a prelimi-
+nary versionof this workand J.-M.Maillet for the interests hown.
+I gratefullyacknowledge support from the ECbythe Marie Cur ie Excellence GrantMEXT-CT-2006-042695.
+2. The Sine-Gordon model
+Weusethissectiontorecallthemainresultsderivedin[1]o nthedescriptionintermsofSOVofthe
+lattice Sine-Gordonmodel. This will be used as the starting point to introducea characterizationof
+the spectrumof the transfermatrix T(λ)which will lead to the constructionof the Q-operatorfrom
+SOV.
+2.1 Definitions
+Thelattice Sine-Gordonmodelcanbecharacterizedbythefo llowingLaxmatrix8:
+LSG
+n(λ) =κn
+i/parenleftigg
+iun(q−1
+2κnvn+q+1
+2κ−1
+nv−1
+n)λnvn−λ−1
+nv−1
+n
+λnv−1
+n−λ−1
+nvniu−1
+n(q+1
+2κ−1
+nvn+q−1
+2κnv−1
+n)/parenrightigg
+,(2.5)
+whereλn≡λ/ξnfor anyn∈ {1,...,N}withξnandκnparameters of the model. For any n∈
+{1,...,N}thecoupleofoperators( un,vn)defineaWeyl algebra Wn:
+unvm=qδnmvmun,whereq=e−πiβ2. (2.6)
+We will restrictourattentiontothecase inwhich qisarootofunity,
+β2=p′
+p, p,p′∈Z>0, (2.7)
+6Thistype of equations werebefore introduced in adifferent framework in [35,36]
+7See [45, 46] for a related model analyzed in the framework of A BA and [47, 48] for the corresponding finite volume
+continuum limit.
+8Thelattice regularization of the Sine-Gordon model that we consider here goes back to [4,50] and is related to formula-
+tions which have morerecently been studied in [51,52,53].5
+withp≡2l+ 1odd andp′even so that qp= 1. In this case each Weyl algebra Wnadmits a
+finite-dimensional representation of dimension p. In fact, we can represent the operators un,vnon
+thespace ofcomplex-valuedfunctions ψ:SN
+p→Cas
+un·ψ(z1,...,z N) =unznψ(z1,...,z n,...,z N),
+vn·ψ(z1,...,z N) =vnψ(z1,...,q−1zn,...,z N).(2.8)
+whereSp={q2n;n= 0,...,2l}is a subset of the unit circle; note that Sp={qn;n= 0,...,2l}
+sinceq2l+2=q.
+Themonodromymatrix M(λ)definedin(1.2)intermsoftheLax-matrix(2.5)satisfiesthe quadratic
+relations:
+R(λ/µ)(M(λ)⊗1)(1⊗M(µ)) = (1⊗M(µ))(M(λ)⊗1)R(λ/µ), (2.9)
+wheretheauxiliary R-matrixisgivenby
+R(λ) =
+qλ−q−1λ−1
+λ−λ−1q−q−1
+q−q−1λ−λ−1
+qλ−q−1λ−1
+. (2.10)
+The elements of M(λ)generate a representation RNof the so-called Yang-Baxter algebra char-
+acterized by the 4Nparametersκ= (κ1,...,κ N),ξ= (ξ1,...,ξ N),u= (u1,...,u N)and
+v= (v1,...,v N); in the present paper we will restrict to the case un= 1,vn= 1,n= 1,...,N.
+The commutation relations (2.9) are at the basis of the proof of the mutual commutativity of the
+T-operators.
+Inthecase ofa latticewith Nevenquantumsites, we havealso tointroducetheoperator:
+Θ =N/productdisplay
+n=1v(−1)1+n
+n, (2.11)
+whichplaystheroleofa gradingoperator inthe Yang-Baxteralgebra:
+Proposition 6 of [1] Θcommuteswiththetransfermatrixandsatisfiesthefollowin gcommutation
+relationswith theentriesofthemonodromymatrix:
+ΘC(λ) =qC(λ)Θ,[A(λ),Θ] = 0, (2.12)
+B(λ)Θ =qΘB(λ),[D(λ),Θ] = 0. (2.13)
+Moreover,the Θ-chargeallowstoexpressthe asymptoticsofthetransferma trixas:
+lim
+logλ→∓∞λ±NT(λ) =/parenleftiggN/productdisplay
+a=1κaξ±1
+a
+i/parenrightigg
+/parenleftbig
+Θ+Θ−1/parenrightbig
+. (2.14)6
+Let us denotewith ΣTthe spectrum(the set of the eigenvaluefunctions t(λ)) of the transfer matrix
+T(λ). By the definitions(1.2) and (2.5), then ΣTis contained9inC[λ2,λ−2](N+eN−1)/2, where we
+haveusedthenotatione N= 0forNoddand1forNeven.
+Notethat inthecase of Neven,the Θ-chargenaturallyinducesthegrading ΣT=/uniontextl
+k=0Σk
+T,where:
+Σk
+T≡/braceleftigg
+t(λ)∈ΣT: lim
+logλ→∓∞λ±Nt(λ) =/parenleftiggN/productdisplay
+a=1κaξ±1
+a
+i/parenrightigg
+(qk+q−k)/bracerightigg
+.(2.15)
+This simply follows by the asymptotics of T(λ)and by its commutativity with Θ. In particular,
+anyt(λ)∈Σk
+Tis aT-eigenvalue corresponding to simultaneous eigenstates of T(λ)andΘwith
+Θ-eigenvalues q±k.
+2.2 CyclicSOVrepresentations
+TheseparationofvariablesmethodofSklyaninisbasedonth eobservationthatthespectralproblem
+forT(λ)simplifies considerablyif one worksin an auxiliaryreprese ntationwherethe commutative
+familyofoperators B(λ)isdiagonal.
+InthecaseoftheSine-Gordonmodelthevectorspace10CpNunderlyingtheSOVrepresentationcan
+beidentifiedwiththespaceoffunctions Ψ(η)definedforηtakenfromthediscreteset
+BN≡/braceleftbig
+(qk1ζ1,...,qkNζN); (k1,...,k N)∈ZN
+p/bracerightbig
+, (2.16)
+onthesefunctions B(λ)actsasa multiplicationoperator,
+BN(λ)Ψ(η) =ηeN
+Nbη(λ)Ψ(η), b η(λ)≡N/productdisplay
+n=1κn
+i[N]/productdisplay
+a=1(λ/ηa−ηa/λ) ; (2.17)
+where[N]≡N−eNandη1,...,η[N]are the zerosof bη(λ). In the case of even Nit turns out that
+we needa supplementaryvariable ηNinordertobeable toparameterizethe spectrumof B(λ).
+In[1]wehaveproventhatforgeneralvaluesoftheparameter sκandξoftheoriginalrepresentation
+it is possible to construct these SOV representationsand mo reoverwe have defined the map which
+fixestheSOVparameter ηintermsoftheparameters κandξ.
+In these SOV representations the spectral problem for T(λ)is reduced to the following discrete
+system ofBaxter-likeequationsin thewave-function Ψt(η) =/a\}bracketle{tη|t/a\}bracketri}htofaT-eigenstate |t/a\}bracketri}ht:
+t(ηr)Ψ(η) =a(ηr)T−
+rΨ(η)+d(ηr)T+
+rΨ(η)∀r∈ {1,...,[N]},(2.18)
+9Herewith C[x,x−1]Mwearedenoting the linear space ofthe Laurentpolynomials o f degreeMin thevariable x∈C.
+10It is always possible to provide the structure of Hilbert spa ce to this finite-dimensional linear space. In particular, t he
+scalar product in the SOVspace is naturally introduced by th e requirement that the transfer matrix is self-adjoint in th e SOV
+representation. Appendix B addresses this issue.7
+whereT±
+raretheoperatorsdefinedby
+T±
+rΨ(η1,...,η N) = Ψ(η1,...,q±1ηr,...,η N),
+whilethe coefficients a(λ)andd(λ)are definedby:
+a(λ) =N/productdisplay
+n=1κn
+iλn(1−iq−1/2λnκn)(1−iq−1/2λn
+κn),d(λ) =qNa(−λq).(2.19)
+Inthecase of Nevenwe haveto addto thesystem(2.18) thefollowingequatio ninthevariable ηN:
+T+
+NΨ±k(η) =q±kΨ±k(η), (2.20)
+fort(λ)∈Σk
+Twithk∈ {0,...,l}.NotethatthecyclicityoftheseSOVrepresentationsisexpr essed
+bytheidentificationof (T±
+j)pwith theidentityforany j∈ {1,...,N}.
+3. SOV characterization of T-eigenvalues
+Let usintroducetheoneparameterfamily D(λ)ofp×pmatrix:
+D(λ)≡
+t(λ)−d(λ) 0 ··· 0 −a(λ)
+−a(qλ)t(qλ)−d(qλ) 0 ··· 0
+0......
+... ···...
+... ···...
+...... 0
+0... 0−a(q2l−1λ)t(q2l−1λ)−d(q2l−1λ)
+−d(q2lλ) 0 ... 0−a(q2lλ)t(q2lλ)
+(3.21)
+wherefornow t(λ)isjust anevenLaurentpolynomialofdegree N+eN−1inλ.
+Lemma 1. Thedeterminant detpDisanevenLaurentpolynomialofmaximaldegree N+eN−1in
+Λ≡λp.
+Proof.Let us start observingthat D(λq)is obtainedby D(λ)exchangingthe first and p-th column
+andafterthefirst and p-throw,so that
+det
+pD(λq) = det
+pD(λ)∀λ∈C, (3.22)
+whichimpliesthat detpDisfunctionof Λ. Let usdevelopthedeterminant:
+det
+pD(Λ) =p/productdisplay
+h=1a(λqh)+p/productdisplay
+h=1a(−λqh)−qNa(λ)a(−λ) det
+2l−1D(1,2l+1),(1,2l+1)(λ)
+−qNa(λq)a(−λq) det
+2l−1D(1,2),(1,2)(λ)+t(λ)det
+2lD1,1(λ), (3.23)8
+whereD(h,k),(h,k)(λ)denotes the (2l−1)×(2l−1)sub-matrix of D(λ)obtained removing the
+rowsandcolumns handkwhileDh,k(λ)denotesthe 2l×2lsub-matrixof D(λ)obtainedremoving
+therowhandcolumn k. Theinteresttowardthisdecompositionof detpD(Λ)isduetothefact that
+the matrices D(1,2),(1,2)(λ),D(1,2l+1),(1,2l+1)(λ)andD1,1(λ)aretridiagonal matrices. Following
+thesamereasoningusedinLemma4toprovethat det2lD1,1(λ)isanevenfunctionof λwecanalso
+showthatthisistruefor det2l−1D(1,2),(1,2)(λ)anddet2l−1D(1,2l+1),(1,2l+1)(λ). Fromtheparityof
+these functionsthe parityof detpD(Λ)followsbyusing(3.23).
+Beinga(λ),d(λ)andt(λ)Laurentpolynomialofdegree Ninλ,inthecaseof Neventhestatement
+ofthelemmaisalreadyproven;so wehavejust toshowthat:
+lim
+logΛ→∓∞Λ±Ndet
+pD(Λ) = 0 (3.24)
+forNoddwhichfollowsobservingthat:
+lim
+logΛ→∓∞Λ±Ndet
+pD(Λ) =i±pNN/productdisplay
+n=1κp
+nξ±p
+ndet
+p/vextenddouble/vextenddouble/vextenddoubleq−(1∓1)N/2δh,k+1−q(1∓1)N/2δh,k−1/vextenddouble/vextenddouble/vextenddouble.
+(3.25)
+The interesttowardthe function detpD(Λ)isdueto the fact thatit allowsthefollowingcharacteri-
+zationofthe T-spectrum:
+Lemma 2. ΣTis the set of all the functions t(λ)∈C[λ2,λ−2](N+eN−1)/2which satisfy the system
+of equations:
+det
+pD(ηp
+a) = 0∀a∈ {1,...,[N]}and(η1,...,η[N])∈BN, (3.26)
+plusin thecaseof Neven:
+lim
+logΛ→∓∞Λ±Ndet
+pD(Λ) = 0. (3.27)
+Proof.The requirement that the system of equations (2.18) admits a non-zerosolution leads to the
+equations(3.26),while theequation(3.27) foreven Nsimplyfollowsbyobservingthat:
+lim
+logΛ→∓∞Λ±Ndet
+pD(Λ) = det
+p/vextenddouble/vextenddouble/vextenddoubleq(1∓1)N/2δi,j−1+q−(1∓1)N/2δi,j+1−(qk+q−k)δi,j/vextenddouble/vextenddouble/vextenddouble
+×(−1)N/productdisplay
+n=1/parenleftbig
+iκnξ±
+n/parenrightbigp= 0. (3.28)
+Note that the above characterization of the T-spectrum ΣTrequires as input the knowledge of BN,
+i.e. the lattice of zeros of the operator B(λ). It is so interesting to notice that this characterization9
+has in fact a reformulation which is independent from the kno wledge of BN. To explain this let us
+notethatLemma1allowsto introducethefollowingmap:
+Dp,N:t(λ)∈C[λ2,λ−2](N+eN−1)/2→ Dp,N(t(λ))≡det
+pD(Λ)∈C[Λ2,Λ−2](N+eN−1)/2.
+(3.29)
+Intermsofthismapwecanintroduceafurthercharacterizat ionofthespectrumofthetransfermatrix
+T(λ).
+Theorem 1. The spectrum ΣTof the transfer matrix T(λ)coincides with the kernel NDp,N⊂
+C[λ2,λ−2](N+eN−1)/2ofthe map Dp,N.
+Proof.The inclusion NDp,N⊂ΣTis trivial thanks to Lemma 2, vice-versa if t(λ)∈ΣTthen
+the function detpD(Λ)is zero in N+eNdifferent values of Λ2which thanks to Lemma 1 implies
+detpD(Λ)≡0,i.e.ΣT⊂ NDp,N.
+That is the set of eigenvalues of the transfer matrix T(λ)is exactly characterized as the subset of
+C[λ2,λ−2](N+eN−1)/2whichcontainsallthesolutionsofthefunctionalequation detpD(Λ) = 0. In
+thenextsectionwewill showthat thisfunctionalequationi s nothingelse thattheBaxterequation.
+Remark 1. Let us note that the same kind of functional equation detD(Λ) = 0 also appears
+in [54, 55, 56]. There it recasts, in a compact form, the funct ional relations which result from the
+truncatedfusionsoftransfermatrixeigenvalues. Itissor elevanttopointoutthatforthe BBS-model11
+in the SOV representation the non-triviality condition of t he solutions of the system of Baxter-like
+equations has been shown [60] to be equivalent to the truncat ion identity in the fusion of transfer
+matrixeigenvalues.
+4. Baxterfunctional equation
+The main consequence of the previous analysis is that it natu rally leads to the complete character-
+ization of the transfer matrix spectrum in terms of polynomi al solutions of the Baxter functional
+equation.
+Theorem2. Lett(λ)∈ΣTthent(λ)definesuniquelyuptonormalizationapolynomial Qt(λ)that
+satisfiestheBaxterfunctionalequation:
+t(λ)Qt(λ) =a(λ)Qt(λq−1)+d(λ)Qt(λq)∀λ∈C. (4.30)
+Proof.The fact that given a t(λ)∈C[λ2,λ−2](N+eN−1)/2there exists up to normalizationat most
+one polynomial Qt(λ)that satisfies the Baxter functional equationhas been prove nin Lemma 2 of
+[1]. So we have to prove only the existence of Qt(λ)∈C[λ]. An interesting point about the proof
+givenhereisthatit isa constructiveproof.
+11TheBBS-model [12, 57,58,59] has been analyzed in the SOVapp roach in aseries of works [60,61,62].10
+Let usnoticethatthe condition t(λ)∈ΣT≡ NDp,Nimpliesthatthe p×pmatrixD(λ)hasrank2l
+foranyλ∈C\{0}. Letusdenotewith
+Ci,j(λ) = (−1)i+jdet
+2lDi,j(λ) (4.31)
+the(i,j)cofactorof the matrix D(λ); then the matrix formedout of these cofactorshasrank 1, i.e.
+all thevectors:
+Vi(λ)≡(Ci,1(λ),Ci,2(λ),...,Ci,2l+1(λ))T∈Cp∀i∈ {1,...,2l+1}(4.32)
+areproportional:
+Vi(λ)/Ci,1(λ) =Vj(λ)/Cj,1(λ)∀i,j∈ {1,...,2l+1},∀λ∈C. (4.33)
+Theproportionality(4.33)oftheeigenvectorsV i(λ)implies:
+C2,2(λ)/C2,1(λ) =C1,2(λ)/C1,1(λ) (4.34)
+which,byusingtheproperty(A.69),canberewrittenas:
+C1,1(λq)/C1,2l+1(λq) =C1,2(λ)/C1,1(λ). (4.35)
+Moreover,thefirst elementinthe vectorialcondition D(λ)V1(λ) =0¯reads:
+t(λ)C1,1(λ) =a(λ)C1,2l+1(λ)+d(λ)C1,2(λ). (4.36)
+Let us note that from the form of a(λ),d(λ)andt(λ)∈ΣTit follows that all the cofactors are
+Laurentpolynomialofmaximaldegree122lNinλ:
+Ci,j(λ) = Ci,jλ−2lN+ai,j4lN−(ai,j+bi,j)/productdisplay
+h=1(λ(i,j)
+h−λ). (4.37)
+In Lemma 5, we show that the equations (4.35) and (4.36) imply that if C 1,1(λ)has a common
+zero with C 1,2(λ)then this is also a zero of C 1,2l+1(λ)and that the same statement holds ex-
+changing C 1,2(λ)with C 1,2l+1(λ). So we can denote with C1,1C1,1(λ),C1,2l+1C1,2l+1(λ)and
+C1,2C1,2(λ)the polynomials of maximal degree 4lNobtained simplifying the common factors in
+C1,1(λ), C1,2l+1(λ)and C1,2(λ). Then,byequation(4.35),theyhavetosatisfythe relation s:
+C1,2l+1(λ) =q¯N1,1C1,1(λq−1),C1,2(λ) =q−¯N1,1C1,1(λq)andC1,2l+1=ϕC1,1,(4.38)
+whereϕ≡C1,1/C1,2and¯N1,1is the degree of the polynomial C1,1(λ). So that equation (4.36)
+assumestheformofa Baxterequationin thepolynomial C1,1(λ):
+t(λ)C1,1(λ) = ¯a(λ)C1,1(λq−1)+¯d(λ)C1,1(λq), (4.39)
+12Theai,jandbi,jare non-negative integers and λ(i,j)
+h/ne}ationslash= 0for anyh∈ {1,...,4lN−(ai,j+bi,j)}.11
+with coefficients ¯a(λ)≡q¯N1,1ϕa(λ)and¯d(λ)≡q−¯N1,1ϕ−1d(λ). Note that the consistence of the
+aboveequationimpliesthat ϕisap-rootoftheunity. Indeed,denotingwith ¯D(Λ)thematrixdefined
+asin(3.21) butwithcoefficients ¯a(λ)and¯d(λ), equation(4.39) implies:
+0 = det
+p¯D(Λ)≡(ϕp−1)/parenleftiggp/productdisplay
+h=1a(λqh)−ϕ−pp/productdisplay
+h=1a(−λqh)/parenrightigg
+. (4.40)
+The expansionfor detp¯D(Λ)in (4.40) is derivedby using the expansion(3.23) for detp¯D(Λ), the
+formulae13:
+det
+2lD1,1(λ) = det
+2lD1,1(λ), (4.41)
+det
+2l−1D(1,2),(1,2)(λ) = det
+2l−1D(1,2),(1,2)(λ), (4.42)
+det
+2l−1D(1,2l+1),(1,2l+1)(λ) = det
+2l−1D(1,2l+1),(1,2l+1)(λ), (4.43)
+andthecondition t(λ)∈ΣT. Finally,if wedefine:
+Qt(λ)≡λaC1,1(λ), (4.44)
+whereq−a=q¯N1,1ϕwitha∈ {0,..,2l},we getthestatementofthetheorem.
+Remark2. Theprevioustheoremimpliesthatforany t(λ)∈ΣTthepolynomialsolution Qt(λ)of
+theBaxterequationcanberelatedtothedeterminantofatri diagonalmatrixoffinitesize p−1. Note
+thatthe spectrumoftheSine-Gordonmodelinthecase ofirra tionalcoupling ¯β2shouldbededuced
+fromβ2=p′/prational in the limit β2→¯β2. In particular, this implies that underthis limit ( p→
++∞)thedimensionoftherepresentationdivergesaswellasthe sizeofthetridiagonalmatrixwhose
+determinant is associated to the solution Qt(λ)of the Baxter equation. It is then relevant to point
+out that in the case of the quantum periodic Toda chain the sol utions of the corresponding Baxter
+equationareexpressedintermsofdeterminantsofsemi-infi nitetridiagonalmatrices[63,13, 64].
+It is worth noticing that the set of polynomials Qt(λ), introducedin the previoustheorem,admitsa
+moreprecisecharacterization:
+Theorem 3. Lett(λ)∈ΣTthent(λ)defines uniquely up to normalization a polynomial solution
+Qt(λ)oftheBaxterfunctionalequation(4.30) ofmaximaldegree 2lN.
+Inthecase Nodd,it results:
+Qt(0)≡Q0/\e}atio\slash= 0,andlim
+λ→∞λ−2lNQt(λ)≡Q2lN/\e}atio\slash= 0. (4.45)
+In the case Neven, the condition (4.45) selects t(λ)∈Σ0
+Twhile fort(λ)∈Σk
+Twithk∈ {1,...,l}
+we havethecharacterization Q0=Q2lN= 0and:
+lim
+λ→0Qt(λq)
+Qt(λ)=q±k,lim
+λ→∞Qt(λq)
+Qt(λ)=q−(N±k). (4.46)
+13They follow from the tridiagonality of these matrices and by using Lemma3.12
+Proof.Thankstoformula(A.74),thecofactor C 1,1(λ)∈C[λ,λ−1]2lNiseveninλandso it admits
+theexpansions:
+C1,1(λ) = C1,1λ−2lN+2˜a1,12lN−(˜a1,1+˜b1,1)/productdisplay
+i=1(λ(1,1)
+i−λ)(λ(1,1)
+i+λ).(4.47)
+Let us note now that by using the properties(A.69) and (A.74) , the relation (4.34) can be rewritten
+as:
+C1,1(λq)C1,1(λ) =qNC1,2(λ)C1,2(−λ). (4.48)
+Usingthat andthegeneralrepresentation(4.37)forthe cof actor C 1,2(λ), weget:
+a1,2= 2˜a1,1≡2a,b1,2= 2˜b1,1≡2b,C2
+1,2=C2
+1,1q−2(N+b)(4.49)
+and:/parenleftig
+λ(1,1)
+i/parenrightig2
+=/parenleftig
+λ(1,2)
+i/parenrightig2
+≡¯λ2
+i,/parenleftig
+λ(1,2)
+i+2lN−(a+b)/parenrightig2
+=/parenleftbig¯λi/q/parenrightbig2(4.50)
+with¯λi/\e}atio\slash= 0for anyi∈ {1,...,2lN−(a+b)}withaandb∈Z≥0. Note that the equation (4.49)
+andthefactthat ϕ≡C1,1/C1,2isap-rootofthe unityimply ϕ=qb+N. Thenwecanwrite:
+C1,1(λ) = Cλ−2lN+2a2lN−(a+b)/productdisplay
+i=1(¯λi+λ)(¯λi−λ), (4.51)
+C1,2(λ) =qaCλ−2lN+2a2lN−(a+b)/productdisplay
+i=1(¯λi+λ)((−1)H(x−i)¯λi−λq), (4.52)
+whereC≡C1,1andH(n)≡ {0forn <0,1forn≥0}is the Heaviside step function. Here, x
+isanon-negativeintegerwhichisfixedtozerothankstoform ula(4.38). Thenthesolution Qt(λ)of
+theBaxter equation(4.30) belongsto C[λ]2lNandhastheform:
+Qt(λ)≡λa2lN−(a+b)/productdisplay
+i=1(¯λi−λ). (4.53)
+Let usshow nowthe remainingstatementsof thetheoremconce rningthe asymptoticsof Qt(λ). To
+thisaimwe computethe limits:
+lim
+logλ→∓∞λ±2lNC1,1(λ) = det
+2l/vextenddouble/vextenddouble/vextenddoubleq−(1∓1)N/2δi,j+1+q(1∓1)N/2δi,j−1−(qk+q−k)δeN,1δi,j/vextenddouble/vextenddouble/vextenddouble
+i/ne}ationslash=1,j/ne}ationslash=1
+×N/productdisplay
+h=1(κhξ±1
+h
+i)2l= (δeN,1(1+(2l+1)δk,0)−1)N/productdisplay
+h=1(κhξ∓1
+h
+i)2l,(4.54)
+whichimply:
+a=b= 0, (4.55)13
+forNoddandNevenwitht(λ)∈Σ0
+T,i.e. thecondition(4.45). Inthe remainingcases, Nevenand
+t(λ)/∈Σ0
+T,the sameformulaimplies:
+a/\e}atio\slash= 0,b/\e}atio\slash= 0, (4.56)
+sothatQ0=Q2lN= 0,whiletheasymptoticsbehaviors(4.46)simplyfollowtaki ngtheasymptotics
+oftheBaxterequationsatisfied by Qt(λ).
+5.Q-operator: Existence andcharacterization
+Let us denote with Σtthe eigenspace of the transfer matrix T(λ)corresponding to the eigenvalue
+t(λ)∈ΣT,then:
+Definition 1. LetQ(λ)betheoperatorfamily definedby:
+Q(λ)|t/a\}bracketri}ht ≡Qt(λ)|t/a\}bracketri}ht ∀|t/a\}bracketri}ht ∈Σtand∀t(λ)∈ΣT, (5.57)
+withQt(λ)the element of C[λ]2lNcorresponding to t(λ)∈ΣTby the injection defined in the
+previoustheorem.
+Under the assumptions ξandκreal or imaginarynumbers, which assure the self-adjointne ssof the
+transfermatrix T(λ)forλ∈R,thefollowingtheoremholds:
+Theorem4. Theoperatorfamily Q(λ)isaBaxter Q-operator:
+(A)Q(λ)satisfieswith T(λ)thecommutationrelations:
+[Q(λ),T(µ)] = [Q(λ),Q(µ)] = 0∀λ,µ∈C, (5.58)
+plusthe Baxterequation:
+T(λ)Q(λ) =a(λ)Q(λq−1)+d(λ)Q(λq)∀λ∈C. (5.59)
+(B)Q(λ)isa polynomialofdegree 2lNinλ:
+Q(λ)≡2lN/summationdisplay
+n=0Qnλn,
+with coefficients Qnself-adjointoperators.
+(C)Inthecase Nodd,the operator Q2lN=idandQ0isaninvertibleoperator.
+(D)Inthecase Neven,Q(λ)commuteswiththe Θ-chargeandtheoperator Q2lNistheorthogonal
+projectionontothe Θ-eigenspacewith eigenvalue1. Q0hasnon-trivialkernel coincidingwith
+theorthogonalcomplementto the Θ-eigenspacewith eigenvalue1.14
+Proof.Note that the self-adjointness of the transfer matrix T(λ)implies that Q(λ)is well defined,
+indeed its action is defined on a basis. The property (A) is a tr ivial consequence of Definition 1.
+Notethat theinjectivityofthemap t(λ)∈ΣT→Qt(λ)∈C[λ]2lNimplies:
+(Qt(λ))∗=Qt(λ∗)∀λ∈C (5.60)
+being(a(λ))∗=d(λ∗)and(t(λ))∗=t(λ∗). So we get the Hermitian conjugation property
+(Q(λ))†=Q(λ∗), i.e. the self-adjointness of the operators Qn. The properties (C) and (D) of
+the operators Q0andQ2lNdirectly follow from the asymptotics of the eigenfunction Qt(λ)while
+thecommutativityof Q(λ)andΘisa directconsequenceofthecommutativityof T(λ)andΘ.
+6. Conclusion
+Intheprevioussectionwehaveshownthatbyonlyusingthech aracterizationofthespectrumofthe
+transfer matrix obtained by the SOV method we were able to rec onstruct the Q-operator. It is also
+interestingto pointoutastheresultsderivedin [1]togeth erwiththoseofthepresentarticleyield:
+Theorem5. Thefamily Qwhichcharacterizesthequantumintegrabilityofthelatti ceSine-Gordon
+model(see definition(1.1)) isdescribedby thetransfermat rixT(λ)fora chainwith Noddnumber
+of siteswhile by T(λ)plustheΘ-chargefora chainwith Nevennumberof sites.
+Proof.LetusstartnoticingthatProposition3andTheorem4of[1]a rederivedonlyusingtheSOV
+method (i.e. without any assumption about the existence of t heQ-operator). So only using SOV
+analysis we have derived that for Nodd the transfer matrix T(λ)has simple spectrum while for
+Neven this is true for T(λ)plus theΘ-charge; i.e. they define a complete family of commuting
+observables and so satisfy the properties(A) and (C) of the d efinition (1.1). In this article we have
+moreover shown that the Q-operator is defined as a function of the transfer matrix whic h implies
+the property(B) of (1.1) recalling that in [1] the time-evol utionoperator Uhas been expressed as a
+functionofthe Q-operator.
+Let us shortly point out the main features required in abstra ct to extend to cyclic representationsof
+other integrable quantum models the same kind of spectrum ch aracterization derived here for the
+lattice Sine-Gordonmodel.
+R1.The model admits an SOV description and the spectrum of the tr ansfer matrix can be charac-
+terizedbyasystem ofBaxter-likeequationsin the T-wave-function Ψ(η) =/a\}bracketle{tη|t/a\}bracketri}ht:
+t(ηr)Ψ(η) =a(ηr)Ψ(η1,...,q−1ηr,...,η N)+d(ηr)Ψ(η1,...,qη r,...,η N),(6.61)
+where(η1,...,ηN)∈BNwithBNtheset ofzerosofthe B-operatorintheSOV representation.
+Here,theparameter qisa rootofunitydefinedasin (2.6) and(2.7).15
+Note that for cyclic representationsof an integrable quant um model the set BNis a finite subset of
+CN. So the coefficients a(ηr)andd(ηr)are specified only in a finite number of points where they
+satisfy thefollowingaveragevaluerelations14:
+A(ηp
+r) =p/productdisplay
+k=1a(qkηr),D(ηp
+r) =p/productdisplay
+k=1d(qkηr). (6.62)
+HereA(Λ)andD(Λ)are the average values of the operator entries A(λ)andD(λ)of the mon-
+odromy matrix. Let us recall that the operator entries of the monodromymatrix are expected to be
+polynomials(orLaurentpolynomials)inthespectralparam eterλsothecorrespondingaverageval-
+uesarepolynomials(orLaurentpolynomials)in Λ≡λp. Itisthennaturaltointroducethefunctions
+a(λ)andd(λ)aspolynomial(orLaurentpolynomial)solutionsofthefoll owingaveragerelations:
+A(Λ)+γB(Λ) =p/productdisplay
+k=1a(qkλ),D(Λ)+δB(Λ) =p/productdisplay
+k=1d(qkλ), (6.63)
+whereB(Λ)istheaveragevalueoftheoperator B(λ)andγandδare constanttobe fixed.
+R2.Let usdenotewith Zf(λ)the set ofthezerosofthefunctions f(λ), then:
+∃λ0∈Za(λ):λ0/∈ ∪2l−1
+h=0Zd(λqh). (6.64)
+R3.Theaveragevaluesofthefunctions aanddarenotcoincidinginallthezerosofthe B-operator:
+A(ηp
+a)/\e}atio\slash=D(ηp
+a)∀a∈ {1,...,[N]}and(η1,...,η[N])∈BN. (6.65)
+The requirement R1yields the introduction of the p×pmatrixD(λ), defined as in (3.21), by
+the functions a(λ)andd(λ)solutions of (6.63). This should allow us to reformulate the spectral
+problem for the transfer matrix as the problem to classify al l the solutions t(λ)to the functional
+equationdetpD(Λ) = 0ina modeldependentclassoffunctions.
+The requirement R2implies that the rank of the matrix D(λ)is almost everywhere 2l. Indeed, the
+condition (6.64) implies C 1,p(λ0)/\e}atio\slash= 0, independently from the function t(λ). Being the cofactor
+C1,p(λ)acontinuousfunctionofthespectralparametertheabovest atementontherankofthematrix
+D(λ)follows. Underthisconditionwecanfollowtheprocedurepr esentedinTheorem2toconstruct
+the solutionsof the Baxter equation. Then the self-adjoint nessof the transfer matrix Tallows us to
+proceedasinsection5to showthe existenceofthe Q-operatorasa functionof T.
+The requirement R3is a sufficient criterion15to show the simplicity of the spectrum of Twhich
+should imply that the full integrable structure of the quant um model should be described by the
+14Theequations in (6.62) are trivial consequences of the SOVr epresentation and of the cyclicity.
+15It is worth noticing that in the case of the Sine-Gordon model the criterion R3does not apply to the representations
+withun=vn= 1. Nevertheless, we have shown the simplicity of Tby using some model dependent properties of the
+coefficients a(λ)andd(λ), see section 5of [1].16
+transfermatrixassoonastheproperty(B)indefinition(1.1 )isshownforthemodelunderconsider-
+ation.
+Following the schema here presented, in a future publicatio n we will address the analysis of the
+spectrumfortheso-called α-sectorsoftheSine-Gordonmodel(see[1]). Theuseofthisapproachi s
+in particularrelevantin these sectorsof theSine-Gordonm odelbecausea direct constructionof the
+Q-operatorleadstosometechnicaldifficulty.
+A. Properties ofthecofactors C i,j(λ)
+Let usconsideran M×Mtridiagonal matrix16O:
+O≡
+z1y10··· 0 0
+x1z2y20··· 0
+0x2z3y3...
+.........
+...... 0
+0...0xM−2zM−1yM−1
+0 0...0xM−1zM
+(A.66)
+i.e. a matrix with non-zero entries only along the principal diagonal and the next upper and lower
+diagonals.
+Lemma 3. The determinantof atridiagonalmatrix is invariantundert he transformation ̺αwhich
+multiplies for αthe entries above the diagonal and for α−1the entries below the diagonal leaving
+theentriesonthediagonalunchanged.
+Proof.Letusnotethatthedeterminantofa tridiagonalmatrixadmi tsthefollowingexpansion:
+det
+MO=z1det
+M−1O1,1+x1y1det
+M−2O(1,2),(1,2), (A.67)
+wherewe haveused thesame notationsintroducedafterformu la(3.23). By usingit, we getthat the
+actionof̺αreads:
+det
+M̺α(O) =z1det
+M−1̺α(O)1,1+x1y1det
+M−2̺α(O)(1,2),(1,2). (A.68)
+Then the statement follows by induction noticing that the tr ansformation ̺αleaves always un-
+changedthedeterminantofa 2×2matrix.
+16An interesting analysis of the eigenvalue problem for tridi agonal matrices is presented in [65].17
+Lemma 4. Thefollowingpropertieshold:
+Ch+i,k+i(λ) =Ch,k(λqi)∀i,h,k∈ {1,...,2l+1}, (A.69)
+and:
+C1,1(λ) =C1,1(−λ)andC2,1(λ) =qNC1,2(−λ). (A.70)
+Proof.Note that by the definition (4.31) of the cofactors C i,j(λ)the equations (A.69) are simple
+consequencesof qp= 1andareprovenexchangingrowsandcolumnsin thedeterminan ts.
+Let us provenow that the cofactor C 1,1(λ) = det 2lD1,1(λ)is an even function of λ. The tridiago-
+nalityofthematrix D1,1(λ)allowsusto usethepreviouslemma:
+C1,1(λ)≡det
+2l/vextenddouble/vextenddoublet(λqh)δh,k−a(λqh)δh,k+1−qNa(−λqh+1)δh,k−1/vextenddouble/vextenddouble
+h>1,k>1
+= det
+2l/vextenddouble/vextenddoublet(λqh)δh,k−qNa(λqh)δh,k+1−a(−λqh+1)δh,k−1/vextenddouble/vextenddouble
+h>1,k>1
+= det
+2l/vextenddouble/vextenddoublet(λqh)δh,k−d(−λqk)δk,h−1−a(−λqk)δk,h+1/vextenddouble/vextenddouble
+h>1,k>1
+≡det
+2l(D1,1(−λ))T=C1,1(−λ). (A.71)
+To provenowthesecondrelationin (A.70) weexpandthe cofac tors:
+C2,1(λ) =2l+1/productdisplay
+h=2a(λqh)+d(λ) det
+2l−1D(1,2),(1,2)(λ), (A.72)
+C1,2(λ) =2l/productdisplay
+h=1d(λqh)+a(λq) det
+2l−1D(1,2),(1,2)(λ). (A.73)
+Byusingthesamestepsshownin(A.71),thetridiagonalityo fthematrixD (1,2),(1,2)(λ)impliesthat
+its determinant is an even function of λfrom which the statement C 2,1(λ) =qNC1,2(−λ)follows
+recallingthat d(λ) =qNa(−λq).
+Remark 3. Inthisarticlewe needonlytheproperties(A.70);however, it isworthpointingoutthat
+theyarespecialcasesofthefollowingpropertiesofthe cof actors:
+Ci,j(λ) =qN(i−j)Cj,i(−λ)∀i,j∈ {1,...,2l+1}. (A.74)
+Theproofof(A.74)canbedonesimilarlytothatof(A.70) but we omitit forsimplicity.
+Let ususe onceagainthenotation Zfforthe set ofthezerosofafunction f(λ), then:
+Lemma 5. Theequations(4.35)and(4.36) imply:
+ZC1,1∩ZC1,2≡ZC1,1∩ZC1,2l+1. (A.75)18
+Proof.The inclusions/parenleftbig
+ZC1,1∩ZC1,2/parenrightbig
+\Za⊂ZC1,1∩ZC1,2l+1and/parenleftbig
+ZC1,1∩ZC1,2l+1/parenrightbig
+\Zd⊂
+ZC1,1∩ZC1,2triviallyfollowbyequation(4.36).
+Let us observe now that C 1,2(λq−1)has no common zero with a(λ)and that C 1,2l+1(λq)has no
+common zero with d(λ). These statements simply follow from (A.73), (A.69)and(A. 72) when we
+recall that a(λ)has no common zero with/producttext2l−1
+h=0d(λqh)and thatd(λ)has no common zero with/producttext2l+1
+h=2a(λqh). So,if/parenleftbig
+ZC1,1∩ZC1,2/parenrightbig
+∩Zaisnotemptyand λ0∈/parenleftbig
+ZC1,1∩ZC1,2/parenrightbig
+∩Za,theequation
+(4.35) computed in λ=q−1λ0implies C 1,2l+1(λ0) = 0being C 1,2(λ0q−1)/\e}atio\slash= 0, i.e.λ0∈
+ZC1,1∩ZC1,2l+1. Similarly,if/parenleftbig
+ZC1,1∩ZC1,2l+1/parenrightbig
+∩Zdisnotemptyand λ0∈/parenleftbig
+ZC1,1∩ZC1,2l+1/parenrightbig
+∩Zd,
+the equation (4.35) computed in λ=λ0implies C 1,2(λ0) = 0being C 1,2l+1(λ0q)/\e}atio\slash= 0, i.e.λ0∈
+ZC1,1∩ZC1,2. So that (4.35) implies the inclusions/parenleftbig
+ZC1,1∩ZC1,2/parenrightbig
+∩Za⊂ZC1,1∩ZC1,2l+1and/parenleftbig
+ZC1,1∩ZC1,2l+1/parenrightbig
+∩Zd⊂ZC1,1∩ZC1,2inthiswaycompletingthe proofofthelemma.
+B. Scalarproduct inthe SOV space
+Here is described as a natural structure of Hilbert space can be provided to the linear space of the
+SOV representationbypreservingtheself-adjointnessoft hetransfermatrix.
+B.1 CyclicrepresentationsoftheWeylalgebra
+Here,we considerthecyclicrepresentationsoftheWeyl alg ebraW(n)
+qinthecase:
+up
+n=vp
+n= 1forβ2=p′/pwithp′evenandp= 2l+1odd. (B.76)
+At anysitenofthechain,weintroducethe quantumspace Rnwithvn-eigenbasis:
+vn|k,n/a\}bracketri}ht=qk|k,n/a\}bracketri}ht ∀|k,n/a\}bracketri}ht ∈Bn={|k,n/a\}bracketri}ht,∀k∈ {−l,...,l}}. (B.77)
+Note that the eigenvaluesof vndescribe the unit circle Sp={qk:k∈ {−l,...,l}},indeedql+1=
+q−l. OnRnisdefinedap-dimensionalrepresentationoftheWeyl algebrabysetting :
+un|k,n/a\}bracketri}ht=|k+1,n/a\}bracketri}ht ∀k∈ {−l,...,l} (B.78)
+with thecyclicitycondition:
+|k+p,n/a\}bracketri}ht=|k,n/a\}bracketri}ht. (B.79)
+B.2 Representationin the SOVbasis
+The analysis developed in [1] define recursively the eigenba sis{|¯η1qh1,...,¯ηNqhN/a\}bracketri}ht}of theB-
+operator in the original representation, i.e. as linear com binations of the elements of the basis
+{|h1,...,hN/a\}bracketri}ht ≡/circlemultiplytextN
+n=1|hn,n/a\}bracketri}ht}, where|hn,n/a\}bracketri}htare the elements of the vn-eigenbasis defined in
+(B.77). To writethischangeofbasisin amatrixformlet usin troducethe followingnotations:
+|yj/a\}bracketri}ht ≡ |¯η1qh1,...,¯ηNqhN/a\}bracketri}htand|xj/a\}bracketri}ht ≡ |h1,...,hN/a\}bracketri}ht (B.80)19
+where:
+j:=h1+N/summationdisplay
+a=2(2l+1)(a−1)(ha−1)∈ {1,...,(2l+1)N}, (B.81)
+notethat thisdefinesa oneto onecorrespondencebetween N-tuples(h1,...,hN)∈ {1,...,2l+1}N
+and integers j∈ {1,...,(2l+1)N}, which just amountsto chose an orderingin the elementsof th e
+two basis. Underthisnotation,wehave:
+|yj/a\}bracketri}ht=W|xj/a\}bracketri}ht=(2l+1)N/summationdisplay
+i=1Wi,j|xi/a\}bracketri}ht, (B.82)
+where we are representing |xj/a\}bracketri}htas the vector |j/a\}bracketri}htin the natural basis in C(2l+1)NandW=||Wi,j||
+is a(2l+1)N×(2l+1)Nmatrix. The matrix Wis defined by recursion in terms of the kernel K
+constructedinappendixCof[1], letususethenotation:
+K({h1,...,hN},k1,{k2,...,kN})≡KN(η|χ2;χ1), (B.83)
+whereweareconsideringthecase N−M = 1. Thenthe recursionreads:
+W(N)
+i,j=2l+1/summationdisplay
+k2,...,kN=1K({h1(j),...,hN(j)},h1(i),{k2,...,kN})W(N−1)
+¯h(i),a(k2,...,kN),(B.84)
+where we have introduced the index (N)and(N−1)in the matrices Wto make clear the step
+of the recursion. Here, (h1(j),...,hN(j))is the unique N-tuples corresponding to the integer j∈
+{1,...,(2l+ 1)N}andh1(i)is the first entry in the unique N-tuples corresponding to the integer
+i∈ {1,...,(2l+1)N}. Moreover,wehavedefined:
+¯h(i) := 1+i−h1(i)
+2l+1∈ {1,...,(2l+1)(N−1)}anda(k2,...,kN) =k2+N/summationdisplay
+a=3(2l+1)(a−2)(ka−1),
+(B.85)
+Remarks:
+a)Underthechangeofbasis {|xj/a\}bracketri}ht} → {|yj/a\}bracketri}ht}thegenericoperatorX transformsforsimilarity:
+XSOV≡W−1XW, (B.86)
+so fromtheactionofthezerooperators ηaandtheshift operators T±
+aontheB-eigenbasis |yj/a\}bracketri}ht:
+ηa|yj/a\}bracketri}ht= ¯ηaqha(j)|yj/a\}bracketri}htandT±
+a|yj/a\}bracketri}ht=|yj±(2l+1)(a−1)/a\}bracketri}ht (B.87)
+we havethat:
+(ηa)SOV= ¯ηa||qha(j)δi,j||and/parenleftbig
+T±
+a/parenrightbig
+SOV=||δi,j±(2l+1)(a−1)||. (B.88)20
+Fromtheaboveexpressionwe have17:
+(ηa)†
+SOV= (ηa)∗
+SOVand/parenleftbig
+T±
+a/parenrightbig†
+SOV=/parenleftbig
+T∓
+a/parenrightbig
+SOV. (B.89)
+b) The known transformation properties of the entries of the monodromy matrix in the original
+representationimply:
+/parenleftbiggDSOV(λ)CSOV(λ)
+BSOV(λ)ASOV(λ)/parenrightbigg
+=/parenleftigg
+G−1(ASOV(λ∗))†G−G−1(BSOV(λ∗))†G
+−G−1(CSOV(λ∗))†G G−1(DSOV(λ∗))†G/parenrightigg
+,(B.90)
+withGisapositiveself-adjointmatrixdefinedby G:=W†W.
+c)Thequantumdeterminantrelationis invariantundersimi laritytransformationsandso we have:
+a(λ)d(λq−1) =ASOV(λ)DSOV(λq−1)−BSOV(λ)CSOV(λq−1), (B.91)
+Lemma 6. Thebasis {|yj/a\}bracketri}ht}isnotanorthogonalbasisw.r.t. thenaturalscalarproduct on{|xj/a\}bracketri}ht}.
+Proof.Note that the condition {|yj/a\}bracketri}ht}is an orthogonal basis is equivalent to the statement Gis
+a diagonal matrix (with positive diagonal entries). Let us r ecall that the Hermitian conjugation
+propertyofB(λ)togetherwiththeYang-Baxtercommutationrelationsimply :
+[B†(λ),B(µ)] = [B(µ),C(λ∗)] =q−q−1
+λ∗/µ−µ/λ∗(A(λ∗)D(µ)−A(µ)D(λ∗))/\e}atio\slash= 0 (B.92)
+that is the operator B(λ)is not a normal operator. Now let us show that the non-normali tyofB(λ)
+impliesthat Gisnotdiagonal. Indeed,wecanwrite:
+[B†(λ),B(µ)] =/parenleftbig
+W†/parenrightbig−1(BSOV(λ))†GBSOV(µ)W−1−WBSOV(µ)G−1(BSOV(λ))†W†
+=W(G−1(BSOV(λ))†GBSOV(µ)−BSOV(µ)G−1(BSOV(λ))†G)W−1.(B.93)
+Notenowthatifweassume Gdiagonal,then Gcommutesbothwith BSOV(λ)andwith(BSOV(λ))†,
+being all diagonal matrices in the SOV representation, whic h implies the absurd [B†(λ),B(µ)] =
+0.
+B.3 Scalarproductin theSOVspace
+The self-adjointness of the family T(λ)implies that the transfer matrix eigenstates are orthogona l
+undertheoriginalscalar product:
+δi,j= (|ti/a\}bracketri}ht,|tj/a\}bracketri}ht), (B.94)
+we have chosen the orthonormal ones. Note that the above equa tion naturally induces a scalar
+productintheSOV representationobtainedunderchangeofb asis:
+(|b/a\}bracketri}ht,|a/a\}bracketri}ht)SOV≡(G|b/a\}bracketri}ht,|a/a\}bracketri}ht) (B.95)
+17Here, weare using the standard notation for the adjoint X†≡(X∗)t.21
+thatisascalarproductforwhichtheadjointofavector |a/a\}bracketri}htisthenaturaladjointtimesthematrix G:
+|b/a\}bracketri}ht†SOV≡ /a\}bracketle{tb|Gwith/a\}bracketle{tb|=/parenleftig
+(|b/a\}bracketri}ht)t/parenrightig∗
+, (B.96)
+andsoforthegenericoperator Xwehave:
+X†SOV≡G−1X†G. (B.97)
+It istrivialtonoticethat:
+Lemma 7. The family of operators TSOV(λ)is self-adjoint w.r.t. †SOVand the eigenstates
+|tj/a\}bracketri}htSOV≡W−1|tj/a\}bracketri}htare orthonormal w.r.t. the scalar product defined in (B.95). Moreover, it
+results:
+/parenleftigg
+(ASOV(λ∗))†SOV(BSOV(λ∗))†SOV
+(CSOV(λ∗))†SOV(DSOV(λ∗))†SOV/parenrightigg
+=/parenleftbiggDSOV(λ)−CSOV(λ)
+−BSOV(λ)ASOV(λ)/parenrightbigg
+.(B.98)
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\ No newline at end of file
diff --git a/1001.0036.txt b/1001.0036.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1aca85bae218d0e9ad3dafa9a5d27d0e9e72cea1
--- /dev/null
+++ b/1001.0036.txt
@@ -0,0 +1,1085 @@
+The Computational Structure of Spike Trains
+Robert Haslinger,1, 2Kristina Lisa Klinkner,3and Cosma Rohilla Shalizi3, 4
+1Martinos Center for Biomedical Imaging,Massachusetts General Hospital, Charlestown MA
+2Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge MA
+3Department of Statistics, Carnegie Mellon University, Pittsburgh PA
+4Santa Fe Institute, Santa Fe NM
+(Dated: September 2008; January 2009)
+Neurons perform computations, and convey the results of those computations
+through the statistical structure of their output spike trains. Here we present a
+practical method, grounded in the information-theoretic analysis of prediction, for
+inferring a minimal representation of that structure and for characterizing its com-
+plexity. Starting from spike trains, our approach nds their causal state models
+(CSMs), the minimal hidden Markov models or stochastic automata capable of
+generating statistically-identical time series. We then use these CSMs to objec-
+tively quantify both the generalizable structure and the idiosyncratic randomness
+of the spike train. Specically, we show that the expected algorithmic informa-
+tion content (the information needed to describe the spike train exactly) can be
+split into three parts describing (1) the time-invariant structure (complexity) of
+the minimal spike-generating process, which describes the spike train statistically ,
+(2) the randomness (internal entropy rate) of the minimal spike-generating process,
+and (3) a residual pure noise term not described by the minimal spike generating
+process. We use CSMs to approximate each of these quantities. The CSMs are in-
+ferred non-parametrically from the data, making only mild regularity assumptions,
+via the Causal State Splitting Reconstruction (CSSR) algorithm. The methods
+presented here complement more traditional spike train analyses by describing not
+only spiking probability, and spike train entropy, but also the complexity of a spike
+train's structure. We demonstrate our approach using both simulated spike trains
+and experimental data recorded in rat barrel cortex during vibrissa stimulation.
+I. INTRODUCTION
+The recognition that neurons are computational devices is one of the foundations of modern neuroscience (McCulloch
+& Pitts, 1943). However, determining the functional form of such computation is extremely dicult, if only because
+while one often knows the output (the spikes) the input (synaptic activity) is almost always unknown. Often, therefore,
+scientists must draw inferences about the computation from its results, namely the output spike trains and their
+statistics. In this vein, many researchers have used information theory to determine, via calculation of the entropy
+rate, a neuron's channel capacity, i.e., how much information the neuron could conceivably transmit, given the
+distribution of observed spikes (Rieke et al., 1997). However, entropy quanties randomness, and says little about
+how much structure a spike train has, or the amount and type of computation which must have, at a minimum, taken
+place to produce this structure. Here, and throughout this paper, we mean \computational structure" information-
+theoretically, i.e., the most compact eective description of a process capable of statistically reproducing the observed
+spike trains. The complexity of this structure is the number of bits needed to describe it. This is dierent from the
+algorithmic information content of a spike train, which is the number of bits needed to reproduce the latter exactly ,
+describing not only its regularities, but also its accidental, noisy details.
+Our goal is to develop rigorous yet practical methods for determining the minimal computational structure necessary
+and sucient to generate neural spike trains. We are able to do this through non-parametric analysis of the directly-
+observable spike trains, without resorting to a priori assumptions about what kind of structure they have. We do this
+by identifying the minimal hidden Markov model (HMM) which can statistically predict the future of the spike train
+without loss of information. This HMM also generates spike trains with the same statistics as the observed train.
+It thus denes a program which describes the spike train's computational structure, letting us quantify, in bits, the
+structure's complexity.
+From multiple directions, several groups, including our own, have shown that minimal generative models of time
+series can be discovered by clustering histories into \states", based on their conditional distributions over future events
+(Crutcheld & Young, 1989; Grassberger, 1986; Jaeger, 2000; Knight, 1975; Littman et al., 2002; Shalizi & Crutcheld,
+2001). The observed time series need notbe Markovian (few spike trains are), but the construction always yieldsarXiv:1001.0036v1  [q-bio.NC]  30 Dec 20092
+the minimal HMM capable of generating and predicting the original process. Following Shalizi (2001); Shalizi &
+Crutcheld (2001), we will call such a HMM a \Causal State Model" (CSM). Within this framework, the model
+discovery algorithm called Causal State Splitting Reconstruction , or CSSR (Shalizi & Klinkner, 2004) is an adaptive
+non-parametric method which consistently estimates a system's CSM from time-series data. In this paper we adapt
+CSSR for use in spike train analysis.
+CSSR provides us with non-parametric estimates of the time- and history- dependent spiking probabilities found by
+more familiar parametric analyses. Unlike those analyses, it is also capable, in the limit of innite data, of capturing all
+the information about the computational structure of the spike-generating process contained in the spikes themselves.
+In particular, the CSM quanties the complexity of the spike-generating process by showing how much information
+about the history of the spikes is relevant to their future, i.e., how much information is needed to reproduce the
+spike train statistically. This is equivalent to the log of the eective number of statistically-distinct states of the
+process (Crutcheld & Young, 1989; Grassberger, 1986; Shalizi & Crutcheld, 2001). While this is not the same as
+the algorithmic information content, we show that CSMs can also approximate the average algorithmic information
+content, splitting it into three parts: (1) The generative process's complexity in our sense; (2) the internal entropy
+rateof the generative process, the extra information needed to describe the exact state transitions the undergone while
+generating the spike train; and (3) the residual randomness in the spikes, unconstrained by the generative process.
+The rst of these quanties the spike train's structure, the last two its randomness.
+Below, we give precise denitions of these quantities, both their ensemble averages ( xII.C) and their functional
+dependence on time ( xII.D). The time-dependent versions allow us to determine when the neuron is traversing states
+requiring complex descriptions. Our methods put hard numerical lower bounds on the amount of computational
+structure which must be present to generate the observed spikes. They also quantify, in bits, the extent to which the
+neuron is driven by external forces. We demonstrate our approach using both simulated and experimentally recorded
+single-neuron spike trains. We discuss the interpretation of our measures, and how they add to our understanding of
+neuronal computation.
+II. THEORY AND METHODS
+Throughout this paper we treat spike trains as stochastic binary time series, with time divided into discrete, equal-
+duration bins steps (typically at one millisecond resolution); \1" corresponds to a spike and \0" to no spike. Our aim is
+to nd a minimal description of the computational structure present in such a time series. Heuristically, the structure
+present in a spike train can be described by a \program" which can reproduce the spikes statistically. The information
+needed to describe this program (loosely speaking the program length) quanties the structure's complexity. Our
+approach uses minimal, optimally predictive HMMs, or Causal State Models (CSMs), reconstructed from the data, to
+describe the program. (We clarify our use of \minimal" below.) The CSMs are then used to calculate various measures
+of the computational structure, such as its complexity.
+The states are chosen so that they are optimal predictors of the spike train's future, using only the information
+available from the train's history. (We discuss the limitations of this below.) Specically the states Stare dened
+by grouping the histories of past spiking activity Xt
+1which occur in the spike train into equivalence classes, where
+all members of a given equivalence class are statistically equivalent in terms of predicting the future spiking X1
+t+1.
+(Xt
+t0denotes the sequence of random observables, i.e., spikes or their absence, between t0andt>t0whileXtdenotes
+the random observable at time t. The notation is similar for the states.) This construction ensures that the causal
+states are Markovian, even if the spike train is not (Shalizi & Crutcheld, 2001, Lemma 6, p. 839). Therefore, at all
+timestthe system and its possible future evolution(s) can be specied by the state St. Like all HMMs, a CSM can
+be represented pictorially by a directed graph, with nodes standing for the process's hidden states and directed edges
+the possible transitions between these states. Each edge is labeled with the observable/symbol emitted during the
+corresponding transition (\1" for a spike and \0" for no spike), and the probability of traversing that edge given that
+the system started in that state. The CSM also species the time-averaged probability of occupying any state (via
+the ergodic theorem for Markov chains).
+The theory is described in more detail below, but at this point examples may clarify the ideas. Figures 1 A and B
+show two simple CSMs. Both are built from simulated 40 Hz spike trains 200 seconds in length (1 msec time bins,
+p= 0:04 IID at each time when spiking is possible). However, spike trains generated from the CSM in Figure 1 B
+have a 5 msec refractory period after each spike (when p= 0), while the spiking rate in non-refractory periods is still
+40 Hz (p= 0:04). The refractory period is additional structure, represented by the extra states. State Arepresents
+the status of the neuron during 40 Hz spiking, outside of the refractory periods. While in this state, the neuron either
+emits no spike ( Xt+1= 0), staying in state A, or emits a spike ( Xt+1= 1) with probability p= 0:04 and moves to
+stateB. The equivalence class of past spiking histories dening state Atherefore includes all past spiking histories
+for which the most recent ve symbols are 0, symbolically f00000g. StateBis the neuron's state during the rst3
+msec of the refractory period. It is dened by the set of spiking histories f1g. No spike can be emitted during a
+refractory period so the transition to state Cis certain and the symbol emitted is always '0'. In this manner the
+neuron proceeds through states CtoFand back to state Awhereupon it is possible to spike again.
+The rest of this section is divided into four subsections. First, we brie
y review the formal theory behind CSMs
+(for details, see Shalizi (2001); Shalizi & Crutcheld (2001)) and discuss why they can be considered a good choice for
+understanding the structural content of spike trains. Second, we describe the Causal State Splitting Reconstruction
+(CSSR) algorithm used to reconstruct CSMs from observed spike trains (Shalizi & Klinkner, 2004). We emphasize
+that CSSR requires no a priori knowledge of the structure of the CSM which is discovered from the spike train. Third,
+we discuss two dierent notions of spike train structure, namely statistical complexity and algorithmic information
+content. These two measures can be interpreted as dierent aspects of a spike train's computational structure, and
+each can be related to the reconstructed CSM. Fourth and nally, we show how the reconstructed CSM can be used
+to predict spiking, measure the neural response and detect the in
uence of external stimuli.
+A. Causal State Models
+The foundation of the theory of causal states is the concept of a predictively sucient statistics . A statistic, ,
+on one random variable, X, is sucient for predicting another random variable, Y, when(X) andXhave the
+same information1aboutY,I[X;Y] =I[(X);Y]. This holds if and only if XandYare conditionally independent
+given(X):P(YjX;(X)) = P(Yj(X)). This is a close relative of the familiar idea of parametric suciency;
+in Bayesian statistics, where parameters are random variables, parametric suciency is a special case of predictive
+suciency (Bernardo & Smith, 1994). Predictive suciency shares all of parametric suciency's optimality properties
+(Bernardo & Smith, 1994). However, a statistic's predictive suciency depends only on the actual joint distribution of
+XandY, not on any parametric model of that distribution. Again as in the parametric case, a minimal predictively
+sucient statistic is one which is a function of every other sucient statistic , i.e.,(X) =h((X)) for some h.
+Minimal sucient statistics are the most compact summaries of the data which retain all the predictively-relevant
+information. A basic result is that a minimal sucient statistic always exists and is (essentially) unique, up to
+isomorphism (Bernardo & Smith, 1994; Shalizi & Crutcheld, 2001).
+In the context of stochastic processes, such as spike trains, is the minimal sucient statistic of the history Xt
+1
+for predicting future of the process, X1
+t+1. This statistic is the optimal predictor of the observations. The sequence
+of values of the minimal sucient statistic, St=(Xt
+1), is another stochastic process. This process is always a
+homogeneous Markov chain, whether or not the Xtprocess is (Knight, 1975; Shalizi & Crutcheld, 2001). Turned
+around, this means that the original Xtprocess is always a random function of a homogeneous Markov chain, whose
+latent states, named the causal states by Crutcheld & Young (1989), are optimal, minimal predictors of the future
+of the time series.
+Acausal state model orcausal state machine is a stochastic automaton or HMM constructed so that its Markov
+states are minimal sucient statistics for predicting the future of the spike train, and consequently can generate spike
+trains statistically identical to those observed.2Causal state reconstruction means inferring the causal states from
+the observed spike train. Following Crutcheld & Young (1989); Shalizi & Crutcheld (2001), the causal states can be
+seen as equivalence classes of spike-train histories Xt
+1which maximize the mutual information between the state(s)
+and the future of the spike train X1
+t+1. Because they are sucient, they predict the future of the spike train as well
+as it can be predicted from its history alone. Because they are minimal, the number of states or equivalence classes
+is as small as it can be without discarding predictive power.3
+Formally, two histories, xandy, are equivalent when P(X1
+t+1jXt
+1=x) =P(X1
+t+1jXt
+1=y). The equiva-
+lence class of xis [x]. Dene the function which maps histories to their equivalence classes:
+(x)[x]
+=
+y:P(X1
+t+1jXt
+1=y) =P(X1
+t+1jXt
+1=x)	
+1See Cover & Thomas (1991) for information-theoretic denitions and notation.
+2Some authors use \hidden Markov Model" only for models where the current observation is independent of all other
+variables given the current state, and call the broader class which includes CSMs \partially observable Markov
+model".
+3There may exist more compact representations, but then the states, or their equivalents, can never be empirically
+identied | see Shalizi & Crutcheld (2001, thm. 3, p. 846), or L ohr & Ay (2009).4
+The causal states are the possible values of , i.e., the equivalence classes; each corresponds to a distinct distribution
+for the future. The state at time tisSt=(Xt
+1). Clearly,(x) is a sucient statistic. It is also minimal, since if 
+is sucient, then (x) =(y) implies(x) =(y). One can further show (Shalizi & Crutcheld, 2001, Theorem
+3) thatis the unique minimal sucient statistic, meaning that any other must be isomorphic to it.
+In addition to being minimal sucient statistics, the causal states have some other important properties which
+make them ideal for quantifying structure (Shalizi & Crutcheld, 2001). (1) As mentioned, fStgis a Markov process,
+and one can write the observed process Xas a random function of the causal state process, i.e., Xhas a natural
+hidden-Markov-model representation. (2) The causal states are recursively calculable; there is a function Tsuch
+thatSt+1=T(St;Xt+1) | see Appendix A. (3) CSMs are closely related to the \predictive state representations" of
+controlled dynamical systems (Littman et al., 2002); see Appendix C.
+B. Causal State Splitting Reconstruction
+Our goal is to nd a minimal sucient statistic for the spike train, which will form a hidden Markov model. As
+stated previously, the states of this model are equivalence classes of spiking histories Xt
+1. In practice, we need an
+algorithm which can both cluster histories into groups which preserve their conditional distribution of futures, and
+nd the history length  at which the past may be truncated while preserving the computational structure of the
+spike train. The former is accomplished by the CSSR algorithm (Shalizi & Klinkner, 2004) for inferring causal states
+from data by building a recursive next-step-sucient statistic.4We do the latter by minimizing Schwartz's Bayesian
+Information Criterion (BIC) over .
+To save space, we just sketch the CSSR algorithm here.5CSSR starts by treating the process as an independent,
+identically-distributed sequence, with one causal state. It adds states when statistical tests show that the current set
+of states is not sucient. Suppose we have a sequence xN
+1=x1;x2;:::xNof lengthNfrom a nite alphabet Aof
+sizek. We wish to derive from this an estimate ^ of the minimal sucient statistic . We do this by nding a set  of
+states, each of which will be a set of strings, or nite-length histories. The function ^ will then map a history xto
+whichever state contains a sux of x(taking \sux" in the usual string-manipulation sense). Although each state
+can contain multiple suxes, one can check (Shalizi & Klinkner, 2004) that the mapping ^ will never be ambiguous.
+The null hypothesis is that the process is Markovian on the basis of the states in ,
+P(XtjXt1
+tL=axt1
+tL+1) =P(Xtj^S= ^(xt1
+tL+1)) (1)
+for alla2A. In words, adding an extra piece of history does not change the conditional distribution for the next
+observation. We can check this with standard statistical tests, such as 2or Kolmogorov-Smirnov. In this paper, we
+used a KS test of size = 0:01.6If we reject this hypothesis, we fall back on a restricted alternative hypothesis , that
+we have the right set of conditional distributions, but have matched them with the wrong histories. That is,
+P(XtjXt1
+tL=axt1
+tL+1) =P(Xtj^S=s) (2)
+for somes2, buts6= ^(xt1
+tL+1). If this hypothesis passes a test of size , thensis the state to which we assign
+the history7. Only if the (2) is itself rejected do we create a new state, with the sux axt1
+tL+1.8
+4A next-step-sucient statistic contains all the information needed for optimal one-step-ahead prediction,
+I[Xt+1;(Xt
+1)] =I[Xt+1;Xt
+1], but not necessarily for longer predictions. CSSR relies on the theorem that
+ifis next-step sucient, and it is recursively calculable, then is sucient for the whole of the future (Shalizi &
+Crutcheld, 2001, pp. 842{843). CSSR rst nds a next-step sucient statistic, and then renes it to be recursive.
+5In addition to Shalizi & Klinkner (2004), which gives pseudocode, some details of convergence, and applications to
+process classication, are treated in Klinkner & Shalizi (2009); Shalizi et al. (2009). An open-source C++ imple-
+mentation is available at http://bactra.org/CSSR/ . The CSMs generated by CSSR can be displayed graphically,
+as we do in this paper, with the open-source program dot(http://www.graphviz.org/
+6For niteN, decreasing tends to yield simpler CSMs with fewer states. In a sense, it is a sort of regularization
+coecient. The in
uence of this regularization diminishes as Nincreases. For the data used in the Results section
+of this paper, varying in the range 0 :001<< 0:1 made little dierence.
+7If more than one such state sexists, we chose the one for which bP(Xtj^S=s) diers least, in total variation
+distance, from bP(Xtjt1
+tL=axt1
+tL+1), which is plausible and convenient. However, which state we chose is irrelevant
+in the limit N!1 , so long as the dierence between the distributions is not statistically signicant.
+8The conceptually similar algorithm of Kennel & Mees (2002) in eect always creates a new state, which leads to
+more complex models, sometimes innitely more complex ones; see Shalizi & Klinkner (2004).5
+The algorithm itself has three phases. Phase I initializes  to a single state, which contains only the null sux ;.
+(That is,;is a sux of any string.) The length of the longest sux in  is L; this starts at 0. Phase II iteratively
+tests the successive versions of the null hypothesis, Eq. 1, and Lincreases by one each iteration, until we reach some
+maximum length . At the end of II, ^ is (approximately) next-step sucient. Phase III makes ^ recursively calculable,
+by splitting the states until they have deterministic transitions. Under mild technical conditions (a nite true number
+of states, etc.), CSSR converges in probability on the correct CSM as N!1 , provided only that  is long enough
+to discriminate all of the states. The error of the predicted distributions of futures P(X1
+t+1jXt
+1), measured by total
+variation distance, decays as N1=2. Section 4 of Shalizi & Klinkner (2004) details CSSR's convergence properties.
+Comparisons of CSSR's performance with that of more traditional expectation maximization based approaches can
+also be found in Shalizi & Klinkner (2004) as can time complexity bounds for the algorithm. Depending upon the
+machine used, CSSR can process an N= 106time series in under a minute.
+1. Choosing 
+CSSR requires no a priori knowledge of the CSM's structure, but does need a choice of of ; here pick it by
+minimizing the BIC of the reconstructed models over , i.e.,
+BIC2 logL+dlogN (3)
+whereLis the likelihood, Nis the data length and dis the number of model parameters, in our case the number
+of predictive states9BIC's logarithmic-with- Npenalty term helps keep the number of causal states from growing
+too quickly with increased data size, which is why we use it instead of the Akaike Information Criterion (AIC). Also,
+BIC is known to be consistent for selecting the order of Markov chains and variable-length Markov models (Csisz ar
+& Talata, 2006), both of which are sub-classes of CSMs.
+Writing the observed spike train as xN
+1, and the state sequence as sN
+0, the total likelihood of the spike train is
+L=X
+sN
+02N+1P(XN
+1=xN
+1jSN
+0=sN
+0)P(SN
+0=sN
+0); (4)
+the sum over all possible causal state sequences of the joint probability of the spike train and the state sequence.
+Since the states update recursively, st+1=T(st;xt+1), the starting state s0and the spike train xN
+1x the entire state
+sequencesN
+0. Thus the sum over state sequences can be replaced by a sum over initial states
+L=X
+si2P(XN
+1=xN
+1jS0=si)P(S0=si) (5)
+with the state probabilities P(S0=si) coming from the CSM. By the Markov property,
+P(XN
+1=xN
+1jS0=si) =NY
+j=1P(Xj=xjjSj1=sj1) (6)
+Selecting  is now straightfoward: for each value of , we build the CSM from the spike train, calculate the
+likelihood using Eq. 5 and 6, and pick the value, and CSM, minimizing Eq. 3. We try all values of  up to a model-
+independent upper bound. For a wide range of stochastic processes, Marton & Shields (1994) showed that the length
+mof subsequences for which probabilities can be consistently and non-parametrically estimated can grow as fast as
+logN=h, wherehis the entropy rate, but no faster. CSSR estimates the distribution of the next symbol given the
+previous  symbols, which is equivalent to estimating joint probabilities of blocks of length m=  + 1. Thus Marton
+and Shield's result limits the usable values of :
+logN
+h1 (7)
+9The number of independent parameters dinvolved in describing the CSM will be (number of states)*(number of
+symbols - 1) since the sum of the outgoing probabilities for each state is constrained to be 1. Thus, for a binary
+alphabet,d= number of states.6
+Using Eq. 7 requires the entropy rate h. The latter can either be upper bounded as the log of the alphabet size (here,
+log 2 = 1), or by some other, less pessimistic, estimator of the entropy rate (such as the output of CSSR with  = 1).
+Use of an upper bound on hresults in a conservative maximum value for . For example, a 30 minute experiment
+with 1 msec time bins lets us use at least 20 by the most pessimistic estimate of h= 1; the actual maximum
+value of  may be much larger. We use  25 in this paper but see no indication that this can't be extended further,
+if need be.
+2. Condensing the CSM
+For real neural data, the number of causal states can be very large | hundreds or more. This creates an interpreta-
+tion problem, if only because it is hard to t such a CSM on a single page for inspection. We thus developed a way to
+reduce the full CSM while still accounting for most of the spike train's structure. Our \state culling" technique found
+the least-probable states and selectively removed them, appropriately redirecting state transitions and reassigning
+state occupation probabilities. By keeping the most probable states, we focus on the ones which contribute the most
+to the spike train's structure and complexity. Again, we used BIC as our model selection criterion.
+First, we sorted the states by probability, nding the least probable state (\remove" state) with a single incoming
+edge from a state (its \ancestor") with outgoing transitions to two dierent states, the remove state and a second,
+\keep" state. We redirected both of the ancestor's outgoing edges to the keep state. Second, we reassigned the remove
+state's outgoing transitions to the keep state. If the outgoing transitions from the keep state were still deterministic (at
+most a single 0 emitting edge and a single 1 emitting edge), we stopped. If the transitions were non-deterministic, we
+merged states reached by emitting 0s with each other (likewise those reached by 1s), repeating this until termination.
+Third, we checked that there existed a state sequence of the new model which could generate the observed spikes. If
+there was, we accepted the new CSM. If not, we rejected the new CSM and chose the next lowest probability state
+from the original CSM to remove.
+This culling was iterated until removing any state made it impossible for the CSM to generate the spike train. At
+each iteration, we calculated BIC (as described in the previous section), and ultimate chose the culled CSM with the
+minimum BIC. This gave a culled CSM for each value of ; the nal one we used was chosen after also minimizing
+BIC over . The CSMs shown below in the results section paper result from this minimizing of BIC over  and state
+culling.
+3. ISI Bootstrapping
+While we do model selection with BIC, we also want to do model checking or adequacy-testing. For the most part,
+we do this by using the CSM to bootstrap point-wise condence bounds on the interspike interval (ISI) distribution,
+and checking their coverage of the empirical ISI distribution. Because this distribution is not used by CSSR in
+reconstructing the CSM, it provides a check on the latter's ability to accurately describe the spike-train's statistics.
+Specically, we generated condence bounds as follows. To simulate one spike train, we picked a random starting
+state according to the CSM's inferred state-occupation probabilities, and then ran the CSM forward for Ntime-steps,
+Nbeing the length of the original spike train. This gives a binary time-series, where a \1" stands for a spike and
+a \0" for no-spike, and gave us a sample of inter-spike intervals from the CSM. This in turn gave an \empirical"
+ISI distribution. Repeated over 104independent runs of the CSM, and taking the 0 :005 and 0:995 quantiles of the
+distributions at each ISI length, gives 99% pointwise condence bounds. (Pointwise bounds are necessary because of
+the ISI distribution often modulates rapidly with ISI length.) If the CSM is correct, the empirical ISI will, by chance,
+lie outside the bounds at 1% of the ISI lengths.
+If we split the data into training and validation sets, a CSM reconstructed from the training set can be used to
+bootstrap ISI condence bounds, which can be compared to the ISI distribution of the test set. We discuss this sort
+of of cross validation, as well as an additional test based on the time rescaling theorem, in Appendix B.7
+C. Complexity and Algorithmic Information Content
+The algorithmic information content K(xn
+1) of a sequence xn
+1is the length of the shortest complete (input-free)
+computer program which will output xn
+1exactly and then halt (Cover & Thomas, 1991)10. In general, K(xn
+1) is
+strictly uncomputable, but when xn
+1is the realization of a stochastic process Xn
+1, the ensemble-averaged algorithmic
+information essentially coincides with the Shannon entropy (\Brudno's theorem"; see Badii & Politi (1997)), re
ecting
+the fact that both are maximized for completely random sequences (Cover & Thomas, 1991). Both the algorithmic
+information and the Shannon entropy can be conveniently written in terms of a minimal sucient statistic Q:
+E[K(Xn
+1)] =H[Xn
+1] +o(n)
+=H[Q] +H[Xn
+1jQ] +o(n) (8)
+The equality H[Xn
+1] =H[Q] +H[Xn
+1jQ] holds because Qis a function of Xn
+1, soH[QjXn
+1] = 0.
+The key to determining a spike train's expected algorithmic information is thus to nd a minimal sucient statistic.
+By construction, causal state models provide exactly this; a minimal sucient statistic for xn
+1is the state sequence
+sn
+0=s0;s1;:::sn(Shalizi & Crutcheld, 2001). Thus the ensemble-averaged algorithmic information content, dropping
+termso(n) and smaller, is
+E[K(Xn
+1)] =H[Sn
+0] +H[Xn
+1jSn
+0]
+=H[S0] +nX
+i=1H[SijSi1] +nX
+i=1H[XijSi;Si1] (9)
+Going from the rst to the second line uses the causal states' Markov property. Assuming stationarity, Eq. 9 becomes
+E[K(Xn
+1)] =H[St] +n(H[StjSt1] +H[XtjSt;St1])
+=C+n(J+R) (10)
+This separates terms representing structure from those representing randomness.
+The rst term in Eq. 10 is the complexity ,C, of the spike-generating process (Crutcheld & Young, 1989; Grass-
+berger, 1986; Shalizi et al., 2004).
+C=H[St] =E[logP(St)] (11)
+Cis the entropy of the causal states, quantifying the structure present in the observed spikes. This is distinct from
+the entropy of the spikes themselves, which quanties not their structure but their randomness (and is approximated
+by the other two terms). Intuitively, Cis the (time-averaged) amount of information about the past of the system
+which is relevant to predicting its future. For example, consider again the IID 40 Hz Bernoulli process of Figure
+1 A. With p= 0:04, this has an entropy of 0 :24 bits/msec, but because it can be described by a single state, the
+complexity is zero. (That state emits either a \0" or a \1", with respective probabilities 0 :96 and 0:04, but either way
+the state transitions back to itself.) In contrast, adding a 5 ms refractory period to the process means six states are
+needed to describe the spike trains (Figure 1 B). The new structure of the refractory period is quantied by the higher
+complexity, C= 1:05 bits.
+The second and third terms in Eq. 10 both describe randomness, but of distinct kinds. The second term, the
+internal entropy rate J, quanties the randomness in the state transitions; it is the entropy of the next state given
+the current state.
+J=H[St+1jSt] =E[logP(St+1jSt)] (12)
+This is the average number of bits per time-step needed to describe the sequence of states the process moved through
+(beyond those given by C). The last term in Eq. 10 accounts for any residual randomness in the spikes which is not
+captured by the state transitions.
+R=H[Xt+1jSt;St+1] =E[logP(Xt+1jSt;St+1)] (13)
+10The algorithmic information content is also called the Kolmogorov complexity. We do not use this term, to avoid
+confusion with our \complexity" Cthe information needed to reproduce the spike train statistically rather than
+exactly (Eq. 11). See Badii & Politi (1997) for a detailed comparison of complexity measures.8
+For long trains, the entropy of the spikes, H[Xn
+1], is approximately the sum of these two terms, H[Xn
+1]n(J+R).
+Computationally, Crepresents the xed generating structure of the process, which needs to be described once, at
+the beginning of the time series, and n(J+R) represents the growing list of details which pick out a particular time
+series from the ensemble which could be generated; this needs, on average, J+Rextra bits per time-step. (Cf. the
+\sophistication" of G acs et al. (2001).)
+Consider again the 40 Hz Bernoulli process. As there is only one state, the process always stays in that state. Thus
+the entropy of the next state J= 0. However, the state sequence yields no information about the emitted symbols
+(the process is IID), so the residual randomness R= 0:24 bits/msec | as it must be, since the total entropy rate is
+0:24 bits/msec. In contrast, the states of the 5 msec refractory process are informative about the process's future. The
+internal entropy rate J= 0:20 bits/msec and the residual randomness R= 0. All of the randomness is in the state
+transitions, because they uniquely dene the output spike train. The randomness in the state transition is conned
+to stateA, where the process \decides" whether it will stay in A, emitting no spike, or emit a spike and go to B. The
+decision needs, or gives, 0 :24 bits of information. The transitions from BthroughFand back to Aare xed and
+contribute 0 bits, reducing the expected J.
+The important point is that the structure present in the refractory period makes the spike train less random,
+lowering its entropy. Averaged over time, the mean ring rate of the process is p= 0:0333. Were the spikes IID, the
+entropy rate would be 0 :21 bits/msec, but in fact J+R= 0:20 bits/msec. This is because a minimal description
+of a long sequence Xt1:::XtN=XtN
+t1, the generating process only needs to be described once (C), while the internal
+entropy rate and randomness need to be updated at each time step ( n(J+R)). Simply put, a complex, structured
+spike train can be exactly described in fewer bits than one which is entirely random. The CSM lets us calculate this
+reduction in algorithmic information, and quantify the structure by means of the complexity.
+D. Time-Varying Complexity and Entropies
+The complexity and entropy are ensemble-averaged quantities. In the previous section the ensemble was the entire
+time series, and the averaged complexity and entropies were analogous to a mean ring rate. The time-varying
+complexity and entropies are also of interest, for example their variation after stimuli. A peri-stimulus time histogram
+(PSTH) shows how the ring probability varies with time; the same idea works for the complexity and entropy.
+Since the states form a Markov chain, and any one spike train stays within a single ergodic component, we can
+invoke the ergodic theorem (Gray, 1988), and (almost surely) assert that
+X
+St;St+1P(St;St+1;Xt+1)f(St;St+1;Xt+1) = lim
+N!11
+NNX
+t=1f(St;St+1;Xt+1)
+= lim
+N!1hf(St;St+1;Xt+1)iN (14)
+for arbitrary integrable functions f(St;St+1;Xt+1).
+In the case of the mean ring rate, the function to time-average is l(t)Xt+1. For the time averaged-complexity,
+internal entropy and residual randomness, the functions (respectively c,jandr) are
+c(t) =logP(St)
+j(t) =logP(St+1jSt)
+r(t) =logP(Xt+1jSt;St+1) (15)
+and time-varying entropy h(t) =j(t) +r(t).
+The PSTH averages over an ensemble of stimulus presentations, rather than time:
+PSTH (t) =1
+MMX
+i=1li(t) =1
+MMX
+i=1Xt+1;i (16)
+withMbeing number of stimulus presentations, and tre-set to zero at each presentation. Analogously, the \PSTH"
+of the complexity is
+CPSTH (t) =1
+MMX
+i=1ci(t) =1
+MMX
+i=1logP(St;i) (17)9
+For the entropies, replace cwithj,rorhas appropriate. Similar calculations can be made with any well-dened
+ensemble of reference times, not just stimulus presentations; we will also calculate cand the entropies as functions of
+the time since the latest spike.
+We can estimate the error these time-dependent quantities as the standard error of the mean as a function of time,
+SEt=st=p
+Mwherestis the sample standard deviation in each time bin tandMis the number of trials. The
+probabilities appearing in the denitions of c(t),j(t),r(t) also have some estimation errors, either because of sampling
+noise or, more interestingly, because the ensemble is being distorted by outside in
uences. The latter creates a gap
+between their averages (over time or stimuli) and what the CSM predicts for those averages. In the next section, we
+explain how to use this to measure the in
uence of external drivers.
+E. The In
uence of External Forces
+If we know that St=s, the CSM predicts that ring probability is (t) =P(Xt+1= 1jSt=s). By means of the
+CSM's recursive ltering property (Appendix A), once a transient regime has passed, the state is always known with
+certainty. Thereafter, the CSM predicts what the ring probability should be at all times, incorporating the eects of
+the spike train's history. As we show in the next section, these predictions give good matches to the actual response
+function in simulations where the spiking probability depends only on the spike history. But real neurons' spiking
+rates generally also depend on external processes, e.g., stimuli. As currently formulated, the CSM is (or, rather,
+converges on) the optimal predictor of the future of the process given its own past. Such an \output only" model
+does not represent the (possible) eects of other processes, and so ignores external covariates and stimuli. Presently,
+determining the precise form of spike trains' responses to external forces is best left to parametric models.
+However, we can use output-only CSMs to learn something about the computation: the PSTH-calculated entropy
+rateHPSTH (t) =JPSTH (t) +RPSTH (t) quanties the extent to which external processes drive the neuron. (The
+PSTH subscript is henceforce supressed.) Suppose we know the true ring probability true(t). At each time step,
+the CSM predicts the ring probability CSM(t). IfCSM(t) =true(t), then the CSM correctly describes the spiking
+and the PSTH entropy rate is
+HCSM(t) =CSM(t) log [CSM(t)](1CSM(t)) log [1CSM(t)] (18)
+However, if CSM(t)6=true(t), then the CSM mis-describes the spiking, because it neglects the in
uence of external
+processes. Simply put, the CSM has no way of knowing when the stimuli happen. The PSTH entropy rate calculated
+using the CSM becomes
+HCSM(t) =true(t) log [CSM(t)](1true(t)) log [1CSM(t)] (19)
+Solvingtrue(t),
+true(t) =HCSM(t) + log [1CSM(t)]
+log [1CSM(t)]log [CSM(t)](20)
+The discrepancy between CSM(t) andtrue(t) indicates how much of the apparent randomness in the entropy rate
+is actually due to external driving. The true PSTH entropy rate Htrue(t) is
+Htrue(t) =true(t) log [true(t)](1true(t)) log [1true(t)] (21)
+The dierence between HCSM(t) andHtrue(t) quanties, in bits, the driving by external forces as a function of the
+time since stimulus presentation.
+
H=HCSM(t)Htrue(t)
+=true(t) logtrue(t)
+CSM(t)
++ (1true(t)) log1true(t)
+1CSM(t)
+(22)
+This stimulus-driven entropy 
His the relative entropy or Kullback-Leibler divergence D(XtruekXCSM) between the
+true distribution of symbol emissions and that predicted by the CSM. Information-theoretically, this relative entropy
+is the error in our prediction of the next state due to assuming the neuron is running autonomously when it's actually
+externally driven. Since every state corresponds to a distinct distribution over future behavior, this is our error in
+predicting the future due to ignorance of the stimulus.11
+11Cf. the informational coherence introduced by Klinkner et al. (2006) to measure of information-sharing between10
+III. RESULTS
+We now present a few examples. (All of them use a time-step of 1 millisecond.) We begin with idealized model
+neurons to illustrate our technique. We recover CSMs for the model neurons using only the simulated spike trains
+as input to our algorithms. From the CSM we calculate the complexity, entropies, and, when appropriate, stimulus-
+driven entropy (Kullback-Leibler divergence between the true and CSM predicted ring probabilities) of each model
+neuron. We then analyze spikes recorded in vivo from a neuron in layer II/III of rat SI (barrel) cortex. We use spike
+trains recorded both with and without external stimulation of the rat's whiskers. See Andermann & Moore (2006)
+for experimental details.
+A. Model neuron with a \soft" refractory period and bursting
+We begin with a refractory, bursting model neuron, whose spiking rate depends only on the time since the last
+spike. The baseline rate is 40 Hz. Every spike is followed by a 2 msec \hard" refractory period, during which spikes
+never occur. The spiking rate then rebounds to twice its baseline, to which it slowly decays. (See dashed line in the
+rst panel of Figure 3 B.) This history dependence mimics that of a bursting neuron, and is, intuitively, more complex
+than the simple refractory period of the model in Figure 1.
+Figure 2 shows the 17-state CSM reconstructed from a 200 second spike train (at 1 msec resolution) generated by
+this model. It has a complexity of C= 3:16 bits (higher than that of the model in Figure 1, as anticipated), an
+internal entropy rate of J= 0:25 bits/msec and a residual randomness of R= 0 bits/msec. The CSM was obtained
+with  = 17 (selected by BIC). Figure 3 A shows how the 99% ISI bounds bootstrapped from the CSM enclose the
+empirical ISI distribution, with the exception of one short segment.
+The CSM is easily interpreted. State Ais the baseline state. When it emits a spike, the CSM moves to state
+B. There are then two deterministic transitions, to Cand thenD, which never emit spikes; this is the hard 2 msec
+refractory period. Once in Dit is possible to spike again, and if that happens, the transition is back to state B.
+However, if no spike is emitted, the transition is to state E. This is repeated, with varying ring probabilities, as
+statesEthroughQare traversed. Eventually, the process returns to Aand so to baseline.
+Figure 3 B plots the ring rate, complexity, and internal entropies as functions of the time since the last spike
+conditional on no subsequent spike emission . This lets us compare the ring rate predicted by the CSM (solid line
+squares) to the specication of the model which generated the spike train (dashed line) and a PSTH calculated by
+triggering on the last spike (solid line). Except at 16 and 17 msec post spike, the CSM-predicted ring rate agrees
+with both the generating model and the PSTH. The discrepancy arises because the CSM only discerns the structure
+in the data, and most of the ISIs are shorter than 16 msec. There is much closer agreement between the CSM and
+the PSTH if ring rates are plotted as a function of time since a spike without conditioning on no subsequent spike
+emission (not shown).
+The second and third panels of Figure 3 plot the time-dependent complexity and entropies. The complexity is
+much higher after the emission of a spike than during baseline, because the states traversed (B-Q) are less probable,
+and represent the additional structures of refractoriness and bursting. The time-dependent entropies (third panel)
+show that just after a spike, the refractory period imposes temporary determinism on the spike train, but burstiness
+increases the randomness before the dynamics return to the baseline state.
+B. Model neuron under periodic stimulation
+Figure 4 shows the CSM for a periodically-stimulated model neuron . This CSM was reconstructed from 200 seconds
+of spikes with a baseline ring rate of 40 Hz ( p= 0:04). Each second, the ring rate rose over the course of 5 msec to
+p= 0:54 spikes/msec, falling slowly back to baseline over the next 50 msec. This mimics the periodic presentation of
+a strong external stimulus. (The exact inhomogeneous ring rate used was (t) = 0:93[et=10et=2] + 0:04 witht
+in msec. See Figure 5 B, rst panel, dashed line.) In this model, the ring rate does not directly depend on the spike
+train's history, but there is a sort of history dependence in the stimulus time-course, and this is what CSSR discovers.
+BIC selected  = 7, giving a 16 state CSM with C= 0:89 bits,J= 0:27 bits/msec and R= 0:0007 bits/msec. The
+baseline is again state Aand if no spike is emitted then the process stays in A. Spikes are either spontaneous and
+neurons, by quantifying the error in predicting the distribution of the future of one neuron due to ignoring its
+coupling with another.11
+random, or stimulus-driven. Because the stimulus is external, it is not immediately clear which of these two causes
+produced a given spike. Thus, if a spike is emitted, the CSM traverses states BthroughF, deciding, so to speak,
+whether or not the spike is due to a stimulus. If two spikes happen within 3 msec of each other, then the CSM decides
+that it is being stimulated and goes to one of states G,HorM. StatesGthroughPrepresent the response to the
+stimulus. The CSM moves between these states until no spike is emitted for 3 msec, when it returns to the baseline,
+A.
+The ISI distribution from the CSM matches that from the model (Figure 5 A). However, because the stimulus
+doesn't depend on spike train's history, the CSM makes inaccurate predictions during stimulation. The rst panel of
+Figure 5 B plots the ring rate as a function of time since stimulus presentation, comparing the model (dashed line)
+and the PSTH (solid line) with the CSM's prediction (line with squares). The discrepancy between these is due to
+the CSM having no way of knowing that an external stimulus has been applied until several spikes in a row have been
+emitted (represented, as we just say, by states B{F)12. Despite this, c(t) shows that something more complex than
+simple random ring is happening (second panel of Figure 5 B), as do j(t) andr(t) (third panel). Further, something
+is clearly wrong with the entropy rate, because it should be upper-bounded by h= 1 bit/msec (when p= 0:5). The
+fact thath(t) exceeds this bound indicates an external force, not fully captured by the CSM, is at work.
+As discussed in Methods ( xII.E), drive from the stimulus can be quantied with a relative entropy (Figure 5 C).
+Stimuli are presented at t= 1 msec, where 
 H(t)>1 bit. It is not until 25 msec post-stimulus that 
 H(t)0 and
+the CSM once again correctly describes the internal entropy rate. Thus, as expected, the stimulus strongly in
uences
+neuronal dynamics immediately after its presentation. The true internal entropy rate Htrue(t) is slightly less than 1
+bit/msec shortly after stimulation, when the true spiking rate has a maximum of pmax= 0:54. The fact that the CSM
+gives an inaccurate value for Jactually lets us nd the number of bits of information gain supplied by the stimulus,
+e.g., 
H > 1 bit immediately after the stimulus is presented.
+C. Spontaneously Spiking Barrel Cortex Neuron
+We reconstructed a CSM from 90 seconds of spontaneous (no vibrissa de
ection) spiking recorded from a layer
+II/III FSU barrel cortex neuron. CSSR, using  = 21, discovered a CSM with 315 states, a complexity of C= 1:78
+bits, and internal entropy rate of J= 0:013 bits/msec. After state culling ( xII.B.2), the reduced CSM, plotted in
+Figure 6, has 14 states, C= 1:02,J= 0:10 bits/msec, and residual randomness of R= 0:005 bits/msec. We focus on
+the reduced CSM from this point onwards.
+This CSM resembles that of the spontaneously-ring model neuron of xIII.A and Fig. 2. The complexity and
+entropies are lower than those of our model neuron because the mean spike rate is much lower, and so simple
+descriptions suce most of the time. (Barrel cortex neurons exhibit notoriously low spike rates, especially during
+anesthesia.) There is a baseline state Awhich emits a spike with probability p= 0:01, i.e., 10 Hz. When a spike is
+emitted, the CSM moves to state Band then on through the chain of states CthroughN, return to Aif no spike is
+subsequently emitted. However, the CSM can emit a second or even third spike after the rst, and indeed this neuron
+displays spike doublets and triplets. In general, emitting a spike moves the CSM to B, with some exceptions that
+show the structure to be more intricate than the model neuron's.
+Figure 7 A shows the CSM's 99% condence bounds almost completely enclosing the empirical ISI distribution.
+The rst panel of Figure 7 B plots the history-dependent ring probability predicted by the CSM as a function of the
+time since the latest spike, according to both the PSTH and the CSM's prediction. They are highly similar in the
+rst 13 msec post-spike, indicating that the CSM gets the spiking statistics right in this epoch. The CSM and PSTH
+the diverge after this, for two reasons. First, as with the model neuron, there are few ISIs of this length. Most of the
+ISIs are either shorter, due to the nueron's burstiness, or much longer, due to the low baseline ring rate. Secondly,
+90 seconds is not very much data. We show in Figure 10 that a CSM reconstructed from a longer spike train does
+capture all of the structure. We present the results of this shorter spike train to emphasize that, as a non-parametric
+method, CSSR only uncovers the statistical structure in the data , no more, no less.
+Finally, the second and third panels of Figure 6 B show, respectively, the complexity and entropies as functions of
+the time since the latest spike. As with the model of xIII.A, the structure in the process occurs after spiking, during
+the refractory and bursting periods. This is when the complexity is largest, and also when the entropies vary most.
+12In eect, this part of the CSM implements Bayes's rule, balancing the increased likelihood of a spike after a stimulus
+against the low a priori probability or base-rate of stimulation.12
+D. Periodically Stimulated Barrel Cortex Neuron
+We reconstructed CSMs from 335 seconds of spike trains taken from the same neuron used above, but recorded
+while it was being periodically stimulated by vibrissa de
ection. BIC selected  = 25, giving the 29-state CSM shown
+in Figure 8. (Before state culling, the original CSM had 1916 states, C= 2:55 andJ= 0:11.) The reduced CSM has a
+complexity of C= 1:97 bits, an internal entropy rate of J= 0:10 bits/msec, and a residual randomness of R= 0:005
+bits/msec. Note that Cis higher when the neuron is being stimulated as opposed to when it is spontaneously ring,
+indicating more structure in the spike train.
+While at rst the CSM may seem to only represents history-dependent refractoriness and bursting, ignoring the
+external stimulus, this is not quite true. Once again, there is a baseline state A, and most of the other states ( B{X)
+comprise a refractory/bursting chain, like this neuron has during spontaneous ring. However, the transition upon A
+emitting a spike is not back to Band then down the chain again, but to either state C1, and subsequently C2, or more
+often to state ZZ. These three states represent the structure induced by the external stimulus, as we saw with the
+model stimulated neuron of xIII.B and Figure 4. (The state ZZis comparable to the state Mof the model stimulated
+neuron: both loop back to themselves if they emit a spike.) Three states are enough because, in this experiment,
+barrel cortex neurons spike extremely sparsely, 0 :1{0:2 spikes per stimulus presentation.
+Figure 9 A plots the ISI distribution, nicely enclosed by the bootstrapped condence bounds. Figure 9 B shows
+the ring rate, complexity and entropies as functions of the time since stimulus presentation (averaged over all
+presentations). These plots look much like those in Figure 7 B. However, there is a clear indication that something
+more complex takes place after stimulation: the CSM's ring-rate predictions are wrong. The stimulus-driven entropy
+
Hturns out to be as large as 0 :02 bits within 5{15 msec post-stimulus. This agrees with the known 5{10 msec
+stimulus propagation time between vibrissae and barrel cortex (Andermann & Moore, 2006). The reason that 
 H
+is so much smaller for the real neuron than the stimulated model neuron of xIII.B is that the former's ring rate is
+much lower. Although the ring rate post-stimulus can be almost twice as large as the CSM's prediction, the actual
+rate is still low, max (t)0:04 spikes/msec. Most of the time the neuron does not spike, even when stimulated, so
+on average, the stimulus provides little information per presentation. For completeness, Figure 10 shows the spike
+probability, complexity and entropies as functions of the time since the latest spike. Averaged over this ensemble, the
+CSM's predictions are highly accurate.
+IV. DISCUSSION
+The goal of this paper was to present methods for determining the structural content of spike trains while making
+minimal a priori assumptions as to the form which that structure takes. We use the CSSR algorithm to build
+minimal, optimally predictive hidden Markov models (CSMs) from spike trains, Schwartz's Bayesian Information
+Criterion to nd the optimal history length  of the CSSR algorithm, and bootstrapped condence bounds on the
+ISI distribution from the CSM to check goodness-of-t. We demonstrated how CSMs can estimate a spike train's
+complexity, thus quantifying its structure, and its mean algorithmic information content, quantifying the minimal
+computation necessary to generate the spike train. Finally we showed how to quantify, in bits, the in
uence of
+external stimuli upon the spike-generating process. We applied these methods both to simulated spike trains, for
+which the resulting CSMs agreed with intuition, and to real spike trains recorded from a layer II/II rat barrel cortex
+neuron, demonstrating increased structure, as measured by the complexity, when the neuron was being stimulated.
+We are unaware of any other practical techniques for quantifying the complexity and computational structure of
+a spike train as we dene them. Intuitively, neither random (Poisson), nor highly ordered (e.g., strictly periodic, as
+in Olufsen et al. (2003)) spike trains should be thought of as complex since they do not possess structure requiring a
+sophisticated program to generate. Instead, complexity lies between order and disorder (Badii & Politi, 1997), in the
+non-random variation of the spikes. Higher complexity means a greater degree of organization in neural activity than
+would be implied by random spiking. It is the reconstruction of the CSM through CSSR which allows us to calculate
+the complexity.
+Our denition of complexity stands in stark contrast to other complexity measures which assign high values to
+highly disordered systems. Some of these, such as Lempel Ziv complexity (Amigo et al., 2002, 2004; Jimenez-Montano
+et al., 2002; Szczepanski et al., 2004) and context free grammar complexity (Rapp et al., 1994) have been applied to
+spike trains. However, both of these are measures of the amount of information required to reproduce the spike train
+exactly , and take on very high values for completely random sequences. These \complexity" measures are therefore
+much more similar to total algorithmic information content and even to the entropy rate than to our sort of complexity.
+Our measure of complexity is the entropy of the distribution of causal states. This has the desired property of
+being maximized for structured, rather than ordered or disordered systems, because the causal states are dened
+statistically, as equivalence classes of histories conditioned on future events. Other researchers have also calculated13
+complexity measures which are entropies of state distributions, but have dened their states dierently. Amigo et al.
+(2002) uses the observables (symbol strings) present in the spike train to dene a k-th order Markov process and calls
+each individual length k string which appears in the spike train a state. Gorse and Taylor (1990) similarly use single
+sux symbol strings to dene the states of a Markov process. In both cases, IID Bernoulli sequences could exhibit
+up to 2kstates (in long enough sequences), and possess an extremely high \complexity". However, all of these states
+make the same prediction for the future of the process. The minimal representation is a single causal state, a CSM
+with a complexity of zero.
+It should be noted that there are also many works which model spike trains using HMMs, but in which the hidden
+states represent macro -states of the system (awake/asleep, Up/Down, etc.), and spiking rates are modeled separately
+in each macro-state (Abeles et al., 1995; Achtman et al., 2007; Chen et al., 2008; Danoczy & Hahnloser, 2005; Jones
+et al., 2007). Although the graphical representation of such HMMs may look like those of CSMs, the two kinds of
+states have very dierent meanings. Finally, there are also state space methods which model the dynamical state of
+the system as a continuous hidden variable, the most well known of which is the linear Gaussian model with Kalman
+ltering. These have been extensively applied to neural encoding and decoding problems (Eden et al., 2007; Smith
+et al., 2004; Srinivasan et al., 2007). Interestingly, for a univariate Gaussian ARMA model in state-space form, the
+Kalman lter's one-step-ahead prediction and mean-squared prediction error are, jointly, minimal-sucient for next-
+step prediction, and since they can be updated recursively they in fact constitute the minimal sucient statistic, and
+hence the causal state in this special case.
+Neurons are driven by their aerent synapses. Although as discussed in Appendix C, there is a parallel \transducer"
+formalism for generating CSMs which take external in
uences into account, this is not yet computationally imple-
+mented, and our current approach reconstructs CSMs only from the spike train. Since the history of the neuron under
+study is typically connected with the history of the network in which it is located, this CSM will, in general, re
ect
+more than a neuron's internal biophysical properties. Nonetheless, in both our model neurons and in the real barrel
+cortex neuron, states not interpretable as simple refractoriness or bursting appeared when a stimulus was present,
+proving we can detect stimulus-driven complexity. Further, we showed that the CSM can be used to determine the
+extent (in bits) to which a neuron is driven by external stimuli.
+The methods presented here complement more established modes of spike-train analysis, which have dierent goals.
+Parametric methods, such as PSTHs or maximum likelihood estimation (Brown et al., 2004; Truccolo et al., 2005)
+generally focus on determining a neuron's ring rate (mean, instantaneous or history-dependent), and on how known
+external covariates modulate that rate. They have the advantage of requiring less data than non-parametric methods
+such as CSSR, but the disadvantage, for our purposes, of imposing the structure of the model at the outset. When
+the experimenter wants to know how a neuron encodes a particular aspect of a covariate, e.g., how neurons in the
+sensory periphery or primary sensory cortices encode stimuli, parametric methods have proved highly illuminating.
+However, in many cases the identity or even existence of relevant external covariates is uncertain. For example, one
+could envision using CSMs to analyze recordings in pre-frontal cortex during dierent cognitive tasks, or to perhaps
+compare spiking structure during dierent attentional states. In both cases, the relevant external covariates are not
+at all clear, but CSMs could still be used to quantify changes in computational structure, for single neurons or for
+groups of them. For neural populations one can envision generating distributions (over the population) of complexities
+and examining how these distributions change in dierent cortical macro-states. This would be entirely analagous to
+analyzing distributions of ring rates or tuning curves.
+In addition to calculations of the complexity, the whole array of mutual-information analyses can be applied to
+CSMs, but instead of calculating mutual information between the spikes and the covariates (which could include other
+spike trains), one can calculate the mutual information between the covariates and the causal states . The advantage
+is that the causal states represent the behavioral patterns of the spike-generating process, and so are closer to the
+actual state of the system than the spikes (output observables) are themselves. Results on calculating the mutual
+information between the causal states of dierent neurons (informational coherence) in a large simulated network
+show that synchronous neuronal dynamics are more eectively revealed than when calculated directly from the spikes
+(Klinkner et al., 2006).
+In closing, our methods provide a way to understand structure in spike trains, and should be considered as comple-
+ments to traditional analysis methods. We rigorously dene structure, and show how to discover it from the data itself.
+Our methods go beyond those which seek to describe the observed variation in the spiking rates by also describing
+the underlying computational process (in the form of a CSM) needed to generate that variation. A CSM can show
+not only that the spike rate has changed, but also show exactly howit has changed.
+Acknowledgments The authors thank Mark Andermann and Christopher Moore for the use of their data. RH thanks
+Emery Brown, Anna Dreyer and Christopher Moore for valuable discussions. CRS thanks Anthony Brockwell, Dave
+Feldman, Chris Genovese, Rob Kass and Alessandro Rinaldo for valuable discussions.14
+APPENDIX A: Filtering with CSMs
+A common diculty with hidden Markov models is that predictions can only be made from a knowledge of the state,
+which must itself be guessed at from the time series, since it is, after all, hidden. This creates the state estimation
+orltering problem. Under strong assumptions (linear Gaussian stochastic dynamics, linearly observed through IID
+additive Gaussian noise) the Kalman lter is an optimal yet tractable solution. For non-linear processes, however,
+optimal ltering essentially amounts to maintaining a posterior distribution over the states and updating it via Bayes's
+rule (Ahmed, 1998). (This distribution is sometimes called the process's \information state".)
+One convenient and important feature of CSMs is that this whole machinery of ltering is unnecessary, because of
+their recursive-updating property. Given the state at time t,St, and the observation at time t+ 1,Xt+1, the state at
+timet+ 1 is xed, St+1=T(St;Xt+1) for some transition function T. Clearly, if the state is known with certainty
+at any time, it will remain known. However, the same recursive updating property also allows us to show that the
+state does become certain, i.e., that after some nite (but possibly random) time ,P(S=sjX
+1) is either 0 or 1 for
+all statess. For Markov chains of order k, clearlyk; under more general circumstances P(t) goes to zero
+exponentially or faster.
+Thus, after a transient period, the state is completely unambiguous. This will be useful to us in multiple places,
+including understanding the computational structure of the process and predicting the ring rate of the neuron. It also
+leads to considerable numerical simplications, compared to approaches which demand conventional ltering. Further,
+recursive ltering is easily applied to a new spike train, not merely the one from which the CSM was reconstructed.
+This helps in cross-validating CSMs, as discussed in the next appendix.
+APPENDIX B: Cross-Validation
+It is often desirable to cross-validate a statistical model by spliting one's data set in two, using one part (generally
+the larger) as a training set for the model and the other part to validate the model by some statistical test. In the
+case of CSMs it is particularly important to check the validity of the BIC used to regularize the  control-setting.
+One possible test is the ISI bootstrapping of xII.B.3. A second, somewhat stronger, goodness-of-t test is based
+on the time rescaling theorem of Brown et al. (2002). This test rescales the interspike intervals as a function of the
+integrated history-dependent spiking rate over the ISI:
+k= 1eRtk+1
+tk(t)dt(B1)
+where theftkgare the spike times and (t) is the history-dependent spiking rate from the CSM. If the CSM describes
+the data well, then rescaled ISI's fkgshould follow a uniform distribution. This can be tested using either a
+Kolmogorov Smirnov test or by plotting the empirical CDF of the rescaled times against the CDF of the uniform
+distribution (Kolmogorov Smirnov or \KS" plot) (Brown et al., 2002).
+Figure 11 gives cross-validation results for the rat barrel cortex neuron, during both spontaneous ring and periodic
+vibrissae de
ection. 90 seconds of spontaneously ring spikes were split into a 75 second training set and a 15 second
+validation set. The 335 seconds of stimulus-evoked ring were split into a 270 second training set and a 65 second
+validation set. Panels A and B show the ISI bootstrapping results for the spontaneous and stimulus evoked ring
+respectively. The dashed lines are 99% condence bounds from a CSM reconstructed from the training set and the
+solid line is the ISI distribution of the validation set. The ISI distribution largely falls within these bounds for both
+the spontaneous and stimulus evoked data.
+Panels C-F display the time rescaling test. Panels C and D show the time rescaling plots for the spontaneous and
+stimulus evoked training data respectively. The dashed lines are 95% condence bounds. The spontaneous KS plot
+largely falls within the bounds. The stimulus-evoked does not, but this is expected because, as discussed, the CSM
+does not completely capture the imposition of the external stimulus. (The jagged \steps" in both plots result from
+the 1 msec temporal discretization.) Panels E and F show the time rescaling plots for, respectively, the spontaneous
+and stimulus evoked validation data. The ts here are somewhat worse. In the stimulated case, this is not surprising.
+In the spontaneous case the cause is likely non-stationarity in the data, a problem shared with other spike train
+analysis techniques, such as the Generalized Linear Model approaches described in the next Appendix. It should be
+emphasized that the point of reconstructing CSMs is not to obtain perfect ts to the data, but instead to estimate
+the structure inherent in the spike train, and the cross-validation results should be viewed in this light.15
+APPENDIX C: Causal State Transducers and Predictive State Representations
+Mathematically, CSMs can be expanded to include the in
uence of external stimuli on the process, yielding causal
+state transducers , which are optimal representations of the history-dependent mapping from inputs to outputs (Shalizi,
+2001, ch. 7). Such causal state transducers are a type of partially-observable Markov decision process, closely related
+to predictive state representations (PSRs) (Littman et al., 2002). In both formalisms, the right notion of \state"
+is a statistic, a measurable function of the observable past of the process. Causal states represent this through an
+equivalence relation on the space of observable histories. For PSRs, the representation is through \tests", i.e., a dis-
+tinguished set input/output sequence pairs; the idea is that states can be uniquely characterized by their probabilities
+of producing the output sequences conditional on the input sequences.
+An algorithm for reconstructing causal state transducers would begin by estimating probability distributions of
+future histories conditioned on both the history of the spikes and the history of an external covariate Y, e.g.
+P(X1
+t+1jXt
+1;Yt
+1), and otherwise be entirely parallel to CSSR. This has not yet implemented.
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+A 0 | 0.96 1 | 0.04
+A0 | 0.96 B 1 | 0.04 C0 | 1 D0 | 1
+E0 | 1
+F0 | 1
+0 | 1A
+B
+FIG. 1 Two simple CSMs reconstructed from 200 sec of simulated spikes using CSSR. States are represented as the nodes
+of a directed graph. The transitions between states are labeled with the symbol emitted during the transition (1 = spike, 0
+= no spike) and the probability of the transition given the origin state. (A) The CSM for a 40 Hz Bernoulli spiking process
+consists of a single state Awhich always transitions back to itself, emitting a spike with probability p= 0:04 per msec. (B)
+CSM for 40 Hz Bernoulli spiking process with a 5 msec refractory period imposed after each spike. State Aagain spikes with
+probability p= 0:04. Upon spiking the CSM transitions through a deterministic chain of states B{F(squares) which represent
+the refractory period. The increased structure of the refractory period requires a more complex representation.18
+A0 | 0.957
+B1 | 0.043
+C0 | 1.000
+D0 | 1.000
+H1 | 0.053
+I0 | 0.9471 | 0.069
+J0 | 0.931F1 | 0.024
+G0 | 0.9761 | 0.039
+0 | 0.9611 | 0.076
+K0 | 0.924E1 | 0.012
+0 | 0.9881 | 0.003
+0 | 0.997
+P1 | 0.069
+Q0 | 0.9310 | 0.933
+1 | 0.067
+O1 | 0.074
+0 | 0.926N1 | 0.078
+0 | 0.922M1 | 0.080
+0 | 0.920L1 | 0.075
+0 | 0.9251 | 0.079
+0 | 0.921
+FIG. 2 CSM reconstructed from a 200 sec simulated spike train with a \soft" refractory/bursting structure. C= 3:16,J= 0:25,
+R= 0. State A(circle) is the baseline 40 Hz spiking state. Upon emitting a spike the transition is to state B. States Band
+C(squares) are \hard" refractory states from which no spike may be emitted. States Dthrough Q(hexagons) compromise a
+refractory/bursting chain from which if a spike is emitted the transition is back to state B. Upon exiting the chain the CSM
+returns to the baseline state A.19
+0 5 10 15 20 25 3000.20.4Entropies0 5 10 15 20 25 300246Complexity (C(t))0 5 10 15 20 25 3000.050.1History Dependent Firing ProbabilityISI Distribution
+0 5 10 15 20 25 30 35 40 45 5000.020.040.060.08
+time since most recent spike (msec)msecspikes/msec bits bitsA
+B
+λCSM(t)
+λPSTH (t)
+J(t)
+R(t)H(t)λmodel (t)
+FIG. 3 \Soft" refractory and bursting model ISI distribution and time dependent ring probability, complexity and entropies.
+(A) ISI distribution and 99% condence bounds bootstrapped from the CSM. (B) First panel: Firing probability as a function
+of time since the most recent spike. Line with squares = ring probability predicted by CSM. Solid line = ring probability
+deduced from PSTH. Dashed line = model ring rate used to generate spikes. Second panel: Complexity as a function of time
+since most recent spike. Third panel: Entropies as a function of time since most recent spike. Squares = internal entropy rate,
+circles = residual randomness, solid line = entropy rate. (overlaps squares)20
+A0 | 0.957
+B1 | 0.043
+C0 | 0.944
+H1 | 0.056D1 | 0.055
+F0 | 0.945
+E0 | 0.885
+M1 | 0.1150 | 0.882
+1 | 0.1180 | 0.948
+G1 | 0.0520 | 0.821
+1 | 0.179
+I0 | 0.791
+1 | 0.209 J0 | 0.839
+K1 | 0.1610 | 0.878
+1 | 0.122
+L0 | 0.677 1 | 0.323
+1 | 0.205
+N0 | 0.795
+1 | 0.392 0 | 0.6081 | 0.366
+O0 | 0.6341 | 0.265
+P0 | 0.7350 | 0.816
+1 | 0.184
+FIG. 4 16-state CSM reconstructed from 200 sec of simulation of periodically-stimulated spiking. C= 0:89,J= 0:27,
+R= 0:0007. State Ais the baseline state. States Bthrough F(octagons) are \decision" states in which the CSM evaluates
+whether a spike indicates a stimulus or was spontaneous. Two spikes within 3 msec cause the CSM to transition to states G
+through P, which represent the structure imposed by the stimulus. If no spikes are emitted within 5 (often fewer) sequential
+msec, the CSM goes back to the baseline state A.21
+0 5 10 15 20 25 30 35 40 45 5000.020.040.060.080.10.12
+0 5 10 15 20 25 30 35 40 45 5000.20.40.6Time Dependent Firing Probability
+0 5 10 15 20 25 30 35 40 45 500510Complexity C(t)
+0 5 10 15 20 25 30 35 40 45 500123EntropiesISI Distribution
+msec
+Time since stimulus (msec)
+Time since stimulus (msec)bits bitsspikes/msec
+Stimulus Driven Entropy  (ΔH(t))A
+B
+CλCSM(t)
+λPSTH (t)
+J(t)
+R(t)
+H(t)λmodel (t)
+0 5 10 15 20 25 30 35 40 45 5000.511.5 bits
+FIG. 5 Stimulus model ISI distribution and time-dependent complexity and entropies. (A) ISI distribution and 99% condence
+bounds. (B) First panel: Firing probability as a function of time since stimulus presentation. Second panel: Time dependent
+complexity. Third panel: time-dependent entropies. (C) The stimulus-driven entropy is >1 bit, indicating strong external
+drive. See text for discussion.22
+A0 | 0.990
+B1 | 0.010
+D
+E0 | 0.9281 | 0.072
+1 | 0.065
+F0 | 0.935C
+0 | 0.9611 | 0.039 0 | 0.9911 | 0.009
+H1 | 0.046 G0 | 0.954
+M1 | 0.042
+N0 | 0.9580 | 0.959
+1 | 0.041
+L1 | 0.036
+0 | 0.964K1 | 0.041
+0 | 0.959J
+1 | 0.0320 | 0.968I1 | 0.047
+0 | 0.9531 | 0.052
+0 | 0.9481 | 0.053
+0 | 0.947
+FIG. 6 14-state CSM reconstructed from 90 sec of spiking recorded from a spontaneously spiking (no stimulus) neuron located
+in layer II/III of rat barrel cortex. C= 1:02,J= 0:10,R= 0:005. State A(circle) is baseline 10 Hz spiking. States Bthrough
+Ncomprise a refractory/bursting chain similar to, but with a somewhat more intricate structure than, that of the model neuron
+in Figure 223
+0 5 10 15 20 25 30 35 40 45 5000.020.040.060.080.1
+0 5 10 15 20 25 3000.020.040.06History Dependent Firing Probability
+0 5 10 15 20 25 3002468Complexity C(t)
+0 5 10 15 20 25 3000.20.4EntropiesISI Distribution
+msecspikes/msecbits bits
+time since most recent spike (msec)BA
+J(t)
+R(t)
+H(t)λCSM(t)
+λPSTH (t)
+FIG. 7 Spontaneously spiking barrel cortex neuron. (A) ISI distribution and 99% bootstrapped condence bounds. (B) First
+panel: Time dependent ring probability as a function of time since most recent spike. See text for explanation of discrepancy
+between CSM and PSTH spike probabilities. Second Panel: Complexity as a function of time since most recent spike. Third
+Panel: Entropy rates as a function of time since most recent spike.24
+A0 | 0.99
+B1 | 0.01
+ZZ
+1 | 0.02
+C2 0 | 0.99C1 | 0.010 | 0.99
+X0 | 0.98
+1 | 0.02L
+1 | 0.03M0 | 0.97
+1 | 0.04N0 | 0.96D0 | 0.95
+C11 | 0.05H2
+1 | 0.050 | 0.95
+1 | 0.06E0 | 0.94
+H1
+1 | 0.01 0 | 0.99
+1 | 0.040 | 0.96
+O1 | 0.030 | 0.97
+1 | 0.03P0 | 0.977J
+1 | 0.04K0 | 0.96
+1 | 0.040 | 0.97H
+1 | 0.04
+I0 | 0.96
+1 | 0.040 | 0.96G
+1 | 0.050 | 0.95F
+1 | 0.050 | 0.95
+1 | 0.060 | 0.94
+0 | 0.991 | 0.01 W
+0 | 0.981 | 0.02V
+1 | 0.020 | 0.98S
+1 | 0.03T0 | 0.97
+1 | 0.03U0 | 0.98Q
+1 | 0.03R0 | 0.97
+1 | 0.030 | 0.97
+1 | 0.030 | 0.97
+1 | 0.020 | 0.98
+FIG. 8 29-state CSM reconstructed from 335 seconds of spikes recorded from a layer II/III barrel cortex neuron undergoing
+periodic (125 msec inter-stimulus interval) stimulation via vibrissa de
ection. C= 1:97,J= 0:11,R= 0:004. Most of the
+states are devoted to refractory/bursting behavior, however states \C1", \C2" and \ZZ" represent the structure imposed by the
+external stimulus. See text for discussion.25
+0 5 10 15 20 25 30 35 40 45 5000.020.040.060.08
+0 5 10 15 20 25 30 35 40 45 5000.020.04Time Dependent Firing Probability
+0 5 10 15 20 25 30 35 40 45 501.522.53Complexity C(t)
+0 5 10 15 20 25 30 35 40 45 5000.20.4EntropiesISI Distribution A
+msecspikes/msecbits bits
+time since stimulus presentation (msec)λCSM(t)
+λPSTH (t)
+J(t)
+R(t)
+H(t)B
+C
+time since stimulus presentation (msec)0 5 10 15 20 25 30 35 40 45 5000.0050.010.0150.020.025Stimulus Driven Entropy  (ΔH(t))bits
+FIG. 9 Stimulated barrel cortex neuron ISI distribution and time-dependent complexity and entropies. (A) ISI distribution
+and 99% condence bounds. (B) First panel: Firing probability as a function of time since stimulus presentation. Second
+panel: Time-dependent complexity. Third panel: time-dependent entropies. (C) The stimulus driven entropy (maximum of
+0:02 bits/msec) is low because the number of spikes per stimulus ( 0:10:2) is very low and hence the stimulus does not
+supply much information.26
+0 5 10 15 20 25 3000.020.040.06History Dependent Firing Probability
+0 5 10 15 20 25 30 35 40 45 500510Complexity C(t)
+0 5 10 15 20 25 3000.20.4Entropiesspikes/msecbits bits
+time since most recent spike (msec)J(t)
+R(t)
+H(t)λCSM(t)
+λPSTH (t)
+FIG. 10 Firing probability, complexity and entropies of the stimulated barrel cortex neuron as a function of time since the
+most recent spike.27
+0 10 20 30 40 5000.020.040.060.080.10.12
+0 10 20 30 40 5000.020.040.060.080.1
+0 0.2 0.4 0.6 0.8 100.20.40.60.81
+0 0.2 0.4 0.6 0.8 100.20.40.60.81
+0 0.2 0.4 0.6 0.8 100.20.40.60.81
+0 0.2 0.4 0.6 0.8 100.20.40.60.81ISI Distrib. Spont. Spiking
+      Validation DataISI Distrib. Stimulated Spiking           Validation Data
+KS Plot, Spont. Spiking      Training Data
+Validation DataKS Plot, Stimulated Spiking          Training Data
+Validation Datamsec msecA B
+C D
+E F
+FIG. 11 Cross validation of CSMs reconstructed from spontaneously ring and stimulus evoked rat barrel cortex on an inde-
+pendent validation training set. (A,B) ISI distribution of spontaneously and stimulus evoked ring validation sets and 99%
+condence bounds bootstrapped from CSM. (C-D) Time rescaling plots of training data sets for spontaneously ring and stim-
+ulus evoked ring respectively. Dashed lines are 95% condence bounds and the solid line is the rescaled ISIs. The solid line
+along the digagonal is for visual comparison to an ideal t. (E-F) Similar time rescaling plots for the validation data sets.
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+arXiv:1001.0038v1  [math.CA]  30 Dec 2009BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC
+SPACES
+ALEXANDER VOLBERG†AND BRETT D. WICK‡
+Abstract. In this paper we study “Bergman-type” singular integral ope rators on Ahlfors regular
+metric spaces. The main result of the paper demonstrates tha t if a singular integral operator on a
+Ahlfors regular metric space satisfies an additional estima te, then knowing the “T(1)” conditions
+for the operator imply that the operator is bounded on L2. The method of proof of the main result
+is an extension and another application of the work originat ed by Nazarov, Treil and the first author
+on non-homogeneous harmonic analysis.
+1.Introduction and Statements of results
+We are interested in Calder´ on-Zygmund operators living on metric spaces. In particular, these
+kernels will live on a metric space of homogeneous type. We br iefly recall these types of metric
+spaces. A metric space of homogeneous type is a space X, a quasi-metric ρ, and a non-negative
+Borel measure νon the space X. The key property that defines these spaces is that all balls B(x,r)
+defined byρare open, and the measure νsatisfies the doubling condition
+ν(B(x,2r))≤Cdoubν(B(x,r))∀x∈X, r∈R+.
+We also require that ν(B(x,r))<∞for allx∈Xandr∈R+. The main example of the metric
+spaces that the reader should keep in mind is the case of Rnwith the standard metric and Lebesgue
+measure. Instead of the standard doubling condition, we wil l impose a slightly stronger condition.
+Let(X,ν,ρ) beaAhlforsregularmetricmeasurespace. By thiswemeanth at (X,ρ) isacomplete
+metric space, ν≥0 is a Borel measure on X, and there exist constants 0 < c1< c2,n >0, such
+that, for all r≥0 andx∈X:
+c1rn≤ν(B(x,r))≤c2rn. (1.1)
+It is easy to see that condition 1.1implies the doubling condition on νwithCdoub=c2
+c12n.
+We next recall the definition of Calder´ on-Zygmund operator s on metric spaces as introduced by
+Christ, [1]. For anyx,y∈X, we set
+λ(x,y) =ν(B(x,ρ(x,y)))≈ρ(x,y)n.
+A simple calculation shows that λ(x,y)≈λ(y,x) because of the doubling condition on ν. Then
+astandard kernel is a function k:X×X\ {x=y} →Csuch that there exists constants CCZ,
+τ,δ>0
+|k(x,y)| ≤CCZ
+λ(x,y)=CCZ
+ρ(x,y)n∀x/\e}atio\slash=y∈X;
+and
+|k(x,y)−k(x,y′)|+|k(x,y)−k(x′,y)| ≤CCZρ(x,x′)τ
+ρ(x,y)τ1
+λ(x,y)=CCZρ(x,x′)τ
+ρ(x,y)τ+n
+†Research supported in part by a National Science Foundation DMS grant.
+‡The second author is supported by National Science Foundati on CAREER Award DMS# 0955432 and an
+Alexander von Humboldt Fellowship.
+12 A. VOLBERG AND B. D. WICK
+provided that ρ(x,x′)≤δρ(x,y). In this situation, we say that the kernel ksatisfies the standard
+estimates. Again, the canonical examples to keep in mind are the usual Calder´ on-Zygmund kernels
+onRn.
+However, we will be interested in kernels that satisfy estim ates as if they lived on a “smaller
+space”. First, suppose that we have another measure µon the metric space X(which need not be
+doubling), but satisfies the following relationship, for so me 0≤m<n
+µ(B(x,r))/lessorsimilarrm∀x∈X,∀r. (H)
+Then, we define a standard kernel of order 0<m≤nas a function k:X×X\{x=y} →C
+such that there exists constants CCZ,τ,δ >0
+|k(x,y)| ≤CCZ
+ρ(x,y)m∀x/\e}atio\slash=y∈X;
+and
+|k(x,y)−k(x,y′)|+|k(x,y)−k(x′,y)| ≤CCZρ(x,x′)τ
+ρ(x,y)τ+m
+provided that ρ(x,x′)≤δρ(x,y). In this situation, we say that the kernel ksatisfies the standard
+estimates. In this case, we then define the Calder´ on-Zygmun d operator associated to µas
+Tµ(f)(x) :=/integraldisplay
+Xk(x,y)f(y)dµ(y).
+For “nice” functions f, this integral is well defined and
+These definitions are motivated by the Calder´ on-Zygmund ke rnels that live in Rn, but satisfy
+estimates as if they lived in Rmwithm≤n. One should think of the measure µas given by the
+m-dimensional Lebesgue measure after restricting to a m-dimensional hyperplane.
+The constants CCZ,τ,δandmwill be referred to as the Calder´ on-Zygmund constants of th e
+kernelk(x,y).
+We will also be interested in the kernels that have the additi onal property that satisfy
+|k(x,y)| ≤1
+max(dm(x),dm(y)),
+whered(x) := dist(x,X\Ω) = inf{ρ(x,y) :y∈X\Ω}and Ω being an open set in X.
+Our main result is the following theorem:
+Theorem 1. Let(X,ρ,ν)be a Ahlfors regular metric space. Let k(x,y)be a Calder´ on-Zygmund
+kernel of order mon(X,ρ,ν), with Calder´ on-Zygmund constants CCZandτ, that satisfies
+|k(x,y)| ≤1
+max(d(x)m,d(y)m),
+whered(x) := dist(x,X\Ω). Letµbe a probability measure with compact support in Xand all
+balls such that µ(B(x,r))>rmlie in an open set Ω. Finally, suppose also that a “ T1Condition”
+holds for the operator Tµ,mwith kernel kand for the operator T∗
+µ,mwith kernel k(y,x):
+/bardblTµ,mχQ/bardbl2
+L2(X;µ)≤Aµ(Q),/bardblT∗
+µ,mχQ/bardbl2
+L2(X;µ)≤Aµ(Q). (1.2)
+Then/bardblTµ,m/bardblL2(X;µ)→L2(X;µ)≤C(A,m,d,τ ).
+The balls for which we have µ(B(x,r))>rmwill be called “non-Ahlfors balls”. The key hypoth-
+esis is that we can capture all the non-Ahlfors balls in some o pen set Ω. To mitigate against this
+difficulty, we will have to suppose that our Calder´ on-Zygmun d kernels have an additional estimate
+in terms of the behavior in terms of the distance to the comple ment of Ω.
+An immediate application of Theorem 1is a new proof of results by the authors in [ 9]. In
+[9] a variant of Theorem 1was obtained in the Euclidean setting, and then is further ex tended
+to Calder´ on-Zygmund kernels in the natural metric associa ted to the Heisenberg group on theBERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 3
+unit ball. This was then used to characterize the Carleson me asures for the analytic Besov–Sobolev
+spaces on the unit ball in Cn. The connection between Carleson measures and a variant of T heorem
+1is provided since a measure is Carleson if and only if a certai n naturally occurring Calder´ on-
+Zygmund operator is bounded on L2. The operator to be studied is amenable to the methods of
+non-homogeneous harmonic analysis.
+Themethod of proof of Theorem 1will beto usethe tools of non-homogeneous harmonicanalysi s
+as developed by F. Nazarov, S. Treil, and the first author in th e series of papers [ 3–6] and further
+explained in the book by the first author [ 8]. We essentially adapt the proof given by the authors in
+[9] to the case of metric spaces considered in this paper. Const ants will bedenoted by Cthroughout
+the paper.
+2.Proof of Theorem 1
+The proof of this theorem will be divided in several parts. We first recall the construction of M.
+Christ of “dyadic cubes” on a metric space of homogeneous typ e, see [1]. The interested reader
+can also consult the paper by E. Sawyer and R. Wheeden, [ 7], where a similar construction is
+performed.
+Theorem 2 (M. Christ, [ 1]).There exists a collection of open sets {Qk
+α⊂X:k∈Z,α∈Ik}and
+constantsκ∈(0,1),a0>0, andη>0andC1,C2<∞such that
+(i)ν(X\/uniontext
+αQk
+α) = 0∀k∈Z;
+(ii)Ifl≥k, then either Ql
+β⊂Qk
+αorQl
+β⊂Qk
+α=∅;
+(iii)For each (k,α)and eachl<kthere is a unique βsuch thatQk
+α⊂Ql
+β;
+(iv)The diameter of Qk
+αis an absolute constant multiple of κk;
+(v)EachQk
+αcontains some ball B(zα,a0κk);
+(vi)ν{x∈Qk
+α: dist(x,X\Qk
+α)≤tκk}/lessorsimilartην(Qk
+α)
+HereIkis a (possibly finite) index set, depending only on k∈Z.
+The construction of these cubes uses only the properties of t he homogeneous space ( X,ρ,ν).
+One can think of the cubes Qk
+αas being cubes or balls of diameter κkand centerzk
+α. We will let D
+denote the collection of dyadic cubes on Xthat exists by the above Theorem.
+We further remark that it is possible to “randomize” this con struction. In a recent paper by
+Hyt¨ onen and Martikainen, [ 2], they studied this construction in and showed that it is pos sible to
+construct several random dyadic grids of the type above. The details of this construction aren’t
+immediately important for the proof of the main results in th is paper, only the existence of these
+randomgrids. We recommendthat thereaderconsultthewell- written paper[ 2] fortheconstruction
+of these grids. In particular, Section 10 of that paper conta ins the necessary modifications of
+Theorem 2to construct the random dyadic lattices in a metric space.
+We also define the dilation of a set E⊂Xby a parameter λ≥1 by
+λE:={x∈X:ρ(x,E)≤(λ−1)diam(E)}.
+2.1.Terminal and transit cubes. We will call the cube Q∈ Daterminal cube if the parent of
+Q(which exists and is unique by (iii) of Theorem 2) is contained in our open set Ω or µ(Q) = 0.
+All other cubes are called transitcubes. Then, denote by DtermandDtranas the terminal and
+transit cubes from D. We first state two obvious Lemmas.
+Lemma 3. IfQbelongs to Dterm, then
+|k(x,y)| ≤1
+κm.
+This follows since Qbelongs to its parent which is a subset of Ω and so for x,y∈Qwe have
+thatd(x)≥κand similarly for y. Another obvious lemma:4 A. VOLBERG AND B. D. WICK
+Lemma 4. IfQbelongs to Dtran, then
+µ(B(x,r))/lessorsimilarrm.
+We assume that F= suppµlies in a grand child cube of Qwhere, this Qis a certain (fixed) unit
+cube. We then take two “random” lattices as constructed by Hy t¨ onen and Martikainen in [ 2]. Now,
+letD1andD2be two such dyadic lattices, that have the property that the u nit cube contains the
+support ofµdeep inside a unit cube of the corresponding lattice. We will decompose our functions
+fandgwith respect to the lattices D1andD2.
+We would like to denote Qjas a dyadic cube belonging to the dyadic lattice Dj. Unfortunately,
+this makes the notation later very cumbersome. So, we will us e the letter Qto denote a dyadic
+cube belonging to the lattice D1and the letter Rto denote a dyadic cube belonging to the lattice
+D2. We will also let s(Q) denote the “size” or “scale” of the cube, namely, what gener ation of the
+construction from Theorem 2the cube belongs to.
+From now on, we will always denote by Qjthe dyadic subcubes of a cube Qenumerated in some
+“natural order”. Similarly, we will always denote by Rjthe dyadic subcubes of a cube RfromD2.
+Next, notice that there are special unit cubes Q0andR0of the dyadic lattices D1andD2
+respectively. They have the property that they are both tran sit cubes and contain Fdeep inside
+them.
+2.2.Projections Λand∆Q.LetDbe one of the dyadic lattices above. For a function ψ∈
+L1(X;µ) and for a cube Q⊂X, denote by /a\}bracketle{tψ/a\}bracketri}htQthe average value of ψoverQwith respect to the
+measureµ, i.e.,
+/a\}bracketle{tψ/a\}bracketri}htQ:=1
+µ(Q)/integraldisplay
+Qψdµ
+(of course, /a\}bracketle{tψ/a\}bracketri}htQmakes sense only for cubes Qwithµ(Q)>0). Put
+Λϕ:=/a\}bracketle{tϕ/a\}bracketri}htQ0.
+Clearly, Λϕ∈L2(X;µ) for allϕ∈L2(X;µ), and Λ2= Λ, i.e., Λ is a projection. Note also, that
+actually Λ does not depend on the lattice Dbecause the average is taken over the whole support
+of the measure µregardless of the position of the cube Q0(orR0).
+Below we will start almost every claim by “Assume (for definit eness) that s(Q)≤s(R)...”.
+Below, for ease of notation, we will write that a cube Q∈ X ∩Y to mean that the dyadic cube Q
+has both property XandYsimultaneously.
+For every transit cube Q∈ D1, define ∆Qϕby
+∆Qϕ/vextendsingle/vextendsingle
+X\Q:= 0, ,∆Qϕ/vextendsingle/vextendsingle
+Qj:=
+
+/bracketleftBig
+/a\}bracketle{tϕ/a\}bracketri}htQj−/a\}bracketle{tϕ/a\}bracketri}htQ/bracketrightBig
+ifQjis transit;
+ϕ−/a\}bracketle{tϕ/a\}bracketri}htQifQjis terminal.
+Observe that for every transit cube Q, we haveµ(Q)>0, so our definition makes sense since no
+zero can appear in the denominator. We repeat the same definit ion forR∈ D2.
+We then have have following Lemma that collects several easy properties of ∆Qϕ. To check these
+properties is left to the reader as an exercise.
+Lemma 5. For everyϕ∈L2(X;µ)and every transit cube Q,
+(1) ∆Qϕ∈L2(X;µ);
+(2)/integraltext
+X∆Qϕdµ= 0;
+(3) ∆Qis a projection, i.e., ∆2
+Q= ∆Q;
+(4) ∆QΛ = Λ∆Q= 0;BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 5
+(5) IfQ,/tildewideQare transit,/tildewideQ/\e}atio\slash=Q, then ∆Q∆eQ= 0.
+We next note that it is possible to decompose functions ϕinto the corresponding projections Λ
+and ∆Q.
+Lemma 6. LetQ0be a transit cube. For every ϕ∈L2(X;µ)we have
+ϕ= Λϕ+/summationdisplay
+Qtransit∆Qϕ,
+the series converges in L2(X;µ)and, moreover,
+/bardblϕ/bardbl2
+L2(µ)=/bardblΛϕ/bardbl2
+L2(µ)+/summationdisplay
+Qtransit/bardbl∆Qϕ/bardbl2
+L2(µ).
+Proof.Note first of all that if one understands the sum
+/summationdisplay
+Qtransit
+as limk→∞/summationtext
+Qtransit:s(Q)>δk, then forµ-almost every x∈X, one has
+ϕ(x) = Λϕ(x)+/summationdisplay
+Qtransit∆Qϕ(x).
+Indeed, the claim is obvious if the point xlies in some terminal cube. Suppose now that this is not
+the case. Observe that
+Λϕ(x)+/summationdisplay
+Qtransit:s(Q)>κk∆Qϕ(x) =/a\}bracketle{tϕ/a\}bracketri}htQk,
+whereQkis the dyadic cube of size κk, containing x. Therefore, the claim is true if
+/a\}bracketle{tϕ/a\}bracketri}htQk→ϕ(x).
+But, the exceptional set for this condition has µ-measure 0. Now the orthogonality of all ∆ Qϕ
+between themselves, and their orthogonality to Λ ϕproves the lemma. /square
+3.Good and bad functions
+We consider the functions fandg∈L2(X;µ). We fix two dyadic lattices D1andD2as before
+and define decompositions of fandgvia Lemma 6,
+f= Λf+/summationdisplay
+Q∈Dtran
+1∆Qf, g= Λg+/summationdisplay
+R∈Dtran
+2∆Rg.
+For a dyadic cube Rwe denote ∪i∈Ik∂RibyskR, called the skeleton ofR. Here theRiare the
+dyadic children of R.
+Letτ,mbe parameters of the Calder´ on-Zygmundkernel k. We fixα=τ
+2τ+2m.
+Definition 7. Fix a small number δ >0 andS≥2 to be chosen later. Choose an integer rsuch
+that
+κ−r≤δS<κ−r+1. (3.1)
+A cubeQ∈ D1is called bad(δ-bad) if there exists R∈ D2such that
+(1)s(R)≥κrs(Q);
+(2) dist(Q,skR)<s(Q)αs(R)1−α.6 A. VOLBERG AND B. D. WICK
+LetB1denote the collection of all bad cubes and correspondingly l etG1denote the collection of
+good cubes. The symmetric definition gives the collection of badcubesR∈ D2, denotes as B2.
+We say, that ϕ=/summationtext
+Q∈Dtran
+1∆Qϕisbadif in the sum only bad Q’s participate in this decompo-
+sition with the same appling to ψ=/summationtext
+Q∈Dtran
+2∆Qψ. In particular, given two distinct lattices D1
+andD2we fix the decomposition of fandginto good and bad parts:
+f=fgood+fbad,wherefgood= Λf+/summationdisplay
+Q∈Dtran
+1∩G1∆Qf.
+The same applies to g= Λg+/summationtext
+R∈Dtran
+2∆Rg=ggood+gbad.
+Theorem 8. One can choose S=S(α)in such a way that for any fixed Q∈ D1,
+P{Qis bad} ≤δ2. (3.2)
+By symmetry P{Ris bad} ≤δ2for any fixed R∈ D2.
+Theproof of this Theorem can befoundin thepaper [ 2]. Theuseof Theorem 8gives usS=S(α)
+in such a way that for any fixed Q∈ D1,
+P{Qis bad} ≤δ2. (3.3)
+We are now ready to prove
+Theorem 9. Consider the decomposition of ffrom Lemma 6. Then one can choose S=S(α)in
+such a way that
+E(/bardblfbad/bardblL2(X;µ))≤δ/bardblf/bardblL2(X;µ). (3.4)
+The proof depends only on the property ( 3.3) and not on a particular definition of what it means
+to be a bad or good function.
+Proof.By Lemma 6(its left inequality),
+E(/bardblfbad/bardblL2(X;µ))≤E/parenleftBig/summationdisplay
+Q∈Dtran
+1∩B1/bardbl∆Qf/bardbl2
+L2(X;µ)/parenrightBig1/2
+.
+Then
+E(/bardblfbad/bardblL2(X;µ))≤/parenleftBig
+E/summationdisplay
+Q∈Dtran
+1∩B1/bardbl∆Qf/bardbl2
+L2(X;µ)/parenrightBig1/2
+.
+LetQbe a fixed cube in D1; then, using ( 3.3), we conclude:
+E/bardbl∆Qf/bardbl2
+L2(X;µ)=P{Qis bad}/bardbl∆Qf/bardbl2
+L2(X;µ)≤δ2/bardbl∆Qf/bardbl2
+L2(X;µ).
+Therefore, we can continue as follows:
+E(/bardblfbad/bardblL2(X;µ))≤δ/parenleftBig/summationdisplay
+Q∈Dtran
+1∩B1/bardbl∆Qf/bardbl2
+L2(X;µ)/parenrightBig1/2
+≤δ/bardblf/bardblL2(X;µ).
+The last inequality uses Lemma 6again (its right inequality). /square
+This theorem can also be found in the paper [ 2].BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 7
+3.1.Reduction to Estimates on Good Functions. We consider two random dyadic lattices
+D1andD2as constructed in [ 2]. Take now two functions fandg∈L2(X;µ) decomposed according
+to Lemma 6
+f= Λf+/summationdisplay
+Q∈Dtran
+1∆Qf, g= Λg+/summationdisplay
+R∈Dtran
+2∆Rg.
+Recall that we can now write f=fgood+fbad,g=ggood+gbad. Then
+(Tf,g) = (Tfgood,ggood)+R(f,g),whereR(f,g) = (Tfbad,g)+(Tfgood,gbad).
+Theorem 10. LetTbe any operator with bounded kernel. Then
+E|R(f,g)| ≤2δ/bardblT/bardblL2(X;µ)→L2(X;µ)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+Remark 11.Notice that the estimate depends on the norm of Tnot on the bound on its kernel.
+Proof.The procedure of taking the good and bad part of a function are projections in L2(X;µ)
+and so they do not increase the norm. Since we have that the ope ratorTis bounded, then
+|R(f,g)| ≤ /bardblT/bardblL2(X;µ)→L2(X;µ)/parenleftbig
+/bardblg/bardblL2(X;µ)/bardblfbad/bardblL2(X;µ)+/bardblf/bardblL2(X;µ)/bardblgbad/bardblL2(X;µ)/parenrightbig
+Therefore, upon taking expectations we find
+E|R(f,g)| ≤ /bardblT/bardblL2(X;µ)→L2(X;µ)/parenleftbig
+/bardblg/bardblL2(X;µ)E(/bardblfbad/bardblL2(X;µ))+/bardblf/bardblL2(X;µ)E(/bardblgbad/bardblL2(X;µ))/parenrightbig
+.
+Using Theorem 9we finish the proof.
+/square
+We see that we need now only to estimate
+|(Tfgood,ggood)| ≤C(τ,m,d,T 1)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ). (3.5)
+In fact, considering any operator Twith bounded kernel we conclude
+(Tf,g) =E(Tf,g) =E(Tfgood,ggood)+ER(f,g).
+Using Theorem 10and (3.5) we have
+|(Tf,g)| ≤C/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ)+2δ/bardblT/bardblL2(X;µ)→L2(X;µ)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+From here, taking the supremum over fandgin the unit ball of L2(X;µ), and choosing δ=1
+4we
+get
+/bardblT/bardblL2(X;µ)→L2(X;µ)≤2C.
+3.2.Splitting (Tfgood,ggood)into Three Sums. First let us get rid of the projection Λ. We fix
+two corresponding dyadic lattices D1andD2. Recall that F= suppµis deep inside a unit cube Q
+of the standard dyadic lattice Das well as inside the shifted unit cubes Q0∈ D1andR0∈ D2. If
+f∈L2(X;µ), we have
+/bardblTΛf/bardblL2(X;µ)=/a\}bracketle{tf/a\}bracketri}htQ0/bardblTχQ0/bardblL2(X;µ)
+≤A1.2/bardblf/bardblL2(X;µ)µ(Q0)1/2
+µ(Q0)µ(Q0)1/2
+=A1.2/bardblf/bardblL2(X;µ).
+So we can replace fbyf−Λfand identically we can repeat this argument with gand from now
+on we may assume further that/integraldisplay
+Xf(x)dµ(x) = 0 and/integraldisplay
+Xg(x)dµ(x) = 0.
+Based on the reductions above, we can now think that fandgare good functions with zero
+averages. We skip mentioning below that Q∈ Dtran
+1andR∈ Dtran
+2, since this will always be the
+case by the convention established above.8 A. VOLBERG AND B. D. WICK
+To study the action of the Calder´ on-Zygmund operator Tonfandg, we split the pairing in
+the following manner,
+(Tf,g) =/summationdisplay
+Q∈G1,R∈G2,s(Q)≤s(R)(∆Qf,∆Rg)+/summationdisplay
+Q∈G1,R∈G2,s(Q)>s(R)(∆Qf,∆Rg).
+The question of convergence of the infinite sum can be avoided here, as we can think that the
+functionsfandgare only finite sums. This removes the question of convergenc e and allows us to
+rearrange and group the terms in the sum in any way we want.
+We need to estimate only the first sum, as the second will follo w by symmetry. For the sake of
+notational simplicity we will skip mentioning that the cube sQandRare good and we will skip
+mentioning s(Q)≤s(R). So, for now on,
+/summationdisplay
+Q,R:other conditionsmeans/summationdisplay
+Q,R:s(Q)≤s(R),Q∈G1,R∈G2,other conditions.
+Remark 12.It is convenient sometimes to think that the summation/summationdisplay
+Q,R:other conditions
+goes over good QandallR. Formally, this does not matter, since the functions fandgare good
+functions, and so this merely reduces to adding or omitting s everal zeros to the sum. For the
+symmetric sum over Q,R:s(Q)> s(R) the roles of QandRin this remark must of course be
+interchanged.
+The definition of δ-badness involved a large integer r, see (3.1). Use this notation to write our
+sum overs(Q)≤s(R) as follows
+/summationdisplay
+Q,R(∆Qf,∆Rg) =/summationdisplay
+Q,R:s(Q)≥κ−rs(R)+/summationdisplay
+Q,R:s(Q)<κ−rs(R)=/summationdisplay
+Q,R:s(Q)≥κ−rs(R),dist(Q,R)≤s(R)+
+/bracketleftbigg/summationdisplay
+Q,R:s(Q)≥κ−rs(R),dist(Q,R)>s(R)+/summationdisplay
+Q,R:s(Q)<κ−rs(R),Q∩R=∅/bracketrightbigg
++/summationdisplay
+Q,R:s(Q)<κ−rs(R),Q∩R/\egatio\slash=∅
+=:σ1+σ2+σ3.
+3.3.Three Potential Estimates of/integraltext
+X/integraltext
+Xk(x,y)f(x)g(y)dµ(x)dµ(y).Recall that the kernel
+k(x,y) ofTsatisfies the estimate
+|k(x,y)| ≤1
+max(d(x)m,d(y)m), d(x) = dist(x,X\Ω),
+Ω being an open set in X, and
+|k(x,y)| ≤CCZ
+ρ(x,y)m∀x/\e}atio\slash=y∈X;
+and
+|k(x,y)−k(x,y′)|+|k(x,y)−k(x′,y)| ≤CCZρ(x,x′)τ
+ρ(x,y)τ+m
+provided that ρ(x,x′)≤δρ(x,y), with some fixed constants numbers CCZ,τ,m.
+First, we will sometimes write
+/integraldisplay
+X/integraldisplay
+Xk(x,y)f(x)g(y)dµ(x)dµ(y) =/integraldisplay
+X/integraldisplay
+X[k(x,y)−k(x0,y)]f(x)g(y)dµ(x)dµ(y)
+using the fact that our fandgwill actually be ∆ Qfand ∆ Rgand so their integrals are zero.
+Temporarily write K(x,y) for either k(x,y) ork(x,y)−k(x0,y).BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 9
+After that we have three logical possibilities to estimate
+/integraldisplay
+X/integraldisplay
+XK(x,y)f(x)g(y)dµ(x)dµ(y).
+(1) Estimate |K|inL∞, andf,ginL1norms;
+(2) Estimate |K|inL∞L1norm, and finL1norm,ginL∞norm (or maybe, do this
+symmetrically);
+(3) Estimate |K|inL1norm, andf,ginL∞norms.
+The third method is widely used for Calder´ on–Zygmund estim ates on homogeneous spaces (say
+with respect to Lebesgue measure), but it is very dangerous t o use in the case of a nonhomogeneous
+measure. Here is the reason. After fandgare estimated in the L∞norm, one needs to continue
+these estimates to have L2norms. There is nothing strange in that as usually fandgare almost
+proportional to characteristic functions. But for fliving onQsuch thatf=cQχQ(cQis a
+constant),
+/bardblf/bardblL∞(X:µ)≤1
+µ(Q)1/2/bardblf/bardblL2(X;µ).
+The same reasoning applies for gonR. Then
+/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+X/integraldisplay
+XK(x,y)f(x)g(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle≤1
+(µ(Q)µ(R))1/2/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+And the nonhomogeneous measure has no estimate from below. H aving two uncontrollable almost
+zeroes in the denominator is a very bad idea. We will never use the estimate of type (3).
+On the other hand, estimates of type (2) are much less dangero us (although requires the care as
+well). This is because, in this case one applies
+/bardblf/bardblL1(X;µ)≤µ(Q)1/2/bardblf/bardblL2(X;µ)and/bardblg/bardblL∞(X;µ)≤1
+µ(R)1/2/bardblg/bardblL2(µ),
+and gets
+/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+X/integraldisplay
+XK(x,y)f(x)g(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle≤/parenleftbiggµ(Q)
+µ(R)/parenrightbigg1/2
+/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+If we choose to use estimate of the type (2) only for pairs Q,Rsuch thatQ⊂Rwe are in good
+shape. This approach is what we will end up going when estimat ingσ3.
+Plan. The first sum is the “diagonal” part of the operator, σ1. The second sum, σ2is the “long
+range interaction”. The final sum, σ3, is the “short range interaction”. The diagonal part will be
+estimated using our T1 assumption of Theorem 1.2, for the long range interaction we will use the
+first type of estimates described above, for the short range i nteraction we will use estimates of types
+(1) and (2) above. But all this will be done carefully!
+4.The Long Range Interaction: Controlling Term σ2
+We first prove a lemma that demonstrates that for functions wi th supports that are far apart,
+we have some good control on the bilinear form induced by our C alder´ on-Zygmund operator T.
+For two dyadic cubes QandR, we set
+D(Q,R) :=s(Q)+s(R)+dist(Q,R).10 A. VOLBERG AND B. D. WICK
+Lemma 13. Suppose that QandRare two cubes in X, such that s(Q)≤s(R). LetϕQ,ψR∈
+L2(X;µ). Assume that ϕQvanishes outside Q, andψRvanishes outside R;/integraltext
+XϕQdµ= 0and, at
+last,dist(Q,suppψR)≥s(Q)αs(R)1−α. Then
+|(ϕQ,TψR)| ≤ACs(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ/radicalbig
+µ(Q)µ(R)/bardblϕQ/bardblL2(X;µ)/bardblψR/bardblL2(X;µ).
+Remark 14.Note that we require only that the support of the function ψRlies far from the cube
+Q; the cubes QandRthemselves may intersect! Such situations will arise when e stimating the
+termσ2.
+Proof.LetxQbe the center of the cube Q. Note that for all x∈Q,y∈suppψR, we have
+ρ(xQ,y)≥s(Q)
+2+dist(Q,suppψR)≥s(Q)
+2+2r(1−α)s(Q)/greaterorsimilars(Q)/greaterorsimilarρ(x,xQ).
+Therefore,
+|(ϕQ,TψR)|=/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+X/integraldisplay
+Xk(x,y)ϕQ(x)ψR(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle
+=/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+X/integraldisplay
+X[k(x,y)−k(xQ,y)]ϕQ(x)ψR(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle
+/lessorsimilars(Q)τ
+dist(Q,suppψR)m+τ/bardblϕQ/bardblL1(X;µ)/bardblψR/bardblL1(X;µ).
+There are two possible cases.
+Case 1: dist(Q,suppψR)≥s(R).Then
+D(Q,R) :=s(Q)+s(R)+dist(Q,R)≤3dist(Q,suppψR)
+and therefore
+s(Q)τ
+dist(Q,suppψR)m+τ/lessorsimilars(Q)τ
+D(Q,R)m+τ/lessorsimilars(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ.
+Case 2:s(Q)αs(R)1−α≤dist(Q,suppψR)≤s(R).ThenD(Q,R)≤3s(R) and we get
+s(Q)τ
+dist(Q,suppψR)m+τ≤s(Q)τ
+[s(Q)αs(R)1−α]m+τ=s(Q)τ
+2s(R)τ
+2
+s(R)m+τ/lessorsimilars(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ.
+Here, key to the proof was the choice of α=τ
+2(τ+m). Now, to finish the proof of the lemma, it
+remains only to note that
+/bardblϕQ/bardblL1(X;µ)≤/radicalbig
+µ(Q)/bardblϕQ/bardblL2(X;µ)and/bardblψR/bardblL1(X;µ)≤/radicalbig
+µ(R)/bardblψR/bardblL2(X;µ).
+/square
+Applying this lemma to ϕQ= ∆QfandψR= ∆Rg, we obtain
+|σ2|/lessorsimilar/summationdisplay
+Q,Rs(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ/radicalbig
+µ(Q)/radicalbig
+µ(R)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ). (4.1)
+To control term σ2the computations above suggest that we will define a matrix op erator, de-
+pending on the cubes QandRand show that it is a bounded operator on ℓ2.
+Lemma 15. Define
+TQ,R:=s(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ/radicalbig
+µ(Q)/radicalbig
+µ(R) (Q∈ Dtr
+1, R∈ Dtr
+2, s(Q)≤s(R)).BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 11
+Then, for any two families {aQ}Q∈Dtr
+1and{bR}R∈Dtr
+2of nonnegative numbers, one has
+/summationdisplay
+Q,RTQ,RaQbR≤AC/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Rb2
+R/bracketrightBig1
+2.
+Remark 16.Note thatTQ,Ris defined for all QandRwiths(Q)≤s(R) and that the conditions
+dist(Q,R)≥s(Q)αs(R)1−α(or even the condition Q∩R=∅) no longer appears as a condition in
+the summation!
+Assuming Lemma 15for the moment, the estimate of σ2then proceeds in an obvious fashion.
+|σ2|/lessorsimilar/summationdisplay
+Q,Rs(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τ/radicalbig
+µ(Q)/radicalbig
+µ(R)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ)
+/lessorsimilar
+/summationdisplay
+Q/bardbl∆Qf/bardbl2
+L2(X;µ)
+1/2/parenleftBigg/summationdisplay
+R/bardbl∆Rg/bardbl2
+L2(X;µ)/parenrightBigg1/2
+/lessorsimilar/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+Here the first line follows by ( 4.1), the second by Lemma 15, and finally the last by Lemma 6. We
+now turn to the proof of Lemma 15.
+Proof.Let us “slice” the matrix TQ,Raccording to the ratios(Q)
+s(R). Namely, let
+T(k)
+Q,R=/braceleftBigg
+TQ,Rifs(Q) =κ−ks(R);
+0 otherwise ,
+(k= 0,1,2,...). To prove the lemma, it is enough to show that for every k≥0,
+/summationdisplay
+Q,RT(k)
+Q,RaQbR≤C2−τ
+2k/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Rb2
+R/bracketrightBig1
+2.
+The matrix {T(k)
+Q,R}has a “block” structure since the variables bRcorresponding to the cubes
+R∈ Dtr
+2for whichs(R) =κjcan only interact with the variables aQcorresponding to the cubes
+Q∈ Dtr
+1, for which s(Q) =κj−k. Thus, to get the desired inequality, it is enough to estimat e each
+block separately, i.e., to demonstrate that
+/summationdisplay
+Q,R:s(Q)=κj−k,s(R)=κjT(k)
+Q,RaQbR≤C/bracketleftBig/summationdisplay
+Q:s(Q)=κj−ka2
+Q/bracketrightBig1
+2/bracketleftBig/summationdisplay
+R:s(R)=κjb2
+R/bracketrightBig1
+2.
+Let us introduce the functions
+F(x) :=/summationdisplay
+Q:s(Q)=κj−kaQ/radicalbig
+µ(Q)χQ(x) and G(x) :=/summationdisplay
+R:ℓ(R)=κjbR/radicalbig
+µ(R)χR(x).
+Note that the cubes of a given size in one dyadic lattice do not intersect (Property (ii) of Theorem
+2), and therefore at each point x∈X, at most one term in the sum can be non-zero. Also observe
+that
+/bardblF/bardblL2(X;µ)=/bracketleftBig/summationdisplay
+Q:s(Q)=κj−ka2
+Q/bracketrightBig1
+2and/bardblG/bardblL2(X;µ)=/bracketleftBig/summationdisplay
+R:s(R)=κjb2
+R/bracketrightBig1
+2.
+Then the estimate we need can be rewritten as/integraldisplay
+X/integraldisplay
+XKj,k(x,y)F(x)G(y)dµ(x)dµ(y)≤C/bardblF/bardblL2(X;µ)/bardblG/bardblL2(X;µ),12 A. VOLBERG AND B. D. WICK
+where
+Kj,k(x,y) =/summationdisplay
+Q,R:s(Q)=κj−k,s(R)=κjs(Q)τ
+2s(R)τ
+2
+D(Q,R)m+τχQ(x)χR(y).
+Again, for every pair of points x,y∈X, only one term in the sum can be nonzero. Since ρ(x,y)+
+s(R)≤3D(Q,R) for anyx∈Qandy∈R, we obtain
+Kj,k(x,y) =Cκ−τ
+2ks(R)τ
+D(Q,R)m+τ
+/lessorsimilarκ−τ
+2kκjτ
+[κj+ρ(x,y)]m+τ=:κ−τ
+2kkj(x,y).
+So, it is enough to check that/integraldisplay
+X/integraldisplay
+Xkj(x,y)F(x)G(y)dµ(x)dµ(y)/lessorsimilar/bardblF/bardblL2(X;µ)/bardblG/bardblL2(X;µ).
+We remind the reader that we called the balls “non-Ahlfors ba lls” if
+µ(B(x,r))>rm.
+According to the Schur test, it would suffice to prove that for e veryy∈X, one has the estimate/integraltext
+Xkj(x,y)dµ(x)/lessorsimilar1 and vice versa (i.e., for every x∈X, one has/integraltext
+Xkj(x,y)dµ(y)/lessorsimilar1). Then
+the norm of the integral operator with kernel kjinL2(X;µ) would be bounded by a constant and
+the proof of Lemma 15would be over. If we assumed a priori that the supremum of radi i of all
+non-Ahlfors balls centered at y∈Rwiths(R) =κj,were less than κj+1, then the needed estimate
+would be immediate. In fact, we can write/integraldisplay
+Xkj(x,y)dµ(x) =/integraldisplay
+B(y,κj+1)kj(x,y)dµ(x)+/integraldisplay
+X\B(y,κj+1)kj(x,y)dµ(x)
+/lessorsimilarκ−jmµ(B(y,κj+1))+/integraldisplay
+X\B(y,κj+1)κjτ
+ρ(x,y)m+τdµ(x)
+/lessorsimilarκ−jmµ(B(y,κj+1))+∞/summationdisplay
+k=0κjτ
+(κkκj+1)m+τµ(B(y;κkκj+1))
+/lessorsimilar/parenleftBig
+1+∞/summationdisplay
+k=01
+κkτ/parenrightBig
+≈1.
+The passage from the second to the third line follows by exhau sting the the space X\B(y,κj+1)
+by “annular regions” and making obvious estimates using con dition (H).
+The difficulty with this approach is that we cannot guarantee t he supremum of the radii of all
+non-Ahlfors balls centered at ybe less than κj+1for everyy∈X. Our measure may not have this
+uniform property.
+So, generally speaking, we are unable to show that the integr al operator with kernel kj(x,y) acts
+inL2(X;µ); but we do not need that much! We only need to check that the corresponding bilin ear
+form is bounded on two givenfunctionsFandG. So, we are not interested in the points y∈X
+for whichG(y) = 0 (or in the points x∈X, for which F(x) = 0). But, by definition, Gcan be
+non-zero only on the transit cubes in D2. Here we used our convention that we omit in all sums
+the fact that QandRare transit cubes, however they are!
+Now let us notice that if (and this is the case for all Rin the sum we estimate in our lemma)
+R∈ Dtran
+2, then the supremum of radii of all non-Ahlfors balls centere d aty∈Ris bounded by
+s(R) for every y∈R. Indeed, this is just Lemma 4. The same reasoning shows that if Q∈ Dtran
+1,
+then the supremum of radii of all non-Ahlfors balls centered atx∈Qis bounded by κj−k+1≤κj+1
+wheneverF(x)/\e}atio\slash= 0, and we are done with Lemma 15. /squareBERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 13
+Now, wehope, thereader will agree that thedecision to decla re thecubescontained inΩterminal
+was a good one. As a result, the fact that the measure µis not Ahlfors did not put us in any real
+trouble – we barely had a chance to notice this fact at all. But , it still remains to explain why we
+were so eager to have the extra condition
+|k(x,y)| ≤1
+max(dm(x),dm(y)), d(x) := dist(x,X\Ω)
+on our Calder´ on–Zygmund kernel. The answer is found in the n ext two sections.
+5.Short Range Interaction and Nonhomogeneous Paraproducts: Controlling
+Termσ3.
+Recall that the sum σ3is taken over the pairs Q,R, for which s(Q)<κ−rs(R) andQ∩R/\e}atio\slash=∅.
+We would like to improve this condition to the demand that Qlie “deep inside” one of the subcubes
+Rj. Recall also that we defined the skeletonskRof the cube Rby
+skR:=/uniondisplay
+j∂Rj.
+We have declared a cube Q∈ D1bad if there exists a cube R∈ D2such thats(R)>κrs(Q) and
+dist(Q,skR)≤s(Q)αs(R)1−α. Now, for every good cube Q∈ D1, the conditions s(Q)<κ−rs(R)
+andQ∩R/\e}atio\slash=∅together imply that Qlies inside one of the children RjofR. We will denote this
+subcube by RQ. The sum σ3can now be split into
+σterm
+3:=/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis terminal(∆Qf,T∆Rg)
+and
+σtran
+3:=/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit(∆Qf,T∆Rg).
+5.1.Estimation of σterm
+3.First of all, write (recall that Rjdenote the children of R):
+σterm
+3=/summationdisplay
+j/summationdisplay
+Q,R:s(Q)<κ−rs(R),Q⊂Rj∈Dterm
+2(∆Qf,T∆Rg).
+Clearly, it is enough to estimate the inner sum for every fixed , and so let us do this for j= 1. We
+have/summationdisplay
+Q,R:s(Q)<κ−rs(R),Q⊂R1∈Dterm
+2(∆Qf,T∆Rg) =/summationdisplay
+R:R1∈Dterm
+2/summationdisplay
+Q:s(Q)<κ−rs(R),Q⊂R1(∆Qf,T∆Rg).
+Recall that the kernel kof our operator Tsatisfies the estimate of Lemma 3
+|k(x,y)|/lessorsimilar1
+s(R)mfor allx∈R1,y∈X. (5.1)
+Hence,
+|T∆Rg(x)|/lessorsimilar/bardbl∆Rg/bardblL1(X;µ)
+s(R)mfor allx∈R1, (5.2)
+and therefore
+/bardblχR1·T∆Rg/bardblL2(X;µ)/lessorsimilar/bardbl∆Rg/bardblL1(X;µ)/radicalbig
+µ(R1)
+s(R)m
+/lessorsimilarµ(R)
+s(R)m/bardbl∆Rg/bardblL2(X;µ)≤AB/bardbl∆Rg/bardblL2(X;µ).14 A. VOLBERG AND B. D. WICK
+This follows because /bardbl∆Rg/bardblL1(X:µ)≤/radicalbig
+µ(R)/bardbl∆Rg/bardblL2(X;µ)andµ(R1)≤µ(R) hold trivially. Ad-
+ditionally, by Lemma 4we have
+µ(R)/lessorsimilars(R)m(5.3)
+becauseR(the father of the cube R1) is a transit cube if R1is terminal.
+Now, recalling Lemma 6, and taking into account that ∆Qf≡0 outsideQ, we get
+/summationdisplay
+Q:Q⊂R1|(∆Qf,T∆Rg)|=/summationdisplay
+Q:Q⊂R1|(∆Qf,χR1·T∆Rg)|
+/lessorsimilar/bardblχR1·T∆Rg/bardblL2(X;µ)/bracketleftBig/summationdisplay
+Q:Q⊂R1/bardbl∆Qf/bardbl2
+L2(X;µ)/bracketrightBig1
+2
+/lessorsimilar/bardbl∆Rg/bardblL2(X;µ)/bracketleftBig/summationdisplay
+Q:Q⊂R1/bardbl∆Qf/bardbl2
+L2(X;µ)/bracketrightBig1
+2.
+So, we obtain/summationdisplay
+R:R1∈Dterm
+2/summationdisplay
+Q:Q⊂R1|(∆Qf,T∆Rg)|
+/lessorsimilar/summationdisplay
+R:R1∈Dterm
+2/bardbl∆Rg/bardblL2(X;µ)/bracketleftBig/summationdisplay
+Q:Q⊂R1/bardbl∆Qf/bardbl2
+L2(X;µ)/bracketrightBig1
+2
+/lessorsimilar/bracketleftBig/summationdisplay
+R:R1∈Dterm
+2/bardbl∆Rg/bardbl2
+L2(X;µ)/bracketrightBig1
+2/bracketleftBig/summationdisplay
+R:R1∈Dterm
+2/summationdisplay
+Q:Q⊂R1/bardbl∆Qf/bardbl2
+L2(X;µ)/bracketrightBig1
+2.
+But the terminal cubes in D2do not intersect! Therefore every ∆Qfcan appear at most once in
+the last double sum, and we get the bound
+/summationdisplay
+R:R1∈Dterm
+2/summationdisplay
+Q:Q⊂R1|(∆Qf,T∗∆Rψ)|
+/lessorsimilar/bracketleftBig/summationdisplay
+R/bardbl∆Rg/bardbl2
+L2(X;µ)/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Q/bardbl∆Qf/bardbl2
+L2(X;µ)/bracketrightBig1
+2/lessorsimilar/bardblf/bardblL2(X;µ)/bardblψ/bardblL2(X;µ).
+Lemma6has been used again in the last inequality.
+5.2.Estimation of σtran
+3.Recall that
+σtran
+3=/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit(∆Qf,T∗∆Rg).
+Split every term in the sum as
+(∆Qf,T∆Rψ) = (∆Qf,T(χRQ∆Rg))+(∆Qf,T∗(χR\RQ∆Rg)).
+Observe that since Qis good,Q⊂R, ands(Q)<κ−rs(R), we have
+dist(Q,suppχR\RQ∆Rg)≥dist(Q,skR)≥s(Q)αs(R)1−α.
+Using Lemma 13and taking into account that the norm /bardblχR\RQ∆Rψ/bardblL2(X;µ)does not exceed
+/bardbl∆Rψ/bardblL2(X;µ), we conclude that the sum
+/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit|(∆Qf,T∗(χR\RQ∆Rg))|BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 15
+can be estimated by the sum ( 4.1). Thus, our task is to find a good bound for the sum
+/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit(∆Qf,T∗(χRQ∆Rg)).
+Recalling the definition of ∆Rψand recalling that RQis atransitcube, we get
+χRQ∆Rg=cR,QχRQ,
+where
+cRQ=/a\}bracketle{tψ/a\}bracketri}htRQ−/a\}bracketle{tg/a\}bracketri}htR
+is aconstant. So, our sum can be rewritten as
+/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transitcRQ(∆Qf,T∗(χRQ)).
+Our next goal will be to extend the function χRQto the function 1 in every term.
+Let us observe that
+(∆Qf,T∗(χX\RQ)) =/integraldisplay
+X/integraldisplay
+X\RQk(x,y)∆Qf(x)dµ(x)dµ(y)
+=/integraldisplay
+X/integraldisplay
+X\RQ[k(x,y)−k(xQ,y)]∆Qf(x)dµ(x)dµ(y).
+Note again that for every x∈Q,y∈X\RQ, we have
+ρ(xQ,y)≥s(Q)
+2+dist(Q,X\RQ)/greaterorsimilars(Q)/greaterorsimilarρ(x,xQ).
+Therefore,
+|k(x,y)−k(xQ,y)|/lessorsimilar/parenleftBiggρ(x,xQ)
+ρ(xQ,y)/parenrightBiggτ1
+ρ(x,y)m/lessorsimilars(Q)τ
+ρ(xQ,y)m+τ,
+and
+|(∆Qf,T(χX\RQb))|/lessorsimilars(Q)τ/bardbl∆Qf/bardblL1(X;µ)/integraldisplay
+X\RQdµ(y)
+ρ(xQ,y)m+τ.
+Now let us consider the sequence of cubes R(j)∈ D2, beginning with R(0)=RQand gradually
+ascending ( R(j)⊂R(j+1),s(R(j+1)) =κs(R(j))) to the starting cube R0=R(N)of the lattice D2.
+Clearly, all these cubes R(j)are transit cubes.
+We have
+/integraldisplay
+X\RQdµ(y)
+ρ(xQ,y)m+τ=/integraldisplay
+R0\RQdµ(y)
+ρ(xQ,y)m+τ=N/summationdisplay
+j=1/integraldisplay
+R(j)\R(j−1)dµ(y)
+ρ(xQ,y)m+τ.
+We call the j-th term of this sum Ij. Note now that, since Qis good and s(Q)< κ−rs(R)≤
+κ−rs(R(j)) for allj, we have
+dist(Q,R(j)\R(j−1))≥dist(Q,skR(j))≥s(Q)αs(R(j))1−α.
+Hence
+Ij≤1
+[s(Q)αs(R(j))1−α]m+τ/integraldisplay
+R(j)dµ.16 A. VOLBERG AND B. D. WICK
+Recalling that α=τ
+2(m+τ), we see that the first factor equals
+1
+s(Q)τ
+2s(R(j))m+τ
+2.
+SinceR(j)is transit, we have /integraldisplay
+R(j)dµ/lessorsimilarµ(R(j))/lessorsimilars(R(j))m.
+Thus,
+Ij/lessorsimilars(Q)τ
+2s(R(j))τ
+2=κ−(j−1)ε
+2s(Q)τ
+2s(R)τ
+2.
+Summing over j≥1, we get
+/integraldisplay
+X\RQ|b(y)|dµ(y)
+ρ(xQ,y)m+τ=N/summationdisplay
+j=1Ij/lessorsimilar1−κ−τ
+21
+s(Q)τ
+2s(R)τ
+2.
+Now let us note that
+|cRQ| ≤/bardbl∆Rg/bardblL1(RQ,µ)
+µ(RQ)≤/bardbl∆Rg/bardblL2(RQ,µ)/radicalbig
+µ(RQ). (5.4)
+We finally obtain
+|(∆Qf,T∗(χX\RQ))|
+/lessorsimilar1
+η(1−κ−τ
+2)/bracketleftbiggs(Q)
+s(R)/bracketrightbiggτ
+2/radicalBigg
+µ(Q)
+µ(RQ)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ)
+and
+/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit|cR,Q|·|(∆Qf,T∗(χX\RQ))|
+/lessorsimilar1
+η(1−κ−τ
+2)/summationdisplay
+j/summationdisplay
+Q,R:Q⊂Rj/bracketleftbiggs(Q)
+s(R)/bracketrightbiggτ
+2/radicalBigg
+µ(Q)
+µ(Rj)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ).
+Lemma 17. For every two families {aQ}Q∈Dtr
+1and{bR}R∈Dtr
+2of nonnegative numbers, one has
+/summationdisplay
+Q,R:Q⊂R1TQ,RaQbR≤1
+1−κ−τ
+2/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Rb2
+R/bracketrightBig1
+2.
+Proof.Let us “slice” the matrix TQ,Raccording to the ratios(Q)
+s(R). Namely, let
+T(k)
+Q,R=/braceleftBigg
+TQ,R,ifQ⊂R1, s(Q) =κ−ks(R);
+0,otherwise
+(k= 1,2,...). It is enough to show that for every k≥0,
+/summationdisplay
+Q,RT(k)
+Q,RaQbR≤κ−τ
+2k/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Rb2
+R/bracketrightBig1
+2.
+The matrix {T(k)
+Q,R}has a very good “block” structure: every aQcan interact with only onebR. So,
+it is enough to estimate each block separately, i.e., to show that for every fixed R∈ Dtran
+2,
+/summationdisplay
+Q:Q⊂R1,ℓ(Q)=κ−kℓ(R)κ−τ
+2k/radicalBigg
+µ(Q)
+µ(R1)aQbR≤κ−τ
+2k/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2bR.BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 17
+But, reducing both parts by the non-essential factor κ−τ
+2kbR, we see that this estimate is equivalent
+to the trivial estimate
+/summationdisplay
+Q:Q⊂R1,s(Q)=κ−ks(R)/radicalBigg
+µ(Q)
+µ(R1)aQ
+≤/bracketleftBig/summationdisplay
+Q:Q⊂R1,s(Q)=κ−ks(R)µ(Q)
+µ(R1)/bracketrightBig1
+2/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2≤/bracketleftBig/summationdisplay
+Qa2
+Q/bracketrightBig1
+2,
+(since cubes Q∈ D1of fixed size do not intersect,/summationtext
+Q:Q⊂R1,s(Q)=κ−ks(R)µ(Q)≤µ(R1)).
+/square
+Remark 18.We did not use here the fact that {aQ},{bR}are supported on transit cubes. We
+actually proved
+Lemma19. The matrix {TQ,R}defined by
+TQ,R:=/bracketleftbiggs(Q)
+s(R)/bracketrightbiggτ
+2/radicalBigg
+µ(Q)
+µ(R1)(Q⊂R1),
+generates a bounded operator in l2.
+We just finished estimating an extra term which appeared when we extendχRQto the whole
+1. So, the extension of χRQto the function 1 does not cause much harm, and we are left with
+estimating the sum/summationdisplay
+Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transitcRQ(∆Qf,T∗1).
+Note that the inner product (∆Qf,T∗1)does not depend onRat all, so it seems to be a good idea
+to sum over Rfor fixedQfirst.
+Recalling that
+cRQ=/a\}bracketle{tg/a\}bracketri}htRQ−/a\}bracketle{tg/a\}bracketri}htR
+and that Λψ= 0⇐⇒ /a\}bracketle{tψ/a\}bracketri}htR0= 0, we conclude that for every Q∈ Dtran
+1that really appears in the
+above sum,/summationdisplay
+R:R⊃Q,s(R)>κms(Q),RQis transitcRQ=/a\}bracketle{tg/a\}bracketri}htRQ.
+Definition. LetR(Q) be the smallest transitcubeR∈ D2containing Qand such that s(R)≥
+κrs(Q).
+So, we obtain the sum
+/summationdisplay
+Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)(∆Qf,T∗1)
+to take care of.
+Remark. Let us recall that we had the convention that says that the cub esQconsidered are
+only good ones (and of course they are only transit cubes). Th e range of summation should be
+Q∈ Dtran
+1,Qis good (default); there exists a cube R∈ Dtran
+2such thats(Q)<κ−rs(R),Q⊂R
+and the child RQ(the one containing Q) ofRis transit. In other words, in fact, the sum is written18 A. VOLBERG AND B. D. WICK
+formally incorrectly. We have to replace R(Q) byRQin the summation. However, the smallest
+transit cube containing Q(this isR(Q)) and the smallest transit child (containing Q) of a certain
+subcubeRofR0(this child is RQ) are of course the same cube, unless R(Q) =R0. Thus the sum
+formally has some extra terms corresponding to R(Q) =R0. But, they all are zeros! In one of the
+first reductions, we were allowed to work only with with gsuch that Λ g= 0 (recall that Λ gmeans
+the average of gwith respect to µ), so/a\}bracketle{tg/a\}bracketri}htR(Q)= 0 ifR(Q) =R0.
+5.3.Pseudo-BMOand special paraproduct. Tointroducetheparaproductoperator, werewrite
+our sum as follows/summationdisplay
+Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)(∆Qf,T∗1) =/summationdisplay
+Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)(f,∆∗
+QT∗1)
+=
+f,/summationdisplay
+Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)∆QT∗1
+.
+We use the fact that ∆∗
+Q= ∆Q. We now introduce the paraproduct operator, which will allo w
+us to control term σtran
+3.
+Definition 20. Given a function F, theparaproduct with symbol Fis the function
+ΠFg(x) :=/summationdisplay
+R∈D2,R⊂R0/a\}bracketle{tg/a\}bracketri}htR/summationdisplay
+Q∈D1,Qgood and transit ,s(Q)=κ−rs(R)∆QF(x).
+As in the case when the metric space is Rd, the behavior of the paraproduct operators will be
+governed by “BMO” conditions on the symbol F. In the case of a metric space though, we face an
+additional wrinkle since we have to overcome the challenge o f dealing with the dyadic cubes, and
+we need an appropriate notion of “dilation” in the metric spa ce.
+Recall that we defined the dilation by the parameter λ≥1 of a setS⊂Xby
+λ·S:={x∈X: dist(x,S)≤(λ−1)diamS}
+Note thatS⊂λ·S.
+Definition 21. A function F∈L2(X;µ) will be called a “pseudo- BMOfunction” if there exists
+Λ>1 such that for any cube Qwithµ(sQ)≤Ksmdiam(Q)m,s≥1, we have
+/integraldisplay
+Q|F(x)−/a\}bracketle{tF/a\}bracketri}htQ|2dµ(x)≤Cµ(ΛQ).
+Lemma 22. Letµ,Tsatisfy the assumptins of Theorem 1. Then
+T∗1∈pseudo-BMO . (5.5)
+HereCdepends only on the constants of Theorem 1.
+Proof.Forx∈Qwe writeT∗1(x) = (T∗χΛQ)(x)+(T∗χX\ΛQ)(x) =:ϕ(x)+ψ(x). First, we notice
+that
+x,y∈Q⇒ |ψ(x)−ψ(y)| ≤C(K,Λ),
+whereKis the constant form our definition above. This is easy:
+|ψ(x)−ψ(y)| ≤/integraldisplay
+X\ΛQ|k(x,t)−k(y,t)|dµ(t) =∞/summationdisplay
+j=1/integraldisplay
+Λj+1Q\ΛjQ|k(x,t)−k(y,t)|dµ(t)≤
+∞/summationdisplay
+j=1diam(Q)τ
+(Λjdiam(Q))m+τK(Λjdiam(Q))m=∞/summationdisplay
+j=1K
+Λjτ≤C(K,Λ,τ).BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 19
+Therefore,/integraldisplay
+Q|ψ(x)−/a\}bracketle{tψ/a\}bracketri}htQ|2dµ(x)/lessorsimilarµ(Q)≤µ(ΛQ).
+But,/integraldisplay
+Q|ϕ(x)−/a\}bracketle{tϕ/a\}bracketri}htQ|2dµ(x)/lessorsimilar/integraldisplay
+Q|T∗χΛQ|2dµ≤Aµ(ΛQ)
+by theT1 assumption of Theorem 1.
+/square
+Lemma 23. Letµ,Tsatisfy the assumptins of Theorem 1. Then
+/bardblΠT∗1/bardblL2(X;µ)→L2(X;µ)≤C. (5.6)
+HereCdepends only on the constants of Theorem 1.
+Proof.LetF=T∗1. In the definition of Π Fall ∆Qare mutually orthogonal. So it is easy to see
+that
+/bardblΠFg/bardbl2
+L2(X;µ)=/summationdisplay
+R∈D2,R⊂R0|/a\}bracketle{tg/a\}bracketri}htR|2/summationdisplay
+Q∈D1,Qgood and transit ,s(Q)=κ−rs(R)/bardbl∆QF/bardblL2(X;µ).
+Put
+aR:=/summationdisplay
+Q∈D1,Qgood and transit ,s(Q)=κ−rs(R)/bardbl∆QF/bardblL2(X;µ).
+By Carleson Embedding Theorem, it is enough to prove that for everyS∈ D2
+/summationdisplay
+R∈D2,R⊂SaR≤Cµ(S). (5.7)
+This is the same as
+/summationdisplay
+Q∈D1,Qtransit,s(Q)≤κ−rs(R),dist(Q,∂R)≥s(Q)αs(R)1−α/bardbl∆QF/bardblL2(X;µ)≤Cµ(R). (5.8)
+Let us consider a Whitney decomposition of Rinto disjoint cubes P, such that 1 .5P⊂R, 1.4P
+have only bounded multiplicity C(d) of intersection. This can be accomplished by modifying the
+arguments found in Section 7 of [ 2].
+Consider the sums
+sP:=/summationdisplay
+Q∈D1,Qtransit,s(Q)≤κ−rs(R),Q∪P/\egatio\slash=∅,dist(Q,∂R)≥s(Q)αs(R)1−α/bardbl∆QF/bardblL2(X;µ).(5.9)
+ThissPcan be zero if there is no transit cubes as above intersecting it. But ifsP/\e}atio\slash= 0 then
+necessarily
+µ(P)≤A(d)s(P)m,
+and moreover
+µ(sP)≤A(d)sms(P)m,∀s≥1.
+In fact, in this case Pintersects a transit cube Q, which by elementary geometry is “smaller”’
+thanP:s(Q)≤c(r,d)s(P). But then the above inequalities follow from the definition oftransit.
+It is also clear that for large rand forQ,Pas above
+Q∩P/\e}atio\slash=∅ ⇒Q⊂1.2P .
+Therefore,
+sP/\e}atio\slash= 0⇒sP≤/summationdisplay
+Q∈D1,Qtransit,s(Q)≤κ−rs(R),Q⊂1.2Pdist(Q,∂R)≥s(Q)αs(R)1−α/bardbl∆QF/bardblL2(X;µ).20 A. VOLBERG AND B. D. WICK
+So
+sP/\e}atio\slash= 0⇒sP≤/integraldisplay
+1.2P|F−/a\}bracketle{tF/a\}bracketri}ht1.2P|2dµ≤Cµ(1.4P).
+The last inequality follows from Lemma 22.
+Now we add all sP’s. We get ≤C/summationtextµ(1.4P). This is smaller than C1µ(R) as 1.4P’s have
+multiplicity C(d)<∞. /square
+6.The Diagonal Sum: Controlling Term σ1.
+To complete the estimate of |(Tfgood,ggood)|in only remains to estimate σ1. But notice that
+/bardbl∆Qf/bardblL1(X;µ)≤ /bardbl∆Qf/bardblL2(X;µ)/radicalbig
+µ(Q) and/bardbl∆Rg/bardblL1(X;µ)≤ /bardbl∆Rg/bardblL2(X;µ)/radicalbig
+µ(R).
+Remember that all cubes QandRin the sums considered at this point are transit cubes. In
+particular, in σ1we have that QandRare close and of the almost same size. If a son of Q,S(Q),
+is terminal, then by Lemma 3
+|(TχS(Q)∆Qf,∆Rg)| ≤/radicalbig
+µ(Q)/radicalbig
+µ(R)
+s(Q)m/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ).
+The sons are terminal, but QandRare transit, so µ(Q)/lessorsimilars(Q)m≈s(R)m. Summing such pairs
+(and symmetric ones, where a son of Ris terminal) we get C(r)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
+We are left with the part of σ1, where we sum over QandRsuch that their sons are transit.
+Then we use pairing
+|(TχS(Q)∆Qf,χS(R)∆Rg)| ≤ |cS(Q)||cS(R)|/radicalbig
+µ(S(Q))µ(S(R)).
+The estimate above follows from our T1 assumption in Theorem 1. Now using ( 5.4), again we
+obtain
+|(TχS(Q)∆Qf,χS(R)∆Rg)| ≤C/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ).
+This completes the proof of Theorem 1.
+References
+[1] Michael Christ, AT(b)theorem with remarks on analytic capacity and the Cauchy int egral, Colloq. Math. 60/61
+(1990), no. 2, 601–628. ↑1,3
+[2] T. Hyt¨ onen and H. Martikainen, Non-Homogeneous Tb Theorem on Metric Spaces , preprint. ↑3,4,6,7,19
+[3] F. Nazarov, S. Treil, and A. Volberg, TheTb-theorem on non-homogeneous spaces , Acta Math. 190(2003), no. 2,
+151–239. ↑3
+[4] ,Accretive system Tb-theorems on nonhomogeneous spaces , Duke Math. J. 113(2002), no. 2, 259–312. ↑3
+[5] ,Weak type estimates and Cotlar inequalities for Calder´ on- Zygmund operators on nonhomogeneous spaces ,
+Internat. Math. Res. Notices (1998), no. 9, 463–487. ↑3
+[6] ,Cauchy integral and Calder´ on-Zygmund operators on nonhomo geneous spaces , Internat. Math. Res. No-
+tices (1997), no. 15, 703–726. ↑3
+[7] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclide an and homogeneous spaces ,
+Amer. J. Math. 114(1992), no. 4, 813–874. ↑3
+[8] A. Volberg, Calder´ on-Zygmund capacities and operators on nonhomogene ous spaces , CBMS Regional Conference
+Series in Mathematics, vol. 100, Published for the Conferen ce Board of the Mathematical Sciences, Washington,
+DC, 2003. ↑3
+[9] A. Volberg and B. D. Wick, Bergman-type Singular Operators and the Characterization of Carleson Measures for
+Besov–Sobolev Spaces on the Complex Ball (2009), preprint. ↑2,3
+Alexander Volberg, Department of Mathematics, Michigan St ate University, East Lansing, MI
+USA 48824
+E-mail address :volberg@math.msu.eduBERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 21
+Alexander Volberg, Department of Mathematics, University of Edinburgh, James Clerk Maxwell
+Building, The King’s Buildings, Mayfield Road, Edinburgh S cotland EH9 3JZ
+E-mail address :a.volberg@ed.ac.uk
+Brett D. Wick, School of Mathematics, Georgia Institute of T echnology, 686 Cherry Street,
+Atlanta, GA 30332-1060 USA
+E-mail address :wick@math.gatech.edu
+URL:http://people.math.gatech.edu/~bwick6/
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+arXiv:1001.0039v1  [astro-ph.IM]  30 Dec 2009Astronomical Data Analysis Software and Systems XIX P45
+ASP Conference Series, Vol. XXX, 2009
+Y. Mizumoto, K.-I. Morita, and M. Ohishi, eds.
+TGCat, The Chandra Transmission Grating Catalog and
+Archive: Systems, Design and Accessibility
+Arik W. Mitschang1, David P. Huenemoerder2, Joy S. Nichols1
+Abstract. The recently released Chandra Transmission Grating Catalog and
+Archive, TGCat, presents a fully dynamic on-line catalog allowing users to
+browse and categorize Chandra gratings observations quickly and easily, gen-
+erate custom plots of resulting response corrected spectra on- line without the
+need for special software and to download analysis ready product s from multi-
+ple observations in one convenient operation. TGCathas been registered as
+a VO resource with the NVO providing direct access to the catalogs in terface.
+The catalog is supported by a back-end designed to automatically fe tch newly
+public data, process , archive and catalog them, At the same time ut ilizing an
+advanced queue system integrated into the archive’s MySQL datab ase allowing
+large processing projects to take advantage of an unlimited numbe r of CPUs
+across a network for rapid completion. A unique feature of the cat alog is that
+all of the high level functions used to retrieve inputs from the Chan dra archive
+and to generate the final data products are available to the user in an ISIS writ-
+ten library with detailed documentation. Here we present a structu ral overview
+of the Systems, Design, and Accessibility features of the catalog a nd archive.
+1. Introduction
+TGCataims to be the definitive end-user source for all Chandra HETG S and
+LETGS observations. In order to achieve this goal the catalo g must both have
+the absolute best processed and calibrated data, as well as h ave an interface
+which makes it easy for users to find, review and download thei r observations
+of choice. The science requirements for accurately process ing gratings data are
+discussed elsewhere ( Huenemoerder et. al. 2010 ). This writ ing will focus on
+the automated system that collects and processes new public data, the interfaces
+used for administrative review, and the interfaces and syst ems provided for user
+access.
+2. Database, Archive, and Subsystems
+TheTGCatprocessing system is comprised of three major components:
+•MySQL database storing meta-data
+•File archive
+•Processing Software
+1Smithsonian Astrophysical Observatory, Cambridge, MA, US A
+2MIT Kavli Institute for Space Research, Cambridge, MA
+12 MITSCHANG, HUENEMOERDER, NICHOLS
+Figure 1. TGCatdata flow
+These components are described in the context of TGCatprocessing in this
+section.
+Tables: Thetables relevant toprocessingincludetheextractions, source, spec-
+tral properties, files, and queue tables. The extractions ta ble has one entry per
+processed extraction, where extraction is taken to mean a si ngle source in a
+uniqueChandra observation ID (ObsID). Any one ObsIDcan have many sources
+and any one source can be in many ObsIDs, the extractions tabl e will store one
+entry for each combination thereof. In order to consolidate all extractions of a
+single source, there is a source table indexed on SIMBAD1identifier, a TGCat
+identifier, and coordinates, that associates entries in the extractions table. The
+files table tracks processing output products and summary im ages allowing the
+webpagestoeasily displayinformation onfilesavailable fo rdownload. Thespec-
+tral properties table stores and indexes spectral properti es in several different
+wavebands, these data are also available in a fits table for do wnload.
+Data Flow: The flow of data through the system is rather simple, as illus-
+trated in Figure 1. A bash script run via cronat regular intervals downloads a
+list of gratings observations from the public chandra archi ve2and compares with
+the list of ObsIDs that have been submitted to TGCat. Any not in TGCat
+will be added to the queue table to be processed in line. Daemo n processes,
+written in python, run on any number of network connected hos ts continuously
+requesting entries from the queue in a FIFOmanner, the first process to ask
+entries from the queue table will retrieve the first entry ( by time of creation
+). They then parse the queue entry which can contain a number o f custom
+processing parameters, setup work spaces and logging, fork off the processing
+which is implemented in ISIS ( Houck & Denicola 2000 ) interac tive library
+functions, which are available for download3and custom use, and finally ingest
+1http://simbad.u-strasbg.fr/simbad
+2http://cda.harvard.edu/chaser/
+3http://space.mit.edu/cxc/analysis/tgcatTGCat 3
+data into the database and archive. During the ingest step, s everal checks are
+done to evaluate whether or not the resultant processed data are a complete
+set and worth being manually reviewed by TGCatscientists. If so, meta data
+are added to the database’s main extractions table which ret urns a unique iden-
+tifier that is then used to tag the file based products. At this t ime linking is
+done between the source table and extractions table. If not, the queue entry is
+marked as an error and notification sent. TGCatoperators have the choice to
+investigate the existing processing workspace or simply re -evaluate parameters
+and re-queue the extraction as new.
+Validation and Verification: Each extraction available for browsing has
+been reviewed by a member of the TGCatscience team in order to confirm
+zeroth order placement, proper masking, etc. This is done vi a an internal web-
+site nearly mimicking the public interface for reviewing ex tractions but with
+the addition of forms for queueing processing and for markin g the extraction as
+“good”, “bad”, or “warning” and optional comment fields whic h are available
+for review by the end user. “bad” extractions are never shown on the public
+site and administrators have the option of rejecting any oth er extraction for
+whatever reason. Because each extraction is tagged with the date of processing,
+the version of TGCatused for processing, and a so called group ID which is
+unique per group of extractions intended to be of the same obj ect for a single
+ObsID, keeping track of accepted sources and avoiding dupli cates is made easy.
+3. Access
+TGCathas several interfaces for data access. Currently three are operational
+includingthewebbrowserinterface, andtwoVirtualObserv atoryaccessprotocol
+interfaces. Each oftheseaccess interfaces areimplemente dasa“plug-in”written
+in php. Plug-in functions to create output are called by a cat alog independent
+generalized query library written in php under TGCatdevelopment dubbed
+“queryLib”. This modular approach makes it trivial to add ac cess interfaces to
+TGCat.
+queryLib: queryLib stores state and type information of an individual query
+in a database table. Each time a query is performed, fields in t his table such as
+awhereclause, sort fields, indexed IDs and columns are populated wi th infor-
+mation specific to that query. As a user interacts with the par ticular interface,
+information in this table entry are updated to reflect the new state of the query
+through object method calls. queryLib initialized a query b y creating a new en-
+try in the table with two unique IDs: one is the primary key ide ntifier assigned
+at creation and can be used to reference the query, the other i s created on the fly
+using a combination of the primary key and a random hash strin g that is very
+unlikely to be guessed. This is done so that a malicious user c annot change the
+state of anyone’s query by simply guessing an ID. The queryLi b speeds up the
+process of redrawing a table of results by storing the indexe d primary keys of
+all entries selected rather than having to rerun a potential ly complicated where
+clause.4 MITSCHANG, HUENEMOERDER, NICHOLS
+Web Catalog Interface: The web interface4toTGCatallows users browse
+extraction and object entries in a way that is most suitable t o the catalog data
+structure. Each extraction has an associated preview page w here a set of stan-
+dard plots produced at the time of processing is available fo r viewing. A table
+of spectral properties providing quick general categoriza tion, and a table of each
+individual file available for the extraction along with a lin k to download that file
+are provided. Perhaps one of the most important features of t he web interface
+is the interactive plotting of individual or combined spect ra directly on the site.
+This is implemented server side by taking the POST request fo r plotting param-
+eters along with a unique file name, creating a small script of the commands for
+loading data and plotting, and piping this script into an ISI S process. Malicious
+useisprevented by checking all parameter inputsfor approp riatevalues, running
+the ISIS process as an unprivileged user, and checking the te mp file name for
+validity before piping. Since the file name is known at the tim e of the request,
+the page simply needs to reload to show the new plot. The comma nds, any error
+output, and an ASCII dump of the plot data are available for do wnload as well.
+Virtual Observatory: TGCatis a registered VO service providing both
+the Simple Cone Search5and Simple Image Access6Protocols. These are both
+implemented as an XML output plug-ins taking input from a php script that
+provides appropriate GET parameters, then creates a query o bject after parsing
+the input data, finally running the query in exactly the same w ay as for the web
+and ASCII interfaces. Error handling is done by the calling s cript and meta
+data is provided by running a query with parameters known to h ave no catalog
+entries ( indexed ID=0 ). VO queries, and any other type, can b e tracked in
+the query table using the type column. In this way we can track statistics on
+requests coming from services such as datascope7.
+Data Package Downloads: Generally, users will want to download more
+than one file at a time for analysis, such as such as spectra and responses or
+productsfrom multiple extractions of the same target. To th is endTGCatruns
+apackager processthatparsesaqueuetablemuchlikethepro cessingqueuetable
+described above. Requested packages are added to the packag e queue table,
+validated and read by the packager which then fetches data to a temp space
+placing them in a directory hierarchy tagged with ObsID and TGCatID. The
+entire hierarchy is then tarred, compressed and provided to the user along with
+file checksums via HTTP.
+Acknowledgements. We would like to thank Dan Dewey for extensive input
+on data organization and interface layout, and Mike Nowak fo r ISIS plotting
+routines and advice on the interactive plotting interface. This work is supported
+4http://tgcat.mit.edu
+5http://tgcat.mit.edu/tgCli.php?OUTPUT=V
+6http://tgcat.mit.edu/tgSia.php
+7http://heasarc.gsfc.nasa.gov/cgi-bin/vo/datascope/i nit.plTGCat 5
+by the Chandra X-ray Center (CXC) NASA contract NAS8-03060. DPH was
+supportedby NASA through the Smithsonian Astrophysical Ob servatory (SAO)
+contract SV3-73016 for the Chandra X-Ray Center and Science Instruments.
+References
+Huenemoerder, et. al. 2010, ( in preparation )
+Houck, J. C., Denicola, L. A. 2000, in ASP Conf. Ser. 216, ADASS IX, ed. N. Manset,
+C. Veillet, & D. Crabtree (San Francisco: ASP), 591
+Williams, Roy, Hanisch, Robert, Szalay, Alex, Plante, Ray 2009, Simple Cone Search
+Version1.03,http://www.ivoa.net/Documents/REC/DAL/ConeSea rch-20080222.html
+Tody, Doug & Plante, Ray 2009, Simple Image Access Specification Ve rsion 1.0,
+http://www.ivoa.net/Documents/SIA/20091008/PR-SIA-1.0-20 091008.html
\ No newline at end of file
diff --git a/1001.0040.txt b/1001.0040.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e986d26d0916c6e6177fd2b7e57c8d09a5668556
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+++ b/1001.0040.txt
@@ -0,0 +1,725 @@
+arXiv:1001.0040v2  [math-ph]  16 Sep 2010COURANT ALGEBROIDS FROM CATEGORIFIED
+SYMPLECTIC GEOMETRY
+CHRISTOPHER L. ROGERS
+Abstract. In categorified symplectic geometry, one studies the cate-
+gorified algebraic and geometric structures that naturally arise on man-
+ifolds equipped with a closed nondegenerate ( n+ 1)-form. The case
+relevant to classical string theory is when n= 2 and is called ‘2-plectic
+geometry’. Just as the Poisson bracket makes the smooth func tions on
+a symplectic manifold into a Lie algebra, there is a Lie 2-alg ebra of
+observables associated to any 2-plectic manifold. String t heory, closed
+3-forms and Lie 2-algebras also play important roles in the t heory of
+Courant algebroids. Courant algebroids are vector bundles which gen-
+eralize the structures found in tangent bundles and quadrat ic Lie alge-
+bras. It is known that a particular kind of Courant algebroid (called an
+exact Courant algebroid) naturally arises in string theory , and that such
+an algebroid is classified up to isomorphism by a closed 3-for m on the
+base space, which then induces a Lie 2-algebra structure on t he space of
+global sections. In this paper we begin to establish precise connections
+between 2-plectic manifolds and Courant algebroids. We pro ve that any
+manifold Mequipped with a 2-plectic form ωgives an exact Courant
+algebroid EωoverMwithˇSevera class [ ω], and we construct an embed-
+ding of the Lie 2-algebra of observables into the Lie 2-algeb ra of sections
+ofEω. We then show that this embedding identifies the observables as
+particular infinitesimal symmetries of Eωwhich preserve the 2-plectic
+structure on M.
+1.Introduction
+The underlying geometric structures of interest in categor ified symplectic
+geometry are multisymplectic manifolds: manifolds equipp ed with a closed,
+nondegenerate form of degree ≥2 [8]. This kind of geometry originated in
+the work of DeDonder [10] and Weyl [24] on the calculus of vari ations, and
+more recently has been used as a formalism to investigate cla ssical field the-
+ories [11, 12, 13]. In this paper, we call a manifold ‘ n-plectic’ if it is equipped
+with a closed nondegenerate ( n+ 1)-form. Hence ordinary symplectic ge-
+ometry corresponds to the n= 1 case, and the corresponding 1-dimensional
+field theory is just the classical mechanics of point particl es. In general,
+examples of n-plectic manifolds include phase spaces suitable for descr ibing
+n-dimensional classical field theories. We will be primarily concerned with
+Date: October 29, 2018.
+This work was partially supported by a grant from The Foundat ional Questions
+Institute.
+12 CHRISTOPHER L. ROGERS
+then= 2case. Thisis the firstreally new case of n-plectic geometry andthe
+corresponding 2-dimensional field theories of interest inc lude bosonic string
+theory. Indeed, just as the phase space of the classical part icle is a mani-
+fold equipped with a closed, nondegenerate 2-form, the phas e space of the
+classical string is a finite-dimensional manifold equipped with a closed non-
+degenerate 3-form. This phase space is often called the ‘mul tiphase space’
+of the string [11] in order to distinguish it from the infinite -dimensional
+symplectic manifolds that are used as phase spaces in string field theory [6].
+In classical mechanics, the relevant mathematical structu res are not just
+geometric, butalsoalgebraic. Thesymplecticformgives th espaceofsmooth
+functions the structure of Poisson algebra. Analogously, i n classical string
+theory, the 2-plectic form induces a bilinear skew-symmetr ic bracket on
+a particular subspace of differential 1-forms, which we call H amiltonian.
+The Hamiltonian 1-forms and smooth functions form the under lying chain
+complex of an algebraic structure known as a semistrict Lie 2 -algebra. A
+semistrict Lie 2-algebra can be viewed as a categorified Lie a lgebra in which
+the Jacobi identity is weakened and is required to hold only u p to isomor-
+phism. Equivalently, it can be described as a 2-term L∞-algebra, i.e. a
+generalization of a 2-term differential graded Lie algebra in which the Ja-
+cobi identity is only satisfied up to chain homotopy [1, 14]. J ust as the
+Poisson algebra of smooth functions represents the observa bles of a system
+of particles, it has been shown that the Lie 2-algebra of Hami ltonian 1-forms
+contains the observables of the classical string [2]. In gen eral, ann-plectic
+structure will give rise to a L∞-algebra on an n-term chain complex of dif-
+ferential forms in which the ( n−1)-forms correspond to the observables of
+ann-dimensional classical field theory [15].
+Many of the ingredients found in 2-plectic geometry are also found in
+the theory of Courant algebroids, which was also developed b y generalizing
+structures found in symplectic geometry. Courant algebroi ds were first used
+by Courant [9] to study generalizations of pre-symplectic a nd Poisson struc-
+tures in the theory of constrained mechanical systems. Roug hly, a Courant
+algebroid is a vector bundle that generalizes the structure of a tangent bun-
+dle equipped with a symmetric nondegenerate bilinear form o n the fibers.
+In particular, the underlying vector bundle of a Courant alg ebroid comes
+equipped with a skew-symmetric bracket on the space of globa l sections.
+However, unlike the Lie bracket of vector fields, the bracket need not satisfy
+the Jacobi identity.
+In a letter to Weinstein, ˇSevera [20] described how a certain type of
+Courant algebroid, known as an exact Courant algebroid, app ears naturally
+when studying 2-dimensional variational problems. In clas sical string the-
+ory, the string can be represented as a map φ: Σ→Mfrom a 2-dimensional
+parameter space Σ into a manifold Mcorresponding to space-time. The
+imageφ(Σ) is called the string world-sheet. The map φextremizes the inte-
+gral of a 2-form θ∈Ω2(M) over its world-sheet. Hence the classical string
+is a solution to a 2-dimensional variational problem. The 2- formθis calledCOURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 3
+the Lagrangian and depends on elements of the first jet bundle of the trivial
+bundleΣ ×M. TheLagrangian isnotunique. A solution φremainsinvariant
+if an exact 1-form or ‘divergence’ is added to θ. It is, in fact, the 3-form dθ
+that is relevant. Inthiscontext, ˇSeveraobserved that the3-form dθuniquely
+specifies (up to isomorphism) the structure of an exact Coura nt algebroid
+overM. The general correspondence between exact Courant algebro ids and
+closed 3-forms on the base space was further developed by ˇSevera, and also
+by Bressler and Chervov [4], to give a complete classificatio n. An exact
+Courant algebroid over Mis determined up to isomorphism by its ˇSevera
+class: an element [ ω] in the third de Rham cohomology of M.
+Just as in 2-plectic geometry, the underlying geometric str ucture of a
+Courant algebroid has an algebraic manifestation. Roytenb erg and Wein-
+stein [16] showed that the bracket on the space of global sect ions induces
+anL∞structure. If we are considering an exact Courant algebroid , then
+the global sections can be identified with ordered pairs of ve ctor fields and
+1-forms on the base space. Roytenberg and Weinstein’s resul ts imply that
+these sections, when combined with the smooth functions on t he base space,
+form a semistrict Lie 2-algebra [23]. Moreover, the bracket of the Lie 2-
+algebra is determined by a closed 3-form corresponding to a r epresentative
+of theˇSevera class [21].
+Thus there are striking similarities between 2-plectic man ifolds and ex-
+act Courant algebroids. Both originate from attempts to gen eralize certain
+aspects of symplectic geometry. Both come equipped with a cl osed 3-form
+that gives rise to a Lie 2-algebra structure on a chain comple x consisting
+of smooth functions and differential 1-forms. In this paper, w e prove that
+there is indeed a connection between the two. We show that any manifold
+Mequipped with a 2-plectic form ωgives an exact Courant algebroid Eω
+withˇSevera class [ ω], and that there is an embedding of the Lie 2-algebra
+of observables into the Lie 2-algebra corresponding to Eω. Moreover, this
+embedding allows us to characterize the Hamiltonian 1-form s as particular
+infinitesimal symmetries of Eωwhich preserve the 2-plectic structure on M.
+2.Courant algebroids
+Here we recall some basic facts and examples of Courant algeb roids and
+then we proceed to describe ˇSevera’s classification of exact Courant alge-
+broids. There are several equivalent definitions of a Couran t algebroid found
+in the literature. In this paper we use the definition given by Roytenberg
+[17].
+Definition 2.1. ACourant algebroid is a vector bundle E→Mequipped
+with a nondegenerate symmetric bilinear form /an}bracketle{t·,·/an}bracketri}hton the bundle, a skew-
+symmetric bracket /llbracket·,·/rrbracketonΓ(E), and a bundle map (called the anchor)
+ρ:E→TMsuch that for all e1,e2,e3∈Γ(E)and for all f,g∈C∞(M)the
+following properties hold:
+(1)/llbrackete1,/llbrackete2,e3/rrbracket/rrbracket−/llbracket/llbrackete1,e2/rrbracket,e3/rrbracket−/llbrackete2,/llbrackete1,e3/rrbracket/rrbracket=−DT(e1,e2,e3),4 CHRISTOPHER L. ROGERS
+(2)ρ([e1,e2]) = [ρ(e1),ρ(e2)],
+(3) [e1,fe2] =f[e1,e2]+ρ(e1)(f)e2−1
+2/an}bracketle{te1,e2/an}bracketri}htDf,
+(4)/an}bracketle{tDf,Dg/an}bracketri}ht= 0,
+(5)ρ(e1)(/an}bracketle{te2,e3/an}bracketri}ht) =/an}bracketle{t[e1,e2]+1
+2D/an}bracketle{te1,e2/an}bracketri}ht,e3/an}bracketri}ht+/an}bracketle{te2,[e1,e3]+1
+2D/an}bracketle{te1,e3/an}bracketri}ht/an}bracketri}ht,
+where[·,·]is the Lie bracket of vector fields, D:C∞(M)→Γ(E)is the map
+defined by /an}bracketle{tDf,e/an}bracketri}ht=ρ(e)f, and
+T(e1,e2,e3) =1
+6(/an}bracketle{t/llbrackete1,e2/rrbracket,e3/an}bracketri}ht+/an}bracketle{t/llbrackete3,e1/rrbracket,e2/an}bracketri}ht+/an}bracketle{t/llbrackete2,e3/rrbracket,e1/an}bracketri}ht).
+The bracket in Definition 2.1 is skew-symmetric, but the first property
+implies that it needs only to satisfy the Jacobi identity “up toDT”. (The
+notation suggests we think of this as a boundary.) The functi onTis often
+referred to as the Jacobiator . (When there is no risk of confusion, we shall
+refer to the Courant algebroid with underlying vector bundl eE→MasE.)
+Note that the vector bundle Emay be identified with E∗via the bilinear
+form/an}bracketle{t·,·/an}bracketri}htand therefore we have the dual map
+ρ∗:T∗M→E.
+Hence the map Dis simply the pullback of the de Rham differential by ρ∗.
+Thereisanalternatedefinitiongiven by ˇSevera[20]forCourantalgebroids
+which uses a bilinear operation on sections that satisfies a J acobi identity
+but is not skew-symmetric.
+Definition 2.2. ACourant algebroid is a vector bundle E→Mtogether
+with a nondegenerate symmetric bilinear form /an}bracketle{t·,·/an}bracketri}hton the bundle, a bilinear
+operation ◦onΓ(E), and a bundle map ρ:E→TMsuch that for all
+e1,e2,e3∈Γ(E)and for all f∈C∞(M)the following properties hold:
+(1)e1◦(e2◦e3) = (e1◦e2)◦e3+e2◦(e1◦e3),
+(2)ρ(e1◦e2) = [ρ(e1),ρ(e2)],
+(3)e1◦fe2=f(e1◦e2)+ρ(e1)(f)e2,
+(4)e1◦e1=1
+2D/an}bracketle{te1,e1/an}bracketri}ht,
+(5)ρ(e1)(/an}bracketle{te2,e3/an}bracketri}ht) =/an}bracketle{te1◦e2,e3/an}bracketri}ht+/an}bracketle{te2,e1◦e3/an}bracketri}ht,
+where[·,·]is the Lie bracket of vector fields, and D:C∞(M)→Γ(E)is the
+map defined by /an}bracketle{tDf,e/an}bracketri}ht=ρ(e)f.
+The “bracket” ◦is related to the bracket given in Definition 2.1 by:
+x◦y=/llbracketx,y/rrbracket+1
+2D/an}bracketle{tx,y/an}bracketri}ht. (1)
+Roytenberg [17] showed that if Eis a Courant algebroid in the sense of
+Definition 2.1 with bracket /llbracket·,·/rrbracket, bilinear form /an}bracketle{t·,·/an}bracketri}htand anchor ρ, thenEis
+a Courant algebroid in the sense of Definition 2.2 with the sam e anchor and
+bilinear form but with bracket ◦given by Eq. 1. Unless otherwise stated, all
+Courant algebroids mentioned in this paper are Courant alge broids in the
+sense of Definition 2.1. We introduced Definition 2.2 mainly t o connect our
+results here with previous results in the literature.COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 5
+Example 1.An important example of a Courant algebroid is the standard
+Courant algebroid E0=TM⊕T∗Mover any manifold Mwith bracket
+/llbracket(v1,α1),(v2,α2)/rrbracket0=/parenleftbigg
+[v1,v2],Lv1α2−Lv2α1−1
+2d(ιv1α2−ιv2α1)/parenrightbigg
+,(2)
+and bilinear form
+/an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1. (3)
+In this case the anchor ρ:E0→TMis the projection map, and for a
+function f∈C∞(M),Df= (0,df).
+The standard Courant algebroid is the prototypical example of anexact
+Courant algebroid [4].
+Definition 2.3. A Courant algebroid E→Mwith anchor map ρ:E→TM
+isexactiff
+0→T∗Mρ∗
+→Eρ→TM→0
+is an exact sequence of vector bundles.
+2.1.TheˇSevera class of an exact Courant algebroid. ˇSevera’s clas-
+sification originates in the idea that choosing a splitting o f the above short
+exact sequence corresponds to defining a kind of connection.
+Definition 2.4. Aconnection on an exact Courant algebroid Eover a
+manifold Mis a map of vector bundles A:TM→Esuch that
+(1)ρ◦A= idTM,
+(2)/an}bracketle{tA(v1),A(v2)/an}bracketri}ht= 0for allv1,v2∈TM,
+whereρ:E→TMand/an}bracketle{t·,·/an}bracketri}htare the anchor and bilinear form, respectively.
+IfAis a connection and θ∈Ω2(M) is a 2-form then one can construct a
+new connection:
+(A+θ)(v) =A(v)+ρ∗θ(v,·). (4)
+(A+θ) satisfies the first condition of Definition 2.4 since ker ρ= imρ∗. The
+second condition follows from the fact that we have by definit ion ofρ∗:
+/an}bracketle{tρ∗(α),e/an}bracketri}ht=α(ρ(e)) (5)
+for alle∈Γ(E) andα∈Ω1(M). Furthermore, one can show that any two
+connections on an exact Courant algebroid must differ (as in Eq . 4) by a
+2-form on M. Hence the space of connections on an exact Courant algebroi d
+is an affine space modeled on the vector space of 2-forms Ω2(M) [4].
+The failure of a connection to preserve the bracket gives a su itable notion
+of curvature:
+Definition 2.5. IfEis an exact Courant algebroid over Mwith bracket /llbracket·,·/rrbracket
+andA:TM→Eis a connection then the curvature is a map F:TM×
+TM→Edefined by
+F(v1,v2) =/llbracketA(v1),A(v2)/rrbracket−A([v1,v2]).6 CHRISTOPHER L. ROGERS
+IfFis the curvature of a connection Athen given v1,v2∈TM, it follows
+from exactness and axiom 2 in Definition 2.1 that there exists a 1-form
+αv1,v2∈Ω1(M) such that F(v1,v2) =ρ∗(αv1,v2). Since Ais a connection,
+its image is isotropic in E. Therefore for any v3∈TMwe have:
+/an}bracketle{tF(v1,v2),A(v3)/an}bracketri}ht=/an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht.
+The above formula allows one to associate the curvature Fto a 3-form on
+M:
+Proposition 2.6. LetEbe an exact Courant algebroid over a manifold M
+with bracket /llbracket·,·/rrbracketand bilinear form /an}bracketle{t·,·/an}bracketri}ht. LetA:TM→Ebe a connection
+onE. Then given vector fields v1,v2,v3onM:
+(1)The function
+ω(v1,v2,v3) =/an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht
+defines a closed 3-form on M.
+(2)Ifθ∈Ω2(M)is a 2-form and ˜A=A+θthen
+˜ω(v1,v2,v3) =/an}bracketle{t/llbracket˜A(v1),˜A(v2)/rrbracket,˜A(v3)/an}bracketri}ht
+=ω(v1,v2,v3)+dθ(v1,v2,v3).
+Proof.The statements in the proposition are proven in Lemmas 4.2.6 , 4.2.7,
+and 4.3.4 in the paper by Bressler and Chervov [4]. In their wo rk they
+define a Courant algebroid using Definition 2.2, and therefor e their bracket
+satisfies the Jacobi identity, but is not skew-symmetric. In our notation,
+their definition of the curvature 3-form is:
+ω′(v1,v2,v3) =/an}bracketle{tA(v1)◦A(v2),A(v3)/an}bracketri}ht.
+In particular they show that ◦satisfying the Jacobi identity implies ω′is
+closed. The Jacobiator corresponding to the Courant bracke t is non-trivial
+in general. However the isotropicity of the connection and E q. 1 imply
+A(v1)◦A(v2) =/llbracketA(v1),A(v2)/rrbracket∀v1,v2∈TM.
+Henceω′=ω, so all the needed results in [4] apply here. /square
+Thus the above proposition implies that the curvature 3-for m of an exact
+Courantalgebroid over Mgives awell-defined cohomology class in H3
+DR(M),
+independent of the choice of connection.
+2.2.Twisting the Courant bracket. The previous section describes how
+to go from exact Courant algebroids to closed 3-forms. Now we describe the
+reverse process. In Example 1 we showed that one can define the standard
+Courant algebroid E0over any manifold M. The total space is the direct
+sumTM⊕T∗M, the bracket and bilinear form are given in Eqs. 2 and 3,
+and the anchor is simply the projection. The inclusion A(v) = (v,0) of the
+tangent bundle into the direct sum is obviously a connection onE0and it
+is easy to see that the standard Courant algebroid has zero cu rvature.COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 7
+ˇSevera and Weinstein [20, 21] observed that the bracket on E0could be
+twisted by a closed 3-form ω∈Ω3(M) on the base:
+/llbracket(v1,α1),(v2,α2)/rrbracketω=/llbracket(v1,α1),(v2,α2)/rrbracket0+ω(v1,v2,·).
+This gives a new Courant algebroid Eωwith the same anchor and bilinear
+form. Using Eqs. 2 and 3 we can compute the curvature 3-form of this new
+Courant algebroid:
+/an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht=/an}bracketle{t/llbracket(v1,0),(v2,0)/rrbracket,(v3,0)/an}bracketri}ht
+=/an}bracketle{t([v1,v2],ω(v1,v2,·)),(v3,0)/an}bracketri}ht
+=ω(v1,v2,v3),
+and we see that Eωis an exact Courant algebroid over MwithˇSevera class
+[ω].
+3. 2-plectic geometry
+We nowgive abriefoverview of 2-plectic geometry. Moredeta ils including
+motivation for several of the definitions presented here can be found in our
+previous work with Baez and Hoffnung [2, 3].
+Definition 3.1. A3-formωon aC∞manifold Mis2-plectic , or more
+specifically a 2-plectic structure , if it is both closed:
+dω= 0,
+and nondegenerate:
+∀v∈TxM, ιvω= 0⇒v= 0
+Ifωis a2-plectic form on Mwe call the pair (M,ω)a2-plectic manifold .
+The 2-plectic structure induces an injective map from the sp ace of vector
+fields on Mto the space of 2-forms on M. This leads us to the following
+definition:
+Definition 3.2. Let(M,ω)be a2-plectic manifold. A 1-form αonMis
+Hamiltonian if there exists a vector field vαonMsuch that
+dα=−ιvαω.
+We sayvαis theHamiltonian vector field corresponding to α. The set
+of Hamiltonian 1-forms and the set of Hamiltonian vector fiel ds on a 2-
+plectic manifold are both vector spaces and are denoted as Ham(M)and
+VectH(M), respectively.
+The Hamiltonian vector field vαis unique if it exists, but note there may
+be 1-forms αhaving no Hamiltonian vector field. Furthermore, two distin ct
+Hamiltonian 1-forms may differ by a closed 1-form and therefor e share the
+same Hamiltonian vector field.
+We can generalize thePoisson bracket of functionsin symple ctic geometry
+by defining a bracket of Hamiltonian 1-forms.8 CHRISTOPHER L. ROGERS
+Definition 3.3. Givenα,β∈Ham(M), thebracket {α,β}is the
+1-form given by
+{α,β}=ιvβιvαω.
+Proposition 3.4. Letα,β,γ∈Ham(M)and letvα,vβ,vγbe the respective
+Hamiltonian vector fields. The bracket {·,·}has the following properties:
+(1)The bracket of Hamiltonian forms is Hamiltonian:
+d{α,β}=−ι[vα,vβ]ω, (6)
+so in particular we have
+v{α,β}= [vα,vβ].
+(2)The bracket is skew-symmetric:
+{α,β}=−{β,α} (7)
+(3)The bracket satisfies the Jacobi identity up to an exact 1-form :
+{α,{β,γ}}−{{α,β},γ}−{β,{α,γ}}=dJα,β,γ (8)
+withJα,β,γ=ιvαιvβιvγω.
+Proof.See Proposition 3.7 in [2]. /square
+4.Lie2-algebras
+Both the Courant bracket and the bracket on Hamiltonian 1-fo rms are,
+roughly, Lie brackets which satisfy the Jacobi identity up t o an exact 1-
+form. This leads us to the notion of a Lie 2-algebra: a categor y equipped
+with structures analogous to those of a Lie algebra, for whic h the usual laws
+involving skew-symmetry and the Jacobi identity hold up to i somorphism
+[1, 19]. A Lie 2-algebra in which the isomorphisms are actual equalities
+is called a strict Lie 2-algebra. A Lie 2-algebra in which the laws govern-
+ing skew-symmetry are equalities but the Jacobi identity ho lds only up to
+isomorphism is called a semistrict Lie 2-algebra.
+Here we define a semistrict Lie 2-algebra to be a 2-term chain c omplex
+of vector spaces equipped with structures analogous to thos e of a Lie al-
+gebra, for which the usual laws hold up to chain homotopy. In t his guise,
+a semistrict Lie 2-algebra is nothing more than a 2-term L∞-algebra. For
+more details, we refer the reader to the work of Lada and Stash eff [14], and
+the work of Baez and Crans [1].
+Definition 4.1. Asemistrict Lie 2-algebra is a2-term chain complex
+of vector spaces L= (L1d→L0)equipped with:
+•a chain map [·,·]:L⊗L→Lcalled the bracket;
+•an antisymmetric chain homotopy J:L⊗L⊗L→Lfrom the chain
+map
+L⊗L⊗L→L
+x⊗y⊗z/mapsto−→[x,[y,z]],COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 9
+to the chain map
+L⊗L⊗L→L
+x⊗y⊗z/mapsto−→[[x,y],z]+[y,[x,z]]
+called the Jacobiator ,
+such that the following equation holds:
+[x,J(y,z,w)] +J(x,[y,z],w) +J(x,z,[y,w]) +[J(x,y,z),w]
++[z,J(x,y,w)] =J(x,y,[z,w]) +J([x,y],z,w)
++[y,J(x,z,w)] +J(y,[x,z],w) +J(y,z,[x,w]).(9)
+We will also need a suitable notion of morphism:
+Definition 4.2. Given semistrict Lie 2-algebras LandL′with bracket and
+Jacobiator [·,·],Jand[·,·]′,J′respectively, a homomorphism fromLto
+L′consists of:
+•a chain map φ= (φ0,φ1) :L→L′, and
+•a chain homotopy φ2:L⊗L→Lfrom the chain map
+L⊗L→L
+x⊗y/mapsto−→[φ(x),φ(y)]′,
+to the chain map
+L⊗L→L
+x⊗y/mapsto−→φ([x,y])
+such that the following equation holds:
+J′(φ0(x),φ0(y),φ0(z))−φ1(J(x,y,z)) =
+φ2(x,[y,z])−φ2([x,y],z)−φ2(y,[x,z])−[φ2(x,y),φ0(z)]′
++[φ0(x),φ2(y,z)]′−[φ0(y),φ2(x,z)]′.(10)
+This definition is equivalent to the definition of a morphism b etween 2-
+termL∞-algebras. (The same definition is given in [1], but it contai ns a
+typographical error.)
+4.1.The Lie 2-algebra from a 2-plectic manifold. Given a 2-plectic
+manifold( M,ω), wecanconstructasemistrictLie2-algebra. Theunderlyi ng
+2-term chain complex is namely:
+L=C∞(M)d→Ham(M)
+wheredis the usual exterior derivative of functions. This chain co mplex is
+well-defined, since any exact form is Hamiltonian, with 0 as i ts Hamiltonian
+vector field. We can construct a chain map
+{·,·}:L⊗L→L,
+by extending the bracket on Ham( M) trivially to L. In other words, in
+degree 0, the chain map is given as in Definition 3.3:
+{α,β}=ιvβιvαω,10 CHRISTOPHER L. ROGERS
+and in degrees 1 and 2, we set it equal to zero:
+{α,f}= 0,{f,α}= 0,{f,g}= 0.
+The precise construction of this Lie 2-algebra is given in th e following the-
+orem:
+Theorem 4.3. If(M,ω)is a2-plectic manifold, there is a semistrict Lie
+2-algebraL(M,ω)where:
+•the space of 0-chains is Ham(M),
+•the space of 1-chains is C∞,
+•the differential is the exterior derivative d:C∞→Ham(M),
+•the bracket is {·,·},
+•the Jacobiator is the linear map JL: Ham(M)⊗Ham(M)⊗Ham(M)→
+C∞defined by JL(α,β,γ) =ιvαιvβιvγω.
+Proof.See Theorem 4.4 in [2]. /square
+4.2.The Lie 2-algebra from a Courant algebroid. Given any Courant
+algebroid E→Mwith bilinear form /an}bracketle{t·,·/an}bracketri}ht, bracket /llbracket·,·/rrbracket, and anchor ρ:E→
+TM, we can construct a 2-term chain complex
+C=C∞(M)D→Γ(E),
+with differential D=ρ∗d. The bracket /llbracket·,·/rrbracketon global sections can be ex-
+tended to a chain map /llbracket·,·/rrbracket:C⊗C→C. Ife1,e2are degree 0 chains then
+/llbrackete1,e2/rrbracketis the original bracket. If eis a degree 0 chain and f,gare degree 1
+chains, then we define:
+/llbrackete,f/rrbracket=−/llbracketf,e/rrbracket=1
+2/an}bracketle{te,Df/an}bracketri}ht
+/llbracketf,g/rrbracket= 0.
+This extended bracket gives a semistrict Lie 2-algebra on th e complex C:
+Theorem 4.4. IfEis a Courant algebroid, there is a semistrict Lie 2-
+algebraC(E)where:
+•the space of 0-chains is Γ(E),
+•the space of 1-chains is C∞(M),
+•the differential the map D:C∞(M)→Γ(M),
+•the bracket is /llbracket·,·/rrbracket,
+•the Jacobiator is the linear map JC: Γ(M)⊗Γ(M)⊗Γ(M)→C∞(M)
+defined by
+JC(e1,e2,e3) =−T(e1,e2,e3)
+=−1
+6(/an}bracketle{t/llbrackete1,e2/rrbracket,e3/an}bracketri}ht+/an}bracketle{t/llbrackete3,e1/rrbracket,e2/an}bracketri}ht+/an}bracketle{t/llbrackete2,e3/rrbracket,e1/an}bracketri}ht).
+Proof.Theproofthat aCourantalgebroid inthesenseofDefinition 2 .1gives
+rise to a semistrict Lie 2-algebra follows from the work done by Roytenberg
+on graded symplectic supermanifolds [18] and Lie 2-algebra s [19]. In partic-
+ular we refer the reader to Example 5.4 of [19] and Section 4 of [18].COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 11
+Onthe other hand, theoriginal construction of Roytenberg a nd Weinstein
+[16] gives a L∞-algebra on the complex:
+0→kerDι→C∞(M)D→Γ(E),
+with trivial structure maps lnforn≥3. Moreover, the map l2(correspond-
+ing to the bracket /llbracket·,·/rrbracketgiven above) is trivial in degree >1 and the map
+l3(corresponding to the Jacobiator JC) is trivial in degree >0. Hence we
+can restrict this L∞-algebra to our complex Cand use the results in [1] that
+relateL∞-algebras with semistrict Lie 2-algebras. /square
+5.The Courant algebroid associated to a 2-plectic manifold
+Now we have the necessary machinery in place to describe how C ourant
+algebroids connect with 2-plectic geometry. First, recall the discussion in
+Section 2.2 on twisting the bracket of the standard Courant a lgebroid E0by
+a closed 3-form. From Definition 3.1, we immediately have the following:
+Proposition 5.1. Let(M,ω)be a2-plectic manifold. There exists an exact
+Courant algebroid EωwithˇSevera class [ω]overMwith underlying vector
+bundleTM⊕T∗M→M, anchor ρ(v,α) =v, and bracket and bilinear form
+given by:
+/llbracket(v1,α1),(v2,α2)/rrbracketω=/parenleftbigg
+[v1,v2],Lv1α2−Lv2α1−1
+2d(ιv1α2−ιv2α1)+ιv2ιv1ω/parenrightbigg
+,
+/an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1.
+More importantly, the Courant algebroid constructed in Pro position 5.1
+not only encodes the 2-plectic structure ω, but also the corresponding Lie
+2-algebra constructed in Theorem 4.3:
+Theorem 5.2. Let(M,ω)be a2-plectic manifold and let Eωbe its corre-
+sponding Courant algebroid. Let L(M,ω)andC(Eω)be the semistrict Lie
+2-algebras corresponding to (M,ω)andEω, respectively. Then there exists
+a homomorphism embedding L(M,ω)intoC(Eω).
+Before we prove the theorem, we introduce some lemmas to ease the
+calculations. In the notation that follows, if α,βare Hamiltonian 1-forms
+with corresponding vector fields vα,vβ, then
+B(α,β) =1
+2(ιvαβ−ιvβα). (11)
+Also by the symbol /anticlockwisewe mean cyclic permutations of the symbols α,β,γ.
+Lemma 5.3. Ifα,β∈Ham(M)with corresponding Hamiltonian vector
+fieldsvα,vβ, thenLvαβ={α,β}+dιvαβ.
+Proof.SinceLv=ιvd+dιv,
+Lvαβ=ιvαdβ+dιvαβ=−ιvαιvβω+dιvαβ={α,β}+dιvαβ.
+/square12 CHRISTOPHER L. ROGERS
+Lemma 5.4. Ifα,β,γ∈Ham(M)with corresponding Hamiltonian vector
+fieldsvα,vβ,vγ, then
+ι[vα,vβ]γ+/anticlockwise=−3ιvαιvβιvγω+2/parenleftbig
+ιvαdB(β,γ)+ιvγdB(α,β)+ιvβdB(γ,α)/parenrightbig
+.
+Proof.The identity ι[v1,v2]=Lv1ιv2−ιv2Lv1and Lemma 5.3 imply:
+ι[vα,vβ]γ=Lvαιvβγ−ιvβLvαγ
+=Lvαιvβγ−ιvβ({α,γ}+dιvαγ)
+=ιvαdιvβγ−ιvβιvγιvαω−ιvβdιvαγ,
+where the last equality follows from the definition of the bra cket.
+Hence it follows that:
+ι[vγ,vα]β=ιvγdιvαβ−ιvαιvβιvγω−ιvαdιvγβ,
+ι[vβ,vγ]α=ιvβdιvγα−ιvγιvαιvβω−ιvγdιvβα.
+Finally, note 2 ιvαdB(β,γ) =ιvαdιvβγ−ιvαdιvγβ. /square
+Lemma 5.5. Ifα,β∈Ham(M)with corresponding Hamiltonian vector
+fieldsvα,vβ, then
+Lvβα−Lvαβ=−2({α,β}+dB(α,β)).
+Proof.Follows immediately from Lemma 5.3 and the definition of B(α,β).
+/square
+Proof of Theorem 5.2. We will construct a homomorphism from L(M,ω) to
+C(Eω). LetLbe the underlying chain complex of L(M,ω) consisting of
+Hamiltonian 1-forms in degree 0 and smooth functions in degr ee 1. Let
+Cbe the underlying chain of C(Eω) consisting of global sections of Eωin
+degree 0 and smooth functions in degree 1. The bracket /llbracket·,·/rrbracketωdenotes the
+extension of the bracket on Γ( Eω) to the complex Cin the sense of Theorem
+4.4. Let φ0:L0→C0be given by
+φ0(α) = (vα,−α),
+wherevαis the Hamiltonian vector field corresponding to α. Letφ1:L1→
+C1be given by
+φ1(f) =−f.
+Finally let φ2:L0⊗L0→C1be given by
+φ2(α,β) =−B(α,β) =−1
+2(ιvαβ−ιvβα).
+Nowweshow φ2isawell-definedchainhomotopyinthesenseofDefinition
+4.2. For degree 0 we have:
+/llbracketφ0(α),φ0(β)/rrbracketω=/parenleftbigg
+[vα,vβ],Lvα(−β)−Lvβ(−α)+1
+2d/parenleftbig
+ιvαβ−ιvβα/parenrightbig
++ιvβιvαω/parenrightbigg
+= ([vα,vβ],−{α,β}+dφ2(α,β)),COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 13
+where the last equality above follows from Lemma 5.5. By Prop osition 3.4,
+the Hamiltonian vector field of {α,β}is [vα,vβ]. Hence we have:
+/llbracketφ0(α),φ0(β)/rrbracketω−φ0({α,β}) =dφ2(α,β).
+Indegree1, thebracket {·,·}istrivial. Henceitfollows fromthedefinition
+of/llbracket·,·/rrbracketωand the bilinear form on Eω(given in Proposition 5.1 ) that
+/llbracketφ0(α),φ1(f)/rrbracketω=−/llbracketφ1(f),φ0(α)/rrbracketω=1
+2/an}bracketle{t(vα,−α),(0,−df)/an}bracketri}ht=φ2(α,df).
+Therefore φ2is a chain homotopy.
+It remains to show the coherence condition (Eq. 10 in Definiti on 4.2) is
+satisfied. We rewrite the Jacobiator JCas:
+JC(φ0(α),φ0(β),φ0(γ)) =−1
+6/an}bracketle{t/llbracketφ0(α),φ0(β)/rrbracket,φ0(γ)/an}bracketri}ht+/anticlockwise
+=−1
+6/an}bracketle{t([vα,vβ],−{α,β}−dB(α,β)),(vγ,−γ)/an}bracketri}ht+/anticlockwise
+=1
+6/parenleftBig
+ι[vα,vβ]γ+ιvγιvβιvαω+ιvγdB(α,β)/parenrightBig
++/anticlockwise
+=−JL(α,β,γ)+1
+2/parenleftbig
+ιvγdB(α,β)+/anticlockwise/parenrightbig
+.
+The last equality above follows from Lemma 5.4 and the definit ion of the
+Jacobiator JL. Therefore the left-hand side of Eq. 10 is
+JC(φ0(α),φ0(β),φ0(γ))−φ1(JL(α,β,γ)) =1
+2/parenleftbig
+ιvγdB(α,β)+/anticlockwise/parenrightbig
+.
+By the skew-symmetry of the brackets, the right-hand side of Eq. 10 can
+be rewritten as:
+(/llbracketφ0(α),φ2(β,γ)/rrbracketω+/anticlockwise)−(φ2({α,β},γ)+/anticlockwise).
+From the definitions of the bracket, bilinear form and φ2we have:
+/llbracketφ0(α),φ2(β,γ)/rrbracketω+/anticlockwise=1
+2/an}bracketle{t(vα,−α),(0,dφ2(β,γ)/an}bracketri}ht+/anticlockwise
+=−1
+2ιvαdB(β,γ)+/anticlockwise,
+and:
+φ2({α,β},γ)+/anticlockwise=−1
+2/parenleftBig
+ι[vα,vβ]γ−ιvγιvβιvαω/parenrightBig
+=−(ιvαdB(β,γ)+/anticlockwise).
+The last equality above follows again from Lemma 5.4. Theref ore the right-
+hand side of Eq. 10 is
+1
+2/parenleftbig
+ιvγdB(α,β)+/anticlockwise/parenrightbig
+.
+Hencethemaps φ0,φ1,φ2give ahomomorphismofsemistrictLie2-algebras.
+/square14 CHRISTOPHER L. ROGERS
+We note that Roytenberg [19] has shown that a Courant algebro id defined
+using Definition 2.2 with the bilinear operation ◦induces a hemistrict Lie
+2-algebra on the complex Cdescribed in Theorem 4.4 above. A hemistrict
+Lie 2-algebra is a Lie 2-algebra in which the skew-symmetry h olds up to
+isomorphism, while the Jacobi identity holds as an equality . We have proven
+in previous work [2] that a 2-plectic structure also gives ri se to a hemistrict
+Lie 2-algebra on the complex described in Theorem 4.3. One ca n show that
+all results presented above, in particular Theorem 5.2, car ry over to the
+hemistrict case.
+6.Hamiltonian 1-forms as infinitesimal symmetries of the
+Courant algebroid
+Givena2-plecticmanifold( M,ω), theLie2-algebraofobservables L(M,ω)
+identifies particular infinitesimal symmetries of the corre sponding Courant
+algebroid Eωvia the embedding described in the proof of Theorem 5.2. To
+see this, we first recall some basic facts concerning automor phisms of exact
+Courant algebroids. The presentation here follows the work of Bursztyn,
+Cavalcanti, and Gualtieri [7].
+Definition 6.1. LetE→Mbe a Courant algebroid with bilinear form /an}bracketle{t·,·/an}bracketri}ht,
+bracket /llbracket·,·/rrbracket, and anchor ρ:E→TM. Anautomorphism is a bundle
+isomorphism F:E→Ecovering a diffeomorphism ϕ:M→Msuch that
+(1)ϕ∗/an}bracketle{tF(e1),F(e2)/an}bracketri}ht=/an}bracketle{te1,e2/an}bracketri}ht,
+(2)F(/llbrackete1,e2/rrbracket) =/llbracketF(e1),F(e2)/rrbracket,
+(3)ρ(F(e1)) =ϕ∗(ρ(e1)).
+Consider the exact Courant algebroid Eωdescribed in Section 2.2 with
+underlyingvector bundle TM⊕T∗M→MandˇSevera class [ ω]∈H3
+DR(M).
+Given a 2-form B∈Ω2(M), one can define a bundle isomorphism
+expB:TM⊕T∗M→TM⊕T∗M
+by
+expB(v,α) = (v,α+ιvB).
+The map exp Bis known as a ‘gauge transformation’. It covers the identity
+id:M→Mand therefore is compatible (in the sense of Definition 6.1) w ith
+the anchor ρ(v,α) =v. Since Bis skew-symmetric, exp Bpreserves the
+bilinear form /an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1. However a simple compu-
+tation shows that exp Bpreserves the bracket /llbracket·,·/rrbracketω(defined in Eq. 2.2) if
+and only if Bis a closed 2-form:
+/llbracketexpB(v1,α1),expB(v2,α2)/rrbracketω= expB/parenleftbig
+/llbracket(v1,α1),(v2,α2)/rrbracketω+dB/parenrightbig
+.
+Given a diffeomorphism ϕ:M→Mof the base space, one can define a
+bundle isomorphism Φ: TM⊕T∗M→TM⊕T∗Mby
+Φ(v,α) =/parenleftBig
+ϕ∗v,(ϕ∗)−1α/parenrightBig
+.COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 15
+ThemapΦsatisfiesconditions1and3ofDefinition6.1butdoes notpreserve
+the bracket in general:
+/llbracketΦ(v1,α1),Φ(v2,α2)/rrbracketω= Φ/parenleftBig
+/llbracket(v1,α1),(v2,α2)/rrbracketϕ∗ω/parenrightBig
+.
+Bursztyn, Cavalcanti, and Gualtieri [7] showed that any aut omorphism F
+of the exact Courant algebroid Eωmust be of the form
+F= ΦexpB, (12)
+where Φ is constructed from a diffeomorphism ϕ:M→Msuch that
+ω−ϕ∗ω=dB. (13)
+This classification of automorphisms allows one to classify the infinitesimal
+symmetries as well. Let
+Ft= ΦtexptB=/parenleftBig
+ϕt∗exptB,(ϕ∗
+t)−1exptB/parenrightBig
+bea 1-parameter family of automorphismsof the Courant alge broidEωwith
+F0= idEω. Letu∈Vect(M) be the vector field that generates the flow ϕ−t.
+Then differentiation gives:
+dFt
+dt(v,α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+t=0= ([u,v],Luα+ιvB).
+Sinceω−ϕ∗
+tω=tdB, it follows that uandBmust satisfy the equality:
+Luω=dB. (14)
+Theseinfinitesimaltransformationsarecalled derivations [7]oftheCourant
+algebroid Eω, sincetheycorrespondtolinearfirstorderdifferential oper ators
+which act as derivations of the non-skew-symmetric bracket :
+(v1,α1)◦ω(v2,α2) =/llbracket(v1,α1),(v2,α2)/rrbracketω+1
+2d/an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht.(15)
+= ([v1,v2],Lv1α2−ιv2dα1+ιv2ιv1ω). (16)
+In general, derivations are pairs ( u,B)∈Vect(M)⊕Ω2(M) satisfying Eq.
+14. They act on global sections ( v,α)∈Γ(Eω) by:
+(u,B)·(v,α) = ([u,v],Luα+ιvB).
+Global sections themselves naturally act as derivations vi a anadjoint
+action[7]. Given ( u,β)∈Γ(Eω) letBbe the 2-form
+B=−dβ+ιuω. (17)
+Define ad (u,β): Γ(Eω)→Γ(Eω) by
+ad(u,β)(v,α) = (u,B)·(v,α) = ([u,v],Luα+ιv(−dβ+ιuω)).(18)
+One can see this is indeed the adjoint action in the usual sens e if one con-
+siders the non-skew-symmetric bracket given in Eq. 15:
+ad(u,β)(v,α) = (u,β)◦ω(v,α).16 CHRISTOPHER L. ROGERS
+Recall that in the proof of Theorem 5.2 we constructed a homom orphism
+of Lie 2-algebras using the map φ0: Ham(M)→Γ(Eω) defined by
+φ0(α) = (vα,−α),
+wherevαis the Hamiltonian vector field correspondingto α. Comparing Eq.
+17 to Definition 3.2 of a Hamiltonian 1-form, we see that a sect ion (u,β)∈
+Γ(Eω) is in the image of the map φ0if and only if its adjoint action ad (u,β)
+corresponds to the pair ( u,0)∈Vect(M)⊕Ω2(M). This implies that ad (u,β)
+preserves the 2-plectic structure on Mand that −βis a Hamiltonian 1-form
+with Hamiltonian vector field u. Also if uis complete, then Eqs. 12 and 13
+imply that the 1-parameter family Ftof Courant algebroid automorphisms
+generated by ad (u,β)correspondsto a1-parameter family of diffeomorphisms
+ϕt:M→Mwhich preserve the 2-plectic structure:
+ϕ∗
+tω=ω.
+In analogy with symplectic geometry, we call such automorph ismsHamil-
+tonian 2-plectomorphisms .
+We provide the following proposition as a summary of the disc ussion
+presented in this section:
+Proposition 6.2. Let(M,ω)be a 2-plectic manifold and let Eωbe its cor-
+responding Courant algebroid. There is a one-to-one correspo ndence be-
+tween the Hamiltonian 1-forms Ham(M)on(M,ω)and sections (u,β)of
+Eωwhose adjoint action satisfies the equality
+ad(u,β)(v,α) = (Luv,Luα).
+7.Conclusions
+We suspect that the results presented here are preliminary a nd indicate
+a deeper relationship between 2-plectic geometry and the th eory of Courant
+algebroids. For example, the discussion of connections and curvature in
+Section 2.1 is reminiscent of the theory of gerbes with conne ction [5], whose
+relationship with Courant algebroids has been already stud ied [4, 20]. In 2-
+plecticgeometry, gerbeshavebeenconjecturedtoplayarol einthegeometric
+quantization ofa2-plecticmanifold[2]. Itwillbeinteres tingtoseehowthese
+different points of view complement each other.
+In general, much work has been done on studying the geometric struc-
+tures induced by Courant algebroids (e.g. Dirac structures , twisted Dirac
+structures). Perhaps this work can aid 2-plectic geometry s ince many geo-
+metric structures in this context are somewhat less underst ood or remain
+ill-defined (e.g. the notion of a 2-Lagrangian submanifold o r 2-polarization).
+On the other hand, n-plectic manifolds are well understood in the role
+they play in classical field theory [11], and are also underst ood algebraically
+in the sense that an n-plectic structure gives an n-termL∞-algebra on a
+chain complex of differential forms [15]. Perhaps these insig hts can aid inCOURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 17
+understanding ‘higher’ analogs of Courant algebroids (e.g . Lien-algebroids)
+and complement similar ideas discussed by ˇSevera in [22].
+8.Acknowledgments
+We thank John Baez, Yael Fregier, Dmitry Roytenberg, Urs Sch rieber,
+James Stasheff and Marco Zambon for helpful comments, questi ons, and
+conversations.
+References
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+the classical string, Comm. Math. Phys. 293(2010), 701–715. Also available as
+arXiv:0808.0246.
+[3] J. Baez and C. Rogers, Categorified symplectic geometry a nd the string Lie 2-algebra,
+available as arXiv:0901.4721.
+[4] P. Bressler and A. Chervov, Courant algebroids, J. Math. Sci. (N.Y.) 128(2005),
+3030–3053. Also available as arXiv:hep-th/0212195.
+[5] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantiz ation,
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+arXiv:1001.0041v2  [math.MG]  2 Jul 2010Almost-Euclidean subspaces of ℓN
+1via tensor
+products: a simple approach to randomness
+reduction
+Piotr Indyk1⋆and Stanislaw Szarek2⋆⋆
+1MITindyk@mit.edu
+2CWRU & Paris 6 szarek@math.jussieu.fr
+Abstract. It has been known since 1970’s that the N-dimensional ℓ1-
+space contains nearly Euclidean subspaces whose dimension isΩ(N).
+However, proofs of existence of such subspaces were probabi listic, hence
+non-constructive, which made the results not-quite-suita ble for subse-
+quently discovered applications to high-dimensional near est neighbor
+search, error-correcting codes over the reals, compressiv e sensing and
+other computational problems. In this paper we present a “lo w-tech”
+scheme which, for any γ >0, allows us toexhibitnearly Euclidean Ω(N)-
+dimensional subspaces of ℓN
+1while using only Nγrandom bits. Our re-
+sults extend and complement (particularly) recent work by G uruswami-
+Lee-Wigderson. Characteristic features of our approach in clude (1) sim-
+plicity (we use only tensor products) and (2) yielding almos t Euclidean
+subspaces with arbitrarily small distortions.
+1 Introduction
+It is a well-known fact that for any vector x∈RN, itsℓ2andℓ1norms are
+related by the (optimal) inequality /⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l1≤√
+N/⌊ar⌈⌊lx/⌊ar⌈⌊l2. However, classical
+results in geometric functional analysis show that for a “substant ial fraction” of
+vectors , the relation between its 1-norm and 2-norm can be made m uch tighter.
+Specifically, [FLM77,Kas77,GG84] show that there exists a subspace E⊂RNof
+dimensionm=αN, and a scaling constant Ssuch that for all x∈E
+1/D·√
+N/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤S/⌊ar⌈⌊lx/⌊ar⌈⌊l1≤√
+N/⌊ar⌈⌊lx/⌊ar⌈⌊l2 (1)
+whereα∈(0,1) andD=D(α), called the distortion ofE, are absolute (notably
+dimension-free) constants.Overthe last few years,such “almos t-Euclidean”sub-
+spaces ofℓN
+1have found numerous applications, to high-dimensional nearest
+neighbor search [Ind00], error-correcting codes over reals and c ompressive sens-
+ing [KT07,GLR08,GLW08], vector quantization [LV06], oblivious dimension ality
+⋆This research has been supported in part by David and Lucille Packard Fellowship,
+MADALGO (Center for Massive Data Algorithmics, funded by th e Danish National
+Research Association) and NSF grant CCF-0728645.
+⋆⋆Supported in part by grants from the National Science Founda tion (U.S.A.) and the
+U.S.-Israel BSF.reduction and ǫ-samples for high-dimensional half-spaces [KRS09], and to other
+problems.
+For the above applications, it is convenient and sometimes crucial th at the
+subspaceEis defined in an explicit manner3. However, the aforementioned re-
+sults do not providemuch guidance in this regard,since they use the probabilistic
+method. Specifically, either the vectors spanning E, or the vectors spanning the
+space dual to E, are i.i.d. random variables from some distribution. As a result,
+the constructionsrequire Ω(N2)independent randomvariablesasstartingpoint.
+Until recently, the largest explicitly constructible almost-Euclidean subspace of
+ℓN
+1, due to Rudin [Rud60] (cf. [LLR94]), had only a dimension of Θ(√
+N).
+During the last few years, there has been a renewed interest in the prob-
+lem[AM06,Sza06,Ind07,LS07,GLR08,GLW08],withresearchersusingide asgained
+from the study of expanders, extractorsand error-correctin gcodes to obtain sev-
+eral explicit constructions. The work progressed on two fronts, focusing on (a)
+fully explicit constructions of subspaces attempting to maximize the dimension
+and minimize the distortion [Ind07,GLR08], as well as (b) construction s using
+limited randomness, with dimension and distortion matching (at least q ualita-
+tively)theexistentialdimensionanddistortionbounds[Ind00,AM06,L S07,GLW08].
+The parameters of the constructions are depicted in Figure 1. Qua litatively,
+they show that in the fully explicit case, one can achieve either arbitr arily low
+distortion or arbitrarily high subspace dimension, but not (yet?) bo th. In the
+low-randomness case, one can achieve arbitrarily high subspace dim ension and
+constant distortion while using randomness that is sub-linear in N; achieving
+arbitrarily low distortion was possible as well, albeit at a price of (super )-linear
+randomness.
+Reference Distortion Subspace dimension Randomness
+[Ind07] 1+ǫ N1−oǫ(1)explicit
+[GLR08] (logN)Oη(logloglog N)(1−η)N explicit
+[Ind00] 1+ǫ Ω(ǫ2/log(1/ǫ))NO(Nlog2N)
+[AM06,LS07] Oη(1) (1−η)N O(N)
+[GLW08] 2Oη(1/γ)(1−η)N O(Nγ)
+This paper 1+ǫ (γǫ)O(1/γ)N O(Nγ)
+Fig.1.The best known results for constructing almost-Euclidean s ubspaces of ℓN
+1. The
+parameters ǫ,η,γ∈(0,1) are assumed to be constants, although we explicitly point
+out when the dependence on them is subsumed by the big-Oh nota tion.
+3For the purpose of this paper “explicit” means “the basis of Ecan be generated
+by a deterministic algorithm with running time polynomial i nN.” However, the
+individual constructions can be even “more explicit” than t hat.Our result In this paper we show that, using sub-linear randomness, one can
+constructasubspacewitharbitrarilysmalldistortionwhilekeepingit sdimension
+proportional to N. More precisely, we have:
+Theorem 1 Letǫ,γ∈(0,1). GivenN∈N, assume that we have at our dis-
+posal a sequence of random bits of length max{Nγ,C(ǫ,γ)}log(N/(ǫγ)). Then,
+in deterministic polynomial (in N) time, we can generate numbers M >0,
+m≥c(ǫ,γ)Nand anm-dimensional subspace of ℓN
+1E, for which we have
+∀x∈E,(1−ǫ)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l1≤(1+ǫ)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2
+with probability greater than 98%.
+In a sense, this complements the result of [GLW08], optimizing the dist ortion
+of the subspace at the expense of its dimension. Our approach also allows to
+retrieve – using a simpler and low-tech approach – the results of [GLW 08] (see
+the comments at the end of the Introduction).
+Overview of techniques The ideas behind many of the prior constructions as
+well as this work can be viewed as variants of the related developmen ts in the
+context of error-correcting codes. Specifically, the construct ion of [Ind07] resem-
+bles the approach of amplifying minimum distance of a code using expan ders
+developed in [ABN+92], while the constructions of [GLR08,GLW08] were in-
+spired by low-density parity check codes. The reason for this stat e of affairs is
+that a vector whose ℓ1norms and ℓ2norms are very different must be “well-
+spread”, i.e., a small subset of its coordinates cannot contain most of itsℓ2
+mass (cf. [Ind07,GLR08]). This is akin to a property required from a g ood error-
+correcting code, where the weight (a.k.a. the ℓ0norm) of each codeword cannot
+be concentrated on a small subset of its coordinates.
+In this vein, our construction utilizes a tool frequently used for (lin ear) error-
+correcting codes, namely the tensor product . Recall that, for two linear codes
+C1⊂ {0,1}n1andC2⊂ {0,1}n2, their tensor product is a code C⊂ {0,1}n1n2,
+such that for any codeword c∈C(viewed as an n1×n2matrix), each column of
+cbelongs toC1and each row of cbelongs toC2. It is known that the dimension
+ofCis a product of the dimensions of C1andC2, and that the same holds
+for the minimum distance. This enables constructing a code of “large ” block-
+lengthNkby starting from a code of “small” block-length Nand tensoring it k
+times. Here, we roughly show that the tensor product of two subs paces yields a
+subspace whose distortion is a product of the distortions of the su bspaces. Thus,
+we can randomly choose an initial small low-distortion subspace, and tensor it
+with itself to yield the desired dimension.
+However, tensoring alone does not seem sufficient to give a subspac e with
+distortionarbitrarilycloseto1.Thisisbecausewecanonlyanalyzeth edistortion
+of the product space for the case when the scaling factor Sin Equation 1 is
+equal to 1 (technically, we only prove the left inequality, and rely on t he general
+relation between the ℓ2andℓ1for the upper bound). For S= 1, however, the
+best achievable distortion is strictly greater than 1, and tensoring can make itonly larger. To avoid this problem, instead of the ℓN
+1norm we use the ℓN/B
+1(ℓB
+2)
+norm, for a “small” value of B. The latter norm (say, denoted by /⌊ar⌈⌊l · /⌊ar⌈⌊l) treats
+the vector as a sequence of N/B“blocks” of length B, and returns the sum of
+theℓ2norms of the blocks. We show that there exist subspaces E⊂ℓN/B
+1(ℓB
+2)
+such that for any x∈Ewe have
+1/D·/radicalbig
+N/B/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤/radicalbig
+N/B/⌊ar⌈⌊lx/⌊ar⌈⌊l2
+forDthat is arbitrarily close to 1. Thus, we can construct almost-Euclide an
+subspaces of ℓ1(ℓ2) of desired dimensions using tensoring, and get rid of the
+“inner”ℓ2norm at the end of the process.
+We point out that if we do not insist on distortion arbitrarily close to 1,
+the “blocks” are not needed and the argument simplifies substantia lly. In par-
+ticular, to retrieve the results of [GLW08], it is enough to combine the scalar-
+valued version of Proposition 1 below with “off-the-shelf” random co nstructions
+[Kas77,GG84] yielding – in the notation of Equation 1 – a subspace E, for which
+the parameter αis close to 1.
+2 Tensoring subspaces of L1
+We start by defining some basic notions and notation used in this sect ion.
+Norms and distortion In this section we adopt the “continuous” notation for
+vectorsand norms. Specifically, considera real Hilbert space Hand a probability
+measureµover [0,1]. Forp∈[1,∞] consider the space Lp(H) ofH-valuedp-
+integrable functions fendowed with the norm
+/⌊ar⌈⌊lf/⌊ar⌈⌊lp=/⌊ar⌈⌊lf/⌊ar⌈⌊lLp(H)=/parenleftbigg/integraldisplay
+/⌊ar⌈⌊lf(x)/⌊ar⌈⌊lp
+Hdµ(x)/parenrightbigg1/p
+In what follows we will omit µfrom the formulae since the measure will be
+clear from the context (and largely irrelevant). As our main result c oncerns
+finite dimensional spaces, it suffices to focus on the case where µis simply the
+normalizedcountingmeasureoverthe discreteset {0,1/n,...(n−1)/n}for some
+fixedn∈N(although the statements hold in full generality). In this setting, t he
+functionsffromLp(H) areequivalent to n-dimensional vectorswith coordinates
+inH.4The advantage of using the Lpnorms as opposed to the ℓpnorms that
+the relation between the 1-norm and the 2-norm does not involve sc aling factors
+that depend on dimension, i.e., we have /⌊ar⌈⌊lf/⌊ar⌈⌊l2≥ /⌊ar⌈⌊lf/⌊ar⌈⌊l1for allf∈L2(H) (note
+that, for the Lpnorms, the “trivial” inequality goes in the other direction than
+for theℓpnorms). This simplifies the notation considerably.
+4The values from Hroughly correspond to the finite-dimensional “blocks” in th e
+construction sketched in the introduction. Note that Hcan be discretized similarly
+as theLp-spaces; alternatively, functions that are constant on int ervals of the type/parenleftBig
+(k−1)/N,k/N/parenrightBig
+can be considered in lieu of discrete measures.We will be interested in lialmost subspaces E⊂L2(H) on which the 1-norm
+and 2-norm uniformly agree, i.e., for some c∈(0,1],
+/⌊ar⌈⌊lf/⌊ar⌈⌊l2≥ /⌊ar⌈⌊lf/⌊ar⌈⌊l1≥c/⌊ar⌈⌊lf/⌊ar⌈⌊l2 (2)
+for allf∈E. The best (the largest) constant cthat works in (2) will be denoted
+Λ1(E). For completeness, we also define Λ1(E) = 0 if noc>0 works.
+Tensor products IfH,Kare Hilbert spaces, H⊗2Kis their (Hilbertian) tensor
+product, which may be (for example) described by the following prop erty: if (ej)
+is an orthonormal sequence in Hand (fk) is an orthonormal sequence in K,
+then (ej⊗fk) is an orthonormal sequence in H ⊗2K(a basis if ( ej) and (fk)
+were bases). Next, any element of L2(H)⊗ Kis canonically identified with a
+function in the space L2(H ⊗2K); note that such functions are H ⊗K-valued,
+but are defined on the same probability space as their counterpart s fromL2(H).
+IfE⊂L2(H) is a linear subspace, E⊗Kis – under this identification – a linear
+subspace of L2(H⊗2K).
+As hinted in the Introduction, our argument depends (roughly) on the fact
+that the property expressed by (1) or (2) “passes” to tensor p roducts of sub-
+spaces, and that it “survives” replacing scalar-valued functions b y ones that
+have values in a Hilbert space. Statements to similar effect of various degrees
+of generality and precision are widely available in the mathematical liter ature,
+see for example [MZ39,Bec75,And80,FJ80]. However, we are not awar e of a ref-
+erence that subsumes all the facts needed here and so we presen t an elementary
+self-contained proof.
+We start with two preliminary lemmas.
+Lemma 1 Ifg1,g2,...∈E⊂L2(H), then
+/integraldisplay/parenleftbig/summationdisplay
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+H/parenrightbig1/2dx≥Λ1(E)/parenleftBig/integraldisplay/summationdisplay
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+Hdx/parenrightBig1/2
+.
+ProofLetKbe an auxiliary Hilbert space and ( ek) an orthonormal sequence
+(O.N.S.) in K. We will apply Minkowski inequality – a continuous version of
+the triangle inequality, which says that for vector valued functions /⌊ar⌈⌊l/integraltext
+h/⌊ar⌈⌊l ≤/integraltext/⌊ar⌈⌊lh/⌊ar⌈⌊l– to the K-valued function h(x) =/summationtext
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊lHek. As is easily seen,
+/⌊ar⌈⌊l/integraltext
+h/⌊ar⌈⌊lK=/⌊ar⌈⌊l/summationtext
+k/parenleftbig/integraltext
+/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊lHdx/parenrightbig
+ek/⌊ar⌈⌊lK=/parenleftbig/summationtext
+k/⌊ar⌈⌊lgk/⌊ar⌈⌊l2
+L1(H)/parenrightbig1/2. Given that gk∈
+E,/⌊ar⌈⌊lgk/⌊ar⌈⌊lL1(H)≥Λ1(E)/⌊ar⌈⌊lgk/⌊ar⌈⌊lL2(H)and so
+/vextenddouble/vextenddouble/vextenddouble/integraldisplay
+h/vextenddouble/vextenddouble/vextenddouble
+K≥Λ1(E)/parenleftBig/integraldisplay/summationdisplay
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+Hdx/parenrightBig1/2
+On the other hand, the left hand side of the inequality in Lemma 1 is exa ctly/integraltext
+/⌊ar⌈⌊lh/⌊ar⌈⌊lK, so the Minkowski inequality yields the required estimate.
+We are now ready to state the next lemma. Recall that Eis a linear subspace
+ofL2(H), andKis a Hilbert space.Lemma 2 Λ1(E⊗K) =Λ1(E)
+IfE⊂L2=L2(R), the lemma says that any estimate of type (2) for scalar
+functionsf∈Ecarries overto their linear combinations with vector coefficients,
+namely to functions of the type/summationtext
+jvjfj,fj∈E,vj∈ K. In the general case,
+any estimate for H-valued functions f∈E⊂L2(H) carries over to functions of
+the form/summationtext
+jfj⊗vj∈L2(H⊗2K), withfj∈E,vj∈ K.
+Proof of Lemma 2 Let (ek) be an orthonormalbasis of K. In fact w.l.o.g. we may
+assume that K=ℓ2and that (ek) is the canonical orthonormal basis. Consider
+g=/summationtext
+jfj⊗vj, wherefj∈Eandvj∈ K. Then also g=/summationtext
+kgk⊗ekfor some
+gk∈Eand hence (pointwise) /⌊ar⌈⌊lg(x)/⌊ar⌈⌊lH⊗2K=/parenleftbig/summationtext
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+H/parenrightbig1/2. Accordingly,
+/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K)=/parenleftbig/integraltext/summationtext
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+Hdx/parenrightbig1/2,while/⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)=/integraltext/parenleftbig/summationtext
+k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
+H/parenrightbig1/2dx.
+Comparing such quantities is exactly the object of Lemma 1, which imp lies that
+/⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)≥Λ1(E)/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K).Sinceg∈E⊗Kwasarbitrary,it follows that
+Λ1(E⊗K)≥Λ1(E). The reverse inequality is automatic (except in the trivial
+case dim K= 0, which we will ignore).
+IfE⊂L2(H) andF⊂L2(K) are subspaces, E⊗Fis the subspace of
+L2(H ⊗2K) spanned by f⊗gwithf∈E,g∈F. (For clarity, f⊗gis a
+function on the product of the underlying probability spaces and is defined by
+(x,y)→f(x)⊗g(y)∈ H⊗K .)
+The next proposition shows the key property of tensoring almost- Euclidean
+spaces.
+Proposition 1. Λ1(E⊗F)≥Λ1(E)Λ1(F)
+ProofLet (ϕj) and (ψk) be orthonormal bases of respectively EandFand let
+g=/summationtext
+j,ktjkϕj⊗ψk.Weneedtoshowthat /⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)≥Λ1(E)Λ1(F)/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K),
+where thep-norms refer to the product probability space, for example
+/⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)=/integraldisplay /integraldisplay/vextenddouble/vextenddouble/summationdisplay
+j,ktjkϕj(x)⊗ψk(y)/vextenddouble/vextenddouble
+H⊗2Kdxdy.
+Rewriting the expression under the sum and subsequently applying L emma 2 to
+the inner integral for fixed ygives
+/integraldisplay/vextenddouble/vextenddouble/summationdisplay
+j,ktjkϕj(x)⊗ψk(y)/vextenddouble/vextenddouble
+H⊗2Kdx=/integraldisplay/vextenddouble/vextenddouble/summationdisplay
+jϕj(x)⊗/parenleftBig/summationdisplay
+ktjkψk(y)/parenrightBig/vextenddouble/vextenddouble
+H⊗2Kdx
+≥Λ1(E)/parenleftBig/integraldisplay/vextenddouble/vextenddouble/summationdisplay
+jϕj(x)⊗/parenleftBig/summationdisplay
+ktjkψk(y)/parenrightBig/vextenddouble/vextenddouble2
+H⊗2Kdx/parenrightBig1/2
+=Λ1(E)/parenleftBig/summationdisplay
+j/vextenddouble/vextenddouble/summationdisplay
+ktjkψk(y)/vextenddouble/vextenddouble2
+K/parenrightBig1/2In turn,/summationtext
+ktjkψk∈F(for allj) and so, by Lemma 1,
+/integraldisplay/parenleftBig/summationdisplay
+j/vextenddouble/vextenddouble/summationdisplay
+ktjkψk(y)/vextenddouble/vextenddouble2
+K/parenrightBig1/2
+dy≥Λ1(F)/parenleftBig/integraldisplay/summationdisplay
+j/vextenddouble/vextenddouble/summationdisplay
+ktjkψk(y)/vextenddouble/vextenddouble2
+Kdy/parenrightBig1/2
+=Λ1(F)/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K).
+Combining the above formulae yields the conclusion of the Proposition .
+3 The construction
+In this section we describe our low-randomness construction. We s tart from a
+recap of the probabilistic construction, since we use it as a building blo ck.
+3.1 Dvoretzky’s theorem, and its “tangible” version
+For general normed spaces, the following is one possible statement of the well-
+known Dvoretzky’s theorem [Dvo61]:
+Givenm∈Nandε>0there isN=N(m,ε)such that, for any norm on RN
+there is an m-dimensional subspace on which the ratio of ℓ1andℓ2norms is
+(approximately) constant, up to a multiplicative factor 1+ε.
+For specific norms this statement can be made more precise, both in describing
+the dependence N=N(m,ε) and in identifying the constant of (approximate)
+proportionality of norms. The following version is (essentially) due to Milman
+[Mil71].
+Dvoretzky’s theorem (Tangible version) Consider the N-dimensional Eu-
+clidean space (real orcomplex) endowed with the Euclidean norm /⌊ar⌈⌊l·/⌊ar⌈⌊l2and some
+other norm /⌊ar⌈⌊l·/⌊ar⌈⌊lsuch that, for some b>0,/⌊ar⌈⌊l·/⌊ar⌈⌊l ≤b/⌊ar⌈⌊l·/⌊ar⌈⌊l2. LetM=E/⌊ar⌈⌊lX/⌊ar⌈⌊l, whereX
+is a random variable uniformly distributed on the unit Euclidean sphere . Then
+there exists a computable universal constant c >0, so that if 0< ε <1and
+m≤cε2(M/b)2N, then for more than 99% (with respect to the Haar measure)
+m-dimensional subspaces Ewe have
+∀x∈E,(1−ε)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤(1+ε)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2. (3)
+Alternative good expositions of the theorem are in, e.g., [FLM77], [MS8 6] and
+[Pis89]. We point out that standard and most elementary proofs yield m≤
+cε2/log(1/ε)(M/b)2N; the dependence on εof orderε2was obtained in the
+important papers [Gor85,Sch89], see also [ASW10].
+3.2 The case of ℓn
+1(ℓB
+2)
+OurobjectivenowistoapplyDvoretzky’stheoremandsubsequent lyProposition
+1 to spaces of the form ℓn
+1(ℓB
+2) for some n,B∈N, so from now on we set/⌊ar⌈⌊l·/⌊ar⌈⌊l:=/⌊ar⌈⌊l·/⌊ar⌈⌊lℓn
+1(ℓB
+2)To that end, we need to determine the values of the parameter
+Mthat appears in the theorem. (The optimal value of bis clearly√n, as in
+the scalar case, i.e., when B= 1.) We have the following standard (cf. [Bal97],
+Lecture 9)
+Lemma 3
+M(n,B) :=Ex∈SnB−1/⌊ar⌈⌊lx/⌊ar⌈⌊l=Γ(B+1
+2)
+Γ(B
+2)Γ(nB
+2)
+Γ(nB+1
+2)n.
+In particular,/radicalBig
+1+1
+n−1/radicalBig
+2
+π√n>M(n,1)>/radicalBig
+2
+π√nfor alln∈N(the scalar
+case) andM(n,B)>/radicalBig
+1−1
+B√nfor alln,B∈N.
+The equality is shown by relating (via passing to polar coordinates) sp heri-
+cal averages of norms to Gaussian means: if Xis a random variable uniformly
+distributed on the Euclidean sphere SN−1andYhas the standard Gaussian
+distribution on RN, then, for any norm /⌊ar⌈⌊l·/⌊ar⌈⌊l,
+E/⌊ar⌈⌊lY/⌊ar⌈⌊l=√
+2Γ(N+1
+2)
+Γ(N
+2)E/⌊ar⌈⌊lX/⌊ar⌈⌊l
+The inequalities follow from the estimates/radicalBig
+x−1
+2<Γ(x+1
+2)
+Γ(x)<√x(forx≥1
+2),
+which in turn are consequences of log-convexity of Γand its functional equation
+Γ(y+1) =yΓ(y). (Alternatively, Stirling’s formula may be used to arrive at a
+similar conclusion.)
+Combining Dvoretzky’s theorem with Lemma 3 yields
+Corollary 1 If0< ε <1andm≤c1ε2n, then for more than 99%of the
+m-dimensional subspaces E⊂ℓn
+1we have
+∀x∈E(1−ε)/radicalbigg
+2
+π√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l1≤(1+ε)/radicalbigg
+1+1
+n−1/radicalbigg
+2
+π√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2(4)
+Similarly, if B >1andm≤c2ε2nB, then for more than 99%of them-
+dimensional subspaces E⊂ℓn
+1(ℓB
+2)we have
+∀x∈E(1−ε)/radicalbigg
+1−1
+B√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2 (5)
+We point out that the upper estimate on /⌊ar⌈⌊lx/⌊ar⌈⌊lin the second inequality is valid
+for allx∈ℓn
+1(ℓB
+2) and, like the estimate M(n,B)≤√n, follows just from the
+Cauchy-Schwarz inequality.
+Since a random subspace chosen uniformly according to the Haar me asure
+on the manifold of m-dimensional subspaces of RN(orCN) can be constructed
+from anN×mrandom Gaussian matrix, we may apply standard discretization
+techniques to obtain the followingCorollary 2 There is a deterministic algorithm that, given ε,B,m,n as in
+Corollary 1 and a sequence of O(mnlog(mn/ǫ))random bits, generates sub-
+spacesEas in Corollary 1 with probability greater than 98%, in time polynomial
+in1/ε+B+m+n.
+We point out that in the literature on the “randomness-reduction” , one typ-
+ically uses Bernoulli matrices in lieu of Gaussian ones. This enables avoid ing the
+discretization issue, since the problem is phrased directly in terms of random
+bits. Still, since proofs of Dvoretzky type theorems for Bernoulli m atrices are
+often much harder than for their Gaussian counterparts, we pre fer to appeal in-
+stead to a simple discretization ofGaussian random variables.We not e, however,
+that the early approach of [Kas77] was based on Bernoulli matrices .
+We are now ready to conclude the proof of Theorem 1. Given ε∈(0,1)
+andn∈N, chooseB=⌈ε−1⌉andm=⌊cε2(1−1
+B)nB⌋ ≥c0ε2nB. Corollary
+2 (Equation 5) and repeated application of Proposition 1 give us a sub space
+F⊂ℓν
+1(ℓβ
+2) (whereν=nkandβ=Bk) of dimension mk≥(c0ε2)kνβsuch that
+∀x∈F(1−ε)3k/2nk/2/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤nk/2/⌊ar⌈⌊lx/⌊ar⌈⌊l2.
+Moreover,F=E⊗E⊗...⊗E, whereE⊂ℓn
+1(ℓB
+2) is a typical m-dimensional
+subspace.Thusin ordertoproduce E, henceF,weonlyneed togeneratea“typi-
+cal”m≈c0ε2(νβ))1/ksubspace of the nB= (νβ))1/k-dimensional space ℓn
+1(ℓB
+2).
+Note that for fixed εandk >1,nBandmare asymptotically (substantially)
+smaller than dim F. Further, in order to efficiently represent Fas a subspace of
+anℓ1-space, we only need to find a good embedding of ℓβ
+2intoℓ1. This can be
+done using Corollary 2 (Equation 4); note that βdepends only on εandk. Thus
+we reduced the problem of finding “large” almost Euclidean subspace s ofℓN
+1to
+similar problems for much smaller dimensions.
+Theorem 1 now follows from the above discussion. The argument give s, e.g.,
+c(ε,γ) = (cεγ)3/γandC(ε,γ) =c(ε,γ)−1.
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+arXiv:1001.0042v2  [hep-th]  16 Apr 2010Topological Gravity in Seven Dimensions
+H. L¨ u†‡and Yi Pang⋆
+†China Economics and Management Academy
+Central University of Finance and Economics, Beijing 100081
+‡Institute for Advanced Study, Shenzhen University, Nanhai A ve 3688, Shenzhen 518060
+⋆Key Laboratory of Frontiers in Theoretical Physics
+Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing 100190
+ABSTRACT
+We obtain new topological gravity in seven dimensions by add ing two topological terms
+to the Einstein-Hilbert action. For certain choice of the co upling constants, these terms
+may have an origin as the R4correction to the 3-form field equation of eleven-dimension al
+supergravity. We derive the full set of the equations of moti on, and obtain large classes of
+solutions including static AdS black holes, squashed seven spheres and Q111spaces.1 Introduction
+There has been considerable interest in topological gauge t heories [1] because of their wide
+application in physics. The most studied example is the thre e-dimensional one. In addition
+to the Einstein-Hilbert term, the theory has the Chern-Simo ns term, given by
+S=1
+µ/integraldisplay
+d3xTr(dω∧ω−2
+3ω∧ω∧ω), (1)
+whereωcan be either a Yang-Mills gauge potential or the connection for gravity. Topo-
+logical Yang-Mills theory can provide a fundamental interp retation for anyons [2]; it can
+also generate Lorentz violation dynamically [3]. Topologi cal gravity [4] becomes dynamical
+with a propagating massive particle, with the mass proporti onal to the coupling constant
+µ. Recently, a cosmological constant is added and the corresp onding boundary conformal
+field theory (CFT) is discussed [5]. The three-dimensional m assive topological gravity is
+conjectured to be unitary for certain parameter region even though the theory has higher
+derivatives in time [6].
+The attention on higher dimensional generalizations is con siderably less. The five di-
+mensional Yang-Mills Chern-Simons term was discussed in [7 ], but there is no gravity coun-
+terpart dueto the fact that the holonomy group SO(1,4) has no invariant rank-3 symmetric
+tensor. In seven dimensions, Yang-Mills Chern-Simons term s arise naturally from N= 4
+supergravity [8]. As in the case of three dimensions, we find t hat such terms in the grav-
+ity sector can be obtained directly from those in the Yang-Mi lls sector by replacing the
+gauge potential Ato the connection Γ. Moreover, as we shall see later, seven-d imensional
+topological gravity has a direct origin in eleven-dimensio nal supergravity, while any higher-
+dimensional origin of the three-dimensional theory remain s unknown.
+In section 2, we present the two topological terms in seven di mensions, and discuss their
+properties. Sincethey arenot manifestly invariant underg eneral coordinate transformation,
+we find it is more convenient to lift the system to eight dimens ions in order to derive the
+equations of motion (EOMs). We obtain the full set. In sectio n 3, we construct large
+classes of solutions including static Anti-de Sitter (AdS) black holes, squashed S7andQ111.
+We emphasize that all the previously-known static (AdS) bla ck holes remain to be solutions
+whenthetopological termsare addedinto theaction. Thisis analogous to threedimensions,
+where the BTZ black hole remains to be a solution in topologic al massive gravity. We
+conclude in section 4.
+22 The theory
+In seven dimensions, there are two topological terms; they a re given by
+S1=µ/integraldisplay
+Ω(7)
+1=µ/integraldisplay
+Tr(Γ∧Θ+1
+3Γ3)∧Tr(Θ2) =µ/integraldisplay
+Ω(3)∧dΩ(3), (2)
+S2=ν/integraldisplay
+Ω(7)
+2=ν/integraldisplay
+Tr(Θ3∧Γ+2
+5Θ2∧Γ3+1
+5Θ∧Γ2∧Θ∧Γ+1
+5Θ∧Γ5+1
+35Γ7),
+with Ω(3)= Tr(dΓ∧Γ−2
+3Γ3). Here, Θ is the curvature 2-form, defined as Θ ≡dΓ−Γ∧Γ,
+andµ,νare two parameters of length dimension 5. (We rescale the tot al action by the
+seven-dimensional Newton constant.) The 3-form Ω(3)has the same structure as the Chern-
+Simons term in D= 3, except that now Γ depends on seven coordinates. Ω(7)
+1and Ω(7)
+2are
+topological in the same sense as Ω(3)being topological in D= 3. We can lift the system to
+D= 8, with the seven-dimensional spacetime as the boundary. T hen, we have
+dΩ(7)
+1=Y(8)
+1≡Tr(Θ∧Θ)∧Tr(Θ∧Θ), dΩ(7)
+2=Y(8)
+2≡Tr(Θ∧Θ∧Θ∧Θ).(3)
+As we have mentioned earlier, these terms can be derived from the Yang-Mills Chern-
+Simons terms in [8] by changing the gauge potential to the con nection.1Note that the
+Pontryagin term is proportional to Y(8)
+1−2Y(8)
+2, corresponding to ν=−2µ. In eleven-
+dimensional supergravity, there is an R4correction to the field equation, namely d∗F(4)=
+1
+2F(4)∧F(4)+X(8), whereX(8)is given by
+X(8)∝Y(8)
+1−4Y(8)
+2. (4)
+Thus forν=−4µ, the topological terms can be obtained from the S4reduction of super-
+gravity inD= 11, and the coupling constant is proportional to the 4-form M5-brane fluxes.
+For large fluxes, this topological term dominates the higher -order corrections.
+To derive the contribution to the EOMs from the Chern-Simons terms, it is necessary
+to perform their variation with respect to the metric. These topological terms are not
+manifestly invariant under the general coordinate transfo rmation, but Y(8)
+1andY(8)
+2are.
+Wefindthataconvenient waytoderivethevariationistolift thesystemtoeightdimensions.
+Let us first consider the variation of S1. In terms of coordinate components, we have
+/integraldisplay
+dΩ(7)
+1=1
+16/integraldisplay
+d8xǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6Rµ4µ3ν7ν8. (5)
+1In [8], the field strength 2-form is defined by F=dB+gB∧B, with gauge coupling g= 2. Then by
+rescaling the field B→B/gandF→F/gand setting g=−2, one can obtain the same expressions as the
+ones given here.
+3Here we use Greek letters to denote the eight-dimensional co ordinates and Latin letters to
+represent the seven-dimensional ones hereafter. We adopt t he convention ǫ12345678= 1.
+For an infinitesimal variation of the metric δg, using the Bianchi identity and the fol-
+lowing relation
+δRµ
+ναβ=δΓµ
+νβ;α−δΓµ
+να;β, (6)
+we find that
+/integraldisplay
+dδΩ(7)
+1=−1
+2/integraldisplay
+d8x√g/parenleftBig1√gǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7/parenrightBig
+;ν8
+≡1
+2/integraldisplay
+d∗J, (7)
+where “;” denotes a covariant derivative and ∗is the Hodge dual. For simplicity, we have
+introduced a 1-form current J=Jαdxα. Its components are given by
+Jα=1√gǫν1ν2ν3ν4ν5ν6ν7αRµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7. (8)
+Clearly, we have d∗J=−√gJα;αd8x, Thus we obtain
+δΩ(7)
+1=1
+2∗J, (9)
+up to a total derivative term. Now restricting the coordinat e indices to seven dimensions
+only, we have
+δS1= 4µ/integraldisplay
+Tr(Θ∧Θ)∧Tr(Θ∧δΓ). (10)
+The variation of S2can be obtained in the same manner, given by
+δS2= 4ν/integraldisplay
+Tr(Θ∧Θ∧Θ∧δΓ). (11)
+Finally, we make use of the variation of the connection
+δΓi
+mj=1
+2gin(δgnm;j+δgnj;m−δgml;n), (12)
+and after integrating by parts, we obtain thecontributions to EOMs from the Chern-Simons
+terms, given by
+Cij
+1=δS1√gδgij=µ
+4√g[ǫij1j2j3j4j5j6(Ri1
+i2j1j2Ri2
+i1j3j4Rjk
+j5j6);k+i↔j],
+Cij
+2=δS2√gδgij=ν
+4√g[ǫij1j2j3j4j5j6(Rk
+i1j1j2Ri1
+i2j3j4Rji2
+j5j6);k+i↔j].(13)
+For the total action S, which is the sum of the Einstein-Hilbert action, cosmologi cal
+constant Λ and S1+S2, the corresponding full set of EOMs is given by
+Rij−1
+2gijR+Λgij+Cij
+1+Cij
+2= 0. (14)
+4It should be remarked that under a large gauge transformatio n Γ→ O−1ΓO−O−1dO,
+the action transforms as S→S+µv(O)+νw(O), where
+v(O) =/integraldisplay
+−1
+3d/parenleftBig
+Tr(O−1dO)3∧Ω(3)/parenrightBig
+;w(O) =−1
+35/integraldisplay
+Tr(O−1dO)7.(15)
+Thevterm is trivial and gives no restriction to the parameter µ, while the wterm should
+be classified by the seventh homotopy group of SO(1,6)
+π7[SO(1,6)]≃π7[SO(6)]≃Z. (16)
+The invariance of eiSrequires that
+ν= 2πn, n = 0,±1,±2.... (17)
+This quantization condition is clearly consistent with the M5-brane quantization, since
+it has a direct origin in D= 11. This result is completely different from that in three
+dimensions, where the SO(1,2) is homotoplically trivial and the mass parameter is not
+quantized. Moreover, since νis quantized, S2will not be renormalized in the quantum
+theory. This suggests some intriguing properties in the cor responding CFT dual.
+3 Solutions
+Spherically-symmetric solutions:
+Having obtained the full set of EOMs for topological gravity in seven dimensions, we are
+in the position to construct solutions. It is clear that the m aximally-symmetric space(time)
+is unmodifiedby theinclusion of the topological terms. Then extsimplest case is to consider
+the spherically-symmetric ansatz, given by
+ds2=−F(r)dt2+dr2
+G(r)+r2dΩ2
+5. (18)
+We find that for this ansatz, the contributions from the topol ogical terms Cij
+1andCij
+2
+vanish identically. This implies that the previously-know n static (AdS) black holes, charged
+or neutral, are still solutions when the topological terms a re added to the action. This is
+analogous to three dimensions, where the BTZ black hole is st ill a solution in massive
+topological gravity. However the thermodynamic quantitie s such as the mass and entropy
+will acquire modifications [9, 10].
+S3bundle over S4:
+5We now turn our attention to the Euclidean theory. In three di mensions, there exists a
+large class of squashed S3or AdS 3[11]. We expect the same in seven dimensions. Without
+loss of generality, we set Λ = 30 so that it can give rise to a uni t roundS7. We first consider
+the squashed S7that can be viewed as an S3bundle over S4. The metric ansatz is given
+by
+ds2=α3/summationdisplay
+i=1(σi−cos2(1
+2θ)˜σi)2+β/parenleftBig
+dθ2+1
+4sin2θ3/summationdisplay
+i=1˜σ2
+i/parenrightBig
+. (19)
+whereσiand ˜σiare theSU(2) left-invariant 1-forms, satisfying dσi=1
+2ǫijkσj∧σkand
+d˜σi=1
+2ǫijk˜σj∧˜σk. The metric is Einstein provided that either α=β=1
+4orα=1
+5β=9
+100.
+The first case corresponds to the round S7and the second is a squashed S7that is also
+Einstein. Now with the contribution from the topological te rms, the EOMs can be reduced
+to
+2α2+4αβ(7β−2)−β2= 0, (20)
+together with
+√α(α−β)3(4(10α+β)µ−(55α+7β)ν)+2β6(20αβ−4α−β) = 0.(21)
+It is clear from (20) that there exists one and only one positi veαfor any positive β. The
+squashing parameter γ≡α/βlies in the range 0 <γ <2+3√
+2. Note that when 2 µ= 3ν,
+the squashed S7that is Eisntein remains Einstein.
+S1bundle over CP3:
+There is another way of squashing an S7, which can be viewed as an S1bundle over
+CP3. The metric ansatz is given by
+ds2=α(dτ+sin2θ(dψ+B))2+βds2
+CP3,
+ds2
+CP3=dθ2+sin2θcos2θ(dψ+B)2+sin2θ/parenleftBig
+d˜θ2+1
+4sin2˜θcos2˜θσ2
+3
++1
+4sin2˜θ(σ2
+1+σ2
+2)/parenrightBig
+,
+B=1
+2sin2˜θσ3. (22)
+It is of a round S7whenα=β= 1. In general, the EOMs imply that
+α=β(8−7β),8µ+ν+β3
+10976(β−1)2√α= 0. (23)
+The squashing parameter γ≡α/βlies in the range (0 ,8).
+Squashed Q111spaces:
+6TheQ111space is an Einstein-Sasaki space of U(1) bundle over S2×S2×S2. We
+consider the following ansatz
+ds2=α/parenleftBig
+dψ+3/summationdisplay
+i=1cosθidφi/parenrightBig2
++β3/summationdisplay
+i=1(dθ2
+i+sinθ2
+idφ2
+i). (24)
+It is ofQ111provided that α=1
+2β= 1/16, and it remains so for ν= 0. In general, we have
+α= 4β(1−7β),8(α−β)(2α−β)µ+α(2α−3β)ν+β5(α−8β+60β2)
+4α3/2= 0.(25)
+Thus the squashing parameter γ≡α/βlies in the range (0 ,4). We expect that many of
+the squashed homogeneous spaces in seven dimensions are now solutions in this new gravity
+theory, and we shall not enumerate them further.
+4 Conclusions
+This work is motivated by studying the classical solutions o f Einstein-Chern-Simons gravity
+with asymptotic AdS structure. In seven dimensions, there a re two topological Chern-
+Simons terms, and we obtain the full set of equations of motio n. We find that spherically-
+symmetric solutions are unmodified by the inclusion of these topological terms. We also
+obtain squashed S7andQ111spaces, where the squashing parameter is related to the cou-
+pling constants of the topological terms. It is intriguing t o see that these known squashed
+homogeneous spaces which appear to have no connection can no w be unified under our new
+gravity theory.
+As in three dimensions, our topological gravity should play an important role in explor-
+ing the AdS 7/CFT6correspondence. The CFT 6that describes the world-volume theory of
+multiple M5-branes is yet to be known, and our solutions prov ide many new gravity dual
+backgrounds. The quantization condition for one of the coup ling constant suggests an un-
+usual property of the CFT 6that is absent in lower dimensions. Additional future direc tions
+include a classification of all topological gravities in (4 k+3) dimensions, investigating the
+linearization of D= 7topological gravity and obtaining the propagating degre es of freedom.
+Acknowledgement
+We are grateful to Chris Pope for useful discussions. Y.P. is supported in part by the NSFC
+grant No.1053060/A050207 and the NSFC group grant No.10821 504.
+7References
+[1] S. Deser, R. Jackiw and S. Templeton, “Topologically mas sive gauge theories,” An-
+nals Phys. 140, 372 (1982) [Erratum-ibid. 185, 406.1988 APNYA,281,409 (1988
+APNYA,281,409-449.2000)].
+[2] F. Wilczek and A. Zee, “Linking numbers, spin, and statis tics of solitons,” Phys. Rev.
+Lett.51, 2250 (1983).
+[3] S.M. Carroll, G.B. Field and R. Jackiw, “Limits on a Loren tz and parity violating
+modification of electrodynamics,” Phys. Rev. D 41, 1231 (1990).
+[4] S. Deser, R. Jackiw and G. ’t Hooft, “Three-Dimensional E instein gravity: dynamics
+of flat space,” Annals Phys. 152, 220 (1984).
+[5] E. Witten, “Three-dimensional gravity revisited,” arX iv:0706.3359 [hep-th].
+[6] W. Li, W. Song and A. Strominger, “Chiral gravity in three dimensions,” JHEP 0804,
+082 (2008) [arXiv:0801.4566 [hep-th]]. E.A. Bergshoeff, O. H ohm and P.K. Townsend,
+“Massive gravity in three dimensions,” Phys. Rev. Lett. 102, 201301 (2009)
+[arXiv:0901.1766 [hep-th]].
+[7] M. Gunaydin, G. Sierra and P.K. Townsend, “Quantization of the gauge coupling
+constant in a five-dimensional Yang-Mills/Einstein superg ravity theory,” Phys. Rev.
+Lett.53, 322 (1984).
+[8] M. Pernici, K. Pilch and P. van Nieuwenhuizen, “Gauged ma ximally extended super-
+gravity in seven-dimensions,” Phys. Lett. B 143, 103 (1984).
+[9] S. Deser and B. Tekin, “Energy in topologically massive g ravity,” Class. Quant. Grav.
+20, L259 (2003) [arXiv:gr-qc/0307073].
+[10] Y. Tachikawa, “Black hole entropy in the presence of Che rn-Simons terms,” Class.
+Quant. Grav. 24, 737 (2007) [arXiv:hep-th/0611141].
+[11] D.D.K. Chow, C.N. Pope and E. Sezgin, “Exact solutions o f topologically massive
+gravity,” arXiv:0906.3559 [hep-th].
+8arXiv:1001.0042v2  [hep-th]  16 Apr 2010Seven-Dimensional Gravity with Topological Terms
+H. L¨ u†‡and Yi Pang⋆
+†China Economics and Management Academy
+Central University of Finance and Economics, Beijing 100081
+‡Institute for Advanced Study, Shenzhen University, Nanhai A ve 3688, Shenzhen 518060
+⋆Key Laboratory of Frontiers in Theoretical Physics
+Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing 100190
+ABSTRACT
+We construct new seven-dimensional gravity by adding two to pological terms to the
+Einstein-Hilbert action. For certain choice of the couplin g constants, these terms may be
+related to the R4correction to the 3-form field equation of eleven-dimension al supergravity.
+We derive the full set of the equations of motion. We find that t he static spherically-
+symmetric black holes are unmodified by the topological term s. We obtain squashed AdS 7,
+and also squashed seven spheres and Q111spaces in Euclidean signature.1 Introduction
+There has been considerable interest in topological gauge t heories [1] because of their wide
+application in physics. The most studied example is the thre e-dimensional one. In addition
+to the Einstein-Hilbert term, the theory has the Chern-Simo ns term, given by
+S=1
+µ/integraldisplay
+d3xTr(dω∧ω+2
+3ω∧ω∧ω), (1)
+whereωcan be either a Yang-Mills gauge potential or the connection for gravity. Topo-
+logical Yang-Mills theory can provide a fundamental interp retation for anyons [2]; it can
+also generate Lorentz violation dynamically [3]. Topologi cally massive gravity [4] becomes
+dynamical with a propagating massive particle, with the mas s proportional to the coupling
+constantµ. Recently, a cosmological constant is added and the corresp onding boundary
+conformal field theory (CFT) is discussed [5]. The three-dim ensional massive topological
+gravity is conjectured to be unitary for certain parameter r egion even though the theory
+has higher derivatives in time [6].
+The attention on higher dimensional generalizations is con siderably less. The five di-
+mensional Yang-Mills Chern-Simons term was discussed in [7 ], but there is no gravity coun-
+terpart dueto the fact that the holonomy group SO(1,4) has no invariant rank-3 symmetric
+tensor. In seven dimensions, Yang-Mills Chern-Simons term s arise naturally from N= 4
+supergravity [8]. As in the case of three dimensions, we find t hat such terms in the grav-
+ity sector can be obtained directly from those in the Yang-Mi lls sector by replacing the
+gauge potential Ato the connection Γ. As we shall see later, these topological terms in
+seven dimensions may be related to the anomaly cancelation t erms in eleven-dimensional
+supergravity.
+In section 2, we present the two topological terms in seven di mensions, and discuss their
+properties. Sincethey arenot manifestly invariant underg eneral coordinate transformation,
+we find it is more convenient to lift the system to eight dimens ions in order to derive the
+equations of motion (EOMs). We obtain the full set. In sectio n 3, we construct large classes
+of solutions. We find that the static spherically-symmetric black holes are unmodified by
+the topological terms. This is analogous to three dimension s, where the BTZ black hole
+remains to be a solution in topologically massive gravity. I n Euclidean signature, we obtain
+squashedS7andQ111spaces. In particular, one of the squashed seven sphere can b e Wick
+rotated to become squashed AdS 7. We conclude in section 4.
+22 The theory
+In seven dimensions, there are two topological terms; they a re given by
+S1= ˜µ/integraldisplay
+Ω(7)
+1= ˜µ/integraldisplay
+Tr(Γ∧Θ−1
+3Γ3)∧Tr(Θ2) = ˜µ/integraldisplay
+Ω(3)∧dΩ(3), (2)
+S2= ˜ν/integraldisplay
+Ω(7)
+2= ˜ν/integraldisplay
+Tr(Θ3∧Γ−2
+5Θ2∧Γ3−1
+5Θ∧Γ2∧Θ∧Γ+1
+5Θ∧Γ5−1
+35Γ7),
+with Ω(3)= Tr(dΓ∧Γ+2
+3Γ3). Here, Θ is the curvature 2-form, defined as Θ ≡dΓ+Γ∧Γ,
+and ˜µ,˜νare two parameters of length dimension 5. (We rescale the tot al action by the
+seven-dimensional Newton constant.) The 3-form Ω(3)has the same structure as the Chern-
+Simons term in D= 3, except that now Γ depends on seven coordinates. Ω(7)
+1and Ω(7)
+2are
+topological in the same sense as Ω(3)being topological in D= 3. We can lift the system to
+D= 8, with the seven-dimensional spacetime as the boundary. T hen, we have
+dΩ(7)
+1=Y(8)
+1≡Tr(Θ∧Θ)∧Tr(Θ∧Θ), dΩ(7)
+2=Y(8)
+2≡Tr(Θ∧Θ∧Θ∧Θ).(3)
+As we have mentioned earlier, these terms can be derived from the Yang-Mills Chern-
+Simons terms in [8] by changing the gauge potential to the con nection.1Note that the
+Pontryagin term is proportional to Y(8)
+1−2Y(8)
+2, corresponding to ˜ ν=−2˜µ. In eleven-
+dimensional supergravity, there is an R4correction to the field equation, namely d∗F(4)=
+1
+2F(4)∧F(4)+X(8), whereX(8)is given by
+X(8)∝Y(8)
+1−4Y(8)
+2. (4)
+Thus for ˜ν=−4˜µ, the topological terms can be obtained from the S4reduction of super-
+gravity inD= 11, and the coupling constant is proportional to the 4-form M5-brane fluxes.
+For large fluxes, this topological term dominates the higher -order corrections.
+To derive the contribution to the EOMs from the Chern-Simons terms, it is necessary
+to perform their variation with respect to the metric. These topological terms are not
+manifestly invariant under the general coordinate transfo rmation, but Y(8)
+1andY(8)
+2are.
+Wefindthataconvenient waytoderivethevariationistolift thesystemtoeightdimensions.
+Let us first consider the variation of S1. In terms of coordinate components, we have
+/integraldisplay
+dΩ(7)
+1=1
+16/integraldisplay
+d8xǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6Rµ4µ3ν7ν8. (5)
+1In [8], the field strength 2-form is defined by F=dB+gB∧B, with gauge coupling g= 2. Then by
+rescaling the field B→B/gandF→F/gand setting g= 2, one can obtain the same expressions as the
+ones given here.
+3Here we use Greek letters to denote the eight-dimensional co ordinates and Latin letters to
+represent the seven-dimensional ones hereafter. We adopt t he convention ǫ12345678= 1.
+For an infinitesimal variation of the metric δg, using the Bianchi identity and the fol-
+lowing relation
+δRµ
+ναβ=δΓµ
+νβ;α−δΓµ
+να;β, (6)
+we find that
+/integraldisplay
+dδΩ(7)
+1=−1
+2/integraldisplay
+d8x√g/parenleftBig1√gǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7/parenrightBig
+;ν8
+≡1
+2/integraldisplay
+d∗J, (7)
+where “;” denotes a covariant derivative and ∗is the Hodge dual. For simplicity, we have
+introduced a 1-form current J=Jαdxα. Its components are given by
+Jα=1√gǫν1ν2ν3ν4ν5ν6ν7αRµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7. (8)
+Clearly, we have d∗J=−√gJα;αd8x, Thus we obtain
+δΩ(7)
+1=1
+2∗J, (9)
+up to a total derivative term. Now restricting the coordinat e indices to seven dimensions
+only, we have
+δS1= 4˜µ/integraldisplay
+Tr(Θ∧Θ)∧Tr(Θ∧δΓ). (10)
+The variation of S2can be obtained in the same manner, given by
+δS2= 4˜ν/integraldisplay
+Tr(Θ∧Θ∧Θ∧δΓ). (11)
+Finally, we make use of the variation of the connection
+δΓi
+mj=1
+2gin(δgnm;j+δgnj;m−δgml;n), (12)
+and after integrating by parts, we obtain thecontributions to EOMs from the Chern-Simons
+terms, given by
+Cij
+1=δS1√gδgij=µ
+4√g[ǫij1j2j3j4j5j6(Ri1
+i2j1j2Ri2
+i1j3j4Rjk
+j5j6);k+i↔j],
+Cij
+2=δS2√gδgij=ν
+4√g[ǫij1j2j3j4j5j6(Rk
+i1j1j2Ri1
+i2j3j4Rji2
+j5j6);k+i↔j].(13)
+For the total action S, which is the sum of the Einstein-Hilbert action, cosmologi cal
+constant Λ and S1+S2, the corresponding full set of EOMs is given by
+Rij−1
+2gijR+Λgij+Cij
+1+Cij
+2= 0. (14)
+4It should be remarked that under a large gauge transformatio n Γ→ OΓO−1−dOO−1,
+the action transforms as S→S+ ˜µv(O)+ ˜νw(O), where
+v(O) =/integraldisplay
+1
+3d/parenleftBig
+Tr(dOO−1)3∧Ω(3)/parenrightBig
+;w(O) =1
+35/integraldisplay
+Tr(dOO−1)7.(15)
+Thevterm is trivial and gives no restriction to the parameter ˜ µ, while the wterm should
+be classified by the seventh homotopy group of SO(1,6)
+π7[SO(1,6)]≃π7[SO(6)]≃Z. (16)
+The invariance of eiSrequires that
+64π4˜ν= 2πn, n = 0,±1,±2.... (17)
+This result is completely different from that in three dimensi ons, where the SO(1,2) is
+homotoplically trivial and the mass parameter is not quanti zed. Moreover, since ˜ νis quan-
+tized,S2will not be renormalized in the quantum theory. This suggest s some intriguing
+properties in the corresponding CFT dual.
+3 Solutions
+Spherically-symmetric solutions:
+Having obtained the full set of EOMs for topological gravity in seven dimensions, we are
+in the position to construct solutions. It is clear that the m aximally-symmetric space(time)
+is unmodifiedby theinclusion of the topological terms. Then extsimplest case is to consider
+the spherically-symmetric ansatz, given by
+ds2=−F(r)dt2+dr2
+G(r)+r2dΩ2
+5. (18)
+We find that for this ansatz, the contributions from the topol ogical terms Cij
+1andCij
+2
+vanish identically. This implies that the previously-know n static (AdS) black holes, charged
+or neutral, are still solutions when the topological terms a re added to the action. This is
+analogous to three dimensions, where the BTZ black hole is st ill a solution in massive
+topological gravity. However the thermodynamic quantitie s such as the mass and entropy
+will acquire modifications [9, 10].
+As we shall discuss presently, there also exist squashed AdS 7solutions.
+S3bundle over S4:
+5We now turn our attention to the Euclidean theory. In three di mensions, there exists a
+large class of squashed S3or AdS 3[11]. We expect the same in seven dimensions. Without
+loss of generality, we set Λ = 30 so that it can give rise to a uni t roundS7. We first consider
+the squashed S7that can be viewed as an S3bundle over S4. The metric ansatz is given
+by
+ds2=α3/summationdisplay
+i=1(σi−cos2(1
+2θ)˜σi)2+β/parenleftBig
+dθ2+1
+4sin2θ3/summationdisplay
+i=1˜σ2
+i/parenrightBig
+. (19)
+whereσiand ˜σiare theSU(2) left-invariant 1-forms, satisfying dσi=1
+2ǫijkσj∧σkand
+d˜σi=1
+2ǫijk˜σj∧˜σk. The metric is Einstein provided that either α=β=1
+4orα=1
+5β=9
+100.
+The first case corresponds to the round S7and the second is a squashed S7that is also
+Einstein. Now with the contribution from the topological te rms, the EOMs can be reduced
+to
+2α2+4αβ(7β−2)−β2= 0, (20)
+together with
+√α(α−β)3(4(10α+β)˜µ−(55α+7β)ν)+2β6(20αβ−4α−β) = 0.(21)
+It is clear from (20) that there exists one and only one positi veαfor any positive β. The
+squashing parameter γ≡α/βlies in the range 0 <γ <2+3√
+2. Note that when 2˜ µ= 3˜ν,
+the squashed S7that is Eisntein remains Einstein.
+S1bundle over CP3:
+There is another way of squashing an S7, which can be viewed as an S1bundle over
+CP3. This example can be generalized to Minkowskian signature t o give rise to squashed
+AdS7[12]. The metric ansatz is given by
+ds2=α(dτ+sin2θ(dψ+B))2+βds2
+CP3,
+ds2
+CP3=dθ2+sin2θcos2θ(dψ+B)2+sin2θ/parenleftBig
+d˜θ2+1
+4sin2˜θcos2˜θσ2
+3
++1
+4sin2˜θ(σ2
+1+σ2
+2)/parenrightBig
+,
+B=1
+2sin2˜θσ3. (22)
+It is of a round S7whenα=β= 1. In general, the EOMs imply that
+α=β(8−7β),8˜µ+ ˜ν+β3
+10976(β−1)2√α= 0. (23)
+The squashing parameter γ≡α/βlies in the range (0 ,8).
+Squashed Q111spaces:
+6TheQ111space is an Einstein-Sasaki space of U(1) bundle over S2×S2×S2. We
+consider the following ansatz
+ds2=α/parenleftBig
+dψ+3/summationdisplay
+i=1cosθidφi/parenrightBig2
++β3/summationdisplay
+i=1(dθ2
+i+sinθ2
+idφ2
+i). (24)
+It is ofQ111provided that α=1
+2β= 1/16, and it remains so for ˜ ν= 0. In general, we have
+α= 4β(1−7β),8(α−β)(2α−β)˜µ+α(2α−3β)˜ν+β5(α−8β+60β2)
+4α3/2= 0.(25)
+Thus the squashing parameter γ≡α/βlies in the range (0 ,4). We expect that many of
+the squashed homogeneous spaces in seven dimensions are now solutions in this new gravity
+theory, and we shall not enumerate them further.
+4 Conclusions
+This work is motivated by studying the classical solutions o f Einstein-Chern-Simons gravity
+with asymptotic AdS structure. In seven dimensions, there a re two topological Chern-
+Simons terms, and we obtain the full set of equations of motio n. We find that spherically-
+symmetric solutions are unmodified by the inclusion of these topological terms. We also
+obtain squashed AdS 7, and squashed S7andQ111spaces in Euclidean signature, where
+the squashing parameter is related to the coupling constant s of the topological terms. It is
+intriguing to see that these known squashed homogeneous spa ces which appear to have no
+connection can now be unified under our new gravity theory.
+As in three dimensions, our topological gravity should play an important role in explor-
+ing the AdS 7/CFT6correspondence. The CFT 6that describes the world-volume theory of
+multiple M5-branes is yet to be known, and our solutions prov ide many new gravity dual
+backgrounds. The quantization condition for one of the coup ling constant suggests an un-
+usual property of the CFT 6that is absent in lower dimensions. Additional future direc tions
+include a classification of all topological gravities in (4 k+3) dimensions, investigating the
+linearization of D= 7topological gravity and obtaining the propagating degre es of freedom.
+Acknowledgement
+We are grateful to Chris Pope for useful discussions. Y.P. is supported in part by the NSFC
+grant No.1053060/A050207 and the NSFC group grant No.10821 504.
+7References
+[1] S. Deser, R. Jackiw and S. Templeton, “Topologically mas sive gauge theories,” An-
+nals Phys. 140, 372 (1982) [Erratum-ibid. 185, 406.1988 APNYA,281,409 (1988
+APNYA,281,409-449.2000)].
+[2] F. Wilczek and A. Zee, “Linking numbers, spin, and statis tics of solitons,” Phys. Rev.
+Lett.51, 2250 (1983).
+[3] S.M. Carroll, G.B. Field and R. Jackiw, “Limits on a Loren tz and parity violating
+modification of electrodynamics,” Phys. Rev. D 41, 1231 (1990).
+[4] S. Deser, R. Jackiw and G. ’t Hooft, “Three-Dimensional E instein gravity: dynamics
+of flat space,” Annals Phys. 152, 220 (1984).
+[5] E. Witten, “Three-dimensional gravity revisited,” arX iv:0706.3359 [hep-th].
+[6] W. Li, W. Song and A. Strominger, “Chiral gravity in three dimensions,” JHEP 0804,
+082 (2008) [arXiv:0801.4566 [hep-th]]. E.A. Bergshoeff, O. H ohm and P.K. Townsend,
+“Massive gravity in three dimensions,” Phys. Rev. Lett. 102, 201301 (2009) arXiv:
+0901.1766 [hep-th].
+[7] M. Gunaydin, G. Sierra and P.K. Townsend, “Quantization of the gauge coupling
+constant in a five-dimensional Yang-Mills/Einstein superg ravity theory,” Phys. Rev.
+Lett.53, 322 (1984).
+[8] M. Pernici, K. Pilch and P. van Nieuwenhuizen, “Gauged ma ximally extended super-
+gravity in seven-dimensions,” Phys. Lett. B 143, 103 (1984).
+[9] S. Deser and B. Tekin, “Energy in topologically massive g ravity,” Class. Quant. Grav.
+20, L259 (2003), gr-qc/0307073.
+[10] Y. Tachikawa, “Black hole entropy in the presence of Che rn-Simons terms,” Class.
+Quant. Grav. 24, 737 (2007), hep-th/0611141.
+[11] D.D.K. Chow, C.N. Pope and E. Sezgin, “Exact solutions o f topologically massive
+gravity,” arXiv:0906.3559 [hep-th].
+[12] P. Hoxha, R.R. Martinez-Acosta and C.N. Pope, “Kaluza- Klein consistency, Killing
+vectors, and Kaehler spaces,” Class. Quant. Grav. 17, 4207 (2000), hep-th/0005172.
+8
\ No newline at end of file
diff --git a/1001.0043.txt b/1001.0043.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f58dda47a5495beb436e5d93094c3f13f34724b5
--- /dev/null
+++ b/1001.0043.txt
@@ -0,0 +1,399 @@
+arXiv:1001.0043v2  [astro-ph.EP]  13 Jan 2010Strong Constraints to the Putative Planet Candidate around VB 10 using
+Doppler spectroscopy1
+Guillem Anglada-Escud´ e
+Department of Terrestrial Magnetism, Carnegie Institution o f Washington
+5241 Broad Branch Road, NW, Washington, DC 20015 USA
+anglada@dtm.ciw.edu
+Evgenya Shkolnik
+Department of Terrestrial Magnetism, Carnegie Institution o f Washington
+5241 Broad Branch Road, NW, Washington, DC 20015 USA
+shkolnik@dtm.ciw.edu
+Alycia J. Weinberger
+Department of Terrestrial Magnetism, Carnegie Institution o f Washington
+5241 Broad Branch Road, NW, Washington, DC 20015 USA
+weinberger@dtm.ciw.edu
+Ian B. Thompson
+The Observatories of the Carnegie Institution of Washington
+813 Santa Barbara Street, Pasadena, CA 91101 USA
+ian@obs.carnegiescience.edu
+David J. Osip
+Las Campanas Observatory, Carnegie Institution of Washington
+Colina El Pino Casilla 601, La Serena, Chile
+dosip@lco.cl
+John H. Debes
+Goddard Space Flight Center, NASA Postdoctoral Program
+8463 Greenbelt Rd, Greenbelt, MD 20770, USA
+john.H.debes@nasa.gov
+ABSTRACT– 2 –
+We present new radial velocity measurements of the ultra-co ol dwarf VB 10,
+which was recently announced to host a giant planet detected with astrometry. The
+new observations were obtained using optical spectrograph s(MIKE/Magellan and ES-
+PaDOnS/CHFT) and cover a 63% of the reported period of 270 day s. We apply Least-
+squares periodograms to identify the most significant signa ls and evaluate their corre-
+spondingFalse Alarm Probabilities. We show that this metho d is the proper generaliza-
+tion to astrometric data because (1) it mitigates the coupli ng of the orbital parameters
+with the parallax and proper motion, and (2) it permits a dire ct generalization to
+include non-linear Keplerian parameters in a combined fit to astrometry and radial ve-
+locity data. In fact, our analysis of the astrometry alone un covers the reported 270 d
+period and an even stronger signal at ∼50 days. We estimate the uncertainties in the
+parameters using a Markov Chain Monte Carlo approach. The no minal precision of the
+new Doppler measurements is about 150 s−1while their standard deviation is 250 ms−1.
+However, the best fit solutions still have RMS of 200 ms−1indicating that the excess
+in variability is due to uncontrolled systematic errors rat her than the candidate com-
+panions detected in the astrometry. Although the new data al one cannot rule-out the
+presence of a candidate, when combined with published radia l velocity measurements,
+the False Alarm Probabilities of the best solutions grow to u nacceptable levels strongly
+suggesting that the observed astrometric wobble is not due t o an unseen companion.
+Subject headings: astrometry, methods: statistical, stars: individual (VB 1 0), tech-
+niques: radial velocities
+1. Introduction
+Pravdo & Shaklan (2009) recently announced the discovery of an astrometric companion to
+VB 10, an ultra-cool dwarf with a mass of ≈0.08 M ⊙. From a Keplerian fit to the motion, they
+determined a mass of 6 M Jand a period of 270 d. Thus VB 10 became the lowest mass star
+known to harbor a planetary companion. The mass ratio betwee n VB 10 and its companion, ∼13,
+also is intriguing. A similar mass ratio for a Solar-type sta r would make the companion a brown
+dwarf, but brown dwarfs as small separation companions to st ars are quite rare. VB 10 is itself the
+secondary in a wide binary with V1428 Aql, a M2.5 star (van Bie sbroeck 1961). At a distance of
+1Based on observations collected with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory,
+Chile, at the W. M. Keck Observatory and the Canada-France-H awaii Telescope (CFHT). The Keck Observatory is
+operated as a scientific partnership between the California Institute of Technology, the University of California, and
+NASA, and was made possible by the generous financial support of the W. M. Keck Foundation. CFHT is operated
+by the National Research Council of Canada, the Institut Nat ional des Sciences de l’Univers of the Centre National
+de la Recherche Scientique of France, and the University of H awaii.– 3 –
+5.8 pc from the Sun, the 74′′separation of this proper motion binary corresponds to a pro jected
+separation of 430 AU.
+Such low-mass stars have not been the target of intensive pre cision radial velocity (PRV)
+monitoring because they have low visual fluxes and high stell ar activity. For example, the dedicated
+HARPS M-dwarf planet search observes stars2only brighter than V=14 and of moderate to low
+activity levels (Bonfils et al. 2007). VB 10 has V mag = 17.3 and is known to be a flare star
+(Berger et al. 2008). PRV and lensing planet searches have so far found only 13 stars under 0.5
+M⊙hosting 18 planets, and of these, more than half have masses b elow 0.1 M J.
+Despite the challenges, searches for planetary companions to low mass stars are of continuing
+interest. Low-mass stars appear less likely to have lower ma ss stellar companions and less likely to
+harbor planets than Solar-mass stars (Cumming et al. 2008). When they do have companions, they
+tend to be stars of nearly equal mass to the primary (Burgasse r et al. 2007). The mass function of
+planets orbiting M dwarfs, and how it differs from the planet ma ss function for higher-mass stars,
+provides a constraint on the planet formation mechanism(s) in general. Disks sufficiently massive
+to form Jupiter-mass planets appear to be rare around brown d warfs, whose disks generally look
+like lower mass versions of T Tauri disks (Scholz et al. 2006) . High mass companions would have
+to form via a binary-like fragmentation mechanism (e.g. Fon t-Ribera et al. 2009). Thus how a 6
+MJplanet could form around an ∼80 MJstar and how common such high-mass ratio companions
+remain important questions (Boss et al. 2009).
+The reported planet’s astrometric orbit predicts a radial v elocity (RV) amplitude of at least 1
+km s−1for a circular orbit and up to several km s−1for an eccentric orbit. This magnitude signal is
+detectable with ordinary RV measurements without requirin g the adoption of precision techniques
+such as an iodine cell or simultaneous thorium reference.
+Although several RV measurements of VB 10 exist in the litera ture before 2009, it is difficult
+to combine the historical RVs (see list in Table 4 of Pravdo & S haklan (2009)), as each observation
+used different calibration techniques and/or RV standards th at introduce zero-point offsets and the
+typical uncertainties are also large ( ∼1.5 km s−1). The most precise measurements in the literature
+were recently published by Zapatero Osorio et al. (2009) (he reafter Z09), but provide only a “hint
+of variability.” These data did little to constrain the orbi tal parameters of the planet beyond what
+the astrometry had already done (Anglada-Escud´ e et al. 200 9).
+Here, we present a more precise set of RV observations over 17 5 days (or 65% of the reported
+orbital period). We also present general techniques for joi nt fitting of astrometric and RV data and
+show how they can be used to constrain the orbit of the candida te planet.
+2http://www.eso.org/sci/observing/proposals/77/gto/h arps/3.txt– 4 –
+2. New Data
+We acquired spectra at 7 epochs in 2009 with the MIKE spectrog raph at the Magellan Clay
+telescope at Las CampanasObservatory(Chile). We usedthe0 .35′′andthe0.5′′slits whichproduce
+a spectral resolution of ≈45,000 and 35,000, respectively, across the 4900 – 10000 ˚A range of the
+red chip. The seeing was in the range from 0 .5 to 1.1′′. These data were reduced using the facility
+pipeline (Kelson 2003).
+We also have in hand a single spectrum of VB 10 taken in 2006 usi ng the HIRES (Vogt et al.
+1994)) on the Keck I 10-m telescope. We used the 0.861′′slit to obtain a spectral resolution of
+λ/∆λ≈58,000 at λ∼7000˚A. We used the GG475 order-blocking filter and the red cross-di sperser
+to maximize throughput in the red orders.
+To increase the phase coverage, an additional spectrum was o btained using the ESPaDOnS on
+the CFHT 3.6-m telescope. ESPaDOnS is fiber fed from the Casse grain to Coud´ e focus where the
+fiber image is projected onto a Bowen-Walraven slicer at the s pectrograph entrance. ESPaDOnS’
+‘star+sky’ mode records the full spectrum over 40 grating or ders covering 3700 to 10400 ˚A at
+a spectral resolution of λ/∆λ≈68,000. The data were reduced using Libre Esprit described in
+Donati et al. (1997, 2007).
+Each stellar exposure is bias-subtracted and flat-fielded fo r pixel-to-pixel sensitivity variations.
+After optimal extraction, the 1-D spectra are wavelength ca librated with a thorium-argon arc. To
+correct for instrumental drifts, we used the telluric molec ular oxygen A band (from 7620 – 7660
+˚A) which aligns the MIKE spectra to 40 m s−1, after which we corrected for the heliocentric
+velocity. Consistency tests with the bluer Oxygen band show s comparable values but with larger
+measurement error.
+The final spectra are of moderate S/N reaching ≈25 per pixel at 8000 ˚A. Each night, spectra
+were also taken of a M-dwarf RV standard, namely GJ 699 (Barna rd’s star; SpT = M4V) and/or
+GJ 908 (SpT = M1V).
+To measure VB 10’s RV, we cross-correlated each of 9 orders be tween 7000 and 9000 ˚A (ex-
+cluding those with strong telluric absorption) where VB 10 e mits most of its optical light, with the
+spectrumofGJ699 and/orGJ908taken onthesamenight usingI RAF’s1fxcorroutine(Fitzpatrick
+1993). Both GJ 699 and GJ 908 have been monitored for planets f or years and none has been found
+within the RV stability level of 0.1 km s−1. Here we use the systemic RVs published by a planet-
+search team (Nidever et al. 2002): RV(GJ 699) = –110.506 km s−1and RV(GJ 908) = –71.147
+km s−1. The zero-point of the absolute RVs is uncertain at the 0.4 km s−1level. We measured
+the RVs from the gaussian peak fitted to the cross-correlatio n function (CCF) of each order and
+adopt the average RV of all orders with a mean standard deviat ion of the individual measurements
+of 0.150 km s−1. The average of all our measurements is 36.02 km s−1with a standard deviation
+1IRAF (Image Reduction and Analysis Facility), http://iraf .noao.edu/– 5 –
+of 0.25 km s−1. An observing log with the measured RVs and uncertainties fo r VB 10 is shown in
+Table 1.
+3. Data Analysis: Combining Astrometry and Radial Velocities
+In this section, we reanalyze the original astrometric data to calculate the likelihood of astro-
+metrically allowed solutions, and then combine the astrome try and RV data sets in a consistent
+framework to quantify how the new RV measurements constrain the possibleorbits of the candidate
+signals observed in the astrometry of VB 10b.
+3.1. Least-squares periodograms
+The most popular method to look for periodicities in data is t he so-called Lomb-Scargle
+periodogram. A version adapted to deal with astrometric two -dimensional data developed by
+Catanzarite et al. (2006) (Joint Lomb Scargle periodogram) was implemented in the discovery pa-
+per of VB 10b (Pravdo & Shaklan 2009). Any method based on the L omb-Scargle periodogram
+performs optimally only under an important implicit assump tion: all other signals (e.g. linear
+trend, an average offset, etc.) can be subtracted from the data without affecting the significance of
+the signal under investigation. This assumption does not ho ld for astrometry because the proper
+motion and the parallax are also a significant part of the sign al and they typically correlate with
+the periodic motion of a planet (see Black & Scargle 1982).
+We use instead a Least-squares periodogram. The weighted Le ast-squares solution is obtained
+by fitting all the free parameters in the model for a given peri od. The sum of the weighted residuals
+divided by Nis the so-called χ2statistic. Then, each χ2
+Pof a given model with kPparameters, can
+be compared to the χ2
+0of the null hypothesis with k0free parameters by computing the power, z,
+as
+z(P) =(χ2
+0−χ2
+P)/(kP−k0)
+χ2
+P/(Nobs−kP)(1)
+where a large zis interpreted as a very significant solution. The values of zfollow a Fisher F-
+distribution with kP−k0andNobs−kPdegrees of freedom (Scargle 1982; Cumming 2004). Even
+if only noise is present, a periodogram will contain several peaks (see Scargle 1982, as an example)
+whoseexistencehavetobeconsideredinobtainingtheproba bilityofaspuriousdetection. Assuming
+Gaussian noise, the probability that a peak in the periodogr am has a power higher than z(P) by
+chance is the so-called False Alarm Probability (FAP) :
+FAP = 1 −(1−Prob[z > z(P)])M(2)
+whereMis the number of independent frequencies. In the case of unev en sampling, Mcan be quite
+large and is roughly the number of periodogram peaks one coul d expect from a data set with only– 6 –
+Gaussian noise and thesame cadence as thereal observations . We adopt the recipe M≈2∆T/Pmin
+given in Cumming (2004, Sec 2.2), where ∆ Tis the time-span of the observations and Pminis the
+minimum period searched. One still has to select Pminarbitrarily. Assuming a Pmin= 20 days, the
+astrometric data alone has M∼300, and the combination of astrometry and RVs has M∼360.
+In our particular problem, the null hypothesis is the basic k inematic model with k0= 6 param-
+eters: 2 coordinates, 2 proper motions, parallax and system ic RV. As a first approach, our simplest
+non-null hypothesis considers circular orbits, astrometr ic data only and one RV measurement. For
+a given period, the number of free parameters is then kP= 10: the 6 kinematic ones plus the four
+Thiele Innes elements A,B,FandG(e.g. Wright & Howard 2009). Since the model is linear in all
+10 parameters, the power can be efficiently computed for many p eriods between 20 days and 4000
+days to obtain a familiar representation of the periodogram that we call a Circular Least-squares
+Periodogram (CLP). The CLP of the astrometric data, shown at top in Figure 1, displays two
+obvious peaks: the reported one at 270 days (Pravdo & Shaklan 2009) and a more significant one
+at 49.9 days, both with high power and very low FAPs.
+To find the full Keplerian solution for both periods and estim ate their FAPs, we perform a
+Least-squares periodogram sampling a grid of fixed eccentri city-period (eP) pairs and fitting all
+other parameters. For each eP pair kPis 11: the null-hypothesis ( X0,Y0,µX,µY,πandv0)
+plus all the other Keplerian elements: Mass of the planet, Ω, ω,i, and the Mean anomaly at the
+initial epoch M0(see Wright & Howard 2009, for a recent review). We analyze bo thastrometry
+onlyandastrometry+RVs . Theχ2of the best fit solution is then used to obtain each FAP as
+previously described. Figure 1 shows the resulting color-c oded FAPs for each eccentricity–period
+pair (eP-map).
+3.1.1. Astrometry only
+A value of M= 300 has been used to obtain the FAP, and our result at 270 d qua litatively
+agrees with Pravdo & Shaklan (2009), however the more signifi cant period is at ∼50 d. For both
+periods, there are regions with FAP <1% spanning all possible eccentricities (second row in Fig-
+ure 1). The best fits and their χ2per degree of freedom (¯ χ2) are summarized in Table 2. The
+obtained results for the 270 d period are in agreement with th ose reported in the discovery paper
+by Pravdo & Shaklan (2009). The best fit solution for the 50 d pe riod has mass ∼15 MJ, which
+would be a very low mass brown dwarf. It is important to point o ut that the best fit inclination is
+close to 90 (edge on) for both solutions. The uncertainties o n the orbital parameters are quantified
+in Sec 3.2.– 7 –
+3.1.2. Astrometry+RVs
+We now fit jointly for the best orbital solution to the astrome try and RVs. Our campaign
+covered about 65% of the 270 d orbit. The standard deviation o f all our RVs measurements is
+250 m s−1(null hypothesis) which is larger than the individual uncer tainties in Table 4. When we
+cross-correlate our standards, we measure a similar RMS of 2 00 m s−1, which indicates that the
+difference is due to an uncontrolled or unmeasured systematic . The RMS of the RVs for the best
+fit solution is 200 m s−1, which is not statistically different from the RMS of the null h ypothesis.
+This is another indication that our measurements contain sy stematic errors at the level of 100 −200
+m/s. Despite of that, we use the nominal errors in the Least-s quares solution as the best estimates
+for the individual uncertainties we can provide. In Figure 2 , we show the best solutions to both
+signals including all the data.
+For the 270 d period, our RV non detection cannot exclude a small region of orbital solutions
+arounde∼0.8 with a FAP between 1%–5% – see Figure 1, third row right panel . We now add
+the RVs measurements by Z09 and solve for a joint solution. A z ero-point offset between datasets
+is added as an additional free parameter. The combined RV mea surements force the eccentricity
+to large values which apparently still provides a reasonabl e fit to the astrometry (see top panels in
+Figure 2). However, the FAPs are now all higher than 10% (Figu re 1, bottom right panel), which
+indicates that the signal can be barely distinguished from t he noise fluctuations. The “hint” of
+detection in Z09 based on one discrepant value at 3 .1−σout of five can be due to random errors
+with a non negligible probability.
+For the 50 d period, there are still several orbits that provi de a decent fit to the combined
+astrometry and the new RV data with a FAP lower than 1%. These o ccupy a small space around
+the best joint solution, with e= 0.90 (see Figure 1, 3rd row, left panel) and an inclination clos e
+to 0. Large eccentricity causes the duration of fast RV varia tion to be very short (and difficult to
+catch); an inclination close to 0 tends to suppress any RVs si gnal. Such inclination is in apparent
+contradiction with the one obtained using the astrometry al one (∼90 deg). The reason is the
+following: while the new fit to the astrometry forced by the RV s is much worse than the one
+obtained from the astrometry alone, such a solution still re presents an improvement compared to
+the null hypothesis. Adding Z09 data to the fit increases the F AP of the most likely solution to 2%,
+an eccentricity of 0 .91 and the inclination close to 0 (see Table 2). This suggests that the signal at
+50 d is also spurious, even though it has slightly better chan ces of survival than the one at 270 d.
+3.2.A Posteriori Probability Distributions
+We adapt the method developed by Ford (2005, 2006) to assess u ncertainties in orbit determi-
+nations by obtaining the a posteriori probability distribution for the parameters using a Markov
+Chain with a Gibbs sampler strategy. Our problem is identica l to the one described by Ford (2005),
+where now the χ2contains both RV and astrometric observations and the model has a few more– 8 –
+free parameters. Several properly adjusted MCMC with 106steps have been computed obtaining
+compatible results. The step sizes of the Gibbs sampler are i nitialized with the formal errors from
+the best fit Least-squares solution, and adjusted to obtain a transition probability between 10%
+and 20%. The first 105steps of each chain are rejected. The final distributions mat ch very well the
+areas of low FAPs in the eP-maps (see Figure 3 as an example) gi ving further proof that the chains
+have converged to the equilibrium distributions. The MCMC c ontains 13 free parameters – the 11
+from the Least-squares periodogram plus eccentricity and p eriod. When the RVs measurements
+from Z09 are included, and additional offset parameter is incl uded.
+Table 2 presents the standard deviations obtained via the MC MC for both the 50 d and 270 d
+periods using astrometry alone and astrometry + all RV data. As an example, we show the two
+dimensional density of states in period-eccentricity spac e in Figure 3 (left) obtained in both cases
+aroundthe 270 d signal. Themarginalized distributionsfor ein theform of histograms are shownin
+Figure 3 (right). For the astrometry-alone case, the distri bution of eis almost uniform. It becomes
+strongly peaked towards high eccentricities when all the RV data are included. Since the best fit
+solution at 270 d is poor (¯ χ2= 1.76), the corresponding χ2minimum is not very deep which is
+reflected in a significant increase in the derived uncertaint ies (See Table 2). The same happens to
+the signal at 50 d with the exception of the inclination that h as a small uncertainty (4 deg) close
+to 0. Even though this solution has a low FAP, the inclination has to be coincidentally very small
+to suppress any RV signal and very different from using astrome try only (94 deg), rasing serious
+doubts of its reality.
+4. Discussion and Conclusions
+The non-detection of a significant RV variation in our data se t already discards most orbital
+configurations allowed by the astrometry. When combined wit h Z09 RVs measurements, there are
+no remaining solutions with a FAP lower than 10% around the 27 0 d period, so the presence of
+a planet candidate at that period is not supported by the obse rvations. For the 50 d period, the
+constraints arealso strongandbecome almost definitive whe ntheZ09data is included. Even highly
+eccentric solutions have a relatively large FAP ( >2%). We find that particular combinations of
+eccentricity, inclination and ωcan fit an almost flat RV curve indicating that the analytic met hods
+applied to estimate FAPs for high eccentricities tend to giv e over optimistic results and that this
+issue should be studied in more detail.
+We have developed and implemented useful tools for detailed analysis of combined astrometric
+and RV data: Circular Least-squares periodogram as the prop er generalization of the classic Lomb-
+Scargle periodogram to deal with astrometric data, eP-maps to visualize the most likely period–
+eccentricity combinations and a Bayesian characterizatio n of the parameter uncertainties based on
+a MCMC approach.
+VB 10 is also part of the Carnegie Astrometric Planet Search p rogram (Boss et al. 2009). RV– 9 –
+measurements with precision techniques in the near-infrar ed (Bean et al. 2009) may provide the
+required accuracy to put even stronger limits to the existen ce of VB10b or find other planets in the
+system. VB 10 will certainly be observed by the space astrome try mission Gaia (Perryman et al.
+2001), which would be capable of finding a planet with a period of 270 d and as small as 0 .2 MJ.
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+Table 1. Log of RVs
+Telescope UT Date HJD Slit Width RV (w/ GJ 699)aRV (w/ GJ 908)a
++Instrument –2450000′′km s−1km s−1
+Keck I + HIRES 2006 Aug 12 3959.57 0.86 – 35.59 ±0.15
+Clay+MIKE 2009 Jun 06 4988.74 0.35 36.23 ±0.13 35.99 ±0.15
+Clay+MIKE 2009 Jun 07 4989.82 0.50 36.22 ±0.13 36.09 ±0.20
+Clay+MIKE 2009 Jun 08 4990.75 0.50 36.15 ±0.12 36.10 ±0.22
+Clay+MIKE 2009 Jun 30 5012.72 0.35 35.72 ±0.11 –
+Clay+MIKE 2009 Jul 25 5037.66 0.50 35.96 ±0.11 36.03 ±0.11
+Clay+MIKE 2009 Sep 04 5078.58 0.50 35.96 ±0.09 36.37 ±0.24
+Clay+MIKE 2009 Oct 15 5119.54 0.50 36.30 ±0.14 36.41 ±0.13
+Clay+MIKE 2009 Oct 26 5130.51 0.50 36.41 ±0.16 36.27 ±0.18
+CFHT+ESPaDOnS 2009 Nov 29 5164.69 –b– 35.74 ±0.20
+aUncertainties are the standard deviation of the 9 orders of t he cross correlation and do not include the 40
+m s−1systematic uncertainty from the telluric wavelength corre ction. Absolute radial velocity determination
+has an uncertainty of 0 .4 km/s but it is not relevant for orbital fitting purposes.
+bESPaDOnS is a fiber fed spectrograph with an effective resolut ion of R ∼68000 in the wavelength range
+of interest
+Table 2. Best fitting valuesa. Uncertainties obtained from a MCMC with 106steps.
+Parameter Astrometry 50 d Astrometry 270 d Astro+ all RV 50 d A stro+ all RV 270 d
+X0(mas) -16.6 ±1.6 -14.1d±3.2 -21.15±2.3 -17.9 ±4.7
+Y0(mas) -408.0 ±1.9 -406.1d±3.5 409.52±2.8 -410.5 ±5.51
+µR.A.(mas/yr) -588.98 ±0.25 -589.08 ±0.25 -588.66±0.29 -589.21 ±0.26
+µDec(mas/yr) -1360.95 ±0.25 -1361.08 ±0.24 -1361.02 ±0.25 -1361.36 ±0.20
+π(mas) 168.3 ±1.51 169.5 ±1.4 169.95±1.37 169.24 ±1.30
+v0(km/s) 35.2 ±1.4 35.4d±1.050 36.06±0.11 36.05 ±0.08
+voffset(km/s) - - 1.5±0.42 1.5 ±0.36
+P(days) 49.7 ±0.5 272.1 ±4.1 49.84±0.11 278.5 ±2.7
+Mass(MJ) 17.5 ±4.4 7.1 ±2.7 13.7±6.4 5.0 ±2.9
+e 0.22c±0.30 0.48c±0.31 0.91±0.13 0.90 ±0.16
+i(deg) 93 ±5 90 ±15 4±5 110c±50
+Ω(deg) 40 ±20 220 ±25 13c±100 40c±66
+ω(deg) 20 ±40 30c±80 122c±60 17c±90
+M0(deg) 270 ±0 170c±108 340c±70 156c±80
+a(AU)d0.12 0.36 0.12 0.36
+¯χ2
+02.28 2.28 2.75 2.75
+¯χ20.87 0.93 1.62 1.76
+aThe mass of VB 10 is assumed to be 0 .078 M ⊙according to Pravdo & Shaklan (2009)
+bLarge uncertainty due to correlation with the eccentricity
+cUnconstrained or poorly constrained
+dDerived quantity using Kepler equations– 12 –
+Fig. 1.— Top panel. Circular Least-squares periodogramshowingthe two most si gnificant periods
+with their corresponding False Alarm Probabilities (FAP). Second row. FAPs obtained for a grid
+of Eccentricity–Period pairs around the 50 d (left) and the 2 70 d (right) when only astrometry is
+considered. Third row. FAPs obtained when our new RV are included to the fit. Bottom row.
+Final FAPs obtained when all published RV data are combined i n a joint fit.– 13 –
+0 0.2 0.4 0.6 0.8 1
+Phase-4-20246810R.A.(mas)
+0 0.2 0.4 0.6 0.8 1-4-20246810
+0 0.2 0.4 0.6 0.8 1
+Phase-4-20246810 Dec (mas)
+0 0.2 0.4 0.6 0.8 1-4-20246810
+0 0.2 0.4 0.6 0.8 1
+Phase-4-20246810R.A.(mas)
+0 0.2 0.4 0.6 0.8 1-4-20246810
+0 0.2 0.4 0.6 0.8 1
+Phase-4-20246810 Dec (mas)
+0 0.2 0.4 0.6 0.8 1-4-20246810
+0 0.2 0.4 0.6 0.8 1
+Phase32333435363738RV (km/s)
+0 0.2 0.4 0.6 0.8 132333435363738
+0 0.2 0.4 0.6 0.8 1
+Phase32333435363738
+MIKE
+HIRES
+CFHT
+NIRSPEC Z09P = 49.84 days P = 278.5 days
+Fig. 2.— The best fit (lowest χ2) joint solutions to the Pravdo & Shaklan (2009) astrometry a nd
+used RVs for the two signals. Top panels contain the astromet ric offsets after the removal of the
+corresponding parallax and the proper motion. The lower pan els contain all RVs used. Each RV
+point represents the weighted average of the values obtaine d using both reference stars if available.
+The best fit offset has been applied to Z09 data (Green triangles ). Phase 0 corresponds to the first
+astrometric epoch at JD 2451438 .64 and the corresponding folding periods are given on the top .
+Fig. 3.— Left: Steps in period-eccentricity space of a Marko v Chain of 106elements applied to the
+astrometry only (black) and to the astrometry+all RV data (b rown). The distribution resembles
+the FAP contours on the eP-maps around 270 days indicating th at the chain has successfully
+converged to the equilibrium distribution. Right: Histogr am reproducing the marginalized density
+distributions in efor the astrometry only and astrometry+RV around the 270 d so lution.
\ No newline at end of file
diff --git a/1001.0044.txt b/1001.0044.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ec95b1f50e50dc12848b6fa1c8c1046bfeaf30eb
--- /dev/null
+++ b/1001.0044.txt
@@ -0,0 +1,1486 @@
+arXiv:1001.0044v3  [math.PR]  31 Mar 2011A law of large numbers approximation for
+Markov population processes with countably
+many types
+A. D. Barbour∗and M. J. Luczak†
+Universit¨ at Z¨ urich and London School of Economics
+Abstract
+When modelling metapopulation dynamics, the influence of a s in-
+gle patch on the metapopulation depends on the number of indi vidu-
+als in the patch. Since the population size has no natural upp er limit,
+this leads to systems in which there are countably infinitely many
+possible types of individual. Analogous considerations ap ply in the
+transmission of parasitic diseases. In this paper, we prove a law of
+large numbers for quite general systems of this kind, togeth er with
+a rather sharp bound on the rate of convergence in an appropri ately
+chosen weighted ℓ1norm.
+Keywords: Epidemic models, metapopulation processes, countably many
+types, quantitative law of large numbers, Markov population proce sses
+AMS subject classification: 92D30, 60J27, 60B12
+Running head: A law of large numbers approximation
+1 Introduction
+There are many biological systems that consist of entities that diffe r in their
+influence according to the number of active elements associated wit h them,
+∗Angewandte Mathematik, Universit¨ at Z¨ urich, Winterthurertra sse 190, CH-8057
+Z¨URICH; ADB was supported in part by Schweizerischer Nationalfond s Projekt Nr. 20–
+107935/1.
+†London School of Economics; MJL was supported in part by a STICE RD grant.
+1and can be divided into types accordingly. In parasitic diseases (Bar bour &
+Kafetzaki 1993, Luchsinger 2001a,b, Kretzschmar 1993), the in fectivity of a
+host depends on the number of parasites that it carries; in metapo pulations,
+the migration pressure exerted by a patch is related to the number of its
+inhabitants (Arrigoni 2003); the behaviour of a cell may depend on the num-
+ber of copies of a particular gene that it contains (Kimmel & Axelrod 2 002,
+Chapter 7); and so on. In none of these examples is there a natura l upper
+limit to the number of associated elements, so that the natural set ting for
+a mathematical model is one in which there are countably infinitely man y
+possible types of individual. In addition, transition rates typically incr ease
+with the number of associated elements in the system — for instance , each
+parasite has an individual death rate, so that the overall death ra te of par-
+asites grows at least as fast as the number of parasites — and this le ads
+to processes with unbounded transition rates. This paper is conce rned with
+approximations to density dependent Markov models of this kind, wh en the
+typical population size Nbecomes large.
+In density dependent Markov population processes with only finitely
+many types of individual, a law of large numbers approximation, in the f orm
+ofasystemofordinarydifferentialequations, wasestablishedbyK urtz(1970),
+together with a diffusion approximation (Kurtz, 1971). In the infinit e di-
+mensional case, the law of large numbers was proved for some spec ific mod-
+els (Barbour & Kafetzaki 1993, Luchsinger 2001b, Arrigoni 2003 , see also
+L´ eonard 1990), using individually tailored methods. A more general result
+was then given by Eibeck & Wagner (2003). In Barbour & Luczak (20 08),
+the law of large numbers was strengthened by the addition of an err or bound
+inℓ1that is close to optimal order in N. Their argument makes use of an
+intermediate approximation involving an independent particles proce ss, for
+which the law of large numbers is relatively easy to analyse. This proce ss is
+then shown to be sufficiently close to the interacting process of act ual inter-
+est, by means of a coupling argument. However, the generality of t he results
+obtained is limited by the simple structure of the intermediate proces s, and
+the model of Arrigoni (2003), for instance, lies outside their scop e.
+In this paper, we develop an entirely different approach, which circu m-
+vents the need for an intermediate approximation, enabling a much w ider
+class of models to be addressed. The setting is that of families of Mar kov
+population processes XN:= (XN(t), t≥0),N≥1, taking values in the
+countable space X+:={X∈ZZ+
++;/summationtext
+m≥0Xm<∞}. Each component repre-
+2sents the number of individuals of a particular type, and there are c ountably
+many types possible; however, at any given time, there are only finit ely
+many individuals in the system. The process evolves as a Markov proc ess
+with state-dependent transitions
+X→X+Jat rate NαJ(N−1X), X∈ X+, J∈ J,(1.1)
+where each jump is of bounded influence, in the sense that
+J ⊂ {X∈ZZ+;/summationdisplay
+m≥0|Xm| ≤J∗<∞},for some fixed J∗<∞,(1.2)
+so that the number of individuals affected is uniformly bounded. Dens ity
+dependence is reflected in the fact that the arguments of the fun ctionsαJ
+are counts normalised by the ‘typical size’ N. Writing R:=RZ+
++, the func-
+tionsαJ:R →R+are assumed to satisfy
+/summationdisplay
+J∈JαJ(ξ)<∞, ξ∈ R0, (1.3)
+whereR0:={ξ∈ R:ξi= 0 for all but finitely many i}; this assumption
+implies that the processes XNare pure jump processes, at least for some
+non-zero length of time. To prevent the paths leaving X+, we also assume
+thatJl≥ −1 for each l, and that αJ(ξ) = 0 ifξl= 0 for any J∈ Jsuch
+thatJl=−1. Some remarks on the consequences of allowing transitions J
+withJl≤ −2 for some lare made at the end of Section 4.
+Thelawoflargenumbersisthenformallyexpressed intermsofthesy stem
+ofdeterministic equations
+dξ
+dt=/summationdisplay
+J∈JJαJ(ξ) =:F0(ξ), (1.4)
+to be understood componentwise for those ξ∈ Rsuch that
+/summationdisplay
+J∈J|Jl|αJ(ξ)<∞,for alll≥0,
+thus by assumption including R0. Here, the quantity F0represents the in-
+finitesimal average drift of the components of the random proces s. However,
+in this generality, it is not even immediately clear that equations (1.4) h ave
+a solution.
+3In order to make progress, it is assumed that the unbounded comp onents
+in the transition rates can be assimilated into a linear part, in the sens e
+thatF0can be written in the form
+F0(ξ) =Aξ+F(ξ), (1.5)
+again to be understood componentwise, where Ais a constant Z+×Z+
+matrix. These equations are then treated as a perturbed linear sy stem
+(Pazy 1983, Chapter 6). Under suitable assumptions on A, there exists a
+measure µonZ+, defining a weighted ℓ1norm/⌊ard⌊l · /⌊ard⌊lµonR, and a strongly
+/⌊ard⌊l·/⌊ard⌊lµ–continuoussemigroup {R(t), t≥0}oftransitionmatriceshaving point-
+wise derivative R′(0) =A. IfFis locally /⌊ard⌊l·/⌊ard⌊lµ–Lipschitz and /⌊ard⌊lx(0)/⌊ard⌊lµ<∞,
+this suggests using the solution xof the integral equation
+x(t) =R(t)x(0)+/integraldisplayt
+0R(t−s)F(x(s))ds (1.6)
+as an approximation to xN:=N−1XN, instead of solving the deterministic
+equations (1.4) directly. We go on to show that the solution XNof the
+stochastic system can be expressed using a formula similar to (1.6), which
+has an additional stochastic component in the perturbation:
+xN(t) =R(t)xN(0)+/integraldisplayt
+0R(t−s)F(xN(s))ds+/tildewidemN(t),(1.7)
+where
+/tildewidemN(t) :=/integraldisplayt
+0R(t−s)dmN(s), (1.8)
+andmNis the local martingale given by
+mN(t) :=xN(t)−xN(0)−/integraldisplayt
+0F0(xN(s))ds. (1.9)
+The quantity mNcanbe expected to be small, at least componentwise, under
+reasonable conditions.
+To obtain tight control over /tildewidemNin all components simultaneously, suf-
+ficient to ensure that sup0≤s≤t/⌊ard⌊l/tildewidemN(s)/⌊ard⌊lµis small, we derive Chernoff–like
+boundsonthedeviations ofthemost significant components, witht hehelpof
+a family of exponential martingales. The remaining components are t reated
+usingsomegeneral a prioriboundsonthebehaviourofthestochasticsystem.
+4This allows us to take the difference between the stochastic and det erministic
+equations (1.7) and (1.6), after which a Gronwall argument can be c arried
+through, leading to the desired approximation.
+The main result, Theorem 4.7, guarantees an approximation error o f or-
+derO(N−1/2√logN) in the weighted ℓ1metric/⌊ard⌊l·/⌊ard⌊lµ, except on an event of
+probability of order O(N−1logN). More precisely, for each T >0, there
+exist constants K(1)
+T,K(2)
+T,K(3)
+Tsuch that, for Nlarge enough, if
+/⌊ard⌊lN−1XN(0)−x(0)/⌊ard⌊lµ≤K(1)
+T/radicalbigg
+logN
+N,
+then
+P/parenleftBig
+sup
+0≤t≤T/⌊ard⌊lN−1XN(t)−x(t)/⌊ard⌊lµ> K(2)
+T/radicalbigg
+logN
+N/parenrightBig
+≤K(3)
+TlogN
+N.(1.10)
+Theerrorboundissharper, byafactoroflog N, thanthatgiveninBarbour&
+Luczak(2008),andthetheoremisapplicabletoamuch widerclassof models.
+However, the method of proof involves moment arguments, which r equire
+somewhat stronger assumptions on the initial state of the system , and, in
+models such as that of Barbour & Kafetzaki (1993), onthe choice of infection
+distributions allowed. The conditions under which the theorem holds c an be
+divided into three categories: growth conditions on the transition r ates, so
+that the a prioribounds, which have the character of moment bounds, can
+be established; conditions on the matrix A, sufficient to limit the growth of
+the semigroup R, and (together with the properties of F) to determine the
+weights defining the metric in which the approximation is to be carried o ut;
+and conditions on the initial state of the system. The a priori bounds are
+derived in Section 2, the semigroup analysis is conducted in Section 3, and
+the approximation proper is carried out in Section 4. The paper conc ludes
+in Section 5 with some examples.
+The form (1.8) of the stochastic component /tildewidemN(t) in (1.7) is very simi-
+lar to that of a key element in the analysis of stochastic partial differ ential
+equations; see, for example, Chow (2007, Section 6.6). The SPDE a rguments
+used for its control are however typically conducted in a Hilbert spa ce con-
+text. Our setting is quite different in nature, and it does not seem cle ar how
+to translate the SPDE methods into our context.
+52 A priori bounds
+We begin by imposing further conditions on the transition rates of th e pro-
+cessXN, sufficient to constrain its paths to bounded subsets of X+dur-
+ing finite time intervals, and in particular to ensure that only finitely ma ny
+jumps can occur in finite time. The conditions that follow have the flav our
+of moment conditions on the jump distributions. Since the index j∈Z+is
+symbolic in nature, we start by fixing an ν∈ R, such that ν(j) reflects in
+some sense the ‘size’ of j, with most indices being ‘large’:
+ν(j)≥1 for allj≥0 and lim
+j→∞ν(j) =∞. (2.1)
+We then define the analogues of higher empirical moments using the q uanti-
+tiesνr∈ R, defined by νr(j) :=ν(j)r,r≥0, setting
+Sr(x) :=/summationdisplay
+j≥0νr(j)xj=xTνr, x∈ R0, (2.2)
+where, for x∈ R0andy∈ R,xTy:=/summationtext
+l≥0xlyl. In particular, for X∈ X+,
+S0(X) =/⌊ard⌊lX/⌊ard⌊l1. Note that, because of (2.1), for any r≥1,
+#{X∈ X+:Sr(X)≤K}<∞for allK >0. (2.3)
+To formulate the conditions that limit the growth of the empirical mom ents
+ofXN(t) witht, we also define
+Ur(x) :=/summationdisplay
+J∈JαJ(x)JTνr;Vr(x) :=/summationdisplay
+J∈JαJ(x)(JTνr)2, x∈ R.(2.4)
+The assumptions that we shall need are then as follows.
+Assumption 2.1 There exists a νsatisfying (2.1)andr(1)
+max,r(2)
+max≥1such
+that, for all X∈ X+,
+/summationdisplay
+J∈JαJ(N−1X)|JTνr|<∞,0≤r≤r(1)
+max,(2.5)
+the case r= 0following from (1.2)and(1.3); furthermore, for some non-
+negative constants krl, the inequalities
+U0(x)≤k01S0(x)+k04,
+U1(x)≤k11S1(x)+k14, (2.6)
+Ur(x)≤ {kr1+kr2S0(x)}Sr(x)+kr4,2≤r≤r(1)
+max;
+6and
+V0(x)≤k03S1(x)+k05,
+Vr(x)≤kr3Sp(r)(x)+kr5,1≤r≤r(2)
+max, (2.7)
+are satisfied, where 1≤p(r)≤r(1)
+maxfor1≤r≤r(2)
+max.
+The quantities r(1)
+maxandr(2)
+maxusually need to be reasonably large, if Assump-
+tion 4.2 below is to be satisfied.
+Now, for XNas in the introduction, we let tXNndenote the time of its
+n-th jump, with tXN
+0= 0, and set tXN∞:= lim n→∞tXNn, possibly infinite. For
+0≤t < tXN∞, we define
+S(N)
+r(t) :=Sr(XN(t));U(N)
+r(t) :=Ur(xN(t));V(N)
+r(t) :=Vr(xN(t)),
+(2.8)
+once again with xN(t) :=N−1XN(t), and also
+τ(N)
+r(C) := inf {t < tXN
+∞:S(N)
+r(t)≥NC}, r≥0,(2.9)
+where the infimum of the empty set is taken to be ∞. Our first result shows
+thattXN∞=∞a.s., and limits the expectations of S(N)
+0(t) andS(N)
+1(t) for any
+fixedt.
+In what follows, we shall write F(N)
+s=σ(XN(u),0≤u≤s), so that
+(F(N)
+s:s≥0) is the natural filtration of the process XN.
+Lemma 2.2 Under Assumptions 2.1, tXN∞=∞a.s. Furthermore, for any
+t≥0,
+E{S(N)
+0(t)} ≤(S(N)
+0(0)+Nk04t)ek01t;
+E{S(N)
+1(t)} ≤(S(N)
+1(0)+Nk14t)ek11t.
+Proof. Introducing the formal generator ANassociated with (1.1),
+ANf(X) :=N/summationdisplay
+J∈JαJ(N−1X){f(X+J)−f(X)}, X∈ X+,(2.10)
+we note that NUl(x) =ANSl(Nx). Hence, if we define M(N)
+lby
+M(N)
+l(t) :=S(N)
+l(t)−S(N)
+l(0)−N/integraldisplayt
+0U(N)
+l(u)du, t ≥0,(2.11)
+7for 0≤l≤r(1)
+max, it is immediate from (2.3), (2.5) and (2.6) that the process
+(M(N)
+l(t∧τ(N)
+1(C)), t≥0) is a zero mean F(N)–martingale for each C >0.
+In particular, considering M(N)
+1(t∧τ(N)
+1(C)), it follows in view of (2.6) that
+E{S(N)
+1(t∧τ(N)
+1(C))} ≤S(N)
+1(0)+E/braceleftBigg/integraldisplayt∧τ(N)
+1(C)
+0{k11S(N)
+1(u)+Nk14}du/bracerightBigg
+≤S(N)
+1(0)+/integraldisplayt
+0(k11E{S(N)
+1(u∧τ(N)
+1(C))}+Nk14)du.
+Using Gronwall’s inequality, we deduce that
+E{S(N)
+1(t∧τ(N)
+1(C))} ≤(S(N)
+1(0)+Nk14t)ek11t,(2.12)
+uniformly in C >0, and hence that
+P/bracketleftBig
+sup
+0≤s≤tS1(XN(s))≥NC/bracketrightBig
+≤C−1(S1(xN(0))+k14t)ek11t(2.13)
+also. Hence sup0≤s≤tS1(XN(s))<∞a.s. for any t, limC→∞τ(N)
+1(C) =∞
+a.s., and, from (2.3) and (1.3), it thus follows that tXN∞=∞a.s. The bound
+onE{S(N)
+1(t)}is now immediate, and that on E{S(N)
+0(t)}follows by applying
+the same Gronwall argument to M(N)
+0(t∧τ(N)
+1(C)).
+The next lemma shows that, if any T >0 is fixed and Cis chosen large
+enough, then, with high probability, N−1S(N)
+0(t)≤Cholds for all 0 ≤t≤T.
+Lemma 2.3 Assume that Assumptions 2.1 are satisfied, and that S(N)
+0(0)≤
+NC0andS(N)
+1(0)≤NC1. Then, for any C≥2(C0+k04T)ek01T, we have
+P[{τ(N)
+0(C)≤T}]≤(C1∨1)K00/(NC2),
+whereK00depends on Tand the parameters of the model.
+Proof. It is immediate from (2.11) and (2.6) that
+S(N)
+0(t) =S(N)
+0(0)+N/integraldisplayt
+0U(N)
+0(u)du+M(N)
+0(t)
+≤S(N)
+0(0)+/integraldisplayt
+0(k01S(N)
+0(u)+Nk04)du+ sup
+0≤u≤tM(N)
+0(u).(2.14)
+8Hence, from Gronwall’s inequality, if S(N)
+0(0)≤NC0, then
+S(N)
+0(t)≤/braceleftbigg
+N(C0+k04T)+ sup
+0≤u≤tM(N)
+0(u)/bracerightbigg
+ek01t.(2.15)
+Now, considering the quadratic variation of M(N)
+0, we have
+E/braceleftBigg
+{M(N)
+0(t∧τ(N)
+1(C′))}2−N/integraldisplayt∧τ(N)
+1(C′)
+0V(N)
+0(u)du/bracerightBigg
+= 0 (2.16)
+for anyC′>0, from which it follows, much as above, that
+E/parenleftBig
+{M(N)
+0(t∧τ(N)
+1(C′))}2/parenrightBig
+≤E/braceleftbigg
+N/integraldisplayt
+0V(N)
+0(u∧τ(N)
+1(C′))du/bracerightbigg
+≤/integraldisplayt
+0{k03ES(N)
+1(u∧τ(N)
+1(C′))+Nk05}du.
+Using (2.12), we thus find that
+E/parenleftBig
+{M(N)
+0(t∧τ(N)
+1(C′))}2/parenrightBig
+≤k03
+k11N(C1+k14T)(ek11t−1)+Nk05t,(2.17)
+uniformlyforall C′. Doob’smaximal inequality appliedto M(N)
+0(t∧τ(N)
+1(C′))
+now allows us to deduce that, for any C′,a >0,
+P/bracketleftBig
+sup
+0≤u≤TM(N)
+0(u∧τ(N)
+1(C′))> aN/bracketrightBig
+≤1
+Na2/braceleftbiggk03
+k11(C1+k14T){ek11T−1}+k05T/bracerightbigg
+=:C1K01+K02
+Na2,
+say, so that, letting C′→ ∞,
+P/bracketleftBig
+sup
+0≤u≤TM(N)
+0(u)> aN/bracketrightBig
+≤C1K01+K02
+Na2
+also. Taking a=1
+2Ce−k01Tand putting the result into (2.15), the lemma
+follows.
+In the next theorem, we control the ‘higher ν-moments’ S(N)
+r(t) ofXN(t).
+9Theorem 2.4 Assume thatAssumptions 2.1are satisfied, andthat S(N)
+1(0)≤
+NC1andS(N)
+p(1)(0)≤NC′
+1. Then, for 2≤r≤r(1)
+maxand for any C >0, we
+have
+E{S(N)
+r(t∧τ(N)
+0(C))} ≤(S(N)
+r(0)+Nkr4t)e(kr1+Ckr2)t,0≤t≤T.(2.18)
+Furthermore, if for 1≤r≤r(2)
+max,S(N)
+r(0)≤NCrandS(N)
+p(r)(0)≤NC′
+r,
+then, for any γ≥1,
+P[ sup
+0≤t≤TS(N)
+r(t∧τ(N)
+0(C))≥NγC′′
+rT]≤Kr0γ−2N−1, (2.19)
+where
+C′′
+rT:= (Cr+kr4T+/radicalbig
+(C′
+r∨1))e(kr1+Ckr2)T
+andKr0depends on C,Tand the parameters of the model.
+Proof. Recalling (2.11), use the argument leading to (2.12) with the martin-
+galesM(N)
+r(t∧τ(N)
+1(C′)∧τ(N)
+0(C)), for any C′>0, to deduce that
+ES(N)
+r(t∧τ(N)
+1(C′)∧τ(N)
+0(C))
+≤S(N)
+r(0)+/integraldisplayt
+0/parenleftBig
+{kr1+Ckr2}E/braceleftBig
+S(N)
+r(u∧τ(N)
+1(C′)∧τ(N)
+0(C))/bracerightBig
++Nkr4/parenrightBig
+du,
+for 1≤r≤r(1)
+max, sinceN−1S(N)
+0(u)≤Cwhenu≤τ(N)
+0(C): define k12= 0.
+Gronwall’s inequality now implies that
+ES(N)
+r(t∧τ(N)
+1(C′)∧τ(N)
+0(C))≤(S(N)
+r(0)+Nkr4t)e(kr1+Ckr2)t,(2.20)
+for 1≤r≤r(1)
+max, and (2.18) follows by Fatou’s lemma, on letting C′→ ∞.
+Now, also from (2.11) and (2.6), we have, for t≥0 and each r≤r(1)
+max,
+S(N)
+r(t∧τ(N)
+0(C))
+=S(N)
+r(0)+N/integraldisplayt∧τ(N)
+0(C)
+0U(N)
+r(u)du+M(N)
+r(t∧τ(N)
+0(C))
+≤S(N)
+r(0)+/integraldisplayt
+0/parenleftBig
+{kr1+Ckr2}S(N)
+r(u∧τ(N)
+0(C))+Nkr4/parenrightBig
+du
++ sup
+0≤u≤tM(N)
+r(u∧τ(N)
+0(C)).
+10Hence, from Gronwall’s inequality, for all t≥0 andr≤r(1)
+max,
+S(N)
+r(t∧τ(N)
+0(C))≤/braceleftBig
+N(Cr+kr4t)+ sup
+0≤u≤tM(N)
+r(u∧τ(N)
+0(C))/bracerightBig
+e(kr1+Ckr2)t.
+(2.21)
+Now, as in (2.16), we have
+E/braceleftBigg
+{M(N)
+r(t∧τ(N)
+1(C′)∧τ(N)
+0(C))}2−N/integraldisplayt∧τ(N)
+1(C′)∧τ(N)
+0(C)
+0V(N)
+r(u)du/bracerightBigg
+= 0,
+(2.22)
+from which it follows, using (2.7), that, for 1 ≤r≤r(2)
+max,
+E/parenleftBig
+{M(N)
+r(t∧τ(N)
+1(C′)∧τ(N)
+0(C))}2/parenrightBig
+≤E/braceleftBigg
+N/integraldisplayt∧τ(N)
+1(C′)∧τ(N)
+0(C))
+0V(N)
+r(u)du/bracerightBigg
+≤/integraldisplayt
+0{kr3ES(N)
+p(r)(u∧τ(N)
+1(C′)∧τ(N)
+0(C))+Nkr5}du
+≤N(C′
+r+kp(r),4T)kr3
+kp(r),1+Ckp(r),2(e(kp(r),1+Ckp(r),2t)−1)+Nkr5T,
+this last by (2.20), since p(r)≤r(1)
+maxfor 1≤r≤r(2)
+max. Using Doob’s
+inequality, it follows that, for any a >0,
+P/bracketleftBig
+sup
+0≤u≤TM(N)
+r(u∧τ(N)
+0(C))> aN/bracketrightBig
+≤1
+Na2/braceleftbiggkr3(C′
+r+kp(r),4T)
+kp(r),1+Ckp(r),2(e(kp(r),1+Ckp(r),2T)−1)+kr5T/bracerightbigg
+=:C′
+rKr1+Kr2
+Na2.
+Takinga=γ/radicalbig
+(C′
+r∨1) and putting the result into (2.21) gives (2.19), with
+Kr0= (C′
+rKr1+Kr2)/(C′
+r∨1).
+Note also that sup0≤t≤TS(N)
+r(t)<∞a.s. for all 0 ≤r≤r(2)
+max, in view of
+Lemma 2.3 and Theorem 2.4.
+In what follows, we shall particularly need to control quantities of t he
+form/summationtext
+J∈JαJ(xN(s))d(J,ζ), where xN:=N−1XNand
+d(J,ζ) :=/summationdisplay
+j≥0|Jj|ζ(j), (2.23)
+11forζ∈ Rchosen such that ζ(j)≥1 grows fast enough with j: see (4.12).
+Defining
+τ(N)(a,ζ) := inf/braceleftBigg
+s:/summationdisplay
+J∈JαJ(xN(s))d(J,ζ)≥a/bracerightBigg
+,(2.24)
+infinite if there is no such s, we show in the following corollary that, under
+suitable assumptions, τ(N)(a,ζ) is rarely less than T.
+Corollary 2.5 Assume that Assumptions 2.1 hold, and that ζis such that
+/summationdisplay
+J∈JαJ(N−1X)d(J,ζ)≤ {k1N−1Sr(X)+k2}b(2.25)
+for some 1≤r:=r(ζ)≤r(2)
+maxand some b=b(ζ)≥1. For this value
+ofr, assume that S(N)
+r(0)≤NCrandS(N)
+p(r)(0)≤NC′
+rfor some constants
+CrandC′
+r. Assume further that S(N)
+0(0)≤NC0,S(N)
+1(0)≤NC1for some
+constants C0,C1, and define C:= 2(C0+k04T)ek01T. Then
+P[τ(N)(a,ζ)≤T]≤N−1{Kr0γ−2
+a+K00(C1∨1)C−2},
+for anya≥ {k2+k1C′′
+rT}b, whereγa:= (a1/b−k2)/{k1C′′
+rT},Kr0andC′′
+rT
+are as in Theorem 2.4, and K00is as in Lemma 2.3.
+Proof. In view of (2.25), it is enough to bound the probability
+P[ sup
+0≤t≤TS(N)
+r(t)≥N(a1/b−k2)/k1].
+However, Lemma 2.3 and Theorem 2.4 together bound this probability by
+N−1/braceleftbig
+Kr0γ−2
+a+K00(C1∨1)C−2/bracerightbig
+,
+whereγais as defined above, as long as a1/b−k2≥k1C′′
+rT.
+If (2.25) is satisfied,/summationtext
+J∈JαJ(xN(s))d(J,ζ)isa.s. bounded on0 ≤s≤T,
+becauseS(N)
+r(s) is. The corollary shows that the sum is then bounded by
+{k2+k1C′′
+r,T}b, except on an event of probability of order O(N−1). Usually,
+one can choose b= 1.
+123 Semigroup properties
+We make the following initial assumptions about the matrix A: first, that
+Aij≥0 for alli/ne}ationslash=j≥0;/summationdisplay
+j/negationslash=iAji<∞for alli≥0,(3.1)
+and then that, for some µ∈RZ+
++such that µ(m)≥1 for each m≥0, and
+for some w≥0,
+ATµ≤wµ. (3.2)
+We then use µto define the µ-norm
+/⌊ard⌊lξ/⌊ard⌊lµ:=/summationdisplay
+m≥0µ(m)|ξm|onRµ:={ξ∈ R:/⌊ard⌊lξ/⌊ard⌊lµ<∞}.(3.3)
+Note that there may be many possible choices for µ. In what follows, it is
+important that Fbe a Lipschitz operator with respect to the µ-norm, and
+this has to be borne in mind when choosing µ.
+Setting
+Qij:=AT
+ijµ(j)/µ(i)−wδij, (3.4)
+whereδis the Kronecker delta, we note that Qij≥0 fori/ne}ationslash=j, and that
+0≤/summationdisplay
+j/negationslash=iQij=/summationdisplay
+j/negationslash=iAT
+ijµ(j)/µ(i)≤w−Aii=−Qii,
+using (3.2) for the inequality, so that Qii≤0. Hence Qcan be augmented to
+a conservative Q–matrix, in the sense of Markov jump processes, by adding a
+coffin state ∂, and setting Qi∂:=−/summationtext
+j≥0Qij≥0. LetP(·) denote the semi-
+group of Markov transition matrices corresponding to the minimal p rocess
+associated with Q; then, in particular,
+Q=P′(0) and P′(t) =QP(t) for all t≥0 (3.5)
+(Reuter 1957, Theorem 3). Set
+RT
+ij(t) :=ewtµ(i)Pij(t)/µ(j). (3.6)
+13Theorem 3.1 LetAsatisfy Assumptions (3.1)and(3.2). Then, with the
+above definitions, Ris a strongly continuous semigroup on Rµ, and
+/summationdisplay
+i≥0µ(i)Rij(t)≤µ(j)ewtfor alljandt. (3.7)
+Furthermore, the sums/summationtext
+j≥0Rij(t)Ajk= (R(t)A)ikare well defined for all
+i,k, and
+A=R′(0)andR′(t) =R(t)Afor allt≥0.(3.8)
+Proof. We note first that, for x∈ Rµ,
+/⌊ard⌊lR(t)x/⌊ard⌊lµ≤/summationdisplay
+i≥0µ(i)/summationdisplay
+j≥0Rij(t)|xj|=ewt/summationdisplay
+i≥0/summationdisplay
+j≥0µ(j)Pji(t)|xj|
+≤ewt/summationdisplay
+j≥0µ(j)|xj|=ewt/⌊ard⌊lx/⌊ard⌊lµ, (3.9)
+sinceP(t) is substochastic on Z+; henceR:Rµ→ Rµ. To show strong
+continuity, we take x∈ Rµ, and consider
+/⌊ard⌊lR(t)x−x/⌊ard⌊lµ=/summationdisplay
+i≥0µ(i)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+j≥0Rij(t)xj−xi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/summationdisplay
+i≥0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleewt/summationdisplay
+j≥0µ(j)Pji(t)xj−µ(i)xi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤(ewt−1)/summationdisplay
+i≥0/summationdisplay
+j≥0µ(j)Pji(t)xj+/summationdisplay
+i≥0/summationdisplay
+j/negationslash=iµ(j)Pji(t)xj+/summationdisplay
+i≥0µ(i)xi(1−Pii(t))
+≤(ewt−1)/summationdisplay
+j≥0µ(j)xj+2/summationdisplay
+i≥0µ(i)xi(1−Pii(t)),
+from which it follows that lim t→0/⌊ard⌊lR(t)x−x/⌊ard⌊lµ= 0, by dominated conver-
+gence, since lim t→0Pii(t) = 1 for each i≥0.
+The inequality (3.7) follows from the definition of Rand the fact that P
+is substochastic on Z+. Then
+(ATRT(t))ij=/summationdisplay
+k/negationslash=iQikµ(i)
+µ(k)ewtµ(k)
+µ(j)Pkj(t)+(Qii+w)ewtµ(i)
+µ(j)Pij(t)
+=µ(i)
+µ(j)[(QP(t))ij+wPij(t)]ewt,
+14with (QP(t))ij=/summationtext
+k≥0QikPkj(t) well defined because P(t) is sub-stochastic
+andQis conservative. Using (3.5), this gives
+(ATRT(t))ij=µ(i)
+µ(j)d
+dt[Pij(t)ewt] =d
+dtRT
+ij(t),
+and this establishes (3.8).
+4 Main approximation
+LetXN,N≥1, beasequence ofpure jumpMarkov processes asinSection 1,
+withAandFdefined as in (1.4) and (1.5), and suppose that F:Rµ→ Rµ,
+withRµas defined in (3.3), for some µsuch that Assumption (3.2) holds.
+Suppose also that Fis locally Lipschitz in the µ-norm: for any z >0,
+sup
+x/negationslash=y:/bardblx/bardblµ,/bardbly/bardblµ≤z/⌊ard⌊lF(x)−F(y)/⌊ard⌊lµ//⌊ard⌊lx−y/⌊ard⌊lµ≤K(µ,F;z)<∞.(4.1)
+Then, for x(0)∈ RµandRas in (3.6), the integral equation
+x(t) =R(t)x(0)+/integraldisplayt
+0R(t−s)F(x(s))ds. (4.2)
+has a unique continuous solution xinRµon some non-empty time interval
+[0,tmax), such that, if tmax<∞, then/⌊ard⌊lx(t)/⌊ard⌊lµ→ ∞ast→tmax(Pazy 1983,
+Theorem 1.4, Chapter 6). Thus, if Awere the generator of R, the function x
+would be a mild solution of the deterministic equations (1.4). We now wish
+to show that the process xN:=N−1XNis close to x. To do so, we need a
+corresponding representation for XN.
+To find such a representation, let W(t),t≥0, be a pure jump path on X+
+that has only finitely many jumps up to time T. Then we can write
+W(t) =W(0)+/summationdisplay
+j:σj≤t∆W(σj),0≤t≤T, (4.3)
+where ∆W(s) :=W(s)−W(s−)andσj,j≥1, denote thetimes when Whas
+its jumps. Now let Asatisfy (3.1) and (3.2), and let R(·) be the associated
+semigroup, as defined in (3.6). Define the path W∗(t), 0≤t≤T, from the
+equation
+W∗(t) :=R(t)W(0)+/summationtext
+j:σj≤tR(t−σj)∆j−/integraltextt
+0R(t−s)AW(s)ds,
+(4.4)
+15where ∆ j:= ∆W(σj). Note that the latter integral makes sense, because
+each of the sums/summationtext
+j≥0Rij(t)Ajkis well defined, from Theorem 3.1, and
+because only finitely many of the coordinates of Ware non-zero.
+Lemma 4.1 W∗(t) =W(t)for all0≤t≤T.
+Proof. Fix any t, and suppose that W∗(s) =W(s) for alls≤t. This is
+clearly the case for t= 0. Let σ(t)> tdenote the time of the first jump
+ofWaftert. Then, for any 0 < h < σ(t)−t, using the semigroup property
+forRand (4.4),
+W∗(t+h)−W∗(t)
+= (R(h)−I)R(t)W(0)+/summationdisplay
+j:σj≤t(R(h)−I)R(t−σj)∆j (4.5)
+−/integraldisplayt
+0(R(h)−I)R(t−s)AW(s)ds−/integraldisplayt+h
+tR(t+h−s)AW(t)ds,
+where, in the last integral, we use the fact that there are no jumps ofW
+between tandt+h. Thus we have
+W∗(t+h)−W∗(t)
+= (R(h)−I)
+
+R(t)W(0)+/summationdisplay
+j:σj≤tR(t−σj)∆j−/integraldisplayt
+0R(t−s)AW(s)ds
+
+
+−/integraldisplayt+h
+tR(t+h−s)AW(t)ds
+= (R(h)−I)W(t)−/integraldisplayt+h
+tR(t+h−s)AW(t)ds. (4.6)
+But now, for x∈ X+,
+/integraldisplayt+h
+tR(t+h−s)Axds= (R(h)−I)x,
+from (3.8), so that W∗(t+h) =W∗(t) for all t+h < σ(t), implying that
+W∗(s) =W(s) for all s < σ(t). On the other hand, from (4.4), we have
+W∗(σ(t))−W∗(σ(t)−) = ∆W(σ(t)), so that W∗(s) =W(s) for alls≤σ(t).
+Thus we can prove equality over the interval [0 ,σ1], and then successively
+over the intervals [ σj,σj+1], until [0 ,T] is covered.
+16Now suppose that Warises as a realization of XN. ThenXNhas transi-
+tion rates such that
+MN(t) :=/summationdisplay
+j:σj≤t∆XN(σj)−/integraldisplayt
+0AXN(s)ds−/integraldisplayt
+0NF(xN(s))ds(4.7)
+is a zero mean local martingale. In view of Lemma 4.1, we can use (4.4) t o
+write
+XN(t) =R(t)XN(0)+/tildewiderMN(t)+N/integraldisplayt
+0R(t−s)F(xN(s))ds,(4.8)
+where
+/tildewiderMN(t) :=/summationdisplay
+j:σj≤tR(t−σj)∆XN(σj)
+−/integraldisplayt
+0R(t−s)AXN(s)ds−/integraldisplayt
+0R(t−s)NF(xN(s))ds.(4.9)
+Thus, comparing (4.8) and (4.2), we expect xNandxto be close, for
+0≤t≤T < tmax, provided that we can show that supt≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµis small,
+where/tildewidemN(t) :=N−1/tildewiderMN(t). Indeed, if xN(0) andx(0) are close, then
+/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ
+≤ /⌊ard⌊lR(t)(xN(0)−x(0))/⌊ard⌊lµ
++/integraldisplayt
+0/⌊ard⌊lR(t−s)[F(xN(s))−F(x(s))]/⌊ard⌊lµds+/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ
+≤ewt/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ
++/integraldisplayt
+0ew(t−s)K(µ,F;2ΞT)/⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµds+/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ,(4.10)
+by (3.9), with the stage apparently set for Gronwall’s inequality, ass uming
+that/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµand sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµare small enough that then
+/⌊ard⌊lxN(t)/⌊ard⌊lµ≤2ΞTfor 0≤t≤T, where Ξ T:= sup0≤t≤T/⌊ard⌊lx(t)/⌊ard⌊lµ.
+Bounding sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµis, however, not so easy. Since /tildewiderMNis not
+itselfamartingale, wecannotdirectlyapplymartingaleinequalitiestoc ontrol
+its fluctuations. However, since
+/tildewiderMN(t) =/integraldisplayt
+0R(t−s)dMN(s), (4.11)
+17we can hope to use control over the local martingale MNinstead. For this
+and the subsequent argument, we introduce some further assum ptions.
+Assumption 4.2
+1. There exists r=rµ≤r(2)
+maxsuch that supj≥0{µ(j)/νr(j)}<∞.
+2. There exists ζ∈ Rwithζ(j)≥1for alljsuch that (2.25)is satisfied
+for some b=b(ζ)≥1andr=r(ζ)such that 1≤r(ζ)≤r(2)
+max, and that
+Z:=/summationdisplay
+k≥0µ(k)(|Akk|+1)/radicalbig
+ζ(k)<∞. (4.12)
+The requirement that ζsatisfies (4.12) as well as satisfying (2.25) for some
+r≤r(2)
+maximplies in practice that it must be possible to take r(1)
+maxandr(2)
+max
+to be quite large in Assumption 2.1; see the examples in Section 5.
+Note that part 1 of Assumption 4.2 implies that lim j→∞{µ(j)/νr(j)}= 0
+for some r= ˜rµ≤rµ+1. We define
+ρ(ζ,µ) := max {r(ζ),p(r(ζ)),˜rµ}, (4.13)
+wherep(·) is as in Assumptions 2.1. We can now prove the following lemma,
+which enables us to control the paths of /tildewiderMNby using fluctuation bounds for
+the martingale MN.
+Lemma 4.3 Under Assumption 4.2,
+/tildewiderMN(t) =MN(t)+/integraldisplayt
+0R(t−s)AMN(s)ds.
+Proof. From (3.8), we have
+R(t−s) =I+/integraldisplayt−s
+0R(v)Adv.
+Substituting this into (4.11), we obtain
+/tildewiderMN(t) =/integraldisplayt
+0R(t−s)dMN(s)
+18=MN(t)+/integraldisplayt
+0/braceleftbigg/integraldisplayt
+0R(v)A1[0,t−s](v)dv/bracerightbigg
+dMN(s)
+=MN(t)+/integraldisplayt
+0/braceleftbigg/integraldisplayt
+0R(v)A1[0,t−s](v)dv/bracerightbigg
+dXN(s)
+−/integraldisplayt
+0/braceleftbigg/integraldisplayt
+0R(v)A1[0,t−s](v)dv/bracerightbigg
+F0(xN(s))ds.
+It remains to change the order of integration in the double integrals , for
+which we use Fubini’s theorem.
+In the first, the outer integral is almost surely a finite sum, and at e ach
+jump time tXN
+lwe havedXN(tXN
+l)∈ J. Hence it is enough that, for each i,
+mandt,/summationtext
+j≥0Rij(t)Ajmis absolutely summable, which follows from Theo-
+rem 3.1. Thus we have
+/integraldisplayt
+0/braceleftbigg/integraldisplayt
+0R(v)A1[0,t−s](v)dv/bracerightbigg
+dXN(s) =/integraldisplayt
+0R(v)A{XN(t−v)−XN(0)}dv.
+(4.14)
+For the second, the k-th component of R(v)AF0(xN(s)) is just
+/summationdisplay
+j≥0Rkj(v)/summationdisplay
+l≥0Ajl/summationdisplay
+J∈JJlαJ(xN(s)). (4.15)
+Now, from (3.7), we have 0 ≤Rkj(v)≤µ(j)ewv/µ(k), and
+/summationdisplay
+j≥0µ(j)|Ajl| ≤µ(l)(2|All|+w), (4.16)
+becauseATµ≤wµ. Hence, puttingabsolutevaluesinthesummandsin(4.15)
+yields at most
+ewv
+µ(k)/summationdisplay
+J∈JαJ(xN(s))/summationdisplay
+l≥0|Jl|µ(l)(2|All|+w).
+Now, in view of (4.12) and since ζ(j)≥1 for allj, there is a constant K <∞
+such that µ(l)(2|All|+w)≤Kζ(l). Furthermore, ζsatisfies (2.25), so that,
+by Corollary 2.5,/summationtext
+J∈JαJ(xN(s))/summationtext
+l≥0|Jl|ζ(l) is a.s. uniformly bounded in
+0≤s≤T. Hence we can apply Fubini’s theorem, obtaining
+/integraldisplayt
+0/braceleftbigg/integraldisplayt
+0R(v)A1[0,t−s](v)dv/bracerightbigg
+F0(xN(s))ds=/integraldisplayt
+0R(v)A/braceleftbigg/integraldisplayt−v
+0F0(xN(s))ds/bracerightbigg
+dv,
+19and combining this with (4.14) proves the lemma.
+We now introduce the exponential martingales that we use to bound the
+fluctuations of MN. Forθ∈RZ+bounded and x∈ Rµ,
+ZN,θ(t) :=eθTxN(t)exp/braceleftBig
+−/integraltextt
+0gNθ(xN(s−))ds/bracerightBig
+, t≥0,
+is a non-negative finite variation local martingale, where
+gNθ(ξ) :=/summationdisplay
+J∈JNαJ(ξ)/parenleftBig
+eN−1θTJ−1/parenrightBig
+.
+Fort≥0, we have
+logZN,θ(t) =θTxN(t)−/integraldisplayt
+0gNθ(xN(s−))ds
+=θTmN(t)−/integraldisplayt
+0ϕN,θ(xN(s−),s)ds, (4.17)
+where
+ϕN,θ(ξ) :=/summationdisplay
+J∈JNαJ(ξ)/parenleftBig
+eN−1θTJ−1−N−1θTJ/parenrightBig
+,(4.18)
+andmN(t) :=N−1MN(t). Note also that we can write
+ϕN,θ(ξ) =N/integraldisplay1
+0(1−r)D2vN(ξ,rθ)[θ,θ]dr, (4.19)
+where
+vN(ξ,θ′) :=/summationdisplay
+J∈JαJ(ξ)eN−1(θ′)TJ,
+andD2vNdenotes thematrixofsecond derivatives withrespect totheseco nd
+argument:
+D2vN(ξ,θ′)[ζ1,ζ2] :=N−2/summationdisplay
+J∈JαJ(ξ)eN−1(θ′)TJζT
+1JJTζ2(4.20)
+for anyζ1,ζ2∈ Rµ.
+Now choose any B:= (Bk, k≥0)∈ R, and define ˜ τ(N)
+k(B) by
+˜τ(N)
+k(B) := inf/braceleftBigg
+t≥0:/summationdisplay
+J:Jk/negationslash=0αJ(xN(t−))> Bk/bracerightBigg
+.
+Our exponential bound is as follows.
+20Lemma 4.4 For anyk≥0,
+P
+sup
+0≤t≤T∧˜τ(N)
+k(B)|mk
+N(t)| ≥δ
+≤2exp(−δ2N/2BkK∗T).
+for all0< δ≤BkK∗T, whereK∗:=J2
+∗eJ∗, andJ∗is as in(1.2).
+Proof. Takeθ=e(k)β, forβto be chosen later. We shall argue by stopping
+the local martingale ZN,θat timeσ(N)(k,δ), where
+σ(N)(k,δ) :=T∧˜τ(N)
+k(B)∧inf{t:mk
+N(t)≥δ}.
+Note that eN−1θTJ≤eJ∗, so long as |β| ≤N, so that
+D2vN(ξ,rθ)[θ,θ]≤N−2/parenleftBigg/summationdisplay
+J:Jk/negationslash=0αJ(ξ)/parenrightBigg
+β2K∗.
+Thus, from (4.19), we have
+ϕN,θ(xN(u−))≤1
+2N−1Bkβ2K∗, u≤˜τ(N)
+k(B),
+and hence, on the event that σ(N)(k,δ) = inf{t:mk
+N(t)≥δ} ≤(T∧˜τ(N)
+k(B)),
+we have
+ZN,θ(σ(k,δ))≥exp{βδ−1
+2N−1Bkβ2K∗T}.
+But since ZN,θ(0) = 1, it now follows from the optional stopping theorem
+and Fatou’s lemma that
+1≥E{ZN,θ(σ(N)(k,δ))}
+≥P/bracketleftBig
+sup
+0≤t≤T∧˜τ(N)
+k(B)mk
+N(t)≥δ/bracketrightBig
+exp{βδ−1
+2N−1Bkβ2K∗T}.
+We can choose β=δN/B kK∗T, as long as δ/BkK∗T≤1, obtaining
+P
+sup
+0≤t≤T∧˜τ(N)
+k(B)mk
+N(t)≥δ
+≤exp(−δ2N/2BkK∗T).
+Repeating with
+˜σ(N)(k,δ) :=T∧˜τ(N)
+k(B)∧inf{t:−mk
+N(t)≥δ},
+21and choosing β=δN/B kK∗T, gives the lemma.
+Theprecedinglemmagivesaboundforeachindividualcomponentof MN.
+We need first to translate this into a statement for all components simulta-
+neously. For ζas in Assumption 4.2, we start by writing
+Z(1)
+∗:= max
+k≥1k−1#{m:ζ(m)≤k};Z(2)
+∗:= sup
+k≥0µ(k)(|Akk|+1)/radicalbig
+ζ(k).(4.21)
+Z(2)
+∗is clearly finite, because of Assumption 4.2, and the same is true for Z(1)
+∗
+also, since Zof Assumption 4.2 is at least # {m:ζ(m)≤k}/√
+k, for each k.
+Then, using the definition (2.24) of τ(N)(a,ζ), note that, for every k,
+/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))h(k)≤/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))h(k)d(J,ζ)
+|Jk|ζ(k)≤ah(k)
+ζ(k),(4.22)
+for anyt < τ(N)(a,ζ) and any h∈ R, and that, for any K ⊆Z+,
+/summationdisplay
+k∈K/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))h(k)≤/summationdisplay
+k∈K/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))h(k)d(J,ζ)
+|Jk|ζ(k)
+≤a
+mink∈K(ζ(k)/h(k)). (4.23)
+From (4.22) with h(k) = 1 for all k, if we choose B:= (a/ζ(k), k≥0), then
+τ(N)(a,ζ)≤˜τ(N)
+k(B) for allk. For this choice of B, we can take
+δ2
+k:=δ2
+k(a) :=4aK∗TlogN
+Nζ(k)=4BkK∗TlogN
+N(4.24)
+in Lemma 4.4 for k∈κN(a), where
+κN(a) :=/braceleftbig
+k:ζ(k)≤1
+4aK∗TN/logN/bracerightbig
+={k:Bk≥4logN/K∗TN},
+(4.25)
+since then δk(a)≤BkK∗T. Note that then, from (4.12),
+/summationdisplay
+k∈κN(a)µ(k)δk(a)≤2Z/radicalbig
+aK∗TN−1logN, (4.26)
+withZas defined in Assumption 4.2, and that
+|κN(a)| ≤1
+4aZ(1)
+∗K∗TN/logN. (4.27)
+22Lemma 4.5 If Assumptions 4.2 are satisfied, taking δk(a)andκN(a)as
+defined in (4.24)and(4.25), and for any η∈ R, we have
+1.P
+/uniondisplay
+k∈κN(a)/braceleftBig
+sup
+0≤t≤T∧τ(N)(a,ζ)|mN(t)| ≥δk(a)/bracerightBig
+≤aZ(1)
+∗K∗T
+2NlogN;
+2.P
+/summationdisplay
+k/∈κN(a)Xk
+N(t) = 0for all0≤t≤T∧τ(N)(a,ζ)
+≥1−4logN
+K∗N;
+3. sup
+0≤t≤T∧τ(N)(a,ζ)
+
+/summationdisplay
+k/∈κN(a)η(k)|Fk(xN(t))|
+
+≤aJ∗
+mink/∈κN(a)(ζ(k)/η(k)).
+Proof. For part 1, use Lemma 4.4 together with (4.24) and (4.27) to give
+the bound. For part 2, the total rate of jumps into coordinates w ith indices
+k /∈κN(a) is
+/summationdisplay
+k/∈κN(a)/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))≤a
+mink/∈κN(a)ζ(k),
+ift≤τ(N)(a,ζ),using(4.23)with K= (κN(a))c,which, combinedwith(4.25),
+proves the claim. For the final part, if t≤τ(N)(a,ζ),
+/summationdisplay
+k/∈κN(a)η(k)|Fk(xN(t))| ≤/summationdisplay
+k/∈κN(a)η(k)/summationdisplay
+J:Jk/negationslash=0αJ(xN(t))J∗,
+and the inequality follows once more from (4.23).
+LetB(1)
+N(a) andB(2)
+N(a) denote the events
+B(1)
+N(a) :=
+
+/summationdisplay
+k/∈κN(a)Xk
+N(t) = 0 for all 0 ≤t≤T∧τ(N)(a,ζ)
+
+;
+B(2)
+N(a) :=
+/intersectiondisplay
+k∈κN(a)/braceleftBig
+sup
+0≤t≤T∧τ(N)(a,ζ)|mN(t)| ≤δk(a)/bracerightBig
+,(4.28)
+and setBN(a) :=B(1)
+N(a)∩B(2)
+N(a). Then, by Lemma 4.5, we deduce that
+P[BN(a)c]≤aZ(1)
+∗K∗T
+2NlogN+4logN
+K∗N, (4.29)
+23of order O(N−1logN) for each fixed a. Thus we have all the components
+ofMNsimultaneously controlled, except on a set of small probability. We
+now translate this into the desired assertion about the fluctuation s of/tildewidemN.
+Lemma 4.6 If Assumptions 4.2 are satisfied, then, on the event BN(a),
+sup
+0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ≤√aK4.6/radicalbigg
+logN
+N,
+where the constant K4.6depends on Tand the parameters of the process.
+Proof. From Lemma 4.3, it follows that
+sup
+0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ (4.30)
+≤sup
+0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊lmN(t)/⌊ard⌊lµ+ sup
+0≤t≤T∧τ(N)(a,ζ)/integraldisplayt
+0/⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµds.
+For the first term, on BN(a) and for 0 ≤t≤T∧τ(N)(a,ζ), we have
+/⌊ard⌊lmN(t)/⌊ard⌊lµ≤/summationdisplay
+k∈κN(a)µ(k)δk(a)+/integraldisplayt
+0/summationdisplay
+k/∈κN(a)µ(k)|Fk(xN(u))|du.
+The first sum is bounded using (4.26) by 2 Z√aK∗T N−1/2√logN, the sec-
+ond, from Lemma 4.5 and (4.25), by
+TaJ∗
+mink/∈κN(a)(ζ(k)/µ(k))≤Z(2)
+∗2J∗/radicalbigg
+Ta
+K∗/radicalbigg
+logN
+N.
+For the second term in (4.30), from (3.7) and (4.16), we note that
+/⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµ≤/summationdisplay
+k≥0µ(k)/summationdisplay
+l≥0Rkl(t−s)/summationdisplay
+r≥0|Alr||mr
+N(s)|
+≤ew(t−s)/summationdisplay
+l≥0µ(l)/summationdisplay
+r≥0|Alr||mr
+N(s)|
+≤ew(t−s)/summationdisplay
+r≥0µ(r){2|Arr|+w}|mr
+N(s)|.
+24OnBN(a) and for 0 ≤s≤T∧τ(N)(a,ζ), from (4.12), the sum for r∈κN(a)
+is bounded using
+/summationdisplay
+r∈κN(a)µ(r){2|Arr|+w}|mr
+N(s)|
+≤/summationdisplay
+r∈κN(a)µ(r){2|Arr|+w}δr(a)
+≤/summationdisplay
+r∈κN(a)µ(r){2|Arr|+w}/radicalBigg
+4aK∗TlogN
+Nζ(r)
+≤(2∨w)Z/radicalbig
+4aK∗T/radicalbigg
+logN
+N.
+The remaining sum is then bounded by Lemma 4.5, on the set BN(a) and
+for 0≤s≤T∧τ(N)(a,ζ), giving at most
+/summationdisplay
+r/∈κN(a)µ(r){2|Arr|+w}|mr
+N(s)|
+≤/summationdisplay
+r/∈κN(a)µ(r){2|Arr|+w}/integraldisplays
+0|Fr(xN(t))|dt
+≤(2∨w)saJ∗
+mink/∈κN(a)(ζ(k)/µ(k){|Akk|+1})
+≤(2∨w)Z(2)
+∗2J∗/radicalbigg
+Ta
+K∗/radicalbigg
+logN
+N.
+Integrating, it follows that
+sup
+0≤t≤T∧τ(N)(a,ζ)/integraldisplayt
+0/⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµds
+≤(2T∨1)ewT/braceleftBigg/radicalbig
+4aK∗TZ+Z(2)
+∗J2J∗/radicalbigg
+Ta
+K∗/bracerightBigg/radicalbigg
+logN
+N,
+and the lemma follows.
+This has now established the control on sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµthat we need,
+in order to translate (4.10) into a proof of the main theorem.
+25Theorem 4.7 Suppose that (1.2),(1.3),(3.1),(3.2)and(4.1)are all satis-
+fied, and that Assumptions 2.1 and 4.2 hold. Recalling the defi nition(4.13)
+ofρ(ζ,µ), forζas given in Assumption 4.2, suppose that S(N)
+ρ(ζ,µ)(0)≤NC∗
+for some C∗<∞.
+Letxdenote the solution to (4.2)with initial condition x(0)satisfying
+Sρ(ζ,µ)(x(0))<∞. Thentmax=∞.
+Fix any T, and define ΞT:= sup0≤t≤T/⌊ard⌊lx(t)/⌊ard⌊lµ. If/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ≤
+1
+2ΞTe−(w+k∗)T, wherek∗:=ewTK(µ,F;2ΞT), then there exist constants c1,c2
+depending on C∗,Tand the parameters of the process, such that for all N
+large enough
+P/parenleftBigg
+sup
+0≤t≤T/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ>/parenleftBigg
+ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+c1/radicalbigg
+logN
+N/parenrightBigg
+ek∗T/parenrightBigg
+≤c2logN
+N. (4.31)
+Proof. AsS(N)
+ρ(ζ,µ)(0)≤NC∗, it follows also that S(N)
+r(0)≤NC∗for all
+0≤r≤ρ(ζ,µ). Fix any T < tmax, takeC:= 2(C∗+k04T)ek01T, and observe
+that, for r≤ρ(ζ,µ)∧r(2)
+max, and such that p(r)≤ρ(ζ,µ), we can take
+C′′
+rT≤/tildewideCrT:={2(C∗∨1)+kr4T}e(kr1+Ckr2)T, (4.32)
+in Theorem 2.4, since we can take C∗to bound CrandC′
+r. In particular,
+r=r(ζ) as defined in Assumption 4.2 satisfies both the conditions on r
+for (4.32) to hold. Then, taking a:={k2+k1/tildewideCr(ζ)T}b(ζ)in Corollary 2.5, it
+follows that for some constant c3>0, on the event BN(a),
+P[τ(N)(a,ζ)≤T]≤c3N−1.
+Then, from (4.29), for some constant c4,P[BN(a)c]≤c4N−1logN. Here,
+the constants c3,c4depend on C∗,Tand the parameters of the process.
+We now use Lemma 4.6 to bound the martingale term in (4.10). It fol-
+lows that, on the event BN(a)∩ {τ(N)(a,ζ)> T}and on the event that
+/⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµ≤ΞTfor all 0≤s≤t,
+/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ≤/parenleftBigg
+ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
+logN
+N/parenrightBigg
++k∗/integraldisplayt
+0/⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµds,
+26wherek∗:=ewTK(µ,F;2ΞT). Then from Gronwall’s inequality, on the
+eventBN(a)∩{τ(N)(a,ζ)> T},
+/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ≤/parenleftBigg
+ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
+logN
+N/parenrightBigg
+ek∗t,
+(4.33)
+for all 0≤t≤T, provided that
+/parenleftBigg
+ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
+logN
+N/parenrightBigg
+≤ΞTe−k∗T.
+This is true for all Nsufficiently large, if /⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ≤1
+2ΞTe−(w+k∗)T,
+which we have assumed. We have thus proved (4.31), since, as show n above,
+P(BN(a)c∪{τ(N)(a,ζ)> T}c) =O(N−1logN).
+We now use this to show that in fact tmax=∞. Forx(0) as above, we
+can take xj
+N(0) :=N−1⌊Nxj(0)⌋ ≤xj(0), so that S(N)
+ρ(ζ,µ)(0)≤NC∗forC∗:=
+Sρ(ζ,µ)(x(0))<∞. Then, by (4.13), lim j→∞{µ(j)/νρ(ζ,µ)(j)}= 0, so it fol-
+lowseasilyusing boundedconvergence that /⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ→0asN→ ∞.
+Hence, for any T < t max, it follows from (4.31) that /⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ→D0
+asN→ ∞, fort≤T, with uniform bounds over the interval, where ‘ →D’
+denotes convergence in distribution. Also, by Assumption 4.2, ther e is a con-
+stantc5such that /⌊ard⌊lxN(t)/⌊ard⌊lµ≤c5N−1S(N)
+rµ(t) for each t, whererµ≤r(2)
+maxand
+rµ≤ρ(ζ,µ). Hence, using Lemma 2.3 and Theorem 2.4, sup0≤t≤2T/⌊ard⌊lxN(t)/⌊ard⌊lµ
+remains bounded in probability as N→ ∞. Hence it is impossible that
+/⌊ard⌊lx(t)/⌊ard⌊lµ→ ∞asT→tmax<∞,implyingthatinfact tmax=∞forsuchx(0).
+Remark . The dependence on the initial conditions is considerably compli-
+cated by the way the constant Cappears in the exponent, for instance in the
+expression for /tildewideCrTin the proof of Theorem 4.7. However, if kr2in Assump-
+tions 2.1 can be chosen to be zero, as for instance in the examples be low, the
+dependence simplifies correspondingly.
+Therearebiologicallyplausiblemodelsinwhichtherestrictionto Jl≥ −1
+is irksome. In populations in which members of a given type lcan fight one
+another, a natural possibility is to have a transition J=−2e(l)at a rate
+proportional to Xl(Xl−1), which translates to αJ=α(N)
+J=γxl(xl−N−1),
+a function depending on N. Replacing this with αJ=γ(xl)2removes the
+27N-dependence, but yields a process that can jump to negative value s ofXl.
+For this reason, it is useful to be able to allow the transition rates αJto
+depend on N.
+Since the arguments inthis paper are not limiting arguments for N→ ∞,
+it does not require many changes to derive the corresponding resu lts. Quan-
+tities such as A,F,Ur(x) andVr(x) now depend on N; however, Theorem 4.7
+continues toholdwithconstants c1andc2thatdo notdepend on N, provided
+thatµ,w,ν, theklmfrom Assumption 2.1 and ζfrom Assumption 4.2 can
+be chosen to be independent of N, and that the quantities Z(l)
+∗from (4.21)
+can be bounded uniformly in N. On the other hand, the solution x=x(N)
+of (4.2) that acts as approximation to xNin Theorem 4.7 now itself depends
+onN, through R=R(N)andF=F(N). IfA(and hence R) can be taken
+to be independent of N, and lim N→∞/⌊ard⌊lF(N)−F/⌊ard⌊lµ= 0 for some fixed µ–
+Lipschitz function F, a Gronwall argument can be used to derive a bound
+for the difference between x(N)and the (fixed) solution xto equation (4.2)
+withN-independent RandF. IfAhas to depend on N, the situation is
+more delicate.
+5 Examples
+We begin with some general remarks, to show that the assumptions are sat-
+isfied in many practical contexts. We then discuss two particular ex amples,
+those of Kretzschmar (1993) and of Arrigoni (2003), that fitte d poorly or
+not at all into the general setting of Barbour & Luczak (2008), th ough the
+other systems referred to in the introduction could also be treate d similarly.
+In both of our chosen examples, the index jrepresents a number of individ-
+uals — parasites in a host in the first, animals in a patch in the second —
+and we shall for now use the former terminology for the preliminary, general
+discussion.
+Transitions that can typically be envisaged are: births of a few para sites,
+which may occur either in the same host, or in another, if infection is b eing
+represented; births and immigration of hosts, with or without para sites; mi-
+gration of parasites between hosts; deaths of parasites; death s of hosts; and
+treatment of hosts, leading to the deaths of many of the host’s pa rasites. For
+births of parasites, there is a transition X→X+J, whereJtakes the form
+Jl= 1;Jm=−1;Jj= 0, j/ne}ationslash=l,m, (5.1)
+28indicating that one m-host has become an l-host. For births of parasites
+within a host, a transition rate of the form bl−mmXmcould be envisaged,
+withl > m, the interpretation being that there are Xmhosts with parasite
+burdenm, each of which gives birth to soffspring at rate bs, for some small
+values of s. For infection of an m-host, a possible transition rate would be
+of the form
+Xm/summationdisplay
+j≥0N−1Xjλpj,l−m,
+since an m-host comes into contact with j-hosts at a rate proportional to
+their density in the host population, and pjrrepresents the probability of a
+j-host transferring rparasites to the infected host during the contact. The
+probability distributions pj·can be expected to be stochastically increasing
+inj. Deaths of parasites also give rise to transitions of the form (5.1),
+but now with l < m, the simplest form of rate being just dmXmforl=
+m−1, though d=dmcould also be chosen to increase with parasite burden.
+Treatment of a host would lead to values of lmuch smaller than m, and
+a rate of the form κXmfor the transition with l= 0 would represent fully
+successful treatment of randomly chosen individuals. Births and d eaths of
+hosts and immigration all lead to transitions of the form
+Jl=±1;Jj= 0, j/ne}ationslash=l. (5.2)
+Fordeaths, Jl=−1, anda typical ratewould be d′Xl. Forbirths, Jl= 1, and
+a possible rate would be/summationtext
+j≥0Xjb′
+jl(withl= 0 only, if new-born individuals
+are free of parasites). For immigration, constant rates λlcould be supposed.
+Finally, for migration of individual parasites between hosts, transit ions are
+of the form
+Jl=Jm=−1;Jl+1= 1;Jm−1= 1;Jj= 0, j/ne}ationslash=l,m,l+1,m−1,
+(5.3)
+a possible rate being γmXmN−1Xl.
+For all the above transitions, we can take J∗= 2 in (1.2), and (1.3) is
+satisfied in biologically sensible models. (3.1) and (3.2) depend on the wa y in
+which the matrix Acan be defined, which is more model specific; in practice,
+(3.1) is very simple to check. The choice of µin (3.2) is influenced by the
+need to have (4.1) satisfied. For Assumptions 2.1, a possible choice o fνis to
+takeν(j) = (j+1) for each j≥0, withS1(X) then representing the num-
+ber of hosts plus the number of parasites. Satisfying (2.5) is then e asy for
+29transitions only involving the movement of a single parasite, but in gen eral
+requires assumptions as to the existence of the r-th moments of the distri-
+butions of the numbers of parasites introduced at birth, immigratio n and
+infection events. For (2.6), in which transitions involving a net reduc tion
+in the total number of parasites and hosts can be disregarded, th e parasite
+birth events are those in which the rates typically have a factor mXmfor
+transitions with Jm=−1, withmin principle unbounded. However, at such
+events, an m-individual changes to an m+sindividual, with the number s
+of offspring of the parasite being typically small, so that the value of JTνr
+associated with this rate has magnitude mr−1; the product mXmmr−1, when
+summed over m, then yields a contribution of magnitude Sr(X), which is al-
+lowable in(2.6). Similar considerations showthat theterms N−1S0(X)Sr(X)
+accommodate the migration rates suggested above. Finally, in orde r to have
+Assumptions 4.2 satisfied, it is in practice necessary that Assumptio ns 2.1
+are satisfied for large values of r, thereby imposing restrictions on the dis-
+tributions of the numbers of parasites introduced at birth, immigra tion and
+infection events, as above.
+5.1 Kretzschmar’s model
+Kretzschmar (1993) introduced a model of a parasitic infection, in which the
+transitions from state Xare as follows:
+J=e(i−1)−e(i)at rate Niµxi, i ≥1;
+J=−e(i)at rate N(κ+iα)xi, i≥0;
+J=e(0)at rate Nβ/summationtext
+i≥0xiθi;
+J=e(i+1)−e(i)at rate Nλxiϕ(x), i ≥0,
+wherex:=N−1X,ϕ(x) :=/⌊ard⌊lx/⌊ard⌊l11{c+/⌊ard⌊lx/⌊ard⌊l1}−1withc >0, and/⌊ard⌊lx/⌊ard⌊l11:=/summationtext
+j≥1j|x|j; here, 0≤θ≤1, andθidenotes its i-th power (our θcorresponds
+to the constant ξin [7]). Both (1.2) and (1.3) are obviously satisfied. For
+Assumptions (3.1), (3.2) and (4.1), we note that equation corresp onding
+to (1.5) has
+Aii=−{κ+i(α+µ)};AT
+i,i−1=iµandAT
+i0=βθi, i≥2;
+A11=−{κ+α+µ};AT
+10=µ+βθ;
+A00=−κ+β, i≥1,
+30with all other elements of the matrix equal to zero, and
+Fi(x) =λ(xi−1−xi)ϕ(x), i≥1;F0(x) =−λx0ϕ(x).
+Hence Assumption (3.1) isimmediate, andAssumption (3.2)holds for µ(j) =
+(j+1)s, for any s≥0, withw= (β−κ)+. For the choice µ(j) =j+1,F
+maps elements of RµtoRµ, and is also locally Lipschitz in the µ-norm, with
+K(µ,F;Ξ) =c−2λΞ(2c+Ξ).
+For Assumptions 2.1, choose ν=µ; then (2.5) is a finite sum for each
+r≥0. Turning to (2.6), it is immediate that U0(x)≤βS0(x). Then, for
+r≥1,
+/summationdisplay
+i≥0λϕ(N−1X)Xi{(i+2)r−(i+1)r} ≤λS1(X)
+S0(X)/summationdisplay
+i≥0rXi(i+2)r−1
+≤r2r−1λSr(X),
+since, by Jensen’s inequality, S1(X)Sr−1(X)≤S0(X)Sr(X). Hence we can
+takekr2=kr4= 0 and kr1=β+r2r−1λin (2.6), for any r≥1, so that
+r(1)
+max=∞. Finally, for (2.7),
+V0(x)≤(κ+β)S0(x)+αS1(x),
+so thatk03=κ+β+αandk05= 0, and
+Vr(x)≤r2(κS2r(x)+αS2r+1(x)+µS2r−1(x)+22(r−1)λS2r−1(x))+βS0(x),
+so that we can take p(r) = 2r+1,kr3=β+r2{κ+α+µ+22(r−1)λ}, and
+kr5= 0 for any r≥1, and so r(2)
+max=∞. In Assumptions 4.2, we can clearly
+takerµ= 1 and ζ(k) = (k+1)7, givingr(ζ) = 8,b(ζ) = 1 and ρ(ζ,µ) = 17.
+5.2 Arrigoni’s model
+Inthemetapopulation model ofArrigoni (2003), thetransitions f romstate X
+are as follows:
+J=e(i−1)−e(i)at rateNixi(di+γ(1−ρ)), i ≥2;
+J=e(0)−e(1)at rateNx1(d1+γ(1−ρ)+κ);
+J=e(i+1)−e(i)at rateNibixi, i ≥1;
+J=e(0)−e(i)at rateNxiκ, i ≥2;
+J=e(k+1)−e(k)+e(i−1)−e(i)at rateNixixkργ, k ≥0, i≥1;
+31as before, x:=N−1X. Here, the total number N=/summationtext
+j≥0Xj=S0(X) of
+patches remains constant throughout, and the number of animals in any one
+patch changes by at most one at each transition; in the final (migra tion)
+transition, however, the numbers in two patches change simultane ously. In
+the above transitions, γ,ρ,κare non-negative, and ( di),(bi) are sequences of
+non-negative numbers.
+Once again, both (1.2) and (1.3) are obviously satisfied. The equatio n
+corresponding to (1.4) can now be expressed by taking
+Aii=−{κ+i(bi+di+γ)};AT
+i,i−1=i(di+γ);AT
+i,i+1=ibi, i≥1;
+A00=−κ,
+with all other elements of Aequal to zero, and
+Fi(x) =ργ/⌊ard⌊lx/⌊ard⌊l11(xi−1−xi), i≥1;F0(x) =−ργx0/⌊ard⌊lx/⌊ard⌊l11+κ,
+where we have used the fact that N−1/summationtext
+j≥0Xj= 1. Hence Assumption (3.1)
+is again immediate, and Assumption (3.2) holds for µ(j) = 1 with w= 0,
+forµ(j) =j+ 1 with w= max i(bi−di−γ−κ)+(assuming ( bi) and (di)
+to be such that this is finite), or indeed for µ(j) = (j+1)swith any s≥2,
+with appropriate choice of w. With the choice µ(j) =j+1,Fagain maps
+elements of RµtoRµ, and is also locally Lipschitz in the µ-norm, with
+K(µ,F;Ξ) = 3ργΞ.
+To check Assumptions 2.1, take ν=µ; once again, (2.5) is a finite sum
+for each r. Then, for (2.6), it is immediate that U0(x) = 0. For any r≥1,
+using arguments from the previous example,
+Ur(x)≤r2r−1/braceleftBigg/summationdisplay
+i≥1ibixi(i+1)r−1+/summationdisplay
+i≥1/summationdisplay
+k≥0iργxixk(k+1)r−1/bracerightBigg
+≤r2r−1{max
+ibiSr(x)+ργS1(x)Sr−1(x)}
+≤r2r−1{max
+ibiSr(x)+ργS0(x)Sr(x)},
+so that, since S0(x) = 1, we can take kr1=r2r−1(maxibi+ργ) andkr2=
+kr4= 0 in (2.6), and r(1)
+max=∞. Finally, for (2.7), V0(x) = 0 and, for r≥1,
+Vr(x)
+≤r2/braceleftBig
+22(r−1)max
+ibiS2r−1(x)+max
+i(i−1di)S2r(x)+γ(1−ρ)S2r−1(x)
++ργ(22(r−1)S1(x)S2r−2(x)+S0(x)S2r−1(x))/bracerightBig
++κS2r(x),
+32so that we can take p(r) = 2r, and (assuming i−1dito be finite)
+kr3=κ+r2{22(r−1)(max
+ibi+ργ)+max
+i(i−1di)+γ},
+andkr5= 0 for any r≥1, andr(2)
+max=∞. In Assumptions 4.2, we can again
+takerµ= 1 and ζ(k) = (k+1)7, givingr(ζ) = 8,b(ζ) = 1 and ρ(ζ,µ) = 16.
+Acknowledgement
+We wish to thank a referee for recommendations that have substa ntially
+streamlined our arguments. ADB wishes to thank both the Institut e for
+MathematicalSciencesoftheNationalUniversityofSingaporeand theMittag–
+Leffler Institute for providing a welcoming environment while part of t his
+work was accomplished. MJL thanks the University of Z¨ urich for th eir hos-
+pitality on a number of visits.
+References
+[1]Arrigoni, F. (2003). Deterministic approximation of a stochastic
+metapopulation model. Adv. Appl. Prob. 35691–720.
+[2]Barbour, A. D. andKafetzaki, M. (1993). A host–parasite model
+yielding heterogeneous parasite loads. J. Math. Biology 31157–176.
+[3]Barbour, A. D. andLuczak, M. J. (2008). Laws of large numbers
+for epidemic models with countably many types. Ann. Appl. Probab. 18
+2208–2238.
+[4]Chow, P.-L. (2007).Stochastic partial differential equations. Chapman
+and Hall, Boca Raton.
+[5]Eibeck, A. andWagner, W. (2003). Stochastic interacting particle
+systems and non-linear kinetic equations. Ann. Appl. Probab. 13845–
+889.
+[6]Kimmel, M. andAxelrod, D. E. (2002).Branching processes in biol-
+ogy.Springer, Berlin.
+33[7]Kretzschmar, M. (1993).Comparison ofaninfinite dimensional model
+for parasitic diseases with a related 2-dimensional system. J. Math. Anal-
+ysis Applics 176235–260.
+[8]Kurtz, T. G. (1970). Solutions of ordinary differential equations as
+limits of pure jump Markov processes. J. Appl. Probab. 749–58.
+[9]Kurtz, T. G. (1971).Limit theorems forsequences ofjumpMarkov pro-
+cesses approximating ordinary differential processes. J. Appl. Probab. 8
+344–356.
+[10]L´eonard, C. (1990).Some epidemic systems are long range interacting
+particle systems. In: Stochastic Processes in Epidemic Theory , Eds J.-P.
+Gabriel, C. Lef` evre & P. Picard, Lecture Notes in Biomathematics 86
+170–183: Springer, New York.
+[11]Luchsinger, C. J. (1999).MathematicalModelsofaParasiticDisease,
+Ph.D. thesis, University of Z¨ urich.
+[12]Luchsinger, C. J. (2001a). Stochastic models of a parasitic infection,
+exhibiting three basic reproduction ratios. J. Math. Biol. 42, 532–554.
+[13]Luchsinger, C. J. (2001b). Approximating the long term behaviour
+of a model for parasitic infection. J. Math. Biol. 42, 555–581.
+[14]Pazy, A. (1983).Semigroups of Linear Operators and Applications to
+Partial Differential Equations. Springer, Berlin.
+[15]Reuter, G. E. H. (1957). Denumerable Markov processes and the
+associated contraction semigroups on l.Acta Math. 97, 1–46.
+34
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+arXiv:1001.0045v1  [hep-ph]  31 Dec 2009EPJ manuscript No.
+(will be inserted by the editor)
+Axial Anomaly and Mixing Parameters of
+Pseudoscalar Mesons
+Yaroslav N. Klopot1,a, Armen G. Oganesian2,b, and Oleg V. Teryaev1,c
+1Joint Institute for Nuclear Research, Bogoliubov Laborato ry of Theoretical Physics, Dubna 141980,
+Russia
+2Institute of Theoretical and Experimental Physics, B.Cher emushkinskaya 25, Moscow 117218, Russia
+Abstract. In this work the analysis of mixing parameters of the system i nvolving
+η,η′mesons and some third massive state Gis carried out. We use the generalized
+mixing scheme with three angles. The framework of the disper sive approach to
+Abelian axial anomaly of isoscalar non-singlet current and the analysis of exper-
+imental data of charmonium radiative decays ratio allow us t o get a number of
+quite strict constraints for the mixing parameters. The ana lysis shows that the
+equal values of axial current coupling constants f8andf0are preferable which
+may be considered as a manifestation of SU(3) and chiral symmetry.
+1 Introduction
+This work is developing the approach of the papers [1,2] and is devot ed to the significant
+problem of mixing of pseudoscalar mesons. It is especially important w ith a number of current
+and planned experiments.
+The problem of η-η′mixing has been studied for many years. The usual approach with on e
+mixingangledominatedfordecades,butintherecentyearsthemor eelaboratedschemesappear
+to be unavoidable [3–8]. In particular, the theoretical ground of th is was based on the recent
+progress in the ChPT [9–11]. On the other hand, it was shown, that t he current experimental
+data cannot satisfactory describe the whole set of experiments w ithin the one-angle mixing
+scheme.
+The mixing schemes are usually enunciated either in terms of SU(3) or quark basis. In
+our paper [2] we construct and use the generalization of SU(3) basis similar to the mixing of
+massive neutrinos. This is because we use the dispersive approach t o axial anomaly ( [12], [13]
+for a review) to find some model-independent and precise restrictio n on the mixing parameters.
+It was shown that any scheme with more than one angle unavoidably d emands an additional
+admixture of higher mass state. If we restrict ourselves to only on e additional state G(denoted
+as a glueball without really specifying its nature) then the general m ixing scheme can be
+described in terms of 3 angles. In particular cases the number of an gles can be reduced to two.
+In the paper [2] the analysis of different conventional (and most ph ysically interesting)
+particular cases was performed (including two-angle mixing schemes ) basing on the dispersive
+representation of axial anomaly from one side and charmonium deca ys ratio from the other
+side.
+ae-mail:klopot@theor.jinr.ru
+be-mail:armen@itep.ru
+ce-mail:teryaev@theor.jinr.ru2 Will be inserted by the editor
+The main conclusion of the paper [2] is that in all considered cases the only reasonable
+solutions appear at f8=f0≃fπ. The main aim of this work is to check whether this relation
+remains valid in the most general case with some specific constraints imposed.
+This paper is organized as follows. In the Sec. 2 we introduce our not ation and the general
+approach to the mixing. In Sec. 3 we derive the basic equations relyin g on the dispersive
+approach to Abelian axial anomaly of isoscalar non-singlet current J8
+µ5and the charmonium
+radiativedecayratio RJ/Ψ, while in Sec. 4 we performthe numerical analysisofthese equations .
+Finally, in Sec. 5 we present the conclusion.
+2 Mixing scheme
+We start with a ( N-component) vector of physical pseudoscalar fields consisting of the fields of
+the lightest pseudoscalar mesons and other fields:
+/tildewideΦ≡
+π0
+η
+η′
+G
+...
+. (1)
+We are not able to specify the physical nature of the other compon ents with higher masses,
+the lowest of which Gcan be either a glueball or some excited state1. Let us also introduce,
+following [16,17], a set of SU(3) fields ϕ3,ϕ8,ϕ0(Φ1,Φ2,Φ3) and complement them with other
+(sterile) fields gi(Φi,i= 4..N)
+Φ=
+ϕ3
+ϕ8
+ϕ0
+g
+...
+. (2)
+The three upper fields ϕ3,ϕ8,ϕ0are the only ones which define the generalized PCAC
+relationforaxialcurrent Ja
+µ5=qγµγ5λa
+√
+2q(nosummationover acontraryto jandkisassumed):
+∂µJa
+µ5=faδ∆L
+δΦa=FajMjkΦk, a= 3,8,0, j,k= 1..N, (3)
+where∆Lis the mass term in the effective Lagrangian with a non-diagonal mass matrixM(as
+fieldsΦkare not orthogonal to each other):
+∆L=1
+2ΦTMΦ, (4)
+andFis a matrix of decay constants2:
+F≡
+f30 0 0...0
+0f80 0...0
+0 0f00...0
+. (5)
+In order to proceed from initial SU(3) fields Φto physical mass fields /tildewideΦthe unitary (real,
+as the CP-violating effects are negligible) matrix Uis introduced
+1Note, that the mixing with the excited states is usually(e.g . [14,15]) supposed to be suppressed.
+2Note, that matrix of decay constants Fis non-square expressing the fact that generally the number
+ofSU(3) currents is less then the number of all possible states in volved in mixing. The similar situation
+takes place (see e.g. [18]) in one of the extensions of the Sta ndard Model – neutrino mixing scenario
+involving sterile neutrinos.Will be inserted by the editor 3
+/tildewideΦ=UΦ (6)
+that diagonalizes the mass matrix
+UMUT=/tildewiderM≡diag(m2
+π0,m2
+η,m2
+η′,m2
+G,...), (7)
+wheremπ,mη,mη′andmGare the masses of the π,η,η′mesons and glueball state G,
+respectively.
+Simple transformations of Eq.(3) read:
+∂µJµ5=FUT/tildewiderM/tildewideΦ (8)
+This formula is close to those obtained in [16,17] (in the limit of small mix ing). When the decay
+constants are equal, it is reduced to formula (3.40) in [19].
+The matrix elements of ∂µJµ5between vacuum state and physical states |/tildewiderΦk∝angbracketright
+∝angbracketleft0|∂µJa
+µ5|/tildewiderΦk∝angbracketright=Fa
+i(UT/tildewiderM)i
+k (9)
+can be compared to the standard definition of the ”physical” couplin g constants of axial cur-
+rents:
+∝angbracketleft0|Ja
+µ5|/tildewiderΦk∝angbracketright=ifa
+kqµ. (10)
+From (9) and (10) follows the relation
+fa
+k=Fa
+i(UT)i
+k=fa(UT)a
+k. (11)
+This expression (recall, that there is no summation over a) clearly shows that fa
+kare obtained
+bymultiplicationofeachlineof UTbyrespectivecoupling faandformanon-diagonal(contrary
+toF) matrix.
+Taking into account the well-known smallness of π0mixing with the η,η′sector [16,17,20]
+and neglecting all higher contributions we restrict our consideratio n to three physical states
+η,η′,Gand two currents J8
+µ5,J0
+µ5. Then the divergencies of the axial currents (recall, that Gis
+a first mass state heavier than η′):
+/parenleftbigg∂µJ8
+µ5
+∂µJ0
+µ5/parenrightbigg
+=/parenleftbigg
+f80 0
+0f00/parenrightbigg
+UT
+m2
+η0 0
+0m2
+η′0
+0 0m2
+G
+
+η
+η′
+G
+. (12)
+Exploring the mentioned similarity of the meson and lepton mixing, we us e the Euler
+parametrization for the mixing matrix U(we use notation ci≡cosθi,si≡sinθi):
+U=
+c8c3−c0s3s8−c3s8−c8c0s3s3s0
+s3c8+c3c0s8−s3s8+c3c8c0−c3s0
+s8s0 c8s0 c0
+. (13)
+In the following consideration we will need the divergency of the octe t current ∂µJ8
+µ5, so let
+us write it out explicitly:
+∂µJ8
+µ5=f8(m2
+ηη(c8c3−c0s3s8)+m2
+η′η′(s3c8+c3c0s8)+m2
+GG(s8s0)). (14)
+As soon as in the chiral limit J8
+µ5should be conserved, from Eq.(14) follows that coefficients
+of the terms m2
+η′,m2
+Gmust decrease at least as ( mη/mη′,G)2. More specifically, we expect the
+following limits for the terms of Eq.(14):
+|s8s0|
+|s3c8+c3c0s8|/lessorsimilar/parenleftbiggmη
+mG/parenrightbigg2
+. (15)4 Will be inserted by the editor
+3 Abelian axial anomaly and charmonium decays ratio
+In our paper the dispersive form of the anomaly sum rule will be exten sively used, so we remind
+briefly the main points of this approach (see e.g. review [13] for det ails).
+Consider a matrix element of a transition of the axial current to two photons with momenta
+pandp′
+Tµαβ(p,p′) =∝angbracketleftp,p′|Jµ5|0∝angbracketright. (16)
+The general form of Tµαβfor a case p2=p′2can be represented in terms of structure
+functions (form factors):
+Tµαβ(p,p′) =F1(q2)qµǫαβρσpρp′
+σ+
+1
+2F2(q2)[pα
+p2ǫµβρσpρp′
+σ−p′
+β
+p2ǫµαρσpρp′
+σ−ǫµαβσ(p−p′)σ],(17)
+whereq=p+p′. The functions F1(q2),F2(q2) can be described by dispersion relations with
+no subtractions and anomaly condition in QCD results in the sum rule:
+∞/integraldisplay
+0Im F1(q2)dq2= 2αNc/summationdisplay
+e2
+q, (18)
+whereeqare quark electric charges and Ncis the number of colors. This sum rule [21] was
+developed by Jiˇ r´ ı Hoˇ rejˇ s´ ı [22], and later generalized [23]. Not ice that in QCD this equation
+does not have any perturbative corrections [24], and it is expected that it does not have any
+non-perturbative corrections as well due to the ’t Hooft’s consist ency principle [25]. It will be
+important for us that as q2→ ∞the function ImF1(q2) decreases as 1 /q4(see discussion
+in Ref. [2]). Note also that the relation (18) contains only mass-indep endent terms, which is
+especially important for the 8th component of the axial current J8
+µ5containing strange quarks:
+J8
+µ5=1√
+6(¯uγµγ5u+¯dγµγ5d−2¯sγµγ5s). (19)
+The general sum rule (18) takes the form:
+∞/integraldisplay
+0Im F1(q2)dq2=2√
+6α(e2
+u+e2
+d−2e2
+s)Nc=/radicalbigg
+2
+3α , (20)
+whereeu= 2/3,ed=es=−1/3,Nc= 3.
+In order to separate the form factor F1(q2), multiply Tµαβ(p,p′) byqµ/q2. Then, taking the
+imaginary part of F1(q2), using the expression for ∂µJ8
+µ5from Eq.(12) and unitarity we get:
+ImF1(q2) =Im qµ1
+q2∝angbracketleft2γ|J(8)
+µ5|0∝angbracketright=
+−f8
+q2∝angbracketleft2γ|[m2
+ηη(c8c3−c0s3s8)+m2
+η′η′(s3c8+c3c0s8)+m2
+GGs8s0]|0∝angbracketright=
+πf8[Aηδ(q2−m2
+η)(c8c3−c0s3s8)+Aη′δ(q2−m2
+η′)(s3c8+c3c0s8)+AGδ(q2−m2
+G)(s8s0)].
+(21)
+If we employ the sum rule (20), we obtain a simple equation:
+(c8c3−c0s3s8)+β(s3c8+c3c0s8)+γ(s8s0) =ξ, (22)
+whereWill be inserted by the editor 5
+β≡Aη′
+Aη=/radicaligg
+Γη′→2γ
+Γη→2γm3η
+m3
+η′, γ≡AG
+Aη=/radicaligg
+ΓG→2γ
+Γη→2γm3η
+m3
+G, (23)
+ξ≡/radicaligg
+α2m3η
+96π3Γη→2γ1
+f2
+8, Γη→2γ=m3
+η
+64πA2
+η. (24)
+Note that if we include higher resonancesin this equation, they will be suppressed as 1 /m2
+res
+by virtue of the mentioned above asymptotic behavior of F1(q2)∝1/q4. For the last two terms
+in (22) we can specify this constraint as follows:
+|s8s0|
+|s3c8+c3c0s8|/lessorsimilarβ
+γ/parenleftbiggmη′
+mG/parenrightbigg2
+. (25)
+As an additional experimental constraint we use, following [26,27], t he data of the decay
+ratioRJ/Ψ= (Γ(J/Ψ)→η′γ)/(Γ(J/Ψ)→ηγ).
+As it was pointed out in [28], the radiative decays J/Ψ→η(η′)γare dominated by non-
+perturbative gluonic matrix elements, and the ratio of the decay ra tesRJ/Ψ= (Γ(J/Ψ)→
+η′γ)/(Γ(J/Ψ)→ηγ) can be expressed as follows:
+RJ/Ψ=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∝angbracketleft0|G/tildewideG|η′∝angbracketright
+∝angbracketleft0|G/tildewideG|η∝angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenleftbiggpη′
+pη/parenrightbigg3
+, (26)
+wherepη(η′)=MJ/Ψ(1−m2
+η(η′)/M2
+J/Ψ)/2. The advantage of this ratio is expected smallness of
+perturbative and non-perturbative corrections.
+The divergencies of singlet and octet components of the axial curr ent in terms of quark
+fields can be written as:
+∂µJ8
+µ5=1√
+6(muuγ5u+mddγ5d−2mssγ5s), (27)
+∂µJ0
+µ5=1√
+3(muuγ5u+mddγ5d+mssγ5s)+1
+2√
+33αs
+4πG/tildewideG. (28)
+Following [26], neglect the contribution of u- and d- quark masses, th en the matrix elements
+of the anomaly term between the vacuum and η,η′states are:
+√
+3αs
+8π∝angbracketleft0|G/tildewideG|η∝angbracketright=∝angbracketleft0|∂µJ(0)
+µ5|η∝angbracketright+1√
+2∝angbracketleft0|∂µJ(8)
+µ5|η∝angbracketright, (29)
+√
+3αs
+8π∝angbracketleft0|G/tildewideG|η′∝angbracketright=∝angbracketleft0|∂µJ(0)
+µ5|η′∝angbracketright+1√
+2∝angbracketleft0|∂µJ(8)
+µ5|η′∝angbracketright. (30)
+Using Eq. (12), (26), (29), (30) we deduce:
+RJ/Ψ=/bracketleftiggf0(−s3s8+c3c8c0)+1√
+2f8(s3c8+c3c0s8)
+f0(−c3s8−c8c0s3)+1√
+2f8(c8c3−c0s3s8)/bracketrightigg2
+×/parenleftbiggmη′
+mη/parenrightbigg4/parenleftbiggpη′
+pη/parenrightbigg3
+.(31)
+4 Analysis
+For further analysis it is convenient to rewrite the equations (22), (31) in terms of angles
+θ1≡θ8+θ3,θ8andθ0:
+1
+2(c1+c2−c0(c2−c1))+β
+2(s1−s2+c0(s1+s2))+γ(s8s0) =ξ. (32)6 Will be inserted by the editor
+RJ/Ψ=/bracketleftiggf0(c1−c2+c0(c1+c2))+1√
+2f8(s1−s2+c0(s1+s2))
+f0(−s1−s2−c0(s1−s2))+1√
+2f8(c1+c2−c0(c2−c1))/bracketrightigg2/parenleftbiggmη′
+mη/parenrightbigg4/parenleftbiggpη′
+pη/parenrightbigg3
+,(33)
+whereθ2≡2θ8−θ1.
+The angles θ1,θ8,θ0have the explicit physical meaning. From the definition (13) of the
+mixing matrix Uone can see that the angle θ1describes the overlap in the η−η′system with
+an accuracy ∼θ2
+0/2 and coincides with their mixing angle as θ0→0. At the same time θ0is
+responsible for the glueball admixture to η−η′system, and s8s0describes the contribution of
+the glueball state Gto the octet component of axial current ∂J8
+µ5only.
+In the further analysis we will use the following assumptions:
+I) As we discussed in Sec. 2, the last term in (14) should be suppress ed as (mη/mG)2. So
+we impose the following constraint:
+|s8s0|
+|s3c8+c3c0s8|/lessorsimilar/parenleftbiggmη
+mG/parenrightbigg2
+. (34)
+II) In sec 3 we found another constraint, which follows from the as ymptotic behavior of
+ImF1(see 25):
+|s8s0|
+|s3c8+c3c0s8|/lessorsimilarβ
+γ/parenleftbiggmη′
+mG/parenrightbigg2
+. (35)
+III) In our numerical analysis we suppose that γcannot exceed 1 (i.e. ΓG→2γ/m3
+G/lessorsimilar
+Γη→2γ/m3
+η). This restriction corresponds to the assumption that 2-photon decay widths of
+pseudoscalar mesons grow like the third power of their masses, or in other words, the glueball
+coupling to quarks is of the same order as for the meson octet stat es.
+IV) We accept that the decay constants obey the relation f8/greaterorsimilarf0/greaterorsimilarfπ(for various kinds
+of justification see, e.g., [3,9]).
+For the purposes of numerical analysis, the values of RJ/Ψ(RJ/Ψ= 4.8±0.6), masses
+and two-photon decay widths of η,η′mesons are taken from PDG [29]. Using the values
+mη,mη′,Γη→2γ,Γη′→2γ, we see that the relation for the constraint (34) is more strict tha n the
+constraint (35). Supposing the minimal mass of the glueball to be of ordermG≃3mη≃1.5
+GeV, we get the estimation:
+|s8s0|/|s3c8+c3c0s8|/lessorsimilar0.1. (36)
+On Fig. 1 the plots of the equations (32) and (33) in the parameter s pace (θ8,θ1) are shown
+for different values of decay constants f8,f0and mixing angle θ0. The dashed curves denote
+experimental uncertainties. The intersection points of the curve s represent the solutions of both
+equations (32),(33). The filled area indicates the region, where the constraint (36) is valid. The
+plotted range of angle θ1is limited to the physically interesting region, where the solution for
+relatively small angles θ0exists. Let us note for completeness, that there is another solut ion for
+θ1∼90◦,θ0/greaterorsimilar50◦which does not seem to have a physical sense.
+The numerical analysis shows, that the solution of the equations (3 2) and (33) satisfying
+the mentioned above constraint is possible only for rather small mixin g angleθ0and for decay
+constants f8,f0closetoeachotherandcloseto fπ:forf8/fπ=f0/fπ= 1.0the possiblerangeof
+mixing angle θ0isθ0= (0÷25)◦(see Fig. 1(a)-1(c) for demonstration), for f8/fπ=f0/fπ= 1.1
+the possible range of mixing angle θ0isθ0= (0÷20)◦.
+There is no solutions for decay constant values f8/fπ= 1.1,f0/fπ= 1.0 for any θ0(see Fig.
+1(d)-1(f) for demonstration), and for any f0/lessorsimilarf8in case of f8/fπ≥1.2. The obtained results
+are quite stable: even if we relax the constraint (36) making its r.h.s. several times larger, all
+the conclusions are preserved.
+Note finally, that this result is in contradiction with the prediction for the decay constant
+f8/fπ= 1.34 [10] obtained in the Large NcChPT.Will be inserted by the editor 7
+/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(a) (f8,f0) = (1.0,1.0)fπ,θ0=
+0◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(b) (f8,f0) = (1.0,1.0)fπ,θ0=
+5◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(c) (f8,f0) = (1.0,1.0)fπ,θ0=
+30◦
+/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(d) (f8,f0) = (1.1,1.0)fπ,θ0=
+0◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(e) (f8,f0) = (1.1,1.0)fπ,θ0=
+5◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
+Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
+(f) (f8,f0) = (1.1,1.0)fπ,θ0=
+30◦
+Fig. 1.The solutions of the Eq. (32) (thin curves, blue online) and ( 33)(thick curves, red online) with
+the experimental uncertainties (dashed curves) for differe nt values of the parameters f8,f0andθ0. The
+shaded area indicates the region, where the relation (36) is valid.
+5 Conclusion
+In this paper we studied what can be learnt about the mixing in the pse udoscalar sector from
+the dispersive approach to axial anomaly.
+Our analysis shows that the equal values of axial current coupling c onstants f8andf0are
+favorablewhich may be considered as a manifestation of SU(3) and chiral symmetry. Moreover,
+with a less definiteness the relation fπ≈f8≈f0[2] is also supported.
+Theanalysisdemands f8<1.2fπwhich deviatesat 10%levelfrom theresultsofcalculations
+within the chiral perturbation theory ( f8= 1.34fπ) [10].
+Thevalueofthe mixingangle θ0,whichisresponsibleforthe glueballadmixturetothe η−η′,
+is limited to θ0<25◦for (f8,f0) = (1.0,1.0)fπand toθ0<20◦for (f8,f0) = (1.0,1.0)fπ.
+The improvement of the experimental data of RJ/Ψcan significantly limit the constraints
+for the parameters θ0,θ8andf8,f0.
+We thank J. Hoˇ rejˇ s´ ı, B. L. Ioffe and M. A. Ivanov for useful co mments and discussions.
+Y. K. and O. T. gratefully acknowledge the organizers of the works hop for hospitality and
+support. This work was supported in part by RFBR (Grants 09-02- 00732,09-02-01149),by the
+funds from EC to the project ”Study of the Strong Interacting M atter” under contract N0.
+R113-CT-2004-506078 and by CRDF Project RUP2-2961-MO-09.
+References
+1. Y. N. Klopot, A. G. Oganesian, O. V. Teryaev, arXiv:0810.1 217 [hep-ph] (2008).8 Will be inserted by the editor
+2. Y. N. Klopot, A. G. Oganesian, O. V. Teryaev, arXiv:0911.0 180 [hep-ph] (2009).
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+arXiv:1001.0046v1  [math.CO]  31 Dec 2009THE CAUCHY-SCHWARZ INEQUALITY IN CAYLEY GRAPH
+AND TOURNAMENT STRUCTURES ON FINITE FIELDS
+STEPHAN FOLDES AND L ´ASZL´O MAJOR
+Abstract. The Cayley graph construction provides a natural grid struc ture
+on a finite vector space over a field of prime or prime square car dinality, where
+the characteristic is congruent to 3 modulo 4, in addition to the quadratic
+residue tournament structure on the prime subfield. Distanc e from the null
+vector in the grid graph defines a Manhattan norm. The Hermiti an inner prod-
+uct on these spaces over finite fields behaves in some respects similarly to the
+real and complex case. An analogue of the Cauchy-Schwarz ine quality is valid
+with respect to the Manhattan norm. With respect to the non-t ransitive order
+provided by the quadratic residue tournament, an analogue o f the Cauchy-
+Schwarz inequality holds in arbitrarily large neighborhoo ds of the null vector,
+when the characteristic is an appropriate large prime.
+1.Manhattan norms and grid graphs
+We consider the finite fields FpandFp2of prime and prime square cardinality,
+wherep≡3 mod 4. The field Fp2has a natural graph structure with the field
+elements as vertices, two distinct vertices u,zbeing adjacent if ( z−u)4= 1. The
+subfieldFpofFp2then induces a subgraph in which xandyare adjacent if and only
+if (y−x)2= 1.The graph Fp2is isomorphic to the Cartesian square C2
+p=Cp/squareCp,
+whereCpis ap-cycle and within Fp2the induced subgraph Fpis itself a p-cycle.
+Clearly the graph Fp2is not planar, but can be drawn as a grid on the torus.
+For any connected graph whose vertex set is a group, the distanc e of any vertex
+zfrom the identity element of the group is called the normofz, denoted N(z).
+In general, distances and norms measured in connected subgraph s induced by sub-
+groups can be larger than distances and norms measured with refe rence to the
+whole graph. However, with respect to the distance-preserving s ubgraph induced
+byFpinFp2, the norm of any z∈Fpis the same as its norm with respect to the
+whole graph Fp2: this is simply the length of the shortest path from 0 to zin the
+cycle induced by Fp.
+Forq=porq=p2, then-dimensional vector space Fn
+qis also endowed with
+the Cartesian product graph structure Fq/square···/squareFqisomorphic to Cn
+porC2n
+p. The
+norm of a vector v= (v1,...,v n) inFn
+qis then equal to the sum N(v1)+···+N(vn)
+and we also write N(v) for this vector norm.
+The Gaussian integers Z[i] also constitute a graph in which uandzare adjacent
+if and only if ( z−u)4= 1.
+Date: Dec 24, 2009.
+1991Mathematics Subject Classification. Primary 05C12, 05C20, 05C25; Secondary 06F99,
+11T99.
+Key words and phrases. Cauchy-Schwarz inequality, triangle inequality, submult iplicativity,
+finite field, quadratic field extension, quadratic residue to urnament, grid graph, Manhattan dis-
+tance, discrete norm, Gaussian integers, graph product, gr aph quotient, Cayley graph.
+12 STEPHAN FOLDES AND L ´ASZL´O MAJOR
+It iseasytoseethatthenorminthis infinite Manhattan grid satisfiesthetriangle
+and submultiplicative inequalities
+N(u+z)≤N(u)+N(z)
+N(uz)≤N(u)N(z)
+To emphasize that the norms on Fp2,Fn
+p2andZ[i] are understood with reference
+to the specific grid graphs defined above, we call these norms Manhattan norms .
+Throughout this paper we think of Fp2as the ring quotient Z[i]/(p).
+2.Graph quotients and Cayley graphs
+Given a graph G(undirected, with possible loops) on vertex set Vand an equi-
+valence relation ≡onV, thequotient graph G/≡is defined as follows: the vertices
+ofG/≡are the equivalence classes of ≡, and classes A,Bare adjacent if for some
+a∈A,b∈B, the elements a,bare adjacent in G. Note that the distance of Ato
+Bin the quotient graph is at most equal to, but possibly less than the m inimum of
+the distances atobfor alla∈A,b∈B. Note also that G/≡can have loops even
+ifGhas not.
+Given a group Gwith identity element eand a set Γ of group elements that
+generates G, the(left) Cayley graph C(G,Γ) ofGwith respect to Γ has vertex set
+G, elements a,b∈Gbeing considered adjacent if ab−1orba−1belongs to Γ. For
+each congruence ≡of the group G, corresponding to some normal subgroup H, Γ
+yields a generating set Γ ≡ofG/≡consisting with those classes of ≡that intersect
+Γ. The graph quotient of C(G,Γ) by the equivalence ≡coincides with the Cayley
+graph of the quotient graph G/≡with respect to Γ ≡. ForR⊆Ginducing a
+connected subgraph [ R] inC(G,Γ), denote by dR(x,y) the distance function of the
+subgraph [ R]. Denoting by xHtheH-coset of any x∈G, this relates to norms in
+C(G,Γ) andC(G,Γ)/≡as follows: for all x∈R,
+dR(x,e)≥N(x)≥N(xH)
+Both inequalities can be strict. However, we have:
+Cayley Graph Quotient Lemma. Let a group Gwith identity ebe generated
+byΓ⊆G, and consider any normal subgroup Hwith corresponding congruence ≡.
+There is a set R⊆Ghaving exactly one element in common with each congruence
+class modulo H, and such that for every x∈R
+dR(x,e) =N(x) =N(xH)
+Proof.We can define the unique (representative) element r(A)∈R∩Afor each
+cosetAby induction on the distance d(H,A) ofAfromHinC(G,Γ)/≡. Let
+r(H) =e. Assuming r(A) defined for all Awithd(H,A)≤m, let a coset Bhave
+distance m+1 from H. Choose any coset Aadjacent to Bwithd(H,A) =mand
+elements a∈A,b∈Bthat are adjacent in C(G,Γ). Letr(B) =ba−1r(A)./square
+We can apply the above lemma in the case where G=Z[i], Γ ={1,i}and
+H=pZ[i] ={pa+pbi:a,b∈Z}foraprimeinteger p≡3 mod 4. Now C(G,Γ)and
+C(G,Γ)/≡are the Manhattan grid graphs on Z[i] andZ[i]/H=Fp2, respectively.
+Referringtothe set Rofrepresentativesinthe lemma, forany H-cosetsX,Yletx,y
+be the unique elements in X∩R,Y∩R. Asxy∈XY, we have N(XY)≤N(xy).
+By the submultiplicative inequality in Z[i] we have N(xy)≤N(x)N(y). Using the3
+lemmawehave N(x)N(y) =N(X)N(Y). Thisyieldsasubmultiplicativeinequality
+inFp2and a similar reasoning on the coset X+Yyields a triangle inequality:
+Triangle and Submultiplicative Inequalities in Fp2.For allu,zinFp2
+N(u+z)≤N(u)+N(z)
+N(uz)≤N(u)N(z)
+/square
+This indicates that Manhattan distance provides a well-behaved not ion of neigh-
+borhood of 0 in the finite fields Fp2.
+3.Squares in Fpand non-transitive order
+For each prime p≡3 mod 4 the quadratic residue tournament onFpis the
+directed graph with vertex set Fpin which there is an arrowfrom vertex xto vertex
+yify−xis a non-zero square in Fp, in which case we write x <py. We write x≤py
+ifx <pyorx=y. The relation ≤pis reflexive, anti-symmetric but not transitive,
+and for every x/ne}ationslash=yexactly one of x≤pyory≤pxholds. Using Dirichlet’s
+theorem on primes in arithmetic progressions, Kustaanheimo showe d [4] that for
+every positive integer k, there is a prime p≡3 mod 4, such that ≤pis a transitive
+(and linear) order relation on {0,1,...,k} ⊆Fp, that is, all positive integers up to
+kare quadratic residues mod p. Obviously kcannot exceed ( p−1)/2. Implications
+of [4] and related questions were investigated by J¨ arnefelt, Kust aanheimo, Quist
+[3, 5], in particular with a view to discrete models in physics, also in subse quent
+application-oriented work between the 1950’s (Coish [1]) and the 198 0’s (Nambu
+[6]). For further references see [2]. In particular [4] implies that for every positive
+integerk, there is a prime p≡3 mod 4, such that all z∈Fp2withN(z)≤kare
+squares in Fp2. (Note that all elements of the prime subfield Fpare squares in Fp2.)
+To emphasise the analogy of the relation ≤pwith the ordinary inequality relation
+≤among numbers, we say that a non-zero z∈Fp2ispositiveifz∈Fpand 0≤pz.
+4.Inner products compared in non-transitive order
+The only non-trivial automorphism of the field Fp2associates to each z∈Fp2
+itsconjugate z. Theinner product v·wof vectors v= (v1,...,v n) andw=
+(w1,...,w n) inFn
+p2is defined as the scalar v1w1+···+vnwn∈Fp2. This inner
+product is left and right distributive over vector addition, satisfies v·w=w·v,
+c(v·w) = (cv)·w=v·(cw) for allc∈Fp2. However, while v·vbelongs to the
+prime subfield Fp,v·vis not necessarily positive, and can be 0 even if v/ne}ationslash=0. Still,
+a conditional version of positive definiteness holds locally:
+Conditional Positive Definiteness. For every k≥1there is a prime p≡3
+mod 4, such that for all n≥1and for all vectors v∈Fn
+p2of Manhattan norm
+N(v)≤k,we have 0≤pv·vwith equality if and only if v=0.
+Proof.By Kustaanheimo’s result in [4] there is a prime integer p≡3 mod 4 such
+that 0,1,...,2k3are all quadratic residues mod p. Forv= (v1,...,v n) inFn
+p2, let
+vj=aj+bji, wherei2=−1. IfN(v)≤kthen for all j,N(aj)≤kandN(bj)≤k,
+vjvj=a2
+j+b2
+jbelongs to the set of squares {0,...,2k2}. Sincevjcan be non-zero
+for at most kindices 1 ≤j≤nonly, the sum of the corresponding terms a2
+j+b2
+j
+belongs to the set of squares {0,1,...,2k3}. /square4 STEPHAN FOLDES AND L ´ASZL´O MAJOR
+Note that for all vectors v,w∈Fn
+p2
+(v·w)(w·v) = (v·w)(v·w)∈Fpand
+(v·v)(w·w)∈Fp
+Ifvandwareproportional , i.e. if there exists a scalar cinFp2such that v=cw
+orw=cv, then the above two products are equal. Generally, they are relat ed in
+the quadratic residue tournament of Fpas follows.
+Cauchy-Schwarz Inequality for Quadratic Residue Tourname nts.For eve-
+ryk≥1there is a prime p≡3 mod 4 , such that for all n≥1and for all vectors
+v,w∈Fn
+p2of Manhattan norm at most k,
+(v·w)(w·v)≤p(v·v)(w·w)
+Proof.Forn= 1 the inequality holds trivially as the two sides are equal. Assume
+n≥2,v= (v1,...,v n),w= (w1,...,w n). Forall1 ≤i≤n,N(vi)≤k,N(wi)≤k.
+By Kustaanheimo’s result [4] there is a prime p≡3 mod 4 such that all positive
+integers up to 4 k6are quadratic residues modulo p. For each of the/parenleftbign
+2/parenrightbig
+pairs
+{i,j} ⊆ {1,...,n},i/ne}ationslash=j, by the triangle and submultiplicative inequalities in Fp2
+N[(viwj−vjwi)(viwj−vjwj)]≤(k2+k2)2= 4k4
+Thus the element
+(viwjviwj+vjwivjwi)−(viwjvjwi+vjwiviwj) = (viwj−vjwi)(viwj−vjwj)
+is a square of Manhattan norm at most 4 k4inFp, and it is non-zero for at most/parenleftbigk
+2/parenrightbig
+≤k2pairs{i,j}. Summing over all pairs {i,j}, all but at most/parenleftbigk
+2/parenrightbig
+≤k2terms
+vanish in the sum/summationdisplay
+[(viwjviwj+vjwivjwi)−(viwjvjwi+vjwiviwj)]
+which therefore has Manhattan norm at most 4 k6and it must also be a square in
+Fp. But this sum is equal to the difference of products
+n/summationdisplay
+i=1vivin/summationdisplay
+j=1wjwj−n/summationdisplay
+i=1viwin/summationdisplay
+j=1vjwj= (v·v)(w·w)−(v·w)(w·v)
+which is consequently a square in Fp. /square
+Remark. From the proof it is clear that, in analogy with the classical Cauchy-
+Schwarz inequality, for vectors v,wof norm not exceeding kinFn
+p2, wherepis
+related to kas stipulated above, the Cauchy-Schwarz inequality with respect t o≤p
+holds with equality if and only if viwj−vjwi= 0 for all i,j, i.e. if and only if v,w
+are proportional.
+We note that the inequality established above is conditional, it holds on ly in a
+specified Manhattan neighborhood of the null vector. Every non- zero element of
+Fpcan be written as a sum of two squares, in particular there are a,b∈Fp, such
+thata2+b2=−1. Forz=a+biwe have zz=−1. As soon as n≥2, inFn
+p2let
+v= (a,b,0,...,0) and w= (bz,−az,0,...,0)
+The inequality ( v·w)(w·v)≤p(v·v)(w·w) fails because the left-hand side is 0
+and the right-hand side is −1. In fact if n≥3, the inequality can be invalidated
+with vectors v,winFn
+pas follows. Taking again a,b∈Fpwitha2+b2=−1, let
+v= (1,a,b,0,...,0) and w= (1,0,0,0,...,0)5
+However,the Cauchy-Schwarzinequalityholdsunconditionallyinthe 2-dimensional
+case for vectors with components in Fp:
+Special case of F2
+p.Letpbe a prime congruent 3modulo4. For all vectors v,w
+inF2
+p
+(v·w)(w·v)≤p(v·v)(w·w)
+Proof.Nowtheconjugationappearinginthe innerproductsisthe identity. Written
+in components,
+(v·v)(w·w)−(v·w)(w·v) = (v2
+1+v2
+2)(w2
+1+w2
+2)−(v1w1+v2w2)2=
+=v2
+1w2
+2+v2
+2w2
+1−2v1w1v2w2= (v1w2−v2w1)2
+/square
+5.Manhattan norm of inner product
+The Manhattan norm can be seen to be submultiplicative not only on th e ring
+Z[i] and its quotient field Fp2, but on all vector spaces Fn
+p2, with respect to the
+inner product:
+Cauchy-Schwarz Inequality for Manhattan Norm on Fn
+p2.Consider any
+primep≡3 mod 4 and letn≥1. For all v,w∈Fn
+p2
+N(v·w)≤N(v)N(w)
+Proof.Letv= (v1,...,v n),w= (w1,...,w n)∈Fn
+p2. Thenv·w=/summationtextvjwj. Clearly
+N(z) =N(z) for any z∈Fp2. By the triangle and submultiplicative inequalities in
+Fp2we have
+N(v·w) =N/parenleftbig/summationtextvjwj/parenrightbig
+≤/summationtextN/parenleftbig
+vjwj/parenrightbig
+≤/summationtextN(vj)N(wj)≤
+≤/summationtextN(vj)/summationtextN(wj) =N(v)N(w)
+/square
+Remark. The inequality N(v·w)≤N(v)N(w) is easily interpreted and continues
+to hold for v,win the module ( Z[i]/mZ[i])nfor any positive integer m. As soon as
+mis composite, or a prime not congruent to 3 modulo 4, the ring Z[i]/mZ[i] fails
+to be an integral domain.
+References
+[1] H.R. Coish, Elementary particles in a finite world geomet ry, Phys. Rev. 114 - 1 (1959) 383-388
+[2] S. Foldes, The Lorentz group and its finite field analogues : local isomorphism and approxima-
+tion, J. Math. Phys. 49, 093512 (2008)
+[3] G. J¨ arnefelt, P. Kustaanheimo, An observation on finite geometries, in Proc. Skandinaviske
+Matematikerkongress i Trondheim 1949, 166-182
+[4] P. Kustaanheimo, A note on a finite approximation of the eu clidean plane geometry, Comment.
+Phys.-Math. Soc. Sc. Fenn. XV. 19 (1950) 1-11
+[5] P. Kustaanheimo, B. Qvist, On differentiation in Galois fi elds. Ann. Acad. Sci. Fennicae. Ser.
+A. I. Math.-Phys. 1952, (1952). no. 137, 12 pp.
+[6] Y. Nambu, Field theory of Galois fields, in Field Theory an d Quantum Statistics, eds. J.A.
+Batalin et.al., Institute of Physics Publishing 1987, pp. 6 25-6366 STEPHAN FOLDES AND L ´ASZL´O MAJOR
+Stephan Foldes
+Institute of Mathematics,
+Tampere University of Technology,
+PL 553, 33101 Tampere, Finland
+E-mail address :sf@tut.fi
+L´aszl´o Major
+Institute of Mathematics,
+Tampere University of Technology,
+PL 553, 33101 Tampere, Finland
+E-mail address :laszlo.major@tut.fi
\ No newline at end of file
diff --git a/1001.0047.txt b/1001.0047.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1922ba2b8a19692f97811e319ce11dfc4323be74
--- /dev/null
+++ b/1001.0047.txt
@@ -0,0 +1,953 @@
+Coulomb interaction and transient charging of excited states in open nanosystems
+Valeriu Moldoveanu,1Andrei Manolescu,2Chi-Shung Tang,3and Vidar Gudmundsson4
+1National Institute of Materials Physics, P.O. Box MG-7, Bucharest-Magurele, Romania
+2Reykjavik University, School of Science and Engineering, Kringlan 1, IS-103 Reykjavik, Iceland
+3Department of Mechanical Engineering, National United University, Lienda, Miaoli 36003, Taiwan
+4Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
+We obtain and analyze the eect of electron-electron Coulomb interaction on the time dependent
+current 
owing through a mesoscopic system connected to biased semi-innite leads. We assume
+the contact is gradually switched on in time and we calculate the time dependent reduced density
+operator of the sample using the generalized master equation. The many-electron states (MES) of
+the isolated sample are derived with the exact diagonalization method. The chemical potentials of
+the two leads create a bias window which determines which MES are relevant to the charging and
+discharging of the sample and to the currents, during the transient or steady states. We discuss the
+contribution of the MES with xed number of electrons Nand we nd that in the transient regime
+there are excited states more active than the ground state even for N= 1. This is a dynamical
+signature of the Coulomb blockade phenomenon. We discuss numerical results for three sample
+models: short 1D chain, 2D lattice, and 2D parabolic quantum wire.
+PACS numbers: 73.23.Hk, 85.35.Ds, 85.35.Be, 73.21.La
+I. INTRODUCTION
+Due to the increasing interest in ultra-fast electron
+dynamics considerable progress occurred recently in the
+theoretical description of time dependent mesoscopic
+transport. New methods and numerical implementations
+are rapidly evolving. Transient currents in open nanos-
+tructures are studied with Green-Keldysh formalism,1,2,3
+scattering theory,4and quantum master equation.5,6,7,8
+Most of the results were obtained for noninteracting elec-
+trons due to the well known computational diculties to
+include time-dependent Coulomb eects.
+It is nevertheless clear that the electron-electron inter-
+action is important in such problems. An eort to incor-
+porate it has been recently done by Kurth et al.9followed
+by My oh anen et al.10who have described correlated time-
+dependent transport in a short 1D chain dened by a
+lattice Hamiltonian. The 1D sample was connected to
+external leads and the current was driven by a time-
+dependent bias. Those authors used a method based
+on the Kadano-Baym equation for the non-equilibrium
+Green's function combined with the time-dependent den-
+sity functional theory to include the Coulomb interac-
+tion in the sample. Once the Green's functions were
+calculated total average quantities of interest could be
+obtained, like charge density or current, both in the tran-
+sitory and in the steady state. However this method does
+not say much about the dynamics of specic internal
+states of the sample system. In view of the spectroscopy
+of excited states11it is important to have a theoretical
+tool for understanding separately the charging and re-
+laxation of the ground states and excited states in meso-
+scopic systems in time-dependent conditions.
+Our alternative is to use the statistical, or density op-
+erator. The complete information about the time evo-
+lution of each quantum state of the sample is captured
+in the reduced density operator (RDO), which is the so-lution of the generalized master equation (GME). Once
+the RDO is dened in the Fock space the inclusion of
+the Coulomb interaction becomes a known computational
+problem: obtaining the many-electron states (MES) of
+the sample. The RDO matrix is then calculated in the
+basis of the interacting MES.
+Let us enumerate some of the previous theoretical
+schemes to treat transport and electron-electron inter-
+action with the master equation. One of the rst at-
+tempts to derive a master equation for an interacting sys-
+tem with time-dependent perturbations belongs to Lan-
+greth and Nordlander for the Anderson model.12Gurvitz
+and Prager started from the time-dependent Shr odinger
+equation for the MES wave functions and ended up with
+Bloch-like rate equations for the density matrix of a quan-
+tum dot.13The electronic currents were calculated in the
+steady state and it was shown that the Coulomb interac-
+tion renormalizes the tunneling rates between the leads
+and the system. In the same context K onig et al.14de-
+veloped a powerful diagrammatic technique by expand-
+ing the RDO of a mesoscopic system in powers of the
+tunneling Hamiltonian. The time-dependence of the sta-
+tistical operator of the coupled and interacting system
+implies a quantum master equation for the so called pop-
+ulations. In this method the Coulomb interactions are
+treated exactly, which makes it appealing for studying
+various correlation eects like cotunneling.15The con-
+nection between the real-time diagrammatic approach of
+K onig et al.14and the Nakajima-Zwanzig approach16,17
+to the generalized master equation (GME) approach was
+made transparent by Timm.18
+More recently Li and Yan19combined the n-resolved
+master equation and the time dependent density-
+functional method to write down a Kohn-Sham master
+equation for the reduced single-particle density matrix.
+Also, Esposito and Galperin,20using the equation of mo-
+tion for the Hubbard operators, have obtained a many-arXiv:1001.0047v1  [cond-mat.mes-hall]  30 Dec 20092
+body description of quantum transport in an open sys-
+tem and established a connection between the GME and
+non-equilibrium Greeen's functions. They studied simple
+systems in the steady state regime: a resonant level cou-
+pled to a a single vibration mode, an interacting dot with
+two spins, and a two-level bridge. Another recent work
+by Darau et al.21implemented the GME for a benzene
+single-electron transistor and used exact MES to compute
+steady state currents within the Markov approximation.21
+The stability diagram and the conductance peaks were
+obtained and a current blocking due to interferences be-
+tween degenerated orbitals was noticed.
+In our previous papers7,8we considered the GME
+method for the RDO of independent electrons in the Fock
+space. We discussed the transient transport through
+quantum dots and quantum wires. The contact between
+the leads and the sample was switched on at a certain ini-
+tial moment t0. We discussed extensively the occupation
+of the states within the bias window and the geometrical
+eects on the transient currents. We described the cou-
+pling between the sample and the leads via a tunneling
+Hamiltonian in which we took into account the spatial
+extension of the wave functions of both subsystems in
+the contact region.
+In spite of earlier or more recent attempts a complete
+description of the Coulomb eects in the time-dependent
+transport is still missing, especially in sample models
+larger than a few sites. In the present work we com-
+bine the GME method with the Coulomb interaction in
+the sample and we analyze the dynamics of the electrons
+starting with the moment when the leads are coupled to
+the sample until a steady state is reached. The Coulomb
+interaction is included in the Hamiltonian of the isolated
+sample and the interacting MES are calculated with the
+exact diagonalization method. This means the Coulomb
+interaction is fully included with no mean eld assump-
+tion or density-functional model. The number of single-
+electron states (SES) used to dene the matrix elements
+of the Hamiltonian of interacting electrons is suciently
+large such that the MES of interest are convergent. Due
+to the nite bias window only a limited number of MES
+participate to the charge transport through the sample,
+i. e.only those energetically compatible with the elec-
+trons in the leads. Hence the MES of interest are selected
+by the chemical potentials in the leads. We calculate the
+RDO matrix elements in the subspace of these MES using
+the GME. The electron-electron interaction in the leads
+is neglected.
+It is well known that the Fock space increases expo-
+nentially with the number of SES. In addition the time
+dependent numerical solution of the GME is also com-
+putational expensive. So at this stage we are limited to
+describe only few electrons in the system: up to ve in a
+small system, but only up to three in a larger one.
+The paper is organized as follows. In Section 2 we
+brie
y describe the GME, the inclusion of the Coulomb
+interaction, and the selection of the MES. Next, in Sec-
+tion 3, we show results for three models: a short 1Dchain, a 2D lattice of 12 10 sites, and a nite quantum
+wire with parabolic lateral connement. Conclusions and
+discussions are presented in Section 4.
+II. GME METHOD AND COULOMB
+INTERACTION
+In this section we summarize the main lines of our
+method. The equations apply both to the lattice and
+continuous models. The time-dependent transport prob-
+lem is considered within the partitioning approach which
+is known both from the pioneering work of Caroli22and
+from the derivation of the GME. Prior to an initial time
+t0the left lead (L) having a \source" role, and the right
+lead (R) having a \drain" role, are not connected to the
+sample and therefore can be characterized by equilibrium
+states with chemical potentials LandRrespectively.
+Our aim is to compute the time dependent currents 
ow-
+ing through the sample and leads starting at moment t0,
+when the three subsystems are connected, until a station-
+ary state is reached.
+The generic Hamiltonian of the total system consisting
+of the sample plus the leads is:
+H(t) =HL+HR+HS+HT(t): (1)
+Hlwithl=L;R are the Hamiltonians of the leads. We
+denote by "qland qlthe single-particle energies and
+wave functions respectively, for each lead. Using the cre-
+ation and annihilation operators associated to the single-
+particle states, cy
+qlandcql, we can write
+Hl=Z
+dq"qlcy
+qlcql: (2)
+HSis the Hamiltonian of the sample. In the absence
+of the interaction the SES have discrete energies denoted
+asEnand corresponding one-body wave functions n(r).
+Using now the creation and annihilation operators for the
+sample SES, dy
+nanddn, we can write
+HS=X
+nEndy
+ndn+1
+2X
+nm
+n0m0Vnm;n0m0dy
+ndy
+mdm0dn0:(3)
+The second term in Eq. (3) is the Coulomb interaction.
+In the SES basis the two-body matrix elements are given
+by:
+Vnm;n0m0=Z
+drdr0
+n(r)
+m(r0)u(rr0)n0(r)m0(r0);
+(4)
+whereu(rr0) is the Coulomb potential.
+The third term of Eq. (1) is the so-called tunneling
+Hamiltonian describing the transfer of particles between
+the leads and the sample:
+HT(t) =X
+l=L;RX
+nZ
+dql(t)(Tl
+qncy
+qldn+h:c:):(5)3
+HTcontains two important elements: (1) The time de-
+pendent switching functions l(t) which open the contact
+between the leads and the sample; these functions mimic
+the presence of a time dependent potential barrier. (2)
+The coupling Tl
+qnbetween a state with momentum qof
+the leadland the state nof the isolated sample, with
+wave function n. The coupling coecients Tl
+qndepend
+on the energies of the coupled states and, maybe more
+important, on the amplitude of the wave functions in the
+contact region. As we have shown in our previous work7,8
+this construction allows us to capture geometrical eects
+in the electronic transfer. A precise denition of the cou-
+pling coecients is however model specic, and will be
+mentioned in the next section.
+The evolution of our system is completely determined
+by the statistical operator W(t) associated to the total
+Hamiltonian H(t) dened in Eq.(1). W(t) is the solution
+of the quantum Liouville equation with a known initial
+value, prior to the coupling of the sample and leads:
+i~_W(t) = [H(t);W(t)]; W (tt0) =LRS;(6)
+The isolated leads are described by equilibrium distribu-
+tions,
+l=e(HllNl)
+Trlfe(HllNl)g; l=L;R; (7)
+and the isolated sample by the density operator S. Af-
+ter the coupling moment the dynamics of the sample is
+conveniently described by the RDO which is dened by
+averaging the total statistical operator over those degrees
+of freedom belonging to the leads:
+(t) = TrLTrRW(t); (t0) =S: (8)
+In the absence of the electron-electron interaction the
+MES eigenvectors of HSare bit-strings of the form ji=
+ji
+1;i
+2;::;i
+n:::i, wherei
+n= 0;1 is the occupation number
+of then-th SES. The set fgis a basis in the Fock space
+of the isolated sample and the RDO can be seen as a
+matrix in this basis. From Eqs. (6)-(8) we obtain in the
+lowest (2-nd) order in the coupling parameters Tl
+qnthe
+GME (see Ref. 7 for details):
+_(t) =i
+~[HS;(t)]
+1
+~2X
+l=L;RZ
+dql(t)([Tql;
+ql(t)] +h:c:);(9)
+where the coupling operator Tqlhas matrix elements
+(Tql)=X
+nTl
+qnhjdyji: (10)
+The operators 
+ qland qlare dened as
+
+ql(t) =eitHSZt
+t0dsl(s)ql(s)ei(st)"qleitHS;
+ql(s) =eisHS
+Ty
+ql(s)(1fl)(s)Ty
+qlfl
+eisHSandflis the Fermi function of the lead l.
+In the presence of the electron-electron interaction in
+the sample the MES which are eigenstates of HSare lin-
+ear combinations of bit-strings: HSj) =Ej), where
+j) =P
+Cji,Cbeing the mixing coecients
+which can be found together with the energies Eby
+diagonalizing HS. (To distinguish better between the
+noninteracting and the interacting MES we use the right
+angular bracket for the former and the regular curved
+bracket for the later.) Using now the set fgas a basis,
+i. e.theinteracting MES, the GME has the same form
+as Eq. (9), where the matrix elements of all operators
+are now dened in the interacting basis and the matrix
+elements of the coupling operators are
+(Tql)=X
+nTl
+qn(jdyj): (11)
+Because the sample is open the number of electrons N
+contained in the sample is not xed. The Hamiltonian
+HSgiven in Eq. (3) commutes with the total \number"
+operatorP
+ndy
+ndn. ThusNis a \good quantum number"
+such that any state j) has a xed number of electrons.
+So the MES can also be labeled as j) =jN;i) with
+i= 0;1;2;:::an index for the ground and excited states
+of the MES subset with Nelectrons. The many-body
+energies can also be written as E=E(i)
+N. In the practical
+calculations Nvaries between 0 (the vacuum state) and
+Nmaxwhich is the total number of SES considered in the
+numerical diagonalization of HS. The total number of
+MES is thus 2Nmax.
+If the coupling between the leads and the sample is
+not too strong we expect that only a limited number of
+MES participate eectively to the electronic transport.
+These states are naturally selected by the bias window
+[R;L]. In the following examples, by selecting suitable
+values of the chemical potentials in the leads, we will
+truncate the basis of interacting MES to a reasonably
+small subset such that we can solve numerically Eq. (9)
+with our available computing resources. To relate the
+bias window with the eective MES we need to consider
+the chemical potential of the isolated sample containing
+Nelectrons,
+(i)
+N=E(i)
+NE(0)
+N1; (12)
+which is the energy required to add the N-th electron on
+top of the ground state with N1 to obtain the i-th MES
+withNparticles.23We expect the current associated to
+the MESjN;i) to depend on the location of the chemical
+potential(i)
+Nrelatively to the bias window. In particular
+it is clear that if at the coupling moment t0the sample is
+empty all MES with (i)
+NLwill remain empty both
+during the transient and the steady states, so they can
+be safely ignored when solving the GME.4
+III. MODELS AND RESULTS
+We have numerically implemented the GME method
+both for lattice and continuous models. The sample mod-
+els are: a short 1D chain with 5 sites, a 2D rectangular
+lattice with 1210 = 120 sites, and a short quantum
+wire withe parabolic lateral connement. In all cases the
+coupling functions have the form
+l(t) = 12
+e
t+ 1(13)
+with
a constant parameter, such that at the initial mo-
+ment, which is t0= 0, we have l(0) = 0 (no coupling),
+and in the steady state, for t!1 ,l= 1 (full coupling).
+A. A toy model: short 1D chain
+In this model the two semi-innite leads are attached
+to the ends of a 1D chain with 5 sites. The coupling
+between a lead state with wave function  qland a sam-
+ple state with wave function nis given by the product
+between the wave functions at the contact site:
+Tl
+qn=Vl 
+ql(0)n(il); (14)
+where 0 is the contact site of the lead l=L;R, the end
+sites of the sample being iL= 1 andiR= 5.
+FIG. 1: (Color online) The equilibrium chemical potentials
+(0)
+Nfor 1N5 as a function of the interaction strength
+U. The dotted lines mark the chemical potentials of the leads
+selected in the transport simulations shown in the next gure,
+i. e.L= 5:25 andR= 4:75.
+The reason to call this a toy model is that we can ob-
+tain the complete set of 25= 32 MES, i. e.we do not
+need to cut the basis of the 5 SES. We also do not need
+to cut the MES basis, all matrix elements of the statisti-
+cal operator can be numerically calculated, even if not all
+of them might be important for the currents. In addition
+we will consider the strength of the Coulomb interaction
+as a free parameter U, whereas in a realistic systems this
+is xed by the electron charge and the dielectric con-
+stant of the material. Our goal is to have a qualitativeunderstanding of the underlying physics, and in particu-
+lar to show the presence of the Coulomb blocking eects
+at certain values of Uor of the chemical potentials of the
+leads. The Coulomb matrix elements dened in Eq. (4)
+are calculated as
+Vnm;n0m0=X
+i6=i0
+n(i)
+m(i0)U
+jii0jn0(i)m0(i0):(15)
+In Fig.1 we show the equilibrium chemical potentials
+(0)
+Ncorresponding to ground states with 1 N5 par-
+ticles against the interaction strength U. One observes
+a linear dependence of (0)
+NonU, with slope increasing
+withN. Obviously the total Coulomb energy increases
+both withUandN.
+Let us now brie
y review the Coulomb blockade
+scenario.24Suppose the isolated sample contains Nelec-
+trons and the chemical potentials of the leads are cho-
+sen such that (0)
+N< R< L< (0)
+N+1. Then the bias
+window [R;L] may include one or more of the excited
+congurations with Nparticles. In general some states
+withNelectrons may have excitation energies exceed-
+ingLor even(0)
+N+1. This situation corresponds to the
+Coulomb blockade phenomenon. Indeed, the addition of
+the (N+ 1)-th electron is energetically forbidden. Con-
+sequently the current in the steady state should vanish.
+However, shorter or longer transient currents are gener-
+ated by all many-body congurations in the vicinity of
+the bias window.
+Fig. 2(a) and 2(b) show the total currents in the left
+lead and the total charge residing in the sample for sev-
+eral values of the interaction strength. Uis measured
+in units of tS, the hopping parameter in the sample,7
+and the time is expressed in units of ~=tSwhile the cur-
+rent is in units of etS=~. The coupling constant in Eq.
+(13) is
= 1. The system is initially empty and thus
+(0) =j00000ih00000j.
+The chemical potentials of the leads, L= 5:25 and
+R= 4:75, are chosen such that in the absence of
+Coulomb interaction, i. e. forU= 0,(0)
+4is located
+within the bias window. In this case we obtain in the
+steady state the mean number of electrons about 3.6 and
+a non-vanishing current in the leads. This is understand-
+able, since (0)
+4=E4= 5, which is the 4-th level of
+the isolated sample. The occupation of this level in the
+steady state is about 0.6, the other states being either full
+or empty. Also in this case, the excited states have small
+contributions to the steady state current as the system
+tends to be in the ground state with N= 3 electrons.
+Those contributions may also depend on the coupling
+strength of individual states with the leads, but in gen-
+eral remain small.25
+The situation may change for U6= 0. For the inter-
+acting system, e. g. forU= 0:3, the system settles down
+in the Coulomb blockade regime, the total current be-
+ing almost suppressed in the steady state. This happens
+because the interaction pushes the chemical potentials
+upwards such that for U= 0:3 both ground states with5
+FIG. 2: (Color online) The total current entering the 5 1
+sample from the left lead as a function of time for the dierent
+values of the interaction strength U. The chemical potentials
+of the leads L= 5:25 andR= 4:75.
+N= 3 andN= 4 electrons are outside the bias win-
+dow and cannot produce steady currents. When the in-
+teraction strength is further increased to U= 0:5 and
+U= 1 the steady state currents are gradually restored.
+This could look surprising, but one can see in Fig.1 that
+by increasing Uthe ground state conguration with 3
+electrons approaches and enters the bias window. Con-
+sequently the transport becomes again possible. Note
+that while the steady state currents are not monotonous
+w.r.t.Uthe charge absorbed in the system continuously
+decreases, Fig. 2(b).
+In transport experiments the strength of the electron-
+electron interaction is indeed xed. The usual way to
+obtain the Coulomb blockade is to vary the chemical po-
+tentials of the leads relatively to the energy levels of the
+sample, or vice versa. In Fig. 3 we show the currents
+in both leads for dierent values of the chemical poten-
+tialR, while keeping xed L= 7. The strength of
+the Coulomb interaction is U= 1 and(0)
+4almost equals
+L. The steady state value of the current decreases as
+Rincreases, because fewer states are included in the
+bias window. The Coulomb blockade onset occurs for
+R>5, when(0)
+3drops below R. We observe that the
+FIG. 3: The time-dependent total currents in the left and
+right leads at dierent values of the chemical potential R.
+The current in the right lead starts at negative values. Other
+parameters: VL=VR= 0:750,U= 1:0.
+maximum value of the total current in the left lead does
+not change much when Rvaries. In contrast, the tran-
+sient current in the right lead is negative and increases
+in magnitude as Rincreases. This means that the right
+lead feeds the many-body congurations that fall below
+R.
+The contribution of the excited states to the transient
+and steady state currents depends strongly on the bias
+window. In Fig. 4 we show the currents entering the sam-
+ple from the left lead, carried by the states with N= 2
+andN= 3 electrons, for R= 3;4;5 (the cases with non-
+vanishing current in the steady state). We also show sep-
+arately the contribution to the currents associated to the
+ground state congurations, related to (0)
+2and(0)
+3, and
+the complementary contribution of all the excited states
+with 2 and 3 particles. In this case the wave vectors of
+the ground states are mostly given by the non-interacting
+wave vectors:j11000iwith weight 97% and j11100iwith
+98% forN= 2 andN= 3 respectively.
+ForR= 3 the steady state current of the ground
+state conguration is vanishingly small and so the total
+negative current associated to two-particle states comes
+mostly from the excited states. In the many-body energy
+spectrum of the isolated sample we obtain 5 excited con-
+gurations with (i)
+22[R;L] = [3;7]. AsRmoves up
+the steady state current of the ground state with N= 2
+becomes also negative. The combined contributions of
+the excited states vanishes at R= 5. As can be seen
+from Fig. 1 R= 5 is well above (0)
+2, but very close to
+(0)
+3. Consequently, the ground conguration with N= 2
+is heavily populated in the steady state, whereas the ex-
+cited states have low probability and thus weak current.
+Actually, as we have checked, all the currents associated
+to each excited state with N= 2 vanish individually. In
+the transient regime however the N= 2 currents in all
+three cases are dominated by the excites states.
+The currents of the excited states having N= 3 elec-6
+FIG. 4: The separate contributions to the current of the
+ground state with Nparticles and of allexcited states with
+Nparticles, for dierent values of R. For completeness we
+also include the total currents JLfor the same congura-
+tions. The discussion is made in the text. Other parameters:
+VL=VR= 0:750,U= 1:0.
+trons are positive at R= 3, but change sign at R= 4.
+ForR= 5 their magnitude exceeds the contribution of
+the ground state which is always positive. A more de-
+tailed analysis of the currents carried by specic excited
+states will be given for the 2D model.
+Finally, both in the transient and in the steady states
+the currents have small periodic 
uctuations determined
+by the permanent transitions of electrons between the
+states in the sample and the states in the leads and
+back.25They are best seen in Fig. 2(a). Such 
uctuations
+have also been obtained very recently by Kurth et al. us-
+ing combination of the non-equilibrium Green's functions
+and the time dependent density-functional theory of the
+Coulomb interaction.26
+B. 2D lattice
+We show now results for a 2D rectangular lattice with
+1210 sites. For a lattice constant of a= 5 nm this
+sample can be seen as a discrete version of a quantum
+dot of 60 nm50 nm. We used the lowest 10 SES of the
+non-interacting sample in the numerical diagonalization
+of the interacting Hamiltonian. This number is sucient
+to produce convergent results for the rst 50 MES for
+an interaction strength U= 0:8. The Coulomb matrix
+elements are calculated in the same way as for the 1Dcase, Eq. (15), except that now the site indices are two-
+dimensional, i. e.i= (ix;iy) andi0= (i0
+x;i0
+y).
+The two contact sites are chosen at diagonally opposite
+corners of the sample. The coupling coecients are cal-
+culated with Eq. (14), like for the 1D chain, and depend
+on the wave function of the particular SES at the con-
+tact sites. These coecients are illustrated in Fig. 5(a).
+The reduced density matrix is calculated using the rst
+50 MES. This allowed us to take into account many-body
+congurations with up to 3 electrons.
+In Fig. 5(b) we show the chemical potentials (i)
+Nfor
+the ground and excited states with N= 1;2;and 3 par-
+ticles. At the initial moment t0= 0 the system is empty.
+Based on the previous example, the main contribution to
+the currents in the steady states is expected from those
+MES with ground state chemical potentials located inside
+the bias window [ R;L]. One also observes excited con-
+gurations with Nparticles having chemical potentials
+larger than (0)
+N+1.
+FIG. 5: (Color online) (a) The coupling amplitudes jTqnj2
+forn= 1;::;5 between single-particle states in the leads with
+momentum qand the lowest 5 single-particle states of the
+isolated dot. (b) The generalized chemical potentials for N-
+particle interacting congurations. The red crosses mark (0)
+N
+while the other ones correspond to generalized potentials (i)
+N
+related to the i-th excited state of the Nparticle system.
+In the following we discuss the currents carried by the7
+various many-body states involved in transport. In a rst
+series of calculations we selected the chemical potential
+R= 0:2 and used two values of the chemical potential
+of the left lead L= 0:4 andL= 0:6. ForR= 0:2 and
+L= 0:4 the bias window contains only the 1-st and the
+2-nd excited congurations with N= 1, Fig. 5(b). The
+ground states for N= 1 andN= 2 are instead located
+below and above the bias window, respectively. Conse-
+quently the steady state current is very small. When
+Lincreases to 0.6 the ground state conguration with
+N= 2 enters the bias window and the current increases,
+Fig. 6(a).
+To analyze the transient regime we split the current
+into contributions given by the ground state and excited
+states with 1 electron (see Fig. 6(b)). When L= 0:4 the
+1-st and 2-nd excited state carry currents exceeding the
+current associated to the ground state, which survive all
+the way to the steady state. The current corresponding
+to the 2-nd excited state is smaller than the current of the
+1-st excited state, but comparable to that of the ground
+state. This is explained by the strength of the coupling
+coecients shown in Fig. 5(a), the 2-nd single-particle
+state being stronger coupled to the leads. The remaining
+higher excited states give oscillating and fast decaying
+transient currents. In Fig. 6(c) L= 0:6 and therefore
+higher excited states enter the bias window; their tran-
+sient currents are still decaying but at a smaller rate.
+Comparing with Fig. 6(a) it in clear that the transient
+regime is dominated by excited states.
+Next we discuss currents associated with states having
+2 and 3 electrons. We keep now xed R= 0:35 and
+again increase Lstarting with 0.6. Fig. 7(a) shows the
+total currents in the left lead for N= 2 andN= 3. As
+the bias increases the transient currents are enhanced,
+but they become comparable as the system approaches
+the steady state. In Fig. 7(b) the total current on three
+particle states shows a dierent behavior: the steady
+states value increases drastically when Lmoves up. To
+explain this one can look again at the diagram of the
+chemical potentials, Fig. 5(b). At L= 0:6 the 3-particle
+congurations are above the bias window and as such
+they contribute less to the current. In contrast, as L
+increases the ground state conguration with N= 3 en-
+ters the bias window, the window is closer to the excited
+states, and thus the total current increases. Actually, for
+L= 0:8 and 0:9 the current for N= 3 does not reach
+the steady state in the time interval considered.
+Now we look at the contribution of the excited states
+withN= 2 for two cases, L= 0:6 andL= 0:9. Again,
+the inspection of the diagram in Fig. 5(b) predicts the
+results of Fig. 8. When L= 0:6 there is just one excited
+conguration within the bias window, in addition to the
+ground state. In Fig. 8(a) we see that in the steady state
+these two congurations give signicant contributions to
+the current, whereas the higher excited states play a role
+only in the transient regime. Fig. 8(b) shows that at
+L= 0:9 the currents of the excited states and of the
+ground state are decreasing, some of them reaching even
+FIG. 6: (a) The total currents in the left and right leads for
+L= 0:6 andL= 0:4, while keeping R= 0:2. (b) The
+partial currents in the left lead for single-particle states when
+L= 0:4 andR= 0:2. (c) The partial currents in the left
+lead for single-particle states when L= 0:6 andR= 0:2
+negative values towards the steady state. This happens
+because the bias window includes now the ground state
+withN= 3 and the excited states with N= 2 deplete in
+the favor of the ground state.
+The sign of the current carried by states with Npar-
+ticles depends on the placement of the corresponding
+ground state chemical potential relatively to the bias win-
+dow. For example if we x L= 1:5 andR= 0:65 we8
+FIG. 7: (a) The total current in the left lead carried by all
+many-body congurations with N= 2, for increasing values
+ofL(i. e.0.6,0.7,0.8 and 0.9) and R= 0:2. (b) The same
+forN= 3.
+obtain(0)
+2< L. Fig. 9(a) shows the N-particle cur-
+rents when the sample initially contains two electrons in
+the ground state. This initial state evolves faster to the
+steady state than the empty system. While for N= 3
+the current in the left lead is positive, for both N= 2
+andN= 1 the currents are negative. The charge re-
+siding on each N-particle state and the total charge are
+shown in Fig. 9(b). Since single-particle congurations
+are unlikely their occupation vanishes. The total charge
+accumulated on the N= 3 states increases up to 2, while
+the total charge on the N= 2 states decreases from 2 to
+0.75. The sign of the current for N= 2 becomes pos-
+itive when Ris lowered to 0.2, Fig. 9(c), and exceeds
+the current carried by the states with N= 3. This is
+because the 1-st and the 2-nd SES practically determine
+the ground state with two electrons and thus (0)
+2, and
+also because the 1-st SES is strongly coupled to the leads.
+However, the current with N= 1 is still negative.
+FIG. 8: (a) The total current in the left lead carried by all
+many-body congurations with N= 2 atL= 0:6. (b) The
+same forL= 0:9. Other parameters R= 0:35.
+C. Parabolic quantum wire
+In this subsection we apply the GME with Coulomb
+interaction to describe the transport through a short
+quantum wire of length Lx= 300 nm with a parabolic
+connement in the y-direction perpendicular to the di-
+rection of transport. The contact ends of the isolated
+wire atLx=2 are described by hard walls. This is
+now a continuous model, where a large functional ba-
+sis is used to expand the eigenfunctions of the system
+in. In a similar manner we use a functional basis with
+complete truncated sets of continuous and discrete func-
+tions to expand the eigenfunctions of the semi-innite
+parabolic leads in. To show that we can describe the com-
+bined geometrical eects imposed on the system by it's
+geometry and an external perpendicular magnetic eld
+we place the quantum wire is in an external magnetic
+eld of strength 1 :0 T. The characteristic connement
+energy is given by ~
+0= 1:0 meV. We assume GaAs
+parameters with m= 0:067me,= 12:4 meV. The
+magnetic length modied by the parabolic connement
+isaw=p
+~=(m
+w), with 
+2
+w= 
+2
+0+!2
+c. and the
+cyclotron frequency !c=eB=(mc). AtB= 1:0 T,
+aw= 23:87 nm. The semi-innite leads having the same9
+FIG. 9: (a) The total current in the left lead carried by N-
+particle states and the total charge. for L= 1:5 and for
+R= 0:65. (b) The occupation number of the N-particle
+states. (c) The total current in the left lead carried by N-
+particle states for L= 1:2 andR= 0:2. (d) The occupation
+number of the N-particle states and the total charge.parabolic connement and being subject to the same ex-
+ternal perpendicular magnetic eld have a continuous en-
+ergy spectrum with discrete Landau sub-bands.
+The Coulomb potential in Eq. (4) in the 2D wire is
+described by
+u(rr0) =e2
+p
+(xx0)2+ (yy0)2+2; (16)
+with the small convergence parameter ( =aw) = 0:01
+to facilitate the two-dimensional numerical integration
+needed for the matrix elements (4).
+After the GME (9) has been transformed to the in-
+teracting many-electron basis by the unitary transfor-
+mation obtained by the diagonalization of HS(3) we
+truncate the RDO (8) to 32 MES. For the bias range
+0:0
=LR1:7 meV used here 10 SES are
+sucient to obtain these lowest 32 states with good accu-
+racy. We will be omitting singly occupied states of high
+energy that should not be relavant for the parameters
+here. The natural strength of the Coulomb interaction
+will only give us MES that are occupied by one or two
+electrons in the energy range 0 to 6 meV covered by the
+32 MES.
+Since in the partitioning approach [ HS;HL] = 0 we
+have to construct Tl
+qnas a non-local overlap ofnand
+ L;R
+qon the contact regions Cl; l=L;R:8
+Tl
+qn=Z
+Cldrdr0
+ 
+ql(r)n(r)gl
+qn(r;r0) +h:c:

+:(17)
+gl
+qn(r;r0) =gl
+0exp
+l
+1(xx0)2l
+2(yy0)2
+expjEn"qlj
+
l
+E
+: (18)
+As before"qlis the energy spectrum of lead l, andEn
+is the energy of the SES numbered by nin the quan-
+tum wire. The quantum number qfor the states in leads
+represents both the discrete Landau band number and a
+continuous quantum number that can be related to the
+momentum of a particular state. Here we use the pa-
+rameters1a2
+w=2a2
+w= 0:25, 
LR
+E= 0:25 meV, and
+gLR
+0= 40 meV for B= 1:0 T. The domain of the over-
+lap integral for the leads is 2awinto the lead or the
+system forxandx0from each end of the wire at Lx=2
+and between4awforyandy0, see Ref. (8) for an exact
+denition. All the SES will be coupled to the leads, but
+the coupling strength will depend on the character of the
+SES, whether it is an edge- or bulk state and other ner
+geometrical details that is brought about by the magnetic
+eld.
+The right chemical potential Ris held at 1 :4 meV
+and the transport properties are calculated for dierent
+values of the bias 
 by varyingL. Figure 10 compares
+the total occupation of all one- electron and two-electron
+MES for the interacting system at two dierent values of
+the bias. At, 
 = 0:2 meV we see that almost solely10
+FIG. 10: (Color online) The total charge residing in one- and
+two-electron states as a function of time for two dierent val-
+ues of the bias 
 .B= 1:0 T,Lx= 300 nm, ~
+0= 1:0
+meV.
+one-electron states are occupied, while for 
 = 1:2 meV
+initially it is likely to have one-electron states occupied,
+but very soon the occupation of the two-electron states
+becomes as probable with the likelihood of the occupa-
+tion of the one-electron states fast reducing with time.
+We also have to admit here that even though the steady
+state value of the total current through the system can
+be deduced by the values of the current at 270 ps, the
+charging of the system takes much longer time, since we
+are using here a very weak coupling to the leads that
+mimics a tunneling regime.
+If we now use the average value of the current in the left
+and right leads at t= 270 ps as a measure of the steady
+state current we get the information displayed in Fig.
+11, where the steady state value of the current is shown
+for the interacting system as a function of the bias and
+compared to the charge in the system. We have a clear
+Coulomb blocking in the interacting system. In the case
+of a non-interacting system the lack of a gap between the
+one- and two electron MES and a strong mixing of the
+energy regimes of two- and three-electron states the two-
+electron plateau only appears as a small shoulder. The
+FIG. 11: The total steady state current for interacting 10
+SES, and the total charge at t= 270 ps. for dierent values
+of the bias 
 .B= 1:0 T,Lx= 300 nm, ~
+0= 1:0 meV.
+32 MES selected here include no three-electron or MES
+with higher number of electrons. It should be mentioned
+here that a dierent choice of the right bias Rcan result
+in the system charging faster and thus at the same time
+the total current through it being smaller. This comes
+from the fact that the states have a dierent coupling to
+the leads and the time range shown here is very much in
+the transient- or it's long exponential decay regime.
+Figure 12 displaying the current in the right lead gives
+an idea how the Coulomb blocking plateau appears after
+the transition regime. The transition regime where the
+FIG. 12: The total current in the right lead for interacting
+and non-interacting 10 SES as a function of the bias 
 and
+time.B= 1:0 T,Lx= 300 nm, ~
+0= 1:0 meV.
+right current goes negative, i. e.where it supplies charge
+to the system is partially truncated from the gure.11
+IV. SUMMARY AND CONCLUSIONS
+We calculated time-dependent currents in open meso-
+scopic systems composed by a sample attached to two
+semi-innite leads, by solving the generalized master
+equation for the reduced density operator acting in the
+Fock space of the sample. This is the natural frame-
+work for including the Coulomb electron-electron inter-
+action in the sample, which is the main achievement of
+this work. The Coulomb interaction is treated in the
+spirit of the exact diagonalization method, i. e.in a pure
+many-body manner. The interacting many-body states
+of the sample are expanded in the basis of non-interacting
+\bit-string" states with unspecied number of electrons.
+We believe our method is a viable alternative to a recent
+approach based on a time-dependent density-functional
+model.9,10,26We used three sample models, a short 1D
+wire with 5 sites, but also a larger 2D lattice with 120
+sites and a continuous model, whereas the cited group
+used much smaller samples even with no structure.26
+Indeed, due to computational limitations we could use
+only a restricted, eective number of many-body states in
+the GME, between 30-50 depending on the model, from
+the bottom of the energy spectrum. We chose the bias
+window [R;L] and the strength of the sample-leads
+coupling parameters VR;Lsuch that only the eective
+states contribute to the transport of electrons through
+the sample, whereas the states with higher energy are
+unreachable by the electrons. Consequently the number
+of electrons in the sample can be only up to 3 or 4.
+We could calculate the contribution to the charge and
+currents in the sample and in the leads respectively, cor-
+responding to any particular many-body state. We use
+the 1D chain as a toy model to emphasize the dominant
+role of the excited states in the transient regime and the
+onset of the Coulomb blockade in the steady state. A
+similar 1D model with 4 sites 1D has been considered
+recently by My oh anen et al.10
+As shown also in our previous works on time-dependenttransport in non-interacting systems the GME method
+includes information on the energy structure of the sam-
+ple, but also on the geometrical properties re
ected in
+the wave functions and sample-lead contacts.7,8,25Here
+we illustrate these aspects, in the interacting case, for two
+nanosystems: a two-dimensional quantum dot described
+by a lattice Hamiltonian and a short parabolic quantum
+wire. The time-dependent occupation of specic many-
+body states was thoroughly analyzed, for dierent values
+of the chemical potentials of the leads. It turned out
+that the excited states with Nelectrons contribute to
+the steady state currents if the ground state conguration
+withN+ 1 particles is not available for transport. How-
+ever, if(0)
+N<Rand at the same time (0)
+N+1lies within
+the bias window the excited states with Nparticles
+are active only in the transient regime and become de-
+populated in the steady state regime. This behavior is of
+interest in the excited-state spectroscopy experiments.11
+To our knowledge the time-dependent currents associated
+to excited states have not been discussed theoretically so
+far.
+Acknowledgments
+The authors acknowledge nancial support from the
+Development Fund of the Reykjavik University Grant
+No. T09001, the Research and Instruments Funds of the
+Icelandic State, the Research Fund of the University of
+Iceland, the Icelandic Science and Technology Research
+Programme for Postgenomic Biomedicine, Nanoscience
+and Nanotechnology, the National Science Council of Tai-
+wan under contract No. NSC97-2112-M-239-003-MY3.
+V.M. also acknowledges the hospitality of the Reykjavik
+University, Science Institute and the partial nancial sup-
+port from PNCDI2 program (grant No. 515/2009) and
+grant No. 45N/2009.
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\ No newline at end of file
diff --git a/1001.0048.txt b/1001.0048.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6a0323eddfb563cf606f8dc2ccce2cfdc8fd9acb
--- /dev/null
+++ b/1001.0048.txt
@@ -0,0 +1,1038 @@
+arXiv:1001.0048v2  [math.AP]  7 Jan 2010Nonlinear stability of periodic traveling wave solutions o f
+viscous conservation laws in dimensions one and two
+Mathew A. Johnson∗Kevin Zumbrun†
+November 12, 2018
+Keywords : Periodic traveling waves; Bloch decomposition; modulate d waves.
+2000 MR Subject Classification : 35B35.
+Abstract
+Extending results of Oh and Zumbrun in dimensions d≥3, we establish nonlin-
+ear stability and asymptotic behavior of spatially-periodic traveling- wave solutions of
+viscous systems of conservation laws in critical dimensions d= 1,2, under a natural
+set of spectral stability assumptions introduced by Schneider in th e setting of reaction
+diffusion equations. The key new steps in the analysis beyond that in d imensionsd≥3
+are a refined Green function estimate separating off translation as the slowest decaying
+linear mode and a novel scheme for detecting cancellation at the leve l of the nonlinear
+iteration in the Duhamel representation of a modulated periodic wav e.
+1 Introduction
+Nonclassical viscous conservation laws arising in multiph ase fluid and solid mechanics ex-
+hibit a rich variety of traveling wave phenomena, including homoclinic (pulse-type) and
+periodic solutions along with the standard heteroclinic (s hock, or front-type) solutions
+[GZ, Z6, OZ1, OZ2]. Here, we investigate stability of period ic traveling waves: specifi-
+cally, sufficient conditions for stability of the wave. Our ma in result, generalizing results of
+Oh and Zumbrun [OZ4] in dimensions d≥3, is to show that strong spectral stability in the
+sense of Schneider [S1, S2, S3] implies linearized and nonli nearL1∩HK→L∞bounded
+stability, for all dimensions d≥1, andasymptotic stability for dimensions d≥2.
+∗Indiana University, Bloomington, IN 47405; matjohn@india na.edu: Research of M.J. was partially sup-
+ported by an NSF Postdoctoral Fellowship under NSF grant DMS -0902192.
+†Indiana University, Bloomington, IN 47405; kzumbrun@indi ana.edu: Research of K.Z. was partially
+supported under NSF grants no. DMS-0300487 and DMS-0801745 .
+11 INTRODUCTION 2
+More precisely, we show that small L1∩Hsperturbations of a planar periodic solution
+u(x,t)≡¯u(x1) (without loss of generality taken stationary) converge at Gaussian rate in
+Lp,p≥2 to a modulation
+(1.1) ¯ u(x1−ψ(x,t)),
+of the unperturbed wave, where x= (x1,˜x), ˜x= (x2,...,xd), andψis a scalar function
+whosex- andt-gradients likewise decay at least at Gaussian rate in all Lp,p≥2, but which
+itself decays more slowly by a factor t1/2; in particular, ψis merely bounded in L∞for
+dimensiond= 1.
+The one-dimensional study of spectral stability of spatial ly periodic traveling waves of
+systems of viscous conservation laws was initiated by Oh and Zumbrun [OZ1] in the “quasi-
+Hamiltonian” case that the traveling-wave equation posses ses an integral of motion, and in
+the general case by Serre[Se1]. An important contribution o f Serre was to point out a larger
+connection between the linearized dispersion relation (th e functionλ(ξ) relating spectra to
+wave number of the linearized operator about the wave) near z ero and the formal Whitham
+averaged system obtained by slow modulation, or WKB, approx imation.
+In [OZ3], this was extended to multi-dimensions, relating t he linearized dispersion rela-
+tion near zero to
+(1.2)∂tM+/summationdisplay
+j∂xjFj= 0,
+∂t(ΩN)+∇x(ΩS) = 0,
+whereM∈Rndenotes the average over one period, Fjthe average of an associated flux,
+Ω =|∇xΨ| ∈R1the frequency, S=−Ψt/|∇xΨ| ∈R1the speeds, andN=∇xΨ/|∇xΨ| ∈
+Rdthe normal νassociated with nearby periodic waves, with an additional c onstraint
+(1.3) curl (Ω N) = curl ∇xΨ≡0.
+As an immediate corollary, similarly as in [OZ1], [Se1] in th e one-dimensional case, this
+yieldedas anecessary condition formulti-dimensional sta bility hyperbolicityoftheaveraged
+system (1.2)–(1.3).
+The present study is informed by but does not directly rely on this observation relating
+Whitham averaging and spectral stability properties. Like wise, the Evans function tech-
+niquesusedin[Se1,OZ3]toestablishthisconnection play n oroleinouranalysis; indeed, the
+Evans function makes no appearance here. Rather, we rely on a direct Bloch-decomposition
+argument in the spirit of Schneider [S1, S2, S3], combining s harp linearized estimates with
+subtle cancellation in nonlinear source terms arising from the modulated wave approxima-
+tion. The analytical techniques used to realize this progra m are somewhat different from
+those of [S1, S2, S3], however, coming instead from the theor y of stability of viscous shock
+fronts through a line of investigation carried out in [OZ1, O Z2, OZ3, OZ4, HoZ]. In partic-
+ular, the nonsmooth dispersion relation at ξ= 0 typical for convection-diffusion equations1 INTRODUCTION 3
+requires different treatment from that of [S1, S2, S3] in the re action diffusion case; see Re-
+mark 2.4. Moreover, we detect nonlinear cancellation in the physicalx-tdomain rather than
+the frequency domain as in [S1, S2, S3]. The main difference bet ween the present analysis
+and that of [OZ4] is the systematic incorporation of modulat ion approximation (1.1).
+1.1 Equations and assumptions
+Consider a parabolic system of conservation laws
+(1.4) ut+/summationdisplay
+jfj(u)xj= ∆xu,
+u∈ U(open)∈Rn,fj∈Rn,x∈Rd,d≥1,t∈R+, and a periodic traveling wave solution
+(1.5) u= ¯u(x·ν−st),
+of periodX, satisfying the traveling-wave ODE ¯ u′′= (/summationtext
+jνjfj(¯u))′−s¯u′with boundary
+conditions ¯u(0) = ¯u(X) =:u0.Integrating, we obtain a first-order profile equation
+(1.6) ¯ u′=/summationdisplay
+jνjfj(¯u)−s¯u−q,
+where (u0,q,s,ν,X )≡constant. Without loss of generality take ν=e1,s= 0, so that
+¯u= ¯u(x1) represents a stationary solution depending only on x1.
+Following [Se1, OZ3, OZ4], we assume:
+(H1)fj∈CK+1,K≥[d/2]+4.
+(H2) Themap H:R×U×R×Sd−1×Rn→Rntaking (X;a,s,ν,q)/mapsto→u(X;a,s,ν,q)−a
+is a submersion at point ( ¯X;¯u(0),0,e1,¯q), whereu(·;·) is the solution operator for (1.6).
+Conditions (H1)–(H2) imply that the set of periodic solutio ns in the vicinity of ¯ uform
+a smooth (n+d+1)-dimensional manifold {¯ua(x·ν(a)−α−s(a)t)}, withα∈R,a∈Rn+d.
+1.1.1 Linearized equations
+Linearizing (1.4) about ¯ u(·), we obtain
+(1.7) vt=Lv:= ∆xv−/summationdisplay
+(Ajv)xj,
+where coefficients Aj:=Dfj(¯u) are now periodic functions of x1. Taking the Fourier
+transform in the transverse coordinate ˜ x= (x2,···,xd), we obtain
+(1.8)ˆvt=L˜ξˆv= ˆvx1,x1−(A1ˆv)x1−i/summationdisplay
+j/negationslash=1Ajξjˆv−/summationdisplay
+j/negationslash=1ξ2
+jˆv,
+where˜ξ= (ξ2,···,ξd) is the transverse frequency vector.1 INTRODUCTION 4
+1.1.2 Bloch–Fourier decomposition and stability conditions
+Following [G, S1, S2, S3], we define the family of operators
+(1.9) Lξ=e−iξ1x1L˜ξeiξ1x1
+operating on the class of L2periodic functions on [0 ,X]; the (L2) spectrum of L˜ξis equal
+to the union of the spectra of all Lξwithξ1real with associated eigenfunctions
+(1.10) w(x1,˜ξ,λ) :=eiξ1x1q(x1,ξ1,˜ξ,λ),
+whereq, periodic, is an eigenfunction of Lξ. By continuity of spectrum, and discreteness of
+the spectrum of the elliptic operators Lξon the compact domain [0 ,X], we have that the
+spectra ofLξmay be described as the union of countably many continuous su rfacesλj(ξ).
+Without loss of generality taking X= 1, recall now the Bloch–Fourier representation
+(1.11) u(x) =/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xˆu(ξ,x1)dξ1d˜ξ
+of anL2functionu, where ˆu(ξ,x1) :=/summationtext
+ke2πikx1ˆu(ξ1+ 2πk,˜ξ) are periodic functions of
+periodX= 1, ˆu(˜ξ) denoting with slight abuse of notation the Fourier transfo rm ofuin the
+full variable x. By Parseval’s identity, the Bloch–Fourier transform u(x)→ˆu(ξ,x1) is an
+isometry in L2:
+(1.12) /ba∇dblu/ba∇dblL2(x)=/ba∇dblˆu/ba∇dblL2(ξ;L2(x1)),
+whereL2(x1) is taken on [0 ,1] andL2(ξ) on [−π,π]×Rd−1. Moreover, it diagonalizes the
+periodic-coefficient operator L, yielding the inverse Bloch–Fourier transform representation
+(1.13) eLtu0=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xeLξtˆu0(ξ,x1)dξ1d˜ξ
+relating behavior of the linearized system to that of the dia gonal operators Lξ.
+Following [OZ4], weassumealongwith(H1)–(H2) the strong spectral stability conditions:
+(D1)σ(Lξ)⊂ {Reλ<0}forξ/ne}ationslash= 0.
+(D2) Reσ(Lξ)≤ −θ|ξ|2,θ>0, forξ∈Rdand|ξ|sufficiently small.
+(D3)λ= 0 is a semisimple eigenvalue of L0of multiplicity exactly n+1.1
+For each fixed angle ˆξ:=ξ/|ξ|, expandLξ=L0+|ξ|L1+|ξ|2L2. By assumption (D3)
+and standard spectral perturbation theory, there exist n+1 smooth eigenvalues
+(1.14) λj(ξ) =−iaj(ξ)+o(|ξ|)
+1The zero eigenspace of L0is at least ( n+1)-dimensional by the linearized existence theory and (H2 ),
+and hence n+ 1 is the minimal multiplicity; see [Se1, OZ3]. As noted in [O Z1, OZ3], minimal dimension
+of this zero eigenspace implies that ( M,NΩ) of (1.2) gives a nonsingular coordinatization of the fami ly of
+periodic traveling-wave solutions near ¯ u.1 INTRODUCTION 5
+ofLξbifurcating from λ= 0 atξ= 0, where −iajare homogeneous degree one functions
+given by |ξ|times the eigenvalues of Π 0L1|KerL0, with Π 0the zero eigenprojection of L0.
+Conditions(D1)–(D3) areexactly thespectralassumptions of[S1,S2,S3], corresponding
+to “dissipativity” of the large-time behavior of the linear ized system. As in [OZ4], we make
+the further nondegeneracy hypothesis:
+(H3) The eigenvalues λ=−iaj(ξ)/|ξ|of Π0L1
+KerL0are simple.
+The functions ajmay be seen to be the characteristics associated with the Whi tham av-
+eraged system (1.2)–(1.3) linearized about the values of M,S,N, Ω associated with the
+background wave ¯ u; see [OZ3, OZ4]. Thus, (D1) implies weak hyperbolicity of (1 .2)–(1.3)
+(reality ofaj), while (H1) corresponds to strict hyperbolicity.
+1.2 Main results
+With these preliminaries, we can now state our main results.
+Theorem 1.1. Assuming (H1)–(H3) and (D1)–(D3), for some C >0andψ∈WK,∞(x,t),
+(1.15)|˜u−¯u(·−ψ)|Lp(t)≤C(1+t)−d
+2(1−1/p)|˜u−¯u|L1∩HK|t=0,
+|˜u−¯u(·−ψ)|HK(t)≤C(1+t)−d
+4|˜u−¯u|L1∩HK|t=0,
+|(ψt,ψx)|WK+1,p≤C(1+t)−d
+2(1−1/p)|˜u−¯u|L1∩HK|t=0,
+and
+(1.16) |˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+t)−d
+2(1−1
+p)+1
+2|˜u−¯u|L1∩HK|t=0
+for allt≥0,p≥2,d= 1, for solutions ˜uof(1.4)with|˜u−¯u|L1∩HK|t=0sufficiently small.
+In particular, ¯uis nonlinearly bounded L1∩HK→L∞stable for dimension d= 1.
+Theorem 1.2. Assuming (H1)–(H3) and (D1)–(D3), for any ε >0, someC >0and
+ψ∈WK,∞(x,t),
+(1.17)|˜u−¯u(·−ψ)|Lp(t)≤C(1+t)−d
+2(1−1/p)|˜u−¯u|L1∩HK|t=0,
+|˜u−¯u(·−ψ)|HK(t)≤C(1+t)−d
+4|˜u−¯u|L1∩HK|t=0,
+|(ψt,ψx)|WK+1,p≤C(1+t)−d
+2(1−1/p)+ε−1
+2|˜u−¯u|L1∩HK|t=0,
+and
+(1.18)|˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+t)−d
+2(1−1
+p)+ε|˜u−¯u|L1∩HK|t=0,
+|˜u−¯u|HK(t),|ψ(t)|HK≤C(1+t)−d
+4+ε|˜u−¯u|L1∩HK|t=0,
+for allt≥0,p≥2,d= 2, for solutions ˜uof(1.4)with|˜u−¯u|L1∩HK|t=0sufficiently small.
+In particular, ¯uis nonlinearly asymptotically L1∩HK→HKstable for dimension d= 2.1 INTRODUCTION 6
+Remark 1.1. In Theorem 1.2, derivatives in x∈R2refer to total derivatives. Moreover,
+unless specified by an appropriate index, throughout this pa per derivatives in spatial variable
+xwill always refer to the total derivative of the function.
+In dimension one, Theorem 1.1 asserts only bounded L1∩HK→L∞stability, a very
+weak notion of stability. The absence of decay in perturbati on ˜u−¯uindicates the delicacy
+of the nonlinear analysis in this case. In particular, it is c rucial to separate off the slower-
+decaying modulated behavior (1.1) in order to close the nonl inear iteration argument.
+Remark 1.2. In dimension d= 1, it is straightforward to show that the results of Theorem
+1.1 extend to all 1 ≤p≤ ∞using the pointwise techniques of [OZ2]; see Remark 3.3.
+Remark 1.3. The slow decay of |˜u−¯u|Lp(t)∼ |ψ(t)|Lpin (1.16) is due to nonlinear
+interactions; as shown in [OZ2, OZ4], the linearized decay r ate is faster by factor (1+ t)−1/2
+(Proposition 2.1). In [OZ4], it was shown that for d≥3, where linear effects dominate
+behavior, (1.16) may be replaced by the stronger estimate |˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+
+t)−d
+2(1−1
+p)|˜u−¯u|L1∩HK|t=0.These distinctions reflect fine details of both linearized es timates
+(Section 3) and nonlinear structure (Sections 4.1–4.2) tha t are not immediately apparent
+from the formal Whitham approximation (1.2)–(1.3).
+1.3 Discussion and open problems
+Linearized stability under the same assumptions, with shar p rates of decay, was established
+ford= 1 [OZ2] and for d≥1 in [OZ4], along with nonlinear stability for d≥3. Theorem
+1.1 completes this line of investigation by establishing no nlinear stability in the critical
+dimensions d= 1,2, a fundamental open problem cited in [OZ1, OZ4].
+This gives a generalization of the work of [S1, S2, S3] for rea ction diffusion equations
+to the case of viscous conservation laws. Recall that the ana lysis of [S1, S2, S3] concerns
+also multiply periodic waves, i.e., waves that are either pe riodic or else constant in each
+coordinate direction. It is straightforward to verify that the methods of this paper apply
+essentially unchanged to this case, to give a corresponding stability result under the analog
+of (H1)–(H3), (D1)–(D3), as we intend to report further in a f uture work. Likewise, the
+extension from the semilinear parabolic case treated here t o the general quasilinear case is
+straightforward, following the treatment of [OZ4].
+On the other hand, as noted in [OZ2], condition (D3) is in the c onservation law setting
+nongeneric, corresponding to the special “quasi-Hamilton ian” situation studied there; in
+particular, it implies that speed is to first order constant a mong the family of spatially pe-
+riodic traveling-wave solutions nearby ¯ u. In the generic case that (D3) is violated, behavior
+is essentially different [OZ1, OZ2], and perturbations decay more slowly at the linearized
+level. Nonlinear stability remains an interesting open pro blem in this setting.
+Our approach to stability in the critical dimensions d= 1,2, as suggested in [OZ4], is,
+loosely following the approach of [S1, S2, S3], to subtract o ut a slower-decaying part of the
+solution describedby anappropriatemodulation equation a ndshowthat theresidualdecays2 BASIC LINEARIZED STABILITY ESTIMATES 7
+sufficiently rapidly to close a nonlinear iteration. It is wor th noting that the modulated
+approximation ¯ u(x1−ψ(x,t)) of (1.1) is not the full Ansatz
+(1.19) ¯ ua(Ψ(x,t)),
+Ψ(x,t) :=x1−ψ(x,t), associated with the Whitham averaged system (1.2)–(1.3) , where ¯ua
+isthemanifoldofperiodicsolutions near ¯ uintroducedbelow(H2), butonlythetranslational
+part not involving perturbations ain the profile. (See [OZ3] for the derivation of (1.2)–
+(1.3) and (1.19).) That is, we don’t need to separate out all v ariations along the manifold
+of periodic solutions, but only the special variations conn ected with translation invariance.
+The technical reason is an asymmetry in y-derivative estimates in the parts of the Green
+function associated with these various modes, something th at is not apparent without a
+detailed study of linearized behavior as carried out here. T his also makes sense formally,
+if one considers that (1.2) indicates that variables a,∇xΨ are roughly comparable, which
+would suggest, by the diffusive behavior Ψ >>∇xΨ, thatais neglible with respect to Ψ.
+However, note that in the case that (D3) holds, hence wave spe ed is stationary along the
+manifold of periodic solutions, the final equation of (1.2) d ecouples to (Ψ x)t= (ΩN)t= 0,
+and could be written as Ψ t= 0 in terms of Ψ alone. Hence, there is some ambiguity in this
+degenerate case which of Ψ, Ψ xis the primary variable, and in terms of linear behavior, the
+decay of variations aand Ψ are in fact comparable [OZ4]; in the generic case, aand Ψxare
+comparable at the linearized level [OZ2]. It would be very in teresting to better understand
+the connection between the Whitham averaged system (or suit able higher-order correction)
+and behavior at the nonlinear level, as explored at the linea r level in [OZ3, OZ4, JZ1, JZB].
+2 Basic linearized stability estimates
+We begin by recalling the basic linearized stability estima tes derived in [OZ4]. We will
+sharpen these afterward in Section 3. By standard spectral p erturbation theory [K], the
+total eigenprojection P(ξ) onto the eigenspace of Lξassociated with the eigenvalues λj(ξ),
+j= 1,...,n+1describedintheintroductioniswell-definedandanalyti cinξforξsufficiently
+small, since these (by discreteness of the spectra of Lξ) are separated at ξ= 0 from the rest
+of the spectrum of L0. Introducing a smooth cutoff function φ(ξ) that is identically one
+for|ξ| ≤εand identically zero for |ξ| ≥2ε,ε >0 sufficiently small, we split the solution
+operatorS(t) :=eLtinto low- and high-frequency parts
+(2.1) SI(t)u0:=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xφ(ξ)P(ξ)eLξtˆu0(ξ,x1)dξ1d˜ξ
+and
+(2.2) SII(t)u0:=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·x/parenleftbig
+I−φP(ξ)/parenrightbig
+eLξtˆu0(ξ,x1)dξ1d˜ξ.2 BASIC LINEARIZED STABILITY ESTIMATES 8
+2.1 High-frequency bounds
+By standard sectorial bounds [He, Pa] and spectral separati on ofλj(ξ) from the remaining
+spectra ofLξ, we have trivially the exponential decay bounds
+(2.3)/ba∇dbleLξt(I−φP(ξ))f/ba∇dblL2([0,X])≤Ce−θt/ba∇dblf/ba∇dblL2([0,X]),
+/ba∇dbleLξt(I−φP(ξ))∂l
+x1f/ba∇dblL2([0,X])≤Ct−l
+2e−θt/ba∇dblf/ba∇dblL2([0,X]),
+/ba∇dbl∂l
+x1eLξt(I−φP(ξ))f/ba∇dblL2([0,X])≤Ct−l
+2e−θt/ba∇dblf/ba∇dblL2([0,X]),
+forθ,C >0, and 0 ≤m≤K(Kas in (H1)). Together with (1.12), these give immediately
+the following estimates.
+Proposition 2.1 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D2), for some θ,
+C >0, and allt>0,2≤p≤ ∞,0≤l≤K+1,0≤m≤K,
+(2.4)/ba∇dbl∂l
+xSII(t)f/ba∇dblL2(x),/ba∇dblSII(t)∂l
+xf/ba∇dblL2(x)≤Ct−l
+2e−θt/ba∇dblf/ba∇dblL2(x),
+/ba∇dbl∂m
+xSII(t)f/ba∇dblLp(x),/ba∇dblSII(t)∂m
+xf/ba∇dblLp(x)≤Ct−d
+2(1
+2−1
+p)−m
+2e−θt/ba∇dblf/ba∇dblL2(x),
+where, again, derivatives in the variable x∈Rdrefer to total derivatives.
+Proof.The first inequalities follow immediately by (1.12) and (2.3 ). The second follows for
+x1derivatives in the case p=∞,m= 0 by Sobolev embedding from
+/ba∇dblSII(t)f/ba∇dblL∞(˜x;L2(x1))≤Ct−d−1
+4e−θt/ba∇dblf/ba∇dblL2([0,X])
+and
+/ba∇dbl∂x1SII(t)f/ba∇dblL∞(˜x;L2(x1))≤Ct−d−1
+4−1
+2e−θt/ba∇dblf/ba∇dblL2([0,X]),
+which follow by an application of (1.12) in the x1variable and the Hausdorff–Young in-
+equality /ba∇dblf/ba∇dblL∞(˜x)≤ /ba∇dblˆf/ba∇dblL1(˜ξ)in the variable ˜ x. The result for derivatives in x1and general
+2≤p≤ ∞then follows by Lpinterpolation. Finally, the result for derivatives in ˜ xfollows
+from the inverse Fourier transform, equation (2.2), and the large|ξ|bound
+|eLtf|L2(x1)≤e−θ|˜ξ|2t|f|L2(x1),|ξ|sufficiently large ,
+which easily follows from Parseval and the fact that Lξis a relatively compact perturbation
+of∂2
+x−|ξ|2. Thus, by the above estimate we have
+/ba∇dbleLt∂˜xf/ba∇dblL2(x)≤C/ba∇dbleLξt|˜ξ|ˆf/ba∇dblL2(x1,ξ)
+≤Csup/parenleftBig
+e−θ|˜ξ|2t|ξ|/parenrightBig
+/ba∇dblˆf/ba∇dblL2(x1,ξ)
+≤Ct−1/2/ba∇dblf/ba∇dblL2(x).
+A similar argument applies for 1 ≤m≤K.2 BASIC LINEARIZED STABILITY ESTIMATES 9
+2.2 Low-frequency bounds
+Denote by
+(2.5) GI(x,t;y) :=SI(t)δy(x)
+the Green kernel associated with SI, and
+(2.6) [ GI
+ξ(x1,t;y1)] :=φ(ξ)P(ξ)eLξt[δy1(x1)]
+the corresponding kernel appearing within the Bloch–Fouri er representation of GI, where
+the brackets on [ Gξ] and [δy] denote the periodic extensions of these functions onto the
+whole line. Then, we have the following descriptions of GI, [GI
+ξ], deriving from the spectral
+expansion (1.14) of Lξnearξ= 0.
+Proposition 2.2 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D3),
+(2.7)[GI
+ξ(x1,t;y1)] =φ(ξ)n+1/summationdisplay
+j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗,
+GI(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)[GI
+ξ(x1,t;y1)]dξ
+=/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
+j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ,
+where∗denotes matrix adjoint, or complex conjugate transpose, qj(ξ,·)and˜qj(ξ,·)are right
+and left eigenfunctions of Lξassociated with eigenvalues λj(ξ)defined in (1.14), normalized
+so that/an}b∇acketle{t˜qj,qj/an}b∇acket∇i}ht ≡1, whereλj/|ξ|is a smooth function of |ξ|andˆξ:=ξ/|ξ|andqjand˜qj
+are smooth functions of |ξ|,ˆξ:=ξ/|ξ|, andx1ory1, withℜλj(ξ)≤ −θ|ξ|2.
+Proof.Smooth dependence of λjand ofq, ˜qas functions in L2[0,X] follow from standard
+spectral perturbation theory [K] using the fact that λjsplit to first order in |ξ|asξis varied
+along rays through the origin, and that Lξvaries smoothly with angle ˆξ. Smoothness of
+qj, ˜qjinx1,y1then follow from the fact that they satisfy the eigenvalue eq uation forLξ,
+which has smooth, periodic coefficients. Likewise, (2.7)(i) is immediate from the spectral
+decomposition of elliptic operators on finite domains. Subs tituting (2.5) into (2.1) and
+computing
+(2.8) /hatwideδy(ξ,x1) =/summationdisplay
+ke2πikx1/hatwideδy(ξ+2πke1) =/summationdisplay
+ke2πikx1e−iξ·y−2πiky1=e−iξ·y[δy1(x1)],
+where the second and third equalities follow from the fact th at the Fourier transform either
+continuous or discrete of the delta-function is unity, we ob tain
+GI(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xφP(ξ)eLξt/hatwideδy(ξ,x1)dξ
+=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·(x−y)φP(ξ)eLξt[δy1(x1)]dξ,2 BASIC LINEARIZED STABILITY ESTIMATES 10
+yielding (2.7)(ii) by (2.6)(i) and the fact that φis supported on [ −π,π].
+Proposition 2.3 ([OZ4]).Under assumptions (H1)-(H3) and (D1)-(D3),
+(2.9) sup
+y/ba∇dblGI(·,t,;y)/ba∇dblLp(x),sup
+y/ba∇dbl∂x,yGI(·,t,;y)/ba∇dblLp(x)≤C(1+t)−d
+2(1−1
+p)
+for all2≤p≤ ∞,t≥0, whereC >0is independent of p.
+Proof.From representation (2.7)(ii) and ℜλj(ξ)≤ −θ|ξ|2, we obtain by the triangle in-
+equality
+(2.10) /ba∇dblGI/ba∇dblL∞(x,y)≤C/ba∇dble−θ|ξ|2tφ(ξ)/ba∇dblL1(ξ)≤C(1+t)−d
+2,
+verifying the bounds for p=∞. Derivative bounds follow similarly, since derivatives fa lling
+onqjor ˜qjare harmless, whereas derivatives falling on eiξ·(x−y)bring down a factor of ξ,
+again harmless because of the cutoff function φ.
+To obtain bounds for p= 2, we note that (2.7)(ii) may be viewed itself as a Bloch–
+Fourier decomposition with respect to variable z:=x−y, withyappearing as a parameter.
+Recalling (1.12), we may thus estimate
+(2.11)sup
+y/ba∇dblGI(x,t;y)/ba∇dblL2(x)=/summationdisplay
+jsup
+y/ba∇dblφ(ξ)eλj(ξ)tqj(·,z1)˜q∗
+j(·,y1)/ba∇dblL2(ξ;L2(z1∈[0,X]))
+≤C/summationdisplay
+jsup
+y/ba∇dblφ(ξ)e−θ|ξ|2t/ba∇dblL2(ξ)/ba∇dblqj/ba∇dblL2(0,X)/ba∇dbl˜qj/ba∇dblL∞(0,X)
+≤C(1+t)−d
+4,
+where we have used in a crucial way the boundedness of ˜ qj; derivative bounds follow simi-
+larly. Finally, bounds for 2 ≤p≤ ∞follow byLp-interpolation.
+Remark 2.4. In obtaining the key L2-estimate, we have used in an essential way the
+periodic structure of qj, ˜qj. For, viewing GIas a general pseudodifferential expression
+rather than a Bloch–Fourier decomposition, we find that the s moothness of qj, ˜qjis not
+sufficient to apply standard L2→L2bounds of H¨ ormander, which require blowup in ξ
+derivatives at less than the critical rate |ξ|−1found here; see, e.g., [H] for further discussion.
+Nor do the weighted energy estimate techniques used in [S1, S 2, S3] apply here, as these also
+rely on the property of smoothness of λj,qj, ˜qjwith respect to ξat the origin ξ= 0. The
+lack of smoothness of the linearized dispersion relation at the origin is an essential technical
+difference separating the conservation law from the reaction diffusion case; see [OZ4] for
+further discussion.
+Remark 2.5. Underlying the above analysis, and also the technically rat her different
+approach of [OZ2], is the fundamental relation
+(2.12) G(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·(x−y)[Gξ(x1,t;y1)]dξ2 BASIC LINEARIZED STABILITY ESTIMATES 11
+which, provided σ(Lξ) is semisimple, yields the simple formula
+G(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·(x−y)/summationdisplay
+jeλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ
+resembling that of the constant-coefficient case, where λjruns through the spectrum of Lξ.
+The basic idea in both cases is to separate off the principal pa rt of the series involving small
+λj(ξ) and estimate the remainder as a faster-decaying residual.
+Corollary 2.6 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D3), for all p≥2,t≥0,
+(2.13) /ba∇dblSI(t)f/ba∇dblLp,/ba∇dbl∂xSI(t)f/ba∇dblLp,/ba∇dblSI(t)∂xf/ba∇dblLp≤C(1+t)−d
+2(1−1
+p)/ba∇dblf/ba∇dblL1.
+Proof.Immediate, from (2.9) and the triangle inequality, as, for e xample,
+/ba∇dblSI(t)f(·)/ba∇dblLp=/vextenddouble/vextenddouble/vextenddouble/integraldisplay
+RdGI(x,t;y)f(y)dy/vextenddouble/vextenddouble/vextenddouble
+Lp(x)≤/integraldisplay
+Rdsup
+y/ba∇dblGI(·,t;y)/ba∇dblLp|f(y)|dy.
+Proposition 2.1 ([OZ4]).Assuming (H1)-(H3), (D1)-(D3), for some C >0, allt≥0,
+p≥2,0≤l≤K,
+(2.14) /ba∇dblS(t)∂l
+xu0/ba∇dblLp≤Ct−l
+2(1+t)−d
+2(1
+2−1
+p)+l
+2t−d
+4−l
+2/ba∇dblu0/ba∇dblL1∩L2.
+Proof.Immediate, from (2.4) and (2.13).
+2.3 Additional estimates
+Lemma 2.7. Assuming (H1)–(H3), (D1)–(D3), for all t≥0,0≤l≤K,
+(2.15) /ba∇dbl∂l
+xSI(t)f/ba∇dblLp(x),/ba∇dblSI(t)∂l
+xf/ba∇dblLp(x)≤C(1+t)−d
+2(1/2−1/p)/ba∇dblf/ba∇dblL2(x).
+Proof.From boundedness of the spectral projections Pj(ξ) =qj/an}b∇acketle{t˜qj,·/an}b∇acket∇i}htinL2[0,X] and their
+derivatives, another consequence of first-order splitting of eigenvalues λj(ξ) at the origin,
+we obtain boundedness of φ(ξ)P(ξ)eLξtand thus, by (1.12), the global bounds
+(2.16) /ba∇dbl∂l
+xSI(t)f/ba∇dblL2(x),/ba∇dblSI(t)∂l
+xf/ba∇dblL2(x)≤C/ba∇dblf/ba∇dblL2(x),
+for allt≥0, yielding the result for p= 2. Moreover, by boundedness of ˜ q,qin allLp(x1),
+we have
+|φ(ξ)P(ξ)eLξtˆf(ξ,·)|L∞(x1)≤Ce−θ|ξ|2t|P(ξ)ˆf(ξ,·)|L∞(x1)≤Ce−θ|ξ|2t|ˆf(ξ,·)|L2(x1),3 REFINED LINEARIZED ESTIMATES 12
+C, θ>0, yielding by SIf=/parenleftBig
+1
+2π/parenrightBigd/integraltextπ
+−π/integraltext
+Rd−1eiξ·xφ(ξ)P(ξ)eLξtˆf(ξ,x1)dξ1d˜ξthe bound
+(2.17)/ba∇dblSI(t)f/ba∇dblL∞(x)≤/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1|φ(ξ)P(ξ)eLξtˆf(ξ,·)|L∞(x1)dξ1d˜ξ
+≤/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1Cφ(ξ)e−θ|ξ|2t|ˆf(ξ,·)|L2(x1)dξ1d˜ξ
+≤C|φ(ξ)e−θ|ξ|2t|L2(ξ)|ˆf|L2(ξ,x1)
+=C(1+t)−d
+4/ba∇dblf/ba∇dblL2([0,X]),
+yielding the result for p=∞,l= 0. The result for p=∞, 1≤l≤Kfollows by a similar
+argument. The result for general 2 ≤p≤ ∞then follows by Lpinterpolation between p= 2
+andp=∞.
+By Riesz–Thorin interpolation between (2.15) and (2.13), w e obtain the following, ap-
+parently sharp bounds between various LqandLp.2
+Corollary 2.8. Assuming (H0)–(H3) and (D1)–(D3), for all 1≤q≤2≤p,t≥0,
+0≤l≤K,
+(2.18) /ba∇dbl∂l
+xSI(t)f/ba∇dblLp,/ba∇dblSI(t)∂l
+xf/ba∇dblLp≤C(1+t)−d
+2(1
+q−1
+p)/ba∇dblf/ba∇dblLq.
+Proposition 2.2. Assuming (H1)-(H3), (D1)-(D3), for some C >0, allt≥0,1≤q≤
+2≤p, and0≤l≤K,
+(2.19) /ba∇dblS(t)∂l
+xu0/ba∇dblLp≤C(1+t)−d
+2(1
+2−1
+p)+l
+2t−d
+2(1
+q−1
+2)−l
+2/ba∇dblu0/ba∇dblLq∩L2.
+Proof.Immediate, from (2.4) and (2.8).
+3 Refined linearized estimates
+The bounds of Proposition 2.1 are sufficient to establish nonl inear stability and asymptotic
+behavior in dimensions d≥3, as shown in [OZ4]. However, they are not sufficient in the
+critical dimensions d= 1,2; see Remark 1, Section 7 of [OZ4]. Comparison with standard
+diffusive stability arguments as in [Z7] show that this is due t o the fact that the full solution
+operator |S(t)∂x|decays no faster than S(t), or, equivalently, Gyno faster than G.
+Following the basic strategy introduced in [ZH, Z1, MaZ2, Ma Z4] in the context of vis-
+cous shock waves, we now perform a refined linearized estimat e separating slower-decaying
+translational modes from a faster-decaying “good” part of t he solution operator. This will
+be used in Section 4 in combination with certain nonlinear ca ncellation estimates to show
+convergence to the modulated approximation (1.1) at a faste r rate sufficient to close the
+nonlinear iteration.
+The key to this decomposition is the following observation.
+2The inclusion of general p≥2 in Lemma 2.7 repairs an omission in [OZ4], where the bounds ( 2.8) were
+stated but not used.3 REFINED LINEARIZED ESTIMATES 13
+Lemma 3.1. Assuming (H1)–(H3), (D1)–(D3), let λj(ξ/|ξ|,ξ),qj(ξ/|ξ|,ξ,·),˜qj(ξ/|ξ|,ξ,·)
+denote the eigenvalues and associated right and left eigenf unctions of Lξ, withqj,˜qjsmooth
+functions of ξ/|ξ|and|ξ|as noted in Prop. 2.2. Then, without loss of generality, q1(ω,0,·)≡
+¯u′, while˜qj(ω,0,·)forj/ne}ationslash= 1are constant functions depending only on angle ω=ξ/|ξ|.
+Proof.Expanding Lξ=L0+|ξ|L1
+ξ/|ξ|+|ξ|2L2
+ξ/|ξ|as in the introduction, consider the con-
+tinuous family of spectral perturbation problems in |ξ|indexed by angle ω=ξ/|ξ|. Then,
+both facts follow by standard perturbation theory [K] using the observations that ¯ u′is in
+the right kernel of L0and constant functions care in the left kernel of L0, with
+/an}b∇acketle{tc,L1¯u′/an}b∇acket∇i}ht=/an}b∇acketle{tc,(ω1(2∂x1−A1)−/summationdisplay
+j/negationslash=1ωjAj))¯u′/an}b∇acket∇i}ht=/an}b∇acketle{tc,ω1∂2
+x1¯u−/summationdisplay
+j/negationslash=1ωj∂x1fj(¯u)/an}b∇acket∇i}ht ≡0,
+where/an}b∇acketle{t·,·/an}b∇acket∇i}htdenotesL2(x1) inner product on the interval x1∈[0,X], that the dimension
+of kerL0by assumption is ( n+ 1), so that the orthogonal complement of ¯ u′in KerL0
+is dimension nso exactly the set of constant functions, and that by (H3) the functions
+qj(ω,0,·) and ˜qj(ω,0) are right and left eigenfunctions of Π 0L1|kerL0(Π0as earlier denoting
+the zero eigenprojection associated with L0).
+Remark 3.2. The key observation of Lemma 3.1 can be motivated by the form o f the
+Whitham averaged system (1.2). For, recalling (Section 1.3 ) that (D3) implies that speed
+sis stationary to first order at ¯ ualong the manifold of nearby periodic solutions, we find
+that the last equation of (1.2) reduces to ( ∇xΨ)t= 0, i.e., the equation for the translational
+variation Ψ decouples from the equations for variations in o ther modes. This corresponds
+heuristically to the fact derived above that the translatio nal mode ¯u′(x1) decouples in the
+first-order eigenfunction expansion.
+Corollary 3.1. Under assumptions (H1)–(H3), (D1)–(D3), the Green functio nG(x,t;y)
+of(1.7)decomposes as G=E+˜G,
+(3.1) E= ¯u′(x)e(x,t;y),
+where, for some C >0, allt>0,1≤q≤2≤p≤ ∞,0≤j,k,l,j+l≤K,1≤r≤2,
+(3.2)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp(x)≤C(1+t)−d
+2(1/2−1/p)t−1
+2(1/q−1/2)|f|Lq∩L2,
+/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞∂r
+y˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp(x)≤C(1+t)−d
+2(1/2−1/p)−1
+2+r
+2
+×t−d
+2(1/q−1/2)−r
+2|f|Lq∩L2,
+/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞∂r
+t˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp(x)≤C(1+t)−d
+2(1/2−1/p)−1
+2+r
+×t−d
+2(1/q−1/2)−r|f|Lq∩L2.3 REFINED LINEARIZED ESTIMATES 14
+(3.3)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞∂j
+x∂k
+t∂l
+ye(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp≤(1+t)−d
+2(1/q−1/p)−(j+k)
+2|f|Lq.
+Moreover,e(x,t;y)≡0fort≤1.
+Proof.We first treat the simpler case q= 1. Recalling that
+(3.4) GI(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
+j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ,
+define
+(3.5) ˜e(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)eλ1(ξ)t˜q1(ξ,y1)∗dξ,
+so that
+(3.6)
+GI(x,t;y)−¯u′(x1)˜e(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
+j=2eλj(ξ)tqj(ξ/|ξ|,0,x1)˜qj(ξ,y1)∗dξ
++/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdn+1/summationdisplay
+j=1eiξ·(x−y)φ(ξ)eλj(ξ)tO(|ξ|)dξ.
+Noting, by Lemma 3.1, that ∂y˜q(ω,0,y)≡constant for j/ne}ationslash= 1, we have therefore
+(3.7)∂r
+y(GI(x,t;y)−¯u′(x1)˜e(x,t;y)) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
+j=1eλj(ξ)tO(|ξ|)dξ,
+which readily gives
+(3.8) |∂r
+y(GI(x,t;y)−¯u′(x1)˜e(x,t;y))|Lp≤C(1+t)−d
+2(1−1/p)−1
+2,
+p≥2, by the same argument used to prove (2.9), and similarly
+(3.9) |∂r
+t(GI(x,t;y)−¯u′(x1)˜e(x,t;y))|Lp≤c(1+t)−d
+2(1−1/p)−1
+2.
+These yield (3.2) by the triangle inequality.
+Defininge(x,t;y) :=χ(t)˜e(x,t;y), whereχisasmoothcutofffunctionsuchthat χ(t)≡1
+fort≥2 andχ(t)≡0 fort≤1, and setting ˜G:=G−¯u′(x1)e(x,t;y), we readily obtain the
+estimates (3.2) by combining (3.9) with bound (2.4) on GII. Bounds (3.3) follow from (3.5)
+by the argument used to prove (2.9), together with the observ ation thatx- ort-derivatives
+bring down factors of |ξ|, followed again by an application of the triangle inequalit y.
+Thecases1 ≤q≤2followsimilarly, bytheargumentsusedtoprove(2.15)and (2.8).4 NONLINEAR STABILITY IN DIMENSION ONE 15
+Remark 3.3. Despite their apparent complexity, the above bounds may be r ecognized
+as essentially just the standard diffusive bounds satisfied fo r the heat equation [Z7]. For
+dimensiond= 1, it may be shown using pointwise techniques as in [OZ2] tha t the bounds
+of Corollary 3.1 extend to all 1 ≤q≤p≤ ∞.
+Note the strong analogy between the Green function decompos ition of Corollary 3.1
+and that of [MaZ3, Z4] in the viscous shock case. We pursue thi s analogy further in the
+nonlinearanalysisofthefollowingsections, combiningth e“instantaneous tracking” strategy
+of [ZH, Z1, Z4, Z7, MaZ2, MaZ4] with a type of cancellation est imate introduced in [HoZ].
+4 Nonlinear stability in dimension one
+For clarity, we carry out the nonlinear stability analysis i n detail in the most difficult,
+one-dimensional, case, indicating afterward by a few brief remarks the extension to d= 2.
+Hereafter, take x∈R1, dropping the indices on fjandxjand writing ut+f(u)x=uxx.
+4.1 Nonlinear perturbation equations
+Given a solution ˜ u(x,t) of (1.4), define the nonlinear perturbation variable
+(4.1) v=u−¯u= ˜u(x+ψ(x,t))−¯u(x),
+where
+(4.2) u(x,t) := ˜u(x+ψ(x,t))
+andψ:R×R→Ris to be chosen later.
+Lemma 4.1. Forv,uas in(4.1),(4.2),
+(4.3) ut+f(u)x−uxx= (∂t−L)¯u′(x1)ψ(x,t)+∂xR+(∂t+∂2
+x)S,
+where
+R:=vψt+vψxx+(¯ux+vx)ψ2
+x
+1+ψx=O(|v|(|ψt|+|ψxx|)+/parenleftBig|¯ux|+|vx|
+1−|ψx|/parenrightBig
+|ψx|2)
+and
+S:=−vψx=O(|v|(|ψx|).
+Proof.To begin, notice from the definition of uin (4.2) we have by a straightforward
+computation
+ut(x,t) = ˜ux(x+ψ(x,t),t)ψt(x,t)+ ˜ut(x+ψ,t)
+f(u(x,t))x=df(˜u(x+ψ(x,t),t))˜ux(x+ψ,t)·(1+ψx(x,t))4 NONLINEAR STABILITY IN DIMENSION ONE 16
+and
+uxx(x,t) = (˜ux(x+ψ(x,t),t)·(1+ψx(x,t)))x
+= ˜uxx(x+ψ(x,t),t)·(1+ψx(x,t))+(˜ux(x+ψ(x,t),t)·ψx(x,t))x.
+Using the fact that ˜ ut+df(˜u)˜ux−˜uxx= 0, it follows that
+(4.4)ut+f(u)x−uxx= ˜uxψt+df(˜u)˜uxψx−˜uxxψx−(˜uxψx)x
+= ˜uxψt−˜utψx−(˜uxψx)x
+where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
+at (x+ψ(x,t),t). Moreover, by another direct calculation, using the fact t hatL(¯u′(x)) = 0
+by translation invariance, we have
+(∂t−L)¯u′(x)ψ= ¯uxψt−¯utψx−(¯uxψx)x.
+Subtracting, and using the facts that, by differentiation of ( ¯u+v)(x,t) = ˜u(x+ψ,t),
+(4.5)¯ux+vx= ˜ux(1+ψx),
+¯ut+vt= ˜ut+ ˜uxψt,
+so that
+(4.6)˜ux−¯ux−vx=−(¯ux+vx)ψx
+1+ψx,
+˜ut−¯ut−vt=−(¯ux+vx)ψt
+1+ψx,
+we obtain
+ut+f(u)x−uxx= (∂t−L)¯u′(x)ψ+vxψt−vtψx−(vxψx)x+/parenleftBig
+(¯ux+vx)ψ2
+x
+1+ψx/parenrightBig
+x,
+yielding (4.3) by vxψt−vtψx= (vψt)x−(vψx)tand (vxψx)x= (vψx)xx−(vψxx)x.
+Corollary 4.2. The nonlinear residual vdefined in (4.1)satisfies
+(4.7) vt−Lv= (∂t−L)¯u′(x1)ψ−Qx+Rx+(∂t+∂2
+x)S,
+where
+(4.8) Q:=f(˜u(x+ψ(x,t),t))−f(¯u(x))−df(¯u(x))v=O(|v|2),
+(4.9) R:=vψt+vψxx+(¯ux+vx)ψ2
+x
+1+ψx,
+and
+(4.10) S:=−vψx=O(|v|(|ψx|).
+Proof.Taylor expansion comparing (4.3) and ¯ ut+f(¯u)x−¯uxx= 0.4 NONLINEAR STABILITY IN DIMENSION ONE 17
+4.2 Cancellation estimate
+Our strategy in writing (4.7) is motivated by the following b asic cancellation principle.
+Proposition 4.3 ([HoZ]).For anyf(y,s)∈Lp∩C2withf(y,0)≡0, there holds
+(4.11)/integraldisplayt
+0/integraldisplay
+G(x,t−s;y)(∂s−Ly)f(y,s)dyds=f(x,t).
+Proof.Integrating the left hand side by parts, we obtain
+(4.12)/integraldisplay
+G(x,0;y)f(y,t)dy−/integraldisplay
+G(x,t;y)f(y,0)dy+/integraldisplayt
+0/integraldisplay
+(∂t−Ly)∗G(x,t−s;y)f(y,s)dyds.
+Noting that, by duality,
+(∂t−Ly)∗G(x,t−s;y) =δ(x−y)δ(t−s),
+δ(·) here denoting the Dirac delta-distribution, we find that th e third term on the righthand
+side vanishes in (4.12), while, because G(x,0;y) =δ(x−y), the first term is simply f(x,t).
+The second term vanishes by f(y,0)≡0.
+Remark 4.1. Forψ=ψ(t), term (∂t−L)¯u′ψin (4.7) reduces to the term ˙ψ(t)¯u′(x)
+appearing in the shock wave case [ZH, Z1, Z4, Z7, MaZ2, MaZ4].
+4.3 Nonlinear damping estimate
+Proposition 4.2. Letv0∈HK(Kas in (H1)), and suppose that for 0≤t≤T, theHK
+norm ofvand theHK(x,t)norms ofψtandψxremain bounded by a sufficiently small
+constant. There are then constants θ1,2>0so that, for all 0≤t≤T,
+(4.13) |v(t)|2
+HK≤Ce−θ1t|v(0)|2
+HK+C/integraldisplayt
+0e−θ2(t−s)/parenleftBig
+|v|2
+L2+|(ψt,ψx)|2
+HK(x,t)/parenrightBig
+(s)ds.
+Proof.Subtracting from the equation (4.4) for uthe equation for ¯ u, we may write the
+nonlinear perturbation equation as
+(4.14) vt+(df(¯u)v)x−vxx=Q(v)x+ ˜uxψt−˜utψx−(˜uxψx)x,
+where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
+at (x+ψ(x,t),t). Using (4.6) to replace ˜ uxand ˜utrespectively by ¯ ux+vx−(¯ux+vx)ψx
+1+ψx
+and ¯ut+vt−(¯ux+vx)ψt
+1+ψx, and moving the resulting vtψxterm to the lefthand side of
+(4.14), we obtain
+(4.15)(1+ψx)vt−vxx=−(df(¯u)v)x+Q(v)x+ ¯uxψt
+−((¯ux+vx)ψx)x+/parenleftBig
+(¯ux+vx)ψ2
+x
+1+ψx/parenrightBig
+x.4 NONLINEAR STABILITY IN DIMENSION ONE 18
+Taking the L2inner product in xof/summationtextK
+j=0∂2j
+xv
+1+ψxagainst (4.15), integrating by parts, and
+rearranging the resulting terms, we arrive at the inequalit y
+∂t|v|2
+HK(t)≤ −θ|∂K+1
+xv|2
+L2+C/parenleftBig
+|v|2
+HK+|(ψt,ψx)|2
+HK(x,t)/parenrightBig
+,
+for someθ >0,C >0, so long as |˜u|HKremains bounded, and |v|HKand|(ψt,ψx)|HK(x,t)
+remain sufficiently small. Using the Sobolev interpolation |v|2
+HK≤ |∂K+1
+xv|2
+L2+˜C|v|2
+L2for
+˜C >0 sufficiently large, we obtain ∂t|v|2
+HK(t)≤ −˜θ|v|2
+HK+C/parenleftBig
+|v|2
+L2+|(ψt,ψx)|2
+HK(x,t)/parenrightBig
+from which (4.13) follows by Gronwall’s inequality.
+4.4 Integral representation/ ψ-evolution scheme
+By Proposition 4.3, we have, applying Duhamel’s principle t o (4.7),
+(4.16)v(x,t) =/integraldisplay∞
+−∞G(x,t;y)v0(y)dy
++/integraldisplayt
+0/integraldisplay∞
+−∞G(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds+ψ(t)¯u′(x).
+Definingψimplicitly as
+(4.17)ψ(x,t) =−/integraldisplay∞
+−∞e(x,t;y)u0(y)dy
+−/integraldisplayt
+0/integraldisplay+∞
+−∞e(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds,
+following [ZH, Z4, MaZ2, MaZ3], where eis defined as in (3.1), and substituting in (4.16)
+the decomposition G= ¯u′(x)e+˜Gof Corollary 3.1, we obtain the integral representation
+(4.18)v(x,t) =/integraldisplay∞
+−∞˜G(x,t;y)v0(y)dy
++/integraldisplayt
+0/integraldisplay∞
+−∞˜G(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds,
+and, differentiating (4.17) with respect to t, and recalling that e(x,s;y)≡0 fors≤1,
+(4.19)∂j
+t∂k
+xψ(x,t) =−/integraldisplay∞
+−∞∂j
+t∂k
+xe(x,t;y)u0(y)dy
+−/integraldisplayt
+0/integraldisplay+∞
+−∞∂j
+t∂k
+xe(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds.
+Equations (4.18), (4.19) together form a complete system in the variables ( v,∂j
+tψ,∂k
+xψ),
+0≤j≤1, 0≤k≤K, from the solution of which we may afterward recover the shif tψvia
+(4.17). From the original differential equation (4.7) togeth er with (4.19), we readily obtain
+short-time existence and continuity with respect to tof solutions ( v,ψt,ψx)∈HKby a
+standard contraction-mapping argument based on (4.13), (4 .17), and and (3.3).4 NONLINEAR STABILITY IN DIMENSION ONE 19
+4.5 Nonlinear iteration
+Associated with the solution ( u,ψt,ψx) of integral system (4.18)–(4.19), define
+(4.20)ζ(t) := sup
+0≤s≤t|(v,ψt,ψx)|HK(s)(1+s)1/4.
+Lemma 4.3. For allt≥0for whichζ(t)is finite, some C >0, andE0:=|u0|L1∩HK,
+(4.21) ζ(t)≤C(E0+ζ(t)2).
+Proof.By (4.9)–(4.10) and definition (4.20),
+(4.22) |(Q,R,S)|L1∩L∞≤ |(v,vx,ψt,ψx)|2
+L2+|(v,vx,ψt,ψx)|2
+L∞≤Cζ(t)2(1+t)−1
+2,
+so long as |ψx| ≤ |ψx|HK≤ζ(t) remains small, and likewise (using the equation to bound t
+derivatives in terms of x-derivatives of up to two orders)
+(4.23) |(∂t+∂2
+x)S|L1∩L∞≤ |(v,ψx)|2
+H2+|(v,ψx)|2
+W2,∞≤Cζ(t)2(1+t)−1
+2.
+Applying Corollary 3.1 with q= 1,d= 1 to representations (4.18)–(4.19), we obtain for
+any 2≤p<∞
+(4.24)|v(·,t)|Lp(x)≤C(1+t)−1
+2(1−1/p)E0
++Cζ(t)2/integraldisplayt
+0(1+t−s)−1
+2(1/2−1/p)(t−s)−3
+4(1+s)−1
+2ds
+≤C(E0+ζ(t)2)(1+t)−1
+2(1−1/p)
+and
+(4.25)
+|(ψt,ψx)(·,t)|WK,p≤C(1+t)−1
+2E0+Cζ(t)2/integraldisplayt
+0(1+t−s)−1
+2(1−1/p)−1/2(1+s)−1
+2ds
+≤C(E0+ζ(t)2)(1+t)−1
+2(1−1/p).
+Using (4.13) and(4.24)–(4.25), we obtain |v(·,t)|HK(x)≤C(E0+ζ(t)2)(1+t)−1
+4. Combining
+this with (4.25), p= 2, rearranging, and recalling definition (4.20), we obtain (4.3).
+Proof of Theorem 1.1. By short-time HKexistence theory, /ba∇dbl(v,ψt,ψx)/ba∇dblHKis continuous
+so long as it remains small, hence ηremains continuous so long as it remains small. By
+(4.3), therefore, it follows by continuous induction that η(t)≤2Cη0fort≥0, ifη0<1/4C,
+yielding by (4.20) the result (1.15) for p= 2. Applying (4.24)–(4.25), we obtain (1.15) for
+2≤p≤p∗for anyp∗<∞, with uniform constant C. Takingp∗>4 and estimating
+|Q|L2,|R|L2,|S|L2(t)≤ |(v,ψt,ψx)|2
+L4≤CE0(1+t)−3
+45 NONLINEAR STABILITY IN DIMENSION TWO 20
+in place of the weaker (4.22), then applying Corollary 3.1 wi thq= 2,d= 1, we obtain
+finally (1.15) for 2 ≤p≤ ∞, by a computation similar (4.24)–(4.25); we omit the detail s of
+this final bootstrap argument. Estimate (1.16) then follows using (3.3) with q=d= 1, by
+(4.26)
+|ψ(t)|Lp≤CE0+Cζ(t)2/integraldisplayt
+0(1+t−s)−1
+2(1−1/p)(1+s)−1
+2ds≤C(1+t)1
+2p(E0+ζ(t)2),
+together with the fact that ˜ u(x,t)−¯u(x) =v(x−ψ,t)+(¯u(x)−¯u(x−ψ),so that|˜u(·,t)−¯u|
+is controlled by the sum of |v|and|¯u(x)−¯u(x−ψ)| ∼ |ψ|. This yields stability for
+|u−¯u|L1∩HK|t=0sufficiently small, as described in the final line of the theore m.
+5 Nonlinear stability in dimension two
+We now briefly sketch the extension to dimension d= 2. Given a solution ˜ u(x,t) of (1.4),
+define the nonlinear perturbation variable
+(5.1) v=u−¯u= ˜u(x1+ψ(x,t),x2,t)−¯u(x1),
+where
+(5.2) u(x,t) := ˜u(x1+ψ(x,t),t)
+andψ:Rd×R→Ris to be chosen later.
+Lemma 5.1. Forv,uas in(5.2),
+(5.3)ut+d/summationdisplay
+j=1fj(u)xj−d/summationdisplay
+j=1uxjxj= (∂t−L)¯u′(x1)ψ(x,t)+d/summationdisplay
+j=1∂xjRj+∂tS+T,
+where
+Rj=O((|v,ψt,ψx)||(v,vx,ψt,ψx)|), S:=−vψx1= (|v|(|ψx|), T:=O(|ψx|3+|(v,ψx)||ψxx|).
+Proof.Similarly as in the proof of Lemma 4.1, it follows by a straigh tforward computation
+Using the fact that ˜ ut+/summationtext
+jdfj(˜u)˜uxj−/summationtext
+j˜uxjxj= 0, it follows that
+(5.4)ut+/summationdisplay
+jdfj(u)uxj−/summationdisplay
+juxjxj= ˜ux1ψt−˜utψx1+/summationdisplay
+j/negationslash=1dfj(˜u)˜ux1ψxj
+−/summationdisplay
+j/negationslash=1˜uxjx1ψxj−/summationdisplay
+j(˜ux1ψxj)xj,
+where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
+at (x+ψ(x,t),t). Moreover, by another direct calculation, using the fact t hatL(¯u′(x1)) = 0
+by translation invariance, we have
+(∂t−L)¯u′(x1)ψ= ¯ux1ψt−¯utψx1+/summationdisplay
+j/negationslash=1dfj(¯u)¯ux1ψxj−/summationdisplay
+j/negationslash=1¯uxjx1ψxj−/summationdisplay
+j(¯ux1ψxj)xj.5 NONLINEAR STABILITY IN DIMENSION TWO 21
+Subtracting, and using (4.5) and
+(5.5)¯uxj+vxj= ˜uxj+ ˜ux1ψxj,
+¯ut+vt= ˜ut+ ˜ux1ψt,
+so that
+(5.6)˜uxj−¯uxj−vxj=−(¯ux1+vx1)ψxj
+1+ψx1,
+˜ut−¯ut−vt=−(¯ux1+vx1)ψt
+1+ψx1,
+we obtain
+ut+/summationdisplay
+jdfj(u)uxj−/summationdisplay
+juxjxj= (∂t−L)¯u′(x1)ψ+vx1ψt−vtψx1
++/summationdisplay
+j/negationslash=1(dfj(˜u)˜ux1−dfj(¯u)¯ux1)ψxj
+−/summationdisplay
+j/negationslash=1(˜uxjx1−¯uxjx1)ψxj−/summationdisplay
+j((˜ux1−¯ux1)ψxj)xj.
+Usingvx1ψt−vtψx1= (vψt)x1−(vψx1)t,
+dfj(˜u)˜ux1=f(u)x1−dfj(˜u)˜ux1ψx1=f(u)x1(1−ψx)−dfj(˜u)˜ux1ψ2
+x1,
+and ˜uxjx1= (˜uxj)x1−˜uxjx1ψx1= (˜uxj)x1(1−ψx1)+ ˜uxjx1ψ2
+x1,and rearranging, we obtain
+ut+/summationdisplay
+jdfj(u)uxj−/summationdisplay
+juxjxj= (∂t−L)¯u′(x1)ψ+(vψt)x1−(vψx1)t
++/summationdisplay
+j/negationslash=1(fj(u)−fj(¯u))x1)ψxj
+−/summationdisplay
+j/negationslash=1f(u)x1ψx1ψxj−/summationdisplay
+j/negationslash=1dfj(˜u)˜ux1ψ2
+x1ψxj
+−/summationdisplay
+j/negationslash=1(˜uxj−¯uxj)x1ψxj+/summationdisplay
+j/negationslash=1(˜uxj)x1ψx1ψxj
++/summationdisplay
+j/negationslash=1˜uxjx1ψ2
+x1ψxj
+−/summationdisplay
+j(vx1ψx1)xj−/summationdisplay
+j/parenleftBig
+(¯ux1+vx1)ψxjψx1
+1+ψx1/parenrightBig
+xj.
+Noting that
+(fj(u)−fj(¯u))x1)ψxj= ((fj(u)−fj(¯u)ψxj)x1−(fj(u)−fj(¯u))ψxjx1,5 NONLINEAR STABILITY IN DIMENSION TWO 22
+f(u)x1ψx1ψxj= (f(u)ψx1ψxj)x1−f(u)(ψx1ψxj)x1,
+and
+(˜uxj−¯uxj)x1ψxj= ((˜uxj−¯uxj)ψxj)x1−(˜uxj−¯uxj)ψxjx1,
+with|fj(u)−fj(¯u)|=O(|v|) and|˜uxj−¯uxj|=O(|v|),we obtain the result
+Proof of Theorem 1.2. The result of Lemma 5.1 is the only part of the analysis that di ffers
+essentially from that of the one-dimensional case. The canc ellation and nonlinear damping
+arguments go through exactly as before to yield the analogs o f Propositions 4.3 and (4.2).
+Likewise, we obtain a Duhamel representation analogous to ( 4.18)–(4.19), forming a closed
+system in variables ( v,ψx,ψt).
+To obtain the analog of Lemma 4.3, completing the proof of non linear stability, we can
+carry out a somewhat simpler argument than in the one-dimens ional case, using Corollary
+3.1 withd= 2,q= 2 for all estimates, not only the final bootstrap argument, g iving in
+place of (4.24) the estimate
+(5.7)
+|v(·,t)|Lp(x)≤C(1+t)−(1−1/p)E0+Cζ(t)2/integraldisplayt
+0(1+t−s)−(1/2−1/p)(t−s)−1
+2(1+s)−1ds
+≤C(E0+ζ(t)2)(1+t)−(1−1/p),
+(5.8)|(ψx,ψt)(·,t)|Lp(x)≤C(1+t)−(1−1/p)−1
+2E0
++Cζ(t)2/integraldisplayt
+0(1+t−s)−(1/2−1/p)(t−s)−1
+2(1+s)−1ds
+≤C(E0+ζ(t)2)(1+t)ε−(1−1/p)−1
+2
+for divergence-form source terms, and
+(5.9)|v(·,t)|Lp(x)≤Cζ(t)2/integraldisplayt
+0(1+t−s)−(1/2−1/p)(1+s)−3
+2ds
+≤C(E0+ζ(t)2)(1+t)−(1−1/p),
+(5.10)|(ψx,ψt)(·,t)|Lp(x)≤C(1+t)−(1−1/p)−1
+2E0
++Cζ(t)2/integraldisplayt
+0(1+t−s)−(1/2−1/p)(t−s)−1
+2(1+s)−3
+2ds
+≤C(E0+ζ(t)2)(1+t)ε−(1−1/p)−1
+2
+for faster-decaying nondivergence-form source terms.
+We omit the details, which are entirely similar to, but subst antially simpler than, those
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+arXiv:1001.0049v1  [astro-ph.CO]  30 Dec 2009Optical Spectral Properties of Swift BAT Hard X-ray Selecte d
+Active Galactic Nuclei Sources
+Lisa M. Winter1,∗, Karen T. Lewis2, Michael Koss3, Sylvain Veilleux3,4, Brian Keeney1,
+Richard F. Mushotzky3,5
+ABSTRACT
+The Swift Burst Alert Telescope (BAT) survey of Active Galac tic Nuclei (AGN) is
+providing an unprecedented view of local AGNs ( < z >≈0.03) and their host galaxy
+properties. In this paper, we present an analysis of the opti cal spectra of a sample of
+64 AGNs from the 9-month survey, detected solely based on the ir 14-195keV flux. Our
+analysis includes both archived spectra from the Sloan Digi tal Sky Survey and our own
+observations from the 2.1-m Kitt Peak National Observatory telescope. Among our
+results, we include line ratio classifications utilizing st andard emission line diagnostic
+plots, [O III] 5007˚A luminosities, and H βderived black hole masses. As in our X-
+ray study, we find the type 2 sources to be less luminous (in [O III] 5007˚A and 14–
+195keV luminosities) with lower accretion rates than the ty pe 1 sources. We find that
+the optically classified LINERs, H II/composite galaxies, and ambiguous sources have
+the lowest luminosities, while both broad line and narrow li ne Seyferts have similar
+luminosities. FromacomparisonofthehardX-ray(14–195ke V)and[O III]luminosities,
+we find that both the observed and extinction-corrected [O III] luminosities are weakly
+correlated with X-ray luminosity. In a study of the host gala xy properties from both
+continuum fits and measurements of the stellar absorption in dices, we find that the
+hosts of the narrow line sources have properties consistent with late type galaxies.
+Subject headings: X-rays: galaxies, galaxies:active
+1. Introduction
+The Swift Burst Alert Telescope (BAT) provides an unprecede nted opportunity to study
+the optical properties of an unbiased sample of AGN. Conduct ing an all-sky mission in the 14–
+*Hubble Fellow
+1Center for Astrophysics and Space Astronomy, University of Colorado, Boulder, CO
+2Department of Physics & Astronomy, Dickinson College, Carl isle, PA
+3Department of Astronomy, University of Maryland, College P ark, MD
+4Max-Planck-Institut f¨ ur extraterrestrische Physik, Pos tfach 1312, D-85741 Garching, Germany
+5NASA Goddard Space Flight Center, Greenbelt, MD– 2 –
+195keV band, the BAT survey has detected 153 AGN in the first 9- months1(Tueller et al. 2008;
+Baumgartner et al. 2008). Since the sources were detected ba sed on 14–195keV flux, with a flux
+limit of 2–3 ×10−11ergss−1cm−2, selection effects due to obscuring material are minimal. Due
+to the unbiased nature of the Swift BAT survey, Suzaku follow -ups of Swift-detected sources
+led to the identification of a new class of “hidden” AGN (Ueda e t al. 2007), heavily obscured
+(NH>1023cm−2) sources that would not likely be identified as AGN based on th eir optical or
+soft X-ray ( E <3keV) properties alone. This class of “hidden” sources was f ound to comprise
+24% of the 9-month BAT AGNs (Winter et al. 2009a), making an an alysis of the collective optical
+properties an important piece in understanding the propert ies of the Swift BAT-detected AGN.
+Currently, great progress is being made in collecting and an alyzing the multi-wavelength prop-
+erties of this uniquely selected, very hard X-ray, 9-month S wift BAT AGN sample. The collec-
+tive properties of the 0.3–10keV X-ray band have been analyz ed and presented in Winter et al.
+(2009a). A comparison of the IR [O IV], optical [O III], and X-ray 2–10keV luminosity are pre-
+sented in Mel´ endez et al. (2008) for a sample of 40 BAT AGNs. S imultaneous optical-to-X-ray
+spectral energy distributions are analyzed for 26 of the BAT AGNs in Vasudevan et al. (2009).
+Additionally, some details of the optical host properties a re presented in Winter et al. (2009a) as
+well as Schawinski et al. (2009). Further, the results a full analysis of the optical colors and mor-
+phologies are being compiled in Koss et al.(in prep) and the Spitzer-based IR properties will be
+presented in Weaver et al.(submitted).In this paper, we present an analysis of the opt ical spectral
+properties of a sub-sample of the AGN from the BAT 9-month cat alog.
+Since the BAT-detected sources are bright ( mV<16) and nearby ( < z >= 0.03), they are
+easily observable with ground-based facilities. Between p ublished optical spectral analyses, the
+publicly available Sloan Digital Sky Survey (SDSS) spectra , and our own follow-up observations
+with the Kitt Peak National Observatory’s (KPNO) 2.1-m tele scope of sources for which optical
+spectra/analyses were not available, we present the optica l emission line properties of 64/153 of
+the SWIFT BAT AGNs. This sample includes 35 broad line (55%) a nd 29 narrow line (45%)
+sources, the same ratio as in the total sample. All selected s ources were chosen based on positions
+viewable from the Kitt Peak Observatory. In this way, our sam ple represents 81% of the non-blazar
+“northern”BAT AGN sources. As in our X-ray study(Winter et a l. 2009a), we excludethebeamed
+sources due to the different physical mechanisms producing th eir spectra (i.e. jets). The missing
+“northern” sources were missed purely due to observation sc heduling and poor weather conditions.
+In the following sections, we describe the observations, da ta analysis, and finally our results.
+1http://heasarc.gsfc.nasa.gov/docs/swift/results/bs9 mon/– 3 –
+2. Observations and Data Reduction
+For our analysis of the optical spectra of the Swift BAT-dete cted AGN, we first obtained
+spectra of our sources that were publicly available from the Sloan Digital Sky Survey (SDSS). We
+supplemented this data set with our own observations at the K itt Peak Observatory. Additionally,
+we included several of the SDSS observed sources as Kitt Peak targets, in order to compare the
+results of our analysis from each observatory.
+Our Kitt Peak observations were obtained on the 2.1-m telesc ope as part of MD-TAC time.
+Over the course of 5 observing trips, from August 2006 – April 2009, we used the GoldCam
+spectrograph to observe the central region of ≈50 objects, including AGN and template galaxies.
+The AGN observed were sources for which we could not find archi ved optical spectra or analyses
+of optical lines in the literature. The template galaxies (1 0) were chosen from non-active templates
+listed in Ho et al. (1997). A majority of the sources were obse rved with two 30-minute exposures in
+both the red (grating 35, which covers 4760–7240 ˚A) and blue (grating 26new, which covers 3660–
+6140˚A), through a 2′′slit. Both of these gratings have a spectral resolution of 3. 3˚A, corresponding
+to a velocity width of 200 kms−1at 5007˚A. The exposure times were chosen in order to achieve a
+S/N of≈70 per pixel for the AGN sources and the dispersion relation f or both gratings corresponds
+to≈1.25˚A pixel−1. However, for some of the faintest sources we used a lower dis persion grating,
+grating 32, which covered a larger wavelength range than the higher dispersion gratings (4280–
+9220˚A, at 2.25 ˚A pixel−1and which has a spectral resolution of 6.7 ˚A). We used this grating because
+of the unknown redshift of many of these sources.
+Initial processingof thedataproceededusingthestandard tasks inIRAFtoextract thespectra
+and remove cosmic rays. The spectra were dispersion correct ed using comparison observations of
+the HeNeAr lamp taken at each telescope position. They were t hen flux calibrated using standard
+stars, from the spectrophotometric standards compiled by M assey et al. (1988), observed on the
+same night as the template/AGN. We then added the medium reso lution red and blue spectra
+together to obtain a single medium resolution spectrum for e ach source.
+In addition to the Kitt Peak observations, we include spectr a from the SDSS data release 7
+(Abazajian et al. 2009). Such spectra were publicly availab le for 24 of the non-blazar BAT AGN
+sources. A list of the BAT AGN 9-month sources for which we ana lyzed Kitt Peak/Sloan spectra
+are listed in Table 1. The KPNO observations (with typical to tal exposure times of 1hr in each
+medium resolution grating) were planned such that we would o btain similar signal-to-noise spectra
+as the SDSS spectra (S/N ≈75), to provide an easy comparison between both sets of spect ra. In
+total, our sample consists of 64 sources, including 40 spect ra from our KPNO observations, 24 with
+SDSS spectra (4 sources having both a KPNO and SDSS spectrum) , and 13 with emission line
+properties listed in the literature (9 of which also have eit her a KPNO or SDSS spectrum). Details
+of the KPNO observations, including the extraction apertur e along the slit, are listed in Table 2
+for the target AGN sources and in Table 3 for the galaxy templa te sources. Details of the SDSS
+observations are listed in Table 4. Based on visual inspecti ons of the AGN spectra, we indicate– 4 –
+in the tables which sources display broad lines with a ‘B’. In total, 33 sources (including 3 with
+emission line properties available in the literature), 55% of the sample, exhibit clear broad lines
+(i.e. broad H αand Hβ).
+3. Data Analysis
+Analysis of the spectra consisted of three steps: de-redden ing the spectrum to correct for
+reddening from the Milky Way, continuum subtraction, and fit ting the emission lines. The spectra
+were de-reddenned using the IDL procedure CCMUNREDfrom the Goddard IDL Astronomy User’s
+Library. This procedure uses the reddening curve of Cardell i et al. (1989), with R V= 3.1, and
+the input value of E B−V. The Milky Way E B−Vvalues (listed in Table 1) were obtained for the
+Kitt Peak and SDSS observed sources from the NASA Extragalac tic Database (NED). Following
+this step, the spectra were de-redshifted to their restfram e wavelengths, using the NED redshifts or
+measured redshift from the [OIII] 5007 ˚Aline. Following the continuum fits ( §3.1), we measured
+the emission ( §3.2) and absorption ( §3.3) line parameters for prominent spectral features. We
+then tested forapertureeffects by comparingemission linean dstellar absorption line measurements
+with redshift, finding no correlations.
+3.1. Continuum Modeling
+In order to fit the emission lines as correctly as possible, gr eat care must be taken in modeling
+the continuum. For an AGN source, we expect the continuum to b e a combination of non-thermal
+emissionfromtheAGNandstellarlightfromthehostgalaxy. Tomodelthecontributionfromstellar
+light, we used the population synthesis models from the GALA XEV package2(Bruzual & Charlot
+2003) in the 3200 ˚A–9300˚A range. The spectral resolution of these models ( ≈3˚A) is directly
+comparable to that in the SDSS and KPNO samples. We assume tha t the galaxy light is the
+sum of bursts of formation at different ages, using stellar pop ulations at 3 different ages (25,
+2500, and 10000 Myr) to determine whether the host is consist ent with a young, intermediate,
+or old population (or any combination of these three). While a three component stellar model
+(young, intermediate, old) does not fully describe the spec tra of all galaxies, we have found (see
+the appendix) that adding more components results in degene rate solutions with different sums
+of the 10 spectral models in the Bruzual & Charlot instantane ous burst models. We thus use
+the 3 component models adopted and recognize that this may no t be a fully accurate description
+of the stellar components of the host galaxies. Additionall y, we used 3 metallicity levels: 0.05Z
+(2.5Z⊙), 0.02Z( Z⊙), and 0.004Z (1
+5Z⊙). We usethesame code describedin Tremonti et al. (2004),
+which was used to measure the continuum in a sample of 53,000 S DSS galaxies. As described in
+Tremonti et al. (2004), the best fit is obtained using a nonneg ative least squares fit using the same
+metallicity for all 3 of the ‘age’ groups, attenuated by dust (which is modeled as a free parameter).
+Theχ2values, using different metallicity populations, are compar ed to find the best fit metallicity– 5 –
+range. We note, however, that these models depend upon neces sary assumptions, such as stellar
+populations created in an instantaneous burst of star forma tion (see Conroy et al. (2009) for a
+discussion of many of the associated uncertainties in singl e stellar population models), which are
+not physical.
+To test the effectiveness of the galaxy continuum fits, we first a pplied the models to our set
+of template galaxies obtained at KPNO. The final input to the T remonti et al. (2004) code is
+the galaxy’s velocity dispersion, a quantity that is unknow n for many of our AGN host galaxies.
+Therefore, we fit each of the templates with a range of dispers ion values to obtain the best-fit.
+These values were then compared to the known galaxy paramete rs, listed in LEDA3(Paturel et al.
+2003). The galaxy type and velocity dispersion, as well as th e fitted values, are listed in Table 5.
+On average, we find that the fitted dispersion velocities (for a Gaussian, FWHM = 2.35×σ) are
+in agreement with the central velocity dispersions listed i n LEDA ( < σM>= 132kms−1, while
+the LEDA values give < vdisp>= 159kms−1). From a comparison of the galaxy type to the light
+fraction (at 5500 ˚A) from the young, intermediate, and old stellar population s, there are no obvious
+contradictions. Our sample includes late spirals through e llipticals and we find that the models
+suggest the light is dominated by intermediate to old stella r populations in most of the galaxies
+(this is consistent with the color analysis of the images fou nd by Koss et al.(in preparation)).
+In Figure 1, we plot examples of the results of the stellar con tinuum fitting. We find that the
+models are particularly accurate at fitting the blue end (bel ow 5000˚A) of the spectra. While the
+addition of more stellar populations (at different ages) woul d provide better fits to the spectra,
+Tremonti et al. (2004) point out that the fits are often degene rate (they use 10 different population
+ages). Therefore, in an effort to get a broad understanding of t he stellar properties of the AGN
+host galaxy properties, we confineour fits to the young, inter mediate, and old populations indicated
+above.
+Additionally, we created a grid of test spectra using differen t combinations of the three stellar
+populations indicated. Random noise was added to the test sp ectra, which were then broadened
+with FWHM = 300kms−1and an instrumental resolution of 75kms−1, and reddened using the
+Charlot & Fall law (Charlot & Fall 2000). The results of conti nuum fits to these test spectra are
+presented in §A. As shown, we find that the velocity dispersion is well-dete rmined for our test
+spectra while the metallicities are not. We can clearly dist inguish young stellar populations from
+the intermediate/old populations, however, there is a dege neracy between the intermediate and old
+populations when they are combined with the young populatio ns. These degeneracies are taken
+into account in the following discussions. We also created a grid of test spectra including a power
+law contribution similar to that of our sample along with the stellar populations, from which we
+found no degeneracy between the power law and stellar compon ents (see appendix).
+2http://www2.iap.fr/users/charlot/bc2003/
+3http://leda.univ-lyon1.fr– 6 –
+In order to subtract a continuum from the KPNO and the SDSS spe ctra, we modified the
+galaxy modeling code to include a non-thermal power law cont ribution from the AGN ( p0×λp1,
+wherep0is constrained to range from 0 to 1 and p1>0). In our model, separate reddening values
+were fitted for both the power law component and the stellar co mponent. Additionally, in our
+fits we masked out regions near prominent emission line posit ions (i.e. H β, [OIII]λ5007˚A) at a
+standard width of 500kms−1and used a larger width of 7000kms−1around H α. For the broad line
+sources (identified as such by visual inspection of the optic al spectra), we masked a larger region
+with a width of 10000kms−1around prominent hydrogen and helium emission features (H γ, Hδ,
+Hβ, Hα, HeI, and He II). The results of these fits are presented in Table 6. Average v alues of
+p1for our sources were 0.67, very similar to the power law slope s found for luminous quasars by
+Richards et al. (2006), with a range of fitted values from 0 to 2 .89. The average value for p0is 0.47,
+with values ranging from 0 to 1. As listed in the Table 6, p0was calculated for the specific flux at
+1˚A and has units of 10−17ergss−1cm−2˚A−1.
+As we show in the appendix, we found no statistical degenerac ies between the power law
+component and stellar continua based upon our simulations. However, the issue of separating
+stellar and non-thermal AGN continua is complex. In order to assess the degree of degeneracy
+in our models, we carefully analyzed the results of our model fits. From our models, 37% of the
+narrow line sources (sources in this category tend to be clas sified as Sy 1.8/1.9 sources by other
+authors) and 38% of the broad line sources have contribution s of 50% or greater from a power
+law. We examined the spectra of these sources in the region fr om 3800–4200 ˚A, which includes
+the important stellar diagnostic lines of Ca H and K as well as the Hδabsorption. For broad line
+sources with high power law contributions, we find that absor ption lines tend to be weak, while
+[NeIII] (at 3869 and 3968 ˚A) and occasionally weaker hydrogen Balmer (H ζ, Hǫ, Hδ) emission lines
+are comparatively strong. For the narrow line sources, sour ces with strong power law contributions
+tend to have weak to no clearly evident absorption features. Nearly half of these narrow line
+spectra have either poor fits to the data ( χ2>>1) or no spectral coverage at the blue wavelengths
+which include important stellar lines like Ca H and K (making the fits less reliable). Therefore,
+the effects of any degeneracies between power law and stellar p opulation models are likely small
+for our purposes (i.e. rough estimates of the continuum).
+In Figures 2 and 3, we show examples of the continuum results. Both the original and contin-
+uum subtracted spectra are plotted in black with the continu um plotted in blue. For the majority
+of sources, we find acceptable fits with the stellar + power law continuum models. Particularly,
+good fits are obtained for the narrow line sources. For the bro ad line sources, the presence of broad
+Balmer lines makes it particularly hard to obtain a good fit to the spectrum below ≈4500˚A (see
+for example the spectrum of MCG +04-22-042).
+To show how the spectra and continuum models for spectra take n at KPNO compare to the
+SDSS spectra, we plot the KPNO and SDSS spectra + continuum fit s for the four sources with
+spectra from both in Figure 4. We chose to show the region from 3700–6200 ˚A, a region which
+includes both prominent emission lines (i.e. H βand [OIII]) and intrinsic absorption features (Ca– 7 –
+H&K, the G-Band, Mg Ib, and Na ID). Both the SDSS and KPNO spectra of Ark 347 are well fit
+with a continuum dominated by an old stellar population at so lar metallicity. The KPNO spectrum
+of Mkn 417 is found to be dominated by a power law, while the SDS S continuum is dominated by
+a solar metallicity old stellar population. For the broad li ne source MCG +04-22-042, neither the
+KPNO or SDSS spectra are fit well at the blue end of the spectrum (due to the hydrogen Balmer
+lines), making it unsurprising that the models do not match.
+Finally, for Mkn 18, different metallicities (low in the SDSS s pectrum and high in the KPNO
+spectrum)andgalaxycontributionsarefound. However, aso urtestmodelsshowed, themetallicities
+are not well-determined with the continuum models. The youn g stellar population contributions
+are similar for both the KPNO and SDSS spectra, leaving the di screpancy in the intermediate
+and old contributions as a likely effect of the degeneracy we fo und in our test models between the
+intermediate and old populations. The difference in the conti nuum flux between the KPNO and
+SDSS spectra of Mkn 18 is an extreme case, likely due to the fac t that Mkn 18 is highly elliptical
+and inclined along the E-W direction of the slit in the KPNO ob servation (15′′), while the circular
+fiber of the SDSS (3′′) misses out on this flux.
+3.2. Emission Line Fitting
+To measure the properties of the emission lines in the KPNO an d SDSS spectra (including the
+FWHM and flux of each line), we adopted two separate methods fo r the narrow line and broad line
+spectra. For the narrow line spectra, we first measured the pr ominent lines in two distinct regions,
+the regions surrounding H βand Hα. At the blue end of the spectrum, we fixed the positions of the
+Hγ, Hβ, and [O III] lines (λ4959 and λ5007), requiring that the velocity offset and FWHM of the
+lines remain the same for all of the lines measured, and fit for the flux and equivalent width. For
+spectra whose wavelength range includes [O II]λ3727, an important diagnostic for distinguishing
+low-ionization narrow emission-line regions (LINERs) (He ckman 1980), we include this line in the
+fits to the blue end of the spectrum. Additionally, we followe d the same procedure to fit the
+prominent emission lines surrounding and including H α, [OI]λ6300, [N II]λ6548, [N II]λ6584,
+[SII]λ6716, and [S II]λ6731. The intensities of the [N II] lines are fixed such that the λ6548
+line is at a 1:2.98 ratio with the λ6584 line, as dictated by atomic physics. For all of the narro w
+line fits, the FWHM was corrected for the instrumental resolu tion (200kms−1at 5007˚A for the
+KPNO spectra and 75kms−1for the SDSS spectra) and we placed the restriction that the F WHM
+values have a lower limit of 50kms−1and an upper limit of 1000kms−1. The results are recorded
+in Table 7. In Table 8, we include the intensity ratios for add itional weaker lines (i.e. H δ, [NI],
+HeI) measured in the spectra.
+For the broad line sources, two complications arise which pr event us from performing the
+same analysis as for the narrow line sources. Firstly, great er uncertainties exist in the continuum
+measurements. Secondly, the lines can not be fit by simple Gau ssians with the same widths. While
+the hydrogen Balmer lines of many of the broad line sources sh ow asymmetries, we chose to fit– 8 –
+both Hαand Hβwith a combination of narrow and broad Gaussians. To ensure t he uniform
+measurements of the lines in our spectra, we used an automate d process which focused on fitting
+lines in a narrow region surrounding both the H βand Hαlines, separately.
+In the Hβregion, defined as the region from 4600–5200 ˚A, we fit a combination of three narrow
+Gaussians to [O III] 5007˚A. The use of three Gaussians allowed us to reproduce the shap e more
+robustly, since this line often shows extended wings. The na rrow line shape, particularly the widths
+of these lines, were applied to the narrow He II4686˚A, Hβ4861˚A, and [O III] 4959˚A lines. Both
+the flux and velocity offset of each line were allowed to vary. Th e continuum was fit with a linear
+function in a region unaffected by the prominent lines. Finall y, these fitted narrow components
+were combined with a broad H βline, which was modeled with a single broad Gaussian compone nt,
+and re-fit. The use of essentially a narrow and broad Gaussian allows us to estimate the flux and
+width of each component, important in estimating the black h ole mass (based on the FWHM in
+the broad component) and emission line ratios (which depend on the ratio of the narrow lines).
+Results of these fits are included in Table 9, including the me asured continuum flux at 5100 ˚A. The
+recorded values of FWHM for the narrow component apply to the strongest narrow line component
+of the three Gaussians used to fit the [O III] 5007˚A line.
+In the Hαregion, defined as the region from 6200–6900 ˚A, we used the narrow [O I] 6300˚A line
+to define the initial guess for the velocity offset of the measur ed lines and the set FWHM of a
+single Gaussian component. The offset velocities and fluxes of the remaining narrow H αline, [N II]
+lines, and [S II] lines were allowed to vary. However, the intensities of the [NII] lines are fixed
+such that the λ6548 line is at a 1:2.98 ratio with the λ6584 line. A linear continuum was fit in a
+region unaffected by the emission lines. The narrow lines were added to a single broad Gaussian
+for broad H αand re-fit. Results from these fits are recorded in Table 10. Ex amples of fits to both
+the Hβand Hαregions are shown in Figure 5. The largest uncertainties inv olved in these fits are
+associated with the measurements of H αand the two [N II] lines, which are blended in our broad
+line spectra, particularly for a source such as MCG +04-22-0 42.
+Additionally, weaker lines that are also present in the spec tra were measured by manually
+selecting a continuum region surrounding the selected emis sion feature. The flux of each of these
+measured lines are included in Table 11. Where broad lines we re present and clearly separable from
+a narrow component, the indicated flux is for the narrow compo nent.
+3.3. Stellar Absorption Features
+As an alternate method of determining ages of the host galaxi es from the stellar continuum
+fits, we measure the strength of stellar absorption features directly from the non-galaxy continuum
+subtracted (both with and without subtraction of the AGN non -thermal component) spectra of our
+sources. This method is analogous to the work measuring Lick -indices by Worthey & Ottaviani
+(1997). However, instead of broadening our spectra to the ve locity dispersion of the Lick/IDS– 9 –
+spectral library (9 ˚A), we follow the procedure outlined in Kauffmann et al. (2003b ) for SDSS spec-
+tra, which instead compares the measured indices to the Bruz ual & Charlot (2003) stellar models.
+For further details on the SDSS analysis, along with a compar ison of the measured indices with
+additional high resolution stellar libraries, see the disc ussion in Kauffmann et al. (2003b).
+Two particularly important indicators of the age of a stella r population were used extensively
+in galaxy studies using SDSS spectra (Kauffmann et al. 2003a,b ; Gallazzi et al. 2005; Kewley et al.
+2006). These are the 4000 ˚A break (measured with Dn(4000)) and the equivalent width of H δ
+absorption (measured with H δA). Among these, the 4000 ˚A break, or Ca IIbreak, is observed as a
+discontinuity in the optical spectrum, caused mainly by the presence of absorption features from
+metals below 4000 ˚A. Since the opacity of metals in young, hot stars is low, this feature is weak in
+young stellar populations and strong in old populations. As a measurement of the Ca IIbreak, we
+use the definition of Balogh et al. (1999) to compute:
+Dn(4000) =/integraltext4000
+4100fλdλ
+/integraltext3850
+3950fλdλ(1)
+While strong Ca IIbreaks indicate old populations, strong equivalent widths of Hδabsorption
+indicate a recent burst of star formation within 0.1–1Gyr (W orthey & Ottaviani 1997). Therefore,
+we measure
+HδA= (4083.50−4122.25)(1−(FI/FC)), (2)
+whereFIis the flux of the line within the bandpass of the feature (4122 .25 – 4083.50) and FCis
+the flux in a pseudo-continuum. The pseudo-continuum is defin ed as the line drawn through the
+average of the flux in the continuum immediately blueward ( λλ4041.60 – 4079.75 ˚A) and redward
+(λλ4128.50 – 4161.00 ˚A) of the H δabsorption feature.
+Half of the spectra show an H δemission line (10 narrow line sources and 16 broad line sourc es),
+while emission from [Ne III] 3869˚A is often present in the pseudo-continuum from which Dn(4000)
+is measured. For the narrow line sources in our sample, we sub tracted the measured narrow lines
+before calculating these age indicators. Such a calculatio n is not straight forward for the broad line
+sources, where broad emission features are often present in the region containing H δA(with Hδ
+emission) and D n(4000) (including [Ne III] + H7λ3968˚A and H δ). In Figure 6, we plot examples
+of spectra for both narrow and broad line sources, where stel lar absorption features are seen.
+In Figure 7, we plot H δAversus D n(4000) for our sources, excluding broad line sources with
+prominent H δemission. We plot the values measured both after subtractin g the power law con-
+tinuum (Table 6; top plot) and from the original dereddened s pectrum (bottom plot). From each
+of these measurements, the D n(4000) break does not change appreciably whether or not the p ower
+law component is subtracted, with a median value of 1.26 for n arrow line sources and 0.91 for broad
+line sources when the power law is subtracted and 1.41 (narro w) and 0.92 (broad) without the sub-
+traction. The H δAvalues are affected, however, for the narrow line sources with median values of
+0.81 (narrow) and -2.15 (broad) with the power law subtracte d and 1.73 (narrow) and -2.15 (broad)– 10 –
+without the subtraction. To test whether any aperture effects influenced our measurements, we
+plotted each of these diagnostic measurements against reds hift. With no correlation in either H δA
+or Dn(4000) with z, we conclude that there are no obvious aperture effects to be ac counted for in
+our measurements.
+The majority of the narrow line sources occupy the area expec ted from our stellar population
+model tests, discussed in §A and plotted in Figure 19. The broad line sources, however, o ccupy a
+region with considerably lower values of H δA. This is true even for sources where an H δemission
+line is not seen in the spectrum (as for the sources plotted). From visual inspection of the H δ
+region of our sources, we find that unlike the narrow line sour ces, we can not clearly identify an H δ
+absorption feature in any of the broad line sources. In most c ases, we see emission features that
+are often broad. The low values of H δAmeasurements for broad line sources are therefore a likely
+effect of emission in this region.
+In addition to these stellar age diagnostics, we measured ad ditional absorption indices for
+common stellar absorption features. These values were meas ured using the same method as used
+for the H δAindex, first subtracting the emission line spectra for the na rrow line sources and
+subtracting the power law component for all of the sources. B andpasses and continuum ranges
+are defined in Worthey et al. (1994) and Worthey & Ottaviani (1 997). In Table 12, we present the
+stellar age indicators (D n(4000) and H δA) along with 6 metallicity indicators, chosen to sample
+indices sensitive to several different elements (i.e. C, N, Ca , Mg, Fe). Two of these indices are
+combinations of other indices, defined in Gonz´ alez (1993):
+[MgFe] =/radicalbig
+Mgb<Fe>and<Fe>=1
+2(Fe5270 +Fe5335) . (3)
+We use the modified form of [MgFe]′, defined by Thomas et al. (2003) as:
+[MgFe]′=/radicalbig
+Mgb(0.72 Fe5270+0 .28 Fe5335) . (4)
+Additionally, tobetterunderstandourresults, wealsomea suredthesestellar absorptionindices
+for a sample of test spectra created from the stellar populat ion models used for the continuum fits.
+We discuss these results, where we used different combination s of stellar ages and metallicities, in
+§A. Of the additional stellar absorption indices recorded in Table 12, H δemission could affect
+the value measured for CN 1. Additionally, He II4686˚A is within the range of C 24668 and [N I]
+5199˚A is within the range of Mgb. Since [N I] 5199˚A is weak in our broad line sources, we expect
+little error in our Mgb measurements. In Figure 8, we plot var ious metallicity indicators and the
+age indicator D n(4000) versus themetallicity indicator Mgb for our target s ources. Comparingwith
+our results from the test spectra, it appears that C 24668 is the most affected by “contaminating”
+emission features. The narrow line sources should be unaffect ed, however, since we have subtracted
+the emission components from their spectra.
+Based on a comparison of the plots in Figure 8 with the test spe ctra values, we find that the
+[MgFe]′vs Mgb and <Fe>vs Mgb plots are the best indicators of the metallicity of the stellar– 11 –
+populations. However, the only clear result is that we do not find old, high-metallicity (2.5 Z ⊙)
+populations within our sample (all of the old population tes t spectra have Mgb /lessorsimilar2, as determined
+from the D n(4000) vs Mgb plot). Since there is little difference in the par ameter space occupied by
+solar and low metallicity populations, we can not discern an ything more from our measured stellar
+absorption indices.
+4. Emission Line Classification
+Emission line diagnostic plots, utilizing the optical line ratios of [O III]λ5007/Hβ, [NII]
+λ6583/Hα, [SII]λλ6716,6731/Hα, [OIII]λ5007/[O II]λ3727 and [O I]λ6300/Hα, are an em-
+pirical method of separating Seyferts, LINERs, and star-fo rming galaxies (Baldwin et al. 1981;
+Veilleux & Osterbrock 1987). The chosen line ratios (1) have small wavelength separations, so that
+the effects of reddening are minimal, and (2) distinguish betw een photo-ionization from O and B
+stars (H IIobjects) and a non-thermal/power law continuum (AGNs). In o rder to construct these
+diagnostic diagrams for our Swift BAT AGNs, we first correcte d the line ratios for reddening.
+To correct our line ratios for extinction, we use the line rat io of the strongest narrow Balmer
+lines (Hα/Hβ) along with the Cardelli et al. (1989) reddening curve. The e ffect of reddening is
+represented as
+I(Hα)
+I(Hβ)=F(Hα)
+F(Hβ)10c[f(Hα)−f(Hβ)](5)
+whereI(λ) is the intrinsic flux, F(λ) is the observed flux, and f(λ) is from the reddening curve. We
+assumeanintrinsicH α/Hβratio(I(Hα)
+I(Hβ))of3.1foroursources, assumingthattheyaredominatedby
+the underlying AGN. Additionally, we assume that RV= 3.1 and therefore E(B−V) = (2.5/3.1)c.
+For 11 of the spectrafromKPNO or SDSS,we findthat the ratio of Hα/Hβis less than the assumed
+intrinsic value, for which we do not apply a reddening correc tion [E(B−V) = 0]. The corrected line
+ratios, along with values found in the literature for an addi tional 13 sources, are shown in Table 13.
+In Figure 9, we plot the distribution of E(B−V) for the narrow and broad line sources.
+Excluding the few outlying observations with measured valu es ofE(B−V)>1.0, we find that the
+broad line sources have a lower average value than the narrow line sources and a smaller range of
+values. We find the average value of E(B−V) = 0.08 with a standard deviation of 0.11 for broad
+line sources and an average value of E(B−V) = 0.29 with a standard deviation of 0.33 for narrow
+line sources. The results of a Kolmogorov-Smirnov comparis on test show that it is unlikely that the
+values are drawn from the same distribution with the maximum difference between the cumulative
+distributions ( D) of 0.375 and a corresponding probability of 0.016. This pro bability is less, but
+still low, when the outlying points are included ( D= 0.301 andP= 0.067). Thus, the narrow lines
+in type 2 objects are more extincted.
+We classify our sources as H IIgalaxies, composites (COMPs), Seyferts, or LINERs using th e
+classification criteria based on the analysis of the emissio n line properties of 85224 SDSS galaxies– 12 –
+presented in Kewley et al. (2006). These criteria include a t heoretical ’maximum starburst line’
+from Kewley et al. (2001), shown as a solid line in the diagram s in Figure 10, which represents a
+boundary between H IIgalaxies and AGNs. Additionally, in the [O III]/Hβvs. [NII]/Hαdiagram,
+a dashed line shows the empirical division between pure star -forming galaxies and Seyfert-H II
+composites from Kauffmann et al. (2003a). Finally, empirical ly derived divisions between LINERs
+and Seyferts, from Kewley et al. (2006), are shown in the [O III]/Hβvs. [SII]/Hα, [OIII]/Hβvs.
+[OI]/Hα, and [O III]/[OII] vs. [O I]/Hαdiagnostic plots. The emission line diagnostic plots are
+shown in Figure 10 and the classifications are shown in Table 1 4.
+Based on these classifications of the narrow line sources (ci rcles and a few squares [values from
+the literature] in Figure 10), 25 spectra are consistent wit h Seyferts, 1 spectrum corresponds to an
+HIIobject, 5 spectra are consistent with LINERs, 1 is a composit e, and 6 are ambiguous. Among
+these, we classify the Ark 347 KPNO spectrum as a Seyfert and N GC 4992 as a LINER. For each
+of these sources, the Veilleux & Osterbrock (1987) diagram i ncluding the [S II]/Hαratio is the
+only diagram with a classification inconsistent with the oth er classification plots. Errors in this
+measurement ([S II]/Hα) could easily place the spectra within the Seyfert or LINER c lassification,
+respectively.
+The LINER sources include NGC 788, NGC 2110, NGC 4992, MCG+04 -48-002, and NGC
+7319. Of these, NGC 4992 is classified as a possible X-ray brig ht optically normal galaxy (XBONG)
+by Masetti et al. (2006), and MCG+04-48-002 was previously c lassified as a starburst/H IIgalaxy
+with a hidden Sy 2 nucleus (Masetti et al. 2006) (in their spec trum the [O I]λ6300 line was not
+detected). All but one of the classified LINERs (NGC 4992) hav e ratios of H α/Hβ <3.1.
+The spectra classified as starburst/H IIgalaxy and composite, respectively, are the SDSS
+spectrum of Mkn 18 and UGC 11871. Finally, the 6 ambiguous sou rces include: 2 spectra with
+COMP/LINER properties (the KPNO spectrum of Mkn 18 and NGC 62 40, a luminous infrared
+galaxy known to show contributions from both the AGN and star bursts (Sanders et al. 1988)), 2
+spectra with Seyfert/H II(both the KPNO and literature spectra of NGC 4102), and 2 spec tra
+with Seyfert/LINER properties (NGC 1275 (which is in the mid dle of a strong emission nebulae
+associated with the cooling flow in the Perseus cluster) and N GC 4138). In general, there is
+good agreement between classifications of sources with mult iple spectra. Both Mkn 417 and Ark
+347 spectra indicate a Seyfert and the NGC 4102 spectra show a n ambiguous source between
+Seyfert/H II. While the Mkn 18 classifications are not the same, they both p oint to having at least
+some H II-like emission line ratios (particularly [S II]/Hα).
+While it is clear that broad line sources are Seyfert 1s, it is of interest to examine how they
+would be classified based on their narrow line ratios. If the p redictions of the unified model are
+true then, if the broad line region is absorbed out, the narro w line ratios should classify these
+objects as Seyferts also. We find, much to our surprise, that a significant fraction of the broad line
+objects have narrow line ratios which lie outside the AGN reg ion based on the Kewley et al. (2006)
+classifications. While the majority (75%) of broad line sour ces have narrow line ratios consistent– 13 –
+with classification as Seyferts (30 spectra), some (in parti cular NGC 931, 1RXSJ193347.6+325422,
+UGC 6728, and IGR21247+5058) are not, being classified as com posites or H IIsources, though H α
+and the [N II] lines were too heavily blended to separate for the latter tw o. Additionally, 7 spectra
+(including the KPNO and SDSS spectra of MCG +04–22–042) have ambiguous classifications.
+There is good agreement between classifications of sources w ith multiple spectra (i.e. MCG +04–
+22–042, NGC 4151, NGC 3227, NGC 3516). The source NGC 4051, cl assified as ambiguous from
+the KPNO spectrum due to the [N II]/Hαdiagram result showing a COMP, should more likely be
+classified as a Seyfert (as in the spectrum analyzed in the lit erature).
+Therefore, the Swift BAT AGN optical classifications are mos tly Seyferts. There are a total of
+29 individual narrow line sources represented, and of these , about 66% are Seyferts, 16% LINERs,
+13% ambiguous, 3% composites, and 3% H IIgalaxies. Of the 35 broad line sources, about 75%
+are Seyferts, 14% are ambiguous, and 11% are composites or H IIgalaxies. We find no broad line
+sources with narrow emission consistent with LINERs.
+Since we are studying in this paper the optical properties of a hard X-ray detected sample, it is
+useful to make a comparison with optically selected samples , in particular the recent results of the
+SDSS. In this comparison, we find that the optically selected emission-line sources from the 85224
+SDSS galaxy sample of Kewley et al. (2006) consist of very diffe rent percentages of the various
+classification categories than our hard X-ray selected samp le. The SDSS sample consists of 75%
+star-forming/H IIgalaxies, 3% Seyferts, 7% LINERs, 7% composites, and 8% ambi guous. It is no
+surprise, that the majority of our 14–195keV X-ray sample co nsists of the much more energetic
+(across multiple bands) Seyferts. However, comparing the S DSS results solely with our narrow line
+sources, we are finding far fewer LINERs than we might expect. In the optically selected SDSS
+sample, the LINER class contains more than twice the number o f sources as Seyferts, while we are
+finding four-times as many Seyferts as LINERs among the narro w line sources.
+There are a few possibilities as to why the hard X-ray sample s elects fewer LINERs. The most
+obvious reason could be that LINERs are less luminous X-ray s ources (we discuss this further in
+§6). Indeed, Kewley et al. (2006) didfindthat LINERshadsubst antially lower reddeningcorrected
+[OIII] 5007˚A luminosities than Seyferts. If L [OIII]is an indicator of bolometric luminosity and
+scales with the Swift BAT luminosity, we may simply not be det ecting many LINERs with BAT
+because their X-ray fluxes are below the current detection th reshold. Further, studies such as
+the Chandra snapshot analysis of Terashima & Wilson (2003) a lso find LINERs as less luminous
+than Seyferts in X-rays. Also, based on the nuclear X-ray lum inosities of local LINER sources
+determined from the Chandra analysis of Flohic et al. (2006) who used the IR-selected LINER
+sample of Sturm et al. (2006), the typical local LINER would h ave a BAT flux far below the flux
+detection limit of the Swift survey.
+It is also possible that LINERs are more absorbed in the X-ray s. In Winter et al. (2009a), we
+have shown that the more X-ray absorbed (i.e. highest neutra l hydrogen column density) sources
+have lower X-ray luminosities, on average. If this is the cas e, we would expect to find a higher– 14 –
+number of LINERs as Swift BAT detects more heavily absorbed a nd less luminous sources. In
+support of this possibility, the average value of the X-ray d erived N H= 6×1023cm−2of our
+LINERs is high (Winter et al. 2009a). This is in contrast, how ever, to the optical reddening, where
+we noted that the ratio H α/Hβis below the accepted value for AGN (3.1) and the theoretical ly
+expected value for case B recombination (2.85) for most of ou r LINERs. Kewley et al. (2006) also
+foundthis in 45% of their LINER sample, which could bethe res ult of a higher nebular temperature
+(Osterbrock 1989) orshocks. Inthesecases, it is unclearho w to relate theoptical Balmer decrement
+to the X-ray derived column density.
+With lower luminosities than typical AGN sources and emissi on line ratios potentially indicat-
+ing a shock origin, it is possible that LINERs are typically n ot powered by accretion processes. As
+Flohic et al. (2006) show, many LINERs do not have any detecte d X-ray emission. Further, recent
+work by the SAURON team (Sarzi et al. 2009) and SEAGAL collabo ration (Cid Fernandes et al.
+2009) indicates that themajority of LINERs are not powered b y AGN but instead by evolved stellar
+populations. Therefore, we would expect to detect few LINER s in the Swift BAT band.
+5. Additional Diagnostic Lines
+Comparisons of the intensities of multiple emission lines f rom the same ion provide important
+diagnostics of the gas in which they are produced. In the opti cal range probed by our spectra, the
+relative population and therefore intensity of [S II]λ6716/λ6731 depends on the density of the gas
+(with only a slight dependence on temperature of the order T1/2
+e). The [O III]λ4363 emission line
+comes from a different upper energy level than the λ4959 and λ5007 lines, where the relative rates
+of excitation to these upper levels is strongly dependent on temperature. An equation relating the
+ratio of the [O III] lines to temperature and density is given in (Osterbrock 19 89) as:
+Iλ4959+Iλ5007
+Iλ4363=7.73exp((3 .29×104)/T
+1+4.5×10−4(Ne/T1/2). (6)
+In Figure 11, we plot the reddening corrected flux ratios for b oth of these diagnostics ([S II]
+and [OIII]). While both intensity ratios do not necessarily probe the same regions of the narrow
+line region, this figure is useful in illustrating the range o f values measured for our sample. One of
+the results of our analysis is that the ratio of [S II]λ6716/λ6731 is similar for both the broad and
+narrow line sources. Using a Kolmogorov-Smirnov compariso n test, we find that both distributions
+are likely to be drawn from the same population with D= 0.22 andP= 0.50. The average and
+standard deviations of these values are 1.12 and 0.27 for the narrow line sources and 1.09 and 0.23
+for the broad line sources. These values of the ratio of [S II]λ6716/λ6731 correspond to electron
+densities of Ne≈103cm−3(assuming Te= 104K as in figure 6.2 of Peterson (1997)). These results
+are consistent with average narrow line region densities of 2000cm−3found by Koski (1978). Thus,
+the hard X-ray detected Swift BAT AGN have the same densities as optically selected AGN in this
+region (which produces the [S II] emission), regardless of whether broad lines are present.– 15 –
+The temperature sensitive diagnostic [O III] (λ4959+λ5007)/λ4363 clearly is not the same for
+the narrow and broad line sources. The Kolmogorov-Smirnov c omparison test yields a P-value of
+0.000. The average and standard deviation of [O III] (λ4959+λ5007)/λ4363 is 166.0 and 193.0 for
+the narrow line sources and 14.53 and 12.71 for the broad line sources. To better illustrate what
+these values mean, in Figure 11 we also plot the relationship of the [O III] temperature diagnostic
+versus electron density for fixed temperatures. The average values of both the narrow and broad
+line sources are indicated with a horizontal line. In the low density limit ( Ne<104cm−3), the
+average temperature of the [O III] emitting gas is approximately 10000K for narrow line sourc es
+and 50000K for broad line sources. Typical temperatures for narrow line regions are between
+10000–25000K, with an average value of 16000K reported in Ko ski (1978).
+If the temperature of the narrow line region in the type 1s and 2s is different, this would
+be a violation of the unified model. However, if the densities are different, this might be due to
+geometrical effects wherein the dense regions in type 2s are bl ocked from our view or have very
+high reddening values. However, there is uncertainty in the measurement of [O III] (λ4959 +
+λ5007)/λ4363 associated specifically with the measurement of the fai nt [OIII]λ4363˚A line, which
+is just 1% of the bright λ4959˚A andλ5007˚A lines. We note that it is particularly hard to measure
+this line in the broad line sources where H γλ4340˚A may be producing a tilted pseudo-continuum.
+The result of broad line sources having lower values of [O III] (λ4959 +λ5007)/λ4363 than
+narrow line sources has been noted before and is attributed t o broad line sources having stronger
+λ4363 emission (Osterbrock 1978). Instead of a higher temper ature in the narrow line regions of
+broad line sources, Osterbrock (1978) suggests densities o f 106–107cm−3in broad line sources and
+/lessorsimilar105cm−3in narrow line sources. To reconcile these high densities wi th lower densities derived in
+the S+emission region, the narrow line region must consist of a ran ge of densities, among which
+low densities are found in low-ionization regions. Under th is interpretation, the temperatures of
+the narrow line region producing O+2are the same for broad and narrow line sources, provided the
+densities differ in this higher ionization region.
+6. [O III] and Hard X-ray Luminosities
+AfundamentalpropertyofanAGN isitspower, measuredthrou ghluminosity. InFigure12, we
+plot the distributions of both the observed and extinction- corrected [O III] 5007˚A luminosities for
+both our narrow line and broad line sources. For sources with multiple measurements, we averaged
+the values together to obtain a single measurement of observ ed and extinction corrected luminosity
+per source (these values are included in Table 15). We find tha t the extinction corrections do not
+significantly change the luminosity measurements, with the corrected values being on average 1.1
+(broad line sources) and 1.3 (narrow line sources) times lar ger than the observed luminosities.
+Themeanvalueforthedistributionof extinction corrected luminosity forthebroadlinesources
+is logL [OIII]= 41.79 with a standard deviation of 0.90, while the narrow line so urces have a mean– 16 –
+value of logL [OIII]= 40.82 with a standard deviation of 1.16. The results of a Kolmogr ov-Smirnov
+comparison test suggest that these values are not drawn from a single population ( D= 0.49 and
+P= 0.001). Therefore, the broad line sources appear to be more lum inous than the narrow line
+sources, on average. This is also true of the observed lumino sities (the averages and standard
+deviations are 41.76, 0.79 (broad line sources) and 40.87, 1 .08 (narrow line sources)) and therefore
+not an effect of incorrect reddening corrections. If the [O III] 5007˚A emission line is indeed an
+estimator of the AGN power (assuming that the contamination from star formation is not great),
+theseresults agree withour X-ray results fortheBAT AGNs. N amely, Winter et al. (2009a) showed
+that the unobscured X-ray sources (presumably optical broa d line sources) in the sample were also
+intrinsically more luminous.
+In§4, we described that previous optical and X-ray studies find L INERs as less luminous
+than Seyferts. Comparing the extinction-corrected [O III] luminosities for the narrow line sources,
+we confirm these results with our unbiased hard X-ray detecte d sample. We find Seyferts have an
+average valueoflogL [OIII]= 41.55 withastandarddeviation of0.85, LINERshaveanaverage v alue
+of logL [OIII]= 40.73 with a standard deviation of 0.60, and sources in other cat egories (including
+ambiguousclassifications, H IIgalaxies, andcomposites) haveanaverage valueoflogL [OIII]= 40.33
+with a standard deviation of 0.65. Of particular importance , we find that the narrow line Seyferts
+have luminosities consistent with those of broad line sourc es.
+Further, we find that the hard X-ray luminosities (in the 14–1 95keV band) show these same
+trends. To illustrate these results, we plot the distributi on of hard X-ray luminosity for our
+sources in Figure 13. For the narrow line sources, we find that the Seyferts have an average
+value of logL 14−195keV= 43.87 with a standard deviation of 0.94, LINERs have an average v alue
+of logL 14−195keV= 43.50 with a standard deviation of 0.16, and sources in other cat egories have
+an average value of logL 14−195keV= 42.69 with a standard deviation of 0.98. Once again, the
+HII/composites/ambiguous sources have the lowest luminositi es while the Seyferts are most lumi-
+nous. Also, the X-ray luminosities of the narrow line Seyfer ts are consistent with those of the broad
+line sources (which have an average value of logL 14−195keV= 43.74 with a standard deviation of
+0.74).
+BasedonX-raysurveys, severalstudieshadfoundthefracti onofobscuredsourcestoincreaseat
+lower 2–10keV luminosities, including those by Ueda et al. ( 2003) and Steffen et al. (2003), as well
+as our own study of the Swift sources (Winter et al. 2009a). Ba sed on an optically selected sample,
+Diamond-Stanic et al. (2009) also found differences in the dis tributions of 2–10keV and [O III]
+λ5007˚A luminosities for Sy 1s andSy 2s in therevised Shapley-Ames sample (Sandage & Tammann
+1987). A clear explanation for the differences in the X-ray sel ected samples is that the lowest
+luminosity X-ray sources, which tend to be absorbed sources , are not optical Seyferts, as found
+in our current study. In an optically defined sample, we would expect both the obscured and
+unobscured Seyferts to have the same luminosity distributi ons. However, in this same optically-
+selected sample Diamond-Stanic et al. (2009) find that the [O IV]λ25.89µm line, an indicator of
+bolometric luminosity (Mel´ endez et al. 2008), does not sho w this difference in distributionsbetween– 17 –
+Sy1sandSy2s. Itisunclearhowtointerprettheseresults. S incethestudyofDiamond-Stanic et al.
+(2009) consists of only sources for which multi-wavelength luminosity measurements are available
+(18 Seyfert 1s and 71 Seyfert 2s), it is potentially biased (c onsidering that X-ray surveys find
+equal numbers of absorbed and unabsorbed sources) compared to the Swift sample. However,
+the Diamond-Stanic et al. (2009) sample also includes a high percentage of Compton-thick sources
+(20%), which the Swift sample is currently not finding (due to the low X-ray flux in the BAT band
+of Compton thick sources).
+Since the hard X-ray luminosities are at high enough energie s to cut through much of the
+gas and dust around the AGN, they are a good estimate of the bol ometric luminosity. In lieu of
+these measurements, the optical [O III] luminosities are often used as a measurement of the AGN
+total power. Further, in support of using the [O III] luminosities as a proxy for the bolometric
+luminosity, Heckman et al. (2005) found a relationship betw een the observed hard X-ray (3-20keV)
+and observed [O III] luminosities for a sample of AGN in the RXTE slew survey. How ever, the
+results from a sample of Swift BAT AGN dispute the claim that [ OIII] luminosities are good
+estimates ofbolometric luminosity. Mel´ endez et al. (2008 ) foundthat [O III] was notwell-correlated
+with the hard X-ray (14–195keV).
+With our larger and more uniformly measured extinction-cor rected [O III] sample than in the
+Mel´ endez et al. (2008) sample (drawn from the 3-month Swift catalog (Markwardt et al. 2005)),
+we tested for a correlation between the BAT and [O III] luminosities. In Figure 14a, we plot the
+results of our comparison. We find weak linear correlations b etween the 14–195keV and [O III]
+luminosities for the broad and narrow line sources. Using th e ordinary least-squares (OLS) bisector
+method (Isobe et al. 1986)), we found L[OIII](corr)∝L1.16±0.13
+14−195keVandR2= 0.34 (P≈0.005) for the
+broad line sources and L[OIII](corr)∝L1.16±0.24
+14−195keVandR2= 0.42 (P≈0.002) for the narrow line
+sources. Here, R2is the correlation coefficient. As further illustrated in the ratio of optical/hard
+X-ray luminosity in Figure 14b, there is a great deal of scatt er in these relationships (e.g. more
+than 2 magnitudes at log L14−195keV). Our results support those of Mel´ endez et al. (2008), show ing
+that even the reddening corrected L[OIII]is affected by extinction. This effect is most pronounced
+for the narrow line sources, which show the greatest amount o f scatter.
+As shown in Rigby et al. (2009), at high levels of absorption t he luminosities measured in the
+Swift BAT band are affected by extinction. Using models from Ma tt et al. (2000), they show the
+difference between the emergent and input BAT flux at a variety o f column densities. For column
+densities less than 1023cm−2, this effect is minimal ( ≤4%). Since none of our targets are Compton
+thick (NH<1024cm−2in the Swift sample, see Winter et al. (2009b) for Suzaku obse rvations of
+heavily obscured sources confirming their Compton thin natu re), the effects on our sample are
+confined to a factor of ≈10−20% for the highest column density sources (25% of the narrow line
+sources). Even with this level of scatter introduced in the B AT luminosities, clearly the scatter
+seen in the narrow line sources in Figure 14 is not accounted f or by a 20% underestimate in BAT
+luminosity.– 18 –
+7. Mass and Accretion Rate Estimates
+Foreachofthebroadlinespectra, wewereabletoderivethem assofthecentral blackholeusing
+the FWHM of the broad component of H βand the continuum luminosity at 5100 ˚A. The continuum
+luminosity at 5100 ˚A was computedfromapower law continuumfit totheH βregion. We calculated
+the Hβmasses using our measurements in Table 9 and equation 5 from V estergaard & Peterson
+(2006). Thecomputedvaluesofextinctioncorrected λLλ(5100˚A)andM HβareincludedinTable15.
+At the resolution of our spectra, we found that the H βline is often more complicated than a simple
+combination of narrow and broad Gaussian profiles. Addition al structure or asymmetries are seen
+in a number of sources, making our measurements an approxima tion of the broad H βline FWHM
+(see Figure 5 for example fits).
+To test how our values of M Hβcompare with other mass estimates, we plot our values versus
+reverberation mapping masses and masses derived from the st ellar K-band light from 2MASS
+photometry in Figure 15. The mass estimates from reverberat ion mapping were obtained for 9
+sources from Peterson et al. (2004) and are listed in Table 15 . As shown, our H βderived masses
+are in good agreement with the reverberation mapping result s (with the exception of NGC 4593).
+There are no systematic offsets between the two methods.
+Not surprisingly, there are larger differences between the IR derived and H βderived masses.
+The 2MASS K sband derived masses (Mushotzky et al. 2008; Winter et al. 200 9a; Vasudevan et al.
+2009) were calculated by subtractingthe central luminosit y of apoint source(the size of the2MASS
+PSF). This presumed AGN luminosity was subtracted from the i ntegrated luminosity of the galaxy
+to determine the luminosity of the stellar bulge. The relati on defined by Novak et al. (2006) was
+then usedto convert the bulgeluminosity to stellar mass. Ap proximately 40% of themass estimates
+from the 2MASS K s-band and H βare within a factor of 2 of each other. A greater majority of th e
+IR masses are higher (typically, by up to a factor of 10).
+Despitethefactthatthe2MASSK sbandderivedmassesarealessaccuratemassdetermination
+than those using reverberation mapping or the H βFWHM method, we find that the 2MASS and
+HβFWHM masses are linearly correlated. Using the OLS bisector method, we find
+logM2MASS= (0.91±0.14)×logMHβ+(1.07±1.13), (7)
+withR2= 0.56. This is encouraging since the 2MASS derived measurement s are the only uniform
+estimates that we have for the narrow line and broad line sour ces. Therefore, we use the 2MASS
+derived masses to compare estimated masses, and later accre tion rates, between all of our sources.
+Unlike our comparison of luminosities (log L[OIII]), we find that the 2MASS derived masses show a
+great probability (from the K-S test) of the narrow and broad line masses being derived from the
+same population ( D= 0.21 andP= 0.71). The mean and standard deviations of log( MIR/M⊙)
+are 8.07 and 0.83 (narrow line sources) and 8.19 and 0.62 (bro ad line sources). With the more
+accurate H βFWHM method, we find that the average mass of our sources (base d on the broad
+line sources) is log M/M⊙= 7.87 with a standard deviation of 0.66. The range of masses, as shown– 19 –
+in Figure 15, is consistent with those found in other AGN surv eys. For example, our values are
+consistent with the range, 106–7×109M⊙, found by Woo & Urry (2002) in a sample of 377 AGNs.
+Our values are also similar to those of nearby PG QSOs derived from H-band host magnitudes
+(Veilleux et al. 2009).
+Since the masses of our narrow and broad line sources are simi lar while the average narrow
+line source luminosities are lower, we expect the values of L [OIII]/LEddto differ. The [O III]
+λ5007˚A luminosity is often used as an estimate of the bolometric lu minosity of AGN, particularly
+for sources detected in the SDSS (see Heckman et al. (2004)). Typical bolometric corrections
+for extinction corrected [O III] luminosities are expected to be between 600–800 for Seyfer t 1s
+(Kauffmann & Heckman 2008). There are, however, problems with using L [OIII]as an estimate of
+bolometric luminosity. In the previous section, we showed t hat the hard X-ray luminosities, which
+are less affected by contamination from star formation and ext inction, are not well-correlated with
+L[OIII], particularly for the narrow line sources. Despite these pr oblems, the ratio of L [OIII]/LEdd
+allows us to compare a rough estimate of the accretion rates o f our broad and narrow sources, which
+we can also compare with the more robust values we obtained in our X-ray study. In Figure 16, we
+plot the results (where L Eddis defined as 1 .38×1038(M/M⊙) and the mass is obtained from the
+2MASS measurements). We find, as expected, that the ratio of L [OIII]/LEddis lower for the narrow
+line sources, with the average and standard deviations corr esponding to 10−5.25±0.81(narrow) and
+10−4.61±0.85(broad). Since only three LINERs and two H II/composite/ambiguous sources have
+available 2MASS-derived masses, we can not test whether sou rces in these categories have lower
+L[OIII]/LEddvalues than Seyferts.
+For the broad line sources, our estimate of the average accre tion rate ( Lbol/LEdd), assuming a
+bolometric correction of 600, is 0.015 with the 2MASS derive d masses or 0.034 with the H βFWHM
+derived masses. Based on our X-ray analysis (Winter et al. 20 09a), the 2MASS derived masses,
+and an assumed 2–10 keV bolometric correction of 35 for unabs orbed sources (Barger et al. 2005),
+we estimate an X-ray derived accretion rate of 0.040. Theref ore, there is very good agreement
+between the optical and X-ray derived accretion rates, in an average sense. Unfortunately, with
+increased uncertainty in the bolometric corrections, it is more difficult to determine these values
+for the narrow line/Sy 2 sources.
+8. Host Galaxy Properties
+Since the Swift BAT-detected AGN are relatively close ( < z >≈0.03) and bright, intrinsic
+stellar absorption features are seen in the majority of the s pectra we analyzed. This allows us
+to determine some of the properties of the host galaxies of ou r target AGN. To do this, we have
+employed two particular methods to analyze the intrinsic st ellar absorption features – one using
+continuum fits and the other measuring stellar absorption in dices. We note, however, that the
+sampling of the host galaxy populations for the BAT-detecte d AGN is not uniform, but a function
+of the aperture size (2–3′′), distance to the source, and both the size and orientation o f the host– 20 –
+galaxy within the slit. It is our intent, in this paper, to det ermine basic conclusions about the
+stellar populations from the optical spectra. More detaile d information on the host galaxies of
+this sample, including star formation rates from Spitzer fo llow-ups and colors from an analysis of
+ground-based optical imaging data, will be presented by our collaborators.
+The first method we used to obtain information about the AGN ho st galaxy populations was
+the continuum model fitting described in §3.1. For each of our sources, we fit the continuum with
+a combination of a power law (representing the non-thermal c ontinuum) and a combination of a
+young, intermediate, and old single stellar population mod el, utilizing three different metallicities.
+Wethencreatedagridoftestcasestotesttheabilityofthec ontinuummodelstoaccurately describe
+the host galaxy spectrum, finding that metallicities could n ot be determined but that young stellar
+populations are clearly distinguished between both the int ermediate and old stellar populations
+(see§A). There is a degree of degeneracy between the intermediate and old populations as well
+which occurs when these populations are in combination with other populations (for example, a
+model of 50% intermediate and 50% young populations can be eq ually well modeled with a best fit
+continuum model that is a combination of young, intermediat e, and old populations).
+The main result of our continuum model fits is that the majorit y of the Swift BAT AGN in our
+sample have either a weak or no contribution from young stell ar populations and are dominated
+by intermediate/old populations. Of the sources with conti nuum light dominated by stellar popu-
+lations, only one source has 50% or more of their light domina ted by a young (25Myr) population
+– NGC 4151, whose host galaxy is a barred spiral (Sab) with a ri ng of star formation. This is in
+contrast with the 15 sources with 50% dominated by intermedi ate (2500Myr) populations and the
+23 with 50% or more dominated by old (10000Myr) populations. To these results, however, we
+must add the caveat that we measured the continua with very si mplified models. It is also possible
+that degeneracies between the power law and young stellar co mponent exist.
+Still, the result of the BAT AGN hosts being largely composed of intermediate to old popula-
+tions, is supported further through an analysis of the H δAand Dn(4000) stellar absorption indices.
+These age sensitive indicators, the former sensitive to rec ent star bursts and the latter to an indi-
+cator of old populations through measurement of the Ca IIbreak, reveal few sources (6 total) in
+the region of the H δA-Dn(4000) parameter space occupied by systems with significant contributions
+from young stellar populations ( /greaterorsimilar30%). Due to contamination of the absorption features from
+AGN emission lines (i.e. [Ne III]λ3869˚A and H δ), this result is based largely on the obscured
+sources.
+Based on an SDSS study by Kauffmann et al. (2003a), low luminosi ty narrow line AGN are
+hosted in old galaxies (as indicated by D n(4000)). This is consistent with the results of our study.
+Additionally, we find that the distribution of our narrow lin e sources in the H δA–Dn(4000) plot is
+consistent with the location of ‘strong’ AGN in the SDSS samp le (Figure 17 in Kauffmann et al.
+(2003a)). Since their definition of strong (3 .85×1040ergss−1in [OIII]λ5007˚A) includes the
+majority of our sources, this shows that our results are cons istent with the SDSS results. As shown– 21 –
+in Kauffmann et al. (2003a), the values of D n(4000) for our narrow line sources are indicative of
+normal late-type galaxies. This is also consistent with our analysis of the morphologies of the
+9-month sample AGNs, as listed in NED. In Winter et al. (2009a ), we had shown that the hosts of
+our sources (both Sy1 and Sy2 sources) were mostly spirals an d irregulars.
+Another result from the Kauffmann et al. (2003a) study, is a con nection between the age
+distribution of host galaxies and the [O III] luminosity of the AGN. In Figure 17, we plot each
+of the stellar age indicators (H δAand Dn(4000)) versus the extinction-corrected [O III] luminosity
+and the ratio L [OIII]/LEddfor both the narrow and broad line AGN. The top plots of this fig ure
+are comparable to Figure 12 of Kauffmann et al. (2003a) (whose L [OIII]measurements are in units
+of L⊙). We find no direct correlation between either of these stell ar absorption indices and either
+[OIII] luminosity or accretion rate ( R2/lessorsimilar0.1). Since our sources include only the equivalents of
+SDSS ‘strong’ AGN, it is not surprising that we do not see a cor relation. Our sample does not
+include weak AGN, which tend to have older populations (asso ciated with early type galaxies).
+Finally, we find a possible indication that the host galaxies of broad and narrow line sources
+may be different. Namely, we see differences in the metallicity i ndicator Mgb. Applying the
+Kolmogorov-Smirnov test, we find a P-value of 0.01, indicating that the populations are likely
+different. For the broad line sources, we find an average value o f 0.84 with a standard deviation of
+1.65 in Mgb. The narrow line sources have a much higher averag e Mgb measurement of 1.96 with
+a standard deviation of 2.27. Based on our simulations, lowe r values of Mgb also correspond to
+younger populations (see the top left panel of Figure 20). Th erefore, there is a degeneracy between
+age and metallicity such that the result of broad line source s having lower values of Mgb could
+indicate their hosts as either having a larger contribution from a younger population or from a
+lower metallicity than the hosts of narrow line sources.
+9. Conclusions
+AGN surveys are typically dominated by two selection effects: (1) dilution by starlight from
+the host galaxy and (2) obscuration by dust and gas in the host galaxy and/or the AGN itself (see
+Hewett & Foltz (1994); Mushotzky (2004)). For these reasons , an unbiased AGN sample is difficult
+to define. The Swift’s BAT AGN survey provides one of the first t ruly unbiased (to all but the
+highest column densities) samples of local AGNs.
+Since the BAT-detected sources are nearby, < z >= 0.03 (Tueller et al. 2008), they are ex-
+cellent targets for multi-wavelength follow-ups. In this p aper, we presented the optical spectral
+properties from sources detected in the first 9-months of the survey. Our analysis includes both the
+emission line properties of the AGN as well as the host proper ties revealed from intrinsic stellar ab-
+sorption features. The sample includes 40 spectra taken at t he 2.1-m KPNO telescope, 24 archived
+SDSS spectra, and the emission line properties of 13 sources presented in the literature. In total,
+this sample covers 81% of the Swift BAT AGN sources viewable f rom KPNO. It is comprised of– 22 –
+55% broad line sources and 45% narrow line sources, in the sam e ratio as the total Swift sample.
+With our unbiased AGN sample, it is important to compile the f undamental properties of the
+sources both as a test to our current understanding of AGN and as a comparison to more biased
+methods of detection (e.g. optical surveys). Using standar d emission line diagnostic plots, we find
+that the majority of our hard X-ray detected sources are opti cally Seyferts (66% of narrow line and
+75% of broad line sources). This contrasts with the opticall y selected SDSS sample examined by
+Kewley et al. (2006), which includes a large (75%) fraction o f HIIgalaxies with few Seyferts (3%).
+Since H IIgalaxies are less luminous than Seyferts in the X-ray band (R analli et al. 2003), it is not
+surprising that our hard X-ray flux limited sample detects th e more luminous local sources, which
+are Seyferts. In the same sense, the optical SDSS sample dete cts more LINERs, which are also less
+luminous sources than Seyferts, than we find in the Swift BAT s ample. In particular, we classify
+16% of the narrow line sources as LINERs and none of the broad l ine sources.
+One of the most fundamental properties of a black hole is its m ass. Under the unified AGN
+model, we expect to find no difference in the mass distribution b etween the broad and narrow line
+sources. Indeed, we find the distributions of our 2MASS deriv ed masses statistically consistent
+with being drawn from the same population. Comparing 2MASS d erived masses with a more
+accurate determination from the FWHM of H βin broad line sources, we find the masses from
+both methods are well correlated. The average value of our ha rd X-ray detected sources is <
+M/M⊙>= 107.87±0.66, with a range of values consistent with those found in previo us studies of
+AGNs (Woo & Urry 2002) and nearby PG QSOs (Veilleux et al. 2009 ).
+Determinations of the reddening from the ratio of narrow H α/Hβ, as well as gas densities and
+temperatures in the narrow line regions from diagnostic emi ssion lines of the same ion, are also
+consistent with both the unified model and previous results f rom optical studies. Under the unified
+model, we expect narrow line sources to have heavier extinct ion (assuming the extinction is on the
+nuclear/galactic scale and not simply from the torus), whil e other narrow line region properties
+like density and temperature to be the same for narrow and bro ad line sources. As expected, we
+find the average distribution of reddening values [E(B-V)] h igher in the narrow line sources. In
+our calculations of the gas density in the S+emission region, we find the same electron densities
+ofNe≈103cm−3for broad and narrow line sources. Superficially, the O+2region appears at a
+higher temperature for the broad line sources. However, as d iscussed in Osterbrock (1978), a likely
+explanation is that the narrow and broad line sources both ha ve similar temperatures (we find
+Te≈10000K), but in the broad line sources we are able to probe [O III]λ4363 emitting gas into
+denser regions of the narrow line region.
+Based on the results of our X-ray study of our unbiased AGN sam ple, we suspect that the
+distributions of luminosities of the Swift AGN conflict with the unified model. Namely, our X-ray
+results (Winter et al. 2009a) showed that the absorbed/type 2 AGNs (X-ray absorbed/optically
+narrow line sources, including optically classified Sy2s, L INERs, and H IIgalaxies) have lower ab-
+sorption corrected 2–10keV luminosities and accretion rat es. These same trends are found among– 23 –
+the optically derived luminosities and accretion rates. Sp ecifically, we find average extinction-
+corrected 5007 ˚A [OIII] luminosities of 1041.74±0.93ergss−1and 1040.94±1.00ergss−1and ratios of
+L[OIII]/LEddof 10−4.61±0.85and 10−5.25±0.81, respectively for broad and narrow line sources. Con-
+trary to the results of Heckman et al. (2005), but in agreemen t with Mel´ endez et al. (2008), we find
+that the 14–195keV BAT luminosities are only weakly correla ted with [O III] luminosity for broad
+and narrow line sources.
+Seemingly, the result of narrow line sources having lower lu minosities and possibly accretion
+rates (depending on the bolometric corrections) poses a cha llenge to the unified model. On closer
+inspection, we find that the narrow line sources with optical classifications as Seyferts have similar
+X-ray and optical luminosities to their broad line, Seyfert 1 counterparts. Instead, it is the sources
+optically classified as LINERs and H II/composite/ambiguous sources which have lower luminosi-
+ties. While these sources are clearly detected AGN based on t heir X-ray properties, modification
+of the unified model to include a luminosity dependence is cle arly required to link these fainter
+non-Seyfert sources with the Seyfert 1s and 2s.
+Finally, through our continuum model fits and measurements o f stellar absorption indices, we
+can make a few general comments on the host galaxy properties of our sources. We find that
+the stellar ages of the hosts include small contributions fr om young populations (0.25Gyr). The
+populations are more consistent with intermediate/old (2. 5–10Gyr) populations. Comparing with
+the results drawn from the SDSS survey, we find that our narrow line sources have the same
+properties as the ‘strong’ narrow line AGN from Kauffmann et al . (2003a). Therefore, their stellar
+absorption properties (from the Ca IIbreak and H δabsorption) are like those of late type galaxies.
+This is also consistent with the NED morphologies of our sour ces (both Sy 1s and Sy 2s), which
+are mostly spirals and irregulars (Winter et al. 2009a).
+The authors would like to greatly thank Christy Tremonti for use of her analysis codes and
+discussions on how to modifythem for application to our AGN s ources. L.W. acknowledges support
+by NASA grant NNX08AC14G. Also, she acknowledges support th rough NASA grant HST-HF-
+51263.01-A, through a Hubble Fellowship from the Space Tele scope Science Institute, which is
+operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA
+contract NAS5-26555. S.V. acknowledges support from a Seni or Award from the Alexander von
+HumboldtFoundation andthanksthehost institution, MPE Ga rching, wheresomeof this work was
+performed. K.T.L. acknowledges support from the NASA Postd octoral Program Fellowship (NNH
+06CC03B). The Kitt Peak National Observatory observations were obtained using MD-TAC time
+as part of the thesis of L.W. at the University of Maryland (fo r programs 0322 and 0107) and M.K.
+(program 0295). Kitt Peak National Observatory, National O ptical Astronomy Observatory, is
+operated by the Association of Universities for Research in Astronomy (AURA) under cooperative
+agreement with the National Science Foundation.
+Facilities: Swift (), KPNO:2.1m (), Sloan ()– 24 –
+A. Galaxy Continuum Spectral Fits
+In this section, we detail additional tests that we conducte d to test the accuracy of the galaxy
+continuum fits. As explained in §3.1, we used a grid of single stellar age population models
+(Bruzual & Charlot 2003) with 3 different ages (25, 2500, and 10 000 Myr) and at 3 different metal-
+licities (2.5Z ⊙, Z⊙, and 0.2 Z ⊙) to fit the continua of our AGN and template galaxy spectra. In
+order to test the accuracy of these models, we constructed a g rid of test spectra, broadening the
+sources by assuming FWHM = 300kms−1(σ≈128kms−1) and adding both random noise and
+reddening ( τ= 1.5) to the stellar population models used in the continuum fits . This grid of
+models includes a young, intermediate, and old population, as well as the following combinations:
+50% young + 50% intermediate, 50% young + 50% old, 50% interme diate + 50% old, and 33%
+young + 33% intermediate + 33% old. In Figure 18, we plot sever al of these test spectra.
+Assuming an error of 10% in the flux, we fit each of the test spect ra with the model spectra
+used to fit the continua of our target spectra. The results of t hese fits are shown in Table 16.
+Our results show that the velocity dispersions are accurate ly measured by the models in all cases.
+The metallicities, however, are not since all of our test spe ctra have solar metallicity but a range of
+values are found from the fitting process. We also find that the young stellar population component
+is measured well, though its contribution is underestimate d by up to about 20%. Finally, we find
+that there is a degeneracy between the intermediate and old p opulations when they are found in
+combination with the young population. This is well illustr ated in Figure 18, where there is little
+difference between the 50% young + 50% intermediate and 50% you ng + 50% old spectra. We do,
+however, find that a 100% intermediate population is disting uishable from a 100% old population.
+Therefore, the main conclusion that we draw from our test spe ctra is that our continuum model
+fits can clearly distinguish between young and intermediate /old populations. See Kauffmann et al.
+(2003b) for more detailed investigations used to study the S DSS host galaxies.
+In a similar manner, we also created test spectra to look for d egeneracies between the stellar
+continuum and power law component. To accomplish this, we us ed the same set of test galaxy
+models as above. For each of these test spectra, we added a pow er law component with an index
+(p1) set to the average value determined from fits to our AGN sourc es (0.67). We constrained the
+values such that the light fraction from both the stellar lig ht and power law contributed 50% of
+the light at 5500 ˚A. Results of these fits are presented in Table 17. We find that t here is no obvious
+degeneracy between any of the stellar population models and the power law component (at least
+at this power law index). The average fitted power law index, 0 .74, is slightly higher than the true
+value while the fitted fraction of light contributed from the power law tends to be slightly lower
+than the true value (most of the values are between 0.41–0.45 instead of 0.50). Generally, we find
+that the fitted values are consistent with the input paramete rs.
+In addition to testing the accuracy of the continuum fits, we a lso used our grid of test galaxy
+spectra (excluding a power law contribution) to interpret t he results of measurements of stellar
+absorption indices in our target spectra. In addition to usi ng the grid of solar models we described– 25 –
+above, we created grids of populations with metallicities 2 .5 and 0.20 times solar abundance. For
+all of these sources, we measured the stellar absorption ind ices in the same manner as for our
+target spectra (see §3.3). In this way, we can use our test spectra, which are of app roximately the
+same signal-to-noise as our target spectra, to understand t he results of an analysis of the stellar
+absorption indices.
+In Figure 19, we plot the H δAindex versus D n(4000) for our test spectra. As we described in
+§3.3, these two indices are commonly used as indicators of the age of stellar populations. As the
+plot shows, metallicity of the stellar population models do es not have a large effect on these stellar
+absorption indices. Further, as expected, there is a clear d ependence on age, where populations
+with a significant (33% or higher) contribution from a young p opulation have both the highest
+values of H δA, associated with recent bursts of star formation, and the lo west values of D n(4000),
+which indicates the strength of the Ca IIbreak. We find that the populations with significant
+contributions of young populations have H δA>2 and D n(4000)<1.2.
+InFigure20,weplotadditionalstellarabsorptionindices oftenusedasmetallicity indicators(as
+wellastheCa IIbreakageindicator)forthetestgalaxyspectra. Eachmetal licity isrepresentedwith
+a different color, with the same grid of different stellar popula tion components mentioned above.
+We point out that, as shown in the D n(4000) versus Mgb plot, that the populations with significan t
+contributionsfromyoungstars( <1.2) tendtohavelower valuesofMgb( /lessorsimilar3). Distinctionsbetween
+intermediate/old higher metallicity (Z ⊙and 2.5 Z ⊙) and low metallicity (0.2 Z ⊙) populations are
+seen in the C 24668 vs. Mgb, [MgFe] vs. Mgb and <Fe>vs. Mgb plots – where higher values in
+x and y parameters are seen for the higher metallicity popula tions. For young populations of any
+metallicity, it is more difficult to distinguish between differ ent metallicity populations.
+B. Notes on Individual Spectra
+In this section, we include notes on the emission line spectr a of the sources examined. These
+notes particularly relate to peculiarities in the spectra o r the fitting procedurefor sources indicated.
+For 10 broad line sources (or ≈1/3 of the broad line sources), absorption lines from the Na ID
+doubletλ5890,5896˚A are seen. These absorption features are seen in Mkn 1018, Mk n 590, MCG
+-01-13-025, Mrk 6, SDSS J090432.19+553830.1, NGC 3227, NGC 3516, NGC 4593, MCG +09-21-
+096, NGC 5548, and RX J2135.9+4728. In the spectrum of MCG +09 -21-096, the absorption line
+is embedded in a broad He I(FWHM ≈2270kms−1) emission line. The Na ID doublet was also
+detected (by eye) in 9 narrow line sources: NGC 788, Mkn 18, SD SS J091129.97+452806.0, SDSS
+J091800.25+042506.2, Ark 347, NGC 4102, NGC 6240, UGC 11871 , and NGC 7319.
+Additionally:
+BROAD LINE SOURCES:– 26 –
+LEDA 138501: Hβhas a “red” wing.
+MCG -01-13-025, Mrk 1018, NGC 3227: Strong intrinsic absorption lines are seen in the
+spectra of these broad line sources. He Iis seen in absorption for both sources.
+IRAS 05589+2828: There is a clear broad component to He IIλ4686˚A. Hβhas a red wing.
+MCG +04-22-042: There is a clear broad component to He IIλ4686˚A.
+SBS 1136+594: There is a clear broad component to He IIλ4686˚A.
+UGC 6728: Two narrow emission lines are present for each of the [O III] emission lines (at
+λ4959˚Aandλ5007˚A).
+NGC 4593: Two narrow emission lines are present for each of the [O III] emission lines (at
+λ4959˚Aandλ5007˚A).
+MCG +09-21-096: The profiles of the broad Balmer lines are complex, with broad “boxy”
+shapes (including H δ, Hγ, Hβ, and Hα).
+Mrk 813: Hβis blended with the nearby [O III] emission lines.
+4C +74.26 : Hβis extremely broad and blended with the nearby [O III] emission lines at this
+resolution.
+NARROW LINE SOURCES:
+Mkn 18: Both the KPNO and SDSS spectra show an additional broad compo nent to H αof
+approximately 370kms−1.
+Ark 347: The Hαregion is quite complex. Six distinct narrow lines are seen i n the region
+including [N II]λ6548˚A, Hα, and [N II]λ6583˚A. The measured wavelengths of these lines are:
+6546.72±0.17,6555.7±0.34,6563.25±0.19,6570.23±0.08,6582.42±0.20, and 6591 .70±0.09˚A.
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+Table 1. SWIFT BAT-detected AGN
+Source RA (h m s) Dec (d m s) Redshift E(B-V)1Type2Host Galaxy2Obs.3
+Mkn 352 00:59:53.3 +31:49:36.8 0.015 0.06 Sy 1 SA0 Lit.
+NGC 788 02:01:06.5 – 06:48:55.8 0.014 0.03 Sy 2 SA(s)0/a KPNO
+Mkn 1018 02:06:16.0 – 00:17:29.2 0.043 0.03 Sy 1.5 S0; merger SDSS
+LEDA 138501 02:09:34.3 +52:26:33.0 0.049 0.16 Sy 1 KPNO
+Mkn 590 02:14:33.6 – 00:46:00.3 0.026 0.04 Sy 1.2 SA(s)a SDSS
+NGC 931 02:28:14.5 +31:18:42.1 0.017 0.10 Sy 1.5 Sbc Lit.
+2MASX J03181899+6829322 03:18:19.0 +68:29:31.6 0.090 0.7 2 Sy 1.9 KPNO
+NGC 1275 03:19:48.2 +41:30:42.1 0.018 0.16 Sy 2 NLRG Lit.
+3C 105 04:07:16.5 +03:42:25.8 0.089 0.48 NLRG KPNO
+3C 111 04:18:21.3 +38:01:35.8 0.049 1.65 Sy 1 N KPNO
+2MASX J04440903+2813003 04:44:09.0 +28:13:01.0 0.011 0.8 5 (Sy2) S KPNO
+MCG -01-13-025 04:51:41.5 – 03:48:33.7 0.016 0.04 Sy 1.2 SAB (s)0+ pec KPNO
+1RXS J045205.0+493248 04:52:05.0 +49:32:45.0 0.029 0.73 S y 1 KPNO
+NGC 2110 05:52:11.4 – 07:27:22.3 0.008 0.38 Sy 2 SAB0- Lit.
+MCG +08-11-011 05:54:53.6 +46:26:22.0 0.021 0.22 Sy 1.5 SB0 KPNO
+IRAS 05589+2828 06:02:10.7 +28:28:22.1 0.033 0.43 Sy 1 KPNO
+Mkn 3 06:15:36.3 +71:02:15.0 0.014 0.19 Sy 2 S0 KPNO
+2MASX J06411806+3249313 06:41:18.0 +32:49:31.4 0.048 0.1 5 Sy 2 KPNO
+Mkn 6 06:52:12.2 +74:25:37.0 0.019 0.14 Sy 1.5 SAB0+ KPNO
+Mkn 79 07:42:32.8 +49:48:34.8 0.022 0.07 Sy 1.2 SBb KPNO
+Mkn 18 09:01:58.4 +60:09:06.2 0.011 0.04 (HII/Ambig.) S? SD SS, KPNO
+SDSS J090432.19+553830.1 09:04:32.2 +55:38:30.3 0.037 0. 02 (Sy1.5) SDSS
+SDSS J091129.97+452806.0 09:11:30.0 +45:28:06.0 0.027 0. 02 (Sy2) SDSS
+SDSS J091800.25+042506.2 09:18:00.3 +04:25:06.2 0.156 0. 04 (Sy2) SDSS
+MCG -01-24-012 09:20:46.3 – 08:03:22.1 0.020 0.03 Sy 2 SAB(r s)c KPNO
+MCG +04-22-042 09:23:43.0 +22:54:32.6 0.033 0.04 Sy 1.2 SDS S, KPNO
+Mkn 110 09:25:12.9 +52:17:10.3 0.035 0.01 Sy 1 Pair? SDSS
+NGC 3227 10:23:30.6 +19:51:54.0 0.004 0.02 Sy 1.5 SAB(s) pec KPNO, Lit.
+Mkn 417 10:49:30.9 +22:57:52.4 0.033 0.03 Sy 2 Sa SDSS, KPNO
+NGC 3516 11:06:47.5 +72:34:07.0 0.009 0.04 Sy 1.5 (R)SB(s) K PNO, Lit.
+1RXS 112716.6+190914 11:27:16.3 +19:09:20.2 0.106 0.02 Sy 1.8 KPNO
+SBS 1136+594 11:39:09.0 +59:11:54.8 0.061 0.01 Sy 1.5 SDSS
+UGC 06728 11:45:16.0 +79:40:53.0 0.007 0.10 Sy 1.2 SB0/a KPN O
+CGCG 041-020 12:00:57.9 +06:48:23.1 0.036 0.02 (Sy2) SDSS
+NGC 4051 12:03:09.6 +44:31:52.7 0.002 0.01 Sy 1.5 SAB(rs)bc KPNO, Lit.
+Ark 347 12:04:29.7 +20:18:58.4 0.022 0.03 Sy 2 S0: pec SDSS, K PNO
+NGC 4102 12:06:23.0 +52:42:39.8 0.003 0.02 LINER SAB(s)b? K PNO, Lit.
+NGC 4138 12:09:29.8 +43:41:07.1 0.003 0.01 Sy 1.9 SA(r)0+ Li t.
+NGC 4151 12:10:32.6 +39:24:20.6 0.003 0.03 Sy 1.5 (R’)SAB(r s)ab KPNO, Lit.
+Mkn 766 12:18:26.5 +29:48:46.3 0.013 0.02 Sy 1.5 (R’)SB(s)a KPNO
+NGC 4388 12:25:46.7 +12:39:42.8 0.009 0.03 Sy 2 SA(s)b SDSS, Lit.
+NGC 4395 12:25:48.9 +33:32:48.7 0.001 0.02 Sy 1.8 SA(s)m SDS S, Lit.
+NGC 4593 12:39:39.4 – 05:20:39.3 0.009 0.03 Sy 1 (R)SB(rs)b K PNO
+MCG +09-21-096 13:03:59.5 +53:47:30.1 0.030 0.02 Sy 1 KPNO
+NGC 4992 13:09:05.6 +11:38:02.9 0.025 0.03 (LINER) Sa SDSS
+NGC 5252 13:38:15.9 +04:32:33.3 0.023 0.03 Sy 1.9 S0 SDSS– 31 –
+Table 1—Continued
+Source RA (h m s) Dec (d m s) Redshift E(B-V)1Type2Host Galaxy2Obs.3
+NGC 5506 14:13:14.9 – 03:12:27.4 0.006 0.06 Sy 1.9 Sa pec SDSS
+NGC 5548 14:17:59.6 +25:08:12.7 0.017 0.02 Sy 1.5 (R’)SA(s) 0/a SDSS, Lit.
+Mkn 813 14:27:25.1 +19:49:51.5 0.111 0.03 Sy 1 KPNO
+Mkn 841 15:04:01.2 +10:26:16.0 0.036 0.03 Sy 1.5 E KPNO
+Mkn 290 15:35:52.4 +57:54:09.5 0.030 0.02 Sy 1 E1? SDSS
+Mkn 1498 16:28:04.0 +51:46:31.0 0.055 0.03 Sy 1.9 KPNO
+NGC 6240 16:52:58.9 +02:24:03.0 0.025 0.08 Sy 2 I0: pec KPNO
+1RXS J174538.1+290823 17:45:38.2 +29:08:22.0 0.111 0.05 ( Sy1) KPNO
+3C 382 18:35:03.4 +32:41:46.8 0.058 0.07 Sy 1 KPNO
+NVSS J193013+341047 19:30:13.3 +34:10:47.0 0.063 0.19 (Sy 1.5) KPNO
+1RXS J193347.6+325422419:33:47.6 +32:54:22.0 0.030 0.27 (BL COMP) KPNO
+3C 403 19:52:15.8 +02:30:24.5 0.059 0.19 NLRG S0 KPNO
+Cygnus A 19:59:28.3 +40:44:02.0 0.056 0.38 Sy 2 S?; Radio gal . KPNO
+MCG +04-48-002 20:28:35.1 +25:44:00.0 0.014 0.45 Sy 2 S KPNO
+4C +74.26 20:42:37.3 +75:08:02.0 0.104 0.44 Sy 1 KPNO
+IGR 21247+5058 21:24:38.1 +50:58:58.0 0.020 2.43 Sy 1 KPNO
+RX J2135.9+4728 21:35:55.0 +47:28:23.2 0.025 0.62 Sy 1 KPNO
+UGC 11871 22:00:41.4 +10:33:08.7 0.027 0.06 Sy 1.9 Sb KPNO
+NGC 7319 22:36:03.5 +33:58:33.0 0.023 0.08 Sy 2 SB(s)bc pec K PNO
+3C 452 22:45:48.8 +39:41:15.7 0.081 0.14 NLRG KPNO
+Mkn 926 23:04:43.5 – 08:41:08.6 0.047 0.04 Sy 1.5 SDSS
+1Milky Way reddening values, E(B-V), obtained from NED.
+2AGN type and host galaxy type from NED, Tueller et al. (2008), and the results of this paper. For AGN types, optical
+identifications are listed, where available. Values in pare ntheses indicate classifications from this paper, where ’Sy 1’ is a source
+with broad emission lines and narrow line emission consiste nt with a Seyfert and ’Sy 2’ is a source without broad emission lines and
+with narrow line emission consistent with a Seyfert. Sub-cl assifications were made (i.e. Sy1.5) following the criteria of Osterbrock
+(1981) (based upon the ratio of the broad to narrow component s of Hαand Hβ. Where “Gal” is indicated, there are no optical
+emission lines indicative of the presence of an AGN. The opti cal spectrum looks like a galaxy spectrum. Additional host g alaxy
+classifications were obtained from the LEDA database. Where “?” is indicated, there is no available classification.
+3Observation type from Sloan Digital Sky Survey archive (SDS S) or our Kitt Peak Observations (KPNO). Sources with line
+ratios that we have obtained in the literature are indicated by (Lit.).
+4This source was initially included in the 9-month catalog ba sed on an earlier method of source selection. However, with
+subsequent analysis it fell below the 4.8 σdetection threshold. It is detected above 5 σand included in the 22-month BAT survey.– 32 –
+Table 2. Details of KPNO Observations
+Source Grating UT Date Exp. (s) x (kpc)†y (kpc)†B?∗
+NGC 788 35 2006-11-20 3600 0.53 4.49
+LEDA 138501 26new 2006-11-17 1800 1.93 5.53 B
+LEDA 138501 35 2006-11-19 1800 1.93 7.53 B
+3C 111 35 2006-11-20 2700 1.90 7.42 B
+2MASX J04440903+2813003 26new 2006-11-17 3380 0.44 2.25
+2MASX J04440903+2813003 35 2006-11-19 3601 0.44 1.71
+MCG -01-13-025 26new 2006-11-18 3600 0.62 2.79 B
+MCG -01-13-025 35 2006-11-19 3601 0.62 2.79 B
+1RXS J045205.0+493248 26new 2006-11-18 3600 1.13 3.21 B
+1RXS J045205.0+493248 35 2006-11-20 3600 1.13 4.27 B
+2MASX J06411806+3249313 35 2006-11-20 5400 1.88 8.71
+Mkn 79 26new 2007-04-14 1200 0.87 3.62 B
+Mkn 79 35 2007-04-15 2401 0.87 3.38 B
+Mkn 18 26new 2006-11-18 3601 0.43 2.87
+Mkn 18 35 2006-11-19 3600 0.43 2.34
+MCG -01-24-012 26new 2007-04-14 1200 0.76 3.77
+MCG -01-24-012 35 2007-04-15 1199 0.76 2.98
+MCG +04-22-042 26new 2007-04-14 1200 1.30 4.32 B
+MCG +04-22-042 35 2007-04-15 1200 1.30 4.16 B
+Mkn 417 35 2006-11-20 3600 1.28 5.00
+1RXS J1127166.6+190914 26new 2007-04-14 1200 4.19 24.62
+1RXS J1127166+190914 35 2007-04-15 2399 4.19 15.32
+UGC06728 26new 2006-11-18 3000 0.25 0.81 B
+UGC06728 35 2006-11-19 3600 0.25 0.89 B
+Ark 347 26new 2007-04-14 1199 0.87 6.05
+Ark 347 35 2007-04-15 1200 0.87 4.45
+NGC 4593 26new 2007-04-14 1200 0.35 1.77 B
+NGC 4593 35 2007-04-15 1200 0.35 2.24 B
+MCG +09-21-096 26new 2007-04-14 2399 1.17 3.90 B
+MGC +09-21-096 35 2007-04-15 2400 1.17 4.56 B
+Mkn 813 26new 2007-06-15 2399 4.40 15.92 B
+Mkn 813 35 2007-06-17 2700 4.40 12.75 B
+Mkn 841 26new 2007-06-15 2399 1.42 5.56 B
+Mkn 841 35 2007-06-17 2700 1.42 4.59 B
+Mrk 1498 26new 2007-06-16 3600 2.15 8.38
+Mrk 1498 35 2007-06-18 3600 2.15 12.31
+NGC 6240 26new 2007-06-15 2400 0.96 3.73
+NGC 6240 35 2007-06-17 2701 0.96 7.65– 33 –
+Table 2—Continued
+Source Grating UT Date Exp. (s) x (kpc)†y (kpc)†B?∗
+1RXS J174538.1+290823 26new 2007-04-14 2400 4.43 17.28 B
+1RXS J174538.1+290823 35 2007-04-15 3601 4.43 17.86 B
+3C 382 26new 2007-06-15 3599 2.28 8.88 B
+3C 382 35 2007-06-17 3601 2.28 27.10 B
+3C 382 35 2007-06-18 2700 2.28 7.31 B
+NVSS J193013+341047 26new 2007-06-16 3600 2.48 8.56 B
+NVSS J193013+341047 35 2007-06-18 3602 2.48 6.60 B
+1RXS J193347.6+325422 26new 2007-06-15 2399 1.17 3.17 B
+1RXS J193347.6+325422 35 2007-06-17 2700 1.17 4.27 B
+3C 403 26new 2006-11-17 3600 2.32 18.93
+3C 403 26new 2006-11-18 1918 2.32 10.21
+3C 403 35 2006-11-19 3599 2.32 9.05
+3C 403 35 2006-11-20 1800 2.32 9.23
+Cyg A 26new 2007-06-16 2401 2.21 8.60
+Cyg A 35 2007-06-18 2699 2.21 13.76
+MCG +04-48-002 26new 2006-11-18 1801 0.54 2.11
+MCG+04-48-002 35 2006-11-19 3601 0.54 2.11
+4C +74.26 26new 2007-06-16 2400 4.13 11.16 B
+4C +74.26 35 2007-06-18 2698 4.13 24.36 B
+IGR 21247+5058 26new 2007-06-15 2400 0.78 3.04 B
+IGR 21247+5058 35 2007-06-17 2701 0.78 2.86 B
+UGC 11871 26new 2006-11-17 1800 1.04 4.73
+UGC 11871 26new 2006-11-18 1800 1.04 4.36
+UGC 11871 35 2006-11-19 1800 1.04 4.84
+UGC 11871 35 2006-11-20 1800 1.04 4.35
+NGC 7319 26new 2007-06-15 1422 0.88 3.42
+NGC 7319 35 2007-06-17 2736 0.88 5.84
+2MASX J03181899+6829322 32 2006-11-21 3599 3.57 15.28
+3C 105 32 2006-11-21 3599 3.53 10.87
+MCG+08-11-011 32 2008-12-04 900 0.80 6.23 B
+IRAS 05589+2828 32 2006-11-21 1800 1.29 1.79 B
+Mkn 3 32 2008-12-04 540 0.53 4.10
+Mkn 6 32 2008-12-04 900 0.73 5.71 B
+NGC 3227 32 2009-04-17 900 0.15 0.46 B
+NGC 3516 32 2009-04-17 900 0.34 1.53 B
+NGC 4051 32 2009-04-17 900 0.09 0.30
+NGC 4102 32 2009-04-17 900 0.11 0.58
+NGC 4151 32 2009-04-17 900 0.13 0.43 B– 34 –
+Table 2—Continued
+Source Grating UT Date Exp. (s) x (kpc)†y (kpc)†B?∗
+Mrk 766 32 2009-04-17 900 0.50 2.29
+NVSS 19013+341047 32 2006-11-21 3600 2.48 11.97 B
+RX J2135.9+4728 32 2006-11-21 3600 0.98 7.31 B
+3C 452 32 2006-11-21 5401 3.21 18.76
+†An estimate of the extraction aperture along the slit in both x and y is given in units
+of kpc. The x value is calculated as the fixed 2′′slit size, converted to kpc using the
+redshift to the source. The y value is obtained from the aperture s ize used to extract the
+individual spectrum.
+∗B indicates the presence of broad lines (particularly H I Balmer lines) f rom a visual
+inspection of the spectra.– 35 –
+Table 3. Details of KPNO Observations of Template Galaxies
+Source Grating UT Date Exp. (s) x (kpc)†y (kpc)†
+NGC 205 26new 2006-11-17 2101 0.03 0.09
+NGC 205 35 2006-11-19 1800 0.03 0.19
+NGC 221 26new 2006-11-18 1800 0.03 0.16
+NGC 221 35 2006-11-20 1799 0.03 0.15
+NGC 628 26new 2006-11-17 3600 0.09 0.65
+NGC 628 35 2006-11-19 1800 0.09 1.01
+NGC 1023 26new 2006-11-17 1799 0.08 0.67
+NGC 1023 35 2006-11-20 1800 0.08 0.68
+NGC 3384 26new 2006-11-17 1799 0.09 0.66
+NGC 3884 35 2006-11-20 1801 0.09 0.82
+NGC 3640 26new 2006-11-17 1800 0.16 1.52
+NGC 3640 35 2006-11-20 1800 0.16 1.91
+NGC 4914 26new 2007-04-14 1200 0.61 3.01
+NGC 4914 35 2007-04-15 1200 0.61 3.71
+NGC 5308 26new 2007-06-16 1801 0.26 1.83
+NGC 5308 35 2007-06-18 1800 0.26 5.50
+NGC 5557 26new 2007-04-14 1200 0.42 3.36
+NGC 5557 35 2007-04-15 1200 0.42 4.21
+NGC 5638 26new 2007-06-16 1800 0.22 1.77
+NGC 5638 35 2007-06-18 1800 0.22 5.57
+NGC 6654 26new 2007-06-16 1800 0.24 1.26
+NGC 6654 35 2007-06-18 1799 0.24 3.53
+†An estimate of the extraction aperture along the slit in both x and
+y is given in units of kpc.– 36 –
+Table 4. Details of SDSS Observations
+Source UT Date Exp. (s) Plate Tile d (kpc)†B?∗
+Mkn 1018 2000-09-25 2700 404 193 2.53 B
+Mkn 590 2003-01-08 4200 1073 9328 1.52 B
+Mkn 18 2007-12-05 4204 1785 1290 0.64
+SDSS J090432.19+553830.1 2000-12-29 9000 450 238 2.17 B
+SDSS J091129.97+452806.0 2002-02-07 4803 832 603 1.58
+SDSS J091800.25+042506.2 2003-03-09 3000 991 763 9.40
+MCG +04-22-042 2005-12-23 4800 2290 1658 1.94 B
+Mkn 110 2001-12-09 4803 767 553 2.05 B
+Mkn 417 2006-12-16 5884 2481 1736 1.94
+SBS 1136+594 2002-05-06 5408 952 724 3.60 B
+CGCG 041-020 2005-01-15 3100 1622 1147 2.11
+Ark 347 2008-01-06 6008 2608 1821 1.29
+NGC 4388 2004-06-10 2400 1615 1140 0.52
+NGC 4395 2006-03-25 3000 2015 1471 0.06
+NGC 4992 2004-04-21 2100 1696 1212 1.46
+NGC 5252 2002-04-10 2701 853 624 1.35
+NGC 5506 2002-04-14 2646 916 688 0.35
+NGC 5548 2006-05-04 2500 2127 1533 0.99 B
+Mkn 290 2002-03-14 3904 615 400 1.76 B
+Mkn 926 2001-12-15 3304 725 503 2.77 B
+†The diameter of the aperture size for SDSS (3′′) in kpc.
+∗B indicates the presence of broad lines (particularly H I Balmer lines) f rom a visual
+inspection of the spectra.– 37 –
+Table 5. Stellar Light Fits to the Galaxy Templates
+Galaxy Type∗vdisp∗FWHM†Z†Lfyoung†Lfinterm†Lfold†
+NGC 205 E5 pec 40.8 300 0 .2Z⊙0.02 0.98 –
+NGC 221 cE2 71.8 170 Z⊙ 0.05 – 0.95
+NGC 628 SA(s)c 72.2 270 Z⊙ 0.07 0.40 0.53
+NGC 1023 SB(rs)0- 204.5 300 2 .5Z⊙– 0.02 0.98
+NGC 3384 SB(s)0- 148.4 300 Z⊙ – – 1.00
+NGC 3640 E3 181.6 430 2 .5Z⊙– 0.10 0.90
+NGC 4914 E+ 223.6 330 2 .5Z⊙– 0.46 0.53
+NGC 5308 S0- 227.2 370 Z⊙ – – 1.00
+NGC 5557 E1 253.0 400 2 .5Z⊙– 0.27 0.73
+NGC 5638 E1 165.0 270 Z⊙ – – 1.00
+NGC 6654 (R’)SB(s)0/a 157.8 270 Z⊙ – 0.05 0.95
+∗The galaxy type was obtained from NED while the central velocity disp ersion was found
+in LEDA. Typical errors on the central velocity dispersion are of th e order 5kms−1. These
+templates were selected from the non-active galaxy templates liste d in Ho et al. (1997)
+†ThefittedvaluesusingthestellarpopulationmodelsofBruzual & Cha rlot(2003)include
+the FWHM (kms−1), metallicity ( Z), and light fractions ( Lf) at 5500 ˚A using populations
+at 25 (young), 2500 (interm), and 10000 (old) Myr. A dash indicate s no contribution from
+the indicated component.– 38 –
+Table 6. Stellar Light Fits to the AGN Sources
+Source FWHM†Z†p0†p1†Lfpow†Lfyoung†Lfinterm†Lfold†χ2/dof
+KPNO Spectra
+NGC 788 200 2.5 Z⊙1.00 0.77 0.73 0.05 0.09 0.12 10.3
+LEDA 138501 400 0.2 Z⊙0.56 0.37 1.00 – – – 1.0
+2MASX J03181899+6829322 200 0.2 Z⊙– – – – 1.00 – 17.9
+3C 105 170 0.2 Z⊙– – – – 1.00 – 25.3
+3C 111 200 2.5 Z⊙0.32 0.96 1.00 – – – 2840
+2MASX J04440903+2813003 460 2.5 Z⊙0.01 1.58 1.00 – – – 101
+MCG -01-13-025 330 2.5 Z⊙– – – 0.06 – 0.94 3.0
+MCG +04-22-042 260 0.2 Z⊙– – – – – 1.00 4.9
+1RXS J045205.0+493248 460 Z⊙0.26 0.75 1.00 – – – 107
+MCG +08-11-011 50 0.2 Z⊙– – – 0.25 0.75 – 60.1
+IRAS 05589+2828 400 2.5 Z⊙0.00‡1.43 0.78 0.22 – – 30.3
+Mkn 3 50 0.2 Z⊙– – – – 1.00 – 85.8
+2MASX J06411806+3249313 200 0.2 Z⊙0.53 0.83 1.00 – – – 2.7
+Mkn 6 430 0.2 Z⊙1.00 0.44 0.16 – 0.84 – 95.6
+Mkn 79 460 Z⊙1.00 0.51 1.00 – – – 3.4
+Mkn 18 460 2.5 Z⊙– – – 0.35 0.58 0.08 4.7
+MCG -01-24-012 400 Z⊙1.00 0.02 0.02 – – 0.98 2.5
+MCG +04-22-042 260 0.2 Z⊙– – – – – 1.00 4.9
+NGC 3227 50 0.2 Z⊙– – – 0.46 0.26 0.28 15.2
+Mkn 417 200 2.5 Z⊙0.11 1.09 0.98 – 0.01 0.01 2.8
+NGC 3516 50 0.2 Z⊙– – – – 0.32 0.68 21.9
+1RXS J1127166+190914 270 2.5 Z⊙0.22 1.06 0.99 – 0.01 – 5.0
+UGC 6728 460 2.5 Z⊙1.00 0.32 0.33 – – 0.67 6.1
+NGC 4051 300 0.2 Z⊙1.00 0.54 0.39 – – 0.61 12.5
+Ark 347 300 Z⊙– – – – – 1.00 1.7
+NGC 4102 50 0.2 Z⊙– – – – 0.28 0.72 21.9
+NGC 4151 50 0.2 Z⊙– – – 0.71 – 0.29 92.7
+Mkn 766 360 0.2 Z⊙1.00 0.45 0.39 – – 0.61 15.4
+NGC 4593 460 Z⊙– – – 0.19 – 0.81 8.2
+MCG +09-21-096 230 0.2 Z⊙– – – – – 1.00 0.8
+Mkn 813 460 2.5 Z⊙1.00 0.19 0.17 0.15 – 0.68 1.1
+Mkn 841 400 Z⊙0.08 0.62 1.00 – – – 2.9
+Mkn 1498 460 Z⊙– – – – – 1.00 1.7
+NGC 6240 460 Z⊙1.00 0.23 0.01 – – 0.99 8.1
+1RXS J174538.1+290823 400 0.2 Z⊙0.00‡2.89 0.94 0.06 – – 3.7
+3C 382 460 2.5 Z⊙0.56 0.00 0.08 – 0.92 – 0.3– 39 –
+Table 6—Continued
+Source FWHM†Z†p0†p1†Lfpow†Lfyoung†Lfinterm†Lfold†χ2/dof
+NVSS J193013+341047 130 0.2 Z⊙1.00 0.45 0.44 – 0.10 0.45 2.2
+1RXS J193347.6+325422 460 Z⊙0.00 2.01 1.00 – – – 12.9
+3C 403 270 2.5 Z⊙1.00 0.53 0.72 – 0.09 0.19 1.8
+Cygnus A 270 Z⊙0.56 0.00 0.01 – – 0.99 6.3
+MCG +04-48-002 400 2.5 Z⊙– – – 0.29 0.53 0.18 17.3
+4C +74.26 430 0.2 Z⊙1.00 0.52 0.07 – – 0.93 90.6
+IGR 21247+5058 230 2.5 Z⊙1.00 0.61 0.22 – 0.78 – 0.4
+RX J2135.9+4728 460 0.2 Z⊙– – – – 1.00 – 1.5
+UGC 11871 430 0.2 Z⊙0.00‡1.73 0.77 0.11 0.11 – 1.1
+NGC 7319 460 Z⊙0.01 1.42 0.93 – – 0.07 0.8
+3C 452 200 0.2 Z⊙– – – – 1.00 – 6.1
+SDSS Spectra
+Mkn 1018 50 0.2 Z⊙0.02 0.51 0.01 – – 0.99 1.4
+Mkn 590 50 2.5 Z⊙0.17 0.46 0.02 – 0.98 – 3.6
+Mkn 18 50 0.2 Z⊙– – – 0.22 0.26 0.52 1.4
+SDSS J090432.19+553830.1 200 0.2 Z⊙1.00 0.39 1.00 – – – 3.0
+SDSS J091129.97+452806.0 50 0.2 Z⊙0.53 0.70 0.63 0.06 – 0.31 1.4
+SDSS J091800.25+042506.2 330 2.5 Z⊙1.00 0.09 0.12 – 0.88 – 2.4
+MCG +04-22-042 460 0.2 Z⊙0.62 0.51 1.00 – – – 7.4
+Mkn 110 50 0.2 Z⊙0.03 0.53 0.03 – – 0.97 11.1
+Mkn 417 50 Z⊙1.00 0.32 0.18 0.03 – 0.79 2.9
+SBS 1136+594 400 0.2 Z⊙1.00 0.27 1.00 – – – 4.9
+CGCG 041-020 50 0.2 Z⊙1.00 0.12 0.01 0.01 0.30 0.67 1.7
+Ark 347 50 Z⊙1.00 0.40 0.12 – – 0.88 6.0
+NGC 4388 50 0.2 Z⊙0.82 0.21 0.01 – – 0.99 19.5
+NGC 4395 50 0.2 Z⊙1.00 0.20 0.23 – 0.77 – 15.5
+NGC 4992 50 Z⊙1.00 0.31 0.10 – – 0.90 2.59
+NGC 5252 50 Z⊙1.00 0.37 0.13 – – 0.87 5.2
+NGC 5506 50 2.5 Z⊙0.00‡2.28 0.96 – 0.04 – 31.8
+NGC 5548 330 Z⊙1.00 0.54 1.00 – – – 4.9
+Mkn 290 50 0.2 Z⊙1.00 0.36 1.00 – – – 2.9
+Mkn 926 400 Z⊙1.00 0.41 1.00 – – – 14.9
+†The fitted values using the stellar population models of Bruzual & Cha rlot (2003) include FWHM (kms−1), metal-
+licity (Z), and light fractions ( Lf) at 5500 ˚A using both a power law and stellar population models with ages of:
+25 (young), 2500 (interm), and 10000 (old) Myr. The values p0andp1are the power law components, defined as
+p0×λp1. The constant factor, p0, is constrained to range from 0 to 1 and is the specific flux at 1 ˚Awith units of– 40 –
+10−17ergss−1cm−2˚A−1. Where a component’s contribution (e.g. power law) was not require d in the best-fit, a dash
+is indicated.
+‡For the indicated sources, the value of p0<0.01 but non-negligible. The parameter p0for the marked sources is:
+9.9×10−5(IRAS 05589+2828), 4 .2×10−12(1RXS J174538.1+290823), 2 ×10−4(UGC 11871), and 1 .6×10−7(NGC
+5506).– 41 –Table 7. Emission Line Properties For Strong Lines (Narrow L ine Sources)
+Source FWHM blue†[OII]λ3727∗Hγ λ4340∗Hβ λ4861.3∗[OIII]λ4959∗[OIII]λ5007∗
+FWHM red†[OI]λ6300∗[NII]λ6548∗[NII]λ6583∗[SII]λ6716∗[SII]λ6731∗log F(Hα)
+KPNO Spectra
+NGC 788 674.2 ±30.6··· ··· 1.82±0.65 2.31 ±0.76 3.79 ±1.53
+177.7±5.0 0.81 ±0.22 0.44 1.33 0.95 0.92 -13.29
+2MASX J03181899+6829322 144.8 ±1.5··· 0.14±0.01 0.37 ±0.01 1.06 ±0.01 2.93 ±0.03
+50.0‡0.09±0.01 0.14 0.43 ±0.01 0.35 ±0.02 0.26 ±0.02 -14.09
+3C 105 243.6 ±2.1··· ··· 0.13±0.01 0.74 ±0.01 2.49 ±0.04
+50.0‡0.22±0.01 0.57 ±0.01 1.71 ±0.03 0.34 ±0.02 0.57 ±0.02 -14.10
+2MASX J04440903+2813003 299.4 ±1.4··· 0.05 0.15 0.21 0.60
+160.5±0.5 -0.13 0.45 1.34 0.52 0.47 -13.17
+Mkn 3 410.5 ±0.5··· 0.08 0.15 0.69 2.21 ±0.01
+153.5±2.8 0.24 0.40 1.20 ±0.01 0.27 0.35 -11.65
+2MASX J06411806+3249313 216.0 ±1.8··· 0.18±0.04 0.32 ±0.01 1.25 ±0.02 3.65 ±0.05
+233.9±5.2 0.24 ±0.01 0.13 0.40 ±0.01 0.26 ±0.01 0.22 ±0.01 -13.97
+Mkn18 478.0 ±31.7 1.79 ±0.99 0.23 ±0.02 0.55 ±0.11 0.41 ±0.06 0.69 ±0.18
+33.3±24.8 0.10 0.33 ±0.05 0.45 ±0.08 0.18 ±0.03 0.19 ±0.02 -12.72
+MCG -01-24-012 541.7 ±23.6 0.64 ±0.02 0.07 0.48 ±0.05 1.05 ±0.13 2.02 ±0.35
+301.9±10.4 0.19 0.31 ±0.01 0.67 ±0.02 0.34 0.17 ±0.07 -13.14
+Mkn 417 107.8 ±3.1··· 0.10±0.05 0.27 ±0.01 0.83 ±0.01 1.97 ±0.02
+50.0‡0.23±0.01 0.27 0.83 ±0.01 0.32 ±0.01 0.31 ±0.01 -13.76
+1RXS J1127166+190914 169.3 ±1.5··· ··· 0.41±0.03 1.35 ±0.13 3.99 ±1.00
+442.5±21.0 0.25 0.42 ±0.02 0.86 ±0.02 0.37 0.21 ±0.01 -13.03
+Ark 347 166.3 ±9.4 0.56 ±0.03 0.14 ±0.00 0.59 ±0.05 1.49 ±0.28 3.60 ±1.58
+355.4±5.1 0.3 ±0.01 0.22 ±0.12 1.11 ±0.45 0.54 ±0.03 0.54 ±0.05 -13.08– 42 –Table 7—Continued
+Source FWHM blue†[OII]λ3727∗Hγ λ4340∗Hβ λ4861.3∗[OIII]λ4959∗[OIII]λ5007∗
+FWHM red†[OI]λ6300∗[NII]λ6548∗[NII]λ6583∗[SII]λ6716∗[SII]λ6731∗log F(Hα)
+NGC 4102 262.2 ±50.5··· ··· 0.29±1.96 0.14 ±1.77 0.39 ±2.21
+334.6±4.6 0.09 ±0.17 0.41 ±0.14 0.93 ±0.02 0.15 ±0.14 0.16 ±0.15 -12.25
+Mkn 1498 321.3 ±16.8 1.14 ±0.08 0.50 ±0.02 1.35 ±0.25 2.47 ±0.47 5.69 ±2.62
+256.3±18.3 0.05 0.19 0.26 ±0.10 0.14 0.11 ±0.00 -13.18
+NGC 6240 425.1 ±23.8 0.35 ±0.01 0.02 0.11 ±0.01 0.06 ±0.01 0.20 ±0.01
+377.4±1.4 0.27 0.33 1.00 0.36 0.52 -12.73
+3C 403 134.6 ±4.2 0.21 ±0.03 0.10 ±0.02 0.32 ±0.02 1.30 ±0.03 3.69 ±0.05
+50.0‡0.16 0.32 0.96 ±0.01 0.30 ±0.01 0.29 ±0.01 -13.87
+Cygnus A 115.8 ±3.1 0.98 ±0.01 0.12 0.27 ±0.01 0.92 ±0.01 2.68 ±0.03
+320.4±6.5 0.26 0.59 1.77 ±0.01 0.51 0.43 -13.04
+MCG +04-48-002 186.0 ±5.9 1.05 ±0.05 0.18 0.60 ±0.02 0.27 ±0.00 0.72 ±0.02
+197.6±16.0 0.19 0.57 ±0.03 0.85 ±0.28 1.00 ±0.06 0.76 ±0.03 -12.99
+UGC 11871 50.0‡0.22 0.05 0.14 0.16 0.31 ±0.01
+279.1±6.7 0.08 0.24 0.67 0.22 0.20 -12.25
+NGC 7319 285.0 ±20.1 2.34 ±0.09 0.17 ±0.03 0.66 ±0.10 1.21 ±0.10 2.18 ±0.16
+239.9±3.6 0.42 ±0.04 0.60 ±0.01 1.81 ±0.03 0.80 ±0.02 0.55 ±0.02 -13.68
+3C 452 235.1 ±7.3··· ··· 0.14±0.01 0.36 ±0.02 0.98 ±0.03
+50.0‡0.21±0.01 0.32 ±0.01 0.95 ±0.03 0.27 ±0.02 0.20 ±0.03 -14.22
+SDSS Spectra
+Mkn 18 91.8 ±1.1··· 0.08 0.20 0.09 0.29
+117.2±0.7 0.05 0.15 0.44 ±0.01 0.20 0.16 -12.98
+SDSS J091129.97+452806.0 140.9 ±3.4··· 0.05±0.01 0.12 ±0.01 0.29 ±0.01 0.90 ±0.02
+118.0±2.3 0.10 ±0.01 0.24 ±0.00 0.72 ±0.02 0.29 ±0.01 0.23 ±0.01 -14.49– 43 –Table 7—Continued
+Source FWHM blue†[OII]λ3727∗Hγ λ4340∗Hβ λ4861.3∗[OIII]λ4959∗[OIII]λ5007∗
+FWHM red†[OI]λ6300∗[NII]λ6548∗[NII]λ6583∗[SII]λ6716∗[SII]λ6731∗log F(Hα)
+SDSS J091800.25+042506.2 176.2 ±0.8 0.56 ±0.01 0.08 0.25 1.00 ±0.01 3.02 ±0.03
+187.7±1.4 0.18 0.22 0.66 ±0.01 0.22 ±0.01 0.20 -14.08
+Mkn 417 196.3 ±6.4 0.43 ±0.02 0.07 ±0.00 0.24 ±0.01 1.08 ±0.18 2.95 ±1.19
+228.8±7.2 0.20 0.20 0.62 ±0.05 0.23 ±0.01 0.23 ±0.01 -13.02
+CGCG 041-020 133.9 ±2.0 0.25 ±0.01 0.07 0.18 ±0.01 0.26 ±0.01 0.73 ±0.01
+120.5±1.4 0.09 ±0.01 0.23 0.68 ±0.01 0.25 ±0.01 0.22 ±0.01 -13.96
+Ark 347 225.6 ±5.0 0.5 0.07 0.27 0.91 ±0.03 2.46 ±0.07
+171.7±0.7 0.11 0.39 1.18 0.30 0.28 -13.27
+NGC 4388 188.3 ±0.4··· 0.10 0.34 ±0.02 1.12 ±0.16 2.67 ±0.66
+280.7±3.8 0.12 ±0.01 0.11 ±0.03 0.53 ±0.10 0.19 ±0.05 0.26 ±0.03 -12.33
+NGC 4395 270.7 ±0.5··· 0.11 0.31 0.74 ±0.01 2.07 ±0.03
+248.2±0.6 0.19 0.07 0.21 0.13 0.16 -12.81
+NGC 4992 113.5 ±5.1 1.42 ±0.70 0.34 ±0.04 0.28 ±0.11 0.31 ±0.48 1.30 ±2.15
+106.9±4.0 0.87 ±0.39 0.86 ±0.31 2.06 ±2.96 0.59 ±0.60 0.30 ±0.22 -14.32
+NGC 5252 186.0 ±0.9··· 0.10 0.24 0.52 ±0.01 1.57 ±0.02
+211.9±1.0 0.34 ±0.01 0.32 0.95 ±0.01 0.45 ±0.01 0.41 ±0.01 -13.32
+NGC 5506 289.1 ±0.6··· 0.04 0.17 ±0.01 0.43 ±0.03 1.24 ±0.24
+333.8±7.0 0.12 0.27 ±0.01 0.70 ±0.06 0.14 ±0.02 0.12 ±0.03 -11.99
+†The FWHM of the lines, in kms−1, are tied together for all of the narrow emission lines liste d in this table.
+∗Ratio of the intensity of the indicated line to the intensity of Hα. The units of the H αflux are ergss−1cm−2. Where errors are not indicated,
+the errors are on the order of 10−3.
+‡Indicated FWHM of the lines was fixed to the narrow velocity va lue of 50kms−1.– 44 –
+Table 8. Emission Line Fluxes For Weaker Lines (Narrow Line S ources)
+Source [Ne III]λ3869∗Hδ [OIII]λ4363∗HeIIλ4686∗[NI]λ5199∗
+HeIλ5876∗[FeVII]λ6087∗[OI]λ6363∗[FeX]λ6375∗[ArIII]λ7136∗
+NGC 788 ··· ··· ··· -14.13±0.16 -14.09 ±0.12
+-14.28±0.14 -14.72 ±0.22 -15.03 ±0.31 ··· -14.71±0.24
+2MASX J03181899+6829322 ··· -14.02±0.16 -14.87 ±0.18 -15.69 ±0.34 -15.75 ±0.39
+··· -15.62±0.32 -16.50 ±0.78 -14.96 ±0.20 -15.87 ±0.54
+3C 105 ··· ··· ··· ··· -15.84±0.43
+··· -15.29±0.24 ··· -15.24±0.27 -14.60 ±0.15
+2MASX J04440903+2813003 ··· ··· ··· -14.71±0.15 -13.85 ±0.06
+··· ··· ··· ··· ···
+Mkn 3 ··· ··· -13.22±0.22 ··· -13.17±0.14
+-13.40±0.16 -13.55 ±0.19 -12.82 ±0.10 -13.83 ±0.29 -12.75 ±0.09
+2MASX J06411806+3249313 ··· ··· -14.64±0.26 -15.37 ±0.33 -15.71 ±0.39
+-15.46±0.29 -15.37 ±0.27 -15.08 ±0.22 -15.65 ±0.39 -15.14 ±0.28
+Mkn 18 -14.19 ±0.17 -14.29 ±0.16 -15.14 ±0.37 -14.84 ±0.25 -14.32 ±0.20
+-14.40±0.21 ··· ··· ··· -14.33±0.13
+MCG -01-24-012 -14.27 ±0.16 -15.46 ±0.55 -14.85 ±0.24 -14.90 ±0.23 -14.98 ±0.31
+-14.82±0.27 -15.31 ±0.46 -15.06 ±0.26 -15.13 ±0.28 ···
+Mkn 417 ··· ··· ··· -15.77±0.54 -15.08 ±0.23
+-14.99±0.20 ··· -15.01±0.20 -16.35 ±0.70 -15.10 ±0.26
+1RXS J1127166+190914 ··· ··· ··· -14.66±0.18 -14.69 ±0.17
+··· -14.82±0.19 -14.67 ±0.17 -14.68 ±0.18 ···
+Ark 347 -14.02 ±0.16 -14.74 ±0.30 -14.83 ±0.31 -14.51 ±0.20 -14.44 ±0.26
+-14.25±0.24 -14.39 ±0.28 -14.68 ±0.23 -15.56 ±0.54 -14.49 ±0.18
+NGC 4102 ··· ··· -13.76±0.25 ··· -13.82±0.20
+··· -16.20±1.17 ··· -15.04±0.68 -14.05 ±0.27
+Mkn 1498 -13.78 ±0.09 -14.36 ±0.15 -14.15 ±0.11 -14.21 ±0.12 -15.01 ±0.36
+-14.87±0.41 -15.35 ±0.66 -15.23 ±0.37 -15.54 ±0.49 -14.61 ±0.23
+NGC 6240 -14.24 ±0.26 -14.46 ±0.29 -15.27 ±0.54 ··· -13.82±0.19
+··· ··· -13.86±0.11 ··· ···
+3C 403 -14.56 ±0.23 -16.38 ±0.92 -15.31 ±0.39 -14.91 ±0.25 -14.87 ±0.23
+-15.07±0.20 -15.07 ±0.17 -15.19 ±0.19 -15.16 ±0.19 -14.52 ±0.18– 45 –
+Table 8—Continued
+Source [Ne III]λ3869∗Hδ [OIII]λ4363∗HeIIλ4686∗[NI]λ5199∗
+HeIλ5876∗[FeVII]λ6087∗[OI]λ6363∗[FeX]λ6375∗[ArIII]λ7136∗
+Cygnus A -13.55 ±0.10 -14.20 ±0.17 -14.24 ±0.15 -14.11 ±0.12 -14.01 ±0.17
+-14.74±0.39 -14.87 ±0.42 -14.12 ±0.11 -14.79 ±0.22 -14.14 ±0.13
+MCG +04-48-002 ··· -15.03±0.37 -14.82 ±0.26 ··· -14.32±0.14
+-14.40±0.14 ··· ··· ··· -14.95±0.22
+UGC 11871 -14.06 ±0.14 -14.68 ±0.24 -14.90 ±0.29 -14.63 ±0.20 -14.31 ±0.32
+-14.44±0.32 -15.89 ±0.90 -14.43 ±0.15 ··· -14.60±0.23
+NGC 7319 -13.98 ±0.22 -14.45 ±0.31 -14.87 ±0.42 -14.83 ±0.37 -14.29 ±0.30
+-14.68±0.42 -14.88 ±0.47 -14.52 ±0.18 -15.69 ±0.57 -14.76 ±0.25
+3C 452 ··· ··· ··· ··· ···
+··· ··· ··· ··· -14.79±0.20
+Mkn 18 -14.64 ±0.15 -14.43 ±0.12 -15.23 ±0.28 -15.35 ±0.32 -14.99 ±0.23
+-14.50±0.14 ··· -15.04±0.27 -16.29 ±0.79 -14.85 ±0.24
+SDSS J091129.97+452806.0 -15.53 ±0.24 -16.02 ±0.36 -16.44 ±0.53 -16.11 ±0.40 -16.31 ±0.48
+-15.75±0.27 -16.52 ±0.57 ··· ··· -16.20±0.48
+SDSS J091800.25+042506.2 -14.70 ±0.08 -15.50 ±0.18 -15.35 ±0.16 -15.27 ±0.15 -15.57 ±0.20
+-15.73±0.24 -15.96 ±0.30 -15.36 ±0.17 -16.22 ±0.43 -15.36 ±0.18
+Mkn 417 -14.26 ±0.08 -15.10 ±0.16 -14.93 ±0.14 -14.82 ±0.13 -15.30 ±0.22
+-15.25±0.20 -15.37 ±0.25 -14.80 ±0.13 -15.76 ±0.36 -14.88 ±0.15
+CGCG 041-020 -15.30 ±0.20 -15.47 ±0.23 -15.78 ±0.33 -15.79 ±0.34 -15.84 ±0.37
+-15.79±0.36 -16.00 ±0.45 ··· -15.95±0.44 ···
+Ark 347 -14.00 ±0.07 -14.69 ±0.12 -14.73 ±0.13 -14.38 ±0.10 -14.88 ±0.17
+-14.65±0.14 -14.34 ±0.11 -14.73 ±0.16 -15.28 ±0.29 -14.26 ±0.11
+NGC 4388 -13.65 ±0.06 -14.18 ±0.08 -14.30 ±0.08 -14.06 ±0.07 -14.55 ±0.11
+-14.28±0.09 -14.55 ±0.12 -14.07 ±0.08 -15.30 ±0.25 -13.86 ±0.07
+NGC 4395 -13.55 ±0.07 -14.04 ±0.08 -14.05 ±0.08 -14.10 ±0.08 -14.74 ±0.12
+-14.37±0.09 -15.19 ±0.17 -14.07 ±0.08 -15.61 ±0.26 -14.18 ±0.08
+NGC 4992 -15.43 ±0.25 -15.75 ±0.34 ··· -15.78±0.38 -15.81 ±0.42
+-16.89±0.90 -15.94 ±0.49 -15.82 ±0.44 ··· ···
+NGC 5252 -13.98 ±0.08 -14.67 ±0.13 -14.81 ±0.17 -14.77 ±0.16 -14.72 ±0.16– 46 –
+Table 8—Continued
+Source [Ne III]λ3869∗Hδ [OIII]λ4363∗HeIIλ4686∗[NI]λ5199∗
+HeIλ5876∗[FeVII]λ6087∗[OI]λ6363∗[FeX]λ6375∗[ArIII]λ7136∗
+-15.08±0.24 -15.40 ±0.37 -14.28 ±0.12 -15.18 ±0.30 -14.74 ±0.19
+NGC 5506 -13.81 ±0.07 ··· -14.57±0.11 -14.19 ±0.08 -14.31 ±0.09
+-14.12±0.07 -14.74 ±0.14 -14.00 ±0.07 -15.82 ±0.44 -13.74 ±0.07
+∗Logarithm of the intensity of the indicated line.– 47 –Table 9. Emission Line Properties For Strong Blue Lines (Bro ad Line Sources)
+Source FWHM (km s−1) H βN∗[OIII]λ4959∗[OIII]λ5007∗HβBFWHM (km s−1) H βB∗F5100˚A
+KPNO Spectra
+3C 111 214.6 ±0.4 -12.70 -11.97 -11.54 4960.5 ±1.6 -11.55 -13.90 ±0.05
+MCG -01-13-025 656.3 ±666.1 -13.76 -13.32 -13.02 8162.8 ±288.6 -13.11 ±0.01 -14.63 ±0.02
+1RXS J045205.0+493248 374.1 ±4.8 -13.01 ±0.01 -12.48 ±0.01 -12.04 ±0.01 7402.1 ±21.7 -12.36 -14.33 ±0.01
+MCG +08-11-011 986.1 ±20.5 -12.15 ±0.07 -11.71 ±0.07 -11.24 ±0.07 3762.3 ±28.6 -11.61 -13.60 ±0.02
+IRAS 05589+2828 563.2 ±33.5 -12.87 ±0.33 -12.81 ±0.33 -12.36 ±0.33 5564.9 ±15.1 -12.34 -14.40 ±0.01
+Mkn 6 750.1 ±212.0 -12.41 ±0.53 -11.97 -11.52 ±0.00 4757.8 ±69.5 -12.19 ±0.01 -13.89 ±0.02
+Mkn 79 1078.6 ±96.9 -13.11 ±0.04 -12.56 ±0.02 -12.09 ±0.02 3940.9 ±54.4 -12.55 ±0.01 -14.66 ±0.03
+MCG +04-22-042 1486.6 -12.72 ±0.20 -12.73 ±0.20 -12.26 ±0.20 2951.4 ±62.5 -12.43 ±0.01 -14.49 ±0.02
+NGC 3227 1445.1 -12.67 ±0.57 -12.11 -11.64 3737.2 ±61.2 -12.17 ±0.01 -13.98 ±0.01
+NGC 3516 315.4 ±57.7 -13.53 -12.48 -12.04 5294.9 ±100.3 -12.18 ±0.01 -13.88 ±0.01
+UGC 6728 327.9 ±612.0 -12.80 -13.13 -12.75 2308.3 ±79.6 -12.68 ±0.02 -14.47 ±0.01
+NGC 4051 1445.1 -12.52 ±0.18 -12.36 ±0.18 -11.87 ±0.18 1498.9 ±35.3 -12.29 ±0.02 -14.05 ±0.01
+NGC 4151 626.3 ±26.5 -11.40 ±0.08 -11.00 -10.51 2653.5 -13.41 ±0.04
+Mkn 766 939.1 ±24.5 -12.66 ±0.02 -12.26 ±0.02 -11.78 ±0.02 2422.6 ±59.0 -12.68 ±0.01 -14.43 ±0.02
+NGC 4593 1486.6 -13.12 ±0.55 -12.71 ±0.54 -12.48 ±0.54 5966.3 ±390.7 -12.40 ±0.04 -14.15 ±0.01
+MCG +09-21-096 485.5 ±351.2 -13.87 ±0.00 -13.51 -13.06 ±0.00 5412.3 ±115.8 -12.93 ±0.01 -14.94 ±0.01
+Mkn 813 1486.6 -13.82 ±0.25 -13.51 ±0.24 -13.15 ±0.24 7072.1 ±207.9 -12.98 ±0.01 -14.94 ±0.01
+Mkn 841 1486.6 -13.23 ±0.96 -12.68 -12.24 ±0.00 4957.7 ±87.3 -12.47 ±0.01 -14.61 ±0.02
+1RXS J174538.1+290823 1001.7 ±36.4 -13.71 ±0.03 -13.19 ±0.03 -12.75 ±0.03 9998.0 -13.68 ±0.01 -15.52 ±0.02
+3C 382 361.6 ±259.8 0.00 -14.63 ±0.76 -14.19 ±0.84 9998.0 -14.02 ±0.08 -15.62 ±0.02
+NVSS J193013+341047 1366.4 ±579.9 -13.43 ±0.74 -12.81 ±0.74 -12.35 4999.5 ±204.5 -13.32 ±0.02 -15.17 ±0.30
+1RXS J193347.6+325422 157.4 ±36.0 -13.08 -12.93 ±0.98 -12.40 ±0.99 3979.2 ±32.3 -12.36 -14.51 ±0.04
+4C+74.26 1428.6 ±814.2 -13.39 ±0.20 -12.66 ±0.17 -12.31 ±0.17 9099.9 ±108.5 -11.90 -13.80
+IGR 21247+5058 734.2 ±356.9 -14.22 ±0.41 -13.93 ±0.41 -13.40 ±0.41 2322.7 ±162.1 -13.66 ±0.04 -15.72 ±0.02
+RX J2135.9+4728 1486.6 -14.58 ±0.74 -14.05 ±0.74 -13.54 ±0.74 5047.7 ±385.9 -14.11 ±0.03 -15.71 ±0.01
+SDSS Spectra
+Mkn 1018 693.8 ±100.3 -13.91 ±0.09 -13.39 ±0.09 -12.91 ±0.09 5857.6 ±130.7 -13.19 ±0.01 -14.64 ±0.01
+Mkn 590 779.9 ±137.8 -13.59 ±0.04 -12.91 ±0.04 -12.46 ±0.04 5402.8 ±130.3 -13.40 ±0.01 -14.70 ±0.01
+SDSS J090432.19+553830.1 200.2 ±5.8 -13.52 ±0.03 -13.35 ±0.03 -12.87 ±0.03 5694.8 ±44.1 -13.25 -15.15 ±0.01
+MCG+04-22-042 318.7 ±7.3 -12.56 ±0.01 -12.67 ±0.01 -12.21 ±0.01 3780.2 ±23.5 -12.41 -14.41 ±0.02
+Mkn 110 483.3 ±11.50 -13.09 ±0.19 -12.64 ±0.19 -12.17 ±0.19 3332.8 ±21.1 -13.21 -15.28 ±0.03
+SBS 1136+594 1498.1 -13.45 ±0.02 -12.87 ±0.01 -12.39 3955.4 ±24.9 -12.76 -14.79 ±0.01
+NGC 5548 247.1 ±15.1 -12.74 ±0.02 -12.14 ±0.02 -11.69 ±0.02 7736.2 ±76.3 -12.45 -14.39 ±0.01
+Mkn 290 659.8 ±24.0 -13.18 ±0.58 -12.60 ±0.58 -12.13 4343.8 ±37.0 -12.61 -14.54 ±0.02– 48 –Table 9—Continued
+Source FWHM (km s−1) H βN∗[OIII]λ4959∗[OIII]λ5007∗HβBFWHM (km s−1) H βB∗F5100˚A
+Mkn 926 1331.7 ±30.3 -13.05 ±0.01 -12.53 ±0.01 -12.05 ±0.01 6993.6 ±93.4 -13.11 ±0.01 -14.83 ±0.01
+∗
+The logarithm of the indicated lines are given in ergss−1cm−2, where H βNindicates the narrow component of H βand HβBindicates the broad component.
+The limits on the velocity offsets of the lines were ±1000kms−1. Where error-bars are not listed, they are on the order of 10−3.– 49 –Table 10. Emission Line Properties For Strong Red Lines (Bro ad Line Sources)
+Source FWHM (km s−1) H αN∗[NII]λ6583∗[SII]λ6716∗[SII]λ6731∗HαBFWHM (km s−1) H αB∗
+KPNO Spectra
+3C111 481.1 ±2.4 -12.14 -12.81 -12.93 -12.98 4589.8 ±0.5 -11.23
+MCG-01-13-025 888.8 ±70.9 -13.21 ±0.01 -13.20 -13.58 ±0.03 -13.48 ±0.02 6381.7 ±27.7 -12.55
+1RXS J045205.0+493248 310.2 ±2.0 -12.48 -12.68 -13.08 -13.09 5706.2 ±2.1 -11.88
+MCG +08-11-011 766.1 ±19.6 -11.51 -11.60 -12.35 -12.19 4214.3 ±9.3 -11.20
+IRAS 05589+2828 785.1 ±67.6 -12.49 -12.85 -13.85 ±0.02 -13.86 ±0.03 5416.2 ±7.1 -12.11
+Mkn 6 848.2 ±25.5 -11.97 -12.26 -12.60 -12.45 6800.9 ±15.1 -11.51
+Mkn 79 395.5 ±38.5 -12.66 -12.74 -13.36 ±0.01 -13.45 ±0.01 3660.3 ±5.5 -12.08
+MCG +04-22-042 365.4 ±30.9 -12.57 -13.84 -13.48 -13.54 2328.1 ±3.1 -11.90
+NGC 3227 601.4 ±26.7 -12.04 ±0.01 -11.93 -12.55 ±0.01 -12.55 ±0.01 3452.9 ±16.4 -11.73
+NGC 3516 528.1 ±182.4 -12.46 ±0.59 -11.27 ±0.93 -13.62 ±0.97 -13.70 ±0.97 4418.8 ±11.2 -11.56
+UGC 6728 207.2 ±55.8 -12.34 0.00 -13.92 -13.88 1288.1 ±1.7 -12.00
+NGC 4051 227.5 ±42.5 -11.97 -12.59 -12.91 -12.93 1627.3 ±8.3 -11.80
+NGC 4151 488.3 ±5.8 -11.12 -11.22 -11.80 -11.73 4745.8 ±7.8 -11.06
+Mkn 766 511.1 ±35.8 -12.10 ±0.04 -12.46 ±0.04 -13.17 ±0.05 -13.15 ±0.05 2327.3 ±13.2 -12.17 ±0.01
+NGC 4593 427.8 ±198.5 -13.34 ±0.71 -13.17 ±0.71 -13.28 ±0.71 -13.27 ±0.71 8259.5 ±62.3 -12.41
+MCG +09-21-096 394.0 ±39.5 -13.54 ±0.07 0.00 -13.79 ±0.07 -13.83 ±0.07 5104.7 ±15.8 -12.42
+Mkn 813 0.0 -13.77 ±0.03 0.00 -14.39 ±0.06 -14.35 ±0.06 6495.0 ±21.9 -12.50
+Mkn 841 120.2 ±19.6 -12.8 -13.20 -13.41 ±0.01 -13.55 ±0.01 4190.3 ±7.8 -12.12
+1RXS J174538.1+290823 590.2 ±72.2 -14.46 ±0.19 -14.22 ±0.19 -13.83 ±0.05 -14.07 ±0.07 7303.9 ±48.7 -12.93
+3C382 0.0 -15.34 ±0.79 -15.01 ±0.77 -15.01 ±0.77 -15.14 ±0.78 1315.8 ±996.1 -14.84 ±0.21
+NVSS J193013+341047 624.9 ±28.4 -12.87 ±0.02 -13.15 ±0.02 -13.71†-13.79†5282.8±15.7 -12.50
+1RXS J193347.6+325422 789.8 ±23.9 -12.11 ±0.83 -12.36 ±0.83 -14.09 ±0.94 -14.39 ±0.93 3269.9 ±3.5 -11.97
+4C+74.26 1486.6 ±0.0 -12.83 ±0.01 -12.56 -13.25 ±0.13 -13.24 ±0.12 9998.0 -11.44
+IGR 21247+5058 295.1 ±292.0 -13.28 ±0.99 0.00 -14.62 ±0.99 -14.69 ±0.99 2122.0 ±9.0 -12.69
+RX J2135.9+4728 620.0 ±99.8 -13.59 ±0.17 -13.72 ±0.17 -14.53 ±0.17 -14.58 ±0.18 4475.1 ±33.2 -13.19
+SDSS Spectra
+Mkn 1018 457.0 ±69.7 -13.46 ±0.20 -13.24 ±0.20 -13.87 ±0.20 -13.91 ±0.20 4847.3 ±28.2 -12.67
+Mkn 590 566.0 ±22.7 -13.00 -12.99†-13.76†-13.75†6850.3±43.0 -12.79
+SDSS J090432.19+553830.1 342.8 ±12.7 -12.95 -13.25 -13.71 -13.77 5190.9 ±10.4 -12.71
+MCG +04-22-042 354.1 ±21.4 -12.21 ±0.01 -12.81 -13.36 ±0.01 -13.42 ±0.01 3059.7 ±8.0 -11.94
+Mkn 110 362.1 ±3.2 -12.47 -13.05 -13.41 -13.46 3069.7 ±6.8 -12.47
+SBS 1136+594 245.6 ±10.0 -12.92 -14.06 -13.84 -13.92 3846.5 ±8.7 -12.35
+NGC 5548 587.4 ±14.3 -12.37†-12.61†-13.07†-13.13†6736.0±18.5 -11.92
+Mkn 290 349.2 ±24.1 -12.70 ±0.01 -13.11 -13.59 ±0.01 -13.65 ±0.01 4480.0 ±13.9 -12.21– 50 –Table 10—Continued
+Source FWHM (km s−1) HαN∗[NII]λ6583∗[SII]λ6716∗[SII]λ6731∗HαBFWHM (km s−1) HαB∗
+Mkn 926 529.8 ±6.6 -12.68 -12.75†-13.14†-13.14†8292.8±20.3 -12.33
+∗
+The logarithm of the indicated lines are given in ergss−1cm−2, where H αNindicates the narrow component of H αand HαBindicates the broad component.
+The [NII]λ6548 line, not shown in the table, was fixed to a ratio of 1:2.98 with [NII]λ6583. Where error-bars are not included, they are on the orde r of 10−3.
+†The indicated value is a lower limit.– 51 –Table 11. Emission Line Properties For Weaker Narrow Lines ( Broad Line Sources)
+Source [O II]λ3727∗[NeIII]λ3869∗[NeIII]λ3968∗Hδ4101∗Hγ4340∗[OIII]λ4363∗HeIIλ4686∗
+[NI]λ5199∗HeIλ5876∗[FeVII]λ6087∗[OI]λ6300∗[OI]λ6363∗[FeX]λ6375∗[ArIII]λ7136∗
+KPNO Spectra
+LEDA 138501 -13.86 ±0.08 -13.57 ±0.08 -13.66 ±0.12 -12.76 ±0.10 -13.39 ±0.06 -13.36 ±0.05 -13.96 ±0.09
+··· ··· ··· ··· ··· ··· ···
+3C 111 ··· ··· ··· ··· ··· ··· ······ ··· -12.85±0.14 -13.01 ±0.17 -13.78 ±0.18 ···
+MCG -01-13-025 -13.23 ±0.17 ··· ··· ··· ··· ··· ··· ···
+-14.44±0.19 ··· ··· -13.49±0.05 -13.74 ±0.05 ··· ···
+1RXS J045205.0+493248 -12.37 ±0.07 -12.59 ±0.12 ··· ··· -13.19±0.10 -12.48 ±0.08 -14.12 ±0.18
+-14.46±0.09 ··· ··· -13.05±0.04 -13.60 ±0.07 ··· -14.21±0.07
+MCG +08-11-011 ··· ··· ··· ··· -11.61±0.03 -12.40 ±0.05 -13.03 ±0.06
+-13.32±0.08 -12.32 ±0.04 -13.32 ±0.10 -12.41 ±0.04 -12.69 ±0.05 ··· -13.13±0.10
+IRAS 05589+2828 ··· ··· ··· ··· -12.49±0.10 -12.84 ±0.15 -13.58 ±0.06
+··· -14.00±0.03 ··· -14.15±0.06 -14.08 ±0.07 -15.83 ±0.18 ···
+Mkn 6 ··· ··· ··· ··· -12.49±0.09 -13.00 ±0.11 -13.66 ±0.10
+-14.19±0.09 ··· ··· -12.61±0.05 -13.39 ±0.10 ··· -13.92±0.15
+Mkn 79 -12.87 ±0.11 -12.97 ±0.14 -13.11 ±0.23 -12.63 ±0.21 -12.72 ±0.07 -13.02 ±0.09 -13.36 ±0.11
+-14.25±0.19 -13.77 ±0.06 -13.71 ±0.05 -13.41 ±0.03 -13.87 ±0.06 -13.87 ±0.05 -13.86 ±0.08
+MCG +04-22-042 -12.95 ±0.10 -13.06 ±0.10 -12.79 ±0.12 -12.35 ±0.11 -12.29 ±0.06 -12.90 ±0.08 -12.59 ±0.07
+··· -12.77±0.04 -13.74 ±0.05 -13.76 ±0.05 -13.46 ±0.06 ··· -14.32±0.08
+NGC 3227 ··· ··· ··· ··· -12.19±0.06 -13.01 ±0.09 -13.69 ±0.11
+-13.79±0.07 ··· ··· -12.66±0.04 -12.87 ±0.07 ··· -12.98±0.08
+NGC 3516 ··· ··· ··· ··· -12.07±0.12 ··· ··· ···
+··· ··· ··· ··· ··· ··· ···– 52 –Table 11—Continued
+Source [O II]λ3727∗[NeIII]λ3869∗[NeIII]λ3968∗Hδ4101∗Hγ4340∗[OIII]λ4363∗HeIIλ4686∗
+[NI]λ5199∗HeIλ5876∗[FeVII]λ6087∗[OI]λ6300∗[OI]λ6363∗[FeX]λ6375∗[ArIII]λ7136∗
+UGC 6728 -13.35 ±0.15 -13.49 ±0.14 -13.13 ±0.15 -12.88 ±0.13 -12.64 ±0.08 -13.43 ±0.12 -13.36 ±0.11
+··· -13.42±0.07 ··· -14.02±0.04 -14.65 ±0.07 ··· ···
+NGC 4051 ··· ··· ··· ··· -12.54±0.11 -13.41 ±0.17 -13.55 ±0.10
+··· -13.18±0.07 -13.75 ±0.08 -13.05 ±0.06 -12.93 ±0.07 ··· -14.04±0.06
+NGC 4151 ··· ··· ··· ··· -11.64±0.06 -11.86 ±0.06 -12.60 ±0.07
+-12.63±0.06 -12.35 ±0.08 -12.22 ±0.06 -11.67 ±0.03 -12.04 ±0.05 ··· -12.17±0.03
+Mkn 766 ··· ··· ··· ··· -13.06±0.14 ··· ··· -13.92±0.10
+··· -13.37±0.09 -13.83 ±0.10 -13.42 ±0.08 -13.58 ±0.12 ··· -13.90±0.06
+NGC 4593 -12.85 ±0.15 -12.99 ±0.14 ··· -12.26±0.21 -12.21 ±0.11 -12.55 ±0.11 ···
+··· -12.10±0.08 -13.10 ±0.12 -13.75 ±0.08 -13.31 ±0.07 ··· ···
+MCG +09-21-096 -13.23 ±0.07 -13.64 ±0.12 ··· ··· -13.35±0.15 -12.69 ±0.12 ···
+··· -12.62±0.05 ··· -13.88±0.06 -14.14 ±0.10 ··· ···
+Mkn 813 -14.39 ±0.16 -13.69 ±0.14 -14.33 ±0.15 -12.88 ±0.06 -13.23 ±0.06 -13.50 ±0.07 -14.62 ±0.12
+··· -12.54±0.06 ··· -13.88±0.05 -14.15 ±0.10 ··· ···
+Mkn 841 -13.06 ±0.08 -13.06 ±0.09 -13.44 ±0.15 -12.79 ±0.15 -13.16 ±0.08 -12.90 ±0.08 -13.52 ±0.11
+-14.35±0.14 -12.52 ±0.08 -14.23 ±0.17 -14.30 ±0.07 ··· ··· -13.97±0.16
+1RXS J174538.1+290823 -13.26 ±0.06 -13.62 ±0.09 -14.06 ±0.09 -14.37 ±0.13 -13.90 ±0.10 -13.92 ±0.10 -14.66 ±0.09
+-15.11±0.09
+NVSS J193013+341047 -13.10 ±0.10 -13.18 ±0.13 -13.63 ±0.24 -13.79 ±0.25 -13.59 ±0.13 -13.37 ±0.11 -14.03 ±0.14
+···
+1RXS J193347.6+325422 ··· -12.77±0.13 -12.68 ±0.12 -12.40 ±0.10 -12.37 ±0.04 -12.81 ±0.05 ···
+-14.68±0.05 -14.63 ±0.10 -14.81 ±0.13 -13.75 ±0.06 -14.41 ±0.10 ··· ···
+4C +74.26 ··· ··· ··· ··· -11.97±0.07 ··· ···– 53 –Table 11—Continued
+Source [O II]λ3727∗[NeIII]λ3869∗[NeIII]λ3968∗Hδ4101∗Hγ4340∗[OIII]λ4363∗HeIIλ4686∗
+[NI]λ5199∗HeIλ5876∗[FeVII]λ6087∗[OI]λ6300∗[OI]λ6363∗[FeX]λ6375∗[ArIII]λ7136∗
+··· ··· ··· ··· ··· ··· ···
+IGR 21247+5058 ··· ··· ··· -14.01±0.24 -13.87 ±0.19 ··· ··· ···
+··· -13.73±0.19 -14.68 ±0.10 -14.84 ±0.12 -14.58 ±0.10 -15.26 ±0.15 ···
+RX J2135.9+4728 ··· ··· ··· ··· ··· ··· ··· -15.26±0.10
+··· -14.61±0.11 -14.09 ±0.10 -14.72 ±0.08 -15.48 ±0.12 -15.65 ±0.18 ···
+SDSS Spectra
+Mkn 1018 -13.61 ±0.06 -14.48 ±0.09 ··· ··· ··· ··· ··· ···
+··· ··· ··· -14.57±0.08 ··· ··· ···
+Mkn 590 -13.42 ±0.16 -13.28 ±0.15 ··· ··· -13.30±0.18 -13.62 ±0.13 ···
+··· ··· -13.97±0.12 -13.54 ±0.06 -14.24 ±0.13 ··· -14.53±0.07
+SDSS J090432.19+553830.1 -13.23 ±0.04 -13.87 ±0.10 ··· -14.67±0.09 -14.03 ±0.06 -13.68 ±0.09 -14.91 ±0.08
+-15.35±0.12 -14.17 ±0.10 ··· -14.16±0.05 -14.66 ±0.10 ··· -14.88±0.07
+MCG +04-22-042 -13.01 ±0.02 -12.99 ±0.08 -12.68 ±0.11 -12.25 ±0.06 -12.25 ±0.05 -12.88 ±0.07 -13.55 ±0.12
+··· -12.63±0.06 -13.58 ±0.07 -13.76 ±0.06 -14.89 ±0.12 -13.51 ±0.07 -14.51 ±0.07
+Mkn 110 -12.87 ±0.04 -13.16 ±0.05 -13.42 ±0.08 -13.53 ±0.12 -13.29 ±0.10 -13.24 ±0.09 -14.03 ±0.07
+-14.44±0.12 -14.03 ±0.06 -14.45 ±0.06 -13.25 ±0.02 -13.66 ±0.03 ··· -14.28±0.04
+SBS 1136+594 -13.17 ±0.06 -13.32 ±0.05 -13.46 ±0.09 -13.84 ±0.09 -13.48 ±0.05 -13.02 ±0.04 -14.02 ±0.09
+··· -13.11±0.03 -15.07 ±0.10 -14.05 ±0.03 -14.55 ±0.06 -14.68 ±0.09 -15.09 ±0.09
+NGC 5548 ··· -12.50±0.05 -13.05 ±0.10 -13.43 ±0.08 -12.91 ±0.08 -12.67 ±0.07 -13.57 ±0.09
+··· -13.59±0.07 -13.17 ±0.05 -12.98 ±0.03 -13.52 ±0.06 ··· -14.24±0.06
+Mkn 290 -13.35 ±0.07 -13.14 ±0.07 -13.42 ±0.12 -14.09 ±0.06 -13.72 ±0.07 -13.45 ±0.06 -13.86 ±0.07
+··· -12.66±0.06 -13.87 ±0.06 -13.85 ±0.05 -14.47 ±0.09 -15.02 ±0.14 -14.62 ±0.07– 54 –Table 11—Continued
+Source [O II]λ3727∗[NeIII]λ3869∗[NeIII]λ3968∗Hδ4101∗Hγ4340∗[OIII]λ4363∗HeIIλ4686∗
+[NI]λ5199∗HeIλ5876∗[FeVII]λ6087∗[OI]λ6300∗[OI]λ6363∗[FeX]λ6375∗[ArIII]λ7136∗
+Mkn 926 -12.55 ±0.03 -13.01 ±0.05 -13.48 ±0.08 -13.52 ±0.11 -13.20 ±0.08 -13.35 ±0.11 -13.90 ±0.14
+-13.77±0.07 -14.20 ±0.10 ··· -13.06±0.03 -13.84 ±0.06 ··· -14.22±0.05
+∗The logarithm of the flux for each indicated line is given in un its of ergss−1cm−2.– 55 –Table 12. Measurements of Intrinsic Stellar Absorption
+Source D n(4000) H δA(˚A) CN 1(mag) Ca 4227 ( ˚A) C2 4668 ( ˚A) Mgb ( ˚A) [MgFe]′(˚A) <Fe>(˚A)
+KPNO Spectra
+NGC 788 ··· ··· ··· ··· -24.12±0.80 -8.50 ±0.14 13.61 ±0.06 -7.05 ±0.11
+LEDA 138501 0.88 ±0.00 -3.25 ±0.16 0.11 ±0.00 0.24 ±0.09 -2.62 ±0.19 -0.09 ±0.12 0.69 ±0.17 -0.25 ±0.09
+2MASX J03181899+6829322 ··· ··· -0.52±0.02 -0.36 ±0.68 -4.64 ±0.45 2.33 ±0.30 ··· 2.19±0.20
+3C 105 ··· ··· ··· ··· 3.63±0.48 3.89 ±0.26 3.66 ±0.16 3.52 ±0.18
+3C 111 ··· ··· ··· ··· 21.49±0.08 -210.88 ±92.86 ··· -98.29±46.43
+2MASX J04440903+2813003 ··· -0.22±0.00 0.00 ±0.00 0.04 ±0.00 -0.21 ±0.00 -0.14 ±0.00 0.16 -0.10 ±0.00
+MCG -01-13-025 1.51 ±0.01 -2.15 ±0.25 0.09 ±0.01 1.05 ±0.12 5.14 ±0.17 4.21 ±0.17 4.32 ±0.09 3.20 ±0.13
+MCG +04-22-042 0.86 ±0.00 -12.30 ±0.18 0.21 ±0.00 0.02 ±0.09 -10.97 ±0.20 -1.49 ±0.33 2.96 ±0.47 -0.26 ±0.23
+1RXS J045205.0+493248 0.78 ±0.00 -0.30 ±0.06 0.12 ±0.00 -0.28 ±0.04 0.08 ±0.09 1.25 ±0.05 0.30 ±0.16 1.17 ±0.04
+MCG +08-11-011 ··· -10.93±0.83 -0.05 ±0.01 0.09 ±0.19 -5.44 ±0.25 0.71 ±0.13 ··· 0.37±0.09
+IRAS 05589+2828 ··· ··· ··· 0.51±0.12 -5.46 ±0.08 -0.06 ±0.06 ··· 0.07±0.04
+Mkn 3 ··· ··· 0.32±0.23 -3.18 ±2.17 -7.11 ±0.84 8.09 ±0.26 ··· 4.24±0.22
+2MASX J06411806+3249313 ··· ··· ··· ··· -0.09±0.05 -0.12 ±0.02 0.10 -0.09 ±0.01
+Mkn 6 ··· -6.79±1.60 -0.00 ±0.02 -0.18 ±0.26 -1.34 ±0.29 0.88 ±0.13 ··· 0.18±0.10
+Mkn 79 0.84 ±0.01 -14.85 ±0.30 0.32 ±0.01 0.48 ±0.15 -8.12 ±0.30 1.85 ±0.43 ··· 1.73±0.32
+Mkn 18 1.10 ±0.01 2.14 ±0.20 -0.02 ±0.01 0.39 ±0.10 2.11 ±0.18 2.15 ±0.21 2.03 ±0.11 1.80 ±0.15
+MCG -01-24-012 1.37 ±0.04 0.09 ±0.99 -0.03 ±0.02 1.50 ±0.36 1.99 ±0.49 3.66 ±0.45 2.52 ±0.22 2.81 ±0.32
+MCG +04-22-042 0.86 ±0.00 -12.30 ±0.18 0.21 ±0.00 0.02 ±0.09 -10.97 ±0.20 -1.49 ±0.33 2.96 ±0.47 -0.26 ±0.23
+NGC 3227 ··· ··· ··· -0.52±0.38 -3.08 ±0.37 1.50 ±0.17 ··· 1.20±0.13
+Mkn 417 ··· ··· ··· ··· -0.41±0.04 -0.26 ±0.01 0.30 -0.20 ±0.01
+NGC 3516 ··· ··· -0.18±0.04 -0.32 ±0.39 2.92 ±0.34 1.99 ±0.16 2.40 ±0.12 1.97 ±0.12
+1RXS J1127166+190914 ··· 0.16±0.07 0.01 ±0.00 -0.08 ±0.02 -0.03 ±0.03 -0.16 ±0.01 0.07 -0.14 ±0.01
+UGC 6728 0.91 ±0.01 -9.14 ±0.26 0.16 ±0.01 0.52 ±0.12 -8.25 ±0.24 -1.08 ±0.28 2.58 ±0.34 -0.59 ±0.20
+NGC 4051 ··· ··· ··· -0.58±0.47 -2.30 ±0.44 0.12 ±0.21 ··· 0.71±0.15
+Ark 347 1.63 ±0.04 -3.75 ±0.62 0.09 ±0.02 0.13 ±0.28 5.35 ±0.36 3.64 ±0.39 4.25 ±0.21 3.16 ±0.28
+NGC 4102 ··· ··· ··· 0.44±0.50 1.96 ±0.44 2.19 ±0.19 1.97 ±0.17 1.82 ±0.14
+NGC 4151 ··· ··· ··· -0.33±0.15 -8.56 ±0.20 2.17 ±0.09 ··· 0.83±0.07
+Mkn 766 ··· ··· 0.16±0.04 -1.20 ±0.54 -8.12 ±0.57 -0.34 ±0.26 0.54 ±1.56 0.20 ±0.20
+NGC 4593 0.91 ±0.01 -5.95 ±0.38 0.17 ±0.01 -0.32 ±0.21 -0.38 ±0.42 1.13 ±0.63 ··· 1.40±0.45
+MCG +09-21-096 0.97 ±0.01 -2.84 ±0.29 0.03 ±0.01 0.52 ±0.17 -3.00 ±0.31 1.74 ±0.40 ··· 1.39±0.30
+Mkn 813 0.86 ±0.01 -1.16 ±0.26 0.06 ±0.01 0.17 ±0.14 0.93 ±0.33 0.37 ±0.45 0.43 ±0.42 0.07 ±0.36
+Mkn 841 0.83 ±0.00 -6.04 ±0.27 0.19 ±0.01 -0.07 ±0.17 -5.94 ±0.29 0.54 ±0.50 ··· 0.60±0.36
+Mkn 1498 0.92 ±0.02 -9.97 ±0.72 0.20 ±0.02 0.26 ±0.34 -7.18 ±0.61 2.34 ±0.72 ··· 1.70±0.63
+NGC 6240 1.40 ±0.06 -0.85 ±1.18 0.04 ±0.03 1.10 ±0.58 2.55 ±0.72 6.07 ±0.76 3.39 ±0.32 3.26 ±0.57– 56 –Table 12—Continued
+Source D n(4000) H δA(˚A) CN 1(mag) Ca 4227 ( ˚A) C2 4668 ( ˚A) Mgb ( ˚A) [MgFe]′(˚A)<Fe>(˚A)
+1RXS J174538.1+290823 0.79 ±0.01 -3.20 ±0.32 0.15 ±0.01 0.18 ±0.17 3.18 ±0.38 0.59 ±0.31 0.86 ±0.47 -0.05 ±0.25
+3C 382 1.11 ±0.04 -0.99 ±1.56 0.09 ±0.05 -0.88 ±0.86 2.24 ±1.15 1.45 ±1.65 1.33 ±1.30 0.28 ±1.79
+NVSS J193013+341047 0.51 ±0.03 -28.46 ±3.36 0.38 ±0.08 3.40 ±1.29 -19.33 ±2.24 -2.59 ±1.76 9.52 ±1.61 -6.34 ±1.97
+1RXS J193347.6+325422 0.84 ±0.00 -6.62 ±0.09 0.19 ±0.00 -0.12 ±0.05 -2.95 ±0.30 0.06 ±0.16 ··· -0.01±0.16
+3C 403 0.72 ±0.02 3.35 ±1.19 -0.25 ±0.03 0.55 ±0.68 -14.05 ±3.19 -22.14 ±1.97 17.25 ±1.38 -20.44 ±5.88
+Cygnus A 0.95 ±0.03 -5.58 ±1.22 0.14 ±0.04 -0.68 ±0.58 -4.52 ±0.72 10.01 ±0.86 ··· 5.46±1.58
+MCG +04-48-002 1.14 ±0.01 4.75 ±0.31 -0.09 ±0.01 0.39 ±0.14 1.65 ±0.22 2.44 ±0.12 1.89 ±0.10 1.93 ±0.09
+4C +74.26 0.86 ±0.00 -0.75 ±0.04 0.05 ±0.00 0.11 ±0.03 1.14 ±0.06 -0.14 ±0.20 ··· -0.20±0.15
+IGR 21247+5058 1.02 ±0.01 1.02 ±0.26 -0.01 ±0.01 0.11 ±0.12 1.12 ±0.19 0.10 ±0.15 0.21 ±0.33 -0.01 ±0.11
+RX J2135.9+4728 ··· ··· 0.35±0.90 2.56 ±5.61 -1.30 ±1.42 0.59 ±0.99 ··· 0.97±0.56
+UGC 11871 1.08 ±0.00 -0.75 ±0.06 0.01 ±0.00 -0.07 ±0.03 -0.03 ±0.05 -0.41 ±0.13 0.11 -0.28 ±0.09
+NGC 7319 1.03 ±0.01 0.40 ±0.19 -0.02 ±0.01 0.07 ±0.09 -0.52 ±0.14 -0.71 ±0.14 0.55 -0.47 ±0.10
+3C 452 ··· ··· ··· ··· 5.30±0.54 1.20 ±0.41 2.83 ±0.31 1.75 ±0.26
+SDSS Spectra
+Mkn 1018 0.96 ±0.00 -1.92 ±0.12 0.05 ±0.00 0.33 ±0.07 1.93 ±0.18 1.98 ±0.11 1.86 ±0.08 1.65 ±0.09
+Mkn 590 1.17 ±0.00 -3.76 ±0.13 0.17 ±0.00 0.88 ±0.06 5.25 ±0.16 3.77 ±0.10 4.20 ±0.06 3.05 ±0.07
+Mkn 18 1.20 ±0.00 3.08 ±0.16 -0.03 ±0.00 0.58 ±0.09 2.64 ±0.22 2.11 ±0.13 2.32 ±0.09 1.98 ±0.10
+SDSS J090432.19+553830.1 0.88 ±0.00 -4.24 ±0.20 0.15 ±0.01 0.59 ±0.11 0.98 ±0.27 2.43 ±0.18 1.50 ±0.14 2.19 ±0.14
+SDSS J091129.97+452806.0 0.99 ±0.00 -0.30 ±0.05 0.01 ±0.00 -0.13 ±0.03 -0.72 ±0.06 -0.39 ±0.03 0.53 -0.38 ±0.02
+SDSS J091800.25+042506.2 1.52 ±0.02 0.32 ±0.47 0.06 ±0.01 0.69 ±0.27 4.51 ±0.48 4.55 ±0.28 4.22 ±0.16 3.46 ±0.20
+MCG +04-22-042 0.78 ±0.00 -11.76 ±0.12 0.21 ±0.00 -0.30 ±0.06 -9.96 ±0.18 -1.13 ±0.11 2.33 ±0.19 -0.09 ±0.08
+Mkn 110 0.72 ±0.00 -15.51 ±0.21 0.34 ±0.01 0.47 ±0.10 -8.30 ±0.26 0.60 ±0.14 0.54 ±0.87 -0.53 ±0.11
+Mkn 417 1.69 ±0.02 0.64 ±0.28 0.05 ±0.01 1.22 ±0.14 7.15 ±0.27 5.53 ±0.17 5.88 ±0.09 4.30 ±0.12
+SBS 1136+594 0.85 ±0.00 -6.08 ±0.13 0.15 ±0.00 0.13 ±0.07 -9.36 ±0.20 -0.48 ±0.12 2.00 ±0.24 -0.39 ±0.11
+CGCG 041-020 1.48 ±0.01 0.91 ±0.25 -0.02 ±0.01 1.08 ±0.13 5.05 ±0.27 3.36 ±0.16 3.99 ±0.10 2.98 ±0.12
+Ark 347 2.24 ±0.02 -1.51 ±0.27 0.11 ±0.01 1.33 ±0.12 7.42 ±0.24 4.86 ±0.14 5.79 ±0.08 4.27 ±0.10
+NGC 4388 1.11 ±0.01 2.97 ±0.35 0.07 ±0.01 -2.10 ±0.19 -3.03 ±0.38 3.83 ±0.18 ··· 2.41±0.13
+NGC 4395 0.92 ±0.01 4.20 ±0.33 0.08 ±0.01 -0.19 ±0.17 -10.31 ±0.46 1.10 ±0.22 ··· 0.08±0.17
+NGC 4992 -1.25 ±0.51 4.54 ··· ··· ··· ··· ··· ···
+NGC 5252 2.06 ±0.02 -0.14 ±0.27 0.13 ±0.01 1.48 ±0.13 7.47 ±0.26 5.85 ±0.15 6.05 ±0.09 4.14 ±0.12
+NGC 5506 1.27 ±0.01 -1.54 ±0.14 0.18 ±0.01 -0.54 ±0.15 -4.09 ±0.32 3.11 ±0.15 ··· 1.17±0.12
+NGC 5548 0.80 ±0.00 -1.42 ±0.17 0.14 ±0.01 0.47 ±0.10 -2.45 ±0.26 2.63 ±0.16 ··· 0.95±0.12
+Mkn 290 0.86 ±0.00 -3.98 ±0.14 0.11 ±0.00 0.16 ±0.08 -4.78 ±0.21 0.06 ±0.13 ··· 0.19±0.10
+Mkn 926 0.79 ±0.00 -4.54 ±0.16 0.25 ±0.00 0.18 ±0.09 2.80 ±0.24 5.02 ±0.14 3.26 ±0.09 2.85 ±0.12– 57 –– 58 –
+Table 13. De-reddened Emission Line Properties
+Source∗Hα/HβE(B - V) int[OIII]/Hβ[OI]/Hα[NII]/Hα[SII]/Hα[OIII]/[OII]
+KPNO Spectra
+NGC 788 0.55 ··· 2.08 0.81 1.33 1.87 ···
+2MASX J03181899+6829322 2.70 ··· 7.92 0.09 0.43 0.61 ···
+3C 105 7.69 0.92 17.07 0.11 0.77 0.39 ···
+3C 111 (B) 3.60 0.15 14.01 0.10 0.19 0.27 ···
+2MASX J04440903+2813003 6.67 0.77 3.63 -0.07 0.68 0.48 ···
+MCG -01-13-025 (B) 3.50 0.12 5.38 0.43 0.93 0.86 1.39
+1RXS J045205.0+493248 (B) 3.43 0.10 9.22 0.17 0.57 0.45 1.89
+MCG +08-11-011 (B) 4.38 0.35 7.72 0.07 0.59 0.25 ···
+IRAS 05589+2828 (B) 2.42 ··· 3.26 0.01 0.43 0.09 ···
+Mkn 3 6.67 0.77 13.37 0.13 0.61 0.30 ···
+2MASX J06411806+3249313 3.12 0.01 11.39 0.24 0.40 0.48 ···
+Mkn 6 (B) 2.74 ··· 7.82 0.17 0.51 0.57 ···
+Mkn 79 (B) 2.80 ··· 10.58 0.08 0.83 0.36 6.10
+Mkn 18 1.82 ··· 1.25 0.10 0.45 0.37 0.39
+MCG -01-24-012 2.08 ··· 4.21 0.19 0.67 0.51 3.16
+MCG +04-22-042 (B) 1.42 ··· 2.89 0.02 0.05 0.23 5.57
+NGC 3227 (B) 4.21 0.31 10.19 0.13 1.00 0.46 ···
+Mkn 417 3.70 0.18 7.13 0.20 0.71 0.53 ···
+NGC 3516 (B) 11.66 1.34 25.72 ··· 4.81 0.04 ···
+1RXS J1127166+190914 2.44 ··· 9.73 0.25 0.86 0.58 ···
+UGC 6728 (B) 2.89 ··· 1.13 0.01 ··· 0.06 3.96
+NGC 4051 (B) 3.50 0.12 4.35 0.07 0.22 0.20 ···
+Ark 347 1.69 ··· 6.10 0.25 1.11 1.08 6.43
+NGC 4102 3.45 0.11 1.33 0.08 0.85 0.28 ···
+NGC 4151 (B) 1.90 ··· 7.70 0.18 0.80 0.46 ···
+Mkn 766 (B) 3.67 0.17 7.45 0.04 0.37 0.15 ···
+NGC 4593 (B) 0.60 ··· 4.33 0.39 1.46 2.30 2.33
+MCG +09-21-096 (B) 2.14 ··· 6.57 0.38 ··· 1.08 1.49
+Mkn 813 (B) 1.13 ··· 4.74 0.07 ··· 0.50 17.55
+Mkn 841 (B) 2.99 ··· 9.86 0.09 0.36 ··· 6.67
+Mkn 1498 0.74 ··· 4.21 0.05 0.26 0.25 4.99
+NGC 6240 9.09 1.09 1.59 0.12 0.39 0.32 0.16
+1RXS J174538.1+290823 (B) 0.18 ··· 9.23 0.33 1.73 6.72 3.28
+3C 382 (B) ··· ··· ··· 0.41 2.13 3.70 ···
+NVSS J193013+341047 (B) 3.63 0.16 11.74 0.09 0.46 0.23 4.72
+1RXS J193347.6+325422 (B) 9.33 1.11 4.16 ··· 0.21 0.01 ···
+3C 403 3.12 0.01 11.52 0.16 0.95 0.59 17.40
+Cygnus A 3.70 0.18 9.70 0.23 1.51 0.79 2.20
+MCG +04-48-002 1.67 ··· 1.20 0.19 0.85 1.76 0.69
+4C +74.26 (B) 3.63 0.16 11.53 0.41 1.62 ··· ···
+IGR 21247+5058 (B) 8.80 1.06 5.81 ··· ··· 0.03 ···
+RX J2135.9+4728 (B) 9.82 1.17 9.64 0.04 0.27 0.07 ···
+UGC 11871 7.14 0.84 1.99 0.04 0.32 0.19 0.51
+NGC 7319 1.52 ··· 3.30 0.42 1.81 1.35 0.93
+3C 452 7.14 0.84 6.30 0.11 0.45 0.21 ···– 59 –
+Table 13—Continued
+Source∗Hα/HβE(B - V) int[OIII]/Hβ[OI]/Hα[NII]/Hα[SII]/Hα[OIII]/[OII]
+SDSS Spectra
+Mkn 1018 (B) 2.82 ··· 9.91 0.12 1.66 0.74 5.00
+Mkn 590 (B) 3.97 0.25 13.10 0.18 0.81 0.28 6.68
+Mkn 18 5.00 0.48 1.36 0.03 0.29 0.23 ···
+SDSS J090432.19+553830.1 (B) 3.75 0.19 4.34 0.05 0.42 0.27 1 .83
+SDSS J091129.97+452806.0 8.33 1.00 6.62 0.05 0.30 0.20 ···
+SDSS J091800.25+042506.2 4.00 0.26 11.70 0.15 0.53 0.33 3.9 6
+MCG +04-22-042 (B) 2.25 ··· 2.22 0.02 0.25 0.13 6.22
+Mkn 110 (B) 4.10 0.28 7.98 0.12 0.21 0.17 3.61
+Mkn 417 4.17 0.30 11.84 0.18 0.64 0.48 4.80
+SBS 1136+594 (B) 3.38 0.09 11.42 0.10 0.07 0.20 5.53
+CGCG 041-020 5.56 0.59 3.77 0.06 0.41 0.27 1.44
+Ark 347 3.70 0.18 8.91 0.10 1.01 0.49 4.05
+NGC 4388 2.94 ··· 7.85 0.12 0.53 0.45 ···
+NGC 4395 3.23 0.04 6.64 0.18 0.20 0.28 ···
+NGC 4992 3.57 0.14 4.56 0.78 1.82 0.78 0.77
+NGC 5252 4.17 0.30 6.30 0.27 0.73 0.65 ···
+NGC 5506 5.88 0.65 6.73 0.07 0.40 0.14 ···
+NGC 5548 (B) 2.34 ··· 11.23 0.20 0.57 0.38 ···
+Mkn 290 (B) 3.02 ··· 11.15 0.07 0.39 0.24 16.56
+Mkn 926 (B) 2.37 ··· 10.08 0.30 0.85 0.69 3.17
+Spectra from the Literature
+MRK 3521(B) 0.95 ··· 18.25 0.007 0.28 0.007 ···
+NGC 9312(B) 6.17 0.70 2.29 0.04 0.21 0.32 ···
+NGC 127514.16b 0.30 14.88b 1.37 1.36 1.38 ···
+NGC 211033.24 0.04 4.79 0.37 1.29 1.12 ···
+NGC 32271(B) 2.90b ··· 5.91b 0.23 1.33 0.68 ···
+NGC 35161(B) 2.31 ··· 9.28 0.15 1.31 0.70 ···
+NGC 40511(B) 3.30 0.06 4.50 0.14 0.64 0.36 ···
+NGC 410218.33 1.00 0.99 0.041 0.92 0.31 ···
+NGC 413813.66 0.17 5.94 0.33 1.47 1.32 ···
+NGC 41511(B) 3.40 0.09 11.56 0.22 0.68 0.54 ···
+NGC 438815.69 0.61 11.15 0.16 0.57 0.61 ···
+NGC 439512.12 ··· 6.22 0.36 0.44 0.96 ···
+NGC 55481(B) 1.28 ··· 10.09 0.36 0.88 0.66 ···
+∗Sources with broad lines (approximately Sy 1 – Sy 1.5) are ind icated with a (B).
+1Reference: Ho et al. (1997)
+2Reference: Veilleux & Osterbrock (1987)
+3Reference: Kewley et al. (2001)– 60 –
+Table 14. Classification
+Source [N II]/Hα[SII]/Hα[OI]/Hα[OIII]/[OII] Class
+KPNO Spectra
+NGC 788 AGN LINER LINER ··· LINER
+2MASX J03181899+6829322 AGN Seyfert Seyfert ··· Seyfert
+3C 105 AGN Seyfert Seyfert ··· Seyfert
+3C 111 (B) AGN Seyfert Seyfert ··· Seyfert
+2MASX J04440903+2813003 AGN Seyfert Seyfert ··· Seyfert
+MCG -01-13-025 (B) AGN Seyfert LINER LINER Ambig.
+1RXS J045205.0+493248 (B) AGN Seyfert Seyfert Seyfert Seyf ert
+MCG +08-11-011 (B) AGN Seyfert Seyfert ··· Seyfert
+IRAS 05589+2828 (B) AGN HII HII ··· Ambig.
+Mkn 3 AGN Seyfert Seyfert ··· Seyfert
+2MASX J06411806+3249313 AGN Seyfert Seyfert ··· Seyfert
+Mkn 6 (B) AGN Seyfert Seyfert ··· Seyfert
+Mkn 79 (B) AGN Seyfert Seyfert Seyfert Seyfert
+Mkn 18 COMP HII LINER LINER Ambig.
+MCG -01-24-012 AGN Seyfert Seyfert Seyfert Seyfert
+MCG +04-22-042 (B) HII HII HII Seyfert Ambig.
+NGC 3227 (B) AGN Seyfert Seyfert ··· Seyfert
+Mkn 417 AGN Seyfert Seyfert ··· Seyfert
+NGC 3516 (B) AGN Seyfert ··· ··· Seyfert
+1RXS J1127166+190914 AGN Seyfert Seyfert ··· Seyfert
+UGC 6728 (B) ··· HII HII HII HII
+NGC 4051 (B) COMP Seyfert Seyfert ··· Ambig.
+Ark 347 AGN LINER Seyfert Seyfert (Seyfert)
+NGC 4102 AGN HII Seyfert ··· Ambig.
+NGC 4151 (B) AGN Seyfert Seyfert ··· Seyfert
+Mkn 766 (B) AGN Seyfert Seyfert ··· Seyfert
+NGC 4593 (B) AGN Seyfert LINER Seyfert Ambig.
+MCG +09-21-096 (B) ··· LINER Seyfert LINER Ambig.
+Mkn 813 (B) ··· Seyfert Seyfert Seyfert Seyfert
+Mkn 841 (B) AGN ··· Seyfert Seyfert Seyfert
+Mkn 1498 AGN Seyfert Seyfert Seyfert Seyfert
+NGC 6240 COMP HII LINER HII Ambig.
+1RXS J174538.1+290823 (B) AGN Seyfert Seyfert Seyfert Seyf ert
+3C 382 (B) ··· ··· ··· ··· ···
+NVSS J193013+341047 (B) AGN Seyfert Seyfert Seyfert Seyfer t
+1RXS J193347.6+325422 (B) COMP HII ··· ··· COMP
+3C 403 AGN Seyfert Seyfert Seyfert Seyfert
+Cygnus A AGN Seyfert Seyfert Seyfert Seyfert
+MCG +04-48-002 AGN LINER LINER LINER LINER
+4C +74.26 (B) AGN ··· Seyfert ··· Seyfert
+IGR 21247+5058 (B) ··· HII ··· ··· HII (?)
+RX J2135.9+4728 (B) AGN Seyfert Seyfert ··· Seyfert
+UGC 11871 COMP HII HII HII COMP
+NGC 7319 AGN LINER LINER LINER LINER
+3C 452 AGN Seyfert Seyfert ··· Seyfert– 61 –
+Table 14—Continued
+Source [N II]/Hα[SII]/Hα[OI]/Hα[OIII]/[OII] Class
+SDSS spectra
+Mkn 1018 (B) AGN Seyfert Seyfert Seyfert Seyfert
+Mkn 590 (B) AGN Seyfert Seyfert Seyfert Seyfert
+Mkn 18 HII HII HII ··· HII
+SDSS J090432.19+553830.1 (B) AGN Seyfert Seyfert Seyfert S eyfert
+SDSS J091129.97+452806.0 AGN Seyfert Seyfert ··· Seyfert
+SDSS J091800.25+042506.2 AGN Seyfert Seyfert Seyfert Seyf ert
+MCG +04-22-042 (B) HII HII HII Seyfert Ambig.
+Mkn110 (B) AGN Seyfert Seyfert Seyfert Seyfert
+Mkn 417 AGN Seyfert Seyfert Seyfert Seyfert
+SBS 1136+594 (B) AGN Seyfert Seyfert Seyfert Seyfert
+CGCG 041-020 AGN Seyfert Seyfert Seyfert Seyfert
+Ark 347 AGN Seyfert Seyfert Seyfert Seyfert
+NGC 4388 AGN Seyfert Seyfert ··· Seyfert
+NGC 4395 AGN Seyfert Seyfert ··· Seyfert
+NGC 4992 AGN Seyfert LINER LINER (LINER)
+NGC 5252 AGN Seyfert Seyfert ··· Seyfert
+NGC 5506 AGN Seyfert Seyfert ··· Seyfert
+NGC 5548 (B) AGN Seyfert Seyfert ··· Seyfert
+Mkn 290 (B) AGN Seyfert Seyfert Seyfert Seyfert
+Mkn 926 (B) AGN Seyfert Seyfert Seyfert Seyfert
+Spectra from the Literature
+MRK 3521(B) AGN Seyfert Seyfert ··· Seyfert
+NGC 9312(B) HII HII HII ··· HII
+NGC 12751AGN Seyfert LINER ··· Ambig.
+NGC 21103AGN LINER LINER ··· LINER
+NGC 32271(B) AGN Seyfert Seyfert ··· Seyfert
+NGC 35161(B) AGN Seyfert Seyfert ··· Seyfert
+NGC 40511(B) AGN Seyfert Seyfert ··· Seyfert
+NGC 41021AGN HII HII ··· Ambig.
+NGC 41381AGN LINER Seyfert ··· Ambig.
+NGC 41511(B) AGN Seyfert Seyfert ··· Seyfert
+NGC 43881AGN Seyfert Seyfert ··· Seyfert
+NGC 43951AGN Seyfert Seyfert ··· Seyfert
+NGC 55481(B) AGN Seyfert Seyfert ··· Seyfert
+Parenthesis are placed around classifications where the pro bable class is noted despite the fact that not all of the crite ria point
+to the same class. This is discussed within the text. The symb ol (B) indicates broad line sources (i.e. approximately Sy 1 –1.5).
+1Reference: Ho et al. (1997)
+2Reference: Veilleux & Osterbrock (1987)
+3Reference: Kewley et al. (2001)– 62 –
+Table 15. Mass and Luminosity
+Source λLλ1MHβ2L[OIII](obs)3L[OIII](corr)3Mreverb2M2MASS2
+NGC 788 40.84±0.15 41.50 ±0.15 8.51
+Mkn 1018 43.61 ±0.01 8.25 ±0.02 41.64 ±0.09 41.68 ±0.09 8.94
+Mkn 590 43.12 ±0.01 7.94 ±0.03 41.66 ±0.04 41.66 ±0.04 7.68 ±0.06 8.87
+2MASX J03181899+6829322 41.59 41.64
+3C 105 41.50±0.01 41.50 ±0.01 7.79
+3C 111 44.47 ±0.05 8.54 ±0.03 43.12 43.12
+2MASX J04440903+2813003 40.00 40.00
+MCG -01-13-025 42.77 ±0.02 8.12 ±0.04 40.67 40.67 8.06
+1RXS J045205.0+493248 43.59 ±0.01 8.45 ±0.01 42.17 ±0.01 42.17 ±0.01
+MCG +08-11-011 44.02 ±0.02 8.07 ±0.02 42.67 ±0.07 42.67 ±0.07
+IRAS 05589+2828 43.63 ±0.01 8.22 ±0.01 41.97 ±0.33 42.06 ±0.33
+Mkn 3 42.24 42.24 8.48
+2MASX J06411806+3249313 41.24 ±0.01 41.24 ±0.01
+Mkn 6 43.66 ±0.02 8.09 ±0.02 42.31 42.36 8.24
+Mkn 79 43.03 ±0.03 7.62 ±0.03 41.89 ±0.02 41.93 ±0.02 7 .72±0.10 8.42
+Mkn 18 40.18±0.05 40.19 ±0.05 7.45
+SDSS J090432.19+553830.1 42.99 ±0.01 7.91 ±0.01 41.56 ±0.03 41.56 ±0.03 7.70
+SDSS J091129.97+452806.0 39.61 ±0.01 39.61 ±0.01 7.53
+SDSS J091800.25+042506.2 42.10 42.10 8.57
+MCG -01-24-012 41.04±0.07 41.19 ±0.07 7.16
+MCG +04-22-042 43.63 ±0.02 7.88 ±0.01 42.07 ±0.20 42.37 ±0.20 8.49
+Mkn 110 42.81 ±0.03 7.36 ±0.02 42.22 ±0.19 42.22 ±0.19 7.40 ±0.09 7.80
+NGC 3227 42.19 ±0.01 7.15 ±0.02 40.83 40.83 7 .63±0.17 7.83
+Mkn 417 41.32±0.08 41.32 ±0.08 8.04
+NGC 3516 43.00 ±0.01 7.86 ±0.02 41.13 41.13 7 .63±0.13 8.13
+1RXS J1127166+190914 42.92±0.10 43.01 ±0.10 9.00
+SBS 1136+594 43.78 ±0.01 8.00 ±0.01 42.48 42.48 7.53
+UGC 6728 42.15 ±0.01 6.71 ±0.03 40.16 40.19 6.81
+CGCG 041-020 40.30±0.01 40.30 ±0.01 8.46
+NGC 4051 41.67 ±0.01 6.10 ±0.03 40.14 ±0.18 40.14 ±0.18 6 .28±0.15 7.27
+Ark 347 41.29±0.09 41.37 ±0.09 8.12
+NGC 4102 39.52±0.82 39.52 ±0.82
+NGC 4151 42.62 ±0.04 7.07 ±0.02 41.81 41.99 7 .12±0.13 7.69
+Mkn 766 42.78 ±0.02 7.06 ±0.03 41.73 ±0.02 41.73 ±0.02 7.85
+NGC 4388 41.24±0.10 41.26 ±0.10 8.53
+NGC 4395 38.79±0.01 38.79 ±0.01 5.30
+NGC 4593 42.75 ±0.01 7.83 ±0.07 40.71 ±0.54 41.33 ±0.54 6 .99±0.06 8.61
+MCG +09-21-096 43.01 ±0.01 7.88 ±0.02 41.18 41.32
+NGC 4992 39.88±0.42 39.88 ±0.42 8.56
+NGC 5252 40.89±0.01 40.89 ±0.01 8.64
+NGC 5506 40.96±0.08 40.96 ±0.08 7.77
+NGC 5548 43.04 ±0.01 8.21 ±0.02 42.03 ±0.02 42.14 ±0.02 7 .83±0.01 8.42
+Mkn 813 44.16 ±0.01 8.69 ±0.03 42.24 ±0.24 42.63 ±0.24
+Mkn 841 43.51 ±0.02 8.05 ±0.02 42.17 42.19 8.15
+Mkn 290 43.42 ±0.02 7.90 ±0.02 42.12 42.13 7.68
+Mkn 1498 42.34±0.16 42.89 ±0.16 8.59– 63 –
+Table 15—Continued
+Source λLλ1MHβ2L[OIII](obs)3L[OIII](corr)3Mreverb2M2MASS2
+NGC 6240 40.64±0.02 40.64 ±0.02
+1RXS J174538.1+290823 43.59 ±0.02 8.70 ±0.01 42.65 ±0.03 43.73 ±0.03 8.75
+3C 382 42.91 ±0.02 8.36 ±0.01 40.63 ±0.84 40.63 ±0.84 9.22
+NVSS J193013+341047 43.43 ±0.30 8.02 ±0.19 42.54 42.54
+1RXS J193347.6+325422 43.44 ±0.04 7.83 ±0.03 41.84 ±0.99 41.84 ±0.99
+3C 403 41.53±0.01 41.53 ±0.01
+Cyg A 42.18 42.18
+MCG +04-48-002 40.44±0.01 40.67 ±0.01
+4C +74.26 45.25 ±0.00 9.45 ±0.01 43.03 ±0.17 43.03 ±0.17 9.00
+IGR 21247+5058 41.88 ±0.02 6.58 ±0.07 40.49 ±0.41 40.49 ±0.41
+RX J2135.9+4728 42.08 ±0.01 7.35 ±0.08 40.54 ±0.74 40.54 ±0.74
+UGC 11871 41.38±0.01 41.38 ±0.01 8.34
+NGC 7319 40.65±0.03 40.92 ±0.03 8.54
+3C 452 40.89±0.01 40.89 ±0.01 8.54
+Mkn 926 43.51 ±0.01 8.36 ±0.02 42.58 ±0.01 42.69 ±0.01 8.95
+1λLλis derived from the power law continuum flux at 5100 ˚A and is the logarithm of the luminosity in units of
+ergss−1.
+2Indicated masses are the logarithm of the mass in solar masse s.
+3The logarithm of [O III]5007˚A luminosity is in units of ergss−1, including the observed luminosity (obs) and
+the extinction corrected values (corr) using values of E(B- V) recorded in Table 13. Where errors are not indicated,
+they are of the order of 10−3.– 64 –
+3500 4000 4500 5000 5500 6000 6500 7000
+Wavelength (
+A)2000400060008000100001200014000Flux (
+
10
+17 ergs s
+1 cm
+2)NGC 221
+3500 4000 4500 5000 5500 6000 6500 7000
+Wavelength (
+A)5001000150020002500300035004000Flux (
+10
+17 ergs s
+1 cm
+	2)NGC 1023
+3500 4000 4500 5000 5500 6000 6500 7000
+Wavelength (
+
+A)400600800100012001400160018002000Flux (
+10
+17 ergs s
+
1 cm
+2)NGC 3640
+3500 4000 4500 5000 5500 6000 6500 7000
+Wavelength (
+A)2004006008001000120014001600Flux (
+10
+17 ergs s
+1 cm
+2)NGC 5308
+Fig. 1.— Plotted are the KPNO spectra of 4 template galaxies ( black) with the best-fit continuum
+model (blue) described in Table 5. Using three simple stella r population models (young, inter-
+mediate, old), we find that we can replicate the spectra well, particularly in the blue end of the
+spectrum. Using additional populations at intervening age s, we could better replicate the spectra.
+However, such fits are degenerate (Tremonti et al. 2004) and w e would lose information about the
+host galaxy, when fitting to our AGN sources (for which the hos t properties are not well-defined).– 65 –
+Fig. 2.— Plotted are the SDSS spectra of 4 AGN, two narrow line sources (top) and two broad
+line sources (bottom), before and after the continnum subtr action (black). The best-fit continuum
+model is plotted in blue (described in Table 6). The continuu m model utilizes three simple stellar
+population models (young, intermediate, old) along with a p ower law model to account for AGN
+emission.– 66 –
+Fig. 3.— Plotted are the best-fit individual components (pow er law and stellar components, mod-
+ulated by reddening) fitted to the spectra shown in Figure 2. T he flux is shown in units of
+10−17ergss−1cm−2˚A−1. The sources shown include NGC 4992 (top left), Mrk 417 (top r ight),
+Mkn 1018 (bottom left), and MCG +04–22–042 (bottom right). T he combined fit is shown in red,
+while individual stellar components and the power law are ea ch shown in blue. Masked regions
+are shown in green. The first three sources have strong galaxy contributions, each dominated by a
+contribution from an old population. The final source is best -fit with a pure reddened power law
+model.– 67 –
+Fig. 4.— Plotted are the SDSS spectra (blue) and KPNO spectra (black) of the 4 AGN sources
+with spectra from both sources, focusing on a region (3700–6 200˚A) which shows both emission
+(i.e. Hβand [O III]) and intrinsic absorption features. The fitted continuum f or each spectrum
+is shown with the dotted lines. Comparison of the two sets of s pectra for each source show good
+agreement between the flux measurements and spectral shape.– 68 –
+Fig. 5.— Plotted are fits to the H βand Hαregions of the broad line spectra of SDSS
+J090432.19+553830.1 (top), Mkn 841 (middle), and MCG +04-2 2-042 (bottom). The narrow com-
+ponents are shown in blue, while the total fit (broad + narrow l ines) is shown in red.– 69 –
+3850 3900 3950 4000 4050 4100
+Wavelength (
+A)51015202530Flux (
+10
+17 ergs s
+1 cm
+2)Dn(4000) =0 .99SDSS J091129.97+452806.0
+3850 3900 3950 4000 4050 4100
+Wavelength (
+A)10152025303540455055Flux (
+10
+17 ergs s
+1 cm
+2)Dn(4000) =1 .48CGCG 041-020
+3850 3900 3950 4000 4050 4100
+Wavelength (
+A)1020304050607080Flux (
+10
+ 17 ergs s
+!1 cm
+"2)Dn(4000) =#1.25NGC 4992
+4040 4060 4080 4100 4120 4140 4160
+Wavelength (
+$A)1015202530Flux (
+%10
+&17 ergs s
+'1 cm
+(2) H)A=*0.30SDSS J091129.97+452806.0
+4040 4060 4080 4100 4120 4140 4160
+Wavelength (
++A)303540455055Flux (
+,10
+-17 ergs s
+.1 cm
+/2)H 0A=0.91CGCG 041-020
+4040 4060 4080 4100 4120 4140 4160
+Wavelength (
+1A)45505560657075Flux (
+210
+317 ergs s
+41 cm
+52)H 6A=4.5NGC 4992
+3850 3900 3950 4000 4050 4100
+Wavelength (
+7A)406080100120140160180Flux (
+810
+917 ergs s
+:1 cm
+;2)Dn(4000) =1 .51MGC -01-13-025
+3850 3900 3950 4000 4050 4100
+Wavelength (
+<A)260280300320340Flux (
+=10
+>17 ergs s
+?1 cm
+@2)Dn(4000) =0 .96Mkn 1018
+3850 3900 3950 4000 4050 4100
+Wavelength (
+AA)8009001000110012001300140015001600Flux (
+B10
+C17 ergs s
+D1 cm
+E2)Dn(4000) =0 .91NGC 4593
+4040 4060 4080 4100 4120 4140 4160
+Wavelength (
+FA)120130140150160170Flux (
+G10
+H17 ergs s
+I1 cm
+J2)H KA= L2.15MGC -01-13-025
+4040 4060 4080 4100 4120 4140 4160
+Wavelength (
+MA)260265270275280285290295300305Flux (
+N10
+O17 ergs s
+P1 cm
+Q2)HRA=S1.92Mkn 1018
+4040 4060 4080 4100 4120 4140 4160
+Wavelength (
+TA)850900950100010501100115012001250Flux (
+U10
+V17 ergs s
+W1 cm
+X2)HYA=Z1.92NGC 4593
+Fig. 6.— Plotted are the spectra, with errors, in the regions whereDn(4000) and H δAare mea-
+sured for representative narrow line (top two panels) and br oad line (bottom two panels) sources.
+Absorption lines of Ca IIH&K and H δare indicated with dashed lines. For the narrow line sources ,
+emission lines in the spectra were subtracted.– 70 –
+0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
+Dn(4000)
+[8
+\6
+]4
+^20246H
+_A
+0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
+Dn(4000)
+`8
+a6
+b4
+c20246H
+dA
+Fig. 7.— Plotted are two age indicators, H δAwhich measures recent bursts of star formation and
+Dn(4000) which measures the Ca IIbreak and is sensitive to old stellar populations. The circl es
+represent narrow line sources and the triangles represent b road line sources. In the top plot, we
+show the values measured after subtracting out the power law components (Table 6). The bottom
+plot shows the values measured directly from the spectra. In both plots, the box in the upper left
+hand corner shows the area where young stellar populations h ad significant ( /greaterorsimilar30%) contributions
+in our test galaxy spectra (see §A, Figure 19).– 71 –
+Mgb0.81.21.62.0Dn(4000)
+Mgb
+e0.40.20.81.4CN1
+Mgb
+f4048Ca 4227
+Mgb
+g10
+h505C2 4668i4j20 2 4 6 8
+Mgb
+k226[MgFe]
+lm4n20 2 4 6 8
+Mgb
+o226<Fe>
+Fig. 8.— Plotted are a selection of stellar absorption indic es indicating stellar age (D n(4000))
+or metallicity of the populations vs. the metallicity indic ator Mgb. The circles represent narrow
+line sources and the triangles represent broad line sources . In the top left plot, we show a line
+representing the division in D n(4000) between populations with a significant contribution from
+young stars ( /greaterorsimilar30%), as determined in Figure 19.– 72 –
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
+E(B-V)051015202530 No. of Narrow Line Sources
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
+E(B-V)051015202530 No. of Broad Line Sources
+Fig. 9.— Plotted are the distributions of E(B-V) for the narr ow line (left) and broad line (right)
+sources. The average value and standard deviation are much s maller for the broad line sources, as
+expected from the unified model.– 73 –
+10-210-1100
+[NII]/Hp10-1100101[OIII]/H
+qAGN
+HIIAGN
+HIIAGN
+HII
+10-210-1100
+[SII]/Hr10-1100101[OIII]/H
+sSeyfert
+HII
+LINERSeyfert
+HII
+LINERSeyfert
+HII
+LINER
+10-210-1100
+[OI]/Ht10-1100101[OIII]/H
+uSeyfert
+HII
+LINERSeyfert
+HII
+LINERSeyfert
+HII
+LINER
+10-210-1100
+[OI]/Hv10-210-1100101102[OIII]/[OII]Seyfert
+HII LINERSeyfert
+HII LINER
+Fig. 10.— Plotted are the emission line diagrams showing nar row line (circles) and broad line
+(triangles) sources from the KPNO or SDSS spectra that we hav e analyzed, as well as values
+extracted from the literature (square). The diagnostic lin es separating H IIgalaxies from AGNs
+are shown in red (Kewley et al. 2001). In the [O III]/Hβversus [N II]/Hαdiagnostic plot, the
+dashedbluelinerepresentsthedivisionbetween H IIgalaxies andcompositesfromKauffmann et al.
+(2003a). The blue dashed lines on the remaining plots repres ent the division between Seyferts and
+LINERs from Kewley et al. (2006).– 74 –
+10-1100101102103
+(I4959+I5007)/I43630.60.81.01.21.41.61.82.02.2I6716/I6731
+103104105106107108
+Ne (cm
+w3)100101102103(I4959+I5007)/I436310000 K
+12500 K
+15000 K
+20000 K
+50000 K
+Fig. 11.— Plotted at the top is a comparison of the ratio of int ensities of [S II]λ6716/λ6731 versus
+the ratio of [O III] (λ4959 +λ5007)/λ4363 for the narrow line (circle) and broad line (triangle)
+sources. A K-S test shows that the ratios of [S II] (an indicator of density) are likely from the
+same population, while the [O III]-temperature diagnostic is not. In the bottom plot, we show the
+average diagnostic value for the narrow (horizontal blue li ne) and broad (horizontal red line) line
+sources versus density for values of constant temperature. This diagnostic points to a much higher
+temperature for the broad line sources, if the densities are low. If the densities are high in this O+2
+emission region for broad line sources (106cm−3/lessorsimilarNe/lessorsimilar107cm−3) and low ( Ne/lessorsimilar104cm−3) for
+narrow line sources, the temperatures are similar.– 75 –
+37 38 39 40 41 42 43 44
+log L[OIII ](obs)0246810 No. of Narrow Line Sources
+37 38 39 40 41 42 43 44
+log L[OIII ](obs)05101520 No. of Broad Line Sources
+37 38 39 40 41 42 43 44
+log L[OIII ](corr)0246810 No. of Narrow Line Sources
+37 38 39 40 41 42 43 44
+log L[OIII ](corr)05101520 No. of Broad Line Sources
+Fig. 12.— Plotted are histograms of the [O III] 5007˚A emission line luminosity for the narrow line
+(Seyferts = blue, LINERs = green, others = hatched) and broad line (red) sources, showing both
+the observed (obs, top plots) and extiction-corrected (cor r, bottom plots) luminosities. The broad
+line sources are more luminous on average than the narrow lin e sources, in both the observed and
+extinction-corrected luminosities. The mean value for the extinction-corrected luminosity distribu-
+tion of broad line sources is logL [OIII]= 41.79, while the narrow line sources have a mean value of
+logL[OIII]= 40.82.– 76 –
+40 41 42 43 44 45 46
+log L14x195keV02468101214 No. of Broad Line Sources
+40 41 42 43 44 45 46
+log L14y195keV024681012 No. of Narrow Line Sources
+Fig. 13.— Plotted are the distributions of the 14–195keV lum inosity for broad line (left) and
+narrow line (right) sources. The narrow line classification s are represented as Seyferts (white),
+LINERs (black), and either ambiguous/H IIgalaxies/composites (hatched). The Seyferts are the
+most luminous of the narrow line sources, both in the hard X-r ay band and the optical (indicated
+by the [OIII] 5007 ˚A luminosities). While the broad line sources have an averag e X-ray luminosity
+higher than the narrow line sources, the distribution of lum inosities for narrow line Seyferts is the
+same as for the broad line sources as a whole.– 77 –
+1040104110421043104410451046
+L14z195keV (ergs s
+{1)1038103910401041104210431044L[OIII ] (obs) (ergs s
+|1)
+1040104110421043104410451046
+L14 }195keV10-510-410-310-210-1100L[OIII ] (obs)/L14
+~195keV
+1040104110421043104410451046
+L14 195keV (ergs s
+€1)1038103910401041104210431044L[OIII ] (corr) (ergs s
+1)
+1040104110421043104410451046
+L14‚195keV10-510-410-310-210-1100L[OIII ] (corr)/L14
+ƒ195keV
+Fig. 14.— Plotted is the relationship between observed (top ) and reddening corrected (bottom)
+[OIII] 5007 ˚A luminosities and the 14–195keV Swift BAT luminosities (le ft) and the ratio of these
+luminosities versus the Swift BAT luminosity (right). In th e plots, broad line sources are indicated
+with red triangles, while the narrow line sources are indica ted with blue circles. As the left plots
+show, L [OIII]is not well correlated with the hard X-ray luminosity. The li nes indicate the weak
+correlations seen for the Seyfert 1s ((corrected) R2= 0.35) and narrow line sources ((corrected)
+R2= 0.38). In the right-hand plots, it is clear that there is a great deal of scatter in the optical/X-
+ray ratio for a given X-ray luminosity.– 78 –
+106107108109
+MH„ (M…)106107108109Mreverb (M
+†)
+106107108109
+MH ‡ (Mˆ)106107108109M2MASS (M
+‰)
+Fig. 15.— Plotted are comparisons of the H βderived masses from the FWHM of the broad
+component of H βandλLλ(5100˚A) and two additional mass estimates. The first comparison is
+with reverberation mapping masses, largely from Peterson e t al. (2004). We find good agreement
+between the H βmasses and this more direct mass measurement. The second com parison is with
+masses derived from the 2MASS K-band stellar magnitudes (Mu shotzky et al. 2008; Winter et al.
+2009a). We also find a correlation between these two mass meas urements (IR and H βderived),
+indicated by the bolder dashed line. The additional dashed l ines on the second plot represent
+values with (from the top to bottom most line) 10 ×difference, 2 ×difference, 1:1 correspondence,
+1/2 difference, or 1/10 difference.– 79 –Š7‹6Œ54Ž32
+Log L[OIII ]/LEdd024681012 No. of Narrow Line Sources7‘6’5“4”3•2
+Log L[OIII ]/LEdd024681012 No. of Broad Line Sources
+Fig. 16.—Plotted arethedistributionsofL [OIII]/LEddforthenarrowline(Seyferts=blue, LINERs
+= green) and broad line (red) sources. The [O III] 5007˚A luminosity scales with the bolometric
+luminosity, making the ratio L [OIII]/LEddan indicator of the accretion rate. While the ratio of
+L[OIII]/LEddislower forthenarrowlinesources, acomparisonoftheaccr etion ratesdependsgreatly
+on the bolometric corrections, which are determined from th e spectral energy distributions and are
+not well known, particularly for the narrow line sources. On ly two of the H II/ambiguous/other
+sources have masses available to calculate L [OIII]/LEdd. The average value for these sources is low,
+with logL [OIII]/LEdd=−5.4, but not as low as the value for the three LINERs with availab le mass
+measurements (-5.9).– 80 –
+0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
+Dn(4000)1038103910401041104210431044Log L[OIII ]–8—6 ˜4™2 0 2 4 6
+HšA1038103910401041104210431044Log L[OIII ]
+0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
+Dn(4000)10-710-610-510-410-3Log L[OIII ]/LEdd›8 œ6 4ž2 0 2 4 6
+HŸA10-710-610-510-410-3Log L[OIII ]/LEdd
+Fig. 17.— Plotted are two age indicators, H δAwhich measures recent bursts of star formation
+and Dn(4000) which measures the Ca IIbreak and is sensitive to old stellar populations versus the
+reddening corrected [O III] 5007˚A luminosity and L [OIII]/LEddfor the narrow line (circles) and
+broad line (triangles) sources.– 81 –
+Table 16. Stellar Light Fits to the Test Spectra
+Test Spectrum FWHM†Z†Lfyoung†Lfinterm†Lfold†
+25 Myr (Y) 300 0 .2Z⊙0.89 0.00 0.11
+2500 Myr (I) 330 2.5 Z⊙0.00 1.00 0.00
+10000 Myr (O) 300 Z⊙0.00 0.00 1.00
+0.5×(Y + I ) 300 0 .2Z⊙0.41 0.28 0.30
+0.5×(Y + O ) 300 Z⊙0.32 0.68 0.00
+0.5×(I + O) 330 2.5 Z⊙0.00 1.00 0.00
+0.33×(Y + I + O) 300 Z⊙0.19 0.81 0.00
+†The fitted values using the stellar population models of Bruzual & Cha rlot
+(2003) include FWHM (kms−1), metallicity ( Z), and light fractions ( Lf) at
+5500˚Ausing populations at 25 (young or Y), 2500 (interm or I), and 1000 0
+(old or O) Myr.
+Table 17. Best-fit Power law + Stellar Light Fits to the Test Sp ectra
+Source FWHM†Z†p0†p1†Lfpow†Lfyoung†Lfinterm†Lfold†
+25 Myr (Y) + pow 300 Z⊙9.3×10−40.77 0.42 0.58 0.00 0.00
+2500 Myr (I) + pow 300 Z⊙3.3×10−30.66 0.44 0.00 0.56 0.00
+10000 Myr (O) + pow 300 Z⊙4.6×10−40.87 0.45 0.00 0.09 0.46
+0.5×(Y + I ) + pow 300 Z⊙8.5×10−40.79 0.41 0.30 0.24 0.05
+0.5×(Y + O ) + pow 300 Z⊙4.2×10−20.41 0.61 0.20 0.00 0.19
+0.5×(I + O) + pow 300 Z⊙6.2×10−40.83 0.44 0.00 0.29 0.28
+0.33×(Y + I + O) + pow 300 Z⊙3.8×10−40.87 0.41 0.19 0.20 0.20
+†The fitted values using the stellar population models of Bruzual & Cha rlot (2003) include FWHM (kms−1),
+metallicity ( Z), and light fractions ( Lf) at 5500 ˚A using both a power law and stellar population models with
+ages of: 25 (young), 2500 (interm), and 10000 (old) Myr. The valu esp0andp1are the power law components,
+defined as p0×λp1. The constant factor, p0, is constrained to range from 0 to 1.– 82 –
+Fig. 18.— Plotted are several of the test spectra created by b roadening combinations of the stellar
+population models to a velocity dispersion of 300 km s−1, adding random noise, and the effects of
+reddening. From top to bottom, plotted are a young, 50% young + 50% intermediate, 50% young +
+50% old, 33% young + 33% intermediate + 33% old, and 50% interm ediate + 50% old population.
+Notice, there is very little difference between the 50% young + 50% intermediate and 50% young
++ 50% old populations. We specifically plot the region surrou nding the 4000 ˚Abreak, a region with
+prominent intrinsic absorption features.– 83 –
+0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
+Dn (4000)
+ 8
+¡6
+¢4
+£20246H
+¤AYoung
+Fig. 19.— Plotted isthestellar absorptionindexH δAversustheD n(4000) indexforthetest spectra.
+Both are commonly used as age indicators of a stellar populat ion. In the plot, our test spectra,
+consisting of combinations of single stellar population mo dels, are shown for three metallicities:
+0.2Z⊙(triangle), Z ⊙(circle), and 2.5 Z ⊙(square). We find that metallicity has little effect on the
+values of thesestellar absorption indices, as expected. We also findthat populations withsignificant
+contributions from young populations (33% or higher) fall w ithin a small parameter space on the
+plot, towards the upper left hand corner.– 84 –
+Mgb1.01.41.82.2Dn(4000)
+Mgb
+¥0.6
+¦0.20.20.6CN1
+Mgb
+§1258Ca 4227
+Mgb0510C2 4668¨4©20 2 4 6 8
+Mgb
+ª1258[MgFe]«4¬20 2 4 6 8
+Mgb
+­1258<Fe>
+Fig. 20.— Plotted are measured stellar absorption indices v ersus the Mgb stellar absorption index
+for the test spectra. With the exception of D n(4000), which is an age indicator, the remaining
+plotted indices are sensitive to abundancesof metals inthe population. Inthe plot, our test spectra,
+consisting of combinations of single stellar population mo dels, are shown for three metallicities:
+0.2Z⊙(yellow), Z ⊙(black), and 2.5 Z ⊙(green). The line in the top left plot indicates the division
+between young populations (below the line) and older popula tions (see Figure 19).
\ No newline at end of file
diff --git a/1001.0050.txt b/1001.0050.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1a713388394641044f330d2ee47ad8d41ed36df7
--- /dev/null
+++ b/1001.0050.txt
@@ -0,0 +1,509 @@
+arXiv:1001.0050v1  [physics.hist-ph]  30 Dec 2009Fundamental times, lengths and physical constants:
+some unknown contributions by Ettore Majorana
+S. Esposito
+Dipartimento di Scienze Fisiche, Universit` a di Napoli “Fe derico II” and I.N.F.N. Sezione di Napoli,
+Complesso Universitario di Monte S. Angelo, Via Cinthia, 80 126 Naples, Italy∗
+G. Salesi
+Facolt` a di Ingegneria, Universit` a di Bergamo, viale Marc oni 5, 24044 Dalmine (BG) Italy
+and I.N.F.N. Sezione di Milano, via Celoria 16, 20133 Milan, Italy†
+We review the introduction in physics of the concepts of an el ementary space length and of a
+fundamental time scale, analyzing some related unknown con tributions by Ettore Majorana. In
+particular, we discuss the quasi-Coulombian scattering in presence of a finite length scale, as well
+as the introduction of an intrinsic (universal) time delay i n the expressions for the retarded elec-
+tromagnetic potentials. Finally, we also review a special m odel considered by Majorana in order to
+deduce the value of the elementary charge, in such a way antic ipating key ideas later introduced in
+quantum electrodynamics.
+INTRODUCTION: THE KNOWN STORY
+The idea of elementary space or time intervals has been quite a recur ring one in the philosophical and scientific
+literature of every time and, as almost all the fundamental physica l ideas, from time to time it has been recovered
+from oblivion. Typical examples range from the ancient Greek period to our days, namely from Zeno’s paradox on
+Achilles and the tortoise to Poincar´ e and Mach, who regarded the c oncept of continuum as a consequence of our
+physiological limitations.1
+As often shown in the widespread literature on the structure of pa rticles (e.g., see below, the “chronon theory”),
+a time-discretization for an elementary object involves an “interna l” (that is, referred to the center-of-mass frame)
+motion associated with microscopic space distances. For such an ele mentary particle, an “extended-like” structure,
+ratherthanapointlikeone,isexpectedandthishasimportantcons equencesevenforthefundamentaltheorydescribing
+thatparticlessince, indeed, theclassicaltheoryofapointlikechar gedparticleleadstowell-knowndivergencyproblems,
+which are only apparently solved by the renormalization group theor y.
+In the realm of particle physics, the concepts of an elementary time and length are substantially equivalent for
+several reasons: in most cases one can pass from time to space ju st by introducing the speed of light. Already in
+1930’s (see below), it was early realized that a fundamental length is directly related to the existence of a “cut-off” in
+the momentum transferred during the particle-detector interac tion, which is required in order to avoid the emergence
+of the so-called ultraviolet catastrophe in quantum field theories. A nother way to provide a small distance cut-off in
+field theory is to formulate it on a discrete lattice: this approach was introduced in 1940 by Wentzel [2], but only
+later it went into detail, becoming a standard technique in modern num erical approach to high energy interactions,
+as e.g. in quantum chromodynamics.
+Space-time discretizations, or fundamental scales, are introduc ed in classical and quantum theories mainly for two
+reasons: to prevent undesired infinities in some physical quantities , or (though less known) to explain the emergence
+of an intrinsic angular momentum for spinning particles.
+The first appearance in physics of a fundamental length was sugge sted by H.A. Lorentz, who introduced the so-called
+classical electron radius Rcl≡e2/mc2≃2.82·10−13cm in his famous classical relativistic model of the electron;2it is
+equivalently known as “Lorentz radius”or “Thomsonscattering len gth”. Roughly speaking, such a radius corresponds
+to the electron size required for obtaining its mass from the electro static potential energy, assuming that quantum
+mechanicaleffects are irrelevant. Actually, such effects playa non -negligiblerolein the behaviorofchargedparticlesat
+very short distances, and the classical electron radius may be con sidered as the length scale at which renormalization
+effects become relevant in quantum electrodynamics. In other wor ds,Rcldenotes the watershed between classical and
+1“...le temps et l’espace ne repr` esentent, au point de vue ph ysiologique, qu’un continue apparent, qu’ils se composent tr` es vraisemblable-
+ment d’elements discontinus, mais qu’on ne peut distinguer nettement les uns des autres.” [1]
+2Lorentz considered electrons just as rigid spheres, albeit contracting when moving in the ether [3].2
+quantum electrodynamics; it appears in the semiclassical theory fo r the non-relativistic Thomson scattering, as well
+as in the relativistic Klein-Nishina formula [4].
+In early semiclassical models, the electron was substantially regard ed as a charged sphere but, unfortunately, the
+electromagnetic self-energy of such a sphere diverges in the limit of pointlike charge. This was, then, the basic
+motivation that led M.Abraham, Lorentz and, later, P.A.M.Dirac, to derive an equation of motion for the electron
+where a finite, non-vanishing size is fully taken into account [5]. In suc h a way a fundamental length was effectively
+introduced in order to properly describe the motion of charged par ticles. Particularly extensive work on extended,
+not pointlike, charges was carried out by A.Sommerfeld [6]. In stud ying the motion of a sphere of radius Rwith
+a uniform surface charge, he obtained the following “retarded-like ” first-order differential-finite difference equation
+(T≡2R/c):
+ma=e(E+v×B)+2
+3e2
+Rc2v(t−T)−v(t)
+T, (1)
+where the electromagnetic field of the charge itself causes a self-f orce, which has a delayed effect on its motion. The
+“delay time” 2 R/cappearing in the Sommerfeld equation, corresponding to the time th e light takes to go across
+the sphere, was one of the earlier elementary times introduced in mic rophysics. Later in the 1920s, J.J.Thomson [7]
+suggested that the electric force may act in a discontinuous way, p roducing finite increments of momentum separated
+by finite intervals of time.
+The relativistic version of Sommerfeld’s non-relativistic approach wa s developed by P.Caldirola [8]. His theory of
+the electron is one of the first and simplest theories which assumes a prioria minimum time interval: it is based on
+the existence of an elementary proper-time interval, the so-called chronon. When applied to electrons in an external
+electromagnetic field, such a finite-difference theory succeeds (a lready at the classical level) in overcoming all the
+known difficulties met by the Abraham-Lorentz-Dirac (ALD) theory , like the so-called pre-acceleration problem and
+the emergence of run-away solutions [9, 10]. It is also able to give a c lear answer to the problem related to some
+ambiguities associated with the hyperbolic motion of the electron [11], as well as to the question of whether a free
+falling charged particle does or does not emit radiation [12]. Furtherm ore, at the quantum level, Caldirola’s chronon
+theory is seemingly able to explain the origin of the “classical (Schwing er’s) part”, e/planckover2pi1/2mc·α/2π=e3/4πmc2, of
+the anomalous magnetic momentum of the electron as well as the mas s spectrum of the charged leptons [8]. In this
+theory, time flows continuously, but, when an external force act s on the electron, the reaction of the particle to the
+applied force is not continuous: the value of the electron velocity uµis supposed to jump from uµ(τ−τ0) touµ(τ)
+only at certain positions snalong its world line. These “discrete positions” are such that the elec tron takes a time τ0
+for traveling from one position sn−1to the next one sn. In principle, the electron is still considered as pointlike, but
+the ALD equation for the relativistic radiating electron is replaced: ( i) by a corresponding finite–difference (retarded)
+equation in the velocity uµ(τ)
+m
+τ0/braceleftbigg
+uµ(τ)−uµ(τ−τ0)+uµ(τ)uν(τ)
+c2[uν(τ)−uν(τ−τ0)]/bracerightbigg
+=e
+cFµν(τ)uν(τ) (2)
+(which reduces to the ALD equation when τ0≪τ) and (ii) by a second equation (the so-called transmission law ),
+connecting the discrete positions xµ(τ) along the world line of the particle among themselves:
+xµ(nτ0)−xµ((n−1)τ0) =τ0
+2{uµ(nτ0)−uµ[(n−1)τ0]}, (3)
+which is valid inside each discrete interval τ0and describes the “internal” motion of the electron. In such equat ions
+the chronon associated with the electron results to beτ0
+2≡θ0=2
+3ke2
+mc3≃6.266·10−24s (k≡1/4πε0), depending,
+therefore, on the particle properties (namely, on its charge eand on its rest mass m).
+A different elementary length appearing in modern physics is the Compton wavelength , introduced in 1923 by
+A.H.Compton [13] in his explanation of the scattering of photons by e lectrons. It is defined as λC≡h/mc, and
+corresponds to the wavelength of a photon whose energy equals t he particle rest mass; for the electron, the Compton
+wavelength is about 2 .42·10−10cm. The meaning of this length is quite clear: it indicates the watershe d between
+classical mechanics and quantum wave-mechanics, since for spatia l distances below h/mca microsystem behaves as
+a very quantum object. This was promptly realized by E.Schr¨ oding er in 1930-31 [14] in his study of the so-called3
+Zitterbewegung , which emerges in the Dirac theory by means of the Darwin term appe aring in the (decomposed)
+Hamiltonian of the Dirac equation, as found by C.G.Darwin [15] already in 1928. Indeed, because of the Zitterbewe-
+gung (see also below) the electron undergoes extremely rapid fluct uations on scales just of the order of the Compton
+wavelength, causing, for example, the electrons moving inside an at om to experience a smeared nuclear Coulomb
+potential (this peculiar form of microscopic kinetic energy is presen tly better known as “quantum potential”of the
+Schr¨ odinger equation [16]).
+The Compton wavelength plays also the role of the spatial scale which naturally arise in any semi-classical model of
+spinning particles, when attempting to formulate a kinematical pictu re of the intrinsic angular momentum. Already in
+the 1920s, another extended-like structure of particles (in addit ion to the electrodynamical one mentioned above) was
+discovered, namely the mechanical (or, rather, kinematical) configuration in the particle’s center-of -mass frame due
+to the spin. Indeed, in 1925 R. Kronig, G. Uhlenbeck and S. Goudsmit [17] introduced the celebrated self-rotating
+electron hypothesis, suggesting a physical interpretation for particles sp inning around their own axis. Now, it is
+interesting to notice that any mechanical model of a self-rotating electron fatally leads to a fundamental length for
+this very peculiar rotator, resulting to be just the Compton wavele ngthλC. Anyway, models with a self-rotating
+spinning charge are very naive and problematic.3In the subsequent literature, these problems were overcome by
+considering kinematical theories of the spin based on the Zitterbew egung [19, 20], where spinning particles, though
+not effectively extended, can be considered as something which is ha lf-way between a point and a rotating body (as,
+e.g., a top). In these models the center of the (even pointlike) char ge is spatially distinct from the center-of-mass, so
+thatvelocityandmomentumarenolongerparallelvectors,andanin ternal(microscopic), rotationalspin motionofthe
+particle arises in addition to the external (macroscopic) translatio nal motion of the center-of-mass. As a consequence,
+in the classical limit the global inertial motion of a free spinning particle turns out to be no longer uniform and
+rectilinear, but rather accelerated and helical, with a radius once ag ain equal to the Compton wavelength.
+A different insight was provided by quantum mechanics with its discret e energy levels and the uncertainty principle.
+This led physicists to speculate about space-time discreteness as e arly as the 1930’s. W. Heisenberg himself [21], for
+example, in 1938 noted that physics musthave a fundamental length scale which, together with handc, might allow
+the derivation of the mass of particles. In Heisenberg’s view, this len gth scale would have been around 10−13cm,
+thus corresponding to the classical electron radius. Later, H.S.S nyder [22] in 1947 introduced a minimum length by
+“quantizing” the spacetime, thus anticipating of more than half a ce ntury the so-called “non-commutative geome-
+try” models (see below). Spacetime coordinates are represented by quantum operators and, in Snyder’s approach,
+these operators have a discrete spectrum, thus entailing a discre te interpretation of spacetime. Subsequently, also
+T.D.Lee [23] introduced an effective time discretization on the basis o f the necessarily finite number of measurements
+performable in any finite interval of time.
+The concept of non-continuous spacetime has recently returned into fashion in GUT’s [24], in string theories [25], in
+quantum gravity [26] and in the approaches regarding spacetime eit her as a sort of quantum ether or as a spacetime
+foam [27], or even as endowed with a non-commutative geometry (like in Deformed or Double Special Relativity
+[28]4). In particular, M-Theory, Loop Quantum Gravity [29, 30] and Ca usal Dynamical Triangulation [31] lead
+to postulate an essentially discrete and quantum spacetime, where (as expected from the uncertainty relations)
+fundamental momentum and mass-energy scales naturally arise, in addition to /planckover2pi1andc. In contemporary physics the
+most important case of an elementary space scale is doubtless the “ Planck length” ℓP=/radicalbigg
+/planckover2pi1G
+c3≃1.62·10−33cm; for
+example, it plays a fundamental role in string theory, where it is defin ed as the minimum length of a typical string.
+As a consequence, any space distance smaller than ℓPis deprived of any physical meaning or possibility of being
+measured. M.Planck himself, early in 1899 [32] proposed a system of units to be used in fundamental physics (a
+“System of Natural Units”), where fundamental units are the Pla nck length, the Planck time, and the Planck mass.
+Although Planck did not know yet quantum mechanics and general re lativity, such scales mark out the regions where
+these theories become, in a sense, reciprocally in contrast, so tha t a proper quantum gravity theory is required. At
+the present the Planck length is the minimum spacetime scale present in physics, at which we expect a substantial
+3It is quite famous a statement by A.O. Barut: “If a spinning pa rticle is not quite a point particle, nor a solid three dimens ional top,
+what can it be?” [18]
+4It is interesting to point out that in the first of Refs.[28] it is stated that “the special role of the time coordinate in the structure of
+k-Minkowski spacetime forces one to introduce an element of d iscretization in the time direction”.4
+“unification” of high energy microphysics and early cosmology.
+This is, in short terms and until now, the known story of the introdu ction in physics of “fundamental” constants
+with the meaning of elementary lengths or times. However, with the t horough study, started in recent years, of the
+copious set of unpublished notes left by another protagonist of th eoretical physics [33], Ettore Majorana [34], we
+have been acquainted with some unknown contributions pointing out his own elaboration about the arguments just
+described above. In the remaining part of this paper, we will then de scribe and comment such contributions, present
+in the so-called Volumetti [35] and Quaderni [36], containing study and research notes which were not published b y
+Majorana.
+In these booklets, among many different topics the author studied extensively several arguments of electrodynamics
+(both at a classical and at a quantum level). Many of these regarde d topics and methods “usually” discussed in
+the scientific workplaces of 1930s, although they were dealt with in a very original manner and, sometimes, with
+extraordinary results (see, for example, Ref. [37]). Here, howev er, we will focus just on three of these notes, which
+result to be particularly relevant for the topics considered here. A characteristic of the Majorana notes is that they
+always concern specific examples rather than generic theoretical issues. In the following section, then, we report
+the study performed by Majorana about the scattering of partic les by a quasi-Coulombian potential, where a non
+vanishing radius for the scatterer is considered. In Sect.3 we inst ead discuss the introduction of an intrinsic time delay
+in the propagation of the (classical) electromagnetic field, consider ed by Majorana in order to take into account the
+possible effect on the electromagnetic potentials of a fundamental constant length. Finally, in Sect.4 we will describe
+an interesting (though leading to incorrect results) attempt to fin d a relation between the fundamental physical
+constants e,/planckover2pi1andc. Our conclusions and outlook then follow.
+QUASI-COULOMBIAN SCATTERING
+The set of Majorana’s Quaderni opens with the study of the problem of the scattering of particles f rom a quasi-
+Coulombian potential of the form
+V(r) =k√
+r2+a2, (4)
+wherekis a positive constant (related to the electric charge of the potent ial source) and ais, according to Majorana,
+the “magnitude of the radius of the scatterer”. We do not know pr ecisely the motivations for such a study (likely,
+for applications to atomic and nuclear physics problems; see below), but, as a matter of fact, the original intention
+of the author was to extend the well-known Rutherford formula fo r the scattering of a beam of particles (with charge
+Z′eend mass m) from a given body (of charge Ze). In this case we have a pure Coulomb scattering, and the cross
+section (number of scattering particles at an angle θper unit time and solid angle) is given by the classical formula
+f(θ) =Z2Z′2e4
+4m2v4sin4θ/2=Z2Z′2e4
+16T2sin4θ/2, (5)
+wherevis the velocity of the incident particles and Tis their kinetic energy. Majorana deduced the above Rutherford
+formulain his Volumetti , usingboth classicalmechanicsargumentsandthe quantum Borna pproximationmethod [35].
+As well-known, Eq.(5) was employed for the first time by E. Rutherf ord in his experiments on αparticles impinging
+on a gold target, aimed at explaining the structure of the atom (with or without a compact nucleus at its center).
+The modification of the Coulomb scattering potential, considered by Majorana in Eq.(4), certainly introduces an
+improvement in the description of the physical phenomenon, with th e introduction of the dimensions of the scattering
+center (in the Rutherford formula assumed to be pointlike, a= 0), but the story does not end here. In fact, the most
+simple approximation in this line of thinking is to consider the scattering center as a uniformly charged sphere of
+radiusabut, in such a case, the potential outside the sphere is strictly Cou lombian (thanks to the Gauss law), and
+Eq.(4) would not apply. Of course, at the time when Majorana perf ormed his calculations, it was known that the
+nucleus of a given atom is not uniformly charged, being formed by indiv idual particles, but, again, such a situation is
+not described by the simple formula in Eq.(4). An example is the Yukaw a potential [38], where the Coulomb potential
+acquires a screening factor with an exponential form ruled by one m ore parameter, giving the range of nuclear forces.
+In any case, it is striking that the only application of Eq.(4) reporte d in the Quaderni [36] was to the hydrogen
+atom, where just one proton forms its nucleus, thus being conside red effectively as a uniformly charged particle. Thus
+the reasoning of Majorana behind Eq.(4) has a different starting p oint, upon which we will briefly speculate below.5
+For the moment, we only note that, from a strictly mathematical po int of view, the introduction of a non-vanishing
+radius for the scattering center has the effect of regularizing the Coulomb potential which, otherwise, would diverge
+forr→0 (however, by contrast to other cases, Majorana did not resto re the full Coulomb potential by taking the
+limita→0 at the end of his calculations, but always maintained in the present c ase a finite value for a).
+In the problem proposed, Majorana studied the deviations from pu re Coulomb scattering by parameterizing it with
+the ratio i/iRof the effective scattering intensity under an angle θ(i.e. the flux of scattered particles per unit surface
+(normal to the incident direction) and per unit time) with respect to that deduced in the Rutherford approximation.
+Besides the radius a, this ratio depends also on the energy and momentum of the incident particles, but Majorana
+preferred to parameterize these by means of the scattering par ameter (or, as denoted by himself, the “minimum
+approach distance”) bin the Coulomb limit, defined by K/b=T, and from the wavelength λof the free particle. By
+settingα=a/(λ/2π),β=b/(λ/2π), he obtained:
+i=f(α,β,θ)iR. (6)
+Of course, in the pure Coulomb limit, the Rutherford formula should b e recovered, so that f(0,β,θ) = 1. The
+calculations then proceed at evaluating the function f(α,β,θ) by keeping αfixed (that is, for fixed scatterer size)
+and considering the limit β→0 (for scattering particles with increasing momentum, approaching nearer and nearer
+the center). The wavefunction of the system is, then, evaluated perturbatively by expanding it in a series ruled by
+the experimental parameter β, at zeroth order ( β= 0) being used the WKB method [39]. After some passages5. He
+obtains the following analytic result in the approximation considered ( ρ=r/(λ/2π))
+f(α,β,θ) = 2sinθ
+2/integraldisplay∞
+0ρ/radicalbig
+ρ2+α2sin/bracketleftbigg
+2ρsinθ
+2/bracketrightbigg
+dρ, (7)
+which can be expressed in term of the Bessel kfunction k(αsinθ/2) [40]. In this approximation Majorana then finds
+that the actual scattering intensity may be substantially different from that of the Rutherford formula, for backward
+scattering and provided that the radius ais appreciably different from zero (the ratio i/iRapproaching zero for θ=π
+and increasing α).
+This remarkable result is not explicitly reported in the Quaderni , but Majorana gives an attempt to tabulate the
+radial wavefunction uℓ, satisfying the equation
+u′
+ℓ+/parenleftBigg
+1−β/radicalbig
+ρ2+α2−ℓ(ℓ+1)
+r2/parenrightBigg
+ue= 0, (8)
+obtained numerically with the method of the particular solutions. Alth ough numerical solutions are searched only for
+the particular case of ℓ= 0 andβ= 0.4 (and not β= 0), here the interesting point is that Majorana explicitly reports
+that “for the hydrogen atom we consider the values β= 0.4,0.5,0.6,0.7andα= 0,0.2,0.4,0.6,0.8,1”. Unfortunately,
+no further discussion is given in the Quaderni ; the key point is, however, the fact that Majorana uses λ/2πas
+the length scale for boththe lengths aandb. Now, it is very natural to measure the scattering parameter (or “the
+minimum approachdistance”) in terms ofthe free particlewavelengt hor, equivalently, (the inverseof) the free particle
+momentum ( λ=h/p), since it is obvious that increasing the momentum (or decreasing th e probe wavelength), the
+incident particle approaches nearer and nearer the scattering ce nter, thus decreasing b. The same would not apply,
+instead, to the radius of the scattering center which should be inde pendent of the incident particle properties, unless
+Majoranaconsiders effective, momentum-dependent, dimensionsforthe scatteringcenter. I n suchacase, theattention
+is evidently shifted from the investigation of an actual, particular ph ysical system considered (particles in a scattering
+potential) to the study of more general properties of the backgr ound field or space. We will come back later on this
+point.
+Finally, we conclude this section by mentioning also another modificatio n of the Coulomb potential considered by
+5In order to avoid convergence problems in the usage of the Gre en method, Majorana assumes that the scattering force acts o nly for
+distances less then a quantity R, and than let R→ ∞at the end of calculations. The potential in Eq.(4) is thus re placed during
+calculations, by k„1√
+r2+a2−1
+R«6
+Majorana [36], reminiscent of the Gamow potential for the descript ion ofα-decay process, namely
+V=
+
+V0,forr < R,
+k
+r,forr > R.(9)
+Here the role of the scattering center radius is played by the range R,α=R/(λ/2π), and one more parameter enters,
+that is the depth V0of the potential, measured with respect to the kinetic energy of th e incident particles, A=V0/T.
+Although, in this case, the calculations are only sketched, the key p oints envisaged above are as well present. Now
+the correction function in Eq.(6) is replaced by
+i
+iR=f/parenleftbiggV0
+T,R
+λ/2π,b
+λ/2π,θ/parenrightbigg
+(10)
+where “for the hydrogen” Majorana considers the same values as before for the α,βparameters (except, obviously,
+α= 0,0.2), while A=2,1.5,1,0.5,0,-0.5,-1,-1.5,-2,-2.5,-3,...,-8. Interestingly, besides negative values for V0(as in the
+Gamow model), Majorana also considers few positive values for the d epth of the potential. As above, however, the
+very notable point is that both the potential parameters V0andRappear to be strictly related to the energy and
+momentum of the probe particles.
+For both the cases studied, in the presence of continuum states ( Eq.(4)) or bound states (Eq.(9)), the reasoning
+is thus the same. This is even more intriguing if compared to the compr ehensive study of the scattering from a pure
+Coulomb potential, performed by Majorana [36] some pages after those discussed here up to now. Indeed, in this case
+the length unit is no more the free particle wavelength, the Bohr rad ius being introduced (and the energy unit is the
+Rydberg), evidently denoting possible applications to atomic problem s. From what discussed above, it is thus clear
+that Majorana’s original motivations for the studies of quasi-Coulo mbian scattering are different from those standard
+related to atomic and nuclear physics.
+RETARDED ELECTROMAGNETIC FIELDS AND THE INTRODUCTION OF A F UNDAMENTAL
+LENGTH
+Some light on this issue may come from other pages of Majorana’s Quaderni [36], where the possibility is considered
+of introducing an intrinsic constant time delay (or, equivalently, an in trinsic space constant) in the expression for the
+retarded electromagnetic fields. Here the treatment is fully classic al, and goes as follows.
+The starting point is the wave equation satisfied by any component o f the electromagnetic potentials, denoted
+generically with f(x,y,z,t), and then the evaluation of the D’Alembert operator for the stan dard retarded field
+denoted by
+ϕ(x,y,x,t) =f/parenleftBig
+x,y,z,t−r
+c/parenrightBig
+≡f(x,y,z,t). (11)
+The known result is, of course, the following:
+/squaref=∇2ϕ+2
+rc˙ϕ+2
+c∂2ϕ
+∂r∂t, (12)
+where a dot indicates time differentiation. Majorana then introduce s an intrinsic space constant ε, so that Eq.(11) is
+replaced by
+ϕ(x,y,z,t) =f/parenleftBigg
+x,y,z,t−√
+r2+ε2
+c/parenrightBigg
+=˜f(x,y,z,t). (13)
+The D’Alembertian term entering into the wave equation is thus
+/tildewider/squaref=∇2ϕ−ε2
+c2(r2+ε2)¨ϕ+2r2+3ε2
+c(r2+ε2)3/2˙ϕ+2r
+c√
+r2+ε2∂2ϕ
+∂r∂t, (14)
+which replacesEq.(12). Majoranaalsointroducesexplicitly atime de layτwhich, in his view, is a“universalconstant”
+taking the value τ= 0 classically; in his calculations, however, he always uses the length
+ε=τc (15)7
+(already introduced), so that we here follow this line of reasoning.
+The wave equation is apparently solved by using the Green method; d enoting with S′the generic source function
+(charge density ρor current density J), the modified generic potential Φ, solution of the wave equation, a ssumes now
+the following from:
+Φ =/integraldisplay1√
+R2+ε2S/parenleftBigg
+x′,y′,z′,t−√
+R2+ε2
+c/parenrightBigg
+dx′dy′dz′(16)
+(withR=|r−r′|, as usual). Majorana then ends his explicit calculations with the writin g of Eq.(16) expanded up
+to second order in εforε→0 (thus approaching the classical limit, as of course coming from exp eriments):
+Φ =/integraldisplay1
+RS/parenleftbigg
+x′,y′,z′,t−R
+c/parenrightbigg
+dx′dy′dz′
+−ε2/braceleftbigg/integraldisplay1
+2R3S/parenleftbigg
+x′,y′,z′,t−R
+c/parenrightbigg
+dx′dy′dz′
++/integraldisplay1
+2R2cS′/parenleftbigg
+x′,y′,z′,t−R
+c/parenrightbigg
+dx′dy′dz′/bracerightbigg
++... . (17)
+Some comments are now in order. Unfortunately, again we do not kn ow the motivations for such a study, but
+it is intriguing the following observation. By taking the ordinary electr ic monopole limit [41] into Eq.(16), with
+S(r,t−r
+c) =ρ(r),|r−r′|∼=|r|=rand/integraldisplay
+ρ(r′)dx′dy′dz′=Qbeing the electric charge, one can easily obtain
+Φ =Q√
+r2+ε2, (18)
+that is exactly the potential considered in Eq.(4).
+Although Eq.(18) is not explicitly reported in the Quaderni , it seems very likely that the topics discussed here
+and in the previous section (and considered in no other place in the Quaderni ) are related. If so, the “magnitude of
+the radius of the scatterer” in the scattering problem considered above, whose interpretation in terms of the physical
+dimensions of the scattering center led to difficulties, should be inter preted instead as the “universal” space constant
+introduced here. In this case Majorana’s original intention for the study of quasi-Coulombian scattering was not that
+of exploring the properties of the scattering particle (an atom or it s nucleus or even other elementary particles) but
+rather that of investigating the underlying properties of the phys ical space. The intrinsic space constant then plays
+the role of a fundamental length, like the one introduced by Planck in 1899 or that conjectured by Heisenberg in
+1938 (see the the Introduction). We do not know if Majorana effec tively thought to a given particular value but, in
+any case, to the best of our knowledge, with Majorana it is the first time that such a detailed study was effectively
+undertaken, as early as in the 1930s.
+A RELATION AMONG THE FUNDAMENTAL CONSTANTS
+The great interest of Majorana for electrodynamics [37], in its class ical or (especially) quantum form, is very well
+documented in his writings (see mainly Refs.[35] and [36]), so that ev en the particular studies described above do
+not seem at all surprising. Nevertheless, the line of thinking of such studies goes well beyond mere applications or
+generalizations of standard electrodynamics, as shown in the prev ious pages, involving questions about fundamental
+constants. It is intriguing that yet another time Majorana reason ed about such fundamental questions, though in an
+(apparently) different context, when he was still a student, as ea rly as in 1928. Indeed, in a page of his Volumetti (see
+section 31 of Volumetto II, in Ref.[35]), Majorana attempted to fin d a relation between the fundamental constants
+e,handc, or, more explicitly, an expression for the elementary charge eappearing in the Coulomb law, describing,
+for example, the electrostatic force between two electrons. The reasoning of Majorana goes as follows. In the region
+of space surrounding the two electrons, which are at a distance ℓapart, an electromagnetic field is present that, “in
+some sense” (according to Majorana’s wording), is quantized. Her e we should point out the first interesting thing
+since, in what follows (see below), it is evident that what is considered “quantized” by Majorana is undoubtedly the
+electromagnetic field acting among the two electrons. Nevertheles s, differently from what appearing elsewhere in his
+notes, Majorana does not refer explicitly to an electromagnetic field, but rather to the ’ether’ surrounding the two8
+electrons, that is (apparently) the physical space itself (in the imp roper language of that time). The two electrons are
+assumed to interact amongthem by means of a point particle (what w e could term the quantum of the electromagnetic
+field) moving with a group velocity equal to the speed of light c. Such particle is further assumed to propagate freely
+from one electron to the other, whereas it inverts its motion when “ colliding” with the electrons. The momentum of
+the quantum particle is deduced by a sort of Sommerfeld quantizatio n rule
+|p|=nh
+2ℓ,
+withn= 1. The “kick” received by the electron is therefore 2 |p|=h/ℓ, and since the number of collisions per unit
+time is 1 /T=c/2ℓ, the magnitude of the force acting on each electron is then F= 2|p|/T=hc/2ℓ2. By equating
+this expression to the Coulomb law, Majorana finally deduces the exp ression he searched for the elementary charge
+e=/radicalbig
+hc/2 which, however, as pointed out explicitly by himself, gives a value “21 times greater than the real one”.
+Though leading to results far from the truth, what discussed by Ma jorana is interesting for many reasons, one of
+which already mentioned above. Indeed, first of all, the mechanical model considered here is the first one to have
+been proposed as an attempt to deduce the value of the electric ch arge (the famous paper by Dirac in Ref.[42] was
+published few years after). However, as it should be evident from w hat reported above, the interpretation given by
+Majorana of the quantized electromagnetic field substantially coincides with that introduced mo re than a decade later
+by R.P. Feynman in quantum electrodynamics [43], the point particles e xchanged by electrons being assumed to be
+the photons. It is then remarkable that fundamental questions r egarding the constants of Nature discussed properly
+only in more recent times, were addressed in pioneering works by Maj orana as early as at the end of 1920s, where,
+on the other hand, the key ideas of quantum field theory were alrea dy present.
+CONCLUSIONS
+The problem of the existence and of the effects of a fundamental le ngth dates back to the early years of the last
+century. As reviewed in the introduction, this concept was introdu ced and rediscovered in physics several times since
+then, in order to solve difficulties of different kind as, e.g., in regularizin g some divergent results. It is, however, quite
+notable the presence of “side” effects associated to (or, rather , postulated about) the existence of such elementary
+space intervals, which could allow, in Heisenberg’s view for example, th e derivation of the mass of the particles, or
+even play a key role in the physical interpretation of the spin variable .
+Strictly related to the introduction of a fundamental length is, on t he other hand, the assumption of the existence
+of a fundamental time, although such hypothesis is less fashioned a mong physicists than the first one.
+In more recent times, the present subject is embedded into the mo re general framework of the quantization of the
+spacetime, where attempts to provide a well-defined quantum theo ry of gravity manifest into different theoretical
+approaches, with the common denominator of a fundamental lengt h scale.
+In this paper we have shown that the known picture about this intrig uing concept has been further enriched by
+several, yet unknown, contributions by Ettore Majorana. As ear ly as the beginning of 1930s, indeed, the Italian
+physicist not only conjectured (as already Planck and, later, othe rs did) the existence of both an elementary length
+and of a fundamental time scale, but also investigated in some detail the immediate physical consequences of such
+hypothesis. In this respect, his contributions once again [33] reve al a farsighted intuition, and result to be quite
+precious pieces of research both on the historical point of view and for present day theoretical investigations.
+Majorana considered, for example, the scattering from electrica lly charged particles when the underlying properties
+of the space are such that an intrinsic length exists, resulting in the modification of the pure Coulomb potential as
+in Eq.(4) or (18). Even just at the classical level, he obtained the s ame effect by introducing an intrinsic time delay,
+which is considered a “universal constant”, in the expression for t he retarded electromagnetic fields.
+At the quantum level, instead, it is quite interesting still another con tribution by Majorana about fundamental
+constants, aimed at finding a relation between e,/planckover2pi1,cor, rather, aimed at deducing the value of the elementary
+charge. Such a scope was pursued, few years later, in a famous pa per by Dirac [42], although based on a different
+theoretical scheme and with different results. Indeed, the partic ularnumerical prediction by Majorana (as opposed to
+the particular theoretical prediction of the existence of magnetic monopoles by Dirac) is far fr om the truth, but the key
+ideabehind it anticipatedofmorethan adecadethe assumption, intr oducedbyFeynmanin quantumelectrodynamics,
+of photon exchange during electromagnetic processes.
+We expect that Majorana’s important ideas and applications recove red from the oblivion through this paper will
+bring some benefit also to the present day abstract theoretical r esearch on the fundamental properties of the physical
+spacetime.9
+Acknowledgments
+The authors warmly thank E.Recami for interesting discussions an d suggestions, while acknowledge the very kind
+collaboration by M.G.Cammarota and M.Longo.
+∗Electronic address: Salvatore.Esposito@na.infn.it
+†Electronic address: salesi@unibg.it
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+arXiv:1001.0051v2  [astro-ph.CO]  22 Apr 2010Draft version October 30, 2018
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+LUKEWARM DARK MATTER: BOSE CONDENSATION OF ULTRALIGHT PART ICLES
+Andrew P. Lundgren1, Mihai Bondarescu2, Ruxandra Bondarescu3, Jayashree Balakrishna4
+Draft version October 30, 2018
+ABSTRACT
+We discuss the thermal evolution and Bose-Einstein condensation o f ultra-light dark matter parti-
+cles at finite, realistic cosmologicaltemperatures. We find that if th ese particles decouple from regular
+matterbeforeStandardmodelparticlesannihilate, theirtempera turewillbeabout0.9K.Thistemper-
+atureissubstantiallylowerthanthetemperatureofCMBneutrinos andthusBigBangNucleosynthesis
+remains unaffected. In addition the temperature is consistent with WMAP 7-year+BAO+H0 obser-
+vations without fine-tuning. We focus on particles of mass of m∼10−23eV, which have Compton
+wavelengths of galactic scales. Agglomerations of these particles c an form stable halos and naturally
+prohibit small scale structure. They avoid over-abundance of dwa rf galaxies and may be favored by
+observations of dark matter distributions. We present numerical as well as approximate analytical
+solutions ofthe Friedmann-Klein-Gordonequationsand study the c osmologicalevolution of this scalar
+field dark matter from the early universe to the era of matter domin ation. Today, the particles in the
+ground state mimic presureless matter, while the excited state par ticles are radiation like.
+Subject headings: (cosmology:) theory, (cosmology:) dark matter, (cosmology:) ear ly universe
+1.INTRODUCTION
+Most matter in the universe is non-luminous. The
+observed flatness of the galactic rotation curves indi-
+cates the presence of dark matter halos around galax-
+ies. Observations of the cosmic microwave background
+anisotropies (Spergel et al.2007) combined with large-
+scale structure and type Ia supernova luminosity data
+(Reisset al.1998, 2004, 2005; Perlmutter et al.1999)
+constrain cosmological parameters finding that visible
+matter contributes only about 4% of the energy den-
+sity of the universe, as opposed to 22% being dark mat-
+ter and 74% dark energy. More recently, a clear sep-
+aration between the center of baryonic matter and the
+total center of mass was observed in the Bullet clus-
+ter (Clowe et al.2006) and later in other galaxy cluster
+collisions (Bradac et al.2008). These observations rein-
+force the claim that dark matter is indeed composed of
+weakly interacting particles and is not a modification of
+gravity.
+In the past few decades numerous dark matter can-
+didates have been suggested including WIMPs, ax-
+ions, and various spin zero bosons (Kamionkowski
+2007; Bertone et al.2005). Fundamental spin zero
+particles represented by scalar fields play an impor-
+tant role in particle physics models (Peccei & Quinn
+1977; Torres et al.2000). These particles could form
+gravitationally stable structures such as boson stars,
+soliton stars, and galactic halos through some type
+of Jeans instability mechanism (Seidel & Suen 1991;
+Urena-Lopez 2002; Alcubierre et al.2003). The stabil-
+ity and gravitational wave signatures of compact scalar
+stars have been studied numerically (Khlopov et. al.
+1985; Seidel & Suen 1990; Balakrishna et al.2006,
+2008).
+Bosonic halos in which scalar particles Bose-condense
+1Syracuse University, Syracuse, NY
+2University of Mississippi, Oxford, MS
+3Pennsylvania State University, State College, PA
+4Harris Stowe State University, St Louis, MOand form gravitationally stable structures are supported
+against collapse by Heisenberg’s uncertainty principle
+like boson stars. Structure formation on scales smaller
+than the spreading of an individual boson (the Comp-
+ton wavelength of one particle) is forbidden by quan-
+tum mechanics (Hu et al.2000; Sahni & Wang 2000;
+Lee & Lim 2008). Halos formed from ultralight scalars
+with Compton wavelength of galactic scales thus do not
+leadto over-abundanceofdwarfgalaxiesunlikecold dark
+matter simulations with heavier bosons (Navarro et al.
+1996; Matos & Urena-Lopez 2001; Alcubierre et al.
+2002; Salucci et al.2003).
+Scalar field halos have been fit to rotation
+curves of spiral galaxies (Schunck & Liddle 1997;
+Guzman & Urena-Lopez 2003; Matos & Urena-Lopez
+2000; Arbey et al.2001; Boehmer & Harko 2007).
+By using the mass as a free parameter to fit ro-
+tation curves Arbey et al.(2001) have obtained
+m= (0.4−1.6)×10−23eV for non-interacting ultra-
+light bosons. In a followup paper, Arbey et al.(2002)
+performed a non-thermal analysis of ultralight bosonic
+halos.
+Urena-Lopez (2009) pointed out that scalarsfield par-
+ticlescanBose-condenseatfinite temperaturesresurrect-
+ing previous work on relativistic Bose-Einstein conden-
+sation by Parker & Zhang (1991, 1993). A condensate
+is considered relativistic when the temperature of the
+condensate is significantly larger than the mass of one
+boson. Parker & Zhang (1993) discuss inflationary ex-
+pansion driven by a relativistic Bose-Einstein condensate
+that then evolves into a radiation dominated universe.
+Inthispaperweperformathermalanalysisofthepost-
+inflationary cosmological behavior of scalar field dark
+matter formed from ultra-lightbosons. We use the quan-
+tum field theory formalism of Parker & Fulling (1974)
+and extend the analysis that Hu (1982) used for a de-
+scription of finite temperature effects in the early uni-
+verse. The bosons are described by a complex scalar
+field to provide a conserved charge.2
+We assume the scalar particles decouple after inflation
+in the early universe, after which the field has a simple
+quadratic potential with no interactions. There are no
+constraintson this field from particle physicsor precision
+tests of gravity, as there would be for an interacting light
+scalar field. Ultralight particles ( m∼10−23eV) form a
+pure ground state Bose-Einstein condensate with a high
+critical temperature behaving like cold dark matter to-
+day. Particles in excited states behave like radiation to-
+dayand hence contribute to the amountof hot darkmat-
+ter in the universe, which is constrained by cosmological
+observations. Our model assumes that the interaction of
+the scalar field with normal matter turns off in the early
+universe, at a temperature where most Standard Model
+particles have not yet annihilated, which yields a scalar
+field temperature TΦ= 0.9 K with no fine-tuning. A
+reasonable choice of TΦstrongly consistent with cosmo-
+logical observations, is TΦ/lessorsimilar1.5 K. Since these estimates
+come from states that behave like radiation, they are in-
+dependent of particle mass.
+In§2 we review relativistic Bose-condensation at high
+temperaturesanddiscussthe temperatureat whichthese
+scalars could decouple. §3 follows the cosmological evo-
+lution of the field. We use units where /planckover2pi1=c= 1.
+2.BOSE-EINSTEIN CONDENSATION AT FINITE
+TEMPERATURE
+We consider a system of ultralight ( m∼10−23eV)
+relativistic bosons represented by complex scalar fields.
+The condition for a relativistic condensate is that the
+temperatureofthe condensate T≫m, which iscertainly
+true up to the present day (Urena-Lopez 2009).
+In the case of a complex field, there is a conserved
+charge, which is required for traditional Bose-Einstein
+condensation (BEC)1. The charge density is defined as
+the excess of particles nover anti-particles ¯ n:
+q=n−¯n. (1)
+For the excited states, this charge density is
+(Mukhanov 2005)
+qex=gµT2
+3, (2)
+wheregisthenumberofdegreesoffreedomofthesystem
+and the chemical potential µ≤m. The maximum qex=
+mT2/3 occurs for µ=m. For ultra-light bosons, the
+excess of bosons over anti-bosons in the excited states
+is small compared to the number density. Any charge
+added to the system when µ=mmust condense to the
+ground state. If‘ qis large, the ground state is populated
+almost exclusively by particles (Mukhanov 2005).
+The criticaltemperature below which condensationoc-
+curs is found in terms of the charge density of the dark
+matter particles (Urena-Lopez 2009; Mukhanov 2005)
+T <T c=/radicalbigg
+3q
+m. (3)
+WhenT <T cthe majorityofthe bosons will condenseto
+the ground state. In the ground state the particles will
+1Recently, Sikivie & Yang (2009) showed that dark matter ax-
+ions can form a BEC as well.behave like non-relativistic matter, while the particles in
+excited states will remain highly relativistic.
+Assuming that BEC occursand that most particles are
+in the ground state, a first approximation to the total
+dark matter density is
+ρDM≈(n+ ¯n)m. (4)
+The density of dark matter today ρ0
+DMis
+ρ0
+DM≈23%ρc, (5)
+whereρc≈4.19×10−11eV4. Sincen≫¯n
+n≈ρDM
+m≈1012eV3(6)
+and
+Tc≈1.7×1017eV∼1021K, (7)
+whichcorrespondstoaverypurecondensatetoday. Note
+thattherequiredchargedensityisveryhighimplyingthe
+necessityofamechanismthatwouldproducesuchalarge
+asymmetry of scalar particles over anti-particles.
+The matter in the excited states is relativistic and con-
+tributes to the density of hot dark matter ρHDM, com-
+monly parameterized by the effective number of neutrino
+speciesNeff. The nominal value due to the three known
+neutrino species plus small cosmological corrections is
+Neff= 3.04. The scalar field temperature TΦcan be
+determined from
+ρHDM= (Neff−3.04)7π2
+120T4
+ν=π2
+15T4
+Φ,(8)
+withTν= 1.95 K the neutrino temperature today.
+The value given by the WMAP 7-year results com-
+bined with measurements of the Baryon Acoustic Os-
+cillation and Hubble parameter (Komatsu et al.2010)
+isNeff= 4.34+0.86
+−0.88(68% confidence level). The 68%
+confidence level on Neffcorresponds to TΦbetween 1.51
+K and 2.25 K.
+Big Bang Nucleosynthesis measurements constrain
+relic abundances of deuterium and4Heputting a 2-
+sigma bound of Neff<3.5 (orNeff<3.3), which
+depends on the uncertainty in the4Heabundance con-
+sidered (Steigman 2007). This bound corresponds to
+TΦ<1.5 K (TΦ<1.35 K). Larger values of Neffallow a
+greater contribution of unknown particles like our scalar
+field toρHDM. This could either allow TΦto be larger
+or allow other particles to contribute to ρHDM.
+We now show that TΦcan be less than the observa-
+tional upper limits and substantially less than Tν, with-
+out fine-tuning. Our model assumes that the scalar field
+interactions with normal matter turned off completely
+in the very early universe. Subsequently, the heavier
+Standard Model particles annihilated, dumping their en-
+ergyintothe photons, neutrinos, electrons, and positrons
+that remained, increasing their temperature relative to
+that of the scalar field. The photons and leptons have
+aboutgℓ= 10.75 degrees of freedom, the complex scalar
+field has 2, and when all the Standard Model particles
+are present g∗≈100. Following a standard textbook
+(Mukhanov 2005), entropy conservation gives a relation
+for the temperatures
+gℓT3
+ν+2T3
+Φ=g∗T3
+Φ. (9)
+This givesTΦ≈0.9K, well within our constraints.3
+3.COSMOLOGICAL EVOLUTION
+The density evolution of the universe must closely fol-
+low the standard ΛCDM model at times later than nu-
+cleosynthesis, at a≈10−10. During radiation domina-
+tion, ourscalarfield must be asubdominant contribution
+to the density of the universe, and during matter dom-
+ination it must be a replacement for dark matter. As
+shown below, the excited states are a subdominant con-
+tribution to the density. The macroscopically-occupied
+groundstatehas ρ∝a−6at earlytimesand mustbe con-
+strained to be less dense than the density of radiation for
+at least all times after nucleosynthesis.
+We first derive equations in a general background, as
+well as expressions for density and pressure, then spe-
+cialize to the epochs of radiation and matter domination.
+We also present numerical solutions to the equations and
+showthat the initial conditionscanbe adjusted to satisfy
+the cosmological constraints.
+3.1.Evolution Equations
+The line element in an expanding universe can be writ-
+ten as the Friedmann-Robertson-Walker (FRW) metric
+ds2=−dt2+a(t)2/parenleftbig
+dx2+dy2+dz2/parenrightbig
+.(10)
+The Klein-Gordon equation
+✷Φ−m2Φ = 0
+is derived from a Lagrangian density of the form
+L=1
+2√−g(gµν∂µΦ∂νΦ−m2Φ).(11)
+The density and pressure can be defined in the usual
+way
+ρ=1
+2/parenleftbig
+∂tΦ†∂tΦ+∂jΦ†∂jΦ+m2ΦΦ†/parenrightbig
+,(12)
+p=1
+2/parenleftbig
+∂tΦ†∂tΦ+∂jΦ†∂jΦ−m2ΦΦ†/parenrightbig
+,
+where Greek indices vary between 1 and 4 and the index
+jvaries between 1 and 3.
+Following Hu (1982), we perform a series of variable
+transformations to expose the conformal properties of
+the scalar field equation. We introduce a conformal time
+coordinate defined by dt=adτ. We also make the sub-
+stitution Ψ = aΦ. The metric is conformally static
+ds2=a(τ)2/parenleftbig
+−dτ2+dx2+dy2+dz2/parenrightbig
+.(13)
+We cannow rewrite the Klein-Gordonequation using the
+flatspaceoperator /tildewide✷≡ −∂2
+τ+∂2
+x+∂2
+y+∂2
+zandremember-
+ing that the d’Alembertian ✷= (√−g)−1∂µ(√−ggµν∂ν)
+1
+a3/tildewide✷Ψ+a′′
+a4Ψ−m2
+aΨ = 0, (14)
+where′denotes the derivativewith respect to τ. The last
+two terms in Eq. (14) break conformal invariance. The
+invariance could be restored, as in Hu (1982), by setting
+the mass to zero and adding a term proportional to the
+four-dimensional Ricci scalar. However, in this case the
+field would no longer be minimally coupled. We choose
+to treat the terms that break conformal invariance as a
+perturbation.The solution to the scalar field equation can be decom-
+posed into modes (Parker & Fulling 1974)
+Ψ(x,τ) =/integraldisplay
+d3/vectorkA/vectorkψk(τ)ei/vectork·/vector x+H.c.,(15)
+where H.c. denotes the Hermitian conjugate and ψksat-
+isfies
+d2ψk
+dτ2+/bracketleftbigg
+k2−a′′
+a+a2m2/bracketrightbigg
+ψk= 0.(16)
+The conserved current in mode kcan be written as
+J0k=1
+i(ψk∂τψ⋆
+k−ψ⋆
+k∂τψk). (17)
+Thecanonicalcommutationrelationofthe field Φand its
+conjugate momentum Π leads to the usual commutation
+relations
+[A/vectork,A/vectork′] = 0,[A/vectork,A†
+/vectork′] =δ(/vectork,/vectork′),(18)
+when the conserved current is chosen to be J0k= 1 for
+particles and J0k=−1 for anti-particles. The opera-
+torA/vectorkcorresponds to physical particles and the num-
+ber density of particles is defined to be n=< A†
+/vectorkA/vectork>
+(Parker & Fulling 1974).
+The commutation relations are automatically satisfied
+if we takeψkof the form
+ψ/vectork(τ) =1√2ωke−i/integraltextτωkdτ′. (19)
+Each mode is now characterized by its eigenfunction ωk
+given by
+−1
+2(a2H)2ωkd2ωk
+da2+3
+4(a2H)2/parenleftbiggdωk
+da/parenrightbigg2
+(20)
+−ω2
+kaHd(a2H)
+da+ωk
+2(a2H)d(a2H)
+dadωk
+da−ω4
+k
++(k2+a2m2)ω2
+k= 0.
+To concomitantly solve the Friedmann equation
+H2=8πG
+3ρ (21)
+weapproximatetheHubbleparameterindifferentepochs
+power laws of the form H=a′/a2=H0a−n. Hereρ=
+ρrad+ρΛ+ρm, the radiation energy density ρrad∝a−4,
+the matter density ρm≈ρΦ∝a−3is dominated by dark
+matter and the dark energy term ρΛ= constant in a
+ΛCDM model. Thus, the exponent is n= 2 during the
+radiation domination era and n= 3/2 during the matter
+domination era.
+3.2.Radiation domination
+In the radiation domination regime ( n= 2), the scalar
+field Eq. (16) reduces to the flat space wave equation
+with an effective mass that varies with the scale factor
+d2ψk
+dτ2+/parenleftbig
+k2+a2m2/parenrightbig
+ψk= 0. (22)
+The mass term is the only perturbation from conformal
+invariance (Since a′′= 0.) We extend the analysis of Hu4
+10−1010−910−810−710−610−510−4100101010201030
+aρ (eV4)
+  
+ρDM − Solution 1
+ρDM − Solution 2
+ρrad
+(a)
+10−810−710−6−1−0.500.51
+aw
+  
+Solution 1
+Solution 2
+(b)
+Fig. 1.— The dark matter density ρDM(along with ρrad) and
+w=pDM/ρDMare displayed as a function of scale factor a.
+(1982) to determine the averagedensity in excited states.
+The density in state kis
+ρk=1
+2/bracketleftbiggH2
+2ωka2+(ω′
+k)2
+8a4ω3
+k+Hω′
+k
+2a3ω2
+k+ωk
+a4/bracketrightbigg
+.(23)
+Modes with k≫amare effectively massless ( ωk≈√
+k2+a2m2). The total density in the excited states
+is
+ρex=1
+π2/integraldisplay∞
+0k2dkρk
+exp[ωk/(Ta)]−1(24)
+=T4π2
+15+T2H2
+12−m2
+12T2
+a2+...,
+wheretheω′
+ktermsinEq.(23)contributetohigherorder.
+WhenT≫H, theT4term dominates the density and
+the excited states behave like radiation.
+Forthegroundstate( k= 0), Eq. (20) canbe rewritten
+as
+−y
+2d2y
+dx2+3
+4/parenleftbiggdy
+dx/parenrightbigg2
+−y4+x2y2= 0,(25)
+whereyandxare dimensionless variables defined by
+ω0=/radicalbig
+H0rmy, a=/radicalbigg
+H0r
+mx, (26)
+whereH=H0ra−2withH0r≈1.4×10−35eV. Note
+thatx= 1 (a≈10−6), which corresponds to H=m, is
+the transition to matter-like behavior for these particles.
+Whenx≪1 (orH≫m), we can neglect the x2y2term.
+Now Eq. (25) has an exact solution of
+y(x) =C0
+1+C2
+0(x−x0)2, (27)whereC0andx0are constants. When x0>1,yis ap-
+proximatelyconstant. When x0/lessorsimilar1,thetypicalbehavior
+is thatyfor smallx, peaks atx=x0with a height C0,
+andthen falls offas C−1
+0x−2(the higher C0, the narrower
+the peak and hence the transition between the constant
+andx−2behaviors is more abrupt). Initially, y= con-
+stant and the density ρ0and pressure p0for the ground
+state are ∝a−6. Whenx≫x0and ifx≫1/|C0|then
+y=C−1
+0x−2. The pressure and the density then have
+two terms
+ρ0=H3/2
+0r
+4a6m1/2C0+m5/2C0
+2H3/2
+0r(28)
+p0=H3/2
+0r
+4a6m1/2C0−m5/2C0
+2H3/2
+0r,
+where thea−6term dominates at early times. The den-
+sity transitions to a cosmologicalconstant with p0=−ρ0
+whena≈H1/2
+0rC−1/3
+02−1/6m−1/2. IfρΦ> ρradthen
+ρΦ∝a−6(Arbeyet al.2002). However, in this regime
+the above derivation becomes invalid.
+3.3.Matter Domination
+In the matter domination regime ( n= 3/2), Eq. (16)
+becomes
+d2ψk
+dτ2+/parenleftbigg
+k2−H2
+0m
+2a+a2m2/parenrightbigg
+ψk= 0,(29)
+whereH=H0ma−3/2withH0m≈7.8×10−34eV.
+Using Eq. (19) we obtain an equation for ωk:
+−ωka
+2d2ωk
+da2−ωk
+4dωk
+da+3a
+4/parenleftbiggdωk
+da/parenrightbigg2
+−ω4
+k
+H2
+0m(30)
++ω2
+k/bracketleftBigg/parenleftbiggk
+H0m/parenrightbigg2
+−1
+2a+/parenleftbiggam
+H0m/parenrightbigg2/bracketrightBigg
+= 0.
+For the ground state ( k= 0), this equation has an exact
+solution
+ω0=am
+C1sin[4ma3/2/(3H0m)+α]+C2,(31)
+whereC1,C2and the phase αare constants with the
+constraintC2
+2−C2
+1= 1. This solution is in agreement
+with Arbey et al.(2002). When C1= 0 the solution
+reduces to ω0=am. Solutions with non-zero C1oscil-
+late around the ω0=amsolution. Eq. (31) can also be
+written in terms of tas
+ω0=am
+C1sin(2mt+α)+C2. (32)
+The oscillations have a period of π/m(∼a few years
+form= 10−23eV). The pressure averages to zero on
+cosmological timescales causing the ground state scalar
+field particles to behave like pressureless matter.
+3.4.Numerical Solution
+We also solve Eq. (20) numerically including the effect
+of both radiation and matter in the Hubble parameter,
+H=/radicalBig
+H2
+0ra−4+H2
+0ma−3. (33)5
+The numerical solutions are fully specified by the value
+of the field and its first derivative at a given a, as well
+as an overall scaling of the density. In both cases, the
+overall scaling of the density was chosen to match the
+observed cosmological density of cold dark matter.
+Two representative solutions are shown in Fig. 1. The
+initial conditions in terms of the xandyvariables intro-
+duced above are y(10−8)≈0.48 andy′(10−8) = 0.49 for
+solution 1, and y(10−8)≈5×103andy′(10−8)≈5.×107
+for solution 2. Fig. 1(a) shows the ground state density
+as a function of scale factor, along with the radiation
+densityρrad. Fig. 1b displays w=pDM/ρDM. The den-
+sity at early times decays like a−6and is determined by
+the massofthe scalarparticleand the requireddensity in
+the groundstate at latetimes. Solution 1 requiresa large
+density at early times which is not compatible with the
+standard ΛCDM model. Solution 2 has a phase where
+the ground state density is constant. This allows the
+initial density to be much lower and therefore compati-
+ble with the standard model up to the the approximate
+time of nucleosynthesis ( a≈10−10). The final density
+is the same, but woscillates at late times around the
+pressureless w= 0 solution.
+4.CONCLUSION
+In this paper we assume dark matter is composed
+of ultralight scalar particles ( m= 10−23eV) with a
+Compton wavelength of galactic scales. Halos formed
+from such particles naturally do not exhibit small scale
+structure, avoiding the over-abundance of dwarf galax-
+ies. Urena-Lopez (2009) showed that for m <10−14
+eV Bose-condensation always occurs. The condensate
+has the correct cosmological behavior today. When
+H <m, particlesin the groundstate behavelike presure-
+less matter while particles in excited states act as radia-tion. When H >m, the dark matter density is initially
+ρΦ∝a−6and then switches to a cosmological-constant
+behavior(Arbey et al.2002). The density of the excited
+states remains radiation-like ρΦ ex∝a−4untilT∼H
+(a∼10−32) and is sub-dominant to the density of known
+radiation.
+We find that if these particles decouple from regu-
+lar matter before Standard Model particles annihilate,
+their temperature today is TΦ≈0.9 K. This tempera-
+ture is substantially lower than the temperature of the
+CMB and neutrinos, leaving nucleosynthesis unaffected.
+It is consistent with cosmological constraints on the
+amount of hot dark matter in the universe from WMAP
+7 year+BAO+H0 observations, which yield a one-sigma
+upper limit of TΦ/lessorsimilar2.25 K that is independent of parti-
+cle mass. After decoupling the scalar field has no inter-
+actions and hence cannot be detected by particle physics
+experiments or precision tests of gravity.
+A challenging requirement for our model is the large
+particle-antiparticle asymmetry that may arise from sta-
+tistical fluctuations in the early universe or from asym-
+metric reactions before decoupling. Understanding this
+better is the subject of future work. A more obvious
+challenge is the low mass, which seems unnatural from a
+particlephysics perspective. However, evenlighter scalar
+fields (m∼10−33eV) have been proposed to explain the
+accelerated expansion of the universe (Ratra & Peebles
+1988; Coble et al.1997).
+ACKNOWLEDGEMENTS
+We thank Duncan Brown, Sam Finn, Ed Seidel, Ira
+Wasserman and Ed Witten for inspiring discussions.
+This research is partially funded by NSF grants Nos.
+PHY 06-53462 and PHY-0847611.
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diff --git a/1001.0052.txt b/1001.0052.txt
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+arXiv:1001.0052v1  [math-ph]  30 Dec 2009NewPhase-IntegralMethodPlatformFunction
+New Phase-Integral Method PlatformFunction
+S.Yngve1,a)andB.Thid ´e2,b)
+1)Department of Physics and Astronomy, Theoretical Physics, Uppsala University, P.O. Box 516, SE-75120 Uppsala,
+Sweden,
+2)SwedishInstitute of Physics,Physics inSpace, P.O.Box 537 , SE-75121Uppsala, Swedenc)
+(Dated: 12December 2009; Revised1November 2018)
+The phase-integralmethod(PIM) is an asymptotic methodof t he geometricaloptics or semi-classical type for solving
+approximately,but in many cases very accurately, a wide cla ss of differential equations in physics. Unlike the related
+(J)WKB method, the higher-order corrections in the PIM can b e generated from a generic, unspecified base function,
+providingaddedsymmetryandflexibility. However,withthe conventionalapproachofusingthenext-to-lowest(third)
+ordercorrectiontotheintegrandinthephaseintegralasap latformforcalculatinghigher(fifth,seventh,ninth,...) order
+corrections,the higher-ordercalculationsveryoftenbec omequitecomplicated.
+We therefore introduce a new platform function, which consi derably simplifies the calculation of the third-order
+contributionfora wide rangeof problems. We also presentdi rectly integrableconditionsforthe phase integral,which
+so farseemtohavegoneunnoticed.
+Foralargenumberofobservables,ouranalysismakesaclear erdistinctionbetweenphysicaland,inasense,unphys-
+ical contributions.
+PACS numbers: 03.65.Sq,02.30.Hq,02.30.Mv,02.60.Nm
+Keywords: WKBmethod,phase-integralapproximations,sem iclassical applications
+I. INTRODUCTION
+In their pioneering 1936 article on the calculation of
+Coulombwavefunctionsinnon-relativisticquantummechan -
+ics, Yost, Wheeler and Breit1commented on the fact that the
+WKB methodin the first-orderapproximationsometimes has
+an unsatisfactory accuracy. Moreover, they found that the r e-
+placement, originally suggested by Kramers,2ofl(l+1)by
+(l+1/2)2wherelis the orbital angular momentumquantum
+number, does not always improve the results obtained in the
+first-order WKB approximation. This situation can mostly
+be remedied by going to the next-order approximation. The
+phase-integral method of Fr¨ oman and Fr¨ oman3provides a
+straightforward structure for calculating the contributi ons of
+order five and higher. For this, Ref 4 introduces a platform
+functionε0(z)which, apart from a factor 1 /2, is the third-
+ordercontribution.
+In the Fr¨ oman and Fr¨ oman PIM, the third-order contribu-
+tionis themost importantcorrection. Hence,it isdesirabl eto
+find ways to simplify the calculationof this contribution. F or
+thispurpose,wehereintroducea moreflexibleplatformfunc -
+tionPs(z)which generalizes ε0(z)in a way that depends on
+thenumberofsingularitiesintheintegrandandwhichhasth e
+potentialto simplifythe evaluationofhigher-ordercorre ction
+terms. As a result, the choice of the so called base function,
+usuallydenoted Q(z), becomesmoretransparent.
+a)Electronic mail: staffan.yngve@fysast.uu.se.
+b)Electronic mail: bt@irfu.se.; Home page: www.physics.irfu.se/
+˜bt.
+c)Also atLOIS Space Centre, V¨ axj¨ o University, SE-35195 V¨ a xj¨ o, Sweden.II. HIGHER-ORDERPHASE-INTEGRAL
+APPROXIMATIONSANDTHE PLATFORMFUNCTION
+Letusconsiderthedifferentialequation
+d2ψ
+dz2+R(z)ψ=0 (1)
+whereR(z)isameromorphicfunctionofthecomplexvariable
+z. Following Ref. 4, we introduce into equation (1) a “small”
+book-kepingparameter λ, ultimately to be set equal to unity,
+suchthatwe obtaintheauxiliarydifferentialequation
+d2ψ
+dz2+/parenleftbiggQ2(z)
+λ2+R(z)−Q2(z)/parenrightbigg
+ψ=0 (2)
+whichgoesoverintoequation(1)when λ=1. Themeromor-
+phic function Q(z)is the unspecified base which determines
+the phase-integral approximationof order one. The auxilia ry
+differential equation (2) has two linearly independent sol u-
+tions
+f+=q−1
+2(z)e+iw(z)(3a)
+f−=q−1
+2(z)e−iw(z)(3b)
+where
+w(z)=/integraldisplayz
+q(z)dz (4)
+and
+q2(z)−q1
+2(z)d2
+dz2q−1
+2(z)=R(z)−/parenleftbigg
+1−1
+λ2/parenrightbigg
+Q2(z)(5)
+TypesetbyREVT EXandAIPNewPhase-IntegralMethodPlatformFunction 2
+or,alternatively,
+/parenleftbiggq(z)λ
+Q(z)/parenrightbigg2
+−λ21
+Q2(z)[q(z)λ]1
+2d2
+dz2[q(z)λ]−1
+2
+−λ2R(z)−Q2(z)
+Q2(z)=1 (6)
+Introducingthecomplexvariable
+ζ=/integraldisplayz
+z0Q(z)dz (7)
+where, typically, {z0:Q2(z0)=0}, one finds, after some cal-
+culations,that equation(6)canberewritten
+/parenleftbiggq(z)λ
+Q(z)/parenrightbigg2
+−λ2/parenleftbiggq(z)λ
+Q(z)/parenrightbigg1
+2d2
+dζ2/parenleftbiggq(z)λ
+Q(z)/parenrightbigg1
+2
+−λ2R(z)−Q2(z)
+Q2(z)−λ21
+Q3
+2(z)d2
+dz21
+Q1
+2(z)=1 (8)
+[cf.equation (2.2.5), with (2.2.1), in Ref. 4]. Using equa-
+tion(7),andtheequality
+1
+Q3
+2(z)d2
+dz21
+Q1
+2(z)=1
+Q(z)d
+dz/parenleftbigg1
+2zsd
+dz1
+zsQ(z)/parenrightbigg
+−/parenleftbigg1
+2zsd
+dz1
+zsQ(z)/parenrightbigg2
++s(s−2)
+4z2Q2(z)(9)
+thiscanberewritten
+/parenleftbiggq(z)λ
+Q(z)/parenrightbigg2
+−λ2/parenleftbiggq(z)λ
+Q(z)/parenrightbigg1
+2d2
+dζ2/parenleftbiggq(z)λ
+Q(z)/parenrightbigg−1
+2
++λ2
+Q2(z)−R(z)−s(s−2)
+4z2
+Q2(z)−dPs(z)
+dζ+P2
+s(z)
+=1 (10)
+where
+Ps(z)=1
+2zsd
+dz1
+zsQ(z)(11)
+This is our new platform function. It allows us to write the
+righthandmemberof(3.5c)inRef. 3as
+R(z)−Q2(z)
+Q2(z)+dP0(z)
+dζ−P2
+0(z) (12)
+In practice, the choice of sinPs(z)more or less determines
+thebasefunction Q(z)aswe shallshownext.
+III. BASEFUNCTION
+To obtain a solution of equation (10), we make the formal
+Ansatz
+q(z)λ
+Q(z)=∞
+∑
+nY2n(z)λ2n(13)TABLE I. Properties of the platform function Ps(z), equation (11),
+for various choices of s.
+sPlatformfunction Base function Typical zrange
+01
+2d
+dz1
+Q(z)Q2(z)=R(z) −∞<z<∞
+11
+2zd
+dz1
+zQ(z)Q2(z)=R(z)−1
+4z20<z<∞
+−2l1
+2z2ld
+dzz2l
+Q(z)Q2(z)=R(z)+l(l+1)
+z20<z<∞
+andfindthat
+Y0=1 (14a)
+Y2=−1
+2P2
+s+1
+2dPs
+dζ−Q2(z)−R(z)−s(s−2)
+4z2
+2Q2(z)(14b)
+Using equations (3) and (13) with λ=1, and truncating
+aftern=2, we obtain the approximate local solutions of the
+originaldifferentialequation(1)as
+ψ±=[Q(z)(1+Y2(z))]−1
+2e±i/integraltextzQ(z)(1+Y2(z))dz(15)
+It should be noted that the integrands of these solutions con -
+tainthetotal derivative
+1
+2Q(z)dPs(z)
+dζ=1
+2dPs(z)
+dz(16)
+which is integrable irrespective of the expression for R(z)in
+the original differential equation (1) and of the choice of t he
+basefunction Q(z). Hence,we canwrite
+ψ±=
+1
+2dPs
+dz+/parenleftBig
+1−1
+2P2
+s/parenrightBig
+Q−Q2−R−s(s−2)
+4z2
+2Q2
+−1
+2
+×e±i/bracketleftBig
+1
+2Ps+/integraltextz/parenleftBig/parenleftbig
+1−1
+2P2s/parenrightbig
+Q−Q2−R−s(s−2)
+4z2
+2Q2/parenrightBig
+dz/bracketrightBig
+(17)
+From this we concludethat an obviouschoice for the (square
+ofthe)basefunctionis
+Q2(z)=R(z)+s(s−2)
+4z2(18)
+For the new platform function (11), three cases can be dis-
+cerned(seealso TableI):
+1.s=0. This corresponds to the case of an unmodified
+basefunction, Q2(z)=R(z). Onecanshowthattheuse
+oftheplatformfunction P0(z)simplifiesthecalculation
+of phase-integral approximations, e.g., for the Weber
+functions.
+2.s=1. This corresponds to the well-known Kramers-
+Langer modification for obtaining the base function
+Q2(z) =R(z)−1/(4z2). One can show that the use
+oftheplatformfunction P1(z)simplifiesthecalculation
+ofphase-integralapproximations, e.g.,fortheCoulomb
+wave functions.NewPhase-IntegralMethodPlatformFunction 3
+3.s=−2l. This corresponds to the omission of the cen-
+trifugal barrier in the base function, i.e., the choice
+Q2(z) =R(z)+l(l+1)/z2. This is (for l/ne}ationslash=0) a less
+known alternative; see, e.g., Chapter 7 in Ref. 3. One
+can show that the use of the platform function P−2l(z)
+simplifies the calculation of phase-integral approxima-
+tions for spherically and cylindrically symmetric prob-
+lems.
+Inall thethreeabovecases, thethird-ordersolutionbecom es
+ψ±=/parenleftbigg1
+2dPs
+dz+/parenleftBig
+1−1
+2P2
+s/parenrightBig
+Q/parenrightbigg−1
+2
+×e±i/bracketleftbig
+1
+2Ps+/integraltextz/parenleftbig
+1−1
+2P2s/parenrightbig
+Qdz/bracketrightbig
+(19)
+which exhibits the simplification attainable if the new plat -
+formfunction Ps(z)is used.
+IV. CONCLUSIONS
+We have derived a platform function Ps(z)for the third-
+order correction of the phase-integral method. Since Ps(z)isreadily generalized to problems involving more than one sin -
+gular point in R(z)and makes computational simplifications
+possible, this new platform function is in many cases prefer -
+abletotheconventionalPIM platformfunction ε0(z).
+ACKNOWLEDGMENTS
+Someofthenewideaspresentedinthispaperwereinspired
+byadiplomaworkbyOscarSt˚ al,whostudiedamodelforthe
+scatteringofaradiobeamoffacylindricalionizationtrai lpro-
+duced by ultra-high-energy cosmic particles interacting w ith
+theEarth’sloweratmosphere. Oneoftheauthors(B.T.)grat e-
+fullyacknowledgesthefinancialsupportfromtheSwedishRe -
+searchCouncil(VR).
+1F. L. Yost, John A. Wheeler, and G. Breit, “Coulomb wave funct ions in
+repulsive fields,” Phys.Rev. 49,174–189 (Jan 1936).
+2H. A. Kramers, “Wellenmechanik und halbzahlige quantisier ung,” Zeitschr.
+Phys.39, 828–840 (Oct1926).
+3Nanny Fr¨ oman and Per Olof Fr¨ oman, JWKB Approximation: Contributions
+to the Theory (North-Holland, Amsterdam, NL,1965).
+4Nanny Fr¨ oman and Per Olof Fr¨ oman, Physical Problems Solved by the
+Phase-IntegralMethod (CambridgeUniversityPress,Cambridge,UK,2002)
+ISBN 0-521-67476-6.
\ No newline at end of file
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new file mode 100644
index 0000000000000000000000000000000000000000..e8645becb66a7f1d120749c4de47af957c173f6a
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@@ -0,0 +1,1539 @@
+arXiv:1001.0053v4  [math.DS]  12 Sep 2011Rotation Vectors for Homeomorphisms of
+Non-Positively Curved Manifolds∗
+Pablo Lessa†
+November 8, 2018
+Abstract
+Rotation vectors, as defined for homeomorphisms of the torus that
+are isotopic to the identity, are generalized to such homeom orphisms of
+any complete Riemannian manifold with non-positive sectio nal curvature.
+These generalized rotation vectors are shown to exist for al most every
+orbit of such a dynamical system with respect to any invarian t measure
+with compact support. The concept is then extended to flows an d, as
+an application, it is shown how non-null rotation vectors ca n be used
+to construct a measurable semi-conjugacy between a given flo w and the
+geodesic flow of a manifold.
+Contents
+1 Introduction 2
+2 Statement of the Main Theorem 4
+3 Rate of Escape 6
+4 Geodesic Escorts 8
+5 Aligned Sequences 10
+6 Alignment of Random Sequences 14
+7 Proof of the Main Theorem 16
+8 Periodic points and Homological Rotation Vectors 16
+9 Past orbits 20
+9.1 Past and future rotation vectors . . . . . . . . . . . . . . . . . . 20
+9.2 An example without a bi-infinite geodesic escort . . . . . . . . . 23
+∗MSC 37E45 37A99
+†CMAT, Facultad de Ciencias, Universidad de la Rep´ ublica, U ruguay
+110 Rotation vectors for flows 24
+11 A Semi-conjugacy result for Flows on Fiber Bundles 25
+12 Examples 30
+12.1 Homeomorphisms of hyperbolic surfaces . . . . . . . . . . . . . . 30
+12.2 Hyperbolic Magnetic Flow . . . . . . . . . . . . . . . . . . . . . . 31
+Acknowledgments 33
+Appendix: Visibility Manifolds 34
+1 Introduction
+Since Poincar´ e introduced rotation numbers to study the dynamic s of homeo-
+morphismsofthecircletherehavebeenseveraltypesof‘rotation objects’defined
+with the purpose of capturing the way the trajectories of a dynam ical system
+‘wind around’ a manifold as time progresses (see [Mis07] for an overv iew).
+Perhaps the most direct generalization of rotation numbers has be en the
+concept of rotation vectors as defined for homeomorphisms isoto pic to the iden-
+tity on the two-dimensional torus. Although the situation is more co mplicated
+than for homeomorphisms of the circle (e.g. there may exist points w ithout
+a well defined rotation vector, and different points may have differe nt vectors
+associated to them) the concept has been successful at least in t he following two
+ways:
+1. All periodic orbits have a well defined rotation vector. And, more gener-
+ally, so do almost all points with respect to any invariant measure.
+2. Simple hypotheses on the set of rotation vectors have strong d ynamical
+consequences. For example, the existence of three non-collinear rotation
+vectors implies that there are infinitely many periodic points and that
+topological entropy is positive (see [LM91] and [Fra89]). Also, in some
+cases where there is a single rotation vector one can show that the re exists
+a semi-conjugacy to a rigid rotation (see [J¨ ag09]).
+Ongeneralmanifolds, asymptoticcyclesforflows(introducedin[Sc h57])and
+homological rotation vectors for homeomorphisms isotopic to the id entity (e.g.
+see [Fra96]) both give information about how orbits ‘wind around’ hom ology
+and have several applications.
+Even on a compact hyperbolic surface, one can construct a flow po ssessing a
+periodicorbitwhich ishomologicallytrivial(and hencedefines anull asy mptotic
+cycle) but homotopically non-trivial. Furthermore, there are cert ain Lagrangian
+dynamical systems for which the critical value of the Abelian coverin g (whose
+group of covering transformations is the first homology group of t he manifold
+over the integers) is strictly greater than that of the universal c overing (see
+[CI99, Section 2-7]). This suggests that between these two energ y levels the
+2trajectories may be exhibiting a behavior which is homologically trivial b ut
+homotopically non-trivial.
+With these two examples in mind, it is natural to ask whether there ex ists a
+useful concept of rotation vector that is applicable to manifolds ot her than tori
+and which measures homotopical information of trajectories.
+A first step in this direction was made in [BM93] where ‘asymptotic hom o-
+topy cycles’ are defined. However, as the authors noted, the co ncept lacks a
+general existence theorem.
+Morerecently, motivated by comparisontheoremsbetween differe nt geodesic
+flows on the same Riemannian manifold, P.Boyland introduced rotation mea-
+sures (see [Boy00]), which arise from associating to each trajecto ry of a flow on
+a fiber bundle (e.g. a Lagrangian flow) a geodesic of the base manifold (which
+is assumed to be compact and hyperbolic). For an invariant measure supported
+on a homotopically non-trivial periodic orbit the associated rotation measure is
+supported on the minimizing geodesic of the same free homotopy clas s. Hence
+rotation measures succeed in capturing homotopical information.
+In this work, we introduce rotation vectors for homeomorphisms w hich are
+isotopic to the identity on any complete Riemannian manifold of non-po sitive
+sectional curvature. In the case of the two-dimensional torus, our definition
+specializes to the usual one. Furthermore, we prove that rotatio n vectors have
+the same existence properties on all manifolds under consideration (i.e. all
+periodic orbits, and almost all points with respect to any invariant me asure
+with compact support, have rotation vectors). This is our main res ult, the
+proof of which occupies the first seven sections of the paper.
+In section 8 we show that, for periodic orbits in free homotopy class es of
+positive length (e.g. all non-trivial homotopy classes on a compact h yperbolic
+manifold) the associated rotation vector is non-null. And in section 9 we study
+the relationship between rotation vectors of a homeomorphism and its inverse
+(which on the torusare simply opposite), and conclude that for almo stall points
+the two are either both zero, or different (one can easily construc t an example
+of a periodic orbit on a compact hyperbolic surface such that the tw o vectors
+aren’t opposite).
+As an application, we extend the concept to flows and flows on fiber b undles
+over complete Riemannian manifolds of non-positive curvature (sec tions 10 and
+11) and prove a semi-conjugacy result which improves on [Boy00, Th eorem 4.1]
+by relaxing the restriction of strict hyperbolicity of the base manifo ld. Here, we
+require the base manifold to satisfy a visibility condition and the examp le given
+in section 9 shows that the result may fail without this hypothesis.
+In section 12 we discuss implications of our results for homeomorphis ms of
+surfaces and illustrate them with a discussion of the ‘magnetic flow’.
+It is noteworthy that, on one extreme, the existence theorem fo r rotation
+vectors on the two-torus is a direct consequence of Birkhoff’s erg odic theorem
+while, on the other extreme, it was observedby the authors of[BM9 3] that there
+doesn’t seem to be an adequate ergodic theorem for establishing th e existence
+of asymptotic homotopy cycles.
+The techniques we use for establishing our main theorem arose by su ccessive
+3generalization of Oseledets’ multiplicative ergodic theorem and were developed
+in [Ka˘ ı87],[KM99] and [KL06]. Since the space of interest for Oseldets’ theorem
+is that of positive definite symmetric real matrices (which has non-p ositive cur-
+vature but also flat totally geodesic submanifolds) these generaliza tions are well
+adapted to non-positively curved manifolds. However, the results in question
+concern orbits of cocycles of isometries and hence are not directly applicable to
+other homeomorphisms (for which, for example, it might be the case that two
+points on the same orbit don’t even have isometric neighborhoods).
+In sections 3 4 and 5, besides a standard application of Kingman’s erg odic
+theorem, wehavebasicallyisolatedKarlssonandMargulis’geometric arguments
+from [KM99] and specialized them to Hadamard manifolds (as opposed to uni-
+formly convex and Busemann non-positively curved metric spaces w hich was
+their original domain of application). In particular a more general ve rsion of
+our Lemma 11 is implicit in the proof of their main theorem. The fact tha t
+neither the result nor the relevant definitions are explicitly stated in the cited
+work is one reason why we include a full proof (another being that in t he case
+of Hadamard manifolds some simplifications are possible).
+In section 6, we prove a new ergodic theorem which is related to [KL0 6,
+Theorem 1.2] but applies to sequences not arising from cocycles of is ometries.
+This is enough to establish the main existence theorems for rotation vectors.
+The arguments of the rest of the paper rely on this result and on ge neral facts
+about non-positively curved manifolds which are presented in an app endix.
+2 Statement of the Main Theorem
+Let us begin by reviewing the definition of a rotation vector for a hom eomor-
+phism of the torus.
+Definition 1 (Rotation Vector for Torus Homeomorphisms) .LetTd=Rd/Zd
+be thed-dimensional torus, and let f:Td→Tdbe a homeomorphism that is
+isotopic to the identity. Also, fix a lift F:Rd→Rdoff.
+The rotation vector vF(x)of a point x∈Tdis the following limit in case it
+exists:
+vF(x) = lim
+n→+∞Fn(˜x)−˜x
+nwhere˜x∈Rdis any lift of x
+The following is a well known consequence of Birkhoff’s Ergodic Theore m.
+Proposition 1. Iff:Td→Tdis isotopic to the identity, F:Rd→Rdis a lift
+off, andµis anf-invariant Borel probability measure. Then the limit vF(x)
+exists for µ-almost every x∈Td.
+A point x∈Tdhas rotation vector vF(x) for a certain lift Fof a home-
+omorphism f:Td→Td, if and only if for any lift ˜ xofxit holds that
+/bardblFn˜x−(˜x+nvF(x))/bardbl=o(n) whenn→+∞. If one considers the usual
+flat metric on Tdthen the fact that the curve t/ma√sto→x+tvF(x) is a geodesic
+shows that the following definition generalizes Definition 1.
+4By a Riemannian covering we mean a covering map π:/tildewiderM→Mbetween
+Riemannian manifolds that is a local isometry at all points. We will denot e the
+distance function on a Riemannian manifold by d.
+Definition 2 (Rotation Vector) .Letπ:/tildewiderM→Mbe a Riemannian covering,
+f:M→Ma homeomorphism, and F:/tildewiderM→/tildewiderMa lift off.
+A rotation vector vF(x)of a point x∈Mis a vector in the tangent space
+TxMsuch that for any lift ˜xofxthe following holds:
+d(˜α(n),Fn˜x) =o(n)whenn→+∞
+where˜αis the lift starting at ˜xof the geodesic α: [0,+∞)→Mdefined by
+α(t) = expx(tvF(x)).
+If the lift Fin the above definition commutes with all covering transforma-
+tions then the existence of a rotation vector vF(x) is a statement about the
+F-orbit of a single lift ˜ xofx. Also, notice that in this case the expression
+ρ(x) =d(˜x,F˜x) where ˜xis a lift of xis a well defined continuous function from
+Mto [0,+∞). Finally, notice that if f:M→Mis isotopic to the identity
+then one can always obtain a lift F:/tildewiderM→/tildewiderMoffthat commutes with all
+covering transformations (e.g. this can be done by lifting the isotop y between
+the identity map on Mandf).
+We recall that a Hadamard manifold is a complete, connected and simp ly
+connected Riemannian manifold with non-positive sectional curvatu re.
+We can now state our main theorem as follows.
+Theorem 2 (Main Theorem) .Let/tildewiderMbe a Hadamard manifold and π:/tildewiderM→M
+a Riemannian covering. For each x∈Mlet˜x∈/tildewiderMdenote an arbitrary lift of
+x.
+Suppose that f:M→Mis a homeomorphism that is isotopic to the identity,
+F:/tildewiderM→/tildewiderMis a lift of fthat commutes with all covering transformations, and
+µis anf-invariant Borel probability measure satisfying the condi tion:
+/integraldisplay
+Md(˜x,F˜x)dµ(x)<+∞
+Then for µ-almost every x∈Mthere exists a unique rotation vector vF(x)∈
+TxM.
+The rotation vectors given by the above theorem depend on the ch oice of a
+liftFwhich commutes with all covering transformations (from now on we w ill
+call such a lift ‘admissible’).
+IfM=Tdthen it is simple to show that all lifts are admissible, any two lifts
+differ by a translation with integer coordinates, and the correspon ding rotation
+vectors satisfy the same relationship.
+For hyperbolic manifolds we have the following result:
+Proposition 3. IfMis a complete hyperbolic manifold with finite volume and
+f:M→Mis isotopic to the identity then there is a unique admissible lift off
+to the universal Riemannian covering space /tildewiderMofM.
+5Proof.LetF1andF2be distinct admissible lifts. It follows that F−1
+1◦F2
+is a covering tranformation which commutes with all others. In part icular the
+maximalnormalAbelian subgroupof π1(M) is non-trivial. By [Ebe96, Theorem
+10.3.10] this implies that /tildewiderMsplits into a Riemannian direct product isometric
+toR×Xfor some simply connected Riemannian manifold X. However by
+taking two distinct points x1,x2∈Xand considering the curves t/ma√sto→(t,xi)
+we see that this would imply the existence of disjoint geodesics which r emain
+at a positive fixed distance. This contradicts the fact that /tildewiderMis isometric to
+hyperbolic space.
+We observe that because its proof proceeds by contradicting the strong visi-
+bilityproperty(seeAppendix) thepropositionisvalidifthisproperty isassumed
+forMinstead of hyperbolicity.
+In general, assuming that Mis complete and has finite volume one obtains
+that either the conclusion of the above proposition holds or /tildewiderMsplits into a Rie-
+mannian direct product with a non-trivial factor isometric to Rd. Furthermore
+ifMis compact it follows from [LY72, The Center Theorem] that Mis foliated
+by isometrically embdedded flat tori. This suggests that a characte rization of
+the dependence of rotation vectors on the choice of lift similar to th e case in
+whichM=Tdmight be possible. However, such a result is unknown to the
+author at the time of writing.
+3 Rate of Escape
+Definition 3 (Rate of Escape) .The rate of escape of a sequence {xn}n≥0⊂X
+in a metric space Xis the value of the following limit if it exists:
+R= lim
+n→+∞d(x0,xn)
+n
+Observe that if M,/tildewiderM,fandFare as in the statement of Theorem 2 and
+there exists a rotation vector vF(x) for a certain x∈Mthen for any lift ˜ xof
+xthe sequence {Fn˜x}n≥0has rate of escape /bardblvF(x)/bardbl(for details see Lemma
+6). In view of this our first task is to show that for almost every x∈Mthe
+sequence {Fn˜x}n≥0has a well defined finite rate of escape.
+The purpose of this section is to show a somewhat more general fac t, i.e.
+that a certain general class of random sequences in a metric space almost surely
+have a well defined finite rate of escape. This is a consequence of Kin gman’s
+Subadditive Ergodic Theorem which we restate in a convenient fashio n.
+Theorem 4 (Kingman’s Subadditive Ergodic Theorem [Kin68]) .If(M,B,µ)
+is a probability space, f:M→Mis a measure preserving measurable function,
+and{a(m,n)}0≤m<n(wherem,n∈Z) is a family of measurable functions from
+MtoRsatisfying:
+1.ax(l,n)≤ax(l,m)+ax(m,n)for allx∈Mand all0≤l < m < n
+62.ax(m+1,n+1) =af(x)(m,n)for allx∈Mand all0≤m < n
+3./integraltext
+M|ax(0,1)|dµ(x)<+∞
+4.limn→+∞/integraltext
+Max(0,n)
+ndµ(x)>−∞
+Then there exists an invariant and integrable function R:M→Rsuch that
+/integraldisplay
+AR(x)dµ(x) = lim
+n→+∞/integraldisplay
+Aax(0,n)
+ndµ(x)
+for every invariant measurable set A⊂M. And for µalmost every x∈Mthe
+following holds:
+R(x) = lim
+n→+∞ax(0,n)
+n
+Afamilyofmeasurablefunctionssatisfyingcondition1iscalledasubad ditive
+process. Condition 2 guarantees that this process is stationary. Conditions 3
+and 4 imply in particular that each function in the process is integrable (this
+is a consequence of subadditivity). Processes satisfying condition 4 are said to
+have “finite time constant”, one can prove Kingman’s Theorem witho ut this
+condition but the pointwise limit Rwill no longer be integrable (e.g. it might
+be equal to −∞on a set of positive probability). Since its first proof in [Kin68],
+the theorem has received many alternative proofs. See [Kre85] f or a general
+reference.
+A typical example of a subadditive process arising from a dynamical s ystem
+is the following: Let f:M→Mbe a volume preserving diffeomorphisms of a
+compact Riemannian manifold Mand define ax(m,n) = log(/bardblDfm(x)fn−m/bardbl).
+In this example Kingman’s theorem gives the existence of the largest Lyapunov
+exponent for almost every orbit (which was previously an independe nt result
+first proved by Furstenberg and Kesten, see [KL06, Introductio n]).
+We will use Kingman’s theorem in the following form.
+Corollary 5 (Rate of Escape for Random Sequences) .Suppose (M,B,µ)is
+a probability space and f:M→Mis a measurable and measure preserving
+transformation.
+LetXbe a metric space and φ:M→XNa measurable function such that
+the family of functions {a(m,n)}n>m≥0that is defined by
+a(m,n) :M→[0,+∞)
+ax(m,n) =d(φ(x)m,φ(x)n)
+satisfies the hypothesis of Theorem 4. Then there exists an in variant and inte-
+grable function R:M→[0,+∞)such that
+/integraldisplay
+AR(x)dµ(x) = lim
+n→+∞/integraldisplay
+Aax(0,n)
+ndµ(x)
+7for every invariant measurable set A⊂M. And for almost every x∈Mthe
+following holds:
+R(x) = lim
+n→+∞ax(0,n)
+n
+In particular, for almost every x∈Mthe sequence φ(x)has a finite rate of
+escape.
+4 Geodesic Escorts
+The main theorems of [Ka˘ ı87] and [KM99] can be restated in terms of the
+following definition. We will prove in this section that it is also connected to
+rotation vectors.
+Definition 4 (Geodesic Escort) .Let(X,d)be a metric space. A sequence
+{xn}n≥0⊂Xis said to be escorted by a geodesic, if there exists a functio n
+α: [0,+∞)→Xthat is either constant or a local isometry onto its image and
+satisfies:
+•α(0) =x0
+•d(xn,α(d(x0,xn))) =o(n)whenn→+∞
+Notice that any sequence with rate of escape equal to zero is esco rted by a
+(constant) geodesic.
+Before giving a proof of a necessary and sufficient condition for the existence
+of a rotation vector we would like to recall the following facts:
+•If/tildewiderMis a Hadamard manifold then it is diffeomorphic to Rdim(/tildewiderM)(see for
+example [Lan99] Theorem 3.8 on page 252).
+•Any two points in /tildewiderMbelong to a unique geodesic (up to reparametriza-
+tions) and therefore any arclength parametrization of this geode sic is a
+(global) isometry onto its image (see [Lan99] p.353 Corollary 3.11).
+Lemma 6. Let/tildewiderMbe a Hadamard manifold and π:/tildewiderM→Ma Riemannian
+covering.
+Suppose that f:M→Mis a homeomorphism that is isotopic to the identity,
+and that F:/tildewiderM→/tildewiderMis a lift of fthat commutes with all covering transforma-
+tions.
+A rotation vector vF(x)exists for a point x∈Mif and only if there exists a
+lift˜xofxsuch that {Fn˜x}n≥0has a well defined rate of escape and is escorted
+by a geodesic.
+In such a case it holds that for any lift ˜xofxthe sequence {Fn˜x}is escorted
+by a geodesic and has rate of escape /bardblvF(x)/bardbl.
+8Proof.First suppose v=vF(x)∈TxMis a rotation vector for x. For any lift ˜ x
+ofxthe geodesic ˜ α: [0,+∞)→/tildewiderMgiven by Definition 2 satisfies:
+d(˜α(n),Fn˜x) =o(n) whenn→+∞
+Since each pair of points in /tildewiderMbelongs to a unique geodesic we have that:
+d(˜α(0),˜α(n)) =d(˜x,˜α(n)) =n/bardbl˜α′(0)/bardbl=n/bardblv/bardbl
+The triangle inequality now implies:
+d(˜x,˜α(n))−d(˜α(n),Fn˜x)≤d(˜x,Fn˜x)≤d(˜x,˜α(n))+d(˜α(n),Fn˜x)
+And this in turn implies:
+n/bardblv/bardbl−o(n)≤d(˜x,Fn˜x)≤n/bardblv/bardbl+o(n)
+Which shows that {Fn˜x}n≥0has rate of escape /bardblv/bardbl.
+If/bardblv/bardbl= 0 then αis constant and is a geodesic escort for {Fn˜x}, otherwise
+letβ: [0,+∞)→/tildewiderMbe the geodesic starting at ˜ xwithβ′(0) = ˜α′(0)//bardbl˜α′(0)/bardbl.
+Since
+β(d(˜x,Fn˜x)) =β(n/bardblv/bardbl+o(n)) = ˜α(n+o(n))
+one has that:
+d(Fn˜x,β(d(˜x,Fn˜x))≤d(Fn˜x,˜α(n))+d(˜α(n),˜α(n+o(n))) =o(n)
+and therefore βescorts the sequence {Fn˜x}n≥0.
+Suppose now that for some lift ˜ xofxone has that {Fn˜x}n≥0has rate of
+escapeRand is escorted by a geodesic β. Letγ:/tildewiderM→/tildewiderMbe a covering
+transformation. Since Fcommutes with γandγis an isometry one has that
+γ◦βis a geodesic escort for the sequence {Fn(γ˜x)}n≥0and this sequence has
+rate of escape R. In particular the rate of escape is independent of the chosen
+lift ˜x, and therefore if R= 0 then v= 0∈TxMis a rotation vector for x.
+On the other hand if R >0, thenβis a non-constant geodesic and the image
+vof the vector Rβ′(0) under the differential of the covering map π:/tildewiderM→M
+is independent of the chosen lift ˜ x. To show that vis a rotation vector for x
+all that is needed is to prove that d(β(Rn),Fn˜x) =o(n). We can obtain this
+directly from d(˜x,Fn˜x) =Rn+o(n) and the fact that βis a geodesic escort, as
+follows:
+d(β(Rn),Fn˜x)≤d(β(Rn),β(d(˜x,Fn˜x)))+d(β(d(˜x,Fn˜x)),Fn˜x) =o(n)
+95 Aligned Sequences
+Definition 5 (Linear Escape to Infinity) .A sequence {xn}n≥0⊂Xin a metric
+spaceXis said to escape linearly to infinity if it has a positive and fi nite rate
+of escape.
+In this section we will give a condition under which a sequence that esc apes
+linearly to infinity will be escorted by a geodesic.
+The simplest such condition known to the author is the following (which we
+will state without proof), valid for d-dimensional hyperbolic space:
+Proposition 7. LetHddenoted-dimensional hyperbolic space. Any sequence
+{xn}n≥0⊂Hdthat escapes linearly to infinity and satisfies
+d(xn,xn+1) =o(n)whenn→+∞
+is escorted by a unique geodesic.
+The condition d(xn,xn+1) =o(n) is almost always satisfied by random se-
+quences, provided that the variables d(xn,xn+1) are identically distributed and
+have finite expectation. Using this fact one can obtain a proof of a s pecial case
+of our main theorem.
+However the sequence in Cdefined by xn=neilog(n)satisfiesd(xn,xn+1) =
+o(n) (where d(x,y) =|x−y|for every x,y∈C) but isn’t escorted by a geodesic.
+This shows that the proposition is false for general non-positively c urved spaces.
+Since our objective is to prove the existence of geodesic escorts f or sequences
+in Hadamard manifolds we will need a stronger hypothesis than d(xn,xn+1) =
+o(n). The following definitions will allow us to formulate such a hypothesis.
+Definition 6 (ǫ-Cone).Let(X,d)be a metric space. If ǫ∈[0,+∞]andx,y∈
+Xtheǫ-cone from xtoyis defined as the following set:
+[x,y]ǫ={z∈X:e−ǫd(x,z)+d(z,y)≤d(x,y)}
+The 0-cone between two points in Cis a segment. The cone [ x,y]+∞is a
+closed disk centered at yand containing x(in particular note that the definition
+is not symmetric in xandy).
+We willnowshowthatinHadamardmanifoldsthe ǫ-conebetweentwopoints
+is close to a geodesic segment for small ǫ.
+We will rely on two properties of Hadamard manifolds which we state be low
+without proof. Lemma 8 is contained in theorems 4.3 and 4.4 of chapte r IX in
+[Lan99]. While the semi-parallelogram law is proved to hold locally in section 3
+of chapter XI of the same reference. In the context of Hadamar d manifolds the
+same proof gives the (global) statement below.
+Lemma 8 (Convexity of geodesic distance) .IfMis a Hadamard manifold
+then for any x∈Mand any pair of geodesics α,β:R→Mthe following two
+functions are convex:
+t/ma√sto→d(x,α(t))
+t/ma√sto→d(α(t),β(t))
+100.25 0.750.2
+0.0 0.50.0
+1.0
+−0.2
+Figure 1: The boundary of [0 ,1]0.2inC.
+Lemma 9 (Semi-Parallelogram Law) .LetMbe a Hadamard manifold and
+letx,y,z∈M. Then if mis the midpoint of the geodesic segment [x,y]0the
+following inequality holds:
+d(x,y)2+4d(m,z)2≤2d(x,z)2+2d(z,y)2
+Lemma 10. LetMbe a Hadamard manifold. For all ǫ >0and allx,y,z∈M
+withz∈[x,y]ǫit holds that:
+d(z,w)2≤4(1−e−2ǫ)d(x,z)2
+wherew=α(d(x,z))andα: [0,+∞)→Mis the unique geodesic parametrized
+by arclength with α(0) =xandα(d(x,y)) =y.
+Proof.Leta=d(x,z) =d(x,w),b=d(x,y)−aandc=d(z,y). Notice that
+b=d(w,y) ifd(x,z)< d(x,y) andb=−d(w,y) otherwise. In both cases |b| ≤c.
+Sincez∈[x,y]ǫwe have:
+e−ǫa+c≤a+b
+which implies
+a+b−c≥e−ǫa
+Letmbe the midpoint of the segment [ z,w]0. Lemma 8 implies that:
+d(y,m)≤max(|b|,c) =c
+and from this we obtain
+d(x,m)≥d(x,y)−d(y,m)≥a+b−c≥e−ǫa
+11The semi-parallelogram law (lemma 9) now gives:
+d(w,z)2+4d(x,m)2≤4a2
+Combining these inequalities we obtain:
+d(w,z)2≤4(a2−d(x,m)2)≤4(1−e−2ǫ)a2= 4(1−e−2ǫ)d(x,z)2
+We will now state a condition that will guarantee the existence of a ge odesic
+escort for a sequence that escapes linearly to infinity. One can inte rpret the
+condition saying either that the sequence eventually stays in arbitr arily small
+ǫ-cones, or that the differences d(x0,xn)−d(xk,xn) are close to their largest
+possible value (i.e. d(x0,xk)).
+Definition 7 (Aligned Sequence) .Let(X,d)be a metric space. A sequence
+{xn}n≥0⊂Xis said to be aligned if for each ǫ >0there exists K∈Nsuch
+that for infinitely many n∈Nthe following holds:
+xk∈[x0,xn]ǫfor allK≤k≤n
+Observation 1.Let (X,d) be a metric space and {xn}n≥0⊂Xa sequence that
+escapes linearly to infinity with rate of escape R. Then{xn}n≥0is aligned if
+and only if L≥Rwhere:
+L= lim
+K→+∞limsup
+n→+∞min
+K≤k≤n{d(x0,xn)−d(xn,xk)
+k}
+Lemma 11. LetHbe a Hadamard manifold. If {xn}n≥0⊂Hescapes linearly
+to infinity and is aligned then it is escorted by a geodesic.
+Proof.Letf: [0,+∞)→[0,+∞) be given by f(ǫ) = 2√
+1−e−2ǫ.
+Foreach n∈Ndefinedn=d(x0,xn)andlet αn: [0,+∞)→Hbetheunique
+geodesic parametrized by arclength such that αn(0) =x0andαn(dn) =xn.
+Also, let R >0 be the rate of escape of the sequence {xn}n≥0.
+By hypothesis for each ǫ >0 there is a natural number Kǫand infinitely
+many values of n(which we will call ǫ-admissible) such that:
+xk∈[x0,xn]ǫfor allKǫ≤k≤n
+Sincedk=Rk+o(k) we may assume that Kǫabove is chosen large enough
+so thatdk≥1 for all k≥Kǫ.
+Also, note that by lemma 10 we have the following inequality for all ǫ-
+admissible nand allkwithKǫ≤k≤n:
+d(αn(dk),xk)≤f(ǫ)dk
+We will prove the lemma in two steps: First, we will show that {αn(1)}n≥0
+is a Cauchy sequence. Second, defining αto be the geodesic ray with α(0) =x0
+andα(1) = lim n→+∞αn(1) we will show that αis ageodesicescortfor {xn}n≥0.
+12Toprovethefirstclaimfix ǫ >0andthecorresponding Kǫ. Forany k,l≥Kǫ
+we can find an ǫ-admissible n≥max(k,l) and for this value of none has:
+d(αk(1),αl(1))≤d(αk(1),αn(1))+d(αn(1),αl(1))
+≤d(xk,αn(dk))
+dk+d(αn(dl),xl)
+dl
+≤2f(ǫ)
+where the second inequality is obtained by applying lemma 8 to αnand either
+αk(in the case of the first summand) or αl(for the second). Here we have used
+the fact that dk,dl≥1.
+Asǫcan be chosensothat f(ǫ) is arbitrarilysmall the aboveargumentshows
+thatαn(1) is a Cauchy sequence and therefore lim n→+∞αn(1) exists. Therefore
+one can define a geodesic ray αas claimed above.
+We will now show that αis a geodesic escort for {xn}n≥0.
+Sinceα(0) =αn(0) for all nandα(1) = lim n→+∞αn(1) it follows that
+α(t) = lim
+n→+∞αn(t)
+for allt1.
+Givenǫ >0 and the corresponding Kǫfor each k≥Kǫone can choose a
+sequence nl→+∞ofǫ-admissible numbers. For this sequence we obtain:
+d(α(dk),xk) = lim
+l→+∞d(αnl(dk),xk)≤f(ǫ)dk
+for allk≥Kǫ.
+Hence for each ǫ >0 one has d(α(dk),xk)≤f(ǫ)dkfor allk≥Kǫ. This
+shows that:
+d(α(dk),xk) =o(dk) whenk→+∞
+so thatαis a geodesic escort as claimed.
+We conclude this section by showing that if an orbit of an isometry or s emi-
+contraction escapes linearly to infinity, then it is aligned. This result is n’t
+necessary for the proof of our main theorem, but will be used later on as a
+source of examples.
+Lemma 12. Let(X,d)be a metric space and {xn}n≥0⊂Xa sequence that
+escapes linearly to infinity and satisfies:
+d(xm+k,xn+k)≤d(xm,xn)for allk,m,n≥0
+Then{xn}n≥0is aligned.
+1By the Arsel` a-Ascoli theorem any subsequence of αnhas a subsequence converging uni-
+formly on compact sets. If βis the limit of such a subsequence then it must be a geodesic
+joiningα(0) and α(1) and it follows that α=β.
+13Proof.LetRbe the rate of escape of {xn}.
+For each ǫ >0 takeKsuch that:
+d(x0,xk)≤eǫ/2Rkfor allk≥K
+Since the sequence d(x0,xn)−e−ǫ/2Rnis unbounded there exist infinitely
+manynsuch that:
+d(x0,xn)−e−ǫ/2Rn > d(x0,xm)−e−ǫ/2Rmfor allm < n
+Hence for infinitely many nit holds that for all K≤k≤n:
+e−ǫd(x0,xk)+d(xk,xn)≤e−ǫ/2Rk+d(x0,xn−k)< d(x0,xn)
+This implies that xk∈[x0,xn]ǫand hence the sequence is aligned.
+6 Alignment of Random Sequences
+The following lemma will enable us to prove that, in a sense, almost all ra ndom
+sequences that escape linearly to infinity are aligned.
+Lemma 13 ([KM99] Lemma 4.1) .Suppose {a(m,n)}0≤m<nsatisfies the hy-
+pothesis of Theorem 4 and that:
+lim
+n→+∞/integraldisplay
+Max(0,n)
+ndµ(x)>0
+Then the probability of the following set is strictly positi ve:
+{x∈M:for infinitely many n∈N,a(0,n)−a(k,n)>0for all1≤k≤n}
+Proof.Chooseǫpositive but smaller than the time constant of {a(m,n)}and
+consider the subadditive cocycle c:N×M→R(overf:M→M) given by:
+c(n,x) =ax(0,n)−ǫnfor allx∈M,n∈N
+The cocycle csatisfies the hypothesis of Lemma 4.1 in [KM99] and the
+conclusion follows.
+We will now establish the ergodic theorem used in the proofs of existe nce of
+rotation vectors throughout this work.
+Theorem 14 (Alignment of Random Sequences) .Suppose (M,B,µ)is a prob-
+ability space and f:M→Mis a measurable and measure preserving transfor-
+mation.
+LetXbe a metric space and φ:M→XNa measurable function such that
+the family of functions {a(m,n)}n>m≥0defined by:
+a(m,n) :M→[0,+∞)
+ax(m,n) =d(φ(x)m,φ(x)n)
+satisfies the hypothesis of Theorem 4. Then for almost every x∈Mif the
+sequence φ(x)escapes linearly to infinity then it is aligned.
+14Proof.LetR:M→[0,+∞) be the function given by Corollary 5. For almost
+everyx∈M, the sequence φ(x) escapes linearly to infinity if and only if R(x)>
+0.
+Consider the measurable function L:M→[0,+∞) defined by:
+L(x) = max(0 ,lim
+K→+∞limsup
+n→+∞min
+K≤k≤n{d(φ(x)0,φ(x)n)−d(φ(x)n,φ(x)k)
+k})
+By Observation 1 it suffices to show that L≥Ron a set of full measure.
+First we will show that the function Lis invariant on a set of full measure.
+For this purpose it suffices to show that L(x)≤L(f(x)) for almost every x,
+since the probability of the set {x∈M:L(x)≥a}is equal to that of the set
+{x∈M:L(f(x))≥a}for every a∈Rbecausefis measure preserving.
+IfL(x) = 0 then trivially L(x)≤L(f(x)). Suppose L(x)≥t >0. This
+implies that for each ǫ >0 there exists K∈Nand infinitely values of n∈N
+such that:
+d(φ(x)0,φ(x)n)−d(φ(x)n,φ(x)k)> te−ǫkfor allK≤k≤n
+for a slightly larger Kthe following holds:
+d(φ(x)1,φ(x)n)−d(φ(x)n,φ(x)k)≥ −d(φ(x)1,φ(x)0)+d(φ(x)0,φ(x)n)−d(φ(x)n,φ(x)k)
+>−d(φ(x)1,φ(x)0)+te−ǫk > te−2ǫ(k−1) for all K≤k≤n
+and therefore L(f(x))≥te−2ǫ. Since this holds for every ǫ >0 we have shown
+the claim that L(x)≤L(f(x)) almost everywhere and therefore Lis invariant
+on a set of full measure.
+To simplify the discussion, modify Lon a set of measure 0 so that it is
+strictly invariant.
+We will now show that L≥Ralmost everywhere.
+This is trivially true in the set where R= 0.
+Suppose that for some ǫ >0 the set A={x∈M:L(x)< e−ǫR(x)}has
+positive measure. Let 1 Abe the function that takes the value 1 on Aand 0
+outside of A, and consider the stationary subadditive process {b(m,n)}0≤m<n
+defined by
+bx(m,n) = (d(φ(x)m,φ(x)n)−(n−m)e−ǫR(x))1A(x)
+Since this process satisfies
+lim
+n→+∞/integraldisplay
+Mbx(0,n)
+ndµ(x) = (1−e−ǫ)/integraldisplay
+AR(x)dµ(x)>0
+by Lemma 13 there is a set of positive probability of x∈Msuch that there
+exist infinitely many nsatisfying the following:
+bx(0,n)−bx(k,n)>0 for all 1 ≤k≤n
+15However since
+bx(0,n)−bx(k,n) = (d(φ(x)0,φ(x)n)−d(φ(x)n,φ(x)k)−ke−ǫR(x))1A(x)
+this would imply that for some x∈Awe have L(x)≥e−ǫR(x) contradicting
+the definition of A. Therefore we must have L≥Ralmost everywhere as
+claimed.
+7 Proof of the Main Theorem
+We will now prove the main theorem (Theorem 2).
+Proof.One can choose ˜ xso that it is a measurable function of x(e.g. apply
+[Kec95, Theorem 12.16 page 78]). By Corollary 5, for almost every x∈Mthe
+sequence {Fn˜x}n≥0has a finite rate of escape R(x).
+IfR(x) = 0 then 0 ∈TxMis a rotation vector for x.
+On the other hand, by Theorem 14, for almost every x∈Min the case
+thatR(x)>0 the sequence {Fn˜x}n≥0is aligned. In this case the existence of a
+geodesic escort follows from Lemma 11. This implies the existence of a rotation
+vector by Lemma 6.
+The uniqueness claim follows from the fact that no two geodesics ˜ α,˜β:
+[0,+∞)→/tildewiderMwith the same starting point satisfy d(˜α(n),˜β(n)) =o(n) when
+n→+∞. This is a direct consequence of Lemma 8.
+8 Periodic points and Homological Rotation Vec-
+tors
+The purpose of this section is to clarify the relationship between hom ological
+rotation vectors and rotation vectors as defined in this paper. We will consider
+rotation vectors associated to periodic points. The result of this s ection is
+illustrated by the following two examples:
+•Let/tildewiderM={(x,y)∈R2:y >0}with the hyperbolic metric, and define
+F:/tildewiderM→/tildewiderMbyF(x,y) = (x+ 1,y). IfMis any quotient of /tildewiderMby
+the group generated by a non-trivial translation with respect to t hex-
+axis,fis the projection of FtoMandµis an invariant probability
+measure supported on the projection of R×{1}toMthen the hypothesis
+ofTheorem2aresatisfied. Direct calculationshowsthatthe rateo fescape
+is equal to 0, however the homological rotation vector of every forbit in
+Mis non-null.
+•LetMbe the two dimensional sphere with two handles (i.e. a double-
+torus) with a hyperbolic metric. There is a simple closed curve (sepa-
+rating both handles) on Mthat is not homotopic to a constant but is
+homologically trivial. Consider a flow φ:R×M→Mwith this curve
+16as a periodic orbit, take f=φ1andµto be the invariant measure for
+the flow supported on the selected periodic orbit. By lifting this flow t o
+the universal covering space and taking the time 1 to be Fwe obtain an
+example with null homological rotation vectors but with positive rate of
+escape.
+In this section we will have to distinguish between the usual relations hip
+of (end-point fixing) homotopy between curves and free-homoto py equivalence
+between closed curves (where there is no fixed base-point). We re call that
+two continuous closed curves α,β: [0,1]→Min a manifold are called freely-
+homotopic if there exists a continuous function γ: [0,1]×[0,1]→Msuch
+that:
+1.γ(0,t) =α(t) for allt.
+2.γ(1,t) =β(t) for allt.
+3.γ(s,0) =γ(s,1) for all s.
+The requirement that both curves be parametrized on [0 ,1] is non-essential and
+is removed by declaring that two arbitrary closed curves α: [a,b]→Mand
+β: [c,d]→Mare freely-homotopic if there exist continuous and increasing
+reparametrizations of them satisfying the above definition.
+Free homotopy is an equivalence relationship among all closed continu ous
+closed curves in M(see [DFN85, Section 17] for a general reference on this
+subject). The equivalence classes of this relationship are called fre e homotopy
+classes.
+On a Riemannian manifold each free homotopy class of closed curves h as an
+associated length as follows:
+Definition 8 (Length of a free homotopy class) .LetMbe a Riemannian man-
+ifold and Cbe a free homotopy class of closed curves in M. The length l(C)
+ofCis defined as: l(C) = inf{|α|:α∈C}where|α|denotes the length of a
+smooth curve α.
+We will need to use the following two facts about free homotopy class es of
+manifolds of non-positive curvature.
+Lemma 15. LetMbe a connected manifold. Two closed curves α,β: [0,1]→
+Mare freely homotopic if and only if there exists γ: [0,1]→Mwithγ(0) =α(0)
+andγ(1) =β(0)such that αis homotopic to γ−1·β·γ(where the dot denotes
+concatenation and γ−1is an orientation-reversing reparametrization of γ).
+Proof.First suppose that αis homotopic to γ−1·β·γfor some γ. It follows
+thatαis freely homotopic to β·γ·γ−1and hence to β.
+Now suppose that αandβare freely homotopic.
+Choose any γ1: [0,1]→Mwithγ1(0) =α(0) andγ1(1) =β(0). Notice that
+γ−1
+1·β·γ1is freely homotopic to βand hence to α. By [DFN85, Theorem 17.3.1]
+thereexists γ2: [0,1]→Mwithγ2(0) =γ2(1) =α(0)suchthat γ−1
+2·γ−1
+1·β·γ1·γ2
+ishomotopicto α. Hencebytaking γ=γ1·γ2wehaveshownthat αishomotopic
+toγ−1·β·γ.
+17Lemma16. Letπ:/tildewiderM→Mbe a Riemannian covering where /tildewiderMis a Hadamard
+manifold. Let Cbe a free homotopy class in Mandρ:/tildewiderM→/tildewiderMbe a covering
+transformation such that the geodesic segment [p,ρ(p)]projects to a curve in
+classCfor some p∈/tildewiderM. Then the length of the class Cis given by:
+l(C) = inf{d(x,ρ(x)) :x∈/tildewiderM}
+Proof.Letα: [0,1]→Mbe the projection of a parametrization the geodesic
+segment [ p,ρ(p)] toM. Ifβ: [0,1]→Mis a closed curve freely homotopic to α
+then there exists γ: [0,1]→Msuch that αis homotopic to γ−1·β·γ. Hence
+by lifting γstarting at pthe other endpoint qsatisfies that the segment [ q,ρ(q)]
+projects to a curve homotopic to β. Hence the length of βis greater than or
+equal to d(q,ρ(q)) and it follows that:
+l(C)≥inf{d(x,ρ(x)) :x∈/tildewiderM}
+On the other hand for any x∈/tildewiderMthe geodesic segment [ x,ρ(x)] projects
+to a curve βwhich is freely homotopic to αas one can see by considering the
+projection γof any curve γbetween pandxand noting that γ−1·β·γis
+homotopic to α. Hence one obtains that d(x,ρ(x)) is greater than or equal to
+l(C), which establishes the claim.
+Once we fix an isotopy between a homeomorphism f:M→Mof a Rie-
+mannian manifold and the identity each periodic orbit is associated to a free
+homotopy class as follows:
+Definition 9 (Freehomotopyclassofa periodicorbit) .LetMbe a Riemannian
+manifold and f: [0,1]×M→Mbe an isotopy with f0equal to the identity
+mapping of M. Ifx∈Mis a periodic point for f1with minimal period pthen
+the free homotopy class C(x)ofxis defined as the free homotopy class of the
+following closed curve:
+α: [0,p]→M
+α(t) =ft−k◦fk
+1ifk≤t≤k+1wherek∈Z
+Notice that if xis a periodic orbit of period nfor a homeomorphism fthen
+µ=1
+n/summationtextn−1
+k=0δfk(x)is an ergodic invariant measure (where δpdenotes the Dirac
+delta at a point p). Any such measure will trivially satisfy the integrability hy-
+pothesisofTheorem2. Henceallperiodicorbitshaverotationvect orsassociated
+to them.
+The theorem below shows the relationship between the norm of the r otation
+vector of a periodic orbit and the length of its free homotopy class.
+Theorem 17. Let/tildewiderMbe a Hadamard manifold and π:/tildewiderM→Ma Riemannian
+covering. Suppose f:M→Mis isotopic to the identity and x∈Mis a
+periodic point for fof minimal period p. Giveng: [0,1]×M→Man isotopy
+between the identity and f, ifF=G1whereG: [0,1]×/tildewiderM→/tildewiderMis the unique
+lift ofgto/tildewiderMstarting at the identity then /bardblvF(x)/bardbl=l(C(x))/p.
+18Proof.To begin we observe that Fcommutes with all covering transformation.
+To see this let γbe a covering transformation and notice that for each tthe
+equation ρt=γ−1◦G−1
+t◦γ◦Gtdefines a covering transformation. Since for all
+y∈/tildewiderMthe curve t/ma√sto→ρt(y) is continuous and starts at yit must be constant.
+Fort= 1 this gives γ◦F=F◦γas claimed.
+Next fix a lift ˜ xofxand letρ:/tildewiderM→/tildewiderMbe the covering transformation such
+thatρ(˜x) =Fp(˜x). Since Fcommutes with ρit holds that Fnp(˜x) =ρn(˜x) so
+that by lemma 6 one has:
+p/bardblvF(x)/bardbl= lim
+n→+∞d(˜x,ρn(˜x))
+n
+For anyy,z∈/tildewiderMone has:
+lim
+nd(y,ρn(y))
+n≤lim
+nd(y,x)+d(x,ρn(x))+d(ρn(x),ρn(y))
+n= lim
+nd(x,ρn(x))
+n
+Hencep/bardblvF(x)/bardblequals the rate of escape of all points in /tildewiderMunder iteration
+ofρ.
+For each n≥1 we define:
+Rn= inf{d(y,ρn(y)) :y∈/tildewiderM}
+We will prove that p/bardblvF(x)/bardbl=R1and by lemma 16 the theorem follows.
+First we establish p/bardblvF(x)/bardbl ≤R1as follows: For each ǫ >0 choose y∈/tildewiderM
+withd(y,ρ(y))≤R1+ǫ. By the triangle inequality one has that
+p/bardblvF(x)/bardbl= lim
+nd(y,ρn(y))
+n≤R1+ǫ
+and by letting ǫgo to zero one establishes the desired inequality.
+On the other hand, if the rate of escape of some point y∈/tildewiderMwas less than
+R1then for all nlarge enough one would have d(y,ρn(y))< nR1. In particular
+this would imply Rn< nR 1for allnlarge enough. Hence to establish the
+equality R1=p/bardblvF(x)/bardblit suffices to show that Rn=nR1whenever nis a
+power of two2.
+We will show R2= 2R1and the general claim follows from applying the
+same argument to iterates of ρ.
+First consider a sequence ynin/tildewiderMwithd(yn,ρ(yn))→R1, it holds that:
+R2≤d(yn,ρ2(yn))≤d(yn,ρ(yn))+d(ρ(yn),ρ2(yn))
+which establishes (by taking limit of the right hand side) the inequality R2≤
+2R1.
+Next consider a sequence ynin/tildewiderMsuch that d(yn,ρ2(yn))→R2whenn→
++∞. Ifznis the midpoint of the geodesic segment [ yn,ρ(yn)] then, because ρis
+2Here any unbounded sequence would do. In fact, once R1is characterized as the rate of
+escape of all points, one can deduce that Rn=nR1for alln.
+19an isometry, ρ(zn) is the midpoint of [ ρ(yn),ρ2(yn)]. Hence by lemma 8 applied
+to the geodesic segments [ ρ(yn),yn] and [ρ(yn),ρ2(yn)] respectively we obtain:
+R1≤d(zn,ρ(zn))≤d(yn,ρ2(yn))
+2→R2
+2whenn→+∞
+From which the result follows.
+9 Past orbits
+9.1 Past and future rotation vectors
+We will begin this section with a lemma that shows in particular that the s et of
+pointsx∈Mfor which vF(x) exists is f-invariant, and that the norm /bardblvF(x)/bardbl
+is constant along each orbit.
+Lemma 18. LetHbe a Hadamard manifold and {xn}n≥0,{yn}n≥0⊂Hbe
+such that:
+d(xn,yn) =o(n)whenn→+∞
+If there exists v∈Tx0Hsuch that
+d(xn,exp(nv)) =o(n)whenn→+∞
+then there exists a unique vector w∈Ty0Hsuch that:
+d(yn,exp(nw)) =o(n)whenn→+∞
+In such a case /bardblw/bardbl=/bardblv/bardbland if/bardblv/bardbl /ne}ationslash= 0thenv//bardblv/bardblis asymptotic to w//bardblw/bardbl
+(see Appendix).
+Proof.By [EO73, Remark 4, p.48] there is a unique vector w∈Ty0Hsuch that:
+d(exp(tv),exp(tw)) =O(1) when t→+∞
+By lemma 8 (see also the discussion following definition 14) wis also the
+unique vector in Ty0Hsatisfying:
+d(exp(tv),exp(tw)) =o(t) whent→+∞
+Notice that:
+t/bardblv/bardbl=d(x0,exp(tv))≤d(x0,y0)+d(y0,exp(tw))+d(exp(tw),exp(tv)) =t/bardblw/bardbl+o(t)
+which implies that /bardblv/bardbl ≤ /bardblw/bardbland hence (by symmetry) that /bardblv/bardbl=/bardblw/bardblas
+required.
+Assuming /bardblv/bardbl=/bardblw/bardbl /ne}ationslash= 0 it follows from the bound on d(exp(tv),exp(tw))
+thatv//bardblv/bardblandw//bardblw/bardblare asymptotic.
+To conclude we estimate d(yn,exp(nw)) as follows:
+d(yn,exp(nw))≤d(yn,xn)+d(xn,exp(nv))+d(exp(nv),exp(nw)) =o(n)
+20Corollary 19. Suppose M,/tildewiderM,fandFsatisfy the hypothesis of Theorem 2
+andx∈Mhas a rotation vector vF(x). ThenvF(fk(x))exists and satisfies
+/bardblvF(fk(x))/bardbl=/bardblvF(x)/bardblfor allk∈Z.
+Proof.Let ˜xbe a lift of x. By definition of vF(x) the lift ˜ αoft/ma√sto→exp(tvF(x))
+starting at ˜ xsatisfies:
+d(˜α(n),Fn˜x) =o(n) whenn→+∞
+As an immediate consequence for any k∈Zone has:
+d(Fn˜x,Fn+k˜x) =o(n) whenn→+∞
+The claim now follows by lemma 18 and the fact (observed in section 2)
+that the existence of vF(fk(x)) is a property of the F-orbit of any single lift of
+fk(x).
+Since we are studying invertible dynamical systems it is natural to as k what
+the relationship is between rotation vectorsfor the system fand its inverse f−1.
+In this direction we will prove the following theorem.
+Theorem 20. LetM,/tildewiderM,f,Fandµsatisfy the hypothesis of Theorem 2. Then
+for almost every x∈Mit holds that /bardblvF(x)/bardbl=/bardblvF−1(x)/bardbland either vF(x) =
+vF−1(x) = 0orvF(x)/ne}ationslash=vF−1(x).
+Proof.Letx/ma√sto→˜xbe a measurable function such that ˜ xprojects to xfor all
+x∈M. Consider for each x∈Mthe Busemann function on /tildewiderMassociated to
+the lift ˜αof the geodesic t/ma√sto→exp(tvF(x)) starting at ˜ x:
+Bx(p,q) = lim
+t→+∞d(p,˜α(t))−d(˜α(t),q)
+Notice that if ˜ αis non-constant then Bxcoincides with the Busemann func-
+tion associated to the vector ˜ α′(0)//bardbl˜α′(0)/bardblas defined in the Appendix. The
+function BxsatisfiesBx(p,q)+Bx(q,r) =Bx(p,r) for allp,q,r∈/tildewiderM. Also, one
+has the inequality |Bx(p,q)| ≤d(p,q).
+With this notation we define a measurable function g:M→Rby the
+formulag(x) =Bx(˜x,F˜x). Since|g(x)| ≤d(˜x,F˜x) the function gis integrable.
+Also, ifρis a covering transformation then, because Fcommutes with ρ, it
+follows that:
+g(x) = lim
+t→+∞d(ρ(˜x),ρ◦˜α(t))−d(ρ◦˜α(t),F(ρ(˜x)))
+so that the value of g(x) is independent of the chosen lift of x.
+We will now show that /bardblvF(x)/bardbl ≤ /bardblvF−1(x)/bardblfor almost all x. IfvF(x) = 0
+this is satisfied so we may assume vF(x)/ne}ationslash= 0 in what follows.
+By lemma 18 and corollary 19 it follows for all k∈Zthat the lift of t/ma√sto→
+exp(tvF(fk(x)) starting at Fk(˜x) is asymptotic to ˜ α. By lemma 28 and the fact
+thatg(fk(x)) may be calculated at any lift of fk(x) this implies that:
+g(fk(x)) =Bfk(x)(/tildewiderfk(x),F/tildewiderfk(x)) =Bx(Fk˜x,Fk+1˜x) for allk∈Z
+21Consider now the Birkhoff averages of g. One has the following equality:
+1
+nn−1/summationdisplay
+k=0g(fk(x)) =1
+nn−1/summationdisplay
+k=0Bx(Fk˜x,Fk+1˜x) =Bx(˜x,Fn˜x)
+n
+By definition of vF(x) one has that
+d(˜α(n),Fn˜x) =o(n)
+from which one obtains the following:
+n/bardblvF(x)/bardbl=Bx(˜x,˜α(n)) =Bx(˜x,Fn˜x)+Bx(Fn˜x,˜α(n)) =Bx(˜x,Fn˜x)+o(n)
+Hence the limit ˜ gof the Birkhoff averages of gequals/bardblvF(x)/bardblalmost surely.
+But by Birkhoff’s theorem one also obtains:
+/bardblvF(x)/bardbl= lim
+n→+∞1
+nn/summationdisplay
+k=1g(f−k(x)) = lim
+n→+∞Bx(F−n˜x,˜x)
+n
+≤lim
+n→+∞d(F−n˜x,˜x)
+n=/bardblvF−1(x)/bardbl
+This establishes that /bardblvF(x)/bardbl ≤ /bardblvF−1(x)/bardbl. By applying the same argument
+toF−1it follows that /bardblvF(x)/bardbl=/bardblvF−1(x)/bardblalmost everywhere.
+Suppose now that for some xwe have vF(x) =vF−1(x). Then Bxis the
+Busemann function associated to the geodesics given by both rota tion vectors.
+In this case the above analysis and the property Bx(p,q) =−Bx(q,p) give:
+/bardblvF(x)/bardbl= lim
+n→+∞Bx(F−n˜x,˜x)
+n=−lim
+n→+∞Bx(˜x,F−n˜x)
+n=−/bardblvF−1(x)/bardbl
+Since both sides arenon-negativethis implies that vF(x) =vF−1(x) = 0.
+On strong visibility manifolds (e.g. manifolds with Anosov geodesic flow,
+see Appendix) we obtain the following corollary.
+Corollary 21. LetM,/tildewiderM,f,Fandµsatisfy the hypothesis of Theorem 2 and
+supposeMis a strong visibility manifold. Then for almost every x∈Mwith
+vF(x)/ne}ationslash= 0it holds that for each lift ˜xofxthere exists a geodesic ˜α:R→/tildewiderM
+such that:
+d(˜α(n),Fn˜x) =o(n)whenn→ ±∞
+Furthermore ˜αis unique up to reparametrizations by time translation.
+Proof.SetR=/bardblvF(x)/bardblandletv+,v−∈T˜x/tildewiderMprojectto vF(x)/RandvF−1(x)/R
+respectively. Consider the geodesic defined by:
+˜α(t) = exp(tRv)
+wherev=πh(v+,v−) andπhis the projection along horospheres (Lemma 31).
+The uniqueness claim follows from the definition of strong visibility (see Ap-
+pendix).
+We will call a geodesic satisfying the properties of the above corollar y a
+bi-infinite geodesic escort for the sequence {Fn˜x}n∈Z.
+229.2 An example without a bi-infinite geodesic escort
+The limitations of preceding corollary are illustrated by the following ex ample
+(see [KM99] Remark 2.4). Consider the metric d s2= (1 +e−y)2dx2+dy2in
+R2. Direct calculation shows that this metric has non-positive curvatu re. Note
+that (compare with Theorem 2.1 in [AB98]), a curve α= (x,y) :R→R2is a
+geodesic if and only if the following Hamiltonian equations are satisfied:
+
+
+x′= (1+e−y)−2p1
+y′=p2
+p′
+1= 0
+p′
+2=−(1+e−y)−3e−yp2
+1
+In particular p1= (1+e−y)2x′is constant along each geodesic. Since E=
+(1 +e−y)2x′2+y′2= (1 +e−y)−2p2
+1+y′2(i.e. the square of the norm of the
+velocity) is also constant along each geodesic one can classify the as ymptotic
+behavior of all geodesics as follows:
+•IfE > p2
+1theny′2is bounded away from 0. This implies that y′has
+constant sign and is bounded away from 0.
+•IfE=p2
+1theny′2= (1−(1 +e−y)−2)E(e.g. it is never 0 unless the
+geodesic is constant). From this one deduces that yis a bijection of the
+real line with y′→0 wheny→+∞and|y′| →+∞wheny→ −∞.
+Depending on sign of y′(0) the function yis decreasing or increasing.
+•IfE < p2
+1thenyis bounded and y′′is bounded from above by a negative
+constant. In particular y→ −∞and|y′| →+∞whent→ ±∞.
+Consider the sequence xn= (n,0)∈R2forn∈Z. It follows from compar-
+ison with the usual Euclidean metric that the rate of escape of this s equence
+greater then or equal to 1. One can show that, in fact, the rate o f escape is
+exactly 1 by calculating the length of the polygonal path through th e points
+(0,0),(0,log(n)),(n,log(n)) and (n,0) (which turns out to be n+2log(n)+1).
+Since{xn}is the orbit of an isometry it is aligned by Lemma 12. Therefore
+one obtains by Lemma 11 that it must have a geodesic escort αstarting at
+(0,0). The considerations above ensure that this geodesic must be of the type
+withE=p2
+1. The geodesic βsymmetric to αwith respect to the y-axis escorts
+the sequence xnwhenn→ −∞(because symmetry with respect to the y-axis is
+an isometry). However there is no bi-infinite geodesic γwithd(γ(n),xn) =o(n)
+whenn→ ±∞, as one can see by checking each of the three types of geodesics
+discussed above.
+Considernow /tildewiderM=R2withthemetricunderdiscussion, Fgivenby F(x,y) =
+(x+1,y)andM=/tildewiderM/ΓwhereΓisthegroupgeneratedbyanynon-trivialtrans-
+lationγ(x,y) = (x+t,y). One can take fto be the projection of FtoMand
+µto be the projection of the uniform probability measure on [0 ,t]×{0}. With
+these definitions one is in hypothesis of Theorems 2 and 20.
+Letpbe the projection to Mof (0,0), the rotation vector vF(p) is the
+projection of α′(0) toTMand the rotation vector vF−1(p) is the projection
+23ofβ′(0). Hence the two vectors are different (since α′(0) has a non-null x-
+component). However one cannot apply Corollary 21 and in fact, as we have
+shown, the conclusions of this corollary do not hold in this case.
+This example also shows that the distance to the geodesic escort giv en by a
+rotation vector can be non-bounded.
+10 Rotation vectors for flows
+We will now develop the concept of rotation vectors for continuous flows. Let
+us begin with the definition.
+Definition 10 (Rotation vector for a flow) .LetMbe a complete Riemannian
+manifold, /tildewiderMits universal covering space, f:R×M→Ma continuous flow,
+andF:R×/tildewiderM→/tildewiderMits lift.
+A rotation vector v(x)of a point x∈Mis a vector in the tangent space
+TxMsuch that for any lift ˜xofxthe following holds:
+d(˜α(t),Ft˜x) =o(t)whent→+∞
+where˜αis the lift starting at ˜xof the geodesic α: [0,+∞)→Mdefined by
+α(t) = expx(tv(x)).
+An important difference between the case of flows and that of home omor-
+phisms is that for flows there is a unique lift which automatically commut es
+with all covering transformations. Also the geodesic escort given b y a rotation
+vector satisfies an asymptotic condition on all t∈[0,+∞) (as opposed to only
+integer values). In order to deduce an existence theorem for flow s from the cor-
+respondingtheoremforhomeomorphisms, it is necessaryto contr olthe variation
+of a trajectory between integer times (compare with the discussio n in [Kin73]
+section 1.4).
+Theorem 22 (Existence of rotation vectors for flows) .LetMbe a complete
+connected Riemannian manifold with non-positive sectiona l curvature and let /tildewiderM
+be its universal covering space. For each x∈Mlet˜x∈/tildewiderMdenote an arbitrary
+lift ofx.
+Suppose that f:R×M→Mis a continuous flow with lift F:R×/tildewiderM→/tildewiderM,
+and that µis anf-invariant Borel probability measure satisfying the condi tion:
+/integraldisplay
+Msup
+0≤s≤t≤1d(Fs˜x,Ft˜x)dµ(x)<+∞
+Then for µ-almost every x∈Mthere exists a unique rotation vector v(x)∈
+TxM.
+Proof.By Theorem 2 for almost every x∈Mthere is a unique vector v(x)∈
+TxMsuch that:
+d(˜α(n),Fn˜x) =o(n) whenn→+∞,n∈Z
+24where ˜αis the lift starting at ˜ xoft/ma√sto→exp(tv(x)).
+The function g:M→[0,+∞) given by g(x) = sup0≤s≤t≤1d(Fs˜x,Ft˜x) is
+well defined and continuous. In order to prove that
+d(˜α(t),Ft˜x) =o(t) whent→+∞
+if suffices to show that for almost all x∈Mit holds that g(fnx) =o(n) when
+n→+∞. This follows from Birkhoff’s ergodic theorem which is applicable
+becausegis integrable.
+11 A Semi-conjugacy result for Flows on Fiber
+Bundles
+In this section we will show how the existence of non-null rotation ve ctors pro-
+vides a measurable semi-conjugacy between a given flow and the geo desic flow
+of a manifold. This generalizes the semi-conjugacy statement of Th eorem 4.1
+in [Boy00] by replacing the restriction of constant negative curvat ure by that of
+strong visibility (see Appendix).
+We will begin by defining rotation vectors through a map between two man-
+ifolds:
+Definition 11. Let/tildewiderMbe a Hadamard manifold, π:/tildewiderM→Ma Riemannian
+covering, f:R×N→Na continuous flow on a manifold Nandh:N→M
+a continuous map.
+A rotation vector vh(x)of a point x∈Nis a vector in the tangent space
+Th(x)Msuch that for any lift ˜pofp=h(x)the following holds:
+d(˜α(t),˜β(t)) =o(t)whent→+∞
+where˜αis the lift starting at ˜pof the geodesic α: [0,+∞)→Mdefined by
+α(t) = expx(tv(x))and˜βis the lift starting at the same point of the curve
+β(t) =h(ft(x)).
+This definition applies in particular to the geodesic flow on TMby taking
+the map h:TM→Mto be the natural projection. In this case the rotation
+vector associated to a point v∈TMis simply v.
+Whenf:R×TM→TMis the geodesic flow of a different metric then
+the one we are considering on Mrotation vectors provide a comparison between
+geodesicsofthetwometrics. Ingeneral,if p:E→Misafiberbundleprojection
+andf:R×E→Eis a continuous flow rotation vectors through pprovide a
+comparison between the trajectories of fon the base manifold, and the geodesic
+flow of this manifold as in [Boy00].
+Asanotherexamplesuppose f:R×S(N)→S(N)isthegeodesicflowonthe
+unit tangent bundle S(N) of Riemannian manifold Nandh:S(N)→Mis the
+composition of the natural projection from S(N) toNand a map h1:N→M.
+In this case the integral of the norm of the associated rotation ve ctors (i.e. the
+25rate of escape) is equal to the ‘intersection’ ih1(gN,gM) of the geodesic flows gN
+andgMonNandMrespectively introduced in [CF90].
+The following existence theorem is a corollary of Theorem 14 by argum ents
+very close to the proof of Theorem 2.
+Theorem 23. Suppose f:R×N→Nis aC1flow on a manifold Npre-
+serving a Borel probability measure µwith compact support. If Mis a con-
+nected complete Riemannian manifold with non-positive sec tional curvature and
+h:N→Mis aC1map then for µ-almost every x∈Nthere exists a unique
+rotation vector vh(x).
+Proof.Letπ:/tildewiderM→Mbe the universal Riemannian covering of Mand let
+L:M→/tildewiderMbe measurable such that π(L(x)) =xfor allx∈M.
+Defineφ:N→/tildewiderMNso that, for all x∈Nandn∈N, ifαx: [0,+∞)→/tildewiderM
+is the lift starting at L(h(x)) of the curve t/ma√sto→h(ftx) thenφ(x)n=αx(n).
+Notice that if X:N→TNis the vector field generating the flow fand
+C= max supp(µ)/bardblDhX/bardbl,where the maximum is taken on the support of µ, then:
+/integraldisplay
+Nd(φ(x)0,φ(x)1)dµ(x)≤/integraldisplay
+N/integraldisplay1
+0/bardblDhX(fsx)/bardbldsdµ(x)≤C
+Also one has that
+d(φ(x)m+1,φ(x)n+1) =d(αx(m+1),αx(n+1)) =d(αf1(x)(m),αf1(x)(n))
+where the second equality is obtained by applying the covering trans formation
+takingαx: [1,+∞)→/tildewiderMtoαf(x): [0,+∞)→/tildewiderM(which are two lifts of
+t/ma√sto→ft(f1(x))).
+Henceφsatisfies the hypothesis of Theorem 14 and in consequence for µ
+almost every x∈N, either because the rate of escape is zero or, if it is positive,
+by Lemma 11, there is a unique geodesic βx: [0,+∞)→/tildewiderMstarting at L(h(x))
+and escorting the sequence αx(n).
+For such a point x∈NletRbe the rate of escape of {αx(n)}n≥0. To
+conclude that the projection of Rβ′(0) toTMis a rotation vector for xthrough
+hit suffices to note that if n≤t≤n+1 where n∈None has:
+d(αx(t),αx(n))≤C
+The uniqueness of vh(x) for the points where it is defined follows directly
+from the uniqueness statement for βx.
+We define the rotation vectors to the past as those for the time-r eversed flow
+gtx=f−tx. In the same spirit as Section 9 we obtain the following:
+Theorem 24. With the hypothesis and notation of Theorem 23 and the ad-
+ditional assumption that µis ergodic, let v−
+h(x)denote the rotation vector to
+the past of a point x∈Nif it exists. For almost every x∈Nit holds that
+/bardblvh(x)/bardbl=/bardblv−
+h(x)/bardbland, either vh(x) =v−
+h(x) = 0orv−
+h(x)/ne}ationslash=vh(x).
+26Proof.Letπ:/tildewiderM→Mbe the universal Riemannian covering of M.
+Notice that, because µis ergodic, there exist constants R+,R−such that
+/bardblvh(x)/bardbl=R+and/bardblv−
+h(x)/bardbl=R−for almost every x∈N. IfR+=R−= 0
+then there is nothing to prove. Otherwise by a linear time change (of the form
+t/ma√sto→ ±Ct) we may assume that R+= 1.
+For each x∈Nwith/bardblvh(x)/bardbl= 1 letαx:R→/tildewiderMbe a lift of the curve
+t/ma√sto→h(ftx). Also, let v(x)∈Tαx(0)/tildewiderMproject to vh(x) and define g:N→Ras
+follows (see Appendix for definition of Bv):
+g(x) =Bv(x)(αx(0),αx(1))
+By noting that BDρv(ρ(p),ρ(q)) =Bv(p,q) for any p,q∈/tildewiderM,v∈S(/tildewiderM) and
+any isometry ρ:/tildewiderM→/tildewiderMit follows that gdoesn’t depend on the chosen lift αx
+and in particular is measurable (by choosing x/ma√sto→αx(0) measurable). This also
+gives the following equalities:
+n−1/summationdisplay
+k=0g(fkx) =n−1/summationdisplay
+k=0Bv(x)(αx(k),αx(k+1)) =Bv(x)(αx(0),αx(n))
+n/summationdisplay
+k=1g(f−kx) =n/summationdisplay
+k=1Bv(x)(αx(−k),αx(−k+1)) =Bv(x)(αx(−n),αx(0))
+Becausev(x)projectsto vh(x)whichhasnorm1itfollowsthat Bv(x)(αx(0),exp(nv(x))) =
+n. Also,
+Bv(x)(αx(0),αx(n)) =Bv(x)(αx(0),exp(nv(x)))) +Bv(x)(exp(nv(x)),αx(n))
+From this and the fact that
+|Bv(x)(exp(nv(x)),αx(n))| ≤d(exp(nv(x),αx(n)) =o(n)
+one obtains that
+lim
+n→+∞1
+nBv(x)(αx(0),αx(n)) = lim
+n→+∞1
+nn−1/summationdisplay
+k=0g(fkx) = 1 =/bardblvh(x)/bardbl(1)
+for allx∈Nwith/bardblvh(x)/bardbl= 1.
+Furthermore one has |g(x)| ≤d(αx(0),αx(1)) which is bounded on the sup-
+port ofµ, hence by Birkhoff’s ergodic theorem and the ergodicity of µ:
+1 = lim
+n→+∞1
+nn/summationdisplay
+k=1g(f−k(x)) = lim
+n→+∞1
+nBv(x)(αx(−n),αx(0)) (2)
+for almost every x∈N.
+Since|Bv(x)(αx(−n),αx(0))| ≤d(αx(−n),αx(0)) for all nthis implies that
+/bardblv−
+h(x)/bardbl ≥1 =/bardblvh(x)/bardblfor almost every x∈N. However by applying the
+27argument to a time reversing reparametrization of fit follows that /bardblvh(x)/bardbl=
+/bardblv−
+h(x)/bardbl= 1 for almost every x∈N.
+To show that vh(x)/ne}ationslash=v−
+h(x) for almost every x∈N, we proceed by con-
+tradiction. If vh(x) =v−
+h(x) with positive probability then for some x∈Nwe
+would have:
+1 =/bardblv−
+h(x)/bardbl= lim
+n→+∞1
+nBv(x)(αx(0),αx(−n))
+=−lim
+n→+∞1
+nBv(x)(αx(−n),αx(0)) =−1
+where the second equality is given by equation 1 applied to the time-re versal of
+fand last equality follows from equation 2.
+In orderto stateour semi-conjugacyresult let us fix the followingd efinitions.
+Definition 12 (Cocyle).Letf:R×N→Nbe a measurable flow on a measur-
+able space (N,BN). A cocycle over fis a measurable function a:N×R→R
+such that:
+a(x,s+t) =a(x,s)+a(fsx,t)for alls,t∈R,x∈N
+The cocycle ais said to be invertible if a(x,·) :R→Ris an increasing homeo-
+morphism for all x.
+Definition 13 (Measurable semi-conjugacy) .A measure preserving flow f:
+R×N→Non a probability space (N,BN,µ)is said to be semi-conjugate to a
+measurable flow g:R×M→Mon a measurable space (M,BM)if there exists
+an invertible cocycle a:N×R→Rand a measurable function φ:N→Msuch
+that:
+φ(ftx) =ga(x,t)φ(x)for allt∈R,x∈N
+The last definition is quite lax, e.g. all flows are semi-conjugate to the
+constant flow on the space containing a single point. However, we will prove a
+semi-conjugacyresultwithgeometriccontentsincetheconjugat ionwillassociate
+to each point a geodesic escort of the image of its orbit through a giv en map.
+Theorem 25 (Semi-conjugacy) .With the hypothesis and notation of Theorem
+24 suppose that Mis a strong visibility manifold and that /bardblvh(x)/bardbl=R >0for
+almost every x∈N.
+Then there exists an invariant Borel set G⊂Nwithµ(G) = 1and such
+thatfrestricted to Gis measurably semi-conjugate to the geodesic flow g:
+R×TM→TMofM. Furthermore, the semi-conjugation φ:G→TMcan
+be chosen so that for all x∈Gthere exist ˜α,˜β:R→/tildewiderMwhich are lifts of
+the curves t/ma√sto→exp(tφ(x))andt/ma√sto→h(ftx)respectively, and have the following
+property:
+d(˜α(t),˜β(t)) =o(t)whent→ ±∞
+28Proof.By linear time change of the flow fwe may suppose R= 1
+LetG⊂Nbe the set of points possessing distinct rotation vectors with
+norm 1 to the past and the future. This set is invariant, and has full measure
+by Theorems 23 and 24.
+We first note that the functions vh,v−
+h:G→TMare measurable. In fact if
+one defines the curve t/ma√sto→vt
+h(x) to be the lift of t/ma√sto→h(ftx) toTh(x)Mvia the
+exponential map, then vt
+his a continuous function of x∈Gand for all x∈G
+one has:
+vh(x) = lim
+t→+∞1
+tvt
+h(x)
+v−
+h(x) = lim
+t→+∞1
+tv−t
+h(x)
+Now fixx∈G, consider any lift y∈/tildewiderMofh(x) and define ˜β:R→/tildewiderMas the
+lift oft/ma√sto→h(ftx) starting at y.
+Letv+,v−∈Ty/tildewiderMproject to vh(x) andv−
+h(x) respectively and let v=
+πh(v+,v−)betheprojectionalonghorospheresof v+,v−(seeLemma31). Defin-
+ing ˜α(t) = exp(tv) it holds that:
+d(˜α(t),˜β(t)) =o(t) whent→ ±∞
+Also, ifγis a covering transformation then one can see that Dγv+remains
+asymptotic (see Appendix) to Dγvand similarly for Dγv−and−Dγv, also the
+base points remain on the same horosphere with respect to BDγvand therefore
+it must be that Dγv=πh(Dγv+,Dγv−). This shows that the projection of v
+toTMdoesn’t depend on the chosen lift y. Hence the equation:
+φ1(x) =Dπ(v)
+defines a measurable function φ1:G→TM.
+For anyt∈Rletv+(t) andv−(t) denote the lifts of vh(ftx) andv−
+h(ftx) to
+T˜β(t)/tildewiderM. Sincev+(t) is asymptotic to v+for allt, Lemma 27 implies that v+(t)
+is a continuous function of t. Similarly v−(t) is a continuous function of t. This
+implies that there exists a unique b(x,t)∈R, which is continuous with respect
+tot, such that:
+φ1(ftx) =gb(x,t)φ1(x)
+From the aboveequation the function b:G×R→Ris a measurable cocycle.
+However the cocycle bis not necessarily invertible.
+To construct an invertible cocycle we notice that since ˜ αand˜βare asymp-
+totic one has:
+lim
+t→+∞b(x,t)
+t= 1 for all x∈G
+And also if C= max supp(µ)/bardblDhX/bardblone has:
+|b(x,t)| ≤/integraldisplayt
+0/bardblDhX(fsx)/bardblds≤Ct
+29so thatbis uniformly Lipshitz with respect to t.
+Under these conditions [Boy00, Lemma 1.4] guarantees the existen ce of a
+measurable function r:G→(0,+∞) such that the equation
+a(x,t) =b(x,t)+r(ftx)−r(x)
+defines an invertible cocycle which satisfies the property:
+lim
+t→+∞a(x,t)
+t= 1 for all x∈G
+Hence by defining φ(x) =gr(x)φ1(x) the theorem follows.
+12 Examples
+12.1 Homeomorphisms of hyperbolic surfaces
+Besides the two-dimensional torus the simplest manifolds to which ou r results
+are applicable are complete hyperbolic surfaces. Self-homeomorph isms of such
+manifolds which are isotopic to the identity form a large family of dynam ical
+systems which have been widely studied with other tools. We will now dis cuss
+some implications of the existence of rotation vectors in this contex t and relate
+them to known results.
+Consider the disk D={z∈C:|z|<1}endowed with the Poincare metric.
+We recall that geodesics of this metric are parametrizations of eith er Euclidean
+diameters of Dor Euclidean circle arcs which are perpendicular to the boundary
+∂D. In particular, we note that any geodesic ray has a well defined end point in
+∂D. The only additional fact we will need about this metric is that the clos ed
+disk centered at a point x∈Dwith hyperbolic radius R≥0 (which we will
+denote by D(x,R)) is in fact an Euclidean closed disk centered at a point on
+the Euclidean segment [0 ,x] (see the proof of [And05, Proposition 4.5]).
+Throughout this section let Mbe a complete hyperbolic surface and f:
+M→Ma homeomorphism which is isotopic to the identity. Fix a Riemannian
+covering π:D→Mand letF:D→Dbe an admissible lift of f. The
+following proposition summarizes the consequences of the existenc e of points
+with non-null rotation vectors.
+Proposition 26. Letx∈Dproject to a point in Mpossessing rotation vectors
+to the future and past which are non-null, distinct, and have equal norm. Then
+there exist two distinct boundary points x−,x+∈∂Dsuch that:
+lim
+n→+∞Fn(x) =x+
+lim
+n→+∞F−n(x) =x−
+30Proof.Letv+,v−∈TxDproject to the rotation vectors to the past and future
+ofπ(x) respectively, and let x+,x−∈∂Dbe the boundary endpoints of the
+geodesic rays with initial condition v+andv−respectively. One has that:
+d(exp(nv+),Fn(x)) =o(n) whenn→+∞
+d(exp(nv−),F−n(x)) =o(n) whenn→+∞
+In particular it follows that d(0,Fn(x)) =/bardblv+/bardbln+o(n) so that any limit
+point of the sequence {Fn(x)}n≥0will belong to the boundary ∂D. Define
+xn= exp(nv+) and notice that Rn=d(xn,0) =n/bardblv+/bardbl+O(1). If follows
+thatFn(x)∈D(xn,Rn) =Dnfor all but finitely many n≥0. Since Dnis
+an Euclidean disk centered on a point in the segment [0 ,xn] and 0 belongs to
+the boundary of Dnit follows that Dnis contained in the closed Euclidean
+disk with diameter [0 ,xn/|xn|]. Since lim n→+∞xn/|xn|=x+one obtains that
+limn→+∞Fn(x) =x+as claimed.
+The same argument establishes the claim for for {F−n(x)}n≥0andx−.
+The conclusion of the above proposition was shown by Michael Hande l to
+hold in some cases in which xis the lift of a periodic orbit of fas discussed
+in [Fra96, Section 3] (compare with Theorems 17 and 20). The existe nce of
+points in the lift whose orbits have distinct limits on the boundary circle when
+n→ ±∞can in certain circumstances imply the existence of fixed points of
+fwith index 1 by application of Handel’s fixed point theorem as in [Fra96,
+Proposition 3.4] (see also [Han99]).
+In particular from the observations above it follows that if a point x∈
+Dprojects to a point with non-null and distinct rotation vectors to t he past
+and future then its orbit cannot remain close to a horocycle since th is would
+implyx+=x−contradicting Proposition 26 (compare with the example at the
+beginning of section 8).
+12.2 Hyperbolic Magnetic Flow
+The differential equation
+α′′(t) =iα′(t) for allt
+whereα:R→R2and multiplication by idenotes rotation by 90 degrees in the
+counter-clockwise direction, determines the trajectory of a cha rged particle in a
+plane orthogonal to a constant magnetic field in Euclidean space.
+It is easyto showthat anysuch curveisperiodic and in fact is aparam etriza-
+tion of an Euclidean circle of radius /bardblα′(0)/bardbl.
+We will now consider a similar dynamical system in the hyperbolic plane,
+that is, we consider the differential equation:
+D
+dtα′(t) =iα′(t) (3)
+31Figure 2: Trajectories parting from 0 in the Poincar´ e disk model of the hyper-
+bolic plane are Euclidean circle arcs and are traversed in the counter -clockwise
+sense.
+whereD/dtdenotes covariant derivation with respect to the canonical conne c-
+tion of the hyperbolic metric. This differential equation defines a flow on the
+tangent bundle of the hyperbolic plane and also on the tangent bund le of any
+oriented hyperbolic surface (this is the so called ‘magnetic flow’ for t he volume
+2-form, e.g. see [Mir07]).
+Fixinganon-constanttrajectoryanddenotingthenormofitsvelo city(which
+remains constant) by vwe will prove the following (see figure 12.2 ):
+•Ifv <1 then the trajectory is periodic and in fact is a hyperbolic circle of
+radiusrwhere tanh( r) =v.
+•Ifv= 1 then the trajectory is a horosphere.
+•Ifv >1 then the trajectory has rate of escape R=√
+v2−1 and describes
+acurvewhichisataconstantdistance r >0fromageodesic. Thedistance
+ris given by the equation tanh( r) = 1/v.
+In particular, if Mis a compact oriented hyperbolic surface and f:R×
+TM→TMis the magnetic flow described above then any ergodic invariant
+measure for fwhich is supported on a set of vectors with constant norm v >1
+has non-null rotation vectors and satisfies the hypothesis of The orem 25.
+Since the set of solutions to equation 3 is invariant by isometries and t he
+description above accounts for all possible values of the initial spee dvall one
+needs to do to establish the claims is calculate that the described cur ves are
+in fact solutions to the equation. This verification amounts to calcula ting the
+covariant derivative of an explicit curve and is simple to do if an approp riate
+coordinate system (which we will now indicate) is used for each case.
+For the case v <1 the claim follows from [Lam02] but one can also verify it
+by calculating in polar coordinates. Concretely consider the plane P={(r,θ)∈
+R2:r >0}with the metric d r2+ sinh(r)2dθ2. Let Γ be the group generated
+by the isometry ( r,θ)/ma√sto→(r,2π+θ). The Riemannian quotient manifold P/Γ
+32is isometric to the hyperbolic plane minus one point (this follows from th e fact
+that the perimeter of the hyperbolic circle of radius ris 2πsinh(r)). Our claim
+amounts to the statement that for each r >0 one has:
+D
+dtα′(t) = (−tanh(r),0)
+where
+α(t) = (r,t/cosh(r))
+For the case v= 1 the verification may be done in the upper half-plane
+model (i.e. H={x= (x,y)∈R2:y >0}with the metric (d x2+dy2)/y2). It
+amounts to verifying that:
+D
+dtα′(t) = (0,1)
+where
+α(t) = (t,1)
+Finally, consider the case v >1. We will use a system of coordinates ( x,r)
+described as follows: Fix a unit speed geodesic γin the hyperbolic plane, the
+point with coordinates ( x,r) is the unique point at distance rfromγ(x) on the
+geodesic ray with initial condition iγ′(x). In these coordinates the hyperbolic
+metric is cosh( r)2dx2+dr2as can be seen by the isometry with the upper half-
+plane model given by φ(x,r) = (extanh(r),ex/cosh(r)). One must verify that
+for each r >0 the following curve is a solution to equation 3:
+α(t) = (t/sinh(r),r)
+In this case calculation yields:
+D
+dtα′(t) = (0,−1/tanh(r))
+The norm of α′(t) is 1/tanh(r) =v. Sinceαis at a constant distance from
+t/ma√sto→(t/sinh(r),0) which is a geodesic it follows that the rate of escape of either
+curve isR=1
+sinh(r). By application of the identity3:
+sinh(arctanh( x)) =x√
+1−x2
+it follows that:
+R=1
+sinh(r)=1
+sinh(arctanh(1 /v))=/radicalbig
+v2−1
+3This identity can be obtained starting with tanh(arctanh( x)) =xby squaring both sides
+and using the fact that cosh( y)2= 1+sinh( y)2.
+33Acknowledgments
+The author would like to thank Gonzalo Contreras, Fran¸ cois Ledra ppier, Rafael
+Potrie, Mart´ ın Sambarino, Juliana Xavier, the participants and org anizers of
+the “Seminario Atl´ antico de Geometr´ ıa 2010”, and the two journa l referee’s
+and board member, for contributing significantly to improving the qu ality and
+clarity of this work.
+Appendix: Visibility Manifolds
+Thepurposeofthis appendix istopresentsomefactsabout Busem annfunctions
+and Visibility Manifolds.
+Definition 14 (Asymptotic tangent vectors) .LetHbe a Hadamard manifold.
+Two vectors v,w∈S(H)are said to be asymptotic if:
+d(exp(tv),exp(tw)) =O(1)whent→+∞
+We observe that if v,w∈S(p) for some p∈Hthenvandware asymptotic
+if and only if v=w. This follows because t/ma√sto→d(exp(tv),exp(tw)) is a non-
+constant convex function (lemma 8) taking the value 0 at t= 0 and therefore
+grows at least linearly. Also, by the same argument, the condition th at the
+distanceinthedefinitionbeboundedisequivalenttoitbeing o(t)whent→+∞.
+The main non-trivial fact we shall use about this equivalence relation ship is the
+following:
+Lemma 27. LetHbe a Hadamard manifold. If {vn}n≥1,{wn}n≥1⊂S(H)and
+v,w∈S(H)are such that:
+vn→vwhenn→+∞
+wn→wwhenn→+∞
+vnis asymptotic to wnfor alln
+thenvis asymptotic to w.
+Proof.Letπ:S(H)→Hdenote the bundle projection and g:R×S(H)→
+S(H) the geodesic flow of H.
+Suppose that there exists t >0 such that:
+d(π(gtv),π(gtw))> d(π(v),π(w))
+Then because gis continuous there exists nsuch that:
+d(π(gtvn),π(gtwn))> d(π(vn),π(wn))
+However by convexity this implies that:
+lim
+s→+∞d(π(gsvn),π(gswn)) = +∞
+Which contradicts the fact that vnandwnare asymptotic.
+34We recall the following definition of Busemann functions (compare wit h the
+definition of fγin [EO73, pg. 56]). See [DPS10] for background on this concept.
+Definition 15 (Busemann function, horosphere) .LetHbe a Hadamard man-
+ifold and v∈S(H). The Busemann function Bv:H×H→Ris defined
+by:
+Bv(x,y) = lim
+t→+∞d(x,exp(tv))−d(exp(tv),y)
+The maximal sets of points L⊂HwithBv(x,y) = 0for allx,y∈Lare
+called horospheres or limit spheres of H.
+Directlyfromthedefinitiononecandeducethatboththeinequality |Bv(p,q)| ≤
+d(p,q) and the equality Bv(p,r) =Bv(p,q) +Bv(q,r) hold for all p,q,r∈H
+and allv∈S(H). We shall also use two more facts.
+Lemma 28. LetHbe a Hadamard manifold. If v,w∈S(H)are asymptotic to
+each other then Bv=Bw.
+Proof.Letv∈S(p) andw∈S(q). If one defines f,g:H→Rby:
+f(x) =Bv(p,x)
+g(x) =Bw(q,x)
+Then by [EO73, Proposition 3.1] one has that f−gis constant. Since:
+Bv(x,y)−Bw(x,y) =f(y)−f(x)−(g(y)−g(x))
+the conclusion follows.
+Lemma 29. LetHbe a Hadamard manifold. If {vn}n≥1⊂S(H)andv∈S(H)
+are such that:
+vn→vwhenn→+∞
+thenBvn→Bvuniformly on compact subsets of H×Hwhenn→+∞.
+Proof.The function C:S(H)×H→Rdefined by:
+C(w,x) = lim
+t→+∞d(exp(tw),x)−t
+is continuous by [Ebe73, Proposition 2.3]. We will show that this is equiv alent
+to our claim.
+For this purpose observe that if w∈S(H) andx,y∈Hthen:
+Bw(x,y) = lim
+t→+∞d(exp(tw),x)−t+t−d(exp(tw),y)
+=C(w,x)−C(w,y)
+It now follows from the the continuity of CthatBvnconverges to Bvpoint-
+wise. Uniformconvergenceoncompactsubsetsisaconsequenceo ftheinequality
+|Bw(x,y)| ≤d(x,y) which is valid for all w∈S(H) andx,y∈H.
+35We will now restrict our attention to manifolds satisfying so called ‘visib ility
+axioms’ 1 and 2. In these manifolds we will find a natural projection f rom the
+setG ⊂S(H)×S(H) of pairs of distinct unit tangent vectors with the same
+base point, to S(H).
+Our definition of ‘strong visibility’ is equivalent to visibility along with Ax-
+iom 2 of [EO73, Definition 4.1] (here we use the fact that H(∞) with the cone
+topology is homeomorphic to S(p) for all p∈Has shown in [EO73, Theorem
+2.10]).
+Definition 16 (Visibility manifold) .A visibility manifold is a Riemannian
+manifold whose universal Riemannian covering is a Hadamard manifold Hwith
+the following property: For each p∈Hand each ǫ >0there exists a number
+R=R(p,ǫ)such that for any geodesic segment [q,r]⊂Hwithd(p,[q,r])≥R
+it holds that ∠p(q,r)≤ǫ.
+We will use the following result:
+Lemma 30 (Corollary4.6of[EO73]) .LetHbe a visibility Hadamard manifold.
+For each compact K⊂Handǫ >0there exists R >0such that any geodesic
+segment [q,r]⊂Hwithd(K,[q,r])≥Rsatisfies∠p(q,r)≤ǫfor allp∈K.
+The existence part of the following definition was shown to hold on any
+visibility Hadamard manifold in [EO73, Proposition 4.4] and the converse im-
+plication was stated without proof.
+Definition 17 (Strong visibility manifold) .A strong visibility manifold is a
+visibility manifold such that its universal Riemannian cov ering space Hsat-
+isfies the following: For each pair (v+,v−)∈ Gthere exists a unique (up to
+reparametrization by time translation) unit speed geodesi cα:R→Hsuch that
+α′(0)is asymptotic to v+and−α′(0)is asymptotic to v−.
+Note that it was shown by T.Ukai that any Hadamard manifold whose
+geodesic flow is of Anosov type is a strong visibility manifold (see [Uka00 ]).
+To conclude we show that αdepends continuously on ( v+,v−) in the above
+definitions. This has been used to construct measurable semi-conj ugating maps
+in section 11.
+Lemma31 (Projectionalonghorospheres) .LetHbe a strong visibility Hadamard
+manifold. There is a unique function πh:G →S(H)such that for all (v+,v−)∈
+Gthe following conditions are satisfied:
+1. Setting v=πh(v+,v−)it holds that vis asymptotic to v+and−vis
+asymptotic to v−.
+2. Ifp,q∈Hare such that v+,v−∈S(p)andv∈S(q)thenBv(p,q) = 0.
+Furthermore, πhis continuous.
+Proof.It follows from Lemma 28 that if v,w∈S(H) are asymptotic then Bv=
+Bw. Also,becausegeodesicsaregloballyminimizing, onehas Bv(exp(sv),exp(tv)) =
+36t−s. Hence, given ( v+,v−)∈ Gwith basepoint p, ifαis the unique unit
+speed geodesic (up to reparametrization by time translation) with α′(0) asymp-
+totic to v+and−α′(0) asymptotic to v−then there is a unique t∈Rwith
+Bv+(α(t),p) = 0. This shows that the function πhexists and is unique (in fact,
+in the above notation, πh(v+,v−) =α′(t)).
+To establish continuity let ( v+,v−)∈ G ∩S(p) withπh(v+,v−) =vand let
+q∈Hbe such that v∈S(q). Consider sequences pnandv+
+n,v−
+n∈S(pn) such
+that:
+(v+
+n,v−
+n)→(v+,v−) whenn→+∞
+Finally, define vn=πh(v+
+n,v−
+n) and let qnbe such that vn∈S(qn). We must
+show that vn→vwhenn→+∞.
+First suppose vn→w∈S(r) whenn→+∞. One has that Bvn(pn,qn) = 0
+for allnand by Lemma 29 this implies that Bw(p,r) = 0. Also by Lemma 27
+one has that wis asymptotic to v+and−wtov−. By the uniqueness of πh,
+which we have already established, it follows that r=qandw=v.
+The above argument implies that any convergent subsequence of vnmust
+converge to v, hence to establish continuity it suffices to show that {vn}is
+bounded.
+For this purpose take 0 < ǫ <infn∠pn(exp(v+
+n),exp(v−
+n)) andK={p} ∪/uniontext
+n{pn}. By Lemma 30 there exists a number Rand a sequence {tn} ⊂Rsuch
+that:
+d(exp(tnvn),pn)≤R
+and therefore |Bv+
+n(pn,exp(tnvn))| ≤R.
+However since
+0 =Bvn(pn,qn) =Bvn(pn,exp(tnvn))+Bvn(exp(tnvn),qn) =O(1)−tn
+one obtains that tn=O(1) when n→+∞and hence {vn}is bounded.
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+39
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+Cryptographic Implications for Artificially Mediate d Games 
+Thomas Meyer 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+12.22.2009 
+ 
+  1 Abstract  
+There is currently an intersection in the research of game theory and 
+cryptography. Generally speaking, there are two asp ects to this partnership. First there is 
+the application of game theory to cryptography. Yet , the purpose of this paper is to focus 
+on the second aspect, the converse of the first, th e application of cryptography to game 
+theory. Specifically, there exist a branch of non-c ooperative games which have a 
+correlated equilibrium as their solution. These equ ilibria  tend to be superior to the more 
+conventional Nash equilibria. The primary condition  for a correlated equilibrium is the 
+presence of a mediator within the game. This is sim ply a neutral and mutually trusted 
+entity. It is the role of the mediator to make reco mmendations in terms of strategy 
+profiles to all players, who then act (supposedly) on this advice. Each party privately 
+provides the mediator with the necessary informatio n, and the referee responds privately 
+with their optimized strategy set. However, there s eem to be a multitude of situations in 
+which no mediator could exist. Thus, games modeling  these sorts of cases could not use 
+these entities as tools for analysis. Yet, if these  equilibria are in the best interest of 
+players, wouldn’t it be rational to construct a mac hine, or protocol, to calculate them? Of 
+course, this machine would need to satisfy some sta ndard for secure transmission 
+between a player and itself. The requirement that n o third party (namely other players) 
+could detect either the input or strategy profile w ould need to be satisfied by this scheme. 
+Here is the synthesis of cryptography into game the ory; analyzing the ability of the 
+players to construct a protocol which can be used s uccessfully in the place of a mediator. 
+ 
+I. Introduction  
+In the past, game theory and cryptography were deve loped independently 
+of one another, deemed different both in their aims  and assumptions. The first discipline 
+models the behavior of firms and other entities in situations such as manufacturing, 
+decisions of collusion, etc. It assumes that each p arty acts rationally, or in its own self-
+interest. Conversely, cryptography is concerned sol ely with secure transactions. The 
+assumption is that the parties transacting with one  another follow, to the point of 
+irrationality, certain security protocols or crypto graphic systems. The cryptanalyst, or 
+hacker, feverishly and arbitrarily attempts to disc over the content of these transactions 
+using methods such as brute-forcing, birthday attac ks, etc. Furthermore, they are believed 
+to have access to extremely sophisticated equipment . No thought is given as to whether 
+or not the utility of the information is greater th an the resources used to illicitly obtain it. 
+Hence, the dogmatic determination of the cryptanaly st is unfounded in certain instances.  
+Despite these differences, there is currently an in tersection in the research 
+of game theory and cryptography. Generally speaking , there are two aspects to this 
+partnership. First there is the application of game  theory to cryptography. Here, we use 
+the assumptions of the former to derive new, possib ly more realistic, security protocols. 
+As the paper by Katz states: 
+“Traditionally, cryptographic protocols are designe d under the assumption 
+that some parties…faithfully follow the protocol, w hile some parties are 
+malicious and behave in an arbitrary fashion. The g ame theoretic 
+perspective, however, is that all parties are simpl y rational and behave in 
+their own best interests. This viewpoint is incompa rable to the 
+cryptographic one: although no one can be trusted t o follow the protocol  2 (unless it is in their own best interests), the [ga me-theoretic] protocol need 
+not prevent ‘irrational’ behavior.” [1] 
+That is, these protocols need not take into account  what is considered as irrational 
+behavior. For instance, if the amount of work requi red to break the standard is greater 
+than the value of any data it protects, the protoco l may be considered as totally secure.  
+This paper is primarily concerned with examining th e second aspect of 
+this union; the application of cryptography to game  theory. Specifically, there exist a 
+branch of non-cooperative games which have a correl ated equilibrium (both will be 
+described in detail in the next section) as their s olution. These equilibria  tend to offer 
+more advantages compared to the more conventional N ash equilibria. Namely, they offer 
+“better payoffs than Nash [equilibria].”[2] Additio nally, these solutions “have payoffs 
+outside the convex hull of all Nash equilibria, and  therefore give more options to 
+players.”[1] The primary condition for a correlated  equilibrium is the presence of a 
+mediator within the game. This is simply a neutral entity, trusted by all parties involved 
+in the game. It is the role of the mediator to make  recommendations in terms of strategy 
+profiles to all players, who then act (supposedly) on this advice. All transactions which 
+the players have with this entity are, of necessity , private. Each party privately provides 
+the mediator with the necessary information, and th e mediator tells them and them alone 
+their optimized strategy set. However, there seem t o be a multitude of situations in which 
+no mediator could exist. Correspondingly, games mod eling these sorts of cases could not 
+use these entities as tools for analysis. Yet, if t hese equilibria are in the best interest of 
+players, it would be rational to simply construct a  machine, or protocol, to calculate them. 
+Of course, for this machine to be used, there would  need to be some standard for secure 
+transmission between player and machine. The requir ement that no third party (namely 
+other players) could detect either the input or str ategy profile of a player would need to be 
+satisfied by this scheme. This is the synthesis of cryptography into game theory; 
+analyzing the ability of the players to construct a  protocol which can be used successfully 
+in the place of a mediator. Specifically, the work of Dodis, Halevi, and Rabin, in this 
+regard will be examined. [3] 
+ 
+II. Basic Game Theoretic Concepts  
+Central to the discussion of this paper is the non- cooperative normal form 
+game. Imagine such a game, ,Χis being played by n players, the ith  of which is denoted 
+by .iP Then, it is entirely described as follows: =Χ }, )( ,){( 1 1n
+iin
+iiuS= = where 
+},...., {
+1 ki ii ssS= and →××=n i SSu... 1 /DoubleCapR. Let iSindicate the strategy set for ,iPand 
+the iiSs
+l∈represent the actual strategies. Hence, an outcome is simply the n-tuple 
+). ,..., (1b a nssv= Additionally, iuis the utility function for iP; )(vuiis the utility derived 
+from the outcome v by i. Because this is a non-cooperative game, the follo wing 
+restriction is imposed: “it is assumed that each pa rticipant acts independently, without 
+collaboration or communication with any of the othe rs.” [4]  
+However, non-cooperative normal form games in and o f themselves are 
+not important to this piece. Instead, their signifi cance is derived from their ability to be 
+extended to what are called mediated games. A media ted game, ,Ψis exactly as was 
+described above. It relies on the presence of a med iator, M, a neutral and mutually trusted  3 figure. The existence of M causes the given non-cooperative normal form game, ,χto be 
+separated into two stages. Firstly, M examines a known probability distribution D over a 
+set of outcomes, which is dependent upon certain in puts from all the players. From this 
+distribution, they select some vector of actions, a n outcome ). ,..., (* *
+1 nssa=  The 
+strategy *
+isis then recommended privately by M to .iPAs is stated by Hart and Mas-Colell 
+“Each player is given-privately-instructions for hi s own play only.” [5] The second stage 
+now occurs. The game χproceeds as before, with each player selecting some  strategy 
+from their strategy set. Of course, the convention is that each party utilizes the 
+recommended strategy. A correlated equilibrium for χoccurs when it is rational for each 
+entity to play their recommendation. Let ) |,(*kssuiii −′be the expected utility for iPof 
+playing iiSs∈′as opposed to the recommendation ,*
+isgiven that all other parties play their 
+recommendations. Then, a correlated equilibrium occ urs only when 
+)|,()|,(** *kssukssuiii iii − −≤′ for all players and elements of their strategy sets . This is 
+echoed in the following theorem *.  
+Theorem One – Define =Χ }. )( ,){( 1 1n
+iin
+iiuS= = Then, a distribution D leads 
+to a correlated equilibrium if and only if for all ),..., (1t jnssk= created from 
+D by M, all ,iPand all ,iiSs∈′the condition )|,()|,(** *kssukssuiii iii − −≤′  
+holds. 
+ 
+III. The benefits of a Correlated Equillibrium  
+The concept of a correlated equilibrium was constru cted by mathematician 
+Aumann partially to address his numerous critiques of Nash equilibria. Among these 
+critiques is a general unattractiveness of the equi librium, strange assumptions, nearly 
+circular reasoning, etc. He lists them succinctly a s follows: 
+“Now why should any player assume that the other pl ayers will play their 
+components of such an n-tuple [(one of) the Nash equilibria of a game], 
+and indeed why should they? This is particularly pe rplexing when, as 
+often happens, there are multiple equilibria; but i t has considerable force 
+even when the equilibrium is unique. Indeed, there are games whose Nash 
+equilibria appear quite unattractive even though th ey are unique… In a 
+two-person game, for example, Player 1 would play h is component only if 
+he believes that 2 will play his; this in turn woul d be justified only by 2’s 
+belief that 1 will indeed play his component…To man y this will sound 
+like a plain old circular argument: consistent, per haps, but hardly 
+compelling. Nash [equilibria do] make sense if one starts by assuming 
+that, for some specified reason, each player knows which strategies the 
+other players are using.”[6] 
+Regardless of the validity of these critiques, corr elated equilibria do in fact offer 
+numerous advantages over the more conventional Nash  equilibria. Firstly, it has been said 
+by many that “correlated equilibrium may be the mos t relevant non-cooperative solution 
+concept.” [5] This characteristic is derived from a n alternative, but equivalent 
+                                                 
+* A rigorous proof of this theorem can be found in [ 1].   4 formulation of the equilibrium. Namely, a game ,Γan extension of ,Χis considered in 
+which there are n players, each of whom receives a private signal (w hich does not 
+necessarily originate from a mediator) at the game’ s beginning. These signals then serve 
+as the basis for a player’s actions, and the game c ontinues as normal. The Nash 
+equilibrium derived for Γis simply the correlated equilibrium for .ΧYet, this calculation 
+is much simpler than finding the Nash equilibrium f or .ΧThis is because “with the 
+exception of well-controlled environments, it is ha rd to exclude a priori that…signals are 
+amply available to the players, and thus find their  way [easily] into a [Nash] equilibrium 
+[for ]Γ.” [5] That is, the Nash equilibrium for Γalmost naturally arises (not so for ), Χ 
+from which the correlated equilibrium to Χmay be derived almost effortlessly. Hence, the 
+correlated equilibria are quire relevant to non-coo perative games in the sense that they 
+are simple to compute.  
+Apart from an ease of computation, there are many i nstances in which 
+these equilibria offer greater imputations to all p layers than the Nash equilibrium might: 
+“In some games there may exist a correlated equilib rium that, for every party ,iPgives a 
+better payoff to iPthan any Nash equilibrium.”[1] Additionally, a larg er range of 
+equilibrium strategies are afforded to each player under this paradigm. This is indicated 
+by the nature of the topological spaces which both sorts of equilibria inhabit. The set of 
+correlated equilibria for a game, ,Ωcreate a convex polytope. The set of Nash 
+equilibria, ,Ξalso defines a convex polytope. However, this set i s of a much more 
+complex and restricted nature, since it consists en tirely of n-tuples which are fixed points. 
+Additionally, let } : )({ Ξ∈ = lli iEπ denote the set of all Nash equilibrium strategies 
+for .iPThen, “the sets ] ,..., ,[21 nEEE of equilibrium strategies in a solvable game are 
+polyhedral convex subsets of the respective mixed s trategy spaces.” †[4] In this, another 
+restriction is imposed upon the possible equilibriu m strategies for a given player under 
+the scheme of Nash equilibria. Yet, none of these c onstraints pertain to the correlated 
+equilibrium or its equilibrium strategies. In brief , it offers “more options to the 
+players.”[1] 
+ 
+IV. The Artificial Mediator  
+Ostensibly, the most necessary requirement for a co rrelated equilibrium is 
+that all parties involved rely upon a mediator. Thi s referee, as was briefly stated above, 
+must have no vested interest in the outcomes of the  game; the mediator must be a 
+mutually trusted, external entity. It is easy to th ink of many non-cooperative games in 
+which the existence of such a body is not possible.  Yet, if it is to the benefit of all players, 
+since it is necessary for the favorable correlated equilibrium, wouldn’t it be rational to 
+simply construct some sort of machine to perform th is task? In other words, “can the 
+trusted mediator be replaced by a protocol [or mach ine] that is run by the parties 
+themselves?”[1] This machine would provide the pres ence of a mediator when one 
+couldn’t be found. In this, it would provide assura nce of a correlated equilibrium, or at 
+the very least, that some of the necessary steps to wards one were being made.  
+                                                 
+† A proof of this can be found in [4].  5 Of course, such a protocol would need to satisfy ce rtain requirements. 
+Chiefly, when given the appropriate input from ,iPthe outcome it computes must yield 
+the same expected imputations as the human mediator . Additionally, all transactions 
+between iPand the device would need to be both private and se cure; no other player could 
+discover either the input of ,iPor the machine’s strategy recommendation for this p layer, 
+this is a non-cooperative game after all.  
+An Iterative Turing machine, denoted ,Π is the best candidate for this 
+artificial mediator. It can be made to satisfy the first of these requirements (shown in the 
+next section), and through further modification, th e Turing machine performs its duties 
+securely. Additionally, one can be reassured that a  correlated equilibrium can be 
+calculated in a practical amount of time. This is b ecause it has been calculated that such a 
+machine can compute a correlated equilibrium for a game in polynomial time. ‡ 
+ 
+V. A Non-Cooperative Game with Artificial Mediator  
+To examine the soundness of this artificial mediato r, and the ability of the 
+players to implement such a protocol, a non-coopera tive game with such a referee is 
+considered. It should be noted that because this an alysis is the realm of cryptography, the 
+game consists of certain more cryptographic convent ions, which an entirely game 
+theoretic model would never employ. Let this game b e represented by .ΘEach of the n 
+players is first given a security parameter, k. The game itself occurs in two stages, much 
+like a game with a human mediator, .Ψ To begin, there is the “cheap talk” phase. Here, 
+the parties run the protocol ,Π which computes the outcome ). ,..., (* *
+1 nrrb=  Next, 
+Πprivately and securely gives iP the strategy recommendation .*
+ir Note that any party 
+may choose to abort the game during this stage; the y may refuse to provide input to the 
+calculating machine.  
+The second phase of the game simply consists of eac h player selecting and 
+utilizing an element of their strategy set, not nec essarily their recommendation. If it is 
+rational for all parties to use these recommended s trategies, then a correlated equilibrium 
+at b ensues. As is mentioned by Katz, “It is not hard t o see that the strategy vector thus 
+specified [ b]…[yields] expected payoffs identical to those in t he original mediated game 
+[the correlated equilibrium vector a in ]. Ψ”[1] Yet, a very large assumption is made 
+in ,Θan assumption which is unacceptable in any rigorous  cryptographic analysis. 
+Namely, it was assumed that the mediator protocol w as secure and completely fair. In this 
+sense, fair refers to the receipt of a strategy rec ommendation by all players which 
+provided input to the machine. That is, all transac tions between a given party and the 
+protocol are successful. But, both security and fai rness could very easily not be the case 
+for the protocol in .ΘFor instance, consider a cryptanalyst (possibly a p layer or a body 
+under the employ of a player) who constructs a bloc k in the communication channel 
+                                                 
+‡ Interestingly enough, Nash equilibria correspond t o the complexity class, NP, on this machine; they a re 
+believed to require quasi-superpolynomial time.   6 between a given player(s) and the machine. Furtherm ore, suppose this block functions to 
+prevent the player(s) receiving a strategy recommen dation. § Denote such a game by .Θ′ 
+Despite these adverse conditions, Dodi, Halevi, and  Rabin, have 
+demonstrated that the outcome of Θ′is still a correlated equilibrium, at least in the 
+following special case. Let Θ′consist of two players, and assume that 2Preceives no 
+output from .. ΠThen, as they prove, **  there is a correlated equilibrium at which 1P 
+plays *
+1rand 2Putilizes the minimax profile, 1m′′ against .1PThe minimax profile against the 
+ith  player is the action i iSm− −∈such that )} ,({max *
+* iiirmru
+i−is minimized. That is, the 
+minimax profile is an element of the strategy set o f all other players, where “ iP [is given] 
+its lowest possible utility, assuming iP plays a best response [calculated by the protocol]  
+to the strategy of the other parties.”[1] This solu tion however does not generalize well 
+into games with more than two players. Thus it cann ot be used as a basis for the claim 
+that a correlated equilibrium exists despite a game  not being fair. In point of fact, this 
+solution is the closest to proving this generalizat ion, despite the fact that it only functions 
+in the aforementioned special case. 
+ 
+VI. Conclusion  
+A good deal can be concluded about the feasibility of a mediator protocol 
+created by the players, despite the fact that no on e is certain as to whether or not 
+correlated equilibria exist in an unfair artificial ly mediated game. Were they to exist, then 
+the player created protocol would be quite advantag eous and therefore feasible. 
+Regardless of its fairness, correlated equilibria w ould occur, and the players would enjoy 
+the corresponding advantages: greater payoffs, etc.  Alternatively, if these solutions did 
+not exist, apart from the special case mentioned ab ove, there would still be impetus 
+enough to create this protocol. However, the player s involved in its creation would need 
+to be quite cautious when approaching the fairness aspect, so as to achieve correlated 
+equilibria. For instance, the players might conside r dividing up the task of securing the 
+protocol among themselves such that no one player k nows enough about the scheme to 
+create unfairness. Additionally, the channels conne cting the parties to the machine would 
+most likely need to involve a cryptographic system( s) widely regarded as impenetrable. 
+These measures are equally necessary for the case i n which correlated equilibria exist in 
+some instances, and not others. That is, they are n eeded to guarantee equilibrium among 
+the players. In brief, because this protocol is ver y capable of producing advantageous 
+results, it is always feasible. Although, it is imp ortant to note that more effort may be 
+required on the part of the players, depending on t he existence of correlated equilibria in 
+the general case.  
+ 
+ 
+ 
+ 
+                                                 
+§ Even though it is deemed unfair, the protocol is s till considered secure. An insecure protocol would imply 
+that a third party could capture and decipher the t ransmissions between a player(s) and the machine. T his is 
+not necessarily the case when a machine is unfair.    
+**  This is established at length in [1] and [3].   7 Works Cited  
+1.  Katz, Jonathan. “Bridging Game Theory and Cryptogra phy: Recent Results and 
+Future Directions.” Theory of Cryptography . 1 ed. Lectures in Computer 
+Science . 4948, R. Canneti. Springer Berlin / Heidelberg, 2 008. 
+2.  Milosavljevic, Nebojsa and Anupam Prakash. “Game Th eory and Cryptography.” 
+CS 294. Department of Electrical Engineering and Co mputer Science. Berkeley, 
+CA. 10 March 2009.  
+3.  Dodis, Yevgeniy, Shai Halevi, and T. Rabin. “A Cryp tographic Solution to a 
+Game Theoretic Problem.” Proceedings of Crypto 2000 . (2000): 113-130.  
+4.  Nash Jr., John F. “Non-Cooperative Games.” Annals of Mathematics  54, no. 2 
+(1951): 286-295  
+5.  Hart, Sergiu and Andreu Mas-Colell. “A Simple Adapt ive Procedure Leading To 
+Correlated Equilibrium.” Econometrica  68, no. 5 (2000): 1127-1150.  
+6.  Aumann, Robert J. “Correlated Equilibrium as an Exp ression of Bayesian 
+Rationality.” Econometrica  55, no. 1 (1987): 1-18.  
\ No newline at end of file
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+arXiv:1001.0055v1  [cond-mat.stat-mech]  31 Dec 2009Fractal dimensions of the Q-state Potts model for
+the complete and external hulls
+David A. Adams1, Leonard M. Sander2, and Robert M. Ziff3
+1Department of Physics, University of Michigan, Ann Arbor MI, 4810 9, USA
+2Department of Physics and Michigan Center for Theoretical Physic s, University of
+Michigan, Ann Arbor MI, 48109, USA
+3Department of Chemical Engineering and Michigan Center for Theor etical Physics,
+University of Michigan, Ann Arbor MI,48109, USA
+Email: davidada@umich.edu, lsander@umich.edu, rziff@umich.edu
+Abstract. Fortuin-Kastelyn clusters in the critical Q-state Potts model are
+conformally invariant fractals. We obtain simulation results for the f ractal dimension
+of the complete and external (accessible) hulls for Q= 1, 2, 3, and 4, on clusters that
+wrap around a cylindrical system. We find excellent agreement betw een these results
+and theoretical predictions. We also obtain the probability distribut ions of the hull
+lengths and maximal heights of the clusters in this geometry and pro vide a conjecture
+for their form.
+Keywords : Percolation problems (Theory), Critical exponents andamplitude s (Theory),
+Conformal field theory, Classical Monte Carlo simulationsFractal dimensions of the Q-state Potts model for the complete and external hulls 2
+1. Introduction
+TheQ-state Potts model [1] is a well-studied system in condensed matte r physics,
+exhibiting a continuous phase transition for Q≤4 [2]. It is a generalization of the
+Ising model [3] with Qdifferent spins, such that like spins interact with a single coupling
+constant K, and has applications to a range of different physical systems: Q= 1 and
+Q= 2 correspond to percolation and the Ising model, respectively, an dQ= 3 has been
+used to represent absorbed rare-gas monolayers on graphite su rfaces [4, 5]. In this
+paper we present results from a numerical study of the fractal d imensions of the hulls
+of the Fortuin-Kastelyn (FK) clusters [6, 7] for Q= 1,2,3,4 using a cylindrical system
+geometry. In addition we look at the probability distribution of the len gths of these
+hulls, and find that it has a simple exponential tail. A related quantity, the probability
+distribution of the height (i.e. the vertical span) of the hulls, also ha s such a tail.
+Fractal clusters enter the Potts model in the following way: for pe rcolation, Q= 1,
+it is well-known that the spanning cluster at the critical point is a frac tal. The
+generalization of this for other Qis that at the critical point, Kc, a subset of clusters of
+like spins, the FK clusters [6, 7], are fractal [8]. FK introduced the clu sters by showing
+that the partition function of the Q-state Potts model can be written as a sum over
+all bond configurations on a lattice with a bond occupation probability p= 1−e−K,
+multiplied byaweight QNcwhereNcisthenumber ofclusters inadistinct configuration.
+To go from a spin configuration to a corresponding bond configurat ion, bonds are added
+between like spins with probability p; the connected clusters are the ones we want. The
+partition function for the Potts model is thus a sum over all possible FK clusters, and
+at the critical point p=pc= 1−e−Kc, the clusters are fractal. The pc(Q) for the
+triangular lattice are given by [9]:
+Q= 2 :pc(2) = 1−1/√
+3≈0.42265,
+Q= 3 :pc(3) = 1−/bracketleftbigg
+1+1
+2√
+3sec/parenleftbiggπ
+18/parenrightbigg/bracketrightbigg−1
+≈0.46791,
+Q= 4 :pc(4) = 1/2.
+(1)
+These clusters are characterized by several different fractal d imensions, including
+DH,DEP,DM,DSC,DG, corresponding to the complete hull, external hull, mass, singly
+connected bonds, and narrow-gate fjords. Most previous work on the fractal properties
+oftheseclustershasfocusedonpercolation( Q= 1)forwhichnumericalvalueshavebeen
+given for DH[8, 10, 11, 12], DEP[13],DSC[14], and DG[15]. Theoretical studies for
+percolation include studies of DSC[16, 17],DH[18],DEP[13] andDG[19]. Theoretical
+values for DM,DH, andDSChave been calculated for all Qvalues up to the upper
+critical dimension Q= 4 [20, 21, 22]. Hull exponents have also been derived by SLE
+theory [23].
+In this paper will present numerical data for DHandDEPand the height and
+length distributions for both types of perimeters. In a recent pap er, Asikainen etFractal dimensions of the Q-state Potts model for the complete and external hulls 3
+al. [15] measured DM,DH,DEP,DSC,DGforQ= 1 through Q= 4 by studying
+individual isolated clusters that do not touch the boundary. In the present work we
+consider a cylindrical geometry and look at the hulls of the clusters t hat wrap around
+it. This method provides an unambiguous measure of the length scale (namely, the
+circumference) and leads to accurate numerical results for the f ractal dimensions. We
+also measure the height and length distributions of the hulls themselv es.
+2. Model
+Two types of simulations are used. To model Q= 1 (percolation), we use the Leath
+algorithm [24] with site percolation. This method starts with a single “a ctive” site,
+which attempts to change its undetermined neighbors into active sit es with a probability
+pc. If a change fails, the site is becomes “inactive”, and can never be m ade active. Each
+new active site attempts to make all of its neighbors active until no m ore active sites
+remain. The resulting structure is a percolation cluster. All cluster s are grown on the
+triangular lattice.
+For all other Q-states, we generate FK clusters [6, 7] using the Swendsen-Wang
+(SW) method [25]. The SW algorithm is as follows: after the bonds hav e been placed
+for a given configuration, each cluster of sites connected by bond s is labeled with a
+randomly chosen spin. Then new bonds are put down with probability pbetween like
+spins, forming the FK clusters. This process is then repeated. Eac h cycle constitutes a
+spin update in that we have flipped entire FK clusters. The SW method allows for fast
+equilibration of critical clusters.
+3. Simulation
+We grew our clusters on a triangular lattice with periodic boundary co nditions along
+bothdirections. Wemadethedimensions ofsystem elongatedsotha tmostlargeclusters
+wrapped around in one direction but not the other. The system, th ough really a torus,
+was effectively a cylinder. Clusters that wrapped around both direc tions were rejected.
+The lattices had an aspect ratio of 100:1 and 8:1 for percolation and F K clusters,
+respectively. We use the width ( W), of the lattice as the characteristic length. For
+Q= 1, clusters were grown for W= 8, 16, 32, 64, 128, and 256. For Q= 2 through 4,
+wealsoconsidered W= 512and1024inordertoreachtheasymptoticfractaldimension.
+For each valid cluster grown, the number of sites on the top and bot tom of the
+cluster were recorded as well as the height (vertical span) of the hulls, defined as the
+highest point on the top hull minus the lowest point on the top hull. The top and
+bottom hulls were considered independently and both were used to c alculate the fractal
+dimension. To record the complete hull, we used the outermost layer of active sites as
+the hull. For the external hull, we used the layer of inactive sites tha t pad (are neighbors
+to) the outermost layer of active sites.
+In the case of FK clusters, the system must be equilibrated before recording canFractal dimensions of the Q-state Potts model for the complete and external hulls 4
+Table 1. The theoretical [20, 21, 22] and measured values for the fracta l dimensions
+for the complete (C) and external (E) hulls of Q-state Potts model
+Qtheory measured
+1C7/4 = 1.750 1.747
+1E4/3 = 1.333 1.330
+2C5/3 = 1.667 1.663
+2E11/8 = 1.375 1.375
+3C8/5 = 1.600 1.602
+3E17/12 = 1.417 1.412
+4C3/2 = 1.500 1.510
+4E3/2 = 1.500 1.534
+start. The equilibration times used are 2 W, 3W, and 16 W, forQ= 2, 3, and 4,
+respectively. We use longer equilibrium times for Q= 4 because of its slower dynamics
+[26]. To determine the equilibration time, we measure the average ene rgy per spin
+and the average largest cluster size as a function of the number of spin updates. We
+find that both obserables relax to their steady-state values expo nentially quickly with
+the same decay time for a given QandW, though the average energy is always closer
+to the steady-state value at a given step. For example, we find the average energy was
+within 1% of the steady-state value within 500 spin updates, wherea s it took the average
+largest cluster size 1500 spin updates to reach 1% of its steady-st ate value for Q= 4 and
+W= 256. Because of its slower relaxation, we believe that the average largest cluster
+size is a better measure of equilibration.
+After the system is equilibrated, we pick a random cluster, and if the cluster wraps
+around the width of the system, its hull length and height are recor ded in a similar
+fashion as with percolation. Note that while the FK clusters are esse ntially bond
+percolation clusters, we consider the hulls on the sites of the cluste rs – that is, we treat
+the sites in the FK clusters as a site percolation problem. To record t he external hulls
+of these clusters a layer of inactive sites are simply added outside of the top and bottom
+hulls. (This is the advantage of using the triangular lattice.) Several spin updates are
+taken between each successive attempt to find a valid cluster. The number of updates
+between measurements are 8, 12, and 64 for Q= 2, 3, and 4, respectively.
+The work of Asikainen et al. [15] differs with ours in several ways. For Q >1,
+they simulated a single system size, 4096 ×4096, and picked random clusters after the
+system was well equilibrated. To equilibrate such a large system, the y needed to use
+a new technique to speed up equilibration. For their system of isolate d clusters, the
+length scale is not as clearly defined as it is in the cylindrical geometry; they used the
+cluster’s radius of gyration for the length scale. Lastly, they used a square lattice for
+their simulations, for which the complete and external hulls are not a s clearly defined as
+in the triangular case because of the corners in the square system [27]. Our results are
+consistent with theirs in regards to the fractal dimensions, but ar e significantly moreFractal dimensions of the Q-state Potts model for the complete and external hulls 5
+32 
+64 
+128 
+256 Q = 1 Q = 2 Q = 3 Q = 1 
+Figure 1. The probability distributions for the lengths of the complete hulls, fo r
+several system widths: W= 32, 64, 128 and 256 for Q= 1 as a function of L/∝angbracketleftL∝angbracketright
+where∝angbracketleftL∝angbracketright= 0.93W7/4(left). The data are put into bins of size C∝angbracketleftL∝angbracketrightwithC= 0.4.
+The black line is the best fit using an exponential with an inverse decay length of λ−1.
+The plot on the right is of the residual (difference) of the fit of expo nentials to the
+different Qcomplete hulls, where different C= 0.3 forQ= 2,3. The best fit for the
+λ−1forQ= 1, 2, and 3 are 1 .4, 1.65, and 1 .7, respectively. The residual data for
+different Qare offset vertically for clarity.
+precise.
+4. Results
+For each state Qwe obtain a large set of values for the lengths of the complete and
+external hulls and hull heights for wide range of system widths. With this data we can
+compute the fractal dimension of the different hulls. For a fractal the length of the hulls
+scales as a power L∝WD, whereDis the fractal dimension. We measure Din the
+conventional way by making a linear fit of ln Las a function of ln W. The accuracy
+can be assessed two ways: (1) the value of the square of the Pear son product-moment
+coefficient, R2, of the fit and (2) the apparent randomness of the residuals. We f ocus
+on the latter as it can be used to determine when finite-size effects a re significant, and
+thus which sizes we can use for the fit. We determined that all measu red widths for the
+complete hull can be used to fit DforQ <4. For the external hull, there are significant
+corrections for W <256, so for those systems, we use 256, 512, and 1024. We use the
+same three sizes for both hulls of Q= 4.
+Table 1 shows a summary of our findings for the fractal dimensions o f the complete
+and external hulls for Q= 1,2,3,4. With the exception of the external hull for Q= 4,
+allthemeasuredvaluesagreewiththeorywithinafractionofaperc ent. Thediscrepancy
+atQ= 4 can be interpreted as arising from logarithmic corrections to sca ling [28]. In
+Ref [15] a correction is made for the slow crossover.
+We now turn to the probability distribution of the quantities that we m easured.
+We first found that the average height of the hulls scales linearly with system width so
+that these are isotropic fractals. We then measured the heights a nd hull lengths of a
+large number of clusters in order to produce the probability distribu tion of hull lengths.
+Figure 1 shows the scaled probability distributions of complete hulls fo rQ= 1 and the
+residuals to the best-fit exponential to the first three values of Q.Q= 4 is not shownFractal dimensions of the Q-state Potts model for the complete and external hulls 6
+32 
+64 
+128 
+256 Q = 1 Q = 2 Q = 3 Q = 1 
+Figure 2. The probability distributions for the lengths of the external hulls, f or
+several system widths: W= 32,W= 64,W= 128,W= 256 for Q= 1 (left) where
+the different system sizes are scaled in the same way as fig. 1. The rig ht shows the
+residuals for Q= 1 through Q= 3. The best fit for λ−1for the different Q’s are 2.5,
+2.4, and 2.35 forQ= 1, 2, and 3, respectively. The residual data for different Qare
+offset vertically for clarity.
+because the scaling does not work for the size systems we used. Fig ure 2 is a similar
+plot for the external hulls, with a finer bin size. All scaled hull-length d istributions have
+exponential tails. The complete and external hulls appear to have f airly similar decay
+lengths for the different values of Q.
+We also calculate the probability distribution of hull heights for the co mplete and
+external hulls, Figures 3 and 4. Again we see that the distributions h ave exponential
+tails.
+The exponential tails are, in some sense, no surprise in this type of p roblem as we
+can see from a simple example. Consider the track of a free random w alker in a channel
+with absorbing walls at x=±W/2. The walker starts in the middle of the channel,
+and we seek the probability distribution of the maximum height attaine d by the walk
+before it hits the side walls. This is more-or-less what we are doing with our spanning
+fractals, though they are not free random walks, of course.
+It is not hard to see why we get exponential tails in this case, by cons idering the
+following steps: first we define an auxiliary problem by putting an abso rbing wall at
+heighthabove the origin. The number of walks, N(h), that ever hit this wall before
+being absorbed on the sides is the number of walks with height greate r thanh. The
+distribution we seek is proportional to dN/dh. The problem can be solved exactly by
+going to the continuum limit and using conformal mapping. However, w e can guess the
+solution easily: the probability for a walker to penetrate a channel f or height his clearly
+exponentially decreasing in h; the conformal map gives e−πh/W. Then Nmust have
+this dependence, along with its derivative.
+For percolation or FK clusters, the probability distribution of the ma ximum height
+can also be found from the derivative of the probability of crossing, and there one also
+expects exponential behavior for large h[29, 30].
+The existence of exponential tails in both the hull length and height d istributions
+may appear contradictory. We know that the average height, h, is proportional to the
+system width, Wwhich is related to hull length, Lthrough a power-law W∼L1/DH.Fractal dimensions of the Q-state Potts model for the complete and external hulls 7
+32 
+64 
+128 
+256 Q = 1 Q = 2 Q = 3 Q = 1 
+Figure 3. The probability distributions for the heights of the complete hulls, fo r
+several system widths: W= 32,W= 64,W= 128,W= 256 for Q= 1 (left). The
+bin sizes are CW, whereC= 0.4 . The right shows the residuals for Q= 1 through
+Q= 3. The best fit for λ−1for the different Q’s are 1.15, 1.0, and 1.05 forQ= 1, 2,
+and 3, respectively. The residual data for different Qare offset vertically for clarity.
+One might guess that this should lead to a stretched exponential ex p(−cL1/DH) rather
+than simple exponential for the distribution of L. In order to clarify this issue, we
+looked at another well-studied system: the percolation hull-walk in an open rectangular
+region [10] for which it is easy to generate a large ensemble of fracta l curves.
+Weperformedsimulationsoftheself-avoidinghullwalker onasquare lattice[10,11].
+The hull walk algorithm is as follows. Every step the walker either turn s right or left. If
+the current site hasn’t been marked, the walker turns left with pro babilityp, marks the
+site ‘active’ and takes a step forward. With probability 1 −pit turns right, marks the
+site inactive, and takes a step forward. If the walker steps onto a active or inactive site
+it always turns left or right, respectively. We used this walk to creat e closed figures that
+have the same fractal dimension as the complete hull of percolation clusters, D= 7/4.
+We recorded the average walk length as a function of the maximum he ight for several
+values of W.
+ForeveryW,thewalklengthisproportionaltothemaximumheight,i.e .,L∼mWh,
+as one would expect for large h. However, we find that the coefficient ,mW, depends on
+WasmW=aW3/4. With these pieces, we get back the known power-law relationship
+between LandW,∝angbracketleftL∝angbracketright ∼mW∝angbracketlefth∝angbracketright ≈aW3/4W=aW7/4, so that we recover the fractal
+dimension of 7 /4.
+5. Conclusion
+In this paper, we measured the fractal dimensions for the complet e and external hull
+lengths,DHandDEPforQ= 1 through Q= 4. We used the Leath algorithm to grow
+critical percolation clusters. For Q= 2,3,4, we used the SW method to generate critical
+FK clusters. All systems used a triangular lattice with periodic bound ary conditions.
+The aspect ratio of the systems were heavily skewed so that spann ing clusters would
+span and one direction and not the other. The smaller length (W) exa ctly determined
+the length scale of the system. We generated a large number of spa nning clusters
+of various system sizes, W= 8,16,32,64,128,256 and additionally W= 512,1024,Fractal dimensions of the Q-state Potts model for the complete and external hulls 8
+32 
+64 
+128 
+256 Q = 1 Q = 2 Q = 3 Q = 1 
+Figure 4. The probabilitydistributionsfortheheightsofthe externalhulls, f orseveral
+system widths: W= 32,W= 64,W= 128,W= 256 for Q= 1 (left). The bin sizes
+areCW, whereC= 0.2 . The right shows the residuals for Q= 1 through Q= 3.
+The best fit for λ−1for the different Q’s are 3.3, 1.8, and 1.7 forQ= 1, 2, and 3,
+respectively. The residual data for different Qare offset vertically for clarity.
+for percolation and Q >1, respectively. We find excellent agreement between our
+results and the associated theories [20, 21, 22]. We also measured t he distributions
+of the hull lengths and heights. We found that the distributions for Q= 1,2,3 have
+exponential tails. The values of the inverse decay lengths for the e xponentials are given
+in the figure captions.We discussed the apparent contradiction bet ween the height and
+length distributions both having exponential tails, as opposed to on e being a stretched
+exponential. We resolved this contradiction by showed that the rela tionship between
+average height and width is in fact linear for a fixed W. We found that the power-law
+scaling formula can be written as ∝angbracketleftL∝angbracketright ∼mW∝angbracketleftH∝angbracketright ≈aW3/4W. This formula clarifies the
+relationships between L,H, andW. We are not aware of any predictions regarding the
+exponential tails in the hull length distributions.
+Acknowledgments
+This work was supported in part by the National Science Foundation through DMS-
+0553487. We would like to thank C. Doering for a useful discussion.
+References
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+[29] J. L. Cardy. Critical percolation in finite geometries. Journal Of Physics A-Mathematical And
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+arXiv:1001.0056v2  [math.AG]  5 Jan 2010QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION
+ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+Abstract. LetGdenote a complex, semisimple, simply-connected group and Bits
+associated flag variety. We identify the equivariant quantu m differential equation
+for the cotangent bundle T∗Bwith the affine Knizhnik-Zamolodchikov connection of
+Cherednik and Matsuo. This recovers Kim’s description of qu antum cohomology of
+Bas a limiting case. A parallel result is proven for resolutio ns of the Slodowy slices.
+Extension to arbitrary symplectic resolutions is discusse d.
+1.Introduction
+The main goal of this paper is to study the quantum cohomology of the Springer
+resolution associated to a semisimple algebraic group G. Since many of the ideas and
+constructions here apply to arbitrary symplectic resoluti ons, much of the introduction
+will discuss this more general context.
+1.1.Geometry of symplectic resolutions. Recall from [26] that a smooth algebraic
+varietyXwith a holomorphic symplectic form ω∈H0(X,Ω2) is called a symplectic
+resolution if the canonical map
+(1) X→X0= SpecH0(X,OX)
+is projective and birational.
+The deformations of the pair ( X,ω) are classified by the image [ ω] of the symplectic
+form inH2
+deRham(X). In many important examples, the universal families
+(2) X/d31/d127/d47/d47
+/d15/d15/tildewideX
+φ
+/d15/d15
+[ω]/d31/d127/d47/d47H2(X,C)
+may be described explicitly, see [26] for further discussio n. The generic fiber of (2) is
+affine. Fibers with algebraic cycles α∨∈H2(X,Z) lie over hyperplanes
+(3) ( ω,α∨) =/integraldisplay
+α∨ω= 0
+in the base. Primitive effective classes α∨as above are called primitive coroots by the
+analogy with the following prominent example.
+12 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+1.2.Grothendieck resolution. LetGbe a complex, semi-simple simply connected
+algebraic group with Lie algebra g. LetBbe the flag variety of Gthat parametrizes,
+among other things, Borel subalgebras b⊂g. Consider
+/tildewideX={(b,ξ), ξ∈[b,b]⊥} ⊂B×g∗,
+and the map
+(4) /tildewideX∋(b,ξ)φ−→ξ/vextendsingle/vextendsingle
+b/[b,b]∈t∗,
+wheretis the Lie algebra of a maximal torus T⊂G.
+The generic fiber of φis a coadjoint orbit of Gwith its canonical Kirillov-Kostant
+formω, while
+X=φ−1(0)∼=T∗B
+with the canonical exact symplectic form of a cotangent bund le. The map
+(5) /tildewideX∋(b,ξ)−→ξ∈g∗
+takesXto the nilpotent cone N⊂g∗. This is known as the Springer resolution ofN,
+while (5) is referred to as the Grothendieck simultaneous re solution.
+Fix a Borel subgroup B⊂Gwith a maximal torus T. A weight λ∈Hom(B,C∗)
+defines an equivariant line bundle
+O(λ) =G×BCλ
+onBand, by pullback, on X. The map
+λ/mapsto→Dλ=c1(O(λ))
+yields an isomorphism
+(6) Λ = Hom( T,C∗)∼=H2(X,Z)
+Applying ⊗ZCto (6), we get an isomorphism t∗∼=H2(X,C) that makes
+(7) T∗B/d31/d127/d47/d47
+/d15/d15/tildewideX
+φ
+/d15/d15
+0/d31/d127/d47/d47t∗
+an instance of the diagram (2) with φas in (4).
+We will denote by R,R+,and Π the set of roots, positive roots, and simple roots
+respectively. A positive coroot α∨defines an SL2-subgroup Gα∨⊂Gand hence a
+rational curve
+Gα∨·[B]⊂B⊂X .
+SinceGis simply-connected, coroots generate the lattice
+(8) Λ∨= Hom(C∗,T)∼=H2(X,Z),
+inwhich thepositive onesgenerate theeffective cone. Itis we ll-known that thisclassical
+notion of a positive coroot agrees with the definition given i n Section 1.1.QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 3
+1.3.Equivariant symplectic resolutions. Our interest lies in enumerative invari-
+ants of curves in X, particularly, in quantum cohomology of X. Since a symplectic
+resolution Xmay be deformed to an affine variety, most Gromov-Witten invar iants of
+Xtrivially vanish. However, equivariant Gromov-Witten invariants of Xmay be very
+interesting.
+In what follows, we assume that a connected reductive algebr aic group Gacts onX
+so that:
+(1)Xgis proper for some g∈G;
+(2)Gscalesωby anontrivial character .
+The second property implies Xdoes not have any G-equivariant symplectic deforma-
+tions sinceany connected groupacts trivially on H2(X). See[24] forgeneral conjectures
+concerning the existence of scaling actions on X.
+For the Springer resolution X=T∗B, we take
+G=G×C∗
+where the second factors acts by scalar operators on g∗,t∗, and the cotangent fibers.
+While all constructions of Section 1.2 are G-equivariant, note that Gacts nontrivially
+on the base of the deformation with a unique fixed point 0 ∈t∗.
+Other examples of equivariant symplectic resolution inclu de: cotangent bundles of
+homogeneous spaces, as well as Hilbert schemes of points and more general moduli of
+sheaves on symplectic surfaces. The largest class of exampl es comes from Nakajima
+quiver varieties, see [17] for an introduction. For a genera l symplectic resolution, we
+denote by G⊂Gthe stabilizer of ω.
+1.4.Goal of the project. There exist powerful heuristic as well as proven principles
+that organize Gromov-Witten invariants of algebraic varie ties into certain beautiful and
+powerful structures. This is loosely known as mirror symmetry , see e.g. [11, 19, 21] for
+an introduction. We believe for symplectic resolutions, th ese structures recover and
+generalize some classical notions of geometric representa tion theory.
+To see this in full generality is the goal of a large joint proj ect which we pursue with
+R. Bezrukavnikov, P. Etingof, V. Toledano-Laredo, and othe rs. It has also many points
+of contact with the ongoing work of N. Nekrasov and S. Shatash vili [41].
+This paper may be seen as a first step, in which we explain what t hese structures
+mean for the most classical symplectic resolution of all — th e Springer resolution. This
+will bereflectedin thepaper’sstructure: thereaderwill no tice that thegeneralities take
+up much more space than the actual geometric computation of q uantum multiplication
+in Section 5.
+1.5.Quantum multiplication. Bydefinition,theoperatorofquantummultiplication
+byα∈H∗
+G(X) has the following matrix elements
+(9) ( α◦γ1,γ2) =/summationdisplay
+β∈H2(X,Z)qβ/a\}b∇acketle{tα,γ1,γ2/a\}b∇acket∇i}htX
+0,3,β,
+where (·,·) denotes the standard inner product on cohomology and the qu antity in
+angle brackets is a 3-point, genus 0, degree βequivariant Gromov-Witten invariant of4 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+X, see Section 4. It is a very general fact that (9) defines a comm utative associative
+deformation of the algebra H∗
+G(X), see e.g. [11].
+We denote by T∨the dual torus, for which the roles of (6) and (8) are reversed .
+Each monomial qβis naturally a function on T∨. In fact, the series (9) will be shown
+to be rational, that is, to lie in C(T∨). Parallel rationality is expected for arbitrary
+symplectic resolutions.
+It is well-known that H∗(B) is generated by divisors and, for an arbitrary symplectic
+resolution X, one may hope that the quantum cohomology is generated by div isors
+even if the classical cohomology is not.
+Themainresultof thepaperis acomputation of operators ofq uantummultiplication
+bydivisorsfortheSpringerresolution. Beforestatingit, wediscussthegeneralstructure
+of the answer.
+1.6.Correspondence algebra and root subalgebras. Consider the fiber product
+(10) X×X0X⊂X×X .
+Itisknownthat all components of (10) havedimension ≤dimXandthoseof dimension
+dimXare Lagrangian, see [17]. Lagrangian components of (10) act by correspondences
+on the Borel-Moore homology of X. We denote by Corr( X)⊂End(H∗(X)) the subal-
+gebra that they generate, known as the correspondence algebra . We will show that the
+quantum contribution to divisor multiplication will lie in Corr(X). More precisely, we
+establish the following refined statement in section 5.
+LetXαbe a generic fiber of (2) over the hyperplane where α∨is algebraic. Then we
+can again consider the fiber product
+(11) Xα×Xα
+0Xα⊂Xα×Xα.
+SinceXα∼Xas a topological G-space, Lagrangian components of (11) act by corre-
+spondences in the Borel-Moore homology of X. We denote Sα⊂Corr(X) the subalge-
+bra that they generate.
+LetD⊂H2(X) be a divisor. By an abstract argument, we show the operator D◦
+of quantum multiplication by Dhas the following form
+(12) D◦= classical + t/summationdisplay
+α∨(D,α∨)Sα/parenleftBig
+qα∨/parenrightBig
+, Sα(z)∈Sα⊗C[[z]],
+where the first term is the classical multiplication by D, the sum is over all positive
+corootsα∨∈H2(X,Z), andtis the weight of the symplectic form ω. See sections 5.1
+and 5.2 for the argument.
+Formula (12) is functorial in the sense that α∨-term in it describes the quantum
+corrections to multiplication in H∗(Xα). This reduces the computation of quantum
+cohomology to varieties with a unique effective curve class, a s we discuss further below.
+We will also see that similar functoriality holds for slices ofX.
+In the case of Nakajima varieties, which come in families, th e root subalgebras Sα
+and the operators of classical multiplication can be extend ed to a representation of
+the so called Yangian algebraY(g) associated to the corresponding Kac-Moody Lie
+algebra, see [48]. Through the work of Nekrasov and Shatashv ili, this has a direct link
+to the classical work of C. N. Yang in quantum integrable syst ems.QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 5
+1.7.Main result. In the Springer case, the action of classical multiplicatio n and the
+root subalgebras has a very concrete description in terms of thegraded affine Hecke
+algebraHt. Recall that Htis generated by the symmetric algebra Sym t∗oft∗and the
+group algebra of Wsubject to the relation
+sαxλ−xsα(λ)sα=t(α∨,λ),
+wheresαis the reflection associated with the simple root αandxλ∈S1t∗corresponds
+toλ∈t∗.
+As before, tis a parameter which may be identified with the weight of the sy mplectic
+formω, that is, with the equivariant parameter of the C∗-factor in G. The algebra Ht
+is graded by deg xλ= 2,degw= 0,degt= 2.
+An action of HtonH∗
+G(T∗B) is due to Lusztig [31]. Its construction is recalled in
+Section 3. In particular, xλacts by classical multiplication by Dλ, whileSαis generated
+by the corresponding reflection in sα∈W. Having introduced the above notations we
+can now state
+Theorem 1.1. The operator of quantum multiplication by the divisor Dλis given by
+(13) Dλ◦=xλ+t/summationdisplay
+α∨∈R∨
++(λ,α∨)qα∨
+1−qα∨(sα−1).
+SinceDλgenerate cohomology, this determines all operators of quan tum multiplica-
+tion. We give a more detailed statement of this theorem in sec tion 3.5. The commuting
+elements (13) are, of course, well-known, as we now recall.
+1.8.The quantum connection. Over
+T∨
+reg={q∈T∨|α(q)/\e}atio\slash= 1 for any α∈R+}.
+consider what is known as the quantum differential equation , namely the following
+connection on the trivial bundle with fiber H∗
+G(X)
+(14) ∇λ=dλ−Dλ◦=dλ−xλ−t/summationdisplay
+α∈R+(λ,α∨)qα∨
+1−qα∨(sα−1).
+Heredλis the derivative in the direction of λ∈t∗= Lie(T∨). It is a general result of
+Dubrovin that quantum connections are always flat, see [11].
+In our case, a simple gauge transformation, see Section 2.2, identifies (14) with the
+affine KZ connection studied by Cherednik [8], Matsuo [32], Felder and Veselov [1 5]
+and others.
+Recall that an integrable connection ∇on an algebraic vector bundle Edefines a
+D-module, that is, a module over differential operators on the b ase. For the quantum
+differential equation, this is known as the quantum D-module. The isomorphism class
+of theD-module determines ∇up to gauge equivalence; in other words, passing from
+the quantum connection to the quantum D-module corresponds to forgetting the fact
+that the quantum connection was defined on the trivial bundle (thus, knowing the
+quantum D-module alone is not enough in order to recover the quantum mu ltiplication
+by divisor classes).6 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+In the Springer case, the quantum D-module coincides with the quantum Calogero-
+MoserD-module for the Langlands dual group. After providing backg round, we will
+give a precise formulation in Section 2.5 and Theorem 3.2. In particular, one can de-
+scribe the corresponding quantum cohomology ring of T∗Bby using the corresponding
+classical Calogero-Moser integrable system . Forg=sln, this was obtained indepen-
+dently by Nekrasov and Shatashvili. A related result is due t o A. Negut [40], see below.
+1.9.The K¨ ahler moduli space. LetK=T∨⊃T∨be the compactification defined
+by the fan of Weyl chambers. The connection (14) extends to Kas a connection
+with first-order poles along a normal crossing divisor. In ac cordance with the mirror
+symmetry nomenclature, we call Kthecompactified K¨ ahler moduli space .
+The expansion (9) is around the point
+qα∨
+1=qα∨
+2=···= 0,
+whereα∨
+iare the simple positive coroots. This is one of the |W|-manyT∨-fixed points
+ofK. Such points are called the large radius points .
+The Weyl group Wacts on both the base and the fiber of ∇and one easily checks
+that
+(15) w∇λw−1=∇w(λ), w∈W .
+In fact, this is equivalent to the commutation relations in Ht.
+Note that T∨is naturally the base of the multiplicative version of Groth endieck
+simultaneous resolution for the Langlands dual group G∨. In fact, we expect these two
+families to be equivariant mirrors and hopeto elaborate on this point in a futurepaper.
+1.10.Monodromy and derived equivalences. In general, a compactified K¨ ahler
+modulispacemayhavelargeradiuspoints m1andm2thatcorrespondtononisomorphic
+varieties X1/\e}atio\slash∼=X2. One always expects, however, that
+Db(X1)∼=Db(X2).
+and, moreover, that the equivalence should depend on a choic e of a path from m1to
+m2, thus giving
+(16) ρ:π1(U)−→AutDb(Xi)
+for a certain open U⊂K. One further expects that Uis the regular locus of ∇and
+that there is a nonzero intertwiner between the monodromy of ∇and the image of ρ
+inK-theory
+ρK:π1(U)−→AutK(X)⊗C.
+See, for example, [22] for an introduction to these ideas.
+Specializing this to theSpringercase, we firstusethe W-equivariance (15) to descend
+∇to a connection on T∨
+reg/W. The fundamental group of T∨
+reg/Wis the affine braid
+groupB/hatwiderW. Bezrukavnikov constructed an action of B/hatwiderWonDb(X) in [2, 3], see also
+the work of Khovanov and Thomas [28] in the case g=sln. On the K-theory, this
+braid group action factors trough the affine Hecke algebra act ion constructed earlier
+by Kazhdan and Lusztig [27] and Ginzburg [9]. And indeed this precisely matchesQUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 7
+Cherednik’s description of the monodromy, see page 64 in [8] and Sections 2.7 and 3.6
+for further discussion.
+A parallel result for X= Hilb(C2,n) is the subject of [4]. In both cases, monodromy
+of∇is irreducible and isomorphic to ρK.
+1.11.Matching of equivariant parameters. One aspect of the correspondence de-
+scribed in Section 1.10 merits a special discussion, namely the identification of equi-
+variant parameters.
+Equivariant K-theoryKG(X)⊗Cis a module over the representation ring C[G]Gof
+GwhileH∗
+G(X,C) is a module over C[g]Gwheregis the Lie algebra of G. Assuming
+that∇depends regularly on a∈g, its monodromy is an entire function of a. Hence,
+to match ρKwith the monodromy of ∇, we need a map
+ζ:C[G]G→Can[g]G,
+whereCanstands for analytic functions. We claim the only natural cho ice is
+(17) ζ(f)(a) =f/parenleftbig
+e2πia/parenrightbig
+.
+This holds in the Springer case and can be argued in general as follows.
+The simplest autoequivalences of Db(X) are tensor products with line bundles L∈
+Pic(X). These are expected to be compatible with ρas follows. Let m∈Kbe a large
+radius point. The intersection of a small neighborhood of mwithUhas an abelian
+fundamental group, namely H2(X,Z). One requires that
+(18) ρ(c1(L)) =L⊗—
+To see the connection with (17), we treat the purely equivari ant classes in H2
+G(X,Z)
+on the same footing as the geometric ones. For a purely equiva riant class, the quantum
+differential equation is a constant coefficient, in fact, scala r equation. In this case (18)
+specializes to (17).
+1.12.Shift operators. Formula (17) predicts the monodromy is a periodic function o f
+equivariant parameters. For theSpringerresolution, this meansthat forany s∈Λ∨⊕Z,
+we have a a shift operator
+S(s)∈End(H∗
+T(X))⊗C(T∨)
+satisfying
+∇(a)S(s) =S(s)∇(a+s),
+wherea∈Lie(G) denotes the equivariant parameters of ∇. And, indeed, such inter-
+twiner may be constructed geometrically as described in Sec tion 6.
+These are the shift operators of Opdam ([43, 20]) that played a very significant role
+in understanding the quantum Calogero-Moser systems.
+1.13.The Toda limit. Recallthattheparameter tin(13)istheequivariantparameter
+for the scaling action in the fibers of T∗B. It is known and explained in Section 8 that
+thet→ ∞limit, taken correctly, recovers the quantum cohomology of the base B.
+The analog of HtforBis a certain nil-version Hnilgenerated by xλforλ∈t∗and
+wforw∈W. The element xλstill acts on by multiplication by a divisor DB
+λ, whilew
+are nilpotent in Hnil. Precise definitions are given in Section 2.3.8 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+Referring to Sections 7 and 8 for the details of the t→ ∞limit, we state the result.
+Introduce the following subset R′
++of the set of R+of positive roots of G. Namely, let
+α∈R+. Denoting by Π the set of simple positive roots, we have an exp ansion
+α=/summationdisplay
+β∈Πaββ.
+Thenα∈R′
++if one of the following conditions is satisfied:
+(1)αis a long root;
+(2)aβ= 0 for all long simple roots β.
+In particular, we see that R+=R′
++ifGis simply laced and that R′
++contains all
+simple roots.
+Theorem 1.2. The operator of quantum multiplication by DB
+λonH∗
+G(B)is equal to
+xλ+/summationdisplay
+α∈R′
++(λ,α∨)qα∨sα.
+The quantum cohomology of Bhas been studied in great detail by many authors,
+see for example [10, 18, 29, 37, 45] for a very incomplete sele ction of references. In
+particular, the above formula may be deduced from Theorem 6. 4 in [37].
+1.14.Generalization to Slodowy slices. To anynin the nilpotent cone N, one
+associates the Slodowy slice Sn⊂N, which is a transversal slice to the G-orbit of nin
+N. This is an affine conical algebraic variety, which has an open smooth symplectic
+subset (equal to the intersection of Snwith the open G-orbit in N). The slice Snis
+defined uniquely up to conjugation by an element of the centra lizerZnofninG. Let
+/tildewideSn=π−1(Sn). Then /tildewideSnis smooth and symplectic; it provides a symplectic resoluti on
+of singularities for Sn. Note that when n= 0 we have Sn=Nand/tildewideSn=T∗B.
+According to [31] and [16] the algebra Htstill acts on H∗
+Zn×C∗(/tildewideSn). Abusingnotation
+we shall denote the restriction of the class Dλto/tildewideSnby the same symbol. Then we have
+Theorem 1.3. Assume that the restriction map H2(X,Z)→H2(/tildewideSn,Z)is an isomor-
+phism. Then the operator of quantum multiplication by DλinH∗
+Zn×C∗(/tildewideSn)is given by
+the same formula as in Theorem 1.1.
+Let us note that the assumption on H2in the formulation of Theorem 1.3 is not very
+restrictive. For example, one can show (cf. the Appendix) th at this assumption always
+holds when Gis simply laced.
+1.15.Reduction to rank 1. As discussed in section 1.6, one can study quantum
+cohomology of Xin terms of that of generic non-affine deformations Xα. In fact, we
+can further simplify this analysis as follows.
+The symplectic form ωinduces a Poisson algebra structure on OXwhich makes X0
+a Poisson variety with finitely many symplectic leaves, see [ 24]. Let Z⊂X0be a
+symplectic leaf of minimal dimension. If one has an isotrivi al fibration X→Zwhich is
+a Poisson map, then, since Zis affine, this reduces quantum cohomology of Xto thatQUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 9
+of the fiber X′. If one has such a structure for Xα, in combination with (12), we can
+reduce further to the fibers
+X′
+α/d31/d127/d47/d47Xα
+/d15/d15
+Zα
+for the simpler varieties XαofX.
+In general, while we may not have a global fibration X→Z, there is a formal result
+of Kaledin [24] which allows us to write, for z∈Z, the formal neighborhood /hatwide(X0)zas
+a product of /hatwideZzand the formal neighborhood of a lower-dimensional symplec tic singu-
+larity. It turns out that, for equivariant symplectic resol utions, this formal statement
+is already sufficient for the reduction step.
+We say that a symplectic resolution Xhas rank 1 if Zis a point and dim H2(X) = 1.
+The above reduction procedure obviously stops once we reach a rank 1 variety X.
+Examples of rank 1 varieties are cotangent bundles T∗Gr(k,n) of the Grassmannians
+and also moduli of framed torsion-free sheaves on C2. Further examples may be found
+among Nakajima quiver varieties.
+IntheSpringercase, only T∗P1appearasfibers X′
+α-heretheformalargumentwillbe
+unnecessary. Their quantum cohomology is very well underst ood. Similarly, Nakajima
+varieties for quivers of finite type lead only to T∗Gr(k,n). They will be discussed
+in forthcoming paper (the corresponding quantum connectio n should presumably be
+related to the trigonometric Casimir connection – cf. [46, 47]). Similarly, for quiver
+varieties ofaffinetypeonehastounderstandthethequantumc ohomology ofthemoduli
+space of framed sheaves on P2of arbitrary rank; this is more involved - cf. [35].
+1.16.Related work. From our viewpoint, the operators of quantum multiplicatio n
+form a family (parametrized by q) of maximal abelian subalgebras in a certain geomet-
+rically constructed Yangian. Long before geometric repres entation theory, Yangians
+appeared in mathematical physics as symmetries of integrab le models of quantum me-
+chanics and quantum field theory. The maximal abelian subalg ebra there is the algebra
+of quantum integrals of motion, that is, of the operators com muting with the Hamilton-
+ian. A profound correspondence between quantum integrable systems and supersym-
+metric gauge theories was discovered by Nekrasov and Shatas hvili, see [41]. It connects
+the two appearances of Yangians and, for example, correctly predicts that the eigenval-
+ues of the operators of quantum multiplication are given by s olutions of certain Bethe
+equations (well-known in quantum integrable systems, but p robably quite mysterious
+to geometers). The integrable and geometric viewpoints are complementary in many
+ways and, no doubt, will lead to new insights on both sides of t he correspondence.
+It is a well-known phenomenon that a particular solution of t he quantum differential
+equation (the so-called J-function) may often be computed as the generating function
+of integrals of certain cohomology classes over different com pactifications of the moduli
+space of maps P1→X. For example for maps to flag varies of g=sln, one can try
+to use the so called Laumon moduli space flags of sheaves on P1. The corresponding
+generating function was identified by A. Negut with the eigen functions of the quantum
+Calogero-Moser system [40].10 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+As a rule, sheaf compactifications admit torus action with is olated fixed points, thus
+expressing J-functions as a certain multivariate hypergeometric serie s. The spectrum
+of the operators of quantum multiplication may be deduced fr om the semiclassical
+asymptotics of this solution. This asymptotic behavior is d etermined by a unique
+maximal term in the hypergeometric series. The spectrum is t hus determined as a
+solution of acertain finite-dimensional variational probl em. Itis one of thecornerstones
+of Nekrasov-Shatashvili theory that this variational prob lem coincides with the Yang-
+Yang variational description of Bethe roots. For flag variet ies, this can be seen very
+explicitly.
+1.17.Organization of the paper. In Section 2 we review necessary facts about
+the graded affine Hecke algebra and the corresponding affine KZ c onnection as well
+as introduce the graded affine nil-Hecke algebra and the corre sponding connection.
+In Section 3 we review various well-known facts about the coh omology of Band its
+cotangent bundle and restate our main result in these terms. In Section 4 we discuss
+equivariant Gromov-Witten invariants and explain general properties of the reduced
+virtual fundamental class. Once these generalities are in p lace, we apply them in
+Section 5 to give an extremely short proof of Theorem 1.1. In S ection 6 we explain the
+geometric construction of the shift operators for the affine K Z connection. In Sections
+7 and 8 we study the t→ ∞limit of the above constructions.
+Finally, in Section 9 we prove the generalization of Theorem 1.1 to the varieties /tildewideSn.
+1.18.Acknowledgments. Numerous discussions with R. Bezrukavnikov, P. Etingof,
+N. Nekrasov, and S. Shatashvili played a very important role in development of the
+ideas presented in this paper. We would also like to thank T. B ridgeland, I. Chered-
+nik, D. Kaledin, T. Lam, E. Opdam, R. Pandharipande, and Z. Yu n for very helpful
+conversations and for providing us with references on vario us subjects.
+A.B. was partially supported by the NSF grant DMS-0901274. D .M. was partially
+supported by a Clay Research Fellowship. A.O. was partially supported by the NSF
+grant DMS-0853560.
+Part of this work was done when A.B. was visiting the program o n ”Algebraic Lie
+Theory” at Isaac Newton Mathematical Institute. He would li ke to thank the program
+organizers for the invitation and the institute stuff for its hospitality.
+2.Hecke algebras, connections and integrable systems
+In this section we recall some standard algebraic backgroun d about graded affine
+Hecke algebras and the affine KZ connection. The geometricall y-minded reader can
+skip this section on a first reading.
+2.1.The graded affine Hecke algebra. Consider the graded Hecke algebra Ht. By
+definition, it is generated by elements xλforλ∈t∗andw∈Wand a central element
+tsuch that
+a)xλdepends linearly on λ∈t∗
+b)xλxµ=xµxλfor anyλ,µ∈t∗
+c) Thew’s form the Weyl group inside Ht;QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 11
+d) For any i∈I,λ∈t∗we have
+sixλ−xsi(λ)si=t(α∨
+i,λ),
+wheresiis the reflection associated with the simple root αi.
+We define a grading on Htin such a way that deg xλ= 2,degw= 0,degt= 2.
+2.2.The affine KZ connection. TheDunkl (or affineKZ)connection is aconnection
+onT∨
+regwith values in Ht. In other words, given a module MoverHt, we may define a
+connection ∇on the trivial bundle over T∨
+regwith fiber M. The connection is defined
+as follows. Any λ∈t∗defines a vector field on T∨and thus on T∨
+reg; we shall denote by
+dλthe derivative in the direction of this vector field. In order to define ∇it is enough
+to define ∇λfor every λ∈t∗, which is given by the following formula:
+(19) ∇λ=dλ−t/summationdisplay
+α>0(λ,α∨)qα∨
+1−qα∨(sα−1)−xλ
+By specifying xλ’s to their values at some point in twe get a family of connections on
+the trivial bundle with fiber MoverT∨
+reg, parametrized by t. This connection is known
+to have regular singularities; moreover, it is clear that it isW-equivariant (with respect
+to the natural action of WonT∨
+regandtrivialW-action on M).
+For the sake of completeness, let us compare our form of the co nnection ∇with the
+connection of [8], (1.1.41). In loc. cit. the author defines a slightly different connection
+∇′given by
+∇′
+λ=dλ−t/summationdisplay
+α>0(λ,α∨)sα
+qα∨−1−xλ.
+Let
+δ=/productdisplay
+α∈R+(qα∨−1).
+Then it is clear that ∇=δ−t∇′δt. In other words, our connection ∇is obtained from
+Cherednik’s connection ∇′by a simple gauge transformation. As a byproduct of this
+remark, we see that ∇is integrable (since the same is known about ∇′).
+2.3.The graded nil-Hecke algebra and the Toda connection. In this subsection
+we’ll be working with the affine nil-Hecke algebra Hnil, defined as follows: it is also
+generated by xλfor every λ∈t∗and by elements wforw∈Wsatisfying the following
+relations:
+a’)xλxµ=xλxµfor every λ,µ∈t∗
+b’)
+(20) w1w2=/braceleftBigg
+w1w2ifℓ(w1w2) =ℓ(w1)+ℓ(w2)
+0 otherwise
+c’)
+(21) sixλ−xsi(λ)si= (α∨
+i,λ).
+The algebras HtandHnilare related in the following way: let /tildewideHtdenote the C[t−1]-
+subalgebra of Ht[t−1] generated by all the xλ’s and by the elements /tildewidew=t−ℓ(w)w. Then12 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+it is clear that the fiber of /tildewideHtatt=∞is naturally isomorphic to Hnil. In other words
+we get a P1-familyHP1of associative algebras whose restriction to A1is isomorphic
+toHtand whose fiber at t=∞is naturally isomorphic to Hnil. Informally we can
+say that Hnilis the limit of Htwhent→ ∞; under this limit the elements xλgo to
+themselves and the limit of /tildewidewis equal to w.
+Let now Mbe anHt-module which is free over C[t] and let /tildewiderMbe an/tildewideHt-submodule
+ofM[t−1] such that /tildewiderM[t] =M[t−1]. In other words, /tildewiderMdefines an extension of Mto
+a sheaf of modules MP1overHP1. The fiber of /tildewiderMatt=∞is anHnil-module which
+we shall denote by M. In the future such P1-families of HP1-modules will be of special
+interest to us. One example of such family is discussed below .
+2.4.Some modules. Here we would like to describe explicitly some modules over Ht
+andHnil, which we are going to use in the future.
+First, we define a module MoverHnil. As a vector space we have M= Sym(t∗) =
+O(t). The action of Hnilon Sym( t∗) is described as follows: the action of xλis just
+given by the multiplication by λ. The action of si(for a simple reflection si) is given
+by
+(22) si(f) =f−fsi
+αi,
+wherefsi(a) =f(si(a)).
+Similarly, we can define a module MtoverHt. Namely, we set Mt= Sym(t∗)[t] =
+O(t×C) as a vector space. The action of Htis defined as follows: the element xλ∈Ht
+as before acts by multiplication by λ;tacts in the obvious way. The action of a simple
+reflection si∈Wis defined by
+si(f(a,t)) =f(a,t)−(f(a,t)−f(si(a),t))(1−t
+αi).
+In other words we have
+(23) (1 −si)f(a,t) = (f(a,t)−f(si(a),t))(1−t
+αi).
+The verification of the relations of Htis straightforward.
+It is also clear that we if we set /tildewidesi=t−1sithen limt→∞/tildewidesiis independent of tand
+is equal to si. Hence when t→ ∞the action of /tildewidew=t−ℓ(w)wbecomes independent of
+tand goes to the the action of won Sym(t∗). In other words, if we set MP1to be the
+trivial bundle over P1with fiber Sym( t∗) thenMP1becomes a sheaf of modules over
+HP1whose restriction to A1isMtand whose fiber at ∞isM.
+Note that the action of Sym( t∗)Wcommutes with the action of HtonMtand with
+the action of HnilonM. For any ξ∈t/Wlet us denote by Cξthe corresponding
+one-dimensional module over Sym( t∗)W. Then we set
+Mξ,t=Mt⊗
+Sym(t∗)WCξ;Mξ=M⊗
+Sym(t∗)WCξ,
+which are modules over HtandHnilrespectively of dimension # W.QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 13
+On the other hand, for each a∈tlet us denote by Cathe one-dimensional Sym( t∗)-
+module supported at a. We define
+Ma,t=Ht⊗
+Sym(t∗)Ca.
+These are often called the principal series representations ofHt. It is clear that
+dimMa= #W.
+The same definitions make sense for Hnilinstead of Ht. Namely, for any a∈twe
+define the principal series representation MaofHnilby
+Ma=Hnil⊗
+Sym(t∗)Ca.
+The following fact is shown in Theorem 1.2.2 of [8] in the case of the algebra Ht; the
+same proof as in loc. cit. works for Hnil.
+Proposition 2.1. (1)There are isomorphisms
+Mt≃Ht⊗
+C[W]C;M≃Hnil⊗
+C[W]C,
+where by C[W]⊂Hnilwe mean the span of all the w.
+For generic (a,t)∈t×Cone has:
+(2)The modules Ma,tandMaare irreducible.
+(3)For any w∈Wthere are isomorphisms
+Ma,t≃Mw(a),t;Ma≃Mw(a).
+(4)Letξbe the image of aint/W. Then there are isomorphisms
+Ma,t≃Mξ,t;Ma≃Mξ.
+2.5.The Calogero-Moser system. The trigonometric Calogero-Moser (or CM for
+short) quantum integrable system is an embedding ηCM,tof the commutative algebra
+Sym(t∗)Winto the algebra D(T∨
+reg) of differential operators on T∨
+reg, which depends on
+a parameter t∈C. This embedding is characterized by the following properti es:
+CM1) For any f∈Sym(t∗)Wthe highest symbol of ηCM,t(f) is equal to f;
+CM2) Let us choose a non-degenerate W-invariant quadratic form on t∗; letCbe
+the corresponding element of Sym2(t∗)Wand let ∆ denote the corresponding Laplacian
+onT∨(this is a W-invariant differential operator of order 2 on T∨with constant
+coefficients). Then
+ηCM,t(C) = ∆−t(t−1)/summationdisplay
+α∈R+(α∨,α∨)
+(qα∨/2−q−α∨/2)2.
+Using the map ηCM,twe may consider D(T∨
+reg)[t] as aD(T∨
+reg)⊗Sym(t∗)W[t]-module:
+hereD(T∨
+reg) acts by left multiplication and any f∈Sym(t∗)Wacts by right mul-
+tiplication by η(f). We shall call this the Calogero-Moser D-module and denote it
+byCMt. For any ξ∈t/Wwe denote by CMξ,tthe specialization of CMttoξ(i.e.
+Mξ,t=Mt⊗
+Sym(t∗)WCξ).14 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+On the other hand, consider O(T∨
+reg)⊗Mt. Since Sym( t∗)Wacts onMtby endomor-
+phisms of the Ht-module structure, by using the connection ∇we may view it as a
+D(T∨
+reg)⊗Sym(t∗)W[t]-module. According to Cherednik (cf. [7] or [8] Theorem 1.2 .11)
+and Matsuo [32] we have the following result:
+Proposition 2.2. There exists an isomorphism
+O(T∨
+reg)⊗Mt≃CMt
+ofD(T∨
+reg)⊗Sym(t∗)W[t]-modules. In particular, for any ξ∈t/Wthere exists an
+isomorphism
+O(T∨
+reg)⊗Mξ,t≃CMξ,t
+ofD(T∨
+reg)-modules.
+Remark. In fact, in [32] Theorem 2.2 is proved for generic ( a,t)∈tforMa,tinstead
+ofMa,t. In [8] Theorem 2.2 is proved for Ma,t(for any a) and it is easy to see that the
+proof ”works in families” (i.e. for Mtitself).
+2.6.The classical Calogero-Moser system. The quantum Calogero-Moser system
+has it quasi-classical analog – the classical Calogero-Moser system . This is a C[t]-linear
+embedding
+ηcl
+CM,t: Sym(t∗)W[t] =C[g]G֒→O(T∗(T∨
+reg)×C) =O(T∨
+reg×t)[t]
+whoseimageconsistsofPoissoncommutingfunctionsandwhi chsatisfiesobviousanalogs
+of the conditions CM1, CM2.
+2.7.The monodromy. Let now Hvdenote the ”usual” affine Hecke algebra; by the
+definition this is a C[v,v−1]-algebra, generated by elements Xλforλ∈Λ andTwfor
+w∈Wsubject to the well-known relations. In particular, Hvis a quotient of the group
+algebra of C[/hatwideBW] of the affine braid group /hatwideBW=π1(T∨
+reg/W) associated with the Weyl
+groupW; also the subalgebra of Hvgenerated by the Xλ’s is just C[Λ] which is the
+same as the algebra of regular functions on T. The subalgebra of Hvgenerated by all
+theTwis thefinite Hecke algebra H′
+v. It is a deformation of the group algebra C[W]
+and we shall denote by 1the corresponding deformation of the trivial C[W]-module.
+For any z∈Twe shall denote by Czthe corresponding 1-dimensional module over
+C[Λ]. Abusing notation, we shall sometimes think about vas an element of C∗rather
+than as a formal variable. Let us set
+Mv=Hv⊗
+H′v1.
+The algebra C[Λ]W=O(T/W) acts on Mvon the right (in fact, this algebra is the
+center of Hvand therefore it acts on every Hv-module); for any z∈T/Wwe shall
+denote by Mz,vthe specialization of Mvatz. This is a finite-dimensional Hv-module
+of dimension # W, which is known to be irreducible for generic z.
+Let us denote Ht−modfthe category of finite-dimensional Ht-modules; similarly,
+let us denote by Hv−modfthe category of finite-dimensional Hv-modules. Cherednik
+(cf. e.g. [8], Theorem 1.2.8) shows the following:QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 15
+Proposition 2.3. There exists a functor I:Ht−modf→Hv−modf, which depends
+on a choice of a point q0∈T∨
+reg/W, satisfying the following conditions:
+(1)I(Mξ,t) =Mz,vwherez=e2πiξandv=e2πit.
+(2)For anyMinHt−modflet us consider the corresponding affine KZ connection,
+as a connection over T∨
+reg/Won the trivial bundle with fiber M. Then the
+monodromy representation of C[π1(T∨
+reg/W,q0)]onMfactors through Hvand
+the resulting Hv-module is I(M).
+In other words, one may say that the functor Iis provided by the monodromy of the
+affine KZ connection.
+3.Affine Hecke algebras via the Steinberg variety
+In this section, we recall how the algebras HtandHnilappear geometrically and
+give a more detailed statement of our main result.
+3.1.The Steinberg variety. Recall that we have the natural proper (Springer) map
+f:T∗B→N. The Steinberg variety Stis defined as follows:
+St=T∗B×
+NT∗B.
+Explicitly, Stparametrises triples ( b1,b2,x) whereb1andb2are two Borel subalgebras
+ingandxis a nilpotent element in b1∩b2. Alternatively, Stcan bedefinedas theunion
+of the conormal bundles to the G-orbits in B×B. In particular, Stis equi-dimensional
+of dimension 2dim Band its irreducible components are parametrised by element s of
+W. The variety Stis endowed with a natural action of the group G=G×C∗where
+Gacts on everything by conjugation and the multiplicative gr oupC∗dilatesx(and
+doesn’t change b1andb2). We refer the reader to Section 3.3 of [9] for further detail s
+aboutSt.
+3.2.Borel-Moore homology and Lusztig’s construction. LetHG
+∗(St) denotethe
+G-equivariant Borel-Moore homology of St. According to [9] the space is endowed with
+a natural structure of an associative algebra. Moreover, th is algebra acts naturally on
+HG(T∗B) as well as on H∗
+Zn×C∗(Sn).
+According to [31] we have a natural isomorphism1
+(24) HG
+∗(St)≃Ht.
+Under this isomoprhism tcorresponds to the generator of H∗
+C∗(pt) (note that HG
+∗(St)
+is a module over H∗
+G(pt)).
+Similarly, one can define an associative algebra structure o nHG
+∗(B×B). In this case
+one has the isomorphism
+(25) HG
+∗(B×B)≃Hnil.
+The construction of the above isomorphism. Let us just note ( cf. [9]) that the algebra
+H∗
+G(St) acts naturally on HG(T∗B) (and moregenerally on H∗
+Zn×C∗(/tildewideSn) for any n∈N)
+and the algebra HG
+∗(B×B) acts on H∗
+G(B).
+1In fact, in [31] this isomorphism is constructed in a much mor e general situation16 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+3.3.Explicit construction: geometry. The isomorphism (24) can be described as
+follows. Namely, according to [31], Section 4, the element xλ∈Htcorresponds just
+to the push-forward of the class Dλunder the diagonal embedding T∗B֒→St. In
+particular, the action of xλonH∗
+G(T∗B) (or, more generally, on H∗
+Zn×C∗(/tildewideSn)) is given
+bymultiplication by Dλ. Thedescriptionoftheimageof WinHG
+∗(St)ismoreinvolved;
+however one can still describe explicitly the image of a simp le reflection si∈W(cf.
+[31], Section 3). Let us recall this construction.
+First of all, for each vertex iof the Dynkin diagram of GletPithe corresponding
+sub-minimal2parabolic and let Pi=G/Pibe the variety parametrizing all subgroups
+ofGwhich are conjugate to Pi. We have the natural projection pi:B→Piwhich
+is a locally trivial G-equivariant P1-fibration. Let Yi=B×
+PiB. ThenYiis a closed
+subvariety of B×Band its fundamental class [ Yi]∈HG
+∗(B×B) is equal to si. The
+corresponding operator on H∗
+G(B) is justp∗
+i(pi)∗.
+Similarly, let us now denote by Withe conormal bundle to B×
+PiB. This is a smooth
+closedG-invariant subvariety of Stand thus its fundamental class [ Wi] is a well defined
+element of HG
+∗(St)whichisequal to si. Inparticular, ifweview Wiasacorrespondence
+fromT∗Bto itself, then its action on G-equivariant cohomology of T∗Bis equal to the
+action of si−1.
+3.4.Explicit construction: algebra. The above actions of HtonH∗
+G(T∗B) and of
+HnilonH∗
+G(B) can be described explicitly in the following sense. First o f all, we have
+the natural isomorphisms
+H∗
+G(B)≃H∗
+B(pt) =H∗
+T(pt) = Sym( t∗) =M.
+Similarly, by using the pull-back map with respect to the pro jectionT∗B→Bwe
+may identify H∗
+G(T∗B) withH∗
+G(B) = Sym( t∗)[t] =Mt. Then we have the following
+Proposition 3.1. The above isomorphism H∗
+G(B)≃Mis an isomorphism of Hnil-
+modules. Similarly, the above isomorphism H∗
+G(T∗B)≃Mtis an isomorphism of
+Ht-modules.
+3.5.Main result revisited. With all this context in place, we are now ready to give
+a reformulation of the main result of this paper.
+Theorem 3.2. (1)The operator of quantum multiplication by DλinH∗
+G(T∗B)is
+equal to
+xλ+t/summationdisplay
+α∨∈R∨
++(λ,α∨)qα∨
+1−qα∨(sα−1),
+where the action of HtonH∗
+G(T∗B)is the one described above.
+(2)TheG-equivariant quantum connection of T∗Bis given by Equation 33.
+(3)TheG-equivariant quantum D-module of T∗Bis isomorphic to the Calogero-
+MoserD-moduleCMt(this is an isomorphism of D(T∨
+reg)⊗Sym(t∗)W[t]-modules).
+2Here and and in the sequel the word ”subminimal” means that th e semi-simple rank of the corre-
+sponding Levi is equal to 1QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 17
+(4)TheG-equivariant quantum cohomology ring of T∗Bis isomorphic to the ring
+of functions on T∗(T∨
+reg)×A1=T∨
+reg×t×A1, where the embedding H∗
+G(pt) =
+C[g]G= Sym(t∗)W[t]֒→O(T∗(T∨
+reg)×A1)is given by the classical Calogero-
+Moser map ηcl
+CM,t.
+The proof of the first assertion of Theorem 3.2 occupies the ne xt two Sections.
+Theorem 3.2(2) is just a reformulation of Theorem 3.2(1). Th eorem 3.2(3) follows from
+Theorem 3.2(2) by Proposition 2.2 and Proposition 3.1. Part 3.2(4) was also known to
+Nekrasov and Shatashvili [41] for G=SL(n); in the general case it follows easily from
+Theorem 3.2(3).
+3.6.Monodromy and derived equivalences. The isomorphism (24) has an analog
+inK-theory (in fact, historically, it was discovered before (2 4)). Namely, for a G-
+schemeYlet us denote by let KG(Y) =K0(CohG(Y))⊗Cthe Grothendieck group
+of the category CohG(Y) ofG-equivariant coherent sheaves on Y. ThenKG(St) has
+a natural structure of an associative algebra (defined by con volution) which acts on
+KG(T∗B). Then we have:
+Proposition 3.3. [27, 9]
+(1)There exists a natural isomorphism
+KG(St)≃Hv.
+(2)KG(T∗B)is isomorphic to Mvas aKG(St) =Hv-module.
+SinceHvis a quotient of the group algebra of the corresponding affine b raid group
+/hatwideBW, Proposition 3.3provides anaction of /hatwideBWonKG(St). Thisaction iscategorified in
+[3]. Namely, in loc. cit. the authors construct an action of /hatwideBWon the derived category
+Db(CohG(St)), which descends to the above action of /hatwideBWon theK-theory. Thus,
+via Proposition 2.3, Proposition 3.1 and Theorem 3.2(2), we can verify the relation
+between monodromy of the quantum connection and derived aut o-equivalences in our
+situation, as discussed in section 1.10.
+4.Preliminaries on Gromov-Witten invariants
+In this section, we review some definitions from Gromov-Witt en theory and sketch
+the basic properties of the reduced virtual class. The latte r is a technical construction
+responsible for many of the nice qualitative properties dis cussed in the introduction.
+Proofs and more detailed discussion of the results here can b e found in [42, 36].
+4.1.Definitions. We first clarify the definitions of the equivariant Gromov-Wi tten
+invariants
+/a\}b∇acketle{tγ1,...,γn/a\}b∇acket∇i}htX
+0,n,β=/integraldisplay
+[M0,n(X,β)]virn/productdisplay
+k=1ev∗
+kγk.
+In the above expression, the integrand is the virtual class o n the moduli space of
+n-pointed stable maps to X, which has expected dimension
+−KX·β+dimX+n−3 = 2dim B+n−3,18 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+and the map ev kis the evaluation map associated to the k-th marked point on the
+domain curve.
+The moduli space of maps is typically noncompact; however th ere is still a well-
+defined pushforward morphism H∗
+G,c(M0,n(X,β))→H∗
+G(pt) (hereH∗
+G,cstands for the
+corresponding equivariant cohomology with compact suppor t). On the other hand, we
+have the map H∗
+G,c(M0,n(X,β))→H∗
+G(M0,n(X,β)). This is a map of H∗
+G-modules,
+which becomes an isomorphism once we tensor everything with the field of fractions
+ofH∗
+G(pt), provided thatthe T-fixed locus is compact. Thus we get a well-defined
+integration on H∗
+G(M0,n(X,β) which takes values in the field of fractions of H∗
+G(pt).
+Explicitly, this integral can be defined using virtual local ization.
+We further observe that for β/\e}atio\slash= 0,
+/a\}b∇acketle{tγ1,Dλ,γ2/a\}b∇acket∇i}htX
+0,3,β= (Dλ·β)/a\}b∇acketle{tγ1,γ2/a\}b∇acket∇i}htX
+0,2,β,
+by the divisor equation. As a result, in order to understand t he non-classical contribu-
+tion to divisor operators, it suffices to study the two-point i nvariants of X.
+4.2.Reduced class. Given a variety Vwith an everywhere-nondegenerate holomor-
+phic symplectic form ω, it is well-known that the usual non-equivariant virtual fu n-
+damental class on Mg,n(V,β) vanishes for β/\e}atio\slash= 0. However, one can correct this phe-
+nomenon by modifying the standard obstruction theory so tha t the virtual dimension
+is increased by 1.
+We explain this modification for the deformation theory of ma ps from a fixed nodal
+curveC; applying this construction relative to the moduli stack of nodal curves gives
+the construction in general. Let MC(V,β) denote the moduli space of maps from Cto
+Vwith target homology class 0 /\e}atio\slash=β∈H2(V,Z). The standard obstruction theory for
+MC(V,β) is defined by the natural morphism
+(26) Rπ∗(ev∗TV)∨→LMC,
+whereLMCdenotes the cotangent complex of MC(V,β) and
+ev :C×MC(V,β)→V,
+π:C×MC(V,β)→MC(V,β).
+are the evaluation and projection maps.
+Letωπdenote the relative dualizing sheaf. There is a map
+ev∗(TV)→ωπ⊗(Cω)∗,
+induced by the pairing with the symplectic form and pullback of differentials. This, in
+turn, yields a map of complexes
+Rπ∗(ωπ)∨⊗Cω→Rπ∗(ev∗(TV)∨)
+and the truncation
+ι:τ≤−1Rπ∗(ωπ)∨⊗Cω→Rπ∗(ev∗(TV)∨).
+The truncation is a trivial line bundle, although will carry a nontrivial weight in the
+equivariant setting if the group action acts by a nontrivial character on ω.QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 19
+There is an induced map
+(27) C(ι)→LMC
+whereC(ι) is the mapping cone associated to ι. Moreover, this map (27) satisfies
+the necessary properties of a perfect obstruction theory. T his is known as the reduced
+obstruction theory and defines the reducedvirtual fundamental class associated to V.
+4.3.One-parameter families. Another characterization of reduced virtual classes is
+given in [36] in terms of one-parameter deformations of the h olomorphic symplectic
+varietyV.
+Suppose we are in the following situation: There is a smooth m apπ:V→B
+whereBis a smooth curve, such that πis topologically trivial as a fibration and Vis
+equipped with a fiber-wise holomorphic symplectic form. The re is a point b∈Bsuch
+thatV=Vb. Furthermore, assume that we have a class β∈H2(V,Z)⊂H2(V,Z)
+satisfying the following conditions:
+•The fiber Vis the unique fiber of πfor which the class βis represented by an
+effective curve.
+•The composition
+(28) TB,b→H1(V,TV)∼− →H1(V,ΩV)β∪− − →C
+is an isomorphism.
+In the above diagram, the first map is the Kodaira-Spencer map , the second map is
+the isomorphism induced by the holomorphic symplectic form onV, and the last map
+is given by cup product with the curve class β.
+The following result is proven in Theorem 1 of [36]
+Proposition 4.1. Given the above hypotheses, the embedding
+M0,n(V,β)→M0,n(V,β)
+is an isomorphism of stacks and we have an equality of virtual classes
+[M0,n(V,β)]red= [M0,n(V,β)]vir,
+where the right-hand side is just the usual virtual fundamen tal class associated to V.
+This statement can be easily generalized in many directions . If we have a fiber-wise
+group action of TonV, then the above equality holds for T-equivariant virtual classes.
+If there are finitely many points on Bfor which βis effective and infinitesimally rigid
+in the sense of equation (28), the statement generalizes in t he obvious way where the
+left-hand side involves a summation over these points.
+5.Proof of Theorem 1.1
+We now use the results of the last section to give a short proof of Theorem 1.1.20 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+5.1.Lagrangian correspondences. We first apply the results of section 4.2 to our
+situation. Since the holomorphic symplectic form on Xhas equivariant weight −twith
+respect to G, the construction there extends to the equivariant setting . In particular,
+the reduced obstruction theory produces a cycle class
+[M0,2(X,β)]red∈HBM,G
+2dimX(M0,2(X,β)
+forβ/\e}atio\slash= 0 such that
+[M0,2(X,β)]vir=t·[M0,2(X,β)]red.
+Thisequality explainsthefactorof tinfrontofthenonclassicalcontributiontoTheorem
+1.1.
+Recall the Springer map
+f:X→N.
+SinceNis affine, any proper curve on Xis contained in a fiber of f, and the evaluation
+map toX×Xfactors through the Steinberg variety
+φ:M0,2(X,β)→St=X×NX.
+The resolution X→Nis semismall, so the irreducible components ZkofX×NXall
+have dimension dim X. In particular, we immediately have the following elementa ry
+but crucial lemma.
+Lemma 5.1.
+(29) φ∗[M0,2(X,β)]red=/summationdisplay
+ak[Zk]∈HBM,G
+2dimX(X×NX),
+whereak∈Qare nonequivariant constants.
+Proof.This follows immediately from dimension constraints, sinc e we are working with
+non-localized coefficients. /square
+Since the correspondence operators defined by [ Zk] give the action of Z[W] on the
+cohomology of X, this explains why the non-classical contribution to Dλis expressed
+in terms of Weyl group operators. To complete the proof of The orem 1.1, it remains
+to match the coefficients akwith the predicted answer. Since these coefficients are
+rational numbers, they are unaffected by specializing to t= 0. Therefore, we only need
+to compute the left hand side of equation 29 in G-equivariant cohomology.
+5.2.Simultaneous resolution. We now apply section 4.3 to rephrase reduced invari-
+ants in terms of one-parameter deformations of X; here it is important that we are only
+working G-equivariantly.
+We obtain such deformations using the Grothendieck resolut ion of section 1.2:
+φ:/tildewideX→t∗.
+Recall that the fiber over the origin is Xand that there is a fiber-preserving G-action
+which cannot be extended to a fiber-preserving G-action. In general, for z∈t∗, the
+monoid of effective curve classes in the fiber φ−1(z) is given by
+Eff(φ−1(z)) = Span {α∨|α∈∆+,(α∨,z) = 0}.QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 21
+Consider a generic linear subspace φ0:A1→t, chosen to intersect each hyperplane
+transversely exactly once at the origin; via pullback, we ha ve a smooth map V0→A1
+whose fiber over the origin is X.
+Lemma 5.2. The family V0→A1satisfies the conditions listed in section 4.3.
+Proof.On the set-theoretic level, this is obvious. The infinitesim al statement follows
+from the transverse assumption, since the cup product pairi ng in equation 28 can be
+identified with the pairing of the root lattice with t. /square
+Proposition 4.1 implies that, for γ1,γ2∈H∗
+T(X,C), we have
+/a\}b∇acketle{tγ1,γ2/a\}b∇acket∇i}htX,red
+0,2,β=/a\}b∇acketle{tγ1,γ2/a\}b∇acket∇i}htV0
+0,2,β,
+where the right-hand side again denotes invariants defined v ia the standard virtual
+class.
+We now deform the variety V0and apply deformation invariance, following the ap-
+proach in [6, 33]. Let D=A1and consider a family of maps φt:A1→t,t∈D, so that
+fort= 0 we recover φ0and for any t/\e}atio\slash= 0 sufficiently near 0, the image of φtintersects
+each hyperplane Hαtransversely at distinct points. We fix t/\e}atio\slash= 0 with this property,
+and letzαbe the distinct intersection points of φtwithHα. The associated smooth
+familyVtwill again satisfy the conditions of section 4.3 at each of th ese intersection
+points.
+Lemma 5.3. Fort/\e}atio\slash= 0, we have the equality of G-equivariant reduced Gromov-Witten
+invariants
+/a\}b∇acketle{tγ1,γ2/a\}b∇acket∇i}htX,red
+0,2,β=/summationdisplay
+α∈∆+/a\}b∇acketle{tγ1,γ2/a\}b∇acket∇i}htXzα,red
+0,2,β
+Proof.The left and right sides of the statement can be identified wit h/a\}b∇acketle{tγ1,γ2/a\}b∇acket∇i}htVs
+0,2,β, for
+s= 0 ands=trespectively. These are equal by deformation invariance of the standard
+virtual class construction appliedto thefamily V→D. Although theassociated moduli
+space of maps relative to D
+M0,2(V/D,β)→D
+is not proper over D, it admits a fiberwise T-action whose fixed loci are proper over D,
+and the results from [1] can be applied. /square
+The arguments in these two sections apply without change to t he general setting of
+the introduction, as discussed in section 1.6. In particula r, we see that the nonclassical
+terms of divisor operators lie in the subalgebras associate d to primitive coroots.
+5.3.Actual calculation. Lemma5.3reducestheargumenttounderstandingthecodi-
+mension one fibersof thesimultaneous resolution. Thesehav e thefollowing description.
+Consider a general point z∈Hαforα∈∆+and letMzdenote its centralizer in G; it
+is easy to see that Mzfits into an exact sequence with its center Z(Mz):
+1→Z(Mz)→Mz→PGL(2)→1.
+This sequence induces an action of MzonT∗P1; the fiber Xzis22 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+Xz=G×MzT∗P1,
+and this identification is compatible with the G-action. This description shows that
+Xzis a fiber bundle over the affine variety Az=G/Mzwith fiber T∗P1; in particular,
+we see the effective curves are multiples of the zero-section o f the fiber, in class α∨. Let
+Yzdenote the affinization of Xz, which is just the contraction of this class.
+AsintheproofofLemma5.1, evaluation onthemodulispaceof mapsfactors through
+the Steinberg construction:
+φ:M0,2(Xz,β)→Xz×YzXz=Xz∪Wz
+whereWzis aP1×P1-bundle over Az.
+Lemma 5.4.
+φ∗[M0,2(Xz,dα∨)]red=1
+d[Wz]
+Proof.LetM0,2(Xz/Az,β) denote the moduli space of stable maps to Xzrelative to
+Az, parametrizing a point a∈Azand a stable map to the fiber over a. Since the map
+Xz→Azis smooth, this moduli space admits a perfect obstruction th eory relative to
+the map to Az. Furthermore, since we have a fiberwise holomorphic symplec tic form on
+Xz, we can construct the reduced obstruction theory relative t oAzand the associated
+reduced virtual class [ M0,2(Xz/Az,β)]red.
+We first claim that
+[M0,2(Xz,β)]red= [M0,2(Xz/Az,β)]red.
+This follows from a comparison of obstruction theories. Fir st, for standard virtual
+classes, there is a map of obstruction spaces associated to m aps from a fixed curve C
+H1(C,f∗TXz/Az)→H1(C,f∗TXz).
+WhenCis a rational curve, this is an isomorphism. For reduced clas ses, we need this
+map to be compatible with the one-dimensional quotient aris ing from the holomorphic
+symplectic form (fiberwise, in the case of Xz/Az). This follows from the fact that the
+holomorphic symplectic form on fibers of Xz/Azare the restrictions of the form on Xz.
+As in Lemma 5.1, dimension constraints imply that
+φ∗[M0,2(Xz,dα∨)]red=c1[Xz]+c2[Wz],
+forc1,c2∈Q.
+We can calculate the coefficients c1,c2after restricting to a fiber of Xz→Az. More
+precisely,
+if we pick a point ι:a→Az,
+φ∗[M0,2(T∗P1,d)]red=φ∗ι![M0,2(Xz,dα∨)]red=
+ι!φ∗[M0,2(Xz,dα∨)]red=c1[T∗P1]+c2[P1×P1].QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 23
+These coefficients are thus determined from the case G= SL2, where Theorem 1.1
+has already been established in this case ([5],[33]). The ac tion of the cycle [ P1×P1] is
+the operator s−1 onH∗(T∗P1,C), so we see that c1= 0 and c2= 1/d.
+/square
+The proof of Theorem 1.1 now follows from the following resul t at the end of section
+3.3.
+Lemma 5.5.
+[Wz] =sα−1
+6.Shift operators
+Wenow explainthegeometric construction oftheshiftopera tors, discussedinsection
+1.12, that intertwine the monodromy representation of the q uantum connection for
+different values of the equivariant parameters.
+Let us choose an integral parameter s∈Λ∨⊕Z. We can associate to sa principal
+T-bundle over P1
+P(s) = (T×(C2\{0})/C∗,
+whereC∗acts onTvia the 1-parameter subgroup defined by s. Let
+X(s) =P(s)×TX→P1
+be the associated fibration over P1with fiber X. There exists a unique lifting of the C∗
+action on P1to an action on X(s) so that the fiber X(s)0of 0∈P1is fixed pointwise
+by the action, giving an action of /tildewideT=T×C∗onX(s). Letzdenote the equivariant
+parameter associated to the extra torus factor.
+Given a weight λ∈Λ, there is again an associated equivariant line bundle Dλ,son
+X(s), compatible with restriction to fibers. Given β∈H2(X(s),Z), we can use these
+line bundles to assign a natural monomial function qβon the dual torus T∨.
+We will only consider curve classes β∈H2(X(s),Z) such that
+π∗β= [P1],
+and the corresponding moduli spaces of stable maps
+M0,2(X(s),β).
+Let
+ι0,ι∞:X ֒→X(s)
+denote the inclusions of the fiber over 0 and ∞respectively. Given γ1,γ2∈H∗
+T(X),
+the two-pointed Gromov-Witten invariant
+/a\}b∇acketle{tγ1,γ2/a\}b∇acket∇i}htX(s)
+0,2,β=/integraldisplay
+[M0,2(X(s),β)]virev∗
+1(ι0,∗γ1)∪ev∗
+2(ι∞,∗γ2)∈FracH∗
+/tildewideT(X).
+In order to make sense of the above expression, for each fiber o ver 0 and ∞, we use the
+identification H∗
+/tildewideT(X) =H∗
+T(X)⊗C[z] to liftγ1andγ2to/tildewideT-equivariant cohomology.
+The corresponding shift operator
+S(s)∈End(H∗
+T(X))⊗C[[Λ]]24 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+is then defined via the pairing
+/a\}b∇acketle{tγ1|S(s)|γ2/a\}b∇acket∇i}ht=/summationdisplay
+β∈H2(X(s),Z)
+π∗β=[P1]qβ/parenleftBig
+/a\}b∇acketle{tγ1,γ2/a\}b∇acket∇i}htX(s)
+0,2,β/parenrightBig
+|z=1.
+The intertwiner property of S(s) can be seen as follows. We first describe the contri-
+bution of the residues associated to classical multiplicat ion. Given x∈XTassociated
+to the Weyl group element wx∈W, let
+φx= (wx,0) :t∗→t∗⊕C= (Lie(T))∗
+denote the associated linear map, which we can view as a multi -valued function on T∨,
+depending on a parameter a∈Lie(T). We define the operator qMonH∗
+T(X) which is
+diagonal in the fixed-point basis, with diagonal entries giv en by the function φx.
+In the neighborhood of the base point
+qα∨
+1=qα∨
+2=···= 0,
+there exists a unique fundamental solution of the quantum di fferential equation, nor-
+malized to have the form
+Ψ(q,a) =qM(Id+O(qα∨
+i)).
+Here, we have made explicit the dependence of Ψ on the equivar iant parameters a∈
+Lie(T). ThereexistsageometricexpressionforΨ( q,a)intermsofdescendentinvariants
+(see, for example, [11]).
+We also define the operator ∆( s) onH∗
+T(X), so that it is diagonal in the fixed-point
+basis, with diagonal entries given by the ratio
+∆(s)p,p=/productdisplay
+tangent weights ρ
+atx∈XΓ(ρ+s(ρ))
+Γ(ρ+1).
+It follows from equivariant localization with respect to /tildewideTthatS(s) can be factored
+in terms of the fundamental solution and ∆.
+(30) S(s) = Ψ(q,a)◦∆(s)◦Ψ−1(q,a+s).
+From here, we see immediately that the operators S(s) satisfy the following inter-
+twiner and composition identities:
+∇(a)S(s) =S(s)∇(a+s) (31)
+S(s1+s2) =S(s1)◦S(s2)|a=a+s1 (32)
+In order to compare the monodromy representation after inte gral parameter shifts
+a/mapsto→a+s,
+we use the following proposition.
+Proposition 6.1. The matrix entries of S(s)are rational functions on C(T∨):
+S(s)∈End(H∗
+T(X))⊗C(T∨).QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 25
+Proof.We will only sketch the proof here. In the case of Hilb( C2) and higher-rank
+framed sheaves, a version of this argument can be found in [34 ] and [35]. We refer the
+reader to these papers for a more detailed exposition.
+Using the composition identity for S(s), it suffices to prove the proposition for
+s= (δ,0),(0,1)∈Λ∨⊕Z,
+whereδis a fundamental weight of T∨. The virtual dimension of the moduli spaces
+M0,0(X(s),β) are, respectively, 2dim Band dim B.
+In the first case, if we choose insertions over 0 and ∞from a basis of Schubert conor-
+mal Lagrangians in X, the relevant two-point invariants will be degree 0. Moreov er,
+there exists a choice of basis for which the space of stable ma ps meeting these cycles
+is proper. As a consequence, the associated matrix entries a re degree 0 equivariant
+polynomials, thus constant. If we specialize to t= 0, we can deform X(s) so that it
+has affine fibers, and only sections of πcontribute. In particular, the answer can be
+calculated directly (e.g. using the case of B=P1).
+The second case is more involved. We consider the subspace Vof vectors v∈
+HT(X)⊗C(T∨) for which S(s)vhas rational function entries. We require two claims
+aboutV:
+•Vcontains the identity element 1 ∈H∗
+T(X)
+•Vis closed under application of the differential operators ∇λ.
+Thefirstclaimfollowsagainviaequivariantspecializatio n. Byconstruction, M0,2(X(s),β)
+is proper - any section of X(s)→P1is forced to lie inside B×P1. Therefore, when
+expressed in a basis of Schubert conormals, every entry in S(s)◦1 is a nonequivari-
+ant constant, so the claim can be checked after setting t= 0. If we apply virtual
+localization, we require the specialization of the fundame ntal solution of the quantum
+connection when t= 0,1. However, it follows from the form of the Calogero-Moser
+system given in section 2.5 that the connection becomes esse ntially trivial at these
+values, yielding the result. The second claim follows immed iately from the intertwiner
+property. Since classical cohomology is generated by divis ors, we then conclude from
+these two claims that V=HT(X)⊗C(T∨). /square
+Corollary 6.2. Fora∈Lie(T)away from a discriminant locus (depending on s), the
+monodromy representation of the quantum differential equat ion∇(a)is isomorphic to
+the monodromy representation for ∇(a+s)
+Notice that the intertwiners S(s) may have acquire singularities as a function of a,
+accounting for the discriminant locus in the statement of th e corollary. It follows from
+irreducibility of the monodromy representation that the ge ometric intertwiners agree
+with the Opdam shift operators, up to a global equivariant co nstant.
+If we consider a general equivariant symplectic resolution as in section 1.1, we expect
+analogous statements to hold; however the arguments given h ere only partially extend.
+While the definition of the geometric intertwiner is complet ely formal, one needs to
+check that the proof in Proposition 6.1 holds in any given geo metry. In particular, the
+simplification of the quantum connection for t= 1 and the generation of the quantum
+ring by divisors are the key statements that need to be establ ished.26 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+7.The limit t→ ∞: algebra
+The purpose of the next two sections is to explain how Theorem 1.1 allows us to
+compute the quantum cohomology of B.
+7.1.The connection ∇.Similarly to the affine KZ connection ∇we can define the
+Toda connection ∇, which is a connection on T∨with values in Hnil. More precisely,
+given a module MoverHnilwe have a connection on the trivial bundle over T∨with
+fiberMoverT∨defined as follows. First, let us recall the subset R′
++ofR+defined in
+section 1.13. Then we define
+(33) ∇λ=dλ−/summationdisplay
+α∈R′
++(λ,α∨)qα∨sα−xλ.
+We are going to show that ∇is integrable a little later. We shall refer to ∇asthe Toda
+connection ; the reason for this name is explained in section 7.2.
+We now claim that under the above circumstances the connecti on∇is ”the limit of
+∇ast→ ∞” in the appropriate sense. More precisely, we have the follo wing
+Proposition 7.1. The limit
+lim
+t→∞t2ρ∇t−2ρ
+exists and is equal to ∇(which is a connection on T∨with values in the trivial bundle
+with fiber M). In other words, ∇is thet→ ∞-limit of ∇if we make the change of
+variables q→t−2ρq.
+Remark. Notethatthesingularitiesoftheconnection ∇onT∨disappearwhen t→ ∞;
+however one can check that in general the connection ∇will have irregular singularities
+at∞. Also, the W-equivariance of ∇is lost under the above limit procedure (this has
+to do with the fact that our change of variables q→t−2ρqis notW-equivariant).
+Proof.¿From equation 19 and from the definition of /tildewidewwe get
+t2ρ∇λt−2ρ=dλ+/summationdisplay
+α>0t−(2ρ,α∨)+1qα∨
+1−t−(2ρ,α∨)qα∨(1−tℓ(sα)/tildewidesα)+xλ.
+Thus, inordertoproveProposition7.1, itisenoughtocheck thevalidity ofthefollowing
+Lemma 7.2. (1)For any α∈R+we have
+ℓ(sα)≤(2ρ,α∨)−1.
+(2)The above inequality is an equality if and only if α∈R′
++.
+Proof.This lemma is essentially proved in Section 1 of [23]. More pr ecisely, it is shown
+in[23](Lemma1.3) that ℓ(sα)≤(2ρ,α∨)−1whenαislongandthat ℓ(sα) = (2ρ∨,α)−1
+whenαis short. Thus it is enough to show that for any short positive rootαwe have
+(34) ( ρ∨,α)≤(ρ,α∨),QUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 27
+where the equality holds if and only if α∈R′
++. Note that if αis short, then α∨is long.
+Let Π∨
+lgdenote the set of simple long coroots and let Π∨
+shdenote the set of simple short
+coroots. Let
+α∨=/summationdisplay
+β∨∈Π∨
+lgaββ∨+/summationdisplay
+γ∨∈Π∨
+shaγγ∨.
+Let us choose a W-invariant inner product on t∗satisfying ( α,α) = 1 for every short
+rootαand set
+r= max
+α∈R+(α,α).
+Then we have ( ρ,α∨) =/summationtextaβ+/summationtextaγand according to Section 1.2 of [23] we have
+(ρ∨,α) =/summationdisplay
+β∨∈Π∨
+lgaβ+1
+r/summationdisplay
+γ∨∈Π∨
+shaγ,
+which implies (34). /square
+/square
+Corollary 7.3. The connection ∇is integrable.
+Proof.This follows immediately from Proposition 7.1 and from the f act that ∇is
+integrable. /square
+7.2.The Toda system. Similarly to section 2.5 one can define the quantum Toda
+integrable system, which is now a map ηT: Sym(t∗)W→D(T∨). The map ηTis
+characterized by the following properties:
+T1) For any f∈Sym(t∗)Wthe highest symbol of ηT(f) is equal to f;
+T2) For any non-degenerate W-invariant quadratic form on t∗as above one has
+ηT(C) = ∆−/summationdisplay
+α∈Π(α∨,α∨)qα∨.
+It is known (cf. Section 7 of [14]) that for any f∈Sym(t∗)Wone has
+(35) lim
+t→∞t2ρηCM,t(f)t−2ρ=ηT(f).
+Using the map ηTwe may view D(T∨) as aD(T∨)⊗Sym(t∗)W-module, which we
+shall denote by T. We shall call Tthe Toda D-module.
+We now want to present an analog of Theorem 2.2 in the case when the Calogero-
+Moser system is replaced by the Toda system. Namely, by using the connection ∇we
+may view O(T∨)⊗Mas aD(T∨)⊗Sym(t∗)W-module. We now claim the following
+analog of the Cherednik-Matsuo theorem for the Toda lattice :
+Theorem 7.4. There exists an isomorphism
+O(T∨)⊗M≃T
+ofD(T∨)⊗Sym(t∗)W-modules.28 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+Remark. It is tempting to say that Theorem 7.4 follows from Theorem 2. 2 by taking
+t→ ∞limit and using Proposition 7.1 together with (35). However ,a prioriit is not
+clear why the limiting procedure in Proposition 7.1 is the sa me as in (35). Hence we
+are going to give an independent proof of Theorem 7.4.
+Proof.Let us define a map ι:T=D(T∨)→O(T∨)⊗M=O(T∨)⊗Sym(t∗) which send
+everyd∈D(T∨) tod(1⊗1) (let usrecall that theaction of D(T∨)onO(T∨)⊗Sym(t∗)W
+is given by ∇).
+Lemma 7.5. The map ιis an isomorphism of vector spaces.
+Proof.The module T=D(T∨) is endowed with a natural filtration by order of a
+differential operator. Similarly, O(T∨)⊗Sym(t∗) is endowed with a filtration coming
+from the natural filtration on Sym( t∗) (by degree of a polynomial). From the definition
+ofιit is clear, that it is compatible with the above filtration it defines between the
+associated graded spaces. This shows that ιis an isomorphism itself.
+/square
+On the other hand, it is obvious that ιis a morphism of D(T∨)-modules. Hence, in
+order to prove Theorem 7.4 it is enough to verify that it is a mo rphism of Sym( t∗)W-
+modules. We claim that it is enough to check this for quadrati c elements of Sym( t∗)W.
+Indeed, applying ι−1to the action of Sym( t∗)WonO(T∨)⊗Sym(t∗) we get an action
+of Sym(t∗)WonT=D(T∨); this action commutes with the left D(T∨)-action and
+thus comes from a homomorphism η′: Sym(t∗)W→D(T∨). We need to show that
+η′=ηT. However, η′obviously satisfies property T1. Thus in order to show proper ty
+the equality η′=ηTwe need to check that η′satisfies T2, which exactly means that ι
+commutes with quadratic elements in Sym( t∗)W.
+Thus we are reduced to checking that for any Cas in T2 we have
+(36) η(C)(1⊗1) = 1⊗C.
+Letλ1,···,λℓbe a basis of t∗and letµ1,···,µℓbe the dual basis with respect to the
+quadratic form corresponding to C. Then∇λi(1⊗1) =−1⊗λi. Thus
+∆(1⊗1) =ℓ/summationdisplay
+i=1∇µi∇λi(1⊗1) =−ℓ/summationdisplay
+i=1∇µi(1⊗λi) =ℓ/summationdisplay
+i=1(1⊗µiλi+
+/summationdisplay
+α∈R′
++(µi,α∨)qα∨(1⊗sα(λi))) = 1⊗C+/summationdisplay
+α∈R′
++qα∨ℓ/summationdisplay
+i=1(µi,α∨)(1⊗sα(λi))
+Note that sα(λi) = 0 if αis not a simple root; for α∈Π we have sα(λi) = (λi,α∨).
+Thus
+/summationdisplay
+α∈R′
++ℓ/summationdisplay
+i=1(µi,α∨)(qα∨⊗sα(λi)) =/summationdisplay
+α∈Πqα∨ℓ/summationdisplay
+i=1(µi,α∨)(λi,α∨) =/summationdisplay
+α∈Π(α∨,α∨)qα∨(1⊗1),
+which implies (36). /squareQUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 29
+8.The limit t→ ∞: geometry
+In this section we explain how we can recover the quantum coho mology of Bby
+studying the t→ ∞limit of the quantum cohomology of X=T∗B. Using the projec-
+tionπ:X→B, we can identify
+H∗
+T(X,C) =H∗
+T(B,C)[t],
+and view the divisor operators DX
+λas acting on the latter space:
+DX
+λ(t,q)∈End(H∗
+T(B,C))[t][[qβ]].
+LetDX
+λ(t,t−c1q) be the operators obtained by the substitution
+qβ/mapsto→t−c1(B)·βqβ.
+We want to compare these operators with the divisor operator sDB
+λ(q) arising from
+the quantum cohomology of B; since the T-action on Bfactors through T, there is no
+t-dependence here.
+Proposition 8.1.
+DB
+λ(q) = lim
+t→∞DX
+λ(t,t−c1q).
+Proof.Matrix elements of DX
+λare obtained from invariants of the form
+/a\}b∇acketle{tγ1·[B],Dλ,γ2/a\}b∇acket∇i}htX
+0,3,β,
+whereγ1,γ2∈H∗
+T(B,C) and [B] denotes the class of the zero-section. Define the
+moduli space of maps
+M0,3(X,p1,β) ={(C,f)∈M0,3(X,β)|f(p1)∈B⊂X}.
+It is a standard fact that M0,3(X,p,β) admits a T-equivariant virtual class for which
+ι∗[M0,3(X,p1,β)]vir= ev∗
+1[B]·[M0,3(X,β)]vir.
+Moreover, since TBis nef,f(p1)∈Bimplies that ffactors through B, and we have
+M0,3(X,p1,β) =M0,3(B,β)
+as stacks, equipped with two different obstruction theories.
+Consider the universal family
+Cf/d47/d47
+π
+/d15/d15B
+M0,3(B,β)σ1/d73/d73
+whereσ1is the section associated to the first marked point, with imag e Σ1⊂C. We
+can define a sheaf KonCby the short exact sequence
+0→K→f∗T∗B→f∗T∗B|Σ1→0,
+where the second map is the restriction map. It is easy to see t hatR0π∗K= 0 and
+that
+E=R1π∗K30 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+is aT-equivariant vector bundle on M0,3(B,β) with rank c1(B)·β.
+The relative obstruction theories are then related by the tr iangle
+E∨[1]→EM(X,p1,β)/M0,3→EM(B,β)/M0,3,
+and we have
+[M0,3(X,p1,β)]vir=ct(E)∩[M0,3(B,β)]vir
+where
+ct(E) =trk(E)+c1(E)trk(E)−1+···+crk(E)(E)
+is theT-equivariant Euler class. This yields
+/a\}b∇acketle{tγ1·[B],Dλ,γ2/a\}b∇acket∇i}htX
+0,3,β=/a\}b∇acketle{tct(E),γ1,Dλ,γ2/a\}b∇acket∇i}htB
+0,3,β.
+If we want to recover the term corresponding to the operator DB
+λ, we can weight this
+invariant by t−rk(E)=t−c1(B)·βand take the limit as t→ ∞. This same argument
+works for all multiplication operators. /square
+Let us now recall that c1(B) =−2ρ. Combining this with Proposition 8.1, Proposi-
+tion 7.1, and Theorem 7.4 we obtain a new proof of the followin g result:
+Theorem 8.2. (1)The operator of quantum multiplication by DλinH∗
+G(B)is
+equal to
+xλ+/summationdisplay
+α∈R′
++(λ,α∨)qα∨sα.
+Herexλis the classical multiplication by Dλ.
+(2)TheG-equivariant quantum connection of Bis given by Equation 33.
+(3)TheG-equivariant quantum D-module of Bis isomorphic to the Toda D-module
+T(as aD(T∨)⊗Sym(t∗)W-module).
+These statements are already well-known in the literature T he first follows from
+Theorem 6.4 in [37]. The second statement is of course just a r eformulation of the first,
+and the third statement follows from the main result of [29], although via a different
+argument.
+9.Slodowy slices
+In this section, we prove Theorem 1.3.
+Givenn∈N, we can associate an sl2-triple (e=n,h,f) and consider the affine
+subspace vn={n+Keradf} ⊂t. The Slodowy slice resolution /tildewideSnis a local complete
+intersection subvariety of Xdefined by the Cartesian diagram
+(37) /tildewideSn/d31/d127ι/d47/d47
+g′
+/d15/d15X
+g
+/d15/d15vn/d31/d127ι/d47/d47tQUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 31
+LetT′⊂Tbethesubtoruswhichpreserves /tildewideSn. Assumethenaturalmap H2(/tildewideSn,Z)→
+H2(X,Z) is an isomorphism. Given β∈H2(/tildewideSn,Z), sincevnandtare affine, the moduli
+space of stable maps M0,2(/tildewideSn,β) fits in the Cartesian diagram
+M0,2(/tildewideSn,β)ι/d47/d47
+h
+/d15/d15M0,2(X,β)
+h
+/d15/d15
+vn/d31/d127ι/d47/d47t
+In order to prove Theorem 1.3, we will show the following.
+Proposition 9.1. There is an equality of T′-equivariant virtual classes
+ι![M0,2(/tildewideSn,β)]vir= [M0,2(X,β)]vir
+Proof.As in section 4.2, we will prove the analogous statement for t he moduli space
+MC(/tildewideSn,β) of maps from a fixed genus 0 nodal curve C; the full statement will then
+follow by applying this statement relative to the moduli sta ck of nodal curves.
+In order to prove the pullback of virtual classes, it is suffici ent to prove that the ob-
+struction theories on MC(/tildewideSn,β) andMC(X,β) are compatible in the sense of Theorem
+5.10 of [1]. To show this, we need to construct a morphism of exac t triangles
+ι∗EMC(X,β)φ/d47/d47
+/d15/d15EMC(/tildewideSn,β)ψ/d47/d47
+/d15/d15h∗Lvn/tχ/d47/d47
+/d15/d15ι∗EMC(X,β)[1]
+/d15/d15
+LMC(X,β)/d47/d47LMC(/tildewideSn,β)/d47/d47LMC(/tildewideSn,β)/MC(X,β)/d47/d47LMC(X,β)[1]
+Here, the bottom row is the standard triangle associated to t he cotangent complex of
+a morphism, and the vertical maps are the data of an obstructi on theory.
+The top row will be induced by the short exact sequence
+(38) 0 →T/tildewideSn→ι∗TX→g∗Nvn/t→0.
+To see this, recall that, in the notation of section 4.2, we ha ve
+EMC(V,β)=Rπ∗(ev∗TV)∨
+forV=Xand/tildewideSn. So if we apply Rπ∗(ev∗(−)∨to the sequence (38) we get the triangle
+ι∗EMC(X,β)→EMC(/tildewideSn,β)→Rπ∗(ev∗g∗Nvn/t)∨[1]→ι∗EMC(X,β)[1].
+Since the genus of Cis 0, we have Rπ∗(ev∗g∗Nvn/t)∨[1] =h∗Lvn/t,and the maps
+obviously commute with the vertical maps associated to each obstruction theory.
+Since the obstruction theories associated to Xand/tildewideSnare compatible, Theorem 5 .10
+in [1] then immediately gives an isomorphism of virtual clas ses after pullback. /square
+We use this proposition to give a short proof of Theorem 1.3.
+Proof.As in the proof of Theorem 1.1, the identification of classica l multiplication
+byDλfollows from the definition of the Hecke algebra action on H∗
+T′(/tildewideSn). For the32 ALEXANDER BRAVERMAN, DAVESH MAULIK AND ANDREI OKOUNKOV
+nonclassical terms, it suffices to calculate the two-point Gr omov-Witten invariants of
+/tildewideSn.
+Consider the Cartesian diagram induced by the evaluation ma ps to the Steinberg
+variety:
+M0,2(/tildewideSn,β)ι/d47/d47
+φ
+/d15/d15M0,2(X,β)
+φ
+/d15/d15
+/tildewideSn×Sn/tildewideSnι/d47/d47X×NX
+The action of the Hecke algebra HtonH∗
+T′(/tildewideSn) is induced by the pullback map
+ι∗:H∗
+T′(X×NX)→H∗
+T′(/tildewideSn×Sn/tildewideSn).
+Since Theorem 1.1 identifies the action of the operator given byφ∗[M0,2(X,β)]vir, the
+corresponding result for /tildewideSnfollows from the equality
+φ∗[M0,2(/tildewideSn,β)]vir=φ∗ι![M0,2(X,β)]vir=ι∗(φ∗[M0,2(X,β)]vir).
+/square
+10.Appendix: second cohomology of Springer fibers
+In this appendix we would like to prove the following theorem :
+Theorem 10.1. LetGbe a simply laced semi-simple group and let as before f:
+T∗B→Nbe the Springer map. Then for any non-regular n∈Nthe restriction
+mapH2(B,Z)→H2(f−1(n),Z)is an isomorphism (note that f−1(n)is embedded in
+Bby the definition of f).
+Theorem 10.1 implies that the restriction map H2(T∗B,Z)→H2(/tildewideSn,Z) is an iso-
+morphism as was announced in the remark after Theorem 1.3.
+Proof.Without loss of generality we may assume that Gis almost simple. We shall
+also write Bninstead of f−1(n). LetOrdenote the regular orbit in N(i.e. the unique
+open orbit) and let Osr⊂Ndenote the subregular nilpotent orbit (i.e. the unique orbi t
+which has codimension 2 in N). Let us first assume that n∈Osr. Then it is well known
+thatf−1(n) is a tree of P1’s of degrees corresponding to all α∨∈Π∨⊂Λ∨; this implies
+thatH2(Bn,Z) = Λ∨andH2(Bn,Z) = Λ. Since H2(B,Z) = Λ the statement of the
+Theorem follows.
+For general n∈Nour strategy will be as follows. First we are going to show tha t
+the natural map H2(Bn,Z)→H2(B,Z) is surjective. Then we going to show that the
+natural map H2(Bn,C)→H2(B,C) is an isomorphism. On the other hand, it is shown
+in [12] that H∗(Bn,Z) has no torsion, which implies the statement of Theorem 10.1 .
+To prove the surjectivity of the map H2(Bn,Z)→H2(B,Z) we are going to show
+that for every simple coroot α∨∈Π∨there exists an embedding P1→Bn, which has
+degreeα∨inB. The required surjectivity follows since H2(B,Z) = Λ∨is generated
+by simple coroots. To show the existence of such a P1let us denote by Pαthe moduli
+space of subminimal parabolics corresponding to α. Then all the P1’s of degree α∨
+inBare fibers of the projection pα:B→Pα. Sincenis not regular, it lies in theQUANTUM COHOMOLOGY OF THE SPRINGER RESOLUTION 33
+closure of Osr. Thus there exists a curve Cwith a point c∈Cand a map η:C→N
+such that η(x)∈Osrifx/\e}atio\slash=candη(c) =n. For any x/\e}atio\slash=cthe Springer fiber Bη(x)
+contains unique P1of degree α∨. In other words, we get a map φ:C\{c} →Pαsuch
+p−1
+α(φ(x))⊂Bη(x). SincePαis proper, the map φextends to the whole of Cand we
+are going to have p−1
+α(φ(c))⊂Bφ(c). Sinceφ(c) =n, we are done.
+To prove that the map H2(Bn,C)→H2(B,C) is an isomorphism, let us consider
+the Springer sheaf Spr = f∗(C[dimN]) (here f∗denotes the derived direct image in
+the category of constructible sheaves). It is well-known (c f. [9]) that it is perverse
+and semi-simple. Moreover, its irreducible direct summand s are all of the form IC( E)
+whereEis aG-equivariant local system on a G-orbitO⊂Nand IC(E) stands for its
+Goresky-Macpherson extension to N. We shall write COfor the constant local system
+onOwith fiber C. Also, for any complex of sheaves FonNwe shall denote by Fnthe
+fiber ofFatn; this is a complex of vector spaces over C.
+Let now n∈Nwhich is not regular and not subregular and let Obe itsG-orbit.
+ThenH2(Bn,C) is the cohomology of the fiber of Spr at nof degree 2 −dimN. Let
+O′⊂Nbe aG-orbit such that ¯Osr⊃¯O′⊃Oand letEbe a local system on O′. Then
+we claim that H2−dimN(IC(E)n) is 0 unless O′=Osr. Indeed, this follows from the
+definition of IC-sheaves and from the fact that if O′/\e}atio\slash=Osrthen dim O′<dimN−2.
+Thus in order the only contribution to H2−dimN(IC(E)n) may come from IC( E) where
+Eis a local system either on OrorOsr. It is well-known that the only EonOrsuch that
+IC(E) appears in Spr is the constant local system and in this case I C(E) is constant
+onNand therefore does not contribute anything to H2−dimN(IC(E)n). Also, if Eis an
+irreducible local system on Osrsuch that IC( E) appears in Spr, then Eis constant and
+Hom(IC( COsr[dimOsr]),Spr) =t∗3. On the other hand, H2−dimN(IC(COsr)) =C¯Osr.
+4ThusH2−dimN(IC(COsr)n) =Cand hence H2−dimN(Sprn) =t∗=H2(B,C).
+/square
+References
+[1] K. Behrend and B. Fantechi, The intrinsic normal cone , Invent. Math. 128(1997), 45–88.
+[2] R.Bezrukavnikov, Noncommutative counterparts of the Springer resolution , International Congress
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+arXiv:1001.0058v1  [math.NT]  31 Dec 2009A SUP-HODGE BOUND FOR EXPONENTIAL SUMS
+CHUNLEI LIU
+Abstract. TheC-function of T-adic exponential sums is studeid. An
+explicit arithmetic bound is established for the Newton pol ygon of the
+C-function. This polygon lies above the Hodge polygon. It giv es a
+sup-Hodge bound of the C-function of p-power order exponential sums.
+1.Introduction
+Letpbe a prime number, Fp=Z/(p),Fpa fixed algebraic closure of Fp,
+andFpkthe subfield of Fpwithpkelements.
+Letq >1beapowerof p,WtheringschemeofWittvectors, Zq=W(Fq),
+Qqthe fraction field of Zq,Qp= lim
+→
+kQpk, andCpthep-adic completion of
+Qp.
+Let△/supersetnoteql{0}be an integral convex polytope in Rn, andIthe set of
+vertices of △different from the origin. Let
+f(x) =/summationdisplay
+u∈△(auxu,0,0,···)∈W(Fq[x±1
+1,···,x±1
+n]) with/productdisplay
+u∈Iau/\⌉}atio\slash= 0,
+wherexu=xu1
+1···xunnifu= (u1,···,un)∈Zn.
+Definition 1.1. Fork∈N, the sum
+Sf(k,T) =/summationdisplay
+x∈(F×
+qk)n(1+T)TrZqk/Zp(f(x))∈Zp[[T]]
+is call a T-adic exponential sum. And the function
+Lf(s,T) = exp(∞/summationdisplay
+k=1Sf(k,T)sk
+k)∈1+sZp[[T]][[s]],
+as a power series in the single variable swith coefficients in the T-adic
+complete field Qp((T)), is called an L-function of T-adic exponential sums.
+Letmbe a positive integer, ζpma primitive pm-th root of unity, and
+πm=ζpm−1. Then Sf(k,πm) is the exponential sum studied by Liu-Wei
+[LWe]. If m= 1, the exponential sum Sf(k,πm) was studied by Adolphson-
+Sperber[AS,AS2]. And, if n= 1, theexponential sum Sf(k,πm) wasstudied
+by Kumar-Helleseth-Calderbank [KHC] and Li [Li].
+This research is supported by NSFC Grant No. 10671015.
+12 CHUNLEI LIU
+Definition 1.2. The function
+Cf(s,T) =Cf(s,T;Fq) = exp(∞/summationdisplay
+k=1−(qk−1)−nSf(k,T)sk
+k),
+as a power series in the single variable swith coefficients in the T-adic
+complete field Qp((T)), is called a C-function of T-adic exponential sums.
+We have
+Lf(s,T) =n/productdisplay
+i=0Cf(qis,T)(−1)n−i+1(n
+i),
+and
+Cf(s,T) =∞/productdisplay
+j=0Lf(qjs,T)(−1)n−1(n+j−1
+j).
+So theC-function Cf(s,T) and the L-function Lf(s,T) determine each
+other. From the last identity, one sees that
+Cf(s,T)∈1+sZp[[T]][[s]].
+LetC(△) be the cone generated by △, andM(△) =M(△)∩Zn. There
+is a degree function deg on C(△) which is R≥0-linear and takes the values
+1 on each co-dimension 1 face not containing 0. For a/\⌉}atio\slash∈C(△), we define
+deg(a) = +∞.
+Definition 1.3. A convex function on [0,+∞]which is linear between con-
+secutive integers with initial value 0is called the infinite Hodge polygon of △
+if its slopes between consecutive integers are the numbers deg(a),a∈M(△).
+We denote this polygon by H∞
+△.
+Liu-Wan [LWa] also proved the following.
+Theorem 1.4 (Hodge bound) .We have
+T−adic NP of Cf(s,T)≥ordp(q)(p−1)H∞
+△,
+where NP is the short for Newton polygon.
+Denote by ⌈x⌉the least integer equal or greater than x, and by {x}the
+fractional part of x.
+Definition 1.5. LetC⊆M(△)be a finite subset. We define
+rC= max
+β(#{a∈C| {deg(pa)}′≥β}−#{a∈C| {deg(a)}′≥β}).
+Definition 1.6. Leta⊆M(△). We define
+̟(a) =⌈pdeg(a)⌉−⌈deg(a)⌉+r{a∈M(△)|deg(u)<deg(a)}∪{a}−r{a∈M(△)|deg(u)<deg(a)}.
+Definition 1.7. The arithmetic polygon p△of△is a convex function on
+[0,+∞]which is linear between consecutive integers with initial v alue0,
+and whose slopes between consecutive integers are the numbe rs̟△(a),a∈
+M(△).A SUP-HODGE BOUND FOR EXPONENTIAL SUMS 3
+One can prove the following.
+Theorem 1.8. We have then
+p△≥(p−1)H∞
+△.
+Moreover, they coincide at the point n!Vol(△).
+LetDbe the least positive integer such that deg( M(△))⊆1
+DZ. The
+main result of this paper is the following.
+Theorem 1.9. Ifp >3D, then
+T−adic NP of Cf(s,T)≥ordp(q)p△.
+From the above theorem we shall deduce the following.
+Theorem 1.10. Ifp >3D, then, for t∈Cpwith0/\⌉}atio\slash=|t|p<1, we have
+t−adic NP of Cf(s,t)≥ordp(q)p△.
+2.TheT-adic Dwork Theory
+In this section we review the T-adic analogue of Dwork theory on expo-
+nential sums.
+Let
+E(t) = exp(∞/summationdisplay
+i=0tpi
+pi) =+∞/summationdisplay
+i=0λiti∈1+tZp[[t]]
+be thep-adic Artin-Hasse exponential series. Define a new T-adic uni-
+formizer πofQp((T)) by the formula E(π) = 1 +T. Letπ1/Dbe a fixed
+D-th root of π. Let
+L={/summationdisplay
+u∈M(△)cuπdeg(u)xu:cu∈Zq[[π1/D]]}.
+Leta/mapsto→ˆabe the Teichm¨ uller lifting. One can show that the series
+Ef(x) :=/productdisplay
+au/ne}ationslash=0E(πˆauxu)∈L.
+Note that the Galois group of QqoverQpcan act on Lbut keeping π1/Das
+well as the variable xfixed. Let σbe the Frobenius element in the Galois
+group such that σ(ζ) =ζpifζis a (q−1)-th root of unity. Let Ψ pbe the
+operator on Ldefined by the formula
+Ψp(/summationdisplay
+i∈M(△)cixi) =/summationdisplay
+i∈M(△)cpixi.
+Then Ψ := σ−1◦Ψp◦Efacts on the T-adic Banach module
+B={/summationdisplay
+u∈M(△)ciπdeg(u)xu∈L,ordT(cu)→+∞if deg(u)→+∞}.
+We call it Dwork’s T-adic semi-linear operator because it is semi-linear over
+Zq[[π1
+D]] .4 CHUNLEI LIU
+Letb= logpq. Then the b-iterate Ψbis linear over Zq[[π1/D]], since
+Ψb= Ψb
+p◦b−1/productdisplay
+i=0Eσi
+f(xpi).
+One can show that Ψ is completely continuous in the sense of Se rre [Se]. So
+det(1−Ψbs|B/Zq[[π1
+D]]) and det(1 −Ψs|B/Zp[[π1
+D]]) are well-defined.
+We now state the T-adic Dwork trace formula [LWa].
+Theorem 2.1. We have
+Cf(s,T) = det(1 −Ψbs|B/Zq[[π1
+D]]).
+Lemma 2.2. The Newton polygon of det(1−Ψbsb|B/Zq[[π1
+D]])coincides
+with that of det(1−Ψs|B/Zp[[π1
+D]]).
+Proof.Note that
+det(1−Ψbs|B/Zp[[π1
+D]]) = Norm(det(1 −Ψbs|B/Zq[[π1
+D]])),
+where Norm is the norm map from Zq[[π1
+D]] toZp[[π1
+D]]. The lemma now
+follows from the equality
+/productdisplay
+ζb=1det(1−Ψζs|B/Zp[[π1
+D]]) = det(1 −Ψbsb|B/Zp[[π1
+D]]).
+/square
+Write
+det(1−Ψs|B/Zp[[π1
+D]]) =+∞/summationdisplay
+i=0(−1)icisi.
+Theorem 2.3. TheT-adic Newton polygon of det(1−Ψbs|B/Zq[[π1
+D]])is
+the lower convex closure of the points
+(m,ordT(cbm)), m= 0,1,···.
+Proof.ByLemma2.2,the T-adicNewtonpolygonofdet(1 −Ψbsb|B/Zq[[π1
+D]])
+is the lower convex closure of the points
+(i,ordT(ci)), i= 0,1,···.
+It is clear that ( i,ordT(ci)) is not a vertex of that polygon if b∤i. So that
+Newton polygon is the lower convex closure of the points
+(bm,ordT(cbm)), m= 0,1,···.
+It follows that the T-adic Newton polygon of det(1 −Ψbs|B/Zq[[π1
+D]]) is
+the lower convex closure of the points
+(m,ordT(cbm)), m= 0,1,···.
+/squareA SUP-HODGE BOUND FOR EXPONENTIAL SUMS 5
+3.The arithmetic bound
+In this section we prove the following.
+Theorem 3.1. We have
+ordT(cbm)≥p△(m).
+Proof.First, we choose a basis of B⊗ZpQp(π1/D) overQp(π1/D) as follows.
+Fix a normal basis ¯ξi,i∈Z/(b) ofFqoverFp. Letξibe their Terchm¨ uller
+lift of¯ξi. The system ξi,i∈Z/(b) is a normal basis of QqoverQp. Then
+{ξixu}u∈M(△),1≤i≤bis a basis of B⊗ZpQp(π1/D) overQp(π1/D).
+Secondly, we write out the matrix of Ψ on B⊗ZpQp(π1/D) with respect
+to the basis {ξixu}u∈M(△),1≤i≤b. Write
+Ef(x) =/summationdisplay
+u∈M(△)γuxu,
+and
+σ−1(ξjγpu−w) =b/summationdisplay
+i=1γ(u,i),(w,j)ξi.
+Then
+Ψ(ξjxw) =/summationdisplay
+u∈M(△)σ−1(ξjγu)Ψp(xu+w)
+=/summationdisplay
+u∈M(△)σ−1(ξjγpu−w)xu
+=/summationdisplay
+u∈M(△)b/summationdisplay
+i=1γ(u,i),(w,j)ξixu.
+So (γ(u,i),(w,j))u,w∈M(△),1≤i,j≤bis the matrix of Ψ on B⊗ZpQp(π1/D) with
+respect to the basis {ξixu}u∈M(△),1≤i≤b.
+Thirdly, we claim that
+ordT(γ(u,i),(w,j))≥ ⌈deg(pu−w)⌉.
+In fact, this follows from the equality
+σ−1(ξjγpu−w) =b/summationdisplay
+i=1γ(u,i),(w,j)ξi,
+and the inequality ord T(γu)≥ ⌈deg(u)⌉.
+Finally, we show that
+ordT(cbm)≥p△(m).
+Note that
+cbm=/summationdisplay
+Adet((γ(u,i),(w,j))(u,i),(w,j)∈A),6 CHUNLEI LIU
+whereAruns over all subsets of M(△)×Z/(b) with cardinality bm. So it
+suffices to show that
+ordT(det(γ(i,u),(j,ω))(i,u),(j,ω)∈A)≥bp△(m).
+Note that
+det(γ(i,u),(j,ω))(i,u),(j,ω)∈A) =/summationdisplay
+τ∈SA/summationdisplay
+a∈Aordπ(γa,τ(a)),
+whereSAis the permutation group of A. So it suffices to show that
+/summationdisplay
+a∈Aordπ(γa,τ(a))≥bp△(m), τ∈SA.
+Since
+ordT(γ(u,i),(w,j))≥ ⌈deg(pu−w)⌉,
+the theorem follows from the following. /square
+Theorem 3.2. Ifp >3D,Ais a subset of M(△)×Z/(b)with cardinality
+bm, andτ∈SA, then
+/summationdisplay
+a∈A⌈deg(pν(a)−ν(τ(a)))⌉ ≥bp△(m),
+whereν(u,i) =u.
+Proof.We have
+/summationdisplay
+a∈A⌈deg(pν(a)−ν(τ(a)))⌉ ≥/summationdisplay
+a∈A⌈deg(pν(a))−ν(τ(a))⌉
+≥/summationdisplay
+a∈A⌈deg(pν(a))⌉−⌈deg(ν(τ(a)))⌉+1{deg(pν(a))}′>{deg(ν(τ(a)))}′
+≥/summationdisplay
+a∈A⌈deg(pν(a))⌉−⌈deg(ν(a))⌉+1{deg(pν(a))}′>{deg(ν(τ(a)))}′.
+Choose a set Bof cardinality |A|such that B∩Ais as big as possible under
+the condition that, for some α,
+{a∈M(△)×Z/(b)|deg(ν(a))< α} ⊆B/subsetnoteql{a∈M(△)×Z/(b)|deg(ν(a))≤α}.
+Choose a permutation τ0onB∩Awhich agrees with τon (B∩A)∩τ−1(B∩
+A). Extend it trivially to B. We have
+/summationdisplay
+a∈A⌈deg(pν(a))⌉−⌈deg(ν(a))⌉ ≥/summationdisplay
+a∈B(⌈deg(pν(a))⌉−⌈deg(ν(a))⌉)+2#(A\B).
+We also have/summationdisplay
+a∈A1{deg(pν(a))}′>{deg(ν(τ(a)))}′≥/summationdisplay
+a∈B1{deg(pν(a))}′>{deg(ν(τ0(a)))}′−2#(A\B).
+It follows that/summationdisplay
+a∈A⌈deg(pν(a))⌉−⌈deg(ν(a))⌉+1{deg(pν(a))}′>{deg(ν(τ(a)))}′A SUP-HODGE BOUND FOR EXPONENTIAL SUMS 7
+≥/summationdisplay
+a∈B⌈deg(pν(a))⌉−⌈deg(ν(a))⌉+1{deg(pν(a))}′>{deg(ν(τ0(a)))}′
+≥/summationdisplay
+a∈B(⌈deg(pν(a))⌉−⌈deg(ν(a))⌉)+rB,
+where
+rB= max
+β(#{a∈B| {deg(pν(a))}′≥β}−#{a∈B| {deg(ν(a))}′≥β}).
+Choose a set Cof cardinality msuch that for some α,
+{a∈M(△)|deg(ν(a))< α} ⊆C/subsetnoteql{a∈M(△)|deg(ν(a))≤α}.
+Recall that
+rC= max
+β(#{a∈C| {deg(pa)}′≥β}−#{a∈C| {deg(a)}′≥β}).
+It is easy to see that rB=brC, and
+/summationdisplay
+a∈B(⌈deg(pν(a))⌉−⌈deg(ν(a))⌉)+rB
+=b/summationdisplay
+a∈C(⌈deg(pa)⌉−⌈deg(a)⌉)+brC=bp△(m).
+The theorem now follows. /square
+References
+[AS] A. Adolphson and S. Sperber, Newton polyhedra and the de gree of the L-function
+associated to an exponential sum, Invent. Math. 88(1987), 5 55-569.
+[AS2] A. Adolphson and S. Sperber, Exponential sums and Newt on polyhedra: cohomol-
+ogy and estimates, Ann. Math., 130 (1989), 367-406.
+[BF] R. Blache and E. F´ erard, Newton straitification for pol ynomials: the open stratum,
+J. Number Theory, 123(2007), 456-472.
+[KHC] P. V. Kumar, T. Helleseth and A. R. Calderbank, An upper bound for some
+exponential sums over Galois rings with applications, IEEE Trans. Inform. Theory
+41 (1995), no.2, 456-468.
+[Li] W.-C. W. Li, Character sums over p-adic fields, J. Number Theory 74 (1999), no.2,
+181-229.
+[LWa] C. Liu and D. Wan, T-adic exponential sums, Algebra & Number theory, Vol. 3,
+No. 5 (2009), 489-509.
+[LWe] C. Liu and D. Wei, The L-functions of Witt coverings, Math. Z., 255 (2007), 95-115.
+[Se] J-P. Serre, Endomorphismes compl´ etement continus de s espaces de Banach p-adiques,
+Publ. Math., IHES., 12(1962), 69-85.
+[Zh1] J. H. Zhu, p-adic variation of L functions of one variab le exponential sums, I. Amer.
+J. Math., 125 (2003), 669-690.
+[Zh2] J. H. Zhu, Asymptotic variation of L functions of one-v ariable exponential sums, J.
+Reine Angew. Math., 572 (2004), 219–233. 1529–1550.
+Department of Mathematics, ShanghaiJiao Tong University, Shanghai200240,
+P.R. China.
+E-mail address :clliu@sjtu.edu.cn
\ No newline at end of file
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+arXiv:1001.0059v1  [astro-ph.CO]  31 Dec 2009TANAMI: Milliarcsecond Resolution Observations of Extrag alactic Gamma-ray
+Sources
+Roopesh Ojha1,2, Matthias Kadler3,4,5,6, M. B¨ ock3,4, R. Booth7, M. S. Dutka8, P. G. Edwards9, A. L. Fey1,
+L. Fuhrmann10, R. A. Gaume1, H. Hase11, S. Horiuchi12, D. L. Jauncey9, K. J. Johnston1, U. Katz4,
+M. Lister13, J. E. J. Lovell14, Cornelia M¨ uller3,4, C. Pl¨ otz15, J. F. H. Quick7, E. Ros10,16, G. B. Taylor17,
+D. J. Thompson18, S. J. Tingay19, G. Tosti20,21, A. K. Tzioumis9, J. Wilms3,4, J. A. Zensus10
+1United States Naval Observatory,
+3450 Massachusetts Ave., NW,
+Washington DC 20392, USA
+2NVI, Inc., 7257D Hanover Parkway,
+Greenbelt, MD 20770, USA
+3Dr. Remeis-Sternwarte, Sternwartstr. 7,
+96049 Bamberg, Germany
+4ECAP, Universit¨ at Erlangen-N¨ urnberg, Germany
+5CRESST/NASA Goddard Space Flight Center,
+Greenbelt, MD 20771, USA
+6USRA, 10211 Wincopin Circle,
+Suite 500 Columbia, MD 21044, USA
+7Hartebeesthoek Radio Astronomy Observatory,
+PO Box 443, Krugersdorp 1740, South Africa
+8The Catholic University of America,
+620 Michigan Ave., N.E.,
+Washington, DC 20064, USA
+9CSIRO Astronomy and Space Science,
+PO Box 76, Epping, NSW 1710 Australia
+10MPIfR, Auf dem H¨ ugel 69, 53121 Bonn, Germany
+11BKG, Univ. de Concepcion,
+Casilla 4036, Correo 3, Chile
+12Canberra Deep Space Communication Complex,
+PO Box 1035, Tuggeranong, ACT 2901, Australia
+13Dept. of Physics, Purdue University,
+525 Northwestern Avenue,
+West Lafayette, IN 47907, USA
+14School of Mathematics & Physics,
+Private Bag 37, Univ. of Tasmania,
+Hobart TAS 7001, Australia
+15BKG, Geodetic Observatory Wettzell,
+Sackenrieder Str. 25, 93444 Bad K¨ otzting, Germany
+16Dept. d’Astronomia i Astrof´ ısica,
+Universitat de Val` encia,
+46100 Burjassot, Val` encia, Spain
+17Dept. of Physics and Astronomy,
+University of New Mexico,
+Albuquerque NM, 87131, USA
+18Astrophysics Science Division,
+NASA Goddard Space Flight Center,
+Greenbelt, MD 20771, USA
+19Curtin Institute of Radio Astronomy,
+Curtin University of Technology,
+Bentley, WA, 6102, Australia
+20Istituto Nazionale di Fisica Nucleare,
+Sezione di Perugia, 06123 Perugia, Italy
+21Dipartimento di Fisica,
+Universit` a degli Studi di Perugia, 06123 Perugia, Italy
+The TANAMI (Tracking AGN with Austral Milliarcsecond Inter ferometry) and associated pro-
+grams provide comprehensive radio monitoring of extragala ctic gamma-ray sources south of decli-
+nation−30 degrees. Joint quasi-simultaneous observations betwee n the Fermi Gamma-ray Space
+Telescope and ground based observatories allow us to discri minate between competing theoretical
+blazar emission models. High resolution VLBI observations are the only way to spatially resolve the2 Early TANAMI Results
+sub-parsec level emission regions where the high-energy ra diation originates. The gap from radio
+to gamma-ray energies is spanned with near simultaneous dat a from the Swift satellite and ground
+based optical observatories. We present early results from the TANAMI program in the context of
+this panchromatic suite of observations.
+I. INTRODUCTION
+Near simultaneous observations across the electro-
+magnetic spectrum are essential to further our un-
+derstanding of the physics of active galactic nuclei
+(AGN). Milliarcsecond resolution Very Long Baseline
+Interferometry (VLBI) observations make a unique
+contribution as they are the only way to resolve the
+regions where the high-energy emission originates.
+VLBI monitoring of AGN jets is the only way to di-
+rectly measure their relativistic motion and calculate
+intrinsic jet parameters. VLBI observations let us
+probe the conditions under which blazars and non-
+blazars emit γ-rays.
+The TANAMI ( TrackingActive Galactic Nuclei
+withAustralMilliarcsecond Interferometry) program
+([14, 15]) provides parsec scale resolution monitoring
+of extragalactic gamma-ray sources south of −30 de-
+grees declination at dual frequency (8.4 and 22GHz)
+by making VLBI observations with the Australian
+Long Baseline Array (LBA[21]) and associated tele-
+scopes in Australia, Antarctica, Chile and South
+Africa. Observations are made at intervals of about
+two months. TANAMI observations are comple-
+mented by arcsecond resolution monitoring across the
+radiospectrum with the AustraliaTelescope Compact
+Array (PI: S. Tingay) and single-dish resolution radio
+monitoring with the Hobart and Ceduna telescopes of
+the University of Tasmania (PI: J. Lovell).
+TANAMI began observations in November 2007
+with an initial sample of 43 sources consisting of
+a radio selected flux-density limited subsample and
+aγ-ray selected subsample of known and candidate
+EGRET detections. This initial sample has since
+been expandedtoinclude newdetections bythe Fermi
+Large Area Telescope (LAT). Details of the sample
+and observations are presented in [13]. Here we out-
+line some early results from the TANAMI program.
+II. PKS1454-354
+On September 4th, 2008, a strong γ-ray flare was
+detected by the Large Area Telescope (LAT) coming
+from the direction of the flat-spectrum radio quasar
+PKS1454 −354 (z= 1.424). This quasar was a pos-
+sible counterpart of the unidentified EGRET source
+3EG J1500 −3509 [6]. The flux rose on a timescale
+of hours before dropping over the following two days.
+TANAMI images provided the first high-resolution,
+high-sensitivity parsec-scale image of the blazar jet
+for the first LAT AGN paper reporting this detectionFIG. 1: Naturally weighted TANAMI image of PKS1454-
+354 at 8.4GHz. The image has an rms noise of ∼0.08 mJy
+beam−1. The hatched ellipse on the bottom left shows the
+restoring beam of 2 .87×0.75 mas at 4◦.
+and its multi-wavelength follow-up [1].
+Figure 1 shows a naturally weighted image of
+PKS1454 −354 observed at 8.4GHz on Nov 10th,
+2007. The image achieves a resolution of 2 .87×0.75
+mas at an rms noise of ∼0.08 mJy beam−1. It shows
+a compact core and a single-sided jet extending to
+∼50 mas at a position angle of ∼300, confirming the
+activity of this source. Model fitting the core with
+an elliptical Gaussian yields a brightness temperature
+limit ofTB= 1.6×1011K. The activity of this source
+continues to be monitored by TANAMI at unprece-
+dented resolution and dynamic range.
+III. CENTAURUS A
+Atadistanceof3.4Mpc[5],CentaurusAistheclos-
+est known radio galaxy making it possible to study it
+at subparsec scales. Not surprisingly, it has been ex-
+tensively observed at many wavelengths and its basic
+VLBI-scale radio structure is well-known to consist of
+a bright jet and a faint counter-jet at a viewing angle
+of 50−80 degrees [19].
+The TANAMI program has produced one of the
+highest resolution images of an AGN jet ever made
+[8]. The uniformly weighted beam with a resolution
+eConf C091122Early TANAMI Results 3
+FIG. 2: The core region of Centaurus A at 8.4GHz. The
+best view yet of a possible γ-ray birthplace.
+FIG. 3: The core region of Centaurus A at 22GHz. Note
+the close agreement with the core region at 8.4GHz shown
+in the previous figure.
+of 0.68×0.41mas happens to be comparable to the
+resolution (0 .8×0.7mas) of the previous highest reso-
+lution image of this object [7]. Though these observa-
+tions are separated by about a decade the structures
+are remarkably similar which would appear to be con-
+trary to the previously reported apparent velocity of
+∼1.4mas [20]. Velocity information obtained from
+continuing multi-epoch TANAMI observations is be-
+ing used to investigate this.
+Figure 2 and Figure 3 show the core region from
+naturally weighted images of Centaurus A at 8.4GHz
+and 22GHz respectively. In both cases the data have
+been self-calibrated using the CLEAN algorithm and
+the clean components in the final model have been
+replaced by Gaussian model components in order to
+parametrize the core region. There is close agree-
+ment between these two figures suggesting that the
+“core” has the same size at both frequencies. Due to
+the presence of trans-oceanic baselines to the TIGO(Chile) and O’Higgins (Antarctica) telescopes only at
+8.4GHz, the lowerfrequencyimagehasthe higherres-
+olution, resolving the core and providing the finest
+view of the putative γ-ray production region seen so
+far. These simultaneous dual-frequency data are part
+of a set of simultaneous broadband data being used to
+model the behavior of Centaurus A [3].
+IV. FIRST EPOCH RESULTS
+First epoch images at 8.4GHz of all 43 sources in
+the initial TANAMI sample have been analyzed and
+theresultsarepresentedin [15]. Theimageshavehigh
+dynamic range and are often the best and sometimes
+the first images of an AGN at milliarcsecond scale
+resolution. Here we conclude this presentation with
+a few highlights from this analysis. Unless explicitly
+stated all discussions below are made with reference
+to the LAT 3-month list [2].
+•Morphology and the Fermiconnection
+Figure 4 shows four sources which represent
+the four morphological classes we have classi-
+fied the TANAMI sample into. All the images
+aremadewithnaturallyweighted,8.4GHzdata.
+The axes are labeled in units of milliarcseconds.
+The restoring beam of each image is shown as
+a hatched ellipse on the bottom left and, for
+the three sources that have known redshifts, a
+bar indicating a linear scale of 10pc or 1pc is
+shown on the bottom right. The root-mean-
+square (rms) noise in the images is typically
+about∼0.5 mJy beam−1. The lowest contour
+level is at 3 times the root-mean-square noise
+level.
+The classification scheme we have adopted is
+that of [9] which makes no assumptions about
+the physical nature of the objects thus sepa-
+rating their description from their interpreta-
+tion. Clockwise from the top left, are examples
+of single-sided (SS), compact (C), double-sided
+(DS) and Irregular(Irr) morphology. The initial
+TANAMI sample has 33 SS sources and 5 DS
+sources with just one example each of the other
+two morphological types. Three sources do not
+have an optical identification. All of the quasars
+and BL Lacertae objects in the sample have an
+SS morphology while all 5 DS sources are galax-
+ies. The lone C source is optically unidenti-
+fied while the only Irr source is a GPS galaxy
+1718−649 which is tentatively detected by LAT
+[4].
+In its first three months of operation Fermide-
+tected 12 of 43 TANAMI sources with >10σ
+significance. All but one of these sources are
+quasars and BLLacertae objects with an SS
+eConf C0911224 Early TANAMI Results
+20 0 −20 −40 −60−200204060
+10 pc0521−365 2007−11−10
+40 20 0 −20 −40−40−2002040
+1501−343 2007−11−10
+20 10 0 −10−1001020
+1 pc1718−649 2008−02−07
+40 20 0 −20 −40−40−2002040
+10 pc1549−790 2008−02−07
+FIG. 4: TANAMI first epoch 8.4GHz images illustrating the diff erent morphological types in the sample, see text for
+description. The scale of both axes of each image is in millia rcseconds and the restoring beam is shown as a hatched
+ellipse on the bottom left.
+morphology (the exception is the nearby radio
+galaxy Centaurus A). This accords well with
+the canonical view that quasars and BL Lacer-
+taeobjectshaveanintrinsicallysymmetrictwin-
+jet structure that is transformed by differential
+Doppler boosting to the observed core-jet mor-
+phology.
+•Are opening angles correlated with γlu-
+minosity?
+Given the expectation that the γ-ray emission
+from AGN is beamed and thus orientation de-
+pendent, a link between γ-ray emission and
+the parsec scale morphology of AGN has been
+sought (e.g. [18]). We fit circular Gaussians to
+the visibility data and measured the angle at
+which the innermost jet component appears rel-
+ative to the position of the core i.e. the opening
+angle. Of the LAT AGN Bright Sample (LBAS)
+sources78% have an opening angle >30 degreeswhile only 27% of non-LBAS sources do. In
+other words, γ-ray bright jets are pointed closer
+to the line of sight than γ-rayfaint jets. This re-
+sult should be treated with great caution as the
+sample size for this analysis is currently small
+but [16] report similar results.
+If confirmed the above result presents two pos-
+sibilities: either the LBAS jets have smaller
+Lorentz factors (since the width of the rela-
+tivistic beaming cone ∼1/Γ) or LBAS jets are
+pointed closer to the line of sight. The former
+scenario appears unlikely, indeed the opposite
+effect is reported by [11, 12].
+•Redshift distribution
+The redshift distribution of the quasars and
+BLLacs in the TANAMI sample is similar to
+those for the LBAS and EGRET blazars. There
+does not appear to be any significant difference
+between the radio- and γ-ray selected subsam-
+eConf C091122Early TANAMI Results 5
+ples.
+•Brightness Temperature
+The core brightness temperature ( TB) limit of
+all initial TANAMI sources was calculated. The
+high end ofthe distribution of calculated bright-
+ness temperatures is dominated by quasars and
+the low end by BLLacertaeobjects and galaxies.
+Of the 43 sources in the sample, 13 have a max-
+imumTBbelow the equipartition value of 1011
+K [17], 29 below the inverse Compton limit of
+1012K [10], putting about a third of the values
+above this limit. Doppler boosting is the most
+likely reason behind these high values though a
+variety of exotic mechanisms are also possible.
+There is no significant difference in the bright-
+nesstemperaturedistribution ofLBAS and non-
+LBAS sources. Many of the highest brightness
+temperature sources are not detected by LAT
+yet which is counterintuitive since they are ex-
+pected to have higher Doppler factors.
+•Luminosities
+The core and the total luminosity was calcu-
+lated for all 38 initial TANAMI sourcesthat had
+publishedredshifts,assumingisotropicemission.
+There is no significant difference in the distri-
+bution of luminosities of LBAS and non-LBAS
+sources. On the other hand, there is a clear re-
+lationship between luminosity and optical type
+with quasars dominating the high luminosity
+end of the distribution, galaxies dominating the
+low luminosity end while the BL Lacertae ob-
+jects fall in between,
+None of the five most distant and most lumi-
+nous sources have been detected by Fermi in its
+first3months ofoperation. Intriguingly, noneof
+the nine most luminous jets (difference of total
+and core luminosities) are detected. If this per-
+sists it would suggest a completely unexpected
+anti-correlation between jet luminosity and γ-
+brightness.V. CONCLUSION
+The TANAMI program is providing high quality,
+highresolutionradiomonitoringforthesouthernthird
+of the sky. Such data is a crucial member of the suite
+of multiwavelength observations that is set to revolu-
+tionize our understanding of the physics of AGN in
+the era of Fermi.
+Data from the TANAMI program is already a part
+of severalindividual AGN studies and statistical anal-
+ysis of the growing sample of TANAMI sources is pro-
+vidinginsightintothenatureandoriginofhighenergy
+radiation from AGN. The availability of core and jet
+component spectra as well as proper motions is going
+to greatly expand the questions that TANAMI can
+help answer.
+VI. ACKNOWLEDGMENTS
+We are grateful to Dirk Behrend, Neil Gehrels,
+Julie McEnery, David Murphy, and John Reynolds,
+who contributed in various ways to the success of the
+TANAMI program so far. Furthermore, we thank the
+Fermi/LAT AGN group for the good collaboration.
+This research has been partially funded by the Bun-
+desministerium f¨ ur Wirtschaft und Technologie under
+Deutsches Zentrum f¨ ur Luft- und Raumfahrt grant
+number 50OR0808. This research has made use of the
+United States Naval Observatory (USNO) Radio Ref-
+erenceFrameImageDatabase(RRFID).Thisresearch
+has made use of data from the NASA/IPAC Extra-
+galactic Database (NED, operated by the Jet Propul-
+sion Laboratory, California Institute of Technology,
+under contract with the National Aeronautics and
+Space Administration); and the SIMBAD database
+(operated at CDS, Strasbourg, France). This research
+has made use of NASA’s Astrophysics Data System.
+[1] Abdo, A., A. et al. 2009 ApJ, 697, 934
+[2] Abdo A. A., et al., 2009, ApJ 700, 597
+[3] Abdo A. A., et al., 2010, In Preparation.
+[4] B¨ ock et al. 2009, These Proceedings
+[5] Ferrarese L. et al. 2007, ApJ, 654, 186
+[6] Hartman, R. C. et al. 1999, ApJS, 123, 79
+[7] Horiuchi, S., Meier, D., Preston, R., Tingay, S. 2005
+PASJ, 58, 211
+[8] Kadler, M. et al. 2010, In Preparation.
+[9] KellermannK.I., VermeulenR.C., ZensusJ.A., Cohen
+M.H. 1998, AJ 115, 1295
+[10] KellermannK.I., Pauliny-Toth, I.I.K.1969, ApJ155,
+L71[11] Kovalev Y.Y., Aller H.D., Aller M.F., et al., 2009,
+ApJ 696, L17
+[12] Lister M.L., Aller H.D., Aller M.F., et al., 2009a, AJ
+137, 3718
+[13] M¨ uller et al. 2009, These Proceedings
+[14] Ojha R. et al., 2008, AIPC, 1053, 395
+[15] Ojha R. et al., 2010, A&A, Submitted
+[16] Pushkarev, A. B., Kovalev, Y. Y., Lister, M. L.,
+Savolainen, T. 2009, A&A, 507, L33
+[17] Readhead, A. C. S. 1994, ApJ, 426, 51
+[18] Taylor, G. B. et al. 2007, ApJ, 671, 1355
+[19] Tingay, S. J. et al. 1998, AJ, 115, 960
+[20] Tingay, S. J., Preston, R. A., Jauncey, D. L. 2001,
+eConf C0911226 Early TANAMI Results
+AJ, 122, 1697
+[21] The Long Baseline Array is part of the Australia Tele-
+scope which is funded by the Commonwealth of Aus-tralia for operation as a National Facility managed by
+CSIRO.
+eConf C091122
\ No newline at end of file
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+arXiv:1001.0060v1  [hep-th]  31 Dec 2009Disk relations for tree amplitudes in minimal coupling theo ry of
+gauge field and gravity
+Yi-Xin Chen,∗Yi-Jian Du,†and Qian Ma‡
+Zhejiang Institute of Modern Physics,
+Zhejiang University, Hangzhou 310027, P. R. China
+(Dated: November 8, 2018)
+Abstract
+KLT relations on S2factorize closed string amplitudes into product of open str ing tree ampli-
+tudes. The field theory limits of KLT factorization relation s hold in minimal coupling theory of
+gauge fieldandgravity. Inthispaper, weconsiderthefieldth eory limits of relations on D2. Though
+the relations on D2and KLT factorization relations hold on worldsheets with di fferent topologies,
+we find the field theory limits of D2relations also hold in minimal coupling theory of gauge field
+and gravity. We use the D2relations to give three- and four-point tree amplitudes whe re gluons are
+minimally coupled to gravitons. We also give a discussion on general tree amplitudes for minimal
+coupling of gauge field and gravity. In general, any tree ampl itude with Mgravitons in addition
+toNgluons can be given by pure-gluon tree amplitudes with N+2Mlegs.
+PACS numbers: 04.60.Cf, 11.25.Db, 11.15.Bt
+Keywords: Gauge-gravity correspondence.
+∗Corresponding author; Electronic address: yxchen@zimp.zju.edu .cn
+†Electronic address: yjdu@zju.edu.cn
+‡Electronic address: mathons@gmail.com
+1I. INTRODUCTION
+Superstring theories are theories containing both gravitational a nd gauge interactions[1,
+2]. They offer a possible way to unify the gauge field and gravity. In st ring theory, gravitons
+and gauge particles can correspond to massless states of closed a nd open strings. Then the
+relations between closed and open strings imply the relations betwee n gravity and gauge
+field.
+There are many relationships between gauge field and gravity in strin g theory such as
+AdS/CFT[3] correspondence and the perturbative relations for a mplitudes on sphere( S2)[4],
+disk(D2)[5–10] and real projective plane( RP2)[5]. The perturbative relations in S2case
+named KLT relation[4] factorize the amplitudes on S2into products of two amplitudes
+for open strings corresponding to the left- and the right-moving s ectors of closed strings.
+However, the amplitudes on D2cannot be factorized into two sectors, because the boundary
+ofD2connect the two sectors into a single one. We should use the new rela tions instead
+of KLT factorization relations on D2. KLT factorization relations and D2relations hold on
+worldsheets with different topologies.
+In field theory limit, KLT factorization relations allow one to obtain gra vity amplitudes
+from gauge theory ones[11–16]. They can also be used in many theor ies of gravity-matter
+couplings[17, 18]. An important application is that the low energy limits o f KLT relations
+can be used to calculate the tree amplitudes for gluons minimally couple d to gravitons. In
+this case, using the relations, the amplitudes with gluons and gravito ns can be factorized
+into products of amplitudes in left- and right-moving sectors. Amplit udes in one sector are
+pure gauge partial amplitudes while those in the other sector are pa rtial amplitudes with
+gluons and scalars. These relations for gauge-gravity minimal coup ling are based on the
+2structure of heterotic strings[1, 2, 19–21]. In fact, in heterotic string theories, gauge degrees
+of freedom are taken by Lorentz singlets in one sector of closed st rings.
+There is another way to incorporate gauge degrees of freedom int o string theory. In
+theories containing open strings such as Type I theory, one can ad d Chan-Paton factors[1,
+2, 22, 23] to the ends of open strings. The interactions between c losed and open strings at
+tree-level are on D2. In the field theory limits of this case, the tree amplitudes for gauge -
+gravity interaction and those for pure gauge field are connected v iaD2relations. Then a
+question arises: In what kind of theory for gauge-gravity interac tions do the field limits
+ofD2relations hold? Or do the field theory alimits of D2relations also hold in minimal
+coupling theory of gauge and gravity?
+In this paper, we study the amplitudes in minimal coupling theory of ga uge field and
+gravity. We find though the KLT factorization relations and the D2relations hold on
+worldsheets with different topologies in string theory, the field theo ry limits of D2relations
+also hold in minimal coupling theory of gauge field and gravity. D2relations in minimal
+coupling theory of gauge field and gravity are based on the disk stru cture. They give a new
+understanding on gauge-gravity interaction.
+The structure of this paper is as follows. In Section II, we give an int roduction to KLT
+relations and D2relations. We also give the low energy limits of D2relations for amplitudes
+with three and four legs. In Section III, we use the low energy limits o fD2relations to
+give the tree amplitudes for gauge-gravity interaction. We find the amplitudes given by D2
+relations are same with those given by KLT factorization relations. T hen theD2relations
+give the amplitudes in minimal coupling theory of gauge field and gravity . In Section IV, we
+consider the D2relations for general tree amplitudes where gluons are minimally coup led to
+gravitons. We first study the mixed amplitudes where all the legs tak e positive helicity or
+3only one leg take negative helicity. In these cases, the D2relations hold trivially. Then we
+study the tree amplitudes with one and two gravitons in addition to Ngluons where N−2
+gluons as well as all the gravitons take positive helicity and two gluons take negative helicity.
+We will show the D2relations also hold in this case. The discussions can be extended to
+tree amplitudes where Ngluons minimally coupled to Mgravitons with arbitrary helicity
+configurations. In general, any tree amplitudes for gauge-gravit y minimal coupling can be
+expressed by partial tree amplitudes with N+2Mgluons via D2relations. Our conclusions
+are given in Section V. Some useful properties of spinor helicity form alism are given in
+Appendix A.
+II. KLT RELATIONS VERSUS D2RELATIONS
+KLT relations[4] in string theory are the relations between amplitude s for closed strings
+onS2and open string tree amplitudes. KLT relations factorize amplitudes for closed strings
+onS2into products of two open string tree amplitudes corresponding to the left- and right-
+moving sectors except for a phase factor1
+M(N)
+S2∼κN−2/summationdisplay
+P,P′A(N)(P)¯A(N)(P′)eiπF(P,P′), (1)
+WhereM(N)
+S2isN-pointamplitude on S2whileA(N)and¯A(N)arethepartialtreeamplitudes
+for open strings corresponding to the left- and right-moving sect ors. The phase factor only
+depends on the permutations PandP′of the legs in left- and right-moving sectors. The
+terms in KLT relations can be reduced by contour deformations. In the reduced form, the
+phase factors become sine functions. After taking the field theor y limitα′→0, the KLT
+1In this paper, we use ∼to omit a proportional factor which does not affect our discussion.
+4relations for three- and four-point amplitudes are given as
+M(1,2,3)∼κA(1,2,3)¯A(1,2,3), (2a)
+M(1,2,3,4)∼κ2(−i)s12A(1,2,3,4)¯A(1,2,4,3). (2b)
+In field theory limits, KLT relations factorize the pure-graviton tre e amplitudes into
+products of tree amplitudes for gluons corresponding to left- and right-moving sectors. The
+tree amplitudes where gravitons are minimally coupled to gluons can als o be factorized by
+KLT relations. In this situation, the two sectors of a graviton with h elicity±2 correspond
+to two gluons with helicity ±1, while the two sectors of a gluon with helicity ±1 correspond
+to one gluon with helicity ±1 and one scalar particle. The gauge degrees of freedom are
+takenbythescalarfieldinonesector. Withthesecorrespondence s, KLTrelationsexpressthe
+amplitudesfor Ngluonsand Mgravitonsbyproductsofamplitudesinleft-andright-moving
+sectors. Theamplitudesintheleft-movingsector arepure-gluonp artialtreeamplitudes with
+N+Mlegswhile the amplitudes inthe right-moving sector arepartialtree a mplitudes for M
+gluons and Nscalar particles. Using the Feynman rules given in [17], one can calcula te the
+partial amplitudes in the left- and right-moving sectors, then the a mplitudes where gluons
+are minimally coupled with gravitons can be given by KLT relations.
+The KLT factorization relations in string theory hold on S2. But they do not hold on D2.
+InD2case, the left- and right-moving sectors of closed strings are con nected into a single
+one[5]. The relations between amplitudes for Nopen strings in addition to Mclosed strings
+onD2and open string tree amplitudes are given as
+M(N,M)
+D2∼gN−2κM/summationdisplay
+PA(N,2M)(P)eiπΘ′(P). (3)
+In the relation (3), we do not introduce the Chan-Paton degrees o f freedom. On D2, one can
+add Chan-Paton factor to the ends of open strings to incorporat e gauge degree of freedom.
+5Then the amplitudes on D2can be given by color decomposed form
+M(1a1
+o,...,NaN
+o,(N+1)c,...,(N+M)c)
+=/summationdisplay
+σTr/parenleftbig
+Taσ(1)...Taσ(N)/parenrightbig
+A(σ(1o),...,σ(No),(N+1)c,...,(N+M)c),(4)
+whereia
+o(i= 1,...,N) denote open string legs with Chan-Paton degrees of freedoms
+andjc(j=N+ 1,...,N+M) denote closed string legs. σruns over the set of non
+cyclic permutations of the open strings. Then D2relations give the partial amplitudes
+A(σ(1o),...,σ(No),(N+1)c,...,(N+M)c) for a given permutation of open strings by pure
+open string amplitudes:
+A(σ(1o),...,σ(No),(N+1)c,...,(N+M)c) =/summationdisplay
+PeiπΘ′(P′′)A(N,2M)(P′′).(5)
+whereP′′are all the permutations of the N+2Mexternal legs which preserve the relative
+positions of the open strings 1 o,...,No. This expression implies that for a given permutation
+of the open strings on the boundary of D2, any closed string can split into two open strings
+inserted on the boundary of D2. Using contour deformations, the relations (5) can also be
+reduced[6]. In the reduced form of the relations, the phase facto rs become sine functions.
+In field theory limits, D2relations give tree amplitudes with Ngluons and Mgravitons
+by pure-gluon amplitudes with N+2Mlegs. They are different from the KLT factorization
+relations which factorize amplitudes with N+Mlegs into products of two amplitudes with
+N+Mlegs. For M= 0,D2relations trivially give the pure-gluon amplitudes. Then we do
+notneedtoconsider M= 0case. Sincethegeneratorsofthegaugegroupsatisfy Tr(Ta) = 0,
+wealsodonotneedtoconsider M= 1case. Aftertakingthefieldtheorylimit α′→0,theD2
+relations for two-gluon one-graviton tree amplitude A(1g,2g,3h), three-gluon one-graviton
+tree amplitude A(1g,2g,3g,4h) and two-gluon two-graviton tree amplitude A(1g,2g,3h,4h)
+6are given as2
+A(1g,2g,3h)∼κs12A(1g,2g,3g,4g), (6a)
+A(1g,2g,3g,4h)∼gκs13A(1g,5g,2g,4g,3g), (6b)
+A(1g,2g,3h,4h)∼κ2[s2
+12A(1g,6g,3g,5g,4g,2g)−s12s13A(1g,3g,5g,4g,2g,6g)],(6c)
+wheresij= 2ki·kj, 3gand 4ghave momentum1
+2k3while 5 gand 6ghave momentum1
+2k4.
+The total amplitude is derived by substitute the relations (6) into Eq . (4).
+So far, we have seen, though KLT and D2relations hold on worldsheets with different
+topologies in string theory. They can both give the amplitudes for ga uge-gravity coupling.
+KLTrelations can given the amplitudes for gauge-gravity minimal cou pling. Then we should
+consider the question: Can the field theory limits of the two different relations in string
+theory hold in a same theory for gauge-gravity coupling? In the nex t two sections, we will
+show the D2relations also hold in minimal coupling theory of gauge field and gravity.
+III.D2RELATIONS FOR THREE- AND FOUR-POINT TREE AMPLITUDES
+In this section, we use the D2relations (6) to give the three- and four-point tree ampli-
+tudes for gauge-gravity coupling. These results are same with tho se given by using KLT
+relations[17]. Thus D2relations also hold in minimal coupling theory of gauge field and
+gravity.
+2In this paper, we use igandjhto denote gluons and gravitons respectively. We do not consider th e
+amplitudes where all the external legs are gravitons. We also do not consider the amplitudes where
+graviton exchanges between gluons, which contribute to higher or der process.
+7A. Three-point tree amplitude
+The only nontrivial three-point partial amplitude needed is A(1g,2g,3h). To give this
+amplitude, we need to calculate the amplitudes A(1g,2g,3g,4g) in which 3 gand 4gcorre-
+spond to the left- and right-moving sectors of the graviton 3 h. Using the color-ordered
+Feynman rules in [24] and replacing the momenta k3andk4by1
+2k3, we can get the ampli-
+tudeA(1g,2g,3g,4g). After substituting it into the relation (6a), the three-point amp litude
+A(1g,2g,3h) is given
+A(1g,2g,3h)∼2[−ǫ1·ǫ2ǫρσ
+3k1ρk2σ+ǫ1·k2ǫ3ρσk1ρǫ2σ+ǫ2·k1ǫρσ
+3ǫ1ρk2σ],(7)
+where the physical conditions ǫ1·k1=ǫ2·k2= 0,ǫ3ρσkρ
+3=ǫ3ρσkσ
+3= 0, momentum conserva-
+tionkµ
+1+kµ
+2+kµ
+3= 0 andthe traceless condition ofthe polarizationtensor of gravito nǫρ
+3ρ= 0
+have been used. This is same with that given by KLT relation(2a). Thu s,D2relation can
+give the three-point tree amplitude where gluons are minimally coupled to graviton.
+B. Four-point tree amplitudes
+In this subsection, we study the four-point amplitudes A(1g,2g,3g,4h) and
+A(1g,2g,3h,4h). We use the spinor helicity formalism[24–26] to consider the four-p oint
+amplitudes. Useful properties of spinor helicity formalism are listed in Appendix A.
+The two gluons corresponding to a graviton with helicity ±2 take helicity ±1. Then
+the four-point tree amplitudes with all legs of positive helicity can be g iven by pure-gluon
+amplitudes with all legs of positive helicity via the relations (6b) and (6c ). Because the
+pure-gluon partial amplitudes in which all gluons take the same helicity vanish, we have
+A(1+
+g,2+
+g,3+
+g,4+
+h) =A(1+
+g,2+
+g,3+
+h,4+
+h) = 0. (8)
+8The tree amplitudes with one gluon of negative helicity and other legs o f positive helicity
+can be given by pure-gluon tree amplitudes where only one leg take ne gative helicity and
+otherlegstakepositivehelicity. Sincethepure-gluontreeamplitude swithonelegofnegative
+helicity and other legs of positive helicity vanish, we have
+A(1−
+g,2+
+g,3+
+g,4+
+h) =A(1−
+g,2+
+g,3+
+h,4+
+h) = 0. (9)
+The tree amplitudes with one graviton of negative helicity and other le gs of positive helicity
+can be given by pure-gluon MHV[27] amplitudes. In this pure-gluon MH V amplitude, the
+two gluons corresponding to the negative helicity graviton take neg ative helicity. Using
+the relation (6b) and the expression of MHV amplitude for gluons (A1 1), the amplitude
+A(1+
+g,2+
+g,3+
+g,4−
+h) is given
+A(1+
+g,2+
+g,3+
+g,4−
+h)∼g2κA(1+
+g,5−
+g,2+
+g,4−
+g,3+
+g) =gκs13i/an}bracketle{t54/an}bracketri}ht4
+/an}bracketle{t15/an}bracketri}ht/an}bracketle{t52/an}bracketri}ht/an}bracketle{t24/an}bracketri}ht/an}bracketle{t43/an}bracketri}ht/an}bracketle{t31/an}bracketri}ht= 0,(10)
+where we have use k5=k4. Similarly, A(1+
+g,2+
+g,3−
+h,4+
+h) = 0.
+Now we consider the MHV amplitudes where two legs take negative helic ity and oth-
+ers take positive helicity. There are three independent amplitudes A(1−
+g,2−
+g,3+
+g,4+
+h),
+A(1−
+g,2−
+g,3+
+h,4+
+h) andA(1−
+g,2+
+g,3−
+h,4+
+h). With the D2relations (6b) and (6c), the first two
+amplitudes can be expressed by five- and six-point pure-gluon MHV a mplitudes respectively.
+After using some properties of the spinor helicity formalism and the f act that the two gluons
+corresponding to one graviton take half of the momentum of the gr aviton, we get the first
+two amplitudes
+A(1−
+g,2−
+g,3+
+g,4+
+h)∼gκs13A(1−
+g,5+
+g,2−
+g,4+
+g,3+
+g)
+=gκs13i/an}bracketle{t12/an}bracketri}ht4
+/an}bracketle{t15/an}bracketri}ht/an}bracketle{t52/an}bracketri}ht/an}bracketle{t24/an}bracketri}ht/an}bracketle{t43/an}bracketri}ht/an}bracketle{t31/an}bracketri}ht
+∼gκ√
+2/an}bracketle{t12/an}bracketri}ht4[12]
+/an}bracketle{t14/an}bracketri}ht/an}bracketle{t24/an}bracketri}ht/an}bracketle{t34/an}bracketri}ht2.(11)
+9A(1−
+g,2−
+g,3+
+h,4+
+h)∼κ2[s2
+12A(1−
+g,6+
+g,3+
+g,5+
+g,4+
+g,2−
+g)−s12s13A(1−
+g,3+
+g,5+
+g,4+
+g,2−
+g,6+
+g)]
+=κ2/bracketleftbigg
+s2
+12i/an}bracketle{t12/an}bracketri}ht4
+/an}bracketle{t16/an}bracketri}ht/an}bracketle{t63/an}bracketri}ht/an}bracketle{t35/an}bracketri}ht/an}bracketle{t54/an}bracketri}ht/an}bracketle{t42/an}bracketri}ht/an}bracketle{t21/an}bracketri}ht−s12s13/an}bracketle{t12/an}bracketri}ht4
+/an}bracketle{t13/an}bracketri}ht/an}bracketle{t35/an}bracketri}ht/an}bracketle{t54/an}bracketri}ht/an}bracketle{t42/an}bracketri}ht/an}bracketle{t26/an}bracketri}ht/an}bracketle{t61/an}bracketri}ht/bracketrightbigg
+= 0.
+(12)
+To calculate A(1−
+g,2+
+g,3−
+h,4+
+h), we need six-point non-MHV tree amplitudes for gluons
+A(1−
+g,6+
+g,3−
+g,5+
+g,4−
+g,2+
+g) andA(1−
+g,3−
+g,5+
+g,4−
+g,2+
+g,6+
+g). Using the tree amplitude with six
+gluons given in[25] and substituting k3,k4andk5,k6byk3
+2andk4
+2correspondingly, we get
+A(1−
+g,6+
+g,3−
+g,5+
+g,4−
+g,2+
+g) =−16i[24]4/an}bracketle{t13/an}bracketri}ht2/an}bracketle{t23/an}bracketri}ht2
+s3
+12s2
+23,
+A(1−
+g,3−
+g,5+
+g,4−
+g,2+
+g,6+
+g) = 16i[24]4/an}bracketle{t13/an}bracketri}ht2/an}bracketle{t23/an}bracketri}ht2
+s12s2
+13s2
+23.(13)
+Then we substitute the amplitudes(13) into the relation(6c). The a mplitude
+A(1−
+g,2+
+g,3−
+h,4+
+h) is given
+A(1−
+g,2+
+g,3−
+h,4+
+h)∼κ2[s2
+12A(1−
+g,6+
+g,3−
+g,5+
+g,4−
+g,2+
+g)−s12s13A(1−
+g,3−
+g,5+
+g,4−
+g,2+
+g,6+
+g)]
+=−16i[24]4/an}bracketle{t13/an}bracketri}ht2/an}bracketle{t23/an}bracketri}ht2
+s12s2
+23−16i[24]4/an}bracketle{t13/an}bracketri}ht2/an}bracketle{t23/an}bracketri}ht2
+s12s13s2
+23
+∼[24]4/an}bracketle{t23/an}bracketri}ht2/an}bracketle{t13/an}bracketri}ht2
+s12s23s13.(14)
+So far, we have given all the independent three- and four-point am plitudes, other three-
+and four-point amplitudes can be derived from these amplitudes by u sing a parity transfor-
+mation or performing an appropriate replacement on the external legs. These results are
+same with those given by KLT relations[17]. Then the three- and four -point tree ampli-
+tudes for gluons minimally coupled to graviton satisfy D2relations. We then expect the D2
+relations hold in all amplitudes where gluons are minimally coupled to grav itons.
+10IV. GENERAL DISCUSSIONS ON D2RELATIONS FOR TREE AMPLITUDES
+IN MINIMAL COUPLING THEORY OF GAUGE FIELD AND GRAVITY
+In this section, we study general tree amplitudes in minimal coupling t heory of gauge
+field and gravity. D2relations can give amplitudes with Ngluons and Mgravitons by sum
+of pure-gluon partial amplitudes with N+2Mlegs except for appropriate factors. If all the
+gluons and gravitons take positive helicity, the amplitude must vanish . This is because the
+pure-gluon amplitudes with all legs of positive helicity vanish.
+The tree amplitudes with one gluon of negative helicity and other legs o f positive helicity
+can be expressed as sum of pure-gluon amplitudes with one leg of neg ative helicity and
+other legs of positive helicity. Then these amplitudes also vanish. The amplitudes with one
+graviton of negative helicity and other N+M−1 legs of positive helicity are expressed
+by sum of MHV ( N+ 2M)-gluon tree amplitudes where the two gluons corresponding to
+the negative helicity graviton take negative helicity. The two negativ e helicity gluons in the
+(N+ 2M)-gluon amplitudes take the same momentum. Considering the antisy mmetry of
+the spinor products (A6), these MHV tree amplitudes for N+ 2Mgluons vanish. Thus
+the amplitudes with one graviton of negative helicity and other N+M−1 legs of positive
+helicity must vanish.
+The results above given by D2relations are same with those given by KLT relations.
+Then the D2relations can give the amplitudes with all the legs of positive helicity for gauge-
+graviton minimal coupling. They also give amplitudes with one leg of nega tive helicity and
+other legs of positive helicity for gauge-gravity minimal coupling.
+Though the trivial cases are easy to consider, D2relations for nontrivial helicity con-
+figurations are not so clear. In the Subsections IVA and IVB, we will give discussions
+11FIG. 1: Positions of the two gluons corresponding to the grav iton for (a) 1 < l < i, (b)i < l≤N
+and (c) the expression independent of helicity configuratio n.
+on amplitudes with one and two gravitons in addition to Ngluons. Here two gluons take
+negative helicity and other legs take positive helicity. In Subsection I VC, we extend the dis-
+cussions to arbitrary tree amplitudes where Ngluons are minimally coupled to Mgravitons.
+Two gluons take negative helicity and the other N+M−2 legs take positive helicity. We
+then extend the D2relations for amplitudes with one graviton in addition to Ngluons to
+relations independent of the helicity configurations. We suggest D2relations should hold for
+any helicity configurations. The D2relation for a given amplitude is not unique. Different
+relations can be related by relations among partial amplitudes of gau ge field.
+12A.D2relations for tree amplitudes with one positive helicity graviton, N−2
+positive helicity gluons and two negative helicity gluons
+The tree amplitudes with two gluons of negative helicity andother legs of positive helicity
+can be expressed by[28, 29]
+A(1−
+g,2+
+g,...,i−
+g,...,N+
+g,(N+1)+
+h,...,(N+M)+
+h)
+=igN−2/parenleftBig
+−κ
+2/parenrightBigM/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}htS(1,i,{h+},{g+}),(15)
+where
+S(i,j,{h+},{g+}) =
+/productdisplay
+m∈{h+}d
+dam
+
+×/productdisplay
+l∈{g+}exp
+/summationdisplay
+n1∈{h+}an1/an}bracketle{tli/an}bracketri}ht/an}bracketle{tlj/an}bracketri}ht[ln1]
+/an}bracketle{tn1i/an}bracketri}ht/an}bracketle{tn1j/an}bracketri}ht/an}bracketle{tln1/an}bracketri}ht×exp
+/summationdisplay
+n2∈{h+},n2/ne}ationslash=n1an2/an}bracketle{tn1i/an}bracketri}ht/an}bracketle{tn1j/an}bracketri}ht[n1n2]
+/an}bracketle{tn2i/an}bracketri}ht/an}bracketle{tn2j/an}bracketri}ht/an}bracketle{tn1n2/an}bracketri}htexp[...]
+
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+aj=0.
+(16)
+InM= 1 case where all the legs take positive helicity except 1 and i, the amplitude is
+reduced to
+A(1−
+g,2+
+g,...,i−
+g,...,N+
+g,(N+1)+
+h)
+=igN−2/parenleftBig
+−κ
+2/parenrightBig/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}ht/summationdisplay
+l∈{g+}/an}bracketle{tl1/an}bracketri}ht/an}bracketle{tli/an}bracketri}ht[l,N+1]
+/an}bracketle{tN+1,1/an}bracketri}ht/an}bracketle{tN+1,i/an}bracketri}ht/an}bracketle{tl,N+1/an}bracketri}ht
+=igN−2/parenleftBigκ
+2/parenrightBig/summationdisplay
+l∈{g+}/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}ht/an}bracketle{tl,N+1/an}bracketri}ht[l,N+1]
+×/an}bracketle{t1l/an}bracketri}ht
+/an}bracketle{t1,N+1/an}bracketri}ht/an}bracketle{tN+1,l/an}bracketri}ht/an}bracketle{tli/an}bracketri}ht
+/an}bracketle{tl,N+1/an}bracketri}ht/an}bracketle{tN+1,i/an}bracketri}ht.(17)
+In the equation above,/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}htis just the MHV amplitude for Ngluons with the per-
+mutation 1 ,...,N. Using (A5), we have /an}bracketle{tl,N+ 1/an}bracketri}ht[l,N+ 1] =sl,N+1. With the eikonal
+identity(A9),/an}bracketle{t1l/an}bracketri}ht
+/an}bracketle{t1,N+1/an}bracketri}ht/an}bracketle{tN+1,l/an}bracketri}htand/an}bracketle{tli/an}bracketri}ht
+/an}bracketle{tl,N+1/an}bracketri}ht/an}bracketle{tN+1,i/an}bracketri}htcan split into sums of terms over all the legs
+13between 1, landl,irespectively. For 1 < l < i,
+/an}bracketle{t1l/an}bracketri}ht
+/an}bracketle{t1,N+1/an}bracketri}ht/an}bracketle{tN+1,l/an}bracketri}ht=l−1/summationdisplay
+r=1/an}bracketle{tr,r+1/an}bracketri}ht
+/an}bracketle{tr,N+1/an}bracketri}ht/an}bracketle{tN+1,r+1/an}bracketri}ht,
+/an}bracketle{tli/an}bracketri}ht
+/an}bracketle{tl,N+1/an}bracketri}ht/an}bracketle{tN+1,i/an}bracketri}ht=i−1/summationdisplay
+t=l/an}bracketle{tt,t+1/an}bracketri}ht
+/an}bracketle{tt,N+1/an}bracketri}ht/an}bracketle{tN+1,t+1/an}bracketri}ht.(18)
+Fori < l≤N,
+/an}bracketle{til/an}bracketri}ht
+/an}bracketle{ti,N+1/an}bracketri}ht/an}bracketle{tN+1,l/an}bracketri}ht=l−1/summationdisplay
+r=i/an}bracketle{tr,r+1/an}bracketri}ht
+/an}bracketle{tr,N+1/an}bracketri}ht/an}bracketle{tN+1,r+1/an}bracketri}ht,
+/an}bracketle{tl1/an}bracketri}ht
+/an}bracketle{tl,N+1/an}bracketri}ht/an}bracketle{tN+1,1/an}bracketri}ht=N/summationdisplay
+t=l/an}bracketle{tt,t+1/an}bracketri}ht
+/an}bracketle{tt,N+1/an}bracketri}ht/an}bracketle{tN+1,t+1/an}bracketri}ht,(19)
+where we define t+1 = 1 for t=N. The amplitude then becomes
+A(1−
+g,2+
+g,...,i−
+g,...,N+
+g,(N+1)+
+h)
+=igN−2/parenleftBigκ
+2/parenrightBig/parenleftBigg/summationdisplay
+1<l<isl,N+1l−1/summationdisplay
+r=1i−1/summationdisplay
+t=l+/summationdisplay
+i<l≤Nsl,N+1l−1/summationdisplay
+r=iN/summationdisplay
+t=l/parenrightBigg
+·/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}ht/an}bracketle{tr,r+1/an}bracketri}ht
+/an}bracketle{tr,N+1/an}bracketri}ht/an}bracketle{tN+1,r+1/an}bracketri}ht/an}bracketle{tt,t+1/an}bracketri}ht
+/an}bracketle{tt,N+1/an}bracketri}ht/an}bracketle{tN+1,t+1/an}bracketri}ht,(20)
+where for t=Nin the sum over t, we define kt+1≡k1. For a given r, the numerator of
+/an}bracketle{tr,r+1/an}bracketri}ht
+/an}bracketle{tr,N+1/an}bracketri}ht/an}bracketle{tN+1,r+1/an}bracketri}htis a spinor product for two adjacent points. Then we can remove /an}bracketle{tr,r+1/an}bracketri}ht
+in/an}bracketle{tr,r+1/an}bracketri}ht
+/an}bracketle{tr,N+1/an}bracketri}ht/an}bracketle{tN+1,r+1/an}bracketri}htand/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}htsimultaneously. Then /an}bracketle{tr,r+ 1/an}bracketri}htin the denominator of
+/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}htis replaced by /an}bracketle{tr,N+ 1/an}bracketri}ht/an}bracketle{tN+ 1,r+ 1/an}bracketri}ht. After this replacement,/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}ht
+becomes/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tr,N+1/an}bracketri}ht/an}bracketle{tN+1,r+1/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}htwhich is the ( N+1)-gluon MHV tree amplitude with
+the permutation 1, 2,..., r,N+1,r+1,...,N. Thus/an}bracketle{tr,r+1/an}bracketri}ht
+/an}bracketle{tr,N+1/an}bracketri}ht/an}bracketle{tN+1,r+1/an}bracketri}htjust insert a gluon with
+momentum kN+1between the two gluons randr+ 1. For a given l(1< l < i), the sum
+overrfor 1≤r≤l−1 becomes sum over all the possible insertions between 1 and l. For
+i < l≤N, the sum over rfori≤r≤l−1 becomes sum over all the possible insertions
+between iandl. After a similar discussion,/an}bracketle{tt,t+1/an}bracketri}ht
+/an}bracketle{tt,N+1/an}bracketri}ht/an}bracketle{tN+1,t+1/an}bracketri}htinsert a gluon with momentum
+kN+1at the positions between l,ifor 1< l < iand between i, 1 fori < l≤N. The two
+14gluons with momentum kN+1just correspond to the left- and right-moving sectors of the
+graviton3. Then/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}htbecomes/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tr,N+1/an}bracketri}ht/an}bracketle{tN+1,r+1/an}bracketri}ht.../an}bracketle{tt,N+1/an}bracketri}ht/an}bracketle{tN+1,t/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}htwhich is the
+MHV tree amplitude with N+2 gluons.
+Thus, the amplitude can be considered as a sum of terms. In each te rm, there is an MHV
+tree amplitude for N+2 gluons. Two of the N+2 gluons take the momentum kN+1. Then
+the amplitude satisfies the relation
+A(1−
+g,2+
+g,...,i−
+g,...,N+
+g,(N+1)+
+h)
+=igN−2/parenleftBigκ
+2/parenrightBig/summationdisplay
+l∈{g+}sl,N+1/summationdisplay
+PAN+2
+MHV(P),(21)
+Where for any given lin{g+}, we sum over permutations P.Pare the permutations in
+which the relative position of the Ngluons is 1 g, 2g,...,Ng, one gluon corresponding to
+the graviton ( N+ 1)hcan be inserted at any position between 1 gandlg, the other gluon
+corresponding to the graviton can be inserted at any position betw eenlgandig(See Fig.
+1(a) and (b)).
+B.D2relations for tree amplitudes with two positive helicity gravitons, N−2
+positive helicity gluons and two negative helicity gluons
+The tree amplitude (15) and (16), with M= 2 can be reduced to
+A(1−
+g,2+
+g,...,i−
+g,...,N+
+g,(N+1)+
+h,(N+2)+
+h) =A+B+C, (22)
+3In this section, we let the gluons corresponding to a graviton take t he momentum of the graviton for
+convenience. This is a little different from in the sections above where each gluon from a graviton take
+half of the momentum of the graviton. However, a redefinition of th e momentum k→1
+2konly contribute
+a constant factor which does not affect our discussions.
+15whereA,BandCare
+A=igN−2/parenleftBig
+−κ
+2/parenrightBig2/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}ht
+×/summationdisplay
+l∈{g+}sN+1,N+2sl,N+1/an}bracketle{t1l/an}bracketri}ht
+/an}bracketle{t1,N+1/an}bracketri}ht/an}bracketle{tN+1,l/an}bracketri}ht/an}bracketle{tli/an}bracketri}ht
+/an}bracketle{tl,N+1/an}bracketri}ht/an}bracketle{tN+1,i/an}bracketri}ht(23)
+×/an}bracketle{t1,N+1/an}bracketri}ht
+/an}bracketle{t1,N+2/an}bracketri}ht/an}bracketle{tN+2,N+1/an}bracketri}ht/an}bracketle{tN+1,i/an}bracketri}ht
+/an}bracketle{tN+1,N+2/an}bracketri}ht/an}bracketle{tN+2,i/an}bracketri}ht,
+B=igN−2/parenleftBig
+−κ
+2/parenrightBig2/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}ht
+×/summationdisplay
+l∈{g+}sN+2,N+1sl,N+2/an}bracketle{t1l/an}bracketri}ht
+/an}bracketle{t1,N+2/an}bracketri}ht/an}bracketle{tN+2,l/an}bracketri}ht/an}bracketle{tli/an}bracketri}ht
+/an}bracketle{tl,N+2/an}bracketri}ht/an}bracketle{tN+2,i/an}bracketri}ht(24)
+×/an}bracketle{t1,N+2/an}bracketri}ht
+/an}bracketle{t1,N+1/an}bracketri}ht/an}bracketle{tN+1,N+2/an}bracketri}ht/an}bracketle{tN+2,i/an}bracketri}ht
+/an}bracketle{tN+2,N+1/an}bracketri}ht/an}bracketle{tN+1,i/an}bracketri}ht,
+C=igN−2/parenleftBig
+−κ
+2/parenrightBig2/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}ht
+×/summationdisplay
+l∈{g+}sl,N+2/an}bracketle{tl1/an}bracketri}ht
+/an}bracketle{tN+2,1/an}bracketri}ht/an}bracketle{tl,N+2/an}bracketri}ht/an}bracketle{tli/an}bracketri}ht
+/an}bracketle{tl,N+2/an}bracketri}ht/an}bracketle{tN+2,i/an}bracketri}ht(25)
+×/summationdisplay
+k∈{g+}sk,N+1/an}bracketle{tk1/an}bracketri}ht
+/an}bracketle{tN+1,1/an}bracketri}ht/an}bracketle{tl,N+1/an}bracketri}ht/an}bracketle{tki/an}bracketri}ht
+/an}bracketle{tk,N+1/an}bracketri}ht/an}bracketle{tN+1,i/an}bracketri}ht.
+We first look at Apart./an}bracketle{t1l/an}bracketri}ht
+/an}bracketle{t1,N+1/an}bracketri}ht/an}bracketle{tN+1,l/an}bracketri}htand/an}bracketle{tli/an}bracketri}ht
+/an}bracketle{tl,N+1/an}bracketri}ht/an}bracketle{tN+1,i/an}bracketri}htcan split into sums of terms as in
+(18) and (19). For a given l, as inM= 1 case, they insert the two gluons corresponding to
+the graviton( N+1)hinto positions between 1, landl,irespectively. After this insertion, we
+consider the insertion of gluons corresponding to ( N+2)h. With the eikonal identity(A9),
+for 1< l < i,/an}bracketle{t1,N+1/an}bracketri}ht
+/an}bracketle{t1,N+2/an}bracketri}ht/an}bracketle{tN+2,N+1/an}bracketri}htand/an}bracketle{tN+1,i/an}bracketri}ht
+/an}bracketle{tN+1,N+2/an}bracketri}ht/an}bracketle{tN+2,i/an}bracketri}htin (23) can be given as
+/an}bracketle{t1,N+1/an}bracketri}ht
+/an}bracketle{t1,N+2/an}bracketri}ht/an}bracketle{tN+2,N+1/an}bracketri}ht=/parenleftbigg/an}bracketle{t1,r/an}bracketri}ht
+/an}bracketle{t1,N+2/an}bracketri}ht/an}bracketle{tN+2,r/an}bracketri}ht+/an}bracketle{tr,N+1/an}bracketri}ht
+/an}bracketle{tr,N+2/an}bracketri}ht/an}bracketle{tN+2,N+1/an}bracketri}ht/parenrightbigg
+,
+/an}bracketle{tN+1,i/an}bracketri}ht
+/an}bracketle{tN+1,N+2/an}bracketri}ht/an}bracketle{tN+2,i/an}bracketri}ht=/parenleftbigg/an}bracketle{tt+1,i/an}bracketri}ht
+/an}bracketle{tt+1,N+2/an}bracketri}ht/an}bracketle{tN+2,i/an}bracketri}ht+/an}bracketle{tN+1,t+1/an}bracketri}ht
+/an}bracketle{tN+1,N+2/an}bracketri}ht/an}bracketle{tN+2,t+1/an}bracketri}ht/parenrightbigg
+,
+(26)
+16FIG. 2: Positions of the gluons corresponding to the two grav itons (N+1)hand (N+2)hfor (a)
+1< l < i, (b)i < l≤NinApart.
+while for i < l≤N,
+/an}bracketle{ti,N+1/an}bracketri}ht
+/an}bracketle{ti,N+2/an}bracketri}ht/an}bracketle{tN+2,N+1/an}bracketri}ht=/parenleftbigg/an}bracketle{ti,r/an}bracketri}ht
+/an}bracketle{ti,N+2/an}bracketri}ht/an}bracketle{tN+2,r/an}bracketri}ht+/an}bracketle{tr,N+1/an}bracketri}ht
+/an}bracketle{tr,N+2/an}bracketri}ht/an}bracketle{tN+2,N+1/an}bracketri}ht/parenrightbigg
+,
+/an}bracketle{tN+1,1/an}bracketri}ht
+/an}bracketle{tN+1,N+2/an}bracketri}ht/an}bracketle{tN+2,1/an}bracketri}ht=/parenleftbigg/an}bracketle{tt+1,1/an}bracketri}ht
+/an}bracketle{tt+1,N+2/an}bracketri}ht/an}bracketle{tN+2,1/an}bracketri}ht+/an}bracketle{tN+1,t+1/an}bracketri}ht
+/an}bracketle{tN+1,N+2/an}bracketri}ht/an}bracketle{tN+2,t+1/an}bracketri}ht/parenrightbigg
+.
+(27)
+The first term of the sum in each line of (26) and (27) can split into sum over adjacent
+points again, for example, the first term in the first line of (26) can b e expressed as
+r−1/summationdisplay
+p=1/an}bracketle{tp,p+1/an}bracketri}ht
+/an}bracketle{tp,N+2/an}bracketri}ht/an}bracketle{tN+2,p+1/an}bracketri}ht. (28)
+For a given r, we have inserted a gluon corresponding to ( N+1)hbetween randr+1, then
+(28) insert a gluon corresponding to ( N+ 2)hat a position between 1 and r. The second
+term in the first line of (26) insert an gluon corresponding to ( N+2)hbetween rand the
+gluon corresponding to ( N+1)h. Thus the first line of (26) just insert a gluon corresponding
+to (N+2)hat the positions between 1 and the gluon corresponding to ( N+1)h, where the
+gluon corresponding to ( N+1)hhave been inserted at positions between 1 and l. In a same
+17way, the second line of (26) insert the other gluon corresponding t o (N+ 2)hat positions
+between the gluon corresponding to ( N+ 1)handi, where this gluon corresponding to
+(N+ 1)hhave been inserted at a position between landi(See Fig. 2 (a)). Following a
+similar discussion, for the case of i < l≤N, we insert the two gluons corresponding to
+(N+1)hat positions between i,landl, 1 respectively. Then insert one gluon corresponding
+to (N+ 2)hat the positions between iand the gluon corresponding to ( N+ 1)hwhich is
+between i,l. Insert the other gluon corresponding to ( N+2)hat the positions between the
+other gluon corresponding to ( N+1)hand 1(See Fig. 2 (b)). The Apart then satisfy the
+relation
+A=/summationdisplay
+l∈{g+}sl,N+1sN+1,N+2/summationdisplay
+P1AN+4
+MHV(P1), (29)
+where for a given l,P1are the possible insertions of the gluons corresponding to the two
+gravitons. These insertions has the form 1, ..., N+2, ...,N+1,...,l, ...,N+1, ...,N+2,
+...,i, ...,Nfor 1< l < iand 1,..., i, ...,N+2, ...,N+1, ...,l, ...,N+1, ...,N+2, ... for
+i < l≤N.
+TheBpart can be derived from Apart by the replacement N+1↔N+2 and is given
+as
+B=/summationdisplay
+l∈{g+}sl,N+2sN+2,N+1/summationdisplay
+P2AN+4
+MHV(P2), (30)
+where for a given l,P2are the possible insertions of the gluons corresponding to the two
+gravitons. These insertions has the form 1, ..., N+1, ...,N+2,...,l, ...,N+2, ...,N+1,
+...,i, ...,Nfor 1< l < i( See Fig. 3 (a)) and 1,..., i, ...,N+1, ...,N+2, ...,l, ...,N+2,
+...,N+1, ... for i < l≤N(See Fig. 3 (b)).
+Now we consider the Cpart. Both the second and the third lines in (25), can split into
+the forms of (18) and (19). Then for a given l, each term in the second line of the expression
+(25) insert the two gluons corresponding to ( N+2)hat the positions between 1, landl,i
+18FIG. 3: Positions of the gluons corresponding to the two grav itons (N+1)hand (N+2)hfor (a)
+1< l < i, (b)i < l≤NinBpart.
+respectively. Then for any given k, each term in the second line of (25) insert the two gluons
+corresponding to ( N+1)hat the positions between 1, kandk,irespectively. The Cpart
+then becomes
+C=/summationdisplay
+l,k∈g+sl,N+2sk,N+1/summationdisplay
+P3AN+4
+MHV(P3). (31)
+whereP3are the insertions of the four gluons corresponding to the two gra vitons (N+2)h
+and (N+1)h. In these insertions, two gluons corresponding to ( N+2)hare inserted at the
+positions between 1, landl,irespectively, then two gluons corresponding to ( N+1)hare
+inserted at the positions between 1, kandk,irespectively( See Fig. 4).
+After considering all the contributions from A,BandC, we give the D2relation
+A(1−
+g,2+
+g,...,i−
+g,...,N+
+g,(N+1)+
+h,(N+2)+
+h)
+=/summationdisplay
+l∈{g+}sl,N+1sN+1,N+2/summationdisplay
+P1AN+4
+MHV(P1)+/summationdisplay
+l∈{g+}sl,N+2sN+2,N+1/summationdisplay
+P2AN+4
+MHV(P2)
++/summationdisplay
+l,k∈g+sl,N+2sk,N+1/summationdisplay
+P3AN+4
+MHV(P3).(32)
+19FIG. 4: Positions of the gluons corresponding to the two grav itons (N+1)hand (N+2)hfor (a)
+1< l < i, 1< k < i, (b)i < l≤N,i < k≤Nin (c)i < l≤N, 1< k < i and (d) 1 < l < i,
+i < k≤NinCpart. Here we first insert the two gluons corresponding to ( N+2)hbetween 1, l
+andl,irespectively. We then insert two gluons corresponding to ( N+1)hbetween 1, kandk,i
+respectively.
+C.D2relations for arbitrary tree amplitudes with Ngluons minimally coupled to
+Mgravitons
+The amplitudes with more gravitons are more complicated. However t he discussions are
+similar with those for amplitudes with one and two gravitons. The amplit ude with Ngluons
+minimally coupled to Mgravitons, where two gluons take negative helicity and other legs
+20take positive helicity (15) and (16) can be given by a sum of terms. Ea ch term has the form
+igN−2/parenleftBig
+−κ
+2/parenrightBigM/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}ht
+×/an}bracketle{tl11/an}bracketri}ht/an}bracketle{tl1i/an}bracketri}ht[l1n1
+1]
+/an}bracketle{tn1
+11/an}bracketri}ht/an}bracketle{tn1
+1i/an}bracketri}ht/an}bracketle{tn1
+1n1
+2/an}bracketri}ht×/an}bracketle{tn1
+11/an}bracketri}ht/an}bracketle{tn1
+1i/an}bracketri}ht[n1
+1n1
+2]
+/an}bracketle{tn1
+21/an}bracketri}ht/an}bracketle{tn1
+2i/an}bracketri}ht/an}bracketle{tn1
+1n1
+2/an}bracketri}ht×...
+×/an}bracketle{tl21/an}bracketri}ht/an}bracketle{tl2i/an}bracketri}ht[l2n2
+1]
+/an}bracketle{tn2
+11/an}bracketri}ht/an}bracketle{tn2
+1i/an}bracketri}ht/an}bracketle{tn2
+1n2
+2/an}bracketri}ht×/an}bracketle{tn2
+11/an}bracketri}ht/an}bracketle{tn2
+1i/an}bracketri}ht[n2
+1n2
+2]
+/an}bracketle{tn2
+21/an}bracketri}ht/an}bracketle{tn2
+2i/an}bracketri}ht/an}bracketle{tn2
+1n2
+2/an}bracketri}ht×...
+...
+×/an}bracketle{tlN1/an}bracketri}ht/an}bracketle{tlNi/an}bracketri}ht[lNnN
+1]
+/an}bracketle{tnN
+11/an}bracketri}ht/an}bracketle{tnN
+1i/an}bracketri}ht/an}bracketle{tnN
+1nN
+2/an}bracketri}ht×/an}bracketle{tnN
+11/an}bracketri}ht/an}bracketle{tnN
+1i/an}bracketri}ht[nN
+1nN
+2]
+/an}bracketle{tnN
+21/an}bracketri}ht/an}bracketle{tnN
+2i/an}bracketri}ht/an}bracketle{tnN
+1nN
+2/an}bracketri}ht×...,(33)
+where each lican be any positive helicity gluon, and n1
+1,n1
+2,...,n2
+1,n2
+2,..., ...,nN
+1,nN
+2,... is
+a permutation of all the gravitons. Following the discussions on amplit udes with one and
+two gravitons, this term can be given by sum of MHV amplitudes with N+2Mgluons with
+appropriate factors. The first line insert two gluons correspondin g ton1
+1between 1, l1andl1,
+irespectively, theninserttwo gluonscorrespondingto n1
+2between 1, onegluoncorresponding
+ton1
+1andn1
+1, the other gluon corresponding to n1
+1respectlvely, ... After inserting all the
+gluons corresponding to gravitons in the first line, we insert two gluo ns corresponding to n2
+1
+between 1, l2andl2,irespectively, then insert two gluons corresponding to n2
+2between 1,
+onegluoncorresponding to n2
+1andn2
+1, theother gluoncorresponding to n2
+1respectively, ... In
+this way, we insert all the 2 Mgluons corresponding to the Mgravitons into the amplitudes.
+The phase factor is sl1,n1
+1sn1
+1,n1
+2...sl2,n2
+1sn2
+1,n2
+2...slN,nN
+1slN,nN
+2.... At last, the amplitudes become
+MHV tree amplitudes with N+2Mgluons, where the two gluons corresponding to a same
+graviton take the same momentum. This is just the D2relations in field theory.
+In string theory, the D2relations are independent of helicity configurations of the legs.
+Then we expect the D2relations should have helicity-independent form. For example, in the
+relation for amplitudes with one graviton and Ngluons given in Subsection IVA, we only
+sum over lcorrespongding to the gluons with positive helicity, and for each l, the two gluons
+21are inserted at the positions between 1, landl,irespectively, the relation (21) depends on
+the relative positions of the two negative helicity gluons. Then we exp ect the relation (21)
+can be extended to that independent of the relative positions of th e two negative helicity
+legs. In fact, in the expression (17), we can sum over l(1< l≤N). This is because
+/an}bracketle{tli/an}bracketri}htvanishes for l=i. Using the eikonal identity (A9) and the identity (A10) implied by
+momentum conservation, the amplitude becomes
+A(1−
+g,2+
+g,...,i−
+g,...,N+
+g,(N+1)+
+h)
+=igN−2/parenleftBigκ
+2/parenrightBig/summationdisplay
+1<l≤N/an}bracketle{t1i/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tN1/an}bracketri}ht/an}bracketle{tl,N+1/an}bracketri}ht[l,N+1]
+×/an}bracketle{t1l/an}bracketri}ht
+/an}bracketle{t1,N+1/an}bracketri}ht/an}bracketle{tN+1,l/an}bracketri}ht/an}bracketle{tl1/an}bracketri}ht
+/an}bracketle{tl,N+1/an}bracketri}ht/an}bracketle{tN+1,1/an}bracketri}ht.(34)
+Then we repeat the discussions above. The amplitude satisfy the re lation
+A(1−
+g,2+
+g,...,i−
+g,...,N+
+g,(N+1)+
+h)
+=igN−2/parenleftBigκ
+2/parenrightBig/summationdisplay
+1<l≤Nsl,N+1/summationdisplay
+P′AN+2
+MHV(P′),(35)
+where we sum over all the external gluons. For any given l,P′in this relation denote all
+the permutations where one of the two gluons corresponding to th e graviton is inserted at
+positions to the left of land right of 1, the other one is inserted to the left of 1 and right
+ofl(See Fig. 1(c)). Since this expression of the amplitude does not dep end on the helicity
+configuration of the legs. We suggest that for any helicity configur ation, the tree amplitude
+withNgluons minimally coupled to one graviton satisfy the relation
+A(1g,2g,...,Ng,(N+1)h)
+=igN−2/parenleftBigκ
+2/parenrightBig/summationdisplay
+1<l≤Nsl,N+1/summationdisplay
+P′AN+2(P′).(36)
+Though this extension will be more complicated, we expect there mus t be such extensions
+to the relations for arbitrary helicity configurations.
+22TheD2relation for a given amplitude may have different expressions as in str ing theory.
+A tree amplitude for gauge-gravity minimal coupling can be expresse d by different sets of
+pure-gluon partial tree amplitudes. The permutations of the legs a nd the factors in different
+expressions aredifferent. However, the partial tree amplitudes o f gluons are not independent
+of each other, there are relations among the partial tree amplitud es with gluons[6, 30]. Then
+the different expressions of the D2relation for a given amplitude can be related by the
+relations among pure-gluon partial tree amplitudes. To see this, we take amplitude with
+three gluons and one graviton as an example. In Section II, the rela tion is given by (6b), the
+factor is in s13-channel. However, in Section IV, the relation is given by (36). For N= 3,
+we have
+A(1g,2g,3g,4h) =g/parenleftBig
+−κ
+2/parenrightBig/bracketleftBigg
+s24A(1g,5g,2g,4g,3g)+s24A(1g,4g,2g,3g,5g)
++s34A(1g,4g,2g,3g,5g)+s34A(1g,2g,4g,3g,5g)/bracketrightBigg
+,(37)
+where we denote the two gluons corresponding to 4 hby 4gand 5g. Then the amplitude is
+given by different expressions. In the last expression, s2,4=s1,3,s24+s34=−s14=−s23
+ands34=s12. The second and the third terms can be given by one term with the fa ctor
+−s23. Using the relations among partial amplitudes, we have
+A(1g,4g,2g,3g,5g) =s13
+s23A(1g,5g,2g,4g,3g),
+A(1g,2g,4g,3g,5g) =s13
+s12A(1g,5g,2g,4g,3g).(38)
+Then the two relations (37) and (6b) are equivalent.
+V. CONCLUSION
+Inthispaper, westudytheamplitudeswheregluonsareminimallycoup ledwithgravitons.
+We find the three- and four-point amplitudes satisfy the field theor y limits of D2relations
+23in string theory. The left- and right-moving sectors are connecte d into a single one.
+We give particular forms of the relations for the amplitude A(1−
+g,2+
+g,...,i−
+g,...,N+
+g,(N+
+1)+
+h), andA(1−
+g,2+
+g,...,i−
+g,...,N+
+g,(N+1)+
+h),(N+2)+
+h. We extend the relation to arbitrary
+helicity configurations for N+ 1 case. The discussions can be extended to arbitrary legs
+with arbitrary helicity configurations. The tree amplitude with Ngluons and Mgravitons
+can be expressed by sum of amplitudes for N+ 2Mgluons with appropriate factors. The
+relation for a given amplitude is not unique, because there are relatio ns among pure-gluon
+partial amplitudes.
+Though the D2relations and KLT factorization relations only hold on D2andS2re-
+spectively in string theory, the field theory limits of both two relation s hold in in minimal
+coupling theory of gauge and gravity. This is because we have two diff erent methods to
+incorporate gauge degree of freedom in string theory.
+Acknowledgement
+We would like to thank C. Cao, Y. Q. Wang and Y. Xiao for useful discus sions. The
+work is supported in part by the NNSF of China Grant No. 90503009, No. 10775116, and
+973 Program Grant No. 2005CB724508.
+Appendix A: Spinor helicity formalism
+Here we given the useful properties of spinor helicity formalism[24– 26] Positive and neg-
+ative helicity spinor
+|i±/an}bracketri}ht ≡ |k±
+i/an}bracketri}ht ≡u±(ki) =v∓(ki),/an}bracketle{ti±| ≡ /an}bracketle{tk±
+i| ≡¯u±(ki) = ¯v±(ki), (A1)
+24whereuandvare positive and negative energy solutions of Dirac equation.
+/an}bracketle{tij/an}bracketri}ht ≡ /an}bracketle{ti−|j+/an}bracketri}ht=/radicalBig
+|sij|eiφij,
+[ij]≡ /an}bracketle{ti+|j−/an}bracketri}ht=/radicalBig
+|sij|e−i(φij+π).(A2)
+Momentum
+/an}bracketle{ti±|γµ|i±/an}bracketri}ht= 2ki. (A3)
+Polarization vector
+ǫ±
+µ(k,q) =±/an}bracketle{tq±|γµ|k∓/an}bracketri}ht√
+2/an}bracketle{tq∓|k±/an}bracketri}ht, (A4)
+whereqis reference momentum, reflecting the freedom of on-shell gauge transformation, k
+is the vector boson momentum.
+Useful properties:
+/an}bracketle{tij/an}bracketri}ht[ji] =sij (A5)
+antisymmetry
+/an}bracketle{tij/an}bracketri}ht=−/an}bracketle{tji/an}bracketri}ht,[ij] =−[ji],/an}bracketle{tii/an}bracketri}ht= [ii] = 0, (A6)
+Fierz rearrangement
+/an}bracketle{ti+|γµ|j+/an}bracketri}ht/an}bracketle{tk+|γµ|l+/an}bracketri}ht= 2[ik]/an}bracketle{tlj/an}bracketri}ht, (A7)
+charge conjugation
+/an}bracketle{ti+|γµ|j+/an}bracketri}ht=/an}bracketle{tj−|γµ|i−/an}bracketri}ht, (A8)
+eikonal identity
+k−1/summationdisplay
+i=j/an}bracketle{ti,i+1/an}bracketri}ht
+/an}bracketle{tiq/an}bracketri}ht/an}bracketle{tq,i+1/an}bracketri}ht=/an}bracketle{tjk/an}bracketri}ht
+/an}bracketle{tjq/an}bracketri}ht/an}bracketle{tqk/an}bracketri}ht. (A9)
+in anN-point amplitude, momentum conservation imply
+n/summationdisplay
+i=1,i/ne}ationslash=k[ji]/an}bracketle{tik/an}bracketri}ht= 0. (A10)
+25The amplitudes with all positive helicity gluons are zero. The amplitudes for gluons with
+one negative helicity and others positive helicity are zero. The amplitu de forNgluons with
+two negative and N−2 positive(MHV) helicities can be given[6]
+A(1+,...,i−,...,j−,...,n+) =i/an}bracketle{tij/an}bracketri}ht4
+/an}bracketle{t12/an}bracketri}ht/an}bracketle{t23/an}bracketri}ht.../an}bracketle{tn1/an}bracketri}ht. (A11)
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\ No newline at end of file
diff --git a/1001.0061.txt b/1001.0061.txt
new file mode 100644
index 0000000000000000000000000000000000000000..b7e8c70be616baadba06ad47d86e52fc0993cc3f
--- /dev/null
+++ b/1001.0061.txt
@@ -0,0 +1,6678 @@
+EIC Science Book
+by the Euclid Imaging Consortium
+EDITED BY:
+Alexandre R´ efr´ egier
+and
+Adam Amara
+Thomas Kitching
+Ana¨ ıs Rassat
+Roberto Scaramella
+Jochen Weller
+1stJanuary 2010
+Euclid Imaging ConsortiumEuclid Imaging Consortium Members
+PI: Alexandre Refregier (CEA Saclay)
+Co-PIs : Ralf Bender (MPE Germany), Mark Cropper (MSSL, UK), Roberto Scaramella (INAF-
+Oss. Roma, Italy), Simon Lilly (ETH Zurich, Switzerland), Nabila Aghanim (IAS Orsay, France),
+Francisco Castander (IEEC Barcelona, Spain), Jason Rhodes (JPL, USA)
+Study Manager : Jean-Louis Augueres (CEA Saclay)
+System engineer : Jerome Amiaux (CEA Saclay)
+Instrument scientists : Mario Schweitzer (MPE), Mark Cropper (MSSL) Olivier Boulade (CEA
+Saclay), Adam Amara (ETH Zurich), Jeff Booth (JPL), Anna Maria di Giorgio (INAF-IFSI)
+Co-Investigators :France : Marian Douspis (IAS Orsay), Yannick Mellier (IAP Paris), Olivier
+Boulade (CEA Saclay), Germany : Peter Schneider (U. Bonn), Oliver Krause (MPIA Heidelberg),
+Frank Eisenhauer (MPE Garching), Italy : Lauro Moscardini (U. Bologna), Luca Amendola
+(INAF-Oss. Roma and University of Heidelberg), Fabio Pasian (INAF-OATS), Spain : Ramon
+Miquel (IFAE Barcelona), Eusebio Sanchez (CIEMAT Madrid), Switzerland : George Meylan
+(EPFL-UniGE), Marcella Carollo (ETH Zurich), Francois Widi (EPFL-UniGE), UK: John
+Peacock (IfA Edinburgh), Sarah Bridle (UCL, London), Ian Bryson (IfA Edinburgh), USA : Jeff
+Booth (JPL), Steven Kahn (Stanford U.)
+Science working group coordinators :
+Weak Lensing : Adam Amara (ETH Zurich), Andy Taylor (IfA Edinburgh)
+Clusters/CMB : Jochen Weller (U. Munich/MPE), Nabila Aghanim (IAS Orsay)
+BAO (photometric) : Francisco Castander (IEEC Barcelona), Anais Rassat (CEA Saclay)
+Supernovae : Isobel Hook (Oxford, and Obs. Rome), Massimo Della Valle (Oss. Capodi-
+monte/ICRA)
+Theory : Luca Amendola (INAF-Oss. Roma), Martin Kunz (U. Sussex/U. Geneva)
+Strong Lensing : Matthias Bartelmann (U. Heidelberg), Leonidas Moustakas (JPL)
+Galaxy Evolution : Marcella Carollo (ETH Zurich), Hans-Walter Rix (MPIA)
+Galactic studies and planets : Eva Grebel (U. Heidelberg), Jean-Philippe Beaulieu (IAP
+Paris),
+Photometric Redshifts : Ofer Lahav (UCL), Adriano Fontana (INAF Oss. Roma)
+Image Simulations : Jason Rodes (JPL), Lauro Moscardini (U. Bologna)
+Communication : Marian Douspis (IAS Orsay)
+Science :
+France : Karim Benabed (IAP), Emmanuel Bertin (IAP), Pascal Bord´ e (IAS Orsay), Frederic
+Bournaud (CEA Saclay), Marian Douspis (IAS Orsay), Herve Dole (IAS Orsay), Raphael
+Gavazzi (IAP), Guilaine Lagache (IAS Orsay), M. Langer (IAS Orsay), Henry McCracken (IAP),
+Christophe, Magneville (CEA Saclay), M. Ollivier (IAS Orsay), Nathalie Palanque-Delabrouille
+(CEA Saclay), Stephane Paulin-Henriksson (CEA Saclay), Sandrine Pires (CEA Saclay), Jean-Luc
+Starck (CEA Saclay), Romain Teyssier (CEA Saclay/U. Zurich)
+Germany : Wolfgang Brandner (MPIA), Hans B¨ ohringer (MPE), Frank Eisenhauer (MPE),
+Natascha Foerster-Schreiber (MPE), Bertrand Goldman (MPIA), Oliver Krause (MPIA), Peter
+Melchior (ITA - Heidelberg), Steffi Phleps (MPE), Roberto Saglia (MPE), Stella Seitz (MPE),
+Peter Schneider (Univ. Bonn), Benjamin Joachimi (Univ. Bonn), Joachim Wambsgans (Univ.
+Heidelberg), David Wilman (MPE)
+Italy : Vicenzo Antonuccio (INAF-Oss. Catania), Carlo Baccigalupi (SISSA), Fabio Bellagamba
+(U. di Bologna), Silvio Bonometto (U. Milano Bicocca), Andrea Grazian (INAF- Oss. Roma),4
+Marco Lombardi (U. di Milano), Roberto Mainini (INAF Oss. Roma), Filippo Mannucci (INAF
+- IRA), Roberto Maoli (U. Roma La Sapienza), Sabino Matarrese (U. di Padova), Alessandro
+Melchiorri (U. Roma La Sapienza), Massimo Meneghetti (INAF- Oss. Bologna), Julian Merten
+(INAF Oss. Bologna), Lauro Moscardini (U. di Bologna), Claudia Quercellini (U. Roma Tor
+Vergata), Mario Radovich (INAF Oss. Capodimonte), Anna Romano (U. Roma La Sapienza),
+Roberto Scaramella (INAF Oss. Roma), Andrea Zacchei (INAF - Oss. Trieste)
+Spain : Ricard Casas (IEEC Barcelona), Martin Crocce (IEEC Barcelona), Pablo Fosalba (IEEC
+Barcelona), Enrique Gaztanaga (IEEC Barcelona), Ignacio Sevilla (CIEMAT Madrid), Eusebio
+Sanchez (CIEMAT Madrid), Juan Garcia-Bellido (UAM Madrid)
+Switzerland : Rongmon Bordoloi (ETH Zurich), Julien Carron (ETH Zurich), Frederic Courbin
+(EPFL Lausanne), Stephane Paltani (ISDC/U. Geneva), Justin Read (U. Zurich/U. Leicester),
+Robert Smith (U. Zurich)
+UK: Filipe Abdalla (UCL), Manda Banerji (UCL), David Bacon (U. Portsmouth), Kirk Donnacha
+(UCL), Ignacio Ferreras (MSSL-UCL), Gert Hutsi (UCL), Alan Heavens (IfA Edinburgh),
+Tom Kitching (IfA Edinburgh), Lindsay King (IoA Cambridge), Ofer Lahav (UCL), Katarina
+Markovic (UCL/Ludwigs-Maximilians U.), Richard Massey (IfA Edinburgh), Michael Schneider
+(U. Durham), Fabrizio Sidoli (UCL), Jiayu Tang (UCL/IPMU), Shaun Thomas (UCL), Lisa
+Voigt (UCL)
+USA : Joel Berge (JPL), Benjamin Dobke (JPL), Richard Ellis (Caltech), David Johnston
+(Northwest. U.), Michael Seiffert (JPL), Ali Vanderveld (JPL)
+Other : Eduardo Cypriano (U. Sao Paulo), Hakon Dahle (U. Oslo), Konrad Kuijken (Leiden)
+Instrument :
+France : Christophe Cara (CEA Saclay), Sandrine Cazaux (CEA Saclay), Arnaud Claret (CEA
+Saclay), Philippe Daniel-Thomas (CEA Saclay), Eric Doumayrou (CEA Saclay), Cydalise
+Dusmesnil (IAS Orsay), Jean-Jacques Fourmond (IAS Orsay), Jean-Claude Leclech (IAS Orsay),
+G. Morinaud (IAS Orsay), Samuel Ronayette (CEA Saclay), Zihong Sun (CEA Saclay), Tristan
+VanDenBerghe (IAS Orsay)
+Germany : Reiner Hofmann (MPE), Rory Holmes (MPIA), Reinhard Katterloher (MPE), Oliver
+Krause (MPIA)
+Italy : Marco Frailis (INAF Oss. Trieste), Pasquale Cerulli Irelli (INAF IFSI), Stefano Gallozzi
+(INAF Oss. Roma), Michele Maris (INAF Oss. Trieste), Renato Orfei (INAF IFSI), Fabio
+Pasian (INAF Oss. Trieste), Andrea Zacchei (INAF - Oss. Trieste)
+Spain : Laia Cardiel (IFAE Barcelona), Francesc Madrid (IEEC Barcelona), Santiago Serrano
+(IEEC Barcelona)
+Switzerland : Adrian Glauser (ETH Zurich), Francois Wildi (ISDC/U. Geneva)
+UK: Eli Atad-Ettedgui (UK-ATC), Ian Bryson (UK-ATC), Richard Cole (MSSL), Jason Gow
+(Open U), Phil Guttridge (MSSL) , Andrew Holland (Open U), Neil Murray (Open U), Kerrin
+Rees (MSSL), Phil Thomas (MSSL), Dave Walton (MSSL)
+USA : Parker Fagrelius (JPL), Kirk Gilmore (Stanford), Steven Kahn (Stanford), Andrew Ras-
+mussen (Stanford), Suresh Seshadri (JPL), Roger Smith (Caltech), Harry Teplitz (IPAC/Caltech)Acknowledgements
+We thank the national agencies and institutes in the EIC member countries for their support
+of the consortium during the assessment phase. We are particularly grateful to CNES, CEA,
+CNRS, CNAP in France, DLR and the Max Planck Society in Germany, STFC in the UK, ASI
+(contract “Euclid-DUNE” No. I/064/08/0) in Italy, ETH Zurich, EPF Lausanne and U. Geneva
+in Switzerland, the Spanish Ministerio de Educacion y Ciencia (grant ESP2007-29984-E) and the
+Spanish Ministerio de Ciencia e Innovacion (grant AYA2008-03347-E/ESP) in Spain, and the Jet
+Propulsion Laboratory, run under a contract for NASA by the California Institute of Technology
+in the US.
+We also thank ESA for their coordination of the Euclid Assessment phase, and the members
+of the Euclid Science Study team and of the ENIS consortium for scientific interactions.
+5Contents
+List of Authors 3
+Acknowledgements 5
+Table of Contents 6
+I INTRODUCTION 12
+1 Overview 13
+1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
+1.2 The Euclid Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
+1.3 The Euclid Imaging Instrument and Surveys . . . . . . . . . . . . . . . . . . . . . 15
+1.4 Primary Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
+1.5 Legacy Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
+1.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
+1.7 Data handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
+2 Publications 20
+2.1 Collaboration Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
+2.2 Science working groups and related papers . . . . . . . . . . . . . . . . . . . . . . 20
+II THEORY 23
+3 Fundamental Cosmology with Euclid: A Theoretical Perspective 24
+3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
+3.2 Phenomenology vs fundamental theory of the dark energy . . . . . . . . . . . . . 26
+3.3 Beyond the background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
+3.4 Extracting the potentials with Euclid . . . . . . . . . . . . . . . . . . . . . . . . . 29
+3.5 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
+3.6 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
+3.7 Synergy of Euclid with LHC, EELT and Gaia . . . . . . . . . . . . . . . . . . . . 32
+6Contents 7
+3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
+III WEAK GRAVITATIONAL LENSING 36
+4 EIC Weak Lensing 37
+4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
+4.2 Basics of Weak Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
+4.3 Developing the Science Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
+4.3.1 Summary of the Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 39
+4.4 Code comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
+4.5 Mitigation of Systematic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
+4.5.1 Setting Systematic Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
+4.5.2 Shape Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
+4.6 Summary of Top Level Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 44
+5 Weak Lensing Survey Optimisation for Dark Energy Measurement 48
+5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
+5.2 Optimising the survey geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
+5.3 Optimising constraints on cosmological parameters . . . . . . . . . . . . . . . . . 50
+5.3.1 Towards a more precise description of dark energy . . . . . . . . . . . . . 52
+5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
+6 Modified Gravity with Lensing 55
+6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
+6.2 Distinguishing between Modified Gravity and Dark Energy . . . . . . . . . . . 55
+6.3 Constraining Modified Gravity with Weak Lensing . . . . . . . . . . . . . . . . . 56
+6.3.1 Parameter constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
+6.3.2 Evolution of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
+6.3.3 Optimising the Survey for Modified Gravity . . . . . . . . . . . . . . . . . 58
+6.4 Using the Non-Linear Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
+7 Dark Matter 62
+7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
+7.2 Particle Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
+7.3 The Large-Scale Distribution of Dark Matter . . . . . . . . . . . . . . . . . . . . 65
+7.4 Dark Matter Sub-Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
+7.5 Dark Matter Halo Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
+7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
+8 Point-Spread Function and Instrument Calibration 74
+8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
+8.2 The number of stars required to calibrate the PSF . . . . . . . . . . . . . . . . . 75
+8.3 Impact of pixellation and PSF shape . . . . . . . . . . . . . . . . . . . . . . . . . 76
+8.3.1 PSF shape assessment simulation . . . . . . . . . . . . . . . . . . . . . . . 76Contents 8
+8.3.2 The nominal PSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
+8.3.3 Current results about the impact of pixellation and PSF shape . . . . . . 78
+8.4 Impact of the Point-Spread-Function Colour Dependence . . . . . . . . . . . . . 80
+8.5 Intra-Pixel Sensibility Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
+8.5.1 Impact of Intra-Pixel Sensibility Variations in the near infrared . . . . . . 82
+8.5.2 Impact of Intra-Pixel Sensibility Variations for shape measurements . . . 84
+9 Charge Transfer Inefficiency 87
+9.1 Origin of CTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
+9.2 Observational consequences of CTI . . . . . . . . . . . . . . . . . . . . . . . . . . 88
+9.3 CTI mitigation by hardware design . . . . . . . . . . . . . . . . . . . . . . . . . . 89
+9.4 CTI mitigation by software post-processing . . . . . . . . . . . . . . . . . . . . . 92
+9.5 A feasible Euclid mission design unimpeded by CTI . . . . . . . . . . . . . . . . 93
+10 Intrinsic Alignments 96
+10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
+10.2 Nulling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
+10.3 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
+IV ADDITIONAL COSMOLOGICAL PROBES & LEGACY SCIENCE 102
+11 Galaxy Clusters 103
+11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
+11.2 Cosmology with the Redshift Distribution of Galaxy Clusters . . . . . . . . . . . 104
+11.3 Selection of Galaxy Clusters with the Euclid Imaging Survey . . . . . . . . . . . 105
+11.4 Forecasting Cosmological Parameters with Clusters . . . . . . . . . . . . . . . . . 107
+12 The Integrated Sachs Wolfe Effect 112
+12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
+12.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
+12.3 ISW requirements: Signal to Noise analysis . . . . . . . . . . . . . . . . . . . . . 115
+12.4 Fisher Matrix analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
+12.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
+13 Galaxy Correlations with Euclid’s Imaging Survey 120
+13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
+13.2 Building Blocks of the Galaxy Power Spectrum . . . . . . . . . . . . . . . . . . . 121
+13.3 Galaxy correlations with photometric survey . . . . . . . . . . . . . . . . . . . . 123
+13.3.1 Projected Spherical Harmonic Method [ C(/lscript)]. . . . . . . . . . . . . . . . 123
+13.3.2 Wiggles only BAO method . . . . . . . . . . . . . . . . . . . . . . . . . . 124
+13.4 Constraints from the Euclid Imaging Survey . . . . . . . . . . . . . . . . . . . . . 126
+13.4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
+13.4.2 Combination with Planck priors . . . . . . . . . . . . . . . . . . . . . . . 127
+13.4.3 Constraints from the Euclid Imaging Survey . . . . . . . . . . . . . . . . . 127Contents 9
+13.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
+14 Euclid’s Imaging Legacy: Galaxy Evolution 131
+14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
+14.2 Understanding reionization, and galaxy formation in the reionization era . . . . . 132
+14.3 Unveiling clusters and massive groups in the early universe . . . . . . . . . . . . 134
+14.4 Linking matter and light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
+14.5 Identifying the physical drivers of galaxy evolution . . . . . . . . . . . . . . . . . 135
+14.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
+15 Strong lensing with Euclid 138
+15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
+15.2 Objects of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
+15.2.1 Individual, mostly massive galaxies . . . . . . . . . . . . . . . . . . . . . . 139
+15.2.2 Galaxies in groups, whose environment is crucial for their strong lensing .140
+15.2.3 Galaxy clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
+15.3 Detection Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
+15.3.1 Searching for clear identifying features in lenses . . . . . . . . . . . . . . . 141
+15.3.2 Specialised algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
+15.4 Astrophysical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
+15.4.1 Galaxy structure and its evolution over cosmic time . . . . . . . . . . . . 141
+15.4.2 Cluster structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
+15.4.3 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
+15.5 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
+15.6 Preparatory Work for Euclid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
+16 Cosmology and Astrophysics with a Euclid Supernova Survey 149
+16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
+16.2 Constraining Dark Energy with Type Ia Supernovae . . . . . . . . . . . . . . . . 149
+16.2.1 Using Euclid near-infrared photometry to improve the accuracy of cosmo-
+logical measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
+16.2.2 Euclid spectroscopy of supernovae type Ia . . . . . . . . . . . . . . . . . . 152
+16.2.3 Cosmology from Core-collapse Supernovae . . . . . . . . . . . . . . . . . . 153
+16.3 Astrophysics from Supernovae of all Types . . . . . . . . . . . . . . . . . . . . . . 153
+16.3.1 Dust extinction and near-infrared searches for supernovae . . . . . . . . . 153
+16.3.2 Supernova rate and galaxy evolution . . . . . . . . . . . . . . . . . . . . . 154
+16.3.3 Type Ia supernova progenitors and stellar evolution . . . . . . . . . . . . 154
+16.3.4 Core Collapse supernova progenitors . . . . . . . . . . . . . . . . . . . . . 155
+16.3.5 Core collapse supernova rates . . . . . . . . . . . . . . . . . . . . . . . . . 155
+17 Euclid Microlensing Planet Hunter 159
+17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
+17.2 Basic microlensing principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
+17.3 A program on board Euclid to hunt for planets . . . . . . . . . . . . . . . . . . . 163Contents 10
+17.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
+18 The Milky Way and the Local Group 168
+18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
+18.2 Milky Way Science from the Wide Survey . . . . . . . . . . . . . . . . . . . . . . 169
+18.3 The Local Neighbourhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
+18.4 A Euclid Galactic Plane Survey (E-GPS) . . . . . . . . . . . . . . . . . . . . . . 170
+18.4.1 Galactic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
+18.4.2 Young Stellar Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
+18.4.3 Brown Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
+18.4.4 Late Stages of Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . 172
+18.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
+V SIMULATIONS 173
+19 Image Simulations 174
+19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
+19.2 Image simulation pipelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
+19.3 Instrumental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
+19.4 Sky background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
+19.5 Analysis of the images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
+19.5.1 Detections vs effective exposure time . . . . . . . . . . . . . . . . . . . . . 179
+19.5.2 Detections vs sky background level . . . . . . . . . . . . . . . . . . . . . . 179
+19.5.3 Redshift distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
+19.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
+20 Photo-z 183
+20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
+20.2 Generation of photo-z from mock photometric catalogues . . . . . . . . . . . . . 185
+20.3 The Near Infra-Red Filter Configuration . . . . . . . . . . . . . . . . . . . . . . . 185
+20.4 Ground-based photometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
+20.4.1 Effect of depth of ground based photometry . . . . . . . . . . . . . . . . . 187
+20.5 Characterization of N(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
+20.5.1 Direct Sampling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
+20.5.2 Estimation of N(z) from photo-z themselves . . . . . . . . . . . . . . . . . 191
+20.6 Internal Calibration of Galactic Foreground Extinction . . . . . . . . . . . . . . . 192
+20.7 Impact on photo-z from blended objects at different redshifts . . . . . . . . . . . 193
+20.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
+21 The EUCLID ETC 199
+21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
+21.2 Layout overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
+21.3 Using the E-ETC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Contents 11
+22 Cosmological Simulations 204
+22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
+22.2 A 70 billion particle N-body simulation . . . . . . . . . . . . . . . . . . . . . . . 205
+22.3 A different strategy: many small simulations . . . . . . . . . . . . . . . . . . . . 205
+22.4 Introducing the baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
+22.5 Disc intrinsinc alignement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
+VI APPENDIX: Letters of Support from Ground-Based Projects 210Part I
+INTRODUCTION
+12Chapter 1
+EIC Science Book Overview
+Authors: Alexandre Refregier(CEA Saclay) for the EIC collaboration
+Abstract
+The energy density of the universe is dominated by dark energy and dark matter, two mysterious
+components which pose some of the most important questions in fundamental science today.
+Euclid is a high-precision survey mission designed to answer these questions by mapping the
+geometry of the dark universe. Euclid’s Visible-NIR imaging and spectroscopy of the entire
+extragalactic sky will further produce extensive legacy science for various fields of astronomy.
+Over the 2008-2009 period, Euclid has been the object of an ESA Assessment Phase in which
+the study of the Euclid Imaging instrument was under the responsibility of the Euclid Imaging
+Consortium (EIC). In this chapter, we provide an overview of the EIC Science Book which
+presents the studies done by the EIC science working groups in the context of this study phase.
+We first give a brief description of the Euclid mission and of the imaging instrument and surveys.
+We then summarise the primary and legacy science which will be achieved with the Euclid imaging
+surveys, along with the simulations and data handling scheme which have been developed to
+optimise the instrument and ensure its science performance.
+1.1 Introduction
+The energy density of the universe is dominated by dark energy and dark matter, two mysterious
+components which pose some of the most important questions in fundamental science today.
+Euclid is a high-precision survey mission designed to answer these questions by mapping the
+geometry of the dark universe. Euclid’s Visible-NIR imaging and spectroscopy of the entire
+extragalactic sky will further produce extensive legacy science for various fields of astronomy.
+The Euclid payload consist of a wide-field imager in the visible and NIR and a NIR spectrometer.
+During the 2008-2009 period, Euclid has been the object of an ESA Assessment Phase during
+which the study of the imaging instrument was under the responsibility of the Euclid Imaging
+Consortium (EIC; see members and organisation on page 3). This EIC Science book presents
+the work done by the EIC science working groups in the context of this Assessment Phase. It
+131.2. The Euclid Mission 14
+Figure 1.1: The Dark Universe. Left: Evolution of the universe from a homogeneous state after
+the big bang through cooling and expansion. Right: Mass-energy budget at our cosmological
+epoch showing that the universe is dominated by two mysterious components, dark energy and
+dark matter, whose nature poses some of the most important questions in fundamental physics.
+is a supporting document within the EIC data pack delivered at the end of the assessment
+phase. Some of this work has appeared in scientific publications which are listed in Chapter 2. A
+description of the overall Euclid mission is presented in the Euclid Assessment Study Report
+(Euclid Yellow Book).
+In this overview of the EIC Science Book, we first give a brief description of the Euclid mission
+and then describe the imaging instrument and surveys. We then summarise the primary and
+legacy science which will be achieved with the imaging instruments, as well as the simulations
+and data handling scheme which have been developed to optimise the instrument and ensure its
+scientific performance.
+1.2 The Euclid Mission
+Euclid is a candidate ESA mission to map the geometry and evolution of the dark universe with
+unprecedented precision (see Figure 1.1). Its primary goal is to place high accuracy constraints
+on Dark Energy, Dark Matter, Gravity and cosmic initial conditions using two independent
+cosmological probes: weak gravitational lensing (WL) and baryonic acoustic oscillations (BAO).
+For this purpose, Euclid will measure the shape and spectra of galaxies over the entire extragalactic
+sky in the visible and NIR, out to redshift 2, thus covering the period over which dark energy
+accelerated the universe expansion ( <10 Billion years). Galaxy clusters and the Integrated
+Sachs-Wolfe effect will be used as secondary cosmological probes. The Euclid datasets will also
+provide unique legacy surveys for the study of galaxy evolution, large-scale structure, the search
+for high redshift objects and for various other fields of astronomy.
+The baseline mission is based on a telescope with a primary mirror of 1 .2 m diameter (see
+Figure 1.2). The payload baseline comprises wide field instruments (0.5 deg2): an imaging
+instrument comprising a visible and a NIR channel, and a NIR spectroscopic instrument. The
+visible channel is used to measure the shapes of galaxies for weak lensing, with a resolution of
+0.18 arcsec in a wide visible red band (R+I+Z, 0 .55−0.92µ). The NIR photometric channel
+provides three NIR bands (Y, J, H, spanning 1 .0−1.6µ) with a resolution of 0 .3 arcsec. The
+baseline for the NIR spectroscopic channel operates in the wavelength range 1 .0–2.0 micron in1.3. The Euclid Imaging Instrument and Surveys 15
+Figure 1.2: Euclid Spacecraft concept from EADS (left) and Thales-Alenia (right).
+slitless mode at a spectral resolution R∼500, employing 0 .5” pixels. The optional spectroscopic
+implementation is slit spectroscopy using digital micro-mirror devices (DMDs).
+The mission will perform a wide survey of the entire extragalactic sky (20,000 deg2) down
+to 24.5 AB magnitude in the visible, thus providing 30–40 resolved galaxies per amin2. For all
+galaxies, photometric redshifts are obtained from the broad-band visible and near-IR measure-
+ments and complementary ground-based observations in other visible bands. For 40–60 million
+galaxies with Hαline flux level >3−4×10−16erg s−1cm−2the slitless spectrometer will
+directly measure the redshift with a success rate in excess of 35%. A deep survey will also be
+performed to monitor the stability of the spacecraft and payload and for legacy science.
+The Euclid mission concept followed from two dark energy related missions which were
+proposed to ESAs Cosmic Vision programme and were jointly selected: the Dark UNiverse
+Explorer (DUNE) and the SPectroscopic All-sky Cosmology Explorer (SPACE). Initial studies
+in 2008 led to the Euclid merged concept. Euclid has been the object of an Assessment Phase
+in 2008-2009 in the context of ESA’s Cosmic Vision programme and is a candidate for the first
+medium mission launch slot in late 2017. A description of the overall Euclid mission is presented
+in the Euclid Assessment Study Report (Euclid Yellow Book).
+1.3 The Euclid Imaging Instrument and Surveys
+The Euclid baseline payload consist of a Korsch telescope with a primary mirror of 1 .2 m diameter.
+The design of the telescope optics is such that it provides a large field of view and that it can feed
+the imaging channels and a spectroscopic channel. The imaging instrument (see Figure 1.3) are
+optimised for weak lensing and consists of a CCD based optical imaging channel (VIS), a NIR
+imaging photometry channel (NIP), a Common Mechanical Assembly (COMA) and a Payload
+Data Handling Unit (PDHU).
+VIS will measure the shapes of galaxies with a resolution of 0.18 arcsec (PSF FWHM) with
+0.1 arcsec pixels in one wide visible band (R+I+Z). NIP contains three NIR bands (Y, J, H),
+employing HgCdTe NIR detector with 0 .3 arcsec pixels. To accomplish the wide and deep surveys
+within the nominal mission lifetime of 5 years, each channel has a large field of view of about 0.51.4. Primary Science 16
+Figure 1.3: The Euclid Imaging instrument showing the visible imaging camera (VIS), the
+NIR imaging camera (NIP), the common mechanical assembly (COMA) and the Payload data
+Handling Unit (PDHU).
+deg2, and the system design is optimised for a sky survey with fast attitude slews to support a
+step-and-stare tiling mode. To meet the survey depth and sensitivity, the telescope has a well
+baffled design and is cooled to minimise background noise. For the NIR detectors, on-board
+on-the-ramp processing will be performed, i.e. combining image frames to lower the noise. The
+NIR related optics and detectors are cooled down to ∼100 K.
+Euclid’s primary wide survey will cover 20,000 deg2, i.e. the entire extragalactic sky, thus
+measuring shapes and redshifts of galaxies to redshift 2 as required for weak lensing and BAO.
+For weak lensing, Euclid will measure the shape of over 2 billion galaxies with a density of
+30–40 resolved galaxies per amin2in one broad visible R+I+Z band (550-920 nm) down to AB
+mag 24.5 (10σextended). The photometric redshifts for these galaxies will reach a precision of
+σ(z)/(1 +z) = 0.03−0.05. They will be derived from three additional Euclid NIR band (Y,J,H
+in the range 0 .92–2.0 micron) reaching AB mag 24 (5 σpoint sources) in each, complemented by
+ground based photometry in visible bands derived through engaged discussions and collaborations
+with the Pan-STARRS, DES and LSST ground based projects (see support letters from the DES
+and Pan-STARRS projects in Part VI). To measure the shear from the galaxy ellipticities a tight
+control is imposed on possible instrumental effects and will lead to the variance of the shear
+systematic errors to be less than 10−7. Euclid will also perform a deep survey, about 2 mag
+deeper than the wide survey and which will cover an area of about 40 deg2. Although unique as
+a self standing survey, the deep survey will also be used to monitor the stability of the spacecraft
+and payload through repeated visits of the same regions.
+1.4 Primary Science
+Over the last decades, a combination of observations has led to the emergence and confirmation
+of the concordance cosmological model (see Figure 1.1). Remarkably, the energy density of the
+resulting universe is dominated by two mysterious components. First, 74% of its energy density1.5. Legacy Science 17
+is in the form of Dark Energy, which is causing the Universe expansion to accelerate. Another
+22% of the energy in the Universe is in the form of dark matter, which exerts a gravitational
+attraction as normal matter, but does not emit light. One possibility to explain one or both of
+these puzzles is that Einstein’s General Relativity, and thus our understanding of gravity, needs
+to be revised on cosmological scales. Together, dark energy and dark matter pose some of the
+most important questions in fundamental physics today.
+The Euclid Imaging instrument is optimised for the Weak gravitational Lensing cosmological
+probe. Weak lensing is a method to map the dark matter and measure dark energy through the
+distortions of galaxy images by mass inhomogeneities along the line-of-sight. The Euclid imaging
+surveys will also make use of several secondary cosmological probes such as the Integrated Sachs
+Wolfe Effect (ISW), galaxy clusters to provide additional measurements of the cosmic geometry
+and structure growth. Weak lensing requires a high image quality for the shear measurements,
+near-infrared imaging capabilities to measure photometric redshifts for galaxies at redshifts z>1,
+a very high degree of system stability to minimize systematic effects, and the ability to survey
+the entire extra-galactic sky. Such a combination of requirements cannot be met from the ground,
+and demands a wide-field Visible/NIR space mission. A central design driver for Euclid is the
+ability to provide tight control of systematic effects in space-based conditions.
+With this capability, the Euclid imaging instrument will contribute to the four Euclid primary
+science objectives in fundamental cosmology: (1) Euclid will measure the dark energy equation
+of state parameters w0andwato a precision of 2% and 10% from the geometry and structure
+growth of the Universe. Euclid will thus achieve a Dark Energy Figure of Merit of 500 (1500)
+without (with) Planck Priors, thus improving by a factor of 50 (150) upon current knowledge. (2)
+Euclid will test the validity of General Relativity against modified gravity theories, and measure
+the growth factor exponent γto an accuracy of 2%. (3) Euclid will study the properties of dark
+matter by mapping its distribution, testing the Cold Dark Matter paradigm and measuring the
+sum of the neutrino masses to a few 0 .01eV in combination with Planck. (4) Euclid will improve
+the constraints on the initial condition parameters by a factor of 2–30 compared to Planck alone.
+Euclid is therefore poised to uncover new physics by challenging all sectors of the cosmological
+model. The Euclid survey can thus be thought as the low-redshift, 3-dimensional analogue
+and complement to the map of the high-redshift universe provided by CMB experiments. The
+primary science which should be achieved by the Euclid Imaging Surveys is described in Parts II,
+IIIandIV
+1.5 Legacy Science
+Beyond these breakthroughs in fundamental cosmology, the Euclid imaging surveys will provide
+unique legacy science in various fields of astrophysics. In the area of galaxy evolution and
+formation, Euclid will deliver high quality morphologies and masses for billions of galaxies out to
+z∼2, over the entire extra-galactic sky, with a resolution 4 times better and 3 NIR magnitudes
+deeper than ground based surveys. The Euclid deep survey will probe the “dark ages” of galaxy
+formation as it is predicted to find thousands of galaxies at z >6, of which about 100 could
+be atz >10 i.e. probing the era of reionisation of the Universe. These high redshift galaxies
+and quasars will be critical targets for JWST and E-ELT. Euclid will also augment the Gaia
+survey of our Milky Way, taking it several magnitudes deeper. Below V= 20, Euclid will provide
+complementary information to Gaia, adding infrared colours for every Gaia star it observes;
+hence breaking the age-metallicity degeneracy, which is critical for the chemical enrichment
+history of our Galaxy. Also, the all-sky coverage of Euclid will detect nearby extremely low
+surface brightness tidal streams of stars thus allowing us to probe the formation and evolution of
+our own Galaxy. The Euclid Imaging surveys will also provide key measurements of the mass1.6. Simulations 18
+

+Figure 1.4: Overview of the EIC simulation tools.
+function of galaxy clusters (esp. in combination with eROSITA, Planck and SZ telescopes), and
+of over 105strong lensing systems, and should find thousands of intermediate-redshift supernovae
+in the near-IR. Euclid could also undertake a programme to detect Earth-mass planets in the
+habitable zone through the microlensing technique. The legacy science which will be achieved by
+the Euclid Imaging surveys is described in Part IV.
+1.6 Simulations
+The Euclid Imaging Consortium has performed simulations at each stage of the performance
+monitoring and prediction chain. A comparison of the key results with our requirements is
+provided. In this book, we set out additional details of our simulations. An overview of the
+entire simulations process is provided in Figure 1.4and we give details of individual components
+in Part V.
+1.7 Data handling
+The processing of the imaging data will be handled through the Euclid data handling architecture
+which is based on existing space-based and ground-based projects (see Figure 1.5for an overview).
+Dedicated teams from the EIC and the spectroscopic consortium (ENIS) will process the data
+during all phases of the mission through and Instrument Operations Centres (IOCs) and several
+Science Data Centres (SDCs). The IOCs will be responsible for the first level standard data
+processing (calibration, removal of the instrumental effects, etc) and for requesting to the SOC
+corrective action in operations. The SDCs will be in charge of second and third level data products
+and the development of simulation pipelines. The data handling system includes a common
+archive, the Euclid Mission Archive (EMA) which will support the sharing of data within the
+project, the reporting of quality controls and a built in redundancy in the key processing tasks. A
+subset of the qualified data of the EMA will form the Euclid Legacy Archive (ELA) which will be1.7. Data handling 19
+ 
+Figure 1.5: Overview of the Euclid Ground segment as proposed in the EIC-ENIS Ground
+Segment document.
+delivered to the astronomical community at large. The high-precision requirements for the weak
+lensing analysis will be achieved within this architecture using redundant cross-check, built-in
+simulations, quality control, and software development from existing weak lensing pipelines.Chapter 2
+Publications
+2.1 Collaboration Papers
+The following are publications from the DUNE/EIC Collaboration:
+•The Dark Universe Explorer (DUNE): Proposal to ESA’s Cosmic Vision ; DUNE collabora-
+tion, Experimental Astronomy, 2008, arxiv:0802.2522
+•Summary of the DUNE Mission Concept ; DUNE collaboration , To appear in Proc. SPIE,
+2008, arxiv:0807.4036
+•The focal plane instrumentation for the DUNE mission ; DUNE collaboration, To appear
+in Proc. SPIE, 2008, arxiv:0807.4037
+•Description of the DUNE concept as defined during the CNES Phase 0 study ; Refregier et
+al., Proc. of the SPIE Symposium, Orlando, 6265,58 (arXiv:0610062)
+•NODI optical design for the DUNE CNES phase 0 concept ; Grange et al., Proc. of the
+SPIE Symposium, Orlando, 6265,129
+2.2 Science working groups and related papers
+The following are publications by EIC Science Working Group members done in the context of
+the Euclid Assessment phase:
+•Optimal Surveys for Weak Lensing Tomography ; Amara, A.; Refregier, A., MNRAS, 381,
+1018, (2007)
+•Systematic Bias in Cosmic Shear: Beyond the Fisher Matrix ; Amara, A.; Refregier, R.,
+MNRAS, 391, 228, (2008)
+•Requirements on Systematics: The Interplay Between Software and Hardware ; Amara A.;
+Refregier A.; Paulin-Henriksson S., submitted to MNRAS, arXiv:0905.3176
+•Photo-z for weak lensing tomography from space: the role of optical and near-IR photometry ;
+Abdalla, F., Amara, A.; Capak, P.; Cypriano, E.; Lahav, O.; Rhodes, J., MNRAS, 387,
+969, (2008)
+202.2. Science working groups and related papers 21
+•Measuring the dark side (with weak lensing) ; Amendola, L.; Kunz M.; Sapone, D.; JCAP,
+4, 013, (2008)
+•Measuring Dark Matter Substructure with Galaxy-Galaxy Flexion Statistics ; Bacon, D.;
+Amara, A.; Read, J., submitted to MNRAS , arXiv:0909.5133
+•Optimal capture of non-Gaussianity in weak lensing surveys: power spectrum, bispectrum
+and halo counts ; Berge, J.; Amara, A.; Refregier, A., submitted to ApJ, arXiv:0909.0529
+•Dark energy constraints from cosmic shear power spectra: impact of intrinsic alignments
+on photometric redshift requirements ; Bridle, S.; King, L.; NJP, 9, 44, (2007)
+•Results of the GREAT08 Challenge: An image analysis competition for cosmological
+lensing ; Bridle, S.; Balan, S.; Bethge, M.; Gentile, M.; Harmeling, S.; Heymans, C.; Hirsch,
+M.; Hosseini, R.; Jarvis, M.; Kirk, D.; Kitching, T.; Kuijken, K.; Lewis, A.; Paulin-
+Henriksson, S.; Scholkopf, B.; Velander, M.; Voigt, L.; Witherick, D.; Amara, A.; Bernstein,
+G.; Courbin, F.; Gill, M.; Heavens, A.; Mandelbaum, R.; Massey, R.; Moghaddam, B;
+Rassat, A.; Refregier, A.; Rhodes, J.; Schrabback, T.; Shawe-Taylor, J.; Shmakova, M.;
+van-Waerbeke, L.; Wittman, D.; submitted to MNRAS, arXiv0908.0945
+•Optimising large galaxy surveys for ISW detection ; Douspis, M.; Castro, P.; Caprini, C.;
+Aghanim, N., A&A, 485, 395, (2008)
+•Gravitational Flexion by Elliptical Dark Matter Haloes ; Hawken, A.; Bridle, S.; submitted
+to PRD, arXiv0903.3870
+•On model selection forecasting, Dark Energy and modified gravity Heavens A.; Kitching, T.;
+Verde, L.; MNRAS, 380, 1029, (2007)
+•Fisher matrix decomposition for dark energy prediction ; Kitching, T.; Amara, A., MNRAS,
+398, 213, (2009)
+•Cosmological Systematics Beyond Nuisance Parameters : Form Filling Functions ; Kitching,
+T.; Amara, A.; Abdalla, F.; Joachimi, B.; Refregier, A., MNRAS, 399, 2107, (2009)
+•Finding Evidence for Massive Neutrinos using 3D Weak Lensing ; Kitching, T.; Heavens,
+A.; Verde, L.; Serra, P.; Melchiorri, A.; PhRvD, 77, 3008, (2008)
+•Systematic effects on dark energy from 3D weak shear ; Kitching, T.; Taylor, A.; Heavens,
+A., MNRAS, 389, 173, (2008)
+•Pixel-based correction for Charge Transfer Inefficiency in the Hubble Space Telescope
+Advanced Camera for Surveys ; Massey, R.; Stoughton, C.; Leauthaud, A.; Rhodes, J.;
+Koekemoer, A.; Ellis, R.; Shaghoulian, E., MNRAS, 1564, (2009)
+•Realistic simulations of gravitational lensing by galaxy clusters: extracting arc parameters
+from mock DUNE images ; Meneghetti, M.; et al., A&A, 462, 403, (2008)
+•Requirements on PSF Calibration for Dark Energy from Cosmic Shear ; Paulin-Henriksson,
+S.; Amara, A.; Voigt, L.; Refregier, A.; Bridle S., A&A, 484, 67. (2009)
+•Optimal PSF modeling for weak lensing : complexity and sparsity ; Paulin-Henriksson S.;
+Refregier A.; Amara A., A&A, 500, 647, (2009)2.2. Science working groups and related papers 22
+•Deconstructing Baryon Acoustic Oscillations: A Comparison of Methods ; Rassat, A.;
+Amara, A.; Amendola, L.; Castander, F.; Kitching, T.; Kunz, M.; Refregier, A.; Wang, Y.;
+Weller, J.; submitted to MNRAS, arXiv0810.0003
+•A halo model for intrinsic alignments of galaxy ellipticities ; Schneider, M.; Bridle, S.,
+submitted to PRD, arXiv0903.3870
+•Constraining Modified Gravity and Growth with Weak Lensing ; Thomas, S.; Abdalla, F.;
+Weller, J., MNRAS, 395, 197, (2009)
+•Full-Sky Weak Lensing Simulation with 70 Billion Particles ; Teyssier, R.; Pires, S.; Prunet,
+S.; Aubert, D.; Pichon, C.; Amara, A.; Benabed, K.; Colombi, S.; Refregier, A.;, Starck,
+J.-L., A&A, 497, 335, (2009)
+•Limitations of model fitting methods for lensing shear estimation ; Voigt, L.; Bridle, S.;
+MNRAS, 1560, (2009)Part II
+THEORY
+23Chapter 3
+Fundamental Cosmology with Euclid:
+A Theoretical Perspective
+Authors: Martin Kunz (University of Geneva), Luca Amendola (University of Heidelberg &
+INAF Rome), Thomas Kitching (University of Edinburgh), Adam Amara (ETH Zurich), David
+Bacon (University of Portsmouth), Alan Heavens (University of Edinburgh), Jochen Weller
+(University of Munich).
+Abstract
+This Chapter provides an overview of the four primary science objectives of Euclid, with a
+special emphasis on the dark energy / modified gravity aspects. We discuss the cosmological
+phenomenology of dark energy and modified gravity at the level of first order perturbation theory.
+We show that in addition to the equation of state parameter w(z) we require two additional
+functions of scale and time to characterise the scalar sector for cosmological observations. We
+argue that measuring these two functions will be an important goal for observational cosmology
+in the next decades, and discuss how Euclid will be able to measure them. We also discuss how
+Euclid can constrain particle properties with a particular emphasis on dark matter and neutrino
+constraints. We highlight the synergy of Euclid with CMB experiments that will measure the
+initial conditions of the perturbations. We finally highlight the synergy between Euclid and
+future large European projects.
+3.1 Introduction
+In the early nineties, the model of the Universe most widely accepted by theoretical cosmologists
+was flat and filled with matter. But slowly doubts increased whether this simple model was
+really an adequate reflection of reality. Early measurements of the large scale structure and of
+velocity fields indicated that there was not enough matter to make the Universe flat. In 1998,
+just over a decade ago, two teams using supernovae to measure luminosity distances made an
+amazing discovery: the expansion rate of the universe was accelerating. Combined with the other
+data and new measurements of the anisotropies in the cosmic microwave background (CMB),
+a coherent picture started to emerge. Even though the Universe appears to be spatially flat,
+matter only makes up 25% of the critical energy today, and the rest is “something else”.
+The best-known candidate for the 75% of “something else” is the cosmological constant Λ,
+originally introduced and then abandoned by Einstein (see e.g. ( Copeland et al. 2006) for a
+review on dark energy models). According to particle physics there should be a cosmological
+243.1. Introduction 25
+constant from the zero point energy of quantum fields. But how big is the zero-point contribution
+to Λ? We know that general relativity (GR), being a classical theory, breaks down at some
+high energy scale. Sensible choices for a high-energy cut-off would be the Planck scale, or if
+SUSY is realised in nature, the supersymmetry breaking scale. SUSY also may lead to an exact
+cancellation of the zero-point energy, this cancellation makes SUSY an interesting concept in
+dark energy research, but from experimental particle physics we know that it is definitely broken
+below 100 GeV. The classical world we observe around us is not supersymmetric, however dark
+matter may be a relic population of the lightest supersymmetric particle.
+Comparing the measured values required to make up the missing 75% of the critical energy
+density today with the theoretical value from quantum field zero-point energy, there is a
+discrepancy of 120 orders of magnitude (when cutting at the Planck scale) or still over 40 orders
+of magnitude (when cutting at the lowest possible SUSY breaking scale). Even worse, the
+quantum contributions from gravity coupling to well-understood physics, like electrons, should
+already be much larger than the observed size of Λ! It is always possible to subtract a “bare”
+value, but there is no guidance from theory. This cancellation may then be a window to physics
+taking place high in the energy scale, where quantum gravity is important, and which we cannot
+probe in any other way. This is one of the reasons why the “something else” in the Universe is
+so exciting. It may change our understanding of the Universe as completely as the development
+of quantum mechanics did in the last century. To highlight just one possibility: maybe our world
+has 10 spatial dimensions, but we only experience 3 of them, and the dark energy is nature’s way
+of telling us about all the others! If so, then nature may have left a few more clues lying around
+in between the stars and galaxies, and one of the main science goals for the Euclid mission is to
+go and look for those clues.
+But it is not only the 75% dark energy that is puzzling. Both big-bang nucleosynthesis (BBN)
+and the CMB indicate that baryons (the “normal” matter) only make up about 4% of the energy
+density today. The remaining 21% are an unknown substance called dark matter because it is
+apparently invisible but clusters at least on large scales like pressureless matter. We do not know
+yet what the dark matter is, but candidates exist in extensions of the standard model of particle
+physics, for example in SUSY. Some aspects of the nature of dark matter is expected to show up
+in the precise way that it clusters to form the halos of galaxies, clusters and generally the large
+scale structure in the Universe. Such matter concentrations bend light and act like gravitational
+lenses, and the Euclid Imaging Instrument is uniquely qualified to detect the weak distortions in
+the shapes of background galaxies due to dark matter concentrations in between those galaxies
+and us.
+Yet another cosmic riddle concerns the origin of the tiny perturbations which have grown
+over time under the influence of gravity to finally become galaxies. Most cosmologists accept the
+inflationary model in which the Universe underwent a phase of extremely rapid expansion early
+in its evolution, similar to what seems to be starting again now. During this expansion, tiny
+quantum perturbations were enlarged and froze once they became larger than the horizon at
+the time. Much later they re-entered the horizon and started to collapse under the influence
+of gravity and to form the observed structure. The large-scale structure therefore retains the
+imprint of what happened in the very early Universe.
+The cosmological standard model is very successful at describing the current measurements,
+however it lacks a physical description for over 95% of the “stuff” in the Universe, and we do not
+know with any certainty how the seeds for the observed structures like galaxies were generated.
+In this chapter we will describe how Euclid in general and the EIC in particular can make major
+advances in our understanding of all those questions.3.2. Phenomenology vs fundamental theory of the dark energy 26
+3.2 Phenomenology vs fundamental theory of the dark energy
+Although a plethora of models have been constructed to explain observed late-time accelerated
+expansion, none of them is really appealing on theoretical grounds. An alternative approach in
+this situation, especially when designing a future experiment, is to ask what can actually be
+measured by cosmological observations, and how we can exploit the data to learn something
+about the nature of the dark energy phenomenon, and generally about cosmology.
+A successful example for such a phenomenological parameterisation in the dark energy context
+is the equation of state parameter of the dark energy component, w≡p/ρ. If we can consider the
+Universe as evolving like a homogeneous and isotropic Friedmann-Lemaˆ ıtre-Robertson-Walker
+(FLRW) universe, then the only observationally accessible quantity is the expansion rate of the
+universeH, given by the Friedmann equation,
+H(a)2=/parenleftbigg˙a
+a/parenrightbigg2
+=8πG
+3(ρm(a) +ρDE(a)). (3.1)
+For simplicity we neglect here radiation and assume that space is flat; we also take the value
+of the speed of light to be c= 1. This equation governs the expansion law of the Universe as
+whole and can be studied with geometrical tests: luminosity and angular diameter distances
+are determined by integrals of 1 /H; andHcan be directly measured by a number of methods,
+among which are radial baryonic acoustic oscillations (BAO) ( Blake & Glazebrook 2003;Seo &
+Eisenstein 2003) or the dipole of the supernova distribution ( Bonvin et al. 2006).
+For any dark energy or modified gravity model, it is possible to compute H(a) and compare it
+to the observational data. However, this is not necessary: anyexpansion history can be created
+through adding a (possibly effective) dark energy component which is parameterised through
+a choice of w(a): The dark energy (or anything else) is described by its homogeneous energy
+densityρDEand the isotropic pressure pDE, corresponding to the T0
+0andTi
+ielements respectively
+of the energy momentum tensor (in the global FLRW frame). Any other non-zero component Tj
+i
+would require us to go beyond the FLRW description of the Universe. The evolution of ρis then
+governed by the “covariant conservation” equation Tν
+µ;ν= 0 which is just
+˙ρDE=−3H(ρDE+pDE) =−3H(1 +w)ρDE. (3.2)
+Instead of working forward from a specific model, this allows us to work backwards from the
+observations. For any model one can predict its own effective w(a) and compare it to observational
+limits. Also, even though this w(a) is a phenomenological quantity, we can immediately draw
+conclusions: if the observed wever deviates significantly from −1 then a cosmological constant
+is ruled out, and if w<−1 then canonical scalar field models of the dark energy are in trouble.
+3.3 Beyond the background
+OnceH(a) has been measured with the needed accuracy, then w(a) can be extracted. This then
+allows us for example to reconstruct a scalar field potential that would give rise to this expansion
+history. However, there is an ambiguity present since it is also possible to construct a function
+fso that a modified gravity model leads to the same expansion history. In other words, it is
+possible to get the same global expansion law by changing either side of Einstein’s equations,
+the metric side or the matter side. The expansion history alone does therefore not allow us to
+distinguish between large classes of models. Can we do better?
+Arguably this is not the right question: we have to do better! A well-known example is
+afforded by the growth-rate of the matter perturbations: once the expansion history H(a) is3.3. Beyond the background 27
+Figure 3.1: Left: Current 95% constraints on w(z) from CMB data (WMAP3), SN-Ia data
+(Union sample) and BAO (SDSS) using the w0,waparametrisation for a quintessence-like model.
+Right: Examples of different growth rates with predicted error bars for Euclid. The two curves
+can be clearly distinguished. DGP with γ≈0.68 would be even more discrepant.
+measured, the dark matter perturbations evolve according to
+¨δm+ 2H˙δm= 4πGρ mδm. (3.3)
+We can therefore predict how the density contrast δm(a) grows. A common parametrisation is in
+terms of a parameter γ(Wang & Steinhardt 1998;Lueet al. 2004),
+dlogδm
+dloga= Ω m(a)γ. (3.4)
+In this case, γis uniquely fixed by the expansion history, which in turn depends on w(a). A
+good fit to the full numerical result is γ≈0.55 + 0.05(1 +w) (Linder 2005). However, in deriving
+Eq. ( 3.3) we have used the Poisson equation, ∆ φ= 4πGρδ ,andwe have assumed both that it is
+valid, and that only matter contributes to the total density perturbation ρδon the right hand
+side. In general this is not the case, so that we need to go beyond the background description
+of the Universe and understand dark energy and modified gravity at the level of perturbation
+theory. Then we will find that they make different predictions for γ, even if they lead to the
+sameH(a)! The same is true for weak lensing, the CMB, redshift space distortions, and so on.
+But how many new parameters are necessary to describe all possible measurements?
+The metric is in principle composed of ten arbitrary functions gµν. But coordinate invariance
+allows us to eliminate four of those, and at the level of first order perturbation theory, the
+remaining six can be split into two scalar, two vector and two tensor degrees of freedom. If we
+work in the Newtonian gauge, and limit ourselves to scalar (density) perturbations, we thus
+have to add two functions φandψto the metric, which play a role very similar to gravitational
+potentials. The line element then becomes
+ds2=a(τ)2/bracketleftbig
+−(1 + 2ψ)dτ2+ (1−2φ)dx2/bracketrightbig
+. (3.5)
+These potentials can be considered as being similar to Hin that they enter the description of the
+space-time geometry (but they are functions of scale/position as well as time). Also the energy
+momentum tensor becomes more general, and ρis complemented by perturbations δρas well as3.3. Beyond the background 28
+a velocityvi. The pressure pnow can also have perturbations δpand there can further be an
+anisotropic stress π.
+The reason why we grouped the new parameters in this way is to emphasise their role: at the
+background level, the evolution of the universe is described by H, which is linked to ρby the
+Einstein equations, and pcontrols the evolution of ρbut is a priori a free quantity describing
+the physical properties of the fluid. Now in addition there are φandψdescribing the Universe,
+and they are linked to δρandvof the fluids through the Einstein equations. δpandπin turn
+describe the fluids. Actually, there is a simplification: the total anisotropic stress πdirectly
+controls the difference between the potentials, φ−ψ.
+This means that a general dark energy component can be described by phenomenological
+parameters similar to w, even at the level of first order perturbation theory. This description
+adds two new parameters δpandπ, which are both functions of scale as well as time. These
+parameters fully describe the dark energy fluid, and they can in principle be measured.
+However, recently much interest has arisen in modifying GR itself to explain the accelerated
+expansion without a dark energy fluid. What happens if we try to reconstruct our parameters in
+this case? Is it possible at all?
+Let us assume that the (dark) matter is three-dimensional and conserved, and that it does
+not have any direct interactions beyond gravity. We assume further that it and the photons
+move on geodesics of the same (possibly effective) 3 + 1 dimensional space-time metric. In this
+case we can write the modified Einstein equations as
+Xµν=−8πGT µν (3.6)
+where the matter energy momentum tensor still obeys Tν
+µ;ν= 0. While in GR this is a consequence
+of the Bianchi identities, this is now no longer the case and so this is an additional condition on
+the behaviour of the matter1.
+In this case, we can construct Yµν=Xµν−Gµν, so thatGµνis the Einstein tensor of the
+3+1 dimensional space-time metric and we have that
+Gµν=−8πGT µν−Yµν. (3.7)
+Up to the prefactor we can consider Yto be the energy momentum tensor of a dark energy
+component. This component is also covariantly conserved since Tis and since Gobeys the
+Bianchi identities. The equations governing the matter are going to be exactly the same, by
+construction, so that the effective dark energy described by Ymimics the modified gravity model
+(Hu & Sawicki 2007;Kunz et al. 2008).
+By looking at Ywe can then for example extract an effective anisotropic stress and an
+effective pressure perturbation and build a dark energy model which mimics the modified gravity
+model and leads to exactly the same observational properties ( Kunz & Sapone 2007). This
+provides a clear target for future experiments: their job is to measure the two additional functions
+describingYas precisely as possible. These functions can then provide clear hints about the
+nature of the dark energy phenomenon. For example, scalar field models have generically a sound
+horizon that could be detected in the data as it suppresses the dark energy perturbations on
+smaller scales ( Weller & Lewis 2003;Bean & Dor´ e 2004;Sapone & Kunz 2009). Modified gravity
+models on the other hand have generically a non-zero effective anisotropic stress, while scalar
+field models usually have π= 0 ( Mukhanov et al. 1992;Boisseau et al. 2000;Kunz & Sapone
+2007). Since the parameters of Yare just effective quantities for a modified gravity model, they
+1This condition could be relaxed due to the dark degeneracy, since all visible components are conserved to the
+best of our current knowledge.3.4. Extracting the potentials with Euclid 29
+will probably look very unnatural, so that Bayesian model comparison methods may provide an
+avenue for distinguishing between different competing possibilities.
+We can choose more or less freely which two additional functions we want to introduce to
+describe the dark energy perturbations or the modifications of gravity at the linear perturbation
+level. However, it is important to understand that we need two. A single extra parameter, for
+example only γ, does not suffice. In this case, we make effectively an additional choice, which
+may not be the one that we wanted. In the following section we take a closer look at the two
+gravitational potentials φ(k,a) andψ(k,a) since they are very close to the measurements and
+have a direct geometric interpretation. An alternative parametrisation used in Chapter 6stays
+close to the potentials, but replaces them with dimensionless quantities Qandηthat allow for
+simpler parameterisations, very similar to wcompared to H. They are defined through
+k2φ=−4πGa2Qρmδm, (3.8)
+ψ= (1 +η)φ. (3.9)
+Since weak lensing probes the combination φ+ψit is often useful to replace one of the two
+parameters by the combination Σ ≡Q(1 +η/2) (Amendola et al. 2008). There is also a relation
+betweenγandQ(1 +η). In the limit of small deviations from the canonical GR values, we have
+(Amendola et al. 2008)
+γ=γs/parenleftbigg
+1 +Q(1 +η)−1
+(w−1)(1−Ωm)/parenrightbigg
+(3.10)
+whereγsis the standard value (i.e. for uncoupled, unclustered dark energy in Einstein gravity).
+A significant deviation of γfromγswould amount to the discovery that either dark energy
+clusters or that gravity is modified.
+3.4 Extracting the potentials with Euclid
+Here we provide a high-level overview on how to extract the potentials from a survey like the
+one planned for the Euclid satellite project. A much more detailed look at how lensing measures
+these quantities is presented in Chapter 6.
+Since we actually observe galaxies, not the dark matter directly, we would really like to
+reconstruct not only the potentials but also the dark matter perturbation variables and the
+biasb. All together, we therefore would like to extract the set {δm,vm,φ,ψ,b }. Herevmis the
+divergence of the dark matter velocity field.
+Since the matter behaves in the usual way, we can use the conservation equations,
+δ/prime
+m= 3φ/prime−vm
+Ha, (3.11)
+v/prime
+m=−vm+k2
+Haψ, (3.12)
+where/prime=d/dlna. These equations determine effectively two of the variables. We therefore need
+three independent measurements. For example, if we could measure vm(k,a) then we would
+immediately know ψas well. This may not be impossible to do: one of our assumptions is that
+galaxies are test-particles that are accelerated by gravity in the same way as the dark matter,
+at least on scales where we can neglect baryonic physics like gas pressure. One of the three
+measurements should therefore be a probe of the velocity field. Apart from direct measurements,
+a promising approach is to extract the redshift space distortions of the galaxy power spectrum in
+redshift space,
+Pz= (1 +βµ2)2Pr (3.13)3.5. Dark Matter 30
+wherePr=b2δ2
+mis the real-space galaxy power spectrum, which is affected by bias and does not
+directly probe the matter power spectrum.
+Using this direction-dependence of the power spectrum, it is possible to separate out the
+velocity power spectrum, for example by comparing the power spectrum for µ= 1 andµ= 0.
+Integrating the conservation equation for δmand neglecting the φ/primeterm, it is then further possible
+to recover separately the matter power spectrum δ2
+mand the bias b. Since light bending is
+described by φ+ψ, we can finally use the lensing signal to recover φ.
+How this is best done in practice is an open and difficult question. Maybe it is preferable to
+parameterise all quantities of interest and to constrain all the parameters simultaneously by the
+data. The main point remains that it is possible in principle for Euclid alone to measure the
+potentials (which encode information about dark energy) and the dark matter perturbations as
+well as the bias, although at least CMB data would be additionally used in any realistic scenario
+to improve the constraints. Further data sets allow to cross-check the results and to test for the
+presence of systematic problems within the Euclid data set.
+3.5 Dark Matter
+The weak lensing probes of Euclid will measure φ+ψwith great accuracy. So far we have
+discussed how to extract the minute changes in the gravitational potentials due to different
+dark energy or modified gravity models. However, the largest contribution to the gravitational
+potential is expected to be due to the clustering of dark matter. For this reason, EIC is really an
+indirect dark matter detection and characterisation machine of tremendous power.
+We have already discussed how the EIC in combination with the Euclid spectrograph will
+measure the large-scale properties of the dark matter, δm(k,t) andvm(k,t), and disentangle them
+from effects due to the dark energy or modified gravity. This enables immediately strong bounds
+on physical properties of the dark matter, for example to put limits on its velocity dispersion
+which would lead to a cut-off in the perturbation spectrum.
+The weak lensing survey allows to go well beyond this global view of the dark matter. It will
+enable us to directly construct a 3D map of the dark matter distribution in the Universe. It
+will also allow us to characterise the dark matter sub-structure of cluster halos, and constrain
+galactic dark matter halo properties.
+Many extensions of the standard model of particle physics predict the existence of new
+long-lived or stable particles that can play the role of the dark matter (or at least of part of it).
+Two of the best-motivated such particles are the axion and the lightest supersymmetric particle
+(LSP). The axion appears in models where the Peccei-Quinn symmetry is introduced to solve
+the so-called strong CP problem of QCD. When this extra symmetry is spontaneously broken,
+a (pseudo-) Nambu-Goldstone boson appears, the axion. This particle would have extremely
+weak interactions and would be very difficult to find, except through its possible conversion to
+photons or with cosmological measurements.
+The LSP is a dark matter candidate in supersymmetric extensions of the standard model
+which also obeys R parity. R-parity prevents any SUSY reactions that could violate lepton/baryon
+number conservation hence SUSY particles can decay into lighter SUSY particles but not into
+standard model particles, so that the lightest SUSY particle (the LSP) is stable under these
+assumptions. Depending on the model parameters, there are various candidates for the LSP that
+include the neutralino, gravitino and axino. The expected mass and cross sectional ranges of
+these particles cover many orders of magnitude.
+If the majority of the dark matter consists of such new fundamental particles then the EIC
+will be able to constrain many of its properties like the mass and the interaction cross section,3.6. Initial Conditions 31
+independently of particle accelerators. Indeed cosmological observations from experiments such
+as Euclid are the only way to verify any candidate as being dark matter.
+The LSP is an example of so-called “cold” dark matter (CDM). These are particles that are
+non-relativistic when they decouple from the photon background. The opposite is called “hot”
+dark matter (HDM). An example of hot dark matter that is known to be present is due to the
+neutrinos. The observed oscillations between different neutrino flavours (for recent reviews see
+(King 2007;Lesgourgues & Pastor 2006;Amsler et al. 2008)) implies that neutrinos are massive
+since the mass eigenstates |νi/angbracketrightand the interaction eigenstates |να/angbracketrightcan be rotated with respect
+to each other,
+|να/angbracketright=Uαi|νi/angbracketright. (3.14)
+In this case, a neutrino flavour which is created in interactions can turn into a different flavour as
+it propagates through space. Neutrino oscillation experiments measure only the (absolute) mass
+differences between the mass eigenstates but are not sensitive to the absolute mass scale. Current
+constraints on the neutrino mass differences are δm23∼0.05 eV and δm12∼0.007 eV. Hence
+the hierarchy (order of m1,m2,m3) and total mass scale of the mass eigenstates is unknown.
+Proposedβdecay experiments (e.g. KATRIN ( KATRIN collaboration 2001;Kristiansen &
+Elgarøy 2008)) are expected to reach a precision of about ∼0.35 eV. But if the neutrino masses
+are of the order of the mass differences (as for the other particle types) then a better accuracy is
+required, for which cosmological probes will be needed. These can detect the cumulative effect
+of this small mass on cosmological scales since relativistic particles diffuse out of overdensities
+and so suppress structure formation on small scales. The EIC will provide strong limits on the
+sum of neutrino masses, and may even be able to determine individual neutrino masses, and as a
+result distinguish between the normal and inverted hierarchies.
+For these reasons there is a natural complementarity between Euclid and the LHC as well as
+dark matter and neutrino direct detection experiments, as we also discuss below. We will discuss
+the EIC dark matter science with a focus on weak lensing in much more detail in Chapter 7.
+3.6 Initial Conditions
+In measuring the properties of the dark matter and the dark energy or modified gravity, Euclid
+studies the evolution of cosmological perturbations. According to our current understanding,
+these perturbations were generated at a very early stage in the history of the Universe, called
+inflation. The very precise measurements of Euclid will not only allow us to characterise the
+evolution of the perturbations, but it will also be possible to study their initial properties. In
+this way, Euclid will probe inflation itself.
+The power spectrum of the initial perturbations is conventionally parametrised through a
+(weakly) scale dependent scalar index ns(k),
+P(k)≡/angbracketleftbig
+|δin(k)|2/angbracketrightbig
+∝kns(k). (3.15)
+If the Universe had been expanding exactly exponentially as the perturbations on observable scales
+were generated, then ns= 1 which is also called a Harrison-Zel’dovich (HZ) or scale-invariant
+spectrum. Inflation models predict small deviations from exact scale invariance, so that a precise
+characterisation of ns(k) will permit to constrain the space of allowed inflation models. Most
+inflation models also lead to a small deviation from a Gaussian distribution of the perturbations,
+and Euclid will be able to put strong limits on such non-Gaussianities. In addition, features in
+ns(k) may be correlated with a deviation from Gaussianity in the initial perturbations.3.7. Synergy of Euclid with LHC, EELT and Gaia 32
+CMB probes like Planck provide an excellent measurement of the initial power spectrum and
+very sensitive tests of non-Gaussianity on large scales. Euclid/EIC will rival the precision of
+Planck and provide independent measurements, allowing a cross-validation with the results of
+Planck and tighter combined limits. It will also extend the measurements to much smaller scales,
+allowing a longer lever arm to determine deviations from scale invariance with greater precision.
+Euclid is therefore an excellent complement to CMB probes.
+3.7 Synergy of Euclid with LHC, EELT and Gaia
+It is a hallmark of cosmology to require the input from many branches of physics and astronomy.
+The same applies to an observational project like Euclid where its scientific objectives will be
+strengthened and complemented by other planned surveys and experiments of the next decade
+and beyond. We mentioned already in passing the Planck mission, that will provide the best
+picture of the early universe, and will allow a firm initial point in the reconstruction of the cosmic
+dynamics. We expect at least three other large projects to interact constructively with Euclid’s
+goals: the Large Hadron Collider (LHC) at CERN, the European Extremely Large Telescope
+(EELT), and the Gaia astrometric mission. This impressive array of European-led projects hold
+the promise to maintain Europe’s decisive role in physics and astrophysics for the next decades.
+Let us discuss briefly the area of overlap with Euclid.
+The primary scientific target for the LHC accelerator is the search for the last missing brick
+of the particle standard’ model, the Higgs boson. Although its discovery could be classified
+among the biggest triumphs of technology and human inventiveness, its existence is taken almost
+for granted by particle physicists and many eyes are already pointed towards the next step, the
+search for new physics at high energy scales, i.e. for features not predicted within the standard
+model. One possibility would be the discovery of supersymmetry. This would have far-reaching
+implications for cosmology, since the lightest supersymmetric particle is seen as one of the main
+candidates for the particles making up the dark matter. In this case, the comparison between
+the collider data of LHC and the cosmological measurements of Euclid would allow to check
+whether this model is consistent and allow for a much better determination of particle properties
+than is possible with either experiment alone. An even more exciting discovery would be the
+existence of extra spatial dimensions, as predicted by e.g. string theory. The energy of the
+LHC beam could for the first time scrape the surface of our three-dimensional world and reveal
+a new extra-dimensional Universe. This would completely change the foundations of particle
+physics and cosmology, providing some scenarios of dark energy (and inflation) based upon
+extra-dimensional physics with a sound experimental basis.
+The EELT, if realised, will be the largest optical telescope on Earth. Its 42 meter primary
+mirror will collect a huge quantity of light from distant sources and provide the astrophysics
+community with many new opportunities in all fields, from extrasolar planet search to high-
+resolution spectroscopy. One of the scientific goals of EELT is the detection of the so-called
+Sandage effect or redshift drift. That is, the detection of slight changes in redshift due to
+cosmological expansion between observations several years apart. Several authors anticipated the
+use of the redshift drift as the ultimate probe to reconstruct the cosmic expansion in real time,
+without assuming standard candles and with very few sources of possible systematics. Moreover,
+observations along two light-cones, instead of one, allows us to distinguish at a fundamental level
+a genuine accelerated expansion in a homogeneous universe from an apparent expansion induced
+by a large-scale inhomogeneity as, for instance, in void models.
+Finally, the Gaia satellite, to be launched in 2011, will provide astrometric positions of up to
+a billion stars and a million quasars with unprecedented accuracy, thereby fixing to exceptional
+precision both the first rung of the distance ladder, the star parallax, and the cosmic reference3.8. Conclusions 33
+frame. Although Gaia is not primarily a cosmological probe, precise quasar astrometry could be
+employed to rule out the possibility that dark energy causes an anisotropic expansion or that we
+live off-center in a huge void. These possibilities, exotic as they might appear, are still relatively
+unconstrained and the very fact that we can still speculate about them shows how little we know
+about dark energy and how important is to integrate all sources of information.
+The joint efforts of Euclid, Gaia and EELT hold the potential to map the full three-dimensional
+cosmic kinematics and dynamics of our Universe for a large fraction of its history and of its
+volume.
+3.8 Conclusions
+We have discussed how both dark energy and modified gravity cosmologies can be described
+at the level of first-order perturbation theory by adding two functions to the equation of state
+parameter w(z). This allows the construction of a phenomenological parametrisation for the
+analysis of e.g. CMB data, weak lensing surveys or galaxy surveys, which depend in essential
+ways on the behaviour of the perturbations.
+Different choices are possible for these two functions. For example, one can directly use the
+gravitational potentials φ(k,t) andψ(k,t) and so try to measure the metric. Alternatively one
+can use the parameters which describe the dark energy: the pressure perturbation δp(k,t) and
+the anisotropic stress π(k,t). The pressure perturbation can be replaced by a sound speed c2
+s,
+which then has to be allowed to depend on scale and time. As argued above, these parameters can
+alsodescribe modified gravity models, in which case they do of course not have a direct physical
+counterpart. Also other choices are possible, like Qand Σ which provide a parametrisation
+similar towcompared to H. The important message is that only two new functions are required
+(although they are functions of scale kand timet). Together with w(z), they span the complete
+model space for both modified gravity and dark energy models in the cosmological context (i.e.
+without direct couplings, and for 3 + 1 dimensional matter and radiation moving on geodesics of
+a single metric). By measuring them, we extract the full information from cosmological data
+sets to first order.
+These phenomenological functions are useful in different contexts. Firstly they can be used to
+analyse data sets and to look for general departures from e.g. a scalar field dark energy model. If
+measured, they can also give clues to the physical nature of whatever makes the expansion of the
+Universe accelerate. Finally, they are useful to forecast the performance of future experiments
+in e.g. allowing one to rule out scalar field dark energy, since for this explicit alternatives are
+needed.
+In order to measure the two functions, we will require at least two independent data sets
+spanning the desired range in scale and redshift. If we use galaxy clustering, we will also need to
+determine the bias of the galaxies, adding a third function. The proposed Euclid satellite mission,
+with its combination of weak lensing andgalaxy clustering (including growth and redshift space
+distortions) is uniquely qualified to measure all of these functions. Especially the weak lensing
+measurements will play a crucial role in distinguishing the two gravitational potentials φandψ
+and in providing limits on the difference between the two.
+Weak lensing is a powerful tool which can probe many more physical phenomena, beyond the
+dark energy question. We have briefly commented on measurements of the properties of the dark
+matter and of the initial perturbations. Other examples include searches for topological defects
+(Thomas et al. 2009) and extrasolar planets based on lensing (see Chapter 17). The combination
+of all Euclid data sets can also measure the curvature at different redshifts ( Clarkson et al. 2008)
+and so test the Copernican principle and constrain large voids and backreaction effects ( Larena
+et al. 2009). Some of these applications are discussed elsewhere in this book. Finally we shouldBIBLIOGRAPHY 34
+not forget that there will be many synergies between EIC, Euclid and other current and future
+projects like Planck, LHC, EELT and Gaia.
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+WEAK GRAVITATIONAL
+LENSING
+36Chapter 4
+Weak Lensing with Euclid Imaging
+Authors: Adam Amara (ETH Zurich), Andrew Taylor (U. Edinburgh), Filipe Abdalla (UCL),
+Nabila Aghanim (IAS), Luca Amendola (INAF - OSS.Roma/U. Heidelberg), Vincenzo Antonuccio
+(INAF - Cantania Astrophys. Obs.), David Bacon (U. Portsmouth), Manda Banerji (UCL/IoA
+Cambridge), Joel Berg´ e (JPL), Rongmon Bordoloi (ETH Zurich), Julien Carron (ETH Zurich),
+Frederic Courbin (EPF Lausanne), Eduardo Cypriano (UCL/U. Sao Paulo), Hakon Dahle (U.
+Oslo), Benjamin Dobke (JPL), Adam Hawken (UCL), Alan Heavens (U. Edinburgh), Benjamin
+Joachimi (U. Bonn/UCL), David Johnston (Nothwestern U.), Lindsay King (IoA Cambridge),
+Donnacha Kirk (UCL), Thomas Kitching (U. Edinburgh), Konrad Kuijken (U. Leiden), Ofer
+Lahav (UCL), Marco Lombardi (U. Milano), Roberto Maoli (U. Roma La Sapienza), Katarina
+Markovic (UCL/Ludwigs-Maximilians U.), Richard Massey (U. Edinburgh), Massimo Meneghetti
+(INAF - Oss. Bologna), St´ ephane Paulin-Henriksson (CEA Saclay), Mario Radovich (INAF -
+Oss. Capodimonte), Ana¨ ıs Rassat (CEA Saclay), Justin Read (U. Zurich/U. Leicester), Alexandre
+R´ efr´ egier (CEA Saclay), Jason Rhodes (JPL), Anna Romano (U. Roma La Sapienza), Roberto
+Scaramella (INAF - Oss. Roma), Michael Schneider (U. Durham), Michael Seiffert (JPL),
+Fabrizio Sidoli (UCL), Robert Smith (U. Zurich), Jiayu Tang (UCL/IPMU), Romain Teyssier
+(CEA Saclay/U. Zurich), Shaun Thomas (UCL), Lisa Voigt (UCL).
+Abstract
+During the Euclid assessment phase, the EIC Weak Lensing Working Group has focused on
+two aspects of the Euclid mission: (1) developing the weak lensing related science case; and (2)
+support the technical studies of the instrument and mission by setting and maintaining the weak
+lensing science requirements. In this section we give a brief overview of this work, some of which
+is discussed in more detail in later chapters.
+4.1 Introduction
+During the Euclid definition phase, the EIC Weak Lensing Working Group (WLWG) has been
+actively and consistently working on the concept of weak lensing from space, with the over-
+arching aim of creating the Euclid weak lensing surveys that will be of unprecedented quality
+and size. This will enable advances to be made in all aspects of cosmology. The WLWG has
+focused on providing a comprehensive science case for the Euclid imaging surveys and supporting
+the technical studies of the instrument. The science case will tackle all of the fundamental
+cosmological questions, including dark energy, dark matter, probing the initial conditions of the
+Universe and testing gravity itself. To enable these science goals, the WLWG has focused on
+374.2. Basics of Weak Lensing 38
+the determination and mitigation of weak lensing related systematic effects, which will be of
+paramount importance for Euclid. To achieve and refine these goals, the WLWG has held weekly
+teleconferences and regular face-to-face meetings.
+4.2 Basics of Weak Lensing
+The light from galaxies is slightly deflected by the intervening matter distribution, causing
+a distortion in the shapes of the galaxies that we measure. Weak lensing is the subtlest of
+the lensing effects (see Figure 4.1), where an intrinsically circular galaxy would appear to us
+as an ellipse. However, since galaxies themselves are intrinsically elliptical, the lensing effect
+cannot be measured on individual galaxies. Instead, weak lensing must be measured statistically
+by measuring the shapes of several millions of galaxies ( Blandford et al. 1991;Bartelmann &
+Schneider 2001;Refregier 2003). The weak lensing effect can be most simply captured in a linear
+distortion of a galaxies image, this can be expressed in a matrix transformation as follows:
+/parenleftbiggx2
+y2/parenrightbigg
+=/parenleftbigg1−γ1−γ2
+−γ21 +γ1/parenrightbigg/parenleftbiggx1
+y1/parenrightbigg
+. (4.1)
+The “shear” has two components: a positive γ1stretches an image along the x-axis and compresses
+along the y-axis; and a positive γ2stretches an image along the diagonal y=xand compresses
+alongy=−x. The coordinates ( x1,y1) denote a point on the original galaxy image (in the
+absence of lensing) and ( x2,y2) denotes the new position of this point on the distorted (lensed)
+image (there is also an isotropic scaling that we do not focus here, but it can also be used to
+probe cosmology; see e.g. Barber & Taylor (2003)). The strength of the shear (lensing) signal
+depends on both the amount of matter and the distances between the observer, lens and source.
+This allows lensing to measure the geometry of the Universe and the growth of structure.
+Figure 4.1: Examples of lensed images. On the far left we see a pre-lensing image. The first
+order lensing effect (weak lensing/cosmic shear) is to stretch the image, thus turning it from
+a circle to an ellipse. As the gravitational lensing signal becomes stronger, higher order shape
+distortions are induced, changing the circle into a banana-shaped arc. This effect is known as
+flexion ( Goldberg & Natarajan 2002). On the far right we see that the strong lensing regime,
+gravitational lensing can be so extreme that multiple images of the same object can be seen.4.3. Developing the Science Case 39
+4.3 Developing the Science Case
+The WLWG has been investigating the wide array of sectors that can be measured through
+gravitational lensing. In Chapter 5, we show how an imaging survey can be designed to maximise
+our constraints on the dark energy and the broader cosmology parameters. We show that for
+all the parameters in the standard concordance model, measurement errors are minimised for
+an ultra wide survey that is able to reach a median redshift of roughly z= 1. In Chapter
+7, we review in more detail how it will be able to study dark matter. We show that the 3D
+matter power spectrum can be constrained to a high precision, which in conjunction with CMB
+measurements can be used to measure the make up of the early Universe, such as the slope and
+roll of the initial (post inflation) power spectrum (see for example Heavens et al. 2006). We also
+show how, on the sub-atomic scales, Euclid will constrain the temperature and mass of dark
+matter particles and neutrinos to a high precision.
+Euclid will also be able to test possible explanations for accelerated expansion beyond dark
+energy. These include a need to modify our model gravity. In Chapter 6, we show how Euclid will
+be able to detect deviations from Einstein gravity (the implications of which have already been
+discussed in Chapter 3). Finally, the large area survey envisioned for Euclid will allow unique
+cross-correlations between Euclid and other other cosmological experiments, such as Planck and
+eROSITA.
+4.3.1 Summary of the Predictions
+Table 4.1shows the projected errors that Euclid will be able to achieve using the techniques
+enabled by the visible and near-IR imaging surveys. For these calculations, we have assumed an
+eight-parameter cosmological model, where curvature has been allowed, with the fiducial values
+given below:
+•Density of matter density: Ω m= 0.25
+•Density of dark energy: Ω Λ= 0.75
+•Density of baryons: Ω b= 0.0445
+•Power spectrum normalisation: σ8= 0.8
+•Hubble parameter: h= 0.7
+•Spectral index: ns= 1
+•Dark energy Equation of state pivot point parameter: wp=−0.95
+•Dark energy Equation of state parameter: wa= 0.0
+For the dark energy component, we use a simple parameterisation where the equation of state
+(w) is modelled as a linear expansion with scale factor ( a) (Linder 2003;Chevallier & Polarski
+2001),
+w(a) =wp+ (ap−a)wa, (4.2)
+withapbeing the scale (redshift) at which the errors on w0andwpare uncorrelated.
+For the lensing survey, we have assumed a 20,000 square degree survey with 30 galaxies per
+square arcminutes and median redshift of 0.8. We have also assumed that the redshift of the4.4. Code comparison 40
+∆wp∆wa∆Ωm∆ΩΛ ∆Ωb ∆σ8 ∆ns ∆hDE FoM
+Current + WMAP 0.13 - 0.01 0.015 0.0015 0.026 0.013 0.013 ∼10
+Planck - - 0.008 - 0.0007 0.05 0.005 0.007 -
+Weak Lensing 0.03 0.17 0.006 0.04 0.012 0.013 0.02 0.1 180
+EIC probes 0.018 0.15 0.004 0.02 0.007 0.009 0.014 0.07 400
+EIC + Planck 0.013 0.08 0.001 0.004 0.0005 0.0016 0.003 0.002 1000
+Table 4.1: The error predictions for an eight-parameter cosmological model achievable with
+the probes enabled by Euclid’s imaging. These are compared against current measurements
+(Komatsu et al. 2009). We also show the results when Euclid constrains are combined with those
+coming from Planck.
+galaxies can be measured to a precision of σ(z) = 0.05(1 +z). This configuration corresponds to
+the minimal survey requirements for Euclid (see Table 4.2). If Euclid is able to meet its goals
+(see EIC Requirements Document) we can expect stronger constraints. From Table 4.1, we see
+that the weak lensing survey alone is able to make significant improvements over today in most
+sectors of the cosmological model. This is especially true of the dark energy sector, where we see
+more than an order of magnitude improvement on the dark energy FoM from the stand alone
+lensing survey (with no prior) as compared to today’s data (which is a combination of CBM,
+SNe, BAO). When all of the imaging probes (galaxy clustering through photoz - Chapter 20,
+cluster counts - Chapter 11and ISW - Chapter 12) are combined we see over a factor of two
+improvement in the FoM. Finally, we see that when the Euclid imaging probes are combined with
+Planck we are able to reach sub-percent level precision on almost all cosmological parameters and
+a factor of five improvement in DE FoM compared to Euclid alone. This shows that although the
+CMB is not able to directly measure dark energy, it provides a data set that works extremely
+well Euclid by boosting the Euclid dark energy measurements due to degeneracy breaking in
+other sectors. We also show in the left panel of Figure 4.2the error ellipses in the equation of
+state parameters space for these configurations. These are shown in relation to distinct regions
+for the dark energy theories. Thus far, these error predictions only consider the statistical errors.
+Below we discuss how we can deal with systematic errors.
+4.4 Code comparison
+Within the WLWG, a number of independent routines have been developed for calculating the
+expected errors for future weak lensing missions. These have largely focused on constraints
+coming from weak lensing tomography. In this approach, galaxies are first divided into redshift
+bins (typically 10 bins are used) and the two point correlation function of shear, both within the
+bin and between different bins, is used to measure cosmological parameters. We have conducted a
+detailed code comparison program using four tomographic routines. These calculate the expected
+errors using the Fisher matrix formalism (e.g. Tegmark et al. 1997), and they have been used in
+a number of published works (these include, among others, Amara & R´ efr´ egier 2007;Bridle &
+King 2007;Thomas et al. 2009;Joachimi & Schneider 2009). For these calculations, we have
+found sub-percent level agreement on fixed error prediction and a better than 10% agreement on
+marginalised errors, which corresponds to the level of accuracy expected from a Fisher matrix
+approach. One of these codes (the package iCosmo ) has been made public ( Refregier et al. 2008),
+along with a user-friendly website ( Kitching et al. 2009a ). Another approach is to measure the
+full 3D correlation function of shear without first binning the galaxies ( Kitching et al. (2007)4.5. Mitigation of Systematic effects 41
+ 
+Figure 4.2: Left Panel: Predictions for constraints on the equation of state parameters w0and
+wacoming from weak lensing and the other imaging probes. We also show the expected errors
+when combined with Planck. The background grey regions show distinct theoretical regions.
+These are: dark grey - phantom models; middle grey - Freeze models; and light grey - Thaw
+models (see Chapter 3for more details). Right Panel: The total error on the equation of state
+parameter, which includes both statistical and systematic, as a function of area and the level of
+systematic floor.
+andHeavens et al. (2006)). Expected errors from this method have also been compared against
+those coming from weak lensing tomography, and we find consistent error predictions with the
+3D lensing method performing slightly better, as expected. Unless stated elsewhere, our fiducial
+cosmology model used the transfer function by Eisenstein & Hu (1998) and a non-linear correction
+calculated using halofit ( Smith et al. 2003). We have found that our predictions are not very
+sensitive to the choice of transfer function and that the choice of non-linear prescription can
+have a 50% effect on the projected errors. This highlights the importance of accurate predictions
+of the non-linear growth of structure, which is discussed in Chapter 22. Code comparison tests
+have also been performed for predictions using higher order correlations, such as the bispectrum
+(see Chapter 5) and modified gravity models (see Chapter 6for more details).
+4.5 Mitigation of Systematic effects
+The potential of weak gravitational lensing as a uniquely powerful way of making precision
+cosmology measurements is now well established ( Peacock et al. 2006;Albrecht et al. 2006).
+The key issue rests on our ability to control sources of systematic errors to the point where
+they remain sub-dominant to the statistical ones. There are a number of potential sources of
+systematic error. From the theory and the modelling side, care is needed to understand: (i)
+the non-linear growth of structure; (ii) the impact of baryons on small scale structure growth;
+and (iii) the intrinsic correlated alignment of galaxy ellipticities even in the absence of lensing.
+Tackling these problems will require a combination of high resolution computer simulations
+to better understand the physical processes involved, as well as the development of statistical
+techniques to minimise the impact of any theoretical uncertainty. In Chapter 22, we discuss the
+numerical simulations that have been performed by EIC members to tackle these issues; and in
+Chapter 10we give a detailed discussion of intrinsic alignment, along with mitigation strategies.4.5. Mitigation of Systematic effects 42
+Given the simplicity of the gravitational lensing effect, these sources of theoretical uncertainty
+will likely be overcome with continued work along the lines highlighted in this document. The
+main remaining challenge for weak lensing studies will then be the control of measurement errors
+to the precision that will be needed to meet the ambitious Euclid targets.
+Weak lensing measurement rely on the accurate measurements of galaxy shapes and redshifts.
+For practical considerations coming from the large number of galaxies that are needed ( ∼2
+billion), the redshifts of galaxies will be measured photometrically. In Chapter 20, we highlight
+the challenges to this approach and ways that our targets can be met. Galaxy shapes are most
+easily measured in the visible, and the process is outlined below in section 4.5.2.
+4.5.1 Setting Systematic Limits
+To set limits on the level of systematics that we can tolerate, we have developed a number of
+methods for propagating these errors. The first approach is to add nuisance parameters to the
+Fisher matrix that is used to perform the error predictions for cosmology parameters. We would
+then marginalise over these nuisance parameters to determine their effect. A good example of
+this comes from setting the requirement on the knowledge of the mean redshift of galaxies in each
+redshift bin. By leaving the mean redshift of the galaxy distributions as a free parameters, we
+effectively try to simultaneously measure these along with the cosmology parameters ( Kitching
+et al. 2008b ;Amara & R´ efr´ egier 2007;Abdalla et al. 2008;Zhang et al. 2009;Schrabback et al.
+2009). Using only shear data, there is a serious degradation of cosmology errors if no prior
+information is known about the mean redshift of the galaxies. To remove this negative effect, we
+have found that the mean redshift of the galaxies in each bin needs to be known to a precision of
+/angbracketleftz/angbracketright= 0.002(1 +z). In Chapter 20we show how this goal can be achieved.
+The difficulty in adding nuisance parameters to the Fisher matrix comes from the fact that we
+have to decide how to parameterise the unknown. Another approach that has been developed by
+members of the EIC is to study the biases that an unknown systematic will induce on cosmological
+parameters and to use this to set tolerance limits on possible uncertainties. Using this approach
+Amara & R´ efr´ egier (2008), we were able to quantify the systematic uncertainty in term of the
+quantityσ2
+sys, which is the induced variance coming from the difference between the measured
+and true correlation function. For a Euclid-like survey, the systematics need to be controlled
+to a level better than σ2
+sys= 10−7. This result was also confirmed by Kitching et al. (2009b ),
+who developed the method further. In Figure 4.2we show how the measurement error of a
+survey depends on the σ2
+sysas a function of survey area. We see that as the area of the survey is
+increased, the statistical error on the equation of state parameter decrease. However, a given
+mission will be limited by the Root Mean Squared Error (RMSE) of the measurement, which
+comes from the quadratic sum of statistical error and biases, RMSE (p) =/radicalbig
+σ2(p) +b2(p). We
+see that depending on the level of the systematics of a survey, the RMSE will plateau at a given
+survey area at which point the mission becomes systematics limited. Given the fact that current
+lensing data is limited to areas <100 square degrees, we have estimated the current systematic
+level to beσ2
+sys∼4×10−6(Paulin-Henriksson & et al. 2009). The Euclid targets are close to two
+orders of magnitude more demanding in σ2
+sys, and this motivates our need to have space-based
+high quality imaging that does not suffer from adverse effects of the atmosphere.
+4.5.2 Shape Measurement
+The process of measuring galaxy shapes has a central significance in securing the weak lensing
+science goals of Euclid. For example, if the ellipticity (or flexion) signal measured is biased or
+offset relative to the true underlying galaxy ellipticity, then this can propagate through the weak
+lensing statistics into a bias in the cosmological parameter estimation, as discussed above. The4.5. Mitigation of Systematic effects 43
+process that galaxy images go through is encapsulated in Figure 4.3. A galaxy is sheared by the
+gravitational potential along the line of sight. This sheared galaxy is then further convolved with
+a PSF, pixelated and is observed in the presence of noise. The shape measurement problem is to
+disentangle the steps subsequent to the shearing process and to measure this shear to a high
+accuracy. Members of the EIC WLWG are at the forefront in finding solutions to this problem,
+with EIC members having developed a number of shape measurement approaches. These includes
+lensfit (Miller et al. 2007;Kitching et al. 2008a ), shapelets ( Refregier & Bacon 2003;Massey
+& Refregier 2005), im2shape ( Bridle & et al. 2004). Shape measurement methods have always
+met the demands of contemporary weak lensing data. However, this is an area in which further
+improvements must be made.
+To improve on the current methods, a roadmap of simulations and testing has been developed.
+InBridle et al. (2008,2009), the first GRavitational lEnsing Accuracy Testing (GREAT) challenge
+was launched. These are a set of simulations in which the shear is introduced in a controlled
+manner. Shape measurement algorithms can then be used on these simulations and their
+performance measured against the input. GREAT08 ran for 6 months during 2008 as a blind
+challenge. During this short time, a factor of 2 improvement was gained over existing methods.
+In some simulated conditions, the most successful methods met and surpassed the requirements
+set by Euclid, but further work is ongoing to broaden this success. The next suite of simulations
+in this challenge is GREAT10 ( Kitching & et al. 2010), which will increase the complexity over
+that of the GREAT08 challenge by introducing variable shear and PSF.
+Intrinsic galaxy(shape unknown)Gravitational lensing causes a shear (g)Atmosphere and telescopecause a convolutionDetectors measurea pixelated imageImage also contains noiseThe Forward Process.Galaxies: Intrinsic galaxy shapes to measured image:
+Stars: Point sources to star images:
+Intrinsic star(point source)Atmosphere and telescopecause a convolutionDetectors measurea pixelated imageImage also contains noise
+Figure 4.3: Illustration of the processes that affect galaxy and star images. The intrinsic shape
+of a galaxy is gravitationally lensed by intervening matter causing the cosmic shear effect that
+we plan to measure. After this, the galaxy image becomes blurred due to the PSF (in space
+this would come only from the instrument), pixelated by the detectors. The final image will
+also have noise. Star images suffer from many of these effects but crucially their images are
+not gravitationally lensed. We are therefore able to use star images to correct galaxy images to
+recover the shear signal. This is discussed in more detail in Chapter 8(Figure taken from Bridle
+et al. 2008)4.6. Summary of Top Level Requirements 44
+4.6 Summary of Top Level Requirements
+In Table 4.2we summarise the top level requirements coming from weak lensing on the sur-
+vey geometry and the level to which the systematics need to be controlled. These top level
+requirements form the inputs to the EIC requirements document. These then flow down into
+lower level requirements, such as PSF profile, etc. The requirements are also used as targets for
+the work described in the sections that follow. In Part V of this science document we describe
+the set of simulations that we have developed to verify that the Euclid survey will fulfil these
+requirements. Specifically we describe image simulations (Chapter 19), photometric redshift
+simulations (Chapter 20), radiometric simulations (Chapter 21) and cosmological simulations
+(Chapter 22).
+Description Quantity Requirement
+Survey Geometry: Area: Errors on dark energy
+parameters depend on the area of the surveyAs>20000 deg2
+Survey Geometry: density of galaxies: And the
+effective number density of galaxies useful for
+gravitational lensing (Neff)Neff>30 gals/amin2
+[Goal:>40]
+Survey Geometry: galaxy redshift: Redshift dis-
+tribution of the lensing galaxieszm>0.8
+Shape Measurement: To reach the above cos-
+mological objectives, systematic effects shall be
+controlled to a level where they do not domi-
+nate over the statistical errors. This is done
+by controlling the variance of the residual shear
+systematics.( σ2
+sys)σ2
+sys<10−7
+Photometric Redshifts statistical: The statisti-
+cal rms error σ(z) in the photo-zs in the range
+0.2<z< 2.0σ(z)/(1 + z)<0.05
+Photometric Redshifts error in the mean: The
+mean of the redshift distribution n(z) of each bin
+must be known to high precisionσ(zi)/(1 +zi)<0.002
+Table 4.2: Summary of the top level requirement for precision measurements with Euclid. The
+area, number density, median redshift and photometric uncertainty set the statistical potential
+of a weak lensing survey (see Chapter 5). To meet this potential, the shape measurement errors
+need to be controlled, as well as having accurate measurements of the mean redshift of the
+galaxies.
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+weak lensing surveys. ArXiv e-prints .Chapter 5
+Weak Lensing Survey Optimisation for Dark
+Energy Measurement
+Authors: Joel Berg´ e (JPL/Caltech), Adam Amara (ETH Zurich), Alan Heavens (ROE Ed-
+inburgh), Benjamin Joachimi (Bonn University), Tom Kitching (ROE Edinburgh), Alexandre
+R´ efr´ egier (CEA Saclay), Andy Taylor (ROE Edinburgh), + EIC Weak Lensing Working Group.
+Abstract
+We describe the work done by the EIC Weak Lensing working group members to optimise the
+survey’s configuration and the extraction of cosmological parameters. We present the weak
+lensing techniques that we use, shear power spectrum, shear tomography, 3D shear, shear-ratio
+geometric test, and how we design a weak lensing to optimise the dark energy measurement using
+those techniques. We then present non-Gaussian measurements, like the weak lensing bispectrum
+and weak-lensing-selected-cluster counts, and show how we use them to optimise the extraction
+of cosmological information.
+5.1 Introduction
+We present how we optimise both the survey geometry and the weak lensing measurements that
+will allow the optimal extraction of cosmological information. Besides the usual shear power
+spectrum tomography, we will use more recent techniques such the full 3D weak lensing and
+the shear-ratio geometric test. Non-Gaussian-sensitive measurements, like the weak lensing
+bispectrum and weak-lensing-selected cluster counts, are complementary to those, and allow us
+to optimise cosmological parameters measurements. Section 5.2presents the survey geometry
+optimisation : we show that to optimise dark energy measurement, for a given observation time,
+we need a large survey with median redshit zm≈1. Section 5.3shows how to optimise dark
+energy measurement by combining complementary weak lensing statistics, such as combining the
+weak lensing power spectrum with the weak lensing bispectrum or weak-lensing-selected cluster
+counts. We parameterise the varying dark energy equation of state w(a) =w0+ (1−a)wa, where
+ais the scale factor.
+5.2 Optimising the survey geometry
+The shear power spectrum, the Fourier transform of the real-space shear two-point correlation
+function, shows a strong dependence on the dark matter density parameter Ω mand on the
+485.2. Optimising the survey geometry 49
+Figure 5.1: Survey geometry optimisation. Left : Gain in FoM, as measured with shear
+tomography, when observation time is dedicated to increasing the survey’s area, galaxy density ng
+or median redshift zm. Increasing the area is synonymous with a wide survey, whereas increasing
+ngandzmis synonymous with a deep survey (from Amara & R´ efr´ egier 2007). Right : Inverse
+FoM’s dependence on the median redshift of the survey, as measured with 3D shear, for a given
+survey area and for 5-band (solid), 9-band (dashed) and 17-band (dot-dashed) photometric
+redshift survey (from Heavens et al. 2006).
+dark matter power spectrum amplitude σ8. It has already shown its efficiency in constraining
+cosmological parameters on current data (e.g. Massey et al. 2007;Fuet al. 2008). The source
+galaxies’ redshift information is usually taken into account when measuring the shear power
+spectrum by binning the galaxies’ population into redshift slices. It has been demonstrated, both
+theoretically (e.g. Hu1999) and observationally ( Massey et al. 2007;Schrabback et al. 2009) that
+this tomographic method helps to tighten constraints on cosmological parameters. Moreover,
+tomography helps control systematics such as galaxy shapes intrinsic alignments. Heavens (2003)
+introduced a 3D cosmic shear analysis (see also Taylor 2001), a technique that takes into account
+the full 3D information, and avoids the loss of information suffered by tomography when binning
+galaxies into redshift. Jain & Taylor (2003) introduced the shear-ratio geometric test. By taking
+the ratio of the galaxy-shear correlation functions at different redshift, one is left with a purely
+geometric quantity that can be efficiently used to measure dark energy.
+Amara & R´ efr´ egier (2007) have investigated the dependence of the precision on the cosmolog-
+ical parameters, as measured with power spectrum tomography, on the survey’s characteristics.
+The left panel of Fig. 5.1shows the gain in the dark energy Figure of Merit (FoM) when changing
+the survey’s characteristics. Increasing the area provides a much more pronounced gain in the
+FoM compared to increasing the number density of galaxies ngand the median redshift zm. A
+wide survey spends the exposure time increasing the area, keeping ngandzmconstant, while a
+deep survey increases them. Therefore, for a given observation time, we should always prefer a
+wide survey rather than a deep survey. Heavens et al. (2006) optimised 3D weak lensing surveys,
+and found that for a fixed time, larger surveys provide better cosmological parameter constraints,
+in particular for dark energy, once a median redshift of z≈1 is reached. The right panel of Fig.
+5.1shows that the dark energy is best constrained as soon as the median redshift zm≈1. They5.3. Optimising constraints on cosmological parameters 50
+Figure 5.2: Ω m-σ8degeneracy left over by the weak lensing power spectrum (from Pires et al.
+2009). The pluses represent the models that they considered (see main text).
+showed that it gives comparable, but a little better, constraints on dark energy than weak lensing
+tomography. Taylor et al. (2007) have adapted and optimised the method around galaxy clusters
+to constrain dark energy and its evolution in large surveys. They showed that a large survey
+yields better constraints on dark energy than a targeted survey of 60 of the largest clusters.
+To optimise dark energy measurement, we will survey the entire extragalactic sky (20,000
+deg2) at an intermediate median redshift ( zm≈1), providing us with ng≈40 galaxies per square
+arcminute.
+5.3 Optimising constraints on cosmological parameters
+Besides optimising the survey configuration, one must also optimise the methods with which
+cosmological information is extracted. For instance, one must investigate how the well-known
+degeneracies left over by the statistics introduced in the previous section, as shown by Fig. 5.2
+for the matter density Ω mand matter power spectrum normalisation σ8, can be broken. This
+can be done with observables sensitive to non-Gaussianity, such as third-order statistics and
+galaxy cluster counts.
+Third-order statistics are the lowest order statistics sensitive to non-Gaussianity. Hence, they
+can be combined with the aforementioned 2-points statistics to break degeneracies left over by
+the latter. For instance, the weak lensing bispectrum is the Fourier transform of the convergence
+3-point correlation function. In the flat-sky approximation, it is defined as
+Bκ,(ijk)(l1,l2,l3) =/integraldisplayχh
+0dχgi(χ)gj(χ)gk(χ)χ−4Bδ(k1,k2,k3;χ) (5.1)
+wheregi(χ) is the weak lensing efficiency function, i,jandkdenote redshift bins, and
+Bδ(k1,k2,k3;χ) is the matter bispectrum, which can be estimated e.g. with the halo model
+or related to the matter power spectrum with scaling relations from the hyper-extended theory
+perturbation (e.g. Scoccimarro & Couchman 2001). We have developed and compared inde-
+pendent codes to assess the constraining power of the weak lensing bispectrum, with Fisher
+matrices analyses. Joachimi et al. (2009) have derived the bispectrum covariance in the flat-sky5.3. Optimising constraints on cosmological parameters 51
+Table 5.1: Discrimination efficiencies (in percent) achieved on wavelet-filtered convergence maps
+with peak counts (from Pires et al. 2009).
+Model 1 Model 2 Model 3 Model 4 Model 5
+Model 1 x 86 100 100 100
+Model 2 87 x 94 100 100
+Model 3 100 92 x 94 100
+Model 4 100 100 93 x 99
+Model 5 100 100 100 100 x
+approximation. We found that the bispectrum gives constraints similar to those provided by the
+power spectrum.
+Clusters of galaxies are the most massive structures evolved from the non-linear gravitational
+evolution of the density field. Assessing their abundance is thus a direct probe of non-Gaussianity,
+and has been shown to be strongly dependent on cosmology (e.g. Oukbir & Blanchard 1992;
+Wang & Steinhardt 1998;Horellou & Berge 2005). Galaxy clusters can be selected in different
+ways. For instance, Euclid will allow us to detect them thanks to their optical signal (e.g. Adami
+et al. 2009) and thanks to their weak lensing signal (e.g. Marian & Bernstein 2006;Berg´ e et al.
+2008). In the following, we show how weak-lensing selected cluster can break the degeneracies
+between cosmological parameters left with the power spectrum.
+Pires et al. (2009) and Berg´ e et al. (2009) have shown how weak lensing alone can measure
+dark energy at the percent level by combining non-Gaussian measures with the power spectrum.
+Using numerical simulations, Pires et al. (2009) have compared how higher order weak lensing
+statistics break the degeneracy between Ω mandσ8. They showed that while third-order statistics,
+such as the skewness, are efficient at this task, counting galaxy clusters provides the best way to
+discriminate between several models. They considered five different cosmological models lying
+along the Ω m-σ8degeneracy, shown by pluses in Fig. 5.2. Table 5.1presents the discrimination
+efficiency of peak counts on a wavelet-filtered weak lensing convergence map (where peaks
+correspond to galaxy clusters).
+Berg´ e et al. (2009) used Fisher matrices to generalise Pires et al. (2009)’s results. They showed
+that, when counting galaxy clusters as signal-to-noise peaks, non-Gaussianities are captured as
+well as when including the redshift and/or mass information in clusters counts. This makes it an
+easy measurement, that does not require any additional information nor nuisance parameters to
+allow for the lack of knowledge in how the observable one uses to estimate a cluster’s mass is
+related to its real mass (e.g. Lima & Hu 2004). Furthermore, they showed that the weak lensing
+bispectrum is as efficient as cluster counts to capture non-Gaussianity. Figure 5.3shows the
+1-σconstraints that will be obtained with Euclid on the varying dark energy equation of state
+parameters w0andwa, when marginalising over the other six parameters. The blue ellipse shows
+the forecasted constraints provided by the power spectrum. Non-Gaussianities are captured when
+combining it with the bispectrum (red ellipse) and with weak-lensing-selected galaxy clusters
+(green ellipse), hence tightening the constraints.
+Besides these weak-lensing-only constraints, one can combine weak lensing measurements
+with other probes, such as the Cosmic Microwave Background (CMB) currently surveyed by
+Planck, Baryon Acoustic Oscillations (BAO), or type Ia supernovae (SNIa). In particular, the
+shear-ratio test provides constraints that are orthogonal to those from CMB : the shear-ratio
+test constrains w0, and its combination with the CMB constrains wa.Heavens et al. (2006) and5.4. Conclusion 52
+Figure 5.3: Forecasted marginalised 1- σconstraints on the varying dark energy equation of
+state provided by Euclid. The blue ellipse shows the constraints from the weak lensing power
+spectrum. The red and green ellipses show the constraints when the weak lensing bispectrum
+and weak-lensing-selected clusters counts are combined with the power spectrum, respectively
+(from Berg´ e et al. 2009).
+Taylor et al. (2007) have shown how 3D weak lensing and the shear-ratio geometric test constrain
+dark energy when combined with observations of the CMB, of BAO, and of SNIa. They found
+that those combinations will yield percent level constraints on the evolving dark energy.
+5.3.1 Towards a more precise description of dark energy
+It is usual to the time evolution of the dark energy equation of state by the first order expansion
+w(a) =w0+ (1−a)wa. However, higher order expansions, using different data sets, will better
+represent the reality. Kitching & Amara (2009) have investigated how we must use Fisher
+matrices when dealing with such high order expansions. They showed that while finding the
+eigenvalues of w(a) from Fisher matrix depends on the basis functions choice and on the order of
+the expansion, Euclid will allow us to put sub-percent errors on the best constrained eigenvalues.
+They also recommend that physical motivated functional forms should be used as basis sets.
+5.4 Conclusion
+To optimally measure dark energy using cosmic shear, the Euclid survey has been optimised
+to be as wide as possible : we will survey the entire extragalactic sky (20,000 deg2), with a
+median redshift zm≈1. The cosmological information will be optimally extracted by first
+measuring the weak lensing power spectrum, then combining it with probes of non-Gaussianities.
+For instance, counting clusters of galaxies, defined as signal-to-noise ratio peaks, will optimally
+break degeneracies. Combining the power spectrum with the bispectrum will then give similar
+constraints, as well as a check on the constraints obtained when combining the power spectrum
+with clusters counts. Combining weak lensing measurements, such as 3D weak lensing or the
+shear-ratio geometric test, with other cosmological probes, like the CMB or BAO, will provideBIBLIOGRAPHY 53
+additional constraints. With these available combinations, Euclid will provide percent level
+accuracy on dark energy.
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+with a Time-varying, Spatially Inhomogeneous Energy Component with Negative Pressure.
+ApJ,508, 483–490.Chapter 6
+Modified Gravity with Lensing
+Authors: David Bacon (ICG Portsmouth), Adam Amara (ETH Zurich), Filipe Abdalla (UCL),
+Luca Amendola (Rome), Emma Beynon (ICG Portsmouth), Sarah Bridle (UCL), Alan Heav-
+ens (Edinburgh), Tom Kitching (Edinburgh), Kazuya Koyama (ICG Portsmouth), Martin
+Kunz (Geneva), Shaun Thomas (UCL), Jochen Weller (U. Munich/MPE) + EIC Weak Lensing
+Working Group.
+Abstract
+Here we describe the work that has been carried out by EIC Weak Lensing group members to
+characterise modified gravity theories, particularly with a view to predicting the capabilities of
+weak gravitational lensing with Euclid to detect and measure deviations from Einstein’s theory.
+6.1 Introduction
+As has been discussed in Chapter 3, it is possible that the indications of dark energy which we
+have observed are actually the signature of something even more radical: the departure of the
+law of gravity from Einstein’s description. In this view, General Relativity is seen as a limiting
+case of a more general theory of gravity. Note that we still want GR as a limit at small scales,
+due to the very stringent tests which GR has passed in our Solar System.
+In this chapter we describe recent work which members of our group have carried out, seeking
+to understand the extent to which these theories can be excluded or distinguished with Euclid
+lensing analyses.
+It should be noted that various group members (Amara, Amendola, Heavens, Kitching,
+and Thomas) have compared their cosmology codes for calculating weak lensing predictions for
+modified gravity, particularly for the γgrowth parameter. Careful notice has been taken of the
+varying assumptions between models, summarised at the end of this chapter in Table 6.1.
+6.2 Distinguishing between Modified Gravity and Dark Energy
+A key concern is how one can distinguish between modified gravity models with a particular set
+of parameters, and dark energy models with a different set of parameters - and indeed a different
+number of parameters. Members of the weak lensing group have addressed this in Heavens et al.
+(2007).
+This paper demonstrates how the Bayesian evidence can be used to distinguish between
+models, using the Fisher matrices for the parameters of interest. In particular, the authors
+556.3. Constraining Modified Gravity with Weak Lensing 56
+Figure 6.1: The value of |lnB|for Euclid (solid), Pan-STARRS (dot-dashed) and DES (dashed),
+in combination with CMB constraints from Planck, as a function of the difference in the growth
+rate between the modified gravity model and GR. The dotted vertical line shows the offset
+of the growth factor for the DGP model. The horizontal lines mark the boundaries between
+‘inconclusive’, ‘significant’, ‘strong’, and ‘decisive’ in Jeffrey’s (1961) terminology.
+calculate the ratio of evidences Bfor a 3D weak lensing analysis of the full Euclid survey, for
+a dark energy model with varying equation of state, and modified gravity with additionally
+varying growth parameter γ; the survey parameters are chosen as in the Yellow Book. The work
+focusses on models such as DGP, for which the relation between the lensing potential and density
+fluctuations is the same (i.e. k2(φ+ψ) is the same).
+The results are shown in Figure 6.1. We see that Euclid can decisively distinguish between e.g.
+DGP and dark energy using the methods of this paper. Moreover, we will be able to distinguish
+any departure from GR which has a difference in γgreater than /similarequal0.03.
+As has been described in Theory, it is possible to construct rather peculiar dark energy
+models which could mimic the modified gravity models considered. So when we refer to the
+discriminatory power of Euclid for modified gravities such as those above, it should be understood
+that we are distinguishing between these and standard dark energy quintessence models. However,
+the Bayesian approach suggested here will disfavour artificial dark energy models which require
+more parameters to fit the data.
+6.3 Constraining Modified Gravity with Weak Lensing
+6.3.1 Parameter constraints
+Going beyond the above discriminatory test, we are interested in the degree to which we can
+constrain modified gravity parameters using Euclid lensing. This is an active topic of research,
+especially for predictions in the linear regime (e.g. Knox et al. 2006;Yamamoto et al. 2007;
+Afshordi et al. 2008;Schmidt 2008;Song & Dore 2008;Tsujikawa & Tatekawa 2008;Zhao et al.
+2009a ,b); a key study has been carried out by members of our group in Thomas et al. (2009).
+In this paper, the authors initially consider a set of models which they call ‘mDGP’ with
+ΛCDM and DGP as extremes, using a Friedmann equation with new parameter α:
+H2−Hα
+r2−αc=8πGρ
+3(6.1)6.3. Constraining Modified Gravity with Weak Lensing 57
+0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8!0.8!0.6!0.4!0.200.20.40.60.8
+!"0l (max) = 10000l (max) = 500
+Figure 6.2: Marginalised γ−Σ0forecast for weak lensing only analysis with Euclid. Black
+contours correspond to lmax= 10000, demonstrating an error of 0.069(1 σ) on Σ 0, whereas the
+red contours correspond to lmax= 500 giving an error of 0.25. In both cases, the inner and outer
+contours are 1 σand 2σrespectively. General Relativity resides at [0.55, 0], while DGP resides at
+[0.68, 0].
+Using the CFHTLS lensing dataset, together with 2dF and SDSS data for BAO and SNLS data
+for supernova information, they show that DGP ( α= 1) can already be excluded, while γis still
+unconstrained.
+On the other hand, by constructing Fisher matrix forecasts for Euclid, with survey parameters
+as given in the Yellow Book, they find that γwill be accurately measured, together with the Σ 0
+parameter; the resulting constraints from Euclid weak lensing alone are shown in Figure 6.2.
+We see that the constraints obtained are dependent upon the maximum wavenumber lmax
+probed by the survey; lmax= 500 corresponds to looking at the linear regime of the power
+spectrum, while lmax= 10000 corresponds to including non-linear power. The latter requires
+the use of the Smith et al. (2003) fitting function, and results for this should be considered as
+indicative only (see below for further progress on this issue). We see that γis not found to be
+very sensitive to lmax, while Σ 0is measured much more accurately in the non-linear regime. In
+both cases, DGP is strongly excluded if the true model is ΛCDM.
+6.3.2 Evolution of parameters
+Members of our team considered an expansion of this parameterisation in Amendola et al. (2008).
+After considering predictions for Euclid accuracy of measurement for γand Σ = 1 + aΣ0in a
+similar fashion to above, the evolution of Σ( zi) was investigated by dividing the Euclid weak
+lensing survey into three redshift bins with equal numbers of galaxies in each bin. Σ( zi) was then
+approximated by Σ( zi) = Σ i, a piece-wise constant in the three bins. Σ 1is set to 1, as there is a
+degeneracy between the overall amplitude of Σ and the amplitude of matter fluctuations. The
+results are shown in Figure 6.3.
+We see that the Σ evolution is well characterised by Euclid lensing; a deviation from unit Σ
+(i.e. General Relativity) of 3% can be detected in our second redshift bin, and a deviation of
+10% is still detected in our furthest redshift bin.6.4. Using the Non-Linear Power 58
+0.94 0.96 0.98 11.02 1.04 1.06
+/CapSigma20.80.911.11.2/CapSigma3
+Figure 6.3: Forecast for Euclid constraints for Σ evolution, with zmean = 0.9 and number density
+of galaxies 35 per square arcmin (outer contour), 50 per square arcmin (middle contour), 75 per
+square arcmin (inner contour).
+6.3.3 Optimising the Survey for Modified Gravity
+We have also examined whether the optimization of the survey for Dark Energy constraints
+also holds for modified gravity studies. Initial indications are positive; using the Beynon et al.
+(2009) models described below, the fiducial Euclid survey with ngal= 35 per square arcmin,
+A=20000 sq deg, zm= 0.9 provides constraints on the growth factor γ= 0.55±0.12(1σ) for a
+ΛCDM model. On the other hand, an equal-time deep survey with ngal= 130 per square arcmin,
+A=1000 sq deg, zm= 1.1, we obtain γ= 0.55±0.34.These results are using the linear regime
+only, marginalising over Ω mandσ8. We see that the wide survey strategy is much preferred for
+obtaining constraints on γ.
+6.4 Using the Non-Linear Power
+A limitation in using weak lensing for modified gravity studies has been that, while much of the
+lensing signal on small scales arises from the non-linear matter power spectrum, we have not had
+good predictions for modified gravity in this regime. This is a crucial area where much theoretical
+work is being carried out, looking at how to probe non-linear power perturbatively (e.g. Koyama
+et al. 2009), via N-body simulations ( Oyaizu et al. 2008;Schmidt 2009), halo statistics ( Schmidt
+et al. 2008) and fitting functions ( Hu & Sawicki 2007). A way of making progress by combining
+several of these insights has been described by members of our group in Beynon et al. (2009).
+This paper has used the fact that the power spectrum must asymptote towards General
+Relativity predictions on small scales, for the particular expansion history in question. We can
+therefore approximate the power spectrum using an interpolation suggested by Hu & SawickiBIBLIOGRAPHY 59
+Figure 6.4: Correlation function predicted for ΛCDM, DGP and f(R) with error estimates for
+Euclid. Models are for the central cosmological parameter values fitting WMAP+BAO+SNe,
+using a ΛCDM background (for ΛCDM and f(R)) and a DGP background (for DGP).
+(2007),
+P(k,z) =Pnon−GR(k,z) +cnl(z)Σ2(k,z)PGR(k,z)
+1 +cnl(z)Σ2(k,z)(6.2)
+with
+Σ2(k,z) =/parenleftbiggk3
+2π2Plin(k,z)/parenrightbiggα1
+, c nl(z) =A(1 +z)α2(6.3)
+i.e. with three new parameters, A,α1andα2. By fitting these parameters to N-body simulations
+of DGP and f(R) models ( Oyaizu et al. 2008;Schmidt 2009), this study obtains weak lensing
+predictions for these models down to small scales, and shows that the discriminatory power of
+Euclid lensing is (as might be expected) stronger than in the linear regime alone. This paper
+restricts itself to the Σ 0= 0 case, and measures the covariance on non-linear scales from the
+Horizon simulations of Teyssier et al. (2009).
+An example of the shear correlation functions that are expected for Euclid following this
+method are shown in Figure 6.4; note the extremely small errors expected on this quantity for
+our survey. This translates into constraints on modified gravity parameters exemplified by Figure
+6.5; hereA,α 1andα2have not been fit but instead are constrained by the lensing data itself.
+We see that, even with this larger range of parameters to fit, Euclid provides a measurement of
+the growth factor γto within 10%, and also allows some constraint on the new α1parameter,
+probing the physics of nonlinear collapse in the modified gravity model.
+Bibliography
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+Extra Dimensions.
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+Journal of Cosmology and Astro-Particle Physics ,4, 13–+.BIBLIOGRAPHY 60
+Figure 6.5: Constraints on γ,α1,α2andAfrom Euclid, using a DGP fiducial model and
+0.4 redshift bins between 0.3 and 1.5 for the central cosmological parameter values fitting
+WMAP+BAO+SNe.
+Includes Marginalised over Assumes Nonlinear
+Planck? # parameters flatness? approach
+Heavens et al. Yes 10 No Smith et al.
+Thomas et al. No 7 No Smith et al.
+Amendola et al. No 8 No Smith et al.
+Beynon et al. No 2 Yes Smith et al., Hu & Sawicki
+Table 6.1: Summary of assumptions in our models and analyses.
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+at non-linear scales. ArXiv e-prints .
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+III: Halo Statistics.
+Smith, R. E., Peacock, J. A., Jenkins, A., White, S. D. M., Frenk, C. S., Pearce, F. R., Thomas,
+P. A., Efstathiou, G., & Couchman, H. M. P. (2003). Stable clustering, the halo model and
+non-linear cosmological power spectra. MNRAS ,341, 1311–1332.
+Song, Y.-S. & Dore, O. (2008). Testing gravity using weak gravitational lensing and redshift
+surveys.
+Teyssier, R., Pires, S., Prunet, S., Aubert, D., Pichon, C., Amara, A., Benabed, K., Colombi, S.,
+Refregier, A., & Starck, J. (2009). Full-sky weak-lensing simulation with 70 billion particles.
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+Zhao, G.-B., Pogosian, L., Silvestri, A., & Zylberberg, J. (2009a). Cosmological tests of GR – a
+look at the principals.
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+Illuminating Dark Matter with Weak
+Lensing from Euclid
+Authors: T. D. Kitching (University of Edinburgh), A. Amara (ETH, Zurich), D. J. Bacon (Uni-
+versity of Portsmouth), S. L. Bridle (University of London), K. Markovic (University of Munich),
+R. Massey (University of Edinburgh), J. Read (University of Zurich) + EIC Weak Lensing
+Working Group.
+Abstract
+In the EIC we have shown that Euclid’s low level of systematics and high precision statistics,
+available only from a space based experiment, can be used to improve our knowledge of all
+aspects of dark matter. Here we focus on the work of this group in during the Euclid Phase
+A activities. The power spectrum of fluctuations in the dark matter distribution, measured
+through the weak lensing power spectrum, can be used to constrain both the growth of large
+scale structure and the geometry of Universe. The distribution and halo shape of dark matter
+from galaxy to cluster scales will be mapped using shear and flexion statistics. The particle
+properties and nature of dark matter will be constrained using cosmic shear observations. Euclid
+will be a unique experiment for the investigation dark matter, and complementary to advanced
+particle physics and direct detection experiments. The large scale distribution, self-interaction
+cross-section and confirmation of any dark matter candidate as being the actual dark matter can
+only be achieved using high precision cosmological observations.
+7.1 Introduction
+The matter content of the Universe is dominantly in the form of a cold non-baryonic component
+that interacts with ordinary baryonic matter only via the gravitational force on cosmic scales. This
+astonishing realisation has been proved by a number of independent and disparate experiments.
+The first evidence for some “missing” (non luminous) mass came from observations of the Coma
+cluster ( Zwicky 1933,1937) and Virgo cluster ( Smith 1936) where the velocity dispersion of the
+galaxies could not be accounted for by the mass implied by the luminous matter. On galactic
+scales a similar line of evidence was discovered when stars in Andromeda ( Babcock 1939;Rubin
+& Ford 1970;Roberts & Whitehurst 1975), NGC3115 ( Oort 1940) and other spiral galaxies
+(Rubin et al. 1985) were found to be rotating about the galactic enters at velocities that could
+not be explained by the luminous matter alone.
+627.2. Particle Properties 63
+10 010 510 10 10 15 10 −30 10 −20 10 −10 10 010 10 
+M [M solar h] dn(M)/dM mWDM  = 1 keV 
+mWDM  = 5 keV 
+CDM 
+10 010 510 10 10 15 0.2 0.4 0.6 0.8 1
+M [M solar h] dn WDM (M)/dM / dn CDM (M)/dM mWDM  = 1 keV 
+mWDM  = 5 keV 
+Figure 7.1: Left: the halo mass functions for Warm (1KeV or 5KeV) and cold dark matter
+particles. Right: the ratio of the mass function in a universe containing warm dark matter
+particles with masses of 1 keV (dashed line) and 5 keV (solid line) to the mass function in a
+universe with standard cold dark matter. The circles mark the halo mass that corresponds to the
+scales suppressed by free-streaming of these two types of warm dark matter particles. particles.
+From Markovic et al. (2009).
+The modern ΛCDM model of cosmology also requires a large reservoir of (cold) dark matter
+to account for the observed energy density of the Universe, and to explain structure formation.
+Slowing the expansion after the Big Bang required much more mass than the baryons could
+provide ( Cocet al. 2004). However, the dark matter must have stopped interacting with photons
+very early to preserve the uniformity of the primordial Cosmic Microwave Background radiation,
+in which fluctuations reach only one part in 10−5(Peebles 1982). Indeed, after decoupling from
+other particles, dark matter could collapse under its own gravity before ordinary matter and
+these dark matter potential wells laid the initial seeds about which structure formation grew
+(Davis et al. 1985;Efstathiou et al. 1985,1990;Springel et al. 2006). The latest measurements of
+the Cosmic Microwave Background and Large-Scale Structure ( Dunkley et al. 2009;Lesgourgues
+et al. 2007) indicate that the Universe contains approximately one hydrogen atom per cubic
+metre, but five times that in the form of dark matter.
+Most significantly gravitational lensing (see Massey et al. 2009, for a recent review by EIC
+members) probes the total matter content along the line of sight so in our Universe, dominated
+by dark matter, gravitational lensing is effectively a direct probe of the dark matter distribution.
+On cluster scales the “bullet cluster” ( Clowe et al. 2007) represents the first empirical observation
+of dark matter and has been used to constrain the dark matter cross-section from lensing
+measurements. We review other dark matter constraints from lensing measurements in the
+following sections.
+Despite the overwhelming abundance of dark matter we do not know what it is, how much
+there is over all scales or how it is distributed in detail. Euclid’s exquisite imaging surveys will
+help to illuminate the nature of this unknown but dominant component of our Universe.
+7.2 Particle Properties
+One explanation for dark matter is that there is some as yet undetected non-baryonic sub-atomic
+particle beyond our standard model of particle physics that, once created in the early Universe
+exists and pervades the Universe until the present time. This Weakly Interacting Massive Particle7.2. Particle Properties 64
+(WIMP) picture has emerged as being preferable over an alternative baryonic explanation, that
+undetected MAssive Compact Halo Objects (MACHOs) surround galaxies and clusters, due
+to substantial evidence from gravitational lensing microlensing events. The MACHO survey
+(Popowski et al. 2003;Alcock et al. 2000), OGLE ( Wyrzykowski et al. 2009;Udalski 2009) and
+POINT-AGAPE ( Calchi Novati et al. 2005) all rule out MACHOs as being the dominant dark
+matter component, in fact after collectively monitoring may tens of millions of stars for over 10
+years only a handful of microlensing events have been observed - whilst many hundreds were
+expected. There are two extensions to the standard model of particle physics both of which
+provide natural dark matter candidates. The axion is a Nambu-Goldstone boson produced by
+the addition of an extra symmetry (the U(1) Peccei-Quinn symmetry) that has been proposed
+to solve the CP problem. These particles may exist in the range of a few eV to a a few GeV.
+The supersymmetric (SUSY) extension to the standard model implies that any fermion/boson
+superpartners to standard boson/fermion particles must obey a condition called R-parity. Lepton
+and baryon number are not conserved in general SUSY models but these conservation rules
+are very well constrained by observations so R parity was introduced to suppress any coupling
+between SUSY particles and any interaction that may violate lepton or baryon number (e.g.
+a SUSY particle decaying into photons). This implies that the lightest SUSY particle cannot
+decay even if it is heavier that most (or all) standard model particles. Lightest SUSY particle
+include the neutralino (which is in fact an admixture of many SUSY particles), the axino and
+the gravitino. These particle are expected to have masses anywhere between a few eV-GeV (for
+the neutralino) and a few eV-TeV (for the gravitino).
+Candidates for particle nature of dark matter may appear in future particle physics exper-
+iments, such as the LHC, or be detected in direct detection experiments. However to prove
+that these candidates are indeed the dark matter that constitutes the majority of mass in the
+Universe we will have to use cosmological observations.
+Euclid will be extremely sensitive to any dark matter particle in the range of a few eV to a few
+hundred eV. A direct effect of dark matter is to change the relative abundance of sub-structure
+in galaxy clusters. For dark matter with a particle mass of 1 keV the majority of substructure
+below a mass of 1011M⊙will be diffused at 5 keV; this limit is reduced to 109M⊙. Using a
+weak lensing flexion variance ( Bacon et al. 2009) such a signature could be distinguished from
+LCDM at a 10- σconfidence.
+There exists a discrepancy between observations of structure on sub-galactic scales and high
+resolution N-body simulations assuming the ΛCDM cosmological scenario. In particular, the
+observed numbers of dwarf galaxies seem to differ from the theoretical prediction by up to an order
+of magnitude. Observations can potentially be reconciled with theory by replacing the standard
+cold dark matter with particles that are mildly relativistic (or ’warm’) at their decoupling. Such
+particles suppress the formation of structure by dampening the density fluctuations on scales
+below their free-streaming length. A fortunate consequence of such a scenario is that measuring
+the precise characteristics of the matter distribution on sub-galactic scales can provide constraints
+on the properties of the dark matter particle. Weak gravitational lensing traces the distribution of
+matter in the universe and so it can be used to calculate the power spectrum of matter structure.
+With the cosmic shear power spectrum from an upcoming survey like Euclid one can calculate
+the matter power spectrum on scales small enough to be of importance in warm dark matter
+calculations. Markovic et al. (2009) calculate a prediction for the upper limit detectable with
+such cosmic shear data providing a lower limit on the warm dark matter particle mass that would
+be allowed if the Euclid survey measured an apparently ΛCDM shear power spectrum. In this
+case, the least energetic, but still detectable neutrino-like, early decoupled dark matter particle
+would have a mass of 2keV. Figure 7.2shows how the halo mass distribution changes with respect
+to the mass of the warm dark matter particle. These are Sheth-Tormen mass functions in which7.3. The Large-Scale Distribution of Dark Matter 65
+the WDM effect was simulated by suppressing the linear power spectrum with the warm dark
+matter transfer function from Viel et al. (2005).
+Dark matter is most likely to contain multiple particle components, one of these components
+which is already known to exist will be massive neutrinos. Euclid will be extremely sensitive to
+massive neutrino properties through the effect that such particles have on structure formation.
+Massive neutrino tend to damp small scale structure by providing an effective pressure at small
+scales, this effect will be detectable by measuring the matter power spectrum through cosmic
+shear. Kitching et al. (2008) have shown that Euclid will be able to constrain the sum of neutrino
+masses and number to 3% and 10% respectively in combination with Planck. De Bernardis
+et al. (2009) have shown that Euclid will be able to distinguish the mass of individual neutrino
+species to percent accuracy and will be able to decisively (Jeffrey’s scale of Bayesian evidence)
+distinguish between the neutrino hierarchies.
+7.3 The Large-Scale Distribution of Dark Matter
+The large-scale distribution of mass can be described statistically in terms of its power spectrum
+P(k), which is shown in Figure 7.2. This is the Fourier transform of the correlation functions and
+measures the amount of clumping on different physical scales. The power spectrum is commonly
+decomposed into several components: the primordial power spectrum, the transfer function that
+maps the primordial fluctuation onto the dark matter power spectrum, and the growth factor
+that evolves the power spectrum as a function of time (or redshift)
+P(k)∝kn(k)T2(k)Growth2, (7.1)
+where the primordial power spectrum n(k) =n+αdlnn
+dlnkcan be written as a power law pa-
+rameterised by a slope nand some running α. A slope of unity is the Harrison-Zeldovitch
+approximation. If the density field is Gaussian, the same information can also be expressed
+as the mass function N(M,z), i.e. the density of haloes of a given mass as a function of that
+mass and cosmological redshift. There has been a substantial amount of work on analysing the
+properties of the dark matter power spectrum, including its growth over time from a power
+spectrum of primordial density fluctuations. In order to compare theoretical models to data, the
+semi-analytic approach of Eisenstein & Hu (1998) is used, predominantly for its simple fitting
+functions to the linear power spectrum, which can be extended (at 5-10% accuracy) into the
+mildy nonlinear high- kregime ( Peacock & Dodds 1996;Smith et al. 2003).
+The matter power spectrum thus encodes a wealth of information on the statistical distribution
+and growth of dark matter on all scales. The cosmic shear power spectrum combines measurements
+of the geometry of the Universe through the lensing effect with measurements of the dark matter
+power spectrum directly. For example the observed strength of the cosmic shear signal as a
+function of distance and angle can be written like
+Cκκ
+/lscript=9
+4/parenleftbiggH0
+c/parenrightbigg4
+Ω2
+m/integraldisplayrH
+0drP δ/parenleftbigg/lscript
+r,r/parenrightbigg/bracketleftbiggW(r)
+a(r)/bracketrightbigg2
+(7.2)
+wherePδis the dark matter power spectrum. This has the advantage over using the galaxy
+distribution to infer the dark matter power spectrum since the direct measure does not include
+any assumptions about the way that galaxies trace the underlying dark matter structure. The
+EIC has shown (see Chapter 4) that using the cosmic shear power spectrum the clumpiness of
+dark matter matter over a wide range of scales can be used to constrain cosmological parameters
+to unprecedented accuracy. The growth of structure contains information on how the dark
+matter structure grows over cosmic time as well as the dark energy component or signatures of7.4. Dark Matter Sub-Structure 66
+Figure 7.2: The statistical distribution of dark matter in Fourier space. Left: The mass power
+spectrum, showing the clumping of dark matter as a function of large scales (low k) to small scales
+(highk) (Tegmark et al. 2004), including some very early weak lensing constraints ( Hoekstra
+et al. 2002). Current constraints are tighter and extend over a wider range of scales in both
+directions. Right: The effect on the power spectrum of 1 MeV mass WIMP dark matter that
+remains coupled to other particles in the early universe ( Hooper et al. 2007); the solid (dashed)
+line assumes a 10 keV (1 keV) kinetic decoupling temperature. The dotted line illustrates the
+similar small-scale damping effect of a component of warm dark matter.
+modified gravity (see Chapter 6), weak lensing observations been used to detect the growth of
+these perturbations (e.g. Massey et al. 2007;Bacon et al. 2005).
+Using gravitational lensing Euclid will map the dark matter distribution. Pires et al. (2009)
+have reviewed the state-of-the-art in dark matter mass map reconstruction within the context of
+Euclid and discussed how to analyse this wealth of information to maximise the cosmological
+information that can be constructed.
+The dark matter distribution can also be mapped in three dimensions using weak lensing
+techniques coupled with photometric redshifts. Simon et al. (2009) has recently shown how the
+dark matter density map in three dimensions can be reconstructed, an extension of the work of
+Taylor et al. (2004) that showed how the gravitational potential of the dark matter field can be
+mapped in three dimensions. Such techniques will allow Euclid to map the three dimensional
+dark matter distribution over the entire extra-galactic sky.
+The large number density of high resolution galaxy images from Euclid will also allow non-
+linear features in the dark matter cosmic web to be observed such as filamentary structures that
+should connect dark matter haloes on large scales. Mead et al. (2009) have developed a number
+of weak lensing techniques that could be used to detect filaments in individual clusters or by
+stacking clusters.
+7.4 Dark Matter Sub-Structure
+The relative and absolute abundance of dark matter sub-structure as well as the distribution of
+dark matter sub-structure contains information on the nature of dark matter. For example if the
+temperature of dark matter was high then the effective pressure created on small scales will act
+to damp the growth of small sub haloes. In conjunction with large N-body simulations, that can
+be used to predict the abundance of small dark matter haloes for a given dark matter candidate,
+lensing measurements are already testing the CDM paradigm.7.4. Dark Matter Sub-Structure 67
+Figure 7.3: Plot adapted from Massey et al. (2007) and Fuet al. (2008). This central panel
+shows the raw ellipticity measurement from the weakly lensing galaxies, the length of the stick os
+proportional to the total ellipticity. Right : this measurement can be used to create dark matter
+maps where the colour represents the dark matter density. Left : this measurement can be used
+to construct a correlation function (or an equivalent power spectrum) from which global dark
+matter properties can be inferred.
+Figure 7.4: Images from Bacon et al. (2009). Left : a simulated cluster, contours show lines of
+constant density, this exhibits a large amount of substructure on all scales and at all radii. Right
+: Dashed are the mean flexion signal for simulations that include sub-structure (upper line) and
+for those that do not (lower line), solid are the flexion variance signal for simulations with (upper
+line) and without (lower line) sub-structure. Using flexion variance sub-structure is detected at
+all scales.7.4. Dark Matter Sub-Structure 68
+We here briefly review weak lensing flexion signal, where the galaxy distortion is measured
+at second order (a small ‘arcness’ in the image), which is particularly sensitive to small scale
+mass distributions including dark matter sub-structure. To measure this effect, even smaller in
+signal-to-noise that cosmic shear, a stable and well characterised PSF is crucial, hence Euclid is
+the ideal experiment to take advantage of this next-level of weak lensing information.
+Gravitational lensing maps a coordinates on the sky θ/prime
+ionto observed coordinates θifor
+i∈ {1,2}via
+θ/prime
+i=Aijθj+1
+2Dijkθjθk+..., (7.3)
+where the usual first-order matrix
+Aij(θ)≡∂θ/prime
+i
+∂θj= (δij−∂i∂jψ(θ)), (7.4)
+A=/parenleftbigg1−κ−γ1 −γ2
+−γ2 1−κ+γ1/parenrightbigg
+encodes shear and magnification, but the next terms in a Taylor series Dijk=∂kAijinclude
+flexion.
+This extension of the lensing formalism is most efficiently explored by first defining a complex
+gradient operator ∂=∂1+i∂2. The lensing deflection angle is obtained by applying this operator
+to the lensing scalar potential ψ
+α=α1+iα2=∂ψ. (7.5)
+The first order lensing observables are combinations of applying the operator twice
+κ=1
+2∂∗α=1
+2∂∗∂ψ (7.6)
+γ=γ1+iγ2=1
+2∂∂ψ, (7.7)
+where the gradient and its complex conjugate gradient ∂∗commute. As shown in figure 7.5,
+these respectively produce spin-zero (magnification) and spin-two (shear) distortions to an image.
+Taking the third derivative of the lensing potential we find the unique combinations
+F=1
+2∂∂∗∂ψ=∂κ=∂∗γ,
+G=1
+2∂∂∂ψ =∂γ. (7.8)
+These are known as the first and second flexions. They introduce spin-one and spin-three
+perturbations to an initially round image. These can be measured in a similar way to weak
+lensing shear methods, and yield constraints on the local derivative of the projected mass
+distribution.
+The use of flexion in the effort to reconstruct sub-structure was first suggested by Massey et al.
+(2007). More recently Bacon et al. (2009) have developed the analysis method by investigating
+the variance of the flexion signal around clusters. The key realisation in this work is that for a
+smooth matter profile the flexion signal increases towards areas of high density but the variance
+of the signal in azimuthal bins is small (for a smooth profile the flexion amplitude is the same
+in all directions). If sub-structure is present then the azimuthal symmetry is broken and the
+variance of the flexion signal increases. Using this technique Bacon et al. (2009) have shown that
+dark matter haloes as small as 109M⊙could be distinguished and that the signature of dark
+matter particles with masses of up to 5 keV could be detected.7.5. Dark Matter Halo Properties 69
+Figure 7.5: Weak lensing distortions with increasing spin values. Here an unlensed Gaussian
+galaxy with radius 1 arcsec has been distorted with 10% convergence/shear, and 0.28 arcsec−1
+flexion. Convergence is a spin-0 quantity; first flexion is spin-1; shear is spin-2; and second flexion
+is spin-3. From Bacon et al. (2006).
+In addition to the flexion signal there are also extra higher order moments of weak lensing
+that can be used as a systematic check. These are known as ‘twist’ and ‘turn’ and are described
+inBacon & Sch¨ afer (2009).
+7.5 Dark Matter Halo Properties
+In the standard cosmological model, dark matter haloes are expected to be significantly non-
+spherical. Measurements of weak gravitational lensing in the Sloan Digital Sky Survey confirm
+that the axis ratio of haloes around isolated galaxy clusters (projected onto the 2D plane of
+the sky) is 0 .48+0.14
+−0.19(Evans & Bridle 2008). This rules out sphericity at 99 .6% confidence and
+is consistent with the ellipticity of the cluster galaxy distribution. However, the dark matter
+haloes around individual galaxies are slightly rounder than the light profiles. By creating a 1D
+analogue of some computationally intensive 2D integrals Hawken & Bridle (2009) have shown
+that constraints on the ellipticity of dark matter haloes should be two orders of magnitude tighter
+from gravitational flexion than those from shear.
+Schneider & Bridle (2009) have extended the standard dark matter halo occupation to include
+the effects that these haloes have on the alignment of galaxies. This new model is based on the
+placement of galaxies into dark matter halos. The central galaxy ellipticity follows the large
+scale potential and, in the simplest case, the satellite galaxies point at the halo center. The
+two-halo term is then dominated by the linear alignment model and the one-halo term provides
+a motivated extension of intrinsic alignment models to small scales. Such effects will need to be7.6. Summary 70
+understood to characterise the intrinsic alignment systematic in weak lensing as well as being a
+direct probe of the local dark matter environment.
+Accurate measurement of the cluster mass function is a crucial element in efforts to constrain
+dark matter structure formation models. Euclid will observe hundreds of thousands of clusters
+and groups from 108–1015M⊙. By measuring the averaged mass profiles of these clusters over
+a wide range of masses and redshifts, the characteristic ‘clumpiness’ and ‘cuspiness’ of dark
+matter structures can be tightly constrained as a function of redshift and mass. Constraining
+the evolution of cluster mass profiles in this way places limits on the overall dark matter content
+of the Universe as well as the temperature and mass properties of the dark matter particles.
+Corless & King (2009) have shown that the effect of ignoring potential tri-axiality of dark matter
+haloes when stacking the lensing signals of many clusters and groups, binned by mass-correlated
+observables such as richness and luminosity, bias 3D virial mass estimates by a few percent. This
+bias affects not only direct lensing constraints on the mass function but can also affect the scatter
+and normalisation of the mass-observable relations derived from lensing that are so crucial to
+constraining the cluster mass function with large samples.
+7.6 Summary
+Euclid is poised to revolutionise the study of the dominant dark matter component of our
+Universe. Euclid will map the universal distribution of dark matter on large scales over the entire
+extra-galactic sky and on cluster and galaxy scales. Euclid will provide the data with which
+galaxy clusters and galactic haloes can be used as laboratories for the study of dark matter.
+Using shear and flexion statistics dark matter haloes will be detected from 108–1015M⊙. The
+constraints on the shapes of dark matter haloes including the cuspiness and tri-axiality will be
+improved by two orders of magnitude over current results. The nature of dark matter will be
+tested, using cosmic shear and flexion Euclid will constrain any eV–KeV dark matter models,
+and in particular constrain the neutrino mass and hierarchy to percent accuracy.
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+Zwicky, F. (1937). On the Masses of Nebulae and of Clusters of Nebulae. ApJ,86, 217–+.Chapter 8
+Point-Spread Function and
+Instrument Calibration
+Authors: St´ ephane Paulin-Henriksson (CEA - Paris), Adam Amara (ETH - Zurich), Sarah
+Bridle (UCL - London), Eduardo Cypriano (UCL - London), Alexandre R´ efr´ egier (CEA - Paris),
+Jason Rhodes (JPL - Los Angeles), Michael Seiffert (JPL - Los Angeles), Mario Schweitzer (MPI
+- Munich), Lisa Voigt (UCL - London) + EIC Weak Lensing Working Group.
+Abstract
+The Point-Spread-Function (PSF) and instrumental calibrations have been identified as critical
+issues to achieve the required accuracy in shape measurements, characterised by the level-1-
+requirement σ2
+sys<10−7. A simple analytical model allows us to estimate that one needs typically
+50 stars to calibrate the PSF. After presenting this, we discuss three effects that could degrade
+the accuracy of shape measurements, but are not taken into account in the analytical model:
+the pixellation, whose the impact is correlated with the PSF shape; the colour dependence of
+the PSF, and the Intra-Pixel Sensitivity Variations (IPSV). The latter effect is also discussed
+in the framework of infra-red detectors. First, to investigate the impact of pixellation, we have
+implemented a PSF assessment pipeline and derived preliminary level-2 requirements on the PSF
+shape. Second, we find the PSF colour dependence is compatible with the required accuracy.
+Third, we investigate the impact of IPSV in a general case for shape measurements in the visible
+and photometry in the near infra-red.
+8.1 Introduction
+Since the Point-Spread-Function (PSF) of an instrument varies on all scales, measuring the shape
+of a galaxy requires the PSF being calibrated using the stars that surround the lensed galaxy.
+Any error on the estimated PSF propagates in a systematic error on the measure of the galaxy
+shape. The required accuracy of the PSF calibration is fixed in the level-1-Top-Level-Science-
+Requirements (EIC.LEV1.4), in terms of an upper limit of 10−7on the variance of the systematic
+effectsσ2
+syswhen estimating the galaxy shear. σ2
+sysis defined in Amara & R´ efr´ egier (2008).
+In Sect. 8.2, we estimate the number of stars that must be considered during the PSF-
+calibration process, to achieve this accuracy, in the simple framework of an analytical propagation
+of uncertainties, for a given wavelength, and for infinitely small pixels. We then review 3 effects
+that could significantly degrade this accuracy, and study them in details through simulations.
+748.2. The number of stars required to calibrate the PSF 75
+Figure 8.1: Illustration of the required number of stars N∗to calibrate the PSF: when measuring
+the shape of a galaxy (indicated by the red spiral), the PSF needs to be calibrated with at least
+N∗nearby stars (black asterisks) contained in the shaded region, to achieve the requirement
+σ2
+sys<10−7. In this example N∗= 11. Therefore, on scales smaller than the shaded region,
+there is not enough information coming from the stars to measure the PSF variations with the
+required accuracy.
+Each of these 3 effects have a dedicated section in the following, that present the current status
+and point out that there is on going work to further develop our requirements:
+•The pixellation and PSF shape . For shape measurements, pixellation induces biases
+and larger statistical errors than in the analytical case. The impact of pixellation is highly
+correlated with the PSF shape. This is discussed in Sect. 8.3.
+•The wavelength dependence of the PSF : In general the PSF depends on the Spectral
+Energy Distribution (SED) of the object, and therefore will not be the same for stars and
+galaxies. Thus, calibrating the PSF on stars to deconvolve galaxies, is a biased approach.
+This point is discussed in Sect. 8.4.
+•Intra-Pixel Sensitivity Variations (IPSV): The IPSV is discussed in the framework
+of the photometric observations performed in the Near Infra-Red (NIR) bands. This is
+presented in Sect. 8.5.
+This chapter does not discuss of the finite accuracy of the analysing, which can be considered
+as another effect that may degrade the accuracy of measures. Indeed, as discussed in Paulin-
+Henriksson et al. (2009), this is a software issue that concerns the data analysis, while this
+chapter is dedicated to instrumental effects.
+8.2 The number of stars required to calibrate the PSF
+Each star provides an image of the PSF that is pixellated and noisy, which means that to reach
+a given accuracy in the knowledge of the PSF, a number of stars is required. This is illustrated
+in Fig. 8.1. But increasing the number of stars does not necessarily imply that the accuracy
+of the PSF calibration will be better. There is a trade-off between the number of stars, their8.3. Impact of pixellation and PSF shape 76
+density, the stability of the PSF, and the accuracy of the interpolation scheme (i.e. the way the
+PSF is interpolated between stars).
+The required number of stars to calibrate the PSF at a given accuracy was first issued in
+Paulin-Henriksson et al. (2008), in the simple case of an analytical propagation of uncertainties,
+for infinitely small pixels and assuming the shear estimator is function of the second order moment
+of the flux. In this paper we find the scaling relation between the number of stars N∗involved in
+the PSF calibration, the value of σ2
+sys, the complexity Ψ of the PSF model (characterising the
+number of degrees of freedom in the model), the Signal-to-Noise Ratio (SNR) of stars, and the
+dilution factor D=Rgalaxy
+RPSF, which is the ratio between the radius of the smallest galaxy and the
+radius of the PSF:
+N∗≈50×/parenleftbiggSNR
+500/parenrightbigg−2
+×/parenleftbiggD
+1.5/parenrightbigg−4
+×/parenleftbiggΨ
+3/parenrightbigg2
+×/parenleftBigg
+σ2
+sys
+10−7/parenrightBigg−1
+(8.1)
+For the simplistic case of a nearly circular Gaussian-like PSF, we have Ψ ≈√
+2. In a realistic case,
+Ψ is at least of few. The accuracy of this scaling relation is limited by two main approximations:
+•first, it is pessimistic in the sense that it assumes the shear estimate is a function of
+unweighted 2ndorder moments of the flux. This assumption was made to allow analytical
+calculations but in practice we expect the required number of stars to be lower;
+•second, it is optimistic in the sense that it does not take into account a number of significant
+factors, such as the pixellation. In other words, the scaling relation 8.1holds for infinitely
+small pixels. In practice, for big pixels or wide observation bands, the required number
+of stars may be larger. Moreover, radiation damages during the mission on the charge
+transfer efficiency may require this number of stars to increase during the mission time.
+To investigate the required number of stars in practice, we performed a number of simulations
+summarised in the following.
+8.3 Impact of pixellation and PSF shape
+The impact of pixellation is highly correlated with the shape of the PSF. That is why it is
+necessary to investigate together pixellation and PSF shape. For this, we have implemented a
+PSF shape assessment pipeline, that is described below (Sect. 8.3.1), and that is used to perform
+large simulation sets. In particular, we focused our attention on a realistic system PSF, noted
+nominal’ PSF because it is derived from the optical design proposed by industry. The nominal
+PSF is described in Sect. 8.3.2, and our current results on the impact of pixellation and PSF
+shape are summarised in Sect. 8.3.3.
+8.3.1 PSF shape assessment simulation
+An overview of the PSF shape assessment simulations is shown in Fig. 8.2. The main steps are
+the following:
+•Simulation of star fields and galaxies: the system PSF is used to simulate N star fields, and
+M 2D-exponential profile galaxies. We currently take into account the dithering strategy
+and the Intra-Pixel Sensitivity Variations.
+•Calibration of the PSF on a star field: we consider the system PSF with few arbitrary
+chosen degrees of freedom (e.g. a rotation, a dilation/compression and a distortion) and fit
+these degrees of freedom on one star field, with a χ2minimisation.8.3. Impact of pixellation and PSF shape 77
+Figure 8.2: Summary of the PSF assessment simulations that estimates the bias on the shear
+measurements, for a given system PSF and galaxy population. The system PSF, in input, is used
+to simulate Nstar fields and Mgalaxies. Then the simulations can be sorted into 2 nodes. First,
+we simulate Ntimes the PSF calibration on the star fields. In output, each star field leads to an
+estimation of the PSF. Second, we estimate the galaxy shears, considering the PSF estimations
+instead of the true PSF. Each calibration of the PSF leads to a bias on the galaxy shear coming
+from the fact that the estimated PSF is not exactly equal to the true one. From the N biases, we
+deriveσsys.
+•Measure of the M galaxy shears, for a given estimated PSF: For a given estimation of the
+PSF, we fit a 2D-exponential profile on each galaxy, with a chi2-minimisation, and derive
+the corresponding fitted ellipticity. There is a bias in the fitted ellipticity, due to the fact
+that the estimated PSF is not infinitely accurate. Each of the N estimations of the PSF
+leads to a different bias.
+•From the N biases, we compute the corresponding δγandσsys.
+These simulations allow us to investigate the impact of number of effects, such as: the PSF and
+the galaxy shapes, the pixel size, the SNR of stars and galaxies, the number of stars involved in
+the PSF calibration, the interpolation scheme between stars, the dithering strategy, and some
+detector effects. We are currently including a detailed simulation of Radiation Damages on the
+Charge Transfer Efficiency.
+8.3.2 The nominal PSF
+The ’nominal’ PSF is derived from the optical design proposed by industry. We calculated it by
+the convolution of 3 components of the PSF:
+•The optical PSF: This is the component that comes from the optical configuration.
+For instance this would include diffraction effects. We have simulated the optical PSF
+(illustrated in Fig 8.3) that corresponds to the optical design proposed by industry with a
+resolution of 5 ×10−3arcsec per step, over a postage stamp of 5 .12×5.12 arcsec2.
+•The AOCS PSF: This component comes from the movement of the instrument during
+an exposure. We adopted a realisation of the AOCS pattern.
+•The detector PSF: This component comes from the detectors effects such as diffusion
+(but this does not include pixellation, which can not be modelled as a convolution). We
+adopted a simple Gaussian description of the detector PSF.8.3. Impact of pixellation and PSF shape 78
+Figure 8.3: PSF at 800nm generated by the nominal optical design. The left-hand panel is
+the optical PSF (with a 3-arm spider of the structure maintaining the central obstruction) in
+logarithmic scale. The middle shows the system PSF (ie. the convolution of the optical PSF
+with the AOCS and the detector patterns). The right-hand panel shows the profile of the system
+PSF. Black line: from the centre toward the top (ie. far from any diffraction spike). Red line:
+from the centre toward the right-hand side (ie. inside a diffraction spike).
+It should be noted that the system PSF under discussion here is the un-pixellated realisation of
+the PSF. The final image that a point source would form is a pixellated version of the system
+PSF, that depends on the position of the point source inside a pixel. The requirements and
+verifications are performed on the un-pixellated system PSF.
+The properties of the nominal PSF are listed in Tab. 8.1. The level-2-requirements are derived
+from the level-1-requirement σ2
+sys<10−7, using the PSF shape assessment simulations presented
+in Sect. 8.3.1. As explained in Sect. 8.3.3, these level-2-requirements are conservative: it is possible
+to achieve the level-1-requirement σ2
+sys<10−7even if the level-2-requirements on the PSF shape
+are marginally achieved. This is the case for the nominal PSF, that is acceptable although the
+FWHM is too small compared to the corresponding level-2-requirement.
+8.3.3 Current results about the impact of pixellation and PSF shape
+Fig.8.4showsσ2
+sysas a function of the PSF FWHM, for pixels of 0 .1×0.1 arcsec2, in 3 cases:
+1.Analytical pixellation-free model for a Gaussian-like PSF (dotted line): This is
+by far the most simple case, that requires no simulation. When applying the analytical
+calculations of Paulin-Henriksson et al. 2008, already mentioned above, in the case of a
+nearly circular Gaussian PSF, one obtains:
+σ2
+sys≈1.33
+D4SNR2N∗(8.2)
+We can see that, for a given star population, σ2
+sysdepends only on the dilution factor D
+(the ratio between the radius of the galaxy and the radius of the PSF): ie. larger the galaxy,
+smallerσ2
+sys. In this example, the dotted line corresponds to the case Rquadrupoles = 2 pixels.
+2.Semi-Analytical model for a Gaussian-like PSF, taking into account the pixel-
+lation (solid line) : We multiply the previous analytical prediction of σ2
+sys(ie. the
+relation 8.2forRquadrupoles = 2 pixels), with a corrective factor that corresponds to the
+impact of pixellation, that was estimated with the node 1 of the PSF shape assessment8.3. Impact of pixellation and PSF shape 79
+FWHM /epsilon1PSF EE50EE90
+EE50averaged shear σ2
+sys
+(arcsec) (arcsec) error |<δγ > |
+level 1
+requirement <10−7
+level 2
+requirements 0.18−0.23<0.1 none <5 none
+(goal) (<0.05) ( <4.5)
+nominal PSF 0.16 0.034 0.22 5.0 2 .7×10−47.3×10−8
+Table 8.1: Properties of the nominal PSF properties, at 800nm, at the centre of the field. The
+nominal PSF is the system PSF derived from the current optical design (illustrated above),
+after convolution with the AOCS and detector patterns. FWHM: Full Width at Half Maximum.
+/epsilon1PSF: ellipticity of the PSF. EE50 and EE90: diameters containing 50 and 90 percents of the
+flux, respectively. The level-1-requirement σsys<10−7is the criteria that drives the level-2-
+requirements. We can see that it is achieved, implying that the nominal PSF is acceptable, even
+if the conservative level-2-requirement on the FWHM is not achieved. This is discussed in the
+text.
+pipeline (see Fig. 8.2). One can see 2 regimes: for a large PSF (right hand side of the plot,
+that corresponds to small pixels compared to the PSF size), the solid line is close to the
+dashed one; one the other hand, when the PSF size decreases (and therefore the pixel size
+increases compared to the PSF size), the impact of pixellation increases and the solid line
+moves away from the dashed line, the optimal PSF size being around 0 .18 arcsec, with a
+nearly flat plateau between 0 .15 arcsec and 0 .18 arcsec. Note that this plateau holds for the
+assumption of a Gaussian PSF. We are not sure of this plateau for a realistic PSF. That is
+why level-2-requirements recommend the FWHM to be between 0 .18 arcsec and 0 .23 arcsec
+(see Tab. 8.1).
+3.Complete simulations with a realistic PSF (colour-shaded region and star): We
+use nodes 1 and 2 of the PSF assessment pipeline to fully simulate σ2
+sysfor a given PSF,
+and characterise the PSF shape with 4 parameters: its FWHM, the diameters EE50
+and EE90 containing 50 and 90 percents of the flux, respectively, and its ellipticity /epsilon1PSF.
+To investigate the impacts of these 4 shape parameters, we perform thousand of times
+the whole simulation set, each time with a different system PSF. For the moment, to
+focus on realistic PSFs, we restrict our study to small variations around the nominal
+PSF (presented in Sect. 8.3.2). And we adopt a PSF model with 5 degrees of freedom:
+a distortion; a rotation; a dilatation; 1 parameter ruling the wings, correlated with the
+centre; and a 2ndwing parameter uncorrelated with the centre. For small galaxies of size
+Rquadrupoles = 1 pixel , the realistic PSFs we have investigated so far, lie in the colour-shaded
+area of the Fig. 8.4. This area is divided in 3 regions: the red one that is not acceptable,
+because the σ2
+sys<10−7requirement is not achieved; the green one that is acceptable,
+and an orange one at the interface. The star shows the position of the nominal PSF,
+that is acceptable. We find that the following preliminary requirements can be adopted
+to achieveσ2
+sys/lessorsimilar10−7: 0.18/lessorsimilarFWHM /lessorsimilar0.23 arcsec,/epsilon1PSF<0.1, and EE90 / EE50 <5.
+We emphasise that these preliminary requirements were derived for some small variations
+around the nominal PSF. On one hand, they are conservative: it possible to achieve the
+level-1-requirement σ2
+sys<10−7, even if these level-2-requirement are marginally achieved.8.4. Impact of the Point-Spread-Function Colour Dependence 80
+This is the case, for instance, of the nominal PSF (see Tab. 8.1). On the other hand, these
+requirements are probably not sufficient to ensure σ2
+sys<10−7in a less specific case.
+8.4 Impact of the Point-Spread-Function Colour Dependence
+In general, the PSF depends on the spectral energy distribution of the object, and therefore will
+not be the same for stars and galaxies. Cosmic shear analyses of Hubble Space Telescope data do
+often use a PSF found from the telescope model due to the stability of the instrument. However
+even in that case the spectral energy distribution of the galaxy and the wavelength dependence
+of the PSF must be taken into account. The simulations and results are shown in more detail in
+Cypriano et al. (2009).
+We use realistic galaxy and stellar populations to quantify the amount by which the PSF
+FWHM is likely to be misestimated. Galaxy SEDs are generated using mock catalogues designed
+to simulate the distribution of redshifts, colours and magnitudes of galaxies in GOODS-N
+(Kinney et al. 1996;Cowie et al. 2004). Template spectra are taken from Kinney et al. (1996)
+and intermediate types obtained by linear interpolation of these templates (see Abdalla et al.
+(2008) for further details). The PSF sizes for stars are estimated using stellar SEDs from the
+Bruzual-Persson-Gunn-Stryker Spectro-photometric Atlas. The catalogue contains 175 different
+SEDs covering a broad range of spectral types.
+Fluxes are obtained for each object in the mock catalogues for the Euclid on-board filters F1
+and Y up to the limiting magnitudes 24.5 and 22.80 respectively (AB magnitudes, 10 σdetections).
+In addition, fluxes are measured in the g, r, i, z and y filters up to the magnitudes 24.45, 23.85,
+23.05, 22.45 and 20.95. These configurations are chosen to simulate data from future ground
+based cosmology surveys such as DES and Pan-STARRS. Such observations could be exploited
+to optimally correct the wavelength-dependence of the PSF.
+We used a template fitting method to predict the PSF FWHM of a galaxy by using all the
+colours available. Using ANNz ( Collister & Lahav 2004) we trained a neural network using
+approximately one fourth of the simulated galaxies available, to predict the redshifts, spectral
+types and reddening of each galaxy, given the fiducial multi-colour information. With this
+information we compute the SED of each object and use a model of the wavelength dependence
+to predict the PSF FWHM for this galaxy. A telescope model and stars will be used to build this
+model for the wavelength dependence, and the accuracy of the model will depend on the stability
+of the wavelength dependence with telescope properties, and the number of stars available to
+calibrate which model to use. In the case of the Hubble Space Telescope, a PSF model taken
+from the telescope design is routinely used in conjunction with calibration from any stars in the
+field to assess the telescope configuration in a given observation. Therefore in this paper we use
+the exact model as given in its Eq. 9 and propagate the noisy and potentially biased galaxy SED
+estimates through to PSF biases and cosmological parameter biases. The comparison between
+this predicted PSF FWHM and the truth for the exact galaxy SED and redshift can be seen in
+the right hand panel of Fig. 8.5.
+We propagate the biases as a function of redshift through to biases on cosmological parameters
+assuming a tomographic cosmic shear survey with the usual seven free parameters and find that
+the residual biases meet our requirements.
+8.5 Intra-Pixel Sensibility Variations
+Intra-Pixel Sensibility Variations (IPSV) are variations of the detection efficiency within pixels.
+These small variations are the result of wavelength dependent absorption depth of CCDs. As8.5. Intra-Pixel Sensibility Variations 81
+Figure 8.4: σ2
+sysaccording to the PSF size, for a fixed pixel size of 0.1 arcsec. Dotted line: analytical
+pixellation-free model for a Gaussian-like PSF, for galaxies of size Rquadrupoles = 2 pixels . Solid line
+(semi-Analytical model for a Gaussian-like PSF, taking into account the pixellation): the previous
+analytical pixellation-free model is corrected for the impact of pixellation, estimated with the node
+1 of the simulations (see Fig 8.2). Note that the nearly flat plateau, for 0.15<FWHM<0.18 arcsec ,
+holds for the simple assumption of a Gaussian-like PSF. This plateau depends on the PSF shape.
+In particular, for a realistic PSF this region 0.15<FWHM<0.18 arcsec is unstable: there can
+be a large increase of σ2
+syswhen decreasing the PSF size below 0 .18 arcsec. That is why level-
+2-requirements, that are conservative, recommend the FWHM being larger than 0 .18 arcsec.
+Colour-shaded area: region where the systematics lay, according to the current results of the PSF
+assessment simulations (see Fig. 8.2), for a realistic PSF close to the nominal one, a PSF model
+with 5 degrees of freedom described in the text, and for a galaxy size of Rquadrupoles = 1 pixel .
+Red corresponds to σ2
+sys>10−7and is not acceptable, green corresponds to σ2
+sys<10−7, and
+orange corresponds to the interface. star: position of the nominal PSF.8.5. Intra-Pixel Sensibility Variations 82
+Figure 8.5: PSF FWHM for galaxies (black dots) and stars (yellow stars) as a function of object
+colour. The left hand panel shows calibration using space-data alone and the middle panel shows
+the result of using a colour composed of the wide Euclid optical filter and a ground based r band
+filter such as that used for photometric redshift estimates. The insets show the dispersion about
+the best fit first or second order polynomial to the stars (yellow line). We see the results using
+r-F1 are significantly better than using F1-Y. The right hand panel shows the results of our
+template fitting method.
+such, they are amenable to the same techniques described to deal with the wavelength dependent
+PSF (see Sect. 8.4). The size of the effect depends on both the resistivity and thickness of the
+CCD (with thickness and resistivity being mitigating factors). Future work will include a study
+of exactly how large this effect is for the chosen Euclid configuration and the development of
+mitigating techniques. Below, we describe a preliminary estimate of the impact of IPSV for the
+infra red photometry, and shape measurements in the visible channel.
+8.5.1 Impact of Intra-Pixel Sensibility Variations in the near infrared
+8.5.1.1 Discussion
+The required relative photometric accuracy of NIP for photo-z (as broken down from science
+requirements) has been defined as follows: Relative Photometric Accuracy for photo-z: the
+final relative photometric accuracy (i.e. post-calibration) δF<0.5. This number applies to the
+uniformity within a field and among areas on the sky. The final calibration includes the off-line
+data processing of the fields, which assumes additional ground based information (secondary
+standard stars) and the possibility of self calibration based on consistencies in the (redundant)
+Euclid data. As long as the relative accuracy is met, the absolute photometric accuracy can be
+constructed from the data afterwards. Enough statistics will be available from standard stars to
+obtain a consistent absolute photometric calibration of TBD final accuracy.
+The size of the PSF FWHM relative to the detector pixel size plays an important role w.r.t.
+to the impact that the detector Pixel Response Function (PRF) will have on the final photometric
+accuracy. In case the PSF FWHM becomes significantly smaller than the detector pixel size
+the PRF may lead to a degradation of the photometric accuracy. To meet the aforementioned
+relative photometric accuracy the following requirement on the PSF sampling has been defined
+on the Near Infra-Red (NIR) spatial resolution: the number of pixels per system PSF FWHM in
+the J band must be greater than 1 (TBC). The PSF FWHM of the current (EIC) optical design
+is less (<13µm) than the NIR detector pixel size of 18 µm. In this case the PRF may lead to a8.5. Intra-Pixel Sensibility Variations 83
+Figure 8.6: PRF from Barron et al. (2007).
+degradation of the photometric performance.
+Inter Pixel Capacity (IPC), lateral charge diffusion, optical cross talk and intra pixel variations
+are shaping the PRF. The PRF has been measured for the H2RG 1 .7µm cut-off detector by
+Barron et al. (2007), see Fig. 8.6. Based on this data the impact of the PSF size on the
+photometric accuracy has been investigated. Instead of using the PSF profile from the zemax
+optical simulation, a Bessel function J1 of first order and of first kind has been used to simulate
+the PSF (IJ actual) in J band :
+IJactual(x,δ) := if
+|x−δ|<10−48,1,
+2J1/bracketleftBig
+π(x−δ)×10−6
+λ2×N/bracketrightBig
+π(x−δ)×10−6
+λ2×N
+2
+, (8.3)
+with f-Number N=10. The wavelength lambda2 has been chosen such that the PSF FWHM
+equals 12.6µm (as determined by the optical simulation). The resulting flux (FJ actual(delta))
+has been calculated by integrating the product of the PSF (IJ actual) and the normalised PRF
+(PRFtotal6) from Fig. 8.6(blue curve) over a photometric aperture of 18 µm:
+FJactual(δ) :=/integraldisplay9+δ
+−9+δdxPRFtotal6( x) IJactual(x,δ) (8.4)
+The resulting flux FJ actual(δ) is a function of δ, which determines the relative shift between the
+centre of the PSF and the PRF. Further, it was assumed that the Euclid observation strategy
+will involve four dither positions so that the total final flux (Ftotal) will be the sum of four
+individual flux measurements resulting from four randomly shifted PSFs w.r.t. to the PRF:
+Ftotal(d,in) :=(FJactual(rin+d) + FJ actual(rbin+d) + FJ actual(rcin+d) + FJ actual(rdin+d))
+4.
+(8.5)
+dis the common shift of the individual PSF’s w.r.t. the PRF; r,rb,rc,rd are random shifts (in a
+range of ±10µm) from a set of random numbers with index ’in’. It was found (see Fig. 8.7) that8.5. Intra-Pixel Sensibility Variations 84
+Figure 8.7: Comparison of flux variations between FJ actual and Ftotal (for 10 different sets of
+random shifts).
+the resulting flux variations are significantly reduced (averaged out) when the actual observing
+scheme is taken into account.
+Using 10 different sets of random shifts for Ftotal on average it was found:
+stddev(Ftotal( d))
+mean(Ftotal( d)))= 0.0014 (8.6)
+8.5.1.2 Conclusion
+It can be concluded that the actual size of the PSF FWHM is not expected to be critical w.r.t. the
+photometric calibration requirements. Dithering allows to reduce (averaging out) the photometric
+error caused by the detector effects to a level which is well acceptable and in line with the
+photometric requirements. We caution that to our knowledge no PRF has been measured for the
+2.5µm detector. In addition it is known that single detector devices can show untypical PRF
+profiles that may seriously degrade the photometric accuracy. For this reason it is assumed that
+each single detector has to be characterised w.r.t. to its PRF performance. In addition it may
+be uncertain to which level the PRF measurements of the intrinsic detector PRF effects may be
+biased by the precision of the measurement itself. If the measurement errors are significant there
+is further room for improvement. If the above mentioned result holds true also for the 2 .5µm
+cut-off detector at operational conditions it is expected that no PSF enlargement for the NIP is
+required.
+8.5.2 Impact of Intra-Pixel Sensibility Variations for shape measurements
+We implemented the IPSV in the simulations presented in Sect. 8.3.1, and performed an initial
+estimation of their impact on σsysin case of a miscalibration of the IPSV pattern. As we do
+not know yet the IPSV pattern of pixels in Euclid CCDs, we have restricted our study to a
+2D elliptical Gaussian pattern and we have investigated the bias on ellipticity measurements,
+implied by such a miscalibration, with respect to the pixel size, with the assumption that all
+the pixels have the same IPSV pattern. To do that, we simulate galaxies with a given IPSV
+pattern of ellipticity /epsilon1p, but we analyse them assuming perfect top-hat pixels. /epsilon1pis interpretedBIBLIOGRAPHY 85
+Figure 8.8: Impacts of Intra Pixel Sensitivity Variations (IPSV) for shape measurements. The
+y-axis shows the bias on one component of the ellipticity. The horizontal dashed lines show the
+acceptable upper limit (corresponding to σ2
+sys= 10−7). The x-axis shows the ellipticity /epsilon1pof
+the 2D-Gaussian IPSV pattern used to simulate galaxies. These galaxies are analysed assuming
+perfect top-hat pixels, and /epsilon1pis interpreted as a miscalibration of the IPSV pattern, noted
+’accuracy of the pixel ellipticity calibration’ in this example.
+as the miscalibration of the IPSV pattern (noted ’accuracy of the pixel ellipticity calibration’
+on the plot). The result is shown in Fig. 8.8. We find that, as expected, the systematic effects
+induced by IPSV for shape measurements, increase rapidly with the pixel size: For 0 .1 arcsec
+pixels, the accuracy of the calibration must be around 0.01, while it must be lower than 5 ×10−3
+for 0.2 arcsec pixels. The study of correlations between IPSV and wavelength dependence of the
+PSF will be subject of further work.
+Bibliography
+Abdalla, F. B., Amara, A., Capak, P., Cypriano, E. S., Lahav, O., & Rhodes, J. (2008).
+Photometric redshifts for weak lensing tomography from space: the role of optical and near
+infrared photometry. MNRAS ,387, 969–986.
+Amara, A. & R´ efr´ egier, A. (2008). Systematic bias in cosmic shear: extending the Fisher matrix.
+MNRAS ,391, 228–236.
+Barron, N., Borysow, M., Beyerlein, K., Brown, M., Lorenzon, W., Schubnell, M., Tarl´ e, G.,
+Tomasch, A., & Weaverdyck, C. (2007). Subpixel Response Measurement of Near-Infrared
+Detectors. PASP ,119, 466–475.
+Collister, A. A. & Lahav, O. (2004). ANNz: Estimating Photometric Redshifts Using Artificial
+Neural Networks. PASP ,116, 345–351.BIBLIOGRAPHY 86
+Cowie, L. L., Barger, A. J., Hu, E. M., Capak, P., & Songaila, A. (2004). A Large Sample of
+Spectroscopic Redshifts in the ACS-GOODS Region of the Hubble Deep Field North. AJ,
+127, 3137–3145.
+Cypriano et al. (2009). Cosmic shear requirements on the wavelength dependence of telescope
+point spread functions. in prep. .
+Kinney, A. L., Calzetti, D., Bohlin, R. C., McQuade, K., Storchi-Bergmann, T., & Schmitt, H. R.
+(1996). Template Ultraviolet to Near-Infrared Spectra of Star-forming Galaxies and Their
+Application to K-Corrections. ApJ,467, 38–+.
+Paulin-Henriksson, S., Amara, A., Voigt, L., Refregier, A., & Bridle, S. L. (2008). Point spread
+function calibration requirements for dark energy from cosmic shear. A&A ,484, 67–77.
+Paulin-Henriksson, S., Refregier, A., & Amara, A. (2009). Optimal point spread function
+modeling for weak lensing: complexity and sparsity. A&A ,500, 647–655.Chapter 9
+Mitigating Charge Transfer Inefficiency
+Authors: Richard Massey (University of Edinburgh), Olivier Boulade (CEA, Saclay), Mark
+Cropper (Mullard Space Science Laboratory, University College London), Jason Rhodes (NASA
+Jet Propulsion Laboratory).
+Abstract
+Charge Transfer Inefficiency (CTI) due to radiation damage above the Earth’s atmosphere
+creates spurious trailing in imaging with any Charge Coupled Device (CCD) detectors. We have
+examined the specific effects of CTI on the demanding galaxy shape measurements required
+by studies of weak gravitational lensing, and constraints on cosmology. We have developed an
+end-to-end method of simulating these effects for any specific CCD model and mission design
+(years at L2, operating temp, shielding, etc), plus a software post-processing method to correct
+images pixel by pixel. These are the tools that will be required during the development phase to
+perform a thorough analysis of the CTI effects on Euclid images to develop a mission design that
+meets requirements. A first pass using these tools has demonstrated that existing technology can
+be configured to implement a five-year Euclid mission throughout which the requirements on
+galaxy shape measurement are always met.
+9.1 Origin of CTI
+Euclid will image the sky using Charge-Coupled Device (CCD) detectors. During an exposure,
+these convert incident photons into electrons, which are stored within a silicon substrate, in a
+pixellated grid of electrostatic potential wells that gradually fill up. At the end of the exposure,
+the electrons are shuffled, row by row, to a readout register at the edge of the device as illustrated
+in figure 9.1. The electrons that emerge are then counted and converted into a digital signal.
+Above the Earth’s atmosphere, a continuous bombardment of high energy particles makes a
+harsh environment for sensitive electronic equipment. Euclid will operate at the Earth-Moon
+Lagrange point L2, outside the Earth’s protective Van Allen belts, where the dominant concern is
+solar radiation ( Barth et al. 2000). While high-energy, ionising particles generally create spurious
+streaks within a detector that impact individual exposures, low energy protons and heavy ions
+create longer-lasting bulk damage by colliding with and displacing atoms from the silicon lattice.
+The dislodged atoms can come to rest in the interstitial space, and the vacancies left behind move
+about the lattice until they combine with interstitial impurities, such as Phosphorus, Oxygen or
+another vacancy ( Janesick 2001). Such defects degrade a CCD’s ability to shuffle electrons, known
+as its Charge Transfer Efficiency (CTE; Charge Transfer Inefficiency CTI=1-CTE). Electrons
+879.2. Observational consequences of CTI 88
+Figure 9.1: Cartoon illustrating imaging with a CCD device, as will be used for Euclid . Incident
+photons generate photoelectrons that are stored in and gradually fill up electrostatic potential
+wells in a grid of pixels. At the end of the exposure, they are shuffled through the device in the
+parallel and serial directions, to a single amplifier and analogue-to-digital converter. Reproduced
+from Rhodes & et al. (2009).
+can become temporarily trapped in the local potential, then released after a delay τthat depends
+upon the properties of the lattice and impurities, and the operating temperature of the device
+(Shockley & Read 1952;Hall1952). Several different species of charge trap may be present in
+any given device, with different characteristic release times.
+The trapping and delayed release of charge mainly causes problems during CCD readout. As
+demonstrated in figure 9.2, if a few electrons are trapped and held while others are moved along,
+the trapped electrons are released as a spurious trail. Regions of the image furthest from the
+readout register are worst affected, because electrons starting there encounter the most charge
+traps during their journey across the device, and the effect gets worse over time, as cumulative
+radiation damage creates more charge traps.
+9.2 Observational consequences of CTI
+The trailed electrons affect weak lensing science by altering the photometry and astrometry
+of faint galaxies, and by adding a spurious, coherent ellipticity in the readout directions. It
+is important to note that the effect is non-linear: faint galaxies are most affected, as a higher
+fraction of their electrons are trailed by a fixed number of charge traps, compared to bright
+stars. It thus closely mimics the effect of cosmic shear, in which distant galaxies are coherently
+elongated and, since the effect is not a convolution, it cannot be dealt with by traditional weak
+lensing measurement methods. CTI was the dominant systematic in the weak lensing
+analysis of the largest survey by the Hubble Space Telescope (Massey & et al. 2007),
+and is one of the main concerns in the ESA Gaia mission.
+To investigate the level of CTI damage in Euclid , and its effect on galaxy shape measurement,
+we have simulated the process of image collection and readout in a software model ( Massey & et
+al.2009). The model is built around physical parameters, and has been verified and calibrated
+using real data from both in-flight Hubble Space Telescope observations and p-channel CCDs9.3. CTI mitigation by hardware design 89
+Figure 9.2: A typical, raw Hubble Space Telescope /Advanced Camera for Surveys image, in units
+of electrons. This 30/prime/prime×15/prime/prime(600×300 pixels) region is at the far side of the CCD to the readout
+register, which lies towards the bottom of the page. It was obtained in May 2005, 1171 days after
+the launch of ACS. Upon close inspection, as illustrated in the zoomed inset, spurious trailing of
+charge is manifest behind (above) every object. Reproduced from Massey & et al. (2009).
+irradiated in a laboratory. By running this software readout model on the simulated Euclid
+images described in part V, we have demonstrated that not all CTI is equally bad for the
+purposes of weak lensing measurement . As shown in figure 9.3, only charge traps with
+characteristic release times of the same order as the clock cycle during readout affect galaxy
+shape measurements. Charge traps with much longer release times steal flux from sources and
+primarily affect photometric redshift estimation.
+Some CTI can be traded off again other. An example of this is presented in figure 9.4, in
+which the operating temperature (which governs the traps’ characteristic release times) and CCD
+clock speed can be adjusted to optimise the operating regime of Euclid to avoid particularly
+problematic trap species. The EIC has now established a comprehensive analysis programme
+tailored to weak lensing measurement from hardware development to shape measurement. We
+have developed the necessary tools and are now working closely with manufacturers e2v to enforce
+specifications beyond the usual single number for CTI.
+9.3 CTI mitigation by hardware design
+Various schemes can be used to mitigate the damaging effect of the radiation flux on the CCD
+hardware:
+•optimising CCD operating temperature and clock speed to avoid “resonance” with the
+trapping speeds of common charge trap species (see figure 9.4);
+•building in radiation shielding;
+•using radiation-tolerant p-channel CCDs;
+•adding more readout registers or altering the CCD’s aspect ratio to decrease the distance
+travelled by electrons within a CCD;9.3. CTI mitigation by hardware design 90
+Figure 9.3: The effect of charge trap characteristic release time τon a measurement of photometry,
+astrometry, size and ellipticity of a typically small, faint galaxy degraded by CTI in Euclid . Each
+yaxis represents the fractional change in that quantity. The absolute values of the yaxes are
+largely irrelevant, depending upon the assumed density of charge traps, CCD well filling model,
+galaxy SN (13), size (FWHM=3.8 pixels) and morphology (circularly symmetric De Vaucouleurs
+profile). However, the trends reveal several lessons for future hardware: notice particularly the
+local maximum in ∆ e1, which implies a worst case clocking time, or readout cadence, for CCDs.
+Reproduced from Rhodes & et al. (2009).9.3. CTI mitigation by hardware design 91
+Euclid parallel clock spee d    
+Euclid serial clock spee d    
+Operating temp erature [K]   Charge trap characteristic release ti me [seconds]
+Traps relevant for shape
+measurement with Euclid
+Figure 9.4: The characteristic release times of charge trap species in n-channel CCDs, which
+depend upon operating temperature. By adjusting the temperature and clock speed, it is possible
+to avoid the effects of several charge trap species, as shown by the target greyscale boxes (whose
+shading reflects the bottom panel of figure 9.3).
+•implementing additional calibration diagnostics, such as pocket pumping, which allow the
+locations and properties of existing traps to be accurately measured on-orbit.
+The use of additional shielding is limited in the Euclid case because of the size of the focal
+plane, and the penetrating nature of the damaging radiation. Nevertheless, shielding is naturally
+available from the surrounding equipment, and some progress may be possible from the payload
+layout.
+There is an ongoing programme of discussions and workshops within the Euclid visible
+imager team in coordination with leading CCD manufacturers e2v and with ESA. An important
+consideration is to maintain a high level of technology readiness level, and the baselined e2v
+CCD203/204 family provides essentially all of the required characteristics. The CCD203/4
+optimisation programme includes the structural options noted above (such as different aspect
+ratios), and in addition considers the reduction of the size of the readout register to improve CTI
+in the serial direction: larger readout registers are used in case CCD rows are summed before
+readout, which will not be the Euclid case. Additional work centres on the clock timing, shaping
+and sequencing, how many phases are used for the integration, voltage bias levels and using
+different readout speeds. This work is being done with flight representative readout electronics,
+which have already been developed in the Phase A, since the Gaia experience has shown how
+critical it is to treat the detector and electronics as a coupled system. The CCD204 includes a
+charge injection structure, which can be used to fill traps on the CCD readout, but with the
+drawback that additional lines of charge are imprinted on the image. The usefulness of such a9.4. CTI mitigation by software post-processing 92
+structure in the Euclid case is still under investigation: in ESA’s Gaia mission charge injection is
+baselined, but there the operation is in drift scan (TDI) mode.
+In addition to these initiatives, there is an ESA-funded test programme on CCD204s irradiated
+with the worst case Euclid mission dose under way at SSTL-Astrium and institutes in the UK
+(Open University, MSSL-UCL) and France (CEA Saclay). The test plan (0128465 SSTL) has
+been constructed using the experience gained from the extensive Gaia campaign. Earlier tests
+have also been conducted on e2v-provided devices at Open University, and are the subject of
+several reports (Open-Euclid-TN-004-02; Open-Euclid-TN-003-01; Minutes of Meeting for Euclid
+CCDs 23 July 2009).
+Alternative p-Channel CCDs are being actively considered for the Euclid programme. These
+devices collect holes rather than electrons and are not affected by the dominant electron trapping
+caused by Phosphorus-vacancy displacement damage. They have been shown to have charge
+transfer efficiency up to an order of magnitude better than several models of n-channel CCDs
+designed for space applications ( Marshall et al. 2004;Lumb 2009;Gow et al. 2009). Such
+devices have been developed at LBNL in the USA (reported more fully in the next section)
+and by e2v, but in neither case is their TRL as high as for well-proven n-channel CCDs. A
+UK-funded test programme on irradiated e2v p-channel devices is under way at Open University
+(Open-P-Chan-TP-002.01; Minutes of Meeting for Euclid CCDs 23 July 2009), and ESA have
+funded a new fabrication run of e2v CCD204 p-channel CCDs, which should become available
+for investigation and characterisation in early 2010.
+Survey strategy can also be used to partially mitigate radiation damage. Since the effects of
+CTI depend on the position of a source on the CCD relative to the readout register, sequences
+of dithered images can be used in principle to decouple its intrinsic shape from spurious CTI
+degradation. Four-step dithering is now established in the Euclid mission baseline.
+9.4 CTI mitigation by software post-processing
+The spurious ellipticity induced in faint galaxies can be measured in large weak lensing surveys
+as a shear signal that depends on chip position, after many pointings are stacked, to average
+away any cosmological signal. Parametric schemes have been developed for several instruments
+to correct this statistically at the catalogue level ( Rhodes & et al. 2007). However, they fail
+to account for the “shielding” of one galaxy by another one nearby, which prefills charge traps
+during readout, and may not account for variation of the spurious ellipticity as a function of
+galaxy size, concentration and morphology.
+Fortunately, the same software algorithm that we developed to mimic CCD readout and add
+charge trailing can also be used to remove it, via an iterative procedure illustrated in table 9.1.
+This attempts to find to find an image that, when read out through software, results in the
+image obtained from the telescope. This is the unblemished image, corrected to appear as it
+would have without CTI. Since charge transfer is one of the last process to occur during data
+acquisition, correction for CTI should be one of the first operations during data analysis.
+To demonstrate the technique, we have removed charge transfer trailing from real imaging
+with the Hubble Space Telescope /Advanced Camera for Surveys . We used warm pixels in in-flight
+science exposures to measure the eight model parameters that encode the densities and release
+times of two charge trap species, and the rate at which potential wells in pixels are filled by
+electrons. Warm pixels also happen to be created by radiation damage. They are short circuits
+in the detector that create perfect delta-functions in images, and deviations from single-pixel
+spikes clearly show trails from charge traps with release times similar to the clock speed. The
+trap density was found to be about the same as that expected in a modern p-channel CCD after
+approximately 30 years at L2. The correction was then applied to the HST COSMOS survey.9.5. A feasible Euclid mission design unimpeded by CTI 93
+True image I Not available
+Downloaded from HST I+δ (A)
+After one extra readout I+ 2δ+δ2(B)
+(A)+(A) −(B) I−δ2(C)
+After another readout I+δ−δ2−δ3(D)
+(A)+(C) −(D) I+δ3(E)
+After another readout I+δ+δ3+δ4(F)
+(A)+(E) −(F) I−δ4etc.
+Table 9.1: Iterative method to remove CTI trailing, using only a forward algorithm that adds
+trailing. The true image Iis desired but not available. Only a version with (a small amount of)
+trailing,I+δ, can be obtained from the telescope. However, by running that image through a
+software version of the readout process, and subtracting the difference, deviations from the true
+image can be reduced to O(δ2). Successive iterations further reduce the trails until an image is
+produced that, when “read out”, reproduces data arbitrarily close to those obtained from the
+telescope. This is the corrected image.
+As shown in figure 9.5, the spurious shear in the faintest galaxies was reduced by an order of
+magnitude. The technique is currently limited by the accuracy of the charge integration and
+readout model. Even better model calibration (and hence CTI correction) should be possible
+with more data: especially if calibration diagnostics like pocket pumping can be enabled by the
+Euclid readout electronics, to bring tailored CTI measurements in flight.
+Other more dedicated software mitigation and calibration schemes are also being considered
+for the Euclid survey. These may use developments of the CDM02 model (GAIA-CH-TN-ESA-
+AS-015-1) baselined in the Gaia programme and modified for the standard exposure mode of
+operation. CDM02 is an analytical charge distortion model based on physical processes within
+the pixel, which predicts the effect on the detected charge distribution as the CCD is read out.
+Operating on an aggregated readout rather than a pixel-pixel transfer level, the model is fast
+enough to be applied to all Euclid visible imager data in real time. The parameters within the
+model are determined from calibrations (stellar sources) and the effect of prior sources read out
+ahead of the profile of interest is implicit. The investigation is ongoing.
+9.5 A feasible Euclid mission design unimpeded by CTI
+Our CCD readout software and end-to-end characterization methods are general; they can be
+applied to any CCDs, once the relevant density and characteristic release times of charge trap
+species are known. Our comprehensive software suite then allows us to couch our results in terms
+of the spurious galaxy ellipticities ∆ eadded to the faintest Euclid galaxies (which dominate the
+weak lensing signal), and hence in terms of the resultant errors on dark energy parameters. To
+benchmark one possible configuration for Euclid , we have modelled the properties of fully-depleted
+LBNL p-channel CCDs. Dawson & et al. (2008) irradiated several of these CCDs at the LBNL
+88-Inch Cyclotron with 12 .5 MeV protons to characterise their CTI properties. Although a
+variety of irradiation levels were tested, we have confirmed ( Rhodes & et al. 2009) that the
+spurious ellipticity induced in faint galaxies depends linearly upon the density of relevant trap
+species, so we only considered data with the maximum irradiation levels of 2 ×1010protons/cm2.
+For a baseline Euclid mission using p-channel CCDs, and averaging over the solar cycle, weBIBLIOGRAPHY 94
+Figure 9.5: Top: The Hubble Space Telescope image from figure 9.2, after CTI correction, in
+units of electrons. Bottom : Difference image. Reproduced from Massey & et al. (2009).
+calculate that bias in the measurement of galaxy shapes furthest from the readout registers
+will increase at a rate of d∆e/dt = (2.65±0.02)×10−4per year at L2. If uncorrected, this
+would consume the entire shape measurement error budget σ2
+sys∼10−7of a dark energy mission
+surveying the entire extra-galactic sky within about 4 years of accumulated radiation damage
+(Amara & R´ efr´ egier 2008). However, software mitigation techniques at an accuracy already
+demonstrated on Hubble Space Telescope data in §9.4can reduce this by a factor of ∼10, bringing
+the effect well below mission requirements.
+We have thus demonstrated that an implementation of the Euclid mission using existing
+technology and proven software post-processing techniques can achieve galaxy shape measurement
+accuracy that is unimpeded by CTI. This conclusion is currently valid only for the radiation-
+tolerant p-channel CCDs we have modelled; CCDs with worse CTI will fare worse and may
+not meet the mission requirements. The EIC is therefore continuing to develop mitigation
+strategies and extend them to detector technologies with higher TRL. All the tools are in place
+to quantitatively test alternative mission configurations during the Euclid development phase.
+Bibliography
+Amara, A. & R´ efr´ egier, A. (2008). Systematic bias in cosmic shear: extending the Fisher matrix.BIBLIOGRAPHY 95
+MNRAS ,391, 228–236.
+Barth, J., Isaacs, J., & Poivey, C. (2000). (the radiation environment for the next generation
+space telescope). NASA SST ISAL , See .
+Dawson, K. & et al. (2008). Radiation Tolerance of Fully-Depleted P-Channel CCDs Designed
+for the SNAP Satellite. IEEE Transactions on Nuclear Science ,55, 1725.
+Gow, J., Murray, N., Holland, A., Burt, D., & Pool, P. (2009). (a comparative study of proton
+radiation damage in p- and n-channel ccds). Proc. SPIE ,7435 , 74350F–1.
+Hall, R. (1952). Electron-hole recombination in germanium. Phys. Rev. ,87, 387.
+Janesick, J. (2001). Scientific Charge Coupled Devices. SPIE (ISBN 0-8194-3698-4) .
+Lumb, D. (2009). . SPIE ,7439A-03 .
+Marshall, C., Marshall, P., Wacynski, A., Polidan, E., Johnson, S., & Campbell, A. (2004).
+(comparison of the proton-induced dark current and charge transfer efficiency responses of n-
+and p-channel ccds). Proc. SPIE ,5499 , 542.
+Massey, R. & et al. (2007). COSMOS: 3D weak lensing and the growth of structure. ApJS ,172,
+239.
+Massey, R. & et al. (2009). The effects of Charge Transfer Inefficiency on galaxy shape measure-
+ments. MNRAS in press (arXiv:0909.0507) .
+Rhodes, J. . & et al. (2007). Modelling and Correcting the Time-Dependent ACS PSF. ApJS ,
+172, 203.
+Rhodes, J. & et al. (2009). Pixel-based correction for Charge Transfer Inefficiency in the Hubble
+Space TelescopeAdvanced Camera for Surveys. PASP submitted .
+Shockley, W. & Read, W. (1952). Statistics of the recombinations of holes and electrons. Phys.
+Rev.,87, 835.Chapter 10
+Intrinsic Alignments of Galaxies
+Authors: Michael Schneider (ICC Durham), Filipe Abdalla (UCL), Adam Amara (ETH Zurich),
+Sarah Bridle (UCL), Julien Carron (ETH Zurich), Benjamin Joachimi (U. Bonn / UCL), Tom
+Kitching (U. Edinburgh), Alexandre R´ efr´ egier (CEA Saclay), + EIC Weak Lensing Working
+Group.
+Abstract
+Intrinsic alignments (IAs) of galaxy shapes are likely to be a significant source of systematic
+error in cosmic shear measurements. EIC members have pursued two approaches to mitigate this
+systematic error: “nulling” methods that subtract certain combinations of tomographic redshift
+bins that are systematics dominated, and “modelling” methods that fit parameterised models for
+IAs that are treated as nuisance parameters when constraining cosmology. In this section we
+describe the relative merits and drawbacks of each method for removing IA systematics from
+cosmic shear measurements. Simulations done by EIC members of the intrinsic alignments of
+galaxy discs and dark matter halos with the large-scale matter distribution are described in the
+section Numerical Simulations.
+10.1 Introduction
+Based on recent low-redshift measurements Brown et al. (2002);Mandelbaum et al. (2006);
+Hirata et al. (2007), IAs are expected to be a non-negligible systematic error for cosmic shear
+(and potentially dominant for constraining the dark energy equation of state).
+Two types of intrinsic alignments can contaminate the shear two-point functions. So-called
+intrinsic-intrinsic (II) alignment occurs when two galaxies at the same redshift are aligned with
+each other because of correlations in the gravitational potential in which they formed and reside.
+On the other hand, shear-intrinsic (GI) alignments occur between galaxies at widely separated
+redshifts . This is caused when one galaxy resides in the lensing potential at intermediate redshift
+that is lensing a background galaxy at even larger redshift. This causes an anti-correlation of
+galaxy shapes because the galaxy in the lensing potential is preferentially aligned parallel to the
+gradient of the potential while the sheared galaxy is flattened tangentially to the gradient Hirata
+& Seljak (2004).
+Because the lensing amplitude increases with increasing distance between the lens and the
+source, II alignments dominate shear measurements at low redshift. Based on both theory Crit-
+tenden et al. (2002);Jing & Suto (2002) and observations Brown et al. (2002);Mandelbaum
+et al. (2006) II alignments are expected to contribute 10% of the observed shear correlations
+9610.2. Nulling 97
+for surveys with median redshift ∼1. The GI alignments, which include lensing effects, become
+more dominant at higher redshifts. Heymans et al. (2004) modelled the GI alignments for central
+galaxies in low-mass halos using N-body simulations and predicted the GI signal to also decrease
+the shear power spectrum amplitude by ∼10%. The analytical models of Hirata & Seljak (2004);
+Schneider & Bridle (2009) are consistent with this level of contamination.
+10.2 Nulling
+Both the II and GI alignments have particular geometric properties that can be exploited to
+remove the IA contamination from cosmic shear observations. The II alignments occur between
+galaxies that are located in the same gravitational potential, which means the galaxies must
+have the same redshift. By downweighting galaxy pairs that are close in redshift and in angular
+separation on the sky, the II contamination can be largely removed (depending on the accuracy
+of the galaxy redshifts) King & Schneider (2002);Heymans & Heavens (2003);King & Schneider
+(2003).
+Joachimi & Schneider (2008) introduced a nulling technique to remove the GI signal from
+the cosmic shear signal. This method is purely geometric in the sense that it only takes the
+characteristic and well-known dependence of the intrinsic alignment signal into account and does
+not rely at all on the existing models of the underlying correlations between matter and the
+intrinsic orientation of galaxies. In the limit of perfect knowledge of galaxy redshifts, nulling is
+capable of eliminating the GI effect completely. Joachimi & Schneider (2009) have demonstrated
+that this technique reduces the GI term to about 10% in presence of photometric redshifts whose
+quality corresponds to the requirements of Euclid. However, the price to pay for the complete
+model independence of this approach is a reduction in the dark energy Figure of Merit of roughly
+an order of magnitude. The degradation of the constraints on a 6-parameter cosmological model
+using this technique are shown in fig. 10.1. The 1-σerrors for the example data set used in
+fig.10.1 are increased by factors of 2-3 after nulling. The biases in the mean values of the
+parameters are simultaneously reduced by factors of 2-10 so that in all cases the biases are much
+less than the size of the 1- σerrors after nulling.
+10.3 Modelling
+A second approach for removing the IAs as a systematic error is to construct a parameterised
+model for the IA correlations and marginalise over the model parameters when constraining
+cosmology. An advantage of this approach is that, with an accurate model, all the information in
+the data is retained. However, without an accurate model for IAs two issues must be considered.
+An insufficiently flexible IA model will leave a systematic bias in the cosmic shear data while
+a model with too many parameters will degrade the cosmological parameter constraints to a
+sub-optimal level.
+Building on the model of Hirata & Seljak (2004),Bridle & King (2007) introduced a tunable
+set of parameters for describing the power spectrum of IAs. By marginalising over the amplitude
+of the IA power spectra binned in wavenumber and in redshift (with varying bin sizes) Bridle &
+King (2007) explored the degradation in cosmological parameter constraints from cosmic shear
+as the number of IA parameters is increased. Figure 10.2 shows the change in the dark energy
+Figure of Merit as a function of the number of wavenumber and redshift bins. For such flexible
+IA models, the Figure of Merit can be degraded by up to a factor of 4 compared to the prediction
+assuming no IA contamination. The Figure of Merit for the maximally flexible IA model is a
+factor of 3 times smaller than the minimally flexible model.10.3. Modelling 98
+Figure 10.1: Parameter constraints before and after nulling the GI contamination from cosmic
+shear tomographics power spectra. Shown are the two-dimensional marginalised 2 σ-errors for the
+original data set (with IA contamination) as solid curves and for the nulled data set as dotted
+curves. The fiducial parameter values are marked by the crosses. The survey has been divided
+into 10 photometric redshift bins. Photometric redshift errors are characterised by a dispersion
+inzof 0.05, catastrophic error fraction 0.05, and bias in the mean of the catastrophic error
+population of ∆ z=±1.0. The linear alignment model, downscaled by a factor of five, was used
+to model the IA contribution. (Taken from Joachimi & Schneider (2009).)10.3. Modelling 99
+Figure 10.2: Degradation in the dark energy Figure of Merit as the number of IA model parameters
+is increased. Top row: assuming perfect photometric redshifts. Bottom row: more realistic
+photometric redshifts (with dispersion in zof 0.05). Left column: including just the II terms.
+Middle column: including just the GI terms. Right column: including both II and GI terms.
+Solid lines: Figure of Merit as a function of number of wavenumber bins in the IA power spectra.
+Lines from top to bottom within one panel: the number of bins in the redshift direction increases
+1, 2, 5. Dotted line: Figure of Merit for GG alone. Dashed line: using the linear alignment
+model with the linear theory matter power spectrum, instead of the nonlinear theory matter
+power spectrum. In every case a free amplitude parameter and unknown power law evolution was
+marginalised over (allowed to be different for each of GI and II). The default intrinsic alignment
+model is shown by a circle and the maximally flexible model is marked by a cross. (Taken from
+Bridle & King (2007).)
+In an effort to build a simultaneously more descriptive and minimally flexible IA model,
+Schneider & Bridle (2009) constructed a model for IAs on small scales by considering the
+alignments of galaxies within individual dark matter halos. This modelling showed that a range
+of assumptions about the small-scale alignments of galaxies have a limited range of effects on the
+IA power spectra. Within the framework considered in Schneider & Bridle (2009) the IA spectra
+can be described by just 2 effective amplitude parameters: one for the large-scale alignments and
+one for the small-scale alignments within halos. This result illustrates the potential for future
+physical modelling of IAs to yield constrained parameterisations of the systematic contamination
+to cosmic shear.
+Using alternate cosmic shear statistics Kitching et al. (2008) showed that IA, photometric
+redshift and shape measurement systematics models can be simultaneously constrained. They
+found that the combination of 3-D cosmic shear and the shear ratio test are very complimentary
+and that with only modest priors on the systematics parameters (10 to a few percent) the
+degradation of the dark energy Figure of Merit is at most a factor of 2.
+Further progress has been made by considering the additional information available in
+the Euclid imaging survey in the form of the galaxy angular positions. In addition to shear
+correlations, one can consider galaxy number density correlations and number density-shear cross
+correlations. The number density of galaxies is determined by the intrinsic galaxy clustering andBIBLIOGRAPHY 100
+the changes due to lensing magnification effects. However, uncertainties in the clustering bias of
+galaxies with respect to the dark matter add additional astrophysical systematics beyond the
+IAs in the cosmic shear signal.
+Joachimi & Bridle (in prep.) have undertaken a joint analysis of the complete set of galaxy
+shape and position correlations, choosing a general parameterisation for both intrinsic alignment
+and galaxy bias with very low model dependence. In their most flexible model over 200 nuisance
+parameters have been marginalised over accounting for uncertainties in galaxy bias, intrinsic
+alignments, galaxy luminosity function, and photometric redshift information. Making the further
+conservative assumption that even this large degree of freedom is not sufficient to describe galaxy
+bias on small non-linear scales, all clustering signals in the substantially non-linear regime were
+discarded.
+With this setup Joachimi & Bridle showed that the total information content (expressed in
+terms of the determinant of the Fisher matrix) of the pure lensing signal and the combined set
+of shape and position signals after marginalisation is the same to good accuracy. Dark energy
+parameters are slightly more affected by the marginalisation process, so that the Figure of Merit
+is reduced by about 40% with respect to the pure lensing signal. However, to obtain these results,
+all priors on the nuisance parameters were increased to non-informative values. Any substantial
+prior information, in particular on the galaxy bias and intrinsic alignments, readily brings the
+Figure of Merit back to pure lensing values and possibly beyond.
+Bibliography
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+of intrinsic alignments on photometric redshift requirements. New J. Phys. ,9(12), 444–444.
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+alignments in galaxy ellipticities. MNRAS ,333, 501–509.
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+& Wake, D. (2007). Intrinsic galaxy alignments from the 2slaq and sdss surveys: luminosity
+and redshift scalings and implications for weak lensing surveys. Monthly Notices RAS ,381,
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+Joachimi, B. & Schneider, P. (2009). The removal of shear-ellipticity correlations from the cosmic
+shear signal: Influence of photometric redshift errors on the nulling technique. ArXiv e-prints .
+King, L. & Schneider, P. (2002). Suppressing the contribution of intrinsic galaxy alignments to
+the shear two-point correlation function. A&A ,396, 411–418.
+King, L. J. & Schneider, P. (2003). Separating cosmic shear from intrinsic galaxy alignments:
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+Kitching, T. D., Taylor, A. N., & Heavens, A. F. (2008). Systematic effects on dark energy from
+3D weak shear. MNRAS ,389, 173–190.
+Mandelbaum, R., Hirata, C. M., Ishak, M., Seljak, U., & Brinkmann, J. (2006). Detection
+of large-scale intrinsic ellipticity-density correlation from the sloan digital sky survey and
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+Schneider, M. D. & Bridle, S. (2009). A halo model for intrinsic alignments of galaxy ellipticities.
+ArXiv e-prints .Part IV
+ADDITIONAL COSMOLOGICAL
+PROBES & LEGACY SCIENCE
+102Chapter 11
+Cosmology with Galaxy Clusters Counts
+Authors: Jochen Weller (Ludwig-Maximilians-Universit¨ at M¨ unchen and Excellence Cluster
+‘Origin and Structure of the Universe’), Filipe Abdalla (University College London), Joel Berg´ e
+(Jet Propulsion Laboratory, California Institute of Technology), Lauro Moscardini (Bologna
+University), Alexandre R´ efr´ egier (CEA Saclay), Stella Seitz (Ludwig-Maximilians-Universit¨ at
+M¨ unchen), Adam Amara (ETH Z¨ urich), Nabila Aghanim (IAS, Orsay), Thomas Kitching (ROE,
+Edinburgh), Marian Doupsis (IAS, Orsay).
+Abstract
+We discuss how the imaging survey of Euclid will be able to identify galaxy clusters and how
+they can be exploited to constrain cosmological parameters. We show the extreme sensitivity
+of galaxy cluster counts on the growth of structures and hence their ability to constrain dark
+energy scenarios and modified gravity models. We discuss how uncertainties in the selection
+function degrade this ability. These uncertainties come in two ways: First the scaling of the
+mass-observable relation and second the scatter in this relation. We show how a self-calibration
+approach can address the first problem and discuss how much prior information is required on
+the scatter in order to obtain competitive constraints from galaxy cluster counts.
+11.1 Introduction
+Clusters of galaxies are the largest over-dense objects in the Universe and bear imprints of
+the history of their formation because they trace: a) the spectrum of initial fluctuations; b)
+the growth of these structures over time and c) the dynamics of the collapse of halos. This
+threefold dependency makes clusters an excellent probe of the growth of structure in the Universe.
+Clusters of galaxies have long been suggested as cosmological probes. They are sensitive to the
+cosmological parameters in three ways: Firstly their distribution in redshift is sensitive to the
+above mentioned processes and hence cosmology, secondly their spatial distribution is a biased
+tracer of the underlying dark matter power spectrum (see Evrard 1989;Bahcall & Bode 2003)
+and thirdly individual clusters can be viewed, under certain conditions, as fair samples of the
+matter content in the Universe as pointed out by White et al. (1993). We are not pursuing
+the last approach, since this requires a resolved observation of a cluster in order to only select
+relaxed galaxy clusters. However, the distribution of galaxy clusters in redshift and space will be
+accessible to Euclid. Currently Vikhlinin et al. (2009) have performed the most detailed study
+of discovered galaxy clusters with x-ray observations. These will be complemented in the near
+future by observations of galaxy clusters via the Sunyaev-Zel’dovich decrement in the cosmic
+10311.2. Cosmology with the Redshift Distribution of Galaxy Clusters 104
+microwave background, as first shown by Sunyaev & Zeldovich (1972), for example with the
+South-pole Telescope (see Ruhl et al. 2004) and the Planck satellite (see Planck Collaboration
+2006). However, there have been already promising results for optically selected clusters as well.
+For example, the Sloan Digital Sky Survey (SDSS) cluster sample of about 130,000 objects has
+been exploited (see Bahcall & Bode 2003;Koester et al. 2007;Rozo et al. 2007b ;Rozo et al.
+2009a ) and provided interesting constraints. Further the Red-Sequence Cluster Survey (RCS)
+has also provided cosmological information, mainly on the amplitude of density fluctuations σ8
+as presented by Gladders et al. (2007). While these constraints are mainly exploiting a relation
+between the mass of a galaxy cluster and the number of member galaxies, a good way to calibrate
+the mass-observable relation is by facilitating the weak lensing properties of clusters as shown in
+a recent series of papers by Sheldon et al. (2009b ,a).Dahle (2006) has shown the first promising
+indications that the distribution of clusters as measured from weak lensing alone yield meaningful
+cosmological constraints.
+The rest of the chapter is organised as follows: We first discuss how the redshift distribution of
+galaxy clusters is sensitive to cosmological models, then we discuss the different selection methods,
+which are relevant for the Euclid imaging survey and self- and cross-calibration techniques. We
+then discuss the prospects for Euclid imaging to constrain cosmological parameters with galaxy
+clusters.
+11.2 Cosmology with the Redshift Distribution of Galaxy Clus-
+ters
+Recent years have seen a plethora of papers forecasting, mainly x-ray and Sunyaev-Zel’dovich,
+galaxy cluster counts and their ability to constrain cosmology (see Holder et al. 2001;Haiman
+et al. 2001;Majumdar & Mohr 2003;Battye & Weller 2003;Hu2003;Majumdar & Mohr 2004;
+Lima & Hu 2004,2005). In order to forecast the number of galaxy clusters in a particular redshift
+bin we have to calculate
+∆N(z) =z+∆z/2/integraldisplay
+z−∆z/2∆Ωd2V
+dzdΩ/integraldisplay∞
+Mlim(z)dn
+dMdM , (11.1)
+with ∆zthe width of the redshift bin, ∆Ω the sky coverage, d2V/dz/d Ω the volume element
+anddn/dM the mass function of dark matter halos. Mlim(z) encodes the selection function of
+the particular survey. Analytically derived mass functions, like Press & Schechter (1974) or its
+extension by Sheth & Tormen (2002), have the advantage that they capture the entire cosmology
+dependence of the mass function. However, fits to numerical simulations, are generally more
+accurate. Currently the most widely used mass function is by Jenkins et al. (2001), although
+this has been surpassed by more precise versions by Warren et al. (2006) and most recently by
+Tinker et al. (2008). In the analysis presented here we use the fit by Jenkins et al. (2001), which
+is given by
+dn
+dM(M,z) =−0.316ρm,0
+MdσM
+dM1
+σMexp/braceleftBig
+− |0.67−log [D(z)σM]|3.82/bracerightBig
+, (11.2)
+whereσMis the variance of a fluctuation of mass M,ρm,0is the density in mass today and D(z)
+is the growth of linear matter fluctuations with redshift. From equations ( 11.1) and ( 11.2) we see
+that the strongest cosmology dependence of the number counts actually comes from the volume
+element and the growth of structures. There is an almost exponential dependence on the growth
+of structures and this becomes an important feature to investigate modified gravity scenarios11.3. Selection of Galaxy Clusters with the Euclid Imaging Survey 105
+Figure 11.1: Left: Mass limits expected from Euclid optical cluster selection (dashed) and weak
+lensing selection thresholds for a 3- σdetection (solid), 5- σ(dash-dotted) and 7- σ(dash-triple
+dotted). Right: Distribution of galaxy clusters in redshift bins of width ∆ z= 0.1 for different
+cosmologies observed on 20,000 deg2. All clusters above a mass limit of 5 ×1013h−1M⊙are
+selected. Lines show a ΛCDM model (solid), a w=−0.9 model (dotted) and a modified gravity
+model (γ= 0.68; dashed). The dot-dashed line is for a ΛCDM model with the mass limit of the
+weak lensing 3- σdetection limit. Note that the Poisson errors ∼√
+Nare of the order of a few
+hundred in most bins for Euclid and hence negligible on this plot.
+as described in Chapter 6. In order to get an understanding of the cosmology dependence of
+the number counts we show in Fig. 11.1the redshift distribution of clusters for three different
+cosmologies. The base line is a concordance ΛCDM model (solid line). The dotted line is for a
+model with an equation of state of w=−0.9 compared to the concordance model with w=−1.
+From the plot we see a different behaviour at low redshift ( z≤1) compared to high redshift. This
+is because the difference from a ΛCDM model is driven at low redshifts by the different volume
+factor, while at high redshift from the difference in the growth of structures. Although, the
+difference between the two models seems miniscule, one has to keep in mind that the statistical
+error for number counts is driven by the Poisson noise. The overall number of clusters for this
+setup is over half a million, so the Poisson noise is tiny, and of the order of a few hundred per bin,
+compared to the overall number of over 10,000 per bin. Hence in order to study the statistical
+significance of the difference we need to understand the systematics, and we will come to this
+later. The dashed line corresponds to a modified gravity model, which we parameterised with
+γ= 0.68 as described in Chapter 6. This is a huge difference, which demonstrates the power of
+galaxy cluster counts to constrain the growth of structures.
+11.3 Selection of Galaxy Clusters with the Euclid Imaging Sur-
+vey
+The Euclid imaging survey can target clusters with three methods. The first is to count all the
+member galaxies of the over-dense structures, the second is to look at the weak lensing signal
+imposed by the massive galaxy clusters on the background galaxies, and the third is by the
+strong lensing signal. The last is discussed in detail in Chapter 15. Here we concentrate on the
+first two possibilities. In recent years the maxBCG method by Koester et al. (2007) has been put
+forward. maxBCG assumes that the Brightest Cluster Galaxy (BCG) sits in the centre of every11.3. Selection of Galaxy Clusters with the Euclid Imaging Survey 106
+cluster. The tight colour-magnitude relation of these objects is used to select candidate BCGs.
+In addition to identifying the BCG, so called ridge-line galaxies are identified with a model,
+by finding them with their radial and color distribution. The two models are maximised as a
+function of redshift and hence provide an estimate of the photometric redshift of the cluster. In
+an iterative scheme the most likely clusters and their satellites are removed from the sample. A
+chain of probabilities, which have been calibrated with mock catalogues, is then applied to infer
+the mass of the cluster. The free parameters in the models, besides the cosmological parameters,
+are the halo model parameters. In terms of systematics or noise the biggest problems are the
+completeness and purity of the sample. For example, projection effects along the line of sight can
+lead to a misestimate of cluster members for galaxies along this line. Euclid’s good photometry
+can in part avoid this problem. For more details see the paper by Rozo et al. (2007a ). This
+method has been successfully applied to the SDSS cluster sample by Rozo et al. (2007b );Rozo
+et al. (2009b ), where they analysed the likelihoods of over 10,000 galaxy clusters and could place
+tight constraints on σ8and Ω m.
+The second possibility to select clusters with the Euclid imaging survey is weak lensing. For
+example, Dahle (2006) describes how the mass function can be estimated entirely from lensing
+masses, albeit for an x-ray selected sample. A generalisation of this for a blind survey was
+presented by Berg´ e et al. (2008) for the XMM and CFHTLS survey in combination. Also the
+cross-correlation of the SDSS photometric cluster sample with weak lensing, for 130,000 objects,
+allowed a calibration of the signal to the ∼10% level (see Sheldon et al. 2009b ;Johnston et al.
+2007;Sheldon et al. 2009a ).
+A large problem for estimating cosmological parameters with galaxy cluster counts is not just
+the statistical uncertainty in the mean mass-observable relation, but even more so the scatter in
+this relation. This in general leads to a shift to larger cluster numbers in the sample, due to
+the shape of the mass-function. In order to analyse galaxy cluster counts in a general set-up we
+adopt the approach proposed by Hu(2003) and Lima & Hu (2004,2005). They suggest to study
+the galaxy cluster counts in cells of redshift andobservable bins, i.e.:
+ni=θobs
+i+1/integraldisplay
+θobs
+idθobs
+θobs/integraldisplaydM
+Mdn
+dlnMp(θobs|M), (11.3)
+where the index ’i’ refers to bins in redshift and the observable θobsshould be viewed as a generic
+observable. This could be x-ray flux, SZ flux, number of member galaxies (richness), etc. The
+selection function is encoded in the probability of a cluster with true mass Mhaving the observed
+“mass”θobsand we choose following Lima & Hu (2005):
+p(θobs|M) =1/radicalBig
+2πσ2
+lnMexp/bracketleftBig
+−x2(θobs)/bracketrightBig
+, (11.4)
+where
+x(θobs)≡lnθobs−lnM−lnMbias
+/radicalBig
+2σ2
+lnM. (11.5)
+Note thatMbiasencodes the mean scaling relation between the observable and the true underlying
+mass. We assume here Gaussian scatter with a variance of σ2
+lnM. Note that this can be easily
+extended to include non-Gaussian likelihoods as discussed by Shaw et al. (2009). In this approach
+byLima & Hu (2005), the scatter and bias can be allowed to take any form, even changing
+completely freely from bin to bin.11.4. Forecasting Cosmological Parameters with Clusters 107
+Rather than knowing the exact scaling relation and scatter a priori, the survey can self-calibrate
+for the unknowns in the mass-observable relation (see Majumdar & Mohr 2003;Lima & Hu 2005).
+However if there is too much freedom there are of course no meaningful cosmological constraints.
+For the analysis presented here we parameterise the bias with lnMbias=Ab+nb(1 +zi) and the
+scatter quadratic in redshift, meaning an additional three parameters. All these parameters are
+fitted for by the survey, in addition to the cosmological parameters. In order to judge if this
+is a meaningful parameterisation, we would require mock catalogs and analyse them with an
+analysis pipeline both adapted and tailored for Euclid’s case. However a quadratic evolution in
+redshift of the scatter is already quite general. As Lima & Hu (2005) have shown it is actually
+not the scatter itself, it is the prior uncertainty on this quantity, which drives cosmological
+uncertainties. This uncertainty can be calibrated by observing clusters with different methods.
+For example Rozo et al. (2009) have constrained the scatter in the mass-richness relation of
+the SDSS maxBCG cluster with complementary weak lensing and x-ray observations and find
+σlnM|N200≈0.45±0.20. It is important to note that the power spectrum of clusters in addition
+to the number counts is also sensitive to these uncertainties and plays an important role for
+the self-calibration of the galaxy cluster observation. We have not yet implemented this in our
+forecast pipeline, which would further tighten our error forecasts. So from this point of view our
+forecasts should be considered conservative.
+11.4 Forecasting Cosmological Parameters with Clusters
+In this section we concentrate on cosmological constraints from Euclid imaging survey clusters
+alone. We want to emphasise that, in particular the weak lensing information on clusters from
+Euclid imaging can play a pivotal role in calibrating scaling relations of galaxy clusters in surveys
+such as Planck or eRosita. This potential has been already indicated by Dahle (2006) and Berg´ e
+et al. (2008). We also want to stress that Euclid is ideally complementary for the aforementioned
+surveys in a sense that it provides a full sky coverage like these surveys. We have performed
+a preliminary forecasting analysis for calibrating SZ selected clusters of the Planck survey, by
+stacking these clusters and constraining the mass - SZ flux relation with Euclid weak lensing.
+This improves the uncertainty in the scaling relation typically by 50% and in turn improves the
+cosmological constraints, compared to a Planck alone analysis of SZ selected clusters (see also
+Cunha 2009). A particular problem, which arises for this type of calibration, is that one has
+to be careful how the different probes are correlated in a joint analysis as shown by Takada &
+Bridle (2007).
+Here we concentrate on a maxBCG like analysis. However we want to stress that a range of
+cluster finding algorithms could be used in the Euclid data besides the more popular maxBCG
+method used extensively in the SDSS data. These would include Voronoi tesselation, matched
+filter techniques, hybrid filter techniques, counts in cells, percolation algorithms such as friends
+of friends in photo-z space, smoothing kernels, adaptive kernels, surface brightness enhancements,
+cut and enhance techniques, just to name a few. Each technique is sensitive to different types
+of structures and there is no consensus within the community which is the best technique for
+dark energy studies or for general cluster studies. Some techniques rely of the red properties of
+galaxy clusters where others could find bluer concentrations of objects or clusters which are not
+necessarily spherically symmetric.
+In the SDSS photometric survey it has been shown that clusters with masses above 1013.5h−1M⊙
+are detectable with high purity and completeness, with more than 10 bright red galaxies out to
+z∼0.3 (see Koester et al. 2007;Johnston et al. 2007). Note that we assume that this still holds
+out to a maximum redshift of zmax= 2. There have been indications that clusters have a robust
+red sequence out to redshift z= 1, but this will eventually break down and detailed observations11.4. Forecasting Cosmological Parameters with Clusters 108
+Figure 11.2: On the left the 1 −σjoint likelihood contour in the w0−waplane after marginalisation
+over all the cosmological and nuisance parameters with the priors mentioned in the text. On
+the right we show the joint likelihoods for constraining modified gravity vs. the ‘dark energy’
+contribution.
+and simulations still have to show this beyond z= 1. However we should note that given the
+filter set and depth chosen for Euclid it will be possible to perform high redshift cluster selection
+with Euclid alone. Given the lack of optical filters the low redshift will not be optimally sampled
+by Euclid, but given the depth of the Y, J and H filters, Euclid will be able to probe the tail of
+the cluster redshift distribution accurately since it will be deep enough to find ridgeline galaxy
+photo-z at high redshift 1 .3<z< 2.0 (cf. Chapter 20) and will also cover the entire sky, hence
+have as little cosmic variance as possible.
+As mentioned above we assume two parameters on the scaling relation we fit for and 3 for
+the scatter. In addition we vary the matter contents with Ω m, the dark energy contents Ω de, the
+two dark energy parameters w0andwa, the tilt of the primordial power spectrum nand the
+amplitude of the fluctuations σ8. Where indicated we also vary the modified gravity parameter γ.
+Note that in the way we set up the survey there is nodependence on the Hubble constant hand
+only very weak dependence on the baryon contents Ω b. This is because the selection function is
+fixed and the cluster mass function depends only on the dark matter power spectrum. For the
+fiducial model we assume Ω m= 0.25, Ω de= 0.75,w0=−0.95,wa= 0,n= 1.0,σ8= 0.8. The
+scaling bias is the mass-observable relation is assumed to be 10% constant, the intrinsic scatter
+in the logarithm of the mass 25%, we use four equally spaced bins in logarithmic mass for the
+self calibration between 1013.5h−1M⊙and 2 ×1015h−1M⊙, and redshift bins of width ∆ z= 0.1
+betweenz= 0−2. For the sky coverage we assume half the sky with ∆Ω = 20 ,000deg2. Finally
+we assume a prior uncertainty on the scatter and bias parameters of 25%, in line with observations
+and simulations. For the presentation of our results we add a prior on the cosmological parameters
+as expected from the Planck mission and described in Rassat et al. (2008).
+In Fig. 11.2 we show the forecasted constraints for Euclid imaging survey clusters, on the left
+for a standard dark energy scenario and on the right for a modified gravity parameterisation.
+Note that for modified gravity we still use a Planck prior without the inclusion of the γparameter.
+A proper treatment is likely to tighten the errors even further. In addition the size of the
+errors is entirely driven by the large uncertainties in the scatter and bias in the mass-observable
+scaling relation. This relation is likely to be much tighter than assumed here, in particular if we
+calibrate with Euclid weak lensing clusters or x-ray observations from surveys such as eRosita.
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+Euclid with eRosita and vice versa will benefit both. In addition the spectroscopic part of Euclid
+bears potential to calibrate some of the clusters masses via velocity dispersion measurements.
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+The Integrated Sachs Wolfe Effect
+Authors: Nabila Aghanim (IAS, Orsay, France), Jochen Weller (Ludwig-Maximilians-Universit¨ at
+M¨ unchen and Excellence Cluster ‘Origin and Structure of the Universe), Marian Douspis (IAS,
+Orsay, France), Ana¨ ıs Rassat (CEA, Saclay, France), Alexandre R´ efr´ egier (CEA, Saclay, France).
+Abstract
+The Integrated Sachs-Wolfe (ISW) effect is an independent probe of dark energy, curvature
+and deviations from General Relativity on large scales. Though difficult to detect directly in
+the cosmic microwave background (CMB), it can be measured using correlations of the CMB
+with tracers of large scale structure (LSS) and has recently been detected using several LSS
+surveys. The Euclid wide survey provides a galaxy survey which is naturally optimised to lead
+to a maximum detection of the ISW effect when correlated with future Planck data. As Euclid
+covers a large redshift range, it will also be possible to perform a tomographic analysis, which
+will tighten constraints on cosmological parameters. As it probes both geometry and growth of
+structure, it is complementary to the primary probes of Euclid. We find that in a flat Universe,
+we can expect to constrain the cosmological constant w0at the 8% level with the ISW and Planck,
+and that with strong priors on other cosmological parameters, we can expect to discriminate
+modified gravity theories by constraining the growth parameter γ. We summarise in this Chapter
+the work done by the Euclid Imaging Consortium ISW working group.
+12.1 Introduction
+To better constrain and understand the present acceleration of the expansion there is a crucial
+need for multiple and complementary observational probes. The Integrated Sachs-Wolfe (ISW)
+effect ( Sachs & Wolfe 1967) imprinted in the CMB and its correlation with the distribution of
+matter at lower redshifts (through the galaxy surveys) is one of them. The ISW effect arises from
+the time-variation of the scalar metric perturbations and offers a promising new way of inferring
+cosmological constraints (for e.g. Corasaniti et al. 2005;Pogosian 2006). There exists both an
+early- and a late-time ISW effect. The early effect is only important around recombination when
+anisotropies can start growing and the radiation energy density is still dynamically important.
+The late ISW effect, on the other hand, originates much later after the onset of matter domination
+from the time derivative of the gravitational potential ( Kofman & Starobinskii 1985;Mukhanov
+et al. 1992;Kamionkowski & Spergel 1994). It is to this latter effect that we refer to as being
+the ISW effect. The late-time ISW effect can be due to dark energy domination at low redshift,
+curvature, or modifications to the growth of structure on large scales. In flat universe with no
+modifications to gravity, detection of the ISW is a direct signature of the presence of dark energy.
+11212.2. Formalism 113
+The ISW effect dominates the largest scales of the CMB temperature power spectrum ( l<30).
+Its importance comes from the fact that it is sensitive to the amount, equation of state and
+clustering properties of the dark energy. Detection of the weak signal in the CMB power spectrum
+is highly limited by cosmic variance. However, since the time evolution of the potential giving
+rise to the ISW effect is probed by observations of large scale structure (LSS), it is also possible
+to detect the ISW effect through the cross-correlation of the CMB with tracers of the LSS
+distribution.
+This idea, first proposed by Crittenden & Turok (1996), has been widely discussed in the
+literature. The recent WMAP data provide high enough quality measurements at large scales
+to be used in combination with LSS tracers to assess the ISW detection (for e.g. Giannantonio
+et al. 2006;Rassat et al. 2007, and references therein). The ISW effect has been detected with
+2-3.9σsignificance which increases to 4.5 σwhen all current surveys are used in combination
+(Giannantonio et al. 2008).
+12.2 Formalism
+The ISW effect is a contribution to the CMB anisotropies that arises in the direction ˆ ndue to
+variations of the gravitational potential, Φ, along the path of CMB photons from last scattering
+until now. It adds power to the CMB temperature anisotropies as:
+∆TISW
+T(ˆ n) =−2/integraldisplayr0
+rLSdr˙Φ(r,ˆ nr) (12.1)
+where ˙Φ≡∂Φ/∂rcan be related to the matter density field δthrough the Poisson equation. The
+variableris the conformal distance, defined today as r0and at the surface of last scattering as
+rLS. In a flat universe, within the linear regime, the gravitational potential does not change with
+time if the expansion of the universe is dominated by matter. Therefore, for most of the time
+since last scattering, matter domination ensured a vanishing ISW contribution. Conversely, a
+detection of an ISW effect would indicate that the effective potential of the universe has changed.
+In fact, detection of an ISW effect can only be attributed to three causes: the presence of dark
+energy at late times, spatial curvature or a modification to General Relativity at large scales. The
+Euclid mission will probe LSS at redshifts z= 0−2, i.e. it probes the dark energy dominated
+era and is therefore ideal to detect changes in the potential due to dark energy. Euclid will also
+be able to detect signatures of spatial curvature and modifications of gravity on large scales (see
+Figure 12.1) by constraining the parameter γ(defined in Chapter 3).
+The 2-point angular cross-correlation of the ISW temperature anisotropies with the galaxy
+distribution field is defined as
+CISW−G
+l=2
+π/integraldisplay
+dkk2Pδδ(k)IISW
+l(k)IG
+l(k), (12.2)
+whereIG
+l(k) =/integraltextr0
+0drWG(k,r)jl(kr) , and the galaxy window function is given by WG(k,r) =
+bG(k,r)nG(r)G(r) . The power spectrum of the over-density fluctuations is described by Pδδ(k).
+The spherical Bessel functions are denoted by jl;bG(k,r) is the bias, and nG(z) =nG(r)/H(z)
+is the normalised redshift distribution of the galaxies, for which we use the following parameteri-
+sation:
+nG(z) =n0
+G/parenleftbiggz
+zmed/parenrightbiggα
+exp/bracketleftBigg
+−/parenleftbiggz
+zmed/parenrightbiggβ/bracketrightBigg
+. (12.3)
+The variables αandβprovide a description of the galaxy distribution at low and at high redshifts
+respectively and zmedis the median redshift. The variable n0
+Gis a normalising constant.12.2. Formalism 114
+Figure 12.1: Left Panel : Prediction of the ISW cross-correlation signal for different values of
+the dark energy density (Ω DE= 0.10, green line; Ω DE= 0.20, red line; Ω DE= 0.30, blue line)
+for universes with flat geometry (solid lines) and universes with open geometry and no dark
+energy. The ISW signal for universes with the same matter density is larger in open universes
+than in flat universes. The signal is calculated for a Euclid-like photometric survey. Right panel :
+The ISW cross-correlation signal for different values of the growth parameter ( γ= 0.44, green,
+dash-dotted line; γ= 0.55, blue dashed line; γ= 0.68, e.g. a DGP model, red short dashed).
+Both figures are taken from Rassat (2007).
+Figure 12.2: Contours for wand Ω DEfrom 4 redshift bins from the Euclid photometric survey
+with roughly equal number of galaxies per bin ( z= [0,0.6],[0.7,1.10],[1.1,1.4],[1.5,2.7]). The
+direction of the degeneracy in the w−ΩDEplane changes with the redshift of the galaxies
+considered. Figure taken from Rassat (2007).12.3. ISW requirements: Signal to Noise analysis 115
+As the Euclid survey will provide a galaxy survey over redshift [0 −2], it will be possible to
+measure the evolution of the ISW signal. This is of special importance to model the evolution of
+the equation of state w(z). Degeneracies between different parameters also evolve with redshift
+(see Figure 12.2), and so a full tomographic analysis is expected to provide tighter constraints on
+cosmological parameters (see Rassat (2007) and Giannantonio et al. (2008)).
+The uncertainty on the bias and most importantly on its redshift evolution might have a
+non negligible impact on the measure of the ISW. Consequently it will affect the cosmological
+parameter estimate, in particular the equation of state ( Schaefer et al. 2009). Assuming a linear
+bias evolution, we find that bias evolution shifts the best fit position by an amount comparable
+to the statistical accuracy in the case of σ8, nsandw. This comes from the fact that σ8affected
+due to the higher average bias at earlier times, n sdue to the change in the shape of the cross-
+correlation spectrum and wsince it is not as well constrained from primary CMB anisotropies.
+Ignoring bias evolution yields less negative values for wintroducing a bias towards dark energy
+models with respect to ΛCDM.
+The galaxy distribution nG(z) will also be subject to redshift distortions ( Kaiser 1987),
+which will affect both the projected galaxy field ( Fisher et al. 1994;Padmanabhan et al. 2007)
+and the galaxy-temperature cross-correlation will be subject to redshift distortions ( Rassat
+2009). This will modify the ISW cross-correlation equation 12.2. To include redshift space
+distortions, the IG
+/lscript(k) term in equation 12.2 should be replace by the following term: IG+z
+/lscript=
+IG
+/lscript(1 +βA/lscript)−βB/lscriptIG
+/lscript−2−βD/lscriptIG
+/lscript−2, whereβ=dlnD
+dlnais the growth factor and A/lscript,B/lscriptandC/lscriptall
+depend on the multipole /lscript. The inclusion of redshift distortions (which will naturally be present
+in any redshift selected galaxy survey) adds a new dependence on the growth, which is useful for
+constraining both dark energy, curvature and modified gravity.
+12.3 ISW requirements: Signal to Noise analysis
+We explored what an ideal survey, optimised for detecting the ISW effect, would resemble, by
+considering the detection level of ISW measured through its signal-to-noise (SN) as in Douspis
+et al. (2008):
+/parenleftbiggS
+N/parenrightbigg2
+=fc
+skylmax/summationdisplay
+l=lmin(2l+ 1)×/bracketleftBig
+CISW−G
+l/bracketrightBig2
+/bracketleftBig
+CISW−G
+l/bracketrightBig2
++/parenleftbig
+CISW
+l+NISW
+l/parenrightbig/parenleftbig
+CG
+l+NG
+l/parenrightbig, (12.4)
+wherefc
+skyis the fraction of sky common to the CMB and the Euclid galaxy survey maps, and the
+cumulative SN is summed over multipoles between lmin= 2 andlmax= 60, which are the scales
+which contribute most to the ISW effect. The spectra CISW−G
+l,CISW
+landCG
+lare the cross- and
+auto- correlation spectra of temperature (ISW) and the galaxy (G) fields. The noise contributions
+in the temperature signal and galaxy surveys are NISW
+landNG
+lwithNISW
+l=CCMB
+l+NCMBexp
+l
+(NCMBexp
+lis the CMB experimental noise for an experiment such as the Planck satellite). The
+galaxy survey noise is defined by the shot noise contribution: NG
+l=1
+NwhereNis the surface
+density of sources per steradian used for the correlation with CMB. The noise part of the SN
+depends then, at first order, on the common sky fraction, on the surface density of sources, and
+on their median redshift through the amplitude of the ISW power spectrum CISW
+land of the
+galaxy auto-correlation signal CG
+l.
+We find that the SN plateaus once Nreaches typically about 10 sources per arcmin2. Above
+this limit for N, the SN ratio is very sensitive to fc
+skyandzmed(see Figure 12.3). For a given
+zmed, the larger fc
+skythe higher the detection level. Conversely, at a given fc
+sky, increasing zmed
+significantly improves the ISW detection only up to zmed∼1. An optimal survey (with a12.4. Fisher Matrix analysis 116
+Figure 12.3: Total signal-to-noise for an ISW detection in the constant equation of state model
+withw=−0.9 as function of the galaxy survey parameters fc
+skyandzmed.
+detection level above 4 σ) should thus be designed so that it has a minimum number density of
+sources of around 10 galaxy per arcmin−2, covers a minimum sky fraction of the order of 0.5 and
+is reasonably deep, with a minimum median redshift of about 0.8. Such a survey, optimised for
+the weak lensing science case, is being planned for the future and satisfies the ISW requirements.
+It is the wide survey of Euclid mission proposed to the ESA’s Cosmic Vision.
+12.4 Fisher Matrix analysis
+We calculate forecasts for the Euclid imaging survey, considering the same fiducial cosmology and
+survey details as in Chapter 13, and consider a tomographic calculation, which includes the effect12.4. Fisher Matrix analysis 117
+Figure 12.4: Expected constraints (marginalised over the other parameters shown in Table 12.1)
+from the ISW probe using the cross-correlation of the Euclid photometric galaxy survey with
+Planck data. Constraints include priors from Planck and allow for curvature.
+of redshift distortions. The Fisher matrix can then be calculated using (see Pogosian 2006):
+FG−ISW
+α,β=fc
+sky/summationdisplay
+/lscript/summationdisplay
+i,j∂CGi−ISW
+/lscript
+∂pαCOV−1/parenleftBig
+CGi−ISW
+/lscriptCGj−ISW
+/lscript/parenrightBig∂CGj−ISW
+/lscript
+∂pβ, (12.5)
+whereGiandGjcorrespond to two separate galaxy redshift bins and αandβto different
+cosmological parameters.
+For the Fisher forecasts we consider two parameter sets, one which allows for curvature and
+a varying equation of state w0=w0(z):
+Θ2= (H0,Ωb,Ωm,σ8,ns,w0,wa,ΩDE). (12.6)
+and another which considers only a flat geometry and a constant equation of state w0:
+Θ1= (H0,Ωb,Ωm,σ8,ns,w0), (12.7)12.5. Conclusions 118
+The cosmological constraints we obtain for the ISW signal are shown in Table 12.1 and the
+marginalised constraints in the w0−waplane are shown in Figure 12.4.
+Table 12.1: Cosmological constraints for Euclid’s Imaging survey (i.e. photometric redshifts) using
+the Integrated Sachs Wolfe effect. The constraints are presented with Planck priors calculated as
+inRassat et al. (2008). The marginalised constraints in the w0−waplane are shown in Figure
+12.4.
+Method w0waΩDE Ωm Ωbnsh σ 8
+Full model 0.59 1.60 0.067 0.065 0.010 0.0038 0.08 0.068
+Flat Universe & constant w0.081 - - 0.0072 0.00091 0.0038 0.0073 0.048
+Despite the constraints from the cross-correlation itself are weak, they play a non negligible
+role in the combination with CMB. This is mainly due to the 6–dimensional shape of the
+likelihood and its degeneracies. The ISW effect does help (as compared to CMB alone) in
+breaking degeneracies among the parameters describing the dark energy model and the other
+cosmological parameters, primarily σ8. The cross-correlation between Planck and Euclid allows
+to place a constraint of the order of 8% on wfor a model with a constant equation of state and
+reduces the errors on the estimation of the parameter wain a linear model. However, it does not
+allow to distinguish among a constant and a dynamical equation of state for the dark energy.
+12.5 Conclusions
+The ISW effect offers a promising new way of inferring cosmological constraints (for e.g. Corasaniti
+et al. 2005;Pogosian 2006) and though it is weaker than other probes, it provide a complementary
+way to probe our cosmological model. In particular it can be used to probe the energy density,
+inhomogeneities and redshift evolution of dark energy, as well as curvature and modifications to
+the large scale growth of structure. By construction, the Euclid photometric survey is ideally
+optimised for an ISW survey, which requires maximum sky coverage, at least 10 galaxies per
+square arcmin, and a redshift range covering that of dark energy domination. The relative depth
+of the Euclid photometric survey will also permit a tomographic analysis, which will improve
+cosmological constraints. We find that with a Planck prior and the assumption of a flat Universe,
+an ISW survey from Euclid can expect to constraint the cosmological constant equation of state
+to 8%. In the case of dark energy and allowing for curvature, the ISW constraints from Euclid
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+cosmology and there exists various methods in the literature which use different features present
+in the galaxy power spectrum, including Baryon Acoustic Oscillations (BAOs). In this Chapter, we
+present the work of the Euclid Imaging Consortium’s (EIC) working group on galaxy correlations,
+which has studied a spherical harmonic approach to the studying the galaxy power spectrum,
+as well as a ‘wiggles only’ method in Fourier space. We find the Euclid Imaging (photometric)
+Survey has the power to constrain different sectors of the cosmological model, which though
+weaker than weak lensing constraints are useful for cross-checking results.
+13.1 Introduction
+Galaxy surveys are often used to trace a feature in the galaxy power spectrum named Baryon
+Acoustic Oscillations (BAOs) - or baryon wiggles, which are considered a powerful standard ruler
+with which to constrain cosmology. However, BAOs are only one feature in a rich statistic - the
+entirety of the statistical distribution of galaxies encodes information on various characteristic
+scales about the underlying dark matter distribution and the nature of dark energy (e.g., Peebles
+1980). As it also probes both the growth of structure as well as the expansion history of the
+Universe, the full statistical distribution of galaxies is a very powerful tool with which to probe
+the composition of the Universe, the nature of dark energy and the behaviour of gravity on large
+scales.
+Large scale galaxy surveys have been used extensively to constrain our cosmological model
+both in the optical (for e.g., Geller et al. 1987;Tegmark et al. 2004;Peacock et al. 2001), as well
+as in the near-infrared (for e.g., Fisher et al. 1994;Heavens & Taylor 1995;Erdo˘ gdu (a) et al.
+2006), and provided early evidence for a cosmological constant ( Efstathiou et al. 1991).
+12013.2. Building Blocks of the Galaxy Power Spectrum 121
+There exists several methods to measure the statistical distribution of galaxies. The first
+variable is the statistics used in the analysis: the measurement can be done in real space
+(Eisenstein et al. 2005,ξ(r)), configuration space ( Loverde et al. 2008,w(θ)), Fourier space
+(Seo & Eisenstein 2003;Amendola et al. 2005, P(k)) or in Spherical Harmonic space ( Dolney
+et al. 2006,C(/lscript)). For a given statistic, it is also possible to use different methods which use
+information from specific scales. In Fourier space, some measure the full power spectrum ( Seo &
+Eisenstein 2003, i.e. including information from the BAOs), whereas others subtract the smooth
+part of the spectrum and focus only on the oscillations ( Blake et al. 2007;Seo & Eisenstein 2007)
+- the latter uses less information but may be more robust with respect to systematic uncertainties.
+Knowledge of the full three-dimensional galaxy distribution can yield very competitive con-
+straints, even if only considering the baryon wiggles feature. However, partial radial information
+- e.g., with photometric redshifts - can still be used to compute a statistical measure of projected
+features in spherical harmonic space ( Dolney et al. 2006;Rassat et al. 2008), and the radial
+degradation can also be included in Fourier space measurements ( Seo & Eisenstein 2007). Fur-
+thermore, radial features such as redshift distortions will also have secondary effects that can
+still be measured in the projected distribution ( Padmanabhan et al. 2006;Rassat et al. 2008).
+In this Chapter we will present the different building blocks which carry information in the
+galaxy distribution and present different measurement methods in both Fourier and projected
+spherical harmonic space. We explore the cosmological constraints we can expect to achieve
+for the Euclid Imaging Survey (i.e., with photometric redshifts) for the three sectors of the
+cosmological model: dark matter, dark energy and initial conditions.
+13.2 Building Blocks of the Galaxy Power Spectrum
+The three-dimensional galaxy power spectrum is a rich statistic, where several features on different
+scales contain cosmological information specific to different sectors of the cosmological model
+(dark energy, dark matter and initial conditions). On linear scales we identify the following three
+main features in the galaxy power spectrum, which are:
+•The broad-band power: Information is contained in the shape, normalisation and time
+evolution of the power spectrum.
+•The Baryon Acoustic Oscillations (BAOs): Information is contained in the tangential and
+radial wavelengths, as well as the wiggles amplitude.
+•The linear redshift space distortions: Even when considering the projected power spectrum,
+this radial information is partially present.
+The Fourier space matter power spectrum P(k) describes the fluctuations of the matter distribu-
+tion and is defined by /angbracketleftδ(k)δ∗(k/prime)/angbracketright= (2π)3δ3
+D(k−k/prime)P(k), whereδ(k) represents the Fourier
+transform of the matter overdensities δ(r)=ρ(r)−(ρ)
+ρ, and the mean density of the Universe is ρ.
+The termδ3
+D(k−k/prime) represents the Dirac delta function.
+Thegalaxy power spectrum is related to the matter power spectrum by:
+Pg(k)∝(1 +βµ2)2b2(k,z)P(k), (13.1)
+where the first term describes the effect of redshift space distortions, modulated by the distortion
+parameter β=1
+b(z)dlnD(a)
+dlnaandµ(the cosine of the angle between the tangential and radial
+modes) and on linear scales a scale independent galaxy bias can be assumed, i.e. b(k,z) =b(z).
+The three building blocks of the observed galaxy power spectrum are plotted separately in
+Figure 13.1 (panels B, C and D), while the observed galaxy power spectrum is plotted in panel13.2. Building Blocks of the Galaxy Power Spectrum 122
+Figure 13.1: Illustration of the main building blocks of the linear Fourier space galaxy power
+spectrum. Panel A : The observed linear galaxy power spectrum. This includes the three
+building blocks of the matter power spectrum: the smooth power spectrum (broad band power),
+baryonic wiggles, linear redshift distortions as well as the linear galaxy bias b(z). In linear theory
+the redshift evolution of the galaxy power spectrum depends solely on the linear growth factor ;
+we illustrate this by comparing the linear galaxy power spectrum at redshift z= 0 (solid blue
+line), and redshift z= 0.5 (dot-dashed red line). Panel B, C and D illustrate the different
+building blocks of the galaxy power spectrum. Panel B: The smooth part of the galaxy power
+spectrum contains information on the shape and normalisation of the power spectrum. Panel C:
+The ratio (blue solid line) of the full power spectrum and the smooth part of the power spectrum
+reveals the residual baryonic wiggles. The dashed line corresponds to no baryonic wiggles. Panel
+D:The ratio (blue solid line) of the radial galaxy power spectrum to the tangential spectrum
+illustrates the scale-independent effect of linear redshift distortions.13.3. Galaxy correlations with photometric survey 123
+A. Each feature depends on different cosmological parameters in complementary ways, and a
+detailed description of the cosmological dependence of each building block on the different sectors
+of the cosmological model is given in Rassat et al. (2008).
+13.3 Galaxy correlations with photometric survey
+It is possible to study the statistical distribution of galaxies using various basis sets. In this
+chapter we consider the projected spherical harmonic decomposition ( C(/lscript)) and a wiggles only
+method (BAO) in Fourier space.
+13.3.1 Projected Spherical Harmonic Method [ C(/lscript)]
+There are several advantages to using the projected spherical harmonic method. For data on a
+sphere, spherical harmonics provide a natural basis set in which transverse and radial modes
+are independent. In this basis, there exists an exact prescription for redshift distortions ( Fisher
+et al. 1994;Heavens & Taylor 1995). This contrasts with the far field approximation necessary
+to describe redshift distortions in Fourier space ( Kaiser 1987).
+The bare observables provided by a galaxy survey ( θ,φ,z ) are also naturally translated into
+a single measurement C(/lscript), independently of cosmology, whereas to measure the Fourier space
+power spectrum, it is first necessary to relate these observables to the wavenumber kwhich
+requires an assumption of the cosmological model.
+Finally, in the case of a photometric survey like the Euclid Imaging Survey, there is not as much
+information lost when using the projected galaxy information (as opposed to a three-dimensional
+decomposition), since the radial information is already weakened by redshift uncertainty. Further-
+more, some radial information from linear redshift distortions can be measured in the projected
+spectrum.
+In this section, we describe the tomographic method we use for the projected C(/lscript) method.
+This method draws on that described in Dolney et al. (2006), but includes the effect of redshift
+distortions as described in Padmanabhan et al. (2006) and Rassat et al. (2008). For full details
+of the tomographic method, the reader is referred to Rassat et al. (2008).
+For a redshift selected galaxy survey, the spherical harmonic power spectrum can only be
+measured in redshift space; this can be theoretically modelled by adding a redshift distortion
+correction term to the real space window function:
+C(/lscript,zi) =/angbracketleftbig
+|a/lscriptm|2/angbracketrightbig
+= 4πb2(zi)/integraldisplay
+dk∆2(k)
+k|Wr(k) +βWz(k)|2
+= 4πb2(zi)/integraldisplay
+dk∆2(k)
+k|Wr+z
+/lscript(k)|2, (13.2)
+wherea/lscriptmrepresent the projected spherical harmonic coefficients and the galaxy bias is taken to
+be constant across the depth of each bin centred at redshift zi. The power spectrum is expressed
+as: ∆2(k) =4π
+(2π)3k3P(k) and is calculated using the publicly available code CAMB ( Lewis et al.
+2000).
+Padmanabhan et al. (2006) showed that the final term in equation 13.2 could be re-written:
+Wr+z
+/lscript(k) =Wr(k) +βWz(k),
+=Wr
+/lscript(k) +β/bracketleftbig
+A/lscriptWr
+/lscript(k) +B/lscriptWr
+/lscript−2(k) +D/lscriptWr
+/lscript+2(k)/bracketrightbig
+, (13.3)13.3. Galaxy correlations with photometric survey 124
+where the terms A/lscript,B/lscriptandD/lscriptdepend only on /lscript. The real space window function Wr
+/lscript(k) is
+given byWr
+/lscript(k) =/integraltext
+drΘ(r)j /lscript(kr)D(r).The normalised galaxy distribution is denoted by Θ( r),
+the termj/lscript(r) refers to the spherical Bessel function of order /lscriptandD(r) is the growth function.
+For a galaxy survey with photometric redshifts, the different galaxy bins will necessarily
+overlap. The correlation functions between two bins i,jcan be taken into consideration for the
+Fisher matrix calculation by:
+Fij
+α,β=/summationdisplay
+/lscript/summationdisplay
+i,j∂Cij
+/lscript
+∂pαCOV−1/bracketleftBig
+Cij
+/lscript/bracketrightBig∂Cij
+/lscript
+∂pβ, (13.4)
+whereCij
+/lscriptis the correlation between two redshift bins iandjand the covariance matrix is given
+by:COV/bracketleftbig
+Cij(/lscript)/bracketrightbig
+=/radicalBig
+2
+(2/lscript+1)fsky/bracketleftbig
+Cij(/lscript) +Ngδij/bracketrightbig
+, so that the shot noise Ngonly intervenes in
+the diagonal elements of the covariance matrix.
+13.3.2 Wiggles only BAO method
+We also consider a ‘wiggles only’ method which uses a Fourier space decomposition. The main
+advantage of using Fourier space is that the measured power spectrum in is physical - not
+projected - coordinates. This means the scale kof the wiggles is constant for every redshift
+bin considered, whereas in spherical harmonic the scale /lscriptof the wiggles will shift as different
+redshift bins are considered. This method is also a three-dimensional method, where the radial
+degradation from photometric redshift is included.
+In the case of the ‘wiggles only’ method, only the size of the radial and tangential BAO scale
+is used - all other information from the power spectrum P(k) (see Figure 13.1) is discarded. This
+method omits all information from the shape and amplitude of the power spectrum, the redshift
+distortions and even the amplitude of the BAO wiggles.
+We follow the method of Seo & Eisenstein (2007), which we briefly summarise here. The
+starting point is the expression to compute the Fisher matrix from the full Fourier space power
+spectrumP(k). The non-linear power spectrum used is extracted from the linear power spectrum
+with a damping term representing the loss of information coming from the non-linear evolutionary
+behaviour. This damping term is approximated by an exponential computed using the Lagrangian
+displacement fields Eisenstein et al. (2006):
+Pnl(k,µ) =Plin(k,µ) exp( −k2Σ2
+nl) (13.5)
+This non-linear damping can be decomposed into the radial and perpendicular directions by:
+Pnl(k,µ) =Plin(k,µ) exp/parenleftBigg
+−k2
+⊥Σ2
+⊥
+2−k2
+/bardblΣ2
+/bardbl
+2/parenrightBigg
+, (13.6)
+where Σ2
+nlvaries with redshift, following Σ nl,⊥= Σ nlD(z) and Σ nl,/bardbl= Σ nlD(z)(1 +f(z)).
+A delta function baryonic peak in the correlation function at the sound horizon scale, s,
+translates into a ‘wiggles only’ power spectrum of the form
+Pb∝sin(ks)
+ks. (13.7)
+This functional form is also obtained if the power spectrum is divided by a smooth version of
+it or if we only use the baryonic part of the power spectrum transfer function (e.g., Eisenstein &13.3. Galaxy correlations with photometric survey 125
+Hu(1999)). However, the baryonic peak is widened due to Silk damping and non-linear effects.
+This Gaussian broadening translates into an exponential decaying factor in Fourier space:
+Pb∝sin(ks)
+ksexp/parenleftbigg−k2Σ2
+2/parenrightbigg
+, (13.8)
+=sin(ks)
+ksexp/bracketleftbig
+−(kΣSilk)1.4/bracketrightbig
+exp/parenleftbigg−k2Σ2
+nl
+2/parenrightbigg
+. (13.9)
+We can further refine the calculation to 2 dimensions, separating the radial and tangential
+location of the baryon acoustic peak in the correlation function. Using the same notation as Blake
+et al. (2006) and Parkinson et al. (2007), the observables are y(z) =r(z)/sandy/prime(z) =r/prime(z)/s
+respectively, where r(z) is the comoving distance to redshift z. Measuring the fractional errors
+on these two observables is equivalent to measuring the fractional errors on H·sandDA/s.
+We can now use the derivatives of the ‘wiggles only’ power spectrum by these quantities (see
+Seo & Eisenstein (2007) for all the details) to compute the Fisher matrix (equation 26 of Seo &
+Eisenstein (2007)). This equation includes the effect of redshift distortions on the baryon wiggles
+in the factor R(µ) = (1 +βµ2)2, which decreases the amount of shot noise.
+The effect of photometric redshift errors can be included in this formalism as an additional
+exponential term in the redshift distortion factor, R(µ) = (1 +βµ2)2exp(−k2µ2Σ2
+z), where Σ zis
+the uncertainty in the determination of photometric redshifts. The effect of photometric redshift
+errors will increase the shot noise, leading to larger errors. We evaluate these integrals to obtain
+the Fisher matrix for the angular diameter distance, DA/s, and the rate of expansion, H·s.
+To compute the errors on the parameters, we follow Parkinson et al. (2007), with some
+changes: Since we know the correlation between the errors on the vector yi={r(zi)/s,r/prime(zi)/s},
+we form a small 2x2 covariance matrix for each redshift bin, which we invert to obtain the
+corresponding Fisher matrix, F(i). While Parkinson et al. (2007) compute the transformation to
+the cosmological parameters analytically, we perform it numerically.
+The above is enough to provide constraints on cosmological parameters. We also use another
+equivalent implementation which adds another step. In this second implementation we form a
+Gaussian likelihood L∝exp(−χ2/2) with
+χ2=/summationdisplay
+i2/summationdisplay
+a,b=1∆aF(i)
+ab∆b (13.10)
+whereiruns over the redshift bins and a,boverr/sandr/prime/s, and ∆ =Xdata(zi)−Xtheory (zi)
+is the difference between the data and the theory. We then compute numerically the matrix of
+second derivatives of χ2at the peak of the likelihood, via finite differentiating in all parameters
+(where it is necessary to divide by 2 when using χ2rather than −lnL).
+The accuracy to which one can constrain cosmological parameters from galaxy clustering
+depends on the accuracy one can measure the galaxy power spectrum. The error on the
+determination of the power spectrum comes from two factors: sampling variance and shot noise.
+The relative error can be written as (Feldman, Kaiser & Peacock 1984):
+∆P
+P∝1√
+V(1 +1
+nP) (13.11)
+where the first term represents the sampling variance term and depends on the volume sampled.
+The second term is the shot noise term and depends on the effective number of galaxies used
+and the value of the power spectrum.13.4. Constraints from the Euclid Imaging Survey 126
+Figure 13.2: Left Panel: Constraints on w0andwafor the projected C(/lscript) and the ‘wiggles
+only’ BAO method using Euclid’s Imaging (photometric) survey. Right Panel: Constraints on
+w0andwafor the projected C(/lscript) method using Euclid’s Imaging (photometric) survey with and
+without Planck priors.
+13.4 Constraints from the Euclid Imaging Survey
+13.4.1 Implementation
+In order to calculate forecasts for the Euclid Imaging Survey, we use a central fiducial model
+similar to WMAP5 results ( Dunkley et al. 2009), i.e. flat universe (though we allow for curvature
+in the Fisher matrix calculations), with no running spectral index and no massive neutrinos.
+The central values are shown in Table 13.1, where we use the parameterisation of Chevallier &
+Polarski (2001) to quantify the redshift evolution of the equation of state w(a):
+w(a) = (1 −a)wa=wp+ (ap−a)wa, (13.12)
+wherea= 1/(1 +z), and the subscript pdenotes the pivot point. We also consider the dark
+energy ‘Figure of Merit’, as defined by the DETF ( Albrecht et al. 2006, Dark Energy Task Force):
+FoM =1
+σ(wp)×σ(wa)(13.13)
+For the ‘wiggles only’ method, we estimate the number of galaxies observed as a function of
+redshift integrating an evolving luminosity function down to Euclid’s imaging survey magnitude
+limit. Together with the area, this also give us the sampled volume. We take the photo-z error13.4. Constraints from the Euclid Imaging Survey 127
+Table 13.1: A flat universe (though we allow for curvature in the Fisher matrix calculations, with
+no running spectral index nor massive neutrinos.
+Parameter Central fiducial Value
+Ωm 0.25
+ΩDE 0.75
+Ωb 0.045
+w0 -0.95
+wa 0
+ns 1
+h 0.7
+σ8 0.8
+from the simulations of Abdalla et al and Banerji et al. We take the bias as a combination of the
+VVDS determination and that obtained from numerical simulations.
+For the spherical harmonic method, we consider an imaging survey covering 20 ,000deg2, with
+median redshift z= 1, which we split into 10 tomographic bins with roughly the same number of
+galaxies in each bin (2 billion galaxies in total). We take the linear bias to be constant across
+each redshift bin and marginalise over these 10 extra parameters for the C(/lscript) method. As we
+assume linearity we do not consider kmodes above kmax= 0.25hMpc−1.
+13.4.2 Combination with Planck priors
+The constraints for both methods can be combined with constraints from other probes. Here we
+also consider priors we expect to achieve with the upcoming Planck experiment. We calculate
+Planck priors, using the Fisher matrix formalism and considering both the temperature and
+E-mode polarisation spectra (i.e., TT,TEandEEspectra; see Appendix B in Rassat et al. 2008,
+for details), and consider scales which are not correlated with the galaxy power spectrum (i.e.,
+we do not use information from the Integrated Sachs Wolfe effect on large scales (see Chapter
+12)). The Planck priors are included, using:
+FCombined =FPlanck +FX, (13.14)
+whereXis eitherC(/lscript) orBAO . Inclusion of Planck priors helps to break degeneracies between
+different cosmological parameters.
+13.4.3 Constraints from the Euclid Imaging Survey
+We consider cosmological forecasts for both the C(/lscript) and the wiggles only methods described
+earlier, with and without Planck priors. The constraints are summarised in Table 13.2.
+As expected, the wiggles only method has less constraining power than the C(/lscript) method for
+all cosmological sectors. It cannot alone constrain parameters related to the primordial power
+spectrum (ns). The parameters Ω bandhare also difficultly constrained though we find that
+using a different parameter set using Ω bh2allows this combination to be constrained - though
+weakly. The C(/lscript) method constrains dark energy parameters better than the wiggles only method,
+though still weaker than the weak lensing method for most parameters as expected (see Chapter
+4), except Ω bandhwhich show similar constraints.. This is mainly due to the fact that the C(/lscript)
+ignores radial information, which is not present in this photometric survey.13.5. Conclusion 128
+Table 13.2: Cosmological constraints for Euclid’s Imaging survey (i.e. photometric redshifts)
+using wiggles only and C(l) method. The constraints are presented with and without Planck
+priors.
+Method wpw0waΩDE Ωm Ωbnsh
+No priors
+BAO 0.48 1.43 7.7 0.40 0.32 - - -
+C(/lscript) 0.19 0.80 2.4 0.19 0.09 0.023 0.19 0.22
+with Planck priors
+BAO 0.063 0.55 1.65 0.047 0.046 0.0072 0.0038 0.055
+C(/lscript) 0.13 0.36 0.82 0.039 0.030 0.0049 0.0037 0.038
+By combining the results from the galaxy correlations with Planck priors, we can improve the
+cosmological constraints on all parameters. The right panel in Figure 13.2 shows the constraints
+onw0andwawith and without Planck priors, and we see the inclusion of priors greatly improves
+the constraints on both parameters. We also find that the difference between the wiggles only
+and the projected spherical harmonic methods is less important (see Table 13.4.2 ). A three-
+dimensional Fourier space analysis on spectroscopic data can provide competitive constraints,
+which will be possible using the Euclid Spectroscopic Survey (see the Euclid Yellow Book).
+13.5 Conclusion
+Over the past decades, galaxy surveys have proved to be powerful probes of our cosmological
+model. The Euclid Imaging survey naturally provides a photometric galaxy survey over the
+redshift ranges z= 0−2. Though lacking in radial precision, such as survey can be used to
+study the tangential modes of the statistical distribution of galaxy.
+In this chapter we overviewed the work done by the Euclid Imaging Consortium’s (EIC)
+galaxy correlations working group. We considered the different methods which exist in the
+literature regarding galaxy correlations, and decided to concentrate on a Fourier space wiggles
+only method ( Seo & Eisenstein 2007) as well as a projected spherical harmonic method [ C(/lscript)]
+which is ideal to study tangential modes on a sphere.
+We found constraints from the spherical harmonic method provided competitive constraints
+on all sectors of the cosmological method, though these constraints were weaker than those from
+weak lensing for this survey, as expected. Constraints from the wiggles only method - which
+only considers the modes of Baryon Acoustic Oscillations (BAOs), were much weaker. We also
+presented our constraints including Planck priors, for which many degeneracies are broken. The
+resulting constraints show that galaxy correlations are a useful and complementary probe for the
+Euclid Imaging Survey.
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+Rassat, A., Amara, A., Amendola, L., Castander, F. J., Kitching, T., Kunz, M., Refregier, A.,
+Wang, Y., & Weller, J. (2008). Deconstructing Baryon Acoustic Oscillations: A Comparison
+of Methods. ArXiv e-prints .
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+from Future Large Galaxy Redshift Surveys. APJ,598, 720–740.
+Seo, H.-J. & Eisenstein, D. J. (2007). Improved Forecasts for the Baryon Acoustic Oscillations
+and Cosmological Distance Scale. APJ,665, 14–24.
+Tegmark, M., Blanton, M. R., Strauss, M. A., Hoyle, F., Schlegel, D., Scoccimarro, R., Vogeley,
+M. S., Weinberg, D. H., Zehavi, I., Berlind, A., Budavari, T., Connolly, A., Eisenstein, D. J.,
+Finkbeiner, D., Frieman, J. A., Gunn, J. E., Hamilton, A. J. S., Hui, L., Jain, B., Johnston, D.,
+Kent, S., Lin, H., Nakajima, R., Nichol, R. C., Ostriker, J. P., Pope, A., Scranton, R., Seljak,
+U., Sheth, R. K., Stebbins, A., Szalay, A. S., Szapudi, I., Verde, L., Xu, Y., Annis, J., Bahcall,
+N. A., Brinkmann, J., Burles, S., Castander, F. J., Csabai, I., Loveday, J., Doi, M., Fukugita,
+M., Gott, J. R. I., Hennessy, G., Hogg, D. W., Ivezi´ c, ˇZ., Knapp, G. R., Lamb, D. Q., Lee,
+B. C., Lupton, R. H., McKay, T. A., Kunszt, P., Munn, J. A., O’Connell, L., Peoples, J., Pier,
+J. R., Richmond, M., Rockosi, C., Schneider, D. P., Stoughton, C., Tucker, D. L., Vanden
+Berk, D. E., Yanny, B., & York, D. G. (2004). The Three-Dimensional Power Spectrum of
+Galaxies from the Sloan Digital Sky Survey. apj,606, 702–740.Chapter 14
+Galaxy Evolution with Euclid’s Imaging
+Authors: C. Marcella Carollo (Physics Department, ETH Zurich), R. Bender (Max Planck
+Munich), C. Lacey (Durham University), S. Lilly (Physics Department, ETH Zurich), J. Pea-
+cock (ROE, Edinburgh), H.-W. Rix (Max Planck Heidelberg), R. Saglia (Max Planck Munich).
+Abstract
+Euclid’s Wide and Deep Imaging Surveys will achieve order of magnitude gains in the combined
+parameter space of area, depth, wavelength coverage and spatial resolution. The Wide Survey
+will span a transverse size of ∼1 comoving Gpc at z∼0.1, up to more than 20 comoving
+Gpc atz∼6, with a total volume in excess of ∼1200 Gpc3returning images for more than
+a billion galaxies. The Y, J and H wavelength coverage down to AB∼24 will probe the
+rest-frame optical light of high-z galaxies and enable photometric redshift measurements to an
+accuracy of ∆ z/(1 +z)∼0.05, and reliable conversion of the observed galaxy properties to
+key physical parameters like stellar mass and star formation rate - out to epochs well beyond
+the peak of the integrated star formation rate in the universe ( Lilly et al. 1995;Madau et al.
+1996). The exquisite <0.2 resolution of the AB∼24.5 optical images will provide detailed
+morphological and structural information e.g., bulges, disks, bars, spiral arms, lopsidedness out
+toz∼2, and fundamental structural information such as size, concentration, at any redshifts. At
+AB∼26.5-26, the Deep optical/NIR images will fill in the unexplored niche of area and depth
+between Euclid’s Wide Imaging Survey and the extraordinarily deep but narrow-beam surveys of
+the James Webb Space Telescope (JWST), and will unveil galaxies and quasars (QSOs) at high
+redshifts that will remain undetected in ground-based surveys. The Euclid imaging surveys will
+therefore be a unique resource to study the co-evolution of galaxies and black holes from the
+reionisation era down to the present time.
+14.1 Introduction
+A crucial advantage of Euclid’s Imaging Wide and Deep Surveys over similar past and planned
+surveys of galaxies will be the sheer number of galaxies ( >109) available with morphological
+information and accurate photometric redshifts. This massive database will be important for
+many areas of galaxy evolution studies, such as the discovery of:
+•The most luminous objects in the very early universe. The first generations of galaxies
+were assembled around redshifts z∼7−10, in the heart of the reionization of the universe.
+Despite great efforts with space- and 8m-mirror ground-based telescopes, not many galaxies (or
+galaxy candidates) have been reliably detected so far at z∼7. Samples are too small to draw
+13114.2. Understanding reionization, and galaxy formation in the reionization era 132
+conclusions on the physical properties of galaxies near the end of the reionization epoch. Euclid’s
+deep imaging survey will dramatically change this situation. Using the Lyman-dropout technique
+(Steidel 1990) in the near-IR, the survey will detect many thousands of the most luminous – and
+thus rarest but of key importance – objects in the early Universe at z>7.
+•The largest scale signposts of structure formation at early epochs. While at present only
+a few massive galaxy clusters at z >1 are known, Euclid’s near-infrared imaging will allow
+for red-sequence determinations, and thus the identification of hundreds of galaxy clusters at
+z>2. The latter are the likely environments in which the peak of QSO activity at z∼2 takes
+place, and hold the empirical key to understanding the heyday of such activity, as well as its
+relationship with the formation and evolution of the massive galaxies that host the QSOs.
+•How the bias varies with galaxy type and cosmic epoch . The bias function that links the
+galaxies to their host dark matter halos is a major unknown in galaxy evolution. Connecting
+Euclid’s weak lensing maps for the total mass density with the stellar mass and luminosity of
+more than a billion galaxies out to z∼3 will solve this mystery.
+•Which physics shapes galaxies – and on what timescales? Euclid’s images will open up the
+study of the z∼1-2 universe, at the peak of cosmic activity, at a similar level of detail that the
+SDSS has achieved on the z= 0 universe.
+14.2 Understanding reionization, and galaxy formation in the
+reionization era
+Understanding galaxy formation at early times is a major challenge of today’s astrophysical
+cosmology. The timescale for the formation of the first stars and first galaxies is unknown, with
+estimates ranging from z∼30 toz∼15 (∼100−300 Myr after recombination). Hints for the
+timescale for galaxy assembly are provided by e.g., WMAP5; these indicate that reionization
+extends from roughly z∼11 toz∼6 (∼400−900 Myr after recombination). This crucial
+period is virtually unexplored to date. The current state-of-the-art is set by the groundbreaking
+images delivered by the new Wide Field Camera-3, recently installed on the HST, which have
+provided of order 20 z∼7 galaxies, a handful of robust candidate z∼8 Ly-break galaxies, and
+possibly a few robust z∼10 candidates ( Bouwens et al. 2009;Oesch et al. 2009;McLure et al.
+2009;Bunker et al. 2009). Not much can be learned from such small numbers of galaxies at
+such fundamental epochs in our universe. Large samples are essential to understand the intrinsic
+physical properties of z>7 galaxies – and the physics of reionization.
+Studying early-epoch galaxies without star-formation, studying the highest redshifts ( z >7)
+and studying the faint and, at early epochs representative part of the luminosity distribution, is
+the realm of Euclid optical and NIR Deep Imaging Survey, whose four-colour imaging will provide
+galaxy populations statistics by itself. Using the Lyman-dropout technique in the near-IR, the
+survey will detect thousands of the most luminous objects in the early Universe at z >6: for
+star formation rates of order and above 30 M⊙/yr, about 104star-forming galaxies at z∼8 and
+up to 103atz∼10.
+High redshift quasars (QSOs) may also play a crucial role in reionising the universe at epochs
+beyondz∼7. To date, no quasars have been found above z= 6.4, and presently planned
+ground-based programmes will struggle to push this beyond z∼7 because of the need for deep,
+wide area NIR imaging and spectra (see e.g., Venemans et al. 2007). Euclid’s visible and NIR
+Wide Imaging Survey will identify likely QSOs by their colours out to redshifts z∼12 and above,
+if they exist. Many of these objects many be difficult to find by spectroscopic techniques, as
+their Lyαlines are often weak or even absent (e.g. Fanet al. 2006). The likely steep luminosity
+function of these objects implies that maximal imaging depth at ∼2πarea is paramount: even a14.2. Understanding reionization, and galaxy formation in the reionization era 133
+Figure 14.1: Figure reported from Bouwens et al. (2009), to show the state of the art knowledge
+on theUVLuminosity Function of galaxies at z∼8 from the present Y-dropout searches in the
+Hubble Ultra Deep Field WFC3/IR images (solid red circles and 1 σerror bars). Also included
+are: (i) theUVLF determination at z∼7 from az-dropout search over the same ultra-deep
+WFC3/IR field (magenta lines and points: Oesch et al. 2009); and (ii) the Bouwens et al. (2007)
+UVLF determinations at z∼4, 5 and 6 (shown in blue, green, and cyan lines and points). The
+dotted red line gives one possible Schechter-like LF that agrees with the observed data points at
+z∼8. Key is however to note that the z∼8 LF is totally unconstrained at the bright end. This
+will be determined with exquisite accuracy by Euclid’s Deep Imaging Surveys - together with
+the physical properties of the z∼8 Lyman-break galaxy population.14.3. Unveiling clusters and massive groups in the early universe 134
+factor of two in flux limit improvement will provide a ten-fold increase in the expected number
+of objects. Such searches, and their follow-ups, may well hold the key on the rate of black
+hole growth at z>8 and hence answer questions about the masses of likely seed-black holes, a
+fundamental open question in galaxy formation. Euclids deep imaging survey should furthermore
+add up to of order a thousand QSOs at z>7, depending on the (unknown) details of the LF at
+such early epochs.
+Detection of these high redshift sources – Lyman-break galaxies and quasars – in Euclid’s
+images will provide crucial targets for the European Extremely Large Telescope (E-ELT) and
+the JWST which will achieve greater depths but on much smaller areas, thus probing a fully
+complementary part of the luminosity functions.
+14.3 Unveiling clusters and massive groups in the early universe
+Euclid’s near-infrared imaging will allow the identification, through their red sequence, of hundreds
+of galaxy clusters at z∼2-3 withM > 1014M⊙i.e., comparable with the Virgo Cluster, and
+thousands of structures with M > 1.0−3.0×1013M⊙, the mass scale of the massive groups.
+Current models of galaxy formation predict that many of the processes that lead to galaxy
+transformations should occur in such massive-group environment. First, massive groups should
+have an enhanced efficiency of tidal forces and dynamical friction. The in-spiral timescale from
+dynamical friction varies as σ3
+halo/ρhalo(withσhaloandρhalorespectively the velocity dispersion
+and density of the dark matter halo), and it is thus minimised in relatively high densities but low
+velocity dispersion groups ( Barnes 1990). Furthermore, state-of-the-art cosmological simulations
+indicate that ram pressure, which creates an induced pressure force proportional to ∝ρσ2, is
+already active at such intermediate mass scales ( Rasmussen et al. 2008). Clear signatures of
+galaxy strangulation have also been observed at the intermediate mass scales of massive galaxy
+groups ( Kawata & Mulchaey 2008). Finally, cold streams of ‘unvirialised’ gas persist in such
+potentials down to intermediate redshifts – and thus sustain enhanced star formation rates over
+substantially longer timescales than major mergers ( Dekel et al. 2009).
+Euclid’s Wide Imaging Survey will thus dramatically advance our understanding of the
+evolution of galaxies in massive clusters and in the intermediate-mass potentials of massive
+groups. Euclid’s optical morphologies at sub-kpc spatial resolution, and deep NIR galaxy images
+for robust stellar mass estimates, will be a fundamental ingredient in mapping the growth of stellar
+mass in the member galaxies, as a function of galactic structural properties and cluster-centric
+distance, over a wide span of galaxy- and cluster/group mass scales. The >1 billion galaxies
+sample with morphological and spectrophotometric diagnostics, over the full range of galaxy
+properties, will allow to disentangle local effects of evolution from effects related to the global
+properties of the clusters, and to identify the physical processes responsible for differential galaxy
+evolution in high-, intermediate- and low-density environments.
+The next generation of wide-area Sunyaev-Zeldovich (e.g., SPT, CCAT) and X-ray (e.g.,
+eROSITA, WFXT) surveys will detect a huge number of distant clusters ( ∼5000 for eROSITA
+to>105for WFXT), but only above the ∼1014M⊙mass scale, and/or only up to z∼2.
+Capitalising on the large area covered down to AB∼24 in the NIR, the Euclid Wide Imaging
+Survey will be uniquely positioned to detect bound structures at higher redshifts, and down to
+one order of magnitude smaller scales.14.4. Linking matter and light 135
+14.4 Linking matter and light
+Euclid’s optical and NIR Imaging Surveys will provide rest-frame optical luminosities and stellar
+masses for the bulk of the galaxy population (to ∼3.5 magnitudes below M∗) out toz∼3, and
+the bulk of the unobscured star formation to z∼6 through rest-frame UV light. Comparison
+between the distribution of light and stellar mass – and weak lensing maps – will provide a
+direct mapping between total mass density and galaxies distribution and properties (stellar mass,
+luminosity, morphological type, etc), the bias, which is at the core of understanding galaxy
+formation. It will furthermore unveil or at least constrain the existence and abundance of ’naked’
+dark matter peaks.
+The Euclid’s images will also uncover a very large number of strong lensing systems: about
+105galaxy-galaxy lenses, 103galaxy-quasar lenses and several thousand strong lensing arcs in
+clusters. Several tens of galaxy-galaxy lenses will be double Einstein rings ( Gavazzi et al. 2008),
+which are exceptionally powerful probes of the cosmological model as they simultaneously probe
+several redshifts. Euclids large-area exploitation of Natures telescope, strong gravitational lensing,
+will assemble an unparalleled sample of intrinsically extremely faint, but highly magnified, objects
+at high redshifts. These are exciting targets for synergetic spectroscopic studies with the JWST
+of the seeds of the massive galaxies that are detected in the universe at later epochs.
+Furthermore, the shapes of galaxies measured on Euclid’s images, in addition to measuring the
+cosmic shear, will reveal possible alignments of galaxy ellipticities induced by tidal interactions.
+The presence of such intrinsic alignments is a source of contamination in Euclid weak lensing
+measurements, which must be carefully calibrated and subtracted from them (e.g., Hirata &
+Seljak 2004). Additional cross-correlation effects of the intrinsic ellipticities and the cosmic shear
+must also be calibrated and subtracted separately. Such alignments would however provide crucial
+information on the hierarchical assembly of structure in the universe. Recent measurements of
+intrinsic ellipticity correlations ( Okumura et al. 2009;Brainerd et al. 2009) show the presence of
+a strong alignment that varies among different galaxy types. On large scales, the mean galaxy
+ellipticity alignment should behave linearly with the tidal field, and observations of the intrinsic
+alignments will provide a basic test for the assumed relationship between the observed galaxy
+density and the tidal field.
+14.5 Identifying the physical drivers of galaxy evolution
+The relative importance of internal vs. external drivers of galaxy evolution, and the relative
+timescales of star formation and mass assembly in galaxies, are still unknown. Galaxy properties
+atz <1 are strongly dependent upon galaxy mass and current star formation rate. Massive
+galaxies (in dense environments) have old stars which passively evolve without further star
+formation. In contrast, less massive galaxies (in less dense environments) are still actively
+forming stars down to today. What physical mechanisms are responsible for quenching the
+star formation in the most massive galaxies (and thus driving to the Hubble sequence and the
+morphology-density relation)? Galaxy formation models can reproduce the ‘red&dead’ z= 0
+massive galaxies by assuming that their host massive dark matter halos produce shock-heated,
+quasi-hydrostatic hot gas halos (e.g., Hopkins et al. 2008). Cold gas accretion from dense
+filaments in the low-mass halos prevents such form of ‘heating’ – and thus ‘quenching’ of star
+formation ( Dekel & Birnboim 2006). Within this paradigm, the critical mass separating ‘hot
+accretion’ halos from ‘cold accretion’ halos should evolve with cosmic time (and was larger at
+earlier epochs). By providing detailed internal galactic structure (bars, spiral arms, lopsidedness,
+etc) for many millions of z <1 galaxies as a function of cluster/group properties, and global
+structural, photometric and physical properties of central (and satellite) galaxies at earlier times,14.6. Conclusions 136
+z∼2, when cosmic star formation rate and QSO activity peak their efficiencies, Euclid’s images
+will directly test key predictions of galaxy formation models.
+14.6 Conclusions
+The last decade has seen unprecedented advances in our understanding of galaxy formation.
+Fundamental questions remain open: how is star formation in galaxies regulated, triggered, and
+quenched? How does the energy released by accreting black holes affect their host galaxies
+and their surroundings? How does the large scale environment affect galaxies? What is the
+relationship between dark matter halo mass and galaxy mass and properties? The next decade
+will be a critical period where these main open questions in astrophysical cosmology will be
+resolved. Euclid’s Imaging Surveys will be instrumental in providing the answers by producing a
+massive legacy of deep images over at least half of the entire sky.
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+Strong lensing with Euclid
+Authors: Matthias Bartelmann (Zentrum f¨ ur Astronomie, Universit¨ at Heidelberg), Leonidas
+Moustakas (JPL, Pasadena), Massimo Meneghetti (Bologna Observatory), Fr´ ed´ eric Courbin (EPF
+Lausanne).
+Abstract
+Euclid will provide an excellent data set not only for weak, but also for strong lensing. The
+most important strongly lensing objects are individual galaxies, galaxies in groups, and galaxy
+clusters, whose strong lensing effects give rise to multiple or highly distorted images. Strong
+lensing sensitively probes the mass distribution in the innermost parts of these objects and thus
+allows to study mass profiles, concentrations, substructures, total masses and their relations
+to observable emission e.g. in the optical or X-ray regimes. Besides information on individual
+objects, strong lensing is cosmologically important because it constrains non-linear structure
+growth. Dedicated search algorithms exist which can scan huge amounts of data automatically
+and efficiently for the rare cases of strong lensing. Euclid is expected to find of order 105galaxies
+lensed by foreground galaxies, /lessorsimilar103multiply-imaged quasars, and ≈5000 clusters containing
+strongly distorted arcs.
+15.1 Introduction
+Strong and weak gravitational lensing are related but distinct phenomena. When the surface mass
+density in a “lens” is above a critical threshold, the appearance of an aligned background object
+can be both distorted as well multiply imaged into several images, arranged in a highly distinctive
+(and easy to identify) pattern. The specific appearance or arrangement of these multiple images is
+sensitive to the underlying cosmology through the relative distances, as well as to the distribution
+of the total lensing mass (meaning the combination of dark plus luminous matter). Thus, while
+weak lensing predominantly probes linear and weakly non-linear structures, strong lensing probes
+into the cores of non-linear structures and provides astrophysical information which cannot
+otherwise be obtained.
+Strong lensing is a phenomenon that can occur across many scales of lensing mass. Individual
+galaxies, groups of galaxies, and clusters of galaxies, can all lens more-distant objects. The
+objects lensed are generally either distant quasars, or extended and faint background galaxies.
+In galaxy clusters, the predominant features are strongly distorted, arc-like images formed from
+background galaxies and multiple galaxy images. Multiple imaging of quasars by clusters does
+not play an important role because it is a very rare phenomenon.
+13815.2. Objects of Interest 139
+Figure 15.1: Example of a quasar, RX J1131 −123, seen quadruply due to the strong lensing
+effect of a foreground galaxy. The host galaxy of the lensed quasar is seen as an almost complete
+Einstein ring that can be used to constrain the mass model of the lens, in the centre of the ring.
+The side length of this HST image from CASTLEs is about 10/prime/prime.
+Euclid will produce an ideal dataset for strong-lensing studies. Strong lensing is rare, thus
+surveying wide areas is essential, and many strongly lensed images are faint, thus depth is similarly
+crucial. Euclid combines both to the degree that will allow substantially increasing samples
+of strongly lensed systems. Equally importantly, these large samples will be homogeneously
+selected and thus statistically complete. The applications of these phenomena to astrophysics
+and fundamental physics are both diverse and powerful, and we explore some of the highlights
+here.
+15.2 Objects of Interest
+A broad variety of scientific topics can be addressed based on a large, homogeneously selected
+data base of strongly lensed objects. Searching for strong-lensing events in the Euclid imaging
+data targets three broad classes of lensing objects, in which strong lensing gives rise to a variety
+of different effects. We explore these in a general sense in this section, and concentrate on specific
+astrophysical applications in a subsequent section.
+15.2.1 Individual, mostly massive galaxies
+Individual lensing galaxies were the first in which gravitational lensing was detected, in the form
+of multiply imaged quasars ( Walsh et al. 1979;Mu˜ noz et al. 1998). Since the light-travel times
+between different images differ, time delays can be observed in systems whose source is variable.15.2. Objects of Interest 140
+For galactic lenses, the time delays are between a few and several hundred days. They have
+attracted considerable attention because they can be used to constrain the Hubble constant and
+thus the scale of the Universe without invoking a distance ladder ( Refsdal 1964). Time delays
+are difficult to measure as they need frequent and long-term observations with high angular
+resolution. Nonetheless, they start to be measured with a few percent accuracy (e.g., Vuissoz
+et al. 2008;Oscoz et al. 2001). In combination with high-resolution imaging, as is currently
+done with the HST (or in the future with the JWST), time delays lead to measurements of the
+Hubble parameter that are fully competitive with, and complementary to, other cosmological
+probes (e.g., Suyu et al. 2009). Euclid will provide hundreds of lenses suitable for time delay
+measurements.
+Background galaxies strongly lensed by foreground galaxies typically appear as highly curved
+arcs or partial rings ( Hewitt et al. 1988;Warren et al. 1996;Impey et al. 1998). They constrain
+galaxy mass models substantially better than individual, point-like images because of their much
+larger number of image points. Some systems contain both point sources and arcs. For these,
+time delays can be measured thanks to the variability of the point source, and the lens model can
+be well constrained based on the arc feature (Fig. 15.1). Strong lensing occurs for sources located
+next to so-called caustic curves. Mathematically, these are catastrophes of the lens mapping.
+Peculiar types of image configurations can arise at catastrophes, more precisely metamorphoses,
+of critical curves ( Blandford & Narayan 1986;Marshall et al. 2005). They are general in the
+sense that they do not depend on details of the lensing mass model and thus allow constraints
+independent of the precise mass distribution.
+15.2.2 Galaxies in groups, whose environment is crucial for their strong lens-
+ing
+These objects are interesting because their lensing effects are enhanced and can be largely
+modified by their environment. They can give rise to complicated configurations of multiple
+images ( Keeton & Zabludoff 2004) or to arcs or (partial) Einstein rings around galaxy groups
+(Cabanac et al. 2007). Both types of image configuration are most useful for studying the mass
+distribution surround galaxies. Studying the effect of groups on the potential well of individual
+galaxies is also relevant for cosmology, as they modify time delays and thus the value inferred for
+the Hubble parameter (e.g., Momcheva et al. 2006). On even larger scales, it has been shown
+that large-scale structures influence the lensing effects of individual galaxies or by groups ( Faure
+et al. 2009).
+15.2.3 Galaxy clusters
+Strong lensing by clusters typically causes spectacular effects. Most notable are the formation
+of large arc-like images ( Fort & Mellier 1994;Fort et al. 1992), so-called (giant) arcs, which
+form either very close to the cluster cores and are then radially oriented, or they are located
+farther away and oriented tangentially to the cluster centre. Near higher-order catastrophes
+or metamorphoses in caustic curves, peculiar types of images can appear, such as straight
+arcs ( Kassiola et al. 1992). Arc systems from sources at multiple redshifts ( Kneib et al. 1993;
+Broadhurst et al. 2005) are very helpful in analysing the core mass distribution of galaxy clusters.15.3. Detection Algorithms 141
+15.3 Detection Algorithms
+Successful searches in the Euclid imaging data require automatic discovery techniques for these
+types of object. Various algorithms have been suggested and demonstrated to be useful:
+15.3.1 Searching for clear identifying features in lenses
+Although strong lens searches have seen enormous success by mining the spectroscopic portion
+of the Sloan Digital Sky Survey (SDSS; e.g., the Sloan Lens ACS Survey, SLACS, Bolton et al.
+2006), searches with Euclid will be based on some combination of morphological and colour
+information from the comprehensive imaging it will acquire. There have been several pioneering
+explorations on the theory and application of designing such searches recently, in both ground-
+and space-based data, all of which will inform how the most effective search strategy should be
+built for Euclid. These include a) identifying multiple-image candidates based on large-separation
+objects in the SDSS ( Oguri et al. 2006;Inada et al. 2003); b) searches for candidates based on a
+combination of multiple or highly distorted objects combined with colour information, as applied
+by the very successful CFHTLS-SL2S survey ( Cabanac et al. 2007); and c) a careful exploration
+of a probabilistic and automated search for lenses in space-quality data, which includes both
+detailed modelling of the candidates, but also inclusion of available colour information ( Marshall
+et al. 2009).
+15.3.2 Specialised algorithms
+Such algorithms were recently proposed and tested to search for highly distorted images. The
+most relevant of these are the algorithm by Cabanac et al. and used within SL2S (Cabanac et al.
+2007) to detect arc-like images mainly around galaxies, the algorithm by Horesh et al. used
+for comparing real and simulated cluster lenses ( Horesh et al. 2005), and the fast algorithm by
+Seidel et al. specifically designed for scanning huge data sets and minimising spurious detections
+(Seidel & Bartelmann 2007). These algorithms have their respective merits. The procedure by
+Cabanac et al. includes colour information into the image selection. Horesh et al. use the widely
+used image-analysis software packages IRAF and SExtractor and parameterise them such as
+to optimise the detection of faint, elongated images, and Seidel et al. devised a novel, highly
+efficient algorithm that avoids convolutions and is sufficiently fast for wide-area surveys.
+15.4 Astrophysical Applications
+Numerous astrophysical applications can be based on sufficiently large and homogeneously
+selected samples of strong lenses. We describe here a representative set.
+15.4.1 Galaxy structure and its evolution over cosmic time
+Modelling the image configuration, in some cases also their fluxes and their morphology, density
+profiles of the lensing galaxies and the haloes they are embedded in can be measured, whether
+the lens is an early-type galaxy or a spiral galaxy. Since gravitational lensing is sensitive to all15.4. Astrophysical Applications 142
+forms of inhomogeneously distributed mass rather than the luminous matter only, strong lensing
+can thus constrain the balance between dark and luminous matter, when combined with galaxy
+scaling relations. Since lenses will be found across many redshifts, or epochs, this mapping will
+be done over some 10 billion years of cosmic time.
+The general mathematical properties of imaging near caustics, their cusps and their higher-
+order metamorphoses, demand certain imaging properties that are independent of the detailed
+mass model. Perhaps the most prominent of those is a mathematical statement on the magnifica-
+tion ratios within image triplets, which is violated in almost all suitable observed cases. These
+so-called anomalous magnification ratios can help quantifying the amount of substructure in
+galaxies. Higher-order catastrophes and metamorphoses, such as umbilics, swallowtails, butter-
+flies, in lensing caustics can be exploited in a similar manner, provided sufficiently many of such
+cases can be detected.
+15.4.2 Cluster structure
+Strong lensing by clusters, in particular the large arcs produced by it, allows inferences on the
+cluster density profiles, in particular by combining radial and tangential arcs with weak-lensing
+maps. Qualitatively, radial arcs constrain the local slope of the density profiles, while tangential
+arcs place limits on the mass enclosed by them. Density profiles of dark matter haloes are a firm
+prediction of the Cold Dark Matter paradigm, thus testing them observationally is cosmologically
+very important. While they are substantially modified on galactic scales by baryonic physics,
+they are expected to be conserved in clusters well into their cores. Core sizes, or equivalently
+concentration parameters, are determined in the CDM paradigm by the formation time of the most
+massive progenitor halo. They can effectively be constrained by strong lensing. Values derived
+so far indicate that strongly lensing clusters seem to be special because their concentrations are
+routinely measured to be much higher than expected theoretically. The morphology of observed
+large arcs helps quantifying the amount of cluster substructure. An obvious example is the
+“straight arc” in the cluster Abell 2390, which indicates the presence of a structure comparable in
+mass to the main cluster body, however much less luminous both in X-rays and in visible light.
+Automatic arc-detection algorithms are now efficient enough to be run over wide-area surveys to
+search for objects strongly lensed by underluminous or dark halos, allowing expectations on the
+mass-to-light ratio of galaxy clusters to be tested.
+15.4.3 Cosmology
+The possibility to test the CDM paradigm through substructure and halo density profiles was
+mentioned before but is of course cosmologically highly important. Similarly, reliable constraints
+on the total mass of dark-matter haloes cannot be obtained by other means than lensing without
+further simplifying and specialising assumptions. Halo masses are important to know on the
+galaxy scale in view of galaxy formation in a universe with low σ8, and on the cluster scale
+because scaling relations between cluster temperature, luminosity and mass need to be externally
+calibrated. Mass determinations by methods other than gravitational lensing suffer from the
+necessity to adopt hydrostatic or virial equilibrium, which can hardly be independently tested.
+Another important subject of cosmological relevance is the growth of non-linear structures15.5. Predictions 143
+Table 15.1: Expected numbers of various strong-lensing events in the Euclid survey area
+Galaxies lensed by galaxies ≈105
+Quasars lensed by galaxies /lessorsimilar103
+Clusters containing strongly lensed arcs
+(with a length-to-width ratio ≥10)≈5000
+throughout cosmic history. Since strong lensing relies on high surface mass densities, it constrains
+the growth of nonlinear structures through the occurrence of sufficiently compact and massive
+halos, which is particularly interesting at high redshift.
+The geometry of the Universe is constrained by strong lensing through time delays between
+several of the images in multiply-imaged quasars. Given a model for the lensing mass distribution,
+such time delays measure the overall scale of the Universe through the inverse of the Hubble
+constant, as well as the overall expansion history of the universe through the angular diameter
+distances that are involved (e.g., Coe & Moustakas 2009).
+Galaxy clusters containing multiple arc systems originating from sources at different redshifts
+also probe the cosmic geometry directly. Such sources experience lensing effects of different
+strength from the same mass distribution, where the difference depends solely on the cosmic
+geometry.
+Statistically, the abundance of strong-lensing events place cosmological constraints through
+the comparison of survey results, in particular distributions of image separation, multiplicity
+and morphology, with expectations from simulations. Moreover, samples of strongly lensing
+galaxy clusters, selected by their lensing effects, provide cosmological information through their
+comparison to samples defined by the occurrence of a red cluster sequence, by their X-ray emission
+or their thermal Sunyaev-Zel’dovich effect.
+15.5 Predictions
+Euclid is expected to find thousands of interesting examples of strong gravitational lensing. The
+estimates given below are based on the following assumptions; The telescope will have a diameter
+of 1.5 metres; The exposure time is of order 1500 s; The FWHM of the PSF is 0 .3/prime/prime; The pixel
+size is 0.1/prime/prime; and The approximate quantum efficiency Q≈0.1. This configuration is close to
+current Euclid baseline of 1.2m diameter, with an exposure time of 1800 seconds and a PSF
+FWHM of 0.18”. The assumptions imply a 10- σpoint-source detection limit of IAB≈25, and a
+3-σdetection limit for extended sources with ≈1/prime/primediameter of IAB≈25.
+Based on current optical-depth calculations for lensing of QSOs and galaxies and of galaxies
+by clusters, and on source counts from the HDFS, we estimate the numbers of lensing events on
+the expected 20,000 square degrees of the Euclid survey given in Tab. 15.1:15.6. Preparatory Work for Euclid 144
+Figure 15.2: Simulated multicolour observation of a cluster core. The simulation assumes here to
+being able to observe the cluster in three bands (B,V,I), although Euclid will work with only one
+optical filter (a broad band RIZ filter, covering the wavelength range 550...920 nm)
+15.6 Preparatory Work for Euclid
+The SL working group has already developed tools which are being tested and particularised
+for a future usage with Euclid data. In particular, several studies have be undertaken, which
+make use of numerical simulations. In synergy with the Image Simulation WG, we have used
+state-of-the-art codes for image simulations for producing mock observations of galaxy cluster
+fields. The simulations take into account the optical design of Euclid, as well as its throughput
+and quantum efficiency of the camera. All the relevant noises have been included to make these
+simulations fully realistic. The galaxy morphologies are modelled using shapelet decompositions
+of real galaxies from the Hubble-Ultra-Deep-Field. Last but not least, the image simulations
+include lensing effects from galaxy cluster mass distributions, which are obtained from numerical
+and hydrodynamical cosmological simulations. An example of such images is shown in Fig. 15.2.
+These images are currently being used for the following applications:
+•Train algorithms for arc detections. In particular, the code by Seidel & Bartelmann (2007)
+is being used and calibrated with such images. A comparison with other instruments is
+also done, highlighting the great capabilities of Euclid for SL studies (Seidel et al. in prep).
+•In a paper by Meneghetti et al. (2008), we show that arcs arising from galaxies down to an
+apparent magnitude of 27 will be detectable with Euclid. We study how the arc properties
+change as a function of the morphology of the sources and we define some empirical methodsBIBLIOGRAPHY 145
+Figure 15.3: Right panel: detection and fitting of a gravitational arc in a simulated Euclid
+observations. The procedure involves fitting an arc through three characteristic points, measure
+the length and the curvature radius of the arc, and finally performing an azimuthal scansion of
+the arc to measure the width profile. Such properties can be used for arc statistics applications.
+Left panel: the measurement of a cluster mass profile. This is done analysing the images with
+several methods used for analysing real data. Among them, our method which combines strong
+and weak lensing constraints (dotted line). The solid line shows the true profile of the simulated
+cluster. The bottom panel shows the ratios between estimated and true masses.
+to measure the length, the width and the curvature radius of each lensed image (see the
+left panel in Fig. 15.3).
+•We have developed and tested with simulations some tools for reconstructing the mass of
+gravitational lenses using both parametric and non-parametric methods ( Comerford et al.
+2006;Cacciato et al. 2006;Merten et al. 2008). Using parametric methods we will be able
+to constrain the projected masses within the Einstein radii with an accuracy of order /lessorsimilar5%.
+Our non-parametric code combines the strong lensing constraints in the cluster centres
+with the weak lensing signal in the external regions, and it allows to measure the mass
+profile from kpc to Mpc scales with an accuracy of order 10% (Meneghetti et al. in prep,
+see left panel of Fig. 15.3).
+•We have developed an automatic image-deconvolution pipeline to unveil multiple point
+sources with separations as small as half the FWHM of the PSF. These algorithms have
+been tested on ground-based data such as the PQUEST survey and are currently used on
+SDSS to find small-separation lenses.
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+Cosmology and Astrophysics with a Euclid
+Supernova Survey
+Authors: Isobel Hook (U. Oxford, UK and INAF Obs. Rome, Italy), Massimo Della Valle (INAF-
+Obs Capodimonte and International Centre for Relativistic Astrophysics, Pescara, Italy), Filippo
+Mannucci (INAF- Obs. Arcetri, Italy), see footnote.1
+Abstract
+During the repeat imaging of the deep survey fields, Euclid will find thousands of distant super-
+novae and other transients. As described below, supernovae have powerful uses as cosmological
+probes of dark energy (in a way that complements WL and BAO), as probes of the star formation
+history of the Universe and of stellar evolution.
+16.1 Introduction
+Supernovae are the violent explosions that occur at the end of the lives of some stars. The
+luminosity of these explosions can be comparable to that of an entire galaxy, allowing them
+to be observed at cosmological distances. In the sections below we describe how a sample of
+distant supernovae studied with Euclid could be used for cosmology and other applications in
+astrophysics.
+16.2 Constraining Dark Energy with Type Ia Supernovae
+Type Ia Supernovae (SNe Ia) provide accurate and well-understood cosmological distance
+indicators for cosmology, and were used to produce the first direct evidence for the accelerating
+expansion of the Universe ( Riess et al. 1998;Perlmutter et al. 1999). Since then, several major
+1Much of this text was initially developed for the Euclid Yellow Book in collaboration with Anne Ealet, Ariel
+Goobar, Bruno Leibundgut, Bob Nichol and Pierre Astier, whose input is gratefully acknowledged. A summarised
+version appears in the Yellow Book.
+14916.2. Constraining Dark Energy with Type Ia Supernovae 150
+surveys have amassed large samples of SNe Ia to further improve constraints on the properties of
+the dark energy driving that acceleration (e.g. see Figure 16.1). Recent results based on ∼250
+distant SNe Ia from the Supernova Legacy Survey, in combination with WMAP-5 data and BAO
+data from SDSS and assuming a constant wand a flat Universe, constrain the equation of state
+parameterwto better than 5% statistical accuracy (Sullivan et al, in prep), and about 7% when
+systematic uncertainties are included. This is the tightest constraint on wto date. Moreover,
+the cosmological constraints from SNe are complementary (i.e. in a different direction in the
+w−waplane) to those from weak lensing and BAO.
+The statistical uncertainty on wfrom SNe Ia is now reduced to the level where systematic
+effects are comparable (e.g. see lower right panel of Figure 16.1, reproduced from Kowalski et al.
+2008). To make major advances it will be necessary to control these systematic effects. The
+Euclid deep survey provides a unique combination of area, stability and depth, particularly in
+the infrared where the sky background is significantly lower than from the ground and where
+the effects of extinction by dust are reduced. This will allow us to create a larger and, more
+importantly, better controlled (in terms of systematic effects) SNe Ia sample for cosmology.
+By the time Euclid enters operation, large, ground-based SNe surveys are expected to have
+collected of order 1000 well-measured SNe with redshifts up to z∼1. Perhaps a few hundred of
+those with redshift z <0.5−0.6 will have near-IR measurements (for example from VISTA),
+which will provide control on systematics. In addition a smaller number (perhaps 50) at higher
+redshifts will have been studied from space using HST.
+Euclid, by repeat imaging of 10-square degree patches within the deep survey, could detect
+and follow 1000 −2000 SNe Ia with z<0.7 in the J band during the five-year mission lifetime
+(see Fig 16.2). This is the redshift range where the acceleration of the Universal expansion (and
+hence the direct effect of dark energy) is strongest. In addition, detections at peak magnitude
+will be available for a further 1000 −2000 SNe Ia up to z∼1.
+The power of such a Euclid SN survey would be greatly enhanced if coordinated with a ground-
+based campaign obtaining optical light curves and spectroscopic redshifts and classification.
+16.2.1 Using Euclid near-infrared photometry to improve the accuracy of
+cosmological measurements
+The key aspect of any Euclid SN survey will be the availability of deep and temporally stable
+data (important when subtracting images) in the NIR. Such data are extremely difficult to
+obtain from the ground. There are two ways in which Euclid near-IR photometry would advance
+cosmological measurements from SNe Ia:
+(a) Better constraints on extinction by dust Accurate cosmological measurements from
+SNe rely on correcting for extinction by dust, which is complicated by the fact that SNe Ia also
+show intrinsic colour variations that correlate with luminosity. In the near-IR not only are the
+effects of dust much reduced, but the combination with data at shorter wavelengths allows the
+two effects to be separated. Extinction measurements towards some well-measured SNe show
+that the dust extinction law (characterised by RV) differs from that of the Milky Way (e.g.
+Krisciunas et al. 2004). Euclid near-IR photometry will allow us to calculate the extinction by
+dust along the line of sight towards each SN individually, thereby reducing the systematic error
+introduced by global assumptions about dust properties.
+(b) Lower intrinsic dispersion in the Hubble diagram Traditional SN cosmology is based
+on the rest-frame B-band Hubble diagram. However, at longer rest-frame wavelengths, SNe
+Ia peak magnitudes are believed to have smaller intrinsic dispersion. The NIR imaging data16.2. Constraining Dark Energy with Type Ia Supernovae 151
+=1X!+M!
+ YearstSNLS 1
+BAO (SDSS)99,7 %
+95 %
+68 %
+M!0 0.1 0.2 0.3 0.4 0.5 0.6w
+-2-1.8-1.6-1.4-1.2-1-0.8-0.6-0.4
+Figure 16.1: Top: (Reproduced from Astier et al. 2006) Constraints in the Ω M−wplane from
+the first year Supernova Legacy Survey data, assuming a flat Universe and a constant w, are
+shown in green. The combined constraints with SDSS BAO results ( Eisenstein 2005) are shown
+in red. The statistical error on wof∼9% improves to better than 5% when the 3rd year SNLS
+data are included (Sullivan et al in prep). Bottom left (reproduced from Kowalski et al. 2008):
+Contours at 68.3%, 95.4% and 99.7% confidence level in the w−waplane from the Union SNe
+Ia compilation (consisting of 307 SNe Ia compiled from multiple surveys) when combined with
+BAO and/or CMB constraints (from the 5-year WMPA data release, Dunkley et al. 2009) and
+assuming a flat Universe. Bottom right : as for previous plot but showing the effect of including
+systematic effects in the supernovae analysis.16.2. Constraining Dark Energy with Type Ia Supernovae 152
+0.0 0.5 1.0 1.5
+Redshift0200400600Number per 0.1 bin in redshiftSN Ia (detection only)SN IaSN Ibc
+SN IInSN IIp
+Figure 16.2: Number of SNe of various types that are expected to be detected by Euclid in the
+J band, as a function of redshift. Estimates for SNe of type Ia (dark blue shaded region), Ibc,
+IIn and IIp were provided by A. Goobar based on assumptions in Goobar et al. (2008), using
+SNe Ia rates from Dahlen et al. (2004) and assuming a 5 year survey that monitors a patch of
+10sq deg at any time. These histograms represent the N(z) for SNe with sufficient sampling to
+measure their lightcurve shapes (i.e. reaching 1 magnitude fainter than the peak brightness). The
+light-blue shaded region shows an independent estimate of the total number of SNe Ia detections
+including those only detected at peak luminosity, i.e. without full lightcurve measurements.
+will allow us to measure distances in the rest-frame I-band where the scatter is only 0.13 mag
+(Freedman et al. 2009).
+Overall we expect that the J-band photometry from the Euclid deep survey will be the
+most sensitive for supernovae, with the Y and H bands providing additional colour information.
+The optical component of the deep survey will also provide useful information (for example
+morphology of SN host galaxies and position of the SN within its host) but the single broad
+optical R+I+Z filter is difficult to calibrate for precision light-curve photometry, hence the
+benefit of a coordinated ground-based optical survey. When combined with ground-based data,
+‘standard’ rest frame B-band distances and rest-frame I-band distances to the same supernovae
+could be compared. The large wavelength coverage can be used to study colour variations and
+in turn reduce the scatter in distance measurements. This would yield a high quality Hubble
+diagram in the rest-frame I-band with thousands of events to z∼0.7.
+In summary, Euclid would provide much-improved extinction corrections up to z∼0.8 (plus
+additional objects with 5-sigma detections to z∼1), and a rest-frame I-band Hubble diagram
+with thousands of objects to z∼0.7.
+16.2.2 Euclid spectroscopy of supernovae type Ia
+Spectroscopy is needed in order to measure the redshift (usually from the host galaxy) and to
+determine the supernova type. Euclid itself, through the deep spectroscopic survey (depth TBD
+in the slitless case, or H(AB)∼24 in the DMD case), could provide spectra for the brightest16.3. Astrophysics from Supernovae of all Types 153
+SNe as well as redshifts for most of the host galaxies below z∼0.5.
+Repeated visits to a deep spectroscopic field (e.g. with a cadence of less than about 5 days),
+correlated with photometry, would provide time series of the brightest SN spectra “for free”, which
+can provide typing information. The possibility of generating SN photometric measurements from
+spectroscopy in a slitless mode could also be investigated. This technique would require accurate
+positioning, dithering capability and high sensitivity and should be explored with simulations.
+16.2.3 Cosmology from Core-collapse Supernovae
+In addition to the type Ia SNe described above, the Euclid deep survey will discover several
+thousand (3000-6000) core-collapse SNe (i.e. SNe of types other than Ia) out to z∼0.5. For
+many years, various methods to derive the distance to type II SNe have been proposed based
+on the expanding photosphere effect which relates the SN luminosity to the geometry of the
+photosphere (e.g. Hamuy et al. 2001;Dessart et al. 2008). Now, tight Hubble diagrams are being
+constructed with the current scatter on the distance of ∼10%. This is somewhat larger than that
+can be currently derived with type Ia SNe, but it is considered to be subject to lower systematics
+because the luminosity is directly related to the geometry of the source. In addition metallicity
+is not expected to have a role as the hydrogen shell is optically thick. Type II SNe detected
+by Euclid will provide a complementary cosmological test compared to type Ia SNe and other
+cosmological probes.
+16.3 Astrophysics from Supernovae of all Types
+SNe have a central role in many other fields of astrophysics. Their study is important both to
+understand the exploding systems and to obtain a complete picture of galaxy formation and
+evolution, processes that are strongly affected by SN explosions. Euclid will have a central role in
+this study because, according to the current scanning strategy of the deep survey, it will discover
+a few thousands of well observed SNe in galaxies whose properties are well known. SN rates will
+be measured for both types of SNe, although the precision of these results will depend on details
+of the scanning strategy.
+For such astrophysical use of the SNe, spectroscopic observations of the SNe themselves
+are not strictly required. We expect that a significant fraction of SNe will be discovered in
+galaxies with a spectroscopic redshift obtained by Euclid itself (depending on the final depth of
+the spectroscopic deep field, which remains TBD), while for most of the others a photometric
+redshift of the host will be available. In this situation, methods of SN classification based on
+repeated multi-wavelength photometric observations show promising results ( Poznanski et al.
+2007;Rodney & Tonry 2009), although simulations are required to demonstrate the application
+of this technique to the Euclid filter set. We note that for the purposes of measuring SN rates,
+precise optical photometry is not required.
+16.3.1 Dust extinction and near-infrared searches for supernovae
+Dust extinction is known to limit the number of SNe detected in the local Universe, especially in
+star-forming galaxies ( Maiolino et al. 2002;Mannucci et al. 2003;Cresci et al. 2007;Mattila &
+Meikle 2001). This is even more important because the majority of the SNe exploding inside
+starburst regions are hidden ( Lonsdale et al. 2003,2006;Smith et al. 1998;Ulvestad 2009). The
+properties of the SNe exploding in starburst regions, which are directly related to dynamical
+feedback, remain unknown. At redshift larger than ∼1, dust extinction becomes so important
+that up to half of the SNe could be hidden from optical searches ( Mannucci et al. 2007). From16.3. Astrophysics from Supernovae of all Types 154
+the ground, near-IR searches are limited by sky background and detector sizes. Euclid will be
+the first large-scale near-IR search for SNe from space. At the long-wavelength end of Euclid’s
+range (2 micron), the near-IR extinction magnitude AHis expected to be ∼7 times smaller than
+in the optical, AV. For example, trSNe hidden by 12 mag of optical extinction (a factor ∼60000)
+and therefore out of reach of any optical SNe search, in the near-IR would lose less than 2 mags
+only (less than a factor of 6). As a consequence, much deeper parts of dusty regions can be
+investigated.
+16.3.2 Supernova rate and galaxy evolution
+Several effects link galaxy evolution with the number and the properties of the SNe they host.
+Both core-collapse and thermonuclear SNe are the main producers of heavy elements in the
+Universe. Chemical enrichment within a galaxy is determined by the SN rate as a function of
+galaxy age, while the cosmic SN rate as a function of redshift determines the chemical evolution
+of the Universe as a whole. Core-collapse SNe are believed to be the main producer of dust at
+high redshifts ( Maiolino et al. 2004;Bianchi & Schneider 2007). Therefore understanding dust
+extinction at high redshifts requires a good knowledge of at least this type of SNe. Both types of
+SNe could contribute and even dominate the feedback processes in galaxy formation and could
+explain the ubiquitous presence of outflows in star forming galaxies. With the measurement of
+SN rates, Euclid will give an important contribution in all these areas.
+16.3.3 Type Ia supernova progenitors and stellar evolution
+SN rates are also an important tool to investigate the nature of the exploding systems. While
+the evolution of SN photometry and spectra and the stratification of the chemical elements
+can constrain the explosion mechanism, SN rates are crucial to constrain the progenitors.
+SNe Ia are believed to be due to the thermonuclear explosion of a C/O white dwarf (WD).
+Considerable uncertainties about the explosion model remain within this broad framework, such
+as the structure and the composition of the exploding WD (He, C/O, or O/Ne), its mass at
+explosion (at, below, or above the Chandrasekhar mass) and flame propagation (detonation,
+deflagration, or a combination of the two). Large uncertainties also remain as to the nature of
+the progenitor system. Usually, a binary system is considered, with the WD dwarf accreting mass
+either from a nondegenerate secondary star (single-degenerate model, SD) or from a secondary
+WD (double-degenerate model, DD). The evolution of the binary system through one or more
+common envelope phases, and its configuration at the moment of the explosion are not known
+(seeYungelson 2005 for a review). Single-star models are also possible ( Tout 2005;Maoz 2008)
+and current observations are unable to solve the problem ( Maoz & Mannucci 2008).
+The key quantity to relate Type Ia SN rate to the parent stellar population is the delay
+time distribution (DTD), i.e. the distribution of the delay time between the formation of the
+progenitor system and its explosion as a SN. In general, an expected DTD can be derived from
+each progenitor model, and this can be compared with observations. Several different estimates
+of the DTD have been proposed, but severe limitations on the SN sample and on the knowledge
+of the parent galaxies do not allow these to be distinguished reliably. Euclid will allow us to
+measure the SN rate accurately. From this data DTD can be estimated in two ways, i.e. either
+by comparing the evolution of type Ia rate with with the cosmic star formation history (SFH), or
+by comparing the number of SNe in each galaxy with the SFH of that particular host, without
+the need of making averages among different galaxies.BIBLIOGRAPHY 155
+16.3.4 Core Collapse supernova progenitors
+Core Collapse (CC) SNe are considered to be due to the gravitational collapse of very massive
+stars (M>8 solar masses), although exactly how massive is still to be defined. This uncertainty
+in the mass range can be reduced by measuring accurate SN rates and the initial mass function
+(IMF) of the parent population. The high spatial resolution of Euclid in the optical will allow us
+to study the position of the exploding SN within its host galaxy. For example, it will be possible
+to study if and how much the different classes of CC SNe are associated with the spiral arms
+of the disk galaxies, or what is the radial distribution of the Ia SNe when compared with the
+host galaxies. This can be done up to a significant distance, although simulations are needed to
+quantify this. This method has proved to be able to put constraints on the mass of the progenitor
+stars. For example, it has been demonstrated that the progenitors of Ib and Ic SNe are more
+massive than for type II SNe ( Hakobyan et al. 2009;Georgy et al. 2009;Raskin et al. 2009;
+Anderson & James 2009).
+16.3.5 Core collapse supernova rates
+The CC rates are an independent probe of the cosmic SFH ( Botticella et al. 2008;Dahlen et al.
+2008). Comparing this with the other probes (UV, H α, far-IR) will help understand the relative
+biases and constrain the IMF. This is true, in particular, if dust effects can be controlled, as
+explained above. When using SNe, the SFH is measured by counting point-sources instead of
+measuring the luminosity of diffuse objects, and this have several advantages, as explained by
+(e.g. Botticella et al. 2008). Euclid is expected to detected a few thousand CC SNe in a well
+controlled sample of galaxies. These SNe will reach z∼0.5, i.e. will cover the about 1/3 of
+cosmic history. Therefore it will give a fundamental contribution in this field.
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+Astrophysics and Space Science Library , 163–173.Chapter 17
+Euclid Microlensing Planet Hunter
+Authors: Jean-Philippe Beaulieu (Institut d’Astrophysique de Paris, France), Jordi Miralda-
+Escud´ e (Institut de Cincies del Cosmos, Barcelona) and Joachim Wambsganss (Astronomisches
+Rechen-Institut (ARI), Heidelberg).
+Abstract
+While gravitational lensing can be used to trace the distribution of matter on cosmic and Galactic
+scales, it can also probe smaller scales, through an effect called microlensing, in order to detect
+exoplanets. There is remarkable synergy between requirements for dark energy probes by cosmic
+shear measurements and planet hunting by microlensing. Euclid therefore offers a unique scientific
+programme linking cosmology and exoplanets. It will use gravity as the tool to explore the full
+range of planet masses not accessible by any other means. Euclid is a 1.2m telescope proposed
+to the ESA Cosmic Vision programme and can be exploited for an exoplanet hunt using two
+microlensing programmes. A 3 month microlensing exoplanet hunt will efficiently detect planets
+down to the mass of Mars at the snow line, free floating terrestrial or gaseous planets and
+habitable super Earths. A 12+ month survey would give a census on habitable Earth planets
+around Sun-like stars. This is the perfect complement to the statistics that will be provided
+by the KEPLER satellite, and the combination of these missions will provide a full census of
+extrasolar planets from hot, warm, habitable, frozen to free floating.
+17.1 Introduction
+In the last fifteen years, astronomers have found over 400 exoplanets, including some in systems
+that resemble our very own solar system ( Gaudi et al. 2008). These discoveries have already
+challenged and revolutionised our theories of planet formation and dynamical evolution. Several
+different methods have been used to discover exoplanets, including radial velocity, stellar transits,
+and gravitational microlensing. Exoplanet detection via gravitational microlensing is a relatively
+new method ( Mao & Paczynski 1991;Gould & Loeb 1992;Wambsganss 1997) and is based on
+Einstein’s theory of general relativity. So far 9 exoplanets have been published with this method.
+While this number is relatively modest compared with those discovered using the radial velocity
+method, microlensing probes a part of the parameter space (host separation vs. planet mass)
+not accessible in the medium term to other methods (see Figure 17.1).
+The mass distribution of microlensing exoplanets has already revealed that cold super-Earths
+(at or beyond the snow line and with a mass of around 5 to 15 M⊕) appear to be common
+(Beaulieu et al. 2006;Gould et al. 2006,2007;Kubas et al. 2008). Microlensing is currently
+15917.1. Introduction 160
+Figure 17.1: Semi major axis as a function of mass for all exoplanets discovered as of September
+2009 (microlensing planets are plotted as red dots) and the planets from our solar system. We
+also plot the sensitivity of KEPLER and of space based microlensing observations.
+capable of detecting cool planets of super-Earth mass from the ground and, with a network of
+wide-field telescopes strategically located around the world, could detect planets with mass as
+low as the Earth. Old, free-floating planets can also be detected; a significant population of such
+planets are expected to be ejected during the formation of planetary systems ( Juri´ c & Tremaine
+2008). Microlensing is roughly uniformly sensitive to planets orbiting all types of stars, as well
+as white dwarfs, neutron stars, and black holes, while other method are most sensitive to FGK
+dwarfs and are now extending to M dwarfs. It is therefore an independent and complementary
+detection method for aiding a comprehensive understanding of the planet formation process.
+Ground-based microlensing mostly probes exoplanets outside the snow line, where the favoured
+core accretion theory of planet formation predicts a larger number of low-mass exoplanets ( Ida &
+Lin2005). The statistics provided by microlensing will enable a critical test of the core accretion
+model.
+Exoplanets probed by microlensing are much further away than those probed with other
+methods. They provide an interesting comparison sample with nearby exoplanets, and allow us
+to study the extrasolar population throughout the Galaxy. In particular, the host stars with
+exoplanets appear to have higher metallicity e.g. ( Fischer & Valenti 2005). Since the metallicity
+is on average higher as one goes towards the Galactic centre, the abundance of exoplanets may
+well be somewhat higher in microlensing surveys.17.2. Basic microlensing principles 161
+Figure 17.2: Euclid will monitor main sequence stars in the galactic bulge searching for those
+magnified by gravitational lensing due to foreground star-planet systems in the disk or in the
+bulge.
+17.2 Basic microlensing principles
+The physical basis of microlensing is the gravitational deflection of light rays by a massive body.
+A distant source star is temporarily magnified by the gravitational potential of an intervening
+star (the lens) passing near the line of sight, with an impact parameter smaller than the Einstein
+ring radius RE, a quantity which depends on the mass of the lens, and the geometry of the
+alignment. For a source star in the Bulge, with a 0.3 M⊙lens,RE∼2 AU, the angular Einstein
+ring radius is ∼1 mas, and the time to transit REis typically 20-30 days, but can be in the range
+5-100 days. The lensing magnification is determined by the degree of alignment of the lens and
+source stars (see Figure 17.2). The closer the alignment the higher the magnification.
+A planetary companion to the lens star will induce a perturbation to the microlensing light
+curve with a duration that scales with the square root of the planet’s mass, lasting typically
+a few hours (for an Earth) to a few days (for a Jupiter). Hence, planets can be discovered by
+dense photometric sampling of ongoing microlensing events ( Mao & Paczynski 1991;Gould &
+Loeb 1992). The limiting mass for the microlensing method occurs when the planetary Einstein
+radius becomes smaller than the projected radius of the source star ( Bennett & Rhie 1996). The
+∼5.5M⊕planet detected by Beaulieu et al. (2006) is near this limit for a giant source star, but
+most microlensing events have G or K-dwarf source stars with radii that are at least 10 times
+smaller than this. High angular enough resolution to resolve dwarf sources of the galactic bulge
+(≤0.5 arcsec) will open the sensitivity below a few Earth masses (Fig. 17.3).
+The inverse problem, finding the properties of the lensing system (planet/star mass ratio,
+star-planet projected separation) from an observed light curve, is a complex non-linear one within
+a wide parameter space. In general, model distributions for the spatial mass density of the Milky
+Way, the velocities of potential lens and source stars, and a mass function of the lens stars are
+required in order to derive probability distributions for the masses of the planet and the lens
+star, their distance, as well as the orbital radius and period of the planet by means of Bayesian
+analysis. With complementary high angular resolution observations, currently done either by
+HST or with adaptive optics, it is possible to get additional constraints to the parameters of the17.2. Basic microlensing principles 162
+Figure 17.3: We illustrate the detection capability of the microlensing technique in the very
+low-mass exoplanet regime. The source star and the lens (a foreground star hosting a planet) are
+both located in the Galactic bulge. The sampling interval is twenty minutes, and the photometric
+precision is one percent. Left Panel : The planetary signal on the expected from an Earth-mass
+planet at 2 AU around a solar star, or from an Earth at 1.2 AU but orbiting an 0 .3M⊙M-dwarf
+star. Right Panel : A planet of the mass of Mars (0 .1M⊕) at 1.2 AU can also be detected
+around such a low-mass host star. Both are typical examples of planets to be detected by Euclid
+Microlensing Survey.
+system, and determine masses to 10 % by directly constraining the light coming from the lens
+and measuring the lens and source relative proper motion ( Bennett et al. 2006,2007b ;Dong et al.
+2009). A space-based microlensing survey can provide the planet mass, projected separation from
+the host, host star mass and its distance from the observer for most events using this method.
+Different papers have presented the future strategies in the near, medium and long term,
+with the ultimate goal of achieving a full census of Earth-like planets with either a dedicated
+space mission or advocating for synergy between dark energy probes and microlensing. There
+is a general consensus in the microlensing community about these milestones. White papers
+have been submitted such as: the ESA-EPRAT ( Beaulieu et al. 2008, ExoPlanetary Roadmap
+Advisory Team), the ExoPTF ( Bennett et al. 2007a ;Gould et al. 2007, US ExoPlanet Task Force),
+the exoplanet forum ( Gaudi et al. 2009b ), the JDEM request for information and Astro2010 PSF
+(Bennett et al. 2009;Gaudi et al. 2009a ).
+We summarise the road map with the following milestones :
+The short term (5 years) : automatic follow up with existing networks
+A) An optimised planetary microlens follow-up network, including feedback from fully-automated
+real-time modelling.
+B) The first census of the cold planet population, involving planets of Neptune to super-Earth
+(fewM⊕to 20M⊕) with host star separations around 2 AU.
+C) Under highly favourable conditions, sensitivity to planets close to Earth mass with host
+separations around 2 AU.
+The medium-term (5-10 years) : wide-field telescope networks
+A) Complete census of the cold planet population down to ∼10M⊕with host separations above
+1.5 AU.
+B) The first census of the free-floating planets population.17.3. A program on board Euclid to hunt for planets 163
+C) Sensitivity to planets close to Earth mass with host separations around 2 AU.
+The longer-term (10+ years) : a space-based microlensing survey
+A) A complete census of planets down to Earth mass with separations exceeding 0.5 AU.
+B) Complementary coverage to Kepler of the planet discovery space.
+C) Potential sensitivity to planets down to 0.1 M⊕, including all Solar System analogues except
+for Mercury.
+D) Complete lens solutions for most planet events, allowing direct measurements of the planet
+and host masses, projected separation and distance from the observer.
+17.3 A program on board Euclid to hunt for planets
+Space based microlensing observations
+The ideal satellite is a 1m class space telescope with a focal plane of 0 .5 square degree or more in
+the visible or in the near infrared. The Microlensing Planet Finder or MPF is an example of such
+a mission (which was proposed to NASA’s Discovery program, and endorsed by the ExoPTF, see
+Bennett et al. 2007a ). Despite the fact that the designs were completely independent, there
+is a remarkable similarity between the requirements for missions aimed at probing
+dark energy via cosmic shear and a microlensing planet hunting mission (Beaulieu
+et al. 2008;Bennett et al. 2007a ,2009). Microlensing benefits from the strong requirement
+from cosmic shear on the imaging channel, and does not add any constraint to the design of Euclid.
+Observing strategy
+We will monitor 2 square degree of the area with highest optical depth to microlensing from the
+galactic Bulge with a sampling rate once every twenty minutes. Observations will be conducted
+in the optical and NIR channel.
+Angular resolution is the key to extend sensitivity below few earth masses
+Microlensing relies upon the high density of source and lens stars towards the Galactic bulge to
+generate the stellar alignments that are needed to generate microlensing events, but this high
+star density also means that the bulge main sequence source stars are not generally resolved
+in ground-based images. This means that the precise photometry needed to detect planets of
+≤1M⊕is not possible from the ground unless the magnification due to the stellar lens is moder-
+ately high. This, in turn, implies that ground-based microlensing is only sensitive to terrestrial
+planets located close to the Einstein ring (at ∼2-3 AU). The full sensitivity to terrestrial planets
+in all orbits from 0 .5AUto free floating comes only from a space-based survey ( 17.1). In fig-
+ure17.3we give examples of simulated detections of an Earth and a Mars-mass planet with Euclid.
+Microlensing from space yields precise star and planet parameters
+The high angular resolution and stable point-spread-functions available from space enable a
+space-based microlensing survey to detect most of the planetary host stars. When combined
+with the microlensing light curve data, this allows a precise determination of the planet and star
+properties for most events (Bennett et al. 2007b).
+Probing a parameter space out of reach of any other technique
+The Exoplanet Task Force (ExoPTF) recently released a report ( Lunine et al. 2008) that evaluated
+all of the current and proposed methods to find and study exoplanets, and they expressed strong
+support for space-based microlensing. Their finding regarding space-based microlensing states17.3. A program on board Euclid to hunt for planets 164
+Figure 17.4: Microlensing planet search with a 1m class telescope and 0.5 square degree with 36
+months vs. Kepler exoplanet depth of survey estimates from the Exoplanet Task Force report
+(Lunine et al. 2008). The shading indicates the number of stars that are effectively searched
+for exoplanets as a function of mass and orbital radius, scaled by stellar luminosity so that the
+HZs of all types of stars are at ∼1 AU. A three months program on board Euclid would yield
+numbers 12 times smaller.
+that: Space-based microlensing is the optimal approach to providing a true statistical census
+of planetary systems in the Galaxy, over a range of likely semi-major axes, and can likely be
+conducted with a Discovery-class mission. A program on board the European M class
+mission Euclid will provide a census of extrasolar planets that is complete (in a
+statistical sense) down to 0.1M⊕at orbital separations ≥0.5 AU , and when combined
+with the results of the Kepler mission a space-based microlensing survey will give a comprehensive
+picture of all types of extrasolar planets with masses down to well below an Earth mass. This
+fundamental exoplanet census data is needed to gain a comprehensive understanding of processes
+of planet formation and migration, and this understanding of planet formation is an important
+ingredient for the understanding of the requirements for habitable planets and the development
+of life on extrasolar planets.
+A subset of the science goals can be accomplished with an enhanced ground-based microlens-
+ing program (Gaudi et al. 2009), which would be sensitive to Earth-mass planets in the vicinity
+of the snow-line. But such a survey would have its sensitivity to Earth-like planets limited to a
+narrow range of semi-major axes, so it would not provide the complete picture of the frequency of
+exoplanets down to 0 .1M⊕that a space-based microlensing survey would provide. Furthermore,
+a ground-based survey would not be able to detect the planetary host stars for most of the
+events, and so it will not provide the systematic data on the variation of exoplanet properties as
+a function of host star type that a space-based survey will provide.
+Duration of the program
+One of the remarkable feature of such microlensing program is its linear sensitivity to allocated
+time and area of the focal plane. The minimal time allocation of three months will
+already give important statistics on planets at the snow line, down to the mass17.4. Conclusions 165
+of mars and of free floating planets. Habitable super Earth will also be probed .
+Expected sensitivity can be extrapolated from Figure 17.4. Longer observing time (12 months of
+galactic bulge observing) would lead to good sensitivity to habitable Earth mass planets.
+17.4 Conclusions
+There is a remarkable synergy between requirements for dark energy probes by cosmic shear
+measurements and planet hunting by microlensing. Employing weak and strong gravitational
+lensing to trace and detect the distribution of matter on cosmic and Galactic scales, but as well
+as to the very small scales of exoplanets is a unique meeting point from cosmology to exoplanets.
+It will use gravity as the tool to explore the full range of masses not accessible by any other
+means. A modest allocation of telescope time on Euclid will already efficiently probe for planets
+down to the mass of Mars at the snow line, for free floating terrestrial or gaseous planets and
+habitable super Earth. This is the perfect complement to the statistics that will be provided
+by the KEPLER satellite, and these missions combined will provide a full census of extrasolar
+planets.
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+The Milky Way and the Local Group
+Authors: Mark Cropper (Mullard Space Science Laboratory, University College London), Eva
+Grebel (ARI, University of Heidelberg), Serena Viti (Department of Physics & Astronomy,
+University College London).
+Abstract
+Euclid will make a huge impact on Milky Way science as a direct result of the Wide Survey. The
+scale of this survey at the visible and near-infrared imager sensitivity, spatial resolution and the
+extension to the infrared will transform studies of how our Galaxy was formed and will constrain
+the cosmological scenarios in which this process could have occurred. Euclid will also enable
+whole new areas of Local Group science. Beyond this, a Galactic Plane survey in an extended
+mission would be the reference dataset for star formation and Galactic structure studies, over
+the entire HR diagram and would very substantially increase the science return from Euclid,
+making it a fundamental astronomical resource. This survey would be technically feasible, and
+would be impacted minimally by the degradations expected in the instrumentation by the end of
+the nominal mission.
+18.1 Introduction
+Galactic astronomy and indeed Galactic archaeology are now experiencing a golden age. Without
+any special effort, Euclid is already primed to make a major contribution to Galactic astronomy.
+At high Galactic latitudes, it will provide a vast point source catalogue of stars in the Milky
+Way and nearby galaxies, enabling structural and stellar population studies with unprecedented
+depth, wavelength coverage, and spatial resolution.
+The history of galaxy formation and mass assembly and the underlying physical mechanisms
+are key questions of modern astrophysics. In order to explore them, we can either turn to the
+high-redshift Universe to analyse the integrated light of distant massive galaxies, or we can
+observe our immediate neighbourhood to analyse the resolved stars in nearby galaxies. Our
+own Milky Way is the prime target for these studies as it allows us to explore the full range
+of stellar masses and ages, along with detailed information about chemistry and kinematics.
+Because of their proximity, our Galaxy and its neighbours are a vital and unique laboratory to
+test cosmological models of galaxy evolution.
+Euclid’s suite of imaging and spectroscopic instrumentation is highly suited to Galactic
+studies. The scale of the impact Euclid will have in this field can be extrapolated from the
+Sloan Digital Sky Survey (SDSS), in which, arguably, many of the most interesting results have
+16818.2. Milky Way Science from the Wide Survey 169
+been in Milky Way and Local Group science. Particular highlights include halo streams and
+new dwarf galaxies, with implications for cosmological structure formation at the lower mass
+extreme, and for how the Milky Way has been assembled. The Euclid Wide Survey covers 20 ,000
+square degrees, twice that of the SDSS. It also reaches into the infrared to faint magnitudes,
+and a factor 10 deeper in the visible. It may also be possible to include a dedicated Galactic
+Plane Survey in an extended Euclid mission. It is clear that Euclid will be a discovery machine
+on an unprecedented scale for the Galaxy, and absolutely fundamental to more detailed and
+comprehensive studies with ALMA, JWST and E-ELT.
+We specifically addressed the imaging capability of Euclid here, but it should be noted that
+the visible and near-infrared imagers will be complemented by the power of the Euclid near
+infrared spectrograph (NIS), which will obtain spectra of every object to Hab= 19.5 in the
+continuum, in regions which are not too crowded.
+18.2 Milky Way Science from the Wide Survey
+By returning a vast catalogue of stars in the Milky Way, the Euclid surveys will enable a treasure
+of legacy science based on structural and stellar population studies with unprecedented depth,
+wavelength coverage, and spatial resolution. Euclid will massively augment the Gaia survey
+of our Milky Way, taking it several magnitudes deeper. Especially when combined with other
+deep surveys by ground-based surveys (SDSS, PanSTARRS, SkyMapper, DES), Euclid will
+provide 4-D information on the positions and velocities of hundreds of millions of medium
+and high latitude stars, highly constraining the integrals of motion when locked into the Gaia
+reference frame. Moreover, the 4 optical/IR colours will provide photometric distances (a 5th
+dimension) allowing us to trace large-scale Galactic streams and structure to much greater
+distances than Gaia will be able to, and accessing a wide and more representative range of the
+HR diagram (rather than relying on giant tracers). Below V∼20, the spatial sampling of Euclid
+will provide complementary information to the Gaia astrometry, photometry and spectroscopy,
+adding infrared colours, and providing infrared spectra for every single Gaia star it observes.
+This will go much of the way in breaking the age-metallicity degeneracy, which is critical for the
+chemical enrichment history of our Galaxy.
+Euclid will permit us to derive Galactic structure and scale heights from the faintest, yet
+most numerous tracers – G and M main-sequence stars and brown dwarfs. Stellar overdensities
+and substructure will be mapped in unprecedented depth and detail, complemented by Gaia
+parallaxes and phase space information for the brighter stellar sources.
+18.3 The Local Neighbourhood
+The full sky coverage of Euclid will also yield the most detailed, most sensitive survey of structure
+and substructure in nearby galaxies, of their outer boundaries, of possible tidal disruption and
+streams, and of intergalactic or intragroup stars, again constraining merger and accretion histories.
+Euclid will provide the most complete census of extremely low-surface-brightness, low-mass dwarf
+galaxies in the Local Group and beyond, much less affected by extinction than any of the ongoing
+surveys, thus aiding in constraining the cosmological missing satellite problem. Euclid will make
+a substantial leap in the definition of both the faintest-end of the luminosity function, and the
+radial distribution of dwarfs, putting stringent constraints on the nature of dark matter, the
+epoch of reionisation, and the properties of star formation in small galaxies.
+Euclid’s imagers will also map the entire Magellanic Cloud/Stream system which is at
+relatively high Galactic latitude, allowing many of the studies discussed below for the Galactic18.4. A Euclid Galactic Plane Survey (E-GPS) 170
+Figure 18.1: The Galactic plane in the infra-red from 2MASS data – the inset above right
+shows details around the Galactic centre, and above left shows starfields in regions of lower
+obscuration (from http://pegasus.astro.umass.edu/gc images/ ). A Euclid Galactic Plane
+Survey (E-GPS) survey would reach ∼9 magnitudes deeper, include a visible band and the
+50 times finer spatial sampling in the infrared and 400 times finer sampling in the visible will
+militate against confusion in all but the most crowded regions.
+plane but in the context of their different metallicities and star formation histories. In addition
+it will be possible to trace induced star formation as a result of the interaction with the Milky
+Way, and to place limits on the Galactic potential.
+Generally, Euclid’s sensitivity will open up a new discovery space for rare stellar and other
+low-temperature objects. For nearby galaxies with resolved stellar populations, Euclid will permit
+us to obtain a complete census of (TP-)AGB and C/M stars, which in turn allow us to derive
+spatially resolved star formation histories for periods of up to 8 or 9 Gyr. Euclid will detect stars
+far from their parent galaxies, important for tracing those galaxy potentials.
+18.4 A Euclid Galactic Plane Survey (E-GPS)
+The Euclid high Galactic latitude surveys will generate large samples of disk stars, as we have
+discussed above. If an extended mission could include surveys at lower Galactic latitudes, the
+harvest of Galactic science would be exceptionally bountiful, given Euclid’s fine spatial sampling
+and red/near-IR capabilities. Before Euclid, the state-of-the-art ground-based large-scale surveys
+will reachJ,H,K s∼20 (see Figure 18.1), with only narrower (10–20 square degrees) surveys
+reaching down to the Euclid near-infrared imager Y,J,H ∼24 limits. It is in the near-IR that
+the increase in scientific capability is most pronounced. Such a survey is not prohibited by the
+current spacecraft operational envelope, and because these observations would be less sensitive to
+the deleterious effects of radiation damage to the detectors, there would be minimal performance
+impact even after the end of the nominal mission. Were the funds to be available to pursue a
+mission extension, a Galactic Plane Survey (E-GPS) should be feasible unless Euclid had by
+then suffered significant other deterioration.18.4. A Euclid Galactic Plane Survey (E-GPS) 171
+18.4.1 Galactic Structure
+E-GPS would probe the Galactic plane with many of the science goals of the currently-planned
+infrared surveys, but surpassing them by orders of magnitude. Such a survey would be the basis
+of understanding the bulge, bar and thin and thick disk structure as a function of age, metallicity
+and position. It will inform among others the chemodynamical evolution of the Galaxy; the
+relative importance of mergers and disk instabilities in bulge formation; the location and longevity
+of spiral arms; the structure and characteristics of the central bar and its connection to the
+spiral arms; the origin and nature of the thin and thick disk; and the disk heating and cooling
+mechanisms – particularly the relative importance of minor mergers, giant molecular clouds and
+globular clusters in the disk heating.
+The Euclid imagers, and particularly the near infra-red imager, will trace disk structure and
+recent star formation through luminous main sequence and giant stars, to much greater distances
+or through more obscuration than existing or planned Galactic plane surveys, tracing the spiral
+arms and allowing the star formation efficiency to be determined as a function of Galactocentric
+distance. Among intermediate mass stars, the AGB phase, and planetary nebulae are extremely
+luminous in the infrared, and can be used as further tracers. Euclid is sufficiently sensitive that
+G-M main sequence stars will be a major resource, especially given the Euclid spatial resolution.
+In short, E-GPS will address the entire field of Galactic science, from star formation processes,
+to dust formation, the recycling of elements in the interstellar medium, the end points of stellar
+evolution and their role in galactic evolution, to identifying stellar clusters and their remnant
+streams and so on. In conjunction with and building on Gaia data, E-GPS will determine stellar
+cluster distances, streaming motions, the stellar initial mas function and cluster luminosity
+functions to exceptionally faint limits. E-GPS will also provide significant samples of rare
+evolutionary stages and exotic astrophysically important systems, in addition to a vast sample
+of every other spectral type. Because the Gaia data will be more limited in the Galactic plane
+through obscuration, the E-GPS resource would be a key tool in determining the 3-D structure
+of the Milky Way disk.
+The E-GPS will also complement the Herschel survey of the Galactic Plane in the far IR and
+will provide essential preparatory science for ALMA. Depending on the sequencing of observations
+within the E-GPS it may be possible to include the time dimension by returning to the same field
+at later times. This would allow the identification of huge numbers of variable stars, important as
+distance indicators and hence critical to our understanding of Milky Way dynamics and structure.
+18.4.2 Young Stellar Objects
+Young Stellar Objects (YSOs), in particular massive ones, shape the evolution and fate of galaxies
+by dominating their radiation budget and determining the ionization and metallicity of the
+intergalactic medium. Star formation and evolution ultimately drive the evolution of the Universe
+from its primordial composition to the present-day chemical diversity. Most YSO will still be
+embedded in their parental molecular clouds, hence observations in the far optical/near-IR are
+essential in order to determine their nature and distribution in the galaxy. The high sensitivity
+and resolution of Euclid will allow the identification of a statistically important number of YSOs.
+The survey will determine the number and spatial extent of YSOs and to measure their ages
+and masses. It will provide essential information for the characterization of the initial mass
+function; the star formation rate, history and efficiency; the formation of clusters and sequential
+star formation processes.18.5. Summary 172
+18.4.3 Brown Dwarfs
+Brown dwarfs, and in general very low mass stars (VLMS), are the most numerous objects
+in the galaxy but they are also some of the most difficult objects to detect, owing to their
+faintness. They emit the bulk of their radiation in the near infrared (1 −3µm). The astrophysical
+importance of identifying a large number of brown dwarfs and very low mass stars comes from
+the fields of cosmology (are some of the VLMS primordial) and Galactic dynamics (how much
+mass is stored in faint VLMS), from the field of star formation (does the initial mass function
+vary among star formation regions, and is there a lower mass limit below which no ‘stars’ form)
+as well as from an area of great current interest, that of extraSolar planets (what distinguishes a
+brown dwarf from a M dwarf, and an extrasolar planet from a brown dwarf). By determining
+their fundamental properties, such as effective temperature and metallicity, the parameters of
+very low mass stars can be derived.
+18.4.4 Late Stages of Stellar Evolution
+The Euclid imagers will be extraordinarily sensitive to late phases of stellar evolution, including
+cool white dwarfs, which are critical laboratories for fundamental physics (such as limits on the
+change ofGand the fine structure constant. In addition, the SDSS experience has shown that
+the survey will identify large numbers of exotic objects, such as accreting white dwarfs, double
+white dwarf systems (which are the brightest sources of gravitational radiation), X-ray binaries
+and even close-by isolated neutron stars. Such systems are important laboratories for the study
+of physical laws and processes in extreme conditions and are the observational driver for exciting
+and ground-breaking astrophysics.
+18.5 Summary
+It is clear that within its nominal mission, and without any special measures, Euclid will
+revolutionise local Universe and Milky Way studies, addressing formation histories, mergers and
+interactions, induced star formation, measuring galactic potentials, and exploring the nature of
+the thick disk and halo. The Solar neighbourhood will be characterised with complete samples to
+considerable distances, even for cool faint objects. If it proves possible to extend the Euclid survey
+to lower Galactic latitudes in a Galactic plane survey, Euclid will address a huge range of Galactic
+science, producing fundamental datasets of the disk and centre of the Galaxy, accumulating
+vast samples of every population of the Galaxy, even of rare objects. This resource will be
+used to address key questions in galactic evolution: chemical enrichment, feedback processes,
+star formation, the role of dynamical and other processes in driving disk structure, the nature
+and origin of the Milky Way bulge and so on. These surveys will identify objects and regions
+for more detailed followup by JWST, ALMA and SKA on a staggering scale, and the Euclid
+datasets will be an absolutely fundamental resource for a wide swathe of astronomical research
+for many years to come. Euclid’s fine spatial resolution, high sensitivity and reach into the
+infra-red, coupled with infrared spectroscopy is the key: nothing of this scale, depth, and spatial
+sampling is anywhere near being matched from any other space- or ground-based facility, existing
+or planned.Part V
+SIMULATIONS
+173Chapter 19
+Image Simulations
+Authors: Massimo Meneghetti (INAF - Osservatorio Astronomico di Bologna), Benjamin
+Dobke (JPL - Pasadena), Jason Rhodes (JPL - Pasadena), Lauro Moscardini (Dipartimento
+di Astronomia - Universit` a di Bologna), St´ ephane Paulin-Henriksson (CEA - Paris), Olivier
+Boulade (CEA - Paris), J´ erˆ ome Amiaux (CEA - Paris), Thomas Kitching (University of
+Edinburgh), Sarah Bridle (UCL London), Lisa Voigt (UCL London), Peter Melchior (Institut
+f¨ ur Theoretische Astrophysik - Zentrum f¨ ur Astronomie Heidelberg), Julian Merten (INAF -
+Osservatorio Astronomico di Bologna), Andrea Grazian (INAF - Osservatorio Astronomico di
+Monteporzio), Richard Massey (University of Edinburgh), David Johnston (Case Western).
+Abstract
+The image simulation working group has developed tools for producing mock observations
+adopting the Euclid optical design. Among the many possible applications, these tools were used
+to estimate the number counts of galaxies in future Euclid observations. We produced images
+assuming different levels of background and changing the effective exposure time, in order to take
+into account possible changes of the instrument design. Our simulations show that an effective
+number density of galaxies between 30 and 40 arcmin−2is easily achievable with the current
+design of Euclid. The median redshift of the detected sources is around unity. These results
+allow the conclusion that Euclid will match all the requirements needed to reach a sub-percent
+accuracy on the estimate of cosmological parameters like the equation of state of dark energy.
+19.1 Introduction
+The precision with which cosmological parameters like the equation of state of dark energy can be
+constrained using weak lensing principally depends on the number of galaxies for which a shape
+measurement is achievable and on the redshift distribution of the sources. The work done by the
+image simulation working group aims at quantifying how many galaxies per square arcminute
+will be detectable in Euclid observations and usable for the weak lensing analysis. We tackle this
+issue by using state-of-the-art simulation pipelines that allows one to produce mock observations
+with specific instrumental set-ups, including all the relevant sources of noise. The images are
+subsequently analysed to compile catalogues of galaxies. By applying proper selection cuts to
+the catalogues, we estimate the effective number density of galaxies useful for lensing.
+Reaching an accuracy of percent level precision on the measurement of dark energy requires
+an effective number density of, on average, 30–40 galaxies per square arcminute over the entire
+Euclid survey. We show here that the current Euclid mission and survey design will meet that
+goal.
+17419.2. Image simulation pipelines 175
+19.2 Image simulation pipelines
+The simulations are made using two pipelines, independently developed by two groups, one in
+Italy ( SkyLens ) and one in the US ( simage ). A detailed description of their implementation can
+be found in Meneghetti et al. (2008) and in Massey et al. (2004) (see also Dobke et al. in prep).
+Both the pipelines use the Hubble Ultra-Deep Field (HUDF; Beckwith et al. 2006) as a basis set
+to generate the galaxy population to be placed in the simulations. About 10,000 galaxies are
+decomposed into shapelets ( Refregier 2003;Melchior et al. 2007;Massey & Refregier 2005) in
+the fourB,V,i , andzbands. The shapelets formalism allows to easily manipulate the galaxy
+shapes. For example, operations like convolutions (or de-convolutions), rotations, mirroring,
+shearing, etc. are straightforward. Moreover, the shapelets provide an analytical description of
+the galaxies. Thus, they can be “painted” on a virtual sky at any desired resolution, with the
+only limitation of the native resolution limit of the HUDF images. Using the HUDF galaxies as
+a reference basis set, ensures the coverage of a broad range of galaxy morphologies and sizes.
+Moreover, the HUDF is the deepest observation available to date. Briefly, the steps required
+to simulate an observation of a patch of the sky are: 1) they generate a population of galaxies
+using the luminosity and the redshift distribution of the galaxies in the HUDF; 2) they prepare
+the virtual observation, receiving from the user the pointing instructions, the exposure time,
+and the filter F(λ) to be used. The pointing coordinates are used to calculate the level of the
+background, i.e. the surface brightness of the sky; 3) they assemble the virtual telescope. This
+implies that the user provides some a set of input parameters, like the effective diameter of the
+telescope, the field of view, the CCD specifications (gain, read-out-noise, pixel scale) and the
+additional information necessary to construct the total throughput function, defined as
+T(λ) =C(λ)M(λ)R(λ)F(λ). (19.1)
+In the previous formula, C(λ) is the quantum efficiency of the CCD, M(λ) is the mirror reflectivity,
+andR(λ) is the transmission curve of the lenses in the optical system; 4) they calculate the
+fluxes in the band of the virtual observation. Then, using the shapelets coefficients of the galaxy
+decompositions, the surface brightness of the sources is calculated at each position on the sky
+and converted into a number of Analogue to digital units (ADUs) on the CCD pixels. Finally,
+noise is added according to the sky brightness and to the Read-Out-Noise (RON) of the CCD.
+The two pipelines follow very similar approaches for modeling the galaxies in the images,
+although significant differences are present in the methods for creating the galaxy populations.
+While SkyLens uses the magnitude and redshift distributions of the HUDF, simage calibrates the
+number counts per magnitude bin such to match other existing surveys like COSMOS ( Leauthaud
+et al. 2007). The pipelines also use different approaches when modeling the SEDs of the sources.
+Given these differences, the two pipelines have been extensively tested against each other. In
+particular, they have been validated by performing simulations of existing HST fields. The
+agreement between the outputs of two pipelines and the real catalogues is remarkable in terms of
+both limiting magnitudes and galaxy number counts. Both the pipeline produce galaxy number
+densities which differ from the observed ones by less than 10%. We conclude that we can safely
+combine the results of the two pipelines for the analyses shown in the next sections.
+19.3 Instrumental set-up
+The simulations used in this paper are based on the parameters listed in Tab. 19.1. Each
+Euclid observation will consist of four 450s-long dithered exposures of the same field. The total
+integration time will then be texp= 1800s. In the present study, we work with images obtained19.4. Sky background 176
+Table 19.1: Key parameters used in the simulations.
+parameter value
+Diameter 1.19m
+Obscuration (diameter) 0.37m
+Pixel scale 0.1”
+refl. of each mirror 0.98
+number of mirrors 5
+refl. of the dichroic 0.98
+filter transmission 0.9
+detector Q.E. (550nm) 0.9
+detector Q.E. (700nm) 0.9
+detector Q.E. (850nm) 0.7
+detector Q.E. (900nm) 0.5
+detector Q.E. (920nm) 0.4
+R.O.N 5 e−
+dark current 0.001 e−/pixel/s−1
+n. of. exposures 4
+tot. exp. time 1800s
+by combining four exposures with different noise patterns, and we do not include and we do not
+use dithering to increase the resolution of the final stacked image.
+The key parameters listed above provide the reference configuration of the instrument.
+However, in order to take into account possible changes of their values, our simulations do
+not stick on a single set of parameters. Instead, we consider a range of possible values for the
+parameters listed above. For example, we allow for possible variations of the mirror and dichroic
+reflectivity.
+Changes in the instrumental set-up are incorporated in the simulations by scaling by a proper
+factor the ADUs in each CCD pixel computed for the reference set-up. The pixel counts are
+proportional to the total throughput, being
+ADU(/vector x)∝texp/integraldisplay
+T(λ)S(/vector x,λ)dλ
+λ, (19.2)
+whereS(/vector x,λ) is the surface brightness (sources and sky) at the position /vector xon the CCD array.
+The above equation shows that we can account for changes in the throughput by defining the
+effective exposure time ,teff. This is defined as teff=texpf, where the factor fcontains all the
+parameter variations with respect to the reference design. In order to explore several solutions,
+we allow for changes in teffby a factor fbetween 0.5 and 1.6.
+The images are convolved with a realistic system PSF model of FWHM=0.180 arcsec, which
+takes into account the optical PSF, the jitter, and the detector effects.
+19.4 Sky background
+The principal background contributor in the wavelength range for Euclid will be the zodiacal
+light. A model for the zodiacal light near the North-Ecliptic-Pole (NEP) has been derived by19.5. Analysis of the images 177
+Figure 19.1: Simulated observation of a patch of the sky, corresponding to a region of 34” ×22”,
+assuming the instrumental set-up of Euclid . The images in the upper panels refer to simulations
+with effective exposure times of 900s (left) and 2880s (right) assuming an “average” sky background
+level of 22.342 mag ABarcsec−2. The images in the bottom panels show two simulations with
+effective exposure time 1800s and sky background magnitudes of 21 .75 mag ABarcsec−2and 22.95
+mag ABarcsec−2.
+Leinert et al. (1998) and later revised by Aldering et al. (2004). This model has been used to
+derive “low”, “average”, and “high” background levels for Euclid observations. In terms of AB
+magnitudes, the model provides the surface brightnesses in the wavelength range [550 ÷920]nm
+of 22.95, 22.35, and 21.75 for the “low”, “average”, and “high” background cases, respectively.
+Thus, we carry out simulated observations assuming a continuous range of sky surface brightness
+values between 21 .75 and 22.95 mag ABarcsec−2.
+19.5 Analysis of the images
+Some examples of our simulations are shown in Fig. 19.1. The upper panels show a patch of the
+sky, which corresponds to a region of 34” ×22”, as it might be seen by Euclid with teff= 900s
+(left panel) and teff= 2880s (right panel) assuming the “average” sky background. Instead, the
+bottom left and right panels show the same patch observed with teff= 1800s and assuming
+the “high” and the “low” sky background levels, respectively. The colour bar on the bottom
+allows to convert the gray levels into ADUs s−1. Obviously the galaxy counts and the source
+signal-to-noises differ significantly in the limiting cases shown here. We now consider several19.5. Analysis of the images 178
+Figure 19.2: Left panel: Effective number density of galaxies in Euclid images as a function of
+effective exposure time, assuming the “average” sky background level. The vertical dashed line
+indicates the planned effective exposure time for this mission. Right panel: Effective number
+density of galaxies in Euclid images as a function of sky brightness, assuming nominal effective
+exposure time of 1800s (co-adding four short exposures of 450s each). The vertical dashed
+lines indicate the “low”, “average”, and “high” sky background levels defined in Sect. 19.4. In
+both panels, the shaded region denotes the number density interval which matches the scientific
+requirements for Euclid. We use different line-styles for the three selection methods discussed in
+this paper. The solid, dotted, and dash-dotted line indicate the results obtained with methods 1,
+2, and 3, respectively.
+selection criteria to show how the galaxy counts depend on both the effective exposure time and
+the sky surface brightness.
+Method 1: Selection by signal-to-noise: Firstly, among the galaxies detected in the images,
+we count those passing some cuts based on their signal-to-noise ratio (SNR) and size. These
+are motivated by the necessity of measuring the shape of the sources, given that the main goal
+is to perform shear measurements. Thus galaxies need to be well resolved. Note that, for this
+selection method, we do not calibrate the SNR limit on the effective capability of measuring
+the galaxy shape with a given accuracy. We pick a limit of SNR min>10, to enable good
+shape measurements with all the selected objects. We further only retain those galaxies whose
+FWHM>1.25×FWHM PSF.
+Method 2: Selection by shear measurability Lens fit is a template fitting shape measure-
+ment method that makes use of Bayesian shape estimation techniques to measure the shear. The
+method is described in Miller et al. (2007) and Kitching et al. (2008) and has been applied to the
+STEP ( Heymans et al. 2006;Massey et al. 2007) and GREAT08 simulations ( Bridle et al. 2008).
+The Bayesian shape estimation procedure is optimised so that in the noisy low signal-to-noise
+regime any bias caused by noise should be removed. For lensfit we make no cuts at the catalogue
+level, every object in the SExtractor catalogues is analysed, and a weight and identification given.
+For these simulations we performed two analyses, the first used the stacked images created by
+stacking the pixels in each individual exposure. For the second analysis we simultaneously esti-
+mated the galaxy shapes from each exposure and stacked the ellipticity probability distributions
+for each galaxy from each exposure. To calculate the effective number density we summed the
+lensfit weights of each galaxy for which we take the mean of the lensing sensitivities (as described19.5. Analysis of the images 179
+inMiller et al. (2007) and Kitching et al. (2008)).
+Method 3: Selection by shear uncertainty An additional shear analysis is done with the
+Im2shape software. Running it on the simulations to measure the galaxy ellipticities, we calculate
+the total number of galaxies with a shear uncertainty smaller than 0 .1 (ignoring shape noise), to
+obtain the effective number of galaxies.
+19.5.1 Detections vs effective exposure time
+In the left panel of Fig. 19.2 we show how the galaxy number counts change as a function of
+the effective exposure time for Euclid-like observations. Considering the selection method 1, the
+expected number density of galaxies with high SNR is ∼33 arcmin−2for an effective exposure
+time of 1800s. We recall that, assuming a survey area of 20,000 sq. degrees, the effective galaxy
+number density required for achieving the Euclid objectives, is between 30-40 arcmin−2. Such
+requirement is satisfied for teff>1500s. The method 1 is the most conservative at estimating the
+effective galaxy number counts, since we intentionally adopt a large SNR limit for the detections.
+However, in many cases the ellipticity measurements are possible down to much lower SNRs
+(/greaterorsimilar5). This is shown by the analyses using the selection methods 2 and 3, whose results are
+given by the dotted and dash-dotted lines in Fig. 19.2. For example, using the lensfit pipeline
+to select those galaxies with achievable ellipticity measurements, we find ∼54 arcmin−2. This
+number decreases to ∼43 arcmin−2when imposing a cut based on the ellipticity error. However,
+in a realistic situation we should consider that a number of detections will be missed due to pixel
+defects or CCD gaps, which may decrease the effective galaxy number counts. Decreasing the
+effective exposure time down to 1000s, i.e. in the most pessimistic case for Euclid, the expected
+number counts drop to ∼22, 28, and 37 arcmin−2for the methods 1, 2, and 3, respectively.
+Increasing the effective exposure time to 2000s, 2500s, and 2900s would improve the number
+counts by ∼10%, ∼20% and ∼30%.
+19.5.2 Detections vs sky background level
+The dependence of the effective galaxy number density in the Euclid images on the level of the sky
+background is shown in the right panel of Fig. 19.2. The analysis reveals that for sky brightnesses
+in the range msky= 21.75–22.95 mag ABarcsec−2the effective galaxy number density varies
+between ∼28 arcmin−2and∼40 arcmin−2using the selection method 1 assuming the nominal
+effective exposure time, teff= 1800s. Thus, we expect variations of the order of ∼15–20% in the
+effective number counts depending on the pointing directions. Similar trends are found with the
+method 2 and 3, for which the galaxy number counts always remain well above the threshold of
+30 arcmin−2. Therefore, the current reference design of Euclid matches the requirements on the
+effective galaxy number density for essentially any plausible level of sky background.
+The zodiacal background can be minimised using a suitable observing strategy. In particular,
+the scanning strategy of Euclid will enable scans in ecliptic latitude, rolling around the Sun
+direction with the telescope pointing at right angle to the Sun. The whole sky outside a stripe of
+galactic latitude |b|<30 degrees will be observed. The histogram in the left panel of Fig. 19.3
+shows the distribution of sky levels obtained by dividing the sky into 3072 pixels of 13 .4 sq.
+degrees, assuming each line of sight is observed at 90 degrees from the Sun and discarding the
+pixels at |b|<30 degrees. In the right panel of Fig. 19.3we show the distribution of ngalexpected
+for the same survey strategy. The results are shown for the method 1, which, we remind again,
+is a very conservative method to define ngal. In this case, we find that /lessorsimilar28% of the survey
+area will have 27 <n gal<30 arcmin−2and/greaterorsimilar38% of the survey area will have 35 <n gal<40
+arcmin−2if observed with Euclid with an effective exposure time of 1800s.19.6. Conclusions 180
+Figure 19.3: Left panel: distribution of sky magnitudes in the area of the sky at galactic latitudes
+|b|>30 degrees, assuming to observe at 90 degrees from the direction to the Sun. Right panel:
+the magnitude distribution on the right is here converted into a distribution of effective galaxy
+number densities by means of the image simulations, using the selection method 1.
+19.5.3 Redshift distribution
+The last but not least important ingredient to establish the accuracy of on the measurements
+of the cosmological parameters is given by the redshift distribution of the sources. For each of
+the galaxies used in the simulations, we know the redshift, which is taken from the photometric
+redshift catalogue by Coeet al. (2006). Thus, we can use this information to determine the
+expected redshift distribution of the sources in the Euclid images. Assuming the reference design
+for the instrument and the “average” sky background level, the redshift distribution of all galaxies
+detected by SExtractor in the simulated images is given by the red dashed histogram in Fig. 19.4.
+If the selection method 1 is used to discard low SNR galaxies, the redshift distribution changes
+as shown by the solid histogram. In these two limiting cases, the median redshifts of the sources
+are 0.99 and 1.17, respectively, thus well matching the requirement zm/greaterorsimilar0.8. Of course, this
+conclusion relies on the strong assumption that the redshift distribution of galaxies in the HUDF
+(shown by the dot-dashed blue histogram) can be extended to the whole sky.
+19.6 Conclusions
+The simulations discussed above show that the current optical design of the Euclid imaging
+channel ensures an average effective number density of galaxies detected at high significance
+/greaterorsimilar33–35 arcmin−2. The effective number density of galaxies which will be available for the
+lensing analysis, i.e. for which it will be possible to obtain a reasonably accurate shear estimate,
+is likely to be significantly larger. Assuming the redshift distribution of the galaxies in the HUDF,
+we find that the median redshift in Euclid observations will be zm∼1. Given these results, we
+conclude that the requirements needed for reaching the desired level of accuracy will be matched
+with the current Euclid design.
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+Photometric redshifts
+Authors: Rongmon Bordoloi (ETH Zurich), Manda Banerji (UCL, IoA Cambridge), Filipe
+Abdalla (UCL), Adam Amara (ETH Zurich), Ofer Lahav (UCL), Simon Lilly (ETH Zurich).
+Abstract
+In this chapter we summarise the work done by the EIC Photo-z Working Group during the
+Euclid definition phase. The work involved contributions from teams in Barcelona, Rome, UCL
+and Zurich. We focus on two studies, presented in Abdalla et al. (2008a ) and Bordoloi et al.
+(2009), that use different approaches to the photo-z measurements: a template fitting method
+ZEBRA ( Feldmann et al. 2006) (ETH Zurich) and a Neural network method AnnZ ( Collister &
+Lahav 2004) (UCL). Through extensive comparisons, we find that these two methods give similar
+performance, which in turn allows us to form robust conclusions. In this chapter, we address
+several of the issues associated with high performance photo-z measurements. These include the
+need for space-based Near Infrared imaging and our requirements for ground-based counterparts.
+We find that in combination with ground-based photometry (e.g. PanStarrs-2 or DES), the
+baseline Euclid mission can reach our requirements of sigma(z) ≤0.05(1 +z) after a simple
+cleaning methods have been applied. We find that the cleaning, which uses only information that
+is available in the photometry, is also able to reduce our catastrophic failure rate to below 0 .25%.
+We also show that there are a number of approaches for reaching the demanding requirement
+that the mean redshift of galaxies in a redshift bin be known to a precision of 0 .002(1 +z).
+Specifically, we show: (i) a case where the mean is measured directly using spectroscopic redshifts
+of a representative subsample; and (ii) a method for measuring the mean using the likelihoods
+coming from the photo-z estimation codes. Finally, we also summarise the effect of an imprecise
+correction for Galactic extinction and the effects of contamination by over-lapping objects in
+photo-z determination. The overall conclusion is that the acquisition of photometrically estimated
+redshifts with the precision required for the Euclid weak lensing science goals will be possible.
+20.1 Introduction
+Application of weak lensing for cosmology requires at least a statistical knowledge of the
+distances, i.e. redshifts, of large numbers of individual galaxies. At IAB<24.5, there are about
+2.5 billion galaxies in the Euclid 2 πsr survey area and so, realistically, reliance must be made on
+photometrically estimated redshifts (hereafter photo-z ). In weak-lensing tomography, redshift
+information is required at two conceptually distinct steps. First, galaxies must be assigned to
+individual redshift bins. The shear signal is then extracted from the cross-correlation of the shape
+18320.1. Introduction 184
+measurements of individual galaxies in different redshift bins, to exclude potential problems such
+as intrinsic alignments between physically associated galaxies. Second, the statistical N(z) (i.e.
+effectively /angbracketleftz/angbracketright) of the galaxies in a given bin is required to then map the results onto cosmological
+distance and thereby extract the cosmological information. It is, of course, possible to do a
+similar correlation analysis with unbinned data ( Castro et al. 2005;Kitching et al. 2008a ), but
+for our purposes this distinction is unimportant.
+The required accuracy of the individual photo-z for the first bin-construction task is set by
+the need to exclude overlaps in the N(z) of individual bins (or the probability distribution for
+individual galaxies) and thereby remove physically close pairs ( King & Schneider 2002). This
+typically sets a requirement on the precision of individual photo-z of about σz= 0.05(1 +z)
+(Bridle & King 2007). This step must be done with photo-z because of the large number of
+galaxies involved (2 .5 billion).
+The required accuracy in the statistical redshift distribution N(z) and the mean redshift /angbracketleftz/angbracketright
+for each of these bins is set by the required accuracy of the cosmological parameters. The Euclid
+scientific goals require a precision of order 0 .002(1 +z) in/angbracketleftz/angbracketright(Amara & R´ efr´ egier 2007;Maet al.
+2006;Kitching et al. 2008b ). There are two approaches to constructing N(z) and /angbracketleftz/angbracketright. The first
+involves the “direct” measurement of large numbers of spectroscopic redshifts for a representative
+set of objects in each bin ( Abdalla et al. 2008a ). The other relies on continued use of the photo-z
+themselves to characterise the bin, which relaxes the requirement on the spectroscopic redshifts
+which are used only to calibrate the photo-z scheme ( Bordoloi et al. 2009).
+These two requirements are both demanding and represent one of the observational challenges
+that lie along the path to enabling precision cosmology with weak lensing. Fortunately, there are
+some mitigating features of weak lensing analysis. For instance, the analysis is robust (aside from
+shot-noise statistics) to the exclusion of individual galaxies, provided only that the exclusion is
+unrelated to their shapes. One is free therefore to reject galaxies that are likely to have poor
+photo-z provided that they can be recognised a priori , i.e. from the photometric data alone.
+This chapter summarises the photo-z work done within the EIC photo-z working group.
+These results appear in Abdalla et al. (2008a ) and Bordoloi et al. (2009). Specifically, we discuss
+here the following topics:
+•The choice of near-infrared filter configuration for Euclid.
+•The required depth of ground-based complements.
+•The photo-z performance on individual objects, emphasising the a priori identification and
+rejection of catastrophic failures.
+•The number and sampling requirements on spectroscopic redshifts for the “direct” con-
+struction of N(z).
+•The construction of N(z) using the photo-z alone, focussing on approaches that yield the
+least biased estimate of N(z) and /angbracketleftz/angbracketrightfor the ensemble.
+•The systematic biases that can enter into the photo-z from an incorrect assumption about
+the level of foreground Galactic reddening, and how the reddening can be estimated from
+the photometric data themselves.
+•The effects of the photometric superposition of two galaxies at different redshifts, leading
+to a mixed spectral energy distribution that may perturb the photo-z.
+Our approach is to try to isolate these problems and to explore each in turn with the aim
+of providing an existence proof that gives a plausible route to achieve the very high photo-z
+performance required for Euclid .20.2. Generation of photo-z from mock photometric catalogues 185
+20.2 Generation of photo-z from mock photometric catalogues
+We have utilised the publicly available “neural network” photo-z package ANNz ( Collister &
+Lahav 2004), and the publicly available template fitting package ZEBRA ( Feldmann et al. 2006).
+It is now widely recognised, and demonstrated by direct comparison early in our work, that these
+two approaches to photo-z estimation give broadly similar results ( Abdalla et al. 2008b ).
+These photo-z codes have been applied to mock photometric catalogues. The mock catalogues
+used at UCL are described in Abdalla et al. (2008a ). The GOODS-N spectroscopic sample
+(Cowie et al. 2004;Wirth et al. 2004) was used. This is a flux limited sample with R< 24.5 and
+redshiftsz <4. The broadband photometry available for this sample in multiple wavebands,
+was used to generate a series of galaxy templates using a method similar to Budav´ ari et. al
+(1999) but assuming a prior set of templates consisting of the observed Coleman et al. (1980)
+templates and the starburst galaxies of Kinney et al. (1996). Any reddening is removed from the
+photometry before template construction and the new templates used to calculate a best-fit SED
+value and reddening for each object assuming the reddening law of Calzetti (1997) as well as
+a correction for the IGM absorption according to the prescription of Madau (1995). Gaussian
+noise is added to the fluxes before calculating magnitudes and errors for the noisy sample in
+order to produce the final catalogue.
+At ETH, the COSMOS mock catalogues were produced by Kitzbichler & White (2007).
+There are a total of 24 mock catalogues each representing 2 deg2and containing approximately
+600,000 objects, the majority of which are at z≤1.To produce the photometric catalogue for a
+customised set of filters, SED templates were identified, from amongst 10,000 available, that well
+matched the quoted photometry for each galaxy in the COSMOS mock catalogue at its known
+redshift. These templates include a range of internal reddening, which ranges from 0 <A v<2
+magnitudes. These templates were then used to compute photometry for the galaxies in all
+other passbands of interest, adding Gaussian noise. Intergalactic absorption in each template is
+compensated for using the Madau law ( Madau 1995). All 24 mocks were combined and a random
+sub-sample used to mimic a survey over a large area in the sky. Each of the simulations contains
+at least 100,000 objects. To match the proposed Euclid experiment objects with IAB≤24.5 were
+considered. Table 20.1 provides the assumed 5 σAB magnitude limits in each of the different
+filters for different surveys that are described in this work. For this study the input photometric
+catalogues were constructed using exactly the same set of approximately ten thousand templates
+as were subsequently used in the ZEBRA photo-z code. This may strike some readers as being
+somewhat circular. However, this approach allows us to eliminate the choice of templates as a
+variable, or uncertainty, in our analysis. This is motivated by the exceptional performance that
+has already been achieved with the same templates coupled with the exquisite observational
+data in COSMOS. This strongly suggests to us that the choice of templates is unlikely to be the
+limiting factor with the degraded photometry that we can realistically expect to have over the
+whole sky within the timescale of a decade or so.
+20.3 The Near Infra-Red Filter Configuration
+InAbdalla et al. (2008a ) we investigated the effect of adding deep NIR photometry to the
+visible bands we can expect from upcoming ground based wide field imaging surveys, namely
+DES, PanSTARRS and LSST. We find that there is a dramatic improvement in the redshift
+measurements when NIR photometry is added (see Figure 20.1). Further to this study we have
+varied the total exposure and find that we are able to meet our requirements of σ(z)<0.05(1+z),
+after cleaning, for the depths shown in Table 20.1.
+We have also studied the effect of choosing different combinations of infrared filters (see20.3. The Near Infra-Red Filter Configuration 186
+Band Survey-A Survey-B Survey-C
+g 24.66 25.53 26.10
+r 24.11 24.96 25.80
+i 24.00 24.80 25.60
+z 22.98 23.54 24.10
+y 21.52 22.01 22.50
+Euclid NIR
+Y 24.00 24.00 24.00
+J 24.00 24.00 24.00
+H/K 24.00 24.00 24.00
+Table 20.1: The assumed filter sensitivities for different survey configurations considered. The
+values quoted here are 5 σerrors in AB magnitude. Survey-A corresponds roughly to an optimistic
+PanSTARRS-1 like survey, Survey B is designed to correspond roughly to a DES/PanSTARRS-2
+like survey and Survey C corresponds to depths achievable with LSST/PanSTARRS-4.
+Figure 20.1: The expected photo-z performance for DES, PanSTARRS-4 and LSST. We show
+here the impact of combining these surveys with deep NIR coming from Euclid. We see that
+adding the NIR bands drastically improves the performance. Details of these calculations can be
+found in Abdalla et al. (2008a ).20.4. Ground-based photometry 187
+0.81  1.21.41.61.800.20.40.60.8
+0.811.21.41.61.8
+x 10−600.20.40.60.8
+0.811.21.41.61.8
+x 10−600.20.40.60.8
+0.811.21.41.61.8
+x 10−600.20.40.60.8
+JH J J
+J1 J2
+HH
+HY
+YY
+Figure 20.2: Four different NIR filter configurations considered for Euclid. These photo-z
+performance for each of these was investigated. For a fixed total exposure time we find that
+comparable results are achieved, with the configuration in the top right giving marginally better
+performance.
+Figure 20.2), trading off the number of filters with the exposure time in each filter so as to
+preserve the total exposure time, and using the Euclid exposure time calculator1(Fontana
+et al. 2006), see chapter 21. It is found that the combination shown in the top right of Figure
+20.2 produces marginally better results and this three-filter combination has been adopted as
+the default NIR configuration for Euclid. However, the trade-off is rather flat, and the photo-z
+performance is determined primarily by the total available exposure time for the near-infrared
+photometry, integrated across the available filters.
+20.4 Ground-based photometry
+20.4.1 Effect of depth of ground based photometry
+As noted above, the basic performance of different photo-z codes is similar. The main challenge in
+reaching the Euclid requirements is in “cleaning” the photo-z of catastrophic failures, i.e. objects
+for which the photo-z is widely discrepant from the real redshift. This must of course be done a
+priori , i.e. without knowledge of the actual redshift from the photometry alone, and inevitably
+results in the loss also of objects with good photo-z. Increased depth in the photometry leads to
+a reduction in the percentage of these outliers and a reduction in the the associated “wastage” of
+good photo-z, as well as an increased precision of individual photo-z.
+InBordoloi et al. (2009) we examined the photo-z performance using the photometric
+catalogues degraded to simulate different surveys as listed in Table 20.1. In Figure 20.3 we show
+the r.m.s. spread ( σz(z)) and the bias (∆ z(z)) for the different survey configurations. The blue
+curves in all the panels give the initial performance of the photo-z code, without attempting to
+remove outliers. As expected, increased depth in the optical ground-based photometry increases
+1http://lbc.oa-roma.inaf.it/cgi-bin/ETC DUNE.pl20.5. Characterization of N(z) 188
+the reliability of the photo-z estimates. However, we always need to clean the photo-z catalogues
+to meet the requirement of σz(z)/(1 +z)≤0.05, especially at the lower redshifts z∼0.5 where
+many the galaxies in fact lie. The green curves show the effect of removing outliers, recognised
+purely photometrically, while the red curve shows the effect of modifying the individual L(z)
+(discussed further below). All these procedures are described in detail in Bordoloi et al. (2009).
+After the a priori removal of doubtful photo-z, the errors in σz(z) and mean bias ∆ z(z) are
+dramatically reduced, as shown by the green lines in Figure 20.3. This is summarised in Table
+20.2. The large improvement in σz(z) and ∆ z(z) comes from rejection of catastrophic failures
+rather than a tightening of the good photo-z. Furthermore, as the depth of the photometry
+increases, it is found that fewer objects need to be rejected to improve the photo-z estimates. In
+the case of survey-A, we find that 23% must be rejected to get below σz(z)≤0.05(1 +z), for
+survey-B it is 12% and for survey-C, only 9%. The trade off between beneficial cleaning and
+the wasteful loss of objects determines the efficiency of the cleaning process. After the above
+cleaning has been performed, the fraction of 5 σoutliers (catastrophic failures) is reduced below
+1% in all the three cases Table 20.3.
+/angbracketleftσz(z)
+1+z/angbracketrightfor different surveys in the range 0 .3≤z≤3.0
+Survey Before Cleaning After Cleaning After Cleaning + Correction
+Survey-A 0.1703 0.0884 0.0675
+Survey-B 0.1164 0.0640 0.0497
+Survey-C 0.0876 0.0492 0.0398
+Table 20.2: The /angbracketleftσz(z)
+1+z/angbracketrightfor the three surveys studied. After cleaning and correction has been
+performed survey-B just about reaches σz(z)/(1 +z)∼0.05 Euclid requirements.
+Percentage of 5 σoutliers (fcat) in
+Survey Before Cleaning After Cleaning
+Survey-A 1.18 0.2300
+Survey-B 0.8820 0.2138
+Survey-C 0.8221 0.1776
+Table 20.3: The percentage of 5 σoutliers in various surveys studied. fcatreduces significantly
+once cleaning of the catalogue is performed, which identifies most of the outliers effectively.
+20.5 Characterization of N(z)
+20.5.1 Direct Sampling approach
+One approach to determining N(z) and /angbracketleftz/angbracketrightis to obtain spectroscopic redshifts for a subset of
+the objects in a given photo-z bin. As only a finite number of galaxies are available to determine
+this distribution, there will be uncertainties associated with its estimate, which will propagate
+into uncertainties in cosmological parameter estimates using weak lensing ( Amara & R´ efr´ egier
+2007;Kitching et al. 2008b ).20.5. Characterization of N(z) 189
+0.050.10.150.20.250.30.35σz(z) /(1+ zphot)
+  
+Survey−ABefore Cleaning
+After Cleaning
+After Cleaning+Correction
+0.5 1 1.5 2 2.5 3−0.0500.05∆z(z) /(1+ zphot)
+zphot
+0.050.10.150.20.250.30.35σz(z) /(1+ zphot)
+  
+Survey−BBefore Cleaning
+After Cleaning
+After Cleaning + Correction
+0.5 1 1.5 2 2.5 3−0.0500.05∆z(z) /(1+ zphot)
+zphot
+0.050.10.150.20.250.3σz(z) /(1+ zphot)
+  
+Survey−CBefore Cleaning
+After Cleaning
+After Cleaning + Correction
+0.5 1 1.5 2 2.5 3−0.04−0.0200.020.04∆z(z) /(1+ zphot)
+zphot
+Figure 20.3: The overall performance of several surveys investigated whose depths are as quoted
+in Table 20.1. The blue lines give the performance without cleaning and the green lines after
+cleaning and the red line gives performance after cleaning and applying correction. For survey-A
+23% for survey-B 13% and for survey-C 9% rejections were made after cleaning.20.5. Characterization of N(z) 190
+z-range µ2µ4Nsfrom mean Nsfrom variance
+0-0.561 0.013 0.0006 3300 7
+0.561-0.741 0.02 0.001 5000 9
+0.741-0.931 0.036 0.0035 8800 30
+0.931-1.181 0.06 0.008 15000 70
+1.181-3.531 0.14 0.07 35000 800
+Table 20.4: The second (variance) and fourth (kurtosis) moments about the mean of the redshift
+distributions in each of the redshift bins and resulting requirements on the size of the spectroscopic
+training set, Ns.
+The two other photometric redshift statistics required for a Euclid weak lensing survey, are
+calibration of the mean of the redshift probability distribution, pi(z) to be less than 0 .002(1 +z),
+and the calibration of variance of pi(z), to be less than 0 .008.
+If we define the mean of the distribution as zand thekthmoment of the distribution, µk, as
+only a finite number of galaxies with spectroscopic redshifts, Nsare available to determine this
+distribution, assuming Poisson statistics, the standard deviation on the mean redshift of each
+bin is given by:
+rms(z) =√µ2√Ns(20.1)
+and similarly the standard deviation of the variance is given by, in the limit of large Nsthis
+becomes:
+rms(µ2) =/radicalbig
+µ4−µ2
+2/radicalbig
+Nspec(20.2)
+One can marginalise over both these uncertainties. The dependence on the number of spectroscopic
+redshifts is really an indirect expression of the scatter in zand inµ2. Using the simulations
+from the previous section the number of spectroscopic training set galaxies required in each
+redshift bin can be calculated. Estimates of Nsfrom equations 20.1 and20.2 are given in Table
+20.4assumingrms(z)<0.002 andrms(µ2)<0.008. As the requirements on the spectroscopic
+training set is roughly the same for any optical survey with Euclid NIR photometry, an average
+value forNsis quoted in Table 20.4across all the surveys.
+The requirements for the calibration of the mean are clearly more demanding than those
+for calibrating the variance and it can be seen that a total of ∼105spectroscopic training set
+galaxies are required for calibration of a weak lensing survey such as Euclid assuming five redshift
+bins ( Abdalla et al. 2008a ).
+Such direct spectroscopic determination of N(z), and it’s moments, requires impressively
+complete spectroscopic redshift determination on very large numbers of faint galaxies. A further
+complication is the need to have a large number of independent survey fields in order to limit
+the effects of cosmic variance in the N(z) within a particular survey field. This aspect is studied
+inBordoloi et al. (2009), by requiring that the effects of large scale structure in the galaxy
+distribution, also known as cosmic variance, are at most equal to the Poisson noise (considered
+above) in determining the error in /angbracketleftz/angbracketright. We approach this by determining, from the COSMOS
+mocks described above, how many galaxies can be observed within a given spectrograph field of
+view before the variation in N(z) from field to field becomes dominated by the effects of large20.5. Characterization of N(z) 191
+Figure 20.4: The minimum number of independent fields required (as a function of individual
+survey field size) in order to ensure that the uncertainty in the mean redshift of a given redshift
+bin is not dominated by large scale structure within the survey fields (i.e. by cosmic variance),
+and is equal to the Poisson variance from 104galaxies. Typically, the number of required fields
+is set by the lower redshifts where the effects of large scale structure are strongest.
+scale structure in the Universe. This then translates to a maximum on the “sampling rate” in
+each spectrograph field of view, and a minimum number of independent survey fields. Both will
+depend on the spectrograph field of view. For typical spectrographs on 8-m class telescopes, such
+as VIMOS on the ESO VLT, this maximum sampling rate is rather small (e.g. 2%), with an
+implied inefficiency in telescope utilization. The minimum number of independent fields that will
+be required in order to attain an uncertainty in the mean redshift of this particular redshift bin
+/angbracketleftz/angbracketrightthat is equivalent to the Poisson variance from 104galaxies is shown in Figure 20.4.
+20.5.2 Estimation of N(z) from photo-z themselves
+Generally a single redshift estimator from the photo-z code (i.e. the maximum likelihood photo-z)
+is used to construct the photo-z bins. Not surprisingly, using these same redshifts, the ∆ /angbracketleftz/angbracketright
+requirement cannot be reached, as clearly shown in Figure 20.3, because the maximum likelihood
+redshifts cannot, by construction, then trace the wings of the N(z) that lie outside of the nominal
+bins, or sample the remaining catastrophic failures which produce distant outliers in the N(z).
+Therefore a more sophisticated approach is required. We find ( Bordoloi et al. 2009) that we can
+characterise N(z) as the sum of the individual L(z) likelihood functions for the galaxies in each
+redshift bin. These individual normalised L(z) are output for each galaxy by the photo-z code.
+The straight sum of the original likelihood functions is already able to characterise the redshift
+distribution well, as seen in Figure 20.5, which shows for survey-C the summed L(z) traces
+(visually) both the the catastrophic failures and the wings of the redshift bins. However, we find
+that this approach must be further modified so as to characterise the N(z) of the bins to the
+required precision. We do this by modifying the individual L(z) using an approach described in
+detail in Bordoloi et al. (2009). In outline, this approach is based on a fundamental statistical
+requirement on the L(z). Defining Pito be the cumulative position within an individual likelihood20.6. Internal Calibration of Galactic Foreground Extinction 192
+0123400.20.40.60.811.21.41.6 Normalized ΣL(z)
+zreal01234
+zreal
+Figure 20.5: N(z) constructed from the/summationtextL(z) function before and after cleaning. Here the
+normalised histogram gives the real redshift distribution in the bin and line is the N(z) constructed
+from the/summationtextL(z) function. The left panel gives the redshift bin before cleaning and right panel
+gives after cleaning. The constructed N(z) clearly traces the catastrophic failures.
+functionLi(z) of the real “spectroscopic” redshift, zi, then the distribution of Piacrossany
+subset of galaxies should be flat across 0 < P < 1. We can therefore use a limited number
+of spectroscopic redshifts, in principle for any subset of galaxies, to determine an empirical
+correction to L(P), that we then apply to all galaxies. This simple ad hoc approach works to
+bring us within the required precision. In Figure- 20.6the bias on the mean of the N(z) is given for
+different redshift bins. The shaded region gives the Euclid requirement of |∆/angbracketleftz/angbracketright/(1 +z)| ≤0.002
+on the mean redshift of the redshift bins. The black dots are for survey-C, which easily reaches
+the Euclid requirements. The red open boxes are for survey-B and it just meets the Euclid
+requirement. The blue stars are for survey-A which do not meet the specifications as given by
+the shaded region. From this analysis we conclude that for a Euclid like survey, using a survey-B
+like ground based complement(similar to PanStarrs-2 or DES) we can characterise the N(z) of
+the tomographic bins to a precision of |∆/angbracketleftz/angbracketright/(1 +z)| ≤0.002 using around 800–1000 random
+spectroscopic redshifts per redshift bin.
+20.6 Internal Calibration of Galactic Foreground Extinction
+Extragalactic photometry is routinely corrected for the effects of foreground Galactic extinction
+using reddening maps and an assumed extinction curve. In practical terms, the effect that we
+should therefore worry about is an error or uncertainty in the AV, i.e. a ∆Av, which may be
+positive or negative. This will cause galaxies to be either too red, or too blue, in the photometric
+input catalogue and will likely lead to systematic biasses in their photo-z, the sign of which may
+change with redshift. However, the photometry of the galaxies themselves can, in principle, be
+used to determine this error in the Galactic reddening, leading to an internal correction of the
+problem.
+This is studied in Bordoloi et al. (2009) using a mock catalogue containing 104objects
+down toIAB∼24.5, mimicing a roughly 0 .1 deg2region of the Euclid survey. We consider20.7. Impact on photo-z from blended objects at different redshifts 193
+∆〈 z〉
+zreal  
+0.40.60.811.21.41.61.822.2−0.02−0.015−0.01−0.00500.0050.010.0150.02
+Survey−C
+Survey−B
+Survey−A
+Figure 20.6: The bias in the mean of the tomographic bins estimates from the normalised/summationtextL(z)
+functions for all surveys investigated. For survey-C, with cleaning for catastrophic failures and
+after applying correction gives |∆/angbracketleftz/angbracketright/(1 +z)| ≤0.002. Here the shaded region gives the Euclid
+requirements. We have introduced a small offset in x-axis values of survey-B and survey-C for
+legibility.
+photometry with the accuracy expected from both survey-A and survey-C, and then perturb these
+catalogues by applying a standard mean reddening law ( Cardelli et al. 1989) with a relatively
+large ∆Av∼0.155, in both the positive and negative directions. We then compare the photo-z
+for the galaxies with and without these ∆ AVoffsets. With this quite large assumed error, the
+effects on the photo-z are much larger than can be tolerated. Furthermore, the effect varies with
+redshift, making it very difficult to isolate this effect from cosmological effects.
+However, we find that the photometry of the galaxies can be used to determine ∆ Avusing
+a simple chi-squared minimization scheme. This works better, as expected, for galaxies with
+relatively high S/N photometry, and if the redshifts of the galaxies are known. To estimate
+the effect of photometric noise in estimating ∆ Avin this internal way, we consider objects in
+magnitude bins in IAB(see Figure- 20.7). We find that it is worth using only galaxies with
+relatively high S/N photometry, i.e. with the adopted survey parameters, down to IAB∼22.
+Below this level, the estimate degrades appreciably. The addition of spectroscopic redshift reduces
+the error bar significantly, but the method is still practicable down to the same magnitude limit
+and yields an error on ∆ AVof order 0.01 (with known redshifts) or 0 .02 (without). This is
+sufficient to bring the effects of uncertain Galactic reddening within the precision requirements
+for Euclid.
+20.7 Impact on photo-z from blended objects at different red-
+shifts
+Sometimes multiple galaxies will overlap on the sky, and the photometry will be a composite
+of the two spectral energy distributions. Even with spectroscopy, such objects have composite
+spectra rendering even their spectroscopic redshift estimation non-trivial. In this final section20.8. Conclusions 194
+20 21 22 23−0.125−0.105−0.095−0.075−0.055−0.035−0.0150.0050.0250.0450.065
+IAB∆ Av
+  
+Using Spectra
+Using Photo−z
+20 21 22 230100200300400500600700800900
+IABNumber of objects per magnitude bin
+Figure 20.7: The left panel gives estimates of ∆ Avfor different magnitude bins (in the con-
+figuration of Survey-C). The red line is obtained using internally computed photo-z, without
+knowledge of the redshifts of the galaxies, and the blue line is using spectroscopic information.
+Here we have introduced a small offset in x- axis on the red line for legibility. The right panel is
+the number of objects per magnitude bin. With spectra, at IAB∼22 magnitude bin around 350
+spectra are sufficient to estimate ∆ Avaccurately.
+we explore the effects of this blending on photo-z estimates. We simulate many such blended
+objects by constructing composite spectral energy distributions constructed from galaxies at
+different redshifts, different colours and a wide range of relative brightnesses, from dominance of
+one through to dominance of the other. For definiteness we look at the photo-z behaviour for a
+survey-C like depth.
+Our general conclusion from this analysis is that the effect on the photo-z of blended objects is
+complex and varies in a rather unpredictable way from pair-to-pair. The photo-z are trustworthy
+only if the second component is at least two magnitudes fainter than the primary (in a waveband
+close to the middle of the spectral range of the photometry). At smaller magnitude differences
+the photo-z can be corrupted in a way that is not always recognizable. Sometimes the best fit
+photo-z varies smoothly between the two real redshifts, sometimes it jumps between the two,
+and sometimes there is an additional local maximum in between. The conclusion is that these
+blended objects should be recognised morphologically from the images, and excluded from the
+analysis. This is unlikely to be a problem, since the shape measurement of these objects would
+anyway be problematical.
+20.8 Conclusions
+In this chapter we have investigated a number of issues that could potentially impact the photo-z
+performance of deep all-sky surveys such as Euclid. In each case, we have found that the most
+basic approaches will not meet the demanding specifications. However, simple and logically
+motivated developments to these exiting techniques, yield a performance that meets the Euclid
+specification. Specifically, the following conclusions can be drawn from this photometric redshift
+analysis:20.8. Conclusions 195
+Figure 20.8: L(z) functions of 42 pairs of blended objects. Here when the low redshift object
+is brighter than the high redshift one and the likelihood function traces low redshift function.
+The degeneracy of the likelihood functions at ∆ m∼0 means that the redshift estimation gets
+completely unreliable at that region. Here the objects are taken for survey-C like depth.
+•We have studied the redshift performance of a number of ground based surveys with and
+without the space based Near-IR that Euclid is able to deliver and find that Near-IR
+photometry is crucial for securing reliable redshifts. The example in Table ( 20.2) shows
+the functioning point that sets our requirements.
+•For a fixed integration time, we find that the photometric redshift estimate is not very
+sensitive to the details of near infra-red filter configuration. Optimising this configuration
+points to a three filter IR survey complemented by a multi band optical survey from the
+ground.
+•Photo-z accuracy of σz(z)/(1 +z)∼0.05 down to IAB≤24.5 is achieved by combination
+of ground-based photometry with depth of Survey-B (similar to PanStarrs-2 or DES) and
+the deep all-sky NIR survey from Euclid itself. This requires the implementation of an a
+priori rejection scheme (i.e. based on the photometry alone, without knowledge of the
+actual redshifts of any galaxies) that rejects 13% of the galaxies and reduces the fraction of
+5σoutliers to below fcat<0.25%. There is a trade off between the rejection of outliers and
+the loss of innocent galaxies with usable photo-z. Deeper photometry improves both the
+statistical accuracy of the photo-z and reduces the wastage in eliminating the catastrophic
+failures, and the combination of Survey-C (similar to PanStaars-4) with Euclid near-infrared
+photometry achieves σz(z)/(1 +z)∼0.04 after 9% rejection.BIBLIOGRAPHY 196
+•If spectroscopic redshifts are to be used for the “direct” measurement of N(z), then ∼105
+redshifts are required, with high completeness and reliability to meet the required precision
+on the mean redshift. Furthermore, these spectroscopic redshifts must be obtained from a
+large number of sparsely sampled fields or else the error on /angbracketleftz/angbracketrightwill be dominated by the
+effects of large scale structure in the spectroscopic fields.
+•Many of the difficulties of the direct spectroscopic determination of N(z) can be circum-
+vented by using the photo-z themselves. The/summationtextL(z) function characterises both the
+wings of the N(z) and the remaining catastrophic failures. However, to reach the required
+performance on the mean of the redshifts |∆/angbracketleftz/angbracketright| ≤0.002(1 +z) with the Survey-B, or
+deeper Survey-C, combination (together with Euclid infrared photometry), an adaptive
+modification scheme on the individual L(z) is implemented. This is based on the spectro-
+scopic measurement of redshifts for a rather small number of galaxies ( ≤1000) with relaxed
+requirements on statistical completeness (and no dependence on Large Scale Structure in
+the spectroscopic survey fields).
+•Uncertainties in foreground Galactic reddening can have a serious effect in perturbing the
+photo-z. However, such errors in Avcan be identified internally from the photometric data
+of galaxies, either with or without spectroscopic redshifts. This procedure is optimum for
+galaxies with relatively high S/N photometry IAB≤22. The required number of galaxies
+suggests that a reddening map on the scales of 0 .1 deg2can be internally constructed
+from the data on galaxies without known redshifts, or from a few hundred galaxies with
+spectroscopic redshifts.
+•The photo-z of composite objects in which the photometry is a mixture of two objects
+at different redshifts behaves in a complex and unpredictable way. The photo-z of the
+composite SED is a good representation of the redshift of the brighter object as long as the
+magnitude difference is large, i.e. ∆ IAB>2.When the galaxies are closer in brightness,
+∆IAB<2, there is a wide-range of irratic behaviour. Our conclusion is that composite
+objects with ∆ IAB<2 should be recognised morphologically from imaging data and
+removed from the photo-z analysis.
+As a consequence of this and other work undertaken by the photometric redshift working
+group, the current imaging component design for Euclid consists of a broad optical RIZ filter
+as well as three near infra-red YJH bands. The general conclusion of our study is that while
+reaching the photo-z performance required for weak-lensing surveys such as Euclid will not be
+trivial, implementation of new techniques, coupled with internal calibration of (e.g. foreground
+reddening from the photometric data itself), will allow this performance to be attained. If more
+reliance can be placed in this way on the photo-z themselves, then this will lead to a major
+simplification of the otherwise challenging requirement for spectroscopic calibration of large scale
+photometric surveys.
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+127, 3121–3136.Chapter 21
+An Exposure Time Calculator for Euclid
+Authors: Andrea Grazian (IAF-OAR), Stefano Gallozzi (IAF-OAR), Adriano Fontana (IAF-
+OAR).
+Abstract
+We describe here the Exposure Time Calculator that has been developed for Euclid. It allows one
+to compute the predicted performances of Euclid under a variety of instrumental and observing
+conditions. The ETC is a Euclid-tailored version of one developed for the Large Binocular Camera
+at LBT, and is available at http://lbc.oa-roma.inaf.it/cgi-bin/calculateETC DUNE.pl
+21.1 Introduction
+The variety of science applications and instrumental configurations developed during the design of
+the Euclid mission are so large that it is necessary to have a dedicated tool that can estimate the
+Euclid predicted performances. To accomplish this task, a dedicated Euclid Imaging Exposure
+Time Calculator (E-ETC in the following) has been developed. The E-ETC is a modified version
+of the one already developed and operational at the Large Binocular Camera of LBT.
+The basic functions provided are typical of ETC’s, with some improvement over standard ones.
+While typical ETC’s allows the user to set the desired signal-to-noise (S/N) or an exposure time,
+the E-ETC allows to define two out of the three actual free parameters (for a given configuration):
+S/N, exposure time and magnitude limit. This flexibility allows the user to explore different
+combinations of parameters, and is very useful when designing a space mission like Euclid, where
+several constraints complicate the planned observations.
+In addition to this an implementation, specific to Euclid, is the possibility of changing several
+instrumental parameters. For instance, the user can create on-the-fly new filters of arbitrary
+bandwidth and efficiency, or chose among several instrumental configurations for electronics.
+In this section, we shall briefly discuss its use.
+21.2 Layout overview
+The front page of the ETC is organized in three panels shown in Fig. 21.1. The main upper panel
+summarises the input parameters, which should be necessary filled by the common user. In
+this section the user can change the instrumental parameters, including the selected detector
+(optical/IR).
+19921.2. Layout overview 200
+Figure 21.1: Overall layout of the front page of the E-ETC.
+The second (left) panel operates with the Total Exposure Time, the Signal to Noise ratio
+and the Magnitude of a given object. In this panel the user can set the global parameters of the
+observations.
+In case of observations resulting from the sum of different exposures, the details of the
+individual exposures can change the overall results. The details of the single exposure can be set
+in the third (right) panel, where the number of exposures, the background and the magnitude of
+saturation can be set.
+In the following, we shall describe the assumed formula that sets the results for the Total
+Exposure Time panel.21.3. Using the E-ETC 201
+21.3 Using the E-ETC
+The key formula used in the ETC is described in the following, and allows to compute the S/N
+when the total exposure time tand the total magnitude of an object Mare given. This is the
+reference formula for the ETC: the first and the third cases of this panel are derived by inverting
+this formula. To enhance the S/N it is possible to modify the total exposure time tand/or on
+the number of exposures n
+S/N =F/radicalbig
+(F+B∗A)/g+n∗A∗(ron/g )2+ Dark. (21.1)
+Fis the flux source (in ADUs), Bis the background, nis the number of individual exposures
+andAis the area of the collecting mirror.
+A full description of the mathematics involved can be found here in http://lbc.oa-roma.
+inaf.it/ETC/ETC help/index.html
+In output, the ETC provides the basic number that describe the observations. The funda-
+mental numbers (S/N, magnitude limits, exposure time and details of individual exposures) are
+shown in the first page, so that the user can easily iterate with few clicks. A more detailed input
+can be found in a separate page, where a simulated image - with the specified morphology and
+noise - is shown and can also be downloaded. (Fig. 21.3)
+It is also possible to see the final efficiency as a function of wavelength, both for filters only
+and including the detectors (Fig. 21.3).21.3. Using the E-ETC 202
+Figure 21.2: An example output page from the E-ETC.21.3. Using the E-ETC 203
+Figure 21.3: An example output page showing the efficiency as a function of wavelength.Chapter 22
+Cosmological Simulations
+Authors: Romain Teyssier (CEA Saclay, Univ. of Zurich), Alexandre R´ efr´ egier (CEA Saclay,
+France), Adam Amara (ETH Zurich), Marcella Carollo (ETH Zurich), Alan Heavens (Uni. of
+Edinburgh).
+Abstract
+In this section, we describe the work done by the EIC regarding cosmological simulations activities.
+N-body and gas dynamics simulations are great tools to compute theoretical expectations from
+the standard ΛCDM scenario, as well as for non-standard models. Given the required accuracy of
+the Euclid mission, these simulations are quite computationally demanding, and have to involve
+gas and galaxy formation physics. We outline recent progresses made by the EIC in running and
+analysing these challenging simulations.
+22.1 Introduction
+We have substantial evidence that dark matter is a collisionless, self-gravitating fluid. Its
+dynamical evolution is therefore described by the well understood Vlassov-Poisson equations,
+for which N-body models are known to capture the main bulk properties quite accurately. The
+challenge here is the dynamical range. Large-scale structure surveys such as Euclid will set very
+tight requirements on our N-body simulations. First, we need to cover a very large volume, in
+order to reproduce the same statistical sample of the forthcoming survey. In order to match
+the observational Universe, we need a box size of 6000 Mpc. Second, we need to describe the
+matter distribution down to very small scales, in order to achieve the angular resolution of those
+surveys (a few arcseconds). This requires a minimal dark halo mass of about 1011Solar masses.
+Since we need at least ∼100 particles to capture the formation of a dark matter halo, these
+requirements translate into a total number of particles in excess of 4 trillion. Obviously, we need
+the largest computers in the world in conjunction with highly optimised software to achieve this.
+Moreover, in order to reach the level of accuracy set by the low signal-to-noise galaxies that will
+be analysed in the the Euclid mission, numerical errors should controlled within a level lower
+than 0.1%, a rather strong requirement on existing N-body codes. Last but not least, at this
+level of precision, gas dynamics and galaxy formation physics, both relatively poorly known,
+should be also included. We now report the work done by our consortium in the last 3 years.
+20422.2. A 70 billion particle N-body simulation 205
+22.2 A 70 billion particle N-body simulation
+We have performed in 2007 the largest N-body simulation so far: it features 70 billion particles
+(Teyssier et al. 2009). This “grand challenge” simulation required deploying the RAMSES code
+(Teyssier 2002) on a totally new hardware in CCRT, the CEA supercomputing centre. This
+rather unique effort was performed in collaboration between the Horizon project and computer
+scientists of CEA and BULL. Many technical bottlenecks have been solved and the simulation was
+completed in 2 months time, for a total of 6 million CPU hours (1000 years). We have analysed
+this 70 billion particle simulations in the context of the Euclid experiment. We have used for
+that purpose a rather advanced Ray-Tracing strategy to compute a high-resolution Healpix map
+(G´ orski et al. 2005) of the weak-lensing convergence over the whole sky. We have used this
+map to test various map-making algorithms that will be applied within the EIC consortium to
+detect point sources and to reconstruct the matter density field. We have shown that one of the
+main challenges in the Euclid All-Sky survey will be dealing optimally with both the large-scale,
+quasi-Gaussian signal and the small-scale, highly non-linear and non-Gaussian fluctuations.
+22.3 A different strategy: many small simulations
+In a completely different spirit, Teyssier & Pires have simulated a large number of small
+simulations, in order to explore various cosmological parameters. This strategy is interesting
+because a reliable statistical sample of weak-lensing maps can be generated without having to
+rely on world-class supercomputers. N-body simulations are only needed to reliably estimate
+the highly non-Gaussian likelihood functions. This series of simulations was performed on the
+Grid5000 system, in collaboration with a group of computer scientists from ENS Lyon. This was
+part of the LEGO project, funded by the French ANR in 2005-2009, and the feasibility of the
+approach was demonstrated in Caniou et al. (2006). This large number of small simulations was
+used to study various aspects of Euclid data processing and cosmological parameters estimation
+(Pires et al. 2009).
+Using yet another approach Kiessling, Lynn, Taylor, Heavens & Peacock have run smaller-
+scale, fast simulations with halo replacement by NFW profiles (Lynn & Peacock, in prep),
+which provides greatly increased dynamic range at a very moderate cost. The method has been
+calibrated once and for all using one expensive, ultra-high-resolution simulation. This allows
+large numbers of reasonably accurate simulations to be run very fast, an essential requirement
+to provide good estimates of covariance properties of lensing statistics (Kiessling et al., in
+preparation).
+22.4 Introducing the baryons
+The other important work related to simulation activities is related to the effect of baryons on
+the predicted power spectrum. This has been investigated first by White (2004) and Zhan &
+Knox (2004). These authors showed that baryons decouple dynamically from dark matter at
+various scales within the halo virial radius. The hot halo gas, first, is known to be spatially
+extended, following the so-called “beta model”. This density profile differs significantly from
+the dark matter profile ( Navarro et al. 1997) and presents a core in the halo centre, while dark
+matter is cuspy. This translates into a deficit of power on intermediate scales (around a few
+arcminutes). The cold gas and the stars, finally, are more concentrated than dark matter on
+small scales. This translates into an excess of power at arcsecond scales. These early theoretical
+calculations have been confirmed by detailed cosmological simulations including galaxy formation22.5. Disc intrinsinc alignement 206
+physics ( Jing et al. 2006;Rudd et al. 2008). Note however that these numerical simulations, like
+most present day galaxy formation simulations, suffer from the so-called over-cooling problem:
+too much baryonic mass ends up in the very centre of the galaxy, so that the effect on the matter
+power spectrum is probably over-estimated by these simulations. It is therefore of paramount
+importance be able to understand these effects, parameterised them in a sensible way and fit the
+baryonic parameters on the high-quality data of the future high-redshift surveys. Recently, we
+have proposed a model to incorporate baryonic effects onto the matter power spectrum (Guillet
+et al, submitted). We have shown that a simple model based on two components (the halo and
+a compact stellar disc in the centre) can account for the total matter distribution with great
+accuracy. The stellar disc has two effects on the matter distribution: first, this central mass
+concentration causes the dark halo to adiabatically contract, so that the dark matter density
+profile is denser and steeper than the original distribution; second, in the very centre, the stellar
+disc dominates the total mass and therefore contributes directly to the weak-lensing signal. These
+effects can be captured using rather simple analytical models, such as the “halo model” ( Seljak
+2000), but the model’s parameters have to be calibrated on detailed galaxy formation simulations
+and/or direct observations of galaxy groups. The model predictions have to go beyond the simple
+2-points correlations function of the weak-lensing signal and incorporate the effect of baryons on
+the bi-spectrum and higher order statistics.
+22.5 Disc intrinsinc alignement
+The other important effect related to baryons physics is the orientation of stellar discs with respect
+to the surrounding large-scale matter distribution. The main hypothesis of the weak-lensing
+analysis is that background galaxies are randomly oriented, so that systematic alignment between
+these galaxies is the signature of gravitational lensing. Any intrinsic alignment of galactic discs
+with the underlying dark matter distribution can be a very serious systematic effect for future
+high-redshift surveys. On intermediate scale, such alignment is predicted by N-body simulations
+of dark matter halos ( Knebe et al. 2008). Surprisingly, however, observational results indicate
+that the orientations of galaxies are correlated up to scales of order 100 times larger than the virial
+radii of their host haloes. Mandelbaum et al. (2008),Okumura et al. (2009) and Faltenbacher
+et al. (2008) detect an alignment of galaxy shapes for the luminous red galaxies (LRGs) in the
+SDSS survey out to the largest probed scales (roughly 80 Mpc). Various effects related to gas
+physics can explain discs alignments on large scales. Ram pressure stripping of gaseous discs
+on the hot gas in filaments could be responsible for a systematic alignment of star-forming
+discs perpendicular to the satellite-central vector. These subtle effects have been explored in a
+detailed galaxy formation simulation by our group (Hahn et al., submitted). Analytic models
+based on the halo model have been also developed that also include prescriptions for small-scale
+satellite-central alignments ( Schneider & Bridle 2009), and that can be tested using this type of
+simulation.
+Bibliography
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+Figure 22.1: Effect of baryons on the total matter power spectrum for various central galaxy
+mass. We see a boost at small scale, from 1% around l=5000 to 50% at l=105(from Guillet et al.
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+210
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+arXiv:1001.0062v2  [hep-th]  18 Apr 2010FIAN/TD/29-09
+Frame-Like Action and Unfolded Formulation for
+Massive Higher-Spin Fields
+D.S. Ponomarev and M.A. Vasiliev
+I.E.Tamm Department of Theoretical Physics, P.N.Lebedev Physical Institute,
+Leninsky prospect 53, 119991, Moscow, Russia
+ponomarev@lpi.ru, vasiliev@lpi.ru
+Abstract
+Unfolded equations of motion for symmetric massive bosonic fields of any spin in
+Minkowski and ( A)dSspaces are presented. Manifestly gauge invariant action fo r a
+spins≥2 massive field in any dimension is constructed in terms of gau ge invariant
+curvatures.Contents
+1 Introduction 1
+2 Forms of definite degree 4
+2.1 General form of curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
+2.2 Field redefinition ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
+2.3 Formal solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
+2.4 Scaling gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
+2.5 Spinssystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
+3 Unfolded equations for massive fields of any spin 12
+3.1 Field pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
+3.2 Unfolded equations via mixing of forms of different degrees . . . . . . . . . . 14
+3.3 Dynamical fields and Singh-Hagen system . . . . . . . . . . . . . . . . . . . 17
+3.3.1 Vanishing of lower-spin supplementary fields . . . . . . . . . . . . . . 1 8
+3.3.2 Dynamical equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
+3.4 Spin 2 example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
+3.5 Massless and partially massless fields . . . . . . . . . . . . . . . . . . . . . . 22
+4 Action 26
+4.1 Action derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
+4.2 Spin 2 example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
+4.3 Massless and partially massless limits . . . . . . . . . . . . . . . . . . . . . . 3 1
+5 Conclusion 32
+1 Introduction
+Although elementary particles of spins higher than two are unlikely to be directly observed
+in the modern high-energy experiments, there are several reaso ns to believe that they play a
+fundamental role at ultra high energies relevant to quantum gravit y. Indeed, infinite towers
+ofmassive higher-spin (HS) fields areimportant for consistency of String Theory [1, 2, 3]. On
+the other hand, existence of nonlinear theories of massless HS field s with unbroken HS sym-
+metries indicates new remarkable structures underlying a theory o f fundamental interactions
+(for recent reviews on HS theory see, e.g., [4, 5, 6, 7, 8, 9]). To un cover relation between the
+two types of theories and, more generally, to understand a spont aneous breakdown mecha-
+nism for HS symmetries that gives rise to a theory of HS massive fields , it is important to
+extend the approach, which underlies nonlinear massless HS theorie s, to massive HS fields
+starting from the linearized level. This is the goal of this paper.
+TherearetwoapproachestoHSfields. Themetric-likeapproach[1 0,11,12,13,14,15,16]
+generalizes metric formulation of gravity. The frame-like formalism [ 17, 18, 19, 20, 21, 22],
+that generalizes Cartan formulation of gravity, deals with different ial forms. It reproduces
+the metric-like formulationina particular gaugeandis most appropria teto control gaugeHS
+symmetries and interactions [23, 24, 25]. The study of the frame-lik e approach in the context
+1of massive HS fields was initiated by Zinoviev who constructed gauge in variant actions in
+terms of the frame-like fields in [22] where, however, the gauge inva riance of the action
+was not manifest, requiring the adjustment of coefficients in the ac tion from the condition
+of gauge invariance. In this paper we propose the manifestly gauge invariant action for
+symmetric massive HS fields formulated in terms of gauge invariant HS curvatures. The
+same time, these curvatures underly the construction of unfolde d formulation of free massive
+HS field equations which is also given in this paper.
+Free equations for symmetric massive fields of any spin are1[26]
+(∂n∂n+m2)φa(s)= 0, ∂nφna(s−1)= 0, φn
+na(s−2)= 0. (1.1)
+Our aim is to reformulate the equations (1.1) in the unfolded form of g eneralized zero-
+curvature equations [27] (for a review see [7])
+Rα(x)def=dWα(x)+Gα(W(x)) = 0, (1.2)
+whered=dxµ∂µis the exterior differential and
+Gα(Wβ)def=∞/summationdisplay
+n=1fα
+β1...βnWβ1...Wβn(1.3)
+is built from the exterior product (which is implicit in this section) of diffe rential forms
+Wβ(x) and satisfies the compatibility condition
+Gβ(W)δLGα(W)
+δWβ≡0. (1.4)
+Here index αenumerates a set of differential forms, which may be infinite.
+Unfolding consists of the equivalent reformulation of a system of pa rtial differential equa-
+tions in the form (1.2), which is always possible by virtue of introducing enough (may be
+infinite towers of) auxiliary and Stueckelberg fields. The fields, that neither can be expressed
+intermsof derivatives of otherfields, norcanbegaugedaway arec alleddynamical .Auxiliary
+fieldsareexpressed via derivatives ofall ordersofthedynamical field φa(s).Stueckelberg fields
+are pure gauge with respect to algebraic shift symmetries. Differen tial conditions on the dy-
+namical fields imposed by the unfolded equations are called dynamical equations . (Note that
+the standard tool for the analysis of physical content of unfolde d equations is provided by
+theσ−-cohomology technics [28, 7].)
+Unfolded field equations (1.2) are manifestly invariant under the gau ge transformation
+with a degree pα−1 differential form gauge parameter εα(x) associated to any degree pα>0
+formWα
+δWα=dεα−εβδLGα(W)
+δWβ(1.5)
+because
+δRα=−RγδL
+δWγ/parenleftbigg
+εβδLGα(W)
+δWβ/parenrightbigg
+. (1.6)
+1For notation see Appendix A
+2The gauge transformations of 0-forms only contain the gauge par ametersεα1associated to
+1-formsWα1in (1.5):
+δCα0=−εα1δLGα0(W)
+δWα1. (1.7)
+In the unfolded dynamics approach, background Minkowski or ( A)dSgeometry is de-
+scribed by the 1-form frame field (vielbein) ea=ea
+µdxµand Lorentz spin-connection 1-form
+ωa,b=ωa,b
+µdxµ,ωa,b=−ωb,a, that obey the equations
+Ta≡dea+ωa,
+b∧eb= 0, (1.8)
+Ra,b≡dωa,b+ωa,
+c∧ωc,b−λ2ea∧eb= 0, (1.9)
+where−λ2is the cosmological constant. The vielbein ea
+µis nondegenerate and relates fiber
+(tangent) and base (world) indices. Eq. (1.8) is the zero-torsion c ondition that expresses the
+Lorentz connection ωa,bvia vielbein ea. The equation (1.9) then describes the AdSdspace
+with the symmetry algebra o(d−1,2) forλ2>0,dSdspace with the symmetry algebra
+o(d,1) forλ2<0 and Minkowski space for λ2= 0.
+Unfolded formulation brings together such important issues as coo rdinate independence
+due to using the exterior algebra formalism and manifest gauge invar iance (Stueckelberg
+in the massive case) that, controls a number of degrees of freedo m in the system. The
+unfolded dynamics approach made it possible to find a full nonlinear sy stem of equations for
+massless fields in four [24] and any [25] dimensions. Extension of the u nfolded description
+to the massive case should shed light on the mechanism of spontaneo us breakdown of HS
+gauge symmetries and, eventually, unfolded formulation of String T heory to establish a
+correspondence between the latter and HS theory.
+Our aim is to find an explicit form of the unfolded system of equations t hat contains a
+dynamical equation (1.1) on the dynamical field φa(s)plus constraints that express auxiliary
+fields in terms of φa(s). The unfolded form of massive field equations was worked out in [28]
+for the case of spin 0. General features of the unfolded descript ion of massive fields of any
+spin were analyzed in [29].
+Inthispaper wepresent theexplicit formoftheunfoldedequations forthevery particular
+case of spin smassive symmetric gauge fields in Minkowski and ( A)dSspace which, as we
+show, can be described by a set of 1-form connections and 0-form Weyl tensors valued in
+various Lorentz tensor representations described by two-row Y oung tableaux. In contrast to
+massless and partially massless HS fields and in agreement with the res ults of [29], gauge
+invariant curvatures for massive fields necessarily involve 0-forms . Using the gauge invariant
+HS curvatures that result from the unfolded formulation of massiv e fields we also present
+the manifestly gauge invariant actions for HS massive fields.
+The layout of the rest of the paper is as follows. Unfolded equations , that describe Weyl
+module of a spin smassive field, are derived in Section 2. Extension of these equations to
+1-form gauge potentials is given in Section 3. Manifestly gauge invaria nt action for massive
+HS fields is constructed in Section 4. Section 5 contains conclusions. Our conventions and
+some useful formulas are given in Appendices A and B, respectively.
+32 Forms of definite degree
+2.1 General form of curvatures
+The frame-like formulation of massless [17, 18, 19, 20] and partially m assless [21] fields op-
+erates with 1-forms valued in irreducible Lorentz tensor spaces de scribed by two-row Young
+tableaux (for more details on Young tableaux see, e.g., [7, 30]; the fa cts relevant to the
+consideration of this paper are given in Appendix A). Let Y(k,l) denote the space of tensors
+described bytwo-rowYoungtableauxwith kcellsinthefirstrowand l≤kcellsinthesecond
+row. In this section we analyze the most general linear unfolded equ ations that can be for-
+mulated in terms of p−formsWa(k),b(l)with some definite p, valued inV=/summationtext
+k≥l≥0⊕Y(k,l).
+Since the analysis is insensitive to p, it is kept arbitrary in this section.
+Unfolded equations formulated for differential forms of a definite d egree, valued in V,
+read as
+0 =Ra(k),b(l)=DWa(k),b(l)+Ga(k),b(l)(W,e), (2.1)
+whereDis the Lorentz covariant derivative
+DCab...=dCab...+ωa,
+cCcb...+ωb,
+cCac...+... . (2.2)
+From (1.9) it follows that
+D2Ca(k),b(l)=λ2(keaecCca(k−1),b(l)+lebecCa(k),cb(l−1)). (2.3)
+The background vielbein 1-form ea=dxµea
+µentersGa(k),b(l)(W,e) linearly in (2.1). Generally
+ea∧W...is a (p+1)-form valued in a reducible Lorentz module. Since vielbein carries o ne
+fiber index, the projection to irreducible components described by two-row Young diagrams
+gives
+lk⊗=σ1
++
+lk+1⊕σ2
++
+l+1k⊕σ1
+−
+lk−1⊕σ2
+−
+l−1k+... ,
+where tensors described by three-row Young tableaux are proje cted out. Hence, there exist
+four linearly independent operators built from the vielbein 1-form. σ1
++andσ2
++(σ1
+−and
+σ2
+−) increase (decrease), respectively, the lengths of the first and the second row by one.
+Operatorsσwill be sometimes referred to as “cell operators”.
+Thus, the curvature (2.1) can be represented in the form
+R(W) =DW+σ1
++(W)+σ2
++(W)+σ1
+−(W)+σ2
+−(W), (2.4)
+where the form of the cell operators is determined up to overall co efficients by the properties
+of traceless two-row Young diagrams
+σ1
+−(k,l)(Wa(k),b(l)) =f(k,l)(emWa(k−1)m,b(l)+l
+k−l+1emWa(k−1)b,b(l−1)m),(2.5)
+σ2
+−(k,l)(Wa(k),b(l)) =F(k,l)(emWa(k),b(l−1)m), (2.6)
+σ1
++(k,l)(Wa(k),b(l)) =g(k,l)(k+1)(eaWa(k),b(l)−k
+d+2k−2emWa(k−1)m,b(l)ηaa−
+4−l
+d+k+l−3emWa(k),mb(l−1)ηab+lk
+(d+2k−2)(d+k+l−3)emWa(k−1)b,b(l−1)mηaa),
+(2.7)
+σ2
++(k,l)(Wa(k),b(l)) =G(k,l)(l+1)(ebWa(k),b(l)−k
+k−leaWa(k−1)b,b(l)−
+−l
+d+2l−4emWa(k),b(l−1)mηbb−k(k−l−1)
+(k−l)(d+k+l−3)emWa(k−1)m,b(l)ηab+
++lk(d+2k−4)
+(k−l)(d+2l−4)(d+k+l−3)emWa(k−1)b,b(l−1)mηab+
++k(k−1)
+(k−l)(d+k+l−3)emWa(k−2)mb,b(l)ηaa−
+−lk(k−1)
+(k−l)(d+k+l−3)(d+2l−4)emWa(k−2)bb,b(l−1)mηaa). (2.8)
+To rule out the terms with Young diagrams Y(k,l) withl > k, that are zero, it is
+convenient to demand
+f(k,l) =F(k,l) =g(k,l) =G(k,l) = 0 for l>k (2.9)
+and
+f(n,n) =G(n,n) = 0. (2.10)
+The compatibility condition (1.4) for the equation (2.4) is
+D2+σ2= 0, σ=σ1
+−+σ2
+−+σ1
+++σ2
++. (2.11)
+Restriction of (2.11) to different types of Young tableaux yields the following conditions
+(σ1
+−)2= 0,(σ2
+−)2= 0,(σ1
++)2= 0,(σ2
++)2= 0, (2.12)
+{σ1
+−,σ2
+−}= 0,{σ1
+−,σ2
++}= 0,{σ2
+−,σ1
++}= 0,{σ1
++,σ2
++}= 0, (2.13)
+D2+{σ1
+−,σ1
++}+{σ2
+−,σ2
++}= 0. (2.14)
+The conditions (2.12) are trivially satisfied due to the antisymmetry o f the exterior product.
+The conditions (2.13) give the following constraints on the coefficient sF(k,l),G(k,l),f(k,l)
+andg(k,l)
+f(k,l−1)F(k,l)
+f(k,l)F(k−1,l)=k−l
+k−l+1:{σ1
+−,σ2
+−}= 0,
+G(k−1,l)f(k,l)
+G(k,l)f(k,l+1)=(k−l−1)(k−l+1)(d+k+l−2)
+(k−l)(k−l)(d+k+l−3):{σ1
+−,σ2
++}= 0,
+F(k+1,l)g(k,l)
+F(k,l)g(k,l−1)=d+k+l−3
+d+k+l−2:{σ2
+−,σ1
++}= 0,
+g(k,l+1)G(k,l)
+g(k,l)G(k+1,l)=k−l+2
+k−l+1:{σ2
++,σ1
++}= 0.(2.15)
+Note that only three of these conditions turn out to be independen t.
+52.2 Field redefinition ambiguity
+Eq. (2.4) has the freedom in the field redefinition
+W→˜W, Wa(k),b(l)=β(k,l)˜Wa(k),b(l), β(k,l)/ne}ationslash= 0, (2.16)
+which induces the following redefinition of the coefficients
+˜g(k,l) =g(k,l)β(k,l)
+β(k+1,l),˜G(k,l) =G(k,l)β(k,l)
+β(k,l+1),
+˜f(k,l) =f(k,l)β(k,l)
+β(k−1,l),˜F(k,l) =F(k,l)β(k,l)
+β(k,l−1).(2.17)
+For the future convenience of the analysis of variation of the actio n in Section 4 we fix this
+ambiguity by demanding the operators σ1,2
++be conjugated to σ1,2
+−up to some factors nand
+Nwith respect to the scalar product
+/an}bracketle{tψa(k),b(l)
+{p}|φa(k),b(l)
+{q}/an}bracketri}ht=/integraldisplay
+ǫl1...ldel5...eldψ{p}l1a(k−1),l2b(l−1)φ{q}l3a(k−1),l4b(l−1),(2.18)
+whereǫl1...ldis the totally antisymmetric tensor. The scalar product /an}bracketle{t...|.../an}bracketri}htis nonzero
+provided that the p-formψa(k),b(l)
+{p}andq-formφa(k),b(l)
+{q}havep+q= 4 and carry equivalent
+representations of the Lorentz group Y(k,l),k≥l≥1 (otherwise the r.h.s. of (2.18) does
+not make sense).
+We impose the following conditions
+f(k+1,l) =n(k,l)(d+k−2)k
+k+1g(k,l), k≥1,l≥0, (2.19)
+F(k,l+1) =N(k,l)(d+l−3)(k−l+1)
+(k−l)G(k,l), k≥1,l≥0,(2.20)
+where the coefficients n(k,l) andN(k,l) will be determined later on. For k≥1 andl≥1
+these conditions follow from
+/an}bracketle{tσ1
+−ψ{p}(k+1,l)|φ{q}(k,l)/an}bracketri}ht=n(k,l)(−1)p/an}bracketle{tψ{p}(k+1,l)|σ1
++φ{q}(k,l)/an}bracketri}ht, k≥1,l≥1,
+/an}bracketle{tσ2
+−ψ{p}(k,l+1)|φ{q}(k,l)/an}bracketri}ht=N(k,l)(−1)p/an}bracketle{tψ{p}(k,l+1)|σ2
++φ{q}(k,l)/an}bracketri}ht, k≥1,l≥1,(2.21)
+while fork≥1,l= 0 they are imposed by hand. Note that eq. (2.20) can be assumed t o
+be satisfied at k=l= 0 as well because F(0,1) =G(1,1) = 0 by (2.9), (2.10). However,
+the condition (2.19) cannot be imposed at k=l= 0 to relate f(1,0) andg(0,0), because it
+would imply that f(1,0) = 0 that is not necessarily true. Instead, it is convenient to impos e
+the condition
+f(1,0) =n(0,0)g(0,0). (2.22)
+Let us note that Eqs. (2.19), (2.20) give
+g(k,l+1)f(k+1,l)G(k,l)F(k+1,l+1)
+g(k,l)f(k+1,l+1)G(k+1,l)F(k,l+1)=N(k+1,l)n(k,l)
+N(k,l)n(k,l+1)(k−l)(k−l+2)
+(k−l+1)2.(2.23)
+6On the other hand, from (2.15) it follows that
+g(k,l+1)f(k+1,l)G(k,l)F(k+1,l+1)
+g(k,l)f(k+1,l+1)G(k+1,l)F(k,l+1)=(k−l)(k−l+2)
+(k−l+1)2. (2.24)
+So, the conditions (2.21) require
+N(k+1,l)n(k,l)
+N(k,l)n(k,l+1)= 1. (2.25)
+Once (2.25) is true, Eq. (2.23) is equivalent to the equation (2.24) wh ich is invariant under
+the transformations (2.17). This explains how it is possible to achieve the two conditions
+(2.19), (2.20) with the help of a single function β(k,l). After imposing (2.19), (2.20), the
+ambiguity due to rescaling (2.16) is fixed up to an overall rescaling with β(k,l) =const.
+2.3 Formal solution
+Let us introduce the following combinations of the coefficients
+h(k,l) =f(k+1,l)g(k,l), H(k,l) =F(k,l+1)G(k,l), (2.26)
+that remain invariant under the redefinitions (2.16) and satisfy the following two equations
+as a consequence of Eqs. (2.15)
+h(k,l−1)
+h(k,l)=k−l+1
+k−l+2d+k+l−2
+d+k+l−3,
+H(k+1,l)
+H(k,l)=d+k+l−2
+d+k+l−1k−l+1
+k−l+2.(2.27)
+The condition (2.14) gives
+h(k,l)d+2k
+d+2k−2−h(k−1,l)−H(k,l)d+2l−2
+(k−l)(d+k+l−3)−λ2= 0,
+h(k,l)d+2k
+(k−l+2)(d+k+l−3)+H(k,l)d+2l−2
+d+2l−4−H(k,l−1)−λ2= 0.(2.28)
+GivenH(k,l) andh(k,l),σis reconstructed by (2.19), (2.20), (2.26)
+f(k+1,l) =/radicalbigg
+n(k,l)(d+k−2)k
+k+1h(k,l), k>0, l≤k, (2.29)
+g(k,l) = sign(h(k,l))/radicalBigg
+k+1
+k(d+k−2)n(k,l)h(k,l), k>0, l≤k,(2.30)
+f(1,0) =/radicalbig
+n(0,0)h(0,0), g(0,0) = sign(h(0,0))/radicalbig
+h(0,0)/n(0,0),(2.31)
+F(k,l+1) =/radicalbigg
+N(k,l)(d+l−3)(k−l+1)
+k−lH(k,l), k>l ≥0,(2.32)
+7G(k,l) = sign(H(k,l))/radicalBigg
+k−l
+(d+l−3)(k−l+1)N(k,l)H(k,l), k>l ≥0.(2.33)
+The formal general solution of the equations (2.27) and (2.28) is
+h(k,l) =−(d+2l0−2)(k0−k)(d+k0+k−1)
+(d+2k)(k−l+1)(d+k+l−2)H(k0,l0)+
++(d+2k0)(k−l0+1)(d+k+l0−2)
+(d+2k)(k−l+1)(d+k+l−2)h(k0,l0)+
+−λ2(k0−k)(d+k0+k−1)(k−l0+1)(d+k+l0−2)
+(d+2k)(k−l+1)(d+k+l−2),(2.34)
+H(k,l) =(d+2k0)(l0−l)(d+l0+l−3)
+(d+2l−2)(d+k+l−2)(k−l+1)h(k0,l0)+
++(d+k0+l−2)(k0−l+1)(d+2l0−2)
+(d+k+l−2)(k−l+1)(d+2l−2)H(k0,l0)+
+−λ2(d+k0+l−2)(k0−l+1)(d+l0+l−3)(l0−l)
+(d+k+l−2)(k−l+1)(d+2l−2).(2.35)
+From Eqs. (2.34), (2.35) we observe that H(k,l) andh(k,l) withk≥l≥0 are determined
+by their values H(k0,l0) andh(k0,l0) at any point ( k0,l0) withk0≥l0≥0. That this should
+have happenfollowsfromformalconsistency ofthe“finitedifferen ce equations” (2.27), (2.28)
+in the integral variables kandl. Note that denominators in (2.34), (2.35) are nonzero for
+k≥l, which is the condition that the second row of a Young diagram is not lo nger than the
+first one.
+An important property of the solution (2.34), (2.35) is expressed b y
+Lemma. IfH(p,s) = 0 orh(s−1,p) = 0 for some pandsthenH(k,s) = 0∀kand
+h(s−1,l) = 0∀l
+H(p,s) = 0 or h(s−1,p) = 0 : ⇒H(k,s) =h(s−1,l) = 0∀k,l. (2.36)
+Proof.Suppose that H(p,s) = 0. Let us use (2.34), (2.35) to express all h(k,l) and
+H(k,l) in terms of H(p,s) andh(p,s) settingk0=pandl0=s. The first term in (2.34)
+is then proportional to H(p,s) and hence vanishes. The other terms contain a factor of
+(k−s+ 1) that is zero for k=s−1. Henceh(s−1,l) = 0. The second term in (2.35)
+is proportional to H(p,s) and hence vanishes. The other terms contain a factor of ( s−l)
+which implies that H(k,s) = 0. That (2.36) is true if h(s−1,p) = 0 is proved analogously.
+✷
+The solution (2.34), (2.35) is formal since it does not take into accou nt (2.10) which
+demands
+H(n,n) =h(n−1,n) = 0. (2.37)
+This can be treated as a boundary condition for Eqs. (2.27)-(2.28) . Generally, (2.37) is not
+true for (2.34), (2.35). It turns out that if we would use the formu la (2.34), (2.35) for all
+handHexcept for those given by (2.37), the compatibility condition (2.11) w ill not be
+satisfied only in the sector of rectangular Young diagrams
+/parenleftbig
+D2+σ2/parenrightbig
+|W/ne}ationslash= 0,/parenleftbig
+D2+σ2/parenrightbig
+|V/W= 0, (2.38)
+8whereWdenotes the set of operators that map rectangular Young diagra ms to rectangular
+ones. For example, σ1
+−(n,n−1)σ2
+−(n,n)∈ Wbecause it maps Y(n,n) toY(n−1,n−1).
+In the general case, the equations (2.1) contain Wa(k),b(l)with allk≥l.H(k,l) and
+h(k,l) are defined by the two-parametric solution (2.34), (2.35). In this case the boundary
+condition (2.37) is not satisfied. Hence the curvatures (2.1) are inc onsistent for generic
+coefficients H(k,l) andh(k,l) that should be chosen appropriately to describe a spin sfield.
+2.4 Scaling gauges
+The coefficients h(k,l) (2.34) and H(k,l) (2.35) can be chosen to be real. In order the
+coefficients f(k,l),g(k,l),F(k,l) andG(k,l) also be real we demand
+sign(n(k,l)) = sign(h(k,l)),sign(N(k,l)) = sign(H(k,l)) (2.39)
+in(2.19), (2.20). The signs in(2.29), (2.32)arechosen so that f(k,l) andF(k,l)arepositive.
+In the practical analysis we will deal with differential forms of degre es 0 and 1 and use
+the two ways of fixation of the field rescaling ambiguity. The type-I scaling gauge is
+n(0)(k,l) =n(1)(k,l) = sign(h(k,l)), N(0)(k,l) =N(1)(k,l) = sign(H(k,l)),(2.40)
+wherethesubscripts(0)or(1)indicateadegreeofthedifferentia l forminquestion. Although
+the type-I scaling is most convenient for the general analysis, it is in applicable to the analysis
+of the flat massless limit where it should be replaced by the type-II scaling gauge
+N(1)(k,l) = sign(H(k,l))/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
+m2+λ2(s−l−1)(s+l+d−4)/vextendsingle/vextendsingle/vextendsingle/vextendsingle, n(1)(k,l) = sign(h(k,l)),
+(2.41)
+N(0)(k,l) = sign(H(k,l)), n(0)(k,l) = sign(h(k,l))/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
+m2+λ2(s−k−2)(s+k+d−3)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
+(2.42)
+It is easy to check that (2.41) and (2.42) respect the condition (2.2 5). SinceN(1)(k,l) is
+k-independent and n(0)(k,l) islindependent, we introduce notation
+N(1)(l) =N(1)(k,l), n (0)(k) =n(0)(k,l). (2.43)
+Note that
+N(1)(l) =−n(0)(l−1). (2.44)
+2.5 Spin ssystems
+It turns out that solutions of (2.34), (2.35), that satisfy (2.36) w ith somes, correspond to a
+spinsfield.
+From the condition h(s−1,l) = 0 and (2.26) it follows that either f(s,l) = 0 org(s−
+1,l) = 0. Consider the case where f(s,l) =g(s−1,l) = 0 for all l. It is easy to see that
+f(s,l) = 0 leads to σ1
+−(s,l) = 0, which means that the connections Wa(u),b(v)withu≥sdo
+not contribute to the curvatures Ra(k),b(l)(2.4) withk≤s−1. Analogously, g(s−1,l) = 0
+leads toσ1
++(s−1,l) = 0, which means in turn that Wa(k),b(l)withk≤s−1 do not contribute
+9Figure 1: Spin sfield, non-critical m/ne}ationslash=mt. Fields in distinct shaded regions can form
+independent unfolded systems (2.4)
+to the curvatures Ra(u),b(v)(2.4) withu≥s. So, if both f(s,l) = 0 and g(s−1,l) = 0
+are satisfied the respective curvatures do not mix Wa(u),b(v)withu≥sandWa(k),b(l)with
+k≤s−1. Hence, in this case Wa(k),b(l)withk≤s−1 (finite triangle region in Fig. 1)
+andWa(u),b(v)withu≥s(both infinite triangular and infinite rectangular regions in Fig. 1)
+form independent subsystems. Relaxing the conditions f(s,l) = 0 org(s−1,l) = 0 leads to
+the mixing of the fields Wa(k),b(l)withk≤s−1 andWa(u),b(v)withu≥s.
+The condition H(k,s) = 0 can be analyzed analogously. It is easy to see that in the case
+wheref(s,l) =g(s−1,l) =F(k,s+1) =G(k,s) = 0 for some sand allkandl, the shaded
+regions in Fig. 1 form independent subsystems that are not mixed in t he curvatures (2.4).
+The subsystem associated to the infinite rectangular region in Fig. 1 containsWa(k),b(l)
+withk≥s,l≤s. In the case of 0-forms ( p= 0), it describes a so-called Weyl module of
+a massive particle of spin sthat encodes nontrivial degrees of freedom of the system. The
+corresponding coefficients H(k,l) andh(k,l) are
+h(k,l) =−/parenleftbig
+m2+λ2(s−k−2)(s+k+d−3)/parenrightbig
+(k−s+1)(d+s+k−2)
+(d+2k)(k−l+1)(d+k+l−2),
+H(k,l) =−/parenleftbig
+m2+λ2(s−l−1)(s+l+d−4)/parenrightbig
+(s−l)(d+s+l−3)
+(d+2l−2)(k−l+1)(d+k+l−2),(2.45)
+where the free parameter midentifies with the mass as one can see by the direct analysis
+of the equations (2.4) for 0-forms Ca(k),b(l)withk≥s,l≤sanalogous to the analysis in
+the 1-form sector carried out in Section 3.3.2. Note that Eq. (2.45) provides an alternative
+parametrization of the general solution (2.34), (2.35) with sinterpreted as an arbitrary (not
+10necessarily integer) parameter.
+Figure 2: Partially massless field, m=mt. The set of fields can be divided into six groups
+Both finite ( k≤s−1) and infinite ( l>s) triangle regions in Fig. 1 do not respect (2.37).
+Nevertheless, in Section 3 we will show that the subsystem associat ed to the finite triangle
+plays important role in the description of massive fields, leading to con sistent unfolded
+equations (2.1) upon introducing an appropriate mixture between 0 - and 1-forms.
+It is easy to see that for special values m=mt(tis integer) in (2.45)
+m2
+t=−λ2(s−t−1)(d+s+t−4), t= 0,1,...,s−1 (2.46)
+the following property takes place
+H(k,t) =h(t−1,l) = 0. (2.47)
+Analogouslytothecondition(2.36), Eq. (2.47)subdivides thesyste mintofurthersubsystems
+as shown in Fig. 2. Since the region 4 does not contain fields valued in Y(n,n) the condition
+(2.37) here is inessential. The regions 2 and 5 both respect (2.37) as a consequence of
+(2.36) and (2.46). To summarize, the regions 2, 4 and 5 respect (2.1 1) leading to consistent
+curvatures, while in the regions 1, 3 and 6 the condition (2.37) is not r espected.
+Thecriticalvaluesofmass(2.46)correspondtopartiallymassless [3 1,13,21]andmassless
+[18, 19] fields for t<s−1 andt=s−1, respectively. Namely, the subsystem associated to
+the region 2 of Fig. 2 was used to describe partially massless fields in te rms of 1-forms in
+[21]. The region 5 describes the corresponding Weyl module.
+Note that letting t,s∈Rin Eqs. (2.45), (2.46) provides an alternative parametrization
+of the general solution (2.34), (2.35).
+11Also, from (2.45) it is easy to see that if m2is given by (2.46) with integer t,t>sand
+λ2/ne}ationslash= 0 then the Weyl module of such a field satisfies (2.47). In this case t he Weyl module
+(the infinite rectangular region in Fig. 1) admits a submodule. It cons ists of 0-forms Ca(k),b(l)
+withs≤k≤t−1. This phenomenon was observed for the case of massive spin 0 field in
+[28]. Such degeneracies usually indicate the coexistence of different dual descriptions of the
+same field theoretical system, and hence may be interesting from v arious perspectives.
+The following comment is now in order. It is well-known (see e.g. [7] and r eferences
+therein) that any consistent unfolded equation for a set of forms of a definite degree implies
+thatthissetspansamoduleoftheglobalsymmetryofthesystem, whichisPoincar´ eor( A)dS
+symmetry in the case of most interest in this paper. That the compa tibility conditions are
+not satisfied only at the boundary of the area of definition of Young tableaux suggests the
+following interpretation. Gauge fields of massless and partially massle ss fields are described
+byfinitedimensional tensor o(d−1,2)-modules T[19,21]. Inthiscase, theparameterofmass
+is discrete, being associated to the length of the second row of the respective Young tableaux
+[21]. These finite dimensional modules can be understood as particula r members of the set
+of modules Vmparametrized by the continuous parameter of mass m. More precisely, for
+the specific values miofmassociated to massless and partially massless fields, Vmiacquire
+infinite dimensional submodules Umisuch thatVmi/Umi=Tiare the finite dimensional
+modules used for the description of (partially) massless fields. That the unfolded equations
+are compatible almost everywhere except for the boundary of the region associated to the
+Young tableaux (i.e. finite dimensional representations of the Lore ntz group) confirms this
+picture. Most likely, the obstruction to the continuation of the con struction at the boundary
+of the Young tableaux region, shown in this section for generic value s of mass, signals that
+the corresponding infinite dimensional module Vmdoes not decompose into a sum of finite
+dimensional modules of the Lorentz algebra as is also anticipated for the submodules Umi. In
+other words, although the module Vmexists, it may admit no interpretation in the standard
+field-theoretical terms of finite-component Lorentz fields.
+3 Unfolded equations for massive fields of any spin
+3.1 Field pattern
+Unfolded equations (2.4) formulated in terms of 0-forms Ca(k),b(l)withk≥sandl≤s
+(infinite rectangular region in Fig. 1) are consistent. They describe the Weyl module of a
+massive HS field, that operates with gauge invariant combinations of gauge fields. To figure
+out how to introduce gauge potentials, it is useful to use the obser vation of [13, 22] that
+the Lagrangian of a spin smassive field can be represented as a sum of the Lagrangians
+for the set of spin s′, 0≤s′≤smassless fields supplemented with certain mass-dependent
+lower-derivative terms that mix massless fields of different spins. He nce, it is natural to
+expect that the unfolded formulation of a massive spin sparticle can be given in terms of
+fields of the unfolded formulation for the set of massless fields of sp ins 0≤s′≤s.
+As shown in [19, 20], a spin s′symmetric massless field is described by the set of 1-form
+connections valued in traceless tensors of o(d−1,1) that have symmetries of two-row Young
+12Figure 3: On this plot the set of fields used for description of a massiv e spinsfield is shown
+fors= 8.“×” denote 1-forms and “ ◦” denote 0-forms. The fields associated to each massless
+field areconnected by a solidline. The set of fields ofa massive spin smodel isa combination
+of the sets of fields of the massless unfolded systems of spins s′= 0,...s.
+tableaux with s′−1 cells in the first row and 0 ≤l≤s′−1 cells in the second row
+1-forms: ls′−1
+,0≤l≤s′−1
+and Weyl 0-forms valued in traceless tensors of o(d−1,1) possessing symmetries of two-row
+Young tableaux with k≥s′cells in the first row and s′cells in the second row
+0-forms: s′k
+, k≥s′.
+Thus the full set of fields appropriate for the description of a spin smassive field should
+contain 1-form connections valued in Y(k,l) withk≤s−1,l≤kand 0-forms valued in
+Y(k,l) withl≤s,k≥l. We observe that the resulting set of 1-forms precisely correspo nds
+to the triangular region of Fig. 3 while the set of 0-forms correspon ds to the unification of
+the infinite rectangular region with the triangular region of Fig. 3. In this section it will
+be shown how the compatibility problems in the triangular region encou ntered in the case
+of forms of definite degree are resolved by a mixture between 1-fo rms and 0-forms of this
+region.
+Note that the gluing the Weyl module to the combined set of 1- and 0- forms in the
+triangle is not necessary for formal consistency but is needed to r elate the 1-forms to the
+propagating degrees of freedom described by the 0-form Weyl mo dule associated to the
+infinite rectangular region.
+133.2 Unfolded equations via mixing of forms of different degre es
+General linear unfolded equations (1.2), (1.3) formulated in terms o f (p+1)-forms Wa(k),b(l)
+andp-formsCa(k),b(l), that possess symmetries of two-rowYoung tableaux, have thes tructure
+0 =Ra(k),b(l)
+{p+2}=dWa(k),b(l)+Ga(k),b(l)
+(p+1)(W,C) =dW+σ(p+1)W+κC,
+0 =Ra(k),b(l)
+{p+1}=dCa(k),b(l)+Ga(k),b(l)
+(p)(W,C) =dC+σ(p)C+αW.(3.1)
+A label in parentheses refers to the differential form degree of a c urvature. Since αcarries
+differential form degree 0, it is independent of the vielbein and does n ot change a tensor
+type,i.e.,
+αY(k,l) =α(k,l)Y(k,l), (3.2)
+whereα(k,l) are some coefficients. (Here we use the shorthand notation Y(k,l) for a tensor
+described by the Young tableau with two rows of lengths k≥l.)
+The operators κare built from two vielbeins, carrying the differential form degree tw o
+and contain several terms
+κ=κ0+κ12
+−−+κ12
++++κ12
++−+κ12
+−+, (3.3)
+κ0:Y(k,l)→Y(k,l), κ12
+−−:Y(k,l)→Y(k−1,l−1),
+κ12
+++:Y(k,l)→Y(k+1,l+1), κ12
+−+:Y(k,l)→Y(k−1,l+1),(3.4)
+κ12
++−:Y(k,l)→Y(k+1,l−1).
+Each of these operators except for κ0is determined by the Young symmetry properties up
+to a (k,l)-dependent coefficient. κ0contains two linearly independent terms, each defined
+up to a (k,l)-dependent coefficient.
+The operators σ(p+1)andσ(p)act on (p+1)-forms and p-forms, respectively and add one
+differential form degree. They still have the form (2.5)-(2.8) but t he compatibility conditions
+(1.4)aremodifiedbecauseofpresenceof κandα. Hencetherespectivecoefficients F(p+1)(k,l)
+(F(p)(k,l)),G(p+1)(k,l)(G(p)(k,l)),f(p+1)(k,l)(f(p)(k,l))andg(p+1)(k,l)(g(p)(k,l))maydiffer
+from those given in (2.26) and (2.34), (2.35).
+In the new framework, the compatibility conditions imply
+ασ(p+1)=σ(p)α, (3.5)
+ακ=D2+(σ(p))2, (3.6)
+κα=D2+(σ(p+1))2, (3.7)
+σ(p+1)κ=κσ(p). (3.8)
+Eqs. (3.6), (3.7) express κin terms of σ.
+From (3.5) it follows that
+h(p+1)(k,l) =h(p)(k,l) if α(k,l)α(k+1,l)/ne}ationslash= 0, (3.9)
+H(p+1)(k,l) =H(p)(k,l) if α(k,l)α(k,l+1)/ne}ationslash= 0. (3.10)
+14Indeed, let us for example derive (3.9). The operators σ1
+−(k+ 1,l) andσ1
++(k,l) forp- and
+(p+1)-forms are related by the conditions (3.5)
+α(k,l)σ(p+1)1
+−(k+1,l) =σ(p)1
+−(k+1,l)α(k+1,l),
+α(k+1,l)σ(p+1)1
++(k,l) =σ(p)1
++(k,l)α(k,l).
+This implies
+α(k,l)f(p+1)(k+1,l) =f(p)(k+1,l)α(k+1,l),
+α(k+1,l)g(p+1)(k,l) =g(p)(k,l)α(k,l). (3.11)
+Multiplying these relations and using (2.26) we obtain (3.9).
+The gauge transformations that follow from (1.5) are
+δW=dξ+σ(p+1)ξ, δC =−αξ. (3.12)
+In addition to the independent scaling redefinitions (2.16) for p- and (p+1)-forms, the
+equations (3.1) possess the ambiguity in the field redefinitions
+W=˜W+ ˜σC (3.13)
+that leads to following redefinition of the operators
+σ(p)→σ(p)+α˜σ, σ (p+1)→σ(p+1)+ ˜σα, κ→κ+ ˜σσ(p)+σ(p+1)˜σ+ ˜σα˜σ.(3.14)
+It is not hard to see that to single out a solution of the compatibility co nditions that
+describes a spin sparticle in the case of p= 0, the following condition, that plays a role
+analogous to that of (2.36), should be imposed
+α(k,l) = 0 for k≥s, α(k,l)/ne}ationslash= 0 fork≤s−1. (3.15)
+Then from (3.5) it follows that
+σ(0)1
++(s−1,l) =σ(1)1
+−(s,l) = 0. (3.16)
+Indeed, the condition (3.15) and the second of the conditions (3.16 ) imply that 1-forms
+Wa(u),b(v)withu≥sdo not contribute to the equations for all 0-forms and 1-forms Wa(k),b(l)
+withk≤s−1. Hence, (3.15) guarantees that the 1-forms in the infinite triang ular and
+rectangular regions of Fig. 3 decouple from all other fields and can b e dropped from the
+system. Since the curvatures for 0-forms valued in Y(u,v) withu≥sdo not contain 1-
+forms, the analysis of Section 2 is applicable to this case. The first of the relations (3.16)
+implies that (2.36) is satisfied. This means that the 0-forms in the infin ite triangular region
+of Fig. 1 also decouple and can be dropped from the system. As a res ult we are left with
+the set of fields of Fig. 3.
+Let us now consider the 1- and 0-forms in the triangular region of Fig . 3. These should
+be mixed by the equations (3.1) to resolve the compatibility problem of the unmixed case
+of Section 2. Together with the Weyl module these fields should lead t o an unfolded system
+15that consistently describes massive HS particles. To ensure that t he Weyl module is glued
+to the sector of mixed 0- and 1-forms we demand
+σ(0)1
+−(s,l)/ne}ationslash= 0. (3.17)
+The coefficients f(0)(s,l) will be specified later.
+Note that (3.14) implies that any σ(0)(orσ(1)related toσ(0)by (3.5)) can be obtained
+by some field redefinition (3.13) in the sector where 1- and 0-forms a re mixed. Our goal
+is to fix this freedom in such a way that, for particular values of mandλthat correspond
+to massless and partially massless fields, the unfolded system decom pose into appropriate
+subsystems. The solution
+H(0)(k,l) =H(1)(k,l) =H(k,l), h (0)(k,l) =h(1)(k,l) =h(k,l) (3.18)
+withHandhgiven by (2.45) satisfies this requirement. The compatibility problems en-
+countered in Section 2 are resolved by the nontrivial mixture of 1- a nd 0-forms. Recall that
+Handh(2.45) solve (2.11) on its restriction to ¯W, while the restriction of D2+σ2toWis
+nonzero (2.38). Taking into account (3.6), we see that with this cho ice ofHandhwe should
+setκ= 0 ifκ∈¯W, whileκ|Wshould be non-zero.
+To summarize, the unfolded equations that describe the Weyl modu le in terms of 0-forms
+Ca(k),b(l)withk≥s,l≤sare
+R{1}=DC+σ(0)C= 0 forRa(k),b(l)
+{1}withk≥s, l≤s. (3.19)
+These fields belong to the infinite rectangular region of Fig. 3. The fin ite triangle region
+contains 1-forms Wa(k),b(l)and 0-forms Ca(k),b(l)withk≤s−1 that satisfy the equations
+R{2}=DW+σ(1)W= 0,forRa(k),b(l)
+{2}withl<k≤s−1,(3.20)
+R{2}=DW+σ(1)W+κC= 0,forRa(n),b(n)
+{2}withn≤s−1,(3.21)
+R{1}=DC+σ(0)C+αW= 0 forRa(k),b(l)
+{1}withk≤s−1.(3.22)
+Bothσ(0)andσ(1)are still defined by (2.26) and (2.45) up to the rescaling ambiguity (2.1 7)
+(independently for 0- and 1-forms). Explicit formulae for the coeffi cients, that satisfy the
+normalization conditions (2.19), (2.20) with arbitrary Nandn(2.21) that respect (2.25),
+are given in Appendix B.
+For the type-I scaling gauge (2.40), f(0)(s,l) is determined by (2.15) up to an overall
+factor which can be fixed in such a way that
+f(0)(s,l) =/radicalBigg
+1
+(s−l)(d+s+l−3). (3.23)
+Since in the type-I scaling gauge σ(0)=σ(1)by (2.40), (3.18), we obtain from (3.5) α(k,l) =
+α(0,0). By the leftover scaling symmetry we can set
+α(k,l) = 1. (3.24)
+16For the type-II scaling gauge (2.41), (2.42), f(0)(s,l) is still defined by (3.23) while the
+expression for α(k,l) in terms of α(0,0) acquires the form
+α(k,l) =α(0,0)/radicaltp/radicalvertex/radicalvertex/radicalbt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN(1)(0)
+/producttextk
+i=lN(1)(i)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
+Theleftoverscalingsymmetryisusedtofix α(0,0)/radicalbig
+|N(1)(0)|= 1, sothatthefinalexpression
+forαis
+α(k,l) =1/radicalbigg/vextendsingle/vextendsingle/vextendsingle/producttextk
+i=lN(1)(i)/vextendsingle/vextendsingle/vextendsingle. (3.25)
+The explicit form of κ(3.4) is
+κ12
+−−(n,n) =1
+α(n,n)σ(1)1
+−(n,n−1)σ(1)2
+−(n,n),
+κ12
+++(n,n) =1
+α(n,n)σ(1)2
++(n+1,n)σ(1)1
++(n,n), (3.26)
+κ0(n,n) =1
+α(n,n)/parenleftbig
+D2+σ(1)1
+−(n+1,n)σ(1)1
++(n,n)+σ(1)2
++(n,n−1)σ(1)2
+−(n,n)/parenrightbig
+,
+whereD2is given in Eq. (2.3). Other components of κin (3.3), (3.4) are zero. Note that the
+conditionκ12
+−−(s,s)/ne}ationslash= 0 is required by (3.17) rather than by the compatibility conditions.
+The gauge transformations for 0- and 1-forms in the triangular re gion of Fig. 3 are given
+by (3.12) with p= 0, while the Weyl 0-forms in the rectangular region are gauge invar iant.
+3.3 Dynamical fields and Singh-Hagen system
+To show that the constructed unfolded system with m/ne}ationslash=mt(the case of partially massless
+fields is special and will be considered in Section 3.5) indeed describes a massive HS field, it
+is convenient to use the σ−cohomology analysis [28, 7] with the σ−complex defined precisely
+as in the massless case m= 0 for the set of massless fields of spins 0 ,1,...s. This implies
+that theσ−cohomological grade for 0(1)-forms identifies with the length of th e first (second)
+row of their Young diagrams. The 1-form fields of the lowest grade d escribed by one-row
+diagrams are called frame-like fields.
+Since so defined σ−is independent of m, such analysis treats m2as a deformation param-
+eter. Hence it treats constraints like Lφ1+m2φ2= 0 as dynamical equation as they are in
+the massless case of m2= 0, rather than constraints that express φ2viaφ1. Such constraints
+should therefore be taken into account after performing the σ−cohomology analysis to show
+that the dynamical field φa(s)satisfies the equations (1.1) while the other fields are either
+gauged away or expressed in terms of dynamical field by the equatio ns of motion.
+Theσ−cohomology analysis for a massless field gives the following results (se e e.g. [7]).
+Dynamical fields for every massless field of spin k+1 contained in the frame-like field Wa(k)
+consist of two irreducible symmetric tensors
+Wa(k)→k+1⊕k−1,0≤k≤s−1. (3.27)
+17A massless spin zero field is described by the scalar 0-form C. This set of fields coincides
+with that proposed by Zinoviev in [13].
+The leftover gauge symmetry is:
+δ(W(k,0))a(k)=D(ξ(k,0))a(k)+(σ(1)1
+−ξ(k+1,0))a(k)+(σ(1)1
++ξ(k−1,0))a(k).(3.28)
+For generic values of m, the first component in (3.27) of each frame-like field except for
+Wa(s−1)can be gauge fixed to zero by the gauge parameter ξa(k+1)using the second term on
+the r.h.s. of (3.28). The remaining gauge parameter ξis used to gauge away the 0-form C.
+After the gauge symmetry is completely fixed, the remaining non-ze ro field projections are
+Wa(k)→k−1,0<k<s−1, (3.29)
+Wa(s−1)→s−2⊕s. (3.30)
+This is the set of fields of Singh and Hagen [10]. All other fields involved in the unfolded
+formulation either are pure gauge or are expressed via derivatives of the Singh-Hagen fields.
+Note that for special values of the mass mthe second term in the transformation law
+(3.28) may degenerate not allowing to gauge fix to zero the highest c omponent field in (3.27)
+for one or another k. This corresponds to the cases of massless and partially massless fi elds
+considered in more detail in Section 3.5 while in this section we assume th at this does not
+happen.
+3.3.1 Vanishing of lower-spin supplementary fields
+Let us show that, in the chosen gauge, all fields (3.29), (3.30) exce pt for traceless symmetric
+projection of Wa(s−1)vanish by virtue ofthe equations ofmotion. Weshall use two equatio ns
+for each frame-like field. The first one expresses the Lorentz-like auxiliary field via the frame-
+like field. The second one imposes a non-trivial condition on the frame -like field. For the
+lower-spin supplementary fields (i.e. those, associated to the mass less spins′fields with
+s′< s) this condition equates them to zero. For the dynamical field assoc iated to the
+massless spin sfield it imposes the dynamical equation.
+The proof will be given by induction. Namely, assuming that Wa= 0 andWa,b= 0 we
+prove that
+Wa(l)= 0, Wa(l),b= 0∀l<k=⇒Wa(k)= 0, Wa(k),b= 0.(3.31)
+Let us first consider the special cases of s′= 0,1,2. Using the gauge conditions
+C(0,0) = 0, W(0,0) = 0, (3.32)
+the lowest equation in the spin zero sector
+0 =DC(0,0)+σ(0)1
+−C(1,0)+α(0,0)W(0,0)
+leads to
+Ca= 0. (3.33)
+18The only non-zero projection of Wais its trace Wρ|ρsince, according to (3.29), all other its
+components have been gauged fixed to zero. From the trace of th e equation
+0 =D(C(1,0))a+(σ(0)1
+−C(2,0))a+(σ(0)1
++C(0,0))a+(σ(0)2
+−C(1,1))a+α(1,0)(W(1,0))a
+(3.34)
+we obtain using (3.32) that Wρ|ρ= 0 and hence
+Wa= 0. (3.35)
+Also it follows from (3.34) that
+Ca,b= 0. (3.36)
+To show that Wa,b= 0, we observe that Wµ|a,bhas three irreducible components
+Wa,b→⊕⊕.(3.37)
+Projecting the equation
+0 =D(W(1,0))a+(σ(1)1
+−W(2,0))a+(σ(1)1
++W(1,0))a+(σ(1)2
+−W(1,1))a(3.38)
+to the symmetry types of the first two terms in (3.37) we obtain tha t the corresponding
+projections of Wa,bare zero. Taking into account (3.36), the trace of the equation
+0 =D(C(1,1))a,b+(σ(0)1
+−C(2,1))a,b+(σ(0)2
++C(1,0))a,b+α(1,1)(W(1,1))a,b
+implies that the third component of Wa,bin (3.37) is also zero. Hence,
+Wa,b= 0. (3.39)
+The equations (3.35), (3.39) provide the first step of induction. Le t us now prove (3.31).
+The projection of the equation
+0 =D(W(k−1,0))a(k−1)+(σ(1)1
+−W(k,0))a(k−1)+
++(σ(1)1
++W(k−2,0))a(k−1)+(σ(1)2
+−W(k−1,1))a(k−1)(3.40)
+to the symmetry type of k−1implies that the second component of Wa(k)in (3.27) is
+zero. Since the first component has been gauge fixed to zero, we o btain
+Wa(k)= 0. (3.41)
+The situation with Wa(k),bis a bit more involved. It has the following irreducible com-
+ponents
+Wa(k),b→k⊕k−1⊕k+1⊕k
+⊕k.
+The component kis pure gauge and can be gauge fixed to zero as mentioned in the
+beginning of Section 3.3. The projections of the equation
+0 =D(W(k,0))a(k)+(σ(1)1
+−W(k+1,0))a(k)+(σ(1)1
++W(k−1,0))a(k)+(σ(1)2
+−W(k,1))a(k)(3.42)
+19to the symmetry types
+k−1
+,k
+,k+1
+imply that the components of Wa(k),bof these symmetry types are zero. The remaining
+component kvanishes by virtue of the rank ksymmetric part of the equation
+0 =D(W(k−1,1))a(k−1),b+(σ(1)1
+−W(k,1))a(k−1),b+(σ(1)1
++W(k−2,1))a(k−1),b+
++(σ(1)2
+−W(k−1,2))a(k−1),b+(σ(1)2
++W(k−1,0))a(k−1),b. (3.43)
+Hence,
+Wa(k),b= 0. (3.44)
+Thereby, all fields Wa(k)andWa(k),bwithk<s−1 are shown to be zero.
+3.3.2 Dynamical equations
+Let us now derive the dynamical equation in Minkowski and ( A)dScases. Projection of
+(3.40) with k=s−1 tos−2implies that Wρρa(s−2)= 0, while the traceless totally
+symmetric part sofWa(s−1)remains non-zero and cannot be gauged away. This is
+the spinsdynamical field denoted φa(s). It is totally symmetric and traceless
+φa(s)=Wa|a(s−1), φρ
+ρa(s−2)= 0. (3.45)
+The projection of (3.42) tok+1
+expressesWa(k),bin terms of dWa(k)which is not
+zero only for k=s−1. Analogously to Subsection 3.2.2, we can use (3.42) and (3.43) with
+k=s−1 to prove thats
+is the only non-vanishing projection of Wa(s−1),b. One
+consequence of this result is that Wa(s−1),bis traceless
+Wρ|
+a(s−1),ρ= 0, Wρ|
+a(s−2)ρ,b= 0. (3.46)
+With the help of (3.45) and (3.46), the contraction of indices aandµof the equation
+0 =Rµν|a(s−1)=Dµφνa(s−1)−Dνφµa(s−1)+F(1)(s−1,1)(Wν|a(s−1),µ−Wµ|a(s−1),ν) (3.47)
+yields
+Dρφρa(s−1)= 0. (3.48)
+The projection of the equation R{2}a(s−1),b= 0 to the symmetry type sgives
+0 =DρWa|a(s−1),ρ+G(1)(s−1,0)(d−2)φa(s). (3.49)
+Expressing Wa|a(s−1),ρin terms of φby means of (3.47) we obtain
+DρWa|a(s−1),ρ=1
+F(1)(s−1,1)(DρDaφρa(s−1)−DρDρφa(s)). (3.50)
+20Using Eqs. (2.3), (3.48) and substituting (3.50) into (3.49) we obtain
+0 =DρDρφa(s)−/parenleftbig
+λ2(d+s−2)+(d−2)H(s−1,0)/parenrightbig
+φa(s), (3.51)
+whereH(s−1,0) is defined in (2.26). Using (2.45) we finally obtain
+0 =DρDρφa(s)+/parenleftbig
+λ2(s2+s(d−6)−2d+6)+m2/parenrightbig
+φa(s). (3.52)
+This is the field equation for a massive spin sfield in the ( A)dSbackground. The mass-like
+term reproduces that obtained in [32]. In the flat case λ= 0 we recover the equation (1.1).
+It is easy to see that, the only fields that remain non-zero on shell in the chosen gauge are
+the 0-forms that constitute the Weyl module and the projections of the 1-forms Wa(s−1),b(l)
+tols
+. All of them are expressed by the unfolded equations via higher der ivatives of
+the dynamical field φa(s).
+3.4 Spin 2 example
+Let us consider the example of a spin 2 field of generic mass, m/ne}ationslash=mt(2.46) fort= 0,1.
+In this case, the full set of fields consists of 1-forms W,Wa,Wa,b, 0-formsC,Ca,Ca,band
+0-formsCa(k),b(l),k≥2≥lof the Weyl module. Unfolded equations (3.20)-(3.22) are
+0 =DC+f(0)(1,0)emCm+α(0,0)W, (3.53)
+0 =DCa+f(0)(2,0)emCam+g(0)(0,0)eaC+F(0)(1,1)emCa,m+α(1,0)Wa,(3.54)
+0 =DCa,b+f(0)(2,1)em(Cam,b+1
+2Cab,m)+G(0)(1,0)(ebCa−eaCb)+α(1,1)Wa,b,(3.55)
+0 =DW+f(1)(1,0)emWm+α−1(0,0)f(0)(1,0)F(0)(1,1)emenCm,n,(3.56)
+0 =DWa+g(1)(0,0)eaW+F(1)(1,1)emWa,m, (3.57)
+0 =DWa,b+G(1)(1,0)(ebWa−eaWb)+2α−1(1,1)G(0)(1,0)g(0)(0,0)ebeaC+
++α−1(1,1)(λ2+G(0)(1,0)F(0)(1,1))(eaemCm,b+ebemCa,m)+ (3.58)
++α−1(1,1)f(0)(2,1)F(0)(2,2)enemCan,bm
+plus equations for the 0-forms in the Weyl module that follow from (3 .53)-(3.58). The
+coefficients in curvatures both for 0- and 1-forms satisfy (2.45), (2.19), (2.20). The gauge
+symmetries (3.12) are
+δC=−α(0,0)ξ, δCa=−α(1,0)ξa, δCa,b=−α(1,1)ξa,b, (3.59)
+δW=Dξ+f(1)(1,0)emξm, (3.60)
+δWa=Dξa+g(1)(0,0)eaξ+F(1)(1,1)emξa,m, (3.61)
+δWa,b=Dξa,b+G(1)(1,0)(ebξa−eaξb). (3.62)
+Analogously to the massless spin 2 case, using (3.61), one fixes ξa,bby requiring the anti-
+symmetric part of Wato be zero. Then we can fix ξandξasettingC= 0 andW= 0. In
+21this gauge Eq. (3.53) implies Cm= 0. Then from (3.54) it follows that Wρ|ρ= 0. Eq. (3.56)
+states that Ca,b= 0. Contracting one fiber and one space-time index of (3.55) we obt ain
+thatWρ|ρ,a= 0. Then from (3.57) it is easy to obtain that the dynamical field φaa=Wa|a
+is transversal. Eq. (3.57) expresses Wa,bin terms of derivatives of φaa. Plugging this ex-
+pression into (3.58), contracting one fiber and one space-time inde x and symmetrizing the
+others we obtain the spin 2 equation in AdSd:
+DρDρφaa+(−2λ2+m2)φaa= 0. (3.63)
+3.5 Massless and partially massless fields
+Let us now discuss peculiarities of the massless and partially massless cases. Using the
+type-II scaling gauge, it will be shown that a massive spin sfield decomposes into the set
+of massless fields of spins s′with 0≤s′≤satm= 0,λ2= 0 and into one spin smassless
+field and one spin s−1 massive field at m= 0,λ2/ne}ationslash= 0. Analogously, for m=mta massive
+spinsfield will be shown to decompose into one partially massless field of spin sand depth
+tand one spin tmassive field.
+As discussed in Section 2, massless and partially massless fields are ch aracterized by the
+special values of mass mt(2.46) for which the condition (2.47) H(k,t) = 0 holds. As follows
+from (2.26) this can be achieved by setting F(k,t+1) = 0 and/or G(k,t) = 0. Analogously,
+h(t−1,l) = 0 can be achieved by setting f(t,l) = 0 and/or g(t−1,l) = 0. In the type-I
+scaling gauge, from (2.40), (2.29)-(2.33) we have
+F(0)(k,t+1) =G(0)(k,t) =f(0)(t,l) =g(0)(t−1,l) =
+=F(1)(k,t+1) =G(1)(k,t) =f(1)(t,l) =g(1)(t−1,l) = 0. (3.64)
+In the type-II scaling gauge we have from (2.41), (2.42), (2.29)-( 2.33)
+F(0)(k,t+1) =G(0)(k,t) =g(0)(t−1,l) = 0, f(0)(t,l)/ne}ationslash= 0,
+G(1)(k,t) =f(1)(t,l) =g(1)(t−1,l) = 0, F(1)(k,t+1)/ne}ationslash= 0. (3.65)
+The main difference between the two scalings is that the type-II sca ling gauge keeps non-
+zeroσ(0)1
+−andσ(1)2
+−which is crucial for the field equations to remain sensible for the spec ial
+values of the parameter of mass.
+Indeed, for λ2= 0, the Weyl module of a spin s′massless field consists of 0-forms valued
+inY(k,s′),k≥s′. In Minkowski space, the unfolded massless equations have the fo rm [20]
+R{1}(k,s′) = 0 =dC(k,s′)+σ(0)1
+−C(k+1,s′), k≥s′, (3.66)
+i.e.,σ(0)1
++=σ(0)2
++=σ(0)2
+−= 0 and consequently H(0)(k,l) =h(0)(k,l) = 0. It is easy to
+see from (2.45) that both H(0)(k,l) andh(0)(k,l) of the massive Weyl module indeed vanish
+atλ2= 0 andm2= 0. The choice of n(0)(k,l) in (2.42) provides non-zero σ(0)1
+−in the flat
+massless limit. H(0)(k,l) = 0 leads to σ(0)2
++=σ(0)2
+−= 0,i.e.,Eq. (3.66) is reproduced and
+the massive Weyl module decomposes into ssets of 0-forms valued in Y(k,s′) with definite
+s′<sand arbitrary k≥sfor each set. Compared to the Weyl module of a massive particle,
+the combined set of the Weyl modules for massless fields of spins s′, 0≤s′≤scontains
+22additional 0-forms that form the finite triangular set in Fig. 3. Let u s discuss the origin of
+this difference in more detail.
+Atλ2= 0,m2= 0 Eq. (2.45) yields H(k,l) =h(k,l) = 0 both for 0- and for 1-forms of
+the triangular region of Fig. 3. In the scaling (2.41), (2.42) this leads to vanishing of all σ
+except forσ(1)2
+−andσ(0)1
+−. As a result, the triangular set of 1-forms decomposes into vertic al
+lines, each containing Wa(s′−1),b(l)withl≤s′−1 and definite s′< s, while the triangular
+set of 0-forms decomposes into horizontal lines, each containing Ca(k),b(s′)withk≥s′and
+definites′< s. Since from (3.25) it follows that α= 0 atm2=λ2= 0, the curvatures for
+0-forms of the triangular region do not contain 1-forms. As a resu lt, these 0-forms form the
+completion of the Weyl modules of the decoupled massless particles. It can be shown that
+κ12
+++andκ0also vanish. κ12
+−−remains non-zero and glues Wa(s′−1),b(s′−1)toCa(s′),b(s′). This
+describes the standard gluing of gauge potentials to the Weyl modu le in the massless case.
+In the case of λ2/ne}ationslash= 0, a massless field is a particular case of t=s−1 of partially massless
+fields that allows us to use Fig. 2 to visualize the decomposition of the W eyl module in the
+massless limit in ( A)dS. So all 0-forms of the region 5 have scells in the second row, while
+the 0-forms of the region 2 have s−1 cells in the first row. The Weyl module of a spin s
+massive particle consists of 0-forms of regions 4 and 5.
+The (A)dSWeyl module of a spin smassless field consists of 0-forms valued in Y(k,s),
+k≥sforming region 5 of Fig. 2. The unfolded equations are
+R{1}(k,s) = 0 =dC(k,s)+σ(0)1
+−C(k+1,s)+σ(0)1
++C(k−1,s), k≥s.
+Decoupling of 0-forms of the Weyl module associated to a massless s pinsfield atλ2/ne}ationslash= 0
+requiresH(0)(k,s−1) = 0. This is indeed true for (2.45) with m2= 0. The remaining
+0-forms, valued in Y(k,l) withk≥s,l < s, that form the region 4 of Fig. 2, constitute a
+part of the Weyl module of a spin s−1 massive field with the specific mass ms(2.46) (recall
+that this valueof mass fulfils the condition H(0)(k,s) =h(0)(s−1,l) = 0 of thedecomposition
+of the massive Weyl module into the two parts forming regions 2 and 4 of Fig. 2).
+The lacking part of the Weyl module of the spin s−1 massive field contains 0-forms
+valued in Y(s−1,l) with any l≤s−1 (i.e. region 2 in Fig. 2). In the case of λ2/ne}ationslash= 0, the
+1-formsWa(s−1),b(l)of a massless spin sfield decouple from the rest 1-forms, being still glued
+to the massless spin sWeyl module. Since from (3.25) it follows that α(s−1,l) = 0 for
+m2= 0, the curvatures for 0-forms Ca(s−1),b(l)do not contain 1-forms. These 0-forms form a
+lacking part of the Weyl module of the massive spin s−1 particle. 0- and 1-forms valued in
+Y(k,l) withk <s−1 are still mixed and form a finite triangle in Fig. 3 of a massive spin
+s−1 particle.
+Let us briefly consider the partially massless case. At m=mt, the condition (2.47) is
+satisfied, thereby (3.65) is fulfilled for the type-II scaling gauge. I t is easy to see from (3.25)
+that
+α(k,l) = 0, l≤t≤k (3.67)
+i.e.,α(k,l) = 0 for ( k,l) from the region 2 of Fig. 2. It follows from (3.65), (3.67) that
+unfolded equations do not mix fields of the following two sets.
+The first set consists of mixed 1- and 0-forms of the region 1 glued t o 0-forms of regions
+2 and 4 of Fig. 2. These fields describe a massive spin tfield of mass ms(2.46).
+23The second set consists of 1-forms of the region 2. Since σ(1)2
+−(k,t+ 1)/ne}ationslash= 0 by (3.65),
+these 1-forms are glued to mixed 1- and 0-forms valued in the region 3. At the same time, 0-
+and 1-forms of the region 3 are glued to the 0-forms of the region 5 becauseσ(0)1
+−(s,l)/ne}ationslash= 0.
+Compared to the approach of [21] this set provides a non-minimal description of a par-
+tiallymassless fieldofspin sanddepthtthatcontainsadditionalfieldsalongwithconstraints
+thatexpress themalgebraicallyintermsofderivatives ofthedynam ical fieldsmoduloStueck-
+elbergdegreesoffreedomthataregaugedawaybyadditionalStu eckelberg gaugesymmetries.
+Indeed, to show that the two descriptions are equivalent we obser ve that the 0-forms of the
+region 3 can be gauge fixed to zero by the gauge transformation (3 .12)
+Ca(k),b(l)= 0, k≤s−1, l≥t+1.
+Then, equating curvatures (3.22) for 0-forms of the region 3 to z ero we obtain Wa(k),b(l)= 0
+fork≤s−2,l≥t+1, whileWa(s−1),b(l)withl≥t+1 are expressed in terms of 0-forms
+that belong to the partially massless Weyl module
+σ(0)1
+−(s,l)C(s,l)+α(s−1,l)W(s−1,l) = 0, l≥t+1. (3.68)
+Taking intoaccount (3.68)for l=t+1, onefindsthattheterm σ(1)2
+−(s−1,t+1)W(s−1,t+1)
+in the curvature Ra(s−1),b(t)
+{2}glues the partially massless Weyl module to the gauge potentials
+just in the same way as in the standard description where the gauge potentials of the region
+2 are glued to the Weyl module via2[33]
+R{2}a(s−1),b(t)=ecedCa(s−1)c,b(t)d.
+Thereby the non-minimal description of a partially massless field redu ces to the standard
+one. Let us note that, although the type-II scaling gauge is most c onvenient for the analysis
+of the massive field decoupling in the partially massless limit, the standa rd description of a
+single partially massless field simplifies in the type-I scaling gauge.
+Let us now analyze the dynamical content of the unfolded field equa tions in the partially
+massless case where σ(1)1
+−(t,l) =σ(1)1
++(t−1,l) = 0by (3.65) and1-formsarevalued in Y(k,l)
+witht≤k≤s−1. Analogously to the massive case we use the σ−cohomology analysis as
+in the massless case for the set of massless fields of spins t+1,t+2,...,s. As a result, we
+are left with a set of frame-like fields
+Wa(k)→k+1⊕k−1, t≤k≤s−1 (3.69)
+(cf (3.27)). The leftover gauge symmetry is still given by (3.28) for k:t≤k≤s−1. The
+first component in (3.69) of each frame-like field except for Wa(s−1)can be gauged away by
+means of a gauge parameter ξa(k+1):
+PTr=0(Wa|a(k)) = 0. (3.70)
+HerePTr=0denotes the projector to the traceless part. The only gauge par ameter that
+cannot be fixed this way is ξa(t). This is the parameter of a differential gauge symmetry.
+2We acknowledge a useful discussion with E. Skvortsov of the specifi cities of gluing the Weyl module in
+the partially massless case.
+24The gauge transformation with the parameter ξa(t)must be accompanied by the gauge
+transformations with the parameters ξa(k)witht < k≤s−1 to satisfy the gauge fixing
+condition (3.70). Indeed, the gauge transformation of PTr=0(Wa|a(t)) is
+PTr=0(δWa|a(t)) =PTr=0(Daξa(t))+F(1)(t+1,0)ξa(t+1)
+and hence
+ξa(t+1)=−PTr=0(Daξa(t))
+F(1)(t+1,0).
+Analogously, one can show that
+ξa(k)= (−1)k−tPTr=0(k−t/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
+Da...Daξa(t))/producttextk
+i=t+1F(1)(i,0). (3.71)
+After the Stueckelberg gauge symmetry is completely fixed, we are left with non-zero
+field projections
+Wa(k)→k−1, t≤k<s−1,
+Wa(s−1)→s−2⊕s (3.72)
+and the differential gauge symmetry (3.28) with gauge parameters satisfying (3.71).
+The only non-zero projection of Wa(t)is
+ψa(t−1)=Wρ|ρa(t−1), δψa(t−1)=Dρξρa(t−1). (3.73)
+It can neither be expressed in terms of other fields nor gauged awa y by a Stueckelberg gauge
+transformation, hence being a dynamical field. It follows from
+0 =D(W(t,0))a(t)+(σ(1)1
+−W(t+1,0))a(t)+(σ(1)2
+−W(t,1))a(t)(3.74)
+that the only non-zero projections of Wa(t),baret−1
+andt. The hook-type
+projection is expressed in terms of derivative of ψby (3.74). Also Eq. (3.74) expresses the
+projection of Wa(t),btot, namelyWρ|a(t),ρ, in terms of DψandWρ|ρa(t). Since there
+are no other restrictions on the fields Wρ|a(t),ρandWρ|ρa(t), one of them, say Wρ|ρa(t), remains
+unrestricted and should be identified with the second dynamical field
+ϕa(t)=Wρ|ρa(t). (3.75)
+Its transformation law follows from (3.28), (3.71)
+δϕa(t)=g(1)(t,0)(d+2t)(d+t−2)
+d+2t−2ξa(t)−
+−(d+2t−2)DρDρξa(t)+t(d+2t−4)DρDaξρa(t−1)−t(t−1)DρDλξρλa(t−2)ηaa
+(t+1)(d+2t−2)F(1)(t+1,0).
+25The further analysis of equations for partially massless case is analo gous to the massive
+case.φa(s)=PTr=0(Wa|a(s−1)) is the third dynamical field. From (3.28), (3.71) it follows
+that
+δφa(s)= (−1)s−t−1PTr=0(s−t/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
+Da...Daξa(t))/producttexts−1
+i=t+1F(1)(i,0). (3.76)
+This is in accordance with the known property that gauge transfor mations of partially mass-
+less fields contain higher derivatives [31, 13, 21].
+4 Action
+Theadvantageoftheunfoldedformulationisthatitoperateswithg augeinvariantcurvatures
+(1.2), (3.19)-(3.22) which can be used to construct gauge invarian t actions. Any action,
+bilinear in these curvatures, is manifestly gauge invariant. The gaug e invariant action that
+describes a unitary theory should satisfy the conditions
+δS
+δWa(k),b(l)≡0, l≥2,δS
+δCa(q),b(r)≡0, q≥2, (4.1)
+which generalize the extra field decoupling condition for massless [18, 19] and partially mass-
+less [21] fields to the massive case. This is equivalent to the condition t hat the action is free
+of derivatives higher than two. Another requirement is the massless decomposition condition
+which demands that the part of the action that contains two deriva tives of the dynamical
+fields do not mix dynamical fields associated to different massless field s. Here we refer to
+the fact, that in agreement with construction of Zinoviev [13], the m assless flat limit of a
+massive action should give the sum of free massless actions for spins from 0 tos. In this
+manner,Wa(k−1),b(l)is associated to the l-th derivative of a spin kdynamical massless field.
+Hence, such fields should not contribute to the action if l≥2. Analogously, in the mass-
+less case, the Weyl 0-form Ca(q),b(r)containsqderivatives of a spin ( q−r) dynamical field.
+Thereby such fields with q≥2 also are not allowed to contribute to a unitary action. The
+extra field decoupling condition (4.1) along with the massless decompo sition condition fix
+the action up to an overall factor and total derivatives, guarant eeing that it contains at most
+two derivatives of the dynamical fields and has correct structure in the massless limit.
+In this section we discard the contribution of the 0-forms from the Weyl module to the
+curvatures (3.21), (3.22) using that this does not violate gauge inv ariance because the Weyl
+tensors are gauge invariant. In practice, this only applies to the cu rvaturesRa(s−1),b(s−1)
+{2}and
+Ra(s−1),b(s−1)
+{1}because other curvatures that appear in this section do not cont ain 0-forms
+that belong to the Weyl module.
+Most of the consideration of this section applies to an arbitrary sca ling gauge (2.19),
+(2.20), unless the conditions (2.41), (2.42) are explicitly referred t o in the analysis of the flat
+massless limit.
+4.1 Action derivation
+To begin with, let us find the part of the action S1,1built from the curvatures for 1-forms,
+that satisfies (4.1) up to C-dependent terms. To compensate the C-dependent terms we will
+26then add appropriate terms S0,0built from 0-forms so that
+S=S1,1+S0,0
+will satisfy (4.1), yielding correct field equations.
+S1,1is constructed in terms of the curvatures R{2}analogously to the frame-like actions
+for massless [18, 19] and partially massless [21] fields
+S1,1=/summationdisplay
+1≤l≤k≤s−1ak,l/an}bracketle{tRa(k),b(l)
+{2}|Ra(k),b(l)
+{2}/an}bracketri}ht, (4.2)
+where/an}bracketle{t...|.../an}bracketri}htis the inner product (2.18) and the coefficients ak,lremain to be determined.
+Using (2.21), the part of the variation of the action, that contains R{2}is
+δS1,1=−2/summationdisplay
+1≤l≤k≤s−1/parenleftBig
+/an}bracketle{tδW(k,l)|σ(1)1
+−R{2}(k+1,l)/an}bracketri}ht(1
+n(1)(k,l)ak+1,l−ak,l)+
++/an}bracketle{tδW(k,l)|σ(1)2
+−R{2}(k,l+1)/an}bracketri}ht(1
+N(1)(k,l)ak,l+1−ak,l)+
++/an}bracketle{tδW(k,l)|σ(1)1
++R{2}(k−1,l)/an}bracketri}ht(n(1)(k−1,l)ak−1,l−ak,l)+
++/an}bracketle{tδW(k,l)|σ(1)2
++R{2}(k,l−1)/an}bracketri}ht(N(1)(k,l−1)ak,l−1−ak,l)/parenrightBig
+.(4.3)
+Obviously, the extra field decoupling condition (4.1) demands
+ak+1,l=ak,ln(1)(k,l), a k,l+1=ak,lN(1)(k,l). (4.4)
+Using (2.40) this gives
+ak,l=z(k,l)as−1,1=z(k,l)a (4.5)
+for the type-I scaling gauge. Here z(k,l) is a sign factor
+z(k,l) =l−1/productdisplay
+i=1sign(H(s−1,i))s−2/productdisplay
+j=ksign(h(j,l)).
+For the type-II scaling gauge (2.41), (2.42) one obtains
+ak,l=z(k,l)as−1,1l−1/productdisplay
+n=1|N(1)(n)|. (4.6)
+As explained in more detail in Section 4.3, to obtain non-zero variation in the flat massless
+limit we should set
+ak,l=z(k,l)al−1/productdisplay
+n=0|N(1)(n)|, (4.7)
+whereais bothm2andλ2independent.
+27The variation (4.3) remains nonzero because the coefficients ak,lare zero at k≤0. As a
+result,
+δS= 2/summationdisplay
+1≤k≤s−1ak,1/parenleftbig
+/an}bracketle{tσ(1)2
++δW(k,0)|R{2}(k,1)/an}bracketri}ht+/an}bracketle{tδW(k,1)|σ(1)2
++R{2}(k,0)/an}bracketri}ht/parenrightbig
+.(4.8)
+Thus the extra field decoupling condition determines S1,1up to an overall factor a. The only
+terms in (4.8) that contain two derivatives of the dynamical fields ar e
+/an}bracketle{tσ(1)2
++δW(k,0)|DW(k,1)/an}bracketri}htand/an}bracketle{tδW(k,1)|σ(1)2
++DW(k,0)/an}bracketri}ht.
+Sincetheydonotmixfieldsassociatedtodifferentmasslessfields, th emasslessdecomposition
+condition is also fulfilled.
+Additional terms in the variation result from the presence of 0-for ms in the curva-
+turesRa(n),b(n)
+{2}(3.21) valued in the rectangular Young diagrams, i.e.,from the terms
+/summationtext
+nan,n/an}bracketle{tRa(n),b(n)
+{2}|Ra(n),b(n)
+{2}/an}bracketri}ht.These give
+∆(δS1,1) = 2/parenleftBig/summationdisplay
+1≤n≤s−1/parenleftbig
+an,n/an}bracketle{tδW(n,n)|κ0R{1}(n,n)/an}bracketri}ht−an,n/an}bracketle{tδC(n,n)|κ0R{2}(n,n)/an}bracketri}ht/parenrightbig
++
+/summationdisplay
+1≤n≤s−1/parenleftbig
+an,n/an}bracketle{tδW(n,n)|κ12
+++R{1}(n−1,n−1)/an}bracketri}ht−
+−an−1,n−1α(n,n)
+α(n−1,n−1)/an}bracketle{tδC(n−1,n−1)|κ12
+−−R{2}(n,n)/an}bracketri}ht/parenrightbig
++ (4.9)
++/summationdisplay
+1≤n≤s−2/parenleftbig
+an,n/an}bracketle{tδW(n,n)|κ12
+−−R{1}(n+1,n+1)/an}bracketri}ht−
+−an+1,n+1α(n,n)
+α(n+1,n+1)/an}bracketle{tδC(n+1,n+1)|κ12
+++R{2}(n,n)/an}bracketri}ht/parenrightbig/parenrightBig
+,
+whereα(k,l) is determined by (3.24) for the type-I scaling and by (3.25) for the type-II
+scaling. Note that the terms
+/an}bracketle{tδW(s−1,s−1)|κ12
+−−R{1}(s,s)/an}bracketri}htand/an}bracketle{tδC(s,s)|κ12
+++R{2}(s−1,s−1)/an}bracketri}ht
+do not appear in (4.9) because the term κ−−Ca(s),b(s), constructed by means of the 0-form
+that belongs to the Weyl module, have been dropped from the curv atureRa(s−1),b(s−1)
+{2}.
+The terms (4.9) can be compensated by terms bilinear in the curvatu resR{1}which have
+the following general form
+S0,0=/summationdisplay
+k,lπk,l/integraldisplay
+ǫl1...ldR{1}l1a(k−1),b(l)R{1}l2a(k−1),b(l)el3...eld+
++/summationdisplay
+k,l̺k,l/integraldisplay
+ǫl1...ldR{1}a(k),l1b(l−1)R{1}a(k),l2b(l−1)el3...eld+
++/summationdisplay
+k,lτk,l/integraldisplay
+ǫl1...ldR{1}l1a(k),l2b(l)R{1}a(k),b(l)el3...eld+
++/summationdisplay
+k,lϕk,l/integraldisplay
+ǫl1...ldR{1}l1a(k−1),b(l)R{1}a(k−1),l2b(l)el3...eld.(4.10)
+28It is useful to observe that almost all terms in (4.10) can be re-writ ten in terms of the
+inner product /an}bracketle{t...|.../an}bracketri}htand some differential form of degree 2 built from the background
+vielbein. For example, the third term in (4.10) is proportional to
+/an}bracketle{tR{1}(k+1,l+1)|σ(1)2
++σ(1)1
++R{1}(k,l)/an}bracketri}ht. (4.11)
+The analysis of terms in this form is much easier than in the general fo rm (4.10) because
+their variation can be easily found by means of Bianchi identities and c onjugation properties
+(2.21). Such arepresentation works nicely provided thatthecurv atures carryenoughindices.
+For example, to vary the term
+/an}bracketle{tσ(1)2
++δW(k,l)|σ(1)1
+−σ(1)1
++R{1}(k,l+1)/an}bracketri}ht
+it is enough to use (2.21) for σ2
++to obtain
+−1
+N(1)(k,l)/an}bracketle{tδW(k,l)|σ(1)2
+−σ(1)1
+−σ(1)1
++R{1}(k,l+1)/an}bracketri}ht.
+Unfortunately, such a representation does not work in the case o fl= 0 where (2.21) is
+inapplicable and the variation of such terms should be analyzed separ ately. Let these special
+terms be denoted as Sspec, while the other regular terms be denoted as Sreg, so that
+S0,0=Sspec+Sreg. (4.12)
+It can be easily checked that the variation of
+Sreg=−/summationdisplay
+reg/an}bracketle{ta
+αR{1}|κR{1}/an}bracketri}ht=−s−1/summationdisplay
+n=2/parenleftbiggan−1,n−1
+α(n−1,n−1)/an}bracketle{tR{1}(n−1,n−1)|κ12
+−−R{1}(n,n)/an}bracketri}ht+
++an,n
+α(n,n)/an}bracketle{tR{1}(n,n)|κ12
+++R{1}(n−1,n−1)/an}bracketri}ht+an,n
+α(n,n)/an}bracketle{tR{1}(n,n)|κ0R{1}(n,n)/an}bracketri}ht/parenrightbigg
+(4.13)
+compensates (4.9) up to the following terms
+2/parenleftbig
+a1,1/an}bracketle{tδW(1,1)|κ0R{1}(1,1)/an}bracketri}ht−a1,1/an}bracketle{tδC(1,1)|κ0R{2}(1,1)/an}bracketri}ht+
++a1,1/an}bracketle{tδW(1,1)|κ++R{1}(0,0)/an}bracketri}ht+a1,1/an}bracketle{tκ++δC(0,0)|R{2}(1,1)/an}bracketri}ht−
+−a2,1
+α(2,2)/an}bracketle{tδC(2,1)|σ(0)2
+−κ++R{1}(1,1)/an}bracketri}ht−a1,1
+α(1,1)/an}bracketle{tδC(1,1)|κ−−σ(0)2
++R{1}(2,1)/an}bracketri}ht/parenrightbig
+,
+which do not satisfy the extra field decoupling condition (4.1) and mas sless decomposition
+condition. Hence they should be compensated by Sspecof the form
+Sspec=a1,1
+α(1,1)/integraldisplay
+ǫl1...ld/parenleftbig
+µR{1}l1,l2R{1}+νR{1}l1,bR{1}l2,
+b/parenrightbig
+el3...eld.(4.14)
+A somewhat involved calculation shows that the proper choice of µandνin (4.14) is
+ν=4
+(d−2)α(1,1)/parenleftbigg
+h(1,1)d+2
+2(d−1)−H(1,0)−λ2/parenrightbigg
+, µ=4G(1)(1,0)g(1)(0,0)
+α(0,0).(4.15)
+29It can be shown that (4.14) nevertheless can be represented in th e regular form
+Sspec=−a1,1
+α(1,1)/an}bracketle{tR{1}(1,1)|κ0R{1}(1,1)/an}bracketri}ht−2a1,1
+α(1,1)/an}bracketle{tR{1}(1,1)|κ12
+++R{1}(0,0)/an}bracketri}ht.(4.16)
+Let us note, that the conjugation relations (2.21) can be used to t ransform (4.13) to the
+form
+Sreg=−/summationdisplay
+reg/an}bracketle{ta
+αR{1}|κR{1}/an}bracketri}ht=
+=s−1/summationdisplay
+n=2/parenleftbigg
+−an,n
+α(n,n)/an}bracketle{tR{1}(n,n)|κ0R{1}(n,n)/an}bracketri}ht−2an,n
+α(n,n)/an}bracketle{tR{1}(n,n)|κ12
+++R{1}(n−1,n−1)/an}bracketri}ht/parenrightbigg
+.
+(4.17)
+Sspec(4.16) has the form of the term in brackets on the r. h. s. of (4.17) withn= 1.
+The manifestly gauge invariant action for symmetric massive fields of any spins≥2 is
+S=/summationdisplay
+1≤l≤k≤s−1ak,l/an}bracketle{tR{2}(k,l)|R{2}(k,l)/an}bracketri}ht−
+−s−1/summationdisplay
+n=1/parenleftbigan,n
+α(n,n)/an}bracketle{tR{1}(n,n)|κ0R{1}(n,n)/an}bracketri}ht+ (4.18)
++2an,n
+α(n,n)/an}bracketle{tR{1}(n,n)|κ12
+++R{1}(n−1,n−1)/an}bracketri}ht/parenrightbig
+.
+For the reader’s convenience let us recall relevant notations. The curvatures Rgiven in
+(3.19)-(3.22) are contracted by means of the scalar product /an}bracketle{t...|.../an}bracketri}ht(2.18). The curvatures
+containoperators σ=σ1
+−+σ1
+++σ2
+−+σ2
++definedby(2.5)-(2.8)bothfor0-and1-formsandop-
+eratorsκgiven in (3.26). The theory has the field redefinition freedom (2.16) in dependently
+for 0- and 1-forms. To fix it one should choose the scaling gauge. Th e type-I scaling gauge is
+(2.40). The type-II scaling gauge is (2.41), (2.42). Since, for the t ype-II scaling, N1(k,l) is
+kindependent and n(0)(k,l) islindependent, we introduce notation N(1)(l) =N(1)(k,l) and
+n(0)(k) =n(0)(k,l). Note that N(1)(l) =−n(0)(l−1). The coefficients in σare determined
+by (2.29)-(2.33) where H(k,l) andh(k,l) are given by (2.45). The coefficients ak,landαare
+given by (4.5), (3.24) for the type-I scaling gauge and by (4.7), (3.2 5) for the type-II scaling
+gauge.
+The variation of (4.18) is
+δS= 2/summationdisplay
+1≤k≤s−1ak,1/parenleftbig
+/an}bracketle{tσ(1)2
++δW(k,0)|R{2}(k,1)/an}bracketri}ht+/an}bracketle{tδW(k,1)|σ(1)2
++R{2}(k,0)/an}bracketri}ht/parenrightbig
++
++a1,1µ
+α(1,1)/integraldisplay
+ǫl1...ldel3...eld/parenleftbig
+−α(0,0)δCl1,l2R{2}(0,0)−α(0,0)δW(0,0)Rl1,l2
+{1}+
++(σ(0)2
++δC)l1,l2R{1}(0,0)−δC(0,0)(σ(0)2
++R{1})l1,l2/parenrightbig
+.(4.19)
+The action (4.18) yields all equations used in the analysis of Section 3.3 of the dynam-
+ical content of the unfolded field equations. Namely, the second te rm of (4.19) yields the
+equations that express the first auxiliary fields in terms of the fram e-like fields (3.40), (3.42),
+(3.47). The first term yields a non-trivial equations on the frame-lik e fields (3.43), (3.49).
+Other terms yield equations for the spin 0 and spin 1 fields analyzed in S ection 3.3.
+304.2 Spin 2 example
+Let us consider the example of spin 2. Recall that terms involving 0-f orms from the Weyl
+module (namely Caa,bbandCaa,b) should be dropped from the action (4.18) which has the
+form
+S=a1,1/an}bracketle{tR{2}(1,1)|R{2}(1,1)/an}bracketri}ht−a1,1
+α(1,1)/an}bracketle{tR{1}(1,1)|κ0R{1}(1,1)/an}bracketri}ht−
+−2a1,1
+α(1,1)/an}bracketle{tR{1}(1,1)|κ12
+++R{1}(0,0)/an}bracketri}ht. (4.20)
+Its variation is
+δS=−2a1,1/parenleftbig
+/an}bracketle{tσ(1)2
++δW(1,0)|R{2}(1,1)/an}bracketri}ht−/an}bracketle{tδW(1,1)|σ(1)2
++R{2}(1,0)/an}bracketri}ht/parenrightbig
++
++a1,1µ
+α(1,1)/integraldisplay
+ǫl1...ldel3...eld/parenleftbig
+−α(0,0)δCl1,l2R{2}(0,0)−α(0,0)δW(0,0)Rl1,l2
+{1}+
++(σ(0)2
++δC)l1,l2R{1}(0,0)−δC(0,0)(σ(0)2
++R{1})l1,l2/parenrightbig
+.(4.21)
+The first term gives the projection of (3.58) to . The second term gives (3.57). Other
+terms yield (3.53), (3.56) and traces of (3.54), (3.55). These are e xactly the equations used
+in Subsection 3.4 to derive dynamical equations.
+4.3 Massless and partially massless limits
+To analyze the massless and partially limits it is convenient to use the ty pe-II scaling gauge
+(2.41), (2.42). Let us fix the coefficient as−1,1in (4.6) in such a way that the product
+ak,1σ(1)2
++(k,0) in (4.19) be independent of mandλ. This insures non-zero variation in the
+flat massless case. Since σ(1)2
++(k,0) is proportional to N(1)−1(0) (2.33), (2.41), (2.45) one
+should demand
+as−1,1=a|N(1)(0)|, (4.22)
+whereais a coefficient independent of mandλ.
+Forλ2/ne}ationslash= 0 andm= 0 the action still has the form (4.18) and its variation amounts to
+(4.19). As discussed in Section 3.5, the 1-form fields valued in Y(s−1,l) with arbitrary
+l≤s−1 decouple from the other fields. The part of the action for these fi elds has the form
+Sm=0=as−1,l/summationdisplay
+1≤l≤s−1/an}bracketle{tR{2}(s−1,l)|R{2}(s−1,l)/an}bracketri}ht (4.23)
+Being formulated in terms of the 1-forms Wa(s−1),b(l)this is the action of [19] for a spin s
+massless particle in ( A)dSd. The leftover part of (4.18) is the action of a massive spin ( s−1)
+particle.
+As discussed in Section 3.5, at λ2= 0, 1-forms decompose into ssets, each constituted
+by the fields valued in Y(s′−1,l) withl≤s′−1 and various fixed s′<s. The action (4.18)
+also decomposes into sparts. The condition (4.22) guarantees that the variation is differe nt
+from zero. Although the coefficients ak,lthat have the form (2.41), (2.42), (4.7)
+ak,l=z(k,l)/vextendsingle/vextendsingle/vextendsingle/producttextl−1
+i=0/parenleftbig
+m2+λ2(s−l−1)(s+l+d−4)/parenrightbig/vextendsingle/vextendsingle/vextendsingle(4.24)
+31are singular in the flat massless limit λ→0,m→0, the related terms in the action (4.18)
+forma totalderivative asisobvious fromthefact that they do not contribute to thevariation
+oftheaction. Droppingtheseterms, theactionremainsgaugeinva riant uptototalderivative
+terms. However, it cannot be written in terms of manifestly gauge in variant curvatures for
+m=λ= 0, giving a combination of the actions found originally in [17].
+In the partially massless limit, the action (4.18) also decomposes into t wo parts in agree-
+ment with the pattern described in Section 3.5. It yields equations fo r a partially massless
+field of spin sand depthtand for a massive spin tfield of mass ms(2.46).
+In the type-II scaling gauge the partially massless part of the actio n contains the coeffi-
+cientsak,lforl≥t+1, that tend to infinity at m→mtalthough the variation (4.19) remains
+finite. Analogously to the flat massless case, the singular terms com bine into total deriva-
+tives and can be dropped from the action. The remaining part of the action is non-singular
+and yields correct equations. However, it cannot be written in term s of the curvatures. Note
+that this does not contradict to the results of [21], where the act ion for a single partially
+massless field is given in terms of curvatures because our descriptio n involves non-minimal
+curvatures that differ from those of [21]. (The main difference is tha tσ(1)2
+−(k,t+1) (3.65)
+is nonzero for the non-minimal curvatures but vanishes in the curv atures of [21].)
+5 Conclusion
+In this paper we found the explicit form of (Stueckelberg) gauge inv ariant linearized cur-
+vatures for symmetric massive HS fields in d-dimensional Minkowski and ( A)dSspace. In
+terms of these curvatures we constructed the manifestly gauge invariant action as well as
+full unfolded field equations for free symmetric massive HS fields.
+An interesting problem for the future is to extend obtained results to general massive
+mixed symmetry fields. Note that the unfolded equations for mixed- symmetry fields have
+been recently analyzed in [29] via radial reduction of the frame-like f ormulation of mixed
+symmetry fields in Minkowski space developed by Skvortsov [34]. The manifestly gauge
+invariant action for mixed-symmetry massive fields in ( A)dSremains to be worked out.
+The challenging problem is to extend obtained results to interacting m assive HS fields
+that may help to establish the correspondence between HS gauge t heory and String Theory.
+Acknowledgments
+We acknowledge with gratitude the collaboration at the early stage o f this work and many
+useful discussions with E. Skvortsov. D.P. is grateful to A. Artsu kevich for fruitful dis-
+cussions. This research was supported in part by, RFBR Grant No 0 8-02-00963, LSS No
+1615.2008.2 and Alexander von Humboldt Foundation Grant PHYS016 7 and by Grant of
+UNK.
+32Appendix A: Conventions
+In this paper we use the following conventions. Space-time is d-dimensional Minkowski or
+(A)dSspace. Both base indices µ,ν,...(i.e., indices of differential forms) and fiber indices
+a,b,...run from 0 to d−1. The fiber space is endowed with the mostly minus Minkowski
+metricηab. Upper and lower indices denoted by the same letter are contracte d. Upper or
+lower symmetrized indices can be also denoted by the same letter. A n umber of symmetrized
+indices is often indicated in brackets by writing e.g. a(k) instead of ( a1...ak). In this paper
+we deal with traceless tensors that possess symmetries of two-r ow Young tableaux. Indices of
+the first and second rows are separated by comma, i.e., the notation Ca(k),b(l)automatically
+implies that
+Cnna(k−2),b(l)= 0, Cna(k−1),nb(l−1)= 0, Ca(k),
+nnb(l−2)= 0, Ca(k),ab(l−1)= 0.
+Sometimes instead of writing indices of tensors that possess symme tries of a two-row Young
+tableau, we use the shorthand notation indicating the lengths of ro ws in brackets. For
+instance
+C(k,l)∼Ca(k),b(l).
+We often deal with similar objects for 0- and 1-forms. To distinguish between these
+objects we endow them with subscripts (0) and (1) which refer to t he degree of a differential
+form they act on. No subscript means that the formula is the same f or 0- and 1-forms.
+Appendix B: Arbitrary scaling
+Here we present some useful expressions valid for arbitrary N(0)(k,l),n(0)(k,l),N(1)(k,l) and
+n(1)(k,l).
+The coefficients f(0)(s,l) inσ(0)1
+−(s,l) (3.17), which are responsible for gluing of the Weyl
+module to the sector of mixed 0- and 1-forms, are defined by (2.15) up to an overall factor
+f(0)(s,l)
+f(0)(s,l−1)=F(0)(s,l)
+F(0)(s−1,l)s−l+1
+s−l, l<s (B.1)
+andf(0)(s,s) = 0 (2.10).
+The compatibility condition (3.5) with p= 0 implies
+α(k,l+1) =α(k,l)/radicalBigg
+N(1)(k,l)
+N(0)(k,l), α(k+1,l) =α(k,l)/radicalBigg
+n(1)(k,l)
+n(0)(k,l).(B.2)
+Eq. (B.2) expresses any α(k,l) in terms of α(0,0)
+α(k,l) =α(0,0)/radicaltp/radicalvertex/radicalvertex/radicalbt/producttextk−1
+i=0n(1)(i,0)/producttextl−1
+j=0N(1)(k,j)
+/producttextk−1
+i=0n(0)(i,0)/producttextl−1
+j=0N(0)(k,j). (B.3)
+(Note that Eqs. (B.2) are consistent by virtue of (2.25).) Eq. (4.4) can be used to express
+ak,lin terms of as−1,1:
+ak,l=as−1,1/producttextl−1
+i=1N(1)(s−1,i)/producttexts−2
+j=kn(1)(j,l). (B.4)
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+35
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+arXiv:1001.0063v1  [cs.AI]  31 Dec 2009Ona modelfor integratedinformation∗
+AlessandroEpasto1EnricoNardelli1
+1. Dept. ofMathematics,Univ. ofRomaTorVergata,Roma,Ita ly.
+Printed: September15,2018
+Abstract
+In this paper we give a thorough presentation of a model propo sed by Tononi et al. for
+modeling integrated information , i.e. how much information is generated in a system tran-
+sitioning from one state to the next one by the causal interac tion of its parts and above and
+beyondthe information given by the sum of its parts. We also provide s a more general for-
+mulation of such a model, independent from the time chosen fo r the analysis and from the
+uniformity of the probability distribution at the initial t ime instant. Finally, we prove that
+integrated information is null for disconnected systems.
+Keywords : integrated information, effective information, informa tion theory, neural networks, proba-
+bilistic boolean networks.
+1 Introduction
+Theterm integrated information (denotedφ,for short) hasbeenintroduced byGiulioTononi [5,6,10]to
+characterize the capacity of asystem tointegrate informat ion acquired byitsparts. Informally speaking,
+theintegrated information ownedbyasystem inagivenstate canbedescribed astheinformation (inthe
+TheoryofInformation sense) generated byasysteminthetra nsition fromonegivenstatetothenextone
+asaconsequence ofthecausalinteractionofitspartsabove andbeyondthesumofinformationgenerated
+independently byeach of its parts.
+Suchatheory wasfirstintroduced asalinear model[6,9–12], thenreformulated asadiscrete one[1,
+2,7] and was aimed at trying to formally capture what is consc iousness in living beings [3,7,8]. Its
+description is not always clear from a mathematical point of view, and to best of our knowledge this is
+the first formal description where all steps of the model are p resented in detail using the framework of
+probabilistic boolean networks.
+Inour presentation wealsoprovides amoregeneral formulat ion of themodel, whichcanbeused for
+analyzing the system at a generic time instant, and which doe s not require the assumption of uniformity
+of the probability distribution at theinitial time instant .
+We also formally prove here, for the first time in the literatu re to the best of our knowledge, that
+integrated information is null for a disconnected system, t hat is a system made up by independent com-
+ponents.
+∗Contact Author: EnricoNardelli, nardelli@mat.uniroma2.it
+1Version1.2.1—Last Revision: 30December2009 2
+The characterization of integrated information is based on another concept, always defined by Tononi
+andcoauthors, named effectiveinformation andmodelinghowmuchinformationisgainedbyanexternal
+observer on the previous state of a system from checking whic h is its current state, with respect to what
+can "a priori" be deduced on the previous state from the known dynamics of the system itself. Given
+this emphasis on the experimental side of the knowledge acqu isition process, wesuggest here to use the
+terms "experimental information" or "Galileian informati on" assynonyms for "effective information".
+Effective information is zero for static systems or uniform ly random systems, which is consistent
+witheverydayscientist’s experience. And,similarly, int egrated information isalsozerofordisconnected
+systems, independently from their kind.
+2 Probabilistic BooleanNetworks
+LetX= (V,E)be a directed graph with nboolean nodes, i.e. taking values in {0,1}. The value taken
+by a node is called also its state. Edge(u,v)∈Emodels the fact that node vgets in input the state of
+u. We assume time runs in discrete steps or instants, and nodes may change their value with the flow of
+timedepending on(the value of) thestates of their input nod es.
+Temporal evolution of state of node iis given by a law fi:{0,1}ni→ {0,1}computing state of i
+at the next time instant as a function only of the current stat e of itsni≤ninput nodes. Self loops are
+admitted. Nodescan all havethesamelaw for eachnode canhave itsspecific law. Inanycaselawsare
+constant withtime.
+We callXas defined above a Deterministic Boolean Network . To put things into context, Random
+Boolean Networks have been defined in the literature since many years, differi ng from the deterministic
+version only in the fact that each fiis randomly chosen when building the network. Random boolea n
+networks have been widely studied as model for gene expressi on inbiological systems.
+Various probabilistic versions of Boolean Networks have al so been defined, different from ours, for
+example [4], where each node at each time instant randomly ch ooses, according to a given probability
+distribution, the lawto beused from afinite domain of admiss ible laws.
+Our version of Probabilistic Boolean Network (PBN, for short) assumes the probabilistic law ri:
+{0,1}ni→[0,1]associated to node iprovides for each configuration of the states of the niinput nodes
+theprobability rithat atthenexttimeinstant node ihas(equivalently, isin)state 1(beingthen 1−rithe
+probability iisinstate 0). Itcan beshownthat this model can describe everynetwork d efined according
+tothe model introduced in[4]. In thefollowing weuse interc hangeably the terms system and network.
+At each time instant ta PBN can be in any of its 2nstates, we assume are provided of some arbitrary
+enumeration {xi}. State of network Xat timetis denoted Xt. A PBN can also be considered as a
+Markov chain witha finite space state .Version1.2.1—Last Revision: 30December2009 3
+APBNiscompletely described by its state transition matrix S, whose elements sijare:
+sij⊜p(Xt+1=xj|Xt=xi)
+thatis,element sijistheprobabilitythatattime t+1thenetworkisinstate xjconditioned tothefactthat
+attimetthenetworkwasinstate xi. Notethatsincetheprobabilistic lawassociated toeachno deistime
+constant, state transition matrix Sis alsotimeconstant, hence wecanspeak of an homogeneous Markov
+chain. Asquare matrix of real numbers is astate transition matrix if0≤sij≤1e/summationtextn
+i=1sij= 1.
+Values of sijcan be easily computed by means of the rkvalues for each node kas it follows. Let
+i=σnσn−1...σ1be the bit string representing the network state at instant t, whereσkrepresent state
+of nodekat instant t. The network state at the next instant t+1isj=σ′
+nσ′
+n−1...σ′
+1whereσ′
+kis the
+stateofnode kcomputedbylaw rkforinstant t+1. Itisσ′
+k= 1withprobability rk(σnσn−1...σ1)and
+σ′
+k= 0withprobability 1−rk(σnσn−1...σ1). Then
+sij=n/productdisplay
+k=1ρk
+whereρk=rk(σnσn−1...σ1)ifσ′
+k= 1andρk= 1−rk(σnσn−1...σ1)ifσ′
+k= 0.
+Let us denote with pt(i) =p(Xt=xi)probability that network is in state xiat instant t. State distribu-
+tion probability at t+1is given by:
+pt+1(xi) =2n/summationdisplay
+j=1pt(xj)sji
+Note that, even if Sis time constant (i.e., stationary), state probability dis tribution is not necessarily so.
+Letptbe the row vector withelements pt(i). Previous formula can be written inamatrix form as
+pt+1=pt·S
+and, denoting with Sithei-th column of S,it is
+pt+1(i) =pt·Si
+If for some tit ispt+1(·) =pt(·)then wesay the network isin the stationary regime . It is then
+p=p·S
+thatispisaneigenvector of Switheigenvalue 1. Notethatnoteveryeigenvector of Scanbeastationary
+probability distribution, since it has to fulfill probabili ty distribution constraints. For example, the null
+eigenvector isnever astationary probability distributio n.
+RowSiof thestate transition matrix provides the conditional pro bability distribution p(Xt+1|Xt=
+xi)describing network state at the instant nextto theone the network isin state xi.Version1.2.1—Last Revision: 30December2009 4
+Network dynamics can also be analyzed backwards in time. Let us assume that we have observed or
+measured that network at instant tis in a given state. Wecan then compute state distribution pr obability
+for instant t−1, that is we can compute the law by which states at instant t−1might have caused the
+stateactually observed ormeasured atinstant t. Thisisprovided bydefininga state backward-transition
+matrixB, describing probabilities obtained inverting through Bay es rule the relations between events.
+Its elements bijare:
+bij(t)⊜p(Xt−1=xj|Xt=xi)
+that can be written as
+bij(t) =p(Xt−1=xj, Xt=xi)
+p(Xt=xi)
+and applying again Bayes rule wehave
+bij(t) =p(Xt=xi|Xt−1=xj)p(Xt−1=xj)
+p(Xt=xi)=sjip(Xt−1=xj)
+p(Xt=xi)=pt−1(j)sji
+pt(i)=pt−1(j)sji
+pt−1·Si
+If at instant t−1state probability distribution is uniform then last formul a becomes
+bij(t) =sji/summationtext
+kski(1)
+Note that if state probability distribution is uniform then state backward-transition matrix Bis a kind of
+transpose ofthestatetransition matrix S. Notealsothatwhile Sistimeconstant, Bisnotso,ingeneral.
+RowBi(t)ofthestatebackward-transition matrix Bprovidestheconditional probabilitydistribution
+p(Xt−1|Xt=xi)describing network state at the instant previousto theone the network isin state xi.
+3 EffectiveInformation
+3.1 Introduction
+Effectiveinformationcanbeinformallydescribedas thequantityofinformationonpossiblepredecessors
+of current states acquiredadditionally from actually measuring the current network state with respect to
+what can be acquired from the knowledge of state transition m atrix only . We propose calling it exper-
+imental information orGalileian information , given the emphasis it gives to experimentally acquired
+knowledge with respect topurely theoretical knowledge. He requantity of information isintended inthe
+standard sense of the Shannon’s Information Theory.
+Themainquestion effectiveinformations answerstois: ifn etworkobservation findsthatitscurrent state
+isxi, which is the additional knowledge provided by this measure with respect to what can be known
+on the network by its state transition matrix only, i.e. with out knowing which is the current state of the
+network?
+Still remaining at the informal level this additional knowl edge can be described as the reduction in
+uncertainty provided by the actual measurement with respec t to the uncertainty existing on the basis of
+the state transition matrix only.Version1.2.1—Last Revision: 30December2009 5
+On one side there are those systems whose regime trajectory i n the space state is a deterministic
+cycle. For such systems the observation provides an effecti ve information of log2kbits1(wherekisthe
+number of the nodes on the cycle, i.e. its length). Since a det erministic closed trajectory of length kin
+the state space corresponds to a suitable subset of krows of the state transition matrix each containing
+exactly one value 1, and since before measuring the system the uncertainty is ma ximum – given that the
+system can be in any of these kstates – while after measuring the systems it is univocally k nown the
+predecessor of the current state, the information acquired through observation is maximum and equal,
+according to thestandard wayof measuring information, to logkbits.
+On the other side there are those systems whose behavior in th e state space is uniformly random,
+that isthosesystemswhereeachstatecanbe,withequal prob ability, thepredecessor ofthecurrent state.
+Measuring the actual current state in these systems provide s an effective information of 0bits since no
+reduction inuncertainty isprovided through the observati on (complete uncertainty both before and after
+themeasurement). Alsofor completely static systems, that issystems whosestateisconstant whiletime
+runs there is no reduction in uncertainty provided through t he observation (no uncertainty either before
+or after the measurement).
+3.2 Formaldefinition
+Wedefine theeffective information obtained by observing th at system Xis instate xiat instant tas
+ei(t,xi)⊜DKL(Bi(t)||Xt−1) (2)
+whereDKLis theKullback-Leibler divergence2. Then
+ei(t,xi) =/summationdisplay
+jbij(t)logbij(t)
+p(Xt−1=xj)
+=−H(Bi(t))−/summationdisplay
+jbij(t)logp(Xt−1=xj)
+Our definition is a generalization of the one provided by Tono ni and coauthors (cfr. equations 1A and
+1B of [1]). Ours in fact allows to study system behavior for ea ch time instant and for each probability
+distribution X0,whilein[1]thetimeinstantunderinvestigation isalways t= 1anditisalwaysassumed
+probability distribution X0istheuniform one. Ourformulation henceallowstomodel bot hthetransient
+and the stationary regime of asystem.
+For thecase whenthe state probability distribution Xt−1is uniform the formula above becomes:
+ei(t,xi) =−H(Bi(t))−/summationdisplay
+jbij(t)log1
+2n
+=−H(Bi(t))+n/summationdisplay
+jbij(t)
+=n−H(Bi(t))
+1from now onall logarithms are tothe base 2
+2TheKullback-Leiblerdivergence(ordistance)ofprobabil itydistribution q(x)fromprobabilitydistribution p(x)isdefined
+asDKL(p||q)⊜/summationtext
+x∈Ωxp(x)logp(x)
+q(x)=/angbracketleftBig
+logp(x)
+q(x)/angbracketrightBig
+pand note it isasymmetric.Version1.2.1—Last Revision: 30December2009 6
+Effectiveinformationintheregimephaseofasystemisprov idedbyconsidering equation(2)inthelimit
+for the instant ttending toinfinity
+ei(xi)⊜DKL(Bi||X∞)
+whereBiedX∞are the stationary probability distributions defined by the limits, if they exist, of the
+probability distributions for instant t,which describe the regime phase of the system. That is:
+p(X∞=xi)⊜lim
+t→∞p(Xt=xi)⊜pi
+and
+p(Bi=xj)⊜p(X∞=xj|X∞=xi) =sjipj
+pi
+hence
+ei(xi) =/summationdisplay
+jbijlogbij
+p(X∞=xj)=−H(Bi)−/summationdisplay
+jbijlogpj
+A system which has a uniformly random behavior in the regime p hase has H(Bi) =n, since state
+probability distribution p(Xt−1|Xt)isp(xj) =1
+2n,hence
+ei(xi) =−n−/summationdisplay
+j1
+2nlog1
+2n=/summationdisplay
+jn
+2n−n=n−n= 0
+A system completely static in the regime phase, i.e. which re mains fixed in a single attraction state xi,
+hasH(Bi) = 0since the unique possible predecessor is xiitself and p(xj) = 0ifi/ne}ationslash=jfrom which we
+have
+ei(xi) = log1 = 0
+Note that sum is computed only on observable states (i.e. whe rep(xj)/ne}ationslash= 0), to avoid the undeterminate
+form0log0
+0.
+Asystemhavingintheregimephaseasinglecyclicattractor containing allstates, i.e. adeterministic
+closedtrajectoryinthespacestatewalkingthroughallsta tes,hasH(Bj) = 0sinceeachstatehasexactly
+one predecessor while p(xj) =1
+2nand hence
+ei(xi) = 0−log1
+2n=n
+The same holds, assuming the stationary state space distrib ution is uniform, when the system has more
+cyclic attractors partitioning all the space state.
+If the system has a single cyclic attractor with k <2nstates (or more cyclic attractors partitioning a
+subset of size k <2nof all states, still assuming a uniform stationary state spa ce distribution) then it is
+ei(xi) = logk.
+The analysis in [1] assumes the maximum uncertainty and unif ormity on the initial systems conditions
+andisfocused oncomputing effectiveinformation intheins tant right after theinitial state. Theformula-
+tion of effective information in [1] istherefore the follow ing particular case of ours:
+ei1(xi) =DKL(Bi(1)||X0)Version1.2.1—Last Revision: 30December2009 7
+Note also that since for this particular case the assumption s used for the derivation of (1) hold, it can be
+written
+bij(1) =sji/summationtext
+kski
+3.3 Effective informationofsubsets
+For the definition of integrated information it is required t o define how to measure effective information
+for subsets of a given network X. LetA⊆X. WhenXis in state xiwedenote with πA(xi) =Axithe
+state ofA. LetAtbe the random variable representing state of Aat instant t. We can define for Astate
+transition matrixASand state backward-transition matrixABin analogy with thegeneral case as
+Asij⊜p(At+1=aj|At=ai)
+and
+Abij(t)⊜p(At−1=aj|At=ai)
+Both can be obtained from Sep(·)after some long but straightforward computations. Intuiti vely and
+informally speaking, the computation is based on summing tr ansition probabilities over all states of X
+which are equivalent withrespect to subset A,averaged withtheir state probabilities.
+Now, all definitions introduced for a network Xcan be applied to any of its subset of nodes Aby
+substituting in theprevious formulas S,B,andXrespectively withAS,AB,andA. Wethen obtain
+ei(t,A,ah)⊜DKL(ABh(t)||At−1) (3)
+4 IntegratedInformation
+Wearenowreadytoformallydefineintegrated information, t hatisthequantity ofinformation generated
+inasystemtransitioning fromonestatetothenextbythecau salinteractionofitsparts,aboveandbeyond
+the quantity of information generated independently by eac h of its parts.
+Given a system XletV⊆Xand{Mk}a partition of Vinmsubsets. Let Mk(t)be the random
+variables describing the state of the k-th component of the partition at instant t. LetXbe in state xi
+at instant t. ThenVat the same instant is in stateVxiand thek-th component is in stateMkxi. In the
+following weuse vhandµkas ashorthand forVxiandMkxi,respectively.
+Partition-dependent integrated information is first define d for a subset Vas a function of partition
+{Mk}, time instant t,and current state vhas
+φ(t,V,{Mk},vh)⊜ei(t,V,vh)−m/summationdisplay
+k=1ei(t,Mk,µk) (4)
+Value computed by this formula clearly depends on the consid ered partition. Tipically, an unbalanced
+partition produces alowervalueof φ(see[1]). Hencethefollowingnormalization functionisin troduced
+N(t,V,{Mk},vh)⊜(m−1)min
+k{H(Mk(t))}Version1.2.1—Last Revision: 30December2009 8
+Then,the MinimumInformation Partition (MIP)isdefinedasthepartition providing theminimumvalue
+for the integrated information after the normalization pro cess, that is
+PMIP(t,V,vh)⊜argmin
+P/braceleftBigφ(t,V,P,v h)
+N(t,V,P,v h)/bracerightBig
+The above formula has been defined by Tononi for generic parti tions, but in all of its papers and here it
+isonly discussed the case of bi-partitions, i.e. partition s intwosubsets.
+Integrated information φfor subset V,instatevhat instant t,isnow formally defined asthe valueof the
+partition-dependent integrated information computed on MIP,that is
+φ(t,V,vh)⊜φ(t,V,PMIP(t,V,vh),vh)
+Andit isnow possible to formally define the value of integrat ed information for the whole system X. A
+subsetV⊆Xhavingφ >0iscalledcomplex. If itisnot aproper subset of another subset withalarger
+φit is called main complex . The value of integrated information of X, in statexiat instant t, is defined
+asthe value of integrated information of its main complex of maximum value.
+φ(t,xi)⊜max
+V⊂Xφ(t,V,PMIP(t,V,vh),vh)
+The value of integrated information averaged over all state s of the system is provided through the state
+distribution probability pt(·), that is
+φ(t)⊜/summationdisplay
+xi∈Xφ(t,xi)pt(i)
+5 Integratedinformationin disconnected systems
+Intuitively, any system having a partition in two independe nt subsets, i.e. that can be partitioned in two
+subsets suchthat nonodeinasubset affectsthestatevalueo fnodesintheother subset, should havezero
+asvalue of its integrated information.
+We now give a formal proof of this property, to the best of our k nowledge never appeared in the
+literature. We consider the value of integrated informatio n assuming at instant t−1the system has a
+uniform state probability distribution, consistently wit hdiscussion in[1]. Rememberthat for asubset V
+of the system Xinstatexhweusevhas ashorthand forVxh, the restriction of xhtonodes in V.
+Theorem 1(Integrated information ina disconnected networ k)LetA′andA′′be two disjoint sub-
+sets of a network X,A′∪A′′=V⊆X. Let us denote with vhthe current state of V, and with a′
+hea′′
+h
+the current states of subsets A′andA′′, respectively.
+For each state vhand timeinstant tit is
+φ(t,V,{A′,A′′},vh) = 0Version1.2.1—Last Revision: 30December2009 9
+Proof.Fromthedefinition (4)of partition-dependent integrated i nformation andthedefinition (3)of the
+effective information for asubset it is
+φ(t,V,{A′,A′′},vh) =ei(t,V,vh)−ei(t,A′,a′
+h)−ei(t,A′′,a′′
+h)
+=DKL(VBh(t)||Vt−1)−DKL(A′Bi(t)||A′
+t−1)−DKL(A′′Bj(t)||A′′
+t−1)(5)
+From the definition of the Kullback-Leibler divergence it is
+DKL(VBh(t)||Vt−1) =−H(VBh(t))−/summationdisplay
+jVbhj(t)logp(Vt−1=vj)
+Remember thatVBh(t)isaconditional probability distribution for the state pre ceding the current one
+p(VBh(t) =vj) =p(Vt−1=vj|Vt=vh)
+=p(A′
+t−1=a′
+j∧A′′
+t−1=a′′
+j|Vt=vh)
+Applying the chain rule of entropy it is
+H(VBh(t)) =H(A′
+t−1|Vt=vh)+H/parenleftBig
+(A′′
+t−1|Vt=vh)/vextendsingle/vextendsingleA′
+t−1/parenrightBig
+and given the independence between A′′andA′it follows that
+H(VBh(t)) =H(A′
+t−1|Vt=vh)+H(A′′
+t−1|Vt=vh)
+=H(A′
+t−1|A′
+t=ah′)+H(A′′
+t−1|A′′
+t=ah′′)
+=H(A′Bh′(t))+H(A′′Bh′′(t))
+From the assumption of uniform state probability distribut ion att−1it is
+DKL(VBh(t)||Vt−1) =−H(VBh(t))+|V|
+DKL(A′Bh(t)||A′
+t−1) =|A′|−H(A′Bh(t))
+DKL(A′′Bh(t)||A′′
+t−1) =|A′′|−H(A′′Bh(t))
+and substituting the above right members for the left ones in equation (5) and considering that |V|=
+|A′|+|A′′|weobtain
+φ(t,V,{A′,A′′},vh) =|V|−|A′|−|A′′|−H(VBh(t))+H(A′Bh′(t))+H(A′′Bh′′(t)) = 0
+✷
+6 Conclusions
+In this paper we have given a thorough presentation of a model proposed by Giulio Tononi [5,6,10]
+for modeling integrated information , i.e. how much information is generated in a system by causal
+interaction of its parts and above and beyond the information given by the sum of its parts. The modelVersion1.2.1—Last Revision: 30December2009 10
+was aimed at trying to formally capture what is consciousnes s in living beings [3,7,8] and the reader is
+referred to Tononi’s papers for detailed motivations of the model.
+Wehave considered the discrete version of the model [1,2,7] . The original papers describing the model
+are not always fully clear in their mathematical formulatio n and here we have given the first formal
+description of such amodel whereall steps aredetailed pres ented.
+In doing so we have provided a more general formulation of suc h a model, which is independent
+fromthetimechosen for theanalysis andfromtheuniformity oftheprobability distribution at theinitial
+timeinstant.
+Finally, we have also given here the first formal proof that a s ystem made up by independent parts
+has avalue of integrated information equal to zero.
+Acknowledgments. Wewould like to thank Luciano Gualà and Guido Proietti for us eful and interest-
+ing discussions related tothe work here described.
+References
+[1] David Balduzzi and Giulio Tononi. Integrated informati on in discrete dynamical systems: Motivation and
+theoreticalframework. PLoSComputationalBiology ,4(6),2008.
+[2] David Balduzzi and Giulio Tononi. Qualia: The geometry o f integrated information. PLoS Computational
+Biology,5(8),2009.
+[3] GeraldM.EdelmanandGiulioTononi. Auniverseofconsciousness: howmatterbecomesimaginatio n. New
+York,NY: Basic Books,1edition,2000.
+[4] IlyaShmulevich,EdwardR.Dougherty,SeungchanKim,an dWeiZhang. Probabilisticbooleannetworks: A
+rule-baseduncertaintymodelforgeneregulatorynetworks .Bioinformatics ,18(2):261–274,2002.
+[5] GiulioTononi. Informationmeasuresforconsciousexpe rience.ArchiviItalianidiBiologia ,139(4):367–371,
+2001.
+[6] GiulioTononi. Aninformationintegrationtheoryofcon sciousness. BMCNeuroscience ,5(1):42,2004.
+[7] Giulio Tononi. Consciousness as integrated informatio n: a provisional manifesto. Biol. Bull. , December
+2008.
+[8] Giulio Tononi and Gerald M. Edelman. Consciousness and c omplexity. Science, 282(5395):1846–1851,
+1998.
+[9] GiulioTononi,GeraldM. Edelman,andSpornsOlaf. Compl exityandcoherency: integratinginformationin
+thebrain. TrendsinCognitiveSciences ,2(11),1998.
+[10] GiulioTononiandOlaf Sporns. Measuringinformationi ntegration. BMCNeuroscience ,4,December2003.
+[11] Giulio Tononi, Olaf Sporns, and Gerald M. Edelman. A mea sure for brain complexity: relating functional
+segregationand integration in the nervous system. Proceedings of the National Academy of Sciences of the
+UnitedStatesofAmerica ,91(11):5033–5037,1994.
+[12] GiulioTononi,OlafSporns,andGeraldM.Edelman. Acom plexitymeasureforselectivematchingofsignals
+by the brain. Proceedings of the National Academy of Sciences of the Unite d States of America , 93:3422,
+1996.
\ No newline at end of file
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+arXiv:1001.0064v2  [math.FA]  17 Jun 2010SSDB spaces and maximal monotonicity
+by
+Stephen Simons
+Abstract
+In this paper, we develop some of the theory of SSD spaces and S SDB spaces, and
+deduce some results on maximally monotone multifunctions o n a reflexive Banach space.
+1 Introduction
+In this paper, we develop enough of the theory of SSD spaces an d SSDB spaces that we
+can obtain some significant results on maximally monotone mu ltifunctions on a reflexive
+Banach space. With a few minor additions, this is a written ve rsion of the lecture with
+the same title delivered at the IX ISORA meeting in Lima, Peru , in October 2009, and
+we will not attempt to give a comprehensive exposition of the theory of SSD spaces and
+SSDB spaces. For this, we refer the reader to [14] and [15], fr om which many of the proofs
+given here are taken.
+Many of the original results on maximally monotone multifun ctions on a reflexive
+Banach space were obtained using Brouwer’s fixed–point theo rem either directly or in-
+directly. The approach given here is based on convex analysi s — more specifically the
+Fenchel duality theorem. In Section 2, we give the three vers ions of the Fenchel duality
+theorem that we will use in this paper.
+In Section 3, we introduce the concepts of SSD space ,q–positive set and the functions
+Φ·, which are the generalizations to SSD spaces of Fitzpatrick functions of monotone
+sets. We also introduce the q–positive sets Pq(·) determined by certain convex functions.
+Despite the fact that this section uses the idea of conjugate function from convex analysis,
+the arguments used are essentially algebraic, apart from th e disguised differentiability
+argument of Lemma 3.7.
+In Section 4, we introduce the concept of SSDB space , which is a SSD space with
+an appropriate Banach space structure. The main results of t his section are the two
+“pos–neg theorems”, Theorem 4.3 and 4.6 and the criterion fo r and properties of maximal
+q–positivity contained in Theorem 4.4.
+For the rest of this paper, we suppose that Eis a reflexive Banach space. In Section
+5, we show how E×E∗can be considered as an SSDB space, and how Theorems 2.4 and
+4.4 lead to results on maximal monotonicity. Theorem 5.1 and Theorem 5.4 will be used
+in Section 6.
+Uptothispoint, our discussion ofmonotonicityhasbeen int ermsofmonotonesubsets
+ofE×E∗. In Section 6 we move the emphasis to monotone multifunction s fromEinto
+E∗. In Theorem 6.3, we show how Theorem 5.1 gives Rockafellar’s surjectivity theorem,
+and in Theorem 6.5, we show how Theorem 5.4 gives sufficient con ditions for the sum of
+maximally monotone multifunctions to be maximally monoton e.
+In the final section of this paper, Section 7, we show how the po s–neg theorem,
+Theorem 4.6, together with the simple properties of the refle ction maps ρ1,ρ2:E×E∗→
+E×E∗defined by ρ1(x,x∗) := (−x,x∗) andρ2(x,x∗) := (x,−x∗) lead to an abstract
+Hammerstein theorem.
+1SSDB spaces and maximal monotonicity
+The author would like to express his appreciation to Radu Ioa n Bot ¸ for some very
+constructive comments on the first version of this paper.
+2 Versions of the Fenchel duality theorem
+All vector spaces in the paper will be real. IfXis a nonzero vector space and f:F/mapsto→
+]−∞,∞]thenwe writedom f:={x∈F:f(x)∈R}. The set dom fistheeffective domain
+off.fis said to be properif domf/\e}atio\slash=∅. We write PC(X) for the set of proper convex
+functions from Xinto ]−∞,∞]. IfXis a nonzero normed space we write PCLSC(X)
+for{f∈ PC(X):fis lower semicontinuous }. IfEandFare nonzero vector spaces,
+/a\}⌊ra⌋ketle{t·,·/a\}⌊ra⌋ketri}ht:E×F→Ris a bilinear form and f∈ PC(E) then the Fenchel conjugate, f∗, off
+with respect to /a\}⌊ra⌋ketle{t·,·/a\}⌊ra⌋ketri}htis defined for y∈Fby
+f∗(y) := supx∈E/bracketleftbig
+/a\}⌊ra⌋ketle{tx,y/a\}⌊ra⌋ketri}ht−f(x)/bracketrightbig
+.
+The commonest (but not the only) use of this notation is when Eis a normed space, Fis
+the dual, E∗, ofEand/a\}⌊ra⌋ketle{t·,·/a\}⌊ra⌋ketri}htis the duality pairing.
+We start off by stating a result that is an immediate consequen ce of Rockafellar’s
+version of the Fenchel duality theorem (see [11, Theorem 1, p . 82–83] for the original
+version and Z˘ alinescu, [18, Theorem 2.8.7, p. 126–127] for more general results):
+Theorem 2.1. LetFbe a nonzero normed space, f:F/mapsto→]−∞,∞]be proper and convex,
+g:F/mapsto→Rbe convex and continuous, and f+g≥0onF. Then there exists z∗∈F∗such
+thatf∗(z∗)+g∗(−z∗)≤0.
+Theorem 2.2 below was first proved by Attouch–Brezis/parenleftbig
+this follows from [1, Corollary
+2.3, pp. 131–132]/parenrightbig
+– there is a somewhat different proof in Simons, [14, Theorem 1 5.1, p.
+66], and a much more general result was established in Z˘ alin escu, [18, Theorem 2.8.6, pp.
+125–126]. We note that the result contained in Simons, [14, T heorem 8.4, p. 46] implies
+both Theorem 2.1 and Theorem 2.2, and that [14, Theorem 7.4, p . 43] gives a sharp lower
+bound on the possible values of /⌊ard⌊lz∗/⌊ard⌊l.
+Theorem 2.2. LetEbe a nonzero Banach space, f, g∈ PCLSC (E),f+g≥0onEand/uniontext
+λ>0λ/bracketleftbig
+domf−domg/bracketrightbig
+be a closed linear subspace of E. Then there exists z∗∈E∗such
+thatf∗(z∗)+g∗(−z∗)≤0.
+Notation 2.3. IfEandFare nonzero normed spaces, we define the dual of E×Fto be
+E∗×F∗under the pairing/angbracketleftbig
+(x,y),(x∗,y∗)/angbracketrightbig
+:=/a\}⌊ra⌋ketle{tx,x∗/a\}⌊ra⌋ketri}ht+/a\}⌊ra⌋ketle{ty,y∗/a\}⌊ra⌋ketri}ht/parenleftbig
+(x,y)∈E×F,(x∗,y∗)∈
+E∗×F∗/parenrightbig
+. We then define the projection maps π1,π2byπ1(x,y) :=xandπ2(x,y) :=y.
+We end this section with a bivariate generalization of Theor em 2.2, which was first
+proved in Simons–Z˘ alinescu, [16, Theorem 4.2, pp. 9–10]. T here was a simpler proof given
+in Simons, [14, Theorem 16.4, pp. 68–69]. The hypothesis of T heorem 2.4 is that h(x,·)
+is the “inf–convolution” of f(x,·) andg(x,·), and the conclusion is that h∗(·,y∗) is the
+“exact inf–convolution” of f∗(·,y∗) andg∗(·,y∗).
+2SSDB spaces and maximal monotonicity
+Theorem 2.4. LetEandFbe nonzero Banach spaces, f, g:E×F/mapsto→]−∞,∞]be
+proper, convex and lower semicontinuous,/uniontext
+λ>0λ/bracketleftbig
+π1domf−π1domg/bracketrightbig
+be a closed linear
+subspace of Eand, for all (x,y)∈E×F,
+h(x,y) := inf/braceleftbig
+f(x,u)+g(x,v):u, v∈F, u+v=y/bracerightbig
+>−∞.
+Then, for all (x∗,y∗)∈E∗×F∗= (E×F)∗,
+h∗(x∗,y∗) = min/braceleftbig
+f∗(s∗,y∗)+g∗(t∗,y∗):s∗, t∗∈E∗, s∗+t∗=x∗/bracerightbig
+.
+3 SSD spaces
+Definition 3.1. We will say that/parenleftbig
+B,⌊·,·⌋/parenrightbig
+is asymmetrically self–dual space (SSD space)
+ifBis a nonzero real vector space and ⌊·,·⌋:B×B→Ris a symmetric bilinear form. In
+this case, we will alwayswriteq(b) :=1
+2⌊b,b⌋(b∈B). (“q” stands for “quadratic”.)
+Now let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be an SSD space and A⊂B. We say that Aisq–positive ifA/\e}atio\slash=∅
+and
+b,c∈A=⇒q(b−c)≥0.
+In this case, since q(0) = 0,
+b∈A=⇒infq(A−b) = 0. (1)
+We then define Φ A:B→]−∞,∞] by
+ΦA(b) := supA/bracketleftbig
+⌊·,b⌋−q/bracketrightbig
+(b∈B). (2)
+ΦAis a generalization to SSD spaces of the “Fiztpatrick functi on” of a monotone set,
+which was originally introduced in [6] in 1988, but lay dorma nt until it was rediscovered
+by Mart´ ınez-Legaz and Th´ era in [8] in 2001. We note then tha t, for all b∈B,
+ΦA(b) =q(b)−infa∈A/bracketleftbig
+q(a)−⌊a,b⌋+q(b)/bracketrightbig
+=q(b)−infa∈Aq(a−b) =q(b)−infq(A−b)./bracerightBigg
+(3)
+From (1),
+ΦA=qonA. (4)
+Thus Φ A∈ PC(B). We say that Aismaximally q–positive ifAisq–positive and Ais not
+properly contained in any other q–positive set. In this case, if b∈Band infq(A−b)≥0
+then clearly b∈A. In other words,/parenleftbig
+b∈B\A=⇒infq(A−b)<0/parenrightbig
+. From (1),
+infq(A−b)≤0 and/parenleftbig
+infq(A−b) = 0⇐⇒b∈A/parenrightbig
+. Thus, from (3)
+ΦA≥qonBand/parenleftbig
+ΦA(b) =q(b)⇐⇒b∈A/parenrightbig
+. (5)
+We make the elementary observation that if b∈Bandq(b)≥0 then the linear span Rb
+of{b}isq–positive.
+3SSDB spaces and maximal monotonicity
+We now give some examples of SSD spaces and their associated q–positive sets. These
+examples are taken from [14, pp. 79–80].
+Example 3.2. LetBbe a Hilbert space with inner product ( b,c)/mapsto→ /a\}⌊ra⌋ketle{tb,c/a\}⌊ra⌋ketri}ht.
+(a) If, for all b,c∈B,⌊b,c⌋:=/a\}⌊ra⌋ketle{tb,c/a\}⌊ra⌋ketri}htthenq(b) =1
+2/⌊ard⌊lb/⌊ard⌊l2and every nonempty subset of
+Bisq–positive.
+(b) If, for all b,c∈B,⌊b,c⌋:=−/a\}⌊ra⌋ketle{tb,c/a\}⌊ra⌋ketri}htthenq(b) =−1
+2/⌊ard⌊lb/⌊ard⌊l2and theq–positive sets are
+the singletons.
+(c) IfB=R3and
+/floorleftbig
+(b1,b2,b3),(c1,c2,c3)/floorrightbig
+:=b1c2+b2c1+b3c3,
+thenq(b1,b2,b3) =b1b2+1
+2b2
+3. Here, if Mis any nonempty monotone subset of R×R
+(in the obvious sense) then M×Ris aq–positive subset of B. The set R(1,−1,2) is a
+q–positive subset of Bwhich is not contained in a set M×Rfor any monotone subset of
+R×R. The helix/braceleftbig
+(cosθ,sinθ,θ):θ∈R/bracerightbig
+is aq–positive subset of B, but if 0 < λ <1 then
+the helix/braceleftbig
+(cosθ,sinθ,λθ):θ∈R/bracerightbig
+is not.
+Example 3.3. LetEbe a nonzero Banach space and B:=E×E∗. For all ( x,x∗) and
+(y,y∗)∈B, we set/floorleftbig
+(x,x∗),(y,y∗)/floorrightbig
+:=/a\}⌊ra⌋ketle{tx,y∗/a\}⌊ra⌋ketri}ht+/a\}⌊ra⌋ketle{ty,x∗/a\}⌊ra⌋ketri}ht. Then/parenleftbig
+B,⌊·,·⌋/parenrightbig
+is an SSD space
+withq(x,x∗) =1
+2/bracketleftbig
+/a\}⌊ra⌋ketle{tx,x∗/a\}⌊ra⌋ketri}ht+/a\}⌊ra⌋ketle{tx,x∗/a\}⌊ra⌋ketri}ht/bracketrightbig
+=/a\}⌊ra⌋ketle{tx,x∗/a\}⌊ra⌋ketri}ht. Consequently, if ( x,x∗),(y,y∗)∈B
+then/a\}⌊ra⌋ketle{tx−y,x∗−y∗/a\}⌊ra⌋ketri}ht=q(x−y,x∗−y∗) =q/parenleftbig
+(x,x∗)−(y,y∗)/parenrightbig
+. Thus if A⊂BthenAis
+q–positive exactly when Ais a nonempty monotone subset of Bin the usual sense, and A
+is maximally q–positive exactly when Ais a maximally monotone subset of Bin the usual
+sense. We point out that any finite dimensional SSD space of th e form described here must
+haveevendimension. Thus cases of Example 3.2 with finite odd dimensio n cannot be of
+this form.
+Example 3.4./parenleftbig
+R3,⌊·,·⌋/parenrightbig
+isnotan SSD space with
+/floorleftbig
+(b1,b2,b3),(c1,c2,c3)/floorrightbig
+:=b1c2+b2c3+b3c1.
+(The bilinear form ⌊·,·⌋is not symmetric.)
+Definition 3.5. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be an SSD space. If f∈ PC(B) andf≥qonB, we write
+Pq(f) :=/braceleftbig
+b∈B:f(b) =q(b)/bracerightbig
+.
+We then note from (5) that
+ifAis maximally q–positive then A=Pq(ΦA). (6)
+Ifg∈ PC(B) andg≥ −qonBthen we write Nq(g) :=/braceleftbig
+b∈B:g(b) =−q(b)/bracerightbig
+.
+We now introduce the concept of intrinsic conjugate for an SSD space.
+4SSDB spaces and maximal monotonicity
+Definition 3.6. If/parenleftbig
+B,⌊·,·⌋/parenrightbig
+is an SSD space and f∈ PC(B), we write f@for the Fenchel
+conjugate of fwith respect to the pairing ⌊·,·⌋, that is to say,
+for allc∈B, f@(c) := supB/bracketleftbig
+⌊·,c⌋−f/bracketrightbig
+.
+The concepts introduced inDefinitions 3.5and3.6 arerelate dinour next result, which
+uses a disguised differentiability argument.
+Lemma 3.7. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be an SSD space, f∈ PC(B)andf≥qonB. Then
+f@=qonPq(f).
+Proof.Leta∈ Pq(f). Letλ∈]0,1[. For simplicity in writing, let µ:= 1−λ∈]0,1[.
+Then, for all b∈B,
+λ2q(b)+λµ⌊b,a⌋+µ2q(a) =q/parenleftbig
+λb+µa/parenrightbig
+≤f(λb+µa)
+≤λf(b)+µf(a) =λf(b)+µq(a).
+Thusλ2q(b) +λµ⌊b,a⌋ ≤λf(b) +λµq(a). Dividing by λand letting λ→0, we have
+⌊b,a⌋ ≤f(b) +q(a), that is to say ⌊a,b⌋ −f(b)≤q(a), and, taking the supremum
+overb,f@(a)≤q(a). On the other hand, f@(a)≥ ⌊a,a⌋−f(a) = 2q(a)−q(a) =q(a),
+completing the proof of Lemma 3.7. /square
+The next result gives a basic property of Φ A@.
+Lemma 3.8. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be an SSD space and Abe a nonempty q–positive subset of
+B. Then ΦA@≥ΦAonB.
+Proof.Letc∈B. Then, from (4),
+ΦA@(c) = supB/bracketleftbig
+⌊c,·⌋−ΦA/bracketrightbig
+≥supA/bracketleftbig
+⌊c,·⌋−ΦA/bracketrightbig
+= supA/bracketleftbig
+⌊c,·⌋−q/bracketrightbig
+= ΦA(c)./square
+We now introduce the important concepts of “BC–function” an d “TBC–function”.
+Definition 3.9. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be a SSD space and f,g∈ PC(B). We say that fis a
+BC–function if
+b∈B=⇒f@(b)≥f(b)≥q(b). (7)
+“BC” stands for “bigger conjugate”. We say that gis aTBC–function if
+b∈B=⇒g@(−b)≥g(b)≥ −q(b). (8)
+“TBC” stands for “twisted bigger conjugate”. Of course, gis a TBC–function if, and only
+if,gis a BC–function with respect to the SSD space/parenleftbig
+B,−⌊·,·⌋/parenrightbig
+Lemma 3.10. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be a SSD space and f∈ PC(B)be a BC–function. Then
+Pq/parenleftbig
+f@/parenrightbig
+=Pq(f).
+Proof.This follows from Lemma 3.7 and the inclusion Pq/parenleftbig
+f@/parenrightbig
+⊂ Pq(f), which is
+immediate from (7). /square
+5SSDB spaces and maximal monotonicity
+Theorem 3.11. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be an SSD space and Abe a maximally q–positive subset
+ofB. Then ΦAis a BC–function and Pq/parenleftbig
+ΦA@/parenrightbig
+=Pq/parenleftbig
+ΦA/parenrightbig
+=A.
+Proof.The first assertion follows from Lemma 3.8 and (5), and the sec ond assertion is
+immediate from Lemma 3.10 and (6). /square
+Remark 3.12. We shall see by combining (9) and Remark 5.5 that there may exi st a
+function that is both a BC–function and a TBC–function but no t of the form Φ Afor any
+nonempty q–positive set A.
+We now give two computational results. In the first of these, w e investigate the
+“translation” of a BC–function. This result will be used in t he “pos–neg” theorems,
+Theorem 4.3 and Theorem 4.6.
+Lemma 3.13. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be a SSD space, f∈ PC(B)be a BC–function and c∈B.
+We define fc∈ PC(B)byfc:=f(·+c)− ⌊·,c⌋ −q(c). Then fcis a BC–function,
+domfc= domf−candPq(fc) =Pq(f)−c.
+Proof.For allb∈B,
+fc@(b) = supd∈B/bracketleftbig
+⌊d,b⌋+⌊d,c⌋+q(c)−f(d+c)/bracketrightbig
+= supe∈B/bracketleftbig
+⌊e−c,b+c⌋+q(c)−f(e)/bracketrightbig
+= supe∈B/bracketleftbig
+⌊e,b+c⌋−⌊c,b⌋−f(e)/bracketrightbig
+−q(c)
+=f@(b+c)−⌊c,b⌋−q(c).
+It follows from (7) that fc@(b)≥f(b+c)−⌊b,c⌋−q(c) =fc(b) and
+fc(b) =f(b+c)−⌊b,c⌋−q(c)≥q(b+c)−⌊b,c⌋−q(c) =q(b).
+Consequently, fcis a BC–function. It is obvious that dom fc= domf−c. Further, since
+b∈ Pq(fc)⇐⇒f(b+c)−⌊b,c⌋−q(c) =q(b)
+⇐⇒f(b+c) =q(b+c)⇐⇒b+c∈ Pq(f),
+we have Pq(fc) =Pq(f)−c, as required. /square
+In the second computational result, we show how we can always obtain a TBC–
+function from a BC–function by an appropriate linear transf ormation.
+Lemma 3.14. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be an SSD space, ρ:B→Bbe a linear surjection such
+that, for all b,c∈B,/floorleftbig
+ρ(b),ρ(c)/floorrightbig
+=⌊b,−c⌋, andg∈ PC(B)be a BC–function. Then
+g◦ρandg◦(−ρ)are TBC– functions, ρdom(g◦ρ) = dom g,ρNq(g◦ρ) =Pq(g),
+−ρdom(g◦(−ρ)) = dom gand−ρNq(g◦(−ρ)) =Pq(g).
+Proof.We give the proof for g◦ρ. The proof for g◦(−ρ) is similar. For all c∈B,
+(g◦ρ)@(−c) = supb∈B/bracketleftbig
+⌊b,−c⌋−g/parenleftbig
+ρ(b)/parenrightbig/bracketrightbig
+= supb∈B/bracketleftbig
+⌊ρ(b),ρ(c)⌋−g/parenleftbig
+ρ(b)/parenrightbig/bracketrightbig
+= supd∈B/bracketleftbig
+⌊d,ρ(c)⌋−g(d)/bracketrightbig
+=g@/parenleftbig
+ρ(c)/parenrightbig
+≥g/parenleftbig
+ρ(c)/parenrightbig
+≥q/parenleftbig
+ρ(c)/parenrightbig
+=1
+2⌊ρ(c),ρ(c)⌋=1
+2⌊c,−c⌋=−q(c).
+6SSDB spaces and maximal monotonicity
+Thusg◦ρis a TBC– function. Further,
+b∈ρdom(g◦ρ)⇐⇒there exists c∈Bsuch that g/parenleftbig
+ρ(c)/parenrightbig
+∈Randb=ρ(c)
+⇐⇒there exists d∈Bsuch that g(d)∈Randb=d
+⇐⇒g(b)∈R⇐⇒b∈domg,
+from which ρdom(g◦ρ) = dom g, and
+b∈ρNq(g◦ρ)⇐⇒there exists c∈Bsuch that g/parenleftbig
+ρ(c)/parenrightbig
+=−q(c) andb=ρ(c)
+⇐⇒there exists c∈Bsuch that g/parenleftbig
+ρ(c)/parenrightbig
+=q/parenleftbig
+ρ(c)/parenrightbig
+andb=ρ(c)
+⇐⇒there exists d∈Bsuch that g(d) =q(d) andb=d
+⇐⇒b∈ Pq(g),
+from which ρNq(g◦ρ) =Pq(g). /square
+Lemma 3.15 is a subtler property of the bilinear form q. In the situation of Example
+3.3, Corollary 3.16 gives us Voisei–Z˘ alinescu [17, Propos ition 1].
+Lemma 3.15. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be an SSD space, f∈ PC(B),f≥qonBandb,c∈B.
+Then
+−q(b−c)≤/bracketleftBig/radicalbig
+(f−q)(b)+/radicalbig
+(f−q)(c)/bracketrightBig2
+.
+Proof.See [15, Lemma 2.6]. /square
+Corollary 3.16. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be an SSD space, f∈ PC(B),f≥qonBandb,c∈B.
+Then
+−q(b−c)≤2(f−q)(b)+2(f−q)(c).
+Proof.This is immediate from Lemma 3.15 and the Cauchy–Schwarz ine quality. /square
+Lemma 3.17 is suggested by Burachik–Svaiter, [5, Theorem 3. 1, pp. 2381–2382] and
+Penot, [9, Proposition 4(h)= ⇒(a), pp. 860–861]. Lemma 3.17 will be used in Theorem 4.4.
+Lemma 3.17. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be an SSD space, f∈ PC(B),f≥qonBandPq(f)/\e}atio\slash=∅.
+ThenPq(f)is aq–positive subset of B.
+Proof.This is immediate from Lemma 3.15, or Corollary 3.16, or by us ing the fact that
+qsatisfies the parallelogram law :x,y∈B=⇒q(x)+q(y) =1
+2q(x+y)+1
+2q(x−y)./square
+The final result of this section can be thought of as a completi on of Lemma 3.8. It
+will not be used in the sequel.
+Lemma 3.18. Let/parenleftbig
+B,⌊·,·⌋/parenrightbig
+be an SSD space and Abe a nonempty q–positive subset of
+B. Then ΦA@≤qonA,ΦA@≥qonB, and ΦA@@= ΦAonB.
+Proof.See [15, Lemma 2.11]. /square
+7SSDB spaces and maximal monotonicity
+4 SSDB spaces
+We now introduce the SSDB spaces, a subclass of the class of SS D spaces. Our treatment
+of the SSD spaces in Section 3 has been essentially nontopolo gical. The additional norm
+structureoftheSSDBspacesisessentiallywhatmakesmaxim allymonotonemultifunctions
+on a reflexive Banach space much more tractable than those on a general Banach space.
+We will return to this issue when we consider Example 3.3 in Se ction 5.
+Definition 4.1. We will say that/parenleftbig
+B,⌊·,·⌋,/⌊ard⌊l·/⌊ard⌊l/parenrightbig
+is asymmetrically self–dual Banach space
+(SSDB space) if/parenleftbig
+B,⌊·,·⌋/parenrightbig
+is a SSD space,/parenleftbig
+B,/⌊ard⌊l·/⌊ard⌊l/parenrightbig
+is a Banach space, and there exists a
+linear isometry ιfromBontoB∗such that, for all b,c∈B,/angbracketleftbig
+b,ι(c)/angbracketrightbig
+=⌊b,c⌋. We note then
+that, for all f∈ PC(B) andc∈B,f@(c) := supB/bracketleftbig
+⌊·,c⌋−f/bracketrightbig
+= supB/bracketleftbig
+/a\}⌊ra⌋ketle{t·,ι(c)/a\}⌊ra⌋ketri}ht−f/bracketrightbig
+=
+f∗/parenleftbig
+ι(c)/parenrightbig
+, that is to say f@=f∗◦ι. It is easy to see that the quadratic form qis
+continuous and, for all b∈B,|q(b)|=1
+2/vextendsingle/vextendsingle⌊b,b⌋/vextendsingle/vextendsingle≤1
+2/⌊ard⌊lb/⌊ard⌊l2.
+Letg0:=1
+2/⌊ard⌊l·/⌊ard⌊l2onB. Then, for all b∈B,
+g0@(b) =g0∗/parenleftbig
+ι(b)/parenrightbig
+=1
+2/⌊ard⌊lι(b)/⌊ard⌊l2=1
+2/⌊ard⌊lb/⌊ard⌊l2=g0(b).
+It follows that
+g0is both a BC–function and a TBC–function. (9)
+Examples 4.2. (a) In Example 3.2(a),/parenleftbig
+B,⌊·,·⌋,/⌊ard⌊l·/⌊ard⌊l/parenrightbig
+is a SSDB space under the Hilbert
+space norm, q=g0andNq(g0) ={0}./parenleftbig
+We recall that the set Nq(g0) was defined in
+Definition 3.5./parenrightbig
+(b) In Example 3.2(b),/parenleftbig
+B,⌊·,·⌋,/⌊ard⌊l·/⌊ard⌊l/parenrightbig
+is a SSDB space under the Hilbert space norm,
+q=−g0andNq(g0) =B.
+(c) In Example 3.2(c),/parenleftbig
+B,⌊·,·⌋,/⌊ard⌊l·/⌊ard⌊l/parenrightbig
+is a SSDB space under the Euclidean norm and
+Nq(g0) ={(b1,b2,b3)∈B:b1+b2= 0, b3= 0}.
+Our next result is the “pos–neg theorem of Rockafellar type” . This will be used
+indirectly in Theorem 5.1, which will be used in turn indirec tly in our proof of Rockafellar’s
+surjectivity theorem, Theorem 6.3, and the sum theorem, The orem 6.5. This result first
+appeared in Simons, [14, Theorem 19.16, p. 83].
+Theorem 4.3. Let/parenleftbig
+B,⌊·,·⌋,/⌊ard⌊l· /⌊ard⌊l/parenrightbig
+be a SSDB space, f∈ PC(B)be a BC–function and
+g:B→Rbe a continuous TBC–function. Then Pq(f)−Nq(g) =B.
+Proof.Letcbe an arbitrary element of B. From Lemma 3.13, fcis a BC–function and
+so, using (7) and (8),
+b∈B=⇒fc(b)+g(b)≥q(b)−q(b) = 0.
+Rockafellar’s version of the Fenchel duality theorem, Theo rem 2.1, and the surjectivity of ι
+now give b∈Bsuch that fc∗/parenleftbig
+ι(b)/parenrightbig
++g∗/parenleftbig
+−ι(b))≤0, that isto say fc@(b)+g@(−b)≤0.
+From (7) and (8) again, fc(b)+g(b)≤0 =q(b)−q(b). From (7) and (8) for a third time,
+fc(b) =q(b) andg(b) =−q(b), that is to say, using Lemma 3.13, b∈ Pq(fc) =Pq(f)−c
+and also b∈ Nq(g). But then c= (c+b)−b∈ Pq(f)− Nq(g). This completes the
+proof of Theorem 4.3. /square
+8SSDB spaces and maximal monotonicity
+Theorem 4.4. Let/parenleftbig
+B,⌊·,·⌋,/⌊ard⌊l·/⌊ard⌊l/parenrightbig
+be a SSDB space. Then:
+(a)Letf∈ PC(B)be a BC–function. Then Pq(f)−Nq(g0) =B.
+(b)LetAbe aq–positive subset of BandA− Nq(g0) =B. Then Ais maximally
+q–positive.
+(c)Letf∈ PC(B)be a BC–function. Then Pq(f)is maximally q–positive. Further-
+more,Pq(f@) =Pq(f).
+(d)LetAbe aq–positive subset of B. Then Ais maximally q–positive if, and only if
+A−Nq(g0) =B.
+Proof.(a) This follows from (9) and Theorem 4.3.
+(b) Suppose that b∈BandA∪ {b}isq–positive. By hypothesis, there exists
+a∈Asuch that a−b∈ Nq(g0). Thus1
+2/⌊ard⌊la−b/⌊ard⌊l2=−q(a−b). Since A∪ {b}is
+q–positive, q(a−b)≥0, and so1
+2/⌊ard⌊la−b/⌊ard⌊l2≤0. Thus b=a∈A.
+(c) From (a) and Lemma 3.17, Pq(f) is nonempty and q–positive. (c) now follows
+from (a), (b) and Lemma 3.10.
+(d) We have already established ( ⇐=) in (b). Suppose, conversely, that Ais maxi-
+mallyq–positive. It follows from Theorem 3.11 that Φ Ais a BC–function and Pq(ΦA) =A.
+Thus, from (a), A−Nq(g0) =Pq(ΦA)−Nq(g0) =B. /square
+Remark 4.5. We note from (9) and Theorem 4.4(c) that Pq(g0) is maximally q–positive
+andPq(g0@) =Pq(g0). Now suppose that a∈ Pq(g0) andb∈B. From the Fenchel–
+Young inequality, ⌊a,b⌋ −q(a) =⌊a,b⌋ −g0@(a)≤g0(b). Taking the supremum over
+a∈ Pq(g0), we deduce that Φ Pq(g0)≤g0onB. In particular, domΦ Pq(g0)=B.
+The next result in this section is the “pos-neg theorem of Att ouch–Brezis type”. A
+stronger result was proved in Simons, [14, Theorem 21.12, p. 93].
+Theorem 4.6. Let/parenleftbig
+B,⌊·,·⌋,/⌊ard⌊l·/⌊ard⌊l/parenrightbig
+beaSSDBspace, f∈ PCLSC (B)beaBC–function, and
+g∈ PCLSC (B)be a TBC–function. Then domf−domg=B⇐⇒ P q(f)−Nq(g) =B.
+Proof.Since the implication ( ⇐=) is trivial, we only have to prove (= ⇒). Letcbe an
+arbitrary element of B. Then, from Lemma 3.13,
+domfc−domg= domf−c−domg=B−c=B,
+and so/uniontext
+λ>0λ/bracketleftbig
+domfc−domg/bracketrightbig
+=B. From Lemma 3.13, fcis a BC–function and so,
+using (7) and (8),
+b∈B=⇒fc(b)+g(b)≥q(b)−q(b) = 0.
+The Attouch–Brezis theorem, Theorem 2.2, and the surjectiv ity ofιnow give b∈Bsuch
+thatfc∗/parenleftbig
+ι(b)/parenrightbig
++g∗/parenleftbig
+−ι(b))≤0, that is to say fc@(b)+g@(−b)≤0. From (7) and (8)
+again,fc(b)+g(b)≤0 =q(b)−q(b). From (7) and (8) for a third time, fc(b) =q(b)
+andg(b) =−q(b), that is to say, using Lemma 3.13, b∈ Pq(fc) =Pq(f)−cand also
+b∈ Nq(g). But then c= (c+b)−b∈ Pq(f)− Nq(g). This completes the proof of
+Theorem 4.6. /square
+9SSDB spaces and maximal monotonicity
+Corollary4.7belowistheforminwhichwewillactuallyappl yTheorem 4.6toabstract
+Hammerstein theorems in Theorem 7.1.
+Corollary 4.7. Let/parenleftbig
+B,⌊·,·⌋,/⌊ard⌊l·/⌊ard⌊l/parenrightbig
+be a SSDB space, f,g∈ PCLSC (B)andf,gbe BC–
+functions. Suppose, further, that ρ:B→Bis a continuous linear bijection such that, for
+allb,c∈B,⌊ρ(b),ρ(c)⌋=⌊b,−c⌋. Then
+domf−ρ−1domg=B⇐⇒ P q(f)−ρ−1Pq(g) =B
+and
+domf+ρ−1domg=B⇐⇒ P q(f)+ρ−1Pq(g) =B.
+Proof.This is immediate from Theorem 4.6 and Lemma 3.14. /square
+5 BC–functions on E×E∗
+From now on, Eis a nonzero reflexive Banach space and E∗is its topological dual space.
+As observed in Example 3.3,/parenleftbig
+E×E∗,⌊·,·⌋/parenrightbig
+is an SSD space with q(x,x∗) =/a\}⌊ra⌋ketle{tx,x∗/a\}⌊ra⌋ketri}ht, and
+ifA⊂E×E∗thenAisq–positive exactly when Ais a nonempty monotone set in the
+usual sense, and Ais maximally q–positive exactly when Ais a maximally monotone set
+of in the usual sense. We norm E×E∗by/vextenddouble/vextenddouble(x,x∗)/vextenddouble/vextenddouble:=/radicalbig
+/⌊ard⌊lx/⌊ard⌊l2+/⌊ard⌊lx∗/⌊ard⌊l2. Then
+/parenleftbig
+E×E∗,/⌊ard⌊l·/⌊ard⌊l/parenrightbig∗= (E∗×E,/⌊ard⌊l·/⌊ard⌊l/parenrightbig
+,
+under the duality/angbracketleftbig
+(x,x∗),(y∗,y)/angbracketrightbig
+:=/a\}⌊ra⌋ketle{tx,y∗/a\}⌊ra⌋ketri}ht+/a\}⌊ra⌋ketle{ty,x∗/a\}⌊ra⌋ketri}ht. Combining this with the formula in
+Example 3.3, we have/angbracketleftbig
+(x,x∗),(y∗,y)/angbracketrightbig
+=/floorleftbig
+(x,x∗),(y,y∗)/floorrightbig
+, and so ι(y,y∗) = (y∗,y).
+It is easily seen from these relationships that/parenleftbig
+E×E∗,⌊·,·⌋,/⌊ard⌊l · /⌊ard⌊l/parenrightbig
+is an SSDB space.
+Further, ( x,x∗)∈ Pq(g0)⇐⇒1
+2/⌊ard⌊lx/⌊ard⌊l2+1
+2/⌊ard⌊lx∗/⌊ard⌊l2=/a\}⌊ra⌋ketle{tx,x∗/a\}⌊ra⌋ketri}ht ⇐⇒ (x,x∗)∈G(J),
+and (x,x∗)∈ Nq(g0)⇐⇒1
+2/⌊ard⌊lx/⌊ard⌊l2+1
+2/⌊ard⌊lx∗/⌊ard⌊l2=−/a\}⌊ra⌋ketle{tx,x∗/a\}⌊ra⌋ketri}ht ⇐⇒ (x,x∗)∈G(−J),
+whereJ:E⇒E∗is the duality map. Thus we obtain the following fundamental
+result:
+Theorem 5.1. LetEbe a nonzero reflexive Banach space. Then:
+(a)Iff∈ PC(E×E∗)andfis a BC–function then Pq(f)is maximally monotone,
+Pq/parenleftbig
+f@/parenrightbig
+=Pq(f)andPq(f)−G(−J) =B.
+(b)IfAis a monotone subset of E×E∗thenAis maximally monotone if, and only if,
+A−G(−J) =E×E∗.
+Proof.(a) is immediate from Theorem 4.4(a,c), and (b) is immediate from Theorem
+4.4(d). /square
+Remark 5.2. Iff∈ PCLSC (E) and, for all ( x,x∗)∈B,h(x,x∗) :=f(x)+f∗(x∗) then
+the Fenchel–Moreau theorem and the Fenchel–Young inequali ty imply that his a BC–
+function on E×E∗. SincePq(h) =G(∂f), Theorem 5.1(a) gives us that ∂fis maximally
+monotone. (Remember that we are assuming that Eis reflexive!)
+We note that the “partial episum theorem” of Theorem 5.3 belo w can also be deduced
+from Penot–Z˘ alinescu, [10, Corollary 3.7].
+10SSDB spaces and maximal monotonicity
+Theorem 5.3. LetEbe a nonzero reflexive Banach space, f, g∈ PCLSC (E×E∗)be
+BC–functions,/uniontext
+λ>0λ/bracketleftbig
+π1domf−π1domg/bracketrightbig
+be a closed linear subspace of Eand, for all
+(x,x∗)∈E×E∗,
+h(x,x∗) := inf/braceleftbig
+f(x,s∗)+g(x,t∗):s∗, t∗∈E∗, s∗+t∗=x∗/bracerightbig
+. (10)
+Thenhis a BC–function, Pq(h@) =/braceleftbig
+(x,s∗+t∗): (x,s∗)∈ Pq(f@),(x,t∗)∈ Pq(g@)/bracerightbig
+,
+and
+Pq(h) =/braceleftbig
+(x,s∗+t∗): (x,s∗)∈ Pq(f),(x,t∗)∈ Pq(g)/bracerightbig
+.
+Proof.Sincef≥qandg≥qonE×E∗, (10) implies that, for all ( x,x∗)∈E×E∗,
+h(x,x∗)≥inf/braceleftbig
+/a\}⌊ra⌋ketle{tx,s∗/a\}⌊ra⌋ketri}ht+/a\}⌊ra⌋ketle{tx,t∗/a\}⌊ra⌋ketri}ht:s∗, t∗∈E∗, s∗+t∗=x∗/bracerightbig
+=/a\}⌊ra⌋ketle{tx,x∗/a\}⌊ra⌋ketri}ht=q(x,x∗),
+and then Theorem 2.4 and the fact that f@≥fandg@≥gonE×E∗give
+h@(x,x∗) =h∗(x∗,x) = min/braceleftbig
+f∗(s∗,x)+g∗(t∗,x):s∗, t∗∈E∗, s∗+t∗=x∗/bracerightbig
+= min/braceleftbig
+f@(x,s∗)+g@(x,t∗):s∗, t∗∈E∗, s∗+t∗=x∗/bracerightbig
+(11)
+≥inf/braceleftbig
+f(x,s∗)+g(x,t∗):s∗, t∗∈E∗, s∗+t∗=x∗/bracerightbig
+=h(x,x∗).
+Thushis a BC–function, the required characterization of Pq(h@) is immediate from (11),
+and the final assertion now follows from three applications o f Theorem 5.1(a). /square
+Theorem 5.4. LetEbe a nonzero reflexive Banach space, f, g∈ PCLSC (E×E∗)be
+BC–functions and/uniontext
+λ>0λ/bracketleftbig
+π1domf−π1domg/bracketrightbig
+be a closed linear subspace of E. Then
+/braceleftbig
+(x,s∗+t∗): (x,s∗)∈ Pq(f),(x,t∗)∈ Pq(g)/bracerightbig
+is a maximally monotone subset of E×E∗.
+Proof.This is immediate from Theorem 5.1(a) and Theorem 5.3. /square
+Remark 5.5. The question arises naturally as to whether there can exist a q–positive
+subsetAofBsuch that Φ A=g0. It would then follow from (4) that g0=qon A, and so
+A⊂ Pq(g0). Thus, for all b∈B, (2) would give
+g0(b) = ΦA(b) = supA/bracketleftbig
+⌊·,b⌋−q/bracketrightbig
+= supA/bracketleftbig
+⌊·,b⌋−g0/bracketrightbig
+≤supPq(g0)/bracketleftbig
+⌊·,b⌋−g0/bracketrightbig
+.(12)
+Now letHbe a nonzero Hilbert space, and consider the SSDB space/parenleftbig
+H×H,⌊·,·⌋,/⌊ard⌊l·/⌊ard⌊l/parenrightbig
+,
+as described at the beginning of this section. Then ( x,x∗)∈ Pq(g0)⇐⇒x∗=x. Let
+y∈H\{0}. Then (12) would give
+/⌊ard⌊ly/⌊ard⌊l2=g0(y,−y)≤supx∈H/bracketleftbig/floorleftbig
+(x,x),(y,−y)/floorrightbig
+−g0(x,x)/bracketrightbig
+= supx∈H/bracketleftbig
+−g0(x,x)/bracketrightbig
+≤0.
+Since this is manifestly impossible, there cannot exist a q–positive subset AofBsuch that
+ΦA=g0. This example is an extension of [14, Example 19.20, p. 85].
+11SSDB spaces and maximal monotonicity
+6 Monotone multifunctions on reflexive Banach spaces
+LetEbe a nonzero reflexive Banach space. If S:E⇒E∗is a multifunction then we use
+the standard notation G(S) :=/braceleftbig
+(x,x∗)∈E×E∗:x∗∈Sx/bracerightbig
+. We also write D(S) :=/braceleftbig
+x∈E:Sx/\e}atio\slash=∅/bracerightbig
+=π1G(S). IfSis monotone and G(S) is nonempty, we define the
+Fitzpatrick functions ,ϕS, associated with Sby
+ϕS(x,x∗) := Φ G(S)(x,x∗) = sup(s,s∗)∈G(S)/bracketleftbig
+/a\}⌊ra⌋ketle{tx,s∗/a\}⌊ra⌋ketri}ht+/a\}⌊ra⌋ketle{ts,x∗/a\}⌊ra⌋ketri}ht−/a\}⌊ra⌋ketle{ts,s∗/a\}⌊ra⌋ketri}ht/bracketrightbig
+.
+The function ϕAwas introduced by Fitzpatrick in [6, Definition 3.1, p. 61]. T he following
+result was first proved in [6, Corollary 3.9, p. 62] and [6, Pro position 4.2, pp. 63–64].
+Theorem 6.1. Ebe a nonzero reflexive Banach space and S:E⇒E∗be maximally
+monotone. Then ϕSis a BC–function and Pq/parenleftbig
+ϕS@/parenrightbig
+=Pq/parenleftbig
+ϕS/parenrightbig
+=G(S).
+Proof.This is immediate from Theorem 3.11. /square
+The following result will be useful in Corollary 6.6.
+Lemma 6.2. Ebe a nonzero reflexive Banach space and S:E⇒E∗be maximally
+monotone. Then D(S)⊂π1domϕS.
+Proof.From Theorem 6.1 and the finite–valuedness of q,G(S) =Pq(ϕS)⊂domϕS, from
+whichD(S) =π1G(S)⊂π1domϕS. /square
+IfS,T:E⇒Fthen, for all x∈E, we use the following standard notation:
+(S+T)x:={y+z:y∈Sx, z∈Tx}.
+Theorem 6.3 is “Rockafellar’ssurjectivity theorem” — see [ 12, Proposition1, p. 77] for
+the original proof depending ultimately on Brouwer’s fixed– point theorem and an Asplund
+renorming. The proof given here is a simplification of that gi ven in Simons–Z˘ alinescu, [16,
+Theorem3.1(b),p.8],andappearedinSimons, [14,Theorem2 9.6,p.119]. Thesereferences
+also provide a number of formulae for the exact value of min/braceleftbig
+/⌊ard⌊lx/⌊ard⌊l:x∈E,(S+J)x∋0/bracerightbig
+in terms of ϕG(S). Here is one of them:
+min/braceleftbig
+/⌊ard⌊lx/⌊ard⌊l:x∈E,(S+J)x∋0/bracerightbig
+=1√
+2supb∈E×E∗/bracketleftBig
+/⌊ard⌊lb/⌊ard⌊l−/radicalbig
+2ϕS(b)+/⌊ard⌊lb/⌊ard⌊l2/bracketrightBig
+∨0.
+Theorem 6.3. LetEbe a nonzero reflexive Banach space and S:E⇒E∗be maximally
+monotone. Then (S+J)(E) =E∗.
+Proof.Lety∗be an arbitrary element of E∗. From Theorem 5.1(b)/parenleftbig
+withA:=G(S)/parenrightbig
+,
+(0,y∗)∈G(S)−G(−J). Thus there exist ( s,s∗)∈G(S) and (x,x∗)∈G(J) such that
+(0,y∗) = (s,s∗)−(x,−x∗). But then x=s, and so
+y∗=s∗+x∗∈Sx+Jx= (S+J)x⊂(S+J)(E). /square
+Remark 6.4. We note that if JandJ−1are single–valued then the converse of Theorem
+6.3 holds, that is to say ( S+J)(E) =E∗=⇒Sis maximally monotone, however this
+fails in general. See Simons, [14, Remark 29.7, pp. 120–121] . It is interesting to observe
+that, even when the converse of Theorem 6.3 fails, the necess ary and sufficient condition
+for maximal monotonicity given in Theorem 5.1(b) remains tr ue.
+12SSDB spaces and maximal monotonicity
+We now come to the “sum theorem”. More general results can be f ound in Bot ¸–
+Csetnek–Wanka, [3] and Bot ¸–Grad–Wanka, [4].
+Theorem 6.5. LetEbe a nonzero reflexive Banach space, S,T:E⇒E∗be maximally
+monotone and/uniontext
+λ>0λ/bracketleftbig
+π1domϕS−π1domϕT/bracketrightbig
+be a closed linear subspace of E. Then
+S+Tis maximally monotone.
+Proof.From Theorem 6.1, ϕSandϕTare BC–functions, Pq/parenleftbig
+ϕS/parenrightbig
+=G(S) andPq/parenleftbig
+ϕT/parenrightbig
+=
+G(T). Theorem 5.4 with f:=ϕSandg:=ϕTnow implies that
+/braceleftbig
+(x,s∗+t∗): (x,s∗)∈G(S),(x,t∗)∈G(T)/bracerightbig
+=G(S+T)
+is a maximally monotone subset of E×E∗. This completes the proof of Theorem 6.5. /square
+The following consequence of Theorem 6.5 is useful in applic ations:
+Corollary 6.6. LetEbe a nonzero reflexive Banach space, S,T:E⇒E∗be maximally
+monotone and/uniontext
+λ>0λ/bracketleftbig
+D(S)−D(T)/bracketrightbig
+=E. ThenS+Tis maximally monotone.
+Proof.This is immediate from Lemma 6.2 and Theorem 6.5. /square
+Our next result is “Rockafellar’s sum theorem”. It follows i mmediately from Corollary
+6.6.
+Corollary 6.7. LetEbe a nonzero reflexive Banach space, S,T:E⇒E∗be maximally
+monotone and D(S)∩intD(T)/\e}atio\slash=∅. ThenS+Tis maximally monotone.
+A number of other sufficient conditions have been given for the sum of maximally
+monotone multifunctions on a reflexive Banach space to be max imally monotone. Many
+of them are contained in the following “ Sandwiched closed su bspace theorem”, which first
+appeared in Simons–Z˘ alinescu, [16, Theorem 5.5, p. 13].
+Theorem 6.8. LetEbe a nonzero reflexive Banach space, S,T:E⇒E∗be maximally
+monotone, and suppose that there exists a closed linear subs paceFofEsuch that
+D(S)−D(T)⊂F⊂/uniondisplay
+λ>0λ/bracketleftbig
+π1domϕS−π1domϕT/bracketrightbig
+.
+ThenS+Tis maximally monotone. Furthermore, for all ε >0,
+D(S)−D(T)⊂π1domϕS−π1domϕT⊂(1+ε)/parenleftbig
+D(S)−D(T)/parenrightbig
+,
+(that is to say, the sets π1domϕS−π1domϕTandD(S)−D(T)are almost identical)
+and /uniondisplay
+λ>0λ/bracketleftbig
+π1domϕS−π1domϕT/bracketrightbig
+=/uniondisplay
+λ>0λ/bracketleftbig
+D(S)−D(T)/bracketrightbig
+.
+7 An abstract Hammerstein theorem
+LetEbe a nonzero reflexive Banach space. We define the reflection maps ρ1,ρ2
+onE×E∗byρ1(x,x∗) := (−x,x∗) andρ2(x,x∗) := (x,−x∗). Our first result is the
+“ρ1–transversality theorem”.
+13SSDB spaces and maximal monotonicity
+Theorem 7.1. LetEbe a nonzero reflexive Banach space, f,g∈ PCLSC (E×E∗)and
+f,gbe BC–functions. Then:
+domf+ρ1domg=E×E∗⇐⇒ P q(f)+ρ1Pq(g) =E×E∗
+and
+domf−ρ1domg=E×E∗⇐⇒ P q(f)−ρ1Pq(g) =E×E∗.
+Proof.This follows from Corollary 4.7 with ρ:=ρ2, and the fact that ρ1=−ρ−1
+2./square
+We give the next two results for completeness, though more ge neral results are known.
+See, for instance, Z˘ alinescu, [19, Theorem 3 and Corollary 4].
+Theorem 7.2. LetEbe a nonzero reflexive Banach space, S,T:E⇒E∗be maximally
+monotone and domϕS+ρ1domϕT=E×E∗. Then (S+T)(E) =E∗.
+Proof.Lety∗be an arbitrary element of E∗. From Theorem 7.1 (with f:=ϕSand
+g:=ϕT) and Theorem 6.1, (0 ,y∗)∈G(S) +ρ1G(T). Thus there exist ( s,s∗)∈G(S)
+and (x,x∗)∈G(T) such that (0 ,y∗) = (s,s∗)+(−x,x∗). But then x=s, and so
+y∗=s∗+x∗∈Sx+Tx= (S+T)x⊂(S+T)(E). /square
+SinceG(J) =Pq(g0), it follows that ϕJ= ΦG(J)= ΦPq(g0). Thus Remark 4.5
+implies that dom ϕJ=E×E∗. Consequently, the following result generalizes Theorem
+6.3. Compare with/bracketleftbig
+7, Theorem 2/parenleftbig
+(1)=⇒(2)/parenrightbig/bracketrightbig
+.
+Corollary 7.3. LetEbe a nonzero reflexive Banach space, S,T:E⇒E∗be maximally
+monotone and domϕT=E×E∗. Then (S+T)(E) =E∗.
+Proof.This is immediate from Theorem 7.2. /square
+Theorem 7.4. LetEbe a nonzero reflexive Banach space, f,g∈ PCLSC (E×E∗)be
+BC–functions, and w∗∈E∗be such that E×{w∗} ⊂domfandπ2domg=E∗. Then:
+(a)Pq(f)+ρ1Pq(g) =E×E∗andPq(f)−ρ1Pq(g) =E×E∗.
+(b)Ifx∈Ethen there exist (y,y∗)∈ Pq(f)and(z,y∗)∈ Pq(g)such that y+z=x.
+(c)Ifx∗∈E∗then there exist (y,y∗)∈ Pq(f)and(y,z∗)∈ Pq(g)such that y∗+z∗=x∗.
+Proof.(a) Let ( x,x∗) be an arbitrary element of E×E∗. Since π2domg=E∗, there
+existy,z∈Esuch that ( y,x∗−w∗)∈domgand (z,w∗−x∗)∈domg. But
+(x+y,w∗)∈domfand (x−z,w∗)∈domf, hence
+(x,x∗) = (x+y,w∗)+ρ1(y,x∗−w∗)∈domf+ρ1domg
+and
+(x,x∗) = (x−z,w∗)−ρ1(z,w∗−x∗)∈domf−ρ1domg.
+Thus we have proved that dom f+ρ1domg=E×E∗and dom f−ρ1domg=E×E∗,
+and (a) follows from Theorem 7.1.
+(b) It follows from (a) that there exist ( y,y∗)∈ Pq(f) and ( z,z∗)∈ Pq(g) such
+that (y,y∗)+(z,−z∗) = (x,0). But then z∗=y∗andy+z=x.
+(c) It follows from (a) that there exist ( y,y∗)∈ Pq(f) and ( z,z∗)∈ Pq(g) such
+that (y,y∗)−(z,−z∗) = (0,x∗). But then z=yandy∗+z∗=x∗. /square
+14SSDB spaces and maximal monotonicity
+We now reverse the direction of T.
+Theorem 7.5. LetEbeanonzero reflexiveBanachspaceand S:E⇒E∗andT:E∗⇒E
+be maximally monotone. Suppose that π1domϕT=E∗and there exists w∗∈E∗such
+thatE×{w∗} ⊂domϕS. Then:
+(a)IfIEis the identity map on E,(IE+TS)(E) =E.
+(b)IfIE∗is the identity map on E∗,(IE∗+ST)(E∗) =E∗.
+Proof.Letf:=ϕSandg:=ϕT−1, so that π2domg=E∗.
+(a) Letxbe an arbitrary element of E. From Theorem 7.4(b) and Theorem 6.1,
+there exist ( y,y∗)∈G(S) and ( y∗,z)∈G(T) such that y+z=x. Thus
+z∈Ty∗⊂TSyandx=y+z∈(IE+TS)(y)⊂(IE+TS)(E).
+(b) Letx∗be an arbitrary element of E∗. From Theorem 7.4(c) and Theorem 6.1,
+there exist ( y,y∗)∈G(S) and ( z∗,y)∈G(T) such that y∗+z∗=x∗. Thus
+y∗∈Sy⊂STz∗andx∗=z∗+y∗∈(IE∗+ST)(z∗)⊂(IE∗+ST)(E∗). /square
+The next result is a considerable generalization of [20, The orem 32.O, p. 909] which,
+in turn, was applied to Hammerstein integral equations. See [14, Remark 30.5, p. 124] for
+a more complete discussion.
+Theorem 7.6. LetEbeanonzero reflexiveBanachspaceand S:E⇒E∗andT:E∗⇒E
+be maximal monotone. Suppose that either π1domϕT=E∗and there exists
+w∗∈E∗such that E× {w∗} ⊂domϕSorπ1domϕS=Eand there exists
+w∈Esuch that E×{w} ⊂domϕT. Then (IE+TS)(E) =E.
+Proof.The first case has already been established in Theorem 7.5(a) , while the
+second case follows from Theorem 7.5(b), with Ereplaced by E∗, and the roles of S
+andTinterchanged. /square
+Remark 7.7. In the recent paper [2], Jonathan Borwein has written a surve y of the
+history of monotonicity (even in the nonreflexive case) over the past fifty years.
+References
+[1] H. Attouch and H. Brezis, Duality for the sum of convex funtions in general Banach
+spaces., Aspects of Mathematics and its Applications, J. A. Barroso , ed., Elsevier
+Science Publishers (1986), 125–133.
+[2] J. M. Borwein, Fifty years of maximal monotonicity ,<http://www.carma.newcastle.
+edu.au/∼jb616/fifty.pdf >.
+[3] R. I. Bot ¸, E. R. Csetnek and G. Wanka, A new condition for maximal monotonicity
+via representative functions , Nonlinear Anal. 67(2007), 2390–2402.
+[4] R. I. Bot ¸, S–M Grad and G. Wanka, Maximal monotonicity for the precomposition
+with a linear operator . SIAM J. Optim. 17(2006), 1239–1252.
+[5] R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality
+product, Proc. Amer. Math. Soc. 131(2003), 2379–2383.
+[6] S. Fitzpatrick, Representing monotone operators by convex functions , Workshop/
+Miniconference on Functional Analysis and Optimization (C anberra, 1988), 59–65,
+15SSDB spaces and maximal monotonicity
+Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra,
+1988.
+[7] J-E. Mart´ ınez-Legaz, Some generalizations of Rockafellar’s surjectivity theor em,
+Pacific J. of Optimization 4(2008), 527–535.
+[8] J.–E. Mart´ ınez-Legaz and M. Th´ era, A convex representation of maximal monotone
+operators , J. Nonlinear Convex Anal. 2(2001), 243–247.
+[9] J.–P. Penot, The relevance of convex analysis for the study of monotonici ty, Nonlinear
+Anal.58(2004), 855–871.
+[10] J.–P. Penot and C. Z˘ alinescu, Some problems about the representation of monotone
+operators by convex functions , ANZIAM J. 47(2005), 1–20.
+[11] R. T. Rockafellar, Extension of Fenchel’s duality theorem for convex function s, Duke
+Math. J. 33(1966), 81–89.
+[12] —–, On the Maximality of Sums of Nonlinear Monotone Operators , Trans. Amer.
+Math. Soc. 149(1970),75-88.
+[13] S. Simons, Minimax and monotonicity , Lecture Notes in Mathematics 1693(1998),
+Springer–Verlag.
+[14] —–, From Hahn–Banach to monotonicity , Lecture Notes in Mathematics, 1693,
+second edition, (2008), Springer–Verlag.
+[15] —–, Banach SSD spaces and classes of monotone sets , http://arxiv/org/abs/
+0908.0383v2, posted August 26, 2009.
+[16] S. Simons and C. Z˘ alinescu, Fenchel duality, Fitzpatrick functions and maximalmono-
+tonicity, J. of Nonlinear and Convex Anal., 6(2005), 1–22.
+[17] M. D. Voisei and C. Z˘ alinescu, Strongly–representable operators , J. of Convex Anal.,
+16(3), 2009.
+[18] C. Z˘ alinescu, Convex analysis in general vector spaces , (2002), World Scientific.
+[19] —–, A new convexity property for monotone operators , J. Convex Anal. 13(2006),
+883–887.
+[20] E. Zeidler, Nonlinear functional analysis and its applications, II/B, Nonlinear mono-
+tone operators , Springer-Verlag, New York, 1990.
+Department of Mathematics
+University of California
+Santa Barbara
+CA 93106-3080
+U. S. A.
+email: simons@math.ucsb.edu
+16
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+arXiv:1001.0065v1  [astro-ph.SR]  31 Dec 2009
+Science in China Series G: Physics, Mechanics & Astronomy
+@2009 SCIENCE IN CHINA PRESS
+Springerwww.scichina.com
+phys.scichina.com
+www.springerlink.com
+EIT waves and coronal magnetic field diagnostics
+CHENPengFei
+Department of Astronomy, Nanjing University,Nanjing 2100 93, China
+Magnetic field in the solar lower atmosphere can be measured b y the use of the Zeeman and Hanle
+effects. In contrast, the coronal magnetic field well above t he solar surface, which directly controls
+various eruptive phenomena, can not be precisely measureed with the traditional techniques. Several
+attempts arebeingmadetoprobethecoronal magneticfield,s uchasforce-freeextrapolationbasedon
+the photospheric magnetograms, gyroresonance radio emiss ions, and coronal seismology based on
+MHDwavesinthecorona. Comparedtothewavestrappedinthel ocalizedcoronalloops,EITwavesare
+the only global-scale wave phenomenon, and thus are the idea l tool for the coronal global seismology.
+In this paper, we review the observations and modelings of EI T waves, and illustrate how they can be
+appliedto probe the global magnetic fieldinthe corona.
+coronalmass ejections (CMEs), magnetic field,diagnostics , magnetohydrodynamics (MHD)
+Magnetic field plays a vital role in various astro-
+physical processes. In the solar atmosphere, mag-
+netic field interacts with the plasma, producing
+abundant eruptive activities, besides the relatively
+quiescent heating in the atmosphere. Therefore,
+the measurement of magnetic field in the solar at-
+mosphere is extremely important. For strong mag-
+netic field, Zeeman effect has long been used in
+the solar photosphere and chromosphere, and is
+being testified for the transition region. For weak
+magnetic field measurement, Hanle effect can be
+utilized[1]. Spectral lines are formed in a relatively
+narrow height range in these atmospheric layers, so
+the local magnetic field can be obtained directly.
+However, for the corona, where magnetic field di-
+rectly controls theplasmakinematics, optical emis-
+sion lines are optically thin, and the traditionaltechniques do not work so well[2]. Although some
+techniques have been developed to extrapolate the
+coronal magnetic field based on photospheric mag-
+netograms, the extrapolation is an ill-posed prob-
+lem anyway. Therefore, it is imperative to develop
+alternative approaches to probe the coronal mag-
+netic field, such as gyroresonance radio emissions.
+Ubiquitous perturbations in the corona would
+propagate in the form of MHD waves in the quiet
+Sunortrigger theoscillations ofcoronal structures,
+such as prominences and coronal loops, caused by
+the trapped MHD waves. The phase speeds of the
+propagating waves or the standing waves are de-
+termined by the local magnetic field and/or tem-
+perature. Thus, either the propagation speed or
+the oscillation period provides effective informa-
+tion to diagnose the magnetic field in the corona,
+Received July 15, 2009; accepted August 15, 2009
+doi: 10.1007/s11433-009-0240-9
+†Corresponding author (email: chenpf@nju.edu.cn)
+Supported by the 973 project (Grant No. 2006CB806302), and t he National Natural Science Foundation of China (Grant Nos. 10933003 and
+10403003)
+Citation: Chen, P. F. EIT waves and coronal magnetic field diagnostics. Sci China Ser G, 2009, 52(11): 1785-1789, doi: 10.1007/s114 33-009-
+0240-9which gives rise to a new discipline, i.e, coronal
+seismology[3].
+In the early 1960s, Moreton & Ramsey[4]discov-
+ered a kind of arc-shaped chromospheric pertur-
+bations propagating away from big flares to a dis-
+tance>5×105km, later called Moreton waves.
+According to Uchida[5], Moreton waves were due
+to coronal fast-mode waves sweeping the chromo-
+sphere, and they can be used to reconstruct the
+Alfv´ en velocity, hence the magnetic field distribu-
+tions in the corona. In contrast to this large-scale
+waves, localized MHD waves were also found to be
+trapped in coronal loops and prominences (see ref.
+[6] for a review).
+Coronal loop/prominence oscillations provide
+the diagnosis of the local magnetic field, whereas
+Moreton waves provide the magnetic field informa-
+tion on a relatively larger scale. Both phenomena
+cover only a small part of the solar surface. How-
+ever, EITwaves, whichwerediscovered inthecoro-
+nal mass ejection (CME) event on 1997 May 12,
+offer a perfect opportunity for the magnetic field
+diagnosis on a global scale.
+1 EIT waves: Observations and model-
+ings
+After thelaunch of theSOHOsatellite inlate 1995,
+one of its payloads, the EUV Imaging Telescope
+(EIT), detected an almost circular front propagat-
+ing outward from the source active region in the
+CME/flare event on 1997 May 12[7]. More fre-
+quently, they are present as propagating patches,
+rather than circular fronts. Such a wave phe-
+nomenon is often referred to as “EIT wave”, but
+occasionally called “coronal wave” bysomeauthors
+(e.g., ref. [8]). EIT wave fronts, with an emission
+enhancement ranging from a few to tens of per-
+cent, are always followed immediately by expand-
+ing EUV dimmings in the base difference images[9].
+Therefore, EIT waves and dimmings are consid-
+ered symbiotic phenomena which require a com-
+mon physical process in any theoretical model[10].
+Since EIT waves and dimmings are found visible
+in several EUV bands, they are believed to result
+mainly from density enhancement and depletion,
+respectively, althoughWills-Davey &Thompson[11]and Chen & Fang[12]pointed out that the temper-
+ature variation also plays a role in determining the
+EUV intensity. The statistical study by Klassen
+et al.[13], using SOHO/EIT observations, indicates
+that the typical velocities of EIT waves range from
+170 to 350 km ·s−1, with a mean value of 271
+km·s−1. Note that the EIT wave velocity can be
+as slow as 50 km ·s−1or even smaller[14]. According
+to recent research by Long et al.[15], who analyzed
+the high-cadence EUV observations by STEREO
+satellite, the low cadence of the SOHO/EIT ob-
+servations smoothed the EIT wave velocity evo-
+lution, i.e, the maximum velocity was underes-
+timated and the low velocity was overestimated.
+Therefore, a renewed statistics of EIT wave veloc-
+ities, based on the high cadence observations of
+STEREO/SECCHI, is strongly recommended.
+The discovery of EIT waves immediately re-
+minded the researchers of the coronal counter-
+parts of H αMoreton waves, and so they were
+explained in terms of fast-mode MHD waves in
+the corona[9,16,17], though there were arguments
+whether the fast-mode waves originate from the
+flare or the CME[18,19]. However, the fast-mode
+wave model can not account for the following ob-
+servational features: (1) Even considering that the
+high-cadence observations may upgrade the statis-
+tical results made by Klassen et al.[13], the EIT
+wave velocities are still significantly smaller than
+those of Moreton waves, which are truly fast-mode;
+(2) EIT waves may stop at the footpoints of the
+magnetic separatrix[20]; (3) EIT wave velocities are
+anti-correlated with the speeds of the correspond-
+ing type II radio bursts[13]; (4) EIT waves acceler-
+ate from ∼20 km·s−1near the source active region
+to more than 400 km ·s−1in the quiet region[15].
+In order to reconcile these discrepancies, Chen et
+al.[21,22]proposed a field-line stretching model, i.e,
+as the strongly twisted flux rope erupts to form a
+CME, a fast-mode piston-driven shock propagates
+outward, which corresponds to the coronal coun-
+terpart of H αMoreton wave. At the same time,
+the erupting flux rope pushes the overlying mag-
+neticloopsuccessively. Thestretchingofeachmag-
+netic loop leads to an EIT wave front on the outer
+sideanddensitydepletioninsidethemagneticloop.
+1786 Chen, P. F. SciChinaSer G-Phys Mech Astron |Nov. 2009 |vol. 52|no. 11|1785-1789Therefore, the model explains the EIT waves and
+dimmings simultaneously. The field-line stretching
+model can account for all the four features men-
+tioned above, and was supported by imaging and
+spectroscopic observations[23].
+3 EIT waves as a tool for coronal seis-
+mology
+Since EIT waves propagate across almost the en-
+tire solar disk in many events, they are the ideal
+tool for the global magnetic field diagnosis. As for
+other wave phenomena, how EIT waves are used
+for coronal seismology depends on the identifica-
+tion of the wave mode. If they are fast-mode MHD
+waves, the magnetic field reconstruction would be
+the same as used for Moreton waves. Within such
+a framework, several groups tried to reconstruct
+theglobal magnetic field distribution[24]. Consider-
+ing the discrepancies between the fast-mode wave
+model and the observations, we have proposed a
+magnetic field-line stretching model, as described
+in Sec. 2. In this section, we discuss how EIT
+waves can be used for coronal seismology in the
+framework of our field-line stretching model.
+The sketch of our model is depicted in Figure 1,
+where the thick curved lines are the initial mag-
+netic field lines. As the flux rope ( the circles )
+moves upward, it pushes the top part of the first
+field line near point A,and this local part expands
+outward. As shown in Figure 1(a), the expan-sion would be transferred along the field line to
+its footpoints CandE,with the Alfv´ enic veloc-
+ityvA. The expansion near the footpoints Cand
+Ecompresses the plasma on the outer side, which
+produces the first EIT wave front. At the same
+time, the local expansion near point Awould also
+betransferredupward to thetop part of thesecond
+magnetic field line, near point B,with the fast-
+mode wave velocity ( vf=/radicalbig
+v2
+A+v2
+s, where vsis
+the sound speed) since it is perpendicular to the
+field lines, as illustrated by the right panel. Thus
+the top part of the second field line near point B
+expands. Again, the local expansion, as a kind of
+perturbation, would be transferred along the sec-
+ond field line to its footpoints DandF,with the
+Alfv´ enic velocity vA. Thereby, a second EIT wave
+front is formed near points DandF.With such a
+repeating circle, progressing EIT wave fronts are
+formed as outer field lines are pushed to expand
+successively. Note that the successive expansion
+of the magnetic loops naturally leads to expanding
+dimming regions behind the EIT wave fronts. The
+apparent velocity of the EIT wave in the model
+is the distance between two successive EIT wave
+fronts, e.g., CD, divided by the time difference
+between the formation of the two successive EIT
+wave fronts, ∆ t. This is to say, the EIT wave
+apparent velocity near points CandDis related
+to the magnetic field by vEIT=CD/∆t, where
+∆t=/integraltextB
+A1/vfds+/integraltextD
+B1/vAds−/integraltextC
+A1/vAds.
+I GA
+C DH
+DH
+G C IA
+EF F EB B
+Figure 1 The sketch of our field-line stretching model for EIT waves[21].
+Chen,P. F. Sci ChinaSerG-Phys Mech Astron |Nov. 2009 |vol. 52|no. 11|1785-1789 1787Figure 2 (a) A magnetic configuration with an active region surrounde d by potential field; (b) the corresponding distributions of the
+fast-mode wave and EIT wave velocities along the surface. Th e unit of the velocity is 900 km ·s−1.
+As demonstrated by Chen et al.[21], if the mag-
+netic field lines are concentric semicircles, the re-
+sultingvEITis about 0.34 vf. For any given mag-
+netic configuration, the EIT wave velocity distri-
+bution can be analogically derived by the above
+formula. For example, Figure 2(a) depicts the
+magnetic configuration for an active region sur-
+rounded by a potential bipolar field. The cor-
+responding EIT wave velocity and the fast-mode
+wave velocity distributions are plotted in the Fig-
+ure2(b). TheEITwaves areseentoaccelerate near
+the boundary of the source active region. This fea-
+ture is consistent with the recent observation[15],
+but which cannot be interpreted by the fast-mode
+wave model.
+We can adjust the coronal magnetic configura-
+tion so that the resulting EIT wave velocity distri-
+bution can best match the observational one. In
+this way, the coronal magnetic field distribution
+can be reconstructed. This procedure, which is
+forward modeling, is demonstrated here to be eli-
+gible. The inverse modeling, i.e., to derive the 3D
+coronal magnetic field directly from the observedEIT wave velocity distribution, similar to the tech-
+niquesusedinhelioseismology, deservessubstantial
+further studies.
+4 Prospects
+As the largest-scale wave phenomenon in the
+corona, EIT waves offer an ideal opportunity to
+probe the coronal magnetic field. As proposed in
+our field-line stretching model, the formation of
+each EIT wave front conveys the magnetic infor-
+mation in 3D space, all the way from the erupting
+flux rope to the top of each field line, and then
+along the field line to its footpoints. Therefore,
+similar to the p-mode waves in the helioseismology,
+EIT waves can be used to probe the 3D distribu-
+tion of the coronal magnetic field, providing that
+high-cadence EUV imaging observations are avail-
+able. Inthis paper, wedemonstrated thefeasibility
+of the forward modeling of the coronal seismology
+based on EIT wave. However, the inverse model-
+ing, i.e., the direct inversion from EIT wave veloc-
+ity evolution tothecoronal magneticfield, deserves
+further investigation.
+1 Sahal-Br´ echot S. The Hanle effect applied to magnetic field
+diagnostics. Space Sci Rev, 1981, 29: 391–401
+2 White S M. Solar and Space Weather Radiophysics, Vol.314.
+Gary D E, Keller C U, eds. Dordrecht: Academic Publisher,
+2004. 89–113
+3 Roberts B, Edwin P M, Benz A O. On coronal oscillations.Astrophys J, 1984, 279: 857–865
+4 Moreton G E, Ramsey H E. Recent observations of dynamical
+phenomena associated with solar flares. Publ Astron Soc Pac,
+1960, 72: 357
+5 Uchida Y. Propagation of hydromagnetic disturbances in th e
+solar corona and Moreton’s wave phenomenon. Sol Phys, 1968,
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+6 Roberts B. Waves and oscillations in the corona. Sol Phys,
+2000, 193: 139–152
+7 Thompson B J, Plunkett S P, Gurman J B et al. SOHO/EIT
+observations of an Earth-directed coronal mass ejection on
+May 12, 1997. Geophys Res Lett, 1998, 25: 2465–2468
+8 Vrˇ snak B, Warmuth A, Temmer M et al. Multi-wavelength
+study of coronal waves associated with the CME-flare event of
+3 November 2003. Astron Astrophys, 2006, 448: 739–752
+9 Thompson B J, Gurman J B, Neupert W M et al. SOHO/EIT
+observations of the 1997 April 7 coronal transient: Possibl e
+evidence of coronal moreton waves. Astrophys J, 1999, 517:
+151–154
+10 Chen P F. Initiation and propagation of CMEs. J Astrophys
+Astron, 2008, 29: 179–186
+11 Wills-Davey M J, Thompson B J. Observations of a propagat-
+ing disturbance in TRACE. Sol Phys, 1999, 190: 467–483
+12 Chen P F, Fang C. EIT waves - A signature of global magnetic
+restructuring in CMEs. IAU Symp, 2005, 226: 55–64
+13 Klassen A, Aurass H, Mann G et al. Catalogue of the 1997
+SOHO-EIT coronal transient waves and associated type II ra-
+dio burst spectra. Astron Astrophys, 2000, 141: 357–369
+14 Thompson B J, Myers D C. Catalog of Coronal “EIT Wave”
+Transients. Astrophys J Suppl Ser, 2009, 183: 225–243
+15 Long D M, Gallagher P T, McAteer R T J et al. The kinemat-ics of a globally propagating disturbance in the solar coron a.
+Astrophys J, 2008, 680: L81-L84
+16 Wu S T, Zheng H, Wang S et al. 3D numerical simulation
+of MHD waves observed by EIT. J Geophys Res, 2001, 106:
+25089-25102
+17 Wang Y M. EIT waves and fast-mode propagation in the solar
+corona. Astrophys J, 2000, 543: L89–L92
+18 Grechnev V V, Uralov A M, Slemzin V A et al. Absorption
+phenomena and a probable blast wave in the 13 July 2004
+eruptive event. Sol Phys, 2008, 253: 263–290
+19 Veronig A M, Temmer M, Vrˇ snak B. High-cadence observa-
+tions of a global coronal Wave by STEREO EUVI. Astrophys
+J, 2008, 681: L113–L116
+20 Delann´ ee C,AulanierG. CMEAssociated withtransequato rial
+loops and a bald patch flare. Sol Phys, 1999, 190: 107–129
+21 Chen P F, Wu S T, Shibata K, et al. Evidence of EIT and
+Moreton waves in numerical simulations. Astrophys J, 2002,
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+22 Chen P F, Fang C, Shibata K. A full view of EIT waves. As-
+trophys J, 2005, 622: 1202–1210
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+arXiv:1001.0066v2  [gr-qc]  7 Apr 2010A matter dominated navigation Universe in accordance with t he Type Ia supernova
+data
+Xin Li1,3∗
+1Institute of Theoretical Physics, Chinese Academy of Scien ces, 100190 Beijing, China
+Zhe Chang2,3†and Minghua Li2,3‡
+2Institute of High Energy Physics, Chinese Academy of Scienc es, 100049 Beijing, China
+3Theoretical Physics Center for Science Facilities, Chines e Academy of Sciences
+We investigate a matter dominated navigation cosmological model. The influence of a possi-
+ble drift (wind) in the navigation cosmological model makes the spacetime geometry change from
+Riemannian to Finslerian. The evolution of the Finslerian U niverse is governed by the same gravi-
+tational field equation with the familiar Friedmann-Robert son-Walker one. However, the change of
+space geometry from Riemannian to Finslerian supplies us a n ew relation between the luminosity
+distant and redshift. It is shown that the Hubble diagram bas ed on this new relation could account
+for the observations on distant Type Ia supernovae.
+PACS numbers: 02.40.-k,98.80.-k
+INTRODUCTION
+Einstein’sgeneralrelativityenablesustocomeup with
+a testable theory of the Universe. Hubble[1] first found
+that the galaxies are receding from us not long after the
+birth of general relativity. The Hubble’s observation in-
+dicates that our Universe is expanding. Over the past
+decade, two groups [2, 3] observing supernovae reported
+that the luminosity distance can not be explained by a
+matter dominated Universe. If one accepts the convinci-
+ble assumption of homogeneity and isotropy of the Uni-
+verse which is approximately true on large scales, then
+the general relativity tells us that we now live in a dark
+energy dominated Universe and the Universe is acceler-
+ated expanding. Dark energy, which has the property of
+negative pressure, is different from the classical matter
+and the found particles. A great amount of models have
+been proposed to study the possible candidates of dark
+energy and their dynamics [4]. The most famous and
+acceptable candidate is the cosmological constant. How-
+ever, the magnitude of cosmological constant 10−3eV4
+is much smaller than the energy density of vacuum in
+quantum field theory.
+The theory of dark energy [5] dominates the modern
+cosmologyin the past decade. This situation is similar to
+theraiseofthetheoryofether, whichhasbeenconsidered
+as direct evidence of the aberration of starlight, an im-
+portant astronomical effect known since eighteenth cen-
+tury. Although the rapid progress in technology makes
+the astronomical observations more and more accurate,
+up to now, there is no direct evidence indicate what the
+dark energy it is. Since the theory of dark energy is con-
+trived, which requires fine tuning and apparently cannot
+be tested in the laboratory or solar system, several mod-
+ified theories of general relativity have been developed as
+the alternative source of cosmological acceleration [6].Einstein first used the Riemann geometry to describe
+the theory of gravitation. In this Letter, we suppose that
+the spacetime in large scale may be described by other
+geometry instead of Riemann geometry. To preserve re-
+deeming feature of general relativity, this geometry must
+take Riemann geometry as its special case. Fortunately,
+Paul Finsler proposed a natural generation of Riemann
+geometry-Finsler geometry.
+Finsler geometry as a more general geometry could
+provide new sight on modern physics. It is of great
+interest for physicists to investigated the violation of
+Lorentz symmetry [7]. An interesting case of Lorentz
+violation, which was proposed by Cohen and Glashow[8],
+is the model of Very Special Relativity (VSR) character-
+ized by a reduced symmetry SIM(2). In fact, Gibbons,
+Gomis and Pope[9] showed that the Finslerian line ele-
+mentds= (ηµνdxµdxν)(1−b)/2(nρdxρ)bis invariant un-
+der the transformations of the group DISIM b(2). In the
+framework of Finsler geometry, modified dispersion rela-
+tion has been discussed[10, 11]. Also, the model based
+on Finsler geometry could explain the recent astronom-
+ical observations which Einstein’s gravity could not. A
+list includes: the flat rotation curves of spiral galaxies
+can be deduced naturally without invoking dark mat-
+ter [12]; the anomalous acceleration[13] in solar system
+observed by Pioneer 10 and 11 spacecrafts should corre-
+spond to Finsler-Randers space [14]; the secular trend in
+the astronomical unit[15, 16] and the anomalous secular
+eccentricity variation of the Moon’s orbit[17] should be
+correspondtothe effect ofthe length changeofunit circle
+in Finsler geometry[18].
+In this Letter, we present a matter dominated navi-
+gation model. The influence of a possible drift (wind)
+in the navigation cosmolgical model makes the expand-
+ing Universe “accelerated”. It is remarkable that the
+navigation cosmological model is described in the frame-
+work of Finsler geometry. We find that the predictions2
+of the navigation cosmological model could account for
+the observations of Riess and Perlmutter[2, 3] on distant
+supernovae.
+FORMULISM
+Instead of defining an inner product structure over the
+tangent bundle in Riemann geometry, Finsler geometry
+is based on the so called Finsler structure Fwith the
+property F(x,λy) =λF(x,y) for all λ >0, where x
+representsposition and yrepresentsvelocity. The Finsler
+metric is given as[19]
+gµν≡∂
+∂yµ∂
+∂yν/parenleftbigg1
+2F2/parenrightbigg
+. (1)
+In Finsler manifold, there exists a unique linear connec-
+tion - the Chern connection[20]. It is torsion freeness and
+almost metric-compatibility,
+Γα
+µν=γα
+µν−gαλ/parenleftBigg
+AλµβNβ
+ν
+F−AµνβNβ
+λ
+F+AνλβNβ
+µ
+F/parenrightBigg
+,
+(2)
+whereγα
+µνis the formal Christoffel symbols of the sec-
+ond kind with the same form of Riemannian connec-
+tion,Nµ
+νis defined as Nµ
+ν≡γµ
+ναyα−Aµ
+νλγλ
+αβyαyβand
+Aλµν≡F
+4∂
+∂yλ∂
+∂yµ∂
+∂yν(F2) is the Cartan tensor (regarded
+as a measurement of deviation from the Riemannian
+Manifold). In terms of Chern connection, the curvature
+of Finsler space is given as
+Rλ
+κ µν=δΓλ
+κν
+δxµ−δΓλ
+κµ
+δxν+Γλ
+αµΓα
+κν−Γλ
+ανΓα
+κµ,(3)
+whereδ
+δxµ=∂
+∂xµ−Nν
+µ∂
+∂yν. The notion of Ricci ten-
+sor in Finsler geometry was first introduced by Akbar-
+Zadeh[21]
+Ricµν=∂2/parenleftbig1
+2F2R/parenrightbig
+∂yµ∂yν, (4)
+whereR=yµ
+FRκ
+µ κνyν
+F. And the scalar curvature in
+Finsler geometry is given as S=gµνRicµν.
+In standard cosmology, following the cosmological
+principle, one gets the Friedmann-Robertson-Walker
+(FRW) metric[22]. In another word, the spatial part of
+theUniverseisaconstantsectionalcurvaturespace. Here
+comes our major assumption, the gravity in large scale
+should be described in terms ofFinsler geometry. In light
+of the cosmological principle, the Finsler structure of the
+Universe should be written in such form
+¯F2=dt2−R2(t)F2, (5)
+whereR(t) is scale factor with cosmic time t, the struc-
+tureFis a constant flag curvature space. By makinguse of the geometrical terms we mentioned above, one
+obtains the components of Ricci tensor
+Ric00=−3¨R
+Rg00, (6)
+Ricij=−/parenleftBigg¨R
+R+2˙R2
+R2+2K
+R2/parenrightBigg
+gij,(7)
+where the dot denotes a derivative with respect to t.
+Thegravitationalfield equationin Finslerspaceshould
+reduce to the Einstein’s gravitationalfield equation while
+the Finsler space reduce to the Riemannian space. Thus,
+the symmetrical tensor Gµν≡Ricµν−1
+2gµνSshould in-
+volve in the gravitational field equation in Finsler space.
+In general, Finsler space is an anisotropic space. There-
+fore, it has less Killing vectors than the Riemannian
+space, and it breaks the Lorentz symmetry[23]. It means
+the angular momentum in Finsler space is not a conser-
+vative quantity. This fact implies that the energy mo-
+mentum tensor is not symmetrical in Finsler space. For
+these reasons, the gravitational field equation in Finsler
+space should be taken in such form
+Gµν+Aµν= 8πG(Ts
+µν+Ta
+µν), (8)
+whereAµνis an asymmetrical tensor and Ts
+µν,Ta
+µνare
+symmetrical part and asymmetrical part of energy mo-
+mentum tensor respectively. This general form is agree
+with the result of Asanov, its gravitational field equa-
+tion contains the asymmetrical term in Finsler space of
+Landsberg type[24]. Dealing with field equation in the
+anisotropicandinhomogeneouscosmologyisnotasimple
+staff, and it is hard to find an exact solution of gravita-
+tional field equation while its energy momentum tensor
+involves the non diagonal terms[25]. At first glance, we
+just deal with the symmetrical part of field equation (8)
+Gµν= 8πGTµν. (9)
+By making use of the equations (6) and (7), the solu-
+tion of equation (9) is the same with the Einstein’s field
+equation deduced by FRW metric. Taking the energy-
+momentum tensor Tµνto be the form of perfect fluid,
+one can see that the evolution of scale factor R(t) is the
+same with the Riemannian case. However, since the lu-
+minosity distant is related to the spatial geometry of the
+Universe, one may expect that it is different from the
+Riemannian luminosity distant.
+Unlike Riemann space, a complete classification of the
+constant flag curvature spaces remain an unsolved prob-
+lem. However, by making use of the Zermolo navigation
+on Riemannian space, Bao et al. [26] gave a complete
+classification of Finsler-Randers space [27] of constant
+flag curvature. The Zermelo navigation problem [28]
+aims to find the paths of shortest travel time in a Rie-
+mannian manifold ( M,h) under the influence of a drift
+(“wind”) represented by a vector field W. In standard3
+cosmology, our Universe is very flat now. We may imag-
+ine that the Universe is a flat Riemannian manifold with
+flat Friedmann metric
+ds2=dt2−R2
+h(t)(dx2+dy2+dz2),(10)
+and it influenced by the “wind” W. The relationbetween
+the Riemannian manifold ( M,h) and the Randers metric
+Fis
+F=1
+λ/parenleftbigg/radicalBig
+λhijyiyj+(Wiyi)2−Wiyi/parenrightbigg
+,(11)
+whereWi=hijWjandλ= 1−h(W,W). One should
+notice that there is a map between the Riemannian
+space which influence by the “wind” and the Randers-
+Finsler space[29]. It means the effect of “wind” already
+accounted in the gravitational field equation in Finsler
+space. The theorem[26] of the classification supplies an
+interesting case where the Randers metric Fhas con-
+stant flag curvature K: forK=−1
+16σ2<0 andhis
+flat. And σsatisfies the constraint LWh=−σh,Lde-
+notes Lie differentiation. We set the vector field to be
+radialW=ǫ(t)(x1∂x1+x2∂x2+x3∂x3). Then the flag
+curvature of Randers metric is K=−1
+4ǫ2. After doing
+coordinate change and taking spherical coordinate, we
+get the Randers metric as
+F=/radicalbigg
+dr2
+1+ǫ2r2+r2(dθ2+sin2θdφ2)−ǫrdr
+1+ǫ2r2.(12)
+After scale change r→2r/ǫ, the metric Fchanges as
+F=/radicalbigg
+dr2
+1+4r2+r2(dθ2+sin2θdφ2)−2rdr
+1+4r2,(13)
+and the cosmic scale factor changes as R(t) =2Rh(t)
+ǫ(t).
+Taking the spatial part of the Finsler structure (5) to be
+of the form (13), we get metric of the Universe in the
+framework of Finsler geometry, and the space curvature
+of the Universe is −1. One can deduce directly from such
+metric that the relation between the redshift zand the
+scale factor R(t) is the same with the Riemannian case
+1+z=R0
+R(t)=1
+a(t), (14)
+where the subscript zero denotes the quantities given at
+thepresentepoch. Sincethenon-radialpartofthemetric
+(13) isthe samewith FRWmetric, the luminosity distant
+in the navigation cosmological model is given as dL=
+R0r(1+z). However, the proper distant is not the case,
+and it is given as
+dp=/integraldisplayr
+0/parenleftbigg1√
+1+4r2−2r
+1+4r2/parenrightbigg
+dr
+=sinh−12r
+2−1
+4ln(1+4r2). (15)The light traveling along the radial direction satisfies the
+geodesic equation
+¯F2=dt2−R2(t)/parenleftbigg1√
+1+4r2−2r
+1+4r2/parenrightbigg2
+dr2= 0.(16)
+Then, we obtain
+dp=1
+R0/integraldisplay1
+(1+z)−1da
+a˙a(17)
+The equations (9) and (17) can be solved. Supposing
+the Universe is matter dominated (no more cosmological
+constant and dynamical dark energy), we obtain
+dp= log(/radicalBig
+1+Ω(0)
+mz−/radicalBig
+1−Ω(0)
+m)(1+/radicalBig
+1−Ω(0)
+m)
+(/radicalBig
+1+Ω(0)
+mz+/radicalBig
+1−Ω(0)
+m)(1−/radicalBig
+1−Ω(0)
+m),
+(18)
+whereH0is the Hubble constant and Ω mis the density
+parameter for matter and satisfies
+1−Ω(0)
+m=1
+H2
+0R2
+0. (19)
+Substituting the equation (18) into (15), we obtain the
+relation between luminosity distant and redshift in the
+navigation cosmological model
+H0dL=1+z
+2/radicalBig
+1−Ω(0)
+m|e2dp−1|
+edp√
+2−e2dp.(20)
+NUMERICAL RESULT
+Here, we present numerical result for the relation be-
+tweenluminositydistantandredshift inthe frameworkof
+the navigation cosmological model. The best fit curve is
+shown in Fig.1 with the matter density parameter taken
+as Ω(0)
+m= 0.92. And the average value of Chisquare is
+1.077588. Thus, our prediction could account for exper-
+iment data given by Riess et al.[2]. It should be noticed
+that the Hubble constant H0we took is different from
+the Hubble constant Hh0measured in flat space. The
+relation for HandHhis
+H=Hh−˙ǫ
+ǫ. (21)
+By making use of the value of the Hubble constant mea-
+sured in flat space Hh0= 73±0.3km·s−1·Mpc−1[31],
+we have
+Hǫ0≡˙ǫ0
+ǫ0= 5.42km·s−1·Mpc−1(22)
+This geometrical parameter Hǫ0represents an “acceler-
+ated” effect provided by the vector field W.4
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s108/s111/s103
+/s49/s48/s40/s72
+/s48/s100
+/s76/s47/s99/s41
+/s122
+FIG. 1: The luminosity distant Log10(H0dL/c) versus the
+redshift zfor the navigational cosmological model. The
+data comes from Riess et al.[2] and Hubble Space Telescope
+(HST)[30]. And the value of Hubble constant is set as
+H0= 67.88km·s−1·Mpc−1.
+CONCLUSION
+Our Letter initiates an exploration of the possibility
+that the empirical success of the observations of Type Ia
+supernovae can be regarded as the influence of a naviga-
+tional wind. The particles move on Riemnnian manifold
+and influenced by a vector field (the “wind”) which pro-
+portiontothe curvature −ǫ2
+4ofthe spaceofthe Universe.
+Its geodesicindeed isaFinsleriangeodesic. Thus, the ob-
+servations of Riess et al.[2] on distant supernovae may be
+explained by the effect of the “wind”. Our numerical
+result could account for astronomical observations. At
+last, we point out that the age of the Universe is about
+9.76Gyr in our model. This is contradicted with the age
+(13.5±2Gyr) of Globular clusters in the MilkyWay[32].
+Inthe navigationcosmologicalmodel, weonlyinvolvethe
+radial “wind”. The non-radial “wind” should be taken
+into account in future work, we expect that the effect of
+the non-radial “wind” may supply us a reasonable age of
+the Universe.
+We would like to thank Prof. C. J. Zhu and X. H. Mo
+for useful discussions. The work was supported by the
+NSF of China under Grant No. 10525522 and 10875129.
+∗Electronic address: lixin@itp.ac.cn†Electronic address: zchang@ihep.ac.cn
+‡Electronic address: limh@ihep.ac.cn
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\ No newline at end of file
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+arXiv:1001.0067v1  [quant-ph]  31 Dec 2009Efficient numerical method to calculate three-tangle of mixe d states
+Kun Cao, Zheng-Wei Zhou, Guang-Can Guo, and Lixin He∗
+Key Laboratory of Quantum Information, University of Scien ce and
+Technology of China, Hefei, 230026, People’s Republic of Ch ina
+(Dated: October 22, 2018)
+We demonstrate an efficient numerical method to calculate thr ee-tangle of general mixed states.
+We construct a “energy function” (target function) for the t hree-tangle of the mixed state under
+certain constrains. The “energy function” (target functio n) is then optimized via a replica exchange
+MonteCarlomethod. Wehaveextensivelytestedthemethodfo rtheexampleswith knownanalytical
+results, showing remarkable agreement. The method can be ap plied to other optimization problems
+in quantum information theory.
+PACS numbers: 03.67.Mn, 03.65.Ud, 02.70.Ns, 02.70.Uu
+One of challenges in present quantum information the-
+ory is the quantification of quantum entanglement. The
+primary studies on entanglement measures focus on bi-
+partite systems. In bipartite entanglementmeasures, for-
+mation of entanglement, presented by Bennett et al., is
+veryfundamental[1]. Thesignificanceofthis measurere-
+lies on not only be able to provide an accurate boundary
+between separable states and entangled states but also
+constructing a general formula for entanglement mea-
+sures of mixed states: once a measure for pure states
+is given, the corresponding measure for mixed states can
+be obtained via the convex-roof extension. This scenario
+canbeevengeneralizedtoquantifymultipartiteentangle-
+ment. The initial attempt is 3-tangle of 3-qubit system,
+provided by Coffman et al [2]. Entanglement measures
+beyond 3-qubit were also presented [3, 4]. Although such
+kind of definition is straightforward, the practical eval-
+uation of the convex-roof extension is a difficult mathe-
+matical problem. To date a general analytical method is
+knownonlyforthe concurrenceoftwo-qubitmixed states
+[5, 6].
+Concurrence of mixed two qubits has been applied to
+study quantum phase transitions. Most of studies show
+that, for general spin 1 /2 lattice models, pairwise entan-
+glement (or its functions) depicted by concurrence of the
+nearest-neighbor two site has special singularity at quan-
+tum critical points (see[7] and its references). A natu-
+ral generalization for this viewpoint is that multipartite
+entanglement beyond two sites will reveal more exten-
+sive characteristics on quantum phase transition. How-
+ever, as far as 3-tangle is concerned, since the convex
+roof is obtained by minimizing the average tangle of a
+givenmixedstateoverallpossibledecompositionsofpure
+states, there is no general method to calculate the three-
+tangle of a mixed state. The analytical method is only
+capabletostudysomeparticularthreequbit states[8–11].
+In this work, we develop a method to calculate the
+three-tangle of any mixed states by numerically mini-
+mize certain target functions. The target functions are
+∗Email address: helx@ustc.edu.cnoptimized by using replica exchange Monte Carlo (MC)
+[12, 13] method. Replica exchange MC has been widely
+usedin condensedmatter physicstosimulate the systems
+with very rough “energy surfaces”, such as spin glasses
+[14], protein systemsand Lennard-Jonesparticles[15, 16]
+etc. The replica exchange MC method is much more ef-
+fective to find the global minium of the target functions
+than the conventional simulating annealing method. By
+applying this method we have successfully obtained the
+three-tangle of mixed (generalized) GHZ and W states.
+We also obtained the three-tangle of GHZ state with
+white noise, whose density matrices are of rank 8.
+The concurrence C(ψAB) measures the biparticle en-
+tanglement between particle A and B in a pure two-qubit
+state|ψAB/an}bracketri}ht. C is defined as
+C= 2|φ00φ11−φ01φ10|. (1)
+in terms of the coefficients {φ00,φ01,φ10,φ11}of|ψAB/an}bracketri}ht
+with respect to an orthonormal basis. The measure
+for three particle entanglement in a three-qubit state
+|ψABC/an}bracketri}hthas been introduced in Ref.[2] as three-tangle
+τ3(ψABC). It can be expressed in terms of the coeffi-
+cients{φ000,φ001,...,φ111}.
+τ3= 4|d1−2d2+4d3| (2)
+d1=φ2
+000φ2
+111+φ2
+001φ2
+110+φ2
+010φ2
+101+φ2
+100φ2
+011(3)
+d2=φ000φ111φ011φ100+φ000φ111φ101φ010
++φ000φ111φ110φ001+φ011φ100φ101φ010
++φ011φ100φ110φ001+φ101φ010φ110φ001(4)
+d3=φ000φ110φ101φ011+φ111φ001φ010φ100(5)
+It can be shown that for any factorized state, the three-
+tangle vanishes. For the GHZ state,
+|GHZ/an}bracketri}ht=1√
+2(|000/an}bracketri}ht+|111/an}bracketri}ht) (6)2
+τ3(GHZ)=1. It also appears a class of entangled three-
+qubit states represented by |W/an}bracketri}htstate withτ3vanished,
+i.e.,
+|W/an}bracketri}ht=1√
+3(|100/an}bracketri}ht+|010/an}bracketri}ht+|001/an}bracketri}ht) (7)
+The three-tangle of mixed state can be obtained via
+convex-roof extension[1, 5, 17]. Suppose the density ma-
+trix of a mixed state ρcan be decomposed into sum of
+some pure states πi, i.e.,
+ρ=/summationdisplay
+ipiπi, (8)
+where,
+πi=|Ψi/an}bracketri}ht/an}bracketle{tΨi|. (9)
+|Ψi/an}bracketri}htis the wave function of a normalized pure state. The
+three-tangle of the mixed state is defined as the average
+pure-state concurrence minimized over all possible de-
+compositions.
+τ3(ρ) = min/summationdisplay
+ipiτ3(πi). (10)
+It is very difficult to develop an universal analytical
+method to calculate the three-tangle of mixed three-
+qubit state. So far, there are very limited examples
+of mixed states whose three-tangle have been obtained
+analytically[8–11]. Alternatively, one could resort to the
+numerical methods. To get the three-tangle of mixed
+states is a constrained minimization problem. The pure
+state wavefunction can be expanded on the orthonormal
+three-particle basis, i.e., |Ψi/an}bracketri}ht=/summationtext
+αciα|Φα/an}bracketri}ht. The first
+constrain is,
+/summationdisplay
+ipici,αc∗
+i,β=ραβ. (11)
+whereραβis the density matrix in the basis of {|Φα/an}bracketri}ht}.
+Furthermore, the coefficients have to satisfy the normal-
+ization conditions,
+/summationdisplay
+α|ciα|2= 1,and/summationdisplay
+ipi= 1,pi≥0.(12)
+The above constrains can be enforced in the simulation
+by explicitly applying normalization factors at each MC
+step, whereas the constrain Eq. 11 can be enforced via
+a penalty function. The “energy function” (target func-
+tion) reads,
+E({pi,Ψi}) =/summationdisplay
+ipiτ3(Ψi)+κR2,(13)
+where{pi,Ψi}realizing the mixed state as given in Eq.9.
+R2is the residual between the searched density matrix
+and the target density matrix ρ, defined as
+R2=Nc/summationdisplay
+α=1Nc/summationdisplay
+β=1
+Np/summationdisplay
+i=1pici,αc∗
+i,β−ραβ
+2
+.(14)κis a large constant, to ensure R2∼0.Nc= 8 is the
+dimension of three-particle basis set. Npis the number
+of partition in the decomposition. We can easily see that
+Npdeterminesthenumberofvariables Nv=Np+2NcNp.
+In Eq.13,κshould be large enough to ensure a small
+residualR2to numerically satisfy the constrain given in
+Eq.11. However, large κwill make the “energy” surface
+ofEvery rough, i.e., E({pi,Ψi}) has many local min-
+ima separated by large “energy” barriers, which causes
+great difficulties in finding the global minima of three
+tangle using traditional constrained simulated annealing
+(CSA) method, because it tends to be trapped at some
+local minima. Here, we adopt the replica exchange (also
+known as parallel tempering [18]) Monte Carlo method
+[12] which simulates Mreplicas simultaneously each at
+a different temperature β0= 1/Tmax< β1< ... <
+βn−2< βM−1= 1/Tmincovering a range of interest.
+Each replica runs independently, except that after cer-
+tain steps, the configurations can be exchanged between
+neighboring temperatures, according to the Metropolis
+criterion,
+w=/braceleftbigg1 ∆H <0
+e−∆Hotherwise,(15)
+where ∆H=−(βi−βi−1)(Ei−Ei−1), in which Eiand
+Ei−1are the energy of the i-th andi-1-th replica. During
+the exchange, the detailed balance in canonical ensemble
+is satisfied. Importantly, the inclusion of high- Tconfig-
+urations ensures that the lower- Tsystems can access a
+broad phase space and avoid becoming trapped at local
+minima. The replica temperatures are adjust so that the
+exchangerate between the replicas are all about 20%[19].
+while keep the highest temperature β0= 1/Tmaxand
+lowest temperature βM−1= 1/Tminfixed. This can be
+done by adaptively adjusting the temperature of each
+replica using a recursion method proposed by Berg[20].
+We slightly revised the the algorithms to give better per-
+formance. Suppose, in the n-th iteration, the temper-
+ature of the i-th replica is βn
+iand the acceptance rate
+betweenβn
+iandβn
+i−1isar,n
+i. In then+1-th iteration,
+βn+1
+ican be updated as,
+βn+1
+i=βn+1
+i−1+(1−c+can
+i)(βn
+i−βn
+i−1),(16)
+where,
+an
+i=ar,n
+i(βn
+M−1−βn
+0)
+/summationtextM−1
+i=1ar,n
+i(βn
+i−βn
+i−1), (17)
+cis an empirical parameter that can be adjusted to ac-
+celerate the convergency. In the present case, we found
+that by choosing proper c <1 (in this case c=0.7) the
+recursionscheme geta suitable βdistribution within tens
+of iterations. We then fixed the replica temperatures for
+the simulations.
+Based on convex-roof extension, Lohmayer et al.have
+provided a complete analysis of mixed three-qubit states
+composed of a GHZ state and a W state, obtaining3
+the optimal decomposition and convex-roof for three-
+tangle[8]. Eltschka et algeneralized this method to treat
+the three-tangle of mixed three-qubit states composed of
+a generalized GHZ state and a generalized W state. The
+explicit expressions for mixed-state three- tangle and the
+corresponding optimal decomposition for the more gen-
+eral form were also presented[9]. We first test our scheme
+by comparing with the available analytical results. We
+constructa seriesoftargetfunctions based onthe density
+matrix given in Refs. 8, 9 as were described in the previ-
+ous sections. Replica exchange MC is then performed to
+find the three-tangle and optimal decomposition which
+are compared with the analytical results.
+In all our minimization process, the temperature range
+is chosen between Tmax=100 andTmin=10−6, such that
+Tmaxis high enough to help the low temperature replica
+to avoid being trapped in the local minimum and Tminis
+low enough to give the final minimized value an accuracy
+of 10−6theoretically. With fixed temperature range, the
+suitable number of replicas is determined by the corre-
+spondingdensitymatrixandweightfactor κ. Usually150
+replicas are enough. In our simulations, κis adjusted to
+be 104−107to keep the final R2<10−10which is con-
+sidered accurate enough in identifying the three-tangleof
+mixed state. In eachsimulation, the numberofpartitions
+Npis fixed. We then increase Npuntil the calculated
+three-tangle converges.
+Let us first look at the mixed GHZ and W states, The
+density matrices of the states are,
+ρ(p) =pπGHZ+(1−p)πW, (18)
+whereπGHZ=|GHZ/an}bracketri}ht/an}bracketle{tGHZ|andπW=|W/an}bracketri}ht/an}bracketle{tW|arethe
+density matrices of GHZ and W states respectively[8]. It
+is known that τ3(GHZ)=1 andτ3(W)=0. According to
+Caratheodory’s theorem, Np=4 is enough to get the op-
+timizedτ3of a mixed states of rank 2[8]. We calculate τ3
+by usingNp=4 - 8. Indeed, we find in our simulations,
+thatNp=4 is enough to give the optimal values. The re-
+sults are shown Fig.1(a), which depicts the numerically
+calculatedτ3of the mixed GHZ and W states (open cir-
+cles), comparedtothe analyticalvalues(solid line)[8]. As
+one can see that the numerical and analytical results are
+in excellent agreement.
+We then calculate τ3for the mixtures of generalized
+GHZandgeneralizedWstates[9], whosedensitymatrices
+are,
+ρ(p) =p|gGHZ/an}bracketri}ht/an}bracketle{tgGHZ|+(1−p)|gW/an}bracketri}ht/an}bracketle{tgW|,(19)
+where the density matrix of a generalized GHZ state is
+|gGHZ/an}bracketri}ht=a|000/an}bracketri}ht+b|111/an}bracketri}ht, (20)
+and density matrix of a generalized W state is
+|gW/an}bracketri}ht=c|001/an}bracketri}ht+d|010/an}bracketri}ht+f|100/an}bracketri}ht.(21)
+The coefficients must satisfy the normalization condition
+|a|2+|b|2=1, and |c|2+|d|2+|f|2=1. For the pure gen-
+eralized GHZ and W states, it is easy to calculate that
+τ3(gW)=0, andτ3(gGHZ) = 4|a2b2|.0.00.20.40.60.81.0τ3numerical
+analytical
+0.000.040.080.120.16τ3numerical
+analytical
+0.0 0.2 0.4 0.6 0.8 1.0
+p0.00.20.40.60.81.0τ3numerical
+analytical
+0.0 0.2 0.4 0.6 0.8 1.0
+p0.00.20.40.60.81.0τ3numerical(a)
+(d)(b)
+(c)
+FIG. 1: The three-tangle calculated with replica exchange
+compared to analytical results, for (a) the mixed GHZ and
+W states, (b) the mixed generalized GHZ and generalized W
+states, (c) the mixedGHZ state, W state andflippedW states
+with n=2, (d) the mixed GHZ state and white noise states.
+TABLE I: Numerically calculated τ3of mixed GHZ, W and
+flipped-W states for n=2 compared to the analytical results
+given in Ref. 10.
+p 0.2 0.75 0.8 0.85 0.9 0.933 0.95
+analytical 0 0 0.1835 0.3775 0.5805 0.7182 0.7897
+numerical 5.7e-6 2e-5 0.1836 0.3781 0.5811 0.7186 0.7901
+The calculated τ3of the mixed states are shown in
+Fig.1(b) for a=0.2,c=0.2,d=0.2. As we see the results
+are in excellent agreement with the analytical results.
+Againwefindthat Np=4isenoughtotheoptimalresults.
+In the above tests, we demonstrate that our method
+is very effective to deal with the three-tangle of rank-2
+mixed states. Recently, Jung at al. has obtained the
+analytical results of the three-tangle of the mixed states
+of GHZ, W, and flipped-W states[10]. These states are
+of rank 3. The flipped-W states is defined as
+|˜w/an}bracketri}ht=1√
+3(|110/an}bracketri}ht+|101/an}bracketri}ht+|011/an}bracketri}ht).(22)
+ThedensitymatrixofmixtureofGHZ,W,andflipped-W
+states are given by,
+ρ(p,q) =p|GHZ/an}bracketri}ht/an}bracketle{tGHZ|+q|W/an}bracketri}ht/an}bracketle{tW|+(1−p−q)|˜W/an}bracketri}ht/an}bracketle{t˜W|,
+(23)
+whereqwasdefinedas q= 1−p/n,nisapositivenumber.
+We then compared our numerical results to the analyti-
+cal values. The results for the case of n=2 are shown in
+Fig.1(c). The numerical results are also in good agree-
+ment with the analytical results. We further calculate
+three-tangleofthemixedstateswhichareofrank4,given
+in Ref.11, obtaining the same accuracy compared to the
+analytical results.
+To give an illustration of the accuracy, the numeri-
+cal values of mixture of GHZ, W and flipped-W state
+with n=2 are also listed in Table I. We can see that
+all the results are within an accuracy of ∼10−4. The4
+8 10 12 14 16 18 20
+Np0.0000.0010.0020.0030.0040.005∆τ3p=0.65
+p=0.7
+p=0.75
+p=0.8
+p=0.9
+FIG. 2: (Color online) The Npdependent∆ τ3calculated with
+replica exchange for density matrix constructed by GHZ stat e
+with white noise, where ∆ τ3=τ3−τmin
+3withτmin
+3the min-
+imum three-tangle for each pvalue.
+accuracy can easily be improved by increasing κand de-
+creasingTmin. For the example given above, in the case
+ofp=0.8, ifwe reduce Tminto 10−10and useκ= 1010, we
+obtainτ3=0.18349906 compared to the analytical value
+0.18349861 in an accuracy ∼10−7. In principle, our nu-
+merical method can achieve arbitrary accuracy. We have
+alsotriedtooptimizeEq. 13viatraditionalCSAmethod.
+Wefindtheperformanceofthetraditionalmethodisvery
+bad, because it can easily be trapped at some local min-
+ima.
+Having demonstrated the ability of the method, we
+then calculate the τ3of the GHZ states mixed with white
+noise, which does not have analytical solutions so far.
+The density matrices of the states are
+ρ(p) =pπGHZ+1−p
+8πE, (24)
+whereπEisaunitmatrix. Inthiscase,thedensitymatri-
+ces are of rank 8. We need at least Np=8 to ensure R2∼0. In practice, we have calculated τ3up toNp=20. We
+find that use Np=15 can converge τ3to less than 10−3
+for this problem. (See Fig. 2). The calculated results
+are shown in Fig.1(d). From Fig. 1(d), we see that τ3
+of the white noise mixed GHZ states is zero for pless
+than about 0.7, then increase almost linearly to 1.0 as
+pincreases. The behavior of τ3looks similar to that of
+mixed GHZ and W states.
+Itisstraightforwardtoapplythemethodtoanymixed
+states. It therefore opens up a way to study the multi-
+particleentanglementeffects in manyimportant systems,
+e.g., in systems with quantum phase transitions, which
+was impossible before, because the analytical solution to
+the tree-tangle of a general mixed state is not available.
+To conclude, we have demonstrate an efficient numeri-
+cal method to calculate the three-tangle of general mixed
+states. We construct a “energy function” (target func-
+tion) for the three-tangle of the mixed state under cer-
+tain constrains. The “energy function” (target function)
+is then optimized via a replica exchange Monte Carlo
+method. We have tested the method for the examples
+with known analyticalresults, showingremarkableagree-
+ment. We further calculate the three-tangle of GHZ
+states mixed with white noise. The work opens a way
+to study three-tangle of general mixed states, and can be
+generalized to solve many other optimization problems
+in quantum information theory.
+L.H. acknowledges the support from the Chinese Na-
+tional Fundamental Research Program 2006CB921900,
+the Innovationfunds and“HundredsofTalents”program
+fromChinese AcademyofSciences, and NationalNatural
+Science Foundation of China, Grant No. 10674124. W.Z
+were supported by National Natural Science Foundation
+of China, Grant No. 10874170and K.C.Wong Education
+Foundation, Hong Kong.
+[1] C. H. Bennett, D. DiVincenzo, J. Smolin, and W. K.
+Wootters, Phys. Rev. A 54, 3824 (1996).
+[2] V. Coffman, J. Kundu, and W. Wootters, Phys. Rev. A
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+Bayesian Methods and Universal Darwinism 
+John Campbell 
+Abstract. Bayesian methods since the time of Laplace have bee n understood by their 
+practitioners as closely aligned to the scientific method. Indeed a recent champion of Bayesian 
+methods, E. T. Jaynes, titled his textbook on the s ubject Probability Theory: the Logic of 
+Science. Many philosophers of science including Kar l Popper and Donald Campbell have 
+interpreted the evolution of Science as a Darwinian  process consisting of a 'copy with selective 
+retention' algorithm abstracted from Darwin's theor y of Natural Selection. Arguments are 
+presented for an isomorphism between Bayesian Metho ds and Darwinian processes. Universal 
+Darwinism, as the term has been developed by Richar d Dawkins, Daniel Dennett and Susan 
+Blackmore, is the collection of  scientific theorie s which explain the creation and evolution of 
+their subject matter as due to the operation of Dar winian processes. These subject matters span 
+the fields of atomic physics, chemistry, biology an d the social sciences. The principle of 
+Maximum Entropy states that systems will evolve to states of highest entropy subject to the 
+constraints of scientific law. This principle may b e inverted to provide illumination as to the 
+nature of scientific law. Our best cosmological the ories suggest the universe contained much less 
+complexity during the period shortly after the Big Bang than it does at present. The scientific 
+subject matter of atomic physics, chemistry, biolog y and the social sciences has been created 
+since that time. An explanation is proposed for the  existence of this subject matter as due to the 
+evolution of constraints in the form of adaptations  imposed on Maximum Entropy. It is argued 
+these adaptations were discovered and instantiated through the operations of a succession of 
+Darwinian processes. 
+Keywords: Darwinism, evolution 
+PACS: 00. 02.50  
+I/glyph1197TRODUCTIO/glyph1197 
+Darwin understood that the algorithmic nature of na tural selection suggested that its 
+abstraction might be useful to the study of the evo lution of other processes such as 
+language. Darwinian processes, (where evolution of some subject matter is described 
+as embodying three steps: replication, variation an d differential selection) are now 
+used as the essential explanatory mechanism in nume rous fields of study in biological 
+and social sciences such as population genetics, ev olutionary epistemology, 
+evolutionary psychology, evolutionary archaeology, memetics and  evolutionary 
+linguistics. The term Universal Darwinism has come to be applied to the collection of 
+scientific theories utilizing Darwinian processes a s their essential explanatory model 
+(Dawkins, 1976; Blackmore, 1995; Dennett, 1995). 
+Prominent philosophers of science including Karl Po pper and Donald Campbell 
+have founded the active the field of Evolutionary E pistemology, the evolution of 
+systems of knowledge (including science), in terms of a Darwinian process (Popper, 
+1972; Campbell, 1965). In this light the evolution of Science might be con sidered in terms of copies of 
+existing theories being made in scientists' minds, often with variations from the 
+original. Those variations that experience preferen tial survival are those that are best 
+supported by the data. 
+Of course ‘best supported by the data’ is a measure  supplied by Bayesian methods. 
+Indeed E.T. Jaynes dubbed Bayesian probability 'the  logic of science'. (Jaynes, 2003). 
+Karl Friston and others, of the Bayesian Brain scho ol of neuroscience, have 
+developed a theory of mind which models aspects of  mental processes as near optimal 
+Bayesian mechanisms that update mental models from experience (Friston, 2007).  
+Given that the brain has evolved over hundreds of m illions of years, through natural 
+selection, to gather knowledge and guide effective actions in the world, we are led to 
+the observation that the Bayesian process of updati ng knowledge was discovered long 
+before the existence of Science or even of human be ings.  Indeed this observation 
+suggests that the Darwinian/Bayesian mechanism of k nowledge evolution is a truly 
+ancient and time tested strategy and, contrary to t he view of post modernists, is of a 
+different stature from other cultural accounts such  as myths or religion. 
+In view of the homology between Darwinian and Bayes ian mechanisms apparent in 
+the evolution of Science, Section One will examine the abstract nature of Darwinian 
+processes postulated to be operating by various the ories included within universal 
+Darwinism and will explore their Bayesian nature. 
+The history of the universe has involved the creati on and evolution of systems 
+possessing greater complexity over time. Much of sc ientific subject matter is 
+composed of levels of complexity created and evolve d since the time of the Big Bang 
+including atomic physics, chemistry, biology and cu lture.  It is something of a puzzle 
+that while the history of the universe conforms to the 2nd Law of Thermodynamics 
+many objects of scientific interest in the universe  have managed to defy this overall 
+trend and maintain their existence in states of low  entropy. 
+The principle of Maximum Entropy tells us that syst ems will always evolve to 
+states of higher entropy unless they are constraine d by scientific law to do otherwise. 
+It follows that systems will only maintain low entr opy states if they are constrained to 
+do so by scientific law. Section Two makes the argu ment that insight into the nature of 
+scientific law might be gained through an understan ding of the processes by which 
+these constraints were created and evolved. 
+In the Discussion a recent theory from physics is e xamined that holds some promise 
+of extending this paradigm to areas of scientific s ubject matter more fundamental than 
+those of biology and culture. 
+Darwinian Processes and Bayesian Methods 
+The algorithmic nature of natural selection allows its essential mechanism to be 
+abstracted and hypothesized as a possible mechanism  operating in the evolution of 
+subject matter often other than biological. Numerous  theories of this type abound in 
+the social sciences and even in the hard sciences s uch as physics.    
+The essential abstraction from natural selection, w hich we are calling a Darwinian 
+process, has been developed in the work of Richard Dawkins, Daniel Dennett and 
+Susan Blackmore (amongst others) to consist of a th ree-step process: 1) Replication of system. 
+2) Inheritance of some characteristics that have va riation amongst the offspring. 
+3) Differential survival of the offspring according  to which variable characteristics 
+they possess. 
+It has been proposed that any system adhering to th is three-step algorithm, 
+regardless of its substrate, must evolve and will e volve in the direction of an increased 
+ability to survive (Dawkins, 1976; Dennett, 1995; B lackmore, 1995). 
+Offspring that inherit a slate of characteristics p roviding them with a substantial 
+survival advantage will have greater reproductive s uccess than their competitors. 
+Characteristics that bestow greater survivability a re called adaptations. Adaptations 
+are usually discovered through processes with a ran dom 'trial and error' component, 
+such as genetic mutation, but the greater survivabi lity they bestow allows them to 
+become widespread within the population. Systems wi th long evolutionary histories 
+come to accumulate many adaptations. Indeed any org anism may be considered as 
+largely an accumulation of adaptations built up ove r evolutionary time. 
+Survival of a complex system is only allowed by the  2nd Law of Thermodynamics 
+if exchanges with the system's environment are achi eved such that the entropy of the 
+system is maintained or decreased to an extent more  than made up for by the increase 
+in environmental entropy. For instance the strategy  of photosynthesis in green plants 
+to improve their survivability (through an intricat e process that transfers free energy 
+from photons in the environment to a chemical form of energy usable by the plant) is 
+dependent on an overall increase in entropy in the sun/earth system. 
+Survivability of complex systems thus depends on fi nding loopholes where 
+environmental entropy increase can be used to decre ase system entropy. In this sense 
+adaptations are highly tuned to their environments and may be considered to model 
+their environments. Henry Plotkin (1997 p.116) has argued that adaptations can be 
+considered the knowledge of specific methods used b y low entropy systems to 
+maintain or decrease their entropy at the expense o f their environments. 
+Bayesian probability is often defined as a measure of a state of knowledge. It is the 
+degree of belief we should have in a proposition gi ven the data available to us. It is a 
+subjective measure in the sense that it is dependen t on the evidence available to a 
+given observer and may differ between observers or with a given observer over time. 
+However it is objective in the sense that a unique probability should be calculated by 
+all observers having the same evidence (Jaynes, 200 3, pp. 44-45). 
+This objective nature of Bayesian probability precl udes it from applying solely to 
+the conscious minds of human observers engaged in c ultural systems of knowledge 
+such as Science. We should expect the principles of  Bayesian probability to be 
+followed by any system having internal models of ex ternal reality that are updated 
+through experiences if those models are to be maint ained in a near-optimal manner. 
+A central theme running through E.T. Jaynes’ great text book on Bayesian 
+probability concerns the necessary design of any ro bot capable of plausible reasoning. 
+His conclusion is that such a robot can only be des igned according to the principles of 
+Bayesian probability (Jaynes, 2003). Any system of knowledge that must be updated 
+to model changes in its environment or to model gre ater details of its environment is 
+performing inference, and Bayesian probability is t he unique method of performing 
+such inference (Jaynes, 1986). Karl Friston has argued that any adaptive system mu st contain internal models of its 
+environment. He argues that the accuracy of these m odels in characterizing their 
+environments is maintained by Bayesian processes th at update the models through 
+inference from data. Friston has proposed a method of understanding many aspects of 
+cortical organization and response that he has call ed the free energy principle. It 
+suggests that organisms attempt to minimize discrep ancies (surprises) between their 
+internal models and actual events in the environmen t (Friston, 2007). The Bayesian 
+Brain School has produced a substantial body of res earch suggesting that Bayesian 
+processes operate at the core of many unconscious m ental functions to update internal 
+models in a manner that keeps them in sync with the ir environments (Friston, 2007). 
+As Friston notes, the price of experiencing surpris e may often be death. 
+The mechanics of updating models in a Bayesian mann er when new data becomes 
+available might be understood as forming an algorit hmic process: 
+1) Copies of the competing models brought forward a long with the relevant prior 
+data. 
+2) Variations in likelihoods amongst the competing models as provided by new 
+data in accordance with Bayes' Theorem. 
+3) Differential survival of competing models accord ing to their Bayesian 
+likelihood. 
+ 
+It can be argued that this algorithm is isomorphic to the algorithm underlying 
+Darwinian processes. 
+If we consider the range of scientific subject matt er that might be included within 
+this Darwinian/Bayesian framework of adaptive syste ms we should consider Science 
+itself, the many branches of social science with 'E volutionary' in their titles (such as 
+Evolutionary Psychology), many aspect of behavioura l and neuroscience and of course 
+most fields of biology involving natural selection.  
+The primary knowledge model employed by all organis ms is coded in their genetics 
+and expressed in their phenotypes. The adaptations composing an organism form a 
+detailed model of what is expected in their environ ment and how to respond to this 
+environment in a manner facilitating survival. At t he level of population genetics the 
+composition of a population's genetics is determine d by generational transformations 
+largely mediated through natural selection (Lewonti n, 1974).  These transformations 
+serve to maintain a close fit between the model inh erent in the population's genome 
+and characteristics exhibited by the population's e nvironment.   
+The genetic plan retained at any given time is sele cted on the basis of the 
+adaptations produced in the phenotype. Adaptations encapsulate knowledge of the 
+environment, specifically knowledge of how to survi ve in the particular environment 
+in which the organism expects to find itself (Plotk in, 1997). If we accept a definition 
+of inference as 'conclusions drawn logically from p remises' and perhaps limit the usual 
+meanings of 'premises' and 'conclusions' to 'facts revealed by experience' and 'models 
+derived from experience' then we might view natural  selection's selective retention of 
+adaptations as analogous to updating of knowledge t hrough inference.  
+Bayesian theory tells us that there is only one mat hematically sound method of 
+updating the plausibility of models. That method is  Bayes' Theorem (Jaynes 2003). 
+Thus we should expect, to the extent that a populat ion's genomic model  
+ 
+ 
+ 
+ 
+ 
+maintains a close fit to its environment, that this  updating is accomplished through 
+mechanisms analogous to Bayesian methods. 
+Since the mid-twentieth century many new fields hav e been introduced to the social 
+sciences which employ Darwinian processes as explan ations for the creation and 
+evolution of their subject matter. These include ev olutionary psychology, evolutionary 
+linguistics, evolutionary archaeology, evolutionary  anthropology, evolutionary 
+sociology and evolutionary economics. These discipl ines often borrow  theoretical 
+constructs directly from evolutionary biology. For instance a number of fields have 
+adopted the biological method of cladistics, with o nly minor alterations. As an 
+example Evolutionary Archeology uses cladistics in the study of artifact evolution 
+such as arrowhead design (O'Brien, 2003). In these scenarios cultural artifacts are 
+understood as cultural adaptations which have evolv ed. 
+Besides applying the Darwinian process to explain e volution of specific cultural 
+entities, theories have also proposed Darwinian pro cesses to be responsible for the 
+overall emergence and evolution of culture itself. Two of the most developed are the 
+Dual Inheritance theory (Henrich, 2007) and Memetic s (Blackmore, 1995). 
+Within Universal Darwinism theories have in common the supposition that the 
+knowledge structures of their subject matter are cr eated and evolve through the 
+operation of Darwinian processes. This knowledge ex ists in the form of adaptations 
+that increase survivability and are passed between generations.  
+  
+Maximum Entropy and Evolution 
+The principle of Maximum Entropy predicts that a sy stem will evolve to a state of 
+highest entropy available, subject to constraints i n the form of prior information 
+applicable to the system. The constraint often repr esents our knowledge in the form of 
+scientific law (Jaynes, 1985). In fact scientific l aw might be viewed as generalized 
+principles that have been derived from prior inform ation. 
+The second law is widely recognized as providing pe rhaps the most fundamental 
+framework for the evolution of the universe. Yet wh en we view the actual evolution of 
+the universe as revealed by our best scientific the ories there is an apparent paradox. 
+The scope of scientific subject matter near the tim e of the Big Bang was limited but 
+since that time there have occurred successive intr oductions of new low-entropy 
+subject matter including atomic physics, chemistry,  biology and culture. All of these 
+complex systems operate in accordance with the seco nd law but they appear to exploit 
+loopholes in order to maintain their local low entr opy status. 
+Thermodynamic entropy in this context should be def ined as the constrained 
+maximum of –k Tr(p ln p) over all density matrices of the macrovariables considered 
+(Jaynes, 1992). Thus macrovariables such as tempera ture impose constraints on FIGURE 1 FIGURE 1 FIGURE 1 FIGURE 1. A schematic for adaptive systems containin g internal models of their 
+environments that are inferred, through Bayesian me thods, from prior data. These models 
+are tested by new data or experience in the environ ment and updated when surprised.  Prior Data 
+ Inference  
+Process  Model of  
+Reality     Reality New Data 
+Surprise entropy. Jaynes wrote that he could only give an an swer to questions regarding the 
+possibility of biological systems violating the sec ond law if the variables regarding the 
+thermodynamic state of the system were completely d efined (Jaynes, 1965). For 
+biological systems those variables would, for examp le, include complexities such as 
+numerous enzymes operating in the cell serving to c onstrain the probability of various 
+chemical pathways occurring and thereby constrainin g the entropy of the system. 
+The principle of Maximum Entropy tells us that the failure of complex systems that 
+make up the bulk of scientific subject matter to ev olve to states of higher entropy is 
+due to scientific law. It is apparent that scientif ic law has accumulated over the course 
+of time along with the creation of scientific subje ct matter. Within biology it is a near 
+consensus that this accumulation is due to natural selection. Numerous theories within 
+the social sciences also explain the creation and e volution of the low entropy states 
+composing their subject matter as due to the operat ion of Darwinian processes. 
+Prior knowledge, as used by the principle of Maximu m Entropy, is not only a 
+subjective attribute of the researcher's mind. To p rovide useful results it must 
+encapsulate all pertinent constraints actually occu ring in nature. Indeed experimental 
+results that do not agree with the predictions of M aximum Entropy indicate the 
+researcher is missing pertinent prior knowledge (Ja ynes, 2003, p. 371).    
+Thus, prior knowledge or scientific law, which form s constraints on Maximum 
+Entropy, is an attribute of objective reality and o ne that has evolved during the history 
+of the universe. An understanding of how such const raints were created and 
+transformed may have been most thoroughly developed  within biology where it forms 
+the subject matter of evolutionary biology. 
+It is proposed that much of scientific law or prior  knowledge may be understood as 
+generalizations of those designs that are capable o f maintaining local states of low 
+entropy. These designs although being rare within t he space of all possible designs 
+have become widely distributed in reality due to th eir discovery and propagation by 
+Darwinian processes. It is suggested that much of s cientific subject matter is the 
+accumulated adaptations discovered over evolutionar y history by Darwinian 
+processes. 
+Theories within Universal Darwinism describing the evolution of their subject 
+matter in terms of information accumulated through the operation of Darwinian 
+processes are closely aligned to the Principle of M aximum Entropy as they may 
+explain the creation and evolution of constraints c entral to Maximum Entropy. 
+Discussion 
+The unreasonable effectiveness of mathematics in th e natural sciences has seemed 
+something of a wonder to many researchers (Wigner, 1960). Why is mathematics so 
+powerful in describing natural processes? Max Tegma rk (2009) has offered a solution 
+that may appear somewhat obvious, suggesting that p hysical reality is isomorphic to a 
+mathematical structure which we are gradually disco vering. 
+Bayesian probability may well lay claim to be the ' logic of science', given that it is 
+the unique mathematical structure providing a valid  extension of logic to areas of 
+incomplete knowledge where degrees of plausibility must be dealt with (Jaynes, 2003, 
+p. xx). Science is one such structure. However this  claim may be extended to areas of the natural world where structures containing knowl edge have evolved. Gaining 
+knowledge requires logical inference which operates  according to Bayesian principles.  
+Outside of the realm of biology and the social scie nce lies much of reality that also 
+persists in states of low entropy, notably atomic p hysics and chemistry. These states 
+seem to owe their persistence as low-entropy entiti es directly to scientific law. 
+Some researchers, inspired by quantum computation, view the universe itself as a 
+quantum computer based on the observation that the fundamental force laws of 
+physics and their mediated interactions may be inte rpreted as quantum computations. 
+The outcomes of the interactions are the results of  the computations (Lloyd, 2007). 
+Some biologists are also calling for a more informa tion-centric focus to their subject 
+(Brenner, 1999).  We might expect that these endeav ors, concerned with information 
+and inference, will be informed by Bayesian methods  and Darwinian processes.   
+Discovery of scientific laws might be most complete  in the fields of physics and 
+chemistry where constraints on entropy production a re perhaps of a simpler design, 
+their operation being quantum. It may seem highly u nlikely that these constraints were 
+also discovered through the action of Darwinian pro cesses. Unfortunately there is 
+currently a shortage of interpretations of quantum processes that are explanatory in the 
+usual sense. Indeed Richard Feynman (1967) famously  remarked that 'no one 
+understands quantum mechanics'. 
+This barrier to understanding, existing for nearly a century, may have recently been 
+breached. Wojciech Zurek, of the Los Alamos Nationa l Laboratory, and collaborators 
+have proposed a resolution to the primary point of misunderstanding known as 'the 
+measurement problem'. Central to this resolution is  a theory they have named 
+Quantum Darwinism. Quantum Darwinism portrays quant um interactions in terms of a 
+Darwinian process where information is copied from the quantum system to its 
+environment. According to this theory only a limite d subset of available information 
+can survive the transfer and forms the 'classical' reality we witness and are composed 
+of, a reality that is selected, in a Darwinian mann er, from the vast array of quantum 
+potentialities (Zurek, 2009). 
+It remains to be seen if Quantum Darwinism can be s uccessfully interpreted in a 
+manner portraying the complexities of atomic physic s and chemistry as adaptations 
+discovered by Quantum Darwinism and following Bayes ian principles.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ ACK/glyph1197OWLEDGME/glyph1197TS 
+I would like to thank Karen Drysdale, Mac Campbell,  Ry Glover and Michael 
+Skrigitil for lively and enjoyable discussions. 
+REFERE/glyph1197CES 
+1.  Blackmore, S. (1995). The Meme Machine.  Oxford: Oxford University Press. 
+2.  Brenner, S. (1999). Theoretical Biology In The Thir d Millennium. Philosophical 
+Transactions: Biological Sciences, Vol 354, HglyphJk7o 1392  . 
+3.  Campbell, D. (1965). Variation and selective retent ion in socio-cultural evolution. In G. I. 
+Herbert R. Barringer, Social change in developing areas: A reinterpretati on of evolutionary 
+theory  (pp. 19-49). Cambridge, Mass.: Schenkman. 
+4.  Dawkins, R. (1976). The Selfish Gene.  Oxford: Oxford University Press. 
+5.  Dennett, D. (1995). Darwin's Dangerous Idea.  New York: Touchstone Publishing. 
+6.  Feynman, R. (1967). The Character of Physical Law.  M.I.T. Press. 
+7.  Friston, K. (2007). Free Energy and the Brain. Synthese  , 159: 417-458. 
+8.  Henrich, J. a. (2007). Dual Inheritance Theory: The  Evolution of Human Cultural Capacities 
+and Cultural Evolution. In e. b. Barrett, Oxford Handbook of Evolutionary Psychology.  Oxford 
+University Press. 
+9.  Jaynes, E. (1986). Bayesian Methods: General Backgr ound. In J. H. (ed.), (174Kb),' in 
+MaximumOEntropy and Bayesian Methods in Applied Sta tistics.  Cambridge: Cambridge Univ. 
+Press. 
+10.  Jaynes, E. (1965). Gibbs vs Boltzmann Entropies. Am. J. Phys.  , 391. 
+11.  Jaynes, E. (2003). Probability Theory: The Logic of Science.  Cambridge, U.K.: Cambridge 
+University Press. 
+12.  Jaynes, E. (1992). The Gibbs Paradox. In P. N. G. E rickson, MaximumOEntropy and Bayesian 
+Methods.  Kluwer, Dordrecht. 
+13.  Jaynes, E. (1985). Where do we go from Here? In J.C . R. Smith and W. T. Grandy, MaximumO
+Entropy and Bayesian Methods in Inverse Problems  (p. p. 21 ). 
+14.  Lewontin, R. C. (1974). The Genetic Basis of Evolutionary Change.  Columbia University 
+Press. 
+15.  Lloyd, S. (2007). Programming the Universe.  Vintage. 
+16.  O'Brien, M. J. (2003). Resolving Phylogeny: Evoluti onary Archaeology's Fundamental Issue. 
+In T. L. VanPool, Essential Tensions in Archaeological Method and The ory,  (pp. 115-135). 
+Salt Lake City: University of Utah Press. 
+17.  Plotkin, H. (1997). Darwin Machines and the HglyphJk7ature of Knowledge.  Cambridge, 
+Massachusettes: Harvard University Press. 
+18.  Popper, K. (1972). Objective Knowledge .  Clarendon Press. 
+19.  S.Kirby, M. (2003). Language Evolution.  New York: Oxford University Press. 
+20.  Tegmark, M. (2009). The Mathematical Universe. Foundations of Physics 38  , 101-50. . 
+21.  Wigner, E. (1960). The Unreasonable Effectiveness o f Mathematics in the Natural Sciences. 
+Communications on Pure and Applied Mathematics 13  , 1–14. 
+22.  Zurek, W. (2009). Quantum Darwinism. HglyphJk7ature Physics, vol. 5  , pp. 181-188 . 
+ 
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+ 
+ 1 
+ 
+Abstract —Physical-layer Network Coding (PNC) makes use of t he additive nature of the 
+electromagnetic (EM) waves to apply network coding arithmetic at the physical layer. With PNC, the 
+destructive effect of interference in wireless netw orks is eliminated and the capacity of networks can  
+be boosted significantly. This paper addresses a ke y outstanding issue in PNC: synchronization 
+among transmitting nodes. We first investigate the impact of imperfect synchronization (i.e., finite 
+synchronization errors) in a 3-node network. It is shown that with QPSK modulation, PNC still 
+yields significantly higher capacity than straightf orward network coding when there are 
+synchronization errors. Significantly, this remains  to be so even in the extreme case when 
+synchronization is not performed at all. Moving bey ond a 3-node network, we propose and 
+investigate a synchronization scheme for PNC in a g eneral chain network. At last, numerical 
+simulation verifies that PNC is robust to synchroni zation errors. In particular, for the mutual 
+information performance, there is about 0.5dB loss without time synchronization and there is at most 
+2dB loss without phase synchronization.  
+ 
+Index Terms — Physical-layer network coding, wireless networks,  synchronization Synchronization Analysis in Physical Layer Network Coding 
+Shengli Zhang, Soung-Chang Liew, and Hui Wang  
+ 2 
+ 
+I.  INTRODUCTION 
+One of the biggest challenges in wireless communica tion is how to deal with the interference at the 
+receiver when signals from multiple sources arrive simultaneously. In the radio channel of the physica l 
+layer of wireless networks, data are transmitted th rough electromagnetic (EM) waves in a broadcast man ner. 
+The interference between these EM waves causes the data to be scrambled. While interference has a 
+negative effect on wireless networks in general, it s detrimental effect on the throughput of multi-hop  ad hoc 
+networks is particularly noticeable [1, 2,3].  
+Most communication system designs try to either red uce or avoid interference (e.g., through receiver 
+design or transmission scheduling [1]). However, in stead of treating interference as a nuisance to be 
+avoided, we can actually embrace interference to im prove throughput performance. To do so in a multi-h op 
+network, we proposed Physical-layer Network Coding (PNC) in [4] to create an apparatus similar to that  of 
+network coding, but which performs network coding a rithmetic at the lower physical layer using the 
+additive property of EM signal reception. Through a  proper modulation-and-demodulation technique at 
+relay nodes, addition of EM signals can be mapped t o GF(2 n) addition of digital bit streams, so that the 
+interference becomes part of the arithmetic operati on in network coding.   
+Two levels of synchronization between the two end-n odes were assumed in PNC in [4], namely 
+symbol-level time synchronization, and carrier- fre quency/phase synchronization. In this paper, we fir st 
+investigate the impact of imperfect synchronization  (i.e., finite synchronization errors) on PNC in a 3-node 
+network with QPSK modulation. It is shown that PNC still yields significantly higher capacity than 
+straightforward network coding when there are synch ronization errors. Significantly, this remains to b e so 
+even in the extreme case when synchronization is no t performed at all. Moving beyond the 3-node linear  
+network, we propose a synchronization scheme for PN C in the N-node linear network. The N-node network 
+can be decomposed into a chain of 3-node PNC units for synchronization purposes. We argue that if 
+channel coding is applied on each of the 3-node PNC  units, then the performance in terms of the end-to -end 
+capacity will be the same for the N-node network and the 3-node network.  
+Related Work : 
+Synchronization has long been an active research pr oblem in wireless networks. Here, we here review 
+prior work on synchronization relevant to the 3-nod e PNC case. First, symbol time and carrier-frequenc y 
+synchronizations, which are needed in PNC, have bee n actively investigated by researchers in the field s of  
+ 3 
+OFDMA, wireless-sensor network, and cooperative tra nsmission. In particular, methods for joint estimat ion 
+of carrier-frequency errors, symbol timing error an d channel response have been proposed for OFDMA 
+networks [5, 6], and methods for symbol synchroniza tion have been proposed in wireless sensor networks  
+[7, 8] All these methods can be borrowed to finish the symbol time and frequency synchronization in PN C. 
+Besides, PNC needs the critical carrier phase synch ronization (which also implies a very accurate carr ier 
+frequency synchronization), which has recently been  studied in the field of coherent cooperation 
+(distributed beam forming ) to synchronize the sepa rately distributed nodes. Among all these schemes, the 
+most direct scheme uses the beacons bounded between  the destination (relay) node and the source (end) 
+nodes to estimate the relative phase offsets, so th at each source compensated its phase according to t his 
+offset before transmitting [9]. To reduce the high feedback rate required in [9], a one bit feed back 
+synchronization scheme is proposed in [10] and show ed good performance with experiments. Removing 
+the iterative information exchanging between the de stination and the source nodes, some open loop 
+algorithms are proposed. For example, positive resu lts have been obtained in [11] with a master-slave 
+architecture to prove the feasibility of the distri buted beam forming technique. In [12, 13], another open 
+loop synchronization scheme, round trip synchroniza tion, were proposed and discussed where a beacon is  
+used to measure round trip phase delays among the t ransmitters and the destination. We can use the ide as in 
+these schemes to synchronize the phase of the two e nd nodes in the three-node PNC schemes. Besides the  
+proposed synchronization algorithms, people also an alyzed the affect of the synchronization errors in the 
+coherent cooperate transmissions. For example, a mo re realistic collaborative communication system tha t 
+includes the influence of AWGN and phase error on t he signal transmission is analyzed in [14]. However , 
+the analysis in [14] is different from the analysis  in our paper since the source nodes in [14] transm it the 
+same signal and the source nodes in our paper trans mit different signals.  
+The rest of this paper is organized as follows. Sec tion II presents the system model and illustrates t he 
+basic idea of PNC with a linear 3-node network unde r the assumption of perfect synchronization. Sectio n 
+III analyzes the performance penalty of non-perfect  synchronization on PNC. Section IV proposes a 
+strategy to extend 3-node synchronization to N-node synchronization in a long PNC chain. Section V 
+studies the performance penalty of non-perfect sync hronization by numerical simulation, and section VI  
+concludes the paper.   
+II.  SYSTEM MODEL AND ILLUSTRATING EXAMPLE  
+  
+ 4 
+A.  System Model: 
+In this paper, we focus on two-way relay channels ( TWRC). A typical TWRC is the three node two way 
+relay channel as shown in Fig. 1. N1 (Node 1) and N3 (Node 3) are nodes that exchange information, but 
+they are out of each other’s transmission range. N2 (Node 2) is the relay node between them. This syst em 
+model has found applications in many scenarios. In satellite communication, the satellite serves as a relay 
+to facilitate information exchange between two mobi le stations on the earth. In wireless mesh networks , 
+wireless nodes may also relay information between i ts neighbors. 
+ 
+Figure 1.  A 3-node linear network 
+We consider frame-based communication in which a ti me slot is defined as the time required for the 
+transmission of one fixed-size frame. In this paper , a frame is denoted by a capital letter and symbol s 
+within a frame are denoted by the corresponding sma ll letter. Each node is equipped with an 
+omni-directional antenna, and the channel is half d uplex so that transmission and reception at a parti cular 
+node must occur in different time slots. We assume that QPSK modulation is employed at all the nodes.  
+B.  Physical-layer Network Coding Scheme 
+Before introducing the PNC transmission scheme, we first describe the traditional transmission 
+scheduling scheme and the “straightforward” network -coding scheme for mutual exchange of a frame in 
+the 3-node network [15, 16]. 
+The traditional transmission schedule is given in F ig. 2. Let Si denote the frame initiated by Ni. N1 first 
+sends S1 to N2, and then N2 relays S1 to N3. After that, N3 sends S3 in the reverse direction. A total of four 
+time slots are needed for the exchange of two frame s in opposite directions.  
+ 
+  
+ 5 
+Figure 2.  Traditional scheduling scheme 
+Ref. [15] and [16] outline the straightforward way of applying network coding in the 3-node wireless 
+network. Fig. 3 illustrates the idea. First, N1 sends S1 to N2 and then N3 sends frame S3 to N2.  After 
+receiving S1 and S3, N2 encodes them to obtain the frame 2 1 3 S S S = ⊕ , where ⊕ denotes bitwise 
+exclusive OR operation being applied over the entir e frames of S1 and S3. N2 then broadcasts S2 to both N1 
+and N3. When N1 (N2) receives S2, it extracts S3 (S1) from S2 using the local information S1 ( S3). A total of 
+three time slots are needed, for a throughput impro vement of 33% over the traditional transmission 
+scheduling scheme.  
+ 
+Figure 3.  Straightforward network coding scheme  
+We now introduce PNC. For the time being, let us al so assume perfect symbol-level time, carrier 
+synchronization (we will remove this assumption in later sections), and the use of power control, so t hat the 
+frames from N1 and N3 arrive at N2 with the same phase and amplitude. The combined pas sband signal 
+received by N2 during one symbol period is  
+2 1 3 
+1 1 3 3 
+1 3 1 3 ( ) ( ) ( ) 
+[ cos( ) sin( )] [ cos( ) sin( )] 
+( )cos( ) ( )sin( ) r t s t s t 
+a t b t a t b t 
+a a t b b t ω ω ω ω 
+ω ω = + 
+= + + + 
+= + + +                      (3) 
+where ( ) is t , i = 1 or 3, is the passband signal transmitted by Ni and 2( ) r t is the passband signal received by 
+N2 during one symbol period, ia and ibare the QPSK modulated information bits (in-phase a nd 
+quadrature-phase respectively) of Ni; and ω is the carrier frequency. Then, N2 will receive two baseband 
+signals, in-phase ( I) and quadrature phase ( Q), respectively, as follows: 
+1 3 
+1 3 I
+Qr a a 
+r b b = + 
+= +                                    (4) 
+Note that N2 cannot extract the individual information transmit ted by N1 and N3, i.e., 1 1 3 3 ,  ,   and a b a b , 
+from its received combined signal rI and rQ. However, N2 is just a relay node and it does not care what the  
+ 6 
+information is. As long as N2 can transmit the necessary information to  N 1 and N3 for extraction of 
+1 1 3 3 ,  ,  ,  a b a b  over there, the end-to-end delivery of information  will be successful. For this, all we need is 
+a special modulation/demodulation mapping scheme, r eferred to as PNC mapping  in this paper, to obtain 
+the equivalence of GF(2) summation of bits from N1 and N3 at the physical layer.  
+Table 1 illustrates the idea of PNC mapping. In Tab le 1, ( ) {0,  1 } I
+js∈  is a variable representing the 
+in-phase data bit of jN and { 1,  1 } ja∈ − is a variable representing the binary modulated bit  of ( ) I
+js  such 
+that ( ) 2 1 I
+j j a s = − . A similar table (not shown here) can also be cons tructed for the quadrature-phase data by 
+letting ( ) {0,  1 } Q
+js∈ be the quadrature data bit of jN, and { 1,  1 } jb∈ − be the binary modulated bit of ( ) Q
+js 
+such that ( ) 2 1 Q
+j j b s = − .   
+Table 1. PNC Mapping: modulation mapping at N1, N2; demodulation and modulation mappings at N3 
+Demodulation 
+mapping at N2 Modulation mapping at N1 
+and N3 
+Input  Output   
+Modulation 
+mapping at N2 Input Output  
+Input  Output  
+( ) 
+1Is ( ) 
+3Is 1a 3a 1 3 a a + ( ) 
+2Is 2a 
+1 1 1 1 2 0 -1 
+0 1 -1 1 0 1 1 
+1 0 1 -1 0 1 1 
+0 0 -1 -1 -2 0 -1 
+ 
+With reference to Table 1, N2 obtains the data bits: 
+( ) ( ) ( ) ( ) ( ) ( ) 
+2 1 3 2 1 3 ;I I I Q Q Q s s s s s s = ⊕ = ⊕                             (5) 
+It then transmits, according to the QPSK modulation  mapping,  
+ 7 
+2 2 2 ( ) cos( ) sin( ) s t a t b t ω ω = +                            (6) 
+Upon receiving 2( ) s t , N1 and N3 can derive ( ) 
+2Isand ( ) 
+2Qs by ordinary QPSK demodulation. The 
+successively derived ( ) 
+2Isand ( ) 
+2Qs bits within a time slot will then be used to form the frame 2S. In other 
+words, the operation 2 1 3 S S S ⊕=  in straightforward network coding can now be reali zed through PNC 
+mapping. 
+As illustrated in Fig. 2, PNC requires only two tim e slots for the exchange of one frame (as opposed t o 
+three time slots in straightforward network coding) . 
+ 
+ 
+Figure 4.  Physical layer network coding 
+In [4], we analyze the bit error rate (BER) of 1 3 S S⊕ at the relay node for PNC scheme and the 
+straightforward network coding scheme. It is shown that PNC slightly outperforms the straightforward 
+network coding scheme, and slightly underperforms t he standard point-to-point BPSK transmission. When 
+the per-hop BER is low, the end-to-end BER for the three schemes is very similar. The main advantage o f 
+PNC, however, is the reduced number of time slots n eeded. In this paper, we showed that this conclusio n is 
+true even without perfect synchronization.  
+III.  PERFORMANCE PENALTY OF SYNCHRONIZATION ERRORS  
+The basic PNC scheme presented thus far requires sy mbol-time, carrier-frequency and carrier-phase 
+synchronizations, although these requirements could  be relaxed for other variations of PNC [17, 18]. W e 
+now consider the performance penalty of synchroniza tion errors on PNC 1. This framework is applicable to 
+situations where synchronization is not perfect (e. g., synchronization may become imperfect with time due 
+the change of channel) as well as where synchroniza tion is not performed at all. The discussion here i s 
+based on the 3-node model in section II. 
+1. Penalty of carrier-frequency/phase synchronizati on errors: 
+                                                        
+1 Our paper tries to show the advantages of PNC over  traditional and straightforward schemes even with synchronization errors. Although the power 
+synchronization error also affects the PNC performa nce, it will affect the traditional and straightfor ward schemes similarly. Therefore, we do not analyz e the 
+power synchronization error penalty in this paper.  
+ 8 
+We first consider carrier-phase and carrier-frequen cy errors. For QPSK modulation, the two received 
+signals from node N1 and N3 can be written as: 
+2 1 1 
+3 3 ( ) [ cos( ) sin( )] 
+[ cos(( ) ) sin(( ) )] r t a t b t 
+a t b t ω ω 
+ω ω θ ω ω θ = + 
++ + ∆ + ∆ + + ∆ + ∆                   (7) 
+where θ∆is the phase offset and ω∆ is frequency offset. Here we assume that the relat ive carrier-phase 
+offset of the two input signals are known to the re ceiver 2. The receiver down-converts the passband signal 
+to the baseband to obtain  
+1 1 3 3 
+1 1 3 3 2 1 3 
+[cos( ) sin( ) 
+[ ] cos( ) sin( )] r s 
+a b a b 
+a b a b s
+T T ω θ ω θ 
+θ θ =
++ + 
++ + +
+= + ∆ + ∆ ∆ + ∆ 
+= +                   (8) 
+where T is the symbol duration and T θ ω θ = ∆ + ∆  is the final phase offset generated by the carrier  
+frequency offset and the carrier phase offset. Here after, we only consider the final phase offset θ without 
+differentiating the contributions of carrier phase and carrier frequency offset. Note that we only nee d to 
+deal with the case when / 4 / 4 π θ π − ≤ < . If '2kπθ θ = + ⋅  and / 4 ' / 4 π θ π − < ≤ , we can simply replace 
+3a with / 2 
+3 3 'ka a e π= ⋅ , and replace θ with ' / 2 k θ θ π = − ; 3'a can be mapped back to 3a in a unique 
+manner after detection. 
+ 
+ 
+Figure 5.  Constellations of superimposed baseband signals in (8) with phase difference θequal to 0 ,15  and 45 ° ° ° . There are 
+four possible values for 1s(i.e., 1 1 s j = ± ± ) as indicated by the intersections of the blue, gr een, red,and black lines) . In each 
+subfigure, we show the four possible values for 2rgiven each of the 1s. The resulting constellation points with the same 
+shape corresponding to the same value of 1 3 s s ⊕  
+ 
+                                                        
+2 Before the adjacent transmitters transmit their da ta concurrently as per PNC, they could first take t urns transmitting a preamble in a non-overlapping m anner. 
+The receiver can then derive the phase difference f rom the two preambles. Frequency and time offsets c an be similarly determined using preambles. Note th at 
+this is different from synchronization, since the t ransmitters do not adjust their phase, frequency an d symbol-time differences thereafter. The receiver simply 
+accepts the synchronization errors the way they are .   
+ 9 
+From Fig. 5, we can see that as the phase differenc e θ increases from 0 to / 4 π (decreases from 0 to 
+/ 4 π− ), a constellation point of 1 3 s s ⊕ may break into several points and the distance bet ween the points 
+of different 1 3 s s ⊕ may increase or decrease. The BER performance of d emodulating 1 3 s s ⊕ from the 
+received signal in Fig. 5 is dominated by the minim um distance between constellation points of differe nt 
+1 3 s s ⊕. As shown in Fig. 5, given the phase difference 0 / 4 θ π ≤ ≤ , the minimum square distance, 
+denoted by d, between the points of different 1 3 s s ⊕ is  
+2 2 2 4(1 cos ) 4(1 sin ) d θ θ = − + −                       (9) 
+We now bound the equivalent power penalty caused by  the phase difference θ by benchmark against a 
+reference system. The transmit power of each source  in the original system is 2. Consider a reference 
+system in which the transmit power of each source i s 2P≤. With perfect phase synchronization, the 
+minimum distance between adjacent points of differe nt 1 3 s s ⊕ of the reference system is 2 / 2 P . We 
+can tune P such that the minimum distance of the reference sy stem is shortened to the minimum distance of 
+the PNC system with phase difference θ, i.e., 2 / 2 P d =. Effectively, the power penalty of the system 
+with phase difference θ is / 2 P  (note that P is the effective power and 2 is the actual power i n the 
+system with phase difference θ). In particular, the BER performance of the PNC sy stem with nonzero θ 
+is no worse than that of the reference system with zero θ and with power thus adjusted. This is because 
+decreasing the transmit power in the reference syst em reduces the distances among all the different 
+constellation points uniformly, while in the origin al system, the phase difference reduces the distanc es 
+between some constellation points and enlarges othe r distances. In other words, the performance loss o f the 
+phase difference θ is upper bounded by a power penalty given as follo ws: 
+2 2 2 ( ) / 2 / 4 (1 cos ) (1 sin | |) P d γ θ θ θ ∆ ≤ = = − + −    4 4 π π θ − ≤ ≤  .                    (10) 
+In Fig. 6, we plot the upper bound in (10) with dif ferent phase offset. We can see that the power pena lty 
+bound is more than 7dB when the phase offset is abo ut / 4 π± . However, we need not be too pessimistic. 
+When there is no synchronization, a reasonable assu mption is that the phase offset is uniformly distri buted 
+over [ / 4, / 4] π π − . In this case, the average upper bounded power pen alty is   
+ 10  
+/ 4 / 4 
+2 2 
+/ 4 0 2 4 ( ) ( ) (1 cos ) (1 sin ) 
+3.4dB d d π π 
+πγ θ γ θ θ θ θ θ π π −∆ = ∆ ≤ − + − 
+= − ∫ ∫        (11) 
+That is, even if carrier phase synchronization is n ot performed at all, the average SNR penalty is upp er 
+bounded by 3.4 dB (the simulation in Fig. 11 shows that the average power penalty is no more than 2dB. ). 
+To avoid the worst-case penalty and to obtain the a verage power penalty performance, the transmitters 
+could intentionally change their phases from symbol  to symbol using a “phase increment” sequence known  
+to the receivers (or intentionally increase their c arrier frequency). If the phase-increment sequences  of the 
+two transmitters are not correlated, then certain s ymbols are received with low error rates and certai n 
+symbols are received with high error rates during a  data packet transmission. With good FEC coding, th e 
+overall packet error rate can be reduced. This esse ntially translates the power penalty to data-rate p enalty.  
+ 
+Figure 6.  Power penalty upper bound of carrier phase and freq uency synchronization errors 
+ 
+Even after taking into account the 3.4dB loss upper  bound due to phase asynchrony, PNC may still have 
+a better throughput in the 1-D network and 2-D netw ork than conventional schemes shown in [19]. For 1- D 
+case, the SIR of PNC in [19, eq. (6)] is about 15.3 dB, and it is about 11.9dB after subtracting the 3. 4dB SIR 
+loss. While the SIR of the traditional 1-D transmis sion scheme is   
+ 11 
+0
+0
+0/
+{2 /[(2 4 ) ] 1/[(3 4 ) ] 1/[(5 4 ) ] } 
+8.5dB lP d SIR 
+P l d l d l d α
+α α α ∞
+==
++ + + + + 
+=∑             (12) 
+where P0 is the transmission power, d is the distance between adjacent nodes, and the fa ding coeffienets α 
+is set to a typical value of 4. For the 2-D case, t he SIR of PNC is 13.5dB with J=5 is the distance between 
+adjacent PNC chains as in [19, Eq. (9)], and it is 10.1dB after subtracting the 3.4dB loss. It still s atisfies the 
+target 10dB SIR requirement as used in traditional wireless networks [19]. 
+2. Penalty of time synchronization errors: 
+Ref. [20] analyzes the impact of time synchronizati on errors on the performance of cooperative MISO 
+systems, and show that the clock jitters as large a s 10% of the bit period actually do not have much 
+negative impact on the BER performance of the syste m. Based on the similar methodology, we can also 
+analyze the impact of time synchronization error to ward the performance of PNC. 
+In this section, we assume perfect carrier phase an d frequency synchronizations for simplicity. In thi s 
+case, the performance of QPSK is the same as that o f BPSK, and therefore we only consider the in-phase  
+signals hence. Let t∆ be the symbol offset of the two input signals. The  two transmitted in-phase signals 
+can be written as: 
+1 1 
+3 3 ( ) [ ]cos(2 ) ( ) 
+( ) [ ]cos(2 ) ( ) l
+ls t a l ft g t lT 
+s t a l ft g t lT t π
+π∞
+=−∞ 
+∞
+=−∞ = − 
+= − − ∆ ∑
+∑                       (13) 
+where, [ ] ja l is the lth  bit of the real part of signal ( ) js t , and ( ) g t is the pulse shaping signal. The baseband 
+signal can be written as 
+1 3 
+1 3 ( ) ( ) ( ) 
+1[ ] ( ) [ ] ( ) 2 lr t r t r t 
+a l g t lT a l g t lT t = + 
+= − + − − ∆ ∑                (14) 
+After the match filter, the receiver samples the si gnal at time instances / 2 t kT t = − ∆ (i.e., at the middle 
+of the offset) 3. We then have 
+                                                        
+3 When the symbol time offset is known to the relay node, another choice for the receiver is to discard  the inter-symbol interfering part and only take th e 
+non-interfering part into account to further improv e the performance.   
+ 12  
+1 3 
+1 3 
+1 3 1 3 
+,[ ] [ ] [ ] 
+1{ [ ] (( ) / 2) [ ] (( ) / 2)} 2
+1( [ ] [ ]) ( / 2) / 2 { [ ] (( ) / 2) [ ] (( ) / 2)} 2l
+l l k r k r k r k 
+a l p k l T t a l p k l T t 
+a k a k p t a l g k l T t a l p k l T t 
+≠= + 
+= − + ∆ + − − ∆ 
+= + ∆ + − + ∆ + − − ∆ ∑
+∑ (15) 
+where, ( ) p t is the response of the receiving filter to the inpu t pulse ( ) g t . As widely used in practice, the 
+raised cosine pulse shaping function, 2 2 2 sin( / )cos( / ) ( ) ( ) / (1 4 / ) t T t T g t p t t T t T π πβ 
+π β = = ⋅ − , is chosen. We see that the time 
+synchronization errors not only decrease the desire d signal power, but also introduce inter-symbol 
+interference (ISI). Therefore, we use SINR (signal over noise and interference ratio) penalty here to 
+evaluate the performance degradation. The SINR pena lty can be calculated as 
+0
+2 2 
+2
+10 10 2( ) ( ) 
+10log ( ( / 2)) 10log ( ) isi n 
+nt SINR t SNR 
+p t γ
+σ σ 
+σ∆ ∆ = ∆ − 
++= ∆ −                      (16) 
+where 2 2 
+1 2 
+,{( [ ] (( ) / 2) [ ] (( ) / 2)) } isi 
+l l k E a l p k l T t a l p k l T t σ
+≠= − + ∆ + − − ∆ ∑  is the variance of the inter-symbol 
+interference. Fig. 12 plots the power penalty versu s /t T ∆ , where the SNR 0 is set to 10dB and the roll 
+factor of the raised cosine function is set to 0.5.  The worst-case SIR penalty is about -2.2 dB. If we  assume 
+the time synchronization error to be uniformly dist ributed over [- T/2, T/2] 4 (, we can calculate the average 
+SIR penalty as 
+0.5 0.5 
+00.5 0.5 ( ) ( ) 1.57 d SINR d SNR dB γ γ τ τ τ τ 
+− − ∆ = ∆ = − = − ∫ ∫            (17) 
+The simulation results in Section V shows that the power penalty due to non-perfect time synchronizati on 
+is less than 1 dB, which is even smaller than the S INR decrease in (17). In other words, our PNC schem e is 
+more sensitive to the phase offset than the symbol time offset. This fact reminds us that we could adj ust the 
+integration time of the match filter at the relay n ode from T to 'T('T T ≤). Then, the phase offset at the 
+relay is changed from T θ ω θ = ∆ + ∆  to a new value ' T θ ω θ = ∆ + ∆ . As a result, we may obtain a smaller 
+phase offset (according to the value of ,ω θ ∆ ∆ ) at the cost of more time synchronization errors. 
+                                                        
+4 This assumption is reasonable since one end node c an intentionally increase each symbol duration by T/N ( N is the number of symbols in one packet) while 
+the other end node keeps its own symbol duration T. As a result, all possible symbol misalignments ar e experienced at the relay node.  
+ 13  
+ 
+Figure 7.  Power penalty of time synchronization errors 
+Based on the discussion in this section, we can con clude that the performance degradation of 3 to 4dB 
+due to various synchronization errors (including la rge carrier phase, frequency, and time synchronizat ion 
+errors in the case where a synchronization mechanis m is not used at all) is acceptable given the small er 
+interference of PNC ([19, Eq. (6) and Eq. (9)]) and  given the more than 100% throughput improvement 
+obtained by PNC.  
+IV.  SYNCHRONIZATION IN N-NODE PNC  CHAIN  
+In the previous section, we argue that PNC detectio n is not very sensitive to synchronization errors i n the 
+case of 3N=. It may appear at first glance that the synchroniz ation problem of the N-node case may cause 
+PNC to break down for large N, due to the propagation of synchronization errors along the chain.  Here, 
+we argue that the detection scheme in PNC does not break down just because N is large. In particular, we 
+argue that if the synchronization errors can be bou nded in the 3-node case, they can also be bounded i n the 
+general N-node case.  
+Synchronization between multiple sources and one de stination has been extensively studied in previous 
+works [5-14]. We assume that the feasibility of syn chronization in a 3-node chain is a given based on these 
+prior results (i.e., the two end nodes are the sour ces and the relay is the destination in the set-up of the 
+3-node chain). Let us consider how the N-node case can make use of 3-node synchronization. A possible 
+approach is to partition the long chain into multip le 3-node local groups, as illustrated in Fig. 8, a nd then 
+synchronize them in a successive manner. Suppose th at the synchronization for 3-node can be achieved 
+with reasonable error bounds for phase, frequency, and time (see Section III, where we argue that PNC  
+ 14  
+detection is not very sensitive to synchronization errors), represented by, say, ,  2 ,  t θ ω ∆ ∆ . An issue is the 
+impact of these errors on the N-node chain. 
+For N-node synchronization, let us divide the time into two parts: the synchronization phase and the 
+data-transmission phase, as shown in Fig. 9. These two phases are repeated periodically, say once ever y PT 
+seconds. The synchronization phase lasts ST seconds and the data transmission phase lasts DT seconds, 
+with S D P T T T + = . The PNC data transmission described in Section II  comes into play only during the data 
+transmission phase. The synchronization overhead is  /S P T T , with ST depending on the synchronization 
+handshake overhead, and PT depending on the speed at which the synchronizatio ns drift as time 
+progresses. That is, the faster the drift, the smal ler the PT, because one will then need to perform 
+resynchronization more often. It turns out that the  N -node case increases the ST required, but not the 
+1/ PT required as compared to the 3-node case, as detail ed below. 
+For the N-node chain, let us further divide the synchronizat ion phase into two sub-phases. The first 
+sub-phase is responsible for synchronizing all the odd-numbered nodes and the second for all the 
+even-numbered nodes. We describe only sub-phase 1 h ere (phase 2 is similar).  With reference to Fig. 8 , 
+we divide the N nodes into ( 1) / 2 M N = −      basic groups (BGs) and denote them by BG j,  where j is 
+index of the BGs. Let BG t∆  be the time needed to synchronizing the two odd no des in one BG (using, say, 
+one of the prior methods proposed by others). Consi der BG1. Let us assume that it is always the case t hat 
+the right node (in this case, node 3) attempts to s ynchronize to the left node (in this case, node 1).  As an 
+example of the synchronization scheme, node 2 may e stimate the frequency difference and phase differen ce 
+of the signals from node 1 and node 3. Node 2 then forwards the differences to node 3 for it to adjust  its 
+own frequency and phase. After this synchronization , the phase, frequency and time errors between node s 1 
+and 3 are ,  2 ,  f t θ∆ ∆ . In the next BG t∆  time, we then synchronize node 5 to node 3 in BG2.  So, a total of 
+time of BG M t ∆  are needed in sub-phase 1. Including sub-phase 2, ( 2) S BG T N t = − ∆ . 
+It turns out that with a cleverer scheme, sub-phase  2 can be eliminated and ST can be reduced roughly 
+by half. But that is not the main point we are tryi ng to make here. The main issue is that with the ab ove 
+method, the bounds of the synchronization errors of  node N with respect to node 1 become 
+,  2 ,  M M M t θ ω ∆ ∆  and these errors grow in an uncontained manner as N increases! In particular, will  
+ 15  
+PNC therefore break down as N increases? 
+Recall that for PNC detection, a receiver receives signals simultaneously from only the two adjacent 
+nodes. By applying a channel coding scheme [21] at all the nodes, the relay node can recover 1 2 S S ⊕  
+from the received signal without any error. Only th e synchronization error penalty within one BG  can 
+affect the PNC performance and the penalty will not  propagate to other BGs. For example, say, N is odd. 
+The reception at node 2 depends only on the synchro nization between nodes 1 and 3; and the reception a t 
+node N-1 only depends on the synchronization of nodes N-2 and N. In particular, it is immaterial that there 
+is a large synchronization error between nodes 1 an d N. So, the fact that the end-to-end synchronization 
+errors have grown to ,  2 ,  M M M t θ ω ∆ ∆  is not important. Only the local synchronization e rrors, 
+,  2 ,  t θ ω ∆ ∆ , are important. The same reasoning also leads us t o conclude that how often synchronization 
+should be performed (i.e., 1/ PT) does not increase with N either, since it is only the drift within 3 nodes 
+that are important as far as PNC detection is conce rned.  
+Of course, ST grows with N, but only linearly. If BG t∆  is small compared with PT, this is not a major 
+concern. In practice, however, we may still want to  impose a limit on the chain size N not just to limit the 
+overhead ST, but also for other practical considerations, such  as routing complexities, network 
+management, etc.  
+ 
+ 
+Figure 8.  Synchronization for multiple nodes 
+ 
+ 
+Figure 9.  Partitioning of time into synchronization phase and  data-transmission phase.  
+ 16  
+ 
+V.  NUMERICAL SIMULATION  
+In this section, we evaluate the performance loss d ue to non-perfect synchronization in PNC by 
+numerical simulation. In our simulation, QPSK modul ation is used. SNR is defined as the transmission 
+power of each end node over the variance of the Gau ssian noise, i.e., 21/ σ.  
+We first compare the BER performance with different  synchronization levels as in Fig. 10. In this figu re, 
+the un-coded BER of 1 2 s s ⊕at the relay node is plotted under different SNRs. The decision rule under 
+perfect synchronization is the same as that in [4],  the decision rule for non-perfect phase synchroniz ation is 
+based on ML detection and the decision rule for non -perfect symbol-time synchronization is also the sa me 
+as that in [4]5. From this figure, we can see that there is only a bout 1dB loss when no symbol-time 
+synchronization is not performed, and the performan ce loss can be ignored when the symbol time offset is 
+randomly distributed in [-0.2 T, 0.2 T]. When there are phase synchronization errors, an SNR loss of more 
+than 4dB can be observed at a BER of 3E-3. It is mo re than the theoretical power penalty of 3.4 dB in (11). 
+The reason is that QPSK is a low order (only 4 cons tellation points) modulation and it can not efficie ntly 
+exploit all the signal information when the phase s ynchronization error is small without the applicati on of 
+channel coding.  
+                                                        
+5 This decision rule for non-perfect time synchroniz ation is not optimal can be further improved. In th at light, the obtained performance in Fig. 9 is onl y a 
+lower bound.   
+ 17  
+ 
+Figure 10.  BER performance of PNC with different synchronizati on levels 
+We then compare the mutual information performance with different synchronization levels as in Fig. 
+11. Mutual information is more closely related to t he channel coded performance than the uncoded BER. 
+For perfect synchronization, we plot 1 2 1( ; ) 2I s s r ⊕  at the relay node, where the coefficient 1/2 
+corresponds to the two dimensions in QPSK modulatio n. For the non-perfect phase synchronization, we 
+plot the mutual information when the phase offset i s randomly distributed in [ / 4, / 4] π π − (this range 
+corresponds to the case of no phase synchronization  as mentioned in section III.) as 
+/ 4 19 
+1 2 1 2 
+0 / 4 2 1 2 ( ; | ) ( ; | 0.05 / 4) 2 kI s s r d I s s r k π
+πθ θ θ π π π = −⊕ = ⊕ = ∑ ∫.           (18) 
+For the non-perfect time synchronization error, we simply plot 1 2 1( ; ) 2I s s r ⊕  with time offset randomly 
+distributed in the given range. From Fig. 11, we ca n see that the SNR loss of no time synchronization is 
+upper bounded by 0.5 dB. This result is much better than the theoretical average SINR loss in (17). The 
+SNR loss of no carrier phase and carrier frequency synchronization is no more than 2dB in the interest ed 
+SNR region of [0dB, 7dB]. It is also much better th an the theoretical upper bound in (11). The simulati on  
+ 18  
+results show that the PNC scheme is more robust to synchronization errors than the analysis as in Sect ion 
+III.  
+ 
+  
+Figure 11.  Mutual information performance of PNC with differen t synchronization levels 
+ 
+VI.  CONCLUSION  
+This paper investigates the synchronization issues i n Physical-layer Network Coding (PNC). We first 
+study the penalty of synchronization errors in PNC.  Both analysis and simulation shows that PNC is ver y 
+robust to the synchronization errors. It has also b een shown that the power penalty due to imperfect 
+synchronization can be compensated by the larger SI R in the PNC transmission system. After that, we 
+propose a new synchronization scheme in an N-node chain which performs PNC transmission. Last b ut not 
+least, we have shown that global synchronization in  PNC can be achieved without detrimental effects fr om 
+synchronization-error propagation.   
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+Transmissions”, IEEE Trans. Commun. , vol. 54, no. 4, pp. 726-736, Apr. 2006. 
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+realization” , in Proc. of Allerton Conference on o n Communication, Control, and Computing, 2006. 
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+ 
\ No newline at end of file
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+arXiv:1001.0070v3  [hep-ph]  14 Jan 2010CPC(HEP & NP), 2009, 33(X): 1—6 Chinese Physics C Vol. 33, No. X, Xxx, 2009
+News on PHOTOSMonte Carlo:
+γ∗→π+π−(γ)andK±→π+π−e±ν(γ)*
+Qingjun Xu1,1)Z. W¸ as1,2,2)
+1 (Institute of Nuclear Physics, PAN, Krak´ ow, ul. Radzikow skiego 152, Poland)
+2 (CERN PH-TH, CH-1211 Geneva 23, Switzerland )
+Abstract PHOTOSMonte Carlo is widely used for simulating QED effects in decay of intermediate particles
+and resonances. It can be easily connected to other main proc ess generators. In this paper we consider decaying
+processes γ∗→π+π−(γ) andK±→π+π−e±ν(γ) in the framework of Scalar QED. These two processes are
+interesting not only for the technical aspect of PHOTOSMonte Carlo, but also for precision measurement of
+αQED(MZ),g−2, as well as ππscattering lengths.
+Key words Monte Carlo Generator, QED Radiative Corrections, Ke4decay
+PACS 13.40.Ks; 13.66.Bc; 13.20.Eb
+IFJPAN-IV-2010-1
+1 Introduction
+In high energy experiments, one of the crucial
+works is to compare new experiment results with pre-
+dictionsfromthetheory. Iftheagreementisobtained,
+the theory is proved to be true. Otherwise one may
+think that the theory calculations turned out to be
+wrong or the effect of new physics appeared. Monte
+Carlo generators, rather than analytical calculations,
+are required to provide theoretical results of real ex-
+periment interest. The PHOTOSMonte Carlo [1, 2] is
+a universal Monte Carlo algorithm that is designed
+for simulating QED radiative corrections in cascade
+decays. It is widely used in low energy experiments
+and high energy experiments as well. The program
+is based on exact multiphoton phase space while the
+matrix element is approximately taken as process in-
+dependent multidimensional kernel. In some cases,
+the exact first order matrix element is employed in
+order to improve the precision.
+From the experience gained in KKMC project
+[3, 4] where spin amplitudes were used, one concludes
+that spin amplitudes are essential for design and tests
+of the program, in particular for choice of the single
+emission kernels. The analysis of the spin amplitudesand tests for the algorithm in case of Zdecay into
+pair of charged fermion was given in Ref. [5]. The
+case of the scalar particle decay into pair of fermions
+is covered in Ref. [6] and the decay of spinless parti-
+cle into pair of scalars is studied in Ref. [7]. The case
+ofWdecay was covered in Ref. [8, 9].
+In this paper we will study spin amplitudes for
+γ∗→π+π−(γ) decay. It not only provides example
+for studies of Lorentz and gauge group properties of
+spin amplitudes and cross sections, but also is impor-
+tant to improve theoretical uncertainty of PHOTOSfor
+this decay. Furthermore, it is of great experiment in-
+terest since its relevance to precision measurement of
+αQED(MZ) andg−2.
+Ke4decay could give the unique information on
+the value of s−andp−waveππscattering lengths.
+The high statistics measurements of Ke4decay has
+been performed by NA48/2 collaboration at CERN
+[10]. Theoretical predictions are well calculated [11]
+and the difference between theoretical prediction and
+experiment value is about 7%. In particular, QED
+corrections to this process are known to be non-
+negligible. They need to be taken into account with
+the help of Monte Carlo because their size depend
+on detector acceptance. In NA48 experiment, to take
+Received 30 Dec. 2009
+∗Supported by EU Marie Curie Research Training Network grant under the contract No. MRTN-CT-2006-0355505 and Polish
+Government grant N202 06434 (2008-2011)
+1)E-mail:qingjun.xu@ifj.edu.pl
+2)E-mail:wasm@mail.cern.ch
+c/circlecopyrt2009Chinese Physical Society and the Institute of High Ener gy Physics of the ChineseAcademy of Sciences and the Institu te
+of Modern Physics of the Chinese Academy of Sciences and IOP P ublishing LtdNo. X Q.Xu et al: News on PHOTOSMC:γ∗→π+π−(γ) andK±→π+π−e±ν(γ) 2
+intoaccountQEDeffects, PHOTOSMonteCarloisused
+together with Coulomb correction (see Ref. [10]). In
+this paper we consider QED radiative corrections to
+Ke4decayandimplementitinto PHOTOSMonteCarlo.
+The paper is organized as follows: process γ∗→
+π+π−(γ) is studied in Section 2, Section 3 will come
+toKe4decay, summary will be in Section 4.
+2γ∗→π+π−(γ)
+In order to match the parton shower and the
+hardbremsstrahlungmatrixelement into interference
+weight, the matrix element of γ∗→π+π−(γ) needs to
+be studied in great detail. Its gauge invariant parts
+need to be identified and relations to amplitudes of
+lower orders have to be found.
+2.1 Spin amplitude
+Consider the process e+e−→γ∗(p)→
+π+(q1)π−(q2)γ(k,ǫ), see Fig. 1.
+e−e+π+
+π−γ
+γ
+e−e+π+
+π−γ γ
+e−e+π+
+π−γ
+γ
+Fig. 1. Feynman diagrams of e+e−→γ∗→π+π−γ
+The amplitudes equivalent to those given in Ref. [12]
+are obtained. It can be written as
+M=VµHµ, (1)
+whereVµ=¯v(p1,λ1)γµu(p2,λ2). Thep1,λ1,p2,λ2are
+momenta and helicities of the incoming electron and
+positron. Vµdefines the spin state ofthe intermediate
+γ∗.
+Let us focus on the part for virtual photon decay.
+Following conventions of Ref. [12], the final interac-
+tion part of the Born matrix element for such process
+is
+Hµ
+0(p,q1,q2)=eF2π(p2)
+p2(q1−q2)µ.(2)
+Herep=q1+q2. If photon is present, this part of the
+amplitude reads:
+Hµ=e2F2π(p2)
+p2/braceleftbigg
+(q1+k−q2)µq1·ǫ∗
+q1·k+
+(q2+k−q1)µq2·ǫ∗
+q2·k−2ǫ∗µ/bracerightbigg
+.(3)It makes sense to rewrite Eq. (3) explicitly as sum of
+two gauge invariant terms:
+Hµ=Hµ
+I+Hµ
+II, (4)
+Hµ
+I=e2F2π(p2)
+p2/parenleftbigg
+(q1−q2)µ+kµq2·k−q1·k
+q2·k+q1·k/parenrightbigg
+/parenleftbiggq1·ǫ∗
+q1·k−q2·ǫ∗
+q2·k/parenrightbigg
+, (5)
+Hµ
+II=2e2F2π(p2)
+p2/parenleftbiggkµ(q1·ǫ∗+q2·ǫ∗)
+q2·k+q1·k−ǫ∗µ/parenrightbigg
+.(6)
+One can easily see that Eq.(5) has a typical form
+for amplitudes of QED exclusive exponentiation [13],
+that is Born-like -expression multiplied by an eikonal
+factor/parenleftBig
+q1·ǫ∗
+q1·k−q2·ǫ∗
+q2·k/parenrightBig
+. The expression in front of the
+factor indeed approaches the Born one in soft pho-
+ton and collinear photon limit. Thus, it is consistent
+with LL level factorization into Born amplitude and
+eikonal factor.
+If one takes separation (4) for the calculation of
+two parts of spin amplitudes, then after spin average,
+the expression for the cross section takes the form:
+/summationdisplay
+λ,ǫ|M|2=/summationdisplay
+λ,ǫ|MI|2+/summationdisplay
+λ,ǫ|MII|2+2/summationdisplay
+λ,ǫMIM∗
+II.(7)
+We should stress that Eq.(7) can have its first term
+even closer to Born-times-eikonal-factor form. For
+that purpose it is enough to adjust normalization
+of the first part of Eq.(7) to Born amplitude times
+eikonal factor, and replace |MI|2with
+|M′
+I|2=|MI|2|/vector q1−/vector q2|2
+Born
+|/vector q1−/vector q2+/vectorkq2·k−q1·k
+q2·k+q1·k|2. (8)
+Compensating adjustment to the remaining parts of
+Eq.(7) is then necessary. Since/summationtext
+λ,ǫ|M′
+I|2is the ex-
+pression used in PHOTOSMonte Carlo in Ref. [7], such
+a modification is of interest. In the next section, we
+willperformournumericalinvestigationswithrespect
+to Ref. [7] which is a reference for us.
+2.2 Numerical results
+We will show results at 2 GeV center of mass en-
+ergy. Comparison of result from/summationtext
+λ,ǫ|M′
+I|2with re-
+sultfrom PHOTOSwithmatrixelementtakenfromRef.
+[7] is shown in Fig.2. One can see that agreement is
+excellent all over the phase space. It is true only for
+the case when distributions are averagedover the ori-
+entation of the whole event with respect to incoming
+beams (or spin state of the virtual photon).
+If instead of/summationtext
+λ,ǫ|M′
+I|2one would use directly/summationtext
+λ,ǫ|MI|2, that is when normalization of Born-like
+factoris not performed, difference with respect to for-
+mulas in Ref. [7] is much larger, see Fig. 3.No. X Q.Xu et al: News on PHOTOSMC:γ∗→π+π−(γ) andK±→π+π−e±ν(γ) 3
+0 0.2 0.4 0.6 0.8 100.20.40.60.81
+110210310410510
+PSfrag replacements
+M2
+π+π−/S
+Fig. 2. Comparison of result using/summationtext
+λ,ǫ|M′
+I|2(green line) with that using
+matrix element taken from Ref. [7] (red line).
+Black line represents their ratio.
+0 0.2 0.4 0.6 0.8 100.511.522.53
+110210310410510
+PSfrag replacements
+M2
+π+π−/S
+Fig. 3. Comparison of result using/summationtext
+λ,ǫ|MI|2(green line) with that using
+matrix element taken from Ref. [7] (red line).
+Black line represents their ratio.
+Finally let us compare result of complete scalar
+QED matrix element with that of matrix element
+takenfromRef. [7], seeFig. 4. At highphoton energy
+0 0.2 0.4 0.6 0.8 100.20.40.60.81
+110210310410510
+PSfrag replacements
+M2
+π+π−/S
+Fig. 4. Comparison of result using complete
+matrix element (green line) with that using
+matrix element taken from Ref. [7] (red line).
+Black line represents their ratio.
+region, there is clear surplus of events with respect
+to formula in Ref. [7]. That contribution should notbe understood as bremsstrahlung, but rather as gen-
+uine process. Anyway in that region of phase space
+scalar QED is not expected to work well. Note that
+the difference between results of Figs. 2 and 4 is only
+0.2% of the total process rate. That is why our de-
+tailed discussion is not important for numerical con-
+clusions, but important for understanding the under-
+lying structure.
+From Fig. 2 one could conclude that the univer-
+sal kernel in Ref. [7], for arbitrary large samples,
+is equivalent to the matrix element/summationtext
+λ,ǫ|M′
+I|2. But
+differences appear in distributions sensitive to initial
+state spin orientation, see Figs. 5 and 6. On these
+plots angular distributions of the photon momentum
+andπ+momentum with respect to the beam line are
+shown, respectively. Regions of phase space giving
+near zero contribution at the Born level are becom-
+ing more populated if approximation for the photon
+radiation matrix element [7] is used.
+-1 -0.5 0 0.5 100.20.40.60.81
+110210310410510
+PSfrag replacements
+cosθγ
+Fig. 5. Comparison of result using/summationtext
+λ,ǫ|M′
+I|2(green line) with that using
+matrix element taken from Ref. [7] (red line).
+Black line represents their ratio.
+-1 -0.5 0 0.5 100.20.40.60.811.21.41.61.8
+110210310410510610
+PSfrag replacements
+cosθπ+
+Fig. 6. Comparison of result using/summationtext
+λ,ǫ|M′
+I|2(green line) with that using
+matrix element taken from Ref. [7] (red line).
+Black line represents their ratio.No. X Q.Xu et al: News on PHOTOSMC:γ∗→π+π−(γ) andK±→π+π−e±ν(γ) 4
+3K±→π+π−e±ν(γ)
+In this section radiative corrections to Ke4decay
+are calculated following approximations explained in
+Ref. [14]. We compare it with Coulomb corrections
+used by NA48/2 collaboration. The hard photon
+bremsstruhlung is calculated analytically with soft
+photon and collinear photon approximation. Numer-
+ical tests with PHOTOSMonte Carlo are performed.
+3.1 QED radiative corrections to Ke4decay
+Consider Ke4decay,
+K±(p)→π+(q+)+π−(q−)+e±(pe)+ν(pν).(9)
+In the framework of Scalar QED but neglecting dia-
+grams with photons emission from hadronic or weak
+blocks, one can calculate the virtual photon correc-
+tions. Note that the electron photon vertex is taken
+fromstandardQED.Contributionofvirtualdiagrams
+reads
+a.) b.)
+Fig. 7. example vertex diagrams [14]
+dΓvirt
+dΓBorn=α
+π/bracketleftbigg
+lnm
+λ/parenleftbigg
+4+L−
+β−−L+
+β+−2ρ−1+β2
+βLβ
++2lnpe·q+
+pe·q−/parenrightbigg
++π21+β2
+2β+ρ2+1
+2ρ
++2ρlnm
+2Ee+9
+4logΛ2
+m2+Kv/bracketrightbigg
+,(10)
+wheremis the charged pion mass, λis photon mass
+used as infrared regulator, Λ is ultraviolet (UV) cut
+off. In Eq.(10) we have defined
+ρ= ln2Ee
+me,
+β=/radicalbigg
+1−4m2
+sπ, L β=ln1+β
+1−β,
+β±=/radicalBigg
+1−m2
+E2
+±, L ±=ln1+β±
+1−β±.(11)
+Kvdepends on masses of particles and kinematics.
+Note that the UV-divergent part in Eq.(10) will can-
+cel if the renormalizedcoupling G2
+FV2
+usis used instead
+of bare ones,
+G2
+FV2
+us→/parenleftbigg
+1−9
+4logΛ2
+m2/parenrightbigg
+(G2
+FV2
+us)bare.(12)The soft photon contribution can be easily ob-
+tained, resulting in expression which is gauge invari-
+ant:
+dΓsoft
+dΓBorn=−α
+4π2/integraldisplayd3k
+ω/parenleftbiggp
+pk+q−
+q−k−q+
+q+k−pe
+pek/parenrightbigg2
+,(13)
+herekis photon momentum, ωis photon energy. It
+is straightforward to integrate out solid angle of pho-
+ton momentum kand over its energy ωup to a limit
+ω<∆ǫ; we obtain:
+dΓsoft
+dΓBorn=α
+π/bracketleftbigg
+ln/parenleftbigg2∆ǫ
+λ/parenrightbigg/parenleftbigg
+−4−L−
+β−+L+
+β+
++2ρ+1+β2
+βLβ−2ln2pe·q+
+2pe·q−/parenrightbigg
++ρ−ρ2+Ks]. (14)
+Function Ksis dependent on masses of particles and
+kinematics. Soft singularity is again regularized with
+the photon mass λ.
+The contribution of soft and virtual photons can
+be easily combined. It reads
+dΓBorn+virt+soft
+dΓBorn= 1+σPδ+πα(1+β2)
+2β
++α
+πKvs, (15)
+where
+Pδ= 2ln∆ǫ
+Ee+3
+2, σ=α
+2π(2ρ−1),(16)
+the expression of Kvsdepends not only on masses
+of particles and kinematics, but also on soft photon
+energy cutoff ∆ ǫ.
+Staring from a certain energy threshold photons
+can be observed, at least in principle. Such contri-
+bution is called hard photon radiation and we will
+assume that it is defined by condition ω >∆ǫ. If
+soft photon matrix element Eq.(14) is used, and max-
+imum energy of photon is taken from kinematical
+constraint Ee, then Eq. (14) can be used to calcu-
+late leading double logarithmic result of hard pho-
+ton radiation. For that reason one need to subtract
+Eq.(14) calculated for Eefrom that for ∆ ǫ. Thus the
+contribution of hard photon emission by soft-photon-
+approximation is obtained,
+dΓHardsoft−like
+dΓBorn=α
+πln/parenleftbigg∆ǫ
+Ee/parenrightbigg/parenleftbigg
+4+L−
+β−−L+
+β+−2ρ
+−1+β2
+βLβ+2ln2pe·q+
+2pe·q−/parenrightbigg
+.(17)
+Thisresulthasto be correctedforsinglelogarithmre-
+lated to collinear photon emission along the charged
+outgoing decay products.No. X Q.Xu et al: News on PHOTOSMC:γ∗→π+π−(γ) andK±→π+π−e±ν(γ) 5
+Photon emission from e±will give collinear singu-
+larity. Remaining charged products, π+andπ−are
+not relativistic. The electron part can be calculated
+withthehelpofcollinear-photon-approximation. The
+squaredcollinearamplitude canbe found in Ref. [15].
+The actual formula for collinear contribution is taken
+from Ref. [16],
+dΓCollinear
+dΓBorn=α
+4π2/integraldisplayd3k
+ωz2
+pe·k/parenleftbigg1+z2
+1−z−m2
+e
+pe·k/parenrightbigg
+,(18)
+herez=Ee
+Ee+ω. First we integrate over solid angle
+of photon direction, later we integrate photon energy
+from ∆ǫto the maximum value Ee. Then the result
+of hard photon emitting from e±is obtained using
+collinear-photon-approximation,
+dΓHarde
+dΓBorn=−σPδ+α
+2π/parenleftbigg
+3−2
+3π2/parenrightbigg
+.(19)
+Eq. (19) includes term proportional to 2 ρ−1 which is
+already present in Eq.(17). This would lead to dou-
+ble counting. That is why we remove this term from
+Eq.(17) and obtain
+dΓHardno−e
+dΓBorn=α
+πln/parenleftbigg∆ǫ
+Ee/parenrightbigg/parenleftbigg
+3+L−
+β−−L+
+β+
+−1+β2
+βLβ+2ln2pe·q+
+2pe·q−/parenrightbigg
+.(20)
+Finally hard (real) photon bremsstrahlung for pho-
+tons of energy above ∆ ǫreads
+dΓHard
+dΓBorn=dΓHarde
+dΓBorn+dΓHardno−e
+dΓBorn.(21)
+If one adds real and virtual photons contribution
+together, one can obtain the expression as following,
+dΓBorn+virt+real
+dΓBorn=1+πα(1+β2)
+2β+α
+πK.(22)
+Complicated, but numerically small function Kis de-
+pendent on masses of particles and kinematics of this
+process. Note that our final analytical result does not
+depend on the large logarithm ln2Ee
+me, or soft photon
+energy cut ∆ ǫ.
+As one can see, Eqs. (15) (21), (22) are obtained
+with the help of approximations. Effectively it was
+assumed that matrix element at Born level can be al-
+ways factorized out and photonic corrections can be
+calculated independently. Further corrections are as-
+sumed to be negligible and not affecting the substan-
+tial nature properties of hard interaction. This may
+be good as starting point, but cannot be left without
+future discussion/improvements∗.Our formulas are based on the same scheme of
+calculation as explained in Ref. [14] and in principle
+they should coincide numerically. Some differences in
+analytical results are still present. Also some numer-
+ical results still remain different. The exact expres-
+sions for Ks,Kv,KvsandK, as well as differences
+between our analytical results and these in Ref.[14]
+will not be listed here for the limit of paper length.
+They will be present elsewhere.
+3.2 Numerical tests
+Let’s start with Eq.(22), the correction for distri-
+bution when photon is integrated out but all other
+kinematic variables are kept. In Figs. 8 and 9
+we show that dominant part of Eq.(22) represents
+Coulomb correction. The difference is much smaller
+Fig. 8. Coulomb correction from Ref. [10].
+Fig. 9. Radiative correction in Eq. (22).
+∗We are grateful to Prof. J. Gasser for stressing this point.No. X Q.Xu et al: News on PHOTOSMC:γ∗→π+π−(γ) andK±→π+π−e±ν(γ) 6
+thantheeffectofCoulombcorrectionitself, seeFig.10
+whereresultsfor1000000Bornleveleventsareplaced
+in the histograms. We may conclude that our numer-
+ical implementation of Eq.(22) works well since its
+dominant part represents Coulomb correction.
+Fig. 10. Difference between radiative correc-
+tion in Eq. (22) and Coulomb correction from
+Ref.[10] calculated event by event.
+Wehavedonenumericaltestswith PHOTOSforsoft
+photons and found the distribution of soft photons
+fromPHOTOSis as same as given in the soft photon
+expression (Eq.(13)). We also have done the simi-
+lar test using hard photon expression (Eq. (21)) and
+found it matchesresultfrom PHOTOSsimulationin the
+soft photon region, too, as it is expected.
+For harder photons, we compare result of PHOTOS,
+with results of Eq.(14) for soft photon and Eq.(21)
+for hard photon. Their ratios are given in Fig. 11,
+where effect as function of upper limit on photon en-
+ergy was used. 100 000 000 PHOTOSevent samples
+were generated using single fixed Born level event.
+PHOTOS/HardPHOTOS/Soft
+E2/sqrtS0.1 0.01 0.001 0.00011.1
+1
+0.9
+0.8
+0.7
+0.6
+0.5
+Fig. 11. Ratios of result from PHOTOSto results
+using soft photon Eq. (14) and hard photon
+Eq. (21). Here E1< Eγ< E2where we set
+E2
+E1=2.5,E2varies from 0 .0002√
+Sto 0.07√
+S.
+Note that the energy of electron Ee≈0.07√
+S.We can conclude that agreement is good, as ex-
+pected. Though differences especially in harder pho-
+ton energy ranges can be seen. However even at the
+endofthespectrum, wheredistributionispoorlypop-
+ulated, differences are at 10 % level only.
+4 Summary
+We have presented the new tests of PHOTOSMonte
+Carlo,wheretheexactmatrixelementof γ∗→π+π−γ
+is implemented and its numerical result is compared
+with the kernel of PHOTOS. QED radiative correction
+to process K±→l±νπ+π−(γ) is also studied ana-
+lytically. Reasonable numerical agreement with sim-
+ulations including Coulomb correction and PHOTOS
+Monte Carlo was found. Since several assumptions
+are employed, further work is necessary. Our result
+is of practical interest for experiments. They con-
+firm that at least on technical level the Monte Carlo
+program works well as expected.
+Acknowledgments
+UsefulandinspiringdiscussionswithB.Blochand
+J. Gasser are acknowledged. We acknowledge also
+discussions with E. Kuraev.
+References
+1 E. Barberio, B. van Eijk, and Z. Was, Comput. Phys. Com-
+mun.66(1991) 115-128.
+2 E. Barberio and Z. Was, Comput. Phys. Commun. 79
+(1994) 291-308.
+3 S. Jadach, B. F. L. Ward, and Z. Was, Phys. Rev.
+D63(2001) 113009, hep-ph/0006359.
+4 S. Jadach, B. F. L. Ward, and Z. Was, Comput. Phys.
+Commun. 130(2000) 260-325, hep-ph/9912214.
+5 P. Golonka and Z. Was, Eur. Phys. J. C50(2007) 53-62,
+hep-ph/0604232.
+6 A.Andonov, S.Jadach, G. Nanava, and Z. Was, Acta Phys.
+Polon.B34(2003) 2665-2672, hep-ph/0212209.
+7 G. Nanava and Z. Was, Eur. Phys. J. C51(2007) 569-583,
+hep-ph/0607019.
+8 G. Nanava and Z. Was, Acta Phys. Polon. B34(2003)
+4561-4570, hep-ph/0303260.
+9 G. Nanava, Qingjun Xu and Z. Was, arXiv:0906.4052 [hep-
+ph]
+10 NA48/2, J. R. Batley et al.,Eur. Phys. J. C54(2008)
+411-423.
+11 G. Colangelo, J. Gasser and H. Leutwyler, Nucl. Phys.
+B603(2001) 125-179, hep-ph/0103088.
+12 H. Czyz, A. Grzelinska, J. H. Kuhn, and G. Rodrigo, Eur.
+Phys. J. C27(2003) 563-575, hep-ph/0212225.
+13 S. Jadach, B. F. L. Ward and Z. Was, Phys. Lett. B449
+(1999) 97-108, hep-ph/9905453.
+14 Yu. M. Bystritskiy, S. R. Gevorkyan, and E. A. Kuraev,
+Eur. Phys. J. C64(2009) 47-54, arXiv:0906.0516 [hep-ph].
+15 V. N. Baier, V. S. Fadin, and V. A. Khoze Nucl. Phys. B65
+(1973) 381-396.
+16 Qingjun Xu, ph.D. thesis, 2006, Technical University of
+Munich, Germany, arXiv:0912.5160 [hep-ph].
\ No newline at end of file
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+arXiv:1001.0071v1  [astro-ph.CO]  31 Dec 2009An all-sky Atlas of Radio/X-ray Associations
+E. FleschA,B
+AP.O. Box 12520, Wellington, New Zealand
+BEmail: eric@flesch.org
+Abstract: An all-sky comprehensive catalogue of calculate d radio and X-ray associations
+to optical objects is presented. Included are X-ray sources fromXMM-Newton ,Chandra and
+ROSAT catalogues, radio sources from NVSS, FIRST and SUMSS catalo gues, and optical
+data, identifications and redshifts from the APM, USNO-A, SD SS-DR7 and the extant
+literature. This ‘Atlas of Radio/X-ray Associations’ inhe rits many techniques from the
+predecessor Quasars.org (2004) catalogue, but object sele ction is changed and processing
+tweaked. Optical objects presented are those which are calc ulated with ≥40% confidence
+to be associated with radio/X-ray detections, totalling 60 2 570 objects in all, including
+23 681 double radio lobe detections. For each of these optica l objects I display the
+calculated percentage probabilities of its being a QSO, gal axy, star, or erroneous radio/X-
+ray association, plus any identification from the literatur e. The catalogue includes 105568
+uninvestigatedobjects listed as 40% to >99% likelyto be a QSO. The catalogue is available
+at http://quasars.org/arxa.htm .
+Keywords: atlases — catalogs — x-rays: general — quasars: general — x-r ays: stars — radio
+continuum: stars
+1 Introduction
+In 2004 we published the all-sky Quasars.org cata-
+logue (QORG: Flesch & Hardcastle 2004) as a com-
+pendium of all radio/X-ray associations and quasars,
+mapped onto an optical background using a uniform
+matching algorithm. This was meant as a comprehen-
+sive resource for investigators, but included X-ray data
+only from ROSAT catalogues. Since then, large X-
+ray source catalogues have been released from XMM-
+NewtonandChandra satellite operations, and large ra-
+dio catalogues have been completed or amended. Also,
+large new optical identification surveys havebeen com-
+pleted, most notablytheSloan Digital SkyLegacy Sur-
+vey (SDSS: Abazajian et al. 2009), and many smaller
+surveys. These developments have left the QORG cat-
+alogue slightlydated; alsoIhaverefinedthepercentage
+calculations and improved the duplications handling.
+I have accordingly included all new data to mid-
+2009 and updated the matching algorithm. Also two
+significant changes in presentation are in order: (1)
+QORG endeavoured to present all quasars as at 2003;
+however, I do not wish to echo the large recent releases
+of new quasars from surveys like SDSS. Therefore I
+constrain the object selection to only those calculated
+as having radio/X-ray associations. (2) QORG pre-
+sented only objects successfully mapped onto an op-
+tical background derived from Cambridge Automatic
+Plate Measuring machine (APM: McMahon & Irwin
+1992) and United States Naval Observatory (USNO-
+A: Monet 1998) data. This seems too restricting now,
+given that SDSS and other surveys publish entirely
+suitable optical data. Therefore I augment this back-
+ground with all published objects and their accompa-
+nying optical data, taking care to avoid duplications.This new presentation constitutes a change in scope
+from QORG.
+I thus present a new comprehensive all-sky cata-
+logue, the ‘Atlas of Radio/X-ray Associations’ (here-
+after: ARXA), which uses all good-resolution radio
+and X-ray source catalogs to mid-2009, and causally
+maps those sources onto all available optical data us-
+ing techniques from QORG, with some enhancements.
+Optical objects presented are those calculated with
+≥40% confidence to be associated with radio/X-ray
+detections; the 40% threshold is chosen as a lean to-
+wards completeness. These optical objects total 602
+570 in all, including 190 393 objects bearing identifi-
+cations from the literature, 294 713 non-identified ob-
+jects previously reported in the QORG catalogue (but
+here recalculated), and 117 320 objects reported here
+for the first time. Using the individual confidences
+of association as expectation values, and taking ob-
+jects identified from the literature as correctly associ-
+ated, this catalogue should yield 500 286 correct asso-
+ciations of radio/X-ray detections to optical objects.
+The individual confidence of associations are divided
+into percentage probabilities of the optical object be-
+ing a QSO, galaxy or star. Accordingly, this catalogue
+displays 105 568 uninvestigated objects listed as 40%
+to>99% likely to be a QSO; 72 522 of these were
+first presented in QORG, and 33 046 are newly pre-
+sented here in ARXA. This catalogue is intended as a
+bulk resource for investigators, and is obtainable from
+http://quasars.org/arxa.htm .
+In this paper I give an overview of ARXA input
+data, focusing on data newly added since QORG, and
+I describe refinements in the processing, and present
+the resultant ARXA catalogue. For input data car-
+ried over from QORG, readers are asked to consult
+1the QORG paper, which also gives full details of the
+matching techniques in its appendix A.
+1.1 X-ray data
+The input X-ray data is comprised of XMM-Newton
+(XMM) and Chandra catalogues published to 2009,
+plus the four ROSAT (ROentgen SATellite) catalogues
+includedanddocumentedin QORG,whichare theAll-
+Sky Survey (RASS: Voges et al. 2000), High Resolu-
+tion Imager (HRI: Voges et al. 1999), Position Sensi-
+tive Proportional Counter (PSPC: Voges et al. 1999),
+and the WGACAT (WGA: White Giommi Angelini
+1994) catalogues. As XMM and Chandra data were
+not featured in QORG, I document them here.
+Three XMM input files are utilized, these are the
+IncrementalSecondXMM-NewtonSerendipitousSource
+Catalogue (2XMMi: Watson et al. 2009), the XMM-
+Newton Slew Survey catalogue release 1.3 (XMMSL:
+Saxton et al. 2008), and the XAssist XMM master
+sourcelist v2009 (XAssist: Ptak & Griffiths 2001). I
+process each of these separately, keeping only those
+with fluxes greater than their uncertainty, matching
+to optical within the XMM positional uncertainty, and
+then combining them into one file; where more than
+one input files match to the same optical object, the
+highest-confidence match is kept. ARXA identifies
+thesethreesourcecatalogues withtheirprefixes2XMM,
+XMMSL or XMMX, respectively. The merged XMM
+data contribute 57 778 X-ray-optical associations to
+ARXA.
+Similarly, three Chandra input files are utilized,
+these are the Chandra Source Catalog v1.01 (CSC:
+http://cxc.harvard.edu/csc), the Chandra Multiwave-
+length Project X-ray Point Source Catalog (ChaMP2:
+Kim et al. 2007), and the XAssist Chandra mas-
+ter sourcelist v2009 (XAssist: Ptak & Griffiths 2001).
+Processing is as with the XMM data. ARXA identifies
+these three source catalogues with their prefixes CXO,
+CXOMP or CXOX, respectively. The merged Chan-
+dra data contribute 32 951 X-ray-optical associations
+to ARXA.
+The numbers sourced from these six input X-ray
+catalogues are shown in Table 1. The 3rd column
+counts associations found per input catalogue, and the
+4th column counts which are used, as ARXA displays
+just one XMM and/or Chandra identification per as-
+sociation. The 5th column counts those associations
+available from only one of the three XMM input cata-
+logues, and similarly for Chandra.
+These input catalogues provide astrometry both
+raw and adjusted via an optical solution. For consis-
+tency, I elect to apply our own optical solutions onto
+the raw source positions, using the probability-based
+technique documented in QORG. 40% of the XMM
+sources are thus shifted 1 to 2 arcsec, with a few of
+3 or 4 arcsec. The Chandra astrometry fits strongly,
+and I find shifts of 1 arcsec for only 4% of its sources,
+with a few outliers of 2 or 3 arcsec. This represents
+an order of improvement over the ROSAT astrometry
+treated in the QORG paper.
+X-ray sources are usually core emissions, but in
+practice offsets result from extended emission and theTable 1: Associationscontributed bythe six XMM
+and Chandra source files.
+# unique Number Number Number
+Source and useful assoc’s assoc’s unique
+catalogue sources found in ARXA assoc’s
+2XMMi 249916 54980 51781 44845
+XMM Slew 7357 1881 1689 1662
+XMMX 63404 11155 4308 1124
+All XMM 57778
+CSC / CXO 93685 25917 21271 13688
+ChaMP2 6512 1936 949 272
+CXOX 94103 18769 10731 6305
+All Chandra 32951
+Table 2: XMM and Chandra sourced associations
+in ARXA, by astrometric offset to optical.
+Astro- Mean Mean
+metric No. of confidence No. of confidence
+offset XMM of XMM Chandra of Chandra
+(arcsec) assoc’s assoc’s (%) assoc’s assoc’s (%)
+0 8088 97.5 7627 98.3
+1 24752 91.9 16478 92.1
+2 16103 85.0 6451 84.8
+3 8479 76.3 1777 75.4
+4 302 75.2 608 69.0
+5 11 74.1 4 56.5
+6 11 73.5 5 79.4
+7 15 75.0 1 60.0
+8 10 59.9
+9 4 74.5
+10 4 78.5
+11 1 60.0
+total 57778 88.4 32951 90.8
+astrometric imprecisioninherentinreducingsmallcounts .
+The input sources are listed with positional uncer-
+tainties, and we accept optical selections only within
+those uncertainties. However, a few farther-offset as-
+sociations are accepted in the course of performing de-
+duplications. Table 2 shows the counts, by astrometric
+offset of X-ray source from optical.
+Inspection of the large-offset outliers of Table 2
+show these are mostly large galaxies or bright stars,
+which often feature multiple signatures. One example
+is CXO J093552.7+612112 which is shown as associ-
+ated to UGC 5101 at an offset of 6 arcsec. Also as-
+sociated to UGC 5101 are XMMX J093551.7+612112,
+2RXPJ093551.8+612110 andFIRSTJ093551.6+612111,
+all at offsets of 0–2 arcsec. Investigation shows there is
+anotherChandrasourceCXOJ093551.6+612111 which
+aligns excellentlywiththeothers, butCXOJ093552.7+6121 12
+was also retained by the processing because it is pre-
+cisely co-positioned onto a component of UGC 5101
+which was deconvolved in our background USNO-A
+data; subsequent de-duplication retained just the one
+Chandra association with the highest confidence of as-
+sociation. Thus these outliers stem from such artefacts
+typical of large data, but their inclusion is valid.
+1.2 Radio data
+The radio data is comprised of the large NRAO VLA
+Sky Survey catalog (NVSS: Condon et al. 1998), the
+2Table 3: Radio/X-ray Associations presented in
+the ARXA.
+# distinct # core # double
+Source astrometric assoc’s lobes
+catalogue sources in ARXA in ARXA
+FIRST 792777 173383 12844
+NVSS 1810664 266148 8308
+SUMSS 259896 59138 2529
+All radio 417075 23681
+XMM 313015 57778
+Chandra 183494 32951
+HRI 56398 15523
+RASS 124730 47486
+PSPC 102005 35607
+WGA 88578 24226
+All X-ray 174337
+re-release (2008) of the Faint Images of the Radio Sky
+at Twenty-cm survey catalog (FIRST: White et al.
+1997), and the completed Sydney University Molonglo
+Sky Survey (SUMSS: Murphy et al. 2007). SUMSS
+is taken to include the Molonglo Galactic Plane Sur-
+vey (MGPS-2: same attribution as SUMSS), and of
+the 61 659 SUMSS sources displayed in ARXA, 3954
+carry the MGPS identifier. Core associations are pre-
+sented from all three source catalogues, which in prac-
+tice come to at most two core associations per object
+because FIRST and SUMSS do not overlap. Double
+lobe associations are also presented, but just one set
+per optical object, for brevity.
+Processing is as detailed in the QORG paper. The
+2008 release of the FIRST catalogue has additions and
+tweaks compared with its 2003 version, and now dis-
+plays a sidelobe probability figure. Frequency analysis
+and FIRST cutout inspection led me to adopt a cutoff
+of 66.7% sidelobe probability – sources more likely to
+be sidelobes were dropped, and retained sources with
+high sidelobe probability are less likely to match to
+optical objects anyway.
+Total counts of these radio and X-ray input cata-
+logues and the ARXA associations derived from them
+are in Table 3.
+1.3 Identifications data
+ARXA is intended as a comprehensive resource for
+investigators, so it behoves us to identify its optical
+objects from the literature where possible. All input
+identification catalogues cited in the QORG paper are
+included, some having since had large updates (most
+notably the SDSS data releases 2 through 7), and also
+newly published spectroscopic and photometric cata-
+logues, and numerous small surveys. ARXA displays
+citations for the name and redshift of each optical ob-
+ject, and a full listing of those citations and respective
+catalogues isintheReadMe(http://quasars.org/arxa/ARX A-ReadMe.txt).
+The primary catalogue used for identification of
+QSOs, AGN and BL Lacs is the Catalogue of Quasars
+and Active Nuclei, 12th edition (Veron: V´ eron-Cetty
+& V´ eron 2006). The Veron catalogue uses an absolute-
+magnitude threshold to differentiate a QSO classifi-
+cation from an AGN classification, and ARXA ad-
+heres to that. QSO identifications are also taken fromTable4: IdentifiedobjectspresentedintheARXA,
+by source catalogue.
+object all radio X-ray
+catalogue type IDs associated associated
+PGC galaxy 54670 49627 7356
+SDSS all 50517 41290 10537
+MegaZ-LRG galaxy 31658 30459 1345
+NBCKDE QSO 24717 12475 12709
+TYCHO star 8646 489 8192
+Veron QSO 8482 4873 5432
+6dFGS galaxy 3003 2330 842
+NPM star 1767 129 1648
+2dFGS galaxy 1431 1178 277
+2QZ QSO 1175 502 695
+all others all 4327 1071 3469
+the NASA/IPAC Extragalactic Database (NED), al-
+though deduplication must be scrupulously done be-
+cause of astrometric mismatches between the Veron
+and NED data. These two catalogues provide histori-
+cal names. Also included are the many large and small
+QSO releases to 2009, notably SDSS-DR7, the 2dF-
+SDSS LRG and QSO catalogue (2SLAQ: Croom et al.
+2009), andthe2dFQSORedshiftSurvey(2QZ:Croom
+etal. 2004). Thetotal countsfoundtobeassociated to
+radio/X-ray sources are 21 735 catalogued QSOs, 6941
+AGN and 803 BL Lacs. 15 241 of these objects stem
+from the SDSS survey. There are also 24 717 SDSS-
+based photometric quasars (NBCKDE: Richards et al.
+2009).
+Our primary source for galaxy identification is the
+PrincipalGalaxyCatalogue (PGC:Patureletal. 2003),
+which has 983 213 objects and historical names. The
+SDSS-DR7 makes spectroscopic identification of 871
+054 galaxies, the 2dF Galaxy Redshift Survey(2dFGS:
+Colless et al. 2001) has 236 078 galaxies, and the 6dF
+Galaxy Survey final release (6dFGS: Jones et al. 2009)
+has 124 575 galaxies. Other galaxy catalogues are doc-
+umented in the QORG paper and ARXA ReadMe.
+The total count presented in ARXA as associated to
+radio/X-ray detections, is 90 006 catalogued galax-
+ies and 31 658 SDSS-based photometric red galaxies
+(MegaZ-LRG: Abdalla et al. 2008). Note that some
+nearby galaxies known to be radio/X-ray emitters are
+missing from ARXAbecause our astrometric matching
+is scaled to objects not so large on the sky.
+The remaining identification is that of star. Many
+large surveys have released star identifications, and I
+use the Tycho catalogue (Hog et al. 2000) as a bright
+star identifier although its objects are formally just
+point sources; the Lick North Proper Motion catalog
+(NPM: Klemola et al. 1987) is also well represented.
+See the QORG paper for discussion of stellar identi-
+fication and listing of many of the input catalogues
+used. The total count presented in ARXA as asso-
+ciated to radio/X-ray detections, is 13 844 stars, for
+which we have about 8x more X-ray associations than
+radio. Only for fainter ( >17 mag) stars are radio as-
+sociations as common as X-ray. Table 4 gives counts
+provided by the largest input identification catalogues.
+This leaves 412 177 radio/X-ray associated optical
+objects without identification from the literature and
+calculated as being ≥40% likely to be associated to
+3Table 5: Non-identified objects presented in the
+ARXA, all and new.
+all all new new
+probability conf of QSO conf of QSO
+percentage assoc’n prob assoc’n prob
+40-49 57480 18662 32110 6735
+50-59 53716 15771 17461 4199
+60-69 46682 13682 10155 3286
+70-79 51748 16839 9968 3090
+80-89 65884 19725 13239 6218
+90-99+ 136523 20889 34387 9518
+total 412033 105568 117320 33046
+radio/X-ray detections. Of these, 294 713 associations
+were first presented in QORG, although recalculated
+here, and 117 320 are newly presented here in ARXA.
+Of all these uninvestigated objects, 105 568 are listed
+as≥40% likely to be a QSO, being 72 522 first pre-
+sented in QORG and 33 046 newly presented here. Ta-
+ble 5 gives counts of these uninvestigated objects, in
+total and also just for newly ARXA presented, binned
+by confidence of association and QSO probability.
+2 Processing issues and mis-
+cellaneous notes
+Our algorithm which calculates probability of associ-
+ation of radio/X-ray sources to optical objects is ex-
+plained in full in the QORG paper and its appendix
+A. In short, it consists of binning the optical objects
+by colour, psf, and astrometric offset to the radio/X-
+ray source, and then comparing the areal density of
+each bin compared with its average over all sky of the
+same overall object density. Twice the average areal
+density means that we expect that half of such objects
+are causally associated, and so on. Double lobe associ-
+ations are calculated heuristically. The QORG paper
+details processing issues; additional issues handled in
+the ARXA processing are described here.
+Large input catalogues derived from surveys typi-
+cally have some rows of lesser quality, whether flagged
+as such or missing some photometric information, or
+having fluxes less than their errors, etc. I exclude
+these to keep false positives out of ARXA. The two
+input photometric catalogues, NBCKDE and MegaZ-
+LRG, assign probabilities to their objects that they
+are indeed QSOs or galaxies. Such probability should
+increase if an association to a radio/X-ray source is
+found. Still, I wish to avoid displaying an erroneous
+identification as a QSO or galaxy, so I apply a cutoff
+threshold. Numerical analysis shows that a probabil-
+ity cutoff of 60% is suitable to accept NBCKDE QSO
+identifications, and 80% for MegaZ-LRG galaxies.
+The Veron and NED QSO catalogues include data
+from older surveys with imprecise astrometry which
+have no obvious match to the optical data. I have
+used a heuristic algorithm to determine the best op-
+tical objects for these. Comparison to finding charts
+(where available) shows accuracy of about 90%.
+The QORG paper, section A.6.1, describes a cor-
+rection applied to those associations with large astro-Table6: SDSS DR2–DR7identificationscompared
+against 36118 QORG associations
+QORG-listed # hits; ie, # misses, ie, hit
+confidence confirmed SDSS says pct
+pct (binned) by SDSS diff optical
+40-42 144 36 80.0
+45 265 84 75.9
+50 335 106 76.0
+55 382 100 79.3
+60 463 98 82.5
+65 502 94 84.2
+70 712 112 86.4
+75 749 118 86.4
+80 1041 144 87.8
+85 1561 130 92.3
+90 2476 176 93.4
+95 4713 215 95.6
+98-100 20929 433 98.0
+total 34272 1846 94.9
+metric offsets ( >6 arcsec) of the radio/X-ray source
+to optical. We applied a linear-with-offset reduction in
+the probability of association, to deter false positives
+at large offsets. It was an arbitrary solution to known
+issues at large offsets, such as increased likelihood of
+multiple optical candidates and true sources being too
+faint for our optical data. We deliberately made the
+correction large, because, as we stated in the paper,
+‘we feel it is more excusable to under-represent true
+far-offset associations than it is to over-represent false
+ones’. Now with passage of 5 years, the subsequent
+publication of SDSSDataReleases 2through 7 enables
+comparison of QORG optical selections for radio/X-
+ray sources with the new SDSS identifications. This
+testispresentedonlineathttp://quasars.org/docs/Test ing-QORG-via-SDSS-DR7.txt,
+and its first table is reproduced here as Table 6.
+This table shows that (1) optical identification ac-
+curacyexceedednominal, and(2)accuracywasatleast
+75%, even for low nominal confidence. These low con-
+fidencebinsare generally those oflarge astrometric off-
+set, and shows that our arbitrary correction (for large-
+offset associations) was too great. Analysis now has
+led me to apply a radio/X-ray survey specific solution,
+with on average about halfthe QORGcorrection. This
+solution is designed to yield performance close to, but
+still exceeding, nominalconfidencepercentagesatlarge
+astrometric offsets. The effect is to include about 20
+000 low-to-medium confidence associations which had
+been excluded from QORG. Numbers of associations
+fromROSAT RASS are most increased by this, up
+about 50%, because far-offset matching is needed to
+accomodate the poor astrometric precision of RASS.
+L´ opez-Corredoiraetal. (LGMGA:L´ opez-Corredoira
+et al. 2008) found that QORG QSO-probabilities un-
+derperformed for brightB <17objects. This iscorrect,
+as optical brightness was not one of our classifiers, and
+the optically brightest outliers of our identifications
+are naturally less likely to be QSOs than the main-
+stream. However, any newly discovered bright QSO
+would be of particular interest, so I have not classified
+by optical brightness in ARXA either; the numbers
+involved are small anyway. LGMGA also identified
+objects near ( <60 arcsec) to bright galaxies as un-
+derperformed in QORG, but we find that the problem
+4is caused by HII/starburst zones and deblending arte-
+facts in the disks of bright galaxies which appear as
+bluish sources in our backgroundoptical APM/USNO-
+A data. Thus the problem exists only as an artefact
+within the visible disks of bright galaxies. These are
+difficult to identify from the data, and can be interest-
+ing anyway (e.g., the possibly background QSO on the
+disk of NGC 7314, J223546.2-260429), so I retain them
+in ARXA; however, I have included any identifications
+from LGMGA.
+Anotherartefact is asmall numberofnominal dou-
+ble lobe declarations in the Galactic dust lane. These
+are unlikely to be true, but there is a residual chance
+that some interesting new or hidden background ob-
+ject could be pinpointed by this; thus I leave these in
+ARXA.
+Processing to identify radio association has been
+done in isolation from the X-ray processing. Thus op-
+tically near neighbours can be identified as a radio
+source on the one hand and an X-ray source on the
+other. De-duplication will combine these only if the
+confidence of association is low for one of them.
+De-duplication of optical objects was done as the
+last step, and the method is different to that used
+for QORG. One type of duplication is when two op-
+tical data points represent the same true optical ob-
+ject. Another type is when two optical objects have
+radio/X-ray associations belonging to just the one true
+optical source. At large astrometric separations, two
+associations can be combined only if one is at very
+low confidence. The heuristic rules were worked out
+with muchtesting against Digitized SkySurveyimages
+and FIRST cutouts. In total, 17,168 optical duplicates
+were removed.
+3 The Atlas of Radio/X-ray
+Associations
+Thecatalogue isavailableathttp://quasars.org/arxa.ht m,
+and is written as one line per optical object. The
+catalogue presents unique ‘best’ associations, so opti-
+cal objects and radio/X-ray sources appear once only.
+The presentation is simplified from thatof QORG:just
+the radio/X-ray association name is displayed, and the
+user can refer to the original radio/X-ray catalogue for
+information on that source. The catalogue is 20Mb
+zipped, and Table 7 shows the first few lines.
+Table 7 shows the data structure of the ARXA
+catalogue, although the right end is truncated to save
+space. The ReadMe file is provided on site which gives
+full file layouts, field definitions and supporting infor-
+mation, plus citations for all input catalogues; I give
+only an overview here. Column 1 displays the opti-
+cal coordinates (epoch J2000) which doubles as the
+IAU-recommended name of the object, e.g., ARXA
+J040904.9-364744. Column 2 gives the name of the
+object where it is identified from the literature. This
+is left blank if previously identified only in QORG (col-
+umn 10 = ‘QO’), in which case the QORG name is e.g.
+QORG J000001.0+012416, using the astrometry from
+column 1. Column 3 summarizes any identification
+of, and associations with, the optical object: R=radiosource, X=X-ray source, 2=double lobe declaration,
+Q=knownquasar, A=AGN,G=galaxy, S=star, B=BL
+Lacobject, q=photometricquasar, g=photometricgalaxy.
+Columns 4 and 5 give the red and blue magnitudes re-
+spectively, and column 6 flags if those magnitudes are
+POSS-I (=’p’) or other photometry, plus other com-
+ments. Columns 7 and 8 give the point spread func-
+tion (psf) classification of the two colours: ’-’=stellar,
+’1’=fuzzy, ’2’=extended, ’n’=nopsfand’x’=objectnot
+seen in this colour. Column 9 gives the redshift, if
+known. Column 10 gives the citation for the name in
+column 2 and/or object type in column 3. The legend
+for these 2-byte citations is given in the ReadMe. Col-
+umn 11 gives the citation for the redshift. Columns
+12-15 give the calculated probability (independent of
+any identification from the literature) that the object
+is turn a QSO, galaxy, star, or erroneous radio/X-ray
+association. Columns 16 and 17 here show radio/X-
+ray association names, but the actual ARXA struc-
+ture is that column 16 is exclusively for NVSS asso-
+ciations, column 17 for FIRST/SUMSS associations
+(which don’t overlap), and so on for XMM, RASS,
+PSPC,WGA,HRIandChandraassociations, andthen
+the two signatures of a double radio lobe declaration.
+4 Summary
+This paper presents the Atlas of Radio/X-ray Associ-
+ations (ARXA), which is intended as grand compila-
+tion of the large good-resolution surveys of the radio
+and X-ray sky overlaid and aligned onto the optical
+sky. It uses the completed ROSAT, NVSS, FIRST
+and SUMSS catalogues and XMM-Newton andChan-
+dradata to 2009. It provides calculated optical associ-
+ations for these together with comprehensive identifi-
+cations of known objects with the intention of present-
+ing an informative map to help formulate and support
+pointed investigations. Its counts are 602 570 associ-
+ated optical objects in total, including 105 568 quasar
+candidates.
+Acknowledgements
+Sincerest thanks to Martin Hardcastle for advice and
+encouragement.
+This research has made use of data obtained from
+theChandraSourceCatalog (ADS/Sa.CXO#CSC2009
+v1.01), provided by the Chandra X-ray Center (CXC)
+as part of the Chandra Data Archive. The National
+Radio Astronomy Observatory is a facility of the Na-
+tional Science Foundation operated under cooperative
+agreementbyAssociatedUniversities, Inc. TheNASA/IPAC
+Extragalactic Database(NED:nedwww.ipac.caltech.edu)
+is operated by the Jet Propulsion Laboratory, Cal-
+ifornia Institute of Technology, under contract with
+the National Aeronautics and Space Administration.
+The Digitized Sky Surveys were produced at the Space
+Telescope ScienceInstituteunderU.S.Governmentgrant
+NAG W-2166.
+5Table 7: Sample lines from the Atlas of Radio/X-ray Associations
+optical J2000 / identification obj optical mag opt PSF sourc e percentages 1st 2nd
+ARXA ID from literature type red blue comR B z ID z qso galstr er rassociation association (etc)
+000000.6+321230 PGC 1985872 GRX12.2 16.0 p 2 2 PG 0 90 2 8NVSS J 000000.1+321233 1RXS J000001.3+321247
+000000.7-083628 R 20.3 20.7 1 - 33 5 2 60FIRSTJ000000.1-083621
+000000.9-200447 RX 18.7 20.3 pv 1 1 QO 4 63 0 33NVSS J000000.2-200448 1RXS J0000 00.9-200444
+000001.0+012416 R 20.3 0 1 x QO 1 98 1 0FIRSTJ000001.0+012415
+000001.3-020200 FIRST J00000-0202 QR 19.7 21.0 p - - 1.356 VE VE 85 7 6 2FIRSTJ000001.2-020200
+000001.3-063114 R 18.6 21.1 1 1 QO 0 95 0 5NVSS J000001.4-063113
+000001.4+111938 X 19.7 0 p - x 19 15 9 57 1RXS J000001.9+111948
+000001.5-092940 SDSS J000001.5-092940 GR 17.3 19.8 p 1 1 0.1 91 SD SD 1 97 1 1NVSS J000001.4-092940 FIRSTJ000001.5-09294 0
+000001.6-251707 X 0 22.0 x - 93 4 0 3 2XMM J000001.6-251706
+000001.6-420429 R 19.4 19.9 - - 89 1 1 9SUMSSJ000001.7-420432
+000001.8-094652 NBCK J000001.88-094652.0 qX 18.7 19.1 p - - 1.000 NB NB 93 1 1 5 CXO J000002.0-094649
+000002.0-152435 R 16.7 19.1 p 1 1 QO 1 92 0 7NVSS J000001.9-152435
+000002.1+155254 CGCG 456-13 GR 11.1 11.4 p 1 1 0.020 PG SD 0 68 1 31NVSS J000001.6+155254
+000002.1-093136 SDSS J000002.1-093136 GR 19.2 22.9 1 2 0.46 6 SD SD 0100 0 0FIRSTJ000002.1-093136
+000002.3-473423 R 20.4 22.5 - - 12 31 1 56SUMSSJ000002.1-473429
+000002.4+051717 PGC 1281052 GR 15.3 18.9 p 2 2 PG 0 99 0 1NVSS J0 00002.3+051717
+000002.4-042804 IRASF 23574-0444 GR 14.2 16.1 p 1 1 0.102 PG 6 d 0 89 0 11NVSS J000002.3-042803
+000002.5+395733 R 16.6 19.9 p 2 2 QO 0 79 0 21NVSS J000003.3+395733
+000002.6-321531 X 20.7 21.1 p - - QO 98 1 1 02XMM J000002.6-321530 2RXP J000002. 8-321527
+000002.7-251137 XMM J00000-2511 QX 0 21.8 x - 1.314 VE VE 93 4 0 3 2XMM J000002.7-251136
+000002.8-233911 IRAS 23574-2356 GR 16.9 18.2 p 1 1 PG 8 79 1 12N VSS J000002.7-233910
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\ No newline at end of file
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+  1 
+ Polya by Examples 
+ 
+KUNG-WEI YANG 
+E-mail: kung-wei.yang@wmich.edu 
+ 
+Polya’s Enumeration Theorem is one of the most usef ul tools dealing with the 
+enumeration of patterns that are symmetric in some ways.  What follows is a procedure for 
+obtaining the results  of Polya's Theorem directly, bypassing the usual p reliminaries ("cycle 
+index,"...).  It is for people who want to use  (and to understand) Polya's Theorem.  There is 
+no proof here.  There is no fancy example here. 
+ “Always begin with the simplest examples.” (Hilber t) 
+We will count the number of distinct color patterns  of the unit disc in the plane 
+partitioned into four quadrants (1, 2, 3, 4) by the  two coordinate axes.  Each quadrant is to 
+be colored in one of three colors: red, white, or  b lue.  To highlight the different symmetry 
+assumptions one may impose on the disc and the grou p action appropriate for each situation 
+(and to help understand the meaning of the Theorem) , we will do the counting in three 
+different settings. 
+ Setting I.  The disc is immovable. 
+ In this case, no question of symmetry is involved.   Each quadrant can be colored in 
+3 ways.  There are 4 quadrants.  The total number o f color patterns is 3 4. 
+ Setting R.  The disc can be freely rotated about its center.  
+ In this situation, for example, the coloring red i n the first quadrant white in the three 
+remaining quadrants, rwww, is indistinguishable fro m the colorings wrww, wwrw, and 
+wwwr. (These last three colorings can be obtained f rom the first one by a rotation of 90 o,    2 
+180 o and 270 o, respectively.)  To find the correct number of dis tinct color patterns, all you 
+need to do is to carry through the following simple  procedure. 
+ Take the cyclic permutation group of order 4 gener ated by the "90 0 rotation" (1234).  
+Express each element of the group in its disjoint c ycle form.  Do not omit 1-cycles. 
+ C = {(1)(2)(3)(4), (1234), (13)(24), (1432)}. 
+ Change every digit to x and simplify.  Keep all pa rentheses.  Then take their average 
+(add up the monomials and divide the sum by the ord er of the group, 4). 
+ (1/4)((x) 4 + (x 2)2 + 2(x 4)). 
+ Substitute (r i + w i + b i) for  x i.  You will get 
+ I C(r, w, b) = (1/4)((r+w+b) 4 + (r 2+w 2+b 2)2 + 2(r 4+w 4+b 4))  
+ =  r 4 + r 3w + br 3 + 2r 2w2 + 3br 2w + 2b 2r2 + rw 3 + 3brw 2 + 3b 2rw + b 3r + w 4 +  
+ bw 3 + 2b 2w2 + b 3w + b 4. 
+ The coefficient of r iwjbk in I C(r, w, b) is the number of distinct color patterns 
+with i quadrants colored red, j quadrants colored w hite, and k quadrants colored 
+blue.  The total number of distinct color patterns is ther efore = I C(1, 1, 1) = 24. 
+Needless to say, there is much more information cont ained in I C(r, w, b). 
+ Setting RF .  The disc can be rotated and in addition it can b e flipped over and the 
+color is visible on both sides (same as a necklace with 4 beads colored red, white, blue). 
+ The only thing we have to do to accommodate the ad ditional symmetry is to change 
+the permutation group.  We use, in place of the cyc lic permutation group of order 4 in 
+Setting R, the dihedral permutation group of order 8 generated by the "90 0 rotation" (1234) 
+and the "flip about the x-axis" (14)(23).  Everythi ng else works exactly the same way. 
+ Disjoint cycles:  D = {(1)(2)(3)(4), (1234), (13)( 24), (1432), (14)(23), (1)(3)(24),  
+   (12)(34), (13)(2)(4)}. 
+ Monomial average:  (1/8)((x) 4 + 3(x 2)2 + 2(x) 2(x 2) + 2(x 4)). 
+Substitution (r i + w i + b i ) /barb2right x i :  I D(r, w, b) = (1/8)((r+w+b) 4 + 3(r 2+w 2+b 2)2 + 
+2(r+w+b) 2(r 2+w 2+b 2) + 2(r 4+w 4+b 4)).   3 
+ The total number of distinct color patterns is = I D(1, 1, 1) = 21.    
+Pick up a book on combinatorics (of some depth).  C hances are good you will find 
+in it a chapter devoted to Polya's theory ([1], [2] , [3], [4], [6]).  An English translation of 
+Polya's original paper is in [5]. 
+I thank Rena Yang for valuable suggestions.  
+ 
+ 
+REFERENCES  
+[1]  C. Berge, Principles of Combinatorics , Academic Press, New York, 1971. 
+[2]  K. P. Bogart, Introductory Combinatorics,  Harcourt/Academic Press, San Diego, 2000. 
+[3]  N. G. deBruijn, Polya's theory of counting, Applied Combinatorial Mathematics (E. F. 
+Beckenbach, ed.), Wiley, New York, 1964. 
+[4]  G. E. Martin, Counting: the Art of Enumerative Combinatorics,  Springer, New York, 
+2001. 
+[5]  G. Polya and R. C. Read, Combinatorial Enumeration of Groups, Graphs and 
+Chemical Compounds,  Springer-Verlag, New York, 1987 
+[6]  A. Tucker, Applied Combinatorics,  John Wiley & Sons, Hoboken, N. J. 2002. 
+ 
+ 
+          
+ 
+ 
+  
+ 
+ 
+  
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+arXiv:1001.0073v1  [math.GN]  4 Jan 2010Nonmeasurability in Banach spaces
+Robert Ra/suppress lowski
+Abstract. We show that for a σ-idealIwith a Steinhaus property defined on
+Banach space, if two non-homeomorphic Banach with the same card inality of the
+Hamel basis then there is a Inonmeasurable subset as image by any isomorphism
+between of them. Our results generalize results from [ 2].
+1. Notation and Terminology
+Throughout this paper, X,Ywill denote uncountable Polish spaces and B(X)
+the Borel σ-algebra of X.We say that the ideal IonXhasBorel base if every
+elementA∈ Iis contained in a Borel set in I.(It is assumed that an ideal is always
+proper.) The ideal consisting of all countable subsets of Xwill be denoted by [ X]≤ω
+and the ideal of all meager subsets of Xwill be denoted by K.Letµbe a continuous
+probability measure on X.The ideal consisting of all µ-null sets will be denoted
+byLµ.By the following well known result, Lµcan be identified with the σ-ideal of
+Lebesgue null sets.
+Theorem 1.1 ([5], Theorem 3.4.23) .Ifµis a continuous probability on B(X),then
+there is a Borel isomorphism h:X→[0,1]such that for every Borel subset Bof
+[0,1], λ(B) =µ(h−1(B)),whereλis a Lebesgue measure.
+Definition 1.1. We say that (Z,I)is Polish ideal space if Zis Polish uncountable
+space and Iis aσ-ideal on Zhaving Borel base and containing all singletons. In
+this case, we set
+B+(Z) =B(Z)\I.
+A subset of Znot inIwill be called a I-positive set; sets in Iwill also be called
+I-null. Also, the σ-algebra generated by B(Z)∪ Iwill be denoted by B(Z),called
+theI-completion of B(Z).
+1991Mathematics Subject Classification. Primary 03E75; Secondary 03E35, 46XYY, 28A05,
+28A99.
+Key words and phrases. Lebesgue measure, Baire property, Banach space, measurable s et,
+algebraic sum, Steinhaus property.
+12 ROBERT RA/suppress LOWSKI
+It is easy to check that A∈B(Z) if and only if there is an I∈ Isuch that A△I
+(the symmetric difference) is Borel.
+Example 1.1. Letµbe a continuous probability measure on X.Then(X,[X]≤ω),
+(X,K),(X,Lµ)are Polish ideal spaces.
+Definition 1.2. A Polish ideal group is 3-tuple (G,I,+)where(G,I)is Polish ideal
+space and (G,+)is an abelian topological group with respect to the Polish to pology
+ofG.
+Now we are ready to recall the crucial property which was introduc ed by Stein-
+hauss see [ 6].
+Definition 1.3. Let(X,+)be any topological group with topology τ. We say that
+idealI ⊂P(X)have Steinhaus property iff
+∀A1,A1∈P(X)∃B1,B2∈ B+(X)∃U∈τ B1⊂A1∧B2⊂A2∧U⊂A1+A2.
+In the same paper [ 6] was proven that ideal of null sets poses the Steinhaus
+property.
+Definition 1.4. Let(X,+)be any topological group and let I ⊂P(X)be any
+invariant σ-ideal with singletons then Ihas strong Steinhaus property iff
+∀B∈ B+∀A∈P(X)\Iint(A+B)/\e}atio\slash=∅.
+It is well known that ideal of meager sets in any topological group ( G,+) has
+strong Steinhaus property see [ 4]. Moreover the ideal of the null sets respect to Haar
+measure on the locally compact topological group ( G,+) has also strong Steinhaus
+property see [ 1].
+Definition 1.5. Let(X,I)be a Polish ideal space and A⊆X. We say that AisI–
+nonmeasurable, if A /∈B(X).Further, we say that Ais completely I–nonmeasurable
+if
+∀B∈ B+(X)A∩B/\e}atio\slash=∅∧Ac∩B/\e}atio\slash=∅.
+Clearly every completely I–nonmeasurable set is I–nonmeasurable. In the lit-
+erature, completely [ X]≤ω–nonmeasurable sets are called Bernstein sets. Also, note
+thatAis completely Lµ–nonmeasurable if and only if the inner measure of Ais zero
+and the outer measure one.
+For any set E,|E|will denote the cardinality of E.
+The rest of our notations and terminology are standard. For othe r notation and
+terminology in Descriptive Set Theory we follow [ 5].
+The main motivation of this paper is the Theorem about the nonmeasu rability
+of the images under isomorphism over Qbetween RnandRmwhenever m/\e}atio\slash=n.
+More precisely in [ 2] the following Theorem was provedNONMEASURABILITY IN BANACH SPACES 3
+Theorem 1.2. LetIis nontrivial invariant σ-ideal of subsets of the group (R,+)
+which has strong Steinhaus property such that [R]ω⊂I. Letf:R2→Rbe any
+linear isomorphism over Q. Then
+(1)IfA⊂R2is a bounded subset of the real plane such that A+T=R2for
+someT∈[R2]ωthenf[A]isI-nonmeasurable subset of the real line.
+(2)IfA⊂R2is a bounded subset of the real plane with non empty interior
+int(A)/\e}atio\slash=∅. Then the set f[A]is completely I–nonmeasurable subset of the
+real line.
+2. Results
+Here we present the main results of this paper.
+Theorem 2.1. LetX,Ybe a Banach spaces and let us suppose that
+(1)I ⊂P(Y)be anσ- ideal with Steinhaus property,
+(2)∀n∈ω∀A∈ In/\e}atio\slash= 0→nA∈ I,
+(3)f:X→Yby any isomorphism between X,Ywhich is not homeomor-
+phism.
+Then the image of the unit ball f[K]isInon-measurable in Y, whereKa unit ball
+of the space Xwith the center equal 0∈X.
+Theorem 2.2. LetX,Ybe any Banach spaces for which there exists linear isomor-
+phismf:X→Ywhich is not continuous then image of the unit ball K⊂Xbyf
+has not Baire property.
+Theorem 2.3. LetX,Ybe a Banach spaces and let us assume that
+(1)I ⊂P(Y)be anκ- complete additive invariant ideal with Steinhaus prop-
+erty,
+(2)letmin{|S|:S∈P(X)∧Sis dense set in X}< κ,
+(3)f:X→Yby any isomorphism between X,Y, which is not homeomor-
+phism.
+Then the image of the unit ball f[K]isInon-measurable in Y, whereKa unit ball
+of the space Xwith the center equal 0∈X.
+3. Proofs o the main results
+Proof. of the Theorem 2.1. Let assume that image f[K]isI-measurable in
+spaceY. First of all let observe that
+Y=f[X] =f[∞/uniondisplay
+n=1nK] =∞/uniondisplay
+n=1f[nK] =∞/uniondisplay
+n=1nf[K]
+then we have that f[K]/\e}atio\slash∈ IisIpositive. Then by Steinhaus property of the ideal I
+there exist unit ball 0∈Uconsisting 0for which U⊂f[K]−f[K] =f[K−K]is
+hold. But fis linear isomorphism then f−1[U]⊂K−Kthenf−1is bounded linear4 ROBERT RA/suppress LOWSKI
+operator (so is continuous) then by Banach Theorem on invert ible operator we have
+thatfis bounded then fis homeomorphism what is impossible by (3). /squaresolid
+Here we give the following Lemma:
+Lemma 3.1. Let(X,/bardbl·/bardbl)be any Banach space then σ-idealKof the meager sets
+has Steinhaus property.
+Proof. LetA,B∈P(X)\Kbe any meager positive subsets of the Banachspace
+then there exists positive radius r >0of two balls K1=K(x,r
+2),K2=K(y,r)⊂X
+with the centers x,y∈Xsuch that A1=A∩K1andB1=B∩K2are comeager
+subsets in the balls K1(x,r)andK2(y,r)respectively. Let z=y−xandr0=r
+2and
+letK0=K(z,r0then we will show that
+∀t∈K0(t+A!)∩B1/\e}atio\slash=∅
+First let observe that if t∈K0ands∈K1then
+/bardbl(t+s)−y/bardbl=/bardbl(t−z+z)+(s−x+x)−y/bardbl=/bardbl(t−z)+(s−x)+z+x−y/bardbl
+=/bardbl(t−z)+(s−x)/bardbl ≤ /bardblt−z/bardbl+/bardbls−x/bardbl< r0+r
+2=r
+thus we have K0+K1⊂K2and then (K+A1)∩B1is comeager in the open set
+(K0+K1)∩K2then(K+A1)∩B1is nonempty set.
+Thus there exists open set K0such that
+∀t∈K0(t+A)∩B/\e}atio\slash=∅
+Finally we have
+∀t∈K0(t+A)∩B/\e}atio\slash=∅ ↔ ∀t∈K0∃a∈A∃b∈Bt+a=b↔ ∀t∈K0t∈B−A
+Thus we have K0⊂B−Awhat finishes proof of this lemma. /squaresolid
+By the above Lemma we can give the Theorem 2.2.
+Proof. (of the Theorem 2.2). By Lemma 3.1 the ideal of the meager sets of the
+spaceYhas Steinhaus property and let observe that for any α∈R\{0}the map
+Y∋y/mapsto→α·y∈Y
+is the homeomorphism then second condition of the Theorem 2. 1 is fullfiled then we
+are getting assertion. /squaresolid
+Immediately we have
+Corollary 3.1. LetX </bardbl · /bardbl)be infinite dimensional Banach space then there
+exists linear automorphism of Xfor which image of the unit ball do not has Baire
+property.
+Proof. LetBis Hamel base which is subset of the unit ball of the space X.
+LetB0={en∈X:n∈ω} ⊂Bbe any countable subset of the our Hamel base B.NONMEASURABILITY IN BANACH SPACES 5
+Let define g:B→Bas follows:
+g(x) =/braceleftbigg
+(n+1)·x x=en∈B0
+x x ∈B\B0
+Now we are ready to define f:X→Xletx∈Xand letA∈[B]<ωand assume
+that
+x=/summationdisplay
+e∈Aαe·e
+thenf(x) =/summationtext
+e∈Aα·g(e). It is easy to see that fis noncontinous linear automor-
+phism of X. Then by Theorem 2.2 proof is finished. /squaresolid
+Proof. of the Theorem 2.3. Let assume that image f[K]isI-measurable in
+spaceY. Letγ < κbe such that γ=|S|and the subset S⊂Xbe dense in space X
+then we have
+Y=f[X] =f[/uniondisplay
+x∈S{x}+K] =/uniondisplay
+x∈Sf[{x}+K] =/uniondisplay
+x∈S{f(x)}+f[K]
+thusf[K]/\e}atio\slash∈ IisIpositive. Then by Steinhaus property of the ideal Ithere exist
+unit ball 0∈Uconsisting 0for which U⊂f[K]−f[K] =f[K−K]is hold. But
+fis linear isomorphism then f−1[U]⊂K−Kthenf−1is bounded linear operator
+(so is continuous) then by Banach Theorem on invertible oper ator we have that fis
+bounded then fis homeomorphism what is impossible by (3). /squaresolid
+Immediately we are getting the following
+Corollary 3.2. LetX,Ybe a Banach spaces with the following properties:
+(1)Xis separable Banach space,
+(2)I ⊂P(Y)be additive invariant σ-ideal with Steinhaus property,
+(3)f:X→Yby any isomorphism between X,Y, which is not homeomor-
+phism.
+Then the image of the unit ball f[K]isInon-measurable in Y, whereKa unit ball
+of the space Xwith the centre equal 0∈X.
+Acknowledgement Author is very indebted to Professor Jacek Cicho´ n for help
+and critical remarks.
+References
+[1] A. Beck, H.H. Corson, A. B. Simon, The interior points of the prod uct of two subsets of a
+locally compact group, Proc. Amer. Math. Soc. 9(1958), 648–652
+[2] J. Cicho´ n, P. Szczepaniak, When is the unit ball nonmeasurable? , accepted to Fund. Math.
+[3] D. H. Fremlin, Measure additive coverings and measurable selecto rs, Dissertationes Math.
+260(1987)
+[4] E. J. McShane, Images of sets satisfying the condition of Baire, Ann. of Math. 51(1950),
+380–386
+[5] S.M. Srivastava, A course on Borel sets, Springer-Verlag, New York, 19986 ROBERT RA/suppress LOWSKI
+[6] H. Steinhaus, Sur les distances des points dans les ensembles de m easure positive, Fund.
+Math.1(1920), 99–104
+Robert Ra/suppress lowski, Institute of Mathematics, Wroc/suppress law Univ ersity of Technol-
+ogy, Wybrze ˙ze Wyspia ´nskiego 27, 50-370 Wroc/suppress law, Poland.
+E-mail address :robert.ralowski@pwr.wroc.pl
\ No newline at end of file
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+arXiv:1001.0074v2  [math.RT]  7 Mar 2010DUALITIES FOR LIE SUPERALGEBRAS
+SHUN-JEN CHENG AND WEIQIANG WANG
+Abstract. We explain how Lie superalgebras of types glandospprovide a natu-
+ral framework generalizing the classical Schur and Howe dua lities. This exposition
+includes a discussion of super duality, which connects the p arabolic categories O
+between classical Lie superalgebras and Lie algebras. Supe r duality provides a con-
+ceptual solution to the irreducible character problem for t hese Lie superalgebras in
+terms of the classical Kazhdan-Lusztig polynomials.
+Contents
+Introduction 1
+1. Lie superalgebra ABC 5
+2. Finite-dimensional modules of Lie superalgebras 10
+3. Schur-Sergeev duality 16
+4. Howe duality for Lie superalgebras of type gl 22
+5. Howe duality for Lie superalgebras of type osp 27
+6. Super duality 32
+References 39
+Introduction
+0.1. ThestudyofLiesuperalgebras,supergroups,andtheir representationswaslargely
+motivated by supersymmetryin mathematical physics, which puts bosons and fermions
+on the same footing. An earlier achievement is the Cartan-Ki lling type classification
+of finite-dimensional simple complex Lie superalgebras by K ac [K1] (also cf. [SNR] for
+an independent classification of the so-called classical Li e superalgebras). The most
+important basic classical Lie superalgebras consist of infi nite series of types sl,osp.
+The basic classical Lie superalgebras afford root systems, Dy nkin diagrams, Cartan
+subalgebras, triangular decomposition, Verma modules, ca tegoryO, and so on. There
+has been much work on representation theory of Lie superalge bras (in particular, basic
+classical) in the last three decades, but conceptual approa ches have been lacking until
+recently.
+0.2. The aim of these lecture notes is to explain three differen t kinds of dualities for
+Lie superalgebras:
+Schur duality, Howe duality, and Super duality.
+12 SHUN-JEN CHENG AND WEIQIANG WANG
+In the superalgebra setting, the first (i.e. Schur) duality w as formulated by Sergeev,
+and the latter two dualities have been largely developed by t he authors and their
+collaborators. These lecture notes are also intended to ser ve as a road map for a
+forthcoming book by the authors.
+The Schur-Sergeev duality is an interplay between Lie super algebras and the sym-
+metric groups which incorporates the trivial and sign modul es in a unified framework.
+On the algebraic combinatorial level, there is a natural sup er generalization of the
+notion of semistandard tableaux which is a hybrid of the trad itional version and its
+conjugate counterpart.
+It has been observed that much of the study of the classical in variant theory on
+polynomial algebras has parallels for exterior algebras, a nd both admit reformulation
+and extension in the theory of Howe’s reductive dual pairs. L ie superalgebras allow a
+uniform treatment of Howe duality on the polynomial and exte rior algebras.
+Super duality has a different flavor. It views representation t heories of Lie super-
+algebras and Lie algebras as two sides of the same coin, and it is an unexpected yet
+powerful approach developed in the past few years which allo w us to overcome various
+superalgebra difficulties. We give an exposition on the new de velopment on super dual-
+ity which is an equivalence between parabolic categories Oof Lie superalgebras and Lie
+algebras. Super duality provides a conceptual solution to t he long-standing irreducible
+character problem for a wide class of modules over (a wide cla ss of) Lie superalgebras
+in terms of Kazhdan-Lusztig polynomials. This is achieved d espite the fact that there
+are no obvious Weyl groups controlling the linkage for super representation theory.
+0.3. In Section 1, we give some basic constructions and struc tures of the general linear
+and the ortho-symplectic Lie superalgebras. We emphasize t he super phenomena that
+are not observed in the ordinary Lie algebra setting, such as odd roots, non-conjugate
+Borel subalgebras, and so on. In Section 2, we present Kac’s c lassification of finite-
+dimensional simple g-modules [K2]. The classification is very easy for type A, but
+nontrivial for osp. In the latter case we explain a new odd reflection approach by Shu
+and the second author [SW], using a more natural labeling of t hese modules by hook
+partitions. We note that odd reflection is also one of the main technical tools in super
+duality. In addition, we present the typicalfinite-dimensional irreducible character
+formula, following [K2].
+0.4. The classical Schur duality relates the representatio n theory of the general linear
+Lie algebras and that of the symmetric groups. In Section 3, w e explain Sergeev’s
+generalization [Sv1] of Schur duality for the general linea r Lie superalgebras gl(m|n)
+(also see Berele and Regev [BeR] for additional insight and d etail). More precisely, we
+establish a double centralizer theorem for the actions of gl(m|n) and the symmetric
+groupSdindletters on the tensor space ( Cm|n)⊗d. We then provide an explicit
+multiplicity-free decomposition of the tensor space into a U(gl(m|n))⊗CSd-modules.
+We further present a simple formula obtained in our latest wo rk with Lam [CLW]
+for extremal weights in a simple polynomial gl(m|n)-module with respect to all Borel
+subalgebras, which has an explicit diagramatic interpreta tion from a Young diagram.DUALITIES FOR LIE SUPERALGEBRAS 3
+0.5. Howe’s theory of reductive dual pairs [H1, H2] can be vie wed as a representation
+theoretic reformulation and extension of the classical inv ariant theory (see Weyl [We]).
+For example, the first fundamental theorem on invariants for classical groups are refor-
+mulated in terms of double centralizer properties of two cla ssical Lie groups/algebras.
+One advantage of Howe duality is that it allows natural gener alizations to classical Lie
+groups/algebras (and superalgebras) other than type A.
+We mainly use two examples of dual pairs to illustrate the mai n ideas of Howe
+duality and the new phenomena of superalgebra generalizati ons. For more detailed
+case study of Howe duality for Lie superalgebras, we refer to the original papers
+[BPT, CW1, CW2, CW3, CL1, CLZ, CZ2, CKW, LZ, Sv2]. In Section 4 , we formulate
+the (gl(m|n),gl(d))-Howe duality and find the highest weight vectors for each i sotyp-
+ical component in the corresponding multiplicity-free dec omposition. In Section 5 we
+present the (Sp( d),osp(2m|2n))-Howe duality and its multiplicity-free decomposition.
+The application of the Howe duality to irreducible characte rs over Lie superalgebras
+follows the simpler approach in our work with Kwon [CKW] (whi ch uses Howe duality
+for infinite-dimensional Lie algebras [Wa]).
+0.6. We recall some truly superphenomenathat have been them ain obstacles towards
+a better understanding of super representation theory:
+(1) There exist odd roots as well as non-conjugate Borel suba lgebras for a Lie
+superalgebra. A homomorphism between Verma modules may not be injective.
+(2) The linkage in category Oof modules for a Lie superalgebra is NOT controlled
+by the Weyl group of g¯0; see e.g. gl(1|1).
+(3) There is no uniform Weyl-type irreducible finite-dimens ional character formula
+for Lie superalgebras.
+(4) The super geometry behind super representation theory i s still inadequately
+developed.
+Inlightofthesesuperphenomena,itwasaratherunexpected discovery[CWZ,CW4],
+which was partly inspired by Brundan [Br1], that there exist s a (conjectural) equiva-
+lence of categories between Lie algebras and Lie superalgeb ras of type A(at a certain
+suitable limit at infinity), which was termed Super Duality . This conjecture in the full
+generality of [CW4] has been proved in [CL2], which in partic ular offers an elementary
+and conceptual solution to the character problem for all fini te-dimensional simple mod-
+ules and for a large class of infinite-dimensional simple hig hest weight modules over Lie
+superalgebras of type A.
+Super duality has been subsequently formulated and establi shed between various Lie
+superalgebras of type ospand thecorrespondingclassical Lie algebras inour very rec ent
+work with Lam [CLW]. This in particular offers a conceptual sol ution of the irreducible
+character problem for a wide class of modules, which include all finite-dimensional irre-
+ducibles, of Lie superalgebras of type ospin terms of Kazhdan-Lusztig polynomials for
+classical Lie algebras [KL, BB, BK] (for more on Kazhdan-Lus ztig theory see Tanisaki’s
+lectures [Ta]). Inaddition, it follows easily from theappr oachof [CL2, CLW] that the u-
+homology groups (or Kazhdan-Lusztig polynomials in the sen se of Vogan [Vo]) match
+perfectly between classical Lie superalgebras and the corr esponding classical Lie al-
+gebras. This generalizes earlier partial results in this di rection from Schur or Howe4 SHUN-JEN CHENG AND WEIQIANG WANG
+duality approach [CZ1, CK, CKW]. The super duality as outlin ed above is explained
+in Section 6.
+Let usputthesuperduality work explained above inperspect ive.Finite-dimensional
+irreducible characters for gl(m|n) have been also obtained earlier in two totally differ-
+ent approaches by [Sva] and [Br1]. The mixed algebraic and ge ometric approach of
+Serganova has been extended very recently in [GS] to obtain a ll irreducible finite-
+dimensional osp-characters. Brundan and Stroppel [BrS] also provided anot her ap-
+proach to the main results of [Br1] and independently proved a special case of the
+super duality conjecture in type Aas formulated in [CWZ]. All these approaches have
+brought new and different insights into super representation theory. Our super duality
+approach has the advantages of explaining the connection wi th classical Lie algebras
+and their Kazhdan-Lusztig polynomials, covering infinite- dimensional irreducible char-
+acters, and being extendable to general Kac-Moody Lie super algebras.
+A list of symbols is added at the end of the paper to facilitate the reading.
+0.7. Let us end the Introduction with some remarks on the inte rrelations among the
+three dualities.
+The (gl(d),gl(n))-Howe duality is equivalent to Schur duality. It follows f rom the
+Schur-Sergeev duality that the characters for irreducible polynomial gl(m|n)-modules
+are given by the so-called hook Schur functions. On the other hand, the irreducible
+character formulas for Lie superalgebras of types glorospobtained from Howe duality
+can be expressed in terms of infinite classical Weyl groups. T he appearance of hook
+Schur functions and infinite Weyl groups in these formulas ar e conceptually explained
+from the viewpoint of super duality.
+Superduality can beinformally interpreted as acategorific ation ofthestandardinvo-
+lution on the ring of symmetric functions. It is well known th at the ring of symmetric
+functions in infinitely many variables admits symmetries wh ich are not observed in
+finitely many variables. Super duality is formulated precis ely at the infinite rank limit.
+On the level of combinatorial parameterizations of highest weights, super duality man-
+ifests itself through (variation of) the conjugate of parti tions.
+Partly due to the time constraint of the lectures, we have lef t out many interesting
+topics on super representation theory. We refer to [BL, J] (a nd more recently [SZ])
+for finite-dimensional irreducible characters of atypical ity one, to [BKN, DS, Ma, Mu,
+Pe, PS] for geometric approaches, to [Br2, CWZ2] for further development of the Fock
+space approach of Brundan for the queer Lie superalgebra q(n) and for osp(2|2n), to
+[CK,CKW,CZ1,Ger,San,Sva,Zou]forsomecohomological asp ects, to[BrK,SW,WZ]
+for prime characteristic, to [JHKT, Su] for related combina torial structures; also see
+[Gor, KW, Naz] for additional work on Lie superalgebras.
+Acknowledgment. This paper is a modified and expanded written account of the
+8 lectures given by the second author at the summer school of E ast China Normal
+University (ECNU), Shanghai, July 2009. We are grateful to N gau Lam for his col-
+laboration and insight. We thank Bin Shu at ECNU for hospital ity and an enjoyable
+summer school.DUALITIES FOR LIE SUPERALGEBRAS 5
+1.Lie superalgebra ABC
+1.1. A vector superspace Vis understood as a Z2-graded vector space V=V¯0⊕V¯1.
+An element a∈Vihas parity |a|=i, and an element in V¯0(respectively, V¯1) is called
+even (respectively, odd).
+Definition 1.1. A Lie superalgebra is a vector superspace g=g¯0⊕g¯1equipped with
+a bilinear bracket operation [ .,.] satisfying [ gi,gj]⊂gi+j,i,j∈Z2, and the following
+two axioms: for Z2-homogeneous a,b,c∈g,
+(1) (Skew-supersymmetry) [ a,b] =−(−1)|a|·|b|[b,a].
+(2) (Super Jacobi identity) [ a,[b,c]] = [[a,b],c] +(−1)|a|·|b|[b,[a,c]].
+Remark 1.2.(1) For a Lie superalgebra g=g¯0⊕g¯1,g¯0is a Lie algebra and g¯1is a
+g¯0-module under the adjoint action.
+(2) (Sign Rule) As explained in Manin’s book [Ma], there is a g eneral heuristic sign
+rule for superalgebras as follows. If in some formula for usual algebra there
+are monomials with interchanged terms, then in the correspo nding formula for
+superalgebra every interchange of neighboring terms, say aandb, is accompanied
+by the multiplication of the monomials by the factor (−1)|a|·|b|.This is already
+manifest in the definition of Lie superalgebra and will persi st throughout the
+paper.
+Example 1.3. (1) LetA=A¯0⊕A¯1beanassociative superalgebra(i.e. Z2-graded).
+Then (A,[.,.]) is a Lie superalgebra, where for homogeneous elements a,b∈A,
+we define
+[a,b] =ab−(−1)|a|·|b|ba.
+(2) A Lie superalgebra gwithg¯1= 0 is just a usual Lie algebra. A Lie superalgebra
+gwith purely odd part (i.e. g¯0= 0) has to be abelian, i.e. [g,g] = 0.
+1.2.Lie superalgebras of type Aand the supertrace. LetV=V¯0⊕V¯1be a
+vector superspace. Then End( V) is naturally an associative superalgebra. The Lie
+superalgebra gl(V) := (End( V),[.,.]) from Example 1.3 (1) is called a general linear
+Lie superalgebra . IfV¯0=CmandV¯1=Cn, we denote VbyCm|n, andgl(V) bygl(m|n).
+Note that both gl(m|0)∼=gl(0|m) are isomorphic to the usual Lie algebra gl(m).
+The Lie superalgebra gl(m|n) consists of block matrices of size m|n:
+(1.1) g=/parenleftbigg
+a b
+c d/parenrightbigg
+.
+Throughout the paper, we choose to parameterize the rows and columns of the matrices
+by the set
+I(m|n) ={1,...,m;1,...,n}
+with a total order
+(1.2) 1< ... <m <0<1< ... < n
+(where 0 is inserted for later convenience). Its even subalg ebra consists of matrices of
+the form /parenleftbigg
+a0
+0d/parenrightbigg6 SHUN-JEN CHENG AND WEIQIANG WANG
+and is isomorphic to gl(m)⊕gl(n).
+Example 1.4. Forg=gl(1|1), let
+e=/parenleftbigg
+0 1
+0 0/parenrightbigg
+, f=/parenleftbigg
+0 0
+1 0/parenrightbigg
+, h 1=/parenleftbigg
+1 0
+0 0/parenrightbigg
+, h 2=/parenleftbigg
+0 0
+0 1/parenrightbigg
+.
+Then
+[e,f] =h1+h2(the identity matrix) .
+Thesupertrace , denoted by str, of (1.1) is defined to be
+str(g) = tr(a)−tr(d).
+Thespecial linear Lie superalgebra is
+sl(m|n) ={x∈gl(m|n)|str(x) = 0}.
+ThedefinitionsofsupertraceandoftheLiesuperalgebra slarejustifiedbythefollowing.
+Exercise 1.5. Show that sl(m|n) = [gl(m|n),gl(m|n)] and in particular sl(m|n) is a
+Lie subalgebra of gl(m|n).
+The notion of simple Lie superalgebras is defined in the same w ay as for Lie algebras.
+We note that sl(n|n) is not a simple Lie superalgebra, as it contains a nontrivia l center
+CI2n.
+1.3.The bilinear form. Lethdenote the Cartan subalgebra of gl(m|n) consisting of
+all diagonal matrices. Note that his an even subalgebra of gl(m|n).
+LetEij, fori,j∈I(m|n), denote the standard basis for gl(m|n). We definea bilinear
+form (·,·) ongby letting
+(a,b) = str(ab), a,b∈g.
+This restricts to a nondegenerate symmetric bilinear form o nh: fori,j∈I(m|n),
+(Eii,Ejj) =
+
+1 if1≤i=j≤m,
+−1 if 1≤i=j≤n,
+0 ifi/\e}atio\slash=j.
+Denoteby {δi,ǫj}i,jthebasisof h∗dualto{Eii,Ejj}i,j, where1 ≤i≤mand1≤j≤n.
+Under the bilinear form ( ·,·), we have the identification δi= (Ei,i,·) andǫj=−(Ejj,·).
+Whenever it is convenient we also use the notation
+(1.3) ǫi:=δi,for 1≤i≤m.
+The form ( ·,·) onhinduces a non-degenerate bilinear from on h∗, which will be
+denoted by the same notation, as follows: for i,j∈I(m|n),
+(ǫi,ǫj) =
+
+1 if1≤i=j≤m,
+−1 if 1≤i=j≤n,
+0 ifi/\e}atio\slash=j.(1.4)DUALITIES FOR LIE SUPERALGEBRAS 7
+1.4.The root system. For the Lie superalgebra gl(m|n), we define the root space
+decomposition, a root system Φ, a set Φ+(respectively, Φ−) of positive (respectively,
+negative) roots, a set Π of simple roots (in Φ+), etc. As this can be done in the same
+way as for semisimple Lie algebras or gl(m), we will merely write down the statements
+for later use.
+Now let us make the super phenomenon explicit. A root αisevenifgα⊂g¯0, and
+it isoddifgα⊂g¯1. Denote by Φ ¯0(respectively, Φ ¯1) the set of all even (respectively,
+odd) roots in Φ. Denote
+Φ±
+i= Φi∩Φ±,Πi= Φi∩Π, i∈Z2.
+With respect to the Cartan subalgebra hthe Lie superalgebra gl(m|n) admits a root
+space decomposition:
+g=h⊕/circleplusdisplay
+α∈Φgα,
+with a root system
+Φ ={ǫi−ǫj|i,j∈I(m|n),i/\e}atio\slash=j}.
+The standard set of simple roots is taken to be Π = Π ¯0∪Π¯1, where
+Π¯0={ǫi−ǫi+1}1≤i≤m−1∪{ǫi−ǫi+1}1≤i≤n−1,Π¯1={ǫm−ǫ1},
+and the associated standard set of positive roots is
+Φ+={ǫi−ǫj|i,j∈I(m|n),i < j},
+where the odd roots are ǫi−ǫjwith indices i <0< j. Clearly, gǫi−ǫj=CEij. It follows
+by (1.4) that
+(δi−ǫj,δi−ǫj) = 0,
+for all the odd roots δi−ǫj, where 1 ≤i≤m,1≤j≤n. An odd root αwith (α,α) = 0
+is called isotropic. The standard Dynkin diagram is:
+/circlecopyrt /circlecopyrt/circlemultiplytext/circlecopyrt /circlecopyrt ··· ···
+δ1−δ2δm−1−δmδm−ǫ1
+ǫ1−ǫ2ǫn−1−ǫn
+where we have used/circlemultiplytextto denote an isotropic odd simple root.
+Remark 1.6.The notion of root systems and Dynkin diagrams makes sense fo r all the
+basic classical Lie superalgebras, which consist of gl(m|n),sl(m|n),osp(m|2n) and three
+exceptional ones (besides the simple Lie algebras).
+1.5.Non-conjugate Borel subalgebras and ǫδ-sequences. Recallthatthebilinear
+form on the real subspace h∗
+Rspanned by the ǫi’s is not positive definite (due to the
+supertrace), and moreover, there exist isotropic odd roots .
+Another distinguished feature of Lie superalgebras is the e xistence of non-conjugate
+Borel subalgebras or non-isomorphic Dynkin diagrams (unde r the Weyl group action).
+Lemma 1.7. Letgbe a Lie superalgebra with triangular decomposition g=n−⊕h⊕n+,
+which corresponds to the root system Φ = Φ+∪Φ−. Letαbe an odd isotropic simple8 SHUN-JEN CHENG AND WEIQIANG WANG
+root. Let b=h+n+. Then, Φ(α)+:= (Φ+\{α})∪{−α}is a new system of positive
+roots, whose corresponding set of simple roots is
+Π(α) ={β∈Π|(β,α) = 0,β/\e}atio\slash=α}∪{β+α|β∈Π,(β,α)/\e}atio\slash= 0}∪{−α}.
+The new Borel subalgebra corresponding to Π( α) will be denoted by b(α).
+Proof.Follows from a straightforward verification. /square
+The process of obtaining Π( α) from Π above will be referred to as an odd reflection ,
+and will be denoted by rα, in accordance with the usual notion of real reflections.
+Example 1.8. Associated to gl(1|2), we have Φ ¯0={±(ǫ1−ǫ2)}, and Φ ¯1={±(δ1−
+ǫ1),±(δ1−ǫ2)}.There are 6 sets of simple roots, that are related by the real a nd odd
+reflections as follows. There are three conjugacy classes of Borel subalgebras, and each
+vertical pair corresponds to such a conjugacy class.
+/circlemultiplytext/circlecopyrt
+δ1−ǫ1ǫ1−ǫ2rδ1−ǫ1⇄/circlemultiplytext /circlemultiplytext
+ǫ1−δ1δ1−ǫ2rδ1−ǫ2⇄/circlecopyrt/circlemultiplytext
+ǫ1−ǫ2ǫ2−δ1
+/circlemultiplytext/circlecopyrt
+δ1−ǫ2ǫ2−ǫ1rδ1−ǫ2⇄/circlemultiplytext /circlemultiplytext
+ǫ2−δ1δ1−ǫ1rδ1−ǫ1⇄/circlecopyrt/circlemultiplytext
+ǫ2−ǫ1ǫ1−δ1/arrowbothvrǫ1−ǫ2 /arrowbothvrǫ1−ǫ2 /arrowbothvrǫ1−ǫ2
+One convenient way to parameterize the conjugacy classes of Borel subalgebras of
+gl(m|n) is viathenotion of ǫδ-sequences. Keeping (1.3) inmind, welist thesimpleroots
+associated to a given Borel subalgebra bin order as ǫi1−ǫi2,ǫi2−ǫi3,...,ǫim+n−1−ǫim+n,
+where{i1,i2,...,im+n}=I(m|n). Switchingtheorderedsequence ǫi1ǫi2...ǫim+ntothe
+ǫδ-notation by (1.3) and then dropping the indices give us the ǫδ-sequence associated
+tob. Note that the total number of δ’s (respectively, ǫ’s) ism(respectively, n).
+For example, the three conjugacy classes of Borels for gl(1|2) above correspond to
+the three sequences δǫǫ,ǫδǫ,ǫǫδ, respectively. In more detail, the first sequence δǫǫis
+obtained by removing the indices of δ1ǫ1ǫ2(read off from the upper-left diagram above)
+orδ1ǫ2ǫ1(from the lower-left diagram above). Also the standard Bore l ofgl(m|n)
+corresponds to the sequence δ···δ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+mǫ···ǫ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+nwhile the opposite Borel to the standard one
+corresponds to ǫ···ǫ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+nδ···δ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+m.
+Exercise 1.9. Let Φ be the roots of gl(m|n) with respect to the Cartan subalgebra
+h. Prove that the sets of simple roots in Φ are in one-to-one cor respondence with the
+ǫδ-sequences with m δ’s andn ǫ’s in total. In particular, there are/parenleftbigm+n
+m/parenrightbig
+conjugacy
+classes of Borel subalgebras.
+1.6. Let Bbe a non-degenerate even supersymmetric bilinear form on a v ector su-
+perspace V=V¯0⊕V¯1. HereBisevenifB(Vi,Vj) = 0 unless i=j∈Z2, andBis
+supersymmetric ifB|V¯0×V¯0is symmetric while B|V¯1×V¯1is skew-symmetric (and hence
+dimV¯1is necessarily even).DUALITIES FOR LIE SUPERALGEBRAS 9
+Fors∈Z2, let
+osp(V)s={g∈gl(V)s|B(g(x),y) =−(−1)s·|x|B(x,g(y)),∀x,y∈V},
+osp(V) =osp¯0(V)⊕osp¯1(V).
+One checks that osp(V) is a Lie superalgebra, whose even subalgebra is isomorphic to
+so(V¯0)⊕sp(V¯1).WhenV=Cℓ|2n, we write osp(V) =osp(ℓ|2n).
+Remark 1.10.One can also define the Lie superalgebra spo(V) as the subalgebra of
+gl(V) which preserves a non-degenerate skew-supersymmetric bi linear form on V(here
+dimV¯0has to be even). When V=C2n|ℓ, we write spo(V) =spo(2n|ℓ).
+Exercise 1.11. Show that Lie superalgebras osp(ℓ|2n) andspo(2n|ℓ) are isomorphic.
+1.7. Define the super transpose st as follows: for a matrix in the block form (1.1), we
+let/parenleftbigg
+a b
+c d/parenrightbiggst
+=/parenleftbigg
+atct
+−btdt/parenrightbigg
+,
+wherextdenotes the usual transpose of the matrix x.
+Define the (2 n+2m+1)×(2n+2m+1) matrix in the ( n|n|m|m|1)-block form
+J2n|2m+1:=
+0In0 0 0
+−In0 0 0 0
+0 0 0 Im0
+0 0 Im0 0
+0 0 0 0 1
+, (1.5)
+whereInis then×nidentity matrix. Let J2n|2mdenote the (2 n+ 2m)×(2n+2m)
+matrix obtained from J2n|2m+1by deleting the last row and column. Then, for ℓ= 2m
+or 2m+ 1, by definition spo(2n|ℓ) is the subalgebra of gl(2n|ℓ) which preserves the
+bilinear form on C2n|ℓwith matrix J2n|ℓrelative to the standard basis of C2n|ℓ, and
+hence
+spo(2n|ℓ) ={g∈gl(2n|ℓ)|gstJ2n|ℓ+J2n|ℓg= 0}.
+By a direct computation, spo(2n|2m+1) consists of the (2 n+2m+1)×(2n+2m+1)
+matrices of the following ( n|n|m|m|1)-block form
+
+d e yt
+1xt
+1zt
+1
+f−dt−yt−xt−zt
+x x1a b −vt
+y y1c−at−ut
+z z1u v 0
+, b,cskew-symmetric ,e,fsymmetric . (1.6)
+TheLiesuperalgebra spo(2n|2m)consistsofmatrices(1.6)withthelastrowandcolumn
+removed. Here and below, the rows and columns of the matrices J2n|ℓand (1.6) (or its
+modification) are indexed by the finite set I(2n|ℓ).
+The standard Dynkin diagrams of spo(2n|2m+1) and spo(2n|2m) are given respec-
+tively as follows:10 SHUN-JEN CHENG AND WEIQIANG WANG
+/circlecopyrt /circlecopyrt ···/circlemultiplytext/circlecopyrt···/circlecopyrt /circlecopyrt=⇒
+δ1−δ2δ2−δ3δn−ǫ1ǫ1−ǫ2ǫm−1−ǫmǫm
+/circlecopyrt /circlecopyrt ···/circlemultiplytext/circlecopyrt···/circlecopyrt/circlecopyrt
+/circlecopyrt
+❅❅δ1−δ2δ2−δ3δn−ǫ1ǫ1−ǫ2ǫm−2−ǫm−1ǫm−1−ǫm
+ǫm−1+ǫm
+Example 1.12. The Lie superalgebra g=osp(1|2) has even subalgebra g¯0∼=sl(2) =
+C/a\}bracketle{te,h,f/a\}bracketri}htandg¯1isomorphic to the 2-dimensional natural sl(2)-module C/a\}bracketle{tE,F/a\}bracketri}ht. More-
+oever, [E,E] = 2e,[F,F] =−2f,[E,F] =h.
+The simple root consists of a (unique) odd non-isotropic roo tδ, twice of which is an
+even root. The Dynkin diagram of osp(1|2) is denoted by/CP(in order to distinguish
+from an odd simple isotropic root/circlemultiplytext).
+Exercise 1.13. Prove the following identities in U(osp(1|2)) (n∈Z+):
+(i) [E,F2n] =−nF2n−1,
+(ii) [E,F2n+1] =F2n(h−n).
+Exercise 1.14. Use Exercise 1.13 to show that the finite-dimensional irredu cible rep-
+resentations of osp(1|2) are parameterized by the set of highest weights {nδ|n∈Z+}.
+Denoting the irreducible module corresponding to nδbyL(nδ), show that
+chL(nδ) =e(n+1
+2)δ−e−(n+1
+2)δ
+e1
+2δ−e−1
+2δ.
+Exercise 1.15. Consider the element
+Ω := 2h2+2h+FE+4fe∈U(osp(1|2)).
+Use Exercise 1.13 to prove that [ F,Ω] = [E,Ω] = 0, and hence Ω is in the center of
+U(osp(1|2)). Conclude from Exercise 1.14 that Ω acts as different scala rs on different
+finite-dimensional irreducible representations and hence that every finite-dimensional
+osp(1|2)-module is completely reducible.
+2.Finite-dimensional modules of Lie superalgebras
+2.1.PBW theorem. Theuniversal enveloping algebra U(g) of a Lie superalgebra
+g=g¯0⊕g¯1is an associative superalgebra characterized by a universa l property exactly
+as for Lie algebras.
+Assume that {x1,...,x r}is a basis for g¯0and{y1,...,ys}is a basis for g¯1. Then
+the universal enveloping algebra U(g) admits a PBW basis
+(2.1) xa1
+1···xarryb1
+1···ybss,∀ai∈Z≥0,∀bj∈ {0,1}.
+Alternatively, if we define the standard PBW filtration {FdU(g)}onU(g) by letting
+FdU(g) be the span of elements (2.1) with Σ iai+Σjbj≤d, then we have the following
+isomorphism for the associated graded of U(g) in terms of the symmetric algebra S(g¯0)
+and exterior algebra ∧(g¯1):
+grFU(g)∼=S(g¯0)⊗∧(g¯1).DUALITIES FOR LIE SUPERALGEBRAS 11
+Example 2.1. Letg=g¯1be a purely odd Lie superalgebra. Then its universal
+enveloping algebra is isomorphic to the exterior algebra ∧(g¯1).
+2.2.Representations of gl(1|1).Recall the basis {e,h1,h2,f}forg=gl(1|1) from
+Example 1.4. Note that the even subalgebra gl(1|1)¯0=gl(1)⊕gl(1) coincides with the
+Cartan subalgebra h, and the Weyl group is trivial in this case.
+Givenλ= (λ1,λ2)∈C2, we denote by Cλ:=Cvλthe one-dimensional h-module by
+lettinghivλ=λivλ, fori= 1,2. Regarding Cλas a module over the Borel subalgebra
+h+Ce, on which eacts trivially, we define the Verma modules (which coincide w ith
+Kac modules defined below for gl(1|1)) over g:
+K(λ) =U(g)⊗U(h+Ce)Cλ.
+By the PBW theorem K(λ) can be identified with the following two-dimensional space
+(where by abuse of notation 1 ⊗vλis denoted by vλ):
+K(λ) =C/a\}bracketle{tvλ,fvλ/a\}bracketri}ht.
+Theg-module K(λ) has a unique simple quotient, denoted by L(λ).
+Theg-module K(λ) is simple if and only if λ1/\e}atio\slash=−λ2. This follows from
+efvλ= (h1+h2)vλ−fevλ= (λ1+λ2)vλ.
+Ifλ1/\e}atio\slash=−λ2, thenK(λ1,λ2) is the unique simple object in its block (in the category
+of finite-dimensional gl(1|1)-modules). This block is semisimple.
+Fora∈C, we have a non-split short exact sequence of g-modules:
+0→L(a−1,1−a)→K(a,−a)→L(a,−a)→0.
+The block containing L(a,−a) is not semisimple, and it contains infinitely many simple
+objectsL(b,−b) forb∈a+Z(the underlying algebra can be described in terms of the
+A∞-quiver).
+2.3.Finite dimensional simple gl(m|n)-modules. Letg=gl(m|n) in this subsec-
+tion.
+Letn+(and respectively, n−) be the subalgebra of strictly upper (and respectively,
+lower) triangular matrices of gl(m|n) with respect to the standard basis. Then we have
+the triangular decomposition
+g=n−⊕h⊕n+.
+The even subalgebra admits a compatible triangular decompo sition
+g¯0=n−
+¯0⊕h⊕n+
+¯0,
+wheren±
+¯0=g¯0∩n±.
+Moreover, the Lie superalgebra gadmits a Z-grading
+g=g−1⊕g¯0⊕g1,
+whereg−1(respectively, g1) is spanned by all Eijwithi >0> j(respectively, i <0<
+j). Note that this Z-grading is compatible with the Z2-grading, i.e., the degree zero
+subspace coincides with the Z2-degree zero subspace. This is equivalent to the fact that
+g¯0is a Levi subalgebra of g(corresponding to the removal of the odd simple root from
+the standard Dynkin diagram of g).12 SHUN-JEN CHENG AND WEIQIANG WANG
+Forλ∈h∗, letL(λ) (respectively, L0(λ)) be the simple module of g(respectively,
+g¯0) of highest weight λ. Define the Kac module overgby
+K(λ) =U(g)⊗U(g¯0⊕g1)L0(λ),
+which can be identified by the PBW theorem with
+(2.2) K(λ) =∧(g−1)⊗L0(λ).
+Proposition 2.2. There exists a surjective g-module homomorphism (unique up to a
+scalar multiple) K(λ)։L(λ). Moreover, the following are equivalent:
+(1)L(λ)is finite-dimensional.
+(2)L0(λ)is finite-dimensional.
+(3)K(λ)is finite-dimensional.
+Proof.Thesurjectivehomomorphismfollowsfromthefactthat K(λ)isahighestweight
+g-module of highest weight λ.
+(1)⇒(2).Note that L0(λ) is a quotient of L(λ) regarded as g¯0-module.
+(2)⇒(3).Follows from (2.2).
+(3)⇒(1).Follows from the surjective map K(λ)։L(λ). /square
+Arguing as for Lie algebras, every finite-dimensional simpl eg-module is a highest
+weight module L(λ) for some λ, and moreover, L(λ)/\e}atio\slash∼=L(µ) ifλ/\e}atio\slash=µ. Hence, the above
+proposition gives a classification of finite-dimensional si mpleg-modules.
+2.4.Typical irreducible characters. Letg=gl(m|n) in this subsection.
+We will denote by P+the set of weights λ∈h∗such that L0(λ) is finite-dimensional.
+Recall that the character of L0(λ) forλ∈P+is given by Weyl’s formula
+chL0(λ) =/summationtext
+σ∈Wsgn(σ)eσ(λ+ρ0)−ρ0
+/producttext
+α∈Φ+
+¯0(1−e−α).
+HereW(∼=Sm×Sn) denotes the Weyl group of g¯0, sgn denotes the sign representation
+ofW, andρ0=1
+2/summationtext
+α∈Φ+
+¯0α.
+Also denote by
+ρ1=1
+2/summationdisplay
+α∈Φ+
+¯1α, ρ =ρ0−ρ1.
+Lemma 2.3. We have
+(1) (ρ,β) =1
+2(β,β),∀β∈Π.
+(2)σ(ρ1) =ρ1,∀σ∈W.
+Proof.Part 1 can be verified directly. Part 2 follows from the fact th atg1is preserved
+by the adjoint group G¯0associated to g¯0under the adjoint action. /square
+Corollary 2.4. We have
+(2.3) chK(λ) =/producttext
+α∈Φ+
+¯1(1+e−α)
+/producttext
+α∈Φ+
+¯0(1−e−α)/summationdisplay
+σ∈Wsgn(σ)eσ(λ+ρ)−ρ.DUALITIES FOR LIE SUPERALGEBRAS 13
+Proof.By (2.2), ch K(λ) = chL0(λ)/producttext
+α∈Φ+
+¯1(1+e−α). Now by Weyl’s character formula
+forL0(λ), we have
+chK(λ) =/producttext
+α∈Φ+
+¯1(1+e−α)
+/producttext
+α∈Φ+
+¯0(1−e−α)/summationdisplay
+σ∈Wsgn(σ)eσ(λ+ρ0)−ρ0.
+This is equivalent to the formula in (2.3) by the second part o f Lemma 2.3. /square
+Exercise 2.5. Show that
+ρ1=n
+2m/summationdisplay
+i=1δi−m
+2n/summationdisplay
+j=1ǫj.
+Again it follows that σ(ρ1) =ρ1,∀σ∈W.
+Definition 2.6. A weight λ∈h∗is called typical, if/producttext
+α∈Φ+
+¯1(λ+ρ,α)/\e}atio\slash= 0. It is called
+atypicalif it is not typical.
+Theorem 2.7 (Kac).Letλ∈P+. Thenλis typical if and only if ch L(λ)is given by
+the right hand side of (2.3), if and only if K(λ)is irreducible.
+The example of gl(1|1) in Section 2.2 fits well with this theorem.
+Sketch of a proof. [K2] Let vλ∈K(λ) be a highest weight vector. Denote by eα,fαthe
+generators of g±α, whereα∈Φ+.
+Assume that Sis a nonzero g-submodule of K(λ). One first shows easily that
+(i)/producttext
+α∈Φ+
+¯1fαvλ∈S.
+It follows that
+(ii)/producttext
+β∈Φ+
+¯1eβ/producttext
+α∈Φ+
+¯1fαvλ∈S.
+Then one further shows that (up to a nonzero scalar multiple)
+(iii)/producttext
+β∈Φ+
+¯1eβ/producttext
+α∈Φ+
+¯1fαvλ=/producttext
+α∈Φ+
+¯1(λ+ρ,α)vλ.
+Step(iii) uses the W-invarianceof Φ+
+¯1and adegreecounting argument amongothers.
+From (i) and (iii) it follows that if λis atypical, then K(λ) is irreducible.
+Now suppose that K(λ) is irreducible. The ad g0-invariance of g±1implies that if
+vλ∈U(g)/producttext
+α∈Φ+
+¯1fαvλ, thenvλis a scalar multiple of/producttext
+β∈Φ+
+¯1eβ/producttext
+α∈Φ+
+¯1fαvλ. Thus by
+(iii)λis typical. /square
+2.5.Odd reflections. As usual, we let {eβ,hβ,fβ}denote the Chevalley generators
+forβ∈Φ+. The following simple and fundamental lemma for odd reflecti ons has been
+used by many authors (e.g. [LSS, Appendix], [PS, Lemma 0.3] a nd [KW, Lemma 1.4];
+also compare [DP, (2.12)] for an unconventional definition) . Recall the new Borel b(α)
+for a simple isotropic odd root αfrom Lemma 1.7.
+Lemma 2.8. LetLbe a simple g-module of b-highest weight λand letvbe ab-highest
+weight vector of L. Letαbe a simple isotropic odd root.
+(1)If/a\}bracketle{tλ,hα/a\}bracketri}ht= 0, thenLis ag-module of b(α)-highest weight λandvis ab(α)-
+highest weight vector.
+(2)If/a\}bracketle{tλ,hα/a\}bracketri}ht /\e}atio\slash= 0, thenLis ag-module of b(α)-highest weight λ−αandfαvis a
+b(α)-highest weight vector.14 SHUN-JEN CHENG AND WEIQIANG WANG
+Proof.We first observe three simple identities:
+(i)eαfαv= [eα,fα]v=hαv=/a\}bracketle{tλ,hα/a\}bracketri}htv.
+(ii)eβfαv= [eβ,fα]v= 0 for any β∈Φ+∩Φ(α)+, since either β−αis not a root
+or it belongs to Φ+∩Φ(α)+.
+(iii)f2
+αv= 0, since αis an isotropic odd root.
+Now, we consider the two cases separately.
+(1) Assume that /a\}bracketle{tλ,hα/a\}bracketri}ht= 0. Then we must have fαv= 0, otherwise fαvwould be a
+b-singular vector in the simple g-module Vby (i) and (ii). This together with Lemma
+1.7 implies that vis ab(α)-highest weight vector of weight λin theg-module V.
+(2) Assume that /a\}bracketle{tλ,hα/a\}bracketri}ht /\e}atio\slash= 0. Then (i), (ii), (iii) and Lemma 1.7 imply that fαvis
+nonzero and it is a b(α)-highest weight vector of weight λ−αinV. /square
+2.6.Finite-dimensional simple osp-modules. The case of g=spo(2m|2n+1) will
+be treated in detail (while the case of g=spo(2m|2n) is similar). As usual, we have
+the triangular decomposition of g(with respect to the standard Borel) g=n−⊕h⊕n+,
+which allows us to define the Verma module ∆( λ) associated to λ=/summationtextm
+i=1λiδi+/summationtextn
+j=1λjǫj∈h∗. Then ∆( λ) admits a unique simple quotient g-module, denoted by
+L(λ).
+As for Lie algebras, a finite-dimensional simple g-module has to be a highest weight
+module (with respect to any Borel), and hence is isomorphic t o someL(λ), andL(λ)/\e}atio\slash∼=
+L(µ) ifλ/\e}atio\slash=µ. However, the classification of finite-dimensional simple g-modules is
+non-trivial, partly because the even subalgebra of gis not a Levi subalgebra. Clearly
+a necessary condition for L(λ) to be finite-dimensional is that λis dominant integral
+with respect to the even subalgebra g¯0=sp(2m)⊕so(2n+1).
+Definition 2.9. A partition µ= (µ1,µ2,...), or simply a hook partition whenm,nare
+implicitly understood, is called an ( m|n)-hook partition ifµm+1≤n.
+µ
+mn
+Given an ( m|n)-hook partition µ, we denote by µ+= (µm+1,µm+2,...) and write its
+transpose, which is necessarily of length ≤n, asν= (µ+)′= (ν1,...,νn). We define
+the weights
+µ♮=µ1δ1+...+µmδm+ν1ǫ1+...+νn−1ǫn−1+νnǫn (2.4)
+µ♮
+−=µ1δ1+...+µmδm+ν1ǫ1+...+νn−1ǫn−1−νnǫn.
+(µ♮
+−is only used for spo(2m|2n) below).DUALITIES FOR LIE SUPERALGEBRAS 15
+Theorem 2.10. Given any (m|n)-hook partition µ, the simple spo(2m|2n+1)-module
+L(µ♮)(with respect to the standard Borel) is finite-dimensional. Moreover, these mod-
+ules form a complete list of non-isomorphic finite-dimensio nal simple spo(2m|2n+1)-
+modules.
+The above theorem is due to Kac [K2] who formulated the condit ions for finite-
+dimensional highest weight simple g-modules in terms of Dynkin labels (instead of hook
+partitions). A different proof using odd reflections is given b y Shu and Wang (which
+also works in characteristic p >0). Let us sketch the idea of a proof following [SW],
+as the argument therein bears some similarity to the argumen t used later on for super
+duality. The same type of argument works for Theorem 2.12 bel ow forspo(2m|2n).
+Sketch of a proof. Letµbe an (m|n)-hook partition. We observe that the simple
+spo(2M|2n+ 1)-module Vof highest weight µ(with respect to the standard Borel)
+is finite-dimensional for M≥ℓ(µ), since it appears as a subquotient of a suitable ten-
+sor product of simple modules of fundamental weights. Then w e use odd reflections to
+change the standard Borel of spo(2M|2n+1) to a Borel which is compatible with the
+standard Borel of spo(2m|2n+1) (which is regarded as a subalgebra of spo(2M|2n+1)).
+We can show that the highest weight of Vwith respect to the new Borel is µ♮, hence
+by restriction to spo(2m|2n+1) we have proved the first part of the theorem.
+A highest weight for a simple finite-dimensional g-module is necessarily of the form
+µ1δ1+...+µmδm+ν1ǫ1+...+νnǫn, where(µ1,...,µ m)and(ν1,...,νn)arepartitionsby
+the dominance condition on the even subalgebra g¯0. To prove the remaining condition
+ν′
+1≤µm, itsufficestoproveitfor spo(2|2n+1) (whichisasubalgebraof spo(2m|2n+1)).
+LetVbe a finite-dimensional simple spo(2|2n+ 1)-module. Via the sequence of odd
+reflections δ1−ǫ1,δ1−ǫ2,...,δ1−ǫnwe change the standard Borel to a Borel with
+an odd non-isotropic simple root δ1. The dominance condition on the simple root δ1
+(or rather on the even root 2 δ1), which is imposed by the finite-dimensionality of V,
+provides the desired necessary condition. /square
+Exercise 2.11. Complete the details in the proof of Theorem 2.10.
+Similarly, we classify the finite-dimensional simple spo(2m|2n)-modules.
+Theorem 2.12. [K2](cf.[SW]) The modules L(µ♮)andL(µ♮
+−)(with respect to the
+standard Borel), where µruns over all (m|n)-hook partitions, are all the non-isomorphic
+finite-dimensional simple spo(2m|2n)-modules.
+Sinceg¯0is not a Levi subalgebra of g, the above definition of the Kac module for
+gl(m|n)-module is no longer valid for g. Nevertheless, the notion of “typical weights”
+in Definition 2.6 still makes sense for osp(or any basic classical Lie superalgebra). The
+following is the ospcounterpart of Theorem 2.7, whose much more involved proof w ill
+be skipped.
+Theorem 2.13. [K2]Letg=osp(ℓ|2n), forℓ= 2mor2m+1. If the weight λ=µ♮∈h∗
+(and in addition λ=µ♮
+−whenℓ= 2m) for an (m|n)-hook partition µis typical, then
+chL(λ)is given by the right hand side of (2.3).16 SHUN-JEN CHENG AND WEIQIANG WANG
+Exercise 2.14. Recall that so(ℓ) may be identified with ∧2(Cℓ), and more generally,
+osp(ℓ|2n) may be identified with ∧2(Cℓ|2n) =∧2(Cℓ)⊕(Cℓ⊗C2n)⊕S2(C2n). Now
+using the invariant bilinear form to identify Cℓ|2nwithCℓ|0⊕C0|n⊕C0|n∗, we define
+aZ-gradation on Cℓ|2nby setting deg v= 0 for v∈Cℓ|0, degv= 1 for v∈C0|n,
+and degv=−1 forv∈C0|n∗. This induces a Z-gradation on ∧2(Cℓ|2n) and hence on
+osp(ℓ|2n). Prove that
+osp(ℓ|2n) =2/circleplusdisplay
+i=−2osp(ℓ|2n)i,
+whereosp(ℓ|2n)0∼=so(ℓ)⊕gl(n); asosp(ℓ|2n)0-modules we have osp(ℓ|2n)1∼=Cℓ⊗Cn
+andosp(ℓ|2n)2∼=C⊗S2(Cn).
+3.Schur-Sergeev duality
+3.1.The formulation. Letg=gl(m|n). Let{ei|i∈I(m|n)}be the standard basis
+for the natural g-module V=Cm|n.
+ThenV⊗dis naturally a g-module by letting
+Φd(g).(v1⊗v2⊗...⊗vd) =g.v1⊗···⊗vd+(−1)|g|·|v1|v1⊗g.v2⊗···⊗vd
++...+(−1)|g|·(|v1|+...+|vd−1|)v1⊗v2⊗···⊗g.vd,
+whereg∈gandvi∈Vare assumed to be Z2-homogeneous.
+On the other hand, the action of the symmetric group SdonV⊗dis determined by
+Ψd((i,i+1)).v1⊗···vi⊗vi+1⊗···⊗vd
+= (−1)|vi|·|vi+1|v1⊗···vi+1⊗vi⊗···⊗vd,1≤i≤d−1,
+where (i,j) denotes a transposition in Sdandvi,vi+1areZ2-homogeneous.
+Lemma 3.1. The actions of (g,Φd)and(Sd,Ψd)onV⊗dcommute with each other.
+Symbolically, we write
+gΦd/archriðhtdownV⊗dΨd/archleftdownSd.
+Exercise 3.2. Verify Lemma 3.1.
+We write λ⊢dfor a partition λofd. Denote by P(d,m|n) the set of all ( m|n)-hook
+partitions of size dand let
+P(m|n) =∪d≥0P(d,m|n) ={λ|λare partitions with λm+1≤n}.
+Also set P(d,m) =P(d,m|0). We denote by L(λ♮) forλ∈P(m|n) the simple g-module
+of highest weight λ♮with respect to the standard Borel subalgebra. For a partiti on
+λ⊢d, we denote by Sλthe Specht module of Sd. For example, S(d)is the trivial
+module, and S(1d)is the sign module sgn. The following superalgebra analog of Schur
+duality was due to Sergeev [Sv1] (also cf. [BeR]).DUALITIES FOR LIE SUPERALGEBRAS 17
+Theorem 3.3 (Schur-Sergeev duality) .(1)The images ΦdandΨd,Φd(U(g))and
+Ψd(CSd), satisfy the double centralizer property, i.e.
+Φd(U(g)) =EndSd(V⊗d),
+EndU(g)(V⊗d) = Ψd(CSd).
+(2)AsU(gl(m|n))⊗CSd-module, we have
+V⊗d∼=/circleplusdisplay
+λ∈P(d,m|n)L(λ♮)⊗Sλ.
+We refer to a U(gl(m|n))⊗CSd-module as a ( gl(m|n),Sd)-module. Similar conven-
+tions also apply to similar setups below.
+Remark 3.4.(1) Forn= 0, the above Sergeev duality reduces to the usual Schur
+duality. Ifinaddition d= 2, (Cm)⊗2=S2(Cm)⊕∧2(Cm).Thisfitswell withthe
+well-known fact that as gl(m)-modules S2(Cm) and∧2(Cm) are, respectively,
+irreducibleof highestweights 2Λ 1andΛ2, whereΛ idenotesthe ithfundamental
+weight.
+(2) Form= 0,λ♮=λ′(the conjugate partition), and Sλ=Sλ′⊗sgn. In this case,
+the Sergeev duality reduces to the version of Schur duality t wisted by the sign
+representation of Sd, i.e., as ( gl(m),Sd)-module,
+(Cm)⊗d∼=/circleplusdisplay
+µ∈P(d,m)L(µ)⊗Sµ′.
+(3) Ifd≤mn+m+n, thenP(d,m|n) ={λ⊢d}, and every simple Sd-module
+appears in the Sergeev duality decomposition.
+(4) Ford= 2, the Sergeev duality reduces to the decomposition ( Cm|n)⊗2=
+S2(Cm|n)⊕ ∧2(Cm|n), where S2and∧2are understood in the super sense.
+In particular, as an ordinary vector spaces,
+S2(Cm|n) =S2(Cm)⊕(Cm⊗Cn)⊕∧2(Cn).
+3.2.Proof of Theorem 3.3. SetA:= Φd(U(g)) andB:= Ψ(CSd). It is clear that
+A⊂EndSd(V⊗d) = (End( V⊗d))Sd∼=Sd(EndV),
+whereSd(−) denotes the dth symmetric tensor. With extra work, one proves that
+Sd(EndV)⊂A(the superalgebra generalization does not cause any extra d ifficulty).
+Hence Φ d(U(g)) = End Sd(V⊗d).
+SinceB=CSdis a semisimple algebra, it follows (cf. [GW, Theorem 3.3.7] ) that
+EndU(g)(V⊗d) = Ψd(CSd).This proves (1).
+LetW=V⊗d. It follows from the double centralizer property and the sem isimplicity
+ofCSdthat we have a multiplicity-free decomposition of the ( gl(m|n),Sd)-module W:
+W≡V⊗d∼=/circleplusdisplay
+λ∈P(d,m|n)L[λ]⊗Sλ,18 SHUN-JEN CHENG AND WEIQIANG WANG
+whereL[λ]is some simple gl(m|n)-module associated to λ, whose highest weight (with
+respect to the standard Borel) is to be determined. Also to be determined is the index
+setP(d,m|n) ={λ⊢d|L[λ]/\e}atio\slash= 0}.
+First we need to prepare some notations.
+Let Cp(m|n) be the set of pairs ν|µof compositions ν= (ν1,...,νm) of length ≤m
+andµ= (µ1,...,µ n) of length ≤n, and let
+Cp(d,m|n) ={ν|µ∈Cp(m|n)|/summationdisplay
+iνi+/summationdisplay
+jµj=d}.
+We have the following weight space decomposition (with resp ect to the Cartan subal-
+gbra of diagonal matrices h⊂gl(m|n)):
+W=/circleplusdisplay
+ν|µ∈Cp(d,m|n)Wν|µ,
+whereWν|µhas a linear basis ei1⊗...⊗eid, with the indices satisfying the following
+equality of sets:
+(3.1) {i1,...,id}={1,...,1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+ν1,...,m,...,m/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+νm;1,...,1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+µ1,...,n,...,n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+µn}.
+LetSν|µ:=Sν1×...×Sνm×Sµ1×...×Sµn. Thespanof thevector eν|µ:=e⊗ν1
+1⊗...⊗
+e⊗νm
+m⊗e⊗µ1
+1⊗...⊗e⊗µnncan be identified with the Sν|µ-module 1 ν⊗sgnµ. SinceSdeν|µ
+spansWν|µwe have a surjective Sd-homomorphism from IndSd
+Sν|µ(1ν⊗sgnµ) ontoWν|µ
+by Frobenius Reciprocity. By counting the dimensions we hav e anSd-isomorphism:
+Wν|µ∼=IndSd
+Sν|µ(1ν⊗sgnµ).
+Let us denote the decomposition of Wν|µinto irreducibles by
+Wν|µ=/circleplusdisplay
+λKλ,ν|µSλ,forKλ,ν|µ∈Z+.
+Letλbe a partition which is identified with its Young diagram. Rec allI(m|n) is
+totally ordered by (1.2). A hook tableau Tof shape λ, or anhookλ-tableau T, is an
+assignment of an element in I(m|n) to each box of the Young diagram λsatisfying the
+following conditions:
+(1) The numbers are weakly increasing along each row and colu mn.
+(2) The numbers from {1,...,m}are strictly increasing along each column.
+(3) The numbers from {1,...,n}are strictly increasing along each row.
+Such aTis said to have contentν|µ∈Cp(m|n) if¯i∈I(m|0) appears νitimes and
+j∈I(0|n) appears µjtimes. Denote by HT(λ,ν|µ) the set of hook λ-tableaux of
+contentν|µ.
+Lemma 3.5. We have Kλ,ν|µ= #HT(λ,ν|µ).
+Proof.Recall that IndSd
+Sν|µ(1ν⊗sgnµ)∼=/circleplustext
+λKλ,ν|µSλ.
+First assume that µ=∅, and we prove the formula by induction on the length
+r=ℓ(ν). A hook (=semistandard) tableau Tof shape λand content νgives rise to aDUALITIES FOR LIE SUPERALGEBRAS 19
+sequence of partitions ∅=λ0⊂λ1⊂...⊂λr=λsuch that λihas the shape given
+by the parts of Twith entries ≤i, andλi/λi−1hasνiboxes for each i. This sets
+up a bijection between HT(λ,ν) and the set of such sequences of partitions. Denote
+d1=d−νrand/tildewideν= (ν1,...,νr−1). We have IndSd1
+S/tildewideν1S˜ν∼=/circleplustext
+ρ⊢d1Kρ,/tildewideνSρ, where
+Kρ,/tildewideν= #HT(ρ,/tildewideν) by induction hyperthesis. Now the induction step is simply the
+Piere’s rule (c.f. [Mac, Chapter 1 (5.16)]): for a partition ρ⊢d1,
+IndSd
+Sd1×Sνr(Sρ⊗1νr)∼=/circleplusdisplay
+λSλ,
+whereλis such that λ/ρis a horizontal strip of νrboxes.
+Then using the above special case as the initial step, we comp lete the proof in the
+general case by induction on the length of µ, in which the induction step is exactly the
+conjugated Piere’s rule. /square
+Lemma 3.6. Letλ⊢dandν|µ∈Cp(d,m|n). ThenKλ,ν|µ= 0unlessλ∈P(d,m|n).
+Proof.By the identity Kλ,ν|µ= #HT(λ,ν|µ), it suffices to prove that if a hook λ-
+tableauTof content ν|µexists, then λm+1≤n.
+By applying the hook tableau condition (2) to the first column ofT, we see that
+the first entry k∈I(m|n) in row ( m+1) satisfies k >0. Applying the hook tableau
+condition (3) to the ( m+1)st row, we conclude that λm+1≤n. /square
+Lemma 3.6 implies that P(d,m|n)⊂P(d,m|n). On the other hand, given λ∈
+P(d,m|n), clearly a hook λ-tableau exists, e.g., we can fill in the numbers 1,...,mon
+the first mrows ofλrow by row downward, and then for the (possibly) remaining
+rows ofλ, we fill in the numbers 1 ,...,ncolumn by column from left to right. This
+distinguished λ-tableau will be denoted by Tλ. Hence, we have proved that
+P(d,m|n) =P(d,m|n).
+For a given λ∈P(d,m|n), we have L[λ]=/circleplustext
+ν|µ∈Cp(d,m|n)L[λ]
+ν|µ. Among all the
+contents of hook λ-tableaux, the one for Tλcorresponds to a highest weight (by the
+three hook tableau conditions). Hence, we conclude that L[λ]=L(λ♮), the simple
+g-module of highest weight λ♮. This completes the proof of Theorem 3.3.
+3.3. Clearly, the character of L(λ♮)
+chL(λ♮) =/summationdisplay
+ν|µ∈Cp(m|n)dimL(λ♮)ν|µ/productdisplay
+i,jxνi
+iyµj
+j
+is a polynomial which is symmetric in x= (x1,...,x m) andy= (y1,...,yn), respec-
+tively. Denote by mν(x) the monomial symmetric polynomial associated to a partiti on
+ν.
+The following can be read off from the above proof of Theorem 3. 3.
+Corollary 3.7. The character of L(λ♮)is given by:
+chL(λ♮) =/summationdisplay
+ν|µ∈Cp(d,m|n)#HT(λ,ν|µ)mν(x)mµ(y),
+whereνandµabove are partitions.20 SHUN-JEN CHENG AND WEIQIANG WANG
+Remark 3.8.The standard basis vectors for S2(Cm|n) areei⊗ej+(−1)|i||j|ej⊗ei, where
+i,j∈I(m|n) satisfy i≤j <0,i <0< j, or 0< i < j(cf. Remark 3.4 (4)). They are
+in bijection with the hook tableaux of shape λ= (2):
+ij
+This is compatible with the identification S2(Cm|n)∼=L(gl(m|n),2δ1).
+Remark3.9.We sketch below a morestandardargument for thestandardSch ur duality
+on (Cn)⊗d, which emphasizes the decomposition of Was ag-module.
+Given a composition (or a partition) µofdof length ≤n, we denote by Wµthe
+µ-weight space of the gl(n)-module. Clearly Wµhas a basis
+(3.2) ei1⊗...⊗eid,where{i1,...,id}={1,...,1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+µ1,...,n,...,n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+µn}.
+As a (gl(n),Sd)-module,
+W∼=/circleplusdisplay
+λL(λ)⊗Uλ,
+whereUλ:= Hom gl(n)(L(λ),W)∼=Wn+
+λ(the space of highest weight vectors in Wof
+weightλ). Onlyλ∈P(d,n) can be as highest weights for gl(n) which appears in the
+decomposition of ( Cn)⊗d, and every such λindeed appears as it appears as a summand
+of∧λ′
+1(Cn)⊗∧λ′
+2(Cn)⊗···.
+Note that P(d,n) has two interpretations: one as the polynomial weights for gl(n)
+and the other as compositions of d. A remarkable fact is that the partial order on
+weights induced by the positive roots of gl(n) coincides with the dominance partial
+order/trianglerightequalon compositions.
+SinceW=⊕µWµ=⊕µ,λ:λ/trianglerightequalµL(λ)µ⊗Uλ, we conclude that, as Sd-module,
+Wµ∼=/circleplusdisplay
+λ/trianglerightequalµdimL(λ)µUλ.
+On the other hand, Sdacts on the basis of Wµ(3.2) transitively and the stabilizer of
+the basis element e⊗µ1
+1⊗...⊗e⊗µnnis the Young subgroup Sµ=Sµ1×...×Sµn.
+Therefore we have
+Wµ∼=IndSd
+Sµ1µ=/circleplusdisplay
+λ/trianglerightequalµKλµSλ,
+whereKλµis the Kostka number which satisfies Kλλ= 1.
+By thedoublecentralizer property, Uλhasto beanirreducible Sd-moduleforeach λ.
+We compare the above two interpretations of Wµin thespecial case when µis dominant
+(i.e., a partition). One by one downward along the dominance order, this provides the
+identification Uµ=Sµfor every µ, and moreover, we obtain the well-known equality
+dimL(λ)µ=Kλµ.
+Definition 3.10. Letµ∈h∗. Thedegree of atypicality ofµis the maximum cardinality
+of a set of pairwise orthogonal α∈Φ+
+¯1such that ( µ+ρ,α) = 0.
+The following observation seems to be new.DUALITIES FOR LIE SUPERALGEBRAS 21
+Proposition 3.11. Letλ∈P(m|n). The degree of atypicality of λ♮equals the minimal
+numberisuch that λcontains a partition of rectangular shape ((m−i)n−i)with0≤
+i≤min{m,n}.
+As a corollary, λ♮is typical if and only if λm≥n, as observed earlier in [BeR].
+Exercise 3.12. Prove Proposition 3.11.
+Remark 3.13.There exists another interesting generalization of Schur d uality for the
+queer Lie superalgebras due to Sergeev [Sv1].
+3.4. Assume that the ǫδ-sequence associated to a Borel subalgebra bofgl(m|n) (cf.
+Section 1.5) is given by a sequence of d1δ’s,e1ǫ’s,d2δ’s,e2ǫ’s,...,drδ’s,erǫ’s
+(possibly d1= 0 orer= 0). Associated to an ( m|n)-hook Young diagram λ, we define
+a weight λb∈h∗as follows. Take the first d1rownumbers of λas the coefficients of the
+firstd1δ’s. Denote by λ1the Young diagram obtained from λwith the first d1rows
+ofλremoved. Take the first e1columnnumbers of λ1as the coefficients of the first e1
+ǫ’s. Denote by λ2the Young diagram obtained from λ1with the first e1columns of λ1
+removed. Then take the first d2rownumbers of λ2as the coefficients of the following
+d2δ’s, and so on, until we reach the empty partition. The resulti ng weight is denoted
+byλb. Below is an example of λbford1=e1= 2 and d2= 1.
+λb
+1
+λb
+2
+λb
+5λb
+4λb
+3
+Recall that by convention L(λ♮) is a highest weight g-module with respect to the
+standard Borel subalgebra bst.
+Theorem 3.14. [CLW]Letλbe an(m|n)-hook partition. Let bbe an arbitrary Borel
+subalgebra of gl(m|n). Then, the b-highest weight of the simple gl(m|n)-module L(λ♮)
+isλb.
+Proof.Let us consider an odd reflection which changes a Borel subalg ebrab1tob2.
+Assume the theorem holds for b1. We observe by Lemma 2.8 that the statement of
+the theorem for b2follows from the validity of the theorem for b1. The statement of
+the theorem is apparently consistent with a change of Borel s ubalgebras induced from
+a real reflection, and all Borel subalgebras of gl(m|n) are linked by a sequence of real
+and odd reflections. Hence, once we know the theorem holds for one particular Borel
+subalgebra, it holds for all. We finally note that the theorem holds for the standard
+Borel subalgebra bst, which corresponds to the sequence of m δ’s followed by n ǫ’s. It
+is clear that λbst=λ♮. /square
+Remark 3.15.A variant of Theorem 3.14 holds for Lie superalgebras of type osp, see
+[CLW].22 SHUN-JEN CHENG AND WEIQIANG WANG
+Example 3.16. Let us describe the highest weights in Theorem 3.14 with resp ect to
+the three Borel subalgebras of special interest.
+(1) As seen above, λbst=λ♮.
+(2) If we take the opposite Borel subalgebra bopcorresponding to a sequence of
+n ǫ’s followed by m δ’s, then λbop= (λ′)♮, where ♮applies to a ( n|m)-hook
+partition (instead of ( m|n)-hook partition). This is as expected from Schur-
+Sergeev duality Theorem 3.3.
+(3) In case when |m−n| ≤1, we may take a Borel subalgebra bowhose simple
+roots are all odd (or equivalently, the corresponding ǫδ- sequence is alternating
+between ǫandδ):
+/circlemultiplytext /circlemultiplytext /circlemultiplytext /circlemultiplytext /circlemultiplytext···
+In this case Theorem 3.14 reduces to [CW3, Theorem 7.1], whic h states the
+coefficients of δandǫinλboare given by the modified Frobenius coordinates
+(pi|qi)i≥1of the partition λ(respectively, λ′), when the first simple root is of
+the form δ−ǫ(respectively, ǫ−δ). Here by modified Frobenius coordinates we
+mean
+pi= max{λi−i+1,0}, qi= max{λ′
+i−i,0}
+so that/summationtext
+i(pi+qi) =|λ|; “modified” here refers to a shift by 1 from the pi
+coordinates defined in [Mac, Chapter 1, Page 3].
+Forλ= (7,5,4,3,1), we have ( p1,p2,p3|q1,q2,q3) = (7,4,2|4,2,1).
+7
+4
+2
+4 2 1
+Exercise 3.17. Letλ= (7,2,2,1,1) be an (1 |2)-hook partition (i.e. λ2≤2). Write
+down all the extremal weights with respect to the six Borel su balgebras of gl(1|2) from
+Example 1.8 for L(gl(1|2),λ♮).
+4.Howe duality for Lie superalgebras of type gl
+4.1.The(gl(m|n),gl(d))-Howe duality. LetV=V¯0⊕V¯1be a vector superspace.
+The symmetric algebra S(V) is understood as S(V) =S(V¯0)⊗∧(V¯1). It follows that
+Sk(V) =/circleplustextk
+ℓ=0Sℓ(V¯0)⊗∧k−ℓ(V¯1).
+The commuting action of gl(m|n) andgl(d) on the super space Cd⊗Cm|ninduces a
+corresponding commuting action on its symmetric algebra S(Cd⊗Cm|n).
+Theorem 4.1. [H1, CW1, Sv2]DUALITIES FOR LIE SUPERALGEBRAS 23
+(1)The images of U(gl(d))andU(gl(m|n))in End/parenleftbig
+S(Cd⊗Cm|n)/parenrightbig
+satisfy the double
+centralizer property.
+(2)As a(gl(d),gl(m|n))-module,
+S(Cd⊗Cm|n) =/summationdisplay
+λ∈P(m|n),ℓ(λ)≤dLd(λ)⊗Lm|n(λ♮).
+Here and below we use subscripts m|nanddto indicate the (super)algebras
+under consideration.
+The part (2) of the theorem is equivalent to
+Sk(Cd⊗Cm|n) =/summationdisplay
+λ⊢k
+λm+1≤n,ℓ(λ)≤dLd(λ)⊗Lm|n(λ♮), k∈N.
+Proof.Let us prove the equivalent version for Sk(Cm|n⊗Cd) for (2). The argument
+below is well-known (see [H2]).
+By the definition of the k-th supersymmetric algebra we have
+Sk(Cd⊗Cm|n)∼=/parenleftig
+(Cd)⊗k⊗(Cm|n)⊗k/parenrightig∆k,
+where ∆ kis the diagonal subgroup of Sk×Skand (−)∆kdenotes the ∆ k-invariant
+subspace. By Theorem 3.3 we have therefore
+Sk(Cd⊗Cm|n)∼=/parenleftig
+(/summationdisplay
+|µ|=kLd(µ)⊗Sµ)⊗(/summationdisplay
+|λ|=kLm|n(λ♮)⊗Sλ)/parenrightig∆k
+∼=/summationdisplay
+|λ|=|µ|=k/parenleftig
+Ld(µ)⊗Lm|n(λ♮)/parenrightig
+⊗(Sµ⊗Sλ)∆k
+∼=/summationdisplay
+|λ|=kLd(λ)⊗Lm|n(λ♮).
+The last equality follows since the Specht module Sλis self-contragredient.
+Part (1) follows from the decomposition in (2). /square
+Remark 4.2.In casen= 0, we have λ♮=λ, and Theorem 4.1 reduces to the classical
+(gl(d),gl(m))-Howe duality on a symmetric tensor space (see [H2, Sectio n 2.1.2]).
+Form= 0, we have λ♮=λ′, and Theorem 4.1 reduces to the classical ( gl(d),gl(n))-
+Howe duality on an exterior space ∧(Cd⊗Cn) (see [H2, Section 4.1.1]).
+Remark 4.3.Similarly, using Sergeev’s duality for queer Lie superalge braq(n) (see
+Remark 3.13), we can establish a ( q(m),q(n))-Howe duality, see [CW2, Naz, Sv2].
+4.2.Hook Schur polynomials as gl(m|n)-characters. Letx= (x1,...,x m) and
+y= (y1,...,yn) be formal variables. For two partitions λandµwithµ⊆λdenote by
+sλ/µ(x) theskew Schurpolynomial associated to theskew Youngdiag ramλ/µsuch that
+ℓ(λ)≤m. The skew Schur polynomials specialize to the (usual) Schur polynomials for
+µ=∅:sλ/∅(x) =sλ(x). We refer to Macdonald [Mac] for more on symmetric function s.24 SHUN-JEN CHENG AND WEIQIANG WANG
+Thehook Schur polynomials hsλ(x,y) in variables xandyare defined as
+hsλ(x,y) =/summationdisplay
+µ⊆λsµ(x)sλ′/µ′(y), λ∈P(m|n). (4.1)
+This is one of the several equivalent definitions, see [BeR]. We should compare this
+definition with the classical identity for (skew) Schur func tions
+(4.2) sλ(x,y) =/summationdisplay
+µ⊆λsµ(x)sλ/µ(y).
+The character of the gl(m|n)-module Lm|n(λ♮) is by definition the trace of the action
+of the diagonal matrix diag( x1,...,x m;y1,...,yn) onLm|n(λ♮). The following theorem
+was obtained in [BeR], where the notion of hook Schur polynom ials was formulated,
+and it is actually equivalent to the combinatorial formula g iven in Corollary 3.7. We
+offer a different proof below using Howe duality (cf. [CL1]).
+Theorem 4.4. Letλ∈P(m|n).The character of Lm|n(λ♮)is given by
+chLm|n(λ♮) =hsλ(x,y).
+Proof.Letu= (u1,...,u d) be another set of formal variables. It is well known that
+sλ(u) is the character of the gl(d)-module Ld(λ), i.e. the trace of the diagonal matrix
+diag(u1,...,u d). Thus, comparing the characters of both sides of Theorem 4. 1 (2), we
+obtain the following combinatorial identity:
+(4.3)/summationdisplay
+λ∈P(m|n),ℓ(λ)≤dsλ(u)chLm|n(λ♮) =/productdisplay
+i≤m,j≤n,k≤d(1−xiuk)−1(1+yjuk).
+LetX= (x1,x2,...)andY= (y1,y2,...)denotetwosets ofinfinitelymanyvariables.
+Recall the classical Cauchy identity
+(4.4)/summationdisplay
+ℓ(λ)≤dsλ(u)sλ(X) =/productdisplay
+k≤d/productdisplay
+i≥1(1−xiuk)−1,
+which readily follows from Howe duality Theorem 4.1 (by sett ingn= 0 and letting
+m→ ∞).
+Consider the involution ωon the ring of symmetric functions in the set of variables
+(xm+1,xm+2,...). Applying ωto (4.4), replacing ( xm+1,xm+2,...) byY, and using
+(4.2), we obtain
+/summationdisplay
+ℓ(λ)≤dsλ(u)hsλ(x,Y) =/productdisplay
+i≤m,k≤d,j≥1(1−xiuk)−1(1+yjuk).
+Now setting yn+1=yn+2=...= 0 in the above identity, we obtain that
+/summationdisplay
+ℓ(λ)≤dsλ(u)hsλ(x,y) =/productdisplay
+i≤m,j≤n,k≤d(1−xiuk)−1(1+yjuk).
+By comparing with (4.3) and noting the linear independence o f{sλ(u)}ford=∞, we
+have proved the theorem. /squareDUALITIES FOR LIE SUPERALGEBRAS 25
+Remark 4.5.It follows that
+hsλ(X,∅) =sλ(X), hsλ(∅,Y) =sλ′(Y), hsλ(Y,X) =hsλ′(X,Y).
+Both the usual Cauchy identity and its dual version are obtai ned from specializations
+of the corresponding character identity of Theorem 4.1.
+4.3.Formulas for highest weight vectors. We lete1,...,eddenote the standard
+basisforthenatural gl(d)-module Cd, andrecallthat ei,i∈I(m|n),denotethestandard
+basis for the natural gl(m|n)-module Cm|n. We set
+(4.5) xi
+a:=ei⊗e¯a(1≤a≤m);ηi
+b:=ei⊗eb(1≤b≤n).
+We will denote by C[x,η] the polynomial superalgebra generated by (4.5). The
+commuting actions on C[x,η] ofgl(d) andgl(m|n) may be realized in terms of first
+order differential operators (4.6) and (4.7) respectively (1 ≤i,i′≤dand 1≤s,s′≤
+m;1≤k,k′≤n):
+Eii′:=m/summationdisplay
+j=1xi
+j∂
+∂xi′
+j+n/summationdisplay
+j=1ηi
+j∂
+∂ηi′
+j, (4.6)
+d/summationdisplay
+j=1xj
+s∂
+∂xj
+s′,d/summationdisplay
+j=1ηj
+k∂
+∂ηj
+k′,d/summationdisplay
+j=1xj
+s∂
+∂ηj
+k,d/summationdisplay
+j=1ηj
+k∂
+∂xj
+s. (4.7)
+In particular, as ( gl(d),gl(m|n))-module, S(Cp⊗Cm|n) is isomorphic to C[x,η]. The
+root vectors corresponding to the simple roots of gl(d) andgl(m|n) are
+m/summationdisplay
+j=1xi−1
+j∂
+∂xi
+j+n/summationdisplay
+j=1ηi−1
+j∂
+∂ηi
+j, (4.8)
+d/summationdisplay
+j=1xj
+s−1∂
+∂xj
+s,d/summationdisplay
+j=1ηj
+k−1∂
+∂ηj
+k,d/summationdisplay
+j=1xj
+m∂
+∂ηj
+1, (4.9)
+respectively.
+Letλbe an (m|n)-hook partition of size kand of length at most d. We are looking
+for the joint highest weight vector (with respect to the stan dard Borel subalgebras), or
+equivalently the vector annihilated by (4.8) and (4.9), for the highest weight module
+Ld(λ)⊗Lm|n(λ♮) ofgl(d)×gl(m|n) appearing in the decomposition of C[x,η]. Such a
+vector is unique up to a scalar multiple, thanks to the multip licity-free decomposition
+in Part 2 of Theorem 4.1.
+For 1≤r≤m, define
+(4.10) ♦r:= det
+x1
+1x1
+2···x1
+r
+x2
+1x2
+2···x2
+r............
+xr
+1xr
+2···xr
+r
+.
+Let us assume for now that d > m(the case of d≤mis trivially included with much
+simplification, in which case we will not need the definition o f♦k,rbelow). Under26 SHUN-JEN CHENG AND WEIQIANG WANG
+this assumption, the condition λm+1≤nis no longer an empty condition. Recall
+λ′= (λ′
+1,λ′
+2,...) denotes the transposed partition of λ. We have d≥λ′
+1≥λ′
+2≥...and
+m≥λ′
+n+1.
+Recall that the row determinant of anr×rmatrix with possibly non-commuting
+entriesA= [aj
+i] is defined to be
+rdetA=/summationdisplay
+σ∈Sr(−1)ℓ(σ)aσ(1)
+1aσ(2)
+2···aσ(r)
+r.
+Form≤r≤d, we introduce the following row determinant:
+(4.11) ♦k,r:= rdet
+x1
+1x2
+1···xr
+1
+x1
+2x2
+2···xr
+2......···...
+x1
+mx2
+m···xr
+m
+η1
+kη2
+k···ηr
+k......···...
+η1
+kη2
+k···ηr
+k
+, k= 1,...,n,
+where the last ( r−m) rows are filled with the same vector ( η1
+k,η2
+k,...,ηr
+k).
+Remark 4.6.The determinant (4.11) is always nonzero. It reduces to (4.1 0) when
+m=r, and, up to a scalar multiple, it reduces to η1
+k···ηr
+kwhenm= 0.
+Now let rbe specified by the conditions λ′
+r> mandλ′
+r+1≤m. Denote by λ≤r
+the subdiagram of the Young diagram λwhich consists of the first rcolumns of λ, i.e.,
+the columns of length > m. A formula for the highest weight vector associated to the
+Young diagram λ≤ris given by the following lemma.
+Lemma 4.7. The vector/producttextr
+k=1♦k,λ′
+kis a highest weight vector for the highest weight
+moduleLd(λ≤r)⊗Lm|n(λ♮
+≤r)in the decomposition of C[x,η].
+Proof.Set♦=/producttextr
+k=1♦k,λ′
+k, andµ=λ≤r. We verify the following.
+(i)♦has weight ( µ,µ♮) with respect to the action of gl(d)×gl(m|n).
+(ii)♦is non-zero.
+(iii)♦is annihilated by the operators in (4.8).
+By Exercise 4.8 below, (i)-(iii) imply that ♦is also a highest weight vector for
+gl(m|n). /square
+Exercise 4.8. Assume that v∈C[x,η] has weights λandλ♮with respect to gl(d) and
+gl(m|n), respectively. Prove that if vis a highest weight vector for gl(d), then so is it
+forgl(m|n).
+Theorem 4.9. [CW1]Letλbe a Young diagram of length ≤dsuch that λm+1≤n.
+Then, a highest weight vector of weight (λ,λ♮)in thegl(d)×gl(m|n)-module C[x,η]is
+given by
+r/productdisplay
+k=1♦k,λ′
+kλ1/productdisplay
+j=r+1♦λ′
+j,DUALITIES FOR LIE SUPERALGEBRAS 27
+whereris defined by λ′
+r> mandλ′
+r+1≤m.
+Proof.Since the expression is a product of two highest weight vecto rs, it is a highest
+weight vector. Also it is easy to verify that it has the correc t weight. /square
+5.Howe duality for Lie superalgebras of type osp
+5.1.General ideas of dual pairs. LetVbe a vector (super)space, and P(V) the
+polynomial (super)algebra on V. LetD(V) denote the Weyl (super)algebra of differen-
+tial operators with polynomial coefficients on V. For example, if x1,...,x m,ξ1,...,ξn
+are the even/odd coordinates on Vrelative to a homogeneous basis (so that V∼=Cm|n),
+thenD(V) is generated by the even generators xi,∂i:=∂/∂xiand the odd generators
+ξj,ðj:=∂/∂ξj(1≤i≤m,1≤j≤n), subject to the only nontrivial (super) commu-
+tation relations among the generators:
+∂ixi−xi∂i= 1,ðjξj+ξjðj= 1.
+Given a reductive group Gacting on V, we have a multiplicity-free decomposition
+over (G,D(V)G):
+P(V) =/circleplusdisplay
+λ∈PL(λ)⊗Uλ,
+whereL(λ) are pairwise non-isomorphic irreducible G-modules and Uλare pairwise
+non-isomorphic irreducible D(V)G-modules, as λruns over some set Pof parameters.
+If there exists a generating set {T1,...,T r}for the (super)algebra D(V)G, such that
+g′:=C-span/a\}bracketle{tT1,...,T r/a\}bracketri}htforms a Lie subalgebra of D(V)G, then the above Uλ’s are
+pairwise non-isomorphic irreducible g′-modules. We refer to [GW] for more detailed
+discussion.
+Remark 5.1.For a suitable choice of VandG, there could exist a very canonical
+candidateforthedualpartner g′, whichissuggested by theFirst Fundamental Theorem
+of invariant theory [H1, GW]. We will refer to ( G,g′) as aHowe dual pair acting on
+S(V).
+OnceG,g′are fixed, the basic problems in ( G,g′)-Howe duality include describing
+theg′-module Uλexplicitly (e.g. describing its highest weight if this make s sense) and
+determining the parameter set P.
+For example, if we let V= (Cd⊗Cn)∗andG= GL(d), thenP(V) =S(Cd⊗Cn) and
+we can choose g′=gl(n). This gives rise to the ( gl(d),gl(n))-duality from Remark 4.2.
+5.2. In the type AHowe dualities, we can replace the Lie group GL( d) by its Lie
+algebragl(d)andviceversawithoutlossofinformation. However formor egeneral Howe
+dualities, it is essential to useLiegroups Gas formulated in Section 5.1. Associated to a
+given Lie algebra g(e.g.so(d)), one associates several different (possibly disconnected )
+groupsG, e.g. SO( d),O(d),Spin(d),Pin(d). The Howe duality formulation favors O( d)
+and Pin( d) in the sense of Remark 5.1.
+To avoid the complication of describing the irreducible rep resentations of different
+covering groups, we will choose to discuss a Howe duality inv olving the Lie group
+Sp(d), where d= 2ℓmust be an even integer. In this case, the finite-dimensional28 SHUN-JEN CHENG AND WEIQIANG WANG
+simple modules over the group Sp( d) and those over the Lie algebra sp(d) correspond
+bijectively, and it makes no difference if we use sp(d) to replace Sp( d). Besides, they
+admit a nice parametrization in terms of partitions. We deno te the Sp( d)-module
+corresponding to the partition λbyL(Sp(d),λ).
+According to [H1], there exists a Howe dual pair (Sp( d),so(2m)) onS(Cd⊗Cm),
+and a Howe dual pair (Sp( d),sp(2n)) on∧(Cd⊗Cn). By fusing these two dual pairs
+together, we obtain a Howe dual pair (Sp( d),osp(2m|2n)) onS(Cd⊗Cm|n). Let us
+explain the commuting actions below.
+As before, we identify S(Cd⊗Cm|n) =C[x,η]. The action of the Lie algebra sp(d)
+as the subalgebra of gl(d) onC[x,η] lifts to an action of Lie group Sp( d). On the other
+hand, the following action of gl(m|n) onC[x,η] is obtained from (4.7) by a shift of
+scalars on the diagonal matrices:
+Exx
+is=d/summationdisplay
+j=1xj
+i∂
+∂xj
+s+d
+2δis, Exη
+ik=d/summationdisplay
+j=1xj
+i∂
+∂ηj
+k, (5.1)
+Eηx
+ki=d/summationdisplay
+j=1ηj
+k∂
+∂xj
+i, Eηη
+tk=d/summationdisplay
+j=1ηj
+t∂
+∂ηj
+k−d
+2δik,
+wherei,s= 1,...,mandk,t= 1,...,n. Introduce the following additional operators
+Ixx
+is=d
+2/summationdisplay
+j=1/parenleftig
+xj
+ixd+1−j
+s−xd+1−j
+ixj
+s/parenrightig
+, Ixη
+ik=d
+2/summationdisplay
+j=1/parenleftig
+xj
+iηj
+k−xd+1−j
+iηd+1−j
+k/parenrightig
+,
+Iηη
+kt=d
+2/summationdisplay
+j=1/parenleftig
+ηj
+kηd+1−j
+t−ηd+1−j
+kηj
+t/parenrightig
+,∆xx
+is=d
+2/summationdisplay
+j=1/parenleftig∂
+∂xj
+i∂
+∂xd+1−j
+s−∂
+∂xd+1−j
+i∂
+∂xj
+s/parenrightig
+,
+∆xη
+ik=d
+2/summationdisplay
+j=1/parenleftig∂
+∂xj
+i∂
+∂ηd+1−j
+k−∂
+∂xd+1−j
+i∂
+∂ηj
+k/parenrightig
+,
+∆ηη
+kt=d
+2/summationdisplay
+j=1/parenleftig∂
+∂ηj
+k∂
+∂ηd+1−j
+t−∂
+∂ηd+1−j
+k∂
+∂ηj
+t/parenrightig
+,(5.2)
+where 1 ≤i < s≤mand 1≤k≤t≤n. It is not hard to see that these operators
+together with (5.1) form a basis for the Lie superalgebra osp(2m|2n). Moreover the
+actions of osp(2m|2n) and of Sp( d) onC[x,η] commute.
+Theorem 5.2. [H1]Letd= 2ℓ. The images of the algebras CSp(d)andU(osp(2m|2n))
+in End/parenleftbig
+S(Cd⊗Cm|n)/parenrightbig
+satisfy the double centralizer property.
+Proof.The proof is based on the fact that the invariants of the class ical group Sp( d) of
+the corresponding dual pair in the endomorphism ring of S(Cd⊗Cm|n) are generated
+by quadratic invariants. /square
+Denote by u+(respectively, u−) the subalgebra of osp(2m|2n) spanned by the ∆
+(respectively, I) operators in (5.2). The highest weight module L(osp(2m|2n),µ) belowDUALITIES FOR LIE SUPERALGEBRAS 29
+is understood to be relative to the Borel subalgebra of osp(2m|2n) corresponding to
+the simple roots listed in the following Dynkin diagram:
+/circlecopyrt
+/circlecopyrt/circlecopyrt···/circlecopyrt/circlemultiplytext/circlecopyrt···/circlecopyrt❅❅
+δ2−δ3 δm−ǫ1 ǫn−1−ǫnδ1−δ2
+−δ1−δ2
+Thengl(m|n) is a Levi subalgebra of osp(2m|2n) corresponding to the removal of
+the simple root −δ1−δ2. We have a triangular decomposition of Lie superalgebra
+osp(2m|2n) =u−⊕gl(m|n)⊕u+.
+Recall that for λ∈P(m|n),λ♮is a weight for gl(m|n). Thenλ♮+ℓ1can be regarded
+as a weight forLie superalgebras gl(m|n) andosp(2m|2n) (which sharethesameCartan
+subalgebra), where
+1=m/summationdisplay
+i=1δi−n/summationdisplay
+j=1ǫj.
+The simplified proof below of the following theorem of [CZ2] i s borrowed from [CKW].
+Theorem 5.3. Letd= 2ℓ. As an(Sp(d),osp(2m|2n))-module,
+S(Cd⊗Cm|n) =/circleplusdisplay
+λ∈P(m|n),ℓ(λ)≤ℓL(Sp(d),λ)⊗L(osp(2m|2n),λ♮+ℓ1).
+Proof.An element f∈C[x,η] is called harmonic , iffis annihilated by the subalge-
+brau+. The space of harmonics will be denoted bySpHand it evidently admits an
+action of Sp( d)×gl(m|n). Furthermore, since S(Cd⊗Cm|n) is a completely reducible
+gl(m|n)-module, L(osp(2m|2n),µ)u+is also completely reducible over gl(m|n), for any
+irreducible osp(2m|2n)-module L(osp(2m|2n),µ) that appears in S(Cd⊗Cm|n). By
+irreducibility of L(osp(2m|2n),µ) we must have
+L(osp(2m|2n),µ)u+∼=Lm|n(µ).
+So, by Theorem 5.2 (Sp( d),gl(m|n)) forms a dual pair on the space of harmonicsSpH.
+Thus, proving the theorem is equivalent to establishing the following decomposition of
+SpHas an Sp( d)×gl(m|n)-module:
+(5.3)SpH∼=/circleplusdisplay
+λ∈P(m|n),ℓ(λ)≤ℓL(Sp(d),λ)⊗Lm|n(λ♮+ℓ1).
+We first consider the limit case n=∞with the space of hamonics denoted by
+SpH∞. Here the only restriction on λisℓ(λ)≤ℓ, and we observe that the vector
+given in Theorem 4.9 associated to such a partition λis indeed annihilated by u+and
+hence is a joint Sp( d)×gl(m|n)-highest weight vector of weight ( λ,λ♮+ℓ1). Hence
+all the summands on the right hand side of (5.3) occur in the sp ace of harmonics,
+and in particular, all irreducible representations of Sp( d) occur. Therefore, we have
+established (5.3) in this case.
+Now consider the finite ncase. We may regard S(Cd⊗Cm|n)⊆S(Cd⊗Cm|∞)
+with compatible actions osp(2m|2n)⊆osp(2m|2∞). From the formulas of the ∆-
+operators in (5.2) we see thatSpH⊆SpH∞. Thus the spaceSpHis obtained from30 SHUN-JEN CHENG AND WEIQIANG WANG
+SpH∞by setting the variables ηi
+k= 0, fork > n. However, it is clear, from the explicit
+formulas of the joint highest vectors inSpH∞, that when setting the variables ηi
+k= 0,
+fork > n, precisely those vectors corresponding to ( m|n)-hook partitions will survive.
+This completes the proof. /square
+5.3.The Howe dual pair (Sp(d),c∞).Setd= 2ℓ. LetC∞bethevector spaceover C
+with basis {ei|i∈Z}. Letgl∞denote the Lie algebra consisting of matrices ( aij)i,j∈Z
+with finitely many non-zero aij’s, andC∞is the natural gl∞-module. Then gl∞has a
+linear basis given by the matrix units Eij(i,j∈Z). Denote by /hatwidegl∞=gl∞⊕CKthe
+central extension of gl∞by a one-dimensional center CKgiven by the 2-cocycle
+τ(A,B) := Tr([J,A]B),
+whereJ=/summationtext
+i≤0Eii. The Lie algebra /hatwidegl∞admits a Z-grading /hatwidegl∞=/circleplustext
+i∈Z/hatwidegl∞,±iby
+letting deg Eij=j−iand degK= 0, and this induces a triangular decomposition of
+/hatwidegl∞, with/hatwidegl∞,±=/circleplustext
+i>0/hatwidegl∞,±i.
+Exercise 5.4. Show that the cocycle τongl∞is a coboundary and hence /hatwideg∞is
+isomorphic to gl∞⊕CKas Lie algebras.
+Letc∞be the subalgebra of gl∞preserving the following bilinear form on C∞:
+(ei|ej) = (−1)iδi,1−j, i,j∈Z.
+Letc∞=c∞⊕CKbe the central extension of c∞determined by the restriction of
+the two-cocycle τabove. Then c∞is an infinite-rank affine Kac-Moody algebra with
+Dynkin diagram as follows:
+/circlecopyrt /circlecopyrt /circlecopyrt ··· =⇒ /circlecopyrt
+α0
+The Lie algebra c∞has a natural triangular decomposition induced from /hatwidegl∞:
+c∞=c∞,−⊕h⊕c∞,+,
+where the Cartan subalgebra of c∞is spanned by Kand
+/tildewideEi:=Eii−E1−i,1−i, i∈N.
+Letǫi∈h∗be so that /a\}bracketle{tǫi,/tildewideEj/a\}bracketri}ht=δij, fori,j∈N. Fori∈Z+, we denote by Λc
+ithei-th
+fundamental weight for c∞. Then Λc
+0∈h∗is determined by /a\}bracketle{tΛc
+0,/tildewideEi/a\}bracketri}ht= 0 fori∈Nand
+/a\}bracketle{tΛc
+0,K/a\}bracketri}ht= 1, and
+Λc
+i= Λc
+0+ǫ1+...+ǫi, i≥1.
+LetL(c∞,µ) denote the irreducible highest weight module with respect to the Borel
+subalgebra h⊕c∞,+of highest weight µ.
+Theorem 5.5. [Wa, Theorem 3.4] There exists a Howe dual pair ( Sp(d),c∞) acting on
+the fermionic Fock space Fℓofℓpairs of complex fermions, where d= 2ℓ. Moreover,
+as an(Sp(d),c∞)-module,
+(5.4) Fℓ∼=/circleplusdisplay
+ℓ(λ)≤ℓL(Sp(d),λ)⊗L(c∞,Λc(λ)),DUALITIES FOR LIE SUPERALGEBRAS 31
+whereΛc(λ) :=d
+2Λc
+0+/summationtext
+k≥1λ′
+kǫk=/summationtextd
+2
+k=1Λc
+λk.
+As all we need later on is the combinatorial identity (5.5) be low, we will skip the
+precise definitions of the fermionic Fock space Fℓand the commuting actions of Sp( d)
+andc∞onFℓ(see [Wa] for detail).
+Recalling (4.6) we set /tildewideEi:=Eii−Ed−i+1,d−i+1, fori= 1,...,d
+2. Computing the
+trace of/producttext
+n∈Nx/tildewideEnn/producttextd
+2
+i=1z/tildewideEi
+ion both sides of (5.4), we have
+(5.5)d
+2/productdisplay
+i=1/productdisplay
+n∈N(1+xnzi)(1+xnz−1
+i) =/summationdisplay
+ℓ(λ)≤ℓchL(Sp(d),λ) chL(c∞,Λc(λ)).
+5.4.Irreducible c∞-characters. LetW(respectively, W0) be the Weyl group of c∞
+(respectively, of the Levi subalgebra of c∞corresponding to the removal of the simple
+rootα0). LetW0
+kbethesetoftheminimallengthrepresentativesoftheright cosetspace
+W0\Wof length k. Then it is standard to write W=W0W0withW0=/unionsqtext
+k≥0W0
+k. It
+follows that, for each w∈W0
+k, we may find a partition λw= ((λw)1,(λw)2,...) such
+that
+(5.6) w(Λc(λ)+ρ)−ρ=d
+2Λc
+0+/summationdisplay
+j>0(λw)jǫj.
+The following is obtained by applying the Kostant homology f ormula for integrable
+modules over Kac-Moody algebras and the Euler-Poincar´ e pr inciple (cf. [CKW, Propo-
+sition 2.5]).
+Proposition 5.6. We have the following character formula:
+chL(c∞,Λc(λ)) =1/producttext
+1≤i≤j(1−xixj)∞/summationdisplay
+k=0/summationdisplay
+w∈W0
+k(−1)ksλw(x1,x2,...).
+Exercise 5.7. Prove the existence of λwfor each w∈W0in (5.6).
+5.5.The irreducible osp(2m|2n)-characters. The following character formula for
+L(osp(2m|2n),λ♮+ℓ1) in a different form was first obtained in [CZ2]. The proof here
+follows [CKW] and is simpler.
+Theorem 5.8. Forλ∈P(m|n)such that ℓ(λ)≤ℓ, we have
+chL(osp(2m|2n),λ♮+ℓ1) =
+/parenleftbiggy1···ym
+x1···xn/parenrightbiggℓ/producttext
+1≤i≤m/producttext
+1≤s≤n(1+yixs)
+/producttext
+1≤i<j≤m/producttext
+1≤s≤t≤n(1−yiyj)(1−xsxt)∞/summationdisplay
+k=0/summationdisplay
+w∈W0
+k(−1)khsλw(x,y).32 SHUN-JEN CHENG AND WEIQIANG WANG
+Proof.Computing the trace of the operator/producttext
+i,jy/tildewideEi
+ix/tildewideE¯j
+j/producttextd
+2
+k=1z/tildewideEk
+kon both sides of the
+isomorphism in Theorem 5.3, where 1 ≤i≤mand 1≤j≤n, we obtain
+d
+2/productdisplay
+k=1/productdisplay
+i,j(1+xiz−1
+k)(1+xizk)
+(1−yjz−1
+k)(1−yjzk)
+=/parenleftbiggy1···ym
+x1···xn/parenrightbigg−ℓ/summationdisplay
+λ∈P(m|n)
+ℓ(λ)≤ℓchL(Sp(d),λ) chL(osp(2m|2n),λ♮+ℓ1). (5.7)
+Replacing ch L(c∞,Λc(λ)) in (5.5) by the expression in Proposition 5.6, we obtain an
+identity of symmetric functions in variables x1,x2,.... Next, we replace xn+ibyyi
+fori∈N, and then apply to this identity the involution on the ring of symmetric
+functions in y1,y2,.... Finally, we put yj= 0 forj≥m+1. Under the composition
+of those maps, we obtain a new identity which shares the same l eft hand side as (5.7).
+Comparing the right hand sides of this new identity and of (5. 7), we obtain the result
+thanks to the linear independence of ch L(Sp(d),λ). /square
+Remark 5.9.The above irreducible character formula can be understood a s an alter-
+nating sum of the characters of parabolic Vermas over a certa in infinite Weyl group. A
+similar observation was made for type Ain [CZ1, Corollary 4.15].
+The Kostant u-cohomology groups for these g-modules can be further computed and
+described in terms of the infinite Weyl groups, and in particu lar, they are multiplicity-
+free as modules over Levi subalgebras [CKW] (also cf. [CK]). In other words, the
+corresponding Kazhdan-Lusztig polynomials via u-cohomology in the sense of Vogan
+[Vo] are monomials.
+6.Super duality
+The goal of the remaining part of the lectures is to formulate precisely a direct con-
+nection between representation theories of Lie algebras an d Lie superalgebras, following
+[CLW] (cf. [CWZ, CW4, CL2]). This is in part supported by the f ollowing.
+(1) The gl(1|1)-principal block can be understood in terms of A∞-quivers, which is
+classical in its own way.
+(2) There exist intimate connections between finite-dimens ional classical Lie super-
+algebras and infinite Weyl groups (cf. Remark 5.9).
+(3) By a purely algebraic approach [Br1], Brundan showed tha t the category of
+finite-dimensional gl(m|n)-modules categorifies the Fock space ∧mV⊗ ∧nV∗,
+whereVdenotes the natural Uq(gl(∞))-module.
+(4) The Brundan-Kazhdan-Lusztig polynomials for some suit able parabolic cate-
+goryOof super type Aare shown in [CWZ, CW4] to coincide with the usual
+Kazhdan-Lusztig polynomials of type A. For the so-called polynomial repre-
+sentations this was already observed in [CZ1]. Moreover, th e super duality
+conjecture [CW4] (generalizing [CWZ]) on a category equiva lence between Lie
+superalgebras and Lie algebras of type Ahas been established recently in [CL2].DUALITIES FOR LIE SUPERALGEBRAS 33
+6.1.Parabolic category O.Assume that kis a (possibly Kac-Moody, and or possibly
+infinite rank) Lie (super)algebra, with Cartan subalgebra hand root system Φ. Let I
+be a set of simpleroots, and bthe correspondingBorel subalgebra. Let Ybea subsetof
+IandlYbe the associated Levi subalgebra of k. Letk=u−⊕lY⊕u+be the generalized
+triangular decomposition. Denote by Pthe weight lattice for k, and by PY⊆Pthe
+subset of weights which are Y-dominant.
+Givenλ∈h∗, we denote by L(lY,λ) the irreducible lY-module of highest weight λ
+with respect to lY∩b, which extends to a ( lY+u+)-module with a trivial action of u+.
+Define the parabolic Verma k-module
+∆(λ) :=U(k)⊗U(lY+u+)L(lY,λ).
+LetL(λ) be the irreducible quotient k-module of ∆( λ).
+LetO=O(k,Y,PY) be the category of g-modules Msuch that Mis a semisimple
+h-module with finite-dimensional weight subspaces Mγ,γ∈h∗, satisfying
+(i)Mdecomposes over lYinto a direct sum of L(lY,µ) forµ∈PY.
+(ii) There exist finitely many weights λ1,λ2,...,λ k∈PY(depending on M) such
+that ifγis a weight in M, thenγ∈λi−/summationtext
+α∈IZ+α, for some i.
+Thechoice of PYshouldbenatural andgeneral enoughsothat L(λ),∆(λ)∈O(k,Y,PY)
+for every λ∈PY. This will be spelled out clearly in the cases of interest lat er on.
+Remark 6.1.In the case when kis a finite-dimensional semisimple Lie algebra, or gl(n),
+or of the four classical infinite-rank affine Kac-Moody Lie alg ebrasa+
+∞,b∞,c∞,d∞,
+each block is controlled by the Weyl group Wofk. For the principal block (in the full
+category O), the transition matrix between [ L(λ)] and [∆( µ)] is given by the Kazhdan-
+Lusztig polynomials associated to W(see Tanisaki’s lectures [Ta]). Other blocks and
+parabolic cases are reduced to the principal block.
+6.2.The master Lie (super)algebras. Let☛
+✡✟
+✠xdenote one of the following four
+classical Dynkin diagrams of types a,b,c,d, which will be referred to as a head diagram :
+/circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt/circlecopyrt ···✡ ✠☛ ✟
+a
+δ1−δ2δ2−δ3 δm−1−δm
+/circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt/circlecopyrt⇐= ···✡ ✠☛ ✟
+b
+−δ1δ2−δ3 δm−1−δm
+/circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt=⇒ ···✡ ✠☛ ✟
+c
+−2δ1δ1−δ2 δm−1−δm
+/circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt/circlecopyrt
+/circlecopyrt❅❅
+···✡ ✠☛ ✟
+d
+−δ1−δ2δ1−δ2
+δ2−δ3 δm−1−δm
+A head diagram☛
+✡✟
+✠xis connected with one of the three tail diagrams to produce
+three “master” Dynkin diagrams as below, whose correspondi ng Lie (super)algebras
+will be denoted by G=Gx,G=Gx, and/tildewideG=/tildewideGx, respectively.
+/circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt✡ ✠☛ ✟x ··· ··· G:
+δm−ǫ1ǫ1−ǫ2 ǫn−ǫn+134 SHUN-JEN CHENG AND WEIQIANG WANG/circlemultiplytext/circlecopyrt /circlecopyrt /circlecopyrt /circlecopyrt✡ ✠☛ ✟x ··· ··· G:
+δm−ǫ1
+2ǫ1
+2−ǫ3
+2ǫn−1
+2−ǫn+1
+2/circlemultiplytext /circlemultiplytext /circlemultiplytext /circlemultiplytext /circlemultiplytext
+✡ ✠☛ ✟x ··· ··· /tildewideG:
+δm−ǫ1
+2ǫ1
+2−ǫ1 ǫn−ǫn+1
+2
+For example, the resulting master Dynkin diagrams for Gare the four infinite-rank
+classical Lie algebras a+
+∞(one-sided infinity), b∞,c∞,d∞.
+6.3.The categories O,O,/tildewideO.LetΠxdenotethestandardset ofsimplerootsassociated
+to the diagram☛
+✡✟
+✠x, and fix Y0⊂Πx. LetT1,T1and/tildewideT1be the subsets of the three
+tail diagrams with the first simple root removed. Let
+Y=Y0∪T1,Y=Y0∪T1,/tildewideY=Y0∪/tildewideT1.
+LetP(respectively, P,/tildewideP) be the lattice which consists of the integral span of ǫi, where
+irunsover I(m|N) (respectively, I(m|1
+2+N),I(m|1
+2N)). Here I(m|N) ={¯1,...,¯m}∪N,
+and the other two sets are similarly defined.
+DefinePY=∪d∈ZPd
+Y, where
+Pd
+Y=/braceleftig
+λ=/summationdisplay
+i∈I(m|N)λiǫi∈P|λisY-dominant; λi=dfori≫0/bracerightig
+. (6.1)
+We define Pd
+Yby changing I(m|N) in the definition of Pd
+YtoI(m|1
+2+N) and changing
+λi=dtoλi=−dfori≫0, and let PY=∪d∈ZPd
+Y.
+Then, for λ∈Pd
+Y,λ+:= (λ1−d,λ2−d,...) is a partition. Also, for λ∈Pd
+Y,
+λ+:= (λ1/2+d,λ3/2+d,...) is a partition. We have a bijection of sets:
+♮:Pd
+Y−→Pd
+Y, λ/mapsto→λ♮,
+which is determined by the conditions
+λi=λ♮
+i,∀i∈I(m|0); (λ♮)+= (λ+)′.
+This further induces a bijection ♮:PY→PY.
+We define another map
+θ:Pd
+Y→Pd
+/tildewideY, λ/mapsto→λθ=/summationdisplay
+i∈I(m|0)λiǫi+/summationdisplay
+i∈N/parenleftbig
+(pi−d)ǫi−1
+2+(qi+d)ǫi/parenrightbig
+,
+where the image λθis determined by the conditions λi=λθ
+ifor alli∈I(m|0) and
+(pi|qi)i≥1is the modified Frobenius coordinates of the partition ( λ+)′(as defined in
+Example 3.16 (3)). The map θis clearly injective, and so we obtain a bijection θ:
+Pd
+Y→Pd
+/tildewideY:=θ(Pd
+Y) for each d∈Z. Letting P/tildewideY=∪d∈ZPd
+/tildewideYwe also obtain a bijection
+θ:PY→P/tildewideY.
+Following the construction of a parabolic category Oin Section 6.1, we obtain the
+categories O:=O(G,Y,PY),O:=O(G,Y,PY), and/tildewideO:=O(/tildewideG,/tildewideY,P/tildewideY). We observe the
+following direct sum decompositions of categories:
+O=/circleplusdisplay
+d∈ZOd,O=/circleplusdisplay
+d∈ZOd,/tildewideO=/circleplusdisplay
+d∈Z/tildewideOd,DUALITIES FOR LIE SUPERALGEBRAS 35
+wheretheindex dindicates additional weight constraints: ( λ,ǫi) =dfori∈Z,i≫0 for
+λ∈PY; (λ,ǫi) =−dfori∈1
+2+Z,i≫0 forλ∈PY; (λ,ǫi) = (−1)2idfori∈1
+2Z,i≫0
+forλ∈P/tildewideY, respectively.
+Remark 6.2.A more conceptual formulation (see [CLW] for details) is to i ntroduce
+central extensions of G,G,/tildewideGrespectively by a one-dimensional center, whose module
+categories at level dare equivalent to Od,Od,/tildewideOdabove, respectively. In this way, the
+additional weight constraints above are transferred to be ( λ,ǫi) = 0 for i≫0, which is
+independent of d.
+6.4.The equivalence of categories. We start with two simple observations:
+(i) The Lie (super)algebras GandGare naturally Lie subalgebras (though not Levi
+subalgebras) of /tildewideG, and the standard triangular decompositions of the three Li e (su-
+per)algebras are compatible with the inclusions G⊂/tildewideGandG⊂/tildewideG. This follows by
+examining the simple roots of the three algebras.
+(ii) We may naturally regard the lattices PandPas sublattices of /tildewideP, by definition
+of the lattices P,P,/tildewideP.
+Given a /tildewideG-module with weight space decomposition /tildewiderM=/circleplustext
+µ∈/tildewideP/tildewiderMµ, we define
+T(/tildewiderM) :=/circleplusdisplay
+µ∈P/tildewiderMµ,andT(/tildewiderM) :=/circleplusdisplay
+µ∈P/tildewiderMµ.
+Clearly, T(/tildewiderM) is aG-module and T(/tildewiderM) is aG-module. We can check that TandT
+send objects in category /tildewideOto objects in category OandO, respectively, and hence we
+obtain functors
+/tildewideO/tildewideT− −−− →O/arrowbtT
+O
+More precisely, we have T:/tildewideOd→OdandT:/tildewideOd→Od.
+We will use ∆(λ) and/tildewide∆(λ) to denote the parabolic Verma modules in the categories
+Oand/tildewideOof highest weights λ, respectively. Similarly, we use L(λ) and/tildewideL(λ) to denote
+the corresponding irreducible modules. We are ready to form ulate the main result of
+[CLW].
+Theorem 6.3. [CLW]We have
+T/parenleftbig/tildewide∆(λθ)/parenrightbig
+= ∆(λ),T/parenleftbig/tildewide∆(λθ)/parenrightbig
+=∆(λ♮),
+T/parenleftbig/tildewideL(λθ)/parenrightbig
+=L(λ),T/parenleftbig/tildewideL(λθ)/parenrightbig
+=L(λ♮).
+Moreover, T:/tildewideO→OandT:/tildewideO→Oare equivalences of categories.
+As a consequence, T:/tildewideOd→OdandT:/tildewideOd→Odare equivalences of categories.
+Idea of a proof. The proof consists of several major steps:36 SHUN-JEN CHENG AND WEIQIANG WANG
+(i)T(L(l/tildewideY,λθ)) =L(lY,λ). Thisfollowsbylocatingahighestweight λforthemodule
+L(l/tildewideY,λθ) with respect to a new Borel of l/tildewideYwhich is compatible with the imbedding
+lY⊆l/tildewideY. Then we check the equality on the character level (which boi ls down to the
+hook-Schur functions.)
+(ii)T(/tildewide∆(λθ)) is a highest weight module of highest weight λwith respect to the
+standard Borel of G. This follows from a weight argument.
+(iii)T/parenleftbig/tildewide∆(λθ)/parenrightbig
+= ∆(λ). Check on the character level, and use (i) and (ii).
+(iv)T/parenleftbig/tildewideL(λθ)/parenrightbig
+=L(λ). Follows by showing that there is no singular vector in
+T/parenleftbig/tildewideL(λθ)/parenrightbig
+of weight different from λ.
+This proves the equality for Tin the theorem. The proof of the equality for Tis
+similar. More work is needed to establish the equivalence of categories. /square
+Corollary 6.4 (Super Duality) .The categories OandOare equivalent.
+Remark 6.5 (History of Super Duality) .First for type A.
+(1) ForY0= Πa, the super duality was conjectured in [CWZ, Conjecture 6.10 ].
+This was motivated by Brundan [Br1] on finite-dimensional si mplegl(m|n)-
+characters.
+(2) For arbitrary Levi subalgebra associated to Y0⊆Πa, the super duality conjec-
+turewasformulatedin[CW4, Conjecture4.18]. Itisshownin [CWZ,CW4]that
+the Brundan-Kazhdan-Lusztig polynomials coincide with th e usual Kazhdan-
+Lusztig polynomials in type A.
+(3) The conjecture of [CWZ] is proved in [BrS], where the unde rlying algebras for
+the two categories were computed and shown to be isomorphic.
+(4) The more general conjecture of [CW4] is independently pr oved in [CL2] essen-
+tially in the form of Theorem 6.3. In particular, this provid es a new proof of
+the main result of [Br1].
+Then for type osp.
+(5) A first supporting evidence for the super duality was prov ided by the com-
+putation in [CKW] of the Kostant u-homology groups with coefficients in the
+modules of classical Lie superalgebras appearing in Howe du ality decomposi-
+tions.
+(6) Theorem 6.3 is formulated and established in [CLW].
+Note that little has been known about representation theory of infinite-dimensional
+Lie superalgebras, such as affine superalgebras, with the exc eption of work of Kac and
+Wakimoto [KW].
+(7) The super duality approach applies to and sheds new insig ht on more general
+Lie superalgebras, including affine superalgebras.
+6.5.Irreducible G-characters. By Remark 6.1, the irreduciblecharacter problem for
+thecategory OissolvedbytheKazhdan-Lusztigconjectures[KL](theorem ofBeilinson-
+Bernstein [BB] and Brylinski-Kashiwara [BK]). Hence by The orem 6.3, the irreducible
+character problem for the module category Ofor Lie superalgebras is solved by the
+same classical Kazhdan-Lusztig polynomials.DUALITIES FOR LIE SUPERALGEBRAS 37
+Recall for x∈ {a,b,c,d}, we have defined a Lie superalgebra G=Gxwhose Dynkin
+diagram is obtained by connecting☛
+✡✟
+✠xwith the type gl(1|∞) tail diagram T. For
+n∈N, we will also consider a Lie superalgebra Gn=Gx
+nwhose Dynkin diagram (see
+below) is obtained by connecting☛
+✡✟
+✠xwith a tail diagram Tnof typegl(1|n):
+(6.2)✡ ✠☛ ✟x/circlemultiplytext/circlecopyrt /circlecopyrt···
+δm−ǫ1
+2ǫ1
+2−ǫ3
+2ǫn−3
+2−ǫn−1
+2✡ ✠☛ ✟
+Gx
+n:
+As☛
+✡✟
+✠xis taken to be one of the four classical Dynkin diagrams, Gnis a classical
+finite-dimensional Lie superalgebra of type either glorosp.
+LetT1
+nbe the subset of Tn, which is obtained from Tnby the removal of the first
+simple root. Let Yn=Y0∪T1
+n, and let
+Pd
+Yn=/braceleftig
+λ=/summationdisplay
+i∈I(m|n−1
+2)λiǫi|λisYn-dominant; λn−1
+2≥ −d/bracerightig
+.
+Ford∈Z, we introduce a category Od
+n:=O(Gn,Yn,Pd
+Yn) ofGn-modules. Note that Od
+n
+is a full subcategory of the (more standard) category On:=O(Gn,Yn,PYn), where
+PYn=/braceleftig
+λ=/summationdisplay
+i∈I(m|n−1
+2)λiǫi|λisYn-dominant/bracerightig
+.
+Moreover, Od
+n⊂Od+1
+nfor alld, andOn=∪d∈ZOd
+n.
+We also introduce truncation functors Trn:Od→Od
+nas follows: Let M∈Odsuch
+thatM=/circleplustext
+µMµ. We define
+Trn(M) =/circleplusdisplay
+{µ|µn+1
+2=−d}Mµ,
+which is clearly a Gn-module. Note that µn+1
+2=−densuresµk+1
+2=−dfor allk≥n.
+Such a truncation functor to a Levi subalgebra (except that w e drop an abelian direct
+summand of the Levi) has its analogue in algebraic group sett ing studied by Donkin
+(cf. e.g. [CW4] for a formulation in type A). The functors Trn:Od→Od
+nhave the
+following key property.
+Proposition 6.6. ForX= ∆orL, we have
+Trn(X(λ)) =/braceleftigg
+Xn(λ(n)),ifλn+1
+2=−d,
+0, otherwise,
+whereλ(n)is obtained from λby ignoring λj, forj > n−1
+2.
+Remark 6.7.By Proposition 6.6 and Theorem 6.3, the classical Kazhdan-L usztig so-
+lution of the irreducible character problem of Oinduces a complete solution of the
+irreducible character problem of Onfor each finite n. It is not hard to show [CLW] that
+every finite-dimensional irreducible modules of every orth o-symplectic Lie superalgebra
+appears in some On.38 SHUN-JEN CHENG AND WEIQIANG WANG
+We end the paper with a list of symbols.
+Symbol Meaning
+|v|Z2-parity of a homogeneous vector vin a vector superspace
+Cm|ncomplex vector superspace of super dimension m|n
+gl(m|n) the general Lie superalgebra
+I(m|n) the totally order set {1<2< ... <m <1< ... < n }
+sl(m|n) the special linear Lie superalgebra
+hCartan subalgebra (of diagonal matrices)
+δidual basis element corresponding to Eii
+ǫjdual basis element corresponding to Ejj
+Φ,Φ¯0,or Φ¯1sets of all, even, or odd roots
+Π set of simple roots
+Φ±
+ǫset of positive/negative roots of parity ǫ=0,1
+osp(m|n) ortho-symplectic Lie superalgebra preserving a
+non-degenerate supersymmetric bilinear form on Cm|n
+spo(m|n) symplectic-orthogonal Lie superalgebra preserving a
+non-degenerate skew-supersymmetric bilinear form on Cm|n
+U(g) universal enveloping algebra of a Lie (super)algebra g
+K(λ) Kac module of highest weight λ
+L(g,λ) orL(λ) irreducible g-module of highest weight λ
+chMcharacter of the h-semisimple module M
+ρ0,ρ1half sums of positive even and odd roots, respectively
+ρ ρ0−ρ1
+Sdsymmetric group in dletters
+SλSpecht module corresponding to the partition λ
+∆(λ) (parabolic) Verma g-module of highest weight λ
+µ′conjugate partition of the partition µ
+P(d,m|n) set of ( m|n)-hook partitions of size d(Definition 2.9)
+P(m|n)/uniontext
+d≥0P(d,m|n)
+sλ(x) Schur function in xfor the partition λ
+hsλ(x,y) hook Schur function for the hook partition λ(4.1)
+♦r,♦k,rrow determinants given in (4.10) and (4.11), respectively
+a+
+∞(one-sided) infinite-rank Kac-Moody Lie algebras of type A
+b∞,c∞,d∞infinite-rank Kac-Moody Lie algebras of types B,C,D
+Λc
+iith fundamental weight of c∞
+λwsee (5.6)
+Gx,Gx,/tildewideGxLie (super)algebras corresponding to master diagrams of
+Section 6.2 of type x; also denoted by G,G,/tildewideG, respectively
+O,O,/tildewideOparabolic categories of G-,G- and/tildewideG-modules, respectively
+(Sections 6.1 and 6.3)
+∆(λ),/tildewide∆(λ) parabolic Verma modules in O,/tildewideO(Section 6.4)
+L(λ),/tildewideL(λ) irreducible modules in O,/tildewideO(Section 6.4)DUALITIES FOR LIE SUPERALGEBRAS 39
+Symbol Meaning
+☛
+✡✟
+✠xhead Dynkin diagrams of types x=a,b,c,d(Section 6.2)
+Pd
+Y,Pd
+Y,Pd
+/tildewideYset of dominant weights for G,G,/tildewideG, respectively (Section 6.3)
+♮,θbijections from Pd
+YtoPd
+YandPd
+/tildewideY(Section 6.3), respectively
+T, orTfunctors from /tildewideOtoO, or toO, respectively (Section 6.4)
+Onparabolic category of a finite-dimensional Lie (super)alge bra
+(Section 6.5)
+Trntruncation functor from OtoOn(Section 6.5)
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+Institute of Mathematics, Academia Sinica, Taipei, Taiwan 10617
+E-mail address :chengsj@math.sinica.edu.tw
+Department of Mathematics, University of Virginia, Charlo ttesville, VA 22904
+E-mail address :ww9c@virginia.edu
\ No newline at end of file
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+arXiv:1001.0075v3  [math.KT]  24 Feb 2013THE PULLBACKS OF PRINCIPAL COACTIONS
+Piotr M. Hajac
+Instytut Matematyczny, Polska Akademia Nauk
+ul.´Sniadeckich 8, Warszawa, 00-956 Poland
+http://www.impan.pl/ /tildewidepmh
+and
+Katedra Metod Matematycznych Fizyki, Uniwersytet Warszaw ski
+ul. Ho˙ za 74, Warszawa, 00-682 Poland
+Elmar Wagner
+Instituto de F´ ısica y Matem´ aticas
+Universidad Michoacana de San Nicol´ as de Hidalgo
+Edificio C-3, Cd. Universitaria, 58040 Morelia, Michoac´ an , M´ exico
+e-mail: elmar@ifm.umich.mx
+Abstract: We prove that the class of principal coactions is closed unde r one-surjective pullbacks
+in an appropriate category of algebras equipped with left an d right coactions. This allows us to
+handle cases of C∗-algebras lacking two different non-trivial ideals. It also a llows us to go beyond the
+category of comodule algebras. As an example of the former, w e carry out an index computation for
+noncommutativelinebundlesoverthestandardPodle´ ssphe reusingtheMayer-Vietoris typearguments
+afforded by a one-surjective pullback presentation of the C∗-algebra of this quantum sphere. To
+instantiate the latter, we define a family of coalgebraic non commutative deformations of the U(1)-
+principal bundle S7→CP3.
+Contents
+1 Introduction and preliminaries 2
+1.1 Pullback diagrams and fibre products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
+1.2 Odd-to-even connecting homomorphism in K-theory . . . . . . . . . . . . . . . . . . . . . . . . . 4
+1.3 Principal coactions and associated projective modules . . . . . . . . . . . . . . . . . . . . . . . . 5
+1.4 Standard Hopf fibration of quantum SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
+2 Principality of one-surjective pullbacks 12
+2.1 Principality of images and preimages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
+2.2 The one-surjective pullbacks of principal coactions are principa l . . . . . . . . . . . . . . . . . . . 16
+3 The pullback picture of the standard quantum Hopf fibration 18
+3.1 Pullback comodule algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
+13.2 Equivalence of the pullback and standard constructions . . . . . . . . . . . . . . . . . . . . . . . 21
+3.3 Index pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
+4 Examples of piecewise principal coalgebra coactions 25
+4.1 Piecewise principal coactions from a noncommutative join constr uction . . . . . . . . . . . . . . . 26
+4.2 Quantum complex projective spaces CP3
+q,s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
+1 Introduction and preliminaries
+The idea of decomposing a complicated object into simpler pieces and c onnecting data is a
+fundamental computational principle throughout mathematics. I n the case of (co)homology
+theory, it yields the Mayer-Vietoris long exact sequence whose sign ificance and usefulness can
+hardly be overestimated. The categorical underpinning of all this a re pullback diagrams: in a
+given category they give a rigorous meaning to putting together tw o objects over a third one.
+The goal of this paper is to prove a general pullback theorem for pr incipal coactions that
+significantly generalizes the main result of [13] restricted to comodu le algebras and pullbacks
+of surjections. More precisely, our main result is that the pullback o f principal coactions
+over morphisms of which at least one is surjective is again a principal c oaction. It may be
+viewed as a non-linear version of the Milnor construction yielding an od d-to-even connecting
+homomorphism in algebraic K-theory [20]. Indeed, linearizing our pullback theorem with the
+help of a corepresentation gives precisely the odd-to-even const ruction of a projective module
+defining the connecting homomorphism in K-theory.
+Weapplythisnewresult intwo cases. Inthefirstcase, wekeep thec omodule-algebrasetting
+but take a one-surjective pullback diagram (only one of the defining morphisms is surjective).
+In the second case, we proceed the other way round, that is, we t ake a pullback diagram given
+by two surjections but take coactions that are not algebra homom orphisms.
+The pullback picture ofthe standardquantum Hopf fibrationgives u s thefirst-case example.
+It provides a new way of computing the index pairing for the associat ed quantum Hopf line
+bundles (cf. [28]). This index pairing was computed in [12] using a nonco mmutative index
+formula, and re-derived in [22]. Here we give yet another method to c ompute it. This simple
+example shows the need to generalize from two-surjective to one- surjective pullback diagrams,
+and the pullback method of index computation seems attractive due to its inherent simplicity.
+To obtain the second-case example, we first show how the piecewise structure [13] of a non-
+commutative joinconstruction[9]allows onetodefine acertainclass ofpiecewise principal coac-
+tions. Althoughthisclassofexamplescanalsobehandledbyearlierme thods, itdefinitelyshows
+that there are interesting piecewise principal coactions that are n ot algebra homomorphisms.
+To obtain a concrete example, we take Pflaum’s instanton bundle S7
+q→S4
+q[23] as the noncom-
+mutative join of SU q(2) and turn it into the coalgebraic quantum principal bundle S7
+q→CP3
+q,s.
+We do it with the help of the canonical surjections π:O(SUq(2))→ O(SUq(2))/Jq,sdetermined
+by the coideals right ideals Jq,s:= (O(S2
+q,s)∩kerε)O(SUq(2)), where S2
+q,sis a generic Podle´ s
+quantum sphere [24] and ker εis the kernel of the counit map.
+2The paper is organized as follows. First, to make our exposition self- contained and to
+establish notation, we recall fundamental concepts that we use la ter on. The key Section 2
+is devoted to the general pullback theorem for principal coactions of coalgebras on algebras,
+Section 3 is on deriving the index pairing for quantum Hopf line bundles a s a corollary to the
+pullback presentation of the standard Hopf fibration of SU q(2), and the final Section 4 presents
+new examples of piecewise principal coactions that go beyond Hopf- Galois theory.
+Throughout the paper, we work with algebras and coalgebras over a field. The unadorned
+tensorproductstandsforthealgebraictensorproductoverth isfield. WeemploytheHeyneman-
+Sweedlertypenotation(withthesummationsymbolsuppressed) f orthecomultiplication∆( c)=
+c(1)⊗c(2)∈C⊗Cand for coactions ∆ V(v) =v(0)⊗v(1)∈V⊗C,V∆(v) =v(−1)⊗v(0)∈C⊗V.
+The convolution product of two linear maps from a coalgebra to an alg ebra is denoted by ∗:
+(f∗g)(c) :=f(c(1))g(c(2)). The set of natural numbers includes 0, that is, N={0,1,2,...}.
+1.1 Pullback diagrams and fibre products
+Thepurposeofthissectionistocollect someelementaryfactsabou tfibreproducts. Weconsider
+the category of vector spaces as it will be the ambient category fo r all our pullback diagrams.
+Letπ1:A1→A12andπ2:A2→A12be linear maps. The fibre product of these maps is defined
+by
+A1×
+(π1,π2)A2:={(a1,a2)∈A1×A2|π1(a1) =π2(a2)}. (1.1)
+Together with the canonical projections
+pr1:A1×
+(π1,π2)A2−→A1,pr2:A1×
+(π1,π2)A2−→A2 (1.2)
+itformsauniversal constructioncompletingtheinitiallygiventwolinea rmapsintothefollowing
+commutative diagram:
+A1×
+(π1,π2)A2pr2− −− →A2
+pr1/arrowbtπ2/arrowbt
+A1π1− −− →A12.(1.3)
+Suchdiagramsarecalled pullback diagrams ,andfibreproductsareoftenreferredtoaspullbacks.
+Next, ifπ1:A1→A12andπ2:A2→A12are morphisms of *-algebras, then the fibre
+productA1×(π1,π2)A2is a *-subalgebra of A1×A2. Furthermore, if we consider the pullback
+diagram(1.3) in the categoryof (unital) C∗-algebras, then A1×(π1,π2)A2with its componentwise
+multiplication and *-structure is a (unital) C∗-algebra. Much the same, if Bis an algebra and
+π1:A1→A12andπ2:A2→A12are morphisms of left B-modules, then the fibre product
+A1×(π1,π2)A2is a leftB-module via the componentwise left action b.(a1,a2) = (b.a1,b.a2).
+31.2 Odd-to-even connecting homomorphism in K-theory
+Consider a pullback diagram
+A
+/d119/d119♣♣♣♣♣♣♣♣♣
+/d39/d39◆◆◆◆◆◆◆◆◆
+A1
+π1 /d38/d38 /d38/d38◆◆◆◆◆◆◆◆ A2
+π2 /d120/d120♣♣♣♣♣♣♣♣
+A12(1.4)
+in the category of unital algebras, and assume that one of the defi ning morphisms (here we
+chooseπ1) is surjective. Then there exists a long exact sequence in algebraic K-theory [20]
+··· −→K1(A12)odd-to-even/d47/d47K0(A)−→K0(A1⊕A2)−→K0(A12). (1.5)
+The mapping K 1(A12)odd-to-even/d47/d47K0(A) is obtained asfollows. First, given left Ai-modules
+Ei,i= 1,2, we obtain left A12-modulesπi∗Eidefined byA12⊗AiEi. SinceA12is unital, there
+are canonical morphisms πi∗:Ei→πi∗Ei,πi∗(e) = 1⊗Aie. The modules Eiandπi∗Eican
+also be considered as left modules over the fibre product algebra Avia the left actions given
+bya.ei= pri(a).ei, forei∈Ei, anda.fi=πi(pri(a)).fi, forfi∈πi∗Ei. Assume now that
+h:π1∗E1→π2∗E2is a morphism of left A12-modules. Then h◦π1∗:E1→π2∗E2and
+π2∗:E2→π2∗E2can be lifted to morphisms of left A-modules, and we can consider their
+pullback diagram in the category of left A-modules:
+E1×
+(h◦π1∗,π2∗)E2
+pr1
+/d120/d120qqqqqqqqqqpr2
+/d38/d38◆◆◆◆◆◆◆◆◆◆
+E1
+π1∗
+/d15/d15E2
+π2∗
+/d15/d15
+π1∗E1h/d47/d47π2∗E2.(1.6)
+In [20, Section 2], it is proven in detail that, if Eiis a finitely generated projective module
+overAi,i= 1,2, andhis an isomorphism, then the fibre product M:=E1×(h◦π1∗,π2∗)E2is a
+finitely generated projective A-module. Furthermore, up to isomorphism, every finitely gener-
+ated projective module over Ahas this form, and the Ai-modulesEiand pri∗M:=Ai⊗AM,
+i= 1,2, are naturally isomorphic. In particular, if E1∼=An
+1andE2∼=An
+2, the isomorphism
+h:π1∗E1→π2∗E2is given by an invertible matrix U∈GLn(A12). Using the canonical
+embedding GL n(A12)⊆GL∞(A12), we get a map
+GL∞(A12)∋U/mapsto−→M∈Proj(A) (1.7)
+given by the pullback diagram
+M
+/d117/d117❦❦❦❦❦❦❦❦❦❦❦❦❦
+/d41/d41❙❙❙❙❙❙❙❙❙❙❙❙❙
+An
+1
+π1/d40/d40 /d40/d40PPPPPPPP An
+2
+π2/d118/d118♥♥♥♥♥♥♥♥
+An
+12U∼=An
+12.(1.8)
+4This map induces an odd-to-even connecting homomorphism on the le vel of both algebraic
+[20] andC∗-algebraic [15] K-theory. An explicit description of the module Mis as follows.
+Assume that π1:A1→A12is surjective. Then there exist liftings c,d∈Matn(A1) such
+that evaluating π1oncanddcomponentwise yields U−1andUrespectively. Applying [11,
+Theorem 2.1] to our situation yields E1×(h◦π1∗,π2∗)E2∼=A2np, where
+p=/parenleftbigg(c(2−dc)d,1) (c(2−dc)(1−dc),0)
+((1−dc)d,0) ((1 −dc)2,0)/parenrightbigg
+∈Mat2n(A). (1.9)
+1.3 Principal coactions and associated projective modules
+Recall first the general definition of an entwining structure. Let Cbe a coalgebra with comul-
+tiplication ∆ and counit ε, and letAbe an algebra with multiplication mand unitη. A linear
+map
+ψ:C⊗A−→A⊗C (1.10)
+is called an entwining structure if and only if it is unital, counital, and distributive with respect
+to both the multiplication and comultiplication:
+ψ◦(id⊗m) = (m⊗id)◦(id⊗ψ)◦(ψ⊗id), ψ◦(id⊗η) = (η⊗id)◦flip,(1.11)
+(id⊗∆)◦ψ= (ψ⊗id)◦(id⊗ψ)◦(∆⊗id),(id⊗ε)◦ψ= flip◦(ε⊗id).(1.12)
+Ifψis an entwining of a coalgebra Cand an algebra A, andMis a rightC-comodule and a
+rightA-module, we call Manentwined module [4] when it satisfies the compatibility condition
+(ma)(0)⊗(ma)(1)=m(0)ψ(m(1)⊗a). (1.13)
+Next, letPbe an algebra equipped with a coaction ∆ P:P→P⊗Cof a coalgebra C.
+Define the coaction-invariant subalgebra of Pby
+B:=PcoC:={b∈P|∆P(bp) =b∆P(p) for allp∈P}. (1.14)
+We call the inclusion B⊆PaC-extension. We call it a coalgebra-Galois C-extension when
+the canonical left P-module right C-comodule map
+can:P⊗
+BP−→P⊗C, p⊗
+Bp′/mapsto−→p∆P(p′), (1.15)
+is bijective [5]. Note that the bijectivity of canallows us to define the so-called translation map
+τ:C−→P⊗
+BP, τ(c) :=can−1(1⊗c). (1.16)
+Moreover, every coalgebra-Galois C-extension comes naturally equipped with a unique entwin-
+ing structure that makes Pa (P,C)-entwined module in the sense of (1.13). It is called the
+canonical entwining structure [5], and is very useful in calculations o r further constructions.
+Explicitly, it can be written as:
+ψ(c⊗p) =can(can−1(1⊗c)p). (1.17)
+5An algebra Pwith a right C-coaction ∆ Pis said to be e-coaugmented if and only if there ex-
+istsagroup-likeelement e∈Csuchthat∆ P(1) = 1⊗e. Wecallthe C-extensionB:=PcoC⊆P
+e-coaugmented. (Much the same way, one defines the coaugmenta tion of left coactions.) For
+thee-coaugmented coalgebra-Galois C-extensions, one can show that the coaction-invariant
+subalgebra defined in (1.14) can be expressed as
+PcoC={p∈P|∆P(p) =p⊗e}. (1.18)
+Indeed, Formula (1.17) allows us to express the right coaction in ter ms of the entwining
+∆P(p) =ψ(e⊗p), (1.19)
+and Equation (1.11) yields the right-in-left inclusion. The opposite inc lusion is obvious.
+Next, ifψis invertible, one can use (1.12) to show that the formula
+P∆(p) :=ψ−1(p⊗e) (1.20)
+defines a left coaction P∆ :P→C⊗P. We define the left coaction-invariant subalgebra
+coCPas in (1.14), and derive the left-sided version of (1.17). Hence, for anye-coaugmented
+coalgebra-Galois C-extension with invertible canonical entwining , the right coaction-invariant
+subalgebra coincides with the left coaction-invariant subalgebra:
+PcoC={p∈P|∆P(p) =p⊗e}={p∈P|P∆(p) =e⊗p}=coCP. (1.21)
+Finally,weneedtoassumeonemoreconditionon C-extensionstoobtainasuitabledefinition
+of a principal coaction: equivariant projectivity . It is a pivotal property that guarantees the
+projectivity of associated modules, and thus leads to index pairings betweenK-theory and
+K-homology. Putting together the aforementioned four conditions , we say that a coalgebra
+C-extensionB⊆Pisprincipal [6] if:
+(i) Thecanonicalmap can:P⊗BP→P⊗C,p⊗Bp′/mapsto→p∆P(p′), isbijective(Galoiscondition).
+(ii) The right coaction is e-coaugmented for some group-like e∈C, i.e. ∆ P(1) = 1⊗e.
+(iii) The canonical entwining ψ:C⊗P→P⊗C,c⊗p/mapsto→can(can−1(1⊗c)p), is bijective.
+(iv) The algebra PisC-equivariantly projective as a left B-module, i.e. there exists a left
+B-linear and right C-colinear splitting of the multiplication map B⊗P→P.
+In the framework of coalgebra extensions, the role of connection s on principal bundles is
+played by strong connections [6]. Let Pbe an algebra and both a left and right e-coaugmented
+C-comodule. (Note that the left and right coactions need not commu te.) Astrong connection
+is a linear map ℓ:C→P⊗Psatisfying
+/tildewidercan◦ℓ=1⊗id,(id⊗∆P)◦ℓ=(ℓ⊗id)◦∆,(P∆⊗id)◦ℓ=(id⊗ℓ)◦∆, ℓ(e) = 1⊗1.(1.22)
+6Here/tildewidercan:P⊗P→P⊗Cis the lifting of cantoP⊗P. Assuming that there exists aninvertible
+entwiningψ:C⊗P→P⊗CmakingPan entwined module, the first three equations of (1.22)
+read in the Heyneman-Sweedler type notation c/mapsto→ℓ(c)/angbracketleft1/angbracketright⊗ℓ(c)/angbracketleft2/angbracketrightas follows:
+ℓ(c)/angbracketleft1/angbracketrightψ(e⊗ℓ(c)/angbracketleft2/angbracketright) =ℓ(c)/angbracketleft1/angbracketrightℓ(c)/angbracketleft2/angbracketright
+(0)⊗ℓ(c)/angbracketleft2/angbracketright
+(1)= 1⊗c, (1.23)
+ℓ(c)/angbracketleft1/angbracketright⊗ψ(e⊗ℓ(c)/angbracketleft2/angbracketright) =ℓ(c)/angbracketleft1/angbracketright⊗ℓ(c)/angbracketleft2/angbracketright
+(0)⊗ℓ(c)/angbracketleft2/angbracketright
+(1)=ℓ(c(1))/angbracketleft1/angbracketright⊗ℓ(c(1))/angbracketleft2/angbracketright⊗c(2),(1.24)
+ψ−1(ℓ(c)/angbracketleft1/angbracketright⊗e)⊗ℓ(c)/angbracketleft2/angbracketright=ℓ(c)/angbracketleft1/angbracketright
+(−1)⊗ℓ(c)/angbracketleft1/angbracketright
+(0)⊗ℓ(c)/angbracketleft2/angbracketright=c(1)⊗ℓ(c(2))/angbracketleft1/angbracketright⊗ℓ(c(2))/angbracketleft2/angbracketright.(1.25)
+Applying id ⊗εto (1.23) yields a useful formula
+ℓ(c)/angbracketleft1/angbracketrightℓ(c)/angbracketleft2/angbracketright=ε(c). (1.26)
+It is worthwhile to observe the left-right symmetry of principal ext ensions. We already
+noted (see (1.21)) the equality of the left and right coaction-invar iant subalgebras. Now let us
+define the left canonical map as
+canL:P⊗
+BP∋p⊗q/mapsto−→p(−1)⊗p(0)q∈C⊗P. (1.27)
+One can check that it is related to the right canonical map canby the formula [7]
+ψ◦canL=can. (1.28)
+Also, ifℓis a strong connection and /tildewidercanL:= (id⊗m)◦(P∆⊗id) is the lifted left canonical
+map, then /tildewidercanL◦ℓ= id⊗1. Hence
+c⊗p/mapsto−→ℓ(c)/angbracketleft1/angbracketright⊗ℓ(c)/angbracketleft2/angbracketrightp (1.29)
+is a splitting of /tildewidercanLjust as
+p⊗c/mapsto−→pℓ(c)/angbracketleft1/angbracketright⊗ℓ(c)/angbracketleft2/angbracketright(1.30)
+is a splitting of /tildewidercan.
+Lemma 1.1. LetPbe an object in the categoryC
+eAlgC
+eof all unital algebras with e-coaugmented
+left and right C-coactions. Assume that there exists an invertible entwini ngψ:C⊗P→P⊗C
+makingPan entwined module. Then, if Padmits a strong connection ℓ, it is principal.
+Proof.Following [6], first we argue that
+σ:P∋p/mapsto−→p(0)ℓ(p(1))/angbracketleft1/angbracketright⊗ℓ(p(1))/angbracketleft2/angbracketright∈B⊗P (1.31)
+is a leftB-linear splitting of the multiplication map. Indeed, m◦σ= id because of (1.26), and
+the calculation
+ψ(e⊗p(0)ℓ(p(1))/angbracketleft1/angbracketright)⊗ℓ(p(1))/angbracketleft2/angbracketright=p(0)ℓ(p(1))/angbracketleft1/angbracketright⊗e⊗ℓ(p(1))/angbracketleft2/angbracketright(1.32)
+obtained using (1.11) proves that σ(P)⊆B⊗P. This splitting is evidently right C-colinear,
+so that its existence proves the equivariant projectivity.
+7Next, let us check that the formula
+can−1:P⊗C−→P⊗
+BP, p⊗c/mapsto−→pℓ(c)/angbracketleft1/angbracketright⊗
+Bℓ(c)/angbracketleft2/angbracketright, (1.33)
+defines the inverse of the canonical map can, so that the coaction of Cis Galois. It follows from
+(1.23) that
+can(can−1(p⊗c)) =pℓ(c)/angbracketleft1/angbracketrightℓ(c)/angbracketleft2/angbracketright
+(0)⊗ℓ(c)/angbracketleft2/angbracketright
+(1)=p⊗c. (1.34)
+On the other hand, taking advantage of (1.26) and (1.31), we see t hat
+can−1(can(p⊗
+Bq)) =pq(0)ℓ(q(1))/angbracketleft1/angbracketright⊗
+Bℓ(q(1))/angbracketleft2/angbracketright=p⊗
+Bq(0)ℓ(q(1))/angbracketleft1/angbracketrightℓ(q(1))/angbracketleft2/angbracketright=p⊗
+Bq.(1.35)
+Thus the conditions (i) and (iv) of the principality of a C-extension are satisfied. Finally,
+Condition(ii)issimplyassumed, andCondition(iii)followsfromtheunique ness ofanentwining
+that makes Pan entwined module.
+Note that, if there exists a strong connection ℓ, then (1.33) yields
+τ(c) =ℓ(c)/angbracketleft1/angbracketright⊗
+Bℓ(c)/angbracketleft2/angbracketright. (1.36)
+In the Heyneman-Sweedler type notation, we write τ(c) =τ(c)[1]⊗Bτ(c)[2]. Then the canonical
+entwining reads
+ψ(c⊗p) =τ(c)[1](τ(c)[2]p)(0)⊗(τ(c)[2]p)(1)=ℓ(c)/angbracketleft1/angbracketright(ℓ(c)/angbracketleft2/angbracketrightp)(0)⊗(ℓ(c)/angbracketleft2/angbracketrightp)(1).(1.37)
+Remark 1.2. In [6], there is the converse statement: if Pis principal, it admits a strong
+connection. Thus principal extensions can be characterized as th ese that admit a strong con-
+nection.
+Recall now that classical principal bundles can be viewed as functor s transforming finite-
+dimensional vector spaces into associated vector bundles. Analog ously, one can prove that a
+principalC-extensionB⊆Pdefines a functor from the category of finite-dimensional left C-
+comodules into the category of finitely generated projective left B-modules [6]. Explicitly, if V
+is a leftC-comodule with coaction V∆, this functor assigns to it the cotensor product
+P✷
+CV:={/summationtext
+ipi⊗vi∈P⊗V|/summationtext
+i∆P(pi)⊗vi=/summationtext
+ipi⊗V∆(vi)}.(1.38)
+In particular, if g∈Cis a group-like element, the formula C∆(1) :=g⊗1 defines a 1-dimen-
+sional corepresentation, and
+P✷
+CC={p∈P|∆P(p)=p⊗g}=:Pg (1.39)
+can be viewed as a noncommutative associated complex line bundle. Th en the general formula
+for computing an idempotent Egof the associated module Pgout of a corepresentation and a
+strong connection becomes a very simple special case of [6, Theore m 3.1]:
+Pg∼=BnEg,(Eg)n
+i,j=1:=/parenleftbig
+gR
+igL
+j/parenrightbign
+i,j=1, ℓ(g) =:n/summationdisplay
+k=1gL
+k⊗gR
+k∈Pg−1⊗Pg.(1.40)
+8A fundamental special case of principal extensions is provided by principal comodule alge-
+bras. One assumes then that C=His a Hopf algebra with bijective antipode S, the canonical
+map is bijective, and Pis anH-equivariantly projective left B-module. This brings us in touch
+with compact quantum groups. Assume that ¯His theC∗-algebra of a compact quantum group
+in the sense of Woronowicz [30,32], and His its dense Hopf *-subalgebra spanned by the ma-
+trix coefficients of the irreducible unitary corepresentations. Let ¯Pbe a unital C∗-algebra and
+δ:¯P→¯P¯⊗¯Han injective C∗-algebraic right coaction of ¯Hon¯P. (See [1, Definition 0.2] for a
+general definition and [3, Definition 1] for the special case of compa ct quantum groups.) Here
+¯⊗denotes the minimal C∗-completion of the algebraic tensor product ¯P⊗¯H.
+To extend Woronowicz’s Peter-Weyl theory [32] from compact qua ntum groups to compact
+quantum principal bundles, one defines [2] the subalgebra Pδ(¯P)⊆¯Pof elements for which the
+coaction lands in ¯P⊗H, i.e.
+Pδ(¯P) :={p∈¯P|δ(p)∈P⊗H}. (1.41)
+One easily checks that it is an H-comodule algebra. We call Pδ(¯P) thePeter-Weyl comodule
+algebraassociated to the C∗-coactionδ. It follows from results of [3] and [25] that Pδ(¯P) is
+dense in ¯P. Also, it is straightforward to verify [2] that the operation ¯P/mapsto→ Pδ(¯P) gives a
+functor commuting with taking fibre products (pullbacks), and tha tPδ(¯P)coHcoincides with
+theC∗-algebra ¯Pco¯H.
+Finally, let us remark that, for a compact Hausdorff topological gro upGand a unital C∗-
+algebraA, we can use the isomorphism A¯⊗C(G)∼=C(G,A) (e.g. see [29, Corollary T.6.17])
+to translate a right C(G)-coactionδinto aG-actionχ:G∋g/mapsto→χg∈Aut(A) as follows:
+δ:A−→A¯⊗C(G)∼=C(G,A), δ(a)(g) =:χg(a). (1.42)
+Thus we can use the terminology of right C(G)-comodule C∗-algebras and G-C∗-algebras syn-
+onymously. It is important to bear in mind that the Peter-Weyl func tor mapsG-equivariant
+*-homomorphisms to colinear homomorphisms of right O(G)-comodule algebras [2].
+1.4 Standard Hopf fibration of quantum SU(2)
+The standard quantum Hopf fibration is given by an action of U(1) on the quantum group
+SUq(2),q∈(0,1). The coordinate ring of O(SUq(2)) is generated by α,β,γ,δwith relations
+αβ=qβα, αγ =qγα, βδ =qδβ, γδ =qδγ, βγ =γβ, (1.43)
+αδ−qβγ= 1, δα−q−1βγ= 1, (1.44)
+and involution α∗=δ,β∗=−qγ. It is a Hopf *-algebra with comultiplication ∆, counit ε, and
+antipodeSgiven by
+∆(α) =α⊗α+β⊗γ,∆(β) =α⊗β+β⊗δ, (1.45)
+∆(γ) =γ⊗α+δ⊗γ,∆(δ) =γ⊗β+δ⊗δ, (1.46)
+ε(α) =ε(δ) = 1, ε(β) =ε(γ) = 0, (1.47)
+S(α) =δ, S(β) =−q−1β, S(γ) =−qγ, S(δ) =α. (1.48)
+9LetO(U(1)) denote the commutative and cocommutative Peter-Weyl H opf *-algebra of
+U(1), and let ustand for its unitary group-like generator. Note that the counit εand the
+antipodeSsatisfyε(u) = 1 andS(u) =u∗. There is a Hopf *-algebra surjection
+π:O(SUq(2))−→ O(U(1)), π(α) =u, π(δ) =u−1, π(β) =π(γ) = 0.(1.49)
+Setting ∆ R:= (id⊗π)◦∆, we obtain a right O(U(1))-coaction on O(SUq(2)). On generators,
+the coaction reads
+∆R(α)=α⊗u,∆R(β)=β⊗u−1,∆R(γ)=γ⊗u,∆R(δ)=δ⊗u−1. (1.50)
+The *-subalgebra of coaction invariants defines the coordinate rin g of the standard Podle´ s
+quantum sphere:
+O(S2
+q) :=O(SUq(2))coO(U(1))={a∈ O(SUq(2))|∆R(a) =a⊗1}. (1.51)
+One can prove (see [24]) that O(S2
+q) is isomorphic to the *-algebra generated by Band
+hermiteanAsatisfying the relations
+AB=q2BA, B∗B=A−A2, BB∗=q2A−q4A2. (1.52)
+Anisomorphism is explicitly given by theformulas A=−q−1βγandB=−βα. The irreducible
+Hilbert space representations of O(S2
+q) are given by
+ρ0(A) =ρ0(B) = 0, ρ0(1) = 1 on H=C, (1.53)
+ρ+(A)en=q2nen, ρ+(B)en=qn(1−q2n)1/2en−1onH=ℓ2(N),(1.54)
+where{en|n= 0,1,...}is an orthonormal basis of ℓ2(N).
+Recall that the universal C∗-algebra of a complex *-algebra is the C∗-completion with
+respect to the universal C∗-norm given by the supremum (if it exists) of the operator norms
+over all bounded *-representations. Let C(S2
+q) denote the universal C∗-algebra generated by A
+andBsatisfying (1.52). From the above representations, it follows that C(S2
+q) is the minimal
+unitalization of K(ℓ2(N)), that is,
+C(S2
+q)∼=K(ℓ2(N))⊕C⊆ B(ℓ2(N)), (1.55)
+(k+α)(k′+α′) = (kk′+α′k+αk′)+αα′, k,k′∈ K(ℓ2(N)), α,α′∈C.(1.56)
+HereK(ℓ2(N)) andB(ℓ2(N)) denote the C∗-algebras of compact and bounded operators respec-
+tively on the Hilbert space ℓ2(N). The isomorphism (1.55) implies that K 0(C(S2
+q))∼=Z⊕Z,
+where one generator of K-theory is given by the class of the unit 1 ∈C(S2
+q), and the other by
+the class of the 1-dimensional projection onto Ce0⊆ℓ2(N).
+Furthermore, K0(C(S2
+q))∼=Z⊕Z. Weidentifyonegeneratorof K-homologywiththeclassof
+thepairofrepresentations [(id ,ε)], where id( k+α) =k+αandε(k+α) =αforallk∈ K(ℓ2(N))
+andα∈C. The other generator can be given by the class of the pair of repre sentations [( ε,ε0)]
+with the (non-unital) representation ε0ofK(ℓ2(N))⊕Cdefined byε0(k+α) =αSS∗, where
+S :ℓ2(N)−→ℓ2(N),Sen=en+1, (1.57)
+10denotes the unilateral shift on ℓ2(N). (See [19] for a detailed treatment of the K-homology and
+K-theory of Podle´ s quantum spheres.)
+We shall also consider the coordinate ring of the quantum disc O(Dq) generated by zand
+z∗with the relation
+z∗z−q2zz∗= 1−q2. (1.58)
+Its bounded irreducible Hilbert space representations are given by
+µθ(z) = eiθonH=C, θ∈[0,2π), (1.59)
+µ(z)en= (1−q2(n+1))1/2en+1onH=ℓ2(N). (1.60)
+It has been shown in [16] that the universal C∗-algebra of O(Dq) is isomorphic to the Toeplitz
+algebra given as the C∗-algebra generated by the unilateral shift S of Equation (1.57). Th e
+representation µdefines then an embedding of O(Dq) intoT.
+LetC(U(1)) denote the C∗-algebra of continuous functions on the unit circle S1, and letu
+be its unitary generator. The Toeplitz algebra gives rise to the follow ing short exact sequence
+ofC∗-algebras:
+0−→ K(ℓ2(N))−→ Tσ−→C(U(1))−→0. (1.61)
+Here the so-called symbol map σ:T →C(U(1)) is given by σ(S) =u. Since S −µ(z) belongs
+toK(ℓ2(N)), it follows in particular that σ(µ(z)) =u.
+Now let us consider the associated quantum line bundles as finitely gen erated projective
+modules. They are defined by the 1-dimensional corepresentation sC∋1/mapsto→uN⊗1,N∈Z, as
+cotensor products (1.39):
+MN:={p∈ O(SUq(2))|∆R(p) =p⊗uN}. (1.62)
+Since ∆ Ris a morphism of algebras, MNis anO(S2
+q)-bimodule. Our next step is to determine
+explicitly projections describing these projective modules.
+Forl∈1
+2Nandi,j=−l,−l+1,...,l, lettl
+i,jdenote the matrix elements of the irreducible
+unitary corepresentations of O(SUq(2)), that is,
+∆(tl
+i,j) =l/summationdisplay
+k=−ltl
+i,k⊗tl
+k,j,l/summationdisplay
+k=−ltl∗
+k,itl
+k,j=l/summationdisplay
+k=−ltl
+i,ktl∗
+j,k=δij. (1.63)
+By the Peter-Weyl theorem for compact quantum groups [31], O(SUq(2)) =⊕l∈1
+2N⊕l
+i,j=−lCtl
+i,j.
+From the explicit description of tl
+i,j[17, Section 4.2.4] and the definition of ∆ R, it follows that
+∆R(tl
+i,j) =tl
+i,j⊗u−2j, so thattl
+i,−j∈M2j. It can be shown [14,26] that t|j|
+i,−j,i=−|j|,...,|j|
+generateM2jas a leftO(S2
+q)-module and M2j∼=O(S2
+q)2|j|+1E2jfor allj∈1
+2Z, where
+E2j=
+t|j|
+−|j|,−j...
+t|j|
+|j|,−j
+/parenleftig
+t|j|∗
+−|j|,−j,···, t|j|∗
+|j|,−j/parenrightig
+∈Mat2|j|+1(O(S2
+q)). (1.64)
+It is clear that E2
+2j=E2jdue to (1.63) and E∗
+2j=E2j, so thatE2jis a projection.
+112 Principality of one-surjective pullbacks
+We begin by defining an ambient category for pullback diagrams appea ring in the second part
+of this section. Let Pbe a unital algebra equipped with both a right coaction ∆ P:P→P⊗C
+and a left coaction P∆ :P→C⊗Pof the same coalgebra C. We do notassume that
+these coactions commute, but we do assume that they are coaugm ented by the same group-like
+elemente∈C, i.e., ∆ P(1) = 1⊗eandP∆(1) =e⊗1. For a fixed coalgebra Cand a group-like
+e∈C, we consider the categoryC
+eAlgC
+eof all such unital algebras with e-coaugmented left and
+rightC-coactions. Here morphisms are bicolinear algebra homomorphisms.
+Since we work over a field, this category is evidently closed under any pullbacks. If
+π1:P1→P12andπ2:P2→P12are morphisms inC
+eAlgC
+e, then the fibre product alge-
+braP:=P1×(π1,π2)P2becomes a right C-comodule via
+∆P(p,q) = (p(0),0)⊗p(1)+(0,q(0))⊗q(1), (2.1)
+and a leftC-comodule via
+P∆(p,q) =p(−1)⊗(p(0),0)+q(−1)⊗(0,q(0)). (2.2)
+Also, it is clear that ∆ P(1,1) = (1,1)⊗eandP∆(1,1) =e⊗(1,1).
+2.1 Principality of images and preimages
+In the following lemma, we prove that any surjective morphism inC
+eAlgC
+ewhose domain is
+a principal extension can be split by a left colinear map and by a right co linear map (not
+necessarily by a bicolinear map). Note that the first part of the lemm a is proved much the
+same way as in the Hopf-Galois case [13, Lemma 3.1]:
+Lemma 2.1. Letπ:P→Qbe a surjective morphism in the categoryC
+eAlgC
+eof unital algebras
+withe-coaugmented left and right C-coactions. If Pis principal, then:
+(i) The induced map πcoC:PcoC→QcoCis surjective.
+(ii) There exists a unital right C-colinear splitting of π.
+(iii) There exists a unital left C-colinear splitting of π.
+(iv)Qis principal.
+Furthermore, if Q′∈C
+eAlgC
+e,Q′⊆Q, is principal, then so is π−1(Q′).
+Proof.It follows from the right colinearity and surjectivity of πthatπ(PcoC)⊆QcoC. To prove
+the converse inclusion, we take advantage of the left PcoC-linear retraction of the inclusion
+PcoC⊆Pthat was used to prove [6, Theorem 2.5(3)]:
+σϕ:P−→PcoC, σϕ(p) :=p(0)ℓ(p(1))/angbracketleft1/angbracketrightϕ(ℓ(p(1))/angbracketleft2/angbracketright). (2.3)
+12Hereℓis a strong connection on Pandϕis any unital linear functional on P. It follows from
+(1.31) that σϕ(p)∈PcoC. Ifπ(p)∈QcoC, thenσϕ(p) is a desired element of PcoCthat is
+mapped by πtoπ(p). Indeed, since π(p(0))⊗p(1)=π(p)(0)⊗π(p)(1)=π(p)⊗e, using the
+unitality of π,ϕ, andℓ(e) = 1⊗1, we compute
+π(σϕ(p)) =π(p(0))π(ℓ(p(1))/angbracketleft1/angbracketright)ϕ(ℓ(p(1))/angbracketleft2/angbracketright) =π(p). (2.4)
+To show the second assertion, let us choose any unital k-linear splitting of π↾PcoCand denote
+it byαcoC. We want to prove that the formula
+αR(q) :=αcoC(q(0)π(ℓ(q(1))/angbracketleft1/angbracketright))ℓ(q(1))/angbracketleft2/angbracketright(2.5)
+defines a unital right colinear splitting of π. Sinceπis surjective, we can write q=π(p). Then,
+using properties of π, we obtain:
+q(0)π(ℓ(q(1))/angbracketleft1/angbracketright)⊗ℓ(q(1))/angbracketleft2/angbracketright=π(p)(0)π(ℓ(π(p)(1))/angbracketleft1/angbracketright)⊗ℓ(π(p)(1))/angbracketleft2/angbracketright
+=π(p(0))π(ℓ(p(1))/angbracketleft1/angbracketright)⊗ℓ(p(1))/angbracketleft2/angbracketright
+=π(p(0)ℓ(p(1))/angbracketleft1/angbracketright)⊗ℓ(p(1))/angbracketleft2/angbracketright. (2.6)
+Now it follows from (1.31) that the above tensor is in QcoC⊗P. HenceαRis well defined.
+It is straightforward to verify that αRis unital, right colinear, and splits π. (Note that, since
+q∈QcoCimpliesq(0)⊗q(1)=q⊗e, we haveαcoC=αR↾QcoC.) The third assertion can be proven
+in an analogous manner.
+To prove (iv), we first show that the inverse of the canonical map canQ:Q⊗QcoCQ→Q⊗C
+(see (1.15)) is given by
+can−1
+Q:Q⊗C−→Q⊗
+QcoCQ, q⊗c/mapsto−→qπ(ℓ(c)/angbracketleft1/angbracketright)⊗
+QcoCπ(ℓ(c)/angbracketleft2/angbracketright). (2.7)
+Using the properties of πandℓ, we get
+(canQ◦can−1
+Q)(π(p)⊗c) =canQ/parenleftig
+π(pℓ(c)/angbracketleft1/angbracketright)⊗
+QcoCπ(ℓ(c)/angbracketleft2/angbracketright)/parenrightig
+=π/parenleftig
+pℓ(c)/angbracketleft1/angbracketrightℓ(c)/angbracketleft2/angbracketright
+(0)/parenrightig
+⊗ℓ(c)/angbracketleft2/angbracketright
+(1)
+=π(p)⊗c. (2.8)
+Similarly,
+(can−1
+Q◦canQ)/parenleftig
+π(p)⊗
+QcoCπ(p′)/parenrightig
+=can−1
+Q/parenleftig
+π(pp′
+(0))⊗p′
+(1)/parenrightig
+=π(pp′
+(0)ℓ(p′
+(1))/angbracketleft1/angbracketright)⊗
+QcoCπ(ℓ(p′
+(1))/angbracketleft2/angbracketright) (2.9)
+=π(p)⊗
+QcoCπ(p′
+(0)ℓ(p′
+(1))/angbracketleft1/angbracketrightℓ(p′
+(1))/angbracketleft2/angbracketright)
+=π(p)⊗
+QcoCπ(p′). (2.10)
+Hereweusedthefactthat π(p′
+(0)ℓ(p′
+(1))/angbracketleft1/angbracketright)⊗ℓ(p′
+(1))/angbracketleft2/angbracketright∈QcoC⊗P. Hencetheextension QcoC⊆Q
+is Galois, and we have the canonical entwining ψQ:C⊗Q→Q⊗C.
+13Our next aim is to prove that ψQis bijective. We know by assumption that the canonical
+entwiningψP:C⊗P→P⊗Cis invertible. To determine its inverse, recall that the left
+and right coactions are given by ψ−1
+P(p⊗e) andψP(e⊗p), respectively. Then apply (1.11) to
+compute
+ψP/parenleftig
+(pℓ(c)/angbracketleft1/angbracketright)(−1)⊗(pℓ(c)/angbracketleft1/angbracketright)(0)ℓ(c)/angbracketleft2/angbracketright/parenrightig
+=pℓ(c)/angbracketleft1/angbracketrightψP/parenleftig
+e⊗ℓ(c)/angbracketleft2/angbracketright/parenrightig
+=pℓ(c)/angbracketleft1/angbracketrightℓ(c)/angbracketleft2/angbracketright
+(0)⊗ℓ(c)/angbracketleft2/angbracketright
+(1)
+=p⊗c. (2.11)
+Henceψ−1
+P(p⊗c) = (pℓ(c)/angbracketleft1/angbracketright)(−1)⊗(pℓ(c)/angbracketleft1/angbracketright)(0)ℓ(c)/angbracketleft2/angbracketright. On the other hand,
+ψQ(c⊗π(p)) =π(ℓ(c)/angbracketleft1/angbracketright)/parenleftbig
+π(ℓ(c)/angbracketleft2/angbracketright)π(p)/parenrightbig
+(0)⊗/parenleftbig
+π(ℓ(c)/angbracketleft2/angbracketright)π(p)/parenrightbig
+(1)
+=π/parenleftbig
+ℓ(c)/angbracketleft1/angbracketright/parenrightbig
+π/parenleftbig
+(ℓ(c)/angbracketleft2/angbracketrightp)(0)/parenrightbig
+⊗(ℓ(c)/angbracketleft2/angbracketrightp)(1)
+= (π⊗id)/parenleftbig
+ψP(c⊗p)/parenrightbig
+, (2.12)
+(id⊗π)/parenleftbig
+ψ−1
+P(p⊗c)/parenrightbig
+= (pℓ(c)/angbracketleft1/angbracketright)(−1)⊗π/parenleftbig
+(pℓ(c)/angbracketleft1/angbracketright)(0)/parenrightbig
+π/parenleftbig
+ℓ(c)/angbracketleft2/angbracketright/parenrightbig
+=/parenleftbig
+π(pℓ(c)/angbracketleft1/angbracketright)/parenrightbig
+(−1)⊗/parenleftbig
+π(pℓ(c)/angbracketleft1/angbracketright)/parenrightbig
+(0)π(ℓ(c)/angbracketleft2/angbracketright)
+=Q∆(π(p)π(ℓ(c)/angbracketleft1/angbracketright))π(ℓ(c)/angbracketleft2/angbracketright). (2.13)
+The second part of the above computation implies that the assignme nt
+ψ−1
+Q:Q⊗C−→C⊗Q, π(p)⊗c/mapsto−→(id⊗π)(ψ−1
+P(p⊗c)) (2.14)
+is well defined. Now it follows from the first part that ψ−1
+Qis the inverse of ψQ:
+ψQ/parenleftig
+ψ−1
+Q/parenleftbig
+π(p)⊗c/parenrightbig/parenrightig
+=ψQ/parenleftig
+(id⊗π)/parenleftbig
+ψ−1
+P(p⊗c)/parenrightbig/parenrightig
+= (π⊗id)/parenleftig
+ψP/parenleftbig
+ψ−1
+P(p⊗c)/parenrightbig/parenrightig
+=π(p)⊗c,(2.15)
+ψ−1
+Q/parenleftig
+ψQ/parenleftbig
+c⊗π(p)/parenrightbig/parenrightig
+=ψ−1
+Q/parenleftig
+(π⊗id)/parenleftbig
+ψP(c⊗p)/parenrightbig/parenrightig
+= (id⊗π)/parenleftig
+ψ−1
+P/parenleftbig
+ψP(c⊗p)/parenrightbig/parenrightig
+=c⊗π(p).(2.16)
+On the other hand, we observe that ( π⊗π)◦ℓis a strong connection on Q. Combined with
+the just proven existence of a bijective entwining that makes Qan entwined module, it allows
+us to apply Lemma 1.1 and conclude the proof of ( iv).
+To prove the final statement of the lemma, note first that π−1(Q′)∈C
+eAlgC
+e. Next, observe
+that, ifℓ′:C→Q′⊗Q′is a strong connection on Q′, then it is also a strong connection on Q.
+Now, it follows from (1.37) that for any q∈Q′
+ψQ(c⊗q) =ℓ′(c)/angbracketleft1/angbracketright/parenleftbig
+ℓ′(c)/angbracketleft2/angbracketrightq/parenrightbig
+(0)⊗/parenleftbig
+ℓ′(c)/angbracketleft2/angbracketrightq/parenrightbig
+(1)∈Q′⊗C. (2.17)
+Much the same way, it follows from the Q-analog of the formula following (2.11) that
+ψ−1
+Q(Q′⊗C)⊆C⊗Q′. Hence to see that ψPandψ−1
+Prestrict to π−1(Q′), we can apply
+(2.12) and (2.14), respectively.
+A key step now is to construct a strong connection on π−1(Q′). LetαRandαLbe, respec-
+tively, right and left colinear unital splittings of π. Their existence is guaranteed by the already
+proven (ii) and (iii). The map ( αL⊗αR)◦ℓ′:C→π−1(Q)⊗π−1(Q) is bicolinear and satisfies
+αL(ℓ′(e)/angbracketleft1/angbracketright)⊗αR(ℓ′(e)/angbracketleft2/angbracketright) = 1⊗1. (2.18)
+14However,
+1⊗c−/parenleftbig/tildewidercan◦(αL⊗αR)◦ℓ′/parenrightbig
+(c) = 1⊗c−αL(ℓ′(c(1))/angbracketleft1/angbracketright)αR(ℓ′(c(1))/angbracketleft2/angbracketright)⊗c(2)/\e}atio\slash= 0.(2.19)
+To solve this problem, we apply to it the splitting of the lifted canonical map given by a strong
+connection ℓ(see (1.30)), and add to ( αL⊗αR)◦ℓ′:
+ℓR(c) := (αL⊗αR)(ℓ′(c))+ℓ(c)−αL(ℓ′(c(1))/angbracketleft1/angbracketright)αR(ℓ′(c(1))/angbracketleft2/angbracketright)ℓ(c/angbracketleft1/angbracketright
+(2))⊗ℓ(c/angbracketleft2/angbracketright
+(2)).(2.20)
+Now/tildewidercan◦ℓR= 1⊗id, as needed. Also, ℓR(e) = 1⊗1 and ((π⊗id)◦ℓR)(C)⊆Q′⊗P. The
+right colinearity of ℓRis clear. To check the left colinearity of ℓR, using the fact that Pis aψP
+entwined and e-coaugmented module, we show that ( mP◦(αL⊗αR)◦ℓ′)∗ℓis left colinear.
+(HeremPis the multiplication of P.) First we note that
+(P∆⊗id)◦((mP◦(αL⊗αR)◦ℓ′)∗ℓ)=(id⊗(mP◦(αL⊗αR)◦ℓ′)∗ℓ)◦∆ (2.21)
+is equivalent to
+αL(ℓ′(c(1))/angbracketleft1/angbracketright)αR(ℓ′(c(1))/angbracketleft2/angbracketright)ℓ(c(2))/angbracketleft1/angbracketright⊗e⊗ℓ(c(2))/angbracketleft2/angbracketright
+=ψP/parenleftig
+c(1)⊗αL(ℓ′(c(2))/angbracketleft1/angbracketright)αR(ℓ′(c(2))/angbracketleft2/angbracketright)ℓ(c(3))/angbracketleft1/angbracketright/parenrightig
+⊗ℓ(c(3))/angbracketleft2/angbracketright.(2.22)
+Sincec(1)⊗αL(ℓ′(c(2))/angbracketleft1/angbracketright)⊗ℓ′(c(2))/angbracketleft2/angbracketright=ψ−1
+P/parenleftbig
+αL(ℓ′(c)/angbracketleft1/angbracketright)⊗e/parenrightbig
+⊗ℓ′(c)/angbracketleft2/angbracketright, we obtain
+ψP/parenleftig
+c(1)⊗αL(ℓ′(c(2))/angbracketleft1/angbracketright)αR(ℓ′(c(2))/angbracketleft2/angbracketright)ℓ(c(3))/angbracketleft1/angbracketright/parenrightig
+⊗ℓ(c(3))/angbracketleft2/angbracketright
+=αL(ℓ′(c(1))/angbracketleft1/angbracketright)ψP/parenleftig
+e⊗αR(ℓ′(c(1))/angbracketleft2/angbracketright)ℓ(c(2))/angbracketleft1/angbracketright/parenrightig
+⊗ℓ(c(2))/angbracketleft2/angbracketright
+=αL(ℓ′(c(1))/angbracketleft1/angbracketright)αR(ℓ′(c(1))/angbracketleft2/angbracketright)ψP/parenleftig
+c(2)⊗ℓ(c(3))/angbracketleft1/angbracketright/parenrightig
+⊗ℓ(c(3))/angbracketleft2/angbracketright
+=αL(ℓ′(c(1))/angbracketleft1/angbracketright)αR(ℓ′(c(1))/angbracketleft2/angbracketright)ψP/parenleftig
+ψ−1
+P/parenleftbig
+ℓ(c(2))/angbracketleft1/angbracketright⊗e/parenrightbig/parenrightig
+⊗ℓ(c(2))/angbracketleft2/angbracketright
+=αL(ℓ′(c(1))/angbracketleft1/angbracketright)αR(ℓ′(c(1))/angbracketleft2/angbracketright)ℓ(c(2))/angbracketleft1/angbracketright⊗e⊗ℓ(c(2))/angbracketleft2/angbracketright. (2.23)
+HenceℓRis a strong connection with the property ℓR(C)⊆π−1(Q′)⊗P. In a similar
+manner, we construct a strong connection ℓLwith the property ℓL(C)⊂P⊗π−1(Q′). Now we
+need to apply the splitting of the left lifted canonical map given by ℓ(see (1.29)) to derive the
+formula
+ℓL:= (αL⊗αR)◦ℓ′+ℓ−ℓ∗(mP◦(αL⊗αR)◦ℓ′). (2.24)
+It is clear that ℓL(e) = 1⊗1 andℓL(C)⊆P⊗π−1(Q′). A computation similar to (2.23)
+shows the right colinearity of ℓL. Since furthermore ψP(1⊗c) =c⊗1 for anyc∈Cand
+/tildewidercan=ψP◦/tildewidercanL, we obtain
+/tildewidercan(ℓL(c)) =ψP/parenleftbig/tildewidercanL(ℓ(c))/parenrightbig
+=ψP(c⊗1) = 1⊗c. (2.25)
+HenceℓLis a desired strong connection. Plugging it into (2.20) instead of ℓ, we get a strong
+connection
+ℓLR= (αL⊗αR)◦ℓ′+ℓL−(mp◦(αL⊗αR)◦ℓ′)∗ℓL (2.26)
+with the property ℓLR⊆π−1(Q′)⊗π−1(Q′). Applying now Lemma 1.1 ends the proof of this
+lemma.
+152.2 The one-surjective pullbacks of principal coactions ar e principal
+Our goal now is to show that the subcategory of principal extensio ns is closed under one-
+surjective pullbacks. Here the right coaction is the coaction definin g a principal extension and
+the left coaction is the one defined by the inverse of the canonical e ntwining (see (1.20)). With
+this structure, principal extensions form a full subcategory ofC
+eAlgC
+e. The following theorem is
+the main result of this paper generalizing the theorem of [13] on the p ullback of surjections of
+principal comodule algebras:
+Theorem 2.2. LetCbe a coalgebra, e∈Ca group-like element, and Pthe pullback of
+π1:P1→P12andπ2:P2→P12in the categoryC
+eAlgC
+eof unital algebras with e-coaugmentedleft
+and rightC-coactions. If π1orπ2is surjective and both P1andP2are principal e-coaugmented
+C-extensions, then also Pis a principal e-coaugmented C-extension.
+Proof.Without loss of generality, we assume that π1is surjective. We first show that Pinherits
+an entwined structure from P1andP2.
+Lemma 2.3. Letψ1andψ2denote the entwining structures of P1andP2, respectively. Then
+Pis an entwined module with an invertible entwining structur e
+ψ=ψ1◦(id⊗pr1)+ψ2◦(id⊗pr2). (2.27)
+Herepr1andpr2are morphisms of the pullback diagram as in (1.3).
+Proof.Our strategy is to construct a bijective map ˜ψ:C⊗(P1×P2)→(P1×P2)⊗C, and to
+show that it restricts to a bijective entwining on C⊗P. We put
+˜ψ:=ψ1◦(id⊗˜ pr1)+ψ2◦(id⊗˜ pr2). (2.28)
+The symbols ˜ pr1and ˜ pr2stand for respective componentwise projections. Their restrict ions to
+Pyield pr1and pr2. It is easy to check that the inverse of ˜ψis given by
+˜ψ−1=ψ−1
+1◦(˜ pr1⊗id)+ψ−1
+2◦(˜ pr2⊗id) (2.29)
+To show that ˜ψ(C⊗P)⊆P⊗Cand˜ψ−1(P⊗C)⊆C⊗P, we note first that P12andπ2(P2)
+are principal by Lemma 2.1(iv). Consequently, their canonical entw iningsψ12andψπ2(P2)are
+bijective. Furthermore, arguing as in the proof of Lemma 2.1, we se e thatψπ2(P2)=ψ12↾C⊗π2(P2)
+andψ−1
+π2(P2)=ψ−1
+12↾π2(P2)⊗C. An advantage of having both summands in terms of ψ12is that we
+can apply (2.12) to compute
+/parenleftbig
+(π1◦˜ pr1−π2◦˜ pr2)⊗id/parenrightbig
+◦˜ψ= (π1◦˜ pr1⊗id)◦ψ1◦(id⊗˜ pr1)−(π2◦˜ pr2⊗id)◦ψ1◦(id⊗˜ pr1)
++ (π1◦˜ pr1⊗id)◦ψ2◦(id⊗˜ pr2)−(π2◦˜ pr2⊗id)◦ψ2◦(id⊗˜ pr2)
+= (π1⊗id)◦ψ1◦(id⊗˜ pr1)−(π2⊗id)◦ψ2◦(id⊗˜ pr2)
+=ψ12◦(id⊗π1)◦(id⊗˜ pr1)−ψπ2(P2)◦(id⊗π2)◦(id⊗˜ pr2)
+=ψ12◦/parenleftbig
+id⊗(π1◦˜ pr1−π2◦˜ pr2)/parenrightbig
+. (2.30)
+Hence˜ψ(C⊗P)⊆P⊗C. Much the same way, using (2.14) instead of (2.12), we show that
+the bijection ˜ψ−1(P⊗C)⊆C⊗P.
+16It remains to verify that the bijection ψ=˜ψ↾C⊗Pis an entwining that makes Pan entwined
+module. The former is proven by a direct checking of (1.11) and (1.12 ). The latter follows from
+the fact that P1andP2are, respectively, ψ1andψ2entwined modules:
+∆P(pq) = ∆ P1(pr1(p)pr1(q))+∆ P2(pr2(p)pr2(q))
+= pr1(p(0))ψ1(p(1)⊗pr1(q))+pr2(p(0))ψ2(p(1)⊗pr2(q))
+=/parenleftbig
+pr1(p(0))+pr2(p(0))/parenrightbig/parenleftbig
+ψ1(p(1)⊗pr1(q))+ψ2(p(1)⊗pr2(q))/parenrightbig
+=p(0)ψ(p(1)⊗q). (2.31)
+This proves the lemma.
+Letα1
+Landα1
+Rbe a unital left colinear splitting and a unital right colinear splitting of π1,
+respectively. Also, let α2
+Rbe a right colinear splitting of π2viewed as a map onto π2(P2). Such
+maps exist by Lemma 2.1. On the other hand, by [6, Lemma 2.2], since P1andP2are principal,
+they admit strong connections ℓ1andℓ2, respectively. For brevity, let us introduce the notation
+α12
+L:=α1
+L◦π2, α12
+R:=α1
+R◦π2, α21
+R:=α2
+R◦π1↾π−1
+1(π2(P2)), L:=mP1◦(α12
+L⊗α12
+R)◦ℓ2,(2.32)
+wheremP1is the multiplication of P1. The situation is illustrated in the following diagram:
+(2.33) C
+L
+/d15/d15P
+pr1
+/d118/d118♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠
+pr2
+/d40/d40◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗
+π−1
+1(π2(P2)) /d69/d101
+/d115/d115❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢
+α21
+R /d43/d43❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳
+P1
+❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖P2.α12
+L/d111/d111
+α12
+R/d111/d111
+♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦
+π2
+/d115/d115 /d115/d115❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣
+π1
+/d39/d39 /d39/d39PPPPPPPPPPPPPPPPP π2(P2)α2
+R/d51/d51❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣
+/d95 /d127
+/d15/d15π2
+/d119/d119♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥
+P12α1
+L/d103/d103❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖α1
+R/d103/d103❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖
+Our proves hinges on constructing a strong connection on Pout of strong connections on
+P1andP2. Roughly speaking, the basic idea is to take a strong connection on P2, induce a
+strong connection on the the common part P12, and prolongate it to P1. To this end, we check
+that (α12
+L+id)⊗(α12
+R+id) is a unital bicolinear map from P2⊗P2toP⊗P. Therefore, as a
+first approximation for constructing a strong connection on P, we choose the formula
+ℓI:=/parenleftbig
+(α12
+L+id)⊗(α12
+R+id)/parenrightbig
+◦ℓ2. (2.34)
+It is a bicolinear map from CtoP⊗PsatisfyingℓI(e) = 1⊗1 as needed. However, it does
+not split the lifted canonical map:
+(/tildewidercan◦ℓI)(c)−1⊗c
+=α12
+L(ℓ2(c)/angbracketleft1/angbracketright)α12
+R(ℓ2(c)/angbracketleft2/angbracketright)(0)⊗α12
+R(ℓ2(c)/angbracketleft2/angbracketright)(1)+ℓ2(c)/angbracketleft1/angbracketrightℓ2(c)/angbracketleft2/angbracketright
+(0)⊗ℓ2(c)/angbracketleft2/angbracketright
+(1)−1⊗c
+=α12
+L(ℓ2(c(1))/angbracketleft1/angbracketright)α12
+R(ℓ2(c(1))/angbracketleft2/angbracketright)⊗c(2)+(0,1)⊗c−1⊗c
+=L(c(1))⊗c(2)−(1,0)⊗c∈P1⊗C. (2.35)
+17We correct it by adding to ℓI(c) the splitting of the lifted canonical map on P1⊗P1afforded
+byℓ1and applied to (1 ,0)⊗c−L(c(1))⊗c(2):
+ℓII(c) =ℓI(c)+ℓ1(c)/angbracketleft1/angbracketright⊗ℓ1(c)/angbracketleft2/angbracketright−L(c(1))ℓ1(c(2))/angbracketleft1/angbracketright⊗ℓ1(c(2))/angbracketleft2/angbracketright
+= (ℓI+ℓ1−L∗ℓ1)(c). (2.36)
+The above approximation to a strong connection on Pis clearly right colinear. Using the fact
+thatP1is aψ1-entwined and e-coaugmented module, we follow the lines of (2.21)–(2.23) to
+show thatL∗ℓ1is left colinear. Hence ℓIIis bicolinear. It also satisfies ℓII(e) = 1⊗1. However,
+the price we pay for having ℓII(c)/angbracketleft1/angbracketrightℓII(c)/angbracketleft2/angbracketright
+(0)⊗ℓII(c)/angbracketleft2/angbracketright
+(1)= 1⊗cis that the image of ℓIIis
+no longer in P⊗P.
+The troublesome term ℓ1−L∗ℓ1takes values in P⊗P1. Now one would like to compose
+id⊗(id+α21
+R) withℓ1−L∗ℓ1toforceittaking values in P⊗P. However, since α21
+Risdefined only
+onπ−1
+1(π2(P2)), we need to replace an arbitrary strong connection ℓ1by a strong connection
+taking values in P1⊗π−1
+1(π2(P2)). Such a strong connection is provided for us by (2.24):
+˜ℓ1:= (α12
+L⊗α12
+R)◦ℓ2+ℓ1−ℓ1∗L. (2.37)
+Inserting ˜ℓ1in place ofℓ1allows us to apply the correction map id ⊗(id+α21
+R) to obtain
+ℓIII=ℓI+/parenleftbig
+id⊗(id+α21
+R)/parenrightbig
+◦(˜ℓ1−L∗˜ℓ1). (2.38)
+To end the proof, let us check that ℓIIIis indeed a strong connection on P. First, since ℓI(C)⊆
+P⊗Pand (id+α21
+R)(π−1
+1(π2(P2)))⊆P, we conclude that ℓIIItakes values in P⊗P. Next, it
+is bicolinear because α21
+Ris right colinear. Also, it is clearly unital. To verify that ℓIIIsplits the
+canonical map, first we note that /tildewidercan◦(id⊗α21
+R)◦(˜ℓ1−L∗˜ℓ1) = 0 because mP1×P2(p1⊗p2) = 0
+for allp1∈P1andp2∈P2. Combining this with the fact that /tildewidercan◦(ℓ′
+1−L∗ℓ′
+1) does not
+depend on the choice of a strong connection ℓ′
+1, we infer that /tildewidercan◦ℓIII=/tildewidercan◦ℓII= 1⊗id.
+ThusℓIIIis a strong connection on P, as desired. Combining this fact with Lemma 2.3 and
+Lemma 1.1 proves the theorem.
+Putting the formulas in the proof of Theorem 2.2 together, we obta in the following strong
+connection on P:
+ℓ=((α12
+L+id)⊗(α12
+R+id))◦ℓ2 (2.39)
++ (η1◦ε−L)∗((id⊗(id+α21
+R))◦(ℓ1−ℓ1∗L+(α12
+L⊗α12
+R)◦ℓ2)).
+3 The pullback picture of the standard quantum Hopf
+fibration
+Recall that the classical Hopf fibration is a U(1)-principal bundle giv en by the maps
+π: S3={(z1,z2)∈C2||z1|2+|z2|2= 1} −→S2∼=CP1, π((z1,z2)) = [(z1:z2)],
+S3×U(1)−→S3,(z1,z2)⊳u= (z1u,z2u). (3.1)
+18To unravel the structure of this non-trivial fibration, we split S3into two disjoint parts:
+S3={(z1,z2)∈C2||z1|2<1,|z2|2= 1−|z1|2} ∪ {(z1,0)||z1|= 1}. (3.2)
+Note that both sets are invariant under the U(1)-action. The sec ond set is U(1), and first set
+is U(1)-equivariantly homeomorphic to the interior of the solid torus D×U(1) equipped with
+the diagonal action. (Here D = {z∈C||z| ≤1}.) By an appropriate U(1)-equivariant gluing
+of the boundary torus of D ×U(1) with U(1), we recover S3with its U(1)-action:
+(3.3) S3
+φ1(z,v) = (z,v/radicalbig
+1−|z|2) φ2(u) = (u,0)
+D×U(1)φ1/d61/d61③③③③③③③③③③③
+U(1)φ2/d94/d94❂❂❂❂❂❂❂❂❂❂
+(ι,id)(u,v) = (u,v) pr1(u,v) =u.
+U(1)×U(1)pr1/d63/d63(ι,id)/d98/d98❊❊❊❊❊❊❊❊❊❊
+However, to view D ×U(1) as a trivial U(1)-principal bundle, we need to gauge the di-
+agonal action to the action on the right slot. This is achieved with the help of the following
+homeomorphism intertwining these two actions:
+Ψ : D×U(1)−→D×U(1),Ψ(x,v) := (xv,v),Ψ(x,vu) = Ψ(x,v)⊳u. (3.4)
+Let us denote the restriction of Ψ to U(1) ×U(1) by the same symbol. Now we can extend the
+above pushout diagram to the commutative diagram
+(3.5) S3
+D×U(1)Ψ/d47/d47D×U(1)φ1/d59/d59✇✇✇✇✇✇✇✇✇✇✇
+U(1)φ2/d96/d96❇❇❇❇❇❇❇❇❇
+U(1)id/d111/d111
+U(1)×U(1)pr1/d62/d62⑥⑥⑥⑥⑥⑥⑥⑥⑥(ι,id)/d99/d99●●●●●●●●●●
+U(1)×U(1),(ι,id)/d102/d102▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲Ψ/d79/d79m/d59/d59✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇
+wheremis the multiplication map. The outer diagram is again a pushout diagram o f U(1)-
+spaces, but now its defining U(1)-spaces are trivial U(1)-principa l bundles. It is the outer
+pushout diagram that we shall use to analyse a noncommutative def ormation of the Hopf
+fibration.
+3.1 Pullback comodule algebra
+We consider the tensor products P1:=T ⊗O(U(1)),P2=C⊗O(U(1))∼=O(U(1)) and P12:=
+C(U(1))⊗ O(U(1)). These algebras are right O(U(1))-comodule algebras for the coaction
+19x⊗uN/mapsto→x⊗uN⊗uN,N∈Z. Moreover, P1andP2are trivially principal with strong
+connections ℓi:O(U(1))→Pi⊗Pigiven byℓi(uN) = (1⊗uN∗)⊗(1⊗uN),i=1,2. To construct
+a pullback of P1andP2, we define the following morphisms of right O(U(1))-comodule algebras:
+π1:T ⊗O(U(1))−→C(U(1))⊗O(U(1)), π1(t⊗w) =σ(t)⊗w, (3.6)
+π2:O(U(1))−→C(U(1))⊗O(U(1)), π2(w) = ∆(w). (3.7)
+Then the fibre product P:=T ⊗O(U(1))×(π1,π2)O(U(1)) defined by the pullback diagram
+T ⊗O(U(1))×
+(π1,π2)O(U(1))
+pr1
+/d118/d118❧❧❧❧❧❧❧❧❧❧❧❧❧❧pr2
+/d40/d40PPPPPPPPPPPPP
+T ⊗O(U(1))
+π1/d41/d41❙❙❙❙❙❙❙❙❙❙❙❙❙❙O(U(1))
+π2/d118/d118❧❧❧❧❧❧❧❧❧❧❧❧❧
+C(U(1))⊗O(U(1))(3.8)
+is a right O(U(1))-comodule algebra. By Proposition 2.2, it is principal.
+Furthermore, define unital respectively left colinear and right colin ear splittings of π1by
+α1
+L(f⊗uN) =α1
+R(f⊗uN) =Tf⊗uN, N∈Z. (3.9)
+Heref∈C(U(1))and Tfdenotes theToeplitz operatorwith symbol f. Inparticular, TuN= SN
+andTu∗N= S∗N. A right colinear splitting of the map π2:O(U(1))→π2(O(U(1))) is given by
+α2
+R(uN⊗uN) =uN, N∈Z. (3.10)
+Inserting thedefinitionsof α1
+L,αi
+Randℓi,i=1,2, into(2.32)and(2.39), weobtainthefollowing
+strong connection on P:
+ℓ(uN) = (S∗N⊗u∗N,u∗N)⊗(SN⊗uN,uN), (3.11)
+ℓ(u∗N) = (SN⊗uN,uN)⊗(S∗N⊗u∗N,u∗N)
++((1−SNS∗N)⊗uN,0)⊗((1−SNS∗N)⊗u∗N,0), N∈N.(3.12)
+Note next that, by construction, we have
+P=/braceleftig/summationtext
+k(tk⊗uk,αkuk)∈/parenleftbig
+T ⊗O(U(1))/parenrightbig
+×O(U(1))/vextendsingle/vextendsingle/vextendsingleσ(tk) =αkuk/bracerightig
+, (3.13)
+whereαk∈C. ForC∆(1) =uN⊗1, let
+LN:=P/square
+O(U(1))C∼={p∈P|∆P(p) =p⊗uN}. (3.14)
+ThenL0=PcoO(U(1)), eachLNis a leftPcoO(U(1))-module and P=/circleplustext
+N∈ZLN. From
+∆P/parenleftbig/summationdisplay
+k(tk⊗uk,αkuk)/parenrightbig
+=/summationdisplay
+k(tk⊗uk,αkuk)⊗uk, (3.15)
+20it follows that
+LN=/braceleftig
+(t⊗uN,αuN)∈/parenleftbig
+T ⊗O(U(1))/parenrightbig
+×O(U(1))/vextendsingle/vextendsingle/vextendsingleσ(t) =αuN/bracerightig
+.(3.16)
+The next proposition shows that L0∼=T ×(σ,1)Cis isomorphic to the C∗-algebra of the
+standard Podle´ s sphere and that
+LN∼=T ×(u−Nσ,1)C, (3.17)
+whereT ×(uNσ,1)Cis given by the pullback diagram
+T ×
+(u−Nσ,1)C
+pr1
+/d120/d120qqqqqqqqqqpr2
+/d38/d38◆◆◆◆◆◆◆◆◆◆
+T
+σ
+/d15/d15C
+α/mapsto→α1
+/d15/d15
+C(U(1))
+f/mapsto→u−Nf/d47/d47C(U(1)).(3.18)
+Proposition 3.1. The fibre product T×(σ,1)Cis isomorphic to the C∗-algebraC(S2
+q), andLN
+is isomorphic to T ×(u−Nσ,1)Cas a leftC(S2
+q)-module with respect to the left C(S2
+q)-action on
+T ×(u−Nσ,1)Cgiven by (t,α)·(h,β) = (th,αβ).
+Proof.ForN= 0, the mappings T ∋t/mapsto→σ(t)∈C(U(1)) and C∋α/mapsto→α1∈C(U(1))
+are morphisms of C∗-algebras, so that T ×(σ,1)Cis aC∗-algebra. Next, recall that C(S2
+q)∼=
+K(ℓ2(N))⊕C(see (1.56)), and define
+φ:T ×
+(σ,1)C−→ K(ℓ2(N))⊕C, φ(t,α) =t, (3.19)
+φ−1:K(ℓ2(N))⊕C−→ T ×
+(σ,1)C, φ−1(k+α) = (k+α,α). (3.20)
+Clearly,φ:T×(σ,1)C→ B(ℓ2(N)) is a morphism of C∗-algebras. Since φ(t,α) = (t−α) +α,
+andσ(t−α) = 0 by the pullback diagram (3.18), it follows from the short exact se quence (1.61)
+thatt−α∈ K(ℓ2(N)), so thatφ(t,α)∈ K(ℓ2(N))⊕C. One easily sees that φ−1is its inverse
+so thatT×(σ,1)C∼=K(ℓ2(N))⊕C.
+The fact that T ×(u−Nσ,1)Cwith the given C(S2
+q)-action is a left C(S2
+q)-module follows from
+the discussion preceding the pullback diagram (1.6) with the free ran k 1 modules E1=T,
+E2=Candπ1∗E1=π2∗E2=C(U(1)). Obviously, LN∋(t⊗uN,αuN)/mapsto→(t,α)∈ T × (u−Nσ,1)C
+defines an isomorphism of left C(S2
+q)-modules.
+3.2 Equivalence of the pullback and standard constructions
+Let us view U(1) as a compact quantum group. We consider its C∗-algebraC(U(1)) of
+all continuous function together with the obvious coproduct, cou nit and antipode given by
+∆(f)(x,y) =f(xy),ε(f) =f(1) andS(f)(x) =f(x−1), respectively. Furthermore, let ¯⊗
+21stand for the completed tensor product of C∗-algebras. In our case it is unique because of the
+nuclearity of the involved C∗-algebras.
+Now letπ2:C(U(1))→C(U(1))¯⊗C(U(1)) be given by the coproduct, i.e., π2(f)(x,y) =
+(∆f)(x,y) =f(xy), and letσ⊗id denote the tensor product of the symbol map σ:T →
+C(U(1)) and id : C(U(1))→C(U(1)). Then ¯P:=T¯⊗C(U(1))×(π1,π2)C(U(1)) is defined by
+the pullback diagram
+T¯⊗C(U(1))×
+(π1,π2)C(U(1))
+pr1
+/d118/d118❧❧❧❧❧❧❧❧❧❧❧❧❧❧pr2
+/d40/d40PPPPPPPPPPPPP
+T¯⊗C(U(1))
+π1=σ⊗id/d41/d41❙❙❙❙❙❙❙❙❙❙❙❙❙❙C(U(1))
+π2=∆/d118/d118❧❧❧❧❧❧❧❧❧❧❧❧❧
+C(U(1))¯⊗C(U(1)) .(3.21)
+With theC(U(1))-coaction given by the coproduct ∆ on the right tensor fac torC(U(1)),
+π1andπ2are morphisms in the category of right C(U(1))-comodule C∗-algebras. Equivalently,
+we can view this diagram as a diagram in the category of U(1)- C∗-algebras (see Section 1.3).
+Therefore, ¯Pinherits the structure of a right U(1)- C∗-algebra. To determine the Peter-Weyl
+comodule algebra P∆(¯P), we first note that O(U(1)) is the dense Hopf *-subalgebra of C(U(1))
+spannedbythematrixcoefficients oftheirreducibleunitarycorepr esentations. Usingthecounit
+ε:C(U(1))→Cand the fact that the Peter-Weyl functor commutes with taking p ullbacks, we
+easily conclude that P∆(¯P) =T ⊗O(U(1))×(π1,π2)O(U(1)), so P∆(¯P) is the comodule algebra
+Pof Section 3.1.
+Consider next the *-representation of O(SUq(2)) onℓ2(N) given by [27]
+ρ(α)en= (1−q2n)1/2en−1, ρ(β)en=−qn+1en,
+ρ(γ)en=qnen, ρ(δ)en= (1−q2(n+1))1/2en+1.(3.22)
+Note thatρ(β), ρ(γ)∈ K(ℓ2(N)). Comparing ρwith the representation µofO(Dq) from (1.60),
+one readily sees that ρ(O(SUq(2)))⊆ T. Furthermore, the symbol map σyieldsσ(ρ(β)) =
+σ(ρ(γ)) = 0. Using an appropriate diagonal compact operator k, we also obtain
+σ(ρ(α)) =σ(ρ(α)−S∗)+σ(S∗) =σ(kS∗)+σ(S∗) =u−1, σ(ρ(δ)) =σ(ρ(α))∗=u.(3.23)
+Thus we obtain a U(1)-equivariant *-algebra morphism ι:O(SUq(2))→Pby setting
+ι(α) = (ρ(α)⊗u,u), ι(γ) = (ρ(γ)⊗u,0). (3.24)
+OneeasilychecksthattheimageofaPoincar´ e-Birkhoff-Wittbasiso fO(SUq(2))remainslinearly
+independent, so that ιis injective and we can consider O(SUq(2)) as a subalgebra of P. In
+particular, we have ι(MN)⊆LNas leftO(S2
+q)-modules. (See Section 1.4 and Section 3.1 for
+the definitions of MNandLN, respectively.)
+The main objective of this section is to establish an U(1)- C∗-algebra isomorphism between
+C(SUq(2)) and ¯P. The universal C∗-algebraC(SUq(2)) ofO(SUq(2)) has been studied in [18]
+22and [31]. Here we shall use the fact from [18, Corollary 2.3] that a fait hful *-representation ˆ ρ
+ofC(SUq(2)) on the Hilbert space ℓ2(N)¯⊗ℓ2(Z) is given by
+ˆρ(α)(en⊗bk) = (1−q2n)1/2en−1⊗bk−1,ˆρ(γ)(en⊗bk) =qnen⊗bk−1,(3.25)
+where{en}n∈Nand{bk}k∈Zdenote the standard bases of ℓ2(N) andℓ2(Z), respectively. To
+compare (3.25) with [18, Corollary 2.3], one has to apply the unitary tr ansformation
+T:ℓ2(N)¯⊗ℓ2(Z)→ℓ2(N)¯⊗ℓ2(Z), T(en⊗bk) =en⊗bk−n. (3.26)
+ArightC(U(1))-coactionon C(SUq(2))isgivenby(id ⊗¯π)◦∆,where∆denotesthecoproductof
+the compact quantum group C(SUq(2)) and ¯πis the extension of the Hopf *-algebra surjection
+π:O(SUq(2))→ O(U(1)) defined in (1.49) to C(SUq(2)). Using the faithfulness of ˆ ρ, we can
+transfer ¯πto ˆρ(C(SUq(2))). In [18], it is shown that ¯ πgives rise to the C∗-algebra extension
+0/d47/d47K(ℓ2(N))¯⊗C(U(1))/d31 /d127/d47/d47ˆρ(C(SUq(2)))¯π/d47/d47C(U(1))/d47/d470.(3.27)
+Theorem 3.2. TheU(1)-C∗-algebrasC(SUq(2))and¯Pare isomorphic.
+Proof.First note that ker(pr1) ={(0,y)∈¯P|π2(y) = ∆(y) = 0}={0}.Hence we can identify
+¯Pwith the image of pr1inT¯⊗C(U(1)). We will prove the theorem by applying the five lemma
+to the following commutative diagram of U(1)- C∗-algebras:
+0/d47/d47K(ℓ2(N))¯⊗C(U(1))
+id
+/d15/d15/d31 /d127/d47/d47ˆρ(C(SUq(2)))¯π/d47/d47
+τ
+/d15/d15C(U(1))
+id
+/d15/d15/d47/d470
+0/d47/d47K(ℓ2(N))¯⊗C(U(1))/d31 /d127/d47/d47pr1(¯P)ω/d47/d47C(U(1))/d47/d470.(3.28)
+To defineτ, we realize C(U(1)) as a concrete C∗-algebra of bounded operators on ℓ2(Z) by
+settingu(bk) =bk−1. Then
+ˆρ(α) =ρ(α)⊗u= pr1(ρ(α)⊗u,u)∈pr1(¯P),ˆρ(γ) =ρ(γ)⊗u= pr1(ρ(γ)⊗u,0)∈pr1(¯P).
+(3.29)
+SinceC(SUq(2)) is generated by αandγ, we takeτto be the inclusion ˆ ρ(C(SUq(2)))⊂pr1(¯P).
+We define the U(1)- C∗-algebra morphism ωby
+ω: pr1(¯P)−→C(U(1)), ω:= (ε◦σ)⊗id, (3.30)
+whereεdenotes the counit of C(U(1)). The surjectivity of ωfollows from uk=ω(ρ(αk)⊗uk)
+andu−k=ω(ρ(α∗k)⊗u∗k) for allk∈Nand taking the closure of O(U(1)) inC(U(1)).
+Toprovetheexactness ofthelower row, notethat K(ℓ2(N))¯⊗C(U(1)) = ker( σ)¯⊗C(U(1))⊂
+ker(ω). Now let f∈pr1(¯P)\ker(σ)¯⊗C(U(1)). Then ( σ⊗id)(f)/\e}atio\slash= 0. By the commutative
+diagram (3.21), there exists a non-zero element g∈C(U(1)) such that ( σ⊗id)(f) = ∆(g).
+Henceω(f) = (ε⊗id)◦∆(g) =g/\e}atio\slash= 0 which proves ker( ω) =K(ℓ2(N))¯⊗C(U(1)).
+It remains to show that the diagram (3.28) is commutative. This is clea r for the left part
+sinceτis just the inclusion. The commutativity of the right part follows from
+ω/parenleftig
+τ/parenleftbig
+ˆρ(α)/parenrightbig/parenrightig
+=ε/parenleftig
+σ/parenleftbig
+ρ(α)/parenrightbig/parenrightig
+⊗u=ε(u)u=u= ¯π(ˆρ(α)), (3.31)
+ω/parenleftig
+τ/parenleftbig
+ˆρ(γ)/parenrightbig/parenrightig
+=ε/parenleftig
+σ/parenleftbig
+ρ(γ)/parenrightbig/parenrightig
+⊗u= 0 = ¯π(ˆρ(γ)) (3.32)
+23by taking limits since ˆ αand ˆγgenerateC(SUq(2)). Therefore, by the five lemma, τis an
+isomorphism of U(1)- C∗-algebra.
+By the final remark of Section 1.3, we conclude from Theorem 3.2 tha t the Peter-Weyl
+comodule algebras P∆(C(SUq(2))) andPare isomorphic. We use this isomorphism to identify
+associatedprojective modules. For N∈Zandtheleft O(U(1))-coactionon Cgivenby C∆(1) =
+uN⊗1, we define a “completed” version of MN(see (1.62)):
+¯MN:=P∆(C(SUq(2)))/square
+O(U(1))C={p∈ P∆(C(SUq(2)))|((id⊗¯π)◦∆)(p) =p⊗uN}.(3.33)
+Now it follows from Equation (3.14) that ¯MN∼=LN. Applying the same arguments as at the
+end of Section 1.4, we infer that ¯MN∼=C(S2
+q)N+1EN, withENbeing the projection matrix of
+Equation (1.64). Taking advantage of these isomorphisms of module s, we prove:
+Lemma 3.3. Identifying C(S2
+q)withK(ℓ2(N))⊕C, we obtain the following isomorphisms of
+leftC(S2
+q)-modules:
+C(S2
+q)N+1EN∼=C(S2
+q)pN, p N:= SNSN∗, N ≥0,(3.34)
+C(S2
+q)|N|+1EN∼=C(S2
+q)2pN, p N:=/parenleftbigg1 0
+0 1−S|N|S|N|∗/parenrightbigg
+, N < 0.(3.35)
+Proof.We apply (1.40) to construct projections PN,N∈Z, from the strong connection given
+in (3.11) and (3.12). For N <0, we obtain
+(PN)11= (S∗|N|⊗uN,uN)(S|N|⊗u|N|,u|N|) = (1⊗1,1), (3.36)
+(PN)12= (PN)∗
+21= (S∗|N|⊗uN,uN)((1−S|N|S∗|N|)⊗u|N|,0) = 0, (3.37)
+(PN)22= ((1−S|N|S∗|N|)⊗uN,0)((1−S|N|S∗|N|)⊗u|N|,0) = ((1−S|N|S∗|N|)⊗1,0).(3.38)
+Analogously, for N≥0, we get
+(PN)11= (SN⊗uN,uN)(S∗N⊗u∗N,u∗N) = (SNS∗N⊗1,1). (3.39)
+Finally, applying the isomorphism (3.19) componentwise to PN,N∈Z, yields the result.
+The projections pNof Lemma 3.3 can also be obtained from the odd-to-even construct ion
+in Section 1.2. First let N <0. SinceLN∼=T×(u−Nσ,1)C(see (3.17)), we can apply the formula
+(1.9) by taking E1=T,E2=C,π1∗E1=π2∗E2=C(U(1)), and choosing hin (1.6) to be
+the isomorphism given by the multiplication with u|N|. As the symbol map σapplied to S is
+u(see (1.61)), we can lift u|N|and its inverse u−|N|to S|N|and S|N|∗respectively. Inserting
+c= S|N|∗andd=S|N|into (1.9) gives T ×(u−Nσ,1)C∼=(T ×(σ,1)C)2QN, where
+QN=/parenleftbigg(1,1) (0 ,0)
+(0,0) (1−S|N|S|N|∗,0)/parenrightbigg
+. (3.40)
+Finally, applying the isomorphism (3.19) yields the projection in (3.35). Similarly, for N≥0,
+we insertc= SNandd= SN∗into (1.9). Since SN∗SN= 1, we obtain T ×(u−Nσ,1)C∼=
+(T×(σ,1)C)2QNwith
+QN=/parenleftbigg(SNSN∗,1) (0,0)
+(0,0) (0,0)/parenrightbigg
+, (3.41)
+which is equivalent to T ×(u−Nσ,1)C∼=C(S2
+q)SNSN∗.
+243.3 Index pairing
+Recall that for a C∗-algebraA, a projection p∈Matn(A), and *-representations ρ+andρ−of
+Aon a Hilbert space Hsuch that [( ρ+,ρ−)]∈K0(A) (e.g. see [10, Chapter 4]), one has the
+following: If the operator Tr Matn(ρ+−ρ−)(p) is trace class, then the formula
+/a\}bracketle{t[(ρ+,ρ−)],[p]/a\}bracketri}ht= Tr H(TrMatn(ρ+−ρ−)(p)) (3.42)
+yields a pairing between K0(A) and K 0(A).
+In this section, we compute the pairing between the K 0-classes of the projective C(S2
+q)-
+modules describing quantum line bundles and the two generators of K0(A). By Lemma 3.3,
+we can take the projections pNas representatives of respective K 0-classes. Their simple form
+makes it very easy to compute the index pairing.
+Theorem 3.4. Let¯MNbe the associated modules of (3.33), and let [(id,ε)]and[(ε,ε0)]denote
+the generators of K0(C(S2
+q))given in Section 1.4. Then, for all N∈Z,
+/a\}bracketle{t[(ε,ε0)],[¯MN]/a\}bracketri}ht= 1,/a\}bracketle{t[(id,ε)],[¯MN]/a\}bracketri}ht=−N. (3.43)
+Proof.LetN≥0. ThenpN= SNSN∗= (SNSN∗−1) + 1, so that ε(pN) = 1 andε0(pN) =
+SS∗. Furthermore, since for any N∈N\{0}, the image of the projection 1 −SNSN∗is
+span{e0,...,e N−1} ⊂ℓ2(N), the projection 1 −SNSN∗is trace class. Moreover, with the help
+of Lemma 3.3 and Formula (3.42), it implies that
+/a\}bracketle{t[(ε,ε0)],[¯MN]/a\}bracketri}ht= Trℓ2(N)(ε−ε0)(pN) = Tr ℓ2(N)(1−SS∗) = 1, (3.44)
+/a\}bracketle{t[(id,ε)],[¯MN]/a\}bracketri}ht= Trℓ2(N)(id−ε)(pN) = Tr ℓ2(N)(SNSN∗−1) =−N. (3.45)
+ForN <0, we have Tr Mat2(pN) = 2−S|N|S|N|∗= 2−p|N|. Combining this with the above
+index pairing for p|N|, the formulas ( ε−ε0)(2) = 2(1 −SS∗) and (id −ε)(2) = 0, and (3.42), we
+obtain
+/a\}bracketle{t[(ε,ε0)],[¯MN]/a\}bracketri}ht= Trℓ2(N)(ε−ε0)(2−p|N|) = Tr ℓ2(N)(1−SS∗) = 1, (3.46)
+/a\}bracketle{t[(id,ε)],[¯MN]/a\}bracketri}ht= Trℓ2(N)(id−ε)(2−p|N|) = Tr ℓ2(N)(1−S|N|S|N|∗) =−N. (3.47)
+This completes the proof.
+The above theorem agrees with the classical situation. Indeed, th e pairing /a\}bracketle{t[(ε,ε0)],[¯MN]/a\}bracketri}ht
+yields the rank of the line bundles, and /a\}bracketle{t[(id,ε)],[¯MN]/a\}bracketri}htcomputes the winding number of the
+mapu−N: S1→S1.
+4 Examples of piecewise principal coalgebra coactions
+We begin by recalling the piecewise structure [13] of a noncommutativ e join construction pro-
+posed by [9]. Then we instantiate it to SU q(2) to obtain a quantum instanton bundle S7
+q→S4
+q
+[23] as a piecewise trivial principal comodule algebra. A key step is the n to replace SU q(2)
+25by quotienting the Hopf algebra O(SUq(2)) by a coideal right ideal ( O(S2
+q,s)∩kerε)O(SUq(2))
+provided by the generic Podle´ s quantum spheres S2
+q,s,s/\e}atio\slash= 0 [24]. The quotient coalgebra is
+isomorphic with O(U(1)) [21]. Applying our main theorem, we will prove that the induced
+right coaction of O(U(1)) is principal.
+4.1 Piecewise principal coactions from a noncommutative jo in con-
+struction
+Let¯Hbe theC∗-algebra of a compact quantum group and Hits Peter-Weyl Hopf algebra
+[30,32]. We take the algebra of norm continuous functions C([a,b],¯H) from a closed interval
+[a,b] to theC∗-algebra ¯H, and define
+P1:={f∈C([0,1
+2],¯H)⊗H|f(0)∈∆(H)}, (4.1)
+P2:={f∈C([1
+2,1],¯H)⊗H|f(1)∈C⊗H}. (4.2)
+Here we identify elements of C([a,b],¯H)⊗Hwith functions [ a,b]→¯H⊗H. ThePi’s are right
+H-comodule algebras for the coaction ∆ Pi= idC([ai,bi],¯H)⊗∆, where ∆ stands for the coproduct
+ofH. The subalgebras of coaction invariants can be identified with
+B1:={f∈C([0,1
+2],¯H)|f(0)∈C},
+B2:={f∈C([1
+2,1],¯H)|f(1)∈C}.
+The comodule algebra P2is evidently the same as B2⊗H. UnlikeP2, the comodule algebra
+P1does not coincide with B1⊗H. However, there is a cleaving map j:H→P1byj(h)(t) :=/parenleftbig
+t/mapsto→h(1)/parenrightbig
+⊗h(2), that is,j(h)(t) := ∆(h) for allt∈[0,1
+2]. Sincejis an algebra homomorphism,
+it identifies the comodule algebra P1with a smash product B1#H.
+Now onecandefine thenoncommutative joinof ¯Hasthe pullback right H-comodulealgebra
+P:={(p,q)∈P1⊕P2|π1(p) =π2(q)} (4.3)
+given by the evaluation maps
+π1:= ev 1
+2⊗id :P1→P12:=¯H⊗H, π 2:= ev 1
+2⊗id :P2→P12:=¯H⊗H, (4.4)
+where ev tis given by the evaluation of functions of C([a,b],¯H) int∈[a,b].
+Our goal now is to replace Hby a quotient coalgebra without loosing principality. Using [7,
+Example 2.29], it is straightforward to verify the following lemma.
+Lemma 4.1. LetHbe a Hopf algebra with bijective antipode, let ∆P:P→P⊗Hbe a coaction
+makingPa rightH-comodule algebra and Ja coideal right ideal of H. ThenC:=H/Jis a
+coalgebra coacting on PviaρR:= (id⊗π)◦∆P,π:H→Cthe canonical surjection, and the
+formula
+Ψ :C⊗P∋¯π(h)⊗p/mapsto−→p(0)⊗π(hp(1))∈P⊗C (4.5)
+defines a bijective entwining making Pan entwined module. The inverse of Ψis given by
+Ψ−1(p⊗π(h)) =π(hS−1(p(1)))⊗p(0), (4.6)
+26and defines a left oaction on Pvia
+ρL:P∋p/mapsto−→Ψ−1(p⊗π(1)) =π(S−1(p(1))⊗p(0)∈C⊗P. (4.7)
+Lemma 4.2. LetPbe a principal H-comodule algebra for ∆P:P→P⊗H. Also, letJbe a
+coideal right ideal of Hdefining a coalgebra C:=H/J, letρR:= (id⊗π)◦∆P,π:H→C
+the canonical surjection, be its right coaction on P, and leti:C→Hbe a unital (i.e.,
+i(π(1)) = 1 )C-bicolinear (for the coactions ∆H:= (id⊗π)◦∆andH∆ := (π⊗id)◦∆)
+splitting (i.e., π◦i= id). ThenPis principal for the coaction ρR.
+Proof.Letℓ:H→P⊗Pbe a strong connection on P. One can easily check that ℓ◦i:
+C→P⊗Pis a strong connection on Pfor the right coaction ρR:= (id⊗π)◦∆Pand the
+left coaction ρL:= (π⊗id)◦P∆, where P∆(p) =S−1(p(1))⊗p(0)is the leftH-coaction on P
+viewed as a principal H-comodule algebra. On the other hand, it follows from Lemma 4.1 that
+Ψ :C⊗P∋π(h)⊗p/mapsto→p(0)⊗π(hp(1))∈P⊗Cis a bijective entwining making Pan entwined
+module. Therefore, since ρR(1) = 1⊗π(1),ρL(1) =π(1)⊗1, andρL(p) = Ψ−1(p⊗π(1)) for
+allp∈Pby (4.7), the principality of Pfor theC-coactionρRfollows from Lemma 1.1.
+Combining Lemma 4.2 with Theorem 2.2 yields the following result.
+Theorem 4.3. Let¯Hbe theC∗-algebra of a compact quantum group, Hits Peter-Weyl Hopf
+algebra,Ja coideal right ideal of Handπ:H→C:=H/Jthe cannonical surjection. Also let
+P1:={f∈C([0,1
+2],¯H)⊗H|(ev0⊗id)(f)∈∆(H)}, (4.8)
+P2:={f∈C([1
+2,1],¯H)⊗H|(ev1⊗id)(f)∈C⊗H} (4.9)
+be right and left C-comodules for the right and left coactions
+ρi
+R:= (id⊗π)◦∆Pi, ρi
+L:= (π⊗id)◦Pi∆, i= 1,2, (4.10)
+respectively. Here ∆Pi:= id⊗∆andPi∆ := (S−1⊗id)◦flip◦∆Pi. Then, if there exists a
+unital bicolinear splitting i:C→Hofπ:H→C, the pullback C-comodule
+P:={(p1,p2)∈P1×P2|(ev1
+2⊗id)(p1) = (ev 1
+2⊗id)(p2)} (4.11)
+is principal.
+Let us now take a closer look at the compatibility of strong connectio ns on principal comod-
+ules as appearing in the above theorem. First we observe that, if bo th ofπ1andπ2defining
+the pullback diagram (2.33) are surjective, then (2.39) simplifies to
+ℓ=((α12
+L+id)⊗(α12
+R+id))◦ℓ2+ (η1◦ε−mP1◦(α12
+L⊗α12
+R)◦ℓ2)∗/parenleftbig(id⊗(id+α21
+R))◦ℓ1/parenrightbig
+.(4.12)
+Indeed, since now α21
+Ris defined on the whole P1, a special constructed connection ˜ℓ1in (2.38)
+can be replaced by any strong connection ℓ1onP1. Note that specializing (4.12) to comodule
+algebras coincides with what was obtained in [13].
+Next, we observe that the formulae
+α1:P12→P1, α1(¯h⊗h) = 2t¯h⊗h+(1−2t)¯ε(¯h)h(1)⊗h(2), (4.13)
+α2:P12→P2, α2(¯h⊗h) = 2(1−t)¯h⊗h+(2t−1)¯ε(¯h)⊗h, (4.14)
+27where ¯εis any unital linear functional on ¯H, define unital C-bicolinear splittings of π1andπ2,
+respectively. Hence we can take α12
+L=α12
+R=α1◦π2andα21
+R=α2◦π1. Combining this with
+the fact that a cleaving map jdefines a strong connection via ℓ:= (j−1⊗j)◦∆, we obtain
+very explicit formulae for strong connections on P1andP2:
+ℓ1:= (j−1
+1⊗j1)◦∆◦i, ℓ2:= (j−1
+2⊗j2)◦∆◦i. (4.15)
+Herej1:H→P1,j1(h) := (t/mapsto→h(1))⊗h(2), andj2:H→P2,j2(h) := 1⊗hare cleaving maps
+forP1andP2, respectively.
+4.2 Quantum complex projective spaces CP3
+q,s
+Finally, we instantiate ¯Hto beC(SUq(2)),H=O(SUq(2)),J= (O(S2
+q,s)∩kerε)O(SUq(2)),
+and ¯ε:C(SUq(2))→Cto the counit. Here O(S2
+q,s) stands for the coordinate algebra of a
+Podle´ s quantum sphere S2
+q,s,s∈[0,1], [8]. (Note that the case s= 0 brings us to the comodule-
+algebra setting.) The most interesting part of this structure is the unital bicolinear splitting of
+π:O(SUq(2))→ O(SUq(2))/Jgiven by [8, Proposition 6.3].
+All this defines a family of noncommutative deformations of the U(1) -principal bundle
+S7→CP3. More precisely, we obtain deformations of a U(1)-principal action on S7given
+by
+(z1,z2,z3,z4)eiϕ= (z1eiϕ,z2e−iϕ,z3eiϕ,z4e−iϕ),|z1|2+|z2|2+|z3|2+|z4|2= 1.(4.16)
+However, this is isomorphic with the diagonal action of U(1) on S7, so that the quotient space is
+againCP3. Thus out of Pflaum’s S7we obtain a family of quantum projective spaces CP3
+q,s. A
+very explicit Mayer-Vietoris type formula for a strong connection o n S7
+q→CP3
+q,sshould allow
+us to study the K-theory aspects of the tautological line bundle over CP3
+q,s, but this is beyond
+the scope of this paper.
+Acknowledgements
+The authors gratefully acknowledge financial support from the fo llowing research grants:
+PIRSES-GA-2008-230836,1261/7.PR UE/2009/7, N201 1770 33, and 189/6.PR UE/2007/7. It
+is also a pleasure to thank Tomasz Brzezi´ nski for helping us with Se ction 2, andUlrich Kr¨ ahmer
+for proofreading the manuscript.
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@@ -0,0 +1,181 @@
+arXiv:1001.0076v1  [physics.ins-det]  31 Dec 2009Giant Liquid ArgonObservatoryfor ProtonDecay, Neutrino
+Astrophysicsand CP-violationin the LeptonSector (GLACIE R)∗
+A.Badertscher, A.Curioni, U.Degunda, L.Epprecht, S. Hori kawa, L.Knecht, C.Lazzaro, D.Lussi,
+A.Marchionni, G.Natterer, P.Otiougova†, F.Resnati, A.Rubbia‡,C.Strabel, J. Ulbricht, and T.Viant
+ETHZurich, 101 Raemistrasse, CH-8092 Zurich, Switzerland
+Abstract
+GLACIER (Giant Liquid Argon Charge Imaging ExpeRiment) is a large un-
+derground observatory for proton decay search, neutrino as trophysics and CP-
+violation studies in the lepton sector. Possible undergrou nd sites are stud-
+ied within the FP7 LAGUNA project (Europe) and along the JPAR C neutrino
+beamincollaboration withKEK(Japan). Theconcept isscala ble toverylarge
+masses.
+1Introduction
+Looking at the future of neutrino long-baseline experiment s, T2K [1] and NOvA [2] can be considered
+as "Phase I" experiments that will improve the sensitivity t o the small mixing angle θ13by one order
+of magnitude over the existing CHOOZ limit (sin22θ13>0.1) [3], but will have limited sensitivity to the
+effects of CPviolation in the neutrino sector. The next gene ration of "Phase II" experiments is expected
+to have significant discovery potential for CP violation, ca pability to establish the mass hierarchy and -
+in caseθ13is small enough to elude "Phase I" experiment - extend the θ13discovery potential. In order
+to do so, a "Phase II" experiment must be designed to have very large statistics, excellent background
+rejection and verygood energy resolution, tomeasure veryp recisely the oscillation pattern asafunction
+ofneutrinoenergy. Thedetectormustthereforebeverymass ive(100,000to1,000,000tonsofactivevol-
+ume, depending on specific detector technology), and must ha ve the capability to accurately reconstruct
+neutrino interactions for energies of about 1 GeV. A liquid a rgon time projection chamber (LAr TPC:
+see for example [4] and references therein) provides high ef ficiency for νecharged current interactions
+("signal" events), with high rejection power against νµneutral and charged currents backgrounds in the
+GeV and multi-GeV region . In particular, excellent e/ π0separation comes from fine sampling (few %
+ofaradiationlength), andatransverse samplingfinerthant hetypical spatial separation ofthe2 γ’sfrom
+theπ0decay. The identification capability for electrons, µ,π, K, and protons is also excellent down to
+energies as low as few tens of MeVs. Embedded in a magnetic fiel d, a LAr TPC gives the possibility
+to measure both wrong sign muons and wrong sign electrons sam ples in a neutrino factory beam [5,6].
+Unlike Water Cherenkov detectors, detection and reconstru ction efficiencies do not depend on the vol-
+ume of the detector, therefore a direct comparison between t he near and far detector is possible (apart
+from fluxextrapolation). Averylarge LArTPCasthe oneneede d in a"Phase II" long baseline neutrino
+experiment wouldalsohaveunprecedented sensitivity topr oton decay[7]andneutrinos fromastrophys-
+ical and terrestrial sources (solar and atmospheric neutri nos, neutrinos from stellar collapse [8,9], or
+neutrinos from DarkMatter annihilation).
+GLACIER(Giant Liquid Argon Charge Imaging ExpeRiment) is a proposed very large LAr TPC
+withawelldefinedconceptualdesign[10,11],andsomeofits characteristicsareoutlinedintherestofthe
+paper. The underground localization of the experiment is be ing investigated along the JPARC neutrino
+beam inCollaboration withKEK(Japan) and inEurope within t he LAGUNAdesign study [12].
+∗Contribution to the Workshop “European Strategy for Future Neutrino Physics”, CERN, Oct. 2009, to appear in the
+Proceedings.
+†Now atUniversityof Zürich, Physik-Institut,CH–8057 Züri ch, Switzerland.
+‡Corresponding author: andre.rubbia@cern.ch2The GLACIER detector concept
+The proposed cryostat for GLACIER is a single module cryo-ta nk based on industrial liquefied natural
+gas (LNG) technology. The shape is cylindrical, which gives an excellent surface-to-volume ratio. The
+detector design is highly scalable, up to a total mass of 100 k ton. One key technical challenge is to drift
+free electrons in liquid argon over a length as large as 20 met ers, which requires special care to achieve
+and maintain high purity in the liquid argon bulk and for the h igh voltage necessary to generate the drift
+field (in the order of a MVolt). The readout of the charge relie s on the novel Large Electron Multiplier
+(LEM) technique: LAr LEM-TPCs operate in double phase with c harge extraction and amplification
+in the vapor phase [14]. The concept has been very successful ly demonstrated on small prototypes:
+ionization electrons, after drifting in the LAr volume, are extracted by a set of grids into the gas phase
+anddrivenintotheholes ofoneor moreLEMs,wherecharge amp lification occurs. EachLEMisathick
+macroscopic hole multiplier, which can be manufactured wit h standard PCB techniques. The electrons
+signal is readout via two orthogonal coordinates. One possi ble configuration, which has been tested
+successfully, is to use the induced signal on the segmented u pper electrode of the LEM itself and the
+other by collecting the electrons on a segmented anode. The i mages obtained with the LAr LEM-TPC
+are of very high quality, as in bubble chambers, owing to the c harge amplification in the LEM, and
+have good measured dE/ dx resolution. Compared to LAr TPCs wi th wires immersed in liquid argon,
+whicharehardtoscale toverylarge sizes duetomechanical a ndcapacitance (electronic noise) issues of
+the long thin wires and by signal attenuation along the drift direction, the LAr LEM-TPC is an elegant
+solution for very large LAr TPCs with long drift paths and mm- sized readout pitch segmentation. In
+GLACIER there is also the possibility to detect the Cherenko v light produced by relativistic particles
+in liquid argon, and allows conceiving to have high Tc superc onducting solenoid in liquid argon, in
+order toobtain amagnetized detector. Therequirement for t heexcavation of theunderground cavern are
+particularly favorable when compared to other detector con cepts, requiring a cavern of about 250’000
+m3and relatively shallow depth (600 m.w.e. overburden).
+3R&D stagedapproach
+Effective extrapolation to the 100 kton scale requires conc rete and well coordinated R&D steps [13].
+A ton-scale LAr LEM-TPC detector is being operated at CERN wi thin the CERN RE18 experiment
+(ArDM, focused on direct Dark Matter searches [15]). In orde r to prove the performance for neutrino
+physics, additional dedicated test beam campaigns are bein g considered for a detector with a mass of
+about10tons[16],totestandoptimizethereadoutmethodsa ndtoassessthecalorimetricperformanceof
+suchdetectors. Beyondtheseefforts,a1kton-scaledevice istheappropriatechoiceforafullengineering
+prototype of a 100 kton detector. The choice of the size for th e prototype is the result of two a priori
+contradictory constraints: (1) having the largest possibl e detector as to minimize the extrapolation to
+100 kton (2) having the smallest possible detector to minimi ze the cost and time requirement to build
+it. A 1 kton detector, which is a full prototype of the GLACIER design, can be built with a diameter
+of 12 m and a vertical drift of 10 m [11]. From the point of view o f the drift path, a mere factor 2
+will be needed to extrapolate from the prototype to the 100 kt on device. Hence, the prototype will be
+the practical demonstration for the long drift of free elect rons in realistic conditions. At the same time,
+the rest of the volume scaling from the 1 kon to the 100 kton is a chieved by increasing the diameter to
+about 70 m, and is relatively straightforward noting that (a ) large LNGtanks with similar diameters and
+aspect ratios already exist (b) the LArLEM-TPCreadout abov e theliquid will bescaled from anarea of
+80 m2(1 kton) to 3800 m2(100 kton). The LAr LEM-TPC is fully modular, and this will no t require
+a fundamental extrapolation of the principle, but rather on ly poses technical challenges of production,
+which can besolved in collaboration withindustry.
+2s)µ time (40 60 80 100 120 140 160 180 200 220amplitude (ADC counts)
+0100200300400500Anode signals (event 10079)
+s)µ time (40 60 80 100 120 140 160 180 200 220amplitude (ADC counts)
+050100150200250300LEM signals (event 10079)
+strip number0 2 4 6 8 10 12 14 16s)µ time (
+406080100120140160180200220
+amplitude (ADC counts)
+-60-40-20020406080100120Anode event display (event 10079)
+strip number0 2 4 6 8 10 12 14 16s)µ time (
+406080100120140160180200220
+amplitude (ADC counts)
+-60-40-20020406080100120LEM event display (event 10079)
+s) time (
+s)µ time (40 60 80 100 120 140 160 180 200amplitude (ADC counts)
+0100200300400500600700Anode signals (event 10248)
+s)µ time (40 60 80 100 120 140 160 180 200amplitude (ADC counts)
+0100200300400500LEM signals (event 10248)
+strip number0 2 4 6 8 10 12 14 16s)µ time (
+406080100120140160180200
+amplitude (ADC counts)
+-60-40-20020406080100120Anode event display (event 10248)
+strip number0 2 4 6 8 10 12 14 16s)µ time (
+406080100120140160180200
+amplitude (ADC counts)
+-60-40-20020406080100120LEM event display (event 10248)
+s)µ time (
+Fig.1:Multi-trackscosmicrayeventsasseen inthe3lt LArLEM-TPC at CERN.
+3.1 LEMreadout R&D
+In order to study the properties of the LEM and the possibilit y to reach high gains in double phase,
+extensive R&D has been performed on several prototypes, bui lt using standard PCB techniques from
+different manufacturers. Double-sided copper-clad FR4pl ates with thicknesses ranging from 0.8 mm to
+1.6 mm are drilled with a regular pattern of 500 µm diameter holes at a relative distance of 800 µm. By
+applying a potential difference on the two faces of the PCB an intense electric field inside the holes is
+produced. A first single stage prototype demonstrated astab le operation in pure Ar at room temperature
+and pressure up to 3.5 bar with a gain of 800 per electron. Meas urements were performed at high
+pressure because thedensity ofArat3.5barisroughly equiv alent totheexpected density ofthevapor at
+the temperature of 87 K (LAr temperature at 1 bar). Double sta ge LEM configurations were then tested
+inpureAratroomtemperature, cryogenictemperatureandin doublephaseconditions. Thedouble-stage
+LEMsystem demonstrated againof ∼103at atemperature of 87Kand apressure of 1bar. Thedouble-
+phase operation of the LEM proved the extraction of the charg e from the liquid to the gas phase. A 3 lt
+active volume LAr LEM-TPC was then constructed and successf ully operated at CERN. In this setup a
+LArdrift volumeof10 ×10cm2crosssection andwithanadjustable depthofupto30cmisfol lowedon
+topbyaLEMamplificationstageintheArvaporatadistanceof ∼1.5fromtheliquidsurface. Ionization
+electrons are drifted upward by a uniform electric fieldgene rated by a system of field shapers, extracted
+fromtheliquidbymeansoftwoextraction gridspositioned a crosstheliquid-vapour interface anddriven
+totheLEMplanes. Theamplifiedcharge isreadout bytwoortho gonal electrodes, oneusing theinduced
+signal on the segmented upper electrode of the LEMitself and the other by collecting the electrons on a
+3segmentedanode. Inthisfirstproduction bothreadoutplane sweresegmentedwith6mmwidestrips, for
+a total of 32 readout channels for a 10 ×10 cm2active area. A gain of about 10 was achieved in realistic
+double-phase conditions; two events are shown are shown in F ig. 1. Transverse segmentations down to
+2-3 mm will be tested in the near future, together with new con figuration of the readout electrodes. As
+a next step of the LEM readout R&D, we foresee the development of the full LEM-based readout for a
+6m3LAr LEM-TPCdesigned for test beam exposure.
+Acknowledgements
+We thank the the Swiss Science National Foundation and ETH Zü rich for supporting this experimental
+programme.
+References
+[1] Y.Itow et al., "TheJHF-Kamioka neutrino project," arXi v:hep-ex/0106019
+[2] D.S.Ayresetal.[NOvACollaboration], "NOvAproposal t obuild a30-kiloton off-axis detector to
+study neutrino oscillations in the Fermilab NuMIbeamline, " arXiv:hep-ex/0503053.
+[3] M.Apollonioetal.[CHOOZCollaboration],Phys.Lett.B 466,415(1999)[arXiv:hep-ex/9907037]
+[4] S. Amerio et al. [ICARUS Collaboration], "Design, const ruction and tests of the ICARUS T600
+detector," Nucl. Instrum. Meth. A 527 (2004)
+[5] A. Rubbia, "Neutrino factories: Detector concepts for s tudies of CP and T violation effects in
+neutrino oscillations," arXiv:hep-ph/0106088.
+[6] A. Ereditato and A. Rubbia, "Conceptual design of a scala ble multi-kton superconducting magne-
+tized liquid argon TPC",Nucl. Phys. Proc. Suppl. 155(2006) 233 [arXiv:hep-ph/0510131].
+[7] A. Bueno et al., "Nucleon decay searches with large liquid argon TPC detect ors at shal-
+low depths: Atmospheric neutrinos and cosmogenic backgrou nds," JHEP 0704(2007) 041
+[arXiv:hep-ph/0701101].
+[8] I. Gil Botella and A. Rubbia, "Decoupling supernova and n eutrino oscillation physics with LAr
+TPCdetectors," JCAP 0408(2004) 001 [arXiv:hep-ph/0404151].
+[9] A.G.Cocco, A.Ereditato, G.Fiorillo, G.Mangano andV.P ettorino, "Supernova relic neutrinos in
+liquid argon detectors," JCAP 0412(2004) 002 [arXiv:hep-ph/0408031].
+[10] A.Rubbia, "Experiments for CP-violation: Agiant liqu id argon scintillation, Cerenkov and charge
+imaging experiment?," arXiv:hep-ph/0402110.
+[11] A. Rubbia, "Underground Neutrino Detectors for Partic le and Astroparticle Science: the Giant
+Liquid Argon Charge Imaging ExpeRiment (GLACIER)," J. Phys . Conf. Ser. 171(2009) 012020
+[arXiv:0908.1286 [hep-ph]].
+[12] D. Angus et al. [LAGUNA consortium], to appear in the Pro ceedings of the Workshop ’European
+Strategy for Future Neutrino Physics’, CERN,Oct. 2009.
+[13] A. Marchionni, to appear in the Proceedings of the Works hop ’European Strategy for Future Neu-
+trino Physics’, CERN,Oct. 2009.
+[14] A.Badertscher etal.,"Operationofadouble-phase pureargonLargeElectronMul tiplierTimePro-
+jection Chamber: comparison of single and double phase oper ation," Nucl.Instr.andMeth.A(2009),
+doi:10.1016/j.nima.2009.10.011 [arXiv:0907.2944 [phys ics.ins-det]].
+[15] A. Rubbia, "ArDM: A ton-scale liquid argon experiment f or direct detection of dark matter in the
+universe," J. Phys.Conf. Ser.39 (2006) 129 [arXiv:hep-ph/ 0510320]
+[16] D. Autiero et al., "Test beam exposure of a Liquid Argon TPC Detector at the CER N SPS North
+Area," Abstract #82, Workshop on New Opportunities in the Ph ysics Landscape at CERN, May
+2009
+4
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+arXiv:1001.0077v1  [physics.ins-det]  31 Dec 2009The LAGUNA design study–towards giant liquid basedundergr ound
+detectors forneutrino physics andastrophysicsand proton decay
+searches∗
+TheLAGUNAconsortium†: D. Angusa, A.Arigab, D.Autieroc, A.Apostud,A.Badertschere,
+T.Bennetf, G.Bertolag, P.F.Bertolag,O.Besidah, A.Bettinii, C.Boothf, J.L.Bornec,I. Brancusd,
+W.Bujakowskyj,J.E.Campagnec,G.Cata Danild, F.Chipesiud, M.Chorowskik,J. Crippsf,
+A.Curionie,S. Davidsonc, Y.Declaisc,U.Drostg, O.Duliul,J. Dumarchezc,T. Enqvistm,A.Ereditatob,
+F.von Feilitzschn, H.Fynboo, T.Gamblef,G. Galvaninp,A.Gendottie, W.Gizickik,M.Goger-Neffn,
+U.Grassling,D.Gurneyq, M.Hakalar, S. Hannestado,M. Haworthq,S. Horikawae,A.Jipal,F.Jugetb,
+T.Kalliokoskis, S. Katsanevasc, M.Keent, J.Kisielu, I.Kreslob,V. Kudryastevf, P.Kuusiniemim,
+L.Labargav,T.Lachenmaiern,J.C.Lanfranchin, I. Lazanul, T.Lewken, K.Loom,P.Lightfootf,
+M.Lindnerw,A.Longhinh,J. Maalampis,M.Marafinic,A.Marchionnie,R.M.Margineanud,
+A.Markiewiczy, T.Marrodan-Undagoitan,J.E.Marteauc,R.Matikainenr, Q.Meindln, M.Messinab,
+J.W.Mietelskiy, B.Mitricad,A.Mordasinig,L.Moscah,U.Moserb, G.Nuijtenr,L.Oberauern,
+A.Oprinad, S.Palingf,S. Pascolia, T.Patzakc, M.Pectud, Z.Pileckij,F.Piquemalc, W.Potzeln,
+W.Pytelx,M.Raczynskix,G.Raffletz,G.Ristainop,M. Robinsonf, R.Rogersq,J. Roinistor,
+M.Romanai, E.RondioA, B.Rossib, A.Rubbiae, Z.Sadeckix, C.Saenzi, A.Saftoiud,J. Salmelainenr,
+O.Simal,J. Slizowskij,K.Slizowskij, J.SobczykB, N.Spoonerf,S. Stoicad, J.Suhonens,R.SulejA,
+M.Szarskay,T.SzeglowskiB, M.Temussip, J.Thompsonq,L.Thompsonf,W.H.Trzaskas,
+M.Tippmannn,A.Tonazzoc,K.Urbanczykj,G.Vasseurh,A.Williamst, J.Wintern,K.Wojutszewskaj,
+M.Wurmn, A.Zalewskay, M.Zampaoloc, M.Zitoh
+(a)UniversityofDurham(UDUR),UniversityOffice,Old Elve t,DurhamDH13HP,UnitedKingdom(b)
+UniversityofBern,4Hochschulstrasse,CH-3012,Bern (c)C entreNationaldela RechercheScientifique,Institut
+NationaldePhysiqueNucléaireet dePhysiquedesParticule s(CNRS/IN2P3),3rueMichel-Ange,Paris75794,
+France(d)HoriaHulubeiNationalInstituteofRD forPhysic sandNuclearEngineering,IFIN-HH,407
+AtomistilorStreet,R-077125,Magurele,jud. ILFOV,PO Box MG-6,postalcodeRO-077125,Romania(e)ETH
+Zurich,101Raemistrasse, CH-8092Zurich(f)TheUniversit yofSheffield(USFD),New SpringHouse231,
+GlossopRoad,SheffieldS102GW,UnitedKingdom(g)Lombardi EngineeringLimited,viaR.Simen,CH-6648,
+Minusio(h)Commissariatà l’EnergieAtomique(CEA)/Direc tiondesSciencesdela Matière,25rueLeblanc,
+Paris75015,France(i)LaboratorioSubterraneodeCanfran c(LSC),Plaza delAyuntamientono. 1,22880
+Canfranc(Huesca),Spain(j)MineralandEnergyEconomyRes earchInstituteofthePolish AcademyofSciences
+(IGSMIE-PAN),Wybickiego7,30-950Krakow,Poland(k)Wroc lawUniversityofTechnology(PWrWroclaw),
+ul. WybrzezeWyspianskiego27,50-370Wroclaw,Poland(l)U niversityofBucarest(UoB),FacultyofPhysics
+Bld.Atomistilornr.405,PhysicsPlatform,Magurele,Ilfo vCounty,RO-077125,MG-11Bucharest-Magurele,
+Romania(m)UniversityofOulu(U-OULU),1Pentti KaiteranK atu,Oulu90014,Finland(n)Technische
+UniversitätMünchen(TUM),21Arcisstrasse,München80333 ,Germany(o)UniversityofAarhus(AU),1 Norde
+Ringgade,AarhusC8000,Denmark(p)AGTIngegneriaSrl, Per ugia,10Avia dellaPallotta,Perugia06126,
+Italy(q)TechnodyneInternationalLtd.,Unit16,Shakespe areBusinessCentreHathawayClose,EastleighUK SO
+504SR,UnitedKingdom(r)KalliosuunnitteluOyRockplanLt d.,2 Asemamiehenkatu,Helsinki00520,Finland
+(s)UniversityofJyväskylä(JyU),9Survontie,Jyväskylä4 0014,Finland(t)ClevelandPotash Limited(CPL),
+BoulbyMine,Loftus,SaltburnCleveland,TS134UZ,UK (u)In stituteofPhysics,UniversityofSilesia
+Uniwersytecka4,40-007Katowice,Poland(v)UniversidadA utonomadeMadrid(UAM),C/Einstein no. 1;
+Rectorado,CiudadUniversitariadeCantoblanco,28049Mad rid,Spain(w)Max-Planck-InstituteforNuclear
+Physics,Heidelberg(x)KGHM CUPRUM LtdResearchandDevelo pmentCentre,Pl. 1Maja,50-136Wrocaw,
+Poland(y)IFJPan,H.NiewodniczaskiInstituteofNuclearP hysicsPAN, Radzikowskiego152,31-342Krakow,
+Poland(z)Max-Planck-InstituteforPhysics,Munich(A)Hi ghEnergyPhysicsDepartment-A.SoltanInstitute
+∗Contribution to the Workshop “European Strategy for Future Neutrino Physics”, CERN, Oct. 2009, to appear in the
+Proceedings.
+1forNuclearStudies(SINS)Hoza6900-681Warsaw,Poland(B) FacultyofPhysicsandAstronomy,Wroclaw
+University,pl M.Borna9,50-204Wroclaw,Poland.
+Abstract
+The feasibility of a next generation neutrino observatory i n Europe is being
+considered within the LAGUNA design study. To accommodate g iant neu-
+trino detectors and shield them from cosmic rays, a new very l arge under-
+ground infrastructure is required. Seven potential candid ate sites in differ-
+ent parts of Europe and at several distances from CERN are bei ng studied:
+Boulby (UK), Canfranc (Spain), Fréjus (France/Italy), Pyh äsalmi (Finland),
+Polkowice-Sieroszowice (Poland), Slanic (Romania) and Um bria (Italy). The
+design study aims at the comprehensive and coordinated tech nical assessment
+of each site, at a coherent cost estimation, and at a prioriti zation of the sites
+within thesummer 2010.
+1Physicsgoals
+Large underground neutrino detectors, for instance SuperK amiokande or SNO, have achieved funda-
+mental results in particle and astro-particle physics. A ne xt-generation very large multipurpose neutrino
+observatory ofatotalmassintherangeof100’000 to1’000’0 00 tonswillprovidenewanduniquescien-
+tificopportunities inthis field, very likely leading tofund amental discoveries. It will aim at asignificant
+improvementinthesensitivitytosearchforprotondecays, pursuingtheonlypossiblepathtodirectlytest
+physicsattheGUTscale,extending theproton lifetimesens itivities upto1035years,arangecompatible
+with several theoretical models; it will measure with unpre cedented sensitivity the last unknown mixing
+angleθ13and unveil the existence of CP violation in the leptonic sect or, which in turn could provide
+an explanation of the matter-antimatter asymmetry in the Un iverse; moreover it will detect neutrinos
+as messengers from astrophysical objects as well as from the Early Universe to give us information on
+processes happening in the Universe, which cannot be studie d otherwise. In particular, it will sense a
+large number of neutrinos emitted by exploding galactic and extragalactic type-II supernovae, allowing
+anaccuratestudyofthemechanismsdrivingtheexplosion. T heneutrinoobservatory willalsoallowpre-
+cision studies of other astrophysical or terrestrial sourc es of neutrinos like solar and atmospheric ones,
+and search for new sources of astrophysical neutrinos, like for example the diffuse neutrino background
+from relic supernovae or those produced inDarkMatter (WIMP )annihilation inthecentre of theSunor
+the Earth.
+2The LAGUNAdesign study
+Theconstruction ofalargescaleneutrino detectorinEurop edevotedtoparticleandastroparticle physics
+was discussed several years ago (see e.g. [1]) and is current ly one of the priorities of the ASPERA
+roadmap, defined in 2008 [2]. Four national underground labo ratories located resp. in Boulby (UK),
+Canfranc (Spain), Gran Sasso (Italy), and Modane (France), are today in operation, hosting detectors
+looking for Dark Matter or neutrino-less double beta decays , or performing long-baseline experiments.
+However,noneoftheseexistinglaboratory islargeenoughf orthenext-generation neutrino experiments.
+The FP7 Design Study LAGUNA [3] (Large Apparatus studying Gr and Unification and Neu-
+trino Astrophysics) is an EC-funded project carrying on und erground sites studies and developments in
+view of such detectors observatories. Three detector optio ns are currently being studied: GLACIER[4],
+LENA[5], and MEMPHYS[6]. The study is evaluating possible e xtensions of the existing deep under-
+ground laboratories, and on top of it, the creation of new lab oratories in the following regions: Umbria
+Region (Italy), Pyhäsalmi (Finland), Sieroszowice (Polan d) and Slanic (Romania). Table 1summarizes
+†LAGUNA Coordinator : andre.rubbia@cern.chsome basic characteristics of the sites under consideratio n, including the distance from CERN,which is
+relevant incase aneutrino beam issent from CERNto the selec ted underground site.
+Table 1:Potentialsitesbeingstudiedwith theLAGUNAdesignstudy.
+Location Type Envisaged depth Distance from Energy1stOsc. Max.
+m.w.e. CERN[km] [GeV]
+Fréjus (F) Road tunnel ≃4800 130 0.26
+Canfranc (ES) Road tunnel ≃2100 630 1.27
+Umbria(IT)aGreen field ≃1500 665 1.34
+Sieroszowice(PL) Mine ≃2400 950 1.92
+Boulby (UK) Mine ≃2800 1050 2.12
+Slanic(RO) Salt Mine ≃600 1570 3.18
+Pyhäsalmi (FI) Mine upto ≃4000 2300 4.65
+a≃1.0◦off axis.
+3Site selection
+Site selection is a complex process involving the optimizat ion and assessment of several parameters,
+encompassing physics performance, technical feasibility , safety and legal aspects, socio-economic and
+environmental impact, costs, etc. As a result, LAGUNA is int erdisciplinary, involving most Euro-
+pean physicists interested in realizing massive neutrino d etectors, as well as geo-technical experts, geo-
+physicists, structural engineers, mining engineers and al so large storage tank engineers. It regroups
+21 beneficiaries, composed of academic institutions from De nmark, Finland, France, Germany, Poland,
+Spain, Switzerland, United Kingdom, as well as industrial p artners specialized in civil and mechanical
+engineering and rock mechanics, commonly assessing the fea sibility of a this Research Infrastructure in
+Europe. Someof the typical issues addressed byLAGUNAare il lustrated below:
+1. assessment of strengths and weaknesses of each site;
+2. rock mechanics and excavation of therequired caverns;
+3. overburden vs. detector options;
+4. design and construction of tanks inrelation tosites;
+5. transport, access, delivery of detector liquids;
+6. safety issues, inroad tunnels and inoperating mines;
+7. environmental impact, e.g. rock removal;
+8. relative costs;
+9. etc.
+The study also gives the opportunity to the members of the con sortium to visit all the potential
+sitesandholdseveral meetings toexchange ideas andworkto gether oncommonareas, solving common
+issues, inacollaborative spirit, although asense of healt hy competition is implied among the sites.
+The results will be summarized in 16 reports (“deliverables ") to be submitted to the EC within
+the fall 2010. A first report on the health, safety and environ mental impact has been prepared during
+the summer 2009. Seven “interim” reports, one for each site, are currently under preparation and should
+be available during the winter 2009/2010. They demonstrate that the site studies are well underway
+with definitions of the infrastructure in advanced stages, p roviding well-defined conceptual designs and
+reliable excavation cost-estimates.
+34The future oflong baselineneutrino physicsinEurope
+The next generation deep underground neutrino detector sho uld be coupled to advanced neutrino beams
+from CERN to complete the understanding of the leptonic mixi ng matrix, in particular to study matter-
+antimatter asymmetry in neutrino oscillations (CP-violat ion), thereby addressing the outstanding puzzle
+of the origin of the excess of matter over antimatter created in the very early stages of evolution of
+theUniverse. Theoutcome of thecurrent international prog ram tomeasure the(small) mixing angle θ13
+(T2KinJapan,DoubleChoozinEurope,DayaBayinChinaandNO vAintheUS)willdefinethestrategy
+to measure CP-violation in the neutrino sector, whether usi ng conventional superbeams or more exotic
+beta-beams orneutrino factories. Inthecasethat θ13isbelowthesensitivity ofT2K/DoubleChooz/Daya
+Bay/NOvA, a new far detector designed to measure CP-violati on coupled to an intensity upgraded con-
+ventional superbeam, is the most natural next step to contin ue exploring neutrino oscillations, since the
+increased statistics yields anorder of magnitude better se nsitivity in θ13.
+Similarplansareemergingworldwide, forinstance inNorth AmericawiththeDeepUnderground
+Science andEngineering Laboratory (DUSEL)at Homestake (S outhDakota), envisioning adeep under-
+ground facility coupled to USaccelerator laboratories loc ated at appropriate distances, for long baseline
+neutrino oscillation experiments [7]. In Asia, the Japanes e High Energy Research Accelerator Research
+Organization (KEK)roadmapforesees extensions oftheJPAR Cneutrino programme beyondthecurrent
+T2K experiment by increasing the neutrino beam intensity an d by constructing a new far detector in
+addition toSuperKamiokande [8].
+This international landscape underlines the “global" natu re of the project, with potential options
+beingconsidered inseveralcontinents, jointlydebatedby theinternational scientificcommunity, making
+all themore likely that only one such facility will be built w orldwide.
+5Conclusions and Outlook
+TheLAGUNAconsortium isstudyingthefeasibilityofanewla rgeunderground infrastructure inEurope
+abletohostnextgeneration neutrinophysicsandastrophys ics andprotondecayexperiments. Sevensites
+are presently being considered. The study aims at the compre hensive and coordinated technical assess-
+ment of each site, at acoherent cost estimation, and at aprio rity list for site selection, by Summer2010.
+After that, a more focused design study, aiming also at detec tor option selection, should be envisaged.
+Future long baseline in Europe should consider a new beam lin e from CERN towards the chosen LA-
+GUNAsite. Thenewlargeunderground infrastructure couldi nthefuturebeoperated inconnection with
+other, moreadvanced neutrino beams like beta-beams or neut rino factories.
+Acknowledgements
+The LAGUNAdesign study is financed by FP7Research Infrastru cture "Design Studies", Grant Agree-
+ment No. 212343 FP7-INFRA-2007-1.
+References
+[1] D. Autiero et al., “Large underground, liquid based detectors for astro-par ticle physics in Europe:
+scientific case and prospects,” JCAP 0711(2007) 011 [arXiv:0705.0116 [hep-ph]].
+[2] ASPERA is a network of national government agencies resp onsible for coordinating and funding
+national research efforts inAstroparticle Physics. See http://www.aspera-eu.org
+[3] For up-to-date information see http://laguna.ethz.ch
+[4] A.Rubbia, “Experiments for CP-violation: Agiant liqui d argon scintillation, Cerenkov and charge
+imaging experiment?,” arXiv:hep-ph/0402110.
+[5] L. Oberauer, F. von Feilitzsch and W. Potzel, “A large liq uid scintillator detector for low-energy
+neutrino astronomy,” Nucl. Phys. Proc. Suppl. 138(2005) 108.
+4[6] A. de Bellefon et al., “MEMPHYS: A large scale water Cerenkov detector at Frejus, ”
+arXiv:hep-ex/0607026.
+[7] Y.K.Kim,“Plans for super-beams inUS”,this report.
+[8] T.Hasegawa, “Plans for super-beams in Japan”, this repo rt.
+5
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+arXiv:1001.0078v2  [quant-ph]  10 Jan 2018Classification of the Entangled States 2 ×M×N
+Jun-Li Liaand Cong-Feng Qiaoa,b∗
+a) Dept. of Physics, Graduate University, the Chinese Academ y of Sciences
+YuQuan Road 19A, 100049, Beijing, China
+b) Theoretical Physics Center for Science Facilities (TPCSF ), CAS
+YuQuan Road 19B, 100049, Beijing, China
+Abstract
+We extend the matrix decomposition method(MDM) in classify ing the 2 ×N×N
+truly entangled states to 2 ×M×Nsystem under the condition of stochastic local
+operations and classical communication. Itis foundthat th e MDM is quite practical
+and convenient in operation for the asymmetrical tripartit e states, and an explicit
+example of the classification of 2 ×6×7 quantum system is presented.
+1 Introduction
+Entanglement is an essential feature of quantum theory, describ ing a quantum cor-
+relation that exhibits nonlocal properties. In the seminal work [1], E instein, Podolsky,
+and Rosen (EPR) demonstrated through a Gedanken experiment t hat the quantum me-
+chanics (QM) can not provide a complete description of the “physica l reality” for two
+spatially separated but quantum mechanically correlated particles s tate which is now
+known as entangled state. The subsequent Bell theorem manifest the nonlocal character
+of the quantum correlation in the violation of Bell’s inequalities [2]. As the quantum
+information science develops, the impact of entanglement goes far beyond the testing of
+the conceptual foundations of QM. Entanglement is now of centra l importance in the
+∗Corresponding author.
+1quantum information theory (QIT) and is thought as the key physic al resource to real-
+ize quantum information tasks, such as quantum cryptography [3, 4], superdense coding
+[5, 6], and quantum computation [7], etc. This necessitates the qualit ative and quanti-
+tative description of the entanglement [8]. However due to the lack o f suitable tools for
+characterizing the entanglement, very limited quantum state spac e was explored in the
+quantum information theory.
+In quantum information processing (QIP), two states are suited t o implement the
+same task if they can be mutually converted by stochastic local ope rations and classical
+communication (SLOCC) [9], and therefore they are said to be in the s ame equivalent
+class. For three qubits, known result is that there are two kinds of true tripartite en-
+tanglement classes for pure state, namely, GHZ and W states [9]. A s the dimensions of
+each party increases nontrivial aspect shows up, i.e., non-local pa rameters may resides
+in the entangled states of 2 ×N×Nsystem when N≥4 [10, 11]. Many investigations
+concerned the classifications of 2 ×M×Nstates has been done in [10, 12, 13]. In the
+Refs.[10, 12], an iterated method was introduced to determine all th e inequivalent classes
+of the entangled states of 2 ×M×Nsystem based on the “range criterion”, where the
+entanglement classification of the low dimension system is a prerequis ite for the high di-
+mensions ones. Practical classifications of dimensions up to 2 ×4×4 and the related
+systems of 2 ×(M+ 4)×(2M+ 4) were given in [10]. With the increasing of dimensions,
+the complexity of the method grows dramatically because of the iter ated nature of their
+inequivalent proof of the entanglement classes. In a recent work [ 11] a novel method of
+classifying the pure state of 2 ×N×Nsystems was introduced in which all the inequiv-
+alent true tripartite entanglement classes can be determined direc tly by using merely the
+elementary operations on the cubic grid form of the state.
+The present work deals with the more general case: quantum stat e of 2×M×N
+systems (pure state if not specified). We show that the method we introduced in [11] can
+be generalized to the classification of true entangled states of 2 ×M×Nsystems. Although
+the main tools are the same for 2 ×M×Nwith that of 2 ×N×N, the generalization
+is nontrivial and the method for 2 ×M×Ncan help the general classification of L×
+M×Nsystems. All the inequivalent classes can be generated directly and no followed-up
+2221211212 213113
+123
+133 132122112
+223
+131121
+231
+233 232Γ1Γ2
+222 224214
+234124114
+134111ψ0
+ψ1ψ2
+Figure 1: The cubic form for 2 ×3×4 state. The node in the grid labeled ijkrepresents the
+matrix elements Γ {i,j,k}.
+inequivalence proof of these classes is needed. The content goes a s follows, in section 2,
+by representing the 2 ×M×Nstate in the form of matrix pairs, the 2 ×M×Nstates are
+divided into inequivalent sets under SLOCC. The detailed classification procedures with
+these inequivalent sets are presented in section 3 and a concrete e xample of classification
+of 2×6×7 system is given. Finally, in section 4 we give some concluding remarks.
+2 Matrix pair representation of 2×M×Nstate
+Adopt the conventions of [11], an arbitrary state of 2 ×M×Ncan be written as
+|Ψ2×M×N/an}bracketri}ht=/summationdisplay
+i,j,kΓ{i,j,k}|i/an}bracketri}htψ0|j/an}bracketri}htψ1|k/an}bracketri}htψ2, (1)
+where,ψ0represents the first qubit, ψ1andψ2has the dimension of MandNseparately;
+Γ{1,j,k}and Γ {2,j,k}areM×Ncomplex matrices (we assume M≤Nwithout loss of
+generalities). Then the state can be written in the following compact form
+|Ψ2×M×N/an}bracketri}ht=/parenleftbiggΓ1
+Γ2/parenrightbigg
+. (2)
+Clearly, to every state of 2 ×M×N, there is a form of Eq.(2) that corresponds to it, and
+a pictorial description of the state is straightforward, see Fig.(1) .
+The reduced density matrix of state Ψ 2×M×Nis defined as ρψi= Tr¬ψi[|Ψ/an}bracketri}ht/an}bracketle{tΨ|], where
+i∈ {0,1,2}. For three-partite systems, true (or genuine [9]) entanglement means that
+3reduced density matrices of each partite have full ranks. Let rdenote the rank of matrix
+hereafter, then r(ρψ0) = 2,r(ρψ1) =M,r(ρψ2) =Nfor the true entangled state of 2 ×
+M×Nsystems. The density matrix in the form of the matrix pairs can be ex pressed as
+ρψ0,ψ1,ψ2= (Γi)jk(Γi′)∗
+j′k′, (3)
+wherei,i′= 1,2;j,j′= 1,2,···,M;k,k′= 1,2,···,N. The reduced density matrix (take
+ψ2as an example) then is
+ρψ2= Trψ0,ψ1(ρψ0,ψ1,ψ2)
+=/summationdisplay
+ij(Γi)∗
+jk′(Γi)jk
+=/summationdisplay
+iÆ
+iΓi. (4)
+Lemma 2.1 ∀i∈ {0,1,2},Det(ρψi) = 0, if and only if the cubic form of the three-partite
+state (see Fig.(1)) can be transformed into a form where at le ast one plane perpendicular
+to axisψiare zero planes (plane with all its coefficients are zeroes) vi a ILOs.
+The proof is presented in Appendix A. Thus if Det( ρψi) = 0,i∈ {0,1,2}, the entanglement
+of 2×M×Nsystem reduces to the case of 2 ×M′×N′withM′<M or/andN′<N
+which should in principle be considered as an entanglement system of 2 ×M′×N′.
+3 Classification of 2×M×NState
+Two 2×M×Nstates/tildewideΨ and Ψ are said to be SLOCC equivalent if they are connected
+via invertible local operators (ILOs). That is, /tildewideΨ is SLOCC equivalent to Ψ if
+|/tildewideΨ2×M×N/an}bracketri}ht=T⊗P⊗Q|Ψ2×M×N/an}bracketri}ht, (5)
+whereT,P,Q are invertible complex matrices of dimension 2 ×2,M×M, andN×Nwhich
+act onψ0,ψ1,ψ2, respectively. Neglecting the extra factor of the determinant of matrices,
+T,P, andQcorrespond to the special linear groups of SL(2,C),SL(M,C),SL(N,C) [9].
+4Takes the wave function |Ψ2×M×N/an}bracketri}htin the matrix pair form [i.e., Eq.(2)], the ILO operators
+T,P,Qin Eq.(5) take the following form
+|/tildewideΨ2×M×N/an}bracketri}ht=/parenleftbiggt11t12
+t21t22/parenrightbigg/parenleftbiggPΓ1Q
+PΓ2Q/parenrightbigg
+, (6)
+wheretijare matrix elements of T. From Eq.(2) and Eq.(6) we can see that the SLOCC
+equivalence of the quantum state turns to the connectivity of the matrix pairs (Γ 1,Γ2)
+under the special linear transformations T,P,Q . Define the set that contains all the
+matrices pair (Γ 1,Γ2) asC. The whole space of Ccan be partitioned into numbers of
+subsets with different n,l
+Cn,l={(Γ1,Γ2)|rmax(α1Γ1+β1Γ2) =n,rmin(α2Γ1+β2Γ2) =l}, (7)
+wherermaxandrminrepresent the the maximum and minimum rank of the matrices
+respectively; αi,βi∈Cand|αi|+|βi| /ne}ationslash= 0;l∈[0,n],n∈[0,M].
+Proposition 3.1 If(Γ1,Γ2)∈Cn,land∃T,P,Q∈ILO,/parenleftbiggΓ′
+1
+Γ′
+2/parenrightbigg
+=/parenleftbiggt11t12
+t21t22/parenrightbigg/parenleftbiggPΓ1Q
+PΓ2Q/parenrightbigg
+,
+then (Γ′
+1,Γ′
+2)∈Cn,l.
+(see Appendix B). This proposition implies that the matrix pairs in subs etsCn,lwith
+differentnorlare SLOCC inequivalent.
+3.1 Classification on sets Cn,lwithn=M
+We start our classification of Cn,lin 2×M×Nsystem from the case n=M. Our aim
+is to construct the subsets cM,l⊂Cn,lwhich: (i), it includes representative states of all the
+inequivalent entanglement classes; (ii), each inequivalent class has o nly one representative
+state incM,l.
+Because∀(Γ1,Γ2)∈CM,l,∃T∈ILO (see Appendix B)
+T
+Γ1
+Γ2
+=
+t11t12
+t21t22
+
+Γ1
+Γ2
+, (8)
+that makes r(t11Γ1+t12Γ2) =M,r(t21Γ1+t22Γ2) =l, so we assume that all the matrix
+pairs inCM,lhave been performed this kind of ILO transformation T. That isr(Γ1) =M
+5andr(Γ2) =l. Two specific ILOs PandQcan transform (Γ 1,Γ2) into the following form
+/parenleftbigg
+Γ1
+Γ2/parenrightbigg
+P,Q− − →/parenleftbigg/parenleftbig
+EM×M0M×(N−M)/parenrightbig
+/parenleftbig
+AM×MBM×(N−M)/parenrightbig/parenrightbigg
+, (9)
+whereEis an unit submatrix of PΓ1Q,0is zero submatrix; AandBare submatrix of
+PΓ2Q, and all of them have the subscripts as their dimensions. We can rep resent the sub-
+matrixBM×(N−M)by matrix theory conventions, i.e., BM×(N−M)= Γ2({1,···,M},{M+
+1,···,N}).
+If (N−M)>M , thenrmax(BM×(N−M)) =M, the right hand of Eq.(9) can be further
+transformed by ILOs into
+/parenleftbigg
+Γ1
+Γ2/parenrightbigg
+P,Q− − →/parenleftbigg/parenleftbig
+EM×M0M×(N−2M)0M×M/parenrightbig
+/parenleftbig
+0M×M0M×(N−2M)EM×M/parenrightbig/parenrightbigg
+, (10)
+In the form of the cubic grid (Fig.(1)), this corresponds to that at least (N−2M) vertical
+planes in the middle of the cube are zero planes, which is actually an ent angled states of
+2×M×2Maccording to lemma 2.1. Thus here we consider the case M≥N/2.
+For arbitrary matrix pair with the form of the right hand of Eq.(9), w e implement the
+following transformation via ILOs
+/parenleftbigg/parenleftbig
+Em×m0m×(n−m)/parenrightbig
+/parenleftbig
+Am×mBm×(n−m)/parenrightbig/parenrightbigg
+stepi− −− →
+/parenleftbiggE1A′01B′01a
+01bE1′01E′/parenrightbigg
+/parenleftbigg
+A′B′02a
+02b02cE′/parenrightbigg
+, (11)
+where the lower-right submatrix of the right hand side Γ 2({m−r(B) + 1,···,m},{m+
+1,···,n}) =E′hasr(E′) =r(B);A′,E1A′are square submatrices with the dimensions
+(m−r(E′))×(m−r(E′)); the rest of the matrices are partitioned accordingly, i.e., 01B′,
+B′have the dimension ( m−r(E′))×r(E′),01a,02ahave the dimension of ( m−r(E′))×
+(n−m),01b,02bhave the dimension r(E′)×(m−r(E′)),E1′,02chave the dimension
+r(E′)×r(E′). After the transformation, Γ 1= (EM×M,0M×(N−M)) being unchanged, Γ 2
+becomes a quasidiagonal matrix and we named this procedure step i.
+Next we repartitioned the matrices on the left hand side of Eq.(11) a s follows
+
+/parenleftbiggE1A′01B′01a
+01bE1′01E′/parenrightbigg
+/parenleftbiggA′B′02a
+02b02cE′/parenrightbigg
+stepii− −− →
+/parenleftbiggE1A′01a01b
+01cE1′01d/parenrightbigg
+/parenleftbiggA′B′02a
+02b02cE′/parenrightbigg
+. (12)
+6This is named as step ii. Consider the submatrix B′, if it is not identically zero we can
+perform the transformation of step ion the left-top submatrices/parenleftbig
+A′B′/parenrightbig
+of Eq.(12)
+/parenleftbigg/parenleftbig
+E1A′01a/parenrightbig
+/parenleftbig
+A′B′/parenrightbig/parenrightbigg
+stepi− −− →
+/parenleftbiggE1A′′01B′′01a′
+01b′E1′′01E′′/parenrightbigg
+/parenleftbiggA′′B′′02a
+02b′02c′E′′/parenrightbigg
+. (13)
+This procedure can be done repeatedly (suppose repeat ntimes), until the r(B(n)) = 0.
+We can get that the matrix pair (Γ 1,Γ2) can be transformed into the following form
+Γ1→
+E1A(n)0 0 ···0 0
+0E1(n−1)0···0 0
+..................
+00 0 ···0 0
+00 0 ···E1′0
+≡/parenleftbiggE0
+0E1/parenrightbigg
+, (14)
+Γ2→
+A(n)B(n)=0 0 ···0 0
+00E(n−1)···0 0
+..................
+00 0 ···E′′0
+00 0 ···0E′
+≡/parenleftbiggSJS−10
+0E2/parenrightbigg
+, (15)
+where the transformed Γ 1is just (EM×M,0M×(N−M)), andE1,2are lower-right submatrices
+defined according to the partition lines; Jis the Jordan form of A(n).
+As a concrete example here we show how this whole procedure is proc eeded on the
+sets ofC4,lof 2×4×6 state. The transformation of Eq.(11) is start with
+/parenleftbigg/parenleftbigE4×404×2/parenrightbig
+/parenleftbig
+A4×4B4×2/parenrightbig/parenrightbigg
+stepi− −− →
+/parenleftbiggE1A′01B′01a
+01bE1′01c/parenrightbigg
+/parenleftbigg
+A′B′02a
+02b02cE′/parenrightbigg
+, (16)
+where
+
+/parenleftbigg
+E1A′01B′01a
+01bE1′01c/parenrightbigg
+/parenleftbiggA′B′02a
+02b02cE′/parenrightbigg
+=
+
+1 0 0 0 0 0
+0 1 0 0 0 0
+0 0 1 0 0 0
+0 0 0 1 0 0
+
+
+× × + + 0 0
+× × + + 0 0
+0 0 0 0 1 0
+0 0 0 0 0 1
+
+. (17)
+7Here, the rank of B4×2must be 2, otherwise the state will not be a true entangled state
+of 2×4×6, similar to the argument below Eq.(10). The step iigoes as follows
+
+/parenleftbigg
+E1A′01B′01a
+01bE1′01c/parenrightbigg
+/parenleftbigg
+A′B′02a
+02b02cE′/parenrightbigg
+step ii− −− →
+/parenleftbiggE1A′01B′01a
+01bE1′01c/parenrightbigg
+/parenleftbiggA′B′02a
+02b02cE′/parenrightbigg
+. (18)
+Next we repeat the step ito the up-left submatrices of the right hand side of Eq.(18).
+This iteration of step idepends on the rank of B′.
+(1),r(B′) = 0. In this case the matrix pair (Γ 1,Γ2) become
+/parenleftbigg
+Γ1
+Γ2/parenrightbigg
+T,P,Q− −− →
+
+1 0 0 0 0 0
+0 1 0 0 0 0
+0 0 1 0 0 0
+0 0 0 1 0 0
+
+
+× × 0 0 0 0
+× × 0 0 0 0
+0 0 0 0 1 0
+0 0 0 0 0 1
+
+. (19)
+And there are three different forms of Γ 2, i.e.,
+(1.1)
+0 0 0 0 0 0
+0 0 0 0 0 0
+0 0 0 0 1 0
+0 0 0 0 0 1
+,(1.2)
+λ0 0 0 0 0
+0 0 0 0 0 0
+0 0 0 0 1 0
+0 0 0 0 0 1
+,(1.3)
+0 1 0 0 0 0
+0 0 0 0 0 0
+0 0 0 0 1 0
+0 0 0 0 0 1
+,
+correspond to two Jordan canonical forms of A′,J=/bracketleftbigg
+λ0
+0 0/bracketrightbigg
+,J=/bracketleftbigg
+0 1
+0 0/bracketrightbigg
+, and a zero
+matrixA′=/bracketleftbigg
+0 0
+0 0/bracketrightbigg
+.
+(2),r(B′) = 1. In this case
+/parenleftbigg
+A′B′0
+0 0E′/parenrightbigg
+stepi− −− →
+× × 0 00
+0 0 0 10
+0 00 0E′
+. (20)
+
+× × 0 00
+0 0 0 10
+0 00 0E′
+stepii− −− →
+A′′B′′0 0
+0 0E′′0
+0 0 0E′
+, (21)
+8whereA′′,B′′are matrices of 1 ×1 andE′′= (0,1). Again apply step ion (A′′B′′) we
+have
+(2.1),r(B′′) = 0
+
+A′′B′′0 0
+0 0E′′0
+0 0 0E′
+stepi− −− →
+×00 0
+0 0E′′0
+0 00E′
+. (22)
+(2.2)r(B′′) = 1
+
+A′′B′′0 0
+0 0E′′0
+0 0 0E′
+stepi− −− →
+0 10 0
+0 0E′′0
+0 00E′
+. (23)
+For Eq.(22), A′′is equivalent to the case of A′′= 0 according to theorem 1 of [11]. For
+Eq.(23), in the next step of step ii,B(3)will be a matrix of dimension zero, and satisfies
+r(B(3)) = 0, thus the procedure is stopped. We get two inequivalent forms of Γ2
+
+0 0 0 0 0 0
+0 0 0 1 0 0
+0 0 0 0 1 0
+0 0 0 0 0 1
+,
+0 1 0 0 0 0
+0 0 0 1 0 0
+0 0 0 0 1 0
+0 0 0 0 0 1
+. (24)
+(3).r(B′) = 2. In this case
+/parenleftbigg
+A′B′0
+0 0E′/parenrightbigg
+stepi− −− →
+0 0 1 00
+0 0 0 10
+0 00 0E′
+. (25)
+Thus here is only one class, where Γ 2has just the form of Eq.(25). In the following, we
+shall see that these six cases correspond to the six inequivalent en tanglement classes in
+2×4×6 systems, which agrees with the result of Ref.[12].
+In all, for every (Γ 1,Γ2)∈CM,l, there exists an ILO transformation that make
+/parenleftbiggΓ′
+1
+Γ′
+2/parenrightbigg
+=T⊗P⊗Q/parenleftbiggΓ1
+Γ2/parenrightbigg
+. (26)
+Here Γ′
+1has the form of Eq.(14), and Γ′
+2=/parenleftbiggJ0
+0E2/parenrightbigg
+has the form of Eq.(15). Eq.(26)
+mapsCM,ltocM,l, wherecM,l⊆CM,land
+cM,l={(Γ1,Γ2)|Γ1=/parenleftbiggE0
+0E1/parenrightbigg
+,Γ2=/parenleftbiggJ0
+0E2/parenrightbigg
+; (Γ1,Γ2)∈CM,l}. (27)
+9Thus we have separated the classification of CM,linto two procedures: (1), the construction
+ofE2matrix; (2), classification of J. And for the second procedure, we have already
+completed the classification in [11]. We have the following theorem
+Theorem 3.2 ∀(Γ1,Γ2)∈cM,l, the setcM,lis of the classification of CM,l.(i)if two
+states are SLOCC equivalent then they can be transformed int o the same matrix vector
+(Γ1,Γ2);(ii)this matrix vector is unique in cM,l, that is if (Γ1,Γ′
+2)is SLOCC equivalent
+with (Γ1,Γ2), then (Γ1,Γ′
+2) = (Γ 1,Γ2),(Γ′
+2= Γ2means that E2=E′
+2and their Jordan forms
+ofJare equivalent under the condition of theorem 1 Ref.[11] )
+Proof:
+(i) The proof of this statement is straightforward, since in every s tep of transformation
+only invertible operators take part in.
+(ii) Suppose
+/parenleftbiggΓ1
+Γ′
+2/parenrightbigg
+=T′⊗P′⊗Q′/parenleftbiggΓ1
+Γ2/parenrightbigg
+. (28)
+It can be proved that the T′transformations can always be replaced by ILO operators
+P−1
+0,Q−1
+0, i.e., (see Appendix C)
+
+t′
+11t′
+12
+t′
+21t′
+22
+/parenleftbigg
+Γ1
+Γ2/parenrightbigg
+=/parenleftbigg
+P−1
+0Γ1Q−1
+0
+P−1
+0Γ2Q−1
+0/parenrightbigg
+. (29)
+Thus Eq.(28) can be rewritten as
+/parenleftbiggΓ1
+Γ′
+2/parenrightbigg
+=P′P−1
+0/parenleftbiggΓ1
+Γ2/parenrightbigg
+Q−1
+0Q′
+=P′′/parenleftbiggΓ1
+Γ2/parenrightbigg
+Q′′, (30)
+which correspond to two matrix equations
+
+
+P′′Γ1Q′′= Γ1
+P′′Γ2Q′′= Γ′
+2. (31)
+10We proceed our proof along the procedure of the construction of the standard form of
+cM,l. When Γ 1has the form of the left hand side of Eq.(11), the invertible transfo rmation
+P′′,Q′′that keep it invariant must be of the form
+Q′′=/parenleftbiggP′′−10
+X Y/parenrightbigg
+, (32)
+where Det(Y)/ne}ationslash= 0. This transformation transform Γ 2of the left hand side of Eq.(11) into
+the follow
+P′′Γ2Q′′=P′′(A,B)Q′′= (P′′AP′′−1+P′′BX,P′′BY), (33)
+whereAis theM×Msubmatrix, and Bis theM×(N−M) submatrix. Since P′′and
+Yboth are ILO operators, the rank of submatrix B, is unchanged and it can be further
+transformed to form of the right hand side of Eq.(11)
+/parenleftbiggA′B′0
+0 0E′/parenrightbigg
+. (34)
+We get that if two states are SLOCC equivalent then E′block of Γ′
+2and Γ 2must be
+identical. In Eq.(34) we see that Eq.(34) can be partitioned as the st epiiin Eq.(12).
+Then we apply the same argument as Eqs.(32,33) on submatrix/parenleftbigA′B′/parenrightbig
+. We can arrive
+that theE′′(E(3),E(4)and so on) must also be identical according to Eq.(31). And
+finally we can get that if (Γ 1,Γ′
+2) is SLOCC equivalent with (Γ 1,Γ2) then Γ′
+2and Γ2have
+the same canonical form in the set of Eq.(27). Q.E.D.
+3.2 Classification on sets Cn,lwithn=M−i
+Here we start by constructing the standard form of the set CM−i,lusing ILOs. It is
+shown that the construction of the entanglement classes cM−i,lcan be realized by apply
+the transformations of cM,lon both columns and rows of the matrix pairs (Γ 1,Γ2).
+∀(Γ1,Γ2)∈CM−i,l, (Γ1,Γ2) can be transformed into the following form
+/parenleftbigg
+Γ1
+Γ2/parenrightbigg
+T,P,Q− −− →/parenleftbigg
+Γ1
+Γ2/parenrightbigg
+=
+/parenleftbiggE(M−i)×(M−i)0 0
+0 0i×i0i×(N−M)/parenrightbigg
+/parenleftbigg×(M−i)×(M−i)× ×
+× 0i×i0i×(N−M)/parenrightbigg
+, (35)
+11where Γ 2is partitioned according to the partitions of Γ 1. Here due to rmax(α1Γ1+β1Γ2) =
+M−i, submatrix Γ 2({M−i+1,···,M},{M−i+1,···,N}) must be zero matrix. After
+this transformation, we can apply the step iin Eq.(11) on the submatrices Γ 2({1,···,M−
+i},{1,···,N}) and Γ 2({1,···,M},{1,···,M−i}) on the right hand side of Eq.(35). Γ 2
+then turns to (see Appendix D)
+Γ2ILO− − →
+× × × 0× × 0 0
+× × × 0× × 0 0
+× × × 0× × 0 0
+× × × 0 0i×i00 0
+0 0 0 0 0 0 Ei×i0
+0 0 0 0 0 0 0E(N−M)×(N−M)
+0 0 0 Ei×i0 0 0i×i0i×(N−M)
+, (36)
+while Γ 1being unchanged. Repartition the above equation as follows
+
+× × × 0× × 0 0
+× × × 0× × 0 0
+× × × 0× × 0 0
+× × × 0 0i×i0 0 0
+0 0 0 0 0 0 Ei×i0
+0 0 0 0 0 0 (N−M)×(N−M)0E(N−M)×(N−M)
+0 0 0 Ei×i0 0 0 i×i0i×(N−M)
+, (37)
+where the lower-right submatrix is (3 i+N−M)×(3i+ 2(N−M)).
+Proposition 3.3 There exists true entanglement state in 2×M×Npure systems if and
+only if (Γ1,Γ2)∈Cn,lwheren≥M+N
+3.
+This proposition reduce to the Eq.(81) of [11] when M=N. Letη={1,···,2M−2i−N},
+ρ={1,···,2M−3i−N,2M−2i−N+ 1,···,M−i}, then
+(Γ1,Γ2)(η,ρ) =
+× × × × ×
+× × × × ×
+× × × × ×
+× × × 0i×i0i×(N−M)
+, (38)
+has the same structure as the right hand side of Eq.(35), where Γ( η,ρ) is the submatrix
+of Γ with the selected rows and columns in sets ηandρ, separately. Then we can apply
+the same procedure as that of Eq.(35).
+12Here presents the 2 ×7×8 state as a demonstration, i.e., CM−1,l=C6,l. The matrix
+pair (Γ 1,Γ2) can be transformed into the following form
+/parenleftbigg
+Γ1
+Γ2/parenrightbigg
+T,P,Q− −−− →
+
+1 0 0 0 0 0 0 0
+0 1 0 0 0 0 0 0
+0 0 1 0 0 0 0 0
+0 0 0 1 0 0 0 0
+0 0 0 0 1 0 0 0
+0 0 0 0 0 1 0 0
+0 0 0 0 0 0 0 0
+
+
+× × × 0c04c050 0
+× × × 0c14c150 0
+× × × 0c24c250 0
+r30r31r320 0 0 0 0
+0 0 0 0 0 0 1 0
+0 0 0 0 0 0 0 1
+0 0 0 1 0 0 0 0
+
+, (39)
+where Γ 2can then be expressed as
+Γ′
+2=
+A0c0
+r00 0
+00 0E
+010 0
+≡/parenleftbiggA c
+r B/parenrightbigg
+. (40)
+The reason why the fifth and sixth entries of the last line in Γ 2are 0 is that otherwise the
+rank of Γ 1can be as large as M, as explained in Eq.(36). Further simplification can be
+proceeded according to the vector(or submatrices) c,r. There are four cases in general,
+i.e., (1), (c= 0,r= 0); (2), ( c/ne}ationslash= 0,r= 0); (3), ( c= 0,r/ne}ationslash= 0); (4), ( c/ne}ationslash= 0,r/ne}ationslash= 0). Here
+c/ne}ationslash= 0 means that r(c)≥1 and different ranks will result in different classes, i.e.,
+Γ00
+2=
+× × × 0 0 0 0 0
+× × × 0 0 0 0 0
+× × × 0 0 0 0 0
+0 0 0 0 0 0 0 0
+0 0 0 0 0 0 1 0
+0 0 0 0 0 0 0 1
+0 0 0 1 0 0 0 0
+,1Γ10
+2=
+× × × 0 0 0 0 0
+× × × 0 0 0 0 0
+0 0 0 0 0 1 0 0
+0 0 0 0 0 0 0 0
+0 0 0 0 0 0 1 0
+0 0 0 0 0 0 0 1
+0 0 0 1 0 0 0 0
+,(41)
+13Γ01
+2=
+× × 0 0 0 0 0 0
+× × 0 0 0 0 0 0
+× × 0 0 0 0 0 0
+0 0 1 0 0 0 0 0
+0 0 0 0 0 0 1 0
+0 0 0 0 0 0 0 1
+0 0 0 1 0 0 0 0
+,1Γ11
+2=
+×0×0 0 0 0 0
+×0 0 0 0 0 0 0
+0 0 0 0 0 1 0 0
+0 1 0 0 0 0 0 0
+0 0 0 0 0 0 1 0
+0 0 0 0 0 0 0 1
+0 0 0 1 0 0 0 0
+,(42)
+2Γ10
+2=
+× × × 0 0 0 0 0
+0 0 0 0 1 0 0 0
+0 0 0 0 0 1 0 0
+0 0 0 0 0 0 0 0
+0 0 0 0 0 0 1 0
+0 0 0 0 0 0 0 1
+0 0 0 1 0 0 0 0
+,2Γ11
+2=
+0 0 0 0 0 0 0 0
+0 0 0 0 1 0 0 0
+0 0 0 0 0 1 0 0
+1 0 0 0 0 0 0 0
+0 0 0 0 0 0 1 0
+0 0 0 0 0 0 0 1
+0 0 0 1 0 0 0 0
+.(43)
+Clearly, analogous with the set cM,lin Section 3.1, we can finally get the following set
+cM−1,l={(Λ,Γ)|r(Γ) =l; Γ =/parenleftbiggJ0
+0B/parenrightbigg
+; (Λ,Γ)∈CM−1,l}. (44)
+Jrepresents the Jordan canonical form.
+Theorem 3.4 ∀(Λ,Γ)∈cM−i,l, the setcM−i,lis of the classification of CM−i,l.(i)
+suppose two states are SLOCC equivalent, they can be transfo rmed into the same matrix
+vector (Λ,Γ);(ii)this matrix vector is unique in cM−i,l, that is suppose (Λ,Γ′)is SLOCC
+equivalent with (Λ,Γ), then (Λ,Γ′) = (Λ,Γ) (Γ′= ΓmeansJs are equivalent under the
+condition of theorem 1 in Ref.[11]andB′=B).
+We give a complete classification of 2 ×(M+ 5)×(2M+ 5) forM= 1, i.e., 2 ×6×7
+state whose classification has not been presented in literature so f ar.
+Classes of sets c6,lof 2×6×7: for all inequivalent classes in c6,l, they have the same form
+14of Γ1in the definition (27)
+Γ1=
+1 0 0 0 0 0 0
+0 1 0 0 0 0 0
+0 0 1 0 0 0 0
+0 0 0 1 0 0 0
+0 0 0 0 1 0 0
+0 0 0 0 0 1 0
+. (45)
+So we only list the form of Γ 2s,
+
+× × × × × 0 0
+× × × × × 0 0
+× × × × × 0 0
+× × × × × 0 0
+× × × × × 0 0
+0 0 0 0 0 0 1
+,
+× × × × 0 0 0
+× × × × 0 0 0
+× × × × 0 0 0
+× × × × 0 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+,
+× × × 0 0 0 0
+× × × 0 0 0 0
+× × × 0 0 0 0
+0 0 0 0 1 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+,(46)
+
+× × 0 0 0 0 0
+× × 0 0 0 0 0
+0 0 0 1 0 0 0
+0 0 0 0 1 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+,
+0 0 0 0 0 0 0
+0 0 1 0 0 0 0
+0 0 0 1 0 0 0
+0 0 0 0 1 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+,
+0 1 0 0 0 0 0
+0 0 1 0 0 0 0
+0 0 0 1 0 0 0
+0 0 0 0 1 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+.(47)
+Here the square matrices of {×}n×nin Eq.(46, 47) consists of all the inequivalent classes
+of setscn,lin 2×n×nstates. For example the first matrix of Eq.(46) is made up by all
+the genuine entanglement classes of the sets c5,lin 2×5×5 state and plus the one with
+{×}5×5= 0, thus there are (26 + 1) [15] inequivalent forms of this matrix.
+Classes of set c5,lof 2×6×7: for all inequivalent classes in c5,l, they has the same form
+of Λ
+Λ =
+1 0 0 0 0 0 0
+0 1 0 0 0 0 0
+0 0 1 0 0 0 0
+0 0 0 1 0 0 0
+0 0 0 0 1 0 0
+0 0 0 0 0 0 0
+. (48)
+15The different Γ 2s are
+
+× × 0 0 0 0 0
+× × 0 0 0 0 0
+0 0 0 0 0 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+0 0 1 0 0 0 0
+,
+0 0 0 0 0 0 0
+0 0 0 0 1 0 0
+0 0 0 0 0 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+0 0 1 0 0 0 0
+,
+0 1 0 0 0 0 0
+0 0 0 0 1 0 0
+0 0 0 0 0 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+0 0 1 0 0 0 0
+,(49)
+
+0 0 0 0 0 0 0
+0 0 0 0 0 0 0
+0 1 0 0 0 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+0 0 1 0 0 0 0
+,
+0 0 0 0 0 0 0
+0 0 0 0 1 0 0
+1 0 0 0 0 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+0 0 1 0 0 0 0
+,
+0 0 0 1 0 0 0
+0 0 0 0 1 0 0
+0 0 0 0 0 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+0 0 1 0 0 0 0
+.(50)
+
+0 0 0 0 0 0 0
+1 0 0 0 0 0 0
+0 1 0 0 0 0 0
+0 0 0 0 0 1 0
+0 0 0 0 0 0 1
+0 0 1 0 0 0 0
+, (51)
+Same as that of c6,l,{×}2×2here has three different forms.
+According to Proposition 3.3, there are no true entangled states in Cn,lwithn<6+7
+3.
+Thus we get (26+1)+(13+1)+(5+1)+(2+1)+1+1+(2+1)+1+1+1+ 1+1+1 = 61
+inequivalent entanglement classes in 2 ×6×7. It is clearly to see that this method is
+simple and effective, meanwhile each entangled state can be read out directly from the
+matrix pairs.
+4 Conclusions
+In summary, we have generalized our method of entanglement class ification under
+SLOCC to the more general case of 2 ×M×Nsystems. Two examples of 2 ×4×6
+and 2×6×7 are given where all their inequivalent entanglement classes are det ermined.
+Because the classification procedure is essentially a constructive a lgorithm, the method
+16can serve as a powerful tool in practical entanglement classificat ions with the aid of
+computers. Most importantly a wide range of state space is explore d which provide a rich
+resource for possible new applications in the quantum information th eory.
+Acknowledgments
+This work was supported in part by the National Natural Science Fo undation of
+China(NSFC) under the grants 10935012, 10928510, 10821063 a nd 10775179, by the CAS
+Key Projects KJCX2-yw-N29 and H92A0200S2, and by the Scientifi c Research Fund of
+GUCAS.
+Appendix
+A Proof of Lemma 2.1
+Necessity : If Det(ρψ2) = 0, there will be ILOs that transform ρψ2toρ′
+ψ2who has at least
+one column and one row of zeros. Without loss of generalities suppos e thekth column of
+ρ′
+ψ2are zeros, for the element ( k,k) ofρ′
+ψ2. From Eq.(4), we have
+(ρ′
+ψ2)kk=/summationdisplay
+ij|(Γ′
+i)jk|2= 0,
+which indicates that (Γ′
+i)jk= 0 for alliandj. So thekth plane perpendicular to partite
+ψ2are all zeros.
+Sufficiency : Suppose the kth (k≤N) plane perpendicular to ψ2can be transformed
+into a zero plane by ILOs: T,P,Q . That is/parenleftbiggΓ′
+1
+Γ′
+2/parenrightbigg
+=T/parenleftbiggPΓ1Q
+PΓ2Q/parenrightbigg
+, where Γ′
+ijk= 0,(j=
+1,2···M,i = 1,2). Then the kth row and kth column of [(Γ′
+i)†Γ′
+i] are all zeroes. Thus
+reduce density matrix of ρψ2:ρψ2=/summationtext
+i(Γ′
+i)†Γi, have thekth row and kth column both
+zeros. So Det( ρψ2) = 0.
+The similar proofs can be applied to ψ0andψ1, then we have Lemma 2.1.
+17B Proof of Proposition 3.1
+First we prove that, in the subsets Cn,l,∃O∈ILO,O/parenleftbigg
+Γ1
+Γ2/parenrightbigg
+=/parenleftbigg
+O11O12
+O21O22/parenrightbigg/parenleftbigg
+Γ1
+Γ2/parenrightbigg
+which makes rmax(O11Γ1+O12Γ2) =n,rmin(O21Γ1+O22Γ2) =l.
+Proof: in the definition Cn,l={(Γ1,Γ2)|rmax(α1Γ1+β1Γ2) =n,rmin(α2Γ1+β2Γ2) =l}, (i)
+if Det/parenleftbigg
+α1β1
+α2β2/parenrightbigg
+/ne}ationslash= 0, thenO=/parenleftbigg
+α1β1
+α2β2/parenrightbigg
+is an ILO. (ii) if Det/parenleftbigg
+α1β1
+α2β2/parenrightbigg
+= 0, then the
+two vectors ( α1,β1),(α2,β2) are linearly dependent. This implies r(Γ1) =r(Γ2) =n=l,
+in this case we can set O=/parenleftbigg
+1 0
+0 1/parenrightbigg
+.
+Proof of Proposition 3.1: Because/parenleftbiggΓ1
+Γ2/parenrightbigg
+∈Cn,l, so∃O∈ILO thatrmax(O11Γ1+O12Γ2) =
+n,rmin(O21Γ1+O22Γ2) =l. Then from/parenleftbiggΓ1
+Γ2/parenrightbigg
+=T/parenleftbiggPΓ′
+1Q
+PΓ′
+2Q/parenrightbigg
+we gave
+rmax(O′
+11PΓ′
+1Q+O′
+12PΓ′
+2Q) =n,
+rmin(O′
+21PΓ′
+1Q+O′
+22PΓ′
+2Q) =l, (52)
+whereO′=OT,T,P,Q are ILOs, so we get (Γ′
+1,Γ′
+2)∈Cn,l.
+C The proof of Eq.(29)
+Γ2in (Γ1,Γ2)∈cM,lhas a form of direct sum of JandE2as shown in the definition
+(27). Thus when the dimension of Jdoes not equal zero, there are no zeroes in pivot of
+T′and the left hand side of Eq.(29) can be separated into two parts
+/parenleftbigg
+1 0
+λ1/parenrightbigg/parenleftbigg
+α β
+0γ/parenrightbigg/parenleftbigg
+E
+J/parenrightbigg
+, (53)
+/parenleftbigg
+1 0
+λ1/parenrightbigg/parenleftbigg
+α β
+0γ/parenrightbigg/parenleftbigg
+E1
+E2/parenrightbigg
+, (54)
+where/parenleftbigg
+1 0
+λ1/parenrightbigg/parenleftbigg
+α β
+0γ/parenrightbigg
+is the LU decomposition of T′[14];E1has the same dimension as
+E2.
+For theJsub-matrix we have proved [11] there exists PJ,QJwhich make
+/parenleftbigg1 0
+λ1/parenrightbigg/parenleftbiggα β
+0γ/parenrightbigg/parenleftbiggPJEQJ
+PJJQJ/parenrightbigg
+=/parenleftbiggE
+J/parenrightbigg
+, (55)
+18For theE1,2parts, there exist operators that
+Py/parenleftbiggE1+λ′E2
+E2/parenrightbigg
+Qy=/parenleftbiggE1
+E2/parenrightbigg
+,
+Px/parenleftbigg
+E1
+E2+λE1/parenrightbigg
+Qx=/parenleftbigg
+E1
+E2/parenrightbigg
+, (56)
+whereλ′=β
+α. It is simple to verify that such kind of Px,y,Qx,ysatisfying the equations does
+exist (see Appendixes of [11] for detailed derivations). Thus PC=PxPyandQC=QyQx
+will make
+/parenleftbigg1 0
+λ1/parenrightbigg/parenleftbiggα β
+0γ/parenrightbigg/parenleftbiggPCE1QC
+PCE2QC/parenrightbigg
+=/parenleftbiggE1
+E2/parenrightbigg
+. (57)
+Combine Eq.(55) and Eq.(57) we can get such P0=PJ⊕PC,Q0=QJ⊕QCthat satisfy
+the following equation
+/parenleftbigg1 0
+λ1/parenrightbigg/parenleftbiggα β
+0γ/parenrightbigg/parenleftbiggP0Γ1Q0
+P0Γ2Q0/parenrightbigg
+=/parenleftbiggΓ1
+Γ2/parenrightbigg
+, (58)
+which is just Eq.(29).
+However there exists the special case that the dimension of Jequals zero, in this case
+there can be zero elements in the pivot of the nonsingular square ma trixT′.T′can then
+be decomposed as decomposed as [14]
+/parenleftbigg
+t′
+11t′
+12
+t′
+21t′
+22/parenrightbigg
+=PT′·/parenleftbigg
+1 0
+λ1/parenrightbigg
+·/parenleftbigg
+α β
+0γ/parenrightbigg
+, (59)
+whereα,β,γ,λ ∈C,PT′=/parenleftbigg
+0 1
+1 0/parenrightbigg
+and both matrices on the righthand side of above
+equation are nonsingular. It can be show that PT′can be compensated by some operators
+Pz,Qzwhich act on Γ 1and Γ2, i.e.,
+/parenleftbigg
+E1
+E2/parenrightbigg
+=PT/parenleftbigg
+PzE1Qz
+PzE2Qz/parenrightbigg
+, (60)
+see Appendixes of [11].
+D Proof of Eq.(36)
+First we prove the following proposition.
+19∀(Γ1,Γ2)∈CM−i,landα,β/ne}ationslash= 0: if (1) Γ 1has the form of Eq.(35) and Γ 2has the
+following structure
+Γ2=
+/bracketleftbigg
+× ×
+× ×/bracketrightbigg
+k×k×
+××
+×0
+0×
+×
+0 0 0I×I0EI×I0
+× × X000
+0 0 0×0×
+0 0 0×0×
+, (61)
+where Γ 2({k+ 1,···,k+I},{l+ 1,···,l+I}) =EI×I; (2) Γ =αΓ1+βΓ2,r(Γ({k+
+I+ 1,···,M},{k+I+ 1,···,N})) =r(Γ1({k+I+ 1,···,M},{k+I+ 1,···,N})) =
+M−i−k−I. Then the rank r(Γ({k+ 1,···,M},{k+ 1,···,N})) is larger when X/ne}ationslash= 0
+thanX= 0.
+Proof: Becauser(Γ({k+I+1,···,M},{k+I+1,···,N})) =r(Γ1({k+I+1,···,M},{k+
+I+ 1,···,N})) =M−i−k−Ithen the column vectors of Γ 2({k+I+ 1,···,M},{k+
+I+ 1,···,N}) are linearly dependent on that of Γ 1. From Eq.(61) we have if X= 0 then
+r(Γ({k+ 1,···,M},{k+ 1,···,N})) =M−i−k; ifX/ne}ationslash= 0 then
+r(Γ({k+ 1,···,M},{k+ 1,···,N})) =M−i−k+r(X)>M−i−k,
+which complete the proof.
+This indicates that if we insist the maximum rank of r(Γ =αΓ1+βΓ2) =M−i
+thenX= 0. Thus ∀(Γ1,Γ2)∈CM−i,l, we haveX= 0. Similarly argument applies to
+[Γ2({1,···,M},{1,···,M−i})]T. Then we can get Eq.(36).
+References
+[1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
+[2] J.S. Bell, Phys. 1, 195 (1964).
+[3] A.K. Ekert, Phys. Rev. Lett. 67, 661 (1991).
+[4] C.H. Bennett, F. Bessette, G. Grassard, L. Salvail, and J. Smolin , J. Cryptology 5,
+3 (1992).
+20[5] C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992).
+[6] K. Mattle, H. Weinfurter, P.G. Kwiat, and A. Zeilinger, Phys. Rev. Lett.76, 4656
+(1996).
+[7] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information
+(Cambridge University Press, Cambridge, England, 2000).
+[8] Ryszard Horodecki, Pawe/suppress l Horodecki, Micha/suppress l Horodecki, and Karol Horodecki, Rev.
+Mod. Phys. 81, 865 (2009).
+[9] W. D¨ ur, G. Vidal, and J.I. Cirac, Phys. Rev. A 62, 062314 (2000).
+[10] Lin Chen, Yi-Xin Chen, and Yu-Xue Mei, Phys. Rev. A 74, 052331 (2006).
+[11] Shuo Cheng, Junli Li, and Cong-Feng Qiao, J. Phys. A 43, 055303 (2010).
+[12] Lin Chen and Yi-Xin Chen, Phys. Rev. A 73, 052310 (2006).
+[13] Marcio F. Cornelio and A.F.R. de Toledo Piza, Phys. Rev. A 73, 032314 (2006).
+[14] Roger A. Horn and Charles R. Johnson, Matrix Analysis , (Cambridge University,
+Cambridge England, 1985).
+[15] Shuo Cheng, Junli Li, and Cong-Feng Qiao, Journal of the Grad uate School of the
+Chinese Academy of Sciences 3, 303 (2009) (in chinese).
+21
\ No newline at end of file
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+arXiv:1001.0079v2  [gr-qc]  3 Dec 2010The imprint of the interaction between dark sectors in galax y clusters
+Jian-Hua He, Bin Wang
+INPAC and Department of Physics, Shanghai Jiao Tong Univers ity, 200240 Shanghai, China
+Elcio Abdalla
+Instituto de Fisica, Universidade de Sao Paulo, CP 66318, 05 315-970, Sao Paulo, Brazil
+Diego Pavon
+Department of Physics, Autonomous University of Barcelona , 08193 Bellaterra, Barcelona, Spain
+Abstract
+Based on perturbation theory, we study the dynamics of how da rk matter and dark energy in the collapsing system approach
+dynamical equilibrium when they are in interaction. We find t hat the interaction between dark sectors cannot ensure the d ark
+energy to fully cluster along with dark matter. When dark ene rgy does not trace dark matter, we present a new treatment
+on studying the structure formation in the spherical collap sing system. Furthermore we examine the cluster number coun ts
+dependence on the interaction between dark sectors and anal yze how dark energy inhomogeneities affect cluster abundanc es. It
+is shown that cluster number counts can provide specific sign ature of dark sectors interaction and dark energy inhomogen eities.
+PACS numbers: 98.80.Cq, 98.80.-k
+1I. INTRODUCTION
+There has been convincing evidence indicating that our universe is ex periencing an accelerated expansion[ 1]. In the
+framework of General Relativity, the impart of the acceleration of the universe is given by a kind of mysterious energy
+called dark energy(DE). One leading candidate of such DE is the cosm ological constant. However, it is difficult to
+understand such a cosmological constant in terms of fundamenta l physics. Its value from observation is far below that
+calculated in quantum field theory, what is referred to as the cosmo logical constant problem. Moreover, using the
+cosmological constant to explain the DE, one is unavoidably led to the coincidence problem, namely, why the vacuum
+and matter energy densities are precisely of the same order today .
+Considering that DE contributes a significant fraction of the conte nt of the universe, it is natural in the framework
+of field theory to consider its interactions with the remaining fields, a s e.g., the other large fraction of the universe,
+the dark matter (DM). The possibility that DE and DM interact with ea ch other has been widely discussed recently
+[2]-[23]. It has been shown that an appropriate interaction between DE an d DM can provide a mechanism to alleviate
+the coincidence problem [ 2–6]. However, it has been suspected that the appearance of the cou pling between dark
+sectors may lead to the curvature perturbation instability [ 7]. This problem was later clarified in [ 8] where it has been
+shown that the stability of curvature perturbation holds while appr opriately choosing the forms of the interaction
+between dark sectors and the equations of state (EOS) of DE. An other possibility to cure the instability was suggested
+in [9]. Observational signatures on the dark sector mutual interactio n have been found in the probes of the cosmic
+expansion history by using the WMAP, SNIa, BAO and SDSS data etc [ 11]-[17]. Interestingly it was disclosed that
+the late ISW effect has the unique ability to give insight into the coupling between dark sectors [ 17]. The small
+positive coupling indicates that there is energy transfer from DE to DM, what can help to alleviate the coincidence
+problem [ 17,18].
+To further unveil the nature of the interaction between DE and DM , we require complementary probes in addition
+to investigating the expansion history of the universe. It has been observed that the growth of cosmic structure can
+be influenced by a nonminimal interaction between dark sectors, wh at leaves a clear change in the growth index due
+to the coupling [ 19,21]. Furthermore, it was suggested that the dynamical equilibrium of c ollapsed structures such as
+clusters would acquire a modification due to the coupling between DE a nd DM [ 22,23]. Comparing the naive virial
+masses of a large sample of clusters with their masses estimated by X -ray and by weak lensing data, small positive
+coupling has been tightly constrained [ 23], what agrees with the results given in [ 17] from CMB.
+The redshift dependence of cluster number counts is another pro mising tool to discriminate different DE models.
+2Discussions can be found in [ 24]-[32]. Similar investigation to test the particular scenario of coupled quint essence
+model by using the cluster number counts was investigated in [ 34–36], where the interaction between quintessence and
+DM was specifically described by some product of the densities of DE a nd DM. The sign they used for the coupling
+between dark sectors just describes the energy decay from DM t o DE and they concentrated on the inhomogeneities
+in the DM fluid propagating in the scalar field and affecting the scalar fie ld evolution. The cluster number counts
+dependence on the amount of DM coupled to DE was analyzed therein .
+In this paper we will investigate the possibility of employing measureme nts from cluster number counts to grasp
+the signature of interactions between DE and DM in some commonly dis cussed phenomenological models. These
+models were proved to have stable curvature perturbations [ 8] and have been tested by confronting CMB [ 17] and
+virial mass of cluster observations [ 23]. In our study we will use the spherical collapse model and consider e ither DE
+being distributed homogeneously or inhomogeneously. We will discuss the dynamics of DM and the inhomogeneous
+DE to virialization. When DE is homogeneous, DM flows to equilibrium in agr eement with that discussed in [ 38] if
+there is no interaction between dark sectors. If the DE is inhomoge neous, we will show that its dynamics is different
+from that of DM in the virialization in the linear perturbation level. This in dicates that DE does not cluster fully
+along with the DM and thus energy is not strictly conserved inside the collapsing system in the linear perturbation
+theory. Our result is a progress in the linear perturbation formalism , which has not been reported before.
+In the following discussion of this work, we will use the Press and Sche chter formalism [ 39] to predict the number
+density of collapsed objects. We will develop a new treatment on stu dying how the structure is formed when DE does
+not trace DM. The Press and Schechter formalism is crude if compar ed with N-body simulations, while here we do
+not seek the precise confrontations with the observational data , but the evolution of the mass function of collapsed
+objects predicted by the formalism will be helpful for us to underst and the influence of the interaction between dark
+sectors and the inhomogeneous DE distribution on cluster number c ounts.
+This paper is organized as follows: In Section II, we review the formalism of the perturbation theory in the presenc e
+of interaction between dark sectors. In Section III, we study the dynamics to describe the flow of the collapsing object
+to virialization. In Section IV, we introduce the spherical collapse model in the presence of coup ling between DE and
+DM and the Press-Schechter formalism for the galaxy number coun ts. In Section V, we show the numerical results
+on how the interaction and inhomogeneous DE influence the cluster n umber counts. In the last section we present
+our summary and discussions.
+3II. PERTURBATION THEORY WHEN DE INTERACTS WITH DM
+In this section, we go over the main results on the linear perturbatio n theory when DE interacts with DM. The
+detailed descriptions can be found in [ 8,19].
+In the spatially flat Friedmann-Robertson-Walker(FRW) backgrou nd, if there is interaction between DE and DM,
+neither of them can evolve independently. The (non)conservation equation are
+ρ′
+m+3Hρm=a2Q0
+m,
+ρ′
+d+3H(1+w)ρd=a2Q0
+d, (1)
+where the subscript “m” denotes the cold DM, “d” denotes the DE a ndQdescribes the interaction between them.
+Since we know neither the physics of DM nor that of DE at the presen t moment, we cannot write out the precise form
+of the interaction between them from first principles (see [ 33] for recent attempts). One has to specify the interaction
+either from the outset[ 3], or determine it from phenomenological requirements [ 5,18]. For the sake of generality, we
+consider the phenomenological description of the interaction betw een DE and DM in the comoving frame [ 8,19]
+Qν
+m=/bracketleftbigg3H
+a2(ξ1ρm+ξ2ρd),0,0,0/bracketrightbiggT
+Qν
+d=/bracketleftbigg
+−3H
+a2(ξ1ρm+ξ2ρd),0,0,0/bracketrightbiggT
+, (2)
+whereHis the global expansion rate in the conformal time, ξ1,ξ2are small dimensionless constants and Tis the
+transpose of the matrix.
+The perturbation equations for DM and DE in the subhorizon scale ha ve been derived in [ 19], having the form
+∆′
+m=−kVm−∆ma2Q0
+m
+ρm+a2δQ0I
+m
+ρm,
+V′
+m=−HVm+kΨ−a2Q0
+m
+ρmVm+a2δQI
+pm
+ρm; (3)
+∆′
+d= 3H(w−C2
+e)∆d−k(1+w)Vd−∆da2Q0
+d
+ρd+a2δQ0I
+d
+ρd,
+V′
+d+H(1−3w)Vd=kC2
+e
+1+w∆d+C2
+a
+1+wρ′
+d
+ρdVd+kΨ−w′
+1+wVd−a2Q0
+d
+ρdVd+a2δQI
+pd
+(1+w)ρd; (4)
+where the prime denotes the derivative with respect to the confor mal time and the gauge-invariant quantities ∆ m,∆d
+represent the density contrast ∆ m≈δρm/ρm=δm, ∆d≈δρd/ρd=δd. The perturbed gauge invariant couplings are
+expressed as [ 8,19],
+a2δQ0I
+m
+ρm≈3H(ξ1∆m+ξ2∆d/r),
+4a2δQ0I
+d
+ρd≈ −3H(ξ1∆mr+ξ2∆d),
+wherer=ρm/ρd.
+It is useful to rewrite these equations in the real space,
+∆′
+m+∇¯x·Vm= 3Hξ2(∆d−∆m)/r ,
+V′
+m+HVm=−∇¯xΨ−3H(ξ1+ξ2/r)Vm; (5)
+∆′
+d+(1+w)∇¯x·Vd= 3H(w−C2
+e)∆d+3Hξ1r(∆d−∆m),
+V′
+d+HVd=−∇¯xΨ−C2
+e
+1+w∇¯x∆d−w′
+1+wVd+3H/braceleftbigg
+(w−C2
+a)+1+w−C2
+a
+1+w(ξ1r+ξ2)/bracerightbigg
+Vd; (6)
+where ¯xrefers to the conformal coordinates.
+Definingσm=δρm,σd=δρd, and assuming that the EOS of DE is constant w′= 0, we can change Eqs.( 5,6)
+into,
+˙σm+3Hσm+∇x(ρmVm) = 3H(ξ1σm+ξ2σd),
+∂
+∂t(aVm) =−∇x(aΨ)−3H(ξ1+ξ2/r)(aVm); (7)
+˙σd+3H(1+C2
+e)σd+(1+w)∇x(ρdVd) =−3H(ξ1σm+ξ2σd),
+∂
+∂t(aVd) =−∇x(aΨ)−C2
+e
+1+w∇x·(a∆d)+3H/bracketleftbigg
+(w−c2
+a)+1+w−c2
+a
+1+w(ξ1r+ξ2)/bracketrightbigg
+(aVd); (8)
+where∇x=1
+a∇¯x. The dot denotes the derivative with respect to the cosmic time. Ψ in dicates the peculiar potential,
+which can be decomposed into Ψ = ψm+ψd, satisfying the Poisson equation [ 26],
+∇2
+λψλ= 4πG(1+3wλ)σλ, (9)
+whereσλrepresents the inhomogeneous fluctuation field and the subscript ”λ” denotes DM or DE, respectively. We
+have included the correction from General Relativity. In a homogen eous and isotropic background <ψλ>= 0, since
+<σλ>= 0. For DE and DM, their peculiar potentials read [ 26]
+ψm=−4πG/integraldisplay
+dV′σm
+|x−x′|, (10)
+ψd=−4πG/integraldisplay
+dV′(1+3w)σd
+|x−x′|. (11)
+III. DERIVATION OF THE LAYZER-IRVINE EQUATION
+In this section we derive the Layer-Irvine equation [ 38] when there is an interaction between DE and DM. Layzer-
+Irvine equation describes how a collapsing system reaches a state o f dynamical equilibrium in an expanding universe.
+5In the presenceof the interactionbetween DE and DM, the Layer- Irvineequation can tell us how the coupling between
+DE and DM affects the process of attaining equilibrium as well as the re spective configuration. Our derivation will
+be based on the perturbation theory given in Sec. II.
+For DM, the rate of change of the peculiar velocity is given by Eqs.( 7)
+∂
+∂t(aVm) =−∇x(aψm+aψd)−3H(ξ1+ξ2/r)(aVm). (12)
+Neglecting the influence of DE and the couplings, the above equation is nothing but the rate of change of the peculiar
+velocity of the DM particle in the expanding universe described by the Newton’s law which was the starting point in
+[38]. To derive the energy equation for local irregularities, we follow the same procedure as [ 38] by forming a product
+of Eqs.(12) withaVmρmˆεand integrating over the volume. Here ˆ ε=a3d¯x∧d¯y∧d¯zis the volume element which
+satisfies∂
+∂tˆε= 3Hˆε. Considering Eqs.( 1), the LHS of Eq.( 12) can be multiplied by aVmand integrated to yield
+∂
+∂t/parenleftbig
+a2Tm/parenrightbig
+−a23H(ξ1+ξ2/r)Tm. (13)
+whereTm=1
+2/integraltext
+V2
+mρmˆεis the kinetic energy of DM associated with peculiar motions of DM part icles.
+The RHS of Eqs.( 12) can be treated by the same token. Using partial integration, the potential part can be
+changed into
+−/integraldisplay
+aVm∇x(aψm+aψd)ρmˆε=a2/integraldisplay
+∇x(ρmVm)ψmˆε+a2/integraldisplay
+∇x(ρmVm)ψdˆε.
+Taking account of the first equation in ( 7), it can turn into
+−/integraldisplay
+aVm∇x(aψm+aψd)ρmˆε=−a2(˙Umm+HUmm)−a2/integraldisplay
+ψd∂
+∂t(σmˆε)
++ 3a2H{ξ1Umd+ξ2Udm+2ξ1Umm+2ξ2Udd} (14)
+whereUmm=1
+2/integraltext
+σmψmˆε,Udm=/integraltext
+σdψmˆε,Umd=/integraltext
+σmψdˆε, andUdd=1
+2/integraltext
+σdψdˆε.
+The second term in the RHS of Eq.( 12) can be changed into,
+−/integraldisplay
+(aVm)23H(ξ1+ξ2/r)ρmˆε=−a26H(ξ1+ξ2/r)Tm (15)
+Combining Eqs.( 13,14,15), we obtain
+˙Tm+˙Umm+H(2Tm+Umm) =−/integraldisplay
+ψd∂
+∂t(σmˆε)−3H(ξ1+ξ2/r)Tm
++3H{ξ1Umd+ξ2Udm+2ξ1Umm+2ξ2Udd}. (16)
+6This equation describes how DM reaches dynamical equilibrium in the co llapsing system in the expanding universe.
+If the DE is distributed homogeneously, σd= 0, this equation reduces to
+˙Tm+˙Umm+H(2Tm+Umm) =−3H(ξ1+ξ2/r)Tm+6Hξ1Umm. (17)
+For a system in equilibrium, ˙Tm=˙Umm= 0, we get the virial condition. If we take ¯ξ1= 3ξ1,¯ξ2= 3ξ2, we have the
+virial condition (2+ ¯ξ1+¯ξ2/r)Tm+(1−2¯ξ1)Umm= 0 [23]. Neglecting the interaction ¯ξ1=¯ξ2= 0, we recover the
+usual virial condition obtained in [ 38]. One can see that the presence of the coupling between DE and DM c hanges
+both the time required by the system to reach equilibrium and the equ ilibrium configuration itself [ 23].
+Now we consider the case that DE is no longer homogeneous and has p erturbation. The rate of change of the
+peculiar velocity of DE is described by the second equation in ( 8). Multiplying both sides of this equation by aVdρdˆε
+and integrating over the volume, on the LHS we have
+∂
+∂t/parenleftbig
+a2Td/parenrightbig
++3a2H(w+ξ1r+ξ2)Td. (18)
+On the RHS the first term reads,
+−/integraldisplay
+aVd∇x(aψm+aψd)ρdˆε=−a2
+1+w(˙Udd+HUdd)−a2
+1+w/integraldisplay
+ψm∂
+∂t(σdˆε)
+−a2
+1+w3H/braceleftbig
+2(C2
+e+ξ2)Udd+2ξ1Umm+ξ1Umd+(C2
+e+ξ2)Udm/bracerightbig
+. (19)
+For the remaining terms, we have
+−c2
+e
+1+w/integraldisplay
+aVd∇x(a∆d)ρdˆε+3H/bracketleftbigg
+(w−c2
+a)+1+w−C2
+a
+1+w(ξ1r+ξ2)/bracketrightbigg/integraldisplay
+(aVd)2ρdˆε
+=−c2
+e
+1+wa2/integraldisplay
+Vd∇x(σd)ˆε+6a2H/bracketleftbigg
+(w−C2
+a)+1+w−c2
+a
+1+w(ξ1r+ξ2)/bracketrightbigg
+Td. (20)
+Combining Eqs.( 18,19,20), we arrive at
+(1+w)˙Td+˙Udd+H[2(1+w)Td+Udd] =−3H/braceleftbig
+2(C2
+e+ξ2)Udd+2ξ1Umm+ξ1Umd+(C2
+e+ξ2)Udm/bracerightbig
+−/integraldisplay
+ψm∂
+∂t(σdˆε)−c2
+e/integraldisplay
+Vd∇x(σd)ˆε+3H/bracketleftbig
+(1+w)(w−2C2
+a)+(1+w−2C2
+a)(ξ1r+ξ2)/bracketrightbig
+Td,(21)
+describing how the DE reaches dynamical equilibrium in the collapsing sy stem in an expanding universe if it partici-
+pates in the structure formation.
+Looking at equations ( 16) and (21), we see that in non-interacting case ( ξ1=ξ2= 0), when C2
+e= 0, DE would
+cluster just like cold DM. Examples of DE models with this property wer e investigated in [ 37]. In interacting case, we
+see that for a collapsing system to reach dynamical equilibrium, the t ime and dynamics required by DE and DM to
+7reach equilibrium are different. This shows that in the collapsing syste m, DE does not fully cluster along with DM.
+Thus, the energy conservation breaks down inside the collapsing sy stem. It would be fair to say that this result was
+obtained in the linear level. It would be more interesting to examine the result in the non-linear perturbation and
+obtain clearer dynamics on how DE participates in the structure for mation.
+IV. SPHERICAL COLLAPSE MODEL AND THE PRESS-SCHECHTER FORMA LISM
+The simplest analytical tool to study the structure formation is th e spherical collapse model. The continuity
+equation for the background DM and DE energy densities are given b y
+˙ρm+3Hρm= 3H(ξ1ρm+ξ2ρd),
+˙ρd+3H(1+w)ρd=−3H(ξ1ρm+ξ2ρd). (22)
+Considering now the spherically symmetric region of radius Rand with the energy density ρcluster
+λ=ρλ+σλ, where
+“λ” denotes DM and DE respectively, it will eventually collapse from its gr avitational pull provided that σλ>0. The
+equation of motion for the collapsing model is governed by Raychaud huri’s equation.
+˙θ=−1
+3θ2−4πG/summationdisplay
+λ(ρλ+3pλ) (23)
+whereθ= 3˙R
+Rand Raychaudhuri’s equation can be rewritten as,
+¨R
+R=−4πG
+3/summationdisplay
+λ(ρλ+3pλ) (24)
+whereRis the local expansion scale factor which is determined by the total e nergy density in the spherical model. If
+the matter distribution is homogenous in the interior of the spherica l region, the local expansion scale factor Rwill
+have the same value inside the spherical region and from the Birkhoff theorem, the behavior of Ris only determined
+by the interior matter and is not affected by the matter distributed outside the spherical region.
+In the spherically symmetric region, supposing the DE distribution is h omogeneous ( σd= 0) and its evolution is
+still described in ( 22), the evolution of the DM energy density in the spherical region beh aves as
+˙ρcluster
+m+3hρcluster
+m= 3H(ξ1ρcluster
+m+ξ2ρd), (25)
+whereh=˙R/R.
+The Raychaudhuri’s equation applied to the spherical region has the form
+¨R=−4πG[1
+3ρcluster
+m+(1
+3+w)ρd]R . (26)
+8whereρdis back ground DE energy density. Converting the time derivative int o the derivative with respect to the
+scale factor a, we can write ¨R= (˙a)2d2R
+da2+¨adR
+daand change the Raychaudhuri’s equation into
+2a2(1+1
+r)R′′−R′[1+(3w+1)/r]a=−R[(3w+1)/r+ζ], (27)
+whereζ=ρcluster
+m/ρm. Therefore, we have
+ζ′=3
+a(1−ξ2/r)ζ−3R′
+Rζ+3
+aξ2/r , (28)
+where the prime denotes the derivative with respective to the scale factoraof the universe.
+In orderto solvethese equations, we set the initial conditions R∼aandR′= 1 at the startingpoint z= 3200which
+approximately corresponds to the epoch of matter-radiation equ ality. At the starting point the cluster is comoving
+with the background expansion. In company with the spherical mod el, we investigate the linear perturbation in the
+subhorizon approximation satisfying [ 19]
+d2lnδm
+dlna2+ [1
+2−3
+2w(1−Ωm)]dlnδm
+dlna+/parenleftbiggdlnδm
+dlna/parenrightbigg2
+=
+−(3ξ1+6ξ2
+r)dlnδm
+dlna−3
+r[ξ2+3ξ1ξ2+3ξ2
+2/r−ξ2dlnr
+dlna+ξ2(dlnH
+dlna+1)]+3
+2Ωm (29)
+and combine with ( 27,28) by starting from the initial condition ζi=δmi+ 1 in Eqs. ( 28) till the point when the
+spherical model collapses R(acoll)≈0 to obtain the critical over density δcabove which objects collapse. This linear
+extrapolated density threshold is useful in the following computatio n on the halo abundances.
+If the DE distribution is not homogeneous, the energy content insid e the spherical region consists of multi-fluids.
+When DE does not fully trace DM, the four velocity of DE and DM are diff erentua
+(d)/ne}ationslash=ua
+(m)and
+ua
+(d)=γ(ua
+(m)+va
+d) (30)
+whereγ= (1−v2
+d)−1/2is Lorentz-boost factor and va
+dis the relative velocity of DE fluid observed by the observer
+rested on the DM frame. If DE fully follows DM, va
+d= 0. Now we consider the non-comoving perfect fluids [ 20],
+Tab
+(m)=ρmua
+(m)ub
+(m)
+Tab
+(d)=ρdua
+(d)ub
+(d)+pdhab
+(d) (31)
+wherehab=gab+uaubis the projection operator [ 20]. Inserting ( 30) into the second equation of ( 31) and denoting
+ua
+(m)byua, the energy momentum tensor for DE reads,
+Tab
+(d)=ρduaub+pdhab+2uaqb
+(d) (32)
+9whereqa= (ρd+pd)va
+dis the energy-flux of DE observed by the observer rested in the DM frame. In the above
+equation, we have already neglected the second order terms of va
+dand assumed that the energy-flux velocity is far
+less than the speed of light va
+d≪1,γ∼1. In the spherical model, we define the top-hat radius as the radiu s of
+the boundary of DM. When DE does not trace DM, DE will not be bound ed inside the top-hat radius and will get
+out of the spherical region. We assume that the leakage of DE is still spherically symmetric and from the Birkhoff
+theorem, in the spherical region, the Raychaudhuri’s equation tak es the same form as Eqs. ( 24). For the energy
+density conservation law, we have,
+∇aTab
+(λ)=Qb
+(λ) (33)
+whereQbis the coupling vector as illustrated in Eqs. 2and “λ” denotes DE and DM respectively. The timelike parts
+of the above equation ubQb
+(λ)give
+˙ρcluster
+m+3hρcluster
+m= 3H(ξ1ρcluster
+m+ξ2ρcluster
+d)
+˙ρcluster
+d+3h(1+w)ρcluster
+d=−ϑ(1+w)ρcluster
+d−3H(ξ1ρcluster
+m+ξ2ρcluster
+d) (34)
+whereϑ=∇xvd. The external term incorporating ϑin the DE density evolution indicates the energy loss caused by
+the leakage of DE out of the spherical region.
+For the spacelike part, only DE has non-zero spatial component an dha
+bQb
+dgives
+˙qa
+(d)+4hqa
+(d)= 0 (35)
+whereqa
+(d)is the dark energy-flux. Assuming that the energy and pressure a re distributed homogenously, we obtain,
+˙ϑ+h(1−3w)ϑ= 3H(ξ1Γ+ξ2)ϑ (36)
+where Γ =ρcluster
+m/ρcluster
+dand we have already used Eqs. ( 34) and kept linear order terms of ϑ. From Eqs. ( 36), ifϑ
+vanishes initially, ϑwill keep to be zero at all the time during the evolution, DE will fully trac e DM. However, in most
+cases, even in linear region there is a small difference between vdandvmthat the initial condition for ϑis non-zero.
+We take the initial conditions for ϑasϑ∼k(vd−vm)∼ −δmi/200<0, which is obtained from the prediction of
+linear equation with k= 1Mpc−1atzi= 3200.ϑis much smaller than 1, |ϑ|<<1, and the negative value of ϑmeans
+that at the initial moment DM expanded faster than that of DE.
+Definingζm=ρcluster
+m/ρm,ζd=ρcluster
+d/ρdand converting the time derivative fromd
+dttod
+dawe have the evolution
+10of DM and DE in the spherical region described by
+ϑ′+R′
+R(1−3w)ϑ=3
+a(ξ1Γ+ξ2)ϑ,
+ζ′
+m=3
+a{1−ξ2/r}ζm−3R′
+Rζm+3
+aξ2ζd/r,
+ζ′
+d=3
+a(1+w+ξ1r)ζd−3(1+w)R′
+Rζd−3
+aξ1ζmr−ϑ(1+w)ζd. (37)
+When DE fully traces along DM, ϑ= 0, only the last two equations above are needed to discuss the str ucture
+formation. The Raychaudhuri’s equation now becomes
+2a2(1+1
+r)R′′−R′[1+(3w+1)/r]a=−R[(3w+1)ζd/r+ζm]. (38)
+Taking the same initial conditions as above for the spherical region o f radiusRand in addition adopting the adiabatic
+initial conditions, δdi= (1+w)δmi, andζmi=δmi+1,ζdi=δdi+1 which lead to ζdi= (1+w)ζmi−w, we can study
+the spherical collapse of DM and DE . However, it is important to note that due to the coupling, ζdmay get negative
+at sometime during the collapse. To avoid this unphysical point, we ma ke a cut-off of the coupling when ζdbecomes
+negative to guarantee ζd>= 0 in our analysis. For linear perturbation, we adopt the equations in the subhorizon
+approximation through [ 19]
+d2lnδm
+dlna2=−/parenleftbiggdlnδm
+dlna/parenrightbigg2
+−/bracketleftbigg1
+2−3
+2w(1−Ωm)/bracketrightbiggdlnδm
+dlna−(3ξ1+6ξ2
+r)dlnδm
+dlna+3ξ2
+rdlnδd
+dlnaexp(lnδd
+δm)
++3[exp(lnδd
+δm)−1]
+r/braceleftbigg
+ξ2+3ξ1ξ2+3ξ2
+2/r+ξ2(dlnH
+dlna+1)−ξ2dlnr
+dlna/bracerightbigg
++3
+2/bracketleftbigg
+Ωm+(1−Ωm)exp(lnδd
+δm)/bracketrightbigg
+,(39)
+d2lnδd
+dlna2=−/parenleftbiggdlnδd
+dlna/parenrightbigg2
+−/bracketleftbigg1
+2−3
+2w(1−Ωm)/bracketrightbiggdlnδd
+dlna+(1+w)3
+2/bracketleftbigg
+Ωmexp(lnδm
+δd)+(1−Ωm)/bracketrightbigg
+−k2C2
+e
+a2H2
++/bracketleftbigg
+3ξ2+6ξ1r+6w−3C2
+a+3(C2
+e−C2
+a)ξ1r+ξ2
+1+w+C2
+e
+1+wdlnρd
+dlna/bracketrightbiggdlnδd
+dlna
+−3rξ1exp(lnδm
+δd)dlnδm
+dlna+3(dlnH
+dlna+1)(w−C2
+e)+3ξ1(dlnH
+dlnar+r+dr
+dlna)−3ξ1/bracketleftbigg
+(dlnH
+dlna+1)r+dr
+dlna/bracketrightbigg
+exp(lnδm
+δd)
++ 3/bracketleftbigg
+w−C2
+e+ξ1r(1−exp(lnδm
+δd))/bracketrightbigg/bracketleftbigg
+(1−3w)−3C2
+e−C2
+a
+1+w(1+w+ξ1r+ξ2)−3(ξ1r+ξ2)−C2
+e
+1+wdlnρd
+dlna/bracketrightbigg
+.(40)
+We see that the DE and DM perturbations are entangled. In the sub horizon approximation, it was observed that the
+influence of the DE perturbation is small if compared with that DM [ 19]. Using ( 39,40) together with ( 37,38), we
+can obtain in the spherical model with inhomogeneous DE distribution the linearly extrapolated density threshold,
+δc(z) =δm(z=zcoll), above which structure collapse.
+The spherical collapse model was used by Press and Schechter[ 39] in providing a formalism to predict the number
+density of collapsed objects. Although this formalism is crude as fou nd in [40] etc., it predicts the evolution of the
+11mass function of collapsed objects well enough for the purpose in t his paper to see how the interaction between DE
+and DM influences cluster number counts.
+The comoving number density of collapsed dark halos of mass Min the mass interval dMat a given redshift of
+collapse is given by [ 39]
+dn(M,z)
+dM=/radicalbigg
+2
+πρm
+3M2δc
+σe−δ2
+c/2σ2/bracketleftbigg
+−R
+σdσ
+dR/bracketrightbigg
+, (41)
+whereρmis the comoving mean matter density at a given redshift. In most cas es it is a constant and equals to the
+present mean matter density, but this is not true when DE interact s with DM [ 35]. The quantity σ=σ(R,z) here is
+the variance smoothed over radius R. It has an explicit form [ 41],
+σ(R,z) =σ8/parenleftbiggR
+8h−1Mpc/parenrightbigg−γ(R)
+D(z), (42)
+whereσ8is the variance over a sphere with radius R= 8h−1Mpc andD(z) is the growth function defined by
+D(z) =δm(z)/δm(0). The index γis a function of the mass scale and the shape parameter Γ of the mat ter power
+spectrum [ 41]
+γ(R) = (0.3Γ+0.2)/bracketleftbigg
+2.92+log10/parenleftbiggR
+8h−1Mpc/parenrightbigg/bracketrightbigg
+. (43)
+We will use Γ = 0 .3 throughout our analysis. The radius Rat givenMcan be calculated by the relation [ 41],
+R= 0.951h−1Mpc/parenleftbiggMh
+1012ρm/ρ0cMs/parenrightbigg1/3
+, (44)
+whereρ0
+cis critical density at present and Msis the solar mass.
+The Press-Schechter formalism presents us the comoving number density of halos, which can be compared with
+astronomical data. In order to do the comparison, we calculate th e all sky number of halos per unit of redshift in the
+mass bin
+dN
+dz=/integraldisplay
+dΩdV
+dzdΩ/integraldisplay
+n(M)dM, (45)
+where the comoving volume element per unit redshift is dV/dzdΩ =r2(z)/H(z) andr(z) is the comoving distance
+r(z) =/integraltextz
+0dz′
+H(z).
+In the next section, we will present numerical results to see the eff ect of the interaction between DE and DM.
+In each situation we will study the cases with homogeneous and inhom ogeneous DE distributions. When the DE is
+distributed inhomogeneously, it will participate the collapse and the s tructure formation. We are not going to seek
+precise confrontations with observational data in this work, but t o understand the influence of coupling between dark
+sectors on cluster number counts.
+12V. CLUSTER NUMBER COUNTS
+In this section we present numerical results on the cluster number counts due to different interactions between
+DE and DM. We limit our attention to three commonly used forms of inte raction between dark sectors, with the
+interaction proportional to the energy densities of DE, DM and tot al dark sectors respectively. In our numerical
+calculation, we consider that all models have the same σ8= 0.8 . In the end of this section we also discuss the results
+by normalizing the halo abundance to the present value.
+A. Interaction proportional to the energy density of DE ( ξ1= 0,ξ2/negationslash= 0)
+When the interaction is proportional to the DE density, it was found that the curvature perturbation is always
+stable for both quintessence and phantom DE EOS [ 8]. When we consider DE EoS w >−1, we have the numerical
+results as shown in Fig 1.
+In the case when DE distribution is homogeneous, the results are sh own in solid lines. With positive coupling
+(ξ2>0 DE decays to DM), we see in Fig 1a that the critical mass density decreases compared with the ΛCDM model.
+This means that with the positive coupling, cluster can be formed eas ier than that of ΛCDM model. However when
+the coupling is negative, with DM decays to DE, δcis higher than that of ΛCDM, which means that it is harder for
+the cluster to be formed. The impact of the coupling between dark s ectors is significant in causing the difference in
+the cluster number counts. For positive coupling ( ξ2>0 DE decays to DM), the cluster number counts are bigger
+than that of ΛCDM model, which is clearly shown in Fig 1b-Fig1d. However when the coupling is negative( ξ2<0
+with DM decays to DE), the situation is opposite and consequently clu ster number counts are smaller than that of
+ΛCDM model.
+When the DE is distributed inhomogeneously, we need to consider the DE perturbation and its participation in
+the structure formation. Results are shown in dotted lines in Fig 1. We see that δcis slightly larger than the results
+without DE perturbation. The difference brought by the DE inhomog eneity is really small if compared with that of
+the interaction.
+For the DE EoS w <−1, the numerical results are shown in Fig 2. When the DE distribution is homogeneous,
+the influence brought by the interaction between dark sectors on the evolution of critical overdensity δcand the
+galaxy number counts agrees well with the situation when w>−1. However when we consider the DE is distributed
+inhomogeneously, the model with clustering DE with w <−1 has a bit more number counts than the homogeneous
+DE distribution appeared in the large mass bin, although the differenc e is very small especially for the case of small
+130 1 2 3 41.21.41.61.82
+W=−0.96, ξ1=0,Ce2=1
+ΛCDM
+ξ2=0.01
+ξ2=0.01
+ξ2=−0.03
+ξ2=−0.03
+ξ2=−0.05
+ξ2=−0.05
+ 
+ 
+zδc
+0 1 2 3 40510152025303540
+W=−0.96, ξ1=0,Ce2=1
+fixed σ8ΛCDM
+ξ2=0.01
+ξ2=0.01
+ξ2=−0.03
+ξ2=−0.03
+ξ2=−0.05
+ξ2=−0.05
+ 
+ 
+1013<M/(h−1Ms)<1014
+zdN/dz/106
+(a) (b)
+0 1 2 3 4050100150
+W=−0.96, ξ1=0,Ce2=1
+fixed σ8ΛCDM
+ξ2=0.01
+ξ2=0.01
+ξ2=−0.03
+ξ2=−0.03
+ξ2=−0.05
+ξ2=−0.05
+ 
+ 
+1014<M/(h−1Ms)<1015
+zdN/dz/104
+0 0.5 1 1.5 2051015202530
+W=−0.96, ξ1=0,Ce2=1
+fixed σ8ΛCDM
+ξ2=0.01
+ξ2=0.01
+ξ2=−0.03
+ξ2=−0.03
+ξ2=−0.05
+ξ2=−0.05
+ 
+ 
+1015<M/(h−1Ms)<1016
+zdN/dz/10
+(c) (d)
+Figure 1: The numerical results for the coupling proportion al to the energy density of DE when the EOS of DE w >−1 .
+The solid lines show the results calculated with homogeneou s DE distribution and the dotted lines are for the inhomogene ous
+distribution of DE.
+mass bin.
+Comparing with the interaction, we see that the influence brought b y the difference between the homogeneous and
+inhomogeneous DE distributions on the critical density δcand the number of galaxy clusters is very small. This in
+fact tells us that the fluctuation of DE field σdis small. In eq.( 21), this small σdleads toUmd∼Udm∼Udd∼Td→0,
+which means that the DE plays very little role in the virialization of the st ructure. The structure formed is almost the
+same as that of the cluster with homogeneous DE distribution. The lit tle effect of the inhomogeneous DE distribution
+140 1 2 3 41.21.41.61.82
+W=−1.04, ξ1=0,Ce2=1
+ΛCDM
+ξ2=0.01
+ξ2=0.01
+ξ2=−0.01
+ξ2=−0.01
+ξ2=−0.03
+ξ2=−0.03
+ 
+  
+zδc
+0 1 2 3 40510152025303540
+W=−1.04, ξ1=0,Ce2=1
+fixed σ8ΛCDM
+ξ2=0.01
+ξ2=0.01
+ξ2=−0.01
+ξ2=−0.01
+ξ2=−0.03
+ξ2=−0.03
+ 
+  
+1013<M/(h−1Ms)<1014
+zdN/dz/106
+(a) (b)
+0 1 2 3 4050100150
+W=−1.04, ξ1=0,Ce2=1
+fixed σ8ΛCDM
+ξ2=0.01
+ξ2=0.01
+ξ2=−0.01
+ξ2=−0.01
+ξ2=−0.03
+ξ2=−0.03
+ 
+  
+1014<M/(h−1Ms)<1015
+zdN/dz/104
+0 0.5 1 1.5 2051015202530
+W=−1.04, ξ1=0,Ce2=1
+fixed σ8ΛCDM
+ξ2=0.01
+ξ2=0.01
+ξ2=−0.01
+ξ2=−0.01
+ξ2=−0.03
+ξ2=−0.03
+ 
+  
+1015<M/(h−1Ms)<1016
+zdN/dz/10
+(c) (d)
+Figure 2: The numerical results for the coupling proportion al to the energy density of DE when the EOS of DE w <−1 . The
+solid lines show the results calculated when DE is distribut ed homogeneously and the dotted lines are for inhomogeneous DE
+model.
+in the structure formation is consistent with the finding that the DE perturbation is small compared with the DM
+perturbation in the subhorizon approximation [ 19].
+B. The interaction proportional to the energy density of DM ( ξ1>0,ξ2= 0)
+In this case, the stable curvature perturbation can only hold when DE EOS is smaller than -1[ 8]. The coupling
+between DE and DM should be positive to avoidthe negativeenergyde nsity ofDE at the earlytime ofthe universe[ 18]
+15The numerical results by choosing this coupling are shown in Fig 3.
+The solid lines are for the homogeneous distribution of DE. For small c oupling, the lines are close to that of ΛCDM
+result. We observethat morepositive couplingleads tomore number counts ofclustersFig 3c. This canbe understood
+fromδcand also the ratio δc(z)/σ8D(z), which are smaller at high redshift for more positive ξ1. This result is also
+consistent with the case when the interaction is proportional to th e DE energy density.
+The dotted lines are for the situations when DE is distributed inhomog eneously and we need to consider the DE
+perturbation and its participation in the structure formation. Fro m the behavior of δcwe see that δcis suppressed at
+low redshift. This suppression is more obvious than the case of homo geneous DE. In the ratio δc(z)/σ8D(z) shown in
+Fig3b, the ratio due to the inhomogeneous DE is smaller than that of the h omogeneous DE. This leads to a larger
+cluster abundance due to the inhomogeneous DE distribution than t hat of homogeneous DE distribution as shown in
+Fig3c.
+C. Interaction proportional to the total energy density of d ark sectors ( ξ=ξ1=ξ2>0)
+Here, again, the curvature perturbation can only be stable when D E EOS is smaller than -1 [ 8] and the coupling has
+to be positive [ 18]. The numerical results are shown in Fig 4. The results are similar to the case when the coupling is
+proportional to the energy density of DM. With more positive couplin g, the critical overdensity is smaller which leads
+to the bigger galaxy number counts than that of the ΛCDM model. Fo r the same coupling, if the DE is clustering, the
+critical overdensity δcandδc(z)/σ8D(z) will be suppressed at small redshift. Therefore it is observed in Fig 4c,Fig4d
+that at small the inhomogeneous DE distribution leads to a larger clus ter number abundance as compared with the
+homogeneous DE distribution.
+D. Normalization
+So far in the above discussions we have fixed σ8= 0.8 as given by present observations [ 42]. Considering that at
+redshift zero all models have the same number density of halos, the normalization is required to be done by adjusting
+σ8in each model such that δc/σ8is equal to the fiducial σ8= 0.8 ΛCDM model
+σ0
+8,model=δc,model(z= 0)
+δc,Λ(z= 0)σ0
+8,Λ. (46)
+We dedicate the rest of this section to discuss the implications on our results by normalizing models to the same
+halo abundance at z= 0. It is interestingly observed that the qualitative structure of c urves changes dramatically
+depending on the choices of the normalizations, which was also found in studying the number counts in homogeneous
+160 1 2 3 41.21.41.61.82
+W=−1.04, ξ2=0,Ce2=1
+ΛCDM
+ξ1=0.0001
+ξ1=0.0001
+ξ1=0.0003
+ξ1=0.0003
+ξ1=0.0005
+ξ1=0.0005
+ 
+ 
+zδc
+0 1 2 3 40246810
+W=−1.04, ξ2=0,Ce2=1 ΛCDM
+ξ1=0.0001
+ξ1=0.0001
+ξ1=0.0003
+ξ1=0.0003
+ξ1=0.0005
+ξ1=0.0005
+ 
+ 
+zδc/ σ8D(z)
+(a) (b)
+0 1 2 3 40510152025303540
+W=−1.04, ξ2=0,Ce2=1
+fixed σ8ΛCDM
+ξ1=0.0001
+ξ1=0.0001
+ξ1=0.0003
+ξ1=0.0003
+ξ1=0.0005
+ξ1=0.0005
+ 
+ 
+1013<M/(h−1Ms)<1014
+zdN/dz/106
+0 1 2 3 4050100150
+W=−1.04, ξ2=0,Ce2=1
+fixed σ8ΛCDM
+ξ1=0.0001
+ξ1=0.0001
+ξ1=0.0003
+ξ1=0.0003
+ξ1=0.0005
+ξ1=0.0005
+ 
+ 
+1014<M/(h−1Ms)<1015
+zdN/dz/104
+(c) (d)
+Figure 3: The numerical results for the coupling proportion al to the energy density of DM. The solid lines shows the resul ts
+calculated with homogeneous DE distribution and the dotted lines are for inhomogeneous DE.
+and inhomogeneous DE models without coupling to DM [ 27].
+Inourmodels, wefindinFig 5athatwhenthecouplingisproportionaltotheenergydensityofDE (ξ1= 0,ξ2/ne}ationslash= 0), in
+the normalization oflocal halos abundance, the departure from th e cosmologicalconstant model due to the interaction
+between dark sectors becomes smaller especially in the low mass bin wh en compared with the results by fixing σ8.
+The major impact due to the coupling emerges in the massive galaxy clu ster as shown in Fig 5b. The signature of
+the DE perturbation is still not obvious which is similar to the result by fi xingσ8.
+When the coupling is proportional to the energy density of DM, we se e from Fig 6that the departure from the
+170 1 2 3 41.21.41.61.82
+W=−1.04, ξ1=ξ2,Ce2=1
+ΛCDM
+ξ=0.0001
+ξ=0.0001
+ξ=0.0003
+ξ=0.0003
+ξ=0.0005
+ξ=0.0005
+ 
+ 
+zδc
+0 1 2 3 40246810
+W=−1.04, ξ1=ξ2,Ce2=1 ΛCDM
+ξ=0.0001
+ξ=0.0001
+ξ=0.0003
+ξ=0.0003
+ξ=0.0005
+ξ=0.0005
+ 
+ 
+zδc/ σ8D(z)
+(a) (b)
+0 1 2 3 40510152025303540
+W=−1.04, ξ1=ξ2,Ce2=1
+fixed σ8ΛCDM
+ξ=0.0001
+ξ=0.0001
+ξ=0.0003
+ξ=0.0003
+ξ=0.0005
+ξ=0.0005
+ 
+ 
+1013<M/(h−1Ms)<1014
+zdN/dz/106
+0 1 2 3 4050100150
+W=−1.04, ξ1=ξ2,Ce2=1
+fixed σ8ΛCDM
+ξ=0.0001
+ξ=0.0001
+ξ=0.0003
+ξ=0.0003
+ξ=0.0005
+ξ=0.0005
+ 
+ 
+1014<M/(h−1Ms)<1015
+zdN/dz/104
+Figure 4: The numerical results for the coupling proportion al to the energy density of total dark sectors. The solid line s shows
+the results calculated with homogeneous DE distribution an d the dotted lines are for inhomogeneous DE.
+cosmological constant model due to the interaction between DE an d DM (solid lines) is suppressed if compared with
+the result by fixing σ8Fig3. This shows that fixing the halo abundance, the coupling does not sh ow up in the galaxy
+number counts. However, in this case, the difference caused by th e clustering DE (dotted lines) is enhanced with large
+deviations from the cosmological constant model. When the coupling is proportional to the energy density of total
+dark sectors, the qualitative results are the same.
+180 1 2 3 40510152025303540
+W=−0.96, ξ1=0,Ce2=1
+Normalized σ8ΛCDM
+ξ2=0.01
+ξ2=0.01
+ξ2=−0.03
+ξ2=−0.03
+ξ2=−0.05
+ξ2=−0.05
+ 
+ 
+1013<M/(h−1Ms)<1014
+zdN/dz/106
+0 0.5 1 1.5 2051015202530
+W=−0.96, ξ1=0,Ce2=1
+Normalized σ8ΛCDM
+ξ2=0.01
+ξ2=0.01
+ξ2=−0.03
+ξ2=−0.03
+ξ2=−0.05
+ξ2=−0.05
+ 
+ 
+1015<M/(h−1Ms)<1016
+zdN/dz/10
+(a) (b)
+Figure 5: The results for the coupling proportional to the en ergy density of DE by fixing the local abundance.
+0 1 2 3 40510152025303540
+W=−1.04, ξ2=0,Ce2=1
+Normalized σ8ΛCDM
+ξ1=0.0001
+ξ1=0.0001
+ξ1=0.0003
+ξ1=0.0003
+ξ1=0.0005
+ξ1=0.0005
+ 
+ 
+1013<M/(h−1Ms)<1014
+zdN/dz/106
+0 1 2 3 4050100150
+W=−1.04, ξ2=0,Ce2=1
+Normalized σ8ΛCDM
+ξ1=0.0001
+ξ1=0.0001
+ξ1=0.0003
+ξ1=0.0003
+ξ1=0.0005
+ξ1=0.0005
+ 
+ 
+1014<M/(h−1Ms)<1015
+zdN/dz/104
+(a) (b)
+Figure 6: The results for the coupling proportional to the en ergy density of DM by fixing the local abundance.
+VI. CONCLUSIONS AND DISCUSSIONS
+Assuming first that DE component is homogeneous, we have investig ated how DM approaches the dynamical
+equilibrium in the collapsing system. In the presence of coupling betwe en DE and DM, the flow of mass and energy
+between the components changes the time required by the system to reach equilibrium and also the equilibrium
+configuration itself. Our discussion is based on the perturbation th eory developed in [ 8,19]. We reproduce the
+19Layzer-Irvine equation [ 38] when the interaction is neglected. In the homogeneous case, the DE component flows
+progressive out of the overdensity which leads the energy conser vation law invalid in the collapsing system [ 43]. It is
+expected in the models of DE interacting with DM, if the DE is distribute d inhomogeneously, it may cluster along
+with the DM and thus we can avoid the energy non-conservation pro blem. To see whether this speculation is correct
+or not, we have further explored the dynamics of the DE that pres ents inhomogeneities at cluster scales and found
+that in the presence of the coupling between DE and DM, the time and dynamics required by the DE in the system
+to reach equilibrium is different from that of DM. This shows that in the collapsing system, DE does not fully cluster
+along with DM, even in the presence of the interaction between them . The energy is not strictly conserved in the
+collapsing system. This result is based on the linear perturbation for malism, where the DE perturbation can be
+neglected in the late universe [19].
+We also developed a novel treatment of the spherical collapse mode l which consists of multi-fluids with different
+collapsing velocities. When DE does not trace DM in the spherical collap se, we studied the dependence of the cluster
+number counts on the coupling between DE and DM and on the DE inhom ogeneities by using the Press-Schechter
+formalism. In most of our work we haveassumed that the models are normalized to have practically the same σ8= 0.8
+as given by present day observations. We have shown that when th e coupling between dark sectors is proportional to
+the energy density of DE, there is a significant dependence of clust er number counts on the interaction. If the energy
+decays from DE to DM, the number of cluster counts increases. Th e dependence of cluster number counts on the DE
+inhomogeneities is negligible compared with the interaction influence. T his in fact shows that the fluctuation of the
+DE field is small and the inhomogeneous DE plays little role in the virializatio n of the structure. For the coupling
+between DE and DM proportional to the energy density of DM and to tal dark sectors, we have observed that more
+energy decay from DE to DM leads more abundance of the cluster nu mber counts compared with the ΛCDM model.
+In addition, the signature of the DE inhomogeneities is enhanced. Th is enhancement can be understood from the
+dynamics in which it shows that the DM gravitational effect pulls the inh omogeneous DE to move and collapse along
+with DM.
+We have further studied these models enforcing the normalization t o reproduce the same abundance of galaxy
+clusters at redshift zero. The qualitative structure changes in ch oosing different normalizations [ 27]. Comparing with
+the interaction effects in fixing σ8, we have verified that the departures from the ΛCDM model are su ppressed, the
+coupling between dark sectors while fixing the local abundance usua lly does not show up in the galaxy number counts,
+except when the interaction is proportional to the energy density of DE in the super mass bin. In fixing the local
+abundance, the influence caused by the inhomogeneous DE is enhan ced with large deviations from the cosmological
+20constant model when the coupling is proportional to the energy de nsity of DM and total dark sectors.
+The Press-Schechter formalism we employed is good enough for the purpose of this paper to illustrate how the
+interaction between dark sectors and the DE inhomogeneities influe nce cluster number counts, but it is still not easy
+to seek precise confrontations with observational data at the pr esent moment. Recently in the uncoupled model,
+the statistical analysis was performed on the sensitivity of cluster counts to DE perturbations by employing a set of
+cosmological parameters using the Fisher matrix method. It was ar gued that the impact of DE fluctuations is large
+enough, which could already be detected by existing instruments[ 32]. It is of great interest to extend their analysis
+to examine the interaction between DE and DM effect and the DE inhom ogeneities influence. Such a study is under
+way.
+Acknowledgments
+This work has been supported partially by NNSF of China and the Natio nal Basic Research Program of China
+under grant 2010CB833000 and brazilian foundations CNPq and FAP ESP.
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+220 0.5 1 1.5 201020304050607080
+W=−1.04, ξ2=0,Ce2=1
+Normalized σ8ΛCDM
+ξ1=0.0001
+ξ1=0.0001
+ξ1=0.0003
+ξ1=0.0003
+ξ1=0.0005
+ξ1=0.0005
+ 
+ 
+1015<M/(h−1Ms)<1016
+zdN/dz/10
\ No newline at end of file
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+arXiv:1001.0081v2  [cond-mat.mes-hall]  22 Mar 2010Electron transport through a quantum interferometer: A
+theoretical study
+Santanu K. Maiti†,‡,1
+†Theoretical Condensed Matter Physics Division, Saha Insti tute of Nuclear Physics,
+1/AF, Bidhannagar, Kolkata-700 064, India
+‡Department of Physics, Narasinha Dutt College, 129 Belilio us Road, Howrah-711 101, India
+Abstract
+In the present work we explore electron transport properties th rough a quantum interferometer attached
+symmetrically to two one-dimensional semi-infinite metallic electrodes , namely, source and drain. The
+interferometer is made up of two sub-rings where individual sub-rin gs are penetrated by Aharonov-Bohm
+fluxesφ1andφ2, respectively. We adopt a simple tight-binding framework to describ e the model and all
+the calculations are done based on the single particle Green’s functio n formalism. Our exact numerical
+calculations describe two-terminal conductance and current as f unctions of interferometer-to-electrode
+coupling strength, magnetic fluxes threaded by left and right sub- rings of the interferometer and the
+difference of these two fluxes. Our theoretical results provide se veral interesting features of electron
+transport across the interferometer, and these aspects may b e utilized to study electron transport in
+Aharonov-Bohm geometries.
+PACS No. : 73.23.-b; 73.63.Rt.
+Keywords : Quantum interferometer; Conductance; I-Vcharacteristic; AB effect; Anti-resonant states.
+1Corresponding Author : Santanu K. Maiti
+Electronic mail: santanu.maiti@saha.ac.in
+11 Introduction
+The study of electronic transport in quantum con-
+fined model systems like quantum rings, quantum
+dots, arrays of quantum dots, quantum dots em-
+bedded in a quantum ring, etc., has become one
+of the most fascinating branch of nanoscience and
+technology. With the aid of present nanotechnolog-
+ical progress [1, 2], these simple looking quantum
+confined systems can be used to design nanodevices
+especially in electronic as well as spintronic engi-
+neering. The key idea of manufacturing nanode-
+vices is based on the concept of quantum interfer-
+ence effect [3, 4, 5], and it is generally preserved
+throughout the sample having dimension smaller or
+comparable to the phase coherence length. There-
+fore, ring type conductors or two path devices are
+ideal candidates to exploit the effect of quantum in-
+terference [6]. In a ring shaped geometry, quantum
+interferenceeffect canbecontrolledbyseveralways,
+and most probably, the effect can be regulated sig-
+nificantly by tuning the magnetic flux, the so-called
+Aharonov-Bohm(AB) flux[7,8, 9, 10], thatthreads
+the ring. This key feature motivates us to widely
+use quantum interference devices in mesoscopic
+solid-state electronic circuits [11]. Using a meso-
+scopic ring we can construct a quantum interferom-
+eter, and here we will show that the interferometer
+exhibits several exotic features of electron trans-
+port which can be utilized in designing nanoelec-
+tronic circuits. To reveal the phenomena, we make
+a bridge system, by inserting the interferometer be-
+tween two electrodes (source and drain), the so-
+called source-interferometer-drain bridge. Follow-
+ing the pioneering work of Aviram and Ratner [12],
+theoretical description of electron transport in a
+bridge system has got much progress. Later, many
+excellent experiments [13, 14, 15, 16, 17, 18, 19]
+have been done in several bridge systems to un-
+derstand the basic mechanisms underlying the elec-
+tron transport. Though extensive studies on elec-
+tron transport have already been done both theo-
+retically[7, 8, 9, 10, 20, 21, 22, 23, 24, 25, 26, 27, 28,29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]
+as well as experimentally [13, 14, 15, 16, 17, 18, 19],
+yet lot of controversies are still present between the
+theory and experiment, and the complete descrip-
+tion of the conduction mechanism in this scale is
+not very well defined even today. Several control-
+ling factors are there which can significantly regu-
+late electron transport in a conducting bridge, and
+all these factors haveto be taken into account prop-
+erly to understand the transport mechanism. For
+our illustrative purposes, here we mention some of
+these issues.
+1. The quantum interference effect [28, 29, 30, 31,
+32, 33, 34] of electronic waves passing through
+different armsofthebridgingmaterialbecomes
+the most significant issue.
+2. The coupling of the bridging material to the
+electrodes significantly controls the current
+amplitude across any bridge system [32].
+3. Dynamical fluctuation in small-scale devices is
+another important factor which plays an ac-
+tive role and can be manifested through the
+measurement of shot noise [36], a direct conse-
+quence of the quantization of charge.
+4. Geometry of the conducting material between
+the two electrodes itself is an important is-
+sue to control electron transmission which has
+been described quite elaborately by Ernzerhof
+et al.[42] through some model calculations.
+Addition to these, several other factors of the
+tight-binding Hamiltonian that describe a system
+also provide important effects in the determination
+of current across a bridge system.
+In this presentation we explore electron trans-
+port properties of a quantum interferometer based
+on the single particle Green’s function formalism.
+Theinterferometerissandwichedbetweentwosemi-
+infinite one-dimensional (1D) metallic electrodes,
+viz, source and drain, and, two sub-rings of the
+interferometer are subject to the Aharonov-Bohm
+2(AB) fluxes φ1andφ2, respectively. The schematic
+view of the bridge system is depicted in Fig. 1.
+A simple tight-binding model is used to describe
+the system and all the calculations are done nu-
+merically, which illustrate conductance-energy and
+current-voltage characteristics as functions of the
+interferometer-to-electrode coupling strength, mag-
+netic fluxes and the difference of these two fluxes.
+Severalexotic featuresareobservedfromthis study.
+These are: (i) semiconducting or metallic nature
+depending on the coupling strength of the inter-
+ferometer to the side attached electrodes, (ii) ap-
+pearance of anti-resonant states [7, 8, 9] and (iii)
+unconventionalperiodic behaviorof typical conduc-
+tance/current as a function of the difference of two
+AB fluxes.
+The scheme of the paper is as follows. Follow-
+ing the introduction (Section 1), in Section 2, we
+describe the model and theoretical formulation for
+the calculation. Section 3 explores the results, and
+finally, we conclude our study in Section 4.
+2 Model and synopsis of the
+theoretical background
+Let us begin with the model presented in Fig. 1.
+A quantum interferometer with four atomic sites
+(N= 4, where Ngives the total number of atomic
+sites in the interferometer) is attached symmet-
+rically to two semi-infinite one-dimensional (1D)
+metallic electrodes, namely, source and drain. The
+atomic sites 2 and 3 of the interferometer are di-
+rectly coupled to each other, and accordingly, two
+sub-rings, left and right, are formed. These two
+sub-rings are subject to the AB fluxes φ1andφ2,
+respectively.
+Consideringlineartransportregime, conductance
+gof the interferometer can be obtained using the
+Landauer conductance formula [43, 44, 45, 46, 47],
+g=2e2
+hT (1)where,Tbecomes the transmission probability of
+an electron across the interferometer. It ( T) can be
+expressed in terms of the Green’s function of the
+interferometer and its coupling to the side-attached
+electrodes by the relation [46, 47],
+T= Tr[Γ SGr
+IΓDGa
+I] (2)
+where,Gr
+IandGa
+Iare respectively the retarded and
+advancedGreen’sfunctionsoftheinterferometerin-
+cluding the effects of the electrodes. Here Γ Sand
+ΓDdescribe the coupling of the interferometer to
+φ
+2φ12
+34 1
+Source Drain
+Quantum Interferometer
+Figure 1: (Color online). Schematic view of a quan-
+tum interferometer with four atomic sites ( N=
+4) attached to two semi-infinite one-dimensional
+metallic electrodes.
+the sourceand drain, respectively. Forthe complete
+system i.e., the interferometer, source and drain,
+Green’s function is defined as,
+G= (E−H)−1(3)
+where,Eis the injecting energy of the source elec-
+tron. To Evaluate this Green’s function, inversion
+of an infinite matrix is needed since the complete
+system consists of the finite size interferometer and
+two semi-infinite electrodes. However, the entire
+system can be partitioned into sub-matrices cor-
+responding to the individual sub-systems and the
+Green’s function for the interferometer can be ef-
+fectively written as,
+GI= (E−HI−ΣS−ΣD)−1(4)
+where,HIis the Hamiltonian of the interferometer
+that can be expressed within the non-interacting
+3picture like,
+HI=/summationdisplay
+iǫic†
+ici+/summationdisplay
+<ij>tij/parenleftBig
+c†
+icjeiθij+c†
+jcie−iθij/parenrightBig
+(5)
+In this Hamiltonian ǫigives the on-site energy for
+the atomic site i, whereiruns from 1 to 4, c†
+i(ci)
+is the creation (annihilation) operator of an elec-
+tron at the site iandtijis the hopping integral
+between the nearest-neighbor sites iandj. For
+the sake of simplicity, we assume that the magni-
+tudes of all hopping integrals ( tij) are identical to
+t. The phase factor θij, associated with the hop-
+ping integral tij, comes due to the fluxes φ1and
+φ2in the two sub-rings. The phase factors ( θij)
+are chosen as, θ12=θ23=θ34=θ41= 2πφ/4φ0,
+θ24= 2π∆φ/2φ0, whereφ=φ1+φ2, ∆φ=φ1−φ2
+andφ0=ch/eis the elementary flux-quantum. Ac-
+cordingly, a minus sign is used for the phases when
+the electron hops in the reverse direction. For the
+two 1D electrodes, a similar kind of tight-binding
+Hamiltonian is also used, except any phase fac-
+tor, where the Hamiltonian is parametrized by con-
+stant on-site potential ǫ0and nearest-neighborhop-
+ping integral t0. The hopping integral between the
+source and interferometer is τS, while it is τDbe-
+tween the interferometer and drain. The parame-
+tersΣSandΣDinEq.(4)representtheself-energies
+due to the coupling of the interferometer to the
+source and drain, respectively, where all the infor-
+mation of this coupling are included into these self-
+energies [46].
+The current passing through the interferometer
+is depicted as a single-electron scattering process
+between the two reservoirs of charge carriers. The
+currentIcan be computed as a function of the ap-
+plied bias voltage Vby the expression [46],
+I(V) =e
+π¯hEF+eV/2/integraldisplay
+EF−eV/2T(E)dE (6)
+whereEFis the equilibrium Fermi energy. Here we
+assume that the entire voltageis dropped acrosstheinterferometer-electrode interfaces, and it is exam-
+ined that under such an assumption the I-Vchar-
+acteristics do not change their qualitative features.
+All the results in this communication are de-
+termined at absolute zero temperature, but they
+should valid even for some finite (low) tempera-
+tures, since the broadening of the energy levels of
+the interferometer due to its coupling to the elec-
+trodes becomes much larger than that of the ther-
+mal broadening [46]. On the other hand, at high
+temperature limit, all these phenomena completely
+disappear. This is due to the fact that the phase
+coherence length decreases significantly with the
+rise of temperature where the contribution comes
+mainly from the scattering on phonons, and accord-
+ingly, the quantum interferenceeffect vanishes. Our
+unit system is simplified by choosing c=e=h= 1.
+3 Numerical results and dis-
+cussion
+Before going into the discussion, let us first assign
+the values of different parameters those are used for
+our numerical calculation. The on-site energy ǫiof
+the interferometer is taken as 0 for all the four sites
+i, and the nearest-neighbor hopping strength tis
+set to 3. On the other hand, for two side attached
+1D electrodes the on-site energy ( ǫ0) and nearest-
+neighbor hopping strength ( t0) are fixed to 0 and 4,
+respectively. The equilibrium Fermi energy EFis
+set to 0.
+Throughoutthe analysiswepresent the basicfea-
+tures of electron transport for two distinct regimes
+of electrode-to-interferometer coupling.
+Case 1:Weak-coupling limit
+This limit is set by the criterion τS(D)<< t. In this
+case, we choose the values as τS=τD= 0.5.
+Case 2:Strong-coupling limit
+This limit is described by the condition τS(D)∼
+t. In this regime we choose the values of hopping
+strengths as τS=τD= 2.5.
+43.1 Interferometric geometry with 4
+atomic ( N= 4) sites
+3.1.1 Conductance-energy characteristics
+In Fig. 2, we plot conductance gas a function of
+the injecting electron energy Efor the interferom-
+eter considering φ= 1, where (a), (b), (c) and (d)
+correspond to ∆ φ= 0.2, 0.4, 0.6 and 0.8, respec-
+/Minus8/Minus4048
+E012g/LParen1E/RParen1/LParen1d/RParen1/Minus8/Minus4048
+E012g/LParen1E/RParen1/LParen1c/RParen1/Minus8/Minus4048
+E012g/LParen1E/RParen1/LParen1b/RParen1/Minus8/Minus4048
+E012g/LParen1E/RParen1/LParen1a/RParen1
+Figure 2: (Color online). g-Ecurves in the weak-
+(black) and strong-coupling (red) limits for the in-
+terferometer with four atomic sites ( N= 4) con-
+sidering φ= 1. (a) ∆ φ= 0.2, (b) ∆ φ= 0.4, (c)
+∆φ= 0.6 and (d) ∆ φ= 0.8.
+tively. Theblackcurvesrepresentthe resultsforthe
+weak-couplinglimit, whiletheresultsforthestrong-
+coupling limit are shown by the red curves. In the
+limitofweak-coupling,conductanceshowsfinereso-
+nant peaks for some particular energies, while it ( g)
+drops to zero almost for all other energies. At theseresonances, conductance reaches the value 2, and
+therefore, the transmission probability Tbecomes
+unity, since the relation g= 2Tis satisfied from
+theLandauerconductanceformula(seeEq.(1)with
+e=h= 1). The transmission probability of getting
+an electron across the interferometer significantly
+depends on the quantum interference of electronic
+waves passing through the different arms of the in-
+terferometer, and accordingly, the probability am-
+plitude becomes strengthened or weakened. Now
+all the resonant peaks in the conductance spec-
+tra are associated with the energy eigenvalues of
+the interferometer, and thus it is emphasized that
+the conductance spectrum reveals itself the elec-
+tronicstructureoftheinterferometer. Thesituation
+becomes quite interesting as long as the coupling
+strength of the interferometer to the electrodes is
+increased from the weak regime to the strong one.
+In the strong-coupling limit, all the resonances get
+substantial widths compared to the weak-coupling
+limit. The contribution for the broadening of the
+resonantpeaks in this strong-couplinglimit appears
+fromtheimaginarypartsoftheself-energiesΣ Sand
+ΣD, respectively [46]. Hence, by tuning the cou-
+pling strength from the weak to strong regime, elec-
+tronic transmissionacrossthe interferometercanbe
+obtained for the wider range of energies, while a
+fine scan in the energy scale is needed to get the
+electron conduction across the bridge in the limit
+of weak-coupling. These results provide an impor-
+tant signature in the study of current-voltage ( I-
+V) characteristics. Another interesting feature ob-
+servedfromtheconductancespectraistheexistence
+of the anti-resonant states. The positions of the
+anti-resonance states can be clearly noticed from
+the red curves, compared to the black curves since
+the widths of these curves are too small, where they
+sharply drop to zero for the respective energy val-
+ues associated with the different values of ∆ φ(see
+Figs. 2(a)-(d)). Such anti-resonant states are spe-
+cific to the interferometric nature of the scattering
+and do not occur in conventional one-dimensional
+scattering problems of potential barriers [7, 8, 9].
+5A clear investigation shows that the positions of
+the anti-resonances on the energy scale are inde-
+pendent of the interferometer-to-electrode coupling
+strength. Since the width of these anti-resonance
+states are too small, they do not provide any sig-
+nificant contribution in the current-voltage ( I-V)
+characteristics.
+3.1.2 Typical conductance gtypas a function
+of∆φ
+The effect of ∆ φ, the difference between two AB
+fluxesφ1andφ2, on the electron transport through
+/Minus4/Minus2024
+/CapDeltaΦ012gtyp/LParen1d/RParen1/Minus4/Minus2024
+/CapDeltaΦ012gtyp/LParen1c/RParen1/Minus4/Minus2024
+/CapDeltaΦ012gtyp/LParen1b/RParen1/Minus4/Minus2024
+/CapDeltaΦ012gtyp/LParen1a/RParen1
+Figure 3: (Color online). gtyp-∆φcurves in the
+strong-coupling limit for the interferometer with
+four atomic sites ( N= 4), where (a) φ= 0.2, (b)
+φ= 0.4, (c)φ= 0.6 and (d) φ= 0.8. The typical
+conductances are calculated at the energy E= 5.
+the interferometer is also an important issue in the
+presentcontext. Tovisualizeit, inFig.3, wedisplay
+the variation of the typical conductance ( gtyp) as afunction of ∆ φfor the interferometer in the limit of
+strong-coupling. Figures 3(a), (b), (c) and (d) cor-
+respond to the results for φ= 0.2, 0.4, 0.6 and 0.8,
+respectively. The typical conductances are calcu-
+lated for the fixed energy E= 5. Very interestingly
+we observe that, for a fixed value of φ, typical con-
+ductance varies periodically with ∆ φshowing 2 φ0
+(= 2, since φ0= 1 in our chosen unit) flux-quantum
+periodicity, associated with the number of atomic
+sites (2) in the vertical line connecting two sub-
+rings of the quantum interferometer. This period
+doubling behavior is completely different from the
+traditionalperiodicnature,sinceinconventionalge-
+ometries we get simple φ0flux-quantum periodicity.
+In the limit of weak-coupling we will also get the
+similar behavior of periodicity (2 φ0) for the typical
+conductance with ∆ φ, and due to the obvious rea-
+son we do not plot the results for this coupling limit
+once gain.
+3.1.3 Current-voltage characteristics
+All these features of electron transfer become
+much more clearly visible by studying the current-
+voltage ( I-V) characteristics. The current Ipass-
+ing through the interferometer is computed from
+the integration procedure of the transmission func-
+tionTas prescribed in Eq. (6) which is not re-
+stricted in the linear response regime, but it is of
+great significance in determining the shape of the
+full current-voltage characteristics. As illustrative
+examples, in Fig. 4, we plot the current-voltage
+characteristics of the interferometer for the three
+different values of φ2, keeping the flux φ1in the
+left sub-ring to a fixed value 0 .2. The red, blue
+and black curves correspond to φ2= 0, 0.1 and
+0.4, respectively. In the limit of weak-coupling (see
+Fig. 4(a)), it is observed that the current exhibits
+staircase-like structure with fine steps as a function
+of the applied bias voltage V. This is due to the
+existence of the sharp resonant peaks in the con-
+ductance spectrum in this coupling limit, since the
+current is computed by the integration method of
+6the transmission function T. With the increase of
+the bias voltage V, the electrochemical potentials
+on the electrodes are shifted gradually, and finally
+cross one of the quantized energy levels of the inter-
+ferometer. Accordingly, a currentchannel is opened
+up which provides a jump in the I-Vcharacteristic
+/Minus12/Minus8/Minus404812
+V0
+/Minus1212
+/Minus6/LParen1b/RParen1
+6I/Minus12/Minus8/Minus404812
+V0
+/Minus.4.4
+/Minus.2/LParen1a/RParen1
+.2I
+Figure 4: (Color online). I-Vcharacteristics of the
+interferometer with four atomic sites ( N= 4) for
+a fixed value of φ1= 0.2, where the red, blue and
+black curves correspond to φ2= 0, 0.1 and 0.4, re-
+spectively. (a) Weak-coupling limit and (b) strong-
+coupling limit.
+curve. The most important feature observed from
+theI-Vcurves for this weak-coupling limit is that,
+the non-zero value of the current appears beyond a
+finite bias voltage, the so-called threshold voltage
+Vth. This is quite analogous to the semiconducting
+nature of a material. Most interestingly, the results
+predict that the threshold bias voltage of electron
+conduction can be controlled very nicely by tuning
+the AB flux φ2. The situation becomes much dif-
+ferent for the strong-coupling case. The results are
+given in Fig. 4(b). In this limit, the current variesalmost continuously with the applied bias voltage
+and achieves much larger amplitude than the weak-
+coupling case. The reason is that, in the limit of
+strong-coupling all the energy levels get broadened
+which provide larger current in the integration pro-
+cedureofthe transmissionfunction T. Thusbytun-
+ing the strength of the interferometer-to-electrode
+coupling, we can achieve very large, even an order
+of magnitude, current amplitude from the very low
+one for the same bias voltage V, which provides an
+important signature in designing nanoelectronic de-
+vices. In contrary to the weak-coupling limit, here
+the electron starts to conduct as long as the bias
+voltage is given i.e., Vth→0, which reveals the
+metallic nature. Thus it can be emphasized that
+the interferometer-to-electrodecouplingis a keypa-
+rameter which controls the electron transport in a
+meaningful way. Additionally, the existence of the
+semiconducting or the metallic behavior of the in-
+terferometer also significantly depends on the AB
+fluxesφ1andφ2. Thenatureofallthese I-Vcurves,
+presented in Fig. 4, will be exactly similar if we plot
+the results for the different values of φ1, keeping φ2
+as a constant.
+3.1.4 Typical current amplitude Itypas a
+function of φ2
+Now, we draw our attention on the variation of the
+typical current amplitude with anyone of these two
+fluxes, when the other one is fixed. To explore it,
+in Fig. 5, we show the variation of the typical cur-
+rent amplitude ( Ityp) withφ2, considering φ1as a
+constant, where (a) and (b) correspond to φ1= 0
+and 0.3, respectively. The black and red lines rep-
+resent the results for the weak- and strong-coupling
+limits, respectively. The typical current amplitudes
+are calculated for the fixed bias voltage V= 1.02.
+Bothforthesetwolimitingcases,thetypicalcurrent
+amplitude varies periodically with φ2, exhibiting φ0
+flux-quantum periodicity, as expected. Similar fea-
+ture is also observed for the Itypvsφ1curves, when
+φ2becomes constant. Here it is also important to
+7note that the variation of Itypwith ∆φis quite sim-
+ilar to that as presented in Fig. 3. The typical cur-
+rent amplitude varies periodically with ∆ φshowing
+/Minus2/Minus1012
+Φ2012Ityp /LParen1b/RParen1/Minus2/Minus1012
+Φ2012Ityp /LParen1a/RParen1
+Figure5: (Coloronline). Ityp-φ2curvesintheweak-
+(black) and strong-coupling (red) limits for the in-
+terferometer with four atomic sites ( N= 4), where
+(a)φ1= 0 and (b) φ1= 0.3. The typical cur-
+rent amplitudes are calculated at the bias voltage
+V= 1.02.
+2φ0flux-quantum periodicity, followingthe gtyp-∆φ
+characteristics.
+With the above description of electron transport
+for a 4-site ( N= 4) quantum interferometer, now
+we can extend our discussion for an interferometer
+with higher number of atomic sites i.e., N >4.
+3.2 Interferometric geometry with N
+atomic ( N >4) sites
+To get an experimentally realizable system, here we
+consider a quantum interferometer with large num-
+berofatomicsitescomparedtoourpresentedmath-
+ematical model with 4 atomic sites. The schematicview of such a quantum interferometer is given in
+Fig. 6, where we set N= 15. The vertical line con-
+necting left and right sub-rings contains 5 atomic
+Drain Source
+Quantum InterferometerΦ Φ1 2
+Figure 6: (Color online). Schematic view of a
+quantum interferometer with 15 atomic sites ( N=
+15) attached to two semi-infinite one-dimensional
+metallic electrodes.
+sites, where the individual sub-rings are penetrated
+by AB fluxes φ1andφ2, respectively. In this inter-
+ferometric geometry, the phase factors ( θij’s) are
+chosen according to our earlier prescription. Along
+the circumference of the ring θij= 2πφ/12φ0and
+alongtheverticalline θij= 2π∆φ/5φ0, whereφand
+∆φcorrespond to the identical meaning as before.
+Forthisbiggerquantuminterferometer( N= 15),
+exactly similar features of conductance-energy and
+current-voltage characteristics are observed as we
+see in the case of a 4-site interferometer. Also, typ-
+ical currentamplitude Itypshowsidentical variation
+withφ2toourpreviousstudy. Onlythe typicalcon-
+ductance gtypvaries in a different way as a function
+of ∆φ. As illustrative examples in Fig. 7 we plot
+gtyp-∆φcharacteristicsfor the quantum interferom-
+eter with N= 15 in the limit of strong-coupling,
+where (a) and (b) correspond to φ= 0.4 and 0.8,
+respectively. The typical conductances are deter-
+mined at the energy E= 1.5. From the spectra
+we notice that for a fixed value of φ, typical con-
+ductance oscillates as a function of ∆ φexhibiting
+5φ0flux-quantum periodicity. This phenomenon is
+completely different from the traditional periodic
+nature. Comparing the results presented in Figs. 3
+and 7 it is manifested that the periodicity of gtyp-
+∆φcurves depends on the total number of atomic
+80 /Minus10/Minus50510
+/RArrow/CapDeltaΦ0.44
+.22/LParen1b/RParen1/RArrowgtyp0 /Minus10/Minus50510
+/RArrow/CapDeltaΦ01.2
+0.6/LParen1a/RParen1/RArrowgtyp
+Figure 7: (Color online). gtyp-∆φcurves in the
+strong-coupling limit for the interferometer with 15
+atomic sites ( N= 15), where (a) φ= 0.4 and (b)
+φ= 0.8. The typical conductances are calculated
+at the energy E= 1.5.
+sites in the vertical line connecting left and right
+sub-rings of a quantum interferometer. Therefore,
+changing the length of the vertical line, periodicity
+can be changed accordingly.
+4 Closing remarks
+To summarize, we have explored electron trans-
+port properties through a quantum interferome-
+ter using the single particle Green’s function for-
+malism. We have adopted a simple tight-binding
+framework to illustrate the bridge system, where
+the interferometer is sandwiched between two elec-
+trodes, viz, source and drain. We have done exact
+numerical calculation to study conductance-energy
+and current-voltage characteristics as functions of
+the interferometer-to-electrode coupling strength,magnetic fluxes φ1andφ2penetrated by left and
+right sub-rings of the interferometer and the differ-
+ence of these two fluxes. Several key features of
+electron transport have been observed those may
+be useful in manufacturing nanoelectronic devices.
+The most exotic features are: (i) existence of semi-
+conducting or metallic behavior, depending on the
+interferometer-to-electrode coupling strength, (ii)
+appearance of the anti-resonant states and (iii) un-
+conventional periodic behavior of the typical con-
+ductance/current as a function of the difference of
+two AB fluxes.
+Throughout our work, we have addressed the
+essential features of electron transport through
+a quantum interferometer with total number of
+atomic sites N= 4. Next, we have extended our
+discussion for an interferometer with higher num-
+ber of atomic sites where we set N= 15 to achieve
+an experimentally realizable system. In our model
+calculations, these typical numbers ( N= 4 and 15)
+are chosen only for the sake of simplicity. Though
+the results presented here change numerically with
+ring size ( N), but all the basic features remain ex-
+actly invariant. To be more specific, it is impor-
+tant to note that, in real situation experimentally
+achievable rings have typical diameters within the
+range 0.4-0.6µm. In such a small ring, very high
+magnetic fields are required to produce a quantum
+flux. To overcome this situation, Hod et al.have
+studied extensively and proposed how to construct
+nanometer scale devices, based on Aharonov-Bohm
+interferometry, those can be operated in moderate
+magnetic fields [48, 49, 50, 51, 52].
+This is our first step to describe the electron
+transport in a quantum interferometer. Here we
+have made several realistic approximations by ig-
+noring the effects of electron-electron correlation,
+electron-phonon interaction, disorder, temperature,
+etc. Over the last few many years people have
+studied a lot to incorporate the effect of electron-
+electron correlation in the study of electron trans-
+port, yet no such proper theory has been well es-
+tablished. Thus the inclusion of electron-electron
+9correlation in the present model is a major chal-
+lenge to us. The presence of electron-phonon inter-
+action in Aharonov-Bohm interferometers provides
+phase shifts of the conducting electrons and due to
+this dephasing process electron transport through
+an AB interferometer becomes highly sensitive to
+the AB flux φwith the increase of electron-phonon
+couplingstrength [53]. In the presentwork, we have
+addressedourresultsconsideringthesiteenergiesof
+all the atomic sites of the interferometer are identi-
+cal i.e., we have treated the ordered system. But in
+real case, the presence of impurities will affect the
+electronicstructureandhencethetransportproper-
+ties. The effect ofthe temperaturehas alreadybeen
+pointed out earlier, and, it has been examined that
+the presented results will not change significantly
+even at finite temperature, since the broadening of
+the energy levels of the interferometer due to its
+coupling to the electrodes will be much larger than
+that of the thermal broadening [46]. At the end, we
+would like to mention that we need further study in
+such systems by incorporating all these effects.
+The importance of this article is mainly con-
+cerned with (i) the simplicity of the geometry and
+(ii) the smallness of the size, and our exact analy-
+sis may be utilized to study electron transport in
+Aharonov-Bohm geometries.
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@@ -0,0 +1,833 @@
+arXiv:1001.0083v4  [hep-ex]  8 Jun 2010Belle Preprint 2009-23
+KEK Preprint 2009-26
+Measurement of the branching fractions and the invariant ma ss
+distributions for τ−→h−h+h−ντdecays
+M. J. Lee,34H. Aihara,42K. Arinstein,1,30V. Aulchenko,1,30T. Aushev,16,11
+A. M. Bakich,37E. Barberio,20A. Bay,16K. Belous,10V. Bhardwaj,32M. Bischofberger,22
+A. Bondar,1,30A. Bozek,26M. Braˇ cko,18,12T. E. Browder,6P. Chang,25A. Chen,23
+P. Chen,25B. G. Cheon,5I.-S. Cho,46Y. Choi,36J. Dalseno,19,39A. Das,38W. Dungel,9
+S. Eidelman,1,30D. Epifanov,1,30M. Fujikawa,22N. Gabyshev,1,30A. Garmash,1,30
+H. Ha,14J. Haba,7B.-Y. Han,14Y. Hasegawa,35K. Hayasaka,21H. Hayashii,22Y. Hoshi,41
+W.-S. Hou,25Y. B. Hsiung,25H. J. Hyun,15T. Iijima,21K. Inami,21R. Itoh,7M. Iwasaki,42
+Y. Iwasaki,7T. Julius,20D. H. Kah,15J. H. Kang,46N. Katayama,7T. Kawasaki,28
+C. Kiesling,19H. J. Kim,15H. O. Kim,15J. H. Kim,36S. K. Kim,34Y. I. Kim,15
+Y. J. Kim,4B. R. Ko,14S. Korpar,18,12P. Kriˇ zan,17,12P. Krokovny,7T. Kumita,43
+A. Kuzmin,1,30Y.-J. Kwon,46S.-H. Kyeong,46J. S. Lange,3S.-H. Lee,14J. Li,6C. Liu,33
+Y. Liu,21D. Liventsev,11R. Louvot,16J. MacNaughton,7F. Mandl,9S. McOnie,37
+H. Miyata,28Y. Miyazaki,21T. Mori,21E. Nakano,31M. Nakao,7H. Nakazawa,23
+Z. Natkaniec,26S. Nishida,7O. Nitoh,44S. Ogawa,40T. Ohshima,21S. Okuno,13
+S. L. Olsen,34,6P. Pakhlov,11G. Pakhlova,11H. Palka,26C. W. Park,36H. Park,15
+H. K. Park,15L. S. Peak,37R. Pestotnik,12M. Petriˇ c,12L. E. Piilonen,45A. Poluektov,1,30
+S. Ryu,34H. Sahoo,6K. Sakai,28Y. Sakai,7O. Schneider,16A. J. Schwartz,2K. Senyo,21
+M. E. Sevior,20M. Shapkin,10V. Shebalin,1,30C. P. Shen,6J.-G. Shiu,25B. Shwartz,1,30
+J. B. Singh,32P. Smerkol,12A. Sokolov,10E. Solovieva,11S. Staniˇ c,29M. Stariˇ c,12
+T. Sumiyoshi,43G. N. Taylor,20Y. Teramoto,31I. Tikhomirov,11S. Uehara,7K. Ueno,25
+Y. Unno,5S. Uno,7P. Urquijo,20Y. Ushiroda,7Y. Usov,1,30G. Varner,6K. Vervink,16
+A. Vinokurova,1,30C. H. Wang,24P. Wang,8Y. Watanabe,13R. Wedd,20E. Won,14
+B. D. Yabsley,37Y. Yamashita,27M. Yamauchi,7C. Z. Yuan,8C. C. Zhang,8
+Typeset by REVT EX 1Z. P. Zhang,33V. Zhilich,1,30V. Zhulanov,1,30T. Zivko,12A. Zupanc,12and O. Zyukova1,30
+(The Belle Collaboration)
+1Budker Institute of Nuclear Physics, Novosibirsk
+2University of Cincinnati, Cincinnati, Ohio 45221
+3Justus-Liebig-Universit¨ at Gießen, Gießen
+4The Graduate University for Advanced Studies, Hayama
+5Hanyang University, Seoul
+6University of Hawaii, Honolulu, Hawaii 96822
+7High Energy Accelerator Research Organization (KEK), Tsuk uba
+8Institute of High Energy Physics, Chinese Academy of Scienc es, Beijing
+9Institute of High Energy Physics, Vienna
+10Institute of High Energy Physics, Protvino
+11Institute for Theoretical and Experimental Physics, Mosco w
+12J. Stefan Institute, Ljubljana
+13Kanagawa University, Yokohama
+14Korea University, Seoul
+15Kyungpook National University, Taegu
+16´Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), Lausan ne
+17Faculty of Mathematics and Physics, University of Ljubljan a, Ljubljana
+18University of Maribor, Maribor
+19Max-Planck-Institut f¨ ur Physik, M¨ unchen
+20University of Melbourne, School of Physics, Victoria 3010
+21Nagoya University, Nagoya
+22Nara Women’s University, Nara
+23National Central University, Chung-li
+24National United University, Miao Li
+25Department of Physics, National Taiwan University, Taipei
+26H. Niewodniczanski Institute of Nuclear Physics, Krakow
+27Nippon Dental University, Niigata
+28Niigata University, Niigata
+29University of Nova Gorica, Nova Gorica
+230Novosibirsk State University, Novosibirsk
+31Osaka City University, Osaka
+32Panjab University, Chandigarh
+33University of Science and Technology of China, Hefei
+34Seoul National University, Seoul
+35Shinshu University, Nagano
+36Sungkyunkwan University, Suwon
+37School of Physics, University of Sydney, NSW 2006
+38Tata Institute of Fundamental Research, Mumbai
+39Excellence Cluster Universe, Technische Universit¨ at M¨ u nchen, Garching
+40Toho University, Funabashi
+41Tohoku Gakuin University, Tagajo
+42Department of Physics, University of Tokyo, Tokyo
+43Tokyo Metropolitan University, Tokyo
+44Tokyo University of Agriculture and Technology, Tokyo
+45IPNAS, Virginia Polytechnic Institute and State Universit y, Blacksburg, Virginia 24061
+46Yonsei University, Seoul
+Abstract
+We present a study of τ−→π−π+π−ντ,τ−→K−π+π−ντ,τ−→K−K+π−ντ, and
+τ−→K−K+K−ντdecays using a 666 fb−1data sample collected with the Belle detector at
+the KEKB asymmetric-energy e+e−collider at and near a center-of-mass energy of 10.58 GeV.
+The branching fractions are measured to be: B(τ−→π−π+π−ντ)= (8.42±0.00+0.26
+−0.25)×10−2,
+B(τ−→K−π+π−ντ)= (3.30±0.01+0.16
+−0.17)×10−3,B(τ−→K−K+π−ντ)= (1.55±0.01+0.06
+−0.05)×10−3,
+andB(τ−→K−K+K−ντ)= (3.29±0.17+0.19
+−0.20)×10−5, where the first uncertainty is statistical
+and the second is systematic. These branching fractions do n ot include contributions from modes
+in which a π+π−pair originates from a K0
+Sdecay. We also present the unfolded invariant mass
+distributions for these decays.
+PACS numbers: 13.35.Dx, 12.15.Hh, 14.60.Fg
+3INTRODUCTION
+The decays of the τlepton into three pseudoscalar particles can provide information on
+hadronic form factors, K∗resonance spectroscopy, and the Wess-Zumino anomaly [1], and
+also can be used for studies of CP violation in the leptonic sector [2]. B y studying decays
+into final states that contain one or three kaons, it is possible to ex tract strange spectral
+functions, which can be used for a direct determination of the stra nge quark mass and
+the Cabibbo-Kobayashi-Maskawa (CKM) matrix element |Vus|[3–5]. These three hadron
+decays have been studied since the initial discovery of the τlepton, but only the decay to
+three charged pions has been closely investigated. Even for this mo de, measurements of its
+branching fraction have only been provided by the CLEO [6] and BaBa r [7] groups. For all
+other modes, there are still many unresolved problems. For examp le, the branching fraction
+forτ−→K−π+π−ντdecay recently measured by the BaBar collaboration [7] is significant ly
+lower than the values from most previous measurements [8]; there is large scatter in the
+measured central values, which is reflected in the large scale facto r (2.1) applied by the
+Particle Data Group when forming their average [8]. Intermediate tw o- and three-body
+resonance states in this decay have been previously studied but on ly with limited statistical
+precision, see, e.g., Refs[9–11]. There is evidence forquite richdyn amics, with strong signals
+for theK1(1270),K1(1400) and K∗(1410) resonances in the three-body final state; other
+resonances can be seen in various two-body sub-systems. Theor etically, two intermediate
+resonances, the ρ(770)0andK∗(892)0(or its excitations), are expected to contribute to τ
+decay into K−π+π−ντ:τ−→K−ρ(770)0(→π+π−)ντandτ−→K∗(892)0(→K−π+)π−ντ
+[12]. These are shown in Figs 1(a) and (b), respectively.
+Here we present new measurements of the branching fractions fo rτ−→π−π+π−ντ,
+τ−→K−π+π−ντ,τ−→K−K+π−ντ, andτ−→K−K+K−ντdecays. (Unless otherwise
+specified, thecharge-conjugatedecayisalso impliedthroughout t hispaper.) Events inwhich
+aπ+π−pair is consistent with a K0
+Sdecay are excluded. Because of particle misidentifica-
+tion, measurements of all four of the above modes are correlated and, thus, are considered
+simultaneously.
+We use a data sample of 666 fb−1collected with the Belle detector at the KEKB
+asymmetric-energy e+e−collider [13] on the Υ(4S) resonance, 10.58 GeV, and 60 MeV
+below it (off-resonance). This sample contains 6 .12×108produced τ-pairs. The Belle de-
+4+e
+-e+τ
+-τ+W
+-W
+τν
+-µ, -eµe, ν) s((u)
++K+π-π
+τν
+(770)ρ
+(a)+e
+-e+τ
+-τ+W
+-W
+τν
+-µ, -eµe, ν(u)) s(
++π-π+K
+τν
+(892)*K
+(b)
+FIG. 1: A schematic view of e+e−→τ+τ−in the center-of-mass system, where one τ(τ+)
+decays to the signal mode ( τ+→K+π−π+ντ) and the other τ(τ−) decays to a leptonic mode
+(τ−→e−ντνeorτ−→µ−ντνµ). Each figure shows one of two possible intermediate resonan t
+states in the signal mode.
+tector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector
+(SVD), a 50-layer central drift chamber (CDC), an array of aero gel threshold Cherenkov
+counters (ACC), a barrel-like arrangement of time-of-flight scint illation counters (TOF),
+and an electromagnetic calorimeter (ECL) comprised of CsI(Tl) cry stals located inside a su-
+perconducting solenoid coil that provides a 1.5 T magnetic field. An iro n flux return located
+outside the coil (KLM) is instrumented to detect K0
+Lmesons and to identify muons. The
+detector is described in detail elsewhere [14].
+SELECTION OF EVENTS
+Three-prong decays of the τare selected from candidate τ+τ−pair events as follows. We
+use events where the number of charged tracks is four and the su m of their charges is zero;
+the transverse momentum of each track in the laboratory frame is required to be larger than
+0.1 GeV/ c, and the tracks should extrapolate back to the interaction point w ithin±5 cm
+along the beam direction and ±1 cm in the transverse plane. The sum of the reconstructed
+5momenta in the center-of-mass (CM) frame is required to be less th an 10 GeV/ c, and the
+sum of energies deposited in the calorimeter is required to be less tha n 10 GeV. The largest
+of the four tracks’ transverse momenta is required to be greate r than 0.5 GeV/ cto reject
+two-photon events, which typically have low transverse momentum tracks. To reject beam-
+related background, we require the position of the reconstructe d event vertex to be less than
+3 cm from the interaction point along the beam direction, and separa ted by less than 0.5
+cm in the transverse plane. The missing mass, M2
+miss= (pinit−/summationtext
+trptr−/summationtext
+γpγ)2, and the
+polar angle of the missing momentum in the CM frame are efficient variab les for rejecting
+two-photon and Bhabha backgrounds. In the definition of the miss ing mass, ptrandpγare
+the four-momenta of measured tracks and photons, respective ly, andpinitis the initial CM
+framefour-momentum ofthe e+e−beams. Werequire that themissing mass belarger than1
+GeV/c2and less than 7 GeV/ c2, and that the polar angle with respect to the beam direction
+in the CM frame be larger than 30◦and less than 150◦. Detailed information on the τ+τ−
+pair selection and background suppression using missing mass can be found in [15].
+Particle identification isused to select τevents that containonelepton (electronor muon)
+and three hadrons (pions or kaons). The magnitude of thrust is ev aluated and is required
+to be larger than 0.9 to suppress two-photon and e+e−→q¯qbackgrounds, where the thrust
+is defined by the maximum of (/summationtext
+i|ˆn·/vector pi|)/(/summationtext
+i|/vector pi|). Here/vector piis the momentum of the i-th
+track and ˆ nis the unit vector in the direction of the thrust axis – the direction ma ximizing
+the thrust. We require that the angle between the total momentu m of the hadrons and the
+lepton momentum in the CM system be larger than 90◦, since the tag-side lepton and signal-
+side hadrons usually lie in opposite hemispheres: the so-called 1–3 pro ng configuration, due
+to the large transverse momenta of τleptons. The invariant mass of charged tracks and
+gamma clusters on each side is required to be less than the τmass. Finally, we require that
+there be no K0
+S,π0, or energetic γon the signal side, where the selection criteria for these
+particles are described below. Figure 2 shows distributions of the va riables used to select the
+1–3 prong sample, where in each case, the selection value is indicated by a vertical line in
+the figure. The surviving events are candidates for τ−→h−
+1h+
+2h−
+3ντ, whereh1,2,3is either a
+pion or a kaon candidate. At this stage, the efficiencies for reconst ructingτ−→π−π+π−ντ,
+τ−→K−π+π−ντ,τ−→K−K+π−ντ, andτ−→K−K+K−ντdecays as τ−→h−h+h−ντ
+are 26.7%, 27.5%, 27.2%, and 24 .8%, respectively, while the reconstruction efficiencies of
+the dominant background modes, τ−→π−π+π−π0ντ,τ−→K0
+Sπ−ντ, ande+e−→q¯q, are
+66.0%, 1.8%, and 0.004%, respectively. The fractions of τ−→π−π+π−π0ντ,τ−→K0
+Sπ−ντ,
+e+e−→q¯qand two-photon backgrounds that are reconstructed as τ−→h−h+h−ντare
+6.47%, 0.34%, 0.35% and 0 .05%, respectively; the two-photon background contribution is
+negligible.
+Angle(signal,tag) [Degree]0 50 100 150Arbitrary units / degree
+00.010.020.03 dataτ
+ MCν π π K → τ
+ MCγ2
+ MCqq
+(a) Angle(signal,tag) [Degree]0 50 100 150Arbitrary units / degree
+00.010.020.03
+|Thrust|0.5 0.6 0.7 0.8 0.9 1Arbitrary units
+00.050.10.150.2
+ dataτ
+ MCν π π K → τ
+ MCγ2
+ MCqq
+(b) |Thrust|0.5 0.6 0.7 0.8 0.9 1Arbitrary units
+00.050.10.150.2
+]2c Invariant mass (signal) [GeV/0 1 2 3 4)2c Arbitrary units / (10MeV/
+00.0050.010.015 dataτ
+ MCν π π K → τ
+ MCqq
+(c) ]2c Invariant mass (signal) [GeV/0 1 2 3 4)2c Arbitrary units / (10MeV/
+00.0050.010.015
+FIG. 2: Distributions of variables used in the event selecti on. (a) The angle between the total
+momentum of the hadronic system on the signal-side and the le pton momentum on the tag-side in
+the CM system. (b) The magnitude of the thrust. (c) The invari ant mass of the hadronic system.
+Thetriangular points show data, the dashed histograms show theτ−→K−π+π−ντsignal MC, the
+solid histograms show the two-photon background, and the do tted histograms show the e+e−→q¯q
+background. The dotted vertical lines show the boundaries u sed to select the signal candidates.
+All samples are normalized to the same number of events.
+An important issue for this analysis is the separation of kaons and pio ns. In Belle, dE/dx
+information from the CDC, hit information from the ACC, and time-of -flight measurements
+from the TOF system are used to construct a likelihood for the kaon (pion) hypothesis,
+L(K) (L(π)). Figure 3 shows the kaon likelihood ratio, L(K)/(L(K)+L(π)), as a function
+7ofmomenta. Clean kaon-pionseparationis evident over theentire m omentum rangerelevant
+for this analysis, where the track momentum ranges from 0.1 GeV/ cto∼5 GeV/c, and the
+average momentum is ∼1.3 GeV/ c. As discussed below, we choose a relatively stringent
+particle identification (PID) requirement for kaons ( L(K)/(L(K)+L(π))>0.9) and a less
+restrictive one for pions ( L(K)/(L(K)+L(π))<0.9). The kaon identification selection has
+an efficiency of ∼73% for kaons and rejects ∼95% of pions. The kaon and pion identification
+efficiencies are calibrated using kaon and pion tracks in data from kine matically identified
+D∗+decays,D∗+→D0(→K−π+)π+. We evaluate the efficiencies and fake rates for
+this calibration sample and compare them to Monte Carlo (MC) expect ations. From this
+comparison, we obtain a correction table as a function of track mom enta (plab) and polar
+angles (θlab), and apply it to the MC. The accuracy of the correction factor, w hich is the
+source of the systematic uncertainty of the evaluation of the bra nching fractionand the mass
+spectra, is limited by the statistical uncertainties of kaon and pion s amples from D∗+decays
+in the certain plabandθlabbins, and the uncertainty of the D∗+signal extraction.
+]c Momentum [GeV/0 1 2 3 4 5))π L(K)/(L(K)+L( Likelihood ratio, 
+00.10.20.30.40.50.60.70.80.91
+FIG. 3: Likelihood ratio for kaon identification, L(K)/(L(K)+L(π)), versus particle momentum.
+The filled and open circles correspond to kinematically iden tified kaons and pions, respectively.
+The kaon identification criterion is determined by maximizing a figure-o f-merit (FOM),
+where the requirement on the kaon identification likelihood ratio is var ied. The figure-of-
+8merit we used is defined as:
+FOM=S√
+S+N, (1)
+whereSis the number of signal events and Nis the number of cross-feed background
+events. For the τ−→K−π+π−ντsignal (S), the cross-feed background component ( N)
+includes contributions from τ−→π−π+π−ντandτ−→K−K+π−ντdecays. The result of
+FOM optimization for the τ−→K−π+π−ντdecay is shown in Fig. 4, where one can see
+that FOM is indeed maximal for the particle identification criteria used in this analysis. The
+same FOM analysis was performed for the τ−→π−π+π−ντandτ−→K−K+π−ντdecay
+modes and resulted in similar particle identification criteria. For electr on identification, the
+likelihood variable is calculated based on the track extrapolation to th e ECL, the ratio of
+the energy deposited in the ECL to the momentum measured in the CD C, the measured
+dE/dxin the CDC, the shower shape in the ECL, and light yield in the ACC [16]. Th e
+extrapolation of tracks from the CDC and SVD to the KLM is used to c onstruct the likeli-
+hood variable for muon identification [17]. The efficiencies and systema tic uncertainties for
+lepton identification are evaluated by using a γγ→e+e−/µ+µ−control sample.
+))π L(K)/(L(K)+L(0.7 0.75 0.8 0.85 0.9 0.95 1Figure of Merit
+373839404142434445
+FIG. 4: Figure-of-merit as a function of the likelihood rati oL(K)/(L(K)+L(π)) requirement, used
+to separate the kaon and pion hypotheses. The uncertainties are evaluated from the comparison of
+the efficiencies and fake rates of kaon identification with tho se of the D∗+→D0(→K−π+)π+con-
+trol sample.
+To reduce the feed-down background and to estimate the remainin g backgrounds that
+9containK0
+Sandπ0mesons, we reconstruct K0
+Sandπ0signals explicitly. K0
+Scandidates are
+formedfromoppositelychargedpairsofpiontracks withinvariant m assM(ππ) within±13.5
+MeV/c2(±5σ) of theK0
+Smass. To improve the K0
+Spurity, the point of closest approach
+to the interaction point along the extrapolation of each track is req uired to be larger than
+0.3 cm in the plane transverse to the beam direction. The azimuthal a ngle between the
+momentum vector and the decay vertex vector of the reconstru ctedK0
+Sis required to be
+less than 0.1 rad. The distance between the two daughter pion trac ks at their point of
+closest approach is required to be less than 1.8 cm, and the flight leng th of the K0
+Sin the
+plane perpendicular to the beam direction is required to be larger tha n 0.08 cm. Finally,
+we require the invariant mass of each K0
+Scandidate to be inconsistent with that of a Λ, ¯Λ,
+and a photon conversion, where the daughter tracks are assume d to be pions and protons
+(antiprotons) or electrons and positrons as appropriate. Candid ateπ0’s are reconstructed
+from pairs of photon clusters. The energy of each photon is requir ed to exceed 50 MeV for
+candidates in the barrel part of the calorimeter, and 100 MeV for c andidates in the endcap
+part of the calorimeter. We also require the invariant mass of the tw o photons, M(γγ), to
+be within ±16 MeV/ c2of theπ0mass. Events containing one or more reconstructed K0
+Sor
+π0are rejected. In addition, in order to reject events containing π0’s, we require that there
+be no photon with energy larger than 0.3 GeV.
+EFFICIENCY ESTIMATION
+To estimate the signal efficiency, 97 million signal MC events for the τ−→π−π+π−ντ
+decay mode and 5 million signal MC events for each of the τ−→K−π+π−ντ,τ−→
+K−K+π−ντ, andτ−→K−K+K−ντmodes were generated using the TAUOLA [18] based
+KKMC [19] generator. The detector response for all MC data sets was simulated with the
+GEANT3 [20] package. The TAUOLA generator provides models for τ−→π−π+π−ντ,
+τ−→K−π+π−ντ, andτ−→K−K+π−ντdecays, where various intermediate resonances
+are taken into account. To check whether there is efficiency bias re lated to the specific decay
+model used, events with phase-space decay distributions were ge nerated using the KKMC
+program. The efficiencies asfunctions ofthe invariant mass for bot hτ−→K−π+π−ντdecay
+models are compared in Fig. 5. The relative difference in efficiency betw een the TAUOLA
+and phase-space decay models is around 1%. Similarly, very small disc repancies between
+10the TAUOLA and phase-space decay model are observed in the τ−→π−π+π−ντandτ−→
+K−K+π−ντdecays. We used the TAUOLA decay model to evaluate the efficiencie s and
+their dependencies on the invariant mass of τ−→π−π+π−ντ,τ−→K−π+π−ντ, andτ−→
+K−K+π−ντdecays. For the τ−→K−K+K−ντdecay mode, the TAUOLA generator does
+not provide any decay model, therefore the efficiency is calculated a ssuming a phase-space
+decay. In practice, the efficiencies are evaluated using a two step p rocedure. First, we
+evaluate the response matrix for unfolding the mass spectra using the TAUOLA decay
+model. Next, we recalibrate the efficiencies using the unfolded spect ra. Details of this
+procedure are described in the next section.
+]2c) [GeV/π πM(K0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8Efficiency
+0.050.10.150.20.250.3
+Efficiency (TAUOLA decay)
+Efficiency (phase space decay)
+FIG. 5: The efficiency for the τ−→K−π+π−ντdecay as a function of M(Kππ). Squares and
+triangular points represent the TAUOLA decay model and phas e-space decay, respectively.
+The average efficiencies and the fractions of the cross-feed back grounds for all three-
+prong decays are summarized in Table I. The probability of reconstr uctingτ−→π−π+π−ντ
+asτ−→K−π+π−ντis relatively low, but due to the large branching fraction of τ−→
+π−π+π−ντdecay, there is a substantial contamination of misidentified τ−→π−π+π−ντ
+events in the τ−→K−π+π−ντsample. For a similar reason, the τ−→K−K+K−ντdecay
+mode has a large cross-feed background from τ−→K−K+π−ντ.
+11BRANCHING FRACTION AND MASS SPECTRUM CALCULATION
+The numbers of candidate events in the τ−→π−π+π−ντ,τ−→K−π+π−ντ,τ−→
+K−K+π−ντ, andτ−→K−K+K−ντmodes, after applying all selection criteria, are sum-
+marized in Table II. Possible backgrounds in these signal candidate s amples include (a)
+cross-feed from the signal modes and (b) other processes such asτ−→π−π+π−π0ντ,
+τ−→K0
+Sπ−ντ, as well as continuum e+e−→q¯q. The fractions of backgrounds coming
+from other processes are 5% to 12%, as summarized in Table II (fou rth column) for each
+signal mode. The fifth column shows the fraction of the main backgr ound component as
+determined fromMC. The mode τ−→π−π+π−π0ντdominates for the τ−→π−π+π−ντand
+τ−→K−π+π−ντmodes, while the background from continuum e+e−→q¯qis dominant for
+theτ−→K−K+π−ντandτ−→K−K+K−ντmodes.
+To take into account the cross-feed between the decay channels , the true yield Ntrue
+i(i
+is the mode number and the values from 1 to 4 correspond to τ−→π−π+π−ντ,τ−→
+K−π+π−ντ,τ−→K−K+π−ντ, andτ−→K−K+K−ντ), is obtained using the following
+equation:
+Ntrue
+i=/summationdisplay
+j(E−1)ij(Nrec
+j−Nother
+j), (2)
+TABLE I: Summary of the efficiencies and the fractions of cross -feed backgrounds. The values in
+parentheses are the differences of the efficiencies or fake rate s from those evaluated directly from
+the MC with the TAUOLA decay model, ( E(2)
+ij−E(1)
+ij)/E(1)
+ijin percent (see text).
+Generated decay mode
+Reconstructed τ−→ τ−→ τ−→ τ−→
+decay mode π−π+π−ντ K−π+π−ντ K−K+π−ντ K−K+K−ντ
+τ−→π−π+π−ντ0.22 (−0.1%) 0.079 (−0.1%) 0.022 (−1.1%)6.4×10−3(+2.6%)
+τ−→K−π+π−ντ0.012 (−0.2%) 0.18 (−0.5%) 0.047 (+1.4%) 0.019 (−3.1%)
+τ−→K−K+π−ντ3.9×10−4(−0.3%)4.7×10−3(−4.4%) 0.12 (−1.0%) 0.051 (−0.4%)
+τ−→K−K+K−ντ5.0×10−6(+4.5%)1.3×10−4(−9.1%)2.3×10−3(−14.0%)0.081 (+1.2%)
+12TABLE II: The number of reconstructed events (second column ), the number of true events (third
+column), the number of background events other than three-p rong cross-feeds (fourth column),
+and the main source of other backgrounds with its fraction of the total (fifth column).
+Decay mode NrecNtrueNotherMain component in Nother
+τ−→π−π+π−ντ8.86×1063.52×1079.35×105(10.6%)τ−→π−π+π−π0ντ(64.2%)
+τ−→K−π+π−ντ7.94×1051.38×1069.60×104(12.1%)τ−→π−π+π−π0ντ(34.4%)
+τ−→K−K+π−ντ1.08×1056.47×1057.16×103(6.66%)e+e−→q¯q(30.3%)
+τ−→K−K+K−ντ3.16×1031.37×1041.71×102(5.41%)e+e−→q¯q(53.0%)
+whereNrec
+jis the number of events for the j-th reconstructed decay mode and Nother
+jis the
+number of background events in the j-th mode. Here Eijis the efficiency for detecting the
+j-th mode as the i-th one. The values of NrecandNotherare listed in the second and third
+columns of Table II, respectively.
+We have determined the efficiency (migration) matrix iteratively. Firs t, it is determined
+using the mass spectra as modeled in the current TAUOLA MC progra m. (We refer to this
+efficiency matrix as E(1)
+ij). As is discussed in the next paragraph, we then unfolded the mass
+spectra of each mode to obtain the true mass distribution. The res ultant unfolded mass
+spectra are then used to determine the efficiency matrix E(2)
+ijusing a weighting procedure.
+The matrix elements E(2)
+ijare given in Table I, where the values given in parentheses are the
+percentage differences between E(1)
+ijandE(2)
+ij; (E(2)
+ij−E(1)
+ij)/E(1)
+ij. Since the efficiency is smooth
+as a function of hadronic mass, the difference between E(1)
+ijandE(2)
+ijis very small ( ∼1%) in
+most cases, except for the off-diagonal entries related to the τ−→K−K+K−ντmode. The
+source of these large discrepancies is the mass dependence of the fake rates, together with
+the considerable differences between the unfolded spectrum and t he generated spectrum.
+The unfolding procedure extracts the true mass spectra for all f our modes, denoted by
+the vector x, by solving the matrix equation ˆAx=b. In this equation, bis the vector of
+the reconstructed mass spectrum with backgrounds other than the three-prong cross-feed
+subtracted, and ˆAis the response matrix.
+Thematrix ˆAtakes intoaccountthecross-feedbetween different modescaus edbyparticle
+misidentification, as well as the effects of finite resolution and the limit ed acceptance of the
+detector. The response matrix is obtained by merging 16 sub-resp onse matrices as shown in
+13Fig. 6. The matrix is determined using the TAUOLA decay model MC simu lation. Clear
+correlations between the measured and generated values are see n along the diagonal parts
+ofˆA.
+-710-610-510-410-310-210-110
+0.41 1.80 0.77 1.80 1.12 1.80 1.48 1.800.411.800.771.801.121.801.481.80
+)π π π(genM ) π π(KgenM ) π(KKgenM (KKK)genM)π π π(recM ) π π(KrecM ) π(KKrecM (KKK)recM
+]2c[GeV/]2c[GeV/
+Bin numbers of M(generated)Bin numbers of M(reconstructed)
+0 100 200 3000100200300
+FIG. 6: Response matrix ˆAobtained by merging 16 sub-response matrices that represen t the
+efficiency or fake rates among the four three-prong modes. The bin numbers correspond to the
+following ranges of the invariant masses: (1) from 0 to 138 , 0 .41 GeV/ c2to 1.8 GeV/ c2ofM(πππ);
+(2) from 139 to 241 , 0.77 GeV/ c2to 1.8 GeV/ c2ofM(Kππ); (3) from 242 to 309 , 1.12 GeV/ c2
+to 1.8 GeV/ c2ofM(KKπ); and (4) from 310 to 341 , 1.48 GeV/ c2to 1.8 GeV/ c2ofM(KKK).
+All bin sizes are 10 MeV/ c2.
+For the unfolding algorithm, we follow the method of the ALEPH Collabo ration, which is
+based onthe Singular ValueDecomposition technique for matrixinver sion [21] and advanced
+14regularization [22]. Two independent τ−→h−h+h−ντsignal MC samples are used to check
+the validity of the unfolding procedure. One sample is used for the de termination of the
+response matrix. We check the reproducibility of the generated M(hhh) mass spectra with
+the other sample, and find, for the whole mass range, that the diffe rences are less than 1%.
+These discrepancies are included in the systematic uncertainties of unfolding (UNF1 below).
+The resulting unfolded normalized mass spectra of the three-pron g decays are shown in
+Fig. 7, where the error bars indicate the statistical uncertainty o nly, and the gray bands
+correspond to the systematic uncertainty. Note that the uncer tainties common for all bins
+are taken into account as the systematic uncertainties on the bra nching fraction (Table IV),
+while the bin-by-bin errors remained in the normalized mass spectra 1 /N(dN/dM) are
+accounted here. The following sources of systematic uncertaintie s in the mass spectra are
+considered: the unfolding procedure (UNF1 and UNF2), the kaon id entification efficiency
+(KID), the background estimation other than the three-prong c ross-feed (BGE), the effect
+ofγveto (GAM), and the track momentum scale (MOM). Table III summa rizes various
+contributions to the systematic uncertainties on the normalized ma ss spectra 1 /N(dN/dM)
+for each decay mode.
+The systematic uncertainty due to the unfolding procedure is dete rmined from MC by
+comparing the generated and unfolded mass spectra (UNF1). Ano ther estimation of the
+uncertainty of the unfolding procedure is obtained by changing the value of unfolding pa-
+rameter that determines the effective rank of the response matr ix (UNF2). This is the most
+important systematic uncertainty for KKKmode. The kaon identification efficiency and
+the fake rate are calibrated by using D∗+sample as described in the previous section. The
+uncertainties due to the particle identification (KID) are evaluated by varying efficiency
+and cross-feed fraction by one standard deviation in the respons e matrix. The systematic
+uncertainty from the background estimation (BGE) is evaluated by varying the branching
+fraction values used in the MC by one standard deviation. The uncer tainty due to the γveto
+(GAM) is evaluated by using different selection criteria for photon en ergy and comparing
+the resultant unfolded spectra. For the uncertainty from the tr ack momentum scale (MOM),
+we reconstruct φ→K+K−andD±→K∓π±π±decays. By comparing the reconstructed
+masses of φandD±mesons with their world average values, we conservatively assign 0.0 1%
+uncertainty for the momentum scale. By applying this variation to th e reconstructed mass
+spectra of τ−→h−h+h−ντdecay and unfolding it, we obtain the uncertainties due to the
+15momentum scale.
+The systematic uncertainty varies as a function of the mass of the hadronic system in
+each mode. The typical uncertainties averaged over all mass bins a re evaluated by summing
+the systematic uncertainties of each bin and ignoring the correlatio ns between bins, and
+the results are 0.7%, 2.2%, 2.2%, and 9.5%, for the τ−→π−π+π−ντ,τ−→K−π+π−ντ,
+τ−→K−K+π−ντ, andτ−→K−K+K−ντdecay modes, respectively. In practice, we have
+obtained the covariance matrix for each source of the systematic uncertainties and added
+them to obtain the total uncertainty. The gray bands in Fig. 7 show the square roots
+of the diagonal components of the covariance matrix. The off-diag onal components of the
+covariance matrix are shown in Fig. 8 in term of the correlation coeffic ientsρSpec(i,j) defined
+byρSpec(i,j) = cov(i,j)//radicalbig
+cov(i,i)·cov(j,j), where iandjare bin numbers.
+Figure 7 shows rather large discrepancies between the mass spect ra of the genera-
+tor and our unfolded spectra except for τ−→π−π+π−ντdecay. For example, for the
+τ−→K−π+π−ντmass spectrum (Fig. 7b), it is evident that there are contributions from
+two resonant peaks, similar to the MC expectations. However, the faster rise of the real
+data as compared with the MC in the 1 .0−1.1 GeV mass region and the fall-off at high
+masses in the τ−→K−π+π−ντmass spectrum may also be suggestive of a substantial
+non-resonant component that is not included in the current MC mod el. The use of the
+unfolded invariant mass distributions in estimating the efficiencies allow s one to compensate
+for model deficiencies when evaluating the branching fractions.
+Inorder to determine the branching fraction, we normalize to the n umber of pure leptonic
+decays, where one τdecays to τ→eννand the other decays to τ→µνν[23] (hereafter such
+events are referred to as {e,µ}events). The branching fraction for the i-th decay mode can
+then be written as:
+Bi=Ntrue
+i·εeµ
+Nsig,eµ·Bτ→eνν·Bτ→µνν
+Bτ→lνν, (3)
+whereNsig,eµ= 8.1×106andεeµ= 0.221±0.003 are the number of {e,µ}events and
+the corresponding detection efficiency, respectively. The uncert ainty in εeµis determined
+from the lepton identification uncertainties. Here Bτ→eνν= 0.178 and Bτ→µνν= 0.174
+are the branching fractions of τ→eννandτ→µννdecays, respectively, and Bτ→lνν=
+16TABLEIII: Summaryoftherelativeerrorsoftheunfoldedmas sspectra, 1 /N(dN/dM) (in%)from
+different sources of uncertainties: the unfolding procedure (UNF1, UNF2), the kaon identification
+(KID), the background estimation (BGE), the γveto (GAM), and the momentum scale (MOM).
+The“average” uncertainties (the2nd, 4-th, 6-th, and8-th c olumns)areevaluated bytaking average
+of errors in all bins. The “peak” uncertainties (the 3rd, 5-t h, 7-th, and 9-th columns) represent
+the errors at the peak position of the spectra. See the text fo r a more detailed description.
+Sources τ→ τ→ τ→ τ→
+of πππν Kππν KKπν KKKν
+uncertainties averagepeakaveragepeakaveragepeakaveragepeak
+(UNF1) 0.50.50.10.10.40.40.60.6
+(UNF2) 0.10.11.61.50.90.97.34.0
+(KID) 0.40.41.00.81.51.11.90.9
+(BGE) 0.20.20.60.50.40.31.91.1
+(GAM) 0.60.40.90.41.20.94.63.2
+(MOM) 0.10.00.40.20.30.32.73.1
+Total systematics 0.70.72.21.92.21.79.56.2
+Statistics 0.20.21.10.81.00.87.14.6
+Total 0.90.82.42.02.41.911.87.7
+Bτ→eνν+Bτ→µνν. The true yields Ntrue
+iobtained from Eq. 2 are given in the third column
+of Table II.
+Thisnormalizationmethodrequiresaprecisemeasurement ofthenu mberof{e,µ}events
+and a careful determination of the corresponding efficiency. Figur e 9 shows a comparison
+of the invariant mass of the electron and muon system and the cosin e of the angle between
+electron and muon, for real {e,µ}events and the sum of MC expectations. The MC
+reproduces the data reasonably well. The background fraction is e stimated to be 5 .9% using
+MC. After subtracting background, we obtain the number of {e,µ}events.
+Figure 10 shows the observed distributions of M(πππ),M(Kππ),M(KKπ), and
+M(KKK). Thebackgrounds arerepresented bythehatchedandthegra yhistogramsforthe
+17]2c)  [GeV/π π πM(0.6 0.8 1 1.2 1.4 1.6 1.8)]2c 1/N (dN/dM) [/10(MeV/
+00.0050.010.0150.020.025
+(a)]2c)  [GeV/π πM(K0.8 1 1.2 1.4 1.6 1.8)]2c 1/N (dN/dM) [/10(MeV/
+00.010.020.030.04
+(b)
+]2c)  [GeV/πM(KK1.2 1.3 1.4 1.5 1.6 1.7 1.8)]2c 1/N (dN/dM) [/10(MeV/
+00.010.020.030.040.05
+(c)]2c M(KKK)  [GeV/1.5 1.55 1.6 1.65 1.7 1.75 1.8)]2c 1/N (dN/dM) [/10(MeV/
+00.020.040.060.08
+(d)
+FIG. 7: Theunfolded mass spectra of the three-prong decays, 1/N(dN/dM): (a)M(πππ) distribu-
+tion ofτ−→π−π+π−ντ, (b)M(Kππ) distributionof τ−→K−π+π−ντ, (c)M(KKπ)distribution
+ofτ−→K−K+π−ντ, and (d) M(KKK) distribution of τ−→K−K+K−ντ. The black points
+correspond to the unfolded mass spectra with statistical un certainties only, and the gray bands cor-
+respondtothesystematic uncertainty. Thesolid histogram sarethemodelpredictionsimplemented
+in the current TAUOLA MC simulation.
+sum of the three-prong cross-feed backgrounds and other bac kground components, respec-
+tively. Note that the three-prong cross-feed backgrounds are determined by the unfolding
+procedure previously described.
+Table IV summarizes various contributions to the systematic uncer tainties on the branch-
+ing fractions. The uncertainty in the track-finding efficiency is estim ated from a comparison
+of real and MC data for the D∗→πD0,D0→ππK0
+S(K0
+S→π+π−) decay sample. The
+uncertainties due to the kaon and pion identification are evaluated b y applying the un-
+certainties in kaon identification efficiencies to the efficiency matrix in T able I. First, we
+18-1-0.75-0.5-0.2500.250.50.751
+0.41 1.80 0.77 1.80 1.12 1.80 1.48 1.800.411.800.771.801.121.801.481.80
+)π π πM( ) π πM(K ) πM(KK M(KKK))π π πM( ) π πM(K ) πM(KK M(KKK)
+]2c[GeV/]2c[GeV/
+inv Bin numbers of Minv Bin numbers of M
+0 100 200 3000100200300
+FIG. 8: The correlation coefficients of the unfolded mass spec tra for three-prong decays. See
+Fig. 6 for the convention of the bin numbers. The pattern is th e reflection of the detailed unfolding
+procedure.
+measure the kaon and pion identification efficiencies and their uncert ainties using the con-
+trol sample of D∗+→D0π+
+sandD0→K−π+events. Since these efficiencies are applied
+to the efficiency matrix, we propagate the uncertainties in the kaon and pion identification
+efficiencies to the covariance matrix of branching fraction measure ment. The kaon identifi-
+cation systematic uncertainties for τ−→π−π+π−ντ,τ−→K−π+π−ντ,τ−→K−K+π−ντ,
+andτ−→K−K+K−ντdecays are 1.3%, 3.9%, 1.9%, and 5.4%, respectively. For lepton
+identification efficiency uncertainties, we use γγ→e+e−/µ+µ−events.
+In order to take into account the effects of the mass spectra on t he branching fraction
+measurements, we have determined the mass spectra of the four dominant decay modes
+iteratively. The uncertainties in branching fractions due to the mas s spectra are evaluated
+19]2c) [GeV/µM(e0 2 4 6 8 10)2c Number of events / (10 MeV/
+0100200300310×
+0 2 4 6 8 100100200300310×
+(a)0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+0 2 4 6 8 100100200300310×
+)µcos(e,-1 -0.8 -0.6 -0.4 -0.2 0Number of events
+0200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+(b)-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+-1 -0.8 -0.6 -0.4 -0.2 00200400600800310×
+FIG. 9: Distributions of the {e,µ}events: (a) Invariant mass of the electron–muon system. (b)
+Cosine of the angle between the electron and muon. The closed circles and the solid histogram rep-
+resent the data and the sum of the MC expectations, respectiv ely. The open histogram represents
+the{e,µ}signal events, while the τ-pair background and the two-photon background are shown
+by the hatched and the gray histograms, respectively.
+by varying the unfolded spectra by one standard deviation of their statistical and systematic
+errors, assuming 100% bin-by-bin correlations for systematic err ors. The trigger efficiency
+for the three-prong modes is ∼86% with an uncertainty of ∼0.6%. This efficiency is obtained
+usingatriggersimulationprogramappliedtothesignalMCsamples. Th euncertaintydueto
+theγvetoisevaluatedbyusingdifferentselectioncriteriaforphotonene rgy. Theuncertainty
+due to the K0
+Sveto is evaluated by removing the veto requirement. The uncertain ties in
+background estimation include the effect of luminosity, the cross se ction for e+e−→τ+τ−,
+and the subtraction of background from non-three-prong τdecays and continuum e+e−→
+q¯q. The uncertainty on the τdecay background subtraction is evaluated by propagating
+the errors from the τbranching fractions other than those with three-prongs. The fr action
+of thee+e−→q¯qbackground is negligible for τ−→π−π+π−ντandτ−→K−π+π−ντ
+modes, and is 2 .0±0.2% and 2 .9±0.3% forτ−→K−K+π−ντandτ−→K−K+K−ντ
+modes, respectively. The uncertainties in the e+e−→q¯qbackground are estimated from
+comparison of the number of events in data and e+e−→q¯qMC, in the mass region above
+theτmass. The effects of the uncertainties in the luminosity and the cros s section for
+e+e−→τ+τ−[24] are negligible because we use {e,µ}events for normalization. The
+uncertainties in the leptonic decay branching fractions for the τare also taken into account.
+20]2c) [GeV/π π πM(0.4 0.6 0.8 1 1.2 1.4 1.6 1.8)2c Events / (10 MeV/
+050100150200310×
+(a)]2c) [GeV/π π πM(0.4 0.6 0.8 1 1.2 1.4 1.6 1.8)2c Events / (10 MeV/
+050100150200310×
+]2c) [GeV/π π πM(0.4 0.6 0.8 1 1.2 1.4 1.6 1.8)2c Events / (10 MeV/
+050100150200310×
+]2c) [GeV/π πM(K0.8 1 1.2 1.4 1.6 1.8)2c Events / (10 MeV/
+05000100001500020000
+(b)]2c) [GeV/π πM(K0.8 1 1.2 1.4 1.6 1.8)2c Events / (10 MeV/
+05000100001500020000
+]2c) [GeV/π πM(K0.8 1 1.2 1.4 1.6 1.8)2c Events / (10 MeV/
+05000100001500020000
+]2c) [GeV/πM(KK1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8)2c Events / (10 MeV/
+01000200030004000
+(c)]2c) [GeV/πM(KK1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8)2c Events / (10 MeV/
+01000200030004000
+]2c) [GeV/πM(KK1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8)2c Events / (10 MeV/
+01000200030004000
+]2c M(KKK) [GeV/1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8)2c Events / (10 MeV/
+050100150
+(d)]2c M(KKK) [GeV/1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8)2c Events / (10 MeV/
+050100150
+]2c M(KKK) [GeV/1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8)2c Events / (10 MeV/
+050100150
+FIG. 10: The reconstructed invariant mass distributions: ( a)M(πππ) distribution for τ−→
+π−π+π−ντ, (b)M(Kππ) distribution for τ−→K−π+π−ντ, (c)M(KKπ) distribution for τ−→
+K−K+π−ντ, and (d) M(KKK) distribution for τ−→K−K+K−ντ. For all histograms, the black
+points with error bars represent the data, the gray histogra ms are the sum of all backgrounds other
+than the three-prong cross-feed, and the hatched histogram s are the sum of three-prong cross-feed
+backgrounds. Note that the cross-feed backgrounds are esti mated from the unfolding of the data.
+The uncertainties shown for the data are statistical only.
+After taking into account the backgrounds, the efficiencies and va rious sources of sys-
+tematic uncertainties discussed above, we obtain the following bran ching fractions for three-
+prong decays:
+21TABLE IV: Summary of the systematic uncertainties on the bra nching fractions (%).
+τ→ τ→ τ→ τ→
+Source πππν Kππν KKπν KKKν
+Tracking efficiency +2.2/−2.0+2.1/−2.0+2.1/−2.0+2.1/−1.9
+Particle ID ±1.9+4.0/−4.1±2.3+5.4/−5.6
+Mass spectrum ±0.1±0.1±0.1+0.5/−0.8
+Trigger efficiency ±0.5±0.5±0.6±0.6
+γveto ±0.9±1.5±1.7±0.5
+K0
+Sveto ±0.2±0.2±0.1±0.3
+Background estimation ±0.3±1.3±0.3±0.4
+Leptonic branching fraction ±0.2±0.2±0.2±0.2
+Total +3.1/−3.0±5.0±3.6+5.9/−6.0
+B(τ−→π−π+π−ντ) = (8.42±0.00(stat.)+0.26
+−0.25(sys.))×10−2,1
+B(τ−→K−π+π−ντ) = (3.30±0.01(stat.)+0.16
+−0.17(sys.))×10−3,
+B(τ−→K−K+π−ντ) = (1.55±0.01(stat.)+0.06
+−0.05(sys.))×10−3,and
+B(τ−→K−K+K−ντ) = (3.29±0.17(stat.)+0.19
+−0.20(sys.))×10−5.
+The covariance matrices for the branching fraction measurement s are given in Table V in
+terms of the correlation coefficients ρBF(i,j) = cov(i,j)//radicalbig
+cov(i,i)·cov(j,j), where iand
+jcorrespond to one of the four three-prong decay modes. There is a strong correlation
+betweenτ−→π−π+π−ντandτ−→K−π+π−ντdecays, which mostly comes from the effect
+of kaon identification.
+Using the results of the branching fractions and the correlation ma trix, the ratios of the
+branching fractions involving kaons to the branching fraction of τ−→π−π+π−ντdecay are
+1The actual value of the statistical uncertainty of B(τ−→π−π+π−ντ) is 0.003×10−2.
+22TABLE V: The correlation coefficients in the branching fracti on measurements.
+τ−→ τ−→ τ−→
+K−π+π−ντK−K+π−ντK−K+K−ντ
+τ−→π−π+π−ντ+0.175 +0 .049 −0.053
+τ−→K−π+π−ντ +0.080 +0 .035
+τ−→K−K+π−ντ −0.008
+estimated:
+B(τ−→K−π+π−ντ)/B(τ−→π−π+π−ντ) = (3.92±0.02+0.15
+−0.16)×10−2,
+B(τ−→K−K+π−ντ)/B(τ−→π−π+π−ντ) = (1.84±0.01±0.05)×10−2,and
+B(τ−→K−K+K−ντ)/B(τ−→π−π+π−ντ) = (3.90±0.20+0.22
+−0.23)×10−4,
+where the first and the second uncertainties are statistical and s ystematic, respectively.
+DISCUSSION
+The results of this analysis for the branching fractions of various t hree-prong modes are
+listed in Table VI together with recent results from BaBar [7]. Note th at for the τ−→
+π−π+π−ντandτ−→K−π+π−ντmodes, the branching fractions listed do not include any
+K0contribution.
+TABLE VI: Comparison of the branching fraction results
+Decay mode BaBar Belle
+τ−→π−π+π−ντ, % 8.83±0.01±0.138.42±0.00+0.26
+−0.25
+τ−→K−π+π−ντ, % 0.273±0.002±0.0090.330±0.001+0.016
+−0.017
+τ−→K−K+π−ντ, %0.1346±0.0010±0.00360.155±0.001+0.006
+−0.005
+τ−→K−K+K−ντ, 10−51.58±0.13±0.123.29±0.17+0.19
+−0.20
+In Fig. 11, our results are compared with the previous measuremen ts. For all modes
+studied, the precision of the branching fractions for both BaBar a nd Belle is significantly
+23higher than before. The accuracy of our results is comparable to t hat of BaBar, but the
+central values show striking differences in all channels other than τ−→π−π+π−ντ. For the
+τ−→π−π+π−ντmode, our result is 1 .4σlower than that of BaBar, while for other modes
+the branching fractions obtained by Belle are higher by 3 .0σ, 3.0σ, and 5.4σthan those
+of BaBar for the τ−→K−π+π−ντ,τ−→K−K+π−ντ, andτ−→K−K+K−ντmodes,
+respectively.
+CONCLUSION
+Using adatasampleof6 .12×108τ+τ−pairscollectedwiththeBelle detector, wemeasure
+the branching fractions for the τ−→π−π+π−ντ,τ−→K−π+π−ντ,τ−→K−K+π−ντ, and
+τ−→K−K+K−ντdecay modes. We have been able to extract unfolded invariant mass
+spectra for all four modes (Fig. 7) by taking into account the cros s-feed effects. A future
+detailed analysis of these spectra as well as those of the intermedia te two-body states will
+allow studies of the decay dynamics to be performed.
+ACKNOWLEDGMENTS
+We thank the KEKB group for the excellent operation of the acceler ator, the KEK cryo-
+genics groupfor theefficient operationofthesolenoid, andtheKEK computer groupandthe
+National Institute of Informatics for valuable computing and SINE T3 network support. We
+acknowledge support from the Ministry of Education, Culture, Spo rts, Science, and Tech-
+nology (MEXT) of Japan, the Japan Society for the Promotion of Sc ience (JSPS), and the
+Tau-Lepton Physics Research Center of Nagoya University; the A ustralian Research Council
+and the Australian Department of Industry, Innovation, Science and Research; the National
+Natural Science Foundation of China under contract No. 1057510 9, 10775142, 10875115 and
+10825524; the Department of Science and Technology of India; th e BK21 and WCU pro-
+gram (grant No. R32-2008-000-10155-0)of the Ministry Educat ion Science and Technology,
+the CHEP SRC program and Basic Research program (grant No. R01 -2008-000-10477-0)
+of the Korea Science and Engineering Foundation, Korea Research Foundation (KRF-2008-
+313-C00177), and the Korea Institute of Science and Technology Information; the Polish
+Ministry of Science and Higher Education; the Ministry of Education a nd Science of the
+248.5 9 9.5CLEO3 03 0.46) %±(9.13
+BABAR 08 0.13) %±(8.83
+This work 0.26) %±(8.42 decayν π π π → τ Branching ratio of 
+2 3 4 5 6-310×DELPHI 97-310×0.80)±(4.90
+ALEPH 98-310×0.47)±(2.14
+CLEO 99-310×0.61)±(3.46
+OPAL 00-310×0.95)±(3.60
+CLEO3 03-310×0.40)±(3.84
+OPAL 04-310×0.66)±(4.15
+BABAR 08-310×0.09)±(2.73
+This work-310×0.17)±(3.30 decayν π π K→ τ Branching ratio of 
+1 1.5 2-310×ALEPH 98-310×0.27)±(1.63
+CLEO 99-310×0.31)±(1.45
+OPAL 00-310×0.69)±(0.87
+CLEO3 03-310×0.11)±(1.55
+BABAR 08-310×0.04)±(1.35
+This work-310×0.06)±(1.55 decayν π KK→ τ Branching ratio of 
+0 10 20 30 40-610×CLEO3 03-510×< 3.70
+BABAR 08-510×0.17)±(1.58
+This work-510×0.26)±(3.29ALEPH 98-410×< 1.9 decayν KKK→ τ Branching ratio of 
+FIG. 11: Summary of the branching fraction measurements of t hree-prong τdecays.
+Russian Federation and the Russian Federal Agency for Atomic Ene rgy; the Slovenian Re-
+search Agency; the Swiss National Science Foundation; the Nation al Science Council and
+the Ministry of Education of Taiwan; and the U.S. Department of Ene rgy. This work is
+supported by a Grant-in-Aid from MEXT for Science Research in a Pr iority Area (”New
+Development of Flavor Physics”), and from JSPS for Creative Scien tific Research (”Evolu-
+tion of Tau-lepton Physics”).
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+26
\ No newline at end of file
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+arXiv:1001.0084v1  [gr-qc]  31 Dec 2009The Schr¨ odinger picture of standard cosmology
+N. Barbosa-Cendejas and M.A. Reyes
+Departamento de F´ ısica, Universidad de Guanajuato,Apdo. Postal E143, 37150 Le´ on,Gto., M´ exico
+November 4, 2018
+We consider a time independent Schr¨ odinger type equation d erived from the equa-
+tions of motion that drives a single scalar field in a standard cosmology model for
+inflation in a flat space-time with a Friedman-Robertson-Wal ker (FRW) metric with
+a cosmological constant. We find that all the 1-dimensional b ound state solutions of
+quantum mechanics lead to at least one exact solution for the dynamical equations
+of standard cosmology, and that these solutions resemble th e most recurrent infla-
+tionary solutions found in the literature. The analogies de rived from this approach
+may be used to realize a deeper understanding of the dynamics of the model.
+Keywords : Cosmology; Inflation; FRW; Schr¨ odinger.
+PACS: 02.30.Hq; 03.65.-w; 11.30.Pb
+1 Introduction
+The current approach of inflation is that it occurs at some stage in t he early universe and
+that its source is one or several scalar fields [1]-[2]. The different mod els of inflationary
+cosmology consider as a general feature a rapid growth of the size of the universe at
+some stage in the early universe [3]-[5], this simple definition of inflation may set the
+initial conditions for the large scale structure of the universe. In t his approach one must
+consider an arbitrary functional form for the scalar field (SF) pot entialV(φ), since there
+is no unique prescription or phenomenology that could help to determ ine it.
+Following this approach, let us consider a homogeneous and isotropic Universe, i.e., a
+model in a FRW background with a scalar homogenous field φ(t) minimally coupled to
+gravity and nonzero cosmological constant
+/integraldisplay
+d4x√−g/bracketleftbigg
+R+Λ+1
+2(∇φ)2+V(φ)/bracketrightbigg
+, (1)
+where (∇φ)2=gµν∂µ∂νandV(φ) is the potential energy of the field. In order to describe
+the dynamics of the scalar field during inflation the usual treatment is performed [1]-[6],
+finally leading to the pair of equations
+3H2=1
+2˙φ2+V(φ)+Λ (2)
+¨φ+3H˙φ=−dV(φ)
+dφ, (3)
+where dot means derivative with respect to time, and we set MPl= 1, ¯h=c= 1. The
+time derivative of eq.(2) is related to eq.(3) through the momentum e quation
+˙H=−1
+2˙φ2. (4)
+1With the use of eqs. (3) and (4) the dynamics of the model may be de scribed by the
+single equation:
+3H2+˙H=V(φ)+Λ. (5)
+which canberecognizedasaRiccatiequationfortheHubbleparame terH(t). TheRiccati
+equation has been one of the most useful equations of mathematic al physics, specially
+in supersymmetric quantum mechanics (SUSY QM.) Its appearance h ere immediately
+suggests a QM approach to inflationary cosmology based on the sec ond order differential
+equation derived from it. This has been proposed earlier, the ansat z being to replace xfor
+tand to assign a(t) =ψ(x), however leading to a nonlinear Schr¨ odinger equation [7]. In
+this article we propose an alternative transformation which leads to a linear Schr¨ odinger
+equation, where the SF potential V(φ) can be easily interpreted as the QM potential
+U(x), a simple scheme whose consequences seem not to have been explo red up to now.
+By simple algebraic comparisons we end up with a powerful method to d erive particular
+exact solutions that may be useful for understanding the inflation ary period. A similar
+approach for the case of cosmology with a perfect fluid has been pr oposed earlier, but in
+the context of classical mechanics ([11, 12].) However, the fact th at in our approach no
+restriction is made on the form of the SF potential allows us to probe a deeper connection
+between QM and inflationary cosmology.
+2 The Schr¨ odinger picture of Standard Cosmology
+2.1 A Schr¨ odinger type equation for inflation
+By defining ψ(t) through
+H≡1
+3˙ψ(t)
+ψ(t), (6)
+theRiccatiequation(5)canbetransformedintotheonedimension alSchr¨ odinger equation
+/bracketleftBigg
+−d2
+dt2+3V(t)/bracketrightBigg
+ψ(t) =−3Λψ(t), (7)
+we shall consider solutions to eq.(7) based only on the fact that the Hubble parameter
+H(t) cannot be a singular function, implying that ψ(t) has to be an at least C1class
+function without zeros, but without any other restriction. For ex ample, we may consider
+all ground state solutions of known exactly solvable bound state pr oblems in QM as
+solutions to eq.(7). Hence, an immediate equivalence arises between the SF potential
+V(φ) and the cosmological constant Λ, with the QM potential U(x) and ground state
+energy eigenvalue Eg,1respectively,
+3V(φ(t))+3Λ↔2U(x)−2Eg (8)
+1For simplicity we shall use m= 1 for the Schr¨ odinger particle.
+2This is indeed a very simple proposal, which shows that all the known ex actly solvable
+stationary problems of 1-dimensional QM must provide at least one e xact solution to the
+cosmological Schr¨ odinger type equation. The general algebraic p rocedure is very simple:
+for any given QM problem, use the ground state eigenfunction ψg(x) and eq.(6) to find
+H(t); then, use eq.(4) to find φ(t), which together with eq.(8) defines V(φ).
+For example, in the QM case of the simple harmonic oscillator (SHO), wh ereU(x) =
+ω2x2/2, the Schr¨ odinger equation
+/bracketleftBigg
+−d2
+dx2+2/parenleftBiggω2
+2x2−En/parenrightBigg/bracketrightBigg
+ψn(x) = 0. (9)
+has the wave functions and energy eigenvalues given by
+ψn(x) =/radicalBigg
+1
+2nn!/radicalbiggω
+πe−w
+2x2Hn(√ωx)En=/parenleftbigg
+n+1
+2/parenrightbigg
+ω (10)
+whereHn(y) are the Hermite polynomials. In the case n= 0, the Hermite polynomial is
+H0(√ωx) = 1, with energy and wave function
+E0=1
+2ω ψ 0(x) =4/radicalbiggω
+πe−ω
+2x2. (11)
+To construct the corresponding cosmological variables, we replac exbytin eq.(11), and
+use eq.(6) to find the associated Hubble’s parameter
+H(t) =−ω
+3t (12)
+and with the use of eq.(4) we obtain the expression of the scalar field
+φ(t) =/radicalBigg
+2w
+3t (13)
+Finally, we use eqs.(8) and (13) to find the SF potential V(φ) and the constant Λ,
+V(φ) =λφ2,Λ =−2
+3λ (14)
+whereλ=ω
+2. As one can see, the scalar field potential derived from the SHO pot ential
+turns out to be λφ2. Surprisingly, one of the most useful and basic potentials of QM
+transforms into one of the most useful potentials in this cosmologic al model (see [1], [13],
+and references there in.) It is even more surprising that other typ ical QM potentials
+resemble typical scalar field potentials in standard cosmology; for e xample, compare the
+results in [8] and [14] to the ones obtained in Table 1.
+32.2 Inflationary solutions from QM central force problems
+In Table 1 we have left some free parameters which can be used to fix the cosmological
+variables strength, even though their overall behavior is alreadyd etermined. For example,
+the cosmological solutions above may be set to correspond to the h alf plane solutions of
+the Schr¨ odinger problem, with the initial condition a(0)∝negationslash= 0 ift= 0 whenx= 0, and
+it is not possible to have as initial condition a= 0. Since in the half plane not only the
+QM ground state function is nodeless, but also the first excited sta te function is, we end
+up with two exact standard cosmology solutions with different initial c onditions derived
+from each 1-dimensional QM problem in the half plane t≥0, the second one with the
+initial condition a(0) = 0.
+Following this discussion, it is obvious that all ground state solutions t o the radial
+problem
+−d2unl
+dr2+2/bracketleftBigg
+U(r)+l(l+1)
+2r2−Enl/bracketrightBigg
+unl= 0 (15)
+are useful to derive exact solutions to the SF equations. Moreove r, since all eigenfunc-
+tions of the radial problems with angular momentum l=n−1 are nodeless, we end up
+with an infinity of exact solutions to the SF dynamics equations, para temeterized by a
+cosmological constant that belongs to the discrete set Λ n=−1
+3En, with the SF potential
+always including the centrifugal barrier term n(n−1)/t2, and with the initial condition
+a(0) = 0.
+As an example, let us consider the Hydrogen like potential U(r) =−Zq2/rwere all
+the nodeless eigenfunctions are given by
+un,n−1(r)∝rne−Zαr/n.
+where the fine structure constant αand the atom number Z, are constants only used to
+determine the strength of the cosmological variables. For the gro und state ( n= 1) the
+cosmological variables are
+H(t) =1
+3t−Zα
+3
+φ(t) =/radicalBigg
+2
+3lnt (16)
+V(φ) =−λe−√
+3/2φ
+Λ =λZ
+2
+whereλ= 2Zα2/3, while for n>1, they become
+H(t) =n
+3t−Zα
+3n
+φ(t) =/radicalBigg
+2n
+3lnt (17)
+4Schr¨ odinger picture Standard Cosmology
+ψ0(x) =4/radicalBig
+ω
+πe−ω
+2x2H(t) =−2λ
+3t,φ(t) = 2/radicalBig
+λ
+3t
+SHO E0=1
+2ω Λ =−2
+3λ
+U(x) =1
+2ω2x2V(φ) =λφ2,λ=ω
+2
+ψ0(x) = 8e−2e−αxe−3
+2αxH(t) =32
+3/radicalBig
+λ
+3/parenleftbigg
+e−16/radicalbig
+λ
+3t−3
+4/parenrightbigg
+,φ(t) =−4√
+3e−4/radicalbig
+λ
+3t
+MOR E0=−9
+8α2Λ = 16λ
+U(x) = 2α2(e−2αx−2e−αx) V(φ) =λφ4−32
+3λφ2,λ=3α2
+64
+ψ0(x) =1
+2λcosλ(αx) H(t) =−αλ
+3tan(αt),φ(t) =/radicalBig
+2λ
+3ln/radicalbigg
+1+sin(αt)
+1−sin(αt)
+PTT E0=α2λ2
+2Λ =−λ2α2
+3
+U(x) =α2
+2λ(λ−1)
+cos2(αx)V(φ) =α2
+3λ(λ−1)cosh2/parenleftBig/radicalBig
+3
+2λφ/parenrightBig
+ψ0(x) =1
+coshαxH(t) =−α
+3tanh(αt),φ(t) =/radicalBig
+2
+3arcsin(tanh( αt))
+PTH E0=−α2
+2Λ =α2
+3
+U(x) =−α2
+cosh2(αx)V(φ) =−2α2
+3cos2/parenleftBig/radicalBig
+3
+2φ/parenrightBig
+un,n−1(r) =(2α)3/2√
+2n4(2n−1)!/parenleftBig
+2αr
+n/parenrightBigne−αr/nH(t) =n
+3t−α
+3n,φ(t) =/radicalBig
+2n
+3lnt
+HA En=−α2
+2n2 Λ =α2
+3n2
+Ueff(r) =−α
+r+n(n−1)
+2r2 V(φ) =n(n−1)
+3/bracketleftbigg
+e−2/radicalbig
+3
+2nφ−2α
+n(n−1)e−/radicalbig
+3
+2nφ/bracketrightbigg
+Table 1: Standard Cosmology exact solutions from six Schr¨ odinger problems: simple
+harmonic oscillator (SHO), Morse potential (MOR), Trigonometric P ¨ oschl Teller (PTT),
+Hyperbolic P¨ oschl Teller (PTH), and Hydrogen Atom (HA).
+V(φ) =n(n−1)
+3/bracketleftBigg
+e2√
+3/2nφ−2Zα
+n(n−1)e−√
+3/2nφ/bracketrightBigg
+Λ =Z2α2
+3n2
+Hence, for n= 1 the SF potential becomes an exponential potential, while for n >1,
+if we choose Zα=n(n−1) it becomes a Morse potential. The appearance of this
+Morse potential is a very interesting feature of this model, since th is potential is very
+slowly varying for t→ ±∞, has a very soft minimum, and later exponentially grows for
+t→ ∓∞, the sign depending on the parameters. Therefore, this potentia l has all the
+desired features to allow a SF φ(t) slowly roll to the minimum of the SF potential V(φ)
+[9]. All these results are also included in Table 1.
+52.3 QM and Standard Cosmology analogies
+Looking at Table 1, the Schr¨ odinger picture of standard cosmolog y seems to be a fruitful
+approach to the construction of exact solutions to the inflationar y equations (2,3), since
+all potentials fromthese known QM problems resemble known SF pote ntials. It may seem
+that there must exist further analogies between these two models of the micro and macro
+cosmos than just an algebraic resemblance.
+In the present analogy ψ(t) =a3(t) describes the way the universe volume is expand-
+ing since in the Schr¨ odinger picture, ψ(x) is related to probability conservation, hence
+the equivalence proposed here points to energy density conserva tion in this expanding
+universe. On the other hand, the only constant term in the QM prob lem is the energy E,
+which therefore determines the cosmological constant Λ of eq.(7) , which could be associ-
+ated with the vacuum energy density. Therefore, the sign of Λ is co mpletely determined
+by the corresponding QM problem from which the SF solution is derived , becoming an
+immediate check for the dynamical characteristics that one wants to determine with the
+proposed SF potential.
+With respect to the scalar field φ(t), following eqs.(4), (5) and (8), we can see that
+wherever ˙a(t)≃0,
+φ(t)≃/integraldisplayt
+dy/radicalBig
+2(E−U(y)) (18)
+which resembles the action S(x) of the quantum theory.
+2.4 Slow Roll and WKB approximations
+One further analogy deserves special attention. Beginning with th e slow roll approxima-
+tion condition /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleV′
+V/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+<1
+where we should do the substitution V→V+Λ to comply with eq.(5), we can see that
+since ¨a/a>0 implies that V+Λ>˙φ2, we have that
+/vextendsingle/vextendsingle/vextendsingle˙V/vextendsingle/vextendsingle/vextendsingle2=/vextendsingle/vextendsingle/vextendsingle˙φV′/vextendsingle/vextendsingle/vextendsingle2<|V+Λ||V′|2<|V+Λ|3
+In our QM analogy, we would have to do the substitutionsdV
+dt→dU
+dxand|V+Λ| →
+|E−U|, giving/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledU
+dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+<|E−U|3
+which is just the WKB approximation
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingled2W
+dx2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle</vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledW
+dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+of the stationary problem, where W(x) =±/radicalBig
+2(E−U) is Hamilton’s principal function.
+63 An ever expanding universe
+All the QM problems considered here lead to two possible initial conditio ns for the scale
+factor,a(0) = 0 or a(0)>0, but with only one final condition, a(t)→0 ast→ ∞,
+if eqs.(2,3) may be used for all the half plane t≥0. If this Big Crunch could not be
+attainable, as observations seem to predict, our simple approach m ay still be useful to
+describe the expected dynamics.
+Ifa(t) is always increasing, then a QM bound state is not the right solution. However,
+as is depicted in Ref.[10], the Schr¨ odinger equation (7) has an infinite number of wave
+functions that diverge to ±∞ast→ ∞, for the continuum set of energies E, with
+En<E <E n+1, as is depicted in Fig.(1) for the case of the SHO.
+ta/LParen1t/RParen1
+Figure 1: Different forms of a(t) obtained from one QM problem.
+In Fig.(1) the dashed curves correspond to a(t) obtained from the ground state and
+first excited state eigenfunctions, ψ0(x) andψ1(x). These two curves have a(t)→0 as
+t→ ∞. On the other hand, the solid curves draw the scale factor for thr ee different
+cosmological constants, derived from the energy eigenvalues, E<E 0,E0<E<E 1and
+E1<E< E 2, whose wave functions diverge to + ∞,−∞and +∞again, the first one
+without nodes and the other with increasing number of nodes. Ther efore, only the wave
+function with E <E 0could lead to a physical solution a(t)>0 for allt, describing a
+cosmological solution for an ever expanding universe.
+4 Conclusion
+In this article we present a simple ansatz that links two completely sep arate models, one
+being a cosmological model for inflation of the macrocosmos, and th e other a quantum
+model for the microcosmos. Surprisingly, we have found that the m ost common potential
+functions in these two models map from one to the other, with proba bility conservation in
+7QM reflecting into energy density conservation in the cosmological m odel, a cosmological
+constant determined by the energy eigenvalue in the former, and t he WKB approximation
+reflecting into the slow roll approximation. All these analogies define deep connection
+between the two models, with respect to their dynamical behaviors . Finally, our initial
+proposal was to look at the bound state solutions of the QM problem s to describe a non
+singular Hubble variable for the cosmological model, but since presen t observations seem
+to indicate that the universe is not receding but on the contrary ac celerates with time,
+we may have to recur to the unbounded solutions of the Schr¨ oding er problem, supporting
+even more the validity of the simple analogy developed in this work.
+5 Acknowledgements
+We would like to thank M. Sabido for useful comments. We also ackwole dge support from
+CONACYT of Mexico, through a scholarship for NBC, and through gr ant No. SEP-2003-
+C0245364.
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+8[11] J. A. S. Lima, J. A. M. Moreira and J. Santos, Gen. Rel. Grav. 30, (1998) 425 .
+[12] J. A. S. Lima, Am. J. Phys. 69(2001) 1245
+[13] L. A. Urena-Lopez and M. J. Reyes-Ibarra, Int. J. Mod. Phy s. D18, 621 (2009).
+[14] T. Matos, J. R. Luevano, I. Quiros, L. A. Urena-Lopez and J. A. Vazquez, Phys.
+Rev. D80, 123521 (2009).
+9
\ No newline at end of file
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+arXiv:1001.0085v2  [math.FA]  20 Aug 2010Refinements of the trace inequality of Belmega, Lasaulce and
+Debbah
+Shigeru Furuichi1∗and Minghua Lin2†
+1Department of Computer Science and System Analysis,
+College of Humanities and Sciences, Nihon University,
+3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Jap an
+2Department of Mathematics and Statistics,
+University of Regina, Regina, Saskatchewan, Canada S4S 0A2
+Abstract. In this short paper, we show a certain matrix trace inequalit y and then give a
+refinement of the trace inequality proven by Belmega, Lasaul ce and Debbah. In addition, we
+give an another improvement of their trace inequality.
+Keywords : Matrix trace inequality and positive definite matrix
+2000 Mathematics Subject Classification : 15A45
+1 Introduction
+Recently, E.-V.Belmega, S.Lasaulce and M.Debbah obtained the following elegant trace inequal-
+ity for positive definite matrices.
+Theorem 1.1 ([1]) For positive definite matrices A,Band positive semidefinite matrices C,D,
+we have
+Tr[(A−B)(B−1−A−1)+(C−D)/braceleftbig
+(B+D)−1−(A+C)−1/bracerightbig
+]≥0. (1)
+In this short paper, we first prove a certain trace inequality for products of matrices, and then
+as its application, we give a simple proof of (1). At the same t ime, our alternative proof gives a
+refinement and of Theorem 1.1. An another improvement of the T heorem 1.1 is also considered
+at the end of the paper.
+2 Main results
+In this section, we prove the following theorem.
+Theorem 2.1 For positive definite matrices A,Band positive semidefinite matrices C,D, we
+have
+Tr[(A−B)(B−1−A−1)+(C−D)/braceleftbig
+(B+D)−1−(A+C)−1/bracerightbig
+]
+≥ |Tr[(C−D)(B+D)−1(A−B)(A+C)−1]|. (2)
+To prove this theorem, we need a few lemmas.
+∗E-mail:furuichi@chs.nihon-u.ac.jp
+†E-mail:lin243@uregina.ca
+1Lemma 2.2 ([1]) For positive definite matrices A,Band positive semidefinite matrices C,D,
+and Hermitian matrix X, we have
+Tr[XA−1XB−1]≥Tr[X(A+C)−1X(B+D)−1].
+Lemma 2.3 For any matrices XandY, we have
+Tr[X∗X]+Tr[Y∗Y]≥2|Tr[X∗Y]|.
+Proof: SinceTr[X∗X]≥0, bythefactthatthearithmetical meanisgreater thantheg eometrical
+mean and Cauchy-Schwarz inequality, we have
+Tr[X∗X]+Tr[Y∗Y]
+2≥/radicalbig
+Tr[X∗X]Tr[Y∗Y]≥ |Tr[X∗Y]|.
+Theorem 2.4 For Hermitian matrices X1,X2and positive semidefinite matrices S1,S2, we
+have
+Tr[X1S1X1S2]+Tr[X2S1X2S2]≥2|Tr[X1S1X2S2]|.
+Proof: Applying Lemma 2.3, we have
+Tr[X1S1X1S2]+Tr[X2S1X2S2]
+=Tr[(S1/2
+2X1S1/2
+1)(S1/2
+1X1S1/2
+2)]+Tr[(S1/2
+2X2S1/2
+1)(S1/2
+1X2S1/2
+2)]
+≥2|Tr[(S1/2
+2X1S1/2
+1)(S1/2
+1X2S1/2
+2)]|
+= 2|Tr[X1S1X2S2]|.
+Remark 2.5 Theorem 2.4 can be regarded as a kind of the generalization of P roposition 1.1 in
+[2].
+Proof of Theorem 2.1 : By Lemma 2.2, we have
+Tr[(A−B)(B−1−A−1)] =Tr[(A−B)B−1(A−B)A−1]
+≥Tr[(A−B)(A+C)−1(A−B)(B+D)−1]
+=Tr[(A−B)(B+D)−1(A−B)(A+C)−1].
+Thus the left hand side of the inequality (2) can be bounded fr om below:
+Tr[(A−B)(B−1−A−1)+(C−D)/braceleftbig
+(B+D)−1−(A+C)−1/bracerightbig
+]
+≥Tr[(A−B)(B+D)−1(A−B)(A+C)−1+(C−D)(B+D)−1(C−D)(A+C)−1]
++Tr[(C−D)(B+D)−1(A−B)(A+C)−1]
+≥2|Tr[(C−D)(B+D)−1(A−B)(A+C)−1]|
++Tr[(C−D)(B+D)−1(A−B)(A+C)−1] (3)
+Throughout the process of the above, Theorem 2.4 was used in t he second inequality. Since we
+have the following equation,
+Tr[(C−D)(B+D)−1(A−B)(A+C)−1]
+=Tr[(C−D)(B+D)−1]−Tr[(C−D)(A+C)−1]
+−Tr[(C−D)(B+D)−1(C−D)(A+C)−1]
+we have Tr[(C−D)(B+D)−1(A−B)(A+C)−1]∈R. Therefore we have
+(3)≥ |Tr[(C−D)(B+D)−1(A−B)(A+C)−1]|.
+23 An another improvement of the inequality (1)
+In this section, we show the following trace inequality.
+Theorem 3.1 For positive definite matrices A,Band positive semidefinite matrices C,D, we
+have
+Tr[(A−B)(B−1−A−1)+4(C−D)/braceleftbig
+(B+D)−1−(A+C)−1/bracerightbig
+]≥0. (4)
+To prove this theorem, we use the following lemmas, which are proven by the similar way of
+Lemma 2.3 and Theorem 2.4 in the previous section.
+Lemma 3.2 For any matrices XandY, any positive real numbers aandb, we have
+a·Tr[X∗X]+b·Tr[Y∗Y]≥2√
+ab·|Tr[X∗Y]|.
+Applying this lemma, we have the following lemma.
+Lemma 3.3 For Hermitian matrices X1,X2, positive semidefinite matrices S1,S2and any pos-
+itive real numbers aandb, we have
+a·Tr[X1S1X1S2]+b·Tr[X2S1X2S2]≥2√
+ab·|Tr[X1S1X2S2]|.
+Proof of Theorem 3.1 : By the similar way to the proof of Theorem 2.1, applying Lemm a 3.2
+asa= 1 and b= 4, the left hand side of the inequality of (4) can be bounded f rom the below:
+Tr[(A−B)(B−1−A−1)+4(C−D)/braceleftbig
+(B+D)−1−(A+C)−1/bracerightbig
+]
+≥Tr[(A−B)(B+D)−1(A−B)(A+C)−1+4(C−D)(B+D)−1(C−D)(A+C)−1]
++Tr[4(C−D)(B+D)−1(A−B)(A+C)−1]
+≥4|Tr[(C−D)(B+D)−1(A−B)(A+C)−1]|
++4·Tr[(C−D)(B+D)−1(A−B)(A+C)−1]≥0,
+sinceTr[(C−D)(B+D)−1(A−B)(A+C)−1]∈R.
+Remark 3.4 Here we note that we have Tr[(A−B)(B−1−A−1)]≥0. However we have
+the possibility that Tr[(C−D)/braceleftbig
+(B+D)−1−(A+C)−1/bracerightbig
+]takes a negative value. Therefore
+Theorem 3.1 is an improvement of Theorem 1.1.
+Corollary 3.5 For positive definite matrices A,B, positive semidefinite matrices C,Dand pos-
+itive real number r, we have
+Tr[(A−B)(B−1−A−1)+4(C−D)/braceleftbig
+(rB+D)−1−(rA+C)−1/bracerightbig
+]≥0. (5)
+Proof: PutA=rA1andB=rB1for positive definite matrices A1andB1, in Theorem 3.1.
+Remark 3.6 In the case of r= 2in Corollary 3.5, the inequality (5) corresponds to the scala r
+inequality:
+(α−β)/parenleftbigg1
+4β−1
+4α/parenrightbigg
++(γ−δ)/parenleftbigg1
+2β+δ−1
+2α+γ/parenrightbigg
+≥0
+for positive real numbers αandβ, nonnegative real numbers γandδ.
+3References
+[1] E.-V. Belmega, S.Lasaulce andM. Debbah, Atrace inequal ity forpositivedefinitematrices,
+J. Inequal. Pure and Appl. Math., Vol.10,No.1(2009), Art. 5 , 4 pp.
+[2] S.Furuichi, K.Kuriyama and K.Yanagi, Trace inequaliti es for products of matrices, Linear
+Alg.Appl., Vol.430(2009),pp.2271-2276.
+4
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+arXiv:1001.0087v3  [nucl-th]  21 Feb 2012Statistical hadronization phenomenology in K/π
+fluctuations at ultra-relativistic energies
+Giorgio Torrieria,Rene Bellwiedb,Christina Markertc,Gary
+Westfalla,d
+aFIAS, J.W. Goethe Universit¨ at, Frankfurt A.M., Germany.
+bWayne State University, Detroit, Michigan 48201, USA
+cUniversity of Texas at Austin, Austin, Texas 78712, USA
+dMichigan State University, East Lansing, Michigan 48824, U SAStatistical hadronization phenomenology in K/πfluctuationsat ultra-relativistic energies 2
+Abstract. We discuss the information that can be obtained from an analy sis
+of fluctuations in heavy ion collisions within the context of the statistical model
+of particle production. We then examine the recently publis hed experimental
+data on ratio fluctuations, and use it to obtain constraints o n the statistical
+properties (physically relevant ensemble, degree of chemi cal equilibration, scaling
+acrossenergies and system sizes)and freeze-out dynamics( amount ofreinteraction
+between chemical and thermal freeze-out) of the system.
+The idea ofmodelling the abundanceof hadronsusing statisticalmec hanics techniques
+has a long and distinguished history [1, 2, 3, 4]. In a sense, any discu ssion of
+the thermodynamic properties of hadronic matter (e.g. the existe nce of a phase
+transition) requiresthat statistical mechanics be applicable to this system ( through
+not necessarily at the freeze-out stage).
+Thatsuchamodelcandescribe quantitatively theyieldofmostparticles,including
+multi-strange ones, has in fact been indicated by fits to average pa rticle abundances
+at AGS,SPS and RHIC energies [5, 6, 7, 8, 9, 10].
+There are, however, several points of contention to this unders tanding: Some
+practitioners,startingfrom[1]interpretthestatisticalmodelre sultsintermsofnothing
+more than phase space dominance: For a process strongly enough interacting with
+enough particles in the final state, dynamics “factors out” into a n ormalization
+constant, and the final state probabilities are dominated by phase space. If this is
+the case, the applicability of the statistical model has nothing to do with a genuine
+equilibration of the system.
+Others [5, 11] believe that the applicability of the statistical model is a sign of a
+phase transition, as the chemical equilibration of hadrons signals a r egime in which
+multi-particle processes and high-lying resonances dominate.
+Still others think that in soft QCD processes particles are “born in e quilibrium”
+[12], and the applicability of the statistical model to even smaller syst ems (itself
+contested [13]) is a fundamental characteristic of QCD.
+How can these scenarios be differentiated? One observable “at the heart” of
+statistical mechanics is event-by-event fluctuations [14, 15]. It is a fundamental
+principle of statistics that variances around averages scale a cert ain way w.r.t.
+averages. In our context “Averages” are particle multiplicities per event and
+fluctuations are event-by-event fluctuations. For macroscopic systems, this principle
+ensures that fluctuations become negligible and the expectation th at the state of the
+system is the maximum entropy one is nearly certain to be realized.
+O(100−1000) particles is not enough for this to be the case, but, if statist ical
+mechanics applies, one should still see that yields, fluctuations and h igher cumulants
+scale in a way calculable from the partition function. There are,howev er, some
+experimental issues specific to heavy ion collisions that need to be ex plored before
+this can be transformed into a quantitative test.
+“standard” fluctuation calculations assume that the volume ‡of the system is
+fixed. Obviously, in heavyioncollisions, the system is notin a box, so th is requirement
+‡The applicability and definition of the term “volume” is itse lf problematic in a dynamical quantum
+system. Statistical particle production assumes a volume, but leaves unexplained how this volume is
+defined, eg if acceptance cuts can be considered as “cuts in vo lume”. If particles are created with a
+boost-invariant flow, vz=z/t, and the thermal parameters are approximately independent of rapidity,
+than cuts in rapidity are equivalent to cuts in volume. No oth er kinematic cuts, eg cuts in pTorφ,
+can be reliably treated in this way, although the effect of the se cuts on fluctuations can be partially
+cancelled out by mixed event subtraction, as discussed late r in this workStatistical hadronization phenomenology in K/πfluctuationsat ultra-relativistic energies 3
+has no reason to hold. The most straight-forward way around this is to construct
+an ensemble where volume can fluctuate while its canonical conjugat e, the pressure,
+stays fixed [16]. This, however, can also present phenomenological difficulties, since
+the origin of volume fluctuations in heavy ion collisions is not necessarily statistical,
+but also geometric (initial state fluctuations) and dynamical (non- zero Knudsen
+number [17]). Statistical mechanics, therefore, must include the se fluctuations in the
+constraints, since it has no ability to describe them.
+Due to our incomplete understanding of how the initial state and dyn amics
+contribute to fluctuations at freeze-out, the best approach is t o choose an observable
+which is insensitive to any volume fluctuations. In the thermodynamic limit, where
+volume becomes a proportionality constant at the level of the part ition function, a
+tempting observable is the scaled variance of the ratioof two particle multiplicities
+measured event by event. That this is in fact a good guess can be se en at the level of
+particle distributions. Assuming we are in the thermodynamic limit, the distribution
+should separate into a component sensitive to volume Vand the other on density,
+N1/V
+F1,2(N1,2) =/integraldisplay
+g(V)f1,2/parenleftbiggN1,2
+V/parenrightbigg
+dV (1)
+heref1,2depend on thermodynamicsand g(V) oninitial geometryand dynamics (note
+that we left it as a general function, and in general we do not really k now what this
+function is). Now, the probability distribution function of a ratio is
+F/parenleftbigg
+R=N1
+N2/parenrightbigg
+=/integraldisplay
+F1(N1)F2(N2)δ/parenleftbiggN1
+N2−R/parenrightbigg
+(2)
+Expanding and substituting α=VN2we get
+F(R) =/integraldisplay
+g(V)2VdV
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+Independent of R/integraldisplay
+f1(αR)f2(α)dα (3)
+hence,N=/integraltext
+g(V)2VdVappearsequallyinallcumulants.
+Thus, quantities such as σN1/N2arestrictly independent of any volume
+fluctuations §. Note that this is not true, for example, for the Kurtosis, recent ly
+proposed as signatures for the critical point [19], since the Kurtos is is not the ratio of
+twocumulants. Theprocedureabove,however,makesiteasytod erivetheappropriate
+volume-independent observable encoding higher cumulants of ratio s (µ4/σof the
+distribution of a ratio, for example, would be volume independent. He reµ4is the
+fourth cumulant and σthe fluctuation).
+The residual dependence of σN1/N2on the average volume /angbracketleftV/angbracketrightcan be in turn
+eliminated, in the grand canonical ensemble, by focusing on Ψ = /angbracketleftN1/angbracketrightσN1/N2
+dyn,
+where/angbracketleftN1/angbracketrightandσdynare to be measured within the same acceptance . Note that
+this independence is specific to the grand-canonical ensemble, so s hould not apply
+to scenarios where the “enhancement of strangeness” in A-A collis ions is due to the
+transitionbetween the canonicallimit in elementaryprocessesand t he grandcanonical
+limit in A-A [20]. In this scenario, the “strangeness correlation volume ” (Vcorrnot in
+general equal to the system volume) should regulate Ψ as in Fig. 1 [21 ]
+§[18] section IV-C makes the opposite claim based on an expans ion around ∆ V//angbracketleftV/angbracketright. The flaw in
+that procedure is that it can not be determined weather terms such as ∆ V//angbracketleftV/angbracketright2are to be included
+in the fluctuation or in higher cumulants. The transport anal ysis in [18] validates our derivation,
+since strictly nodependence of ratio fluctuations on multiplicity fluctuatio ns is foundStatistical hadronization phenomenology in K/πfluctuationsat ultra-relativistic energies 4
+Thermodynamic limitsmall Vcorr
+Figure 1. (left) Fluctuations dependence in canonical scenario [21] (right)
+Strangeness fluctuations in the canonical scenario [20]
+A different problem is the effect of a detectors limited acceptance ( P article
+(mis)identification, Limited rapidity and momentum resolution, momen tum cuts
+necessary to eliminate jets etc) on fluctuation observables. Thes e are much more
+difficult to model than averages,and once again an observable need s to be constructed
+insensitivetothem. Hence, thenecessityofmixedeventsubtract ion[22]. Mixedevents
+here, are definedas events where no physical correlation from the original event are
+left in. This means that any correlation seen is due to the imperfectio n of the detector
+(the fact that detector acceptance excludes particles of pT>1 GeV for both event A
+and B creates a correlation between AandB). To a good approximation, examined
+in the next paragraph, the measured fluctuation is σ2=σ2
+physics+σ2
+acceptance and
+the mixed event one is σ2
+mix=σ2
+trivial+σ2
+acceptance . Therefore, concentrating on
+σ2
+dyn=σ2−σ2
+mixshouldeliminate acceptance effects.
+Two considerations are needed at this point: The first is that this me thod is
+not perfect, since limited acceptance spoils not just fluctuations b ut correlations, and
+this will notbe corrected with mixed event subtraction. For example, in a narro w
+acceptance detector, resonance kinematics modifies σ2
+physics(weakens the correlation
+due to the resonance) rather than introducing an additional σ2
+acceptance (HBT would
+be another source of such physical correlations, although the sm allness of the relevant
+relative momentum makes this largely irrelevant here). There is no pe rfect solution:
+One empirical fix is to vary the acceptance, until its large enough th at Ψ reaches an
+asymptotic limit Fig. 2. This limit is the physical value of Ψ. Such a method is
+not perfect, but it relies on the fact that, for a good event mixing p rocedure (defined
+above) limited acceptance should only destroycorrelations, not createthem. The
+highest value of the correlation seen by varying the detector acce ptance is therefore
+the physical one.Statistical hadronization phenomenology in K/πfluctuationsat ultra-relativistic energies 5
+The second consideration is that, by the definition above ( no physical correlations
+in mixed events), the current algorithms used by experimental colla borations [23, 24]
+for mixed event definition are notperfect. For example, the total multiplicity of
+mixed events is determined according to the experimentally measure d multiplicity
+distribution, containing all correlations of real events (eg, the π+π−correlation
+inducedbythe ρ). Sincetheabundanceofallspeciesaregreaterinahighermultiplicit y
+event, the autocorrelations contained in the experimentally measu red multiplicity
+distribution will translate into a correlation in the mixed event ratio of anytwo
+particles (see Fig 3 for an illustration). This correlation is physical ra ther than
+acceptance-induced, and hence should not be there in a “good” mix ed event.
+A more sound procedure,therefore, would be to normalize π,Kand protons
+separately and construct the mixed event bottom-up. In other words, for e ach mixed
+eventπ,Kand p multiplicities should be determined according to the respective
+experimentally measured multiplicity distributions (uncorrected for acceptance).
+Then the event should be constructed (the appropriate number o fπ,Kand proton
+tracks obtained from different events). As physical autocorrela tions within the the
+same particle species of relative momenta larger than O(100)MeV are negligible, this
+procedure should yield a much better (according to the above defin ition) mixed event
+sample. We do not think the multiplicity auto-correlation in mixed event s is large,
+but it is there, and it could be responsible for the deviation between s tatistical model
+and the data shown in in the next paragraphs. It will be interesting t o see how
+implementing the mixed event definition presented here will change th e results.
+corrections neededAcceptance∆Larger   y∆
+Ψ∼SHM valueSmall   y
+y∆Ψ
+Acceptance 
+corrections small
+Figure 2. Correlation as a function of acceptance
+We now proceed to discuss specific results and conclusions that can be obtained
+from a study of fluctuation data. We refer to [5, 6, 7, 8, 9, 10, 15 ] for a review of the
+statistical model. For this work, it is sufficient to repeat that the pa rticle abundances
+and fluctuations can be calculated from the first and second deriva tive’s a particle’s
+partition function. In the Grand-Canonical limit, this partition func tion is strictly
+proportional to volume. Because of this, the fluctuation/angbracketleftbig
+(∆N)2/angbracketrightbig
+and the yield /angbracketleftN/angbracketright
+should both scale linearly with volume, and hence the fluctuation of a r atio
+σN1/N2=/angbracketleftbig
+(∆N1)2/angbracketrightbig
+/angbracketleftN1/angbracketright2+/angbracketleftbig
+(∆N2)2/angbracketrightbig
+/angbracketleftN2/angbracketright2−/angbracketleft∆N1∆N2/angbracketright
+/angbracketleftN1/angbracketright/angbracketleftN2/angbracketright+O/parenleftBigg
+∆N
+/angbracketleftN/angbracketright2/parenrightBigg
+(4)
+should scale inversely to the volume /angbracketleftV/angbracketright(and be strictly independent of higher
+cumulants of V, by the earlier derivation). Hence, provided the chemical paramet ers
+do not change across systems, ΨN1
+N1/N2=/angbracketleftN1/angbracketrightσN1/N2
+dynshould be strictly independent
+of centrality and system size. This requirement is not satisfied by th e SPS scan [24]Statistical hadronization phenomenology in K/πfluctuationsat ultra-relativistic energies 6
+N1
+N2
+Figure 3. The experimental multiplicity measure used to normalized m ixed
+events (left panel) and an example of the systematic correla tion this procedure
+generates (right panel) in a ratio between two generic N1andN2particles
+0 100 200 300 400 500600 700 800
+dN/dy00.10.2K*/K-
+From fluctuations
+DirectK/πSHM expectation
+for (dN/dy) σdyn
+q,s
+q,sγ  >1
+γ  =1T=140 MeV 
+T=170 MeV 
+Figure 4. Left panel: Scaling of K/πfluctuations at RHIC 200 GeV. Right
+panel:K∗/Kimplied from fluctuations and directly measured. Star fluctu ations
+(left) and resonances (right) results, taken respectively from [23, 27]
+discussed in the last part of the paper, since there µ/Tdoes vary considerably /bardbl. It is
+however satisfied by the RHIC upper energies. Thus, the Grand-C anonical statistical
+model predicts a flat dependence with system size at these energie s. The canonical
+model, on the other hand, would predict a “kink” within the same cent rality as when
+the strangeness correlation volume approaches the thermodyna mic limit (Fig. 1).
+The results can be seen on Fig. 4 left panel. None of the models come o ut
+perfectly, but the Grand-Canonical model is qualitatively much mor e similar to the
+data than the canonical one, as no kink is visible there. The discrepa ncy in Ψ between
+200 GeV and 62 GeV, the slight upward trend in centrality, and the lac k of scaling
+between A-A and Cu-Cu should however be closely watched, as the s light increase of
+fluctuations with multiplicity can not be accounted for by anystatistical model (Since
+we are dealing with ratio fluctuations, a volume fluctuation tuned to K NO scaling, as
+in [25], will not help here).
+Before passing judgement on this topic, however, it will be interest ing to see how
+muchofthistrendisduetothepreviouslydescribedmultiplicitycorre lationretainedin
+mixed events. Correcting for this effect using an improved mixing pro cedure would go
+in the right direction (since physical correlationsdue to volume would not be cancelled
+/bardblA scan in centrality and system size within the same energy sh ould however produce the same
+approximately horizontal bands in Ψ as are seen at RHIC: Each value of√s, and hence µ/T, should
+produce a band when scanned in centrality and system sizeStatistical hadronization phenomenology in K/πfluctuationsat ultra-relativistic energies 7
+out). An analysis is ongoing to determine their extent. The scaling of fluctuations
+between RHIC and the LHC, where µ/Tis virtually the same, should provide further
+light on this.
+Quantitatively, as reported earlier, Ψ is modelled much better with th e inclusion
+of the light-quark non-equilibrium parameter γq, due to Bose-Einstein enhancement
+of fluctuations [14]. It remains to be seen weather the measuremen t of fluctuations of
+moreparticles (Fig. 5 left panel) will corroborate this conclusion.
+We now turn to a potentially important model-independent use of fluc tuations,
+the constraint on resonance reinteraction between chemical fre eze-out and thermal
+freeze-out [26]. The former can be estimated, to a good approxima tion, by
+comparing the fluctuations of same-charged particles (uncorrela ted) and opposite
+charged particles (correlated by K∗)
+K∗0
+K−≃3
+4/parenleftBig
+Ψπ−
+K−/π−−Ψπ−
+K+/π−/parenrightBig
+(5)
+Note that both Ψπ−
+K−/π−and Ψπ−
+K+/π−are equally affected by the volume correlations
+present in mixed events, so these should cancel out when the differ ence is taken.
+Comparing the fluctuation estimate ofK∗0
+K−to a direct measurement should yield
+the amount of K0∗destroyed by rescattering or regenerated through pseudo-ela stic
+interactions.
+The results are shown in Fig. 4 (right panel, resonance values are ta ken from
+[27]). Rescattering and regeneration have,within error bar little or n o effect on the
+final abundance of K∗s. Either chemical and thermal freeze-out proceed very close
+to each other, so the amount of reinteraction is negligible (as was co ncluded in [28]
+based on [29, 30]), or rescattering and regeneration of detectab le resonances cancel
+each other out to the degree of approximation allowed by the error bars (10 −15%).
+This margin appears somewhat below the estimates from transport models, which are
+∼40% [31, 32]. This discrepancy is not too surprising, since, within the f ramework
+of a collectively expanding system, a balance between rescattering and regeneration is
+unnatural: If the number of reinteractions is small, rescattering g enerally dominates
+as was assumed in [29, 30]. If the number of rescattering is large, de tailed balance
+would mean K∗/Kwould reequilibrate from the higher chemical freeze-out to the
+lower thermal freeze-out temperature.
+It would therefore be very interesting to investigate weather a tr ansport model
+such as uRQMD [33] or HSD [18] could be tuned to simultaneously repro duce the
+fluctuation inference and direct measurement ofK∗0
+K−.
+Unfortunately, the baroqueness of the resonance decay tree s everely limits the
+feasibilness of such graphic methods, as the right panel of Fig 5 sho ws.K,πis a
+special pair of particles in that there is only one type of resonances that decays into
+both, the K∗, and its lightest state is considerably lighter than the heavier ones.
+No similar definition is possible for Σ∗/Λ,ρ/πandφ/K, since the resonance decays
+equally into all pairs of decay products. For other resonances, cr oss-contamination
+destroys any value of fluctuations as a graphictool of reinteraction time.
+This does not mean, however, that such resonances are useless f or constraining
+reinteraction time, since both fluctuations [14] and resonances [29 , 30] provide
+a tight constraint on all statistical models. If a statistical model c onsistently
+describes fluctuations, but over-predicts resonance abundanc es, it could be taken as
+an indication of a long reinteraction times with detailed balance. A casu al look atStatistical hadronization phenomenology in K/πfluctuationsat ultra-relativistic energies 8
+experimental data [27] shows this does not seem to be the case, sin ce most resonances
+areunder-predicted by statistical models.
+00.10.20.30.40.5
+K*/K-
+Ξ∗0/Ξ Λ∗/Λ φ/K-∆++/p Σ∗+/Λ ρ0/π−-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.8γq=1.,T=160 MeV
+γq=1.6,T=140 MeV
+Ψp,π+Ψp,Κ− ΨΛ,π+ΨΞ0,π+ ΨK+/K- Ψπ+,π−ΨΝ1,Ν2=Ν2(σdyn
+N1/N2)2
+ΨK-,π−ΨK+,π− ΨΞ0,π− Ψp,Κ+Resonances
+Fluctuations
+Figure 5. Equilibrium and non-equilibrium predictions from fluctuat ions (left
+panel) and resonances (middle panel) with the statistical p arameters taken from
+[39] (The γq= 1 values are similar to [5]). In the right panel, Full symbol s show
+theN∗/Nratio inferred from the correlation in the two models, while empty
+symbols show, wherever possible, the estimate from σdyncomparisons
+We now turn to the energy scan seen at SPS energies. We first warn the reader
+not to take any quantitative statements we make here too serious ly, since to study
+the scaling with volume requires the same acceptance region from all particles (see
+footnote on page 2), as well as for the yield and fluctuation measur ements. The
+analysis in [24] does not meet this requirement, since the acceptanc e forKandπis
+considerably different, and the acceptance for yields is different st ill (note that the
+earlier analysis in [34] failed to consider this). We await for measureme nts where this
+issue is resolved (for experimental reasons they are easier to per form at collider energy
+scans [35, 36, 37]) to see how big it is of an issue for the SPS results.
+6 8 10 12 14 16 18 20
+s1/2 (GeV)00.20.40.60.8 α Non-equilibrium, scaled from lowest energy
+Equilibrium, scaled from lowest energy
+Non-equilibrium, scaled from highest energy
+Equilibrium,scaled from highest energy
+s1/2 (GeV)05101520σdynamical (%)Data (Na49 and STAR)
+Poisson Scaling
+Part. Number Scaling
+γq,s fitted
+γ,s=1
+Geometric Scaling
+68 10 12 14 1618 20
+s1/20.011100Yield in fluctuation acceptanceΛ(1520)K*(892)Λ
+φp
+Ξ−
+Ω−
+Figure 6. Leftand center panel: Scaling of K/πfluctuations with yields. The left
+panel shown the αexponent, defined as in Eq. 6. Middle panel: The fluctuations
+observed and calculated with such scaling. Right panel: Yie lds calculated in the
+equilibrium and non-equilibrium models assuming the norma lization used in the
+respective calculations corresponds to the physical volum e
+[38] has done a very useful job of showing that the scaling of fluctu ations with√s
+reflects, to a good approximation, that of yields. Of course, in the statistical model an
+exactscaling of σK/π
+dynwith/angbracketleftK/angbracketright,/angbracketleftπ/angbracketrightin different energy systems is more complicatedStatistical hadronization phenomenology in K/πfluctuationsat ultra-relativistic energies 9
+than any of the models studied in [38], since it reflects changes in T,µand possibly
+γq,swith√s. To try to see how the statistical model fits within the analysis of [38 ]
+we parametrize the best fit set of parameters in [39, 40, 41] and [6] (in the second case
+T,µB,in the first T,µB,γs,qusing the exponent α
+/angbracketleftπ/angbracketright1/2−α/angbracketleftK/angbracketright1/2+ασK/π
+dyn=C=/angbracketleftπ/angbracketright1/2/angbracketleftK/angbracketright1/2σK/π
+dyn/vextendsingle/vextendsingle/vextendsingle
+Reference(6)
+whereCis a constant adjusted from data (Note that all volume dependenc e cancels
+out in the definition of α, as expected in the statistical model). αis simply extracted
+out of the SHARE estimates [15]
+α=lnC−lnσK/π
+dyn−(lnπ+lnK)/2/vextendsingle/vextendsingle/vextendsingle
+SHARE
+ln/angbracketleftK/angbracketright−ln/angbracketleftπ/angbracketright|SHARE(7)
+We have performed this procedure with SPS data, taking the param eters from the
+equilibrium and non-equilibrium values of the analysis in [39], and using the /angbracketleftK/angbracketrightand
+/angbracketleftπ/angbracketrightexpectations from [38] to fix the normalization.
+The result is seen in Fig. 6 left panel, with full symbols using the lowest
+SPS energy as reference and empty symbols using the maximum SPS e nergy. The
+conclusions to be drawn from this exercise are two-fold: The first is that none of the
+scalings examined in [38] is really compatible with the statistical model. T he rapid
+changein µB,γq,sreallymakesthestatisticalscalingacross√snon-trivial. Thesecond
+is that the scaling is still close enough to Poissonian that the depende nce of the fit
+parameters on√swashes away if just K,πare considered. Note that, however, as Fig
+6 shows, these scaled values will be somewhat different from the SHA RE prediction.
+Note, however,that in this workwe did not considerabundancesof particlesother
+thanK,π. Since, as shown in [14], a yield and a fluctuation are enough to remove any
+ambiguity between Tandγq. Had the acceptance window considered here been the
+same for /angbracketleftK/angbracketrightandπmeasurements, this would yield a tight prediction for the value of
+protonsandhyperons. Takingthe normalizationconstantwefitte d with [15](We want
+to emphasize this is nota reliable assumption, we did this for illustrative purposes) we
+computed the particle yields using the equilibrium and non-equilibrium pa rameters
+in [39, 40, 41, 6]. The results are shown in the right hand panel of Fig. 6. As can
+be seen, a quantitative test of both yields and fluctuations differen tiates between the
+equilibrium and non-equilibrium model much more than a test based on y ields and
+fluctuations alone. We look forward, therefore, to perform this a nalysis on data-sets
+where the acceptance of the two particles in the ratio is the same, a nd hence protons
+and hyperons can also be measured within this acceptance.
+In conclusion, we have provided an overview of the current results of fluctuations
+of particle ratios, and tried to interpret them within the framework of the statistical
+model. None of the models considered here does a perfect job of de scribing the
+data, through models based on Grand-Canonicalensembles bette r capture the scaling,
+and the introduction of γqbring the theory closer to experiment. A different mixed
+event subtraction is however needed to confirm these results. Co mparing the effect
+of resonances inside particle fluctuations with the directly measure d resonances shows
+that the resonance abundance present at chemical freeze-out seems to remain until the
+final decoupling, in contrast to current estimates from transpor t models. While the
+approximate Poissonian scaling found in the data is expected of the s tatistical model
+for all thermal values, the strong variation of µ/Twith√sprevents an exact scaling
+with energy. A centrality scan at lower√s, as well as a uniform acceptance for bothStatistical hadronization phenomenology in K/πfluctuationsat ultra-relativistic energies 10
+particles in the measured ratio are necessary before the scaling of fluctuations with
+energy can be unamibiguously interpreted.
+This work was supported in part by the Offices of NP and HEP within the
+U.S. DOE Office of Science, the U.S. NSF. G.T. acknowledges the financ ial support
+receivedfromtheHelmholtzInternationalCenterforFAIRwithint heframeworkofthe
+LOEWE program (Landesoffensive zur Entwicklung Wissenschaftlich -¨Okonomischer
+Exzellenz) launched by the State of Hesse. G.W. acknowledges the fi nancial support
+given by the Alexander Von Humboldt foundation research award, a nd the hospitality
+provided by the JW Goethe Universitat during the time this work was p erformed. We
+thank M.Hauer and Tim Schuster for very helpful discussions, and T im Schuster and
+Volker Koch for providing the file used to make Fig. 6.
+[1] E. Fermi Prog. Theor. Phys. 5, 570 (1950).
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+[6] J. Cleymans, H. Oeschler, K. Redlich and S. Wheaton, arXi v:hep-ph/0607164.
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+Nucl. Phys. Cosmol. 18, 1, and references therein
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+Phys. Commun. 167, 229 (2005)
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+024907 (2009) [arXiv:0811.3089 [nucl-th]].
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+[20] S. Hamieh, K. Redlich and A. Tounsi, Phys. Lett. B 486, 61 (2000) [arXiv:hep-ph/0006024].
+[21] V. V. Begun, M. Gazdzicki, M. I. Gorenstein and O. S. Zozu lya, Phys. Rev. C 70, 034901 (2004)
+[22] C. Pruneau, S. Gavin and S. Voloshin, Phys. Rev. C 66, 044904 (2002) [arXiv:nucl-ex/0204011].
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+[hep-ph]].
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+[arXiv:nucl-ex/0604019].
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+069902 (2002)] [arXiv:nucl-th/0104042].
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+[32] S. Vogel, J. Aichelin and M. Bleicher, arXiv:0908.3811 [hep-ph].
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+PhD thesis
+[34] G. Torrieri, Int. J. Mod. Phys. E 16, 1783 (2007) [arXiv:nucl-th/0702062].
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\ No newline at end of file
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+ 1  Three +1 Faces of Invariance 
+ 
+Moses Fayngold 
+ 
+Department of Physics, New Jersey Institute of Technology, Newark, NJ 07102 
+ 
+ 
+  A careful look at an allegedly well-known century-old concept reveals interest ing aspects in it 
+that have generally avoided recognition in literature. There are four differ ent kinds of physical 
+observables known or proclaimed as relativistic invariants under space-time rot ations. Only 
+observables in the first three categories are authentic invariants, whereas  the single “invariant” – 
+proper length – in the fourth category is actually not an invariant. The proper lengt h has little if 
+anything to do with proper distance  which is a true invariant. On the other hand, proper distance, 
+proper time, and rest mass have more in common than usually recognized, and particularl y, mass 
+– time analogy opens another view of the twin paradox.    
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  2   The importance of invariant characteristics of physical systems or proce sses cannot be 
+overstated. The role invariants play in the description of the world is best captured b y E. Taylor 
+and J. A. Wheeler [1] in the statement:  
+                  “ In relativity, invariants are diamonds. Do not throw away diamonds !”  
+Invariants are an indispensible working tool of every professional physicist and an unavoidable 
+topic in almost any Physics textbook.  
+   And yet the concept of invariance and invariant characteristics turns out to be mor e subtle than 
+usually perceived. There are some important aspects of invariance which, to m y knowledge, have 
+not been explicitly stated in literature. In particular, what is usually re ferred to as just 
+“invariants” actually fall into a few quite different categories. Ignor ing this fact, as shown below, 
+leads to confusion, and frequently causes misleading or downright wrong statements . This paper 
+outlines four distinct kinds of invariants – four faces of Lorentz-invariance – and shows t hat one 
+of them is just a misconstruction reflecting a widely spread misconception, whi ch is still being 
+disseminated among generations of Physics students.  
+  Sorting out the known invariant characteristics of physical systems by their  properties also 
+helps elucidate very close analogies between some of them. Specifically, an a nalogy between 
+mass and time – the two observables describing totally different aspects of reality – enables us to 
+see the twin paradox from a different angle.   
+        
+I.  FACE 1  
+  The first category can be called the operational invariance. Most physical c haracteristics of a 
+system or process (e.g., mass, size, time, etc.) can be measured using an ex ecutable experimental  
+procedure that can be performed in any inertial reference frame (RF) and/or for  any state of 
+motion of the studied system. If such a procedure finds a certain quantity numeric ally the same 
+irrespective of  the RF used, then this quantity is an operational invariant .  
+   The simplest example is a number N of stable particles in an isolated system, which is the same 
+for all observers 1 .  
+   Another invariant in this category is the electric charge Q.  Its independence from velocity is 
+evident in many known experiments. As an example, consider the electron-positron annihilation 
+e e γ γ ++ → +   under two different conditions: first, in a positronium “atom” when the 
+velocities of both particles are negligible; and second, in a high-energy collis ion with only one 
+particle (“target”) stationary and the other one moving with an ultra-relat ivistic speed. In both 
+cases the net charge of the system after annihilation is zero, which (assum ing the net charge is 
+conserved) shows that the charge of the bombarding particle is the same as in its  stationary state. 
+By reciprocity this also means its independence from the observer’s state of mot ion.  This puts 
+the charge into the category of operational invariants.  
+                                                                   
+                                                              II.   FACE 2 
+   Next we consider the most famous of all invariants – the speed of light in vacuum. It has  a 
+rather subtle distinction from the operational invariants. On the one hand, it seems to be 
+determinable as the former through a procedure which is the same in all RF (e. g., the Michelson 
+experiment) and gives the same result in all of them. But on the other hand, the subject of  direct 
+measurement (free photons with a definite momentum) does not have a rest frame. And it is not a 
+                                                           
+1 We assume “regular” particles – photons and/or tardyons; for tachyons, the s ituation may be different  
+[2].  3 quantity like charge Q or occupancy N whose independence from speed is established in this 
+experiment, but rather the speed itself that is the same for all photons regardl ess of their 
+momentum and for all observers. Moreover, it turns out to be the same for all  objects of nature, 
+if we, instead of 3-velocity v, measure their 4-velocity u defined as [3, 4] 
+ 
+                                                   , 0, 1, 2, 3 j
+j dx u c j ds ≡ =  (1) 
+ 
+Here ds  is the norm of an incremental 4-displacement along the world line of the correspondi ng 
+object (for the reasons discussed in [5],  a more meaningful definition would be using ds  
+instead of ds ). The speed of light turns out to be the norm of 4-velocity of any  object – unlike the 
+operational invariants, it satisfies the identity: 
+ 
+                                                      2 2 2 
+0j
+j c u u u ≡ − = u  (2) 
+ 
+Here subscripts and superscripts on the right stand respectively for the co- and c ontravariant 
+components of a 4-vector, and the Einstein summation rule is assumed. In terms of 4-veloc ity, all  
+objects of nature flow through space-time with the speed of light, and the only differe nce 
+between a photon, a rocket, and a stationary rock in this regard is in the tilt of their re spective 
+world lines. Thus, the speed of light is more than just an invariant – it is a universal const ant, the 
+same for any object. Therefore it can be called an absolute or universal invar iant, which, unlike 
+N or Q , is a single-valued quantity.  
+                                             
+III.  FACE 3 
+     The invariants in the third category are the rest mass, proper time, and proper dis tance.   
+   A: Rest mass .  Traditionally (before 1905) mass was defined as the ratio /f a  of a net external 
+force f to resulting acceleration a. Measurements at v c <<  did not show any noticeable change 
+of mass in a moving object. In this respect, mass was thought to be in one company with el ectric 
+charge.   
+   Relativity changed all this. Both – the theory and new experiments – showed that mass  as a 
+measure of body’s inertness is velocity-dependent. Moreover, this dependence is  different for 
+different orientations between f and v. Measuring mass as the ratio /f a  when ⊥f v , that is, 
+/ m f a ⊥ ⊥ = , gives the well-known expression   
+                                          1/ 2 2
+0 2( ) , ( ) 1 vm m v v 
+cγ γ −  = ≡ −                              (3) 
+  
+For f v /parallelto, that is, for the ratio /f a /parallelto/parallelto, measurements give 3
+0( ) m m v γ = . In both expressions 0m 
+is the rest mass of the object.  
+  Mass (3) determined when ⊥f v , is known as the transverse mass  and is identical to the 
+relativistic  mass  discussed below; mass determined when f v /parallelto, is the longitudinal mass [3, 4].   
+  For an arbitrary orientation of f and v, the resulting acceleration is not even parallel to the force, 
+and f cannot generally be represented as f  = m a with a scalar-valued coefficient m, even if as a  4 function ( ) mv[5, 6].   Nevertheless, the relativistic generalization of the second law can be 
+formulated not only in terms of Minkowski’s 4-force, but also in terms of 3-vectors f and a for 
+any orientation between f and v [3 - 5]. Such formulation involves “anisotropic mass” as a 
+second-rank 3 3 × tensor, which can be linked to the spatial part of the 4 4 ×energy-momentum 
+tensor, and is especially convenient in relativistic mechanics of continuous medi ums [7 - 10].      
+  Here it is sufficient to restrict ourselves to case (3). It has been exte nsively studied for charged 
+particles in a magnetic field, for which the condition ⊥f v  is automatically satisfied. The  
+experiments showed an increase of mass with velocity [11] even before the advent of the  theory 
+of relativity, and later [12] they confirmed Eq. (3) unambiguously. The results we re applied to 
+the development of synchrotron accelerators [13, 14].  
+    As mentioned above, Eq. (3) also defines relativistic mass . Apart from and independently  of 
+(3), relativistic mass is defined as the ratio /p v , where p is particle’s momentum. We thus have 
+two different procedures yielding the same result, and accordingly two equivalent definitions : 
+relativistic mass (3) defined either as /f a ⊥ ⊥  or as /p v ; it is a speed-dependent and therefore not 
+invariant characteristic. This is an experimental  fact  that cannot be dismissed.  
+  The expression (3) for  relativistic mass is consistent with the general m ass-energy equivalence 
+ 
+                                                                  2E m c =   ,                                        (4)  
+ 
+where the relativistic energy E is not invariant either, but is only the temporal component of  4-
+momentum.  
+  Speed dependence of the relativistic mass was, perhaps, one of the factors that t riggered the 
+question of whether such characteristic should be considered at all. There is a f ashionable trend 
+to consider only the rest mass as a legitimate description of an object, denying  the relativistic 
+mass the status of a meaningful characteristic [15 – 17].  According to this view , there is only the 
+rest mass 0m, and the relation (4) must be applicable only to the rest mass and rest energy, 
+respectively, that is, we must reduce it to  
+                                                                   2
+0 0 E m c =           (5) 
+ 
+ (here and hereafter the subscript “0” stands for a quantity measured in the re st frame of an 
+object). This “truncation” of Eq. (4) is merely a suggestion which cannot invalidate the  equation 
+itself. Indeed, expressing 0Ein (5) in terms of E from the Lorentz-transformation of 4-
+momentum,  
+                        ( )0 0 0 0 
+00( ) ( ) , ( ) pEE V E Vp V E E Vγ γ γ == ± = =  (6) 
+ 
+one immediately recovers (4) with 0 ( ) m V m γ= . As Feynman explicitly emphasized in his 
+“Lectures on Physics” [18],   “The total energy of a particle is its mass in motion times  c2 ( 2E mc = ), 
+and when the body stops, its energy is its rest mass times  c2 ( 2
+0 0 E m c = )”   
+ The universal nature of mass-energy relation was emphasized in many sourc es as exemplified in 
+[19]:  “ Physical manifestations of the aspects of matter corresponding to mass and e nergy, respectively, 
+are different; but the quantitative characteristics of these aspects are  universally proportional to one 
+another. It is this universal proportionality that allows one to speak about the mass -energy equivalence ”.   5   Summarizing this part, we can say that the rest mass of an object is a scalar- valued coefficient 
+converting its 4-velocity u into 4-momentum P, whereas its relativistic mass is a scalar-valued 
+(but speed-dependent) coefficient converting its 3-velocity v into 3-momentum p.     
+  Already the mere fact that we have to distinguish between just mass and the r est mass shows 
+that the rest mass is not   an operational invariant defined in part I. The existence of at least one 
+verifiable experiment recording velocity dependence of mass (or any other  observable) 
+automatically takes it out of  domain I. Within I, we do not talk about the  “ rest ” charge and  
+“relativistic ” charge . Unlike charge, the rest mass is (up to the factor c) the norm of a 4-
+momentum ( ) / , E c = p P : 
+                                    2
+2 2 
+0 2, 0, 1, 2, 3 j
+jEm c p p j c= − = = 2p     (7) 
+ 
+   This is used for determining  0m when it cannot be measured directly in its rest fr ame (e.g., in 
+high-energy physics). We measure instead E and p, and then calculate 0m from (7). The 
+measurements must be very accurate since in the ult ra-relativistic case the computed value of m0 
+comes out as a small difference between two large n umbers.  
+  It is argued sometimes that the norm (7) (divided  by c) must be taken as the general definition  
+of mass. One cannot dispute a definition, but one c an dispute its consistence with other elements 
+of reality. In the real world, an entity called mas s manifests itself through its inertia which is 
+measurable under various conditions, and a special but important subset of conditions ( )/f a ⊥ ⊥  
+gives the result (3) called the relativistic mass. This result is consistent with both – ( )/p v  and 
+the general Eq. (4). So definition (7) determines t he rest mass only.   
+    Thus, the rest mass is a relativistic invariant  without being an operational  invariant. But its 
+value measured in its rest frame  can also be computed as the norm of the particle’s 4-momentum 
+measured in an arbitrary  RF. This kind of invariant can be called a rotational invariant or a 
+norm-invariant  since it is not affected by rotations in space-tim e. 
+   This becomes self-evident if we use the “forbidd en” relationship (4) to rewrite (7) as  
+ 
+                                                         2
+2 2 
+0 2m m c= − p  (8) 
+ 
+Since 2 2 p=p  and /p m v =, we can also write this as 
+ 
+                                                    2 2 
+2 2 
+0 2 2 2 1( ) p m m m m c v γ  = − =       (9) 
+ 
+ Eq-s (7) - (9) clearly show that m0 (not m!) is the norm of a 4-vector. The quantities m and /cp 
+on the right of (8) are the “temporal” and “spatial ” projections of this 4-vector in the energy-
+momentum (or mass-momentum) space. In this interpre tation, the velocity dependence of the 
+relativistic mass is a natural geometrical effect , since different velocities correspond to differen t 
+4-rotations of a given RF with respect to the rest frame of the object, and accordingly to different 
+values of the temporal projection of its 4-momentum . Thus, we have the energy-momentum  6 relation (7) or mass-momentum relation (8), but bot h terms, albeit expressing different aspects of 
+matter, are equally legitimate, and which one to us e is a matter of taste but not the matter of 
+principle, in view of the total mathematical equiva lence of (7) and (8).    
+  B: Proper time .  Consider a time-like 4-displacement between two events: 
+ 
+                                                  2 2 2 2 
+AB AB AB 0 s c t = − > r    (10) 
+ 
+Here A and B label the respective end-points of the  interval AB s, AB B A t t t ≡ −  is the time 
+between the events in a given RF, and AB B A = − r r r  is the spatial displacement between them.  
+In the proper  frame K 0, where both events happen at one place ( AB 0=r ), A and B are on its 
+temporal axis. Setting AB 0=r  in (10) gives 
+ 
+                                                               AB 
+AB s
+cτ=                                                  (1 1)  
+  
+Here AB τis the time lapse between the events in the proper frame, which is the definition of the 
+proper time  between the events. For AB 0≠r , we can use (11) to rewrite (10) in the form 
+analogous to (8), (9):   
+ 
+                            2 2 
+2 2 AB AB AB 
+AB AB AB AB 2 2 
+AB ; ( ) ; ( ) tt t V c V t τ τ γ γ= − = = ≡ r r V  (12) 
+ 
+  The right side of this expression describes the t ime dilation effect. As seen from (12), the  
+possibility of writing AB τ as the norm of the corresponding time-like 4-displ acement rests on this 
+effect. Like the rest mass, the proper time can be measured directly  in K 0, or computed  as the 
+norm of the corresponding 4-interval, which puts it  into the category of norm-invariants.   
+   There is perhaps a less known analogy between th e rest mass of an object and the proper time 
+of a process, which, in my opinion, deserves a sub- title. 
+  
+Rest mass – proper time analogy  
+  This is an analogy between a system of non-intera cting particles, on the one hand, and a 
+succession of consecutive processes within a single  moving object, on the other. The analogy is 
+easily seen and yet has evaded recognition in liter ature.  
+    Start with the rest mass. For a system  of moving non-interacting particles, its rest mass  0Mis 
+the sum of the relativistic  masses jmof the particles in the system’s rest frame ( not  the sum of 
+their rest masses 0jm) [3, 5, 6, 15, 20]  
+ 
+                                 0 0 0 0 ( ) j j j j 
+j j j M m m v m m γ = = ≥ = ∑ ∑ ∑      (13) 
+(the latter sum amounts to the rest mass 0mof the system with all particles at rest).  
+  Many textbooks on Relativity express this propert y by saying that the rest mass is a non-
+conserved characteristic of a system. Such statemen t is totally misleading since the term  7 “conserved” is related to something that remains un changed during the time evolution  of an 
+isolated system. In this context, “non-conservation ” of the rest mass would mean, by virtue of 
+Eq. (5), non-conservation of the rest energy of  su ch a system.  
+   The appropriate statement about (13) is that the  rest mass is a non -additive  characteristic of a 
+system [5]. The geometry of this non-additivity is shown in Fig. 1.  
+  Consider now, instead of system of masses, a sing le moving object T (equipped with a clock), 
+starting from a certain point O in space (event A) and later returning to the same point (event B).  
+If we choose event A as the origin of the correspon ding frame K 0, then both A and B lie on the 
+time axis of this frame (Fig. 2). Let p denote the net momentum  of the system of masses (13) and 
+r – the net spatial  displacement of  T between A and B. Just as settin g 00 = = p p  determines the 
+rest frame  of the system of masses, setting 00 = = r r  determines the proper inertial frame  K 0 for 
+process  AB. The term “proper” here reflects the fact that K 0 is the rest frame of another object S 
+remaining at  O during the whole round trip of T. This draws a s harp distinction between frame 
+K0 thus defined and frame K co-moving with T, which i s not  inertial. The net  4-displacement 
+AB sof T (the geometrical sum of its incremental 4-disp lacements) is identical to that of S, but 
+their respective proper times are different. One ca n see from Fig. 2 that AB s is equal (up to the 
+constant c) to proper time of S between events A and B. In ot her words, it is the proper time 
+AB (S) τ  along the world line of S. Alternatively, this pro per time can be obtained as the sum of 
+dilated  times of the individual sub-processes in T as obse rved from K 0 
+ 
+                              AB AB (S) ( ) (T) j j j j 
+j j t v τ τ γ τ τ = ∆ = ∆ ≥ ∆ = ∑ ∑ ∑       (14) 
+ 
+The algebraic sum of their proper  times jτ∆ on the right of Eq. (14) amounts to the proper tim e 
+AB (T) τ  along the world line of T.  
+  The Eq-s (13) and (14), while describing the beha vior of quite different physical characteristics, 
+are mathematically identical. Accordingly, the beha vior of proper time is totally analogous to 
+that of the rest mass. Like  the rest mass, the pro per time is a non-additive characteristic of  an 
+arbitrary process as measured from K 0. Explicit formulation of this analogy gives anothe r view 
+of and another way for explaining the twin paradox.  It may be helpful in demystifying the 
+paradox when teaching it to students. From the view point of (14) this “paradox” ( dependence of 
+proper  time AB τon path connecting  A and  B [1, 3])  is just another manifestation of non-
+additivity of proper time, similar to non-additivit y of the rest mass. In the example illustrated in 
+Fig. 2, the net proper time measured by the S-clock  is the proper time of a Stationary  twin (Sam) 
+residing at O. The algebraic  sum of the proper times of the incremental sub-pro cesses measured 
+by the T-clock is the proper time of the Traveling  twin (Tom). Just as the rest mass of a system is 
+greater than the sum of individual rest masses of i ts moving constituents, the proper time of the 
+stationary twin is greater than the sum of the cons ecutive proper times (net proper time) of 
+traveling twin between their parting and reunion. J ust  as the rest mass of a system is exactly 
+equal to the sum of relativistic  masses of its moving parts, the proper time AB (S) τ  is exactly 
+equal to the sum of dilated  times of the consecutive incremental sub-processes  within moving 
+object T. The relation (3) between the relativistic  mass of an object and its rest mass is identical 
+to relation (12) between the dilated time of a proc ess and its proper time; this is quite natural in  8 view of the fact that mass is the temporal componen t of 4-momentum just as time is the temporal 
+component of 4-displacement.    
+  The above-mentioned trend to discard the concept of relativistic mass might be one of the 
+factors that blocked this beautiful analogy from vi ew even though it was exposed since the onset 
+of the theory of relativity, begging for recognitio n.   
+ 
+   C: Proper distance .  Consider now two events A and B separated by a s pace-like interval, that 
+is 
+                                                           2 2 2 2 
+AB AB AB 0 s c t = − < r                                  (15)  
+ 
+In such case we can find a RF where both events hap pen simultaneously, and the interval (15) 
+reduces to pure distance AB rbetween them. In this RF we have AB 0 t= and get 
+ 
+                                                                AB AB s=r                                               (16)  
+ 
+  The value AB r defines the proper distance  between the two events [1, 15]. The corresponding 
+inertial frame K 0 where AB 0 t= can be called the proper  frame. 
+  Combining (15) and (16) shows that the proper dis tance is the norm of the corresponding 4-
+vector AB AB AB ( , ) s ct = r: 
+                                                     2 2 2 2 
+AB AB AB c t = − r r                          (17) 
+ 
+We can write it as  
+                                         2
+2 2 AB 
+AB AB 2
+AB 1 , r cr u c t u  = − = >     r   (18) 
+ 
+Here u can be considered as the speed of a fictitious sup erluminal particle connecting the end 
+points of a space-like interval (15).  
+   The expression (18) seems to be different from E q-s (9) and (12) for the rest mass and the 
+proper time. But this is only an apparent differenc e, since (18) is expressed in terms of 
+superluminal speed of a fictitious particle. We can  also express it in terms of the relative speed V 
+between frames K and K 0. From Lorentz transformation between these frames,  one obtains the 
+simple relation between u and V  [2, 5] 
+                                                                        2uV c =                       (19) 
+Using this, we can bring (18) to the form 
+ 
+                                                             2
+2 AB 
+AB 2; ( ) ( ) AB AB rr V Vγγ= = r r               (20) 
+ 
+Again, we see the total mathematical equivalence be tween (17), (20)  and (8),(9) ,  
+(10, 12). The proper distance can be either measure d directly in the proper frame K 0 or computed 
+as the norm of AB s through (17); alternatively, it can be computed fr om AB rand V using (20). 
+Thus, the proper distance is in one company with th e rest mass and proper time. And this  9 analogy also extends onto the situations when the i nterval (17) is the geometric sum of a number 
+of sub-intervals js∆(Fig. 3). Then we can express the net proper distan ce AB ras the sum of the 
+spatial components jr∆ of these sub-intervals:  
+ 
+                                      AB ( ) j j j j 
+j j j r V γ = ∆ = ∆ ≠ ∆ ∑ ∑ ∑ r r r                                 (21) 
+ 
+Here jV is relative velocity between the proper frame K 0 of AB s and the proper frame K j of a 
+sub-interval js∆. As seen from Fig. 3, the above-considered analogy  between the rest mass and 
+the proper time can be extended to the proper dista nce: the latter, while being an invariant, is not 
+an additive characteristic of a system of events.    
+   Summarizing this part, we can say that each of t he rotational invariants – 0m, AB τ, AB r – can 
+be computed as the norm of its respective 4-vector from its components. These invariants are 
+different from c (which is also invariant under 4-rotations) in tha t the latter is the norm of a 4-
+velocity of any  object; and by virtue of being single-valued it do es not require any computing.  
+ 
+IV.   “FACE” 3 + 1 
+  Finally, we turn to the fourth category. The quot ation mark in its name reflects the fact that it is  
+merely a mask rather than a real face, with no actu al invariant under it. It contains the proper 
+length  of an object. This well-known observable has been proclaimed  an invariant due to a 
+general oversight. The proper length  has nothing to do  with the proper distance described in the 
+previous section. There is a widely spread confusio n between these two different concepts. It has 
+reached such a scale that we can describe it by wor ds of D. Hestenes (said on a different but 
+related occasion [21])  as “the conceptual virus” s pread in the Physics community. This virus is 
+manifest in numerous misleading or just wrong state ments even in some authoritative sources. 
+Here is one taken from Wilkipedia:    
+     “In relativistic  physics , proper length  is an invariant  quantity which is the rod  distance  between 
+spacelike -separated events  in a frame of reference  in which the events are simultaneous .”  
+ 
+     The first part of this statement claims that i t is about proper length . The second part is the 
+definition of the proper distance . The combination of two, as we will see later, mak es the whole 
+statement, using the Pauli famous expression, not e ven wrong. In particular, as the rod’s state of 
+motion is not specified here, it may as well be a moving  rod whose edges are instantly coincident 
+with two events simultaneous in the given frame, in  which case the quoted statement defines the  
+proper distance as the Lorentz-contracted  length of the rod rather than its proper length. 
+     Here are two more quotations, this time from t he Forum of Physics Educators [22]:  
+ 
+    “…the Lorentz-contracted length is merely a spatial projection of  a 4-di splacement with its norm 
+being the proper  length .”   
+    "The" length of a ruler is invariant under ordinary rotations, even thoug h the projection of the ruler 
+onto this-or-that reference frame will in general change.  In a profoundly a nalogous way, "the" length of 
+rulers and "the" timing of clocks is invariant under rotations in the  XT plane, i.e. boosts, i.e. changes in 
+velocity.  It is only the shadow cast on this-or-that reference frame th at changes. Do not confuse a shadow 
+with the real object that casts the shadow ”. 
+  10    Such statements come from and are shared by a sig nificant part if not majority, of the physics 
+community. Both statements are false, and, when tau ght to students, promote the above-
+mentioned viral infection to the status of the worl d-wide pandemic.  
+   That the first statement is false, is immediately e vident from a very simple observation: in  
+Lorentzian geometry, the norm of a space-like 4-dis placement is less  than its spatial projection 
+(see (18) or (20)), whereas the proper length of a rod is greater  than its Lorentz-contracted 
+length. Therefore, the proper length is not  the norm of a 4-vector, and the Lorentz-contracted  
+length is not its “shadow”.  
+   For the same reason the second statement, making  the proper length “profoundly analogous” to 
+the invariant length under ordinary 3-rotations, is  also false. And the claim that proper length  is 
+in one company with proper time  makes it double-false. 
+   One of the possible sources of confusion between  the two characteristics is the fact that both –  
+Lorentz contraction and time dilation – are consequ ences of the relativity of time, and either of 
+them implies the other one [2, 5]. This logical equ ivalence, however, does not imply identical 
+physical behavior, and indeed, we know that proper time, in contrast to proper length, is shorter 
+than its “shadow” – the dilated time. Nevertheless,  the logical equivalence of the two effects was 
+misconstrued by many into physical  equivalence between them, which promoted the prope r 
+length to the same status as the proper time. This automatically led to identifying the proper 
+length with proper distance, which is, indeed, in o ne category (Part III, C) with proper time.    
+   Probably the best cure against the confusion bet ween these totally different concepts would be 
+reformulation of physics in the language of Geometr ic Algebra [21, 23 - 25], but this requires 
+time and concerted effort.  Here we will just show that the above misconceptions can be easily 
+clarified within the framework of the existing form ulation. We consider a few examples 
+illustrating the difference between proper distance  and proper length.  
+   Take a solid rod of constant proper length 0l. If we are in the rest frame K 0 of the rod and want 
+to measure its length, it is not  necessary that we mark the end-points simultaneous ly in K 0. 
+Moreover, if we are to consider the 0 l l ↔  (length – proper length) relationship for a fixed 4-
+displacement with simultaneous marking of the rod’s  ends in a moving frame K, then Relativity 
+demands  that the same events be not simultaneous in K 0. Thus, we can mark one end now (at a 
+moment 1t) and the other end later (at a moment 2 1 t t >) in K 0, and then measure the distance 
+between the marks. We can even consider such measur ement as a possible operational definition 
+of proper length. From the viewpoint of experimenta l physics, the requirement that the marks be 
+made simultaneously is redundant for a stationary o bject of constant shape and size, and it can in 
+this case be dropped. The distance between the ends  of the stationary rod is its proper length 
+regardless  of the time lapse between the two markings. But th is time lapse, together with the 
+spatial separation l0, determines the 4-displacement between the two cor responding events (Fig. 
+4). Since l0 is fixed while the time lapse is allowed to be arb itrary in K 0, we have an infinite set 
+of possible different 4-displacements with common s patial projection l0. In other words, the same 
+proper length  can be a spatial projection of an infinite number of different  4-vectors with 
+different temporal components and accordingly diffe rent norms. By the same token, it is not  the 
+proper distance between the marking events if the m arks are not made simultaneously in K 0. 
+Moreover, we can make the time separation between t he markings so big, that the corresponding 
+4-displacement becomes light-like or even time-like ! Consider a stationary rigid rod of 1 m 
+proper length. In its rest frame, we mark its left end now and its right end one million years from 
+now. The 4-displacement between the markings is def initely time-like, in which case it cannot 
+even be assigned a proper distance. There is no suc h thing as a proper distance for a time-like  11  interval! And yet we can measure the spatial  dista nce between the marks and obtain exactly 1 m, 
+which is, according to definition, the proper lengt h of the rod. Thus,  the proper length here is 
+neither the proper distance (which does not exist f or OC and OD in Fig. 4), nor the norm of the 
+corresponding 4-displacement, which is time-like!     
+  Consider now a reciprocal situation: let the rod move with velocity V along the x-axis of a 
+reference frame K, with its length along the direct ion of motion. As before, we can measure the 
+length of the rod in K by marking its edges and the n measuring the distance between the marks; 
+but now, since the rod is moving, it is absolutely imperative that the instant positions of  the 
+edges be marked simultaneously in K. In this case, the spatial separation AB l= r  between the 
+marks is, by definition, the proper distance  between the marking events; but it is not  the proper 
+length of the rod! Indeed, the described procedure constitutes the length measurement of the 
+moving and accordingly Lorentz-contracted rod; the rod’s proper  length 0l (assuming its speed 
+is known) is obtained from l as:  
+ 
+                                                       0 AB AB ( ) ( ) l V l V γ γ = = ≠ r r                                           (22)  
+ 
+Again, the proper length here is not the norm of a 4-vector.  
+  The non-trivial relationship between proper lengt h and proper distance can be seen in more 
+details from the space-time diagram of the moving r od (Fig. 5). The diagram shows the world 
+sheet of such a rod, with segment  
+                                                           0 A OA / ( ) l l V x γ = = ≡   (23) 
+ 
+being its Lorentz-contracted length. The primed axe s ct ′ and x′ are now the temporal and 
+spatial axes, respectively, of the RF K 0 co-moving with the rod. Since events O and L happe n at 
+the end points of the rod which is stationary in K 0, the norm of the 4-displacement OL is equal to 
+its proper length. Since these events are simultane ous in K 0, the norm is also equal to the proper 
+distance between O and L. This is one of the few sp ecial cases when the proper length is 
+coincident with proper distance. But even in this c ase the Lorentz-contracted length OA, contrary 
+to the cited statement from [22]  is not the spatial projection of  OL. Apart from b eing 
+immediately evident from Fig. 5, this can easily be  proved quantitatively. Assume as usual that 
+the local clocks at the respective origins of both frames K and K 0 read the zero time when they 
+are instantly coincident (event O). Denoting the sp atial projection of OL as OC Lx≡ and the time 
+coordinate of event L in K as Lt, we have  
+                                                                  L A L x x V t = +      (24) 
+ 
+Lorentz-transformed time-coordinate of this event i n K 0 is  
+ 
+                                                          L L L 2( ) Vt V t x cγ  ′= −                        (25) 
+ 
+Since x′is the line of simultaneity in K 0, we have L0t′=, which gives 
+  12                                                                L L 2Vt x c=  (26) 
+ 
+Putting this into (24) and solving for Lxyields, in view of (23) 
+ 
+                                                               L 0 ( ) x V l γ=  (27) 
+ 
+As was already mentioned, not only is Lx greater than A l x = , it is greater than l0. 
+As a by-product of this derivation we see that the proper length (represented by OB in K) is the 
+geometric mean of the Lorentz-contracted length and  the spatial projection of 4-displacement 
+OL. Thus, the spatial projection 12 r of a proper distance 12 r is, as it should be in Lorentzian 
+geometry, greater  than 12 r , and the Lorentz-contracted length is not a projec tion of the proper 
+length. The latter cannot  be computed from l and t in the same way as (17).  
+   Strictly speaking, we cannot even claim that the  relativistic length contraction is a purely 
+geometric effect. It is rather a combination of geo metry and dynamics, which is especially 
+evident when we consider length measurements of acc elerated objects or objects of varying size 
+[5, 26, 27].   
+  Thus, the proper distance and the proper length d escribe quite different characteristics of a 
+process or an object. The former relates to a pair of events in space-time, which are connected by 
+a space-like interval; the latter describes geometr ical properties of a material object observed in 
+its rest frame. It is not an invariant of category III (let alone I or II !). It could be named the 
+“conditional invariant” merely by convention: when we measure the Lorentz-contracted length l 
+of a moving object, we just remember in the back of  our mind that the observer sitting on this 
+object would record the length l0. In other words, we mentally substitute the actual  length l of the 
+object in a given RF by its length l0 in the co-moving RF. If we know V, we can compute l0 from 
+l using (23), but this computation has nothing to do  with those in Part III. If we apply 
+consistently the logics of described “promotion” (n aming the “invariant” any characteristic of an 
+object in its rest frame) then nearly all physical quantities will become “invariants”; in which 
+case the mere concept of invariance will be strippe d of its meaning. But once the proper length 
+has been universally acclaimed as an invariant, we must at least be very careful to separate it 
+from true invariants of type I, II, and III, and pu t it into a separate category. Naming things as 
+they are, we should call this type the bogus  invariant. More politely (until the proper length is 
+disqualified from its current status) we can call i t conventional (or conditional ) invariant .     
+  
+ 
+ 
+ 
+V.  CONCLUSION 
+Three distinct types of Lorentz-invariance under 4- rotations, plus one included into this family 
+by collective mistake, are summarized in the tables  1 and 2 below: 
+             
+ 
+ 
+ 
+  13  Table 1:  True invariants 
+  
+Invariant Type  
+ Physical  
+characteristic 
+  
+          Computational algorithm 
+ 
+I  
+Operational 
+(frame-independent by 
+measurement ) 
+  
+Number of particles N 
+ 
+Electric charge Q 
+  
+ 
+Unknown or not identified 
+ 
+II  
+ Absolute (universal)  
+Speed of light  c           
+                 Satisfies the identity: 
+              2 2 2 
+0j
+j c u u u ≡ − = u  
+ 
+ 
+ 
+III   
+ Rotational 
+(directly measurable in 
+the rest (proper) frame 
+and computable as the 
+norm of a 4-vector in an 
+arbitrary inertial RF )  
+Rest mass m0 
+ 
+Proper time τ0 
+  
+Proper distance 0r  
+( )2 2 2 
+0 0 or / ( ) m m c m m v m γ = − = ≤ p/  
+      
+( )2 2 2 
+0 0 or / ( ) t c t v t τ τ γ = − = ≤ r/  
+ 
+     ( )2 2 2 
+0 0 or / ( ) ct r v r γ = − = ≤ r r r  
+ 
+ 
+ 
+ 
+                                                              Table II:  False invariants 
+   
+Invariant Type 
+   
+Physical   characteristic 
+       
+Computational algorithm 
+ 
+            Bogus 
+(does not  satisfy the 
+definition of an invariant )   
+Proper length l0  
+(must be expelled from 
+the invariants’ family)  
+  
+ 
+      2 2 2 
+0 0 ( ) ( ) ! l ct l l v l γ − = ≥ ≠r  
+ 
+Summary : Operational invariants (I) are directly measurabl e and remain the same for a system at 
+rest and in motion. Absolute invariant c (II) is directly measurable for an object in motio n if 
+00 m= or computable  as the norm of its 4-velocity in any  state of motion (including rest) if 
+00 m≠. Rotational or norm-invariants (III) are directly measurable in a proper or rest frame of a 
+system and computable as the norm of the correspond ing 4-vector characterizing the system.    
+      
+ 
+ 
+ 
+ 
+  14  Figures 
+ 
+ 
+ 
+ 
+                                                                                m   
+ 
+ 
+ 
+ 
+                                                                      m0 
+                                                                                  ∆m05 
+   ∆m5 
+ 
+                                                                     ∆m4     ∆m04    
+                                                                                   
+                                                             
+                                                           ∆m03 ∆m3  
+                                                                     
+                                                                                 
+                                                                 ∆m02      ∆m2 
+ 
+                                                                    ∆m01   ∆m1 
+ 
+ 
+                                                                                                                                                                                           
+                                                             0                                                             p/c 
+  
+ 
+ 
+Fig. 1  
+The invariant (rest) mass m0 of an object as the norm of its 4-momentum vector.   
+  The vector is normalized to mass and shown here i n the object’s rest frame. In case when the 
+object consists of non-interacting parts, its rest mass m0 is the geometric  (not algebraic!) sum 
+(represented by segment 0 m0) of the rest masses ∆m0j  of individual parts, or, which is the same, 
+the algebraic sum of their projections onto the m-axis (relativistic masses ∆mj). The vectors 
+representing individual invariant masses ∆m0j are generally not parallel to the m-axis because 
+these masses are in motion with respect to the rest  frame of the whole object.   
+  The dashed lines represent photons’ trajectories in the momentum space (or, which is the same, 
+their zero rest masses). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+                                    
+  15                                   
+                                                                      ct    
+ 
+ 
+ 
+                       AB (S) τ    B      
+                                                                    ∆t5          ∆τ05 
+                            AB ( ) Tτ  
+ 
+                                                                     ∆t4                   ∆τ04    
+                                                                   
+                 
+                                                             ∆t3             ∆τ 03  
+                                                            
+                                                                    ∆t2        ∆τ02               
+                                                                      
+                                                                    ∆t1      ∆τ01  
+ 
+ 
+                                                                                                                                                                                           
+                                                                A                                                          x 
+  
+ 
+Fig. 2  
+  The proper time of a process in an inertial frame K 0 in which the initial and final moment of the 
+process (events A and B) happen in the same place ( proper frame). 
+  The intermediate stages of the process can happen  at other places, as is the case for a process 
+within a moving object returning in the end to its starting point. This is just a generalized version 
+of the twin (or clock) paradox with one clock S sta tionary in K 0 and the other one (T) making a 
+round trip. The proper time AB (S) τ  read by S is greater than the proper time AB (T) τ of T between 
+the events A and B. The net proper time AB (S) τ  is the geometric  (not algebraic!) sum of the 
+proper times ∆τ0j  of individual stages of T’s trip, or, which is the same, the algebraic  sum of 
+their projections (dilated times ∆τj) onto the ct -axis. The vectors representing individual proper 
+times ∆τ0j are generally not parallel to the ct -axis because T is moving in K 0 during the 
+corresponding stages.   
+  The dashed lines are the world lines of photons p assing thorough the origin. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+                                                                   16   
+                                                    ct    
+ 
+ 
+ 
+                                                                        
+ 
+                                                                                   
+ 
+ 
+ 
+                                                                                   
+                                                             
+                                                                      
+                                                                     
+                                                                                 
+                                                                     
+ 
+                                                                     
+ 
+                                                      A           1r∆      2r∆     3r∆     4r∆      5r∆    B    
+                                                                                                                                 0 AB ≡r r              x                                  
+                                                    01 ∆r                                       05 ∆r                                   
+                                                       02 ∆r 03 ∆r      04 ∆r 
+                                                                             
+                                                                    
+Fig. 3 
+ The proper distance between the ends A and B of a s pace-like 4-displacement.  
+   It is the distance between A and B in the inerti al frame K 0 (proper frame) in which these events 
+are simultaneous. For a displacement consisting of a number of incremental  displacements, the 
+net proper distance 0 AB ≡r r  is the modulus of geometric  (not algebraic!) sum of these 
+displacements, or, which is the same, the algebraic  sum of modulae of their projections ∆rj onto 
+the x-axis. 
+  The dashed lines represent space-time trajectorie s of the photons passing through the origin. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  17   
+                                                                    
+                        ct 
+ 
+                           P                 Q                       O′ 
+                                                      D 
+                                                
+ 
+ 
+ 
+                                          0   C       
+                                                
+                                              B 
+                          O                 A 
+                                                                                    x 
+                                                                                 
+          l0 l0   
+      
+                                      l0           
+                                 
+ 
+Fig. 4 
+ The world sheet of a stationary rod, with a proper length OA = l0 aligned along the x-axis.  
+   OP and AQ are the world lines of the end points of the rod. OO ′ is the world line of a photon 
+passing through the origin. The width of the sheet (the proper length l0) is the common spatial 
+projection of space-like 4-displacements OA and OB,  light-like 4-displacement OC and time-like 
+4-displacement OD. In case OA the corresponding 4-d isplacement is coincident with the proper 
+length.  In neither case (except for OA) is the pro per length equal to the norm of the 
+corresponding 4-displacement. In particular, the no rm of OC is zero. And in neither case, except 
+for OA, is the proper length identical to proper di stance even numerically, let alone conceptually. 
+Thus, 4-displacement OC has in the described case t he proper length l0 as its spatial projection, 
+but no proper distance is associated with it (there  is no RF in which the events O and C would be 
+simultaneous). The same is true about the time-like  4-displacement OD.    
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  18   
+ 
+ 
+                                                                    
+                            ct 
+                                                                                            O′ 
+                                                                 
+                                                            ct ′                           
+                                                       
+                                                       P              Q 
+ 
+ x′ 
+ 
+                                                                                                    
+                                                      L  
+                                               
+                                                 
+                             O          A   B    C                             x 
+                                                                                  
+   
+       
+Fig. 5 
+  The same rod as in Fig. 4, but now in a reference frame K where it is moving along the x-axis 
+with a speed V.  
+   O-ct ′ and O- x′ are the temporal and spatial axes, respectively, o f the rest frame K 0 of the rod 
+(as observed from K), and OAPQ is its world sheet i n K.   
+0 OA / ( ) l v γ = is the Lorentz-contracted length of the rod. 
+OB represents the proper length of the rod if it we re stationary in K.  
+OC is the spatial projection of the 4-displacement OL between two markings of the rod’s ends 
+made simultaneously in its rest frame K 0.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  19  References 
+1.    E. Taylor,  J. A. Wheeler,  Exploring Black Holes : Introduction to General Relativity , 
+      Addison Wesley Longman, San Francisco (2000),  p. 1-13.  
+2.   M. Fayngold, Special Relativity and Motions Faster than Light , Wiley-VCH (2002) 
+3.   L. Landau and E. Lifshits, The Classical Theory of Fields , Butterworth-Heinemann (1998) 
+4.   P. G. Bergmann, Introduction to the Theory of Relativity , Dover Publications, Ch. 6 (1974) 
+5.   M. Fayngold, Special Relativity and How It Works , Wiley-VCH, Weinheim (2008) 
+6.   T. Sandin, In Defense of Relativistic Mass, Am . J. Phys ., 59  (11), 1032 – 1036 (1991)  
+7.   C. Moeller, The Theory of Relativity , Oxford Univ. Press (1962), Ch. 6 
+8.     R. C.  Tolman, Relativity , Thermodynamics , and Cosmology , Clarendon Press, Oxford (1962) 
+9.   M. A. Tonnelat , The Basis of Electromagnetism and Relativity Theory  (1962) 
+10.  L. Landau, E. Lifshits, Hydrodynamics , Butterworth-Heinemann (1999), Ch. 15, Sec. 133  
+11.  K. Kaufmann, Die Electromagnetische Masse des Electrons., Phys . Zeitschr . 4, p. 54 (1902) 
+12.  G. Neumann, Die trage Masse schnell bewegter E lectronen, Ann . Phys . 45 , p. 529 (1914)     
+13.  E. M. McMillan, The Synchrotron – A Proposed H igh Energy Particle Accelerator, 
+       Phys . Rev . 68  (5 – 6),  143 – 144  (1945)  
+14.  V. Veksler, Concerning Some New Methods of Acc eleration of Relativistic Particles, 
+        Phys . Rev . 69 , p. 244 (1946)  
+15.  E. F. Taylor,  J. A. Wheeler, Spacetime Physics , Freeman & Co. (1963), Sec. 13   
+16.  G. Oas, On the abuse and use of relativistic m ass,  arXiv: Physics /0504110v2 (2005) 
+17.  L. B. Okun, The Virus of Relativistic Mass in the Year of Physics.  
+       In the “ Gribov Memorial Volume : Quarks , Hadrons , and Strong Interactions ”,  
+       World Scientific, Singapore (2006)    
+18.  R. Feynman, The Feynman Lectures on Physics ,  
+       Vol. I, Addison – Wesley (1963), Ch. 16,  Se c. 5 
+19.    V. A.  Fock, Theory of Space, Time, and Gravitation ,  
+       Pergamon Press – Macmillan Co., 2-nd Ed. (19 64) 
+20.  F. Wilczek, Mass Without Mass , I, II.  Phys. Today , Nov. 1999, p. 11; 
+                                                                      Phys. Today , Jan. 2000, p. 14   
+21.  D. Hestenes, Oersted Medal Lecture 2002: Refor ming the mathematical language of     
+       physics, Am. J. Phys . 71  (2),  104 – 121 (2003)  
+22.  “ Teaching Special Relativity ”, Forum for Physics Educators, Phys – 1,  
+       phys-1@carnot.physics.buffalo.edu  , (July 30, 2009)        
+23.  D. Hestenes,  Spacetime physics with geometric  algebra,  
+       Am. J. Phys ., 71  (9), 691 – 714 (2003)    
+24.  D. Hestenes, Space-Time Algebra , Gordon and Breach (1966) 
+25.   A.  Lasenby and C. Doran, Geometric Algebra for Physicists , Cambridge Univ. Press (2003)                                              
+26.  M. Fayngold,  Dynamics of relativistic length contraction and the Ehrenfest paradox,  
+       arXiv: 0712.3891v1 
+27.  M. Fayngold,  Two permanently congruent rods m ay have different proper lengths, 
+       arXiv: 0807.0881 (July 2008)                          
+     
\ No newline at end of file
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+arXiv:1001.0089v1  [quant-ph]  31 Dec 2009Environment Assisted Precision Measurement
+G. Goldstein1, P. Cappellaro1,2,3, J.R. Maze1, J. S. Hodges1,3, L. Jiang1,5, A. S. Sørensen4, M. D. Lukin1,2
+1Department of Physics, Harvard University,2Harvard-Smithsonian Center for Astrophysics, Cambridge M A 02138 USA
+3Nuclear Science and Engineering Dept., Massachusetts Inst itute of Technology, Cambridge MA 02139 USA
+4QUANTOP, Niels Bohr Institute, Copenhagen University, DK 2 100, Denmark and
+5Institute of Quantum Information, California Institute of Technology, Pasadena CA 91125 USA
+We describe a method to enhance the sensitivity of precision measurements that takes advantage
+of a quantum sensor’s environment to amplify its response to weak external perturbations. An
+individual qubit is used to sense the dynamics of surroundin g ancillary qubits, which are in turn
+affected by the external field to be measured. The resulting se nsitivity enhancement is determined
+by the number of ancillas that are coupled strongly to the sen sor qubit; it does not depend on the
+exact values of the coupling strengths and is resilient to ma ny forms of decoherence. The method
+achieves nearly Heisenberg-limited precision measuremen t, using a novel class of entangled states.
+We discuss specific applications to improve clock sensitivi ty using trapped ions and magnetic sensing
+based on electronic spins in diamond.
+Precision measurement is among the most important
+applications of resonance methods in physics. For exam-
+ple, quantum control of atomic systems forms the physi-
+cal basis of the world’s best clocks. Ideas from quantum
+information science have been used to demonstrate that
+quantum entanglement can enhance these measurements
+[1,2]. Atthe sametime awide rangeofquantum systems
+have been recently developed aimed at novel realizations
+of solid state qubits. Potentially such systems can be
+used as quantum measurement devices such as magnetic
+sensors with a unique combination of sensitivity and spa-
+tial resolution [3, 4, 12, 33]. In this Letter, we describe a
+novel technique that makes use of the sensor’s local en-
+vironment as a resource to amplify its response to weak
+perturbations. We shall use solid state sensors and ion
+clocks as examples.
+The purpose of quantum metrologyis to detect a small
+external field, coupled to the sensor by a Hamiltonian:
+HS
+b=b(t)κSz, whereSzis the spin operatorof the quan-
+tum sensor. Here, b(t) can be an external magnetic field
+or the detuning of a laser from a clock transition while κ
+is the spin’s coupling to the field. The working principle
+ofalmost any quantum metrologyscheme can be reduced
+to a Ramsey experiment [18, 19], where the field is mea-
+suredviathe induced phasedifference betweentwostates
+of the quantum sensor. The figure of merit for quantum
+sensitivity is the smallest field δbminthat can be readout
+duringa totaltime T. Fora singlespin 1/2, ifthe sensing
+time is limited to τ(e.g. by environmental decoherence)
+then:δbmin≃1
+κ√
+Tτ.
+In many cases the external field also acts on the sen-
+sor’s environment, which normally only induces decoher-
+ence and limits the sensitivity. Here we show that in
+some cases the environment can instead be used to en-
+hance the sensitivity. For generality we will illustrate the
+key ideas using the central spin model (Fig. 1a). In this
+model a central spin (which can be prepared in a well de-
+fined initial state, coherently manipulated and read out)
+is coupled to a bath of darkspins that can be polar-ized and collectively controlled, but cannot be directly
+detected. The system is described by the Hamiltonian
+H=Hb+Hint, with
+Hb=b(t)/parenleftBig
+κSz+ξ/summationdisplay
+Ii
+z/parenrightBig
+, Hint=|1∝an}bracketri}ht∝an}bracketle{t1|/summationdisplay
+λiIi
+z,(1)
+whereλiare the couplings between sensor and environ-
+ment spins, while κandξare couplings to the external
+field of the central and dark spins respectively. Here |0∝an}bracketri}ht,
+|1∝an}bracketri}htandSzrefer to the centralspin while |↑∝an}bracketri}ht,|↓∝an}bracketri}ht,Ii
+zto the
+dark spins (and we set /planckover2pi1= 1). We consider two cases.
+In the first one Hintcan be turned on and off at will
+and is much larger than any other interaction in the sys-
+tem (e.g. alaser-mediatedioninteraction). Inthe second
+case,Hintis intrinsic to the system and ofthe same order
+of magnitude as the relevant sensing time (e.g. dipole-
+dipole interactionsbetween solid state spins). In all cases
+we will assume collective control over the dark spins.
+To illustrate the sensing method we consider first the
+idealized case where the couplings between the central
+and the dark spins can be turned on and off at will and
+the dark spins are initialized in a pure state |↑↑...↑∝an}bracketri}ht.
+|/a2up〉
+|/a2up〉b) Scrambling Phase Acquistion Echo a) 
+X
+λ1Ix|0 〉+ |1 〉
+λnIxλ1Ix
+λnIxbI zbS z
+bI z
+−i τHint −itH bxe e−i τHint xe/radical_old2
+... 
+FIG. 1: Ideal Model. (a) A central spin coupled to a spin
+bath. (b) Ideal measurement procedure. Gates labeled λIx
+represent controlled rotations e−iλIxof the dark spins, ob-
+tained via the Hamiltonian Hint. The gates bIzrepresent
+rotations e−ibIzdue to the external field b. The central spin
+undergoes a spin-echo before measurement.
+ConsiderthecircuitinFig. 1(b). First, thecentralspin
+is prepared in an equal superposition of the two internal
+states∼ |0∝an}bracketri}ht+|1∝an}bracketri}ht. ThenHintis rotated to the x-axis and2
+applied for a time τ. This induces controlled rotations of
+the dark spins, resulting in an entangled state
+(|0∝an}bracketri}ht| ↑...↑∝an}bracketri}ht+|1∝an}bracketri}ht|ϕ1...ϕN∝an}bracketri}ht)/√
+2, (2)
+where|ϕi∝an}bracketri}ht ≡cos(ϕi)|↑∝an}bracketri}ht −isin(ϕi)|↓∝an}bracketri}htwithϕi=λiτ.
+This state is then used to sense the magnetic field. We
+assume for now κ= 0 and define θd=ξ/integraltextτ
+0dtb(t). Un-
+der the action of the magnetic field the state evolves to
+(|0∝an}bracketri}ht|↑...∝an}bracketri}ht+|1∝an}bracketri}ht|ψ1...ψN∝an}bracketri}ht)/√
+2 with
+|ψi∝an}bracketri}ht= cosϕi|↑∝an}bracketri}hti−isinϕie2iθd|↓∝an}bracketri}hti
+≈ei2θdsin2ϕi|ϕi∝an}bracketri}ht+θdsin(2ϕi)
+2/vextendsingle/vextendsingleϕ⊥
+i/angbracketrightbig
+,
+where/angbracketleftbig
+ϕi|ϕ⊥
+i/angbracketrightbig
+= 0. The central spin is then flipped by
+aπ-pulse and another control operation with Hintalong
+xis applied. If the field (and thus θd) were zero the
+interaction between central and dark spins would then
+be refocused, correspondingtodecouplingthesensorspin
+from the environment as in a spin-echo. For metrology
+purposes, to first order, the effect of a small field is to
+introduceaphasedifferenceΦ ≈2θd/summationtext
+isin2(ϕi)between
+the states of the dark spins, depending on the state of
+the central spin (while the terms ∝/vextendsingle/vextendsingleϕ⊥
+i/angbracketrightbig
+only contribute
+to second order in θd). After the sensor spin is rotated
+around the x-axis, this yields an additional contribution
+to the probability of finding the spin in the |1∝an}bracketri}ht-state:
+P1≈(1+Φ+θs)/2+O(b2) (where we reintroduced the
+phaseacquiredby the sensorspin alone, θs=κ/integraltextτ
+0dtb(t),
+whichcanbesimplyaddedas HS
+bcommuteswiththerest
+of the Hamiltonian).
+Whilethesignalisenhancedbyafactor ∝Φ, thequan-
+tum projection noise remains the same as we still read
+out one spin only. The minimum field that can be mea-
+sured in a total time Tis then:
+δbmin=/radicalbiggτ
+T1
+Φ+θs≈1
+nξ√
+Tτ, (3)
+wherenis the total number of dark spins. The linear
+scaling innof the phase Φ can be achieved in principle
+for any distribution of λi’s, since we can always choose
+a duration τsuch that/angbracketleftbig
+sin2(λiτ)/angbracketrightbig
+≥1
+2, leading to or-
+der one contribution from each spin. Thus we are able to
+perform Heisenberg-limited spectroscopy despite the fact
+that the precise form of the entangled state (2) is uncon-
+trolled, and may not even be known to us [20]. This
+considerably relaxes the requirements for entanglement
+enhanced spectroscopy as compared to known strategies
+involving squeezed or GHZ-like states. We next discuss
+two experimental implementations that approximatethis
+idealized scheme: quantum clocks with trapped ions and
+spin-based magnetometry.
+To reach high precision in quantum clocks, the ions
+must posses several characteristics: a stable clock transi-
+tion, acoolingcyclingtransition, goodinitialstateprepa-
+ration and reliable state detection. It is then convenientto use two species of ions in the same Paul trap [24, 30]:
+Thespectroscopy ions (e.g.27Al+) provide the clock
+transition while the logicion (e.g.9Be+) fulfills the
+other requirements. Although inspired by a similar idea
+as the one proposed here, experiments using two ion
+species have been so far limited to just one spectroscopy
+ion [24, 28–30]; with our method the number of spec-
+troscopy ( dark) ions can be increased.
+Specifically, by usingmultichromaticgates[21] one can
+implement the Hamiltonian Hintin Eq. (1) [26] (such
+multichromatic gates are known to be much more robust
+to heating noise than Cirac-Zoller gates [21] that have
+been used so far). We can then use the method pre-
+sented here to transfer the phase difference due to the
+detuning of several spectroscopy ions onto a single logic
+ion, which can be read out by fluorescence. In princi-
+ple, we achieve Heisenberg-limited sensing of the clock
+transition without individual addressability of the spec-
+troscopy ions or producing GHZ states. Importantly, we
+achieve this even if the spectroscopy ions have different
+couplings to the logic ion, which will be the case in a trap
+with different ion species due to the absenceof a common
+center of mass mode.
+We next briefly discuss the effects of decoherence on
+this method. It has been argued (see e.g. [31]), that the
+coherence time reduction for a n-spin entangled state re-
+duces the sensitivity to roughly that of spectroscopy per-
+formed onnindividual spin, scalingas√n, asopposed to
+the ideal scaling ∝nderived here (See Ref. [32] for a dif-
+ferent scaling in the case of atomic clocks). Our scheme
+would obtain an improvement ∝√nwith respect to cur-
+rent experimental realizations where only a single ancil-
+lary ion is available, even in the decoherence model of
+Ref. [31], where each spin undergoes individual Marko-
+vian dephasing. Furthermore, this decoherence model
+is not so relevant in present setups, as technical noise
+during the gates and imperfect rotations are dominant
+for traps with many ions. Our method is highly robust
+to static repetitive imperfections during the gates, lead-
+ing to further improvement depending on the exact noise
+specifications [26].
+In many physical situations short bursts of controlled
+rotations, as used above, are not available. Instead, the
+couplings between the central and dark spins are always
+on and their exact strength is unknown. Examples of
+such systems are solid-state spin systems used for mag-
+netometry [3, 4, 12, 33]. Still, it is possible to achieve
+nearly Heisenberg-limited metrology even for these sys-
+tems.
+Specifically, we will consider magnetic sensing using
+a single Nitrogen Vacancy (NV) center in diamond [8],
+surrounded by darkspins associated with Nitrogen elec-
+tronic impurities [10, 11]. We focus on NV centers since
+their electronic spins (S=1) can be efficiently initialized
+into theSz= 0 state by optical pumping and measured
+via state selective fluorescence. By applying an exter-3
+nal (static) magnetic field that splits the degeneracy be-
+tweenSz=±1 states and working on resonance with the
+(0↔1) transition, the NV center can be reduced to an
+effective two-level system [4]. Then, the system compris-
+ing one NV center and several N spins is well described
+by Hamiltonian (1).
+In Fig. 2 we introduce a control sequence that yields
+an effect equivalent to the circuit in Fig. 1.b. The action
+of the pulse sequence can be best understood using the
+well known equivalence between Ramsey spectroscopy
+and Mach-Zehnder interferometry [18, 19], where the in-
+terferometer arms describe the central spin state. It is
+sufficient to consider the evolution of each arm sepa-
+rately, replacing Szby its eigenvalues ms={0,1}and
+describing the evolution in the interaction frame defined
+bythecontrolpulses[17]. Hamiltonian(1)becomestime-
+dependent, with darkspinsalternatingbetween Ii
+zandIi
+x
+as shown in Fig. 2. Then, for different halves of the spin
+echo sequence, the coupling Hamiltonian in each arm is
+zero (ms= 0) while for the other halves it has identical
+forms. In the absence of a magnetic field the evolution
+is thus the same along each arm of the interferometer.
+Adding an external field creates a phase shift between
+the two arms. For small field strengths we can then eval-
+uate the phase difference acquired between the two arms
+as a perturbation. For a finite polarization Pof the dark
+spins we find
+Φ =θs/bracketleftbigg
+1+2P/summationdisplayθd
+θssin2(ϕi/4)/bracketrightbigg
+,(4)
+withθs=κ
+τ/bracketleftBig/integraltextτ
+2
+0dtb(t)−/integraltextτ
+τ
+2dtb(t)/bracketrightBig
+,θd=ξ
+τ/integraltext3τ
+4τ
+2dtb(t).
+Compared to spin-echo based magnetometry [3], the sig-
+nal is increased by the factor in the square bracket while
+the measurement noise is the same, as we still read out
+onespinonly. Notethatallthedarkspinscontributepos-
+itively. For values of the couplings such that |λiτ| ≥π,
+orstrongly coupled environment spins, we obtain a con-
+tribution ∝2nsc∝an}bracketle{tsin2(λiτ
+4)∝an}bracketri}ht ≈nsc. Each of the nsc
+strongly coupled spins thus gives a contribution of order
+one, irrespective of the sign or exact value of the cou-
+pling.Weakly coupled dark spins ( λiτ≤1) contribute
+insteadwithafactor( λiτ)2/8andweobtainatotalphase
+Φ≈θs/bracketleftBig
+1+θd
+θsP/parenleftBig
+nsc+1
+8/summationtext′
+(λiτ)2/parenrightBig/bracketrightBig
+,wheretheprimed
+sum is on the weakly coupled spins. In general the sen-
+sitivity enhancement scales as ∼Pnsc. We thus achieve
+nearly Heisenberg-limited sensing of the external field.
+We next take into account decoherence resulting from
+the interaction with the environment spins as well as de-
+coherenceofthe darkspinsthemselves. Toevaluatethese
+effects, we compare the sensitivity achievable with the
+proposed pulse sequence to that obtained with a spin-
+echo sequence, considering the same system (a sensor
+spin surrounded by the same spin bath) and including
+the effects of decoherence (external perturbations and
+couplings between the dark spins). Once the central spina) b) 
+−π
+2y −π
+2xπxSensor 
+−π
+2y −π
+2y−−π 
+2yBath 
+τ/2 τ/4 τ/4 τ/4 τ/4 
+0 τField |0 〉|0 〉
+|0 〉Iz Iz Ix Ix
+Iz Iz Ix Ix|1 〉
+|1 〉
+FIG. 2: Method. a) Mach-Zehnder interferometer, showing
+the central spin state and the effective Hamiltonian of the
+dark spins along each arm. b) Pulse sequence for environment
+assisted magnetometry.
+looses phase coherence due to interactions with the bath,
+it is no longer possible to use it for magnetometry. This
+limits the sensingtime and consequentlythe magnetome-
+ter sensitivity. Spin echo (as well as more sophisticated
+decoupling techniques [16, 17, 26, 30]) can be used to
+prolong the phase coherence of the central spin. Under
+realistic assumptions the coherence time for the pulse se-
+quence presented in this Letter is on the same order of
+thesensorcoherencetimeunderspin-echo Ts
+2. Inessence,
+the decoherencerate ofan entangled state ofthe form (2)
+is dictated by the internal evolution of the strongly cou-
+pled spins, and the relevant decoherence time is thus the
+time it takes before any of these spins have decohered.
+The same is, however, the case for spin echo sequences:
+if a central spin is strongly coupled to nscdark spins, a
+spin flip of a single dark spin will lead to different evo-
+lutions in the two halves of the spin echo sequence, thus
+decohering the state of the central spin. As the source of
+both decoherence processes is the dipole-dipole interac-
+tion among dark spins, the coherence times of spin-echo
+and our procedure are on the same order and the signal
+amplification obtained with our strategy (Eq. 4) thus
+yields a sensitivity enhancement [34].
+To be more quantitative, we analyze the evolution due
+to dipole-dipole couplings in the bath, described by the
+Hamiltonian
+H=|1∝an}bracketri}ht∝an}bracketle{t1|/summationdisplay
+λiIi
+z+/summationdisplay
+i<jκij/parenleftBig
+3Ii
+zIj
+z−− →Ii·− →Ij/parenrightBig
+(5)
+A short time expansion shows that the state fidelity
+at the end of the pulse sequence is given by 1 −
+τ4
+8·/summationdisplay
+i<j/parenleftbig
+1−P2/parenrightbig
+κ2
+ij(λi−λj)2+..., and 1−τ4
+128/summationdisplay
+i<jκ2
+ij×
+/parenleftbig
+7(λ2
+i+λ2
+j)+12λiλj+P2(λi+2λj)(2λi+λj)/parenrightbig
++...,for
+the spin-echo and modified pulse sequences, respectively.
+For experimentally relevant values of polarizations, sim-
+ilar reductions in fidelity occur for both pulse sequences
+[35]. To go beyond the short time expansions we have
+simulated the signal decay for both spin-echo and the
+proposed pulse sequence (see Fig. 3). We compared
+these results to the signal decay when no control se-4
+2 0 4 6 80.2 0.4 0.6 0.8 1.0 
+2 0 4 6 80.2 0.4 0.6 0.8 1.0 0 2 4 60.2 0.4 0.6 0.8 1.0 
+0 2 4 60.2 0.4 0.6 0.8 1.0 
+nc=8 n c=12 
+nc=25 n c=50 
+time time 
+time time 
+FIG. 3: Simulations. Normalized signal decay for the pro-
+posed (red, solid lines), spin-echo sequence (blue, dashed
+lines) and no control (black, dotted lines). We assumed per-
+fect delta function pulses. A leading order cluster expansi on
+was used [23]. 20 dark spins were randomly placed in a cube
+of side-length3√
+20 with the sensor spin at the center. We
+setgµB≡1[m]3/2[s]−1/2to use dimensionless quantities.
+WAHUHA sequences [17] with nc=8, 12, 25 and 50 cycles per
+echo interval were simulated. Each curve is an average over
+10 Monte Carlo simulations. For simplicity we set P= 0.
+quence is applied (for the same environment the decay
+is now described by the dephasing time T∗
+2). We sim-
+ulated a spin bath composed of spin 1/2 paramagnetic
+impurities, undergoing a WAHUHA sequence (which is
+designed to refocus the dipole-dipole coupling of the en-
+vironment spins, but does not cancel out the coupling to
+the external field [17, 26]).
+Adifferentlimitationonthesensingtime τissetbythe
+factthattheorientationofthedarkspinwillnotbestatic
+as assumed. Dipole-dipole couplings among dark spins
+during each τ/4 period of free evolution rotate each spin
+awayfrom the initial direction. This rotation means that
+the spins ceaseto build up a phase difference between the
+twoarmsoftheinterferometerfortime scalescomparable
+to the correlation time of the dark spin bath τd
+c. The
+optimum sensing time is thus T∼min/braceleftbig
+Ts
+2, τd
+c/bracerightbig
+. Since
+in most systems τd
+c≥Ts
+2[17], the optimum sensing time
+of this pulse sequence is comparable to that of spin-echo
+based magnetometry, thus the sensitivity enhancement is
+roughly the same as the signal strength enhancement.
+In conclusion, we proposed a scheme to enhance pre-
+cision measurement by exploiting the possibility to co-
+herently control the ancillary qubits. In solid state
+implementations we are able to exploit dark spins in
+the bath while preserving roughly the same coherence
+times as in spin-echo based magnetometry. Thus sig-
+nal enhancement leads directly to sensitivity enhance-
+ment. For trapped ion implementations we can use im-
+perfect phase gates and still achieve Heisenberg-limited
+sensitivity. Our method has the potential to be applied
+more generally, using different systems and more sophis-
+ticated pulse sequences [26]. It opens the possibility to
+use a broad class of partially entangled states to achieve
+Heisenberglimited metrology, evenin the presenceofdis-ordered couplings, partial control and decoherence.
+Acknowledgments . This work was supported by the
+NSF, ITAMP, DARPA and the Packard Foundation.
+[1] P. O. Schmidt, et al., Science 309, 749 (2005)
+[2] M. H. Schleier-Smith et al., Arxiv 0810.2583
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+[4] J. R. Maze, et al., Nature 455, 644 (2008)
+[5] G. Balasubramanian, et al., Nature 445, 644(2008)
+[6] P. Neumeann, et al., Science 320, 1326 (2008)
+[7] V. Jacques, et al., Phys. Rev. Lett. 102, 0547403 (2009)
+[8] R. J. Epstein, et al., Nat. Phys. 1, 94 (2005)
+[9] M. V. G. Dutt, et al., Science 316, 1312 (2007)
+[10] R. Hanson, et al., Phys. Rev. Lett. 97, 087601 (2006)
+[11] T. Gaebel, et al., Nat. Phys. 2, 408 (2006)
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+220501 (2008)
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+[15] K. Khodjasteh and D. A. Lidar, Phys. Rev. A 75, 062310
+(2007)
+[16] P. Mansfield Journal Physics C 4, 1444 (1971)
+[17] U. Haeberlen, High Resolution NMR in Solids Selective
+Averaging (Academic Press Inc., 1976)
+[18] D. J. Wineland, et al., Phys. Rev. A. 50, 67 (1994)
+[19] B. Yurke, et al., Phys. Rev. A. 33, 4033 (1986)
+[20] P. Cappellaro, et al., Phys. Rev. Lett. 94, 020502 (2005)
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+[22] W.A. Coish and D. Loss, Phys. Rev. B 72, 125337 (2005)
+[23] W. M. Witzel and S. D. Sarma, Phys. Rev. B 74, 035322
+(2006)
+[24] T. Rosenband et al., Phys. Rev. Lett. 98, 220801 (2007)
+[25] S. Boixo and R. D. Somma, Phys. Rev. A 77, 052320
+(2008)
+[26] G. Goldstein et al., In preparation.
+[27] M. Kitagawa and M. Ueda, Phys Rev A 47, 5138 (1993)
+[28] D. Liebfried et al., Science 304, 1476 (2004)
+[29] D. Liebfried et al., Nature 438, 638 (2005)
+[30] D. Liebfried et al., Nature 422, 412 (2003)
+[31] Note that the present method scales favorably with dark
+spins polarization ( P) As seen, the sensitivity scales as
+(P N)−1while it would scale as ( PNN)−1for a GHZ
+state and as1−P2
+√
+Nfor spin squeezed states.
+[31] S. F. Huelga et al., Phys Rev Lett 79, 3865 (1997)
+[32] A. Andre et al., Phys Rev Lett 92, 230801 (2004)
+[33] L-S. Bouchard, V. M. Acosta, E. Bauch, D. Budker,
+arXiv:0911.2533 (2009)
+[34] Although our pulse sequence is more sensitive to dark
+spin dephasing (since it creates superposition states of
+thedarkspins), thisnoise sourceis notrelevantincurrent
+experiments, where noise is dominated by single species
+dipole-dipole interactions [28–30].
+[35] For P≈1 spin-echo achieves longer coherence times
+than the proposed sequence, since flip-flops are quenched
+in a fully polarized bath, they would still be allowed
+in the proposed pulse sequence. However the improve-
+ment in coherence times for spin echo scales poorly as
+∼4√
+1−P2and homonuclear decoupling sequences such
+as WAHUHA can be used to reduce the flip-flop effects.
\ No newline at end of file
diff --git a/1001.0090.txt b/1001.0090.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2f3846e9348c5bbe2c77ba052b69e4583bc769e2
--- /dev/null
+++ b/1001.0090.txt
@@ -0,0 +1,756 @@
+arXiv:1001.0090v1  [cond-mat.supr-con]  31 Dec 2009Isothermal reentrant effect in a mesoscopic cylindrical str ucture of a superconductor
+coated with a normal metal layer
+G. A. Gogadze∗and S. N. Dolya
+B. Verkin Institute for Low Temperature Physics and Enginee ring of the National
+Academy of Sciences of Ukraine 47, Lenin Ave., Kharkov 61103 , Ukraine.
+(Dated: November 10, 2018)
+The coherent phenomena in mesoscopic cylindrical normal me tal (N) - superconductor (S) struc-
+tures have been investigated theoretically. The magnetic m oment (persistent current) of such a
+structure has been calculated numerically and (approximat ely) analytically. It is shown that the
+current in the N-layer corresponding to the free energy mini mum is always diamagnetic. As the field
+increases, the magnetic moment (current) exhibits jumps at certain values of the trapped magnetic
+flux and the NS structure changes to a state with smaller absol ute value of the diamagnetic moment.
+This occurs when the persistent current is unable to screen t he external field. The magnetic moment
+increase stepwise and the system changes into a new stable st ate. The magnetic field penetrates
+into a larger volume of the N-layer. The new state has smaller absolute value of the diamagnetic
+moment. Experimentally, this is interpreted as the presenc e of a paramagnetic addition in the sys-
+tem (paramagnetic reentrant effect). The results obtained a re in qualitative agreement with the
+experiments conducted by P. Visani, A. C.Mota, and A. Pollin i, Phys. Rev. Lett. 65, 1514 (1990).
+PACS numbers: 74.45.+C, 74.50.+r
+I. INTRODUCTION
+The Meissner effect in a mesoscopic cylindrical struc-
+ture consisting of a superconductor coated with a pure
+normal-metal layer has interesting features due to the
+coherent quantum effects in the normal metal. They are
+observable when there is a good contact between S and
+N constituents. This problem within the quasiclassical
+Eilenberger formalism was first investigated theoretically
+by Zaikin [1].
+Recent advanced technologies of preparation of pure
+samples have enabled to investigate the coherence prop-
+erties of mesoscopic samples taking a proper account of
+the proximity effect [2]. The samples were superconduct-
+ing Nb wires with a radius Rof tens of microns coated
+with a thin layer dof high-purity Cu, Ag or Au. Mota
+and co-workers [3] detected surprising behavior of the
+magnetic susceptibility χof a cylindrical NS structure
+(N and S are for the normal metal and the superconduc-
+tor, respectively) at very low temperatures ( T <100
+mK ) in an external magnetic field parallel to the NS
+boundary. Most intriguingly, a decrease in the sample
+temperature below a certain point Tr(at a fixed field)
+produced a paramagnetic reentrant effect: the decrease
+of magnetic susceptibility of the structure is changed to
+an unexpected grow. A similar behavior was observed in
+the isothermal reentrant effect in a field decreasing to a
+certain value H rbelow which the susceptibility started
+to grow sharply. It is emphasized in Ref. [4] that the de-
+tected magnetic response of the NS structure is similar
+to the properties of the persistent currents in mesoscopic
+normal rings.
+∗Electronic address: gogadze@ilt.kharkov.uaThere have been numerous attempts to explain the
+paramagnetic reentrant effect theoretically [5–7]. How-
+ever,thepredictedamplitudesoftheeffectweretoosmall
+( except for [6] ) to account for the experimental facts.
+Fauchere, Belzig and Blatter [6] explain the large param-
+agnetic effect assuming strong repulsive electron-electron
+interaction in noble metals. The proximity effect in the
+N metal induces an order parameter will be shifted by
+πfrom the order parameter ∆ of the bulk superconduc-
+tor. This generates the paramagnetic instability of the
+Andreev states, and the density of states of the NS struc-
+ture exhibits a singlepeak near zero energy. The theory
+developed in Ref. [6] essentially relies on the assumption
+of the repulsive electron interaction in the normal region.
+Maki and Haas [7] made the assumption that in no-
+ble metals (Ag, Au) p-wave superconductors may occur
+with a transition temperature of order 10 mK. Below Tc
+p -wavetriplet superconductivity emerges aroundthe pe-
+riphery of the cylinder. The diamagnetic current flowing
+in the periphery is compensated by a quantized param-
+agnetic current in the opposite direction thus providing a
+simple explanation for the reentrant effect. As in [5], the
+authors of [7] also allow for paramagnetic current in the
+system, which flows in the opposite direction to the dia-
+magnetic current. Its amplitude is sufficient to explain
+the reentrant effect, but this theory says nothing about
+the temperature and field dependences of magnetic sus-
+ceptibility at ultra-lowtemperaturesand in low magnetic
+fields.
+Atheoreticalbasisforunderstandingtheparamagnetic
+reentrant effect has been proposed in [8–10]. The theory
+is essentially based on the properties of the quantized
+levels of the NS structure. The Meissner effect is rather
+special in a superconducting cylinder coated with a pure
+normal-metal layer. The applied magnetic field gener-
+ates superconducting current in the surface layer whose2
+ 
+FIG. 1: Two classes of trajectories in the normal metal of
+the NS structure in a magnetic field: trajectories forming
+the Andreev levels (a): trajectories colliding only with th e
+dielectric boundary (b ). This figure has been taken from [9].
+thickness is equal to the field penetration depth δ. Si-
+multaneously, the Aharonov-Bohm effect generates per-
+sistent current (through the mechanism of the Andreev
+scattering of quasiparticles) in the normal layer near the
+NS boundary. If N and S metals are separated by a di-
+electriclayerdestroyingtheAndreevscattering,theaddi-
+tional current disappears, and the Meissner effect returns
+to its usual form. The levels with energies no more than
+∆ (2∆ is the gap of the superconductor) appear inside
+the normal metal bounded by the dielectric (vacuum) on
+one side and the superconductor on the other side. Be-
+cause of the Aharonov-Bohm effect [11], the spectrum of
+the NS structure in a weak field is a function of the mag-
+netic flux. The specific feature of the Andreev quantum
+levels of the structure is that by varying field H (or tem-
+perature T) each level in the well periodically comes intocoincidence with the chemical potential of the metal. As
+a result, the state of the system suffer strong degeneracy,
+and the density of states on the energy of the NS sample
+experiences resonance spikes [9]. This contributes signif-
+icantly to the magnetic moment and causes a reentrant
+effect. Note that in [9] the calculation was performed for
+orbital susceptibility. In [6] the explanation involves the
+spin (Pauli) susceptibility of the system.
+In this study we calculated the free energy of the NS
+structure and its magnetic moment (current of magneti-
+zation) in the magnetic field. An approximate analytical
+calculationwassupplementedwithanumericalonebased
+on the exact spectrum of Andreev levels [12] in the NS
+contact. Our approach is not based on application of the
+Eilenberger equation. We calculate the thermodynamic
+potential to obtain the magnetic moment.
+II. THEORY AND RESULTS
+A. Spectrum of Quasiparticles of the NS Structure
+Consider a superconducting cylinder with the radius
+Rwhich is concentrically embedded with a thin layer d
+of a pure normal metal. The structure is placed in a
+magnetic field /vectorH(0,0,H) oriented along the symmetry
+axis of the structure. It is assumed that the field is weak
+to the extent that the effect of twisting of quasiparticle
+trajectoriesbecomesnegligible. It actuallyreducesto the
+Aharonov-Bohm effect [11], i.e., allows for the increment
+in the phase of the wave function of the quasiparticle
+moving along its trajectory in the vector potential field.
+We proceed with a simplified model of NS structure in
+which the order parameter magnitude changes stepwise
+at the NS boundary ( ∆( x) = Θ(−x), Im(∆) = 0, a
+bulk superconductor in the region x <0 and a normal
+metal layer in the region 0 ≤x≤d). It is also assumed
+that the magnetic field does not penetrate into the su-
+perconductor. The coherent properties observed in the
+pure normal metal can be attributed to its large ”coher-
+ence” length ξN(T) =/planckover2pi1·vF
+π·kBT( vFis the Fermi velocity,
+kBis the Boltzmann constant) at very low temperatures
+(ξN(T)≫d). Besides, the spectrum of quasiparticles
+was obtained assuming a negligible curvature of the NS
+boundary.
+One can easily distinguish two classes of trajectories,
+insidethe normalmetal. Oneofthem includesthetrajec-
+tories which collide in succession with the dielectric and
+NS boundaries ( Fig.1). The quasiparticles moving along
+these trajectories have energies |E|<∆ and are local-
+ized insidethe potentialwell bounded bya highdielectric
+barrier( ≃1eV) on one side and by the superconducting
+gap ∆ on the other side (∆ = 3 .56·kBTc/2, ∆(Nb) ≈
+1.42 meV). On its collisions, the quasiparticle is reflected
+specularly from the dielectric and experiences the An-
+dreev scattering at the NS boundary [12]. We introduce
+an angle αat which the quasiparticle hits the dielectric
+boundary. The angle is measured in the positive direc-3
+0 3 6 9 12 15 0.00 0.25 0.50 0.75 1.00 1.25 
+σ = 0.644 
+αc = 36 0
+Δ(Nb) = 1.42 meV NbAu ν(Φ) / ν(Φmax )
+ΦΦmax 
+ 
+FIG. 2: The dependence of the density of states of the NS
+structure on the magnetic flux Φ ( E=EF= 0 ). Normal-
+ization was performed for the flux Φ maxcorresponding to the
+highest value of ν(Φ) ( Φ max≈2.175 ).
+tion from the normal to the boundary ( Fig.1). In this
+case the first class contains the trajectories with αvary-
+ing within the range −αc≤α≤αc(αcis the angle at
+which the trajectory touches the NS boundary, sin( αc)=R
+R+d). Another class includes the trajectories whose
+spectra are formed by collisions with the dielectric only,
+i.e., the trajectories with α > αc. The two groups of tra-
+jectories produce significantly different spectra of quasi-
+particles. The distinctions are particularly obvious in
+the presence of the magnetic field. The trajectories with
+α/lessorsimilarαcform a spectrum of Andreev levels which con-
+tains an integral of the vector potential field. The spec-
+trum characterizes the magnetic flux through the area of
+the triangle between the quasiparticle trajectory and the
+part of the NS boundary. It also determines the mag-
+nitude of the screening current produced by ”particles”
+and ”holes” in the N layer. These states are responsible
+for the reentrant effect. The trajectories with α > αcdo
+not collide with the NS boundary. The states induced by
+these trajectories are practically similar to the ”whisper-
+ing gallery” type of states appearing in the cross section
+of a solid normal cylinder in a weak magnetic field [13],
+[14]. The size of the caustic of these trajectories is of the
+order of the cylinder radius, i.e., they correspond in high
+magnetic quantum numbers. The spectrum thus formed
+carries no information about the parameters of the su-
+perconductor, and it is impossible to meet the resonance
+condition in this case. These states make a paramagnetic
+contribution in the thermodynamics of the NS structure
+but their amplitude is small ( ∼1/(kF·R) ). It is there-
+fore discarded from further consideration. Our interest
+will be concentrated on the trajectories with |α| ≤αc.
+The spectrum of quasiparticlesof the NS structure can
+be obtained easily using the multidimensional quasiclas-
+sicalmethod generalizedforthe caseofthe Andreev scat-
+teringin the system [15], [16]. After collisionwith the NS
+boundary the ”particle” transforms into a ”hole”. The
+”hole” travels practically along the path of the ”particle”
+but in the reverse direction ( Fig.1).
+The spectrum was derived by quantizing the adia-
+batic invariant1
+2π/contintegraltext/vectorP·d/vector s, where /vectorP=/vector p+e
+c/vectorA,/vectorA=
+(0,Ay(x),0),/vectorP0=/vector p0−|e|
+c/vectorAfor a ”particle” and /vectorP1=
+/vector p1+|e|
+c/vectorAfor a ”hole”. Note that each collision with the
+NS boundary multiplies the wave function amplitude of
+the quasiparticle by a factor of exp( −iarccos(E/∆)).
+LetL0be the length of the quasiparticle trajectory be-
+tween the collisions at the boundaries of the N layer. We
+thus [8], [9] arrive at the expression for the spectrum of
+the Andreev levels in the NS structure:
+En(q,α,Φ) =π/planckover2pi1vL(q)
+L0/bracketleftbigg
+n+1
+πarccos/parenleftbiggEn(q,α,Φ)
+∆/parenrightbigg
+−tan(α)
+πΦ/bracketrightbigg
+. (1)
+Here v L(q) =/radicalbig
+p2
+F−q2/m∗,L0is length of the quasi-
+particle trajectory, pFis the Fermi momentum, qis the
+quasiparticle momentum component along the cylinderaxis (|q| ≤pF),m∗is the effective mass of the quasipar-
+ticle, and Φ 0=hc/2eis the superconducting flux quan-
+tum. The factor Φ appearing in the last term in Eq. (1)4
+0 3 6 9 12 15 0.0 0.3 0.6 0.9 1.2 
+σ = 0.644 
+αc = 36 0
+Δ(Nb) = 1.42 meV 
+T = 10 mK NbAu Ω(Φ) / Ω(Φmax )
+ΦΦmax 
+ 
+FIG. 3: Free energy (normalized per value Ω(Φ max)) as a
+function of the flux Φ.has the meaning of ”phase”
+Φ =2π
+Φ0d/integraldisplay
+0Ay(x)dx (2)
+which is dependent on the vector potential field /vectorA=
+(0,Ay(x),0). The spectrum of Eq. (1) is similar to
+Kulik’s spectrum [17] for the current state of the SNS
+contact. However, Eq. (1) includes an angle-dependent
+magnetic flux instead of the phase difference of the con-
+tacting superconductors.
+The length of the quasiparticle trajectory ( 2 AB) is
+readily found from Fig.1using the sine and cosine theo-
+rems:
+AB=d·
+cos(α)−/radicalBig
+sin(αc)2−sin(α)2
+1−sin(αc)
+(3)
+where sin( αc) =R
+R+d,−αc≤α≤αc. The spec-
+trum in Eq. (1) was derived assuming that the mean
+free path of the quasiparticles was much longer than the
+cross-section perimeter of the cylinder and the require-
+mentd≪Rwas obeyed. In this limit L0= 2AB∼=
+2d/cos(α)/parenleftbigg
+lim
+αc→π/2AB=d/cos(α)/parenrightbigg
+, i.e., the radius
+Rdrops out from the expression for the spectrum. Al-
+though the boundary curvature of the sample is disre-
+garded, the information about its cylindrical geometry
+is retained through a correct choice of the limits of in-
+tegration for the angle α:−αc≤α≤αc. Putting L0
+= 2d/cos(α), we obtain the following expression for the
+spectrum ( as in [9] ):
+En(q,α,Φ) =π/planckover2pi1vL(q)cos(α)
+2d/bracketleftbigg
+n+1
+πarccos/parenleftbiggEn(q,α,Φ)
+∆/parenrightbigg
+−tan(α)
+πΦ/bracketrightbigg
+. (4)
+The spectrum in Eq. (4) has an important feature
+[8], [9]. As the ”phase” Φ Eq. (2) changes, the density
+of states exhibits resonance spikes. Every time when the
+Andreevlevelcoincideswith thechemicalpotentialofthe
+metal, the state of the NS structure suffers strong degen-
+eracy showing up spikes. The dependence of the density
+of states upon the magnetic flux calculated numerically
+for the NS system is illustrated in Fig.2.
+Note that in [1], [18] the diamagnetic current of NS
+structure was calculated using αc=π/2 instead of theupper limit of integration for the angle ( αc< π/2 ) and
+assumingimplicitlyaninfinitelylargenumberofAndreev
+levels. Therefore, these resultscannot be conformed with
+the experimental findings. The reason is not only that
+the calculation was made for a flat geometry rather than
+foracurvedNSboundary. Numericalanalysisshowsthat
+an adequate interpretation of the experimental magnetic
+moment-field dependence is possible only with a proper
+choice of the upper limit of integration with respect to
+α. Ifα > α corαc=π/2, the consideration includes5
+effectively the states unrelated to the Andreev levels.
+B. Self-consistent equation
+To calculate the ”phase” Φ( T,H) from Eq. (2), we
+should know the distribution of the vector potential field
+inside the normal metal. Zaikin [1] has shown that the
+proximity effect caused the Meissner effect leads to an
+inhomogeneous distribution of the vector potential field
+over the N layer of the structure:
+Ay(x) =µ0Hx+µ0j·x(d−x/2) (5)
+whereµ0is the permeability of free space ( the SI sys-
+tem of units is employed, the geometry of the proxim-
+ity model system is the same as in [19] Fig 1). This
+expression can be obtained from the Maxwell equa-
+tionrot/parenleftBig
+rot/parenleftBig
+/vectorA/parenrightBig/parenrightBig
+=µ0/vectorj= (0,µ0j,0) assuming that
+the current density is uniform over the cross-section of
+the conductor and the boundary condition /vectorA/vextendsingle/vextendsingle/vextendsingle
+x=0= 0,
+rot/parenleftBig
+/vectorA/parenrightBig/vextendsingle/vextendsingle/vextendsingle
+x=d=/vectorB(0,0,µ0H) is met. The fact that the
+current density is constant in the N-layer follows from
+spatial homogeneity of the density of Andreev levels over
+the whole thickness of the N-layer. In cylindrical geome-
+tries, if the N-layer thickness is not thin compared to the
+radius ( d/greaterorsimilarR), the current density is not constant in
+space [20].
+The magnetic moment per unit length of the N-layer
+M(H) =−1
+µ0dΩ(T,Φ(H))
+dH( z-component ) and the current
+density/vectorjare related via a relation
+M(T,H) =1
+2/integraldisplay
+VN/bracketleftBig
+/vector r×/vectorj(/vector r)/bracketrightBig
+zdV=−1
+µ0dΩ
+dH(6)
+whereVNisavolumeoftheN-layerunityheight, Ω( T,Φ)
+is the free energy per unit length. Then, the current
+appears from Eq. (6) is a function of the magnetic flux
+Φ and temperature:
+j=−1
+πR2d·µ0dΩ(T,Φ(H))
+dH(7)
+We can write down the self-consistent equation for
+Φ(T,H) using Eqs. (2), (5) and (7):
+Φ(T,h) = h+ η·M∗(Φ)∂Φ(T,h)
+∂h(8)
+where h =H
+H0, H0=Φ0
+πd2·µ0,η=d2
+3R2Φ0H0,M∗(Φ) =
+−dΩ
+dΦ[21]. To describe the field effect on the magnetic
+moment
+M(T,H) =M∗(T,Φ)
+µ0H0·∂Φ(T,h)
+∂h(9)
+of a NS structure, it is necessary to find the dependence
+Φ(T,h) from Eq. (8). After calculating the free energy0 3 6 9 12 15 -1 012 σ = 0.644 
+αc = 36 0
+Δ(Nb) = 1.42 meV 
+T = 10 mK 
+NbAu M*(Φ)/|M *(Φmax )|
+ΦΦmax 
+ 
+FIG. 4: The magnitude of M∗(Φ)/|M∗(Φmax)|as a function
+of the flux Φ.
+Ω(T,Φ) from the spectrum of Eq. (4), we can estimate
+the magnetic moment Eq. (9) using a solution of the dif-
+ferential equation (8): M(T,H) =1
+η·µ0H0(Φ(T,h)−h).
+In this article, we used the ”thermodynamic” approach
+Eq. (7) whichleads tothe first orderdifferential equation
+(8) for the function Φ( T,h). However, another approach
+based on of the Eilenberger formalism ( [1], [18] ) yields
+analgebraic self-consistent equation ( Eq. 22 in ([18])
+) for the ”phase” Φ( T,h):
+Φ(T,h) = h+ const·j(Φ) (10)
+( in notation Eq., (8) ). In this equation the function
+j(Φ)isdescribedbytheexpressionEq.,13in[1]. Clearly,
+both approaches (8), (10) will lead to quite different de-
+pendencies of M(T,H) on magnetic field and tempera-
+ture. In our point of view, the self-consistent equation
+(10) cannot be applied to the cylindrical NS structures.6
+While deriving the expression for a current [1], the au-
+thor assumed that αc=π/2 (the case of a plane). This
+means account for the contribution non-Andreev states
+(α > α c). For the calculation of thermodynamic po-
+tential in equation (8) we shall use the actual magnitude
+of the parameter αc(sin(αc) =R/(R+d) ). The ap-
+proximation d << R was used by us in the derivation
+of the spectrum (4) only. In other words, the neglect of
+the curvature of cylindrical samples (i.e. the path length
+of quasiparticles was choosen as d/cos(α) (3)) does not
+entail the need of account for the states with α > α cfor
+the cylindrical NS of structures ( Fig.1).
+C. Analytical estimation of the magnetic moment
+of the NS structure
+We proceed from the expression for the free energy of
+a NS contact:
+Ω =−kBT/summationdisplay
+n,q,α,sln/parenleftBig
+1+e−En(q,α)
+kBT/parenrightBig
+(11)where the summation is over the spin variable s=±1
+andallthestatesrelatedtothequasiparticlestrajectories
+with|α| ≤αcin Eq. (4). Then, we obtain the following
+expression for the free energy per unit length L
+Ω(Φ) =−R·kBT
+π/planckover2pi12n=∞/summationdisplay
+n=−∞/integraldisplayαc
+−αc/integraldisplaypF
+−pFln/parenleftbigg
+1+exp/parenleftbigg
+−En(q,α,Φ)
+kBT/parenrightbigg/parenrightbigg/radicalBig
+p2
+F−q2cos(α)dqdα (12)
+where the energy En(q,α,Φ) is given by the exact ex-
+pression for the spectrum in Eq. (4). For simplification,
+weintroduce the dimensionlessquantities εn=En(q,α,Φ)
+∆,
+σ=/planckover2pi1·pF
+2d·∆·m∗,−1≤εn≤1, ( ∆ is the superconduct-
+ing gap ) and perform the change of variables {q,α} →
+{u,v}:
+
+
+u=σ·/radicalbigg
+1−/parenleftBig
+q
+pF/parenrightBig2
+·cos(α)
+v=σ·/radicalbigg
+1−/parenleftBig
+q
+pF/parenrightBig2
+·sin(α).(13)
+The spectrum and the free energy become
+εn= [nπ+arccos( εn)]·u−Φ·v, (14)
+Ω(Φ) =c1Tn=∞/summationdisplay
+n=0/integraldisplay/integraldisplay
+Suln/parenleftBig
+2cosh/parenleftBig
+εn
+2c2T/parenrightBig/parenrightBig
+dudv
+√
+σ2−u2−v2(15)
+wherec1=−2R·∆·c2
+π·/parenleftbigpF
+σ·/planckover2pi1/parenrightbig2,c2=kB
+∆, 0≤u≤σ,
+−σsin(αc)≤v≤σsin(αc),εndef=εn(u,v,Φ), an inte-
+gration domain Sis a sector of a circle ofradius σ. In the
+expression (15) we also took into account the symmetry
+of the spectrum in Eq. (14):
+ε−|n|(u,v,Φ) =−ε|n|−1(u,−v,Φ). (16)Making use of the relation
+dεn
+dΦ=−v·/radicalbig
+1−ε2n
+u+/radicalbig
+1−ε2n(17)
+we evaluate the derivative of the free energy with respect
+to the flux M∗(Φ) =−dΩ
+dΦ:
+M∗(Φ) =c3n=∞/summationdisplay
+n=0/integraldisplay/integraldisplay
+Suvtanh/parenleftBig
+εn
+2c2·T/parenrightBig/radicalbig
+1−ε2ndudv
+/parenleftBig
+u+/radicalbig
+1−ε2n/parenrightBig√
+σ2−u2−v2
+(18)
+wherec3=R·∆
+π/parenleftbigpF
+σ·/planckover2pi1/parenrightbig2. Eqs. (9), (8) and (18) fully de-
+termine the non-linear magnetic response of a cylindrical
+NS structure to an externally applied magnetic field H.
+The integral expression of Eq. (18) suggests that
+M∗(Φ) is the odd function of the flux Φ: M∗(Φ) =
+−M∗(−Φ). A linear term of the function M∗(Φ) has
+been determined from an approximate estimation of the
+integral in Eq. (18). This calculation is similar to that
+in the Attachment of [9]. The final expression for the
+magnetic moment is
+M(T,h)≃M0n0/summationdisplay
+n=0lncosh/parenleftbigTA·˜n
+2T/parenrightbig
+˜n3/bracketleftbigg
+1+/parenleftBig
+Φ(T,h)
+π·˜n/parenrightBig2/bracketrightbigg3
+2(19)7
+where ˜n=n+1
+2,TA=/planckover2pi1·vF
+2πd·kBis Andreev temperature,
+M0=−c3·σ2·/parenleftBig
+T
+TA/parenrightBig
+·Φ(T,h)·∂Φ(T,h)
+∂h, the ”phase”
+Φ(T,h) is a solution of the differential equation (8), n0
+is the number of Andreev levels in the potential well (
+n0≈Φ(T,h)
+πtan(αc) ). Eq. (19) shows that the mag-
+netic moment is diamagnetic in the range of small fields
+( Φ(h) = const·h,const >0 ) and allows for the contri-
+butions of ”particles” and ”holes”.
+D. Numerical results
+Let uscomparetwoapproachesdescribedaboveforthe
+calculation of the magnetic moment of the NS structure.
+Fig.5shows the function M∗(Φ) and dependency of the
+current on the magnetic flux obtained in the Green’s
+function approach. For comparison, we obtained the de-
+pendence M∗(Φ) at the same value αc=π/2 as was
+used in the derivation of the formula j(Φ) in [18]. In
+the initial part (linear in the Φ) both curves coincide. In
+this approximation (Φ(h) = const·h) the self-consistent
+equation (9) turns into Eq. (10). Thus, at small values
+of the magnetic field we would obtain the same field de-
+pendence of the magnetic moment M(T,h) for the NS
+structure in both approaches. However, in large fields
+the behavior M(T,h) is quite different. To calculate
+M(T,h) from Eq. (9), we have used the following physi-
+cal values of the NS structure: R= 8.3µm,d= 3.2µm,
+(αc= 360), vF(Au) = 1 .4·108cm/s, ∆(Nb) = 1 .42 meV
+(σ= 0.644,η·c3= 5.3·103, H0= 51A/m = 0 .64 Oe
+). The selected parameters are close to those used in the
+experiment [3, 4, 22].
+The results of calculation according to formulas (15)
+and (18) are illustrated in Fig. 3, 4 . While plotting
+Fig. 3, the nonzero quantity Ω(Φ = 0) was omitted.
+The dependence M∗(Φ) =−dΩ
+dΦ(Fig. 4) crosses the ab-
+scissa thereby determining singular points of the differ-
+ential equation (8). The dependence Φ(h ,T) calculated
+through numerical solution of the self-consistency equa-
+tion (8) exhibits jumps and is illustrated in Fig.6afor
+the branches correspondingto the minimum of the Gibbs
+free energy [23]:
+G(T,H) = Ω(T,H)+1
+2µ0/integraldisplay
+VN/parenleftBig
+/vectorB−µ0/vectorH/parenrightBig2
+dV(20)
+where/vectorB=rot/parenleftBig
+/vectorA/parenrightBig
+,/vectorH=/vectorH(0,0,H). The magnetic mo-
+mentM(h) and the free energy Ω(h) as functions of the
+magnetic field are shown in Fig. 7a, Fig. 6b. Each
+jump ∆Ω of the free energy (see Fig. 6b ) is accompa-
+nied by the jump of the magnetic moment ∆ M(seeFig.
+7a) in such a way that the Gibbs free energy (20) is a
+continuous function of versus the magnetic field h. We
+have not performed an analysis of behavior Gibbs free
+energy (20) near the points where the magnetic moment
+has jumps because it is beyond the semi-classicalapprox-
+imation adopted in this article.0 1 2 3 4-3 -2 -1 0
+( )
+( )1 . 0**
+MMΦ
+ ( )
+( )1 . 0jjΦ
+ A. D. Zaikin at el., [18]
+NbAu 
+ σ = 0.644, αc = 90 o, 
+T = 10 mK 
+Φ
+ 
+FIG. 5: The magnitudes M∗(Φ)/|M∗(0.1)|and
+j(Φ)/|j(0.1)|as the functions of the flux Φ (for expla-
+nation see text).
+III. CONCLUSIONS
+The goal of our study was to interpret the experiments
+performed by A. C. Mota et al., [3, 4], who detected an
+anomalous behavior of the magnetic susceptibility of the
+NS structure in a weak magnetic field at millikelvin tem-
+peratures. Previously [8, 9], the anomalous behavior of
+the NS structure was attributed to the properties of the
+quantized Andreev levels depending on the magnetic flux
+that varies with the temperature and magnetic field. We
+used the ”thermodynamic” approach for the calculation
+ofthemagneticmomentofnormalregionofNSstructure.
+Within the framework of the self-consistent equation (8),
+we have managed to trace the role of the parameter αc
+in thermodynamics of NS of structures ( Fig. 4andFig.
+5). Failure connected with the use of the quasi-classical8
+0 3 6 9 12 15 0246810 12 
+0 3 6 9 12 15 0.0 0.4 0.8 1.2 H0 = 51 A/m 
+σ = 0.644, 
+αc = 36 0
+T = 10 mK. 
+a)NbAu Φ( h ) 
+h = H / H 0hmax 
+hmax b)NbAu Ω( h ) / Ω( h max  ) 
+h = H / H 0ΔΩ 
+ 
+FIG. 6: a) the dependence of Φ( T,h) on the magnetic field h.
+b) the relative free energy Ω(h) /Ω(hmax) (Φ(h max) = Φ max)
+as a function of the magnetic field h.
+Green-function technique [1] for the explanation of the
+experimental data (Mota et al) is a consequence of ac-
+count for states which are non-Andreev ones ( α > α c
+) for the cylindrical NS structures. Geometrically, this
+can be seen from figure Fig. 1. The quasiparticle tra-
+jectories ( α > α c) hitting the dielectric boundary only
+are responsible for the paramagnetic current of a small
+amplitude ( ∼1/(kF·R)) (we neglected this current).
+The proximity effect is crucial for the reentrant effect.
+The amplitude of the resonance spikes in the density of
+statesstronglydepends onthe probabilityofthe Andreev
+reflection at the NS boundary. It is therefore assumed
+that the normal metal and the superconductor are in a
+good electric contact. The spectrum Eq. (4) was ob-
+tained by the method of multidimensional quasi-classical
+approach [15, 16]. In doing so, we assumed (i) the condi-0 10 20 30 40 50 60 -15 -12 -9 -6 -3 00 3 6 9 12 -1.5 -1.0 -0.5 0.0 
+H2exp H1exp b)T = 7 mK 
+d = 3.2 µm
+R = 8.3 µm
+αc = 36 0
+NbAu 4πMdc ( G ) 
+H ( Oe ) H2calc H1calc 
+σ = 0.644 
+αc = 36 0
+T = 10 mK 
+H0 = 51 A/m NbAu M(h) / |M(h max )|
+ha)
+ 
+FIG. 7: a) The magnetic moment of the NS structure versus
+the magnetic field h. b) Isothermal dc-magnetization curve
+atT= 7 mK for the sample 41AuNb [22]. Used by courtesy
+of A. C. Mota.
+tion of smallness of N-layer thickness in comparison with
+the radius of the cylindrical superconductor, (ii) the va-
+lidityofthemodelofastepwise-varingorderparameterof
+thestructure,(iii)theindependenceof∆onthemagnetic
+field. This permitted us to pass over from the curved NS
+boundary to a flat one. The information about the cylin-
+drical geometry of the sample was retained because the
+critical angle at which a quasiparticle hits the dielectric
+boundary αis smaller than αc. The problem was further
+simplified by assuming that the reflected quasiparticle
+performed a reciprocating motion, i.e., a ”particle” and
+a ”hole” pass along the same trajectory but in opposite
+direction. Actually, here there exists a lot of quasi recip-
+rocating trajectories with the energies near ε∼0. These
+trajectories lead to the spikes in the density of states and
+were taken into account for numerical computation. The9
+numerical calculation shows that the nonlinearity of flux
+Φ(T,h)−field dependence ( T = constant ) ( Fig.6a)
+gives rise to quite interesting features of M(T,H). The
+magneticmomentintheNlayerappearstobealwaysdia-
+magnetic. Theparamagneticcontributiontocurrent(the
+paramagnetic reentrant effect) was not detected. How-
+ever, we have obtained the stepwise change in absolute
+value of the magnetic moment with increasing magnetic
+field (Fig.7a). This behavior can be interpreted as an
+appearance in the magnetic moment the paramagnetic
+additives. A behaviour of the NS structure changes from
+one stable state to another and the magnetic field pene-
+trates further into a bulk of the N layer. The new state
+has smaller absolute value of the diamagnetic moment,
+which is interpreted experimentally as an evidence of a
+paramagnetic addition in the system ( Fig.7b). When
+the field grows further, |M(H)|increases again until its
+new value makes the system to jump to a next stable
+state with a smaller absolute value of the diamagnetic
+moment, and the magnetic field penetrates deeper inside
+the normal metal ( Fig.7a). The number of the moment
+jumps depends on the number of Andreev levels in the
+NS structure. Under the isothermal condition, the values
+of the magnetic field at which jumps occur, do not coin-
+cide when the magnetic field changesfrom small to largervalues and in the opposite direction because of a special
+dependence of the Gibbs free energy on the field. This
+sort of hysteresis was observed experimentally in [4, 22]
+(Fig.7b).
+Numerical comparison between data presented at
+Fig.7bandFig.7ashows that M(T,H) (Fig. 7a )
+gives the qualitative description of the experimental data
+which obey the scaling rule: Hexp
+2/Hexp
+1≃Hcalc
+2/Hcalc
+1≃
+5/2. For the quantitative description of the temperature
+and magnetic field dependence of the magnetic moment
+NSofstructures, it willbeimportanttotakeintoaccount
+the exact spectrum of Andreev levels, the latter is sup-
+posedly possible within the framework of the Bogoliubov
+– de Gennes equations only.
+Note that our consideration was entirely based on
+the model of free electrons without account of strong
+electron-electron repulsion.
+IV. ACKNOWLEDGEMENT
+The authors thank A. N. Omelyanchouk, I. O. Kulik,
+I. V. Krive and G. I. Japaridze for useful discussions.
+[1] A. D. Zaikin, Solid State Comm. 41, 533 (1982).
+[2] A. C. Mota, P. Visani, and A. Pollini, J. Low Temp.
+Phys.76, 465 (1989).
+[3] P. Visani, A. C. Mota, and A. Pollini, Phys. Rev. Lett.
+65, 1514 (1990).
+[4] A. C. Mota, P. Visani, A. Pollini, and K. Aupke. Physica
+B197, 95 (1994).
+[5] C. Bruder, and Y.Imry, Phys.Rev.Lett. 80, 5782 (1998).
+[6] A. L. Fauchere, W. Belzig, and G. Blatter, Phys. Rev.
+Lett.82, 3336 (1999).
+[7] K. Maki and S. Haas, Physica C 341-348, 2667 (2000).
+[8] G. A. Gogadze, Fiz. Nizk. Temp. 31,120 (2005) [Low
+Temp. Phys. 31, 94 (2005)].
+[9] G. A. Gogadze, Fiz. Nizk. Temp. 32, 716 (2006) [Low
+Temp. Phys. 32, 546 (2006)].
+[10] G. A. Gogadze, Fiz. Nizk. Temp. 34, 225 (2008) [Low
+Temp. Phys. 34, 173 (2008)].
+[11] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
+[12] A. F. Andreev, Zh. Eksp. Teor. Fiz. 46,1823 (1964) [Sov.
+Phys. JETP 9, 1228 (1964)]
+[13] E. N. Bogachek and G. A. Gogadze, Zh. Eksp. Teor. Fiz.
+63, 1839 (1972) [Sov.Phys.JETP 36, 973 (1973)][14] N. B. Brandt, E.N. Bogachek, N. D. Gitsu, G.A.
+Gogadze, I. O. Kulik, A. A. Nikolaeva, and Ya. G. Pono-
+marev, Fiz. Nizk.Temp. 8, 718(1982) [Sov.J.LowTemp.
+Phys.8, 358 (1982)]
+[15] J. B. Keller and S.I. Rubinow, Ann. Phys ( N. Y. ) 9, 24
+(1960)
+[16] G. A. Gogadze, R. I. Shekhter, and M. Jonson, Fiz. Nizk.
+Temp.27, 1237(2001) [LowTemp.Phys. 27, 913(2001)].
+[17] I. O. Kulik, Zh. Eksp. Teor. Fiz. 57, 1745 (1969) [Sov.
+Phys. JETP 30, 944 (1969)]
+[18] A. V. Galaktionov and A. D. Zaikin, Phys. Rev. B
+67,184518 (2003).
+[19] W. Belzig, C. Bruder and A. L. Fauch` ere, Phys. Rev. B
+58, 14531 (1998).
+[20] W. Belzig, C. Bruder, and G. Schon, Phys. Rev. B 53,
+5727 (1996).
+[21] G. A. Gogadze S. N. Dolya, Fiz. Nizk. Temp. 35, 932
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+[22] F. B. M¨ uller-Allinger, A. C. Mota, Phys. Rev. B 62, 6120
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\ No newline at end of file
diff --git a/1001.0091.txt b/1001.0091.txt
new file mode 100644
index 0000000000000000000000000000000000000000..860bfc02b2e6d1bf6b74b770be0d95e11bf62063
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+++ b/1001.0091.txt
@@ -0,0 +1,1098 @@
+arXiv:1001.0091v3  [math-ph]  23 Jul 2010RigidSymmetriesandConservationLaws
+in Non-LagrangianFieldTheory
+D.S. Kaparulin, S.L. Lyakhovichand A.A. Sharapov
+Department of Quantum Field Theory, TomskState University ,
+Leninave. 36, Tomsk634050, Russia
+ABSTRACT . Making use of the Lagrange anchor construction introduced earlier
+toquantizenon-Lagrangianfieldtheories,weextendtheNoe thertheorembeyond
+theclass ofvariationaldynamics.
+∗This work was partially supported by the RFBR grant 09-02-00 723-a, by the grant from Russian Federation President Progr amme
+of Support for Leading Scientific Schools no 3400.2010.2., b y the State Contract no 02.740.11.0238 from Russian Federal Agency
+for Science and Innovation, and also by Russian Federal Agen cy of Education under the State Contracts no P1337 and no P22. SLL
+appreciates partial support from theRFBR grant 08-01-0073 7-a.SYMMETRIES AND CONSERVATION LAWS 2
+Contents
+1 Introduction 3
+2 Classical fields 6
+3 Symmetries, identities andconservation laws 9
+4 TheLagrange anchor 15
+5 Proper symmetries andproperdeformations 18
+6 Field-theoretical examples 25
+6.1p-form fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
+6.2 Self-dual p-form fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
+6.3 Chiral bosons in twodimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
+7 Conclusion 31SYMMETRIES AND CONSERVATION LAWS 3
+1 Introduction
+In thispaperweextendtheNoethertheorembeyondtheclasso fvariationaldynamics.
+The classical field theory is completely defined by equations of motion. While the
+least action principle is not vital at classical level, it pr ovides a number of useful tools for
+studying classical dynamics. In particular, given an actio n functional enjoying infinites-
+imal symmetries, one can derive conserved currents. For non -variational equations, the
+infinitesimalsymmetriesdo notnecessarily result in theco nservationlaws and viceversa.
+Allthesymmetriesaredividedintogaugeandrigid. Thegaug esymmetriesareunambigu-
+ously connected to Noether’s identities (also called the st rong conservation laws) as far
+as equations are variational. If the equations do not come fr om the least action principle,
+this relationship (sometimes called the second Noether the orem) is generally invalid: It is
+possible to have gauge invariant field equations without Noe ther’s identities or equations
+possessingstrongconservationlawswithoutbeing gaugein variant.
+In the gauge field theory, the existence of action is a key prer equisite for constructing
+the standard BRST theory either in theHamiltonianBFV form [ 1] orin theBV field-anti-
+field formalism [2] (see [3] for review). The BRST theory, in i ts turn, provides the most
+general tools for quantizing gauge theories [1, 2, 3]. Also, the BRST formalism serves
+for the study of various classical problems such as construc ting consistent interactions in
+gaugemodels[4]oridentifyingthenontrivialconservatio nlawsand rigidsymmetries[5].
+Now,itisknown[6]thattheBRSTtheorycanbeextendedbeyon dtheclassofdynam-
+ical systems whose equations of motion are variational. Cla ssically, the BRST complex
+can always be constructed for any regular system offield equa tions, variational or not [3],
+[6], [7]. The quantization, however, requires an extra ingr edient, besides the field equa-
+tions. In the framework of the deformation quantization it i s theweak Poisson structure
+[6] (see also [8]). The existence of the weak Poisson structu re is much less restrictive
+for the evolutionary equations than the requirement to be Ha miltonian. In the framework
+of the path-integral quantization, the corresponding extr a ingredient was first introduced
+in [7] and called the Lagrange structure . Again, the existence of the Lagrange structure
+is much less restrictive for the equations than the requirem ent to be variational or admit
+an equivalent variational reformulation. Whenever the Lag range structure is known for
+a given system of equations, the theory can be consistently q uantized in three equivalent
+ways. First,theoriginalnon-Lagrangianfieldtheoryin ddimensionscanbeconvertedinto
+equivalentLagrangiantopologicalfieldtheoryin d+1. Thelattercanbethenquantizedby
+theusualBVprocedure[7]. Second,thegeneratingfunction alofGreen’sfunctionscanbeSYMMETRIES AND CONSERVATION LAWS 4
+defined through the generalized Schwinger-Dyson equation [ 9], [10]. Third, the original
+non-Lagrangianfieldtheorycanbeembeddedintoanaugmente dLagrangiantheoryinthe
+sameddimensions[11].
+Besidesprovidingthebasisforquantizingnon-Lagrangian fieldtheories,theLagrange
+structure binds together gauge symmetries and the strong co nservation laws, although the
+relationship is more relaxed than for the variational case [ 7]. In the first-order formalism,
+for example, the constraints and gauge symmetries remain fu lly unrelated [12] unless the
+equations are Hamiltonian. A certain correlation is set up b y the weak Poisson structure,
+although it is not so rigid as the relation between the first-c lass constraints and gauge
+symmetriesintheHamiltonianconstraineddynamics.
+In this paper we will show that the Lagrange structure allows one to connect the rigid
+symmetries with the conservation laws. This relationship r eveals itself in a different way
+than the link between the gauge symmetries and the strong con servation laws in non-
+Lagrangian theories equipped with the Lagrange structure. In the next sections we give
+accurate definitions for symmetries, identities and conser vation laws and, by making use
+of the Lagrange structure, establish a relationship betwee n them. The general construc-
+tion is then exemplified by a class of non-Lagrangian models, where some fundamental
+symmetriesand/orconservedcurrents can beexplicitlyfou nd andcompared.
+To give a preliminary impression of the connection between t he symmetries and con-
+servationlaws,belowintheIntroduction,wereformulatet hestandardNoethertheoremin
+the way that illuminatesthe points which are important for e xtendingthe theorem beyond
+theclass ofLagrangiandynamics.
+LetMdenote the configuration space of fields ϕi. Hereinafter we use De Witt’s con-
+densednotation[17]wherebytheindex i,labellingthefields,includesalsothespace-time
+coordinates {xµ}. Thesmoothfunctions f(ϕ)onMareidentifiedwiththelocalfunction-
+als of fields. In terms of the condensed notation, the Lagrang ian equations of motion are
+givenbythecomponentsofanexact1-form dSonM,whered =δϕiδ
+δϕiisthevariational
+differential and Sis the action functional. In general, the field equations can well be the
+componentsofasectionofanotherbundleover M,notnecessarily T∗M. Forexample,the
+tensortypeofequationscandifferfromthatoffields(seeth enextsection). Here,however,
+we restrict our discussionto equations whoseleft hand side s are givenby thecomponents
+of a local 1-form T=Ti(ϕ)δϕionM. Then the necessary condition for the existence of
+an action forthefield equations Ti(ϕ) = 0requiresTtobeaclosed 1-form
+dT= 0. (1.1)
+This is the well-known Helmholtz criterion from the inverse problem of the calculus ofSYMMETRIES AND CONSERVATION LAWS 5
+variations.
+Recall that a local vector field Ψ = Ψi(ϕ)δ
+δϕionMis called a characteristic of the
+equationsTi= 0if
+d(iΨT)≡0, i ΨT≡ΨiTi. (1.2)
+Since the value iΨTis annihilated by the variational derivative, it must be the integral of
+a total divergence ∂µjµ. By definition, the latter vanishes on the equations of motio n, and
+hencejµisa conservedcurrent.
+Every vector field Ψgenerates a transformation δϕi= Ψi(ϕ)of the space of fields M
+and wesay that Ψisapropersymmetryoftheequationsofmotionif
+LΨT= 0, (1.3)
+whereLΨistheLiederivativealong Ψ. Forthevariationalequationsofmotion T= dSthe
+proper symmetries are clearly the symmetries in the usual se nse. Making use of Cartan’s
+formula
+LΨT=iΨ(dT)+d(iΨT), (1.4)
+we see that the proper symmetry Ψis a characteristic if iΨ(dT) = 0. For the variational
+equations of motion, satisfying the Helmholtz condition (1 .1), the last formula identifies
+symmetrieswithcharacteristics. Itisthestatementwhich isknownastheNoethertheorem.
+Let us summarize the lessons that can be learned from the abov e formulation of the
+Noether theorem towards its extension beyond the scope of va riational dynamics. The
+construction of characteristics from symmetries involves , besides the equations of mo-
+tion, one more crucial ingredient, namely, the variational exterior differential dthat maps
+the variational p-forms to the (p+ 1)-forms. In the particular case discussed above, the
+equations of motion are given by 1-forms. Three main propert ies of the differential dare
+relevantforestablishingthelinkbetween thesymmetriesa nd conservationlaws. First,the
+field equations are d-closed (1.1). Second, the exterior differential and the Li e derivative
+are related to each other through Cartan’s formula (1.4). Th ird, the kernel of the varia-
+tional differential is constituted by the integrals of tota l divergencies ∂µjµ. These three
+facts taken togetherimplythatthesymmetriescan beidenti fied withthecharacteristics.
+The general field equations need not be a section of the cotang ent bundleT∗M. They
+may well be associated with another vector bundle E →Mover the configuration space
+of fieldsM, in which case the differential ddoes not work any more. Thus, one needs
+a more general operator dEto replace the variational exterior differential in the cas e of
+non-Lagrangian equations of motion. The aforementioned La grange structure providesSYMMETRIES AND CONSERVATION LAWS 6
+such an operator that satisfies the first two properties. The t hird one is obeyed, in a sense,
+only by weakly transitive Lagrange structures. Thus, whene ver the non-Lagrangian field
+equations enjoy a weakly transitive Lagrange structure, th e Noether theorem can be still
+formulatedinfull. Intheintransitivecase,aweakerpropo sitioncanbeproved,connecting
+rigid symmetrieswithcharacteristics inamorerelaxedway .
+2 Classicalfields
+Asiswellknown,thegeometryunderlyingthecinematicsofl ocalfieldtheoryisthatofjet
+bundles. Sowebeginthesectionwithrecallingsomebasicsf romthejettheoryrelevantfor
+our purposes. For a systematic expositionof the subject the reader may consult the books
+[13, 14,15, 16].
+The starting point of any local field theory is a locally trivi alfiber bundle π:Y→X,
+which base Xis usually identified with the space-time manifold and which sections, i.e.,
+the smooth maps ϕ:X→Ysatisfyingp◦ϕ= idX, are called fieldsorhistories. The
+spaceofallhistories(=sectionsof Y)willbedenotedby M. Thetypicalfiberof Y,being
+givenbyU≃π−1(x),x∈X, isusuallyreferred to asthe targetspaceoffields .
+Ther-th jet prolongation of the fiber bundle Y→Xis the bundle πr:JrY→X,
+whose points are r-jets. By definition, an r-jetjr
+xϕis the equivalence class of a local sec-
+tionϕofY, where two local sections ϕandϕ′are considered to be equivalent if they
+have the same Taylor development of order ratx∈Xin some (hence any) pair of co-
+ordinate charts centered at xandϕ(x). It follows from the definition that each section ϕ
+ofYinduces the section jrϕofJrYby the rule (jrϕ)(x) =jr
+xϕ. IfYis coordinatized
+by the numbers {xµ,ui}, where{xµ}and{ui}are local coordinates in XandU, then
+(xµ,ui,ui
+µ1,ui
+µ1µ2,...,ui
+µ1···µr)are local coordinates in JrYand the section jrϕis given
+in thesecoordinates by
+x/mapsto→(x,ϕi(x),∂µ1ϕi(x),∂µ1∂µ2ϕi(x),...,∂ µ1...∂µrϕi(x)). (2.1)
+Classical dynamics on Y→Xare specified by imposing differential equations. An
+r-thorderdifferentialequationis,bydefinition,aclosedi mbeddedsubmanifold S⊂JrY.
+A solution,or truehistory ϕ, is a section of Ysatisfyingjrϕ∈S. The truehistoriesform
+asubspace Σinthespaceofallhistories M. Inclassicalfieldtheory,thedifferentialequa-
+tions are mostlydefined by (nonlinear) differential operat ors associated to vectorbundles.
+LetE→Xbe a vector bundle. The r-th orderE-valued differential operator is a mapSYMMETRIES AND CONSERVATION LAWS 7
+T:M→Γ(E)such that the value of the section T(ϕ)atx∈Xis fully determined
+byjr
+xϕ. This definition assumes the existence of a bundle morphism Θ :JrY→Esuch
+thatT(ϕ) = Γ(Θ) ◦jr(ϕ). The differential equation S∈JrYis then identified with the
+pre-image ofthebase X⊂E, considered as thezero section,under thebundlemap Θ. In
+each local coordinate chart on E⊕JrY, the condition that ϕbelongs to the kernel of the
+mapTtakestheformofpartial differentialequations
+Ta(x,ϕi,∂µ1ϕi,∂µ1∂µ2ϕi,...,∂ µ1...∂µrϕi) = 0, (2.2)
+where{Ta}are componentsofthesection T(ϕ)withrespect to someframe {ea}inE.
+Though rigor and geometrically transparent, the language o f jets appears to be unduly
+cumbersomeindevelopinggeneralfield-theoreticalconstr uctionswherethelocalstructure
+of fields, while important, is out of focus. For this reason we will use a little bit loose
+but much more handy notation known in the physical literatur e as De Witt’s condensed
+notation [17]. According to this notation, the fields ϕi(x)are treated as local coordinates
+on the infinite dimensional manifold M; in so doing, the index ialso includes the local
+coordinates {xµ}onXso thatϕi≡ϕi(x). Similarly, the sections s=sa(x)eaof the
+vector bundle E→Xare considered to be linear coordinates sa≡sa(x)on the infinite
+dimensional vector space Γ(E). Then one can view the differential operator T:M→
+Γ(E)asasectionofthetrivial,infinitedimensionalvectorbund leE=M×Γ(E)overM.
+Of a particular importance for our consideration will be dif ferential operators with values
+in differential forms on X, this is the case of E=∧•T∗X. We will denote the space of
+all such operators by Ω•. The scalar differential operators are, by definition, the e lements
+of the space Ω0. Letνbe a fixed volume form on X. By smooth functions on Mwe
+understandthelocal functionalsoffields. Thesearegivenb yintegrals
+f[ϕ] =/integraldisplay
+XνF(ϕ) (2.3)
+of scalar differential operators Fevaluated for fields ϕ∈Mof compact support. The
+relation between localfunctionalsand scalardifferentia loperators isnot generallyone-to-
+one. Forexample,if ∂X=∅andFandF′aretwoscalaroperatorssuchthat ν(F′−F) =
+djfor somej∈Ωn−1,n= dimX, thenf′=f. We say that two local functionals fand
+f′areequivalentmoduloboundaryterms, f.=f′,iftheydifferbyalocalfunctionalofthe
+form/integraltext
+Xdj(ϕ). By theStokes theoremtheequivalenceimpliesthat f′−f=/integraltext
+∂Xj(ϕ).
+In the sequel, for thesake ofsimplicity,we willassumethat Y→Xis atrivial vector
+bundle over a contractible domain in Rn. Then, one has a simple criterion for the local
+functionalftovanishmoduloboundaryterms [3], [5], [15]:
+df≡/integraldisplay
+Xνδf
+δϕiδϕi= 0∀ϕifff.= 0. (2.4)SYMMETRIES AND CONSERVATION LAWS 8
+Of course, in a topologically nontrivial situation only the “if” part of the criterion holds
+true. Following the finite dimensional pattern, we can inter pret the variations dfof local
+functionals as exact 1-forms on M, with{δϕi}being a local frame in T∗M. The compo-
+nentsofthe1-form dfdefinetheEuler-Lagrangedifferentialoperator δf/δϕi. Inasimilar
+manner, onecanintroducethetangentbundle TMas thevectorbundlewhichsectionsare
+evolutionaryvector fields
+V=/integraldisplay
+XνVi(x,ϕ,...,∂ µ1...∂µrϕ)δ
+δϕi(x). (2.5)
+Every evolutionary vector field Vgenerates a flow ΦV
+ton the space of all histories M,
+which isdescribed bytheevolution-typesystem ∂tϕi=ViofPDEs, hencethename.
+Given a fiber bundle Y→X, the correspondence E/mapsto→ Eis quitenatural in the sense
+thatitallowsonetoextendalltheusualtensoroperationsf romthefinitedimensionalvector
+bundles to the corresponding infinite dimensional ones. In p articular, one can define the
+dualofthevectorbundle Easthevectorbundle E∗correspondingto E∗. Thereisanatural
+pairing betweensectionsof EandE∗defined by theintegral
+/an}bracketle{tS,T/an}bracketri}ht=/integraldisplay
+Xν/an}bracketle{tS(ϕ),T(ϕ)/an}bracketri}ht ∀S∈Γ(E∗),∀T∈Γ(E). (2.6)
+As is seen the result of pairing is a smooth function on M. In accordance with De Witt’s
+convention,wecanwritethelastexpressionjustasthesum SaTa,wheretherepeatedindex
+aimpliesalsointegrationover X.
+Each homomorphism h:E→E′induces the homomorphism H:E → E′of the
+associated vector bundles through the action on their fibers ,Γ(h) : Γ(E)→Γ(E′). But
+the infinite dimensional vector bundles E →Madmit more general homomorphisms. By
+definition,ageneral homomorphism H∈Hom(E,E′)isgivenby adifferentialoperator
+H: Γ(Y⊕E)→Γ(E′) (2.7)
+such that the section H(ϕ,s)∈Γ(E′), whereϕ∈Mands∈Γ(E), depends R-linearly
+on the second argument. As usual the homomorphism Hinduces the homomorphism
+on sections. Namely, Γ(H)takes a section (ϕ,T(ϕ))∈Γ(Y⊕E)ofEto the section
+H(ϕ,T(ϕ))ofE′.
+Wedefine thetransposeofahomomorphism H∈Hom(E1,E2)as auniquehomomor-
+phismH∗∈Hom(E∗
+2,E∗
+1)satisfyingtheproperty
+/an}bracketle{tΓ(H)(W),P/an}bracketri}ht˙ =/an}bracketle{tW,Γ(H∗)(P)/an}bracketri}ht (2.8)SYMMETRIES AND CONSERVATION LAWS 9
+forallW∈Γ(E1)andP∈Γ(E∗
+2).
+Thedirectproduct E1⊕E2isdefinedtobethebundlecorrespondingto E1⊕E2. Finally,
+wedefine thetensorproduct
+E0⊗E1⊗···⊗E m
+as thetrivialvectorbundleover Mwhosesectionsaredifferentialoperators
+H: Γ(Y⊕E∗
+1⊕···⊕E∗
+m)→Γ(E0) (2.9)
+that are linear in the sections of Γ(E∗
+k),k= 1,...,m. In casem= 1, we have the usual
+isomorphism Γ(E0⊗ E1)≃Hom(E∗
+1,E0). Applying this construction to the cotangent
+bundleT∗M, we define the p-th exterior power of the cotangent bundle ∧pT∗M, whose
+sections are p-forms onM. The operator of variational derivative mentioned above gi ves
+riseto theexteriordifferential d : Γ(∧pT∗M)→Γ(∧p+1T∗M)withtheproperty d2= 0.
+One can view the theory of jet bundles as a “differential geom etry with higher deriva-
+tives”. The advantage of the condensed notation overjets is that it brings the subject back
+into themorefamiliargeometricframework withouthigherd erivativesat thecost ofpass-
+ing to infinite dimensional manifolds. Of course, care must b e exercised when using the
+standard differential-geometric constructions in the infi nite dimensional setting. It is par-
+ticularly important to keep in mind that unlike the finite dim ensional case, the space of
+sectionsΓ(E)is not a moduleover the space of smooth functions C∞(M)(the latter con-
+sists of local functionals). Although it is possible to thin k ofΓ(E)as a module over the
+scalar differential operators Ω0with respect to the pointwise multiplication of functions
+onX, the induced action of Hom(E,E′)onΓ(E)is onlyR-linear, not Ω0-linear. Finally,
+Γ(E1⊗E2)/ne}ationslash= Γ(E1)⊗Γ(E2).
+3 Symmetries,identitiesandconservationlaws
+The discussion of the previous section can be summarized by s aying that the classical
+dynamics of fields are fully specified by a section Tof some vector bundle E →Mover
+the space of all histories. For this reason we call Ethedynamics bundle . The subspace of
+truehistories Σ⊂Misdefined to bethezero locusof T,
+Σ ={ϕ∈M|T(ϕ) = 0}. (3.1)
+Usingthephysicalterminology,wewillreferto Σastheshell. Thefieldequations T(ϕ) =
+0aresupposedtosatisfythestandard regularityconditions usuallyassumedforPDEs, seeSYMMETRIES AND CONSERVATION LAWS 10
+e.g. [3], [5], [14]. These conditions ensure that any sectio nS∈Γ(E′)vanishing on Σis
+representablein theform
+S= Γ(H)(T) (3.2)
+forsomehomomorphism H∈Hom(E,E′).
+The present section contains no original results, we just ex plain our terminology and
+briefly run through the notions of symmetry, identities and c onservation laws, which are
+studiedin thenextsections.
+Tobeginwith,weintroducethehomomorphism J:TM→ Edefined by1
+/an}bracketle{tJ(X),Ψ/an}bracketri}ht=X·/an}bracketle{tT,Ψ/an}bracketri}ht ∀X∈Γ(TM),∀Ψ∈Γ(E∗), (3.3)
+Ψbeing understood as a field independent section of E∗. The homomorphism Jis called
+[16]theuniversallinearization ofthenonlineardifferentialoperator T. Wealso introduce
+the sign of weak equality S≈S′borrowed from Dirac’s constrained dynamics. It means
+thattwosections SandS′ofsomevectorbundleover Mcoincideonshell,i.e.,S|Σ=S′|Σ.
+An evolutionary vector field Xis said to be a symmetry of the classical system if it
+preserves theshell Σ. Thisamountstotheweak equality
+J(X)≈0. (3.4)
+The symmetries form a subalgebra Sym′(T)⊂Γ(TM)in the Lie algebra of all vector
+fields onM.
+A section Ψ∈Γ(E∗)issaid tobean identityoftheclassical systemif
+/an}bracketle{tΨ,T/an}bracketri}ht.= 0. (3.5)
+As adifferentialconsequenceofthelastequalitywehave
+J∗(Ψ)≈0. (3.6)
+Denoteby Id′(T)⊂Γ(E∗)thesubspaceofall identities.
+The notions of symmetry and identity can be further elaborat ed on. Observe that any
+vector field Xthat vanishes on the shell, i.e., X≈0, is a symmetry in the sense of (3.4).
+The on-shell vanishing symmetries are called trivial. They are present in any classical
+theory, containing no valuable information about the struc ture of dynamics. Fortunately,
+1By abuse of notation, we will use the same symbol to denote a ve ctor bundle homomorphism and the
+inducedhomomorphismonsections. Inparticular,wewill wr iteJinsteadofmorepedantic Γ(J).SYMMETRIES AND CONSERVATION LAWS 11
+being proportional to the equations of motion, the trivial s ymmetries constitute an ideal
+Sym0(T)⊂Sym′(T)in the Lie algebra of all symmetries and can thus be systemati cally
+disregarded. So in what follows, by a symmetry we will actual ly mean an element of the
+factor algebra Sym(T) = Sym′(T)/Sym0(T).
+A symmetry X∈Sym(T)is called a gauge symmetry if there exists a vector bundle
+F →Mtogetherwithasection ε∈Γ(F)and ahomomorphism R∈Hom(F,TM)such
+thatImR⊂Sym(T)andX=R(ε). Ifthevectorbundle Fisbigenoughtoaccommodate
+any (nontrivial) gauge symmetry with a fixed R, then we refer to Fas thegauge algebra
+bundle. Notice that J◦R≈0. The homomorphism Rdefines an (over)completebasis of
+gauge generators. Clearly, the vector distribution ImR⊂TMis on-shell involutive, and
+hence it defines a foliation of Σ. The leaves of this foliation are called gauge orbits . Two
+pointsp,q∈Σare considered to be (gauge) equivalent, q∼p, if they belong to the same
+gaugeorbit. Thequotientspace Σ/∼istermedasa covariantphasespace . It isthespace
+ofall physicalstates ofthesystem.
+Since the gauge symmetries form an ideal GSym(T)⊂Sym(T), one can define the
+factor algebra RSym(T) = Sym(T)/GSym(T) , which is naturally identified with theLie
+algebra of all rigid symmetries . By definition, the rigid symmetries are transverse to the
+gaugeorbitsandthustheyinduceanontrivialactionon thep hasespaceofphysicalstates.
+Equation (3.5) admits a lot of trivial solutions that can be c onstructed as follows.
+Take an arbitrary section of K∈Γ(E∗∧ E∗)and consider it as a homomorphism from
+Hom(E,E∗). ThenΨ =K(T)satisfies (3.5). The identities constructed in such a way
+are of no physical significance and, therefore, they should b e regarded as trivial. Notice
+that each trivial identityvanishes on shell. The conversei mplicationis also true [3]: Each
+on-shell vanishing identity has the form Ψ =K(T)for some skew-symmetric K. The
+proofexploitstheregularityassumptionfortheequations ofmotion. Thenontrivialidenti-
+ties are then defined as the elements of the quotient space Id(T) = Id′(T)/Id0(T), where
+Id0(T)⊂Id(T)isthesubspaceoftrivialidentities.
+An identity Ψ∈Id(T)is calledNoether’s identity if there exists a vectorbundle G →
+Mtogether with a section ξ∈Γ(G)and a homomorphism Z∈Hom(G,E∗)such that
+ImZ⊂Id(T)andΨ =Z(ξ). Denote by NId(T)⊂Id(T)the subspace of all Noether’s
+identities and define the quotient space Char(T) = Id(T)/NId(T), whose elements are
+calledcharacteristics . Again,onecanchoosethevectorbundle Gtobebigenoughso that
+NId(T) = ImZ. In thiscase, Zis said to define an (over)completebasis of thegenerators
+of Noether’s identities. Notice that Z∗(T) = 0, whereZ∗is transpose of Z. It is quite
+natural tocall GthebundleoftheNoether identities .SYMMETRIES AND CONSERVATION LAWS 12
+All we have said above can be concisely reformulated in terms of the following se-
+quenceofhomomorphisms
+0/d47/d47Γ(F)R/d47/d47Γ(TM)J/d47/d47Γ(E)Z∗/d47/d47Γ(G∗)/d47/d470 (3.7)
+and itstranspose
+0Γ(F∗)/d111/d111 Γ(T∗M)R∗/d111/d111 Γ(E∗)J∗/d111/d111 Γ(G)Z/d111/d1110/d111/d111. (3.8)
+Uponrestrictionto Σthesesequencesmakecochaincomplexes;theproperties Z∗◦J≈0
+andJ∗◦Z≈0follow from the differential consequence of the identity Z∗(T) = 0.
+Thespacesofallrigidsymmetriesandcharacteristicsaret hennaturallyidentifiedwiththe
+cohomologygroups
+RSym(T)≃KerJ|Σ
+ImR|Σ,Char(T)≃KerJ∗|Σ
+ImZ|Σ. (3.9)
+Intimately related to the notion of an identity is the notion of aconservation law . The
+latter is identified with a form-valued differential operat orj∈Ωn−1taking true histories
+to closed (n−1)-forms,
+dj≈0. (3.10)
+Inviewoftheregularityassumptions,the n-formdjhastobeproportionaltotheequations
+ofmotion,that is /integraldisplay
+Xdj=/an}bracketle{tΨ,T/an}bracketri}ht (3.11)
+for some Ψ∈Γ(E∗). Comparing the last equality with (3.5), we see that every on -shell
+closed(n−1)-formgivesrisetosome(perhapstrivial)identityandvice versa. Therelation
+(Identities) ↔(ConservationLaws) (3.12)
+is far from being a bijection, since neither left nor right ha nd sides of (3.11) are defined
+byjandΨunambiguously. For instance, one may add to jany closed (and hence exact)
+formdiwithoutchanging Ψ. The exact forms di∈Ωn−1are characterized by zero charge
+and, therefore, one should regard them as trivial. Another s ource of triviality is related to
+the currents jthat vanish on shell. Taking into account either of possibil ities, we identify
+the space of nontrivial conservation laws CL(T)with the equivalence classes of on-shell
+closedforms j∈Ωn−1. Twosuchformsareconsideredasequivalentwhenevertheyd iffer
+by an on-shellexactform2:
+j∼j′⇔j−j′≈di. (3.13)
+2This definition does not exclude the possibility that the cha rge, being related to a nontrivial conserved
+current,vanishesidentically. Fordiscussionofthisphen omenon,see [20].SYMMETRIES AND CONSERVATION LAWS 13
+The space CL(T)is also known as the characteristic cohomology ofΣin form degree
+n−1, see[18], [19]and [25].
+Turning to the right hand side of (3.11), one can see that triv ial identities give rise to
+trivialconservationlaws. Thisisanadditional(infactth emain)reasontocalltheon-shell
+vanishing identities trivial. Furthermore, the Noether id entities correspond to the trivial
+conservationlawsas well.
+Considering(3.12)modulotrivialities,onearrivesat the followingrelation
+Char(T)↔CL(T). (3.14)
+Thislastrelationisinessenceanisomorphism[15],[5],so thattheproblemofconstructing
+nontrivialconservationlowsboilsdowntofinding outchara cteristics.
+Example . Generally it is a rather hard problem to identify the spaces of all non-trivial
+symmetries and characteristics for a given set of equations . The problem, however, is
+considerablysimplifiedfortheordinarydifferentialequa tionsinnormalform,whereeither
+of spaces admits a fairly explicit description. By this reas on we will use this class of
+dynamical systems to exemplify all the general notions and c onstructions throughout the
+paper. Without loss of generality we can restrict ourselves to the systems of first-order
+ODEs
+Ti≡˙xi(t)+vi(t,x) = 0. (3.15)
+Here the overdot stands for the derivative with respect to th e independent variable t∈R
+and{xi}arelocalcoordinateson Y. Considering Y×Rasatrivialfiberbundleover R,one
+can thoughtof vasaverticalvectorfieldon Y×R. Equations(3.15)areneitherreducible
+norgaugeinvariantandthedynamicsbundleisnaturallyide ntifiedwiththetangentbundle
+TMof the trajectory space M= Map(R,Y). By definition, the characteristics of (3.15)
+liveinthespace Γ(T∗M)spannedby 1-forms
+Ψ =/integraldisplay
+dtψi(t,x,˙x,...,(m)x)δxi(t). (3.16)
+The system (3.15) being regular, one can exclude all the deri vativesfrom the integrand of
+(3.16)by meansoftheequationsofmotion,lookingforchara cteristics oftheform
+Ψ =/integraldisplay
+dtψi(t,x)δxi(t), (3.17)
+whereψ=ψi(t,x)dxiis a vertical 1-form on Y×R. Relation (3.5) results in two condi-
+tions
+ψ=/tildewidedf, ∂ tf=v·f, (3.18)SYMMETRIES AND CONSERVATION LAWS 14
+wherefis a function on Y×Rand/tildewidedis the exterior differential on Y. LetC∞
+v(Y×
+R)denote the space of all smooth functions satisfying equatio ns (3.18). These equations
+specifyfup to an additiveconstant and each such fis conserved. Thus we are led to the
+followingisomorphisms:
+Char(˙x+v)≃C∞
+v(Y×R)/R,CL(˙x+v)≃C∞
+v(Y×R). (3.19)
+Let us now turn to the symmetries of (3.15). The general evolu tionary vector field on
+Mhas theform
+W=/integraldisplay
+dtwi(t,x,˙x,...,(m)x)δ
+δxi(t). (3.20)
+Being interested in symmetries, we can exclude all the deriv atives from the integrand of
+(3.20)withthehelpoftheequationsofmotion. Thisreduces theansatz(3.20)to
+W=/integraldisplay
+dtwi(t,x)δ
+δxi(t), (3.21)
+wherew=wi(t,x)∂/∂xiis a vertical vector field on Y×R. Verifying (3.4) one can find
+thatWis asymmetryiff
+∂tw= [v,w]. (3.22)
+Denoting by Xv(Y×R)the space of all vertical vector fields satisfying (3.22), we can
+write
+Sym(˙x+v) =Xv(Y×R). (3.23)
+Let us suppose that the vector field vis complete. Then, considering (3.18) and (3.22)
+as systems of the first-order ODEs for fandw, respectively, we conclude that the spaces
+of conservation laws and symmetries are isomorphic to the sp aces of initial data C∞(Y)
+andX(Y). Thus the space of symmetries appears to be “much larger” tha n the spaces of
+characteristics and conservationlawswhenever dimY >1.
+The form of equations (3.4) and (3.6) shows a certain duality between the concepts of
+symmetry and conservation law. The dual nature of these conc epts is also supported by
+the celebrated Noether theorem for Lagrangian equations of motion. The long time use
+of this theorem has lead to a widespread belief that any conse rvation law or, what is the
+same, characteristic is a reflection of some symmetry. Beyon d the scope of Lagrangian
+dynamics, this is not always true. Not any conservation law c omes from some symmetry,
+nor does each symmetrycorrespond to a conservationlaw. The re is no canonical relation-
+ship between thesetwoconcepts in general. This fact has a si mplegeometricexplanation:
+The characteristics Ψ’s and the symmetries X’s belong to different vector bundles unlessSYMMETRIES AND CONSERVATION LAWS 15
+E∗=TM. This suggests that (i) any systematic procedure for conver ting symmetries to
+characteristics and vice versa has to involve an additional geometric structure on Mand
+(ii)whateverthisstructuremaybe,itwillresultinalinea rrelationbetweenthedualofthe
+dynamics bundle E∗and the tangent bundle of M. It is then quite reasonable to start with
+some homomorphism V:E∗→TMand to look for a suitableset of conditions ensuring
+thatV(Ψ)is a symmetry whenever Ψis a characteristic. Actually, such a homomorphism
+Vhas been already identified as a key structure needed for the p ath-integral quantization
+ofnon-Lagrangian gaugetheories[7]. Thishomomorphismis called aLagrangeanchor .
+4 TheLagrangeanchor
+Definition 4.1. A vector bundle homomorphism V:E∗→TMis called a Lagrange
+anchor,ifthediagram
+Γ(TM)J/d47/d47Γ(E)
+Γ(E∗)J∗/d47/d47V/d79/d79
+Γ(T∗M)V∗/d79/d79(4.1)
+commutes uponrestrictionto theshell,i.e.,
+J◦V≈V∗◦J∗. (4.2)
+Since the last relation is linear in V, the Lagrange anchors form a vectorspace, which
+we denote by An′(T). This space is clearly nonempty as each on-shell vanishing h omo-
+morphismVobeys (4.2). The on-shell vanishingLagrange anchors are of no significance
+fromtheperspectiveofconvertingconservationlawstorig idsymmetriesandwerulethem
+out by passing to the quotient An(T) = An′(T)/An0(T), whereAn0(T)is the subspace
+ofall on-shellvanishingLagrangeanchors.
+Proposition 4.1. The Lagrange anchor takes identities to symmetries, trivia l identities to
+trivialsymmetriesandtheNoether identitiesto thegauges ymmetries.
+Proof.IfΨisan identity,then J∗(Ψ)≈0and
+J◦V(Ψ)≈V∗◦J∗(Ψ)≈0. (4.3)
+HenceX=V(Ψ)is a symmetry. It is also clear that X≈0whenever Ψ≈0. If now
+Ψ =Z(ξ)forsomeξ∈Γ(G), then
+J◦V(Ψ)≈V∗◦J∗◦Z(ξ)≈0∀ξ∈Γ(G). (4.4)SYMMETRIES AND CONSERVATION LAWS 16
+Thus,X=V◦Z(ξ)isagaugesymmetry. Sincethegenerators Rareassumedtoforman
+(over)completebasis inthegaugealgebra bundle,thereexi stsahomomorphism W:G →
+Fsuchthat thediagram
+Γ(F)R/d47/d47Γ(TM)
+Γ(G)Z/d47/d47W/d79/d79
+Γ(E∗)V/d79/d79(4.5)
+commutesuponrestrictiontotheshell,i.e., V◦Z≈R◦W.
+Letus combineallthediagrams(3.7), (3.8), (4.1), (4.5)in tothefollowingone:
+0/d47/d47Γ(F)R/d47/d47Γ(TM)J/d47/d47Γ(E)Z∗/d47/d47Γ(G∗)/d47/d470
+0/d47/d47Γ(G)W/d79/d79
+Z/d47/d47Γ(E∗)V/d79/d79
+J∗/d47/d47Γ(T∗M)V∗/d79/d79
+R∗/d47/d47Γ(F∗)W∗/d79/d79
+/d47/d470(4.6)
+Fromtheprevioussectionweknowthattherowsofthisdiagra mmakecochaincomplexes
+upon restriction to the shell. Then, the on-shell commutati vity of the squares implies that
+theupwardarrowsdefineacochinemap. Itisthestandardfact ofhomologicalalgebrathat
+each cochain map induces a well defined homomorphism in cohom ology. In the case at
+hand thisgivesthehomomorphism
+H(V) : Char(T)→RSym(T) (4.7)
+from thespaceofcharacteristicsto thespaceofrigidsymme tries.
+A natural question arises about identifying the different a nchors from An(T)which
+induce the same homomorphism (4.7). An appropriate algebra ic concept for examining
+the question is that of homotopy. We say that two anchors V,˜V∈An(T)are equivalent,
+V∼˜V,ifthecorrespondingcochainmapsarehomotopic. Thelatte rimpliestheexistence
+ofhomomorphisms GandKsuchthat
+˜V−V≈G◦J∗+R◦K, ˜V∗−V∗≈J◦G+K∗◦R∗,
+˜W−W≈K◦Z, ˜W∗−W∗≈Z∗◦K∗.(4.8)
+All therelevantmapsare depictedin thefollowingdiagram:
+0/d47/d47Γ(F)R/d47/d47Γ(TM)J/d47/d47Γ(E)Z∗/d47/d47Γ(G)/d47/d470
+0/d47/d47Γ(G∗)/d97/d97/d67/d67/d67/d67/d67/d67/d67/d67/d67˜W/d79/d79
+W/d79/d79
+Z/d47/d47Γ(E∗)K/d74/d74/d74/d74/d100/d100/d74/d74/d74/d74
+˜V/d79/d79
+V/d79/d79
+J∗/d47/d47Γ(T∗M)/d106/d106
+G/d76/d76/d76/d76/d101/d101/d76/d76/d76/d76˜V∗/d79/d79
+V∗/d79/d79
+R∗/d47/d47Γ(F∗)/d106/d106
+K∗/d75/d75/d75/d75/d101/d101/d75/d75/d75/d75
+˜W∗/d79/d79
+W∗/d79/d79
+/d47/d470/d97/d97/d67/d67/d67/d67/d67/d67/d67/d67/d67(4.9)SYMMETRIES AND CONSERVATION LAWS 17
+Theweak equalitiesin thefirst lineof(4.8)3implythat
+J◦(G−G∗)≈0. (4.10)
+This is the only condition on the homomorphisms GandK. Taking into account the fact
+of completeness of the gauge algebra generators, we can rewr ite (4.10) in the following
+equivalentform:
+G−G∗≈R◦U (4.11)
+forsomeU:T∗M→ Fsatisfying
+R◦U+U∗◦R∗≈0. (4.12)
+In the diagram above, the homomorphisms Uand−U∗are depicted by the dotted arrows,
+sothatthecorrespondingparallelogramofmapsappearstob eon-shellcommutative. Since
+thehomotopiccochainmapsareknowntoinducethesamehomom orphismincohomology,
+weconcludethat H(V) =H(˜V)whenevertheanchors Vand˜Vareequivalent.
+Denote by HAn(T)the space of the homotopy classes of Lagrange anchors. This c an
+bethoughtofas thequotientof An′(T)bytheLagrangeanchors oftheform
+Vtriv=R◦K+G◦J∗+V0, (4.13)
+whereV0≈0andG≈G∗. Wehave
+Vtriv(Ψ) =R(ε)−X0, (4.14)
+whereε=K(Ψ)andX0=G◦J∗(Ψ)+V0(Ψ)≈0. In otherwords, thetrivialLagrange
+anchorVtrivtakes each identity(trivialornot)to atrivialrigidsymme try.
+Example (continuation). Consider the dynamical system (3.15). The n the most general
+local ansatzfortheLagrangeanchor V:T∗M→TMlookslike
+/an}bracketle{tV(W),P/an}bracketri}ht˙ =/an}bracketle{tW,V∗(P)/an}bracketri}ht=N/summationdisplay
+n=0/integraldisplay
+dtwi(t)αij
+n(t,x,˙x,...,(m)x)(n)pj(t),(4.15)
+whereW=wiδxiandP=piδxiare arbitrary 1-forms on M. The homomorphism
+J:TM→TMisgivenbythematrixfirst-orderdifferentialoperator
+J=/parenleftbigg
+δi
+jd
+dt+∂vi
+∂xj/parenrightbigg
+δ(t−t′) (4.16)
+3TheseequivalencerelationsfortheLagrangeanchorsorigi nallyappearedin[9]. Asusual,thehomotopy
+equivalenceclassesofcochaintransformationscanbeequi valentlyreinterpretedasa cohomologyclasses of
+correspondingcochaincomplex[7]. In the contextof local fi eld theorythe equivalenceclasses of Lagrange
+anchorswererecentlydiscussedin[21].SYMMETRIES AND CONSERVATION LAWS 18
+The peculiar form of the operator Jallows one to absorb all the derivativesof p’s into the
+trivial anchor (4.13) for an appropriate G. Using then the equations of motion or, what
+is the same, adding an appropriate on-shell vanishing ancho rV0, one can also exclude
+all the derivatives of x’s from the integrand of (4.15). We are led to conclude that ea ch
+equivalence class of the Lagrange anchors for equations (3. 15) contains the anchor of the
+form
+V(δxi(t)) =αij(t,x)δ
+δxj. (4.17)
+Substitutingthelastexpressionto thedefiningcondition( 4.2), wefinally get
+αij=−αji, ∂ tα=Lvα, (4.18)
+Thus,thereisaone-to-onecorrespondencebetweenthespac eofnon-equivalentLagrange
+anchorsAn(˙x+v)andtheverticalbivectorfields α=αij(t,x)∂i∧∂jonY×Rsatisfying
+(4.18). If the vector field vis complete, then equation (4.18) defines an isomorphism
+An(˙x+v)≃Γ(∧2TY). This classification result is in agreement with the conclus ions of
+paper[21]. In[21],itisproventhatanyLagrangeanchorfor theAKSZ-typesigmamodels
+[32]alwaysreduces to apurely algebraicoperatorwhosecoe fficients donotdependenton
+thespace-timederivativesoffields.
+5 Propersymmetriesandproperdeformations
+We begin with an alternative geometric interpretation of th e Lagrange anchor as a “Lie
+algebroid with relaxed integrability condition”. It was fir st introduced in [7], and termed
+theLagrange structure . To fix ideas, let us think for a while of E →Mas if it were
+an ordinary vector bundle over a finite dimensional manifold4. The way of adapting this
+constructiontothecontextoflocalfield theory willbeobvi ous.
+Definition 5.1. Given aclassicalsystem (E,T),aLagrangestructureisahomomorphism
+dE: Γ(∧•E)→Γ(∧•+1E)obeyingtwo conditions:
+(i)dEisa derivationofdegree 1,i.e.,
+dE(A∧B) = dEA∧B+(−1)pA∧dEB,
+foranyA∈Γ(∧pE)andB∈Γ(∧•E);
+4SeecommentsattheendofSec. 2.SYMMETRIES AND CONSERVATION LAWS 19
+(ii)dET= 0.
+Here weidentify Γ(∧0E)withC∞(M).
+Due to the Leibnitz rule (i), in each trivializing chart U⊂Mthe operator dEis com-
+pletelydeterminedbyitsactiononthecoordinatefunction sϕiandtheframesections {ea}
+ofE|U:
+dEϕi=Vi
+a(ϕ)ea,dEea=1
+2Ca
+bc(ϕ)eb∧ec. (5.1)
+Applying dEtothesection T=Taea, onecan seethat theproperty (ii)is equivalenttothe
+structurerelations
+dET=1
+2(Vi
+a∂iTb−Vi
+b∂iTa+Cc
+abTc)ea∧eb= 0. (5.2)
+Clearly, the derivation dEdefines a bundle homomorphism V:E∗→TM. The shell
+Σbeing a regularly imbedded submanifold, equation (5.2) is e quivalent to the defining
+condition(4.2). Thus, VisaLagrangeanchor.
+Dualizingrelations(5.1)onegets abracket
+[·,·] : Γ(E∗)∧Γ(E∗)→Γ(E∗). (5.3)
+On framesections eaofE∗|Uit reads
+[ea,eb] =Cd
+abed (5.4)
+and extendstoarbitrary Ψ,Ψ′∈Γ(E∗)by theLeibnitzrule
+[fΨ,Ψ′] =f[Ψ,Ψ′]+(V(Ψ)·f)Ψ′,∀f∈C∞(M). (5.5)
+Generally thebracket violatestheJacobi identity,theref oreitis notaLie-algebrabracket.
+Definition 5.2. A Lagrangestructure (E,T,dE)issaidto beintegrableif d2
+E= 0.
+ComparingDefinitions5.1and5.2withthedefinitionofa Liealgebroid (seee.g. [22],
+[23]), one can see that the integrable Lagrange structure is nothing else but the Lie alge-
+broidoverMwithanchor V:E∗→TM,Liebracket(5.3),andadistinguishedsection T.
+Thisparticularyetimportantcaseaccountsforthenameof V–“Lagrangeanchor”. Foran
+integrable Lagrange structure the bracket (5.3) satisfies t he Jacobi identity and the anchor
+distribution ImV⊂TMisintegrable.SYMMETRIES AND CONSERVATION LAWS 20
+To illustrate (as well as motivate) the definitions above let us consider a Lagrangian
+theory withaction S(ϕ). Theequationsofmotionread
+T≡dS= 0, (5.6)
+so that the dynamics bundle Eis given by the cotangent bundle T∗Mof the space of
+all histories. The canonical Lagrange structure is given by the exterior differential d :
+Γ(∧•T∗M)→Γ(∧•+1T∗M)andthedefining condition(5.2)takestheform
+dT= d2S= 0. (5.7)
+The identity d2= 0means integrability. The Lagrange anchor is given by the ide ntical
+mapV= id :TM→TMand the Lie bracket (5.3) coincides with the commutator of
+vectorfields.
+Definitions5.1and5.2suggesttoviewtheLagrangestructur easa“Liealgebroidwith
+relaxedintegrabilitycondition”inthesensethatthe“str ong”integrabilitycondition d2
+E= 0
+is replaced here by the weaker one d2
+ET= 0. In spite of weakened integrability, it is still
+possibleto utilizemanyoftheusualdifferential-geometr icconstructionsassociated toLie
+algebroids. Inparticular,wecanendowtheexterioralgebr aofE-differentialforms Γ(∧•E)
+withtheoperationsofinnerdifferentiationand Liederiva tive.
+By definition, the inner differential iΨassociated with a section Ψ∈Γ(E∗)is a differ-
+entiationof Γ(∧•E)ofdegree-1 whichactionon A∈Γ(E)isgivenby
+iΨA=/an}bracketle{tΨ,A/an}bracketri}ht. (5.8)
+FollowingtheanalogywithCartan’scalculus,theLiederiv ativealong Ψisdefinednowby
+LΨ=iΨ◦dE+dE◦iΨ. (5.9)
+Itisadifferentiationof Γ(∧•E)ofdegree0. Onecan easilyverifythefollowingidentities:
+{iΨ,iΨ′}= 0, [LΨ,dE] = [iΨ,d2
+E],
+[LΨ,iΨ′] =i[Ψ,Ψ′],[LΨ,LΨ′] =L[Ψ,Ψ′]+{iΨ′,[iΨ,d2
+E]},(5.10)
+where the braces stand for anticommutators. We can also defin e the action of the Lie
+derivativeonthecontravariant E-tensorsbysetting
+LΨΨ′= [Ψ,Ψ′]∀Ψ,Ψ′∈Γ(E∗). (5.11)
+This definition extends to arbitrary E-tensor fields by the usual formulas of differential
+geometry. From thispointon wereturn toinfinite-dimension alsettingofSec. 2.SYMMETRIES AND CONSERVATION LAWS 21
+Definition5.3. GivenaLagrangestructure (E,T,dE)andasection Ψ∈Γ(E∗),thevector
+fieldV(Ψ)issaidto bea propersymmetrygeneratedby ΨifLΨT= 0.
+Notice that the proper symmetry V(Ψ)is a symmetry in the usual sense. Indeed,
+LΨT≈J◦V(Ψ)andLΨT= 0impliesV(Ψ)∈Sym′(T). For variational equations
+of motion equipped with the canonical Lagrange structure (5 .7) the proper symmetries
+coincidewiththesymmetriesofaction.
+We emphasize that the property of being proper symmetry is no t homotopy invariant:
+IfV(Ψ)is a proper symmetry and V∼V′, thenV′(Ψ)may not be a symmetry at all,
+unlessΨ∈Id′(T).
+The image ImV⊂TMof the Lagrange anchor defines a generalized distribution on
+Min the sense of Sussmann [24]. Recall that a generalized dist ributionVis said to be
+involutive ,ifitisclosedundertheLiebracket,thatis [V,V]⊂ V. Theinvolutiveclosure of
+Vis,bydefinition,theminimalinvolutivedistribution Vcontaining Vasasubdistribution.
+Clearly,Visspannedbytheiteratedcommutatorsof Vandtheequality V=Vamountsto
+involutivityof V.
+Definition 5.4. A Lagrange anchor V:E∗→TMis said to be transitiveif ImV=TM
+and weaklytransitiveif ImV=TM.
+For example, the canonical Lagrange anchor V= id :TM→TMfor Lagrangian
+equationsofmotion T= dS= 0is bothintegrableand transitive.
+Proposition 5.1. The Lagrange anchor V:E∗→TMtakes identities to proper symme-
+tries. If the Lagrange anchor is weakly transitive, then eac h proper symmetry comes from
+someidentity.
+Proof.Thefirst statementfollowsfrom theidentity
+LΨT= dE(iΨT)+iΨ(dET) = 0 ∀Ψ∈Id′(T). (5.12)
+Thisidentityalsoimpliesthat
+dE(iΨT) = 0 (5.13)
+for any generator of proper symmetry Ψ. If the function iΨTis annihilated by the anchor
+distribution ImV, then it has to be annihilated by the involutive closure ImV. Since for
+transitive Lagrange anchors ImV=TM, we have d(iΨT) = 0. Thus,/an}bracketle{tΨ,T/an}bracketri}ht.= 0and
+Ψ∈Id′(T).SYMMETRIES AND CONSERVATION LAWS 22
+LetPSym′
+V(T)⊂Sym(T)denote the subspace of all proper symmetries and let
+PSymV(T)be the quotient space of all proper symmetries by the trivial and gauge sym-
+metries. According to this definition PSymV(T)is a subspace in RSym(T). Combining
+Proposition5.1withProposition4.1, wearriveat thefollo wingstatement.
+Proposition5.2. Thebundlemap V:E∗→TMinduces awell defined homomorphism
+H(V) : Char(T)→PSymV(T). (5.14)
+If the equivalence class H(V)contains a weakly transitive Lagrange anchor, then the
+homomorphism(5.14)issurjective.
+The symmetries of a classical system form the Lie algebra wit h respect to the vector-
+field commutator. On the other hand, given a Lagrange anchor, the dual of the dynamics
+bundle carries the bracket operation (5.3). Having in mind t he Lagrangian case, where
+characteristics coincide with the symmetries of action, on e may expect a certain relation-
+ship between the aforementioned multiplicative structure s inΓ(TM)andΓ(E∗). This is
+detailed inthefollowingproposition.
+Proposition 5.3. IfV(Ψ)andV(Ψ′)are two proper symmetries, then V([Ψ,Ψ′])is also
+a proper symmetry. Furthermore, the subspace of identities Id′(T)⊂Γ(E∗)is closed
+with respect to the bracket (5.3), the trivial and Noether id entities forming an ideal in
+Id′(T). IftheLagrangestructureisintegrable,then V: Γ(E∗)→Γ(TM)isaLiealgebra
+homomorphismand thebracket(5.3) makes Id′(T)andChar(T)into theLiealgebras. In
+thelattercase, wehavea well defined Liealgebrahomomorphi sm(5.14).
+Proof.Applyingthefourthidentity(5.10)to Tandusing(5.13),wefindthat L[Ψ,Ψ′]T= 0.
+Hence[Ψ,Ψ′]∈Γ(E∗)isa propersymmetrygenerator.
+Applyingnowthethird identity(5.10)to T, we geti[Ψ,Ψ′]T= 0providedthat Ψ,Ψ′∈
+Id′(T). So thebracket oftwoidentitiesisan identityagain. Suppo seforamomentthat Ψ′
+is given by a sum of the Noether and trivial identities, i.e., Ψ′=Z(ξ)+K(T), for some
+K∈Γ(E∗∧E∗)andξ∈Γ(G). We have
+[Ψ,Ψ′] =LΨK(T)+LΨZ(ξ)
+= (LΨK)(T)+Z(V(Ψ)ξ)+(LΨZ)(ξ).(5.15)
+Bydefinition,therighthandsideisanidentity. Clearly,th efirsttermisatrivialidentityand
+thesecondoneisaNoetheridentity. Thenthethirdterm,bei ngproportionaltoanarbitrary
+sectionξ, has to define a Noether identity as well. We thus conclude tha t the trivial andSYMMETRIES AND CONSERVATION LAWS 23
+Noether identities constitute an ideal in Id′(T)with respect to the bracket multiplication.
+So wehaveawell defined bracket on thequotientspace Char(T).
+It followsfromthefourthidentityin(5.10)that
+[V(Ψ),V(Ψ′)]·f−V([Ψ,Ψ′])·f=W(Ψ,Ψ′)·f∀f∈C∞(M),(5.16)
+where
+W(Ψ,Ψ′)·f≡ /an}bracketle{tΨ∧Ψ′,d2
+Ef/an}bracketri}ht. (5.17)
+By construction, the vector field W(Ψ,Ψ′)is a symmetry, not necessary proper though.
+It vanishes for integrable Lagrange structures, when the br acket (5.3) enjoys the Jacobi
+identity. Inthiscase,relation(5.16)saysthat V: Γ(E∗)→Γ(TM)isaLiealgebrahomo-
+morphism. Thishomomorphismtakesthesubalgebra Id′(T)tothesubalgebra PSym′
+V(T).
+Passingto thequotient,wegeta welldefined Liealgebrahomo morphism(5.14).
+Example (continuation). Besides the homomorphism V:T∗M→TM, the Lagrange
+anchor(4.17)defines abracket (5.3)on thecotangent bundle T∗M. Thelatterisgivenby
+[δxi(t),δxj(t′)] =δxk(t)∂kαij(t,x(t))δ(t−t′). (5.18)
+Formulas (4.17), (4.18) and (5.18) taken together define the most general Lagrange struc-
+tureassociated tothefirst-orderdifferentialequations( 3.15).
+Clearly, the Lagrange anchor (4.17) is transitive if for any t∈Rthe bivector αt∈
+Γ(∧2TY)isnondegenerate. Inthiscase,eachfiber (Y,αt)ofY×Risanalmostsymplectic
+manifold. One can also check [7] that the integrability of th e Lagrange anchor (4.17)
+amountstotheJacobiidentity
+1
+6[α,α]≡αim∂mαjk∂i∧∂j∧∂k= 0. (5.19)
+Here the square brackets denote the Schouten commutator of p olyvector fields. In the
+integrable case, the bundle map αt:T∗Y→TYdefines the Lie algebroid of the Poisson
+manifold (Y,αt). If the Lagrange anchor (4.17) is both transitive and integr able, then
+(Y,αt)isasymplecticmanifold.
+Supposenowthatforany t∈Rtheinvolutiveclosureofthevectordistribution Imαt⊂
+TYcoincideswiththewholetangentbundle TM. ThentheLagrangeanchor(4.17),(4.18)
+isweaklytransitive. Theconditionforthe1-form(3.17)to beapropersymmetryof(3.15)
+gives
+iX(/tildewidedψ) = 0, i X(∂tψ+/tildewidedψ(v)) = 0 (5.20)SYMMETRIES AND CONSERVATION LAWS 24
+for any vertical vector field X=α(w), withwbeing a 1-form. It follows from the first
+equationin (5.20)that
+LX/tildewidedψ= 0⇒/tildewidedψ= 0, (5.21)
+sincethevectorfields X=αt(w)generatethewholeLiealgebra X(Y). Ifthefirstgroupof
+theDeRhamcohomologyof Yistrivial,then ψ=/tildewidedf′forsomefunction f′∈C∞(Y×R)
+and thesecondequationin (5.20)takes theform
+X·(∂tf′+v·f′) = 0. (5.22)
+Again, the transitivity implies that ∂tf′+v·f′= ˙g(t)for someg∈C∞(R). Setting
+f=f′−g, one can see that the 1-form ψ=/tildewidedfdefines a characteristic for (3.15) in
+accordance with (3.18). So, every proper symmetry for the tr ansitive Lagrange anchor
+(4.17) comes from some characteristic. This is in agreement with Proposition 5.2. In the
+case beingconsideredthehomomorphism(5.14)isactuallya n isomorphism. Indeed, if X
+is the proper symmetry corresponding to a characteristic Ψ = d/integraltext
+f(t,x)dt, thenX= 0
+implies that α(w)·f= 0for any 1-form w. In view of the weak transitivity, fhas to be
+a function of talone. Then, Ψ = 0. We thus arrive at the following generalization of the
+Noethertheorem forthefirst-order ODEs:
+LetVbe the weakly transitive Lagrange anchor (4.17) associated to the differential
+equations(3.15)and avertical bivectorfield αonY×RwithH1(Y) = 0, then
+Char(˙x+v)≃PSymα(˙x+v). (5.23)
+Consider now an integrable Lagrange structure (E,T,dE). In the integrable case, the
+dual of the dynamics bundle E∗→Mcarries the structure of Lie algebroid and the dif-
+ferentialdE: Γ(∧•E)→Γ(∧•+1E)makes the exterior algebra of E-differential forms
+into a cochain complex. Let H•(E) = Kerd E/ImdEdenote the corresponding cohomol-
+ogy groups. Classical dynamics on Mare specified by a 1-cocycleT. IfT′is another
+representativeofthe dE-cohomologyclass [T]∈H1(E), then
+T′=T+dES, (5.24)
+forsomeS∈C∞(M). Werefertothelastrelationbysayingthattheequations T′= 0are
+obtained by a proper deformation of the equations T= 0; the function Sis called a twist.
+The relevance of this notion to the context of conservation l aws and proper symmetries
+is evident from the following simple observation. If Sis invariant with respect to the
+characteristic Ψ∈Char(T),
+LΨS.= 0, (5.25)SYMMETRIES AND CONSERVATION LAWS 25
+thenΨ∈Char(T′). Indeed,
+iΨT′=iΨ(T+dES).= 0. (5.26)
+Notice that the conserved current of the deformed equations T′can be different from that
+oftheequations T,eventhoughthecharacteristicisthesame.
+In general the proper deformation (5.24) can be quite nontri vial and by no means re-
+duces to equivalence transformations of the original equat ionsT, like taking linear com-
+binations of the equations or change of variables. To exempl ify the non-triviality of the
+deformation(5.24), itis sufficienttoconsideraproperdef ormationsofthesimplestpossi-
+bledifferentialequations
+Ti≡˙xi(t) = 0. (5.27)
+As is known, these equations enjoy an integrable Lagrange st ructure given by a vertical
+Poissonbivector α=αij(t,x)∂i∧∂jonY×R. Takingthetwist Sintheform
+S=−/integraldisplay
+dtH(t,x(t)), (5.28)
+whereH(t,x)is an arbitrary function,we gettheHamiltonianequations
+T′i=Ti−αijδS
+δxj= ˙xi−{xi,H}= 0. (5.29)
+So, the proper deformation is quite nontrivial as it can tran sform the Hamiltonian equa-
+tions with a given Hamiltonian into equations with any other Hamiltonian and the same
+Poisson bracket. The characteristics for the original equa tions (5.27) have the form Ψ =
+d/integraltext
+f(x(t))dt, wheref(x)is an arbitrary function. If the twist (5.28) is invariant wi th
+respect to Ψ, then{f,H}= ˙g(t)andthevalue f−gisan integralofmotionof(5.29).
+6 Field-theoreticalexamples
+To further illustrate the concept of proper symmetry we cons ider here some well-known
+examples of non-Lagrangian field equations. In addition, we will see what the Lagrange
+anchorlookslikeinlocal field theory.
+6.1p-formfields
+In this subsection, we consider the theory of free p-form fields in the strength-tensor for-
+malism. We will show that the notion of Lagrange anchor enabl es one to relate all theSYMMETRIES AND CONSERVATION LAWS 26
+space-timesymmetrieswithconservedcurrentsexpressibl ethroughtheenergy-momentum
+tensor. In the Lagrangian framework, relied on the notion of gauge potential, this rela-
+tionship is just an immediate consequence of the Noether the orem. Our consideration,
+however,doesnot supposetheexistenceofanyaction functi onal.
+Given ann-dimensional Riemannian manifold (X,g), consider a p-form field F∈
+Λp(X)subject toequations
+T1(F)≡dF= 0, T 2(F)≡d∗F= 0. (6.1)
+Here∗: Λp(X)→Λn−p(X)is the Hodge operator associated to the metric g. The
+equationsenjoyno gaugesymmetryobeyingtheNoetherident ities
+dT1= 0, dT 2= 0. (6.2)
+Thelast fact pointsup thenon-Lagrangian natureofthefield equations(6.1).5
+To make contact with the general definitions of the previous s ections let us mention
+that theconfigurationspace offields Mis givenhere by thespace Λp(X)and thesections
+of the dynamics bundle, in particular the equations of motio nT= (T1,T2), take values in
+the space Λp+1(X)⊕Λn−p+1(X). SinceMis a linear space, we can identify the tangent
+spaceTFMat a pointF∈Mwith thespace M= Λp(X)itself. Furthermore, by making
+useofthestandardinnerproducton Λ(X),
+(A,B) =/integraldisplay
+XA∧∗B, (6.3)
+weidentifythevectorspaces Λp(X)andΛp+1(X)⊕Λn−p+1(X)withtheirdual spaces.
+One can easily check that the field equations (6.1) admit the t wo-parameter family of
+Lagrange anchors Vdefined by
+/an}bracketle{tV(W),P/an}bracketri}ht˙ =/an}bracketle{tW,V∗(P)/an}bracketri}ht=a(W1,dP)+b(W2,d∗P), (6.4)
+for allW= (W1,W2)∈Λp+1(X)⊕Λn−p+1(X),P∈Λp(X), anda,b∈R. This family
+is a straightforward generalization of the Lagrange anchor proposed in [7] for Maxwell’s
+electrodynamicsin thestrength-tensorformalism.
+Since the equations of motion (6.1) are linear and the anchor (6.4) does not depend
+on fields, the bracket (5.3) is zero and the corresponding Lag range structure appears to be
+5Ofcourse,onecansolvethefirstequation(6.1)intermsofag augepotential, F=dA,followingwhich
+the second equation(6.1) becomesLagrangian,but we are int erested in treating these equationsas they are,
+i.e.,withoutpassingtoanyequivalentLagrangianformula tion.SYMMETRIES AND CONSERVATION LAWS 27
+integrable. To investigateits transitivitywe should cons iderthe kernel of theoperator V∗.
+By definition,a p-formPbelongsto kerV∗if
+/an}bracketle{tW,V∗(P)/an}bracketri}ht= 0∀W. (6.5)
+Supposethat ab/ne}ationslash= 0, thenthelastconditionis equivalenttotheequations
+dP= 0, d∗P= 0, (6.6)
+which coincideinform withtheequationsofmotion(6.1)for thefieldF. Sinceequations
+(6.6) enjoy no gauge symmetry,their general solution Pcannot depend on arbitrary func-
+tionalparameters6. Inparticular,itcannotdependonthefield F. Werefertothissituation
+by sayingthattheLagrangeanchor(6.4)is almosttransitive forab/ne}ationslash= 0.
+NoticealsothattheLagrangeanchor(6.4)isnontrivialwhe nevera/ne}ationslash=b. Incasea=b,
+wecan write(6.4)as
+(W1,T1(aP))+(W2,T2(aP)). (6.7)
+Thelast expressioncorrespondsto thetrivialLagrangeanc hor(4.13) with G(P) =aP.
+Recallthatavectorfield ξ∈X(X)iscalleda Killingvector ofthemetric gifLξg= 0.
+Each Killingvectorgivesriseto boththesymmetrytransfor mation
+δξF=LξF (6.8)
+ofthefield equations(6.1)andthecharacteristic Ψ = (Ψ 1,Ψ2)∈Λp+1(X)⊕Λn−p+1(X).
+Thelatterisgivenby
+Ψ1= (−1)(n−p)(p−1)∗iξ∗F,Ψ2= (−1)p−1∗iξF. (6.9)
+It is easyto verifythat
+/an}bracketle{tΨ,T/an}bracketri}ht= (Ψ1,T1)+(Ψ 2,T2) =/integraldisplay
+Xdj, (6.10)
+where theon-shellclosed (n−1)-formjreads
+j=1
+2/parenleftbig
+(iξF)∧∗F+(−1)p−1F∧(iξ∗F)/parenrightbig
+. (6.11)
+In terms of local coordinates {xµ}onXthe corresponding conserved current is given by
+the1-form
+∗j=ξµTµν(x)dxν, (6.12)
+6For instance, if Xis a compact manifold without boundary, then the solutions t o Eqs. (6.6) are the
+harmonic p-formson X. Theseforma finitedimensionalvectorspaceisomorphicto Hp(X).SYMMETRIES AND CONSERVATION LAWS 28
+Tµν=Tνµbeingthesymmetricenergy-momentumtensorofthe p-form fieldF.
+InviewofProposition5.1,thecharacteristic Ψisboundtobethegeneratorofaproper
+symmetry. A simplecalculationgives
+δξF=V(Ψ)≈(a−b)LξF. (6.13)
+So, up to an overall constant, which can be included in ξ, the transformation (6.13) co-
+incides with the Lie derivative of the p-formFalong the Killing vector ξ. As would be
+expected, thetransformationbecomestrivialwhen a=b(thecase oftrivialanchor(6.7)).
+To summarize, each infinitesimal isometry of the Riemannian manifold (X,g)gives
+rise to a conservation law with the current (6.12). The Lagra nge anchor (6.4) allows us
+to consider this conservation law as coming from the proper s ymmetry (6.13) of the field
+equations(6.1). Moreover,as theLagrangeanchorisalmost transitivefor ab/ne}ationslash= 0, onecan
+be sure that each proper symmetry V(Ψ)results in a conservation law provided that the
+generator Ψdepends on F.
+All the above formulas hold true for conformal Killing vecto rsξin the critical di-
+mensionn= 2p. This follows from the conformal invariance of the Hodge ope rator
+∗: Λp(X)→Λp(X).
+6.2 Self-dual p-formfields
+Let(X,g)be a(4k+2)-dimensional pseudo-Riemannian manifold of Lorentz signa ture.
+Then the Hodge operator ∗: Λ2k+1(X)→Λ2k+1(X)squares to +1on the middle forms
+and we have the direct sum decomposition Λ2k+1(X) = Λ2k+1
++(X)⊕Λ2k+1
+−(X), where
+Λ2k+1
+±(X)are the subspaces of (anti-)self-dual (2k+ 1)-forms onX. The subspaces
+Λ2k+1
+±(X)areknowntobeisotropicwithrespecttothestandardinnerp roduct(6.3). There-
+fore, theinnerproductdefinesanon-degeneratepairingbet weenthespaces Λ2k+1
++(X)and
+Λ2k+1
+−(X), sothat wecan regardthesespaces as beingdualto each other .
+Considernowa self-dual (2k+1)-formfieldH+subjecttotheclosednesscondition
+T(H+)≡dH+= 0. (6.14)
+Regarding this condition as equations of motion, we see that the sections of the corre-
+spondingdynamicsbundle E →Mtakevaluesin Λ2k+2(X)andMisgivenby Λ2k+1
++(X).
+Thetangentspacetoeachfieldconfiguration H+∈Mcanbeidentifiedwith Masbefore.
+Equations (6.14) are known to be non-Lagrangian unless one i ntroduces auxiliary fieldsSYMMETRIES AND CONSERVATION LAWS 29
+or breaks the manifest covariance [26, 27, 28, 29, 30, 31]. Th ere is the Noether identity
+dT= 0butnogaugesymmetry. Itwasshownin[11]thatthetheory(6. 14)admitsanatural
+Lagrange structuredefined by theanchor
+/an}bracketle{tV(W),P/an}bracketri}ht˙ =/an}bracketle{tW,V∗(P)/an}bracketri}ht= (W,dP), (6.15)
+whereW∈Λ2k+2(X)andP∈Λ2k+1
+−(X). Again, the Lagrange anchor is almost transi-
+tive,sincethecondition dP= 0impliesthattheanti-self-dualform Pdoesnotdependon
+thefieldH+.
+Ifξ∈X(X)is a conformal Killing vector of the metric g, thenΨ =− ∗iξH+∈
+Λ2k+2(X)is a characteristic for (6.14). Indeed, using the identity H+∧LξH+= 0, one
+can readilyfind
+/an}bracketle{tΨ,T/an}bracketri}ht=−(∗iξH+,dH+) =/integraldisplay
+Xdj, j =1
+2iξH+∧H+.(6.16)
+By analogy with the previous case we can define the energy-mom entum tensor Tµνof the
+self-dual field H+through the conserved current ∗j=ξµTµν(x)dxν. It is easy to see that
+thetensorTµνis symmetricand traceless.
+According to Proposition 5.1, the anchor (6.15) takes the ch aracteristic Ψto a proper
+symmetry of the equations of motion (6.14). As one would expe ct, the corresponding
+variation of H+is given by the Lie derivativealong the conformal Killing ve ctor modulo
+on-shellvanishingterms:
+δξH+=V(Ψ)≈LξH+. (6.17)
+Conversely, one could start with the proper symmetry transf ormation (6.17), which
+presence is evident from the very definition of the theory (6. 14), and then argue that the
+generator Ψis nothing else but a characteristic to the field equations si nce the Lagrange
+anchor (6.15) is almost transitive. This gives the desired r elation between the space-time
+symmetriesofthemodeland theconservationlaws.
+6.3 Chiral bosons intwo dimensions
+Consider now a multiplet of Nself-dual 1-forms H+
+a,a= 1,...,N, in two dimensional
+Minkowskispace R1,1. Asabove,thefieldequationshavetheformofclosednesscon dition
+Ta(H+)≡dH+
+a= 0. (6.18)SYMMETRIES AND CONSERVATION LAWS 30
+Besidesthespace-timesymmetries,consideredintheprevi oussubsection,equation(6.18)
+have internal rigid symmetries corresponding to the genera l linear transformations in the
+target space offields:
+H+
+a→GabH+
+b,(Gab)∈GL(N,R). (6.19)
+The case of two dimensions is rather special at least for thre e reasons. First, the sub-
+spacesofself-dualandanti-self-dual1-formsadmitanice geometricvisualizationasapair
+of transversal isotropic distributionsdefining the light c one at each point of R1,1. Second,
+thedynamicalfields H+
+aare conservedcurrent themselvesforifweset ja=H+
+a,then
+dja=Ta≈0. (6.20)
+Third, and most important, the Lagrange anchor (6.15) admit s a non-abelian generaliza-
+tion7, which can be used to connect some of the rigid symmetries (6. 19) with the conser-
+vation laws (6.20). To define this generalization we interpr et theN-dimensional internal
+spaceoffields H+asthelinearspaceunderlyingasemi-simpleLiealgebra Gwithabases
+{ta}and thecommutationrelations
+[ta,tb] =fab
+ctc. (6.21)
+(Of course, such an interpretation imposes certain restric tions on the possible values of
+N. For example, it excludes N= 2.) Then we can combine the fields H+
+ainto a single
+G-valued 1-form H+=H+
+atasubject to theequationof motion dH+= 0. Geometrically,
+onecanthinkof H+asasectionofthetrivial SO(1,1)×G-vectorbundleover R1,1,where
+Gis aLiegroup withtheLiealgebra G.
+Considernowthefollowingnon-abeliangeneralizationoft heLagrangeanchor(6.15):
+/an}bracketle{tV(W),P/an}bracketri}ht˙ =/an}bracketle{tW,V∗(P)/an}bracketri}ht= (Wa,dPa+g[P,H+]a). (6.22)
+HereP=Patais aG-valued anti-self-dual 1-form and gis an arbitrary constant; all the
+Lie algebra indices are raised and lowered with the help of th e Killing metric on G. In
+caseg= 0, we haveNcopies of the Lagrange anchors (6.15). For g/ne}ationslash= 0, the Lagrange
+structure remains integrable, but the corresponding Lie br acket (5.3) on the dual of the
+dynamicsbundlebecomes nontrivial:
+[ea(x),eb(x′)] =−gfc
+abec(x)δ2(x−x′), e a∈Λ2(R1,1).(6.23)
+7It should be stressed that this generalization has nothing t o do with a non-abelian deformation of the
+original(abelian)equationsofmotion(6.18); thefieldequ ationsremainthe same.SYMMETRIES AND CONSERVATION LAWS 31
+Theconservedcurrents(6.20)correspondtothe G-valuedcharacteristic Ψ =−∗εata:
+/an}bracketle{tΨ,T/an}bracketri}ht=−(∗εa,dH+
+a) =/integraldisplay
+Xdj, j =εaH+
+a. (6.24)
+By Proposition5.1thischaracteristicgenerates a propers ymmetrytransformation
+δεH+=V(Ψ) =−g[ε,H+], ε=εata. (6.25)
+Thus, weseethat thepropersymmetriesconstituteasubgrou pAd(G)⊂GL(N,R)inthe
+group ofallinternalsymmetries(6.19)oftheequationsofm otion.
+As to the relativistic symmetries of the system (6.18), thes e are proper and have the
+sameform as intheabeliancase (6.17).
+7 Conclusion
+Let us make some concluding remarks on the paper results and n otice the open questions
+left forthefuturestudies.
+In the work [7], theconcept of Lagrangeanchor was introduce d to formulatethepath-
+integral quantization for not necessarily Lagrangian field theories. In the present paper,
+we show that another important property of the Lagrangian dy namics - the relationship
+between rigid symmetries and conservation laws - extends to non-Lagrangian field equa-
+tions whenever they are endowed with a Lagrange anchor. It wa s also shown in [7] that
+every Lagrange anchor gives rise to and can be related with a c ertain BRST complex on
+theghost-extendedconfigurationspaceoffields. IntheLagr angiangaugetheory,thiscom-
+plex boilsdownto thewell-knownBRST complexassociated wi ththeBatalin-Vilkovisky
+master action. The unique existence of the local BV master ac tion was proven long ago
+[33]undertheassumptionsthattheoriginalLagrangianwas localandthegaugesymmetry
+was finitely reducible. Unlike the Lagrangian theory, the lo cality of both the Lagrange
+anchor and the non-Lagrangian equations does not necessari ly result in the locality of the
+corresponding quantum BRST differential. Given a local Lag range anchor for local field
+equations,theobstructionstotheexistenceofalocalBRST complexwillbestudiedinour
+next paper. Remarkably, these obstructions, being crucial for the quantum theory, do not
+matter for the study of classical dynamics. At the classical level, the existence of a local
+Lagrange anchor is sufficient to link the conservation laws w ith rigid symmetries, as it is
+seen from thepresentpaper.SYMMETRIES AND CONSERVATION LAWS 32
+In the next work we also plan to develop the cohomological des cription of the rigid
+symmetriesandconservationlaws,makinguseoftheBRSTfor malismfornon-Lagrangian
+gauge theories [7]. This will extend the cohomological form ulation of the Noether theo-
+rem, well studied in the Lagrangian setting [5], to not neces sarily Lagrangian dynamics.
+In the entire BRST cohomology, we will identify the characte ristic cohomology as well
+as the cohomology group capturing the rigid symmetries of no n-Lagrangian equations of
+motion. UnliketheLagrangian theory,thesetwo groups ofth eBRST cohomologyare not
+to begenerallyisomorphictoeach other.
+One can see that many important ingredients of the Lagrangia n dynamics split into
+different categories in the non-Lagrangian setting. The lo cal Lagrange anchor connects
+these categories, although not always making them identica l. As a result, the Lagrange
+anchor endows the non-Lagrangian local field theory with som e important features of the
+Lagrangian one. In particular, it offers a path-integral qu antization of classical dynamics
+and establishes a (partial) connection between the rigid sy mmetries and the conservation
+laws.
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\ No newline at end of file
diff --git a/1001.0092.txt b/1001.0092.txt
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@@ -0,0 +1,720 @@
+arXiv:1001.0092v2  [hep-ph]  9 Jun 2010TU-853, UT-HET 030
+Probing the Majorana nature of TeV-scale radiative seesaw
+models at collider experiments
+Mayumi Aoki1,∗and Shinya Kanemura2,†
+1Department of Physics, Tohoku University,
+Aramaki, Aoba, Sendai, Miyagi 980-8578, Japan
+2Department of Physics, University of Toyama,
+3190 Gofuku, Toyama 930-8555, Japan
+Abstract
+A general feature of TeV-scale radiative seesaw models, in w hich tiny neutrino masses are gen-
+erated via loop corrections, is an extended scalar (Higgs) s ector. Another feature is the Majorana
+nature;e.g., introducing right-handed neutrinos with TeV-scale Major ana masses under the dis-
+crete symmetry, or otherwise introducing some lepton numbe r violating interactions in the scalar
+sector. We study phenomenological aspects of these models a t collider experiments. We find that,
+while properties of the extended Higgs sector of these model s can be explored to some extent, the
+Majorana nature of the models can also be tested directly at t he International Linear Collider via
+the electron-positron and electron-electron collision ex periments.
+PACS numbers: 12.60.Fr, 14.60.St, 14.80.Fd
+Keywords: Radiative seesaw models, Higgs physics, Flavor physics
+∗Electronic address: mayumi@tuhep.phys.tohoku.ac.jp
+†Electronic address: kanemu@sci.u-toyama.ac.jp
+1I. INTRODUCTION
+The neutrino data show that neutrinos have tiny masses as compar ed to the electroweak
+scale. This is clear evidence for physics beyond the standard model (SM). The data also
+indicate that the structure of flavor mixing for neutrinos is largely d ifferent from that for
+charged leptons. These facts would suggest that, while charged le ptons have Dirac type
+masses, the neutrino masses are of the Majorana type. The tiny M ajorana masses of left-
+handed neutrinos are generated from the dimension-five effective operators
+L=cij
+2Λνci
+Lνj
+Lφ0φ0, (1)
+where Λ represents a mass scale, cijare dimensionless coefficients, and φ0is the Higgs
+boson. After electroweak symmetry breaking, the mass matrix Mij
+νfor left-handed neutrinos
+appears as Mij
+ν=cij/angbracketleftφ0/angbracketright2/Λ. As the vacuum expectation value (VEV) /angbracketleftφ0/angbracketrightof the Higgs
+boson is O(100) GeV, the observed tiny neutrino masses ( Mij
+ν<∼0.1 eV) are realized when
+(cij/Λ)∼ O(10−14) GeV−1. It has been aninteresting problem how we can naturally explain
+such a small number with less fine tuning.
+If the operators in Eq. (1) appear at the tree level in the low energ y effective theory,
+Λ has to be as large as O(108)− O(1014) GeV for cijbeingO(10−6)− O(1) to describe
+the data. For example, in the tree-level seesaw scenario where rig ht-handed neutrinos are
+introduced, their Majorana masses have to beset much higher tha nthe electroweak scale [1],
+corresponding to the scale Λ in Eq. (1). Although the scenario is simp le, it requires another
+hierarchy between the mass of right-handed neutrinos and the ele ctroweak scale, and in
+addition, physics at such a large mass scale is difficult to be tested at c ollider experiments.
+Quantum generation of neutrino masses is an alternative way to obt ain (cij/Λ)∼
+O(10−14) GeV−1. Due to the loop suppression factor, Λ in these models can be lower
+as compared to that in the tree-level seesaw models. Consequent ly, the tiny neutrino masses
+would be explained in a natural way by the TeV-scale dynamics without introducing very
+high mass scales. The original model of this line was proposed by Zee [2 ], where neutrino
+masses were generated at the one-loop level. Some variations were considered [3–7], for
+example, by Zee and Babu [3], Kraus-Nasri-Trodden [4], Ma [5], and the model in Ref. [6].
+The last three models contain dark matter (DM) candidates with the odd quantum number
+under the discrete Z2symmetry. It must be a charming point in these TeV-scale radia-
+2tive seesaw models that they are directly testable at the collider exp eriments such as Large
+Hadron Collider (LHC) and the International Linear Collider (ILC).
+A general feature in radiative seesaw models is an extended Higgs se ctor, whose detail is
+strongly model dependent. The discovery of these extra Higgs bo sons and detailed measure-
+ments of their properties at current and future collider experimen ts can give partial evidence
+for the radiative seesaw models. In the literature [8–14], phenomen ology of these radiative
+seesaw models has already been studied extensively. Such previous works mainly discuss
+constraints on the flavor structure from the current data for s uch as neutrino physics and
+DM, and also study collider phenomenology of the Higgs sectors [15–1 7, 19–24].
+Another common feature in radiative seesaw models is the Majorana nature. In order to
+induce tiny Majorana masses for left-handed neutrinos, we need t o introduce its origin such
+as lepton number violating interactions in the scalar sector [2, 3] or r ight-handed neutrinos
+with TeV-scale Majorana masses [4–6]. When the future data would in dicate an extended
+Higgs sector predicted by a specific radiative seesaw model, the dire ct detection of the
+Majorana property at collider experiments should be a fatal probe to identify the model.
+In this Letter, we study the phenomenology in TeV-scale radiative s eesaw models, in
+particular, apossibility ofdetecting theMajorananatureatcollider experiments. Wemainly
+discussthreetypicalradiativeseesawmodelsasreferencemodels ; themodelbyZeeandBabu
+where neutrino masses are generated at the two-loop level [3], tha t by Ma with one-loop
+neutrino mass generation [5], and that in Ref. [6] where neutrino mas ses are generated at
+the three-loop level. Typical parameter regions where the data ca n be satisfied have been
+already studied in each model in the literature. We here study collider phenomenology in
+such typical parameter regions in each model, and discuss the discr imination of these models
+by measuring the details of the Higgs sector and the Majorana natu re at the LHC and the
+ILC.
+II. RADIATIVE SEESAW MODELS
+A. The Zee-Babu model
+In the model proposed in Ref. [3] (we refer to as the Zee-Babu mod el), in addition to
+singly-charged singlet scalar bosons ω±, doubly-charged singlet scalar fields k±±are intro-
+3k!! !!!!
+"Li"Lj
+ºvg’kl 
+ºveL! k eR! l eR! k eL! l "
+fik fjl º ºv v
+º!LjH"H"
+NRc !!Li
+hj!hi!eR" i eR" j S"S" yi yj#0
+κ κº ºv v
+º !Lj"0r,i "0r,i 
+NRc !!Li
+hj! hi!"5
+ʈ ʈ
+FIG. 1: Feynman diagrams for neutrino masses in the model by Z ee-Babu [3] (left), that by Ma [5]
+(center) and that in Ref. [6] (right).
+duced, both of which carry the lepton number of two-unit, and the ir interactions are given
+by
+Lint=fab(Lci
+aLLj
+bL)ǫijω++g′
+ab(ℓc
+aRℓbR)k++−µk++ω−ω−+H.c., (2)
+whereLLis the left-handed lepton doublet and ℓRis the right-handed lepton singlet. The
+matrices fijandg′
+abare respectively an anti-symmetric and a symmetric couplings and th e
+lepton number is violated by the interaction with the parameter µ.
+The neutrino mass matrix is generated at the two-loop level via the d iagram in Fig. 1
+(left);
+Mν
+ij=3/summationdisplay
+k,ℓ=1/parenleftbigg1
+16π2/parenrightbigg24µ
+m2
+ωfik(yℓkgkℓyℓℓ)fℓjv2I1(m2
+k/m2
+ω), (3)
+whereyi[=√
+2mi/v(i=e,µ,τ)] are the SM Yukawa coupling constants of charged leptons
+with the masses miand the VEV v(≃246 GeV), gijare defined as gii=g′
+iiandgij= 2g′
+ij
+(i/negationslash=j),mωandmkare masses of ω±andk±±, and
+I1(r) =−/integraldisplay1
+0dx/integraldisplay1−x
+0dy1
+x+(r−1)y+y2lny(1−y)
+x+ry, (4)
+whereI1(r) takes the value of around 3 - 0.2 for 10−2<∼r<∼102. The universal scale
+of neutrino masses is determined by the two-loop suppression fact or 1/(16π2)2and the
+lepton number violating parameter µ. The charged lepton Yukawa coupling constants yℓi
+(ye≪yµ≪yτ<∼10−2) give an additional suppression factor. Thus, any of fijorgijcan be
+ofO(1) when mωandmkare at the TeV scale. The flavor structure of the mass matrix is
+determined by the combination of the coupling constants fijandyigijyj.
+4Theflavoroff-diagonalcouplingconstants fijandgijinduceleptonflavorviolation(LFV).
+From the results of µ→eγ,τ→eγandτ→µγ,|fµτfτe|,|fτµfµe|and|fτefeµ|are
+respectively constrained as a function of mω. The data of rare decays of µ→eee,τ→µµe
+andτ→µeeare also used to constrain the combinations |gµegee|,|gτegµµ+gτµgµe|and
+|gτegµe+gτµgee|, respectively, depending on mk. Theg−2 data can also be used to constrain
+a combination of these coupling constants with mωandmk1.
+In the scenario with hierarchical neutrino masses, fijsatisfyfeµ≃feτ≃fµτ/2. The
+typical relative magnitudes among the coupling constants gijcan begµµ:gµτ:gττ≃1 :
+mµ/mτ: (mµ/mτ)2. Forgµµ≃1, the neutrino data and the LFV data give the constraints
+such asmk>∼770 GeV and mω>∼160 GeV [10]. On the other hand, the constraints on
+the couplings and masses are more stringent for the inverted neut rino mass hierarchy. The
+current data then gives mω≃825 GeV for gµµ≃1 [10]. One of the notable things in this
+case is the lower bound on sin22θ13, which is predicted as around 0.002 [9].
+B. Models with TeV-scale Right-handed Neutrinos with a discrete Z2symmetry
+Similar to the tree-level seesaw model, tiny masses of left-handed n eutrinos would also
+come from Majorana masses MNα
+Rof gauge-singlet right-handed neutrinos Nα
+Rin the radia-
+tive seesaw scenario [4–6]. One simple way to realize the absence of t he tree-level Yukawa
+interaction νi
+L˜ΦNα
+Ris introduction of a discrete Z2symmetry, with the assignment of the
+odd quantum number to Nα
+Rand the even to the SM particles. To obtain the dimension five
+operator in Eq. (1) at the loop level, we need to introduce additional Z2-odd scalar fields.
+The lightest of all the Z2-odd particles can be a candidate of DM if it is electrically neutral.
+The original model of the radiative seesaw model with such a discret e symmetry is proposed
+by Krauss, Nasri, and Trodden [4], in which neutrino masses are indu ced at the three-loop
+level. In the following, we consider two variant models of the Kraus-N asri-Trodden (KNT)
+model.
+1If we take gee= 0, then mkis unbounded from the µ→eeeandτ→ℓeeresults (ℓ=eorµ), so that
+relatively light k±±(mk∼100−200 GeV) are possbile.
+51. The Ma model
+The model in Ref. [5], which we here refer to as the Ma model, is the simp lest radiative
+seesaw model with right-handed neutrinos Nα
+R, in which the discrete Z2symmetry is intro-
+duced and its odd quantum number is assigned to Nα
+R. The Higgs sector is composed of
+two Higgs doublet fields, one of which (Ξ) is Z2odd. As long as the Z2symmetry is exact,
+the neutral components of Ξ do not receive VEVs. We have one SM- like Higgs boson h,
+and four physical Z2-odd scalar states; ξ0
+r(CP-even), ξ0
+i(CP-odd) and ξ±as physical scalar
+states. This Z2odd Higgs doublet is sometimes called as the inert Higgs doublet [15] or the
+dark scalar doublet [17]. The LEP II limits are studied in this model in Ref . [18].
+The neutrino masses are generated at the one loop level via the diag ram depicted in
+Fig. 1 (center), in which Z2odd particles, ξ0andNα
+R, are in the loop. The mass matrix is
+calculated as
+Mν
+ij=−3/summationdisplay
+α=1/parenleftbigg1
+16π2/parenrightbiggˆhα
+iˆhα
+jλ5v2
+MNα
+R1
+1−rα/parenleftbigg
+1+1
+1−rαlnrα/parenrightbigg
+, (5)
+whereˆhα
+iare the Yukawa coupling constants of νi
+L/tildewideΞNα
+R,MNα
+Ris the Majorana mass of
+theα-th generation right-handed neutrino Nα
+R,λ5= (m2
+ξi−m2
+ξr)/v2,rα=m2
+0/M2
+Nα
+Rwith
+m0= (mξi+mξr)/2, where mξrandmξiare masses of ξ0
+randξ0
+i, respectively. The universal
+scale for neutrino masses is determined by the one-loop suppressio n factor 1 /(16π2),λ5and
+MNα
+R. The flavor structure in Mν
+ijis realized by the combination of ˆhα
+iˆhα
+j/MNα
+R. Therefore,
+forMNα
+R∼ O(1) TeV, the combination of the coupling constants would be |λ5|(ˆhα
+i)2∼10−9.
+In this model, there are two scenarios with respect to the DM candid ate; i.e., the lightest
+right-handed neutrino N1
+Ror the lightest Z2-odd neutral field ( ξ0
+rorξ0
+i). For both cases,
+there are parameter regions where the neutrino data are adjust able without contradicting
+other phenomenological constraints [13]. In this Letter, we cons ider the case where the dark
+doublet component ξ0
+ris the DM candidate2. When the mass of the DM is around 50 GeV,
+the typical value of λ5∼10−2for the neutrino masses gives the mass difference between
+ξ0
+randξ0
+iabout 10 GeV.3The relic abundance of such DM is consistent with the WMAP
+data [19].
+2In Ref. [13], the scenario where the lightest right-handed neutrino is DM is explored.
+3In order to avoid constraint from the DM direct search results, it is required that |λ5|>10−6.
+62. The AKS model
+In the model in Ref. [6], which we here refer to as the AKS model, it is in tended that not
+only the tiny neutrino masses and DM but also baryon asymmetry of U niverse are explained
+at the TeV scale. In addition to the TeV-scale right-handed neutrin osNα
+R(α= 1,2), the
+Higgs sector is composed of Z2-even two Higgs doublets Φ i(i= 1,2) andZ2-odd charged
+singletsS±and aZ2-odd neutral real singlet η0. Therefore the physical states in the Z2-even
+sector are H(CP-even), A(CP-odd), H±andh(CP-even).
+The neutrino mass matrix is generated at the three-loop level via th e diagram in Fig. 1
+(right), and is expressed as
+Mν
+ij=2/summationdisplay
+α=1/parenleftbigg1
+16π2/parenrightbigg3(yℓihα
+i)(yℓjhα
+j)(κtanβ)2v2
+MNα
+RI2(mH±,mS±,mNα
+R,mη), (6)
+wheremH±,mS±,mNα
+Randmηare themasses of the doublet originatedcharged Higgsboson
+H±,S±,Nα
+Randη0, respectively; hα
+iandκvare the coupling constants of Nα
+Rei
+RS+and
+H+S−η0, respectively; tan β=/angbracketleftΦ0
+2/angbracketright//angbracketleftΦ0
+1/angbracketright, and
+I2(x,y,z,w) =−4z2
+z2−w2/integraldisplay∞
+0udu/braceleftbiggB1(−u;x,y)−B1(−u;0,y)
+x2/bracerightbigg2/parenleftbiggz2
+u+z2−w2
+u+w2/parenrightbigg
+,(7)
+whereB1is the tensor coefficient function in the Passarino-Veltman’s formalis m [25]. Al-
+thoughtheHiggssectorisrathercomplicatedtomakeitpossiblefor theelectroweakbaryoge-
+nesis scenario, theflavor structure isdetermined only bythe comb inationof hα
+iandmNα
+Rjust
+asintheMamodel. Themassmatrixhasthethreeloopfactor1 /(16π2)3withadditionalsup-
+pression factor by yi. They are enough to reproduce the neutrino mass scale. Thus, th e elec-
+tronassociated coupling constants h1,2
+eandthescalar coupling κareofO(1) form1,2
+NR∼ O(1)
+TeV. The Yukawa coupling constants hα
+iare hierarchical as h1,2
+e(≃ O(1))≫h1,2
+µ≫h1,2
+τ.
+The parameter sets which satisfy the current data from neutrino oscillation, LFV, relic
+abundances of DM and the condition for strongly first order electr oweak phase transition
+are studied in Ref. [6, 14]. To reproduce the neutrino data, the mas s ofH±should be 100
+- 200 GeV. This is an important prediction of the model. In order to av oid the constraint
+fromb→sγ, the Yukawa interaction for the doublet fields takes the form of so -called Type-
+X [20],4where only one of the doublets couples to leptons and the rest does to quarks. The
+4Type-X is referred to as Type-IV in Ref. [26] and Type-I’ in Ref. [27 ].
+7physics of the Type-X two Higgs doublet model (THDM) shows many d istinctive features
+from the other type of extended Higgs sectors. For example, HandAdecay mainly into
+τ+τ−when tan β>∼3 and sin( β−α)≃1 [20]. There are basically two DM candidates, η0
+andNα
+R. The mass of S±is strongly constrained by the current data and the requirement
+for strongly first order phase transition [6, 14]. The coupling const ant ofS+S−his required
+to be ofO(1), whose indirect effect appears in the quantum correction to th ehhhcoupling
+constant as a large deviation from the SM prediction [14, 28]. As long as kinematically
+allowed, S±decays via S±→H±η0by 100%.
+III. PHENOMENOLOGY IN RADIATIVE SEESAW MODELS AT THE LHC
+The existence of the extra Higgs bosons such as charged scalar bo sons, which are a com-
+mon feature of radiative seesaw models, can be tested at the LHC. Details of the properties
+of such extra Higgs bosons are strongly model dependent, so tha t we can distinguish models
+via detailed measurements of extra Higgs bosons. In addition, as th e (SM-like) Higgs boson
+his expected to be detected, its mass and decay properties are tho roughly measured [29].
+The radiative seesaw models with a DM candidate can also be indirectly t ested via the in-
+visible decay of has long as its branching ratio is more than about 25% for mh= 120 GeV
+withL= 30 fb−1[30]. The phenomenological analyses at the LHC in each model are in th e
+literature [9–12, 14, 16, 17, 20–24]. We here review some remarkab le features.
+At the LHC, ω±andk±±in the Zee-Babu model can be produced in pair, via the Drell-
+Yans-channel processes q¯q→ω+ω−andq¯q→k++k−−. The direct detection of k±±can
+be a signature for this model. The σ(q¯q→k++k−−) is around 0 .1 fb formk∼800 GeV.
+Ifk±±mainly decay into 4 µ(or 4e), the number of the signal event with L=100 fb−1[31]
+is enough for the discovery for mk<∼800 GeV [9]. The doubly-charged Higgs bosons are
+however also predicted in the models with complex triplet scalar fields, ∆ = (∆±±,∆±,∆0).
+The gauge coupling W±∆±∆∓∓induces the single doubly-charged Higgs production q¯q′→
+W±∗→∆±±∆∓, whose cross section is comparable to that of q¯q→γ∗,Z∗→∆++∆−−[32].
+The absence of the gauge coupling W±ω±k∓∓is an important distinctive feature of the Zee-
+Babu model with gauge singlet doubly-charged Higgs bosons from th e triplet model. The
+singly-charged Higgs boson ω±would be more difficult to see the signal at the LHC because
+the final state from ω+ω−isℓ+ℓ−plus a missing energy.
+8In the Ma model, if the mass of the DM candidate satisfies mξr,i< mh/2, the SM Higgs
+boson with mh<∼2mWdecays mainly into the DM pairs [16, 17]. For mξr= 50 GeV and
+mξi= 60 GeV, the branching ratio of the invisible decay h→ξ0
+rξ0
+rreaches to 70% around
+mh∼120 GeV, which can be observed at the LHC. The h→γγmode is suppressed by
+an additional contributions from ξ±[17]. In Ref. [17], the discovery potential of the dark
+scalar doublet is also analyzed by pp→ξ0
+iξ0
+rfor the benchmark points, mξr= 50 GeV, mξi
+= 60 - 80 GeV, and mξ±= 170 GeV. The decay branching ratio of the CP-odd dark scalar,
+ξ0
+i→Z∗ξ0
+r→ℓ¯ℓξr, is about 0.09 (0.07) for mξi−mξr= 10 (30) GeV. A signature ℓℓplus a
+missing energy in the benchmark scenario may be discovered by the o ptimal cuts.
+In the AKS model, the invisible decay of the SM-like Higgs boson h→η0η0can also
+open if kinematically allowed. For the typical scenario in Ref. [6, 14], t he branching ratio of
+the invisible decay can amount to B(h→η0η0)≃36 (34) % for mη= 48 GeV, mh= 120
+GeV and tan β= 3 (10), so it would be testable at the LHC. As the coupling of hS+S−
+is strong, the partial width of Γ( h→γγ) deviates from the SM prediction. The lepton
+specific decays of extra Higgs bosons A,HandH±are discriminative feature of the Type-X
+THDM [20–24]. The dominant decay mode of H(A) isH→τ+τ−;B(H(A)→τ+τ−)≃1
+for tanβ>∼3. The decay into µ+µ−is suppressed by a factor of ( mµ/mτ)2. These neutral
+bosons can be seen by gg→h,A,H→τ+τ−(µ+µ−) [20, 22, 24]. The doublet originated
+charged Higgs boson H±(as well as extra neutral ones) is as light as 100 - 200 GeV, so that
+the property of the Type-X THDM can also be tested by pp→AH±→τ+τ−τ±ντ[20, 23]
+andpp→AH→4τ[20, 21]. On the other hand, the mass of the Z2-odd charged Higgs
+bosonsS±is around 400 GeV in the typical scenario in Ref. [6, 14]. They are pro duced
+in pair via the Drell-Yan process, and decay as S+S−→H+H−η0η0→τ+τ−ννη0η0. The
+event rate is about 0.5 fb for mS±= 400 GeV when tan β>∼2. Separation of the S+S−
+signal from the H+H−event and also the SM backgrounds seems to be challenging.
+At the LHC, via the physics of extra scalar bosons such as (singly an d/or doubly) charged
+Higgs bosons and CP-even Higgs bosons, the structure of the ext ended Higgs sector can be
+clarified to some extent. In addition, the invisible decay of the SM-like Higgs boson and the
+mass spectrum of the extra Higgs bosons would give important indica tion for a possibility
+to a radiative seesaw scenario. However, although they would be a s trong indication of
+radiative seesaw models, one cannot conclude that such Higgs sect or is of the radiative
+seesaw models. In order to further explore the possibility to such m odels, we have to explore
+9the other common feature of radiative seesaw models, such as the Majorana nature. In the
+next section, we discuss a possibility of testing the Majorana natur e at ILC experiments.
+IV. PHENOMENOLOGY IN RADIATIVE SEESAW MODELS AT THE ILC
+At the ILC, properties of the Higgs sector can be measured with mu ch better accuracy
+thanattheLHC, sothatwewouldbeabletoreconstruct theHiggsp otentialinanyextended
+Higgssectorifkinematicallyaccessible. Invisible decays oftheHiggsb osoncanalsobetested
+when the branching ratio B(h→invisible) is larger than a few % [33]. Furthermore, the
+Majorana nature in radiative seesaw models; i.e.,the existence of TeV scale right-handed
+Majorana neutrinos or that of lepton number violating interaction, would also be tested at
+the ILC.
+A. Electron-positron collisions
+In the pair production of charged scalar bosons at the e+e−collision, which appear in
+the radiative seesaw models ( ω+ω−in the Zee-Babu model, ξ+ξ−in the Ma model, and
+S+S−(andH+H−) in the AKS model), there are diagrams of the t-channel exchange of
+left-handed neutrinos or right-handed neutrinos in addition to the usual Drell-Yan type s-
+channel diagrams. The contribution of these t-channel diagrams is one of the discriminative
+features of radiative seesaw models, and no such contribution ent ers into the other extended
+Higgs models such as the THDM.5Theset-channel effects show specific dependences on
+the center-of-mass energy√sin proportion to log sin the production cross section, and
+enhances the production rates of the signal events for higher va lues of√s. The final states
+of produced charged scalar boson pairs arequite model dependen t but with missing energies;
+e+e−→ω+ω−→ℓ+
+Lℓ−
+LνLνL, [The Zee−Babu model] (8)
+e+e−→ξ+ξ−→W+(∗)W−(∗)ξ0
+rξ0
+r→jjjj(jjℓLνL)ξ0
+rξ0
+r,[The Ma model] (9)
+e+e−→S+S−→H+H−η0η0→τ+
+Rτ−
+RνLνLη0η0, [The AKS model] (10)
+5In the minimal supersymmetric standard model (MSSM) selectron p air production can have similar t-
+channel contributions (the Bino exchange). In such a case, the fi nal state would be something like a e+e−
+pair plus a missing energy. Therefore, we can discriminate it from the radiative seesaw models.
+10-1 -0.5 0 0.5 1
+cos θµ01101001000Diff. cross section [fb]SignalBGZee-Babu model √s = 1TeV
+µ−τ+ + missing energy
+WW
+ω+ω−
+FIG. 2: The differential cross section of e+e−→ω+ω−→µ−τ+(+missing energy) as a function
+of the angle between the outgoing muon and the beam axis in the Zee-Babu model for√s= 1
+TeV. The rate of µ−τ+(+missing energy) from main background e+e−→W+W−is also shown.
+where underlined parts in the final states are observed as missing e nergies.
+The Zee-Babu model
+In the Zee-Babu model, the decay branching pattern of ω±is determined by the relative
+magnitudes of the coupling constants fij. As a reference scenario, we take a parameter set
+mω= 300 GeV , mk= 1200 GeV , µ= 800 GeV ,
+feµ=feτ= 0.013, fµτ= 0.027,
+gee= 0.17, gµµ= 1.8, gττ= 0.0061, geµ= 5.7×10−5, geτ= 0.011, gµτ=−0.081,(11)
+which satisfies the neutrino data for the normal mass hierarchy. I n this scenario, the rate
+in final states ℓ+ℓ−=e+e−,e±µ∓,e±τ∓,µ+µ−,µ±τ∓,τ+τ−is given by 2, 13, 13, 19, 36,
+19, respectively. In Fig. 2, the differential cross section dσ/dcosθµfor the signal e+e−→
+ω+ω−→µ−τ+(+missing energy) is shown for√s= 1 TeV as a function of cos θµ, whereθµ
+is the angle between the outgoing muon and the beam axis. The main ba ckground comes
+from the Wpair production, which is also plotted. The angle cut ( e.g.cosθµ<−0.5)
+improves the ratio of the signal and the background.6
+6Although in this Letter we mainly discuss the case where mkis at the TeV scale, we just comment on the
+case of lighter k±±. In such case, the pair production of k++k−−can be a clear signature of this model,
+whose signal is the like-sign dilepton pairs with opposite direction [3, 9 ].
+1110 20 30 40 506070
+M(jj)  [GeV]0.10.20.30.4Diff. cross section [fb/GeV]Signal
+BGMa model
+W+W−ξ+ξ−2j µ + missing energy√s=500 GeV
+jjµµmξ± = 100 GeV
+0 100 200 300 400 500
+M(jjjj)  [GeV]00.050.1Diff. cross section [fb/GeV]SignalBGMa model
+tt
+WWνν4j+ missing energy√s=500 GeV
+ξ+ξ−mξ± = 150 GeV
+FIG. 3: The jets invariant mass distributions of the product ion rates of the signal in the Ma
+model at√s= 500 GeV. left: The di-jet invariant mass M(jj) distribution of the signal
+e+e−→ξ+ξ−→jjµνξ0
+rξ0
+rformξ±= 100 GeV. right:M(jjjj) distribution of e+e−→ξ+ξ−→
+W+W−ξ0
+rξ0
+r→jjjjξ0
+rformξ±= 150 GeV. In addition to the rate from the signal process, tho se
+for main backgrounds are also shown.
+At current and future LFV experiments, the coupling constants fijandgijcan be further
+tested via the LFV rare decays such as ℓ→ℓ′γandℓ→ℓ′ℓ′ℓ′′. The same operators as in
+ℓ−→ℓ′∓e−e±would also be tested directly at the ILC via e±e−→ℓ−ℓ′±. In the scenario
+in Eq. (11), we estimate that σ(e+e−→µ±τ∓)∼5 fb for√s= 1 TeV. When we take
+gee= 0.4,geτ= 0.01,mk=µ= 1.2 TeV and mω= 400 GeV, which also satisfy all the
+current data, we obtain σ(e+e−→τ±e∓)∼0.76 (1.7) fb for√s= 500 GeV (1 TeV).
+The Ma model
+In the Ma model, the coupling constants ˆhα
+e(α= 1,2)7are strongly constrained from
+neutrino data and LFV data. As a typical choice of parameters, we consider8
+mξr= 50 GeV , mξi= 60 GeV , mξ±∼100 GeV, mN1
+R=mN2
+R= 3 TeV,
+λ5=−1.8×10−2,ˆhα
+e,ˆhα
+µ,ˆhα
+τ∼10−5, (12)
+in which the normal neutrino mass hierarchy is realized. Because ˆhα
+eare very small for a
+7Here we consider the minimal case of two generations for the right- handed neutrino.
+8The relatively large mass difference between ξ±andξ0
+r,iimplies a significant deviation from the custodial
+symmetry in the Higgs sector, which affects the allowed mass mhof the SM like Higgs boson h. The
+largermhis favored for larger mass difference of mξ±−mξr,i.
+12TeV scale mNα
+R, the contribution of the t-channel diagrams to the signal e+e−→ξ+ξ−is
+much smaller than that from Drell-Yan type diagrams. For most of th e possible values of
+ˆhα
+ℓandmNα
+Rwhich satisfy the LFV and the neutrino data, the contribution of th et-channel
+diagramsisnegligible. Theproductioncross section ofacharged Higg spairξ+ξ−istherefore
+similar to that in the usual THDM: about 92 (10) fb for mξ±= 100 (150) GeV at√s= 500
+GeV. The produced ξ±decay into W±(∗)ξ0
+r,i.9
+In Fig. 3 ( left), we show the invariant mass distribution of the di-jet jjof the production
+cross section of the signal, e+e−→ξ+ξ−→W+∗W−∗ξ0
+rξ0
+r→jjµνξ0
+rξ0
+rformξ±= 100 GeV.
+The main backgrounds come from WW. Thejjµµevents from ZZ,γγ, andZγcan also
+be the backgrounds. A factor of 0.1 is multiplied to the rate of the jjµµbackgrounds for
+the miss-identification probability of a muon. The signal is significant a roundM(jj)∼30
+GeV. The invariant mass cut (such as 15 GeV < M(jj)<40 GeV) is effective to reduce the
+backgrounds. For the numerical evaluation, we have used a packa ge CalcHEP 2.5.4 [34].
+Formξ±> mW+mξr, on the other hand, the signal W+W−ξ0
+rξ0
+rcan be measured by
+detecting the events of four jets with a missing energy. The main ba ckground comes from
+W+W−ννandtt. By the invariant mass cuts of two-jet pairs at the Wboson mass, the
+biggest background from WWcan be eliminated. In Fig. 3 ( right), we show the invariant
+mass distribution of jjjjof the production cross sections of the signal and the backgroun ds
+without any cut. A factor of 0.1 is multiplied to the rate of ttbackground, by which the
+probability of the lepton from a Wthat escapes from detection is approximately taken into
+account. The signal is already significant. The invariant mass cut ( M(jjjj)<300 GeV)
+gives an improvement for the signal/background ratio.
+The AKS model
+For the AKS model, we take a typical successful scenario for the n eutrino data with the
+the normal mass hierarchy, the LFV data and the DM data as well as the condition for
+strongly first order phase transition [6, 14];
+mη= 50 GeV , mH±= 100 GeV , mS±= 400 GeV , mN1
+R=mN2
+R= 3 TeV,
+h1
+e=h2
+e= 2≫h1
+µ,h2
+µ≫h1
+τ,h2
+τ, κ∼ O(1),sin(β−α) = 1,tanβ= 10.(13)
+9Theξ0
+rξ0
+iproduction can also be interesting. The final state should be two je ts (or dilepton) plus a
+missing energy. The cross section for e+e−→ξ0
+rξ0
+i→ξ0
+rξ0
+rjjis about 40 fb at√s= 500 GeV.
+13200 400 600 800 1000 1200
+√ s   [GeV]101001000Cross section [fb]H+H- WWνeνeSignalτττ+τ−+ missing energy
+WWντντ√s = 1 TeV
+S+S-AKS model
+-1 -0.5 0 0.5 1
+cosθτ-10100Diff. cross section [fb] H+H-ττνeνeSignalτττ+τ−+ missing energy
+ττντντ√s = 1 TeV
+S+S-AKS model
+FIG. 4: left: The cross sections of the signal, e+e−→S+S−→τ+τ−(+ missing energy), in the
+AKS model as a function of the collision energy√s.right: The differential cross section of the
+signal for√s= 1 TeV as a function of the angle of the direction of the outgoi ngτ−and the beam
+axis of incident electrons. In addition to the rate from the s ignal, those from backgrounds such as
+τ+τ−,τ+τ−ννandH+H−are also shown.
+Because h1,2
+e∼ O(1), the contribution from the t-channel Nα
+Rexchange diagrams to the
+production cross section of S+S−dominate that from the Drell-Yan diagrams [14]. The
+cross section is about 87 fb for mS±= 400 GeV at√s= 1 TeV. As the decay branching
+ratio ofS±→H±η0is 100% and that of H±→τ±νis also almost 100% because of the
+Type-X THDM interaction for tan β= 10, the final state of the signal is τ+τ−ννη0η0with
+almost the same rate as the parent S+S−production. The main SM backgrounds are τ+τ−
+andτ+τ−νν. The pair production of the doublet like charged Higgs boson H+H−can also
+be the background. As the signal rate dominantly comes from the t-channel diagram, it
+becomes larger for larger√s, while the main backgrounds except for ττνeνeare smaller
+because they are dominantly s-channel processes (Fig. 4 ( left)). At√s= 1 TeV, the rate of
+the signal without cut is already large enough as compared to those of the backgrounds. It is
+expected that making appropriate kinematic cuts will improve the sig nal background ratio
+to a considerable extent. The√sscan will help us to confirm that the signal rate comes
+from the t-channel diagrams. Fig.4 ( right) shows the differential cross section of the signal
+at√s= 1 TeV as a function of cos θτ−, whereθτ−is the angle between the direction of the
+outgoing τ−and the beam axis of incident electrons. The distribution of the back ground
+fromττis asymmetric, so that the angle cut for larger cos θτ−reduces the backgrounds.
+14B. Electron-electron collisions
+As already stated, the ILC has a further advantage to test radia tive seesaw models via
+the experiment at the e−e−collision option, where dimension five operator of e−e−φ+φ+,
+which is the sub-diagram of the loop diagrams for neutrino mass matr ix. This direct test of
+the dimension five operator is essential to identify the radiative see saw models.
+The Majorana nature in the Zee-Babu model is in the lepton number v iolating coupling
+constant µofk++ω−ω−, which generates the dimension five operator of e−e−ω+ω+at the
+tree level via the s-channel k−−exchange diagram. The cross section of e−e−→ω−ω−is
+given by
+σ(e−e−→ω−ω−) =1
+8π/radicalbigg
+1−4m2ω
+sµ2g2
+ee
+(s−m2
+k)2+m2
+kΓ2
+k, (14)
+where the total width Γ kofk±±is computed as about 168 GeV in our scenario in Eq.(11).
+On the other hand, in the Ma model and the AKS model, the operator comes from the
+t-channel right-handed neutrino exchange diagram. The cross se ction is evaluated as
+σ(e−e−→φ−φ−) =/integraldisplaytmax
+tmindt1
+128πs/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay
+α=1(|cα|2mNα
+R)/parenleftBigg
+1
+t−m2
+Nα
+R+1
+u−m2
+Nα
+R/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+,(15)
+wherenis the number of generation of right-handed neutrinos, φ−represents the Z2-odd
+charged scalar boson ξ−in the Ma model and S−in the AKS model. The constants cα
+represent ˆhα
+eorhα
+ein the Ma model or the AKS model, respectively. We note that due to
+the Majorana nature of the t-channel diagram, we obtain much larger cross section in the
+e−e−collision than in the e+e−collision in each model assuming the same collision energy.
+The mass matrix of left-handed neutrinos is generated at the one, two and three loop
+levels in the Ma model, the Zee-Babu model and the AKS model, respec tively. Therefore,
+the coupling constants can be basically hierarchical among the mode ls, so are the cross
+sections. For the typical scenarios in these models, the cross sec tions are shown in Fig 5.
+The rate of ω−ω−in the Zee-Babu model can be larger than several times 100 fb for 8 00
+GeV<∼√s<∼1.5 TeV. It becomes maximal (several times pb) at√s∼mk, and above that
+asymptotically reduces by 1 /s. The maximal value of the cross section is sensitive to the
+value of geeandµ. In the parameter sets where these coupling constants are smalle r the
+cross section becomes smaller. The signal should be like-sign dilepton with a missing energy.
+On the other hand, in the Ma model, production cross sections of e−e−→ξ−ξ−are smaller
+15500 1000 1500
+√s  [GeV]101102103104105Cross section [fb]AKS   e−e−→S−S−
+Zee-Babu   e−e−→ω−ω−
+SM   e−e−→W−W−νeνe
+FIG. 5: The cross sections of like-sign charged Higgs pair pr oductions in the Zee-Babu model
+(ω−ω−) and in the AKS model ( S−S−) are shown as a function of the collision energy√s. The
+parameters in the Zee-Babu and the AKS model are taken as in Eq . (11) and Eq. (13), respectively.
+than 10−4fb because the coupling constants ˆhα
+iare very small in the parameters in Eq. (12).
+Allowing some fine tuning, ˆhα
+imay be at most 0.01 for heavier Nα
+R. In any case, the cross
+section of e−e−→ξ−ξ−is smaller than 10−3fb. Hence, most of the successful scenarios in
+the Ma model the process e−e−→ξ−ξ−is difficult to be seen. In the AKS model, the cross
+section of e−e−→S−S−is large, and its value amounts to about 15 pb at√s= 1 TeV in
+the scenario given in Eq. (13). Above the threshold, the magnitude of the cross sections are
+not sensitive to√s, so that even if mS±would be at the TeV scale, we might be able to test
+it at future multi-TeV linear colliders, such as the Compact Linear Collid er [35]. Because
+B(S±→η0H±)≃B(H±→τ±ν)≃100 %, the signal should be τ−τ−ννηηwith almost the
+same rate as long as mS±< mNα
+R.
+The background mainly comes from W−W−νeνe, and the cross section is about 2.3 fb (22
+fb) for√s= 500 GeV (1 TeV). The branching ratio for the leptonic decay of Wbosons is
+30%, so that the rate of the final state ℓℓ′ννννis at most 2 fb or less. Therefore, the signal
+in the AKS model and in the Zee-Babu model can be seen.
+Apart from the TeV-scale radiative seesaw models, there are many models with lepton
+number violating interactions or right-handed Majorana neutrinos . Atwood et al. have
+discussed the signature of heavy Majorana neutrinos in the model withoutZ2symmetry via
+16chargedHiggspairproductionat e+e−ande−e−collisions [36].10Insupersymmetric models,
+Majorana particles also appear, and their effects also give similar t-channel contributions to
+the above models in the slepton pair production through the gauge c ouplings, e−e−→˜e−˜e−,
+whose cross section is of O(100) fb. The final state would be e−e−χ0χ0for example.
+Thee−e−collision experiment is useful to test the Majorana nature of radiat ive seesaw
+models such as the Zee-Babu model and the AKS model via like-sign pa ir production of
+charged scalar bosons. The cross section can be significant and hie rarchical among these
+models. The signal can be observed as a model dependent final sta te, by which we can
+discriminate the models. Although in this Letter we did not explicitly disc uss the KNT
+model that also contains TeV-scale right-handed neutrinos Nα
+R, we found that the cross
+section of e−e−→S−
+2S−
+2(S−
+2is theZ2-odd isosinglet charged scalar boson) is very small
+because the coupling constants for eRNα
+RS−
+2(α= 1,2) are tiny [12].
+V. CONCLUSION
+We have discussed general features of TeV-scale radiative seesa w models. They are char-
+acterized by an extended scalar (Higgs) sector and the Majorana nature. We have mainly
+discussed the concrete models with neutrino mass generation at on e-loop (the Ma model),
+two-loop (the Zee-Babu model), and at three-loop (the model in Re f. [6]). Various phe-
+nomenological aspects of these models have been discussed espec ially in experiments at the
+LHC and at the ILC. We have found that, while the extended Higgs se ctor can be explored
+at the LHC, the Majorana nature of the models can directly be test ed at the ILC via the
+pair production of the charged scalar bosons at the electron-pos itron and electron-electron
+collision experiments. The detailed realistic simulation has to be done els ewhere.
+Acknowledgments
+We like to thank Ernest Ma for useful discussions. The work of SK wa s supported, in
+part, by Grant-in-Aid for scientific research (C), Japan Society f or the Promotion of Science
+10Recently, chargedHiggspairproductionat e−e−colliderisstudiedinaspecificmodelwith threeTeV-scale
+right-handed neutrinos and four Higgs doublets in Ref. [37].
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+19
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+arXiv:1001.0093v1  [astro-ph.CO]  31 Dec 2009Mon. Not.R. Astron. Soc. 000, 1–??(2009) Printed 1 November 2018 (MN L ATEXstyle file v2.2)
+AJointChandra andXMM-Newton ViewofAbell3158:Massive
+Off-CentreCoolGasClumpAsARobustDiagnosticofMerger
+Stage
+Yu Wang1, Haiguang Xu1, LiyiGu1, JunhuaGu1, ZhenzhenQin1,
+JingyingWang1, ZhongliZhang2andXiang-Ping Wu3
+1Department ofPhysics, Shanghai Jiao Tong University, 800 D ongchuan Road, Shanghai 200240, PRC.E-mail: wenyu −wang@sjtu.edu.cn; hgxu@sjtu.edu.cn
+2Max-Planck-Institute ofAstrophysics, Karl-Schwarzschi ld-Str. 1,Postfach 1317 D-85741, Garching, Germany
+3National Astronomical Observatories, Chinese Academy ofS ciences, 20A Datun Road, Beijing 100012, PRC
+Received October 22 /Accepted 30 December 2009
+ABSTRACT
+Byanalysingthe Chandra andXMM-Newton archiveddataofthenearbygalaxyclusterAbell
+3158, which was reported to possess a relatively regular, re laxed morphology in the X-ray
+band in previous works, we identify a bow edge-shaped discon tinuity in the X-ray surface
+brightness distribution at about 120h−1
+71kpc west of the X-ray peak. This feature is found to
+beassociatedwithamassive,off-centrecoolgasclump,and actuallyformsthewestboundary
+ofthecoolclump.Bycalculatingthethermalgaspressuresi nthe coolclumpandinthe free-
+stream region, we determine that the cool gas clump is moving at a subsonic velocity of
+700+140
+−340km s−1(M= 0.6+0.1
+−0.3) toward west on the sky plane. We exclude the possibility
+thatthiscoolclumpwasformedbylocalinhomogeneousradia tivecoolingintheintra-cluster
+medium, due to the effectiveness of the thermal conduction o n the time-scale of ∼0.3Gyr.
+SincenoevidenceforcentralAGNactivityhasbeenfoundinA bell3158,andthiscoolclump
+bearsmanysimilaritiestotheoff-centrecoolgasclumpsde tectedinothermergingclustersin
+terms of their mass, size, location,and thermal properties (e.g. lower temperatureand higher
+abundance as compared with the environment), we speculate t hat the cool clump in Abell
+3158 was caused by a merger event, and is the remnant of the ori ginal central cool-core of
+the main cluster or the infalling sub-cluster. This idea is s upported not only by the study of
+line-of-sightvelocitydistributionofthe clustermember galaxies,butalso bythestudyofgas
+entropy-temperature correlation. This example shows that the appearance of such massive,
+off-centrecoolgasclumpscanbeusedtodiagnosethedynami calstateofacluster,especially
+whenprominentshocksandcoldfrontsareabsent.
+Key words: galaxies: clusters: individual (Abell 3158) – X-rays: gala xies: clusters– inter-
+galacticmedium– kinematicsanddynamics
+1 INTRODUCTION
+In the frame of hierarchical clustering scenario, galaxy cl usters
+grow in size by merging with subunits, with each major merger
+event lasting for 2−5Gyr (e.g. Roettiger, Loken & Burns 1997;
+Ascasibar & Markevitch 2006; Poole et al. 2006). In such majo r
+mergers both ram pressure stripping and slingshot cangener ate re-
+markableX-raysubstructuresingasdensityandtemperatur edistri-
+butions,whichareoftenaccompanied byshocks and/or coldf ronts
+thatexhibitarc-shaped oredge-like morphologies (e.g.Ma rkevitch
+& Vikhlinin 2007 and references therein). These substructu res are
+expected to contain valuable information not only on merger dy-
+namics itself, but also on thermal and chemical evolutions o f thehotintra-clustermedium(ICM),whichhelpsunderstandthe nature
+of darkmatter (Markevitch et al.2004; Randall et al.2008).
+Theoretical studies and numerical N-body simulations (e.g .
+Roettiger et al. 1997; Ricker 1998; Poole et al. 2006) showed that
+inatypicalmajormergereventtemperature substructures m aysur-
+viveforarelativelylongtimeof tthermal≃ngaskT/n2
+gasΛ,where
+ngas,T, andΛare gas density, temperature, and cooling func-
+tion, respectively. The derived tthermal(up to about 4 Gyr in some
+cases) is much longer than the lifetimes of shocks and cold fr onts,
+since shocks and cold fronts are usually smeared out quickly on
+dynamic time-scalescharacterizedbythesound crossingti me.Ac-
+tually, high-quality Chandra andXMM-Newton observations have
+shown that in merging clusters there is a prevailing existen ce of
+c/circleco√yrt2009 RAS2Yu Wanget al.
+massive,off-centrecoolgasclumps,whichareabout 50−300kpc
+in size, reside within the central ( /lessorsimilar600 kpc) region of the cluster,
+and possess an average temperature 1−4keV lower than the am-
+bient gas.
+In 1E0657−56(the Bullet cluster; Markevitch et al. 2002),
+Abell 520 (Govoni et al. 2004), Abell 2256 (Sun et al. 2002), a nd
+Abell 3667 (Vikhlinin,Markevitch & Murray 2001), the off-c entre
+cool gas clump is found to be moving adiabatically behind a bo w
+shock and/or a cold front, and thus can be directly identified as
+the remnant of the original central cool-core of the infalli ng sub-
+cluster. On the other hand, in clusters such as Abell 754 (Mar ke-
+vitch et al. 2003), Abell 2065 (Chatzikos, Sarazin & Kempner
+2006), and Abell 2255 (Sakelliou & Ponman 2006), the off-cen tre
+cool gas clump is not escorted by either a shock or a cold front .
+However, unambiguous evidence obtained inradio,optical, andX-
+ray bands still indicates that the cool gas clump was driven o ut of
+the core region of either the main cluster (Abell 754; Markev itch
+et al. 2003; Henry, Finoguenov & Briel 2004) or the infalling sub-
+cluster bythe merger.
+In fact, the off-centre cool gas clumps in above two types of
+clusters share many similarities in terms of their sizes, lo cations,
+and gas masses ( ∼1011M⊙). Therefore, an interesting ques-
+tion arises: can the appearance of such massive, off-centre cool
+gas clumps be safely used to diagnose the dynamical state of a
+cluster, even when there is a lack of violent merger signatur es,
+such as prominent shocks, cold fronts, and other associated ra-
+dio/opticalsubstructures?Clearly,inordertoaddressth isissue,we
+need to examine gas temperature distribution in more galaxy clus-
+ters that show only weak merger signatures and have evolved i nto
+late mergerstages (usuallymean post core passages).
+In this work we present a detailed joint Chandra andXMM-
+Newtonstudy of the nearby rich galaxy cluster Abell 3158 (rich-
+ness class 2, Quintana & Halven 1979; z= 0.0597, Struble &
+Rood 1999), which contains 105 member galaxies within the ce n-
+tral2.4h−1
+71Mpc (Biviano et al. 2002). In literature (e.g. Irwin,
+Bregman& Evrard1999; Lokas et al.2006; Chenet al.2007), th is
+cluster is usually referred as a relaxed, non-cool-core sys tem, be-
+causeneithersignificantX-raysubstructuresnordiffuser adioemis-
+sionhasbeenreportedinthepreviousworks(Kuetal.1983;M ohr,
+Mathiesen & Evrard 1999; Schuecker et al. 2001; Mauch et al.
+2003).Intheopticalband,however,theclustershowssomeu nusual
+properties that can be possibly related to a recent merger ev ent, as
+are summarized as follows. First, the cluster hosts three lu minous
+elliptical galaxies that are brighter than 9×1010LB,⊙(Paturel et
+al. 2003), all classified as a cD galaxy by S ´ ersic (1974). Two of
+them, the brightest member PGC 13641 (E0, LB= 1.45×1011
+LB,⊙) and the S0 galaxy PGC 13652 ( LB= 9.39×1010LB,⊙),
+constitute a closely aligned central galaxy pair, whose pro jected
+separation is 81.3h−1
+71kpc, and jointlydominate the cluster (Quin-
+tana & Havlen 1979). The second luminous member PGC 13679
+(S0,LB= 9.48×1010LB,⊙) is located at about 417.8 h−1
+71kpc
+southeast of the central galaxy pair, and dominates a local s ub-
+cluster whose recessional velocity is about 1 300 km s−1larger
+than the mean recessional velocity of the main cluster (17 50 0 km
+s−1;Luceyetal.1983;Kolokotronisetal.2001).Second,assho wn
+in figure 7 of Smith et al. (2004), the line-of-sight velocity distri-
+bution of the cluster’s member galaxies exhibits a plateau a t the
+high-velocity end, which is often seen in merging systems. T hird,
+Kolokotronis et al. (2001) reported that in this cluster the re is an
+offsetofabout 200h−1
+71kpc betweentheopticalflux-weightedcen-
+troidandtheopticaldensitypeak.Basedonthesewespecula tethatAbell 3158 is evolving at a late merger stage and is thus an ide al
+targetfor our investigation.
+Throughout the paper, we adopt the cosmological parameters
+H0= 71km s−1Mpc−1,ΩM= 0.27, andΩΛ= 0.73, so that
+1′′corresponds to about 1.14h−1
+71kpc at the redshift of the cluster.
+We utilize the solar abundance standards given by Grevesse a nd
+Sauval (1998), where the iron abundance relative to hydroge n is
+3.16×10−5innumber. Unless statedotherwise,the quoted errors
+are the 90 %confidence limits.
+2 OBSERVATIONSANDDATA REDUCTIONS
+2.1Chandra
+Exceptforanextremelyshortexposure on2007September16( 5.1
+ks, ObsID 7688), which is not used in this work, Abell 3158 has
+also been observed with Chandra on 2002 June 19 (31.4 ks, Ob-
+sID 3712) and June 21 (25.1 ks, ObsID 3201), respectively, wi th
+chips 0, 1, 2, 3, and 6 of the Advanced CCD Imaging Spectrom-
+eter (ACIS) operating in VFAINT mode. We used the Chandra
+dataanalysispackage CIAOversion4.1and CALDBversion4.1.2to
+process the archived data in the standard way, by starting wi th the
+level-1 event files. Corrections for the charge transfer ine fficiency
+and time dependent gain have been applied. We kept events wit h
+ASCAgrades 0, 2, 3, 4, and 6, and removed all the bad pixels, bad
+columns, and columns adjacent to bad columns and node bound-
+aries. Weexamined the 0.3−10.0keV lightcurves extracted from
+the background regions defined on ACIS chips 0 and 1, and found
+that there are no strong flares that increase the background c ount
+rate to>120%of the mean quiescent value. The obtained net ex-
+posures are 30.9 ks and 24.8 ks for the two observations, resp ec-
+tively.Inthe spectralanalysis,we extractedthe Chandra spectra in
+0.7−8.0keVfromthetwoobservationsseparately,andfittedthem
+simultaneously with the same model, except that their norma liza-
+tions were left free. Background spectra were extracted fro m the
+Chandra blank-sky fields; a crosscheck based on the use of local
+background yielded essentiallythe same results.
+2.2XMM-Newton
+Abell 3158 was observed with XMM-Newton on 2005 November
+22(22.4ks,ObsID0300210201) and2006 January18(9.4ks,Ob -
+sID0300211301), respectively, withtheEuropeanPhotonIm aging
+Camera(EPIC)operatinginPrimeFullWindow(MOS1andMOS2)
+and PrimeFullWindowExtended (pn) modes. Although during t he
+secondobservationtheReflectionGratingSpectrometer(RG S)was
+also turned on, in this work we limited our analysis to the firs t ob-
+servationonlybyusingthelatestversionof SAS(version8.0.1)and
+its standard filterings. We kept the events with FLAG=0 and PAT-
+TERNs0−12for MOS1 and MOS2, and the events with FLAG=0
+andPATTERN s0−4forpn.Weremovedallthetimeintervalscon-
+taminatedbysoftprotonflares,duringwhichthe 10−12keVcount
+rate exceeds the 2σlimit of the mean quiescent value (see, e.g.
+Katayamaetal.2004).ThecleanedMOS1,MOS2,andpndataset s
+have effective exposure times of 20.9 ks, 20.6 ks, and 12.2 ks , re-
+spectively.Inthespectralanalysis,weremovedallthebri ghtX-ray
+point sources and adopted conservative energy cuts at 0.5 ke V and
+8.0 keV. The obtained MOS1, MOS2, and pn spectra were fitted
+simultaneously using the same model, except that their norm aliza-
+tions were left free. Backgrounds spectra were generated fr om the
+blank-skyeventlists,whichwerefilteredinadvanceusingt hesame
+c/circleco√yrt2009 RAS, MNRAS 000,1–??ChandraandXMM-Newton Observationsof Abell3158 3
+selectioncriteria( PATTERN ,FLAG,etc.)asusedforthesourcespec-
+tra.
+3 X-RAYANALYSISAND RESULTS
+3.1 X-raysurface brightnessdiscontinuity
+In Fig.1aand1b, we show the Chandra ACIS and combined
+XMM-Newton EPIC images of the central 1.5h−1
+71Mpc×1.5h−1
+71
+Mpc (21′.9×21′.9) region of Abell 3158 in 0.3−2.0keV, re-
+spectively, which have been adaptively smoothed and correc ted
+for both exposure and vignetting. The XMM-Newton image was
+generated by combining the MOS1, MOS2, and PN events to-
+gether using the SAS‘image’ script. The X-ray peak of the clus-
+ter (R.A.=03h42m52.7s Dec= −53d37m37.2s J2000) is found at
+about17.8h−1
+71kpc (0′.26) northwest of the optical centroid of
+PGC 13641 (R.A.= 03h42m52.9s Dec= −53d37m52.2s; Katgert et
+al.1998),thebrightermemberofthecentralgalaxypair.Th ecorre-
+sponding optical Digital Sky Survey(DSS) image is shown inF ig.
+1c,onwhichthe Chandra intensitycontours obtainedfromFig. 1a
+are overlaid.
+On bothChandra andXMM-Newton images it can be clearly
+seen that the X-ray isophotes are elongated in nearly the eas t-west
+directionwithanellipticityof ≃0.3. Andonthe Chandra image a
+bowedgecanbeidentifiedatabout 120h−1
+71kpc(1′.88)westofthe
+X-raypeak.Accordingtoitsshapeandlocationonthedetect or,the
+possibility of the feature being associated with the artifa cts caused
+byCCD gaps or badcolumns can be excluded.
+In order to examine the significance of the bow edge in a
+quantitative way, we define two sets of partial elliptical an nuli,
+which are confined in the east and west sector regions shown in
+Fig.2a, and extract the exposure-corrected X-ray surface bright-
+ness profiles (SBPs) with both Chandra ACIS and XMM-Newton
+EPIC (Fig. 2b). Because the gas temperature of Abell 3158 is suf-
+ficientlyhigh( 5.8keV; Reiprich&B ¨ ohringer2002), byfollowing,
+e.g. Markevitch et al. (2000), we restricted the SBP extract ions in
+0.3−2.0keV to minimize the dependence of X-ray emissivity on
+possible temperature fluctuations (see §3.2). We attempt to apply
+the standard βmodel to describe the extracted SBPs, which is ex-
+pressed as
+S(r) =S0[1+(r/rc)2]0.5−3β+Sbkg (1)
+(Jones &Forman1984), where rcisthe coreradius, βistheslope,
+Sbkgis the background, and S0is the normalization. We find that
+intheeastsectortheSBPscanbe fittedverywellwiththe βmodel
+(Table 1), whereas in the west sector the SBPs show a significa nt
+centralexcessbeyondthebest-fit βmodelfortheouter( >120h−1
+71
+kpc) regions. Within about 90h−1
+71−115h−1
+71kpc (1′.32−1′.68),
+wherethebowedge isidentifiedvisually,thereexistsasudd endis-
+continuity, across which the Chandra andXMM-Newton surface
+brightness increases inwards by a factor of ≃1.6and≃1.3, re-
+spectively; this difference may be caused by the different d istri-
+butions of the point spread function (PSF) of Chandra ACIS and
+XMM-Newton EPIC.Theseresultsconfirmtheexistenceofthebow
+edge showninFig. 1a,andindicate thatitis not relatedtothe cen-
+tralsurfacebrightnessexcessusuallyseeninthecDgalaxi es(Mak-
+ishima et al. 2001), which are not associated witha surface b right-
+ness discontinuity. Also, we note that the X-ray surface bri ghtness
+discontinuity in Abell 3158 is similar to, although not as sh arp as,
+those found in 1E 0657−56and Abell 3667, in which the jumps
+of the surface brightness at the edges are 2−3within≃10h−1
+71kpc and thus the edges have been confirmed as cold fronts cause d
+bymergers (Markevitch etal.2002; Vikhlininet al.2001).
+3.2 Anunusualoff-centre cool gas clump
+In order to investigate the thermal properties of the bow edg e, we
+calculate the two dimensional gas temperature distributio n in the
+central0.6h−1
+71Mpc×0.6h−1
+71Mpc (8′.77×8′.77) region ofAbell
+3158, by following the approach of, e.g. O’Sullivan et al. (2 005)
+andGuetal.(2009).Inthecalculationonlythe XMM-Newton EPIC
+datasets are used, because the data statistics of the Chandra ACIS
+datasets isnotgood enough for ustomapthetemperature dist ribu-
+tion. To be specific, we first define about 4000 cells in the 0.6h−1
+71
+Mpc×0.6h−1
+71Mpc region, whose centres ( ri,i= 1,2,3...) are
+randomly distributed with a separation of <10′′between the cen-
+tres of any two adjacent cells. Each cell is apportioned with an
+adaptiveradiusof 0′.3−0′.8,sothatitencloses >3000photonsin
+0.5−8.0keVafterthepointsourcesareexcluded.Thenweextract
+theMOS1,MOS2,andpnspectrafromeachcellandfitthemsimul -
+taneously with an absorbed APECmodel coded in XSPECv12.4.0.
+Theredshiftandabsorptionarefixedto z= 0.0597andtheGalac-
+tic value NH= 1.62×1020cm−2(Dickey & Lockman 1990),
+respectively. Aftertheobtained best-fitgastemperature i sassigned
+to the corresponding cell ( Tc(ri)), and this process is repeated for
+allcells,wecalculatetheprojectedtemperatureatanypos itionras
+T(r) =/summationdisplay
+ri[Gri(Rr,ri)Tc(ri)]//summationdisplay
+riGri(Rr,ri), Rr,ri< s(ri)
+(2)
+(Guetal.2009),where Rr,riisthedistancebetween randthecen-
+treofthecell i,s(ri)isdefinedastheradiusofcell i,andGriisthe
+Gaussian kernel with a scale parameter σ=s(ri). The obtained
+temperature map, along withthe 1σerror map that is calculated in
+nearly the same way as the temperature map, are shown in Fig. 3a
+and3b, respectively.
+We find that, despite the relatively regular appearance of th e
+X-ray images (Fig. 1), the spectral distribution of the proj ected
+gastemperature is highlyasymmetric,inferringthat thecl uster has
+not recovered from a violent event on cluster scales. In part icu-
+lar, we find that there exists a cool gas clump at about 84h−1
+71kpc
+(1′.23) west of the X-ray peak, whose linear scale is about 80h−1
+71
+kpc (1′.17). The projected gas temperature within this cool clump
+ranges from about 4.4 keV to 5.0 keV, which is apparently lowe r
+than that of the ambient gas ( 5.2−6.0keV). Most interestingly,
+the west boundary of the cool gas clump coincides with the bow
+edge perfectly, indicating that they are likely to have the s ame ori-
+gin.
+To clarify the significance of the existence of the cool gas
+clump, we define four sector regions inFig. 3a, twoof which have
+the same angles as those definedearlier inFig. 2a, anddivide each
+of these sector regions into a series of wider partial ellipt ical an-
+nuli.Weextractboththe Chandra ACISand XMM-Newton MOS1,
+MOS2, and pn spectra from these annuli, and fit them with an
+absorbed APECmodel when the redshift and absorption are fixed
+again as above; allowing the absorption to vary does not impr ove
+thefits.Inthefittings,the ChandraACISspectraextractedfromthe
+two observations are treated as group one, and the XMM-Newton
+MOS1,MOS2andpnspectraaretreatedasgrouptwo.Eachgroup
+of spectra are fitted simultaneously with the same model para m-
+eters, except for that the normalizations are left free. The derived
+projectedtemperature profilesinthefoursectorsareplott edinFig.
+4a−4d.Itcanbeclearlyseenthatthe Chandra andXMM-Newton
+c/circleco√yrt2009 RAS,MNRAS 000, 1–??4Yu Wanget al.
+temperaturesareconsistentwitheachother(68%confidence level),
+and in the west sector the temperature in 45.4−95.3h−1
+71kpc
+(4.85±0.17keV), where the cool gas clump is located, is lower
+thanthoseoftheadjacentregionsat68%confidencelevel(se ealso
+Table 2); no such temperature drop is found in other three dir ec-
+tions. This is further confirmed by the results of the deproje cted
+analysis of the XMM-Newton spectra for the west sector (Table 2
+and Fig. 4e−4f), which shows that the gas temperature of the
+cool gas clump ( 3.98+0.43
+−0.41keV for W2 region) is actually lower
+than those of the innermost region ( 5.71+1.04
+−0.71keV for W1 region)
+and the next outer region ( 5.38+0.74
+−0.51keV for W3 region) at the
+90%confidence level, and cannot be ascribed to the uncertaintie s
+indetermining the metalabundance.
+3.3 FaintColdfront associated withthecool gas clump
+To study the nature of the cool gas clump, we carry out a simple
+hydrodynamic study to determine the velocity of the edge in t he
+environment. Byapplying thebest-fitdeprojectedspectral parame-
+ters (Table 2), here we revisit the X-ray surface brightness profiles
+of the west sector (Fig. 2b) and fit it again with two models. In
+the first model (model A), the spatial distribution of gas den sity is
+described withtwo βcomponents as
+ngas(R) ={n2
+g,1[1+(R/Rc1)2]−3β1+n2
+g,2[1+(R/Rc2)2]−3β2}1/2,
+(3)
+whereRis the 3-dimensional radius, Rcis the core radius, and β
+is the slope. In the second model (model B), the gas density di stri-
+bution is broken at the truncation radius Rcut, described with one
+truncatedpower-lawcomponent andonetruncated βcomponentas
+ngas(R) =/braceleftbiggng,1(R/Rcut)−αR < R cut
+ng,2[1+(R/Rc)2]−3β/2R/greaterorequalslantRcut.(4)
+Acceptable fits to both the Chandra andXMM-Newton SBPs can
+beobtainedwithmodelBonly(Table1).The Chandrabest-fitsug-
+gests a density jump by a factor of fjump= 1.7±0.1atRcut=
+111.0±4.4h−1
+71kpc (roughly the position of the edge), and the
+XMM-Newton best-fitgivesconsistentvaluesof fjump= 1.5±0.1
+andRcut= 104.9±2.2h−1
+71kpc.
+Following Vikhlinin et al. (2001), we use the best-fit gas den -
+sitiesacrosstheedge(Eq.(4))andgastemperaturesforW2a ndW3
+regions (§3.2 and Fig. 4 e) to calculate the thermal gas pressures in
+thecool gasclump( P0)andinthe free-streamregion( P1),respec-
+tively.Theformerpressureisassumedtobeinpressureequi librium
+with the gas pressure at the stagnation point (denoted as the place
+where the relative gas velocity vanishes). The pressure rat io is es-
+timated to be P0/P1= 1.3±0.2(68% confidence level). This
+allows us to determine the Mach number ( M) of the cool clump
+tobe0.6+0.1
+−0.3,whichcorresponds toavelocityof 700+140
+−340kms−1.
+Thissubsonicvelocityoftheclump,alongwiththeresultso btained
+in§3.1 (i.e., surface brightness discontinuity) and §3.2 (i.e., gas
+temperature and density discontinuities), allow us to conc lude that
+there exists a faint cold front at the west boundary of the coo l gas
+clump.
+4 VELOCITYPLATEAU
+In early works of Biviano et al. (1997 and 2002), Girardi et al .
+(1996), and Kolokotronis et al. (2001), both the two dimensi onal
+spatial distributionand the line-of-sight velocitydistr ibutionof the
+member galaxies in Abell 3158 were studied, and no significan tsubstructure wasreported.However, aftertighterandmore reliable
+selection criterion for member galaxies, which was based on the
+galaxy colour-magnitude relation (e.g. Colless & Dunn 1996 ; Fer-
+rari et al. 2003; Maurogordato et al. 2008), was applied, Smi th et
+al.(2004)re-constructedtheline-of-sightvelocitydist ributionwith
+higher fidelity and found that there exists a visible plateau beyond
+the Gaussian distribution at the high-velocity side. Since Smith et
+al. (2004) did not use a quantitative approach to assess the s ignifi-
+canceofthehigh-velocityplateau,inwhichthemergerinfo rmation
+might be contained, here we draw the velocity data of the memb er
+galaxiesinAbell3158from Smithetal.(2004), andre-analy se the
+line-of-sight velocitydistributionprofile. Afterthe mod el fittingof
+this distribution is finished, we also attempt to investigat e the 2-
+dimensional spatialdistributionof thegalaxiesinthehig h-velocity
+plateau.
+First, we plot the line-of-sight velocity distribution of t he 68
+membergalaxiesidentifiedbySmithetal.(2004)inFig. 5a,which
+shows that the high-velocity plateau is located at about 19 000km
+s−1. We fit the observed distribution with a single Gaussian pro-
+file, and calculate the Kolmogorov −Smirnov statistic for the ob-
+served distributionagainst the best-fit Gaussian model ( χ2/dof=
+31.6/9). We find that the observed distribution has a probability
+of<10%of being Gaussian. We then attempt to fit the observed
+distribution with a two-Gaussian model. The best-fit ( χ2/dof=
+9.8/6) gives anaveragevelocityof < v1>= 17380 ±50km s−1
+and a corresponding variance of σv,1= 520±50km s−1for the
+main Gaussian component, and < v2>= 19 120 ±130km s−1
+andσv,2= 510±100kms−1forthehigh-velocityplateau.Byap-
+plying the F-test and the Kaye’s Mixture Model (KMM; McLach-
+lan& Basford 1988; Ashman, Bird& Zepf 1994) test,the latter of
+which is based on a maximum likelihood algorithm, we find that
+the second Gaussian component is required at 94.3% confidenc e
+levelandpreferredatasignificantprobabilityof90%,resp ectively.
+Toinvestigateifthegalaxiesinthehigh-velocityplateau form
+a real substructure in the cluster, we divide the member gala xies
+into two subgroups: one low-velocity ( 15 300−18 500km s−1)
+subgroup and one high-velocity ( 18 500−20 000km s−1) sub-
+group,whichconsistof51and17galaxies,respectively.Ac cording
+to the best-fit two-Gaussian model, these two subgroups roug hly
+corresponds tothetwoGaussiancomponents, respectively, withup
+to about two of the galaxies in the high-velocity subgroup co ming
+from the main Gaussian component. We find that the galaxies be -
+longing to the high-velocity subgroup are distributed most ly in the
+central500h−1
+71kpc (7′.3), as well as the west part of the cluster,
+with a geometric centroid of the high-velocity galaxies is l ocated
+at about355h−1
+71kpc (5′.2) northwest of the X-ray peak, which is
+not coincident with any X-ray substructures or luminous mem ber
+galaxies.Thehigh-velocitysubgrouproughlyincludesthe member
+galaxisesofthePGC13679sub-cluster,sincethemembergal axies
+and the boundary of the latter are difficult to be determined b y 2-
+dimensional spatial distribution alone (e.g. Colless & Dun n 1996;
+Boschinetal.2006).Thegalaxiesinthelow-velocitysubgr oupthat
+isdominatedbythecentralPGC13641-PGC13652galaxypair, on
+the other hand, are scattered symmetrically in the field. The se re-
+sults suggest that the galaxy velocity separation (Fig. 5a; see also
+Smith et al. 2004) has a dynamical nature, which is most likel y to
+be a major merger, with a merger mass ratio of about 1 : 1−3, as
+estimated from the velocity variance ratio( σv,2/σv,1≃1 : 1) and
+the galaxynumber ratio( ≃17/51 = 1 : 3 )of the twosubgroups.
+InliteratureAbell3158wasclassifiedasa‘single-compone nt’
+and ‘relaxed’ system (Kolokotronis et al. 2001; Lokas et al. 2006),
+whichpossesses arelativelyregular morphology inthe X-ra yband
+c/circleco√yrt2009 RAS, MNRAS 000,1–??ChandraandXMM-Newton Observationsof Abell3158 5
+(Ku et al. 1983; Mohr et al. 1999; Schuecker et al. 2001). How-
+ever, considering none of violent merger signatures is dete cted in
+the cluster, the existence of the high-velocity subgroup in dicates
+that the cluster is evolving into the late stage of a major mer ger
+event. Thismay explainthe originof theoff-centre cool gas clump
+detected in the west sector region, which is moving at a subso nic
+velocity( §3.2−§3.3; see also §5.1).
+5 DISCUSSION
+5.1 Mergingorigion of themassive, off-centrecool gas clum p
+The most prominent X-ray feature detected in Abell 3158 is th e
+massive,off-centrecoolgasclump,whichismovingbehinda faint
+cold front. It does not resemble the cool substructures caus ed by
+AGN activity (e.g. Forman et al. 2005; Nulsen et al. 2002), wh ich
+are usually tightly associated with the low-density region s such as
+theAGN-inducedradiolobesandX-raycavities,sincenoevi dence
+for AGN activityand its remnants has been reported inAbell 3 158
+according to the existing multi-band observations. In such a sense,
+we speculate that the cool clump was formed either by the inho -
+mogeneous radiative cooling in the ICM, or by the ram pressur e
+stripping or slingshot during a merger event. The first possi bility
+can be ruled out immediately, because given the best-fit spec tral
+parametersforthecoolclump,theradiativecoolingtimeof thegas
+inthe clump(cf. Sarazin1986) is
+tcool= 3.5×1010yr×/parenleftbiggngas
+2×10−3cm−3/parenrightbigg−1
+×/parenleftbiggkT
+10keV/parenrightbigg1/2
+≃9.8 Gyr, (5)
+where continuum and line emissions are both included. This i s
+close to the Hubble time ( ≃12.9Gyr) of the cluster, and is sig-
+nificantlylonger than the Coulomb conduction time
+tcond∼9.2×106yr×/parenleftbiggngas
+2×10−3cm−3/parenrightbigg
+×/parenleftbiggl
+100 kpc/parenrightbigg2/parenleftbiggkTcentre
+10 keV/parenrightbigg−5/2
+≃0.06 Gyr (6)
+(Markevitch et al. 2003), where l= 80h−1
+71kpc is the linear scale
+ofthecoolclump, kTcentre= 5.5±0.1keVistheaveragegastem-
+perature for the cluster’s central 200 h−1
+71kpc (2′.92) region. If the
+conduction is suppressed by a factor of 5due to the tangled mag-
+netic fieldinthe ICM(e.g.Narayan & Medvedev 2001; Zakamska
+& Narayan 2003; Chandran & Maron 2004), the effective conduc -
+tion time increases to 0.3Gyr. Consequently, even if the cool gas
+clump was formed through radiative cooling, it would have be en
+destroyedveryquicklybyconduction,whichindicatesthat thecool
+gas clump is most likely caused by merger. The same conclusio ns
+also hold for off-centre cool gas clumps found in other mergi ng
+clusters (Table 3).
+Assuming that the cool gas clump in Abell 3158 has a spher-
+ical geometry with a diameter of 80−110h−1
+71kpc (Fig. 3 a),
+and has a constant density, the gas mass is estimated to be abo ut
+3.0−7.7×1010M⊙. This value is comparable to masses of the
+off-centre cool clumps identified in other merging clusters , which
+rangefromabout 2.0×1010M⊙to1.3×1012M⊙(Table3).Italso
+falls into the gas mass range of the cool-core regions of poor clus-
+ters and massive groups (e.g. Mulchaey et al. 1996; Gastalde llo et
+al.2007;Baldietal.2009).Infact,asshowninTable3,noto nlythegas mass,but alsothe size,temperature gradient, andthe re latively
+high metal abundance of the cool clump in Abell 3158 resemble
+those of other off-centre cool clumps detected in 1E 0657−56
+(Markevitchetal.2002),Abell754(Markevitchetal.2003) ,Abell
+2065 (Chatzikos et al. 2006), Abell 2255 (Sakelliou & Ponman
+2006), Abell 2256 (Sun et al. 2002), and Abell 3667 (Vikhlini n
+et al. 2001). Note that most of the off-centre cool clumps in t hese
+merging systems are considered tobe the remnant of the cool- core
+of either the main cluster or the infalling sub-cluster. Bas ed on all
+thesesimilarities,itisnaturaltospeculatethatthecool gasclumpin
+Abell 3158 was formed by the ram pressure stripping or slings hot
+duringa merger event.
+The above speculation is enhanced by the study of the cor-
+relation between gas temperature and entropy, which is defin ed as
+S=kT/n2/3
+gasintermsofgastemperature kTanddensity ngas.By
+quoting the data of Zhang et al. (2007) and Sanderson, O’Sull ivan
+& Ponman (2009), we re-calculate the redshift-corrected ga s en-
+tropyat0.1R500(R500istheradiusatwhichthemeanover-density
+is500withrespecttothecriticaldensityoftheuniverse)f orasam-
+pleof 13cool-core clustersanda sampleof 19non-cool-core clus-
+ters.Wealsocalculatethegasentropyofthesevencoolgasc lumps
+listed in Table 3. The entropy-temperature distribution is plotted
+in Fig. 6, where the strong correlations for the cool-core an d non-
+cool-core clusters are characterized by two power-law mode ls, re-
+spectively (Sanderson et al. 2009). It can be clearly seen th at cool
+gas clumps in Abell 3158 and most of other merging clusters fo l-
+lowtherelationforcool-coreclusters,whichrevealsthei rnatureas
+cool-core remnants. Note that the gas entropy of the cool clu mp in
+1E 0657-56 is significantly below the relation for cool-core clus-
+ters atleast 1 σlimit(Fig.6), which ispossibly due tothat the cool
+clump is compressed extremely as it experiences the ram pres sure
+stripping force and passes through the deep potential of the main
+cluster during this drastic high-velocity merger event (Ma rkevitch
+etal. 2002).
+Based on above data analysis, calculations, and discussion s,
+we propose that the appearance of the massive off-centre coo l gas
+clumpscanberegardedasarobustdiagnosticofmergerstate .Com-
+bining the study of cold fronts (e.g. Owers et al. 2009), whic h sur-
+vivelongerthantheshocks,withthestudyofsuchcoolgascl umps,
+we maybe able toplace tight constraints onthe merger histor y.
+5.2 Possiblemerging scenarios
+In Abell 3158, the X-ray bow edge of the faint cold front is poi nt-
+ing towards west in projection ( §3.1−§3.3), which indicates that
+the symmetrical axis of the merging process is reasonably cl ose to
+thewest-eastdirection,althoughtheprojectedeffectlow ersthede-
+tectabilityoftheedge.Thedirectionisalsoroughlyconsi stentwith
+the extending direction of the three cD galaxies (Fig. 1 a), as well
+as the galaxy distribution of the high-velocity subgroup (F ig. 5b).
+Straightforwardly, thehigh-velocity subgroup has alread ymade its
+closest approach to the core of the main cluster roughly from east
+to west. In this scenario, the apparent discrepancy between the
+moving velocity of the cool gas clump ( 700+140
+−340km s−1;§3.3)
+andtherelativeline-of-sightvelocityofthehigh-veloci tysubgroup
+(1740±180km s−1;§4) demonstrates the gas component of the
+high-velocity subgroup possibly has been significantly slo wed by
+the ram pressure stripping force. Regarding the late stage o f the
+majormerger( §4),perhaps thehigh-velocitysubgroupisinitssec-
+ondcorepassage. Ontheotherhand, because thethreecDgala xies
+all lie well behind the cold front (Fig. 1 a), perhaps the cluster is
+c/circleco√yrt2009 RAS,MNRAS 000, 1–??6Yu Wanget al.
+inanother mergingscenario, inwhichthe cool gas clump has b een
+slingshot out from the centre of the main cluster, as expecte d af-
+ter core passage, although the optical centroid of the high- velocity
+subgroup isnot behind the faint cold front edge.
+6 SUMMARY
+By analysing the Chandra andXMM-Newton data of the nearby
+galaxy cluster Abell 3158, we identify a massive, off-centr e cool
+gas clump, whichis moving at a velocity of M= 0.6+0.1
+−0.3, toward
+west onthe skyplane behind a faint coldfront. Thepossibili ties of
+this substructure being formed by inhomogeneous radiative cool-
+ingand by central AGN activityare excluded. Based onthe res ults
+obtained in the X-ray and optical analysis, we speculate tha t the
+cool gas clump is the remnant of the original central cool-co re of
+the main cluster or the infalling sub-cluster during a major merger
+event. This case shows that the appearance of such massive, o ff-
+centrecoolgasclumpscanbeusedtodiagnose thedynamical s tate
+ofa cluster,especiallywhenviolent merger signatures (e. g.promi-
+nent shocks and coldfronts) are absent.
+ACKNOWLEDGMENTS
+We thank the Chandra andXMM-Newton teams for making this
+research. This work was supported by the National Science Fo un-
+dation of China (Grant No. 10673008, 10878001 and 10973010) ,
+the Ministry of Science and Technology of China (Grant No.
+2009CB824900/2009CB24904), and the Ministry of Education of
+China (the NCETProgram).
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+EastSectora
+Model Data S0 β r c ... ... ... χ2/dof
+(cnts cm−2s−1arcsec−2) ( h−1
+71kpc)
+singleβChandra 8.85±0.26×10−80.73±0.02197.2±5.4 ... ... ... 28.5/27
+singleβXMM 7.94±0.07×10−80.74±0.02187.0±1.8 ... ... ... 41.9/27
+WestSectorb
+Model Data ng,1 β1(α)Rc1(Rcut) ng,2 β2 Rc2 χ2/dof
+(10−3cm−3) ( h−1
+71kpc) ( 10−3cm−3) ( h−1
+71kpc)
+A Chandra 3.95±0.12 4 .50±0.12 243 .1±4.0 2.60±0.04 0.54±0.01 257 .2±2.3115.1/15
+A XMM 2.90±0.06 2 .88±0.06 209 .1±4.0 3.08±0.04 0.50±0.01 182 .3±1.272.5/24
+B Chandra 4.51±0.09 0 .07±0.04 111 .0±4.4 3.16±0.05 0.63±0.01 238 .9±3.320.7/15
+B XMM 4.72±0.08 0 .03±0.03 104 .9±2.2 3.92±0.05 0.66±0.01 200 .0±2.133.5/24
+Table1.Best-fits to theexposure-corrected surfacebrightness pro files (Fig.2 b),which areextracted in 0.3−2.0keVfromthepartial elliptical annuli defined
+in the east and west sector regions (Fig. 2a).aA single βmodel is sufficient in fitting the SBPs extracted in the east se ctor (§3.1).bBest-fit deprojected
+spectral parameters obtained in §3.2 (see also Table 2 and Fig. 4e) and two gas density models ( §3.3; model A =βcomponent+ βcomponent (Eq. 3); model
+B=truncated power-law component+truncated βcomponent (Eq.4)) are used to fitthe Chandra andXMM-Newton profiles extracted in the westsector.
+RegionaRadiusbData kT Z χ2/dof
+(h−1
+71kpc) (keV) (Z ⊙)
+PROJECTEDc
+W1 0.0 −45.4 Chandra 5.46+0.48
+−0.310.75+0.27
+−0.25119.2/124
+XMM 5.43+0.41
+−0.300.90+0.43
+−0.27124.7/120
+W2 45.4 −95.3Chandra 4.96+0.19
+−0.190.69+0.14
+−0.14199.2/206
+XMM 4.85+0.17
+−0.170.81+0.13
+−0.13192.5/207
+W3 95.3 −136.1 Chandra 5.42+0.33
+−0.230.78+0.19
+−0.18229.4/205
+XMM 5.53+0.29
+−0.230.93+0.18
+−0.17166.2/165
+W4 136.1 −190.5Chandra 5.17+0.21
+−0.210.69+0.15
+−0.14212.9/206
+XMM 5.54+0.30
+−0.320.48+0.14
+−0.14187.6/187
+W5 190.5 −258.5Chandra 4.86+0.21
+−0.210.58+0.15
+−0.14163.5/126
+XMM 5.16+0.19
+−0.190.63+0.14
+−0.13195.5/216
+DEPROJECTEDd
+W1 0.0 −45.4 XMM 5.71+1.04
+−0.710.99+0.87
+−0.70896.3/893
+W2 45.4 −95.3 XMM 3.98+0.43
+−0.410.58+0.37
+−0.31...
+W3 95.3 −136.1 XMM 5.38+0.74
+−0.511.24+0.59
+−0.50...
+W4 136.1 −190.5 XMM 5.46+1.24
+−1.170.02+0.53
+−0.02...
+Table 2. Gas temperature and abundance distributions in the west sec tor region.aWe divide the west sector region into wider partial elliptic al annuli,
+i.e., W1−W5 regions (Fig. 3 a).bEquivalent inner and outer radii for each partial elliptica l annulus, which are defined as r=√
+ab, where a and b are the
+correspondingsemi-majorandsemi-minoraxes,respective ly.cAnabsorbed APECmodelisusedtofitthe ChandraACISand XMM-Newton MOS1+MOS2+pn
+spectra extracted in the W1 −W5 regions. The redshift and absorption are fixed to z= 0.0597and the Galactic value NH= 1.62×1020cm−2(Dickey &
+Lockman 1990), respectively. Errors are quoted at the 68% co nfidence.dThePROJCTmodel coded in XSPECv12.4.0 is used to deproject the XMM-Newton
+MOS1+MOS2+pn spectra. Errors are quoted at the90% confidenc e.
+Hostcluster 1E0657-56 A754 A2065 A2255 A2256 A3158 A3667
+Clump’s scale l (h−1
+71kpc) 70 120 60 100 160 80−110 300
+Projected distance to centre d (h−1
+71kpc) 650 200 60 130 130 85 400
+Gasdensity ngas (10−3cm−3) 24 4.6 7.2 2.6−4.3 4.3 4.5 3.8
+Gastemperature T (keV) 7.0±1.0 6.2±0.1 3.4±0.6 5.2±0.7 4.5±1.0 4.0±0.4 4.1±0.2
+Ambient temperature Tamb (keV) 13.6 7.4 5.5 6.9 6.7 5.5 7.7
+Cluster temperature Tcluster (keV) 13.6±1.0 10.0±0.3 5.5±0.2 6.9±0.3 6.7±0.2 5.5±0.1 7.3±0.2
+Metal abundance Z (Z⊙) ... 0.54 1 .36 ... 0.84 0 .81 0 .81
+Cooling time tcool (109yr) 2.4 12.0 5.7 11.7−19.410.9 9.8 11.8
+Effective conduction time tcond (109yr) 0.1 0.3 0.3 0.2−0.3 0.7 0.3−0.61.5
+Gasmass Mgas (1011M⊙) 1.1 1.0 0.2 0.3−0.6 2.3 0.3−0.813.4
+Gasentropy S (keV cm2)84±12 224 ±4 91 ±16 236 ±80 170 ±38 147 ±15 168 ±8
+Table 3.Properties of the massive, off-centre cool gas clumps, whic h aredetected in 1E0657-56 (Markevitch et al. 2002), Abell 7 54 (Markevitch et al. 2003;
+Henry etal. 2004), Abell 2065 (Chatzikos etal. 2006), Abell 2255 (Sakelliou &Ponman 2006), Abell2256 (Sun etal. 2002), and Abell 3667 (Vikhlinin etal.
+2001; Briel et al. 2004), respectively. Cooling times of the cool clumps are calculated from Eq (5), and effective conduc tion times are esitmated from Eq (6)
+by assuming a tangled magnetic effect. Gas masses of the cool clumps are calculated by assuming that the gas density, temp erature, and metal abundance are
+constant in the clump. Gas entropies are given by S=kT/n2/3
+gas.
+c/circleco√yrt2009 RAS, MNRAS 000,1–??ChandraandXMM-Newton Observationsof Abell3158 9
+Figure 1. (a)–(b) Adaptively smoothed and exposure-corrected Chandra ACIS-I (ObsID 3712) and XMM-Newton EPIC (ObsID 0300210201) images of the
+central 1.5 h−1
+71Mpc×1.5h−1
+71Mpc (21′.93×21′.93) region of Abell 3158. Both images are extracted in 0.3 −2.0keV and plotted in logarithmic scale. One
+small box and three crosses are used to mark the peak of the X-r ay halo and the centres of luminous member galaxies PGC 13641 , PGC 13652, and PGC
+13679, respectively. ( c) Optical DSS B-band image for the same sky field as ( a) and (b), on which the Chandra X-ray intensity contours (square root scale)
+are plotted.
+Figure 2. (a) East and west sector regions used to extract the X-ray SBPs, which are shown in ( b). Here the Chandra ACIS image is used as the background
+image. (b)Chandra ACIS and XMM-Newton EPIC 0.3−2.0 keV SBPs extracted from the partial elliptical annuli de fined in the two sector regions. For SBPs
+extracted in the east sector, a single βmodel can give an acceptable fitto the observed SBPs (dashed l ines;§3.1 and Table 1). For SBPs extracted in the west
+sector, dashed, dotted, and solid lines are used to show the b est-fits obtained with the single βmodel (§3.1), two- βmodel, and truncated powerlaw+ βmodel
+(§3.3 and Table1), respectively.
+c/circleco√yrt2009 RAS,MNRAS 000, 1–??10Yu Wanget al.
+Figure3. (a) Projected gastemperature map for thecentral 0.6 h−1
+71Mpc×0.6h−1
+71Mpc(8′.77×8′.77)region, along with the68% error map( b), which are
+calculated from the XMM-Newton MOS1+MOS2+pn data. Partial elliptical annuli defined in the north, east, south, and west sector regions are shown in ( a),
+which are used to extract both the Chandra andXMM-Newton spectra to beanalysed in §3.2.
+c/circleco√yrt2009 RAS, MNRAS 000,1–??ChandraandXMM-Newton Observationsof Abell3158 11
+Figure4. (a)−(d)Projected gastemperature profilesobtained with Chandra ACIS(cross)and XMM-Newton MOS1+MOS2+pn (diamond)dataforthenorth,
+east, south, and westsectors defined in Fig. 3 a, along with the 68% errors. ( e) Deprojected XMM-Newton temperature profiles for the west sector, along with
+the68% (solid) and 90% (dashed) errors (see also Table2). ( f)Fit-statistic contours of temperature and abundance atth e 68% (∆χ2= 2.30;solid) and 90%
+(∆χ2= 4.61; dashed) confidence levels for the inner three partial annul i of the west sector, all obtained in the XMM-Newton deprojected analysis.
+c/circleco√yrt2009 RAS,MNRAS 000, 1–??12Yu Wanget al.
+Figure5. (a)Line-of-sightvelocity distribution of68membergalaxie s identified inAbell3158.Originalvelocity dataaredrawnf romSmithetal.(2004).The
+distribution is fitted with a two-Gaussian model (solid) and a single Gaussian model (dashed; §4), respectively. ( b) DSS optical image for the central 3.0 h−1
+71
+Mpc×3.0h−1
+71Mpc (43′.86×43′.86) region of Abell 3158, where thelow-velocity ( v <18 500km s−1) and high-velocity ( v >18 500km s−1) member
+galaxies are marked with small and large circles, respectiv ely. The X-ray peak is marked with a small box and the geometri c centroid of the high-velocity
+subgroup is marked with a cross.
+Figure 6. Redshift-corrected gas entropy measured at 0.1R500(roughly50−150kpc) vs average gas temperature for cool-core clusters (cro sses) and non-
+cool-core clusters (diamonds), which are re-calculated ba sed on the data of Zhang et al. (2007) and Sanderson et al. (200 9). The corresponding best-fit power
+law models given by Sanderson et al. (2009) are also shown as d ashed and dotted lines, respectively. The entropy-tempera ture distribution of the off-centre
+cool gas clumps (Table 3) is also plotted.
+c/circleco√yrt2009 RAS, MNRAS 000,1–??
\ No newline at end of file
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+arXiv:1001.0094v3  [math.PR]  12 Feb 2010STOCHASTIC MONGE-KANTOROVICH PROBLEM AND ITS DUALITY∗
+XICHENG ZHANG
+Abstract. In this article we prove the existence of a stochastic optim al transference plan for a
+stochastic Monge-Kantorovichproblem by measurable selec tion theorem. A stochastic version
+of Kantorovichduality and the characterizationof stochas tic optimal transference plan are also
+established. Moreover,Wasserstein distancebetweentwop robabilitykernelsarediscussedtoo.
+1. Introductionand MainResults
+LetXbe a Polish space and P(X) the total of probability measures on ( X,B(X)), where
+B(X)istheBorel σ-field. Itiswellknownthat P(X)isaPolishspacewithrespecttotheweak
+convergence topology. Let B(P(X)) be the associated Borel σ-field. LetYbe another Polish
+space and c:X×Y→[0,∞] be a lower semicontinuous function called cost function. F or
+µ∈P(X)andν∈P(Y), considertheclassical Monge-Kantorovichproblem
+Cdeter(c,µ,ν) :=inf
+π∈Π(µ,ν)/integraldisplay
+X×Yc(x,y)π(dx,dy), (1)
+whereΠ(µ,ν)denotesthesetofalljointprobabilitymeasureson X×Ywithmarginaldistribu-
+tionsµandν. The history and the background of Monge-Kantorovich probl em are refereed to
+[4, 6] etc. The element in Π(µ,ν) is called transference plan; those achieving the infimum ar e
+called optimal transference plan. We remark that the existe nce of optimal transference plan is
+easilyobtainedbythecompactnessof Π(µ,ν)inP(X×Y). Moreover,thefollowingKantorovich
+dualityformulaholds(cf. [4]or[6, Theorem5.10])
+Cdeter(c,µ,ν)= sup
+(ψ,φ)∈L1(µ)×L1(ν);φ−ψ/lessorequalslantc/parenleftigg/integraldisplay
+Yφ(y)ν(dy)−/integraldisplay
+Xψ(x)µ(dx)/parenrightigg
+. (2)
+WenowturntothedescriptionofstochasticversionsofMong e-Kantorovichproblemandits
+duality. Let (Ω,F,P) be a probability space and µa probability kernel from ΩtoX. Here, by
+aprobabilitykernel µfromΩtoX, wemean thatamapping µ:Ω×B(X)→[0,1]satisfies
+(i)for each ω∈Ω,µω∈P(X);(ii)foreach B∈B(X),ω/mapsto→µω(B)isF-measurable.
+LetYbeanotherPolishspaceand νaprobabilitykernelfrom ΩtoY. Letc:Ω×X×Y→[0,∞]
+be a measurable function called stochastic cost function. C onsider the following stochastic
+Monge-Kantorovichproblem:
+Cstoch(c,µ,ν) :=inf
+π∈K(µ,ν)E/integraldisplay
+X×Yc(ω,x,y)πω(dx,dy), (3)
+whereK(µ,ν) is the set of all probability kernels from ΩtoX×Ywith marginal probability
+kernelsµandν,i.e., foraπω∈K(µ,ν),
+πω(·,Y)=µω, πω(X,·)=νω.
+∗Thisworkis supportedbyNSFs ofChina(Nos. 10971076;10871 215).
+1Ifπopt∈K(µ,ν) attains the infimum for the minimization problem (3), we cal l it a stochastic
+optimaltransferenceplan. Unlikethedeterministicprobl em(1),itseemstobehardtoprovethe
+existence of a stochastic optimal transference plan by a dir ect compactness argument. In fact,
+when the cost function cis deterministic, the existence of πopt
+ωhas been obtained by Zhang [7]
+(see also [6, Corollary 5.22]). On the other hand, one may als o expect the following stochastic
+Kantorovichdualityformulaholds:
+Cstoch(c,µ,ν)= sup
+(ψ,φ)∈L1(µω×P)×L1(νω×P);φ−ψ/lessorequalslantcE/parenleftigg/integraldisplay
+Yφ(ω,y)νω(dy)−/integraldisplay
+Xψ(ω,x)µω(dx)/parenrightigg
+,(4)
+whereL1(µω×P)denotesthesetofallmeasurablefunctions ψwithE/integraltext
+X|ψ(ω,x)|µω(dx)<+∞,
+andφ−ψ/lessorequalslantcmeansthatφ(ω,y)−ψ(ω,x)/lessorequalslantc(ω,x,y)forallω,x,y.
+Ourfirst result isabouttheexistenceofstochasticoptimal transference plans.
+Theorem 1.1. Assume that for each ω,(x,y)/mapsto→c(ω,x,y)is continuous, and for each (x,y)∈
+X×Y,ω/mapsto→c(ω,x,y)isF-measurableand satisfies
+E/integraldisplay
+X×Yc(ω,x,y)µω(dx)νω(dy)<+∞. (5)
+Then thereexistsa stochasticoptimaltransferenceplan πopt∈K(µ,ν)suchthat
+Cstoch(c,µ,ν)=E/integraldisplay
+X×Yc(ω,x,y)πopt
+ω(dx,dy)<+∞. (6)
+Moreover,ω/mapsto→Cdeter(c(ω),µω,νω)isF-measurableand wehave
+Cstoch(c,µ,ν)=E/parenleftigg
+inf
+π∈Π(µω,νω)/integraldisplay
+X×Yc(ω,x,y)π(dx,dy)/parenrightigg
+=E/parenleftig
+Cdeter(c(ω),µω,νω)/parenrightig
+.(7)
+Remark1.2. Forfixedω∈Ω,letXω⊂Π(µω,νω)bethesetofalloptimaltransferenceplansfor
+deterministicproblem(1). It is well known that X ωis a nonemptycompact subset of P(X×Y).
+For proving Theorem 1.1, we have to carefully choose a measur able function ω→πopt
+ωso that
+foreachω,πopt
+ω∈Xω. Thisseems notto betrivialas shownin [7].
+Oursecond resultis aboutthestochasticKantorovichduali ty.
+Theorem 1.3. Keeping thesameassumptionsasin Theorem 1.1,we furtherha ve
+Cstoch(c,µ,ν)= sup
+(ψ,φ)∈L1(µω×P)×L1(νω×P);φ−ψ/lessorequalslantcE/parenleftigg/integraldisplay
+Yφ(ω,y)νω(dy)−/integraldisplay
+Xψ(ω,x)µω(dx)/parenrightigg
+= sup
+(ψ,φ)∈Lipω
+b(X)×Lipω
+b(Y);φ−ψ/lessorequalslantcE/parenleftigg/integraldisplay
+Yφ(ω,y)νω(dy)−/integraldisplay
+Xψ(ω,x)µω(dx)/parenrightigg
+,(8)
+where Lipω
+b(X)is the space of all bounded measurable functions ψ(ω,x)onΩ×Xwhich is
+Lipschitzcontinuousin xforeach ω, similarlyfor Lipω
+b(Y).
+Our third result is about the characterization of stochasti c optimal transference plan, which
+correspondsto [6, Theorem5.10(ii)](seealso [1, 5]).
+Theorem 1.4. In the situation of Theorem 1.1, for any π∈K(µ,ν), the following statements
+areequivalent:
+(a)πisa stochasticoptimaltransferenceplan;
+(b)foralmostall ω∈Ω, thesupportof πωisa c(ω)-cyclicallymonotoneset;
+2(c)thereexist apairof measurablefunctions (φ,ψ)onΩ×YandΩ×Xsuchthat
+φ(ω,y)−ψ(ω,x)/lessorequalslantc(ω,x,y),∀(ω,x,y)∈Ω×X×Y,
+andfor each ω∈Ω,ψ(ω)isc(ω)-convex and
+Γω:={(x,y) :φ(ω,y)−ψ(ω,x)=c(ω,x,y)}⊂∂cψ(ω)
+hasπω-fullmeasure,where ∂cψ(ω)denotesthec (ω)-subdifferentialofψ(ω,·).
+Moreover, the measurable set Γ:={(ω,x,y) : (x,y)∈Γω}defined from (c) may be indepen-
+dent of the choice of optimal plan π. More precisely, let ˜πbe another stochastic optimal plan,
+then˜πωisconcentratedon Γωfor almostall ω.
+Remark 1.5. In these theorems, if we assume that c is lower semi-continuo us and approxi-
+mate it by the usual Lipscitz continuous functions (see (16) below), then we shall encounter a
+very subtle issue about the measurabilityof an uncountable infimum of lower semi-continuous
+functions(cf. [6,p.70-72] ).
+These three theorems will be proved in Section 3 by measurabl e selection theorem. For this
+aim, we give some necessary preliminaries in Section 2. In Se ction 4, we shall give a defi-
+nition of Wasserstein distance between two probability ker nels and discuss the corresponding
+properties. It is hoped that the results of the present paper can be used to the study of Markov
+processes.
+2. Preliminaries
+LetCbethetotal ofall nonnegativecontinuouscostfunctions c:X×Y→[0,∞), which is
+endowedwithametricas follows:
+dC(c1,c2) :=∞/summationdisplay
+m=12−m1∧sup
+(x,y)∈Bm
+X(x0)×Bm
+Y(y0)|c1(x,y)−c2(x,y)|,
+where(x0,y0)∈X×Yis fixed and
+Bm
+X(x0) :={x∈X:dX(x,x0)/lessorequalslantm},Bm
+Y(y0) :={y∈Y:dY(y,y0)/lessorequalslantm}.
+It iseasy tosee that( C,dC)isacompletemetricspace. Let Mbedefined by
+M:=/braceleftigg
+(c,µ,ν)∈C×P(X)×P(Y) :/integraldisplay
+X×Yc(x,y)µ(dx)ν(dy)<+∞/bracerightigg
+.
+Then itisametricspace(maybenotcompleteand separable)u nder
+dM((c1,µ1,ν1),(c2,µ2,ν2)) :=dC(c1,c2)+dP(X)(µ1,µ2)+dP(Y)(ν1,ν2),
+wheredP(X)anddP(Y)areweak convergencemetricin P(X)andP(Y)respectively. Wehave:
+Lemma 2.1. Let{(cn,µn,νn)∈M,n∈N}satisfythat
+sup
+n∈N/integraldisplay
+X×Ycn(x,y)µn(dx)νn(dy)/lessorequalslantM.
+Assumethat (cn,µn,νn)converges to (c,µ,ν)inM. Then
+/integraldisplay
+X×Yc(x,y)µ(dx)ν(dy)/lessorequalslantM.
+3Proof.By Urysohn’s lemma, there exist continuous functions fm
+X:X→[0,1] andfm
+Y:Y→
+[0,1]such that
+fm
+X(x)=1,x∈Bm
+X(x0),fm
+X(x)=0,x/nelementBm+1
+X(x0)
+and
+fm
+Y(y)=1,y∈Bm
+Y(y0),fm
+Y(y)=0,y/nelementBm+1
+Y(y0).
+Thus,by themonotoneconvergencetheorem,wehave
+/integraldisplay
+X×Yc(x,y)µ(dx)ν(dy)=lim
+m→∞/integraldisplay
+X×Yc(x,y)∧m·fm
+X(x)fm
+Y(y)µ(dx)ν(dy)
+=lim
+m→∞lim
+n→∞/integraldisplay
+X×Yc(x,y)∧m·fm
+X(x)fm
+Y(y)µn(dx)νn(dy).
+Sincecn→cinC, wehave
+lim
+n→∞sup
+(x,y)∈Bm+1
+X(x0)×Bm+1
+Y(y0)|c(x,y)−cn(x,y)|=0.
+Hence,/integraldisplay
+X×Yc(x,y)µ(dx)ν(dy)=lim
+m→∞lim
+n→∞/integraldisplay
+X×Ycn(x,y)∧m·fm
+X(x)fm
+Y(y)µn(dx)νn(dy)/lessorequalslantM.
+Theproofiscomplete. /square
+We recall the following definitions of cyclical monotonicit y andc-convexity (cf. [6, Defini-
+tions5.1,5.2]).
+Definition 2.2. LetX,Ybe two arbitrary set and c :X×Y→(−∞,∞]be a function. A
+subsetΓ⊂X×Yis said to be c-cyclically monotone if for any N ∈Nand any family
+(x1,y1),···,(xN,yN)ofpointsinΓ, thefollowinginequalityholds:
+N/summationdisplay
+i=1c(xi,yi)/lessorequalslantN/summationdisplay
+i=1c(xi,yi+1),yN+1=y1.
+A functionψ:X→(−∞,+∞]issaidto bec-convex ifitis notidentically +∞, andthereexists
+ζ:Y→[−∞,+∞]suchthat
+ψ(x)=sup
+y∈Y(ζ(y)−c(x,y)),∀x∈X.
+Then itsc-transformisdefined by
+ψc(y) :=inf
+x∈X(ψ(x)+c(x,y)),∀y∈Y,
+anditsc-subdifferentialdefinedby
+∂cψ:={(x,y)∈X×Y:ψc(y)−ψ(x)=c(x,y)}
+isa c-cyclicallymonotoneset.
+Wefirst provethefollowingslightextensionof[5,Theorem 3 ]and [6,Theorem 5.20].
+Theorem 2.3. Assume that (cn,µn,νn)→(c,µ,ν)inM. Letπnbe an optimal transference
+plan for problem (1) associated with c n,µn,νn. Then there exists a subsequence still denoted
+by n such that πnweakly converges to some π∈Π(µ,ν)andπis an optimal transference plan
+associatedwithc ,µ,ν.
+4Proof.First of all, by [6, Lemma 4.4], ( πn)n∈Nis tight, and so there exists a subsequence still
+denotedby nweaklyconvergingtosome π∈Π(µ,ν).
+By[6,Theorem5.10], πnisconcentratedonsome cn-cyclicallymonotoneset Γn. ForN∈N,
+letCn(N)⊂(X×Y)⊗Nbedefined by
+N/summationdisplay
+i=1cn(xi,yi)/lessorequalslantN/summationdisplay
+i=1cn(xi,yi+1),yN+1=y1,
+where(xi,yi)N
+i=1∈(X×Y)⊗N. Thenπ⊗N
+nisconcentrated on Γ⊗N
+n⊂Cn(N).
+Foranyε∈[0,1], letCε(N)⊂(X×Y)⊗Nbedefined by
+N/summationdisplay
+i=1c(xi,yi)/lessorequalslantN/summationdisplay
+i=1c(xi,yi+1)+ε,yN+1=y1,
+where(xi,yi)N
+i=1∈(X×Y)⊗N. Sincecn→cinC, foranyε∈(0,1]andN,m∈N, thereexistsa
+n0∈Nsuchthat forall n/greaterorequalslantn0
+Cn(N)∩(Bm
+X(x0)×Bm
+Y(y0))⊗N⊂Cε(N)∩(Bm
+X(x0)×Bm
+Y(y0))⊗N=:Am
+ε(N).
+Sincecis continuous, Am
+ε(N) is closed. Hence,
+π⊗N(Am
+ε(N))/greaterorequalslantlim
+n→∞π⊗N
+n(Am
+ε(N))/greaterorequalslantlim
+n→∞π⊗N
+n(Cn(N)∩(Bm
+X(x0)×Bm
+Y(y0))⊗N).
+In viewthat π⊗N
+nisconcentrated on Cn(N), byletting ε↓0,wefurtherhave
+π⊗N(Am
+0(N))/greaterorequalslantlim
+n→∞[πn(Bm
+X(x0)×Bm
+Y(y0))]N/greaterorequalslant/bracketleftigg
+1−lim
+n→∞(µn((Bm
+X(x0))c)+νn((Bm
+Y(y0))c))/bracketrightiggN
+.(9)
+Noticingthat( µn)n∈Nand (νn)n∈Naretight,wehave
+lim
+m→∞sup
+n∈Nµn((Bm
+X(x0))c)=0,lim
+m→∞sup
+n∈Nνn((Bm
+Y(y0))c)=0.
+Therefore, letting m→∞forbothsides of(9), we obtainthat
+π⊗N(C0(N))=1,∀N∈N,
+whichleads to
+(supportof π)⊗N=supportofπ⊗N⊂C0(N),∀N∈N,
+So thesupportof πisc-cyclicallymonotone. Since ( c,µ,ν)∈M, wehave
+Cdeter(c,µ,ν)/lessorequalslant/integraldisplay
+X×Yc(x,y)µ(dx)ν(dy)<+∞.
+By [6, Theorem 5.10]again, πis anoptimaltransference planassociatedwith c,µ,ν./square
+Thefollowinglemmawillbeused intheproofofTheorem 1.3.
+Lemma 2.4. AssumethatC b(X×Y)∋cn↑cinthesenseof pointwise. Then
+Cdeter(c,µ,ν)/lessorequalslantlim
+n→∞Cdeter(cn,µ,ν).
+Proof.Withoutlossofgenerality,we assumethat
+α:=lim
+n→∞Cdeter(cn,µ,ν)<+∞.
+In particular,there existsasubsequencestilldenotedby nsuch that
+lim
+n→∞Cdeter(cn,µ,ν)=α.
+5Letπn∈Π(µ,ν) be the optimal transference plan associated with cn,µ,ν. SinceΠ(µ,ν) is
+weakly compact, there exists another subsequence nksuch thatπnkweakly converges to some
+π0∈Π(µ,ν). By themonotonicityof cn, wehaveforeach m∈N,
+/integraldisplay
+X×Ycm(x,y)π0(dx,dy)=lim
+k→∞/integraldisplay
+X×Ycm(x,y)πnk(dx,dy)
+/lessorequalslantlim
+k→∞/integraldisplay
+X×Ycnk(x,y)πnk(dx,dy)
+=lim
+k→∞Cdeter(cnk,µ,ν)=α.
+On theotherhand,by themonotoneconvergencetheorem,weha ve
+Cdeter(c,µ,ν)/lessorequalslant/integraldisplay
+X×Yc(x,y)π0(dx,dy)=lim
+m→∞/integraldisplay
+X×Ycm(x,y)π0(dx,dy).
+Theresultnowfollows. /square
+We also recall the following measurability theorem for mult ifunctions (cf. [2] or [3, p.26,
+Theorem 2.3]).
+Theorem 2.5. Let(W,W)be a measurablespace and Xa Polish space. Let X :W→Fbe a
+multifunctions,where Fis thetotalof allclosedsetsin X. Considerthefollowingstatements:
+(1)foranyclosed A ⊂X.
+{w:X(w)∩A/nequal∅}∈W;
+(2)foranyopenset A ⊂X
+{w:X(w)∩A/nequal∅}∈W;
+(3)thereexistsa sequence (ξn)n∈Nofmeasurableselectionsof X suchthatforeach w ∈W
+X(w)={ξn(w),n∈N}.
+Then itholdsthat(1) ⇒(2)⇔(3).
+Thefollowinglemmais useful.
+Lemma 2.6. The Borelσ-fieldB(P(X))coincides with the σ-field generated by the mapping
+µ/mapsto→µ(B), where B∈B(X).
+Proof.LetFbeaclosed setinX. Define
+fn(x) :=1
+(1+dX(x,F))n.
+Thenfn(x)↓1F(x). So, forany r∈[0,1]
+{µ∈P(X) :µ(F)<r}=∪n∈N{µ∈P(X) :µ(fn)<r}∈B(P(X)).
+Theresultnowfollowsby amonotoneclass argument. /square
+63. ProofsofMainTheorems
+InthissectionwegivetheproofsofTheorems1.1,1.3and1.4 . First,weproveTheorem1.1.
+Proofof Theorem 1.1. Define amulti-valuedmap:
+M∋(c,µ,ν)/mapsto→Φ(c,µ,ν)⊂P(X×Y),
+whereΦ(c,µ,ν)isthetotalofalloptimaltransferenceplan associatedwi thc,µ,ν.
+ByTheorem2.3,foreach( c,µ,ν)∈M,Φ(c,µ,ν)isanonemptycompactsubsetof P(X×Y),
+and foranyclosed set A⊂P(X×Y)
+{(c,µ,ν)∈Mm:Φ(c,µ,ν)∩A/nequal∅}is aclosedsubsetof M,
+whereMm:=/braceleftig
+(c,µ,ν)∈M:/integraltext
+X×Yc(x,y)µ(dx)ν(dy)/lessorequalslantm/bracerightig
+. Indeed,let( cn,µn,νn)∈Mmconverge
+to (c,µ,ν). By Lemma 2.1, we have ( c,µ,ν)∈Mm. Letπn∈Φ(cn,µn,νn) weakly converge to
+someπ∈Π(µ,ν). By Theorem 2.3, π∈Φ(c,µ,ν). SinceAis closed,πalsobelongsto A.
+Notethat
+{(c,µ,ν)∈M:Φ(c,µ,ν)∩A/nequal∅}=∪m∈N{(c,µ,ν)∈Mm:Φ(c,µ,ν)∩A/nequal∅}.
+By Theorem 2.5, there exists a B(M)/B(P(X×Y))-measurable selection ( c,µ,ν)/mapsto→π(c,µ,ν)
+such thatforeach ( c,µ,ν)∈M
+π(c,µ,ν)∈Φ(c,µ,ν)⊂Π(µ,ν).
+Wenowdefine
+πopt
+ω:=π(c(ω),µω,νω).
+Sinceω/mapsto→(c(ω),µω,νω)isF/B(M)-measurablebyLemma2.6, wethushave
+ω/mapsto→πopt
+ωisF/B(P(X×Y))-measurable . (10)
+In particular,
+ω/mapsto→/integraldisplay
+X×Yc(ω,x,y)πopt
+ω(dx,dy)=Cdeter(c(ω),µω,νω)
+isF-measurableand
+Cstoch(c,µ,ν)/lessorequalslantE/parenleftig
+Cdeter(c(ω),µω,νω)/parenrightig
+.
+Theoppositeinequalityisclear. Thus,wecompletetheproo fof(6)and(7). /square
+WenowproveTheorem1.3.
+Proofof Theorem 1.3. Wedividetheproofintothreesteps.
+(Step 1): First ofall, forany π∈K(µ,ν), wehave
+sup
+(ψ,φ)∈L1(µω×P)×L1(νω×P);φ−ψ/lessorequalslantcE/parenleftigg/integraldisplay
+Yφ(ω,y)νω(dy)−/integraldisplay
+Xψ(ω,x)µω(dx)/parenrightigg
+= sup
+(ψ,φ)∈L1(µω×P)×L1(νω×P);φ−ψ/lessorequalslantcE/parenleftigg/integraldisplay
+X×Y(φ(ω,y)−ψ(ω,x))πω(dx,dy)/parenrightigg
+/lessorequalslantE/parenleftigg/integraldisplay
+X×Yc(ω,x,y)πω(dx,dy)/parenrightigg
+. (11)
+Thus,weobtainonesideinequality:
+sup
+(ψ,φ)∈L1(µω×P)×L1(νω×P);φ−ψ/lessorequalslantcE/parenleftigg/integraldisplay
+Yφ(ω,y)νω(dy)−/integraldisplay
+Xψ(ω,x)µω(dx)/parenrightigg
+/lessorequalslantCstoch(c,µ,ν).
+7(Step 2): In thisstep,weassumethat c(ω,x,y)isboundedand Lipschitzcontinuousin( x,y)
+foreachω.
+Letπopt
+ωbe the stochastic optimal transference plan constructed in Theorem 1.1. Let Γωbe
+thesupportof πopt
+ω,ac(ω)-cyclicallymonotoneset. Notethat foranyopen set A⊂X×Y,
+{ω:Γω∩A/nequal∅}={ω:πω(A)>0}∈F.
+By Theorem 2.5, there exists a sequence ( ξn(ω),ηn(ω))n∈Nof measurable selections of Γωsuch
+thatforeach ω∈Ω
+Γω={(ξn(ω),ηn(ω)),n∈N}. (12)
+Define foreach ( ω,x)∈Ω×X,
+ψ(ω,x) :=sup
+m∈Nsup
+(x1,y1),···,(xm,ym)∈Γω/braceleftig
+[c(ω,ξ1(ω),η1(ω))−c(ω,x1,η1(ω))]
++[c(ω,x1,y1)−c(ω,x2,y1)]+···+[c(ω,xm,ym)−c(ω,x,ym)]/bracerightig
+.(13)
+Arguingas in[6, p.65,Step 3], weknowthat
+ψ(ω,ξ1(ω),η1(ω))=0
+and
+ψ(ω) isc(ω)-convex.
+Sincec(ω,x,y)is continuouswithrespect to( x,y), by(12)wemay write
+ψ(ω,x)=sup
+m∈Nsup
+(x1,y1),···,(xm,ym)∈{(ξn(ω),ηn(ω)),n∈N}/braceleftig
+[c(ω,ξ1(ω),η1(ω))−c(ω,x1,η1(ω))]
++[c(ω,x1,y1)−c(ω,x2,y1)]+···+[c(ω,xm,ym)−c(ω,x,ym)]/bracerightig
+.(14)
+Hence, for each x∈X,ω/mapsto→ψ(ω,x) isF-measurable. Moreover, since cis Lipschitzcontinu-
+ousin(x,y), itiseasy toseethat foreach ω∈Ω,x/mapsto→ψ(ω,x) isalsoLipschitzcontinuous. Let
+ψc(ω,y)bethec-transformof ψdefined by
+ψc(ω,y) :=inf
+x∈X/parenleftig
+ψ(ω,x)+c(ω,x,y)/parenrightig
+.
+Then for each y∈Y,ω/mapsto→ψc(ω,y) is also F-measurable, and for each ω∈Ω,y/mapsto→ψc(ω,y) is
+Lipschitz continuous. Since cis bounded, as in [6, p.66, Step 4], ψcandψare bounded. Note
+that(cf. [6, p.65,Step 3])
+ψc(ω,y)−ψ(ω,x)=c(ω,x,y)onΓω. (15)
+So /integraldisplay
+Xψc(ω,y)νω(dy)−/integraldisplay
+Yψ(ω,x)µω(dx)=/integraldisplay
+X×Yc(ω,x,y)πopt
+ω(dx,dy),
+whichthen givesthat
+Cstoch(c,µ,ν)=E/parenleftigg/integraldisplay
+Xψc(ω,y)νω(dy)−/integraldisplay
+Yψ(ω,x)µω(dx)/parenrightigg
+.
+(Step 3): Forgeneral c(ω,x,y), definefor n∈N
+cn(ω,x,y) :=inf
+(x′,y′)∈X×Y/braceleftig
+min(c(ω,x′,y′),n)+n/bracketleftbigdX(x,x′)+dY(y,y′)/bracketrightbig/bracerightig
+. (16)
+It iseasy tosee that cnis Lipschitzcontinuous,and
+cn(ω,x,y)/lessorequalslantmin(c(ω,x,y),n)
+8and foreach ( ω,x,y)∈Ω×X×Y
+cn(ω,x,y)↑c(ω,x,y)n→∞.
+Thus,by (7), Lemma2.4and Fatou’slemma,wehave
+Cstoch(c,µ,ν)=E/parenleftig
+Cdeter(c(ω),µω,νω)/parenrightig
+/lessorequalslantE/parenleftigg
+lim
+n→∞Cdeter(cn(ω),µω,νω)/parenrightigg
+/lessorequalslantlim
+n→∞E/parenleftig
+Cdeter(cn(ω),µω,νω)/parenrightig
+=lim
+n→∞E/parenleftigg/integraldisplay
+Xφn(ω,y)νω(dy)−/integraldisplay
+Yψn(ω,x)µω(dx)/parenrightigg
+, (17)
+whereφn=ψc
+n∈Lipω
+b(Y)andψn∈Lipω
+b(X)constructedinStep 2satisfy
+φn(ω,y)−ψn(ω,x)/lessorequalslantcn(ω,x,y)/lessorequalslantc(ω,x,y). (18)
+Theproofisthuscompleteby combiningwithStep 1. /square
+Lastly,weproveTheorem1.4.
+Proofof Theorem 1.4. (a)⇒(b): Letπ∈K(µ,ν) be a stochastic optimal transference plan, and
+let (φn,ψn)n∈Nbeas in(17). By (11)and (17), wehave
+lim
+n→∞E/parenleftigg/integraldisplay
+X×Y[c(ω,x,y)−φn(ω,y)+ψn(ω,x)]πω(dx,dy)/parenrightigg
+=0.
+Ifnecessary, by extractingasubsequenceand by(18), there is anΩ0∈FwithP(Ω0)=1 such
+thatforeach ω∈Ω0,
+lim
+n→∞/integraldisplay
+X×Y[c(ω,x,y)−φn(ω,y)+ψn(ω,x)]πω(dx,dy)=0.
+Fix such an ω. Up to choosing a subsequence (possibly depending on ω), we can assume that
+forπω-almostall ( x,y)∈X×Y,
+lim
+n→∞φn(ω,y)−ψn(ω,x)=c(ω,x,y).
+ForN∈N, by passingtothelimitintheinequality
+N/summationdisplay
+i=1c(ω,xi,yi+1)/greaterorequalslantN/summationdisplay
+i=1[φn(ω,yi+1)−ψn(ω,xi)]=N/summationdisplay
+i=1[φn(ω,yi)−ψn(ω,xi)],
+wefind that π⊗N
+ωisconcentrated on theclosed set
+Cω(N) :=(xi,yi)N
+i=1∈(X×Y)⊗N:N/summationdisplay
+i=1c(ω,xi,yi+1)/greaterorequalslantN/summationdisplay
+i=1c(ω,xi,yi).
+So thesupportof πωisc(ω)-cyclicallymonotone.
+(b)⇒(c): Fixπ∈K(µ,ν) and set ˆΓω:=supp(πω). Since we can redefine πon aP-negligible
+set,withoutlossofgenerality,wecanassumethatforall ω∈Ω,ˆΓωisc(ω)-cyclicallymonotone.
+Define ac(ω)-convex function ψ(ω,x) as in (13) in terms of ˆΓω. From (14), we know that ψis
+9anF×B(X)-measurable function and for each ω,x/mapsto→ψ(ω,x) is lower semicontinuous. Let
+ψc(ω)bethec(ω)-transform of ψ(ω), i.e.,
+ψc(ω,y) :=inf
+x∈X/parenleftig
+ψ(ω,x)+c(ω,x,y)/parenrightig
+.
+Sinceψcis the infimum of uncountably many measurable functions, it i s not known whether
+ψcisF×B(Y)-measurable. As in [1, p.133, Step 2] or [6, p.72], we can mod ifyψcon a
+νω(dy)P(dω)-negligiblesetsothatitbecomesmeasurable. First,wedi sintegrateπω(dx,dy)P(dω)
+asπω(dx|y)νω(dy)P(dω) anddefine an F×B(Y)-measurable function
+ˆφ(ω,y) :=/integraldisplay
+X[ψ(ω,x)+c(ω,x,y)]·1ˆΓω(x,y)πω(dx|y).
+Sinceπω(ˆΓω)=1andˆΓω⊂∂cψ(ω)(see(15)),thereexistsameasurableset A∈F×B(Y)with /integraltext
+Aνω(dy)P(dω)=1 suchthat forall( ω,y)∈A,
+ˆφ(ω,y)=ψc(ω,y)/integraldisplay
+X1ˆΓω(x,y)πω(dx|y)=ψc(ω,y).
+Let usdefine an F×B(Y)-measurable functionby
+φ(ω,y) :=ˆφ(ω,y)=ψc(ω,y),(ω,y)∈A;
+−∞, (ω,y)/nelementA.
+Then, itiseasy to check that( φ,ψ)has thedesiredproperties.
+(c)⇒(a): Arguingas in [5, Theorem 2]or [6, p.72, (d) ⇒(a)], we can proveit by a truncation
+argument.
+Moreover,let ˜ πbeanotherstochasticoptimalplan,asin[6,p.73,(a) ⇒(e)],wecanprovethat
+E/integraldisplay
+X×Y[c(ω,x,y)−φ(ω,y)+ψ(ω,x)]˜πω(dx,dy)=0.
+Hence, foralmostall ω, ˜πωis concentrated on
+Γω:={(x,y)∈X×Y:φ(ω,y)−ψ(ω,x)=c(ω,x,y)}.
+Thewholeproofisfinished. /square
+4. Wasserstein Metricbetween TwoProbability Kernels
+In this section, we define the Wasserstein metric in the space of all probability kernels and
+discuss its properties. Let ( X,dX) be a metric space. For p/greaterorequalslant1, letKp(X) be the space of all
+probabilitykernels from ΩtoXwith
+E/integraldisplay
+XdX(x,x0)pµω(dx)<+∞
+forsome x0∈X(hence forall x0∈X). Letus definefor µ,ν∈Kp(X)
+Wp(µ,ν) :=/parenleftigg
+inf
+π∈K(µ,ν)E/integraldisplay
+X×XdX(x,y)pπω(dx,dy)/parenrightigg1/p
+,
+whichis called p-Wassersteindistance. By Theorem1.1, wehave
+Wp(µ,ν)=/parenleftig
+EWp(µω,νω)p/parenrightig1/p, (19)
+whereWp(µω,νω)=Cdeter(dp
+X,µω,νω)1/pis the usual Wasserstein distance between probability
+measuresµωandνω.
+10Thefollowingresult isadirect consequenceof(19) and[6, T heorem 6.18].
+Theorem4.1. Let(X,dX)beacompleteandseparablemetricspace,and (Ω,F,P)aseparable
+probability space. Then for any p /greaterorequalslant1,(Kp(X),Wp)is also a complete and separable metric
+space.
+We now consider the case of p=1. In this case, Wasserstein distance is usually called
+Kantorovich-Rubinsteindistance. Wehave:
+Theorem 4.2. Foranyµ,ν∈K1(X),
+W1(µ,ν)=sup
+/bardblψ(ω)/bardblLip/lessorequalslant1E/parenleftigg/integraldisplay
+Xψ(ω,x)νω(dx)−/integraldisplay
+Xψ(ω,x)µω(dx)/parenrightigg
+,
+where
+/bardblψ(ω)/bardblLip:=sup
+x,x′∈X|ψ(ω,x)−ψ(ω,x′)|
+dX(x,x′).
+Proof.By Theorem1.3,it onlyneeds toprovethat
+sup
+(ψ,φ)∈Lipω
+b(X)×Lipω
+b(X);φ−ψ/lessorequalslantdXE/parenleftigg/integraldisplay
+Yφ(ω,y)νω(dy)−/integraldisplay
+Xψ(ω,x)µω(dx)/parenrightigg
+(20)
+=sup
+/bardblψ(ω)/bardblLip/lessorequalslant1E/parenleftigg/integraldisplay
+Xψ(ω,x)νω(dx)−/integraldisplay
+Xψ(ω,x)µω(dx)/parenrightigg
+. (21)
+Assumethat φ(ω,y)−ψ(ω,x)/lessorequalslantdX(x,y). Then
+φ(ω,y)/lessorequalslantinf
+x∈X(ψ(ω,x)+dX(x,y))=:ψd(ω,y)
+and
+ψ(ω,x)/greaterorequalslantsup
+y∈X(ψd(ω,y)−dX(x,y))=:ψdd(ω,x).
+Thus,
+(20)/lessorequalslantsup
+ψ∈Lipω
+b(X)E/parenleftigg/integraldisplay
+Yψd(ω,y)νω(dy)−/integraldisplay
+Xψdd(ω,x)µω(dx)/parenrightigg
+.
+On theotherhand,it iseasy toverify
+/bardblψd(ω)/bardblLip/lessorequalslant1,
+and so,
+ψd(ω,x)=ψdd(ω,x).
+Hence, (20)/lessorequalslant(21). Moreover,(20) /greaterorequalslant(21)is obvious. Theproofiscomplete. /square
+Acknowledgements:
+Theauthorisvery grateful to Professor FuqingGao fortelli ngmetheexistenceofreference
+[7]. Thiswork issupportedbyNSFs ofChina(Nos. 10971076;1 0871215).
+11References
+[1] Ambrosio, L. and Pratelli, A.: Existence and stability r esults in the L1theorey of optimal transportation.
+CIME Course,LectureNotesinMathematics,Vol.1813,Sprin ger-Verglag,2003.
+[2] Castaing,C.andValadier,M.: Convexanalysisandmeasu rablemultifunctions.Vol.580,Lect.NotesMath.,
+Springer,Berlin, 1977.
+[3] Molchanov,I.: TheoryofRandomSets. Springer-Verlag, Berlin,2005.
+[4] Rachev, S.T. and R¨ uschendorf, L.: Mass transportation problems, Vol I., Probability and its Applications,
+Springer-Verlag,New York,1998.
+[5] Schachermayer, W. and Teichmann, J.: Characterization of optimal transport plans for the Monge-
+Kantorovichproblem.Proc.Amer.Math.Soc.137(2009),519 -529.
+[6] Villani, C.: OptimalTransport: OldandNew.Springer-V erlag,Berlin,2009.
+[7] Zhang, S.: Existence and application of optimal Markovi an coupling with respect to non-negative lower
+semi-continuousfunctions.Acta Math.Sin.(Engl.Ser.)16 (2000),no.2,261–270.
+XichengZhang: Departmentof Mathematics ,HuazhongUniversityof Scienceand Technology ,Wuhan,Hubei
+430074,P.R.China,email: XichengZhang@gmail.com
+12
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+arXiv:1001.0095v2  [math.CO]  4 Mar 2010Asymptoticvariance of random symmetric digital search
+trees
+Hsien-KueiHwang
+Institute of Statistical Science
+Academia Sinica
+Taipei 115
+TaiwanMichael Fuchs
+Departmentof Applied Mathematics
+National ChiaoTungUniversity
+Hsinchu300
+Taiwan
+Vytas Zacharovas
+Institute ofStatistical Science
+Academia Sinica
+Taipei 115
+Taiwan
+October 30,2018
+Abstract
+Asymptotics of the variances of many cost measures in random digital search trees are often no-
+toriously messy and involved to obtain. A new approach is pro posed to facilitate such an analysis for
+several shape parameters on random symmetric digital searc h trees. Our approach starts from a more
+careful normalization at the level of Poisson generating fu nctions, which then provides an asymptoti-
+callyequivalent approximation tothevariance inquestion . Several newingredients arealsointroduced
+such as a combined use of the Laplace and Mellin transforms an d a simple, mechanical technique for
+justifying the analytic de-Poissonization procedures inv olved. The methodology we develop can be
+easily adapted to many other problems with an underlying bin omial distribution. In particular, the less
+expected and somewhat surprising n(logn)2-variance for certain notions of total path-length is also
+clarified.
+Key words : Digital search trees, Poisson generating functions, Pois sonization, Laplace transform, Mellin
+transform, saddle-pointmethod,Collessindex,weightedp ath-length
+Dedicated tothe60thbirthdayof PhilippeFlajolet
+1 Introduction
+The variance of a distribution provides an important measur e of dispersion of the distribution and plays
+a crucial and, in many cases, a determinantal rˆ ole in the lim it law1. Thus finding more effective means
+1The first formal use of the term “variance” in its statistical sense is generally attributed to R. A. Fisher in his 1918 pape r
+(see [20]orWikipedia’swebpageonvariance),althoughitspractic aluse indiversescientific disciplinespredatedthisbya fe w
+centuries(includingclosely-definedtermssuchasmean-sq uarederrorsandstandarddeviations).
+1of computing the variance is often of considerable significa nce in theory and in practice. However, the
+calculation of the variance can be computationally or intri nsically difficult, either because of the messy
+procedures orcancellationsinvolved,or becausethedepen dencestructureis toostrongorsimplybecause
+no simple manageable forms or reductions are available. We a re concerned in this paper with random
+digital trees for which asymptotic approximations to the va riance are often marked by heavy calculations
+andlong,messyexpressions. Thispaperproposesageneral a pproachtosimplifynotonlytheanalysisbut
+alsotheresultingexpressions,providingnewinsightinto themethodology;furthermore,itisapplicableto
+manyotherconcretesituationsandleadsreadilytodiscove rseveralnewresults,sheddingnewlightonthe
+stochasticbehaviorsoftherandomsplittingstructures.
+A binomial splitting process. The analysis of many splitting procedures in computer algor ithms leads
+naturallyto astructuraldecomposition(interms ofthecar dinalities)oftheform
+structure
+of sizen
+substructure
+of sizeBnsubstructure
+of size¯BnHereBn≈
+Binomialand
+Bn+¯Bn≈n.
+whereBnis essentially a binomial distribution (up to truncation or small perturbations) and the sum of
+Bn+¯Bnisessentially n.
+Concrete examples in the literature include (see the books [ 15,28,44,50,62] and below for more
+detailed references)
+•tries,contention-resolutiontreealgorithms,initializ ationproblemindistributednetworks,andradix
+sort:Bn=Binomial(n;p)and¯Bn=n−Bn,namely, P(Bn=k) =/parenleftbign
+k/parenrightbig
+pkqn−k(hereandthroughout
+thispaper,q:= 1−p);
+•bucketdigitalsearchtrees(DSTs),directeddiffusion-li mitedaggregationonBethelattice,andEden
+model:Bn=Binomial(n−b;p)and¯Bn=n−b−Bn;
+•Patriciatriesand suffix trees: P(Bn=k) =/parenleftbign
+k/parenrightbig
+pkqn−k/(1−pn−qn)and¯Bn=n−Bn.
+Yet anothergeneral form arisesin theanalysisofmulti-acc essbroadcast channelwhere
+/braceleftbiggBn=Binomial(n;p)+Poisson(λ),
+¯Bn=n−Binomial(n;p)+Poisson(λ),
+see[19,33]. Forsomeothervariants,see[ 2,6,25]. Onereasonofsuchaubiquityofbinomialdistribution
+issimplyduetothebinaryoutcomes(eitherzeroorone,eith eronoroff,eitherpositiveornegative,etc.) of
+manypractical situations,resultingin thenatural adapta tionoftheBernoullidistributionin themodeling.
+2PoissongeneratingfunctionandthePoissonheuristic. Averyuseful,standardtoolfortheanalysisof
+thesebinomialsplittingprocessesis thePoissongenerati ng function
+˜f(z) =e−z/summationdisplay
+k≥0ak
+k!zk,
+where{ak}isa givensequence, onedistinctivefeaturebeingthe Poissonheuristic ,whichpredicts that
+Ifanissmoothenough,then an∼˜f(n).
+Inmoreprecisewords,ifthesequence {ak}doesnotgrowtoofast(usuallyatmostofpolynomialgrowth)
+or does not fluctuate too violently, then anis well approximated by ˜f(n)for largen. For example, if
+˜f(z) =zm,m= 0,1,...,thenan∼nm;indeed, insuch asimplecase, an=n(n−1)···(n−m+1).
+NotethatthePoissonheuristicisitselfaTauberiantheore m fortheBorel meaninessence; an Abelian
+typetheoremcan befound inRamanujan’sNotebooks(see[ 3, p. 58]).
+From an elementary viewpoint, such a heuristic is based on th e local limit theorem of the Poisson
+distribution(oressentiallyStirling’sformulafor n!)
+nk
+k!e−n∼e−x2/2
+√
+2πn/parenleftbigg
+1+x3−3x
+6√n+···/parenrightbigg
+(k=n+x√n),
+wheneverx=o(n1/6). Sinceanis smooth,wethenexpect that
+˜f(n)≈/summationdisplay
+k=n+x√n
+x=O(nε)ake−x2/2
+√
+2πn≈an/integraldisplay∞
+−∞e−x2/2
+√
+2πdx=an.
+Ontheotherhand,by Cauchy’sintegralrepresentation,wea lsohave
+an=n!
+2πi/contintegraldisplay
+|z|=nz−n−1ez˜f(z)dz
+≈˜f(n)n!
+2πi/contintegraldisplay
+|z|=nz−n−1ezdz
+=˜f(n),
+since the saddle-point z=nof the factor z−nezis unaltered by the comparatively more smooth function
+˜f(z).
+The Poisson-Charlier expansion. The latter analytic viewpoint provides an additional advan tage of
+obtainingan expansionby usingtheTaylorexpansionof ˜fatz=n, yielding
+an=/summationdisplay
+j≥0˜f(j)(n)
+j!τj(n), (1)
+where
+τj(n) :=n![zn](z−n)jez=/summationdisplay
+0≤ℓ≤j/parenleftbiggj
+ℓ/parenrightbigg
+(−1)j−ℓn!nj−ℓ
+(n−ℓ)!(j= 0,1,...),
+3and[zn]φ(z)denotes the coefficient of znin the Taylor expansion of φ(z). We call such an expansion the
+Poisson-Charlierexpansion sincetheτj’s areessentiallytheCharlier polynomials Cj(λ,n)defined by
+Cj(λ,n) :=λ−nn![zn](z−1)jeλz,
+so thatτj(n) =njCj(n,n). Forothertermsusedin theliterature,see[ 28,29].
+Thefirstfew termsof τj(n)are givenas follows.
+τ0(n)τ1(n)τ2(n)τ3(n)τ4(n)τ5(n) τ6(n)
+10−n2n3n(n−2)−4n(5n−6)−5n(3n2−26n+24)
+It is easilyseen that τj(n)isa polynomialin nofdegree ⌊j/2⌋.
+The meaning of such a Poisson-Charlier expansionbecomes re adily clear by the followingsimplebut
+extremelyuseful lemma.
+Lemma 1.1. Let˜f(z) :=e−z/summationtext
+k≥0akzk/k!. If˜fis an entire function, then the Poisson-Charlier expan-
+sion(1)providesanidentityfor an.
+Proof.Since˜fis entire,wehave
+/summationdisplay
+n≥0an
+n!zn=ez˜f(z) =ez/summationdisplay
+j≥0˜f(j)(n)
+j!(z−n)j,
+and thelemmafollowsby absoluteconvergence.
+Two specific examples are worthy of mention here as they speak volume of the difference between
+identityandasymptoticequivalence. Takefirst an= (−1)n. ThenthePoissonheuristicfailssince (−1)n/\⌉}atio\slash∼
+e−2n, but,byLemma 1.1, wehavetheidentity
+(−1)n=e−2n/summationdisplay
+j≥0(−2)j
+j!τj(n).
+See Figure 1foraplotoftheconvergenceoftheseries to (−1)n.
+204060801000.20.40.60.81.0
+20406080100
+−1−0.8−0.6−0.4−0.200.2
+Figure 1: Convergence of e−2n/summationtext
+j≤k(−2)jτj(n)/j!to(−1)nforn= 10(left) andn= 11(right) for
+increasingk.
+Nowifan= 2n, then2n/\⌉}atio\slash∼en, butwestillhave
+2n=en/summationdisplay
+j≥0τj(n)
+j!.
+4SowhenisthePoisson-Charlierexpansionalsoanasymptoti cexpansionfor an,inthesensethatdrop-
+pingalltermswith j≥2ℓintroducesanerroroforder ˜f(2ℓ)nℓ(whichintypicalcasesisoforder ˜f(n)n−ℓ)?
+Manysufficientconditionsarethoroughlydiscussedin[ 36],althoughthetermsintheirexpansionsareex-
+pressed differently;seealso[ 62].
+Poissonizedmeanandvariance. Themajorityofrandomvariablesanalyzedinthealgorithmi cliterature
+are at most of polynomial or sub-exponential (such as ec(logn)2orecn1/2) orders, and are smooth enough.
+Thus the Poisson generating functions of the moments are oft en entire functions. The use of the Poisson-
+Charlier expansion is then straightforward, and in many sit uations it remains to justify the asymptotic
+natureoftheexpansion.
+For convenience of discussion, let ˜fm(z)denote the Poisson generating function of the m-th moment
+oftherandomvariablein question,say Xn. Thenby Lemma 1.1, wehavetheidentity
+E(Xn) =/summationdisplay
+j≥0˜f(j)
+1(n)
+j!τj(n),
+and forthesecondmoment
+E(X2
+n) =/summationdisplay
+j≥0˜f(j)
+2(n)
+j!τj(n), (2)
+providedonlythatthetwoPoissongenerating functions ˜f1and˜f2are entirefunctions.
+Theseidentitiessuggestthatagoodapproximationto theva rianceofXnbegivenby
+V(Xn) =E(X2
+n)−(E(Xn))2≈˜f2(n)−˜f1(n)2,
+which holds true for many cost measures, where we can indeed r eplace the imprecise, approximately
+equal symbol“ ≈” by themoreprecise, asymptoticallyequivalentsymbol“ ∼”. However,for a large class
+ofproblemsforwhichthevarianceisessentiallylinear,me aning roughlythat
+lim
+n→∞logV(Xn)
+logn= 1, (3)
+the Poissonized variance ˜f2(n)−˜f1(n)2is not asymptotically equivalentto the variance. This is th e case
+for the total cost of constructing random digital search tre es, for example. One technical reason is that
+there are additional cancellations produced by dominant te rms. The next question is then: can we find a
+betternormalizedfunctionso thatthevarianceisasymptot icallyequivalentto itsvalueat n?
+Poissonizedvariancewithcorrection. Thecrucialstepofourapproachthatisneededwhenthevaria nce
+is essentiallylinearisto consider
+˜V(z) :=˜f2(z)−˜f1(z)2−z˜f′
+1(z)2, (4)
+and itthen turnsoutthat
+V(Xn) =˜V(n)+O((logn)c),
+in all cases we consider for some c≥0. The asymptotics of the variance is then reduced to that of ˜V(z)
+for largez, which satisfies, up to non-homogeneous terms, the same type of equation as ˜f1(z). Thus the
+sametoolsusedforanalyzing themean can beappliedto ˜V(z).
+5Toseehowthelastcorrectionterm z˜f′
+1(z)2appears,wewrite ˜D(z) :=˜f2(z)−˜f1(z)2,sothat˜f2(z) =
+˜D(z)+˜f1(z)2, and weobtain,bysubstitutingthisinto( 2),
+V(Xn) =E(X2
+n)−(E(Xn))2
+=/summationdisplay
+j≥0˜f2(j)(n)
+j!τj(n)−/parenleftBigg/summationdisplay
+j≥0˜f1(j)(n)
+j!τj(n)/parenrightBigg2
+=˜D(n)−n˜f′
+1(n)2−n
+2˜D′′(n)+smaller-orderterms .
+Now take ˜f1(n)≍nlogn. Then the first term following ˜D(n)is generally not smaller than ˜D(n)
+because
+n˜f′
+1(n)2≍n(logn)2,
+while˜D(n)≍n(logn)2, at least for the examples we discuss in this paper. Note that the variance is in
+such a case either of order nlognor of ordern. Thus to get an asymptotically equivalent approximation
+to thevariance, weneed at least an additionalcorrection te rm,which isexactly n˜f′
+1(n)2.
+Thecorrectionterm n˜f′
+1(n)2alreadyappearedinmanyearlypapersbyJacquetandR´ egnie r(see[34]).
+A viewpoint from the asymptotics of the characteristic func tion.Most binomial recurrences of the
+form
+Xnd=XBn+X∗
+¯Bn+Tn, (5)
+as arising from the binomial splitting processes discussed above are asymptotically normally distributed,
+a property partly ascribable to the highly regular behavior of the binomialdistribution. Here the (X∗
+n)are
+independent copies ofthe (Xn)and therandom or deterministicnon-homogeneouspart Tnis often called
+the “toll-function,” measuring the cost used to “conquer” t he two subproblems. Such recurrences have
+been extensivelystudiedin numerouspapers;see[ 36,52,58,59] andthereferences therein.
+Thecorrectiontermweintroducedin( 4)forPoissonizedvariancealsoappearsnaturallyinthefol low-
+ingheuristic,formalanalysis,whichcanbejustifiedwhenm orepropertiesareavailable. Bydefinitionand
+formal expansion
+e−z/summationdisplay
+n≥0E/parenleftbig
+eXniθ/parenrightbigzn
+n!=/summationdisplay
+m≥0˜fm(z)
+m!(iθ)m
+= exp/parenleftBigg
+˜f1(z)iθ−˜D(z)
+2θ2+···/parenrightBigg
+,
+where˜D(z) :=˜f2(z)−˜f1(z)2, wehave
+E/parenleftBig
+e(Xn−˜f1(n))iθ/parenrightBig
+≈n!
+2πi/contintegraldisplay
+|z|=nz−n−1exp/parenleftBigg
+z+/parenleftBig
+˜f1(z)−˜f1(n)/parenrightBig
+iθ−˜D(z)
+2θ2+···/parenrightBigg
+dz.
+Observethatwith z=neit, wehavethelocal expansion
+neit−nit+/parenleftBig
+˜f1(neit)−˜f1(n)/parenrightBig
+iθ−˜D(neit)
+2θ2=n−nt2
+2−n˜f′
+1(n)tθ−˜D(n)
+2θ2+···,
+6forsmallt. It followsthat
+E/parenleftBig
+e(Xn−˜f1(n))iθ/parenrightBig
+≈n!n−nen
+2πexp/parenleftBigg
+−˜D(n)
+2θ2/parenrightBigg/integraldisplayε
+−εexp/parenleftbigg
+−nt2
+2−n˜f′
+1(n)tθ/parenrightbigg
+dt
+∼exp/parenleftbigg
+−θ2
+2/parenleftBig
+˜D(n)−n˜f′
+1(n)2/parenrightBig/parenrightbigg
+,
+by extendingtheintegralto ±∞and bycompletingthesquare. Thisagainshowsthat n˜f′
+1(n)2istheright
+correction term forthevariance. Formorepreciseanalysis ofthistype, see[ 36].
+A comparison of different approaches to the asymptotic vari ance.What are the advantages of the
+Poissonized variance with correction? In the literature, a few different approaches have been adopted for
+computingtheasymptoticsofthevarianceofthebinomialsp littingprocesses.
+•Second moment approach: this is the most straightforward me ans and consists of first deriving
+asymptotic expansions of sufficient length for the expected value and for the second moment, then
+considering the difference E(X2
+n)−(E(Xn))2, and identifying the lead terms after cancellations of
+dominant terms in both expansions. This approach is often co mputationally heavy as many terms
+have to be cancelled; additional complication arises from fl uctuating terms, rendering the resulting
+expressionsmoremessy. Seebelowformorereferences.
+•Poissonizedvariance: theasymptoticsofthevarianceisca rriedoutthroughthatof ˜D(n) =˜f2(n)−
+˜f1(n)2. Thedifferencebetweenthisapproachandthepreviousonei sthatnoasymptoticsof ˜f2(n)is
+derivedorneeded,andonealwaysfocusesdirectlyonconsid eringtheequation(functionalordiffer-
+ential)satisfiedby ˜D(z). Aswediscussedabove,thisdoesnotgiveinmanycasesanasy mptotically
+equivalent estimate for the variance, because additional c ancellations have to be further taken into
+account; seeforinstance[ 34,35,36].
+•Characteristic function approach: similar to the formal ca lculations we carried out above, this ap-
+proach tries to derive a more precise asymptoticapproximat ion to the characteristic function using,
+say complex-analytic tools, and then to identify the right n ormalizing term as the variance; see the
+survey[36]andthepapers cited there.
+•Schachinger’s differencing approach: a delicate, mostly e lementary approach based on the recur-
+rence satisfied by the variance was proposed in [ 58] (see also [ 59]). His approach is applicable to
+very general “toll-functions” Tnin (5)butat thepriceofless preciseexpressions.
+The approach we use is similar to the Poissonized variance on e but the difference is that the passage
+through˜D(z)is completelyavoided and wefocusdirectly onequationssat isfiedby ˜V(z)(defined in ( 4)).
+In contrast to Schachinger’s approach, our approach, after starting from defining ˜V(z), is mostly ana-
+lytic. Ityieldsthenmorepreciseexpansions,butmoreprop ertiesofTnhavetobeknown. Thecontrasthere
+between elementary and analytic approaches is thus typical ; see, for example, [ 7,8]. See also Appendix
+fora briefsketchoftheasymptoticlinearityofthevarianc eby elementaryarguments.
+Additional advantages that our approach offer include comp aratively simpler forms for the resulting
+expressions,includingFourierseriesexpansions,andgen eralapplicability(couplingwiththeintroduction
+ofseveralnew techniques).
+7Organization of this paper. This paper is organized as follows. We start with the varianc e of the total
+path-length of random digital search trees in the next secti on, which was our motivating example. We
+then extend the consideration to bucket DSTs for which two di fferent notions of total path-length are
+distinguished, which result in very different asymptotic b ehaviors. The application of our approach to
+several other shape parameters are discussed in Section 4. Table1summarizes the diverse behaviors
+exhibitedbythemeansand thevariancesoftheshapeparamet erswe considerinthispaper.
+Shapeparameters mean variance
+Internal PL nlognn
+Key-wisePL∗nlognn
+Node-wisePL∗nlognn(logn)2
+Peripheral PL n n
+#(leaves) n n
+DifferentialPL nnlogn
+Weighted PL n(logn)m+1n
+Table 1:Orders of the means and the variances of all shape parameters in this paper; those marked with
+an∗areforb-DSTs withb≥2. Here PLdenotespath-lengthand m≥0.
+Applications of the approach we develop here to other classe s of trees and structures, including tries,
+Patriciatries, bucketsort,contentionresolutionalgori thms,etc., willbeinvestigatedin afuturepaper.
+2 DigitalSearchTrees
+We start in thissection with a briefdescription ofdigitals earch trees (DSTs), list majorshapeparameters
+studied in the literature, and then focus on the total path-l ength. The approach we develop is also very
+useful for other linear shape measures, which is discussed i n a more systematic form in the following
+sections.
+2.1 DSTs
+DSTswerefirstintroducedbyCoffmanandEvein[ 9]intheearly1970’sunderthenameofsequencehash
+trees. They can be regarded as the bit-version of binary sear ch trees (thus the name); see [ 44, p. 496et
+seq.]. Givena sequence of binary strings, we place the first in the root node; thosestarting with “ 0” (“1”)
+are directed to the left (right) subtree of the root, and are c onstructed recursively by the same procedure
+but withtheremovaloftheirfirst bitswhen comparisonsare m ade. See Figure 2foran illustration.
+While the practical usefulness of digital search trees is li mited, they represent one of the simplest,
+fundamental, prototype models for divide-and-conquer alg orithms using coin-tossing or similar random
+devices. Ofnotableinterestisitscloseconnectiontothea nalysisofLempel-Zivcompressionschemethat
+has found widespread incorporation into numerous software s. Furthermore, the mathematical analysis is
+often challenging and leads to intriguing phenomena. Also t he splitting mechanism of DSTs appeared
+naturallyin afew problemsin otherareas; someoftheseare m entionedinthelastsection.
+Random digital search trees. The simplest random model we discuss in this paper is the inde pendent,
+Bernoulli model. In this model, we are given a sequence of nindependent and identically distributed
+8010111
+101011
+100001
+011011
+111110
+110111
+010011
+011110
+000100010111
+101011
+100001011011
+111110
+110111010011
+0111100001001
+00
+1
+01
+10
+Figure2: A digitalsearchtreeof ninebinarystrings.
+random variables, each comprising an infinity sequence of Be rnoulli random variables with mean p,0<
+p<1. TheDST constructedfromthegivenrandomsequenceofbinar ystringsiscalled a randomDST . If
+p= 1/2, the DST is said to be symmetric ; otherwise, it is asymmetric . We focus on symmetric DSTs in
+thispaperforsimplicity;extensiontoasymmetricDSTs isp ossiblebutmuchharder.
+Stochastic properties of many shape characteristics of ran dom DSTs are known. Almost all of them
+fall into one of the two categories, according to their growt h order being logarithmicor essentially linear
+(in thesenseof( 3)), which wesimplyrefer toas “log shapemeasures”and “line arshapemeasures”.
+Logshapemeasures. Thetwomajorparametersstudiedinthiscategoryare depth,whichisthedistance
+of the root to a randomly chosen node in the tree (each with the same probability), and height, which
+counts the number of nodes from the root to one of the longest p aths. Both are of logarithmic order in
+mean. Depth provides a good indication of the typical cost ne eded when inserting a new key in the tree,
+whileheightmeasurestheworstpossiblecostthatmay benee ded.
+Depth was first studied in [ 45] in connection with the profile, which is the sequence of numbers, each
+enumerating the number of nodes with the same distance to the root. For example, the tree has
+the profile {1,2,3,2,3}. For other papers on the depth of random DSTs, see [ 11,12,13,37,38,39,44,
+46,47,50,55,60,61]. TheheightofrandomDSTs isaddressed in[ 13,14,43,50,55].
+Linear shape measures. These includethe total internal path-length, which sums th e distancebetween
+the root and every node, and the occurrences of a givenpatter n (leaves or nodes satisfyingcertain proper-
+ties); see[ 24,26,30,31,35,40,42,44].
+The profile contains generally much more information than mo st other shape measures, and it can to
+some extent be regarded as a good bridge connecting log and li near measures; see [ 15,17,45,46] for
+knownpropertiesconcerning expectedprofileofrandomDSTs .
+NodesofrandomDSTswith p= 1/2aredistributedinanextremelyregularway,asshowninFigu res3
+and4.
+2.2 Known and newresults forthe totalinternal path-length
+Throughoutthissection,wefocuson Xn, thetotalpathlengthofarandomdigitalsearchtreebuiltf romn
+binary strings. By definitionand by ourrandom assumption, Xncan becomputedrecursivelyby
+Xn+1d=XBn+X∗
+n−Bn+n,(n≥0) (6)
+9with the initial condition X0= 0, since removing the root results in a decrease of nfor the total path
+length (each internal node below the root contributes 1). HereBn∼Binomial(n;1/2),Xnd=X∗
+n, and
+Xn,X∗
+n,Bnareindependent.
+102
+4
+8
+16
+32
+64
+120
+158
+79
+15
+1n= 500
+2
+4
+8
+16
+32
+64
+128
+243
+332
+201
+61
+7
+1n= 1100
+Figure3: Two typical randomDSTs.
+112
+4
+8
+16
+32
+64
+128
+240
+306
+159
+37
+3
+n= 1000
+2
+4
+8
+16
+3264128240294161455
+n= 1000
+Figure4: TworandomDSTsof 1000nodesrendereddifferently. Formoregraphicalrenderings ofrandom
+DSTs, seethefirstauthor’swebpage algo.stat.sinica.edu.tw.
+12Knownresults. It isknownthat(see [ 26,30,57])
+E(Xn) = (n+1)log2n+n/parenleftbiggγ−1
+log2+1
+2−c1+̟1(log2n)/parenrightbigg
++γ−1/2
+log2+5
+2−c1+̟2(log2n)+O/parenleftbig
+n−1logn/parenrightbig
+,(7)
+whereγdenotes Euler’s constant, c1:=/summationtext
+k≥1(2k−1)−1, and̟1(t),̟2(t)are1-periodic functions with
+zero meanwhoseFourierexpansionsaregivenby( χk:= 2kπi/L,L:= log2)
+̟1(t) =1
+L/summationdisplay
+k/\e}atio\slash=0Γ(−1−χk)e2kπit, (8)
+̟2(t) =−1
+L/summationdisplay
+k/\e}atio\slash=0/parenleftBig
+1−χk
+2/parenrightBig
+Γ(−χk)e2kπit,
+respectively. Here Γdenotes the Gamma function. Thus we see roughly that random digital search trees
+under the unbiased Bernoulli model are highly balanced in sh ape. An important feature of the periodic
+functions is that they are marked by very small amplitudes of fluctuation: |̟1(t)| ≤3.4×10−8and
+|̟2(t)| ≤3.4×10−6. Such a quasi-flat (or smooth) behavior may in practice be ver y likely to lead to
+wrong conclusionsas theyare hardlyvisiblefromsimulatio nsofmoderatesamplesizes.
+2 3 4 5 6 7 8 90.10.2−0.1
+−0.2
+−0.3
+1 10V(Xn)/n
+E(Xn)/(n+1)−log2n
+Figure 5: A plot of E(Xn)/(n+ 1)−log2nin log-scale (the decreasing curve using the y-axis on the
+right-handside),and thatof V(Xn)/ninlog-scale(theincreasingcurveusingthe y-axison theleft-hand
+side).
+Let
+Qk:=/productdisplay
+1≤j≤k/parenleftbigg
+1−1
+2j/parenrightbigg
+,andQ(z) :=/productdisplay
+j≥1/parenleftBig
+1−z
+2j/parenrightBig
+. (9)
+In particular, Q(1) =Q∞. The variance was computed in [ 42] by a direct second-moment approach and
+theresultis
+V(Xn) =n(Ckps+̟kps(log2n))+O(log2n),
+13where̟kps(t)isagaina 1-periodic,zero-mean functionand themeanvalue Ckpsisgivenby ( L:= log2)
+Ckps=−28
+3L−39
+4+π2
+2L2+2
+L2−2Q∞
+L−2/summationdisplay
+ℓ≥1ℓ2ℓ
+(2ℓ−1)2+2
+L/summationdisplay
+ℓ≥11
+2ℓ−1
+−2
+L/summationdisplay
+ℓ≥3(−1)ℓ+1(ℓ−5)
+(ℓ+1)ℓ(ℓ−1)(2ℓ−1)
++2
+L/summationdisplay
+ℓ≥1(−1)ℓ2−(ℓ+1
+2)/parenleftBigg
+L(1−2−ℓ+1)/2−1
+1−2−ℓ−/summationdisplay
+r≥2(−1)r+1
+r(r−1)(2r+ℓ−1)/parenrightBigg
++/summationdisplay
+ℓ≥3/summationdisplay
+2≤r<ℓ/parenleftbiggℓ+1
+r/parenrightbiggQr−2Qℓ−r−1
+2ℓQℓ/summationdisplay
+j≥ℓ+11
+2j−1−2/bracketleftBig
+̟[1]
+1̟[2]
+2/bracketrightBig
+0−/bracketleftBig
+(̟[1]
+1)2/bracketrightBig
+0
++2/summationdisplay
+ℓ≥21
+2ℓQℓ/summationdisplay
+r≥0(−1)r2−(r+1
+2)
+QrQr+ℓ−2×
+×/braceleftBigg
+−/summationdisplay
+j≥11
+2j+r+ℓ+2−1/parenleftBigg
+2ℓ−ℓ−2+/summationdisplay
+2≤i<ℓ/parenleftbiggℓ+1
+i/parenrightbigg1
+2r+i−1−1/parenrightBigg
++1
+(1−2−ℓ−r)2+ℓ+1
+(1−21−ℓ−r)2−1
+L(1−21−ℓ−r)
+−/summationdisplay
+2≤j≤ℓ+1/parenleftbiggℓ+1
+j/parenrightbigg1
+2r+j−1−1+1
+L/summationdisplay
+1≤j≤ℓ+1/parenleftbiggℓ+1
+j/parenrightbigg1
+2r+j−1
++1
+L/summationdisplay
+0≤j≤ℓ+1/parenleftbiggℓ+1
+j/parenrightbigg/summationdisplay
+i≥1(−1)i
+(i+1)(2r+j+i−1)/bracerightBigg
+.
+Here[̟1̟2]0denotes the mean value of the function ̟1(t)̟0(t)over the unit interval. The long expres-
+sionobviouslyshowsthecomplexityoftheasymptoticprobl em.
+We show that this long expression can be largely simplified. B efore stating our result, we mention
+that the asymptotic normality of Xn(in the sense of convergence in distribution) was first prove d in [35]
+by a complex-analytic approach; for other approaches, see [ 59] (martingale difference), [ 31] (method of
+moments),[ 52] (contractionmethod).
+A new asymptoticapproximation to V(Xn).Define
+G2(ω) =Q∞/summationdisplay
+j,h,ℓ≥0(−1)j2−(j+1
+2)+j(ω−2)
+QjQhQℓ2h+ℓϕ(ω;2−j−h+2−j−ℓ), (10)
+where for 0<ℜ(ω)<3andx>0
+ϕ(ω;x) :=/integraldisplay∞
+0sω−1
+(s+1)(s+x)2ds,
+which, bytherelation
+/integraldisplay∞
+0sω−1
+s+1ds=π
+sin(πω)= Γ(ω)Γ(1−ω) (0<ℜ(ω)<1),
+14can berepresented as
+ϕ(ω;x) =
+
+π(1+xω−2((ω−2)ξ+1−ω)
+(x−1)2sin(πω),ifx/\⌉}atio\slash= 1;
+π(ω−1)(ω−2)
+2sin(πω), ifx= 1.
+The last expression provides indeed a meromorphic continua tion ofϕ(ω;x)into the whole complex ω-
+planewhenever x>0. In particular,
+ϕ(2;x) :=
+
+x−logx−1
+(x−1)2,ifx/\⌉}atio\slash= 1;
+1
+2, ifx= 1.
+Theorem 2.1. Thevarianceofthetotalpath-lengthofrandomDSTs of nnodessatisfies
+V(Xn) =n(Ckps+̟kps(log2n))+O(1), (11)
+where
+Ckps=G2(2)
+log2=Q∞
+log2/summationdisplay
+j,h,ℓ≥0(−1)j2−(j+1
+2)
+QjQhQℓ2h+ℓϕ(2;2−j−h+2−j−ℓ),
+and̟kpshastheFourierseries expansion
+̟kps(t) =1
+log2/summationdisplay
+k∈Z\{0}G2(2+χk)
+Γ(2+χk)e2kπit,
+which isabsolutelyconvergent.
+One can derive more precise asymptotic expansions for V(Xn)by the same approach we use. We
+content ourselveswith( 11)forconvenienceofpresentation.
+Notethat
+G2(2+χk)
+Γ(2+χk)= Γ(−1−χk)Q∞/summationdisplay
+j,h,ℓ≥0(−1)j2−(j+1
+2)
+QjQhQℓ2h+ℓλk(2−j−h+2−j−ℓ),
+where
+λk(t) :=
+
+1−tχk(1+χk(1−t))
+(1−t)2,ift/\⌉}atio\slash= 1;
+χk(χk−1)
+2, ift= 1.
+Thus theFourierseries isabsolutelyconvergent bytheorde restimate(see[ 18])
+|Γ(c+it)|=O/parenleftbig
+|t|c−1/2e−π|t|/2/parenrightbig
+(|t| → ∞). (12)
+Numerically, Ckps≈0.266003645405936 ..., in accordance withthat givenin [ 42]. Also|̟kps(t)| ≤
+1.9×10−5.
+15Sketch of our approach. Following the discussions in Introduction, we first prove th at the Poisson-
+Charlierexpansionforthemeanandthatforthesecondmomen tarenotonlyidentitiesbutalsoasymptotic
+expansions. For that purpose, it proves very useful to intro duce the following notion, which we term JS-
+admissible functions (following the survey paper [ 35]). This is reminiscent of the classical H-admissible
+(dueto Hayman)orHS-admissible(duetoHarris-Schoenfeld ) functions;see[ 28,§VIII.5].
+Once we provethe asymptoticnature of the Poisson-Charlier expansionsfor the mean and the second
+moment,itremains,accordingagaintothediscussionsinIn troduction,toderivemorepreciseasymptotics
+for the function ˜V(as defined in ( 4)), for which we will use first the Laplace transforms, normal ize the
+Laplacetransformproperly,andthenapplytheMellintrans form. Suchanapproachwillturnouttobevery
+effective and readily applicable to more general cases such as bucket DSTs, which is discussed in details
+in the next section. The approach parallels closely in essen ce that introduced by Flajolet and Richmond
+in [24], which starts from the ordinary generating function, foll owed by an Euler transform, a proper
+normalization and the Mellin transform, and then conclude b y singularity analysis; see also [ 10]. The
+path we take, however, offers additional operational advan tages, as will be clear later. See Figure 7for a
+diagrammaticillustrationofthetwo analyticapproaches.
+2.3 Analytic de-Poissonizationand JS-admissibility
+Thefundamentaldifferential-functionalequationsforth eanalysisofrandomDSTs isoftheform
+˜f(z)+˜f′(z) = 2˜f(z/2)+ ˜g(z),
+with suitably given initial value f(0)and˜g. For such functions, it turns out that the asymptotic nature
+of the Poisson-Charlier expansions for the coefficients (or de-Poissonization ) can be justified in a rather
+systematicway bytheintroductionofthenotionofJS-admis siblefunctions.
+Here and throughout this paper , the generic symbol ε∈(0,1)always represents an arbitrarily small
+constantwhosevalueisimmaterialand maydifferfromoneoc currence toanother.
+Definition 1. An entire function ˜fis said to be JS-admissible, denoted by ˜f∈JS, if the following two
+conditionsholdfor |z| ≥1.
+(I)There existα,β∈Rsuchthatuniformlyfor |arg(z)| ≤ε,
+˜f(z) =O/parenleftbig
+|z|α(log+|z|)β/parenrightbig
+,
+wherelog+x:= log(1+x).
+(O)Uniformlyfor ε≤ |arg(z)| ≤π,
+f(z) :=ez˜f(z) =O/parenleftbig
+e(1−ε)|z|/parenrightbig
+.
+For convenience, we also write ˜f∈JSα,βto indicate the growth order of ˜finside the sector
+|arg(z)| ≤ε.
+Note that if ˜fsatisfies condition (I), then, by Cauchy’s integral representation for derivative s (or by
+Ritt’stheorem;see[ 54, Ch. 1,§4.3]), wehave,
+˜f(k)(z) =O/parenleftbigg/contintegraldisplay
+|w−z|=ε|z||w|α|(log+|w|)β
+|w−z|k+1|dw|/parenrightbigg
+=O/parenleftbig
+|z|α−k(log+|z|)β/parenrightbig
+.
+16Proposition 2.2. Assume˜f∈JSα,β. Letf(z) :=ez˜f(z). Then the Poisson-Charlier expansion ( 1) of
+f(n)(0)is alsoanasymptoticexpansioninthesensethat
+an:=f(n)(0) =n![zn]f(z) =n![zn]ez˜f(z)
+=/summationdisplay
+0≤j<2k˜f(j)(n)
+j!τj(n)+O/parenleftBig
+nα−k(logn)β/parenrightBig
+,
+fork= 1,2,....
+Proof.(Sketch) Starting from Cauchy’s integral formula for the co efficients, the lemma follows from a
+standard application of the saddle-point method. Roughly, condition (O)guarantees that the integral over
+thecirclewithradius nandargumentsatisfying ε≤ |arg(z)| ≤πisnegligible,whilecondition (I)implies
+smoothestimatesforall derivatives(and thuserrorterms) .
+Thepolynomialgrowthofcondition (I)issufficientforallouruses;see[ 36]formoregeneralversions.
+The real advantage of introducing admissibility is that it o pens the possibility of developing closure
+properties as wenowdiscuss.
+Lemma 2.3. Letmbea nonnegativeintegerand α∈(0,1).
+(i)zm,e−αz∈JS.
+(ii) If˜f∈JS, then˜f(αz),zm˜f∈JS.
+(iii) If˜f,˜g∈JS, then˜f+ ˜g∈JS.
+(iv) If˜f∈JS, thentheproduct ˜P˜f∈JS, where˜Pisa polynomialof z.
+(v) If˜f,˜g∈JS, then˜h∈JS, where˜h(z) :=˜f(αz)˜g((1−α)z).
+(vi) If˜f∈JS, then˜f′∈JS, andthus ˜f(m)∈JS.
+Proof.Straightforwardand omitted.
+Specific toourneed fortheanalysisofDSTs isthefollowingt ransferprinciple.
+Proposition2.4. Let˜f(z)and˜g(z)beentirefunctionssatisfying
+˜f(z)+˜f′(z) = 2˜f(z/2)+ ˜g(z), (13)
+withf(0) = 0. Then
+˜g∈JSif andonlyif ˜f∈JS.
+Proof.Assume˜g∈JS. Wecheck firstthecondition (O)for˜f. Letf(z) :=ez˜f(z)andg(z) :=ez˜g(z).
+By (13),
+f′(z) = 2ez/2f(z/2)+g(z).
+Consequently,since f(0) = 0,
+f(z) =/integraldisplayz
+0/parenleftbig
+2et/2f(t/2)+g(t)/parenrightbig
+dt=z/integraldisplay1
+0/parenleftbig
+2etz/2f(tz/2)+g(tz)/parenrightbig
+dt. (14)
+17Nowdefine
+B(r) := max
+z∈Cr,ε|f(z)|,
+where
+Cr,ε:={z:|z| ≤r,ε≤ |arg(z)| ≤π},(r≥0;0<ε<π/ 2).
+Then, by( 14), wehave
+B(r)≤r/integraldisplay1
+0/parenleftbig
+2etrcos(ε)/2B(tr/2)+|g(tr)|/parenrightbig
+dt
+=/integraldisplayr
+0/parenleftbig
+2etcos(ε)/2B(t/2)+O/parenleftbig
+e(1−ε)t/parenrightbig/parenrightbig
+dt
+≤Cercos(ε)/2B(r/2)+O/parenleftbig
+e(1−ε)r/parenrightbig
+,
+whereC= 4/cosε>1. Thissuggeststhatwedefineamajorantfunction K(r)ofB(r)byK(r) =O(1)
+forr≤1andforr≥1
+K(r) =Cercos(ε)/2K(r/2)+h(r),
+wherehis an entire function satisfying h(r) =O(1)forr≤1andh(r) =O/parenleftbig
+e(1−ε)r/parenrightbig
+forr≥1. Let
+˜K(r) :=e−rcos(ε)K(r)and˜h(r) :=e−rcos(ε)h(r). Thensince cosε−1+ε>0forε∈(0,1),weobtain
+˜K(r) =C˜K(r/2)+˜h(r),˜h(r) =O(1).
+Thus if we choose m=⌈log2r⌉such that 2m≥rand iteratemtimes the functional equation, then we
+obtaintheestimate
+˜K(r) =/summationdisplay
+0≤k≤mCk˜h(r/2k)+Cm+1˜K(r/2m+1)
+=O
+/summationdisplay
+r/2k>1Ck+Cm
+
+=O/parenleftbig
+rlog2C/parenrightbig
+.
+Thus
+B(r) =O/parenleftbig
+rlog2Cercosε/parenrightbig
+.
+which establishescondition (O).
+Ourprooffor ˜fsatisfying (I)proceeds in asimilarmannerand startsagainfrom ( 14)butoftheform
+˜f(z) =z/integraldisplay1
+0e−(1−t)z/parenleftBig
+2˜f(tz/2)+ ˜g(tz)/parenrightBig
+dt.
+Now,define
+˜B(r) := max
+z∈Sr,ε|˜f(z)|,
+where
+Sr,ε:={z:|z| ≤r,|arg(z)| ≤ε},(r≥0;0<ε<π/ 2).
+18Then
+˜B(r)≤r/integraldisplay1
+0e−(1−t)rcosε/parenleftBig
+2˜B(tr/2)+|˜g(tr)|/parenrightBig
+dt
+=/integraldisplayr
+1/parenleftBig
+2e−(r−t)cosε˜B(t/2)+O/parenleftbig
+e−(r−t)cosεtα(log+t)β/parenrightbig/parenrightBig
+dt+O(1)
+≤C˜B(r/2)+O/parenleftbig
+rα(log+r)β+1/parenrightbig
+,
+whereC= 2/cosε>2. Thesamemajorizationargumentused abovefor (O)then leadsto
+˜B(r) =
+
+O(rlog2C), ifα<log2C;
+O(rlog2C(log+r)β+1),ifα= log2C;
+O/parenleftbig
+rα(log+r)β/parenrightbig
+,ifα>log2C.
+Thisproves (I)for˜f.
+ThenecessitypartfollowstriviallyfromLemma 2.3.
+The estimates we derived of asymptotic-transfer type are in deed over-pessimistic when 1≤α≤
+log2C, but they are sufficient for our use. The true orders are those withε→0, which can be proved by
+theLaplace-Mellin-de-Poissonizationapproach weuselat er.
+Lemma2.3and Proposition 2.4provide very effective tools for justifying the de-Poisson ization of
+functionssatisfyingtheequation( 13), whichisoften carried outthroughtheuseoftheincreasin g-domain
+argument(see[ 36]). Thelatterargumentisalsoinductiveinnatureandsimil artotheonewearedeveloping
+here, althoughit isless“mechanical”and lesssystematic.
+2.4 Generating functions andintegraltransforms
+Since our approach is purely analytic and relies heavily on g enerating functions, we first derive in this
+subsection the differential-functional equations we will be working on later. Then we apply the de-
+PoissonizationtoolswedevelopedtothePoissongeneratin gfunctionsofthemeanandthesecondmoment
+and justify the asymptotic nature of the corresponding Pois son-Charlier expansions. Then we sketch the
+asymptotictoolswewillfollowbasedon theLaplaceand Mell intransforms.
+Generating functions. In terms of the moment generating function Mn(y) :=E(eXny), the recurrence
+(6)translates into
+Mn+1(y) =eny2−n/summationdisplay
+0≤j≤n/parenleftbiggn
+j/parenrightbigg
+Mj(y)Mn−j(y),(n≥0), (15)
+withM0(y) = 1.
+Nowconsiderthebivariateexponentialgeneratingfunctio n
+F(z,y) :=/summationdisplay
+n≥0Mn(y)
+n!zn.
+Then by ( 15),
+∂
+∂zF(z,y) =F/parenleftbiggeyz
+2,y/parenrightbigg2
+,
+19and thePoissongeneratingfunction ˜F(z,y) :=e−zF(z,y)satisfiesthedifferential-functionalequation
+˜F(z,y)+∂
+∂z˜F(z,y) =e(ey−1)z˜F/parenleftbiggeyz
+2,y/parenrightbigg2
+, (16)
+with˜F(0,y) = 1. No exact solution of such a nonlinear differential equatio n is available; see [ 35] for an
+asymptoticapproximationto ˜Fforynearunity.
+Mean andsecond moment. Let now
+˜F(z,y) :=/summationdisplay
+m≥0˜fm(z)
+m!ym,
+where˜fm(z)denotesthePoissongeneratingfunctionof E(Xm
+n). Then wededucefrom ( 16)that
+˜f1(z)+˜f′
+1(z) = 2˜f1(z/2)+z, (17)
+˜f2(z)+˜f′
+2(z) = 2˜f2(z/2)+2˜f1(z/2)2+4z˜f1(z/2)+2z˜f′
+1(z/2)+z+z2, (18)
+withtheinitialconditions ˜f1(0) =˜f2(0) = 0.
+Proposition 2.5. The Poisson-Charlier expansion for the mean and that for the second moment are both
+asymptoticexpansions
+E(Xn) =/summationdisplay
+0≤j<2k˜f(j)
+1(n)
+j!τj(n)+O/parenleftbig
+n−k+1/parenrightbig
+,
+E(X2
+n) =/summationdisplay
+0≤j<2k˜f(j)
+2(n)
+j!τj(n)+O/parenleftbig
+n−k+2(logn)2/parenrightbig
+,
+fork= 1,2,....
+Proof.(Sketch)ByLemma 2.3andProposition 2.4,weseethatboth ˜f1,˜f2∈JS,andthuswecanapply
+Proposition 2.2. Indeed the proof of Proposition 2.4provides already crude bounds for the growth order
+of˜f1,˜f2. The more precise estimates ˜f1(z)≍ |z||logz|and˜f2(z)≍ |z|2|logz|2forzinside the sector
+{z:|arg(z)| ≤ε}willbeprovidedlaterinthenexttwosubsections.
+AnasymptoticapproachbasedonLaplaceandMellintransfor ms.Oncethede-Poissonizationsteps
+are justified, all that remains for the proof of Theorem 2.1is to derive more precise asymptotic approxi-
+mations to ˜f1and˜V(as defined in ( 4)). The approach we use beginswith a more precise characteri zation
+of˜f1(z). Both˜f1and˜Vsatisfyadifferential-functionalequationoftheform
+˜f(z)+˜f′(z) = 2˜f(z/2)+ ˜g(z),
+with theinitialcondition ˜f(0) = 0. To derivetheasymptoticsof ˜ffor large complex z, we proceed along
+thefollowingprincipalsteps;seealso[ 10].
+Laplacetransform :TheLaplacetransform of ˜fsatisfies
+(s+1)L[˜f;s] = 4L[˜f;2s]+L[˜g;s], (19)
+which existsand defines an analyticfunctionif ˜ggrowsat mostpolynomiallyforlarge |z|.
+20Normalizingfactor :Dividingbothsidesof( 19)byQ(−2s) =/producttext
+j≥0(1+s/2j)givesafunctionalequation
+oftheform
+¯L[˜f;s] = 4¯L[˜f;2s]+L[˜g;s]
+Q(−2s),
+where¯L[˜f;s] :=L[˜f;s]/Q(−s).
+Mellintransform :TheMellintransformof ¯Lthensatisfies
+M[¯L[˜f1;s];ω] =1
+1−22−ωM/bracketleftbiggL[˜g;s]
+Q(−2s);ω/bracketrightbigg
+.
+Inverting theprocess. We first derive the local behavior of ˜L[˜f;s]for smallsby the Mellin inversion
+(often by calculus of residues after justification of analyt ic properties), and then the asymptotic
+behaviorof ˜f(z)forlargezisderivedbytheLaplaceinversion,similarto singularity analysis.
+2.5 Expected internal path-length of random DSTs
+We considerin detailsinthissubsectiontheexpectedvalue µn:=E(Xn)ofthetotalinternal path-length,
+paving the way for the asymptotic analysis of the variance. S tarting from either the equation ( 17) or the
+recurrence
+µn+1= 21−n/summationdisplay
+0≤j≤n/parenleftbiggn
+j/parenrightbigg
+µj+n(n≥0)
+withµ0:= 0,thereareseveralapproachestotheasymptoticsof µn. Wewillbrieflydescribetheoneusing
+integral representation of finite differences (or Rice’s in tegrals) and then present the Laplace and Mellin
+transforms we will use, which, as will become clear, is essen tially the Flajolet-Richmond approach (see
+[24]).
+Rice’s integral representation. By (17), wehave, with ˜µn:=n![zn]˜f1(z),
+˜µn+1=−/parenleftbig
+1−21−n/parenrightbig
+˜µn(n≥0),
+with˜µ0= 0,which byiterationyields
+˜µn= (−1)nQn−2, Qn:=/productdisplay
+1≤j≤n/parenleftbig
+1−2−j/parenrightbig
+. (20)
+Thus byRice’s formula([ 27])
+µn:=E(Xn) =/summationdisplay
+2≤j≤n/parenleftbiggn
+j/parenrightbigg
+˜µj
+=1
+2πi/integraldisplay
+(3
+2)Γ(n+1)Γ(−s)
+Γ(n+1−s)·Q(1)
+(1−21−s)Q(21−s)ds,
+where the integration path/integraltext
+(c)is along the vertical line with real part equal to candQis defined in ( 9).
+We thenobtain( 7)by standardarguments;see[ 26]or[50]fordetails.
+This approach readily gives the approximation ( 7) for the mean and can be refined to obtain a full
+asymptoticexpansion. However,itsextensiontothevarian cebecomesextremelymessy,asshownin[ 42].
+21Laplace transform. We first show that the asymptotics of ˜f1(z)can be derived through a direct use of
+the Laplace and Mellin transforms, which relies on several a d hoc steps that are not easily extended. A
+moregeneral procedurewillbedevelopedbelow.
+By (17), weseethattheLaplacetransformof f1(z)satisfies thefunctionalequation
+(s+1)L[˜f1;s] = 4L[˜f1;2s]+s−2, (21)
+which existsand isanalyticin C\(−∞,0].
+By dividingbothsides by s+1and byiteration,weget
+L[˜f1;s] =1
+s2/summationdisplay
+j≥01
+(s+1)···(2js+1). (22)
+On theotherhand, from( 20), wehave
+L[˜f1;s] =/integraldisplay∞
+0e−sz/summationdisplay
+n≥0˜µn
+n!zndz
+=/summationdisplay
+n≥0(−1)nQns−n−3.
+Thisimpliestheidentity/summationdisplay
+n≥0(−1)nQn
+sn+1=/summationdisplay
+j≥01
+(s+1)···(2js+1).
+However,neitherform isuseful forourasymptoticpurpose.
+Nowbypartial fraction expansion,weobtain
+1
+(s+1)···(2js+1)=/summationdisplay
+0≤ℓ≤j(−1)j−ℓ2−(j−ℓ+1
+2)−ℓ
+(s+2−ℓ)QℓQj−ℓ.
+Thus
+L[˜f1;s] =1
+s2/summationdisplay
+j≥0/summationdisplay
+0≤ℓ≤j(−1)j−ℓ2−(j−ℓ+1
+2)−ℓ
+(s+2−ℓ)QℓQj−ℓ
+=1
+s2/summationdisplay
+ℓ≥01
+Qℓ(2ℓs+1)/summationdisplay
+j≥0(−1)j2−(j+1
+2)
+Qj.
+Notethat/summationdisplay
+j≥02js
+(s+1)···(2js+1)= 1.
+By theEuleridentity
+/summationdisplay
+j≥0q(j
+2)zj
+(1−q)···(1−qj)=/productdisplay
+k≥0/parenleftbig
+1+qkz/parenrightbig
+,
+wesee that
+/summationdisplay
+j≥0(−1)j2−(j+1
+2)
+Qj=Q(1) =Q∞≈0.28878809...
+22Thisgives
+L[˜f1;s] =Q∞
+s2/summationdisplay
+ℓ≥01
+Qℓ(2ℓs+1),
+and then
+˜f1(z) =Q∞/summationdisplay
+ℓ≥02ℓ
+Qℓ/parenleftBig
+e−z/2ℓ−1+z
+2ℓ/parenrightBig
+. (23)
+Consequently,
+µn=Q∞/summationdisplay
+ℓ≥02ℓ
+Qℓ/parenleftbig
+(1−2−ℓ)n−1+2−ℓn/parenrightbig
+.
+Asymptotically,wehave, by( 23)and theidentity
+1
+Q(z)=/productdisplay
+j≥11
+1−z/2j=/summationdisplay
+ℓ≥0zℓ
+Qℓ2ℓ(|z|<2), (24)
+theMellinintegralrepresentation
+˜f1(z) =1
+2πi/integraldisplay
+(−3/2)Q(1)Γ(s)z−s
+(1−2s+1)Q(2s+1)ds,
+from whichwederivetheasymptoticapproximation
+˜f1(z) = (z+1)log2z+z/parenleftbiggγ−1
+log2+1
+2−c1+̟1(log2z)/parenrightbigg
++O(1), (25)
+uniformlyfor |z| → ∞and|arg(z)| ≤π/2−ε,where̟1isgivenin( 8). (Asusual,weusetheasymptotic
+estimate( 12)fortheGammafunction.)
+LaplaceandMellintransforms. Wenowre-dotheanalysisfor ˜f1(z)inamoregeneralwaythatcanbe
+easily extendedtoothercases.
+Weagainstart from( 21)and consider
+¯L[˜f1;s] :=L[˜f1;s]
+Q(−s),
+whereQ(z)is defined in( 9). Dividingbothsidesof( 21)byQ(−2s)yields
+¯L[˜f1;s] = 4¯L[˜f1;2s]+1
+Q(−2s)s2. (26)
+WenowapplytheMellintransform. Notethatwehave,bythefa ctthatX0=X1= 0andtheproofof
+Proposition 2.4,
+˜f1(z) =/braceleftBigg
+O(z2),ifz→0+;
+O(z1+ε),ifz→ ∞.
+Then
+L[˜f1;s] =/braceleftBigg
+O(s−2−ε),ass→0+;
+O(s−3),ass→ ∞.
+23On theotherhand, bytheMellintransform,
+logQ(−2s) =/summationdisplay
+j≥0log/parenleftBig
+1+s
+2j/parenrightBig
+=1
+2πi/integraldisplay
+(−1
+2)πs−w
+(1−2w)wsinπwdw
+=(logs)2
+2log2+logs
+2+/summationdisplay
+k∈Zqks−χk+O(|s|−1) (27)
+uniformlyfor |s| → ∞and|arg(s)| ≤π−ε,whereχk:= 2kπi/log2,
+q0=log2
+12+π2
+6log2
+and
+qk=1
+2ksinh(2kπ/log2)(k/\⌉}atio\slash= 0).
+Thisasymptoticexpansion,togetherwiththeTaylorexpans ion
+Q(−2s) = 1+O(|s|),(|s| →0),
+givesriseto
+¯L[˜f1;s] =/braceleftBigg
+O(s−2−ε),ass→0+;
+O(s−M),ass→ ∞,
+whereM >0is an arbitrary real number. Consequently, the Mellin trans form of ¯L[˜f1;s], denoted by
+M[¯L;ω],existsinthehalf-plane ℜ(ω)≥2+ε. ThenbyapplyingtheMellintransformto( 26),weobtain
+M[¯L;ω] =G1(ω)
+1−22−ω,(ℜ(ω)>2),
+where
+G1(ω) :=/integraldisplay∞
+0sω−3
+Q(−2s)ds=πQ(2ω−2)
+Q(1)sinπω=Q(2ω−2)
+Q(1)Γ(ω)Γ(1−ω), (28)
+forℜ(ω)>2;see[24].
+Inverse Mellin and inverse Laplace transforms. We can now apply successively the inverse Mellin
+and then Laplace transforms to derive the asymptotics of ˜f1(z). Observe that G1(ω)has a simple pole at
+ω= 2. By (28) orProposition5 in[ 22], weobtain
+|G1(c+it)|=O/parenleftbig
+e−(π−ε)|t|/parenrightbig
+,
+forlarge|t|andc∈R. Then by thecalculus ofresidues,
+¯L[˜f1;s] =1
+s2log21
+s+1
+s2
+1
+2−c1+1
+log2/summationdisplay
+k∈Z\{0}G1(2+χk)s−χk
++O(|s|−1),
+24uniformlyfor |s| →0and|arg(s)| ≤π−ε. Usingtheexpansion
+Q(−s) = 1+s+(|s|2) (|s| ∼0),
+wesee that
+L[˜f1;s] =1+s
+s2log21
+s+1
+s2
+1
+2−c1+1
+log2/summationdisplay
+k∈Z\{0}G1(2+χk)s−χk
++O(|s|−1),
+uniformlyfor |s| →0and|arg(s)| ≤π−ε.
+Finally, we consider the inverse Laplace transform. The fol lowing simpleresult is very useful for our
+purposes.
+Proposition 2.6. Let˜f(z)be a function whose Laplace transform exists and is analytic inC\(−∞,0].
+Assumethat
+L[˜f;s] =
+
+O(|s|−α|log|s+1||m),
+cs−ω(−logs)m,
+o(|s|−α|log|s+1||m),(29)
+uniformlyfor |s| →0and|arg(s)| ≤π−ε, whereα∈R,ω∈Candm= 0,1,.... IfL[˜f;s]satisfies
+|L[˜f;s]|=O/parenleftbig
+|s|−1−ε/parenrightbig
+, (30)
+as|s| → ∞in|arg(s)| ≤π−ε, then
+˜f(z) =
+
+O(|z|α−1(log|z|)m),
+czω−1/summationdisplay
+0≤j≤m/parenleftbiggm
+j/parenrightbigg
+(logz)m−j∂j
+∂ωj1
+Γ(ω),
+o(|z|α−1(log|z|)m),
+respectively, where the O-ando-termsholduniformlyfor |z| → ∞and|arg(z)| ≤π/2−ε.
+Proof.Let˜L(s) =L[˜f;s]. Then bytheinverseLaplacetransform,
+˜f(z) =1
+2πi/integraldisplay
+(1)ezs˜L(s)ds=1
+2πi/integraldisplay
+Hezs˜L(s)ds,
+whereHis the Hankel contour consisting of the two rays te±iε±i/|z|,−∞< t≤0and the semicircle
+exp(iϕ)/|z|,−π/2≤ϕ≤π/2;seeFigure 6.
+Assumefrom nowon |z|is sufficientlylargeand liesinthesectorwith |arg(z)| ≤π/2−ε. Weprove
+only theO-case, the other two cases being similar. For simplicity, we consider only the case m= 0, the
+othercases beingeasily extended.
+Wesplittheaboveintegralalong Hintotwoparts
+1
+2πi/integraldisplay
+Hezs˜L(s)ds=1
+2πi/integraldisplay
+H>ezs˜L(s)ds+1
+2πi/integraldisplay
+H⊃ezs˜L(s)ds,
+whereH>comprises the two rays teiε±i/|z|,−∞< t≤ −TwithT >1a fixed constant and H⊃
+represents theremainingcontour.
+25ℜ(s)ℑ(s)
+εH
+1
+|z|
+Figure6: The contour H.
+Theintegralalong H>iseasilyestimated
+1
+2πi/integraldisplay
+H>ezs˜L(s)ds=O/parenleftbigg/integraldisplay−T
+−∞eℜ(|z|eiarg(z)(teiε+i/|z|))|t|−1−εdt/parenrightbigg
+=O/parenleftbigg/integraldisplay∞
+Tt−1−εe−|z|tcos(arg(z)+ε)dt/parenrightbigg
+=O/parenleftbig
+|z|εe−c|z|T/parenrightbig
+,
+theO-term holding uniformly for |z| → ∞provided that |arg(z)|+ε < π/2, wherec >0is a suitable
+constant.
+Forthesecond integral,weuse( 29). Then theintegralalongthesemicircleis boundedas follo ws.
+1
+2π|z|/integraldisplayπ/2
+−π/2ezeiθ/|z|+iθ˜L(eiθ/|z|)dθ=O/parenleftbig
+|z|α−1/parenrightbig
+,
+uniformlyfor |z| → ∞. Fortheremainingpart t±i/|z|,−T <t≤0,wehave
+1
+2πi/integraldisplay0
+−Tez(t±i/|z|)˜L(t±i/|z|)dt=O/parenleftbigg
+|z|α/integraldisplay0
+−Tec|z|t
+(|z|2t2+1)α/2dt/parenrightbigg
+=O/parenleftbigg
+|z|α−1/integraldisplay∞
+0e−cu
+(u2+1)α/2du/parenrightbigg
+=O/parenleftbig
+|z|α−1/parenrightbig
+,
+uniformlyfor |z| → ∞,wherec>0isasuitableconstant. Thiscompletestheproof.
+Note that the inverse Laplace transform of s−2log(1/s)iszlogz−(1−γ)z. This, together with a
+combineduseofProposition 2.6, leadsto ( 25).
+Thejustificationoftheestimate( 30)is easilyperformed byusingtherelation( 31)below.
+TheFlajolet-Richmondapproach[ 24].InsteadofthePoissongeneratingfunction,thisapproachs tarts
+from theordinarygeneratingfunction A(z) :=/summationtext
+nµnzn.
+– Then theEulertransform2
+ˆA(s) :=1
+s+1A/parenleftbigg1
+s+1/parenrightbigg
+2Forabettercomparisonwith theapproachweuse, our ˆAdiffersfromthe usualEulertransformbyafactorof s.
+26satisfies
+(s+1)ˆA(s) = 4ˆA(2s)+s−2,
+identicalto ( 21).
+– Thenormalizedfunction ¯A(s) :=ˆA(s)/Q(−s)satisfies
+¯A(s) = 4¯A(2s)+1
+s2Q(−2s),
+againidenticalto ( 26).
+– TheMellintransformof ¯Asatisfies(ℜ(ω)>2)
+M[¯A;ω] =G1(ω)
+1−22−ω,
+whereG1(ω)isas defined in ( 28).
+Theninverttheprocess byconsideringfirst theMellininver sion,derivingasymptoticsof
+¯A(s) =1
+2πi/integraldisplay
+(5/2)s−ωG1(ω)
+1−22−ωdω,
+ass→0inC. Then deduceasymptoticsof
+A(z) =1
+zˆA/parenleftbigg1
+z−1/parenrightbigg
+,
+asz→1. Finally,apply singularityanalysis(see[ 23])toconcludetheasymptoticsof µn.
+EGF
+f(z)Laplace
+transform
+ofe−zf(z)=Euler
+trans-
+form of
+A(z)OGF
+A(z)
+Laplace
+Q(−s)Euler
+Q(−s)
+Mellin
+trans-
+formasymptotics
+ofLaplace
+Q(−s)asymptotics
+of˜f(z)as
+|z| → ∞
+de-Poi by
+saddle-
+pointasymptotics
+ofEuler
+Q(−s)asymptotics
+ofA(z)
+asz∼1
+singularity
+analysis
+Figure 7: A diagrammaticcomparison of the major steps used in the Lapl ace-Mellin (left-half) approach
+and the Flajolet-Richmond (right-half) approach. Here EGF denotes “exponential generating function”,
+OGF standsfor“ordinarygeneratingfunction”andde-Poi is theabbreviationforde-Poissonization.
+Thecrucialreasonwhythetwoapproachesareidenticalatce rtainstepsisthattheLaplacetransformof
+aPoissongeneratingfunctionisessentiallyequaltotheEu lertransformofanordinarygeneratingfunction;
+27orformally,
+/integraldisplay∞
+0e−sz/summationdisplay
+n≥0an
+n!zndz=/summationdisplay
+n≥0an(s+1)−n−1
+=1
+s+1A/parenleftbigg1
+s+1/parenrightbigg
+. (31)
+Thus the simple result in Proposition 2.6closely parallels that in singularity analysis. While iden tical at
+certainsteps,thetwoapproachesdivergeintheirfinaltrea tmentofthecoefficients,andthedistinctionhere
+is typically that between the saddle-point method and the si ngularity analysis, a situation reminiscent of
+theusebeforeand afterLagrange’s inversionformula;seef orinstance[ 28].
+The relation ( 31) implies that the order estimate ( 30) for the Laplace transform at infinity can be
+easily justified for all the generating functions we conside r in this paper since A(0) = 0, implying that
+A(z) =O(|z|)as|z| →0.
+This comparison also suggests the possibility of developin g de-Poissonization tools by singularity
+analysis,which willbeinvestigatedindetailselsewhere.
+2.6 Variance of the internal path-length
+In this section, weapply theLaplace-Mellin-de-Poissoniz ationapproach to the Poissonizedvariance with
+correction
+˜V(z) :=˜f2(z)−˜f1(z)2−z˜f′
+1(z)2,
+aiming at proving Theorem 2.1. The starting point of focusing on ˜Vinstead of on ˜f2removes all heavy
+cancellationsinvolvedwhendealingwiththevariance, ake ystep differingfromall previousapproaches.
+Laplace andMellintransform. Thefollowinglemmawillbeuseful.
+Lemma 2.7. If/braceleftbigg˜f1(z)+˜f′
+1(z) = 2˜f1(z/2)+˜h1(z),
+˜f2(z)+˜f′
+2(z) = 2˜f2(z/2)+˜h2(z),
+whereallfunctionsareentirewith ˜f1(0) =˜f2(0) = 0,thenthefunction ˜V(z) :=˜f2(z)−˜f1(z)2−z˜f′
+1(z)2
+satisfies
+˜V(z)+˜V′(z) = 2˜V(z/2)+ ˜g(z),
+with˜V(0) = 0, where
+˜g(z) =z˜f′′
+1(z)2+˜h2(z)−˜h1(z)2−z˜h′
+1(z)2−4˜h1(z)˜f1(z/2)−2z˜h′
+1(z)˜f′
+1(z/2)−2˜f1(z/2)2.
+Proof.Straightforwardand omitted.
+Byusingthedifferential-functionalequations( 17)and(18)for˜f1(z)and˜f2(z),wesee,byLemma 2.7,
+that
+˜V(z)+˜V′(z) = 2˜V(z/2)+z˜f′′
+1(z)2, (32)
+with˜V(0) = 0.
+Before applyingthe integraltransforms,we need roughesti mates of ˜V(z)nearz= 0andz=∞. We
+have
+˜V(z) =/braceleftBigg
+O(z2),asz→0+;
+O(z1+ε),asz→ ∞.(33)
+28Theseestimatesfollowfrom
+z˜f′′
+1(z)2=/braceleftBigg
+O(|z|),as|z| →0;
+O(|z|−1),as|z| → ∞,(34)
+whichinturnresultfrom X0=X1= 0and(25)(bytheproofofcondition (I)ofProposition 2.4). Indeed,
+theproofthereshowsthat thesameboundsholduniformlyfor z∈Cwith|arg(z)| ≤π/2−ε.
+WenowapplytheLaplacetransformtobothsidesof( 32). First,observethattheLaplacetransformof
+˜V(z)existsand isanalyticin C\(−∞,0]. Then, by( 32),
+(s+1)L[˜V;s] = 4L[˜V;2s]+ ˜g⋆(s),
+where˜g⋆(s) :=L[z˜f′′
+1;s]. NextthenormalizedLaplacetransform
+¯L[˜V;s] :=L[˜V;s]
+Q(−s)
+satisfies
+¯L[˜V;s] = 4¯L[˜V;2s]+˜g⋆(s)
+Q(−2s).
+By (33), weobtain
+L[˜V;s] =/braceleftBigg
+O(s−2−ε),ass→0+;
+O(s−3),ass→ ∞.
+From this and the asymptotic expansion ( 27) ofQ(−2s), it follows that the Mellin transform of ¯L[˜V;s]
+existsinthehalf-plane ℜ(ω)≥2+ε. Consequently,
+M[¯L[˜V;s];ω] =G2(ω)
+1−22−ω,(ℜ(ω)>2),
+where
+G2(ω) :=M/bracketleftbigg˜g⋆(s)
+Q(−2s);ω/bracketrightbigg
+=/integraldisplay∞
+0sω−1
+Q(−2s)/integraldisplay∞
+0e−zsz˜f′′
+1(z)2dzds. (35)
+By (23), wehave
+z˜f′′
+1(z)2=Q2
+∞/summationdisplay
+h,ℓ≥01
+QhQℓ2h+ℓze−z/2h−z/2ℓ.
+Substitutingthisand thepartial fractionexpansion
+1
+Q(−2s)=1
+Q∞/summationdisplay
+j≥0(−1)j2−(j
+2)
+Qj(s+2−j),
+into(35), weobtain( 10).
+29Inverse Mellin and inverse Laplace transforms. For the Mellin inversion, we need more precise ana-
+lyticproperties of G2(ω). By (34), wededucethattheLaplacetransform ˜g⋆(s)ofz˜f′′
+1(z)2satisfies
+˜g⋆(s) =/braceleftBigg
+O(|logs|),as|s| →0;
+O(|s|−2),as|s| → ∞
+uniformlyinthecone |arg(s)| ≤π−ε. Thus,bytheasymptoticexpansion( 27)forQ(−2s)andProposi-
+tion5 in[ 22], wehave
+|G2(c+it)|=O/parenleftbig
+e−(π−ε)|t|/parenrightbig
+,
+for large|t|andc>0. Also theMellintransform G2of˜g⋆(s)/Q(−2s)existsin thehalf-plane ℜ(ω)>0.
+Consequently,bystandard calculusofresidues,
+¯L[˜V;s] =1
+log2/summationdisplay
+k∈ZG2(2+χk)s−2−χk+O(|s|−ε),
+uniformlyfor |s| →0and|arg(s)| ≤π−ε. Thisin turnyieldsthefollowingexpansionfor L[˜V;s]
+L[˜V;s] =1
+log2/summationdisplay
+k∈ZG2(2+χk)s−2−χk+1
+log2/summationdisplay
+k∈ZG2(2+χk)s−1−χk+O(|s|−ε),
+again uniformlyfor |s| →0and|arg(s)| ≤π−ε.
+Finally,standard Laplaceinversiongives
+˜V(z) =z
+log2/summationdisplay
+k∈ZG2(2+χk)
+Γ(2+χk)zχk+1
+log2/summationdisplay
+k∈ZG2(1+χk)
+Γ(1+χk)zχk+O(|z|ε−1), (36)
+uniformlyfor |z| → ∞and|arg(z)| ≤π/2−ε.
+Since˜f2(z) =˜V(z)+˜f1(z)2+z˜f′
+1(z)2, weseefrom( 36)and (25)that
+˜f2(z)≍˜f1(z)2≍ |z|2log2|z|(|arg(z)| ≤π/2−ε).
+ThisprovesProposition 2.5andTheorem 2.1bystraightforwardexpansion. Morerefinedcalculationsgi ve
+V(Xn) =˜V(n)−n
+2˜V′′(n)−n2
+2˜f′′
+1(n)2+O(n−1),
+the two terms following ˜V(n)being bothO(1)and periodic in nature. It is possible to further extend
+the same idea and derive a full asymptotic expansion, which h as also its identity nature; details will be
+presented ina futurepaper.
+3 BucketDigitalSearchTrees
+In this section, we extend the same approach to bucket digita l search trees ( b-DSTs) in which each node
+can hold up to bkeys. The constructionrule is the same as DSTs, except that k eys keep staying in a node
+aslongasitscapacityremainslessthan b;seeFigure 8forasimpleexamplewith b= 2. DSTscorrespond
+tob= 1.
+Notethat when b≥2we can distinguishtwo different types of total path-length : the totalpath-length
+ofall keys(summingthedistancebetween each keytotheroot overall keys),which willbereferred toas
+30thetotalkey-wisepath-length (KPL)andthetotalpath-lengthofallnodes(summingthedis tancebetween
+eachnodetotherootoverallnodes,regardlessofthenumber ofkeysineachnode),referredtoasthe total
+node-wisepath-length (NPL). When b= 1thetwototalpath-lengthscoincide. Forsimplicity,wewil luse
+KPL and NPL, dropping the collective adjective “total”. Whi le the expected values of both TPLs are of
+ordernlognunderthe sameindependent Bernoulli model, theirvariance s surprisinglyturn out to exhibit
+very differentbehavior;see Table 1.
+010111
+101011
+100001
+011011
+111110
+110111
+010011
+011110
+000100010111010111
+101011
+100001 011011100001
+111110
+110111011011
+010011
+011110 0001001 0
+1 1 0
+Figure 8: A2-DST with nine keys. The total key-wise path-length is equal to4×1+3×2 = 10and the
+totalnode-wisepath-lengthequals 2×1+3×2 = 8.
+3.1 Key-wise path-length (KPL)
+We assume the same independent Bernoulli model for the input strings. Let Xndenote the KPL in a
+randomb-DST builtfrom nrandomstings. Then bydefinitionand theindependencemodel assumption
+Xn+bd=XBn+X∗
+n−Bn+n,(n≥0) (37)
+with the initial conditions X0=···=Xb−1= 0. HereBn∼Binomial(n,1/2),Xnd=X∗
+n, and
+Xn,X∗
+n,Bnareindependent.
+Known and new results. Hubalek [ 30] showed, by the Flajolet-Richmond approach, that the mean
+satisfies
+E(Xn) = (n+b)log2n+n(c2+̟3(log2n))+c3+̟4(log2n)+O/parenleftbig
+n−1logn/parenrightbig
+,
+wherec2,c3are effectively computable constants and ̟3and̟4are very smooth periodic functions. He
+also provedthat thevarianceis asymptoticallylinear
+V(Xn) =n(Ch+̟h(log2n))+O((logn)2),
+whereChis expressedintermsofavery long,involvedexpressionand ̟hisaperiodicfunction.
+We improve this estimate by deriving a much simpler expressi on for the periodic function, including
+itsaverage value Ch. To stateourresult,wedefine thefollowingfunctions. Let
+˜g(z) :=/parenleftBigg/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+1(z)/parenrightBigg2
++z/parenleftBigg/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j+1)
+1(z)/parenrightBigg2
+−/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg/parenleftbigg/parenleftBig
+˜f2
+1(z)/parenrightBig(j)
++/parenleftBig
+z˜f′
+1(z)2/parenrightBig(j)/parenrightbigg
+.(38)
+31It is easilyseen that ˜g(z)isoftheform
+˜g(z) =/summationdisplay
+2≤i1,i2≤b˜gi1,i2˜f(i1)
+1(z)˜f(i2)
+1(z)+z/summationdisplay
+2≤i1,i2≤b+1˜g′
+i1,i2˜f(i1)
+1(z)˜f(i2)
+1(z), (39)
+where˜gi1,i2,˜g′
+i1,i2≥0are givenexplicitlyby
+˜gi1,i2=/parenleftbiggb
+i1/parenrightbigg/parenleftbiggb
+i2/parenrightbigg
+−/parenleftbiggb
+i1/parenrightbigg/parenleftbiggb−i1
+i2/parenrightbigg
+−(b−i1+1)/parenleftbiggb
+i1−1/parenrightbigg/parenleftbiggb−i1
+i2−1/parenrightbigg
+,
+˜g′
+i1,i2=/parenleftbiggb
+i1−1/parenrightbigg/parenleftbiggb
+i2−1/parenrightbigg
+−/parenleftbiggb
+i1−1/parenrightbigg/parenleftbiggb−i1+1
+i2−1/parenrightbigg
+,
+bothcoefficients beingsymmetricin i1andi2. Define
+G2(ω) =/integraldisplay∞
+0sω−1
+Q(−2s)b/integraldisplay∞
+0e−zs˜g(z)dzds,
+which iswell-defined for ℜ(ω)>0, as wewillseelater.
+Theorem 3.1. Thevarianceofthetotalkey-wise path-lengthofrandom b-DSTs ofnstringssatisfies
+V(Xn) =n(Ch+̟h(log2n))+O(1), (40)
+where
+Ch=G2(2)
+log2=1
+log2/integraldisplay∞
+0s
+Q(−2s)b/integraldisplay∞
+0e−zs˜g(z)dzds,
+and
+̟h(t) =1
+log2/summationdisplay
+k∈Z\{0}G2(2+χk)
+Γ(2+χk)e2kπit.
+By straightforward truncations, expansions and approxima tions, we obtain the following numerical
+valuesforb= 1,...,5.
+b1 2 3 4 5
+Ch0.266000.132600.090040.069580.05781
+Morepowerfulmeans areneeded to bedevelopedifmoredegree ofprecisionis required.
+Generating functions. From (37), it follows that the moment generating function Mn(y) :=E(eXny)
+can berecursivelycomputedbytherelation
+Mn+b(y) =eny
+2n/summationdisplay
+0≤j≤n/parenleftbiggn
+j/parenrightbigg
+Mj(y)Mn−j(y) (n≥0),
+withMn(y) = 1for0≤n < b. The bivariate exponential generating function F(z,y)then satisfies the
+equation
+∂b
+∂zbF(z,y) =F/parenleftbiggeyz
+2,y/parenrightbigg2
+,
+32withF(j)(0,y) = 1for0≤j <b, andwehavethenonlinearequationforthePoissongenerati ngfunction
+˜F(z,y) :=e−zF(z,y)
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜F(j)(z,y) =e(ey−1)z˜F/parenleftbiggeyz
+2,y/parenrightbigg2
+, (41)
+with˜F(0,y) = 1.
+From this form, the asymptotic analysis of the mean value and that of the variance proceed along
+exactly the same line we developed in the previous section. T hus we briefly sketch the principal steps of
+theanalysis,leavingthedetailstotheinterestedreader.
+The expected value of Xn.From (41), we derive the following differential-functional equati on for the
+Poissongenerating functionofthemean
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+1(z) = 2˜f1(z/2)+z,
+withtheinitialconditions ˜f(j)
+1(0) = 0for0≤j <b.
+Before applying the Laplace-Mellin approach, we need first a transfer-type result similar to Proposi-
+tion2.4.
+Proposition3.2. Let˜f(z)and˜g(z)beentirefunctionssatisfying
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)(z) = 2˜f(z/2)+ ˜g(z), (42)
+withf(0) = 0. Then
+˜g∈JS⇐⇒˜f∈JS.
+Proof.(Sketch) The same proof as that for Proposition 2.4appliesmutatis mutandis to (42). The only
+difference isthatwenowhave
+f(b)(z) = 2ez/2f(z/2)+g(z),
+wheref(z) :=ez˜f(z)andg(z) :=ez˜g(z), so that(14)has theextendedrepresentation
+f(z) =1
+(b−1)!/integraldisplayz
+0(z−t)b−1/parenleftbig
+2et/2f(t/2)+g(t)/parenrightbig
+dt
+=zb
+(b−1)!/integraldisplay1
+0(1−t)b−1/parenleftbig
+2etz/2f(tz/2)+g(tz)/parenrightbig
+dt,
+and
+˜f(z) =zb
+(b−1)!/integraldisplay1
+0(1−t)be−(1−t)z/parenleftBig
+2˜f(tz/2)+ ˜g(z)/parenrightBig
+dt.
+All requiredestimatescan bederivedby thesameargumentsu sedthere.
+TheLaplacetransformof ˜f1nowsatisfies thefunctionalequation
+(s+1)bL[˜f1;s] = 4L[˜f1;2s]+s−2,
+33forℜ(s)>0. From thisequation,weobtain
+L[˜f1;s] =1
+s2/summationdisplay
+j≥01
+(s+1)b···(1+2js)b,
+whichextends( 22). Fromthisseriesandpartialfractionexpansions,wecand eriveaclose-formexpression
+for˜f1(z), which becomes messy especially for large b. Define as before ¯L[˜f1;s] :=L[˜f1;s]/Q(−s)b.
+Then weobtain
+¯L[˜f1;s] = 4¯L[˜f1;2s]+1
+Q(−2s)bs2.
+This relation is almost the same as ( 26). Thus the same Mellin analysis given there carries over and we
+deducethat
+L[˜f1;s] =1
+s2log21
+s+1
+s2
+1
+2+c4
+log2+1
+log2/summationdisplay
+k∈Z\{0}G1(2+χk)s−χk
+
++b
+slog1
+s+O(|s|−1),
+uniformlyfor |s| →0and|arg(s)| ≤π−ε,where
+G1(ω) :=/integraldisplay∞
+0sω−3
+Q(−2s)bds,
+and
+c4:= lim
+ω→2/parenleftbigg
+G1(ω)−1
+ω−2/parenrightbigg
+=/integraldisplay1
+01
+s/parenleftbigg1
+Q(−2s)b−1/parenrightbigg
+ds+/integraldisplay∞
+11
+sQ(−2s)bds.
+Consequently,bytheLaplaceinversion,
+˜f1(z) = (z+b)log2z+z
+1
+2+G1(2)+γ−1
+log2+1
+log2/summationdisplay
+k∈Z\{0}G1(2+χk)
+Γ(2+χk)zχk
++O(1),(43)
+uniformlyfor |z| → ∞and|arg(z)| ≤π/2−ε. From thisand Propositions 3.2and2.2, weobtain
+E(Xn) =/summationdisplay
+0≤j<2k˜f(j)(n)
+j!τj(n)+O/parenleftbig
+n−1+k/parenrightbig
+,
+foranyk= 1,2,.... Finally,
+E(Xn) = (n+b)log2n+n
+1
+2+G1(2)+γ−1
+log2+1
+log2/summationdisplay
+k∈Z\{0}G1(2+χk)
+Γ(2+χk)nχk
++O(1).
+34Variance of Xn.The analysis here is again similar to that for the mean. Let ˜f2(z)denote the Poisson
+generating functionofthesecond moment E(X2
+n). Then,by ( 41),
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+2(z) = 2˜f2(z/2)+2˜f1(z/2)2+4z˜f1(z/2)+2z˜f′
+1(z/2)+z+z2,
+withthefirst bTaylorcoefficients zero. Defineagain
+˜V(z) =˜f2(z)−˜f1(z)2−z˜f′
+1(z)2.
+Then˜V(z)satisfies
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜V(j)(z) = 2˜V(z/2)+ ˜g(z),
+where˜g(z)is givenin ( 38).
+By therepresentations( 39)and (43), wehave
+˜g(z) =/braceleftBigg
+O(|z|),as|z| →0;
+O(|z|−1),as|z| → ∞,
+uniformly in the sector |arg(z)| ≤π/2−ε. This is similar to the corresponding estimate ( 34) in the
+analysisofthevarianceintheprevioussection. Thesamepr ocedurethereappliesand wededuce( 40).
+3.2 Node-wise path-length (NPL)
+We considerin this section thetotal node-wisepath-length (NPL). Under thesameindependent Bernoulli
+model,westilluse XntodenotetheNPLinarandom b-DSTofnbinarystringswithnodecapacity b≥2.
+AlsoletNnstandforthetotalnumberofnodes(spacerequirement)inra ndomb-DSTofnstrings. Despite
+its being one of the most natural shape measures for b-DSTs, the consideration of Xnhere seems to be
+new. ForNn,itisknownthatthedistributionisasymptoticallynormal withthemeanandthevarianceboth
+asymptotically ntimes a different smooth periodic function; see [ 31]. In contrast to ( 40) for the variance
+ofKPL, what isunexpectedand surprisinghere isthat thevar ianceofXnisofordern(logn)2.
+Theorem 3.3. Assumeb≥2. Themean of Nnand thatofXnsatisfythefollowingasymptoticrelations.
+/braceleftBigg
+E(Nn) =nP1,0(log2n)+O(1),
+E(Xn) =n(log2n)P1,0(log2n)+nP[2]
+0,1(log2n)+(log2n)P[3]
+0,1(log2n)+O(1);(44)
+and thevariances of NnandXnsatisfy
+
+
+V(Nn) =nP2,0(log2n)+O(1),
+Cov(Nn,Xn) =n(log2n)P2,0(log2n)+nP[2]
+1,1(log2n)+(logn)P[3]
+1,1(log2n)+O(1),
+V(Xn) =n(log2n)2P2,0(log2n)+n(log2n)P[2]
+0,2(log2n)+nP[3]
+0,2(log2n)
++(logn)2P[4]
+0,2(log2n)+(log2n)P[5]
+0,2(log2n)+O(1),(45)
+where theP·,·’sareall computable,smooth, 1-periodicfunctions.
+35Intuitively,thatthevarianceofNPLislargerthanthatofK PLcan beseen fromthedefinitionofNPL,
+whichdependsontherandomvariable Nn(see(46)), whileontheotherhand,KPLdependson nonly(in
+addition to on the two subtrees). The following figure shows t he first few values of the variance of NPL
+and thatofKPL.
+46810121416182012345
+NPL
+KPL
+We seethatthevarianceofNPL increases fasterthanthat ofK PL.
+Note that the periodic functions of the dominant terms are al l equal, implying that the correlation
+coefficient of NnandXnisasymptotically 1.
+Ontheotherhand,themeanvalue c1,0ofP1,0(t)isgivenby
+c1,0=1
+log2/integraldisplay∞
+0(s+1)b−1
+Q(−2s)bds;
+numerical approximationsto c1,0forthefirst few baregivenas follows.
+b12 3 4 5 6
+c1,010.574700.406980.315940.258490.21885
+Notethat when b= 1
+c1,0=1
+log2/integraldisplay∞
+0ds
+Q(−2s)= 1,
+by (28), whichisconsistentwiththefact that Nn≡nin thiscase.
+Whenb= 2,weseethatabout 42.5%ofnodesonaveragecontaintwokeysand 14%ofnodesasingle
+key. Thestorageutilizationis thusnotvery bad.
+From(44)andthesenumericalvalues,weseethat,incontrasttothee xpectedKPL,whichisasymptotic
+tonlog2nfor allb, the expected NPL provides a better indication of the “shape variation” of random b-
+DSTs.
+Ouranalysisis based onthefollowingstraightforwarddist ributionalrecurrences
+/braceleftBigg
+Nn+bd=NBn+N∗
+n−Bn+1,
+Xn+bd=XBn+X∗
+n−Bn+NBn+N∗
+n−Bn,(n≥0), (46)
+with the initial conditions N0= 0,N1=···=Nb−1= 1andX0=···=Xb−1= 0. Here again
+Bn∼Binomial(n,1/2),Nnd=N∗
+n,Xnd=X∗
+nandXn,X∗
+n,Bnaswell asNn,N∗
+n,Bnare independent.
+36Generating functions. DefineMn(u,v) =E(eNnu+Xnv). Then (46)translatesintotherecurrence
+Mn+b(u,v) =eu2−nn/summationdisplay
+j=0/parenleftbiggn
+j/parenrightbigg
+Mj(u+v,v)Mn−j(u+v,v),(n≥0),
+withM0(u,v) = 1,M1(u,v) =···=Mb−1(u,v) =eu. Next,let
+F(z,u,v) :=/summationdisplay
+n≥0Mn(u,v)
+n!zn.
+Then therecurrence relationgives
+∂b
+∂zbF(z,u,v) =euF/parenleftBigz
+2,u+v,v/parenrightBig2
+,
+and thePoissongeneratingfunction ˜F(z,u,v) :=e−zF(z,u,v)satisfies
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg∂j
+∂zj˜F(z,u,v) =eu˜F/parenleftBigz
+2,u+v,v/parenrightBig2
+, (47)
+withtheinitialconditions ˜F(z,u,v) = 1+(eu−1)/summationtext
+1≤j<b(−1)j−1zj/j!+···.
+Forthemoments,ifweexpand ˜F(z,u,v)in termsofuandv,
+˜F(z,u,v) =/summationdisplay
+m≥01
+m!/summationdisplay
+0≤j≤m/parenleftbiggm
+j/parenrightbigg
+˜fj,m−j(z)ujvm−j,
+then˜fj,m−j(z)is the Poisson generating function of E(Nj
+nXm−j
+n). Thus all moments of XnandNnor
+their products can be computed by taking suitable derivativ es of (47) with respect to uandvand then
+substituting u=v= 0.
+Expected number of nodes and expected node-wise path length .By taking first derivatives of ( 47),
+weobtain
+
+
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+1,0(z) = 2˜f1,0(z/2)+1,
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+0,1(z) = 2˜f0,1(z/2)+2˜f1,0(z/2),(48)
+theinitialconditionsbeing ˜f1,0(0) = 0,˜f(j)
+1,0(0) = (−1)j−1for1≤j <band˜f(j)
+0,1(0) = 0for0≤j <b.
+Wecan applytheLaplace-Mellinapproach as before, startin gfrom themean of Nn. Notethat
+L[˜f(j);s] =sjL[˜f;s]−/summationdisplay
+0≤ℓ<jsℓ˜f(j−1−ℓ)(0) (j= 0,1,...),
+providedthattheLaplacetransformexistsfor ℜ(s)>0. Thisgives
+(s+1)bL[˜f1,0;s] = 4L[˜f1,0;2s]+ ˜g⋆
+1,0(s),
+37where
+˜g⋆
+1,0(s) :=1
+s+/summationdisplay
+0≤ℓ≤b−2sℓ/summationdisplay
+ℓ≤j≤b−2/parenleftbiggb
+j+2/parenrightbigg
+˜f(j+1−ℓ)
+1,0(0)
+=1
+s+/summationdisplay
+1≤j<b/parenleftbiggb−1
+j/parenrightbigg
+sj−1
+=s−1(s+1)b−1.
+Unlike all previous cases, iterating this functional equat ion leads to a divergent series. Although this
+problem can be solved by subtracting a sufficient number of in itial terms of ˜f1,0(z), the approach we use
+does notrely onthisand avoidscompletelysuchaconsiderat ion.
+Let¯L[˜f1,0;s] :=L[˜f1,0;s]/Q(−s)b. Then
+M[¯L[˜f1,0;s];ω] =G1,0(ω)
+1−22−ω,(ℜ(ω)>2),
+where
+G1,0(ω) :=/integraldisplay∞
+0sω−2
+Q(−2s)b(s+1)b−1ds,
+forℜ(ω)>1.
+Fromthis,wededucethat
+˜f1,0(z) =zP1,0(log2z)+O(1), (49)
+uniformly for |z| → ∞and|arg(z)| ≤π/2−ε, whereP1,0(t)is a periodic function with the Fourier
+series representation
+P1,0(t) :=1
+log2/summationdisplay
+k∈ZG1,0(2+χk)
+Γ(2+χk)e2kπit,
+theseries beingabsolutelyconvergent. Fromthiswededuce thefirst approximationof( 44).
+Wenowturnto theexpected NPL E(Xn). By (48), wehave
+(s+1)bL[˜f0,1;s] = 4L[˜f0,1;2s]+4L[˜f1,0;2s].
+Let¯L[˜f0,1;s] :=L[˜f0,1;s]/Q(−s)b. Then
+M[¯f0,1;ω] =22−ωG1,0(ω)
+(1−22−ω)2,(ℜ(ω)>2).
+From thiswededucethat
+˜f0,1(z) =z(log2z)P[1]
+0,1(log2z)+zP[2]
+0,1(log2z)+(log2z)P[4]
+0,1(log2z)+O(1), (50)
+uniformly for |z| → ∞and|arg(z)| ≤π/2−ε, whereP[1]
+0,1(t),P[2]
+0,1(t),P[4]
+0,1(t)are smooth, 1-periodic
+functionswhoseFouriercoefficients aregivenby
+P[1]
+0,1(t) =P1,0(t) =1
+log2/summationdisplay
+k∈ZG1,0(2+χk)
+Γ(2+χk)e2kπit,
+P[2]
+0,1(t) =−1
+(log2)2/summationdisplay
+k∈ZG′
+1,0(2+χk)ψ(2+χk)−G1,0(2+χk)
+Γ(2+χk)e2kπit,
+P[4]
+0,1(t) =b
+log2/summationdisplay
+k∈ZG1,0(2+χk)
+Γ(1+χk)e2kπit.
+38Hereψ(z)denotes thederivativeof logΓ(z)andall series areabsolutelyconvergent. Thisproves( 44).
+Variance. Takingsecond derivativesin( 47)and substituting u=v= 0gives
+
+
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+2,0(z) = 2˜f2,0(z/2)+2˜f1,0(z/2)2+4˜f1,0(z/2)+1,
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+1,1(z) = 2˜f1,1(z/2)+2˜f2,0(z/2)+2(˜f1,0(z/2)+˜f0,1(z/2))(˜f1,0(z/2)+1),
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+0,2(z) = 2˜f0,2(z/2)+4˜f1,1(z/2)+2˜f0,2(z/2)+2(˜f1,0(z/2)+˜f0,1(z/2))2,
+with the initial conditions ˜f(j)
+2,0(0) = (−1)j−1for1≤j < band˜f2,0(0) =˜f(j)
+1,1(0) =˜f(j)
+0,2(0) = 0, for
+0≤j <b.
+The remaining calculations follow the same pattern of proof we used above but become much more
+involved. We beginwith
+
+
+˜V(z) =˜f2,0(z)−˜f1,0(z)2−z˜f′
+1,0(z)2,
+˜U(z) =˜f1,1(z)−˜f1,0(z)˜f0,1(z)−z˜f′
+1,0(z)˜f′
+0,1(z),
+˜W(z) =˜f0,2(z)−˜f0,1(z)2−z˜f′
+0,1(z)2.
+Then wededuce 
+
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜V(j)(z) = 2˜V(z/2)+ ˜g2,0(z),
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜U(j)(z) = 2˜U(z/2)+ ˜g1,1(z),
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜W(j)(z) = 2˜W(z/2)+ ˜g0,2(z),
+39where
+
+˜g2,0(z) =/parenleftBigg/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+1,0(z)/parenrightBigg2
++z/parenleftBigg/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j+1)
+1,0(z)/parenrightBigg2
+−/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg/parenleftBig
+˜f1,0(z)2+z˜f′
+1,0(z)2/parenrightBig(j)
+,
+˜g1,1(z) = 2˜V(z/2)+/parenleftBigg/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+1,0(z)/parenrightBigg/parenleftBigg/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+0,1(z)/parenrightBigg
++z/parenleftBigg/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j+1)
+1,0(z)/parenrightBigg/parenleftBigg/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j+1)
+0,1(z)/parenrightBigg
+−/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg/parenleftBig
+˜f1,0(z)˜f0,1(z)+z˜f′
+1,0(z)˜f′
+0,1(z)/parenrightBig(j)
+,
+˜g0,2(z) = 4˜U(z/2)+2˜V(z/2)+/parenleftBigg/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)
+0,1(z)/parenrightBigg2
++z/parenleftBigg/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j+1)
+0,1(z)/parenrightBigg2
+−/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg/parenleftBig
+˜f0,1(z)2+z˜f′
+0,1(z)2/parenrightBig(j)
+.
+Theinitialconditionsare ˜V(0) =˜U(j)(0) =˜W(j)(0) = 0for0≤j <band
+˜V(j)(0) = (−1)j/parenleftbig
+1+(j−2)2j−1/parenrightbig
+,(1≤j≤b).
+From (49), (50)and Ritt’stheorem (see[ 54]), wehave
+
+
+˜g2,0(z) =O/parenleftbig
+|z|−1/parenrightbig
+,
+˜g1,1(z)−2˜V(z/2) =O/parenleftbig
+|z|−1/parenrightbig
+,
+˜g0,2(z)−4˜U(z/2)−2˜V(z/2) =O/parenleftbig
+|z|−1/parenrightbig
+,
+uniformlyfor |z| → ∞and|arg(z)| ≤π/2−ε. Let¯L[˜A;s] :=L[˜A;s]/Q(−s)b,where˜A∈ {˜V,˜U,˜W}.
+Then weobtain,for ℜ(ω)>2,
+
+
+M[¯L[˜V;s];ω] =G2,0(ω)
+1−22−ω,
+M[¯L[˜U;s];ω] =22−ωG2,0(ω)
+(1−22−ω)2+G1,1(ω)
+1−22−ω,
+M[¯L[˜W;s];ω] =25−2ωG2,0(ω)
+(1−22−ω)3+22−ω(2G1,1(ω)+G2,0(ω))
+(1−22−ω)2+G1,1(ω)+G0,2(ω)
+1−22−ω,
+where
+
+G2,0(ω) :=/integraldisplay∞
+0sω−1
+Q(−2s)b/parenleftbigg
+L[˜g2,0;s]+(s+1)b−1−(−1)b(2b−3+(b−1)s)
+(s+2)2/parenrightbigg
+ds,
+G1,1(ω) :=/integraldisplay∞
+0sω−1
+Q(−2s)b/integraldisplay∞
+0e−sz/parenleftBig
+˜g1,1(z)−2˜V(z/2)/parenrightBig
+dzds,
+G0,2(ω) :=/integraldisplay∞
+0sω−1
+Q(−2s)b/integraldisplay∞
+0e−sz/parenleftBig
+˜g0,2(z)−2˜V(z/2)−4˜U(z/2)/parenrightBig
+dzds,
+40withall functionsanalyticfor ℜ(ω)>0. Consequently,wededuce( 45).
+4 Digitalsearchtrees. II.More shapeparameters.
+We consider in this section four additionalexamples on DSTs whose variances are essentiallylinear. The
+sametoolsweusereadilyapplyto b-DSTs,butwefocusonDSTsbecausetheresultsareeasiertos tateand
+theasymptoticbehaviorsdonotdifferinessencewiththose forthemoregeneral b-DSTsthecorresponding
+expressionsofwhicharehowevermuchmessier.
+Thefirstparameterweconsideristheso-called w-parameter(see[ 16]),whichisthesumofthesubtree-
+sizeoftheparent-nodeofeachleaf(overallleaves)3. Insteadofw-parameter,wecallitthe totalperipheral
+path-length (PPL), since it measures to some extent the fringe ampleness of the trees. Also this is in
+consistencywiththetwo previousnotionsofpath-lengthwe distinguished.
+Then we consider the number of leaves, which has previously b een studied in details in [ 26,31,39]
+and which iswellconnected to PPL. Ourexpressionforthevar iancesimplifiesknownones.
+Yet another notion of path-length we consider here is the so- calledColless index in phylogenetics,
+which isthesumoftheabsolutedifferenceofthetwosubtree -sizesofeach node(overallnodes). Wecall
+this index the total differential path-length (DPL) as it clearly indicates the balance or symmetry of the
+tree. Anotherwidelyusedmeasureofimbalanceinphylogene ticsisthe Sackinindex ,whichisnothingbut
+theexternalpath-length.
+The last example we consider is the weighted path-length (WPL), which often arises in coding, opti-
+mizationand manyrelatedproblems.
+Theordersofthemeansandthevariancesexhibitedbyallthe shapeparameterswestudyinthispaper
+are listedinTable 1.
+4.1 Peripheral path-length (PPL)
+ThePPL(or w-parameter)wasintroducedin[ 16],themotivationsarisingfromtheanalysisofcompression
+algorithms. We start from the fringe-size of a leaf node λ, which is defined to be the size of the subtree
+rooted at its parent-node; see Figure 9. The PPL of a tree is then defined to be the sum of the fringe-siz es
+of all leaf-nodes. Let Xndenote the PPL in a DST built from nrandom binary strings under our usual
+independentBernoulli model.
+T...
+T...
+Figure 9: The two possible configurationsof the fringe of a leaf: the fr inge-size (or w-parameter) equals
+|T|+2. NotethatTmaybeempty.
+Drmotaet al. showedin [ 16]that
+E(Xn) =n(Cw+̟w(log2n))+o(n), (51)
+3Theleavesorleaf-nodes ofa treearenodeswithoutanydescendants.
+41where
+Cw:=/summationdisplay
+ℓ≥0(ℓ+1)(ℓ−2)
+Qℓ2ℓ/parenleftBigg/summationdisplay
+k≥11
+2ℓ+k−1−1/parenrightBigg
++1
+log2/summationdisplay
+ℓ≥02ℓ−1
+Qℓ2ℓ.
+Notethatby ( 24), wehavetheidentities
+/summationdisplay
+ℓ≥0(ℓ+1)(ℓ−2)
+Qℓ2ℓ=1
+Q∞
+/summationdisplay
+j≥11
+(2j−1)2+/parenleftBigg/summationdisplay
+j≥11
+2j+1/parenrightBigg2
+−2
+,
+/summationdisplay
+ℓ≥02ℓ−1
+Qℓ2ℓ=1
+Q∞/parenleftBigg/summationdisplay
+j≥12
+2j−1−1/parenrightBigg
+.
+The asymptotic behavior ( 51) is to be compared with the nlogn-order exhibited by most other log-trees
+such as binary search trees and recursive trees; see [ 16]. It reflects that most fringes of random DSTs are
+small in size; see Figure 3. Indeed, since the expected number of leaves is also asympto tic tontimes a
+periodicfunction,theresult( 51)impliesthat theaveragesizeofafringein randomDSTs isbo unded. We
+showthatthestandard deviationisalso small.
+Define
+˜g2(z) :=z˜f′′
+1(z)2−z
+16e−z/parenleftbig
+z4+4z3+16z2−8z+64/parenrightbig
+−z
+4e−z/2/parenleftBig
+4(z+4)˜f1(z/2)−2(z2+2z+8)˜f′
+1(z/2)−(z+2)(z+8)/parenrightBig
+,(52)
+where˜f1(z)represents as usual the Poisson generating function of E(Xn). LetG2(ω)denote the Mellin
+transform of L[˜g2;s]/Q(−2s).
+Theorem 4.1. Themean andthevarianceof thetotalPPL XnofrandomDSTsof nstringssatisfy
+E(Xn) =n(Cw+̟w(log2n))+O(1),
+V(Xn) =nPw(log2n)+O(1), (53)
+wherePw(t)isa smooth, 1-periodicfunctionwiththeFourierseries expansion
+Pw(t) =1
+log2/summationdisplay
+k∈ZG2(2+χk)
+Γ(2+χk)e2kπit,
+theseriesbeing absolutelyconvergent.
+Weprovideonlythemajorstepsoftheproofsinceitfollowst hesameapproach wedevelopedabove.
+Recurrence and generating functions. By definition and by conditioning on the size of one of the
+subtrees oftheroot, wehavethefollowingdifferentconfigu rations
+n−1 n−1 n−2 n−2 k n−1−k
+42from whichwederivetherecurrence forthePPL
+Xnd=
+
+Xn−1, withprobability 22−n;
+n+Xn−2,withprobability (n−1)22−n;
+Xk+X∗
+n−1−k,withprobability 21−n/parenleftbign−1
+k/parenrightbig
+,2≤k≤n−3,
+whereX0=X1= 0,X2= 2andX3hasthedistribution
+X3=/braceleftBigg
+6,withprobability 1/2;
+2,withprobability 1/2.
+Fromthisrecurrence, it followsthatthebivariatePoisson generatingfunction
+˜F(z,y) :=e−z/summationdisplay
+n≥0E(eXny)
+n!zn
+satisfies thenonlinearequation
+˜F(z,y)+∂
+∂z˜F(z,y) =˜F/parenleftBigz
+2,y/parenrightBig2
++ze2y+eyz/2−z˜F/parenleftbiggeyz
+2,y/parenrightbigg
+−ze−z/2˜F/parenleftBigz
+2,y/parenrightBig
++z2
+4e−z/parenleftbig
+e3y−1/parenrightbig2,(54)
+withtheinitialcondition ˜F(0,y) = 1.
+TheexpectedPPL. By(54),weobtainthedifferential-functionalequationfor ˜f1(z)bytakingderivative
+withrespect to yand thensubstituting y= 1, giving
+˜f1(z)+˜f′
+1(z) = 2˜f1(z/2)+z(2+z/2)e−z/2, (55)
+withf1(0) = 0. TheLaplacetransform of ˜f1satisfies
+L[˜f1;s] =4
+s+1L[˜f1;s]+16
+(1+2s)3
+= 16/summationdisplay
+k≥04k
+(s+1)···(2k−1s+1)(2k+1s+1)3.
+Then astraightforwardapplicationoftheLaplace-Mellin- de-Poissonizationapproach yields
+E(Xn) =n
+log2/summationdisplay
+k∈ZG1(2+χk)
+Γ(2+χk)nχk+O(1),
+where
+G1(ω) := 16/integraldisplay∞
+0sω−1
+Q(−s)(2s+1)3ds(ℜ(ω)>0).
+TheO(1)-term can be further refined by the same analysis. In particul ar, we get an alternativeexpression
+forCw
+Cw=G1(2)
+log2=16
+log2/integraldisplay∞
+0s
+Q(−s)(2s+1)3ds≈1.1030266959 ···.
+That the two expressions of Cware identical can be proved by standard calculus of residues ; see [24] for
+similardetails.
+43The variance of the PPL. Again from ( 54), we derivethe equation for the Poisson generating functio n
+˜f2(z)ofthesecond momentof Xn
+˜f2(z)+˜f′
+2(z) = 2˜f2(z/2)+2˜f1(z/2)2+9
+2z2e−z
++ze−z/2/parenleftbigg
+(z+4)˜f1(z/2)+z˜f′
+1(z/2)+z2+10z+16
+4/parenrightbigg
+,(56)
+with˜f2(0) = 0.
+Let˜V(z) =˜f2(z)−˜f1(z)2−z˜f′
+1(z)2. Then, by( 55), (56)and Lemma 2.7,
+˜V(z)+˜V′(z) = 2˜V(z/2)+ ˜g2(z),
+with˜V(0) = 0,where˜g2isdefined in ( 52).
+Applying again the Laplace-Mellin-de-Poissonization app roach, we deduce ( 53). In particular, the
+mean valueoftheperiodicfunction Pwisgivenby
+G2(2)
+log2=1
+log2/integraldisplay∞
+0s
+Q(−2s)/integraldisplay∞
+0e−zs˜g2(z)dzds.
+4.2 The number ofleaves
+The leaves of a tree are the locations where the nodes holding new-coming keys will be connected; thus
+different types of data fields can be used to savememory, nota bly forb-DSTs. The numberof leaves then
+provides a quick and simpler look at the “fringes” of a tree. S uch nodes are sometimes referred to as the
+external-internalnodes orinternal endnodesinthelitera ture;see[ 16,26,41,56].
+LetXndenotethenumberofleavesinarandom DST of nkeys. ThenXnsatisfies therecurrence
+Xn+1d=XBn+X∗
+n−Bn(n≥1), (57)
+withX0= 0andX1= 1, whereBn∼Binomial(n;1/2).
+Flajoletand Sedgewick[ 26], solvingan open questionraisedby Knuth,showedthat
+E(Xn) =n(Cfs+̟fs(log2n))+O(n1/2),
+where̟fs(t)isasmooth, 1-periodicfunctionand
+Cfs= 1+/summationdisplay
+k≥1k
+Qk2k/summationdisplay
+1≤j≤k1
+2j−1−1
+Q∞
+1
+log2+/parenleftBigg/summationdisplay
+k≥11
+2k−1/parenrightBigg2
+−/summationdisplay
+k≥11
+2k−1
+
+≈0.3720486812 ···.
+A finer approximation,togetherwiththealternative(and nu mericallybetter)expression
+Cfs= 1+/summationdisplay
+k≥11
+2k−1−1
+Q∞/parenleftBigg
+1
+log2+/summationdisplay
+k≥1(−1)kk
+Qk(2k−1)2k(k+1)/2/parenrightBigg
+,
+was derived by Kirschenhofer and Prodinger [ 39]; see also [ 56]. They provedadditionallythe asymptotic
+linearityofthevariance
+V(Xn)∼n(Ckp+̟kp(log2n)),
+44where̟kpisasmooth, 1-periodicfunctionwithmeanzeroandalong,complicatedex pressionisgivenfor
+theleadingconstant Ckp. Wederivedifferentformsfor thesetwo asymptoticapproxi mations.
+Define
+˜g2(z) =z˜f′′
+1(z)2+e−z/parenleftBig
+1−e−z(1+z)+2z˜f′
+1(z/2)−4˜f1(z/2)/parenrightBig
+, (58)
+where˜f1(z) :=e−z/summationtext
+n≥0E(Xn)zn/n!.
+Theorem 4.2. Themean andthevarianceofthenumberof leaves arebothasym ptoticallylinearwiththe
+approximations
+E(Xn) =n
+log2/summationdisplay
+k∈ZG1(2+χk)
+Γ(1+χk)nχk+O(1),
+V(Xn) =n
+log2/summationdisplay
+k∈ZG2(2+χk)
+Γ(2+χk)nχk+O(1),
+where thetwo seriesareabsolutelyconvergent with G1,G2defined by
+G1(ω) =/integraldisplay∞
+0sω−1
+(s+1)Q(−2s)ds,
+G2(ω) =/integraldisplay∞
+0sω−1
+Q(−2s)/integraldisplay∞
+0e−zs˜g2(z)dzds,
+forℜ(ω)>0.
+Weseeinparticularthat
+Cfs=1
+log2/integraldisplay∞
+0s
+(s+1)Q(−2s)ds,
+Ckp=1
+log2/integraldisplay∞
+0s
+Q(−2s)/integraldisplay∞
+0e−zs˜g2(z)dzds. (59)
+Sketch of proof. From (57), we derive the equation for the bivariate generating funct ion˜F(z,y) :=
+e−z/summationtext
+n≥0E(eXny)zn/n!
+˜F(z,y)+∂
+∂z˜F(z,y) =˜F/parenleftBigz
+2,y/parenrightBig2
++(ey−1)e−z,
+with˜F(0,y) = 1. Then thePoissongeneratingfunctionsofthefirst two momen tssatisfy
+˜f1(z)+˜f′
+1(z) = 2˜f1(z/2)+e−z, (60)
+˜f2(z)+˜f′
+2(z) = 2˜f2(z/2)+2˜f1(z/2)2+e−z,
+with˜f1(0) =˜f2(0). Consequently,thefunction ˜V(z) :=˜f2(z)−˜f1(z)2−z˜f′
+1(z)2satisfies
+˜V(z)+˜V′(z) = 2˜V(z/2)+ ˜g2(z),
+with˜V(0) = 0,where˜g2isgivenin( 58). Theremaininganalysisfollowsthesamepatternasabovea ndis
+omitted.
+45We provide instead some details for the numerical evaluatio n of the constant Ckpas defined in ( 59),
+which issimilartothecaseofinternalpath-lengthofDSTs.
+By applyingtheLaplacetransform tobothsidesof( 60)and byiteration,weget
+L[˜f1;s] =/summationdisplay
+k≥04k
+(s+1)(2s+1)···(2k−1s+1)(2ks+1)2.
+Since theinverse Laplace transform derived from the partia l fraction expansion of this series is divergent,
+weconsiderthefunction ˆf1(z) :=˜f1(z)−z+z2/2forwhich theequation( 60)becomes
+ˆf1(z)+ˆf′
+1(z) = 2ˆf1(z/2)−1+z+z2
+4+e−z,
+withˆf1(0) = 0,and wehave
+L[ˆf1;s] =1
+2s3/summationdisplay
+k≥03·2ks+1
+2k(s+1)···(2k−1s+1)(2ks+1)2.
+Thenby thepartialfraction expansion
+3·2ks+1
+(s+1)···(2k−1s+1)(2ks+1)2=/summationdisplay
+0≤ℓ<k(−1)k−ℓ(3·2k−ℓ−1)2−(k−ℓ+1
+2)
+(2k−ℓ−1)QℓQk−ℓ·1
+2ℓs+1
++1
+Qk/parenleftBigg
+3+2/summationdisplay
+1≤j≤k1
+2j−1/parenrightBigg
+1
+2ks+1−2
+Qk(2ks+1)2,
+weobtain
+L[ˆf1;s] =1
+2s3/summationdisplay
+ℓ≥01
+2ℓQℓ/parenleftbiggδℓ
+2ℓs+1−2
+(2ℓs+1)2/parenrightbigg
+,
+where
+δℓ= 3+2/summationdisplay
+1≤j≤ℓ1
+2j−1+/summationdisplay
+j≥1(−1)j(3·2j−1)2−(j+1
+2)
+(2j−1)2jQj.
+Obviously, limℓ→∞δℓ= 4. Now,by theinverseLaplace transform,
+ˆf1(z) =1
+2/summationdisplay
+ℓ≥01
+Qℓ/parenleftBigg
+2ℓδℓ/parenleftbigg
+1−z
+2ℓ+z2
+22ℓ+1−e−z/2ℓ/parenrightbigg
+−2ℓ+1/parenleftbigg
+3−z
+2ℓ−1+z2
+22ℓ+1−3e−z/2ℓ/parenrightbigg
++2ze−z/2ℓ/parenrightBigg
+,
+which convergesforall z; also from[ 26]wehave
+ˆf1(z) =/summationdisplay
+n≥3(−1)n−1zn
+n!Q(n−2)/summationdisplay
+0≤j≤n−21
+Q(j).
+46Then thefirst andthesecond derivativesaregivenby
+ˆf′
+1(z) =1
+2/summationdisplay
+ℓ≥01
+Qℓ/parenleftBig
+δℓ/parenleftBig
+−1+z/2ℓ+e−z/2ℓ/parenrightBig
++4−z
+2ℓ−1−4e−z/2ℓ−z
+2ℓ−1e−z/2ℓ/parenrightBig
+,
+ˆf′′
+1(z) =1
+2/summationdisplay
+ℓ≥01
+2ℓQℓ/parenleftBig
+δℓ/parenleftBig
+1−e−z/2ℓ/parenrightBig
+−2+2e−z/2ℓ+z
+2ℓ−1e−z/2ℓ/parenrightBig
+.
+Nowtheconstant Ckpcan beexpressedin termsoftheintegralsof ˆf1as follows.
+(log2)Ckp=/integraldisplay∞
+0s
+Q(−2s)(s+1)(s+2)2ds+/integraldisplay∞
+0s
+Q(−2s)/integraldisplay∞
+0e−zsz(ˆf′′
+1(z)−1)2dzds
++2/integraldisplay∞
+0s
+Q(−2s)/integraldisplay∞
+0e−z(s+1)/parenleftbigg
+z−1
+s+1/parenrightbigg
+(ˆf′
+1(z/2)−z)dzds.
+And weget Ckp≈0.034203···.
+A general weighted sum of node-types for b-DSTs. Forb≥2, we can consider X[j]
+n,1≤j≤b, the
+numberofleavescontaining jrecordsinarandom b-DST withbucketcapacity bbuiltfromnrecords. Let
+alsoX[b+1]
+nbethenumberofinternal(non-leaf)nodes. Define
+Xn=/summationdisplay
+1≤j≤b+1ajX[j]
+n,
+wherea1,...,a b+1are arbitrary real numbers. By astraightforwardcomputati on
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg∂j
+∂zj˜F(z,y) =eab+1y˜F/parenleftBigz
+2,y/parenrightBig2
++e−z(eaby−eab+1y),
+with˜F(0,y) = 1. Then our approach can be applied and leads to the sametypeof results as Theorem 4.2
+with different G1andG2; the resulting expressions for the variance are more explic it and simpler than
+thosegivenin[ 31].
+4.3 Colless index: the differential path-length (DPL)
+TheDPLofatreeisdefinedtobethesumoverallnodesoftheabs olutedifferenceofthetwosubtree-sizes
+ofeach nodeas depicted below.
+Tleft TrightDPL=/summationtext
+all nodes|Tleft− Tright|
+Properties of such a path length in random binary search tree s have long been investigated in the
+systematicbiologyliterature;see [ 4]andthereferences therein.
+LetXndenote the DPL of a random DST of ninput-strings. Then by definition and by our indepen-
+dence assumption,wehavetherecurrence forthemomentgene ratingfunction
+Mn+1(y) = 2−n/summationdisplay
+0≤j≤n/parenleftbiggn
+j/parenrightbigg
+Mj(y)Mn−j(y)e|n−2j|y(n≥0), (61)
+47withM0(y) = 1.
+Letalso
+˜g2(z) :=z˜f′′
+1(z)2+z−˜h1(z)2−z˜h′
+1(z)2−4˜h1(z)˜f(z/2)−2z˜h′
+1(z)˜f′
+1(z/2)+4˜hc(z),
+where˜f1(z)isthePoissongeneratingfunctionof E(Xn)and˜hc(z)is defined by
+˜hc(z) :=e−z/summationdisplay
+n≥0(z/2)n
+n!/summationdisplay
+0≤k≤n/parenleftbiggn
+k/parenrightbigg
+E(Xk)|n−2k|. (62)
+Theorem 4.3. Themean andthevarianceof theDPLof randomDSTs satisfythe asymptoticrelations
+E(Xn) =nPd,µ(log2n)−√
+2n√π(√
+2−1)+O(1), (63)
+V(Xn) =/parenleftbigg
+1−2
+π/parenrightbigg
+nlog2n+nPd,σ(log2n)+O(n1/2), (64)
+wherePd,µandPd,σareexplicitlycomputable,smooth, 1-periodicfunctions.
+These results are to be compared with the known results for ra ndom binary search trees for which the
+DPL has mean oforder nlognand varianceoforder n2;see[4].
+Expected DPL. Theapproach wefollowhereforderivingthedifferential-f unctionalequationssatisfied
+bythePoissongeneratingfunctionsofthefirsttwomomentsi sslightlydifferentfromtheoneweusedsince
+the corresponding nonlinear equation for the bivariategen erating function F(z,y) :=/summationtext
+n≥0Mn(y)zn/n!
+is veryinvolvedas givenbelow.
+∂
+∂zF(z,y)−1 =F/parenleftbiggeyz
+2,y/parenrightbigg
+F/parenleftbigge−yz
+2,y/parenrightbigg
++1
+2πi/contintegraldisplay
+|w|=r>0F/parenleftBigwz
+2,y/parenrightBig/parenleftbiggF(eyz/2,y)−w−1e−yF(z/(2w),y)
+w−e−y
+−F(e−yz/2,y)−w−1eyF(z/(2w),y)
+w−ey/parenrightbigg
+dw,
+withF(0,y) = 1.
+Weuseinsteadamoreelementary argument. From therecurren ce (61), weobtain,with µn:=E(Xn),
+µn+1= 21−n/summationdisplay
+0≤k≤n/parenleftbiggn
+k/parenrightbigg
+µk+2−n/summationdisplay
+0≤k≤n/parenleftbiggn
+k/parenrightbigg
+|n−2k|(n≥1),
+theinitialconditionbeing µ0= 0. Then thePoissongenerating functionof Xnsatisfies theequation
+˜f1(z)+˜f′
+1(z) = 2˜f1(z/2)+˜h1(z),
+with˜f1(0) = 0,where˜h1is givenby
+˜h1(z) =e−z/summationdisplay
+n≥0(z/2)n
+n!/summationdisplay
+0≤k≤n/parenleftbiggn
+k/parenrightbigg
+|n−2k|
+=ze−z(I0(z)+I1(z)),
+48where weusedtheidentity
+/summationdisplay
+0≤k≤n/parenleftbiggn
+k/parenrightbigg
+|n−2k|=2n!
+⌊n/2⌋!(⌈n/2⌉−1)!(n≥1),
+andIα(z)denotesthemodifiedBessel functions
+Iα(z) :=/summationdisplay
+n≥0(z/2)2n+α
+n!Γ(n+α+1).
+Itis known(see [ 63])that,as |z| → ∞,
+Iα(z) =
+
+ez
+√
+2πz/parenleftbig
+1+O(|z|−1)/parenrightbig
+,if|arg(z)| ≤π/2−ε,
+O/parenleftbig
+|z|−1/2/parenleftbig
+eℜ(z)+e−ℜ(z)/parenrightbig/parenrightbig
+,if|arg(z)| ≤π,, (65)
+theO-termholdinguniformlyin zineach case. Thus,by( 65),˜h1∈JSand
+˜h1(z) =/radicalbigg
+2z
+π/parenleftbig
+1+O(|z|−1)/parenrightbig
+,
+for|z| → ∞in|arg(z)| ≤π/2−ε. Also
+L[˜h1;s] = (s+2)−1/2s−3/2(ℜ(s)>0).
+Thus wecan applythesameapproach and deducethat
+E(Xn) =n
+log2/summationdisplay
+k∈ZG1(2+χk)
+Γ(2+χk)nχk−√
+2n√
+2π(√
+2−1)+O(1),
+whereG1(ω)istheMellintransformof L[˜h1;s]/Q(−2s)
+G1(ω) =/integraldisplay∞
+0sω−5/2
+Q(−2s)√s+2ds(ℜ(ω)>3/2).
+This proves ( 63). Numerically, the mean value of the dominant periodic func tion isG1(2)/log2≈
+1.3390746494 .
+The varianceofDPL. Againfrom ( 61), we havetherecurrence forthesecondmoment sn:=E(X2
+n)
+sn+1= 2−n/summationdisplay
+0≤k≤n/parenleftbiggn
+k/parenrightbigg/parenleftbig
+sk+sn−k+(n−2k)2+2µkµn−k+4µk|n−2k|/parenrightbig
+,
+forn≥1withs0=s1= 0. Since
+2−n/summationdisplay
+0≤k≤n/parenleftbiggn
+k/parenrightbigg
+(n−2k)2=n, (66)
+wesee thatthePoissongenerating functionof snsatisfiesthenonlinearequation
+˜f2(z)+˜f′
+2(z) = 2˜f2(z/2)+2˜f1(z/2)2+z+4˜hc(z),
+with˜f2(0) = 0,where˜hc(z)isdefined in( 62).
+49Lemma 4.4. Thefunction ˜hcis JS-admissibleandsatisfies
+˜hc(z) =˜h1(z)˜f1(z/2)+O(|z|1/2), (67)
+in thesector |arg(z)| ≤π/2−ε.
+Proof.Observefirst that
+h1(z) =/summationdisplay
+k≥01
+k!/parenleftBigz
+2/parenrightBigk/summationdisplay
+n≥0|n−k|
+n!/parenleftBigz
+2/parenrightBign
+= 2/summationdisplay
+k≥01
+k!/parenleftBigz
+2/parenrightBigk/summationdisplay
+0≤j≤kk−j
+j!/parenleftBigz
+2/parenrightBigj
+=2
+2πi/contintegraldisplay
+|w|=rez(w+1/w)/2
+(w−1)2dw(r<1),
+since/summationdisplay
+0≤j≤kk−j
+j!/parenleftBigz
+2/parenrightBigj
+= [wk]wezw/2
+(w−1)2.
+Ontheotherhand,since f1(z) =/summationtext
+n≥0µnzn/n!, wehave, bythesameargument,
+hc(z) :=ez˜hc(z)
+=/summationdisplay
+k≥0µk
+k!/parenleftBigz
+2/parenrightBigk/summationdisplay
+n≥0|n−k|
+n!/parenleftBigz
+2/parenrightBign
+=/summationdisplay
+k≥0µk
+k!/parenleftBigz
+2/parenrightBigk/parenleftBigg/summationdisplay
+0≤n≤kk−n
+n!/parenleftBigz
+2/parenrightBign
++/summationdisplay
+n≥kn−k
+n!/parenleftBigz
+2/parenrightBign/parenrightBigg
+=/summationdisplay
+k≥0µk
+k!/parenleftBigz
+2/parenrightBigk/summationdisplay
+0≤n≤kk−n
+n!/parenleftBigz
+2/parenrightBign
++/summationdisplay
+n≥01
+n!/parenleftBigz
+2/parenrightBign/summationdisplay
+0≤k≤n(n−k)µk
+k!/parenleftBigz
+2/parenrightBigk
+=1
+2πi/contintegraldisplay
+|w|=r<1f1/parenleftBigz
+2w/parenrightBigewz/2
+(w−1)2dw+1
+2πi/contintegraldisplay
+|w|=r<1f1/parenleftBigwz
+2/parenrightBigez/(2w)
+(w−1)2dw.
+Toprovecondition (O), westart withchangesofvariables,giving
+hc(z) =z
+2πi◦/contintegraldisplay
+|w|=|z|f1/parenleftBigw
+2/parenrightBigez2/(2w)
+(w−z)2dw+z
+2πi◦/contintegraldisplay
+|w|=|z|f1/parenleftBigw
+2/parenrightBigez2/(2w)
+(w−z)2dw,
+wherethefirstintegrationcircleisindentedtotherightto avoidthepolarsingularity w=z,andthesecond
+to theleft. By splittingeach integrationcontourintotwo p arts,weobtain
+hc(z) =z
+2πi/parenleftbigg
+⊃/integraldisplay
++⊂/integraldisplay/parenrightbigg
+f1/parenleftBigw
+2/parenrightBigez2/(2w)
+(w−z)2dw+O/parenleftbigg
+ε−2/integraldisplay
+ε≤|θ|≤π/vextendsingle/vextendsingle/vextendsingle/vextendsinglef1/parenleftbigg|z|eiθ
+2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglee|z|(cosθ)/2dθ/parenrightbigg
+,
+wheretheintegrationcontour ⊃/integraltext
+isanypathconnectingthetwoendpoints |z|e±iεandindentedtotheright,
+and⊂/integraltext
+denotesthecorrespondingsymmetriccontourwithrespectt ow=z(andindentedtotheleft). Since
+˜f1∈JS, condition (O)for˜hc(z)is readilychecked.
+50Forcondition (I), itsuffices toprove( 67). Forthatpurpose,weusetherepresentation
+˜hc(z) =e−z
+2πi/contintegraldisplay
+|w|=r<1ez(w+1/w)/2
+(w−1)2/parenleftBig
+˜f1/parenleftBigz
+2w/parenrightBig
++˜f1/parenleftBigwz
+2/parenrightBig/parenrightBig
+dw
+=˜f1/parenleftBigz
+2/parenrightBige−z
+2πi/contintegraldisplay
+|w|=r<1ez(w+1/w)/2
+(w−1)2dw+e−z
+2πi/contintegraldisplay
+|w|=r<1ez(w+1/w)/2Rz(w)dw
+=˜f1(z/2)˜h1(z)+e−z
+2πi/contintegraldisplay
+|w|=1ez(w+1/w)/2Rz(w)dw,
+where
+Rz(w) :=1
+(w−z)2/braceleftBigg/parenleftbigg
+˜f1/parenleftBigz
+2w/parenrightBig
+−˜f1/parenleftBigz
+2/parenrightBig
+−˜f′
+1/parenleftBigz
+2/parenrightBigz(w−1)
+2/parenrightbigg
++/parenleftbigg
+˜f1/parenleftBigwz
+2/parenrightBig
+−˜f1/parenleftBigz
+2/parenrightBig
++˜f′
+1/parenleftBigz
+2/parenrightBigz(w−1)
+2/parenrightbigg/bracerightBigg
+is analytic at w=z. The error term of ˜hc(z)−˜h1(z)˜f1(z/2)can be estimated by a similar argument as
+that usedforchecking condition (O). ThiscompletestheproofoftheLemma.
+Theremaininganalysisis nowroutine. Let ˜V(z) :=˜f2(z)−˜f1(z)2−z˜f′
+1(z)2. Then
+˜V(z)+˜V′(z) = 2˜V(z/2)+ ˜g2(z),
+where, byLemma 2.7,
+˜g2(z) =z−˜h1(z)2+4/parenleftBig
+˜hc(z)−˜h1(z)˜f1(z/2)/parenrightBig
+−z˜h′
+1(z)2−2z˜h′
+1(z)˜f′
+1(z/2)+z˜f′′
+1(z)2
+=/parenleftbigg
+1−2
+π/parenrightbigg
+z+O(|z|1/2),
+for|arg(z)| ≤π/2−ε. From thisand theanalyticpropertiesofthefunctionsinvo lved,wededuce( 64).
+Remark. The same approach can be extended to more general differenti al path-length of the form/summationtext
+all nodes|Tleft− Tright|mwithm≥2. Interestingly, when m= 2, themean is identical to the total in-
+ternal path-length in view of ( 66) and the variance is asymptotic to 4n2. Form >2, the mean and the
+varianceare asymptoticto
+2m/2Γ((m+1)/2)√π(1−21−m)nm/2,2m(Γ(m+1/2)−π−1/2Γ((m+1)/2)2)√π(1−21−m)nm,
+respectively.
+4.4 A weighted path-length (WPL)
+Weightedpath-lengthsoftheform Wn:=/summationtext
+1≤j≤nwjℓjappearofteninapplications,where ℓjdenotesthe
+distance of the j-th node (arranged in an appropriate manner, say first level- wise and then left-to-right or
+51in their incomingorder) to the root and wjtheweight attached to the j-th node. The calculation of Wnin
+thecaseofrandom DSTscan becarried outrecursivelyby
+Wn+1d=WBn+W∗
+n−Bn+/summationdisplay
+2≤j≤n+1wj,
+assuming that the root is labelled 1. We consider in this section the case when wj= (logj)m,m≥1.
+From atechnical pointofview,itsuffices toconsidertheran domvariables
+Xn+1d=XBn+X∗
+n−Bn+(n+1)(log(n+1))m(n≥0),
+withX0= 0, sincethepartialsum/summationtext
+2≤j≤n(logj)misnothingbut
+/summationdisplay
+2≤j≤n(logj)m= [zn]L0,m(z)
+1−z,
+where
+La,m(z) :=/summationdisplay
+k≥1n−a(logk)mzm(a/\⌉}atio\slash= 1,2,...),
+on whoseanalyticpropertiesouranalyticapproach heavily relies.
+The random variables Xnrepresent the sole example on DSTs we discuss in this paper wi th non-
+integral values; they also exhibit an interesting phenomen on in that the mean is of order n(logn)m+1but
+thevarianceisasymptoticto ntimesaperiodicfunction,in contrasttotheorders ofDPL.
+Theorem 4.5. Themean andthevarianceof theweighted path-length Xnareasymptoticto
+E(Xn) =n(logn)m+1
+(m+1)log2+n/summationdisplay
+1≤j≤mcm,j(logn)j+nPw,µ(log2n)+O/parenleftbig
+(logn)m+1/parenrightbig
+,
+V(Xn) =nPw,σ(log2n)+O/parenleftbig
+(logn)2m+2/parenrightbig
+,
+respectively, where the cm,j’s are constants depending on m, andPw,µandPw,σare1-periodic, smooth
+functions.
+That the variance is linear is well-predicted by the deep the orem of Schachinger derived in [ 58] since
+theseconddifferenceofthesequence n(logn)miso(n−1/2−ε). Ourapproachhastheadvantageofprovid-
+ing morepreciseapproximations.
+Thenewingredientweneed isincorporated inthefollowingl emma.
+Lemma 4.6 ([21]).Thefunction La,m(z)can analyticallybecontinuedintothecut-plane C\[1,∞)with
+a solesingularityat z= 1near whichitadmitstheasymptoticapproximation
+La,m(e−s) = Γ(1−a)sa−1(−logs)m+O(1),
+theO-termholdinguniformlyfor |arg(s)| ≤π−ε.
+Indeed,thetoolsdevelopedin[ 21]canalsobeeasilyextendedtosimilar“toll-functions”su chasnHm
+n.
+Details areleft fortheinterested readers.
+525 Conclusionsandextensions
+We showed in this paper, through many shape parameters on ran dom DSTs that the crucial use of the
+normalization ˜V(z) :=˜f2(z)−˜f1(z)2−z˜f′(z)2at the level of Poisson generating function is extremely
+helpful in simplifying the asymptotic analysis of the varia nce as well as the resulting expressions. The
+same idea can be applied to a large number of concrete problem s with a binomial splitting procedure.
+These and some related topics and extensions will be pursued elsewhere. We briefly mention in this final
+section someextensionsand related properties.
+Central limit theorems. All shape parameters we considered in this paper are asympto ticallynormally
+distributed in the sense of convergence in distribution. We describe the results in this section and merely
+indicate the methods of proofs. The only case that requires a separate study is NPL of random b-DSTs
+withb≥2(a bivariateconsiderationofthelimitlawsisneeded), det ailsbeinggiveninafuturepaper.
+Theorem 5.1. The internal path-length, the peripheralpath-length, the number of leaves, the differential
+path-length, the weighted path-length of random DSTs, and t he key-wise path-length of random b-DSTs
+withb≥2areallasymptoticallynormallydistributed
+Xn−E(Xn)/radicalbig
+V(Xn)d−→N(0,1),
+whereXndenotesanyoftheseshapeparameters,d−→standsforconvergenceindistribution,and N(0,1)
+is astandardnormaldistributionwithzeromean andunitvar iance.
+SeeFigure 10foraplotofthehistogramsofDPL.
+The method of moments applies to all these cases and establis hes the central limit theorems; similar
+details are given as in [ 31] (the asymptotic normality of the number of leaves being alr eady proved there
+as aspecial case).
+In aparallelway, contractionmethodalsoworkswell forall theseshapeparameters; see[ 51,52,53].
+On the other hand, Schachinger’s asymptotic normality resu lts cover the IPL, PPL, number of leaves
+and WPL, butnot PPL and KPL on b-DSTs, althoughhisapproach maybemodifiedfor thatpurpose .
+Finally,the complex-analyticapproach used in [ 35] for internal path-lengthmay beextended to prove
+some of these cases, but the proofs are messy, although the re sults established are often stronger (for
+example,withconvergencerate).
+Thedepth. Theasymptoticanalysisweusedinthispapercanalsobeexte ndedtothedepth(thedistance
+between a randomly chosen internal node and theroot) althou gh it is of logarithmicorder. Let Xndenote
+thedepthofarandom DST of nnodes. Thestartingpointis toconsidertheexpected profile polynomial
+Pn(y) :=/summationdisplay
+0≤k<nnP(Xn=k)yk,
+wherenP(Xn=k)is nothing but the expected number of internal nodes at dista ncekto the root. Then
+wehavetherecurrence
+Pn+1(y) = 1+y2−n/summationdisplay
+0≤k≤n/parenleftbiggn
+k/parenrightbigg
+(Pk(y)+Pn−k(y)) (n≥0),
+5320 40 60 801000.10.30.50.70.9n= 20
+n= 30
+n= 40n= 50
+Figure10: Thehistogramsof DPLfor n= 20,30,40and50,normalizedbytheir standarddeviations.
+withP0(y) = 0. From this relation, we obtain the equation for thePoisson g enerating function ˜F(z,y)of
+Pn(y)
+˜F(z,y)+∂
+∂z˜F(z,y) = 2y˜F/parenleftBigz
+2,y/parenrightBig
++1,
+with˜F(0,y) = 0. It follows, by taking coefficients of znon both sides and by solving the resulting
+recurrence, that
+Pn(y) =/summationdisplay
+1≤k≤n/parenleftbiggn
+k/parenrightbigg
+(−1)k−1/productdisplay
+0≤j≤k−2/parenleftBig
+1−y
+2j/parenrightBig
+(n≥1);
+see [44, p. 504] for a different proof. Asymptoticapproximation to Pn(y)can then be obtained by Rice’s
+integralformula
+Pn(y) =n−y−1
+2πi/integraldisplay
+(3/2)Γ(n+1)Γ(−s)Q(y)
+Γ(n+1−s)(1−21−sy)Q(21−sy)ds,
+for|y−1| ≤ε. Moreprecisely,if t∈Cliesin asmallneighborhoodoftheorigin,then
+E(eXnt) =Pn(et)
+n
+=(et−1)Q(et)
+Q(1)log2/summationdisplay
+k∈ZΓ/parenleftbigg
+−1−t
+log2−χk/parenrightbigg
+nt/log2+χk/parenleftbig
+1+O/parenleftbig
+n−1/parenrightbig/parenrightbig
++O(n−1),(68)
+uniformly for |t| ≤ε. Alternatively, one can also apply the Laplace-Mellin-de- Poissonization approach
+and obtain thesametypeof result for not only DSTs but also fo r more general b-DSTs. See [ 48,49] for a
+moregeneral and detailedtreatment(by adifferent approac h).
+The estimate ( 68) leads to effective asymptotic estimates for all moments of Xn−log2nby standard
+arguments;see[ 32]. In particular, weobtain
+E(Xn) = log2n+γ−1
+log2+1
+2−/summationdisplay
+k≥11
+2k−1+̟1(log2n)+O/parenleftbig
+n−1logn/parenrightbig
+,
+V(Xn) =1
+12+1
+(log2)2/parenleftbigg
+1+π2
+6/parenrightbigg
+−/summationdisplay
+k≥12k
+(2k−1)2+̟5(log2n)+O/parenleftbig
+n−1log2n/parenrightbig
+,
+where theestimateforthemean isexactly( 7)with̟1givenin ( 8)and̟5isasmoothperiodicfunction.
+54Ananalyticextension. Fromapurelyanalyticviewpoint,theunderlyingdifferent ial-functionalequation
+(13)forthemomentscan beextendedto an equationoftheform
+/summationdisplay
+0≤j≤b/parenleftbiggb
+j/parenrightbigg
+˜f(j)(z) =α˜f/parenleftbiggz
+β/parenrightbigg
++ ˜g(z) (α>0;β >1),
+forwhich ourapproach stillapplies,leadingto thefunctio nalequationfortheLaplacetransform
+(s+1)bL[˜f;s] =αβL[˜f;βs]+L[˜g;s].
+Thenatural normalizingfunctionisthen providedby
+Qβ(−s) :=/productdisplay
+j≥1/parenleftbigg
+1+s
+βj/parenrightbiggb
+,
+and thecorrespondingLaplace-Mellinasymptoticanalysis issimilar.
+In particular, the case when α=β=mcorresponds to a straightforward extension of binary DSTs
+tom-ary DSTS (and the binary unbiased Bernoulli random variabl e to the uniform distribution over
+{0,1,...,m−1}). Thestochasticbehaviorsofallshapeparametersonsucht reesfollowthesamepatterns
+as showedinthispaper.
+Yet anotherconcrete instancearises in theso-called Eden m odelstudiedby Dean and Majumdar[ 10],
+which corresponds to α=mandβ >1. The model is constructed in the followingway. We start at ti me
+t= 0at whichwehavean emptynode. Thenat time t=T,whereT∼Exponential (1), wefill theempty
+node and attach to it mdifferent empty nodes. The process then continues independ ently for each empty
+node by thefollowingrecursiverule. Once an emptynode ofde pthjis attached to atree at time t=t′, it
+is thenfilled at timepoint t′+T, whereT∼E(βj),andmnew emptynodesare attached toit.
+The mean and the variance of the number of filled nodes at a larg e time of such trees are studied in
+detailsin[ 10]. Sincethemodeliscontinuous,thereisnoneedtode-Poiss onizetoderivetheasymptoticsof
+thecoefficient;asaconsequence,nocorrectiontermasweus edinthispaperisrequiredfortheasymptotics
+ofthevariance.
+Other DST-type recurrences. While the technique of Poissonized variance with correctio n remains
+usefulforthenaturalcasewhentheBernoullirandomvariab leisnolongersymmetric,theLaplace-Mellin
+approach does not apply directly. Other asymptotic ingredi ents are needed such as a direct manipulation
+oftheMellintransforms;see[ 49]and thereferences therein.
+DST-typestructuresandrecurrencesalsoariseinothersta tisticalphysicalmodelssuchasthediffusion-
+limitedaggregation;see[ 1,5].
+Acknowledgement
+We thankthereferee foropportunehelpfulcommentsand them oreprecise title.
+Appendix. An Elementary Approach to the Asymptotic Lineari ty of
+theVariance.
+We describe briefly here a direct elementary approach to the v ariance of random variables satisfying the
+recurrence
+Xn+1d=XBn+X∗
+n−Bn+Tn,
+55where
+πn,k:=P(Bn=k) =/parenleftbiggn
+k/parenrightbigg
+2−n(0≤k≤n).
+Thestartingpointistoconsidertherecurrence satisfiedby thevariance vn:=V(Xn)
+vn+1=/summationdisplay
+0≤k≤nπn,k(vk+vn−k)+un+V(Tn),
+whereµk:=E(Xn)and
+un:=/summationdisplay
+0≤k≤nπn,k(µk+µn−k−µn+1+E(Tn))2.
+In mostcases, wehavetheestimate µk=˜f1(k)+O(kε). This,togetherwiththeGaussianapproximation
+ofthebinomialdistribution,impliesthat
+un≈/summationdisplay
+|k−n/2|=o(n2/3)
+k=n/2+x√n/2πn,k/parenleftBig
+˜f1/parenleftBign
+2+x
+2√n/parenrightBig
++˜f1/parenleftBign
+2−x
+2√n/parenrightBig
+−˜f1(n+1)+E(Tn)/parenrightBig2
+≈/summationdisplay
+|k−n/2|=o(n2/3)
+k=n/2+x√n/2πn,k/parenleftBig
+2˜f1/parenleftBign
+2/parenrightBig
+−˜f1(n)−˜f′
+1(n)+E(Tn)/parenrightBig2
+≈/parenleftBig
+2˜f1/parenleftBign
+2/parenrightBig
+−˜f1(n)−˜f′
+1(n)+E(Tn)/parenrightBig2
+.
+But then (see( 13)below)
+2˜f1/parenleftBign
+2/parenrightBig
+−˜f1(n)−˜f′
+1(n)+E(Tn) =E(Tn)−˜h1(n),
+where
+˜h1(z) :=e−z/summationdisplay
+j≥0E(Tj)
+j!zj.
+The order of the difference E(Tn)−˜h1(n)≈n|˜h′′
+1(n)|are expected to be small, roughly O(nε)in all
+cases we considerhere. Consequently, the variance is asymp toticallylinear; see [ 31,58] for more precise
+details.
+We see clearly that the smallness of the variance results nat urally from the high concentration of the
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diff --git a/1001.0096.txt b/1001.0096.txt
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@@ -0,0 +1,347 @@
+arXiv:1001.0096v1  [hep-ph]  31 Dec 2009January 2010
+Relevance of the CDMSII events for mirror dark matter
+R. Foot1
+School of Physics,
+University of Melbourne,
+Victoria 3010 Australia
+Mirror dark matter offers a framework to explain the existing dark m atter direct detec-
+tion experiments, including the impressive DAMA annual modulation sig nal. Here we
+examine the implications of mirror dark matter for experiments like CD MSII/Ge and
+XENON10 which feature higher recoil energy threshold than the DA MA NaI experi-
+ments. We show that the two events seen in the CDMSII/Ge experim ent are consistent
+with the interactions of the anticipated heavy ∼Fe′component. This interpretation
+of the CDMSII/Ge events is a natural one given that a) mirror dark matter predicts
+an event rate which is sharply falling with respect to recoil energy an d b) that the two
+observed events are in the low energy region near threshold. Impo rtantly this interpre-
+tation of the CDMSII events can be checked by on-going and futur e experiments, and
+we hereby predict that the bulk of the events will be in the ER<∼18 keV region.
+1E-mail address: rfoot@unimelb.edu.auRecently the CDMS collaboration[1] have announced two dark matte r candidate
+events fromtheirfinalanalysiscontaining612kg-daysofrawexpo sureonaGermanium
+target. These two events have recoil energies of 12.3 keV and 15.5 keV, which belong
+to the energy region just above the experimental threshold ener gy of 10 keV. The
+backgroundestimateforthe entire10-100keVregionisaround0.8events. However, the
+fact that bothevents are in the low energy part of the spectrum provides an inter esting
+hint that these events are dark matter induced. In fact, theore tical studies within
+the mirror dark matter framework predicted[2] that any positive s ignal obtained by
+CDMSandsimilarexperiments shouldbeinthelowrecoilenergyregionn earthreshold.
+Moreover the background for this low energy (10-20 keV) ‘signal’ r egion is likely 0.2
+events or less for the 612 kg-day exposure, and thus the observ ation of two events in
+the 10-20 keV recoil energy range can be viewed as interesting evid ence for mirror dark
+matter.
+Recall, mirror dark matter posits that the inferred dark matter in t he Universe
+arises from a hidden sector which is an exact copy of the standard m odel sector[3]
+(for a review and more complete list of references see ref.[4])2. That is, a spectrum of
+darkmatter particles of known masses arepredicted: e′,H′,He′,O′,Fe′,...(withme′=
+me,mH′=mH,etc). The galactic halo is then presumed to be a spherically distribute d
+self interacting mirror particle plasma comprising these particles. Su ch a plasma would
+radiatively cool on a time scale of a few hundred million years unless a sig nificant heat
+source exists. It turns out that ordinary supernova can supply t he required energy
+if ordinary and mirror particles interact with each other via (renorm alizable) photon-
+mirror photon kinetic mixing[3, 6]:
+Lmix=ǫ
+2FµνF′
+µν (1)
+whereFµν(F′
+µν)istheordinary(mirror) U(1)gaugebosonfieldstrengthtensor. Match-
+ing the rate of energy lost in the galactic halo due to radiative cooling w ith the energy
+supplied by ordinary supernova’s (the fundamental process being kinetic mixing in-
+ducedplasmondecay into e′+e′−inthesupernova core) gives anestimate[7]of ǫ∼10−9.
+Importantly, a mirror sector with kinetic mixing of this magnitude is co nsistent with
+all known laboratory, astrophysical and cosmological constraint s[8], and crucially such
+values of ǫmake the theory experimentally testable via dark matter direct det ection
+experiments.
+The most sensitive dark matter direct detection experiment to mirr or dark matter
+happens to be the impressive DAMA/NaI[9] and DAMA/Libra experime nts[10]. The
+low recoil energy threshold (2 keV) of these experiments makes th em sensitive to the
+elastic scattering of a putative ∼O′component off the target nuclei. It turns out
+that the annual modulation signal obtained by the DAMA experiment s can be fully
+explained and yields a measurement of ǫ[2]:
+ǫ/radicalBigg
+ξO′
+0.1= (1.0±0.3)×10−9(2)
+2Note that successful big bang nucleosynthesis (BBN) and succes sful large scale structure (LSS)
+requires effectively asymmetric initial conditions in the early Universe ,T′≪Tandnb′/nb≈5. See
+ref.[5] for further discussions.
+1whereξA′≡nA′mA′/(0.3GeV/cm3) is the halo mass fraction of the species A′. Other,
+but more tentative evidence for mirror dark matter has emerged r ecently from an
+analysis of the CDMS electron scattering data[11]. It was shown[12] that the CDMS
+electron scattering data can be interpreted in terms of e′scattering on electrons, and
+suggests
+ǫ≈0.7×10−9. (3)
+However CDMSII[1] and XENON10[13] are the most sensitive experim ents to a heavier
+∼Fe′component and a positive signal for CDMS and similar experiments has been
+anticipated[14]. We will show that interpreting the two events identifi ed in the CDM-
+SII/Ge analysis as an Fe′signal is consistent with all other experiments, and yields
+the estimate:
+ǫ/radicalBigg
+ξFe′
+10−3≈10−9. (4)
+The interaction rate for an experiment like CDMS depends on the cro ss-section,
+dσ/dE R, andhalovelocitydistribution, f(v). Thephoton-mirrorphotonkineticmixing
+enables a mirror nucleus (with mass and atomic numbers A′, Z′and velocity v) to
+interact with an ordinary nucleus (presumed at rest with mass and a tomic numbers
+A, Z) via Rutherford elastic scattering. The cross section is given by[2]:
+dσ
+dER=λ
+E2
+Rv2(5)
+where
+λ≡2πǫ2Z2Z′2α2
+mAF2
+A(qrA)F2
+A′(qrA′) (6)
+andFX(qrX) (X=A,A′) are the form factors which take into account the finite size of
+the nuclei and mirror nuclei. [The quantity q= (2mAER)1/2is the momentum transfer
+andrXis the effective nuclear radius]3. A simple analytic expression for the form
+factor, which we adopt in our numerical work, is the one given by Helm [15, 16]:
+FX(qrX) = 3j1(qrX)
+qrXe−(qs)2/2(7)
+withrX= 1.14X1/3fm,s= 0.9 fm and j1is the spherical Bessel function of index 1.
+The halo distribution function is given by a Maxwellian distribution,
+fi(v) =e−1
+2miv2/T
+=e−v2/v2
+0[i](8)
+where the index ilabels the particle type [ i=e′,H′,He′,O′,Fe′...]. The halo mir-
+ror particles form a self interacting plasma at temperature T. The dynamics of the
+3Unless otherwise specified, we use natural units, ¯ h=c= 1 throughout.
+2mirror particle plasma has been investigated previously[7], where it wa s found that
+the condition of hydrostatic equilibrium implied that the temperature of the plasma
+satisfied:
+T≃1
+2¯mv2
+rot, (9)
+where ¯m=/summationtextnimi//summationtextni[i=e′,P′,He′,O′....] is the mean mass of the particles in
+the plasma, and vrot≈254 km/s is the local rotational velocity for our galaxy[17].
+Assuming the plasma is completely ionized, a reasonable approximation since it turns
+out that the temperature of the plasma is ≈1
+2keV, then:
+¯m
+mp=1
+2−5
+4YHe′(10)
+whereYHe′is theHe′mass fraction. Mirror BBN studies[18] indicate YHe′≈0.9, which
+is the value we adopt in our numerical work. Clearly, eqs.(8,9) imply tha t the velocity
+dispersion of the particles in the mirror matter halo depends on the p articular particle
+species and satisfies:
+v2
+0[i] =2T
+mi
+=v2
+rotm
+mi. (11)
+Notethatif mi≫m, thenv2
+0[i]≪v2
+rot. Consequently mirrornuclei havetheirvelocities
+(and hence energies) relative to the earth boosted by the Earth’s (mean) rotational
+velocity aroundthegalacticcenter, ≈vrot. Thisallowsamirror nuclei withmass mA′=
+18±4 GeV [consistent with a putative O′component], to provide a significant annual
+modulation signal in the energy region probed by dama (2 < ER/keV <6). It turns
+out such a mirror dark matter component has the right properties to fully account[2]
+for the data presented by the DAMA collaboration[10] including the o bserved energy
+dependence of the annual modulation amplitude. Furthermore the narrow velocity
+distributionimplied by v2
+0≪v2
+rotsuppresses thesignalforhigher thresholdexperiments
+like CDMS. The result is an explanation for the DAMA annual modulation signal
+completely consistent with all the other experiments.
+The interaction rate for an experiment like CDMS is given by4:
+dR
+dER=NTnA′/integraldisplaydσ
+dERfA′(v,vE)
+k|v|d3v
+=NTnA′λ
+E2
+R/integraldisplay∞
+|v|>vmin(ER)fA′(v,vE)
+k|v|d3v (12)
+whereNTis the number of target nuclei per kg of detector and nA′=ρdmξA′/mA′is
+the number density of halo mirror nuclei A′at the Earth’s location (we take ρdm=
+4Detector resolution effects are incorporated by convolving this ra te with a Gaussian distribution
+whereσ/keV= 0.2 for CDMSII and σ/keV= 0.579/radicalbig
+ER/keV+0.021ER/keVfor XENON10[19]
+30.3GeV/cm3). Also,fA′(v,vE)/k=exp[−(v+vE)2/v2
+0]/kis theA′velocity distribu-
+tion (k≡[πv2
+0(A′)]3/2is the normalization factor). Here, vis the velocity of the halo
+particles relative to the Earth, and vEis the Earth’s velocity relative to the galactic
+halo. [The bold font is used to indicate that the quantities are vector s]. Note that the
+lower velocity limit, vmin(ER), is given by the kinematic relation:
+vmin=/radicaltp/radicalvertex/radicalvertex/radicalbt(mA+mA′)2ER
+2mAm2
+A′. (13)
+The velocity integral in Eq.(12) is standard and can easily be evaluate d in terms of
+error functions assuming a Maxwellian dark matter distribution[16],
+/integraldisplay∞
+|v|>vmin(ER)fA′(v,vE)
+k|v|d3v=1
+2v0y[erf(x+y)−erf(x−y)] (14)
+where
+x≡vmin(ER)
+v0, y≡|vE|
+v0. (15)
+The Earth’s velocity relative to the galactic halo, vE, has an estimated mean value of
+∝angbracketleftvE∝angbracketright ≃vrot+12 km/s, where vrot≈254 km/s, the local rotational velocity[17].
+Let us examine interactions of A′=O′andA′=Fe′on a Ge target. From Eq.(13)
+we find:
+vmin
+vrot≃1.77/radicalBigg
+ER
+10 keV,forA′=O′,
+vmin
+vrot≃0.74/radicalBigg
+ER
+10 keV,forA′=Fe′. (16)
+Given that the velocity dispersion v0(A′=O′,Fe′)≪vrot, it follows that the veloc-
+ities of the halo particles relative to the Earth are distributed narro wly around the
+mean≈vrot. Therefore the interactions of the O′component will be exponentially
+suppressed because only O′particles in the tail of the Maxwellian distribution can lead
+to recoils with energies greater than the 10 keV threshold. The O′interaction rate can
+be determined using the DAMA measurement of ǫ√ξO′, Eq.(2), and we have checked
+that the event rate in CDMSII/Ge for the O′component is indeed negligible, typically
+<∼10−2events for their 1009.8 kg-days of total raw exposure. [The tota l CDMSII
+exposure consists of 397.8 kg-days for their first results[20] and 612 kg-days for their
+final exposure[1]]. However for a heavier component, such as A′=Fe′, there is no
+exponential suppression at or near the CDMS threshold since vmin/vrot<∼1. Thus, the
+CDMSII experiment is sensitive to a heavier component, which we tak e asA′=Fe′.5
+5Demanding vmin/vrot<∼1 atER= 10keV implies mA′>∼35mP. That is CDMSII/Ge is primarily
+sensitive to mirror elements heavier than about chlorine. By far the most abundant ordinary element
+heavier than chlorine is iron, and this motivates our assumption that the dominant mirror element
+heavier than chlorine in the mirror sector is Fe′.
+4The most exciting possibility is that CDMSII has seen this heavy compo nent,
+which has been anticipated for a while[14]. The CDMSII/Ge event rate isR=/integraltextdR
+dERE(ER)dER, whereE(ER)≃0.18 + 0.007ERis the detection efficiency for the
+CDMSII low energy region[1]. Adjusting this rate to give around 2 eve nts for the total
+raw exposure of 1009.8 kg-days, gives a determination of ǫ√ξFe′:
+ǫ/radicalBigg
+ξFe′
+10−3≈10−9. (17)
+Theresultingrecoilenergyspectrumisgiveninfigure1. Asthefigur edemonstrates, the
+‘signal’ region is in the recoil energy range 10-20 keV, which is precise ly where the two
+eventsseenbyCDMSwerelocated. Observethatfor ER<∼18keVtheratefallsroughly
+as 1/E2
+R, which is due to thedσ
+dER∝1/E2
+Rof the Rutherford elastic scattering cross-
+section. For ER>∼18 keV, the rate begins to fall even faster as vmin/vrotmoves into the
+tail of the Maxwellian velocity distribution. If the CDMSII experiment has really seen
+Fe′interactions, then this possibility can be checked by future measur ements, with the
+bulk of the events predicted to lie in the energy region<∼18 keV.
+ 0 0.2 0.4 0.6 0.8 1
+ 6  8  10  12  14  16  18  20  22  24Counts/keV
+Recoil Energy [keV]CDMS THRESHOLD
+ 0 0.2 0.4 0.6 0.8 1
+ 6  8  10  12  14  16  18  20  22  24Counts/keV
+Recoil Energy [keV]CDMS THRESHOLD
+Figure 1: Recoil energy spectrum,dR
+dERE(ER)T, predicted for the CDMSII detector for
+the total raw exposure of T = 1009.8 kg-days, where E(ER) is the CDMS detection
+efficiency. We have taken parameters: ǫ/radicalBigξFe′
+1.2×10−3= 10−9,vrot= 254 km/s which leads
+to around two expected events above threshold.
+5The XENON10 experiment[13], with their published raw exposure of 31 6 kg-days,
+has a sensitivity comparable to the CDMSII experiment. In the origin al XENON10
+analysis[13], their threshold was assumed to be 4.5 keV, however in a s ubsequent
+study[21]alowerthresholdof2keVwasused. However, thecalibra tionoftheXENON10
+detector depends on the relative scintillation efficiency Leffwhich was originally taken
+to be constant, Leff= 0.19 in ref.[13, 21], but recent measurements[22, 23] have shown
+that in fact Leffis energy dependent. The most recent measurements[23] sugges t
+Leff= 0.10±0.03 atER≃8 keV and Leff= 0.07±0.03 atER≃4 keV. Given that
+the threshold energy is inversely proportional to Leff, it follows that the new measur-
+ments of Leffraise the threshold from 2 keV to around 5 .4 keV, with an uncertainty
+of roughly 30%. In figure 2 we plot the expected count rate for Fe′interactions in
+XENON10 for the same parameters used in figure 1: ǫ/radicalBigξFe′
+1.2×10−3= 10−9,vrot= 254
+km/s. As the figure suggests, for an energy threshold of 5 .4 keV, we might have ex-
+pected a couple of events to have been seen in XENON10, however, with such small
+statistics and also the significant uncertainty in the XENON10 energ y calibration,
+there is no clear disagreement between the two experiments. Impo rtantly XENON10
+has been superseded by the XENON100 experiment with more than a n order of mag-
+nitude improvement in sensitivity. XENON100 is currently in operation and should
+have their first results during 2010. One might reasonably expect X ENON100 to see
+around a dozen or more events in the low energy region ER<∼18 keV if the mirror
+dark matter interpretation of the two CDMSII events is correct.
+ 0 0.2 0.4 0.6 0.8 1
+ 4  6  8  10  12  14  16  18  20Counts/keV
+Recoil Energy [keV]XENON THRESHOLD
+ 0 0.2 0.4 0.6 0.8 1
+ 4  6  8  10  12  14  16  18  20Counts/keV
+Recoil Energy [keV]XENON THRESHOLD
+Figure 2: Recoil energy spectrum,dR
+dERE(ER)T, predicted for the XENON10 detector
+with raw exposure of T= 316 kg-days, where E(ER)≃0.4 is the XENON overall
+detection efficiency[13]. We have taken the same parameters as figu re 1:ǫ/radicalBigξFe′
+1.2×10−3=
+10−9,vrot= 254 km/s.
+6We have checked that the addition of an Fe′component with ǫ/radicalBig
+ξFe′
+10−3≈10−9,
+does not significantly change the good fit to the DAMA/Libra annual modulation
+signal given by the O′component. The DAMA experiments remain the most sensitive
+existing experiments to the ∼O′component, while the CDMSII and the other higher
+threshold experiments are the most sensitive probes of a heavier ∼Fe′component.
+Thus, the CDMS and similar experiments have an important role in prob ing the heavy
+∼Fe′component which is complimentary to experiments such as DAMA/NaI and
+DAMA/Libra which probe the lighter O′component. Note that our estimate of ǫ√ξFe′
+from the CDMSII events can be combined with the ǫ√ξO′value inferred from the
+DAMA/Libraexperiment toyield ξFe′/ξO′≈10−2. Itisinteresting thatthisisthesame
+order of magnitude as the corresponding quantity for ordinary ma tter in our galaxy
+and demonstrates that our combined interpretation of the DAMA/ Libra experiment
+and the two CDMSII events is plausible.
+In conclusion, we have examined the possibility that the two events s een in the
+CDMSIIexperiment areinfactmirrordarkmatterinteractionsfro mtheanticipated[14]
+heavy∼Fe′component. This is actually a natural possibility given that a) mirror
+dark matter predicts an event rate which is sharply falling with respe ct to recoil energy
+and that b) the two observed events are in the low energy region ne ar threshold.
+Importantly the mirror dark matter interpretation of the CDMSII /Ge events can be
+checked by on-going and future experiments, such as SuperCDMS , XENON100, and
+others, with the bulk of the events predicted to arise in the ER<∼18 keV region.
+Acknowledgments
+This work was supported by the Australian Research Council.
+References
+[1] Z. Ahmed et al.(CDMS Collaboration), arXiv: 0912.3592 (2009).
+[2] R. Foot, Phys. Rev. D78, 043529 (2008) [arXiv: 0804.4518]. See also: R. Foot,
+Phys. Rev. D69, 036001 (2004) [hep-ph/0308254]; astro-ph/04 03043; Mod. Phys.
+Lett. A19, 1841(2004)[astro-ph/0405362]; Phys. Rev. D74, 02 3514(2006)[astro-
+ph/0510705].
+[3] R. Foot, H. Lew and R. R. Volkas, Phys. Lett. B272, 67 (1991); Mod. Phys. Lett.
+A7, 2567 (1992).
+[4] R. Foot, Int. J. Mod. Phys. D13, 2161 (2004) [astro-ph/0407 623].
+[5] Z. Berezhiani, D. Comelli and F. L. Villante, Phys. Lett. B503, 362 (2001) [hep-
+ph/0008105]; L. Bento and Z. Berezhiani, Phys. Rev. Lett. 87, 23 1304 (2001)
+[hep-ph/0107281]; A. Yu. Ignatiev and R. R. Volkas, Phys. Rev. D6 8, 023518
+(2003) [hep-ph/0304260]; R. Foot and R. R. Volkas, Phys. Rev. D6 8, 021304
+(2003) [hep-ph/0304261]; Phys. Rev. D69, 123510 (2004) [hep-p h/0402267]; Z.
+7Berezhiani, P. Ciarcelluti, D. Comelli and F. L. Villante, Int. J. Mod. Ph ys. D14,
+107 (2005) [astro-ph/0312605]; P. Ciarcelluti, Int. J. Mod. Phys. D14, 187 (2005)
+[astro-ph/0409630]; Int. J. Mod. Phys. D14, 223 (2005) [astro- ph/0409633].
+[6] R. Foot and X-G. He, Phys. Lett. B267, 509 (1991).
+[7] R. Foot and R. R. Volkas, Phys. Rev. D70, 123508 (2004) [astro -ph/0407522].
+[8] R. Foot, A. Yu. Ingatiev and R. R. Volkas, Phys. Lett. B503, 35 5 (2001)
+[arXiv: astro-ph/0011156]; R. Foot, Int. J. Mod. Phys. A19 3807 (2004) [astro-
+ph/0309330]; R. Foot and Z. K. Silagadze, Int. J. Mod. Phys. D14, 143 (2005)
+[astro-ph/0404515]; P. Ciarcelluti and R. Foot, Phys. Lett. B679 , 278 (2009)
+[arXiv: 0809.4438]. See also, S. Davidson, S. Hannestad and G. Raffe lt, JHEP 5,
+3 (2000) [arXiv: hep-ph/0001179].
+[9] R. Bernabei et al. (DAMA Collaboration), Riv. Nuovo Cimento. 26, 1 (2003)
+[astro-ph/0307403]; Int. J. Mod. Phys. D13, 2127 (2004); Phys . Lett. B480, 23
+(2000).
+[10] R. Bernabei et al. (DAMA Collaboration), Eur. Phys. J. C56: 333 (2008)
+[arXiv:0804.2741].
+[11] Z. Ahmed et al.(CDMS Collaboration), ArXiv: 0907.1438 (2009); Phys. Rev.
+Lett. 103: 141802 (2009) [ArXiv: 0902.4693].
+[12] R. Foot, Phys. Rev. D80, 091701 (2009) [arXiv:0909.3126].
+[13] J. Angle et al. (XENON Collaboration), Phys. Rev. Lett. 100, 021303 (2008)
+[arXiv:0706.0039].
+[14] See e.g. R. Foot, arXiv: 0907.0048 (2009).
+[15] R. H, Helm, Phys. Rev. 104, 1466 (1956).
+[16] J. D. Lewin and P. F. Smith, Astropart. Phys. 6, 87 (1996).
+[17] M. J. Reid et al., arXiv: 0902.3928 (2009).
+[18] P. Ciarcelluti and R. Foot, work in progress.
+[19] C. Savage, G. Gelmini, P. Gondolo and K. Freese, arXiv: 0808.360 7 (2008).
+[20] Z. Ahmed et al.(CDMS Collaboration), Phys. Rev. Lett. 102.011301 (2009)
+[arXiv: 0802.3530].
+[21] J. Angle et al.(XENON10 Collaboration), arXiv: 0910.3698 (2009).
+[22] E. Aprile et al., arXiv: 0810.0274 (2008).
+[23] A. Manzur et al., arXiv: 0909.1063 (2009).
+8
\ No newline at end of file
diff --git a/1001.0097.txt b/1001.0097.txt
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@@ -0,0 +1,350 @@
+Order In Chaos: Definite Rules T hat Govern The Drift Of Moon 
+Away From The Earth  
+Sohail Alam  and Dr. B. K. Sharma  
+Department of Electronics and Communication Engineering  
+National Institute of Technology , Patna, India  
+Sa4250mnpo70@yahoo.com  , electronics@nitp.ac.in  
+ 
+ 
+Abstract  
+When the Moon was formed it was much closer to the Earth than it is today. It just needed about 20 days 
+then to  go around the Earth. Now it takes the Moon 29.5 days to make one revolution. In order to follow 
+the conservation of angular momentum the Moon had to either move closer to the Earth or recede from 
+Earth. The data from the Lunar Laser Ranging Experiment con firms it to be moving away and the velocity 
+of recession of the Moon have been found to be 3.8 cm/year. This rate is not constant though. At present 
+the Moon's orbit has a radius of 384,000km.  
+But what is not yet known to all is that the orbital motion of the moon is actually a spiral motion. The 
+moon is spiralling out and a very simple mathematical equation can describe the actual spiral motion of the 
+Moon. The generalisation of this differenti al equation is the basic aim of this paper  here.  
+ 
+Keywords - lunar  radial expansion; geo -synchronous orbit; synchronous orbit; LOD Curve;  
+ 
+Introduction  
+ 
+The rock samples brought from our Moon during 
+Apollo 11 to Apollo 17 Mission an d during Luna 
+16 and Luna 20 Mission conclusively prove that 
+Earth and Moon had been formed from the disc of 
+accretion about 4.53 billion years ago but they 
+were never a single body. The Age of Moon has 
+been found incorrect by recent findings. The new 
+age is around 4.467 billion years.  Out of the 
+many theories finally in 1984 at the International 
+Conference held at Kona, Hawaii, Giant Impact 
+Theory  was accepted as the most consistent 
+theory regarding Moon’s origin . When the Moon was formed it was much closer 
+to the Earth than it is today.  It just needed about 
+20 days then to go around the Earth.  Now i t takes 
+the Moon 29.5 days to make one revolutio n. In 
+order to follow the conservation of angular 
+momentum the Moon had to eithe r move closer to 
+the Earth or recede from Earth. The data from the 
+Lunar Laser Ranging Experiment confirms  it to 
+be moving  away and the velocity of recession of 
+the Moon have been found to be 3.8 cm/year. 
+This rate is not constant  though . At present the 
+Moon's orbit has a radius of 384,000km.  The first laser ranging retro  reflector was 
+positioned on the Moon in 1969 by the Apollo 11 
+astronauts. By beaming laser pulses at the 
+reflector from Earth, scientists have been able to 
+determine the round -trip travel time that gives the 
+distance between the two bodies at any time to 
+accuracy  of about 3 centimetres .  
+The reason for the increase is that the Moon 
+raises tides on the Earth. Because the side of the 
+Earth that faces the Moon is closer, it feels a 
+stronger pull of gravity than the centre of the 
+Earth. Similarly, the part of t he Earth facing away 
+from the Moon feels less gravity than the centre 
+of the Earth. This effect stretches the Earth a bit, 
+making it a little bit oblong. We call the parts that 
+stick out "tidal bulges." The actual solid body of 
+the Earth is distorted a few  centimetres, but the 
+most noticeable effect is the tides raised on the 
+ocean. Now, all mass exerts a gravitational force, 
+and the tidal bulges on the Earth exert a 
+gravitational pull on the Moon. Because the Earth 
+rotates faster (once every 24 hours) than  the 
+Moon orbits (once every 27.3 days) the bulge 
+tries to "speed up" the Moon, and pull it ahead in 
+its orbit. The Moon is also pulling back on the 
+tidal bulge of the Earth, slowing the Earth's 
+rotation. Tidal friction, caused by the movement 
+of the tidal  bulge around the Earth, takes energy 
+out of the Earth and puts it into the Moon's orbit, 
+making the Moon's orbit bigger (but, a bit 
+paradoxically, the Moon actually moves slower!). 
+The Earth's rotation is slowing down because of 
+this. One hundred years fr om now, the day will be 2 milliseconds longer than it is now. This same 
+process took place billions of years ago --but the 
+Moon was slowed down by the tides raised on it 
+by the Earth. That's why the Moon always keeps 
+the same face pointed toward the Earth. Because 
+the Earth is so much larger than the Moon, this 
+process, called tidal locking, took place very 
+quickly, in a few tens of millions of years. Many 
+physicists considered the effects of tides on the 
+Earth -Moon system. However, George Howard 
+Darwin (Cha rles Darwin's son) was the first 
+person to work out, in a mathematical way, how 
+the Moon's orbit would evolve due to tidal 
+friction, in the late 19th century. He is usually 
+credited with the invention of the modern theory 
+of tidal evolution.  
+There is a tim e lag between Earth -Moon 
+interaction and Earth's deformation. It is this time 
+lag and belated deformation of Earth which 
+causes a retarding torque on Earth. The two 
+bulges act as brake -shoes as these brake -shoes act 
+in Cycle wheel.  
+ 
+ 
+ Problem Definition  
+It was established a century ago by George 
+Howard Darwin that Earth -Moon interaction is 
+causing Lunar Orbital Radius expansion and 
+Secular Lengthening of Day (L.O.D). This paper  intends  to study the  generalisation of the  
+theoretical formulation of Lunar Orbital Radius 
+Expansion.  
+ 
+Details about the Problem  
+When a capacitance is being charged from a 
+constant voltage source usi ng a resistance,  the 
+capacitance voltage exponentially grows or 
+decays with the time constant of growth or decay 
+given by T = RC. Is there some such formulation 
+in the orbital radius expansion or contraction 
+curve? Is there a similar time constant here? A nd 
+if indeed there is time constant what does it 
+depend upon? By the studies conducted till now 
+we have defined a time constant T = 𝑎𝐺2−𝑎𝐺1
+𝑉𝑚𝑎𝑥 
+where a G1 is inner geosynchronous orbit, a G2 is 
+outer geosynchronous orbit  and V max is the 
+maximum velo city of recession as a result of  
+gravitational sling shot effect.  
+But what is not yet known to all is that the orbital 
+motion of the moon is actually a spiral motion. 
+The moon is spiralling out and a very simple 
+mathematical equation can describe the actu al 
+spiral motion of the Moon. The generalisation  of 
+this differential equation is the basic aim of this 
+paper  here. It has been shown in the thesis of Dr. 
+Bijay Kumar Sharma, named as “THE 
+DYNAMICS OF PLANETARY SATELLITES 
+AND THE GENERALIZATION OF THE SAME 
+PROVIDES A NEW THEORY OF THE BIRTH 
+AND EVOLUTION OF OUR SOLAR 
+SYSTEM”  that in any two body system there are two Geo -synchronous orbits 
+1Ga  and 
+2Ga . 
+When the secondary is tidally locked with the 
+primary as all the natural satellites are, then they 
+are called synchronous motion.  Moon is in 
+synchronous motion. Hence we see the  same face 
+of Moon eternally. Whereas , Charo is tidally 
+interlocked with Pluto. This is analogous to geo -
+synchrony. Charon spin period= Charon's orbital 
+period= Pluto’s  spin period. If Moon was tidally 
+interlocked with Earth then Moon would be 
+permanently fixed over a given point say Patna 
+(the capital of Bihar, INDIA) as a geo -
+synchronous satellite is. Plus when secondary 
+body is in sub -synchronous orbit, angular 
+momentum is transferred from Moon to Earth 
+(secondary to primary body). Therefore Moon 
+gets tr apped in a contracting spiral orbit. If it was 
+in extra synchronous orbit then angular 
+momentum would be transferred from Earth to 
+Moon as presently it is happening and Moon is 
+receding at the rate of 3.8cm. The Geo -
+Synchronous orbits may be defined as that  where 
+the spin period of the secondary body  equals that 
+of the Primary  as well and the orbital period of 
+the Secondary  equals the spin period of the 
+Primary . If the accreting body or the captured 
+body, as is the case for Martian Satellite Phobos, 
+lies wit hin 
+1Ga  then the Satellite is gravitationally launched on an inward spiral path 
+towards its sure doom. If the accreting zone lies 
+beyond 
+1Ga  as is the case with our Moon or the 
+captured body lies beyond 
+1Ga  as is the case for 
+the Martian Satellite Deimos, then the said 
+satellite experiences an impulsive, gravitational 
+sling shot effect because of Conservation of 
+Energy Principle.  It has rightly  been  assumed by Dr. B.K. Sharma  
+that the trajectory of evolution obeys the  rules 
+analogous  to Fourier expansion series having the 
+form  𝐀ⅇ−𝐭
+𝟑𝐓+𝐁ⅇ−𝐭
+𝟐𝐓+𝐂ⅇ−𝐭
+𝐓+⋯  where A, B, 
+C … are some constants . Here we approximate  
+the differential equation to be of third order only, 
+(it can be of higher order  but the mathematics get 
+quite involved above order eight or nine, but the 
+basic formulation is the same).  
+ 
+ 
+The Figure represents the Spiral Trajectory of Moon traversed from the birth to the present Orbit of 
+3.844x10^8m over  4.53Gy of time period. (Ref. 4)  
+ 
+All these mathematical formulation has been 
+done using Wolfram Mathematica 7 software . 
+Let us take the case of a polynomial whose roots 
+are -1/T, -1/2T and -1/3T. 
+𝐏𝐨𝐥𝐲𝐧𝐨𝐦𝐢𝐚𝐥=(𝒙+1
+𝑻)(𝒙+1
+2𝑇)(𝒙+1
+3𝑇) On expansion using the Expand  command we get 
+the following result  
+𝟏
+𝟔𝑻𝟑+𝒙
+𝑻𝟐+𝟏𝟏𝒙𝟐
+𝟔𝑻+𝒙𝟑 
+Now observing the above expression we 
+construct a differential equation as follows  
+ 
+Consider 𝐫𝐋 to be the radial distance of the Moon 
+from Earth and 𝐚𝐠𝟏 and 𝐚𝐠𝟐 are the first and 
+second Geo -synchronous orbit respectively. Then 
+consider  
+𝐱=𝐚𝐠𝟐−𝐫𝐋
+−𝐚𝐠𝟏+𝐚𝐠𝟐 
+ 
+Now solving the differential equation using 
+DSolve  command  
+𝐃𝐒𝐨𝐥𝐯𝐞 𝐱′′′ 𝐭 =
+=−𝟏𝟏 𝟔𝐓  𝐱′′ 𝐭 −𝟏𝐓𝟐 𝐱′ 𝐭 
+−𝟏 𝟔𝐓𝟑  𝐱 𝐭 ,𝐱 𝐭 ,𝐭  
+The solution is  
+𝐚𝐠𝟐−𝐫𝐋
+−𝐚𝐠𝟏+𝐚𝐠𝟐=𝐂 𝟏 ⅇ−𝐭
+𝟑𝐓+𝐂 𝟐 ⅇ−𝐭
+𝟐𝐓+𝐂 𝟑 ⅇ−𝐭
+𝐓  
+where C[1],C[2],C[3] are the three constants of 
+integration.   
+This solution is exactly what we have assumed to 
+be.  
+There  are four  unknowns  here  C[1],C[2],C[3],T 
+Also any of the following boundary conditions 
+may be used  
+rL= a G2 at t=infinity  and  
+rL = a G1 at t= 0  
+And in case of Earth -Moon system the two Geo -
+synchronous orbits have been found to be  
+ag1= 1.46257569*107 m 
+ag2= 5.52887891*108 m 
+We know from Laser Lunar Ranging Experiment 
+that the Moon is at a distance of 3.844*108 m 
+from the Earth at present, after 4.53*109 years 
+after the Giant Impact.  
+Hence, rL= 3.844*108 at t= 4.53*109 
+ 
+                                                                                 
+ 
+Results and discussion  
+All we have to do is find the constants of 
+integration and the Time Constant of Evolution.  
+The Time Constant was found out to be 
+T=3.90028×109, by solving the following 
+equation  
+𝐒𝐨𝐥𝐯𝐞 𝟏−ⅇ−𝟒.𝟓𝟑×𝟏𝟎𝟗𝐓 ==𝟎.𝟔𝟖𝟔𝟗𝟕,𝐓  
+Now satisfying different points in space at which 
+the Moon was present at time t  (after the Giant 
+Impact ), we get simultaneous equations containing  the constants of integration C[1],  C[2],  
+C[3].  
+For example satisfying the following points  
+rL  = 2.227*108 at t=1.73*109,  rL= 3.844*108 at t= 
+4.53*109 , and rL = a G1 at t= 0 and solving for 
+C[1],  C[2],  C[3] we get  
+C[1]=  1.88839, C[2]=  -2.72107, C[3]=  1.83268  
+ 
+ 
+Now putting these values of T,  C[1],  C[2] and 
+C[3] and plotting the radial distance with respect 
+to time we get the following curve : 
+ Also on plotting the different points  in space at 
+any given time where the moon would have been 
+possibly present  using the data generated by Prof. 
+B.K.Sharma we get the experimental curve as 
+follows : 
+ 
+ 
+Summary  
+It can be seen that the m oon actually is moving 
+outward and is approaching for its outer geo -
+synchronous orbit. Also when the two plotted 
+graphs are compared we find a very accurate 
+match . We need to study the orbital radius  
+expansion or contraction and  see if they fit in a 
+general formulation.  Also these formulations will 
+the give some very significant idea about the 
+Lengthening of Day (L.O.D).  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ Future Directions  
+There are several causative factors for the 
+deviations in the actual sidereal day L.O.D.  
+Curve  with respect to the theoretical L.O.D. curve 
+as predicted on the basis of Earth -Moon Tidal 
+Interaction. Because it has a causative factor 
+hence the deviations a re not random but chaotic. 
+If they are indeed chaotic they should be carrying 
+the signatures of the various causative factors 
+such as plate -tectonics, ice -caps expansion and 
+thawing, electromagnetic coupling between core 
+and mantle, El -Nino Ocean Currents and global 
+wind pattern interactions with various mountain 
+ranges. If the various signatures are identified 
+then we may isolate the precursors of Earth -quake and sudden Volcanic Eruptions in L.O.D. curve 
+fluctuations. With the advent of quantum gyro 
+develo ped at University of California, Berkeley, 
+it should not be difficult to identify the 
+fluctuations which characterize the plate -tectonic 
+movements. These fluctuations will be 
+categorized according to the magnitude of 
+tremors. Certain categories will act as  the 
+precursors of Earthquakes also.  
+We must also study the orbital radius expansion 
+or contraction  of satellites  and see if they fit in a 
+general formulation. Among the natural satellites 
+we will cover Moon, Phobos, Deimos, Charon, 
+Titan and a few other l arge sized satellites.  
+ 
+References  
+ 
+1. Dr. BijayKumar  Sharma, The 
+Dynamics of Planetary Satellites and 
+the Generalization of the same 
+provides  a new theory of Birth and 
+Evolution of Our Solar System . 
+2. Measuring the Moon's Distance  
+(Apollo Laser Ranging Experiments 
+Yield Results ), From LPI Bulletin, 
+No. 72, August, 1994  
+http://eclipse.gsfc.nasa.gov/SEhelp/Ap
+olloLaser.html   3. Moon Mechanics: What Really Makes 
+Our World Go 'Round 
+http://www.space.com/scienceastrono
+my/moon_mechanics_0303018.html  
+4. "Software for generating the Spiral 
+Trajectory of our Mo on", BKS et al, 
+Journal of Advances in Space 
+Research, Vol.43,  (2009),  pp460 -466, 
+2nd Feb 2009.  
+ <The actual Spiral Trajectory can be 
+found here.>  
+ 
+---------------  
+……..  
+ 
\ No newline at end of file
diff --git a/1001.0098.txt b/1001.0098.txt
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@@ -0,0 +1,1438 @@
+arXiv:1001.0098v1  [cond-mat.quant-gas]  31 Dec 2009Light cone dynamics and reverse Kibble-Zurek mechanism in t wo-dimensional
+superfluids following a quantum quench
+L. Mathey1and A. Polkovnikov2
+1Joint Quantum Institute, National Institute of Standards a nd Technology and University of Maryland, Gaithersburg, MD 20899
+2Department of Physics, Boston University, 590 Commonwealt h Ave., Boston, MA 02215
+(Dated: November 15, 2018)
+We study the dynamics of the relative phase of a bilayer of two -dimensional superfluids after the
+two superfluids have been decoupled. We find that on short time scales the relative phase shows
+“light cone” like dynamics and creates a metastable superflu id state, which can be supercritical. We
+also demonstrate similar light cone dynamics for the transv erse field Ising model. On longer time
+scales the supercritical state relaxes to a disordered stat e due to dynamical vortex unbinding. This
+scenario of dynamically suppressed vortex proliferation c onstitutes a reverse-Kibble-Zurek effect . We
+study this effect both numerically using truncated Wigner ap proximation and analytically within
+a newly suggested time dependent renormalization group app roach (RG). In particular, within RG
+we show that there are two possible fixed points for the real ti me evolution corresponding to the
+superfluid and normal steady states. So depending on the init ial conditions and the microscopic
+parameters of the Hamiltonian the system undergoes a non-eq uilibrium phase transition of the
+Kosterlitz-Thouless type. The time scales for the vortex un binding near the critical point are
+exponentially divergent, similar to the equilibrium case.
+PACS numbers:
+I. INTRODUCTION
+The technological advances of trapping and manip-
+ulating ultra-cold atom systems provide an opportu-
+nity to study many-body dynamics with unprecedented
+clarity. The realization of Bose-Einstein condensates
+in ultra-cold atom systems1, the Mott insulator transi-
+tion2, the BEC-BCS transition3, the Kosterlitz-Thouless
+transition4–6, demonstrated that this technology can be
+usedasaquantum simulatorofmany-bodyphases. Here,
+the static state of a system in equilibrium is created
+and studied. Various dynamical aspects of ultra-cold
+atom systems have also been probed, such as dipole
+oscillations7, vortex excitations8, and soliton dynamics9,
+absence of equilibration in one-dimensional bosonic sys-
+tems10, spontaneous formation of vortices in spinor con-
+densates11and many others (see Ref. [12] for a recent
+review). In Ref. [13], vortices excitations were created
+via laser stirring. In these experiments, the dynamics
+of only a few degrees of freedom were studied, such as
+the center of mass motion, or the dynamical evolution
+of a vortex. These experimental developments stimu-
+lated a considerable theoretical interest in understanding
+non-equilibrium quantum dynamics including analysis of
+dynamics following sudden quenches14, studying connec-
+tions between dynamics and thermodynamics15, dynam-
+ics through quantum critical points16.
+The focus of this paper is a detailed analysis of the
+full many body dynamics following the quench in a two-
+dimensional quantum rotor model. Physically we imag-
+ine the situation where two initially strongly coupled su-
+perfluids are suddenly separated and we are interested
+in the evolution of the relative phase between the two
+superfluids. In particular, we will be interested in the
+question of how the system relaxes to the equilibriumstate. We note that experiments in a similar setup in-
+volving separation of two 1D superfluids were reported
+in Ref. [22] and the corresponding theoretical analysis
+was done in Refs. [23–25]. Unlike the 2D case, phonon
+fluctuations in 1D result in the exponential decay of the
+correlation functions and nonlinear effects in the form of
+phase slips do not bring qualitative changes to the be-
+havior of the correlation functions at least at low initial
+temperatures24.
+In equilibrium for the uncoupled layers, there are two
+possible phases. At low temperatures atoms in each layer
+(which we regard as identical) form a (quasi-)superfluid
+phase while at high temperatures they form a normal
+Bose gas. These phases can be distinguished by the long-
+range behavior of the single particle correlation function
+G(x) =/angb∇acketleftb†(0)b(x)/angb∇acket∇ight ≈ρ/angb∇acketleftexp[i(φ(x)−φ(0))]/angb∇acket∇ight, whereb(x)
+is the single particle operator, ρis the atom density,
+andφis the phase. We note that a rotor representa-
+tion of bosons b(x)∼/radicalbig
+ρ(x)exp[iφ(x)] is possible when
+thehealinglengthcharacterizingthecharacteristiclength
+scale of density fluctuations is short compared to other
+length scales in the problem. Under the same conditions
+the density fluctuations are negligible if we are interested
+in long distance physics. In the superfluid phase this
+function shows algebraic scaling, G(x)∼ |x|−τ/4, where
+the scalingexponent τis proportionalto the temperature
+τ≈T/Tc, withTcbeing the Kosterlitz-Thouless temper-
+ature. At the transition point we have G(x)∼ |x|−1/4.
+Above the transition, the correlation function shows ex-
+ponential scaling G(x)∼exp(−|x|/ξ), with some corre-
+lation length ξwhich diverges near the transition tem-
+perature. The algebraic scaling of the superfluid phase is
+due the thermally excited phonon (Bogoliubov’s) modes.
+In two dimensions these fluctuations generate quasi-long
+rangeorder, rather than true long rangeorder. The tran-2
+sition to the exponential regime is due to vortex excita-
+tions. Above the transition, vortex-antivortex pairs are
+deconfined so that vortices and anti-vortices become un-
+bound. These excitations generate a much more disor-
+dered phase field, which leads to exponential scaling of
+the correlation function.
+If we couple the two superfluids with a hopping term
+in the temperature regime of the critical temperature,
+the system forms a phase-locked state, see Refs. [20,21].
+Here the correlation function of the relative phase scales
+asG(x)∼exp(−|x|/ξi)+C, whereC/negationslash= 0, and ξiis the
+correlation length of the phase-locked state. In this state
+the relative phase is well-aligned over long distances; its
+fluctuations are strongly suppressed. We then turn off
+the hopping and study the evolution of the system.
+In this paper we show that the relaxational dynam-
+ics occurs in two stages. The first fast stage, which we
+will term light cone relaxation, establishes a metastable
+quasi-equilibrium state of phonons (or Bogoliubov exci-
+tations) characterized by effective non-equilibrium tem-
+perature (in principle this metastable state can be com-
+pletely non-thermal). During this stage the correla-
+tions between two arbitrary points in space x1andx2,
+G(x1,x2,t), where tis the time after the quench, ini-
+tially decay in time, independent of their spatial separa-
+tionx=|x1−x2|because these points are not causally
+connected: G(x1,x2,t)∼1/tα, whereαis a power-law
+exponent related to the parameters of the system. At a
+later time t⋆, when the condition 2 vt⋆=xis fulfilled,
+these correlations (approximately) freeze in time so that
+G(x1,x2,t)∼1/xα. The exponent αthus defines the
+non-equilibrium phonon temperature in the system. Be-
+cause this first stage of dynamics involves only phonons,
+the exponent αcan exceed the maximally allowed equi-
+librium value of one fourth, leading to a non-equilibrium
+super-critical metastable state, which can be thought of
+as a supercritical superfluid. It is analogous to an over-
+heated classical liquid, for which a liquid state can be
+sustained above the critical temperature if the creation
+of defects is avoided. We find that the power-law can be
+substantially above the critical scaling, and furthermore,
+that this metastable can be very long-lived.
+At longer time scales, vortex-antivortex pairs emerge
+andproliferateleadingtothetrueequilibriumstate. This
+process occurs at much longer time scales. We describe
+this thermalization process both numerically, using trun-
+cated Wigner approximation (TWA) and analytically.
+In particular, we show that thermalization (here cor-
+responding to the process of vortex-antivortex prolifer-
+ation) can be understood by extending renormalization
+groupideasto realtime dynamics. By doingpartial aver-
+agingoverfast oscillatinghigh energydegreesoffreedom,
+we can rewrite the equations of motion of slower degrees
+of freedom through renormalized coupling constants. As
+in the case of equilibrium systems we observe two pos-
+sible scenarios corresponding to vortex-antivortex pairs
+being irrelevant (superfluid phase) or relevant (normal
+phase). Thus we are able to see how the system relaxesto one of the phases in real time. Divergent time (and
+length) scalesin equilibrium systems translateinto diver-
+gent relaxation times required to reach thermalization in
+the non-equilibrium case.
+Physicallythis decayofthe metastablesuperfluid state
+to the new equilibrium is very reminiscent of the Kibble-
+Zurek (KZ) effect. The latter describes a ramp across
+a phase transition, starting on the disordered side. If
+the ordered state supports topological excitations, like
+vortices, then one expects very slow relaxation of the re-
+sulting state to the equilibrium due to vortex-antivortex
+recombination. This scenario is illustrated in Fig. 1
+a): In the disordered phase we have excitations such as
+phonons, as well as topological defects. When we ap-
+ply a fast ramp across the phase transition, the phonon
+excitations thermalize on very short time scales, while
+topological defects can exist on much longer time scales.
+The mechanism of relaxation in our case is exactly com-
+plimentary and can be termed as reverse Kibble-Zurek
+effect. Here, the ramp across a phase transition starts
+from the ordered side, as illustrated in Fig. 1 b). In the
+ordered phase both phonon excitations and vortices are
+suppressed. When the system is ramped across the tran-
+sition, phonons are generated on a fast time scale. How-
+ever, vortices are generated at much longer time scales
+leading to the long-lived supercritical superfluid state.
+We point outthat in thermallyisolatedsystems(like cold
+atom systems) it is much easier to observe reverse KZ ef-
+fect because the disordered phase usually corresponds to
+a higher temperature. In isolated systems it is relatively
+easy to increase temperature by quenching some param-
+eter, while decreasing temperature requires much more
+effort and can be done only in open systems.
+This paper is organized as follows: In Sect. II we in-
+troduce the numerical method that we use and find that
+at short time scales the system shows light cone dynam-
+ics. In Sect. III we consider the linearized dynamics of
+the bilayersystem. Within this approximationboth light
+cone dynamics and the emerging superfluid state can be
+understood. In Sect. IV we study the light cone dynam-
+ics of a solvable model, the transverse Ising model. In
+Sect. V we study dynamical vortex unbinding both with
+truncated Wigner approximation, and with a renormal-
+ization group approach. We note that a short version of
+this paper with some of the results was published ear-
+lier18. Here we expand the earlier treatment, present ad-
+ditional results and derivations, and formulate the real
+time renormalization group approach which explains the
+numerical results.
+II. MICROSCOPIC MODEL AND THE
+TRUNCATED WIGNER APPROXIMATION
+In this section we present the model that we use in our
+numerical approach. We consider two two-dimensional
+(quasi-)condensates that are aligned in parallel to each
+other, that are coupled by a hopping term which is then3
+Kibble-Zurek 
+t t0
+reverse Kibble-Zurek 
+t t0
+FIG. 1: Illustration of the Kibble-Zurek (KZ) mechanism,
+which describes ramping across a phase transition from the
+disordered phase, and its counterpart, the reverse-Kibble -
+Zurek (rKZ) effect. The latter describes ramping across a
+transition from the ordered side. Its defining feature is the
+dynamical suppression of vortex unbinding, which happens
+on a much longer time scale than the appearance of phononic
+excitations. We propose to study the rKZ in a bilayer of 2D
+superfluids of ultra-cold atoms, by decoupling the superflui ds
+and measuring the dynamics of the relative phase.
+turned off. This can be achieved by increasing the po-
+tential between the two condensates. The coarse-grained
+Hamiltonian describing the relative phase φiof the two
+superfluids corresponds to an XY model, to which we
+add a hopping term to describe the phase-locking in the
+initial state:
+H= Ω0/parenleftBig
+−/summationdisplay
+<ij>κ
+πcos(φi−φj)+π
+2κ/summationdisplay
+in2
+i
+−V(t)/summationdisplay
+icos(√
+2φi)/parenrightBig
+, (1)
+where Ω 0is an overall (Josephson) energy scale, κde-
+scribes the ratio of kinetic and potential energies. We
+can formally replace these parameters by Ω 0κ/π= 2Jn,
+πΩ0/κ=U(so that Ω 0=√
+2JnU,κ=π/radicalbig
+2Jn/U)
+andV(t) = 2J⊥(t)n/Ω0, which gives a representation of
+two coupled Bose-Hubbardsystemsin the quantum rotor
+limit33. In this limit the Bose operators are replaced by
+the phase-density representation and the fluctuations of
+density are assumed to be small. In the Bose-Hubbard
+modelJis the in-plane hopping amplitude, Uis the on-
+site interaction energy, nis the filling number, i.e. the
+number of particles per site, and J⊥is the inter-layer
+hopping amplitude J⊥. This representation gives at best
+a qualitative idea of how the model parameters relate
+to the parameters in experiment, but gives a more in-
+tuitive picture. We note that one can think about the
+continuum limit as discrete, where the lattice constant
+is approximately given by the zero-temperature healing
+FIG. 2: We simulate the dynamics of the relative phase of
+two 2D superfluids by solving the equations of motion and by
+averaging over the Wigner distribution of the initial state . A
+single run is shown here, for V= 100,κ= 10 and T= 2,
+at the times t= 0,5,10,20,40,100. Vortices are marked red,
+anti-vortices blue.
+length in the system, i.e. the length over which density
+fluctuations are suppressed.
+We emphasize that despite the BKT transition being
+classicalin origin, i.e. driven by thermalfluctuations, the
+mechanism of vortex or phonon creation in the process
+we consider comes from quantum fluctuations. Indeed
+when the superfluids are strongly coupled together the
+density (which plays the role of momentum conjugate of
+the phase) strongly fluctuates because of the zero point
+motion. The heating mechanism of this system can be
+thought of as enhancement of this zero point motion fol-
+lowing the quench.
+It is convenient to introduce the rescaled quantities
+˜t= Ω0t//planckover2pi1,˜φ=/radicalbigκ
+πφ, and ˜n=/radicalbigπ
+κn. In terms of these,
+the classical equations of motion (EOMs) are
+d˜φi
+d˜t=−˜ni (2)
+d˜ni
+d˜t=−√
+2
+β/summationdisplay
+jisin/parenleftBigβ(˜φji−˜φi)√
+2/parenrightBig
++V(t)βsinβ˜φi,(3)
+where we defined β=/radicalbig
+2π/κ. The indices jidescribe
+the four neighboring sites of site i.4
+We model the relative phase using a numerical im-
+plementation of the truncated Wigner approximation
+(TWA) (see Ref. [29] for a review): The expectation
+of any quantity at some time t >0 can be determined
+by sampling over a Wigner distribution at time t= 0,
+and solving the classical equations of motion from 0
+tot. This approximation is guaranteed to be accurate
+at short times30,31. This approximation is also exact
+for any quadratic theory so we expect it to be accu-
+rate in the first (light-cone) stage of dynamics primar-
+ily driven by phonon excitations. In our case we expect
+that TWA is also valid at longer times because when
+vortex-antivortex pairs start to emerge the system al-
+ready reached metastable state corresponding to finite
+effective temperature. At this point quantum fluctua-
+tions become suppressed by much stronger thermal fluc-
+tuations driving the slow vortex dynamics.
+We solve these EOMs for initial conditions that are
+distributedaccordingtothe Wignerdistributionat t= 0.
+We can calculate this distribution under the assumption
+thatJ⊥is larger than the other energy scales at t= 0.
+In this limit the phase fluctuations are small and can be
+described within the Bogoliubov approximation, where
+the system reduces to a sum of oscillators. The Fourier
+modes˜φqand ˜nqatt= 0 are distributed according to
+(see Ref. [31])
+W∼exp/parenleftBig
+−˜φ2
+q
+2σ2qrq−2σ2
+q˜n2
+q
+rq/parenrightBig
+, (4)
+whereσ= 1//radicalbig2ωq,rq= coth(ωq/2T0), andωq=/radicalbig
+4sin(qx/2)2+4sin(qy/2)2+Vβ2withT0being the
+initial temperature. Note that formally ωqdiverges at
+V→ ∞. This divergence is unphysical, being an artifact
+of using Hamiltonian (1) in the number phase represen-
+tation. In reality when J⊥becomes very large the trans-
+verse Josephson frequency saturates at ω≈2J⊥. This
+happens at V∼nor equivalently J⊥∼Un. So for very
+strong initial coupling one can still use distribution (4)
+withV→n.
+To visualize our simulations we show an example for a
+single run of the system on a 20-by-20 lattice in Fig. 2.
+The direction of the arrowson each lattice point describe
+the phase φi. We show ’snapshots’ at various times. The
+plaquettes around which there is a phase winding of ±2π
+are marked as vortices and anti-vortices. We see that at
+t= 0, the phases are well aligned due to the coupling be-
+tween the layers, with some small quantum fluctuations
+described by the Wigner function. The coupling is then
+turned off, vortices and anti-vortices are created pair-
+wise, and unbind on a long time scale, as we will discuss
+further on. To extract expectation values of our observ-
+ables from our simulations, we have to averagethem over
+many realizations of initial fluctuations.
+We use this method to extract the equal time correla-
+tion function:
+G(x,t) =/angb∇acketleftexp[i√
+2φj(t)−i√
+2φj+x(t)]/angb∇acket∇ight,(5)
+FIG. 3: Plot ofshort-time behavior of thecorrelation funct ion
+as afunction of time andspace, at temperature T= 3, forκ=
+10 andV= 20. The dynamics separates into instantaneous,
+damped oscillatory behavior, and a ’light cone’ like pulse.
+wherexis an integer separation between the points and t
+is the time after decoupling (see Fig. 3). Because we are
+using periodic boundary conditions G(x,t) depends only
+on the separation between the points xand does not de-
+pend on j. Note that this correlation function (or rather/integraltextx
+0dx′G(x′,t)) can be directly measured in interference
+experiments5,19,22. Weindeedseeveryclearemergenceof
+the light cone thermalization: At separations larger than
+2vt, wherevis characteristic phonon velocity, G(x,t) is
+almostxindependent - it uniformly decreases in time.
+Once 2vt > x the correlations freeze in time and de-
+pend only on x. The quantities in the system have been
+rescaled such that the phonon velocity is set to 1.
+We find that the state that emerges within the light
+cone shows algebraic scaling, and therefore can be re-
+ferred to as a superfluid.
+III. LINEARIZED DYNAMICS
+In this section we study the linearized dynamics of the
+system. Within this description, both the light cone dy-
+namics and the metastable SF state can be understood.
+The quadratic Hamiltonian describing the relative phase
+of two coupled superfluids reads:
+H0=/integraldisplay
+d2r/parenleftBig
+−v
+2r0(∇φ)2+g⊥
+2φ2+vr0
+2n2/parenrightBig
+.(6)
+vis the phonon velocity of the SF, approximately given
+byv=/radicalbig
+gn/m.r0is the short-range cut-off of the
+system, of the order of the healing length. J=v/r0
+is the KT energy. The term g⊥φ2/2 is created by the
+hopping term of the bilayer. It is approximately given
+byg⊥= 4J⊥n. We note that when this Hamiltonian is5
+put on a lattice with a lattice constant rlwe obtain
+H1= Ω0/summationdisplay
+i/parenleftBig
+−rl
+r0/summationdisplay
+ji(φi−φji)2
+2+g⊥r2
+l
+2Ω0φ2
+i+r0
+2rln2
+i/parenrightBig
+.
+(7)
+This expression can be also obtained directly linearizing
+the original Hamiltonian (1). The index jihere describes
+the four neighboring sites of the site i, andniis the
+filling fraction, related to the density via ni=nr2
+l. Ω0is
+relatedtothephononvelocityasΩ 0=v/rl. Wetherefore
+find that the squeezing parameter κ/πis given by κ/π=
+rl/r0, i.e. it is the ratio of the discretization length scale
+rland the short-range cut-off r0of the system. g⊥is
+related to V(t) byg⊥r2
+l/2 = Ω0V(t).
+We now consider the time evolution of φandnunder
+(6). It is convenient to go to momentum representation
+where different modes decouple from each other. Assum-
+ing also that we are interested in momenta smaller than
+1/rl, wherethelatticeeffectsarenotimportantweobtain
+the following equations of motion:
+d
+dtnk=−Ω0/parenleftBigrlǫ2
+k
+r0+2V/parenrightBig
+φ−k (8)
+d
+dtφk= Ω0r0
+rln−k. (9)
+whereǫ2
+k= 4sin2kx/2 + 4sin2ky/2, andkis dimen-
+sionless, k=−π...π. We rescale the time variable as
+˜t= Ω0t//planckover2pi1. The initial dispersion is then given by
+ω2
+k,0=ǫ2
+k+2Vr0/rl. (10)
+The dispersion ωkafter the quench is simply ω2
+k=ǫ2
+k,
+in these units. We solve these equations and calculate
+the equal-time correlation function at time tafter the
+quench. We use
+G(x,t) =/angb∇acketleftexp(iφ(0,t))exp(−iφ(x,t))/angb∇acket∇ight(11)
+= exp(−/angb∇acketleftδφ2/angb∇acket∇ight/2), (12)
+whereδφ=φ(0,t)−φ(x,t). The averaging is now triv-
+ially done using the Wigner distribution (4). If we put
+the system back to the lattice we then find
+/angb∇acketleftδφ2/angb∇acket∇ight=/summationdisplay
+k(2−2coskx)
+×/parenleftBigrk,0
+2ωk,0cos2(ωkt)+rk,0ωk,0
+2ω2
+ksin2(ωkt)/parenrightBig
+.(13)
+The quantities rk,0andωk,0are defined as before.
+We now calculate the Green’s function in the lin-
+earized regime numerically using Eqs. (12) and (13). We
+choose the discretization rl=r0, the initial tempera-
+tureT/Ω0= 1, and the initial coupling Vβ2= 20. In
+Fig. 4 we plot /angb∇acketleftδφ2/angb∇acket∇ightand in Fig. 5 we plot the correla-
+tion function. In both plots the light-cone dynamics is
+clearly visible. Because of the translational invariance
+the correlation function that emerges in the light-cone
+only depends on the relative distance is given by
+G(x,t)≈C1|x|−T∗/4TKT(14)
+FIG. 4:/angbracketleftδφ2/angbracketrightof the linearized system, for T= 1 and Vβ2=
+20, as function of the lattice site, and vt.
+FIG. 5: The correlation function of the linearized system, f or
+T= 1 and Vβ2= 20, as function of the lattice site, and vt.
+forx≪2vt, whereT∗is an effective temperature that is
+estimated below, and C1is a numerical prefactor. Out-
+side of the light cone ( x≫2vt) the function G(x,t) only
+depends on time tbut not on the distance x:
+G(x,t) =C2|t|−T∗/4TKT, (15)
+whereT∗is the same effective temperature. At the light
+cone boundary x≈2vtthe two asymptotics for the cor-
+relation function (14) and (15) approximately coincide.
+However, we note that the prefactor C2is in general dif-
+ferent from C1v−T∗/4TKTas it is evident from the exis-
+tence of a wavefront that is visible in Figs. 4 and 5.
+Thetemperaturethatemergesinsidethelightconecan
+be estimated by considering the quadratures of φat long
+times:
+/angb∇acketleftφ2
+k(t→ ∞)/angb∇acket∇ight=rk,0
+4ωk,0+rk,0ωk,0
+4ω2
+k.(16)
+We find that the whole Wigner function in the non-
+interacting evolution remains Gaussian. It means that6
+for each mode the Wigner function is equivalent to that
+of a harmonic oscillator at finite ’temperature’ T∗
+k, which
+is in general mode-dependent:
+r∗
+k
+ωk=rk,0
+2ωk,0+rk,0ωk,0
+2ω2
+k(17)
+wherer∗
+k= 1/tanh(ωk/2T∗
+k). Solving for T∗
+kgives
+T∗
+k=ωk
+2tanh−1/parenleftBig
+2ωkωk,0
+ω2
+k+ω2
+k,0tanh(ωk,0/2T)/parenrightBig.(18)
+For large Vβ2, which corresponds to initially strong cou-
+pling between two superfluids, this simplifies to a single
+value, independent of k:
+T∗=/radicalbig
+Vβ2
+4tanh(/radicalbig
+Vβ2/2T). (19)
+For small initial temperatures Twe have T∗≈/radicalbig
+Vβ2/4
+(T⋆= 2J⊥/Jin terms of the original Hubbard param-
+eters), that is, the temperature is fully determined by
+the initial coupling energy. The coupling energy be-
+tween the two layers is transferred into the in-plane ki-
+netic energy. We remind again that this result is valid
+as long as J⊥/lessorsimilarUn, otherwise the dependence of T⋆on
+J⊥saturates and for the infinite coupling limit we have
+T⋆∼Un/J. For large Twe have T∗≈T/2. This result
+is a reflection of the doubling of the degrees of freedom
+when two layers are uncoupled. In Fig. 6 a) we show
+dependence T⋆(T) evaluated according to Eq. (19), for
+V= 20, and for κ= 1,3,10 corresponding to lowering
+J⊥. Forκ= 1,T∗is always above the critical tempera-
+tureTc=π/2, forκ= 10, it crosses it. We therefore ex-
+pect to see very little vortex formation for small temper-
+atures for κ= 10, and many vortices for all temperatures
+forκ= 1. The intermediate value κ= 3 approximately
+describes the transition between these limits. To make
+this point more clear in Fig. 6 b)–d) we plot full non-
+linear TWA simulation for each of the three cases. We
+run the quench for different temperatures T, and plot
+the number of vortices nvin the system as a function
+of time. This number is obtained by counting the vor-
+tices (indicated by red plaquettes in Fig. 2), and then
+by averaging over many runs. We find that for κ= 1
+the number of vortices is virtually unchanged implying
+that the dynamics is completely dominated by quantum
+fluctuations, whereas for κ= 10 this number drops to
+zero when the temperature is lowered. These results are
+consistent with the emergent temperature T∗obtained
+within the linearized approach.
+IV. LIGHT-CONE DYNAMICS IN THE
+TRANSVERSE ISING MODEL
+In this section we demonstrate that light cone dynam-
+ics is not just characteristic for the system we are inter-
+ested in, which is characterized by low energy bosonic0510
+010 20 30 020 40 60
+0510
+010 20 30 020 40 60
+0510
+010 20 30 0510 15 20T0
+T0 T0 t ttnv
+nvnv02468123456T*
+Tc
+T0a) b) 
+d) c) 10 
+10 10 10 
+10 10 10 
+20 
+20 
+20 20 20 20 
+5
+5
+5 5
+00000
+0
+00030 40 40 
+30 30 
+60 60 
+15 
+FIG. 6: a) T∗, as given in Eq. 19, for κ= 1,3,10, from top
+to bottom, and for V= 20. The line Tc=π/2 was added
+to indicate the critical temperature. b) – d) Simulations fo r
+these values of κandV. We plot the number of vortices nv
+as a function of time tand initial temperature T0.
+wave excitations. The same mechanism of reaching
+a steady state is much more general and is likely re-
+lated to the existence of the maximum group velocity in
+Schr¨ odinger systems as was proven by Lieb and Robin-
+son32. In this section we demonstrate the presence of
+the light-cone dynamics in another solvable model, the
+transverse Ising chain26,33described by the Hamiltonian
+HI=−JI/summationdisplay
+i(σx
+iσx
+i+1+gσz
+i), (20)
+whereJIis an overall energy scale, gdescribes the
+strength of the transverse field, and σx,zare the Pauli
+matrices. We follow the calculational procedure in
+Ref. [33]. First, we use a Jordan-Wigner transformation
+σz
+i= 1−2ni (21)
+σx
+i=/productdisplay
+j<i(1−2nj)(ci+c†
+i), (22)
+whereciare Fermi operators, and ni=c†
+ici. This
+transformation leads to a fermionic representation of the
+Hamiltonian, that can be further diagonalized using the
+Bogoliubov transformation
+γk,g=uk,gck−ivk,gc†
+−k(23)
+whereckis the Fourier transform of ci,uk,gandvk,g
+are given by cos( θk,g/2) and sin( θk,g/2), where θk,g=
+arctan(sin k/(g−cosk)). The resulting dispersion is
+ǫk,g= 2JI/radicalbig
+g2−2gcosk+1. (24)7
+FIG. 7: The correlation function /angbracketleftσz
+0(t)σz
+r(t)/angbracketrightfor a quench
+fromg= 3 tog′= 1, in (a), and from g= 3 tog′= 0.5, in
+(b), as a function of time t, specifically of 2 Jt, and the spatial
+distance r.We consider a time dependent g(t). Fort <0 we have
+g(t) =g, and we assume the system to be in equilibrium.
+We then assume that for t >0,g(t) jumps to the value
+g′. The equal-time correlation function of σz
+ican be cal-
+culated exactly by expressing it in terms of the operator
+ni, i.e./angb∇acketleftσz
+i(t)σz
+j(t)/angb∇acket∇ight= 1−4/angb∇acketleftni(t)/angb∇acket∇ight+4/angb∇acketleftni(t)nj(t)/angb∇acket∇ight. It can
+be shownthat the averagedensity fermionic density (cor-
+responding to the z-component of the magnetization) is
+given by
+/angb∇acketleftni(t)/angb∇acket∇ight=/angb∇acketleftni(0)/angb∇acket∇ight+1
+M/summationdisplay
+kFg,g′(k,t) (25)
+with
+Fg,g′(k,t) =(cos(2ǫk,g′t//planckover2pi1)/2−1/2)(g′−g)sin2k/radicalbig
+g2−2gcosk+1(g′2−2g′cosk+1)
+(26)
+and
+/angb∇acketleftni(0)/angb∇acket∇ight=1
+2−1
+2M/summationdisplay
+kg−cosk/radicalbig
+g2−2gcosk+1.(27)
+In turn the density-density correlation function
+/angb∇acketleftni(t)nj(t)/angb∇acket∇ightreads
+/angb∇acketleftni(t)nj(t)/angb∇acket∇ight=1
+M2/summationdisplay
+k1,k2/parenleftBig
+exp(−i(k1−k2)(ri−rj))/parenleftBig
+(v2
+k1,g+Fg,g′(k1,t))(u2
+k2,g−Fg,g′(k2,t))
++(uk1,gvk1,g+Gg,g′(k1,t))(uk2,gvk2,g+G∗
+g,g′(k2,t))/parenrightBig
++(v2
+k1,g+Fg,g′(k1,t))(v2
+k2,g+Fg,g′(k2,t))/parenrightBig
+,(28)
+where
+Gg,g′(k,t) =/parenleftBig
+isin(2ǫk,gt//planckover2pi1)+1
+2(cos(2ǫk,g′t//planckover2pi1)−1)(g′−cosk)/radicalbig
+g′2−2g′cosk+1/parenrightBig/parenleftBig(g′−g)sink/radicalbig
+(g2−2gcosk+1)(g′2−2g′cosk+1)/parenrightBig
+.(29)
+Using these expressions we can easily analyze the quench
+dynamics. In Fig. 7 we show two examples showing spin-
+spin(density-density)correlationfunctionafteraquench.
+The first example corresponds to the ramp from g= 3 to
+g′= 1, i.e. a quench to the quantum critical point. At
+this point the dispersion (24) becomes gapless and linearat small energies. Then the light cone dynamics is an-
+ticipated because there is a well defined “speed of light”
+characterizing the propagation of excitations, which is
+equal to 2 J. Indeed Fig. 7a) shows clear signature of
+such dynamics. There is a clearly visible “light cone”,
+which separates into an instantaneous part (connecting8
+not causally connected points) that is independent of the
+distance, and a spatially dependent (causal) part, that
+expands in the form of a wave front. In Fig. 7 b) we use
+g′= 0.5, where at low energies the spectrum of excita-
+tions is gapped. Although the dispersion in this model
+is linear (relativistic) only at sufficiently high energies
+above the gap we still see a clear light-cone structure.
+The expansion velocity of the “light cone” in this case is
+consistent with twice the maximum of the group velocity
+vgr(k) =dǫ/dkgiven by
+vgr,max=/braceleftbigg
+2Jgfor|g|<1
+2Jfor|g| ≥1.(30)
+So forg′= 1 we find 2 vgr,max= 4J, and for g′= 0.5
+we find 2 vgr,max= 2J. These are indeed the expansion
+velocities that we see in Fig. 7.
+V. DYNAMICAL VORTEX UNBINDING
+In this section we address the important question of
+how the supercritical state relaxes to the ground state,
+i.e. the second stage of the dynamics. As we mentioned
+inthe introductiontheanticipatedmechanismforthisre-
+laxation is vortex unbinding. This process is intrinsically
+nonlinear and requires a more sophisticated treatment
+than that of the noninteracting “light cone” dynamics.
+In this work we use two complimentary approaches. In
+Sec. VA we use a numerical implementation of the TWA
+to simulate the dynamics in the system. In Sec. VB we
+generalize a renormalization group approach to analyti-
+cally describe the process of relaxation in real time.
+A. Numerical approach
+Within TWA we need to solve the full nonlinear equa-
+tions of motion (3) subject to the initial conditions dis-
+tributed according to the Wigner function (4). Then the
+equal-time correlation functions or other observables are
+found byaveragingthe Weylsymbolofthecorresponding
+observable computed at time tover the fluctuating ini-
+tial conditions. Note that since we are interested only in
+phase-phase correlation function the corresponding Weyl
+symbolis obtainedby simply substituting the Heisenberg
+quantum operator corresponding to the phase with the
+classical phase31. In Fig. 8 we show the result of such
+simulations. We can observe how the metastable super-
+fluid state relaxes to the disordered state. For that, we
+show the correlation functions of the system on a much
+longer time scale than in Fig. 3. The exponent of the al-
+gebraic scaling gradually decreases. Eventually the cor-
+relation function is more accurately approximated by an
+exponential fitting function, signalling that the thermal
+Bose gas phase has been reached. Because this is the
+phase of deconfined vortices, and because the intermedi-
+ate superfluid phase is well described by a phonon-only
+FIG. 8: Long-time behavior of the correlation function for
+T= 1,κ= 8, and V= 80. The correlation function first
+develops algebraic scaling, so the system forms a metastabl e
+quasi-superfluid state. On longer time scales the correlati on
+function shows exponential decay. The coherence is lost due
+to dynamical vortex unbinding.
+description, we conclude that the dynamical transition
+that we observe is due to vortex unbinding. The exam-
+ple of a single run shown in Fig. 2 is consistent with
+this picture: Defects are created soon after the quench,
+but they only gradually separate on a much longer times
+scale. It is this process that we refer to as the reverse
+Kibble-Zurek mechanism.
+To better characterize the process of vortex unbinding
+further we fit the correlation function G(x,t) to either
+algebraic or exponential fitting functions. Such choice is
+motivated by the two possible regimes of the equilibrium
+system and is supported by the analytic renormalization
+group results presented in the next section. The alge-
+braic fitting function we use is c(L/π|sin(πx/L)|)−τ/4
+and the exponential function is cexp(−|sin(πx/L)|/x0).
+Note that in the fitting functions we use the conformal
+distance L/π|sin(πx/L)|), which is more appropriate in
+finite systemswith periodicboundary conditions(see e.g.
+Ref. [33]). In equilibrium the algebraic exponent τwould
+be the relative temperature T/Tc. Any value above 1
+is therefore supercritical. The parameter x0defines the
+length scale of the exponential decay. The parameter c
+in both functions gives an overall scale.
+Using these fitting functions we analyze four different
+situations corresponding to the same initial temperature
+T= 1 and the same parameter κ= 8, but with dif-
+ferent initial couplings Vbetween the planes. The first
+(I) case corresponding to V= 80 is identical to the one
+plotted in Fig. 8. The other three curves correspond
+toV= 70,50,20 (II–IV). In Fig. 9 we show the expo-
+nentτextracted from the fit as a function of time for
+these situations. In all of them at short times G(x,t)
+develops algebraic scaling when the light-cone dynamics
+reaches the system boundaries. For the cases I–III the
+emerging scaling exponent τis well above the critical ex-9
+20 40 60 0 80 100 125
+0 0125
+tτ
+I
+II 
+III 
+IV 
+FIG. 9: Time dependence of the exponent τextracted from
+fitting the long-time correlation function G(x,t), for different
+initial couplings. Inall four examples we use T= 1 and κ= 8.
+The initial couplings Vare chosen as V= 80,70,50, and 20,
+corresponding to curves I to IV. The curve I corresponds to
+the example shown in Fig. 8. In this case the correlation
+function can be well fitted with an algebraic function for up
+tot≈60, after that G(x,t) is better fitted by an exponen-
+tial function with a decay length of the order of the lattice
+constant. For the other cases, G(x,t) is well fitted with an
+algebraic function throughout the whole time interval.
+ponent. After that, the exponent gradually increases on
+much longer time scales. During this process, the decay
+ofthe correlationfunction isstillfitted wellwith the alge-
+braic function. Eventually the algebraic scalings breaks
+down and G(x,τ) develops exponential scaling, indicat-
+ing vortex unbinding. This regime of exponential scaling
+is reached for V= 80 (I) within the time interval shown
+in Fig. 9. For V= 70 (II) and V= 50 (III) the time
+scale of the vortex unbinding is longer then the time in-
+terval shown. For V= 20 (IV) the system equilibrates to
+the superfluid state. Because in this case the exponent τ
+is less than one, vortices never unbind and the algebraic
+scaling persists at all times. We conclude from these ex-
+amples that there can be a sizeable range of initial values
+ofVwhichgeneratesthescenarioofasupercriticalsuper-
+fluid, and of dynamically suppressed vortex unbinding.
+Furthermore, the algebraic scaling exponents that can
+occur in the metastable state are well above criticality,
+and should be easily distinguishable from subcritical val-
+ues. These supercritical exponents can be detected using
+interference experiments along the lines of Refs. 5,19.
+B. Renormalization group approach
+In this section we develop the renormalization group
+(RG) approach to dynamical vortex unbinding. We find
+that the dynamical evolution of the system can be re-
+lated to the RG flow of the equilibrium system. The idea
+of RG in real time is quite similar in spirit to the RG in
+imaginary time. Namely our goal is to eliminate high en-ergy,highmomentumdegreesoffreedom. Inequilibrium,
+this is done by the means of usual perturbation theory
+(or Gaussian integration), which is justified because of
+the large energy gap separating high energy states from
+the low-energy degrees of freedom we are interested in.
+In real time the idea of renormalization is quite similar.
+High energy (momentum) phonons are not very sensi-
+tive to slow processes leading to vortex formations. Thus
+these phonons can be well treated within the linearized
+approach. However, due to nonlinearities such phonons
+slightly renormalize the parameters governing dynamics
+of low energy degrees of freedom. This renormalization
+is precisely what we are interested in. Note that techni-
+cally in the RG procedure we perform averaging of the
+equations of motion over short times. Then odd powers
+of highly oscillating fields average to zero while averag-
+ing of the even powers gives some constant contribution.
+This contribution is precisely what renormalizescoupling
+constants governing the low temperature dynamics.
+We point out that typical RG flow diagrams contain
+mostly non-equilibrium points, in fact, all except for
+the fixed points. As we have seen in the previous sec-
+tions, one can associate an effective temperature to the
+metastable state that emerges after the dephasing of the
+phonon modes. In turn with this effective parameter we
+can associate a location of the transient state in the RG
+flow of the equilibrium system. This effective tempera-
+ture can then either gradually increase, until the system
+starts to show exponential scaling, or the system can al-
+waysremain superfluid, ifthe algebraicscalingis subcrit-
+ical and the effective temperature always remains below
+TKT. This behavior resembles the equilibrium RG flow
+of a Kosterlitz-Thouless transition (which now occurs in
+real, not imaginary, time), on which we elaborate in this
+section.
+Instead of directly analyzing the rotor model to de-
+scribetheKosterlitz-Thoulessphysicsandvortexunbind-
+ing, we will work with the dual Z1clock model (or equiv-
+alently 2D sine-Gordon model), described by the action
+S=/integraldisplay
+d2r/parenleftBigλ
+2(∂xθ)2−g
+a2cosθ/parenrightBig
+.(31)
+Forthe details ofthe duality transformationseeRef. [17].
+The parameters of this model can be related to those of
+the XY model by
+λ=1
+8πT
+TKT=1
+4π2T
+JKT(32)
+g
+2= exp(−Sc), (33)
+whereEc=ScTis the vortex core energy, and λis a
+measure of the relative temperature. We note that the
+action in Eq. 31 has a high-momentum cut-off Λ, which
+is the inverse of the short-range cut-off a, i.e. we set
+Λa= 1. To describe the dynamics of this model we use
+the effective 2D sine Gordon Hamiltonian:
+H/T=/integraldisplay
+d2r/parenleftBigµ
+2p2−λ
+2(∂xθ)2+g
+a2cosθ/parenrightBig
+.(34)10
+Here the parameter µis chosen, so that the dispersion of
+the linearized XY model is recovered:
+µ=ω2
+k
+λk2T2, (35)
+which we also write as ωk=v|k|, where the velocity v
+is given by v=/radicalbig
+µλT2. The nonlinear term cos θin
+Eq. (34) describes the vortex field. If this term is im-
+portant (large g) then the field θlocalizes corresponding
+to a highly disordered phase of the dual field φ, i.e. to
+the normal state. Conversely small gcorresponds to the
+superfluid algebraic regime. The starting point of our
+RG analysis will be supercritical superfluid state which
+emerges after short time light cone dynamics. Because
+the Kosterlitz-Thouless transition is classical in nature
+occuring at high temperatures the quantum fluctuations
+are no longer expected to be important and instead of
+Wigner function as the new initial condition we can use
+its classical Boltzmann’s limit. The initial state for the
+vortex dynamics, described by the effective temperature
+T, is thus fully characterized by the quadratures of the
+spectrum:
+/angb∇acketleftθ∗
+kθk/angb∇acket∇ight=1
+λk2, (36)
+/angb∇acketleftp∗
+kpk/angb∇acket∇ight=1
+µ=λT2k2
+ω2
+k. (37)
+The equations of motion corresponding to the Hamilto-
+nian (34) are given by
+d
+dtp=λT∂2
+xθ+gT
+a2sinθ, (38)
+d
+dtθ=µTp. (39)
+We now apply the following renormalization procedure
+to these equations. We rescale the spatial and temporal
+variables as r→r(1+dΛ/Λ) andt→t(1+dΛ/Λ), and
+thep-field asp→p(1−dΛ/Λ). This implies that the mo-
+mentumcut-offΛisrescaledasΛ →Λ′≡Λ(1−dΛ/Λ),so
+the momentum degrees of freedom between Λ′and Λ are
+removed. Without the non-linear term in Eq. 38 these
+rescalings leave the equations of motion invariant. The
+linear dynamical evolution can therefore be considered to
+be the non-interactingfixed point of the RG. We now ask
+the question, how this dynamical evolution is affected by
+the non-linear term. Specifically we want to determine
+how the equations of motion behave at long times and
+distances. For this, we go beyond the bare rescaling and
+correct for the integrated-out degrees of freedom up to
+second order in g. The resulting flow equations are of
+the well-known BKT form:
+dg
+dl=/parenleftBig
+2−1
+4πλ/parenrightBig
+g (40)
+dλ
+dl=αg2
+λ. (41)(r,t) 
+(r’,t’) x
+tdl = dt/t = dr/r 
+dl 2vt=|x |˙ p (r,t) = −∂qH(r,t, λ, g)
+˙θ(r,t) = ∂pH(r,t, λ, g)
+˙ p (r′,t′) = −∂qH(r′,t′, λ ′,g′)
+˙θ(r′,t′) = ∂pH(r′,t′, λ ′,g′){
+{
+FIG. 10: Schematic representation of a renormalization ste p
+in the real-time RG approach. In each step we renormalize
+simultaneously the space and time variables. This ’moves’ t he
+equations of motion from ( r,t) to (r′,t′). We correct for the
+integrated-out degrees of freedom to second order in g, which
+renormalizes the parameters gandλaccording to Eqs. 40
+and 41.
+wherel= lnΛ, and αis a non-universal prefactor. The
+RG step generated the equations ofmotion at time t′and
+distance r′from the equations at time tand distance r,
+with renormalized coefficients, according to Eqs. 40 and
+41. Therefore the time dependence of the coefficients can
+be read off the solution of the RG flow, by realizing that:
+dt/dl=tort=t0el. In Fig. 10 we show a schematic
+representation of our RG process. In the Appendix we
+discuss the derivation ofthe flow equations and givetheir
+more complete form.
+One conclusion from Eqs. 40 and 41 is that the crit-
+ical exponent of the dynamical process is equal to the
+one of the equilibrium system. We see from Eq. 40 that
+the critical value of λisλc= 1/8π, which corresponds to
+T=TKTascanbe seenfrom Eq. 32. Another important
+observation is that the RG equations (40) and (41) pre-
+dict a non-equilibrium analogue of the BKT transition,
+where depending on the initial fluctuations in the sys-
+tem, the vortex-antivortex pairs can either unbind in the
+long time limit or remain bounded. This transition, as
+in the equilibrium case, is characterized by exponentially
+divergent time and length scales. Physically these diver-
+gencies correspond to a very slow process of equilibration
+of vortices near the nonequilibrium phase transition.
+We can also use the RG flow to determine the time
+scale of vortex unbinding by using
+g(t∗)∼1. (42)
+WhenTis well above TKT, the time scale can be deter-
+mined from Eq. 40,
+t∗∼exp/parenleftBigEc/2
+T−TKT/parenrightBig
+(43)
+whereEc=ScT. Away from the transition, the time
+scale of vortex unbinding is therefore exponentially in-
+creased, because of the energy cost given by the vortex
+core energy. Very close to the transition t∗scales as:
+t∗∼exp(exp( −Sc/2)//radicalbig
+1−TKT/T).(44)11
+Thetimescaleisrenormalizedbecauseofthecriticalscal-
+ing in the vicinity of the transition.
+VI. CONCLUSIONS
+In conclusion, we have studied the dynamics of the rel-
+ativephaseofabilayerofsuperfluidsin2D,afterthehop-
+ping between them has been turned off rapidly. We find
+that on short time scales the dynamics of the correlation
+function shows a ’light-cone’-like behavior. Depending
+on the parameters of the system, the light cone dynam-
+icscanresultinaphasethatshowssupercriticalalgebraic
+scaling, and can therefore be thought of as a superheated
+superfluid. On long time scales the system relaxes to a
+disordered state via vortex unbinding, which constitutes
+areverse-Kibble-Zurekmechanism. Thepropertiesofthe
+dynamical process can be understood with a renormal-
+ization group approach. We find that the dynamical evo-
+lution of the system resembles the RG flow of the equi-
+librium system. In particular, using the RG equations
+we found two possible scenarios of the system reaching
+the steady state: (i) if initial quantum and thermal fluc-
+tuations are weak the vortices are irrelevant and long
+time long distance behavior is governed by the algebraic
+fixed point. The only role of vortices is then renormal-
+ization of the superfluid stiffness and the sound velocity.
+(ii) If the initial fluctuations are strong then the vor-
+tices become relevant and proliferate resulting in a nor-
+mal (non-superfluid) steady state. In this case RG gives
+the time scale of vortex unbinding, which exponentially
+diverges as the system approaches the non-equilibrium
+phase transition. The behavior of the relative of phase of
+two superfluids can be accurately studied by interference
+experiments of ultra-cold atom systems, and therefore
+our predictions are of direct relevance to experiment.
+Acknowledgments
+We thank A. Castro Neto for useful discussions. Work
+of A.P. was supported by NSF DMR-0907039, AFOSR,
+and Sloan Foundation. L.M. acknowledges support from
+NRC/NIST, NSF Physics Frontier Grant PHY-0822671
+and Boston University visitor’s program.
+Appendix A
+In this Appendix we derive the RG Eqs. 40 and 41,
+which can also be written as a second order differential
+equation for θ
+1
+µd2
+dt2θ=λ△θ+g
+a2sinθ. (A1)
+To simplify the derivation, here and throughout the Ap-
+pendix, we formally change notations λT→λ,µT→µ,andgT→g. The idea of momentum shell RG is that we
+treat high momentum components of θandp(or equiv-
+alently˙θ) perturbatively, while not making any approx-
+imations about the low momentum components. Our
+goal is to find renormalization of the equations of motion
+governing the low momentum components. So we split
+θ(r,t) =θ<(r,t)+θ>(r,t), (A2)
+where the Fourier expansionof θ>(r,t) onlycontains mo-
+menta in the shell Λ′≡Λ−δΛ<|k|<Λ andθ<(r,t)
+contains all other Fourier components:
+θ<(r) =1√
+V/summationdisplay
+k<Λ′exp(ikr)θk (A3)
+θ>(r) =1√
+V/summationdisplay
+Λ′<k<Λexp(ikr)θk.(A4)
+We will treat θ>(and correspondingly p>) pertur-
+bartivelyin gsincethenonlineartermshouldonlyweakly
+couple to the high frequency field. We expand the high-
+momentum field as
+θ>(k,t) =θ>
+0(k,t)+θ>
+1(k,t). (A5)
+Hereθ>
+0(k,t) is the solution of the equations of motion,
+withgset to zero:
+θ>
+0(k,t) =µ
+ωΛp>
+0,ksin(ωΛt)+θ>
+0,kcos(ωΛt),(A6)
+whereωk=v|k|, and the velocity visv=√λµ. In the
+next leading order we have
+θ>
+1(k,t) =gωΛ
+λ/integraldisplayt
+0dτF1(k,τ)sin(ωΛ(t−τ)),(A7)
+where
+F1(k,τ) =/integraldisplay
+d2rexp[−ikr]sin(θ<
+0(r,τ)),(A8)
+and we used Λ a= 1. Note that in the last equation in
+the argument of the sinus we changed θ0toθ<
+0because
+the contribution from θ>
+0is smaller by the factor δΛ/Λ.
+So we see that in the leading order in gthe high mo-
+mentum component of θoscillates with time at very high
+frequency ωΛ. In the next order in gthe high momentum
+component also acquires a low frequency component (as
+we will discuss below).
+Next we consider the equation of motion (A1) expand-
+ing it up to the second order in θ>:
+1
+µd2
+dt2θ(r,t)≈λ∆θ(r,t)+g
+a2cos(θ<(r,t))θ>(r,t)
++g
+a2sinθ<(r,t)/parenleftbigg
+1−(θ>(r,t))2
+2/parenrightbigg
+.(A9)
+Because of the nonlinearity high-momentum modes cou-
+ple to the low momentum modes leading to the renor-
+malization of the couplings governing the dynamics of12
+the latter. The idea of RG is to average equations of
+motion for low-momentum (slow) components over the
+fast oscillations. The averaging is trivially done in the
+last term of Eq. (A9). There it is sufficient to use zeroth
+order in θ>. Using that sin2(ωΛt),cos2(ωΛt) = 1/2 we
+find that averaging of the last term simply renormalizes
+the coupling g:
+g→g/parenleftbigg
+1−EΛ
+4πλδΛ
+Λ/parenrightbigg
+, (A10)
+whereEkis the average energy of the mode kover the
+period (we used the fact that λk2|θk|2=Ek). We note
+that in a Boltzmann ensemble, we would have Ek= 1,
+because the energies here are in units of the tempera-
+tureT. With this assumption we would recover the flow
+equation of the equilibrium case.
+Instead of this assumption, we proceed by noting
+thatunderRG transformationscouplingconstantsslowly
+changeintime. Thisimpliesthattheadiabaticinvariants
+per each mode are approximately conserved, as discussed
+in Ref. 34. For an oscillator the adiabatic invariant is
+Ik=Ek/ωk. Thus we see that the energy of the mode
+is proportional to the frequency. Noting that at initial
+timeEk(t= 0) = 1 (in non-rescaled units this would be
+Ek(t= 0) =T0, whereT0is the initial non-equilibrium
+temperature), one can rewrite Eq. (A10) as follows:
+g/parenleftbigg
+1−1
+4πv01
+KδΛ
+Λ/parenrightbigg
+, (A11)
+where we introduced the analogue of the Luttinger-
+Liquid parameter K=/radicalbig
+λ/µ.vis the velocity which
+is now given by v=√λµ(note that in the original, not
+rescaled units, v=T√λµ).
+Next let us consider the second term in Eq. (A9). This
+term is more subtle since if we use θ>
+0the average over
+fast fluctuations will give zero. So we need to use the
+first correction θ>
+1, which would be a correction at sec-
+ond order in g. We note that the linear term in Eq. (A9)can also be expanded to second in g, generating simi-
+lar contributions. However, when written as Eq. (A1),
+such a term a cancelled by a corresponding term from
+expanding d2θ/dt2. Alternatively we can view Eq. (A9)
+as written for the θ<component, this automatically en-
+sures that only the nonlinear term is responsible for the
+renormalization.
+Let us look closer into the Eq. (A7). We are dealing
+with the integral over the fast oscillating function of τ:
+sin(ωΛ(t−τ)) and the slow oscillating function F. This
+integral can be evaluated by integrating by parts:
+/integraldisplayt
+0dτF1(τ)sin(ωΛ(t−τ)) =F1(τ)cos(ωΛ(t−τ))
+ωΛ/vextendsingle/vextendsingle/vextendsingle/vextendsinglet
+0
+−1
+ωΛ/integraldisplayt
+0dτdF1(τ)
+dτcos(ωΛ(t−τ)). (A12)
+Note that the second integral contains a large denomina-
+tor 1/ωΛ. In the first term only the limit τ=tgives a
+nonoscillating contribution to the integral. We can con-
+tinue the expansion in powers of 1 /ωΛ. Note that the
+next term proportional to ˙F1will contain only highly os-
+cillatory part and can be neglected. So up to the third
+order in 1 /ωΛwe find:
+/integraldisplayt
+0dτF1(τ)sin(ωΛ(t−τ))≈F1(t)
+ωΛ−1
+ω3
+Λd2F1(t)
+dt2.(A13)
+Combining Eqs. (A7), (A8), and (A13) we find
+θ>
+1(k,t)≈g
+λ/integraldisplay
+d2rexp[−ikr]sin(θ<(r,t))
+−g
+λω2
+λ/integraldisplay
+d2rexp[−ikr]cos(θ<(r,t))¨θ<(r,t),(A14)
+where we used Λ a= 1 again. Here we neglected by the
+term proportional to ( ˙θ<(r,t))2, because it leads to a
+subdominant (in the RG sense) contribution. From this
+we find that
+θ>
+1(r,t)≈g
+λ/integraldisplay
+shelld2k
+(2π)2eikr/integraldisplay
+d2xe−ikxsin(θ<(x,t))−g
+λ2Λ2µ/integraldisplay
+shelld2k
+(2π)2eikr/integraldisplay
+d2xe−ikxcos(θ<(x,t))¨θ<(x,t).(A15)
+We now consider the term ( g/a2)cos(θ<(r,t))θ>
+1(r,t):
+g
+a2cos(θ<(r,t))θ>
+1(r,t)≈g2
+λa2/integraldisplay
+shelld2k
+(2π)2eikr/integraldisplay
+d2xe−ikx1
+2sin(θ<(x,t)−θ<(r,t))
+−g2
+λ2µ/integraldisplay
+shelld2k
+(2π)2eikr/integraldisplay
+d2xe−ikx1
+2¨θ<(x,t), (A16)
+where we neglected terms such as sin(2 θ<(r,t)). Because we are integrating over high momentum shell this integr al
+will be suppressed unless xis close to r. This suggests change of variables x=r+ξand Taylor expanding θ<(x,t)13
+in powers of , i.e. θ<(x,t)≈θ<(r,t)+ξ∇θ<(r,t)+1
+2ξαξβ∂θ<(r,t)
+∂rα∂rβ. Then
+g
+a2cos(θ<(r,t))θ>
+1(r,t)≈C2
+8πg2
+λδΛ
+Λ△θ−C1
+4πg2
+λ2µδΛ
+Λ¨θ, (A17)
+where
+C1
+Λ2=/integraldisplay
+d2ξJ0(Λξ),C2
+Λ4=/integraldisplay
+d2ξξ2J0(Λξ).(A18)
+When we need to substitute these expressions back into
+Eq. (A9), we find that the term containing △θrenormal-
+izes the coupling λas
+λ→λ+C2
+8πg2
+λδΛ
+Λ. (A19)
+In addition there is an extra term proportional to ¨θgen-
+erated in Eq. (A9), which renormalizes µ:
+1
+µ→1
+µ+C1
+4πg2
+λ2µδΛ
+Λ. (A20)
+Finally we restore the cutoff by rescaling k→k(1−
+δΛ/Λ),r→r(1 +δΛ/Λ),t→t(1 +δΛ/Λ), and p→
+p(1−δΛ/Λ). This rescaling additionally renormalizes
+the coupling g:g→g(1 + 2δΛ/Λ). Combining this re-
+sult with Eqs. (A11), (A19), (A20) we find the following
+renormalization group equations:
+dg
+dl=g/parenleftbigg
+2−1
+4πv01
+K/parenrightbigg
+(A21)
+dK
+dl=1
+16πKg2
+v2(C2+2C1), (A22)
+dv
+dl=g2
+16πvK2(C2−2C1). (A23)wherel= lnΛ. We can read off from Eq. A23, that
+if the system contains a fixed velocity, for example in
+relativisticsystems, weneedtohave C2= 2C1, toenforce
+that the velocity is invariant under the flow.
+Note that if the initial system is already close to the
+critical point then the RG equations above simplify to
+dg
+dl≈g/parenleftbigg
+2−1
+4πλ/parenrightbigg
+, (A24)
+dλ
+dl≈C2
+8πg2
+λ, (A25)
+which are equivalent to Eqs. (40) and (41). Note that a
+more complete set ofRG equations (A21) - (A23) has the
+same universal predictions of the dynamical phase tran-
+sitions and exponential divergence of the time scales as
+the simplified equations above. Also note that the real
+RG equations bear close analogy to the flow equations in
+imaginarytime characterizingtheequilibrium Kosterlitz-
+Thouless transition35. Thus the non-equilibrium KT
+transition discussed here is characterized by exponen-
+tially divergent length and time scales. Physically these
+long scales characterize very slow process of vortex un-
+binding and equilibration at long distances. Note that
+the RG equations (A21) - (A23) also implicitly take into
+account renormalization of the temperature in the sys-
+tem. This comes from the fact that creating vortex-
+antivortex pairs removes the energy from the phonon de-
+grees of freedom. We are going to investigate this issue
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+22S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, J.
+Schmiedmayer, Nature 449, 324 (2007).
+23A. A. Burkov, M. D. Lukin, E. Demler, Phys. Rev. Lett.
+98, 200404 (2007).
+24R. Bistritzer and E. Altman, PNAS 104, 9955 (2007).
+25I. E. Mazets, J. Schmiedmayer, arXiv:0806.4431
+26P. Calabrese and J. Cardy, Phys. Rev. Lett. 96, 136801
+(2006); J. Stat. Mech. - Theor. and Exp., P06008 (2007).
+27T. W. B Kibble, J. Phys. A 9, 1387 (1976); Physics Today
+60, 47 (20007).
+28W. H. Zurek, Nature 317, 505 (1985).
+29P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and
+C. W. Gardiner, Advances in Physics 57, 363 (2008).
+30A. Polkovnikov, Phys. Rev. A 68, 053604 (2003).
+31A. Polkovnikov, arXiv:0905.3384.
+32E. H. Lieb and D. W. Robinson, Comm. Math. Phys. 28,
+251 (1972).
+33S. Sachdev, Quantum Phase Transitions , (Cambridge Uni-
+versity Press, Cambridge, 1999).
+34L.D.LandauandE.M.Lifshitz, Mechanics , (Butterworth-
+Heinemann, Oxford, 1982).
+35T. Giamarchi, Quantum Physics in One Dimension ,
+(Clarendon Press, Oxford, 2004).
\ No newline at end of file
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@@ -0,0 +1,329 @@
+12/31/09  1 Imaging ferroelectric hysteresis and domain wall pinning  
+ 
+S. Bhattacharya 1 and M.J. Higgins 2 
+1 Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai – 400005, India  
+2 Princeton High School, Princeton, New Jersey 08540, USA  
+ 
+Simultaneous imaging of the piezoresponse phase, amplitude and bare surface 
+topography of displacive ferroelectric thin films by scanning probe microscopy directly 
+shows the nature of domain wall pinning and its relation to morphological disorde r. 
+Strong and stable pinning of walls occurs at grain boundaries while weak and unstable 
+pinning occurs within the grains. The results show that polarization reversal consists of 
+both nucleation of domains and spreading of domain walls, the latter being th e 
+dominant process in multi -grain regions. A characteristic “single -grain” switching in 
+faceted grains is observed as the motion of a single domain wall parallel to one of the 
+facets.  
+ 
+PACS numbers: 68.37Ps, 77.80Dj, and 77.80Fm  12/31/09  2 An understanding of domain  wall dynamics associated with hysteresis phenomena in 
+condensed matter ground states such as a ferroelectric or a ferromagnet, is important 
+in the general area of statistical physics of elastic media in a random pinning 
+environment [1]. The generically metastable  equilibrium behavior of such systems and 
+their non -equilibrium evolution under an external driving force, leading to hysteresis, 
+are of current interest because of their relevance in a variety of condensed matter 
+systems such as vortices in superc onductors [2], sliding charge density waves [3], 
+electron lattices in semiconductor heterostructures [4], two -fluid interfaces in random 
+porous media [5], in addition to the general problem of sliding friction [6].  
+ Furthermore, ferroelectricity, like ferr omagnetism, forms the basis of non -volatile 
+memory through the hysteresis phenomenon where the spontaneous polarization of a 
+polarized ferroelectric is retained even after the polarizing field is turned off [7]. The 
+utility of these materials as memory dep ends critically on their ability to (a) switch 
+polarization repeatedly and robustly from one state to another, (say “up” to “ down”) at 
+suitably small polarizing fields and (b) retain this polarization against thermal or other 
+randomizing influences. There fore, an understanding of the dynamics of domain walls 
+[8] is central to both the basic physics involved and the principles for designing robust 
+memory. Historically, these phenomena were studied through bulk measurements, 
+whose interpretation required ass umptions not easy to ascertain apriori.  Recently, 
+however, more direct scanning probe imaging studies in the nanoscale have begun to 
+provide incisive information about the complex process leading to repeatable hysteresis, 
+or the absence thereof [9].  
+ In this letter we report on a scanning probe microscopy -based study of a 
+ferroelectric thin film that focuses on a central issue, namely, the nature of domain wall 
+pinning. By simultaneous imaging  of the magnitude and direction of the polarization, 
+together wi th the topography, we probe the stability of domain walls, the types of 
+pinning and their connection to quenched morphological disorder. A broad 
+characterization of pinning, weak or strong, random or correlated, point -like or 
+extended, elucidates connectio ns with other condensed matter ground states mentioned 
+above and to the theoretical problem of disordered elastic media, in general. All 12/31/09  3 measurements reported here are on c -axis oriented thin films of lead -zirconate/ titanate 
+(PZT) with Zr/Ti ratio of 20/8 0 of typical thickness of 50 nm, grown by sol -gel method, 
+on a platinum bottom electrode grown on Si -substrates. The imaging studies were made 
+with a Digital 3000 atomic force microscope suitably modified to perform piezo -response 
+studies using metallic ti ps and an external but standard phase -sensitive lock -in 
+detection assembly.   
+Piezo-response Force Microscopy (PFM) is a standard method [10] of visualizing 
+ferroelectric domains of different polarizations. The ferroelectric moment is proportional 
+to the p iezoelectric coefficient; thus for a c -axis oriented sample with an effective 
+electric field also along the c -axis, the technique measures the piezoelectric coefficient 
+d33. The strain measured is given by 
+  ~ 
+ d33(z)E 3dz (where E 3 is an oscillatory elect ric 
+field of frequency 11kHz in the experiment), integrated over the effective thickness of 
+the sample. The order parameter P is then proportional to |R| exp (i
+ ) where |R|, the 
+amplitude of the resulting oscillatory strain, is a continuous variable, summ ed over the 
+thickness of the sample. But the phase 
+  is discrete, 0 and 
+ , for the so -called c+ and c- 
+domains, corresponding to up or down polarization. In our experiments, the metallic tip 
+of the AFM was used to “write” or polarize the sample using a vol tage V dc, which is 
+greater than the local coercive voltage V c. The tip is rastered along a predetermined 
+path to generate the desired “writing”. This written pattern is then read by using the 
+PFM mode with an ac reading voltage V ac significantly smaller th an V c. Typical results of 
+such writing and reading are shown in Figure 1 where the topography, amplitude and 
+phase, measured simultaneously, are plotted in panels a, b and c respectively. A square 
+region was first poled by the tip to “down” polarization. A  “cross” was then written in 
+the center, with the “up” polarization and the written pattern was “read” using the 
+piezo-response microscopy method. The amplitude signal is large inside the domains 
+regardless of the sign of the polarization but is small at t he boundary representing the 
+domain wall (DW), across which the polarization switches, i.e., the dark lines are 
+convenient markers of the DW, consistent with previous studies [11].  
+ The unintended jaggedness  of the domain wall is an indicator of the qualit y of 
+“writing”.  It limits the density of memory elements and is of central importance in 12/31/09  4 memory design. The physical origin of this limitation is illustrated in panel d which is 
+generated by a weighted sum of the raw signals in the amplitude and topograph y 
+images in panels (a) and (b).  As is obvious in panel (d), the boundaries of the written 
+pattern closely follow the contour of contiguous grain boundaries (GB).  
+ Importantly, the DW’s appear in two varieties, sharp and fuzzy.  The sharp walls 
+correspond precisely to the GB’s. The width of the sharp walls is typically about 20nm, 
+which is determined from independent estimates to be the effective size of the tip and 
+does not represent the intrinsic width of the DW’s, which is likely to be significantly 
+smaller [10]. However, it shows unambiguously that the GB’s, which are extended 
+defects, are the favored pinning sites for the domain wall.  This is easily understood: 
+since the ferroelectric order parameter is strongly coupled to lattice distortion, the GB is  
+where the order parameter is suppressed and is thus energetically the best place to 
+locate the DW. We note that the c -axis oriented films have a columnar morphology and 
+thus the GB’s represent extended columnar defects, analogous to columnar pins in 
+vortex phases of superconductors [12].  The image in (b) is not bi -modal as in (c) and 
+shows a large variation of the magnitude of R within regions with the same phase 
+  as 
+in the upper panel.  Even so, the amplitude is remarkably uniform within a grain but 
+shows large grain -to-grain variation for polarization of the same sign. Thus, the 
+correlation length 
+ R for the continuous variable R is typically given by the grain size, 
+while the correlation length 
+  for the discrete variable 
+  can be much larger, spanning 
+multi-grain regions in the poled state.  
+ The fuzzy domain walls, in contrast with the sharp walls, are located in the intra -
+granular space; two of them are encircled on each panel in Figure 1. The y are very 
+wide, covering almost the entire grain, nearly an order of magnitude larger (~100nm) 
+than the sharp walls and are not limited by the tip -size.  These fuzzy walls are 
+characterized by (1) phase signal varying randomly by 
+  from one pixel to the n ext and 
+(2) by a large fluctuation in the overall small amplitude signal within these wall areas . A 
+more detailed analysis shows that a significant source of the fuzziness is temporal in 
+nature, i.e. the signal is temporally noisy, the details of which wil l be reported 
+elsewhere [13].  But the fuzziness clearly implies “weak” pinning, (presumably due to 12/31/09  5 random point -defects such as oxygen vacancies or substitutional impurities) in the 
+intra-granular space.  
+ Using the same methods, we have imaged the switchi ng of a multigrain region, as 
+described below.  A specific area is first written using a dc bias voltage V dc by the tip.  
+As the tip moves over to the next area to be written, the previously written area relaxes 
+towards its remanent state (V dc = 0).  Once the entire area is thus written on and then 
+allowed to relax, we read the resulting domain structure with the voltage applied to the 
+tip is V dc + V ac  , where V ac is the piezo -detection voltage (V ac << V dc). This is equivalent 
+to executing a “minor hystere sis loop” commonly used in ferromagnets: the state to be 
+read is thus closer to the original hysteresis loop than to the remanent state.  
+ Figure 2 shows a composite  of phase and topography images of switching of a 
+500nmx500nm area.  The upper left panel re presents the remanent state after the area 
+was poled “up” and then allowed to relax.  Every subsequent panel marks the V dc value 
+at which the state is “read”.  The images show a grain -by-grain spreading of the 
+polarization reversal from up to down among co ntiguous grains, often preceded by a 
+sliver of switched region extending along the grain boundary and then spreading 
+through a grain. One such spreading front is marked by the black arrows for a grain 
+enclosed by a dark square on the first panel. This proc ess is analogous to the easy 
+movement of vortices in oxide superconductors along twin boundaries but not across 
+them [14]. Isolated nucleation also occurs [15], as shown in panel (c) by the encircled 
+area, but is a secondary contribution to the net polariz ation.  
+ Combining these results, we conclude that the switching process is dominated by 
+DW-motion in c -axis oriented polycrystalline films that are realistic candidates for non -
+volatile memory.  The DW is stable due to the strong pinning by the correlated disorder 
+provided by the GBs.  The intra -granular space, on the other hand, presents weak and 
+presumably collective pinning that results in unstable DW’s that are seldom stable 
+enough to be captured easily in an image. They are mostly visible as fluctuatin g fuzzy 
+DWs. The implication of these results is obvious: In systems such as these, the memory 
+cells should be grain -size matched [16] for optimal stability and large polarization. The 
+fundamental reason for this is the coupling of the polarization to latt ice distortion that 12/31/09  6 makes ferroelectricity extremely sensitive to extended morphological disorder such as 
+the GBs.  
+ Finally, we demonstrate switching observed in a single grain.  The process of 
+single grain switching is typically fast and occurs within a v ery narrow range of bias 
+values so that at a given dc bias, the imaging technique captures few DWs in intra -
+granular space. However, if the grains are faceted, i.e., the GB’s correspond to 
+crystallographic axes, the evolution of the single grain switching spans a large enough 
+bias range to be captured by these imaging techniques.  A remarkable example is 
+shown in Fig.3.  In this case, a large square area was written in “up” polarization. But 
+the central grain remained polarized in the “down” direction, impl ying a larger local 
+coercivity. The relaxed state is found to be clearly faceted, as in epitaxial samples [17]; 
+a few of the facets are marked by dotted lines in panel(a).  The subsequent panels are 
+obtained at higher dc biases aimed at switching the centr al grain back to the “up” 
+polarization.  The left and right panels show the amplitude and phase images, 
+respectively.  As V dc increases, we observe the remarkable movement of a single 
+domain wall, marked by the black arrow in the amplitude panels, through the grain.  
+Very significantly, DW moves in a way parallel to one of the facets, marked by the white 
+arrow, as it sweeps across and becomes rounded only at the very late stages of 
+switching. At the beginning of the switching process, the interior of the gr ain has large 
+amplitude as do the neighboring grains.  As V dc increases, the DW broadens as it moves 
+towards the interior of the grain depleting the amplitude ahead of it and enhancing the 
+amplitude behind, while the back wall, marked by the white arrow, r emains pinned to 
+the initial configuration.  In the last panel (h) the back wall becomes rounded as the 
+domain wall approaches collapse with the switching approaching completion. Similarly, 
+the phase fluctuates in the wall region as seen in Fig1(c), but in  contrast with Fig.1, the 
+fluctuating walls are now stable enough to be captured reliably in the images.  
+ The movement of the domain wall with increasing dc bias implies a large gradient 
+in the local coercivity. The faceted grain boundaries, an analog of both twin boundaries 
+and columnar defects in superconductors, provide the largest local coercivity, a measure 
+of the local pinning strength, which can be as much as an order of magnitude larger 12/31/09  7 than non -faceted ones. Moreover, the local coercivity varies s ignificantly from one facet 
+to another. As the dc bias is increased, the back wall remains pinned to the faceted GB, 
+which presumably has the strongest pinning.  The front wall moves through the intra -
+granular space where uncorrelated local point defects p rovide weaker pinning than the 
+extended GB.  This explains the motion of the domain wall but not the specific direction 
+of motion, nearly parallel to the pinned back wall, nor their stability. We tentatively 
+explain it in the following way. The back wall i s the strongest pinning GB; the long -
+range elastic interaction is responsible for creating a decreasing gradient of local 
+coercivity away from the back wall.  The leading edge of the moving domain wall climbs 
+this gradient as the bias is increased. Thus, t he pinning in the GB’s controls the DW -
+motion in the intra -granular space and accounts for its stability. We also note that the 
+observed motion of the domain wall is remarkably similar to the motion of the boundary 
+between an ordered and a disordered vorte x phase in superconductors where the 
+system has a large gradient in the “local critical current” (the analog of a local coercive 
+field) between the edge of a sample and the interior [18].  
+ The imaging results shown above provide a broad scenario of polariz ation-reversal 
+in device -quality c -axis oriented ferroelectric films.  Due to the strong coupling between 
+the order parameter (ferroelectric moment) and the lattice distortion, lattice defects 
+suppress the ferroelectric order parameter becoming natural pin ning centers for the 
+DW.  The results yield the following conclusions. (I) Extended columnar defects such as 
+the grain boundaries are strong pinning sites for the DW.  (II) The switching is largely 
+“collective” in the weak -pinning intragranular space that produces unstable and noisy 
+domain walls, characterized by fuzzy images and are thus unsuitable for stable and 
+nonvolatile memory. Although the phase correlation length can be larger depending on 
+the poling field, the amplitude correlation length is typica lly set by the grain size. FE -
+based memory in polycrystalline material should thus be grain -size matched. (III) 
+Switching of multi -grain areas occurs through spreading of domain walls across 
+contiguous grains as the dominant contributor to the net polariza tion, whereas local 
+nucleation of disconnected domains is a secondary process. (IV) Single -grain switching 
+for faceted grains proceeds via the motion of a single DW. In this case the GB pinning 12/31/09  8 due to long -range elastic interaction controls the DW dynamics  in intra -granular space. 
+These results provide a characterization of the ferroelectric DW motion in the same 
+common conceptual framework of a classification and hierarchy of pinning for the 
+dynamics of disordered elastic media as has been done for other c losely related systems 
+[1,2,3]. We note that similar jaggedness for both ferroelectric and ferromagnetic 
+domain walls, correlated with morphological disorder, has recently been reported in 
+multiferroic systems as well [19]. These results are thus of import ance in memory 
+design principles in a wider class of systems of current interest. Clearly, a 
+comprehensive theoretical framework is needed for domain wall pinning in ferroelectrics 
+with a coupling between polarization and lattice distortion, which would pr ovide 
+estimates of local coercivities, intrinsic domain wall widths as well as of correlation 
+lengths for both phase and amplitude of the order parameter and their dependence on 
+external field, in order to aid and guide quantitative interpretations of expe rimental 
+results such as reported here.  
+ The authors acknowledge helpful discussions with P. Chandra, S. Duttagupta, S. 
+Ghosh, A. Krishnan, P.B. Littlewood, V.R. Palkar, R. Ramesh, J.F. Scott and M.M.J. 
+Treacy. One of us (SB) especially acknowledges insigh tful discussions with A. I. Larkin 
+about ferroelectric domain wall dynamics and its relation to other disordered elastic 
+media.12/31/09  9 References:  
+1. D.S. Fisher, in Nonlinearity in Condensed Matter, edited by A.R. Bishop, D.K. 
+Campbell, P. Kumar and S.E. Trullin ger (Springer -Verlag, Berlin, 1987).  
+2. G. Blatter et al., Rev. Mod. Phys. 66, 1125 (1994)  
+3. G. Gruner, Rev. Mod. Phys. 60, 1129 (1988)  
+4. Y.P. Li et al, Phys. Rev. Lett. 67, 1630 (1991)  
+5. J.P Stokes et al, Phys. Rev. Lett. 65, 1885 (1990)  
+6. B.N.J. Pers son, Sliding Friction, Physical Principles and Applications (Springer, 
+2000)  
+7. O. Auciello, J.F. Scott and R. Ramesh, Physics Today, 51, 22 (July 1998).  
+ J.F. Scott, Ferroelectric Memories (Springer, 2000).  
+8. S. Lemerle et al, Phys. Rev. Lett. 80, 849 ( 1998); P. Paruch et al, Phys.Rev.Lett. 
+ 95, 197601 (2005), and references therein.  
+9. Nanoscale Characterization of Ferroelectric Materials, Scanning Probe Microscopy 
+Approach, Edited by M. Alexe and A. Gruverman (Springer, 2004)  
+10. See Chapter 1 by S.V. Kalinin and D.A. Bonnell in Reference 9 for more details of 
+the imaging technique.  
+11. For imaging studies of similar kinds see, e.g., B.J. Rodriguez et al, Appl. Phys. 
+Lett. 86, 012906 (2005), S.V. Kalinin et al, ibid, 85, 795 (2004), M. Kelman et al, 
+J. Appl. Phys. 94 5210 (2003) and references therein.  
+12. R.C. Budhani et al, Phys. Rev. Lett. 72, 566 (1992)  
+13.  S. Bhattacharya and M.J. Higgins, Unpublished.  12/31/09  10 14. W.K. Kwok et al., Phys.Rev.Lett. 64, 966 (1990)  
+15.  See Figure 5 in A. Gruverman and A. Khol kin, Rep. Prog. Phys. 69, 2443 (2006).  
+16. For a depoling study is consistent with the same conclusion, see M.J. Higgins et al, 
+Appl. Phys. Lett. 88, 3373 (2002)  
+17. C.S. Ganpule et al, Phys. Rev. B 65, 014101 (2002)  
+18.  M. Marchevsky et al, Nature 409, 5 91 (2001) and Phys. Rev. Lett. 88,087002 
+(2002)  
+19. V.R. Palkar et al, Appl. Phys. Lett. 90, 172901(2007)12/31/09  11  
+Figure captions:  
+1. AFM images of (a) topography, (b) piezo -amplitude and (c) piezo phase signal of 
+a 1
+m x 1
+ m region which is poled in the down con figuration after which a “cross” 
+is written by a dc bias field which reverses the polarization of that spatial region.  
+The domain wall separating up and down polarization has depleted amplitude, 
+marked by the dark line in (b), consists of sharp and fuzzy regions. Panel (d) 
+represents a composite image of topography and amplitude, which shows that the 
+sharp domain wall precisely follows the contour of the contiguous grain 
+boundaries. The “fuzzy” grain boundaries,  two of which are encircled in each 
+panel, ha ve depleted amplitude over nearly the entire grain and the phase 
+fluctuates randomly between 0 and 
+  seen in (c) in the form of fluctuating color.  
+2. Switching of a multigrain area shown by composite images of phase and 
+topography of a 500nm x 500nm area at various polarizing dc voltages marked 
+under each panel. Panel (a) represents the remanent state  of the area after it was 
+poled “down”. Subsequent panels show the progressive switching of the region as 
+the poling dc bias is increased. Most of the switching occurs via the motion of 
+domain walls across contiguous grains, although independent nucleation  also 
+occurs, one of which is encircled in panel (c). The upper left corner in panel (a) 
+shows a marked square area spanning a grain. The following panels show the 
+progression of switching in this grain: first a sliver of the opposite phase enters 
+and subs equently spreads across the grain in panels (c) through (i). A black arrow 
+in the panels marks the moving front.  
+3. Single grain switching sequence in a faceted grain. The right columns are the 
+phase images and the left columns are the amplitude images.  T he corresponding 
+bias voltages are noted in the respective panels. Image area is 250nm x 250 nm. 
+White dotted lines mark a few of the facets in panel (a). As the bias voltage 
+increases, the back wall, marked by the white arrow, remains pinned while the 12/31/09  12 front wall, marked by the black arrow, moves in a direction roughly parallel to the 
+back wall, climbing a large coercivity -gradient that exists between the two walls. 
+See text for discussions.  
+ 12/31/09  13 Figure 1
+ 
+ 
+ 
+ 
+1
+m
+(a) (b) (c) (d)
+50 100 150 200 25050
+100
+150
+200
+250
+-1-0 .8-0 .6-0 .4-0 .200 .20 .40 .60 .8112/31/09  14 Figure 2  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+1
+m
+(a) (b) (c) (d)
+50 100 150 200 25050
+100
+150
+200
+250
+-1-0 .8-0 .6-0 .4-0 .200 .20 .40 .60 .81
+0.0V 0.3V 0.4V 0.5V 0.6V
+0.7V 0.8V 0.9V 1.0V 1.2V(a) (b) (d) (e)
+(f) (g) (h) (i) (j)(c)
+500nm12/31/09  15 Figure 3  
+0V
+0.45V
+0.60V
+0.65V0.72V
+0.82V
+0.83V
+0.85V
+(a)
+(b)
+(c)
+(d)(e)
+(f)
+(g)
+(h)180Þ
+0Þ 0max 250nm
\ No newline at end of file
diff --git a/1001.0100.txt b/1001.0100.txt
new file mode 100644
index 0000000000000000000000000000000000000000..354b3d0f68e408e61b8085bb54fd9f95d1186c2b
--- /dev/null
+++ b/1001.0100.txt
@@ -0,0 +1,167 @@
+arXiv:1001.0100v1  [math.NT]  3 Jan 2010THE QUADRATIC CHARACTER OF 1+√
+2AND AN
+ELLIPTIC CURVE
+YU TSUMURA
+Abstract. Whenp≡1 (mod 8), we have a criterion of the qua-
+dratic character of 1 +√
+2, which is related to the class number
+ofQ(√−p). In this paper, we obtain a similar criterion using an
+elliptic curve, which contrasts to the proof using algebraic number
+theory for the old one.
+1.Introduction.
+Letp≡1 (mod 8) be a prime number. Then by the quadratic
+reciprocity law, 2 is a square mod p, hence there exists√
+2∈Fp. Also
+we know that p≡1 (mod 8) can be expressed as p=c2+8d2, where
+c,dare integers.
+In1969, Pierre Barrucand and Harvey Cohn provided the criterion
+of quadratic character of 1+√
+2. Let/parenleftBig
+∗
+p/parenrightBig
+be the Jacobi symbol mod
+p. Then they proved the following.
+Theorem 1.1. Letp=c2+ 8d2be a prime number, where c,dare
+integers. Then we have
+/parenleftBigg
+1+√
+2
+p/parenrightBigg
+= 1⇐⇒d≡0 (mod 2) ⇐⇒h(−4p)≡0 (mod 8) ,
+whereh(−4p)denotes the class number of Q(√−p).
+Proof.See [1] and Proposition 5.15 in [2]. /square
+This was proved by the method of algebraic number theory. In this
+article, we provide a slightly different criterion with a different taste
+proof using properties of an elliptic curve defined by E:y2=x3−x.
+This proof is simple and more accessible than the old one for those who
+are familiar with the basics of elliptic curves.
+2010Mathematics Subject Classification. Primary 11G20; Secondary 11E25.
+12 YU TSUMURA
+2.A new criterion
+Again let p≡1 (mod 8) be a prime number. Then we can write
+p=a2+b2, wherea,bare integers. Further we can assume that ais
+odd,bis even and a+b≡1 (mod 4). Then we show the following.
+Theorem 2.1. Let notation be as above. Then we have
+/parenleftBigg
+1+√
+2
+p/parenrightBigg
+= 1⇐⇒(a−1)2+b2≡0 (mod 32)
+Proof.LetE(Fp) be an elliptic curve defined by y2=x3−xover a
+finite field Fp. We know that # E(Fp) = (a−1)2+b2. (See Theorem
+4.23, page 115 in [3].)
+Sincep≡1 (mod 4), −1 is a square mod pby the quadratic reci-
+procity law. So let i∈Fpbe a square root of −1. Then the action of i
+on a point ( x,y)∈E(Fp) defined by [ i]·(x,y) = (−x,iy) is easily seen
+to bea homomorphism. Actually, E(Fp) has complex multiplication by
+Z[i]. (See Example 10.2 in [3].) Let us denote η= 1+i. We describe
+the action of ηexplicitly. Let η·(x,y) = (x0,y0). We have
+η·(x,y) = [1+i]·(x,y) = (x,y)+[i]·(x,y) = (x,y)+(−x,iy).
+Hereiin they-coordinate is a square root of −1 inFp. Then by the
+elliptic curve addition, it is equal to
+(2.1)/parenleftBigg/parenleftbigg(1−i)y
+2x/parenrightbigg2
+,y0/parenrightBigg
+(2.2) =/parenleftbiggx2−m
+2ix,y0/parenrightbigg
+,
+wherey0=/parenleftBig
+(1−i)y
+2x/parenrightBig
+(x−x0)−y. Note that by the equation (2.1), the
+x-coordinate x0ofη·(x,y) is a square. Also by the equation (2.2),
+we have deg( η) = 2, hence it is easy to see that ker( η) ={∞,(0,0)},
+where∞denotes the point at infinity (the identity).
+Letx(S),S⊂E(Fp) be the set of x-coordinate of ( x,y)∈S\{∞}.
+Now, using (2.2), we see that x(η−1(∞)) ={0},x(η−2(∞)) ={±1},
+x(η−3(∞)) ={±i},x(η−4(∞)) ={±1±√
+2}.
+Now we prove a claim that ( x0,y0)∈E(Fp) has a preimage by ηin
+E(Fp) if and only if x0is a square mod Fp. Suppose x0is a square
+modp, then we solve equation (2.2). Comparing x-coordinate we have
+x0= (x2−m)/(2ix). Solving for x, we have x=x0i+/radicalbig
+m−x2
+0. SinceTHE QUADRATIC CHARACTER OF 1+√
+2 AND AN ELLIPTIC CURVE 3
+/parenleftBig
+x0
+p/parenrightBig
+= 1, we have
+/parenleftbiggm−x2
+0
+p/parenrightbigg
+=/parenleftbigg−y2
+0x−1
+0
+p/parenrightbigg
+=/parenleftbigg−1
+p/parenrightbigg/parenleftbiggy2
+0
+p/parenrightbigg/parenleftbiggx0
+p/parenrightbigg
+= 1.
+Hence√m−x0∈Fpand therefore, we have x∈Fp. Now comparing
+y-coordinate, we have y=y0/braceleftbig/parenleftbig1−i
+2x/parenrightbig
+(x−x0)−1/bracerightbig−1∈Fp. Then we
+haveη·(x,y) = (x0,y0) for (x,y)∈E(Fp).
+The converse follows from the equation (2.1).
+We finish the consideration of the action of ηand now we move to
+prove the theorem. Suppose/parenleftBig
+1+√
+2
+p/parenrightBig
+= 1. Then since
+(2.3)/parenleftBigg
+1+√
+2
+p/parenrightBigg/parenleftBigg
+1−√
+2
+p/parenrightBigg
+=/parenleftbigg−1
+p/parenrightbigg
+= 1,
+we have/parenleftBig
+1−√
+2
+p/parenrightBig
+= 1. Hence all points whose x-coordinate is one of
+±1±√
+2 have preimages by ηinE(Fp) by the above claim. Hence
+η−5(∞) is a subgroup of E(Fp). Since # η−5(∞) = 32 (note that
+#ker(η) = 2), we have 32 |#E(Fp) = (a−1)2+b2, hence we have
+(a−1)2+b2≡0 (mod 32).
+Conversely, suppose ( a−1)2+b2≡0 (mod 32). Since 32 |#E(Fp) =
+(a−1)2+b2andE(Fp) is in general cyclic or a product of two cyclic
+groups, we have an element P∈E(Fp) of order 8. Then the x-
+coordinate of η(P) orη2(P) is one of ±1±√
+2, say it is Q. Hence
+Qhas a preimage of η. As we saw in the above claim, this means
+that the x-coordinate of Qis a square mod p. So one of ±1±√
+2 is
+a square. Hence by (2.3), all of them are squares. Especially, we ha ve/parenleftBig
+1+√
+2
+p/parenrightBig
+= 1.
+/square
+Comparing these two theorems, we immediately get the following,
+which seems to difficult to prove elementarily.
+Corollary 2.2. Letp≡1 (mod 8) be a prime number. Let p=a2+
+b2=c2+ 8d2, wherea,b,c,dare integers and ais odd,bis even and
+a+b≡1 (mod 4) . Then
+(a−1)2+b2≡0 (mod 32) ⇐⇒d≡0 (mod 2) .
+References
+1. PierreBarrucandandHarveyCohn, Note on primes of type x2+32y2, class num-
+ber, and residuacity , J. Reine Angew. Math. 238(1969), 67–70. MR MR0249396
+(40 #2641)4 YU TSUMURA
+2. Franz Lemmermeyer, Reciprocity laws , Springer Monographs in Mathematics,
+Springer-Verlag, Berlin, 2000, From Euler to Eisenstein. MR MR1761 696
+(2001i:11009)
+3. Lawrence C. Washington, Elliptic curves , second ed., Discrete Mathematics and
+its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2008,
+Number theory and cryptography. MR MR2404461 (2009b:11101)
+Department of Mathematics, Purdue University 150 North Uni ver-
+sity Street, West Lafayette, Indiana 47907-2067
+E-mail address :ytsumura@math.purdue.edu
\ No newline at end of file
diff --git a/1001.0101.txt b/1001.0101.txt
new file mode 100644
index 0000000000000000000000000000000000000000..bacfc8f116b34ccb88dd7b3010ed47ea7a1afeac
--- /dev/null
+++ b/1001.0101.txt
@@ -0,0 +1,122 @@
+arXiv:1001.0101v1  [hep-ph]  31 Dec 2009SPIN LIGHT MODE OF MASSIVE NEUTRINO RADIATIVE DECAY IN
+MATTER
+A. GRIGORIEVa
+Skobeltsyn Institute of Nuclear Physics, Moscow State Univ ersity,
+119992 Moscow, Russia
+A. LOKHOVb
+Department of Quantum Statistics and Quantum Field Theory, Moscow State University,
+119992 Moscow, Russia
+A. STUDENIKINc
+Department of Theoretical Physics, Moscow State Universit y,
+119992 Moscow, Russia
+Introduction. The calculations of the radiative neutrino decay performed by several au-
+thors1have shown that the process characteristics are substantia lly changed if the presence of
+a medium is taken into account. Note, the influence of the back ground matter was considered
+only in the vertex (it is the influence on the neutrino magneti c moment). Here we are going to
+discuss the impact of the medium onto the state of neutrino it self in the process which is similar
+to the so called spin light of neutrino ( SLν)2, a new mechanism of the electromagnetic radiation
+the detailed quantum treatment of which was evaluated in a se ries of our papers3. Lately, there
+appeared several investigations4,5of the same effect but in connection to the electron which can
+bethe source of the spin light of electron ( SLe). Themechanism of SLis based on helicity states
+energy difference of the particle arising due to weak interact ion with the background matter, in
+addition the energy deposit depends on the difference between the total energies of neutrinos
+(or electrons) in the initial and final states. In neutron mat ter, a more significant case is the
+SLνprocess with participation of antineutrinos and thus it is w hat we will study here. However
+for the convenience in what follows we will still speak about neutrinos.
+We will consider the transition of one neutrino mass state ν1into another mass state ν2
+assuming that m1> m2, and restrict ourselves with only these two neutrino specie s and accord-
+ingly with two flavour neutrinos (like in the previous works) . Since the interactions of flavour
+neutrinos with neutron star matter, composed predominantl y of neutrons, are the same and
+governed by the neutron density we will take equal interacti ons for the initial and final massive
+neutrinos with the matter.
+Modified Dirac equation. The system “neutrino ⇔dense matter” depicted above can
+be circumscribed mathematically in different ways. Here we us e the powerful method of exact
+solutions developed in3,4. In the frame of this method neutrino states in matter are des cribed
+aax.grigoriev@mail.ru
+blokhov.alex@gmail.com
+cstudenik@srd.sinp.msu.ruby the modified Dirac equation: {iγµ∂µ−1
+2γµ(1+γ5)fµ−m}Ψ(x) = 0,where for the case of
+unpolarized and stationary matter fµ=GF√
+2(n,0). For the neutron star, the matter density can
+reach the values of n≈1037÷1040cm−3. The corresponding energy spectrum of neutrino is
+given by Eε=ε/radicalbig
+(p−sαmν)2+m2ν+αmν,whereε=±1 defines the sign of the energy, sis
+the neutrino helicity, pis the neutrino momentum and α=1
+2√
+2GFn
+mν. The exact form of the
+solution of the modified Dirac equation can be found in3,4.
+Spin light mode of massive neutrino decay. Using the above-mentioned results we can
+now compute the process itself. From the energy-momentum co nservation law we get for the
+emitted photon energyd
+w=K(p+∆)−pcosθ+/radicalbig
+(K(p+∆)−pcosθ)2−(K2−1)(∆2+2p∆)
+K2−1,
+whereK=E−pcosθ+αm1
+αm1, ∆ =m2
+1−m2
+2
+2αm1and where θis the angle between the direction of the
+initial neutrino momentum pand the direction of the SLνphoton emission. Using the exact
+neutrino wave functions in matter we calculate the rate of th e process, Γ ν1ν2=/integraltextπ
+0dΓ
+dcosθdcosθ.
+Results and discussion. It is worth to investigate the asymptotical behavior of the r ate
+Γν1ν2in three most significant limiting cases keeping only the firs t infinitesimal order of small
+parameters. So we have,
+1) Γν1ν2= 4µ2α3m3
+1/parenleftBig
+1+3
+2m2
+1−m2
+2
+αm1p+p
+αm1/parenrightBig
+,1≪p
+m1≪αm1
+p(for ultrahigh density case),
+2) Γν1ν2= 4µ2α2m2
+1p/parenleftBig
+1+αm1
+p+m2
+1−m2
+2
+αm1p+3
+2m2
+1−m2
+2
+p2/parenrightBig
+,m2
+1
+p2≪αm1
+p≪1 (for high density case),
+3) Γν1ν2∼µ2m6
+1
+p3,αm1
+p≪m1
+p≪1, m1≫m2(for quasi-vacuum case).
+Note that µis the effective transition magnetic moment of the neutrino6.
+The results 1) and 2) obtained within the generalization of t he method of exact solutions to
+the case when the masses of the initial and final neutrinos are different, in the limit m1=m2
+reproduce exactly the results of the SLνcalculation2,3. The asymptotic estimation 3) gives the
+contribution of the spin light mode to the neutrino radiativ e decay rate.
+Acknowledgements. The authors (A.L. and A.S.) are thankful to Jean Tran Thanh Va n,
+Jacques Dumarchez and Ludwik Celnikier for the kind invitat ion to attend the 21st Rencontres
+de Blois and to all of the organizers for their hospitality in Blois. The authors are also thankful
+to Carlo Giunti for stimulating discussions.
+References
+1. S. Petcov, Sov.J.Nucl.Phys 25, 340 (1977), G. T. Zatsepin, A. Yu. Smirnov, Yad. Fiz. 28, 1569
+(1978)Yad. Fiz. 28, 1569 (1978), P.B. Pal and L. Wolfenstein, Phys. Rev. D25, 766 (1982),
+J.C.D’Olivo, J.F.Nieves, and P.B. Pal, Phys. Rev. Lett. 64, 1088 (1990), C. Giutni, C.W. Kim,
+and W.P. Lam, Phys. Rev. D43, 164 (1990)
+2. A. Lobanov and A. Studenikin, Phys. Lett. B564, 27 (2003), A. Lobanov and A. Studenikin,
+Phys. Lett. B601, 171 (2004), M. Dvornikov, A. Grigoriev, A. Studenikin, Int.J.Mod.Phys. D
+14, 308 (2005)
+3. A. Studenikin, A. Ternov, Phys. Lett. B608, 107 (2005), A. Grigorev, A. Studenikin , A. Ternov,
+Phys. Lett. B622, 199 (2005), A. Grigoriev, A. Studenikin, A. Ternov, Grav.Cosmol. 11, 132
+(2005), A. Grigoriev, A. Studenikin, A. Ternov, Phys.Atom.Nucl. 69, 1940 (2006)
+4. A. Studenikin, Ann.Fond.Louis de Broglie 31, 289 (2006), A. Studenikin, J.Phys. A: Math. Gen.
+39, 6769 (2006), A. Studenikin, J.Phys. A: Math. Theor. 41, 164047 (2008)
+5. A. Grigoriev, S. Shinkevich, A. Studenikin, A. Ternov, I. Trofimo v,Grav.Cosmol. 14, 248 (2008)
+6. C. Giunti, A. Studenikin, Phys.Atom.Nucl. 72, 2151 (2009).
+dRecalling the results on the SLν2,3,4for the initial and final neutrino states we take s1=−1,s2= 1.
\ No newline at end of file
diff --git a/1001.0102.txt b/1001.0102.txt
new file mode 100644
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@@ -0,0 +1,460 @@
+arXiv:1001.0102v1  [cond-mat.supr-con]  31 Dec 2009Chemical potential jump between hole- and electron-doped s ides of ambipolar high- Tc
+cuprate
+M. Ikeda,1M. Takizawa,1T. Yoshida,1A. Fujimori,1Kouji Segawa,2and Yoichi Ando2
+1Department of Complexity Science and Engineering and Depar tment of Physics,
+University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-003 3, Japan
+2Institute of Scientific and Industrial Research, Osaka Univ ersity, Mihogaoka 8-1, Ibaraki, Osaka 567-0047, Japan
+(Dated: November 3, 2018)
+In order to study an intrinsic chemical potential jump betwe en the hole- and electron-doped
+high-Tcsuperconductors, we have performed core-level X-ray photo emission spectroscopy (XPS)
+measurements of Y 0.38La0.62Ba1.74La0.26Cu3Oy(YLBLCO), into which one can dope both holes
+and electrons with maintaining the same crystal structure. Unlike the case between the hole-doped
+system La 2−xSrxCuO4and the electron-doped system Nd 2−xCexCuO4, we have estimated the true
+chemical potential jump between the hole- and electron-dop ed YLBLCO to be ∼0.8 eV, which is
+much smaller than the optical gaps of 1.4-1.7 eV reported for the parent insulating compounds.
+We attribute the reduced jump to the indirect nature of the ch arge-excitation gap as well as to the
+polaronic nature of the doped carriers.
+PACS numbers: 74.72.-h, 71.20.-b, 79.60.-i, 74.25.Jb
+In condensed matter physics, chemical potential is one
+of the most fundamental properties and plays many im-
+portant roles. For an insulator, the chemical potential
+is considered to show a jump between hole doping and
+electron doping. In a conventional intrinsic semiconduc-
+tor (or band insulator), the chemical potential should
+be positioned at the midpoint between the conduction-
+band minimum (CBM) and the valence-band maximum
+(VBM). When a small number of holes (electrons) are
+doped into the intrinsic semiconductor, the chemical po-
+tential should move to the VBM (CBM) or to an im-
+purity level formed in the vicinity of the VBM (CBM).
+This leads to a chemical potential jump of the magni-
+tude nearly equal to the band gap of the undoped semi-
+conductor. In a Mott insulator, however, the situation
+is far from clear. For light hole or electron doping, “in-
+gap states” may be formed within the band gap in a
+nontrivial manner, and the chemical potential may be
+positioned within the gap. If this is the case, the chemi-
+cal potential jump would become smaller than the band
+gap. In high- Tcsuperconductors (HTSCs), this issue
+has been controversial for a long time [1–4]. So far,
+although chemical potential shifts as functions of car-
+rier dopings in HTSCs have been studied in various sys-
+tems [3, 5–10], those studies have been made either on
+the hole-doped side or on the electron-doped side and
+have not been able to unambiguously address the ques-
+tion of the chemical potential jump between the hole-
+doped and electron-doped HTSCs. This is because, un-
+like some conventional semiconductors, one has not been
+able to dope both holes and electrons into the same par-
+ent Mott insulator. Hence, the study of the chemical
+potential jump in HTSCs has been performed in a rather
+indirect way of comparing different materials, namely,
+the hole-doped system La 2−xSrxCuO4(LSCO) and the
+electron-doped system Nd 2−xCexCuO4(NCCO). As a
+result, the chemical potential jump between La 2CuO4
+(LCO) and Nd 2CuO4(NCO) has been found to be negli-giblysmallorassmallas ∼0.15-0.4eVfromthecore-level
+X-ray photoemission spectroscopy (XPS) [7] or valence-
+band photoemission spectroscopy studies [2, 11]. Angle-
+resolved photoemission spectroscopy (ARPES) studies
+showedthatthechemicalpotentialsinLCOandNCOare
+located above the Zhang-Rice singlet (ZRS) band maxi-
+mum by ∼0.4 eV [12, 13] and ∼1.2 eV [14], respectively,
+implying a chemical potential jump in the range of 0.8
+eV. According to optical studies [15, 16], on the other
+hand, the band gap of LCO and NCO are found to be
+as large as ∼2.0 eV and ∼1.5 eV, respectively, meaning
+that the chemical potential jump is much smaller than
+the optical gaps. These results suggest that some elec-
+tronic states may be formed within the optical gap upon
+carrier doping and pin the chemical potential. However,
+because LCO and NCO have different chemical compo-
+sitions and different crystal structures (so-called T- and
+T′-structures,respectively), additionaleffectssuchasdif-
+ferences in Madelung potential may also complicate the
+comparison. Therefore, it is strongly desirable to study
+arealchemical potential jump by investigating HTSCs
+that allow both hole and electron dopings in the same
+parent Mott insulator.
+Recently, both electron and hole dopings have
+become possible in a new high- Tccuprate system
+Y0.38La0.62Ba1.74La0.26Cu3Oy(YLBLCO) [17], where
+someoftheYandBaionsinYBa 2Cu3Oy(YBCO)arere-
+placedbyLaions, maintainingthe samecrystalstructure
+as YBCO. Owing to the large in-plane lattice constant
+compared with that of YBCO, one can dope the sys-
+tem not only with holes but also with electrons through
+varying the oxygen content. In this Letter, we report on
+a core-level XPS study of hole- and electron-doped YL-
+BLCO. The observed chemical potential jump between
+them is found to be ∼0.8 eV, much smaller than the
+reported optical gap of 1.4-1.7 eV of insulating YBCO
+[18, 19]. We shall discuss the origin of the difference be-
+tween the optical gap and the chemical potential jump.2
+High-qualitysinglecrystalsofYLBLCOweregrownby
+the flux method. Details can be found in Ref. [17]. An
+electron-doped sample ( y= 6.22) and hole-doped sam-
+ples (y= 6.70,6.80, and 6.95) were prepared. Their car-
+rier concentrations in the CuO 2plane were 1%, 0.3%,
+3%, and 7%, respectively, according to Hall-effect mea-
+surements. The changes in the carrier concentration are
+much smaller than those expected from the changes in
+yprobably because most of holes are removed from or
+enter the Cu-O chains. The XPS measurements were
+performed using the Mg Kα(hν= 1253.6 eV) line and
+a SCIENTA SES-100 electron-energy analyzer. The to-
+tal energy resolution was ∼0.8 eV. However, owing to
+the highly stable power supply of the analyzer, it was
+possible to determine the binding energy shifts with the
+accuracy of ∼50 meV. Samples were cleaved in situun-
+der an ultrahigh vacuum of 10−10Torr to obtain clean
+surfaces. We measured spectra at different tempera-
+tures and found that the spectra of the y= 6.22 and
+y= 6.70 samples, which had high electrical resistivities,
+were shifted to higher binding energies with decreasing
+temperature. We attributed these shifts to the charging
+effect and, therefore, we used only the spectra at ∼250
+K for these samples. For the y= 6.80 andy= 6.95
+samples, on the other hand, we found no charging effect
+down to ∼80 K, and, therefore, we used spectra taken
+both at∼100 and at ∼250 K, as shown in Fig. 1(a). In
+fact, the observedtemperature-dependent shifts forthese
+samples were in the opposite direction to what would be
+expected for the charging effect and should reflect the in-
+trinsic temperature dependence of the chemical potential
+in the hole-doped compounds [20]. The 4 fcore level of
+gold was used to determine the Fermi level ( EF) position
+before and after each set of measurements.
+Figure 1(b)-(g) shows the core-level XPS spectra of
+the electron-doped y= 6.22 and hole-doped y= 6.70
+YLBLCO samples taken at 250 K. In Fig. 1(g), the
+main peak of the Cu 2 p3/2spectrum (corresponding to
+2p53d10Lfinal states, where Lrepresents a ligand hole)
+is located at −933.5 eV close to those of LCO and NCO
+in the previous studies [21–23]. The main peak width
+for they= 6.22 sample is narrower than that for the
+y= 6.70 sample. Since the main peak of Cu+is known
+tobesharperthanthatofCu2+, thevariationofthe peak
+width is attributed to the replacement of Cu2+by Cu+.
+Also, the intensity of the charge-transfer satellite around
+−943 eV (2 p53d9final state) for y= 6.22 is weaker than
+that for y= 6.70. Since the satellite peak has been ob-
+served in the Cu 2 pXPS spectra of Cu2+compounds but
+not in those of Cu+compounds [24], the suppression also
+suggests the replacement of Cu2+by Cu+. These results
+indicate that electrons are indeed doped into the system.
+The large change in the Cu valence for a small change
+in the carrier concentration of the CuO 2plane indicates
+that the holes doped by oxygen are localized in the Cu-O
+chain. The XPS spectra of the Y 3 d, La 3d, O 1s, Ba 3d,
+and Ba 4 dcore levels of the y= 6.22 sample are shifted
+to higher binding energies than those of the y= 6.70-950 -945 -940 -935 -930Cu 2p53d9Cu 2p Cu 2p53d10L-840 -836 -832La 3d 
+-94-92-90-88-86Ba 4d -160 -156Y 3d      y =
+ 6.22
+ 6.70
+-784 -780 -776Ba 3d -530 -526O 1s 
+Energy relative to EF (eV)Intensity (arb. units)(a)
+(f) (e)(d)(b)
+(c)
+(g)Carrier concentration/Cu (%)Temperature (K)
+0 73AF100250
+SC
+1Electron
+      dopingHole doping
+FIG. 1: (Color online) (a) Schematic phase diagram
+of Y0.38La0.62Ba1.74La0.26Cu3Oy(YLBLCO). Measurements
+were made at filled circles. Shaded regions represent the su-
+perconducting (SC) and antiferromagnetic (AF) phases. (b) -
+(g) XPS spectra of electron-doped y= 6.22 and hole-doped
+y= 6.70 YLBLCO at 250 K for the Y 3 d, La 3d, O 1s, Ba
+3d, Ba 4d, and Cu 2 pcore levels. For the Cu 2 pXPS spectra,
+the peak around −933.5 eV is the main peak due to 2 p53d10L
+final states, where Lrepresents the ligand hole, and the broad
+peak around −943 eV is the charge-transfer satellite due to
+2p53d9final states.
+sample nearly by the same amount. As for the Ba 3 d, Ba
+4d, and Cu 2 pcore levels, their line shapes change with
+y, indicating that it is difficult to estimate the chemical
+potential shift using these core-level spectra. Therefore,
+the Y 3d, La 3d, and O 1 score levels are more suitable
+for estimating the chemical potential shift.
+Figure 2(a) summarizes the energy shift ∆ Eof each
+core level relative to that of the y= 6.70 YLBLCO sam-
+ple. We have estimated the core-level shifts using both
+the peak position and the low binding energy side of the
+peak for each core level to check internal consistency.
+While the Y 3 d, La 3d, and O 1 score levels show similar
+shifts, the Cu 2 p, Ba 3d, and Ba 4 dcore levels show dif-
+ferentshifts, duetothechangesin thespectra-lineshapes
+as shown in Fig. 1. When the band filling is varied, the
+changeinthecore-levelenergyrelativeto EF, ∆E, isgen-3
+-1.0-0.8-0.6-0.4-0.20.00.20.40.6Relative Energy (eV)
+840-4Carrier concentration/Cu (%) Y 3d 
+ La 3d 
+ O 1s 
+ Ba 3d 
+ Ba 4d 
+ Cu 2p T = 250 K
+Electron
+ doping Hole doping1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+-0.2
+-0.4
+-0.6Chemical potential shift /s68/s109 (eV)
+3020100-10-20
+Carrier concentration/Cu (%) YLBLCO (250 K)
+ YLBLCO (100 K)
+ LSCO
+ Bi2212
+ Na-CCOC
+ YBCO
+ NCCO
+Electron dopingHole doping(a) (b)
+FIG. 2: (Color online) (a) Energy shift of each core level
+in YLBLCO relative to y= 6.70 at 250 K as a func-
+tion of carrier concentration. (b) Chemical potential shif ts
+in YLBLCO at 250 and 100 K, compared with those in
+La2−xSrxCuO4(LSCO) [6], Bi 2Sr2Ca1−x(Pr, Eu) xCu2O8+δ
+(Bi2212) [9], Na xCa2−xCuO2Cl2(Na-CCOC) [10], YBCO
+[25], and Nd 2−xCexCuO4(NCCO) [7]. Note that the shift
+for NCCO relative to the hole-doped systems are rather arbi-
+trary [7].
+erally given by ∆ E=−∆µ−K∆Q−∆VM+∆ER[6, 7],
+where ∆ µis the change in the chemical potential, ∆ Q
+is the change in the number of valence electrons on the
+atomand Kisaconstant,with −K∆Qpresentingtheso-
+calledchemical shift, ∆ VMis the changein the Madelung
+potential, and ∆ ERis the change in the extra-atomic re-
+laxation energy. When carriers are doped into the CuO 2
+plane,K∆Qchanges in proportion to the carrierconcen-
+tration. We observed such a behavior only for Cu 2 p3/2
+and could ignore K∆Qfor Y 3d, La 3d, and O 1 score
+levels. ∆ VMis negligibly small as in other transition-
+metal oxides [26] because the core levels of cations (Y 3 d
+and La 3 d) and an anion (O 1 s) move in the same di-
+rection by the same amount. ∆ ERis due to the change
+in the screening of the core hole potential and, therefore,
+should be larger for atoms in the metallic CuO 2plane
+than for those in the insulating block layers. The same
+shifts of the Y 3 d, La 3d, and O 1 score levels suggest
+that the ∆ ERterm is also negligibly small. Indeed, in
+the previous reports on the chemical potential shift in
+filling control oxides [6, 7, 10], the effect of ∆ VMand
+∆ERcould be ignored. Therefore, we consider that the
+core-levelshifts ofY3 d, La3d, andO1 sreflectthe chem-
+ical potential shift ∆ µ, and ∆µfor YLBLCO has been
+estimated from the average of the Y 3 d, La 3d, and O 1 s
+core-level shifts.
+Figure 2(b) shows the chemical potential shift of YL-
+BLCO at 250 K and 100 K thus deduced as well as
+those of La 2−xSrxCuO4(LSCO) [6], Nd 2−xCexCuO4
+(NCCO) [7], Bi 2Sr2Ca1−x(Pr, Eu) xCu2O8+δ(Bi2212)
+[9], Na xCa2−xCuO2Cl2(Na-CCOC) [10], and YBCO
+[25]. On the hole-doped side, the behavior of YLBLCO
+is similar to those of Bi2212, Na-CCOC, and YBCO,
+but is different from that of LSCO. This difference has
+been attributed to the small next-nearest-neighbor hop-  Upper
+Hubbard
+  BandYLBLCO LCO NCO
+IndirectIndirect Direct
+~2.0 eV
+Chemical
+potential
+   jump
+~0.8 eV~0.8 eVElectron doping
+ Hole doping
+~0.4 eV ~0.4 eV~1.2 eVµ µµ µ  Upper
+Hubbard
+  Band  Upper
+Hubbard
+  Band
+  ZRS
+ Band  Direct
+~1.5 eV
+      Direct
+~1.4-1.7 eV  ~(π/2, π/2)
+  ~(π/2, π/2)  ~(π, 0)
+  ZRS
+ Band  ZRS
+ Band
+FIG. 3: (Color online) Schematic band diagram of YLBLCO
+and those of LCO and NCO. Orange (thick gray) curves rep-
+resent the schematic dispersions of energy distribution cu rve
+peaks. The positions of the chemical potential µare indi-
+cated by horizontal dashed lines. The maximum positions of
+the Zhang-Rice singlet (ZRS) band are aligned. For LCO and
+NCO, the energy separation between the ZRS band and the
+chemical potential was obtained from angle-resolved photo e-
+mission studies [12–14]. The magnitude of the direct gap was
+estimated from optical studies [15, 16]. For YLBLCO, the
+maximum position of the ZRS band relative to the chemical
+potential is assumed to be the same as those of lightly-doped
+LSCO and Na-CCOC [12, 13, 27]. The direct gap of YL-
+BLCO is estimated from the optical studies of YBCO [18]
+and PrBa 2Cu3Oy[19].
+pingt′or stripe effects in LSCO as discussed previously
+[6,28,29], indicatingthat YLBLCOhasnoeffect ofsmall
+t′or stripes. Between the hole-doped and electron-doped
+sides of YLBLCO, a chemical potential jump of ∼0.6 eV
+is observed at 250 K. However, since a chemical poten-
+tial shift of ∼0.1 eV was observed between 250 K and
+100 K in some hole-doped samples, the chemical poten-
+tial jump would be larger at low temperatures. If the
+temperature dependent shift is symmetric between the
+electron-doped and hole-doped sides, the chemical po-
+tential jump in YLBLCO at 100 K would be estimated
+to be larger than that at 250 K by ∼0.2 eV, that is, ∼0.8
+eV. This value is small compared with the gap ( ∼1.4-1.7
+eV) estimated from the optical studies [18, 19], but is
+comparable to the activation gap ( ∼0.9 eV) determined
+from high-temperature transport [30]. The chemical po-
+tential jump of YLBLCO is somewhat larger than that
+estimated for LCO and NCO from the core-level XPS
+study [7] and from the energy difference of the band
+structure ( ∼0.3 eV) [31]. On the other hand, the en-
+ergy difference between the ZRS band maximum and the4
+chemical potential for LCO and NCO was found to be
+∼0.4 eV [12, 13] and ∼1.2 eV [14], respectively. There-
+fore, if the energy positions of the ZRS band are taken as
+the energy reference, the chemical potential jump would
+be∼0.8 eV. Generally, the magnitude of the optical gap
+in the charge-transfer-type compounds is influenced by
+the Madelung potential. The fact that the optical gap in
+LCO is larger than that in NCO by ∼0.5 eV [15, 16] is
+caused by the Madelung-potential difference due to the
+different crystal structures, meaning that ∆ VMis not
+negligible in the estimation of ∆ µfrom ∆E. Thus, we
+conclude that the intrinsic chemical potential jump be-
+tween hole- and electron-doped HTSCs is ∼0.8 eV, and
+the different Madelung potential between LCO and NCO
+makes it difficult to estimate the chemical potential jump
+between them.
+Finally, we summarize the energy diagram of YL-
+BLCO, LCO, and NCO in Fig. 3. The energy difference
+of∼0.4 eV between the VBM and the chemical potential
+forLCO hasbeen attributed to apolaroniceffect [32, 33],
+which shifts the (multi-phonon) peak in energy distribu-
+tion curves toward higher binding energies through the
+Frank-Condon principle. We should also note that the
+minimum charge-transfer gap is not a direct gap whichwas detected in the optical studies but an indirect one as
+demonstrated by resonant inelastic X-ray scattering ex-
+periments [34] combined with extended Hubbard-model
+calculation [35], according to which the indirect gap from
+/vectork∼(π/2,π/2) to∼(π,0) is found to be smaller than the
+direct one ∼(π/2,π/2).
+In conclusion, we have performed core-level XPS mea-
+surementsofYLBLCOin ordertostudy the chemicalpo-
+tential jump between the hole-doped and electron-doped
+sides. UnlikethecaseofLCOandNCO,theYLBLCOre-
+sults yield the true chemical potential jump between the
+hole- and electron-doped HTSCs. The deduced chemi-
+cal potential jump ( ∼0.8 eV) is much smaller than that
+of the optical gap ( ∼1.4-1.7 eV), but is comparable to
+the activation gap ( ∼0.9 eV) determined from the high-
+temperaturetransport. We attribute the smallnessofthe
+jump to the indirect nature of the charge-excitation gap
+as well as to a polaronic effect on the doped carriers.
+We are grateful to H. Yagi for discussion. This work
+was supported by KAKENHI 19204037. M. I. was sup-
+ported by the Global COE Program, MEXT, Japan, Y.
+A. by KAKENHI 19674002 and 20030004, and K. S. by
+KAKENHI 20740196.
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\ No newline at end of file
diff --git a/1001.0103.txt b/1001.0103.txt
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+arXiv:1001.0103v1  [astro-ph.HE]  31 Dec 20092009 Fermi Symposium, Washington, D.C., Nov. 2-5 1
+The Cosmological Evolution of Blazars and the Extragalacti c Gamma-Ray
+Background in the Fermi Era
+Yoshiyuki Inoue, Tomonori Totani
+Department of Astronomy, Kyoto University, Kitashirakawa , Sakyo-ku, Kyoto 606-8502, Japan
+Susumu Inoue
+Department of Physics, Kyoto University, Kitashirakawa, S akyo-ku, Kyoto 606-8502, Japan
+Masakazu A. R. Kobayashi
+Optical and Infrared Astronomy Division, National Astrono mical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
+Jun Kataoka
+Waseda University, 1-104 Totsukamachi, Shinjuku-ku, Toky o 169-8050, Japan
+Rie Sato
+Department of High Energy Astrophysics, Institute of Space and Astronautical Science (ISAS),
+Japan Aerospace Exploration Agency (JAXA),
+3-1-1 Yoshinodai, Sagamihara, 229-8510, Japan
+The latest determination of the extragalactic gamma-ray ba ckground (EGRB) radiation by Fermi
+is compared with the theoretical prediction of the blazar co mponent by Inoue & Totani (2009; here-
+after IT09). The FermiEGRB spectrum is in excellent agreement with IT09, indicati ng that blazars
+are the dominant component of the EGRB, and contributions fr om any other sources (e.g., dark
+matter annihilations) are minor. It also indicates that the blazar SED (spectral energy distribu-
+tion) sequence taken into account in IT09 is a valid descript ion of mean blazar SEDs. The possible
+contribution of MeV blazars to the EGRB in the MeV band is also discussed. In five total years
+of observations, we predict that Fermiwill detect ∼1200 blazars all sky down to the correspond-
+ing sensitivity limit. We also address the detectability of the highest-redshift blazars. Updating
+our model with regard to high-redshift evolution based on SD SS quasar data, we show that Fermi
+may find some blazars up to z∼6 during the five-year survey. Such blazars could provide a ne w
+probe of early star and galaxy formation through GeV spectra l attenuation signatures induced by
+high-redshift UV background radiation.
+I. INTRODUCTION
+The origin of the extragalactic diffuse gamma-ray
+background (EGRB) is a long-standing puzzle. First
+discovered by the SAS 2satellite [1, 2], its exis-
+tence was subsequently confirmed up to ∼50 GeV
+by EGRET (Energetic Gamma-Ray Experiment Tele-
+scope) on board the Compton Gamma Ray Obser-
+vatory. EGRET measured a flux of about 1 ×10−5
+photons cm−2s−1sr−1above 100 MeV and an ap-
+proximatelypower-lawspectrum with photon indexof
+∼ −2 over the wide range of 30 MeV – 50 GeV [3, 4].
+Very recently, Ref. [5] reported Fermiobservations of
+the EGRB spectrum up to 100 GeV, which connects
+smoothly to the EGRET results below ∼200 MeV
+[4]. However, above this energy, the spectrum is dis-
+crepant from EGRET, being a single power-law with
+index∼ −2.4, and showing no signs of a GeV excess.
+Although different types of gamma-ray sources
+(e.g., clusters of galaxies or dark matter annihila-
+tion) have been proposed to be significant contrib-
+utors to the EGRB [see, e.g., Ref. [6] and refer-
+ences therein], active galactic nuclei (AGNs) of the
+blazar class are considered the primary candidates,since almost all of the extragalactic sources detected
+by EGRET were blazars. The blazar contribution to
+the EGRB has been estimated by numerous authors,
+who have reached different conclusions using different
+approaches, ranging from 20% to 100 % of the ob-
+served flux [6–18].
+In most past studies, the spectral energy distri-
+butions (SEDs) of the blazars were assumed to be
+power laws for all objects. In such models, it is ob-
+vious that the predicted EGRB spectrum is mainly
+determined by the assumed power-law indices. How-
+ever, multi-wavelength observational studies indicate
+the existence of a non-trivial trend among blazar
+SEDs: the energies of the two characteristic spectral
+peaks in blazar SEDs (each likely reflecting the syn-
+chrotron and inverse Compton emission) systemati-
+cally decrease as the bolometric luminosity increases
+[19–25]. This is often referred to as the blazar SED
+sequence. Althoughitsvalidityiscurrentlystillamat-
+ter of debate (e.g., Ref. [24, 25, 27]), one can make
+a non-trivial prediction of the EGRB spectrum if this
+blazar sequence is assumed.
+Here we compare the latest FermiEGRB results
+with our model predictions, previously published in
+eConf C0911222 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
+Ref. [26] (IT09) before the former was announced.
+IT09 calculated the EGRB from blazars by construct-
+ing a blazar gamma-ray luminosity function (GLF)
+model that is consistent with the flux and redshift dis-
+tributions of the EGRET blazars, accounting for the
+blazar sequence as well as the luminosity-dependent
+density evolution (LDDE) scheme that describes well
+the evolution of the X-ray luminosity function (XLF)
+of AGNs. By introducing the blazar SED sequence,
+wewereableto makeareasonableandnon-trivialpre-
+diction of the EGRB spectrum that can be compared
+with observations including the new Fermidata.
+Recently, Ref. [28] has shown that the EGRB in
+the MeV band can be naturally explained by normal
+(i.e., non-blazar) AGNs that compose the cosmic X-
+ray background. If so, they should also contribute to
+the EGRB at <∼1 GeV. We will investigate what frac-
+tion of the observed EGRB can be explained by the
+sum of the blazar and non-blazar AGN components.
+The contribution of blazars to the EGRB in the MeV
+band is also discussed and compared with a recent
+study on the possible relevance of the so-called “MeV
+blazars” [29].
+We will then make quantitative predictions for
+theFermiGamma-ray Space Telescope [30] for its
+five-year survey. The first Fermicatalog for bright
+gamma-raysources including AGNs has recently been
+released [31, 32]. The number of currently detected
+blazars is about 100, larger than that of EGRET by
+a modest factor of about 2. We limit the discussion
+of the blazar data to only the EGRET sample in this
+work,to beconsideredasatheoreticalpredictionfrom
+the pre-Fermiera. In the future, a much larger num-
+ber of blazars should be detected with Fermi, which
+can be compared further with our predictions.
+We also discuss the highest-redshift blazars de-
+tectable by Fermi, by updating the IT09 GLF model
+with respect to high-redshift evolution. We briefly
+mention the possibility of utilizing such blazars to
+obtain information on the high-redshift extragalactic
+background light and early galaxy formation through
+spectral absorption features.
+Throughout this paper, we adopt the standard cos-
+mological parameters of ( h,ΩM,ΩΛ)=(0.7,0.3,0.7).
+II. THE BLAZAR SED SEQUENCE
+Ref. [19, 20, 23] constructed an empirical blazar
+SED model to describe the SED sequence, based on
+fittings to observed SEDs from radio to γ-ray bands.
+These models are comprised of the two components
+(synchrotron and IC), and each of the two is de-
+scribed by a linear curve at low photon energies and
+a parabolic curve at high energies.
+We construct our own SED sequence model mainly
+basedon the SEDmodel [23], becausethere is amath-
+ematical discontinuity in the original model of Ref. 41 42 43 44 45 46 47 48 49
+-5  0  5  10 8  13  18  23Log10 νLν[erg s-1]
+Log10 Eγ [eV]Log10 ν [Hz]
+FIG. 1: The blazar SED sequence. The data points
+are the average SED of the blazars studied by Ref.
+[20, 23]. The solid curves are the empirical SED se-
+quence models constructed and used in this paper. The
+model curves corresponds to the bolometric luminosities
+of log10(P/erg s−1) = 49.50, 48.64, 47.67, 46.37, and 45.99
+(from top to bottom).
+[23]. Our own SED sequence formula is described in
+the Appendix of Ref. [26] in detail. In Fig.1, we show
+this empirical blazar SED sequence model in compar-
+ison with the observed SED data [20, 23].
+III. THE MODEL OF GAMMA-RAY
+LUMINOSITY FUNCTION OF BLAZARS
+The cosmological evolution of AGN XLF has been
+investigated intensively [33–36]. These studies re-
+vealed that AGN XLF is well described by the LDDE
+model, in which peak redshift of density evolution
+increases with AGN luminosity. Here we construct
+blazar GLF models based on the two XLFs derived
+by Ref. [33] (hereafter U03; in hard X-ray band) and
+Ref. [34] (hereafter H05; in soft X-ray band). The use
+of LDDE in blazar GLF has been supported from the
+EGRET blazar data [6].
+We simply assume that the bolometric luminosity
+of radiation from jet, P, is proportional to disk X-ray
+luminosity, LX. We relate these two by the parameter
+q, asP= 10qLX. Here, we define the disk luminosity
+LXto be that in the rest-frame 2–10 and 0.5–2 keV
+bands for the U03 XLF and the H05 XLF , respec-
+tively. Thus, the luminosity at rest 100 MeV, Lγ, and
+LXhave been related through P.
+The blazar GLF ργis then obtained from the AGN
+XLF,ρX, asργ(Lγ,z) =κdLX
+dLγρX(LX,z), where ργ
+andρXare the comoving number densities per unit
+gamma-ray and X-ray luminosity, respectively. The
+parameter κis a normalization factor, representing
+the fraction of AGNs observed as blazars. See the §3
+in Ref. [26] for a detail.
+We use the maximum likelihood method to search
+for the best-fit model parameters of the blazar GLF
+eConf C0911222009 Fermi Symposium, Washington, D.C., Nov. 2-5 3
+to the distributions of the observed quantities of the
+EGRET blazars (gamma-ray flux and redshift). See
+Ref. [26] for a detail. We take qand the faint-end
+slope index of XLF γ1in addition to qas free param-
+eters . These are hereafter called as U03( q), H05(q),
+U03(q,γ1) and H05( q,γ1) fits, respectively. Figure
+2 shows the distributions of redshift and gamma-ray
+luminosity predicted by the best-fit models, in com-
+parison with the EGRET data.
+We performed the Kolmogorov-Smirnov (KS) test
+to see the goodness of fits for the best-fit results of
+each of the four fits, and the chance probabilities of
+getting the observed KS deviation. Since the best KS
+test value is obtained for the U03( q,γ1) GLF model,
+we use this GLF model as the baseline below.
+IV. THE EGRB SPECTRUM
+We calculate the EGRB spectrum by integrating
+our blazar SED sequence model using the blazar GLF
+derived in §III. Since EGRET has already resolved
+bright gamma-ray sources, we should not include
+those sources in the EGRB calculation. Therefore,
+the maximum gamma-ray luminosity in the integra-
+tion should be Lγ(FEGRET,z), which is the luminosity
+of the blazar having a flux Fγ=FEGRETat a given
+z, whereFEGRET= 7×10−8photons cm−2s−1is the
+EGRET sensitivity above 100 MeV.
+High energy photons ( >∼20 GeV) from high redshift
+are absorbed by the interaction with the cosmic in-
+frared background (CIB) radiation [37–40]. We adopt
+the model of Ref. [38] for the CIB optical depth. We
+alsotakeinto accountthe cascadingemission to calcu-
+late EGRB spectrum, considering only the first gener-
+ation of created pairs [17]. We found that the amount
+of energy flux absorbed and reprocessed in intergalac-
+tic medium is only a small fraction of the total EGRB
+energy flux , and hence the model dependence of CIB
+or the treatment of the cascading component does not
+have serious effects on our conclusions in this work
+[26].
+In addition to blazars, we take into account non-
+blazar AGNs as the source of EGRB. Ref. [28] (here-
+after ITU08) has shown that non-blazar AGNs are
+a promising source of EGRB at ∼1–10 MeV. In this
+scenario, a nonthermal power-law component extends
+from hard X-ray to ∼10 MeV band in AGN spec-
+tra, because of nonthermal electrons that is assumed
+to exist in hot coronae around AGN accretion disks.
+The existence of such nonthermal electrons is theo-
+retically reasonable, because magnetic reconnection is
+the promising candidate for the heating source of the
+hot coronae [46] where hard X-ray photons are emit-
+ted. Magnetic reconnections should produce nonther-
+mal particles, as is well known in e.g., solar flares or
+the earth magnetosphere. Although several sources
+have been proposed as the origin of the MeV back-ground, this model givesthe most natural explanation
+for the observed MeV background spectrum that is a
+simple power-law smoothly connected to CXB.
+Theoretically, there is no particular reason to ex-
+pect a cut-off of the nonthermal emission around 10
+MeV, and it is well possible that this emission ex-
+tends beyond 10 MeV with the same power index.
+Then, this component could make some contribution
+to GeV EGRB. The EGRB spectra from the non-
+blazar AGNs are calculated based on the model of
+ITU08. The model parameters of ITU08 have been
+determined to explain the EGRB in the MeV band.
+There is some discrepancy between the MeV EGRB
+data of SMM [43] and COMPTEL [44], and the rea-
+son for this is not clear. Here we use two model pa-
+rameter sets of (Γ ,γtr) = (3.5,4.4) and (3 .8,4.4) in
+ITU08, where Γ is the power-law index of nonthermal
+electron energy distribution and γtris the transition
+electron Lorentz factor above which the nonthermal
+component becomes dominant. These two parameter
+sets are chosen so that they fit to the COMPTEL and
+SMM data, respectively.
+A. On the Origin of the EGRB: Comparison
+with the FermiEGRB Spectrum
+Fig. 3 compares our model predictions with the ob-
+served EGRB data, including the very recent Fermi
+results [5] [54]. Since the flux thresholds of EGRET
+andFermiare different, we must be careful when
+directly comparing with the Fermidata. However,
+when the sensitivity threshold of Fermiis set to be
+the same as that of EGRET, the EGRB flux changes
+by only∼10 % [47]. Therefore, our predicted EGRB
+spectrum can be considered to be in excellent agree-
+ment with the Fermidata. This agreement strongly
+suggests that blazars are the principal sources of the
+EGRB above 1 GeV, and the contribution from other
+sources like dark matter annihilation is minor. This
+also implies that use of the blazar sequence is a valid
+assumption.
+The EGRB prediction using the ITU08 non-blazar
+background model with Γ = 3 .5 also agrees nicely
+with the observed data from X-rays to 100 GeV. In
+this case, the predicted EGRB flux above 100 MeV
+can account for 80% of the observed flux, consider-
+ably higher than in previous studies [6, 12, 13]. It
+should be noted that the contribution to the EGRB
+fluxabove100MeVfromblazarsaloneis ∼45%, while
+the remaining ∼35 % comes from non-blazar AGNs.
+Based on these results and considerations, we con-
+clude that most, and probably all, of the EGRB flux
+can be explained by blazars together with non-blazar
+AGNs, whose luminosity functions are consistent with
+the EGRET blazar data and X-ray AGN surveys.
+eConf C0911224 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
+ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
+ 0.01  0.1  1dN/d(log10 z)
+Redshift zU03(q)
+U03(q,γ1)
+H05(q)
+H05(q,γ1)
+EGRET blazars
+ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
+ 43 44 45 46 47 48 49 50dN/d(log10 Lγ)
+log10 (Lγ [erg s-1])U03(q)
+U03(q,γ1)
+H05(q)
+H05(q,γ1)
+EGRET blazars
+FIG. 2: Redshift and gamma-ray luminosity ( νLνat 100 MeV) distributions of EGRET blazars. The histogram is the
+binned EGRET data, with one sigma Poisson errors. The four mo del curves are the best-fits for the GLF models of
+U03(q), H05(q), U(q,γ1), and H05( q,γ1), as indicated in the figure. Bottom panels: the same as the to p panels, but for
+cumulative distributions.
+10-410-310-210-1
+10-210-1100101102103104105106E2 dN/dE  [MeV2 cm-2 s-1 MeV-1 sr-1]
+Photon Energy [MeV]Blazar: U03 (q, γ1)
+Non-blazar ( Γ=3.5): ITU08
+Non-blazar ( Γ=3.8): ITU08
+Blazar + Non-blazar ( Γ=3.5)
+Blazar + Non-blazar ( Γ=3.8)
+EGRET (S98)
+EGRET (S04a)
+EGRET (S98+S08)
+Fermi
+FIG. 3: The EGRB spectrum from non-blazar AGNs and
+blazars fortheGLFmodelofU03( q,γ1). Themodelcurves
+for the blazar component (absorbed+cascade), non-blazar
+AGN component, and the sum of the two are shown. Note
+that two models are plotted for the non-blazar component
+with different values of Γ. The observed data of HEAO-
+1 [42],Swift/BAT [41], SMM [43], COMPTEL [44] and
+EGRET [3, 4] are shown. We also plot an alternative ver-
+sion of the EGRET data denoted as “S98+S08” which is
+the original EGRET results [3] modified with the correc-
+tion factors of Ref. [45]. The FermiEGRB data is also
+shown [5].
+B. Non-blazar AGNs and Blazars
+As can be seen in Fig. 3, the spectrum of the cos-
+mic MeV background connects smoothly with that of
+the cosmic X-ray background (CXB), while the spec-
+tral index of the EGRB changes abruptly around ∼10
+MeV. This indicates that the sources of the EGRB
+below 10 MeV is the same population as that of the
+CXB (i.e. non-blazar AGNs), and the sources of the
+EGRB above 10 MeV is a different population (i.e.
+blazars).Recently, based on observations of Swift/BAT-
+selected blazars at 15–55 keV, Ref. [29] suggested
+that MeV blazars contribute significantly to the MeV
+background. However, this conclusion is based on
+an extrapolation of blazar SEDs from 55 keV to the
+MeV band, and some fine tuning may be required in
+the MeV blazar SED to reproduce the EGRB spec-
+trum that connects smoothly with the CXB. More-
+over, the spectral break in the EGRB around 10 MeV
+implies that such MeV blazars are a distinct popula-
+tion from those contributing to the EGRB above 10
+MeV. Therefore, it seems more natural that the dom-
+inant contribution to the MeV background is coming
+from non-blazar AGNs as proposed by [28]. Further
+observationalstudies aredesiredtorevealthe true ori-
+gin of the MeV background.
+V. PREDICTIONS FOR THE FERMI MISSION
+A. Expected Number of Blazars and Non-blazar
+AGNs
+The left panel of Figure 4 shows the cumulative dis-
+tribution of >100 MeV photon flux of blazars, non-
+blazar AGNs, and the total of the two. The GLF
+predicts that about 750 and 1200 blazars should be
+detected by Fermiwhere we assumed the Fermisensi-
+tivitytobe Flim= 3×10−9and1×10−9photonscm−2
+s−1above 100 MeV. These sensitivities correspond to
+∼1 and∼5 years survey sensitivities.
+The model of Ref. [11], and the PLE and LDDE
+models [6] predicted ∼10000, 5000, and 3000 blazars
+for the 1-year sensitivity limit, respectively. It is re-
+markable that the GLF models in this work predict
+significantly smaller numbers of blazars than most of
+previous studies.
+About 4–50 non-blazar AGNs would be detected
+eConf C0911222009 Fermi Symposium, Washington, D.C., Nov. 2-5 5
+10-1100101102103104105106107108109
+10-1510-1410-1310-1210-1110-1010-910-810-710-610-5N (>Flux) [(4 π sr)-1]
+Flux (>100MeV) [photons cm-2 s-1]EGRETFermi
+Blazar: U03(q, γ1)
+Non-blazar ( Γ=3.5)
+Non-blazar ( Γ=3.8)
+Blazar + Non-blazar ( Γ=3.5)
+Blazar + Non-blazar ( Γ=3.8)
+10-1110-1010-910-810-710-610-5
+10-1510-1410-1310-1210-1110-1010-910-810-710-6Fγ dN/d(log10 Fγ) [photons cm-2 s-1 sr-1]
+Flux (>100MeV) [photons cm-2 s-1]EGRETFermi
+Blazar: U03 (q, γ1)
+Non-blazar ( Γ=3.5)
+Non-blazar ( Γ=3.8)
+Blazar + Non-blazar ( Γ=3.5)
+Blazar + Non-blazar ( Γ=3.8)
+FIG. 4: Left: the cumulative flux distributions of blazars, non-blazar A GNs, and the total of the two. The detection
+limits of EGRET and Fermiare also shown. Right: the same as the left panel, but showing differential flux dist ribution
+multiplied by flux Fγ, to show the contribution to EGRB per logarithmic flux interv al. The two models of non-blazar
+AGNs with different values of Γ are shown. The total of blazar a nd non-blazar counts is also shown.
+byFermiwith 5-year survey, by their soft nonthermal
+emission from nonthermal electrons in coronae of ac-
+cretion disks, giving an interesting test for our model.
+We estimated the >100 MeV gamma-ray flux from
+theobservedhardX-rayfluxofNGC4151(the bright-
+est Seyfert galaxy in all sky [35]) using the ITU08
+model [26]. Then, NGC 4151 is marginally detectable
+when Γ = 3 .8, and easily detectable with Γ = 3 .5 with
+one year survey sensitivity.
+The right panel of Figure 4 shows the differential
+flux distribution ofgamma-rayblazarsand non-blazar
+AGNs multiplied by flux, showing the contribution to
+the EGRB per unit logarithmic flux interval. EGRB
+from blazars should practically be resolved into dis-
+crete sources. We find that the fraction of EGRB
+flux that should be resolved is 98% against the to-
+tal blazar EGRB flux in 5-year survey. On the other
+hand, EGRBfromnon-blazarAGNsishardlyresolved
+into discrete sources by Fermi. The predicted fraction
+ofnon-blazarEGRBfluxresolvedby Fermiis∼0.01%
+with 5-yearsurvey, respectively, against the total non-
+blazar EGRB flux. These results are in sharp contrast
+to those for blazars, although the contributions to the
+total EGRB by the two populations are comparable
+at around 100 MeV.
+The above results mean that a considerable part
+of EGRB at around 100 MeV will remain unresolved
+even with the Fermisensitivity, because there is a
+considerable contribution from non-blazar AGNs to
+EGRB at ∼100 MeV. However, the contribution from
+non-blazars should rapidly decrease with increasing
+photon energy, and almost all of the total EGRB flux
+at>∼1 GeV should be resolved into discrete blazars
+byFermi, if there is no significant source contribut-
+ing to EGRB other than blazars. It should be noted
+that these predictions can be tested by Fermirela-
+tively easily, without follow-up or cross check at other
+wavebands, once measurements of source counts and10-1100101102103
+ 0  2  4  6  8 10  12N(>z)
+Redshift zDashed : without SDSS constraint
+Solid : with SDSS constraint3×10-9 ph/cm2/s
+1×10-9 ph/cm2/s
+FIG. 5: Expected cumulative redshift distribution of
+blazars detectable by Fermiabove 100 MeV. Red and
+blue curves correspond to the flux sensitivity limits of Flim
+(>100MeV) = 3 ×10−9and 1×10−9in units of photons
+cm−2s−1. Solid and dashed curves correspond to whether
+or not SDSS quasar constraints are taken into account.
+EGRB flux have been done by the Fermidata. There-
+fore this gives a simple and clear test for our theoret-
+ical model [26].
+B. High Redshift FermiBlazars
+SinceFermiis the most sensitive GeV gamma-ray
+observatory to date, it has the potential to discover
+many new high-redshift blazars. GeV photons from
+high-redshift sources are expected to suffer intergalac-
+tic absorption due to e±pair production interactions
+withthecosmicUVbackgroundradiation. Theresult-
+ing features in the spectra of such blazars could there-
+forebeakeyprobeofthepoorly-understoodUVback-
+groundradiation and provide valuable insight into the
+cosmic dark ages [50–52].
+eConf C0911226 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
+The high-redshift evolution of the GLF is uncer-
+tain, as there are no appropriate samples of gamma-
+ray blazars and X-ray AGNs above z= 3. Thus we
+constructed new GLFs by combining the U03 XLF
+with evolutionary constraints from the Sloan Digital
+Sky Survey (SDSS) quasar luminosity function data
+[53]. The SDSS quasar luminosity function is well
+constrained up to z∼5.
+Fig. 5 shows the expected cumulative redshift
+distribution of blazars that are detectable by Fermi
+above 100 MeV. Two sets of curves are plotted corre-
+sponding to the one- and five-year survey sensitivity.
+Solid and dashed curves refer to the cases of whether
+or not the SDSS quasar constraint is taken into ac-
+count. Note that intergalactic absorption effects are
+not included here, as they are not expected to be im-
+portant below 1 GeV [50–52]. The results imply that
+Fermimay detect some blazarsup to z∼6 during the
+five-year survey. Measurements of the spectra of such
+blazars above 1 GeV through subsequent, deep obser-
+vations should provide interesting constraints on the
+high-z UV background. More details will be presented
+in a forthcoming publication.
+VI. CONCLUSIONS
+We have compared our prediction of the EGRB
+spectrum [26] with the recent Fermidata, and made
+quantitative predictions for the next four years of the
+Fermimission. In Ref. [26], we constructed a model
+of the blazar gamma-ray luminosity function (GLF),
+taking into account the blazar SED sequence and the
+LDDE luminosity function inferred from X-ray obser-
+vations of AGNs. The GLF model parameters are
+constrained by fitting to the observed flux and red-
+shift distribution of the EGRET blazars. With this
+model, we can predict the EGRB spectrum in a non-
+trivial way, rather than assuming simple power-law
+spectra.
+The contribution from non-blazar AGNs to the
+EGRB is also considered, using the nonthermal coro-
+nal electron model [28]. This model gives a natu-
+ral explanation for the observed cosmic MeV back-
+ground, and we examined whether the X-ray to GeV
+gamma-ray background radiation can be accounted
+for by the two population model including blazarsand
+non-blazar AGNs.
+Our predicted blazar EGRB spectrum matched
+very well with the latest FermiEGRB data. This
+indicates that blazars are the dominant population of
+the EGRB and that other components such as dark
+matterannihilationdonotcontributetoEGRBsignif-
+icantly. This also indicates that the use of blazar SED
+sequence wasvalid. The predicted spectrum including
+blazars and non-blazar AGNs is also in good agree-
+ment with the observed data from X-ray to 100 GeV.
+Therefore, blazarsand non-blazarAGNs are likely theprimary sources of the cosmic X-ray and gamma-ray
+background radiation. Our model predicts that it is
+possible to account for 80 % of the observed EGRB
+flux above 100 MeV with the sum of the blazar and
+non-blazar AGN components, where blazars alone ac-
+count for about 45 %. The two components are com-
+parable at ∼100 MeV, with blazars and non-blazar
+AGNs dominating at higher and lower energies, re-
+spectively.
+From the viewpoint of the EGRB spectrum from
+MeV to GeV, we argue that the dominant component
+contributing to the MeV background is non-blazar
+AGNs that are also responsible also for the CXB,
+rather than MeV blazars as proposed by Ref. [29].
+We predicted the flux distribution of blazars by our
+model, and we found that 750 and 1200 blazars in
+all sky should be detected by Fermi, assuming a sen-
+sitivity limit of Flim= 3×10−9and 1×10−9pho-
+tons cm−2s−1above 100 MeV corresponding to one
+and five year survey sensitivity. This number is sig-
+nificantly lower than the predictions by many of the
+previous studies. About 4–50 of non-blazar AGNs are
+expected to be detected by Fermiwith 5-year sur-
+vey.Fermishould resolvealmost all of the EGRB flux
+from blazars into discrete blazars, with percentages of
+>∼98% with 5-year survey. On the other hand, less
+than 0.1% of the EGRB flux from non-blazar AGNs
+can be resolved into discrete sources. Therefore, we
+have a clear prediction: Fermiwill resolve almost all
+of the EGRB flux into discrete sources at photon en-
+ergies>∼1 GeV where blazars are dominant, while a
+significant fraction of the EGRB flux will remain un-
+resolved in the low energy band of <∼100 MeV where
+non-blazarAGNshaveasignificantcontribution. This
+prediction can easily be tested, only with the source
+counts and the EGRB estimates by Fermidata.
+We also made predictions for future detections of
+high-redshift blazars. Since the GLF is uncertain at
+z≥3 due to lack of direct observational data, we con-
+strained the redshift evolution of our original AGN
+XLF using SDSS quasar data. With this assumption,
+we find that Fermimay find some blazars up to z∼6
+during the five year survey. Such high-redshift GeV
+blazars will offer a new probe of the formation histo-
+ries of early stars and galaxies via GeV spectral at-
+tenuation signatures caused by the high-redshift UV
+background radiation.
+Acknowledgments
+This work was supported by the Grant-in-Aid for
+the Global COE Program ”The Next Generation of
+Physics, Spun from Universality and Emergence” and
+Scientific Research (19047003, 19047004, 19740099)
+from the Ministry of Education, Culture, Sports, Sci-
+ence and Technology (MEXT) of Japan. YI acknowl-
+eConf C0911222009 Fermi Symposium, Washington, D.C., Nov. 2-5 7
+edgessupportbytheResearchFellowshipoftheJapan
+Society for the Promotion of Science (JSPS).
+The authors wish to thank JACoW for their guid-
+ance in preparing this template. Work supported byDepartment ofEnergycontractDE-AC03-76SF00515.
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+arXiv:1001.0104v3  [hep-ph]  5 Feb 2010Possible indication to the QCD evolution of double parton di stributions?
+A.M. Snigirev
+M.V. Lomonosov Moscow State University, D.V. Skobeltsyn In stitute of Nuclear Physics, 119991, Moscow, Russia
+(Dated: October 25, 2018)
+For the first time the process-independent parameter of doub le parton scattering, σexp
+eff, has been
+measured newly in the D0 experiment at the three different res olution scales. If we interpret the
+measurement as a decrease of the effective cross section with a growth of the resolution scale it can
+indicate the QCD evolution of double parton distributions.
+PACS numbers: 12.38.-t
+I. INTRODUCTION
+The presence of multiple parton interactions in high
+energy hadron collisions has been convincingly demon-
+strated by the AFS [1], UA2 [2] and CDF [3] Collab-
+orations using events with the four-jet final state and
+later by the CDF Collaboration [4] using events with the
+γ+3 jets final state. Recently the D0 Collaboration has
+measured the process-independent parameter of double
+parton scattering, σexp
+eff, using the γ+ 3 jets events at
+the three different resolution (energy) scales [5] provid-
+ing new and complementary information on the proton
+structure. The possibility of observing two separate hard
+collisions has been proposed since long [6], and from that
+has also developed in a number of works [7, 8, 10–13]. A
+brief review of the current situation and some progress
+in the modeling with account of correlated flavor, color,
+longitudinal and transverse momentum distributions can
+be found in Ref. [10].
+Multiple interactions require an ansatz for the struc-
+ture of the interacting hadrons, i.e. correlations between
+the constituent partons. As a simple ansatz, usually, the
+two-parton distributions (of the parton momentum frac-
+tion) are supposed to be the product of two single-parton
+distributions times a momentum conserving phase space
+factor. In recent papers [14] it has been shown that this
+hypothesisis in some contradictionwith the leadingloga-
+rithm approximationof perturbative QCD (in the frame-
+workofwhichapartonmodel, asamatter offact, wases-
+tablished inthe quantum fieldtheories[15]). Namely, the
+two-parton distribution functions being the product of
+two singledistributions at some referenceresolutionscale
+become dynamically correlated at any different scale of a
+hard process.
+The main purpose of the present letter is to show that
+these QCD dynamical correlations result effectively in
+a dependence of the experimentally measured effective
+cross section σexp
+effon the resolution scale unlike naive
+accepted expectations.
+The paper is organized as follows. Section II is de-
+voted to what is known from perturbative QCD theory
+on the two-parton distribution functions. The possible
+manifestation of correlations induced by QCD evolution
+is discussed in Sec. III. We summarize and conclude in
+Sec. IV.II. DOUBLE PARTON DISTRIBUTIONS IN
+THE LEADING LOGARITHM
+APPROXIMATION
+In order to introduce the denotations and to be clear
+let us recall that, for instance, the inclusive differential
+cross section for the four-jet production due to the si-
+multaneous interaction of two parton pairs within one p¯p
+collision, σdp, can be given by [7]
+σdp=/summationdisplay
+q/g/integraldisplayσ12σ34
+2σeffDp(x1,x3)D¯p(x2,x4)dx1dx2dx3dx4,
+(1)
+whereσijstands for the two-jet production cross section.
+The dimensional phenomenological parameter σeffin the
+denominator is a factor characterizing a size of the ef-
+fective interaction region of the hadron (the factor 2 is
+introduced due to the identity of the two parton subpro-
+cesses). The relatively small value of ( σeff)CDFmeasured
+by the CDF Collaboration [4] with respect to the naive
+expectation was, in fact, considered [8, 9] as evidence
+of nontrivial correlation effects between partons in the
+transverse space. But, apart from these correlations, the
+longitudinal momentum correlations can also exist and
+they were investigated in Ref. [14]. The factorization
+ansatz is just applied to the two-parton distributions en-
+tering Eq. (1):
+Dp(xi,xj,Q2) =Dp(xi,Q2)Dp(xj,Q2)(1−xi−xj),(2)
+whereDp(xi,Q2) are the single quark/gluon momentum
+distributions at the scale Q2(determined by a hard pro-
+cess).
+However multi-parton distribution functions sat-
+isfy the generalized Gribov-Lipatov-Altarelli-Parisi-
+Dokshitzer (GLAPD) evolution equations derived for the
+first time in Refs [16, 17] as well as single parton dis-
+tributions satisfy more known and cited GLAPD equa-
+tions [15, 18]. Under certain initial conditions these gen-
+eralized equations lead to the solutions, which are iden-
+tical with the jet calculus rules proposed originally for
+multiparton fragmentation functions by Konishi-Ukawa-
+Veneziano [19] and are in some contradiction with the
+factorization hypothesis (2). Here one should note that
+at the parton level this is a strict assertion within the
+leading logarithm approximation.2
+After introducing the natural dimensionless variable
+t=1
+2πbln/bracketleftBigg
+1+g2(µ2)
+4πbln/parenleftBigg
+Q2
+µ2/parenrightBigg/bracketrightBigg
+=1
+2πbln/bracketleftBiggln(Q2
+Λ2
+QCD)
+ln(µ2
+Λ2
+QCD)/bracketrightBigg
+,
+whereb= (33−2nf)/12πin QCD, g(µ2) is the running
+coupling constant at the reference scale µ2,nfis the
+number of active flavors, Λ QCDis the dimensional QCD
+parameter, the GLAPD equations read [15, 18]
+dDj
+i(x,t)
+dt=/summationdisplay
+j′1/integraldisplay
+xdx′
+x′Dj′
+i(x′,t)Pj′→j/parenleftBigg
+x
+x′/parenrightBigg
+.(3)
+Theydescribethescalingviolationofthepartondistribu-
+tionsDj
+i(x,t) insideadressedquarkorgluon( i,j=q/g).
+We will not write the kernels Pexplicitly and will not
+derive the generalized equations for two-parton distribu-
+tionsDj1j2
+i(x1,x2,t), representing the probability that
+in a dressed constituent ione finds two bare partons of
+typesj1andj2with the given longitudinal momentum
+fractions x1andx2(referring to [14–18] for details). We
+note only that their solutions can be represented as the
+convolution of single distributions [16, 17]. This convolu-
+tioncoincideswiththejetcalculusrules[19]asmentioned
+aboveandisthegeneralizationofthe well-knownGribov-
+Lipatov relation installed for single functions [15] (the
+distribution of bare partons inside a dressed constituent
+is identical to the distribution of dressed constituents in
+the fragmentation of a bare parton in the leading loga-
+rithm approximation). The obtained solution shows also
+that the double distribution of partons is correlated in
+the leading logarithm approximation:
+Dj1j2
+i(x1,x2,t)/negationslash=Dj1
+i(x1,t)Dj2
+i(x2,t).(4)
+Of course, it is interesting to find out the phenomeno-
+logical issue of this parton level consideration. This can
+be done within the well-known factorization of soft and
+hard stages (physics of short and long distances). As a
+result, the equations (3) describe the evolution of parton
+distributions in a hadron ( h) witht(Q2), if one replaces
+the index iby index honly. However, the initial condi-
+tions for new equations at t= 0 (Q2=µ2) are unknown
+a priori and must be introduced phenomenologically or
+mustbeextractedfromexperimentsorsomemodelsdeal-
+ing with physics of long distances [at the parton level:
+Dj
+i(x,t= 0) = δijδ(x−1);Dj1j2
+i(x1,x2,t= 0) = 0].
+Nevertheless the solution of the generalized GLAPD evo-
+lution equations with a given initial condition may be
+written as before via the convolution of single distribu-
+tions [14, 17]. This result shows that if the two-parton
+distributions are factorized at some scale µ2, then the
+evolution violates this factorization inevitably at any
+different scale ( Q2/negationslash=µ2), apart from the violation due
+to the kinematic correlations induced by the momentum
+conservation, which is the analogue of the momentum
+conserving phase space factor in Eq. (2).For a practical employment it is interesting to know
+the degree of this violation. Partialy this problem was
+investigated theoretically in Refs. [17, 20] and for the
+two-particle correlations of fragmentation functions in
+Ref. [21]. That technique is based on the Mellin trans-
+formation of distribution functions and the asymptotic
+behavior can be estimated. Namely, with the growth of
+t(Q2) the correlation term becomes dominant for finite
+x1andx2[20] and thus the two-partondistribution func-
+tions“forget”the initialconditionsunknown a prioriand
+the perturbatively calculated correlations appear.
+The asymptotic prediction “teaches” us a tendency
+only and tells nothing about the values of x1,x2,t(Q2)
+beginning from which the correlations are significant.
+Naturally numerical estimations can give an answer to
+this specific question using the CTEQ fit [22] for single
+distributions as an input. The nonperturbative initial
+conditions Dj
+h(x,0) are specified in a parametrized form
+at a fixed low-energy scale Q0=µ= 1.3 GeV. The par-
+ticular function forms and the value of Q0are not cru-
+cial for the CTEQ global analysis at a flexible enough
+parametrization. The results of numerical calculations
+were obtained in Ref. [14] for the ratio:
+R(x,t) =Dgg
+p(QCD,corr.)(x1,x2,t)
+Dg
+p(x1,t)Dg
+p(x2,t)(1−x1−x2)2/vextendsingle/vextendsingle/vextendsingle
+x1=x2=x(5)
+At the hard process scale of the CDF measurement [4]
+(Q∼6 GeV) the ratio (5) is nearly 10% and increases
+right up to 30% at much higher scale ( Q∼100 GeV)
+for the longitudinal momentum fractions x≤0.1 acces-
+sible to these measurements. For the finite longitudinal
+momentum fractions x∼0.2÷0.4 the correlations may
+increase right up to 90%. They become important for
+almost all xwith growing tin accordance with the pre-
+dictedQCDasymptoticbehavior[17,20]. Themorecom-
+prehensive analysis of double parton distributions based
+on the direct numerical integration of generalized equa-
+tions can be found in the quite recent paper [23] (see
+also Ref. [11] for estimation therein of joint interaction
+probability due to evolution in comparison with the fac-
+torization contribution).
+III. POSSIBLE MANIFESTATION OF
+CORRELATIONS INDUCED BY QCD
+EVOLUTION
+As an effect of evolution, the double distribution func-
+tions become strongly correlated in longitudinal momen-
+tum fractions at large Q2and finite xas mentioned
+above. On the other hand, the indications from the ex-
+perimental observation of double scatterings at CDF [4]
+are not in favor of strong correlations effects in longi-
+tudinal momentum fractions. The most likely reason
+is that the studied kinematical domain, with relatively
+smallxvalues and low resolution (energy) scale, is far
+from the asymptotic QCD prediction. The possibility of
+testing double collisions at much higher resolution scales3
+will open an opportunity of probing the correlations pre-
+dicted by the QCD evolution directly. For this purpose,
+the double parton cross sections of the equal sign Wpair
+production (with a high resolution scale) in ppcollisions
+at 1 TeV ≤√s≤14 TeV are considered in Ref. [24]. As
+a main result, the contribution of the term with corre-
+lations in the equal sign Wpairs production might con-
+tribute almost 40% to the cross section at√s= 1 TeV
+and about 20% at 14 TeV.
+Here we would like to bring attention to the D0 mea-
+surementoftheprocess-independentparameterofdouble
+parton scattering, σexp
+eff, done as a function of the second
+(ordered in the transverse momentum pT) jetpT,pjet2
+T,
+that can serve as a resolution scale. It is shown in Fig. 1
+(see also Fig. 11 from Ref. [5]) and can be considered as
+a firstinderectmanifestation of the evolution effect since
+σexp
+effshows a tendency to be dependent on the resolution
+scale. At first glance the dimensional parameter σeffen-
+tering into Eq. (1) contains the information related with
+the non-perturbative structure of the proton and corre-
+sponds to the overlap of the matter distributions in the
+colliding hadrons and therefore it is independent of the
+hard process scale a priori. However the experimental
+effective cross section σexp
+effis not measured directly but
+is calculated (extracted) using the normalization to the
+product of two single cross sections:
+σγ+3j
+DPS
+σγjσjj= [σexp
+eff]−1(6)
+in both the CDF and D0 experiments. Here σγjandσjj
+are the inclusive γ+ jet and dijets cross sections, σγ+3j
+DPS
+is the inclusive cross section of the γ+3 jets events pro-
+duced in the double parton process. The factor 2 before
+σexp
+effis not needed in this case of distinguishable scat-
+terings unlike the four-jet process. It is worth noticing
+that the CDF and D0 Collaborations extract σexp
+effwith-
+out any theoretical predictions on the γ+ jet and dijets
+crosssections, bycomparingthenumberofobserveddou-
+ble parton γ+ 3 jets events in one hard p¯pcollision to
+the number of γ+ 3 jets events with hard interactions
+occurring in two separate p¯pcollisions.
+At such a normalization (6) and the presence of corre-
+lationsinthetwo-partondistributionstheexperimentally
+extracted dimensional factor σexp
+effwill be different from
+oneσeffincoming in Eq. (1). Indeed, in accordance with
+QCD evolution instead of the factorization ansatz (2) we
+can write [14, 24]:
+Dij
+p(xi,xj,t) =Di
+p(xi,t)Dj
+p(xj,t)(1−xi−xj)[1+
+Rij
+1(xi,xj,t)+Rij
+2(xi,xj,t)],(7)
+where
+Di
+p(xi,t)Dj
+p(xj,t)(1−xi−xj)[1+Rij
+1(xi,xj,t)] (8)
+is the solution of the homogeneous generalized GLAPD
+evolution equation with the given initial condition(GeV)jet2
+Tp15 20 25 30(mb)effσ
+0510152025-1 = 1.0 fbintD0, LGraph
+FIG.1: Effectivecross section σexp
+effmeasuredin thethree pjet2
+T
+bins at the D0 experiment [5]. The solid ( k= 0.5) and dashed
+(k= 0.1) lines are the results from Eq. (11) at pjet2
+T0= 22.5
+GeV and σ0
+eff= 16.3 mb.
+Di
+p(xi,0)Dj
+p(xj,0)(1−xi−xj) which is supposed to be
+factorized at the reference scale t= 0 (Q2=µ2) and
+Di
+p(xi,t)Dj
+p(xj,t)(1−xi−xj)Rij
+2(xi,xj,t) (9)
+is a particular solution of the complete equation with
+zero initial condition Rij
+2(xi,xj,0) = 0. At t >0 this
+solution is always positive as it is the integral convolu-
+tion of positive single distributions with the kernels of
+nonhomogeneous part of the evolution equation. These
+kernels (probabilities) are defined without negative δ-
+function regularization and therefore are always positive
+unlike the kernels of homogeneous part which have neg-
+ative contributions. In the kinematical range of interest
+for the actual case (we never exceed x= 0.1 practically)
+the contribution of the term Rij
+1(xi,xj,t) to Eq. (7) is
+negligible [24]. Substituting Eq. (7) in Eq. (1), and ne-
+glecting the factor (1 −xi−xj) as it is usually done at
+relatively small xi,xj, and using Eq. (6) we derive
+[σexp
+eff]−1≃[σeff]−1[1+∆(t)], (10)
+where ∆( t) is a some positive contribution induced by
+the correlation term Rij
+2(xi,xj,t). This contribution in-
+creases[14,24]withagrowthoftheresolutionscale t(Q2)
+and can be right up 40% at Q=MW= 80.4 GeV fol-
+lowing Ref. [24]. Taking into account that namely σeff
+is the true process-independent dimensional parameter
+we conclude: the experimentally extracted effective cross
+sectionσexp
+effmust decrease with the growth of the reso-
+lution scale. The experimental data in Fig. 1 show just
+such a tendency in spite of large experimental uncertain-
+ties which do not allow the D0 Collaboration to make
+this conclusion.4
+Unfortunately, the calculation of correlation contri-
+bution ∆( t) to the effective cross section σexp
+effis not
+yet possible in the framework of some kind of existing
+event generators. For instance, the double parton scat-
+terings are implemented in the Monte-Carlo generator
+PYTHIA [26, 27] taking into account some correlations
+which, however, are not quite adequate for the case of
+evolution effects under consideration (unlike the theoret-
+ical investigations in Ref. [11]). The implementation of
+the QCD evolution of two-parton distribution functions
+in some Monte-Carlo generator, as this was done for sin-
+gle distributions, is not a trivial task. However we can
+“predict” a functional form of pjet2
+T-dependence of σexp
+eff:
+σexp
+eff=σ0
+eff[1+kln(pjet2
+T/pjet2
+T0]−1(11)
+inspired by the explicit expression for the correlation
+term [14] and the evolution variable t. The results from
+Eq. (11) are also shown in Fig. 1 at k= 0.1 (dashed
+line) and k= 0.5 (solid line) to illustrate the two possi-
+ble slopes of σexp
+effin the observation region: visually im-
+perceptible (practically constant) and noticeable falling
+slopes. The normalization point pjet2
+T0= 22.5 GeV with
+σ0
+eff= 16.3 mb is fixed to reproduce the experimental
+value in the central pjet2
+Tbin.
+IV. CONCLUSIONS
+We argue that the QCD dynamical correlations result
+effectively in the dependence of the experimentally ex-
+tractedσexp
+effon the resolution scale unlike naive accepted
+expectations. The measurements covering a larger range
+of the resolution scale variation with a smaller uncer-tainty are needed to see the evolution effect more dis-
+tinctly.
+In order to investigate the more delicate characteris-
+tics of double parton scatterings (distributions over var-
+ious kinematic variables with various kinematic cuts) it
+is also desirable to implement the QCD evolution of the
+two-parton distribution functions in some Monte Carlo
+event generator as this was done for the single distri-
+butions, for instance, within PYTHIA [26]. It is worth
+noticing once more that the evolution of the two-parton
+distribution functions has the same confidence status as
+the well-established evolution of the single distribution
+functions in the framework of the leading logarithm ap-
+proximation of perturbative QCD. Therefore the exper-
+imental direct or indirect observation of this evolution
+effect is significant to answer to many challenging ques-
+tions of yet poorly-understood aspects of QCD.
+Acknowledgments
+Discussions with D.V. Bandurin (especially),
+E.E. Boos, M.N. Dubinin, V.A. Ilyin, V.L. Ko-
+rotkikh, L.N. Lipatov, I.P. Lokhtin, S.V. Molodtsov,
+S.V. Petrushanko, A.S. Proskuryakov, V.I. Savrin,
+T. Sjostrand, P. Skands, D. Treleani, G.M. Zinovjev
+and N.P. Zotov are gratefully acknowledged. This work
+is partly supported by Russian Foundation for Basic
+Research (grants No 08-02-91001 and No 08-02-92496),
+Grants of President of Russian Federation for support
+of Leading Scientific Schools (No 1456.2008.2) and
+Russian Ministry for Education and Science (contract
+02.740.11.0244).
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\ No newline at end of file
diff --git a/1001.0105.txt b/1001.0105.txt
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@@ -0,0 +1,241 @@
+arXiv:1001.0105v4  [cond-mat.str-el]  5 Mar 2010Itinerancy-localization duality of quasiparticles revea led by strong
+correlation
+Byung Gyu Chae
+Basic Research Laboratory, ETRI, Daejeon 305-350, South Kore a
+Abstract
+The strong interaction between electrons reveals the duali ty of the itinerancy
+and the localization of quasiparticles. The physical pheno mena corresponding
+to each component of the duality could be realized and coexis t within the
+category of the uncertainty principle of the carrier dynami cs, which can be a
+strong reason for the complexity appearing in the strongly c orrelated system.
+A possiblemechanism for thehigh-temperaturesuperconduc tivity is proposed
+on the basis of the interplay between the renormalized expec tation quantities
+of both parts.
+Typeset using REVT EX
+1The dynamics of interacting electrons in solids is expected to be very different from that
+of free electrons, and nevertheless, the quasiparticle concept f rom the Fermi liquid theory
+has been used as a successful tool in describing the elementary ex citation of the interacting
+system.1,2However, in the strong correlation regime, there appears to be an omalous prop-
+erties away from the Fermi liquid behaviors,3,4so that it remains to be a question whether
+this description can be valid for explaining the interaction to some ext ent. Up to now,
+many approaches5–7within this scheme have been carried out in order to analyze the stro ng
+correlation effects on the electrodynamics of the carriers.
+In general, the interaction between electrons impedes their motion . Quasiparticles in
+the interacting system evolve through a non-stationary eigensta te, which indicates a weakly
+localized state with a damping rate. Correlated metal sometimes exh ibits localized property
+together with itinerant behavior, and at the ultimate interaction, it becomes an insulator
+with fully localized carriers, known as the Mott insulator. Spectral r epresentation of the
+propagatorfor interacting systems shows a duality of the quasipa rticle stateandthe incoher-
+ent background fully localized in space.8,9In the strongly correlated regime, the incoherent
+background, aswell astheweaklocalizationofthequasiparticlesta te, canbemeaningfuland
+can play a role in physical phenomenon. We note that a strong corre lation reveals a duality
+of the localization and itinerancy of the charge carrier dynamics, an d it is believed that the
+correlated system can be explained by considering the duality compo nent simultaneously.
+Let us investigate the carrier dynamics by using the propagator, w hich is represented
+as the Green function. The single-particle Green function has a gen eral form,G(xt,x′t′) =
+−i/angbracketleftΨo|T[ˆψ(xt)ˆψ†(x′t′)]|Ψo/angbracketright, where the ˆψ(xt) andˆψ†(x′t′) are the field operators that create
+(annihilate) the electron. By taking the Fourier transform, the Gr een function is expressed
+in energy-momentum representation.
+G(k,ω) =/summationdisplay
+m|/angbracketleftΨo|ˆψ|Ψm/angbracketright|2
+ω−ǫk±iη(1)
+The spectral density function is obtained from the identity, A(k,ω) =−(1/π)ImG(k,ω),
+resulting in/summationtext
+m|/angbracketleftΨo|ˆψ|Ψm/angbracketright|2δ(ω−ǫk). In a non-interacting system, the Green function has
+2one pole with free particle energy ǫk, where the spectral density is delta-function. But, the
+interacting system has spectral function that spreads out from the superposition of many
+poles. The representation of the Green function in time-domain is fa vorable for investigating
+the itinerancy of the quasiparticle, G(k,t) =/integraltextG(k,ω)e−iωtdω. By the contour integral, the
+function can be split into a coherent quasiparticle state and an incoh erent background with
+no pole.2
+G(k,t) =−iZke−iεkt−Γkt+Ginch (2)
+The first part of the equation shows that the state propagates lik e a wave packet with
+an energyεkand damping rate Γ k. The amplitude of the wave packet is the residue of the
+function, known as the renormalization constant Zk. Figure 1(a) depicts the dynamics of
+the quasiparticle in the lattice model that has the strong interactio n. In general, the overlap
+of the atomic orbitals in lattices removes the degeneracy of each or bital to form a band,
+which makes it possible for the electron to freely migrate within the ba nd. As shown in Fig.
+1(a), in the case of a weak overlap of the orbital, the electron is no lo nger a free particle and
+rather, feels the potential barrier to move around, where the ca rrier state is the wave packet
+that is slightly localized in space.
+The Green function in the interacting system can be expressed in te rms of the self-energy
+Σ(k,ω), as likeG(k,ω) = 1/(ω−εk−Σ(k,ω)).10From this notation, the renormalization
+constant and the elementary excitation energy are expressed as Zk= (1−∂ReΣ)−1and
+εk=Zk(ǫk+ReΣ), respectively. In Fig. 1(b), the spectral function shows a Lorentzian form
+with a height Zkand width Γ k, where the latter is proportional to [( πkBT)2+(εk−ǫF)2].
+The Fermi liquid description is valid when it is close to the Fermi surface . On the other
+hand, when the renormalization constant approaches zero, the q uasiparticle is considered
+to be fully localized in the lattice site. In Eq. (2), only the incoherent p art of the Green
+function remains at the zero value of the renormalization constant . It is ensured that Ginch
+is associated with the localization of the quasiparticle in the lattice site .
+The strong many-body interaction increases the self-energy and causes the elementary
+3excitation energy to not be a negligible value. The quasiparticle may no t be a good, effective
+single-particle state, and in addition, the incoherent part of the Gr een function has a value
+that cannot be ignored. In this situation, the interacting particle c an behave in two ways;
+one is a mobile quasiparticle state and the other is an incoherent back ground localized in
+the lattice site. In the limit t →0, the Green function presents a momentum distribution
+in Fig. 1(c), where Zbecomes the discontinuity at the Fermi surface.11We consider that
+the incoherent part of the Green function is closely related to the p article-hole excitation,
+from the curve of the momentum distribution. This is depicted in a dot ted line in Fig. 1(b),
+which can be a strong clue of the collective excitation in the correlate d system. This is a
+different concept from the collective modes derived by the instability of the nesting vector
+of the mobile carriers or in the Luttinger liquid in one dimension.
+According to the sum rules, the spectral density function is repre sented as the quasi-
+particle part,/integraltextAch(k,ω)dω=Zand the incoherent background,/integraltextAinch(k,ω)dω= 1−Z.
+The valueZbecomes a measure to evaluate the itinerancy of the quasiparticle. We simply
+extract that the coherent part of the Green function is weighted by the itinerancy factor
+Z, and the incoherent part by the factor 1 −Z, becauseA(k,ω) is only the probability of
+adding and removing particle to the many-body system. Namely, the Green function has
+the form,G=ZG′
+ch+(1−Z)G′
+inch.
+The interacting system is described by the basic Hamiltonian with a kine tic energy
+and Coulomb energy. We infer that the Hamiltonian can be clearly divide d into coherent
+and incoherent parts because both parts have independent eigen sates in time. Considering
+that the physical observable is represented as the expectation q uantity, it is convenient
+to assume that the operators of both parts are themselves weigh ted by factors√
+Zand
+√1−Z, respectively, which imply the weight factor, as like c†
+k=√
+Zc′†
+k. In the coherent
+part described by operators, c†andck, the state propagates with a quasiparticle eigenvalue
+εkand weighting factor Z. Effective Hamiltonians Hchof coherent part is expressed as
+follows,
+4Hch=/summationdisplay
+kεkc†
+kck+/summationdisplay
+k,k′,qVqc†
+k+qc†
+k′−qck′ck (3)
+whereεkandVqare the dressed kinetic energy and interaction between quasipart icles.
+In a similar way, the operators of incoherent part can describe the collective excitation with
+a weighting factor 1 −Z. Effective Hamiltonian Hinchcorresponds to the Hubbard model
+except for the electron localization,12,13where the electron can not move to the nearest site
+due to the on-site interaction Ularger than the hopping energy t. The hopping energy plays
+a role only in the spread of the excitation bands, so that this may well describe the collective
+excitation of incoherent bands, in Fig. 1(b).
+From the above propagator formalism, we find that the many-body correlation reveals
+a duality in the itinerancy and localization of the quasiparticle in the str ongly correlated
+system. This is based on the wave-particle duality in the basic concep t of quantum mechan-
+ics. However, there is a crucial point here, namely the duality arises from not an external
+measuring operation but a strong correlation in the system. It is ve ry interesting to know
+whether the physical phenomenon corresponding to each compon ent of the duality could be
+simultaneously realized and coexist in the system, and further, inte rplays with each other
+to make a new phenomenon.
+Considering the splitting of the coherent and incoherent states, t he instability exists in
+knowing which component is dominantly revealed in the system, espec ially when two com-
+ponents have comparable values. This closely resembles the observ ing process in quantum
+phenomenon, that is, the physical quantity in possible states can b e uncovered only after
+an external observing process. Here, we can guess the spatial e xtent of the quasiparticle
+state from the uncertainty principle, ∆ x∆p≥¯h
+2. The coherent part of the propagator is ob-
+tained from the summation of the possible excited states of an addin g particle and thus, the
+particle-like peak spreads out in the momentum space. Namely, the p lane wave transforms
+into a wave packet slightly localized in space, and then evolves over a lif etime. Assuming
+that the lifetime of the quasiparticle is about several decades meV, the ability to confirm
+the location of the quasiparticle state extends over several deca des˚A. On the other hand, as
+5mentioned previously, the incoherent background can make the co llective excitation. There-
+fore, electronic inhomogenity will appear in the strongly correlated system.
+Our approachis applicable to theunderstanding of the high-temper ature superconductor
+known as the representative of the strongly correlated system. The parent material of
+this system is the Mott insulator with antiferromagnetic ordering, a nd by deviating from
+the half-filling, collective excitations appear.4Elementary excitation is generally analyzed
+by using the Hubbard model in the mean-field basis.14However, this analysis does not
+consider the renormalization factor. Our study describes the sup erconductor and density
+wave parameters, ∆ SCand ∆ DWas the renormalized quantities by the itinerancy factor Z.
+Doping of the holes (electrons) to the correlated system leads to t he complexity pre-
+sumably including various collective modes and in this regime, electronic inhomogenity ap-
+pears, which makes it difficult to analyze apparently physical proper ties.3,4According to
+our scheme, this can be interpreted as natural results by the itine rancy-localization duality
+of the quasiparticles. Here, the pseudogap states is a kind of collec tive excitations of the
+incoherent background. This is in agreement with the midinfrared ex citation of the optical
+spectroscopy,4,15leading to closure of the charge gap before close to the metal stat e.
+Another important property is the possibility of the pairing of the qu asiparticles. Charge
+carriers move in collective excitations of the incoherent backgroun d that can be considered
+as bosonic glue. Even under weak attraction, fermion particles at t he Fermi surface are
+unstable to form the bound state.16We consider the system of electrons (holes) and bosons
+with mutual interaction.
+Hel−b=/summationdisplay
+k,qWq(b†
+−q+bq)c†
+k+qck (4)
+whereWqis the interaction matrix element and b†
+−qthe boson operator. The effective
+interaction between two electrons is obtained by canonical transf ormation, which arises from
+the exchange of the boson such as collective modes in analogy with th e phonon exchange.
+Veff=/summationdisplay
+k,k′,q|Wq|2ω2
+q
+ω2−ω2qc†
+k+qc†
+k′−qck′ck (5)
+6Forω < ω q, the effective interaction is attractive. The pairing of the carrier is possible
+at low frequencies, leading to the superconductor. According to t he BCS formalism, the gap
+equation is expressed as below,
+∆SC= 2¯hωDexp(−1
+NV) (6)
+whereNis the density of states at the Fermi surface and ωDis the cutoff frequency of the
+collective excitation of incoherent background. The spin wave can b e formed as a possible
+candidate of the collective modes, because in Eq. (5), the strong o n-site interaction with
+comparedtothehoppingenergyleadstotheHeisenberg hamiltonian , sothatthefinite-range
+antiferromagnetic ordering exists. This is very analogous to the sp in-bag concept17, but
+an important difference is that the expectation values of two pheno mena, the quasiparticle
+pairingandcollectivemodearelinkedbytheitinerancyfactor. Sincet hegapparameter, ∆ SC
+is represented as an anomalous Green function, e−2µt/angbracketleftck↑c−k↓/angbracketright, whereµis the Fermi energy,
+it implies the factor Zdue to the symmetry of the many-body system. The energy, ¯ hωD
+comes from the expectation quantity of the spin wave. Therefore , the critical temperature
+can be written as, Tc≃Z(1−Z). Figure 2 shows the change of the critical temperature
+with respect to the itinerancy factor Z. Assuming the itinerancy factor is proportional to
+the doping carrier, Tcvariation is similar to the experimental results.4This relation can be
+valid for the both states of electron and hole doping.
+In conclusion, the possibility to interpret the complexity of the stro ngly correlated sys-
+tem is proposed within the itinerancy-localization duality of the quasip article. The strong
+correlation makes the system reveal the duality of the quasipartic le and new physical phe-
+nomena through the interplay between them. This scheme is applicab le to the analysis of
+other physical phenomena induced by the strong interaction betw een carriers.
+7REFERENCES
+†E-mail:bgchae@etri.re.kr
+1L. D. Landau, Zh. Eksp. Teor. Fiz. 30, 1058 (1956) [Soviet Phys. JEPT 3, 920 (1957)].
+2A. A. Abrikosov, L. P. Gorkov, andI. E. Dzialoshinski, Methods of Q uantum FieldTheory
+in Statistical Physics (Pergamon, Elmsford, New York, 1965).
+3E. Dagotto, Science 309, 257 (2005).
+4P. A. Lee, N. Nagaosa, and X. G. Wen, Rev. Mod. Phys. 78, 17 (2006).
+5D. Vollhardt, Rev. Mod. Phys. 56, 99 (1984).
+6A. Georges, G. Kotliar, W. Krauth, M. J. Rozenberg, Rev. Mod. Ph ys.68, 13 (1996).
+7A. E. Ruckenstein, P. J. Hirschfeld, and J. Appel, Phys. Rev. B 36, 857 (1987).
+8H. Lehman, Nuovo Cimento 11, 342 (1954).
+9A. Damascelli, Z. Hussain, and Z. X. Shen, Rev. Mod. Phys. 75, 473 (2003).
+10A.L.FetterandJ.D.Walecka, QuantumTheoryofMany-Particle S ystems (McGraw-Hill,
+New York, 1971).
+11A. B. Migdal, Zh. Eksp. Teor. Fiz. 32, 339 (1957) [Soviet Phys. JEPT 5, 333 (1957)].
+12J. Hubbard, Proc. Roy. Soc. London A 276, 238 (1963).
+13W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 4302 (1970).
+14R. Micnas, J. Ranninger, S. Robaszkiewicz, and S. Tabor, Phys. Re v. B37, 9410 (1988).
+15M. M. Qazilbash, M. Brehm, Byung-Gyu Chae, P.-C. Ho, G. O. Andree v, Bong-Jun Kim,
+Sun Jin Yun, A. V. Balatsky, M. B. Maple, F. Keilmann, Hyun-Tak Kim, a nd D. N.
+Basov, Science 318, 1750 (2007).
+16J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
+817J. R. Schrieffer, X. G. Wen, and S. C. Zhang, Phys. Rev. Lett. 60, 944 (1988).
+9FIGURES
+Ginch(k,t)Gch(k,t)
+Quasiparticle
+peak
+Incoherent backgroundA(k,Ȧ)
+Ȧ İkīk
+n(k)
+k kFZk(b)
+(c)(a)
+FIG. 1. (a) A schematic diagram to visualize the duality of th e itineracy and the localization
+of quasiparticles. The Green function with a many-body corr elation can be split into a mobile
+quasiparticle state, Gchand the incoherent background localized in the lattice site ,Ginch. In
+the Fermi liquid state, (b) the spectral density clearly sho ws a quasiparticle peak and incoherent
+background and (c) the momentum distribution has discontin uity at the Fermi surface.
+100.0 0.2 0.4 0.6 0.8 1.0 Temperature (K)
+Itinerancy factor  ZSCPseudogap
+stateFermi liquid
+metal
+FIG. 2. Change in the critical temperature of the supercondu ctivity with respect to the itin-
+erancy factor Z. The straight line proportional to 1 - Zindicates a pseudogap temperature from
+the incoherent background.
+11
\ No newline at end of file
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+arXiv:1001.0106v1  [cond-mat.mtrl-sci]  31 Dec 2009Pentacene multilayersonAg(111) surface
+ErsenMete,∗,†˙IlkerDemiro ˘glu,‡M.FatihDanı¸ sman,‡and ¸ SinasiEllialtıo ˘glu¶
+Deparmentof Physics,BalıkesirUniversity,Balıkesir101 45,Turkey, Departmentof Chemistry,
+MiddleEast TechnicalUniversity,Ankara06531,Turkey, an dDepartmentofPhysics,Middle
+EastTechnicalUniversity,Ankara06531,Turkey
+E-mail: emete@balikesir.edu.tr
+Phone: +90 (0)2666121000. Fax: +90(0)266 6121215
+∗To whomcorrespondenceshouldbeaddressed
+†DeparmentofPhysics,BalıkesirUniversity,Balıkesir 101 45,Turkey
+‡DepartmentofChemistry,MiddleEastTechnicalUniversity ,Ankara06531,Turkey
+¶DepartmentofPhysics,MiddleEast TechnicalUniversity,A nkara06531,Turkey
+1Abstract
+The structural profiles and electronic properties of pentac ene (C 22H14) multilayers on
+Ag(111) surface has been studied within the density functio nal theory (DFT) framework. We
+haveperformedfirst-principletotalenergycalculationsb asedontheprojectoraugmentedwave
+(PAW)method toinvestigate theinitial growth patterns of p entacene (Pn)on Ag(111) surface.
+Inits bulk phase, pentacene crystallizes withatriclinic s ymmetry while athin film phase hav-
+inganorthorhombic unitcell isenergetically lessfavorab le by0.12eV/cell. Pentacene prefers
+tostayplanar onAg(111) surfaceandalignsperfectly along silverrowswithoutanymolecular
+deformation at a height of 3.9 Å. At one monolayer (ML) covera ge the separation between
+the molecular layer and the surface plane extends to 4.1 Å due to intermolecular interactions
+weakening surface–pentacene attraction. While the first ML remains flat, the molecules on a
+second full pentacene layer deposited on the surface rearra nge so that they become skewed
+withrespect toeach other. Thisadsorption modeisenergeti cally morepreferable than the one
+for which the molecules form a flat pentacene layer by an energ y difference similar to that
+obtained for bulk and thin film phases. Moreover, as new layer s added, pentacenes assemble
+to maintain this tilting for 3 and 4 ML similar to its bulk phas e while the contact layer always
+remains planar. Therefore, our calculations indicate bulk -like initial stages for the growth pat-
+tern.
+2Introduction
+Due to its use in thin film transistor (TFT) applications pent acene is continuing to enjoy being
+the subject of extensive research. Since most TFT’s employ S iO2as the dielectric layer, initial
+studies on pentacene focused mainly on gaining a thorough un derstanding of structural1–4and
+electronic5–8properties of thin films of this molecule on SiO 2surfaces and in turn achieving the
+best device performance by optimizing9–11these properties. As a result of this heavy research
+effort in the last decade, fabricating pentacene TFT’s with hole mobilities of more than 1 cm2/Vs
+have become an almost routine process.12Nevertheless there are still, fundamental issues to be
+resolved such as the dependence of the charge mobility on the film thickness13and areas open to
+improvementlikemodificationof thesubstratesurfaces wit hbuffer layers7orthepentacene itself
+withfunctionalgroups.7,14
+Another very critical but relatively less well understood s ubject is the growth mechanism of
+pentacene films on metal substrates. Understanding the pent acene film growth on gold and silver
+surfacesisparticularlyimportantsincethesemetalscons titutetheelectrodematerialinmostTFT’s
+and the device performance is directly related to the charge transfer efficiency between the elec-
+trodes and theorganicfilm. Though bothexperimental15–29and theoretical29,31,32research in this
+field has been recently intensified, there are contradictory results in the literature and the growth
+modes of pentacene thin films on Au(111) and Ag(111) surfaces are continuing to be a matter of
+debate. This is mostly due to the relatively strong (when com pared with SiO 2) interaction of the
+metalsurfaceswiththepentacenemolecule. Asaresultofth isstronginteractionpentaceneadopts
+many different monolayer and multilayer phases on metal sur faces which are energetically and
+structurallyvery closeto each other. ForexampleonAu(111 )severaldifferent lowdensitymono-
+layer phases and an identical full coverage phase have been r eported by different groups.15–18,20
+However, in case of the multilayers while Kang et al.15,16report a layer by layer growth of ly-
+ing down pentacene molecules, Beernink et al.19report strong dewettingstarting from thesecond
+layer and growth of bulk like pentacene crystals. For pentac ene films on Ag(111) surfaces, while
+Eremtchenko et al.24and Dougherty et al.26report a bilayer film formation, where an ordered
+3(second) layer (which follows the symmetry of the Ag(111) su rface) forms on top of a disordered
+(2D gas phase) first layer at room temperature, Käfer et al.21report the formation of bulk-like
+pentacene structures immediatelyafter the first monolayer . So on both surfaces the growth mech-
+anism of pentacene films is still not completely clear. In add ition, if the above mentioned 2D gas
+phase mechanism for Ag(111) is really correct then question s like “Why does pentacene behave
+completelydifferently on seeminglysimilarsurfaces, Ag( 111)and Au(111)?” and “How does the
+symmetryofthesubstrateaffect thebilayerfilm structure? ” arise.
+In spite of this richness of experimental studies and points in need of clarification, pentacene
+films on Ag(111) or Au(111) surfaces have not, yet, been studi ed theoretically. Theoretical work
+regarding pentacene films were mostly performed on other met al surfaces, such as Cu(001),33
+Cu(110),34Ag(110),29at semi-empiric level and Al(100),35Cu(100),36Cu(119),37Fe(100),38
+Au(001)31,32at DFT level. These studies in general addressed two points c oncerning the first
+layer of pentacene film/molecule: (1) Determination of the m ost stable adsorption site/geometry,
+and (2) determination of the strength of electronic interac tion/coupling between the substrate and
+the molecule. In these studies either the most stable configu ration was found to be pentacene ly-
+ing flat on the surface29or the calculations were started with this assumption. In te rms of the
+electronic interactions, the DFT studies performed using G GA functionals found considerable
+aromatic- π-system metal substrate interaction36,37on Cu surfaces, hinting at chemisorption. On
+Au(111)32and Al(100)35however, while LDA functionals resulted in strong interact ions, in the
+form of broadening and splitting of π-molecular orbitals, GGA functionals are reported to resul t
+in much weaker interactions, in accord with a physisorption mechanism. Theoretical studies con-
+cerning the further stages of pentacene film growth on metal s urfaces, however, like second layer
+structure/energetics or the thin film crystal/electronic s tructure, are very few and at semi-empiric
+level.33Instead, theoretical works regarding pentacene films are pr imarily focused on the elec-
+tronic structure of different pentacene polymorphs observ ed mainly on SiO 2surfaces, one being
+thefamous“thinfilm phase”.8,39–42
+Hence, a theoretical study of growth mechanism and electron ic properties of pentacene films
+4on Ag(111) and Au(111) may, (1) help resolve the experimenta l contradictions mentioned above
+and (2) fill a gap in the theoretical literature and enable a co mparison of these systems with other
+pentacene films. As a first attempt to this end, here we present the results of our work on the
+structural and electronic properties of monolayerand mult ilayerfilms of pentacene on Ag(111)at
+DFT level. First we discuss full coverage monolayer film in th e light of experimental results. We
+compare the adsorption geometries and the corresponding de nsity of states we found with the ex-
+perimentalresultsreportedsofar. Thenwepresentourresu ltsregardingtwoandthreemonolayers
+of pentacene film and discuss how the crystal and electronic s tructure of the Ag(111)/pentacene
+interfaceandthefilmevolveswithcoverage. Weconcludewit hanoverallsummaryanddiscussion
+oftheresults.
+Method
+Weperformedtotalenergydensityfunctionaltheory(DFT)c alculationsusingtheprojector-augmented
+wave(PAW)method43,44withinthegeneralized gradient approximation(GGA)byemp loyingthe
+Perdew–Burke–Ernzerhoff(PBE)45exchange–correlationenergyfunctionalasimplementedin the
+ViennaAbInitioSimulationPackage (VASP).46
+For consistency, we used a kinetic energy cutoff of 370 eV for the plane wave expansion of
+singleparticlewavefunctionsinall thecalculations. Ele ctronicgroundstateshas been determined
+by requiring a total-energy convergence up to a tolerance va lue smaller than 0.1 meV. We used a
+conjugate-gradientalgorithm,in all geometry optimizati oncalculations, based on the reductionin
+theHellman–Feynmanforces on each constituentatomto less than 10meV/Å.
+We examined two previously known polymorphs of Pn lattice: t he bulk47and the thin film39
+phases. Bulk phasecorrespondstoatriclinicunitcellwhic h containstwoC 22H14formulaewitha
+setofparameters: a=7.90Å,b=6.06Å,c=16.01Å,α=101.9◦,β=112.6◦,andγ=85.8◦.47
+Our calculated values of a=7.90 Å,b=6.06 Å,c=16.01 Å,α=102.0◦,β=112.6◦, and
+γ=85.5◦shows a very good agrement with the experimental results of C ampbellet al.For the
+5thinfilmphase,weobtainedanorthorhombicunitcellwith a=7.42Å,b=5.87Å,and c=16.21
+Å. These parameters agree with Parisse et al.’s39theoretical results of a=7.60 Å,b=5.90 Å,
+andc=15.43 Å except for the last one, corresponding to the longitudin al size of the unit cell.
+Wecalculatedthelengthofanisolatedmoleculetobe14.5Å. Therefore, thedisagreement,inthis
+phase, stemsfromthePn–Pn separationin themultilayers.
+In order to compare the predictions of different DFT functio nals with the experimental results
+for key structural and electronic parameters like the latti ce constants and binding energies, we re-
+peated our calculations using PW9148parametrization within both generalized gradient and loca l
+density approximations (LDA). We calculated the lattice pa rameter of the silver ccpbulk struc-
+ture inFm¯3msymmetry group to be 4.00, 4.15, and 4.17 Å using LDA-PW91, GG A-PW91, and
+GGA-PBE functionals respectively. These compare well with the experimental49value of 4.09
+Å, slightly better than the previous theoretical50result of 4.20 Å. The Ag(111) surface has been
+modeledinafour-layerslabgeometryseparatedfromtheirp eriodicimagesby ∼15Åofavacuum
+spaceand3 ×6×1gridforthecasesof1MLto4MLdeposition,whereasintheca sesofisolatedPn
+adsorptionalargercellisneededandthereforeathree-lay er-slabgeometryfortheAg(111)surface
+and 1×2×1 grid was used. For this metallic system, the number of layer s has been found to be
+sufficiently large to represent the Ag(111) surface structu re such that the geometry optimization
+calculationsdonot disturbthesubsurfacelayeratomsfrom theirbulklatticepositions.
+ResultsandDiscussion
+Single isolatedpentacene onAg(111)
+In order to study the formation of ordered Pn layers on Ag(111 ) we first considered a single Pn
+on the surface. Isolation of the molecules was achieved by us ing a 8×5 silver surface unit cell
+which sets 9.5 Å tip-to-tip and 7.9 Å side-to-side separatio ns between the periodic images. We
+determined the minimum energy Pn/Ag(111) geometry by inves tigating all possible adsorption
+siteswith a numberof orientations,compatiblewith thelat ticesymmetry,at each siteas shownin
+6Figure 1: Single isolated pentacene on different adsorptio n sites of Ag(111) surface. Planar Pn
+adsorptionwiththecentralcarbonringontopofasilverato maligningparalleltooneofthelattice
+directions is abbreviated as “Top-0” (a). Top-30 in (b) refe rs to the adsorption at the top site with
+Pnmajoraxismakinganangleof30◦withanyofthesilverrows. Hollow-0(c)andHollow-30(d)
+follow the same molecular alignments as the top cases, but ce ntered on a triangle whose corners
+definedbyAgatoms,i.e. atthehollowsite. Bridge-0(e),Bri dge-30(f),Bridge-60(g),andBridge-
+90 (h), describe the cases where central ring of Pn lies over a n Ag–Ag bond making the referred
+angles with any of the lattice lines. The minimum energy geom etry, Bridge-60 (g), is depicted in
+ball-and-stickfashionwhiletheothersare allshowninsti cksonly.
+7[figure][1][]1. (Labeling conventions are described in figu re caption.) In addition to these planar
+caseswealsoinvestigatedthepossibilityofstanding-upa dsorptionconfigurationswhichappeared
+to be around 0.15 eV less favorable than the planar ones. In ge ometry optimization calculations
+Pn developsaweak interactionwithAg(111)whereveritisin itiallyplaced onthesurface. In fact,
+as presented in [table][1][]1, the comparison of the total e nergies of these adsorption cases show
+differenceswhicharenogreaterthan36meVfromeachother. Theflatnessofthepotentialenergy
+surface (PES) is indicated by the existence of such small bar riers which might make Pn diffusion
+overthesurfacepossibleinagreementwiththeexperimenta lobservationsthatthecontactlayerPn
+molecules are mobile at the Pn/Ag(111) interface.24,26Similarly, during image acquisition STM
+tiphas been observedtodrag Pn moleculeswhich arephysisor bedon Au(111).51
+Table 1: Calculated values for geometrical and electronic s tructure of Pn/Ag(111) systems
+shown in [figure][1][]1. The lateral height of isolated Pn mo lecule from the Ag(111) surface
+dzinÅ, thebinding energy Ebandthe relativetotal energy ETineV.
+dzEbET
+Top-0 3.90 −0.125 0.030
+Top-30 3.89 −0.119 0.036
+Hollow-0 3.88 −0.147 0.008
+Hollow-30 3.87 −0.128 0.027
+Bridge-0 3.87 −0.124 0.031
+Bridge-30 3.88 −0.124 0.031
+Bridge-60 3.87 −0.155 0.000
+Bridge-90 3.88 −0.129 0.026
+The adsorption configurations (in [figure][1][]1) where the isolated pentacene follows the lat-
+ticesymmetryso that themolecularcharge densitymatches b etter withthe surface charge density
+of silver rows are energetically more preferable. As a resul t, the total energy of the bridge-60 is
+smallerfromthatofthehollow-0byonly8meV.Thisalsoindi catesthattheflatness ofthePES is
+relativelymorepronouncedalongthelatticedirections.
+Single isolated pentacene molecule finds its minimum energy configuration at the bridge-60
+position as depicted in ball-and-stick form in [figure][1][ ]1g. For this adsorption geometry, it is
+almost flat with a negligible bending at a height of 3.87 Å whic h gives a weak binding energy
+8of−0.155 eV. In geometry optimizationcalculations, for all poss ibleinitial adsorption configura-
+tions, both GGA functionals predict a weak interaction betw een thePn moleculeand theAg(111)
+surface where LDA overbinds. (see [table][2][]2). In parti cular, GGA-PBE predicts that an iso-
+lated pentacene with a tilt about 15 degrees off the surface p lane is only 3 meV unfavorable than
+the lowest energy flat geometry. This barrier is so small that the tilted pentacene does not relax
+back toplanargeometry.
+Although GGA functionals result in a weak pentacene–silver interaction, the degree of this
+weakness is overestimated. Since, pure DFT results depend o n the choice of the exchange–
+correlation functional, a hybrid-DFT with corrected excha nge with dispersive interaction energy
+would give an improved description of the binding character istics of such a weakly bound sys-
+tem.52
+Table 2: Calculated values for electronic and geometrical s tructure of Ag and Pn/Ag(111)
+systemsindifferentexchange–correlationfunctionals. L atticeparameterofAg(111)slab aAg
+in Å, lateral heights dz(Å) and the binding energies Eb(eV) of isolated and 1 ML Pn on the
+Ag(111)surface.
+clean slab isolatedPn 1 MLPn
+aAgdzEbdzEb
+LDA-PW91 4.000 2.46 −1.925 2.48 −1.753
+GGA-PW91 4.145 3.69 −0.234 3.94 −0.093
+GGA-PBE 4.174 3.87 −0.155 4.12 −0.078
+Full monolayer
+Full monolayer has been derived from the previously optimiz ed bridge-60 configuration using an
+experimentally observed 6 ×3 silver surface unit cell.22,23,26The relaxed geometry of this contact
+layer is shown in [figure][3][]3a–c. The molecules follow th e surface symmetry and are aligned
+parallel to the silver rows. The distance between Pn and Ag(1 11) at the interface varies with
+varying intermolecular interaction strengths. For instan ce, in the case of GGA-PBE, an isolated
+Pn stays 3.9 Å above the surface while this value extends to 4. 1 Å in the case of 1 ML coverage
+as presented in [table][2][]2. Corresponding binding ener gy at 1ML is calculated to be 0.08 eV
+9forGGA-PBE.Thisshowsaweakbindingsimilartotheexperim entalobservations.22–24,26GGA-
+PW91 gives a slightly better description of both the Ag subst rate and Pn contact layer geometries
+relative to GGA-PBE results. We also calculated the superce ll total energies as a function of Pn–
+Ag(111)distanceasshownin[figure][2][]2forisolatedand fullmonolayercases. LDAincorrectly
+gives strong binding for this system since the charge densit ies are well localized around atoms
+leavingnearly emptyinteratomicregions.
+–2–101234
+–1 0 1–2–101–1 0 1
+single isolated
+1ML
+Relative height ( ˚A)Relative height ( ˚A)Energy (eV)
+Energy (eV)
+GGA-PBE⊙
+⊙⊙⊙⊙⊙⊙⊙⊙⊙⊙
+⊙
+GGA-PW91×
+×
+×××××××××
+×
+LDA-PW91⊙
+⊙⊙⊙⊙ ⊙⊙⊙⊙⊙⊙×
+×
+××××××××
+Figure2: Bindingenergy versuspentaceneheightonAg(111) relativetotheminimumenergy po-
+sition (bridge-60), calculated with different exchange–c orrelation functionals and approximations
+bothforasingleisolatedPn and for1MLPn coverage.
+OurDFTcalculationsshowthatasmalltiltangleoftheconta ctlayerwillnotyieldasignificant
+increaseinthetotalenergywithrespecttoflatgeometry. Th isisduetotheoverestimatedweakness
+ofpentacene–silverinteraction. Based on theseDFT result swe can not reject thepossibilityofan
+averagetiltatthe1MLaswellastheisolatedsinglepentace necasedependingontheexperimental
+conditions. However,ourcalculationsdonotsuggestastro ngbindingbetweenthepentacenelayer
+and the surface. Therefore, our calculations indicate a flat 1 ML physisorptionrather than a tilted
+chemisorbedPnlayerwhichwasconcludedbyKäfer etal.21basedontheirNEXAFSandthermal
+desorptionsignatures.
+10Figure 3: Side views (a, b) and top views (c, d) of (1ML, 4ML) pe ntacene, respectively,adsorbed
+on the Ag(111) surface. No perspective depth is used in the pr oduction of the figures, hence the
+actual tiltsare seen in(b)and (d)forthe4MLcase.
+Multilayers
+We considered two different initial geometries for the seco nd pentacene layer on the already op-
+timized 1ML Pn/Ag(111) structure. For the first case, the sec ond layer molecules are flat on the
+first monolayer where molecular axes follow the surface symm etry. In addition, a second layer
+pentacenestays aboveinbetween thetwo moleculesundernea th. Thesecondcase initialstructure
+is the same as the first one except the molecular planes of only the second layer pentacenes are
+tilted around their long axes as observed in experimental st udies.21–24,26Geometry optimization
+calculationsresultedinaverysmalldifferenceof0.046eV inthetotalenergies infavorofthelat-
+ter case in which second layer molecules slightly misaligne d from the surface lattice direction by
+6.4◦inadditiontothemolecularplanetiltingof18◦aspresentedin[table][3][]3. Thesmallnessof
+theenergydifferencebetweenthetwocasescanbeaddressed totheenergydifferencebetweenthe
+different phases of pentacene. For instance, we calculated the difference in the total energies be-
+tweenthebulkandthethinfilmphasesofpentacenetobe0.12e Vforacellhavingtwomolecular
+formulaeunits.
+11Figure4: Thechargedensitycontourplotof4MLpentacene–A g(111)interfaceonaplanenormal
+tothesurface andcut throughasilverrow.
+For the 2ML structure, the tilted layer (the second one) rela xes to a height of 3.6 Å above the
+contactlayerwhichisseparatedfromthesilversurfaceby3 .8Å.Resultingheightofthefirsttilted
+pentacene layer from silver surface becomes 7.4 Å. In the cas e of 3ML and 4ML structures this
+heightconvergesto7.2Åwhichisslightlylowerthantheexp erimentalvalueof7.8Åreportedby
+Danisman et al..
+Table 3: Calculated values for geometrical structure of Pn/ Ag(111) systems where dzis the
+distancebetweenfirst-layerpentacene andAg(111)surface ,dn−misthedistancebetween nth
+andmth Pn layers (all in Å). θandαnare the tilt angles of the nth layermolecules about the
+(111)-axisandabout their majoraxes,respectively.
+dzd1−2d2−3d3−4θ α2α3α4
+isolated 3.9 – – – 0.0 – – –
+1ML 4.1 – – – 0.0 – – –
+2ML 3.8 3.6 – – 6.4 18 – –
+3ML 3.7 3.5 3.6 – 6.0 25 −17 –
+4ML 3.7 3.5 3.5 3.6 4.0 20 −23 15
+The flat and tilted pentacene configurations above the first la yer has also been considered for
+the 3ML and 4ML initial structures. The difference in the tot al energies has been calculated to be
+0.127 eV and 0.125 eV in favor of the tilted molecules on the fir st layer for 3ML and 4ML cases,
+respectively. Therefore, bulk-likepentaceneformationo nAg(111)surfaceabovethecontactlayer
+ismorepreferable thanflat lyingmultilayers.
+12Inthecaseofmultilayers,theseparationofthetoplayerfr omthelayerunderneathis3.6Åand
+alltheinnerinterlayerdistancesbecome3.5Åwhilethehei ghtofthecontact layerconverges toa
+valueof3.7ÅafterthethirdML.Inaddition,ourcalculatio nsshowadecreaseinthemisalignment
+of pentacenes abovethe contact layer as new layers deposite d on Ag(111)surface up to 4ML. All
+multilayer geometries can be seen through [figure][3][]3b– d, which only show views of the 4ML
+structure,sincetheinterlayerdistancesandanglesdonot changesignificantlyasnewlayersadded.
+Wealso presentthecorrespondingelectronicdensityconto urplotofpentacene–Ag(111)interface
+at 4ML in [figure][4][]4 which indicates charge localizatio n around pentacene molecules and in
+thesubstratesuggestingweak bindingofthecontactlayer.
+Consequently, the thin film formation starting from the seco nd ML indicates that the inter-
+molecular interactions are relatively stronger than the pe ntacene–silver interaction. Substantia-
+tively, the misalignment of pentacenes above the contact la yer can also be attributed to the weak-
+ness of Pn–Ag(111) interaction. In fact, this observation i s in parallel with the experimental ob-
+servations of Danisman et al.53whereupon desorption of the pentacene multilayers on a step ped
+Ag(111) surface a new monolayer phase was observed. In addit ion, tilted second layer struc-
+ture was also reported by both Eremthcenko et al.24and Käfer et al.21Furthermore, molecular
+rearrangements involving such topological phase transfor mations from flat to buckled pentacene
+multilayers mimicking thin film formation on Ag(111) can be e xpected to give the same order of
+energydifferencesasthatofthinfilmandthebulkpentacene phases. Onefinalpointtobestressed
+hereisthat,although(i)ourresultsmaybehelpfulforthec omparisonofstabilityofflat andtilted
+multilayers, and (ii) the more stable multilayer configurat ions we found resemble the experimen-
+tally observed pentacene phases21–23,41,47(i.e., the tilt angles are very close to that of pentacene
+bulk and thin film phases), a direct comparison of our results with the experimentally observed
+structures may not be very meaningful. This is because the mu ltilayer and monolayer in-plane
+unit-cell dimensions which are actually different had to be chosen the same due to computational
+restrictions.
+In order to investigatethecouplingof the frontiermolecul arorbitals ofpentacene to thesilver
+13Figure 5: Calculated STM image of 1ML Pn on Ag(111) (a) for the occupied states around −0.9
+eV and (b)fortheunoccupiedstatesaround 1.3eV vicinityof theFermi energy.
+substrate states, we obtained the STM pictures by using Ters off–Hamann approximation.54The
+calculated STM imagesfortheappliedvoltagesof −0.9 Vand 1 .3V in [figure][5][]5 resembleto
+theHOMOandLUMOchargedensitiesofanisolatedPnsimilart oLeeetal.’sresult.31Ourresults
+also agree well with the recent differential conductance im ages obtained with low-temperature
+STM experiments for seemingly similar physisorption syste m of pentacene/Au(111).51These
+STMimagesin[figure][5][]5areconsistentandarealsoappa rentfromthePDOSof1MLPn/Ag(111)
+presented in[figure][6][]6. Thefirst PDOS peak ofthePn laye rabout 0.5 eV belowtheFermi en-
+ergy comes from the HOMO’s of the molecules. The sharpness of this peak substantiates that the
+frontierorbitalsofthePnmoleculesmixesveryweaklywith the5sstatesofthesurfaceAgatoms.
+Hence, STM calculation coveringan energy range of0.9 eV bel owthe Fermi level showsHOMO
+14charge density for Pn layer over the slab as shown in [figure][ 5][]5 where the small Ag 5 scontri-
+bution was suppressed for visual convenience. Similarly, t he first peak due to Pn layer about 0.7
+eV above the Fermi energy corresponds to the LUMO’s of the mol ecules which are also weakly
+mixingwiththevalencebands of Ag(111). At thispoint, in or derto commenton thereliabilityof
+our method, we compare the experimental and theoretical res ults of Pn–Cu(100). Ferretti et al.,36
+on thissystem, have found a significant broadening of pentac ene HOMO–LUMO bands and mix-
+ing with the substrate levels using the same computational p rocedure. This, in combination with
+theexperimentalresults,wereinterpretedasaninteracti onclosetochemisorption. InPn–Ag(111)
+case, however,itisclear fromourresultsthatthepicturei smorecloseto physisorption.
+-3.0 -2.0 -1.0 0.0 1.0 2.0Flat Pn layerAg(111)1ML Pn/Ag(111)2ML Pn/Ag(111)3ML Pn/Ag(111)4ML Pn/Ag(111)Pn bulk
+-3.0 -2.0 -1.0 0.0 1.0 2.0Flat Pn layerAg(111)1ML Pn/Ag(111)2ML Pn/Ag(111)3ML Pn/Ag(111)4ML Pn/Ag(111)Pn bulk
+-3.0 -2.0 -1.0 0.0 1.0 2.0Flat Pn layerAg(111)1ML Pn/Ag(111)2ML Pn/Ag(111)3ML Pn/Ag(111)4ML Pn/Ag(111)Pn bulk
+-3.0 -2.0 -1.0 0.0 1.0 2.0Flat Pn layerAg(111)1ML Pn/Ag(111)2ML Pn/Ag(111)3ML Pn/Ag(111)4ML Pn/Ag(111)Pn bulk
+-3.0 -2.0 -1.0 0.0 1.0 2.0Flat Pn layerAg(111)1ML Pn/Ag(111)2ML Pn/Ag(111)3ML Pn/Ag(111)4ML Pn/Ag(111)Pn bulk
+-3.0 -2.0 -1.0 0.0 1.0 2.0Flat Pn layerAg(111)1ML Pn/Ag(111)2ML Pn/Ag(111)3ML Pn/Ag(111)4ML Pn/Ag(111)Pn bulk
+-3.0 -2.0 -1.0 0.0 1.0 2.0Flat Pn layerAg(111)1ML Pn/Ag(111)2ML Pn/Ag(111)3ML Pn/Ag(111)4ML Pn/Ag(111)Pn bulk
+Figure6: (coloronline)CalculatedPDOSforPn/Ag(111)str uctures. Theabscissaistheenergy,in
+eV,relativetotheFermilevelforthecleanAg(111)surface . Pentacenecontributionsareindicated
+by gray(red) whiletotalDOSis indark gray (blue).
+Thebottompanelof[figure][6][]6showsthecalculatedDOSf ortheflatpentacenelayerwhich
+is obtained by removingthe silversubstratefrom 1ML/Ag(11 1)([figure][3][]3a–c) structure. The
+sharppeaks,havinglessstructure,ratherlooklikeanener gyleveldiagramduetoverylowoverlap
+between molecularorbitals through tip to tip pentacene con tacts overthe layer. Metallicnature of
+15the bare Ag(111) surface is also presented in the succeeding panel of [figure][6][]6 where Fermi
+level is set as the origin of the energy axis. The DOS structur e corresponding to full pentacene
+monolayer on Ag(111) is shown in the third panel from the bott om in [figure][6][]6. DOS peaks
+stemming from pentacene show no shift in energy with respect to those of the flat Pn layer in
+the absence of the metal substrate. In addition, the broaden ing of each peak is localized over a
+small number of silver states indicating weak semiconducto r–metal coupling. As a result of this
+weakinteraction,thecontactlayershowsbulk-likeHOMO–L UMOcontributionstothetotalDOS
+around theFermi energy. Therefore, STM experimentscaptur ethesefrontiermolecularorbitals.
+As new pentacene layers deposited on the first full monolayer the corresponding PDOS con-
+tribution starts to form localized satellite structures at around flat Pn layer peak positions. Their
+broadening is larger than the broadening in PDOS peaks obtai ned for 1ML/Ag(111). This indi-
+cates that the interlayer molecular orbital overlap is rela tively stronger than the coupling between
+thecontactlayerandthemetalsubstrate. Moreover,theseP DOSsatellitesmatchperfectlywiththe
+bulkpentaceneDOSwhichispresentedinthetoppanelof[figu re][6][]6. Therefore, energetically
+preferable thinfilmpentacenephaseon Ag(111)up to4MLposs essesbulk-likeDOSproperties.
+Our DOS calculationsshow that pentacene has no electronicc ontribution at the Fermi energy.
+Evidently, highly ordered pentacene multilayers on Ag(111 ), considered in this study, does not
+exhibitbandtransport. Inaddition,thesemultilayersocc urinbulk-likephasewheretheoverlapof
+themolecularorbitalsbetweennearestneighborpentacene syieldlarge π-conjugationlengthalong
+the molecular axis. Therefore, our results suggest a hoppin g mechanism between the localized
+statesforthecarrier transport.
+Conclusion
+Inconclusion,wehaveinvestigatedthegeometricandelect ronicstructureofpentaceneonAg(111)
+surfaceupto 4MLcoverageat theDFT levelwhereGGA function alsperform better. Atthemost
+stable configuration a single isolated pentacene lies flat at 3.9 Å above Ag(111) surface on the
+16so called Bridge-60 position with a weak binding energy of −0.155 eV. For the full monolayer
+coverage,moleculesalignperfectlywiththesilverrowson thesurfacewhiletheMLheightextends
+slightly to 4.1 Å due to intermolecular interactions. Calcu lated binding energies as well as STM
+and PDOSstructures indicateweak pentacene–substratecou pling.
+Pentacenes abovethecontactlayerfavorthethinfilmmultil ayerstructureovertheplanarcon-
+figuration with a slight energy difference which can be addre ssed to the small energy barriers
+betweendifferentphasesofpentacene. Moreover,thesligh tmisalignmentofpentacenemolecules
+above the first layer from the surface silver rows indicate th at a bulk-like thin film phase starting
+from thesecond layerisadsorbed ontheAg(111)surface thro ughacontact layerattheinterface.
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+21
\ No newline at end of file
diff --git a/1001.0107.txt b/1001.0107.txt
new file mode 100644
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+++ b/1001.0107.txt
@@ -0,0 +1,2322 @@
+arXiv:1001.0107v2  [cs.IT]  15 Apr 20101
+Exact Regeneration Codes for Distributed
+Storage Repair Using Interference Alignment
+Changho Suh and Kannan Ramchandran
+Wireless Foundations, Department of EECS
+University of California at Berkeley
+Email:{chsuh, kannanr }@eecs.berkeley.edu
+Abstract
+The high repair cost of (n,k)Maximum Distance Separable (MDS) erasure codes has recentl y
+motivateda newclassofcodes,called RegeneratingCodes ,thatoptimallytradeoffstoragecostforrepair
+bandwidth. On one end of this spectrum of Regenerating Codes are Minimum Storage Regenerating
+(MSR) codes that can match the minimum storage cost of MDS cod es while also significantly reducing
+repair bandwidth. In this paper, we describe Exact-MSR codes which allow for anyfailed nodes
+(whether they are systematic or parity nodes ) to be regenerated exactlyrather than only functionally or
+information-equivalently.We showthat Exact-MSRcodes comewith nolossof optimality with respectto
+random-network-codingbased MSR codes (matchingthe cutse t-based lower boundon repair bandwidth)
+for the cases of: (a)k/n≤1/2; and(b)k≤3. Our constructive approach is based on interference
+alignment techniques, and, unlike the previous class of random-netwo rk-coding based approaches, we
+provide explicit and deterministic coding schemes that req uire a finite-field size of at most2(n−k).
+Index Terms
+Interference Alignment, Minimum Storage Regenerating (MS R) Codes, Repair Bandwidth
+I. INTRODUCTION
+In distributed storage systems, maximum distance separabl e (MDS) erasure codes are well-
+known coding schemes that can offer maximum reliability for a given storage overhead. For an
+(n,k)MDS code for storage, a source file of size Mbits is divided equally into kunits (of
+sizeM
+kbits each), and these kdata units are expanded into nencoded units, and stored at n
+October 29, 2018 DRAFT2
+nodes. The code guarantees that a user or Data Collector (DC) can reconstruct the source file by
+connecting to any arbitrary knodes. In other words, any (n−k)node failures can be tolerated
+with a minimumstorage cost ofM
+kat each of nnodes. While MDS codes are optimal in terms of
+reliability versus storage overhead, they come with a signi ficant maintenance overhead when it
+comes to repairing failed encoded nodes to restore the MDS sy stem-wide property. Specifically,
+consider failure of a single encoded node and the cost needed to restore this node. It can be
+shown that this repair incurs an aggregate cost of Mbits of information from knodes. Since
+each encoded unit contains onlyM
+kbits of information, this represents a k-fold inefficiency with
+respect to the repair bandwidth.
+This challenge has motivated a new class of coding schemes, c alledRegenerating Codes [1],
+[2], which target the information-theoreticoptimal trade off between storage cost and repair band-
+width. On one end of this spectrum of Regenerating Codes are M inimum Storage Regenerating
+(MSR) codes that can match the minimum storage cost of MDS cod es while also significantly
+reducing repair bandwidth. As shown in [1], [2], the fundame ntal tradeoff between bandwidth
+and storage depends on the number of nodes that are connected to repair a failed node, simply
+called the degee dwherek≤d≤n−1. The optimal tradeoff is characterized by
+(α,γ) =/parenleftbiggM
+k,M
+k·d
+d−k+1,/parenrightbigg
+, (1)
+whereαandγdenote the optimal storage cost and repair bandwidth, respe ctively for repairing
+a single failed node, while retaining the MDS-code property for the user. Note that this code
+requires the same minimal storage cost (of sizeM
+k) as that of conventional MDS codes, while
+substantially reducing repair bandwidth by a factor ofk(d−k+1)
+d(e.g., for (n,k,d) = (31,6,30),
+there is a 5x bandwidth reduction). In this paper, without loss of gener ality, we normalize the
+repair-bandwidth-per-link (γ
+d)to be 1, making M=k(d−k+ 1). One can partition a whole
+file into smaller chunks so that each has a size of k(d−k+1)1.
+WhileMSRcodes enjoysubstantialbenefitsoverMDScodes,th eycomewithsomelimitations
+in construction. Specifically, the achievable schemes in [1 ], [2] that meet the optimal tradeoff
+bound of (1) restore failed nodes in a functional manner only, using a random-network-coding
+based framework. This means that the replacement nodes main tain the MDS-code property (that
+1In practice, the order of a file size is of 103(Kb)∼109(Gb). Hence, it is reasonable to consider this arbitrary siz e of the
+chunk.
+October 29, 2018 DRAFT3
+anykout ofnnodes can allow for the data to be reconstructed) but do not exactlyreplicate the
+information content of the failed nodes.
+Mere functional repair can be limiting.First, in many appli cations of interest, there is a need to
+maintain the code in systematic form, i.e., where the user data in the form of kinformation units
+are exactly stored at knodes and parity information (mixtures of kinformation units) are stored
+attheremaining (n−k)nodes.Secondly,underfunctionalrepair, additionalover headinformation
+needs to be exchanged for continually updating repairing-and-decoding rules whenever a failure
+occurs. This can significantly increase system overhead. A t hird problem is that the random-
+network-coding based solution of [1] can require a huge finit e-field size, which can significantly
+increase the computational complexity of encoding-and-de coding2. Lastly, functional repair is
+undesirable in storage security applications in the face of eavesdroppers. In this case, information
+leakage occurs continually due to the dynamics of repairing -and-decoding rules that can be
+potentially observed by eavesdroppers [3].
+These drawbacks motivatethe need for exactrepair of failed nodes. This leads to the following
+question: is there a price for attaining the optimal tradeof f of (1) with the extra constraint of
+exact repair? The work in [4] sheds some light on this questio n: specifically, it was shown that
+underscalar linear codes3, whenk
+n>1
+2+2
+n, thereisa price for exact repair. For large n, this
+case boils down tok
+n>1
+2, i.e., redundancy less than two. Now what about fork
+n≤1
+2? This paper
+resolves this open problem and shows that it is indeed possible to attain the optimal tradeo ff of
+(1) for the case ofk
+n≤1
+2(andd≥2k−1), while also guaranteeing exact repair . Furthermore,
+we show that for the special case of k≤3, there is no price for exact repair, regardless of the
+value ofn. The interesting special case in this class is the (5,3)Exact-MSR code4, which is not
+covered by the first case ofk
+n≤1
+2.
+Our achievable scheme builds on the concept of interference alignment , which was introduced
+2In [1], Dimakis-Godfrey-Wu-Wainwright-Ramchandran tran slated the regenerating-codes problem into a multicast com muni-
+cation problem where random-network-coding-based scheme s require a huge field size especially for large networks. In s torage
+problems, the field size issue is further aggravated by the ne ed to support a dynamically expanding network size due to the need
+for continual repair.
+3In scalar linear codes, symbols are not allowed to be split in to arbitrarily small sub-symbols as with vector linear code s.
+This is equivalent to having large block-lengths in the clas sical setting. Under non-linear and vector linear codes, wh ether or
+not the optimal tradeoff can be achieved for this regime rema ins open.
+4Independently, Cullina-Dimakis-Ho in [5] found (5,3)E-MSR codes defined over GF(3), based on a search algorithm.
+October 29, 2018 DRAFT4
+in the context of wireless communication networks [6], [7]. The idea of interference alignment
+is to align multiple interference signals in a signal subspa ce whose dimension is smaller than the
+number of interferers. Specifically, consider the followin g setup where a decoder has to decode
+one desired signal which is linearly interfered with by two s eparate undesired signals. How
+many linear equations (relating to the number of channel use s) does the decoder need to recover
+its desired input signal? As the aggregate signal dimension spanned by desired and undesired
+signals is at most three, the decoder can naively recover its signal of interest with access to
+three linearly independent equations in the three unknown s ignals. However, as the decoder is
+interested in only one of the three signals, it can decode its desired unknown signal even if it
+has access to only two equations, provided the two undesired signals are judiciously aligned in
+a 1-dimensional subspace. See [6], [7], [8] for details.
+We will show in the sequel how this concept relates intimatel y to our repair problem. At a
+high level, the connection comes from our repair problem inv olving recovery of a subset (related
+to the subspace spanned by a failed node) of the overall aggre gate signal space (related to the
+entire user data dimension). There are, however, significan t differences some beneficial and some
+detrimental. On the positive side, while in the wireless pro blem, the equations are provided by
+nature(in the form of channel gain coefficients), in our repair prob lem, the coefficients of the
+equations are man-made choices, representing a part of the overall design space. On the flip side,
+however, the MDS requirement of our storage code and the mult iple failure configurations that
+need to be simultaneously addressed with a single code desig n generate multiple interference
+alignment constraints that need to be simultaneously satis fied. This is particularly acute for
+a large value of k, as the number of possible failure configurations increases withn(which
+increases with k). Finally, another difference comes from the finite-field co nstraint of our repair
+problem.
+We propose a common-eigenvector based conceptual framework (explained in Section IV) that
+covers all possible failure configurations. Based on this fr amework, we develop an interference
+alignment design technique for exact repair. We also propos e another interference alignment
+scheme for a (5,3)code5, which in turn shows the optimality of the cutset bound (1) fo r the
+casek≤3. As in [4], our coding schemes are deterministic and require a field size of at most
+5The finite-field nature of the problem makes this challenging .
+October 29, 2018 DRAFT5
+Functionalrepair
+Partiallyexactrepair
+Exact repair
+ 
+Fig. 1. Repair models for distributed storage systems. In ex act repair, the failed nodes are exactly regenerated, thus r estoring
+lost encoded fragments with their exact replicas. In functi onal repair, the requirement is relaxed: the newly generate d node can
+contain different data from that of the failed node as long as the repaired system maintains the MDS-code property. In par tially
+exact repair, only systematic nodes are repaired exactly, w hile parity nodes are repaired only functionally.
+2(n−k). This is in stark contrast to the random-network-coding bas ed solutions [1].
+II. CONNECTION TO RELATEDWORK
+As stated earlier, Regenerating Codes, which cover an entir e spectrum of optimal tradeoffs
+between repair bandwidth and storage cost, were introduced in [1], [2]. As discussed, MSR
+codes occupy one end of this spectrum corresponding to minim um storage. At the other end of
+the spectrum live Minimum Bandwidth Regenerating (MBR) cod es corresponding to minimum
+repair bandwidth. The optimal tradeoffs described in [1], [ 2] are based on random-network-
+coding based approaches, which guarantee only functional r epair.
+The topic of exact repair codes has received attention in the recent literature [9], [10], [4], [5],
+[11]. Wu and Dimakis in [9] showed that the MSR point (1) can be attained for the cases of:
+k= 2andk=n−1. Rashmi-Shah-Kumar-Ramchandran in [10] showed that for d=n−1, the
+optimalMBRpointcan beachieved withadeterministicschem erequiring asmallfinite-field size
+and zero repair-coding-cost. Subsequently, Shah-Rashmi- Kumar-Ramchandran in [4] developed
+partially exact codes for the MSR point corresponding tok
+n≤1
+2+2
+n, where exact repair is limited
+to the systematic component of the code. See Fig. 1. Finding t he fundamental limits under exact
+repair ofallnodes (including parity) remained an open problem. A key con tribution of this paper
+is to resolve this open problem by showing that E-MSR codes co me with no extra cost over
+the optimal tradeoff of (1) for the case ofk
+n≤1
+2(andd≥2k−1). For the most general case,
+October 29, 2018 DRAFT6
+finding the fundamental limits under exact repair constrain ts for all values of (n,k,d)remains
+an open problem.
+The constructive framework proposed in [4] forms the inspir ation for our proposed solution in
+this paper. Indeed, we show that the code introduced in [4] fo r exact repair of only the systematic
+nodes can also be used to repair the non-systematic (parity) node failures exactly provided repair
+construction schemes are appropriately designed . This design for ensuring exact repair of all
+nodes is challenging and had remained an open problem: resol ving this for the case ofk
+n≤1
+2
+(andd≥2k−1) is a key contribution of this work. Another contribution of our work is the
+systematic development of a generalized family of code stru ctures (of which the code structure
+of [4] is a special case), together with the associated optim al repair construction schemes. This
+generalized family of codes provides conceptual insights i nto the structure of solutions for the
+exact repair problem, whilealso opening up a much larger con structivedesign space of solutions.
+III. INTERFERENCE ALIGNMENT FOR DISTRIBUTED STORAGE REPAIR
+Linear network coding [12], [13] (that allows multiple mess ages to be linearly combined at
+network nodes) has been established recently as a useful too l for addressing interference issues
+even in wireline networks where all the communication links are orthogonal and non-interfering.
+This attribute was first observed in [9], where it was shown th at interference alignment could
+be exploited for storage networks, specifically for minimum storage regenerating (MSR) codes
+having small k(k= 2). However, generalizing interference alignment to large v alues ofk(even
+k= 3) proves to be challenging, as we describe in the sequel. In or der to appreciate this better,
+let us first review the scheme of [9] that was applied to the exa ct repair problem. We will then
+address the difficulty of extending interference alignment for larger systems and describe how
+to address this in Section IV.
+A. Review of (4,2)E-MSR Codes [9]
+Fig. 2 illustrates an interference alignment scheme for a (4,2)MDS code defined over GF(5).
+First one can easily check the MDS property of the code, i.e., all the source files can be
+reconstructed from any k(= 2)nodes out of n(= 4)nodes. Let us see how failed node 1
+(storing(a1,a2)) can be exactly repaired. We assume that the degree d(the number of storage
+October 29, 2018 DRAFT7
+a1+b11
+1b2b1 1
+1a2a1
+a1
+a2
+b1
+b2a1
+a2
+b1+b2
+12a1+a2+b1+b22a2+b2
+2a1+b1
+a2+b21a1+ 2a2+b1+b2AsourcefileEncoded packets How to repair?
+Interference alignmentfailed node1
+node2
+node3
+node4(parity node 1)
+(parity node 2)
+ 
+Fig. 2. Interference alignment for a (4,2)E-MSR code defined over GF(5)[9]. Choosing appropriate projection weights, we
+can align interference space of (b1,b2)into one-dimensional linear space spanned by [1,1]t. As a result, we can successfully
+decode 2 desired unknowns (a1,a2)from 3 equations containing 4 unknowns (a1,a2,b1,b2).
+nodes connected to repair a failed node) is 3, and a source file size Mis4. The cutset bound (1)
+then gives the fundamental limits of: storage cost α= 2; and repair-bandwidth-per-linkγ
+d= 1.
+The example illustrated in Fig. 2 shows that the parameter se t described above is achievable
+using interference alignment. Here is a summary of the schem e. First notice that since the
+bandwidth-per-link is 1, two symbols in each storage node ar e projected into a scalar variable
+with projection weights. Choosing appropriate weights, we get the equations as shown in Fig.
+2:(b1+b2);a1+2a2+(b1+b2);2a1+a2+(b1+b2). Observe that the undesired signals (b1,b2)
+(interference) are aligned onto an 1-dimensional linear su bspace, thereby achieving interference
+alignment . Therefore, we can successfully decode (a1,a2)with three equations although there
+are four unknowns. Similarly, we can repair (b1,b2)when it has failed.
+B. Matrix Notation
+We introduce matrix notation that provides geometric inter pretation of interference alignment
+and isuseful forgeneralization. Let a= (a1,a2)tandb= (b1,b2)tbe2-dimensionalinformation-
+unit vectors, where (·)tindicates a transpose. Let AiandBibe2-by-2encoding matrices for
+parity node i(i= 1,2), which contain encoding coefficients for the linear combin ation of the
+October 29, 2018 DRAFT8
+A1B−1
+1vα1
+A2B−1
+1vα1
+rank

+A1B−1
+1vα1A2B−1
+2vα1
+=2vα1B1vα2B2vα3
+vα2=B−1
+1vα1
+vα3=B−1
+2vα1vα1
+atA1+btB1
+atA2+btB2at
+btvα2
+vα3 InterferencealignmentProjection
+Vectorsnode1
+node2
+node3
+node4(paritynode1 )
+(paritynode2 )
+vt
+α1b
+(A1vα2)ta+(B1vα2)tb
+(A2vα3)ta+(B2vα3)tb
+=
+0
+(A1vα2)t
+(A2vα3)t
+a+
+vt
+α1
+(B1vα2)t
+(B2vα3)t
+b
+ 
+Fig. 3. Geometric interpretation of interference alignmen t. The blue solid-line and red dashed-line vectors indicate linear
+subspaces with respect to “ a” and “b”, respectively. The choice of vα2=B−1
+1vα1andvα3=B−1
+2vα1enables interference
+alignment. For the specific example of Fig. 2, the correspond ing encoding matrices are A1= [1,0;0,2],B1= [1,0;0,1].
+A2= [2,0;0,1],B2= [1,0;0,1].
+components of “ a” and “b”. For example, parity node 1 stores information in the form o f
+atA1+btB1, as shown in Fig. 3. The encoding matrices for systematic nod es are not explicitly
+defined since thoseare triviallyinferred. Finally we define 2-dimensionalprojection vectors vαi’s
+(i= 1,2,3).
+Let us consider exact repair of systematic node 1. By connect ing to three nodes, we get:
+btvα1;at(A1vα2)+bt(B1vα2);at(A2vα3)+bt(B2vα3). Recall the goal, which is to decode 2
+desired unknowns out of 3 equations including 4 unknowns. To achieve this goal, we need:
+rank
+
+(A1vα2)t
+(A2vα3)t
+
+= 2;rank
+
+vt
+α1
+(B1vα2)t
+(B2vα3)t
+
+= 1. (2)
+The second condition can be met by setting vα2=B−1
+1vα1andvα3=B−1
+2vα1. This choice
+forces the interference space to be collapsed into a one-dim ensional linear subspace, thereby
+achieving interference alignment. With this setting, the fi rst condition now becomes
+rank/parenleftbig/bracketleftbig
+A1B−1
+1vα1A2B−1
+2vα1/bracketrightbig/parenrightbig
+= 2. (3)
+It can be easily verified that the choice of Ai’s andBi’s given in Figs. 2 and 3 guarantees the
+above condition. When the node 2 fails, we get a similar condi tion:
+rank/parenleftbig/bracketleftbig
+B1A−1
+1vβ1B2A−1
+2vβ1/bracketrightbig/parenrightbig
+= 2, (4)
+October 29, 2018 DRAFT9
+wherevβi’s denote projection vectors for node 2 repair. This conditi on also holds under the
+given choice of encoding matrices.
+C. Connection with Interference Channels in Communication Problems
+Observe the three equations shown in Fig. 3:
+
+0
+(A1vα2)t
+(A2vα3)t
+a
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+desired signals+
+vt
+α1
+(B1vα2)t
+(B2vα3)t
+b
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+interference.
+Separating into two parts, we can view this problem as a wirel ess communication problem,
+wherein a subset of the information is desired to be decoded i n the presence of interference.
+Note that for each term (e.g., A1vα2), the matrix A1and vector vα2correspond to channel
+matrix and transmission vector in wireless communication p roblems, respectively.
+There are, however, significant differences. In the wireles s communication problem, the chan-
+nel matrices are provided by nature and therefore not controllable. The transmission strategy
+alone (vector variables) can be controlled for achieving in terference alignment. On the other
+hand, in our storage repair problems, both matrices and vect ors are controllable, i.e., projection
+vectors and encoding matrices can be arbitrarily designed, resulting in more flexibility. However,
+our storage repair problem comes with unparalleled challen ges due to the MDS requirement and
+the multiple failure configurations. These induce multiple interference alignment constraints that
+need to be simultaneously satisfied. What makes this difficul t is that the encoding matrices, once
+designed, must be the same for allrepair configurations. This is particularly acute for large
+values of k(evenk= 3), as the number of possible failure configurations increase s withn
+(which increases with k).
+IV. A P ROPOSED FRAMEWORK FOR EXACT-MSR C ODES
+We propose a common-eigenvector based conceptual framework to address the exact repair
+problem. This framework draws its inspiration from the work in [4] which guarantees the exact
+repair of systematic nodes, while satisfying the MDS code pr operty, but which does not provide
+exact repair of failed parity nodes. In providing a solution for the exact repair of allnodes,
+we propose here a generalized family of codes (of which the co de in [4] is a special case).
+October 29, 2018 DRAFT10
+vα1
+vα2
+vα3
+vα4
+vα5Interferencealignment
+vα3=B−1
+1vα1
+vα4=B−1
+2vα1
+vα5=B−1
+3vα1at
+bt
+ct
+(C1B−1
+1)vα1
+(C2B−1
+2)vα1(C3B−1
+3)vα1vα2Goal:  rank=3  rank=1      rank=1 
+atA1+btB1+ctC1
+atA2+btB2+ctC2
+atA3+btB3+ctC3
+0
+0
+(A1vα3)t
+(A2vα4)t
+(A3vα5)t
+a+
+vt
+α1
+0
+(B1vα3)t
+(B2vα4)t
+(B3vα5)t
+b+
+0
+vt
+α2
+(C1vα3)t
+(C2vα4)t
+(C3vα5)t
+c
+ 
+Fig. 4. Difficulty of achieving interference alignment simu ltaneously.
+This both provides insights into the structure of codes for e xact repair of all nodes, as well
+as opens up a much larger design space for constructive solut ions. Specifically, we propose a
+common-eigenvector based approach building on a certain elementary matrix property [14], [15]
+for the generalized code construction. Moreover, as in [4], our proposed coding schemes are
+deterministic and constructive, requiring a symbol alphab et-size of at most (2n−2k).
+Our framework consists of four components: (1) developinga family of codes6for exact repair
+of systematic codes based on the common-eigenvector concep t; (2) drawing a dualrelationship
+between the systematic and parity node repair; (3) guarante eing the MDS property of the code;
+(4) constructing codes with finite-field alphabets. The fram ework covers the case of n≥2k(and
+d≥2k−1). It turns out that the (2k,k,2k−1)code case contains the key design ingredients
+and the case of n≥2kcan be derived from this (see Section VI). Hence, we first focu s on
+the simplest example: (6,3,5)E-MSR codes. Later in Section VI, we will generalize this to
+arbitrary (n,k,d)codes in the class.
+6Interestingly, the structure of the code in [4] turns out to w ork for the exact repair of both systematic and parity nodes
+provided appropriate repair schemes are developed.
+October 29, 2018 DRAFT11
+A. Code Structure for Systematic Node Repair
+Fork≥3(more-than-two interfering information units), achievin g interference alignment for
+exact repair turns out to be significantly more complex than t hek= 2case. Fig. 4 illustrates this
+difficulty through the example of repairing node 1 for a (6,3,5)code. By the optimal tradeoff
+(1), the choice of M= 9givesα= 3andγ
+d= 1. Leta= (a1,a2,a3)t,b= (b1,b2,b3)tand
+c= (c1,c2,c3)t. We define 3-by-3 encoding matrices of Ai,BiandCi(fori= 1,2,3); and
+3-dimensional projection vectors vαi’s.
+Consider the 5 (=d)equations downloaded from the nodes:
+
+0
+0
+(A1vα3)t
+(A2vα4)t
+(A3vα5)t
+a+
+vt
+α1
+0
+(B1vα3)t
+(B2vα4)t
+(B3vα5)t
+b+
+0
+vt
+α2
+(C1vα3)t
+(C2vα4)t
+(C3vα5)t
+c.
+In order to successfully recover the desired signal compone nts of “a”, the matrices associated
+withbandcshouldhaverank 1, respectively,whilethematrix associat ed withashouldhave full
+rank of 3. In accordance with the (4,2)code examplein Fig. 3, if one were to set vα3=B−1
+1vα1,
+vα4=B−1
+2vα2andvα5=B−1
+3vα1, then it is possible to achieve interference alignment with
+respect to b. However, this choice also specifies the interference space ofc. If theBi’s and
+Ci’s are not designed judiciously, interference alignment is not guaranteed for c. Hence, it is
+not evident how to achieve interference alignment at the same time .
+In order to address the challenge of simultaneous interfere nce alignment, we invoke a common
+eigenvector concept. The idea consists of two parts: (i) designing the (Ai,Bi,Ci)’s such that v1
+is a common eigenvector of the Bi’s andCi’s, but not of Ai’s7; (ii) repairing by having survivor
+nodesprojecttheir data onto a linear subspace spanned by this common eige nvectorv1. We
+can then achieve interference alignment for bandcat the same time , by setting vαi=v1,∀i.
+As long as [A1v1,A2v1,A3v1]is invertible, we can also guarantee the decodability of a. See
+Fig. 5.
+7Of course, five additional constraints also need to be satisfi ed for the other five failure configurations for this (6,3,5)code
+example.
+October 29, 2018 DRAFT12
+Goal:  rank=3      rank=1      rank=1 at
+bt
+ctv1
+atA1+btB1+ctC1
+atA2+btB2+ctC2
+atA3+btB3+ctC3
+Idea :
+0
+0
+(A1v1)t
+(A2v1)t
+(A3v1)t
+a+
+vt
+1
+0
+(B1v1)t
+(B2v1)t
+(B3v1)t
+b+
+0
+vt
+1
+(C1v1)t
+(C2v1)t
+(C3v1)t
+cv1
+v1
+v1
+v1
+A1v1
+v1 v1
+A2v1A3v1
+(i) DesignAi’s,Bi’s andCi’s s.t.v1is a common eigenvector of
+theBi’s andCi’s, but not of the Ai’s.
+(ii) Repair by having survivor nodes project their data onto a linearnode1
+node2
+node3
+node4
+node5
+node6(parity node 1 )
+(parity node 2 )
+(parity node 3 )
+subspace spanned by this common eigenvector v1. 
+Fig. 5. Illustration of exact repair of systematic node 1 for (6,3,5)E-MSR codes. The idea consists of two parts: (i) designing
+(Ai,Bi,Ci)’s such that v1is a common eigenvector of the Bi’s andCi’s, but not of Ai’s; (ii) repairing by having survivor
+nodes project their data onto a linear subspace spanned by th is common eigenvector v1.
+The challenge is now to design encoding matrices to guarante e the existence of a common
+eigenvector while also satisfying the decodability of desi red signals. The difficulty comes from
+the fact that in our (6,3,5)code example, these constraints need to be satisfied for allsix
+possible failure configurations. The structure of elementary matrices [14], [15] (generalized
+matrices of Householder and Gauss matrices) gives insights into this. To see this, consider a
+3-by-3 elementary matrix A:
+A=uvt+αI, (5)
+whereuandvare 3-dimensional vectors. Here is an observation that moti vates our proposed
+structure: the dimension of the null space of vis 2 and the null vector v⊥is an eigenvector of
+October 29, 2018 DRAFT13
+A, i.e.,Av⊥=αv⊥. This motivates the following structure:
+A1=u1vt
+1+α1I;B1=u1vt
+2+β1I;C1=u1vt
+3+γ1I
+A2=u2vt
+1+α2I;B2=u2vt
+2+β2I;C2=u2vt
+3+γ2I
+A3=u3vt
+1+α3I;B3=u3vt
+2+β3I;C3=u3vt
+3+γ3I,(6)
+wherevi’s are 3-dimensional linearly independent vectors and so ar eui’s. The values of the
+αi’s,βi’s andγi’s can be arbitrary non-zero values. First consider the simp le case where the vi’s
+areorthonormal . This is for conceptual simplicity. Later we will generaliz e to the case where
+thevi’s need not be orthogonal but only linearly independent: nam ely,bi-orthogonal case. For
+the orthogonal case, we see that for i= 1,2,3,
+Aiv1=αiv1+ui,
+Biv1=βiv1,
+Civ1=γiv1.(7)
+Importantly, notice that v1is a common eigenvector of the Bi’s andCi’s, while simultaneously
+ensuring that the vectors of Aiv1are linearly independent. Hence, setting vαi=v1for alli, it is
+possible to achieve simultaneous interference alignment w hile also guaranteeing the decodability
+of the desired signals. See Fig. 5. On the other hand, this str ucture also guarantees exact repair
+forbandc. We use v2for exact repair of b. It is a common eigenvector of the Ci’s andAi’s,
+while ensuring [B1v2,B2v2,B3v2]invertible. Similarly, v3is used for c.
+We will see that a dual basis property gives insights into the general bi-orthogonal case where
+{v}:= (v1,v2,v3)is not orthogonal but linearly independent. In this case, de fining a dual basis
+{v′}:= (v′
+1,v′
+2,v′
+3)gives the solution:
+v′t
+1
+v′t
+2
+v′t
+3
+:=/bracketleftig
+v1v2v3/bracketrightig−1
+.
+The definition gives the following property: v′t
+ivj=δ(i−j),∀i,j.Using this property, one can
+see thatv′
+1is a common eigenvector of the Bi’s andCi’s:
+Aiv′
+1=αiv′
+1+ui,
+Biv′
+1=βiv′
+1,
+Civ′
+1=γiv′
+1.(8)
+October 29, 2018 DRAFT14
+So it can be used as a projection vector for exact repair of a. Similarly, we can use v′
+2andv′
+3
+for exact repair of bandc, respectively.
+B. Dual Relationship between Systematic and Parity Node Rep air
+We have seen so far how to ensure exact repair of the systemati c nodes. We have known
+that if{v}is linearly independent and so {u}is, then using the code-structure of (6) together
+withprojection direction enables repair, for arbitrary values of (αi,βi,γi)’s. A natural question
+is now: will this code structure also guarantee exact repair of parity nodes? It turns out that
+for exact repair of all nodes, we need a special relationship between{v}and{u}through the
+correct choice of the (αi,βi,γi)’s.
+We will show that parity nodes can be repaired by drawing a dualrelationship with systematic
+nodes. The procedure has two steps. The first is to remap parit y nodes with a′,b′, andc′,
+respectively:
+a′
+b′
+c′
+:=
+At
+1Bt
+1Ct
+1
+At
+2Bt
+2Ct
+2
+At
+3Bt
+3Ct
+3
+
+a
+b
+c
+.
+Systematic nodes can then be rewritten in terms of the prime n otations:
+at=a′tA′
+1+b′tB′
+1+c′tC′
+1,
+bt=a′tA′
+2+b′tB′
+2+c′tC′
+2,
+ct=a′tA′
+3+b′tB′
+3+c′tC′
+3,(9)
+where the newly mapped encoding matrices (A′
+i,B′
+i,Ci)’s are defined as:
+
+A′
+1A′
+2A′
+3
+B′
+1B′
+2B′
+3
+C′
+1C′
+2C′
+3
+:=
+A1A2A3
+B1B2B3
+C1C2C3
+−1
+. (10)
+With this remapping, one can dualize the relationship betwe en systematic and parity node repair.
+Specifically, if all of the A′
+i’s,B′
+i’s, andC′
+i’s areelementary matrices and form a similar code-
+structure as in (6), exact repair of the parity nodes becomes transparent.
+The challenge is now how to guarantee the dual structure. In L emma 1, we show that a special
+relationship between {u}and{v}through ( αi,βi,γi)’s can guarantee this dual relationship of
+(13).
+October 29, 2018 DRAFT15
+a′tA′
+1+b′tB′
+1+c′tC′
+1
+A′t
+3a′+B′t
+3b′+C′t
+3c′
+Goal:  rank=3     rank=1      rank=1 u1
+b′t
+c′ta′t
+0
+0
+(A′
+1u1)t
+(A′
+2u1)t
+(A′
+3u1)t
+a′+
+ut
+1
+0
+(B′
+1u1)t
+(B′
+2u1)t
+(B′
+3u1)t
+b′+
+0
+ut
+1
+(C′
+1u1)t
+(C′
+2u1)t
+(C′
+3u1)t
+c′a′tA′
+2+b′tB′
+2+c′tC′
+2
+A′
+1u1
+u1u1
+u1
+u1
+u1
+u1
+A′
+2u1A′
+3u1
+Weprovidesufficientconditionstoensurethe dualstructure:at
+bt
+ct
+atA1+btB1+ctC1
+atA2+btB2+ctC2
+atA3+btB3+ctC3remapping
+(i)κ[u1,u2,u3]=[v′
+1,v′
+2,v′
+3]
+α1α2α3
+β1β2β3
+γ1γ2γ3
+;
+(ii)
+α1α2α3
+β1β2β3
+γ1γ2γ3
+isinvertible.
+ 
+Fig. 6. Exact repair of a parity node for (6,3)E-MSR code. The idea is to construct the dualcode-structure of (13) by
+remapping parity nodes and then adding sufficient condition s of (11) and (12).
+Lemma 1: Suppose
+M:=
+α1α2α3
+β1β2β3
+γ1γ2γ3
+is invertible . (11)
+Also assume
+κU=V′M. (12)
+whereU= [u1,u2,u3],V′= [v′
+1,v′
+2,v′
+3],{v′}:={v′
+1,v′
+2,v′
+3}is the dual basis of {v}, i.e.,
+v′t
+ivj=δ(i−j)andκis an arbitrary non-zero value s.t. 1−κ2/ne}ationslash= 0. Then, we can obtain the
+October 29, 2018 DRAFT16
+dual structure of (6) as follows:
+A′
+1=1
+1−κ2/parenleftbig
+v′
+1u′t
+1−κ2α′
+1I/parenrightbig
+;B′
+1=1
+1−κ2/parenleftbig
+v′
+1u′t
+2−κ2α′
+2I/parenrightbig
+;C′
+1=1
+1−κ2/parenleftbig
+v′
+1u′t
+3−κ2α′
+3I/parenrightbig
+A′
+2=1
+1−κ2/parenleftbig
+v′
+2u′t
+1−κ2β′
+1I/parenrightbig
+;B′
+2=1
+1−κ2/parenleftbig
+v′
+2u′t
+2−κ2β′
+2I/parenrightbig
+;C′
+2=1
+1−κ2/parenleftbig
+v′
+2u′t
+3−κ2β′
+3I/parenrightbig
+A′
+3=1
+1−κ2/parenleftbig
+v′
+3u′t
+1−κ2γ′
+1I/parenrightbig
+;B′
+3=1
+1−κ2/parenleftbig
+v′
+3u′t
+2−κ2γ′
+2I/parenrightbig
+;C′
+3=1
+1−κ2/parenleftbig
+v′
+3u′t
+3−κ2γ′
+3I/parenrightbig
+,
+(13)
+where{u′}is the dual basis of {u}, i.e.,u′t
+iuj=δ(i−j)and(α′
+i,β′
+i,γ′
+i)’s are the dual basis
+vectors, i.e., <(α′
+i,β′
+i,γ′
+i),(αj,βj,γj)>=δ(i−j):
+
+α′
+1β′
+1γ′
+1
+α′
+2β′
+2γ′
+2
+α′
+3β′
+3γ′
+3
+:=
+α1α2α3
+β1β2β3
+γ1γ2γ3
+−1
+. (14)
+Proof:See Appendix A.
+Remark 1: The dual structure of (13) now gives exact-repair solutions for parity nodes. For
+exact repair of parity node 1, we can use vector u1(a common eigenvector of the B′
+i’s and
+C′
+i’s), since it enables simultaneous interference alignment forb′andc′, while ensuring the
+decodability of a′. See Fig. 6. Notice that more conditions of (11) and (12) are a dded to ensure
+exact repair of all nodes, while these conditions were unnec essary for exact repair of systematic
+nodes only. Also note these are only sufficient conditions.
+Remark 2: Note that the dual structure of (13) is quite similar to the pr imary structure of
+(6). The only difference is that in the dual structure, {u}and{v}are interchanged to form a
+transpose-like structure. This reveals insights into how to guarantee exac t repair of parity nodes
+in a transparent manner.
+C. The MDS-Code Property
+The third part of the framework is to guarantee the MDS-code p roperty, which allows us to
+identifyspecificconstraintsonthe (αi,βi,γi)’sand/or( {v},{u}).Considerfourcases, associated
+in the Data Collector (DC) who is intended in the source file da ta: (1) 3 systematic nodes; (2)
+3 parity nodes; (3) 1 systematic and 2 parity nodes; (4) 1 syst ematic and 2 parity nodes.
+The first is a trivial case. The second case has been already ve rified in the process of forming
+the dual code-structure of (13). The invertibility conditi on of (11) together with (12) suffices to
+October 29, 2018 DRAFT17
+ensure the invertibility of the composite matrix. The third case requires the invertibility of all
+of each encoding matrix. In this case, it is necessary that th eαi’s,βi’s andγi’s are non-zero
+values; otherwise, each encoding matrix has rank 1. Also the non-zero values together with (12)
+guarantee the invertibility of each encoding matrix. Under these conditions, for example, the
+inverse of A1is well defined as:
+A−1
+1=1
+α1/parenleftbigg
+−1
+α1+vt
+1u1u1vt
+1+I/parenrightbigg
+=1
+α1/parenleftbigg
+−κ
+α1(κ+1)u1vt
+1+I/parenrightbigg
+.
+where the second equality follows from vt
+1u1=α1
+κdue to (12).
+The last case requires some non-trivial work. Consider a spe cific example where the DC
+connects to nodes (3,4,5). In this case, we first recover cfrom node 3 and subtract the terms
+associated with cfrom nodes 4 and 5. We then get:
+/bracketleftig
+atbt/bracketrightig
+A1A2
+B1B2
+=/bracketleftig
+atbt/bracketrightig
+u1vt
+1+α1I u2vt
+1+α2I
+u1vt
+2+β1I u2vt
+2+β2I
+. (15)
+Using a Gaussian elimination method, we show that the sub-co mposite matrix is invertible if
+M2:=
+α1α2
+β1β2
+is invertible . (16)
+Here is the Gaussian elimination method:
+
+u1vt
+1+α1I u2vt
+1+α2I
+u1vt
+2+β1I u2vt
+2+β2I
+(a)∼
+vt
+1+α1u′t
+1α2u′t
+1
+vt
+2+β1u′t
+1β2u′t
+1
+α1u′t
+2vt
+1+α2u′t
+2
+β1u′t
+2vt
+2+β2u′t
+2
+α1u′t
+3α2u′t
+3
+β1u′t
+3β2u′t
+3
+(b)∼
+α′
+1vt
+1+β′
+1vt
+2+u′t
+1 0t
+α′
+2vt
+1+β′
+2vt
+2 u′t
+1
+u′t
+2 α′
+1vt
+1+β′
+1vt
+2
+0tα′
+2vt
+1+β′
+2vt
+2+u′t
+2
+u′t
+3 0t
+0tu′t
+3
+.
+where(a)following from multiplying [u′t
+1,0t;0t,u′t
+1;u′t
+2,0t;0t,u′t
+2;u′t
+3,0t;0t,u′t
+3]to the left; (b)
+follows from multiplying [M−1
+2,0,0;0,M−1
+2,0;0,0,M−1
+2]to the left. Here (α′
+i,β′
+i)’s are the
+dual basis vectors of (αi,βi)’s. Note that the resulting matrix is invertible, since {u′}is a dual
+basis.
+October 29, 2018 DRAFT18
+Considering the above 4 cases, the following condition toge ther with (11) and (12) suffices
+for guaranteeing the MDS-code property:
+Any submatrix of Mof (11) is invertible. (17)
+D. Code Construction with Finite-Field Alphabets
+The last part is to design Mof (11) and {v}:= (v1,v2,v3)in (6) such that {v}is linearly
+independent and the conditions of (12) and (17) are satisfied . First, in order to guarantee (17),
+we can use a Cauchy matrix, as it was used for the code introduc ed in [4].
+Definition 1 (A Cauchy Matrix [16]): A Cauchy matrix Mis anm×nmatrix with entries
+mijin the form:
+mij=1
+xi−yj,∀i= 1,···,m,j= 1,···n,xi/ne}ationslash=yj,
+wherexiandyjare elements of a field and {xi}and{yj}are injective sequences, i.e., elements
+of the sequence are distinct.
+The injective property of {xi}and{yj}requires a finite field size of 2sfor ans×sCauchy
+matrix. Therefore, in our (6,3,5)code example, the finite field size of 6 suffices. The field size
+condition for guaranteeing linear independence of {v}is more relaxed.
+E. Summary
+Using the code structure of (6) and the conditions of (11), (1 2) and (17), we can now state
+the following theorem.
+Theorem 1 ( (6,3,5)E-MSR Codes): SupposeMof (11) is a Cauchy matrix, i.e., every sub-
+matrix of is invertible. Each element of Mis inGF(q)andq≥6. Suppose encoding matrices
+form the code structure of (6), {v}:= (v1,v2,v3)is linearly independent, and {u}satisfies the
+condition of (12). Then, the code satisfies the MDS property a nd achieves the MSR point under
+exact repair constraints of allnodes.
+Remark 3: Note that the code introduced in [4] is a special case of Theor em 1, where V=
+[v1,v2,v3] =I.
+October 29, 2018 DRAFT19
+V. EXAMPLES
+We provide two numerical examples: (1) an orthogonal code ex ample where V= [v1,v2,v3]
+is orthogonal, e.g., V=I; (2) an bi-orthogonal code example where Vis not orthogonal but
+invertible. As mentioned earlier, the code in [4] belongs to the case of V=I.
+We will also discuss the complexityof repair constructions chemes for each of these examples.
+It turns out that the first code has significantly lower comple xity for exact repair of systematic
+nodes,ascompared tothatofparitynodes.Ontheotherhand, forthesecondbi-orthogonalcodes,
+the specific choice of V=κ−1MtgivesU=I, thereby providing much simpler parity-node
+repair schemes instead. Depending on applications of inter est, one can choose an appropriate
+code among our generalized family of codes.
+A. Orthogonal Case
+We present an example of (6,3,5)E-MSR codes defined over GF(4)whereV=Iand
+M=
+1 1 1
+1 2 3
+1 3 2
+,U=κ−1V′M= 2
+1 1 1
+1 2 3
+1 3 2
+=
+2 2 2
+2 3 1
+2 1 3
+,
+whereUis set based on (12) and κ= 2−1. We use a generator polynomial of g(x) =x2+x+1.
+Notice that we employ a non-Cauchy -type matrix to construct a field-size 4 code (smaller than
+6 required when using a Cauchy matrix). Remember that a Cauch y matrix provides only a
+sufficient condition for ensuring the invertibility of any s ubmatrices of M. By (6) and (13), the
+primary and dual code structures are given by
+G=
+3 0 0 3 0 0 3 0 0
+2 1 0 3 1 0 1 1 0
+2 0 1 1 0 1 3 0 1
+1 2 0 2 2 0 3 2 0
+0 3 0 0 1 0 0 2 0
+0 2 1 0 1 2 0 3 3
+1 0 2 3 0 2 2 0 2
+0 1 2 0 3 3 0 2 1
+0 0 3 0 0 2 0 0 1
+;G−1=
+2 1 1 3 0 0 3 0 0
+0 3 0 1 2 1 0 3 0
+0 0 3 0 0 3 1 1 2
+2 3 2 2 0 0 1 0 0
+0 3 0 1 1 2 0 1 0
+0 0 3 0 0 2 1 3 3
+2 2 3 1 0 0 2 0 0
+0 3 0 1 3 3 0 2 0
+0 0 3 0 0 1 1 2 1
+.(18)
+October 29, 2018 DRAFT20
+
+1
+1
+1
+
+
+1
+1
+1
+a1
+a2
+a3
+b1
+b2
+b3
+c1
+c2
+c3
+3a1+2a2+2a3+b1+c1
+a2+2b1+3b2+2b3+c2
+a3+b3+2c1+2c2+3c3
+3a1+3a2+a3+2b1+3c1
+a2+2b1+b2+b3+3c2
+a3 +2b3+2c1+3c2+2c3
+3a1+a2+3a3+3b1+2c1
+a2+2b1+2b2+3b3+2c2
+a3+3b3+2c1+c2+1c32a′
+1 +2b′
+1 +2c′
+1
+a′
+1+3a′
+2+3b′
+1+3b′
+2+2c′
+1+3c′
+2
+a′
+1+3a′
+3+2b′
+1+3b′
+3+3c′
+1+3c′
+3
+3a′
+1+a′
+2+2b′
+1+b′
+2+c′
+1+c′
+2
+2a′
+2 +b′
+2+3c′
+2
+a′
+2+3a′
+3+2b′
+2+2b′
+3+3c′
+2+c′
+3
+3a′
+1+a′
+3+b′
+1+b′
+3+2c′
+1+c′
+3
+3a′
+2+a′
+3+b′
+2+3b′
+3+2c′
+2+2c′
+3
+2a′
+3+3b′
+3 +c′
+3
+a′
+1a′
+2
+a′
+3
+b′
+1
+b′
+2
+b′
+3
+c′
+1
+c′
+2
+c′
+3c′
+1+c′
+2+c′
+3b′
+1+b′
+2+b′
+3
+1
+1
+1
+
+
+1
+1
+1
+
+
+1
+1
+1
+2a′
+1+3a′
+2+3a′
+3
++3b′
+1+3b′
+2+3b′
+3
++3c′
+1+3c′
+2+3c′
+3
+3a′
+1+2a′
+2+3a′
+3
++2b′
+1+2b′
+2+2b′
+3
++c′
+1+c′
+2+c′
+3
+3a′
+1+3a′
+2+2a′
+3
++b′
+1+b′
+2+b′
+3
++2c′
+1+2c′
+2+2c′
+3
+(a)Exactrepairofsystematicnode 1(b)Exactrepairofparitynode1 
+Fig. 7. Orthogonal case: Illustration of exact pair for a (6,3,5)E-MSR code defined over GF(4)where a generator polynomial
+g(x) =x2+x+ 1. The projection vector solution for systematic node repair is quite simple: vαi=v1= (1,0,0)t,∀i. We
+download only the first equation from each survivor node; For parity node repair, our new framework provides a simple sche me:
+setting all of the projection vectors as 2−1u1= (1,1,1)t. This enables simultaneous interference alignment, while guaranteeing
+the decodability of a.
+where
+G:=
+A1A2A3
+B1B2B3
+C1C2C3
+;G−1=
+A′
+1A′
+2A′
+3
+B′
+1B′
+2B′
+3
+C′
+1C′
+2C′
+3
+.
+Fig. 7 shows an example for exact repair of (a)systematic node 1 and (b)parity node 1.
+Note that the projection vector solution for systematic nod e repair is quite simple: vαi=v1=
+(1,0,0)t,∀i. We download only the first equation from each survivor node. Notice that the
+downloaded five equations contain only five unknown variable s of(a1,a2,a3,b1,c1)and three
+equations associated with aare linearly independent. Hence, we can successfully recov era.
+On the other hand, exact repair of parity nodes seems non-str aightforward. However, our
+October 29, 2018 DRAFT21
+framework providesquite a simplerepair scheme: setting al l of the projection vectors as 2−1u1=
+(1,1,1)t. This enables simultaneous interference alignment, while guaranteeing the decodability
+ofa. Noticethat (b′
+1,b′
+2,b′
+3)and(c′
+1,c′
+2,c′
+3)arealignedinto b′
+1+b′
+2+b′
+3andc′
+1+c′
+2+c′
+3,respectively,
+while three equations associated with a′are linearly independent.
+As one can see, the complexity of systematic node repair is a l ittle bit lower than that of
+parity node repair, although both repair schemes are simple . Hence, one can expect that this
+orthogonal code is useful for the applications where the com plexity of systematic node repair
+needs to be significantly low.
+B. Bi-Orthogonal Case
+Weprovideanotherexampleof (6,3,5)E-MSRcodeswhere Visnotorthogonalbutinvertible.
+We use the same field size of 4, the same generator polynomial a nd the same M. Instead we
+choose non-orthogonal Vso that the complexity of parity node repair can be significan tly low.
+Our framework provides a concrete guideline for designing t his type of code. Remember that the
+projection vector solutions are u1,u2andu3for exact repair of each parity node, respectively.
+For low complexity, we can first set U=I. The condition (12) then gives the following choice:
+V=Mtκ−1=
+2 2 2
+2 3 1
+2 1 3
+,
+where we use κ= 2−1. By (6) and (13), the primary and dual code structures are giv en by
+G=
+3 2 2 1 0 0 1 0 0
+0 1 0 2 3 2 0 1 0
+0 0 1 0 0 1 2 2 3
+3 3 1 2 0 0 3 0 0
+0 1 0 2 1 1 0 3 0
+0 0 1 0 0 2 2 3 2
+3 1 3 3 0 0 2 0 0
+0 1 0 2 2 3 0 2 0
+0 0 1 0 0 3 2 1 1
+;G−1=
+2 0 0 2 0 0 2 0 0
+1 3 0 3 3 0 2 3 0
+1 0 3 2 0 3 3 0 3
+3 1 0 2 1 0 1 1 0
+0 2 0 0 1 0 0 3 0
+0 1 3 0 2 2 0 3 1
+3 0 1 1 0 1 2 0 1
+0 3 1 0 1 3 0 2 2
+0 0 2 0 0 2 0 0 1
+.(19)
+October 29, 2018 DRAFT22
+
+1
+1
+1
+
+
+1
+1
+1
+a1
+a2
+a3
+b1
+b2
+b3
+c1
+c2
+c3
+3a1 +3b1+3c1
+2a1+a2+3b1+b2+c1+c2
+2a1+a3+b1+b3+3c1+c3
+a1+2a2+2b1+2b2+3c1+ 2c2
+3a2 +b2+2c2
+2a2+a3+b2+2b3+3c2+3c3
+a1+2a3+3b1+2b3+2c1+ 2c3
+a2+2a3+3b2+3b3+2c2+c3
+3a3+2b3 +c3b1+b2+b3
+c1+c2+c3
+a′
+1a′
+2
+a′
+3
+b′
+1
+b′
+2
+b′
+3
+c′
+1
+c′
+2
+c′
+33a′
+3+b′
+3+c′
+1+2c′
+2+c′
+33a′
+2+b′
+1+3b′
+2+3b′
+3+2c′
+22a′
+1+2a′
+2+3a′
+3+b′
+1+2c′
+13a′
+2+b′
+1+b′
+2+2b′
+3+c′
+2
+3a′
+3+2b′
+3+c′
+1+3c′
+2+2c′
+32a′
+1+3a′
+2+2a′
+3+2b′
+1+c′
+13a′
+3+3b′
+3+c′
+1+c′
+2+2c′
+33a′
+2+b′
+1+2b′
+2+b′
+3+3c′
+22a′
+1+a′
+2+a′
+3+3b′
+1+3c′
+1
+
+1
+1
+1
+
+
+1
+1
+1
+
+
+1
+1
+1
+3a1+a2+a3
++b1+b2+b3
++c1+c2+c3
+a1+3a2+a3
++2b1+2b2+2b3
++3c1+3c2+3c3
+a1+a2+3a3
++3b1+3b2+3b3
++2c1+2c2+2c3
+(b)Exactrepairofparitynode1 (a)Exactrepairofsystematicnode 1 
+Fig. 8. Bi-Orthogonal case: Illustration of exact repair fo r a(6,3,5)E-MSR code defined over GF(4)where a generator
+polynomial g(x) =x2+x+1. We use U=I. For parity node repair, the solution for projection vector s is much simpler. We
+download only the first equation from each survivor node; Sys tematic node repair is a bit involved: setting all of the proj ection
+vectors as 2−1v1= (1,1,1)t.
+Notice that the matrices of (19) have exactly the transpose s tructure of the matrices of (18).
+Hence, this code of (19) is a dual solution of (18), thereby pr oviding switched projection vector
+solutions and lowering the complexity for parity node repai r.
+Fig. 8 shows an example for exact repair of (a)systematic node 1 and (b)parity node 1.
+Reverse to the previous case, exact repair of parity nodes is now much simpler. In this example,
+by downloading only the first equation from each survivor nod e, we can successfully recover
+a′. On the contrary, systematic node repair is a bit involved: a projection vector solution is
+2−1v1= (1,1,1)t.Usingthisvector,wecanachievesimultaneousinterferen cealignment,thereby
+decoding the desired components of a′.
+October 29, 2018 DRAFT23
+VI. GENERALIZATION :n≥2k;d≥2k−1
+Theorem 1 gives insights into generalization to (2k,k,k−1)E-MSR codes. The key observa-
+tionisthatassuming M=k(d−k+1),storagecostis α=M/k=d−k+1 =kandthisnumber
+is equal to the number of systematic nodes and furthermore ma tches the number of parity nodes.
+Notice that the storage size matches the size of encoding mat rices, which determines the number
+of linearly independent vectors of {v}:={v1,···}. In this case, therefore, we can generate k
+linearly independent vectors {v}:={v1,···,vk}and corresponding {u}:={u1,···,uk}
+through the appropriate choice of Mto design (2k,k,k−1)E-MSR codes.
+A. Case: n= 2k
+Theorem 2 ( (2k,k,k−1)E-MSR Codes): LetMbe a Cauchy matrix:
+M=
+m(1)
+1m(2)
+1···m(k)
+1
+m(1)
+2m(2)
+2···m(k)
+2
+............
+m(1)
+km(2)
+k···m(k)
+k
+,
+where each element m(i)
+j∈GF(q), whereq≥2k. Suppose
+V= [v1,···,vk]is invertible and
+U=κ−1V′M,(20)
+whereV′= (Vt)−1andκis an arbitrary non-zero value ∈Fqsuch that 1−κ2/ne}ationslash= 0. Also assume
+that encoding matrices are given by
+G(1)
+1=u1vt
+1+m(1)
+1I,···,G(1)
+k=u1vt
+k+m(1)
+kI,
+.........
+G(k)
+1=ukvt
+1+m(k)
+1I,···,G(k)
+k=ukvt
+k+m(k)
+kI,(21)
+whereG(i)
+lindicates an encoding matrix for parity node i, associated with information unit
+l. Then, the code satisfies the MDS property and achieves the MS R point under exact repair
+constraints of all nodes.
+Proof:See Appendix B.
+Remark 4: Note that the minimum required alphabet size is 2k. As mentioned earlier, this
+is because we employ a Cauchy matrix for ensuring the inverti bility of any submatrices of M.
+One may customize codes to find smaller alphabet-size codes.
+October 29, 2018 DRAFT24
+B. Case: n≥2k;d≥2k−1
+Now what if kis less than the size (=α=d−k+1)of encoding matrices, i.e., d≥2k−1?
+Note that this case automatically implies that n≥2k, sincen≥d+1. The key observation in
+this case is that the encoding matrix size is bigger than k, and therefore we have more degrees
+of freedom (a larger number of linearly independent vectors ) than the number of constraints.
+Hence, exact repair of systematic nodes becomes transparen t. This was observed as well in [4],
+where it was shown that for this regime, exact repair of syste matic nodes only can be guaranteed
+by judiciously manipulating (2k,k,k−1)codes through a puncturing operation.
+We show that the puncturing technique in [4] (meant for exact repair of systematic nodes and
+for a special case of our generalized codes) together with ou r repair construction schemes can
+also carry over to ensure exact repair of allnodes even for the generalized family of codes. The
+recipe for this has two parts:
+1. Constructing a target code from a larger code through the p uncturing technique.
+2. Showing that the resulting target code indeed ensures exa ct repair of all nodes as well as
+the MDS-code property for our generalized family of codes.
+The first part contains the following detailed steps:
+1(a) Using Theorem 2, construct a larger (2n−2k,n−k,2n−2k−1)code with a finite field
+size ofq≥2n−2k.
+1(b) Remove all the elements associated with the (n−2k)information units (e.g., from the
+(k+ 1)th to the (n−k)th information unit). The number of nodes is then reduced by
+(n−2k)and so are the number of information units and the number of de grees. Hence,
+we obtain the (n,k,n−1)code.
+1(c) Prune the last (n−1−d)equations in each storage node and also the last (n−1−d)
+symbols of each information unit, while keeping the number o f information units and
+storage nodes. We can then get the (n,k,d)target code.
+Indeed, based on our framework in Section IV, it can be shown t hat the resulting punctured code
+described above guarantees exact repair of all nodes and MDS -code property for our generalized
+family of codes. Hence, we obtain the following theorem. The proof procedure is tedious and
+mimics that of Theorem 2. Therefore, details are omitted.
+Theorem 3 (k
+n≤1
+2,d≥2k−1):Underexactrepairconstraintsofallnodes,theoptimaltra de-
+October 29, 2018 DRAFT25
+a1
+a2
+a3
+b1
+b2
+b3
+c1
+c2
+c3
+3a2 +b2 +2c2
+3a3 +2b3 +c32a1+a3+b1+b3+3c1+c32a1+a2 +3b1+b2+c1+c23a1 +3b1 +3c1
+2a2+a3+b2+2b3+3c2+3c3a1+2a2+2b1+2b2+3c1+ 2c2
+a2+2a3+3b2+3b3+2c2+c3a1+2a3+3b1+2b3+2c1+ 2c3(6,3,5) E-MSR code(5,2,3)
+E-MSRcodeRemove (c1,c2,c3), (a3,b3)
+andassociatedelements.
+3a2+b22a1+a2+3b1+b23a1+3b1
+a1+2a2+2b1+2b2of each storage node .Also remove the third equation
+a2+3b2a1+3b1a1
+a2
+b1
+b2
+ 
+Fig. 9. Bi-Orthogonal case: Illustration of the constructi on of a(5,2,3)E-MSR code from a (6,3,5)code defined over GF(4).
+For a larger code, we adopt the (6,3,5)code in Fig. 8. First, we remove all the elements associated w ith the last (n−2k) = 1
+information unit (“ c”). Next, we prune symbols (a3,b3)and associated elements. Also we remove the last equation of each
+storage node. Finally we obtain the (n,k,d) = (5,2,3)target code.
+off of (1) can be attained with a deterministic scheme requir ing a field size of at most 2(n−k).
+Example 1: Fig. 9 illustrates how to construct an (n,k,d) = (5,2,3)target code based on the
+above recipe. First construct the (2n−2k,n−k,2n−2k−1) = (6,3,5)code, which is larger
+than the(5,2,3)target code, but which belongs to the category of n= 2k. For this code, we
+adopt the bi-orthogonal case example in Fig. 8. For this code , we now remove all the elements
+associated with the last (n−2k) = 1information unit, which corresponds to (c1,c2,c3). Next,
+prune the last symbol (a3,b3)of each information unit and associated elements to shrink t he
+storage size into 2. We can then obtain the (5,2,3)target code. Exact repair and the MDS-code
+property of the resulting code can be verified based on the pro posed framework in Section IV.
+October 29, 2018 DRAFT26
+VII. G ENERALIZATION :k≤3
+As a side generalization, we consider the case of k≤3. The interesting special case of the
+(5,3)E-MSR code8will be focused on, since it is not covered by the above case ofk
+n≤1
+2.
+For this case, we propose another interference alignment te chnique building on an eigenvector
+concept.
+Theorem 4 ( k≤3):The MSR point can be attained with a deterministic scheme req uiring a
+finite-field size of at most 2n−2k.
+Proof:The case of k= 1is trivial. By Theorems 2 and 3, we prove the case of k= 2.
+However, additional effort is needed to prove the case of k= 3. By Theorems 2 and 3, (n,3)
+forn≥6can be proved. But (5,3)codes are not in the class. In Section VII-A, we will address
+this case to complete the proof.
+Remark 5: In order to cover general n, we provide a looser bound on the required finite-field
+size:q≥2n−2k. In fact, for the (5,3)code (that will be shown in Lemma 2), a smaller finite-
+field size of q= 3(<4 = 2n−2k) is enough for construction. We have taken the maximum of
+the required field sizes of all the cases.
+A.(5,3)E-MSR Codes
+We consider d= 4andM= 6. The cutset bound (1) then gives the fundamental limits of:
+storage cost α= 2and repair-bandwidth-per-link=1; hence, the dimension of encoding matrices
+is 2-by-2. Note that the size is lessthan the number of systematic nodes. Therefore, our earlier
+framework does not cover this category. In fact, the (5,3)code is in the case of n+ 1 = 2k,
+where it was shown in [4] that there exist codes that achieve t he cutset bound under exact repair
+of systematic nodes only (not including parity nodes).
+We propose an eigenvector-based interference alignment te chnique to prove the code existence
+under exact repair of allnodes. Let a= (a1,a2)t,b= (b1,b2)tandc= (c1,c2)t. For exact
+repair, we connect to 4(=d)nodes to download a one-dimensional scalar value from each n ode.
+Fig. 10 illustrates exact repair of node 1. We download four e quations from survivor nodes:
+btvα1;ctvα2;at(A1vα3)+bt(B1vα3)+ct(C1vα3);at(A2vα4)+bt(B2vα4)+ct(C2vα4). The
+approach is different from that of our earlier proposed fram ework. Instead an idea here consists
+8Independently, the authors in [5] found (5,3)codes defined over GF(3), based on a search algorithm.
+October 29, 2018 DRAFT27
+C1B−1
+1vα1
+vα2
+rank

+A1B−1
+1vα1A2B−1
+2vα1
+=2 vα1:aneigenvectorof B2C−1
+2C1B−1
+1vα3=B−1
+1vα1vα2=C1B−1
+1vα1vα1
+A1B−1
+1vα1
+vα4=B−1
+2vα1C2B−1
+2vα1
+A2B−1
+2vα1vα1at
+bt
+atA1+btB1+ctC1
+atA2+btB2+ctC2vα2
+vα3ct
+vα4node1
+node2
+node3
+node4
+(paritynode1)
+(paritynode2 )node5
+0
+0
+(A1vα3)t
+(A2vα4)t
+a+
+vt
+α1
+0
+(B1vα3)t
+(B2vα4)t
+b+
+0
+vt
+α2
+(C1vα3)t
+(C2vα4)t
+c
+ 
+Fig. 10. Eigenvector-based interference alignment for (5,3)E-MSR codes. First we align interference “ b” by setting vα3=
+B−1
+1vα1andvα4=B−1
+2vα1. Next, partially align interference of “ c” by setting vα2=C1B−1
+1vα1. Finally, choosing vα1as
+an eigenvector of B2C−1
+2C1B−1
+1, we can achieve interference alignment for c.
+of three steps: (1) choosing projection vectors for achievi ng interference alignment; (2) gathering
+all the alignment constraints and the MDS-code constraint; (3) designing the encoding matrices
+that satisfy all the constraints. Notice the design of encod ing matrices is the last part.
+Here are details. Note that there are 6 unknown variables: 2 d esired unknowns (a1,a2)and 4
+undesired unknowns (b1,b2,c1,c2). Therefore, it is required to align (b1,b2,c1,c2)onto at least
+2-dimensional linear space. We face the challenge that appe ared in the (6,3,5)code example
+in Fig. 4. Projection vectors vα3andvα4affect interference alignment bandcsimultaneously.
+Therefore, we need simultaneous interference alignment. T o solve this problem, we introduce
+an eigenvector-based interference alignment scheme.
+First choose vα3andvα4such that vα3=B−1
+1vα1andvα4=B−1
+2vα1, thereby achieving
+interference alignment for “ b”. Observe the interfering vectors associated with “ c”:
+vα2;C1B−1
+1vα1;C2B−1
+2vα1.
+The first and second vectors can be aligned by setting vα2=C1B−1
+1vα1. Now what about for the
+following two vectors: C1B−1
+1vα1andC2B−1
+2vα1? Suppose that the associated matrices ( C1B−1
+1
+andC2B−1
+2) and the projection vector vα1are randomly chosen. Then, the these two vectors
+October 29, 2018 DRAFT28
+are not guaranteed to be aligned. However, a judicious choic e ofvα1makes it possible to align
+them. The idea is to choose vα1as an eigenvector of B2C−1
+2C1B−1
+1. Sincevα1can be chosen
+arbitrarily, this can be easily done. Lastly consider the co ndition for ensuring the decodability
+of desired signals: rank/parenleftbig/bracketleftbig
+A1B−1
+1vα1A2B−1
+2vα1/bracketrightbig/parenrightbig
+= 2.
+We repeat the procedure for exact repair of “ b” and “c”. For parity nodes, we employ the
+remapping technique described earlier:
+
+a′
+b′
+c′
+:=
+At
+1Bt
+1Ct
+1
+At
+2Bt
+2Ct
+2
+0 0 I
+
+a
+b
+c
+,
+A′
+1A′
+20
+B′
+1B′
+20
+C′
+1C′
+2I
+:=
+A1A20
+B1B20
+C1C2I
+−1
+.(22)
+We gather all the conditions that need to be guaranteed for ex act repair of all nodes:
+rank/parenleftbig/bracketleftbig
+A1B−1
+1vα1A2B−1
+2vα1/bracketrightbig/parenrightbig
+= 2,
+rank/parenleftbig/bracketleftbig
+B1C−1
+1vβ1B2C−1
+2vβ1/bracketrightbig/parenrightbig
+= 2,
+rank/parenleftbig/bracketleftbig
+C1A−1
+1vγ1C2A−1
+2vγ1/bracketrightbig/parenrightbig
+= 2,
+rank/parenleftbig/bracketleftbig
+A′
+1B′−1
+1vα′1A′
+2B′−1
+2vα′1/bracketrightbig/parenrightbig
+= 2,
+rank/parenleftbig/bracketleftbig
+B′
+1C′−1
+1vβ′1B′
+2C′−1
+2vβ′1/bracketrightbig/parenrightbig
+= 2,(23)
+where
+vα1:an eigenvector of B2C−1
+2C1B−1
+1,
+vβ1:an eigenvector of C2A−1
+2A1C−1
+1,
+vγ1:an eigenvector of A2B−1
+2B1A−1
+1,
+vα′1:an eigenvector of B′
+2C′−1
+2C′
+1B′−1
+1,
+vβ′1:an eigenvector of C′
+2A′−1
+2A′
+1C′−1
+1.(24)
+Note that eigenvectors may not exist for the finite Galois fiel d. However, the existence is
+guaranteed by carefully choosing the encoding matrices. We provide an explicit coding scheme
+in the following lemma.
+Lemma 2 ( (5,3)E-MSR Codes): Letα,β∈GF(3)and be non-zero. Suppose encoding ma-
+October 29, 2018 DRAFT29
+a1
+a2
+b1
+c1
+c2b2
+2c1+c2
+2a1+2a2+b1+2c1+c2
+a2+2b1+2b2+2c2
+2a1+a2+b1+c1+2c2b1
+2a1+2a2
++b1+(2c1+c2)
+2a2+2b1+b2+2c22a1+a2
++b1+2(2c1+c2)

+1
+0
+
2
+1
+
1
+0
+

+1
+0
+ 
+Fig. 11. Illustration of exact repair of node 1 for a (5,3)E-MSR code defined over GF(3). The eigenvector-based interference
+alignment scheme enables to decode 2 desired unknowns (a1,a2)from 4 equations containing 6 unknowns. Notice that
+interference “ b” and “c” are aligned simultaneously although the same projection v ectorsvα3andvα4are used.
+trices are given by
+A1=
+2α0
+2β β
+,B1=
+α2α
+0 2β
+,C1=
+2α0
+β2β
+,
+A2=
+2α0
+β2β
+,B2=
+α2α
+0β
+,C2=
+α0
+2β2β
+.(25)
+Then, the code satisfies the MDS property and achieves the MSR point (1) under exact repair
+constraints of all nodes.
+Proof:See Appendix C.
+Remark 6: Note that encoding matrices are lower-triangular or upper- triangular. This structure
+has important properties. Not only does this structure guar antee invertibility, it can in fact
+guarantee the existence of eigenvectors. It turns out the st ructure as above satisfies all of the
+conditions needed for the MDS property and exact repair.
+Example 2: Fig. 11 illustrates exact repair of node 1 (a1,a2)for a(5,3)E-MSR code defined
+October 29, 2018 DRAFT30
+overGF(3). Notice that interference “ b” and “c” are aligned simultaneously. One can check
+exact repair of the remaining four nodes based on our propose d method.
+VIII. C ONCLUSION
+We have systematically developed interference alignment t echniques that attain the cutset-
+based MSR point (1) under exact repair constraints of allnodes. Based on the proposed frame-
+work, we provided a generalized familyof codes for thecases :(a)k
+n≤1
+2;(b)k≤3, for arbitrary
+n≥k. This generalized family of codes provides insights into a d ual relationship between the
+systematic and parity node repair, as well as opens up a large r constructive design space of
+solutions. For (5,3)codes which do not satisfyk
+n≤1
+2, we have developed an eigenvector-based
+interferencealignmenttoshowtheoptimalityofthecutset bound.Unlikewirelesscommunication
+problems,ourstoragerepairproblemshavemoreflexibility indesigningencodingmatrices which
+correspond to wireless channel coefficients (provided by na ture) in communication problems.
+Exploiting this fact, we developed interference alignment techniques for optimal exact repair
+codes in distributed storage systems.
+APPENDIX A
+PROOF OF LEMMA1
+It suffices to show that
+A′
+1A′
+2A′
+3
+B′
+1B′
+2B′
+3
+C′
+1C′
+2C′
+3
+
+A1A2A3
+B1B2B3
+C1C2C3
+=
+I 0 0
+0 I 0
+0 0 I
+.
+Using (6) and (13), we compute:
+(1−κ2)(A′
+1A1+A′
+2B1+A′
+3C1) =/parenleftbig
+v′
+1u′t
+1−κ2α′
+1I/parenrightbig
+(u1vt
+1+α1I)
++/parenleftbig
+v′
+2u′t
+1−κ2β′
+1I/parenrightbig
+(u1vt
+2+β1I)+/parenleftbig
+v′
+3u′t
+1−κ2γ′
+1I/parenrightbig
+(u1vt
+3+γ1I)
+(a)= (v′
+1vt
+1+v′
+2vt
+2+v′
+3vt
+3)+(α1v′
+1+β1v′
+2+γ1v′
+3)u′t
+1−κ2u1(α′
+1v1+β′
+1v2+γ′
+1v3)t−κ2I
+(b)= (v′
+1vt
+1+v′
+2vt
+2+v′
+3vt
+3)+κu1u′t
+1−κ2u1(α′
+1v1+β′
+1v2+γ′
+1v3)t−κ2I
+(c)= (v′
+1vt
+1+v′
+2vt
+2+v′
+3vt
+3)−κ2I
+(d)= (1−κ2)I
+October 29, 2018 DRAFT31
+where(a)follows from α1α′
+1+β1β′
+1+γ1γ′
+1= 1due to (11); (b)follows from (12); (c)follows
+fromu′
+1= 2(α′
+1v1+β′
+1v2+γ′
+1v3)(See Claim 1); and (d)follows from the fact that v′
+1vt
+1+
+v′
+2vt
+2+v′
+3vt
+3=I, since(v′
+1,v′
+2,v′
+3)are dual basis vectors.
+Similarly, one can check that B′
+1A2+B′
+2B2+B′
+3C2=IandC′
+1A3+C′
+2B3+C′
+3C3=I.
+Now let us compute one of the cross terms:
+(1−κ2)A′
+1A2+A′
+2B2+A′
+3C2=/parenleftbig
+v′
+1u′t
+1−κ2α′
+1I/parenrightbig
+(u2vt
+1+α2I)
++/parenleftbig
+v′
+2u′t
+1−κ2β′
+1I/parenrightbig
+(u2vt
+2+β2I)+/parenleftbig
+v′
+3u′t
+1−κ2γ′
+1I/parenrightbig
+(u2vt
+3+γ2I)
+(a)= (α2v′
+1+β2v′
+2+γ2v′
+3)u′t
+1−κ2u2(α′
+1v1+β′
+1v2+γ′
+1v3)t
+(b)= 0
+where(a)follows from u′t
+iuj=δ(i−j)and<(α′
+1,β′
+1,γ′
+1),(α2,β2,γ2)>= 0;(b)follows from
+(12) and Claim 1. Similarly, we can check that the other cross terms are zero matrices. This
+completes the proof.
+Claim 1: For alli,u′
+i=κ(α′
+iv1+β′
+iv2+γ′
+iv3).
+Proof:By (12), we can rewrite
+[u1,u2,u3] =1
+κ[v′
+1,v′
+2,v′
+3]
+α1α2α3
+β1β2β3
+γ1γ2γ3
+.
+Using the fact that (u′
+1,u′
+2,u′
+3)are dual basis vectors, we get
+u′t
+1
+u′t
+2
+u′t
+3
+=κ
+α′
+1β′
+1γ′
+1
+α′
+2β′
+2γ′
+2
+α′
+3β′
+3γ′
+3
+
+vt
+1
+vt
+2
+vt
+3
+.
+This completes the proof.
+APPENDIX B
+PROOF OF THEOREM 2
+For generalization, we are forced to use some heavy notation but only for this section and
+the related appendices. Let wjbe ak-dimensional message vector for information unit j. Let
+w′
+jbe the newly mapped information unit after remapping. Let G(i)
+jbe an encoding matrix for
+parity node i, associated with the jth information unit. Let G′(i)
+jbe the newly mapped entity.
+October 29, 2018 DRAFT32
+A. Exact Repair of Systematic Nodes
+For exact repair of systematic node i, we have each survivor node project their data
+G(i)
+lv′
+i=m(i)
+lv′
+i+ul,
+G(j)
+lv′
+i=m(j)
+lv′
+i.
+Therefore, we can achieve simultaneous interference align ment for non-intended signals, while
+guaranteeing the decodability of desired signals.
+B. Exact Repair of Parity Nodes
+The idea is the same as that of Theorem 1. The detailed procedu res are as follow. First we
+remap parity nodes into new variables:
+
+w′
+1
+w′
+2
+...
+w′
+k
+:=
+G(1)t
+1G(1)t
+2···G(1)t
+k
+G(2)t
+1G(2)t
+2···G(2)t
+k
+............
+G(k)t
+1G(k)t
+2···G(k)t
+k
+
+w1
+w2
+...
+wk
+.
+Define the newly remapped encoding matrices as:
+
+G′(1)
+1G′(2)
+1···G′(k)
+1
+G′(1)
+2G′(2)
+2···G′(k)
+2
+............
+G′(1)
+kG′(2)
+k···G′(k)
+k
+:=
+G(1)
+1G(2)
+1···G(k)
+1
+G(1)
+2G(2)
+2···G(k)
+2
+............
+G(1)
+kG(2)
+k···G(k)
+k
+−1
+. (26)
+We can now apply the generalization of Lemma 1 to obtain the du al structure:
+G′(1)
+1=1
+1−κ2/parenleftig
+v′
+1u′t
+1−κ2m′(1)
+1I/parenrightig
+,···,G′(1)
+k=1
+1−κ2/parenleftig
+v′
+1u′t
+k−κ2m′(k)
+1I/parenrightig
+,
+.........
+G′(k)
+1=1
+1−κ2/parenleftig
+v′
+ku′t
+1−κ2m′(1)
+kI/parenrightig
+,···,G′(k)
+k=1
+1−κ2/parenleftig
+v′
+ku′t
+k−κ2m′(k)
+kI/parenrightig
+,
+where the dual basis vectors are defined as:
+
+m′(1)
+1m′(1)
+2···m′(1)
+k
+m′(2)
+1m′(2)
+2···m′(2)
+k
+............
+m′(k)
+1m′(k)
+2···m′(k)
+k
+:=
+m(1)
+1m(2)
+1···m(k)
+1
+m(1)
+2m(2)
+2···m(k)
+2
+............
+m(1)
+km(2)
+k···m(k)
+k
+−1
+.
+October 29, 2018 DRAFT33
+Let us check exact repair of parity node i. We choose projection vectors as ui. Then,∀l=
+1,···,k, we get:
+(1−κ2)G′(i)
+lui=−κ2m(l)
+iui+v′
+l,
+(1−κ2)G′(j)
+lui=−κ2m(l)
+jui.
+Therefore, we can achieve simultaneous interference align ment for non-intended signals, while
+guaranteeing the decodability of desired signals.
+C. The MDS-Code Property
+We check the invertibility of a composite encoding matrix wh en a Data Collector connects to
+isystematic nodes and (k−i)parity nodes for i= 0,···,k. The main idea is to use a Gaussian
+elimination method as we did in Section IV-C. The verificatio n is tedious and therefore details
+are omitted.
+D. Minimum Required Finite-Field Size
+Note that the dimension of an encoding matrix is k-by-k. Therefore, the minimum finite-field
+size required to generate a Cauchy matrix is 2k, i.e.,q≥2k.
+APPENDIX C
+PROOF OF LEMMA2
+A. Exact Repair
+With the Gaussian elimination method, we get
+A′
+1=
+1
+α1
+β
+1
+α0
+,B′
+1=
+1
+α2
+β
+1
+α0
+,C′
+1=
+02α
+β
+2β
+α1
+,
+A′
+2=
+01
+β
+1
+α1
+β
+,B′
+2=
+02
+β
+1
+α2
+β
+,C′
+2=
+02α
+β
+2β
+α1
+.(27)
+Using this, we can easily check the the existence of eigenvec tors (24) and decodabiity of desired
+signals (23). This completes the proof.
+October 29, 2018 DRAFT34
+B. The MDS-Code Property
+Obviously, all the encoding matrices are invertible due to t heir lower-triangular or upper-
+triangular structure. We consider three cases where a Data C ollector connects to (1) 3 systematic
+nodes; (2) 2 systematic nodes and 1 parity node; and (3) 1 syst ematic node and 2 parity nodes.
+The first is a trivial case where the composite matrix associa ted with information units is an
+identity matrix. The second case is also trivial, since each encoding matrix is invertible so that
+the composite matrix is invertible as well. For the last case , we consider
+
+A1A20
+B1B20
+C1C2I
+=
+2α02α00 0
+2β ββ2β0 0
+α2αα2α0 0
+0 2β0β0 0
+2α0α01 0
+β2β2β2β0 1
+. (28)
+It is easy to check the invertibility of this matrix via the Ga ussian elimination method. The
+invertibility for all the cases guarantees the MDS property .
+ACKNOWLEDGMENT
+We gratefully acknowledge Prof. P. V. Kumar (of IISc) and his students, N. B. Shah and K.
+V. Rashmi, for insightful discussions and fruitful collabo ration related to the structure of exact
+regeneration codes.
+REFERENCES
+[1] A. G. Dimakis, P. B. Godfrey, Y. Wu, M. Wainwright, and K. R amchandran, “Network coding for distributed storage
+systems,” IEEE INFOCOM , 2007.
+[2] Y.Wu,A.G.Dimakis,andK.Ramchandran, “Deterministic regeneratingcodes fordistributedstorage,” AllertonConference
+on Control, Computing and Communication , Sep. 2007.
+[3] S. Pawar, S. E. Rouayheb, and K. Ramchandran, “On secure d istributed data storage under repair dynamics,” submitted to
+IEEE ISIT , 2010.
+[4] N. B. Shah, K. V. Rashmi, P. V. Kumar, and K. Ramchandran, “ Explicit codes minimizing repair bandwidth for distribute d
+storage,” IEEE ITW, Jan. 2010, online avaiable at arXiv:0908.2984v2 , Sep. 2009.
+[5] D. Cullina, A. G. Dimakis, and T. Ho, “Searching for minim um storage regenerating codes,” Allerton Conference on
+Control, Computing and Communication , Sep. 2009.
+October 29, 2018 DRAFT35
+[6] M. A. Maddah-Ali, S. A. Motahari, and A. K. Khandani, “Com munication over MIMO X channels: Interference alignment,
+decomposition, and performance analysis,” IEEE Transactions on Information Theory , vol. 54, pp. 3457–3470, Aug. 2008.
+[7] V. R. Cadambe and S. A. Jafar, “Interference alignment an d the degree of freedom for the K user interference channel,”
+IEEE Transactions on Information Theory , vol. 54, no. 8, pp. 3425–3441, Aug. 2008.
+[8] C. Suh and D. Tse, “Interference alignment for cellular n etworks,” Allerton Conference on Control, Computing and
+Communication , Sep. 2008.
+[9] Y. Wu and A. G. Dimakis, “Reducing repair traffic for erasu re coding-based storage via interference alignment,” Proc. of
+IEEE ISIT , 2009.
+[10] K. V. Rashmi, N. B. Shah, P. V. Kumar, and K. Ramchandran, “Explicit construction of optimal exact regenerating code s
+for distributed storage,” Allerton Conference on Control, Computing and Communicati on, Sep. 2009.
+[11] Y. Wu, “A construction of systematic mds codes with mini mum repair bandwidth,” arXiv:0910.2486 , Oct. 2009.
+[12] R. Koetter and M. Medard, “An algebraic approach to netw ork coding,” IEEE/ACM Transactions on Networking , vol. 11,
+no. 5, Oct. 2003.
+[13] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Networ k information flow,” IEEE Transactions on Information
+Theory, vol. 46, no. 4, pp. 1204–1216, Jul. 2000.
+[14] A. S. Householder, The Theory of Matrices in Numerical Analysis . Dover, Toronto, Cananda, 1974.
+[15] A. A. Bubrulle, “Work notes on elementary matrices,” Tech. Rep. HPL-93-69, Hewlett-Packard Laboratory , 1993.
+[16] D. S. Bernstein, Matrix mathematics: Theory, facts, and formulas with appli cation to linear systems theory . Princeton
+University Press, 2005.
+October 29, 2018 DRAFT
\ No newline at end of file
diff --git a/1001.0108.txt b/1001.0108.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a13973bbc7098b2531eccf7a0e47b1a84050c57d
--- /dev/null
+++ b/1001.0108.txt
@@ -0,0 +1,3061 @@
+Surface plasmon on the metal nanofilament excitation by a 
+quantum oscillator 
+ 
+ 
+V.V.Lidsky 
+ 
+The P.N.Lebedev Physical Institute of RAS 
+119991 Moscow, Russia 
+vlidsky@sci.lebedev.ru  
+ 
+Surface waves on a thin metal filament are described in 
+terms of quantum electrodynamics. The interaction of surface waves with a quantum osc illator is discussed in the 
+dipole approximation. The increase in the spontaneous emission  rate of the excited quantum oscillator, the so called Purcell factor, is evaluate d to be as high as ten to the 
+five times. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 1Возбуждение  поверхностных  плазмонов  на металлическом  
+наноцилиндре  квантовым  излучателем  
+ 
+В.В.Лидский  
+ 
+Физический  институт  им. П. Н. Лебедева  РАН 
+119991 Москва , Ленинский  пр., 53 
+vlidsky@sci.lebedev.ru 
+ 
+ 
+А Н Н О Т А Ц И Я 
+ 
+Рассмотрены  поверхностные  волны на тонком  
+металлическом  цилиндре  в рамках  формализма  
+квантовой  электродинамики . Рассмотрено  в дипольном  
+приближении  взаимодействие  поверхностных  
+плазмонов  с квантовым  излучателем , расположенным  в 
+непосредственной  близости  от поверхности  цилиндра . 
+Показано , что вероятность  спонтанного  перехода  с 
+излучением  плазмона  может  на пять  порядков  
+превосходить  вероятность  дипольного  перехода  в 
+свободном  пространстве . 
+ 
+ 
+ 
+1. После  появления  в печати  сообщений  о запуске  нанолазеров  или SPAZER’ ов 
+/1,2/, использующих механизм возбуждения  поверхностных  волн  молекулами  
+красителя , возник  вопрос  о построении  последовательной  квантовой  теории  
+взаимодействия  поверхностных  волн  с квантовым  излучателем . Дело в том, что 
+вероятность  спонтанного  перехода  квантовой  системы  существенным  образом 
+зависит от структуры  пространства , окружающего  излучатель . Перселл  заметил , что 
+вероятность  спонтанного  перехода  возрастает на несколько  порядков , если  вблизи  
+излучателя  находится  микроскопическая  частица  /3/. В /4/ эффект  Перселла  был 
+привлечен  для объяснения  наблюдаемого  увеличения  на 14 порядков  сечения  
+комбинационного  рассеяния   ⎯  явления  гигантского  комбинационного  рассеяния  
+(SERS) /5,6/. Теория  SPAZER’ а была  предложена  в недавно  появившейся  работе  /7/, 
+где показано , что и здесь учет  эффекта  Перселла  позволяет в принципе  преодолеть  
+трудность , вызванную  исключительно  сильным  затуханием  поверхностных  волн .  
+В данной  работе  мы будем  следовать  методу  расчета вероятности , 
+использованному  нами в /8/. Мы  рассмотрим  поверхностные  волны , 
+распространяющиеся  по тонкому  металлическому  стержню  субволнового  диаметра , 
+окруженному  диэлектрической  средой . Затем  мы разложим  поле  на элементарные  
+моды  и определим  операторы  вторичного  квантования , с помощью  которых  поле  
+поверхностной  волны может быть  описано  с точки  зрения  квантовой  
+электродинамики . Затем  рассмотрим  квантовый  осциллятор , помещенный  вблизи  
+стержня  и вычислим  вероятность  излучения  этим  осциллятором  поверхностного  
+плазмона . Полученную  величину  сравним  с известной  формулой  для вероятности  
+спонтанного  излучения  фотона  в свободном  пространстве  и придем  к формулировке  
+эффекта  Перселла , применительно  к излучателю  вблизи  субволнового  цилиндра . 
+ 2ρ 
+x
+z
+y 2. Рассмотрим  металлический  цилиндр  малого  радиуса  ρ в диэлектрической  
+среде  (рис. 1). Диэлектрическую  проницаемость  среды  будем  считать  равной  hε. 
+Диэлектрическую  проницаемость  металла  будем  считать  связанной  с частотой  
+формулой  Друдэ : 2 21ω ω εpl m− = .  
+Мнимая  часть  диэлектрической  
+проницаемости  в данной  работе  
+учитываться  не будет . Нас будет  
+интересовать  диапазон  частот  не 
+превышающих  плазменную  частоту , 
+таких  что соответствующая  длина  
+волны в диэлектрике  много больше  
+радиуса  цилиндра : εh 
+εm 
+ρωπ>>c2 . 
+Магнитную  проницаемость  металла  
+цилиндра  будем  считать  равной  1. 
+На первом  шаге  наша  задача  определить  собственные  поверхностные  волны  
+такого  пространства . Уравнения  Максвелла  при отсутствии  внешних  токов  запишем  в 
+виде : 
+E H rottG G
+∂ ⋅ =ε  H E rottG G
+−∂ =  (2.1) 
+0=H divG
+ 0=E divG
+ (2.2) 
+Мы видим , что  уравнения  (2.2) выполнены , если  выполнены  (2.1).  
+Аналогично  тому , как это делается  в свободном  пространстве  /9/, выберем  
+конечный  участок  цилиндра  длиной  L. Разложим  поля в ряды Фурье  по координатам  z 
+и ϕ цилиндрической  системы , ось  z которой  совпадает  с осью цилиндра . 
+∑∑∞
+−∞ =∞
+−∞ =⋅ + ⋅ ⋅ =
+knk nk in z ih r t E E ) exp( ) , ( ϕG G
+ ∑∑∞
+−∞ =∞
+−∞ =⋅ + ⋅ ⋅ =
+knk nk in z ih r t H H ) exp( ) , ( ϕG G
+  
+(2.3) 
+Зависимость  от времени  решений  уравнений  (2.1) будем  искать  в виде : 
+t i
+nk nk e r t H r t Eω−∝) , ( ), , (GG
+ (2.4)
+причем  частота  ω определяется  из известного  характеристического  уравнения , 
+которое  будет  выписано  ниже  (см.(3.9-10)). Общий  вид решения  получим  суммируя  
+решения  для каждого  корня  характеристического  уравнения . В этих  предположениях  
+уравнения  Максвелла  (2.1) приводят к уравнениям  для компонент  ) , ( ), , (r t H r t Enk nkGG
+:  
+r z E i H ihr Hrin⋅ − = ⋅ −ωεϕ  (2.5) 
+ϕωεE i HrHrih
+z r r⋅ − = ∂ −1 (2.6) 
+()z r r E i HrinH rr⋅ − = ⋅ − ∂ωεϕ2 1 (2.7) 
+r z H i E ihr Erin⋅ = ⋅ −ωϕ  (2.8) 
+ϕωH i ErErih
+z r r⋅ = ∂ −1 (2.9) 
+()z r r H i ErinE rr⋅ = ⋅ − ∂ωϕ2 1 (2.10)
+ 3Для удобства  записи  мы опустили у компонент  напряженностей  индексы  , 
+указывающие  по каким  переменным  сделано  преобразование  Фурье . Волновое  
+уравнение  приводит к следующему  соотношению  для спектральных  компонент : nk
+()z z r r E h Ern
+r⋅ − =⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− ∂ + ∂ ε ω2 2
+22
+21
+ (2.11) 
+И аналогичному  уравнению для . zH
+3. Решение  этого  уравнения  качественно  зависит  от знака  величины  . 
+Внутри  цилиндра  ε ω2 2−h
+0<mε , следовательно  регулярным  при 0=r  решением   уравнения  
+(2.11) окажется  модифицированная  функция  Бесселя : 
+) (pr I const En z⋅=  (3.1) 
+где  
+2 2ω ε⋅ − =m h p  (3.2) 
+Вне диэлектрика  возможны  два варианта . При малых  значениях  волнового  числа   
+имеет  место  соотношение   и решением  (2.11) окажутся  осциллирующие   
+цилиндрические  функции  . Эти решения  описывают  волны , уносящие  
+энергию  в пространство . В данной  работе  мы их рассматривать  не будем . Наше  
+исследование  посвящено  процессам  излучения  поверхностных  волн , имеющих  
+большие  волновые  числа , так что в дальнейшем  будем  предполагать  , а, 
+следовательно , величину   h
+02 2< −h hε ω
+) ( ), (qr Y qr Jn n
+02 2> −h hε ω
+2 2ω ε⋅ − =h h q  (3.3) 
+будем  предполагать  вещественной . Убывающим  на бесконечности  решением  
+уравнения  (2.11) оказывается  функция  Макдональда : 
+) (qr K const En z⋅=  (3.4) 
+Таким  образом , мы приходим  к решениям  (2.11) в металле  и диэлектрике : 
+) ( ) ( ) , (1) (ρωp I pr I e C t r En nt m
+z ⋅ ⋅ =− 
+) ( ) ( ) , (2) (ρωq K qr K e C t r En nt h
+z ⋅ ⋅ =− 
+) ( ) ( ) , (3) (ρωp I pr I e C t r Hn nt m
+z ⋅ ⋅ =− 
+) ( ) ( ) , (4) (ρωq K qr K e C t r Hn nt h
+z ⋅ ⋅ =−  
+ 
+(3.5) 
+Из (3.5) и (2.5)-(2.10) легко  находятся  компоненты   (см. Приложение  A): ϕ ϕH E,
+) () ( '
+) () () , (1 3 2 2) (
+ρωε
+ρω ω
+ϕp Ipr Ie Cr pi
+p Ipr Ie Cp rnht r H
+nn t m
+nn t m⋅ ⋅ ⋅⋅− ⋅ ⋅⋅=− − 
+) () ( '
+) () () , (2 4 2 2) (
+ρωε
+ρω ω
+ϕq Kqr Ke Cr qi
+p Kpr Ke Cq rnht r H
+nn t h
+nn t h⋅ ⋅ ⋅⋅− ⋅ ⋅⋅=− − 
+) () ( '
+) () () , (3 1 2 2) (
+ρω
+ρω ω
+ϕp Ipr Ie Cp ri
+p Ipr Ie Cp rnht r E
+nn t
+nn t m⋅ ⋅⋅+ ⋅ ⋅⋅=− − 
+) () ( '
+) () () , (4 2 2 2) (
+ρω
+ρω ω
+ϕq Kqr Ke Cq ri
+q Kqr Ke Cq rnht r E
+nn t
+nn t h⋅ ⋅⋅+ ⋅ ⋅⋅=− −  
+ 
+ 
+ 
+(3.6) 
+ 4Условие  непрерывности  тангенциальных  составляющих  полей на границе  цилиндра  
+ρ=r  приводит к системе  уравнений : 
+2 2 4 2 1 2 3 2C DqiCqnhC DpiCpnh
+Kh
+Im⋅ ⋅ − ⋅ = ⋅ ⋅ − ⋅ωε ωε 
+4 2 2 2 3 2 1 2C DqiCqnhC DpiCpnh
+K I ⋅ ⋅ + ⋅ = ⋅ ⋅ + ⋅ω ω 
+2 1C C=  
+4 3C C=  
+ 
+ 
+ 
+(3.7) 
+где введены  обозначения  
+) () ( '
+ρρρ
+p Ip I pD
+nn
+I⋅ ⋅=  ) () ( '
+ρρρ
+q Kq K qD
+nn
+K⋅⋅=  (3.8) 
+Равенство  нолю  определителя  системы  (3.7) приводит к характеристическому  
+уравнению ( см. Приложение  B):  
+() ( ) ()02 2 2 2 2 2 2 2= − ⋅ ⋅ − ⋅ − ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅h m I K I m K h h n D q D p D q D p ε ε ω ε ε  (3.9) 
+Особо  следует рассматривать  случай  0=n , тогда  из (3.9) видно , что  
+0) () ( '
+) () ( '
+00
+00=⋅⋅ −⋅⋅ρρερρεp I pp I
+q K qq K
+m h  (3.10) 
+4. Итак, для каждой  пары  значений   с помощью  характеристического  
+уравнения  (3.9) определяется  частота  поверхностной  волны k n,
+ω, а затем  по формулам  
+(3.2) и (3.3) вычисляются  величины  qp,, определяющие  зависимость  напряженностей  
+поля от расстояния  до оси цилиндра . Поскольку  частота  ω входит  в (3.9) в квадрате , 
+то частота  может  принимать  два значение  ⎯ положительное  и отрицательное , что 
+соответствует  волнам , бегущим  в сторону  положительных  и отрицательных  . z
+При выполнении  (3.9) система  уравнений  имеет решение , зависящее  от одной  
+произвольной  комплексной  константы . Вычислив  величины   несложно  
+найти  компоненты  полей 4 3 2 1, , ,C C C C
+nk nkH EGG
+,  (см. Приложение  C): 
+ 
+Из (3.5) находим : 
+γ ρ⋅ ⋅ = ) ( ) ( ) () (
+, p I pr I e r En n zm
+nk z  
+γ ρ⋅ ⋅ = ) ( ) ( ) () (
+, q K qr K e r En n zh
+nk z  
+γ ρ⋅ ⋅ = ) ( ) ( ) () (
+, p I pr I h r Hn n zm
+nk z  
+γ ρ⋅ ⋅ = ) ( ) ( ) () (
+, q K qr K h r Hn n zh
+nk z   
+ 
+(4.1) 
+Где  
+( )K I z D p D q e⋅ − ⋅ =2 2 
+) (m h z nh i h εεω−⋅⋅⋅ − =   
+(4.2) 
+ 
+ 5Для азимутальной  и радиальной  компонент  вычисления  приводят к выражениям : 
+ 
+γρωε
+ρϕ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅− ⋅ ⋅⋅=) () ( '
+) () () (2 2) (
+,p Ipr Ier pi
+p Ipr Ihp rnhr H
+nn
+zm
+nn
+zm
+nk  
+γρωε
+ρϕ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅− ⋅ ⋅⋅=) () ( '
+) () () (2 2) (
+,q Kqr Ker qi
+q Kqr Khq rnhr H
+nn
+zh
+nn
+zh
+nk  
+γρω
+ρϕ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅+ ⋅ ⋅⋅=) () ( '
+) () () (2 2) (
+,p Ipr Ihp ri
+p Ipr Iep rnhr E
+nn
+z
+nn
+zm
+nk  
+γρω
+ρϕ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅+ ⋅ ⋅⋅=) () ( '
+) () () (2 2) (
+,q Kqr Khq ri
+q Kqr Keq rnhr E
+nn
+z
+nn
+zh
+nk   
+ 
+ 
+ 
+ 
+ 
+(4.3) 
+ 
+γρ ρω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅=) () ( '
+) () (
+2) (
+,p Ipr Iepih
+p Ipr Ihp rnE
+nn
+z
+nn
+zm
+nk r  
+γρ ρω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅=) () ( '
+) () (
+2) (
+,q Kqr Keqih
+q Kqr Khq rnE
+nn
+z
+nn
+zh
+nk r  
+γρ ρω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅− =) () ( '
+) () (
+2) (
+,p Ipr Ihpih
+p Ipr Iep rnH
+nn
+z
+nn
+zm m
+nk r  
+γρ ρω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅− =) () ( '
+) () (
+2) (
+,q Kqr Khqih
+q Kqr Keq rnH
+nn
+z
+nn
+zh h
+nk r   
+ 
+ 
+ 
+ 
+ 
+(4.4) 
+Определим  вектор -потенциалы  поля: 
+A rot HGG
+=  (4.5) 
+ϕ∇ − −∂ =A EtGG
+ (4.6) 
+Выберем  калибровку  так, чтобы  0=ϕ . Обеспечим  выполнение  равенства  (4.6), 
+выбрав  потенциал  для компоненты   частоты  k n, ω в виде : 
+nk nk EiAGG
+ω1=  (4.7) 
+С помощью  (2.1) легко  видеть , что  и (4.5) также  оказывается  выполненным . 
+ 
+5. Обратимся  теперь  к вычислению  энергии , переносимой  каждой  модой . Плотность  
+энергии   и поток  энергии  поля выражаются  известными  формулами : 
+()2 2
+81H E w+ ⋅ =επ []H E SGG G
+× ⋅ =π41 (5.1) 
+При этом , как легко  убедиться  из уравнений  Максвелла  (2.1) имеет место  закон  
+сохранения  энергии : 
+ 6σd n S dV wdtd
+V⋅ ⋅ − = ⋅ ∫∫∫ ∫∫
+ΣGG
+  
+(5.2) 
+Где  -- поверхность , ограничивающая  объем  V.  Σ
+Для вычисления  полной энергии , переносимой  рассматриваемой  волной в 
+расчете  на участок  цилиндра  длины  L вычислим  интеграл  по объему  из (5.2).  
+()2 2
+02
+0 0 41H E rdr d dz dV w WL
+V+ ⋅ = ⋅ =∫ ∫ ∫ ∫∫∫∞
+ε ϕππ
+  
+(5.3) 
+Где  должны быть  выражены  с помощью  разложения  (2.3). При  интегрировании  
+по координате   произведений  вида  H EG G
+,
+dz ) ' exp( ) exp(z ih ihz⋅  ненулевой  вклад  дадут  
+только  члены  с h h−=' .  Аналогично , при интегрировании  по ϕd следует   
+суммировать  произведения  с . После  чего получаем :  n n− ='
+∑∑∫∞
+−∞ =∞
+−∞ =∞
++ ⋅ =
+nknk nk nk nk H H E E rdrLW ) (4* *
+0GGGG
+ε   
+(5.4) 
+Здесь  принято  во внимание , что  поскольку  H EGG
+,величины  вещественные , то 
+) , ( ) , ( ), , ( ) , (*
+, ,*
+, , t r H t r H t r E t r Ek n k n k n k n − − − − = =G G G G
+ (5.5) 
+Что легко  получить  перейдя  к комплексно  сопряженным  выражениям  в (2.3). 
+Характеристическое  уравнение  для каждой  пары   имеет  два корня  k n, ωω− +, .  Эти  
+значения  физически  соответствуют  волнам , бегущим  в положительном  и 
+отрицательном  направлении  оси , соответственно .  Мы  будем  считать , что каждой  
+паре  значений   соответствуют  две независимые  моды . Таким  образом , энергия  
+распадается  на сумму  энергий , переносимых  каждой  модой: z
+k n,
+∑∑∞
+−∞ =∞
+−∞ ==
+nknkW W   
+(5.5) 
+где 
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
++ + + ⋅ = ∫ ∫∞
+) ( ) (4* ) ( ) ( * ) ( ) ( * ) ( ) ( * ) ( ) (
+0h
+nkh
+nkh
+nkh
+nk hm
+nkm
+nkm
+nkm
+nk m nk H H E E rdr H H E E rdrLWG G G G G G G G
+ε ε
+ρρ
+  
+(5.6) 
+ 
+6. Интегралы (5.6) вычислены  в Приложениях  E1, E2. 
+()
+() ()*
+4* * *
+2* **
+2 22 2
+22 2
+* * ) (
+21
+4214
+nk nkm
+z z z z nk nk I z z z z mnk nkI I m
+z z z z mm
+nk
+pinhe h e hLDph h e eLpD D n
+ph h e eLW
+γ γω εγ γ εγ γρρ ω εε
+⋅ ⋅⋅⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ++ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⋅⋅ − −+ ⋅⋅ ⋅⋅ ⋅ + ⋅ ⋅ ⋅ − =
+  
+(6.1) 
+()
+() ()*
+4* * *
+2* **
+2 22 2
+22 2
+* * ) (
+2 4214
+nk nkh
+z z z z nk nkK
+z z z z hnk nkK K h
+z z z z hh
+nk
+qinhh e h eL
+qDh h e eLqD D n
+qh h e eLW
+γ γω εγ γ εγ γρρ ω εε
+⋅ ⋅⋅⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ −⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛ − −+ ⋅⋅ ⋅⋅ ⋅ + ⋅ ⋅ ⋅ =
+  
+(6.2) 
+ 
+Полная  энергия , переносимая  волной равная  сумме  выражений  (6.1) и (6.2). 
+Определим  канонические  переменные  
+ 7ωγ γω ω nk t i
+nkt i
+nk nk e e qΘ⋅ + ⋅ =−) (* nkt i
+nkt i
+nk nk e e i pΘ ⋅ − ⋅ − =−) (*ω ωγ γ   
+(6.3) 
+где величина   выбирается  так, чтобы   nkΘ
+* ) ( ) (2nk nk nkh
+nkm
+nk W W γ γ⋅ ⋅ Θ ⋅ = + (6.4) 
+Очевидно , что   
+nk nkp q=  nk nk nk q p2ω− =  (6.5) 
+Теперь  энергия , колебаний  на данной  моде  выражается  через  : nk nkq p,
+( )2 2 2
+21
+nk nk nk nk p q W + ⋅ =ω  (6.6) 
+Функция  Гамильтона  системы  поверхностных  волн  имеет вид: 
+( ) ∑ ⋅ + =
+k nnk nk nkq p H
+,2 2 2
+21ω  (6.7) 
+ 
+ 
+7. При переходе  к квантовой  теории  мы должны рассматривать  канонические  
+переменные   как операторы с правилом  коммутации : nk nkq p,
+=i p q q pnk nk nk nk−=−  (7.1) 
+Теперь  мы можем выразить  через  операторы k kp q, компоненты  напряженности  и 
+потенциала  электромагнитного  поля . Учитывая  (6.3) находим : 
+) (
+2) () , () (
+) (
+nk nk nk
+nku
+nk u
+nk q p ir ez t E G Kω+ ⋅
+Θ=  
+) (
+2) () , () (
+) (
+nk nk nk
+knu
+nk u
+nk q p ir hz t H GG
+ω+ ⋅
+Θ=   
+ 
+ 
+(7.2) 
+ 
+) (
+2) () , () (
+) (
+nk nk nk
+nku
+nk u
+nk q p ir ar t A G G
+ω+ ⋅
+Θ=   
+(7.3) 
+ 
+где величины  , ) () (r eu
+nkG) () (r hu
+nkG
+ вычисляются  из (4.1)-(4.4). Ниже  нам понадобится  
+значение  потенциала  вне стержня . Найдем  ) () (r ah
+nkK с помощью  (4.7): 
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅ =) () ( '
+) () ( 1) (2) (
+,ρ ρω
+ω q Kqr Keqih
+q Kqr Khq rn
+ir a
+nn
+z
+nn
+zh
+nk r  
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅+ ⋅ ⋅⋅⋅ =) () ( '
+) () ( 1) (2 2) (
+,ρω
+ρ ωϕq Kqr Khq ri
+q Kqr Keq rnh
+ir a
+nn
+z
+nn
+zh
+nk 
+) ( ) ( ) () (
+, ρωq K qr Kier an nz h
+nk z ⋅ =   
+ 
+ 
+(7.4) 
+ 
+8. Гамильтониан  системы  плазмонов  имеет вид вполне  аналогичный  (6.7): 
+()∑ ⋅ + =
+kk k kq p H2 2 2
+21  ω   
+(8.1) 
+ 8 Гамильтониан  распадается  на сумму  гармонических  осцилляторов. Собственные  
+значения  такого  гамильтониана  хорошо известны : 
+k n n Eω=⋅⎟⎠⎞⎜⎝⎛+ =21 (8.2) 
+А для матричных  элементов  операторов k kp q, переходов  между  собственными  
+состояниями  гамильтониана  можно  получить  соотношения :   
+k
+kk k k k k k n n q n n q n ⋅ = − = −ω21 1=   
+kk
+k k k k k k n i n p n n p n⋅ = − − = −21 1ω=    
+ 
+(8.3) 
+ 
+Вместо  канонических  операторов k kp q, удобно  использовать  их линейные  
+комбинации , имеющие  только  по одному  отличному  от ноля матричному  элементу  
+для переходов  между  собственными  состояниями  гамильтониана  (6.4): 
+()k k k
+kk p i q c
+=+ ⋅ =ω
+ω21 (k k k
+kk p i q c )
+=− ⋅ =+ω
+ω21  
+(8.4) 
+Несложно  показать , что   
+k k k k n n c n= −1  k k k k n n c n= −+1 (8.5) 
+Правило  коммутации  для  +
+k kc c, принимает вид: 
+1= −+ +
+k k k kc c c c (8.6) 
+Таким  образом  мы можем  рассматривать  операторы +
+k kc c, как операторы  
+уничтожения  и рождения  плазмона .  
+Оператор  вторично  квантованного  потенциала  поля плазмона  принимает вид: 
+()∑ + − − ⋅ ⋅ + − + ⋅ ⋅Θ=+
+nkh
+nkh
+nk
+nknk ht i in ihz r a c t i in ihz r a c A ) exp( ) ( ) exp( ) (2* ) ( ) ( ) (ω ϕ ω ϕω G G=G (8.6) 
+ 
+9. Вычислим  вероятность  излучения  плазмона  в единицу  времени  излучателем , 
+расположенным  вблизи  металлического  цилиндра . Размеры  излучателя  будем  считать  
+малыми , сравнительно  с длиной  волны  плазмона , и вероятность  вычислим  в 
+дипольном  приближении . Будем  следовать  методу  расчета  вероятности  спонтанного  
+излучения , изложенному  в /8,§45/ для  случая  излучателя  в свободном  пространстве . 
+Взаимодействие  электромагнитного  поля с зарядом  описывается  оператором : 
+∫∫∫⋅ ⋅ =μ
+μj A dV e V  (9.1) 
+Здесь   ⎯ заряд  электрона , eμ
+μj A
+,⎯ вторично  квантованные  операторы потенциала  
+электромагнитного  поля  и “плотности  тока электрона ”. Плотность  тока выражается  
+через  оператор  волновой  функции и матрицы  Дирака : 
+ψ γ ψμ μ=j  (9.2) 
+Интегрирование  в (9.1) выполняется  по всему  3-пространству . 
+Согласно  теории  возмущений  вероятность  перехода  в единицу  времени  
+системы  из состояния , описываемого  набором  квантовых  чисел  i, в состояние  f  
+выражается  формулой : 
+ 9) (2 2ω δπ==− − ⋅ =i f fi fi E E V w  (9.3) 
+Матричный  элемент  оператора  возмущения   вычисляется  с помощью  
+невозмущенных  волновых  функций  состояний  i и f. Будем  считать  для простоты , что 
+как излучатель , так и поле  имеют дискретный  спектр  состояний . fiV
+В дипольном  приближении  считают  величины   медленно  
+меняющимися  в пределах  характерных  размеров  излучателя  и выносят  их из-под 
+знака  интеграла  (9.1). В результате  матричный  элемент  оператора  (9.1) приобретает  
+вид: ) , , (z y x Aμ
+∫∫∫⋅ ⋅ ⋅ − =i f nk em nk nk nk dV r A e i V f ψ γ γ ψκG GG 0 *0 ) ( 1 0 1  (9.4) 
+здесь nk nkA0 1G
+ ⎯ матричный  элемент , соответствующий  спонтанному  излучению : 
+) exp( ) (20 ) ( 1* ) (t i in ihz r a r Aemh
+nk
+nknk
+nk em nk ω ϕω+ − − ⋅ ⋅Θ=G G = GG
+  
+(9.5) 
+а через   обозначены  координаты  точки , где находится  излучатель . Интеграл  по 
+объему  в (9.4) в нерелятивистском  приближении  есть матричный  элемент  скорости  
+электрона , вычисляемый  по нерелятивистским  собственным  функциям . Матричный  
+элемент  скорости  связан  с матричным  элементом  координаты  и частотой  перехода : 
+. Вводя  дипольный  момент  системы  emrG
+fi fi r i vG Gω− = r e dGG
+= , находим  для квадрата  модуля  
+матричного  элемента: 
+nknk
+emh
+nk fi nk nk r a d i V fΘ⋅ ⋅ =2) ( 0 132* ) (2 ω=GGG  (9.6) 
+ 
+10. Будем  считать , что излучатели  вблизи  цилиндра  ориентированы  хаотично . 
+Усредним  выражение  (9.6) по направлению  дипольного  момента  излучателя .  
+() ( )z z y y x x z z y y x x emh
+nk fi a d a d a d a d a d a d r a d* * * * * *2* ) () ( + + ⋅ + + = ⋅GGG
+  
+(10.1) 
+Учитывая , что  
+0* * *= = =z y z x y xd d d d d d  2* * *
+31
+fi z z y y x x d d d d d d dG
+⋅ = = =  (10.2) 
+Находим  для среднего  значения : 
+2* ) (2 2* ) () (31) (emh
+nk fi emh
+nk fi r a d r a dGG G GG
+⋅ ⋅ = ⋅   
+(10.3) 
+Итак, для  вероятности  перехода  получаем : 
+) ( ) (32* ) (23
+, ω δω π=GG− − ⋅ ⋅Θ ⋅⋅=i f emh
+nk fi
+nknk fi E E r a d w  (10.4) 
+ 
+11. Для вычисления полной  вероятности  перехода  с излучением  плазмона  
+просуммируем  величину  (10.3) по  всем  модам , определяемым  набором  . Сумму  
+ряда  по дискретному  набору  состояний  заменим  интегралом , учитывая  при этом , что k n,
+ 10на каждый  интервал  значений   приходится  hΔ) / 2 (LhNnπΔ= Δ  состояний  поля, где  
+⎯ длина  выбранного  отрезка  металлического  цилиндра . L
+Таким  образом  мы приходим  к выражению  для вероятности  спонтанного  
+перехода : 
+) ( ) (3 22* ) (23
+, ω δω π
+π=G G G
+− − ⋅ ⋅ ⋅Θ⋅ ⋅⋅=∫∞
+∞ −i f emh
+nk fi
+nkn fi E E r a ddh Lw   
+(11.2) 
+Переходя  в (11.2) к интегрированию по ωdи вычисляя  интеграл  δ-функции , найдем  
+2* ) (32
+, ) (6emh
+nk
+nkfi n f r ahdLwGG G
+=⋅Θ⋅⎟⎠⎞⎜⎝⎛
+∂∂⋅ ⋅ =ω
+ω  
+(11.3) 
+Сравним  полученное  выражение  с известной  формулой  вероятности  
+дипольного  излучения  в свободном  пространстве  /8,§45/: 
+23
+34
+fi ph d wG
+=⋅ =ω  
+(11.4) 
+Введем  фактор  , характеризующий  возрастание  вероятности  спонтанного  
+излучения  плазмона  с азимутальным  числом   вблизи  тонкого  металлического  
+цилиндра , относительно  вероятности  излучения  фотона : F
+n
+2* ) ( ,) (81
+emh
+nk
+nk phn fi
+n r aL h
+wwFG G⋅Θ⋅⎟⎠⎞⎜⎝⎛
+∂∂⋅ = =ω  
+(11.5) 
+Ниже  приведены  дисперсионные  кривые  ) (kω  и значения  величины   при 
+различных  значениях  параметров . nF
+Автор  благодарен  В.С.Зуеву  и А.М.Леонтовичу  за внимание  к данной  работе  и 
+ряд замечаний . 
+ 
+     (a)                                                                                         (b) 
+Рис. 2. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+Волновой  вектор  в единицах  , частота  в . При 
+расчете  приняты  значения :  F
+1 , 0=n ] [1−nm ] [eV
+. 5 ; 2 ; 1 . 9nm V eh pl = = = ρ ε ω
+ 
+ 11 
+     (a)                                                                                         (b) 
+Рис. 3. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+Волновой  вектор  в единицах  , частота  в . 
+При расчете  приняты  значения : F
+. 5 , 4 , 3 , 2=n ] [1−nm ] [eV
+. 5 ; 2 ; 1 . 9nm V eh pl == = ρε ω  
+ 
+   (a)                                                                                         (b) 
+ 
+Рис. 4. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+. Волновой  вектор  в единицах  , частота  в . При 
+расчете  приняты  значения : F
+1 , 0=n ] [1−nm ] [eV
+. 20 ; 2 ; 1 . 9nm V eh pl == = ρε ω  
+ 
+ 
+ 
+ 12 
+ 
+ 
+     (a)                                                                                         (b) 
+Рис. 5. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+Волновой  вектор  в единицах  , частота  в . 
+При расчете  приняты  значения : F
+. 5 , 4 , 3 , 2=n ] [1−nm ] [eV
+. 20 ; 2 ; 1 . 9nm V eh pl == = ρε ω  
+ 
+     (a)                                                                                         (b) 
+Рис. 6. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+Волновой  вектор  в единицах  , частота  в . 
+При расчете  приняты  значения :  F
+. 5 , 4 , 3 , 2=n ] [1−nm ] [eV
+. 200 ; 2 ; 1 . 9nm V eh pl = = = ρ ε ω
+ 
+ 13   (a)                                                                                         (b) 
+ 
+Рис. 7. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+. Волновой  вектор  в единицах  , частота  в . При 
+расчете  приняты  значения : F
+1 , 0=n ] [1−nm ] [eV
+. 2 ; 6 ; 1 . 9nm V eh pl == = ρε ω  
+ 
+   (a)                                                                                         (b) 
+ 
+Рис. 8. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+. Волновой  вектор  в единицах  , частота  в . 
+При расчете  приняты  значения : F
+5 , 4 , 3 , 2=n ] [1−nm ] [eV
+. 2 ; 6 ; 1 . 9nm V eh pl == = ρε ω  
+ 
+ 14   (a)                                                                                         (b) 
+ 
+Рис. 9. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+. Волновой  вектор  в единицах  , частота  в . При 
+расчете  приняты  значения :  F
+1 , 0=n ] [1−nm ] [eV
+. 5 ; 6 ; 1 . 9nm V eh pl = = = ρ ε ω
+ 
+   (a)                                                                                         (b) 
+ 
+Рис. 10. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+. Волновой  вектор  в единицах  , частота  в . 
+При расчете  приняты  значения :  F
+5 , 4 , 3 , 2=n ] [1−nm ] [eV
+. 5 ; 6 ; 1 . 9nm V eh pl = = = ρ ε ω
+ 
+ 15 
+   (a)                                                                                         (b) 
+Рис. 11. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+. Волновой  вектор  в единицах  , частота  в . При 
+расчете  приняты  значения :  F
+1 , 0=n ] [1−nm ] [eV
+. 20 ; 6 ; 1 . 9nm V eh pl = = = ρ ε ω
+   (a)                                                                                         (b) 
+ 
+Рис. 12. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+. Волновой  вектор  в единицах  , частота  в . 
+При расчете  приняты  значения : F
+5 , 4 , 3 , 2=n ] [1−nm ] [eV
+. 20 ; 6 ; 1 . 9nm V eh pl == = ρε ω  
+ 
+ 
+ 16 
+   (a)                                                                                         (b) 
+Рис. 13. Зависимость  от волнового  вектора  частоты  резонансного  
+плазмона  (а) и фактора   (b) для цилиндрических  гармоник  с 
+. Волновой  вектор  в единицах  , частота  в . При 
+расчете  приняты  значения :  F
+1 , 0=n ] [1−nm ] [eV
+. 200 ; 6 ; 1 . 9nm V eh pl = = = ρ ε ω
+ 
+ 
+ 
+ 
+Литература . 
+ 
+1. M.A.Noginov, G.Zhu, A.M.Belgrave, R.Ba kker, V.M.Shalaev, E.E.Narimanov, S.Stout, 
+E.Herz, T.Suteewong, U.Wiesner. Nature, v.460,1110-1113(2009) 
+2. R.F.Oulton, V.J.Sorger, T.Zentgraf, R.-M.M a, C.Gladden, L.Dai, G.Bartal, X.Zhang. 
+Nature, Advance online pu blication nature08364 
+3. E. M. Purcell, Phys. Rev. v.69(11-12), p.681(1946). 
+4. В.С.Зуев. Оптика  и спектроскопия , т.102, стр . 809-820 (2007) 
+5. Kneipp K., Wang Y., Kneipp H., Itzkan I., Dazari R.R., Feld M.S. , Phys. Rev. Lett. 1996. 
+V. 76. P. 2444. 
+6. Nie S., Emory S.R. // Science. 1997. V. 275. P. 1102–1106. 
+7. В. С. Зуев , arxiv:0910.2394 13 October 2009. 
+8. V.V.Lidsky ,  arxiv:0911.1545 8 November 2009. 
+9. В.Б.Берестецкий , Е.М.Лифшиц , Л.П.Питаевский , Квантовая  электродинамика , 
+М.:”Наука ”,1989. 
+ 17 
+Приложение  A. 
+Из (2.5) и (2.9) находим : 
+ϕ ϕ ωωεH i Er iH ihr Hrin
+rih
+z r z ⋅ = ∂ −−⋅⎟⎠⎞⎜⎝⎛⋅ −1 1 
+ϕ ϕ ωωεH r i EhH ihr Hrin
+z r z ⋅ − = ∂ + ⋅ ⎟
+⎠⎞⎜
+⎝⎛⋅ −  
+ϕ ϕ ε ω ωε H r i E H r ih Hrinh
+z r z ⋅ − = ∂ ⋅ + ⋅ −2 2 
+ϕε ω ωε H r h i E Hrinh
+z r z ⋅ ⋅ − = ∂ ⋅ +) (2 2 
+z r z Er hiHh rnhH ∂ ⋅⋅ −−− ⋅=) ( ) (2 2 2 2 2ε ωωε
+ε ωϕ  
+) (
+2) (
+2 2) ( m
+z rm m
+zmEr piHp rnhH ∂ ⋅⋅−⋅=ωε
+ϕ  
+) ( ' ) (1 3 2 2) (pr I Cr pipr I Cp rnhHnm
+nm⋅ ⋅⋅− ⋅⋅=ωε
+ϕ  
+) ( ' ) (2 4 2 2) (qr K Cr qiqr K Cq rnhHnh
+nh⋅ ⋅⋅− ⋅⋅=ωε
+ϕ  
+ 
+Из (2.6) и (2.8): 
+ϕ ϕ ωεωE i HrE hr Ern
+rih
+z r z ⋅ − = ∂ − ⎟⎠⎞⎜⎝⎛⋅ −1 
+ϕ ϕ ε ω ωrE H i E hr Ernhz r z ⋅ − = ∂ + ⎟
+⎠⎞⎜
+⎝⎛⋅ −2 
+()ϕε ω ω rE h H i Ernh
+z r z ⋅ − = ∂ +2 2 
+() ()z r z Hh riEh rnhE ∂− ⋅+− ⋅=ε ωω
+ε ωϕ 2 2 2 2 2 
+) (
+2) (
+2 2) ( m
+z rm
+zmHp riEp rnhE ∂⋅+⋅=ω
+ϕ  
+) ( ' ) (3 1 2 2) (pr I Cp ripr I Cp rnhEn nm⋅⋅+ ⋅⋅=ω
+ϕ  
+ 18) ( ' ) (4 2 2 2) (qr K Cq riqr K Cq rnhEn nh⋅⋅+ ⋅⋅=ω
+ϕ  
+ 
+Из (2.5) и (2.8)  находим : 
+r z E i H ihr Hrin⋅ − = ⋅ −ωεϕ  
+r z H i E ihr Erin⋅ = ⋅ −ωϕ  
+⎟⎠⎞⎜⎝⎛⋅ + − ⋅ =ϕωεH hr HrnEz r1 
+ϕω ωEhrErnHz r ⋅ − = 
+Подставляя  (2.9) и (2.6): 
+ϕ ωε H hr HrnEz r ⋅ + − = ⋅ 
+ϕ ω E hr ErnHz r ⋅ − = 
+⎟⎠⎞⎜⎝⎛∂ − ⋅ + − = ⋅z r r z r ErErih
+ihrHrnE1
+ωωε  
+⎟⎠⎞⎜⎝⎛∂ − ⋅−− =z r r z r HrHrih
+ihrErnH1
+ωεω  
+z r r z r E h E ih Hrn iE i ∂ ⋅ − + − = ⋅2 2 ωε ω  
+( )z r r z r H h H ih Ern iH i ∂ − −−= −2 2ωεεω  
+()z r z r E h Hrn iE h i ∂ ⋅ + = ⋅ −ωε ω2 2 
+()z r z r H h Ern iH h i ∂ +−= −ωεεω2 2 
+ 
+Приложение  B. 
+Преобразуем  систему  (3.7) 
+01 2 2 3 2 2= ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ − ⋅ + ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− C DpiDqiCqnh
+pnh
+Im
+Khωε ωε 
+ 1901 2 2 3 2 2= ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− + ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ − ⋅ Cqnh
+pnhC DqiDpi
+K Iω ω 
+ 
+()( ) 012 2
+32 2= ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ + ⋅ − ⋅C D q D p i C p q nhI m K h ε ε ω  
+( )() 012 2
+32 2= ⋅ − ⋅ + ⋅ ⋅ − ⋅ ⋅C p q nh C D p D q iK Iω  
+ 
+() ( ) 012 2
+3 = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ + ⋅ − ⋅ ⋅C D q D p i C nhI m K h h m ε ε ε ε ω  
+( ) () 01 32 2= ⋅ − ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅C nh C D p D q ih m K I ε ε ω  
+() ( ) ()02 2 2 2 2 2 2 2= − ⋅ ⋅ − ⋅ − ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅h m I K I m K h h n D q D p D q D p ε ε ω ε ε  
+ 
+Приложение  C. 
+Преобразуем  второе  из уравнений   (3.7), с учетом  третьего  и четвертого  
+ 
+ 3 2 1 2 3 2 1 2C DqiCqnhC DpiCpnh
+K I ⋅ ⋅ + ⋅ = ⋅ ⋅ + ⋅ω ω 
+( ) 0 ) (32 2
+12 2= ⋅ ⋅ − ⋅ ⋅ + ⋅ − ⋅C D p D q i C p q nhK Iω  
+( )32 2
+1) ( C D p D q i C nhK I m h ⋅ ⋅ − ⋅ ⋅ = ⋅ − ⋅ ⋅ ε ε ω  
+( )32 2
+1) ( C D p D q i C nhK I m h ⋅ ⋅ − ⋅ ⋅ = ⋅ − ⋅ ⋅ ε ε ω  (С.1) 
+Решение  (С.1) выберем  в виде  
+γεεω ⋅−⋅⋅⋅ − = = ) (4 3 m h nh i C C  
+( )γ⋅ ⋅ − ⋅ = =K I D p D q C C2 2
+2 1   
+(С.2) 
+Из (3.5) находим : 
+γ ρω ⋅ ⋅ = ) ( ) ( ) () (
+, p I pr I e r En n zm
+nk z  
+γ ρω ⋅ ⋅ = ) ( ) ( ) () (
+, q K qr K e r En n zh
+nk z  
+γ ρω ⋅ ⋅ = ) ( ) ( ) () (
+, p I pr I h r Hn n zm
+nk z  
+γ ρω ⋅ ⋅ = ) ( ) ( ) () (
+, q K qr K h r Hn n zh
+nk z   
+ 
+(C.3) 
+Где  
+ 
+( )K I z D p D q e⋅ − ⋅ =2 2  
+(С.4) 
+ 20) (m h z nh i h εεω−⋅⋅⋅ − =  
+Из (3.6) легко  находятся  компоненты  : ϕ ϕH E,
+γρωε
+ρω ϕ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅− ⋅ ⋅⋅=) () ( '
+) () () (2 2) (
+,p Ipr Ier pi
+p Ipr Ihp rnhr H
+nn
+zm
+nn
+zm
+nk  
+γρωε
+ρω ϕ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅− ⋅ ⋅⋅=) () ( '
+) () () (2 2) (
+,q Kqr Ker qi
+q Kqr Khq rnhr H
+nn
+zh
+nn
+zh
+nk  
+γρω
+ρω ϕ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅+ ⋅ ⋅⋅=) () ( '
+) () () (2 2) (
+,p Ipr Ihp ri
+p Ipr Iep rnhr E
+nn
+z
+nn
+zm
+nk  
+γρω
+ρω ϕ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅+ ⋅ ⋅⋅=) () ( '
+) () () (2 2) (
+,q Kqr Khq ri
+q Kqr Keq rnhr E
+nn
+z
+nn
+zh
+nk   
+ 
+ 
+ 
+(С.5) 
+ 
+Из (2.5) и (2.8) находим :  
+r z E i H ihr Hrin⋅ − = ⋅ −ωεϕ    
+r z H i E ihr Erin⋅ = ⋅ −ωϕ  
+ϕεω εωHhrHrnEz r ⋅ +⋅− =    
+ϕω ωEhrErnHz r ⋅ − =  
+(С.6) 
+ 
+γρωε
+ρ ω ε ρ ω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅− ⋅ ⋅⋅⋅ + ⋅ ⋅⋅− =) () ( '
+) () (
+) () (
+2 2) (
+p Ipr Ier pi
+p Ipr Ihp rnh hr
+p Ipr IhrnE
+nn
+zm
+nn
+z
+m nn
+z
+mm
+r  
+γρ ρ ω ε ρ ω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅ + ⋅ ⋅⋅− =) () ( '
+) () (
+) () (
+2 2) (
+p Ipr Iepih
+p Ipr Ihp rnh hr
+p Ipr IhrnE
+nn
+z
+nn
+z
+m nn
+z
+mm
+r  
+γρ ρ ω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ − =) () ( '
+) () (122
+) (
+p Ipr Iepih
+p Ipr Ihrn
+phE
+nn
+z
+nn
+z
+mm
+r  
+γρ ρω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅=) () ( '
+) () (
+2) (
+p Ipr Iepih
+p Ipr Ihp rnE
+nn
+z
+nn
+zm
+r  
+ 
+ 
+ 21γρωε
+ρ ω ε ρ ω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅− ⋅ ⋅⋅⋅ + ⋅⋅− =) () ( '
+) () (
+) () (
+2 2) (
+q Kqr Ker qi
+q Kqr Khq rnh hr
+q Kqr KhrnE
+nn
+zh
+nn
+z
+h nn
+z
+hh
+r  
+γρ ρ ω ε ρ ω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅ + ⋅⋅− =) () ( '
+) () (
+) () (
+2 2) (
+q Kqr Keqih
+q Kqr Khq rnh hr
+q Kqr KhrnE
+nn
+z
+nn
+z
+h nn
+z
+hh
+r  
+γρ ρ ω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ − =) () ( '
+) () (122
+) (
+q Kqr Keqih
+q Kqr Khrn
+qhE
+nn
+z
+nn
+z
+hh
+r  
+γρ ρω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅=) () ( '
+) () (
+2) (
+q Kqr Keqih
+q Kqr Khq rnE
+nn
+z
+nn
+zh
+r  
+ 
+γρω
+ρ ω ρ ω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅+ ⋅ ⋅⋅⋅ − ⋅ =) () ( '
+) () (
+) () (
+2 2) (
+p Ipr Ihp ri
+p Ipr Iep rnh hr
+p Ipr IernH
+nn
+z
+nn
+z
+nn
+zm
+r  
+γρ ρ ω ρ ω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅ − ⋅ =) () ( '
+) () (
+) () (
+2 2) (
+p Ipr Ihpih
+p Ipr Iep rnh hr
+p Ipr IernH
+nn
+z
+nn
+z
+nn
+zm
+r  
+γρ ρ ω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− =) () ( '
+) () (122
+) (
+p Ipr Ihpih
+p Ipr Iern
+phH
+nn
+z
+nn
+zm
+r  
+γρ ρω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅− =) () ( '
+) () (
+2) (
+p Ipr Ihpih
+p Ipr Iep rnH
+nn
+z
+nn
+zm m
+r  
+ 
+γρω
+ρ ω ρ ω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅+ ⋅ ⋅⋅⋅ − ⋅ ⋅ =) () ( '
+) () (
+) () (
+2 2) (
+q Kqr Khq ri
+q Kqr Keq rnh hr
+q Kqr KernH
+nn
+z
+nn
+z
+nn
+zh
+r  
+γρ ρ ω ρ ω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅ − ⋅ ⋅ =) () ( '
+) () (
+) () (
+2 2) (
+q Kqr Khqih
+q Kqr Keq rnh hr
+q Kqr KernH
+nn
+z
+nn
+z
+nn
+zh
+r  
+γρ ρ ω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− =) () ( '
+) () (122
+) (
+q Kqr Khqih
+q Kqr Kern
+qhH
+nn
+z
+nn
+zh
+r  
+γρ ρω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅− =) () ( '
+) () (
+2) (
+q Kqr Khqih
+q Kqr Keq rnH
+nn
+z
+nn
+zh h
+r  
+ 
+Окончательно , находим : 
+ 
+ 22γρ ρω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅=) () ( '
+) () (
+2) (
+p Ipr Iepih
+p Ipr Ihp rnE
+nn
+z
+nn
+zm
+r  
+γρ ρω⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅=) () ( '
+) () (
+2) (
+q Kqr Keqih
+q Kqr Khq rnE
+nn
+z
+nn
+zh
+r  
+γρ ρω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅− =) () ( '
+) () (
+2) (
+p Ipr Ihpih
+p Ipr Iep rnH
+nn
+z
+nn
+zm m
+r  
+γρ ρω ε⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅⋅− =) () ( '
+) () (
+2) (
+q Kqr Khqih
+q Kqr Keq rnH
+nn
+z
+nn
+zh h
+r   
+ 
+ 
+ 
+(С.7) 
+ 
+Приложение  D. 
+Рассмотрим  компоненты  вектора  Пойнтинга  
+[]H E SG G G
+× ⋅ =π41 
+[][] ()nk nk nk nk nk H E H E SGGG G G
+× + × ⋅ =* *
+41
+π 
+()* *Re42
+z r r zH E H ErS ⋅ − ⋅ ⋅ =πϕ  
+Как видно  из (4.2) и (4.4) при , . 0=n 0=ϕS
+()* *Re42
+ϕ ϕπrH E H rE Sz z r ⋅ − ⋅ =  
+()* *Re42
+ϕ ϕπrH E H rE Sz z r ⋅ − ⋅ =  
+из (4.1),(4.2),(4.3) видно , что  ⎯ вещественны , а ⎯ мнимы. Значит  
+. zE E,ϕ zH H,ϕ
+0=rS
+Приложение  E1. 
+Вычисление  интеграла  (5.6) в металле . 
+*
+, ,*
+, ,2 *
+, ,* ) ( ) (
+nk z nk z nk nk nk r nk rm
+nkm
+nk E E E E r E E E E⋅ + ⋅ ⋅ + ⋅ = ⋅ϕ ϕGG
+ 
+* *
+2*
+2 2 22* *
+2 2* 2 * ) ( 2 * ) ( ) (
+) ( ' ) ( ) ( ' ) () ( ') () ( ') () ( ) (
+nk nkn
+zn
+zn
+zn
+zn zn
+z n zn
+znk nk nm
+z z nm
+nkm
+nk
+rpr Ihpi
+rpr Iepnh
+rpr Ihpi
+rpr Iepnhrpr I epih
+rpr Ihpnpr I epih
+rpr Ihpnpr I e e p I E E
+γ γω ωω ωγ γ ρ
+⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ + ⋅ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ + ⋅ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅ +⋅ ⋅ ⋅ ⋅ = ⋅ ⋅GG
+ 
+ 23()
+()⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ + ⋅ ⋅ ⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ + ⋅ ⋅ ⋅ +⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅−
+) ( ') () ( ') () ( ') () ( ') () ( ) (
+2 2* *2
+22
+2 22
+22 2
+* 2
+22
+2 22
+22 2
+*2 *1* 2 * ) ( ) (
+pr Ip rpr I
+pnhpr Iph
+rpr I
+pni h e e hpr Ip r ppr I
+pnh h pr Iph
+r ppr I
+ph ne epr I e e p I E E
+nn
+nn
+z z z znn
+z z nn
+z zn z z nk nk nm
+nkm
+nk
+ω ωω ωγ γ ρG G
+ 
+ 
+()
+()⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ + ⋅ ⋅ +⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+) ( ') (2) ( ') () ( ) (
+2* *2
+2 22
+2
+22
+*
+22
+*2 *1* 2 * ) ( ) (
+pr Ip rpr I
+pnhi h e h epr Ir ppr Inph hphe epr I e e p I E E
+nn
+z z znn
+z z z zn z z nk nk nm
+nkm
+nk
+ωωγ γ ρG G
+ 
+ 
+*
+, ,*
+, ,2 *
+, ,) ( ) (
+nk z nk z nk nk nk r nk rm
+nkm
+nk H H H H r H H H H⋅ + ⋅ ⋅ + ⋅ = ⋅ϕ ϕGG
+ 
+()
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ + ⋅ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅ ⋅+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ + ⋅ ⋅⋅ ⋅− ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅⋅ ⋅−+ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+rpr Iepi
+rpr Ihpnh
+rpr Iepi
+rpr Ihpnhrpr I hpih
+rpr Iepnpr I hpih
+rpr Iepnh pr I h p I H H
+n
+zm n
+zn
+zm n
+zn zn
+zm
+n zn
+zmz n z nk nk nm
+nkm
+nk
+) ( ' ) ( ) ( ' ) () ( ') () ( ') () ( ) (
+*
+2*
+2 2 22* *
+2 2* 21* 2 ) ( ) (
+ωε ωεω ε ω εγ γ ρGG
+ 
+()
+()⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ + ⋅⋅⋅ ⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⋅+ ⋅⋅ ⋅⋅ ⋅+ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+) ( ') () ( ') () ( ') () ( ') () ( ) (
+2 2* *2
+22
+2 22
+22 2
+* 2
+22 2
+2 22
+22 2 2
+** 21* 2 ) ( ) (
+pr Iph
+rpr I
+pnpr Ip rpr I
+pnhi e h e hpr Iph
+r ppr I
+ph nh h pr Ip r ppr I
+pne eh pr I h p I H H
+nn m
+nm n
+z z z znn
+z z nm n m
+z zz n z nk nk nm
+nkm
+nk
+ωε ωεω ε ω εγ γ ρG G
+()
+() ) ( ') (2) ( ') () ( ) (
+2* *2
+2 22
+2
+22
+* ) ( ) (
+22 2
+*2 *1* 2 ) ( ) (
+pr Ip rpr I
+pnhi e h e hpr Ir ppr Inphh hpe epr I h h p I H H
+nm n
+z z z znn m
+zm
+zm
+z zn z z nk nk nm
+nkm
+nk
+⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ +⋅⋅ ⋅+ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+ωεω εγ γ ρG G
+ 
+ 
+() ()( )
+() ) ( ') (4) ( ') () ( ) (
+2* *2
+2 22
+2
+22
+*
+22
+*
+22
+*
+22 2
+*2 * *1* 2 ) ( ) ( * ) ( ) (
+pr Ip rpr I
+pnhi e h e hpr Ir ppr Inph hphe ephh hpe epr I h h e e p I H H E E
+nm n
+z z z znn m
+z zm
+z z z zm
+z zn z z z z m nk nk nm
+nkm
+nkm
+nkm
+nk m
+⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛ ⋅⋅ ⋅ +⋅⋅ ⋅ + ⋅ ⋅ +⋅⋅ ⋅+ ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅−
+ωεω ε ε ω εε γ γ ρ εG G G G
+ 
+ 24( )( )
+()
+()*
+2* ** 2
+2 22
+2
+22 2
+* ** 2 * * 2 ) ( ) ( * ) ( ) (
+) ( ') (4) ( ') () ( ) (
+nk nk nn m
+z z z znk nk nn m
+z z z z mnk nk n z z z z m nm
+nkm
+nkm
+nkm
+nk m
+pr Irpr I
+p pnhi e h e hpr Ir ppr Inphh h e epr I h h e e p I H H E E
+γ γωεγ γω εεγ γ ε ρ ε
+⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ++ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅+ ⋅⋅ ⋅ + ⋅ ⋅+ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ + ⋅ ⋅GG G G
+  
+ 
+(E1.1) 
+ 
+ 
+Вычислим  интегралы  
+Прямым  дифференцированием  несложно  убедиться , что  
+[] ) (21) ( '2) (2
+22
+2 22
+02ax Ianx ax Ixdz az I zn nx
+n ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ + − = ⋅ ⋅ ∫ 
+Используем  известные  соотношения  между  бесселевыми  функциями : 
+) (2) ( ) (1 1 z Iznz I z In n n ⋅ = −+ −  
+) ( ' 2 ) ( ) (1 1 z I z I z In n n ⋅ = ++ −  
+ () ) ( ) ( ) (2) (1 122
+02ax I ax I ax Ixdz az I zn n nx
+n + −⋅ − ⋅ = ⋅ ⋅ ∫ 
+ 
+ 
+ 
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ + − ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ = ∫ ∫) ( 1 ) ( '2) (1) (2
+2 22
+22
+02
+2
+02
+1 ρρρρρ ρ
+p Ipnp I dz z I zpdr pr I r Fn np
+n nn
+I  
+() ) ( ) ( ) (2) (1) (1 122
+02
+2
+02
+1 ρ ρ ρρρ ρ
+p I p I p I dz z I zpdr pr I r Fn n np
+n nn
+I + −⋅ − ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ = ∫ ∫ 
+() ) ( ) (21) ( ') ( 2
+12
+12
+2 22
+2pr I pr I pr Ir ppr Inn n nn
++ −+ ⋅ = + ⋅  
+()∫ ∫⋅ + ⋅ = ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅ =+ −ρ ρ p
+n n nn n
+I dz z I z I zpdr pr Ir ppr In r F
+02
+12
+1 2
+02
+2 22
+2
+2 ) ( ) (21) ( ') ( 
+[] []⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛++ − + ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛−+ − − =+ + − − ) () 1 (1 ) ( ' ) () 1 (1 ) ( '42
+1 22
+2
+12
+1 22
+2
+12
+2 z Iznz I z Iznz I Fn n n nn
+Iρ 
+ 25⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛++ + ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛−+ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎥⎦⎤
+⎢⎣⎡+ ⋅+− +⎥⎦⎤
+⎢⎣⎡+ ⋅−− =
++ −+ −
+) () 1 (1 ) () 1 (14) ( ) () 1 () ( ) () 1 (
+4
+2
+1 22
+2
+1 22 22
+12
+12
+2
+z Iznz Iznz I z Iznz I z IznF
+n nn n n nn
+I
+ρρ
+ 
+() ) ( ) (4) ( 2 ) ( ) () 1 (2 ) ( ) () 1 (24
+2
+12
+122
+1 12
+2
+z I z Iz I z I z Iznz I z IznF
+n nn n n n nn
+I
++ −+ −
++ +⎟⎠⎞⎜⎝⎛⋅ + ⋅ ⋅+⋅ − ⋅ ⋅−⋅ − =
+ρρ
+ 
+() ) ( ) (4) ( ) ( 2 ) (2) ( 2 ) (2) ( 24
+2
+12
+121 1 1 12
+2
+z I z Iz I z I z Izz Iznz Izz IznF
+n nn n n n n nn
+I
++ −+ + − −
++ +⋅⎟⎠⎞⎜⎝⎛⋅ + ⋅ − ⋅ ⋅ − ⋅ − ⋅ ⋅ − =
+ρρ
+ 
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ +⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ + ⋅ − ⋅ − =
+) (2) ( ' 24) ( ) ( 2 ) ( '4) (4
+4
+2
+22
+2222 2
+2
+z Iznz Iz I z I z Izz IznF
+n nn n n nn
+I
+ρρ
+ 
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− + ⋅ ⋅ − − = ) ( ' ) ( ) ( ) ( '2) (22 2 2
+22 2
+2 z I z I z I z Izz IznFn n n n nn
+Iρ 
+) (21) ( ' ) (1) ( ') (2
+0 03 ρρ ρ
+p Ipdz z I z Ipdr pr Irpr Ir Fnp
+n n nn n
+I = ⋅ ⋅ = ⋅ ⎟⎠⎞⎜⎝⎛⋅ ⋅ = ∫ ∫ 
+========-==========-==============-= 
+Умножим  обе части  (E1.1) на r и проинтегрируем  от 0 до  ρ. 
+Из (5.6) видим , что  
+()
+()
+()*
+3 2* **
+2 22 2
+* **
+1* * 2 ) (
+4) (4
+nk nkn
+Im
+z z z znk nkn
+Im
+z z z z mnk nkn
+I z z z z m nm
+nk
+Fp pnhi e h e hFphh h e eF h h e e p I WL
+γ γωεγ γω εεγ γ ε ρ
+⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ++ ⋅ ⋅ ⋅+ ⋅⋅ ⋅ + ⋅ ⋅+ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅
+ 
+() ( )
+()
+()n
+Im
+z z z zn
+Im
+z z z z mn
+I z z z z m nk nk nm
+nk
+Fp pnhi e h e hFphh h e eF h h e e p I WL
+3 2* *2 22 2
+* *1* *1* 2 ) (
+4) (4
+⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ++ ⋅+ ⋅⋅ ⋅ + ⋅ ⋅+ ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+ωεω εεε γ γ ρ
+ 
+ 26() ( )
+()
+() ) (214) ( ' ) ( ) ( ) ( '2) (2) ( 1 ) ( '2) (4
+2
+2* *2 2 2
+2 22 2
+22 2
+* *2
+2 22
+22
+* *1* 2 ) (
+ρωερ ρ ρ ρρρρρ ω εερρρρε γ γ ρ
+p Ip p pnhi e h e hp I p I p I p Ipp Ipn
+phh h e ep Ipnp I h h e e p I WL
+nm
+z z z zn n n n nm
+z z z z mn n z z z z m nk nk nm
+nk
+⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− + ⋅ ⋅ − ⋅+ ⋅⋅ ⋅ + ⋅ ⋅ −−⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ + − ⋅ ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+() ( )
+()
+() () ) ( 2 ) ( ) ( '2
+2) ( ' ) ( ) (2) ( 1 ) ( '2) (4
+2
+2 2* *2
+22 2
+* *2 2 2
+2 22 2
+22 2
+* *2
+2 22
+22
+* *1* 2 ) (
+ρωερ ρρρ ω εερ ρ ρρρ ω εερρρρε γ γ ρ
+p Ip pnhi e h e h p I p Ip phh h e ep I p I p Ipn
+phh h e ep Ipnp I h h e e p I WL
+nm
+z z z z n nm
+z z z z mn n nm
+z z z z mn n z z z z m nk nk nm
+nk
+⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ + ⋅ ⋅ ⋅ ⋅+ ⋅⋅ ⋅ + ⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− + ⋅+ ⋅⋅ ⋅ + ⋅ ⋅ −−⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ + − ⋅ ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+ 
+()
+()
+() () ) ( 2 ) ( ) ( ') ( ' ) ( ) (21) (4
+2
+2 2* *
+32 2
+* *2 2 2
+2 22 2
+22 2
+* *1* 2 ) (
+ρωερ ρ ρω εερ ρ ρρρ ω εεγ γ ρ
+p Ip pnhi e h e h p I p Iphh h e ep I p I p Ipn
+phh h e ep I WL
+nm
+z z z z n nm
+z z z z mn n nm
+z z z z mnk nk nm
+nk
+⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ + ⋅ ⋅ ⋅+ ⋅⋅ ⋅ + ⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− + ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛−+ ⋅⋅ ⋅ + ⋅ ⋅ −= ⋅ ⋅ ⋅ ⋅−
+ 
+()
+()
+() () ) ( 2 ) ( ) ( ') ( ' ) ( ) () (4
+2
+2 2* *
+32 2
+* *2 2 2
+2 22
+22 2
+* *1* 2 ) (
+ρωερ ρ ρω εερ ρ ρρρ ω εεγ γ ρ
+p Ip pnhi e h e h p I p Iphh h e ep I p I p Ipn
+ph h e ep I WL
+nm
+z z z z n nm
+z z z z mn n nm
+z z z z mnk nk nm
+nk
+⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ + ⋅ ⋅ ⋅+ ⋅⋅ ⋅ + ⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− +⋅ ⋅⋅ ⋅ + ⋅ ⋅ −= ⋅ ⋅ ⋅ ⋅−
+ 
+ 
+ 
+ 
+ 
+() ( )
+() ()2 2* *
+32 2
+* *22
+2 22
+22 2
+* *1* ) (
+2) () ( ') () ( '14
+p pnhi e h e hp Ip I
+phh h e ep Ip I
+pn
+ph h e e WL
+m
+z z z z
+nn m
+z z z z mnn m
+z z z z m nk nkm
+nk
+ωε
+ρρρω εερρ
+ρρ ω εε γ γ
+⋅ ⋅ ⋅ ⋅ − ⋅ + ⋅ ⋅+ ⋅⋅ ⋅ + ⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− +⋅ ⋅⋅ ⋅ + ⋅ ⋅ − = ⋅ ⋅ ⋅−
+ 
+() ( ) ()
+() ()2 2* *
+42 2
+* *2 2 2 2
+42
+* *1* ) (
+24
+p pnhi e h e h Dphh h e eD p nph h e e WL
+m
+z z z z Im
+z z z z mIm
+z z z z m nk nkm
+nk
+ωε ω εερω εε γ γ
+⋅ ⋅ ⋅ ⋅ − ⋅ + ⋅+ ⋅⋅ ⋅ + ⋅ ⋅ ++ − +⋅⋅ ⋅ + ⋅ ⋅ − = ⋅ ⋅ ⋅−
+ 
+ 27() ( ) ()
+() ()2 2* *
+42 2
+* *2 2 2 2
+42
+* *1* ) (
+224
+p pnhi e h e h Dpph h e eD p nph h e e WL
+m
+z z z z Im
+z z z z mIm
+z z z z m nk nkm
+nk
+ωε ω εερω εε γ γ
+⋅ ⋅ ⋅ ⋅ − ⋅ + ⋅+ ⋅ ⋅⋅ ⋅ + ⋅ ⋅ ++ − +⋅⋅ ⋅ + ⋅ ⋅ − = ⋅ ⋅ ⋅−
+ 
+() ( ) ()
+() ()2 2* *
+2* *2 2 2 2
+42
+* *1* ) (
+2124
+p pnhi e h e h Dph h e eD D p nph h e e WL
+m
+z z z z I z z z z mI Im
+z z z z m nk nkm
+nk
+ωεερω εε γ γ
+⋅ ⋅ ⋅ ⋅ − ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ ++ ⋅ − − +⋅⋅ ⋅ + ⋅ ⋅ − = ⋅ ⋅ ⋅−
+ 
+() ()
+() ()*
+2 2* * *
+2* ** 2 2 2 2
+42
+* * ) (
+241
+424
+nk nkm
+z z z z nk nk I z z z z mnk nk I Im
+z z z z mm
+nk
+p pnhi e h e hLDph h e eLD D p npLh h e e W
+γ γωεγ γ εγ γ ρω εε
+⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ++ ⋅ ⋅ ⋅ − − +⋅⋅ ⋅ ⋅ + ⋅ ⋅ − =
+ 
+()
+() ()*
+4* * *
+2* **
+2 22 2
+22 2
+* * ) (
+21
+4214
+nk nkm
+z z z z nk nk I z z z z mnk nkI I m
+z z z z mm
+nk
+pinhe h e hLDph h e eLpD D n
+ph h e eLW
+γ γω εγ γ εγ γρρ ω εε
+⋅ ⋅⋅⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ++ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⋅⋅ − −+ ⋅⋅ ⋅⋅ ⋅ + ⋅ ⋅ ⋅ − =
+ 
+ 
+Приложение  E2. 
+Вычисление  интеграла  (5.6) в  диэлектрике . 
+ 
+*
+, ,*
+, ,2 *
+, ,*
+nk z nk z nk nk nk r nk r nk nk E E E E r E E E E⋅ + ⋅ ⋅ + ⋅ = ⋅ϕ ϕGG
+ 
+()
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⋅− ⋅ ⋅⋅⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⋅+ ⋅ ⋅⋅⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ + ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ − ⋅ ⋅ ++ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+) ( ' ) ( ) ( ' ) () ( ') () ( ') () ( ) (
+* *
+2 2 2 22* *
+2 22 *1* 2 * ) ( ) (
+qr K hq riqr K eq rnhqr K hq riqr K eq rnhrqr K eqih
+rqr Khqnqr K eqih
+rqr Khqnqr K e e q K E E
+n z n z n z n zn zn
+z n zn
+zn z z nk nk nh
+nkh
+nk
+ω ωω ωγ γ ρGG
+ 
+ 
+()
+() ()
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ⋅ − ⋅ ⋅⋅− ⋅ ⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ⋅ + ⋅ ⋅⋅⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ + ⋅ ⋅ ⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ + ⋅ ⋅ ⋅ ++ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+2 2*2 2*2
+22
+22
+42 2
+* 2
+22
+22
+42 2
+*2 * ) ( ) (1* 2 * ) ( ) (
+) () ( ' ) ( ') () () ( ' ) ( ') () ( ') () ( ') () ( ) (
+qnh
+rqr Kqr Kqiqr Krqr K
+qih n
+qe hqnh
+rqr Kqr Kqiqr Krqr K
+qih n
+qe hqr Kqh
+rqr K
+qh ne e qr Kq rqr K
+qnh hqr K e e qK E E
+n
+n nn
+z zn
+n nn
+z znn
+z z nn
+z znh
+zh
+z nk nk nh
+nkh
+nk
+ω ωω ωω ωγ γ ρGG
+ 
+ 28()
+() ()
+()⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅⋅ ⋅ − ⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+⋅⋅ ⋅ ⋅ ⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+⋅⋅ ⋅ ⋅ ⋅ ++ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+) ( ') (2) ( ') () ( ') () ( ) (
+2) ( * ) ( * ) ( ) (2
+2 22
+2
+22
+* 2
+2 22
+2
+22
+*2 *1* 2 * ) ( ) (
+qr Krqr K
+qh n
+qe h e h iqr Kr qqr Knqhe e qr Kr qqr Knqh hqr K e e q K E E
+nn h
+zh
+zh
+zh
+znn
+z z nn
+z zn z z nk nk nh
+nkh
+nk
+ωωγ γ ρGG
+ 
+()
+()
+() ) ( ') (2) ( ') () ( ) (
+2* *2
+2 22
+2
+22
+*
+22
+*2 *1* 2 * ) ( ) (
+qr Krqr K
+qh n
+qe h e h iqr Kr qqr Knqhe eqh hqr K e e q K E E
+nn
+z z z znn
+z z z zn z z nk nk nh
+nkh
+nk
+⋅ ⋅⋅⋅ ⋅ − ⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+⋅⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ + ⋅ ⋅ ++ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+ωωγ γ ρGG
+ 
+ 
+*
+, ,*
+, ,2 *
+, ,) ( ) (
+nk z nk z nk nk nk r nk rh
+nkh
+nk H H H H r H H H H⋅ + ⋅ ⋅ + ⋅ = ⋅ϕ ϕGG
+ 
+()
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅+ ⋅ ⋅⋅⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⋅− ⋅ ⋅⋅⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ + ⋅ ⋅⋅− ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ − ⋅ ⋅⋅− ++ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+) ( ' ) ( ) ( ' ) () ( ') () ( ') () ( ) (
+* *
+2 2 2 22* *
+2 22 * ) ( ) (1* 2 ) ( ) (
+qr K eq riqr K hq rnhqr K eq riqr K hq rnhrqr K hqih
+rqr Keqnqr K hqih
+rqr Keqnqr K h h q K H H
+n zh
+n z n zh
+n zn zn
+zh
+n zn
+zhnh
+zh
+z nk nk nh
+nkh
+nk
+ωε ωεω ε ω εγ γ ρGG
+()
+()
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ⋅ + ⋅ ⋅ ⋅⋅⋅ ⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ⋅ − ⋅ ⋅ ⋅⋅− ⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ + ⋅ ⋅ ⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⋅⋅ ⋅ ++ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+rqr K
+qnhqr Kqiqr Kqih
+rqr K
+qnh erqr K
+qnhqr Kqiqr Kqih
+rqr K
+qnh eqr Kqh
+q rqr K
+qh nh h qr Kq rqr K
+qne eqr K h h q K H H
+n
+nh
+nn h
+z zn
+nh
+nn h
+z znn
+z z nh n h
+z zn z z nk nk nh
+nkh
+nk
+) () ( ' ) ( ') () () ( ' ) ( ') () ( ') () ( ') () ( ) (
+2 2*2 2*2
+22
+2 22
+22 2
+*2 2
+2*2 *1* 2 ) ( ) (
+ωε ω εωε ω εωε ω εγ γ ρG G
+ 
+()
+() ()
+()⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ⋅⋅⋅ ⋅ − ⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅ ⋅ ⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ++ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+) ( ') (2) ( ') () ( ') () ( ) (
+2* *2
+2 22
+2
+22
+* 22 2
+*2 *1* 2 ) ( ) (
+qr Kqh
+rqr K
+qnh e h e iqr Kq rqr Knqhh h qr Kqrqr Knqe eqr K h h q K H H
+nn h
+z z z znn
+z z nn h
+z zn z z nk nk nh
+nkh
+nk
+ω εωεγ γ ρGG
+()
+()
+()⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ⋅⋅⋅ ⋅ − ⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⋅ ⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ++ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+) ( ') (2) ( ') () ( ) (
+2* *2
+2 22
+2
+22
+*2
+*2 *1* 2 ) ( ) (
+qr Kqh
+rqr K
+qnh e h e iqr Kq rqr Knqhh hqe eqr K h h q K H H
+nn h
+z z z znn
+z zh
+z zn z z nk nk nh
+nkh
+nk
+ω εωεγ γ ρG G
+ 
+ 
+ 29() () ( )
+()
+()⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ⋅⋅⋅ ⋅ − ⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅ ⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅ ⋅ ++ ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅−
+) ( ') (4) ( ') () ( ) (
+2* *2
+2 22
+2
+22
+22
+*2
+22
+*2 * *1* 2 ) ( ) ( * ) ( ) (
+qr Kqh
+rqr K
+qnh e h e iqr Kq rqr Knqh
+qh hq qhe eqr K h h e e q K H H E E
+nn h
+z z z znn
+h z zh
+h z zn z z z z h nk nk nh
+nkh
+nkh
+nkh
+nk h
+ω εωεωεεε γ γ ρ εG G G G
+ 
+ 
+() ()( )
+() ()
+()⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ ⋅ ⋅⋅⋅ ⋅ − ⋅ ⋅ ++⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅ ⋅ + ⋅ ⋅ ++ ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅−
+) ( ') (4) ( ') () ( ) (
+2* *2
+2 22
+2
+22
+22
+* *2 * *1* 2 ) ( ) ( * ) ( ) (
+qr Kqh
+rqr K
+qnh e h e iqr Kq rqr Knqh
+qh h e eqr K h h e e q K H H E E
+nn h
+z z z znn
+h z z z z hn z z z z h nk nk nh
+nkh
+nkh
+nkh
+nk h
+ω εωε εε γ γ ρ εG G G G
+  
+ 
+(E2.1)
+ 
+ 
+Вычислим  интегралы 
+[] ∫∞
+⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ − = ⋅ ⋅
+xn n n ax Kanx ax Kxdz az K z ) (21) ( '2) (2
+22
+2 22
+2 
+()⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ − ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ = ∫ ∫∞ ∞
+) ( 1 ) ( '2) (1) (2
+2 22
+22
+2
+22
+1 ρρρρ
+ρ ρp Kpnp K dz z K zpdr pr K r Fn n
+pn nn
+K  
+()∫ ∫∞
++ −∞
+⋅ + ⋅ = ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅ =
+ρ ρ pn n nn n
+K dz z K z K zpdr pr Kr ppr Kn r F ) ( ) (21) ( ') ( 2
+12
+1 22
+2 22
+2
+2  
+ρpz=  
+() ()⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛++ − + ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛−+ − ⋅ =+ + − − ) () 1 (1 ) ( ' ) () 1 (1 ) ( '42
+1 22
+2
+12
+1 22
+2
+12
+2 z Kznz K z Kznz K Fn n n nn
+Kρ 
+Воспользуемся  рекуррентными  соотношениями : 
+) (2) ( ) (1 1 z Kznz K z Kn n n ⋅ − = −+ −  
+) ( ' 2 ) ( ) (1 1 z K z K z Kn n n ⋅ − = ++ −  
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛++ + ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛−+ ⋅ −⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎠⎞⎜⎝⎛⋅++ + ⎟⎠⎞⎜⎝⎛⋅−− ⋅ =
++ −+ −
+) () 1 (1 ) () 1 (14) (1) ( ) (1) (4
+2
+1 22
+2
+1 22 22
+12
+12
+2
+z Kznz Kznz Kznz K z Kznz K F
+n nn n n nn
+K
+ρρ
+ 
+ 30() ) ( ) (4) (1) ( 2 ) ( ) (1) ( 2 ) (4
+2
+12
+1212
+122
+2
+z K z Kz Kznz K z K z Kznz K z K F
+n nn n n n n nn
+K
++ −+ −
++ ⋅ −⎟⎠⎞⎜⎝⎛⋅+⋅ ⋅ + + ⋅−⋅ − ⋅ =
+ρρ
+ 
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ −⋅⎟⎠⎞⎜⎝⎛⋅++ + ⋅−− ⋅ =+ −
+) (2) ( ' 24) ( ) (1) ( ) (1
+2
+2
+22
+221 12
+2
+z Kznz Kz K z Kznz K z KznF
+n nn n n nn
+K
+ρρ
+ 
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ −⋅⎟⎠⎞⎜⎝⎛⋅ + ⋅ + + ⋅ + ⋅ − ⋅ =+ − + −
+) ( ) ( '2) ( ) (1) (1) ( ) ( ) (2
+2
+22
+221 1 1 12
+2
+z Kznz Kz K z Kzz Kzz K z Kznz KznF
+n nn n n n n nn
+K
+ρρ
+ 
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ −⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅ − + ⋅ ⋅ =
+) ( ) ( '2) ( ) ( '2) ( ) (2
+2
+2
+22
+2222 2
+2
+z Kznz Kz K z Kzz K z KznF
+n nn n n nn
+K
+ρρ
+ 
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− ⋅ ⋅ − ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ = ) ( ' ) ( ) ( '2) ( 122 2
+22 2
+2 z K z K z Kzz KznFn n n nn
+Kρ 
+) (21) ( ' ) (1) ( ') ( 2
+2 2 3 ρ
+ρ ρq Kqdz z K z Kqdr qr Krqr Kr Fn
+qn n nn n
+K = ⋅ ⋅ = ⋅ ⋅ = ∫ ∫∞ ∞
+ 
+Из (5.6) и  (E2.1) видим , что  
+() ( )
+()
+()n
+K z z z zn
+K h z z z z hn
+K z z z z h nk nk nh
+nk
+F h e h e iFqh
+qh h e eF h h e e q K WL
+3* *2 22
+22
+* *1* *1* 2 ) (
+4) (4
+⋅ ⋅ − ⋅ ⋅ ++ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅ ⋅ + ⋅ ⋅ ++ ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+ωε εε γ γ ρ
+ 
+ 
+() ( ) ()
+()
+() ) (214) ( ' ) ( ) ( '2) ( 12) ( 1 ) ( '2) (4
+2
+2 2* *2 2
+2 22 2
+22
+22
+* *2
+2 22
+22
+* *1* 2 ) (
+ρω ερ ρ ρρρρρ ωε ερρρρε γ γ ρ
+q Kq qnhh e h e iq K q K q Kqq Kqn
+qh
+qh h e ep Kpnp K h h e e q K WL
+nh
+z z z zn n n n h z z z z hn n z z z z h nk nk nh
+nk
+⋅⋅⋅ ⋅ − ⋅ ⋅ +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− ⋅ ⋅ − ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅ ⋅ + ⋅ ⋅+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ − ⋅ ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+ 
+ 
+ 31 
+ 
+() ( ) ()
+()
+()
+() ) (214) ( ) ( '2
+2) ( ' ) ( 12) ( 1 ) ( '2) (4
+2
+2 2* *2
+22
+22
+* *2 2
+2 22 2
+22
+22
+* *2
+2 22
+22
+* *1* 2 ) (
+ρω ερ ρρρ ωε ερ ρρρ ωε ερρρρε γ γ ρ
+q Kq qnhh e h e iq K q Kq qh
+qh h e eq K q Kqn
+qh
+qh h e ep Kpnp K h h e e q K WL
+nh
+z z z zn n h z z z z hn n h z z z z hn n z z z z h nk nk nh
+nk
+⋅⋅⋅ ⋅ − ⋅ ⋅ +⋅ ⋅ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅ ⋅ + ⋅ ⋅ −⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅ ⋅ + ⋅ ⋅+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ − ⋅ ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−
+() ( )
+()
+() ) ( ) ( '2
+2) ( ' ) ( 121) (214 ) (4
+2
+22
+22
+* *2 2
+2 22 2
+22
+22
+* *2
+2 2* *1* 2 ) (
+ρ ρρρ ωε ερ ρρρ ωε ερωεγ γ ρ
+q K q Kq qh
+qh h e eq K q Kqn
+qh
+qh h e eq Kq qnhh e h e i q K WL
+n n h z z z z hn n h z z z z hnh
+z z z z nk nk nh
+nk
+⋅ ⋅ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅ ⋅ + ⋅ ⋅ −⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− + ⋅ ⋅ ⋅ + ⋅ ⋅ ++ ⋅⋅⋅ ⋅ − ⋅ ⋅ + = ⋅ ⋅ ⋅ ⋅−
+ 
+() ( )
+()
+() ) ( ) ( ') ( ' ) ( 1) (2) (4
+32 2
+* *2 2
+2 22
+2
+22
+* *2
+2* *
+21* 2 ) (
+ρ ρ ρω εερ ρρρω εερωεγ γ ρ
+q K q Kqhh h e eq K q Kqn
+qh h e eq Kqnhh e h eqiq K WL
+n nh
+z z z z hn nh
+z z z z hnh
+z z z z nk nk nh
+nk
+⋅ ⋅ ⋅+ ⋅⋅ ⋅ + ⋅ ⋅ −⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ⋅ ⋅⋅⋅ ⋅ + ⋅ ⋅ ++ ⋅⋅⋅ ⋅ − ⋅ ⋅ + = ⋅ ⋅ ⋅ ⋅−
+ 
+() ( )
+()
+()) () ( ') () ( '12 4
+32 2
+* *22
+2 22
+2
+22
+* ** *
+2 21* ) (
+ρρρω εερρ
+ρρω εεωεγ γ
+q Kq K
+qhh h e eq Kq K
+qn
+qh h e eh e h eqnh
+qiWL
+nn h
+z z z z hnn h
+z z z z hz z z zh
+nk nkh
+nk
+⋅ ⋅+ ⋅⋅ ⋅ + ⋅ ⋅ −⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− + ⋅ ⋅⋅⋅ ⋅ + ⋅ ⋅ ++ ⋅ − ⋅ ⋅⋅⋅ + = ⋅ ⋅ ⋅−
+ 
+()
+()
+()*
+32 2
+* **
+22
+2 22
+2
+22
+* ** * *
+2 2) (
+) () ( '
+4) () ( '142
+4
+nk nk
+nn h
+z z z z hnk nk
+nn h
+z z z z hnk nk z z z zh h
+nk
+q Kq K
+qhh h e eLq Kq K
+qn
+qh h e eLh e h eqnh
+qi LW
+γ γρρρω εεγ γρρ
+ρρω εεγ γωε
+⋅ ⋅ ⋅ ⋅+ ⋅⋅ ⋅ + ⋅ ⋅ ⋅ −⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛− + ⋅ ⋅⋅⋅ ⋅ + ⋅ ⋅ ⋅ ++ ⋅ ⋅ ⋅ − ⋅ ⋅⋅⋅ ⋅ + =
+ 
+ 32()
+()
+()*
+42 2
+* **
+2 22 2
+2
+22
+* ** * *
+2 2) (
+2
+4142
+4
+nk nk Kh
+z z z z hnk nkK h
+z z z z hnk nk z z z zh h
+nk
+Dqqh h e eLqD n
+qh h e eLh e h eqnh
+qi LW
+γ γω εεγ γρρω εεγ γωε
+⋅ ⋅ ⋅+ ⋅⋅ ⋅ + ⋅ ⋅ ⋅ −⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛−+ ⋅ ⋅⋅⋅ ⋅ + ⋅ ⋅ ⋅ ++ ⋅ ⋅ ⋅ − ⋅ ⋅⋅⋅ ⋅ + =
+ 
+()
+() ()*
+4* * *
+2* **
+2 22 2
+22 2
+* * ) (
+2 4214
+nk nkh
+z z z z nk nkK
+z z z z hnk nkK K h
+z z z z hh
+nk
+qinhh e h eL
+qDh h e eLqD D n
+qh h e eLW
+γ γω εγ γ εγ γρρ ω εε
+⋅ ⋅⋅⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ −⋅ ⋅⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛ − −+ ⋅⋅ ⋅⋅ ⋅ + ⋅ ⋅ ⋅ =
+ 
+ 
+ 
+ 33
\ No newline at end of file
diff --git a/1001.0109.txt b/1001.0109.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6a0b510e4a5c29b81733304ff06d48928f9eff11
--- /dev/null
+++ b/1001.0109.txt
@@ -0,0 +1,864 @@
+arXiv:1001.0109v2  [astro-ph.EP]  8 Jan 2010Astronomy& Astrophysics manuscriptno.12870 c/circleco√yrtESO 2018
+October26,2018
+The Validityof the Super-Particle Approximation during
+Planetesimal Formation
+Hanno Rein, Geoffroy Lesur, and Zo¨ eM.Leinhardt
+Universityof Cambridge, Department of AppliedMathematic s andTheoretical Physics,
+Centre for Mathematical Sciences, Wilberforce Road, Cambr idge CB30WA,UK
+e-mail:hr260@cam.ac.uk
+Submitted: 11July2009 - Revised: 18 December 2009 - Accepte d: 22December 2009
+ABSTRACT
+The formation mechanism of planetesimals in protoplanetar y discs is hotly debated. Currently, the favoured model invo lves the
+accumulationof meter-sizedobjectswithina turbulentdis c,followedbya phase ofgravitationalinstability.Atbest ,onecansimulate
+a few million particles numerically as opposed to the severa l trillion meter-sized particles expected in a real protopl anetary disc.
+Therefore, single particles are often used as super-partic les to represent a distribution of many smaller particles. I t is assumed that
+small-scale phenomena do not play a role and particle collis ions are not modelled. The super-particle approximation is not always
+validwhenappliedtoplanetesimal formationbecause thesy stemcanbemarginallycollisional(oforder onecollisionp erparticleper
+orbit). The super-particle approximation can only be valid in a collisionless or strongly collisional system, althoug h, in many recent
+numerical simulations thisis not the case.
+In this work, we present new results from numerical simulati ons of planetesimal formation via gravitational instabili ty. A scaled
+system is studied that does not require the use of super-part icles. This system is simplified for computational practica lity and proper
+identificationofimportantprocesses:1)theevolutionofp articlesisstudiedinalocalshearingbox;2)theparticle- particleinteractions
+suchasgravity,physicalcollisions,andgasdragaresolve ddirectlyassumingaconstantbackground shearflowwithout anyfeedback
+from the particles. We findthat the scaled particles canbe us ed to model the initial phases of clumping if the properties o f the scaled
+particles are chosen such that all important timescales in t he system are equivalent to what is expected in a real protopl anetary disc.
+Constraints are given forthe number of particles needed ino rder toachieve numerical convergence.
+We compare this new method to the standard super-particle ap proach. We find that the super-particle approach produces un reliable
+results that depend on artifacts such as the gravitational s oftening in both the requirement for gravitational collaps e and the resulting
+clumpstatistics.Our results show that short-range intera ctions (collisions) have tobe modelled properly.
+Key words. planet formation–planetesimals – N-body simulations –convergence –accretiondisc
+1. Overview
+Extrasolar planets have been observed around a variety of pa r-
+ent stars from pulsars to solar-type stars to M-dwarfs (see e .g.,
+Chauvinet al. 2004; Wolszczan&Frail 1992) indicating that
+planet formation is common and successful in a broad range of
+environments. However, the process of planet formation its elf
+is not directly observable, leaving theory and numerical si mu-
+lations to fill in the blanks between observationsof hot circ um-
+stellardiscsaroundyoungstarsandplanetsorbitingmatur estars.
+Oneofthemostimportantunansweredquestionsinthetheory of
+planet formationis what is the mechanismfor planetesimal f or-
+mation,i.e., the processby whichthe buildingblocksof pla nets
+areformed.
+There are two main theories for planetesimal formation:
+mutual collisions (e.g., Hayashiet al. 1977) and gravitati onal
+instability in the dust layer (Goldreich&Ward 1973). In the
+first hypothesis, dust particles grow as the result of accret ion-
+dominated collisions. Although the formation of planetesi mals
+by mutual collisions is consistent with meteoritic evidenc e, the
+collision speed between dust particles (or aggregates) mus t be
+much slower than the typical velocity dispersion in a standa rd
+protostellar disc to avoid destructive collisions (Blum&W urm
+2008). In addition, the planetesimal formation process is s o
+slow that meter-sized particles are in danger of spiraling i nto
+the star before growing large enough to decouple from the gas(Weidenschilling 1977b). Even if km-sized planetesimals w ere
+able to form, they would be in danger of being ground down
+againbymutualcollisions(Stewart& Leinhardt2009).
+Gavitational instability is often considered to be a soluti on
+to most of these problems because the intermediate sizes are
+avoidedalltogether.Inthistheory,thedustlayerbecomes dense
+enough for the Keplerian shear and velocity dispersion of th e
+dust particles to be unable to support the dust against its ow n
+self gravity. The dust then collapses into clumps that event u-
+ally cool via drag forces and mutual collisions into planete si-
+mals. Manydifferentauthorshave workedon this subject. Until
+recently, the focus has been on quiet, non-turbulent, and lo w
+density regions of the accretion disc (see e.g., Michikoshi et al.
+2009, 2007; Tangaet al. 2004). However, the turbulent gas in
+the protoplanetarynebula stirs the dust, which increases t he ve-
+locity dispersion of the dust particles. Several ideas have been
+proposedtoovercometheturbulence-inducedmixingofthed ust
+particles and create localized clumps. For example, Cuzzie t al.
+(2008) suggest that the same turbulence that stirs the dust o n
+larger scales may also collect the dust particles on small sc ales.
+AsimilarideawasproposedbyJohansenet al.(2007),inwhic h
+dust particles are localized into clumps using both turbule nce
+and the streaming instability (Youdin&Goodman 2005). The
+clumpsthenbecomegravitationallyunstable.Athirdhypot hesis
+suggests that large structures, such as vortices, may be abl e to2 HannoReinet al.:The validityof the super-particle appro ximation
+collect and protect dust particlesfrom the turbulentbackg round
+(Barge&Sommeria1995).Inthispaper,we focusonthe gravi-
+tationalcollapsein averydenseandturbulentregionofthe pro-
+toplanetarydisc. The overdensityin the dust layer, which i s ap-
+proximately two orders of magnitude higher than the standar d
+minimum-masssolarnebula,mighthavebeencreatedbyanyof
+theprocessesdescribedabove.
+Numerical simulations must be used to test models of plan-
+etesimal formation. However, the separation of scales in th e
+problemishuge.A meter-sizedobjectis morethan12 orderso f
+magnitudesmallerthantheprotoplanetarydiscthicknessf orcing
+numericists to use super-particles . These super-particles have a
+massmuchhigherthanthemassofasingledustparticle,butt he
+forces (e.g., drag forces) acting on them are equivalent to t hose
+ofasingledustparticle.Tangaet al.(2004)wereabletouse this
+approximationtosuccesfullymodelthegravitationalcoll apsein
+aregimeinwhichcollisionsareunimportant.However,when the
+surfacedensityisincreasedbytwoordersofmagnitude,asm en-
+tionedabove,thesystemdoesbecomemarginallycollisiona land
+thereforethe super-particleapproachbreaksdown.
+Michikoshietal.(2009)performasimilarsetofsimulation s,
+also ina low density,non-turbulentregionof thedisc. Alth ough
+the authors describe their particles as super-particles, t he way
+in which they accurately treat the collisions is the same as t he
+method presented in this paper. Other approaches use Monte
+Carlo type methodsto resolve statistically the outcomeof o ver-
+lappingparticles(seee.g.,Lithwick&Chiang2007).
+In the following,we lookcarefullyat thenumericalrequire -
+ments of modelling gravitational instability accurately a nd test
+the validity of using super-particles in a high density regi on.
+For simplicity, we begin with a system without turbulence an d
+assume that the surface density has already been increased b y
+turbulence or some equivalent process. We find that the super -
+particle approach gives erratic results when collisions be come
+important.
+We therefore go on and present a method that does not use
+super-particles. We simulate a scaled system in which the nu m-
+ber of particles is dramatically lower than in a real protopl ane-
+tary disc. To keep the surface density the same, the mass of in -
+dividual particles has to be increased. The drag force produ ced
+by the surrounding gas is scaled accordingly, so that the sto p-
+ping time remainsconstant.Furthermore,we increase the ph ys-
+ical size of each particle. This allows us to keep the collisi on
+timescales in simulations of di fferent particle numbers exactly
+thesame.Theserequirementsleadtoresultsthatareindepe ndent
+ofthenumberofparticles,whichistheonlyfreeparameteri nthe
+simulation,andwecallthosesimulationsconverged.There sults
+agreewithothersimulationsperformedinthecontextofSat urn’s
+rings(Goldreich&Tremaine1982;Daisaka& Ida1999).Inthi s
+paper, we argue that this method should also be used in curren t
+simulationsofplanetesimalformation.
+The outline of this paper is as follows. In Sect. 2, we de-
+fine the important timescales in the problem and show that the
+super-particleapproachbreaksdown in high density region s.In
+Sect. 3, we discuss the numerical method and the initial cond i-
+tionsofoursimulations.Results ofnumericalsimulations using
+boththesuper-particleapproachandourscalingmethodare pre-
+sented in Sect. 4. We conclude the paper with resolution con-
+straints for numerical simulations and discuss the implica tions
+for the planetesimal formationprocess throughgravitatio nal in-
+stabilityin Sect.5.2. Ordersofmagnitude
+2.1. Definitionof timescales
+The aim of this work is to simulate the dynamics of dust (or
+particles)interactinggravitationallyinsideaprotopla netarydisc.
+Eachparticleissubjectto fivedi fferentphysicalprocesses,each
+operatingonvarioustypicaltimescales:
+Stoppingtimeτs
+Each particle feels the e ffects of the surrounding gas
+through a linear drag force. This force can be written as
+Fdrag=mp
+τs(vg−vp), whereτsis the stopping time of a
+particle of mass mp,vgis the velocity of the gas, and vpis
+thevelocityoftheparticle(Weidenschilling1977a).
+Physicalcollisiontimescale τc
+Dust particles will su ffer a physical collision with another
+dust particle on a timescale of τc=(σc¯ vpn)−1, whereσc
+is the geometrical cross-section of the particles, ¯ v pis the
+particle velocity dispersion, and nis the number density of
+particles.
+Orbital timescaleτe
+The timescale associated with the particles orbitingthe ce n-
+tral object in a local shearing patch is the epicyclic period
+τe=2πΩ−1, whereΩis the angular velocity at the semi-
+majoraxisofthe shearingbox.
+In this paper, we consider two limits for the gravitational i nter-
+actionsbetweendustparticles,long-rangeandshort-rang einter-
+actions. The long-range interaction can be seen as a collect ive
+process involving interactions between clouds of particle s. On
+the other hand, the short-range interaction is important fo r re-
+solving close approaches between pairs of particles (i.e., gravi-
+tationalscattering).Forthesakeofclarity,weseparatet hesetwo
+processes.
+Gravitationalcollapsetimescale τGl
+We introducea length scale λ≫δr, whereδris the average
+distance between neighbouring dust particles. In this limi t,
+which we call the long-range interaction, the dust particle
+distribution can be approximated by a continuous density
+ρ, and one can define a gravitational timescale, which is
+of the order of the free fall time of the system, defined by
+τGl=1/√GρwhereGisthegravitationalconstant.
+Gravitationalscatteringtimescale τGs
+Short-range gravitational interactions can be seen as an in -
+teraction between a pair of single particles that can result in
+a scatteringevent.As with physicalcollisions, the timesc ale
+for a gravitational interaction depends on the cross-secti on,
+τGs=(σG¯ vpn)−1,whereσGisthegravitationalcross-section
+of the particle. Using Kepler’s laws, one finds that σG∼
+G2m2
+p/¯ v4
+p. The gravitational cross-section is velocity depen-
+dent, which makes it qualitatively di fferent from physical
+(billard ball) collisions between two particles. We note th at
+gravitational focusing can also lower the physical collisi on
+timescale.
+As mentioned earlier (Sect. 1), the particles might also be a f-
+fected by excitation from a turbulent background state on a
+timescale defined by the turbelence itself. In this paper, we
+model this excitation in a simplified way, described in Sect.
+3. In our case, the turbulence is scale independent and has no
+timescaleassociatedwith it.HannoReinet al.:The validityof the super-particle approx imation 3
+2.2. The realphysical system
+To identify the dominant physical processes in a protoplane tary
+disc, one has to quantify and compare the relevant timescale s
+as defined above. In the following discussion and in the numer -
+ical simulations, we assume R0=1 AU, a gas disc thickness
+ofH/R0=0.01, and a minimum mass solar nebula (MMSN)
+with surface gas density Σ=890 g cm−2. One usually assumes
+asolidtogasratioof ρs/ρg=0.01.We,however,assumeasolid
+to gas ratio of unity. As mentioned above, this value is justi -
+fied by numerical simulations (Johansenet al. 2007, 2009) th at
+demonstrate overdensities of the order of 101−102can easily
+occur because of the interaction with a turbulent gas disc, t he
+streaming instability, or vertical settling. We note that s ignifi-
+cantly different models of the solar nebula have been proposed
+(Desch 2007). However, all our results can easily be scaled t o
+differentscenarios.
+We simplify the system by assuming that all solid compo-
+nents of the disc are in meter-sized particles. We assume tha t
+these boulders have a typical velocity dispersion caused by gas
+turbulence ¯ v p∼0.05cs∼30m/s, where csis the local sound
+speed, in accordance with numerical results (Johansenet al .
+2007). Since ¯ v p≪cs, the particles tend to sediment toward the
+midplane, forming a finite thickness dust layer due to the non -
+zerovelocitydispersion.
+For meter-sized boulders,the physical cross-section is σc∼
+1m2, whereas the gravitational scattering cross-section is σG∼
+10−21m2showing clearly that gravitational scattering is irrel-
+evant to the dynamics of dust particles embedded in a disc.
+However, all other timescales are roughly equivalent. Mete r-
+sized boulders are weakly coupled to the gas with a stopping
+timeτs∼τe∼Ω−1(Weidenschilling 1977a). The physical col-
+lision timescale isτc=7.3Ω−1andthe long-rangegravitational
+interactiontimescalesis τGl∼Ω−1.
+This physical system is expected to become gravitationally
+unstable,accordingtothe Toomrecriterion(Toomre1964). The
+instability occurs when the gravitational collapse timesc aleτGl
+is shorter than the transit time due to random particle motio ns
+λ/¯ vpand the orbital timescale. Assuming that the particles can
+bemodelledbyanisotropicgas1withasoundspeed ¯ v p,thesys-
+tembecomesunstablewhen
+Q≡¯ vpΩ
+πGΣ<Qcrit≃1. (1)
+Inthatcase, themostunstablewavelengthisgivenby
+λT=2π2GΣ
+Ω2. (2)
+In the following, we compare the physical parametersfrom th is
+section to their counterpart in numerical simulations. We fi rst
+summarize the super-particle approach before presenting t he
+scalingmethod.
+2.3. Super-particleapproximation
+A super-particlerepresents many smaller particles. The dy nam-
+icsofthesmallparticlesarenotcalculatedexactly.Itisa ssumed
+that all particlesbehavesimilarly andin a collectivemann er.To
+simulate gravitational interaction between super-partic les, one
+has to use a softened potential. Without this, the gravitati onal
+scatteringcross-sectionofsuper-particlesbecomestool argeand
+1This would be true for a strongly collisional system, but is n ot for-
+mallyvalidinthe studiedregime.super-particles undergo gravitational scattering events , which
+is unphysical since these events never occur in a real system .
+Individualparticle-particlecollisionsare notmodelled .
+There are various examples where this approach is used
+successfully. For example, smoothed particle hydrodynami cs
+(SPH) uses the super-particle approximation to simulate ma ny
+gasmolecules(Lucy1977).Thesesystemsareoftenassumedt o
+bestrongly collisional to ensure thermodynamical equilibrium
+inside each super-particle. One can therefore assign colle ctive
+properties to clouds of particles such as pressure and tempe ra-
+ture. Another example is the evolution of galaxies. When two
+galaxies collide, individual particles (stars) will usual ly not un-
+dergo gravitational scattering events or physical collisi ons. In
+thatcase,thesuper-particleapproachmodelsa collisionless sys-
+tem inwhichcollectivedynamicsis theonlyimportantphysi cal
+process.Inbothcases,thesuper-particleapproximationi savalid
+approach for simulating the system numerically, but will br eak
+downassoonasthesystemis marginally collisional.
+Asanexample,weconsidertwocloudsofparticlesundergo-
+inga“collision”.Inthestronglycollisionalcase,theclo udswill
+slowly merge and the thermodynamic variables (e.g., temper a-
+ture) will diffuse between the clouds, the particles inside each
+cloud followinga random walk trajectory due to numerouscol -
+lisions. On the other hand, in the collionless regime, the tw o
+clouds simply do not see each other because there is no short-
+range interaction present between the particles. In a margi nally
+collisional regime, some particles will collide with parti cles of
+theothercloud,leadingtoapartialthermalisationofthev elocity
+distribution,but some otherparticleswill not have anycol lision
+atallandwillfollowapproximatelyastraightline.Eviden tly,the
+outcomeofthisevent cannotbedescribedusingasuper-particle
+approach.
+2.4. Scalingmethod
+Theideaofourscalingmethodisthatoneshouldkeepallimpo r-
+tanttimescalesinanumericalsimulationasclosetothoseo fthe
+realphysicalsystemaspossibleandmodelallparticlecoll isions
+explicitly.
+The numerical system consists of Nnumparticles in a box
+with length H, simulating a patch of the disc at a fixed radius
+R0. The particle mass is mnum=ΣH2/Nnum(His the box size
+andonescaleheight).Thus,thedensityintheboxandthelon g-
+range gravitational interactions are unchanged compared t o the
+realsystem.
+The gravitational scattering cross-section of the numeric al
+systemisthengivenby
+σG,num=G2Σ2H4
+N2num¯ v4p. (3)
+Fortheinitialsurfacedensityandvelocitydispersionuse dinthe
+simulation(Sect.2.2),onefinds
+σG,num≃H2
+N2num≃0.0001AU2
+N2num. (4)
+The physical collision cross-section in the simulation is
+σc,num=πa2
+numwhereanum, is the radius of the particles in
+the simulation.We derivetwo constraintsfromthe physical and
+gravitationalcollisioncross-sections:
+1.Samemeanfreepathinsimulationandrealphysicalsystems
+That means Nnumσc,num=Nσc, whereσcandNare the
+geometrical cross-section and the number of particles in a4 HannoReinet al.:The validityof the super-particle appro ximation
+regionofsize H3in a real disc, respectively.In otherwords,
+this condition ensures that the physical collision timesca le
+in thesimulationis exactlythesameasin therealsystem.
+2.Negligiblegravitationalscattering
+Whentwoparticlesapproacheachother,theoutcomeshould
+be a physicalcollision,whichmeansthat σc,num≫σG,num.
+The gravitational scattering timescale remains long com-
+paredtothephysicalcollisiontimescale.
+The first conditionplaces a constraint on the particle size i n the
+simulation.With N=4.7·1018anda particle radiusof a=1m
+as found in a real disc within a box of volume H3, using the
+parametersfromabovefindsthat
+anum=/radicalbiggσc,num
+π=/radicalbigg
+Nσc
+Nnumπ≃0.014√NnumAU. (5)
+We note that once this condition is satisfied in the initial co ndi-
+tionsitwillbeautomaticallysatisfiedatalltimes.Insimu lations,
+we typicallyfindthat thecollisiontimescale isreducedbym ore
+thanoneorderofmagnitudeduringthegravitationalcollap se.
+The secondconditionis thensatisfied bychangingthe num-
+ber of particles. An interesting result is that if Nnum>10, the
+second condition is easily satisfied if the first one is satisfi ed.
+We note however that σG,numdepends strongly on the velocity
+dispersion. In particular, a velocity dispersion 5 times sm aller
+than the initial value (as found in some simulations) leads t o
+an increase by a factor of 600 in the gravitational scatterin g
+cross-section. Therefore, we suggest using a large safety f actor
+(Nnum>105)toensurethatthesecondconditionisalwayssatis-
+fied,evenforsignificantlysmaller ¯ v p.Thisconditionalsoallows
+ustoestimatewhenourapproachbreaksdown,namelywhenthe
+number of clumps in the system is so low ( N<10∼100) that
+gravitationalscatteringbecomesimportant.
+Althoughit wouldbe helpfulasa furthersimplification,it i s
+not possible to perfom the above mentioned simulation in two
+dimensions. In that case, the filling factor, which is defined to
+be the ratio of the volume (or area) of all particles to the tot al
+volume(orarea),isoftheorderofonefortheaboveparamete rs.
+We notethat in2D anincreasein particlenumber(anddecreas e
+inparticlesizeasrequiredbyEq.5)doesnotdecreasethefil ling
+factorif thecollisionallifetimeremainsconstant.
+3. Methods
+Twodifferentkindsofsimulationare consideredinthispaper:
+1. Particles areassumed to bepoint massesand havenophysi-
+cal size, the gravitationalfield ofthe particlesbeingappr ox-
+imated with a smoothing length to avoid numerical diver-
+gences(see Sect.3.1).
+2. Particleshaveaphysicalsizeand,therefore,nogravita tional
+smoothing is required but physical collisions must be in-
+cluded.
+We refer to the particles as super-particles and scaled part icles,
+respectively.
+We perform our simulations in a cubic box with shear peri-
+odic boundaryconditionsin the radial ( x) and perdiodicbound-
+ary conditions in the azimuthal ( y) and vertical ( z) directions,
+as illustrated in Fig.1 (Wisdom&Tremaine 1988). In the loca l
+approximation, the force per mass on each particle is a sum of
+thecontributionsfromHill’sequationsandtheinteractio nterms.
+Hill’sequationcanbewrittenas
+Fhill=−2Ωez×vp+3Ω2xex−Ω2zez. (6)The interaction term Fintis divided into components related to
+self gravity, physical collisions between particles, or dr ag and
+excitationforces:
+Fint=Fgrav+Fcol+Fdrag+Fturb. (7)
+Wesolvetheresultingequationsofmotionwithaleapfrog(k ick
+drift kick) time stepping scheme. In the following subsecti ons,
+we describe the numerical methods used to compute each of
+thesetermsandtheirphysicalrelevance.
+The box size of 0 .01 AU was chosen such that there are
+always several unstable modes in the box, when the system is
+pushedintotheregimeofgravitationalcollapse,asestima tedby
+Eq.2.
+ghost boxesy
+xtostarghost boxes
+Fig.1.Shearing box, simulating a small patch of the protoplan-
+etary disc. The grey box represents the shearing box of inter -
+est, the dashed boxessurroundingthe central box represent one
+ghostring.
+3.1. Selfgravity
+In theN-body problem that we consider, we have to solve
+Newton’s equations of universal gravitation for a large num ber
+ofparticles N.Thegravitationalforceonthe i-thparticleisgiven
+by
+Fgrav=/summationdisplay
+j/nequaliGmimj/parenleftBig
+rij+b/parenrightBig2eij, (8)
+wherebis the smoothing length used to avoid divergences in
+numericalsimulations(insimulationsthatincludephysic alcolli-
+sionsb=0)andeijistheunitvectorinthedirectionofthegrav-
+itational force between the i−th andj−th particle. Calculating
+the gravity for each particle from each other particle resul ts in
+O/parenleftBig
+N2/parenrightBig
+operations.To reducethe numberof operations,onecan
+use different approximations to Eq. 8. In this paper, we use a
+Barnes-Hut(BH) treecode(Barnes& Hut1986).
+TheBHtreeinthreedimensionsdividestheoriginalboxinto
+eight smaller cells with half the length of the original box. ThisHannoReinet al.:The validityof the super-particle approx imation 5
+process is continued recursively until there is only one par ticle
+per cell left. The depth of the tree in a homogeneousmediumis
+approximatelylog8N.Onethencalculatesthetotalmassandthe
+centreofmassforeverycellat everylevelofthetree.
+To calculate the force acting on a particle, one starts at the
+top of the tree and descends into the tree as far as necessary t o
+achievea givenaccuracy.Ifthe currentcell isfaraway from the
+particle for which the force is calculated, then the detaile d den-
+sitystructurewithinthiscellisnotimportant.All thatma ttersis
+the box’s monopole moment (total mass and centre of mass).
+One therefore does not have to descend into this tree branch
+any further. The BH tree reduces the number of calculations t o
+O/parenleftbigNlogN/parenrightbig.
+We use 8 ringsof ghost boxes in the radial and azimuthal
+direction(Fig.1showsonering).Aghostboxissimplyashif ted
+copy of the main box. The gravity on each particle is then cal-
+culated by summing over contributions from each (ghost) box .
+This setup approximates a medium of infinite horizontal exte nt
+and avoids large force discontinuities at the boundaries. W e do
+not need any ghost boxes in the vertical direction because th e
+discis stratified.
+Afinitenumberofghostringscanonlyactasanapproxima-
+tionofaninfinitemediumandthegravitationalforcewillte ndto
+concentrate particles horizontally in the centre of the box . Due
+tothe8ghostrings,theasymmetryofthegravitationalforc ebe-
+tween the centre and the faces of the box is reduced by a factor
+of20comparedtousingnoghostringsatall.Wenote,however ,
+that a small asymmetry remains in our simulations and leads
+to higher concentrations in the box centre. In other simulat ions
+such as Tangaetal. (2004), this e ffect is not seen because the
+system is initially gravitationally unstable and integrat ed only
+for approximately one orbit. The system that we are interest ed
+inismarginalgravitationallyunstableandthisslightasy mmetry
+will become important after several orbits. In the beginnin g of
+oursimulations,we integrateformanyorbitsinorderto rea cha
+stable equilibrium. We then push the system into a gravitati on-
+allyunstableregime(seeSect.3.5fordetails).Duringthe stable
+phase,horizontalover-concentrationsaretobeexpecteda ndare
+indeedobservedbecauseofthisasymmetry.However,sincei na
+realprotoplanetarydiscthegravitationalinstabilityis considered
+toappearlocally(e.g.,insideavortexorasimilarstructu re),this
+effect of having a preferred location for gravitational instab ility
+is not unphysical and does not a ffect any conclusions made in
+thispaper.Wetestedthisbyapplyingalinearcut-o fftothegrav-
+itationalforceatadistanceofoneboxlength(A.Toomre,pr ivate
+communication).In thiscase, therewas nopreferredlocati onin
+theboxandthesimulationevolvedin exactlythesameway.
+3.2. Physicalcollisions
+Physical collisions between particles are treated in the fo llow-
+ing way. After each timestep, we check whether any two par-
+ticles are overlapping. Using the already existing tree str ucture
+from the gravity calculation, one can again reduce the compu -
+tational costs from O/parenleftBig
+N2/parenrightBig
+toO/parenleftbigNlogN/parenrightbig. In these simulations,
+a collision between particles is defined as an overlap betwee n
+two particles that are approaching each other. When a collis ion
+is detected, it is resolvedassuming energyand momentumcon -
+servation(perfectlyelasticcollisions).We notethatthe timestep
+has to be small enough such that no collision is missed and the
+overlapisalwayssmall comparedtotheparticlesize.
+TheparticleradiusisgivenbyEq.5.Inordertokeepthecol-
+lisiontimescaleτcclosetounity,whichisnumericallytheworstcase scenario because all timescales are equivalent,we inc rease
+theradiusbyafactorof4inall simulations.
+3.3. Drag force
+Each particle feelsa dragforce.The backgroundvelocityof the
+gasvgis assumedto beasteadyKeplerianprofile
+vg=−3
+2Ωxey. (9)
+Althoughaccretionflows are turbulent,we choose to use a sim -
+ple velocity profile so that the behaviourof self-gravitati ngpar-
+ticles can be understood in conditions that can be easily con -
+trolled.
+3.4. Randomexcitation
+The system of particles described above is dynamically unst a-
+bleevenwithoutself-gravity,asthecoe fficientofrestitutionand
+the stopping time do not depend on the particle velocities2. The
+particles can either settle down in the midplane and create a
+razor-thin disc if the stopping time is too short, or they can ex-
+pandverticallyforeverifthestoppingtimeisabovesomecr itical
+value and the excitation mechanism is provided only by colli -
+sions. Dust particles in accretion discs are found in the set tling
+regime. However, a complete settling never occurs in accret ion
+disksbecausethebackgroundflowisalwaysturbulentduetot he
+Kelvin-Helmoltz instability (Johansenetal. 2006), the MR I, or
+other hydrodynamic instabilities (see e.g., Lesur& Papalo izou
+2009) whichdiffuse particlesvertically(Fromang& Papaloizou
+2006). This stirring process of turbulence is not present in the
+gas velocity field of the simulations presented here (see Eq. 9).
+To approximate the turbulent mixing, we added a random ex-
+citation (white noise in space and time) to the particles in t he
+simulation. This allows us to have a well defined equilibrium
+in which the system is stable rather than starting from unsta ble
+initial conditionsthat mightinfluencethefinal state.
+We perturb the velocity components of each particle after
+each timestep on a scale ∆vi=√
+δtξ, whereδtis the current
+timestep andξis a random variable with a normal distribution
+around0andvariance s.Thisexcitationmechanism heatsupthe
+particlesand allows usto have a well defined threedimension al
+equilibrium as shown in Sect. 4. In our simulations, we use a
+value ofs=1.3·1015m2s−3ands=8.9·1014m2s−3for the
+simulationswithoutandwithcollisions,respectively.Th eseval-
+ueswerechosensuchthattheequilibriumstateisapproxima tely
+equivalentforbothtypesofsimulations.
+3.5. Initialconditions
+All particles, which have equal mass, smoothing length, and
+physical size, are placed randomly inside the box in the x−
+yplane. We note that particles have either a physical size or a
+smoothing length associated with them, depending on the typ e
+of simulation we perform(Sect. 3). In the z-direction,the p arti-
+cles are placed in a layer with an initially Gaussian distrib ution
+aboutthemidplaneandastandarddeviationof0.05H.Weallo w
+thesystemtoreachequilibriumbyintegratingit until t=30Ω−1
+(4.8yrs).
+2This instability is qualitatively similar to the instabili ty described
+byGoldreich &Tremaine (1978)forSaturnrings,althoughin thelatter
+case the drag forces are absent.6 HannoReinet al.:The validityof the super-particle appro ximation
+10 m/s15 m/s20 m/s25 m/s
+30 Ω-135 Ω-140 Ω-145 Ω-150 Ω-1velocity dispersion
+timeN=80000
+N=160000
+N=320000
+N=160000 LS
+Fig.2.Super-particles:Velocitydispersionas a functionof time
+in four differentruns. The simulationlabeled LShasa ten times
+larger smoothing length. The curvesdo not overlap because t he
+simulationsarenotconverged.
+10 m/s15 m/s20 m/s25 m/s30 m/s35 m/s
+30 Ω-135 Ω-140 Ω-145 Ω-150 Ω-1velocity dispersion
+timeN=160000
+N=320000
+N=640000
+Fig.3.Scaled-particles:Velocitydispersionasafunctionoftim e
+inthreedifferentrunsthatincludephysicalcollisions.Allcurves
+overlapbecausethesimulationsareconverged.
+Wetestedvariouswaysofpushingthesystemintotheunsta-
+bleregime,eitherbyreducingtheturbulencestirringorby short-
+eningthestoppingtime,butdidnotseeanyqualitativedi fference
+as long as we start from a well defined equilibrium state. In th e
+simulationspresentedinthispaper,weswitcho fftheturbulence
+stirring.Followingthismodification,thesystem becomesg ravi-
+tationallyunstableandboundclumpsformwithina feworbit s.
+4. Results
+4.1. Superparticles
+We first present simulations without physical collisions (s uper-
+particles) which rely on the smoothing length bto avoid any
+divergencies. Although we use a tree code and therefore a
+smoothed potential for the force calculation is assumed, th e
+results are equivalent to an FFT-based method where the gridlengthactsasaneffectivesmoothinglength.Ingeneral,tocheck
+the numerical resolution, the particle number Nnumis increased
+and the smoothing length bis reduced independently. A simu-
+lation is resolved when the result is independent of both Nnum
+andb. This turns out to be impossible in the present situation.
+The main reason is that the smallest scale in a gravitational
+collapse will ultimately depend on the smoothing length. By
+varying both parameters at the same time, an empirical scal-
+ing ofb∼1/√Nnumworks fairly well if Nnumis large enough.
+However, this procedure is not justified and is unphysical. T he
+smoothing length was introduced to avoid divergent terms an d
+not to model any small-scale physical process. Therefore, i t
+shouldnothaveanyimpactonthe physicaloutcomeofthe sim-
+ulation. Incidentally, the existence of this dependency in dicates
+thatshort-rangeinteractionsareimportant fortheresultofthese
+simulationsandshouldbe modelledwith care.
+InFig.2,weplotthevelocitydispersionasafunctionoftim e.
+The simulations begin from the stable equilibrium describe d in
+Sect. 3. After t=30Ω−1, we switch offthe excitation mecha-
+nism. Because the system continuesto cool,it becomesgravi ta-
+tionally unstable within one orbit, as estimated by Eq. 1. On ce
+clumpingoccurs,thevelocitydispersionbeginstoriseaga in.All
+simulationsusetheparticleradiusgivenbyEq.5asasmooth ing
+length except the ones labeled LS, which use a ten times larger
+value. The larger softening length is approximately a 128-t h of
+theboxlengthandillustratesthekindofevolutionexpecte dfrom
+anFFT methodusinga 1283grid.
+Snapshotsof the particle distribution of two simulations a re
+shown in Fig.4. Both simulations use 160 000 particles and al l
+parameters, except the smoothing length, are the same. The t op
+row is the simulation with a smoothing length given by Eq. 5,
+whereas the bottom row uses a smoothing length that is ten
+times larger. The simulations correspond to the blue (mediu m
+dashed)andred(solid)curveinFig.2.Weshowthesnapshots to
+illustrate the importance of the smoothinglength in simula tions
+withoutphysicalcollisions.Onecanseethatthesimulatio nsdif-
+fer already before clumps form. Stripy structures appear on a
+scale given by Eq. 2. As soon as we enter the unstable regime
+att∼32Ω−1(i.e., the velocity dispersion begins to rise, see
+also Fig.2) the simulations evolve very di fferently. The simula-
+tiononthetoprowofFig.4formsmanyclumpsatanearlytime,
+whereasthebottomrowsimulationformsonlyafew,moremas-
+siveclumpsatlater times.
+Thiscanbeconfirmedbylookingat spectraofthesametwo
+simulationsas shown in Fig.6. These spectra were generated by
+mapping the particles onto a 128 ×128 grid in the x−yplane
+and computing the fast Fourier transform of the mapping in th e
+xdirection. The resulting spectra were finally averaged in th e
+ydirection (see Tangaet al. 2004, for a complete description
+of the procedure). As suggested by the snapshots, the nonlin -
+ear dynamics of the gravitational instability strongly dep ends
+on the smoothing length used. In particular, one observes th at
+smaller scales are amplified more slowly for larger smoothin g
+lengths. The resulting spectra at t=40Ω−1also differ signif-
+icantly. With a small enough smoothing length, the spectrum
+looks almost flat, whereas a large smoothing length introduc es
+a cutoffatk/2π≃10.We also notethat the smoothinglengthin
+the latter case is of the order of the grid size ( k/2π∼100). The
+cutoffobserved clearly demonstrates that the smoothing length
+modifies the dynamics on scales up to 10 times larger than the
+smoothinglengthitself.HannoReinet al.:The validityof the super-particle approx imation 7
+Fig.4.Super-particles: Snapshots of the particle distribution i n thexyplane. Both simulations use 160 000 particles with di fferent
+smoothing lengths. The simulation on the bottom uses a ten ti mes larger smoothing length than the one on the top. The snaps hots
+were taken (fromleft to right)at t=0,30,37,40Ω−1. With a largesmoothinglength,the outcomelooksverydi fferent,the system
+ismorestable, morestripystructurecanbeseen andclumpsf ormlater,ifat all.
+Fig.5.Scaled particles: Snapshots of the particle distribution i n thexyplane. The simulations (from top to bottom) use 40 000,
+320000,640000particleswiththeirphysicalsizegivenbyE q.5.Inallsimulations,thecollisiontimescalewaskeptco nstant.The
+snapshots were taken (from left to right) at t=0,30,37,40Ω−1. The simulation with 40 000 particles has a large filling fact or by
+thelastframe,whichpreventsclumpsfromforming.Theinte rmediateresolutionsimulation(middlerow)andthehighes tresolution
+simulation(bottomrow)havemoreandsmaller particlesand thusa smallerfilling factor.Theresults(i.e.,numberofcl umpsinthe
+last frame)areverysimilar intheintermediateandhighres olutionsimulations.8 HannoReinet al.:The validityof the super-particle appro ximation
+4.2. Scaledparticles
+The velocity dispersion evolution of simulations with N=
+160000,320000,640000particlesincludingphysicalcollisions
+is presented in Fig.3. In these runs, no gravitational smoot hing
+lengthisneeded.Again,thesimulationsstartfromastable equi-
+libriumandafter t=30Ω−1, we changethesame parametersas
+in the non-collisional case to push the system into the gravi ta-
+tionally unstable regime. Snapshots of the particle distri butions
+for three runs are also plotted in Fig.5. The middle and botto m
+rowsofsnapshotscorrespondtothepurple(mediumdashed)a nd
+darkblue(longdashed)linesofFig.3.
+One can see in Fig.3 as well as Fig.5 that the intermedi-
+ate and high resolution runs produce very similar results. W e
+call those simulations converged, as a change in particle nu m-
+ber does not change the outcome. An additional, lower resolu -
+tion run ( N=40000) is not converged because the filling fac-
+tor is too large in dense regions, preventing clumps from bei ng
+gravitationally bound. The problem does not occur in the non -
+collisional cases because particles are allowed to overlap .Since
+the filling factor scales with particle number as 1 /√Nnum, this
+issue is resolved for runs with more than a few hundred thou-
+sand particles. We note that the velocity dispersion might d iffer
+slightly at later stages for the converged runs because the fi nal
+clump radius is still determined by the physical particle ra dius
+andthereforebytheparticlenumber.
+WeshowthedensitydistributionspectruminFig.7asafunc-
+tion of the numberof particles. The resulting spectra do not de-
+pend on the number of particles in the system, as expected. In
+comparison with Fig.6, the simulations with collisions are con-
+vergedsince the dynamicis not controlledby the numericalp a-
+rameterN, whichistheonlyfreeparameterinthe system.
+At very late times ( t/greaterorsimilar38Ω−1), the spectra begin to show
+a systematic trend towards more structure on smaller scales for
+runswith larger N. Thisisexpectedandduetothefillingfactor,
+already mentioned above. As soon as one clump in the simu-
+lation is only determined by the size of the individual parti cles
+it consists out of, our approach breaks down. One can also see
+these clumps forming by looking at the velocity dispersion i n
+Fig.3. The particles inside a clump begin to dominate the ve-
+locity dispersion over the background after t∼38Ω−1. A clear
+indication of this is the spiky structure with a typical corr ela-
+tiontimeof∼Ω−1correspondingtodi fferentclumpsinteracting
+and mergingwith each other. One way around this issue, which
+will be consideredin future work, is to allow particles to me rge
+(Michikoshietal. 2009). Using this accretion model , the mass
+oftheclumpcanbeusedasanupperlimit.
+5. Discussionand implications
+In this paper, we have shown that convergence in N-body sim-
+ulations of planetesimal formation via gravitational inst ability
+can be achieved when taking into account all relevant physic al
+processes.Itisabsolutelyvitaltosimulategravity,damp ing,ex-
+citation, and physical collisions simultaneously, as the r elated
+timescales are of the same order and therefore all e ffects are
+stronglycoupled.
+A set of simulationsis definedto be convergedwhenthe re-
+sultsdonotdependontheparticlenumberoranyotherartific ial
+numericalparametersuchasasmoothinglength.Asatestcas e,a
+boxwith shearperiodicboundaryconditionswas used,conta in-
+ing hundreds of thousands of self-gravitating (super-)par ticles
+in a stratified equilibrium state. The particles were then pu shed
+into a self gravitatingregime that eventuallyled to gravit ational 1e-05 0.0001 0.001 0.01
+ 1  10  100Power
+Wavelength 1/LN=160000
+t=38
+N=160000 LS
+t=38N=160000
+t=36N=160000 LS
+t=36
+N=160000 LS
+N=160000
+t=30
+Fig.6.Super-particles: Density distribution spectrum with
+160000particlesandtwodi fferentsmoothinglengthsthatdi ffer
+by a factor 10 at times t=30 (red, solid curve), t=36 (green,
+longdashedcurve),and t=38 (blue,short dashedcurved).The
+spectra at similar times are not on top of each other because t he
+simulationsarenotconverged.
+ 1e-05 0.0001 0.001 0.01
+ 1  10  100Power
+Wavelength 1/LN=160000
+N=320000
+N=640000t=30t=36t=38
+ 1e-05 0.0001 0.001 0.01
+ 1  10  100Power
+Wavelength 1/LN=160000
+N=320000
+N=640000t=30t=36t=38
+Fig.7.Scaled particles: Density distribution spectrum with
+160000,320000,and640000particles(colorsareequivalen tto
+Fig.6). At each time, the spectra are converged,i.e., the sp ectra
+are on top of each other and independent of the particle num-
+ber, besides the noise level on very small scales. At late tim es
+(t/greaterorsimilar38, blue curve), one begins to see more structure on small
+scalesinsimulationswithlargernumberofparticles(seet ext).
+collapse. Simulationswith and without physicalcollision s were
+studiedfora rangeofparticlenumbersto test convergence.
+Incases withoutphysicalcollisions, convergencecannot b e
+achieved.However,it was possible to changemultiplefree s im-
+ulationparametersatthesametime,namelytheparticlenum ber
+Nand the smoothing length b, such that in special cases the re-
+sults did not depend on the particle number. We note however
+that there is a free parameterin the simulation(i.e., the sm ooth-
+ing length) that effects the outcome and makes it impossible to
+findtherealphysicalsolution.HannoReinet al.:The validityof the super-particle approx imation 9
+Intheprotoplanetarydisc,physicalcollisionsdominateo ver
+gravitational scattering. In the case of planetesimal form ation,
+thesystemismarginallycollisional.Physicalcollisions willpar-
+tially randomise the particle distribution, whereas a smoo thing
+length does not because the softening length is very large co m-
+paredtothegravitationalcross-section.Evenifthegravi tational
+particle-particle scattering becomes important, it can be seen
+from Eq. 3 that the velocity dependence of the cross-section
+σGdiffers fundamentally from the velocity independent cross-
+sectionσC. We therefore argue that the super-particle approach
+shouldnotbeusedwhencollisionsbecomeimportant.
+With collisions, the simulation outcome is both quantita-
+tively and qualitatively very di fferent from simulations with no
+physicalcollisions.Theinitialsizeoftheclumpsislarge randthe
+number of clumps is smaller. As there is no smoothing length,
+there is effectively one fewer free parameter. Changing the par-
+ticle number while keeping the collision time constant does not
+change the outcome. We therefore call these simulations con -
+verged.
+Additional tests including inelastic collisions with a nor mal
+coefficientofrestitutionof0 .25havebeenperformedtoconfirm
+that theresultsdo notdependonthe way physicalcollisions are
+modelled.Asexpected,thequalitativeoutcomeintermsofn um-
+ber of clumps and clump size is di fferent because the physical
+properties of the system have been changed. However, once in -
+dividual collisions are resolved the results converge in ex actly
+thesamemannerasinthosesimulationspresentedabove(whi ch
+haveacoefficientofrestitutionof1).
+We have also exploredother gravity solversby comparinga
+fast Fourier(FFT) gravitysolver with the BH tree code(usin ga
+smoothing length) and confirmed our previousresults. The gr id
+in the FFT code acts as a smoothing length and a direct com-
+parisonbetweenthetwogravitysolversgaveanapproximate ef-
+fectivesmoothinglengthofaquarterofthegridlength.Bec ause
+onehastosolveindividualparticle-particleinteraction s,thegrid
+size has to be smaller than the physical size of the particles . As
+a result, the FFT method is always slower than a tree code. In
+addition,thetreestructurecanbereusedtosearche fficientlyfor
+collisions.
+This paper focused on the numerical requirements to study
+gravitational instability and the formation of planetesim als in
+protoplanetary discs. The initial conditions were chosen s uch
+that the equilibrium is a well defined starting point for the c on-
+vergence study. The collision, damping, collapse and orbit al
+timescales are all of the same order, which is e ffectively the
+worst case scenario. To provideany constraintson planetes imal
+formationitself, one would have to properlysimulate the tu rbu-
+lentbackgroundgasdynamicswhichisbeyondthescopeofthi s
+paper.
+However, we were able to determine the numerical resolu-
+tion needed to resolve dust particles in a protoplanetary di sc.
+Assuming that one wishes to simulate dust particles realist i-
+cally, including collisions, and that the particles are uni formly
+distributed in a box of base L2and height H, the size of each
+particle is determined by the collision rate in the real disc (see
+Eq.5)
+anum=/radicalbigg
+N
+Nnuma, (10)
+whereaandNarethesizeandnumberofparticlesinthevolume
+L2Hof the real disc. The number of particles in the simulation
+Nnumhas to be large enough to ensure that the filling factor is
+low until clumps form.The requirementthat the filling facto r islessthanunityat t=0isgivenby
+4
+3π(anum)3·Nnum≤CfL2H, (11)
+whereCfis a safety factor. Although Cf/lessorsimilar1 would be enough
+toresolvecollisionsinitially,itisinsu fficienttosimulatethecol-
+lapse phase. During the collapse, the filling factor rises ra pidly.
+Assuming that one wishes to resolve collisions correctly wh en
+theparticleshavecontractedbyoneorderofmagnitude,one has
+to include a safety factor of Cf=10−3. If the collapse occurs
+mainlyintwodimensions,asinthesimulationspresentedhe re,a
+factorofCf=10−2is sufficient.Furthermore,onehasto ensure
+that the box contains at least a few unstable modes (see Eq. 2) .
+Allthistogetherplacestightnumericalconstraintsonnum erical
+simulationsof planetesimal formationvia gravitationali nstabil-
+ity.
+In future work, we will expand the discussion to include
+more physics, namely a proper treatment and feedback of the
+background gas turbulence. This will allow us to further cla r-
+ify the behaviour of planetesimals in the early stages of pla net
+formationandeventuallytest thedi fferentformationscenarios.
+Acknowledgements. We thank J. Papaloizou, G. Ogilvie, S.-I. Inutsuka, A.
+Johansen, Y. Lithwick, and A. Youdin for helpful discussion s and com-
+ments. Hanno Rein is supported by an Isaac Newton Studentshi p and St
+John’s College Cambridge. Geo ffroy Lesur acknowledges support from STFC.
+Zo¨ e M. Leinhardt is an STFC postdoctoral fellow. Simulatio ns were per-
+formed on the Astrophysical Fluids computational faciliti es and on Darwin,
+the Cambridge University HPC facility. The authors are grat eful to the Isaac
+Newton Institute forMathematical Sciences inCambridgewh erethefinalstages
+of this work were carried out during the Dynamics of Discs and Planets re-
+search programme. A simplified version of the simulation use d in this pa-
+per can be downloaded free of charge as an application for the iPhone at
+http://itunes.com/apps/gravitytree .
+References
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\ No newline at end of file
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+arXiv:1001.0110v1  [physics.chem-ph]  31 Dec 2009On the Dimers of Pseudoisocyanine
+P.O.J. Scherer∗
+Physics Department T38, TU Munich, 85748 Garching
+Abstract
+The self organisation of pseudoisocyanine-dimers in dilut e aqueous solutions is studied by classical
+MD simulations. The electronic structure of the dimer is eva luated with the semiempirical ZINDO
+method to determine the fluctuations of site energies and exc itonic coupling. We study different
+dimer conformations with blue or red shifted absorption max ima as models for H and J-aggregates.
+The width of the absorption bands is mainly explained by low f requency vibrations whereas the
+fluctuations of site energies are less important.
+1I. INTRODUCTION
+Since the discovery of the so called J-band1,2, an unusual sharp absorption band
+which is characteristic for the aggregation of the classica l sensitizing dye 1,1’-diethyl-2,2’-
+cyaninchloride (pseudoisocyanine) a large amount of exper imental and theoretical work ad-
+dressed the investigation of molecular aggregates and thei r red shifted J-bands. At lower
+concentration blue shifted H-bands were observed which wer e attributed to molecular dimers,
+the smallest possible aggregates. Molecular modelling of t he J-aggregates is difficult due to
+the fact that the PIC molecule is a cation and therefore the Co ulombic interactions as well
+as dielectric shielding have to be taken into account carefu lly. For the formation of larger
+aggregates the counterions are important whereas this seem s not the case for the smaller
+H-aggregates3. Therefore we started the simulation of PIC aggregates by a d etailed inves-
+tigation of the dimer. From the analysis of the experimental spectrum in water4it was
+deduced that both excitonic components contribute with an i ntensity ratio of 2:1. This
+can not be explained5by dimer models4,6where the dipole moments are almost parallel or
+antiparallel as it is the case for the common brickwork or lad der models which are found
+in the literature for the J-aggregate7,8. Another focus of our investigations concerns the
+contribution of local Coulombic interactions to the inhomo geneous broadening of the site
+energies and its importance in comparison to intramolecula r vibrations.
+II. METHODS
+For the classical MD simulation we used the model of rigid rot ors which can be easily
+combined with quantum calculations to obtain electronic ex citations and coupling matrix
+elements9. Since also the position of the ethyl groups is fixed we have to distinguish not
+only two stereoisomers but also a fully C2-symmetric form an d another form where the
+ethyl-groups break this symmetry. A possible interconvers ion between these conformations
+was not taken into account.
+We simulated a cube containing a pair of PIC (positively char ged) molecules and 2150
+TIP510water molecules. We did not use periodic boundary condition s to avoid artefacts from
+the Coulombic interaction with the mirror images. Instead r eflecting boundaries kept the
+molecules from escaping the box by reversing the normal velo city component whenever the
+2center of mass of one of the molecules encountered the bounda ry . The boundary distance
+of 36Å was adjusted to reproduce the experimental density of water at room temperature.
+The equations of motion were solved using an implicit quater nion method11for the rotations
+and a Leap frog method for the translations. The timestep use d was 1 fsec.
+In our simulation we neglected electrostatic interactions with solvent outside the cube.
+We calculated the missing contribution to the solvation ene rgy from a simple PCM model12.
+The simulated box was put into a cubic cavity in a dielectric c ontinuum and the contribution
+to the solvation energy was calculated from the interaction between the charges within the
+box and the induced surface charges. It gave 13% of the total s olvation energy. This value
+stayed rather constant along the trajectory. Therefore we a ssume that the essential changes
+of interaction with solvent molecules in the immediate surr oundings are taken into account
+sufficiently.
+The force field was designed to reproduce the local electrost atic interactions properly,
+which is especially important for the large sized PIC molecu le with its extended π-electron
+system. It is based on a simplified version of the effective fra gment model13–15. The charge
+distribution is approximated by distributed multipoles wh ich were calculated with GAMESS
+on the basis of TZV/HF wavefunctions16. For the simulation only point charges qiand
+dipoles/vector piwere used which are centered at the positions of the nuclei an d the bond centers.
+The Coulombic interaction energy is
+VCoul
+ij=qiqj
+4πǫ0Rij+/vectorRij(qi/vector pj−qj/vector pi)
+4πǫ0R3
+ij+R2
+ij/vector pi/vector pj−3(/vectorRij/vector pi)(/vectorRij/vector pj)
+4πǫ0R5
+ij(1)
+The values of point charges and dipoles are given for the symm etry unique atoms in
+Table I.
+The electronic spectra were calculated on the ZINDO/CI-S le vel17including the point
+monopoles and dipoles of all water molecules. The two lowest excited states of the dimer
+are to a large extent linear combinations of the lowest monom er excitations with only very
+small admixture of higher monomer excitations and of charge resonance states. Therefore
+we use a simple 2-state model to analyze the delocalized dime r states in terms of the local
+excitations A∗andB∗. The interaction matrix is
+
+E−∆/2V
+VE+∆/2
+ (2)
+3with the average monomer transition energy E= (EA∗+EB∗)/2, their splitting ∆ =EB∗−
+EA∗and the excitonic coupling V. Its eigenvectors are the two delocalized dimer excitation s
+which are written with mixing coefficients cosγ,sinγas
+|1>= cosγ|A∗>+sinγ|B∗>|2>=−sinγ|A∗>+cosγ|B∗> (3)
+The transition dipoles of these two states
+/vector µ1= cosγ/vector µA∗+sinγ/vector µB∗/vector µ2= sinγ/vector µA∗−cosγ/vector µB∗ (4)
+are linear combinations of the transition dipoles /vector µA∗,B∗of the two monomers which are
+assumed to be independent from the excitonic interaction. T herefore the mixing angle γ
+can be determined from a least square fit of the two dimer trans ition dipoles to (4). Then
+the elements of the interaction matrix are calculated from t he transition energies of the two
+dimer excitations as
+E=E1+E2
+2(5)
+∆ = (cos( γ)2−sin(γ)2)(E2−E1) (6)
+V= cos(γ)sin(γ)(E2−E1) (7)
+We want to emphasize that this analysis is based on the deloca lized dimer orbitals. It does
+not involve any kind of multipole expansion, especially the calculated excitonic coupling is
+not of the dipole-dipole type which would be quite questiona ble at such short intermolecular
+distances. To check the quality of the ZINDO method, we compa red the results for a selected
+sandwich dimer configuration with a much more elaborate 631G ** HF/CI calculation. The
+calculated transition dipoles of the two lowest singlet exc itations were very similar (3 and 16
+Debyes from ZINDO , 2 and 14 Debyes from HF/CI ), the excitonic splitting was somewhat
+larger for the HF/CI method (0.54eV as compared to 0.37eV for ZINDO). Both methods
+placed the lowest charge resonance states at about 0.65 eV ab ove the upper excitonic band.
+III. RESULTS
+We determined the vibronic coupling parameters for the PIC m onomer as described in
+our earlier paper18. Application of ab initio methods16improved the quality of the results
+4so that a direct comparison with the profile of the absorption spectrum becomes feasible.
+The normal modes were calculated on the 6-31G/MP2 level and t he coupling to the optical
+transition on the CI/SD level. Using these couplings and the displaced harmonic oscillator
+model the lineshape was calculated as the Fourier transform ed time correlation function. In
+the low frequency region the largest vibronic couplings are found for normal modes at 40 and
+46cm−1which contribute significantly to the broadening of the abso rption band. Another
+important contribution from modes around 1500 cm−1which are also known from Raman
+spectra is the origin of the observed vibrational progressi on. Further modes between 50
+and 1400 cm−1form a rather dense continuum of coupling states. The simula ted spectrum
+(fig. 2) largely resembles the experimental absorption profi le. The width of the simulated
+bands is somewhat too small and the intensity of the prominen t stretching modes is slightly
+overestimated. This could be possibly further improved by t aking frequency changes and
+mode coupling into account.
+The MD simulations were started from several plausible dime r structures. First the
+PIC molecules were kept fixed and the solvent was equilibrate d for 50 psec. Then the
+restraints were removed and the system was simulated for ano ther 50 psec. The distance and
+orientation of the two PIC molecules were analyzed to identi fy periods of relative stability.
+Starting from a sandwich structure, a rather stable structu re evolved within 10 psec
+(fig.3a). It is not symmetric but still there is almost no spli tting of the calculated site
+energies (Table 2) which show rapid fluctuations with compon ents down to 20fsec. Such
+fast fluctations are well known from experimental and theore tical work on the dynamics of
+dephasing and solvation in molecular liquids19. They have been attributed to the inertial
+motion of the solvent molecules, which show up as the Gaussia n shaped rapid initial decay
+of the solvation time correlation function20,21. In our simulations the orientational time
+correlation function of the water molecules can be describe d by a Gaussian with a correlation
+time of 60 fs at short times. The time correlation of the elect rostatic potential decays faster.
+The initial Gaussian decay with 20 fs is very similar to that o f the correlation function of
+the transition energies. Probably collective librational motions contribute more efficiently
+to the electronic dephasing than the motion of the individua l molecules.19,22
+The center of the site energies is shifted by 0.06 eV to lower e nergies as compared to
+a monomer in vacuum and the variance of the site energies is co mparable to that of a
+monomer. The two transition dipoles are almost parallel and the lower transition carries
+5only 2% of the total oscillator strength. The excitonic coup ling shows fluctuations similar
+to the site energies. Its variance, however, amounts to only 7% of the average value of
+0.25eV. Hallermeier et al4deduced a smaller excitonic coupling of 0.078 eV. Most proba bly
+their dimer spectrum has some admixture of the monomer spect rum. We assume that the
+absorption maximum at 520nm is due to monomers and the maximu m of the real dimer
+spectrum is at 480nm. This would be consistent with an H-aggr egate with an excitonic
+coupling of 0.2eV.
+We studied also brickwork structures as a model for the J-agg regates with a red shifted
+absorption. We found a relative stable structure which is sh own in fig. 3b. The structural
+fluctuations are much larger than for the sandwich model but t he fluctuations of site energies
+and excitonic coupling are even somewhat smaller. The coupl ing of -0.064eV is close to the
+value of -0.078eV which was used to simulate the vibronic spe ctrum of the J-aggregates18.
+Acknowledgments
+This work has been supported by the Deutsche Forschungsgeme inschaft (SFB 533)
+∗Electronic address: philipp.scherer@ph.tum.de
+1G.Scheibe, Angew. Chemie 49, 563 (1936)
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+J.Phys.Chem.A 105, 293 (2001)
+14P.N.Day, J.H.Jensen,M.S.Gordon and S.P.Webb, W.J.Steven s and M.Krauss, D.Garmer,
+H.Basch and D.Cohen J.Chem.Phys. 105, 1968 (1996)
+15W.Chen, M.S.Gordon, J.Chem.Phys., 105, 11081(1996)
+16GAMESS - M.W.Schmidt, K.K.Baldridge, J.A.Boatz, S.T.Elbert , M.S.Gordon, J.J.Jensen,
+S.Koseki, N.Matsunaga, K.A.Nguyen, S.Su, T.L.Windus, M.D upuis, J.A.Montgomery
+J.Comput.Chem. 14, 1347 (1993)
+17M.A.Thompson. M.C.Zerner, J.Am Chem.Soc. 112, 7828 (1990)
+18P.O.J.scherer, in: J-Aggregates, ed. T.Kobayashi, World s cientific, p.95 (1996)
+19E.T.J.Nibbering,D.A.Wiersma,K.Duppen, Chem.Phys. 183, 167 (1994)
+20M.Maroncelli, J.Chem.Phys. 94,2084 (1991)
+21E.A.Carter, J.T.Hynes,J.Chem.Phys. 94,5961 (1991)
+22T.Hayashi, T. la Cour Jansen,Wei Zhuang, and S.Mukamel, J.P hys.Chem. A 109, 64 (2005)
+7IV. TABLE CAPTIONS
+Table I:
+coordinates, atomic charges and dipoles for PIC
+Table II:
+mean values and standard deviation of distances, orientati on angles, excitation energies and
+excitonic couplings for the two structures. The long axis is defined by the vector connecting
+the two nitrogen atoms R(N13)−R(N14), the short axis by the vector R(N14)−R(C20)+
+R(N13)−R(C19).
+8Table I:
+labelx(Bohr) y(Bohr) z(Bohr) qpx(e•Bohr)py(e•Bohr)pz(e•Bohr)
+C 1 -9.565 1.113 -3.478 0.973 0.000 0.044 0.092
+C 3 -7.152 0.889 -2.403 0.929 0.038 -0.017 0.029
+C 5 -6.682 1.814 0.029 0.928 0.046 0.017 0.005
+C 7 -8.685 2.931 1.361 0.857 -0.014 -0.025 -0.090
+C 9-11.028 3.132 0.263 1.055 0.105 -0.062 -0.039
+C11-11.494 2.232 -2.174 0.992 0.101 -0.009 0.026
+N13 -4.247 1.613 1.055 0.472 -0.038 -0.096 -0.101
+C15 -2.340 0.409 -0.158 0.997 -0.061 0.124 0.024
+C17 -2.850 -0.583 -2.637 0.738 -0.052 0.121 0.041
+C19 -5.129 -0.334 -3.709 1.046 0.019 0.001 0.116
+C21 -3.738 2.886 3.487 0.859 -0.053 -0.089 -0.108
+C23 -4.351 1.265 5.791 0.893 0.020 -0.005 -0.027
+C25 0.000 0.000 1.050 0.421 0.000 0.000 -0.310
+H27 -1.393 -1.638 -3.559 0.393 0.072 0.038 0.027
+H29 -5.484 -1.128 -5.546 0.341 0.003 -0.020 -0.040
+H31 -8.451 3.626 3.244 0.331 -0.016 0.024 0.046
+H33-12.532 3.997 1.319 0.347 -0.033 0.016 0.015
+H35-13.340 2.406 -2.997 0.351 -0.036 0.003 -0.016
+H37 -9.869 0.384 -5.351 0.331 -0.028 -0.008 -0.037
+H39 -1.778 3.440 3.479 0.319 0.024 0.010 -0.006
+H41 -4.812 4.618 3.507 0.313 -0.023 0.024 -0.001
+H43 -6.323 0.721 5.833 0.299 -0.033 -0.015 0.012
+H45 -3.944 2.325 7.497 0.315 0.002 0.013 0.035
+H47 -3.236 -0.453 5.827 0.271 0.003 -0.036 0.020
+9label x(Bohr) y(Bohr) z(Bohr) qpx(e•Bohr)py(Bohr)pz(e•Bohr)
+BO31 -8.358 1.001 -2.940 -0.591 -0.156 0.084 0.045
+BO53 -6.917 1.351 -1.187 -0.737 -0.091 0.047 -0.017
+BO75 -7.683 2.372 0.695 -0.642 -0.075 -0.008 0.003
+BO97 -9.856 3.032 0.812 -0.820 -0.051 0.026 -0.020
+BO111 -10.529 1.672 -2.826 -0.889 -0.029 0.021 0.054
+BO119 -11.261 2.682 -0.956 -0.693 0.049 -0.074 -0.055
+BO135 -5.464 1.713 0.542 -0.348 -0.128 0.109 -0.133
+BO1513 -3.293 1.011 0.449 -0.450 0.093 -0.118 -0.042
+BO1715 -2.595 -0.087 -1.398 -0.495 0.080 -0.050 -0.035
+BO193 -6.140 0.277 -3.056 -0.488 -0.014 -0.194 0.034
+BO1917 -3.989 -0.459 -3.173 -0.905 0.001 -0.013 0.010
+BO2113 -3.993 2.250 2.271 -0.125 0.086 0.192 0.103
+BO2321 -4.045 2.076 4.639 -0.341 -0.016 -0.003 0.074
+BO2515 -1.170 0.204 0.446 -0.609 -0.082 -0.320 0.223
+BO2625 0.000 0.000 2.056 -0.447 0.000 0.000 -0.209
+BO2717 -2.121 -1.111 -3.098 -0.506 -0.137 0.205 0.099
+BO2919 -5.306 -0.731 -4.627 -0.525 0.037 0.121 0.276
+BO317 -8.568 3.279 2.303 -0.537 -0.046 -0.114 -0.263
+BO339 -11.780 3.565 0.791 -0.547 0.237 -0.131 -0.159
+BO3511 -12.417 2.319 -2.586 -0.551 0.285 -0.028 0.126
+BO371 -9.717 0.748 -4.415 -0.538 0.061 0.109 0.278
+BO3921 -2.758 3.163 3.483 -0.555 -0.314 -0.072 0.041
+BO4121 -4.275 3.752 3.497 -0.529 0.200 -0.282 0.023
+BO4323 -5.337 0.993 5.812 -0.543 0.350 0.089 -0.009
+BO4523 -4.148 1.795 6.644 -0.510 -0.074 -0.199 -0.303
+BO4723 -3.794 0.406 5.809 -0.518 -0.218 0.302 -0.004
+10Table II:
+sandwich brickwork
+excitation energy Ea2.522±0.014eV2.565±0.010eV
+excitation energy Eb2.523±0.014eV2.561±0.010eV
+excitonic coupling Vexc0.25±0.017eV−0.064±0.008eV
+center-center distance Rab4.25±0.08Å8.33±0.24Å
+angle between long axes 6.0±1.7o34.0±4.0o
+angle between short axes 174.0±2.6o122.0±3.9o
+distance of charge centers Rcc5.42Å 7.57Å
+V. FIGURE CAPTIONS:
+Figure 1:
+The atom numbering for PIC is shown as it is used to tabulate th e parameters
+Figure 2:
+The experimental absorption spectrum4of monomeric PIC (dots) is compared with a
+calculated spectrum (full line) from the displaced oscilla tor model. The calculated
+spectrum was shifted to reproduce the absorption maximum at 19100cm−1
+Figure 3:
+Starting from a sandwich (a) or brickwork (b) structure rela tive stable dimer
+configurations evolved. The figure shows representative sna pshots.
+11Figure 1:
+C1H26
+C3C5C7
+C9
+C11N13
+C15C17C25
+H27
+H29 H37H35H33H31C21C23
+H39 H41H43H45
+H47
+C19
+C20C4C6N14
+C16C18
+H28
+H30 H38C2C12
+H36H34
+C10C8H32H42 H40C22C24H44H46
+H48
+12Figure 2:
+18000 20000 22000
+transition frequency (cm-1)10203040506070absorption (au)
+13Figure 3:
+(a)
+3.6
+6.73.4(b)
+14
\ No newline at end of file
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@@ -0,0 +1,807 @@
+arXiv:1001.0111v1  [cond-mat.soft]  31 Dec 2009Breakdown of The Excess Entropy Scaling for the Systems with Thermodynamic
+Anomalies
+Yu. D. Fomin, N. V. Gribova, and V. N. Ryzhov
+Institute for High Pressure Physics, Russian Academy of Sci ences, Troitsk 142190, Moscow Region, Russia
+(Dated: October 31, 2018)
+This articles presents a simulation study of the applicabil ity of the Rosenfeld entropy scaling to
+the systems which can not be approximated by effective hard sp heres. Three systems are studied:
+Herzian spheres, Gauss Core Model and soft repulsive should er potential. These systems demon-
+strate the diffusion anomalies at low temperatures: the diffu sion increases with increasing density
+or pressure. It is shown that for the first two systems which be long to the class of bounded poten-
+tials the Rosenfeld scaling formula is valid only in the infin ite temperature limit where there are
+no anomalies. For the soft repulsive shoulder the scaling fo rmula is valid already at sufficiently low
+temperatures, however, out of the anomaly range.
+PACS numbers: 61.20.Gy, 61.20.Ne, 64.60.Kw
+I. INTRODUCTION
+It is well known that some liquids (for example, water,
+silica, silicon, carbon, and phosphorus) show anomalous
+behavior in the vicinity of their freezing lines [1–18]. The
+water phase diagrams have regions where a thermal ex-
+pansion coefficient is negative (density anomaly), a self-
+diffusivity increasesupon pressuring(diffusion anomaly),
+and the structural order of the system decreases upon
+compression (structural anomaly) [6, 7]. The regions
+where these anomalies take place form nested domains
+in the density-temperature [6] (or pressure-temperature
+[7]) planes: the density anomaly region is inside the dif-
+fusion anomaly domain, and both of these anomalous
+regions are inside the broader structurally anomalous re-
+gion. It is natural to relate this kind of behavior with the
+orientational anisotropy of the potentials, however, there
+are a number of studies which demonstrate the water-
+like anomalies in fluids that interact through spherically
+symmetric potentials [19–31, 33–45].
+It was shown [17, 18] that the thermodynamic and ki-
+netic anomaliesmaybe linked throughexcess entropy. In
+particular, in Refs. 17, 18 the authors propose that en-
+tropy scaling relations developed by Rosenfeld [46, 47]
+can be used to describe the the regions of diffusivity
+anomaly.
+Rosenfeld based his arguments on the approximations
+of liquid by an effective hard spheres system. In this
+approach the kinetic coefficients are expressed in re-
+duced units based on the mean length related to den-
+sity of the system d=ρ−1/3and thermal velocity vth=
+(kBT/m)1/2. The reduced diffusion coefficient D∗, vis-
+cosityη∗and thermal conductivity κ∗are written in the
+form
+D∗=Dρ1/3
+(kBT/m)1/3(1)η∗=ηρ−2/3
+(mkBT)1/2(2)
+κ∗=κρ−2/3
+kB(kBT/m)1/2(3)
+Rosenfeld suggested that the reduced transport coef-
+ficients can be connected to the excess entropy of the
+systemSex= (S−Sid)/(NkB) through the formula
+X=aX·ebXSex, (4)
+whereXisthetransportcoefficient, and aXandbXare
+constants which depend on the studying property [47].
+Interestinglythecoefficients aandbshowextremelyweak
+dependence on the material and can be considered as
+universal.
+Another expression for relating diffusion coefficient to
+the excess entropy was suggested by Dzugutov [49]. In
+this approach the natural parameters of the system were
+chosen to be the particle diameter σand the Enskog col-
+lision frequencyΓ E= 4σ2g(σ)ρ√πkBT
+m, whereg(σ) is the
+value of radial distribution function at contact. In case
+of continuous potentials the value of σcorresponds to
+the distance of the first maximum of radial distribution
+function. Defining the reduced diffusion coefficient as
+D∗
+D=D
+ΓEσ2. Dzugutov suggested the following formula
+for it
+D∗
+D= 0.049es2, (5)
+wheres2is a pair contribution to the excess entropy
+[49]. It was shown that this relation holds for many sim-
+ple liquids. At the same time this equation is not strictly
+valid forliquid metals. In the work[50] it wasshownthat
+in this case it is necessary to replace the pair entropy s22
+by the full excess entropy Sex. It was also shown that
+Dzugutov formula does not work for silica modelled with
+an angulardependent potential [50]. It allows to say that
+Rosenfeld relation is more general.
+Remember that original idea underlying the Rosen-
+feld relation is to refer the system under investigation
+to the hard spheres system. In this respect it is inter-
+esting whether the Rosenfeld scaling relation is valid for
+the systems essentially different from the hard spheres.
+One of the examples of such systems is the system with
+potentials with negative curvature [19, 20]. It was shown
+in many publications that the behavior of such systems
+is very complex [41, 44, 45, 51–57]. In particular such
+systems can form complicated structures, like cluster liq-
+uids or different crystal phases. They can demonstrate
+maximum on the melting line and reentrant melting and
+many other unusual properties. In particular systems
+with negative curvature can demonstrate anomalous be-
+havior [17, 26, 30–32, 39, 45, 58].
+It was suggested that the Rosenfeld relations can hold
+even in the case of anomalous diffusion [17, 18]. For
+example, in the paper [18] the dependence of both ex-
+cess entropy and diffusion coefficient on density are re-
+ported for the core-softening potential that consists of a
+combination of a Lennard-Jones potential plus a Gaus-
+sian well. This potential can represent a whole family
+of two length scales intermolecular interactions, from a
+deep double-well potential to a repulsive shoulder. Ac-
+cordingly to these dependencies both excess entropy and
+diffusion have non monotonic behavior which allows to
+preserve exponential dependence of the diffusion coef-
+ficient on the excess entropy. It means that the ther-
+modynamically anomalous regions are characterized by
+anomalous behavior of the excess entropy which induces
+anomalous diffusion as well.
+Another example of systems which can not be ap-
+proximated by a hard sphere model is the systems with
+bounded potentials [59–61, 67]. Since these potentials
+have no singularity in the origin the behavior of such
+system is strongly different from the behavior of hard
+spheres.
+One of the most common model with bounded poten-
+tial is the Gaussian Core Model (GCM). This system is
+defined by the potential
+UG(r) =εe−r2/σ2. (6)
+ThispotentialwasintroducedbyStillinger[59]forsim-
+ulationoftheplasticcrystalssystem. Thephasediagram
+of the GCM demonstrates two crystal phases - fcc and
+bcc [62]. Starting from the densities around ρσ3≈0.25
+the melting curve has a negative slope. It was also shown
+that GCM demonstrates liquid state anomalies: density
+anomaly [63, 64], diffusion anomaly [64, 65] and struc-
+tural anomaly [64]. Interestingly Stockes-Einstein rela-
+tion is also violated in the GCM system [66].In the article [67] an extensive study of another model
+with bounded potential was reported. This work is con-
+cerned to the Herzian spheres system which is defined by
+the interparticle potential of the form
+Φ(r) =/braceleftbigg
+ε(1−r/σ)5/2, r≤σ
+0, r > σ(7)
+The phase diagram of the Herzian spheres system
+demonstrates very complex behavior, including many
+crystal phases and reentrant melting. Anomalous dif-
+fusion is also reported [67].
+Taking into account that the behavior of the systems
+with negative curvature potentials and the systems with
+bounded potentials is rather different from the behavior
+of hard spheres a question arises if the Rosenfeld scaling
+relations are applicable for such systems.
+The purpose of this article is to analyze the validity
+of the entropy scaling for the systems with anomalous
+behavior. For our analysis we have chosen the diffusion
+coefficient since it is the simplest transport coefficient
+to calculate in simulation. The article is organized as
+follows. In the second section we describe the models in-
+vestigated in the present work and the simulation setup.
+Section III gives the results of the simulations and the
+discussion of these results. Finally the section IV repre-
+sents our conclusions.
+II. THE SYSTEMS AND METHODS
+Three systems were studied in the present work:
+Herzian spheres, Gauss core model and a soft repulsive
+shoulder system.
+For the simulation of the Herzian spheres we used
+a system of 1000 particles in a cubic simulation box.
+NVE MD simulation was carried. Equations of mo-
+tion were integrated by velocity Verlet algorithm. The
+time step was set to dt= 0.0005. The equilibration run
+was 5·105time steps and the production run 1 .5·106
+time steps. During the equilibration the velocities were
+rescaled to keep the temperature constant. The diffu-
+sions were computed via the Einstein relation for the
+densities from ρ= 1.0 tillρ= 15.0 with the step
+∆ρ= 0.5. Additional simulations were done for com-
+puting the equation of state for the densities less then
+unity. Free energy of the liquid was calculated by inte-
+gratingthepressurealonganisotherm[68]andtheexcess
+entropy was obtained from the relation Sex=U−Fex
+NkBT,
+whereUis the internal potential energy of the liquid.
+The simulations were done for the set of ten isotherms:
+T= 0.01;0.02;0.03;0.05;0.1;0.15;0.2;0.25;0.3;0.5.
+In the case of GCM the system consisted of 2000
+particles. The time step was set to dt= 0.05.
+The equilibration and production runs were 4 ·105
+and 1·106time steps respectively. The diffusion3
+was computed for the densities from ρ= 0.1 to
+ρ= 1.0 with the step of 0 .1 for the isotherms
+T= 0.04;0.05;0.06;0.07;0.1;0.2;0.3;0.4;0.5;1.0;2.0.
+The calculation of the diffusion coefficient and excess en-
+tropy was carried in the same way as for Herzian spheres.
+The last system considered in the present work is the
+continuous repulsive shoulder system introduced in the
+article [41]. The potential of this system has the form
+U(r) = (σ
+r)14+1
+2ε·[1−tanh(k0{r−σ1})],(8)
+wherek0= 10.0. As it was reported in the paper [45]
+this system demonstrates anomalous behavior due to its
+quasibinary nature. Here we extend the investigation of
+the diffusion anomaly in the repulsive shoulder system
+withσ1= 1.35 and check the Rosenfeld relations for this
+system in the anomalous region.
+For the simulation of the repulsive shoulder system we
+used parallel tempering technic [68]. The details of the
+simulation were described in [45]. We computed the dif-
+fusion coefficients along different isochors starting from
+the density ρ= 0.3 toρ= 0.8 with the step 0 .05. A
+set of 24 temperatures between T= 0.2 andT= 0.5
+was simulated. Taking into account the exchange of the
+temperatures at the same density more then a hundred
+runs at the same isochor was done. This allowed us to
+collect a good statistics on the temperature dependence
+of the diffusion coefficient. The diffusion coefficient along
+an isochor was approximated by a 9 −th order polynome
+of the temperature. The excess entropy was calculated
+in the way described above.
+Usually excess entropy can be well approximated by
+the pair contribution only: Sex=Spair+S3+...≈Spair,
+where
+Spair=−1
+2ρ/integraldisplay
+dr[g(r)ln(g(r))−(g(r)−1)],(9)
+whereρis the density of the system and g(r) is the
+radial distribution function. We did not use the pair con-
+tribution to the excess entropy for the GCM and Herzian
+spheres because of the considerable overlap of the parti-
+cles for the bounded potentials.
+Since the potentials studied in the present work have
+negative curvature regions or are bounded they can not
+be approximated by an one component hard spheres sys-
+tem. It allows us to pose a question about the appli-
+cability of the entropy scaling to these systems both in
+Rosenfeld and Dzugutov forms. Note that Dzugutov re-
+lation (Eq. (5)) involves the size of the particles σwhich
+is ill defined for the negative curvature and bounded po-
+tentials systems. This makes problematic to apply the
+Dzugutov scaling rule to them. Because of this only
+Rosenfeld relations were used in this work.In this paper we use the dimensionless quantities: ˜r=
+r/σ,˜P=Pσ3/ε,˜V=V/Nσ3= 1/˜ρ,˜T=kBT/ε. Since
+we use only these reduced units we omit the tilde marks.
+III. RESULTS AND DISCUSSION
+This section reports the simulation results for the dif-
+fusion coefficient and excess entropy of the three models
+described above and checks the validity of the Rosenfeld
+relation for these systems.
+Herzian spheres
+Lowtemperaturebehaviorofthediffusioncoefficientof
+Herzian spheres system was already reported in the work
+[67]. As it is seen from this publication the diffusivity
+shows even two anomalous regions at the temperature
+T= 0.01 where diffusion coefficient grows with growing
+density. In the present work the dependence of the dif-
+fusion coefficient on density along several isotherms was
+monitored. The simulation data are presented on the
+Fig. 1 (a) - (b).
+One can see from these figures that at low tempera-
+tures (Fig. 1(a)) the diffusion is non monotonic, while at
+high temperatures it monotonically decays with increas-
+ing density (Fig. 1(b)) and comes to a constant value
+(see inset of the Fig. 1(b)).
+It is worth to note that the melting temperatures of
+Herzian spheres reported in the work [67] are of the or-
+der of 10−3, so the temperatures about 0 .1 are extremely
+highforthismodel. ThisiseasilyseenfromtheFig. 2(a)
+- (b) where the radial distribution functions for the den-
+sityρ= 6.0 are shown for the same set of temperatures.
+One can see that at T= 0.01 the liquid has short range
+structure which rapidly decays with increasing tempera-
+ture. At the temperature T= 0.1 the liquid looks almost
+like an ideal gas since g(r) comes to unity very quickly.
+Excess entropy also shows non monotonic dependence
+on density along an isotherm (Fig. 3(a) -(b)). One can
+see from these figures that at low temperatures excess
+entropy has two minima and a maxima in the investi-
+gated density range while at high temperature the first
+minima is depressed and the curves just change the slope
+smoothly.
+Now we turn to the Rosenfeld relation for the Herzian
+spheres system. The dependence of the reduced diffu-
+sion (see formula (1)) on the excess entropy along some
+isotherms is shown in the Fig. 4 (a) - (b). Looking at
+the curve for T= 0.01 (Fig. 4 (a)) one can divide it into
+three distinct regions with different slopes which we de-
+note as ’1’, ’2’ and ’3’. The density increase corresponds
+tomovingalongthe curvesfrom rightto left, i.e. region2
+corresponds to the higher densities then 1, and 3 - higher
+densities then 2. As is seen from the plots the region 34
+/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52/s48/s46/s48/s53/s48/s46/s48/s54/s32/s84/s61/s48/s46/s48/s49
+/s32/s84/s61/s48/s46/s48/s50
+/s32/s84/s61/s48/s46/s48/s51/s68/s40/s97/s41
+/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s49/s50/s51
+/s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54
+/s32/s84/s61/s48/s46/s49
+/s32/s84/s61/s48/s46/s50
+/s32/s84/s61/s48/s46/s51/s68/s40/s98/s41
+FIG. 1: Diffusion coefficient of Herzian spheres for a set of
+isotherms. (a) - T= 0.01;0.02 and 0 .03; (b) - T= 0.1;0.2
+and 0.3. The inset of the figure (b) showes the high density
+behavior of diffusion coefficient.
+rapidly disappears with increasing the temperature. Al-
+ready at T= 0.03 this region is negligibly small. Recall
+from the figures 1 and 3 that at low temperatures both
+diffusion and excess entropy behave non monotonically
+while at growing the temperature this effect disappears.
+This leads to the depression of the region 3 in the Fig. 4
+(a).
+The region 2 is rather stable. As one can see from the
+Fig. 4 (b) this region also becomes less developed with
+increasing the temperature, but it still preserves even for
+high temperatures. It makes the excess entropy scaling/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s103/s40/s114/s41
+/s114/s32/s84/s61/s48/s46/s48/s49
+/s32/s84/s61/s48/s46/s48/s50
+/s32/s84/s61/s48/s46/s48/s51/s40/s97/s41
+/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s40/s98/s41/s32/s84/s61/s48/s46/s49
+/s32/s84/s61/s48/s46/s50
+/s32/s84/s61/s48/s46/s51/s103/s40/s114/s41
+/s114
+FIG. 2: Radial distribution functions of Herzian spheres at
+ρ= 6.0 and a set of temperatures. (a) - T= 0.01;0.02 and
+0.03; (b) - T= 0.1;0.2 and 0.3.
+curve consisting from two parts of different slope and a
+cross region.
+Fig. 5 summarizes all the results obtained for Herzian
+spheres system. Ten different isotherms are shown there.
+As one can see only at the temperature as high as 0 .5 the
+Rosenfeld linear relation between the logarithm of the
+reduced density and the excess entropy becomes valid.
+Remember that the melting temperature is of the order
+of 0.005, that is 100 times smaller. It allows to say that
+the Rosenfeld relation for diffusion coefficient of Herzian
+spheres is valid only in the infinitely high temperature
+limit.5
+/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s45/s51/s46/s53/s45/s51/s46/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s40/s97/s41/s32/s84/s61/s48/s46/s48/s49
+/s32/s84/s61/s48/s46/s48/s50
+/s32/s84/s61/s48/s46/s48/s51/s83
+/s101/s120
+/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s45/s48/s46/s55/s45/s48/s46/s54/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s40/s98/s41/s32/s84/s61/s48/s46/s49
+/s32/s84/s61/s48/s46/s50
+/s32/s84/s61/s48/s46/s51/s83
+/s101/s120
+FIG. 3: Excess entropy along and a set of isotherms. (a) -
+T= 0.01;0.02 and 0 .03; (b) - T= 0.1;0.2 and 0.3.
+Gaussian Core Model
+The results for the GCM are qualitatively similar to
+the case of Herzian spheres. Because of this we do not
+explain them in detail. Fig. 6(a) and (b) shows the dif-
+fusion coefficient for the GCM system for a set of six
+isotherms. One can see that for low temperatures start-
+ing from the densities approximately 0 .3 the diffusion
+coefficient demonstrate anomalous growth with increase
+of the density. We expect that at higher densities the
+curve bends downward but in the present study we have
+not measured so high densities for this model. At high
+temperatures the diffusion monotonically decreases with/s45/s51/s46/s48 /s45/s50/s46/s53 /s45/s50/s46/s48 /s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53/s45/s51/s46/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53
+/s51
+/s50
+/s49/s40/s97/s41
+/s32/s84/s61/s48/s46/s48/s49
+/s32/s84/s61/s48/s46/s48/s50
+/s32/s84/s61/s48/s46/s48/s51/s108/s110/s40/s68/s49/s47/s51
+/s47/s84/s49/s47/s50
+/s41
+/s83
+/s101/s120
+/s45/s48/s46/s54 /s45/s48/s46/s53 /s45/s48/s46/s52 /s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s40/s98/s41
+/s32/s84/s61/s48/s46/s49
+/s32/s84/s61/s48/s46/s50
+/s32/s84/s61/s48/s46/s51/s108/s110/s40/s68/s49/s47/s51
+/s47/s84/s49/s47/s50
+/s41
+/s83
+/s101/s120
+FIG. 4: The dependence of reduced diffusion on the excess
+entropy (see formulas (1) and (4)) along some isotherms (a)
+-T= 0.01;0.02 and 0 .03; (b) - T= 0.1;0.2 and 0.3.
+increasing the density.
+The structure of the liquid rapidly decays with the
+temperature increase. This is shown in the Fig. 7(a) -
+(b). One can see that at the temperature T= 1.0g(r) is
+equal to unity almost in the whole range of the distances
+r.
+Like the diffusion coefficient the excess entropy has
+a minimum at the low temperatures, but with increas-
+ing the temperature it becomes monotonicallydecreasing
+function of the density (Fig. 8(a) - (b)).
+Fig. 9(a) - (b) demonstrates the Rosenfeld relation for
+the GCM for the same set of isotherms. Comparing to6
+/s45/s51/s46/s53 /s45/s51/s46/s48 /s45/s50/s46/s53 /s45/s50/s46/s48 /s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51
+/s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48/s48/s49/s50/s51/s32/s84/s61/s48/s46/s48/s49
+/s32/s84/s61/s48/s46/s48/s50
+/s32/s84/s61/s48/s46/s48/s51
+/s32/s84/s61/s48/s46/s48/s53
+/s32/s84/s61/s48/s46/s49
+/s32/s84/s61/s48/s46/s49/s53
+/s32/s84/s61/s48/s46/s50
+/s32/s84/s61/s48/s46/s50/s53
+/s32/s84/s61/s48/s46/s51
+/s32/s84/s61/s48/s46/s53/s108/s110/s40/s68/s49/s47/s51
+/s47/s84/s49/s47/s50
+/s41
+/s83
+/s101/s120
+FIG. 5: Rosenfeld excess entropy scaling of diffusion coeffi-
+cient for a set of ten isotherms for Herzian spheres system.
+The inset shows several high temperature isotherms in the
+enlarged scale.
+the case of Herzian spheres there is no region ’3’ in the
+low temperature curves of the GCM. We suppose that
+this region corresponds to the higher densities which we
+do not consider in the present study. At low tempera-
+tures most of the points belong to the region ’2’. How-
+ever this region depresses with the temperature increase.
+Although even at the temperature as high as T= 2.0
+which is around 200 times higher then melting tempera-
+ture of GCM the curve still demonstrates a bend from a
+straight line at high densities.
+Finally Fig. 10 shows the whole set of isotherms in-
+vestigated in the present study. As one can see from this
+figure the system comes more close to the Rosenfeld re-
+lation with increasing the temperature. One can expect
+that at infinitely high temperatures the excess entropy
+relation is valid for the GCM, however in the tempera-
+tures rangestudied here, i.e. to approximately200 ·Tmels
+a deviation from the linear behavior is still observed.
+Soft Repulsive Shoulder Model
+The last model considered in the present work is the
+soft repulsiveshoulderpotential(8). In the works[41,45]
+it has already been shown that this system demonstrates
+anomalous behavior of diffusion at low temperatures and
+densitiescorrespondingtotheregionwhereacompetition
+between the characteristic length scales σandσ1takes
+place. This competition gives the great complexity of the
+phase diagram of the system [41, 45], so one can expect/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48/s48/s46/s51/s53/s48/s46/s52/s48
+/s32/s84/s61/s48/s46/s48/s52
+/s32/s84/s61/s48/s46/s48/s55
+/s32/s84/s61/s48/s46/s49/s68/s40/s97/s41
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48 /s40/s98/s41
+/s32/s84/s61/s48/s46/s53
+/s32/s84/s61/s49/s46/s48
+/s32/s84/s61/s50/s46/s48/s68
+FIG. 6: Diffusion coefficient of GCM for a set of isotherms.
+(a) -T= 0.04;0.07 and 0 .1; (b) -T= 0.5;1.0 and 2.0.
+that the thermodynamic quantities, in particularentropy
+which is of the interest of the present study, also have
+a complex behavior in this region of densities. Taking
+into account the complex behavior of both entropy and
+diffusioncoefficientitisinterestingtochecktheRosenfeld
+relation for this system.
+Fig. 11 represents the diffusion coefficient for the soft
+repulsive shoulder system with σ1= 1.35 for a set of
+temperaturesanddensities. Onecanseethatat T= 0.25
+aninflectionpointinthe diffusioncoefficientcurveoccurs
+which then develops into a loop ( T= 0.2).
+Both pair and full excess entropies were considered for
+this potential. Fig. 12 (a) and (b) show the behavior7
+/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s40/s97/s41
+/s32/s84/s61/s48/s46/s48/s52
+/s32/s84/s61/s48/s46/s48/s55
+/s32/s84/s61/s48/s46/s49/s103/s40/s114/s41
+/s114
+/s48 /s50 /s52 /s54 /s56/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s40/s98/s41
+/s32/s84/s61/s48/s46/s53
+/s32/s84/s61/s49/s46/s48
+/s32/s84/s61/s50/s46/s48/s103/s40/s114/s41
+/s114
+FIG. 7: Radial distribution functions of GCM for a set of
+isotherms at the density ρ= 0.5 (a) -T= 0.04;0.07 and 0 .1;
+(b) -T= 0.5;1.0 and 2.0.
+of the entropies along two isotherms. One can see that
+both at high and low temperature the difference between
+excess entropy and pair contribution to it is rather large.
+This discrepancy is small at low densities, but greatly
+increases at the density about 0 .4. Note that this density
+corresponds to a character distance l∼1/ρ1/3≃1.35,
+that isl≃σ1. It allows to conclude that the interplay of
+the distances starts at this density and it is this interplay
+which makes the excess entropy and pair excess entropy
+difference to increase rapidly.
+Fig. 13 shows the diffusion coefficient scaling with the
+pairpartoftheexcessentropy. Asisseenfromthefigures/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s49/s46/s52/s45/s49/s46/s50/s45/s49/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s40/s97/s41/s32/s84/s61/s48/s46/s48/s52
+/s32/s84/s61/s48/s46/s48/s55
+/s32/s84/s61/s48/s46/s49/s83
+/s101/s120
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s40/s98/s41/s32/s84/s61/s48/s46/s53
+/s32/s84/s61/s49/s46/s48
+/s32/s84/s61/s50/s46/s48/s83
+/s101/s120
+FIG. 8: Excess entropy of GCM for a set of isotherms. (a) -
+T= 0.04;0.07 and 0 .1; (b) -T= 0.5;1.0 and 2.0.
+even at high temperatures the curve is not straight while
+at low temperatures the curve becomes very strange.
+Definitely the exponential relation between the diffusion
+coefficient and pair excess entropy is not valid.
+The diffusion scaling with the full excess entropy is
+showninthefigure14(a)and(b). Onecanseefromthese
+pictures that the scaling rule works good for the temper-
+aturesT= 0.5 andT= 0.4 but already for T= 0.35
+the deviation from the linear behavior occurs. This de-
+viation develops more as the temperature decreases. At
+T= 0.25 a self crossing loop occurs. This loop even en-
+larges at lower temperatures. It is worth to note that the
+curve at low temperature T= 0.2 consists of two linear8
+/s45/s49/s46/s52 /s45/s49/s46/s51 /s45/s49/s46/s50 /s45/s49/s46/s49 /s45/s49/s46/s48 /s45/s48/s46/s57 /s45/s48/s46/s56 /s45/s48/s46/s55 /s45/s48/s46/s54 /s45/s48/s46/s53 /s45/s48/s46/s52/s45/s49/s46/s52/s45/s49/s46/s50/s45/s49/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48
+/s50
+/s49/s40/s97/s41/s32/s84/s61/s48/s46/s48/s52
+/s32/s84/s61/s48/s46/s48/s55
+/s32/s84/s61/s48/s46/s49/s108/s110/s40/s68/s49/s47/s51
+/s47/s84/s49/s47/s50
+/s41
+/s83
+/s101/s120
+/s45/s48/s46/s49/s54 /s45/s48/s46/s49/s50 /s45/s48/s46/s48/s56 /s45/s48/s46/s48/s52 /s48/s46/s48/s48/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48/s50/s46/s52/s50/s46/s56/s51/s46/s50
+/s40/s98/s41/s32/s84/s61/s48/s46/s53
+/s32/s84/s61/s49/s46/s48
+/s32/s84/s61/s50/s46/s48/s108/s110/s40/s68/s49/s47/s51
+/s47/s84/s49/s47/s50
+/s41
+/s83
+/s101/s120
+FIG. 9: Excess entropy of GCM for a set of isotherms. (a) -
+T= 0.04;0.07 and 0 .1; (b) -T= 0.5;1.0 and 2.0.
+parts connected by the self crossing loop. It means that
+the Rosenfeld formula is valid in some narrow regions,
+but not in the whole range of densities.
+In order to see the reason for the occurrence of the
+self crossing loop we compare the qualitative behavior
+of the diffusion coefficient and excess entropy. For this
+reasons we measure the densities corresponding to the
+minimum and maximum of the diffusion ( ρD
+minandρD
+max
+correspondingly) and excess entropy ( ρS
+minandρS
+max) at
+the lowest temperature we study T= 0.2. The values we
+obtain are: ρD
+min= 0.44,ρD
+max= 0.55,ρS
+min= 0.40 and
+ρD
+max= 0.535. One can see that there is a mismatch in
+the locationofthe extremalpointsofthesetwoquantities/s45/s49/s46/s50 /s45/s48/s46/s56 /s45/s48/s46/s52 /s48/s46/s48/s45/s49/s48/s49/s50/s51/s32/s84/s61/s48/s46/s48/s52
+/s32/s84/s61/s48/s46/s48/s53
+/s32/s84/s61/s48/s46/s48/s54
+/s32/s84/s61/s48/s46/s48/s55
+/s32/s84/s61/s48/s46/s49
+/s32/s84/s61/s48/s46/s50
+/s32/s84/s61/s48/s46/s51
+/s32/s84/s61/s48/s46/s52
+/s32/s84/s61/s48/s46/s53
+/s32/s84/s61/s49/s46/s48
+/s32/s84/s61/s50/s46/s48/s108/s110/s40/s68/s49/s47/s51
+/s47/s84/s49/s47/s50
+/s41
+/s83
+/s101/s120
+FIG. 10: Reduced diffusion coefficient (formula (1)) of GCM
+for a set of eleven isotherms
+/s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48
+/s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52/s48/s46/s48/s53
+/s32/s84/s61/s48/s46/s53/s48
+/s32/s84/s61/s48/s46/s52/s48
+/s32/s84/s61/s48/s46/s51/s53
+/s32/s84/s61/s48/s46/s51/s48
+/s32/s84/s61/s48/s46/s50/s53
+/s32/s84/s61/s48/s46/s50/s48/s68
+FIG. 11: Diffusion coefficient for soft repulsive shoulder sys -
+tem with σ1= 1.35 along several isotherms. The inset en-
+larges the isotherms T= 0.5 (squares) and T= 0.25 (circles).
+and therefore there are some regions where the qualita-
+tive behavior of diffusion and excess entropy is opposite.
+Taking into account exponential dependence supposed in
+theRosenfeld formulaonecanexpectthat evensmalldis-
+crepancies in the qualitative behavior can lead to large
+errors in the scaling relation.9
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s45/s50/s46/s50/s53/s45/s49/s46/s53/s48/s45/s48/s46/s55/s53/s48/s46/s48/s48
+/s32/s83
+/s101/s120
+/s32/s115
+/s50/s83/s40/s97/s41
+/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57/s45/s52/s45/s51/s45/s50/s45/s49/s48
+/s32/s83
+/s101/s120
+/s32/s115
+/s50/s83
+/s101/s120/s40/s98/s41
+FIG. 12: Excess entropy and pair excess entropy for the soft
+repulsive shoulder system for (a) T= 0.5 and (b) T= 0.2.
+CONCLUSIONS
+Thisarticlespresentsasimulationstudyoftheapplica-
+bilityoftheRosenfeldentropyscalingtothesystemswith
+negativecurvatureandboundedpotentials. It wasshown
+that the excessentropyscalingcannot be applied to such
+systems at low enough temperatures. Interestingly all
+of the systems considered here demonstrate anomalous
+diffusion behavior in some regions of temperatures and
+densities. It makes questionable if the Rosenfeld rela-
+tion is applicable for the systems with diffusion anomaly.
+TheseresultsareincontradictionwiththeresultsofRefs.
+17, 18. This contradiction may be attributed to the dif-/s45/s50/s46/s52 /s45/s50/s46/s50 /s45/s50/s46/s48 /s45/s49/s46/s56 /s45/s49/s46/s54 /s45/s49/s46/s52 /s45/s49/s46/s50 /s45/s49/s46/s48 /s45/s48/s46/s56/s45/s51/s46/s48/s45/s50/s46/s56/s45/s50/s46/s54/s45/s50/s46/s52/s45/s50/s46/s50/s45/s50/s46/s48/s45/s49/s46/s56/s45/s49/s46/s54
+/s32/s84/s61/s48/s46/s53/s48
+/s32/s84/s61/s48/s46/s52/s48
+/s32/s84/s61/s48/s46/s51/s53/s108/s110/s40/s68/s49/s47/s51
+/s47/s84/s49/s47/s50
+/s41
+/s115
+/s50/s40/s97/s41
+/s45/s51/s46/s48 /s45/s50/s46/s53 /s45/s50/s46/s48 /s45/s49/s46/s53/s45/s52/s46/s48/s45/s51/s46/s53/s45/s51/s46/s48/s45/s50/s46/s53
+/s32/s84/s61/s48/s46/s50/s48
+/s32/s84/s61/s48/s46/s50/s53
+/s32/s84/s61/s48/s46/s51/s48/s108/s110/s40/s68/s49/s47/s51
+/s47/s84/s49/s47/s50
+/s41
+/s115
+/s50/s40/s98/s41
+FIG. 13: The diffusion scaling with the pair contribution to
+the excess entropy at (a) high and (b) low temperatures for
+the soft repulsive shoulder system.
+ferencesofconsideredpotentialsandsimulationmethods,
+however, this question requires further investigation and
+will be a topic of a subsequent publication.
+One can suppose that the excess entropy scaling is in-
+valid for the systems with negative curvature potentials
+such as repulsive shoulder potential. A possible reason
+for this can be related to the fact that such systems are
+effectively quasibinary [41, 45]. As it was mentioned in
+the introduction, the original Rosenfeld idea was based
+on the connection of a liquid under investigation to the
+effective hard spheres liquid. At the same time liquids
+with negative curvature potentials may be approximated10
+/s45/s51/s46/s50 /s45/s50/s46/s56 /s45/s50/s46/s52 /s45/s50/s46/s48 /s45/s49/s46/s54 /s45/s49/s46/s50/s45/s50/s46/s56/s45/s50/s46/s52/s45/s50/s46/s48/s45/s49/s46/s54
+/s32/s84/s61/s48/s46/s51/s48
+/s32/s84/s61/s48/s46/s51/s53
+/s32/s84/s61/s48/s46/s52/s48/s108/s110/s40/s68/s49/s47/s51
+/s47/s84/s49/s47/s50
+/s41
+/s83
+/s101/s120/s40/s97/s41
+/s45/s52/s46/s52 /s45/s52/s46/s48 /s45/s51/s46/s54 /s45/s51/s46/s50 /s45/s50/s46/s56 /s45/s50/s46/s52 /s45/s50/s46/s48/s45/s52/s46/s48/s45/s51/s46/s54/s45/s51/s46/s50/s45/s50/s46/s56/s45/s50/s46/s52
+/s32/s84/s61/s48/s46/s50/s48
+/s32/s84/s61/s48/s46/s50/s53
+/s32/s84/s61/s48/s46/s51/s48/s108/s110/s40/s68/s49/s47/s51
+/s47/s84/s49/s47/s50
+/s41
+/s83
+/s101/s120/s40/s98/s41
+FIG. 14: The diffusion scaling with the excess entropy at (a)
+high and (b) low temperatures for the soft repulsive shoulde r
+system.
+by a mixture of hard spheres of two different sizes. The
+concentration of components of such mixture is pressure
+and temperature dependent. As it was shown in liter-
+ature (see, for example, [69]) the excess entropy scaling
+holds for binary mixtures too. But in the case of quasi-
+binary mixture since the effective concentration depends
+on the pressure and temperature the behavior becomes
+more complex. This brings to the breakdown of the scal-
+ing rules for this case.
+Obviously, the systems with bounded potentials can
+not be approximated by hard sphere potentials too. It
+seems that this may be the reason of violation of Rosen-feld entropy scaling for these systems.
+We thank V. V. Brazhkin and Daan Frenkel for stim-
+ulating discussions. Our special thanks to Prof. Ch.
+Chakravarty who attracted our attention to the prob-
+lems considered here. The work was supported in part
+by the Russian Foundation for Basic Research (Grant No
+08-02-00781).
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\ No newline at end of file
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+arXiv:1001.0112v1  [hep-th]  31 Dec 2009G¨ oteborg preprint
+December, 
+D=11supergravity
+with manifestsupersymmetry
+Martin Cederwall
+Fundamental Physics
+Chalmers University of Technology
+SE 412 96 G¨ oteborg, Sweden
+Abstract: The complete supersymmetricactionforeleven-dimensionalsuper gravity
+is presented. The action is polynomial in the scalar fermionic pure spin or super-
+field, and contains only a minor modification to the recently proposed three-point
+coupling./CT/D1/CP/CX/D0/BM /D1/CP/D6/D8/CX/D2/BA
/CT/CS/CT/D6/DB/CP/D0/D0
 /CW/CP/D0 /D1/CT/D6 /D7/BA/D7 /CT ...................... M. Cederwall: “D=11 supergravity with manifest supersymme try”
+1. Introduction
+Eleven-dimensionalsupergravity[ ] is the low-energylimit ofthe (not yet defined) M-theory,
+and hence of a strong-coupling limit of string theory. Having maximal supersymmetry, a
+traditional superspace description [ ] puts the theory on-shell. Recently, a programme was
+initiated to formulate eleven-dimensional supergravity with manifes t supersymmetry using
+a pure spinor superfield. A simple three-point interaction was propo sed [].
+Purespinorsuperfieldsprovideapowerfultoolforformulatingsu persymmetricfield and
+stringtheories[ -].Inmodelswithmaximalsupersymmetry,theconstraintontheord inary
+superfield, which enforces the equations of motion, is encoded in a c ohomologicalequation of
+the typeQΨ+...= 0, which is the equation of motion for the pure spinor superfield. Pu re
+spinor superfield theory inevitably leads to a Batalin–Vilkovisky (BV) f ormalism [,].
+In the present paper, we will show that the deformation of the fre e action represented
+by the three-point interaction of ref. [ ] is almost the whole answer. Due to the simple
+properties of the operators involved, higher order interactions a re essentially absent. Pure
+spinor superfield formulationstend to have some remarkable prope rties, as an extra bonus in
+addition to the manifest supersymmetry. The action for D= 10 super-Yang–Mills is Chern–
+Simons-like, and has only a cubic interaction [ ]. The conformal models in D= 3, whose
+component actions contain couplings of six scalar fields, simplify enor mously in the pure
+spinor framework, where the matter superfields only have a minimal coupling to the Chern–
+Simons field [ ,]. Higher order interactions arise when auxiliary fields are eliminated (in
+both cases the fermionic component of the gauge connection on su perspace). This type of
+simplification is shared by D= 11 supergravity, surprisingly to the extent that the action
+becomes polynomial.
+The organisation of the paper is as follows. In Section , we review the construction
+of ref. []. The full action is given in Section , where we also expand the action around a
+background. Section contains conclusions and a discussion, where we focus on identifying
+future directions of research.
+2. Eleven-dimensional supergravity with pure spinors
+The relevant pure spinors in D= 11 satisfy ( λγaλ) = 0. It has been known for some time
+that the cohomology of a scalar fermionic superfield under the the B RST operator q= (λD)
+gives the linearised supergravity multiplet [ ,,]. The fermionic derivatives anticommute
+to give torsion{Dα,Dβ}=−TαβcDc, withTαβc=−2γc
+αβ. This holds for any background
+satisfying the equations of motion, but for all purposes in this pape rDαwill be the flat
+covariant derivative. The fermionic scalar field Ψ has dimension −3 and ghost number 3,M. Cederwall: “D=11 supergravity with manifest supersymme try”...................... 
+and its lowest component is the third order ghost for the tensor fie ld. The physical fields of
+ghost number 0 sit in the field as λαλβλγCαβγ(x,θ), whereλαis the pure spinor and Cαβγ
+the lowest-dimensional part of the superspace 3-form C. There is a natural measure on the
+pure spinor space, and it is straightforward to write an action/integraltext
+ΨQΨ giving the linearised
+equations of motion [ ]. The integrand has ghost number 7 and dimension −6. We refer to
+ref. [] for details and conventions.
+In refs. [,] it was shown that there is also another field, Φa, of dimension−1 and
+ghost number 1, that contains the linearised multiplet. This field has t he additional gauge
+symmetry Φa≈Φa+ (λγaρ), and has as its lowest component the diffeomorphism ghost.
+The physical fields sit in λαhαa, wherehαais the linearised supervielbein.
+It is necessary, both for having a non-degenerate measure and in order to write the
+relevant operators, to work with non-minimal pure spinors [ ]. In addition to the pure
+spinorλ, one has the pure spinor ¯λand the fermionic spinor rwhich is pure relative to ¯λ,
+(¯λγar) = 0. The non-minimal BRST operator is Q=q+s= (λD)+(r∂
+∂¯λ).
+2.1. The three-point coupling
+In our recent paper [ ], we constructed a three-point coupling for eleven-dimensional su per-
+gravity. This was done by constructing the BRST-invariant operat orRarelating the two
+fields according to Φa=RaΨ. It takes the form
+Ra=Ra
+0+Ra
+1+Ra
+2
+=η−1(¯λγab¯λ)∂b+η−2(¯λγab¯λ)(¯λγcdr)(λγbcdD)
+−16η−3(¯λγa[b¯λ)(¯λγcdr)(¯λγe]fr)(λγfbλ)(λγcdew).(.)
+Some alternative ways of writing the last term are given in Appendix A. Here, the invariant
+ηis defined as η= (λγabλ)(¯λγab¯λ)
+The action of ref. [ ] is
+S3=/integraldisplay
+[dZ]/bracketleftbig1
+2ΨQΨ+1
+6(λγabλ)ΨRaΨRbΨ/bracketrightbig
+. (.)
+In order to show that the BV master equation is fulfilled to third orde r in the field, we only
+need to use
+(λγabλ)[Q,Rb] = 0, (.) ...................... M. Cederwall: “D=11 supergravity with manifest supersymme try”
+which was how Rawas constructed in ref. [ ]. (There,Rawas viewed as an operator from
+the space of scalar functions Ψ to the space of vectorial function s Φawith the extra gauge
+invariance Φa≈Φa+(λγa̺). The factor ( λγabλ) in eq. (.) encodes this invariance.)
+2.2. An example: The Chern–Simons term
+In ref. [], it was shown that one of the terms contained in the three-point co upling gave the
+ghost couplings appropriate for the diffeomorphism algebra. We wou ld like to complement
+that example with one clearly displaying how a known interaction among physical super-
+gravity fields, namely the supergravity Chern–Simons term/integraltext
+C∧H∧H, is generated. The
+cohomology for the C-field is []
+ΨC∼(λγiθ)(λγjθ)(λγkθ)Cijk+... , (.)
+where the ellipsis denotes terms with H=dC(which are higher order in θ). Acting with
+Ra
+0givesRa
+0ΨC∼η−1(¯λγai¯λ)(λγjθ)(λγkθ)(λγlθ)∂iCjkl+.... The integrand in the coupling
+term becomes
+∼η−1(¯λγij¯λ)(λγkθ)(λγlθ)(λγmθ)Cklm
+×(λγnθ)(λγpθ)(λγqθ)∂iCnpq(λγrθ)(λγsθ)(λγtθ)∂jCrst.(.)
+We now use the identity [ ] (λγi1θ)...(λγi9θ)∼εii...i9ab(λγabλ)N, where Nis the scalar
+cohomology at λ7θ9used in the measure. We also replace ∂iCjklbyHijklin eq. (.),
+since the only way to form a scalar is “ C∧H∧H”. Inserting this in ( .) directly gives∼
+Nεii...i11Ci1i2i3Hi4i5i6i7Hi8i9i10i11,withoutthe needofaddingany q-exactterms.Thisshows
+that the supergravity Chern–Simons term, and hence by supersy mmetry all supergravity 3-
+point couplings, are contained in the interaction term.
+3. The complete dynamics
+3.1. The full action
+We will now examine the master equation to higher order. The master equation reads
+(S,S) = 0, (.)M. Cederwall: “D=11 supergravity with manifest supersymme try”...................... 
+where the antibracket is defined as
+(A,B) =/integraldisplay
+A←δ
+δΨ(Z)[dZ]→δ
+δΨ(Z)B . (.)
+We begin by performing a variation of the action ( .):
+δS3=/integraldisplay
+[dZ]δΨ/bracketleftbig
+QΨ+1
+6(λγabλ)RaΨRbΨ+1
+3Ra/parenleftbig
+(λγabλ)ΨRbΨ/parenrightbig/bracketrightbig
+=/integraldisplay
+[dZ]δΨ/bracketleftbig
+QΨ+1
+2(λγabλ)RaΨRbΨ+1
+3ΨRa/parenleftbig
+(λγabλ)RbΨ/parenrightbig/bracketrightbig
+.(.)
+The remainder from the antibracket is
+(S3,S3) =1
+3/integraldisplay
+[dZ](λγabλ)RaΨRbΨΨRc((λγcdλ)RdΨ). (.)
+Here, we have already used ( λγabλ)(λγcdλ)RaΨRbΨRcΨRdΨ = 0, which follows from the
+pure spinor constraint. If the last term would vanish, i.e., ifRa(λγabλ)Rb= 0, the action
+S3would be the full action. We will now show that this is not true, but almo st so, in the
+sense thatRa((λγabλ)RbΨ) is a cohomologically trivial field.
+The detailed calculation is performed in Appendix B. It is a bit lengthy, b ut once it is
+performed it leads to very simple properties for the operators. Th e calculation in Appendix
+B shows that
+Ra(λγabλ)Rb=1
+2(λγabλ)[Ra,Rb] =3
+2{Q,T}. (.)
+where we have defined the fermionic operator Twith dimension 3 and ghost number −3 as
+T= 8η−3(¯λγab¯λ)(¯λr)(rr)Nab. (.)
+Note thatTΨis bosonic and has dimension as well as ghost number zero. It seems likely that
+the fieldTΨ is connected to the trace of the metric fluctuation, and thus to t he determinant
+of the metric. So, the operator Ra(λγabλ)Rbis zero in the cohomology. In addition, the
+operatorThas very nice properties. Since it contains a multiplicative factor ( ¯λr), it squares
+to zero, even when two T’s act on different fields, TATB= 0. Consequently, TA{Q,T}B+
+{Q,T}ATB= 0 (for fermionic A), and in particular TΨ{Q,T}Ψ = 0.Tcommutes with
+Ra, and of course with the regularisation factor in the measure (since Radoes). It does not
+commute with ( λγabλ), but as long as there are contraction with RaandRbthe commutator
+gives zero:RaARbB[T,(λγabλ)]C= 0. ...................... M. Cederwall: “D=11 supergravity with manifest supersymme try”
+The remaining term in the master equations may now be written
+(S3,S3) =1
+2/integraldisplay
+[dZ](λγabλ)Ψ{Q,T}ΨRaΨRbΨ. (.)
+This term is cancelled by the antibracket between a term −1
+4/integraltext
+[dZ](λγabλ)ΨTΨRaΨRbΨ
+and the kinetic term in the action:
+S=/integraldisplay
+[dZ]/bracketleftbig1
+2ΨQΨ+1
+6(λγabλ)(1−3
+2TΨ)ΨRaΨRbΨ/bracketrightbig
+. (.)
+With this slight modification of the action given in ref. [ ], the master equation is exactly
+satisfied. Due to the simple algebraic properties of T, only terms to linear order in TΨ
+appear, and no new terms of higher order in Ψ are generated in ( S,S) (see below for an
+explicit demonstration of this fact using the equation of motion). So mewhat surprisingly,
+we thus find that the full supergravity action in the pure spinor sup erfield formulation is
+polynomial.
+The factor 1−3
+2TΨ in the coupling term may be removed by performing a field redefi-
+nition Ψ = (1+1
+2T˜Ψ)˜Ψ (which due to the nilpotency of gives T˜Ψ =TΨ and the inversion
+˜Ψ = (1−1
+2TΨ)Ψ). This leads to a non-canonical kinetic term:
+S=/integraldisplay
+[dZ]/bracketleftBig
+1
+2(1+T˜Ψ)˜ΨQ˜Ψ+1
+6(λγabλ)˜ΨRa˜ΨRb˜Ψ/bracketrightBig
+=/integraldisplay
+[dZ]/bracketleftBig
+1
+2eT˜Ψ˜ΨQ˜Ψ+1
+6(λγabλ)˜ΨRa˜ΨRb˜Ψ/bracketrightBig
+.(.)
+This action is probably closer related to the geometric formulation th an the action ( .).
+Note that the field redefinition is not canonical with respect to the a ntibracket, which can
+be calculated usingδ
+δΨ= (1−T˜Ψ)δ
+δ˜Ψ+1
+2˜ΨTδ
+δ˜Ψ.
+The equation ofmotionfollowingfrom the redefined action enjoysca ncellationsbetween
+terms from the kinetic term and the coupling term and reads
+(1+3
+2T˜Ψ)Q˜Ψ+1
+2(λγabλ)Ra˜ΨRb˜Ψ = 0, (.)
+while the equation of motion for the canonical field is
+QΨ+1
+2Ψ{Q,T}Ψ+1
+2(λγabλ)(1−2TΨ)RaΨRbΨ = 0. (.)M. Cederwall: “D=11 supergravity with manifest supersymme try”...................... 
+Using the equation of motion, it is easy to show explicitly that the mast er equation
+is indeed satisfied to all orders. The master equation is equivalent to the vanishing of the
+integral of the square of the equation of motion (for the canonica l field). The square of each
+of the three terms in eq. ( .) gives zero, and the cross terms are
+(S,S) =/integraldisplay
+[dZ]/bracketleftbig
+QΨΨ{Q,T}Ψ+(λγabλ)QΨ(1−2TΨ)RaΨRbΨ
++1
+2(λγabλ)Ψ{Q,T}ΨRaΨRbΨ/bracketrightbig(.)
+Using the algebraic properties of the operators above, it is straigh tforward to show that the
+terms both at third and fourth order in Ψ combine into total derivat ives.
+Since the operators involve negative powers of η= (λγabλ)(¯λγab¯λ), the singular prop-
+erties arη= 0 must be checked. In ref. [ ], it was shown that the number of negative powers
+of (λγabλ) or (¯λγab¯λ) must be smaller than 12 for an integral to converge. Each Rahas at
+most 4 negative powers, while Thas 5. The expression ( .) needs regularisation in order to
+be well defined. This can probably be achieved using a method similar to that of ref. [ ],
+but it is not obvious to what extent the algebraic properties of RaandTwill be preserved
+by such a regularisation. The form ( .) of the action, on the other hand, is well defined
+without regularisation.
+3.2. Expansion around a background
+Let Ψ 0be a solution to the equation of motion ( .) and let Ψ = Ψ 0+ψ. We choose
+to expand the “canonical” action ( .), since the field ψis canonical in the sense that the
+antibracket is
+(A,B) =/integraldisplay
+A←δ
+δψ(Z)[dZ]→δ
+δψ(Z)B , (.)
+which allows us to compared the expanded and original actions direct ly. An expansion of the
+non-canonical action ( .) requires letting ˜Ψ =˜Ψ0+(1−1
+2T˜Ψ0)˜ψ−˜Ψ0T˜ψ, but can also be
+obtained from rescaling of result below in terms of ψ. Expanding around the solution gives
+S=S[Ψ0]+/integraldisplay
+[dZ]/bracketleftbig1
+2ψQ′ψ+1
+6(λγabλ)(1−3
+2Tψ)ψR′aψR′bψ/bracketrightbig
+,(.)
+where
+Q′=Q+2QΨ0T+(1−2TΨ0)/bracketleftbig
+(λγabλ)RaΨ0Rb+1
+2Ψ0{Q,T}/bracketrightbig
+,
+R′a= (1−TΨ0)Ra−2RaΨ0T .(.)
+It may be checked directly that Q′2= 0 when Ψ 0fulfills the equation of motion. The
+commutators ( λγabλ)[Q′,R′b] (being 0 for Ψ 0= 0) and (λγabλ)R′aR′b(equalling3
+2{Q,T} ...................... M. Cederwall: “D=11 supergravity with manifest supersymme try”
+forΨ0= 0), seemto becomemorecomplicated,however.Obviously,the ma sterequationwill
+hold,butwehavenotcheckedthisexplicitlyusingtheprimedoperato rs.Themasterequation
+implies relations e.g.(λγabλ)[Q′,R′a]ψR′bψ= 0, which are implied by the previousones (the
+ones in a flat background) but may be weaker in the sense that they need contractions with
+fields.
+It is nice to verify that there is a kind of weak background invariance , in the sense
+that the action in any background is given by the same formal expre ssion, given by eq.
+(.), with background dependent operators fulfilling the same relation s independent of
+background (although weaker relations than the ones used in the fl at background). Since
+the model contains gravity, it is natural that the BRST operator e ncodes information about
+the background geometry. We have not yet been able to examine th e connection between
+the action in a background and the corresponding construction st arting from a solution in
+superspace. There, one would construct the BRST operator as Q=λαDα=λαEαM∂M,
+EAMbeing the inverse supervielbein of the background. It is reasonable to expect a relation
+(equality?)betweenthe operators QandQ′ofeq.(.),andalsobetweeninteractionterms.
+In order to establish such a relation, one should perform the analog ous construction to the
+one in the present paper and ref. [ ] but in a curved background superspace. The calculation
+of Appendix B relies on the algebra of flat superspace derivatives, a nd has to be revised in
+other backgrounds. We find it likely that the calculation will stay form ally unchanged with
+the flat derivatives replaced by covariant derivatives in other back grounds, but this remains
+to be seen.
+The fact that the action is polynomial of course gives a small hope of finding a truly
+background independent formulation. The solution Ψ = 0 correspon ds to flat space. In a
+background independent formulation the expectation value 0 for t he field would be a non-
+geometric situation, and flat space would arise through an expecta tion value of the field.
+4. Conclusions
+Contrary to the expectations expressed in ref. [ ], the interaction term derived there turned
+out to be almost the complete answer. The action for eleven-dimens ional supergravity turns
+out to be polynomial, and only contains up to four-point couplings (or three-point, after a
+field redefinition). Once the algebraic relations between the operat ors used in the construc-
+tion are derived, the construction encodes the full nonlinear stru cture of the supergravity in
+an extremely simple way. The efficiency of the pure spinor formalism in r educing the com-
+plexity of supersymmetric dynamics, already demonstrated for D= 10 super-Yang–Mills
+theory [] and the BLG and ABJM models in D= 3 [,], turns out to be present also
+for supergravity. We have no clear understanding why this happen s.M. Cederwall: “D=11 supergravity with manifest supersymme try”...................... 
+The formulation was made specifically in a flat background, although it was shown in
+Section.that the action takes the same formal expression in any backgrou nd. We have
+however not yet been able to relate that action to one obtained fro m the supergeometry of
+the background. The geometric status of the supersymmetric ac tion is somewhat unclear. It
+shouldbestressed,though,thatthefullgaugeinvariance,cons istingofsuperdiffeomorphisms
+and tensor gauge symmetry (together with an infinite number of co homologically trivial
+symmetries), is present, if not completely covariant. The precise r elation of the pure spinor
+formulation and the geometric formulation needs to be clarified. Suc h a relation would
+hopefully resolve the issue of background invariance. Maybe some im provement of the pure
+spinor action could make it more geometric and simplify the comparison . One possibility
+may be the introduction of a field Ωabcontaining a spin connection, making the Lorentz
+symmetry local. On the other hand, it seems to some extent to be th e “de-geometrisation”
+of the action that allows for the supersymmetric formulation.
+It may not be as strange as it sounds to have a polynomial action for gravity, once
+auxiliaryfieldsareincluded. Recallthe firstorderformulationofgra vitywith anindependent
+spin connection, S∼/integraltext
+εa1...adea1∧...∧eaD−2∧RaD−1aD(ω). The equation of motion from
+varyingthe spin connectionis the the torsion-freecondition on the vielbein, which eliminates
+the spin connection as an independent field. The dynamics becomes n on-polynomial when
+the torsion constraint is solved, since the solution involves the inver se vielbein. Something
+similar may be happening in the pure spinor formalism. There is more tha n enough room
+in the superfield at ghost number 0 to accommodate a spin connectio n.
+The formulation treats metric and tensor degrees of freedom in a d emocratic way. This
+mayopenforasimpleproofofU-dualityindimensionalreductionsoft hemodel.Weenvisage
+two possible ways of dealing with U-duality. One possibility is to try to inc orporate the
+compact subgroup of the U-duality group as an enlarged structur e group, which will involve
+new types of pure spinors and new cohomology. Another would be to try to realise U-duality
+operators on the field Ψ as ghost number 0 operators constructe d with non-minimal pure
+spinors.
+Thequantumpropertiesof N= 8supergravityarenotcompletelyunderstood.Aformu-
+lation with manifest supersymmetry would provide a good starting po int. Some calculations
+have already been made for D= 11 supergravity using pure spinors in a superparticle for-
+malism [], but having a field-theoretic action from which amplitudes are derive d will put
+the formalism on more solid ground. In order to use the action for ca lculating amplitudes
+one needs to perform gauge fixing. An essential part of this is to fin d theb-ghost, with the
+property{Q,b}=/square. Theb-ghost for pure spinor superfields in D= 10 was given in ref. [ ].
+It is singular when ( λ¯λ) = 0,i.e., at the tip of the the pure spinor cˆ one. The r-independent
+part of that operator, b0∼(λ¯λ)−1(¯λγaD)∂adoes not work in D= 11, in that it does not
+satisfy{s,b0}+{q,b1}= 0 for any b1. We envisage that the singular behaviour instead ..................... M. Cederwall: “D=11 supergravity with manifest supersymme try”
+comes with negative powers of η= (λγabλ)(¯λγab¯λ), like in the operator Ra. Work on gauge
+fixing is under way.
+Gauge fixing of a BV action amounts to ordinary gauge fixing of the ph ysical fields
+together with elimination of the antifields. A condition bΨ = 0 is not the whole story, since
+one would like the antifields for the metric and tensor fields not to be s et to zero, but
+to be related to the corresponding “antighosts” in a (field-theore tically) non-minimal BV
+formalism [ ]. The standard procedures for gauge fixing (which treat fields and antifields
+asymmetrically) are not applicable in a setting where all fields and antifi elds reside in the
+single field Ψ. It is likely that extra fields need to be introduced, conta ining Nakanishi–
+Lautrup fields and antighosts. These aspects have to our knowled ge not been addressed for
+pure spinor superfield theory, and should be investigated.
+Finally,therewillbeneedtoregulariseoperatorswhichdivergeonso mesubspaceofpure
+spinor space. We have not dealt with this problem yet, since at least o ne of the alternative
+forms of the action turned out to be well defined without regularisa tion.Hopefully, a method
+similar to the one in ref. [ ] will work.
+Acknowledgements: The author would like to thank Nathan Berkovits for discussions.
+References
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+sional supergravity” , J. High Energy Phys. ()[hep-th/ ].
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+Phys.()[arXiv:.].
+[] M.Cederwall, “Superfield actions for N=8 and N=6 conformal theories in thr eedimensions” , J.HighEnergy
+Phys.()[arXiv:.].
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+[] A. Fuster, M. Henneaux and A. Maas, “BRST-antifield quantization: a short review” , hep-th/ .
+[] N. Berkovits, “Towards covariant quantization of the supermembrane” , J. High Energy Phys. ()
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+[] N. Berkovits and N.Nekrasov, “Multiloop superstring amplitudes from non-minimal pure s pinor formalism” ,
+J. High Energy Phys. ()[hep-th/ ].
+[] L. Anguelova, P.A. Grassi and P. Vanhove, “Covariant one-loop amplitudes in D=11” , Nucl. Phys. B
+()[hep-th/ ]. ..................... M. Cederwall: “D=11 supergravity with manifest supersymme try”
+Appendix A: Spinor and pure spinor identities in D= 11
+We will list some identities that have been useful for calculations.
+Fierz rearrangements are always made between spinors at the righ t and left of two
+spinor products. The general Fierz identity reads
+(AB)(CD) =5/summationdisplay
+p=01
+32p!(Cγa1...apB)(Aγap...a1D) ( A.)
+(with appropriate signs for statistics of operators). For bilinears in a pure spinor λthis
+reduces to
+(Aλ)(λB) =−1
+64(λγabλ)(AγabB)+1
+3840(λγabcdeλ)(AγabcdeB). (A.)
+From the constraint on the spinor r, (¯λγar) = 0, one derives
+(¯λγ[ij¯λ)(¯λγkl]r) = 0. (A.)
+Other useful relations among the non-minimal variables include
+(¯λγik¯λ)(¯λγjkr) = (¯λγij¯λ)(¯λr),
+(¯λγikr)(¯λγjkr) = (¯λγijr)(¯λr)+1
+2(¯λγij¯λ)(rr),
+(¯λγik¯λ)(¯λγklr)(¯λγljr) = 0,
+(¯λγikr)(¯λγklr)(¯λγljr) = 0.(A.)
+These can used to show quite directly that ( λγabλ)[Ra
+1,Rb
+1] = 0 (see Appendix B).
+The symmetry of ( ¯λγa[b¯λ)(¯λγcdr)(¯λγe]fr) is (af), and no contraction is allowed, so this
+expression lies in the irreducible module (20010).M. Cederwall: “D=11 supergravity with manifest supersymme try”..................... 
+Various useful identities for a pure spinor λinclude
+(γjλ)α(λγijλ) = 0,
+(γiλ)α(λγabcdiλ) = 6(γ[abλ)α(λγcd]λ),
+(γijλ)α(λγabcijλ) =−18(γ[aλ)α(λγbc]λ),
+(γijkλ)α(λγabijkλ) =−42λα(λγabλ),
+(γijλ)α(λγabcdijλ) =−24(γ[abλ)α(λγcd]λ),
+(γiλ)α(λγabcdeiλ) =λα(λγabcdeλ)−10(γ[abcλ)α(λγde]λ),(A.)
+Alternative forms for Ra
+2are:
+Ra
+2=−16η−3(¯λγa[b¯λ)(¯λγcdr)(¯λγe]fr)(λγfbλ)(λγcdew)
+=4
+3η−3(¯λγab¯λ)(¯λγcdr)(¯λγefr)(λγbcdegλ)Nfg
+−2
+3η−3(¯λγab¯λ)(¯λγcdr)(¯λr)(λγbcdefλ)Nef
+= 2η−3/bracketleftbig
+η(¯λγabr)−2φ(¯λγab¯λ)/bracketrightbig
+(¯λγcdr)(λγbcdw)
+={s,η−2(¯λγab¯λ)(¯λγcdr)}(λγbcdw),(A.)
+whereNab= (λγabw) andφ= (¯λγijr)(λγijλ).
+Appendix B: Calculation of a commutator
+In this Appendix, we will calculate the operator Ra(λγabλ)Rbappearing in the master
+equation afterthe three-point couplingis introduced,and thus go verninghigher interactions.
+We can write Ra(λγabλ)Rb= [Ra,(λγabλ)]Rb+1
+2(λγabλ)[Ra,Rb].
+Consider the first term. The only non-vanishing contribution comes fromRa
+2. Using the
+form from Appendix A, Ra
+2= 2η−3/bracketleftbig
+η(¯λγabr)−2φ(¯λγab¯λ)/bracketrightbig
+(¯λγcdr)(λγbcdw), one gets
+[Ra,(λγbcλ)] = 4η−3/bracketleftbig
+η(¯λγair)−2φ(¯λγai¯λ)/bracketrightbig
+(¯λγjkr)(λγijkbcλ). (B.)
+If two of the indices are contracted, this gives zero thanks to ( ¯λγ[ij¯λ)(¯λγkl]r) = 0.
+The second term takes some more work. Examine first the terms in [ Ra,Rb] coming
+from the commutator of wwith one of the prefactors η−k. Using eq. ( B.) again, we get
+[Ra,η] =−4/bracketleftbig
+η(¯λγair)−2φ(¯λγai¯λ)/bracketrightbig
+(¯λγjkr)(λγijkbcλ)(¯λγbc¯λ) = 0.(B.) ..................... M. Cederwall: “D=11 supergravity with manifest supersymme try”
+Now, the only remaining things to check are the terms from the antic ommutator of the two
+D’s inR1and from the commutator of the winR2withλ’s inR1andR2(except inη).
+Anticommuting the two D’s inR1gives
+(λγabλ)[Ra
+1,Rb
+1] =η−3(¯λγai¯λ)(¯λγbcr)(¯λγjkr)(λγabcγmγijkλ)∂m. (B.)
+Expanding the product of γ-matrices,
+(λγabcγmγijkλ)∂m= 3(λγabc[ijλ)∂k]+3(λγ[abijkλ)∂c]−9δ[i
+[a(λγbc]jk]mλ)∂m.(B.)
+The corresponding three terms in eq. ( B.) vanish individually due to the identities ( A.)
+and (A.) in Appendix A.
+Similarly, the commutator between R1andR2gives
+2(λγabλ)[Ra
+2,Rb
+1] = 4η−5(λγaiλ)/bracketleftbig
+η(¯λγabr)−2φ(¯λγab¯λ)/bracketrightbig
+(¯λγcdr)
+×(¯λγij¯λ)(¯λγklr)(λγbcdγjklD).(B.)
+Here, we encounter the first non-vanishing contribution. Again us ing the relations from
+Appendix A, the second term in the square brackets vanishes. In t he first term, there is
+the possibility to contract two indices between two separate pairs o f matrices ( λγabλ) or
+(λγabr), thus avoiding the zeroes of the last two identities in eq. ( A.). The result is
+2(λγabλ)[Ra
+2,Rb
+1] = 24η−3(¯λγab¯λ)(¯λr)(rr)(λγabD). (B.)
+(From BRST invariance, it also necessary that the part with the sma llest number of r’s
+does not contain ( λγ(4)D) or (λγ(6)D).) Finally, if we write Ra
+1=Mabcd(λγbcdD) and
+Ra
+2={s,Mabcd}(λγbcdw), the above result may be written
+2(λγabλ)[Ra
+2,Rb
+1] = 2(λγabλ){s,Maijk}Mblmn(λγijkγlmnD)
+= [q,24η−3(¯λγab¯λ)(¯λr)(rr)(λγabw)],
+(λγabλ)[Ra
+2,Rb
+2] = 2(λγabλ){s,Maijk}{s,Mblmn}(λγijkγlmnw)
+= [s,24η−3(¯λγab¯λ)(¯λr)(rr)(λγabw)],(B.)
+and thus
+(λγabλ)[Ra,Rb] = [Q,24η−3(¯λγab¯λ)(¯λr)(rr)(λγabw)]. (B.)
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+arXiv:1001.0114v1  [math.AG]  31 Dec 2009TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES OF
+ALMOST-COMPLEX FOURFOLDS (I)
+JULIEN GRIVAUX
+Abstract. In this article, we study topological properties of Voisin’ s punctual Hilbert schemes
+of an almost-complex fourfold X. In this setting, we compute their Betti numbers and constru ct
+Nakajima operators. We also define tautological bundles ass ociated with any complex bundle
+onX, which are shown to be canonical in K-theory.
+Contents
+2. Introduction 1
+3. The Hilbert scheme of an almost-complex compact fourfold 3
+3.1. Voisin’s construction 3
+3.2. G¨ ottsche’s formula 4
+3.3. Variation of the relative integrable structure 5
+4. Incidence varieties and Nakajima operators 8
+4.1. Construction of incidence varieties 8
+4.2. Nakajima operators 9
+5. Tautological bundles 13
+5.1. Construction of the tautological bundles 13
+5.2. Tautological bundles and incidence varieties 15
+References 17
+2.Introduction
+Our aim in this paper is to extend some properties of the cohom ology of punctual Hilbert
+schemes on smooth projective surfaces to the case of almost- complex compact manifolds of
+dimension four.
+LetXbe a smooth complex projective surface. For any integer n∈N∗, the punctual Hilbert
+schemeX[n]is defined as the set of all 0-dimensional subschemes of Xof lengthn. A theorem
+of Fogarty [Fo] states that X[n]is a smooth irreducible projective variety of complex dimen sion
+2n. The Hilbert-Chow map HC:X[n]/d47/d47SnXdefined by HC(ξ) =/summationtext
+x∈supp(ξ)ℓx/parenleftbig
+ξ/parenrightbig
+xis a
+desingularization of the symmetric product SnX. This implies that the varieties X[n]can be
+seen as smooth compactifications of the sets of distinct unor deredn-tuples of points in X.
+Voisin constructed in [Vo 1] punctual Hilbert schemes X[n]whenXis only supposed to be a
+smooth almost-complex compact fourfold. This constructio n produces almost-complex Hilbert
+schemesX[n]which are differentiable manifolds of dimension 4 nendowed with a stable almost-
+complex structure. Moreover there exists a continuous Hilb ert-Chow map HC:X[n]/d47/d47SnX
+whose fibers are homeomorphic to the fibers of the Hilbert-Cho w map in the integrable case.2 JULIEN GRIVAUX
+Using ideas of Voisin concerning relative integrable struc tures, we generalize to the almost-
+complex setting some results already known in the integrabl e case. In this paper, we will mainly
+focus on the additive structure of the cohomology groups of X[n]with rational coefficients. We
+first expose Voisin’s construction and we study the local top ological structure of the Hilbert-
+Chow map. This allows us to compute the Betti Numbers of X[n], thus extending G¨ ottsche’s
+formula [G¨ o], [G¨ o-So] to the almost-complex case.
+Theorem 1. Let(X,J)be an almost-complex compact fourfold and, for any positive integer
+n, let/parenleftbig
+bi(X[n])/parenrightbig
+i=0,...,4nbe the sequence of Betti numbers of the almost-complex Hilbe rt scheme
+X[n]. Then the generating function for these Betti numbers is give n by the formula
+∞/summationdisplay
+n=04n/summationdisplay
+i=0bi/parenleftbig
+X[n]/parenrightbig
+tiqn=∞/productdisplay
+m=1/bracketleftbig
+(1+t2m−1qm)(1+t2m+1qm)/bracketrightbigb1(X)
+(1−t2m−2qm)(1−t2m+2qm)(1−t2mqm)b2(X)·
+The proof of Theorem 1 relies on a topological version of the d ecomposition theorem of
+[De-Be-Be-Ga] for semi-small maps, which is due to Le Potier [LP].
+The second part of the paper is devoted to the definition and th e study of the Nakajima op-
+eratorsqi(α) of an arbitrary almost-complex compact fourfold X. These operators are obtained
+as actions by correspondence of incidence varieties, const ructed in the almost-complex setting.
+The incidence varieties are stratified topological spaces l ocally modelled on analytic spaces. We
+prove in this context the Nakajima commutation relations [N a]:
+Theorem 2. For any pair (i,j)of integers and any pair (α,β)of cohomology classes in
+H∗(X,Q)we have
+/bracketleftbig
+qi(α),qj(β)/bracketrightbig
+=iδi+j,0/parenleftbigg/integraldisplay
+Xαβ/parenrightbigg
+id.
+It follows from Theorems 1 and 2 that the Nakajima operators p roduce an irreducible represen-
+tation of the Heisenberg super-algebra H(H∗(X,Q)) on/circleplustext
+n∈NH∗(X[n],Q).
+In the last part, we explain how to construct tautological co mplex bundles E[n]on thealmost-
+complex Hilbert schemes X[n]starting from a complex vector bundle EonX. To do so, we use
+variable holomorphic structures on Ein the same spirit as the variable holomorphic structures
+onXused in Voisin’s construction to define X[n].
+IfXis projective, Nakajima’s theory as well as the tautologica l bundles are the fundamental
+tools to describethe ring structure of H∗(X[n],Q) (see [Le]). In a forthcoming paper, we will use
+the analogous almost-complex objects we have constructed h ere to compute the ring structure
+ofH∗(X[n],Q) whenXis a compact symplectic fourfold with vanishing first Betti n umber.
+Acknowledgement. I want to thank my advisor Claire Voisin whose work on the almo st-
+complex punctual Hilbert scheme is at the origin of this arti cle. Her deep knowledge of the
+subject and her numerous advices have been most valuable to m e. I also wish to thank her for
+her kindness and her patience.TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 3
+3.The Hilbert scheme of an almost-complex compact fourfold
+3.1.Voisin’s construction. In this section, we recall Voisin’s construction of the almo st-
+complex Hilbert scheme and establish some complementary re sults. Let ( X,J) be an almost-
+complex compact fourfold. The symmetric product SnXwill be endowed with the sheaf C∞
+SnX
+ofC∞functions on Xninvariant under Sn. Let us introduce the incidence set
+(1) Zn={(x,p)∈SnX×X,such thatp∈x}.
+Definition 3.1. Forε>0, letBεbe the set of pairs ( W,Jrel) such that
+(i)Wis a neighbourhood of ZninSnX×X,
+(ii)Jrelisarelativeintegrablecomplex structureonthefibersofpr1:W /d47/d47SnXdepending
+smoothly on the parameter xinSnX,
+(iii) ifgis a fixed riemannian metric on X, sup
+x∈SnX,p∈Wx||Jrel
+x(p)−Jx(p)||g≤ε.
+Forεsmall enough,Bεis connected and weakly contractible (i.e. πi(Bε) = 0 fori≥1). We
+choose such a εand writeBinstead ofBε.
+Letπ: (W[n]
+rel,Jrel) /d47/d47SnXbe the relative Hilbert scheme of ( W,Jrel) overSnX. The fibers
+ofπare the smooth analytic sets ( W[n]
+x,Jrel
+x),x∈SnX. LetHCrel:W[n]
+rel/d47/d47Sn
+relWbe the
+relative Hilbert-Chow morphism over SnX.
+Definition 3.2. Thetopological Hilbert scheme X[n]
+Jrelis (π,pr2◦HCrel)−1/parenleftbig
+∆SnX/parenrightbig
+, where ∆SnX
+is the diagonal of SnX. More explicitely,
+X[n]
+Jrel={(ξ,x) such that x∈SnX, ξ∈(W[n]
+x,Jrel
+x) andHC(ξ) =x}.
+To put a differentiable structure on X[n]
+Jrel, Voisin uses specific relative integrable structures
+which are invariant by a compatible system of retractions on the strata of SnX. These relative
+structures are differentiable for a differentiable structure DJonSnXwhich depends on Jand
+is weaker than the quotient differentiable structure, i.e. DJ⊆C∞
+SnX. The main result of Voisin
+is the following:
+Theorem 3.3. [Vo 1],[Vo 2]
+(i)X[n]is a4n-dimensional differentiable manifold, well-defined modulo diffeomorphisms
+homotopic to identity.
+(ii)The Hilbert-Chow map HC:X[n]/d47/d47/parenleftbig
+SnX,DJ/parenrightbig
+is differentiable and its fibers HC−1(x)
+are homeomorphic to the fibers of the usual Hilbert-Chow morph ism for any integrable
+structure near x.
+(iii)X[n]can be endowed with a stable almost-complex structure, henc e defines an almost-
+complex cobordism class. When Xis symplectic, X[n]
+Jrelis symplectic.
+The first point is the analogue of Fogarty’s result [Fo] in the differentiable case. In this article
+we will not use differentiable properties of X[n]but only topological ones, which allows us to4 JULIEN GRIVAUX
+work withX[n]
+Jrelfor anyJrelinB. Without any assumption on Jrel, the point (i) in Theorem
+3.3 has the following topological version:
+Proposition 3.4. IfJrel∈B,X[n]
+Jrelis a4n-dimensional topological manifold.
+Proof.Letx0∈SnX. There exist holomorphic relative coordinates ( zx,wx) forJrel
+xin a neigh-
+bourhoodof x0which dependsmoothly on x. For every xnearx0, the mapp /d47/d47(zx(p),wx(p))
+is a biholomorphism between ( Wx,Jrel
+x) and its image in C2with the standard complex struc-
+ture. Let us write ( zx(p),wx(p)) =φ(x,p), whereφis a smooth function defined for xnear a
+liftx0ofx0, invariant under the action of the stabilizer of x0inSn.
+We writex0= (y1,...,y1,...,yk,...,yk) where the points yjare pairwise distinct and each
+yjappearsnjtimes. We will identify small distinct neighbourhoodsof yjinXwith distinct balls
+B(yj,ε) inC2.φis defined on B(y1,ε)n1×···×B(yk,ε)nk×∪k
+j=1B(yj,ε) and isSn1×···×Snk
+invariant. We can also suppose that φ(x0, .) = id. We introduce new holomorphic coordinates
+by the formula /tildewideφ(x,p) =φ(x,p)−D1φ(x0,yj)(x−x0) ifp∈B(yj,ε). Let
+Γ:B(y1,ε)n1×···×B(yk,ε)nk /d47/d47/parenleftbig
+C2/parenrightbign
+be defined by
+Γ(x1,...,xn) = (/tildewideφ(x1,...,xn,x1),...,/tildewideφ(x1,...,xn,xn)).
+The map Γ is Sn1×···×Snkequivariant and has for differential at x0the identity map, so
+it induces a local homeomorphism γofSnXaroundx0. The image of the chart of ( X[n],Jrel)
+over a neighbourhood of x0will be the classical Hilbert scheme ( C2)[n]over a neighbourhood of
+x0. The chart and its inverse are given by the formulae ϕ(ξ) =φ(x, .)∗, whereHC(ξ) =x, and
+ϕ−1(η) = (φ(y, .)−1)∗η, wherey=γ−1(HC(η)). /square
+Remark 3.5. LetJrel
+0andJrel
+1be two relative integrable complex structures, and let φ0,φ1,
+γ0andγ1be defined as above. Then X[n]
+Jrel
+0andX[n]
+Jrel
+1are homeomorphic over a neighbourhood
+ofx0. Ifφ(x,p) =φ−1
+1(γ−1
+1γ0(x),φ0(x,p)) andγ(x) =γ−1
+1γ0(x), then there is a commutative
+diagram
+X[n]
+Jrel
+1
+HC
+/d15/d15HC−1(Vx0) /d63/d95 /d111/d111φ∗
+∼/d47/d47
+/d15/d15HC−1(/tildewideVx0)
+/d15/d15/d31/d127/d47/d47X[n]
+Jrel
+1
+HC
+/d15/d15
+SnX ⊇Vx0γ
+∼/d47/d47/tildewideVx0⊆SnX
+andγis a stratified isomorphism.
+3.2.G¨ ottsche’s formula. We will now turn our attention to the cohomology of X[n]
+Jrel. The
+first step is the computation of the Betti numbers of X[n]. We first recall the proof of G¨ ottsche
+and Soergel ([G¨ o-So]) and then we show how to adapt it in the n on-integrable case.
+LetXandYbe irreducible algebraic complex varieties and f:Y /d47/d47Xbe a proper mor-
+phism. We assume that Xis stratified in such a way that fis a topological fibration over each
+stratumXν. We denote by dνthe real dimension of the largest irreducible component of t he
+fiber. IfYν=f−1(Xν),Lν=Rdνf∗QYνwill be the associated monodromy local system on Xν.TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 5
+Definition 3.6. – The map fis asemi-small morphism if for allν, codimXXν≥dν.
+– A stratum Xνisessential if codimXXν=dν.
+Theorem 3.7. [De-Be-Be-Ga] IfYis rationally smooth and f:Y /d47/d47Xis a proper semi-small
+morphism, there exists a canonical quasi-isomorphism Rf∗QY∼/d47/d47/circleplustext
+νessentialjν∗IC•
+Xν(Lν)[−dν]
+in the bounded derived category of Q-constructible sheaves on X, whereIC•
+Xν(Lν)is the inter-
+section complex on Xνassociated to the monodromy local system Lνandjν:Xν/d47/d47Xis the
+inclusion. In particular, Hk(Y,Q) =/circleplustext
+νessentialIHk−dν(Xν,Lν).
+Remark 3.8. A simple proof of Theorem 3.7 is done in [LP] and can be found in [Gri]. Further-
+more, this proof shows that Rf∗QY≃/circleplustext
+νessentialjν∗IC•
+Xν(Lν)[−dν] under the following weaker
+topological hypotheses: Yis a rationally smooth topological space, Xis a stratified topological
+space andf:Y /d47/d47Xis a proper map which is locally equivalent over Xto a semi-small map
+between complex analytic varieties.
+IfXis a quasi-projective surface, the Hilbert-Chow morphism i s semi-small with irreducible
+fibers (see [Br]), so that the monodromy local systems are tri vial, andX[n]is smooth. The
+decomposition theorem gives G¨ ottsche’s formula for bi(X[n]). We now show that G¨ ottsche’s
+formula holds more generally for almost-complex Hilbert sc hemes.
+Theorem 3.9 (G¨ ottsche’s formula) .If(X,J)is an almost-complex compact fourfold, then for
+any integrable complex structure Jrel,
+∞/summationdisplay
+n=04n/summationdisplay
+i=0bi/parenleftbig
+X[n]
+Jrel/parenrightbig
+tiqn=∞/productdisplay
+m=1/bracketleftbig
+(1+t2m−1qm)(1+t2m+1qm)/bracketrightbigb1(X)
+(1−t2m−2qm)(1−t2m+2qm)(1−t2mqm)b2(X)·
+Proof.By Remark 3.8, it suffices to check that HC:X[n]
+Jrel/d47/d47SnXis locally equivalent to a
+semi-small morphism. The proof of Proposition 3.4 shows tha tHC:X[n]
+Jrel/d47/d47SnXis locally
+equivalent to HC:U[n]/d47/d47S[n]UwhereUis an open set in C2. Thus the decomposition
+theorem applies and the computations are the same as in the in tegrable case. /square
+3.3.Variation of the relative integrable structure. Theorem 3.9 implies in particular that
+the Betti numbers of X[n]
+Jrelare independent of Jrel. We now prove a stronger result, namely
+that the Hilbert schemes corresponding to different relative integrable complex structures are
+homeomorphic.
+Proposition 3.10.
+(i)Let/parenleftbig
+Jrel
+t/parenrightbig
+t∈B(0,r)⊆Rdbe a smooth path in B. Then the associated relative Hilbert scheme
+/parenleftbig
+X[n],/braceleftbig
+Jrel
+t/bracerightbig
+t∈B(0,r)/parenrightbig
+overB(0,r)is a topological fibration.
+(ii)IfJrel
+0,Jrel
+1∈B, then there exist canonical isomorphisms H∗/parenleftbig
+X[n]
+Jrel
+0,Q/parenrightbig
+≃H∗/parenleftbig
+X[n]
+Jrel
+1,Q/parenrightbig
+andK/parenleftbig
+X[n]
+Jrel
+0/parenrightbig
+≃K/parenleftbig
+X[n]
+Jrel
+1/parenrightbig
+.
+In order to prove Proposition 3.10, we first establish the fol lowing result:6 JULIEN GRIVAUX
+Proposition 3.11. Let/parenleftbig
+Jrel
+t/parenrightbig
+t∈B(0,r)⊆Rdbe a family of smooth relative complex structures in a
+neighbourhood of Znvarying smoothly with t. Then there exist ε>0, a neighbourhood WofZn
+inSnX×Xand a continuous map ψ: (t,x,p)/d31/d47/d47ψt,x(p)fromB(0,ε)×WtoXsuch that:
+(i)ψ0,x(p) =p,
+(ii)For fixed (t,x),ψt,xis a biholomorphism between a neighbourhood of xand a neighbour-
+hood ofSnψt,x(x), endowed with the structures Jrel
+0,xandJrel
+t,ψt,x(x),
+(iii)For alltinB(0,ε), the mapx /d47/d47Snψt,x(x)is a homeomorphism of SnX.
+Proof.We can choose a family of maps θtvarying smoothly with tsuch that for all xinSnX
+andtinB(0,r),θt,xis a biholomorphism between two neighbourhoods of xendowed with the
+structuresJrel
+t,xandJrel
+0,x, and such that for all xinSnX,θ0,x= id. We take, as in the proof of
+Proposition 3.4, a system/parenleftbig
+φi
+x/parenrightbig
+1≤i≤Nof holomorphic relative coordinates for Jrel
+0with respect
+to a covering/parenleftbig/tildewideUi/parenrightbig
+1≤i≤NofSnXsuch thatx /d47/d47Snφi
+x(x) is a homeomorphism between /tildewideUiand
+its image /tildewideViinSnC2. We define holomorphic relative coordinates/parenleftbig
+φi
+t,x/parenrightbig
+1≤i≤NforJrel
+tby the
+formulaφi
+t,x(p) =φi
+x/parenleftbig
+θt,x(p)/parenrightbig
+. For small t, after shrinking /tildewideUiif necessary, x /d47/d47Snφi
+t,x(x) is
+still a homeomorphism: indeed the map x /d47/d47Snφi
+t,x(x) is obtained from the Sn1×···×Snk
+equivariant smooth map ( x1,...,xn) /d47/d47/parenleftbig
+φt(x1,...,xn,x1),...,φt(x1,...,xn,xn)/parenrightbig
+.Then we
+use the fact that a sufficiently small smooth perturbation of a smooth diffeomorphism remains
+a smooth diffeomorphism.
+Let/tildewideZn⊆SnC2×C2betheincidencevarietyof C2. Themap ˇφi
+t: (x,p) /d47/d47/parenleftbig
+Snφi
+t,x(x),φi
+t,x(p)/parenrightbig
+is a homeomorphism between two neighbourhoods of Znand/tildewideZnover/tildewideUiand/tildewideVi. If we define
+ˇψt: (x,p) /d47/d47/parenleftbig
+Snψt,x(x),ψt,x(p)/parenrightbig
+, the condition (ii) of the proposition means that ˇφi
+t◦ˇψt◦/parenleftbigˇφi
+0/parenrightbig−1
+is of the form ( y,p) /d47/d47/parenleftbig
+Snut,y(y),ut,y(p)/parenrightbig
+wherey∈/tildewideViandut,yis a biholomorphism between
+a neighbourhood of yand its image (both endowed with the standard complex struct ure of
+C2), varying smoothly with tandy. The condition (i) means that u0,y= id. Thus/parenleftbig
+ψt/parenrightbig
+||t||≤ε
+can be constructed on small open sets of SnX. Since biholomorphisms close to identity form
+a contractible set, we can, using cut-off functions, glue tog ether the local solutions on SnXto
+obtain a global one. The map x /d47/d47Snψt,x(x) is induced by a smooth Sn-equivariant map of
+XnintoXn(and is a small perturbation of the identity map if ||t||is small enough), thus a
+Sn-equivariant diffeomorphism of Xn. We have therefore defined a family of maps/parenleftbig
+ψt/parenrightbig
+||t||≤ε
+satisfying the conditions (i), (ii) and (iii). /square
+We can now prove Proposition 3.10.TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 7
+Proof of Proposition 3.10. (i) We have
+/parenleftbig
+X[n],/braceleftbig
+Jrel
+t/bracerightbig
+t∈B(0,r)/parenrightbig
+=/braceleftBig
+(ξ,x,t) such that x∈SnX, t∈B(0,r), ξ∈/parenleftbig
+W[n]
+x,Jrel
+t,x/parenrightbig
+, HC(ξ) =x/bracerightBig
+·
+A topological trivialization of this family over B(0,r) near zero is given by the map
+Γ:X[n]
+Jrel
+0×B(0,ε) /d47/d47/parenleftbig
+X[n],/braceleftbig
+Jrel
+t/bracerightbig
+t∈B(0,ε)/parenrightbig
+defined by Γ( ξ,x,t) =/parenleftbig
+ψt,x∗ξ,ψt,x(x),t/parenrightbig
+, whereψis given by Proposition 3.11. This proves that
+the relative Hilbert scheme is locally topologically trivi al overB(0,r).
+(ii) ThesetBbeing connected, point (i) shows that X[n]
+Jrel
+0andX[n]
+Jrel
+1are homeomorphic. Since
+π1(B) = 0, if we consider two paths/parenleftbig
+Jrel
+0,t/parenrightbig
+0≤t≤1and/parenleftbig
+Jrel
+1,t/parenrightbig
+0≤t≤1betweenJrel
+0andJrel
+1, we can
+find a smooth family/parenleftbig
+Jrel
+s,t/parenrightbig
+0≤s≤1
+0≤t≤1which is an homotopy between the two paths. The relative
+associated Hilbert scheme over [0 ,1]×[0,1] is locally topologically trivial, hence globally trivia l
+since [0,1]×[0,1] is contractible. This shows that the homeomorphisms betw eenX[n]
+Jrel
+0andX[n]
+Jrel
+1constructed by the procedure above belong to a canonical hom otopy class. /square
+As a consequence of this proposition, there exists a ring H∗(X[n],Q) (resp.K(X[n])) such
+that for any Jrelclose toJ,H∗(X[n],Q) (resp.K(X[n])) andH∗/parenleftbig
+X[n]
+Jrel,Q/parenrightbig
+(resp.K/parenleftbig
+X[n]
+Jrel/parenrightbig
+)
+are canonically isomorphic.
+In the sequel, we will deal with products of Hilbert schemes. We can of course consider
+products of the type/parenleftbig
+X[n]
+Jrel
+n/parenrightbig
+×/parenleftbig
+X[m]
+Jrel
+m/parenrightbig
+, but in practice it is necessary to work with pairs of
+relative integrable complex structures parametrized by ( x,y) inSnX×SmX. Let us introduce
+the incidence set
+(2) Zn×m=/braceleftbig
+(x,y,p) in/parenleftbig
+SnX×SmX/parenrightbig
+×Xsuch thatp∈x∪y/bracerightbig
+·
+LetWbe a small neighbourhood of Zn×mand letJ1,relandJ2,relbe two relative integrable
+complex structures on the fibers of pr1:W /d47/d47SnX×SmX.
+Definition 3.12. The product Hilbert scheme/parenleftbig
+X[n]×[m],J1,rel,J2,rel/parenrightbig
+is defined by
+/parenleftbig
+X[n]×[m],J1,rel,J2,rel/parenrightbig
+=/braceleftBig
+(ξ,η,x,y) such that x∈SnX, y∈SmX, ξ∈/parenleftbig
+W[n]
+x,y,J1,rel
+x,y/parenrightbig
+,
+η∈/parenleftbig
+W[m]
+x,y,J2,rel
+x,y/parenrightbig
+, HC(ξ) =x, HC(η) =y/bracerightBig
+·
+The same definition holds for products of the type/parenleftBig
+X[n1]×···×[nk],J1,rel,...,Jk,rel/parenrightBig
+.
+If there exist two relative integrable complex structures Jrel
+nandJrel
+min neighbourhoods of
+ZnandZmsuch thatJ1,rel
+x,y=Jrel
+n,xandJ2,rel
+x,y=Jrel
+m,xin small neighbourhoods of xandy, we
+have/parenleftbig
+X[n]×[m],J1,rel,J2,rel/parenrightbig
+=X[n]
+Jrel
+n×X[m]
+Jrel
+m.
+If/parenleftbig
+J1,rel
+t,J2,rel
+t/parenrightbig
+t∈B(0,r)is a smooth family of relative integrable complex structure s, it can be
+shownasinPropositions3.11and3.10thatthefamily/parenleftBig
+X[n]×[m],/braceleftbig
+J1,rel
+t/bracerightbig
+t∈B(0,r),/braceleftbig
+J2,rel
+t/bracerightbig
+t∈B(0,r)/parenrightBig
+is topologically trivial over B(0,r). Thus the product Hilbert schemes/parenleftbig
+X[n]×[m],J1,rel,J2,rel/parenrightbig
+is8 JULIEN GRIVAUX
+isomorphic to products X[n]
+Jrel
+n×X[m]
+Jrel
+mof usual Hilbert schemes. If the structures J1,relandJ2,rel
+are equal,/parenleftbig
+X[n]×[m],J1,rel,J2,rel/parenrightbig
+consists of pairs of schemes of given support (parametrized by
+SnX×SmX) for the sameintegrable structure. These product Hilbert schemes are th erefore
+well adapted for the study of incidence relations.
+4.Incidence varieties and Nakajima operators
+4.1.Construction of incidence varieties. IfJis an integrable complex structure on X, the
+incidence variety X[n′,n]is classically defined by
+X[n′,n]=/braceleftbig
+(ξ,ξ′) such that ξ∈X[n], ξ′∈X[n′]andξ⊆ξ′/bracerightbig
+·
+The incidence variety X[n′,n]is never smooth unless n′=n+ 1 (see [Ti]). We have three
+mapsλ:X[n′,n]/d47/d47X[n],ν:X[n′,n]/d47/d47X[n′]andρ:X[n′,n]/d47/d47Sn′−nXgiven byλ(ξ,ξ′) =ξ,
+ν(ξ,ξ′) =ξ′andρ(ξ,ξ′) = supp/parenleftbig
+Iξ/Iξ′/parenrightbig
+. Note that, by definition, ( λ,ν) is injective.
+IfJis not integrable, we can define X[n′,n]using the relative construction. Let Jrel
+n×(n′−n)be
+a relative integrable complex structure in a neighbourhood ofZn×(n′−n)inSnX×Sn′−nX×X
+(see (2)).
+Definition 4.1. The incidence variety/parenleftbig
+X[n′,n],Jrel
+n×(n′−n)/parenrightbig
+is defined by
+/parenleftbig
+X[n′,n],Jrel
+n×(n′−n)/parenrightbig
+=/braceleftBig
+(ξ,ξ′,x,y) such that x∈SnX, y∈Sn′−nX, ξ∈/parenleftbig
+W[n]
+x,Jrel
+n×(n′−n),x,y/parenrightbig
+,
+ξ′∈/parenleftbig
+W[n′]
+x∪y,Jrel
+n×(n′−n),x,y/parenrightbig
+, ξ⊆ξ′, HC(ξ) =x, ρ(ξ,ξ′) =y/bracerightBig
+·
+LetJrel
+n×n′be a relative integrable complex structure in a neighbourho od ofZn×n′such that
+for everyu∈SnXandv∈Sn′−nX,Jrel
+n×n′,u,u∪v=Jrel
+n×(n′−n),u,v. Then
+/parenleftbig
+X[n′,n],Jrel
+n×(n′−n)/parenrightbig
+⊆/parenleftbig
+X[n]×[n′],Jrel
+n×n′,Jrel
+n×n′/parenrightbig
+.
+If/braceleftbig
+Jrel
+t,n×n′/bracerightbig
+t∈B(0,r)is a smooth family of relative complex structures, we can tak e, as in Propo-
+sition 3.10, a topological trivialization of/parenleftbig
+X[n]×[n′],/braceleftbig
+Jrel
+t,n×n′/bracerightbig
+t∈B(0,r),/braceleftbig
+Jrel
+t,n×n′/bracerightbig
+t∈B(0,r)/parenrightbig
+. If we
+defineJrel
+t,n×(n′−n)in a neighbourhood of Zn×(n′−n)by the formula Jrel
+t,n×(n′−n),u,v=Jrel
+t,n×n′,u,u∪v,
+then we can choose the trivialization so that the subfamily/parenleftbig
+X[n′,n],/braceleftbig
+Jrel
+t,n×(n′−n)/bracerightbig
+t∈B(0,r)/parenrightbig
+is sent
+to the product U[n′,n]×B(0,ε), whereUis an open set of C2. This means that the family
+/braceleftBig/parenleftBig
+X[n′,n],/braceleftbig
+Jrel
+n×(n′−n)/bracerightbig
+t∈B(0,r)/parenrightBig
+,/parenleftBig
+X[n]×[n′],/braceleftbig
+Jrel
+n×n′/bracerightbig
+t∈B(0,r),/braceleftbig
+Jrel
+n×n′/bracerightbig
+t∈B(0,r)/parenrightBig/bracerightBig
+is locally, hence globally topologically trivial over B(0,r).
+The natural morphism from/parenleftbig
+X[n′,n],Jn×(n′−n)/parenrightbig
+toSnX×Sn′−nXis locally equivalent over
+SnX×Sn′−nXto the natural morphism U[n′,n]/d47/d47SnU×Sn′−nU.This enables us to define a
+stratification on X[n′,n]by patching together the analytic stratifications of a colle ction ofU[n′,n]
+i.TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 9
+In this way,/parenleftbig
+X[n′,n],Jrel
+n×(n′−n)/parenrightbig
+becomes a stratified CW-complex such that for each stratum S,
+dim(S\S)≤dimS−2. In particular, each stratum has a homology class.
+Let us introduce the following notations:
+(i) The inverse image of the small diagonal of SnXbyHC:X[n]
+Jrel
+n/d47/d47SnXwill be denoted
+by/parenleftbig
+X[n]
+0,Jrel
+n/parenrightbig
+.
+(ii) Theinverse image of the small diagonal of Sn′−nXbyρ:/parenleftbig
+X[n′,n],Jrel
+n×(n′−n)/parenrightbig/d47/d47Sn′−nX
+will be denoted by/parenleftbig
+X[n′,n]
+0,Jrel
+n×(n′−n)/parenrightbig
+.
+In the integrable case, X[n′,n]
+0is stratified by analytic sets/parenleftbig
+Zl/parenrightbig
+l≥0defined by
+(3) Zl=/braceleftbig
+(ξ,ξ′)∈X[n′,n]
+0such that if x=ρ(ξ,ξ′), ℓx(ξ) =l/bracerightbig
+;
+Z0is irreducible of complex dimension n′+n+1, and all the other Zlhave smaller dimensions
+(see [Le]). By the same argument as above, this stratificatio n also exists in the almost complex
+case. We prove the topological irreducibility of Z0in the following lemma:
+Lemma 4.2. Let/bracketleftbig
+Z0/bracketrightbig
+be the fundamental homology class of Z0. Then
+H2(n′+n+1)/parenleftbig
+X[n′,n]
+0,Jrel
+n×(n′−n),Z/parenrightbig
+=Z./bracketleftbig
+Z0/bracketrightbig
+.
+Proof.It is enough to prove that the Borel-Moore homology group Hlf
+2(n′+n+1)(Z0,Z) isZ, since
+all the remaining strata/parenleftbig
+Zl/parenrightbig
+l≥1have dimensions smaller than 2( n′+n+1)−2. Let
+/tildewideZ0=/braceleftBig
+(ξ,η,x,p) such that x∈SnX, p∈X, ξ∈/parenleftbig
+W[n]
+x,(n′−n)p,Jrel
+n×(n′−n),x,(n′−n)p/parenrightbig
+, HC(ξ) =x,
+η∈/parenleftbig
+W[n′−n]
+x,(n′−n)p,Jrel
+n×(n′−n),x,(n′−n)p/parenrightbig
+, HC(η) = (n′−n)p/bracerightBig
+·
+There is a natural inclusion Z0/d31/d127/d47/d47/tildewideZ0given by/parenleftbig
+ξ,ξ′,x,(n′−n)p/parenrightbig/d47/d47(ξ,ξ′
+|p,x,p) . Remark
+that/tildewideZ0is compact. Since dim( /tildewideZ0\Z0)≤4n+ 2(n′−n−1) = 2(n′+n−1), it suffices
+to show that H2(n′+n+1)(/tildewideZ0,Z) =Z./tildewideZ0is a product-type Hilbert scheme homeomorphic to
+X[n]
+Jrel
+n×/parenleftbig
+X[n′−n]
+0,Jrel
+n′−n/parenrightbig
+. Since/parenleftbig
+X[n′−n]
+0,Jrel
+n′−n/parenrightbig
+is by Brian¸ con’s theorem [Br] a topological
+fibration on Xwhose fiber is homeomorphic to an irreducible algebraic vari ety of complex
+dimensionn′−n−1, we obtain the result. /square
+4.2.Nakajima operators. We are now going to construct Nakajima operators qn(α) in the
+almost-complex context. If n′>n, let us define
+(4) Q[n′,n]=Z0⊆/parenleftbig
+X[n]×[n′]×X,Jrel
+n×n′,Jrel
+n×n′/parenrightbig
+×X,
+wherethemapon thelast coordinate is given by therelative r esidual morphismand Z0is defined
+by (3). Since the pair/parenleftbig
+Q[n′,n],X[n]×[n′]×X/parenrightbig
+is topologically trivial when Jrel
+n×n′varies, the cycle
+class/bracketleftbig
+Q[n′,n]/bracketrightbig
+∈H2(n′+n+1)/parenleftbig
+X[n]×X[n′]×X,Z/parenrightbig
+is independent of Jrel
+n×n′.10 JULIEN GRIVAUX
+Definition 4.3. Letα∈H∗(X,Q) andj∈N∗. We define the Nakajima operators qj(α) and
+q−j(α) as follows:
+qj(α) :/circleplustext
+n∈NH∗(X[n],Q) /d47/d47/circleplustext
+n∈NH∗(X[n+j],Q)
+τ/d31 /d47/d47PD−1/bracketleftBig
+pr2∗/parenleftbig/bracketleftbig
+Q[n+j,n]/bracketrightbig
+∩(pr∗
+3α∪pr∗
+1τ)/parenrightbig/bracketrightBig
+q−j(α) :/circleplustext
+n∈NH∗(X[n+j],Q) /d47/d47/circleplustext
+n∈NH∗(X[n],Q)
+τ/d31 /d47/d47PD−1/bracketleftBig
+pr1∗/parenleftbig/bracketleftbig
+Q[n+j,n]/bracketrightbig
+∩(pr∗
+3α∪pr∗
+2τ)/parenrightbig/bracketrightBig
+where pr1, pr2and pr3are the projections from X[n]×X[n+j]×Xto each factor and PDis the
+Poincar´ e duality isomorphism between cohomology and homo logy. We also set q0(α) = 0
+Remark 4.4. Let|α|bethedegreeof α, thenqj(α) mapsHi(X[n],Q) toHi+|α|+2j−2(X[n+j],Q).
+We now prove the following extension to the almost-complex c ase of Nakajima’s theorem
+[Na]:
+Theorem 4.5. Fori,j∈Zandα,β∈H∗(X,Q)we have
+qi(α)qj(β)−(−1)|α||β|qj(β)qi(α) =iδi+j,0/parenleftbigg/integraldisplay
+Xαβ/parenrightbigg
+id
+.
+Proof.WeadaptNakajima’s prooftooursituation. Themostinteres tingcaseisthecomputation
+of [q−i(α),qj(β)] wheniandjare positive. We introduce the classes/bracketleftbig
+Pij/bracketrightbig
+, resp./bracketleftbig
+Qij/bracketrightbig
+in
+H∗/parenleftbig
+X[n],Q/parenrightbig
+⊗H∗/parenleftbig
+X[n−i],Q/parenrightbig
+⊗H∗/parenleftbig
+X[n−i+j],Q/parenrightbig
+⊗H∗/parenleftbig
+X,Q/parenrightbig
+⊗H∗/parenleftbig
+X,Q/parenrightbig
+,resp.
+H∗/parenleftbig
+X[n],Q/parenrightbig
+⊗H∗/parenleftbig
+X[n+j],Q/parenrightbig
+⊗H∗/parenleftbig
+X[n−i+j],Q/parenrightbig
+⊗H∗/parenleftbig
+X,Q/parenrightbig
+⊗H∗/parenleftbig
+X,Q/parenrightbig
+,
+as follows:
+/bracketleftbig
+Pij/bracketrightbig
+:=p13∗/bracketleftBig
+p∗
+124/bracketleftbig
+Q[n,n−i]/bracketrightbig
+·p∗
+235/bracketleftbig
+Q[n−i+j,n−i]/bracketrightbig/bracketrightBig
+,resp.
+/bracketleftbig
+Qij/bracketrightbig
+:=p13∗/bracketleftBig
+p∗
+124/bracketleftbig
+Q[n+j,n]/bracketrightbig
+·p∗
+235/bracketleftbig
+Q[n+j,n−i+j]/bracketrightbig/bracketrightBig
+,
+whereQ[r,s]is defined in (4). Then qj(β)q−i(α), resp.q−i(α)qj(β), is given by
+τ/d31 /d47/d47PD−1/bracketleftBig
+pr3∗/parenleftbig/bracketleftbig
+Pij/bracketrightbig
+∩/parenleftbig
+pr∗
+5β∪pr∗
+4α∪pr∗
+1τ/parenrightbig/parenrightbig/bracketrightBig
+,resp.
+τ/d31 /d47/d47PD−1/bracketleftBig
+pr3∗/parenleftbig/bracketleftbig
+Qij/bracketrightbig
+∩/parenleftbig
+pr∗
+5α∪pr∗
+4β∪pr∗
+1τ/parenrightbig/parenrightbig/bracketrightBig
+.
+First we deform all the relative integrable complex structu res into a single one parametrized by
+SnX×Sn−iX×Sn−i+jX×S2X. LetJrel
+n×(n−i)×(n−i+j)×2be a relative integrable structure in a
+neighbourhood of Zn×(n−i)×(n−i+j)×2, and let
+Y=/parenleftBig
+X[n]×[n−i]×[n−i+j]×[1]×[1],Jrel
+n×(n−i)×(n−i+j)×2←5 times/parenrightBigTOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 11
+be the product Hilbert scheme obtained by taking the same str uctureJrel
+n×(n−i)×(n−i+j)×2five
+times (see Definition 3.12), where Jrel
+n×(n−i)×(n−i+j)×2is identified with its pullback by
+µ:SnX×Sn−iX×Sn−i+jX×X×X /d47/d47SnX×Sn−iX×Sn−i+jX×S2X.
+Then, via the canonical isomorphism
+H∗/parenleftbig
+Y,Q/parenrightbig
+≃H∗/parenleftbig
+X[n],Q/parenrightbig
+⊗H∗/parenleftbig
+X[n−i],Q/parenrightbig
+⊗H∗/parenleftbig
+X[n−i+j],Q/parenrightbig
+⊗H∗/parenleftbig
+X,Q/parenrightbig
+⊗H∗/parenleftbig
+X,Q/parenrightbig
+,
+since incidence varieties vary trivially in families, the c lassp∗
+124/bracketleftbig
+Q[n,n−i]/bracketrightbig
+is the homology class
+of the cycle
+A=/braceleftBig
+(ξ,ξ′,ξ′′,s,t) such that ξ′⊆ξandρ(ξ′,ξ) =s/bracerightBig
+·
+In the same way, p∗
+235/bracketleftbig
+Q[n−i+j,n−i]/bracketrightbig
+is the homology class of the cycle
+B=/braceleftBig
+(ξ,ξ′,ξ′′,s,t) such that ξ′⊆ξ′′andρ(ξ′,ξ′′) =t/bracerightBig
+·
+We now study the intersection of the cycles AandB. Letp∈A∩B. We choose relative
+holomorphic coordinates φx,y,z,s,tforJrel
+n×(n−i)×(n−i+j)×2such that
+(x,y,z,s,t)/d31 /d47/d47/parenleftbig
+Snφx,y,z,s,t(x),Sn−iφx,y,z,s,t(y),Sn−i+jφx,y,z,s,t(z),φx,y,z,s,t(s),φx,y,z,s,t(t)/parenrightbig
+is a local homeomorphism. The associated map given by
+(ξ,x,ξ′,y,ξ′′,z,s,t)/d31 /d47/d47/parenleftbig
+φx,y,z,s,t∗ξ,φx,y,z,s,t∗ξ′,Sn−i+jφx,y,z,s,t∗ξ′′,φx,y,z,s,t(s),φx,y,z,s,t(t)/parenrightbig
+is a homeomorphism from a neighbourhood of pto its image in ( C2)[n]×(C2)[n−i]×(C2)[n−i+j]×
+C2×C2which maps AandBto the classical cycles p−1
+124Q[n,n−i]andp−1
+235Q[n−i+j,n−i]. In the
+integrable case, we know that in the open set {s/\e}atio\slash=t},p−1
+124Q[n,n−i]andp−1
+235Q[n−i+j,n−i]inter-
+sect generically transversally. Using relative holomorph ic coordinates as above, this property
+still holds in our context. If/parenleftbig
+A∩B/parenrightbig
+s/\e}atio\slash=t=Cij, we can write [ A]·[B] =/bracketleftbig
+Cij/bracketrightbig
++ι∗Rwhere
+ι:Y{s=t}/d31/d127/d47/d47Yis the natural injection and R∈H2(2n−i+j+2)/parenleftbig
+Y{s=t},Q/parenrightbig
+. We can do the
+same in
+Y′=/parenleftBig
+X[n]×[n+j]×[n−i+j]×[1]×[1],Jrel
+n×(n+j)×(n−i+j)×2←5 times/parenrightBig
+with the cycles A′andB′defined by
+A′=/braceleftBig
+(ξ,ξ′,ξ′′,s,t) such that ξ⊆ξ′, ρ(ξ,ξ′) =s/bracerightBig
+B′=/braceleftBig
+(ξ,ξ′,ξ′′,s,t) such that ξ′′⊆ξ′, ρ(ξ′′,ξ′) =t/bracerightBig
+·
+We putDij=/parenleftbig
+A′∩B′/parenrightbig
+s/\e}atio\slash=t. Then [A′]·[B′] =/bracketleftbig
+Dij/bracketrightbig
++ι′
+∗R′, whereι′:Y′
+{s=t}/d31/d127/d47/d47Y′is the
+injection and R′∈H2(2n−i+j+2)/parenleftbig
+Y′
+{s=t},Q/parenrightbig
+. The class R(resp.R′) can be chosen supported in
+A∩B∩Y{s=t}(resp. inA′∩B′∩Y′
+{s=t}).
+The following lemma describes the situation outside the dia gonal{s=t}.
+Lemma 4.6. p1345∗/parenleftBig/bracketleftbig
+Cij/bracketrightbig
+∩/parenleftbig
+pr∗
+5β∪pr∗
+4α/parenrightbig/parenrightBig
+= (−1)|α||β|p1345∗/parenleftBig/bracketleftbig
+Dij/bracketrightbig
+∩/parenleftbig
+pr∗
+5α∪pr∗
+4β/parenrightbig/parenrightBig
+.12 JULIEN GRIVAUX
+Proof.Let us introduce the incidence varieties
+T=/braceleftBig
+(x,y,z,s,t)∈SnX×Sn−iX×Sn−i+jX×X×Xsuch thatx=y+is, z=y+jt/bracerightBig
+T′=/braceleftBig
+(x,y,z,s,t)∈SnX×Sn+jX×Sn−i+jX×X×Xsuch thaty=x+js=z+it/bracerightBig
+Let Ω, Ω′be two small neighbourhoods of TandT′andWa neighbourhood of Zn×(n−i+j)×2
+such that if ( x,y,z,s,t)∈Ω (resp. Ω′),y∈Wx,z,s,t. LetJrel
+n×(n−i+j)×2be a relative integrable
+complex structure on W. After shrinking Ω and Ω′if necessary, we can consider two relative
+structuresJrel
+n×(n−i)×(n−i+j)×2andJrel
+n×(n+j)×(n−i+j)×2such that
+
+
+∀(x,y,z,s,t)∈Ω, Jrel
+n×(n−i)×(n−i+j)×2,x,y,z,s,t=Jrel
+n×(n−i+j)×2,x,z,s,t
+∀(x,y,z,s,t)∈Ω′, Jrel
+n×(n+j)×(n−i+j)×2,x,y,z,s,t=Jrel
+n×(n−i+j)×2,x,z,s,t
+LetU(resp.U′) be the points of Y(resp.Y′) lying over Ω (resp. Ω′). We define two maps u
+andvas follows:
+u:U /d47/d47/parenleftbig
+X[n]×[n−i+j]×[1]×[1],Jrel
+n×(n−i+j)×2←4 times/parenrightbig
+,(ξ,ξ′,ξ′′,s,t)/d31 /d47/d47(ξ,ξ′′,s,t),
+v:U′ /d47/d47/parenleftbig
+X[n]×[n−i+j]×[1]×[1],Jrel
+n×(n−i+j)×2←4 times/parenrightbig
+,(ξ,ξ′,ξ′′,s,t)/d31 /d47/d47(ξ,ξ′′,s,t)
+If we take homeomorphisms between X[n]×X[n−i]×X[n−i+j]×X2,X[n]×X[n+j]×X[n−i+j]×X2,
+X[n]×X[n−i+j]×X2andX[n]×[n−i]×[n−i+j]×[1]×[1],X[n]×[n+j]×[n−i+j]×[1]×[1],X[n]×[n−i+j]×[1]×[1],
+uandvcan be extended to global maps which are in the homotopy class ofp1345. As in the
+integrable case, thereis anisomorphism φ:Cij≃/d47/d47Dijgiven as follows: if ( ξ,ξ′,ξ′′,s,t)∈Cij
+withHC(ξ′) =y,HC(ξ) =y+is,HC(ξ′′) =y+jt, thenφ(ξ,ξ′,ξ′′,s,t) = (ξ,/tildewideξ,ξ′′,t,s) where
+/tildewideξis defined by /tildewideξ|p=ξ′
+|pifp∈y,p/\e}atio\slash∈{s,t},/tildewideξ|s=ξ|sand/tildewideξ|t=ξ′′
+|t. All these schemes
+are considered for the structure Jrel
+n×(n−i+j)×2,x,z,s∪t. Let∂Cij=Cij\Cij,∂Dij=Dij\Dijand
+S=u(∂Cij) =v(∂Dij). We define π:Y′ /d47/d47Y′byπ(ξ,ξ′,ξ′′,s,t) = (ξ,ξ′,ξ′′,t,s). We have
+the following diagram, where all the maps are proper:
+Y\∂Cij⊇Cijφ
+≃
+u/d38/d38/d76/d76/d76/d76/d76/d76/d76/d76/d76/d76/d76/d76/d76Dij⊆Y′\∂Dij
+v◦π/d120/d120/d113/d113/d113/d113/d113/d113/d113/d113/d113/d113/d113/d113/d113
+X[n]×[n−i+j]×[1]×[1]\S.
+Thus we obtain in the Borel-Moore homology group Hlf
+2(2n−i+j+2)/parenleftbig
+X[n]×[n−i+j]×[1]×[1]\S,Q/parenrightbig
+the
+equality
+u∗/parenleftbig
+[Cij]∩(pr∗
+5β∪pr∗
+4α)/parenrightbig
+=v∗/parenleftbig
+[Dij]∩(pr∗
+4β∪pr∗
+5α)/parenrightbig
+.
+Since dimS≤2(2n−i+j+2)−2, we get
+p1345∗/parenleftBig/bracketleftbig
+Cij/bracketrightbig
+∩/parenleftbig
+pr∗
+5β∪pr∗
+4α/parenrightbig/parenrightBig
+= (−1)|α||β|p1345∗/parenleftBig/bracketleftbig
+Dij/bracketrightbig
+∩/parenleftbig
+pr∗
+5α∪pr∗
+4β/parenrightbig/parenrightBig
+.
+/squareTOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 13
+By this lemma, in/bracketleftbig
+q−i(α),qj(β)/bracketrightbig
+, the terms coming from CijandDijcancel out. It remains
+to deal with the excess intersection components along the di agonalsY{s=t}andY′
+{s=t}. We
+introduce the locus
+Γ =/braceleftBig/parenleftbig
+ξ, x,ξ′′,z, s, t/parenrightbig
+∈X[n]×[n−i+j]×[1]×[1]such thats=t, ξ|p=ξ′′
+|pforp/\e}atio\slash=s
+andHC(ξ′′) =HC(ξ)+(j−i)sifj≥i, HC(ξ) =HC(ξ′′)+(i−j)sifj≤i/bracerightBig
+·
+Γ contains u(A∩B) andv(A′∩B′). As before, the dimension count can be done as in the
+integrable case: dimΓ <2(2n−i+j+ 2) ifi/\e}atio\slash=jand ifi=j, Γ contains a 2(2 n+ 2)-
+dimensional component, namely ∆X[n]×∆X. All other components have lower dimensions.
+Thus, ifi/\e}atio\slash=j,p1345∗(ι∗R) = 0 and p1345∗(ι′
+∗R′) = 0 since they are supported in Γ and have
+degree 2(2n−i+j+ 2). Ifi=j, thenp1345∗(ι∗R) andp1345∗(ι′
+∗R′) are proportional to the
+fundamental class of ∆
+X[n]×∆X. Nowp45∗/parenleftbig/bracketleftbig
+∆X[n]×∆X/bracketrightbig
+∩/parenleftbig
+pr∗
+4α∪pr∗
+5β/parenrightbig/parenrightbig
+=/integraldisplay
+Xαβ./bracketleftbig
+∆X[n]/bracketrightbig
+and we obtain/bracketleftbig
+q−i(α),qi(β)/bracketrightbig
+=µ/integraldisplay
+Xαβ.id whereµ∈Q. The computation of the multiplicity
+µis a local problem on Xwhich is solved in [Gro], [El-St]. It turns out that µ=−i. /square
+Remark 4.7. The proof remains quite similar for i>0,j >0. There is no excess term in this
+case. Indeed, Y=X[n]×[n+i]×[n+i+j]×[1]×[1], Γ =X[n+i+j,n]⊆X[n]×[n+i]×[n+i+j]×[1]×[1]and
+dimΓ = 2(2 n+i+j+1)<2(2n+i+j+2).
+Theorem 4.5 gives a representation in H:=/circleplustext
+n∈NH∗(X[n],Q) of the Heisenberg super-algebra
+H(H∗(X,Q)).
+Proposition 4.8. His an irreducibleH(H∗(X,Q))-module generated by the vector 1.
+This a consequence of Theorem 4.5 and G¨ ottsche’s formula (T heorem 3.9), as shown by
+Nakajima [Na].
+5.Tautological bundles
+5.1.Construction of the tautological bundles. Our aim in this section is to associate
+to any complex vector bundle Eon an almost-complex compact fourfold Xa collection of
+complex vector bundles E[n]onX[n]which generalize the tautological bundles already known
+in the algebraic context. The vector bundles E[n]are constructed using an auxiliary relative
+holomorphic structure on E. However, the classes E[n]inK(X[n]) are canonical. Finally, we
+compare the classes E[n]andE[n+1]inK(X[n]) andK(X[n+1]) through the incidence variety
+X[n+1,n].
+In the classical case, let Ebe a algebraic vector bundle on an algebraic surface X. Forn∈N,
+letp:X[n]×X /d47/d47X[n]andq:X[n]×X /d47/d47Xbe the two projections and let Yn⊆X[n]×X
+be the incidence locus. Then p|Yn:Yn/d47/d47X[n]is finite. The tautological vector bundle E[n]is
+defined by E[n]=p∗/parenleftbig
+q∗E⊗OYn/parenrightbig
+and satisfies: for all ξinX[n],E[n]
+|ξ=H0/parenleftbig
+ξ,i∗
+ξE/parenrightbig
+. Our first
+aim is to generalize this construction in the almost-comple x case.14 JULIEN GRIVAUX
+Let (X,J) be an almost-complex compact fourfold, Zn⊆SnX×Xthe incidence locus,
+Wa small neighbourhood of ZnandJrel
+na relative integrable structure on W. The fibers of
+pr1:W /d47/d47SnXare smooth analytic sets. We endow Wwith the sheafAWof continuous
+functions which are smooth on the fibers of pr1. We can consider the sheaf A0,1
+W,relof relative
+(0,1)-forms on W. There exists a relative ∂-operator∂rel:AW/d47/d47A0,1
+W,relwhich induces for
+eachx∈SnXthe usual operator ∂:A
+Wx/d47/d47A0,1
+Wxgiven by the complex structure Jrel
+n,xonWx.
+Definition 5.1. LetEbe a complex vector bundle on X.
+(i) Arelative connection ∂rel
+EonEcompatible with Jrel
+nis aC-linear morphism of sheaves
+∂E:AW/parenleftbig
+pr∗
+2E/parenrightbig/d47/d47A0,1
+W/parenleftbig
+pr∗
+2E/parenrightbig
+satisfying∂rel
+E(ϕs) =ϕ∂rel
+Es+∂relϕ⊗sfor all sections
+ϕofAWandsofAW/parenleftbig
+pr∗
+2E/parenrightbig
+.
+(ii) A relative connection ∂rel
+Eisintegrable if/parenleftbig
+∂rel
+E/parenrightbig2= 0.
+If∂rel
+Eis an integrable connection on Ecompatible with Jrel
+n, we can apply the Kozsul-
+Malgrange integrability theorem with continuous paramete rs inSnX(see [Vo 3]). Thus, for ev-
+eryx∈SnX,E|Wxis endowed with the structureof a holomorphic vector bundle over/parenleftbig
+Wx,Jrel
+n,x/parenrightbig
+and this structure varies continuously with x. Furthermore, ker ∂rel
+Eis the sheaf of relative holo-
+morphic sections of E. Therefore, there is no difference between relative integrab le connections
+onEcompatible with Jrel
+nand relative holomorphic structures on Ecompatible with Jrel
+n.
+Taking relative holomorphic coordinates for Jrel
+n, we can see that relative integrable connec-
+tions exist on Wover small open sets of SnX. By a partition of unity on SnX, it is possible to
+build global ones. The space of holomorphic structures on a c omplex vector bundle over a ball
+inC2is contractible. Therefore the space of relative holomorph ic structures on Ecompatible
+withJrel
+nis also contractible.
+We proceed now to the construction of the tautological bundl eE[n]onX[n]
+Jrel
+n. Let∂rel
+Ebe a
+relative holomorphic structure on Eadapted to Jrel
+n. Taking relative holomorphic coordinates,
+we get a vector bundle E[n]
+reloverW[n]
+relsatisfying: for each xinSnX,E[n]
+rel|W[n]
+x=E[n]
+|Wx, where
+E
+|Wxis endowed with the holomorphic structure given by ∂rel
+E,x.
+Definition 5.2. Leti:X[n]
+Jrel
+n/d47/d47W[n]
+relbe the canonical injection. The complex vector bundle
+/parenleftbig
+E[n],Jrel
+n,∂rel
+E/parenrightbig
+onX[n]
+Jrel
+nis defined by E[n]=i∗E[n]
+rel.
+In the sequel, we consider the class of E[n]inK/parenleftbig
+X[n]/parenrightbig
+, which we prove below to be indepen-
+dent of the structures used in the construction.
+Proposition 5.3. The class of E[n]inK/parenleftbig
+X[n]/parenrightbig
+is independent of/parenleftbig
+Jrel
+n,∂rel
+E/parenrightbig
+.
+Proof.Let/parenleftbig
+Jrel
+0,n,∂rel
+E,0/parenrightbig
+and/parenleftbig
+Jrel
+1,n,∂rel
+E,1/parenrightbig
+be two relative holomorphic structures on E,/parenleftbig
+Jrel
+t,n,∂rel
+E,t/parenrightbig
+be a smooth path between them, and W[n]
+relbe the relative Hilbert scheme over SnX×[0,1] forTOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 15
+the family/parenleftbig
+Jrel
+t,n/parenrightbig
+0≤t≤1. There exists a vector bundle/parenleftbig/tildewideE[n]
+rel,/braceleftbig
+Jrel
+t,n/bracerightbig
+0≤t≤1,/braceleftbig
+∂rel
+E,t/bracerightbig
+0≤t≤1/parenrightbig
+overW[n]
+rel
+such that for all tin [0,1],/tildewideE[n]
+rel|W[n]
+rel,t=/parenleftbig
+E[n]
+rel,Jrel
+t,n,∂rel
+E,t/parenrightbig
+. IfX=/parenleftbig
+X[n],/braceleftbig
+Jrel
+t,n/bracerightbig
+0≤t≤1/parenrightbig
+⊆W[n]
+rel
+is the relative Hilbert scheme over [0 ,1], then /tildewideE[n]
+rel|Xis a complex vector bundle on Xwhose
+restriction to Xtis/parenleftbig
+E[n],Jrel
+t,n,∂rel
+E,t/parenrightbig
+. NowXis topologically trivial over [0 ,1] by Proposition
+3.10. Since K(X0×[0,1])≃K(X0), we get the result. /square
+IfT=X×Cis the trivial complex line bundle on X, the tautological bundles T[n]already
+convey geometric informationson X[n]. Let∂X[n]⊆X[n]betheinverseimageofthebigdiagonal
+ofSnXby the Hilbert-Chow morphism. We have dim ∂X[n]= 4n−2 andH4n−2(∂X[n],Z)≃Z
+(this can be proved as in Lemma 4.2).
+Lemma 5.4. c1(T[n]) =−1
+2PD−1/parenleftbig/bracketleftbig
+∂X[n]/bracketrightbig/parenrightbig
+inH2/parenleftbig
+X[n],Q/parenrightbig
+.
+Proof.LetU=/braceleftbig
+(x1,...,xn)∈X[n]such that for all ( i,j) withi/\e}atio\slash=j, xi/\e}atio\slash=xj/bracerightbig
+. Then
+X[n]\∂X[n]is canonically isomorphic to U/Sn. Ifσ:U /d47/d47X[n]\∂X[n]is the associated
+quotient map, σ∗T[n]≃n/circleplustext
+i=1pr∗
+iT, so thatσ∗T[n]is trivial. Since σis a finite covering map,
+c1(T[n])
+|X[n]\∂X[n]is a torsion class, so it is zero in H2/parenleftbig
+X[n]\∂X[n],Q/parenrightbig
+. This implies that
+c1(T[n]) =µPD−1/parenleftbig
+[∂X[n]]/parenrightbig
+whereµ∈Q. To compute µ, we argue locally on SnXaround a
+point in the stratum
+S=/braceleftbig
+x∈SnXsuch thatxi/\e}atio\slash=xjexcept for one pair {i,j}/bracerightbig
+·
+This reduces the computation to the case n= 2. ThenU[2]=Bl∆(U×U)/Z2, whereU⊆Xis
+endowed with an integrable complex structure and ∆ is the dia gonal ofU. IfE⊆Bl∆(U×U) is
+the exceptional divisor and π:Bl∆(U×U) /d47/d47U[2]is the projection, then π∗/parenleftbig
+[∂U[2]]/parenrightbig
+= 2[E]
+andπ∗c1(T[2]) =c1(π∗T[2]) =c1/parenleftbig
+O(−E)/parenrightbig
+=−[E] inH2/parenleftbig
+Bl∆(U×U),Z/parenrightbig
+. This gives the value
+µ=−1/2. /square
+5.2.Tautological bundles and incidence varieties. We want to compare the tautological
+bundlesE[n]andE[n+1]through the incidence variety X[n+1,n]. In the integrable case, X[n+1,n]
+is smooth. If D⊆X[n+1,n]is the divisor Z1(see (3)), we have an exact sequence (see [Da], [Le]):
+(5) 0 /d47/d47ρ∗E⊗O
+X[n+1,n](−D) /d47/d47ν∗E[n+1]/d47/d47λ∗E[n]/d47/d470,
+whereλ:X[n+1,n]/d47/d47X[n],ν:X[n+1,n]/d47/d47X[n+1]andρ:X[n+1,n]/d47/d47Xare the two natural
+projections and the residual map.
+In the almost-complex case, X[n+1,n]is a topological manifold of dimension 4 n+ 4. If we
+choose a relative integrable structure Jrel
+n+1with additional properties as given in [Vo 1], X[n+1,n]
+can be endowed with a differentiable structure, but we will not need it here.16 JULIEN GRIVAUX
+LetJrel
+nandJrel
+n+1be two relative integrable structures in small neighbourho ods ofZnand
+Zn+1. We extend them to relative structures ˇJrel
+nandˇJrel
+n+1in small neighbourhoodsof Zn×(n+1).
+Then/parenleftbig
+X[n]×[n+1],ˇJrel
+n,ˇJrel
+n+1/parenrightbig
+=X[n]
+Jrel
+n×X[n+1]
+Jrel
+n+1. IfJrel
+n×(n+1)is a relative integrable structure in
+a small neighbourhood of Zn×(n+1)andJrel
+n×1is defined by Jrel
+n×1,x,p=Jrel
+n×(n+1),x,x∪p, then we
+have a diagram:
+X[n+1]
+Jrel
+n+1
+X[n+1,n]
+Jrel
+n×1ν/d48/d48
+/d31/d127/d47/d47
+λ
+/d46/d46/parenleftbig
+X[n]×[n+1],Jrel
+n×(n+1),Jrel
+n×(n+1)/parenrightbigΦ
+≃/d47/d47/parenleftbig
+X[n]×[n+1],ˇJrel
+n,ˇJrel
+n+1/parenrightbigpr1/d79/d79
+pr2
+/d15/d15
+X[n]
+Jrel
+n
+where Φ is a homeomorphism uniquely determined up to homotop y. We will denote by D
+the inverse image of the incidence locus of SnX×Xby the map X[n+1,n]/d47/d47SnX×X,so
+thatD=Z1whereZ1is defined by (3). The cycle Dhas a fundamental homology class in
+H4n+2/parenleftbig
+X[n+1,n],Z/parenrightbig
+. Furthermore, there exists a unique complex line bundle FonX[n+1,n]such
+thatPD/parenleftbig
+c1(F)/parenrightbig
+=−[D].
+Proposition 5.5. InK/parenleftbig
+X[n+1,n]/parenrightbig
+, the following identity holds: ν∗E[n+1]=λ∗E[n]+ρ∗E⊗F.
+Proof.Let∂rel
+E,n×1,∂rel
+E,nand∂rel
+E,n+1be relative holomorphic structures on Ecompatible with
+Jrel
+n×1,Jrel
+nandJrel
+n+1. For each ( x,p)∈SnX×X, we consider the exact sequence (5) on
+/parenleftbig
+Wx,p,Jrel
+n×1,x,p/parenrightbig
+for the holomorphic vector bundle/parenleftbig
+E
+|Wx,p,∂rel
+E,n×1,x,p/parenrightbig
+. Putting these exact
+sequences in families over SnX×X,and restricting it to X[n+1,n], we get an exact sequence
+0 /d47/d47ρ∗E⊗G /d47/d47A /d47/d47B /d47/d470,whereGis a complex line bundle on X[n+1,n]andAand
+Bare two vector bundles on X[n+1,n]such that for all ( x,p) inSnX×X:
+(6)A|ξ,ξ′,x,p=/parenleftBig
+E[n+1]
+|ξ′,∂rel
+E,n×1,x,p,Jrel
+n×1,x,p/parenrightBig
+, B|ξ,ξ′,x,p=/parenleftBig
+E[n]
+|ξ,∂rel
+E,n×1,x,p,Jrel
+n×1,x,p/parenrightBig
+.
+Now, Φ is given by Φ( ξ,ξ′,u,v) =/parenleftbig
+φu,v∗ξ,Snφu,v(u),ψu,v∗ξ′,Sn+1ψu,v∗(v)/parenrightbig
+. Thus
+ν∗E[n+1]
+|ξ,ξ′,x,p=/parenleftBig
+E[n+1]
+|ψx,x∪p∗ξ′,∂rel
+E,n+1,Sn+1ψx,x∪p(x∪p),Jrel
+n+1,Sn+1ψx,x∪p(x∪p)/parenrightBig
+,
+λ∗E[n]
+|ξ,ξ′,x,x=/parenleftBig
+E[n]
+|φx,x∪p∗ξ,∂rel
+E,n,Snφx,x∪p(x),Jrel
+n,Snφx,x∪p(x)/parenrightBig
+.
+As in Proposition 5.3, the classes AandBinK/parenleftbig
+X[n+1,n]/parenrightbig
+are independent of the structures used
+to definethem. If Jrel
+n×(n+1)=ˇJrel
+n+1andfor all ( x,p) inSnX×X,∂rel
+E,n×1,x,p=∂rel
+E,n×1,x∪p, wecanTOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 17
+takeψu,v= idinaneighbourhoodof v. ThusA=ν∗E[n+1]. Ontheotherway, if Jrel
+n×(n+1)=ˇJrel
+n
+and for all ( x,p) inSnX×X,∂rel
+E,n×1,x,p=∂rel
+E,n,xin a neighbourhood of x, we can take φu,v= id
+in a neighbourhood of u. ThusB=λ∗E[n]. This proves that ν∗E[n+1]−λ∗E[n]=ρ∗E⊗G
+inK/parenleftbig
+X[n+1,n]/parenrightbig
+. IfTis the trivial complex line bundle on X,ν∗T[n+1]≃λ∗T[n]⊕ρ∗Ton
+X[n+1,n]\D. ThusGis trivial outside D. This yields PD(c1(G)) =µ[D], whereµ∈Qand the
+computation of µis local, as in Lemma 5.4, so that µ=−1. /square
+IfXis a projective surface, the subring of H∗(X[n],Q) generated by the classes chk(E[n])
+(whereEruns through all the algebraic vector bundles on X) is called the ring of algebraic
+classes ofX[n]. If (X,J) is an almost-complex compact fourfold, we can in the same ma nner
+consider the subringof H∗(X[n],Q) generated by the classes chk(E[n]), whereEruns through all
+the complex vector bundles on X. IfXis projective, this ring is much bigger than the ring of the
+algebraic classes. In a forthcoming paper, we will show that it is indeed equal to H∗(X[n],Q) if
+Xis a symplectic compact fourfold satisfying b1(X) = 0, and we will describe the ring structure
+ofH∗(X[n],Q).
+References
+[Br] J. Brianc ¸on, Description de HilbnC{x,y}, Invent. Math., 41, (1977), 45–89.
+[De-Be-Be-Ga] P. Deligne, A. Beilinson, J. Bernstein, O. Gabber, Faisceaux pervers, Ast´ erisque, Soc.
+Math. France, 100, (1983).
+[Da] G. Danila, Sur la cohomologie d’un fibr´ e tautologique sur le sch´ ema de Hilbert d’une surface, J.
+Algebraic Geom. 10, (2001), 247–280.
+[El-St] G. Ellingsrud, S.-A. Strømme, On the cohomology of the Hilbert schemes of points in the
+plane, Invent. Math., 87, (1987), 343–352.
+[Fo] J. Fogarty, Algebraic families on an algebraic surface, Am. J. Math., 10, (1968), 511–521.
+[G¨ o] L. G¨ottsche, The Betti numbers of the Hilbert scheme of points on a smooth p rojective surface,
+Math. Ann., 286, (1990), 193–207.
+[G¨ o-So] L. G¨ottsche, W. Soergel, Perverse sheaves and the cohomology of Hilbert schemes on sm ooth
+algebraic surfaces, Math. Ann., 296, (1993), 235–245.
+[Gri] J. Grivaux, Th` ese de Doctorat. Universit´ e Pierre et Marie Curie, Paris (2009).
+[Gro] I. Grojnowski, Instantons and affine algebras I: the Hilbert scheme and verte x operators Math.
+Res. Letters, 3, (1996), 275–291.
+[Le] M. Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent.
+Math.,136, (1999), 157–207.
+[LP] J. Le Potier, La formule de G¨ ottsche, Unpublished lecture notes.
+[Na] H. Nakajima, Heisenberg algebra and Hilbert schemes on projective surfa ces, Ann. Math, 145,
+(1997), 379–388.
+[Ti] A. Tikhomirov, The variety of complex pairs of zero-dimensional subscheme s of an algebraic
+surface, Izvestiya RAN: Ser. Mat 61, (1997) 153–180. = Izvestiya: Mathematics, 61(1997), 1265–
+1291.
+[Vo 1] C. Voisin, On the Hilbert scheme of points on an almost complex fourfold , Ann. Inst. Fourier,
+50,2, (2000), 689–722.
+[Vo 2] C. Voisin, On the punctual Hilbert scheme of a symplectic fourfold, Con temporary Mathematics,
+(2002), 265–289.
+[Vo 3] C. Voisin, Th´ eorie de Hodge et g´ eom´ etrie alg´ ebrique complexe, Soc i´ et´ e Math´ ematique de France,
+(2002).
+Institut de Math ´ematiques de Jussieu, UMR 7586, Case 247, Universit ´e Pierre et Marie Curie,
+4, place Jussieu, F-75252 Paris Cedex 05, France
+E-mail address :jgrivaux@math.jussieu.fr
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+Developing Articial Herders Using Jason
+Niklas Skamriis Boss, Andreas Schmidt Jensen, and Jrgen Villadsen?
+Department of Informatics and Mathematical Modelling
+Technical University of Denmark
+Richard Petersens Plads, Building 321, DK-2800 Kongens Lyngby, Denmark
+Abstract. This paper gives an overview of a proposed strategy for the
+\Cows and Herders" scenario given in the Multi-Agent Programming
+Contest 2009. The strategy is to be implemented using the Jason plat-
+form, based on the agent-oriented programming language AgentSpeak.
+The paper describes the agents, their goals and the strategies they should
+follow. The basis for the paper and for participating in the contest is a
+new course given in spring 2009 and our main objective is to show that we
+are able to implement complex multi-agent systems with the knowledge
+gained in an introductory course on multi-agent systems.
+1 Introduction
+This paper describes the work with a multi-agent system consisting of articial
+herders attempting to catch cows. The agents will compete in the Multi-Agent
+Programming Contest 2009 (the scenario \Cows and Herders"). One of our main
+objectives in the contest has been to gain experience with the development of
+multi-agent systems using Jason .
+Our basis for participating in the contest is the course \Articial Intelligence
+and Multi-Agent Systems" given in spring 2009 at the Technical University of
+Denmark. The course provides an introduction to multi-agent systems using
+Jason as the implementation platform. We hope to show that this introduction
+is sucient to be able to implement a more complex multi-agent system, such
+as the \Cows and Herders" scenario given in the contest.
+2 System Analysis and Design
+Our system consists of three kinds of agents: a herder, a scout and a leader.
+The leader and the scout are basically herders with extra responsibilities. The
+scout will initially explore the environment and subsequently act as an ordinary
+herder. The leader will delegate targets to each of the herders { including himself.
+?Contact: jv@imm.dtu.dk
+1arXiv:1001.0115v1  [cs.MA]  31 Dec 2009Our system was designed using the Prometheus methodology as a guideline.
+By this we mean that we have adapted relevant concepts from the methodology,
+while not following it too strictly (as stated in [3]). It has allowed us to quickly
+identify the goals and what agents are needed to complete them.
+Fig. 1. Overview of the system.
+Figure 1 gives an overview of the system. The diagram distinguishes between
+the three types of agents, even though the leader and the scout are actually
+special cases of the herder. This has been done to easily see the dierent roles
+each agent plays. All agents know their own position and how many steps of the
+match that have elapsed. This is used to revise targets, since we do not want
+them to blindly follow a target. An agent gets a new target by fullling the goal
+getnewtarget . The herders will tell the leader to delegate a target based on
+the agents current position, while the scout will autonomously decide where to
+go.
+We distinguish between the following types of targets. While the agents do
+not really have an understanding of each concept, it is helpful for us to be able
+to tell the targets apart.
+Exploration targets are targets in an area which has yet to be explored. Such
+target is delegated to the scout, when he has not explored the entire envi-
+ronment, or a herder, whenever he does not fulll the criteria for a receiving
+another type of target.
+Formation targets are targets behind cows, but within a certain distance from
+both cows and other herders, so that the group of cows can be controlled
+and moved (or herded) towards the corral.
+2A switch target is a target next to a switch. The reason for this is that an
+agent should stand next to a switch in order to trigger it. This target will
+be delegated whenever an agent is near a closed switch and it is reasonable
+to open it. This is the case if one or more cows are near the fence or another
+agent is on one side, while having a target on the other side (thus needing
+another agent to open the fence, since one agent cannot pass a fence alone).
+The scenario is quite dynamic since cows are continuously moving and fences
+can be opened and closed, and all of this must be taken into account.
+3 Software Architecture
+Our strategy and agents are implemented using the Jason platform, which is
+an implementation of the AgentSpeak language, written in Java. Jason is an
+eective platform for creating multi-agent systems with a variable number of
+agents. Combined with internal actions, we have a strong foundation for building
+a multi-agent system, which not only uses the features of logic programming, but
+allows us to develop imperative extensions as well.
+The use of custom architectures in Jason allows us to implement a local
+simulation, as described in [1]. This eases the testing, as it can be done much
+faster.
+As reference implementation we have used an implementation of the 2008
+contest made by the authors of Jason . This has helped us getting started, even
+though the scenario diers in many ways from last year.
+Our solution to the contest was developed using Eclipse. The implementation
+will have great focus on the advantages of object oriented programming. This
+would also ease future expansion of more agents etc. Shared memory could also
+be modelled by use of references to shared objects used by multiple agents.
+4 Agent Team Strategy
+The agents will be moving around in a partially known environment. At the
+beginning of a match everything is unknown, except for what lies within the
+agents' eld of view, and as the agents move around they gain knowledge of the
+environment. The entire map is represented by a graph, where each node in the
+graph represents a cell in the environment. When objects such as obstacles or
+cows are discovered etc. the corresponding cell in the graph will be assigned a
+value of that kind of object.
+When agents move around they follow paths calculated by our navigation
+algorithm. We have chosen to represent the environment as a graph, since it
+makes it is easy to use a graph search algorithms for navigation. The actual
+paths are calculated using the A* algorithm, which basically is an advanced
+best-rst search as it uses a heuristic to guide the search for optimal paths.
+A part of our strategy is to try to keep clustered cows together. This means
+that the agents will have to move around a group of cows to avoid splitting
+3them up. This is ensured be assigning weight to the dierent cell in the graph.
+By assigning higher weights to cells occupied by cows and cells adjacent to cows,
+agents will navigate around a cluster instead of through it. Obstacles are handled
+slightly dierent. The algorithm is implemented so that it does not consider cells
+containing obstacles as valid cells for a path. This ensures that agents do not
+try to move through obstacles.
+To optimize the movement of our agents the paths are continuously calcu-
+lated. This is done since all agents can add new knowledge of obstacles etc. to the
+graph as they perceive the environment. This ensures that if one agent discovers
+that a corridor is blocked, then the other agents will try to move around it to
+get to their target.
+Experiments have shown that it is more ecient to herd cows in groups. To
+ensure this the leading agent makes great use of a clustering algorithm. The
+algorithm works be examining the surroundings of each cow; adjacent cows are
+grouped together.
+The strategy for herding the cows will be taken care of by the leading agent.
+The team leader will coordinate the herding, ensuring that the cows are 
eeing
+the right way and that an agent will open the fence at an appropriate time.
+Our strategy is mainly towards maximizing our own score. This means that
+our agents will not try to capture cows already being herded by the opponent
+deliberately, but it might happen if the leading agent estimates that they are
+the cows closest to the corral.
+An agent's beliefs consist of what they perceive and what others tell them to
+believe. Optimally, we would like that every agent knows the same, i.e. they all
+have the same beliefs. Unfortunately, since agents can only see a limited area of
+the environment, this is not directly possible.
+To ensure that every agent knows the same, any new belief an agent perceives
+is sent to every other agent. All beliefs are shared immediately, since it does not
+create much overhead and it is more ecient to share it than consider whether it
+should be shared. When an agent discovers a static obstacle, every agent should
+know this, so that their navigation can be adjusted to this new knowledge.
+If an agent fails to achieve a given goal then we will use the Jason failure
+handling feature. This is done by implementing a deletion event -!g, which will
+be executed if a given plan fails [2]. After recovering from a failed plan, we will
+attempt to reintroduce the goal ( +!g) again.
+5 Discussion
+Our strategy is quite dynamic because of our use of path nding and clustering
+algorithms, which allow the herders to fulll their goals in any given scenario.
+However, some of the choices we have made are made on assumptions which may
+prove to be mistaken when the competition is held.
+We have decided to have a maximum cluster size (i.e. limit the number of
+cows in a single cluster), because we believe that the agents may have a hard
+4time herding larger clusters. This may not be true, though, since it could be
+more ecient to herd as many cows as possible as long as they are clustered.
+To compute an optimal search it is important to move agents in patterns
+so that the largest possible area is explored. For example, agents should never
+move side by side towards the same location, since this would not exploit the full
+potential of the agents' eld of view. Likewise it could prove useful to move agents
+in patterns that ensures that no cow can remain undetected in the explored area.
+However, we need to carefully design our algorithms so that they do not take
+too long time to compute, since the duration of a turn is limited.
+At the time of writing this article our implementation is complete. However,
+the contest has been postponed until after the deadline of the article, so we are
+unable to discuss the results. We have managed to play a single training match
+against another team, which we won. This match gave us an opportunity to see
+how our team plays against others.
+Generally we are quite satised with our system, which is able to fulll the
+goals of the scenario. Our strategy with a single leader delegating targets lead to
+a less autonomous approach, but the Jason framework has allowed us to easily
+implement agents with certain goals and a way to implement plans for handling
+these goals.
+6 Conclusion
+As discussed our primary strategy will be to maximize our own score rather
+than prohibiting the opposing team from scoring points. This has been done by
+optimizing the search for cows and guiding the cows into the corral by using
+cooperating agents. Likewise all agents will take the positions of the opponents
+into account when choosing a target.
+Throughout the project we have considered problems such as navigation,
+search for objects using multiple start points, clustering, cooperation between
+agents and multi-agent planning. All planning was implemented using Agent-
+Speak, while external algorithms such as A*, our clustering algorithm and target
+delegation were implemented in Java.
+Despite our limited experience with AgentSpeak and programming intelli-
+gent multi-agent systems, we have managed to implement a fairly reasonable
+system, with agents which fulll the goals of the contest. The ability of Jason
+to implement custom architectures was a great help during the work.
+References
+1. Rafael H. Bordini, Jomi F. H ubner, and Daniel M. Tralamazza. Developing a Team
+of Gold Miners Using Jason . Springer-Verlag LNAI 4908, pages 241{245, 2008.
+2. Rafael H. Bordini, Michael Wooldridge, and Jomi Fred H ubner. Programming Multi-
+Agent Systems in AgentSpeak Using Jason . John Wiley & Sons, 2007.
+3. Lin Padgham and Michael Winiko. Developing Intelligent Agent Systems: A Prac-
+tical Guide. John Wiley & Sons, 2007.
+5
\ No newline at end of file
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+arXiv:1001.0116v1  [quant-ph]  31 Dec 2009One Dimensional Magnetized TG Gas Properties in an External Magnetic Field
+Zhao Liang Wang∗and An Min Wang†
+Department of Modern Physics, University of Science and Tec hnology of China, Hefei, Anhui, China.
+With Girardeau’s Fermi-Bose mapping, we have constructed t he eigenstates of a TG gas in an
+external magnetic field. When the number of bosons Nis commensurate with the number of
+potential cycles M, the probability of this TG gas in the ground state is bigger t han the TG gas
+raised by Girardeau in 1960. Through the comparison of prope rties between this TG gas and
+Fermi gas, we find that the following issues are always of the s ame: their average value of particle’s
+coordinate and potential energy, system’s total momentum, single-particle density and the pair
+distribution function. But the reduced single-particle ma trices and their momentum distributions
+between them are different.
+PACS numbers: 05.30.Jp, 03.65.Ge
+I. INTRODUCTION
+The problem of a one-dimensional (1D) hard-core gas
+was first studied classically by Tonks [1]. Girardeau[2, 3]
+continued the research in quantum, and put forward the
+Fermi-Bose mapping to solve energy spectrum and the
+correspondingwave function of a one dimensional Tonks-
+Girardeau (TG) gas. After that, Lieb and Liniger [4]
+considered NBose particles interacting via a repulsive
+δ-function potential with a coupling constant γ. In fact,
+TG gas can be considered as the Lieb-Liniger gas when
+γ→ ∞.
+During the past decade, the study on TG gas has un-
+dergone a rapid development in theoretical and experi-
+mental [5–17]. In particular, A. Lenard [5] indicated that
+the off-diagonalparts ofthe one-bodydensity matrix and
+the momentum distribution show distinct differences in
+both the TG gas and the Fermi gas. G. J. Lapeyre et
+al. [8] studied the momentum distribution of a harmon-
+ically trapped gas. The most important progress is the
+realization of the strongly interacting bosons [16] in a
+one-dimensional optical-lattice trap and the TG gas [17]
+by trapping87Rb atoms by trapping them with a combi-
+nation of two light traps.
+We have known some properties of one dimensional
+TG gas, and we are interested in its properties and fea-
+tures in external magnetic field. Consequently, we first
+reveal some properties of both gases in the same exter-
+nal magnetic field and compared them. With the Fermi-
+Bose mapping [2, 3], we obtained all the eigenenergies
+and eigenstates of this TG gas. When the number of
+bosonsNis commensurate with the number of potential
+cyclesM, the TG system in an external magnetic field
+will be in the ground state with a bigger probability than
+theTGgasraisedbyGirardeauin1960. Manyproperties
+of the TG gas and the Fermi gas are always of the same
+even if it’s time-dependent, such as their averagevalue of
+particle’s coordinate and potential energy, system’s total
+∗Electronic address: wzlcxl@mail.ustc.edu.cn
+†Electronic address: anmwang@ustc.edu.cnmomentum, single-particle density and the pair distri-
+bution function. But both their reduced single-particle
+matrices and momentum distributions are different.
+II. TG GAS IN AN EXTERNAL MAGNETIC
+FIELD
+In a TG gas, the boson is assumed to have an “impen-
+etrable” hard core characterized by a radius of a. From
+Girardeau’s work, with the hard core radius a→0, the
+interparticle interaction is given by
+U(xi,xj) =/braceleftigg
+0, xi∝ne}ationslash=xj,
+∞, xi=xj.(1)
+Such an interparticle interaction could be represented
+by the following subsidiary condition on the wave func-
+tionψ:
+ψ(x1,···,xN,t) = 0 ifxi=xj,1/lessorequalslanti<j/lessorequalslantN.(2)
+In order to let the TG gas wave function, which must
+be symmetric with respect to the permutations of any xi
+andxj, satisfy this condition (2), we need the ideas of
+Fermi-Bose mapping proposed by Girardeau [2, 3]. That
+is, using the fact that the Fermi wave function, denoted
+byψF, satisfies condition (2) naturally and is antisym-
+metric, a bosonic wave function ψBof a TG gas can be
+constructed by
+ψB(x1,···,xN,t) =A(x1,···,xN)ψF(x1,···,xN,t)
+(3)
+in which
+A(x1,···,xN)≡N/productdisplay
+i>jsgn(xi−xj), (4)
+and
+sgn(x)≡x
+|x|=/braceleftigg
+1, x> 0,
+−1, x<0.2
+As the groundstate of a bose system is non-negative[19],
+mapping (3) for the stationary ground state of a system
+reduces to a simplified form [2]
+ψB(x1,···,xN) =|ψF(x1,···,xN)|(5)
+To illustrate our general ideas, we now study the case
+that magnetized bosons in an external magnetic field
+B(x) =−Bcos(2ωx). For simplicity, we take B(x) inde-
+pendent of t. therefore,
+ˆH=N/summationdisplay
+i=1/bracketleftbigg
+−/planckover2pi12
+2m∂2
+∂x2
+i+V(xi)/bracketrightbigg
+. (6)
+in which
+V(x) =/braceleftigg
+µBcos(2ωx),0/lessorequalslantxi/lessorequalslantL
+ω,
+0, else.(7)
+We suppose L=Mπ, whereMis an integer and µis
+the magnetic moment of atom. For the sake ofsimplicity,
+we assume that both NandMare odd (As a result of
+the introduction of A(x1,···,xN),ψBis periodic if Nis
+odd and antiperiodic if Nis even [2, 12]). From Bloch
+theory, the corresponding energy spectrum of system in
+periodic potentials has the energy band structure. We
+use the notation nas the band index and kas the Bloch
+wave vector in the first Brillouin zone. For convenience,
+we useα={n,k}to denote both the band index and
+the Bloch wave vector. Not considering the interparticle
+interaction, every particle in x∈[0,L/ω] is governed by
+/bracketleftbigg
+−/planckover2pi12
+2m∂2
+∂x2+µBcos(2ωx)/bracketrightbigg
+ϕα(x) =Eαϕα(x).(8)
+Substitutez=ωx,q=mµB
+/planckover2pi12ω2andλ=2mE
+/planckover2pi12ω2into equa-
+tion (8), then this equation becomes
+d2ϕ(z)
+dz2+[λ−2qcos(2z)]ϕ(z) = 0, (9)
+z∈[0,L].
+It is called Mathieu’s Differential Equation, which can
+be solved with Randall B. Shirts’s [18] method. Using
+Bloch’s theorem, we first set
+ϕ(z) = exp(iνz)u(z), (10)
+ϕ(z) =ϕ(z+π), (11)
+u(z) =u(z+L), (12)
+ν=2l
+M,/parenleftbigg
+l∈0,±1,···,±M−1
+2/parenrightbigg
+.(13)
+Sinceu(z) is periodic with period π, it can be expanded
+in Fourier series:
+ϕ(z) = exp(iνz)/summationdisplay
+ncnexp(i2nz).(14)Then substitute (14) into (9), we obtain an infinite
+symmetric tridiagonal matrix equation, that is, eigen
+equation.
+
+...··· ··· ··· ··· ··· ···
+···(ν−4)2q0 0 0 ···
+···q(ν−2)2q0 0 ···
+···0q ν2q 0···
+···0 0 q(ν+2)2q···
+···0 0 0 q(ν+4)2···
+··· ··· ··· ··· ··· ···...
+
+×
+...
+c−2
+c−1
+c0
+c1
+c2
+...
+=λ
+...
+c−2
+c−1
+c0
+c1
+c2
+...
+.(15)
+Bytruncatingthe matrixin eachdirection(centeredat
+the smallest diagonal element ν2) at sufficiently large di-
+mensions, approximations to the desired eigenvalues can
+be obtained to any desired precision. We could obtain
+the eigenvalue and eigenfunction easily using Mathemat-
+ica or Matlab. In practice, the eigenvalue and eigenfunc-
+tion converge quickly as the dimension increases when
+qis not too large. For example, if we choose ν= 6/7
+andq= 1, the relative difference of the lowest eigenvalue
+between truncating the matrix at 21-D and 2001-D is
+only 10−38. For large values of qand for high orders, we
+can use asymptotic expansions instead [21, 22]. since the
+matrixdimensionthatneeded togetaccurateeigenvalues
+and eigenfunctions becomes large.
+With Girardeau’s Fermi-Bose mapping (3), the eigen-
+states of the hard-core Bose system is given by the Slater
+determinant
+ψB(x1,···,xN) =A(x1,···,xN)√
+N!
+×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleϕα1(x1)ϕα2(x1)···ϕαN(x1)
+ϕα1(x2)ϕα2(x2)···ϕαN(x2)
+............
+ϕα1(xN)ϕα2(xN)···ϕαN(xN)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(16)
+The total energy of the system is
+E=N/summationdisplay
+i=1Eαi. (17)
+As a result of the Bose-Fermi mapping, the energy spec-
+trum of the Bose and corresponding Fermi system are
+identical. When the temperature is 0, the system will
+be on the ground state — Nparticles on the Nlowest
+eigenstates, respectively.3
+Calculationsshowthattheenergygap △Ebetweenthe
+first and the second Bloch bands decreases with the de-
+crease ofM, and when Mis large, the energy gap △Eis
+independent of M. The energy gap between the first and
+the second Bloch bands as a function of magnetic field B
+and frequency ωare plotted in Fig. 1, respectively. At
+B= 0, raised by Girardeau [2], the energy gap of TG
+gas is the smallest (not 0). The energy gap increases as
+Bandωincrease. As the Boltzman Distribution Law
+notes,nk
+nk′∝exp(−△E
+kT), with△E=Ek−Ek′,nkand
+nk′are particle numbers on the corresponding states. So
+the system have a bigger probability in the ground state
+with largeBandωthan the case B= 0 when the to-
+tal number N=M. That is to say, one could realize
+ground-stated TG system more simple with large qand
+ωthanq= 0 when the total number N=M.
+0
+5
+10
+15
+20B/LParen1T/RParen1
+0510
+Ω/LParen1109m/Minus1/RParen10246
+/CaΠDeltaE/LParen110/Minus23J/RParen1
+FIG. 1: Plots of the energy gap between the first and the
+second Bloch bands as a function of magnetic induction B
+and frequency ω.M= 9,m= 1.44×10−25kg,µ= 9.274×
+10−24J·T−1.
+III. PROPERTIES OF THE TG GAS
+COMPARED WITH THE FERMI GAS
+In this section, we further study properties of the 1D
+Nmagnetized hard-coreBosonsin the external magnetic
+field (7), and then compared with the properties of 1D
+Nmagnetized Fermions in the “same” state which hard-
+core Bosons occupy.
+A. Reduced single-particle density matrix and
+Single-particle density
+The reduced single-particle density matrix with nor-
+malization/integraltext
+ρ(x,x)dx=Nis given by
+ρB(x,x′,t)≡N/integraldisplay
+ψB∗(x,x2,···,xN,t)
+×ψB(x′,x2,···,xN,t)dx2···dxN.(18)For the Fermi gas, the reduced single-particle density
+matrix can be simplified to
+ρF(x,x′,t) =N/integraldisplay
+ψF∗(x,x2,···,xN,t)
+×ψF(x′,x2,···,xN,t)dx2···dxN
+=/summationdisplay
+α=allstates
+occupyedϕα(x,t)∗ϕα(x′,t).(19)
+FIG. 2: Density-plots of ρ(z,z′) of the hard-core TG gas
+(a,b,c) and Fermi gas ( a′,b′,c′),M= 7. Abscissa is z1, Ordi-
+nate isz2. (a,a′)N= 3; (b,b′)N= 5; (c,c′)N= 7.
+Both the reduced single-particle matrices of hard-core
+bosons and fermions in the ground state in the exter-
+nal magnetized field (7) are plotted separately in Fig. 2.
+The multidimensional integral in equation (18) is evalu-
+atednumericallybyMonteCarlointegrationusingMath-
+ematica. Because the relationship between xandzis
+proportional, for the sake of simplicity, we take zas a
+variable. From the figure we can see the off-diagonal ele-
+ments of the matrix of both the two kinds of gases decay
+quickly asNincreases; The bright diagonal stripe of TG
+gas is thinner than Fermi gas under the same condition.
+This reflects the condensate properties of Bose gas; And4
+the shockof the off-diagonalelements ρ(z,Mπ−z)ofTG
+gas is smaller than Fermi gas. To see this more clearly,
+we have plotted the off-diagonal elements ρ(z,Mπ−z)
+in Fig. 3. Sudden rise in the border is the result of using
+periodic boundary conditions.
+5101520z0.020.040.060.080.100.120.14Ρ3B/LParen1z,MΠ/Minusz/RParen1
+5101520z
+/Minus0.050.050.100.15Ρ3F/LParen1z,MΠ/Minusz/RParen1
+5101520z0.050.100.150.200.25Ρ5B/LParen1z,MΠ/Minusz/RParen1
+5101520z
+/Minus0.050.050.100.150.200.25Ρ5F/LParen1z,MΠ/Minusz/RParen1
+5101520z0.050.100.150.20Ρ7B/LParen1z,MΠ/Minusz/RParen1
+5101520z0.10.20.3Ρ7F/LParen1z,MΠ/Minusz/RParen1
+FIG. 3: Plots of ρ(z,Mπ−z),M= 7.ρB
+n(z,Mπ−z)
+stands for ρ(z,Mπ−z) ofnhard-core Boson particles, so
+asρF
+n(z,Mπ−z)
+.
+Using the mapping (3), we can prove that the single-
+particle density ρ(x,t), normalized to N, of both hard-
+core Bosons and Fermions are equal, no matter whether
+they are on the ground-state or not.
+ρB(x,t) =N/integraldisplay
+|ψB(x,x2,···,xN,t)|2dx2···dxN
+=N/integraldisplay
+|ψF(x,x2,···,xN,t)|2dx2···dxN
+=/summationdisplay
+α=allstates
+occupyed|ϕα(x,t)|2
+=ρF(x,t)(20)
+ρ(x,t) is just the element of the reduced single-particle
+matrixρ(x,x′,t) whenx=x′. Fig. 2 shows that ρ(x) is
+cyclical, which indicates that particles tend to stay cycle
+at where its potential energy is low.
+B. Pair distribution function
+The pair distribution function, which is the probability
+of finding a second particle as a function of distance froman initial particle, normalized to N(N−1), is defined as
+D(x1,x2,t)
+=N(N−1)/integraldisplay
+|ψB(x1,x2,···,xN,t)|2dx3···dxN
+=N(N−1)/integraldisplay
+|ψF(x1,x2,···,xN,t)|2dx3···dxN=
+1
+2/summationdisplay
+α,α′=allstates
+occupied|ϕα(x1,t)ϕα′(x2,t)−ϕα(x2,t)ϕα′(x1,t)|2.
+(21)
+From the above equation we can see that the pair distri-
+bution functions of both hard-core Boson gas and Fermi
+gas are identical. In fact, the average of particle’s coor-
+dinate and potential of both the two gases are identical,
+too. Because they involve absolute values of the wave
+functions only. Also we can say this feature is deter-
+mined by the fact that the single particle density of both
+hard-core Boson gas and Fermi gas are exactly of the
+same. For the ground state of a system with Nparticles
+governed by Hamilton (6), the pair distribution function
+is
+D(x1,x2) =1
+2N−1/summationdisplay
+α,α′=0|ϕα(x1)ϕα′(x2)−ϕα(x2)ϕα′(x1)|2.
+(22)
+Figure 4 shows density-plots of the pair distribution
+function of the ground state for different number of par-
+ticles. when x1=x2,D(x1,x2) = 0. It is the result
+of the impenetrable hard-core interparticle interaction.
+Just like the single particle density, the pair distribution
+function reflects the same periodicity and particles tend-
+ingtotarryatthelowpotentialenergy. Asthenumberof
+particle increases, the black strip becomes thinner. One
+of the reason is that the averaged distance between par-
+ticles reduces as the number of particles increases while
+the scope of the potential field is certain. In two distant
+regions, away from the diagonal in Fig. 4, D(z1,z2) is
+cyclical. This reflects the hard-core interaction between
+particles is local.
+C. Momentum distribution
+The momentum distribution, related to the reduced
+density matrix, is defined as
+n(k) =1
+2π/integraldisplay
+ρ(x,x′)e−ik(x−x′)dxdx′,(23)
+with the normalization
+/integraldisplay
+n(k)dk=N.
+Actually, the momentum distribution is just the Fourier
+transformation of the reduced density matrix (18). We5
+FIG. 4: Density-plots of the pair distribution function
+D(z1,z2). Abscissa is z1, Ordinate is z2.M= 7. (a) N= 2;
+(b)N= 3; (c) N= 5; (d) N= 7.
+know that the Fourier transform of the original function
+and the transformed function is one-to-one, as the differ-
+ence of the reduced density matrix ρ(x,x′) of the hard-
+core Bose gas and Fermion gas, the momentum distri-
+bution is surely not the same. Yuan Lin and Biao Wu
+[10] have plotted the momentum distributions for both
+the TG gas and the free Fermi gas in a periodic Kronig-
+Penney potential, which shows the difference — Even
+a free Fermi gas has a broader momentum distribution
+than the most strongly interacting boson gas. We would
+like to add that, although their momentum distribution
+functions are different, their total momentum are of the
+same. With the equation
+(∂
+∂x1+∂
+∂x2)sgn(x2−x1) = 0, (24)
+we can prove
+N/summationdisplay
+i=1∂
+∂xiA(x1,···,xN) = 0. (25)And then, go ahead, we can obtain
+¯P(t)B=¯P(t)F,whereP=−i/planckover2pi1N/summationdisplay
+i=1∂
+∂xi.(26)
+IV. SUMMARY AND CONCLUSIONS
+On solving the eigenfunction of the TG gas in the ex-
+ternal magnetic field, we found that the TG system with
+largeBandωhas a bigger probability in the ground
+state than the TG gas raised by Girardeau in 1960 when
+the number of bosons Nis commensurate with the num-
+ber of potential cycles M. It reveals that we can realize
+ground-statedTG system more easier. And then we have
+studied properties of magnetized TG gas and Fermi gas
+in an external magnetic field. comparing with each other
+when they are in the “same” state reveals that it is im-
+possible to distinguish them just from their averagevalue
+of particle’s coordinate and potential energy, system’s
+otal momentum, single-particle density and pair distri-
+bution function. But we could distinguish them from
+their reduced single-particlematricesor their momentum
+distributions. In order to illustrate our results, we have
+plottedtheirreducedsingle-particlematricesandthemo-
+mentum distributions when they are in the ground state.
+Although their momentum distribution functions are dif-
+ferent, their total momentum are of the same. These
+results will be deeply helpful for understanding the TG
+gas.
+Acknowledgment
+This workhas been supported by the National Natural
+Science Foundation of China under Grant No. 10975125.
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\ No newline at end of file
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+arXiv:1001.0117v2  [cs.CC]  3 Feb 2010Symposium on TheoreticalAspects of Computer Science 2010( Nancy, France), pp. 429-440
+www.stacs-conf.org
+COLLAPSING AND SEPARATING COMPLETENESS NOTIONS UNDER
+AVERAGE-CASE AND WORST-CASE HYPOTHESES
+XIAOYANG GU1AND JOHN M. HITCHCOCK2AND A. PAVAN3
+1LinkedIn Corporation
+2Department of Computer Science, University of Wyoming
+3Department of Computer Science, Iowa State University
+Abstract. This paper presents the following results on sets that are co mplete for NP.
+(i) If there is a problem in NP that requires 2nΩ(1)time at almost all lengths, then every
+many-one NP-complete set is complete under length-increas ing reductions that are
+computed by polynomial-size circuits.
+(ii) If there is a problem in co-NP that cannot be solved by pol ynomial-size nondetermin-
+istic circuits, then every many-one complete set is complet e under length-increasing
+reductions that are computed by polynomial-size circuits.
+(iii) If there exist a one-way permutation that is secure aga inst subexponential-size cir-
+cuits andthereis ahardtally language in NP ∩co-NP, thenthere is aTuringcomplete
+language for NP that is not many-one complete.
+Our first two results use worst-case hardness hypotheses whe reas earlier work that showed
+similar results relied on average-case or almost-everywhe re hardness assumptions. The use
+of average-case and worst-case hypotheses in the last resul t is unique as previous results
+obtaining the same consequence relied on almost-everywher e hardness results.
+1. Introduction
+It is widely believed that many important problems in NP such as satisfiability, clique,
+and discrete logarithm are exponentially hard to solve. Exi stence of such intractable prob-
+lems has a bright side: research has shown that we can use this kind of intractability to
+our advantage to gain a better understanding of computation al complexity, for derandom-
+izing probabilistic computations, and for designing compu tationally-secure cryptographic
+primitives. For example, if there is a problem in EXP (such as any of the aforementioned
+problems) that has 2nΩ(1)-size worst-case circuit complexity (i.e., that for all suffi ciently
+largen, no subexponential size circuit solves the problem correct ly on all instances of size
+Key words and phrases: computational complexity, NP-completeness.
+Gu’s research was supported in part by NSF grants 0652569 and 0728806.
+Hitchcock’s research was supported in part by NSF grants 051 5313 and 0652601 and by an NWO travel
+grant. Part of this research was done while this author was on sabbatical at CWI.
+Pavan’s research was supported in part by NSF grants 0830479 and 0916797.
+c/circlecopyrtX. Gu,J.M. Hitchcock,andA.Pavan
+CC/circlecopyrtCreativeCommonsAttribution-NoDerivsLicense430 X. GU, J. M. HITCHCOCK, AND A. PAVAN
+n), then it can be used to construct pseudorandom generators. Using these pseudoran-
+dom generators, BPP problems can be solved in deterministic quasipolynomial time [23].
+Similar average-case hardnessassumptionson thediscrete logarithm andfactoring problems
+have important ramifications in cryptography. While these h ardness assumptions have been
+widely used in cryptography and derandomization, more rece ntly Agrawal [1] and Agrawal
+and Watanabe [2] showed that they are also useful for improvi ng our understanding of NP-
+completeness. In this paper, we provide further applicatio ns of such hardness assumptions.
+1.1. Length-Increasing Reductions
+A language is NP-complete if every language in NP is reducible to it. While there are
+several ways to definethe notion of reduction, the most commo n definition uses polynomial-
+time computable many-one functions. Many natural problems that arise in practice have
+been shown to be NP-complete using polynomial-time computa ble many-one reductions.
+However, it hasbeenobserved that all knownNP-completenes s results holdwhenwerestrict
+the notion of reduction. For example, SAT is complete under p olynomial-time reductions
+that are one-to-one and length-increasing. In fact, all kno wn many-one complete problems
+for NP are complete under this type of reduction [9]. This rai ses the following question: are
+therelanguages that arecompleteunderpolynomial-time ma ny-onereductionsbutnotcom-
+plete under polynomial-time, one-to-one, length-increas ing reductions? Berman [8] showed
+that every many-one complete set for E is complete under one- to-one, length-increasing re-
+ductions. Thus for E, these two completeness notions coinci de. A weaker result is known for
+NE. Ganesan and Homer [17] showed that all NE-complete sets a re complete via one-to-one
+reductions that are exponentially honest.
+For NP, until recently there had not been any progress on this question. Agrawal [1]
+showed that if one-way permutations exist, then all NP-comp lete sets are complete via one-
+to-one, length-increasing reductions that are computable by polynomial-size circuits. Hitch-
+cock and Pavan [20] showed that NP-complete sets are complet e under length-increasing
+P/poly reductions under the measure hypothesis on NP [26]. R ecently Buhrman et al. im-
+proved the latter result to show that if the measure hypothes is holds, then all NP-complete
+sets are complete via length-increasing, P /-computable functions with loglog nbits of ad-
+vice [10]. More recently, Agrawal and Watanabe [2] showed th at if there exist regular
+one-way functions, then all NP-complete sets are complete v ia one-one, length-increasing,
+P/poly-computable reductions. All the hypotheses used in t hese works requirethe existence
+of analmost-everywhere hard language or an average-case hard language in NP.
+In the first part of this paper, we consider hypotheses that on ly concern the worst-case
+hardness of languages in NP. Our first hypothesis concerns the determi nistic time complex-
+ity of languages in NP. We show that if there is a language in NP for which every correct
+algorithm spends more than 2nǫtime at almost all lengths, then NP-complete languages
+are complete via P/poly-computable, length-increasing re ductions. The second hypothe-
+sis concerns nondeterministic circuit complexity of langu ages in co-NP. We show that if
+there is a language in co-NP that cannot be solved by nondeter ministic polynomial-size
+circuits, then all NP-complete sets are complete via length -increasing P/poly-computable
+reductions. For more formal statements of the hypotheses, w e refer the reader to Section 3.
+We stress that these hypotheses require only worst-case har dness. The worst-case hardness
+is of course required at every length, a technical condition that is necessary in order to build
+a reduction that works at every length rather than just infini tely often.COLLAPSING AND SEPARATING COMPLETENESS NOTIONS 431
+1.2. Turing Reductions versus Many-One Reductions
+In the second part of the paper we study the completeness noti on obtained by allowing
+a more general notion of reduction—Turing reduction. Infor mally, with Turing reductions
+an instance of a problem can be solved by asking polynomially many (adaptive) queries
+about the instances of the other problem. A language in NP is T uring complete if there is
+a polynomial-time Turing reduction to it from every other la nguage in NP. Though many-
+one completeness is the most commonly used completeness not ion, Turing completeness
+also plays an important role in complexity theory. Several p roperties of Turing complete
+sets are closely tied to the separation of complexity classe s. For example, Turing complete
+sets for EXP are sparse if and only if EXP contains polynomial -size circuits. Moreover, to
+capture our intuition that a complete problem is easy, then t he entire class is easy, Turing
+reductions seem to be the “correct” reductions to define comp leteness. In fact, the seminal
+paper of Cook [13] used Turing reductions to define completen ess, though Levin [25] used
+many-one reductions.
+This raises the question of whether there is a Turing complet e language for NP that
+is not many-one complete. Ladner, Lynch and Selman [24] pose d this question in 1975,
+thus making it one of the oldest problems in complexity theor y. This question is completely
+resolvedforexponentialtimeclassessuchasEXPandNEXP[3 3,12]. Weknowthatforboth
+these classes many-one completeness differs from Turing-com pleteness. However progress
+on the NP side has been very slow. Lutz and Mayordomo [27] were the first to provide
+evidence that Turing completeness differs from many-one comp leteness. They showed that
+if the measure hypothesis holds, then the completeness noti ons differ. Since then a few
+other weaker hypotheses have been used to achieve the separa tion of Turing completeness
+from many-one completeness [3, 30, 31, 21, 29].
+All the hypotheses used in the above works are considered “st rong” hypotheses as they
+require the existence of an almost everywhere hard language in NP. That is, there is a
+language Lin NP and every algorithm that decides Ltakes exponential-time an all but
+finitely many strings. A drawback of these hypotheses is that we do not have any candidate
+languages in NP that are believed to be almost everywhere har d.
+It has been open whether we can achieve the separation using m ore believable hypothe-
+ses that involve average-case hardness or worst-case hardn ess. None of the proof techniques
+used earlier seem to achieve this, as the they crucially depe nd on the almost everywhere
+hardness.
+In this paper, for the first time, we achieve the separation be tween Turing completeness
+and many-one completeness using average-case and worst-ca se hardness hypotheses. We
+consider two hypotheses. The first hypothesis states that th ere exist 2nǫ-secure one-way
+permutations and the second hypothesis states that there is a language in NEEE ∩coNEEE
+that can not be solved in triple exponential time with logari thmic advice, i.e, NEEE ∩
+coNEEE /ne}ationslash⊆EEE/log. We show that if both of these hypothesis are true, then th ere is a
+Turing complete language in NP that is not many-one complete .
+The first hypothesis is an average-case hardness hypothesis and has been studied exten-
+sively in past. The second hypothesis is a worst-case hardne ss hypothesis. At first glance,
+this hypothesis may look a little esoteric, however, it is on ly used to obtain hard tally lan-
+guages in NP ∩co-NP that are sufficiently sparse. Similar hypotheses invol ving double and
+triple exponential-time classes have been used earlier in t he literature [7, 15, 19, 14].432 X. GU, J. M. HITCHCOCK, AND A. PAVAN
+We use length-increasing reductions as a tool to achieve the separation of Turing com-
+pleteness from many-one completeness. We first show that if o ne-way permutations ex-
+ist then NP-complete sets are complete via length-increasi ng, quasipolynomial-time com-
+putablereductions. We thenshow that ifthesecond hypothes is holds, thenthereis a Turing
+complete language for NP that is not complete via quasi polyn omial-time, length-increasing
+reductions. Combining these two results we obtain our separ ation result.
+2. Preliminaries
+In the paper, we use the binary alphabet Σ = {0,1}. Given a language A,Andenotes
+the characteristic sequence of Aat length n. We also view Anas a boolean function from
+Σnto Σ. For languages AandB, we say that A=ioB, ifAn=Bnfor infinitely many
+n. For a complexity class C, we say that A∈ioCif there is a language B∈ Csuch that
+A=ioB.
+For a boolean function f: Σn→Σ,CC(f) is the smallest number ssuch that there is
+circuit of size sthat computes f. A function fis quasipolynomial time computable (QP-
+computable) if can be computed deterministically in time O(2logO(1)n). We will use the
+triple exponential time class EEE = DTIME(222O(n)
+), and its nondeterministic counterpart
+NEEE.
+A language Lis in NP/poly if there is a polynomial-size circuit Cand a polynomial
+psuch that for every x,xis inLif and only if there is a yof length p(|x|) such that
+C(x,y) = 1.
+Our proofs make use a variety of results from approximable se ts, instance compression,
+derandomization and hardness amplification. We mention the results that we need.
+Definition 2.1. A language Aist(n)-time 2-approximable [6] if there is a function f
+computable in time t(n) such that for all strings xandy,f(x,y)/ne}ationslash=A(x)A(y).
+A language Aisio-lengthwise t(n)-time 2-approximable if there is a function fcom-
+putable in time t(n) such that for infinitely many n, for every pair of n-bit strings xandy,
+f(x,y)/ne}ationslash=A(x)A(y).
+Amir, Beigel, Gasarch [4] proved that every polynomial-tim e 2-approximable set is in
+P/poly. Their proof also implies the following extension for a superpolynomial function
+t(n).
+Theorem 2.2 ([4]).IfAis io-lengthwise t(n)-time 2-approximable, then for infinitely many
+n,CC(An)≤t2(n).
+Given a language H′in co-NP, let Hbe{/an}bracketle{tx1,···,xn/an}bracketri}ht | |x1|=···=|xn|=n,xi∈H′}.
+Observe that a n-tuple consisting of strings of length ncan be encoded by a string of length
+n2. From now we view a string of length n2as ann-tuple of strings of length n.
+Theorem 2.3 ([16, 11]) .LetHandH′be defined as above. Suppose there is a language
+L, a polynomial-size circuit family {Cm}, and a polynomial psuch that for infinitely many
+n, for every x∈Σn2,xis inHif and only if there is a string yof length p(n)such that
+C(x,y)is inL≤n. ThenH′is inioNP/poly.
+The proof of Theorem 2.3 is similar to the proofs in [16, 11]. T he difference is rather
+than having a polynomial-time many-one reduction, here we h ave a NP /poly many-one
+reduction which works infinitely often. The nondeterminism and advice in the reductionCOLLAPSING AND SEPARATING COMPLETENESS NOTIONS 433
+can beabsorbedintothefinal NP /poly decision algorithm. TheNP /poly decision algorithm
+works infinitely often, corresponding to when the NP /poly reduction works.
+Definition 2.4. A function f:{0,1}n→ {0,1}miss-secureif for every δ <1, everyt≤δs,
+and every circuit C:{0,1}n→ {0,1}mof sizet, Pr[C(x) =f(x)]≤2−m+δ. A function
+f:{0,1}∗→ {0,1}∗iss(n)-secureif it iss(n)-secure at all but finitely many length n.
+Definition 2.5. Ans(n)-secure one-way permutation is a polynomial-time computable
+bijection π:{0,1}∗→ {0,1}∗such that |π(x)|=|x|for allxandπ−1iss(n)-secure.
+Under widely believed average-case hardness assumptions a bout the hardness of the RSA
+cryptosystem or the discrete logarithm problem, there is a s ecure one-way permutation [18].
+Definition 2.6. Apseudorandom generator (PRG) family is a collection of functions G=
+{Gn:{0,1}m(n)→ {0,1}n}such that Gnis uniformly computable in time 2O(m(n))and for
+every circuit of Cof sizen,/vextendsingle/vextendsingle/vextendsingle/vextendsinglePr
+x∈{0,1}n[C(x) = 1]−Pr
+y∈{0,1}m(n)[C(Gn(y)) = 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1
+n.
+There are many results that show that the existence of hard fu nctions in exponential
+time implies PRGs exist. We will use the following.
+Theorem 2.7 ([28, 23]) .If there is a language AinEsuch that CC(An)≥2nǫfor all
+sufficiently large n, then there exist a constant kand a PRG family G={Gn:{0,1}logkn→
+{0,1}n}.
+3. Length-Increasing Reductions
+In this section we provide evidence that many-one complete s ets for NP are complete
+via length-increasing reductions. We use the following hyp otheses.
+Hypothesis 1. There is a language Lin NP and a constant ǫ >0 such that Lis not in
+ioDTIME(2nǫ).
+Informally, this means that every algorithm that decides Ltakes more than 2nǫ-time
+on at least one string at every length.
+Hypothesis 2. There is a language Lin co-NP such that Lis not in ioNP/poly.
+This means that every nondeterministic polynomial size cir cuit family that attempts to
+solveLis wrong on on at least one string at each length.
+We will first consider the following variant of Hypothesis 1.
+Hypothesis 3. There is a language Lin NP and a constant ǫ >0 such that for all but
+finitely many n,CC(Ln)>2nǫ.
+We will first show that Hypothesis 3 holds, then NP-complete s ets are complete via
+length-increasing reductions. Then we describe how to modi fy the proof to derive the
+same consequence under Hypothesis 1. We do this because the p roof is much cleaner with
+Hypothesis 3. To useHypothesis1 wehave to fixencodings of bo olean formulas with certain
+properties.434 X. GU, J. M. HITCHCOCK, AND A. PAVAN
+3.1. IfNPhas Subexponentially Hard Languages
+Theorem 3.1. If there is a language LinNPand anǫ >0such that for all but finitely
+manyn,CC(Ln)>2nǫ, then all NP-complete sets are complete via length-increasing,
+P/polyreductions.
+Proof.LetAbe a NP-complete set that is decidable in time 2nk. LetLbe a language in NP
+that requires 2nǫ-size circuits at every length. Since SAT is complete via pol ynomial-time,
+length-increasing reductions, it suffices to exhibit a lengt h-increasing, P /poly-reduction
+from SAT to A.
+Letδ=ǫ
+2k. Consider the following intermediate language
+S=/braceleftBig
+/an}bracketle{tx,y,z/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle|x|=|z|,|y|=|x|δ,MAJ[L(x),SAT(y),L(z)] = 1/bracerightBig
+.
+ClearlySis in NP. Since Ais NP-complete, there is a many-one reduction ffromS
+toA. We will first show that at every length nthere exist strings on which the reduction
+fmust be honest. Let
+Tn=/braceleftBig
+/an}bracketle{tx,z/an}bracketri}ht ∈ {0,1}n×{0,1}n/vextendsingle/vextendsingle/vextendsingleL(x)/ne}ationslash=L(z),∀y∈ {0,1}nδ|f(/an}bracketle{tx,y,z/an}bracketri}ht)|> nδ/bracerightBig
+Lemma 3.2. For all but finitely many n,Tn/ne}ationslash=∅.
+Assuming that the above lemma holds, we complete the proof of the theorem. Given
+a length m, letn=m1/δ. Let/an}bracketle{txn,zn/an}bracketri}htbe the first tuple from Tn. Consider the following
+reduction from SAT to A: Given a string yof length m, the reduction outputs f(/an}bracketle{txn,y,zn/an}bracketri}ht).
+Givenxnandynas advice, this reduction can be computed in polynomial time . Sincenis
+polynomial in m, this is a P/poly reduction.
+By the definition of Tn,L(xn)/ne}ationslash=L(zn). Thus y∈SAT if and only if /an}bracketle{txn,y,zn/an}bracketri}ht ∈S,
+and soyis in SAT if and only if f(/an}bracketle{txn,y,zn/an}bracketri}ht) is inA. Again, by the definition of Tn, for
+everyyof length m, the length of f(/an}bracketle{txn,y,zn/an}bracketri}ht) is bigger than nδ=m. Thus there is a
+P/poly-computable, length-increasing reduction from SAT toA. This, together with the
+proof of Lemma 3.2 we provide next, complete the proof of Theo rem 3.1.
+Proof of Lemma 3.2. Suppose Tn=∅for infinitely many n. We will show that this yields
+a length-wise 2-approximable algorithm for Lat infinitely many lengths. This enables us
+to contradict the hardness of L. Consider the following algorithm:
+(1) Input x,zwith|x|=|z|=n.
+(2) Find a yof length nδsuch that |f(/an}bracketle{tx,y,x/an}bracketri}ht)| ≤nδ.
+(3) If no such yis found, Output 10.
+(4) Ifyis found, then solve the membership of f(/an}bracketle{tx,y,z/an}bracketri}ht) inA. Iff(/an}bracketle{tx,y,z/an}bracketri}ht)∈A, then
+output 00, else output 11.
+We first bound the running time of the algorithm. Step 2 takes O(2nδ) time. In Step
+4, we decide the membership of f(/an}bracketle{tx,y,z/an}bracketri}ht) inA. This step is reached only if the length of
+f(/an}bracketle{tx,y,z/an}bracketri}ht) is at most nδ. Thus the time taken to for this step is (2nδ)k≤2nǫ/2time. Thus
+the total time taken by the algorithm is bounded by 2nǫ/2.
+Consider a length nat which Tn=∅. Letxandzbe any strings at this length.
+Suppose for every yof length nδ, the length of f(/an}bracketle{tx,y,z/an}bracketri}ht) is at least nδ. Then it must
+be the case that L(x) =L(z), otherwise the tuple /an}bracketle{tx,z/an}bracketri}htbelongs to Tn. Thus if the above
+algorithm fails to find yin Step 2, then L(x)L(z)/ne}ationslash= 10.COLLAPSING AND SEPARATING COMPLETENESS NOTIONS 435
+Suppose the algorithm succeeds in finding a yin Step 2. If f(/an}bracketle{tx,y,z/an}bracketri}ht)∈A, then at
+least one of xorzmust belong to L. ThusL(x)L(z)/ne}ationslash= 00. Similarly, if f(/an}bracketle{tx,y,z/an}bracketri}ht)/∈A,
+then at least one of xorzdoes not belong to L, and so L(x)L(z)/ne}ationslash= 11.
+ThusLis 2-approximable at length n. If there exist infinitely many lengths n, at
+whichTnis empty, then Lis infinitely-often, length-wise, 2nǫ/2-time approximable. By
+Theorem 2.2, Lhas circuits of size 2nǫat infinitely many lengths.
+Now we will describe how to modify the proof if we assume that H ypothesis 1 holds.
+LetLbe the hard language guaranteed by the hypothesis. We will wo rk with 3-SAT. Fix
+an encoding of 3CNF formulas such that formulas with same num bers of variables can be
+encoded as strings of same length. Moreover, we require that the formulas φ(x1,···,xn)
+andφ(b1,···,bi,xi+1,···,xn) can be encoded as strings of same length, where bi∈ {0,1}.
+Fix a reduction ffromLto 3-SAT such that all strings of length nare mapped to formulas
+withnrvariables, r≥1. Let 3-SAT′= 3-SAT ∩∪rΣnr. It follows that that if there is an
+algorithm that decides 3-SAT′such that for infinitely many nthe algorithm runs in 2nǫ
+time on all formulas with nrvariables, then Lis inioDTIME(2nǫ).
+Now the proof proceeds exactly same as before except that we u se 3-SAT′instead of L,
+i.e, our intermediate language will be
+{/an}bracketle{tx,y,z/an}bracketri}ht |MAJ[3-SAT′(x),SAT(y),3-SAT′(z)]}= 1.
+Consider the set Tnas before. It follows that if Tnis empty at infinitely many lengths,
+then for infinitely many n, 3-SAT′is 2-approximable on formulas with nrvariables. Now
+we can use the disjunctive self-reducibility of 3-SAT′to show that there is a an algorithm
+that solves 3-SAT′and for infinitely many n, this algorithm runs in DTIME(2nǫ)-time on
+formulas with nrvariables. This contradicts the hardness of L. This gives the following
+theorem.
+Theorem 3.3. If there is a language in NPthat is not in ioDTIME(2nǫ), then all NP-
+complete sets are complete via length-increasing P/polyreductions.
+3.2. Ifco-NPis Hard for Nondeterministic Circuits
+In this subsection we show that Hypothesis 2 also implies tha t all NP-complete sets are
+complete via length-increasing reductions.
+Theorem 3.4. If there is a language Linco-NPthat is not in ioNP/poly, thenNP-complete
+sets are complete via P/poly-computable, length-increasing reductions.
+Proof.We find it convenient to work with co-NP rather than NP. We will show that all
+co-NP-complete languages are complete via P/poly, length- increasing reductions.
+LetH′be a language in co-NP that is not in ioNP/poly. Let Hbe
+{/an}bracketle{tx1,···,xn/an}bracketri}ht | ∀1≤i≤n,[xi∈H′and|xi|=n]}.
+Note that every n-tuple that may potentially belong to Hcan be encoded by a string of
+lengthn2.
+LetS= 0H′∪1SAT. It is easy to show that Sis in co-NP and Sis not in ioNP/poly.
+Observe that Sis co-NP-complete via length-increasing reductions. Let Abe any co-NP-
+complete language. It suffices to exhibit a length-increasin g reduction from StoA.436 X. GU, J. M. HITCHCOCK, AND A. PAVAN
+Consider the following intermediate language:
+L={/an}bracketle{tx,y,z/an}bracketri}ht | |x|=|z|=|y|2,MAJ[x∈H,y∈S,z∈H] = 1}.
+Clearly the above language is in co-NP. Let fbe a many-one reduction from LtoA.
+As before we will first show at every length nthat there exits strings xandzsuch that for
+everyyinSthe length of f(/an}bracketle{tx,y,z/an}bracketri}ht) is at least n.
+Lemma 3.5. For all but finitely many n, there exist two strings xnandznof length n2
+withH(xn)/ne}ationslash=H(zn)and for every y∈Sn,|f(/an}bracketle{txn,y,zn/an}bracketri}ht)|> n.
+Proof.Suppose not. Then there exist infinitely many lengths nat which for every pair of
+strings (of length n2)xandzwithH(x)/ne}ationslash=H(z), there exist a yof length nsuch that
+|f(x,y,z)| ≤n.
+From this we obtain a NP /poly-reduction from HtoAsuch that for infinitely many n,
+for every xof length n2,|f(x)| ≤n. By Theorem 2.3, this implies that H′is inioNP/poly.
+We now describe the reduction. Given nletznbe a string (of length n2) that is not in H.
+(1) Input x,|x|=n2. Advice: zn.
+(2) Guess a string yof length n.
+(3) If|f(/an}bracketle{tx,y,zn/an}bracketri}ht)|> n, the output ⊥.
+(4) Output f(/an}bracketle{tx,y,zn).
+Suppose x∈H. Sincezn/∈H, there exists a string yof length nsuch that y∈Sand
+|f(/an}bracketle{tx,y,zn/an}bracketri}ht)| ≤n. Consider a path that correctly guesses such a y. Sincezn/∈H, and
+y∈S,/an}bracketle{tx,y,zn/an}bracketri}ht ∈L. Thusf(/an}bracketle{tx,y,zn/an}bracketri}ht)∈A≤n. Thus there exists at least one path on
+which the reduction outputs a string from L∩Σ≤n. Now consider the case x /∈H. On any
+path, the reduction either outputs ⊥or outputs f(/an}bracketle{tx,y,zn/an}bracketri}ht). Since both znandxare not
+inH,/an}bracketle{tx,y,z/an}bracketri}ht/∈L. Thusf(/an}bracketle{tx,y,zn/an}bracketri}ht)/∈Afor anyy.
+Thus there is a NP /poly many-one reduction from HtoLsuch that for infinitely many
+n, the output of the reduction, on strings of length n2, on any path is at most n. By
+Theorem 2.3, this places H′inioNP/poly.
+Thus for all but finitely many lengths n, there exist strings xnandznof length n2with
+H(xn)/ne}ationslash=H(zn) and for every y∈Sn, the length of f(/an}bracketle{txn,y,zn/an}bracketri}ht) is at least n.
+This suggests the following reduction hfromStoA. The reduction will have xnandzn
+as advice. Given a string yof length n, the reductions outputs f(/an}bracketle{txn,y,zn/an}bracketri}ht). This reduction
+is clearly length-increasing and is length-increasing on e very string from S. Thus we have
+the following lemma.
+Lemma 3.6. Consider the above reduction hfromStoA, for ally∈S,|h(y)|>|y|.
+Now we show how to obtain a length-increasing reduction on al l strings. We make the
+following crucial observation.
+Observation 3.7. For all but finitely many n, there is a string ynof length nsuch that
+yn/∈Sand|f(/an}bracketle{txn,yn,zn/an}bracketri}ht)|> n.
+Proof.Suppose not. This means that for infinitely many n, for every yfromS∩Σn, the
+length of f(/an}bracketle{txn,y,zn/an}bracketri}ht) is less than n. Now consider the following algorithm that solves S.
+Given a string yof length n, compute f(/an}bracketle{txn,y,zn/an}bracketri}ht). If the length of f(/an}bracketle{txn,y,zn/an}bracketri}ht)> n, then
+acceptyelse reject y.
+The above algorithm can be implemented in P/poly given xnandznas advice. If
+y∈S, then we know that that the length of f(/an}bracketle{txn,y,zn/an}bracketri}ht) is bigger than n, and so theCOLLAPSING AND SEPARATING COMPLETENESS NOTIONS 437
+above algorithm accepts. If y /∈S, then by our assumption, the length of f(/an}bracketle{txn,y,zn/an}bracketri}ht) is
+at mostn. In this case the algorithm rejects y. This shows that Sis inioP/poly which in
+turn implies that H′is inioP/poly. This is a contradiction.
+Now we are ready to describe our length increasing reduction fromStoA. At length n,
+this reduction will have xn,ynandznas advice. Given a string yof length n, the reduction
+outputs f(/an}bracketle{txn,y,zn/an}bracketri}ht) if the length of f(/an}bracketle{txn,y,zn/an}bracketri}ht) is more than n. Else, the reduction
+outputsf(/an}bracketle{txn,yn,zn/an}bracketri}ht).
+SinceH(xn)/ne}ationslash=H(zn),y∈Sif and only if f(/an}bracketle{txn,y,zn/an}bracketri}ht)∈A. Thus the reduction
+is correct when it outputs f(/an}bracketle{txn,y,zn/an}bracketri}ht). The reduction outputs f(/an}bracketle{txn,yn,zn/an}bracketri}ht) only when
+the length of f(/an}bracketle{txn,y,zn/an}bracketri}ht) is at most n. We know that in this case y /∈S. Sinceyn/∈S,
+f(/an}bracketle{txn,yn,zn)/∈A.
+Thus we have a P/poly-computable, length-increasing from StoA. Thus all co-NP-
+complete languages are complete via P/poly, length-increa sing reductions. This immedi-
+ately implies that all NP-complete languages are complete v ia P/poly-computable, length-
+increasing reductions.
+4. Separation of Completeness Notions
+In this section we consider the question whether the Turing c ompleteness differs from
+many-one completeness for NP under two plausible complexit y-theoretic hypotheses:
+(1) There exists a 2nǫ-secure one-way permutation.
+(2) NEEE ∩coNEEE /ne}ationslash⊆EEE/log.
+It turns out that the first hypothesis implies that every many -one complete language for
+NP is complete under a particular kind of length-increasing reduction, while the second
+hypothesis provides us with a specific Turing complete langu age that is not complete under
+the same kind of length-increasing reduction. Therefore, t he two hypotheses together sep-
+arate the notions of many-one and Turing completeness for NP as stated in the following
+theorem.
+Theorem 4.1. If both of the above hypotheses are true, there is is a language that is
+polynomial-time Turing complete for NPbut not polynomial-time many-one complete for
+NP.
+Theorem 4.1 is immediate from Lemma 4.2 and Lemma 4.3 below.
+Lemma 4.2. Suppose 2nǫ-secure one-way permutations exist. Then for every NP-complete
+language Aand every B∈NP, there is a quasipolynomial-time computable, polynomial-
+bounded, length-increasing reduction reduction ffromBtoA.
+A function fispolynomial-bounded if there is a polynomial psuch that the length of
+f(x) is at most p(|x|) for every x.
+Lemma 4.3. IfNEEE∩coNEEE /notsubseteqlEEE/log, then there is a polynomial-time Turing
+complete set for NPthat is not many-one complete via quasipolynomial-time com putable,
+polynomial-bounded, length-increasing reductions.
+The proof of Lemma 4.2 will appear in the full paper. The remai nder of this section
+is devoted to proving Lemma 4.3. It is well known that any set Aover Σ∗can be encoded
+as a tally set TAsuch that Ais worst-case hard if and only if TAis worst-case hard. For438 X. GU, J. M. HITCHCOCK, AND A. PAVAN
+our purposes, we need an average-case version of the this equ ivalence. Below we describe
+particular encoding of languages using tally sets that is he lpful for us and prove the average-
+case equivalence.
+Lett0= 2,ti+1=t2
+ifor alli∈N. LetT=/braceleftbig
+0ti|i∈N/bracerightbig
+. For each l∈N, let
+Tl=/braceleftbig
+0ti/vextendsingle/vextendsingle2l−1≤i≤2l+1−2/bracerightbig
+. Observe that T=/uniontext∞
+l=0Tl. Given a set A⊆ {0,1}∗,
+letTA=/braceleftBig
+022rx/vextendsingle/vextendsingle/vextendsinglex∈A/bracerightBig
+,whererxis the rank index of xin the standard enumeration of
+{0,1}∗. It is easy to verify that for all l∈Nand every x,
+x∈A∩{0,1}l⇐⇒0trx∈TA∩Tl. (4.1)
+Lemma 4.4. LetAandTAbe as above. Suppose there is a quasipolynomial time algorit hm
+Asuch that for every l, on anǫfraction of strings from Tl, this algorithm correctly decides
+the membership in TA, and on the rest of the strings the algorithm outputs “I do not know”.
+There is a 222k(l+1)
+-time algorithm A′for some constant kthat takes one bit of advice and
+correctly decides the membership in Aon1
+2+ǫ
+2fraction of the strings at every length l.
+We know several results that establish worst-case to averag e-case connections for classes
+such as EXP and PSPACE [34, 5, 22, 23, 32]. The following lemma establishes a similar
+connection for triple exponential time classes, and can be p roved using known techniques.
+Lemma 4.5. IfNEEE∩coNEEE /ne}ationslash⊆EEE/log, then there is language LinNEEE∩coNEEE
+such that no EEE/logalgorithm can decide L, at infinitely many lengths n, on more than
+1
+2+1
+nfraction of strings from {0,1}n.
+Now we are ready to prove Lemma 4.3.
+Proof of Lemma 4.3. By Lemma 4.5, there is a language L∈(NEEE∩coNEEE) −EEE/log
+such that no EEE /log algorithm can decide Lcorrectly on more than a1
+2+1
+nfraction of
+the inputs for infinitely many lengths n.
+Without loss of generality, we can assume that L∈NTIME(222n
+)∩coNTIME(222n
+)
+Let
+TL=/braceleftBig
+022rx/vextendsingle/vextendsingle/vextendsinglex∈L/bracerightBig
+.
+Clearly,TL∈NP∩coNP.
+Defineτ:N→Nsuch that τ(n) = max {i|ti≤n}. Now we will define our Turing
+complete language. Let
+SAT0=/braceleftbig
+0x/vextendsingle/vextendsingle0tτ(|x|)/∈TLandx∈SAT/bracerightbig
+,
+SAT1=/braceleftbig
+1x/vextendsingle/vextendsingle0tτ(|x|)∈TLandx∈SAT/bracerightbig
+.
+LetA= SAT 0∪SAT1. SinceLis in NP ∩co-NP,Ais in NP. The following is a Turing
+reduction from SAT to A: Given a formula x, ask queries 0 xand 1x, and accept if and only
+if at least one them is in A. ThusAis polynomial-time 2-tt complete for NP.
+Suppose Ais complete via length-increasing, polynomial-bounded, q uasipolynomial-
+time reductions. Then there is such a reduction ffrom{0}∗toA. There is a constant d
+such that fisnd-bounded and runs in quasipolynomial time.
+The following observation is easy to prove.
+Observation 4.6. Lety∈ {0,1}∗andb∈ {0,1}besuchthat f(0ti) =by. Then0tτ(|y|)∈TL
+if and only if b= 1.COLLAPSING AND SEPARATING COMPLETENESS NOTIONS 439
+Fix a length l. We will describe a quasipolynomial-time algorithm that wi ll decide the
+membership in TLon at least1
+logdfraction of strings from Tl, and says “I do not know” on
+other strings. By the Lemma 4.4, this implies that there is EE E/1 algorithm that decides
+Lon more than1
+2+1
+2logdfraction of strings from {0,1}l. This contradicts the hardness of
+Land completes the proof.
+Lets= 2l−1 andr= 2l+1−2. Recall that Tl=/braceleftbig
+0ti|s≤i≤r/bracerightbig
+. Divide Tlin
+setsT0,T2,···TrwhereTk=/braceleftbig
+0ti|s+klogd≤r+(k+1)logd/bracerightbig
+. This gives at least2l
+logd
+sets. Consider the following algorithm that decides TLon strings from Tl: Let 0tjbe the
+input. Say, it lies in the set Tk. Compute f(0ts+klogd) =by. Iftτ(|y|)/ne}ationslash=tj, then output
+“I do not know”. Otherwise, accept 0tjif and only if b= 1. By Observation 4.6 this
+algorithm never errs. Since fis computable in quasipolynomial time, this algorithm runs
+in quasipolynomial time. Finally, observe that tτ(|y|)lies between ts+klogdandts+(k+1)logd.
+Thus for every k, 0≤k≤r, there is at least one string from from Tkon which the above
+algorithm correctly decides TL. Thus the above algorithm correctly decides TLon at least
+1
+logdfraction of strings from Tl, and never errs.
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+arXiv:1001.0118v4  [hep-th]  24 Oct 2012Chiral algebras of (0 ,2) models
+Junya Yagi
+Department of Physics, National University of Singapore
+2 Science Drive 3, Singapore 117551
+yagi@nus.edu.sg
+Abstract
+We explore two-dimensional sigma models with (0 ,2) supersymmetry through
+their chiral algebras. Perturbatively, the chiral algebra s of (0,2) models have a
+rich infinite-dimensional structure described by the cohom ology of a sheaf of chi-
+ral differential operators. Nonperturbatively, instantons can deform this structure
+drastically. We show that under some conditions they even an nihilate the whole
+algebra, thereby triggering the spontaneous breaking of su persymmetry. For a cer-
+tain class of K¨ ahler manifolds, this suggests that there ar e no harmonic spinors on
+their loop spaces and gives a physical proof of the H¨ ohn–Sto lz conjecture.
+1 Introduction
+Supersymmetric sigma models in two dimensions have played central r oles in a number of
+important physical and mathematical developments during the pas t few decades. A key
+concept underlying much of the developments is that of chiral rings of sigma models with
+(2,2) supersymmetry [1]. These finite-dimensional cohomology rings ar e basic ingredients
+of topological sigma models [2, 3], and intimately connected to, among other things,
+Gromov–Witten invariants [2, 4], Floer homology [5, 6, 7, 8, 9], and mir ror symmetry
+[1, 10, 11].
+While the chiral rings of (2 ,2) models are clearly very interesting, we also know that
+some of the beautiful structures of two-dimensional supersymm etric sigma models arise in
+essentially infinite-dimensional contexts. For example, the elliptic ge nera encode infinite
+series of topological invariants of the target space [12, 13]. It is th en natural to ask
+whether there are infinite-dimensional analogs of the chiral rings. The answer is “yes”.
+They are the chiral algebras of sigma models with (0 ,2) supersymmetry [14, 15].
+Thechiral algebraofa(0 ,2)model isthecohomologyoflocaloperatorswithrespect to
+oneof the supercharges, graded by the right-moving R-chargea nd equipped with operator
+product expansion (OPE). As a consequence of (0 ,2) supersymmetry, its elements vary
+holomorphically on the worldsheet and form a structure analogous t o the chiral algebra
+of a conformal field theory (CFT), the operator product algebra of holomorphic fields.
+It is well known that one can twist (2 ,2) models to obtain topological field theories
+1characterized by their chiral rings. For (0 ,2) models, one can perform a similar quasi-
+topological twisting that turns them into holomorphic field theories c haracterized by their
+chiral algebras.
+Classically, the chiral algebra of a twisted (0 ,2) model is isomorphic, as a graded vec-
+tor space, to the direct sum of the Dolbeault cohomology groups of a certain infinite series
+of holomorphic vector bundles over the target space. Quantum me chanically, this isomor-
+phism gets deformed by quantum corrections. Like the chiral rings , the chiral algebra is
+independent of the choice of the metric on the target space, henc e can be computed in
+the large volume limit where the theory is weakly coupled and the path in tegral localizes
+to instantons. But unlike the chiral rings, it receives perturbative corrections as well as
+instanton corrections, because the contributions from bosonic a nd fermionic fluctuations
+do not cancel due to the lack of left-moving supersymmetry. This c omplication leads to
+the interesting subject of perturbative chiral algebras.
+At the level of perturbation theory, the physics of a sigma model is governed by the
+local geometry of the target space. On the other hand, we can alw ays deform the target
+space metric to make it locally flat without affecting the chiral algebra . Combining these
+observations, we reach a surprising conclusion: the perturbative chiral algebra can be
+described locally by a sigma model with flat target space, therefore reconstructed by
+gluing locally defined freetheories globally over the target space.
+This fact was exploited by Witten [16] to show that in the absence of le ft-moving
+fermions, the perturbative chiral algebra of a twisted (0 ,2) model can be formulated as
+the cohomology of a sheaf of chiral differential operators on the t arget space, a notion
+introduced earlier in mathematics by Malikov et al. [17]. In this picture, the moduli of the
+perturbative chiral algebra are encoded in the different possible wa ys of gluing relevant
+free CFTs, whereas the anomalies of the theory manifest themselv es in the obstructions
+to doing so consistently. When left-moving fermions are present an d take values in the
+tangent bundle of the target space, it was shown by Kapustin [18] t hat the perturbative
+chiral algebra is given by the cohomology of of the chiral de Rham com plex [17]. The
+theory of perturbative chiral algebras has been further develop ed along these lines by Tan
+[19, 20, 21, 22].
+Nonperturbatively, instantons can change the picture radically [16 , 23, 24, 25]. A
+particularly striking example is the model with no left-moving fermions whose target
+space is the flag manifold G/Bof a complex simple Lie group G. The perturbative chiral
+algebra of this model is infinite-dimensional and has the structure o f a/hatwideg-module of critical
+level [17, 26, 27]. In the presence of instantons, however, the eq uation 1 = 0 holds in the
+cohomology and the chiral algebra vanishes.
+One clue to the existence of such a nonperturbative phenomenon lie s in a conjecture
+made independently by H¨ ohn and Stolz [28] in the mid 1990s. The H¨ o hn–Stolz conjecture
+asserts that the elliptic genus of a supersymmetric sigma model with no left-moving
+fermions vanishes if the target space Madmits a Riemannian metric of positive Ricci
+curvature.
+Stolz gave a heuristic argument for his conjecture based on the ge ometry of the loop
+spaceLM, the space of smooth maps from the circle S1toM. It goes as follows. Let us
+assume that the scalar curvature of LMis given, at each loop γ∈ LM, by the integral
+of the Ricci curvature of Malongγ. Then,LMhas positive scalar curvature if Mhas
+positiveRiccicurvature. ByanalogywiththeLichnerowicztheorem , thiswouldimplythat
+2LMhas no harmonic spinors. Meanwhile, supersymmetric states of the theory may be
+identifiedwithharmonicspinorson LM. Hence, thetheorywouldhavenosupersymmetric
+states, and since the elliptic genus counts the number of bosonic su persymmetric states
+minus the number of fermionic ones at each energy level, it would vanis h then.
+If Stolz’s reasoning is correct, the positivity of the Ricci curvatur e implies not only the
+vanishing of the elliptic genus, but also the spontaneous breaking of supersymmetry. Flag
+manifoldshavepositiveRicci curvature, sosupersymmetry should bebrokeninthemodels
+intothesespaces. Supersymmetry breakingisindeedtriggeredwh enever thechiralalgebra
+vanishes —instantons tunnel between infinitely many perturbative supersymmetric states
+and lift all of them at once.
+In fact, what happens for the flag manifold model is a special case o f a more general
+phenomenon: the chiral algebra of a (0 ,2) model with no left-moving fermions vanishes
+nonperturbatively if the target space is a compact K¨ ahler manifold with positive first
+Chern class and contains an embedded P1with trivial normal bundle.
+This vanishing theorem — or rather, “theorem” with quotation mark s — is the main
+result of this paper, which we will “prove” by a physical argument. A s a “corollary”,
+the “theorem” implies that supersymmetry is spontaneously broke n. Therefore, for this
+particular class of K¨ ahler manifolds, it suggests that there are no harmonic spinors on
+their loop spaces and gives a physical proof of the H¨ ohn–Stolz con jecture.
+The paper is organized as follows. In Section 2, we introduce the chir al algebras of
+(0,2) models and discuss their general properties. Section 3 is devote d to the sheaf theory
+ofperturbative chiral algebras. Finally, in Section4, we establish th e vanishing “theorem”
+and explain its relation to supersymmetry breaking and the geometr y of loop spaces.
+2 Chiral algebras of (0 ,2) models
+We now begin our study of the chiral algebras of (0 ,2) models. The focus of this section
+is on general properties of these algebras that do not depend on s pecific details of the
+target space geometry.
+2.1 (0,2) models
+Two-dimensional sigma models have (0 ,2) supersymmetric extension when the target
+space is strong K¨ ahler with torsion (strong KT) [16]. A Hermitian man ifold is called
+strong KT if the (1 ,1)-formωassociated to the Hermitian metric satisfies ∂¯∂ω= 0.
+K¨ ahler manifolds are strong KT since the K¨ ahler form satisfies ∂ω=¯∂ω= 0. In this
+paper we will only consider (0 ,2) models with K¨ ahler target spaces. Let us review how
+these models are constructed.
+Let Σ be a Riemann surface and Xa K¨ ahler manifold of complex dimension d. The
+bosonic sigma model with worldsheet Σ and target space Xis a quantum field theory of
+mapsφ: Σ→X. Imposing (0 ,2) supersymmetry requires the introduction of two right-
+moving fermions, ψ+and¯ψ+. These areworldsheet spinors withvaluesintheholomorphic
+and antiholomorphic tangent bundles of X:
+ψ+∈Γ(K1/2
+Σ⊗φ∗TX),¯ψ+∈Γ(K1/2
+Σ⊗φ∗TX). (2.1)
+HereK1/2
+Σis a square root of the antiholomorphic canonical bundle of Σ. The sim plest
+(0,2) model is constructed with this minimally (0 ,2) supersymmetric field content.
+3In the field space, (0 ,2) supersymmetry is realized as the transformation
+δφi=−ǫ−ψi
++, δφ¯ı= ¯ǫ−¯ψ¯ı
++,
+δψi
++=i¯ǫ−∂¯zφi, δ¯ψ¯ı
++=−iǫ−∂¯zφ¯ı,(2.2)
+whereǫ−and ¯ǫ−are sections of K−1/2
+Σ. We define the right-moving supercharges Q+and
+Q+so that−iǫ−Q++i¯ǫ−Q+acts by this transformation. The supercharges then satisfy
+{Q+,Q+}={Q+,Q+}= 0,
+{Q+,Q+}=−i∂¯z=1
+2(H−P),(2.3)
+and generate the (0 ,2) supersymmetry algebra together with the generators H,Pof
+translations, Mof rotations, and FRof the right-moving U(1) R-symmetry. Under the
+last symmetry, ψ+has charge −1 and¯ψ+has charge +1; thus Q+has charge −1 andQ+
+has charge +1.
+To construct a (0 ,2) supersymmetric action, we choose a K¨ ahler metric gonXand
+make the operator gi¯ψi
++∂zφ¯of R-charge −1. Then the action
+S=/integraldisplay
+Σd2z{Q+,gi¯ψi
++∂zφ¯} (2.4)
+is invariant under the R-symmetry and, by virtue of the relation Q2
++= 0, under the
+symmetry generated by i¯ǫ−Q+provided that ¯ ǫ−is antiholomorphic and so commutes
+with the∂zinside. This action is also invariant under −iǫ−Q+for antiholomorphic ǫ−, as
+becomes clear if we rewrite it as
+S=/integraldisplay
+Σd2z{Q+,gi¯∂zφi¯ψ¯
++}. (2.5)
+Expanding the anticommutators and using the K¨ ahler condition, on e can check that the
+two expressions (2.4) and (2.5) both coincide with
+S=/integraldisplay
+Σd2z(gi¯∂¯zφi∂zφ¯+igi¯ψi
++Dz¯ψ¯
++). (2.6)
+Here the covariant derivative Dzis the∂operator coupled to the pullback of the Levi-
+Civita connection Γ on X. Explicitly, Dz¯ψ¯ı
++=∂z¯ψ¯ı
+++∂zφ¯Γ¯ı
+¯¯k¯ψ¯k
++.
+We can add a topological invariant to the action. For a closed two-fo rmBonX, the
+functional
+SB=/integraldisplay
+Σφ∗B (2.7)
+depends only on the cohomology class of Band the homotopy class of φ. As such, it
+is invariant under any continuous transformations, especially the s upersymmetry trans-
+formation and the R-symmetry. The topological invariant SBvanishes at the level of
+perturbation theory, where one deals with homotopically trivial map s, but affects the
+dynamics nonperturbatively.
+We have obtained a (0 ,2) supersymmetric action. To complete the construction, we
+need to make sure that a sensible quantum theory based on this act ion exists. It turns
+4out thatXmust satisfy two topological conditions for that. First, Xmust be spin, or
+equivalently, its first Chern class must be even:
+c1(X)≡0 (mod 2) . (2.8)
+As we will see, this condition ensures that the fermion parity ( −1)FRis well defined.
+Second, the second Chern character ch 2(X) =c1(X)2/2−c2(X), which is also a half of
+the first Pontryagin class p1(X), must be zero:
+1
+2p1(X) = 0. (2.9)
+This is the condition for the absence of sigma model anomaly [29, 30], the obstruction to
+finding a well-defined path integral measure. From the viewpoint of t he loop space LX,
+the first condition means that LXis orientable [31, 32], while the second condition (given
+the first) is interpreted as the condition for LXto admit spinors [33].
+There is also a geometric condition. The renormalization group gener ates a flow of
+the target space metric. At the one-loop level, the metric g(µ) renormalized at an energy
+scaleµobeys the equation
+µd
+dµgi¯(µ) =1
+2πRi¯/parenleftbig
+g(µ)/parenrightbig
+, (2.10)
+whereRi¯(g) is the Ricci curvature of g[34, 35, 36, 37]. From this equation, we see that
+g(µ) gets larger as µgets larger if the Ricci curvature is positive. Since the theory is
+weakly coupled when the target space has large volume, we have asy mptotic freedom in
+this case. In contrast, if the Ricci curvature is negative, the the ory is strongly coupled
+for largeµand does not have a well-defined ultraviolet limit. Hence, for the micro scopic
+theory to exist, the Ricci curvature should be semipositive.
+When theabove conditions aresatisfied, the action(2.6)plus (2.7) d efines thesimplest
+version of (0 ,2) models. The supercharges are given by
+Q+=/contintegraldisplay
+d¯zgi¯ψi
++∂¯zφ¯,Q+=/contintegraldisplay
+d¯zgi¯∂¯zφi¯ψ¯
++ (2.11)
+and satisfy the reality condition Q†
++=Q+.
+If one has a holomorphic vector bundle EoverX, one can extend the model by adding
+left-moving fermions with values in E. This extended model is called the heterotic model.
+Since the left-movers contribute to sigma model anomalies in the opp osite way as the
+right-movers do, their presence changes the anomaly cancellation condition to
+1
+2p1(X) =1
+2p1(E). (2.12)
+This is trivially satisfied if E=TX, in which case the model actually has (2 ,2) su-
+persymmetry. It is also possible to add superpotentials [38]. For bre vity, we will not
+consider these extensions in this paper. We refer to Tan [19] for pe rturbative aspects of
+the heterotic model.
+52.2 Chiral algebras
+Having formulated (0 ,2) models, let us introduce the notion of their chiral algebras.
+Among the two supercharges we will use Q+exclusively, so simply write Q=Q+. Then
+Q†=Q+.
+Consider the action of Qon local operators Ogiven by supercommutator [ Q,O}. The
+Q-action increases the R-charge of Oby one, and squares to zero. Therefore, given a
+(0,2) model, we can define the Q-cohomology of local operators graded by the R-charge.
+Since∂¯z∝H−PisQ-exactbythe(0 ,2)supersymmetry algebra, itactstriviallyinthe
+Q-cohomology: if OisQ-closed, then ∂¯zOisQ-exact. Thus Q-cohomology classes vary
+holomorphicallyonΣ. Moreover, two classes canbemultiplied by [ O]·[O′] = [OO′]. From
+these facts, it follows that the Q-cohomology of local operators inherits a holomorphic
+OPE structure from the underlying theory:
+[O(z)]·[O′(w)]∼/summationdisplay
+kck(z−w)[Ok(w)]. (2.13)
+The coefficient functions ck(z−w) are holomorphic away from the diagonal z=wwhere
+there can be poles.
+The holomorphic Q-cohomology of local operators, equipped with this natural OPE
+structure, is the chiral algebra of the (0 ,2) model. We will denote the chiral algebra by A,
+and its R-charge qsubspace by Aq. As is clear from the construction, the chiral algebra
+of a (0,2) model has the structure of a chiral algebra in the sense of CFT, except that
+the grading by conformal weight is missing. We will see later that it car ries a similar (but
+possibly reduced to Zn) grading after the theory is twisted.
+The chiral algebra forms a closed sector of a (0 ,2) model in the following sense. Con-
+sider then-point function of Q-closed local operators:
+/angbracketleftbig
+O1(z1,¯z1)···On(zn,¯zn)/angbracketrightbig
+. (2.14)
+If one of the operators is Q-exact,Oi= [Q,O′
+i}, then the n-point function becomes
+±∝a\}bracketle{t[Q,O1···O′
+i···On}∝a\}bracketri}ht. Computed with a Q-invariant action and path integral measure,
+this is the integral of a “ Q-exact form” on the field space and vanishes. The n-point
+function (2.14) thus depends only on the Q-cohomology classes [ Oi]. In particular, it is a
+holomorphic (more precisely, meromorphic) function of the insertio n points.
+So far, we have considered the chiral algebra of a fixed (0 ,2) model, defined by a
+fixedQ-invariant action. Of course, different choices of the action lead to different chiral
+algebras in general. Imagine deforming the theory by perturbing th e action:
+S→S+δS. (2.15)
+For this deformation to preserve the Q-invariance, δSmust beQ-closed. If, however, δSis
+Q-exact, the chiral algebra remains unchanged. To see this, expre ss the matrix elements
+ofQas path integrals on a cylinder of infinitesimal length, with a contour in tegral of the
+supercurrent sandwiched between various boundary conditions. One can show that for a
+Q-exact perturbation, the matrix elements of Qin the deformed theory are equal to those
+ofQ+ [Q,δA] in the original theory for some operator δA. But this latter operator is
+related toQby the conjugation
+Q→e−δAQeδA, (2.16)
+6hence defines an isomorphic chiral algebra. The n-point function (2.14) is also unchanged
+because the perturbation just introduces Q-exact insertions.
+Looking back at the action (2.4) of our model, we see that deformat ions of the target
+space metric just give Q-exact perturbations. Therefore, the chiral algebra is independ ent
+of the K¨ ahler structure of the target space. It does depend on the complex structure,
+however. For this enters the very definition of the supersymmetr y transformation.
+2.3 Instantons
+An important property of the chiral algebra is that it receives cont ributions only from
+instantons and small fluctuations around them. In the case of our model, instantons obey
+{Q,ψi
++}=∂¯zφi= 0, (2.17)
+so they are holomorphic maps from Σ to X. It is this localization principle that makes
+the chiral algebra effectively computable.
+To prove the localization, we rescale the target space metric and ma ke it very large.
+In this large volume limit, the path integral localizes to the zeros of th e bosonic action
+/integraldisplay
+Σd2zgi¯∂¯zφi∂zφ¯, (2.18)
+namely holomorphic maps, or instantons, as promised. Incidentally, there is another
+situation in which the same localization arises. That is when the path int egral computes
+the correlation function of Q-closed operators [3]. This situation is not really relevant for
+us, though. In order to determine the chiral algebra, we have to a sk, in the first place,
+whether a given local operator is Q-closed or not.
+The space Mof holomorphic maps from Σ to Xis called the instanton moduli space.
+It decomposes into disconnected components labeled by the homolo gy classβthat the
+image of Σ represents in X:
+M=/circleplusdisplay
+β∈H2(X,Z)Mβ. (2.19)
+Correspondingly, the path integral splits into distinct sectors eac h of which integrates over
+the neighborhood of a single connected component of M. In sigma model perturbation
+theory, one expands in the inverse volume of the target space in th e zero-instanton sector.
+Instantons of β= 0 are constant maps, whose moduli space M0∼=X.
+Whenc1(X)∝\e}atio\slash= 0, instantons induce two important nonperturbative effects. On e is the
+violation of the R-charge conservation, which breaks the R-symme try down to a discrete
+subgroup. The other is the appearance of powers of a dynamical s cale Λ generated via
+dimensional transmutation, which allows objects of different scaling dimensions to show
+up as quantum corrections.
+The anomaly in the R-symmetry is due to a nontrivial transformation of the fermionic
+path integral measure under this symmetry. The complex conjuga te of¯ψ+is a section of
+K1/2
+Σ⊗φ∗TX, whileψ+can be identified with a (0 ,1)-formwith values in the same bundle.
+Thus, we can define D=D¯zd¯zand expand these fields in the eigenmodes of D∗Dand
+DD∗:
+ψ+=/summationdisplay
+sbs
+0v0,s+/summationdisplay
+nbnvn,¯ψ+=/summationdisplay
+rcr
+0¯u0,r+/summationdisplay
+ncn¯un, (2.20)
+7Here, ¯u0,r,v0,sare zero modes, ¯ un,vnnonzero modes, and bs
+0,cr
+0,bn,cnanticommuting
+coefficients. The fermionic path integral measure is the formal pro duct
+/productdisplay
+r,s,ndbs
+0dcr
+0dbndcn. (2.21)
+The nonzero mode part of the measure is neutral under the R-sym metry because of the
+pairing of nonzero modes. The zero mode part has R-charge equal to the number of ψ+
+zero modes minus the number of ¯ψ+zero modes, i.e., minus the index of the Doperator.
+For a compact Riemann surface Σ, the index is given by
+/integraldisplay
+Σφ∗c1(X) (2.22)
+which (recalling c1(X)≡0 mod 2) is equal to 2 kfor some integer k. The R-charge is
+violated by this amount in an instanton background. We see that the R-symmetry is
+broken to a Z2nsubgroup in the presence of instantons, where nis the greatest common
+divisor of the integers k.
+Instantons produce powers of Λ when c1(X)∝\e}atio\slash= 0 because the Bfield is renormalized
+as
+[B(µ)] = [B0]+lnµ
+Λc1(X), (2.23)
+with [B0] a fixed class in H2(X,C). We will derive this formula at the end of this section.
+Accordingly, instanton corrections violating R-charge by 2 kunits are weighted by the
+topological factor
+e−SB=/parenleftBigΛ
+µ/parenrightBig2k
+exp/parenleftBig/integraldisplay
+Σφ∗B0/parenrightBig
+. (2.24)
+An extra factor of µ2kshould come from somewhere else to cancel the dependence on
+µ. Altogether, these corrections are proportional to Λ2kand can relate objects of scaling
+dimensions differing by 2 k.
+Forinstanton corrections to beunder control, we shouldchoose B0such that thefactor
+(2.24) is exponentially suppressed for nonconstant holomorphic ma ps. The K¨ ahler form
+is a good choice, for example.
+2.4 Twisting
+The action of our model is invariant under supersymmetries whose t ransformation param-
+eters are antiholomorphic sections of the bundle K−1/2
+Σ. If Σ is topologically nontrivial,
+this bundle may not admit any global antiholomorphic sections. In tha t case the super-
+symmetries exist at best locally on Σ, hence so does the chiral algebr a. However, what
+we need to define the chiral algebra is really one of the two supersym metries, not both.
+So it would be nice if we can somehow globalize one in return for giving up t he other.
+This is achieved by twisting the theory.
+Let us modify the spins of the fermionic fields so that ψ+becomes a (0 ,1)-form on Σ,
+while¯ψ+becomes a zero-form. To make this point clear, we rename them as
+−ψi
++→ρi
+¯z,−i¯ψ¯ı
++→α¯ı. (2.25)
+8The action of Qis then given by
+[Q,φi] = 0,[Q,φ¯ı] =α¯ı,
+{Q,ρi
+¯z}=−∂¯zφi,{Q,α¯ı}= 0.(2.26)
+ThusQbecomes a scalar and generates a global supersymmetry. On the o ther hand,Q+
+becomes a (0 ,1)-form and in general does not exist globally. In this way, we obtain a
+theory with global supersymmetry on any Riemann surface Σ, desc ribed by the action
+S=/integraldisplay
+Σd2z{Q,gi¯ρi
+¯z∂zφ¯}+/integraldisplay
+Σφ∗B. (2.27)
+Discarding half the original supersymmetries also allows us to conside r complex target
+spaces which may or may not be K¨ ahler.
+Besides globalizing the supersymmetry, the twisting does one more im portant thing:
+it makes the components Tz¯zandT¯z¯zof the energy-momentum tensor Q-exact. This is
+because the global supersymmetry commutes with infinitesimal diffe omorphisms δz=vz,
+δ¯z=v¯zup to a quantity involving ∂¯zvz, but not∂¯zv¯z,∂zvz, or∂zv¯z. Hence, one can take
+the variation of the action inside the Q-commutator when computing these components.
+Quantum mechanically the action gets renormalized, but the renorm alization can be done
+byQ-exact local counterterms and the conclusion is unchanged.1
+The generators Tz¯z,T¯z¯zbeingQ-exact, antiholomorphic reparametrizations act triv-
+ially in the Q-cohomology. Therefore after the twisting the chiral algebra has no anti-
+holomorphic degrees of freedom — it defines a holomorphic field theor y.
+A local operator Ois said to have dimension ( n,m) if inserted at the origin it trans-
+forms as
+O(0)→λ−n¯λ−mO(0) (2.28)
+under the rescaling z→λz, ¯z→¯λ¯z. An immediate consequence of the decoupling of
+antiholomorphic degrees of freedom is that the chiral algebra is sup ported by local opera-
+tors withm= 0, because those with m∝\e}atio\slash= 0 transform nontrivially under antiholomorphic
+rescalings. The holomorphic dimension, n, is then equal to the spin n−m, so it is an
+integer and protected from quantum corrections (assuming that they are small). Thus,
+the chiral algebra of the twisted model is graded by the dimension as well as the R-charge.
+Ifc1(X)∝\e}atio\slash= 0, the grading by the R-charge is violated by instantons as we discu ssed
+already. The same is true of the grading by dimension. Instanton co rrections are accom-
+panied with powers of Λ2. When multiplying a local operator in the untwisted model,
+Λ2is best thought of as a section of KΣ⊗KΣ, having dimension (1 ,1). In the twisted
+model, it is more natural think of it as a section of KΣwith dimension (1 ,0), compen-
+sating the change in the dimension of the fermionic path integral mea sure. Instanton
+corrections violating R-charge by 2 kunits are proportional to Λ2k, so violate dimension
+byk. The grading by dimension is therefore reduced to Znnonperturbatively when that
+by the R-charge is reduced to Z2n.
+1This is not a precise statement. When Σ is curved, one needs to intro duce a worldsheet metric to
+impose a meaningful cutoff length. This produces an anomalous term inTz¯zproportional to the Ricci
+curvature of Σ [37]. Such a c-number anomaly does not affect the st ructure of the chiral algebra.
+9With the grading by dimension at hand, we can now write the OPE of the chiral
+algebra in the form
+[Oi(z)]·[Oj(w)]∼∞/summationdisplay
+n=0/summationdisplay
+kΛ2nc(n)
+ijk[Ok(w)]
+(z−w)ni+nj−nk−n, (2.29)
+where [Oi] have dimension niandc(n)
+ijkare constants. The Q-cohomology group of di-
+mensionnvanishes for n <0 since there are no local operators of negative dimension
+classically.
+TheQ-cohomology classes of dimension zero are special: they have regula r OPEs,
+as one can see by setting ni=nj= 0 above and noting nk≥0. Hence, they form an
+ordinary ring, much like the chiral rings of (2 ,2) models. This ring is called the chiral
+ring of the (0 ,2) model [16, 39].
+We have seen advantages of the twisting. It has one drawback, ho wever. Twisting the
+fermions amounts to tensoring with K−1/2
+Σto which the ∂operator couples. This changes
+the anomaly cancellation condition: Σ and Xmust now satisfy
+1
+2p1(X) =1
+2c1(Σ)c1(X) = 0. (2.30)
+Herec1(Σ)andc1(X)arepulledbacktoΣ ×X. Thus, thetwistingintroducesanadditional
+anomaly which affects the choice of Σ. If c1(X)∝\e}atio\slash= 0, we must choose Σ with c1(Σ) = 0,
+i.e.,KΣmust be trivial. This was actually implicit in our discussion when we interpr eted
+Λ2as a nowhere vanishing section of KΣ.
+2.5 Classical chiral algebra and quantum deformations
+To determine the chiral algebra of a given model, what one can usually do is to first
+identify its elements in the classical limit, then include quantum correc tions order by
+order. What guarantees the validity of this procedure is the followin g principle: small
+quantum corrections can only annihilate, and never create, Q-cohomology classes.
+This statement is justified as follows. Local operators that are no tQ-closed at a given
+order cannot become Q-closed by higher-order corrections. So the kernel of Qcan never
+become larger. At the same time, the image of Qcan never become smaller, because
+adding quantum corrections order by order defines an injection fr om the image of Qat
+a given order to the image of Qat the next order. Then the cohomology, which is the
+kernel modulo the image, can only become smaller.
+In this respect, the chiral algebra is similar to the space of supersy mmetric ground
+states (of supersymmetric quantum mechanics, say). There, sm all quantum corrections
+can lift supersymmetric ground states by giving a small positive ener gy, but not create
+ones because they cannot push positive energy states down to ze ro energy as long as there
+is a gap in the beginning. Likewise, small quantum corrections can “lift ”Q-cohomology
+classes by making them Q-exact or no longer Q-closed, but not create ones because of a
+“gap” to the Q-closedness or non- Q-exactness.
+Theanalogygoesfurther. Supersymmetry yields a one-to-oneco rrespondence between
+thebosonicpositiveenergystatesandthefermionicones, sosupe rsymmetricgroundstates
+are always lifted in boson-fermion pairs. The same principle applies her e: small quan-
+tum corrections always annihilate Q-cohomology classes in boson-fermion pairs. More
+10precisely, one can show that when a Q-cohomology class [ O] is annihilated by quantum
+corrections at some order, there is another class [ O′] that is annihilated together at the
+same order either via the relation [ Q,O} ∝ O′orO ∝[Q,O′}.2
+Keeping the above principles in mind, let us study the chiral algebra of the twisted
+model in a little more detail.
+The chiral algebra is supported by local operators whose antiholom orphic dimension
+m= 0. Since ρand ¯z-derivatives of any fields have m>0, these do not enter the chiral
+algebra. Furthermore, we can replace ∂zαby other fields using the equation of motion
+Dzα¯ı= 0. Thus, relevant local operators are linear combinations of oper ators of the form
+O(φ,¯φ)j···k···¯l···¯m···¯ı1···¯ıq∂zφj···∂2
+zφk···∂zφ¯l···∂2
+zφ¯m···α¯ı1···α¯ıq.(2.31)
+Identifying α¯ıwith the differential dφ¯ı, we can regard such operators with R-charge qand
+dimensionnas (0,q)-forms with values in a certain holomorphic vector bundle VX,nover
+X.
+For example, an n= 0 operator O¯ı1···¯ıqα¯ı1···α¯ıqis a (0,q)-form onX, henceVX,0= 1,
+thetrivialbundleofrank1. At n= 1,therearetwotypesofoperators, Oj¯ı1···¯ıq∂zφjα¯ı1···α¯ıq
+andOj¯ı1···¯ıqgj¯k∂zφ¯kα¯ı1···α¯ıq. These are(0 ,q)-formswithvaluesin T∗
+XandTX, thusVX,1=
+TX⊕T∗
+X. Atn= 2, we have five types, giving VX,2=TX⊕T∗
+X⊕S2TX⊕(TX⊗T∗
+X)⊕S2T∗
+X.
+HereSkis the symmetric kth power. In general, VX,nis given by the series
+∞/summationdisplay
+n=0qnVX,n=∞/circlemultiplydisplay
+k=0Sqk(TX)∞/circlemultiplydisplay
+l=0Sql(T∗
+X), (2.32)
+whereSt(V) = 1+tS(V)+t2S2(V)+···. We will write
+VX=∞/circleplusdisplay
+n=0VX,n. (2.33)
+At the classical level, we can easily find the action of Qon these operators. As an
+example, take Oj¯ı1···¯ıq∂zφjα¯ı1···α¯ıq. Acting on it with Qgivesα¯ı∂¯ıOj¯ı1···¯ıq∂zφjα¯ı1···α¯ıq.
+On ann= 1 operator of the other type, Oj¯ı1···¯ıqgj¯k∂zφ¯kα¯ı1···α¯ıq, acting with Qgives
+α¯ı∂¯ıOj¯ı1···¯ıqgj¯k∂zφ¯kα¯ı1···α¯ıqplusOj¯ı1···¯ıqgj¯kDzα¯kα¯ı1···α¯ıq, but the latter vanishes by the
+equation of motion. From these examples, we see that Qacts as the ¯∂operator.3Clas-
+sically, the qthQ-cohomology group of dimension nis therefore isomorphic to the qth
+2To prove this, let ǫbe a small parameter that controls the strength of quantum effec ts, andOa
+local operator that is nontrivial in the Q-cohomology to order ǫk−1. First, suppose that Ois no longer
+Q-closed at order ǫk; thus [Q,O}=ǫkO′for some O′, and no corrections to Ocan make it Q-closed
+again. Then O′isQ-closed, but cannot be Q-exact at order ǫlfor anyl < k. For if there exists O′′such
+thatǫlO′= [Q,O′′}+O(ǫl+1), then [Q,O−ǫk−lO′′}=O(ǫk+1) andO−ǫk−lO′′isQ-closed to order ǫk.
+Next, suppose that Obecomes Q-exact at order ǫk; thusǫkO= [Q,O′}+O(ǫk+1) for some O′. ThenO′
+isQ-closed to order ǫk−1, but cannot be Q-exact to the same order. For if there exist O′′andO′′′such
+thatǫlO′= [Q,O′′}+ǫl+1O′′′for some l < k, thenǫk−1O= [Q,O′′′}+O(ǫk) andOis already Q-exact
+at order ǫk−1.
+3General local operators of dimension greater than one are const ructed using covariant derivatives.
+On such operators, Qacts by¯∂plus terms involving the curvature of the target space. Classically, these
+additional terms which vanish for a flat metric do not change the Q-cohomology. This point will become
+clear when we develop a sheaf theory approach to the perturbativ e chiral algebra in Section 3.
+11Dolbeault cohomology group Hq
+¯∂(X,VX,n), and
+A∼=d/circleplusdisplay
+q=0∞/circleplusdisplay
+n=0Hq
+¯∂(X,VX,n) (2.34)
+as graded vector spaces. It is clear from this formula that the chir al algebra is generally
+infinite-dimensional.
+Quantum mechanically, the action of Qis deformed by quantum corrections. Because
+of this deformation, some of the classical Q-cohomology classes can disappear.
+A notable example is the disappearance of the energy-momentum te nsor. Classically,
+our model is conformally invariant and TzzisQ-closed on-shell. We expect that it is no
+longerQ-closedperturbativelyiftheRiccicurvatureisnonzero, sincecon formalinvariance
+is broken at one loop in that case. To determine [ Q,Tzz], we note that Qcommutes with
+∂z∝H+P. Thus
+/bracketleftBig
+Q,/contintegraldisplay
+dzTzz−/contintegraldisplay
+d¯zTz¯z/bracketrightBig
+=/contintegraldisplay
+dz[Q,Tzz] = 0, (2.35)
+where we have used the fact that Tz¯zisQ-exact. This suggests
+[Q,Tzz] =∂zθ (2.36)
+for someθ. Presumably, θis aQ-closed local operator of R-charge one and dimension
+one, constructed from the target space metric. To leading order ,Q=¯∂and for a generic
+choice of the metric there is only one possibility:
+θ∝Ri¯∂zφiα¯. (2.37)
+So, as expected, Tzzwould cease to be Q-closed unless the Ricci curvature vanishes. In
+Section 3.3, we will compute [ Q,Tzz] explicitly in perturbation theory and find that it is
+indeed proportional to ∂z(Ri¯∂zφiα¯) up to higher-order corrections.
+Ifc1(X) = 0, then θisQ-exact and we can add corrections to Tzzthat make it Q-
+closed again. But if c1(X)∝\e}atio\slash= 0, there is no way to do this, so [ Tzz] is annihilated together
+with [∂zθ] by perturbative corrections. The chiral algebra is a little exotic in t his case: it
+is analogous to the chiral algebras of CFTs, but lacks an energy-mo mentum tensor and
+hence invariance under holomorphic reparametrizations.
+Except for the possible lack of energy-momentum tensor, pertur batively the structure
+of the chiral algebra is not very different from its classical descript ion. The basic fact
+about perturbative corrections is that they are local on the targ et space, because one
+only considers fluctuations around constant maps in perturbation theory. Moreover, the
+gradings by the R-charge and dimension are not violated perturbat ively. Thanks to these
+properties, at the perturbative level the action of Qstill defines differential complexes
+··· −→VX,n⊗∧qT∗
+XQ−→VX,n⊗∧q+1T∗
+X−→ ···, (2.38)
+and the chiral algebra is given by the direct sum of their cohomology g roups. How to
+compute these cohomology groups using a sheaf of free CFTs on Xis the subject of the
+next section.
+12Beyond perturbationtheory, the physics is no longer local on the t arget space. Rather,
+it is local on the instanton moduli space Msince quantum fluctuations localize to in-
+stantons. So presumably the chiral algebra can be formulated non perturbatively as a
+cohomology theory on M, but exactly how this should be done is not clear. At any rate,
+in principle one can always compute the instanton corrections to the action ofQby path
+integral and determine the exact chiral algebra. Instanton effec ts often lead to surpris-
+ing results. Later we will see examples where the whole chiral algebra is annihilated by
+instanton corrections.
+Let us tie up the loose ends from Section 2.3 by explaining why the Bfield should
+be renormalized as asserted there. Consider the case where the t wisted and untwisted
+models are isomorphic. Suppose that instantons annihilate perturb ativeQ-cohomology
+classes [O] and [O′] through a relation of the form [ Q,O} ∼ O′. If this relation violates
+R-charge by 2 kunits, the left-hand side contains 2 kmoreαfields than the right-hand side
+does, while the both sides have antiholomorphic dimension equal to ze ro. Viewed in the
+untwisted model, thismeansthattheseinstantons relateoperato rswhoseantiholomorphic
+dimensions differ by k. Since the dynamical scale Λ is the only dimensional parameter
+available and has antiholomorphic dimension 1 /2, to match the scaling dimensions a
+factor of Λ2kmust appear in the right-hand side. For that, the Bfield must obey the
+renormalization group equation (2.23).
+3 Sheaf theory of perturbative chiral algebras
+Aswehaveexplainedinthelastsection, thechiralalgebracanbeund erstoodasaquantum
+deformation of the Dolbeault cohomology of an infinite-dimensional h olomorphic vector
+bundle over the target space. At the perturbative level, there is a nalternative formulation
+of the chiral algebra, which involves a sheaf of free CFTs. The goal of this section is to
+develop the sheaf theory of perturbative chiral algebras.
+3.1 Perturbative chiral algebra from free CFTs
+The chiral algebra of the twisted model is classically the Dolbeault coh omology of the
+holomorphic vector bundle VX. By the ˇCech–Dolbeault isomorphism, this cohomology is
+isomorphic to the cohomology of the sheaf of holomorphic sections o fVX. Perturbatively
+Qgets corrected, but still acts as a differential operator on the ta rget space due to the
+locality of sigma model perturbation theory. In such a situation, th ere is an analog of
+theˇCech–Dolbeault isomorphism: the perturbative Q-cohomology is isomorphic to the
+cohomologyofthesheafofperturbatively Q-closedsections of VX. Theproofiscompletely
+parallel to the classical case.4
+This “ˇCech-Qisomorphism” may seem to have little practical value. To compute the
+sheaf cohomology, first of all one needs to know the general form of perturbatively Q-
+closed local sections of VX. But this requires understanding beforehand how perturbative
+corrections deform the classical expression Q=¯∂precisely, which is generally very hard
+4The key ingredients are the Q-Poincar´ e lemma and the existence of partitions of unity on the she af
+of (0,q)-forms with values in VX. The former follows from the ¯∂-Poincar´ e lemma since small quantum
+corrections cannot create Q-cohomology classes. As for the latter, “quantum” partitions of u nity can
+always be constructed by renormalizing “classical” partitions of unit y.
+13if not impossible. However, we can circumvent this difficulty if we adopt a different
+approach.
+Let us recast the ˇCech-Qisomorphism in a slightly more abstract form. Choose a good
+cover{Uα}onX; thus all nonempty finite intersections of Uαare diffeomorphic to Cd. On
+eachUα, the space of perturbatively Q-closed sections of VXis isomorphic to the chiral
+algebra of the twisted model into Uα. For, the zeroth Q-cohomology group is isomorphic
+to this space, while the higher Q-cohomology groups vanish as Uαis topologically trivial.
+SinceˇCech cohomology is defined by taking the direct limit as the open cover becomes
+finer and finer, and every open cover has a refinement by a good co ver, it follows that
+the perturbative chiral algebra can be computed via the cohomolog y of the sheaf of chiral
+algebras /hatwideAonX:
+Aq∼=Hq(X,/hatwideA). (3.1)
+Rewriting the ˇCech-Qisomorphism this way makes it clear that the perturbative chiral
+algebra can be formulated without reference to any globally defined metric on the target
+space.
+When onecomputes thecohomologyof /hatwideA, say using a goodcover {Uα}, onecanendow
+Uαwith any metric. In practice, we will put gi¯=δi¯on all theUα. The point is that
+the twisted models with the flat target spaces Uαare free theories — their chiral algebras
+receive no quantum corrections.
+In the free twisted model Q=¯∂exactly, so sections of /hatwideA(Uα) are represented by
+local operators of the form O(φ,∂zφ,...,∂ z¯φ,∂2
+z¯φ,...). Introducing bosonic fields βiof
+dimension one and γiof dimension zero by
+βi= 2πδi¯∂z¯φ¯, γi=φi, (3.2)
+we can conveniently write them as O(γ,∂zγ,...,β,∂ zβ,...). These are observables of the
+freeβγsystem, a free CFT with action
+S=1
+2π/integraldisplay
+Σd2zβi∂¯zγi(3.3)
+whose OPEs are
+βi(z)γj(w)∼ −δj
+i
+z−w, βi(z)βj(w)∼0, γi(z)γj(w)∼0. (3.4)
+Hence,/hatwideAmaybeconsidered asasheaf of free βγsystems. Weconclude thattheperturba-
+tive chiral algebra can be reconstructed by gluing free βγsystems over Xand computing
+theˇCech cohomology.
+This construction is exact to all orders in perturbation theory sinc e the spaces /hatwideA(Uα)
+arefreeofquantumcorrections. Wheredidtheperturbativecor rections go? Theyarenow
+encoded in the transition functions /hatwidefαβfor the sheaf /hatwideA. Instead of equipping flat metrics
+on theUα, one may as well start with a globally defined, curved metric on X. One can
+then flatten it over each Uαto obtain a free βγsystem. In this process the perturbative
+correctionsdisappearlocally, butthetransitionfunctionschange by/hatwidefαβ→e−Aα/hatwidefαβeAβfor
+someoperators Aα. Theseoperatorscarrytheinformationontheperturbativecor rections.
+But then, how can we find the Aα? Here we come back to the original problem: to do
+that, we need detailed knowledge of the perturbative corrections . Instead, what we can do
+14is take a collection of locally defined free βγsystems and glue them using various choices
+of the transition functions. In doing so, we are effectively paramet rizing the theory by
+the way this gluing is done. This is the strategy we will adopt.
+3.2 Gluing βγsystems
+We are thus led to the problem of listing the possible sets of transition functions for /hatwideA.
+The first step is to classify the automorphisms of the free βγsystem. Automorphisms are
+generated by currents of dimension one. There are two types of s uch currents.
+LetVbe a holomorphic vector field on Cd. ThenJV=−Vi(γ)βiis a good current.
+(Here and below, normal ordering is implicit in expressions containing b othβiandγi.)
+From the OPEs
+JV(z)γi(w)∼Vi(w)
+z−w, JV(z)βi(w)∼ −∂iVjβj(w)
+z−w, (3.5)
+we see that JVgenerates the infinitesimal diffeomorphism δγ=Vandβtransforms as a
+(1,0)-form under this symmetry. We denote the corresponding cons erved charge by KV.
+With a holomorphic one-form BonCd, we can also make JB=Bi(γ)∂zγi. This has
+the OPEs
+JB(z)γi(w)∼0, JB(z)βi(w)∼ −Bi(w)
+(z−w)2+Cij∂zγj(w)
+z−w(3.6)
+withC=∂B, so generates the transformation δβ=−i∂zγC. For any closed holomorphic
+two-formC, there is a holomorphic one-form Bsuch thatC=∂B. Moreover, δβ= 0 if
+and only if C= 0. The automorphisms of this type are therefore labeled by the clo sed
+holomorphic two-forms C. We denote their charges by KC.
+Onecanreadilyworkoutthecommutatorsbetweentheconserved charges. Computing
+the relevant OPEs, one finds
+[KV,KV′] =K[V,V′]+KC(V,V′),
+[KV,KC] =KLVC,
+[KC,KC′] = 0.(3.7)
+Here,C(V,V′) =∂i∂kVl∂j∂lV′kdφi∧dφj. The last two ofthese relationsshow thatthe KC
+generate an abelian subalgebra on which holomorphic reparametriza tions act naturally.
+Now, suppose thatthecomplex manifold Xisbuiltupbygluingopenpatches Uα∼=Cd
+using transition functions fαβ. Transition functions for /hatwideAare constructed by lifting fαβ
+to automorphisms /hatwidefαβof theβγsystem compatible with the cocycle condition. So we
+choose/hatwidefαβsuch that they act on γbyfαβand satisfy /hatwidefαβ/hatwidefβγ/hatwidefγα= 1 onUα∩Uβ∩Uγ.
+Naively, one might think that the cocycle condition is satisfied if one pic ks a cocycle
+{Cαβ}of closed holomorphic two-forms and let /hatwidefαβact onβby the pullback f∗
+αβfollowed
+by exp(KCαβ). The situation is actually more complicated. The commutator betwe en two
+KVs differs from the expected form by a KCterm, implying
+/hatwidefαβ/hatwidefβγ/hatwidefγα= exp(KCαβγ) (3.8)
+forsomeCαβγ. Weneed toadjust /hatwidefαβso thatCαβγdisappear. Thefreedomatourdisposal
+is to transform /hatwidefαβ→exp(KC′
+αβ)/hatwidefαβ, which shifts
+Cαβγ→Cαβγ+(δC′)αβγ. (3.9)
+15UnlessCαβγcan be canceled by an appropriate choice of C′
+αβ, the gluing cannot be carried
+out consistently. In other words, there may be an obstruction to the existence of a sheaf
+ofβγsystems.
+The obstruction has a natural physical interpretation. It is not h ard to show that
+Cαβγare totally antisymmetric in α,β,γand obey (δC)αβγδ= 0, hence define a cocycle.
+The freedom (3.9) then means that the obstruction is encoded in th e cohomology class
+[Cαβγ]∈H2(X,Ω2,cl
+X), (3.10)
+where Ω2,cl
+Xis the sheaf of closed holomorphic two-forms on X. It can be shown [40]
+that this class is mapped to p1(X)∈H4(X,R) under the ˇCech–Dolbeault isomorphism.
+Thus, the obstruction vanishes if and only if p1(X) = 0. This is the condition for the
+perturbative cancellation of sigma model anomaly. Since the obstru ction arises in lifting
+the diffeomorphisms used to construct the underlying manifold X, in this way we see
+that the sigma model anomaly causes a gravitational anomaly on the target space. The
+cancellation of the anomaly by adjusting /hatwidefαβis a kind of the Green–Schwarz mechanism.
+Givenfαβ, the choice of /hatwidefαβis not unique. Suppose that we choose different tran-
+sition functions, /hatwidef′
+αβ. Then the two sets of transition functions are related by /hatwidef′
+αβ=
+exp(KC′
+αβ)/hatwidefαβfor some cocycle {C′
+αβ}. If this cocycle is exact, C′
+αβ= (δC′′)αβ, then we
+have/hatwidef′
+αβ= exp(−KC′′α)/hatwidefαβexp(KC′′
+β) and these define the same gluing. Hence, inequiva-
+lent choices of the transition functions are parametrized by
+H1(X,Ω2,cl
+X). (3.11)
+The origin of this moduli space can be understood as follows. For any closed form H
+of type (3,0)⊕(2,1), locally we can find a (2 ,0)-formTsuch that H=dT.5Thus, we
+have the short exact sequence
+0−→Ω2,cl
+X−→ A2,0
+Xd−→ Z3,0
+X⊕Z2,1
+X−→0, (3.12)
+whereAp,q
+XandZp,q
+Xare the sheaves of ( p,q)-forms and closed ( p,q)-forms on X, respec-
+tively. Since Hp(X,A2,0
+X) = 0 forp>0, the long exact sequence of cohomology implies
+H1(X,Ω2,cl
+X)∼=H0(X,Z3,0
+X⊕Z2,1
+X)/dH0(X,A2,0
+X). (3.13)
+This is the space of closed forms Hof type (3,0)⊕(2,1) modulo those that can be written
+asH=dT′withagloballydefined(2 ,0)-formT′. Itactuallyparametrizestheconformally
+invariantQ-closed term
+/integraldisplay
+Σd2z{Q,Tijρi
+¯z∂zφj}=/integraldisplay
+Σd2zα¯kH¯kijρi
+¯z∂zφj+i/integraldisplay
+Σφ∗T, (3.14)
+which depends on Tonly through Hperturbatively. We can add this term to our action.
+Then the chiral algebra is deformed unless Tis globally defined. Still, we can always set T
+to zero locally by subtracting a globally defined T′. Combined with flattening the metric,
+5By the Poincar´ e lemma, locally H=d(U+V) for some (2 ,0)-formUand¯∂-closed (1 ,1)-formV. By
+the¯∂-Poincar´ e lemma, locally V=¯∂Wfor some (1 ,0)-formW. ThenT=U+V−dW.
+16this turns the action locally into the free theory form. Therefore, the effect of this term
+can be treated — and is necessarily included — in the present framewo rk.
+Up to this point, we have implicitly worked in some coordinate neighborh ood of Σ.
+The sheaf /hatwideAof chiral algebras in this case (or when Σ ∼=C) is known as a sheaf of chiral
+differential operators [17, 40]. In order to reconstruct the chira l algebra globally on Σ, one
+has to glue sheaves of chiral differential operators patch by patc h over Σ. This amounts
+to gluing free βγsystems over Σ ×X. The obstruction thus takes values in
+H2(Σ×X,Ω2,cl
+Σ×X). (3.15)
+A part of it depends on both Σ and X, and corresponds to the c1(Σ)c1(X)/2 anomaly.
+Likewise, the moduli are parametrized by
+H1(Σ×X,Ω2,cl
+Σ×X). (3.16)
+3.3 Conformal anomaly
+Previously, weclaimedthatthechiralalgebralacksinvarianceunder holomorphicreparametriza-
+tions when c1(X)∝\e}atio\slash= 0, arguing that in that case perturbative corrections would annih ilate
+the classical Q-cohomology class [ Tzz] together with another class [ ∂zθ] via the relation
+[Q,Tzz] =∂zθ. However, we could not really exclude the possibility that θturns out to
+be zero and TzzremainsQ-closed. Let us check that this does not happen using the tool
+developed in this section.
+Let{Uα}be a good cover of X. On eachUα, we put a free βγsystem with energy-
+momentum tensor Tα=−βαi∂zγi
+α. The OPE
+JV(z)Tα(w)∼ −∂iVi(w)
+(z−w)3−(∂z∂iVi+Viβi)(w)
+(z−w)2−1
+2∂2
+z(∂iVi)(w)
+z−w(3.17)
+showsthatTαtransformas δTα=−∂2
+z∂iVi/2underinfinitesimaldiffeomorphisms δγ=V.
+The finite form of this transformation is [40]
+Tβ−Tα=−1
+2∂2
+zlogdet∂γβ
+∂γα. (3.18)
+Here∂γβ/∂γαis the Jacobian matrix. We define a cocycle {θαβ}by
+θαβ=−1
+2∂zlogdet∂γβ
+∂γα(3.19)
+so that it satisfies Tβ−Tα=∂zθαβ. Via the ˇCech-Qisomorphism, this equation translates
+to [Q,Tzz] =∂zθfor some local operator θ.
+The explicit form of θcan be found as follows. Write θαβ=Wβ−Wα, whereWα
+is the quantity W=∂zφi∂ilogdetg/2 evaluated in the coordinate patch Uα. SinceWα
+transform by holomorphic transition functions, ¯∂Wis globally defined. This gives θ
+as a global section of the sheaf of free βγsystems (on which Qacts by ¯∂). Noting
+Ri¯=−∂i¯∂¯logdetg, we find
+θ=−1
+2Ri¯∂zφiα¯. (3.20)
+17As a globally defined local operator of the original theory, this form ula gets higher-order
+corrections since Q=¯∂only to leading order. The formof θis in accord with the previous
+discussion.
+We introduced the perturbative Q-cohomology class [ θ] through the action of Qon
+Tzz. We can also construct it as follows, mimicking the definition of the firs t Chern class.
+Letfαβbe transition functions of K∗
+X. The cocycle condition fαβfβγfγα= 1 implies that
+(δlogf)αβγare integer multiples of 2 πi. Thus, by applying δlog/2πionfαβ, we obtain
+an element of H2(X,Z). It isc1(X). To obtain an element of the chiral algebra, we apply
+∂zlog/2 instead. This gives ∂zlogfαβ/2 =θαβ, which represents [ θ]∈H1(X,/hatwideA).
+3.4P1model
+To conclude the discussion of the sheaf theory approach, let us co mpute the perturbative
+chiral algebra of the twisted model with target space X=P1for the first few dimensions.
+We will work locally on Σ.
+The target space P1∼=C∪ {∞}is covered by two patches, U=P1\ {∞}with
+coordinate γandU′=P1\{0}with coordinate γ′, related to each other by
+γ′=1
+γ. (3.21)
+Classically βtransforms as β′=−γ2β, but this formula gets corrected quantum mechan-
+ically. The quantum transformation law turns out to be
+β′=−γ2β+2∂zγ. (3.22)
+The additional term is needed to keep the ββOPE regular under this transformation.
+Since the moduli space H1(P1,Ω2,cl
+P1) = 0, this is essentially the only way to glue the two
+freeβγsystems.
+We first look at the zeroth Q-cohomology group A0. The elements of A0are repre-
+sented by global sections of the sheaf /hatwideAof chiral algebras on P1.
+At dimension zero, relevant local operators are holomorphic funct ions. Since a holo-
+morphic function on a compact complex manifold must be constant, t he dimension zero
+subspace of A0is one-dimensional and generated by the cohomology class [1], repre sented
+by the identity operator 1.
+At dimension one, the possible local operators are those of the for mB(γ)∂zγand
+V(γ)β, whereBis a holomorphic one-form and Vis a holomorphic vector field. There
+are no global holomorphic one-forms on P1. For holomorphic vector fields, we have three
+independent ones, ∂,−γ∂, and−γ2∂. Hence, at the classical level we have three cohomol-
+ogy classes, represented by β,−γβ, and−γ2β. These survive to the perturbative chiral
+algebra. Their quantum counterparts are
+J−=β=−γ′2β′+2∂zγ′,
+J3=−γβ=γ′β′,
+J+=−γ2β+2∂zγ=β′,(3.23)
+18generating the affine Lie algebra /hatwidesl2at the critical level −2:
+J3(z)J3(w)∼ −1
+(z−w)2,
+J3(z)J±(w)∼ ±J±(w)
+(z−w)2,
+J+(z)J−(w)∼ −2
+(z−w)2+2J3(w)
+z−w.(3.24)
+The existence of these currents in the perturbative chiral algebr a is a reflection of the fact
+thatP1admits an SL 2-action.
+We now turn to the first Q-cohomology group A1. The elements of A1are represented
+by sections of /hatwideA(U∩U′) that cannot be written as the difference of a section of /hatwideA(U) and
+a section of /hatwideA(U′). Extended over the whole target space P1, such sections necessarily
+have poles at both 0 and ∞.
+Meromorphic functions with poles at 0 and ∞can always be split into a part regular
+at 0 and a part regular at ∞. Thus the dimension zero subspace of A1is zero.
+At dimension one, we can try operators of the form β/γn, which have a pole at 0.
+However, they are all regular at ∞. The other possibilities are ∂zγ/γn. Requiring they
+have a pole at ∞, we find that only ∂zγ/γcan represent a nontrivial cohomology class.
+Indeed, it does. This cohomology class is [ θ] since∂zγ/γ=−∂zlog(∂γ′/∂γ)/2.
+At dimension two, sections with poles at both 0 and ∞are linear combinations of
+∂2
+zγ/γ,∂2
+zγ/γ2, (∂zγ)2/γ, (∂zγ)2/γ2, (∂zγ)2/γ3, andβ∂zγ/γ. Among these, the combi-
+nations∂2
+zγ/γ2−2(∂zγ)2/γ3andβ∂zγ/γ+(∂zγ)2/γ3are regular at ∞, so vanish in the
+cohomology. (In verifying this assertion, one should keep in mind tha t the latter operator
+is normal ordered.) Moreover, ∂z(∂zγ/γ) also vanishes due to the perturbative relation
+[Q,Tzz] =∂zθ. Thus the dimension two subspace of A1is at most three-dimensional.
+From the cohomology classes we already have, we can construct th ree: [J−θ], [J3θ], and
+[J+θ].
+Let us summarize. The dimension zero subspace of A0is generated by [1] and the
+dimension one subspace of A1is generated by [ θ], whereas the dimension one subspace
+ofA0is generated by [ J−], [J3], [J+] and the dimension two subspace of A1is generated
+by [J−θ], [J3θ], [J+θ]. Therefore, for the first two nontrivial dimensions, we find an
+isomorphism A0∼=A1given by the map
+[O]∝mapsto−→[Oθ]. (3.25)
+It has been shown [17] that this isomorphism persists in higher dimens ions. This is as
+though [1] and [ θ] are vacua of CFT — both of them are annihilated by ∂z— and the
+elements of A0are creation operators acting on these classes to generate the r est of the
+Q-cohomology classes. It is remarkable that such a structure emer ges despite the lack of
+conformal invariance. In order to understand where this struct ure comes from, we must
+go beyond perturbation theory.
+4 Nonperturbative vanishing of chiral algebras
+In the previous sections we have seen that quantum corrections d eform the classical chiral
+algebra, but perturbatively the deformation can be understood w ithin the framework of
+19a cohomology theory on the target space. This is because in pertur bation theory, one
+considers fluctuations localized around constant maps.
+Instantons are not quite like constant maps, but have a finite size in the target space.
+Their presence may therefore lead to deformations of different kin ds. In this section, we
+will see a particularly striking example: instantons annihilate all of the perturbative Q-
+cohomology classes, making the chiral algebra trivial nonperturba tively. The existence of
+suchaphenomenon wasfirst predicted byWitten[16], andsubseque ntly confirmed byTan
+and the author [23]; see Arakawa and Malikov [41] for a mathematical interpretation of
+Witten’s prediction. Here we generalize theresults of[23, 24, 25] an destablish avanishing
+“theorem” for chiral algebras. We then explain how the vanishing of the chiral algebra
+implies the spontaneous breaking of supersymmetry and the absen ce of harmonic spinors
+on the loop space of the target space.
+4.1P1Model, with instantons
+The simplest example of a vanishing chiral algebra is provided by the P1model which we
+studied in Section 3.4. As we saw there, the perturbative chiral alge bra of the P1model
+has the structure of a Fock space: [1] and [ θ] play the role of “ground states”, on which
+infinite towers of bosonic and fermionic “excited states” are const ructed by acting with
+“creation operators”, namely bosonic Q-cohomology classes. In view of this suggestive
+structure, one may expect that instantons “tunnel” between th e “ground states” and
+“lift” them out of the chiral algebra. This is indeed the case.
+We now show that the perturbative Q-cohomology classes [1] and [ θ] are annihilated
+together by instantons via the relation
+{Q,θ} ∝1. (4.1)
+This relation says that the equation 1 = 0 holds in the Q-cohomology. Therefore the
+chiral algebra of the P1model vanishes nonperturbatively. A half of the perturbative
+Q-cohomology classes are annihilated because their representative s becomeQ-exact. For
+classes [O] in this category, we have {Q,Oθ} ∝ O. Thus [O] are annihilated together with
+[Oθ], and the latter should constitute the other half of the perturbat ive chiral algebra.
+This explains the observed Fock space structure.
+Sincec1(P1) = 2, instantons of degree k(which wrap the target space ktimes) violate
+R-charge by 2 kand dimension by k. Then the instanton corrections to the action of Q
+onθtake the form
+{Q,θ}=∞/summationdisplay
+k=1Λ2ke−kt0Ok, (4.2)
+withOkbeing local operators of R-charge 2 −2kand dimension (1 −k,0). Here,t0is
+the topological invariant SBevaluated for instantons of degree one at the dynamical scale
+Λ. There are no local operators of negative dimension, hence Ok= 0 fork >1. The
+remaining operator, O1, is perturbatively Q-closed. From the fact that for R-charge zero
+and dimension zero, [1] is the only perturbative Q-cohomology class and there are no
+perturbatively Q-exact local operators, it follows O1∝1. So if we can show {Q,θ} ∝\e}atio\slash= 0,
+we establish the relation {Q,θ} ∝1.
+To see whether {Q,θ}is zero or not, we put it on a disk (embedded in the worldsheet)
+and evaluate the path integral for suitable boundary conditions. F or our purpose, we
+20can compactify the disk to a sphere and consider those boundary c onditions that can be
+represented by vertex operators at ∞. We will therefore compute the correlation function
+/angbracketleftBig
+O(∞)/contintegraldisplay
+d¯zG(¯z)θ(0)/angbracketrightBig
+(4.3)
+on Σ = P1for some local operators O. We anticipate that {Q,θ}, expressed here as
+the contour integral of the supercurrent Garoundθ, will be replaced by 1 in the final
+result. To obtain a nonvanishing answer, then, we should take Oto be local operators of
+R-charge zero and dimension zero, i.e., functions on X.
+The contributions to this correlation function should come from inst antons of degree
+one. These are biholomorphic maps from Σ = P1toX=P1, whose moduli space
+M1∼=PGL2(C). Atφ0∈ M1, the number of ¯ψ+andψ+zero modes are, respectively,
+given by the zeroth and first Hodge numbers of the bundle K1/2
+Σ⊗φ∗
+0TX∼=OP1(1). Note
+that we have untwisted the theory before compactifying, becaus e the twisted P1model
+is anomalous on P1. Sinceh0(OP1(1)) = 2 and h1(OP1(1)) = 0, there are two ¯ψ+zero
+modes and no ψ+zero modes. These zero modes can be absorbed by the ¯ψ+fields in
+Gandθwithout bringing down interaction terms. Then, up to the ratio of th e bosonic
+and fermionic determinants, the correlation function is computed t o leading order by
+dropping quantum fluctuations and integrating over the fermion ze ro modes as well as
+the instantons:
+e−SB(φ0)/integraldisplay
+dM1dc1
+0dc2
+0O(∞)/contintegraldisplay
+d¯zG(¯z)θ(0)/vextendsingle/vextendsingle/vextendsingle
+φ=φ0. (4.4)
+HeredM1is the measure on M1, andc1
+0,c2
+0are the zero mode coefficients of the mode
+expansion of ¯ψ+. We can ignore subleading contributions.
+Of course, we must evaluate the contour integral before droppin g quantum fluctua-
+tions. (Without short distance singularities this would vanish!) It tur ns out that this
+seemingly straightforward task is actually very tricky. We are lookin g for an antiholomor-
+phic single pole 1 /¯zin the OPE
+G(¯z)θ(0) = (gφ¯φ∂¯zφ¯ψ+)(¯z)/parenleftBig
+−1
+2Rφ¯φ∂zφ¯ψ+/parenrightBig
+(0). (4.5)
+It may appear that one can obtain such a pole by contracting ∂¯zφwithRφ¯φ. However,
+this does not work because the residue is just the classical action o fQonθand vanishes.
+We must find additional antiholomorphic poles that emerge nonpertu rbatively.
+At this point, we recall that the fermionic fields take values in the pullb ack of the tan-
+gent bundle of Xby the bosonic field. Thus, the eigenmodes in which they are expande d
+depend on the bosonic field, which is itself subject to quantum fluctu ations. As a result,
+the fermion modes — even the zero modes — can produce short dista nce singularities
+when the bosonic field is present at the same location.
+We can try to extract this bosonic dependence of the fermionic field s as follows. Con-
+sider a tubular neighborhood N1ofM1, diffeomorphic to the normal bundle of M1in the
+space of maps from Σ to X. Let{xα}be local coordinates on M1and parametrize the
+normal directions by coordinates {yβ}such thatyβ= 0 onM1. Forφ(z,¯z;x,y)∈ N1,
+we denote its projection to M1byφ0(z;x). An instanton φ0∈ M1maps the points of
+Σ =P1to the points of X=P1in a one-to-one manner, so we can invert φ0(z;x) to
+21obtainz(φ0;x) and write φ(z,¯z;x,y) =φ(φ0(z;x),¯φ0(¯z;x);x,y). Computing [ iQ,¯φ] with
+this last expression, we find
+¯ψ+=∂¯φ
+∂¯φ0[iQ,¯φ0]+···=∂¯z¯φ
+∂¯z¯φ0¯ψ+0(φ0)+···, (4.6)
+where¯ψ+0(φ0) = [iQ,¯φ0] is the zero mode part of ¯ψ+evaluated at φ0. So we have
+extracted, partially, the dependence of ¯ψ+on the bosonic fluctuations.
+In fact, the leading term of the formula (4.6) is all we need. The reas on is that the
+fermion nonzero modes will be discarded in our computation and, up t o the nonzero
+modes and the equation of motion for the bosonic field, the leading te rm represents the
+zero mode part of ¯ψ+. Indeed, it correctly reduces to ¯ψ+0(φ0) atφ=φ0and the equation
+of motion implies
+Dz/parenleftBig∂¯z¯φ
+∂¯z¯φ0¯ψ+0(φ0)/parenrightBig
+=R¯φ¯φφ¯φ∂z¯φ
+∂¯z¯φ0ψ+¯ψ+¯ψ+0(φ0), (4.7)
+which vanishes if the fermion nonzero modes are dropped since ther e are noψ+zero
+modes.
+As desired, ∂¯zφfromGcan now be contracted with ∂¯z¯φin the¯ψ+field fromθto
+produce an antiholomorphic double pole. This gives
+/contintegraldisplay
+d¯zG(¯z)θ(0) =/parenleftBigi
+2Rφ¯φ∂zφ
+∂¯z¯φ0∂¯z¯ψ+¯ψ+0(φ0)/parenrightBig
+(0). (4.8)
+The result may look strange, but will become more natural after th e¯ψ+zero modes are
+integrated out.
+To proceed, we need to specify the path integral measure. On inst antons, the action
+ofQis realized as the superconformal transformation ¯ z∝mapsto→¯z+¯ǫ−(c1
+0+c2
+0¯z). Thus the zero
+mode part of ¯ψ+can be expanded as
+¯ψ+0(φ0) =c1
+0∂¯z¯φ0+c2
+0¯z∂¯z¯φ0. (4.9)
+If we choose dM1to be conformally invariant, it is Q-invariant up to terms involving dc1
+0
+ordc2
+0. Then the product dM1dc1
+0dc2
+0is aQ-invariant measure since dc1
+0dc2
+0isQ-invariant
+by itself. There is a unique conformally invariant measure on M1up to a factor. In terms
+of the points X0,X1,X∞∈Xto which 0, 1, ∞ ∈Σ are mapped by instantons, it is given
+by
+dM1=d2X0d2X1d2X∞
+|X0−X1|2|X1−X∞|2|X∞−X0|2. (4.10)
+The parametrization of M1byX0, X1,X∞provides a compactification of M1to (P1)3,
+butdM1is singular on the compactified moduli space.
+Let us return to the integral (4.4). After the integration over th e¯ψ+zero modes, the
+contour integral
+/integraldisplay
+dc1
+0dc2
+0/contintegraldisplay
+d¯zG(¯z)θ(0)/vextendsingle/vextendsingle/vextendsingle
+φ=φ0=/parenleftBigi
+2Rφ¯φ∂zφ0∂¯z¯φ0/parenrightBig
+(0). (4.11)
+This is the pullback of the Ricci form by instantons. Using the formula
+∂zφ0(0) =(X∞−X0)(X0−X1)
+X1−X∞, (4.12)
+22what is left can be written as
+i
+2e−SB(φ0)/integraldisplay
+P1d2X0Rφ¯φ(X0,X0)/integraldisplay
+(P1)2d2X1d2X∞O(X∞,X∞)
+|X1−X∞|4.(4.13)
+TheX0-integral is the evaluation of 2 πc1(X) on [X], which gives 4 π. TheX1-integral
+diverges, reflecting the noncompactness of M1. One way to regularize it is to impose a
+lower bound l >0 on the distance between X1andX∞measured with the target space
+metric. For l≪1, this restricts the domain of X1to the region where gφ¯φ(X∞,X∞)|X1−
+X∞|2≥l2. The regularized integral is
+/integraldisplay
+P1d2X1
+|X1−X∞|4=π
+l2gφ¯φ(X∞,X∞). (4.14)
+From this and the renormalization group equation (2.23) for the Bfield, we find that the
+integral (4.13) contains a factor (Λ /lµ)2e−t0. We can take a limit such that µ→ ∞and
+l→0 keepinglµfinite. The end result is
+Λ2e−t0/integraldisplay
+P1gφ¯φd2X∞O, (4.15)
+up to an overall numerical factor.
+We see that the resulting X∞-integral is performed over M0∼=Xwith respect to a
+naturalvolumeform. Therefore, theone-instantoncomputatio nofthecorrelationfunction
+(4.3) has reduced to the zero-instanton computation of the expe cted one-point function:
+/angbracketleftbig
+O(∞){Q,θ(0)}/angbracketrightbig
+∝Λ2e−t0/angbracketleftbig
+O(∞)/angbracketrightbig
+. (4.16)
+This shows {Q,θ} ∝1.
+4.2 Vanishing “theorem”
+The vanishing of the chiral algebra is not unique to the P1model. It seems to be a feature
+shared by many (0 ,2) models. In fact, we have the following vanishing “theorem”. Let
+Xbe a compact spin K¨ ahler manifold with p1(X)/2 = 0 andc1(X)>0. Suppose that
+there is an embedding P1⊂Xwith trivial normal bundle. Then, the chiral algebra of the
+(0,2) model with target space Xvanishes nonperturbatively in the absence of left-moving
+fermions.
+As before, we will establish the vanishing by demonstrating that inst antons annihilate
+the perturbative Q-cohomology classes [1] and [ θ] together. Since c1(X)>0, every
+instanton violates R-charge by 2 kand dimension by kfor some integer k >0. Then [θ]
+can only be annihilated by instantons with k= 1, in which case it is paired with a class
+of R-charge zero and dimension zero. Such a class must be proport ional to [1] on the
+compact complex manifold X. Thus, it suffices to show {Q,θ} ∝\e}atio\slash= 0.
+Consider the correlation function (4.3) again. The leading contribut ions to this func-
+tion come from instantons that give precisely two ¯ψ+zero modes and no ψ+zero modes.
+These are embeddings P1֒→Xwhose normal bundle is trivial. (The normal bundle
+NC/Xof a curveC⊂Xis a holomorphic vector bundle over Cdefined by the short exact
+23sequence 0 →TC→TX|C→NC/X→0.) To see this, note that given instantons wrap-
+ping aP1⊂Xonce, there are already the right amount of zero modes from the t angent
+direction. So there should be no additional zero modes from the nor mal directions, which
+is the case if and only if the normal bundle of the P1is trivial.
+But if the normal bundle is trivial and the fermion zero modes come on ly from the
+tangent direction, the contribution from each such P1can be computed essentially in the
+same way as in the P1model; the result is a function on Xsupported on the P1. By
+assumption, there is at least one P1that makes a contribution, hence the correlation
+function is nonzero.6It follows that {Q,θ} ∝\e}atio\slash= 0 and the chiral algebra vanishes.
+One may wonder how the existence of a single P1with trivial normal bundle, whose
+contribution to {Q,θ}is confined on the P1itself, can possibly tell something about
+the behavior of {Q,θ}in the other region of X. This point can be understood if we
+consider deformations of the P1. Infinitesimal deformations are given by holomorphic
+sections of the normal bundle. Since the normal bundle is trivial, the P1in question can
+be moved in every direction in the target space. Then the family of P1s generated by
+deformations sweep out the whole target space, and their contrib utions can add up to a
+nonzero constant.
+4.3 Flag manifold model
+An example in which the above deformation argument is beautifully dem onstrated is the
+flag manifold G/Bof a complex simple Lie group Gwith Borel subgroup B[24, 25]. For
+X=G/B, we havep1(X)/2 = 0 andc1(X) = 2(x1+···+xr) withr= rankG. Thus,
+theG/Bmodel is well defined and has the R-symmetry broken to Z2nonperturbatively.
+The simplest case is when G= SL2, for which G/B∼=P1.
+What makes the G/Bmodel interesting is that its perturbative chiral algebra contains
+currents generating the affine Lie algebra /hatwidegof critical level [17, 26, 27]. The critical level
+makes an appearance here because the Sugawara construction m ust fail; otherwise, it
+would contradict the fact that the chiral algebra lacks an energy- momentum tensor when
+c1(X)∝\e}atio\slash= 0. Nonperturbatively, these currents disappear from the chira l algebra along
+with everything else.
+Pick aP1⊂Xand call it C. Choose linearly independent normal vectors V1,...,
+Vd∈NC/X|g0at a point g0∈C. Underg0∝mapsto→gg0∈C, these are mapped to g∗V1,
+...,g∗Vd∈NC/X|gg0, which are again linearly independent. Varying g, we obtain a global
+frameofNC/X. Therefore NC/Xistrivial, andthechiralalgebravanishes. Inthisexample,
+theG-action generates a family of P1s with trivial normal bundle that covers the target
+space.
+4.4 Supersymmetry breaking and the geometry of loop spaces
+The vanishing of the chiral algebra of a (0 ,2) model has important implications for the
+dynamics of the theory and the geometry of the loop space LXof the target space:
+supersymmetry is spontaneously broken and there are no harmon ic spinors on LX. These
+conclusions are obtained by studying the Q-cohomology of states rather than operators.
+6In case the contributions from all the instantons add up to zero, w e can insert delta functions
+supported on a particular P1at various points on Σ so that the target space effectively becomes thatP1.
+The correlation function is still nonzero.
+24SinceQhas R-charge one and satisfies Q2= 0, one can consider the Q-cohomology
+graded by the R-charge in the Hilbert space of states. This is natur ally a module over
+the chiral algebra: on elements [ |Ψ∝a\}bracketri}ht] of theQ-cohomology of states, [ O]∈ Aacts by
+[O]·[|Ψ∝a\}bracketri}ht] = [O|Ψ∝a\}bracketri}ht]. (4.17)
+As a graded vector space, it is isomorphic to the space of supersym metric states, the
+kernel of {Q,Q†} ∝H−Pwhich is generally infinite-dimensional since Pis not bounded.
+Ifc1(X) = 0, it is further isomorphic to the Q-cohomology of local operators by the
+state-operator correspondence.
+To unravel the geometric meaning of the Q-cohomology of states, take Σ to be a
+cylinderS1×Rwith coordinates ( σ,τ) and regard τas time. The theory may now be
+viewed as supersymmetric quantum mechanics on LX, so let us canonically quantize it
+and see what we get. The fermionic fields are quantized (for z=σ+iτand with respect
+to a local orthonormal frame) to obey
+{ψa
++(σ,τ),¯ψ¯b
++(σ′,τ)}=δabδ(σ−σ′). (4.18)
+ThisisaloopspaceversionoftheCliffordalgebra,withthecontinuous indexσparametriz-
+ing the direction along the loop. States are thus spinors on LX. The supercharge is
+quantized as
+Q=−i/integraldisplay
+dσ¯ψ¯ı
++/parenleftBigD
+δφ¯ı−B¯ıj∂σφj−B¯ı¯∂σφ¯/parenrightBig
+, (4.19)
+whereD/δφis the covariant functional derivative on LX. From this expression, we see
+thatQis almost a half of the Dirac operator on LX. But not quite, since it has extra
+pieces coupled to ∂σ.
+We can eliminate these extra pieces without changing the Q-cohomology. To do this,
+we define a functional AB:LX→Cas follows.7First, we pick a base loop in each
+connected component of LX. Then, given φ∈ LX, wechoose ahomotopy /hatwideφ: [0,1]×S1→
+Xfrom the base loop of the relevant component to φ. Finally, we set
+AB(φ) =/integraldisplay
+[0,1]×S1/hatwideφ∗B. (4.20)
+Under a variation of the end point φ→φ+δφ, this functional changes by
+δAB=/integraldisplay
+dσ/parenleftbig
+δφi(Bij∂σφj+Bi¯∂σφ¯)−δφ¯ı(B¯ıj∂σφj+B¯ı¯∂σφ¯)/parenrightbig
+.(4.21)
+WritingQ0for a half of the Dirac operator obtained by dropping the extra piec es from
+Q, we have
+Q=e−ABQ0eAB. (4.22)
+Therefore, the Q-cohomology of states is isomorphic to the Q0-cohomology, which is the
+cohomology of the spinor bundle over LX.
+7This functional is not single valued if the cohomology class [ B] does not vanish on some two-cycles.
+However, one can always go to a covering space of LXin which it becomes single valued, and define the
+theory in this space. See [42] for more discussion on this point.
+25Now suppose that the chiral algebra is trivial: [1] = 0. Then
+[|Ψ∝a\}bracketri}ht] = [1]·[|Ψ∝a\}bracketri}ht] = 0 (4.23)
+for anyQ-cohomology classes [ |Ψ∝a\}bracketri}ht], so theQ-cohomology of states is also trivial. Since
+supersymmetric states are in one-to-one correspondence with t heQ-cohomology classes,
+there are none. In other words, supersymmetry is spontaneous ly broken. Furthermore,
+we just saw that the Q-cohomology is nothing but the cohomology of the spinor bundle
+overLX. Hence, there are no harmonic spinors on LXeither.
+Acknoweldgments
+This work is based on the author’s Ph.D. thesis. First and foremost, I would like to
+thank the Department of Physics and Astronomy at Rutgers Unive rsity for providing a
+stimulating environment during my graduate studies. I am also grate ful to the Center for
+Frontier Science at Chiba University for their support while this pape r was written. I am
+indebted to my advisor Gregory Moore, whose deep understanding and original thinking
+have always been a great inspiration to me. Last but not least, my gr atitude goes to
+Meng-Chwan Tan for being such a wonderful colleague and friend.
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+28
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+arXiv:1001.0119v1  [math.AG]  2 Jan 2010TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES OF
+ALMOST-COMPLEX FOURFOLDS (II)
+JULIEN GRIVAUX
+Abstract. In this article, we study the rational cohomology rings of Vo isin’s punctual Hilbert
+schemes X[n]associated to a symplectic compact fourfold X. We prove that these rings can
+be universally constructed from H∗(X,Q) andc1(X), and that Ruan’s crepant resolution con-
+jecture holds if c1(X) is a torsion class. Next, we prove that for any almost-compl ex compact
+fourfold X, the complex cobordism class of X[n]depends only on the cobordism class of X.
+Contents
+1. Introduction 2
+2. The almost-complex Hilbert scheme 3
+2.1. Definitions and basic properties 3
+2.2. Incidence varieties and Nakajima operators 4
+3. The boundary operator 5
+3.1. Lehn’s formula in the almost-complex case 5
+3.2. Holomorphic curves in symplectic fourfolds 7
+3.3. Computation of the excess term in the symplectic case 8
+4. The ring structure of H∗/parenleftbig
+X[n],Q/parenrightbig
+9
+4.1. Geometric tautological Chern characters 9
+4.2. Virtual tautological Chern characters 10
+4.3. The ring structure and the crepant resolution conjectu re 11
+5. The cobordism class of X[n]12
+5.1. The relative incidence sheaf 14
+5.2. Computation of TX[n]inK-theory 17
+5.3. Comparison of TX[n]andTX[n+1]via the incidence variety X[n+1,n]18
+5.4. Cohomological computations 20
+6. Appendix 23
+6.1. Relative analytic spaces 23
+6.2. Relatively coherent sheaves 24
+6.3. Analytic K-theory for relatively coherent sheaves 25
+6.4. Topological classes 29
+References 312 JULIEN GRIVAUX
+1.Introduction
+The punctual Hilbert schemes of a smooth quasi-projective s urface have been intensively
+studied in the past twenty years, starting with the works of G ¨ ottsche, Nakajima, Grojnowski,
+Lehn and others (see e.g. [G¨ o 1]). These Hilbert schemes pre sent a strong geometric interest
+because, among many other properties, they are smooth crepa nt resolutions of the symmetric
+products of the surfaces. If Xis aK3 surface,X[n]are hyperk¨ ahler varieties by a result
+of Beauville [Be] and Ruan’s conjecture predicts that for al l positive integer n,H∗/parenleftbig
+X[n],C/parenrightbig
+is isomorphic as a ring to the orbifold cohomology ring of SnXdefined by Chen and Ruan
+[Ch-Ru], [Ad-Le-Ru]. This is obtained by putting together r esults of Lehn-Sorger on the Hilbert
+scheme side [Le-So-1] with the computations done in [Fa-G¨ o ] and [Ur] on the orbifold side.
+The isomorphisms between H∗/parenleftbig
+X[n],C/parenrightbig
+andH∗
+CR/parenleftbig
+SnX,C/parenrightbig
+made it clear that the cohomology
+rings of punctual Hilbert schemes of a K3 surface depend only on the deformation class of the
+complex structure on Xin the space of almost-complex ones, since Chen-Ruan cohomo logy is a
+purely almost-complex theory. This is more generally the ca se for arbitrary projective surfaces:
+the description of the cohomology ring of Hilbert schemes do ne in [Le] and [Li-Qi-Wa-2] shows
+thatH∗/parenleftbig
+X[n],Q/parenrightbig
+depends only on the ring H∗(X,Q) and on the first Chern class of Xin
+H2(X,Q). In the same spirit, it is proved in [El-G¨ o-Le] that the com plex cobordism class of
+X[n]depends only on the complex cobordism class of X. These facts received an explanation
+by a recent construction of Voisin [Vo 1]: for any almost-com plex compact fourfold Xand any
+positive integer n, it is possible to construct a punctual Hilbert scheme X[n]which is a stable
+almost-complex differentiable manifold of real dimension 4 n. Besides,X[n]is symplectic if Xis
+symplectic [Vo 2].
+In Nakajima’s theory [Na1], the correspondence action of th e incidence schemes on the coho-
+mology groups of Hilbert schemes is the main ingredient to un derstand their additive structure
+via representation theory. This approach has been carried o ut for the almost-complex case in
+[Gri1]. Our first aim in this paper is now to study the multipli cative structure of the rational
+cohomology rings of almost-complex Hilbert schemes, gener alizing the work of Lehn [Le] and
+Li-Qin-Wang [Li-Qi-Wa-2]. We obtain a satisfactory result in the symplectic case:
+Theorem 1.1. If(X,ω)is a symplectic compact four-manifold, the rings H∗/parenleftbig
+X[n],Q/parenrightbig
+can be
+constructed by universal formulae from the ring H∗(X,Q)and the first Chern class of Xin
+H2(X,Q).
+WhenXis projective, Lehn uses in an essential way algebraic curve s onXvia their sym-
+metric products imbedded in the Hilbert schemes to determin e some excess intersection terms
+arising in correspondences computations. This argument fa ils a priori as soon as Xis not alge-
+braic, since there is no pseudo-holomorphic curve on Xfor a generic almost-complex structure
+on it. However, Donaldson’s theorem on symplectic submanif olds [Do] allows to produce many
+paseudo-holomorphic curves for arbitrary small perturbat ions of a fixed almost-complex struc-
+ture. As a corollary of an effective version of Theorem 1.1, we o btain the cohomological crepant
+resolution conjecture for symplectic fourfolds with torsi on first Chern class:
+Theorem 1.2. Let(X,ω)be a symplectic four-manifold with zero first Chern class in H2(X,Q).
+Then for any positive integer n,H∗/parenleftbig
+X[n],C/parenrightbig
+is isomorphic as a ring to H∗
+CR/parenleftbig
+SnX,C/parenrightbig
+.
+ThisresultprovidesmanyexampleswhereRuan’sconjecture holdsinthesymplecticcategory.TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 3
+The second part of this article deals with the cobordism clas ses of almost-complex Hilbert
+schemes. The main result is the following:
+Theorem 1.3. Let(X,J)be an almost-complex compact four-manifold. For any positi ve integer
+n, the complex cobordism class of X[n]given by its stable almost-complex structure depends only
+on the complex cobordism class of X.
+The proof of [El-G¨ o-Le] for projective surfaces is rather d elicate to adapt here because coher-
+ent sheaves, contrary to algebraic cycles or vector bundles , have no equivalent in almost-complex
+geometry. However, it turns out that the classical proof can be carried out in a relative context
+on spaces fibered in smooth analytic sets over a singular differ entiable basis.
+Acknowledgement. I want to thank my advisor Claire Voisin whose work on the almo st-
+complex punctual Hilbert scheme is at the origin of this arti cle. Her deep knowledge of the
+subject and her numerous advices have been most valuable to m e. I also wish to thank her for
+her kindness and her patience.
+2.The almost-complex Hilbert scheme
+In this section, we recall definitions and fundamental prope rties of almost-complex Hilbert
+schemes for the reader’s convenience. More details can be fo und in [Gri1], [Vo 1] and [Vo 2].
+2.1.Definitions and basic properties. Let (X,J) be an almost-complex compact fourfold.
+For alln∈N∗, we introduce the incidence set
+Zn=/braceleftbig
+(x,p) inSnX×Xsuch thatp∈X/bracerightbig
+·
+Let us fix a Riemannian metric gonX, andεsufficiently small.
+Definition 2.1. LetBbe the set of pairs ( W,Jrel) such that
+(i)Wis a neighbourhood of ZninSnX×X.
+(ii)Jrelis a relative integrable structure on the fibers of pr1:W /d47/d47SnXdepending
+smoothly on the parameter xinSnX.
+(iii) sup
+x∈SnX, p∈Wx/vextendsingle/vextendsingle/vextendsingle/vextendsingleJrel
+x(p)−Jx(p)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+g<ε.
+Forεsmall enough,Bεis nonempty and weakly contractible.
+Definition 2.2. Let (W,Jrel) be a relative integrable structure in B. Thetopological Hilbert
+schemeX[n]
+Jrelis the subset of W[n]
+reldefined by X[n]
+Jrel=/braceleftbig
+(x,ξ) inW[n]
+relsuch thatHC(ξ) =x/bracerightbig
+,
+whereHC:W[n]
+x/d47/d47SnWxis the usual Hilbert-Chow morphism.
+To obtain a differentiable structure on X[n]
+Jrel, Voisin uses relative integrable structures in a
+contractible subset B′ofBsatisfying some additional geometric conditions. Her main result is
+the following:
+Theorem 2.3. [Vo 1],[Vo 2]LetJrelbe a relative integrable structure in B′. Then
+(i)X[n]
+Jrelhas a natural differentiable structure. Furthermore, if J′relis another relative
+integrable structure in B′,X[n]
+JrelandX[n]
+J′relare canonically isomorphic modulo diffeo-
+morphisms isotopic to the identity.4 JULIEN GRIVAUX
+(ii)There is a canonical Hilbert-Chow map HC:X[n]
+Jrel/d47/d47SnXwhose fibers are homeomor-
+phic to the fibers of the usual Hilbert-Chow morphism for proje ctive surfaces.
+(iii)X[n]
+Jrelcan be endowed with a stable almost-complex structure.
+(iv)IfXis symplectic and Jrelis compatible with the symplectic structure, X[n]
+Jrelis also
+symplectic.
+For arbitrary relative integrable structures, this theore m has the following topological form:
+Theorem 2.4. [Gri1]
+(i)LetJrelbe a relative integrable structure in B. ThenX[n]
+Jrelis a topological manifold.
+(ii)If/braceleftbig
+Jrel
+t/bracerightbig
+t∈B(0,r)⊆Rdis a smooth path in B, then the associated relative topological Hilbert
+scheme/parenleftbig
+X[n],/braceleftbig
+Jrel
+t/bracerightbig
+t∈B(0,r)/parenrightbig
+overB(0,r)is a topological fibration.
+(iii)For anyxinSnXand any integrable structure in a neighbourhood Uxofxsufficiently
+close toJ, the Hilbert-Chow morphism HC:X[n]/d47/d47SnXis locally isomorphic over a
+neighbourhood of xto the classical Hilbert-Chow morphism HC:U[n]
+x/d47/d47SnUx.
+Point (ii) implies that for any couple/parenleftbig
+Jrel,J′rel/parenrightbig
+of relative integrable structures in B,X[n]
+Jrel
+andX[n]
+J′relare homeomorphic and the isotopy class of this homeomorphis m is canonical. There-
+fore, there exist canonical rings H∗/parenleftbig
+X[n],Q/parenrightbig
+andK/parenleftbig
+X[n]/parenrightbig
+such that for every relative integrable
+structureJrelinB,H∗/parenleftbig
+X[n],Q/parenrightbig
+is canonically isomorphic to H∗/parenleftbig
+X[n]
+Jrel,Q/parenrightbig
+andK/parenleftbig
+X[n]/parenrightbig
+to
+K/parenleftbig
+X[n]
+Jrel/parenrightbig
+.
+Point (iii) implies that G¨ ottsche’s classical formula for the Betti numbers of punctual Hilbert
+schemes also holds in the almost-complex case (see [Gri2]).
+2.2.Incidence varieties and Nakajima operators. Ifn,m∈N∗, let
+Zn×m=/braceleftbig
+(x, y, p) inSnX×SnX×Xsuch thatp∈x∪y/bracerightbig
+·
+Relative integrable structures in a neighbourhood of Zn×mwill be denoted by Jrel
+n×m.
+Definition 2.5.
+(i) If/parenleftbig
+W1,J1,rel
+n×m/parenrightbig
+and/parenleftbig
+W2,J2,rel
+n×m/parenrightbig
+are two relative integrable structures in neighbourhoods
+ofZn×m, theproduct Hilbert scheme/parenleftbig
+X[n]×[m],J1,rel
+n×m,J2,rel
+n×m/parenrightbig
+is defined by
+/parenleftbig
+X[n]×[m],J1,rel
+n×m,J2,rel
+n×m/parenrightbig
+=/braceleftbig
+(ξ, ξ′, x, y) inW[n]
+1,rel×SnX×SmXW[m]
+2,rel
+such thatHC(ξ) =xandHC(ξ′) =y/bracerightbig
+·
+(ii) Ifm > nand/parenleftbig/tildewiderW,Jrel
+n×(m−n)/parenrightbig
+is a relative integrable structure in a neighbourhood of
+Zn×(m−n), theincidence variety is defined by
+/parenleftbig
+X[m,n],Jrel
+n×m/parenrightbig
+=/braceleftbig
+(ξ, ξ′,x, y) in/tildewiderW[n]
+rel×SnX×SmX/tildewiderW[m]
+rel
+such thatξ⊂ξ′, HC(ξ) =xandHC(ξ′) =x∪y/bracerightbig
+·TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 5
+As it is thecase fortopological Hilbert schemes, theproduc tHilbert scheme andtheincidence
+varieties are uniquely defined up to homeomorphisms isotopi c to the identity.
+Let/parenleftbig
+W,Jrel
+n×m/parenrightbig
+,/parenleftbig/tildewiderW,Jrel
+n×(m−n)/parenrightbig
+,/parenleftbig
+W′,Jrel
+n/parenrightbig
+and/parenleftbig
+W′′,Jrel
+m/parenrightbig
+be relative integrable structures in
+neighbourhoodsof Zn×m,Zn×(m−n),ZnandZm. We introducethe following set of compatibility
+conditions of relative analytic spaces (see Section 6.1):
+(1)W×SnX×SmX(SnX×Sm−nX)⊆/tildewiderW, where the base change map is ( x,y)/d31/d47/d47(x,x∪y).
+(2)W′×SnX(SnX×Sm−nX)⊆/tildewiderW, where the base change map is the first projection.
+(3)W′′×SmX(SnX×Sm−nX)⊆/tildewiderW, where the base change map is ( x,y)/d31/d47/d47x∪y.
+– If (1) holds, there is a natural embedding of/parenleftbig
+X[m,n],Jrel
+n×(m−n)/parenrightbig
+into/parenleftbig
+X[n]×[m],Jrel
+n×m,Jrel
+n×m/parenrightbig
+.
+– If (2) holds, there is a natural morphism λfrom/parenleftbig
+X[m,n],Jrel
+n×(m−n)/parenrightbig
+to/parenleftbig
+X[n],Jrel
+n/parenrightbig
+.
+– If (3) holds, there is a canonical morphism νfrom/parenleftbig
+X[m,n],Jrel
+n×(m−n)/parenrightbig
+to/parenleftbig
+X[m],Jrel
+m/parenrightbig
+.
+Unfortumately, conditions (2) and (3) cannot hold at the sam e time. However, since X[n]×[m]is
+canonically homeomorphic up to isotopy to X[n]×X[m], one still has morphisms from X[m,n]to
+X[n]andX[m]whose homotopy classes are canonical. They will still be den oted byλandν.
+The incidence varieties X[m,n]are locally homeomorphic to the integrable model U[m,n]where
+Uis an open set of C2. This allows to put a stratification on X[m,n]. In this way, X[m,n]is a
+stratified topological space locally modelled over an analy tic space, so it has a fundamental
+homology class.
+The construction by Nakajima and Grojnowski of representat ions of the Heisenberg super-
+algebra ofH∗(X,Q) intoH:=⊕n∈NH∗/parenleftbig
+X[n],Q/parenrightbig
+via correspondence actions of incidence vari-
+eties done in [Na1] and [Gr] also holds in the almost-complex setting:
+Theorem 2.6. [Gri1]If(X,J)is an almost-complex compact fourfold, Nakajima operators
+qi(α),i∈Z,α∈H∗(X,Q), can be constructed. They depend only on the deformation clas s of
+Jand satisfy the Heisenberg commutation relations :
+∀i,j∈Z,∀α,β∈H∗(X,Q),/bracketleftbig
+qi(α),qj(β)/bracketrightbig
+=iδi+j,0/parenleftBig/integraldisplay
+Xαβ/parenrightBig
+idH.
+Furthermore, these operators induce an irreducible repres entation ofH/parenleftbig
+H∗(X,Q)/parenrightbig
+inHwith
+highest weight vector 1.
+3.The boundary operator
+The aim of this section is to carry out for symplectic fourfor lds Lehn’s computation of the
+boundaryoperator donein [Le]. Thefirstpart of Lehn’s argum ent can beadapted to thealmost-
+complex case as in the proof of Nakajima relations done in [Gr i1]. This is explained in Section
+3.1. The result of Section 3.2, based on Donaldson’s symplec tic Kodaira’s theorem allows to
+carry out in section 3.3 the second part of Lehn’s argument wh enXis symplectic.
+3.1.Lehn’s formula in the almost-complex case. Let (X,J) be an almost-complex com-
+pact fourfold and let Tbe the trivial complex line bundle on it. If Jrelis a relative structure
+in a neighbourhood of Zn, there is a relative tautological bundle T[n]
+relonW[n]
+relwhose restriction6 JULIEN GRIVAUX
+to the fibers/parenleftbig
+W[n]
+x,Jrel
+x/parenrightbig
+are the usual tautological bundles/parenleftbig
+O
+W[n]
+x/parenrightbig[n]. IfT[n]=T[n]
+rel|X[n],T[n]
+satisfies the following properties (see [Gri1]):
+(i) The class of T[n]inK/parenleftbig
+X[n]/parenrightbig
+is independent of Jrel.
+(ii)−2c1/parenleftbig
+T[n]/parenrightbig
+is Poincar´ e dual with Q-coefficients to/bracketleftbig
+∂X[n]/bracketrightbig
+, where
+∂X[n]=/braceleftbig
+(ξ,x) inX[n]such that there exists pinxwith lengthp(ξ)≥2/bracerightbig
+is the so-called boundary of X[n].
+(iii) Ifλ:X[n+1,n]/d47/d47X[n]andν:X[n+1,n]/d47/d47X[n+1]are the usual homotopy classes, Dis
+the exceptional divisor in X[n+1,n]andFis the complex line bundle on X[n+1,n]whose
+first Chern class is Poincare dual to −[D], thenλ∗T[n+1]−ν∗T[n]=FinK/parenleftbig
+X[n+1,n]/parenrightbig
+.
+We recall Lehn’s definition of the boundary operator:
+Definition 3.1. LetH=/circleplustext
+n≥0H∗/parenleftbig
+X[n],Q/parenrightbig
+.
+(i) Theboundary operator d:H /d47/d47His defined by d/bracketleftbig
+(αn)n≥0/bracketrightbig
+=/parenleftbig
+c1/parenleftbig
+T[n]/parenrightbig
+∪αn/parenrightbig
+n≥0.
+(ii) IfAis an endomorphism of H,the derivative A′ofAis defined by the formula
+A′= [d,A] =d◦A−A◦d.
+We now state the following result:
+Theorem 3.2. Let(X,J)be an almost-complex compact fourfold. There exist classes/parenleftbig
+en/parenrightbig
+n≥0
+inH2(X,Q)such that
+∀n,m∈Z,∀α,β∈H∗(X,Q),/bracketleftbig
+q′
+n(α),qm(β)/bracketrightbig
+=−nmqn+m(αβ)+δn+m,0/parenleftBig/integraldisplay
+Xe|n|αβ/parenrightBig
+idH.
+Proof.Letn, mbepositive integers such that m≥n,Jrel
+n×(m−n)a relative integrable structure in
+a neighbourhood of Zn×(m−n), and letI[m,n]
+T,relbe the complex bundle on W[m,n]
+relwhose restriction
+to the fibers W[m,n]
+x,yare the kernels of the natural surjective morphisms from/parenleftbig
+OW[m,n]
+x,y/parenrightbig[m]to
+/parenleftbig
+OW[m,n]
+x,y/parenrightbig[n]. We putI[m,n]
+T=I[m,n]
+T,rel|X[m,n]. ThenI[m,n]
+T=ν∗T[n+1]−λ∗T[n]inK/parenleftbig
+X[m,n]/parenrightbig
+, the
+proof being similar to [Gri1, Proposition 3.5]. This allows to prove that correspondences behave
+well under derivations:
+Lemma 3.3. Letube a class in H∗/parenleftbig
+X[m,n],Q/parenrightbig
+andu∗:H∗/parenleftbig
+X[n],Q/parenrightbig/d47/d47H∗/parenleftbig
+X[m],Q/parenrightbig
+be the
+associated correspondence map. Then/parenleftbig
+u∗/parenrightbig′=/bracketleftbig
+u∩c1/parenleftbig
+I[m,n]
+T/parenrightbig/bracketrightbig
+∗.
+Proof.Letλ:X[m,n]/d47/d47X[n]andν:X[m,n]/d47/d47X[m]be the usual maps. The identity
+ν∗T[m]−λ∗T[n]=I[m,n]
+TTOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 7
+is satisfied in K/parenleftbig
+X[m,n]/parenrightbig
+. Thus
+/parenleftbig
+u∗/parenrightbig′τ=c1/parenleftbig
+T[m]/parenrightbig
+∪u∗τ−u∗/parenleftbig
+c1/parenleftbig
+T[n]/parenrightbig
+∪τ/parenrightbig
+=PD−1/bracketleftBig/parenleftbig
+ν∗(u∩λ∗τ)/parenrightbig
+∩c1/parenleftbig
+T[m]/parenrightbig
+−ν∗/parenleftbig
+u∩λ∗/parenleftbig
+c1/parenleftbig
+T[n]/parenrightbig
+∪τ/parenrightbig/parenrightbig/bracketrightBig
+=PD−1/bracketleftBig
+ν∗/parenleftBig
+u∩/bracketleftbig/parenleftbig
+ν∗c1/parenleftbig
+T[m]/parenrightbig
+−λ∗c1/parenleftbig
+T[n]/parenrightbig/parenrightbig
+∪λ∗τ/bracketrightbig/parenrightBig/bracketrightBig
+=PD−1ν∗/parenleftBig/bracketleftbig
+u∩c1/parenleftbig
+I[m,n]
+T/parenrightbig/bracketrightbig
+∩λ∗τ/parenrightBig
+.
+/square
+By this lemma, to prove the theorem it suffices to compute the co mmutator of two corres-
+pondences. We adapt Lehn’s proof exactly as in [Gri1] for the Nakajima relations. This yields
+(see [Gri2] for a detailed exposition):
+∗For allm,ninZwithm/ne}ationslash=n,/bracketleftbig
+q′
+n(α),qm(β)/bracketrightbig
+=µn,mqn+m(αβ), whereµn,m∈Z.
+∗For allninZ,/bracketleftbig
+q′
+n(α),q−n(β)/bracketrightbig
+=/parenleftbig/integraltext
+Xe|n|αβ/parenrightbig
+idH, wheretheclasses e|n|belongtoH2(X,Q).
+The terms µn,mandenarethe excess contributions . The multiplicity µn,mcan computed
+locally onX, so that Lehn’s proof applies and gives µn,m=−nm. However, this is not the case
+for the class enwhich involves the globalgeometry of X. /square
+Corollary 3.4. Whenαruns through a basis of H∗(X,Q), the operators dandq1(α)generate
+Hfrom the vector 1.
+Proof.This is a consequence of the relations/bracketleftbig
+q′
+1(α),qm(1)/bracketrightbig
+=−mqm+1(α),m∈N. /square
+3.2.Holomorphic curves in symplectic fourfolds. Until now, we have only considered
+integrable structures in small open sets of ( X,J). To compute the excess classes en, we will
+construct pseudo-holomorphic curves in Xfor perturbed almost-complex structures. To do so
+we use the following theorem of Donaldson [Do], which is a sym plectic version of Kodaira’s
+imbedding theorem:
+Theorem 3.5. [Do]Let(V,ω)be a symplectic manifold of dimension 2nsuch thatωis an
+integral class. If his a lift ofωinH2(V,Z), then, fork≫0, the classes PD(kh)inH2n−2(V,Z)
+can be realized by homology classes of symplectic submanifo lds. More precisely, if Jis an almost-
+complex structure on Vcompatible with ω, it is possible to write PD(kh) =/bracketleftbig
+Sk/bracketrightbig
+forklarge
+enough, where Skis aJk-holomorphic codimension 2submanifold in Vand||Jk−J||C0≤C/√
+k.
+We apply this theorem to our situation:
+Corollary 3.6. Let(X,ω)be a symplectic compact fourfold and Jan adapted almost-complex
+structure on X. Then there exist almost-complex structures/parenleftbig
+Ji/parenrightbig
+1≤i≤Narbitrary close to Jand
+Ji-holomorphic curves/parenleftbig
+Ci/parenrightbig
+1≤i≤Nsuch that :
+(i)For alli,Jiis integrable in a neighbourhood of Ci.
+(ii)The classes/bracketleftbig
+Ci/bracketrightbig
+spanH2(X,Q).
+Proof.We pickα1,...,αNinH2(X,R) such that the ω+αi’s are symplectic forms in H2(X,Q)
+which generate H2(X,Q). There exist almost-complex structures/parenleftbig/tildewideJi/parenrightbig
+1≤i≤Nadapted to ( ω+
+αi)1≤i≤Narbitrary close to Jif theαi’s are small enough. For suitable sufficiently large values8 JULIEN GRIVAUX
+ofm, the symplectic forms m(ω+αi) are integral and Donaldson’s theorem applies. We obtain
+Ji-holomorphic curves ( Ci)1≤i≤NwhereJiis arbitrary close to /tildewideJiand/bracketleftbig
+Ci/bracketrightbig
+=PD/parenleftbig
+ki(ω+αi)/parenrightbig
+in
+H2(X,Q). LetUibe a small neighbourhood of the zero section of NCi/X. We identify Uiwith
+a neighbourhood of Ciin X. Since dim Ci= 2,Ji|Ciis integrable. Since CiisJi-holomorphic,
+NCi/Xis naturally a complex vector bundle over the complex curve Ci, so that we can put a
+holomorphic structure on it. This gives an integrable struc tureJ′
+ionUisuch thatJ′
+i|Ci=Ji|Ci.
+We glue together JiandJ′
+iin an annulus around Ci. The resulting almost-complex structure
+can be chosen arbitrary close to JiifUiis small enough. /square
+3.3.Computation of the excess term in the symplectic case. Our aim in this section is
+to prove the following theorem:
+Theorem 3.7. If(X,ω)is a symplectic compact fourfold and Jis any almost-complex structure
+compatible with ω, the excess contributions enof Theorem 3.2 are given by
+∀n∈Z, en=1
+2n2(|n|−1)c1(X).
+This means that for all n,minZand for all α,βinH2(X,Q),
+/bracketleftbig
+q′
+n(α),qm(β)/bracketrightbig
+=−nm/braceleftBig
+qn+m(αβ)−|n|−1
+2δn+m,0/parenleftBig/integraldisplay
+Xc1(X)αβ/parenrightBig
+idH/bracerightBig
+·
+Proof.IfXis a smooth projective surface, this theorem is due to Lehn ([ Le], Th. 3.10). First
+we sketch his argument, then we adapt it in the symplectic cas e.
+In this proof the notation [ Z] will be used to denote the cohomological cycle class of a cyc le
+Z, and its the homology class as it was previously the case.
+IfXis a smooth projective surface and Cis a smooth algebraic curve on X, results of
+Grojnowski and Nakajima ([Gr], [Na2]) describe explicitel y the class/bracketleftbig
+C[n]/bracketrightbig
+inH2n/parenleftbig
+X[n],Q/parenrightbig
+in
+terms of the classes q
+i1/parenleftbig
+[C]/parenrightbig
+...q
+iN/parenleftbig
+[C]/parenrightbig
+.1, wherei1,...,iNare positive integers of total sum n.
+LetX[n]
+0be the set of schemes in X[n]whose support is a single point. If ∂C[n]=C[n]∩∂X[n],
+the termI=/integraldisplay
+X[n]/bracketleftbig
+X[n]
+0/bracketrightbig
+./bracketleftbig
+∂C[n]/bracketrightbig
+can be computed in two different ways:
+(i) The integral Iis equal to q−n(1)/parenleftbig/bracketleftbig
+∂C[n]/bracketrightbig/parenrightbig
+. SinceC[n]and∂X[n]intersect generically
+transversally,/bracketleftbig
+∂C[n]/bracketrightbig
+=/bracketleftbig
+∂X[n]/bracketrightbig
+./bracketleftbig
+C[n]/bracketrightbig
+=−2c1(Tn)./bracketleftbig
+C[n]/bracketrightbig
+=−2d/bracketleftbig
+C[n]/bracketrightbig
+. Therefore, Iis a li-
+nearcombination oftermsoftheform q−n(1)qi1/parenleftbig
+[C]/parenrightbig
+...q′
+ik/parenleftbig
+[C]/parenrightbig
+...qiN/parenleftbig
+[C]/parenrightbig
+.1, wherei1,...,iN
+are positive integers of total sum n. These terms vanish except for:
+–N= 1,i1=n. Thenq−n(1)q′
+n([C]).1 =−/integraldisplay
+Xen.[C].
+–N= 2,i1+i2=n. Thenq−n(1)qk([C])q′
+n−k([C]).1 = 0 and
+q−n(1)q′
+k([C])qn−k([C]).1 =−nkqk−n([C])qn−k([C]).1 =nk(n−k)[C]2.
+This computation gives I=1
+n/integraldisplay
+Xen.[C]+/parenleftbiggn
+2/parenrightbigg
+[C]2.TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 9
+(ii) The cycle C[n]intersect transversally X[n]
+0in its smooth locus and C[n]∩X[n]
+0=C[n]
+0≃C.
+ThereforeI=/integraltext
+X[n]/bracketleftbig
+X[n]
+0/bracketrightbig
+./bracketleftbig
+C[n]/bracketrightbig
+./bracketleftbig
+∂X[n]/bracketrightbig
+= degC/bracketleftbig
+O
+X[n]/parenleftbig
+∂X[n]/parenrightbig/bracketrightbig
+= degC/bracketleftbig
+O
+C[n]/parenleftbig
+∂C[n]/parenrightbig/bracketrightbig
+, which
+is−n(n−1) degCKXby direct computation.
+In the algebraic case, the excess terms enlie in the Neron-Severi group of Xso that it is
+enough to prove that for every smooth algebraic curve C,/integraldisplay
+X/bracketleftBig
+en−1
+2n2(n−1)c1(X)/bracketrightBig
+.[C] = 0.
+This is proved by comparison of the two expressions obtained forI.
+Let us now suppose that ( X,ω) is a symplectic compact fourfold and that Jis an adapted
+almost-complex structure on X. Ifγ∈H∗(X) is a class of even degree, we define the vertex
+operators/parenleftbig
+Sm(γ)/parenrightbig
+m≥0by the formula/summationtext
+m≥0Sm(γ)tm= exp/parenleftBig/summationtext
+n>0(−1)n−1
+nqn(γ)tn/parenrightBig
+. Sinceγ
+is of even degree, the operators/parenleftbig
+qi(γ)/parenrightbig
+i>0commute in the usual sense, and the definition of
+Sm(γ) makes sense.
+Lemma 3.8. ([Gr], [Na2] in the integrable case). Let/tildewideJbe an almost-complex structure close to
+J, and letCbe a/tildewideJ-holomorphic curve. Suppose that /tildewideJis integrable in a neighbourhood of C.
+Then/bracketleftbig
+C[n]/bracketrightbig
+=Sn/parenleftbig
+[C]/parenrightbig
+.1.
+Proof.Since the Nakajima operators are invariant by deformation o f the almost-complex struc-
+ture, we can make the computations with almost-complex stru ctures equal to /tildewideJwhen all the
+points are near C. LetUbea small neighbourhoodof Csuchthat /tildewideJis integrable in U. Then, for
+any positive integers nandmsuch thatm>n, theHilbert schemes X[m],X[n]and the incidence
+varietyX[m,n]are the usual integrable ones over SmU,SnUandSnU×Sn−mUrespectively.
+Since Lemma 3.8 holds in H2n
+c/parenleftbig
+U[n],Q/parenrightbig
+, we are done. /square
+If (C,/tildewideJ) satisfies the hypotheses of Lemma 3.8, Lehn’s computations recalled above apply
+verbatim and give/integraldisplay
+X/bracketleftBig
+en−1
+2n2(|n|−1)c1(X)/bracketrightBig
+.[C] = 0. By Corollary 3.6, H2(X,Q) is spanned
+by cohomology classes of such holomorphic curves. This give s the result. /square
+Thederivativeof theNakajimaoperatorscan beexplicitely computedintermsof theVirasoro
+operators Ln(α) defined in [Le, Section 3.1]:
+Corollary 3.9. If(X,ω)is a symplectic compact fourfold and Jis a compatible almost-complex
+structure, then for all ninZ,q′
+n(α) =nLn(α)−1
+2n(|n|−1)qn/parenleftbig
+c1(X)α/parenrightbig
+.
+For the proof, see [Le, p. 180].
+4.The ring structure of H∗/parenleftbig
+X[n],Q/parenrightbig
+4.1.Geometric tautological Chern characters. Let (X,J) be an almost-complex compact
+fourfold,λ:X[n+1]/d47/d47X[n], ν:X[n+1,n]/d47/d47X[n+1], ρ:X[n+1,n]/d47/d47Xthe three associated
+maps, which are canonical up to homotopy, and D⊆X[n+1,n]the exceptional divisor.
+IfEisa complex vector bundleon X, it is possibletoassociate to Easequenceof tautological
+vector bundles/parenleftbig
+E[n]/parenrightbig
+n>0onX[n]. These tautological bundles are constructed in [Gri1] usin g
+relative holomorphic structures on E, and their classes in K-theory are shown to be independent
+of these auxiliary structures. This defines tautological mo rphisms from K(X) toK/parenleftbig
+X[n]/parenrightbig
+.10 JULIEN GRIVAUX
+LetFbe the complex line bundle on X[n+1,n]such thatc1(F) is Poincar´ e dual to −[D].
+Then (see [Gri1, 3.2]), ν∗E[n+1]=λ∗E[n]+ρ∗E⊗F inK/parenleftbig
+X[n+1,n]/parenrightbig
+. This gives the relation
+ν∗/parenleftbig
+ch/parenleftbig
+E[n+1]/parenrightbig/parenrightbig
+=λ∗/parenleftbig
+ch/parenleftbig
+E[n]/parenrightbig/parenrightbig
++ρ∗ch(E).c1(F) inHeven/parenleftbig
+X[n+1,n],Q/parenrightbig
+.
+Lemma 4.1. For every class αinHeven(X,Q)and everyninN∗, there exists a unique class
+G(α,n)inHeven/parenleftbig
+X[n],Q/parenrightbig
+such thatG(α,1) =α, and for all ninN∗,
+ν∗G(α,n+1)−λ∗G(α,n) =ρ∗α.c1(F).
+Proof.By the degeneration of the Atiyah-Hirzebruch spectral sequ ence fromK-theory to coho-
+mologywith Q-coefficients, wehaveisomorphismsch: K/parenleftbig
+X[n]/parenrightbig
+⊗ZQ∼/d47/d47Heven/parenleftbig
+X[n],Q/parenrightbig
+.There-
+fore, we can define the classes G(α,n) inHeven/parenleftbig
+X[n],Q/parenrightbig
+as follows: if Eis the unique class in
+K(X)⊗ZQsuch that ch( E) =α, thenG(α,n) = ch/parenleftbig
+E[n]/parenrightbig
+. Furthermore, G(α,n) is unique since
+ν∗ν∗=1
+n+1id. /square
+4.2.Virtual tautological Chern characters. This section is somehow technical and can be
+omitted if we suppose that b1(X) = 0, that is if Hodd(X,Q) = 0. The purpose here is to extend
+Lemma 4.1 to odd-dimensional cohomology classes. We adapt t he method originally developed
+in the projective case by Li, Qin and Wang in [Li-Qi-Wa-2].
+Proposition 4.2. For every class αinH∗(X,Q)and everyninN∗, there exists a unique class
+G(α,n)inH∗/parenleftbig
+X[n],Q/parenrightbig
+such thatG(α,1) =αand for all n∈N∗,
+ν∗G(α,n+1)−λ∗G(α,n) =ρ∗α.c1(F).
+Remark 4.3. IfXis projective and Yn⊆X[n]×Xis the incidence locus, it is proved in
+[Li-Qi-Wa-2] that G(α,n) = pr∗
+1/bracketleftbig
+ch(O
+Yn).pr∗
+2α.pr∗
+2td(X)/bracketrightbig
+.
+Proof.We use the relative trick and the machinery of relative coher ent sheaves developed in the
+appendix. Since these methods are going to be explained thor oughly in the second part of the
+paper with similar computations (e.g. in Section 5.4), we sk ip some lengthy details.
+IfXis a relative smooth analytic space and Za relative analytic subspace of X(see Definition
+6.1), we will denote lim←−
+X′H∗
+X′∩Z(X′,Q) byH←∗
+Z(X,Q), where in the projective limit X′runs through
+all open relatively compact relative analytic subspaces of X.
+IfWis a neighbourhood of ZninSnX×X,Ynis the relative incidence locus and Orel
+nis the
+relative incidence sheaf on W[n]
+rel×SnXW(see Definition 5.3 and Definition 5.4), the topological
+class ofOrel
+nlies in lim←−
+X′K
+X′∩Yn(X′). Letπ:W /d47/d47X,p:W[n]
+rel×SnXW /d47/d47W[n]
+relbe the natural
+projections. We define the following cohomology classes:
+∗µrel
+nis the Chern character of Orel
+ninH←∗
+Yn/parenleftbig
+W[n]
+rel×SnXW,Q/parenrightbig
+.
+∗G(α,n)rel=p∗/bracketleftbig
+µrel
+n.π∗α.π∗td(X)/bracketrightbig
+inH←∗/parenleftbig
+W[n]
+rel,Q/parenrightbig
+.
+∗G(α,n) =G(α,n)rel
+|X[n].TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 11
+We take the notations introduced at the beginning of Section 5.3, so that Wwill be from now on
+a neighbourhood of Zn×1inSnX×X. Letπ:W /d47/d47X, p:W[n+1,n]
+rel×SnX×XW /d47/d47W[n+1,n]
+rel
+andq:W[n]
+rel×SnX×XW /d47/d47W[n]
+relbethenatural projections. We definenew cohomology classes :
+∗/tildewideµrel
+n= ch/parenleftbig/tildewideOrel
+n/parenrightbig
+inH←∗
+/tildewideYn/parenleftbig
+W[n]
+rel×SnXW,Q/parenrightbig
+, where/tildewideYn= supp/parenleftbig/tildewideOrel
+n/parenrightbig
+.
+∗/tildewideG(α,n)rel=q∗/bracketleftbig
+/tildewideµrel
+n.π∗α.π∗td(X)/bracketrightbig
+.
+Thenψ∗
+W/tildewideG(α,n+ 1)rel−φ∗
+W/tildewideG(α,n)rel=p∗/bracketleftbig
+(ψ∗
+W/tildewideµrel
+n+1−φ∗
+W/tildewideµrel
+n).π∗α.π∗td(X)/bracketrightbig
+. Lemma
+5.13 shows that this quantity is equal to p∗/bracketleftbig
+p∗ch(L).ρ∗
+rel,Wch/parenleftbig
+Orel
+∆rel/parenrightbig
+.π∗α.π∗td(X)/bracketrightbig
+, which
+is equal toch(L)ρ∗
+relr∗/bracketleftbig
+ch/parenleftbig
+Orel
+∆rel/parenrightbig
+.π∗α.π∗td(X)/bracketrightbig
+viathesecond diagram oftheproofof Propo-
+sition 5.14. Using the same argument as in proof of Lemma 5.15 (iv),
+/bracketleftbig
+ψ∗
+W/tildewideG(α,n+1)rel−φ∗
+W/tildewideG(α,n)rel/bracketrightbig
+|X[n+1,n]= ch(L)
+|X[n+1,n]ρ∗/bracketleftbig
+pr1∗ch/parenleftbig
+C∞
+∆X/parenrightbig
+.α.td(X)/bracketrightbig
+which is equal, by the Grothendieck-Riemann-Roch theorem o f [At-Hi] applied to the diagonal
+injection, to c1(F).ρ∗(α). To conclude the proof, it suffices to show that
+ψ∗
+W/tildewideG(α,n+1)rel
+|X[n+1,n]=ν∗G(α,n+1) and φ∗
+W/tildewideG(α,n)rel
+|X[n+1,n]=λ∗G(α,n).
+This is performed exactly as in Lemma 5.15, (i)–(iii). /square
+4.3.The ring structure and the crepant resolution conjecture. In this section, Xwill
+be a symplectic compact fourfold endowed with a compatible a lmost-complex structure.
+We introduce operators acting on H=/circleplustext
+n∈NH∗/parenleftbig
+X[n],Q/parenrightbig
+by cup-product with the compo-
+nents of the virtual tautological Chern characters constru cted in Section 4.2.
+Definition 4.4. Letα∈H∗(X,Q) be a homogeneous cohomology class. Then
+(i)Gi(α,n) denotes the (|α|+2i)-th component of G(α,n).
+(ii)Si(α) denotes the operator Hwhich acts on H∗(X[n],Q) by cup product with Gi(α,n).
+Then the following result, originally proved by Lehn for geo metric tautological Chern char-
+acters and generalized by Li, Qin and Wang for the virtual one s, is still valid:
+Proposition 4.5. For allα,βinH∗(X,Q)and for all kinN,/bracketleftbig
+Sk(α),q1(β)/bracketrightbig
+=1
+k!q(k)
+1(αβ).
+Proof.This is a consequence of Lemma 3.3 and Proposition 4.2 (see [L e, Theorem 4.2]). /square
+We can now state some significant results on the cohomology ri ngs of Hilbert schemes of sym-
+plectic fourfolds. These results are known in the integrabl e case and are formal consequences of
+the various relations between qn(α),d,Ln(α) andSi(α) (see e.g. Theorem 2.1 in [Li-Qi-Wa-3]),
+even though there is a lot of nontrivial combinatorics invol ved in the proofs. Thus the following
+results are formal consequences of Theorem 3.7, Corollary 3 .9 and Proposition 4.5.
+Theorem 4.6. If0≤i<nandαruns through a fixed basis of H∗(X,Q), the classes Gi(α,n)
+generate the ring H∗/parenleftbig
+X[n],Q/parenrightbig
+.
+This result was initially proved in [Li-Qi-Wa-2] using vert ex algebras. For other proofs, see
+[Li-Qi-Wa-1] and [Li-Qi-Wa-3].12 JULIEN GRIVAUX
+Theorem 4.7. For every integer n, the ringH∗/parenleftbig
+X[n],Q/parenrightbig
+can be built by universal formulae
+from the ring H∗(X,Q)and the first Chern class of XinH2(X,Q).
+For the proof as well as an effective statement, see [Li-Qi-Wa- 3].
+There is a geometrical approach to the ring structure of H∗/parenleftbig
+X[n],Q/parenrightbig
+through orbifold coho-
+mology. IfJisanadaptedalmost-complex structureon X,SnXisanalmost-complex Gorenstein
+orbifold. We can therefore consider the associated Chen-Ru an (ororbifold) cohomology ring
+H∗
+CR/parenleftbig
+SnX,Q/parenrightbig
+which is Z-graded and depends only on the deformation class of J(see [Ch-Ru],
+[Ad-Le-Ru], [Fa-G¨ o]).
+After works by Lehn-Sorger and Li-Qin-Wang, Qin and Wang dev eloped a set of axioms
+which characterize H∗
+CR/parenleftbig
+SnX,Q/parenrightbig
+as a ring (see [Ad-Le-Ru]):
+Theorem 4.8. [Qi-Wa] LetAbe a graded unitary ring and (X,J)an almost-complex compact
+fourfold. We suppose that
+(i)Ais an irreducibleH/parenleftbig
+H∗(X,C)/parenrightbig
+-module and 1is a highest weight vector.
+(ii)For allαinH∗(X,C)and for all iN, there exist classes Oi(α,n)∈A|α|+2isuch that if
+Di(α)is the operator of product by/circleplustext
+nOi(α,n)andD1(1) =dis the derivation,
+(1)∀α,β∈H∗(X,C),∀k∈N,/bracketleftbig
+Dk(α),q1(β)/bracketrightbig
+=q(k)
+1(αβ).
+(2)IfδXis the class in H∗(X,Q)⊗3mapped by the K¨ unneth isomorphism to the cycle
+class of the diagonal in X3,/summationdisplay
+l1+l2+l3=0:ql1ql2ql3: (δX) =−6d.
+ThenAis isomorphic as a ring to H∗
+CR/parenleftbig
+SnX,C/parenrightbig
+.
+In (2), we used the physicists’ normal ordering convention
+:ql1ql2ql3:=ql′
+1ql′
+2ql′
+3,where{l1,l2,l3}={l′
+1,l′
+2,l′
+3}andl′
+1≤l′
+2≤l′
+3.
+We apply this theorem to prove Ruan’s conjecture for the symm etric products of a symplectic
+fourfold with torsion first Chern class.
+Theorem 4.9. Let(X,ω)be a symplectic compact fourfold with vanishing first Chern cl ass
+inH2(X,Q). Then Ruan’s crepant conjecture holds for SnX, i.e. the rings H∗/parenleftbig
+X[n],Q/parenrightbig
+and
+H∗
+CR/parenleftbig
+SnX,Q/parenrightbig
+are isomorphic.
+Proof.LetOk(α,n) =k!Sk(α,n). Then (1) is exactly Proposition 4.5. The relation (2) is a f or-
+malconsequenceoftheNakajimarelationsandoftheformula e/bracketleftbig
+q′
+n(α),qm(β)/bracketrightbig
+=−nmqn+m(αβ),
+q′
+n(α) =nLn(α). /square
+5.The cobordism class of X[n]
+In this section, ( X,J) is an almost-complex compact fourfold, and no symplectic h ypothe-
+ses are required. The almost-complex Hilbert schemes X[n]are endowed with a stable almost
+complex structure (see [Vo 1]), hence define almost-complex cobordism classes. By a funda-
+mental result of Novikov [No] and Milnor [Mi], the almost-co mplex cobordism class of X[n]
+is completely determined by the Chern numbers/integraldisplay
+X[n]P/bracketleftbig
+c1(X[n]),...,c2n(X[n])/bracketrightbig
+wherePrunsTOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 13
+through all polynomials PinQ/bracketleftbig
+T1,...,T2n/bracketrightbig
+of weighted degree 4 n, each variable Tkhaving
+degree 2k. We intend to prove the following result:
+Theorem 5.1. The almost-complex cobordism class of X[n]depends only on the almost-complex
+cobordism class of X.
+This means that if Pis a weighted polynomial in Q/bracketleftbig
+T1,...,T2n/bracketrightbig
+of degree 4n, there exists a
+weighted polynomial /tildewideP/bracketleftbig
+T1,T2/bracketrightbig
+of degree 4, depending only on Pandn, such that
+/integraldisplay
+X[n]P/bracketleftbig
+c1(X[n]),...,c2n(X[n])/bracketrightbig
+=/integraldisplay
+X/tildewideP/bracketleftbig
+c1(X),c2(X)/bracketrightbig
+.
+ThisresulthasbeenprovedbyEllinsgrud, G¨ ottscheandLeh nin[El-G¨ o-Le]when Xisprojective.
+Let us describe briefly the main steps of the argument and the t ools we need.
+LetJrelbe a relative integrable structure in a neighbourhood WofZn. Then the relative
+Hilbert scheme/parenleftbig
+W[n]
+rel,Jrel/parenrightbig
+is fibered in smooth analytic spaces over SnX, so that we can
+consider its relative tangent bundle TrelW[n]
+relwhich is a continuous complex vector bundle on
+W[n]
+rel. It will turn out that the class/bracketleftbig
+TrelW[n]
+rel/bracketrightbig
+|X[n]inK(X[n]) is exactly the class of the
+tangent bundle TX[n], whereX[n]is endowed with the differentiable and stable almost-complex
+structures constructed by Voisin in [Vo 1]. Therefore, the K-theory class of TX[n]can be
+understood via the tangent bundle of classical Hilbert sche mes.
+The main idea in [El-G¨ o-Le] is to relate X[n+1]andX[n]×Xvia the smooth incidence Hilbert
+schemeX[n+1,n]. The essential step in their argument is the explicit compar ison inK/parenleftbig
+X[n+1,n]/parenrightbig
+of the classes of TX[n]andTX[n+1]. This is carried out using the explicit description of TX[n]
+as pr1∗Hom/parenleftbig
+Jn,On/parenrightbig
+, whereJnis the ideal sheaf of the incidence locus Yn⊆X[n]×XandOn
+is the structure sheaf of Yn. So it appears necessary to consider coherent sheaves and no t only
+locally free ones.
+In the almost-complex setting, the coherent sheaves have no equivalent. Instead of working
+directly on the almost-complex Hilbert schemes and the asso ciated incidence varieties, we use
+the corresponding relative objects, which can be considere d as homotopically equivalent to the
+original ones, but possess a much stronger structure: they a re fibered in analytic spaces over
+a singular basis. Each fiber consists of the initial object (H ilbert scheme, incidence variety,...)
+associated to an open set of Xwith an integrable structure on it. The almost-complex Hilb ert
+schemeX[n]will be for instance replaced by the fibration W[n]
+reloverSnXassociated to a relative
+integrable complex structure Jrel: the corresponding fibers are the integrable Hilbert scheme s
+/parenleftbig
+W[n]
+x,Jrel
+x/parenrightbig
+x∈SnX.
+In the appendix (Section 6), we develop a general formalism f or relative coherent sheaves on
+spacesX/Bfibered in smooth analytic sets over a differentiable basis Bwith quotient singular-
+ities, such as W[n]
+rel. These relative smooth analytic spaces carry a sheaf Orel
+Xwhich is the sheaf
+of smooth functions holomorphic in the fibers (see Definition 6.1).
+Intuitively, a relatively coherent sheaf Fon a relative smooth analytic space XoverBis a
+family of coherent sheaves/parenleftbig
+Fb/parenrightbig
+b∈Bon/parenleftbig
+Xb/parenrightbig
+b∈Bvarying smoothly with band locally trivially on
+X. If we take relative holomorphic coordinates on X, the local model for XisZ×V, whereZ
+is a smooth analytic set and Vis an open subset of the base B. Then the local model for a14 JULIEN GRIVAUX
+relatively coherent sheaf on Z×Vis pr−1
+1G⊗
+pr−1
+1OZOrel
+Z×V, whereGis a coherent analytic sheaf
+onZ(see Definition 6.5).
+The usual operations on coherent sheaves (such as internal H om, tensor product, dual, pull-
+back, push-forward and the associated derived operations) can be performed on relatively cohe-
+rent sheaves forsmoothmorphismsholomorphicinthefiberss atisfyingsometriviality conditions
+(see Definitions 6.3 and 6.7). For the push-forward, we will o nly have to deal with the situation
+where the map is finite on the support of the sheaf, which is tec hnically much simpler than the
+general case.
+In this context, it is possible to construct a relative versi on of the usual analytic K-theory
+for coherent sheaves (see Definitions 6.10 and 6.12) as well a s associated operations.
+This being done, it will be necessary to consider relatively coherent sheaves as elements in
+topological K-theory. This is achieved by the following proposition:
+Proposition 5.2. IfFis a relatively coherent sheaf on a relative smooth analytic spaceXover
+BandX′is relatively compact in X, thenF∞:=F⊗Orel
+XC∞
+Xadmits a resolution on X′by
+complex vector bundles/parenleftbig
+Ei/parenrightbig
+1≤i≤N. Besides, the element [F∞] :=/summationtextN
+i=1(−1)i−1[Ei]inK(X′)is
+independent of E•.
+According to this proposition, it is possible to associate t o any relatively coherent sheaf Fon
+a relative smooth analytic space Xatopological class [F∞] in lim←−
+X′⊂⊂XK(X′) and therefore Chern
+classes in lim←−
+X′⊂⊂XH∗(X′,Z). Besides, the class [ F∞] depends only on the relative class of Fin
+Krel(X) by Proposition 6.16 (i).
+This device enables us to carry out the proof of [El-G¨ o-Le] i n a relative context.
+5.1.The relative incidence sheaf. In this section, we introduce the relative incidence sheaf
+Orel
+nand we compute its Chern classes. We will use the following no tations.
+–Wis a small neighbourhood of the incidence locus ZninSnX×X.
+–Jrel
+nis a relative integrable structure on Wparametrized by SnX.
+Definition 5.3. Therelative incidence locus Ynis the relative singular analytic subspace of
+W[n]
+rel×SnXWdefined by
+Yn=/braceleftbig
+(ξ, w, x) inW[n]
+rel×SnXWsuch thatw∈supp(ξ)/bracerightbig
+·
+The fibers/parenleftbig
+Yn,x/parenrightbig
+x∈SnXare the usual incidence loci in W[n]
+x×Wx.
+Definition 5.4. Therelative incidence sheaf Orel
+nis the relatively coherent sheaf Orel
+Ynon
+W[n]
+rel×SnXW. The associated ideal sheaf Jrel
+Ynwill be denoted by Jrel
+n.
+By Proposition 5.2, Orel
+ndefines a topological class in lim←−
+X′⊂⊂W[n]
+rel×SnXWK(X′) and therefore by
+restriction a class in K/parenleftbig
+X[n]×X/parenrightbig
+which is independent of Jrel
+nby Proposition 6.16 (ii).
+Definition 5.5. The classes µi,ninH2i/parenleftbig
+X[n]×X,Q/parenrightbig
+are defined by µi,n=ci/parenleftbig
+Orel
+n/parenrightbig
+|X[n]×X.TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 15
+– Let/tildewiderWbe a small neighbourhood of the incidence locus Zn,1inSnX×XandJrel
+n,1a relative
+integrable structure parametrized by SnX×X. For (x,p)∈SnX×X,/tildewiderW[n+1,n]
+x,pis smooth ([Ch],
+[Ti]). Therefore, /tildewiderW[n+1,n]
+relis a relative smooth analytic space over SnX×X.
+– LetDrelbe the relative exceptional divisor in /tildewiderW[n+1,n]
+reldefined by
+Drel=/braceleftbig
+(ξ, w, x, p) in/tildewiderW[n+1,n]
+relsuch thatw∈supp(ξ)/bracerightbig
+·
+ThenDrelis a relative singular analytic divisor of /tildewiderW[n+1,n]
+rel, and we get an associated rela-
+tive holomorphic line bundle Orel/parenleftbig
+−Drel/parenrightbig
+on/tildewiderW[n+1,n]
+rel. Letρ:X[n+1,n]/d47/d47Xbe the residual
+morphism and σ:X[n+1,n]/d47/d47X[n]×Xbe the morphism ( λ,ρ).
+Definition 5.6. We define a class linH2/parenleftbig
+X[n+1,n],Q/parenrightbig
+byl=c1/bracketleftbig
+Orel(−Drel)/bracketrightbig
+|X[n+1,n].
+The classldetermines the Chern classes of the relative incidence shea f in the following way:
+Proposition 5.7. For alli,ninN∗,µi,n=σ∗(li).
+Proof.We recall well known facts from the classical theory (see [Da ], [Ch], [Ti]). If Xis a
+quasi-projective surface and Ynis the incidence locus in X[n]×X, then:
+(i)J
+Ynadmits a locally free resolution of length 2.
+(ii)X[n+1,n]≃P/parenleftbig
+JYn/parenrightbig
+andX[n+1,n]is smooth.
+(iii) If 0 /d47/d47A /d47/d47B /d47/d47JYn/d47/d470 is a resolution of JYn,π:P(B) /d47/d47X[n]×Xis the
+projection and sis the associated section of π∗A∗(1), thensis transverse to the zero
+section.
+Property (iii) follows from (ii) since the vanishing locus o fs(with its schematic structure) is
+isomorphic to P/parenleftbig
+JYn/parenrightbig
+. Note that we use here Grothendieck’s convention for projec tive bundles.
+We adapt now these properties to the relative setting as foll ows. First we can suppose that
+W,/tildewiderW,Jrel
+nandJrel
+n×1satisfy the compatibility condition: W×SnX/parenleftbig
+SnX×X/parenrightbig
+⊆/tildewiderWas relative
+smooth analytic spaces. Let /tildewideYnbe the relative singular analytic subspace of /tildewiderW[n]×[1]
+reldefined by
+/tildewideYn=/braceleftbig
+(ξ,w,x,p) in/tildewiderW[n]×[1]
+relsuch thatw∈suppξ/bracerightbig
+·
+If we consider the following diagram
+W[n]
+rel×SnXW
+/d15/d15/parenleftbig
+W[n]
+rel×SnXW/parenrightbig
+×SnX/parenleftbig
+SnX×X/parenrightbig
+/d15/d15/d111/d111 /d31/d127/d47/d47/tildewiderW[n]×[1]
+rel
+/d15/d15
+SnX SnX×Xpr1/d111/d111 SnX×X
+X[n]×Xembeds into the three relative smooth analytic spaces in a co mpatible way with the two
+morphisms on the first line. Therefore µi,n=ci/parenleftbig
+Orel
+/tildewideYn/parenrightbig
+|X[n]×X. We will denote the two relative
+smooth analytic spaces /tildewiderW[n+1,n]
+reland/tildewiderW[n]×[1]
+relbyXandX′respectively.16 JULIEN GRIVAUX
+Let us take a family of charts/braceleftbig
+φi: Ωi×Vi∼/d47/d47/tildewiderWi/bracerightbig
+on/tildewiderW, where the Ωi’s are (possibly non
+connected) open sets in C2. They provide relative holomorphic charts φ[n+1,n]
+iandφ[n]×[1]
+iofX
+andX′. For each index i, we pick a locally free resolution 0 /d47/d47Ai/d47/d47Bi/d47/d47JYn,i/d47/d470 of
+length 2 ofJYn,i, whereYn,iis the incidence locus in Ω[n]
+i×Ωi. By the very construction of global
+smooth resolutions for relatively coherent sheaves (Propo sition 6.14 (i)), there exists a global
+resolution 0 /d47/d47A /d47/d47B /d47/d47Jrel,∞
+/tildewideYn/d47/d470 of length 2 of Jrel,∞
+/tildewideYn⊗Orel
+X′C∞
+X′such that for all i
+0 /d47/d47A∞
+i/d47/d47B∞
+i/d47/d47Jrel,∞
+/tildewideYn/d47/d470 is a sub-resolution of the global one. Let π:P(B) /d47/d47X′
+bethe projective bundleof Bandsthesection of π∗A∗⊗C∞
+P(B)C∞
+P(B)(1) obtained by the morphism
+π∗A /d47/d47π∗B /d47/d47C∞
+P(B)(1) . Then we have the following result:
+Lemma 5.8.
+(i)The vanishing locus Z (s)ofsis canonically isomorphic to X.
+(ii)After performing base change from SnX×XtoXn×X, the section sis transverse to
+the zero section.
+(iii)Ifjis the embedding of XintoP(B), thenOrel
+X(−Drel)⊗Orel
+XC∞
+X≃j∗C∞
+P(B)(1).
+Proof.(i) We argue locally on X′. In relative coordinates ( z,v), the local resolutions of Jrel,∞
+/tildewideYn
+is of the form 0 /d47/d47TrM/d47/d47Tr+1,whereT=C∞
+XandMis a (r+1)×rmatrix with holomor-
+phic coefficients in the variable z. We can locally split the injection of A∞
+iinA. The global
+resolution ofJrel,∞
+/tildewideYnis therefore locally isomorphic to 0 /d47/d47Tr⊕Tmφ/d47/d47Tr+1⊕Tmwhere
+φ=/parenleftbigg
+M0
+0 id/parenrightbigg
+·Over a small open set of X′,P(B) is the trivial projective bundle with fiber
+Pr+m(C)∗andsis given in coordinates by s(u,z,v)(α,β) =u/parenleftbig
+M(z)α,β/parenrightbig
+, whereu∈Pr+m(C)∗
+and (α,β)∈Cr+mare the coordinates in the fibers of π∗A. Therefore,
+Z(s) =/braceleftbig
+(u,z,v) such that u|Cm= 0 and for all α∈Cr, u/parenleftbig
+M(z)α/parenrightbig
+= 0/bracerightbig
+·
+This implies that if we consider the local embedding P(B∞
+i)/d31/d127/d47P(B) overX′given by the
+splitting of the local resolution in the global one, then Z( s) lies in P(B∞
+i). It can be seen
+straightforwardly that the embedding of Z( s) intoP(B∞
+i) is independant of the splitting. Let
+/tildewideπ:P(Bi) /d47/d47Ω[n]
+i×Ωibe the projective bundle of Biand/tildewidesthe section of /tildewideπ∗A∗
+i(1) given by the
+morphism /tildewideπ∗Ai/d47/d47/tildewideπ∗Bi/d47/d47OP(Bi)(1).Then Z(/tildewides) = Ω[n+1,n]
+iand/tildewidesis transverse to the zero
+section. This proves that over Ω[n]
+i×Ωi×V, Z(s) = Z(/tildewides)×V= Ω[n+1,n]
+i×Vso that Z(s) is
+abstractly isomorphic over /tildewiderW[n]×[1]
+ito/tildewiderW[n+1,n]
+i.
+(ii) In local coordinates, if for any u∈Pr+m(C)∗we defineu1=u|Cr+1andu2=u|Cm, then
+sis given near its zero locus by s(u,z,v) : (α,β) /d47/d47/parenleftbig
+/tildewides(u1,z)(α),u2(β)/parenrightbig
+. After base change,
+the variable vlies in the smooth manifold Xn×Xandsis clearly transverse to the zero section
+since/tildewidesis.TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 17
+(iii) We have a chain of morphisms /tildewiderW[n+1,n]
+i/d31/d127/d47/d47P(B∞
+i)/d31/d127/d47/d47P(B)
+|/tildewiderWiwhere the last
+morphism is only defined locally on P(B∞
+i). To conclude, notice that Orel
+P(B∞
+i)(1) is canonically
+isomorphic toOrel
+X/parenleftbig
+−Drel/parenrightbig
+|/tildewiderWiand that the restriction of C∞
+P(B)|/tildewiderWitoP(B∞
+i) is canonically iso-
+morphic toC∞
+P(B∞
+i)(1). /square
+SinceP(B) =/bracketleftbig
+P(B)×SnX×X(Xn×X)/bracketrightbig
+/Sn, Lemma 5.8 (ii) implies that the homology
+class of the vanishing locus of sinH8n+8(P(B),Q) is Poincar´ e dual to the top Chern class of
+π∗A∗⊗C∞
+P(B)C∞
+P(B)(1). Let us consider the following diagram:
+P(B)
+π
+/d15/d15X⊇Drelj/d111/d111
+σrel /d122/d122/d117/d117/d117/d117/d117/d117/d117/d117/d117/d117
+X′
+Ifε=c1/parenleftbig
+C∞
+P(B)(1)/parenrightbig
+∈H2(P(B),Q), then by Lemma 5.8 (iii), l=j∗ε
+|X[n+1,n]. Now
+σrel∗(j∗εi) =π∗([X].εi) =d/summationdisplay
+k=0ck(A∗)π∗(εd+i−k) =d/summationdisplay
+k=0ck(A∗)si−k(B∗)
+=ci(A∗−B∗) = (−1)ici/parenleftbig
+Jrel,∞
+/tildewideYn/parenrightbig
+= (−1)ici/parenleftbig
+Orel,∞
+/tildewideYn/parenrightbig
+sincei≥1.
+We have used here the Gysin morphism σrel∗withQ-coefficients, which is possible since Xand
+X′are rationally smooth. To conclude, we consider the diagram
+X
+σrel
+/d15/d15X[n+1,n]
+σ
+/d15/d15/d63/d95 /d111/d111
+X′X[n]×X /d63/d95 /d111/d111
+and get:
+σ∗li=σ∗(j∗εi)
+|X[n+1,n]=/bracketleftbig
+σrel∗(j∗εi)/bracketrightbig
+|X[n]×X= (−1)ici/parenleftbig
+Orel,∞
+/tildewideYn/parenrightbig
+|X[n]×X= (−1)iµi,n.
+/square
+5.2.Computation of TX[n]inK-theory. LetWbe a neighbourhood of the incidence locus
+ZninSnX×XandJrel
+na relative integrable structure on it.
+Definition 5.9. We defineκnas theclass of thelocally Orel
+W[n]
+rel-free sheafTrelW[n]
+relinKrel/parenleftbig
+W[n]
+rel/parenrightbig
+.
+The topological class of κncan be described as follows:
+Lemma 5.10. The restriction to X[n]of the topological class associated to κnis the class of the
+complex vector bundle TX[n]inK(X[n]).18 JULIEN GRIVAUX
+Proof.IfJrel
+nsatisfies the conditions ( C) listed in [Vo 1] page 711, then X[n]
+Jrel
+nis smooth. Be-
+sides, the construction of the almost-complex structure do ne in [Vo 1] shows that TX[n]and
+TrelW[n]
+rel|X[n]have the same class in K(X[n]). An arbitrary Jrel
+ncan be joined by a smooth path
+/braceleftbig
+Jrel
+n,t/bracerightbig
+t∈[0,1]to another relative integrable structure satisfying the co nditions ( C). By rigidity of
+the topological K-theory, the class of Trel/bracketleftbig
+W[n]
+rel,Jrel
+n,t/bracketrightbig
+|X[n]inK/parenleftbig
+X[n]/parenrightbig
+is independent of t. This
+yields the result. /square
+Remark 5.11. Foranarbitrary Jrel
+n,X[n]
+Jrel
+nisonlyatopological manifold(see[Gri1]). Therefore,
+the advantage of using TrelW[n]
+relinstead ofTX[n]
+Jrel
+nis that this complex vector bundle is defined
+foranyrelative integrable complex structure.
+Proposition 5.12. InKrel/parenleftbig
+W[n]
+rel/parenrightbig
+, the following identity holds:
+κn=p∗/parenleftbig
+Orel
+n+Orel∨
+n−Orel
+n.Orel∨
+n/parenrightbig
+.
+Proof.We haveTW[n]
+rel=p∗Hom/parenleftbig
+Jrel
+n,Orel
+n/parenrightbig
+. Since supp/parenleftbig
+Orel
+n/parenrightbig
+has relative codimension 2 in
+W[n]
+rel×SnXW,Exti/parenleftbig
+Orel
+n,Orel/parenrightbig
+= 0 fori <2. Besides,Jrel
+nlocally admits a free resolution of
+length 2. Hence we get the following equalities in Krel
+Yn/parenleftbig
+W[n]
+rel×SnXW/parenrightbig
+:
+Jrel∨
+n.Orel
+n=Hom/parenleftbig
+Jrel
+n,Orel
+n/parenrightbig
+−Ext1/parenleftbig
+Jrel
+n,Orel
+n/parenrightbig
++Ext2/parenleftbig
+Jrel
+n,Orel
+n/parenrightbig
+=Hom/parenleftbig
+Jrel
+n,Orel
+n/parenrightbig
+−Ext2/parenleftbig
+Orel
+n,Orel
+n/parenrightbig
+=Hom/parenleftbig
+Jrel
+n,Orel
+n/parenrightbig
+−Ext2/parenleftbig
+Orel
+n,Orel/parenrightbig
+=Hom/parenleftbig
+Jrel
+n,Orel
+n/parenrightbig
+−Orel∨
+n
+so thatp∗Hom/parenleftbig
+Jrel
+n,Orel
+n/parenrightbig
+=p∗/bracketleftbig/parenleftbig
+Orel−Orel∨
+n/parenrightbig
+.Orel
+n+Orel∨
+n/bracketrightbig
+/square
+5.3.Comparison of TX[n]andTX[n+1]via the incidence variety X[n+1,n].LetWbe a
+neighbourhood of Zn×1inSnX×X×XandJrel
+n×1be a relative integrable complex structure
+onW. Let us consider the following morphisms over SnX×X:
+∗ρrel:W[n+1,n]
+rel/d47/d47Wis the residual map.
+∗jrel=/parenleftbig
+id,ρrel/parenrightbig
+:W[n+1,n]
+rel/d47/d47W[n+1,n]
+rel×SnX×XW.
+∗pis the first projection W[n+1,n]
+rel×SnX×XW /d47/d47W[n+1,n]
+rel.
+∗Iff:X /d47/d47X′is a morphism over SnX×X, we define
+fW:=f×SnX×XidW:X×SnX×XW /d47/d47X′×SnX×XW.
+∗ψ:W[n+1,n]
+rel/d47/d47W[n+1]
+relandφ:W[n+1,n]
+rel/d47/d47W[n]
+relare the canonical projection
+maps.
+∗σrel=/parenleftbig
+φ,ρrel/parenrightbig
+:W[n+1,n]
+rel/d47/d47W[n]
+rel×SnX×XW.
+We introduce the following relatively coherent sheaves:TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 19
+∗/tildewideOrel
+nand/tildewideOrel
+n+1are the relative incidence structure sheaves on W[n]
+rel×SnX×XWand
+W[n+1]
+rel×SnX×XW.
+∗L=Orel(−Drel), whereDrel⊆W[n+1,n]
+relis the relative exceptional divisor.
+∗∆relis the relative diagonal in W×SnX×XWandOrel
+∆relis the associated structure sheaf.
+Then we have the following properties which are immediate co nsequences of the same results in
+the integrable case:
+Lemma 5.13.
+(i)OnW[n+1,n]
+rel×SnX×XWwe havejrel∗L=p∗L⊗ρ∗
+rel,WOrel
+∆rel.
+(ii)We have an exact sequence on W[n+1,n]
+rel×SnX×XW:
+0 /d47/d47jrel∗L /d47/d47ψ∗
+W/tildewideOrel
+n+1/d47/d47φ∗
+W/tildewideOrel
+n/d47/d470.
+(iii)ρ∗
+rel,WOrel
+∆rel=ρ!
+rel,WOrel
+∆rel,ψ∗
+W/tildewideOrel
+n+1=ψ!
+W/tildewideOrel
+n+1andφ∗
+W/tildewideOrel
+n=φ!
+W/tildewideOrel
+n.
+We suppose that there exist a neighbourhood /tildewiderWofZn+1inSn+1X×Xand a relative integrable
+complex structure Jrel
+n+1on it such that W=/tildewiderW×Sn+1X(SnX×X). This means that for all
+(x,p) inSnX×X,Wx,p=/tildewiderWx∪pandJrel
+n×1,x,p=Jrel
+n+1,x∪p. Then we get two weak morphisms
+/tildewideψ:W[n+1,n]
+rel/d47/d47W[n+1]
+reland/tildewideψW:W[n+1,n]
+rel×SnX×XW /d47/d47W[n+1]
+rel×Sn+1XW
+(see Definition 6.3 (ii)) obtained by composing ψandψWwith the basechange map from Sn+1X
+toSnX×Xgiven by (x,p)/d31/d47/d47x∪p. Then /tildewideψ∗
+W/tildewideOrel
+n+1=ψ∗
+WOrel
+n+1. Let/tildewideκnbe the class of
+TrelW[n]
+relinKrel/parenleftbig
+W[n]
+rel/parenrightbig
+.
+Proposition 5.14. InKrel/parenleftbig/tildewiderW[n+1,n]
+rel/parenrightbig
+we have
+/tildewideψ!κn+1=φ!/tildewideκn+L+L∨.ρ!
+relKrel∨
+W−ρ!
+rel/parenleftbig
+Orel
+W−TrelW+Krel∨
+W/parenrightbig
+−L.σ!
+rel/tildewideOrel∨
+n−L∨.ρ!
+relKrel∨
+W.σ!
+rel/tildewideOrel
+n.
+Proof.Let/tildewidep:/tildewiderW[n+1]
+rel×Sn+1X/tildewiderW /d47/d47/tildewiderW[n+1]
+relbetheprojectiononthefirstfactor. ByProposition
+5.12,/tildewideψ!κn+1=/tildewideψ!/tildewidep∗/parenleftbig
+Orel
+n+1+Orel∨
+n+1−Orel
+n+1.Orel∨
+n+1/parenrightbig
+. Let us consider the cartesian diagram
+W[n+1,n]
+rel×SnX×XW/tildewideψW/d47/d47
+p
+/d15/d15/tildewiderW[n+1]
+rel×Sn+1X/tildewiderW
+/tildewidep
+/d15/d15
+W[n+1,n]
+rel/tildewideψ/d47/d47/tildewiderW[n+1]
+rel
+where the first column lies over SnX×Xand the second one over Sn+1X. Since/tildewidepis finite
+on supp/parenleftbig
+Orel
+n+1/parenrightbig
+, we obtain by Proposition 6.13 (iv) and Lemma 5.13 (i) and (ii ) the following20 JULIEN GRIVAUX
+relations in Krel/parenleftbig
+W[n+1,n]
+rel/parenrightbig
+:
+/tildewideψ!κn+1=p∗/tildewideψ!
+W/parenleftbig
+Orel
+n+1+Orel∨
+n+1−Orel
+n+1.Orel∨
+n+1/parenrightbig
+=p∗φ!
+W/parenleftbig/tildewideOrel
+n+/tildewideOrel∨
+n−/tildewideOrel
+n./tildewideOrel∨
+n/parenrightbig
++p∗/bracketleftBig
+p!L.ρ!
+rel,WOrel
+∆rel+p!L∨.ρ!
+rel,WOrel∨
+∆rel−p!/parenleftbig
+L.L∨/parenrightbig
+.ρ!
+rel,W/parenleftbig
+Orel
+∆rel.Orel∨
+∆rel/parenrightbig
+−p!L.ρ!
+rel,WOrel
+∆rel.φ!
+W/tildewideOrel
+n−p!L∨.ρ!
+rel,WOrel∨
+∆rel. φ!
+W/tildewideOrel
+n/bracketrightBig
+.
+Ifq:W[n]
+rel×SnX×XW /d47/d47W[n]
+relis the projection on the first factor, then by Proposition 5.1 2,
+/tildewideκn=q∗/parenleftbig/tildewideOrel
+n+/tildewideOrel∨
+n−/tildewideOrel
+n./tildewideOrel∨
+n/parenrightbig
+so that, since qis finite on supp/parenleftbig
+Orel
+n/parenrightbig
+, Proposition 6.13
+(iv) yields: φ!/tildewideκn=p∗φ!
+W/parenleftbig/tildewideOrel
+n+/tildewideOrel∨
+n−/tildewideOrel
+n./tildewideOrel∨
+n/parenrightbig
+. We also consider the diagram
+W[n+1,n]
+rel×SnX×XWρrel,W/d47/d47
+p
+/d15/d15W×SnX×XW
+r
+/d15/d15
+W[n+1,n]
+relρrel/d47/d47W
+where all the terms are over SnX×X. Sinceris injective on ∆rel, by Proposition 6.13 (iv)
+again,
+/tildewideψ!κn+1=φ!/tildewideκn+L.ρ!
+relr∗Orel
+∆rel+L∨.ρ!
+relr∗Orel∨
+∆rel−ρ!
+relr∗/parenleftbig
+Orel
+∆rel.Orel∨
+∆rel/parenrightbig
+−L.p∗/parenleftbig
+jrel∗Orel
+W[n+1,n]
+rel.φ!
+W/tildewideOrel∨
+n/parenrightbig
+−L∨.p∗/bracketleftBig/parenleftbig
+ρ!
+rel,WOrel
+∆rel/parenrightbig∨.φ!
+W/tildewideOrel
+n/bracketrightBig
+.
+Nowr∗Orel
+∆rel=Orel
+W,r∗Orel∨
+∆rel=Krel∨
+Wand ifδ:W /d47/d47W×SnX×XWis the diagonal injection,
+ρ!
+rel,WOrel∨
+∆rel=ρ!
+rel,Wδ∗Krel∨
+W=jrel∗ρ!
+relKrel∨
+W, thanks to Proposition 6.13 (v) and to the
+diagram
+W[n+1,n]
+reljrel/d47/d47
+ρrel
+/d15/d15W[n+1,n]
+rel×SnX×XW
+ρrel,W/d15/d15
+Wδ/d47/d47W×SnX×XW
+/square
+5.4.Cohomological computations. We are now going to consider the identity weobtained in
+Section 5.3 from a cohomological point of view. Recall that t he classesµi,ninH2i/parenleftbig
+X[n]×X,Q/parenrightbigTOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 21
+are defined by µi,n=ci/parenleftbig
+Orel
+n/parenrightbig
+|X[n]×X. Let us consider the following maps:
+X
+X[n+1,n]ρ/d79/d79
+λ
+/d119/d119/d112/d112/d112/d112/d112/d112 ν
+/d39/d39/d80/d80/d80/d80/d80/d80
+σ
+/d15/d15X[n]X[n+1]
+X[n]×X
+Lemma 5.15.
+(i)ci/parenleftbig
+φ!/tildewideκn/parenrightbig
+=λ∗ci/parenleftbig
+X[n]/parenrightbig
+.
+(ii)ci/parenleftbig
+φ!
+W/tildewideOrel
+n/parenrightbig
+= (λ,id)∗µi,n.
+(iii)ci/parenleftbig
+σ!
+rel/tildewideOrel
+n/parenrightbig
+|X[n+1,n]=σ∗µi,n.
+(iv)ci/parenleftbig
+ρ∗
+relTrelW/parenrightbig
+|X[n+1,n]=ρ∗ci(X).
+(v)ci/parenleftbig
+ρ!
+rel,WOrel
+∆rel/parenrightbig
+|X[n+1,n]×X= (ρ,id)∗ci/parenleftbig
+C∞
+∆X/parenrightbig
+.
+Proof.By Lemma 6.16 (ii), the classes φ!/tildewideκn,φ!
+W/tildewideOrel
+nandσ!
+rel/tildewideOrel
+nin topological K-theory are
+independent of the relative integrable structure Jrel
+n×1. Therefore, we can suppose that there
+exist a neighbourhood ˘WofZninSnX×Xand a relative integrable structure Jrel
+non it such
+that˘W×SnX(SnX×X)⊆W. This means that for ( x,p) inSnX×X,˘Wx⊆Wx,pand
+Jrel
+n×1,x,p|˘Wx=Jrel
+n,x. Then (i), (ii) and (iii) are straightforward.
+For (iv), let /hatwiderW=W×SnXX, where the base change morphism is given by the diagonal
+injection of XinSnX. We consider the following diagram:
+W[n+1,n]
+relρrel/d47/d47W /hatwiderW/d63/d95 /d111/d111
+X[n+1,n]/d63/d31/d79/d79
+ρ/d47/d47X/d63/d31/d79/d79
+X/d63/d31/d79/d79
+Then,ci/parenleftbig
+ρ∗
+relTrelW/parenrightbig
+|X[n+1,n]=ρ∗ci/parenleftbig
+Trel/hatwiderW/parenrightbig
+|X. Since/hatwiderWis a neighbourhood of ∆XinX×X,
+Trel/hatwiderW|X≃TX, so thatci/parenleftbig
+Trel/hatwiderW/parenrightbig
+|X=ci(X).
+For (v), we extend the structure Jrel
+n×1to a relative integrable structure Jrel
+n×1×1in a neigh-
+bourhoodWofZn×1×1such that for any ( x,p,q)∈SnX×X×Xnear the incidence locus p=q,
+the equality Jrel
+n×1×1,x,p,q=Jrel
+n×1,x,pholds. This means that W×SnX×X(SnX×X×X)⊆W.
+We define /hatwiderWby/hatwiderW=W×SnX×X×X(X×X), the base change morphism from SnX×X×X
+toX×Xbeing given by ( x,x′)/d31/d47/d47(δ(x),x,x′) , whereδ:X /d47/d47SnXis the diagonal injection.22 JULIEN GRIVAUX
+Then we consider the diagram:
+W[n+1,n]
+rel×SnX×XWρrel,W/d47/d47
+/d127/d95
+/d15/d15W×SnX×XW
+/d127/d95
+/d15/d15
+W[n+1,n]
+rel×SnX×X×XWρrel,W/d47/d47W×SnX×X×XW /hatwiderW×X×X/hatwiderW /d63/d95 /d111/d111
+X[n+1,n]×X
+(ρ,id)/d47/d47/d63/d31/d79/d79
+X×X/d63/d31/d79/d79
+X×X/d63/d31ι/d79/d79
+If ∆rel
+/hatwiderWis the relative diagonal in /hatwiderW×X×X/hatwiderW, thenι∗/bracketleftbig
+C∞
+∆rel
+/hatwiderW/bracketrightbig
+=/bracketleftbig
+C∞
+∆X/bracketrightbig
+inK(X×X). This gives
+the result. /square
+Remark 5.16. Letdibe thei-th Chern class of C∞
+∆XinH2i(X×X,Q). By [At-Hi], diis a
+universal polynomial in c1(X) andc2(X) with rational coefficients.
+Proposition 5.17.
+(i)For alli,ninN∗,(ν,id)∗µi,n+1−(λ,id)∗µi,n=i/summationdisplay
+k=0pr∗
+1lk.(ρ,id)∗di−k, whereµi,nandl
+are defined in 5.5 and 5.6.
+(ii)For alli,ninN∗,ν∗ci/parenleftbig
+X[n+1]/parenrightbig
+−λ∗ci/parenleftbig
+X[n]/parenrightbig
+is a universal polynomial in the classes l,
+ρ∗ci(X)andσ∗µi,n.
+Proof.This is a consequence of Lemma 5.15, Lemma 5.13 (i) and (ii), a nd of Proposition 5.14.
+/square
+We are now going to perform the induction step.
+Proposition 5.18. Ifn,mare positive integers, let Pbe a polynomial in pr∗
+0ci/parenleftbig
+X[n+1]/parenrightbig
+,
+pr∗
+0kµi,n+1,pr∗
+kldi,pr∗
+kci(X) (1≤k,l≤m), which are cohomology classes on X[n+1]×Xm.
+Then there exists a polynomial /tildewidePdepending only on P, in the classes analogously defined on
+X[n]×Xm+1, such that/integraldisplay
+X[n+1]×XmP=/integraldisplay
+X[n]×Xm+1/tildewideP.
+Proof.We consider the incidence diagram
+X[n+1,n]×Xm
+X[n+1]×Xm/parenleftbig
+X[n]×X/parenrightbig
+×Xm(ν,id) ( σ,id)
+Since (ν,id) and (σ,id) are generically finite of degrees n+1 and 1,/integraldisplay
+X[n+1]×XmP=1
+n+1/integraldisplay
+X[n]×Xm+1(σ,id)∗(ν,id)∗P.TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 23
+By Proposition 5.17 (ii), ( ν,id)∗pr∗
+0ci/parenleftbig
+X[n+1]/parenrightbig
+−(σ,id)∗pr∗
+0ci/parenleftbig
+X[n]/parenrightbig
+is a polynomial in the
+classespr∗
+0l, (σ,id)∗pr∗
+1ci(X) and (σ,id)∗pr∗
+01µi,n.
+By Proposition 5.7, ( σ,id)∗li= (−1)ipr∗
+01µi,n. Thus, (σ,id)∗(ν,id)∗pr∗
+0ci/parenleftbig
+X[n+1]/parenrightbig
+is a
+polynomial in pr∗
+01µi,nandpr∗
+1ci(X).
+By Proposition 5.17 (i), ( ν,id)∗pr∗
+0kµi,n+1−(σ,id)∗pr∗
+0,k+1µi,nis a polynomial in the classes
+pr∗
+0land (σid)∗pr∗
+1kdi. Then we can use Proposition 5.7 again.
+Toconclude, weusetherelations ( ν,id)∗prkldi= (σ,id)∗prk+1,l+1diand(ν,id)∗pr∗
+kci(X) =
+(σ,id)∗pr∗
+k+1ci(X). /square
+We can now finish the proof of Theorem 5.1. We write/integraldisplay
+X[n]P/parenleftBig
+c1/parenleftbig
+X[n]/parenrightbig
+,...,c2n/parenleftbig
+X[n]/parenrightbig/parenrightBig
+=/integraldisplay
+X[n−1]×X/tildewideP1=/integraldisplay
+X[n−2]×X2/tildewideP2=···=/integraldisplay
+Xn/tildewideP
+where/tildewidePis a polynomial in the classes pr∗
+kci(X) andpr∗
+kldi. Sincediis a polynomial in c1(X)
+andc2(X), we are done.
+/square
+6.Appendix
+Our aim in this appendix is to develop a general framework for sheaves on spaces which
+are fibered in smooth analytic spaces over a differentiable orb ifold. This formalism is somehow
+heavy but necessary to carry out the computations of [El-G¨ o -Le] in a relative setting.
+6.1.Relative analytic spaces. LetMbe a smooth manifold and let Gbe a finite subgroup of
+Diff(M) (in fact, any effective differentiable orbifold would do, but n o such generality is required
+here). Let B=M/G. Ifp:M /d47/d47Bis the projection, Bwill be endowed with the sheaf of
+ringsC∞
+B:=p∗/parenleftbig
+C∞
+M/parenrightbigG.
+Definition 6.1.
+(i) Arelative smooth analytic space over Bis a ringed topological space/parenleftbig
+X,Orel
+X/parenrightbig
+endowed
+with a surjective morphism π:/parenleftbig
+X,Orel
+X/parenrightbig/d47/d47/parenleftbig
+B,C∞
+B/parenrightbig
+such that/parenleftbig
+X,Orel
+X/parenrightbig
+is locally on X
+isomorphic over Bto a ringed topological space of the type/parenleftbig
+Z×V,Orel
+Z×V/parenrightbig
+, whereVis
+an open subset of B,Zis a smooth analytic space and Orel
+Z×Vis the sheaf of differentiable
+functions on Z×Vwhich are holomorphic on the slices Z×{v},v∈V.
+(ii) Let/parenleftbig
+X,Orel
+X/parenrightbig
+be a relative smooth analytic space over Band letZbe a subset of X.
+We say that Zisa relative analytic subspace of Bif for allx∈Xand for all relative
+holomorphic chart φ:Z×V∼/d47/d47Ux,φ−1/parenleftbig
+Z|Ux/parenrightbig
+=Z′×V, whereZ′is a (possibly
+singular) analytic subset of Z.
+Remark 6.2.
+– If/parenleftbig
+X,Orel
+X/parenrightbig
+is a relative smooth analytic space over B, the fibers/parenleftbig
+Xb/parenrightbig
+b∈Bare smooth
+analytic sets, but they do not form in general a fibration over B, since the projection map πis
+not proper. However, they locally vary in a trivial way on X.
+– The main example of a relative smooth analytic space we can k eep in mind is W[n]
+relover
+SnXwhereWis a neighbourhood of the incidence set in SnX×X, endowed with a relative
+integrable structure parametrized by SnX.24 JULIEN GRIVAUX
+We now introduce morphisms between relative smooth analyti c spaces.
+Definition 6.3.
+(i) LetXandX′be two relative smooth analytic spaces over Bandf:X /d47/d47X′be a
+continuous map over B. We say that fis amorphism if for allx∈Xwe can find
+trivializations of XandX′aroundxandf(x) in which f:Z×V /d47/d47Z′×Vhas the
+form (z,v)/d31/d47/d47(g(z),v) , whereg:Z /d47/d47Z′is holomorphic.
+(ii) LetXandX′be two relative smooth analytic spaces over BandB′andf:X /d47/d47X′
+be a continuous map. We say that fis aweak morphism if there exist a smooth map
+u:B /d47/d47B′and a morphism /tildewidef:X /d47/d47X′×B′BoverBsuch thatfis obtained by
+composing /tildewidefwith the base change morphism u:X′×B′B /d47/d47X′.
+Remark 6.4.
+– IfXandX′are two relative smooth analytic spaces over BandB′, a continuous map
+f:X /d47/d47X′over a smooth map u:B /d47/d47B′which is holomorphic in the fibers is not a weak
+morphism in general.
+– Weak morphisms can be characterized by their expression in suitable charts as it is the
+case for morphims. Indeed, f:X /d47/d47X′is a weak morphism if for all x∈Xthere exist
+trivializations of XandX′aroundxandf(x) in which f:Z×V /d47/d47Z′×V′has the form
+(z,v)/d31/d47/d47(g(z),u(v)), whereg:Z /d47/d47Z′is holomorphic and u:V /d47/d47V′is smooth.
+– The term “morphism” has to be carefully understood because morphisms cannot a priori
+be composed in this context.
+6.2.Relatively coherent sheaves. We introduce now the main object of our study.
+Definition 6.5. LetXbe a relative smooth analytic space over B. A sheafFofOrel
+X-modules
+will berelatively coherent if for allx∈X, ifφ:Z×V∼/d47/d47Uxis a trivialization of Xin a
+neighbourhood Uxofx, there exists a coherent sheaf FonZsuch that
+φ−1F
+|Ux≃pr−1
+1F⊗pr−1
+1OZOrel
+Z×V
+as sheaves ofOrel
+Z×V-modules.
+Remark 6.6.
+– IfXis a relative smooth analytic space over BandZis a relative analytic subspace of X,
+thenOrel
+Zis a relatively coherent sheaf on X.
+– A relatively coherent sheaf on Xcan also be defined by glueing conditions: it is given
+by a family of relative holomorphic charts/braceleftbig
+φi:Zi×Vi∼/d47/d47Ui/bracerightbig
+iwith transition functions
+φji:Zij×Vij∼/d47/d47Zji×Vji,and a family of coherent sheaves/braceleftbig
+Fi/bracerightbig
+ion{Zi}together with
+isomorphisms
+φ−1
+ji/bracketleftBig/parenleftbig
+Fj⊗pr−1
+1OZjOrel
+Zj×Vj/parenrightbig
+|Zji×Uji/bracketrightBig
+≃/parenleftbig
+Fi⊗pr−1
+1OZiOrel
+Zi×Vi/parenrightbig
+|Zij×Uij
+of sheaves ofOrel
+Zij×Vij-modules satisfying the usual cocycle condition.
+– LetFbe a relatively coherent sheaf on Xgiven by a family of sheaves {Fi}i∈I. Then, for
+anyb∈B, ifJ={i∈Isuch thatb∈Vi}, the sheaves{Fi}i∈Jon{Zi×b}i∈Jpatch together
+into a coherent sheaf on Xb, which we will denote by Fb.TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 25
+The glueing procedure allows to perform standard operation s for coherent sheaves in the
+relative setting.
+Definition 6.7.
+(i) LetFandGbe relatively coherent sheaves on Xgiven by two families {Fi}and{Gi}.
+The sheavesExtk(F,G) andTork(F,G) are defined by the families/braceleftbig
+Extk
+OZi(Fi,Gi)/bracerightbig
+and/braceleftbig
+Tork
+OZi(Fi,Gi)/bracerightbig
+.We putHom(F,G) =Ext0(F,G) andF⊗G=Tor0(F,G).
+(ii) Letf:/parenleftbig
+X,Orel
+X/parenrightbig/d47/d47/parenleftbig
+X′,Orel
+X′/parenrightbig
+be a weak morphism between relative smooth analytic
+spaces. Let{φi}i∈Iand{φ′
+i}j∈Jbe two families of charts such that:
+–I⊆Jandf(Ui)⊆U′
+i.
+– In the charts φiandφ′
+i,fhas the form ( z,v)/d31/d47/d47(gi(z),ui(v)).
+IfGis a relatively coherent sheaf on X′given by a family {Gj}j∈J, the sheavesTork(G,f)
+are defined by the families/braceleftBig
+Tork
+g−1
+iOZ′
+i/parenleftbig
+g−1
+iGi,OZi/parenrightbig/bracerightBig
+i∈I. We putf∗G=Tor0(G,f).
+(iii) Letf:/parenleftbig
+X,Orel
+X/parenrightbig/d47/d47/parenleftbig
+X′,Orel
+X′/parenrightbig
+be a morphism of relative smooth analytic spaces over B
+andFa relatively coherent sheaf on Xsuch that for all b∈B,fbis finite on supp(Fb).
+Let{φi,Ui}i∈Iand{φ′
+j,U′
+j}j∈Jbe two families of charts such that
+–J⊆Iandf−1(U′
+j)∩supp(F)⊆Uj.
+– IfWjis a neighbourhood of f−1(U′
+j)∩supp(F),f:Wj/d47/d47Ujhas the form
+(z,w)/d31/d47/d47(gj(z),v) .
+IfFis given by a family {Fi}i∈I, the sheaff∗Fis defined by the family {gj∗Fj}j∈I.
+Remark 6.8.
+– For allbinB,
+Extk(F,G)b=Extk
+OXb(Fb,Gb), Tork(F,G)b=Tork
+OXb(Fb,Gb),
+Tork(G,f)b=Tork
+f−1
+bOX′
+u(b)/parenleftbig
+f−1
+bGu(b),OXb/parenrightbig /parenleftbig
+f∗F/parenrightbig
+b=/parenleftbig
+fb/parenrightbig
+∗Fb.
+– The finiteness hypothesis will be verified for the computati ons of Sections 5.2 and 5.3.
+Therefore, we need not construct direct images in full gener ality.
+– It is straightforward that Hom(F,G) =HomOrel
+X(F,G),F⊗G=F⊗Orel
+XG, f∗G=
+f−1G⊗f−1Orel
+X′Orel
+Xand thatf∗Fis the usual direct image of Fbyfvia the morphism
+Orel
+X′/d47/d47f∗Orel
+X. The associated derived results are also true and are conseq uences of Lemma
+6.9, (i). Thanks to the finiteness hypothesis, there are no hi gher direct images Rif∗.
+6.3.AnalyticK-theory for relatively coherent sheaves. We are now going to define mor-
+phisms of relatively coherent sheaves. The natural idea is t o consider the relatively coherent
+sheaves on/parenleftbig
+X,Orel
+X/parenrightbig
+as a full subcategory of Mod/parenleftbig
+Orel
+X/parenrightbig
+. It is not appropriate because this
+category would be non-abelian. Before giving the definition , we start with a preliminary flatness
+lemma which will be essential in the sequel.
+Lemma 6.9. LetUbe an open set of Rn,Ga finite group acting smoothly on U,V=U/G
+and letZbe a smooth analytic set. Then26 JULIEN GRIVAUX
+(i)Orel
+Z×Vis flat over pr−1
+1OZ,
+(ii)C∞
+Z×Vis flat over pr−1
+1OZ.
+Proof.(i) Letδ:U /d47/d47Vbe the projection and Mbe a sheaf of pr−1
+1OZ-modules. Then
+(δ,id)−1/parenleftbig
+M⊗pr−1
+1OZC∞
+Z×V/parenrightbig
+= (δ,id)−1M⊗pr−1
+1OZ/parenleftbig
+C∞
+Z×U/parenrightbigG≃/bracketleftbig
+(δ,id)−1M⊗pr−1
+1OZC∞
+Z×U/bracketrightbigG.
+Since the functor F/d31/d47/d47FGfrom ModG/parenleftbig
+C∞
+Z×U/parenrightbig
+to Mod/bracketleftbig/parenleftbig
+C∞
+Z×U/parenrightbigG/bracketrightbig
+is exact, it suffices to
+prove thatC∞
+Z×Uis smooth over pr−1
+1OZ. IfYis a real-analytic manifold, we will denote by Cω
+Y
+the sheaf of real-analytic functions on Y. Then,Cω
+Zis flat overOZ,Cω
+Z×Uis flat over pr−1
+1Cω
+Z
+andC∞
+Z×Uis flat overCω
+Z×Uby [Ma, Th 2].
+(ii) As in (i), it suffices to prove that Orel
+Z×Uis flat over pr−1
+1OZ. Letk=⌊(n+1)/2⌋. Then
+U×Rkcan be seen as an open subset /tildewideUinC(n+k)/2. By [Ma, Th 2 bis], Orel
+Z×/tildewideUis flat overOZ×/tildewideU
+andOZ×/tildewideUis flat over pr−1
+1OZ. ThusOrel
+Z×/tildewideYis flat over pr−1
+1OZ. Ifq:Z×/tildewideU /d47/d47Z×Uis the
+projection, q−1Orel
+Z×Uis a direct factor of Orel
+Z×/tildewideU(as pr−1
+1OZ-modules), so that Orel
+Z×Uis flat over
+pr−1
+1OZ. /square
+This being done, the definition of a morphism of relatively co herent sheaves runs as follows:
+Definition 6.10. LetFandGbe relatively coherent sheaves on a relative smooth analyti c
+space/parenleftbig
+X,Orel
+X/parenrightbig
+. A morphism u∈HomOrel
+X(F,G) will be said to be strictif for every x∈X
+and any trivialization φ:Z×V∼/d47/d47Uxin a neighbourhood of x, there exist two isomorphisms
+φ−1F|Ux≃pr−1
+1F⊗pr−1
+1OZOrel
+Z×Vandφ−1G|Ux≃pr−1
+1G⊗pr−1
+1OZOrel
+Z×V, whereFandGare
+coherent on Z, andvin Hom
+OZ(F,G), such that the following diagram commutes:
+φ−1Fφ−1u/d47/d47
+∼
+/d15/d15φ−1G
+∼
+/d15/d15
+pr−1
+1F⊗
+pr−1
+1OZOrel
+Z×Vv⊗id/d47/d47pr−1
+1G⊗
+pr−1
+1OZOrel
+Z×V
+Remark 6.11. Let/parenleftbig
+X,Orel
+X/parenrightbig
+be a relative smooth analytic space. Lemma 6.9 (i) implies th at
+the category Cohrel(X) of relatively coherent sheaves on Xwith strict morphisms is an abelian
+subcategory of Mod/parenleftbig
+Orel
+X/parenrightbig
+.
+Definition 6.12. Let/parenleftbig
+X,Orel
+X/parenrightbig
+be a relative smooth analytic space.
+(i) We define the relative analytic K-theory of Xby
+Krel(X) = lim←−
+X′K/parenleftbig
+Cohrel(X′)/parenrightbig
+,
+whereX′runs through relatively compact open analytic subsets in X.
+(ii) In the same way, if Zis a relative sub-analytic space of X, the relative analytic K-theory
+ofXwith suppoort in Zis defined as
+Krel
+Z(X) = lim←−
+X′K/parenleftbig
+Cohrel
+Z∩X′(X′)/parenrightbig
+,TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 27
+where Cohrel
+Z∩X′(X′) is the abelian category of relatively coherent sheaves on X′supported
+inZandX′runs through relatively compact open analytic subsets in X.
+As for coherent sheaves, we can define usual operations on rel ative analytic K-theory. Here
+is a list of these operations:
+(i)The product. A product from Krel(X)⊗ZKrel(X) toKrel(X) is defined by
+F.G=/summationdisplay
+k≥0(−1)kTork(F,G).
+Similarly, if Zisarelativeanalyticsubspaceof X, aproductwithsupportfrom Krel(X)⊗Z
+Krel
+Z(X) toKrel
+Z(X) is given by the same formula.
+(ii)The dual morphism. It is an involution of Krel(X) given by
+F∨=/summationdisplay
+k≥0(−1)kExtk/parenleftbig
+F,Orel
+X/parenrightbig
+.
+(iii)The pull-back. Iff:/parenleftbig
+X,Orel
+X/parenrightbig/d47/d47/parenleftbig
+X′,Orel
+X′/parenrightbig
+is a weak morphism, the pull-back map
+f!:Krel(X′) /d47/d47Krel(X) is defined by
+f!G=/summationdisplay
+k≥0(−1)kTork(G,f).
+IfZ′is a relative analytic subspace of X′, we also have a pull-back morphism with
+supportsf!:Krel
+Z′(X′) /d47/d47Krel
+f−1(Z′)(X′) given by the same formula.
+(iv)The Gysin map. Iff:/parenleftbig
+X,Orel
+X/parenrightbig/d47/d47/parenleftbig
+X′,Orel
+X′/parenrightbig
+is a morphism and Zis a relative an-
+alytic subspace of Xsuch that for every binB,fb|Zbis finite, the Gysin morphism
+f∗:Krel
+Z(X) /d47/d47Krel(X′) is induced by the exact functor f∗: Cohrel
+Z(X) /d47/d47Coh(X′).
+We now list all the properties we need relating the operation s introduced above.
+Proposition 6.13.
+(i)If/parenleftbig
+X,Orel
+X/parenrightbig
+is a relative smooth analytic space, Krel(X)is a unitary ring. Furthermore,
+ifZis a relative analytic subspace of X,Krel
+Z(X)is a module over Krel(X).
+(ii)The pull-back morphism in relative K-theory is contravariant and the Gysin map is
+covariant.
+(iii)The projection formula holds. More precisely, if f:X /d47/d47X′is a morphism, Za relative
+analytic subspace of Xsuch thatf|Zis finite,Fa relatively coherent sheaf on Xsupported
+inZandGa relatively coherent sheaf on X′, thenf∗/parenleftbig
+F.f!G/parenrightbig
+=f∗F.G.
+(iv)LetXbe a relative smooth analytic space over B,X′and∆be two relative smooth
+analytic spaces over B′andf:X /d47/d47X′a weak morphism; finduces a weak morphism
+f∆:X×B/parenleftbig
+∆×B′B/parenrightbig/d47/d47X′×B∆.LetZ′be a relative analytic subspace of X′×B∆such
+that the projection q:X′×B∆ /d47/d47X′is finite on Z′, and let Z=f−1
+∆(Z′). We consider28 JULIEN GRIVAUX
+the diagram
+X×B/parenleftbig
+∆×B′B/parenrightbigf∆/d47/d47
+p
+/d15/d15X′×B∆
+q
+/d15/d15
+Xf/d47/d47X′
+as well as the following pull-back and push-forward operati ons:
+f!:Krel(X′) /d47/d47Krel(X) f!
+∆:Krel
+Z′(X′×B∆) /d47/d47Krel
+Z/bracketleftbig
+X×B/parenleftbig
+∆×B′B/parenrightbig/bracketrightbig
+p∗:Krel
+Z/bracketleftbig
+X×B/parenleftbig
+∆×B′B/parenrightbig/bracketrightbig/d47/d47Krel(X)q∗:Krel
+Z′(X′×B∆) /d47/d47Krel(X′)
+Then, for everyGinCohZ′(X′×B∆)we havef!q∗G=p∗f!
+∆GinKrel(X).
+(v)LetX,∆be two relative smooth analytic spaces over B,X′a relative smooth analytic
+subspace of ∆andf:X /d47/d47X′a morphism. Consider the cartesian diagram
+X(id,i◦f)/d47/d47
+f
+/d15/d15X×B∆
+f∆/d15/d15
+X′
+(id,i)/d47/d47X′×B∆
+as well as the following pull-back and push-forward operati ons:
+f!:Krel(X′) /d47/d47Krel(X) f!
+∆:Krel
+X′(X′×B∆) /d47/d47Krel
+X(X×B∆)
+(id,i)∗:Krel(X′) /d47/d47Krel
+X′(X′×B∆) (id ,i◦f)∗:Krel(X) /d47/d47Krel
+X(X×B∆)
+Then for every relatively coherent sheaf GonX′, we havef!
+∆(id,i)∗G= (id,i◦f)∗f!Gin
+Krel
+X(X×B∆).
+Proof.(i) IfF,GandHare relatively coherent sheaves on X, for each X′open and relatively
+compact in X, we have a spectral sequence such that
+
+Ep,q
+2=Torp
+Orel
+X′/parenleftbig
+Torq
+Orel
+X′(F,G),H/parenrightbig
+Ep,q
+∞= GrpTorp+q
+Orel
+X′(F,G,H).
+andEp,q
+2= 0except for finitely many couples ( p,q). Furthermore, by Lemma6.9 (i), thesheaves
+Ep,q
+2are relatively coherent on X′and the morphisms dp,q
+2are strict. Therefore, for all r≥2,
+the sheaves Ep,q
+rare relatively coherent and the morphisms dp,q
+2are strict so that
+/summationdisplay
+p,q≥0(−1)p+qEp,q
+2=/summationdisplay
+n≥0(−1)nTorn
+Orel
+X′(F,G,H)
+inKrel(X′). This gives the result.
+(ii) The proof is similar to the proof of (i), using spectral s equences associated to the com-
+position of two functors.
+The proofs of (iii), (iv) and (v) are done in the same way. We wi ll prove (iv).TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 29
+(iv) Letx∈X. We take charts Ux≃Z×VandUf(x)≃Z′×V′such thatf:Ux/d47/d47Uf(x)has
+the form (z,v)/d31/d47/d47(g(z),u(v)) , whereg:Z /d47/d47Z′is holomorphic and u:V /d47/d47V′is smooth.
+Letδ1,...,δN∈∆ such that q−1(f(x))∩Z=∪N
+i=1/parenleftbig
+f(x),δi/parenrightbig
+. We take a chart Uδ1,...,δN≃Y×V′
+in a neighbourhood of the δi’s. Then the diagram looks locally on X′as follows:
+Z×Y×V(g,id,u)/d47/d47
+pr13
+/d15/d15Z′×Y×V′
+pr13
+/d15/d15
+Z×V
+(g,u)/d47/d47Z′×V′
+Furthermore, Z=Z×VwhereZis an analytic subset of Z×Yand pr1|Zis finite. Then for any
+analytic sheafGonZ′×Y, we have
+(g,u)∗pr13∗/parenleftbig
+pr−1
+12G⊗
+pr−1
+12OZ′×YOrel
+Z′×Y×V′/parenrightbig
+≃pr13∗(g,id,u)∗/parenleftbig
+pr−1
+12G⊗
+pr−1
+12OZ′×YOrel
+Z′×Y×V′/parenrightbig
+.
+Taking the derivative with respect to Gand using Lemma 6.9 (i), we obtain the result. /square
+6.4.Topological classes. In Sections 6.2 and 6.3, we have constructed a theory for rela tive
+coherent sheaves as well as associated operations. It remai ns to obtain cohomological informa-
+tions about these objects. To do so, we will construct global smooth resolutions for relatively
+coherent sheaves. We start with a general result about smoot h resolutions.
+Proposition 6.14. LetMbe a smooth manifold, Ga finite group of Diff(M)andY=M/G.
+LetHbe a sheaf ofC∞
+Y-modules which admits a finite free resolution in a neighbour hood of any
+pointy∈Y. Then
+(i)Hadmits a finite locally free resolution in a neighbourhood of any compact set of Y.
+(ii)Two resolutions of Hin a neighbourhood of a compact set are sub-resolutions of a t hird
+one.
+Proof.We will use several times the following lemma:
+Lemma 6.15. LetY=M/Gand let0 /d47/d47F /d47/d47G /d47/d47H /d47/d470be an exact sequence of C∞
+Y-
+modules on an open set U⊆Ysuch thatHis locally free. Then this exact sequence globally
+splits onU.
+Proof.It is sufficient to prove that Ext1
+C∞
+Y(U,H,F) = 0. TheExt=⇒Ext spectral sequence
+satisfies 
+
+Ep,q
+2=Hp/bracketleftBig
+U,Extq
+C∞
+Y/parenleftbig
+H,F/parenrightbig/bracketrightBig
+Ep,q
+∞= GrpExtp+q
+C∞
+Y(U,H,F).
+SinceHis locally free,Extq(H,F) = 0 forq>0 and sinceHomC∞
+Y(H,F) is a finesheaf, Ep,0
+2= 0
+forp≥1. Therefore all the terms Ep,q
+2vanish except E0,0
+2. This implies Ext1
+C∞
+Y(U,H,F) = 0./square
+(i) LetK⊆Ybe a compact. We choose a finite covering/parenleftbig
+Ui/parenrightbig
+1≤i≤dofKand open sets
+/parenleftbig
+Vi/parenrightbig
+1≤i≤dsuch thatUi⊆ViandH|Viadmits a finite free resolution. Using smooth cut-off
+functions, we obtain for each ia complex of sheaves
+0 /d47/d47/parenleftbig
+C∞
+Y/parenrightbigniN /d47/d47······ /d47/d47/parenleftbig
+C∞
+Y/parenrightbigni1πi/d47/d47H /d47/d47030 JULIEN GRIVAUX
+which is exact in Ui. IfE=d/circleplustext
+i=1/parenleftbig
+C∞
+Y/parenrightbigni1andπ:=d/circleplustext
+i=1πi:E /d47/d47H, the morphism πis surjective.
+LetNi= kerπiandN= kerπ. We have an exact sequence:
+0 /d47/d47Ni/d47/d47N|Ui/d47/d47/circleplustext
+j/ne}ationslash=i/parenleftbig
+C∞
+Y/parenrightbignj1 /d47/d470.
+ThusN|Uiis locally isomorphic to Ni⊕/parenleftbig
+C∞
+Y/parenrightbig/summationtext
+j/negationslash=inj1. FurthermoreNiadmits a finite free
+resolution of length N−1. ThusNadmits a finite free resolution of length at most N−1 in
+a neighbourhood of every point in Kand we can start the argument again. After at most N
+steps, the kernel will be locally free.
+(ii) Let/parenleftbig
+Ei/parenrightbig
+1≤i≤Nand/parenleftbig
+Fi/parenrightbig
+1≤i≤Nbe two finite locally free resolutions in a neighbourhood of
+K. Suppose that we have constructed/parenleftbig
+Gi/parenrightbig
+1≤i≤kand injections E•/d31/d127/d47/d47G•andF•/d31/d127/d47/d47G•
+LetQk=Gk/EkandRk=Gk/Fk. The sheaves Q1,...,Qk,R1,...,Rkare locally free. Let
+Nk= ker/parenleftbig
+Ek/d47/d47Ek−1/parenrightbig
+,N′
+k= ker/parenleftbig
+Fk/d47/d47Fk−1/parenrightbig
+,N′′
+k= ker/parenleftbig
+Gk/d47/d47Gk−1/parenrightbig
+,/tildewideQk= ker/parenleftbig
+Qk/d47/d47Qk−1/parenrightbig
+and/tildewideRk= ker/parenleftbig
+Rk/d47/d47Rk−1/parenrightbig
+. Then /tildewideQkand/tildewideRkare locally free. We have two exact sequences
+0 /d47/d47Nk/d47/d47N′′
+k/d47/d47/tildewideQk/d47/d470 and 0 /d47/d47N′
+k/d47/d47N′′
+k/d47/d47/tildewideRk/d47/d470.By Lemma 6.15, N′′
+k≃
+Nk⊕/tildewideQk≃N′
+k⊕/tildewideRk, and we define Gk+1=/parenleftbig
+Ek+1⊕/tildewideQk/parenrightbig
+⊕/parenleftbig
+Fk+1⊕/tildewideRk/parenrightbig
+. We putGN+1=N′′
+N
+to end the resolution G•. /square
+We apply now this result in our context. Let Fbe a relatively coherent sheaf on a relative
+smooth analytic space XoverB, whereB=M/G. ThenX=/parenleftbig
+X×BM/parenrightbig
+/GandX×BMis
+smooth. Furthermore, by Lemma 6.9 (ii), the sheaf F∞:=F⊗Orel
+XC∞
+Xlocally admits finite
+C∞
+X-free resolutions. By Proposition 6.14 (i), F∞admits a globally locally C∞
+X-free resolution
+E•on any relatively compact open analytic subset X′ofX. Besides, Proposition 6.14 (ii) implies
+that the element/summationtextN
+i=1(−1)i−1/bracketleftbig
+Ei/bracketrightbig
+inK(X′) is independent of the chosen resolution E•.
+In conclusion, we can associate to each relatively coherent sheafFonXa topological class/bracketleftbig
+F∞/bracketrightbig
+in lim←−
+X′K(X′), where X′runs through all open relatively compact analytic subspace s ofX.
+We now state properties of this topological class:
+Proposition 6.16. LetXbe a relative smooth analytic space over B.
+(i)The topological class map from Cohrel(X)tolim←−
+X′K(X′)factors through Krel(X).
+(ii)LetFbe a relatively coherent sheaf on X×B(B×[0,1])and for all t∈[0,1], let
+it:X×B(B×{t}) /d47/d47X×B×{t}(B×[0,1])be the associated base change morphism.
+Then the class/bracketleftbig
+i∗
+tF∞/bracketrightbig
+inlim←−
+X′K(X′)is independent of t.
+Proof.(i) We must show that if 0 /d47/d47F /d47/d47G /d47/d47H /d47/d470 is a strict exact sequence of rela-
+tively coherent sheaves, then [ F∞]−[G∞] + [H∞] = 0. This sequence is locally isomorphic
+to
+0 /d47/d47pr−1
+1F⊗pr−1
+1OZOrel
+Z×V/d47/d47pr−1
+1G⊗pr−1
+1OZOrel
+Z×V/d47/d47pr−1
+1H⊗pr−1
+1OZOrel
+Z×V/d47/d470,TOPOLOGICAL PROPERTIES OF PUNCTUAL HILBERT SCHEMES 31
+and is obtained by extension of thestructuresheaves froman exact sequenceof coherent analytic
+sheaves 0 /d47/d47F /d47/d47G /d47/d47H /d47/d470 onZ. Wecanconstructlocally freeresolutions E
+F,•,E
+G,•,
+E
+H,•ofF,G,Hrelated by an exact sequence 0 /d47/d47E
+F,•/d47/d47E
+G,•/d47/d47E
+H,•/d47/d470.Using cut-
+off functions again, we patch these exact sequences together step by step and obtain resolutions
+EF∞,•,EG∞,•,EH∞,•ofF∞,G∞,H∞onX′related by an exact sequence
+0 /d47/d47EF∞,•/d47/d47EG∞,•/d47/d47EH∞,•/d47/d470.
+(ii) LetX′be an open relatively compact analytic subset of XandE•be a resolution of F∞
+onX′. The class α(t) =/summationtextN
+i=1(−1)i−1/bracketleftbig
+i∗
+tEi/bracketrightbig
+inK(X′) is independent of t. SinceF∞is flat over
+C∞
+B×[0,1], it is also flat over C∞
+[0,1], so thatα(t) =/bracketleftbig
+i∗
+tF∞/bracketrightbig
+inK(X′). /square
+Remark 6.17. IfZis a relative analytic subspace of X, there is also a topological class map
+with support from Cohrel
+Z(X) to lim←−
+X′KZ∩X′(X′) which factors through Krel
+Z(X).
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+arXiv:1001.0120v1  [physics.atom-ph]  31 Dec 2009Final-State Spectrum of3He after β−Decay of Tritium Anions T−
+Alexander Stark and Alejandro Saenz∗
+AG Moderne Optik, Institut f¨ ur Physik, Humboldt-Universi t¨ at zu Berlin, Newtonstr.15, D–12489 Berlin, Germany
+(Dated: November 14, 2018)
+The final-state spectrum of βdecaying tritium anions T−was calculated. The wavefunctions de-
+scribing the initial T−ground state and the final3He states were obtained by the full configuration-
+interaction method. The transition probability was calcul ated within the sudden approximation.
+The transition probability into the electronic continuum i s extracted from the complex-scaled resol-
+vent and is shown to converge for very high-energies to an app roximate analytical model probability
+distribution.
+PACS numbers: 31.15.-p, 23.40.Bw, 14.60.Pq
+I. INTRODUCTION
+The neutrino rest mass is a very important parameter
+for cosmology, astrophysics, and the standard model of
+elementary particles. The existence of neutrinos, already
+postulated by Pauli and put into a mathematical frame-
+work ofβdecay by Fermi [1] long time ago, was verified
+by Reines and Cowan in 1956 [2]. However, despite the
+high solar neutrinos flux of about billions per m−2s−1on
+earth the answer to the question about their rest mass is
+one of the big unknowns in physics. Since neutrino-flavor
+oscillations have been observed in the late nineties at the
+Super-Kamiokande experiment [3] a non-vanishing neu-
+trino rest mass has to be expected. Unfortunately, this
+typeofexperimentsrevealsonlymassdifferencesbetween
+neutrino flavors.
+The presently constructed KATRIN (Karlsruhe tri-
+tium neutrino-mass) experiment with an expected sensi-
+tivity of about 0 .2eV (90% C.L.) should have the ability
+to determine the absolute value for one of the flavors or
+at least a new upper limit to it [4]. This so called next-
+generation tritium β-decay experiment is only based on
+kinematic relations and energy and momentum conser-
+vation. Thus KATRIN provides a model-independent
+direct measurement of the antineutrino rest mass m¯νe
+(more accurately the mass of the antineutrino in a given
+mass-flavor mixture mostly attributed to the electronic
+neutrino). In more detail, m2
+¯νewill be extracted in a fit
+procedure from the shape of the βspectrum. Besides the
+precise measurement of the β-electron energy spectrum
+it is crucial for the mass extraction to know how the β
+spectrum is modified by the final-state spectrum of the
+decay product. As in the previous most recent tritium
+neutrino-mass experiments in Mainz and Troitsk the T 2
+molecule is chosen as tritium source. T 2comprises a
+compromise between experimental accessibility and the-
+oreticaltreatability. Thefinal-statespectrum ofitsdecay
+product3HeT+wasthereforesubject ofanumber ofvery
+detailed calculations [5–13], finally accumulating in the
+one covering the whole energy regime [14]. Recently, the
+∗Electronic address: alejandro.saenz@physik.hu-berlin. despectrum was further adapted to specific needs (isotope
+distribution and temperature) of the KATRIN experi-
+ment [15].
+Although a high purity of the molecular tritium source
+isexpectedforKATRIN,theproduced βelectronscanin-
+teract with other gas molecules and thus produce tritium
+species different from T 2. One of the expected processes
+is the dissociative attachment
+e−+T2→T−+T (1)
+where T−formation occurs. Despite the relative small
+cross section compared to, e.g., the one for vibrational
+excitation of the T 2molecule, this process is very impor-
+tant. The reason is the higher endpoint energy of the β
+spectrum for the decay of T−compared to the one of
+T2. Due to this fact the occurrence of T−ions leads to
+a systematical error and hence to a possible limitation of
+the sensitivity ofKATRIN, if it is not properlyaccounted
+for [16].
+To the authors’ knowledge, there exist so far only two
+theoretical predictions for the final-state spectrum fol-
+lowingβdecay of T−. However, in [17] only transition
+probabilities to 4 final states are reported. Furthermore,
+the results in [17] disagree substantially from the ones
+given in an earlier work [18] that was, however, also lim-
+ited to 10 final states. The aim of this work was thus
+to provide a complete final-state spectrum for the decay
+process
+T−→3He +e−+νe. (2)
+and to shed some light on the disagreeing earlier results.
+II. METHOD AND COMPUTATIONAL
+DETAILS
+The calculation of the non-relativistic eigenstates of
+the atomic systems is performed within the approxi-
+mation of an infinitely heavy mass of the nuclei, i.e.
+T≈∞H and3He≈∞He. This is justifiable due to
+the large mass difference of the nucleus and the elec-
+trons. The calculation of the final-state spectrum can be
+performed analytically for neutralT atoms and reveals a2
+negligible mass dependence. Hence a large mass depen-
+dence is also not expected in the case of tritium anions.
+Thenon-relativisticHamiltonianforthetwo-electronsys-
+tem hastheform(atomicunits with me= 1,e= 1,/planckover2pi1= 1
+are used throughout, if not specified otherwise):
+ˆH =−1
+2(∆1+∆2)−Z/parenleftbigg1
+r1+1
+r2/parenrightbigg
++1
+|r1−r2|(3)
+whereZis the charge of the nucleus and rithe posi-
+tion vector of the ith electron. Due to the fact that the
+only bound state of T−is a singlet state with angular
+momentum L=M= 0 [19], only symmetric spatial con-
+figuration state functions (CSF) are important,
+|Φ(+)
+k/angbracketright=/braceleftBigg
+2−1/2(|φi/angbracketright|φj/angbracketright+|φj/angbracketright|φi/angbracketright)i/negationslash=j
+|φi/angbracketright|φj/angbracketright i=j.(4)
+To determine the eigenstates and corresponding energy
+eigenvalues a simple expansion in Slater-type orbitals
+(STO) is used,
+/angbracketleftr|φi/angbracketright=/radicalBigg
+(2ζi)2n+1
+(2n)!exp(−ζir)rn−1Ym
+l(ϑ,ϕ).(5)
+Then,l,mare integer parameters with limitations anal-
+ogously to the ones for the hydrogen quantum numbers
+andtheYm
+lrepresentthesphericalharmonics. The ζiare
+positive real parameters. An appropriate choice of these
+parameters allows for the achievement of an in principle
+complete coverageofthe Hilbert space ofthe one-particle
+part of Hamiltonian (3). In the full configuration-
+interaction (CI) method the eigenstates are expressed as
+a linear superposition of all possible symmetry-adapted
+CSFs
+|Ψj(r1,r2)/angbracketright=/summationdisplay
+kcjk|Φ(+)
+k/angbracketright (6)
+that can be formed with the aid of the chosen STO basis.
+The expansion coefficients cjkare determined by solving
+the generalized eigenvalue problem obtained from insert-
+ing the wavefunction ansatzof Eq.(6) into the eigenvalue
+equation of the Hamiltonian (3).
+The final-state spectrum of He is calculated within
+the sudden approximation [20] that is based on the fact
+that the escaping βelectron has a much higher veloc-
+ity than the bound electrons. In the analysis of tritium
+neutrino-mass experiments like KATRIN only the βelec-
+tronswith an energynear the endpoint ofthe βspectrum
+at18.6keVareused. Theirvelocityisclearlymuchlarger
+than the average speed of the bound electrons in T−. In
+fact, the validity of the sudden approximation has been
+demonstrated for T 2in [10, 11, 21] where the first-order
+correction terms were derived and explicitly calculated.
+From those results it is apparent that also for T−the
+sudden approximation is expected to be valid within the
+accuracy required for the analysis of an experiment likeKATRIN. Nevertheless a brief discussion of possible ef-
+fects beyond the sudden approximation on the final-state
+spectrum is given at the end of this work.
+A basis set of 555 STOs yielding 3481 CSFs in the
+full CI calculation was used to obtain the final results
+shown in this work. This STO basis set contains all pos-
+sible kinds of orbitals (with restrictions on landmas
+mentioned above) up to the angular quantum number
+l= 7,−7≤m≤7. For the optimization of the pa-
+rameters ζia genetic and several other algorithms [22]
+were tested . However, none of those algorithms lead to
+completely convincing results. Therefore, the parame-
+ters were finally optimized by hand. The difficulty of the
+parameter optimization is due to the requirement to con-
+struct a basis set with high coverage of the Hilbert space
+whileavoidinginaccuraciesduetonumericallycausedlin-
+ear dependencies. With the aim to achieve a uniform de-
+scription of the possible He final states it is favorable to
+obtain a homogeneous and a high density of states in the
+continuum as well as a large number of bound states. If
+a large number of CFSs is used, the optimization of the
+individual ζivalues becomes less important, since the
+full CI method leads to a sufficient mixing of the Hilbert
+space covered by the various STOs. Therefore, the pa-
+rameters ζiwere chosen to start in an interval between 2
+and3andtodecreaseinvalueforincreasing n(foragiven
+l). This procedure avoids numerical problems and allows
+the construction of a huge, but linearly independent ba-
+sis set. This basis set is used for both the ground state of
+T−and all final states of He. The chosen basis leads for
+T−to the ground-state energy ET−
+0=−14.3602eVthat
+is only 0.8meVabovethe very accuratevalues in [23, 24].
+In the case of He the adopted basis set yields 16 states
+below the ionization continuum. Out of those 16 states
+15 are identified as true physical states, while the 16th
+state is a pseudo state that resembles the remaining in-
+finite number of Rydberg states as a consequence that a
+finite basis set is adopted.
+Within the sudden approximation the transition prob-
+ability for T−decays into bound states of He is simply
+given by the squared overlap
+Pn=/vextendsingle/vextendsingle/vextendsingle/angbracketleftΨHe
+n|ΨT−
+i/angbracketright/vextendsingle/vextendsingle/vextendsingle2
+(7)
+of the initial state |ΨT−
+i/angbracketright, i.e. the T−ground state, and
+the final state |ΨHe
+n/angbracketright, i.e. the n-th bound state of He.
+To calculate the transition-probability density into
+continuum states the complex scaling method is used.
+It is based on the mathematical development by Aguilar,
+Balslev, and Combes [26, 27] as well as Simon [28]. The
+application of this method leads in practice to a sim-
+ple but powerful modification of the Hamiltonian ˆH in
+Eq. (3),
+ˆH(θ) = exp(−2iθ)ˆT+exp( −iθ)ˆV. (8)
+In Eq. (8) ˆT andˆVarethe usual kinetic and potential en-
+ergy operators of He, respectively. The complex-scaling3
+TABLE I: Population probabilities Pnof the1Sbound states
+of helium after the βdecay of a T−anion. Also given are the
+corresponding energies En(in atomic units) obtained in the
+present work.
+n E n Pn(%) Pn(%) Pn(%)
+(this work)a(this work) (Ref. 17) (Ref. 18)
+1−2.9034572 22 .98998 22 .993764 19 .147
+2−2.1459 527 46 .86960 46 .867404 21 .149
+3−2.0612 659 0 .01320 0 .135 0 .27
+4−2.03358 41 0.18363 0 .21 0 .143
+5−2.02117 49 0.09220 − 0.07
+6−2.01456 04 0.05262 − 0.039
+7−2.01062 35 0.03275 − 0.024
+8−2.00809 09 0.02175 − 0.016
+9−2.00636 79 0.01522 − 0.011
+10−2.00514 07 0.01111 − 0.008
+11−2.00423 55 0.00848 − −
+12−2.00355 07 0.00666 − −
+13−2.00302 05 0.00513 − −
+14−2.0025951 0 .00469 − −
+15−2.0022 570 0 .00273 − −/summationtextPn 70.30975 70 .206168 40 .877
+aThe bold digits agree with the results in Ref.25.
+angleθcan in principle be chosen arbitrarily within
+0◦≤θ≤45◦. In the limit of an infinite basis all observ-
+ables calculated with the aid of complex scaling should
+become independent of θ. Since only finite basis sets can
+be applied in practice, only approximate eigenstates can
+be obtained that may depend on θ. The angle θcan thus
+be understood as a variational parameter that modifies
+the adopted basis as can be seen from the inverse rela-
+tion between basis-set exponents and the scaling angle
+discussed, e.g., in [29]. A diagonalization of the Hamil-
+tonian (8) in the basis described by the Eqs. (4) and (5)
+yields the complex-scaled energies Ej(θ) and wavefunc-
+tions Ψ j(θ) where the latter are still defined by Eq. (6),
+but with complex coefficients cjk(θ).
+With the aid of the complex-scaled energies and wave-
+functions the transition-probability density into the elec-
+tronic continuum can be extracted from the complex-
+scaled resolvent according to [9]
+P(E,θ) =
+1
+πIm/braceleftBigg/summationdisplay
+k/angbracketleftΨT−
+i(θ∗)|ΨHe
+k(θ)/angbracketright/angbracketleftΨHe
+k(θ∗)|ΨT−
+i(θ)/angbracketright
+EHe
+k(θ)−E/bracerightBigg
+.(9)
+The/angbracketleftΨ(θ∗)|is the biorthonormal eigenstate to |Ψ(θ)/angbracketright. It
+is obtained from the latter by a transposition and com-
+plexconjugationoftheangularpart, whiletheradialpart
+is only transposed but not complex conjugated. The sum
+overkincludes all complex-scaled eigenstates and eigen-
+valuescalculatedbysolvingthegeneralizedcomplexsym-
+metric, but non-hermitian eigenvalue problem. As dis-cussed above in the limit of exact eigenstates the density
+P(E,θ) becomes independent of the complex-scaling pa-
+rameter θ. A variation of θfor approximate eigenstates
+providesthe possibilitytodetermineanoptimal θoptwith
+highest stability. The best approximation of P(E,θ) is
+then obtained according to
+∂P(E,θ)
+∂θ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+θopt= min.→P(E) :=P(E,θopt).(10)
+Furthermore, the θdependence of the spectra gives an
+indication for the convergence of the results.
+III. RESULTS
+In Table I the calculated transition probabilities for
+151S bound states of He are listed. The results reveal
+that almost every second T−decay will end in the first
+excited state of He. The next probable final state is
+the He ground state with nearly 23%. With a summed
+probability of 0 .45% the higher excited He states are
+rarely populated after βdecay of T−. The sum over
+all calculated bound states yields 70 .3%. The summa-
+tion over all calculated states (discrete and discretized
+continuum states) yields the expected value of 100 .00%,
+since the same basis is used for initial and all final states,
+but indicates the proper numerical implementation. The
+excellent agreement of the energy eigenvalues at the or-
+der ofµhartree with the very accurate data in [25] as-
+sures on the other hand the high quality of the basis set
+adopted in the present work and its ability to describe
+many states simultaneously with high precision. A closer
+view on the energies shows that the degree of accuracy of
+the present results followsthe expected trends. First, the
+accuracy increases with n, since the importance of cor-
+relation decreases, if the state becomes more asymmet-
+ric and the two electrons have smaller spatial overlap.
+For even higher values of nthe states become increas-
+ingly diffuse and thus it is very difficult to describe them
+properly without running into numerically caused linear
+dependencies.
+A comparison to the final-state probabilities reported
+by Frolov [17] and Harston and Pyper [18] is also given
+in Table I. Especially for the highly populated ground
+and first excited states the results of this work confirm
+the expectedly very accurate results of Frolov [17] that
+were obtained with explicitly correlated basis functions.
+The agreement for the third excited state ( n= 4) is,
+however, less good, and for the second excited state ( n=
+3) there is even an order of magnitude difference. All
+attempts to improve the basis set for this state failed to
+yield a better agreement. This could be an indication for
+a typographical error (a missing zero after the decimal
+point) in [17].
+The comparison with the results in [18] that were
+obtained with a relativistic MCDF (multi-configuration
+Dirac-Fock)method showsontheotherhandpronounced4
+TABLE II: Discretized final-state probability distributio nP(Ei) for He following the βdecay of a T−anion. The mean
+excitation energies Eiare given relative to the ground state of3He .
+Ei(eV) P(Ei)(%) Ei(eV) P(Ei)(%) Ei(eV) P(Ei)(%) Ei(eV) P(Ei)(%)
+25.084 0.36869 50.096 0.07054 75.086 0.09926 186.05 0.0096 0
+26.081 0.32908 51.097 0.07403 76.085 0.09172 191.06 0.0087 3
+27.081 0.28924 52.100 0.08005 77.085 0.08480 196.06 0.0079 6
+28.081 0.25425 53.104 0.09041 78.086 0.07863 201.06 0.0072 7
+29.082 0.22462 54.112 0.10943 80.950 0.31969 206.07 0.0066 6
+30.082 0.19976 55.125 0.14982 85.967 0.23208 211.08 0.0061 2
+31.083 0.17887 56.158 0.26863 90.980 0.17471 232.06 0.0347 1
+32.084 0.16126 57.305 1.44117 95.991 0.13529 272.24 0.0199 8
+33.084 0.14632 57.863 16.80345 101.00 0.10714 313.03 0.012 50
+34.085 0.13358 58.911 0.15914 106.01 0.08644 352.94 0.0083 2
+35.085 0.12267 59.972 0.02281 111.01 0.07084 392.90 0.0058 1
+36.085 0.11328 61.109 0.00924 116.02 0.05883 433.04 0.0042 1
+37.086 0.10517 62.095 3.16161 121.02 0.04942 472.98 0.0031 4
+38.086 0.09816 63.026 0.13662 126.03 0.04194 513.67 0.0023 9
+39.087 0.09209 64.113 0.13095 131.03 0.03590 553.22 0.0018 6
+40.087 0.08684 65.091 0.09512 136.03 0.03098 591.30 0.0014 8
+41.088 0.08233 66.100 0.11217 141.03 0.02692 630.58 0.0011 9
+42.088 0.07847 67.099 0.12245 146.04 0.02355 654.79 0.0010 1
+43.089 0.07523 68.105 0.13639 151.04 0.02071 664.12 0.0008 9
+44.089 0.07256 69.171 0.26130 156.04 0.01832 663.99 0.0008 1
+45.089 0.07046 70.210 0.20446 161.04 0.01629 666.03 0.0007 3
+46.085 0.06892 71.109 0.10126 166.05 0.01454 688.91 0.0006 3
+47.084 0.06803 72.101 0.12307 171.05 0.01303 776.21 0.0004 8
+48.090 0.06792 73.077 0.12211 176.05 0.01173 1550.1a0.00358a
+49.098 0.06870 74.083 0.10784 181.05 0.01059/summationtextP(Ei) 29.65399
+aFor energies above 904eV the model tail in Eq. (13) was used.
+differences. The deviation is most remarkably for the
+first excited state that according to the present work and
+[17]shouldbe populatedwith about47%probabilityand
+thus should clearly dominate the final-state distribution.
+However, in the MCDF results in [18] its probability is
+found to be about 21%. For the other states, except
+n= 3, the results in [18] are always smaller than the
+present ones. The deviation increases rather uniformly
+from about 17 to 28% for nvarying between 1 and 10.
+Since relativistic effects are expected to be small for light
+nuclei like T−and3He, it appears very likely that the
+main reason for the difference of the results in [18] to
+the present ones (as well as the ones in [17]) is due to
+the small number of configurations used in the MCDF
+method compared with the present full CI method. Un-
+fortunately, no details (like energies) of the MCDF cal-
+culation in [18] are available to further clarify this issue,
+but any realistic estimate of the size of relativistic effects
+excludes their responsibility for the large discrepancy be-
+tween the results in [18] compared to the non-relativistic
+calculations of this work or the one in [17].
+The calculated transition-probability density into the
+electronic continuum of3He is presented for three dif-30 40 50 60 70
+Energy [eV]10-610-510-410-310-210-1100Probability Density  [eV-1]
+FIG. 1: (Color online) Final-state continuum probability d en-
+sity of He after βdecay of T−forθ= 24◦(red), 30◦(blue),
+and 36◦(black). (The energy scale is given relative to the He
+ground state.)5
+ferent complex-scaling angles ( θ= 24◦,30◦,and 36◦) in
+Fig. 1. The overall spectrum is practically independent
+ofθ. This indicates the high quality of the adopted ba-
+sis set also for describing the electronic continuum. As
+is usually the case, (higher lying) resonances are most
+sensitive to the choice of θ. This is due to the fact that
+it is difficult to find a single value of θthat is equally
+appropriate for describing a certain resonance and the
+underlying background continuum.
+The continuum probability density is dominated by
+a peak corresponding to the first doubly excited singlet
+state 2s2. About 19% of the T−decays ends up in the
+energy interval between 54 .5eV and 60eV. Above the
+65.4eV threshold the higher-lying doubly excited states
+2sns and in the regime up to 79eV (with diminishing im-
+portance) the 3s nspeaks canbe identified. The complex-
+scaling method provides the probability density P(E) at
+any value of Eand thus as a continuous function. In
+view of the sharp resonant structures and in accordance
+with the experimental needs, the final-state distribution
+is given in a discretized form as in [14]. For this pur-
+pose, the probability distribution P(E) has been divided
+into small bins covering an energy range of 1 .0eV (up
+to a transition energy of 78 .59eV), 5 .0eV (from 78 .59 to
+214eV), and 40 .0eV (from 214 to 904eV). For each bin
+the averageexcitation energy Eiand the integrated tran-
+sition probability P(Ei) were calculated and are given in
+Table II.
+For the high-energy continuum states (above 904eV)
+an approximate model tail is introduced, similar to the
+case of T 2[11, 14]. However, the situation is more com-
+plicated for T−. In T 2βdecay the high-energy tail was
+derived based on the idea that for sufficiently large ener-
+gies of the escaping (formerly bound) electron the effec-
+tive potential of the remaining3HeT2+ion can be well
+approximated by a point charge Z= 2. In fact, the
+remaining electron and tritium nucleus may be viewed
+as pure spectators and thus the transition probability
+should approach for high energies the one obtained for a
+β-decaying tritium atomfor which an analytical result is
+known. Due to the existence of two equivalent electrons,
+the atomic result is simply multiplied by a factor of two
+[11].
+While for T 2a hydrogenic wavefunction is a reason-
+able first-order approximation for the initial state, this
+is not the case for T−. In fact, within independent-
+particle models T−is unstable. As a consequence, the
+fast electron in the final state may be well represented
+by a Coulomb wavefunction for a point charge Z= 1
+(formed by the remaining He+ion), but the modeling
+of the initial state is less obvious within an independent
+particle model. This is also evident from the alternative
+point of view that a description of the remaining T nu-
+cleusandboundelectronasaspectatorwouldcorrespond
+for the active electron to an initial state with Z= 0 and
+thus no bound state. In order to obtain an atomic-like
+high-energy tail the initial state is thus modeled as a
+hydrogen-like state with variable exponent. This expo-nent is then obtained by fitting the model spectrum to
+the ab initio spectrum of the full two-electroncalculation
+in the energy range between 500eV and 10,000eV.
+The initial T−ground-state wavefunction (omitting
+the spin part for better readability)is then approximated
+as
+|˜ΨT−
+i/angbracketright=|1sZe1sZe/angbracketright (11)
+where/angbracketleftr|1sZe/angbracketrightis an STO with ( n,l,m) = (1,0,0), i.e.
+an atomic hydrogen 1s orbital with effective charge Ze.
+In the spirit of the sudden approximation the spectator
+electron remains in its orbital and the final state of the
+He+ion is modeled as
+|˜ΨHe(E)/angbracketright= 2−1
+2/parenleftbig
+|1sZeφc(E)/angbracketright+|φc(E)1sZe/angbracketright/parenrightbig
+(12)
+where|φc(E)/angbracketrightis the Coulombic continuum wavefunction
+for energy Eand charge Z= +1. Using these model
+wavefunctions the analytic expression
+˜P(Ze,E) =
+2/parenleftBigg
+8(1−Ze)Z3/2
+ee−2arctan( κ/Ze)
+κ/radicalbig
+1−e−2π
+κ(κ2+Z2e)2/parenrightBigg2
+dE
+Ry(13)
+withκ=/radicalbig
+(E+2)/Ry and 1Ry = 13 .60585972eV is
+obtained for the probability density, i.e. for the model
+tail for the high-energy continuum states. A fit to the
+ab initio spectrum yielded Ze= 1.3074. As is evident
+from (13) the probability density decays exponentially
+for high energies which is the most essential property.
+(In fact, it has been verified that all subsequent conclu-
+sions are unchanged, if a different model tail is used in
+which the initial-state charge is fixed to the one of the
+tritium nucleus ( Z= 1) and the effective charge of the
+final Coulomb wave is used as fit parameter.)
+Fig. 2 compares the calculated final-state probability
+density with the analytical model tail and confirms the
+applicabilityofthelatterforhighenergies,infactalready
+starting from about 200 to 250eV, similarly as for T 2.
+The integrated probability density (including the model
+tail for energies above 904eV) yields a total probability
+of 29.65% for ionization of He following βdecay of T−.
+Themeanexcitationenergy Erelativetotheelectronic
+ground-state energy of T−can be obtained from
+E=/summationdisplay
+nEnPn+
+/integraldisplay904eV
+24.59eVEP(E)dE+/integraldisplay∞
+904eVE˜P(E)dE.(14)
+Insertion of the final-state probability distribution cal-
+culated in this work in (14) yields E= 27.487eV for the
+mean excitation energy of the decay product. This result
+may be compared to the one obtained by the alternative
+relation
+E=/angbracketleftΨT−
+i|ˆH(He)|ΨT−
+i/angbracketright
+=E0(T−)−2/angbracketleftΨT−
+i|1
+r|ΨT−
+i/angbracketright.(15)6
+100 1000
+Energy [eV]10-910-810-710-610-510-410-310-210-1100Probability Density  [eV-1]
+50 10000.004
+FIG. 2: Final-state continuum probability density of He aft er
+βdecay of T−(solid) and a model tail with Ze= 1.3074
+(dashed). The inset shows the probability density on a linea r
+scale.
+With theexpectationvalueof/angbracketleftbig
+r−1/angbracketrightbig
+andthe groundstate
+energyE0(T−) reported by Frolov in [17] the mean ex-
+citation energy calculated with (15) is E= 27.469eV.
+Frolov also reported the expectation values for the case
+of a finite mass of the tritium anion nuclei. Again us-
+ing (15) for this case one obtains E′= 27.479eV. This
+comparison confirms the quality of the final-state distri-
+bution obtained in this work and validates furthermore
+the use of the approximation of an infinitely heavy nu-
+cleus approximation.
+Frolov noted in [17] that his calculated bound-state
+probabilityappears to imply a continuum contributionof
+about 30%. Since this value is about 10 time larger than
+the continuum probabilityknownfor neutralT atoms, he
+speculated that in fact a large number of decays (about
+15to20%)mayend upintripletstatesofhelium, leaving
+a much smaller fraction in the singlet continuum. The
+reasoning in [17] is that the more diffuse ground state
+of T−is not sufficient to explain such a large continuum
+contribution, as follows from a comparison to results ob-
+tained for Rydberg states of neutral T atoms. On the
+other hand, triplet states may be populated by the (vir-
+tual) interaction with the βelectron omitted in the sud-
+den approximation. This argumentation is, however, er-
+roneous. As has been discussed in detail in [21], the sum
+rule for the sudden approximation gives always unity, in-
+dependently on higher-order corrections to it. This is
+also (as discussed above) fulfilled by the present calcu-
+lation which indeed confirms the about 30% continuum
+probability indirectly found but rejected in [17].
+Finally, the exchange interaction is expected to be
+much smaller than the direct one (in [30] it was found
+for atomic tritium to be by a factor η2smaller), but al-ready the direct term (first term beyond the sudden ap-
+proximation) is by a factor η2smaller than the sudden
+approximation. Close to the end point of tritium βdecay
+one finds for the Sommerfeld parameter η≈ −0.0271and
+for T2it was explicitly shown that the first-order correc-
+tion to the sudden approximation is itself of the order
+ofη2and thus of the order of 0.01% [11], in accordance
+with corresponding system-independent sum rules given
+in [21].
+IV. CONCLUSION
+In this work the complete final-state probability distri-
+bution of He following the nuclear β−decay of tritium
+anions has been calculated. For the small number of
+bound states considered previously in [17] the agreement
+is very good for the dominant ground and first excited
+states, but especially for the (very weakly populated)
+n= 3 state a deviation by about an order of magnitude
+was found. It is preliminarily attributed to a possible
+typographical error. The agreement with an earlier rel-
+ativistic multi-configuration Dirac-Fock calculation [18]
+is on the other hand very poor. Since such a large size
+of relativistic effects is not expected, especially not for
+the light nuclei involved, this deviation is attributed to
+a possibly too small basis set used in [18]. Nevertheless,
+the present study may stimulate further theoretical work
+to clarify the discrepancies to [17] and of both works to
+the relativistic ones in [18].
+In order to further test the accuracy of the present cal-
+culation, the bound-state energieswere comparedto very
+accurate literature data and were found to agree very ac-
+curately with them. Furthermore, the mean excitation
+energy obtained from the complete final-state spectrum
+was compared to the value predicted on the basis of clo-
+sure. Again, very good agreement was found. There-
+fore, the results of this work should be reliable and of
+direct importance for the tritium neutrino-mass experi-
+ment KATRIN that is presently under construction. In
+order to allow the use in the experimental analysis and
+for predicting how much a possible T−admixture to the
+T2source spoils the extracted neutrino mass, the contin-
+uum transition probability is given in binned form, but
+also available numerically on request. Finally, it may be
+noted that a controlled admixture of T−to the tritium
+source may in fact be used for the analysis of the experi-
+mental sensitivity to the atomic and molecularfinal-state
+spectrum.
+Acknowledgments
+The authors acknowledge financial support from the
+Stifterverband f¨ ur die Deutsche Wissenschaft and the
+Fonds der Chemischen Industrie .7
+[1] E. Fermi, Z. Phys. 88, 161 (1934).
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+K. Szalewicz, Phys. Rev. Lett. 55, 1388 (1985).
+[7] B. Jeziorski, W. Ko/suppress los, K. Szalewicz, O. Fackler, and
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+(1988).
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+242 (2000).
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+[16] KATRIN Collaboration, KATRIN Design Report 2004
+(2005), URL http://www-ik.fzk.de/ ~katrin .
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+[18] M. R. Harston and N. C. Pyper, Phys. Rev. A 48, 268
+(1993).
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+[20] A. B. Migdal and V. P. Krainov, Approximation Methods
+in Quantum Mechanics (W. A. Benjamin Inc, New York,
+USA, 1969).
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+[22] R. Saha, P. Chaudhury, and S. P. Bhattacharyya, Phys.
+Lett. A291, 397 (2001).
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+4589 (1997).
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+(1983).
\ No newline at end of file
diff --git a/1001.0121.txt b/1001.0121.txt
new file mode 100644
index 0000000000000000000000000000000000000000..dcb67f21cf7c814d4fefe6751bd41e3c364a2a49
--- /dev/null
+++ b/1001.0121.txt
@@ -0,0 +1,1617 @@
+To be submitted to Phys.  Rev. B 
+ 
+Finite s ize and intrinsic field effect on the polar-active pro perties o f the ferro electric-
+semiconductor heterostructures 
+ 
+A.N. Morozovska1*, E.A. Eliseev1,2, S.V. Svechnikov1,  V.Y. Shur3, A.Y. Borisevich4, 
+P. Maksymovych4, and S.V. Kalinin4†, 
+ 
+1Institute of Semiconductor Physics, National  Academy of Sciences of Ukraine,  
+03028, Kiev, Ukraine 
+ 
+2Institute for Problems of Materials Science, National Academy of Sciences of Ukraine,  
+03142, Kiev, Ukraine 
+ 
+3Institute of Physics and Applied Mathematics, U ral State University, 
+ 620083, Ekaterinburg, Russia 
+ 
+4Oak Ridge National Laboratory, 37831, Oak Ridge, Tennessee, USA 
+ 
+Abstract  
+Using Land au-Ginzburg -Devonshire approach  we calculated the equ ilibrium  distributions  
+of electric field, polariza tion and space charge in the ferroel ectric-sem iconductor heterostructures 
+containing proper or incipient ferro electric thin f ilms. The role of  the polarization gradient and 
+intrinsic surface energy , inte rface dipoles and  free charg es on polarization dy namics are 
+specifically  explored.  The intrinsic field eff ects, which originated at the ferroelectric-
+semiconductor in terface,  lead to th e surface b and bendin g and result in to th e form ation of 
+depletion space-ch arge layer near the sem iconductor su rface. During th e local polarizatio n 
+reversal (caused by the inhom ogeneous electric  field induced by the nanosized tip of the 
+Scanning P robe Micro scope (SP M) probe) the th icknes s and charge of the interface lay er 
+                                                 
+* morozo@i.com .ua 
+† sergei2@ornl.gov  
+ 1drastically changes, it particul ar the sign of the screening carriers is dete rmined by the 
+polarization direction. Obtained analytical solutions could be extended to analyze polarization-
+mediated electronic transport.   
+ 
+1. Introduction  
+Polar d iscontinuity a t the interf aces induced  either by tra nslational s ymmetry breaking of a 
+ferroelectric m aterial or ioni c charge m ismatch between co mponent can produce intriguing  
+modification  of the of the interf acial electron ic states and po larization of the ad jacen t materials  
+[1, 2, 3, 4]. The representative in terfacial phenom ena aris ing from the interpla y between  polarity 
+and electro nic structure include two-dim ensional electro n gases at the in terface of band 
+(LaAlO 3/SrTiO3) [2] or band and Mott insulators [5 ] and polarization-controlled electron 
+tunneling across ferroelectric-sem iconduc tor or bad-m etal interfaces (PbTiO 3/(La,Sr)MnO 3, 
+BaTiO 3/SrRuO 3). [6, 7, 8, 9] These novel physical phenom ena e merging in oxide m aterials at the 
+nanom eter scale hold strong poten tial for novel devices. Correspondi ngly, the theoretical insight 
+into th e ep itaxial interfaces of norm al and in cipient ferroelectrics  with bad  metals and 
+semiconductors and interplay between atom istic phenom ena at in terfaces and m esoscopic 
+potential and field distribu tions is acutely needed.  
+As an illus trative exam ple, the no torious pr oblem  of the Schottky ba rrier in ferro electric 
+films is still widely debated, with the key question of whether sub-100 nm  film s are fully 
+depleted, or that the width of the depletion regi ons is in the sub-10 nm  range due to the overall  
+high density >1020 cm-3 of shallo w and deep donor and/or accep tor levels in the film , and 
+particula rly in the  interfacial regions. [10] More importantly, only severa l previou s works, such 
+as the concept of a fer roelectric Schottky diode [11] and the dielectric non-linearities in the 
+epitaxial PZ T film s [12, 13], have em phasized the effect of spa ce-charge  layers on polarization 
+distribution and dom ain switching dyna mics inside ferroelectric film s. 
+In the current paper we present analytical calculation s of the polar-ac tive pro perties  
+(including local polarization revers al) in the pr oper and inc ipient ferroelectric -dielectric thin films 
+within Landau-Ginzburg-Devonshire phenom enologica l approach, with a speci al attention to the 
+polarization  gradient and intrinsic surface en ergy, in terface dipo les and free charges.  We 
+analyzed th e influence of finite s ize effect on the intrin sic electric field and polar-active 
+properties of the ferroelectric- semiconductor heterostructures.  Although  the stab ility of  the 
+ 2spontaneous polarization in the sy stem  ferroelectric film  /insula tor/sem iconductor was previously  
+studied within LGD approach [14], the band st ructure of the system , interface ch arges and 
+dipoles, th e polarizatio n gradient and intrinsi c surface energy of ferroelectric fil m were 
+previously ignored. Hence, this work provi des a fram ework to link the m esoscopic LGD-
+semiconductor theory to the first–pr inciple calculations that can reve al the electrosta tic details of 
+the in terface structure. 
+The paper is organized as follows. After st ating the problem  in Section 2, analytical 
+solution s for pola rization, elec tric potential, f ield and space charge dist ributions in the m odel 
+heterostructure are presented in the Section 3.1. The results of the stable ground state calculations  
+for SrTi O3/(La,Sr)MnO 3 (STO/LSMO) and BiFeO 3/(La,Sr)MnO 3 (BFO/LSMO) heterostructures 
+are presented Section 3.2. Metastable states ar e considered in Section 3.3. The effect of the  
+incom plete screen ing a nd inte rface charge on  the local polarization reversal and dom ain 
+formation caused by the electric field of the SPM  tip is studied in S ection 4. The tunneling 
+current density is estimated, prov iding an analytical app roach to  quantify recent experim ental 
+measure ments of polarization- controlled tunneling [6,7,8,9].  
+ 
+2. The problem statement 
+Here we consider an as ymm etric he teros tructure consisting of a narrow-gap (or m etallic) 
+semiconductor and a thin ferroelectric film  of thickness L. We will con sider the tw o cases of  the 
+proper and incipient ferroelectric films, both either a wide-gap se miconductor or a dielectric (i.e. 
+semiconducting properties of ferroelectric are neglected).  In the initial state, the external bias is 
+absent and the free ferroelectric surface z = −L is com pletely screened by the am bient sluggis h 
+charges [Fig. 1a]. Then inhom ogeneous external bias U(x,y) is applied to the tip electrode. The 
+bias in creas e may cause local polarization reversal below the tip that f inally results in to 
+cylind rical dom ain formation in thin f erroele ctric f ilm [see the f inal state in Fig. 1c]. Note that 
+while we consider the tip-indu ced polarization switching, the obtai ned solutions are applicable 
+for capacitor geom etry in the lim it of uniform  potential. 
+ 
+ 3 
+ 
+L ϕ(z) 
+WS 0 z(Incipient)  
+ferroelectric film semiconductorUb
+−Lϕ=0 
+ϕ=0 Ub σf
+σS
+1 L P3 P3 x   ϕ 
+semiconductor elec trode  
+2 
+H 
+L z 
+P3 P3 x
+ferroelectric   ϕ 
+semiconductor elec trode conducting SPM probe tip  
+Udielect ric gap 3 
+2 
+domain  1 (incipient) ferroelectric film Initial  state 
+Ambient screen ing charges σf 
+z 
+(a) (b)
+(c) H
+ztip 
+Ferroelectric 
+film 
+gapL (d)
+ Final  state 
+Fig. 1.  (Color online). (a) The initial state of the con sidered h eteros tructure : 
+semiconductor/(incipient or proper) ferro electric-dielectric fil m of thickness L. P3 is the 
+ferroele ctric pola rization. (b)  Sketch of  the  elec trostatic poten tial distr ibution:  ϕ is the  
+electrostatic poten tial, U is the contact potential differe nce on ferroelectric-sem iconductor 
+interface, Wb
+S is the electric field penetration depth into  the sem iconductor, i.e.  the thickness of the 
+space charg e depletion layer (shown  by dashed and solid lines for different charge signs). (c) Th e 
+final state o f the ferroelectric film  is the loca l polarization  reversal ca used by the biased SPM  
+probe. (d) D uring the tip-induced polarization reversal an ultra-th in dielectric gap H between the 
+tip electrod e and the ferroelectric surface coul d exist. Sq uares denote the screening surface 
+charges σf and the opposite ch arge acco mmodated at the tip surface. 
+ 
+The contact potential difference Ub at the interface z = 0 originates from  the band 
+mismatch between ferroelectric and sem iconductor,  the interface bonding e ffect and interfacial 
+ 4polarity [Fig. 1b]. The band bending (or intrinsic field  effec t) in the sem iconductor leads to the 
+depletion (o r accum ulation) charg ed layers of th ickness WS with space ch arge ρS.  
+ The screening interface charg e σS could originate at z = 0 self-consis tently in the c ase of 
+the bad screening from  the sem iconductor side (i.e . for thick depletion layer created by the m inor-
+type carriers). The non-ideal screening that ca uses the strong depolarizat ion field controls the 
+self-consistent m echanism. The field  decreases th e polarization insid e the ferroelectric film  and 
+incre ases th e free energ y of the system , since depol arization f ield energy is always po sitive. As a  
+result, the strong field effect m ay lead to the bend bending at z = 0 and appearance of charge 
+states at the interface.  The interface charges  σS of appropriate sign prov ide effective screen ing of 
+the spontaneous polarization, m ake it m ore hom ogeneous and thus decr ease the depolarization 
+field, which  in turn  self -consisten tly decr eases  the system free en ergy. The den sity of th e 
+interface ch arge σS depends on energetic positi on of chemical poten tial µ at the s urface that 
+modifies the Shottky barrier. The potential µ is manly determ ined by the interface lay ers with  the 
+energy density NS (per unit energy ) of quasi-co ntinuous surface states and Ferm i level EF at the 
+neutral surface [15, 16, 17]. 
+We assum e that in the initial state the slugg ish surface ch arges σf com pletely screen the 
+electric dis placem ent outside the film , i.e. ),,( ),,(3 LyxD Lyxf −−=−σ , ()0 ,,=−ϕ Lyx  and 
+. This behavior is analyzed in Section 3. In  contras t, rech arging of th e surface 
+charges σ0) (3=−<L zD
+f should appear during the polarization reve rsal. The ultra-thin dielectric gap of 
+thickne ss H models the resistive pro perties of the sluggish surface charg es σf, contam ination or  
+dead layer. Corresponding free im age charges −σf are acco mmodated at th e condu cting SPM  tip 
+surface. W ithout loss of generality one can assu me that th e equilibriu m dom ain structure is 
+almost cylin drica l for the case of  complete loc al polar izatio n revers al in thin f erroelectric f ilm. 
+The assum ptions sign ificantly s implify the pro blem considered in the  Section 4, and allows 
+developing the analytical descri ption for the dom ain form ation. 
+Hereinafter we assum e that the time of ex ternal field changing is sm all enough for the 
+valid ity of  the quasi- static app roxim ation 0≈Erot . Maxwell's equations for the quasi-static  
+electric field  E  and displacem ent D inside sem iconductors have the for m: ϕ−∇=
+() )(0 ϕρ=+ε= PE Ddiv div .     (1) 
+ 5Here the  electrostatic po tential ϕ(x,y,z) is determ ined by ex ternal b ias as well as  by contact and 
+surface effects. The potential dete rmines the free carrier density )(ϕρ  determ ined by the 
+concentratio n of holes in the valence band, el ectrons in th e conduction band, and accepto rs and 
+donors at their respective leve ls in the band gap [16]. 
+The ferroelectric film  that occupies the reg ion 0<<− zL  is transv ersally isotropic,  i.e. 
+perm ittivity ε11 = ε22 at zero electric field. W e further a ssum e that the dependence of in-plane 
+polarization com ponent s on  can be linearized  as 2,1E ()2,11E11 0 2,1P −εε≈  (ε is the universal 
+diele ctric co nstant), while the  pola rization com ponent  nonlinearly depends on external field. 
+Thus corresponding polarizatio n vector acquires the form:  0
+3P
+()() () ()() ( )3E33 0 3 11 0 1 11 0 1 , , ,1 P Eb−εε+ −εε−εε= rE rP21E  [18]. 
+ Within the  framework of  the  Landau- Ginzburg-Devonshire (LGD) theory, quasi-
+equilibrium  polarization distribution  P3(x,y,z) in the f erroelectric f ilm with the spatial dispers ion 
+should be found from  the Euler-Lagrange boundary problem : 
+
+
+
+=
+−=
+
+
+∂∂λ− −=
+=
+
+
+∂∂λ+∂ϕ∂−=
+
+
+
+∂∂+∆−β+α⊥
+.0 ,
+0,
+3
+2 33
+1 33 22
+3
+3 3
+LzzPP P
+zzPPzP
+zg P P
+b  (2) 
+Hereinafter we introduced a tr ansverse Laplace operator 22
+22
+y x∂∂+
+∂∂=∆⊥ .  
+The tem perature-d epend ent co efficient α is pos itive f or incipient ferroelectric and  proper 
+ferroele ctrics in para electric phase,  while α<0 for proper ferroele ctrics in ferroe lectric phas e, 
+β > 0 for the second order ferroelectrics co nsidered  hereinafter,  gradient coefficien t g > 0. 
+Extrapolation lengths λ1,2 originate from  the surface energy co efficients in the LGD-free energy.  
+Inhom ogene ity Pb describes the effect of the inte rface polarization s temming from the 
+interface bo nding effect and associated in terface dipole [ 19, 20]. More generally, the translation 
+symmetry b reaking  inevitab ly pres ent in th e vicin ity of the any  interface will give rise to 
+inhom ogeneity in the bo undary cond itions (2) [ 21, 22]. 
+ Eqs. (1)-(2) yield the co upled syste m: 
+ 6.0 ),(,0 ,)(1, ,0
+22
+03
+011 22
+3322
+>ϕρ−=
+
+
+ϕ∆+∂ϕ∂εε<<−
+
+
+ϕρ−∂∂
+ε=ϕ∆ε+∂ϕ∂ε−<<−−=
+
+
+ϕ∆+∂ϕ∂
+⊥⊥⊥
+zzzLzP
+zL zLHz
+S Sfb                       (3) 
+The background dielectric perm ittivi ty of (incipient) ferroelectric ε (typically ε ); b
+33b
+33 33ε>>Sε is 
+the sem iconductor (bare) lattice perm ittivity.  
+Eqs.(3) are supplem ented with the boundary conditions at z = −L−H, z = −L, z = 0 and 
+z = +∞, nam ely  
+( ) ),( ,, yxU HLyxe=−−ϕ ,  ()()0 ,, 0 ,, −−ϕ=+−ϕ Lyx Lyx , ( )0 ,, =∞→ϕ zyx , (4a) 
+()()bU yx yx =−ϕ−+ϕ 0,, 0,, ,      (4b) 
+),()0,,()0,,()0,,(
+0 3 330 yx
+zyxyxP
+zyx
+S Sbσ=
+∂+ϕ∂εε−−−
+∂−ϕ∂εε ,  (4c) 
+),()0 ,,()0 ,,()0 ,,(
+033 3 330 yxzLyxLyxPzLyx
+fg bσ=∂−−ϕ∂εε++−+∂+−ϕ∂εε− . (4d) 
+Whe re Ub is the contact potential difference at  the dielectric-sem iconductor in terface.   is the 
+dielectric co nstant of th e diel ectric gap between the tip  and ferroelectric su rface. The poten tial 
+distribution  produced by the SPM tip is assum ed to be almost constant in the surface  
+spatial r egion much larg er then the f ilm thickness . g
+33ε
+),(yxUe
+ 
+3. Solution for polariz ation, electric potent ial, field and space ch arge distributions  
+3.1. Analytical solution s for the on e-dimension al case 
+Here we calculate the potential and polarizat ion distribution in the initial ground state in 
+the one-d imensional cas e Ue = const that corresponds to the plai n electrodes. The case is realized 
+in paraelectric or incipient fe rroelectric film  as well as in the m onodom ain state of the proper 
+ferroele ctric thin f ilm. 
+The space charge density inside the doped p-ty pe (or n-type) sem i-infinite sem iconductor 
+has the form 
+ 7()()()() ( )
+() ()
+() () . ,, ,, )(
+00
+
+
+
+ ϕ−−⋅=ϕ
+
+
+ ϕ−−⋅=ϕ
+
+
+ ϕ+−⋅=ϕ
+
+
+ ϕ+−⋅=ϕϕ−ϕ−ϕ+ϕ=ϕρ
+−+− +
+Tkq E EFN NTkq E EFN nTkq E EFN NTkq E EFN pN n N pq
+BF a
+a a
+BF C
+eBd F
+d d
+BV F
+pa d S
+                   (5) 
+Where  is th e Ferm i-Dirac distribution function, q is the abso lute va lue of  the 
+carrier elem entary ch arge. E()()(11 exp−+θ=θF
+ρS
+())
+F, EV, EC, Ed and Ea are the energies of Ferm i level, valence band, 
+conductance band, dono r and accep tor levels in  the quasi-n eutral regio n of the sem iconductor 
+correspondingly. Since  in the quasi-neutral region of the sem iconductor, where  
+, the identity 0)(→ϕ
+() 0→ϕ ()()00 0 =−−
+aN n
+const Na≈−
+E EV F0−+
+dN
+const0+p
+Nd≈+
+Tk qB F>>ϕ− should b e valid. Th e iden tity along with  
+typical assum ption ,  and Boltzm an approxim ation for electrons 
+ or holes E EC− Tk qB>>ϕ+−  lead to expressions 
+
+
+
+
+−
+
+
+ϕ− 1 exp0
+Tkq
+BS≈ρ qpS  or 
+
+−1
+
+
+
+
+ϕ− exp0
+Tkqqn
+BS≈ρS  correspondingly, where  and  are 
+equilibrium  concentrations of holes and el ectrons in the quasi-n eutral region of the  
+semiconductor [23].  0
+Sp0
+Sn
+Then in depletion layer (or abrup t junction ) approxim ation the space ch arge density  near 
+the interface of the strongly doped p-type (or n- type) sem i-infinite sem iconductor has the form   
+
+><<ρ
+≈ϕρ
+SS S
+SWzWz
+,00,
+)(0
+                                                               (5b) 
+Hereinafter the choice o f the charge density ρ  and depth W0 0
+S Sqp=Sp SW=  (or ρ  and 
+) is determ ined by the sign of potential (i.e. by the sign of charge in depletion layer). 
+More rigorously, for the definite type of car riers the thickn esses of  the depletion layers  W0 0
+S S qn−=
+Sn SW W=
+S (i.e. 
+the field penetration depths) should be determ ined self-consistently from  the exact solution of the 
+system  (3)-(5).  
+The ferroelectric film is regarded as a wide-gap proper sem iconductor or alm ost 
+diele ctric, so its space -charge dens ity is neg ligibly sm all: 0)(≈ϕρf  at 0<<− zL . The nonzero 
+1D-solution of Eqs. (3) with respect to the boundary conditions (4 a) and (4c-e) is 
+ 8()()
+() ( )
+()
+
+
+
+−<≤−−ρ−σ+σ
+εε+++<≤−ρ−σ+σ
+εε++
+
+
+
+εεσ−ρ+−
+εε>−θ−
+εερ−
+≈ϕ∫
+−
+. ,,0 ,~)~(,0 ,
+2
+)(
+0
+3300
+330 3300
+33032
+00
+L z HL WzLHUzL WHUWzLzdzPz z W z W
+z
+S S f S g eS S f S g e bS S Sz
+LbS S
+SS
+     (6) 
+Here θ(z) is the step-function. So the sem iconductor pot ential and space charge  are distributed in 
+the lay er  and zero outside. Approxim ate expressions (6) for the potential ϕ 
+correspond to parabolic approximation valid in the depletion/accumulation lim it, at that 
+ or W  depending of the m ain carriers n or p-type. Note,  that in th e 
+accum ulation regim e the interface ch arge σSWz<<0
+Sn>>Sp Sn W W<<SpW
+S is lo calized in th e thin dep letion layer  of several nm  
+(for oxide electrodes) thickness WS that is occupied by the m ain-type carriers. The opposite case  
+of charged layers created by the m inor-type carr iers, which  can appear during the polarization 
+reversal, could lead the strong band bending and accommodation of σS.  
+ From  Eq.(6) we deri ved the e lectric field E3 and electrical displacem ent D3 distributions  
+in the pa rabolic app roxim ation: 
+()() ()() .0 at, )( , )(,0 at, )( ,)()(
+0
+3
+00
+30
+3
+3300
+3303
+3
+> −θ−ρ= −θ−εερ≈<<−σ−ρ=εεσ−ρ+εε−≈
+z z W Wz zD z W Wz zEzL W zDW zPzE
+S S S S S
+SSS S S bS S S
+b
+         (7) 
+Note, that the free screening charg e )(3L Df−=σ−  should also originate at another interface 
+z = −L. Thus the condition for electroneut rality of the whole system  is . 00=σ+ρ+σ−f S S S W
+ Allowing for Eqs.(6)-(7) polarization distribution  P3(z) was found from  the Euler-
+Lagrange boundary problem  (2) as described in Appendix A. The polarization distribution 
+acquires the for m: 
+()() 0 ,1)(0
+3 ≥−θ⋅−ρε−ε= z z W Wz zPS S S
+SS,    (8a) 
+()() 0 ),,( ,
+1 32)(
+2
+3 3300 3
+3 330
+3 ≤<− ⋅−
++β+αεεσ−ρ+βεε= zL LzbPLzf
+PW PzPbbS S Sb
+. (8b) 
+ 9Param eter 3Pin Eq.(8b) is the polarization averaged over the film  depth: ∫
+−≡0
+3 3~)~(1
+LzdzP
+LP . 
+Its spatial distribution is governed by the functions f and b: 
+()()() ()()()() ( )
+() ()()()ξ λ+λξ+ξ λλ+ξξ−ξ+ξ+ξλ+ξ+ λξ−=L Lz zL z zLLzfcosh sinhsinh sinh cosh cosh1 ,
+2 1 2121 2,      (9a) 
+()()()()()
+() ()()()ξ λ+λξ+ξ λλ+ξξ+ξ+ξ+ ξλ=L LzL zLLzbcosh sinhsinh cosh,
+2 1 2122
+2.                               (9b) 
+Characteris tic length gb
+330εε≈ξ .  
+ The average polarizatio n 3P and depths WSn,p should be deter mined self -consistently 
+from  the spatial averaging of Eq.(8b) and the boundary conditions ( 4b). After elem entary 
+transform ations we obtained the system  of two c oupled algebraic equations: 
+()
+()
+
+
+εε−
+
+
+
+εεσ−ρ=−β+
+
+
+
+εε+α
+
+
+ρ−σ+σεε++εε−ερε+σ−ρ=
+. 231,2
+330 3300
+3
+3 3
+3300
+330330 20
+33 0
+3
+bPfWf P PWHU ULWLW P
+bb
+bS S S
+bS S f S g e bb
+S
+SSb
+S S S
+                   (10) 
+Where the average values ( )
+()( )212
+2 12 1221λλ+ξ+λ+λξλ+λ+ξξ−≈Lf  and ()12
+λ+ξξ≈Lb . 
+In particular case of narrow-gap m etallic  sem iconductor and thick enough ferroelectric 
+film the strong inequality W  is valid. This  leads to  the line ar approxim ation LS<<
+()
+
+εεσ
+++
+
+
+
+εε+σ−ρ≈gf
+e b gb
+S S SH
+U
+LHW P
+330 3333 0
+3 1
+εε−b
+U
+L330 and th e cub ic equation  for the av erage 
+polarization: 
+() ()(HLE HLE f P PH LfLf
+ef
+b b gg
+b, , 23 11 3
+3 3
+33 3333
+330+ =−β+
+
+
+
+
+
+
+
+ε+εε−εε+α )
+).           (11a) 
+Where the b uilt-in electric field  and extern al field (HLEf
+b, ()HLEf
+e, are in troduced as: 
+() bP H
+U
+H LfHLEbb
+gf
+b b gg
+f
+b
+330 330 33 3333,
+εε−
+
+
+
+εεσ
++
+ε+εε= ,                (11b) 
+ 10()H LUfHLEb geg
+f
+e
+33 3333,ε+εε= .                                              (11c) 
+Note that Eq.(11a) th is is equ ivalent to  the equation  for ferroelectric polarization 
+hysteresis loop in the uniform  electri c field, while the built-in field  determ ines the 
+horizontal imprint of the loop. Thus for the case considered by Eqs.(11) the symm etric intrinsic 
+coercive fields (HLEf
+b,)
+βα−±=3
+332b
+cE  of a bulk m aterial [24] becom e asymmetric and has the form. 
+() ()()()3
+33 3333
+33011
+231
+332, ,,
+
+
+
+
+
+
+
+ε+εε−εε+α−β−± −=±
+H LfLTfHLE THLEb gg
+bf
+b c .       (12) 
+Renorm alization of the coefficient α in Eq.(11 ), i.e. th e term 
+
+
+
+ε+εε−
+εε H LfL
+b gg
+b
+33 3333
+33011, 
+quantitatively reflects th e “ex trinsic contribution” (factor 1 0
+33 3333<
+ε+εε<
+H LL
+b gg
+) orig inated f rom 
+the depolarization field produced by the finite gap H and the “intrinsic contribution” (factor 
+1 0≤<f ) origina ted from the fini te extrapolation lengths λi < ∞ and intrins ic polar ization 
+gradient ( 1=f  for either g=0 or both λi = ∞). Thus Eq.(11) allows ri gorous estim ations of both 
+extrinsic an d intrin sic c ontribu tions  into the re norm alizatio n of the transition tem peratu re into 
+parae lectric phase (com pare th e resu lt with Ref.[25]). 
+ For proper ferroelectrics the coefficient ()*)(c T TT T −α=α , where T is th e absolute  
+temperature and T is the  Curie tem peratu re poss ibly r enorm alized by m isfit strain originated 
+from  the film  and sem iconductor substr ate lattice m ismatch. For instance, *
+c
+1111 12 * 2
+cqc quT T
+Tm
+c c−
+α+=1112c, where um is a m isfit stra in, the s tiffness tensor cijkl is pos itively 
+defined, qijkl stands for th e electros triction stress tensor.  
+Hence, the critica l thic kness (as well as th e critical tem peratu re) of the size-ind uced  
+transition of the ferroelectric film into par aelectric phase can be found from  the condition 
+0 11
+33 3333
+330=
+
+
+
+ε+εε−εε+αH LfL
+b gg
+b as: 
+ 11()( )
+()( )
+
+
+
+ε+
+λλ+ξ+λ+λξελ+λ+ξξ
+αε−≈g b crH
+TTL
+33 212
+2 1 332 12
+02 1)( ,                                        (13) 
+()()
+()() 
+
+
+
+
+
+
+
+λλ+ξ+λ+λξλ+λ+ξξ−ε+εε−εεα−≈
+212
+2 12 12
+33 3333
+330* 21 11
+L H LLTLTb gg
+b
+Tc cr .             (14) 
+ For unstrain ed incip ient ferroelectric film the coefficient α(T) is pos itive up to zero 
+temperature s and typ ically is gi ven by Barret’s for mula, thus the critical tem perature (as well as  
+the cr itical thickne ss) d oes not ex ist, sinc e the film rem ained parae lectric up to ze ro Kelvin. 
+However f or the str ained film it m ay becom e positiv e, indica tive o f the transition to th e 
+ferroelectric state [26]. 
+For particular case H = 0 (gap is absent) the build-in field Ebf is inversely proportional to 
+the f ilm thickness, () LB LEf
+b≈0, , while its value depends on the built-in polarization Pb, 
+surface ch arge σf, and contact pot ential dif ference Ub [see Eq.(11b) and the dashed alm ost 
+straight line  in Fig. 2a]. The build-in field Ebf leads to the vertical asymmetry and horizontal 
+imprint of the polarization hysteresis loops in ferro electric film s of thickne ss m ore than critical 
+ [see regions 2 and 3 in Fig.2a and Fig. 2b]. Field-indu ced p olarized s tate appears at 
+film thickness less than the critic al one )(TLLcr>
+)(TLLcr<  [see regions 5 and 6 in Fig.2a and Fig. 2c].  
+For typic al ferroelectric m aterial pa rameters approxim ate expressions for the right and left 
+coercive b iases are ()()()()( )31 ,0, LTL TELB TLEcrb
+c c − ±−≈± [see Eq.(12)]. Thus solid 
+curves in Fig. 2a look like an asymm etric “bird beak” with the tip ()(cr cr c LB LLE −≈=±)
+() and 
+asym ptotes . The built-in field sign and thickness dependence determ ine 
+the beak «up» or «down» asymm etry and shape correspondingly.  () TE L LEb
+c cr c ±→>>±
+  
+ 12 
+ 
+0.2 0.4 0.6 0.8 1 
+-1.5  -1 -0.5 +0.5 +1 1.5 
+0 Up state is stable  
+Down st ate is metastable Eef/Ecb 
+Right coercive field   Ec+ 
+Left coercive field   Ec− 
+Down state is absolutel y stable
+Up st ate is unstable Up state is absolute ly stable.  
+Down state is unstable 
+Down st ate is stable  
+Up state is metastable Imprint field  −Ebf
+Lcr/L Up state  
+5 6 4 
+3 
+2 
+1
+Eef 4
+3 2 
+1 56
+Eef Up  P3 
+Down P3 Ebf P3 P3 (a)
+(c) (b)Down state  
+ Built-in field-induced 
+polarization 
+(but no hy stere sis) 
+Fig.2.  (Colo r online).  (a) Diagra m in coordin ates “field − inverse thickn ess”, 
+
+
+LL
+EEcr
+b
+cf
+e, , that 
+shows the stability of the “up” ( 03>P ) and “down” ( 03<P ) polarization states in a 
+ferroelectric fil m. Dashed curve is  the thickness dependence of the built-in field ()LEf
+b−  
+calculated from  Eq.(11b); solid curves are thickness dependences of right and left coercive fields 
+ calcu lated from  Eq.( 11b) and (12) at H = 0, fixed temperature and extrapolation lengths. 
+ is the ab solute v alue of the bulk c oerciv e field, L()LEc±
+b
+cE cr is the  film critic al thickness  given by 
+Eq.(13). (b-c) Hysteresis loops schem atics in a fe rroelectric phase (b) and field-induced polarized 
+state (c). 
+ 
+ Note, that there m ay be two possible soluti ons corresponding to th e polarization up and 
+down. Region 4 in Fig.2a corresponds to the stable “up” polarization states ( 03>P ), metastab le 
+ 13states are absent here. The situation is vise versa in the region 1. Re gion 6 corresponds to the 
+“up” polarization. The situation is  vise versa in the region 5. Hyst eresis loops are absent in the 
+thickness regions 5 and 6, since at thicknesses crLL<  the loop s never ex ist, h ere renorm alized 
+coefficient α becom es positiv e and coerc ive bia s given by E q.(12) is co mplex value .  
+Bistability of polarization stat es exists only in the regions 2 and 3. “Up” states are 
+absolutely s table in th e region 3, while th e “dow n” state is m etastable h ere. The situ ation is v ise 
+versa in the region 2. The depletion lengths WSn,p are at lea st sever al tim es dif ferent f or to the 
+“up” and “down” polarization states. 
+In the next s ection s we will show how the prese nce of  build-in f ield Ebf the sm ear the size-
+induced phase transition and i nduces polarization in the inci pient ferroelectric f ilms. 
+ 
+3.2. Calculations of the stable grou nd state for typical heterostru ctures (1D-case, U e = 0) 
+Here we calculate the potential an d polarization distribution in the absolutely s table 
+ground state, m etastable states will be discussed in the next secti on. F irst, we derive the field 
+structure for the cas e when the ex ternal b ias is  absent ( Ue = 0). This one-dim ensional case is 
+realized in paraelectric or  incipient ferroelectric film  as well  as in the monodom ain state of the  
+proper ferro electric th in film . We assum e that the surface ch arge fσ localized at  should 
+provide the full screening of th e spontaneous polarization outsi de the film  and m inimize the  
+depolarization field energy, i. e. they acts as  a perfect electrode and  thus provid e L z−=
+,,() 0=−ϕ , 
+ and . Thus we put H = 0 for the ground states calculations in 
+Sections 3.2 and 3.3. Lyx
+f L D σ−=−)(3 0) (3=−<L zD
+ From relations (10) one can determ ine th e thickness dependence of penetration depths 
+WSp,n and polarization 3P for different extrapo lation lengths, since the quantities Ub, nS0, pS0 εS 
+and ε33b can be regarded as known material pa rameters. LGD-expansion coefficients α, β and the  
+gradien t coefficient g are tabulated for the m ajority of proper and incip ient ferroelectrics. 
+Mater ial parameters use d in the calc ulations of the heterostructures SrTiO 3/(La,Sr) MnO3 
+(STO/LSMO) and BiFeO 3/(La,Sr) MnO3 (BFO/LSMO) polar propertie s are listed in T able 1.  
+ 
+ 
+ 14Table 1.  
+Mater ial Permi-
+ttivity  Carries 
+concentratio n 
+ (cm-3) Band gap 
+(eV) LGD-expansion 
+coefficients for 
+ferroele ctrics 
+LaSrMnO 3 
+(LSMO) 
+half-m etal εS = 30 
+[27] 1.83 1022  [28] 
+1.65 1021 [29] 
+(La 0.7Sr0.3MnO 3) 1 
+p-type 
+[30, 31] Non f erroele ctric 
+BiFeO 3 
+(BFO) 
+ferroele ctric ε33b = 30 
+ wide band-gap 
+semiconductor 3 
+ αT = 9.8⋅105 m/(F K) 
+Tc = 1103 K 
+β = 13⋅108 m5/(C2F) 
+g = 10-8 m3/F 
+SrTiO 3 
+(STO) 
+incip ient 
+ferroele ctric ε33b = 43 
+[32] diele ctric 3  
+(without 
+impurities) αT = 1.26⋅106 m/(F K), 
+Tc = 30 K, 
+β = 6.8⋅109 m5/(C2F) 
+g = 10-7−10-9 m3/F 
+ 
+Extrapolation lengths λi and in terface charge σS valu es are not lis ted in the table, since 
+they strongly depend on the interfacial states. Extr apolation length values could be extracted fr om 
+the po larization d istribution in f erroelectric nanosystem s (film s, wires,  etc) obtained either 
+experim entally [33, 34] or determ ined by t he firs t principle calculations [19, 20]. Extrem ely sm all 
+and extrem ely high values of extr apolation lengths describe the two lim iting cases of  the surf ace 
+energy contr ibution to th e tota l free energy. Extrem ely s mall values of λi correspond to com plete 
+suppression of the polarization on th e surface, while  extremely large ones – to the absence of the 
+surface en ergy dependence on po larization (so ca lled natu ral boundary conditions ). Allowing for 
+the rem ark below we consider two lim iting ca ses of the sm all and high values of  extrapolation 
+lengths.  
+The polar interface m ay produce a positive ionic charge lead ing to  displacem ents in  
+LSMO. Fi g. 3 shows  z-distributions of the electric polarization  in the heteros tructure 
+“ferroelectric BiFeO 3 film − half-m etal LaSrMnO 3” (BFO/LSMO) for different values  of 
+interface polarization Pb and interface charge σS. It is seen that the polarization distribution near 
+z = 0 and its asymm etry are the main effects of the in terface pol arization, while the in terface 
+charge creates the hom ogeneous electric field in side the ferroel ectric film. That is why the effect 
+of Pb on the polarization distribution is much weaker th an th e effect of σS. Below we consider  the 
+case Pb=0. 
+ 
+ 15 
+ 
+-20 -10 0 10 -0.3  0. 0.3  
+-20 -10 0 10 -0.3 0. 0.3  
+-20 -10 0 10 -0.3 0. 0.3 (a) Polarization  P3 (C/m2) 
+(b)
+Distance z  (nm) (c) (d)
+Distance z  (nm) Distance z  (nm) Distance z  (nm) LSMO 
+Polarization  P3 (C/m2) Polarization  P3 (C/m2) 
+Polarization  P3 (C/m2) 
+ 
+-20 -10 0 10 0 0.1 0.2 
+ BFO  
+Fig.3.  Polarization depth distribution (z in nm ) for the BFO/LSMO heteros tructure. Contact 
+potential d ifference at z=0 is Ub=0 V, interface polarizatio n Pb = −0.3, −0.2, −0.1, 0, 0.1, 0.2,  
+0.3 C/ m2 (curves from top to bottom ). Carriers  concentration in LSMO is m26 010=Sp-3, BFO 
+thickne ss L = 25 nm and the gradien t coefficient g = 10-8 m3/F. (a) Extra polation len gths λi=0 nm 
+and interface charge den sity σS = 0; (b) σS = 0.0 5 C/m2, λi=0 nm ; (c) σS = 0.1 C/m2, λi=0 nm ; (d) 
+σS = 0, λi=5 nm . 
+ 
+Fig. 4 shows z-dependence of the electric po larization, potential, field and bulk charge 
+density in the heterostructure of BFO/LS MO for di fferent  interface polarizatio n Pb, fixed 
+extrapo lation length and  interface ch arge σS.  
+ 
+ 16 
+ 
+-20 -10 0 10 -20  
+-20 -10 0 10 -8 -4 0 4  
+-20 -10 0 10 -30 3  
+-20 -10 0 10 -0.3 0. 0.3 (a) Polarization  P3 (C/m2) 
+(b)
+Distance z  (nm) Field  E3 (MV/ cm) 
+Char ge  (107C/m3) 
+(c) 
+Potentia l   ϕ  (V) 
+(d)
+Distance z  (nm) Distance z  (nm) Distance z  (nm) LSMO 
+ BFO  
+Fig.4.  (a) Polarization,  (b) po tentia l, (c) ele ctric field and (d) ele ctric cha rge density  z-
+distributions for the BFO/LSMO he terostru cture. Contact potential difference at z=0 is Ub=0 V, 
+interface ch arge density  σS = 0.05 C/m2 and interface polarization Pb = −0.3, −0.2, −0.1, 0, 0.1,  
+0.2, 0.3 C/m2 (curv es from top to bottom ). Carrie rs concen tration in  LSMO is m26 010=Sp-3, BFO 
+thickne ss L = 25 nm  and the grad ient coefficient g = 10-8 m3/F. 
+ 
+Figs. 5-6 show the spatial distribution of el ectric polarization, pot ential, field and bulk 
+charge den sity in th e heteros tructure BFO/LSMO for zero interface charge σS=0. W hen 
+generating the plots in Figs. 5 we put λi=0. Plots in Figs. 6 correspond to high enough λi values.  
+As anticipated the polarization distribution is almost hom ogeneous for the high extrapolation 
+length. For a  small extra polation len gth the polarization profile is m ore inhom ogeneous. 
+ 
+ 17 
+ 
+- 40 - 20 0 20 - 5 0 5 10  
+- 40 - 20 0 20 0 10 20  
+- 40 - 20 0 20 0 10 20 30  
+- 40 - 20 0 20 0 0.25  0.5 (a) Polarization  P3 (C/m2) 
+(b)
+Distance z  (nm) Field  E3 (MV/ cm) 
+Char ge  (107C/m3) (c) 
+Potentia l   ϕ  (V) 
+(d)
+Distance z  (nm) Distance z  (nm) Distance z  (nm) LSM O 
+ BFO  
+Fig.5.  (a) Polarization,  (b) po tentia l, (c) ele ctric field and (d) ele ctric cha rge density  z-
+distributions for the BFO/LSMO he terostru cture. Contact potential difference at z=0 is Ub=1 V, 
+interface polarization Pb = 0 and charge density σS = 0. Carrie rs concentr ation in LSMO is 
+m26 010=Sp-3. Black, red, violet and blue curves corr espond to different va lues of BFO fil m 
+thickne ss L = 20, 40, 80, 160 nm wi th extrapolation lengths λi≈0 nm  and the gradient coefficient 
+g = 10-8 m3/F. 
+ 
+ 18 
+ 
+- 40 - 20 0 20 - 0.5 0. 0.5  
+- 40 - 20 0 20 0 10 20 30  
+- 40 - 20 0 20 0 0.1 0.2 0.3 0.4 0.5 0.6 (a) Polarization  P3 (C/m2) 
+(b)
+Distance z  (nm) Field  E3 (MV/ cm) 
+Char ge  (107C/m3) 
+(c) 
+Potentia l   ϕ  (V) 
+(d)
+Distance z  (nm) Distance z  (nm) Distance z  (nm) LSMO 
+ 
+- 40 - 20 0 20 0 10 20 
+ BFO  
+Fig.6.  (a) Polarization,  (b) po tentia l, (c) ele ctric field and (d) ele ctric cha rge density  z-
+distributions for the BFO/LSMO he terostru cture. Contact potential difference at z=0 is Ub=1 V, 
+interface polarization  Pb = 0 and charge density σS = 0. Carrie rs concentr ation in LSMO is 
+m26 010=Sp-3. Black, red, violet and blue curv es correspond to different thickness L = 20, 40, 80, 
+160 nm  of BFO fil m with  extrapolation lengths λi=30 nm and the gradient coefficient g = 10-8 
+m3/F.  
+ 
+Figs. 7-8 illustrate z-distribu tions  of the e lectric pola rization, potential, field and bulk 
+charge den sity in the  hetero structure “incipient SrT iO3 film − half-m etal LaSrMnO 3” 
+(STO/LSMO) for zero interface charge σS=0. When generating the plots in Figs. 2 we put λi=0. 
+Plots in Figs. 6 correspond to high enough λi values. The resulting built-in field 
+ 19
+
+
+
+εε− bPfLUEbb b f
+b
+330~  (see Eq.(11b)) induces the electric polarization in the incipient 
+ferroele ctric films. 
+ 
+ 
+ 
+- 40 - 20 0 - 0.5 0. 0.5  
+- 40 - 20 0 0 1 2  
+- 40 - 20 0 - 1 - 0.5 0.  
+- 40 - 20 0 0 0.01  0.05  
+0.02  0.03  0.04  (a) Polarization  P3 (C/m2) 
+(b)
+Distance z  (nm) Field   E3 (MV/ cm) 
+Charge  (107C/m3) (c) 
+Potential   ϕ  (V) 
+(d)
+Distance z  (nm) Distance z  (nm) Distance z  (nm) LSMO 
+ STO  
+Fig.7.  (a) Polarization,  (b) po tential, (c) ele ctric field  and (d) electr ic charge density z-
+distributions for the STO/LSMO he terostru cture. Contact potential difference at z=0 is Ub=1 V, 
+interface polarization  Pb = 0 and charge density σS = 0. Carriers co ncentration in LSMO is 
+m26 010=Sp-3. Black, red, violet and blue curv es correspond to different thickness L = 20, 40, 80, 
+160 nm  of STO film  with extrapolation lengths λi≈0 nm  and the gradient coefficient g = 10-8 
+m3/F. 
+ 
+ 20 
+ 
+- 40 - 20 0 -0.50. 
+-0.2
+-0.4-0.1
+-0.3 
+- 40 - 20 0 0 1 2  
+- 40 - 20 0 - 1 - 0.5 0. 0.5  
+- 40 - 20 0 0 0.03 0.06 (a) Polarization  P3 (C/m2) 
+(b)
+Distance z  (nm) Field  E3  (MV/ cm) 
+Charge  (107C/m3) Potentia l   ϕ  (V) 
+Distance z  (nm) Distance z  (nm) Distance z  (nm) LSM O 
+ 
+- 5 0 0 0.05 
+(d) 
+- 20 - 10 0 0 0.1 0.2 0.3 0.4 
+(c) 
+ STO 
+Fig.8.  (a) Polarization,  (b) po tential, (c) ele ctric field  and (d) electr ic charge density z-
+distributions for the STO/LSMO he terostru cture. Contact poten tial difference at z=0 is Ub=1 V, 
+interface polarization  Pb = 0 and charge density σS = 0. C arriers concentr ation in LSMO  is 
+m26 010=Sp-3. Black, red, violet and blue curv es correspond to different thickness L = 20, 40, 80, 
+160 nm  of STO fil m with  extrapolation lengths λi=30 nm and the gradient coefficient g = 10-8 
+m3/F. 
+ 
+The distributions shown in Figs. 7-8 co rrespond to the stable  ground state of the 
+heterostru cture incip ient ferroelectric  STO/LSMO without interface charge ( σS = 0), i.e. when the 
+free carriers are abundant in LSMO and there is no need in the screening interface charge. The 
+characteristic feature of th e interface charge abs ence is  the thin depletion layer WS (sev eral nm  for 
+LSMO) that is occupied  by the m ain-type carriers. The opposite case  of depletion layers created 
+ 21by the m inor-type carriers, which can appear dur ing the polarization reversal in the proper  
+ferroele ctric film, will be conside red in the nex t section. 
+ 
+3.3. Calculations of the meta stable states for typical he terostructures (1D-case)  
+As it was mentioned in Sections 2-3, when the free carriers are abunda nt there is no need 
+in the s creening interf ace charg e (i.e. for thin  depletion layer created by the m ain-type carriers  
+and thick layer created by the m inor typ e carries). In  the opposite  case of depletion layers created 
+only by th e minor-typ e carriers (i.e. without inte rface charge states lo cated at z = 0) the 
+penetration depth WS is higher (up to tens of nanom eters as shown b y the bo ttom curves in 
+Figs. 9) and corresponding screen ing of the spontaneous polarizat ion appeared weaker (com pare 
+bottom  curves in Fig. 9a for m etastable polariz ation with the top curves for the stable ground 
+state).  
+Fig. 10 shows the hysteresis loops of the average polarization in  the BFO fil m and 
+correspond ing field pen etration dep th in LSMO  under the ab sence of the interface ch arge σS and 
+two different values of the m ajor-type carriers in  LSMO. The loops asymme try increases with the  
+increase of the carriers  concen tration [com pare Figs.10a,c with  Figs.10b,d]. The asymmetry of 
+the loops, both horizontal im print and vertica l shift, are  caused by the charge effects provided by 
+the m ajor- type carr iers for positiv e biases ()0>+b eU U  and m inor-type carriers for negative 
+biases (  respe ctively resulting in the a ppearanc e of the build- in field. )0<+b eU U
+The weak s creening ca uses s trong elec tric fi elds, which res ulting into the suppre ssion of 
+polar ization  inside th e ferroelec tric film and increase o f the system ’s free energy, since 
+depolarization field energy is always  positive. As a result, the str ong field effect may lead to the 
+bend bendin g at z = 0 an d appearan ce of interface charge states.  
+Fig. 11 shows the influence of the interf ace ch arge on the thickne ss dependence of the 
+stable and metas table (if any) st ates o f the averag e polarizatio n 3P in ferroelectric BFO film .  
+ 
+ 22 
+ 
+-20 0 20 40 -0.4  -0.2  0  
+-20 0 20 40 -40 4  
+-20  0 20 40 0 0.25 0.5 0.75 
+Charge (C/cm3) 
+Distance z  (nm) LSM O
+Distance z  (nm) Polariz ation  P3 (C/m2) 
+Ub=1 V (a)
+(b)
+Distance z  (nm) LSM O
+ 
+-20  0 20 40 0 1 2 3 
+LSMO
+Distance z  (nm) (c)
+(d)Field  E3 (MV/ cm) 
+Potentia l   ϕ  (V) 
+LSMO 
+ BFO  BFO 
+BFO  
+BFO 
+Fig. 9.  (a) P olarization, (b) poten tial, (c) e lectric field and (d) bulk charge density z-distributions 
+near the BF O/LSMO interface at zero external bias Ue = 0. Contact potential difference at z=0 is 
+Ub=1 V, interface polarization Pb = 0 and charge density σS = 0. BFO fil m thickness L = 100 nm, 
+gradien t coefficient g = 10-8 m3/F, λi≈0 nm  (solid curves) and λi=30 nm  (dashed curves). Carriers  
+concentratio n in LSMO is  m27 010=Sp−3 (upper curve s) and  m25 010=Sn−3 (bottom  curves). 
+Upper curves (positive P3) correspond to the stable ground states, bottom ones (negative P3) 
+correspond to the reversed polari zation (m etastable states).  
+ 
+It is seen from  Fig. 11 that the interface charges σS of appropriate sign increases the 
+average polarization and s mear th e size-induced phase transition point for the stable states 
+(com pare upper cu rves 1-5). Also  the interface charges lead to  the strong asymm etry of the 
+average polarization  values in  the s table “up” a nd m etastab le “dow n” states, which exis t not for 
+ 23all con sidered values  of σS (com pare up and bottom  curves in  Figs.11). The interface charges σS 
+act as the co ntribu tion in to the bu ilt-in f ield in the right-hand-side of Eqs.(10). 
+ 
+  
+-20 0 20 -0.250 0.25  0.5 0.75  
+ 
+-20 0 200 50 100  
+-20 0 20-0.2 0 0.2 0.4 0.6 (a) Polarization  P3 (C/m2) (b)
+Voltage  Ue+Ub  (V) Penetration depth  WS (nm) BFO
+LSMO 
+-20  0 20 0 50 100  
+Voltage  Ue+Ub  (V) (c)(d)BFO
+LSMO
+ps=1026 m-3 ps=1027 m-3 
+ 
+Fig. 10.  Bias dependence of the average BFO polarizat ion (a, b) and LSMO penetration depth (c , 
+d). BFO fil m thickness L = 100 nm , gradient coefficient g = 10-8 m3/F, extrapolation lengths λi≈0 
+nm (solid cu rves) and λi=30 nm  (dashed curves).  Interface po larization Pb = 0 and charge density 
+σS = 0. LSMO m ajor-type carrier concentratio n is  m26 010=Sp−3 (a, c) and   m27 010=Sp−3(b, d); 
+and  m25 010=Sn−3 for m inor-type carriers resp ectiv ely. Gap is absen t (H = 0). 
+ 
+As expected , the in terface charges σS of appropriate sign provide m ore effective screening 
+of the spontaneous polarization th an the exte nded space-charge lay er. The screening by the  
+interface charges σS makes pola rization m ore hom ogeneous, subsequently decreases the 
+depolarization field, which in turn self-consiste ntly decreases the system free energy. So the 
+absolutely s table p rofiles of reverse d polar ization shown in Figs.12a by the bottom  curves are 
+more energ etically  pref erable th an the ones  shown by the bottom  curves in F igs.9a for zero 
+ 24interface ch arges (σS = 0). Thus th e interface charge m ay originate in the case of the weak  
+polarization screening from  the semiconductor side (i.e. for thick depletion layer created by the 
+minor-type carriers). 
+ 
+  
+0 20 40 60 80 100  -0.5 0. 0.5 
+13 45
+1Stable stat es  
+Metastable stat es  
+ 
+0 20 40 60 80 100  -0.5 0. 0.5 
+134
+51 45Stable stat es  
+Meta stable  states  Polarization 〈P3〉 (C/m2) 
+Film  thickness  L  (nm ) Film  thickness  L  (nm ) Polarization 〈P3〉 (C/m2) 
+ps=1026 m-3 (a)
+(b)
+ps=1027 m-3 
+ 2
+2
+2
+Fig. 11.  The average BFO polarization  thickness depen dence for d ifferent in terface charg e 
+density σS = 0, -0.1, -0.2 , -0.3, -0.6 C/m2 (curves 1, 2, 3, 4, 5 resp ectively). Extrapolation lengths  
+λi=0 nm , carriers concentration in sem iconductor is  m26 0 010==S Sp n−3 (a) and 
+  m27 0 010==S Sp n−3 (b). Other param eters are the s ame as in Fi gs.9. Upper curves are the stable 
+polarization states. Metastable states corresponding to negative pol arization values are shown by 
+the bottom  curves (if any exist for def inite σS). 
+ 25 
+ 
+ 
+-40 -20 0 20 -150 15 
+1 
+34
+5 
+-40 -20 0 20 -0.5  0. 0.5 
+1 3 
+4 5 
+-40 -20 0 20 -50 5 1 
+3 
+4 5 
+ 
+-40 -20 0 20 -25 0 25 
+1 3 
+4 5 (a)Polariz ation  P3 (C/m2) 
+(b)
+Distance z  (nm ) Field  E3 (MV/ cm) 
+Charge  (107C/m3) 
+(c)
+Potentia l   ϕ  (V) 
+(d)
+Distance z  (nm ) Distance z  (nm) Distance z  (nm) LSM O 
+ 
+-10 0 10 20 -0.8 0.0.8
+1
+3 45
+1 
+ 22 2
+2BFO  
+2
+Fig.12.  (a) Polariz ation, (b) pote ntial, (c) e lectric field  and electric charge d ensity (d) z-
+distributions for the STO/LSMO he terostructure at zero ex ternal b ias Ue = 0. Contact potential 
+difference at z=0 is Ub=1 V, interface polarization  Pb = 0 and different interface charge density 
+σS = 0, +0.35, -0.35, +0.75, -0.75 C/m2 (curves 1, 2, 3, 4, 5 respectively). Carriers concentration 
+in LSMO  is m26 010=Sp-3. BFO  film thickness L = 50 nm , extrapolation lengths λi ≈ 0 nm  and the 
+gradien t coefficient g = 10-8 m3/F. 
+ 
+4. Effect of the incomp lete screenin g and interface charge on the local  polariz ation reversal 
+and charge transport  
+In the initial ground state the sluggish surface charges σf completely screen the electric 
+displacem ent outside the film . In contrast to the ground states considered in the S ection 3, the 
+recharging o f surface ch arges σf should appear d uring the lo cal polarization reversal. The ultra-
+thin dielectric gap of thickness H models the separation betw een the  tip elec trode and the 
+sluggish charges inside a contam ination (or dead) layer.  
+ 26The equilibrium  dom ain structure is alm ost cylindrical for the case of local po larization  
+revers al cau sed by the localized potential U  applied to the SPM-tip in thin f erroele ctric 
+film [Fig. 1c]. As a se quence the spontaneous polarization distribution,  the potential and the 
+interface ch arges dens ity variations  can be expressed in { x,y} - Fourie r repres entation. In th e 
+Fourier k),(yxe
+1,2-domain Eqs.(2, 3) im mediately sp lit into the tw o system s of differential equ ations. 
+The first system  corresponds  to the sm ooth components {})(),(3zPzϕ  was solved  in the prev ious 
+section. The second system  for the modulati ng com ponents is listed in Appendix B.  
+Approxim ate analytical solution (6) m ay be us ed in p articular case  of the d isk-like tip  
+apex of radius R >> L , i.e. until 0),(≈ ∆⊥ yxUe  in the region of pol arization revers al. 
+It was shown earlier [35] that the trans verse polarization gradient could be neglected f or 
+the case of the strong inequality R g<<α2  valid f or all consider ed ferroelectrics at room  
+temperature.  Thus polarization ()Lzf zyxP , ~),,(3  averaged ov er the film thickness can be  
+determ ined from  the Eqs.(10).  
+Using the coercive volum e conception of polari zation reversal form ulated in Ref.[35], the 
+domain late ral sizes { x,y} can be es timated from the equa tion: 
+(HLE yxU
+H Lf
+c e b gg
+, ),(
+33 3333 ±=
+ε+εε).   (15) 
+The intrinsic coercive fi elds are given by Eq.(11b). 
+ For a typical tip potential distribution 
+2 2)(
+drUdre+≈ U  (d is the effective tip size)  the 
+domain radius 2 2y x r+=  depends on applied bias U as 
+1 )(2
+3333 33 2−
+
+
+
+εε+ε=−
+±
+c gb g
+E
+fH LUd Ur . 
+In general case the current density inside fe rroelectric film consists of the conductivity, 
+diffusion and displacem ent, tunneling, Schott ky and Frenkel-Poole em ission currents [36].  
+During the  local po larization reve rsal the thickn ess and electric charge o f the space charge layer 
+changes, it particular the positive charge can be substituted by the nega tive one or vise versa  
+depending o n the polarization d irection. Such  transform ations should b e accom panied by th e 
+peaks of dis placem ent currents, which shape an d am plitude depend on the dom ain shape and 
+ 27sizes. However, for nan oscale contact a d isplacem ent current is significantly sm aller and faster  
+then a cons tant leakage curre nt, and hence is ignored. 
+Derived analytical expressions also allow es timations of the tunneli ng current between the  
+tip apex and  ferroelectric surface (if any exist). A ssum ing that the u ltra-thin gap H is transp arent 
+for the tunneling electrons, tunne ling and field em ission currents could flow between the tip apex 
+and ferroelectric film surface. In the Fowler-N ordheim  transport regim e [36,  37] the tunneling 
+current density 
+
+
+ϕ −∫
+−−0
+*)( 22exp~
+HLt zqm dz J
+h is deter mined by the potential distribution ϕ(z) 
+given by Eq.(6) and corresponding pe netration depth in se miconductor, W  is 
+determ ined self -consiste ntly f rom Eqs.(10).  () HLUe S ,,,
+ 
+5. Summar y  
+Using Landau-Ginzburg-Devonshire approach  we have calculate d the equilibrium 
+distributions of electri c field, polarization and space charge  in the ferroelectric-sem iconductor 
+heterostructures containi ng proper or incipient ferroelectric thin  films. In particula r, it is show n 
+that space charge effects introduce strong size -effect on spontaneous polarization in 20-40 nm 
+epitax ial film s of BiFeO 3 on (LaS r)MnO3, and can induce st rong polarization in incipient 
+ferroele ctric SrTiO 3. 
+We obtained analytical expressions for th e cylind rical dom ain sizes appeared i n 
+ferroelectric film  under  the local polarization reversal, which is caused by the electric field 
+induced by the nanosized tip of the SPM probe. The SPM tip can be separated from  the 
+ferroelectric surface covered with sluggish screening charges by the ultra-thin dielectric layer that 
+models either geom etric gap or/and contam inatio n layer res istance.  
+The intrinsic field effects, which originated  at the ferroelectric-sem iconductor interface, 
+lead to th e surface ban d bending and result in  the for mation of depletion/accum ulation space-
+charge layer near the sem iconduc tor surface. W e calculated  how the b uild-in field s smear the 
+size-induced  phase tran sition, induc e polar ization in the incip ient f erroelectric films and lead  to 
+the polarization hysteresis loops vertical and ho rizontal asymm etry in ferroelectric film s. 
+ 
+ 28Acknow ledgements 
+Authors are grateful to P rof. E. Tsymbal for valuable critical rem arks. Research is sponsored by 
+Ministry of Science and Education of Ukraine and National Science Foundation (Materials W orld 
+Network, DMR-0908718). SVK and AB ac knowledge the DOE SISGR program . 
+ 
+Appendix A. 
+ Allowing for Eq.(7), pol arization distribution  P3(z) should be found from  the Eul er-
+Lagrange boundary problem  (2) as: 
+
+
+
+=
+−=
+
+λ− −=
+=
+
+λ+εεσ−ρ= −β+
+
+
+
+εε+α
+.0 ,
+0,1
+3
+2 33
+1 33300
+3 22
+3
+3 3
+330
+LzdzdPP P
+zdzdPPWP
+dzdg P P
+bbS S S
+b
+  (A.1) 
+Let us look for the solution of the problem  (A.1) in the for m ()()zp P zP +=3 3 , where the 
+average polarization ∫
+−≡0
+3 3~)~(1
+LzdzP
+LP  is introduced. The variation p average value is zero : 
+0≡p . So, the pro blem (A.1) acquires the f orm: 
+()
+
+
+
+−=
+−=
+
+λ− −−=
+=
+
+λ+β+α−
+εε−σ−ρ=−β+β+
+
+
+
+εε+β+α
+. ,
+0, 313
+3 2 3 13
+3 3
+33030
+22
+3 2
+3
+3302
+3
+P
+Lzdzdpp P P
+zdzdppP PP W
+dzpdg p pP p P
+bbS S S
+b
+    
+(A.2) 
+Since alway s 01
+330>>
+εε+αb (as w ell as 013
+3302
+3 >>
+εε+β+αbP ) for both proper and incipient 
+ferroele ctrics, Eq.(A .2) can be line arized with respec t to the devia tion p and then  solved by  
+standard m ethods.  
+After elem entary transfo rmations, polar ization distribution acquires the form : 
+()()
+()() 
+− ≥−θ⋅−−− ≥−θ⋅−
+ε−ε=
+carriers typepof depletion,0 ,carriers typenof depletion,0 , 1)(
+00
+3z z W Wznz z W Wzp
+q zP
+Sn Sn SSp Sp S
+SS     (A.3) 
+ 29()() 0 ),,( ,
+1 32)(2
+3 3300 3
+3 330
+3 ≤<− ⋅−
++β+αεεσ−ρ+βεε= zL LzbPLzf
+PW PzPbbS S Sb
+. (A.4) 
+The space distribu tion is governed by  the function s f and b: 
+()()() ()()()() ( )
+() ()()()ξ λ+λξ+ξ λλ+ξξ−ξ+ξ+ξλ+ξ+ λξ−=L Lz zL z zLLzfcosh sinhsinh sinh cosh cosh1 ,
+2 1 2121 2,      (A.5a) 
+()()()()()
+() ()()()ξ λ+λξ+ξ λλ+ξξ+ξ+ξ+ ξλ=L LzL zLLzbcosh sinhsinh cosh,
+2 1 2122
+2.                               (A.5b) 
+Characteris tic length ()g
+Pgb
+bb
+330 2
+3 330330
+1 3εε≈
++β+αεεεε=ξ .  
+ Then the average po larization 3P and depths WSn,p should be determ ined self-
+consistently from  the spatial averaging of Eq.(8b) , 
+()bPf
+PW PPbbS S Sb
+⋅−
++β+αεεσ−ρ+βεε=
+1 32
+2
+3 3300 3
+3 330
+3 , and boundary conditions (4b,c).  
+For particular cas e H = 0 this gives the system  of coupled algebraic equations: 
+()
+
+
+
+
+
+
+εε+β+α⋅−
+
+
+
+εεσ−ρ=
+
+
+
+εε+−β+α=
+
+
+
+εε+σ−
+εερ+
+εερ−
+.13123,
+2
+3302
+3
+3300
+3302
+3 33303
+3300
+02 0
+b b bS S S
+bb bS
+bS S
+SS S
+P bPfWf P PUP WLW
+     (A.6) 
+Where ()() ()() ()
+() ()() () ()()
+()()212
+2 12 12
+212
+2 12 1221sinh coshsinh 1 cosh21λλ+ξ+λ+λξλ+λ+ξξ−≈ξ λλ+ξ+ξ λ+λξξ λ+λ+−ξξξ−=L L L LL Lf  and 
+() () ()
+() ()()() ()()
+() () ()12
+2 1 21222
+2 1 2123
+22
+cosh sinhcosh1 sinh
+λ+ξξ=λ+λξ+λλ+ξλ+ξξ≈ξ λ+λξ+ξ λλ+ξξ−ξ−ξ λξ=L L L L LL Lb . 
+The system  (A.6) reduces to the relations 
+()
+
+
+εε−
+
+
+
+εεσ−ρ=−β+
+
+
+
+εε+αεε−
+ερε+ρ+σ−=
+. 231,
+2
+330 3300
+3
+3 3
+33033020
+33 0
+3
+bb
+bS S S
+bb b
+S
+SSb
+S S S
+bPfWf P PLUW
+LW P
+                   (A.7) 
+ 30Since 
+LUW
+LW Pb b
+S
+SSb
+S S S 33020
+33 0
+32εε−
+ερε+ρ+σ−=  and 01
+0 330 3 >
+ρ
+
+εε+σ+
+Sb b
+SLUP, the second 
+of Eqs.(14) reduces to six order algebraic equation for the built-in field determ ination.  
+The polar ization con tribution in to the relative ato mic displacem ent u3 can be estim ated as 
+)( )(3 3 zP
+QVzuB
+SS≈  at z > 0 and )( )(3 3 zP
+QVzuB
+IFFE≈
+29104−⋅ at –L < z < 0. Here Vj is the vo lume of the 
+corresponding unit cell, QB is the Born effective charge of th e lightest atom  “B”. For  perovskites  
+considered h ereinafter V m, .6≈SFE3 
+ 
+Appendix B. 
+ The potentia l applied to  the SPM-tip  is highly-lo caliz ed, i.e. 
+. As a s equence the spontaneous polarization 
+distribution can be approxim ated as , 
+the potential ϕ  and the interface charges  
+density variation  can be expressed in Fourier 
+repres entation (see).  (∫∫∞
+∞−∞
+∞−−− = yikxik udk dk yxU2 1 2 1 exp)( ),( k
+PS(
+∫∫∞
+∞−∞
+∞−+ϕ= dk dk z zyxf 1 )( ),,(
+∫∫∞
+∞−∞
+∞−σ =δσ dk dk yxS 2 1~),()
+() ∫∫∞
+∞−∞
+∞−−− += yikxik z pdk dk zPzyxS 2 1 2 1 3 exp),( )( ),, k
+( )−− ϕ yikxik z2 1 2 exp),(~k
+( )−− yikxikS 2 1 exp)(k
+In the Fourier k1,2-dom ain Eqs.(3) im mediately split into  the two syste ms of differentia l 
+equations. T he first system  corresponds to the smooth com ponents {})(),(3zPzϕ  was s olved in the  
+previous section. The second system  for the m odulating components is: 
+.0 ,0~~,0 ,),( 1~~, ,0~~
+2
+2202
+11 22
+332
+22
+>=ϕ−ϕ<<−ε=ϕε−ϕε−<<−−=ϕ−ϕ
+z kdzdzLzdz dpkdzdL zLH kdzd
+S b k   (B.1) 
+ 31Where . Rewritten for the m odulating com ponents, th e bo undary con ditions  (4) 
+acquire the f orm: 2
+22
+12k k k+=
+()
+() ()
+()()
+() .0 ,~,00,~0,~),(~)0,()0,(~)0,(~),(~),()0 ,(~)0 ,(~,00 ,~0 ,~),( ,~
+33 033 33 0
+=∞→ϕ≈−ϕ−+ϕσ= −
+
+ −ϕε−+ϕεεσ=−−
+
+ −−ϕε−+−ϕεε≈−−ϕ−+−ϕ=−−ϕ
+zkk k kp
+dzkd
+dzkdLkp
+dzLkd
+dzLkdLk Lkku HLk
+S S Sbf Sg b
+kk       (B.2)  
+Where the g ap dielectric  perm ittivity  ε is introduced.  g
+33
+ LGD-equation (2) for determ ination of the m odulation pS(k,z) acqui res the form: 
+[]
+[]
+0 ,0
+0.)'()' (' )(3)"()"' (" )'(' ),(,),( ),(~),( )( 3
+2 1322
+2
+32
+=
+−=
+
+λ− =
+=
+
+λ+− + −− =β−ϕ−=
+
+
+−β++α
+∫ ∫∫∞
+∞−∞
+∞−∞
+∞−
+Lzdzdpp
+zdzdppp pdzP p pd pd z pQz pQ z
+dzdz p
+dzdgzP kg
+S
+SS
+SS S S S S SS S
+k kk k k kkk k k k kk k k
+ (B.3) 
+The value  should be determ ined self -consistently. In the final state the distribution of  
+the surface charge ),(L pS−k
+)(~kfσ  localized at L z−=  and the interface charge )(~kSσ  localized at 0=z  
+should provide the m ost effective screening of th e spontaneous polarizati on outside the film  and 
+minimal depolarization field energy.  
+The express ion for the bias-dep endent barrier height r elated with a pplied bias difference  
+and polarization changes is given by expression 
+( )
+
+
+−−−−ϕ ≈Φδ ∫∫∞
+∞−∞
+∞−),( exp),(~),(2 1 2 1 yxUyikxik L dk dkq yxB k .   (B.4) 
+ 
+References  
+                                                 
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+scale p erovskite titanate superlattices . Nature 419, 378, (2002). 
+ 32                                                                                                                                                              
+2 A. Ohtom o and H.Y. Hwang, A high-m obility e lectron gas a t the LaAlO 3/SrTiO3 
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+Mater. Res. Soc. Bull. 31, 28–35 (2006).  
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+(Springer Verlag, 2000) 248 p.   ISBN: 978-3-540-66387-4. 
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+Phys. 100, 114112 (2006). 
+13 P. Zubko, D.J. Jung, and J.F. Scott, Electrical  Characterization of PbZr 0.4Ti0.6O3 Capacitors. 
+J. Appl. Phys. 100, 114113 (2006).  
+14 Y. W atanabe. Theoretical stability of the polar ization in a thin sem iconducting ferroelectric. 
+Physical Review B 57, 789 (1998) 
+ 33                                                                                                                                                              
+11 3E15 J. Bardeen, Surface States and Rectificati on at Metal Sem i-Conductor Contact. Ph ys. Rev. 71, 
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+(1980). p.119 
+17 M.A. Itskovsky, Fiz. Tv. Tela, 16, 2065 (1974).  
+18 The dependence of ε on  is absen t for uniaxia l ferroelectrics. It may be essentia l for 
+perovskites with high coupling constant. 
+19 Chun-Gang Duan, Renat F. Sabirianov, W ai-Ning Mei, Sitaram  S. Jaswal, and Evgeny Y. 
+Tsym bal. Interface Effect on Ferroelectricity at th e Nanoscale.  Nano Letters 6, 483 (2006) 
+20 Chun-Gang Duan, S. S. Jaswal, and E.Y. Tsym bal. Predicted Magnetoelectric Effect in 
+Fe/BaTiO 3 Multilaye rs: Ferroelectric Control of Magnetism . Phys. Rev. Lett. 97, 047201 (2006) 
+21 M.D. Glinchuk, A.N. M orozovska, J. Phys.: Condens. Matter. 16, 3517 (2004). 
+22 M.D. Glinchuk, A.N. M orozovska, and E.A. Eliseev, J. Appl. Phys. 99, 114102 (2006) 
+23 S. M. Sze, Physics of Sem iconductor Devices , 2nd ed. (W iley-Interscience, New York, 1981), 
+Chap. 7, p. 382.  
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+25 A. K. Taga ntsev, G. Gerra, and N. Setter, Shor t-range and long-range cont ributions to the s ize 
+effect in m etal-ferroelectric-m etal he terostructu res. Phys. Rev. B 77, 174111 (2008). 
+26 E.A. Eliseev, M.D. Glinchuk, А.N. Morozovsk a. Appearan ce of fe rroelectr icity in thin f ilms 
+of incipien t ferroele ctric. Phys. stat. sol. (b) 244, № 10, 3660–3672 (2007). 
+27 J. L. Cohn, M. Peterca, and J. J. Neum eier, Phys. Rev. B 70, 214433 (2004). 
+28 A. Tiwari, C. Jin, D. Kum ar, and J. Narayan. Appl. Phys. Lett. 83, 1773 (2003). 
+29 A. Ruotolo, C.Y. La m, W.F. Cheng, K.H. W ong, and C.W . Leung. Phys. Rev. B 76, 075122 
+(2007). 
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+31 Kui-juan Jin, Hui-bin Lu, Qing-li Z hou, Kun Zhao, Bo-lin Cheng, Zheng-hao Chen, Yue-liang 
+Zhou, and Guo-Zhen Yang. Phys. Rev B 71, 184428 (2005). 
+32 G.A. Smolenskii, V.A. Bokov, V.A.  Isupov, N.N Krainik, R.E. Pasynkov, A.I. Sokolov, 
+Ferroelectrics and Related Materials (Gordon an d Breach, New York, 1984). P. 421 
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+33 C.-L. Jia, Valanoor Nagarajan, J.-Q. He, L. Houben, T. Zhao, R. Ra mesh, K. Urban, R. 
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+M. Hrabovsky. Optical refraction index and polarization profile of ferroelectric thin film s. 
+Integrated ferroelectrics 38, 101-110, (2001). 
+35 A.N.  Morozovska, E.A. Eliseev, Yulan Li, S.V. Svechnikov, V.Ya. Shur, P. Maksym ovych,  
+V. Gopalan, Long-Qing Chen, S.V. Kalinin. T hermodyna mics of nanodom ain form ation and 
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+ 35
\ No newline at end of file
diff --git a/1001.0122.txt b/1001.0122.txt
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+arXiv:1001.0122v2  [cond-mat.mes-hall]  22 Apr 2010Stark effect, polarizability and electroabsorption in sili con nanocrystals
+Ceyhun Bulutay,1,∗Mustafa Kulakci,2and Ra¸ sit Turan2
+1Department of Physics, Bilkent University, Ankara 06800, T urkey
+2Department of Physics, Middle East Technical University, A nkara 06531, Turkey.
+Demonstratingthequantum-confinedStarkeffect(QCSE)insi licon nanocrystals (NCs)embedded
+in oxide has been rather elusive, unlike the other materials . Here, the recent experimental data from
+ion-implanted Si NCs is unambiguously explained within the context of QCSE using an atomistic
+pseudopotential theory. This further reveals that the majo rity of the Stark shift comes from the
+valence states which undergo a level crossing that leads to a nonmonotonic radiative recombination
+behavior with respect to the applied field. The polarizabili ty of embedded Si NCs including the
+excitonic effects is extracted over a diameter range of 2.5–6 .5 nm, which displays a cubic scaling,
+α=cD3
+NC, withc= 2.436×10−11C/(Vm), where DNCis the NC diameter. Finally, based
+on intraband electroabsorption analysis, it is predicted t hatp-doped Si NCs will show substantial
+voltage tunability, whereas n-doped samples should be almost insensitive. Given the fact that bulk
+silicon lacks the linear electro-optic effect as being a cent rosymmetric crystal, this may offer a viable
+alternative for electrical modulation using p-doped Si NCs.
+PACS numbers: 73.22.-f, 78.67.Bf, 71.70.Ej
+I. INTRODUCTION
+The Stark effect has evolved within the previous cen-
+tury into a powerful spectroscopic tool for the solids.1
+In the case of bulk semiconductors, shallow free excitons
+become easily ionized which limits the strength of the
+applied fields. A more robust variant of the Stark effect
+is the so-called quantum-confined Stark effect (QCSE)
+where the carriers are trapped in a quantum well or a
+lowerdimensionalstructure.2Inthisregard,thequantum
+dots or nanocrystals (NCs) are preferred so as to take
+advantage of the full three-dimensional confinement.3
+As a matter of fact, the electroabsorption studies were
+initiated quite early with CdS xSe1−xNCs embedded
+in a glass matrix.4The QCSE activity in group-II-VI
+NCs5–7was soon extended to group-III-As NCs.8,9A re-
+lated noteworthy achievement was registering photolu-
+minescence (PL) from a single quantum dot within an
+ensemble,10followed by probing the QCSE from a single
+dot.11
+On the technologically important front of group-IV
+materials, a recent breakthrough was the announcement
+of QCSE in germanium multiple quantum wells sand-
+wiched between SiGe barrier layers.12The drawback of
+this structure is the small band offset of the barrier re-
+gions which limits the applied reverse bias before car-
+rier tunneling sets in. Furthermore, it suffers from the
+polarization-dependent response discriminating between
+TE and TM polarizations; both of these shortcomings
+are inherently carried over to Si/Ge self-assembled quan-
+tum dots.13Si NCs embedded in oxide, not only offer
+remedy to both of these problems, but also due to its in-
+sulating host matrix it can withstand very high electric
+fields. Surprisingly, even though nanosilicon has become
+an established field,14the QCSE activity in this system
+has been quite overlooked. In some of the early electro-
+luminescence and photoluminescence studies on Si NCs,
+the precursors of QCSE was reported as a small redshiftwhich was however taken over by a strong blueshift.15,16
+As another indirect measurement of QCSE in Si NCs,
+Linet al.announced a 11 nm redshift under strong illu-
+mination, but without an external bias, which they at-
+tributed to a build up of an internal electric field due
+to capture of carriers in NCs.17Only very recently, the
+direct measurement of QCSE under an external field
+in Si NCs was achieved on ion-implanted samples that
+yielded aslargeasa 40nm redshift at cryogenictempera-
+tures, which remainedto be easilydetectable at the room
+temperature.18This much delayed progress may never-
+theless become crucial for the electronically controllable
+silicon-based photonics and especially for optical modu-
+lators; thelatterhasbeen arealchallenge,asbulksilicon,
+being a centrosymmetric crystal, lacks the Pockels effect
+which leaves the plasma effect as the main route for elec-
+trical modulation.19,20Very recently, a GeSi electroab-
+sorption modulator has been announced that makes use
+of the bulk Franz-Keldysh effect of germanium enhanced
+under tensile strain.21Amidst these developments, the
+present understanding on the QCSE and electroabsorp-
+tion in Si NCs remains to be quite insufficient so as to
+address whether it can offer a viable alternative to the
+existing and emerging ones.
+In this work, we aim for an assessment of these
+prospects from a rigorous atomistic point of view, start-
+ing with the recent QCSE experiment and extending our
+analysis to both fundamental as well as applied direc-
+tions. First, we theoretically show that, the highly pro-
+nounced luminescence shifts as measured in Ref. 18 un-
+ambiguously originates from the Stark effect. In so do-
+ing, the importance of the excitonic effects is emphasized
+for larger NCs. The detailed explanation of the emis-
+sion strength as a function of Stark field reveals the in-
+tricate interplay of the single-particle Stark shifts, their
+level crossings and the electric field dependence of the
+oscillator strengths. From a fundamental point of view,
+in this context the most important physical quantity is2
+their polarizability. However, this is a subject which has
+not been discussed so far in the literature. Therefore,
+we provide the polarizability of embedded Si NCs for the
+useful 2.5–6.5 nm diameter range and furthermore, our
+exhaustive computations are expressed in simple expres-
+sions to enable their use by other researchers. Finally, we
+completeourtheoreticalanalysiswiththe intrabandelec-
+troabsorptionpropertiesofSi NCsunder astrongelectric
+field, where we predict the marked discrepancy between
+then- andp-doped NCs.
+Our computational framework is a semiempirical
+pseudopotential-based atomistic Hamiltonian22in con-
+junction with the linear combination of bulk bands
+(LCBB) as the expansion basis.23,24The strong Stark
+field is included directly to the Hamiltonian without any
+perturbative approximation. This is the current state-
+of-the-art theory for this system size, which is far ad-
+vanced compared to effective mass and envelope func-
+tions approaches25–27(for a critical account of the latter,
+see Ref. 28) and moreover is not amenable by ab ini-
+tiotechniques.29The competence of this technique has
+been well tested; in the case of embedded Si and Ge
+NCs, this has been employed to study interband and
+intraband optical absorptions,31the Auger recombina-
+tion, and carrier multiplication;32furthermore, predic-
+tions for the third-order nonlinear optical properties us-
+ing the same theoretical approach33have been indepen-
+dently verified experimentally.34The paper is organized
+as follows. A description of the theoretical method is
+given in Sec. II. The QCSE, polarizability, and intraband
+electroabsorption are analyzed in Sec. III. Main conclu-
+sions are presented in Sec. IV.
+II. THEORY
+In the LCBB approach,24the NC wave function with
+a state index jis expanded in terms of the bulk Bloch
+band (n) and the wave vector ( /vectork) as,
+ψj(/vector r) =1√
+N/summationdisplay
+n,/vectork,µCµ
+n,/vectork,jei/vectork·/vector ruµ
+n,/vectork(/vector r),(1)
+whereNis the number of primitive cells within the
+computational supercell, Cµ
+n,/vectork,jis the expansion coeffi-
+cient set to be determined, and µis the constituent bulk
+material label pointing to the NC core and embedding
+medium.uµ
+n,/vectork(/vector r) is the cell-periodic part of the Bloch
+states which can be expanded in terms of the reciprocal-
+lattice vectors, {/vectorG}as
+uµ
+n,/vectork(/vector r) =1√Ω0/summationdisplay
+/vectorGBµ
+n/vectork/parenleftBig
+/vectorG/parenrightBig
+ei/vectorG·/vector r, (2)
+where Ω 0is the volume of the primitive cell. The Hamil-
+tonian has the usual kinetic and the ionic potential parts,
+thelatterfordescribingtheatomisticenvironmentwithinthe pseudopotential framework, given by
+ˆH=ˆT+ˆVPP
+=−/planckover2pi12∇2
+2m0+/summationdisplay
+µ,/vectorRj,αWµ
+α(/vectorRj)υµ
+α/parenleftBig
+/vector r−/vectorRj−/vectordµ
+α/parenrightBig
+,(3)
+wherem0is the free electron mass, Wµ
+α(/vectorRj) is the atomic
+identity coefficient that takes values 0 or 1 depending on
+the type of atom at the position /vectorRj−/vectordµ
+α, here/vectorRjis the
+primitive cell coordinate and /vectordµ
+αis the displacement of
+this particular atom within the primitive cell. υµ
+αis the
+screened spherical pseudopotential of atom αof the ma-
+terialµ, the latter distinguishes the NC and the matrix
+regions.
+The formulation can be cast into the following gener-
+alized eigenvalue equation,24,35
+/summationdisplay
+n,/vectork,µHn′/vectork′µ′,n/vectorkµCµ
+n,/vectork,j=Ej/summationdisplay
+n,/vectork,µSn′/vectork′µ′,n/vectorkµCµ
+n,/vectork,j,(4)
+where,
+Hn′/vectork′µ′,n/vectorkµ≡/angbracketleftBig
+n′/vectork′µ′|ˆT+ˆVPP|n/vectorkµ/angbracketrightBig
+,
+/angbracketleftBig
+n′/vectork′µ′|ˆT|n/vectorkµ/angbracketrightBig
+=δ/vectork′,/vectork/summationdisplay
+/vectorG/planckover2pi12
+2m/vextendsingle/vextendsingle/vextendsingle/vectorG+/vectork/vextendsingle/vextendsingle/vextendsingle2
+Bµ′
+n′/vectork′/parenleftBig
+/vectorG/parenrightBig∗
+Bµ
+n/vectork/parenleftBig
+/vectorG/parenrightBig
+,
+/angbracketleftBig
+n′/vectork′µ′|ˆVPP|n/vectorkµ/angbracketrightBig
+=/summationdisplay
+/vectorG,/vectorG′Bµ′
+n′/vectork′/parenleftBig
+/vectorG′/parenrightBig∗
+Bµ
+n/vectork/parenleftBig
+/vectorG/parenrightBig
+×/summationdisplay
+µ′′,αVµ′′
+α/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vectorG+/vectork−/vectorG′−/vectork′/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg
+×Wµ′′
+α/parenleftBig
+/vectork−/vectork′/parenrightBig
+ei(/vectorG+/vectork−/vectorG′−/vectork′)·/vectordµ′′
+α,
+the overlap part in the generalized eigenvalue equation is
+of the form
+Sn′/vectork′µ′,n/vectorkµ≡/angbracketleftBig
+n′/vectork′µ′|n/vectorkµ/angbracketrightBig
+.
+The Si NC is intended to be embedded in silica, repre-
+sentedbyanartificialwidebandgaphostmatrixthathas
+the same band-edge line up and the dielectric constant,
+but otherwise lattice-matched with the diamond struc-
+ture of Si.31We refer to Ref. 31 for the other technical
+details of the implementation of the electronic structure,
+including the form of the pseudopotentials for the NC
+and matrix media.
+For the study of QCSE, we treat the strong external
+field in the same level as the other terms of the atom-
+istic Hamiltonian (i.e., nonperturbatively). At variance
+with the electrostatic model used in Ref. 18, we assume
+that an individual spherical Si NC under consideration is
+embedded in a uniform medium having a constant per-
+mittivity for silicon rich oxide. This is justified by the3
+FIG. 1. (Color online) The electrostatic setting of the embe d-
+ded NC under a dc external field.
+spatial distribution of the light-emitting centers in Si-
+implanted SiO 2.36In connection to the actual samples,
+we inherently assume that the NCs are well separated
+which applies safely to NC volume filling factors of about
+10% or less. The basic electrostatic construction of the
+problem is presented with the assumption that the NCs
+are well separated in Fig. 1. If we denote the uniform
+applied electric field in the matrix region asymptotically
+away from the NC as F0, then the solution for electro-
+static potential is given in spherical coordinates by37
+Φ(r,θ) =/braceleftBigg−3
+ǫ+2F0rcosθ, r ≤a
+−F0rcosθ+/parenleftBig
+ǫ−1
+ǫ+2/parenrightBig
+F0a3
+r2cosθ, r>a,
+(5)
+whereǫ≡ǫNC/ǫmatrixis the ratio of the permittivities
+of the inside and outside of the NCs. Hence, this ex-
+pression accounts for the surface polarization effects due
+to dielectric inhomogeneity which partially screens the
+external (i.e., dc Stark) field. The effect of this exter-
+nal field can be incorporated by adding the Vext=eΦ
+term to the potential-energy matrix elements. A com-
+putationally convenient recipe in the context of QCSE
+is to assume that external potential is relatively smooth
+so that its Fourier transform can be taken to be band
+limited to the first Brillouin zone of the underlying unit
+cell.35This results in the followingLCBB matrix element
+/angbracketleftBig
+n′/vectork′µ′|ˆVext|n/vectorkµ/angbracketrightBig
+=eΦ(/vectork−/vectork′)/summationdisplay
+/vectorG,/vectorG′Bµ′
+n′/vectork′/parenleftBig
+/vectorG′/parenrightBig∗
+Bµ
+n/vectork/parenleftBig
+/vectorG/parenrightBig
+×Rect/vectorb1,/vectorb2,/vectorb3(/vectorG+/vectork−/vectorG′−/vectork′),(6)
+here Φ(/vectork) is the Fourier transform of Φ( /vector r) as given in
+Eq. (5), Rect /vectorb1,/vectorb2,/vectorb3is the rectangular pulse function,
+which yields unity when its argument is within the first
+Brillouin zone defined by the reciprocal-lattice vectors,
+{/vectorb1,/vectorb2,/vectorb3}, and zero otherwise.
+As will be supported by our followingresults, concomi-
+tantwiththeStarkredshiftofthesingle-particleenergies,
+the segregation of the electron and hole wave functions
+gives rise to a blueshift that partially negates the QCSE.
+To account for this effect, we include perturbatively38
+the so-called diagonal direct Coulomb term, Jvv,ccbe-
+tween the valence-state wave function, ψv(/vector r), and the
+conduction-state wave function, ψc(/vector r), using the expres-sion
+Jvv,cc=−/integraldisplay
+d3r1d3r2e2|ψc(/vector r1)|2|ψv(/vector r2)|2
+ǫ(/vector r1,/vector r2)|/vector r1−/vector r2|,(7)
+where,eis the electronic charge, 1 /ǫ(/vector r1,/vector r2) is the inho-
+mogeneous inverse dielectric function for which we use
+the mask function approach of Ref. 40. This is the only,
+but by far the dominant Coulomb term that is included
+in this study. A more elaborate approach for the elec-
+trostatic as well as the nondiagonal Coulomb terms can
+be found in Ref. 41. Furthermore, we ignore the spin-
+orbit coupling and hence, the NC states in this work
+are doubly spin degenerate. This coupling is particu-
+larly weak in silicon with its small atomic number and as
+a matter of fact, it forms the basis for spin-based silicon
+quantum computing proposals.42Accordingly, no spin-
+flip process is considered in the carrier relaxation follow-
+ing the optical excitation so that only spin-triplet exci-
+tons are formed for which there is no exchange Coulomb
+contribution. Evenifspin flipswereto beallowed, forthe
+NC size ranges of this study, their contribution which de-
+cay with the third power of diameter25would become to-
+tally negligible compared to the direct Coulomb and the
+single-particle Stark energies. Obviously, future studies
+can avoid some of these simplifications in this work.
+III. RESULTS
+A. Stark effect
+In the recent experimental demonstration of QCSE in
+Si NCs,18the wavelength for the peak-emitted intensity
+occursat 780nm. Basedon our priortheoretical study,31
+the corresponding diameter of the NC that matches with
+this optical gap is extracted as 5.6 nm. For this size of
+a NC, we display in Fig. 2 the evolution of the single-
+particle states with applied Stark field. This clearly re-
+veals that valence states are more prone to Stark shifts
+which was also observed to be the case in InP quan-
+tum dots.43Indeed, Fig. 3 vividly demonstrates that
+the highest-occupied molecular orbital (HOMO) wave-
+function distribution is significantly shifted by the Stark
+field and gets spatially squeezed between the high Stark
+field and the spherical NC interface. On the other hand,
+the lowest-unoccupied molecular orbital (LUMO) state
+encounters only a slight displacement, which is in the
+opposite direction with respect to HOMO as expected.
+According to stronger confinement of the valence states
+under the electric field, the interlevel separations become
+wider than those in the conduction states, as can be
+checked from Fig. 2. However, it needs to be reminded
+that the size quantization energy of the electrons is much
+larger than holes in Si NCs.44
+In Fig. 4 we compare the experimental Stark redshift
+data18at 30 K with our present theoretical results. To
+correlate with this PL experiment and account for the4
+FIG. 2. (Color online) Single-particle energylevels of a5. 6 nm
+diameter Si NC for different internal NC electric fields.
+thermal effects as well as the brightness of each excited-
+state recombination, we use the following Boltzmann
+factor-averaged and oscillator strength-averaged radia-
+tive recombination (i.e., emission) energy
+¯Eemission=/summationtext
+c,vEcve−β(Ecv−ELH)fcv/summationtext
+c,ve−β(Ecv−ELH)fcv,(8)
+here,Ecv=Ec−Ev,ELHare the conduction ( c) to
+valence (v) state and LUMO to HOMO transition en-
+ergies, respectively; fcvis the Cartesian-averaged oscil-
+lator strength of the transition31andβ= 1/(kT) with
+kbeing the Boltzmann constant. We apply exactly the
+same averaging on the direct diagonal Coulomb energy
+by replacing the first Ecvterm in the numerator with
+Jvv,cc. The necessity for the Coulomb term is justified
+by the excellent agreement with the experimental data
+in Fig. 4, whereas without it (i.e., at the single-particle
+level) Stark shifts become significantly overestimated.
+Since our model does not incorporate any size inhomo-
+geneity, interface states or the strain effects, its success
+also supports the atomistic quantum confinement frame-
+FIG. 3. (Color online) HOMO (upper row) and LUMO (lower
+row) wave function isosurface profiles of a 5.6 nm diameter
+Si NC under no (left column) and 0.6 MV/cm (right column)
+internalelectricfields. Theoppositesigns ofthewavefunc tion
+are represented by blue (dark) and red (light) colors. The
+electric field is horizontally directed from right to left.
+work as the main source of the luminescence in these
+particular Si NC samples. Furthermore, we note that as
+in the experimental work,18we do not observe a dipolar
+term that gives rise to a linear dependence to the electric
+field.
+The inset of Fig. 4 shows the evolution of the va-
+lence and conduction single-particle states of a 5.6 nm-
+diameter Si NC with respect to dc electric field. It in-
+dicates that there exists a level crossing between the
+HOMO and the HOMO-1 states around an internal NC
+field of 150 kV/cm.
+Figure 5 contrasts the experimental PL peak intensity
+with the theoretical radiative recombination rate, both
+as a function of applied electric field. The nonmonotonic
+behavior of the experimental data as well as the peak in-
+tensity appearingaround0.5MV/cm and the subsequent
+falloffbeyondareallreproducedbythetheory. However,
+for small Stark fields, the 300 K emission comes out to
+be stronger in the theoretical estimation. This may be
+due to thermally activated nonradiative processes that
+degrade the emission rate at higher temperatures. This
+deviation shows that further work is required to properly
+account for all thermal aspects of this problem.
+B. Polarizability of Si NCs
+The theoretical results up to now were restricted to a
+single diameter of 5.6 nm as extracted from the PL peak
+of the experimental data.18We have ignored the size dis-
+tribution of the NCs in the actual samples.36Next, we
+extract the size dependence of the polarizability of Si
+NCs, defined as ∆ E=−(1/2)αF2
+NC, where ∆Eis the5
+/s48/s46/s53/s48/s48/s46/s53/s49
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52/s45/s49/s46/s48/s56/s45/s49/s46/s48/s54/s45/s49/s46/s48/s52
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48
+/s32/s32
+/s32/s76/s85/s77/s79/s43/s49
+/s32/s76/s85/s77/s79
+/s32/s32/s83/s105/s110/s103/s108/s101/s45/s80/s97/s114/s116/s105/s99/s108/s101/s32/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41
+/s69/s108/s101/s99/s116/s114/s105/s99/s32/s70/s105/s101/s108/s100/s32/s105/s110/s32/s78/s67/s32/s40/s77/s86/s47/s99/s109/s41/s32/s72/s79/s77/s79
+/s32/s72/s79/s77/s79/s45/s49
+/s32/s72/s79/s77/s79/s45/s50
+/s32/s72/s79/s77/s79/s45/s51
+/s32/s32
+/s84/s104/s101/s111/s114/s121
+/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s84/s104/s101/s111/s114/s121
+/s119/s47/s111/s32/s67/s111/s117/s108/s111/s109/s98
+/s51/s48/s32/s75/s83/s116/s97/s114/s107/s32/s82/s101/s100/s115/s104/s105/s102/s116/s32/s40/s109/s101/s86/s41
+/s69/s108/s101/s99/s116/s114/s105/s99/s32/s70/s105/s101/s108/s100/s32/s105/s110/s32/s78/s67/s32/s40/s77/s86/s47/s99/s109/s41
+FIG. 4. (Color online) The comparison of theoretical and
+experimental18Stark redshifts of a 5.6 nm diameter Si NC at
+30 K. The dotted line shows the theoretical curve without the
+direct Coulomb term included. Lines are solely for guiding
+the eye. Inset shows the single-particle Stark shifts of the
+band-edgestates for theconduction(upperpanel)andvalen ce
+states (lower panel).
+/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48
+/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s52/s54/s56/s49/s48/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116
+/s51/s48/s48/s32/s75/s51/s48/s32/s75
+/s32/s32/s80/s76/s32/s80/s101/s97/s107/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41
+/s84/s104/s101/s111/s114/s121
+/s32/s32
+/s51/s48/s32/s75/s51/s48/s48/s32/s75/s69/s109/s105/s115/s115/s105/s111/s110/s32/s82/s97/s116/s101/s32/s40/s97/s46/s117/s46/s41
+/s69/s108/s101/s99/s116/s114/s105/s99/s32/s70/s105/s101/s108/s100/s32/s105/s110/s32/s78/s67/s32/s40/s77/s86/s47/s99/s109/s41
+FIG. 5. (Color online) The experimental PL peak intensity18
+(upper plot) and the theoretical emission rate (lower plot) for
+Si NCs at 30 and 300 K. Lines are solely for guiding the eye.
+overall Stark shift in the energy and FNCis the elec-
+tric field insidethe NC. Here, the polarizability, α, is
+taken as scalar which is in general a rank-2 tensor, how-
+ever, we have observed that the variation in Stark shift
+with respect to relative orientation of the electric field
+and the crystallographic planes of the NC or the c-axis
+of theC3vpoint group of the NCs,31give rise to less
+than 10 meV changes for the highest applied fields. In
+Fig. 6 we show the excitonic polarizability (i.e., with di-/s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48 /s54/s46/s53/s48/s46/s48/s50/s46/s53/s53/s46/s48/s55/s46/s53/s49/s48/s46/s48/s49/s50/s46/s53
+/s48/s50/s52/s54/s56/s32
+/s51/s48/s32/s75
+/s101/s120/s99/s105/s116/s111/s110/s105/s99/s115/s105/s110/s103/s108/s101/s32/s112/s97/s114/s116/s105/s99/s108/s101/s32/s40/s49/s48/s45/s51/s54
+/s32/s67/s32/s109/s50
+/s47/s86/s41
+/s68/s105/s97/s109/s101/s116/s101/s114/s32/s40/s110/s109/s41/s32/s91/s49/s48/s45/s52
+/s32/s109/s101/s86/s47/s40/s107/s86/s47/s99/s109/s41/s50
+/s93
+FIG. 6. (Color online) Theoretically computed polarizabil ity
+for SiNCsbased onthesingle-particle andexcitonic (i.e., with
+directCoulombtermincluded)Starkshifts at30K.Solidlin es
+are solely for guiding the eye. The dashed lines are cubic fits
+to the data (see text).
+rect Coulomb term included) at 30 K. For comparison
+purposes, the single-particle polarizability is also pro-
+vided where this estimate becomes highly exaggerated
+for larger diameters. Both of these curves display a non-
+monotonic behavior with respect to size which exists in
+other physical properties as well; such variations occur as
+the states, such as HOMO and LUMO, acquire different
+representations of the C3vpoint group for different NC
+diameters.31As in the basic dipole polarizability, their
+overall trend can be easily fitted by a cubic dependence
+α=cD3
+NC, whereDNCis the NC diameter, and with
+c= 2.436×10−11and 4.611×10−11C/(Vm), for the ex-
+citonic and single-particle cases, respectively; in another
+unit system they are expressed as c= 1.521×1015and
+2.878×1015meV/[(kV)2cm], respectively.
+C. Intraband electroabsorption of Si NCs
+In Fig. 7 the intravalence and intraconduction state
+electroabsorption of 4 nm and 6 nm diameter Si NCs
+are shown, under 0 and 0.6 MV/cm internal NC elec-
+tric fields. Using an anisotropic effective mass model
+and focusing only on the conduction band, de Sousa et
+al.27have also modeled the intraband electroabsorption
+in Si NCs and as in this work, they obtained a blueshift.
+However, as expected from the rigidity of the conduc-
+tion states under the electric field (cf., Figs. 2 and 3),
+and as observed in Fig. 7, the electroabsorption effect
+can be best utilized in p-doped Si NCs. Unlike the inter-
+bandtransitions, weobtaina blueshift inthe spectrawith
+the applied electric field; observe from Fig. 2 that as the
+electric field increases, both the valence and conduction
+states individually fans out, whereas the optical band
+gap gets redshifted. For the 6 nm case, the first peaks in6
+FIG. 7. (Color online) The intraconduction and intravalenc e
+state electroabsorption curves for 4 nm and 6 nm diameter Si
+NC at 300 K. For amplitude comparison, all four curves are
+drawn to scale among themselves.
+the intravalence electroabsorption spectra shift close to38meVunderaNCfieldof0.6MV/cm; thisshift reduces
+to about 14 meV for the 4 nm case. Given that bulk sili-
+con has very poor Franz-Keldysh and Kerr effect efficien-
+cies for electro-optic modulation,45these results can be
+encouraging for the consideration of nanocrystalline Si-
+based infrared electroabsorption modulators. However,
+we should also remark that the doping of Si NCs has its
+own challengescompared to bulk.46Finally we should re-
+mark that as mentioned within the context of the Stark
+field, the surface polarization effect, also known as local
+field effect, due to dielectric discontinuity, plays a role in
+the optical absorption as well.47–49However, this effect
+becomes quite insignificant for Si NCs with diameters
+larger than about 4 nm and furthermore it only affects
+the amplitude of the absorption without modifying its
+spectral profile.31
+IV. CONCLUSION
+In conclusion, we show that the recent QCSE data18
+for the embedded Si NCs under a strong Stark field can
+essentially be explained very well with an atomistic pseu-
+dopotential model which otherwise does not incorporate
+any size inhomogeneity, interface states, or the strain
+effects. In this context, the importance of the direct
+Coulomb interaction is demonstrated, and a simple ex-
+pressionforthe SiNC polarizabilityisextracted. Finally,
+incompliancewiththefactthatthevalencestatesdisplay
+much moreStarkshift, it is shownthat intravalenceband
+electroabsorption enjoys wider voltage tunability which
+can be harnessed for Si-based electroabsorption modula-
+tor intentions.
+ACKNOWLEDGMENTS
+The authors gratefully acknowledge the support by
+The Scientific and Technological Research Council of
+Turkey (T ¨UB˙ITAK) under Projects No. 106T048 and
+No. 106M549. C.B. would like to thank Can U˘ gur Ayfer
+and the Bilkent University Computer Center for the crit-
+ical computing service they provide.
+∗bulutay@fen.bilkent.edu.tr
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\ No newline at end of file
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+arXiv:1001.0123v1  [cond-mat.mes-hall]  31 Dec 2009Quantum transport in honeycomb lattice ribbons with zigzag
+edges: A theoretical study
+Santanu K. Maiti1,2,1
+1Theoretical Condensed Matter Physics Division, Saha Insti tute of Nuclear Physics,
+1/AF, Bidhannagar, Kolkata-700 064, India
+2Department of Physics, Narasinha Dutt College, 129 Belilio us Road, Howrah-711 101, India
+Abstract
+We explore electron transport properties in honeycomb lattice ribb ons with zigzag edges coupled to two
+semi-infinite one-dimensional metallic electrodes. The calculations ar e based on the tight-binding model
+andtheGreen’sfunctionmethod, whichnumericallycomputethecon ductance-energyandcurrent-voltage
+characteristics as functions of the lengths and widths of the ribbo ns. Our numerical results predict that
+for such a ribbon an energy gap always appears in the conductance spectrum across the energy E= 0.
+With the increase of the size of the ribbon, the gap gradually decrea ses but it never vanishes. This
+clearly manifests that a honeycomb lattice ribbon with zigzag edges a lways exhibits the semiconducting
+behavior, and it becomes much more clearly visible from our presente d current-voltage characteristics.
+PACS No. : 73.63.-b; 73.63.Rt.
+Keywords : Honeycomb lattice ribbon; Zigzag edges; Conductance; I-Vcharacteristic.
+1Corresponding Author : Santanu K. Maiti
+Electronic mail: santanu.maiti@saha.ac.in
+11 Introduction
+The electronic transport in graphene nanoribbons
+has opened up new challenges in nanoelectronics.
+A graphene nanoribbon (GNR) is a monolayer of
+carbon atoms arranged in a two-dimensional hon-
+eycomb lattice structure1−4and can be regarded
+as the basic building blocks for graphitic materials.
+Graphene based materials have potential applica-
+tions in several branch of nanoelectronics and due
+to their special electronic and physical properties
+they exhibit several novel properties like high car-
+rier mobility,3unconventional quantum Hall effect5
+and many others. The high carrier mobility in
+graphene demonstrates the idea for fabrication of
+high speed switching devices those have widespread
+applications in different fields. Recently GNRs are
+also used extensively in designing field-effect tran-
+sistors and this idea has been predicted in some re-
+cent nice papers.6−8It has many potential applica-
+tions and providesahuge interest in the community
+of nanoelectronics device research. Not only that,
+GNRs can be used to construct MOSFETs which
+perform much better than conventional Si MOS-
+FETs. In other experiment9it has been proposed
+that a narrow strip of graphene, the so-called a
+graphene nanoribbon, exhibits semiconducting be-
+havior due to its edge effects, unlike carbon nan-
+otubes of larger sizes which are mixtures of both
+metallic and semiconducting materials. The rea-
+son is that, in a narrow graphene sheet a band gap
+appears across the energy E= 0, while the gap
+gradually disappears with the increase of the size
+of the ribbon. It reveals a transformation from the
+semiconducting to the metallic material. This phe-
+nomenon has been studied in detail in a very recent
+theoreticalworkby the sameauthorofthis paper.10
+The situation becomes quite different for the caseof
+honeycomb lattice ribbons with zigzag edges. Our
+study predicts that for a ribbon with zigzag edges
+always there exists an energy gap in the conduc-
+tance spectrum across the energy E= 0. The gap
+gradually decreases with the increase of the size of
+the ribbon (as expected), but it never vanishes even
+for much larger ribbons. This clearly reveals that
+a honeycomb lattice ribbon with zigzag edges al-
+ways exhibits the semiconducting behavior, unlike
+the lattice ribbons with armchair edges where both
+the semiconducting and the metallic phases are ob-
+served by tuning the size of a ribbon. All these
+edge effects in graphene ribbons provide many key
+informations in designing nanoelectronic devices.
+The aim of the present paper is to provide a qual-itative study of electron transport in honeycomb
+lattice ribbons with zigzag edges attached to two
+semi-infinite one-dimensional metallic electrodes
+(see Fig. 1). The theoretical description of elec-
+tron transport in a bridge system has been devel-
+oped based on the pioneering work of Aviram and
+Ratner.11Later, many significant experiments12−13
+have been done in several bridge systems to un-
+derstand the basic mechanisms underlying the
+electron transport. Though in literature many
+theoretical14−25aswellasexperimentalpapers12−13
+on electron transport are available, yet lot of con-
+troversies are still present between the theory and
+experiment even today. Several controlling factors
+are there which can regulate the electron transport
+in a conducting bridge significantly, and all these
+factors have to be taken into account properly to
+understand the transport mechanism. For our il-
+lustrative purposes, here we mention some of these
+issues.
+(i) The geometry of the conducting material be-
+tween the two electrodes itself is an important is-
+sue to control the electron transmission which has
+been described quite elaborately by Ernzerhof et
+al.26through some model calculations.
+(ii) The coupling of the bridging material to the
+electrodes significantly controls the current ampli-
+tude across any bridge system.27
+(iii) The quantum interference effect27−31of elec-
+tron waves passing through different arms of the
+bridging material becomes the most significant is-
+sue.
+(iv) The dynamical fluctuation in the small-scale
+devices is another important factor which plays an
+active role and can be manifested through the mea-
+surement of shot noise ,32−33a direct consequence
+of the quantization of charge.
+In addition to these, several other factors of the
+Hamiltonianthatdescribeasystemalsoprovideim-
+portant effects in the determination of the current
+across a bridge system.
+Here we adopt a simple tight-binding model to
+describe the system and all the calculationsare per-
+formed numerically. We narrate the conductance-
+energy and current-voltage characteristics as func-
+tions of the lengths and widths of ribbons. Our re-
+sults clearly predicts that a honeycomb lattice rib-
+bon with zigzag edges always shows the semicon-
+ducting nature irrespective of its length and width.
+The paper is arranged in this way. Following the
+introduction (Section 1), in Section 2, we present
+the model and the theoretical formulations for our
+calculations. Section 3 discusses the significant re-
+2sults, and the summary of this work is available in
+Section 4.
+2 Model and synopsis of the
+theoretical background
+Let us start with Fig. 1, where a honeycomb lat-
+tice ribbon with zigzag edges is attached to two
+semi-infinite one-dimensional metallic electrodes,
+viz, source and drain. As the electron transport
+properties are significantly influenced by the quan-
+Source
+Drain
+Honeycomb lattice ribbon
+Figure 1: (Color online). Schematic view of a hon-
+eycomb lattice ribbon with zigzag edges attached
+to two semi-infinite one-dimensional metallic elec-
+trodes, viz, source and drain. Filled circles corre-
+spond to the position of the atomic sites.
+tum interference effects, throughout the study, we
+contact the electrodes at the two extreme ends of
+nanoribbons (see Fig. 1) to make these interference
+effects uniform.
+Calculation of the conductance:
+We use the Landauer conductance formula34−35
+to calculate the conductance gof the ribbon. At
+very low temperature and bias voltage it ( g) can be
+presented in the form,
+g=2e2
+hT (1)
+whereTgives the transmission probability of an
+electron in the ribbon which can be expressed in
+terms of the Green’s function of the ribbon and its
+coupling to the two electrodes by the relation,34−35
+T= Tr[Γ SGr
+ribΓDGa
+rib] (2)
+whereGr
+ribandGa
+ribare respectively the retarded
+and advanced Green’s functions of the ribbon in-
+cluding the effects of the electrodes. The parame-
+ters ΓSand ΓDdescribe the coupling of the ribbonto the source and drain respectively, and they can
+be defined in terms of their self-energies. For the
+complete system i.e., the ribbon, source and drain,
+the Green’s function is defined as,
+G= (E −H)−1(3)
+whereE=E+iδ.Eis the injecting energy of the
+source electron and δgives an infinitesimal imagi-
+nary part to E. To Evaluate this Green’s function,
+the inversion of an infinite matrix is needed since
+the complete system consists of the finite ribbon
+and the two semi-infinite electrodes. However, the
+entire system can be partitioned into sub-matrices
+corresponding to the individual sub-systems and
+the Green’s function for the ribbon can be effec-
+tively written as,
+Grib= (E −Hrib−ΣS−ΣD)−1(4)
+whereHribis the Hamiltonian of the ribbon which
+can be written in the tight-binding model within
+the non-interacting picture like,
+Hrib=/summationdisplay
+iǫic†
+ici+/summationdisplay
+<ij>t/parenleftBig
+c†
+icj+c†
+jci/parenrightBig
+(5)
+In the above Hamiltonian ( Hrib),ǫi’s are the site
+energies, c†
+i(ci) is the creation (annihilation) oper-
+ator of an electron at the site iandtis the nearest-
+neighbor hopping integral. A similar kind of tight-
+binding Hamiltonian is also used to describe the
+two semi-infinite one-dimensionalperfect electrodes
+where the Hamiltonian is parametrized by constant
+on-site potential ǫ0and nearest-neighbor hopping
+integral t0. In Eq. 4, Σ S=h†
+S−ribgShS−riband
+ΣD=hD−ribgDh†
+D−ribare the self-energy opera-
+tors due to the two electrodes, where gSandgD
+correspond to the Green’s functions of the source
+and drain respectively. hS−ribandhD−ribare the
+coupling matrices and they will be non-zero only
+for the adjacent points of the ribbon, and the elec-
+trodes respectively. The matrices Γ Sand Γ Dcan
+be calculated through the expression,
+ΓS(D)=i/bracketleftBig
+Σr
+S(D)−Σa
+S(D)/bracketrightBig
+(6)
+where Σr
+S(D)and Σa
+S(D)are the retarded and ad-
+vanced self-energies respectively, and they are con-
+jugate with each other. These self-energies can be
+written as,36
+Σr
+S(D)= ΛS(D)−i∆S(D) (7)
+where Λ S(D)are the real parts of the self-energies
+whichcorrespondtotheshift oftheenergyeigenval-
+ues of the ribbon and the imaginary parts ∆ S(D)of
+3the self-energies represent the broadening of these
+energy levels. Since this broadening is much larger
+than the thermalbroadening, we restrictourall cal-
+culationsonlyat absolutezerotemperature. All the
+information about the ribbon-to-electrode coupling
+are included into these two self-energies.
+Calculation of the current:
+The current passing across the ribbon can be de-
+picted as a single-electron scattering process be-
+tween the two reservoirs of charge carriers. The
+currentIcan be computed as a function of the ap-
+plied bias voltage Vthrough the relation,34
+I(V) =2e
+hEF+eV/2/integraldisplay
+EF−eV/2T(E)dE (8)
+whereEFis the equilibrium Fermi energy. Here we
+make a realistic assumption that the entire voltage
+is dropped across the ribbon-electrode interfaces,
+and it is examined that under such an assumption
+theI-Vcharacteristics do not change their qualita-
+tive features. This assumption is based on the fact
+that, the electric field inside the ribbon especially
+for narrow ribbons seems to have a minimal effect
+on the conductance-voltage characteristics. On the
+other hand, for quite larger ribbons and high bias
+voltagesthe electricfield inside the ribbonmayplay
+a more significant role depending on the internal
+structure and size of the ribbon,36but the effect
+becomes too small.
+3 Numerical results and dis-
+cussion
+To have a better insight in the present problem
+i.e., the dependence of the electron transport on
+the lengths and widths of ribbons, we focus our at-
+tention only on the perfect systems rather than any
+disordered one. To get these perfect systems we
+fix the on-site energies ǫi= 0 for all the sites iof
+the honeycomb lattice ribbons. The values of the
+other parameters are taken as follow. The nearest-
+neighbor hopping integral tin the ribbon is set to
+1, the on-site energy ǫ0and the hopping integral t0
+for the two electrodes are fixed to 0 and 1 respec-
+tively. The parameters τSandτDare set as 0 .75,
+where they correspond to the hopping strengths of
+the ribbon to the source and drain respectively. In
+addition to these, to describe the size of a ribbon
+we introduce two other parameters NandMwherethey correspond to the width and length of the rib-
+bon respectively. Thus, for example, a nanorib-
+bon with N= 1 and M= 3 represents a linear
+chain of three hexagons. Hence the parameter M
+determines the total number of hexagons in a single
+chain. Followingthisrule,ananoribbonwith N= 3
+andM= 4 corresponds to three linear chains at-
+/Minus2/Minus1012
+E0.01.0/LParen1b/RParen12.0g/LParen1E/RParen1/Minus2/Minus1012
+E0.01.0/LParen1a/RParen12.0g/LParen1E/RParen1
+Figure 2: (Color online). Conductance gas a func-
+tion of the energy Efor some lattice ribbons with
+fixed width N= 3 and varying lengths where (a)
+M= 6 and (b) M= 8.
+tached side by side (see Fig. 1) where each chain
+contains four hexagons. For simplicity, throughout
+our study we set the Fermi energy EF= 0 and
+choose the units where c=e=h= 1.
+Let us begin our discussion with the variation of
+the conductance gas a function of the energy E.
+In Fig. 2 we plot the conductance-energy charac-
+teristics for some typical lattice ribbons with fixed
+widthN= 3 and varying lengths where (a) and (b)
+correspondtothe lengths M= 6and 8respectively.
+The sharp resonant peaks in the conductance spec-
+tra are observed almost for all energies, while for
+some other energies either the conductance ggets
+much small value or drops to zero. At the reso-
+nances the conductance gets the value 2, and there-
+fore, the transmission probability Tbecomes unity
+4since the relation g= 2Tfollows from the Lan-
+dauer conductance formula, Eq. 1, with e=h= 1.
+Now the reduction of the transmission probability
+(T <1) for some particular energies can be ex-
+plained by considering the quantum interference ef-
+fects of the electronic waves passing through the
+different arms of the ribbon. The interpretation is
+as follow. During the motion of the electrons from
+/Minus2/Minus1012
+E0.01.0/LParen1b/RParen12.0g/LParen1E/RParen1/Minus2/Minus1012
+E0.01.0/LParen1a/RParen12.0g/LParen1E/RParen1
+Figure 3: (Color online). Conductance gas a func-
+tion of the energy Efor some lattice ribbons with
+fixed length M= 4 and varying widths where (a)
+N= 5 and (b) N= 10.
+the source to drain through the lattice ribbon, the
+electron waves propagating along the different pos-
+sible pathways can get a phase shift among them-
+selves, according to the result of quantum interfer-
+ence. Therefore, the probability amplitude of get-
+ting an electron across the ribbon either becomes
+strengthened or weakened. This causes the trans-
+mittance cancellationsandprovidesanti-resonances
+in the conductance spectrum. Thus it can be em-
+phasized that the electron transmission is strongly
+affected by the quantum interference effects, and
+hence the ribbon to electrode interface structure.
+Now all these resonant peaks are associated with
+the energy eigenvalues of the ribbon, and accord-
+ingly, we can say that the conductance spectrummanifests itself the electronic structure of the rib-
+bon. Due to the large number of energy levels,
+associated with the size of the ribbons, the reso-
+nant peaksalmost overlapwith eachotherand form
+quasi-continuous spectra in the g-Echaracteristics.
+Themostsignificantfeatureobservedfromthespec-
+246810121416
+N0.00.10.20.30.40.50.6∆E
+Figure 4: (Color online). Variation of the central
+energygap δEasafunctionofthewidth Nforsome
+lattice ribbons with fixed lengths M. The red and
+blue curvescorrespondto M= 2and4respectively.
+tra, given in Fig. 2, is that a central energy gap
+(δE) appears across the energy E= 0. With the
+increase of the length of the ribbon some more reso-
+nant peaks appear around E= 0, and accordingly,
+the gap decreases which is clearly observed from
+Figs. 2(a) and (b). Thus for a fixed width, the cen-
+tral gap can be controlled by tuning the length of
+the ribbon.
+In the same fashion, to characterize the depen-
+dence of the electron transport on the widths of the
+ribbons for a fixed length, in Fig. 3 we plot the re-
+sultsforsometypicallatticeribbonsconsideringthe
+lengthM= 4. The spectra shown in Figs. 3(a) and
+(b) correspond to the ribbons with widths N= 5
+and10respectively. Quitesimilartotheabovecase,
+here also a gap appears across E= 0 and it de-
+creases with the increase of the width of the ribbon
+though the reduction is much small. Thus from the
+results described in Figs. 2 and 3 we can empha-
+size that the width of the central energy gap always
+decreases with the increase of the size (length and
+width) of the ribbon.
+To illustrate the dependence of the gap δEwith
+other system sizes, in Fig. 4, we display the varia-
+tion ofδEas a function of the width Nfor some
+typical lattice ribbons with fixed lengths. The red
+and blue curves correspond to the lengths M= 2
+and 4 respectively. Quite interestingly we see that,
+5the gap δEgradually decreases with the increase
+of the width N, and beyond a certain value of N,
+the rate of decrease of this gap becomes much small
+and eventually it ( δE) becomes almost a constant.
+Quite similar feature is also observed if we plot the
+variation of the energy gap as a function of the
+lengthMkeeping the width Nas a constant, and
+due to the obvious reason we do not plot these re-
+sults further in the present description. These re-
+/Minus4/Minus2024
+Voltage/LParen1V/RParen1/Minus3.03.0
+/Minus1.5/LParen1b/RParen1
+1.5
+0Current/LParen1I/RParen1/Minus4/Minus2024
+Voltage/LParen1V/RParen1/Minus3.03.0
+/Minus1.5/LParen1a/RParen1
+1.5
+0Current/LParen1I/RParen1
+Figure 5: (Color online). Current Ias a function
+of the bias voltage Vfor some lattice ribbons with
+fixed width N= 3 and varying lengths where (a)
+M= 3 and (b) M= 6.
+sults provide us an important signature which con-
+cern with the variation of the energy gap by tuning
+the size of the ribbon, and we can emphasize that a
+honeycomb lattice ribbon with zigzag edges always
+exhibits the semiconducting (finite energy gap) be-
+havior.
+All these basic features of electron transfer can
+be much more clearly explained from our investi-
+gation of the current-voltage ( I-V) characteristics
+rather than the conductance-energy spectra. The
+currentIis determined from the integration proce-
+dure of the transmission function ( T) (see Eq. 8),
+where the function Tvaries exactly similar to the
+conductance spectra, differ only in magnitude by afactor 2, since the relation g= 2Tholds from the
+Landauer conductance formula (Eq. 1). As an il-
+lustration, in Fig. 5, we present the current-voltage
+(I-V) characteristics for some lattice ribbons with
+fixed width N= 3 and varying lengths where (a)
+and (b) correspond to the lengths M= 3 and 6 re-
+spectively. In the same footing, in Fig. 6, we plot
+the variation of the current Ias a function of the
+bias voltage Vforsome typicallattice ribbonskeep-
+/Minus4/Minus2024
+Voltage/LParen1V/RParen1/Minus3.03.0
+/Minus1.5/LParen1b/RParen1
+1.5
+0Current/LParen1I/RParen1/Minus4/Minus2024
+Voltage/LParen1V/RParen1/Minus3.03.0
+/Minus1.5/LParen1a/RParen1
+1.5
+0Current/LParen1I/RParen1
+Figure 6: (Color online). Current Ias a function
+of the bias voltage Vfor some lattice ribbons with
+fixed length M= 4 and varying widths where (a)
+N= 2 and (b) N= 3.
+ingthelengthasfixed( M= 4)andvarythewidths,
+where (a) and(b) representthe ribbonswith widths
+N= 2 and 3 respectively. The sharpness in the I-
+Vcharacteristics and the current amplitude solely
+depend on the coupling strengths of the ribbon to
+the side attached electrodes, viz, source and drain.
+It is observed that, in the limit of weak coupling,
+defined by the condition τS(D)<< t, current shows
+staircase like structure with sharp steps. While, in
+the strong coupling limit, described by the condi-
+tionτS(D)∼t, current varies quite continuously
+with the bias voltage Vand achieves large current
+amplitudecomparedtotheweak-couplinglimit. All
+these coupling effects have already been explained
+6in many theoretical as well as experimental papers
+in the literature. The key feature observed from
+theseI-Vcharacteristics is that for all such ribbons
+the electron starts to conduct beyond some finite
+bias voltage, the so-called threshold bias voltage
+Vth, and it decreases very slowly with the change of
+the size of the ribbon. Our study reveals that the
+threshold bias voltage never drops to zero, even for
+much larger systems, and accordingly, we can pre-
+dict that a honeycomb lattice ribbon with zigzag
+edges exhibits only the semiconducting behavior.
+4 Concluding remarks
+To summarize, we have addressed electron trans-
+port properties in honeycomb lattice ribbons with
+zigzag edges attached to two semi-infinite one-
+dimensional metallic electrodes within the tight-
+binding framework. We havenumericallycomputed
+theconductance-energyandcurrent-voltagecharac-
+teristics concerning the dependence on the lengths
+and widths of the ribbons. The results have pre-
+dicted thatforsuchribbonsi.e., ribbonswith zigzag
+edges a central energy gap always exists across the
+energyE= 0 in the conductance spectrum. The
+gap decreases gradually with the increase of the
+size of the ribbon but it never vanishes. This phe-
+nomenon clearly manifests that a honeycomb lat-
+tice ribbon with zigzag edges always shows semi-
+conducting nature, unlike the lattice ribbons with
+armchair edges where both the semiconducting and
+the metallic phases are observed by controlling the
+size of the ribbon. This semiconducting behavior of
+the lattice ribbons with zigzagedges has been much
+more clearly addressed from our presented current-
+voltage characteristics. It has been observed that
+the current starts to appear beyond some finite bias
+voltage i.e., the threshold bias voltage Vthhas a
+non-zero value. This Vthdoesn’t change apprecia-
+blywiththeincreaseofthesizeoftheribbonandwe
+have also examined that the threshold bias voltage
+never reduces to zero even for much larger systems,
+which predicts the semiconducting nature only.
+This is our first step to describe how the elec-
+tron transport properties in honeycomb lattice rib-
+bons with zigzag edges depends on the size of the
+ribbons. We have made several realistic assump-
+tions by ignoring the effects of the electron-electron
+correlation, disorder, interaction with a substrate,
+temperature, finite width of the electrodes, bound-
+ary of the ribbons, etc. Here we discuss very briefly
+about these approximations. The inclusion of the
+electron-electron correlation in the present modelis a major challenge to us, since over the last few
+years people have studied a lot to incorporate this
+effect, butnosuchpropertheoryhasyetbeendevel-
+oped. In this work,wehavepresentedallthe results
+only for the ordered systems. But in real samples,
+the presence of impurities will affect the electronic
+structure and hence the transport properties. Be-
+sidethese, inexperiments,thegraphenenanoribbon
+is deposited on an insulating substrate which has
+also not been included in our present study, and,
+it has been observed from first-principles calcula-
+tions that the effect of the substrate is too week.37
+The effect of the temperature has already been
+pointed out earlier, and, it has been examined that
+the presented results will not change significantly
+even at finite temperature, since the broadening of
+the energy levels of the ribbon due to its coupling
+with the electrodes will be much larger than that
+of the thermal broadening.34The other important
+assumption is that here we have chosen the linear
+chains instead of wider leads, since we are mainly
+interested about the basic physics of the ribbon.
+Though the results presented here change with the
+increase of the thickness of the leads, but all the ba-
+sic featuresremainquite invariant. The effect ofthe
+boundary is also an important issue in this context.
+Here we have considered only the perfect geometry
+of the nannoribbons. Several interesting features
+will be observed for the nanoconstrictions with dif-
+ferent shapes38like, square-shaped, wedge-shaped
+nanoconstrictions, etc. Finally, we would like to
+say that we need further study in such systems by
+incorporating all these effects.
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\ No newline at end of file
diff --git a/1001.0124.txt b/1001.0124.txt
new file mode 100644
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+arXiv:1001.0124v2  [gr-qc]  14 Jul 2018Geon black holes and quantum field theory
+Jorma Louko
+School of Mathematical Sciences, University of Nottingham , Nottingham NG7 2RD, UK
+E-mail:jorma.louko@nottingham.ac.uk
+Abstract. Black hole spacetimes that are topological geons in the sens e of Sorkin can be
+constructed by taking a quotient of a stationary black hole t hat has a bifurcate Killing horizon.
+We discuss the geometric properties of these geon black hole s and the Hawking-Unruh effect
+on them. We in particular show how correlations in the Hawkin g-Unruh effect reveal to an
+exterior observer features of the geometry that are classic ally confined to the regions behind the
+horizons.
+Talkgivenatthe FirstMediterraneanConference onClassical andQuantumGr avity, Kolymbari
+(Crete, Greece), September 14–18, 2009.
+This contribution is dedicated to Rafael Sorkin in celebrat ion of his influence, inspiration and
+friendship.
+1. Introduction
+A geon, short for “gravitational-electromagnetic entity” , was introduced in 1955 by John
+Archibald Wheeler [1] as a configuration of the gravitationa l field, possibly coupled to other
+zero-mass fields such as massless neutrinos [2] or the electr omagnetic field [3], such that a
+distant observer sees the curvature to be concentrated in a c entral region with persistent large
+scale features. The configuration was required to be asympto tically flat [4], allowing the mass to
+be defined by what are now known as Arnowitt-Deser-Misner met hods. The examples discussed
+in [5] indicate that the configuration was also understood to have spatial topology R3, excluding
+black holes. These geons are however believed to be unstable , owing to the tendency of massless
+fields either to disperse to infinity or to collapse into a blac k hole [6].
+In 1985, Sorkin [7] generalised Wheeler’s geon into a topological geon by allowing the spatial
+topology to be nontrivial, as well as dropping the a priori re quirement of persistent large scale
+ΣFigure 1. A spatial
+slice Σ of a topological
+geon. The geometry of
+Σ is asymptotically Eu-
+clidean, but the topology
+of Σ does not need to
+beR3, as illustrated by
+the handle drawn.features. The central regions may thus have complicated top ology, as illustrated in Figure 1,
+and the time evolution of the initial data may lead into singu larities and black holes [8, 9].
+In particular, topological geons include quotients of stat ionary black hole spacetimes with a
+bifurcate Killing horizon, such that the two exteriors sepa rated by the Killing horizon become
+identified [10, 11]. Such geon black holes are an intermediat e case between conventional
+stationary black holes and dynamical black holes, in the sen se that the nonstationary features
+are confined behind the horizons.
+As the nontrivial spatial topology in a geon black hole is pre sent since arbitrarily early times,
+one does not expect an astrophysical star collapse to result into a geon black hole. Instead,
+the interest of geon black holes is in their quantum mechanic al properties. Because the Killing
+horizon now does not separate two causally disconnected ext eriors, it is not possible to arrive
+at a thermal density matrix for a quantum field by the usual pro cedure of tracing over the
+second exterior. Nevertheless, as we shall discuss, a quant um field on a geon black hole does
+exhibit thermality in the standard Hawking temperature, al though only for a restricted set of
+observations. We shall in particular see how the quantum cor relations in the Hawking-Unruh
+effect reveal to an exterior observer features of the geometry that are confined to the regions
+behind the horizons.
+We start by introducing in Section 2 the showcase example, th eRP3geon, formed as a
+quotient of Kruskal. Section 3 discusses a number of general isations, including geons with
+angular momenta and gauge charges. Thermal effects in quantum field theory on geons are
+addressedin Section 4. Section 5 concludes by discussingth eprospectsof computingtheentropy
+of a geon from a quantum theory of gravity.
+2. Showcase example: RP3geon
+Before the Kruskal-Szekeres extension of the Schwarzschil d solution was known, it had been
+observed that the time-symmetric initial data for exterior Schwarzschild can be extended into a
+time-symmetric wormhole initial data, connecting two asym ptotically flat infinities, illustrated
+in Figure 2. Misner and Wheeler [5] noted that this wormhole a dmits an involutive freely-acting
+isometry that consists of the radial reflection about the wor mhole throat composed with the
+antipodal map on the S2of spherical symmetry. The Z2quotient of the wormhole by this
+isometry, illustrated in Figure 3, is hence a new time-symme tric spherically symmetric initial
+data for an Einstein spacetime.
+The new time-symmetric initial data has topology RP3\{point}, the omitted point being at
+the asymptotically flat infinity [12]. The time evolution is a topological geon spacetime, called
+theRP3geon [10], and it can be obtained as a Z2quotient of Kruskal as shown in Figures 4
+and 5. A general discussion of quotients of Kruskal can be fou nd in [13].
+TheRP3geon is a black and white hole spacetime, it is spherically sy mmetric, and it is
+time and space orientable. It inherits from Kruskal a black h ole singularity and a white hole
+singularity, but the quotient has introduced no new singula rities. The distinctive feature is that
+is has only one exterior region, isometric to exterior Schwa rzschild.
+As is well known, Kruskal admits a one-parameter group of iso metries that coincide with
+Schwarzschild time translations in the two exteriors. Thes e isometries do however not induce
+globally-defined isometries on the RP3geon, becausethe quotienting map changes the sign of the
+corresponding Killing vector, as shown in Figures 6 and 7. As a consequence the geon exterior
+has a distinguished surface of constant Schwarzschild time , shown in Figure 5, even though one
+needs to probe the geon geometry beyond the exterior to ident ify this surface. The existence of
+this distinguished spacelike surface will turn out signific ant with the Hawking-Unruh effect in
+Section 4.throat
+Figure 2. Time-symmetric wormhole initial
+data for Kruskal manifold, with the radial
+dimension drawn vertical and the two-
+spheres of spherical symmetry drawn as
+horizontal circles. The wormhole connects
+two asymptotically flat infinities. It admits a
+freely-acting involutive isometry that consists
+of the radial reflection about the wormhole
+throat composed with the antipodal map on
+theS2.RP2
+Topological geon!
+Figure 3. Time-symmetric initial data for
+theRP3geon, obtained as the Z2quotient of
+the wormhole by the freely-acting involutive
+isometry. The topology of the wormhole is
+S2×R≃S3\ {2 points}and that of the
+quotient is/parenleftbig
+S3\ {2 points}/parenrightbig
+/Z2≃RP3\
+{point}, each of the omitted points being at
+an asymptotically flat infinity.
+η
+ξ
+Figure 4. Conformal diagram of Kruskal
+manifold, with the S2orbits of spherical
+symmetry suppressed and the Kruskal time
+and space coordinates ( η,ξ) shown. The
+wormhole of Figure 2 is the horizontal line
+η= 0. The freely-acting involutive isometry
+that preserves time orientation and restricts
+to that of the wormhole is JK:/parenleftbig
+η,ξ,θ,ϕ/parenrightbig
+/mapsto→/parenleftbig
+η,−ξ,P(θ,ϕ)/parenrightbig
+, wherePis theS2antipodal
+map.RP2
+Figure 5. Conformal diagram of the RP3
+geon, obtained as the Z2quotient of Kruskal
+by the freely-acting involutive isometry JK.
+The suppressed orbits of spherical symmetry
+areS2except on the dashed line where they
+areRP2. The initial data of Figure 3 is shown
+as the horizontal line.Figure 6. Orbits of Killing time translations
+on Kruskal. In the exteriors these isometries
+are translations in Schwarzschild time.Figure 7. Orbits of the local RP3geon
+isometry inherited from the Killing time
+translations on Kruskal. The isometry is not
+globally defined because the corresponding
+Killing vector has a sign ambiguity in the
+black and white hole interiors.
+3. Other geons
+3.1. Static
+The quotienting of Kruskal into the RP3geon generalises immediately to the higher-dimensional
+spherically symmetric generalisation of Schwarzschild [1 4]. As the antipodal map on a sphere
+preserves the sphere’s orientation in odd dimensions but re verses the sphere’s orientation in even
+dimensions, the geon is space orientable precisely when the spacetime dimension is even.
+A wider class of generalisations occurs when the sphere is re placed by a (not necessarily
+symmetric) Einstein space. In D≥4 spacetime dimensions, the vacuum solution can be written
+in Schwarzschild-type coordinates as [15]
+ds2=−∆dt2+dr2
+∆+r2d¯Ω2
+D−2,∆ =˜λ−µ
+rD−3−˜Λr2, (3.1)
+whered¯Ω2
+D−2is the metric on the ( D−2)-dimensional Einstein space MD−2,˜λis proportional
+to the Ricci scalar of MD−2,˜Λ is proportional to the cosmological constant and µis the mass
+parameter. Suppose that ∆ has a simple zero at r=r+>0 and ∆ >0 forr > r+. The solution
+has then a Kruskal-type extension across a bifurcate Killin g horizon at r=r+, separating two
+causally disconnected exteriors. If now MD−2admits a freely-acting involutive isometry, a geon
+quotient can be formed just as in Kruskal [11]. For a negative cosmological constant these geons
+are asymptotically locally anti-de Sitter.
+ForD= 3 and a negative cosmological constant, the above Z2quotient generalises
+immediately to the nonrotating Ba˜ nados-Teitelboim-Zane lli (BTZ) hole [16]. The BTZ hole
+admits however also geon versions that arise as other quotie nts of anti-de Sitter space, even with
+angular momentum [17, 18, 19].
+If a spacelike asymptopia at r→ ∞is not assumed, further quotient possibilities arise, for
+example by quotienting de Sitter space or Schwarzschild-de Sitter space [20, 21, 22]. We shall
+not consider these quotients here.
+3.2. Spin
+Including spin is delicate. To see the problem, consider the Kerr black hole, shown in Figure 8.
+The two exteriors are isometric by an isometry that preserve s the global time orientation, but
+this isometry inverts the direction of the rotational angle and has hence fixed points in the black−J J
+Figure 8. Conformal diagram of Kerr. The
+angular momenta at the opposing infinities
+are equal in absolute value but opposite in
+sign.−Q Q
+Figure 9. Conformal diagram of Reissner-
+Nordstr¨ om. The electric charges at the
+opposing infinities are equal in absolute value
+but opposite in sign.
+hole region and in the white hole region. Kerr does therefore not admit a geon quotient. A way
+to think about this is that if the angular momentum at one infin ity isJ, the angular momentum
+at the opposing infinity is −J.
+The generalisation of Kerr(-AdS) to D≥5 dimensions [23, 24, 25, 26] has [( D−1)/2]
+independent angular momenta. Each of these angular momenta has a different sign at the
+opposing infinities. This suggests that if the angular momen ta are arranged into multiplets,
+such that within each multiplet the magnitude is constant bu t the signs alternate, there might
+exist a geon quotient by a map that suitably permutes the rota tion planes within each multiplet.
+This turns out to work when Dis odd and not equal to 7, but not for other values of Dat least
+in a way that would be continuously deformable to the nonrota ting case [11].
+For a negative cosmological constant it is possible to define angular momenta also on a torus,
+rather than on a sphere. The geon quotient must then again inv olve a permutation of angular
+momenta with equal magnitude and opposite signs [11].
+Finally, the five-dimensional Emparan-Reall black ring [27 ] has a geon quotient [28], despite
+having just one nonvanishing angular momentum.
+3.3.U(1)charge
+Including the electromagnetic field is also delicate. To see the problem, consider electrically-
+charged Reissner-Nordstr¨ om, shown in Figure 9. Given the g lobal time orientation, Gauss’s law
+says that if the charge at one infinity equals Q, the charge at the opposing infinity equals −Q.
+Whilethere is an involutive isometry by which to take a geon q uotient of thespacetime manifold,
+this isometry reverses the sign the field strength tensor, an d a quotient is thus not possible with
+the usual understanding of the electromagnetic field as a U(1 ) gauge field [5]. The situation is
+the same for the electrically-charged generalisations of ( 3.1) [15].
+There is however a way around this minus sign problem. Maxwel l’s theory has a discrete
+involutive symmetry that reverses the sign of the field stren gth tensor as well as the signs of all
+charges: charge conjugation. The usual formulation of elec tromagnetism as a U(1) gauge theory
+treats charge conjugation as a global symmetry rather than a s a gauge symmetry. However, it
+is possible to promote charge conjugation into a gauge symme try: the gauge group is then no
+longer U(1) ≃SO(2) but O(2) ≃Z2⋉U(1), where in the semidirect product presentation the
+nontrivial element of Z2acts on U(1) by complex conjugation [29]. The disconnected c omponent
+of the enlarged gauge group then reverses the sign of the field strength tensor. A geon quotient is
+now possible provided the quotienting map includes a gauge t ransformation in the disconnected
+component of the enlarged gauge group [11]. The geon’s charg e is well-defined only up to theoverall sign, but detecting the sign ambiguity would requir e observations behind the horizons.
+Within the magnetically-charged generalisations of (3.1) [15], the options depend on the
+(D−2)-dimensional Einstein space MD−2[11]. In some cases (including magnetic Reissner-
+Nordstr¨ om in even dimensions) the situation is just as with electric charge. In other cases
+(includingmagnetic Reissner-Nordstr¨ om in odd dimension s) the field strength tensor is invariant
+under the relevant involutive isometry and the usual U(1) fo rmulation of Maxwell’s theory
+suffices. Further options arise if the spacetime has both elec tric and magnetic charge [11].
+3.4.SU(n)charge
+Black holes with a non-Abelian gauge field also admit a geon qu otient. For generic static,
+spherically symmetric SU( n) black holes with n >2 [30, 31, 32] this requires enlarging the
+gauge group to Z2⋉SU(n), where the nontrivial element of Z2acts on SU( n) by complex
+conjugation and the quotienting map includes a a gauge trans formation in the disconnected
+component[33,34]. Bycontrast, staticSU(2) black holeswi thspherical[35,36,37,38,39,40,41]
+or axial [42, 43] symmetry admit a geon quotient without enla rging the gauge group [11, 33, 34].
+4. Quantum field theory
+The geon black holes that we have discussed are genuine black and white holes, and the exterior
+region is isometric to that in an ordinary stationary black h ole with a bifurcate Killing horizon.
+Does the geon also have a Hawking temperature, and if so, in wh at sense?
+4.1. Real scalar field on the RP3geon
+As the prototype, consider a real scalar field on the RP3geon.
+Recall first the usual setting for a quantised scalar field on K ruskal. There is a distinguished
+vacuumstate |0K/a\}bracketri}ht, theHartle-Hawking-Israel (HHI) vacuum[44, 45], which is theuniqueregular
+state that is invariant under all the continuous isometries [46].|0K/a\}bracketri}htdoesnotcoincide with the
+Boulware vacuum state |0K,B/a\}bracketri}ht[47], which static observers in the exteriors see as a no-par ticle
+state. Instead, we have the expansion [44, 45]
+|0K/a\}bracketri}ht=/summationdisplay
+ij...fij...a†
+R,ia†
+L,i/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+corra†
+R,ja†
+L,j/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+corr···|0K,B/a\}bracketri}ht, (4.1)
+wherea†
+R,ianda†
+L,iare creation operators of particles seen by the static obser vers in respectively
+theright exterior Randthe left exterior L, andistands for the quantumnumberslabelling these
+particles. The creation operators occur in correlated pair s, with one particle in each exterior,
+as shown in Figure 10. From the properties of the expansion co efficients fij...it follows that
+measurements in (say) Rsee the pure state |0K/a\}bracketri}htas a Planckian density matrix, in a temperature
+that at asymptotically large radii equals the Hawking tempe rature
+TH=1
+8πM, (4.2)
+whereMis the mass, and at finite radii is corrected by the gravitatio nal redshift factor.
+To summarise, observers in an exterior region of Kruskal see the HHI vacuum |0K/a\}bracketri}htas a
+thermal equilibrium state in the Hawking temperature (4.2) .
+Consider now a scalar field on the RP3geon. By the quotient from Kruskal, the HHI vacuum
+|0K/a\}bracketri}htinduces on the geon a vacuum state |0G/a\}bracketri}ht, which we call the geon vacuum. Again, |0G/a\}bracketri}ht
+does not coincide with the geon’s Boulware state |0G,B/a\}bracketri}ht, which static observers in the exterior
+see as a no-particle state. Instead, |0G/a\}bracketri}htcan be expanded as in (4.1), with creation operators of
+exterior particles acting on |0G,B/a\}bracketri}ht, and these creation operators come again in correlated pair s,R
+correlationL
+Figure 10. A correlated pair of exterior
+particles in the expansion (4.1) of the Kruskal
+HHI vacuum |0K/a\}bracketri}htas excitations on the
+Boulware vacuum. The two particles are in
+the opposite exteriors. From the invariance of
+|0K/a\}bracketri}htunder Killing time translations it follows
+that if the particle in Ris localised in the far
+future, the correlated particle in Lis localised
+in the far past, and vice versa, regardless the
+precise sense of the localisation.correlation
+Figure 11. A correlated pair of exterior
+particles in the expansion of the RP3geon
+HHI vacuum |0G/a\}bracketri}htas excitations on the
+Boulware vacuum. Both particles are in
+the one and only exterior. If one particle
+is localised in the far future, the correlated
+particle is localised in the far past, and vice
+versa.
+−Q R L Q
+Figure 12. A correlated pair of exterior
+particles in the expansion of the electrically-
+charged Reissner-Nordstr¨ om HHI vacuum as
+excitations ontheBoulwarevacuum. Thetwo
+particles have opposite charge.±Q
+Figure 13. A correlated pair of exterior
+particles in the expansion of the electrically-
+charged Reissner-Nordstr¨ om geon HHI vac-
+uum as excitations on the Boulware vacuum.
+The two particles have the samecharge.as illustrated in Figure 11. For generic observations in the exterior region, the geon vacuum
+state does hence not appear as a mixed state, let alone a therm al one.
+However, the correlated particle pairs in the geon exterior have a special structure: if one
+particle is localised at asymptotically late times, then th e correlated particle is localised at
+asymptotically early times, and vice versa. Here, “early” a nd “late” are defined with respect to
+the distinguished constant Schwarzschild time hypersurfa ce shown in Figure 5. This means that
+|0G/a\}bracketri}htappears as a mixed state when probed by observers at asymptot ically late (or early) times,
+and this mixed state is again Planckian in the Hawking temper ature (4.2) [48].
+To summarise, observers in the geon exterior see the geon vac uum|0G/a\}bracketri}htas thermal in the
+Hawking temperature (4.2) at asymptotically late (or early ) times. At intermediate times the
+non-thermal correlations in |0G/a\}bracketri}htcan be observed by local measurements.
+4.2. Generalisations
+When the scalar field is replaced by a fermion field, a new issue arises in the choice of the spin
+structure[49]. As Kruskal has only one spinstructure, ther eis only oneHHI vacuumon Kruskal.
+TheRP3geon has however two spin structures, and each of them inheri ts a HHI vacuum from
+that on Kruskal. Observations made at asymptotically late ( or early) exterior times see each of
+these vacua as a thermal state in the usual Hawking temperatu re, and are in particular unable
+to distinguish the two vacua. The vacua, and their spin struc tures, can however be distinguished
+by observations of the non-thermal correlations at interme diate times.
+When the geon has a gauge field, a new issue arises from its coup ling to the quantised
+matter field. For concreteness, consider a complex scalar fie ld on electrically-charged Reissner-
+Nordstr¨ om [50, 51]. The HHI vacua on the two-exterior hole a nd on the geon again contain
+exterior particles in correlated pairs as shown in Figures 1 2 and 13, and the sense of thermality
+is as in the uncoupled case but with an appropriate chemical p otential term [52]. However, while
+on the two-exterior hole the correlated pairs consist of par ticles with opposite charge, on the
+geon the pairs consist of particles with the samecharge. This is a direct consequence of the
+charge conjugation that appears in the quotienting map. A ge on exterior observer can hence
+infer from the non-thermal correlations at intermediate ti mes that charges on the spacetime are
+not globally defined, even though this phenomenon only arise s from the geometry of the geon’s
+gauge bundle behind the horizons.
+When the geon has anti-de Sitter asymptotics, the geometry i nduces a state for a quantum
+field that lives on the geon’s conformal boundary, via gauge- gravity correspondence [53, 54].
+The boundary state exhibits thermality for appropriately r estricted observations, and the
+non-thermal correlations in this state bear an imprint of th e geometry behind the geon’s
+horizons [16, 55, 56, 57].
+5. Concluding remarks: geon entropy?
+We have seen that a geon black hole, formed as a Z2quotient of a stationary black hole with
+a bifurcate Killing horizon, has a Hartle-Hawking-Israel v acuum that appears thermal in the
+usual Hawking temperature at asymptotically late (and earl y) times. Does the geon have also
+an entropy?
+For a late time exterior observer, laws of classical black ho le mechanics do not distinguish
+between a geon, a conventional eternal black hole with a bifu rcate Killing horizon, or indeed a
+black hole formed in a star collapse. Combining the laws of cl assical black hole mechanics with
+the late time Hawking temperature, the late time observer wi ll hence conclude that the geon
+has the usual Bekenstein-Hawking entropy.
+Could the geon entropy be obtained directly from a quantum th eory of gravity? If the theory
+is formulated in a way that allows focussing on the quantum sp acetime in the distant future,
+state-counting computations for the geon should proceed es sentially as for the hole with twoexteriors. This is in particular the case in loop quantum gra vity, where the counting of black
+hole microstates is expressly set up on the future horizon [5 8, 59, 60, 61]. By contrast, it is not
+clear whether Euclidean path-integral methods allow a loca lisation at late times. While the geon
+hasaregularEuclidean-signaturesection, ana¨ ıveapplic ation ofEuclideanpath-integral methods
+tothis section yieldsfortheentropy aresultthat isonlyha lfof thelatetimeBekenstein-Hawking
+entropy [48, 62].
+It would be of interest to develop a geon entropy computation within the gauge-gravity
+correspondence. Case studies [16, 55, 56, 57] suggest that a n appropriate formalism for this
+could be late time entanglement entropy in the boundary field theory. The challenge would be
+to give a concrete implementation in which the gauge theory s ide of the correspondence is well
+understood.
+Acknowledgments
+I am indebted to many colleagues, including John Barrett, Da vid Bruschi, John Friedman, Gary
+Gibbons, Nico Giulini, Bernard Kay, George Kottanattu, Kir ill Krasnov, Paul Langlois, Robb
+Mann, Don Marolf, Simon Ross, Kristin Schleich, Bernard Whi ting and Jacek Wi´ sniewski, for
+discussionsandcollaboration ongeons. Iamgratefultothe MCCQGorganisersfortheinvitation
+to participate and to present this talk. This work was suppor ted in part by STFC (UK) grant
+PP/D507358/1.
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\ No newline at end of file
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+arXiv:1001.0125v2  [math.CO]  6 Jan 2011Min-Cost Multiflows in
+Node-Capacitated Undirected Networks
+Maxim A. Babenko∗Alexander V. Karzanov†
+November 16, 2018
+Abstract
+We consider an undirected graph G= (VG,EG)with a set T⊆VGof terminals,
+and with nonnegative integer capacities c(v)and costs a(v)of nodes v∈VG. A
+path inGis aT-path if its ends are distinct terminals. By a multiflow we mean
+a function Fassigning to each T-pathPa nonnegative rational weightF(P), and
+a multiflow is called feasible if the sum of weights of T-paths through each node v
+does not exceed c(v). The value ofFis the sum of weights F(P), and the costofF
+is the sum of F(P)times the cost of Pw.r.t.a, over all T-pathsP.
+Generalizing known results on edge-capacitated multiflows , we show that the
+problem of finding a minimum cost multiflow among the feasible multiflows of maxi-
+mum possible value admits half-integer optimal primal and dual solutions. Moreover,
+we devise a strongly polynomial algorithm for finding such op timal solutions.
+Keywords : minimum cost multiflow, bidirected graph, skew-symmetric graph
+∗Department of Mechanics and Mathematics, Moscow State Univ ersity; Leninskie Gory, 119991
+Moscow, Russia; email: max@adde.math.msu.su .
+†Institute for System Analysis of the RAS; 9, Prospect 60 Let O ktyabrya, 117312 Moscow, Russia;
+email:sasha@cs.isa.ru
+11 Introduction
+1.1 Multiflows
+For a function ϕ:X→R+and a subset A⊆X, we write ϕ(A)to denote/summationtext
+x∈Aϕ(x).
+Theincidence vector ofAinRXis denoted by χA, i.e.χA(e)is 1 fore∈Aand 0 for
+e∈X−A(usually Xwill be clear from the context). When Ais a multiset, χA(e)
+denotes the number of occurrences of einA.
+In an undirected graph G, the sets of nodes and edges are denoted by VGandEG,
+respectively. When Gis a directed graph, we speak of arcs rather than edges and wri te
+AGinstead of EG. A similar notation is used for paths, cycles, and etc.
+AwalkinGis meant to be a sequence (v0,e1,v1,...,ek,vk), where each eiis an edge
+(or arc) and vi−1,viare its endnodes; when Gis a digraph, eiis directed from vi−1tovi.
+Edge-simple (or arc-simple) walks are called paths.
+We consider an undirected graph Gand a distinguished subset T⊆VGof nodes,
+called terminals . Nodes in VG−Tare called inner. AT-path is a path PinGwhose
+endnodes are distinct terminals; we usually assume that all the other nodes of Pare
+inner. The set of T-paths is denoted by P. Amultiflow is a function F:P →Q+.
+Equivalently, one may think of Fas a collection
+(1.1) {(α1,P1),...,(αn,Pm)}
+(for some m), where the PiareT-paths and the αiare nonnegative rationals, called
+weights of paths. Sometimes (e.g., in [IKN98]) such a multiflow Fis called freeto
+emphasize that allpairs of distinct terminals are allowed to be connected by flo ws. The
+valueval(F)ofFis/summationtext
+PF(P). For a node v, define
+(1.2) /hatwideF(v) :=/summationdisplay
+(F(P):P∈P,v∈VP);
+the function /hatwideFonVGis called the (node) load function . Letc:VG→Z+be a nonneg-
+ative integer function of node capacities . We say that Fisfeasible if/hatwideF(v)≤c(v)for all
+v∈VG.
+Suppose we are given, in addition, a function a:VG→Z+ofnode costs . Then the
+costa(F)of a multiflow Fis the sum/summationtext
+Pa(P)F(P), wherea(P)stands for the cost
+a(VP)of a path P.
+In this paper we consider the following problem:
+(N)GivenG,T,c,a as above, find a multiflow Fof minimum possible cost a(F)among
+all feasible multiflows of maximum possible value.
+1.2 Previous results
+When|T|= 2, (N) turns into the undirected min-cost max-flow problem under n ode
+capacities and costs, having a variety of applications; see , e.g., [FF62, La76]. It admits
+integer optimal primal and dual solutions [FF62].
+In the special case a≡0, we are looking simply for a feasible multiflow of maximum
+value. Such a problem has half-integer optimal primal and du al solutions, due to results
+2of Pap [Pa07] and Vazirani [Va01], respectively. Also it is s hown in [Pa07] that the
+problem is solvable in strongly polynomial time by using the ellipsoid method.
+An edge-capacitated version of ( N) has been well studied. In this version, denoted
+by (E),candaare functions on EGrather than VG. For a multiflow F, itsedge load
+function is defined similarly to (1.2):
+(1.3) /hatwideF(e) :=/summationdisplay
+(F(P):P∈P,e∈EP)for alle∈EG,
+and its cost is defined to be/summationtext
+Pa(EP)F(P). Problem ( E) is reduced to ( N) by adding
+an auxiliary node on each edge, but no converse reduction is k nown.
+An old result is that ( E) has a half-integer optimal solution [Ka79]. Also it is show n
+in [Ka94] that ( E) has a half-integer optimal dual solution and that half-int eger primal
+and dual optimal solutions can be found in strongly polynomi al time by using the ellip-
+soid method. A “purely combinatorial” weakly polynomial al gorithm, based on cost and
+capacity scaling, is devised in [GK97].
+In the special case of ( E) witha≡0, the half-integrality results are due to
+Lov´ asz [Lo76] and Cherkassky [Ch77], and a strongly polyno mial combinatorial algo-
+rithm is given in [Ch77] (see also [IKN98] for faster algorit hms).
+1.3 New results
+In this paper we prove that ( N) always admits a half-integer optimal primal and dual
+solutions. In particular, this implies all half-integrali ty results mentioned in the previous
+subsection.
+Similar to [Ka94], we introduce a parametric generalization of ( N), study properties
+of geodesics (shortest T-paths with respect to some length function), and reduce the
+parametric problem to a certain single-commodity flow probl em. However, the details of
+this construction are more involved. In particular, the red uced problem concerns integer
+flows in a bidirected graph.
+The second goal is to explore the complexity of ( N). We show that half-integer
+optimal primal and dual solutions to the parametric problem (and therefore to ( N)) can
+be found in strongly polynomial time by using the ellipsoid m ethod.
+2 Preliminaries
+2.1 Parametric problem and its dual
+Instead of ( N), it is convenient to consider a more general problem, namel y:
+(Nλ)GivenG,T,c,a as in (N) and, in addition, λ∈Z+, find a feasible multiflow F
+maximizing the objective function Φ(F,a,λ) :=λ·val(F)−a(F).
+We will prove the following
+Theorem 2.1 For any λ∈Z+, problem ( Nλ) has a half-integer optimal solution.
+(Note that Φ(F,qa,qλ) =q·Φ(F,a,λ)for any multiflow Fandq∈Q+. Therefore, the
+optimality of a multiflow in the parametric problem preserve s when both aandλare
+3multiplied by the same positive factor q. This implies that the theorem is generalized
+to arbitrary a:VG→Q+andλ∈Q+(but keeping the integrality of c). However, we
+prefer to deal with integer-valued aandλin what follows.)
+By standard linear programming arguments, ( N) and (Nλ) become equivalent when
+λis large enough (moreover, the existence of a half-integer o ptimal solution for ( Nλ)
+easily implies that taking λ:= 2c(VG)a(VG)+1is sufficient).
+Problem ( Nλ) can be viewed as a linear program with variables F(P)∈Q+assigned
+toT-pathsP. Assign to a node v∈VGa variable l(v)∈Q+. Then the linear program
+dual to ( Nλ) is:
+(Dλ)Minimize c·lprovided that the following holds for every T-pathP:
+(2.1) l(P)≥λ−a(P).
+2.2 Translating to edge lengths
+The above dual problem ( Dλ) involves lengths of paths (namely, l(P)anda(P)) deter-
+mined by “lengths” of nodes ( landa, respectively). It is useful to transform lengths of
+nodes into lengths of edges. To do so, for w:VG→Q+, we define the function won
+EGby
+(2.2) w(e) :=αuw(u)+αvw(v)fore=uv∈EG,
+whereαx:=1
+2ifx∈VG−T, andαx:= 1ifx∈T. This provides the correspondence
+(2.3) w(P) =w(P)for each T-pathP
+(wherew(P)stands for w(VP), andw(P)forw(EP)). Fora,las above, define
+(2.4) ℓ:=l+a.
+Letdistℓ(u,v)denote the ℓ-distance between vertices uandv, i.e. the minimum ℓ-length
+ℓ(P)of au–vpathPinG. Then, in view of (2.3) and (2.4), the constraints in ( Dλ) can
+be rewritten as
+(2.5) distℓ(s,t)≥λ for alls,t∈T,s/ne}a⊔ionslash=t.
+By the linear programming duality theorem applied to ( Nλ) and (Dλ), a feasible
+multiflow Fand a function l:VG→Q+satisfying (2.5) are optimal solutions to ( Nλ)
+and (Dλ), respectively, if and only if the following (complementar y slackness conditions)
+hold:
+(2.6) ifPis aT-path and F(P)>0, thenℓ(P) =λ; in particular, Pisℓ-shortest;
+(2.7) ifv∈VGandl(v)>0, thenvissaturated byF, i.e./hatwideF(v) =c(v).
+In the rest of the paper, to simplify technical details, we wi ll always assume that the
+input costs aof all nodes are strictly positive . Then the edge lengths ℓdefined by (2.4)
+are strictly positive as well. This assumption will not lead to loss of generality in essence,
+since the desired results for a nonnegative input cost funct ionacan be obtained by
+applying a perturbation technique in spirit of [Ka94, pp. 32 0–321] (by replacing aby an
+appropriate strictly positive cost function).
+42.3 Geodesics
+Condition (2.6) motivates the study of the structure of ℓ-shortest T-paths in G. To this
+aim, set p:= min{distℓ(s,t)|s,t∈T,s/ne}a⊔ionslash=t}. AT-pathPsuch that ℓ(P) =pis called
+anℓ-geodesic (or just geodesic ifℓis clear form the context). When a multiflow FinG
+is given as a collection (1.1) in which all paths Piareℓ-geodesics, we say that Fis an
+ℓ-geodesic multiflow.
+Next we utilize one construction from [Ka79, Ka94], with min or changes. Consider
+a nodev∈VG. Define the potential π(v)to be the ℓ-distance from vto the nearest
+terminal, i.e. π(v) := min{distℓ(v,t)|t∈T}. SetVGℓ:={v∈VG|π(v)≤1
+2p}(in
+particular, T⊆VGℓ). Fors∈T, defineVs:={v∈VG|distℓ(s,v)<1
+2p}. Also define
+V♮:={v∈VG|π(v) =1
+2p}. We refer to Vsas the zoneof a terminal s∈T, and to V♮
+as the set of central nodes (w.r.t. ℓ). The sets Vs(s∈T) andV♮are pairwise disjoint
+and give a partition of VGℓ.
+The following subset of edges is of importance:
+EGℓ:={uv∈EG|∃s∈T:u∈Vs,v∈Vs∪V♮,|π(u)−π(v)|=ℓ(uv)}
+∪{uv∈EG|∃s,t∈T,s/ne}a⊔ionslash=t:u∈Vs,v∈Vt,π(u)+π(v)+ℓ(uv) =p}.
+One can see that the subgraph Gℓ:= (VGℓ,EGℓ)ofGcontains all ℓ-geodesics.
+Moreover, a straightforward examination shows that the str ucture of ℓ-geodesics possesses
+the properties as expressed in the following lemma (which is , in fact, a summary of
+Claims 1–3 from [Ka94] and uses the strict positivity of ℓ).
+Lemma 2.2 LetPbe anℓ-geodesic running from s∈Ttot∈T. ThenPis contained
+inGℓand exactly one of the following takes place:
+1.Pcontains no central nodes and can be represented as the conca tenation P1◦
+(u,e,v)◦P2, whereu∈Vs,v∈Vt,s/ne}a⊔ionslash=t, ande∈EGℓ.
+2.Pcontains exactly one central node w∈V♮and can be represented as the concate-
+nationP=P1◦(u,e1,w,e2,v)◦P2, whereu∈Vs,v∈Vt,s/ne}a⊔ionslash=t, ande1,e2∈EGℓ.
+In both cases, parts P1andP2are contained in the induced subgraphs Gℓ[Vs]andGℓ[Vt],
+respectively. The potentials πare strictly increasing as we traverse P1fromstou, and
+strictly decreasing as we traverse P2fromvtot.
+Conversely, any T-path inGℓobeying the above properties is an ℓ-geodesic.
+3 Primal half-integrality
+3.1 Auxiliary bidirected graph
+In this subsection we introduce an auxiliary bidirected gra ph, which will be the cor-
+nerstone of our approach both for proving half-integrality results and for providing a
+polynomial-time algorithm.
+GivenG,T,c,aandλas above, let lbe an optimal solution to ( Dλ). Form the edge
+lengthsℓ:=l+a, the potential π, the subgraph Gℓ, and the sets Vs(s∈T) andV♮, as
+in Subsection 2.3. One may assume that p:= min{distℓ(s,t)|s,t∈T,s/ne}a⊔ionslash=t}=λ(since
+p≥λ, by (2.5), and if p > λ thenF= 0is an optimal solution to ( Nλ), by (2.6)).
+5For further needs, we reset c:= 2c, making all node capacities even integers. Now our
+goal is to prove the existence of an integer optimal multiflow Fin problem ( Nλ) (which
+is equivalent to proving the half-integrality w.r.t. the in itialc).
+Recall that in a bidirected graph (or a BD-graph for short) edges of three types
+are allowed: a usual directed edge, or an arc, that leaves one node and enters another
+one; an edge directed from both of its ends; and an edge directed to both of its ends
+(cf. [EJ70, Sc03]). When both ends of an edge coincide, the ed ge becomes a loop. For our
+purposes we admit no loop entering and leaving its end node si multaneously. Sometimes,
+to specify the direction of an edge e=uvat one or both of its ends, we will draw arrows
+above the corresponding node characters. For example, we ma y write− →uvifeis directed
+fromutov(a usual arc),− →u← −vifeleaves both u,v,← −u− →vifeenters both u,v, and− →uvif
+eleavesu(and either leaves or enters v).
+Awalkin a BD-graph is an alternating sequence
+P= (s=v0,e1,v1,...,ek,vk=t)
+of nodes and edges such that each edge eiconnects nodes vi−1andvi, and for i=
+1,...,k−1, the edges ei,ei+1form a transit pair atvi, which means that one of ei,ei+1
+enters and the other leaves vi. As before, an edge-simple walk is referred to as a path.
+Now we associate to Gℓa BD-graph Hwith edge capacities c:EH→Z+, as follows
+(see Fig. 1 for an illustration). Each noncentral node v∈VGℓ−V♮generates two nodes
+v1,v2inH. They are connected by edge (arc) evgoing from v1tov2and having the
+capacity equal to c(v). We say that evinherits the capacity of the node v. Fors∈T,
+the setVs:={v1,v2|v∈Vs}inHis called the zoneofs, similar to VsinG.
+Consider an edge e=uv∈EGℓ. Letu,v∈Vsfor some s∈Tand assume for
+definiteness that π(u)< π(v)(note that ℓ(uv)>0impliesπ(u)/ne}a⊔ionslash=π(v); this is where
+the strict positivity of the cost function ais important). Then egenerates in Han edge
+(arc) going from u2tov1, and we assign infinite capacity to it. (By “infinite capacity”
+we mean a sufficiently large positive integer.) Now let u∈Vsandv∈Vtfor distinct
+s,t∈T. Thenegenerates an infinite capacity edge− →u2← −v2(leaving both u2andv2).
+The transformation of central nodes is less straightforwar d. Each w∈V♮generates
+inHa so-called gadget , denoted by Γw. It consists of|T|+1nodes; they correspond to
+wand the elements of Tand are denoted as θwandθw,s,s∈T. The edges of Γware: a
+loopewleavingθw(twice) and, for each s∈T, an edge ew,sgoing from θw,stoθw, called
+thes-legin the gadget. Each edge in Γwis endowed with the capacity equal to c(w).
+Each gadget Γwis connected to the remaining part of Has follows. For each edge
+of the form vwinGℓ, we know that v∈Vsfor some s∈T(by the construction of Gℓ).
+Thenvwgenerates an infinite capacity edge (arc) going from v2toθw,s.
+Finally, we add to Han extra node q, regarding it as the source , and for each s∈T,
+draw an infinite capacity edge (arc) from qtos1.
+The obtained BD-graph Hcaptures information about the ℓ-geodesics in G. Namely,
+eachℓ-geodesic Pgoing from stotinduces a unique closed q–qwalkPinH. The first
+and the last edges of Pare− →qs1and← −t1← −q, respectively. For a noncentral node vinP,P
+traverses the edge− →v1− →v2. An edge uv∈EPwithπ(u)< π(v)inside a zone induces the
+edge− →u2− →v1inP. An edge uv∈EPconnecting different zones (if any) induces the edge
+− →u2← −v2inP. Finally, suppose Ptraverses a central node w∈V♮and letuw,wv∈EGℓ
+6p
+ab
+cs
+d
+w
+ef
+g
+rV♮VpVs
+Vr
+(a) Graph Gℓ.p1
+p1
+a1
+a2b1
+b2
+c1
+c2s1
+s2
+d1
+d2
+θwθw,p
+θw,s
+θw,r
+e1e2f1f2
+g1g2
+r1r2Γw
+(b) Bidirected graph H.
+Figure 1: Constructing graph H. HereT={p,s,r},Vp={p,a,b},Vs={s,c,d},
+Vr={r,e,f,g},V♮={w}. (The source qis not shown.) Bidirected edges leaving one
+endpoint and entering the other are indicated by ordinary di rected arcs. Marked are one
+ℓ-geodesic Pand its image P.
+be the edges of Pincident to w. By Lemma 2.2, u∈Vsandv∈Vtfor some s/ne}a⊔ionslash=t.
+Then the sequence of nodes u,w,v inPgenerates the subpath in Pwith the sequence of
+edgesu2θw,s,ew,s,ew,ew,t,θw,tv2.
+The resulting walk Pis edge-simple, so it is a closed path. Conversely, let Qbe
+a (nontrivial) q–qwalk inH. One can see that Qwithqremoved is concatenated as
+Q1◦Q′◦Q2, whereQ1is a directed path within a zone Vs,Q2is reverse to a directed
+path within a zone Vt(with possibly s=t), andQ′either (i) is formed by an edge
+− →u2← −v2connecting these zones (in which case s/ne}a⊔ionslash=t), or (ii) is the walk with the sequence
+of edges ew,s,ew,ew,t, for some central node wofGℓ. Moreover, the image in Gof each
+ofQ1,Q2is anℓ-shortest path. When s=thappens in case (ii), Qtraverses the edge
+ew,stwice. In all other cases, Qis edge-simple and its image in Gis anℓ-shortest T-path
+(aλ-geodesic).
+These observations show that there is a natural bijection be tween the ℓ-geodesics in
+Gand the (nontrivial) q–qpaths in H.
+We will refer to the BD-graph Hdescribed above as the compact BD-graph related to
+Gℓ; it will be essentially used to devise an efficient algorithm f or solving ( Nλ) in Section 4.
+Besides, in the proof of the primal integrality (with ceven) in Section 5, we will deal with
+7a modified BD-graph. It is obtained from Has above by replicating each gadget Γwinto
+c(w)copiesΓwi,i= 1,...,c(w), called the 1-gadgets generated by w. More precisely,
+to construct Γwi, we make i-th copy θwiof the node θw,i-th copy ewiof the loop ew
+leavingθwi(twice), and i-th copy ewi,sof each leg ew,s,s∈T, whereewi,sgoes from
+θw,stoθwi(soθw,s,s∈T, are the common nodes of the created 1-gadgets). All edges in
+these 1-gadgets are endowed with unitcapacities.
+We keep notation Hfor the constructed graph and call it the expensive BD-graph
+related to Gℓ. Also we keep notation cfor the edge capacities in H. There is a natural
+relationship between the q–qwalks (paths) in both versions of H. The 1-gadgets created
+from the same central node wofGℓare isomorphic, and for any i,j= 1,...,c(w), there
+is an automorphism of Hwhich swaps θwiandθwjand is invariant on the other nodes.
+3.2 Bidirected flows
+LetΓbe a bidirected graph. Like in usual digraphs, δin(v)andδout(v)denote the sets
+of edges in Γentering and leaving v∈VΓ, respectively. A loop eatv, if any, is counted
+twice in δin(v)ifeentersv, and twice in δout(v)ifeleavesv; henceδin(v)andδout(v)
+are actually multisets. (Recall that we do not allow a loop wh ich simultaneously enters
+and leaves a node.)
+Letqbe a distinguished node with δin(q) =∅inΓ(thesource ) and let the edges of Γ
+have integer capacities c:EΓ→Z+. Abidirected q-flow, or a BD-flow for short, is a
+function f:EΓ→Q+satisfying divf(v) = 0 for all nodes v∈VΓ−{q}; and the value
+offis defined to be divf(q)(cf. [GK04]). Here
+(3.1) divf(v) :=f(δout(v))−f(δin(v))
+is the divergence offatv. Note that if eis a loop at vthenecontributes±2f(e)in
+divf(v). Iff(e)≤c(e)for alle∈EΓthenfis called feasible . In addition, if fis integer-
+valued on all edges then we refer to fas an integer bidirected q-flow, or an IBD-flow .
+One can see that finding a fractional (resp. integer) BD-flow of the maximum value is
+equivalent to constructing a maximum fractional (resp. int eger) packing of closed q–q
+walks (they leave qtwice).
+Return to an optimal solution lto (Dλ), and let ℓ:=a+l. Consider the (expensive
+or compact) BD-graph Hrelated to Gℓ, and the capacity function con the edges of
+H(constructed from the node capacities cofG). The above correspondence between
+ℓ-geodesics in Gandq–qpaths in His extended to ℓ-geodesic multiflows in Gand
+certainq-flows in H(whereqis the source in Has before). More precisely, let Fbe a
+(fractional) ℓ-geodesic multiflow in Grepresented by a collection of ℓ-geodesics Piand
+weightsαi:=F(Pi),i= 1,...,m (cf. (1.1)). Then each Pidetermines a q–qpathPi
+inH, andf:=α1χEP1+...+αmχEPmis a BD-flow in H; we say that fisgenerated
+byF(note that val(f) = 2val( F)). Furthermore, fis feasible if Fis such, and for each
+central node w∈V♮, the following relations hold:
+(3.2)/summationdisplay
+s∈Tf(ew,s) = 2f(ew)andf(ew,s)≤f(ew)for each s∈T.
+Considering an arbitrary BD-flow finH, we say that fisgoodif it satisfies (3.2) for
+allw∈V♮(here the second relation in (3.2) is important, while the fir st one obviously
+holds for any BD-flow). The following assertion is of use.
+8Lemma 3.1 Letfbe a good BD-flow in H. Then fis generated by an ℓ-geodesic
+multiflow FinG. Moreover, if fis integral, then it is generated by an integer ℓ-geodesic
+multiflow F. In both cases, Fcan be found in O(|EH|)time.
+Proof. Suppose there is a central node w∈V♮such that f(ew)>0. Let us say that
+p∈Tdominates atw(w.r.t.f) iff(ew,p) =f(ew). From (3.2) it follows that there exist
+distincts,t∈Tsuch that f(ew,s),f(ew,t)>0and none of p∈T−{s,t}dominates at w.
+Choose such s,t. Build in Ha maximal walk Qstarting with θw,ew,s,θw,s,...and such
+thatf(e)>0for all edges eofQ. It is easily seen from the construction of HthatQ
+is edge-simple, terminates at q, and have all vertices in Vs, except for θw,θw,s. Build a
+similar walk (path) Q′starting with θw,ew,t,θw,t. Then the concatenation of the reverse
+toQ, the loop ewand the path Q′is aq–qpath and its image PinGis anℓ-geodesic
+(fromstot).
+Assign the weight of Pto be the maximum number αsubject to two conditions:
+(i)α≤f(e)for each e∈EP, and (ii) the flow f′:=f−αχPis still good. If αis
+determined by (i), we have |supp(f′)|<|supp(f)|(where supp (ϕ) :={x|ϕ(x)/ne}a⊔ionslash= 0}),
+whereas if αis determined by (ii), there appears p∈Tdominating at w(w.r.t.f′). If
+f′(ew)>0, repeat the procedure for f′andw, otherwise apply the procedure to f′and
+anotherw′∈V♮, and so on. (Note that if, in the process of handling w, a current weight
+αis determined by (ii), then the weights of all subsequent pat hs through eware already
+determined by (i); this will provide the desired complexity .)
+Eventually, we come to a good flow /tildewidefwith/tildewidef(ew) = 0 for allw∈V♮. This/tildewidefis
+decomposed into a sum of flows along q–qpaths in a straightforward way, in O(|EH|)
+time (like for usual flows in digraphs). Taking together the i mages in Gof the constructed
+weighted q–qpaths, we obtain a required ℓ-geodesic multiflow F. The running time of
+the whole process is O(|EH|), and iffis integral, then the weights αof all paths are
+integral as well. (Integrality of a current weight αsubject to integrality of a current flow
+fis obvious when αis determined by (i), and follows from the fact, implied by (3 .2),
+that for any p∈T,/summationtext
+s/ne}ationslash=pf(ew,s)−f(ew,p)is even, when αis determined by (ii).) /square
+Remark 3.2 In case of an expensive BD-graph H, any feasible IBD-flow fis good.
+Indeed, for any 1-gadget ΓwiinH, we have c(ewi) = 1 and, therefore, f(ew)∈{0,1}.
+The second relation in (3.2) is trivial when f(ew) = 0, and follows from the constraints
+f(ew,s)≤c(ew,s) = 1(s∈T) whenf(ew) = 1.
+Define the following subset of edges in H:
+(3.3) E0:={ev|v∈VG, l(v)>0}.
+For an optimal (possibly fractional) solution Fto (Nλ) and a node v∈VGwithl(v)>0,
+we have /hatwideF(v) =c(v)(by (2.7)); so the edge evofHcorresponding to vis saturated by
+the BD-flow fgenerated by F, i.e.f(ev) =c(ev). We call the edges in E0locked .
+Thus, the graph Hadmits a (fractional) good feasible BD-flow saturating the lo cked
+edges. The following strengthening is crucial.
+Proposition 3.3 There exists a good feasible IBD-flow in Hthat saturates all locked
+edges.
+9A proof of this proposition involves an additional graph-th eoretic machinery and will
+be given in Section 5. Assuming its validity, we immediately obtain Theorem 2.1 from
+Lemma 3.1.
+4 Solving ( Nλ) in strongly polynomial time
+In this section we devise a strongly polynomial algorithm fo r solving the primal parametric
+problem ( Nλ). As before, we assume that aandλare integral and that the node capacities
+care even, so our goal is to find an integer optimal multiflow.
+The algorithm starts with computing a (fractional) optimal dual solution land con-
+structing the BD-graph Hw.r.t. the length function ℓ:=a+l. Then it finds a good IBD-
+flowfinHsaturating the locked edges (assuming validity of Proposit ion 3.3). Applying
+the efficient procedure as in the proof of Lemma 3.1 to decompos efinto a collection of
+paths with integer weights, we will obtain an integer optima l solution to ( Nλ).
+To provide the desired complexity, we shall work with Hgiven in the compact form
+(defined in Subsection 3.1). The core of our method consists i n finding the load function
+of some integer optimal multiflow FinG(without explicitly computing Fitself). This
+function will just generate the desired IBD-flow in H. We describe the stages of the
+algorithm in the subsections below.
+4.1 Constructing an optimal dual solution
+Problem ( Dλ) straightforwardly reduces to a “compact” linear program, as follows. Be-
+sides variables l(v)∈Q+(v∈VG), assign a variable ϕs(v)∈Qto each terminal s∈T
+and node v(a sort of “distance” of vfroms). Consider the following problem (where l
+andaare defined according to (2.2)):
+(4.1) Minimize c·lsubject to the following constraints:
+ϕs(u)−ϕs(v)≤a(e)+l(e)
+ϕs(v)−ϕs(u)≤a(e)+l(e)for each e=uv∈EG;
+ϕs(t)−ϕs(s)≥λfor alls,t∈T,s/ne}a⊔ionslash=t.
+Lemma 4.1 Programs ( Dλ) and (4.1) are equivalent.
+Indeed, if (l,ϕ)is a feasible solution to (4.1) then, obviously, lis a feasible solution
+to (Dλ). Conversely, let lbe a feasible solution to ( Dλ). Forv∈VGands∈T, define
+ϕs(v) := dist ℓ(s,v), whereℓ:=l+a. It is easy to check that (l,ϕ)is a feasible solution
+to (4.1).
+The size of the constraint matrix in (4.1) (written in binary notation) is polynomial in
+|VG|. Therefore, ( Dλ) is solvable in strongly polynomial time by use of Tardos’s v ersion
+of the ellipsoid method. (This remains valid when aandλare nonnegative rational
+numbers.)
+104.2 Computing the load function of an optimal multiflow
+The following fact is of importance.
+Lemma 4.2 One can find, in strongly polynomial time, a function g:VG→Z+such
+thatg(v) =/hatwideF(v)for allv∈VG, whereFis some integer optimal multiflow in ( Nλ).
+Proof. We explain that in order to construct the desired g, it suffices to compare
+two optimal objective values: one for the original (integer ) costsaand the other for
+certain perturbed costs aε. These values are computed by solving the corresponding dua l
+problems by the method described in the previous subsection .
+More precisely, let v1,...,vnbe the nodes of G. LetU:= max ic(vi) + 2, define
+ε(vi) := 1/Ui+1,i= 1,...,n , and define the cost function aεonVGto bea+ε. Then
+for any integer feasible multiflow F, we have
+0≤Φ(F,a,λ)−Φ(F,aε,λ) =/summationdisplay
+i/hatwideF(vi)ε(vi)< U−1+U−2+...+U−n<1.
+This and the fact that Φ(F,a,λ)is an integer (as F,a,λ are integral) imply that if F
+is optimal for aε, thenFis optimal for aas well. (An integer optimal multiflow for
+even capacities cexists by Theorem 2.1.) Moreover, for such an optimal F, the number
+r:=/summationtext
+i/hatwideF(vi)ε(vi)is computed in strongly polynomial time, since it is equal to c·l−c·lε,
+wherelandlεare optimal dual solutions for aandaε, respectively. Here we use the LP
+duality equalities Φ(F,a,λ) =c·landΦ(F,aε,λ) =c·lε. Also the size of binary encoding
+ofaεis bounded by that of atimes a polynomial in n, so the dual problem with aεis
+solved in strongly polynomial time w.r.t. the original data .
+Hence, we have rUn+1=/summationtext
+i/hatwideF(vi)Un−i. The number rUn+1is an integer and, in
+view of/hatwideF(vi)≤c(vi)< Ufor each i, thencoefficients in its base Udecomposition (the
+representation via degrees of U) are just /hatwideF(v1),...,/hatwideF(vn), thus giving g. /square
+Recall that together with a node load function each multiflow Falso induces its edge
+counterpart (see (1.3)). Lemma 4.2 can be strengthened as fo llows.
+Lemma 4.3 One can find, in strongly polynomial time, a function g:VG∪EG→Z+
+such that g(v) =/hatwideF(v)for allv∈VGandg(e) =/hatwideF(e)for alle∈EG, whereFis some
+integer optimal multiflow Fin (Nλ).
+Proof. Split each edge e=uvofGinto two edges uxe,xevin series and assign to each
+new node xethe capacity c(xe) := min{c(u),c(v)}and the cost a(xe) := 0 . Clearly this
+transformation does not affect the problem in essence. The no de load function, which
+can be found in strongly polynomial time by Lemma 4.2, yields the desired node and
+edge load functions in the original graph G. /square
+4.3 Constructing an optimal primal solution
+Now we explain how to find, in strongly polynomial time, an int eger optimal multiflow
+solving (Nλ) (for a graph G, even node capacites c, rational node costs a, and an integer
+parameter λ) by using an optimal dual solution land a function gas in Lemma 4.3. For
+thisg, there exists an integer ℓ-geodesic multiflow F′inGsatisfying F′(v) =g(v)for all
+11v∈VGandF′(e) =g(e)for alle∈EG, whereℓ:=a+l. Our goal is to construct one
+of such multiflows explicitly.
+To do this, we consider the compact BD-graph Hrelated to Gℓ(see Subsection 3.1)
+and put fto be the function on EHcorresponding to g. More precisely, let E′be the
+subset of edges of Hneither incident to qnor contained in the gadgets Γw(w∈V♮). By
+the construction of H, there is a natural bijection γofE′to the set (VGℓ−V♮)∪EGℓ.
+For each e∈E′, we setf(e) :=g(γ(e)). In their turn, the values of fon the edges of a
+gadgetΓware assigned as follows: for the loop ewatθw, setf(ew) :=g(w), and for each
+s∈T, setf(ew,s) :=/summationtext(g(e):e∈Es,w), whereEs,wis the set of edges in Gℓconnecting
+Vsandw. Finally, for each s∈T, we setf(qs) :=h(s).
+Using the fact that the function gonVG∪EGis determined by some integer optimal
+(ℓ-geodesic) multiflow F′, it follows that the obtained function fonEHis integer-valued
+and has zero divergency at all nodes different from q. Sofis an IBD-flow in H. Moreover,
+fis generated by F′as above; in particular, fis good (i.e. satisfies (3.2)). By Lemma 3.1,
+we can find, in strongly polynomial time, an integer ℓ-geodesic multiflow Fgenerating f.
+ThenFandF′have the coinciding node and edge load functions, and the opt imality of
+F′implies that Fis an integer optimal solution to ( Nλ) as well, as required.
+5 Proof of Proposition 3.3
+To complete the proof of Theorem 2.1 it remains to prove Propo sition 3.3, which claims
+the existence of an IBD-flow saturating the “locked” edges. We eliminate the lower
+capacity constraints (induced by the locked edges) by reduc ing the claim to the existence
+of an IBD-flow with a certain prescribed value.
+5.1 Maximum IBD-flows
+LetΓbe a bidirected graph with a distinguished source qand edge capacities c:EΓ→Z+,
+as described in Subsection 3.2. The classic max-flow min-cut theorem states that the
+maximum flow value is equal to the minimum cut capacity. A bidi rected version of
+this theorem involves a somewhat more complicated object, c alled an odd barrier . In this
+subsection we give its definition and state the crucial prope rties (in Theorems 5.1, 5.2, 5.3)
+that will be used in the upcoming proof of Proposition 3.3. Th ese properties are nothing
+else than translations, to the language of bidirected graph s, of corresponding properties
+established for integer symmetric flows in skew-symmetric g raphs, as we will explain in
+the Appendix.
+LetX⊆VΓ−{q}. The flipat (the nodes of) Xmodifies Γas follows: for each node
+v∈Xand each edge eincident to v, we reverse the direction of eatv(while preserving
+the directions of edges at nodes in VG−X). For example, if e=− →u← −vandu,v∈X
+thenebecomes← −u− →v, and ife=− →uvandu/ne}a⊔ionslash∈X∋vthenebecomes− →u← −v. BD-graphs Γ
+andΓ′are called equivalent if one is obtained by a flip from the other. Note that flips do
+not affect bidirected walks in Γin essence and do not change the maximum value of an
+IBD-flow in it. We will essentially use flips to simplify requir ements in the definition of
+odd barriers below.
+Next, we employ a special notation to designate certain subs ets of edges. For X,Y⊆
+VΓ, let[X,Y]denote the set of edges with one endpoint in Xand the other in Y. We
+12q
+A
+M B1 Bk
+Figure 2: A bidirected odd barrier. Grayed edges correspond to odd capacity constraints
+(w.r.t.Γ′).
+will often add arrows above Xand/orYto indicate the subset of edges in [X,Y]directed
+in one or another way. For example, [← −X,− →Y]denotes the set of edges that enter both X
+andY,[− →X,Y]denotes the set of edges leaving Xand having the other endpoint in Y
+(where the direction is arbitrary), and [− →X,← −X]denotes the set of edges that leave Xat
+both endpoints (including twice leaving loops). When Y=VΓ−X, the second term in
+the brackets may be omitted: [X],[− →X], and[← −X]stand for [X,VΓ−X],[− →X,VΓ−X],
+and[← −X,VΓ−X], respectively. Finally, for a function ϕon the edges, we write ϕ[X,Y]
+(rather than ϕ([X,Y])) for/summationtext
+e∈[X,Y]ϕ(e).
+A tupleB= (Γ′|A,M;B1,...,B k), whereΓ′is some BD-graph equivalent to Γ, is
+called an odd barrier forΓif the following conditions hold with respect to Γ′(see Fig. 2):
+(5.1) (i) A,M,B 1,...,B kgive a partition of VΓ′=VΓ, andq∈A.
+(ii) For each i= 1,...,k ,c[− →A,Bi]is odd.
+(iii) For distinct i,j= 1,...,k ,c[Bi,Bj] = 0.
+(iv) For each i= 1,...,k ,c[Bi,M] = 0.
+Thecapacity ofBis defined to be
+(5.2) c(B) := 2c[− →A,← −A]+c[− →A]−k.
+Theorem 5.1 (Max IBD-Flow Min Odd Barrier Theorem) ForΓ,c,qas above,
+the maximum IBD-flow value is equal to the minimum odd barrier capacity. A (feasible)
+IBD-flow gand an odd barrier B= (Γ′|A,M;B1,...,B k)forΓhave maximum value and
+minimum capacity, respectively, if and only if the followin g conditions hold with respect
+toΓ′:
+(i)g[− →A,← −A] =c[− →A,← −A]andg[← −A,− →A] = 0;
+(ii)g[− →A,M] =c[− →A,M]andg[← −A,M] = 0;
+13(iii)for eachi= 1,...,k , eitherg[− →A,Bi] =c[− →A,Bi]−1andg[← −A,Bi] = 0, org[− →A,Bi] =
+c[− →A,Bi]andg[← −A,Bi] = 1.
+Note that there may exist many minimum capacity odd barriers forΓ,c,q. It is well-
+known that in a usual edge-capacitated digraph with a source sand a sink t, the set
+of nodes reachable by paths from sin the residual digraph of a maximum s–tflowf
+determines a minimum capacity s–tcut. Moreover, this minimum cut does not depend
+on the choice of fand, therefore, may be regarded as the canonical one.
+A similar phenomenon takes place for maximum IBD-flows and min imum odd barriers
+(and we will essentially use this in the proof of Proposition 3.3). To describe this, consider
+an IBD-flow ginΓ. The residual BD-graph Γgendowed with the residual capacity
+cg:EΓg→Z+is constructed in a similar way as for usual flows. More precis ely,VΓg=
+VΓand the edges of Γgare as follows:
+(5.3) (i) each edge e∈EΓwithg(e)< c(e)whose residual capacity is defined to be
+cg(e) :=c(e)−g(e), and
+(ii) the reverse edgeeRto each edge e∈EΓwithg(e)>0; the directions of
+eRat the endpoints are reverse to those of eand the residual capacity is
+cg(eR) :=g(e).
+A bidirected walk PinΓgis called cg-simple ifPpasses each edge eat mostcg(e)
+times. If Pis acg-simple closed q–qwalk leaving its end qtwice, we can increase the
+value of ginΓby 2, by sending one unit of flow along P. So the existence (in Γg) of
+such a walk P, which is called (Γ,g)-residual , implies that gis not maximum. A converse
+property holds as well.
+Theorem 5.2 An IBD-flow ginΓis maximum if and only if there is no (Γ,g)-residual
+walk.
+When we are given a maximum IBD-flow g, a certain minimum odd barrier can be
+constructed by considering the residual graph Γg. Namely, let− →RΓ,g(resp.← −RΓ,g) be the
+set of nodes vthat are reachable by a (Γ,g)-residual q–vwalk that leaves qand enters v
+(resp. leaves both qandv). Thenq /∈← −RΓ,g, by the maximality of g.
+Theorem 5.3 Letgbe a maximum IBD-flow for Γ,c,q. Define A:= (− →R−← −R)∪(← −R−− →R)
+andM:=VΓ−(− →R∪← −R), where− →R:=− →RΓ,gand← −R:=← −RΓ,g. LetB1,...,B kbe the node
+sets of weakly connected components of the underlying undir ected subgraph of Γginduced
+by− →R∩← −R. Define Γ′to be the BD-graph obtained from Γby flipping the set← −R−− →R
+(contained in A). ThenBg:= (Γ′|A,M;B1,...,B k)is a minimum odd barrier.
+An important fact is that the minimum odd barrier Bfdoes not depend on g, and we
+refer to it as the canonical odd barrier for Γ,c,q.
+Theorem 5.4 The sets← −RΓ,gare same for all maximum IBD-flows ginΓ, and similarly
+for the sets− →RΓ,g, the minimum odd barriers Bf, and the graphs Γ′obtained from Γby
+flipping← −RΓ,g−− →RΓ,g.
+145.2 Proof of Proposition 3.3
+In fact, we have freedom of choosing any of the two forms (expe nsive or compact) of H
+to prove Proposition 3.3 in full, as it is easy to see that the c laims in these cases are
+reduced to each other. We prefer to deal with the expensive fo rm, taking advantage from
+certain nice structural features arising in this case. One r eason for our choice is that any
+IBD-flow in the expensive His automatically good, as explained in Remark 3.2.
+We know that there exists a good fractional bidirected q-flowfinHthat saturates
+the setE0of locked edges, and our goal is to show the existence of an IBD- flow saturating
+E0. Recall that any edge e∈E0is generated by some node vofG, i.e.e=ev.
+It will be convenient for us to construct the desired IBD-flow w ithout explicitly im-
+posing the “lower capacities” on the locked edges. For this p urpose, we modify Has
+follows.
+First, we add a loop← −q− →qwith infinite capacity (entering qtwice). Also we add to H
+a nodez, which is regarded as a new source.
+Second, let E0contain an edge ev=− →v1− →v2generated by a vertex v∈VGℓin some
+zoneVs,s∈T. We delete evfromHand, instead, add two edges− →v1← −zand− →z− →v2with
+capacity c(v)each.
+Third, let E0contain the loops ewi(i= 1,...,c(w)) for some central node wofGℓ.
+We replace each ewi(having unit capacity) by edge− →z←−θwiwith capacity 2; we call it the
+root edge atθwi.
+We denote the resulting BD-graph by H1and keep the previous notation cfor its
+edge capacities. The above q-flowfis transformed, in an obvious way, into a feasible
+z-flow inH1, denoted by fas before. Note that this fsaturates all edges created from
+those in E0(i.e. from evandewias above); these edges leave zand the value of fis
+maximum among the feasible z-flows in H1and is equal to 2c(E0).
+Letgbe a maximum IBD-flow in H1. We are going to prove that val(f) = val(g).
+This would imply that the corresponding IBD-flow in Hsaturates E0as required. To
+this aim, consider the canonical odd barrierB= (H2|A,M;B1,...,B k)forH1,c,z(see
+Theorem 5.4). Here H2is the BD-graph (with the source z) obtained from H1according
+to Theorem 5.3 (i.e. H2:= Γ′forΓ :=H1). From now on, speaking of edge directions,
+the capacities cand the flow g, we mean those in H2, unless explicitly stated otherwise.
+We have (cf. (5.2))
+(5.4) val(g) =c(B) = 2c[− →A,← −A]+c[− →A]−k.
+The following assertion is crucial.
+Lemma 5.5 For each p= 1,...,k :
+(i)Bp={θwi}for some w∈V♮andi∈{1,...,c(w)};
+(ii)ewiis not locked (so H1contains the loop ewibut not the root edge at θwi);
+(iii)among the edges (legs) connecting AandBp, one edge leaves Aand the other edges
+enterA.
+15Proof. By the constructions of HandH1, for any w∈V♮and distinct i,j= 1,...,c(w),
+there is an automorphism π=πw,i,jofH1that swaps θwiandθwjand is invariant on
+the other nodes. Also πrespects the capacities in H1, and the function /tildewideginduced by g
+underπ(i.e./tildewideg(e) :=g(π(e))) is again a maximum IBD-flow in H1. SinceBis canonical,
+it follows from Theorem 5.4 that
+(5.5)− →RH1,g=− →RH1,/tildewideg,← −RH1,g=← −RH1,/tildewideg,− →RH1,g∪← −RH1,g=A∪B1∪...∪Bk.
+The nodes in← −RH1,g−− →RH1,gareflipped when constructing H2fromH1. Then (5.5) and
+Theorem 5.3 imply that for i,jas above,
+(5.6) (a) θwiis flipped if and only if θwjis flipped;
+(b)θwi∈Aif and only if θwj∈A.
+(c)θwi∈B1∪...∪Bkif and only if θwj∈B1∪...∪Bk.
+Letp∈{1,...,k}. Since the capacity c[− →A,Bp](inH2) is odd, the set [− →A,Bp]contains
+an edgeewithc(e)odd. Any edge in H2having an odd capacity is either a loop ewior
+a legewi,s(regarding “infinite” capacities as even ones).
+Obviously, no loop can be “responsible” for the oddness of c[− →A,Bp].
+Soe=ewi,s=θwiθw,sfor some w∈V♮,1≤i≤c(w)ands∈T. Let/hatwideedenote the
+edge ofH1corresponding to e. Then (see Fig. 1(b)) /hatwideeleavesθw,sand enters θwi. Due to
+flips, however, this may not be the case for einH2.
+Suppose θwi∈A(andθw,s∈Bp). Then eleavesθwi, whence θwiis a flipped
+node inA. Now (5.6)(a,b) imply that all θwjare flipped nodes belonging to Aand that
+ewj,s∈[− →A,Bp]for allj= 1,...,c(w). But then ecannot be “responsible” for the oddness
+ofc[− →A,Bp]sincec(w)is even.
+So we have θw,s∈Aandθwi∈Bp. Theneleavesθw,s. The edge /hatwideeleavesθw,sas
+well. Hence θw,sis not flipped. Since c(w)is even, there must be j∈{1,...,c(w)}such
+that the leg ewj,s=θw,sθwjis not in [− →A,Bp](for otherwise one may pick another pair
+w′,i′). Then θwjis not in Bp. In view of (5.6)(c), θwjbelongs to a B-set inBdifferent
+fromBp. Considering the automorphisms π=πw,i′,j′for all distinct i′,j′= 1,...,c(w)
+and using the fact that the canonical barrier Bpreserves under π(in view of (5.5)), we
+can conclude that the nodes θw1,...,θwc(w)belong to different B-sets inB. Since these
+B-sets are pairwise disjoint and each automorphism πswaps two copies of θw,and do
+not move the remaining nodes in H2, each of these B-sets can contain only a single node.
+Thus,Bp={θwi}, yielding (i) in the lemma.
+Next we show (ii). From the construction of H2it follows that
+(5.7) A=− →RH2,g−← −RH2,/tildewidegandB1∪...∪Bk=− →RH2,g∩← −RH2,g.
+By the first equality, any (H2,g)-residual walk ending at a node v∈Aentersv, and by
+the second equality, there exist an (H2,g)-residual walk Ptoθ:=θwithat enters θand
+an(H2,g)-residual walk Qtoθthat leaves θ. Recall that the residual walks leave the
+sourcez. Leta=u− →θbe the last edge of P, andb=v← −θthe last edge of Q. Define E′
+(E′′) to be the set of legs e=ewi,swithg(e) = 0 (resp.g(e) = 1). Note that (cf. (5.3))
+16(5.8) ife∈E′theneR/∈EH2
+gandeentersθinH2
+g, and ife∈E′′thene /∈EH2
+gandeR
+leavesθinH2
+g.
+Supposing the existence of the root edge r=− →z← −θ(inH1andH2), we can come to a
+contradiction as follows. Since there is no loop at θ, both nodes u,vare inA. Note that
+the edge ais different from r(which leaves θ) and from rR(which enters z). Then (5.8)
+implies that a∈E′. Furthermore, ais of the form− →u− →θ. For ifaentersuthen the edge
+ofPpreceding aleavesu, whence the part of Pfromztouforms an (H2,g)-residual
+walk leaving u, which is impossible since u∈A.
+Soa∈[− →A,Bp]andg(a) = 0 = c(a)−1. Theng[← −A,Bp] = 0, by Theorem 5.1(iii).
+This implies that E′′⊆[− →A,B]. But then the last edge b=v← −θof the walk Qas above
+cannot be reverse to any edge in E′′; for otherwise bentersv, implying that the part of
+Qfromztovleavesv. Alsobis neither reverse to an edge in E′(cf. (5.8)), nor equal
+tor. The latter is because r∈[− →A,Bp], and therefore, g(a)< c(a)impliesg(r) =c(r)
+(cf. Theorem 5.1(iii)), whence r /∈EH2
+g. Thus, Qdoes not exist. This contradiction
+yields (ii).
+It remains to show (iii). By (ii), we have ewi∈EH2, and[A,Bp]is exactly the set
+of legs at θ:=θwi. Suppose d:=|[− →A,Bp]|/ne}a⊔ionslash= 1. Thend≥3, sincec[− →A,Bp] =dis
+odd. Hence g[− →A,Bp]≥d−1≥2(by Theorem 5.1(iii)). Also the fact that all legs
+enterθtogether with divg(θ) = 0 andg(ewi)≤1implies that the only possible case is
+wheng(ewi) = 1,g[− →A,Bp] = 2 andg[← −A,Bp] = 0. Now take an (H2,g)-residual walk Q
+toθthat leaves θ, and let bbe its last edge. Then bis neither the loop ewi(which is
+saturated), nor reverse to a leg e=v− →θwithg(e)>0. Indeed, if b=eRthenbentersv
+(in view of v∈Aande∈[− →A,Bp]), and hence the part of Qfromztovleavesv, which
+is impossible since v∈A. This contradiction yields (iii) and completes the proof of the
+lemma. /square
+Based on Lemma 5.5, we now finish the proof of Proposition 3.3. C onsiderBp={θwi}
+and letewi,sbe the unique edge in [− →A,Bp]. Then[← −A,Bp] ={ewi,t|t∈T−{s}}. Consider
+the maximum fractional BD-flow fas before. By the goodness of f(see (3.2)), we have
+f[− →A,Bp]−f[← −A,Bp] =f(ewi,s)−/summationdisplay
+t∈T−{s}f(ewi,t)
+=f(ewi,s)−/parenleftbig
+2f(ewi)−f(ewi,s)/parenrightbig
+= 2/parenleftbig
+f(ewi,s)−f(ewi)/parenrightbig
+≤0 =c[− →A,Bp]−1.
+Using this and (5.4), we have
+val(f) = div f(z) =/summationdisplay
+v∈Adivf(v) =/parenleftBig
+2f[− →A,← −A]−2f[← −A,− →A]/parenrightBig
++/parenleftBig
+f[− →A]−f[← −A]/parenrightBig
+=/parenleftBig
+2f[− →A,← −A]−2f[← −A,− →A]/parenrightBig
++/parenleftBig
+f[− →A,M]−f[← −A,M]/parenrightBig
++/summationdisplayk
+p=1/parenleftBig
+f[− →A,Bp]−f[← −A,Bp]/parenrightBig
+≤2c[− →A,← −A]+c[− →A,M]+/summationdisplayk
+p=1/parenleftBig
+c[− →A,Bp]−1/parenrightBig
+=c[− →A]+2c[− →A,← −A]−k=c(B) = val(g).
+Thus, we obtain the desired relation val(f)≤val(g)(which, in fact, holds with equality).
+This completes the proof of Proposition 3.3.
+176 Dual half-integrality
+6.1 Polyhedral approach
+Theorem 6.1 Leta:VG→Z+andp∈Z+. Then problem ( Dλ) has a half-integer
+optimal solution.
+Proof. The proof follows easily from Theorem 2.1 and the general fac t that the “to-
+tally dual 1/k-integrality” implies the “totally primal 1/k-integrality”, which is a natural
+generalization of a well-known result on TDI-systems due to Edmonds and Giles [EG77].
+More precisely, we utilize the following simple fact (see, e .g., [Ka89, Statement 1.1]):
+Lemma 6.2 LetAbe a nonnegative m×n-matrix, ban integral m-vector, and ka
+positive integer. Suppose that the program D(c) := max{yb|y∈Qm
++, yA≤c}has
+a1/k-integer optimal solution for every nonnegative integral n-vectorcsuch that D(c)
+has an optimal solution. Then for every nonnegative integra ln-vectorc, the program
+P(c) := min{cx|x∈Qn
++, Ax≥b}has a1/k-integer optimal solution whenever it has
+an optimal solution.
+In our case, we set k:= 2and take as A(resp.b) the constraint matrix (resp. the right
+hand side vector) of ( Dλ). Then bis integral, D(c)becomes ( Nλ),P(c)becomes ( Dλ),
+and the half-integrality for the former implies that for the latter. /square
+This proof is not “constructive” and does not lead directly t o an efficient method for
+finding a half-integer optimal solution lto (Dλ). Below we devise a strongly polynomial
+algorithm.
+6.2 The algorithm
+It has as the input arbitrary (rational-valued) optimal sol utionslandFto (Dλ) and (Nλ),
+respectively, and outputs a half-integer optimal solution /hatwidelto (Dλ). (Such landFcan be
+found in strongly polynomial time as described in Section 4. )
+As before, we set ℓ:=a+l, and in what follows, speaking of a geodesic, we always
+mean an ℓ-geodesic in G, i.e. aT-pathPwithℓ(P) =a(P)+l(P) =λ. Our goal is to
+construct /hatwidel:VG→1
+2Z+satisfying the following conditions:
+(6.1) (i) a(P)+/hatwidel(P)≥λfor anyT-pathPinG;
+(ii)a(P)+/hatwidel(P) =λfor each geodesic P;
+(iii) forv∈VG, ifl(v) = 0 then/hatwidel(v) = 0.
+Then (6.1) and the complementary slackness conditions (2.6 )–(2.7) imply that the node
+length/hatwidelforms an optimal solution to ( Dλ).
+We construct an undirected graph Γand endow it with edgelengthsµ:EΓ→Z+as
+follows. We first include in Γthe terminal set Tand all nodes and edges of Gcontained
+in geodesics. Also we add to Γthe edges of Gwith both ends lying on geodesics or T.
+The edges eof the current Γare called regular and we define µ(e) := 0 .
+Next we add to Γadditional edges, which are related to constraints due to pa rts of
+GoutsideΓ. More precisely, we scan all pairs of nodes u,v∈VΓnot connected by a
+18(regular) edge and such that there exists a path QinGhaving all nodes in VG−VΓ
+and whose first node is adjacent to u, and the last node to v. We add to Γedgee=uv,
+referring to it as a virtual edge, and define its length µ(e)to be the minimum value of
+a(Q)among such paths Q. The construction of Γ,µreduces to a polynomial number of
+usual shortest paths problems in G.
+Note that l(v) = 0 holds for each node v∈VG−VΓ(by (2.6) and (2.7)). We assign
+/hatwidel(v) := 0 for these nodes vand will further focus on finding values of /hatwidelon the nodes in Γ.
+For a path PinΓ, letΛ(P)denote its full length a(P)+l(P)+µ(P). Clearly Λ(P)≥λ
+holds for any T-pathPinΓ, and for each T-pathQinG, there exists a shortcut pathP
+inΓsuch that Λ(P)≤a(Q)+l(Q).
+The desired lengths /hatwidelonVΓwill be extracted from a system of linear constraints
+described below. For a node v∈VΓ, letTv(Πv) denote the set of terminals s∈T(resp.
+pairss,t∈T) such that vbelongs to a geodesic from s(resp. connecting sandt). When
+a terminal sbelongs to no geodesic, we set by definition Ts:={s}. For each v∈VΓand
+s∈Tv, we introduce two variables ρ−
+s(v)andρ+
+s(v)and impose the following constraints:
+(6.2) (i) For each s∈T,ρ−
+s(s) = 0.
+(ii) For each v∈VΓands∈Tv,
+ρ+
+s(v)−ρ−
+s(v) =a(v)ifl(v) = 0,
+≥a(v)ifl(v)>0.
+(iii) For each v∈VΓand{s,t}∈Πv,ρ+
+s(v)+ρ−
+t(v) =λ(andρ+
+t(v)+ρ−
+s(v) =λ).
+(iv) Ife=uv∈EΓands∈Tu∩Tv, then
+ρ−
+s(v)−ρ+
+s(u)≤µ(e),
+ρ−
+s(u)−ρ+
+s(v)≤µ(e).
+Moreover, if there exists a geodesic from scontaining both u,vin this order
+(resp. in the order v,u), then the former (resp. the latter) inequality is
+replaced by equality. (Note that in this case µ(e) = 0.)
+(v) Ife=uv∈EΓ,s∈Tu,t∈Tv, ands/ne}a⊔ionslash=t, thenρ+
+s(u)+ρ+
+t(v)≥λ−µ(e).
+The meaning of these variables becomes evident from the proo f of the next statement.
+Lemma 6.3 System (6.2) has a solution.
+Proof. For ap–qpathPinΓ, define its pre-length to beΛ(P)−(a(q) +l(q))(i.e.
+compared with the full length, we do not count the last node). Forv∈VΓands∈Tv,
+defineρ−
+s(v)(resp.ρ+
+s(v)) to be the minimum pre-length (resp. the minimum full length )
+of ans–vpath inΓ. Then (6.2)(i)–(iii) follow from the construction. Condit ion (6.2)(iv)
+represents a sort of triangle inequalities (giving one equa lity ifebelongs to a geodesic
+froms). Finally, condition (6.2)(v) holds since the full length o f anys–tpath inΓis at
+leastλ. /square
+19We observe that in linear system (6.2), each constraint cont ains at most two variables,
+each occurring with the coefficient 1 or –1, and that the R.H.S. in it is an integer. A
+well-known fact is that a linear system with such features is totally dual half-integral;
+therefore, it has a half-integer basis solution (whenever i t has a solution at all), and such
+a solution can be found in strongly polynomial time (cf., e.g ., [EJ70, Sc03]).
+Given a half-integer solution (ρ−,ρ+)to (6.2), we define half-integer node lengths /hatwidel
+as follows:
+/hatwidel(v) :=ρ+
+s(v)−ρ−
+s(v)−a(v)for allv∈VΓands∈Tv.
+Now the desired algorithmic result is provided by the follow ing
+Lemma 6.4 /hatwidelis well-defined and satisfies (6.1).
+Proof. We first show that for any v∈VΓands,t∈Tv,
+(6.3) ρ+
+s(v)−ρ−
+s(v) =ρ+
+t(v)−ρ−
+t(v).
+This is trivial when Πv=∅(since in this case v∈TandTv={v}). LetΠv/ne}a⊔ionslash=∅.
+If{s,t}∈Πv, then (6.3) follows from (6.2)(iii). Now (6.3) with any two s,t∈Tvis
+implied by the fact that the graph whose nodes and edges are th e elements of TvandΠv,
+respectively, is connected (as it is easy to see that for {s,t},{p,q}∈Πv, at least one of
+{s,p},{s,q}is inΠvas well). So /hatwidelis well-defined.
+Property (6.1)(iii) is immediate from (6.2)(ii).
+To see (6.1)(ii), consider an s–tgeodesic P. Going along Pstep by step and apply-
+ing (6.2)(ii),(iv), we observe that for each node vonP, thes–vpartP′ofPsatisfies
+a(P′)+/hatwidel(P′) =ρ+
+s(v)−ρ−
+s(s). When reaching t, we obtain a(P)+/hatwidel(P) =ρ+
+s(t)−ρ−
+s(s),
+and now (6.1)(ii) follows from (6.2)(iii) and ρ−
+s(s) =ρ−
+t(t) = 0 (by (6.2)(i)).
+Finally, consider an arbitrary T-pathQinΓ, fromptoqsay. To conclude with (6.1)(i),
+it suffices to show that
+(6.4) ∆(Q) :=a(Q)+/hatwidel(Q)+µ(Q)≥λ.
+Represent Qas the concatenation Q′·Q′′, whereQ′is a part of a geodesic from p.
+We prove (6.4) by induction on the number |Q′′|of edges in Q′′. When|Q′′|= 0,Qis
+a geodesic, and we are done. Assuming this is not the case, tak e the first edge e=uv
+ofQ′′, whereuis the end of Q′. By reasonings above, ∆(Q′) =ρ+
+p(u). Ifv∈T(and
+therefore, v=q), (6.4) immediately follows from (6.2)(v) (with s:=qandt:=q). And
+ifv /∈T, thenvbelongs to some s–tgeodesic L. W.l.o.g., one may assume that s/ne}a⊔ionslash=p
+andt/ne}a⊔ionslash=q. Applying (6.2)(v) to s,p,e, we have
+ρ+
+p(u)+ρ+
+s(v)+µ(e)≥λ.
+Comparing this with ρ−
+s(v)+ρ+
+t(v) =λand using ρ+
+s(v)−ρ−
+s(v) =a(v)+/hatwidel(v), one can
+conclude that ∆(Q)≥∆(R), whereRis thet–qpath being the concatenation of the t–v
+part of (the reverse of) Land thev–qpartR′′ofQ. Since|R′′|=|Q′′|−1, we can apply
+induction and obtain ∆(Q)≥∆(R)≥λ, as required. /square
+20Appendix: Skew-symmetric graphs and flows
+In this section we recall the notions of skew-symmetric grap hs and integer skew-symmetric
+flows, review known results on such graphs and flows, and then u se them to derive
+necessary results on bidirected graphs and flows to which we a ppealed in Section 5.
+7.1 Skew-symmetric graphs
+Askew-symmetric graph , or an SK-graph for short, is a digraph G= (VG,EG), with
+possible parallel arcs, endowed with two bijections σV,σAsuch that: σVis an involution
+on the nodes (i.e. σV(v)/ne}a⊔ionslash=vandσV(σV(v)) =vfor each node v);σAis an involution
+on the arcs; and for each arc afromutov,σA(a)is an arc from σV(v)toσV(u).
+For relevant results on SK-graphs and a relationship betwee n SK- and BD-graphs, see
+[Tu67, GK96, GK04, BK07]. For brevity σVandσAare combined into one mapping
+σonVG∪AG, which is called the symmetry (or skew-symmetry, to be precise) of G.
+For a node (arc) x, its symmetric node (arc) σ(x)is also called the mate ofx, and we
+usually use notation with primes for mates, denoting σ(x)byx′. Although Gis allowed
+to contain parallel arcs, when it is not confusing, an arc fro mutovmay be denoted as
+(u,v)or− →uv.
+Observe that if Gcontains an arc afrom a node vto its mate v′, thena′is also an
+arc from vtov′(i.e.a′is parallel to a).
+The symmetry σis extended in a natural way to walks, subgraphs and other obj ects
+inG. In particular, two walks are symmetric to each other if the e lements of one of
+them are symmetric to those of the other and go in the reverse o rder: for a walk P=
+(v0,a1,v1,...,a k,vk), the symmetric walk P′=σ(P)is(v′
+k,a′
+k,v′
+k−1,...,a′
+1,v′
+0).
+Next we explain a relationship between skew-symmetric and b idirected graphs. Given
+an SK-graph G, choose an arbitrary partition π= (V1,V2)ofVGsuch that V2=σ(V1).
+ThenGandπdetermine the BD-graph G∗withVG∗=V1whose edges correspond to
+the pairs of symmetric arcs in G. More precisely, arc mates a,a′ofGgenerate one edge
+eofG∗connecting nodes u,v∈V1such that: (i) egoes from utovif one of a,a′goes
+fromutov(and the other goes from v′tou′inV2); (ii)eleaves both u,vif one of a,a′
+goes from utov′(and the other from vtou′); (iii)eenters both u,vif one of a,a′goes
+fromu′tov(and the other from v′tou). Note that ebecomes a loop if a,a′connect a
+pair of symmetric nodes.
+Conversely, a BD-graph G∗determines an SK-graph Gwith symmetry σas follows.
+Make a copy σ(v)of each element vofV∗:=VG∗, forming the set (V∗)′:={σ(v)|
+v∈V∗}. PutVG:=V∗⊔(V∗)′. For each edge eofG∗connecting nodes uandv,
+assign two “symmetric” arcs a,a′inGso as to satisfy (i)–(iii) above (where u′=σ(u)
+andv′=σ(v)). An example is depicted in Fig. 3.
+Remark 7.1 Note that one BD-graph generates one SK-graph, by the second construc-
+tion. On the other hand, one SK-graph generates a set of BD-gr aphs, depending on the
+partition πofV, by the first construction. Namely, for each pair of symmetri c mates
+{v,v′}inGone may distribute v,v′betweenV1,V2so that either v∈V1,v′∈V2or, re-
+versely,v∈V2,v′∈V1. The resulting BD-graphs are obtained from one other by maki ng
+corresponding flips (defined in Subsection 5.1).
+21x yz t
+(a) BD-graph G∗.xx′
+yy′zz′t
+t′
+(b) Corresponding SK-graph G.
+Figure 3: Related bidirected and skew-symmetric graphs.
+There is essentially a one-to-one correspondence between t he walks in G∗andG.
+More precisely, let τbe the natural mapping of VG∪AGtoVG∗∪EG∗. Each walk
+P= (v0,a1,v1,...,ak,vk)inG(whereai= (vi−1,vi)) induces the sequence
+τ(P) := (τ(v0),τ(a1),τ(v1),...,τ(ak),τ(vk))
+of nodes and edges in G∗. One can see that τ(P)is a walk in G∗(i.e.τ(ai),τ(ai+1)form
+a transit pair at τ(vi), for each i) and that τ(P′)is the walk reverse to τ(P). Moreover,
+for any walk P∗inG∗, there is exactly one walk PinGsuch that τ(P) =P∗(considering
+Pup to replacing an arc a∈APby its mate a′whena,a′are parallel, i.e. correspond
+to a loop in G∗).
+7.2 Skew-symmetric flows
+We call a function ϕon the arcs of an SK-graph G(self-)symmetric ifϕ(a) =ϕ(a′)for
+alla∈AG. Lets∈VGbe a designated source ; its mate s′is regarded as the sink.
+Aninteger skew-symmetric s–s′flow, or an ISK-flow for short, is a symmetric function
+f:AG→Z+being an s–s′flow in a usual sense: divf(v) = 0 for allv∈VG−{s,s′},
+anddivf(s)≥0. The value offisval(f) := div f(s). Heredivf(v)denotes the usual
+divergence (given by (3.1), where δin(v)andδout(v)are the sets of arcs entering and
+leavingv, respectively).
+For a capacity function c:AG→Z+, a flowfis said to be feasible iff(a)≤c(a)for
+alla∈AG. We refer to a feasible ISK-flow of maximum possible value as a maximum
+ISK-flow .
+The above correspondence between BD- and SK-graphs is natura lly extended to flows.
+More precisely, if fis a symmetric s–s′flow inG, then transferring the values of ffrom
+the pairs of arc mates of Gto the edges of the BD-graph G∗:=τ(G), we obtain a
+τ(s)-flow inG∗, denoted as f∗. The converse correspondence is evident as well.
+ForX,Y⊆VG, let(X,Y)denote the set of arcs going from XtoY. Also (ac-
+commodating notation from Section 5 to digraphs) we denote b y[− →X]the set of arcs
+leavingX.
+22ss′
+AA′
+M B1 Bk
+Figure 4: A skew-symmetric odd-barrier. Grayed arcs corres pond to odd capacity con-
+straints.
+Letc:AG→Z+be a symmetric capacity function. Then a tuple B=
+(A,M;B1,...,B k)of subsets of VGis called a (skew-symmetric) odd barrier (w.r.t.
+the source s) if the following conditions hold (see Fig. 4):
+(7.1) (i) the sets A,A′=σ(A),M,B1,...,B kgive a partition of VG, eachBiis self-
+symmetric ( B′
+i=Bi), ands∈A;
+(ii) For each i,c(A,Bi)is odd.
+(iii) For distinct i,j,c(Bi,Bj) = 0.
+(iv) For each i,c(Bi,M) =c(M,Bi) = 0.
+Thecapacity ofBis defined to be
+(7.2) c(B) :=c[− →A]−k.
+Odd barriers in skew-symmetric graphs are related to their b idirected counterparts
+introduced in Section 5. Indeed, consider a BD-graph G∗with integer edge capacities
+c:EG∗→Z+and a source s. Construct the related SK-graph GwithVG=V⊔V′,
+whereV:=VG∗. Edge capacities cinG∗induce symmetric arc capacities in G, also
+denoted by c. The source sinG∗gives the source sand the sink s′inG. Consider a
+skew-symmetric odd barrier B= (A,M;B1,...,B k)inG.
+This barrier gives rise to the following odd BD-barrier B∗inG∗obeyingc(B∗) =c(B).
+We first construct a new BD-graph from Gby taking a bipartition (V1,V2=σ(V1))of
+VGsuch that A⊆V1andV1−(A∪A′) =V−(A∪A′); cf. Remark 7.1. The resulting
+BD-graph H∗is equivalent to G∗. Moreover, H∗is obtained from G∗by flipping a subset
+of nodes within A.
+The node subsets M,B1,...,B kinGare self-symmetric and induce subsets
+M∗,B∗
+1,...,B∗
+kinG∗andH∗in a natural way; namely, M∗:=M∩V=M∩V1
+and similarly for B∗
+i. DefineB∗:= (H∗|A∗,M∗;B∗
+1,...,B∗
+k), whereA∗:= (A∪A′)∩V.
+A straightforward examination shows that the properties in (7.1) imply their bidirected
+counterparts in (5.1). To see that c(B∗) =c(B), defineZ:=M∪B1∪...∪Bk. Note
+thatc[− →A] =c(A,A′)+c(A,Z). The capacity c(A,A′)is equal to 2c[− →A∗,← −A∗](inH∗) since
+23(A,A′)consists of pairs of arc mates, each pair corresponding to an edge in[− →A∗,← −A∗].
+And the capacity c(A,Z)is equal to c[− →A∗]since the (symmetric) set Zcorresponds to
+M∗∪B∗
+1...B∗
+kinH∗.
+In light of these observations, Theorem 5.1 is a consequence of the following Tutte’s
+theorem. (For shorter proofs of this and next theorems, see a lso [GK04].)
+Theorem 7.2 ( Max ISK-Flow Min Odd Barrier Theorem [Tu67]) ForG,c,s as above,
+the maximum ISK-flow value is equal to the minimum odd barrier capacity. An ISK-
+flowfand an odd barrier B= (A,M;B1,...,B k)have maximum value and minimum
+capacity, respectively, if and only if the following hold:
+(i)f(A,A′∪M) =c(A,A′∪M)andf(A′∪M,A) = 0;
+(ii)for eachi= 1,...,k , eitherf(A,Bi) =c(A,Bi)−1andf(Bi,A) = 0, orf(A,Bi) =
+c(A,Bi)andf(Bi,A) = 1.
+Next we establish additional correspondences. Consider an ISK-flow finG. The
+residual SK-graph Gfendowed with the residual capacities cf:AG→Z+is constructed
+in a standard fashion: VGf=VG, and the arcs of Gfare:
+(7.3) (i) each arc a∈AGwithf(a)< c(a)whose residual capacity is defined to be
+cf(a) :=c(a)−f(a), and
+(ii) the reverse arcaR= (v,u)to each arc a= (u,v)∈AGwithf(a)>0; its
+residual capacity is cf(aR) :=f(a)
+(cf. (5.3)). A path PinGfis called cf-regular ifcf(a) =cf(a′)≥2holds for each pair
+of arc mates a,a′occurring in P. (In other words, the bidirected image of PinG∗
+fis a
+cf-simple walk.) If Pis acf-regulars–s′path, we can increase the value of fby 2 (by
+sending one unit of flow along Pand one unit of flow along P′). So the existence of such
+aPimplies the non-maximality of f. A converse property is valid as well.
+Theorem 7.3 ( [Tu67]) An ISK-flow fis maximum if and only if there is no cf-regular
+s–s′path inGf.
+This implies Theorem 5.2 for IBD-flows.
+Given a maximum ISK-flow f, a certain minimum odd barrier can be constructed
+by considering the residual graph Gf. The construction described in the proof of Theo-
+rem 3.5 in [GK04] (relying on Lemma 2.2 in [GK96]) is as follow s.
+Theorem 7.4 Letfbe a maximum ISK-flow. Let R=Rfbe the set of nodes reachable
+fromsbycf-regular paths in Gf. Define A:=R−R′andM:=VG−(R∪R′). Let
+B1,...,B kbe the node sets of weakly connected components of the subgra phGinduced by
+R∩R′. ThenBf:= (A,M;B1,...,B k)is a minimum odd barrier. /square
+This subset Rof nodes in Gcorresponds to two sets− →R=− →RG∗,f∗and← −R=← −RG∗,f∗in
+G∗(defined just before Theorem 5.3; here Γ =G∗andg=f∗). More precisely, assuming
+that each node v∈VG∗corresponds to node mates v,v′inG(cf. Subsection 7.1), one
+can realize that− →R(resp.← −R) is the set of nodes v∈VG∗such that v∈R(resp.v′∈R).
+Finally, the last theorem in Subsection 5.1 (Theorem 5.4) is implied by the following
+assertion.
+24Theorem 7.5 The sets Rfin Theorem 7.4 are equal for all maximum ISK-flows f.
+Therefore, the minimum odd barriers Bfare equal as well.
+This fact can be extracted from reasonings in [GK04], yet it i s not formulated there
+explicitly. For this reason, we give a direct proof.
+Letfbe a maximum ISK-flow such that the set Rfis inclusion-wise minimal and let
+Bf:= (A,M;B1,...,B k). Consider another maximum ISK-flow g(if any). Then
+(7.4)c[− →A]−k=c(B) = val(g) =/summationdisplay
+v∈Adivg(v)
+=g/an}bracke⊔le{⊔A,A′/an}bracke⊔ri}h⊔+g/an}bracke⊔le{⊔A,M/an}bracke⊔ri}h⊔+g/an}bracke⊔le{⊔A,B1/an}bracke⊔ri}h⊔+...+g/an}bracke⊔le{⊔A,Bk/an}bracke⊔ri}h⊔,
+where for disjoint subsets X,Y⊂VG,g/an}bracke⊔le{⊔X,Y/an}bracke⊔ri}h⊔denotesg(X,Y)−g(Y,X). Fori=
+1,...,k , we have: g(A,Bi)≤c(A,Bi);c(A,Bi)is odd; and g/an}bracke⊔le{⊔A,Bi/an}bracke⊔ri}h⊔is even (the latter
+is due to a result in [Tu67]; see also [GK04, Corollary 3.2]). Therefore, c(A,Bi)−
+g/an}bracke⊔le{⊔A,Bi/an}bracke⊔ri}h⊔ ≥1. Alsog/an}bracke⊔le{⊔A,A′/an}bracke⊔ri}h⊔ ≤c(A,A′)andg/an}bracke⊔le{⊔A,M/an}bracke⊔ri}h⊔ ≤c(A,M). Comparing these
+relations with (7.4), we conclude that:
+(i) all arcs in (A,A′∪M)are saturated by g, while all arcs ain(A′∪M,A)are free
+ofg(i.e.g(a) = 0);
+(ii) for each i,g/an}bracke⊔le{⊔A,Bi/an}bracke⊔ri}h⊔=c(A,Bi)−1.
+In terms of the residual graph Gg, (i) and (ii) mean that the sets AandVG−A
+are connected in Ggby exactly karcsa1,...,ak, eachaigoing from AtoBiand having
+the residual capacity 1. By symmetry, A′andVG−A′are connected by only the arcs
+a′
+1,...,a′
+k(eacha′
+igoes from BitoA′, andcg(ai) = 1). Also by (7.1)(iii),(iv), no arc in
+Ggconnects different sets among M,B1,...,B k. Therefore, the set Rgof nodes in Gg
+reachable from sbycg-regular paths is contained in Rf. By the minimality of Rf, we
+haveRf=Rg, as required. /square
+Acknowledgements. We thank the anonymous referee for useful suggestions.
+References
+[BK07] M.A. Babenko and A.V. Karzanov. Free multiflows in bidir ected and skew-
+symmetric graphs. Discrete Appl. Math. , 155(13):1715–1730, 2007.
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+graphs. Ann. Discrete Math. , 1:185–204, 1977.
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+Combinatorica , 16(3):353–382, 1996.
+[GK97] A.V. Goldberg and A.V. Karzanov. Scaling methods for finding a maximum
+free multiflow of minimum cost. Math. Oper. Res. , 22(1):90–109, 1997.
+[GK04] A.V. Goldberg and A.V. Karzanov. Maximum skew-symme tric flows and match-
+ings.Math. Program. , 100(3):537–568, 2004.
+[IKN98] T. Ibaraki, A.V. Karzanov, and H. Nagamochi. A fast a lgorithm for finding a
+maximum free multiflow in an inner Eulerian network and some g eneralizations.
+Combinatorica , 18(1):61–83, 1998.
+[Ka79] A.V. Karzanov. A problem on maximum multiflow of minim um cost. In: Com-
+binatorial Methods for Flow Problems (Inst. for System Studies Press, Moscow,
+1979, issue 3), pp. 138–156, in Russian.
+[Ka89] A.V. Karzanov. Polyhedra related to undirected mult icommodity flows. Linear
+Algebra and its Applications , 114/115:293–328, 1989.
+[Ka94] A.V. Karzanov. Minimum cost multiflows in undirected networks. Math. Pro-
+gram., 66(3):313–325, 1994.
+[La76] E.L. Lawler. Combinatorial Optimization: Networks and Matroids . Holt, Rein-
+hart, and Winston, NY, 1976.
+[Lo76] L. Lov´ asz. On some connectivity properties of euler ian graphs. Acta Math.
+Akad. Sci. Hung. , 28:129–138, 1976.
+[Pa07] G. Pap. Some new results on node-capacitated packing of a-paths. In STOC ’07:
+Proceedings of the thirty-ninth annual ACM symposium on The ory of computing ,
+pages 599–604, New York, NY, USA, 2007. ACM.
+[Sc03] A. Schrijver. Combinatorial Optimization . Springer, Berlin, 2003.
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+26
\ No newline at end of file
diff --git a/1001.0126.txt b/1001.0126.txt
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+arXiv:1001.0126v3  [nucl-th]  1 Apr 2013Searching for squeezed particle-antiparticle correlatio ns in high energy heavy ion
+collisions
+Sandra S. Padula1and O. Socolowski, Jr.2
+1Instituto de F´ ısica Te´ orica-UNESP, C. P. 70532-2, 01156- 970 S˜ ao Paulo, SP, Brazil
+2IMEF - FURG - C. P. 474, 96201-900, Rio Grande, RS, Brazil
+(Dated: November 13, 2018)
+Squeezed correlations of particle-antiparticle pairs wer e predicted to exist if the hadron masses
+were modified in the hot and dense medium formed in high energy heavy ion collisions. Although
+well-established theoretically, they have not yet been obs erved experimentally. We suggest here a
+clear method to search for such signal, by analyzing the sque ezed correlation functions in terms of
+measurable quantities. We illustrate this suggestion for s imulated φφpairs at RHIC energies.
+DOI: 10.1103/PhysRevC.82.034908
+INTRODUCTION
+In the late 90’s, it was shown [1] that the hadronic
+mass-modification in hot and dense media could lead
+to a a novel type of correlation between bosons and
+their antiparticles. These squeezed back-to-back cor-
+relations (BBC) are the result of a quantum mechani-
+cal transformation relating in-medium quasi-particles to
+two-mode squeezed states of their free, observable coun-
+terparts [1]. This is achieved by means of a Bogolioubov-
+Valatin (BV) transformation linking the asymptotic cre-
+ation (annihilation) operators, ˆ a†
+k(ˆak), of the observed
+bosons with momentum kµ= (ωk,k), to the operators,
+ˆb†
+k(ˆbk), corresponding to thermalized quasi-particles in
+the medium. This transformation is given by ˆ ak=
+ˆbkck+ˆb†
+−ks∗
+−k; ˆa†
+k=ˆb†
+kc∗
+k+ˆb−ks−k,beingck=
+cosh(fk) andsk= sinh(fk). For conciseness, we keep
+here the short-handnotationintroduced in Ref.[1], where
+(−k) denotes an opposite sign in the spacial compo-
+nents of the momenta. Since the BV transformation be-
+tween these operators is equivalent to a squeezing oper-
+ation, the coefficient of this transformation, fki,kj(x) =
+1
+2log/bracketleftBigωki(x)+ωkj(x)
+Ωki(x)+Ωkj(x)/bracketrightBig
+iscalledsqueezingparameter; ω2
+ki=
+m2+ki2and Ω2
+ki=m2
+∗+ki2are, respectively the disper-
+sion relation in terms of the asymptotic mass, m, and in
+terms of the in-medium modified mass, m∗. A complete
+description of the phenomenon should include a param-
+eterization for m∗depending on the particles’ momenta
+and on their coordinates in the hot and dense system.
+For proposing the means to search for it experimentally,
+however, it suffices to asume a linear relation between
+the masses, i.e., m∗=m±δm, as was also considered in
+[1, 2, 5, 6].
+After the publication in Ref. [1], a similar BBC be-
+tween fermion-antifermion pairs was demonstrated to ex-
+ist [2], if the masses of these particles were modified in-
+medium. Both the fermionic (fBBC) and the bosonic
+(bBBC) back-to-backsqueezedcorrelationsaredescribed
+byanalogousformalisms,beingbothpositivecorrelationswithunlimited intensity. This behavior is in contrast
+to what is observed in femtoscopy, or Hanbury-Brown
+and Twiss effect (HBT), a quantum statistical correla-
+tions among identical particles. In the HBT realm, two
+bosons with similar momenta are positively correlated,
+with intensity ranging from 1 to 2, whereas two fermions
+with similar momenta are anti-correlated, with intensity
+between 0 and 1. Besides, the fBBC and the bBBC con-
+stitute a direct probe of hadronic mass shift in the hot
+and dense media, contradicting naive expectations that
+such information would vanish at the freeze-out surface,
+leaving no trace on correlations.
+In the remainder of this paper, we will focus on the
+bosonic case only, more especifically, on bosons that are
+their own antiparticles, such as φφorπ0π0. The two-
+particle correlation function is written as
+C2(k1,k2) =N2(k1,k2)
+N1(k1)N1(k2), (1)
+wherethe numeratoris the two-particlejoint distribution
+and the denominator is the product of the two single-
+inclusive distributions.
+The numerator in Eq. (1) is proportionalto the expec-
+tation value of the four-operator, /an}bracketle{tˆa†
+k1ˆa†
+k2ˆak2ˆak1/an}bracketri}ht. After
+applying a generalization of Wick’s theorem to locally
+equilibrated systems [3, 4], the complete two-particle dis-
+tribution in such cases can be written as N2(k1,k2) =
+ωk1ωk2/an}bracketle{tˆa†
+k1ˆa†
+k2ˆak2ˆak1/an}bracketri}ht=ωk1ωk2/bracketleftBig
+/an}bracketle{tˆa†
+k1ˆak1/an}bracketri}ht/an}bracketle{tˆa†
+k2ˆak2/an}bracketri}ht+
+/an}bracketle{tˆa†
+k1ˆak2/an}bracketri}ht/an}bracketle{tˆa†
+k2ˆak1/an}bracketri}ht+/an}bracketle{tˆa†
+k1ˆa†
+k2/an}bracketri}ht/an}bracketle{tˆak2ˆak1/an}bracketri}ht/bracketrightBig
+.
+For estimating the above expectation values the
+asymptotic operators (ˆ a,ˆa†) are first written in terms
+of the ones in-medium, ( ˆb,ˆb†). These last two opeartors
+diagonalize the full, in-medium Hamiltonian, i.e., ˆHm=
+ˆH0+ˆH1=/integraltext
+d3kΩkˆb†
+kˆbk; the asymptotic Hamiltonian is
+ˆH0=/integraltext
+d3k ωkˆa†
+kˆakandˆH1is proportional do the mass-
+shift. These thermal averages are calculated by means of
+the density matrix operator, ˆ ρ, as/an}bracketle{tˆO/an}bracketri}ht=Tr(ˆρˆO), where
+ˆρ=1
+Zexp/parenleftbig
+−1
+TV
+2π3/integraltextd3kΩkˆb†
+kˆbk/parenrightbig
+,Z=Tr(ˆρ) andT
+is the temperature. The resulting correlation function,
+given by the ratio in Eq. (1), can be written as2
+C2(k1,k2) = 1+|Gc(k1,k2)|2
+Gc(k1,k1)Gc(k2,k2)+|Gs(k1,k2)|2
+Gc(k1,k1)Gc(k2,k2),
+(2)
+wherethe denominatorsrepresentthe productofthe two
+spectral distributions, Gc(ki,ki) =N1(ki) =ωkid3N
+dki=
+ωki/an}bracketle{ta†
+kiaki/an}bracketri}ht. In the second term, the numerator is writ-
+ten in terms of the chaotic amplitude, Gc(k1,k2) =√ωk1ωk2/an}bracketle{tˆa†
+k1ˆak2/an}bracketri}ht, and is originated in the indinstin-
+guibility of the two identical mesons (either φφor
+π0π0, in the current discussion), reflecting their quan-
+tum statistics. In normal conditions the third term,
+whose numerator is proportional to the squeezed am-
+plitude, Gs(k1,k2) =√ωk1ωk2/an}bracketle{tˆak1ˆak2/an}bracketri}ht, does not con-
+tribute. However, if the interactions in the hot and dense
+medium lead to mass-modification, the third term trig-
+gers this striking particle-antiparticle correlation. The
+three terms in Eq. (2) contribute together in the
+case of neutral bosons that are their own antiparti-
+cles, such as φφorπ0π0. For charged bosons , such
+asπ±orK±, these terms split in two separate corre-
+lation functions, envolving different pairs of particles.
+Thus, the first two terms in Eq.(2) correspond to the
+HBT correlation, Cc(k1,k2) = 1 +|Gc(k1,k2)|2
+Gc(k1,k1)Gc(k2,k2),
+for identical particle pairs ( π±π±orK±K±). On
+the other hand, the sum of the first and the last
+terms leads to the squeezed particle-antiparticle corre-
+lation,Cs(k1,k2) = 1 +|Gs(k1,k2)|2
+Gc(k1,k1)Gc(k2,k2), for particle-
+antiparticle pairs ( π±π∓orK±K∓).
+RESULTS ON SQUEEZED CORRELATIONS
+Initial studies of the problem were performed for a
+static, infinite medium [1, 2]. Later, it was extended
+to the case of finite-size systems expanding with mod-
+erate radial flow [5]. For the sake of simplicity a non-relativistic treatment with flow-independent squeezing
+parameter was considered there, which allowed to obtain
+analytical expressionsfor both the squeezed and the fem-
+toscopic correlation functions [5]. However, those studies
+focussedon the behaviorofthe maximumofthe squeezed
+correlation function, Cs(k,−k,m∗), in terms of modi-
+fied mass, m∗, for particle-antiparticle pairs with exactly
+back-to-back momenta, k1=−k2=k[5, 6]. In studies of
+the HBT effect, this investigation would correspond to
+focusing on the behavior of the λ-parameter, i.e., the in-
+tercept of the correlation function, for identical particles
+with exactly identical momenta.
+Although important for theoretically understanding
+the finite size and flow effects on the squeezed corre-
+lation function, the systematic study of Cs(k,−k,m∗)
+does not represent a practical tool to look for the BBC’s
+experimentally. In reality, the momenta of the two de-
+tected particle are never exactly back-to-back and the
+in-medium shift in the hadronic mass is not a quantity
+measurable in the detector. For an empirical search of
+the BBC signal, and considering the non-relativistic con-
+text ofRef. [5], wesuggestthe following. First, select the
+particle and the antiparticle from the same event, with
+momenta( k1,k2), andcombinethem toformthepairav-
+erage and relative momenta, respectively, K=1
+2(k1+k2)
+andq= (k1−k2). Then, analyze the squeezed correla-
+tionfunctionintermsofthesevariables,similarlytowhat
+is done in HBT. The maximual value of the BBC effect
+is reached for exactly back-to-back pairs, k1=−k2=k,
+being located around K12≈0. Therefore, the squeezed
+correlation function should then be investigated by vary-
+ingK12in the region where it is small, for several values
+ofq12, i.e.,Cs(k1,k2)→Cs(2K12,q12).
+The squeezed correlation function for φφpairs is ob-
+tained by inserting, in Eq.(2), the squeezing amplitude
+and the spectra extracted from the results in Ref.[5], and
+rewritten in terms K12andq12, respectively, as
+Gs(k1,k2) =E1,2c0s0
+(2π)3
+2/braceleftBig
+R3e−2R2K2
+12+2n∗
+0R3
+∗e−2R2
+∗K2
+12exp/bracketleftBig
+−q2
+12
+8m∗T/bracketrightBig
+exp/bracketleftBig
+−K2
+12
+2m∗T∗/bracketrightBig
+exp/bracketleftBig
+−im/an}bracketle{tu/an}bracketri}htR
+2m∗T∗(2K12)2/bracketrightBig/bracerightBig
+,(3)
+Gc(ki,ki) =Ei,i
+(2π)3
+2/braceleftBig
+|s0|2R3+2n∗
+0R3
+∗(|c0|2+|s0)|2)exp/bracketleftBig
+−(K12±1
+2q12)2
+2m∗T/bracketrightBig/bracerightBig
+, (4)
+whereK12±1
+2q12=ki, withi= 1,2 are the individ-
+ual momenta, and considering the region where the con-
+tribution of the middle term (HBT part) in Eq. (2) is
+negligible. The flow-modified radius and temperature in
+Eq.(3) and (4), are given, respectively, as R∗=R/radicalbig
+T/T∗andT∗=T+m2/angbracketleftu/angbracketright2
+m∗, as in Refs. [5, 6]. We adopt natural
+units, ¯h=c= 1, througout this article.
+Extending the analogy with usual procedures adopted
+in HBT, we could think that background could be chosen
+experimentally by combining particle-antiparticle pairs3
+from different events. This choice corresponds to con-
+sider an uncorrelated pair, free from identical particle
+exchange effects. However, this is different in the BBC
+case, since the squeezing factor appears in the denomi-
+nator of the correlation function as well. Therefore, on
+searching for the effect of squeezing, we should consider
+the product of the spectra of the particle and the an-
+tiparticle in the denominator of Cs(2K12,q12), each one
+writen as in Eq. (4).
+Naturally, the analysis in terms of the variable 2 K
+would not be suited for a genuine relativistic treatment.
+Inthis case, arelativisticfour-momentumvariablecanbe
+constructedas Qµ
+back= (ω1−ω2,k1+k2) = (q0,2K), first
+introduced in [6]. In fact, it is preferable to redefine this
+variable as Q2
+bbc=−(Qback)2= 4(ω1ω2−KµKµ), since
+its non-relativistic limit is Q2
+bbc→(2K)2, recovering the
+average momentum of the pair introduced above.
+In Ref.[6], we presented some introductory results on
+φφsqueezed correlations in terms of |K12|and|q12|, but
+most of the plots shown there were obtained by attribut-
+ing precise values to the variables in Eq. (2), (3), and(4).
+In the current analysis, however, we follow the above
+procedure in a more realistic estimate, in which the mo-
+menta are generated in a simulation and then combined
+to form |K12|and|q12|. The binning in such a simu-
+lation should reflect the finite experimental resolution in
+momentum, and experimental acceptance cuts could also
+be introduced, whenever available. It should be stressed
+that the choice of φmesons considered here was made
+as a means to illustrate the proposed method, for two
+main reasons: φ’s are their own antiparticles and their
+large mass validates the non-relativistic approximation
+considered here.
+10-32510-22510-1251001/(2mT)d2N/dmT(GeV-2)
+0.0 0.5 1.0 1.5 2.0 2.5
+mT-m(GeV)meson
+PHENIX(0-10%)
+Tfo=109MeV, T=0.77,Teff=302MeV
+Tfo=140MeV,T=0.50,Teff=395MeV
+Tfo=157MeV,T=0.40,Teff=320MeV
+FIG. 1: (Color online) Generated transverse mass distribu-
+tions (arbitrary normalization) are compared with PHENIX
+data for three sets of fit parameters: i) the freeze-out tem-
+perature Tfo= 109 MeV and flow velocity βT= 0.77, ii)
+Tfo= 157 MeV and βT= 0.4, used in PHENIX simulation,
+andTfo= 140 MeV and βT=< u >= 0.5, adopted here.
+In the simulation we generated the momenta of the φmesonswiththefollowingprescription. Forroughlymim-
+icking the experimental cuts, based on PHENIX data on
+φ’s [7], we introduced some simple geometrical selection
+criteriaintheirgeneration. Thiswasdonebymerelycon-
+sidering the cuts in azimuthal angle and in the pseudo-
+rapidity, as well as by selecting the momentum region.
+In Fig. 1, we show the transverse momenta generated in
+the simulation, as compared to experimental data points,
+for three sets of parameterscorrespondingto the temper-
+ature,T, and to the radial flow velocity, /an}bracketle{tu/an}bracketri}ht. All three
+sets are in reasonable agreement with the measured dis-
+tributions. In the remainder of this work, we fix T= 140
+MeV.
+We then combine the momenta of the φφparticle-
+antiparticle pairs to calculate the squeezed correlation
+function, as explained above. The top two plots corre-
+spond to the simulated Cs(m∗,q12), keeping the average
+momentumofthepairinasmallinterval, |K12| ≤1MeV.
+In the top most plot no cuts were considered, but in the
+middle plot of Fig. 2 (a), a rough versionofthe cuts from
+Ref.[7] were introduced in the simulation. No sensitiv-
+ity to those cuts is apparent. At last, for cross-checking
+the simulationcode, wecomparethe squeezedcorrelation
+functions estimated with the generated pairs, with that
+obtained by considering pairs with exactly back-to-back
+momenta. This is seen in the plot at the bottom of Fig.
+2(a), showing the histogram obtained by attributing ex-
+act values to q12, and also fixing K12≡0, similarly to
+what was shown in Ref. [2, 5, 6]. This plot in the bot-
+tom shows close similarity with the first two, as would
+be required.
+Since the strength of the squeezed correlation is ex-
+pected to be significant in the low- |K12|region, we also
+suggest to plot Cs(K12,q12) as a function 2 K12, for en-
+larging the average momentum region where the corre-
+lation intensity can be significantly above unity. Fig.
+2 (b) shows the result for the squeezed correlation func-
+tion from simulation, obtained consideringthe static case
+(/an}bracketle{tu/an}bracketri}ht= 0), on top, and the case with radial flow ( /an}bracketle{tu/an}bracketri}ht=
+0.5), inthemiddle. Inboth, weconsideredthat theparti-
+cles were emitted in a finite interval, ∆ t= 2 fm/c, during
+which the emission decreases due to a Lorentzian distri-
+bution in time, |Fs(∆t)|2= [1+(ω1+ω2)2∆t2]−1[1, 2, 5, 6],
+whichmultipliesthethirdterminEq. (2). Thisfactorre-
+duces the signal by almost three orders of magnitude, as
+comparedtoainstantaneousemission(∆ t= 0). Thiscan
+be seen by comparing the plot in the middle of Fig. 2(b),
+with the one in the bottom. We note that, although the
+reduction of the strength caused by a finite emission pe-
+riod is dramatic, the intensity of the φφBBC correlation
+is still sizable, suggesting that its experimental search
+is indeed promising. Another particular emission time
+distribution will be discussed below.
+From Figs. 2(b) we see that, in the absence of flow, the
+squeezed correlation grows faster from smaller to higher
+values|q12|than in the presence of flow. However, this4
+11.522.533.544.55
+0 0.05 0.1
+11.522.533.544.55
+0 0.05 0.1Cs(2*K,q) Cs(2*K,q)
+FIG. 2: (Color online) Part (a) shows the squeezed correlati on function in terms of the shifted mass, m∗, and the pair relative
+momentum, |q12|. Results from simulation (top two plots) are compared with t hose from calculation, with fixed K12= 0
+(bottom). Part (b) shows the advertised plots for the squeez ed correlation function to be searched for experimentally, with
+m∗= 1GeV. The striking effect of finite emission times is shown by comparing the middle and the bottom plots in part (b).
+Part (c) shows the inverse width of the BBC function reflectin g the radius of the squeezing region, for R= 7 fm (top) and
+R= 3 fm (bottom).
+last one is stronger even in the low |q12|region, show-
+ing that the presence of flow could enhance the signal’s
+intensity over a wider region of the ( |K12|,|q12|)-plane.
+The size of the squeezing region is reflected in the in-
+verse width of the curves as a function of 2 |K12|, being
+narrower (broader) for larger (smaller) radii. In Figs.
+2(c), we compare the squeezed correlation functions con-
+sidering that the system radius is R= 7 fm, also used
+in all the above calculation, to the case of a system with
+smaller extent of the squeezing region, with R= 3 fm.
+Returning to the discussion about the time emission
+distribution, we should emphasize that we do not know a
+priori how this emission should proceed, mainly in the
+simple model adopted here. On the other hand, the
+lack of knowledge of such distribution, naturally does
+not invalidate the experimental search of the hadronic
+squeezed correlations. The squeezing is a fundamental
+phenomenon already detected in quantum optics and it
+should be empirically observed in relativistic heavy ion
+collisions, if the hadron masses are modified in-medium
+by some mechanism. Therefore, as a common practice
+in Physics, we should look for the effect experimentally
+and, once it is discovered, wecould tryto explainits time
+emission process by means of a suitable model. Never-
+theless, motivated by an analysis made by the PHENIX
+Collaboration[8] we investigated in Ref. [9, 10] a dif-
+ferent emission distribution in time, by considering the
+effects of a a symmetric, α-stable L´ evy distribution, i.e.,
+|F(∆t)|2= exp{−[∆t(ω1+ω2)]α},onthe squeezedcorre-
+lationfunction of K+K−pairs. This functional form had
+been fitted to two- and three-particle Bose-Einstein cor-
+relation functions. The values fitted to date different val-
+ues of the distribution index, α= 1.0 orα= 1.35, werefitted to data, depending on the region investigated of
+the particles’ transverse momentum or transverse mass.
+Briefly summarizing those effects, we concluded that, for
+α= 1, such a time factor acting on the squeezing corre-
+lation function reduces its intensity even more dramat-
+ically than the Lorentzian factor discussed here. How-
+ever, it would still lead to measurable quantities, mainly
+if the emission lasted a short period of time, of about
+∆t= 1 fm/c. However, if Nature favors the higher value,
+α= 1.35, this would result in a very small deviation from
+unity, not detectable by the method proposed here. Nat-
+urally, in the case of φpairs, even for α= 1 and ∆ t= 1
+fm/c the intensity would be negligibly small for a L´ evy-
+type distribution, due to their large asymptotic mass. In
+spite of that, as discussed above, we do not know a priori
+the preferredformchosenbyNature forthe emissionpro-
+cess, which in itself does not invalidate the experimental
+search of this phenomenon. For this reason and for con-
+tinuing the illustration of the proposed method to search
+for the hadronic squeezed states, in the remainder of this
+paper we attain our discussion to the Lorentzian time
+distribution, comparing it to the instantaneous emission
+process only.
+EFFECTS OF SQUEEZING ON φφHBT
+CORRELATIONS
+We next discuss how the HBT correlation function
+could be affected by the in-medium mass-shift. Contra-
+dicting early expectations that the thermalization would
+wash out any trace of mass-shift in this type of correla-
+tion, we find its in-print in HBT, reflecting the presence5
+of the squeezing factor, fi,j(m,m∗), in the chaotic am-
+plitude. This can be inferred from the analytical results
+in previous papers[1, 2, 5]. However, the strength of the
+squeezing effect on the two-identical particle correlationwas not carefully investigated in those references.
+The analytical form of the HBT correlation function is
+obtained by substituting the chaotic amplitude[5],
+Gc(k1,k2) =E1,2
+(2π)3
+2/braceleftBig
+|s0|2R3e−1
+2R2q2
+12+n∗
+0R3
+∗(|c0|2+|s0|2)e−1
+2R2
+∗q2
+12e−K2
+12
+2m∗T∗e−q2
+12
+8m∗Texp/bracketleftBigim/an}bracketle{tu/an}bracketri}htR
+m∗T∗K12.q12/bracketrightBig/bracerightBig
+,(5)
+together with the spectrum given in Eq. (4), into Eq.(2).
+The finite emission time factor, multiplying the square
+modulus of Eq. (5), is now Fc(∆t) = [1+( ω1−ω2)2∆t2]−1.
+For stressing the HBT effects in the φφcase, we selected
+the region (small |q12|) where the particle-antiparticle
+correlation is not significant and the HBT is relevant to
+Eq.(2).
+11.21.41.61.82
+0 0.05 0.1
+11.21.41.61.82
+0 0.05 0.1C(k1,k2)
+FIG. 3: The plots show the HBT correlation function in the
+absence of squeezing (top) and when it is present (bottom).
+This is seen in Fig. 3. The top part shows the ef-
+fect of radial flow alone on the HBT correlation function,
+while in the bottom, the joint effects of flow and squeez-
+ing are shown. We see that, without squeezing, the flow
+broadens the correlation curves, as expected, since the
+expansion reduces the size of the region accessible to in-
+terferometry. When the squeezing effects are present,
+they seem to oppose to the flow effects, almost canceling
+the broadening of the correlation function due to flow for
+large|K|, another striking indication of in-medium mass
+modification.
+CONCLUSIONS
+In this work we suggest an effective way to search for
+the squeezed bosonic correlations in heavy ion collisionsat RHIC and, soon, at the LHC. We argue that the suit-
+able variable to experimentally search for the squeezed
+correlation function is the average momentum of the
+pair, 2|K12|, the non-relativistic limit of the relativis-
+tic variable, Qbbc= 2/radicalbig
+(ω1ω2−KµKµ) [6]. We show
+that, in the presence of flow, the signal is expected to
+be stronger over the momentum regions shown in the
+plots, i.e., roughly for 0 <∼|2K|<∼100 MeV/c (depend-
+ing onR) and 500 <∼|q|<∼1500−2000 MeV/c, sug-
+gesting that flow may enhance the strength of the BBC
+signal, facilitating its experimental discovery. Another
+important result found within our simple non-relativistic
+model is that the squeezing could also distort the HBT
+correlation function, leading to effects opposing those of
+flow, almost neutralizing it for large values of |K12|. For
+emphasizing the dramatic effects induced by in-medium
+hadronic mass modification on the correlation functions,
+we chose a constant mass-shift that leads to the maximal
+intensity, based on results of Fig. (2). For φφmesons,
+this corresponds to m∗ ≈1 GeV, roughly a 2% reduc-
+tion in the φmass, as compared to its asymptotic mass
+(m= 1.02 GeV). As stressed before, a more realistic
+treatment should consider a detailed prescription for the
+mass modification, based on models that predict its de-
+pendence on the particles’ momenta and its distribution
+in the hot and dense system.
+The above procedure is also applicable to other parti-
+cles, such as kaons. The correspondingresults [9] are dis-
+cussed in Ref. [10]. Finally, it is important to note that
+allthe effects shownhere should exist onlyif the particles
+havetheirmassmodifiedinthehotanddensemedium. If
+no modification happens, the squeezed correlation func-
+tions would be flat unity, and the HBT correlation func-
+tions would behave as usual. However, if the particles’
+masses are indeed modified, the experimental discovery
+of squeezed particle-antiparticle correlation (and the dis-
+tortions pointed out in the HBT correlations) would be
+an unequivocal signature of in-medium modifications by
+means of hadronic probes!6
+ACKNOWLEDGEMENTS
+We are grateful to T. Cs¨ org˝ o and M. Nagy, who pro-
+posedthe Qµ
+backvariable,forfruitful discussionsandcom-
+ments on the text. We are also thankful to H. Takai for a
+careful reading and suggestions on the manuscript. OSJ
+acknowledges funding from CNPQ.
+[1] M. Asakawa, T. Cs¨ org˝ o and M. Gyulassy, Phys. Rev.
+Lett.83, 4013 (1999).
+[2] P. K. Panda, T. Cs¨ org˝ o, Y. Hama, G. Krein and Sandra
+S. Padula, Phys. Lett. B 512, 49 (2001).
+[3] M. Gyulassy, S. K. Kaufmann, and L. W. Wilson, Phys.
+Rev.C 20, 2267 (1979).
+[4] A. Makhlin and Yu Sinyukov, Sov. J. Nucl. Phys. 46, 354(1987); Yu Sinyukov, Nucl. Phys. A566, 589c (1994).
+[5] Sandra S. Padula, G. Krein, T. Cs¨ org˝ o, Y. Hama, and P.
+K. Panda, Phys. Rev. C 73, 044906 (2006).
+[6] Sandra S. Padula, O. Socolowski Jr., T. Cs¨ org˝ o and M.
+Nagy, Proc. Quark Matter 2008, J. Phys. G: Nucl. Part.
+Phys.35, 104141 (2008); Sandra S. Padula, D. M. Dudek
+and O. Socolowski Jr., Proc. WPCF 2008, A. Phys. Pol.
+B 40, N. 4, 1225 (2009).
+[7] S. S. Adler et al., PHENIX Collaboration, Phys. Rev. C
+72, 014903 (2005).
+[8] M. Csan´ ad [PHENIX Collaboration], Proc. Quark Mat-
+ter 2005, Nucl. Phys. A 774, 611 (2006).
+[9] Danuce M. Dudek, Squeezed Hadronic Correlations of
+K+K−pairs in Relativistic Heavy Ion collisions , Master
+Dissertation presented to the Instituto de F´ ısica Te´ oric a
+- UNESP (March/2009).
+[10] Danuce M. Dudek and Sandra S. Padula, Phys. Rev, C
+82, 034905 (2010) [ArXiv: 1006.5899 (nucl-th)].
\ No newline at end of file
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+arXiv:1001.0127v1  [gr-qc]  31 Dec 2009On the “scattering law” for Kasner parameters in the
+model with one-component anisotropic fluid
+V.D. Ivashchuk1and V.N. Melnikov2
+Center for Gravitation and Fundamental Metrology, VNIIMS, 46
+Ozyornaya ul., Moscow 119361, Russia
+Institute of Gravitation and Cosmology, Peoples’ Friendsh ip University of
+Russia, 6 Miklukho-Maklaya ul., Moscow 117198, Russia
+Abstract
+A multidimensional cosmological type model with 1-compone nt
+anisotropic fluid is considered. An exact solution is obtain ed. This
+solution is defined on a product manifold containing nRicci-flat fac-
+tor spaces. We singled out a special solution governed by the function
+cosh. It is shown that this special solution has Kasner-like asym p-
+totics in the limits τ→+0 and τ→+∞, whereτis a synchronous
+time variable. A relation between two sets of Kasner paramet ersα∞
+andα0is found. This formula (of “scattering law”) is coinciding w ith
+that obtained earlier for the S-brane solution (when scalar fields are
+absent).
+1ivashchuk@mail.ru
+2melnikov@vniims.ru
+11 Introduction
+In this paper we continue our investigations (started in [1]) of multid imen-
+sional solutions defined on product of several Ricci-flat factor sp aces which
+have two Kasner-like asymptotical regions.
+Here we recall that Kasner-like solutions with a chain of nRicci-flat
+factor-spaces ( Mi,g(i)) have the following form [2]
+g=wdτ⊗dτ+n/summationdisplay
+i=1A2
+iτ2αig(i), (1.1)
+wherew=±1,τ >0,
+n/summationdisplay
+i=1diαi= 1, (1.2)
+n/summationdisplay
+i=1di(αi)2= 1, (1.3)
+and for any i= 1,...,n(n≥2):Ai>0 is constant, g(i)is a Ricci-flat
+metric defined on the manifold Mi(forw=−1 see [2]).
+These solutions with non-Milne-type sets of Kasner parameters ar e sin-
+gular since the Riemann tensor squared is divergent as τ→+0 [3]. For
+Milne-type sets of parameters, i.e. when di= 1 and αi= 1 for some i
+(αj= 0 for all j/ne}ationslash=i) the metric is regular as τ→+0, when either i)
+g(i)=−wdyi⊗dyi,Mi=R(−∞< yi<+∞), or ii)g(i)=wdyi⊗dyi,
+Miis circle of length Li(0< yi< Li) andAiLi= 2π(i.e. when the cone
+singularity is absent).
+In this paper we consider an exact cosmological type solution with 1-
+component “perfect” fluid (Section 2). (For earlier publications on multidi-
+mensional cosmological models with perfect fluid see [4]-[14] and refe rences
+therein.) This solution is defined on a product manifold containing nRicci-
+flat factor spaces. It is derived in the Appendix. For w=−1 it was found in
+[11, 12, 13] and generalized in [14] for the case when a scalar field was added.
+A special case of this solution with a Λ-term component was obtained in
+[15] (see also [16] for scalar field generalization).
+We write the solution in a so-called “minisuperspace-covariant” form that
+significantly simplifies the forthcoming analysis. In Section 3 we single o ut a
+special solution governed by the coshfunction. We show that this solution
+2has a Kasner-like asymptotics in both limits τ→+0 and τ→+∞, where
+τis the synchronous time variable. We also find a relation between two s ets
+of Kasner parameters α∞= (αi
+∞)∈Rnandα0= (αi
+0)∈Rn:
+αi
+∞=αi
+0−2U(α0)Ui(U,U)−1
+1−2U(α0)(U,UΛ)(U,U)−1, (1.4)
+i= 1,...,n. Here U= (Ui) is a co-vector corresponding to the fluid
+component, ¯U= (Ui) is dual vector and UΛis a co-vector, corresponding
+to the Λ-term. All these vectors and the scalar product ( .,.) are defined
+below (see Section 2). Here U(α0) =Uiαi
+0>0 andU(α∞) =Uiαi
+∞<0.
+A relation analogous to (1.4) (“scattering law” formula) was obtaine d
+earlier for S-brane solution with one brane in [1]. We note that in [1] the
+geometrical sense of the scattering law was clarified for n >2. Namely, the
+scattering law transformation for a brane U-vector (obeying ( U,UΛ)<0)
+was expressed in terms of a function mapping a “shadow” part of th e Kasner
+sphereSn−2onto “illuminated” one. The shadow and illuminated parts
+of the Kasner sphere were defined w.r.t. a point-like source of light lo cated
+outsidetheKasnersphere Sn−2. (Fordetailsofthisgeometricalconstruction
+see [1]).
+The relation (1.4) appears also when the billiard approach to multicom-
+ponent anisotropic fluid is considered [18, 19, 20, 21]. It may be show n (as
+it was done in [23, 24] for S-brane solutions) that after the collision with a
+billiard wall (corresponding to the fluid component) the set of Kasne r param-
+eters, is defined by the Kasner set before the collision through the formula
+analogous to (1.4), see [26]. For the billiard approach in models with sca lar
+field and fields of forms see [22, 23, 25, 26] and refs. therein.
+2 Model with anisotropic fluid and its exact
+solution
+2.1 The set-up
+Now, we consider a cosmological type solution to Einstein equations w ith an
+anisotropic (perfect) fluid matter source
+RM
+N−1
+2δM
+NR=k2TM
+N (2.1)
+3defined on D-dimensional manifold
+M=R.×M1×M2×...×Mn, (2.2)
+with block-diagonal metric
+g=we2γ(u)du⊗du+n/summationdisplay
+i=1e2βi(u)g(i). (2.3)
+HereR.= (u−,u+) is an interval, w=±1 andn≥2. Manifold Mi
+with the metric g(i)is a Ricci-flat space of dimension di:Rmini[g(i)] = 0,
+i= 1,2,...,n, andκ2is a multidimensional gravitational constant.
+Energy-momentum tensor of anisotropic fluid is adopted in the follow ing
+form:
+(TM
+N) = diag( −ˆρ,ˆp1δm1
+k1,...,ˆpnδmn
+kn), (2.4)
+where ˆρand ˆpiare “density” and “pressures”, respectively, depending upon
+radial variable u.
+In the cosmological case when w=−1 and all metrics g(i)have Eu-
+clidean signatures, ˆ ρ=ρis a density and ˆ pi=piis a pressure in i-th
+space. For static configurations with w= 1,g(1)=−dt⊗dtand all metrics
+g(i),i >1, having Euclidean signatures, the physical density and pressure s
+are related to the effective (“hat”) ones by formulas: ρ=−ˆp1,pu=−ˆρ,
+pi= ˆpi,(i/ne}ationslash= 1), where puis the pressure in u-th direction.
+We also impose the following equation of state
+ˆpi=/parenleftbigg
+1−2Ui
+di/parenrightbigg
+ˆρ, (2.5)
+whereUiare constants, i= 1,2,...,n.
+In what follows we use a scalar product
+(U,U′) =GijUiU′
+j=n/summationdisplay
+i=1UiU′
+i
+di+1
+2−D(n/summationdisplay
+i=1Ui)(n/summationdisplay
+j=1U′
+j),(2.6)
+forU= (Ui),U′= (U′
+i)∈Rn, where
+Gij=δij
+di+1
+2−D(2.7)
+4are components of dual minisuperspace metric. Recall that ( Gij) = (Gij)−1,
+where
+Gij=diδij−didj, (2.8)
+are components of minisuperspace metric [17].
+We also define a co-vector
+UΛ= (di), (2.9)
+corresponding to the Λ-term and the vector ¯U= (Ui)
+Ui=GijUj=Ui
+di+1
+2−Dn/summationdisplay
+j=1Uj, (2.10)
+which is dual to U.
+2.2 Exact solution
+Hereweconsider anexactcosmologicalsolutiontoHilbert-Einsteine quations
+(2.1) defined on the manifold (2.2). We impose the following restriction on
+theU-vector in (2.5)
+K= (U,U) =n/summationdisplay
+i=1U2
+i
+di+1
+2−D(n/summationdisplay
+i=1Ui)2/ne}ationslash= 0. (2.11)
+(The case K= 0 will be considered in a separate publication.)
+The solution has the following form (see Appendix C)
+g=|f(u)|−2h(U,UΛ)exp(2c0u+2¯c0)wdu⊗du+ (2.12)
+n/summationdisplay
+i=1|f(u)|−2hUiexp(2ciu+2¯ci)g(i),
+k2ˆρ=−wA|f(u)|2h(U,UΛ)−2exp(−2c0u−2¯c0), (2.13)
+wherew=±1,h=K−1,g(i)is a Ricci-flat metric on Mi, and
+(U,UΛ) =/summationtextn
+i=1Ui
+2−D, (2.14)
+i= 1,...,n.
+5The moduli function freads
+f(u) =Rsinh(√
+C(u−u0)), C >0, KA <0; (2.15)
+Rsin(/radicalbig
+|C|(u−u0)), C <0, KA <0; (2.16)
+Rcosh(√
+C(u−u0)), C >0, KA >0; (2.17)
+|2AK|1/2(u−u0), C= 0, KA <0, (2.18)
+whereR=|2AK/C|1/2, andC,u0are constants. (In (2.12) and (2.13)
+f(u)/ne}ationslash= 0 is assumed for all u∈(u−,u+).)
+Vectorsc= (ci) and ¯c= (¯ci) obey the following constraints:
+U(c) =Uici= 0, U(c) =Ui¯ci= 0 (2.19)
+CK−1+Gijcicj= 0, (2.20)
+whereGijcicj=/summationtextn
+i=1di(ci)2−(/summationtextn
+i=1dici)2.
+In (2.12) and (2.13) we also denote
+c0=UΛ(c) =n/summationdisplay
+i=1dici,¯c0=UΛ(¯c) =n/summationdisplay
+i=1di¯ci. (2.21)
+The special solution with C=ci= 0 (for all i) andw=−1 was
+considered in detail in [28, 29]. For U=UΛandA >0 it contains a special
+solution with di= 1,gi=dyi⊗dyi(i= 1,...,n), describing either (a
+part of) de-Sitter space (for w=−1) or (a part of) anti-de-Sitter space (for
+w= 1).
+Minisuperspace-covariant form of solution.
+This solution is derived in Appendix C in terms of “minisuperspace-
+covariant” notations for functions γ(u),βi(u) appearing in metric (2.3).
+Solution for β= (βi(u)) reads as follows:
+βi(u) =−Ui
+(U,U)ln|f(u)|+ciu+¯ci, (2.22)
+wheref(u) was defined in (2.15)-(2.18) and
+γ=γ0≡n/summationdisplay
+i=1diβi=UΛ
+iβi(2.23)
+anduis the harmonic variable.
+63 Scattering law for Kasner parameters
+Now we restrict our consideration by a special solution with C >0,K=
+(U,U)>0 andA >0. In this case the solution is governed by moduli func-
+tionf(u) =Rcosh(√
+C(u−u0)),u∈(−∞,+∞), and has two Kasner-like
+asymptotics in the limits τ→+0 and τ→+∞, whereτis a synchronous
+time variable (see below).
+Another case, when there are two Kasner-like asymptotical regio ns, takes
+place when C >0,K= (U,U)<0 andA <0 (this will be a subject of a
+separate paper).
+3.1 Kasner-like behaviour
+Let us consider our solution in a synchronous time:
+τ=ε/integraldisplayu
+u0d¯ueγ0(¯u), (3.1)
+whereε=±1, and
+eγ0(u)=|f(u)|−h(UΛ,U)exp(c0u+¯c0) (3.2)
+is a lapse function.
+Due to
+f∼R
+2exp(±√
+C(u−u0)), (3.3)
+foru→ ±∞, we get asymptotical relations for the lapse function
+eγ0∼constexp( b±√
+Cu), (3.4)
+asu→ ±∞, with
+b±=∓h(UΛ,U)+c0
+√
+C. (3.5)
+Using relations (2.21) and h= (U,U)−1, we could rewrite parameters b±
+in a minisuperspace-covariant form:
+b±=∓(UΛ,U)
+(U,U)+(s,UΛ), (3.6)
+where
+s= (si) = (Gijcj/√
+C) (3.7)
+7is a co-vector, obeying relations
+(s,U) = 0, (3.8)
+1
+(U,U)+(s,s) = 0, (3.9)
+following just from (2.19) and (2.20). In derivation of (3.6) we used t he
+relation
+c0= (s,UΛ)√
+C, (3.10)
+following from (2.21) and (3.7).
+In what follows we will use the inequality
+|(s,UΛ)|>|(UΛ,U)|
+(U,U), (3.11)
+proved in Appendix C. The proof used relations (3.8), (3.9) and ( U,U)>0.
+The parameter c0is a non-zero one (otherwise the relation (2.20) would
+be incompatible with the conditions C >0,K >0).
+It follows from (3.11) that b±are also non-zero and
+sign(b±) = sign(( s,UΛ)) = sign( c0). (3.12)
+It may be verified that due to (3.11) the lapse function eγ0(u)is monoton-
+ically increasing from +0 to + ∞forc0>0 and monotonically decreasing
+from +∞to +0 for c0<0.
+We define a synchronous-like variable to be
+τ=/integraldisplayu
+−∞d¯ueγ0(¯u)(3.13)
+forc0>0 and
+τ=/integraldisplay+∞
+ud¯ueγ0(¯u)(3.14)
+forc0<0. Then, τ=τ(u) is monotonically increasing from +0 to + ∞
+forc0>0 and monotonically decreasing from + ∞to +0 for c0<0.
+We have the following asymptotical relations for τ=τ(u)
+τ∼constb−1
+±exp(b±√
+Cu), (3.15)
+asu→ ±∞.
+8Forβ= (βi) from (2.22) we get (see (3.3))
+βi(u)∼ ∓Ui√
+Cu
+(U,U)+ciu+ˆci(3.16)
+asu→ ±∞, where ˆciare constants. Hence, due to (3.15), we are led to
+Kasner-like asymptotics
+βi∼αi
+±lnτ+βi
+± (3.17)
+foru→ ±∞, whereβi
+±are constants and
+αi
+±= [∓Ui
+(U,U)+si]/b± (3.18)
+are Kasner-like parameters corresponding to u→ ±∞.
+Asymptotical relations(3.17)couldbealsorewrittenintheformofp roper
+time asymptotics, i.e.
+βi∼αi
+0lnτ+βi
+0,asτ→+0, (3.19)
+βi∼αi
+∞lnτ+βi
+∞,asτ→+∞. (3.20)
+Here
+αi
+0=αi
+−, αi
+∞=αi
++ (3.21)
+forc0>0 and
+αi
+0=αi
++, αi
+∞=αi
+− (3.22)
+forc0<0 andβi
+0,βi
+∞are constants.
+It follows from definitions of Kasner parameters (3.18) that
+Gijαi
+±αj
+±= 0, (3.23)
+U(α±) =Uiαi
+±=∓1
+b±, (3.24)
+UΛ(α±) = 1, (3.25)
+see (3.6), (3.8) and (3.9).
+In components relations (3.23) and (3.25) read as
+n/summationdisplay
+i=1diαi
+±=n/summationdisplay
+i=1di(αi
+±)2= 1. (3.26)
+9Thus, we are led to Kasner-like relations (1.2) and (1.3) for α±= (αi
+±).
+Hence,α0= (αi
+0) andα∞= (αi
+∞) also obey relations (1.2) and (1.3).
+So, we obtained a Kasner-like asymptotical behaviour of our specia l solu-
+tion (with C >0,K >0 andA >0) for i) τ→+0 and for ii) τ→+∞,
+aswell. TheKasner-like behaviour inthecasei)isinagreement withthe gen-
+eral result of the billiard approach from [22]. The the case ii) was con sidered
+in [26].
+Using (3.12) and (3.24) we get
+U(α0) =Uiαi
+0>0, (3.27)
+U(α∞)<0. (3.28)
+3.2 Scattering law
+Now, we derive a relation between Kasner sets α0andα∞.
+We start with formulae:
+b+α+−b−α−=−2¯U
+(U,U)(3.29)
+and
+b+−b−=−2(UΛ,U)
+(U,U), (3.30)
+following from (3.18) and (3.6), respectively. (Recall that ¯U= (Ui). ) Using
+these relations and (3.24) we get
+αi
+±=αi
+∓−2UiU(α∓)(U,U)−1
+1−2U(α∓)(U,UΛ)(U,U)−1. (3.31)
+This formula gives a scattering law formula for Kasner parameters in our
+case (see definitions (2.10), (3.21) and (3.22)) or
+α∞=α0−2¯UU(α0)(U,U)−1
+1−2U(α0)(U,UΛ)(U,U)−1=S(α0). (3.32)
+coinciding with the scattering law formula (1.4) derived in [1] for anoth er
+S-brane solution when scalar fields are absent and Uis coinciding with the
+braneU-vector.
+Due to (3.31) the inverse function S−1is given by just the same relation
+α0=α∞−2¯UU(α∞)(U,U)−1
+1−2U(α∞)(U,UΛ)(U,U)−1=S−1(α∞). (3.33)
+103.3 Geometric meaning of the scattering law
+Here we analyze the geometric meaning of the scattering for n >2 as it was
+done in [1] for the S-brane solution.
+The Kasner-like relations (1.2) and (1.3) describe an ellipsoid isomorph ic
+to a unit ( n−2)-dimensional sphere Sn−2belonging to Rn−1. The sets
+of Kasner parameters αmay be parametrized by vectors /vector n∈Sn−2, i.e.
+α=α(/vector n).
+For (U,UΛ)/ne}ationslash= 0 (or, equivalently, when/summationtextn
+i=1Ui/ne}ationslash= 0, see (2.14)) the
+scattering law formula (1.4) in terms of /vector n-vectors reads as in [1]
+/vector n∞=(/vector v2−1)/vector n0+2(1−/vector v/vector n0)/vector v
+(/vector v−/vector n0)2(3.34)
+where/vector vis a vector belonging to Rn−1with|/vector v|>1.
+Here
+/vector v/vector n0<1/vector v/vector n∞>1, (3.35)
+for (U,UΛ)<0 (or, equivalently, when/summationtextn
+i=1Ui>0) and
+/vector v/vector n0>1/vector v/vector n∞<1, (3.36)
+for (U,UΛ)>0 (or, equivalently, when/summationtextn
+i=1Ui<0).
+The vector /vector v= (vi)∈Rn−1is defined by the formula
+vi=−ˆUi/ˆU0, (3.37)
+i= 1,...,n−1, where
+ˆUa=ei
+aUi, (3.38)
+and the invertible matrix ( ea
+i) satisfies the relations
+ηab=ea
+iGijeb
+j, (3.39)
+a,b= 0,...,n−1, with
+e0
+i=q−1UΛ
+i, (3.40)
+and
+q= [−(UΛ,UΛ)]1/2= [(D−1)/(D−2)]1/2. (3.41)
+(Here (ηab) = (ηab) =diag(−1,+1,...,+1).)
+This implies
+ˆU0=−q−1(U,UΛ) (3.42)
+11and hence ˆU0/ne}ationslash= 0 when ( U,UΛ)/ne}ationslash= 0.
+Relations (3.34), (3.35) and (3.36) could be readily proved from (3.31 ),
+(3.27) and (3.28) if the following “frame” Kasner-like parameters
+ˆαa=ea
+iαi, (3.43)
+with
+ˆα0=q−1,ˆαi=q−1ni, (3.44)
+i= 1,...,n−1,areused(see[1]). Animportantrelationhereisthefollowing
+one
+U(α) =UAαA=ˆUaˆαa=q−1ˆU0(1−/vector v/vector n). (3.45)
+Thus, for ( U,UΛ)/ne}ationslash= 0 we get just a modified inversion with respect
+to a point vlocated outside the Kasner sphere Sn−2(see Fig. 1). For
+(U,UΛ)<0 the function (3.34) maps a shadow part of the Kasner sphere
+Sn−2onto illuminated one, while for ( U,UΛ)>0 this function maps an
+illuminated part of the Kasner sphere Sn−2onto shadow one. Here the
+shadow and illuminated parts of the Kasner sphere are defined w.r.t. a
+point-like source of light located at v.
+For (U,UΛ) = 0 (or, equivalently,
+Figure 1: The graphical represen-
+tation of the modified inversion S
+w.r.t. a point Vforn= 3, and
+(U,UΛ)<0:N′=S(N).when/summationtextn
+i=1Ui= 0) the main formula
+(1.4) in terms of /vector n-vectors reads
+/vector n∞=/vector n0−2(/vectorb/vector n0)/vectorb, (3.46)
+where/vectorb= (bi) is a unit vector belong-
+ing toRn−1(|/vectorb|= 1) with compo-
+nents
+bi=ˆUi/(n−1/summationdisplay
+j=1ˆU2
+j)1/2,(3.47)
+i= 1,...,n−1. The inequalities on
+Kasner-likeparameters(3.27)and(3.28)
+in this case reads as follows
+/vectorb/vector n0>0,/vectorb/vector n∞<0.(3.48)
+Thus, for ( U,UΛ) = 0 the function (3.34) is just a reflection with respect
+to a hyperplane {/vector y:/vectorb/vector y= 0}, which contains a center of the Kasner sphere.
+12Relations(3.47)and(3.48)maybeobtainedfrom(3.34), (3.35)and( 3.36)
+by means of the limiting procedure: ˆU0→ ±0 (|/vector v| →+∞.).
+It should be noted that all formulas presented above are also valid f or
+n= 2. In this case the zero-dimensional Kasner sphere S0={−1,1}should
+be considered.
+4 Example: n= 2
+Here we consider the simplest case of the solution with C >0,K >0,
+whenn= 2. We put U1/ne}ationslash= 0 and U2= 0, i.e. ˆ p1=w1ˆρwithw1/ne}ationslash= 1 and
+ˆp2= ˆρ.
+For Kasner set α= (α1,α2) we get from (1.2) and (1.3) [16, 30]
+α±= (α1
+±,α2
+±) =1
+d1+d2/parenleftbigg
+1±r
+d1,1∓r
+d2/parenrightbigg
+, (4.1)
+wherer=/radicalbig
+d1d2(d1+d2−1). (The number r >0 is integer one when
+d1= 1 ord2= 1 and also for ( d1,d2) = (3,6),(5,5),(2,8),(13,13) etc [30].)
+Letd2>1. Then α1
++>0 andα1
+−<0. Due to U2= 0:U(α) =U1α1
+and hence U(α+)>0 andU(α−)<0 forU1>0 (w1<1) andU(α+)<0
+andU(α−)>0 forU1<0 (w1<1).
+It follows from (3.27) and (3.28) that
+α0=α+, α ∞=α−. (4.2)
+forU1>0 and
+α0=α−, α ∞=α+. (4.3)
+forU1<0.
+Relation Uici=U1c1= 0 implies c1= 0. Here c0=d2c2. Due to (3.21)
+and (3.22) we should put c2<0 forU1>0 andc2>0 forU1<0. In this
+case the sets α±given by (3.18) are coinciding with those given by (4.1).
+This may be also verified by straightforward calculations using the fo llowing
+relations
+U1=(d2−1)U1
+d1(D−2), U2=U1
+(2−D)= (U,UΛ), (4.4)
+K=U1U1, C=Kd2(d2−1)c2
+2, (4.5)
+13whereD=d1+d2+1>3.
+Accelerated expansion of 3-dimensional factor-space. After re-
+placingτ→τ0−0, where τ0is constant, we get for w=−1 two asymp-
+totical Kasner type metrics
+gas=−dτ⊗dτ+2/summationdisplay
+i=1A2
+i(τ0−τ)2αig(i), (4.6)
+where either αi=αi
+0(Ai=Ai,0>0 ) asτ→τ0−0, orαi=αi
+∞
+(Ai=Ai,∞>0 ) asτ→ −∞.
+LetM1be a flat 3-dimensional factor space ( d1= 3), with the metric
+g(1)=dy1⊗dy1+dy2⊗dy2+dy3⊗dy3. Then, due to relations (4.2), (4.3)
+andα∞<0 ford2>1, we get an asymptotical accelerated expansion of
+our 3-dimensional factor space M1either as τ→τ0−0 forU1<0,c2>0
+or asτ→ −∞forU1>0 andc2<0.
+Milne-type asymptotics. Now we put d1= 1. We get
+α+= (1,0), α −=1
+1+d2(1−d2,2). (4.7)
+ForM1=R,g(1)=−wdy1⊗dy1,−∞< yi<+∞, we get a Milne-type
+(flat) asymptotic:
+i) asτ→+0 forU1>0 andc2<0;
+ii) asτ→+∞forU1<0 andc2>0.
+Both cases correspond to u→+∞.
+ForM1=S1,g(1)=wdy1⊗dy1, 0< yi<+2π, we may get either
+non-singular (static) solution in the case i) ( τ=ρ) or asymptotically flat
+(static) solution in the case ii).
+5 Conclusions and discussions
+In this paper we have considered the exact cosmological type solut ion with 1-
+componentanisotropicfluid. Thissolutionisdefinedontheproductm anifold
+(2.2) containing nRicci-flat factor spaces M1,...,M n.
+We have singled out a special solution governed by the coshmoduli
+function and shown that this solution has Kasner-like asymptotics in the
+limitsu→ ±∞, whereuis the harmonic variable, or, equivalently, in the
+limitsτ→+0 and τ→+∞, whereτis the synchronous type variable.
+14We have found a relation between two sets of Kasner parameters α∞
+andα0. The relation between them α∞=S(α0) is coinciding with the
+“scattering law” formula obtained for the S-brane solution from [1] when
+scalar fields are absent and the fluid U-vector is equal to the brane one.
+The function S(defined on the set of Kasner vectors obeying U(α)>0)
+is bijective. The inverse function S−1(defined on the set of Kasner vectors
+obeying U(α)<0) is given by the same formula as the function S. The
+function Sdepends upon the co-vector U= (Ui). It is invariant upon
+the replacement: U/ma√sto→λU, where λ >0 (see [26]). The transformation
+U/ma√sto→ −Uimplies the replacement S/ma√sto→S−1.
+We have also analyzed the geometric meaning of the scattering law fo r-
+mula in terms of transformation of the Kasner sphere Sn−2,n≥2. For
+(U,UΛ)/ne}ationslash= 0 (or, equivalently, when/summationtextn
+i=1Ui/ne}ationslash= 0) we get just a modified
+inversion with respect to a point vlocated outside the Kasner sphere Sn−2,
+while for ( U,UΛ) = 0 (or, equivalently, when/summationtextn
+i=1Ui/ne}ationslash= 0) we are led to a
+reflection with respect to a hyperplane which contains a center of t he Kasner
+sphere.
+The scattering law formula may be applied for the solutions with Kasne r-
+like asymptotical behaviours (written in a slightly different form)
+gas=−dτ⊗dτ+n/summationdisplay
+i=1A2
+i(τ0−τ)2αig(i), (5.1)
+where either τ→τ0−0, orτ→ −∞. In this case the metric (5.1) may
+describe an asymptotical accelerated expansion of flat 3-dimensio nal factor
+spaceM1ifd1= 3,g(1)=dy1⊗dy1+dy2⊗dy2+dy3⊗dy3andα1<0.
+Another application of the scattering law formula appears when d1= 1
+and one of the asymptotical Kasner set of parameters in (1.1) is of Milne
+type:α= (1,0,...,0), e.g. when static non-singular solutions ( w= +1,
+M1=S1) or cosmological solutions ( w=−1,M1=R) with a horizon (for
+τ→+0) are considered. (Compare with flux-brane and S-brane solutions
+[31, 32]). These topics (mentioned above) may be a subject of sepa rate
+publications.
+15Appendix
+A Solution for Liouville system
+Let
+L=1
+2<˙x,˙x >−Aexp[2< b,x >] (A.1)
+be a Lagrangian, defined on V×V, whereV=Rn,A/ne}ationslash= 0, and <·,·>
+is non-degenerate real-valued quadratic form on V. (Here ˙x=dx
+dtetc.)
+Let< b,b >/ne}ationslash= 0. Then, the Euler-Lagrange equations for the Lagrangian
+(A.1)
+¨x+2Abexp[2< b,x >] = 0 (A.2)
+have the following solution [13]
+x(t) =−b
+< b,b >ln|f(t−t0)|+tα+β, (A.3)
+whereα,β∈V,
+< α,b >=< β,b >= 0, (A.4)
+and
+f(τ) =Rsinh(/radicalbig
+Cτ), C > 0, < b,b > A < 0,
+Rsin(/radicalbig
+|C|τ), C < 0, < b,b > A < 0,
+Rcosh(/radicalbig
+Cτ), C > 0, < b,b > A > 0,
+|2A < b,b > |1/2τ, C= 0, < b,b > A < 0,(A.5)
+whereR=|2A<b,b>
+C|1/2andC,t0are constants.
+The energy
+E=1
+2<˙x,˙x >+Aexp[2< b,x >] (A.6)
+calculated for the solution (A.2) reads
+E=C
+2< b,b >+1
+2< α,α > . (A.7)
+16B Lagrange representation
+The Einstein equations (2.1) imply the conservation law
+∇MTM
+N= 0. (B.8)
+that due to relations (2.3) and (2.4) may be written in the following for m
+˙ˆρ+n/summationdisplay
+i=1di˙βi(ˆρ+ ˆpi) = 0. (B.9)
+Using the equation of state (2.5) we get
+κ2ˆρ=−wAe2Uiβi−2γ0, (B.10)
+whereγ0(β) =/summationtextn
+i=1diβi, andAis constant.
+The Einstein equations (2.1) with the relations (2.5) and (B.10) impose d
+are equivalent to the Lagrange equations for the Lagrangian (for w=−1
+see [14])
+L=1
+2e−γ+γ0(β)Gij˙βi˙βj−eγ−γ0(β)V, (B.11)
+where
+V=Ae2Uiβi, (B.12)
+is the potential and the components of the minisupermetric Gijare de-
+fined in (2.8).
+Forγ=γ0(β), i.e. when the harmonic time gauge is considered, we get
+the set of Lagrange equations for the Lagrangian
+L=1
+2Gij˙βi˙βj−V, (B.13)
+with the zero-energy constraint imposed
+E=1
+2Gij˙βiβj+V= 0. (B.14)
+17C The solution
+Theexact solutions fortheLagrangian(B.13)withthepotential (B .12)could
+be readily obtained using the relations from Appendices AandB.
+The solutions read:
+βi(u) =−Ui
+(U,U)ln|f(u)|+ciu+¯ci, (C.15)
+whereu0is constant. Function f(u) in (C.15) is the following:
+function reads
+f(u) =Rsinh(√
+C(u−u0)), C >0, KA <0; (C.16)
+Rsin(/radicalbig
+|C|(u−u0)), C <0, KA <0; (C.17)
+Rcosh(√
+C(u−u0)), C >0, KA >0; (C.18)
+|2AK|1/2(u−u0), C= 0, KA <0, (C.19)
+whereK= (U,U),R=|2AK/C|1/2andC,u0are constants.
+Vectorsc= (ci) and ¯c= (¯ci) satisfy the linear constraint relations (see
+(A.4) in Appendix A)
+U(c) =Uici= 0, (C.20)
+U(¯c) =Ui¯ci= 0. (C.21)
+The zero-energy constraint reads (see (A.6) in Appendix A)
+E=C
+2(U,U)+1
+2Gijcicj= 0. (C.22)
+D Proof of the inequality (3.11)
+Let us prove the inequality (3.11)
+|(s,UΛ)|>|(UΛ,U)|
+(U,U)>0,
+for a vector s= (sA)∈Rnobeying relations ( s,U) = 0, ( s,s) =
+−1/(U,U). Herethescalar-product ( U,U′) =GijUiU′
+j,whereGij=δijd−1
+i+
+(2−D)−1. We also use here the following relations ( U,U)>0,UΛ= (di)
+and (UΛ,UΛ)<0.
+18Proof.Let us define the vector
+U1=U−(U,UΛ)
+(UΛ,UΛ)UΛ. (D.23)
+It is clear that ( U1,UΛ) = 0 and
+(U1,U1) = (U,U)−(U,UΛ)2
+(UΛ,UΛ)>0. (D.24)
+since (U,U)>0 and (UΛ,UΛ)<0. Let us define vectors:
+s0=(s,UΛ)
+(UΛ,UΛ)UΛ, (D.25)
+s1=(s,U1)
+(U1,U1)U1, (D.26)
+s=s−s0−s1. (D.27)
+s0,s1ands2are mutually orthogonal and hence
+(s,s) = (s0,s0)+(s1,s1)+(s2,s2). (D.28)
+For the first two terms in r.h.s. of (D.28) we get
+(s0,s0) =(s,UΛ)2
+(UΛ,UΛ),(D.29)
+(s1,s1) =(s,U1)2
+(U1,U1)=(s,UΛ)2
+(UΛ,UΛ)(U,UΛ)2
+[(U,U)(UΛ,UΛ)−(U,UΛ)2](D.30)
+that implies
+(s,s) =(s,UΛ)2(U,U)
+(U,U)(UΛ,UΛ)−(U,UΛ)2+(s2,s2). (D.31)
+For the third term in r.h.s. of (D.28) the following inequality is valid
+(s2,s2)≥0, (D.32)
+Indeed, due to ( s2,UΛ) = 0, or, equivalently,/summationtextn
+i=1si
+2di= 0 , we obtain
+(s2,s2) =Gijsi
+2sj
+2=n/summationdisplay
+i=1(si
+2)2di≥0. (D.33)
+19Using this inequality, (D.31), ( UΛ,UΛ)<0 and (s,s) =−1/(U,U) we get
+(s,UΛ)2= [(U,UΛ)2
+(U,U)−(UΛ,UΛ)][(U,U)−1+(s2,s2)]>(U,UΛ)2
+(U,U)2>0,(D.34)
+that is equivalent to the inequality (3.11). Thus, (3.11) is proved.
+Acknowledgments
+This work was supported in part by the Russian Foundation for Basic
+Research grants Nr. 09-02-00677-a.
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+23
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+arXiv:1001.0129v2  [math.RT]  31 Jul 2010Repr´ esentation de Weil et β-extensions
+Corinne Blondel
+8 juillet 2010
+R´ esum´ e
+Nous ´ etudions les β-extensions dans un groupe classique p-adique et obtenons une
+relation entre certaines β-extensions ` a l’aide d’une repr´ esentation de Weil. Nous e n
+donnons une application ` a l’´ etude des points de r´ eductib ilit´ e de certaines induites
+paraboliques.
+Mathematics Subject Classification (2000): 22E50
+Introduction
+SoitFun corps local non archim´ edien de caract´ eristique r´ esiduelle impa ire. Soit G0un
+groupe symplectique, sp´ ecial orthogonal ou unitaire sur F(sur le corps des points fixes
+d’une involution sur Fdans le cas unitaire) et, pour n∈N, soitGnle groupe classique de
+mˆ eme nature que G0ayant un sous-groupe parabolique maximal Pnde facteur de Levi Mn
+isomorphe ` a GL( n,F)×G0. Soitσune repr´ esentation supercuspidale irr´ eductible de G0et
+πune repr´ esentation supercuspidale irr´ eductible de GL( n,F).
+L’´ etude des points de r´ eductibilit´ e de la repr´ esentation induite indGn
+Pnπ|det|s⊗σ, o` utradi-
+tionnellement le param` etre sest r´ eel, joue un rˆ ole important en th´ eorie des repr´ esentatio ns,
+aussi bien, depuis plusde trente ans, dansla classification des repr ´ esentations lisses irr´ educti-
+bles deGnet la d´ etermination de son dual unitaire, que plus r´ ecemment dans l’´ etude de
+la correspondance de Langlands. La repr´ esentation induite ci-de ssus, normalis´ ee, est tou-
+jours irr´ eductible si πn’est pas autoduale (au sens appropri´ e du paragraphe 3.2) ; si π
+est autoduale, dans presque tous les cas ses points de r´ eductibilit ´ e sont soit s= 0, soit
+s=±1
+2. D’apr` es les travaux de Mœglin, l’ensemble Red( σ) des paires ( π,s), form´ ees d’une
+repr´ esentation supercuspidale autodualeirr´ eductible πd’ungroupeGL( n,F) et d’unnombre
+r´ eels≥1, telles que la repr´ esentation induite indGn
+Pnπ|det|s⊗σsoit r´ eductible d´ etermine
+leL-paquet auquel appartient σet son image dans la correspondance de Langlands ; cet
+ensemble est fini et sa taille est connue.
+Orlath´ eoriedestypesetpairescouvrantes est unoutilpuissant d’´ etudedetellesinduites,
+qui permet de traiter le cas, ´ echappant aux m´ ethodes tradition nelles, des repr´ esentations non
+g´ en´ eriques, et qui surtout devrait aboutir ` a une description e xplicite de Red( σ), donc du L-
+paquet de σet de son image dans la correspondance de Langlands, en fonction d u type (ou
+12 Corinne Blondel
+d’un type) attach´ e ` a σ. Ce programme, initi´ e dans l’article fondateur [6], est techniqueme nt
+assez difficile puisque pour obtenir Red( σ) il faut partir d’un type ( Jσ,λσ) pourσ, d’un
+type (˜Jπ,˜λπ) pour une repr´ esentation supercuspidale autoduale irr´ eductib leπd’un groupe
+GL(n,F) et d´ eterminer :
+(i) une paire couvrante ( J,λ) de (˜Jπ×Jσ,˜λπ⊗λσ) dansGn;
+(ii) la structure de l’alg` ebre de Hecke associ´ ee H(Gn,λ) ;
+(iii) les points de r´ eductibilit´ e complexes de l’induite ci-dessus via des ´ equivalences de
+cat´ egories transformant l’induction parabolique en induction des H(Mn,˜λπ⊗λσ))-
+modules enH(Gn,λ)-modules (voir le paragraphe 3.2, essentiellement ind´ ependant du
+reste de l’article). Un type pour πest insensible ` a la torsion par un caract` ere non
+ramifi´ e, c’est pourquoi le param` etre svarie ici dans C; cela revient ` a ´ etudier ensemble
+deux repr´ esentations induites ` a param` etre r´ eel ( §3.2).
+La faisabilit´ e de ce programme est attest´ ee par [1] qui le m` ene ` a son terme pour les sous-
+groupes de Levi GL(1 ,F)×Sp(2,F) et GL(2 ,F) de Sp(4 ,F). Dans un travail en cours avec
+Guy Henniart et Shaun Stevens, nous le mettons en œuvre pour d´ e terminer explicitement
+lesL-paquets de Sp(4 ,F) en termes de types, ` a partir de la liste des types de repr´ esenta tions
+supercuspidales g´ en´ eriques et non g´ en´ eriques de Sp(4 ,F) ´ etablie dans [4] (Gan et Takeda
+d´ ecrivent ces L-paquets dans [8] par une m´ ethode diff´ erente ne permettant pa s une telle
+explicitation).
+Dans un groupe classique plus g´ en´ eral et pour une repr´ esenta tion induite de la forme ci-
+dessus, laconstructiond’unepairecouvrante(i)a´ et´ eeffectu´ ee danscertainscasparticuliers :
+le sous-groupe de Levi de Siegel d’un groupe symplectique dans [2], le sous-groupe de Levi
+de Siegel d’un groupe classique dans [10] o` u la structure th´ eoriqu e de l’alg` ebre de Hecke
+(ii) est d´ etermin´ ee, une repr´ esentation supercuspidale autod uale de niveau z´ ero du Levi de
+Siegel d’un groupe symplectique ou sp´ ecial orthogonal dans [13] q ui donne en outre l’´ etude
+de r´ eductibilit´ e (iii). Plus g´ en´ eralement, des paires couvrantes accompagn´ ees de la structure
+th´ eorique de l’alg` ebre de Hecke associ´ ee sont fournies par [21], sous des hypoth` eses assez
+larges, en particulier celles du paragraphe 3.1 ci-dessous, que nous supposerons v´ erifi´ ees
+dans la suite de cette introduction : il s’agit essentiellement de deman der queπetσsoient
+attach´ ees ` a des strates suffisamment “disjointes” du point de vue de la th´ eorie des types et
+` a des caract` eres semi-simples compatibles (cf. remarque 3.18).
+Dans presque tous les cas qui nous int´ eressent ici, la structure t h´ eorique de l’alg` ebre
+de HeckeH(Gn,λ) obtenue est la mˆ eme : il s’agit d’une alg` ebre de convolution ` a deux
+g´ en´ erateurs T0etT1sur un groupe de Weyl affine qui ne d´ epend que de π. Le g´ en´ erateur
+Ti,i= 0,1, est l’image par une injection d’alg` ebres du g´ en´ erateur d’une alg ` ebre de Hecke
+Li=H(Gi,ρi) de dimension 2, l’alg` ebre d’entrelacements de l’induite parabolique d’u ne
+repr´ esentation cuspidale d’un sous-groupe de Levi maximal Midans un groupe r´ eductif fini.
+La connaissance de Gietρidoit permettre de calculer la relation quadratique satisfaite par
+Ti` a l’aide des travaux de Lusztig (voir [13]). Ces deux relations quadra tiques suffisent ` a
+d´ eterminerlespartiesr´ eellesdespointsder´ eductibilit´ edelarep r´ esentationinduiteconsid´ er´ eeRepr´ esentation de Weil et β-extensions 3
+(§3.2) et donc ` a pr´ edire s’il s’agit d’une situation “ordinaire” avec r´ e ductibilit´ es de parties
+r´ eelles 0 ou±1
+2, ou d’une situation fournissant un ´ el´ ement de l’ensemble Red( σ) cherch´ e.
+Mais le diable est dans les d´ etails : ρiest “la partie de niveau z´ ero du type λ”, notion
+` a laquelle on ne peut donner un sens que “relativement ` a Gi” (en prenant un peu de libert´ e
+avec le vocabulaire). En effet ρiest d´ etermin´ ee par une ´ ecriture du type λsous la forme
+κi⊗ρio` uκiest uneβ-extension relativement ` a Gi[21]. Il en r´ esulte que ρin’est a priori
+connue qu’` a un caract` ere (´ eventuellement muni de propri´ et ´ es suppl´ ementaires) pr` es et la
+structure deLipeut en d´ ependre (voir les exemples 2.36 et 3.16 et le paragraphe 3.4 ). C’est
+pourquoi la r´ ealisation du programme ci-dessus doit passer par un e analyse approfondie de
+la notion de β-extension, dont le pr´ esent article constitue une ´ etape.
+Une description plus pr´ ecise des alg` ebres Liqui, redisons-le, d´ eterminent les parties
+r´ eelles des points de r´ eductibilit´ e de indGn
+Pnπ|det|s⊗σ, fait apparaˆ ıtre que leur d´ ependance
+enσr´ eside exclusivement dans la d´ etermination de ρi, c’est-` a-dire dans la d´ etermination des
+β-extensions attach´ ees ` a la situation, l’effet de σ´ etant simplement de modifier le choix de
+ρipar torsion par un caract` ere. Il est naturel dans ces condition s de comparer les parties
+r´ eelles des points de r´ eductibilit´ e des induites indGn
+Pnπ|det|s⊗σet indHn
+Qnπ|det|s, o` uHnest
+le groupe classique de mˆ eme nature que G0ayant un sous-groupe parabolique maximal Qn
+de facteur de Levi isomorphe ` a GL( n,F). Comme on vient de le voir, cette comparaison se
+ram` ene ` a une comparaison entre repr´ esentations cuspidales d ’un mˆ eme groupe fini pouvant
+diff´ erer d’une torsion par un caract` ere.
+C’est la d´ etermination de ce caract` ere, intrins` equement li´ e ` a la notion de β-extension, qui
+est l’enjeu du travail qui suit. Il s’agit en r´ ealit´ e de comparer β-extensions dans le groupe Gn
+etβ-extensions dans son sous-groupe Hn×G0. La solution est donn´ ee dans le th´ eor` eme 2.35,
+r´ esultat principal de l’article. Elle est assez simple (si l’on omet d’´ en oncer les hypoth` eses et
+notations) : les β-extensions consid´ er´ ees dans Gnet dansHn×G0diff` erent d’un caract` ere
+qui est la signature d’une permutation explicite, l’action par conjuga ison sur un groupe fini
+attach´ e ` a la situation.
+Revenons pour terminer ` a la motivation initiale : la description explicite de Red(σ).
+On s’attend ` a ce que les repr´ esentations autoduales πintervenant dans Red( σ) pr´ esentent
+une forte affinit´ e, du point de vue des types, avec σ(comme dans l’exemple final de [3] o` u
+la paire couvrante relative ` a π⊗σest attach´ ee ` a une strate simpleet o` u l’on obtient un
+point de r´ eductibilit´ e de partie r´ eelle 1). Or nous nous sommes plac ´ es dans une situation o` u
+au contraire, les strates sous-jacentes ` a πetσsont suppos´ ees “disjointes” (hypoth` eses du
+paragraphe 3.1, cf remarque 3.18). De fait c’est dans le cas o` u πest un caract` ere autodual
+de GL(1,F) et o` u la strate semi-simple sous-jacente ` a σn’a pas de composante nulle que nos
+r´ esultats trouvent leur premi` ere application. Dans le cas symple ctique, chaque alg` ebre Li
+est l’alg` ebre d’entrelacement d’un caract` ere quadratique ou triv ial du sous-groupe de Borel
+de SL(2,kF) et l’on sait ` a quel point sa structure d´ epend de ce caract` ere ( §3.4) : la signature
+de la permutation du th´ eor` eme 2.35 d´ etermine ici la ramification du caract` ere autodual π
+figurant dans Red( σ).
+L’article est divis´ e en trois parties : mise en place des propri´ et´ es desβ-extensions ; cons-
+truction d’une repr´ esentation de Weil permettant d’obtenir le car act` ere cherch´ e ; application
+` a l’´ etude de r´ eductibilit´ e.4 Corinne Blondel
+La premi` ere partie est technique, centr´ ee sur les notions fond amentales de bijection cano-
+nique et de β-extension d´ efinies par Stevens dans [21]. Apr` es avoir rappel´ e les d´ efinitions
+essentielles (§1.1), on donne au paragraphe 1.2 une caract´ erisation de la bijectio n canonique
+entermesd’entrelacement. Leparagraphe1.3est consacr´ e` ala miseenplaced’unfoncteurde
+restriction de Jacquet rPdans une situation l´ eg` erement plus g´ en´ erale que celle de [21] (d ont
+il s’inspire largement, voir loc. cit.§5.3 et 6.1), o` u le sous-groupe parabolique Pest attach´ e
+` a une d´ ecomposition qui est subordonn´ ee ` a la strate consid´ e r´ ee, sans lui ˆ etre proprement
+subordonn´ ee. Le r´ esultat essentiel (proposition 1.20) est d´ e montr´ e au paragraphe 1.4 : il
+s’agit de la compatibilit´ e entre le foncteur rPet la bijection canonique sur laquelle repose la
+notion de β-extension.
+La deuxi` eme partie, que l’on d´ ecrit ici dans le contexte simplifi´ e ci- dessus, part d’un
+caract` ere semi-simple θdansGnconvenablement d´ ecompos´ e par rapport au sous-groupe
+Hn×G0etrelielesrepr´ esentationsassoci´ ees ηetκdansGn` adesobjetsanalogues η′etκ′dans
+Hn×G0(§2.1). La relation obtenue est pr´ ecis´ ee au paragraphe 2.2 ` a l’aide d ’un foncteur de
+restriction de Jacquet relatif ` a un sous-groupe parabolique de Le vi GL(n,F)×G0. On cons-
+truit alors une repr´ esentation “de Weil” Wtelle que la famille finie des repr´ esentations κet
+celle des κ′soient reli´ ees par κ≃W⊗κ′(§2.3). Grˆ ace aux propri´ et´ es de cette repr´ esentation
+de Weil, on montre, lorsque le caract` ere θest attach´ e ` a un ordre autodual maximal, que
+la repr´ esentation κest uneβ-extension si et seulement si κ′en est une (§2.4). L’usage de
+la compatibilit´ e entre restriction de Jacquet et bijection canonique montr´ ee en 1.4 aboutit
+au th´ eor` eme 2.35, qui d´ ecrit le caract` ere par lequel diff` eren t lesβ-extensions dans Gnet
+Hn×G0.
+La troisi` eme partie se rapproche de la motivation initiale en expliquan t comment les
+r´ esultats obtenus s’appliquent ` a l’´ etude de r´ eductibilit´ e. Nou s commen¸ cons par r´ esumer les
+r´ esultats de Stevens [21] dans le contexte simplifi´ e d’une d´ ecomp osition autoduale en trois
+morceaux (§3.1, ind´ ependant des deux premi` eres parties). On y voit (corolla ire 3.7) que la
+notion de β-extension est cruciale pour trouver les g´ en´ erateurs de l’alg` eb re de Hecke de la
+pairecouvrante consid´ er´ ee. Nous d´ etaillonsensuite lelien entre les relations quadratiques sa-
+tisfaitesparlesdeuxg´ en´ erateursd’unealg` ebredeHeckecorr espondant` anotresituationetles
+points de r´ eductibilit´ e des repr´ esentations induites associ´ ee s (§3.2, proposition 3.12). Il reste
+` atirerquelquesconclusions: lacomparaisondes β-extensionsdans GnetHn×G0faitedansle
+th´ eor` eme 2.35 permet de comparer les alg` ebres de Hecke assoc i´ ees, on passe des g´ en´ erateurs
+de l’une ` a ceux de l’autre par torsion par des caract` eres donn´ es par ce th´ eor` eme ( §3.3,
+proposition 3.17). Nous terminons par des exemples dans un groupe symplectique (§3.4),
+illustrant la relation entre la notion essentielle de β-extension et les points de r´ eductibilit´ e
+d’induites paraboliques.
+Remerciements. J’aimerais remercier Shaun Stevens pour m’avoir expliqu´ e certaines subtilit´ es
+des caract` eres semi-simples et β-extensions et pour ses commentaires d’une premi` ere versi on de ce
+manuscrit, et Laure Blasco, Guy Henniart, Colette Mœglin et Shaun Stevens pour des discussions
+tr` es stimulantes ` a diff´ erents stades de ce travail.Repr´ esentation de Weil et β-extensions 5
+Notations
+On travaille dans cet article sur les objets d´ efinis par Shaun Steven s dans [21] et des articles
+ant´ erieurs, et qui trouvent leur origine dans [5] et [7]. On respect e autant que possible les
+notations de [21], en les simplifiant parfois comme indiqu´ e ci-dessous.
+SoitFun corps local non-archim´ edien de caract´ eristique r´ esiduelle imp airepmuni d’un
+automorphisme not´ e x/mapsto→¯xd’ordre 1 ou 2 et de points fixes F0. On note kFson corps
+r´ esiduel, de cardinal qF=q. On consid` ere le groupe classique G+des automorphismes d’un
+espace vectoriel Vdedimension finiesur Fconservant une forme ǫ-hermitienne (relativement
+` a l’automorphisme x/mapsto→¯x, avecǫ=±1) non d´ eg´ en´ er´ ee hsurV. Dans le cas orthogonal on
+d´ efinitGcomme le groupe sp´ ecial orthogonal des ´ el´ ements de G+de d´ eterminant 1, dans
+les autres cas on pose G=G+. On note aussi g/mapsto→¯gl’involution adjointe sur End F(V)
+associ´ ee ` a h, de sorte que G+={g∈GLF(V)/¯g=g−1}. En g´ en´ eral on notera avec des
+tildes les objets relatifs au groupe lin´ eaire ˜G= GLF(V), sans tilde ceux relatifs ` a G, et avec
+un exposant + ceux relatifs ` a G+(dont les p-sous-groupes sont contenus dans G).
+SoitMunsous-groupedeLevide Getσunerepr´ esentationsupercuspidaleirr´ eductiblede
+M; on noteR[σ,M](G) la sous-cat´ egorie pleine des repr´ esentations lisses complexes d eGdont
+chaque sous-quotient irr´ eductible est sous-quotient d’une repr ´ esentation induite parabolique
+d’une repr´ esentation de Mtordue de σpar un caract` ere non ramifi´ e.
+SoitJun sous-groupe compact ouvert de Getλune repr´ esentation lisse irr´ eductible de J
+d’espace E, on noteH(G,λ) l’alg` ebre de Hecke de λdansG. C’est l’alg` ebre de convolution
+(relativement ` a la mesure de Haar sur Gdonnant ` a Jle volume 1) des fonctions lisses ` a
+support compact f:G−→EndC(E) v´ erifiant :
+∀x,y∈J,∀g∈G,f(xgy) =λ(x)f(g)λ(y).
+On note Mod-H(G,λ) la cat´ egorie des modules ` a droite unitaux sur cette alg` ebre.
+L’objet de base de cet article est une strate gauche semi-simple [Λ,n,0,β] de End F(V)
+[21, Definition 2.5] dont on note V=V1⊥···⊥ Vlla d´ ecomposition orthogonale de V
+associ´ ee. Rappelons l’essentiel :
+•βest un ´ el´ ement de End F(V) v´ erifiant ¯β=−βet somme de ses restrictions βi` aVi,
+´ el´ ements de End F(Vi) tels que F[βi] =Eisoit un corps pour 1 ≤i≤l. De plus
+E=F[β] est la somme E=E1⊕···⊕Elet le centralisateur Bdeβdans End F(V)
+est la somme des commutants Bideβidans End F(Vi), 1≤i≤l. On note ˜GE=B×,
+G+
+E=B×∩G+etGE=B×∩G.
+•Λ est une suite autoduale de oF-r´ eseaux de Vv´ erifiant pour tout r∈Z: Λ(r) =
+⊕l
+i=1Λ(r)∩Vi, et la suite de r´ eseaux ΛideVid´ efinie par Λi(r) = Λ(r)∩Viest une
+suite autoduale de oEi-r´ eseaux, 1≤i≤l. On note a0(Λ) le fixateur de la suite Λ
+dans End F(V), muni d’une filtration par les an(Λ),n≥1 entier, form´ es des ´ el´ ements
+d´ ecalant la suite de n; de mˆ eme pour b0(Λ) =a0(Λ)∩Betbn(Λ) =an(Λ)∩B. Onnote
+enfin :/tildewideP(Λ) =a0(Λ)×,P+(Λ) =/tildewideP(Λ)∩G+,P(Λ) =/tildewideP(Λ)∩G,P+(ΛoE) =P+(Λ)∩B,
+P(ΛoE) =P(Λ)∩B,P1(ΛoE) =P(ΛoE)∩(1+b1(Λ)) etP0(ΛoE) l’image inverse dans
+P(ΛoE) de la composante neutre du quotient P(ΛoE)/P1(ΛoE) [21,§2.1].6 Corinne Blondel
+Les groupes ˜H1(β,Λ),˜J1(β,Λ),˜J(β,Λ) d´ esignent les sous-groupes de ˜Grelatifs ` a une strate
+semi-simple d´ efinis dans[20, §3](apr` es[5]et [7]). Pour unestratesemi-simple gauche onnote
+H1(β,Λ),J1(β,Λ),J(β,Λ) leurs intersections avec G, et on note J+(β,Λ) =˜J(β,Λ)∩G+.
+On pose enfin J0(β,Λ) =P0(ΛoE)J1(β,Λ) – rappelons que J(β,Λ) =P(ΛoE)J1(β,Λ). On
+raccourcira en g´ en´ eral H1(β,Λ) etc. en H1(Λ) etc., puisque la plupart du temps, dans les
+strates gauches semi-simples [Λ ,n,0,β] que l’on consid´ erera, l’´ el´ ement βsera fix´ e. De mˆ eme
+on utilisera de fa¸ con abusive la notation oE, comme dans oE-r´ eseaux ou P+(ΛoE) [21,§2],
+en omettant les indices ou exposants auxquels le Epourrait pr´ etendre, de fa¸ con ` a ´ eviter
+d’´ ecrire “ oE(0)-r´ eseaux” ou “ P+(ΛoE(j))”.
+Dans toute la suite on d´ esigne par [Λ ,n,0,β], [Λ′,n′,0,β], [Λm,nm,0,β], [ΛM,nM,0,β]
+des strates gauches semi-simples ayant en commun l’´ el´ ement β, donc aussi la d´ ecomposition
+deVenV=V1⊥···⊥Vl, et telles que
+•b0(Λm)⊆b0(Λ) ;b0(Λ) =b0(Λ′) ;b0(Λ)⊆b0(ΛM) ;
+•b0(ΛM) est unoE-ordre autodual maximal.
+On fixe un caract` ere semi-simple gauche θdeH1(Λ) [20,§3.6] et on note θ′,θm,θMles
+caract` eres de H1(Λ′),H1(Λm) etH1(ΛM) qui lui correspondent par transfert [21, Proposition
+3.2]. Onnote η(resp.η′,ηm,ηM)l’uniquerepr´ esentation irr´ eductible de J1(Λ)(resp. J1(Λ′),
+J1(Λm),J1(ΛM)) contenant θ(resp.θ′,θm,θM).
+Si[Λ⋆,n⋆,0,β],o` u⋆d´ esignen’importequel symbole, estuneautretellestrate, onutilis era
+de la mˆ eme fa¸ con les notations θ⋆,η⋆etc., ou bien aussi θ(Λ⋆),η(Λ⋆) etc.
+1 Quelques propri´ et´ es des β-extensions
+Pour la commodit´ e de la r´ edaction on rappelle avant de commencer u n fait bien connu et
+´ el´ ementaire mais tr` es utile. Soit HetH′des sous-groupes ouverts compacts d’un groupe
+l.c.t.d.Ketρetρ′des repr´ esentations de dimension finie de HetH′respectivement. Alors
+HomH∩H′(ρ,ρ′)/ne}ationslash={0}=⇒HomK(IndK
+Hρ,IndK
+H′ρ′)/ne}ationslash={0}. (1.1)
+1.1 Faits essentiels
+Rappelons les r´ esultats suivants de [21], qui jouent un rˆ ole crucia l dans la suite.
+Proposition 1.2 ([21] Proposition 3.7). Il existe une unique repr´ esentation irr´ eductible
+η(Λm,Λ)deJ1(Λm,Λ) =P1(Λm
+oE)J1(Λ)v´ erifiant :
+(i)η(Λm,Λ)|J1(Λ)=η;
+(ii)Pour toute suite de oE-r´ eseaux Λ′′v´ erifiant b0(Λ′′) =b0(Λm)eta0(Λ′′)⊆a0(Λ)on a
+IndP1(Λ′′)
+J1(Λ′′)η′′≃IndP1(Λ′′)
+P1(ΛmoE)J1(Λ)η(Λm,Λ).Repr´ esentation de Weil et β-extensions 7
+D´ emonstration. Il ne s’agit que d’´ etendre les notations de [21, Proposition 3.7], qui suppose
+a0(Λm)⊆a0(Λ), au cas g´ en´ eral b0(Λm)⊆b0(Λ). Cette extension est implicite dans [21]
+puisque l’auteur d´ efinit η(Λ′′,Λ) (ηm,Mdans les notations de [21]) et remarque que cette
+repr´ esentation et son groupe de d´ efinition P1(Λ′′
+oE)J1(Λ) ne d´ ependent que de b0(Λ′′) et non
+de la suite Λ′′elle-mˆ eme. Il suffit donc ici de d´ efinir η(Λm,Λ) =η(Λ′′,Λ). On rappelle que
+des suites Λ′′v´ erifiant les conditions de l’´ enonc´ e existent toujours [21, Lemm e 2.8]. /squaresolid
+Proposition 1.3 ([21] Lemma 4.3). Il y a une bijection canonique BΛmΛentre l’ensemble
+des prolongements κmdeηm` aJ+(Λm)et l’ensemble des prolongements κdeη(Λm,Λ)` a
+P+(Λm
+oE)J1(Λ). Sia0(Λm)⊆a0(Λ)cette bijection associe ` a κml’unique repr´ esentation κde
+P+(Λm
+oE)J1(Λ)telle que κetκminduisent des repr´ esentations (irr´ eductibles) ´ equiva lentes de
+P+(Λm
+oE)P1(Λm).
+On notera parfois simplement Bla bijection canonique d´ efinie ci-dessus. C’est cette
+bijection qui permet de d´ efinir la notion de β-extension dans [21] :
+D´ efinition 1.4 ([21] Theorem 4.1, Definition 4.5). (i) Cas maximal. Une repr´ esen-
+tationκMdeJ+(ΛM)est uneβ-extension deηMsiκMest un prolongement de
+η(Λm,ΛM), pour une suite autoduale ΛmdeoE-r´ eseaux telle que b0(Λm)soit unoE-
+ordre autodual minimal contenu dans a0(ΛM). La condition est alors v´ erifi´ ee pour toute
+telle suite Λm. Il existe des β-extensions et deux d’entre elles diff` erent d’un caract` er e
+deP+(ΛM
+oE)/P1(ΛM
+oE)trivial sur les sous-groupes unipotents.
+(ii) Cas g´ en´ eral. SoitκMuneβ-extension de ηM` aJ+(ΛM). Laβ-extension de η` aJ+(Λ)
+relative ` a ΛMet compatible ` a κMest la repr´ esentation κdeJ+(Λ)telle que BΛΛM(κ)
+soit la restriction de κM` aP+(ΛoE)J1(ΛM).
+On dira que κest uneβ-extension de η` aJ+(Λ)si elle en est une relativement ` a une
+suiteΛMconvenable.
+Les faits suivants sont des cons´ equences imm´ ediates, voire tau tologiques, de [21, §4].
+Lemme 1.5. (i)Siκcorrespond ` a κ′par la bijection canonique BΛΛ′, alorsκest uneβ-
+extension de ηrelative ` a ΛM(compatible ` a κM) si et seulement si κ′est uneβ-extension
+deη′relative ` a ΛM(compatible ` a κM).
+(ii)La bijection canonique BΛmΛ, entre prolongements de ηm` aJ+(Λm)et prolongements
+deη(Λm,Λ)` aP+(Λm
+oE)J1(Λ), est compatible ` a la torsion par les caract` eres du groupe
+P+(Λm
+oE)/P1(Λm
+oE).
+D´ emonstration. Le premier fait provient de ce que la compos´ ee de deux bijections ca no-
+niques est une bijection canonique (voir aussi la d´ emonstration de [21, Lemma 4.3]). Le
+second est ´ el´ ementaire. /squaresolid
+1.2 Une propri´ et´ e caract´ eristique de la bijection canon ique
+On d´ emontre dans ce paragraphe la propri´ et´ e suivante :8 Corinne Blondel
+Proposition1.6. Supposonsque a0(Λm)⊆a0(Λ). Soitκmun prolongementde ηm` aJ+(Λm)
+etκun prolongement de η(Λm,Λ)` aP+(Λm
+oE)J1(Λ). Alors
+κ=BΛmΛ(κm)⇐⇒HomJ+(Λm)∩P+(ΛmoE)J1(Λ)(κm,κ)/ne}ationslash={0}
+D´ emonstration. Elle s’appuie sur deux lemmes qu’on ´ etablit d’abord.
+Lemme 1.7. Il existe un vecteur vde l’espace de κmtel que
+/contintegraldisplay
+H1(Λ)∩J+(Λm)θ(x−1)κm(x)vdx/ne}ationslash= 0.
+D´ emonstration. CommeH1(Λ)⊆P1(Λ)⊆P1(Λm), l’int´ egrale calcul´ ee est en fait
+X(v) =/contintegraldisplay
+H1(Λ)∩J1(Λm)θ(x−1)ηm(x)vdx.
+L’application v/mapsto→X(v) est un projecteur de l’espace de ηmsur son sous-espace isotypique
+de typeθsousH1(Λ)∩J1(Λm). Il s’agit donc de montrer que celui-ci est non nul, et comme
+l’induite de θm` aJ1(Λm) est multiple de ηmil suffit d’´ etablir
+HomH1(Λ)∩J1(Λm)(θ,IndJ1(Λm)
+H1(Λm)θm)/ne}ationslash={0}.
+On d´ eveloppe la restriction de IndJ1(Λm)
+H1(Λm)θm` aH1(Λ)∩J1(Λm) selon la formule de Mackey ;
+le premier terme est IndH1(Λ)∩J1(Λm))
+H1(Λ)∩H1(Λm)θm, or par r´ eciprocit´ e de Frobenius on a bien
+HomH1(Λ)∩J1(Λm)(θ,IndH1(Λ)∩J1(Λm))
+H1(Λ)∩H1(Λm)θm)≃HomH1(Λ)∩H1(Λm)(θ,θm)/ne}ationslash={0}
+puisqueθmest le transfert de θ[21, Proposition 3.2]. /squaresolid
+Lemme 1.8. La multiplicit´ e de ηdansIndP+(Λm
+oE)P1(Λm)
+P+(ΛmoE)J1(Λ)κest1.
+D´ emonstration. De nouveau on a J1(Λ)⊆P1(Λ)⊆P1(Λm), il suffit donc de consid´ erer la
+restriction de la repr´ esentation induite ` a P1(Λm), soit Φ = IndP1(Λm)
+P1(ΛmoE)J1(Λ)κ|P1(ΛmoE)J1(Λ), qui
+est irr´ eductible [21, Proposition 3.7]. On a
+HomJ1(Λ)(η,Φ)≃HomP1(Λm)(IndP1(Λm)
+P1(ΛmoE)J1(Λ)IndP1(Λm
+oE)J1(Λ)
+J1(Λ)η,Φ)
+Remarquons que IndP1(Λm
+oE)J1(Λ)
+J1(Λ)η≃η(Λm,Λ)⊗IndP1(Λm
+oE)
+P1(ΛoE)1≃⊕χ∈Zmχη(Λm,Λ)⊗χ, o` uZ
+d´ esigne l’ensemble des classes d’isomorphisme de facteurs irr´ educ tibles de IndP1(Λm
+oE)
+P1(ΛoE)1 et
+mχla multiplicit´ e de χ∈Zdans cette induite.
+Pourχ∈Z,I(χ) = IndP1(Λm)
+P1(ΛmoE)J1(Λ)η(Λm,Λ)⊗χest irr´ eductible car l’entrelacement
+dansP1(Λm) deη(Λm,Λ)⊗χest contenu dans celui de η:J1(Λ)GEJ1(Λ)∩P1(Λm) =
+P1(Λm
+oE)J1(Λ). Ainsi I(χ) et Φ sont entrelac´ ees si et seulement si elles sont isomorphes, s i et
+seulement si χ= 1 par d´ efinition de η(Λm,Λ). Par ailleurs m1vaut 1 par Frobenius. /squaresolidRepr´ esentation de Weil et β-extensions 9
+Montrons maintenant la proposition. Par d´ efinition, les repr´ esen tationsκetκmse corres-
+pondent par la bijection canonique si et seulement si il existe un isom orphismeEentre les
+repr´ esentations induites :
+IndP+(Λm
+oE)P1(Λm)
+P+(ΛmoE)J1(Λ)κE−→IndP+(Λm
+oE)P1(Λm)
+J+(Λm)κm.
+La condition Hom J+(Λm)∩P+(ΛmoE)J1(Λ)(κm,κ)/ne}ationslash={0}est suffisante par1.1 (les induites sont
+irr´ eductibles). Pour montrer qu’elle est n´ ecessaire nous avons b esoin des lemmes ci-dessus.
+On composeEavec l’injection canonique Jdeκdans son induite ` a droite, avec la projection
+canoniquePde l’induite de κmsurκm` a gauche : on obtient un entrelacement P◦E◦J de
+κdansκmdont on va montrer qu’il est non nul.
+D’apr` es le lemme 1.8, l’image de κparJest la composante isotypique de type θde
+l’induite de κ, qui s’envoie injectivement par Esur la composante isotypique de type θde
+l’induite de κm. Il suffit alors de montrer que Pn’est pas identiquement nulle sur cette
+composante.
+L’espacedel’induitede κmpeutsevoircommel’espacedesfonctions fdeP+(Λm
+oE)P1(Λm)
+dans l’espace de κmv´ erifiant f(gx) =κm(g)f(x) pourg∈J+(Λm) et on aP(f) =f(1).
+L’application f/mapsto→fθd´ efinie par fθ(y) =/contintegraltext
+H1(Λ)θ(x−1)f(yx)dxest un projecteur sur la
+composante de type θsousH1(Λ). Prenons fdans l’image canonique de κm, c’est-` a-dire de
+supportJ+(Λm). On a
+P(fθ) =fθ(1) =/contintegraldisplay
+H1(Λ)θ(x−1)f(x)dx=/contintegraldisplay
+H1(Λ)∩J+(Λm)θ(x−1)κm(x)f(1)dx.
+L’existence de f(1) tel que cette int´ egrale soit non nulle est donn´ ee par le lemme 1.7 ./squaresolid
+Corollaire 1.9. Sia0(Λm)⊆a0(Λ), alorsη(Λm,Λ)est l’unique prolongement de η` a
+P1(Λm
+oE)J1(Λ)v´ erifiant
+HomJ1(Λm)∩P1(ΛmoE)J1(Λ)(ηm,η(Λm,Λ))/ne}ationslash={0}.
+1.3 D´ ecompositions subordonn´ ees et restriction de Jacqu et
+Rappelonsles d´ efinitions de[21, §5]. SoitV=⊕k
+j=−kW(j)une d´ ecomposition de Vautoduale
+– c’est-` a-dire que l’orthogonal dans VdeW(j), pour−k≤j≤k, est⊕s/negationslash=−jW(s)– telle
+que, pour tout j,−k≤j≤k:
+•W(j)=⊕l
+i=1W(j)∩Vi;
+•W(j)∩Viest unEi-sous-espace de Vipour tout i, 1≤i≤l.
+Une telle d´ ecomposition est subordonn´ ee ` a la strate [Λ ,n,0,β] si on a pour tout entier r:
+Λ(r) =⊕k
+j=−kΛ(r)∩W(j); on notera Λ(j)la suite de r´ eseaux de W(j)d´ efinie par Λ(j)(r) =
+Λ(r)∩W(j). Elle est proprement subordonn´ ee ` a [Λ,n,0,β] si de plus, pour tout i, 1≤i≤l,10 Corinne Blondel
+chaque saut de la suite de r´ eseaux Λi(r) intervient dans un et un seul des sous-espaces
+W(j)∩Vi(voir [5,§7] et [21,§5]).
+Fixons une d´ ecomposition autoduale V=⊕k
+j=−kW(j)deVproprement subordonn´ ee ` a
+[Λm,n,0,β] ; elle est alors subordonn´ ee ` a [Λ ,n,0,β]. SoitMle sous-groupe de Levi de G+
+fixant la d´ ecomposition V=⊕k
+j=−kW(j)etPun sous-groupe parabolique de G+de facteur
+de LeviMet radical unipotent U; on note P−etU−les oppos´ es de PetUpar rapport
+` aM. Il est montr´ e dans [21, §5.3] que sous l’hypoth` ese de subordination, les sous-groupes
+H1(Λ) etJ1(Λ) ont une d´ ecomposition d’Iwahori par rapport ` a ( P,M) et l’on peut d´ efinir :
+•un prolongement θPdeθ` aH1
+P(Λ) =H1(Λ)(J1(Λ)∩U), trivial sur J1(Λ)∩U;
+•l’unique repr´ esentation irr´ eductible ηPdeJ1
+P(Λ) =H1(Λ)(J1(Λ)∩P) contenant θP;
+sa restriction ` a H1
+P(Λ) est multiple de θP. Elle v´ erifie en outre :
+ηPest la repr´ esentation de J1
+P(Λ) dans les J1(Λ)∩U-invariants de ηobtenue par
+restriction de η` aJ1
+P(Λ) etη≃IndJ1(Λ)
+J1
+P(Λ)ηP.
+Dans les notations de [21, §5.3] on a H1(Λ)∩M=H1(Λ(0))×/producttextk
+j=1/tildewideH1(Λ(j)) ; le caract` ere
+semi-simple θest trivial sur H1(Λ)∩UetH1(Λ)∩U−et a pour restriction ` a H1(Λ)∩M
+un produit θ(0)⊗/circlemultiplytextk
+j=1(˜θ(j))2o` uθ(0)est un caract` ere semi-simple gauche et les ˜θ(j), pour
+j/ne}ationslash= 0, sont des caract` eres semi-simples. De mˆ eme ηP|J1(Λ)∩Mest une repr´ esentation de
+J1(Λ)∩M=J1(Λ(0))×/producttextk
+j=1/tildewideJ1(Λ(j)) de la forme η(0)⊗/circlemultiplytextk
+j=1˜η(j), o` uη(0)est l’unique
+repr´ esentation irr´ eductible de J1(Λ(0)) contenant θ(0)et ˜η(j), pourj/ne}ationslash= 0, est l’unique
+repr´ esentation irr´ eductible de /tildewideJ1(Λ(j)) contenant ( ˜θ(j))2.
+Sous l’hypoth` ese de subordination propre, ici valide pour Λm, on a de tels r´ esultats
+jusqu’au niveau de J+[21,§5.3] : le groupe J+(Λm) a aussi une d´ ecomposition d’Iwahori
+et tout prolongement κmdeηm` aJ+(Λm) est induit de la repr´ esentation κm
+PdeJ+
+P(Λm) =
+H1(Λm)(J+(Λm)∩P) obtenue en prenant les J1(Λm)∩U-invariants de κm; la repr´ esentation
+κm
+Pprolonge ηm
+P. Noter qu’alors J+(Λm)∩U=J1(Λm)∩U.
+Enl’absence desubordination propre, onpeut n´ eanmoins utiliser les techniques de J1∩U-
+invariants pour ´ etudier des prolongements de η` a un sous-groupe convenable de J+(Λ). Gar-
+dons les notations J+
+P(Λ) =H1(Λ)(J+(Λ)∩P),κPprolongement de ηP` aJ+
+P(Λ), dans ce con-
+texte ´ elargi ; on v´ erifie facilement (crit` ere de Mackey) que IndJ+(Λ)
+J+
+P(Λ)κPn’est pas irr´ eductible
+siJ+(Λ) n’est pas ´ egal ` a J1(Λ)J+
+P(Λ). Mais ce sous-groupe lui-mˆ eme est digne d’int´ erˆ et. Il
+a une d´ ecomposition d’Iwahori relative ` a ( M,P) et :
+Lemme 1.10. Pour tout prolongement κdeη` a
+J1(Λ)J+
+P(Λ) = (J1(Λ)∩U−)(J+(Λ)∩M)(J+(Λ)∩U)
+le sous-espace des invariants par J1(Λ)∩Ud´ efinit une repr´ esentation κPdeJ+
+P(Λ) =
+(P+(ΛoE)∩P)J1
+P(Λ)prolongeant ηPet telle que
+κ≃IndJ1(Λ)J+
+P(Λ)
+J+
+P(Λ)κP.Repr´ esentation de Weil et β-extensions 11
+D´ emonstration. Il suffit de v´ erifier l’´ egalit´ e suivante :
+J+(Λ)∩P= (P+(ΛoE)∩P) (J1(Λ)∩P). (1.11)
+CommeJ1(Λ) a une d´ ecomposition d’Iwahori par rapport ` a ( M,P), tout ´ el´ ement de J+(Λ)
+peut s’´ ecrire j=xj−1
+−jPavecx∈P+(ΛoE),jP∈J1(Λ)∩Petj−∈J1(Λ)∩U−. Alorsj
+appartient ` a Psi et seulement si xj−1
+−appartient ` a P. Ecrivons dans ce cas x=umj−avec
+u∈U,m∈M. Par unicit´ e de la d´ ecomposition d’Iwahori on voit que, puisque xcommute
+` aF[β]×qui est contenu dans M, alorsj−commute aussi ` a F[β]×. Ainsij−appartient ` a
+P1(ΛoE)∩U−etxj−1
+−appartient ` a P+(ΛoE)∩P, c.q.f.d.
+Ce point acquis, la d´ emonstration reprend sans difficult´ e celle de [21 ,§5.3] ou de [5,§7.2].
+Noter que dans cette situation d’une d´ ecomposition subordonn´ e e ` a [Λ,n,0,β] sans lui ˆ etre
+proprement subordonn´ ee, on n’a aucune raison d’avoir ´ egalit´ e e ntreJ+(Λ)∩UetJ1(Λ)∩U.
+/squaresolid
+Corollaire 1.12. Pla¸ cons-nous dans les hypoth` eses suivantes :
+/braceleftBigg
+V=⊕k
+j=−kW(j)est proprement subordonn´ ee ` a [Λm,n,0,β];
+P+(Λm
+oE) = (P+(ΛoE)∩P)P1(ΛoE).(1.13)
+Alors le sous-groupe J+
+ΛmΛ=P+(Λm
+oE)J1(Λ)co¨ ıncide avec (J+(Λ)∩P)J1(Λ)et poss` ede
+une d´ ecomposition d’Iwahori par rapport ` a (P,M).
+Pour tout prolongement κdeη` aP+(Λm
+oE)J1(Λ)le sous-espace des invariants par J1(Λ)∩
+Ud´ efinit une repr´ esentation κPdeJ+
+P(Λ) = (P+(ΛoE)∩P)J1
+P(Λ)prolongeant ηPet telle que
+κ≃IndP+(Λm
+oE)J1(Λ)
+J+
+P(Λ)κP.
+Bien entendu on s’int´ eressera particuli` erement aux prolongeme nts deηprolongeant de
+plusη(Λm,Λ). Or les techniques de J1∩U-invariants permettent, dans nos hypoth` eses, de
+pr´ eciser la structure de cette repr´ esentation (comme dans [21 ], d´ emonstration de la Propo-
+sition 6.3). Rappelons la d´ ecomposition ηP|J1(Λ)∩M≃η(0)⊗/circlemultiplytextk
+j=1˜η(j)et notons :
+–η(Λm(0),Λ(0)) le prolongement de η(0)` aJ1(Λm(0),Λ(0)) donn´ e par la proposition 1.2 ;
+–/tildewideη(Λm(j),Λ(j)) le prolongement de /tildewideη(j)` a/tildewideJ1(Λm(j),Λ(j)) donn´ e par [21, Proposition 3.12].
+Proposition 1.14. Sous les hypoth` eses (1.13), soit ηP(Λm,Λ)la repr´ esentation du groupe
+J1
+P(Λm,Λ) = (P1(Λm
+oE)∩P)J1
+P(Λ)obtenue par restriction de η(Λm,Λ)au sous-espace de ses
+J1(Λ)∩U-invariants. On a η(Λm,Λ)≃IndJ1(Λm,Λ)
+J1
+P(Λm,Λ)ηP(Λm,Λ). De plus ηP(Λm,Λ)est d´ efinie
+par :
+ηP(Λm,Λ) =ηPsurJ1
+P(Λ),
+ηP(Λm,Λ) = 1surP1(Λm
+oE)∩U,
+ηP(Λm,Λ) =η(Λm(0),Λ(0))⊗k/circlemultiplydisplay
+j=1/tildewideη(Λm(j),Λ(j))
+surJ1
+P(Λm,Λ)∩M=J1(Λm(0),Λ(0))×k/productdisplay
+j=1/tildewideJ1(Λm(j),Λ(j)).12 Corinne Blondel
+D´ emonstration. La premi` ere assertion d´ ecoule du corollaire 1.12. Les d´ efinitions donn´ ees
+d´ eterminent bien une repr´ esentation φdeJ1
+P(Λm,Λ) : elles recouvrent tout le groupe, elles
+sont compatibles et on v´ erifie ais´ ement, grˆ ace au fait que θest normalis´ e par P+(ΛoE) qui
+contient P1(Λm
+oE), queφest un homomorphisme.
+D’autre part J1(Λm,Λ) =J1(Λ)J1
+P(Λm,Λ) (une seule double classe) car P1(Λm
+oE) =
+[(P+(ΛoE)∩P)P1(ΛoE)]∩P1(Λm
+oE) = (P1(Λm
+oE)∩P)P1(ΛoE). Ainsi l’induite Φ de φ` a
+J1(Λm,Λ) est irr´ eductible et prolonge ηpuisque
+ResJ1(Λ)Φ = Res J1(Λ)IndJ1(Λm,Λ)
+J1
+P(Λm,Λ)φ= IndJ1(Λ)
+J1
+P(Λ)ηP=η.
+Reste ` a voir que Φ est bien le prolongement de ηd´ efini ` a la proposition 1.2. Soit donc Λ′′
+une suite de oE-r´ eseaux telle que b0(Λ′′) =b0(Λm) eta0(Λ′′)⊆a0(Λ) ; il nous faut montrer
+que IndP1(Λ′′)
+J1(Λ′′)η′′≃IndP1(Λ′′)
+P1(ΛmoE)J1(Λ)Φ. Ces deux induites sont irr´ eductibles (voir preuve du
+lemme 1.8) et par (1.1) il suffit de prouver
+HomJ1
+P(Λ′′)∩J1
+P(Λm,Λ)(η′′
+P,φ)/ne}ationslash={0}.
+Les groupes J1
+P(Λ′′) etJ1
+P(Λm,Λ) ont une d´ ecomposition d’Iwahori par rapport ` a ( P,M) et
+les deux repr´ esentations consid´ er´ ees sont triviales sur les inte rsections avec UetU−. Enfin,
+les restrictions ` a J1
+P(Λ′′)∩MetJ1
+P(Λm,Λ)∩Msont ´ evidemment entrelac´ ees par d´ efinition
+deη(Λm(0),Λ(0)) et/tildewideη(Λm(j),Λ(j)) et via le Corollaire 1.9, c.q.f.d. /squaresolid
+Terminons par une variante de la Proposition 1.6:
+Proposition 1.15. Pla¸ cons-nous dans les hypoth` eses (1.13) et supposons a0(Λm)contenu
+dansa0(Λ). Soitκmun prolongement de ηm` aJ+(Λm)etκun prolongement de η(Λm,Λ)` a
+P+(Λm
+oE)J1(Λ). Alors
+κ=BΛmΛ(κm)⇐⇒HomJ+
+P(Λm)∩J+
+P(Λ)(κm
+P,κP)/ne}ationslash={0}
+⇐⇒HomJ+
+P−(Λm)∩J+
+P(Λ)(κm
+P−,κP)/ne}ationslash={0}
+D´ emonstration. Lesimplications⇐=sont´ el´ ementaires vialaproposition1.6, encombinant
+r´ eciprocit´ e de Frobenius et d´ ecomposition de Mackey (dont le pr emier terme suffit).
+Reprenons maintenant la d´ emonstration de 1.6 en posant P′=PouP−et avec les
+mˆ emes notations. On veut montrer que la compos´ ee suivante est non nulle :
+κPJP֒→κJ֒→IndP+(Λm
+oE)P1(Λm)
+P+(ΛmoE)J1(Λ)κE−→IndP+(Λm
+oE)P1(Λm)
+J+(Λm)κmP−→κmPP′−→κm
+P′
+Les ingr´ edients sont les mˆ emes :
+– La multiplicit´ e de ηPdans IndP+(Λm
+oE)P1(Λm)
+P+(ΛmoE)J1(Λ)κest 1 par r´ eciprocit´ e de Frobenius puisque
+celle deη≃IndJ1(Λ)
+J1
+P(Λ)ηPest 1 (lemme 1.8). L’image de E◦j◦jPest donc la composante
+isotypique de type θPde IndP+(Λm
+oE)P1(Λm)
+J+(Λm)κm.Repr´ esentation de Weil et β-extensions 13
+– Il existe un vecteur vde l’espace de κm
+P′tel que
+/contintegraldisplay
+H1
+P(Λ)∩J+
+P′(Λm)θP(x−1)κm
+P′(x)vdx/ne}ationslash= 0.
+En effet, il suffit (par Frobenius et Mackey comme pour le lemme 1.7) de montrer que
+HomH1
+P(Λ)∩H1
+P′(Λm)(θP,θm
+P′)/ne}ationslash={0}. L’intersection des groupes H1
+P(Λ) etH1
+P′(Λm) se calcule
+terme ` a terme sur leur d´ ecomposition d’Iwahori. Or les caract` er es sont triviaux sur les
+intersections avec UetU−et sont tranferts l’un de l’autre sur l’intersection avec M./squaresolid
+1.4 Bijections canoniques et restriction de Jacquet
+SoitV=⊕k
+j=−kW(j)une d´ ecomposition autoduale proprement subordonn´ ee ` a [Λ ,n,0,β] ;
+elle est aussi proprement subordonn´ ee ` a [Λ′,n′,0,β]. SoitMle sous-groupe de Levi de G+
+fixant la d´ ecomposition V=⊕k
+j=−kW(j)etPun sous-groupe parabolique de G+de facteur
+de LeviMet radical unipotent U; on note P−etU−les oppos´ es de PetUpar rapport ` a
+M.
+Lemme 1.16. L’application rP:κ/mapsto→κP|J+(Λ)∩Mest une bijection de l’ensemble des pro-
+longements de η` aJ+(Λ)sur l’ensemble des prolongements de ηP|J1
+P(Λ)∩M` aJ+
+P(Λ)∩M.
+D´ emonstration. D´ ecoule de J+(Λ)∩U=J1(Λ)∩UetJ+
+P(Λ) = (J+(Λ)∩M)J1
+P(Λ)./squaresolid
+La bijection canonique Bde [21, Lemma 4.3] (Proposition 1.3 ci-dessus) est compatible
+` a la bijection rP:
+Proposition 1.17. Le diagramme suivant est commutatif :
+{Prolongements de η` aJ+(Λ)}rP−→ {Prolongements de ηP|J1(Λ)∩M` aJ+(Λ)∩M}
+BΛΛ′/arrowbothv B/arrowbothv
+{Prolongements de η′` aJ+(Λ′)}rP−→ {Prolongements de η′
+P|J1(Λ′)∩M` aJ+(Λ′)∩M}
+De plus, les bijections BetrPci-dessus sont compatibles ` a la torsion par les caract` ere s
+deP+(ΛoE)/P1(ΛoE) =P+(Λ′
+oE)/P1(Λ′
+oE).
+La bijection canonique Bdu cˆ ot´ e droit est bien sˆ ur d´ efinie comme le produit de B(0)=
+BΛ(0)Λ′(0)et des bijections canoniques /tildewideB(j)entre prolongements de ˜ η(j)et de˜η′(j)[21,§4.3].
+En effet le caract` ere semi-simple θ′est un produit θ′(0)⊗/circlemultiplytextk
+j=1(˜θ′(j))2comme ci-dessus, o` u
+θ′(0)est le transfert de θ(0)et chaque ˜θ′(j), pourj >0, le transfert de ˜θ(j).
+D´ emonstration. Grˆ ace ` a la technique de [21, §4], bas´ ee sur [21, Lemme 2.10], il suffit de
+d´ emontrer cette assertion si a0(Λ)⊆a0(Λ′), ce que nous supposons d´ esormais.
+Lesapplicationsconsid´ er´ ees sontdesbijections, ilsuffitdoncde montrer quesi κP|J+(Λ)∩M
+etκ′
+P|J+(Λ′)∩Mse correspondent par la bijection canonique, alors κ′etκaussi. L’hypoth` ese14 Corinne Blondel
+se traduit en Hom J+(Λ)∩J+(Λ′)∩M(κP,κ′
+P)/ne}ationslash={0}(Proposition 1.6). Or J+
+P(Λ) etJ+
+P(Λ′)
+ont tous deux une d´ ecomposition d’Iwahori et κ′
+PetκPsont toutes deux triviales sur les
+intersections avec UetU−(commeη′
+PetηP). Ainsi HomJ+
+P(Λ)∩J+
+P(Λ′)(κP,κ′
+P)/ne}ationslash={0}, d’o` u
+HomP+(ΛoE)P1(Λ)(IndP+(ΛoE)P1(Λ)
+J+
+P(Λ)κP,IndP+(ΛoE)P1(Λ)
+J+
+P(Λ′)κ′
+P)/ne}ationslash={0}par (1.1), et le r´ esultat. /squaresolid
+Pour ´ etudier le cas o` u la subordination n’est pas propre, nous nou s pla¸ cons d´ esormais
+sous les hypoth` eses (1.13) qui permettent de d´ efinir κPpour tout prolongement de η` a
+P+(Λm
+oE)J1(Λ), et d’´ enoncer des variantes du lemme et de la proposition ci-de ssus.
+Lemme 1.18. Sous les hypoth` eses (1.13), l’application rP:κ/mapsto→κP|J+
+P(Λ)∩Mest une bi-
+jection de l’ensemble des prolongements de η(Λm,Λ)` aP+(Λm
+oE)J1(Λ)sur l’ensemble des
+prolongements ` a J+
+P(Λ)∩Mde la repr´ esentation
+ηP|M(Λm,Λ) =ηP(Λm,Λ)|J1
+P(Λm,Λ)∩M.
+D´ emonstration. Certainement κ/mapsto→κPest une bijection de l’ensemble des prolongements
+deη` aP+(Λm
+oE)J1(Λ) sur l’ensemble des prolongements de ηP` aJ+
+P(Λ) (Corollaire 1.12).
+C’est pour conserver une bijection en restreignant ` a J+
+P(Λ)∩Mqu’il faut travailler sur les
+prolongements de η(Λm,Λ). Notons d’abord les propri´ et´ es indispensables :
+J+(Λ)∩M=J+
+P(Λ)∩M= (P+(ΛoE)∩M) (J1(Λ)∩M)
+J+
+P(Λ) = ( H1(Λ)∩U−) (J+
+P(Λ)∩M) (P+(ΛoE)∩U) (J1(Λ)∩U)
+P+(ΛoE)∩U=P+(Λm
+oE)∩U=P1(Λm
+oE)∩U
+P+(ΛoE)∩M=P+(Λm
+oE)∩M(1.19)
+qui d´ ecoulent de l’hypoth` ese, du paragraphe pr´ ec´ edent et d u fait que la d´ ecomposition
+est proprement subordonn´ ee ` a Λm. L’injectivit´ e s’en d´ eduit car ηP(Λm,Λ) est triviale sur
+H1(Λ)∩U−(commeηP), surJ1(Λ)∩U(commeηP) et surP1(Λm
+oE)∩U(Proposition 1.14).
+Pour la surjectivit´ e, on remarque que J+
+P(Λ) = (J+
+P(Λ)∩M)J1
+P(Λm,Λ). Il faut v´ erifier
+que siξest un prolongement de ηP|M(Λm,Λ) ` aJ+
+P(Λ)∩M, alorsφ(xy) =ξ(x)ηP(Λm,Λ)(y),
+x∈J+
+P(Λ)∩M,y∈J1
+P(Λm,Λ), d´ efinit une repr´ esentation de J+
+P(Λ). Cela revient ` a montrer
+queηP(Λm,Λ)(y)ξ(x) =ξ(x)ηP(Λm,Λ)(x−1yx) (x∈P+(Λm
+oE)∩M,ycomme ci-dessus).
+C’est imm´ ediat, grˆ ace ` a la trivialit´ e de ηP(Λm,Λ)(y) poury∈P1(Λm
+oE)∩U(Proposition
+1.14). /squaresolid
+Proposition 1.20. Sous les hypoth` eses (1.13), le diagramme suivant est commu tatif :
+{Prolongements de ηm` aJ+(Λm)}rP−→ {Prolong. de ηm
+P|J1(Λm)∩M` aJ+(Λm)∩M}
+BΛmΛ/arrowbothv B/arrowbothv
+{Prolong. de η(Λm,Λ)` aP+(Λm
+oE)J1(Λ)}rP−→{Prolong. de ηP|M(Λm,Λ)` aJ+(Λ)∩M}
+D´ emonstration. La d´ emonstration est celle de 1.17, on utilise les d´ ecompositions d’Iw ahori
+(1.19) et le fait que les repr´ esentations κm
+PetκPsont triviales sur les intersections avec Uet
+U−, comme ηm
+PetηP(Λm,Λ) (proposition 1.14). /squaresolidRepr´ esentation de Weil et β-extensions 15
+2 Une repr´ esentation de Weil
+2.1 Des sous-groupes remarquables
+Soit [Λ,n,0,β] une strate gauche semi-simple et V=V1⊥···⊥ Vlla d´ ecomposition de
+Vassoci´ ee. Soit V=Z1⊥···⊥ Ztune d´ ecomposition moins fine de V: chaque Zj,
+1≤j≤t, est somme de certains Vi. La strate [Λ ,n,0,β] est alors somme directe de strates
+semi-simples [ Sj= Λ∩Zj,nj,0,βj=β|Zj] dans chaque End F(Zj). On souhaite examiner en
+d´ etail la structure de η, l’unique repr´ esentation irr´ eductible de J1(Λ) contenant le caract` ere
+semi-simple gauche θdeH1(Λ), et de ses prolongements κ` aJ+(Λ), en la comparant ` a celle
+des objets analogues d´ efinis dans le groupe
+G+
+int=G+∩/productdisplay
+1≤j≤tGLF(Zj)
+` a partir des strates [ Sj,nj,0,βj] et des caract` eres semi-simples gauches correspondant ` a θ.
+Remarquons que le commutant G+
+EdeβdansG+v´ erifie
+G+
+E⊆G+∩/productdisplay
+1≤i≤lGLF(Vi)⊆G+
+int.
+Consid´ erons le groupe J1
+int(Λ) =J1(Λ)∩G+
+int. La d´ ecomposition V=Z1⊥···⊥Ztest
+subordonn´ ee ` a la strate [Λ ,n,0,β] et les r´ esultats de [21, §5.1,§5.2] s’appliquent :
+H1(Λ)∩G+
+int=/productdisplay
+1≤j≤tH1(Sj), θ|H1(Λ)∩G+
+int=⊗1≤j≤tθ(Sj),
+o` u lesθ(Sj) sont des caract` eres semi-simples gauches attach´ es aux stra tes [Sj,nj,0,βj] et
+transferts de θau sens de loc. cit., Proposition 5.5, et
+J1
+int(Λ) =/productdisplay
+1≤j≤tJ1(Sj).
+SoitJ1
+ext(Λ) l’orthogonal de H1(Λ)J1
+int(Λ) pour la forme bilin´ eaire altern´ ee non d´ eg´ en´ er´ ee
+usuelle (x,y)/mapsto−→< x,y > =θ([x,y]) surJ1(Λ)/H1(Λ). La restriction de cette forme aux
+sous-espaces orthogonaux H1(Λ)J1
+int(Λ)/H1(Λ) etJ1
+ext(Λ)/H1(Λ) reste non d´ eg´ en´ er´ ee et
+l’on d´ efinit comme d’habitude la repr´ esentation irr´ eductible ηintdeJ1
+int(Λ) (resp. ηextde
+J1
+ext(Λ))commel’uniquerepr´ esentation irr´ eductible(` aisomorphisme pr` es)dontlarestriction
+` aH1(Λ)∩G+
+int(resp.H1(Λ)) contient θ(et en est multiple). On reconnaˆ ıt en outre via
+loc.cit.queηintest isomorphe ` a⊗1≤j≤tη(Sj).
+CommeJ1
+int(Λ)J1
+ext(Λ) =J1(Λ) et (H1(Λ)J1
+int(Λ))∩J1
+ext(Λ) =H1(Λ), on obtient une
+r´ ealisation de la repr´ esentation ηen posant pour g∈J1(Λ) :
+η(g) =ηext(gext)⊗ηint(gint) sig=gextgintavecgext∈J1
+ext(Λ), gint∈J1
+int(Λ).(2.21)
+D´ efinissons dans /tildewideG=GL(V) et/tildewideGint=/producttext
+1≤j≤tGLF(Zj) les sous-groupes analogues
+/tildewideJ1
+int(Λ) =/tildewideJ1(Λ)∩/tildewideGintet son orthogonal /tildewideJ1
+ext(Λ) pour la forme ˜θ([x,y]), o` u˜θest un caract` ere16 Corinne Blondel
+semi-simple gauchede /tildewideH1(Λ)derestriction θ` aH1(Λ). Soit Qunsous-groupeparaboliquede
+/tildewideGde facteur de Levi ˜Gint,Nson radical unipotent et N−le radical unipotent du parabolique
+oppos´ e par rapport ` a ˜Gint. D’apr` es [21, Proposition 5.2, Lemma 5.6(iii)] on a :
+/tildewideJ1
+ext(Λ) = (/tildewideJ1(Λ)∩N−)(/tildewideJ1(Λ)∩N)/tildewideH1(Λ).
+Lemme 2.22. (i)On aJ1
+ext(Λ) =/tildewideJ1
+ext(Λ)∩G, ind´ ependant de θ.
+(ii)On a pour tout g∈G+
+E:
+J1(Λ)∩gJ1(Λ)g−1= (J1
+ext(Λ)∩gJ1
+ext(Λ)g−1).(J1
+int(Λ)∩gJ1
+int(Λ)g−1).
+D´ emonstration. (i) DeJ1
+int(Λ) =/tildewideJ1
+int(Λ)∩Gon d´ eduit par orthogonalit´ e : H1
+ext(Λ)⊆
+/tildewideJ1
+ext(Λ)∩G⊆J1
+ext(Λ). L’´ egalit´ e d´ ecoule alors de /tildewideJ1(Λ)∩G= (/tildewideJ1
+int(Λ)∩G)(/tildewideJ1
+ext(Λ)∩G)
+dont la d´ emonstration est un cas particulier de celle de (ii) ci-dessou s.
+(ii) La difficult´ e technique est que G+
+intn’est pas un sous-groupe de Levi de G+. Il faut
+remonter dans /tildewideG=GL(V) et utiliser la d´ ecomposition d’Iwahori de /tildewideJ1(Λ) relativement ` a
+Qpour calculer l’intersection /tildewideJ1(Λ)∩g/tildewideJ1(Λ)g−1terme ` a terme. Comme gappartient ` a G+
+E
+contenu dans G+
+int, on obtient ainsi l’´ egalit´ e analogue dans /tildewideG. Pour revenir ` a G+on prend
+les points fixes de l’involution adjointe : on applique l’argument de Steve ns [19, Theorem
+2.3], en posant H=/tildewideJ1
+int(Λ)∩g/tildewideJ1
+int(Λ)g−1,U=/tildewideJ1
+ext(Λ)∩g/tildewideJ1
+ext(Λ)g−1, et en remarquant que
+/tildewideJ1
+ext(Λ) et/tildewideJ1
+int(Λ) se normalisent mutuellement puisque [ /tildewideJ1(Λ),/tildewideJ1(Λ)]⊂/tildewideH1(Λ). /squaresolid
+Proposition 2.23. (i)Les repr´ esentations ηextetηintsont entrelac´ ees par G+
+E.
+(ii)Soitg∈G+
+Eet soitEext(g), resp.Eint(g), un op´ erateur d’entrelacement de ηext, resp.
+ηint, eng. AlorsE(g) =Eext(g)⊗Eint(g)est un op´ erateur d’entrelacement de ηeng.
+Tout op´ erateur d’entrelacement de ηengest de cette forme.
+(iii)Les espaces d’entrelacement de ηextet deηinteng∈G+
+Esont de dimension 1.
+D´ emonstration. G+
+Eentrelace θ, doncηextetηint. Pour le deuxi` eme point, comme la dimen-
+sion de l’espace d’entrelacement de ηengest 1, il suffit de montrer que Eext(g)⊗Eint(g)
+entrelace effectivement η. Cela n’est qu’une v´ erification ´ etant donn´ e le lemme ci-dessus. Le
+troisi` eme point d´ ecoule du deuxi` eme vu la propri´ et´ e semblab le deη. /squaresolid
+Le groupe P+(ΛoE), qui est contenu dans G+
+int, normalise J1(Λ) etH1(Λ) et entrelace le
+caract` ere θ; il normalise donc H1(Λ)J1
+int(Λ) et son orthogonal J1
+ext(Λ) pour la forme <,>
+surJ1(Λ)/H1(Λ). Consid´ erons un prolongement κdeη` aJ+(Λ) =P+(ΛoE)J1(Λ) et un
+prolongement κintdeηint` aP+(ΛoE)J1
+int(Λ) =/producttext
+1≤j≤tJ+(Sj) : il en existe par [21, §4.1].
+D’apr` es la proposition 2.23, pour tout g∈P+(ΛoE), l’op´ erateur d’entrelacement κ(g) s’´ ecrit
+de fa¸ con unique sous la forme κ(g) =Eext(g)⊗κint(g) et les op´ erateurs Eext(g) ainsi d´ efinis
+fournissent une repr´ esentation de P+(ΛoE) dans l’espace de ηext, triviale sur P1(ΛoE) (qui
+est contenu dans H1(Λ)). Autrement dit :Repr´ esentation de Weil et β-extensions 17
+Corollaire 2.24. Il existe des repr´ esentations de P+(ΛoE)/P1(ΛoE)dans l’espace de ηexten-
+trela¸ cant ηext; elles diff` erententre elles d’un caract` ere. Tout choix d’ une telle repr´ esentation
+Xd´ etermine une bijection IntXentre prolongements de η` aJ+(Λ)et prolongements de ηint
+` aP+(ΛoE)J1
+int(Λ), donn´ ee par :
+κ(g) =X(g)⊗IntX(κ)(g), g∈P+(ΛoE).
+Une telle repr´ esentation X´ etant fix´ ee, on notera ´ egalement Int Xla bijection reliant de
+la mˆ eme fa¸ con prolongements de η` aTJ1(Λ) et prolongements de ηint` aTJ1
+int(Λ), pour tout
+sous-groupe TdeP+(ΛoE) contenant P1(ΛoE).
+2.2 Mise en place de la restriction de Jacquet
+Donnons-nous ` a pr´ esent, pour 1 ≤j≤t, des d´ ecompositions des sous-espaces Zjsubor-
+donn´ ees aux strates [ Sj,nj,0,βj] et autoduales ([21, §5.3]) :
+Zj=⊕mj
+r=−mjW(r)
+j,
+o` u l’orthogonal de W(r)
+jdansZjest⊕s/negationslash=−rW(s)
+j. On demande que W(0)
+jsoit non nul
+pour au plus un jet l’on somme ces d´ ecompositions de fa¸ con disjointe, c’est-` a-dir e que
+la d´ ecomposition de Vobtenue, soit V=⊕m
+i=−mW(i),v´ erifie que pour tout i,−m≤i≤m,
+il y a un unique j, 1≤j≤t, et un unique r,−mj≤r≤mj, tels que W(i)=W(r)
+jet
+W(−i)=W(−r)
+j. On obtient une d´ ecomposition autoduale de Vsubordonn´ ee ` a la strate
+[Λ,n,0,β].
+SoitMle sous-groupe de Levi de G+stabilisant cette d´ ecomposition et Pun sous-groupe
+parabolique de G+de facteur de Levi M. On note Ule radical unipotent de PetU−son
+oppos´ e par rapport ` a M. D’apr` es [21, Corollaire 5.10] les groupes H1(Λ) etJ1(Λ) ont une
+d´ ecomposition d’Iwahori par rapport ` a ( M,P). Puisque Mest contenu dans G+
+intet que
+les d´ ecompositions induites sur les Zjsont subordonn´ ees aux strates [ Sj,nj,0,βj], le groupe
+H1(Λ)J1
+int(Λ) a lui aussi une d´ ecomposition d’Iwahori par rapport ` a ( M,P).
+Rappelons enfin ([5, Proposition 7.2.3], [21, Lemma 5.6]) que l’on a une d´ e composition
+de l’espace symplectique J1(Λ)/H1(Λ) en
+J1(Λ)
+H1(Λ)=J1(Λ)∩M
+H1(Λ)∩M⊥/parenleftbiggJ1(Λ)∩U−
+H1(Λ)∩U−⊕J1(Λ)∩U
+H1(Λ))∩U/parenrightbigg
+dans laquelle les deux derniers sous-espaces sont totalement isotr opes, en dualit´ e ; on a une
+d´ ecomposition analogue pour H1(Λ)J1
+int(Λ)/H1(Λ). On en d´ eduit que J1
+ext(Λ) a lui aussi
+une d´ ecomposition d’Iwahori par rapport ` a ( M,P) et que :
+•J1
+ext(Λ) = ( J1
+ext(Λ)∩U−) (H1(Λ)∩M) (J1
+ext(Λ)∩U).
+•J1(Λ)∩U−= (J1
+ext(Λ)∩U−)(J1
+int(Λ)∩U−),J1(Λ)∩U= (J1
+ext(Λ)∩U)(J1
+int(Λ)∩U),
+•(H1(Λ)∩U)(J1
+int(Λ)∩U) est l’orthogonal de J1
+ext(Λ)∩U−dansJ1(Λ)∩U,18 Corinne Blondel
+•J1
+ext(Λ)∩Uest l’orthogonal de ( J1
+int(Λ)∩U−)(H1(Λ)∩U−) dansJ1(Λ)∩U.
+Examinons ` a pr´ esent les J1(Λ)∩U-invariants de ηdans le mod` ele (2.21). Notons Xext
+l’espace de ηext,Xintl’espace de ηintetX=Xext⊗Xintcelui de η. L’espace XJ1
+ext(Λ)∩U
+ext
+est de dimension 1 car H1(Λ)(J1
+ext(Λ)∩U)/H1(Λ) est totalement isotrope maximal dans
+J1
+ext(Λ)/H1(Λ) etθ|H1(Λ)∩U= 1. Le choix d’un vecteur non nul xext∈XJ1
+ext(Λ)∩U
+ext nous donne
+un isomorphisme :
+XJ1(Λ)∩U=XJ1
+ext(Λ)∩U
+ext⊗XJ1
+int(Λ)∩U
+intΦ−→XJ1
+int(Λ)∩U
+int
+xext⊗v/mapsto→v(2.25)
+L’espace XJ1(Λ)∩Uest une repr´ esentation de
+J1
+P(Λ) =H1(Λ)(J1(Λ)∩P) =H1(Λ)(J1
+int(Λ)∩P)(J1
+ext(Λ)∩U)
+et l’isomorphisme (2.25) commute ` a l’action de J1
+int(Λ)∩P. D’autre part P+(ΛoE)∩P
+normalise J1(Λ)∩U,J1
+ext(Λ)∩UetJ1
+int(Λ)∩Udonc agit sur leurs invariants ; en particulier,
+pour tout choix de repr´ esentation Xcomme dans le corollaire 2.24, P+(ΛoE)∩Pagit sur
+xextpar un caract` ere (trivial sur P1(ΛoE)∩P).
+Donnons-nous alors un prolongement κdeη` aJ1(Λ)J+
+P(Λ) et un prolongement κintde
+ηint` aJ1
+int(Λ)J+
+int,P(Λ). Le lemme 1.10 permet d’en ´ etudier les J1∩U-invariants ou J1
+int∩U-
+invariants, qui fournissent des repr´ esentations κPdeJ+
+P(Λ) = (P+(ΛoE)∩P)J1
+P(Λ) etκint,P
+deJ+
+int,P(Λ) = (P+(ΛoE)∩P)J1
+int,P(Λ). (On pr´ ef` ere la notation simplifi´ ee Xint,P` a la notation
+exacteXint,PintavecPint=P∩Gint.) Rappelons d’autre part (cf 1.11) que J+
+int,P(Λ)/J1
+int,P(Λ)
+est canoniquement isomorphe ` a P+(ΛoE)∩P/P1(ΛoE)∩P, ce qui permet de consid´ erer tout
+caract` ere du second groupe comme un caract` ere du premier. A vec ces conventions, on a en
+r´ esum´ e :
+Proposition 2.26. SoitXune repr´ esentation de P+(ΛoE)/P1(ΛoE)entrela¸ cant ηext. No-
+tonsχUle caract` ere de P+(ΛoE)∩Ppar lequel ce groupe agit sur les J1
+ext(Λ)∩U-invariants de
+ηext. SoitJ+
+int,P(Λ) = (P+(ΛoE)∩P)J1
+int,P(Λ)et soitκun prolongement de η` aJ1(Λ)J+
+P(Λ).
+Alors
+κP|J+
+int,P(Λ)=χUIntX(κ)P.
+2.3 Une repr´ esentation ` a la Weil
+Il est une situation particuli` ere dans laquelle on peut ´ elucider le ca ract` ereχUintroduit en
+2.26, en construisant une repr´ esentation entrela¸ cant ηext. Soit donc V=W(−1)⊕W(0)⊕W(1)
+une d´ ecomposition autoduale de Vsubordonn´ ee ` a la strate [Λ ,n,0,β] et telle que W(0)soit
+somme de certains Vi; posons Z1=W(−1)⊕W(1),Z2=W(0). SoitPle sous-groupe
+parabolique de G+stabilisant le drapeau {0}⊂W(−1)⊂W(−1)⊕W(0)⊂V,Uson radical
+unipotent, Mle fixateur de la d´ ecomposition, P−etU−les oppos´ es de PetUpar rapport
+` aM.
+Pour simplifier les notations, on raccourcit J1
+ext(Λ),H1(Λ) etc. en J1
+ext,H1etc. Le
+caract` ere θdeH1, trivial sur H1∩UetH1∩U−, se prolonge en un caract` ere θUdeRepr´ esentation de Weil et β-extensions 19
+H1(J1
+ext∩U) trivial sur J1
+ext∩U. On peut alors r´ ealiser la repr´ esentation ηextdeJ1
+extcomme
+l’action par translations ` a droite dans l’espace de fonctions
+W={f:J1
+ext−→C/ f(ht) =θU(h)f(t), h∈H1(J1
+ext∩U), t∈J1
+ext},(2.27)
+isomorphe par restriction ` a l’espace des fonctions de J1
+ext∩U−/H1∩U−dansC.
+On v´ erifie ais´ ement que pour g∈P+(ΛoE)∩P, l’op´ erateur Ω( g) d´ efini par
+Ω(g)(f)(x) =f(g−1xg),(f∈W, x∈J1
+ext, g∈P+(ΛoE)∩P)
+conserveW. Il existe (§2.1) une repr´ esentation projective g/mapsto→Ω(g) deP+(ΛoE)/P1(ΛoE)
+dansWentrela¸ cant ηextetprolongeant cettevraierepr´ esentation de P+(ΛoE)∩P. Ladonn´ ee
+de Ω va nous permettre de construire une vraie repr´ esentation p ar l’´ el´ egante m´ ethode de
+Neuhauser ([16] ; voir aussi [22]). On d´ efinit tout d’abord les sous- espacesW+etW−de
+Wform´ es des fonctions paires ( ∀v∈J1
+ext∩U−,f(v) =f(v−1)) pour le premier, impaires
+(∀v∈J1
+ext∩U−,f(v) =−f(v−1)) pour le second. On pr´ etend que
+Lemme 2.28.W+etW−sont stables par les op´ erateurs Ω(g),g∈P+(ΛoE)/P1(ΛoE).
+D´ emonstration. L’´ el´ ement ǫ:ǫ|Z1=IZ1,ǫ|Z2=−IZ2, appartient au centre de P+(ΛoE). Un
+bref calcul matriciel montre que, pour tout u∈J1
+ext∩U−, (ǫuǫ−1)uappartient ` a Gintdonc ` a
+son intersection avec J1
+ext∩U−, soit :ǫuǫ−1∈(H1∩U−)u−1. Les sous-espaces W+etW−
+sont donc les sous-espaces propres de Ω( ǫ) pour les valeurs propres 1 et −1 respectivement,
+d’o` u le r´ esultat (certes il s’agit d’une repr´ esentation projectiv e, cependant un isomorphisme
+Ω(g) ne peut ´ echanger W+etW−dont les dimensions diff` erent de 1). /squaresolid
+SoitΩ+larestrictiondeΩ` a W+. D’apr` es[16,Theorem4.3],o` ul’onexaminelesd´ eterminants
+des repr´ esentations projectives Ω et Ω+, on d´ efinit une vraie repr´ esentation en posant :
+W(g) =detΩ(g)
+detΩ+(g)2Ω(g), g∈P+(ΛoE)/P1(ΛoE).
+Calculons le caract` ere wUcorrespondant. Dans le mod` ele ci-dessus, la droite des J1
+ext∩U-
+invariants de ηexta pour base la fonction f1de support H1(J1
+ext∩U) et valeur 1 en 1, fix´ ee
+par les op´ erateurs Ω( u),u∈P+(ΛoE)∩P. D’autre partWa pour base l’ensemble des
+fonctions fv,v∈J1
+ext∩U−/H1
+ext∩U−, d´ efinies par leur support H1(J1
+ext∩U)vet leur valeur
+1 env.
+Lemme 2.29. Soitu∈P+(ΛoE)∩U,x∈J1
+ext(Λ)∩Uetv∈J1
+ext(Λ)∩U−. Le commutateur
+[u,x]appartient ` a H1(Λ)∩G+
+intet le commutateur [u,v]appartient ` a H1(Λ)(J1
+ext(Λ)∩U).
+D´ emonstration. Le commutateur de deux ´ el´ ements de Uappartient ` a G+
+int(v´ erification
+matricielle imm´ ediate) donc [ u,x] appartient ` a J1
+ext(Λ)∩U∩G+
+int⊂H1(Λ)∩U∩G+
+int. Le
+commutateur[ u,v]appartient` a J1
+ext(Λ),ilsuffitdemontrerqu’ilestorthogonal` a J1
+ext(Λ)∩U.
+On calcule donc pour x∈J1
+ext(Λ)∩U:
+θ([uvu−1v−1,x]) =θ([uvu−1,x][v−1,x]) =θ([v,u−1xu])θ([v,x−1]) =θ([v,u−1xux−1])
+qui vaut 1 puisque [ u−1,x] appartient ` a H1(Λ). /squaresolid20 Corinne Blondel
+On a donc pour u∈P+(ΛoE)∩U: Ω(u)fv=θU([u−1,v])fv. En outre Ω( u) envoie fonction
+paire sur fonction paire, d’o` u : θU([u−1,v]) =θU([u−1,v−1]) pourv∈J1
+ext∩U−. Il en r´ esulte
+detΩ(u) = detΩ+(u)2, doncW(u) est ´ egal ` a Ω( u) et fixef1.
+Etudions maintenant les op´ erateurs Ω( x) pourx´ el´ ement de P+(ΛoE)∩M, qui normalise
+J1
+ext∩U−etH1
+ext∩U−. On a Ω( x)fv=fxvx−1, autrement dit ces op´ erateurs agissent
+par permutation de la base des fvet leur d´ eterminant est la signature de la permutation
+correspondante. Il en est de mˆ eme pour W+, de base les fv+fv−1, donc detΩ+(x)2= 1. En
+conclusion :
+Proposition 2.30. SoitwUle caract` ere de P+(ΛoE)∩P, trivial sur P+(ΛoE)∩U, qui ` ax∈
+P+(ΛoE)∩Massocie la signature de la permutation v/mapsto→xvx−1deJ1
+ext(Λ)∩U−/H1(Λ)∩U−.
+Il existe une repr´ esentation WdeP+(ΛoE)/P1(ΛoE)dans l’espace de ηext, entrela¸ cant ηext,
+dont la restriction ` a P+(ΛoE)∩Pest donn´ ee, dans le mod` ele (2.27) de ηext, par :
+W(g)(f)(x) =wU(g)f(g−1xg) (g∈P+(ΛoE)∩P, f∈W, x∈J1
+ext).
+En particulier, le caract` ere wUest l’action de P+(ΛoE)∩Psur la droite des J1
+ext∩U-
+invariants de ηext.
+Remarque 2.31. On obtient un caract` ere wU` a valeurs dans{±1}, donc trivial sur tous
+les sous-groupes d’ordre impair, en particulier sur tous les p-sous-groupes. Rappelons aussi
+queJ1
+ext(Λ) ne d´ epend pas de θ(Lemme 2.22), donc wUn’en d´ epend pas non plus.
+Remarque 2.32. Au lieu d’´ etudier directement la repr´ esentation de P+(ΛoE)/P1(ΛoE)
+comme ci-dessus, on aurait pu utiliser l’homomorphisme naturel de ce groupe dans le groupe
+symplectiqueSdeJ1
+ext∩U−/H1∩U−⊕J1
+ext∩U/H1∩Uet se ramener ` a la repr´ esentation
+de Weil deStelle qu’elle est d´ ecrite par G´ erardin [9]. Cette approche s’av` ere p lus com-
+pliqu´ ee. D’une part il n’est pas clair que l’action naturelle de P+(ΛoE)/P1(ΛoE) sur le groupe
+d’Heisenberg correspondant nous donne une section conjugu´ ee ` a la section standard (de S
+dans le groupe des automorphismes du groupe d’Heisenberg). D’aut re part la connaissance
+du caract` ere du stabilisateur de J1
+ext∩U/H1
+ext∩UdansSpar lequel il agit sur un invariant
+de son radical unipotent ne suffit pas ` a d´ ecrire sa restriction ` a P+(ΛoE)∩M.
+2.4 Lien avec les β-extensions
+Nous allons ` a pr´ esent appliquer le corollaire 2.24 pour comparer β-extensions dans G+et
+dansG+
+int. Les deux ingr´ edients n´ ecessaires sont la restriction de Jacque t, qui interviendra
+via la proposition 2.26 et qui commute aux bijections canoniques (pro position 1.20), et un
+bon choix de la repr´ esentation X(proposition 2.30). Nous nous pla¸ cons dans des hypoth` eses
+communes ` a ces r´ esultats :
+(i)V=W(−1)⊕W(0)⊕W(1)est une d´ ecomposition autoduale de Vproprement subor-
+donn´ ee ` a la strate [Λ ,n,0,β], donc subordonn´ ee ` a [ΛM,nM,0,β], et telle que W(0)soit
+somme de certains Vi;Z1=W(−1)⊕W(1),Z2=W(0).
+La strate [Λ ,n,0,β] est donc somme directe des strates semi-simples [ S1,n1,0,β1] et
+[S2,n2,0,β2] avecSj= Λ∩Zj,j= 1,2. Il en est de mˆ eme pour la strate [ΛM,nM,0,β],
+somme directe des strates semi-simples [ SM
+1,nM
+1,0,β1] et [SM
+2,nM
+2,0,β2].Repr´ esentation de Weil et β-extensions 21
+(ii)Pest le sous-groupe parabolique de G+stabilisant le drapeau {0}⊂W(−1)⊂W(−1)⊕
+W(0)⊂W(−1)⊕W(0)⊕W(1),Uest son radical unipotent, Mest le fixateur de la
+d´ ecomposition, P−etU−sont les oppos´ es de PetUpar rapport ` a M.
+(iii) On demande P+(ΛoE) = (P+(ΛM
+oE)∩P)P1(ΛM
+oE).
+Rappelons que b0(ΛM) est un oE-ordre autodual maximal. On applique les r´ esultats du
+paragraphe pr´ ec´ edent ` a ΛMen ajoutant l’exposant Maux notations si n´ ecessaire. Alors :
+Proposition 2.33. SoitWMla repr´ esentation de P+(ΛM
+oE)dans l’espace de ηM
+extd´ efinie dans
+la proposition 2.30 et soit IntWMla bijection correspondante. Alors κM, prolongement de
+ηM` aJ+(ΛM), est une β-extension si et seulement si IntWM(κM), prolongement de ηM
+int` a
+J+
+int(ΛM), est une β-extension.
+D´ emonstration. Il suffit de montrer, pour une suite autoduale ΛmdeoE-r´ eseaux telle que
+b0(Λm) soit un oE-ordre autodual minimal contenu dans b0(ΛM), que les conditions suivantes
+sont ´ equivalentes :
+(i)κMest un prolongement de η(Λm,ΛM) ;
+(ii) Int WM(κM) est un prolongement de ηint(Λm,ΛM).
+On choisit, comme on le peut, une suite Λmtelle que la d´ ecomposition V=W(−1)⊕W(0)⊕
+W(1)soit proprement subordonn´ ee ` a la strate [Λm,nm,0,β] et telle que b0(Λm)⊂b0(Λ).
+•L’hypoth` ese(iii)entraˆ ıne P+(ΛoE) = (P1(ΛM
+oE)∩U−)(P+(ΛM
+oE)∩M)(P+(ΛM
+oE)∩U)donc
+P1(Λm
+oE)∩U−⊂P1(ΛM
+oE)∩U−. Comme P1(ΛM
+oE)⊂P1(Λm
+oE), onaenfait P1(Λm
+oE)∩U−=
+P1(ΛoE)∩U−=P1(ΛM
+oE)∩U−, contenu dans J1(Λ).
+•Rappelons que η(Λm,ΛM) est une repr´ esentation de P1(Λm
+oE)J1(ΛM), poss` edant une
+d´ ecomposition d’Iwahori relative ` a ( P,M) et d’intersection avec U−:J1(ΛM)∩U−.
+On peut comme au paragraphe 1.3 d´ efinir la repr´ esentation ηP(Λm,ΛM) du groupe
+(P1(Λm
+oE)∩P)J1
+P(ΛM) : c’est l’action sur les J1(ΛM)∩U-invariants de η(Λm,ΛM). On
+v´ erifie que η(Λm,ΛM) est la repr´ esentation induite de ηP(Λm,ΛM).
+•Le groupe P+(ΛoE)J1(ΛM) contient P+(Λm
+oE)J1(ΛM), de sorte que la condition voulue
+surκMpeutsevoircommeuneconditionsursarestriction` a P+(ΛoE)J1(ΛM),` alaquelle
+le lemme 1.10 s’applique. Ainsi κMprolonge η(Λm,ΛM) si et seulement si κM
+Pprolonge
+ηP(Λm,ΛM).
+•Les mˆ emes constructions s’appliquent dans G+
+intaux objets induits dans les groupes
+d’isom´ etries de Z1etZ2; on continue d’utiliser la notation Pau lieu d’utiliser une
+notation sp´ ecifique pour le parabolique P∩G+
+int. Elles aboutissent ` a : Int WM(κM)
+prolonge ηint(Λm,ΛM) si et seulement si Int WM(κM)Pprolonge ηint(Λm,ΛM)P.
+•On peut d´ ecrire ηP(Λm,ΛM) de fa¸ con analogue ` a la proposition 1.14 ci-dessus et ` a
+[21], preuve de la proposition 6.3 : c’est la repr´ esentation de ( P1(Λm
+oE)∩P)J1
+P(ΛM)22 Corinne Blondel
+prolongeant ηP(ΛM) qui est triviale sur P1(Λm
+oE)∩Uet ´ egale ` a η(Λm(0),ΛM(0))⊗
+/tildewideη(Λm(1),ΛM(1)) surP1(Λm
+oE)∩M. Il en est de mˆ eme pour ηint(Λm,ΛM)Pavec les
+modifications ´ evidentes.
+Lesconditions“ κM
+Pprolonge ηP(Λm,ΛM)”et “Int WM(κM)Pprolonge ηint(Λm,ΛM)P”se lisent
+surJ+
+int,P(ΛM). Par la proposition 2.26, la restriction de κM
+P` aJ+
+int,P(ΛM) est ´ equivalente ` a
+wM
+UIntWM(κM)P. Comme wM
+Uest trivial sur le ( p-)groupe de d´ efinition de ηint(Λm,ΛM)P,
+on a le r´ esultat annonc´ e. /squaresolid
+Remarque 2.34. On montre de la mˆ eme fa¸ con que κMprolonge η(Λ,ΛM) si et seulement
+si IntW(κM) prolonge ηint(Λ,ΛM).
+Nous pouvons enfin rassembler tous les r´ esultats et d´ ecrire le lien entreβ-extensions dans
+G+et dansG+
+inten termes de l’application rPdu paragraphe 1.4 et du caract` ere wM
+Ude la
+proposition 2.30 appliqu´ ee ` a la suite ΛM. Le caract` ere wM
+UdeP+(ΛM
+oE)∩P/P1(ΛM
+oE)∩P
+s’identifie ` a un caract` ere de P+(ΛoE) = (P+(ΛM
+oE)∩P)P1(ΛM
+oE), trivial sur P1(ΛoE). On
+rappelle que P+(ΛoE) = (P1(ΛoE)∩U−)(P+(ΛoE)∩M)(P1(ΛoE)∩U).
+Th´ eor` eme 2.35. Soitκ(Λ)un prolongement de η(Λ)` aJ+(Λ)etκ(S1)⊗κ(S2)un prolonge-
+ment de η(S1)⊗η(S2)` aJ+
+int(Λ) =J+(S1)×J+(S2), reli´ es par la condition
+rP(κ(Λ)) =rP(κ(S1)⊗κ(S2)).
+SoitwM
+Ule caract` ere de P+(ΛoE), trivial sur P1(ΛoE), qui ` ax∈P+(ΛoE)∩Massocie la
+signature de la permutation v/mapsto→xvx−1deJ1
+ext(ΛM)∩U−/H1(ΛM)∩U−.
+La repr´ esentation κ(Λ)est uneβ-extension de η(Λ)relativement ` a ΛMsi et seulement si la
+repr´ esentation wM
+U(κ(S1)⊗κ(S2))est uneβ-extension de η(S1)⊗η(S2)relativement ` a ΛM.
+D´ emonstration. Elle s’appuie sur le diagramme suivant :
+{Prolongements de η(Λ) ` aJ+(Λ)}B←→{Pr. deη(Λ,ΛM) ` aP+(ΛoE)J1(ΛM)}
+rP↓ rP↓
+{Pr. deηP(Λ)|J1(Λ)∩M` aJ+(Λ)∩M}B←→{Pr. deηP|M(Λ,ΛM) ` aJ+(ΛM)∩M}
+↓ Φ↓
+{Pr. deηP(S1)|J1(S1)∩M⊗η(S2) ` aJ+
+int(Λ)∩M}B←→{Pr. deηP|M(S1,SM
+1)⊗η(S2,SM
+2)
+` aJ+
+int(ΛM)∩M}
+rP↑ rP↑
+{Pr. deη(S1)⊗η(S2) ` aJ+
+int(Λ)}B←→{Pr. deη(S1,SM
+1)⊗η(S2,SM
+2)
+` aP+(ΛoE)J1
+int(ΛM)}
+Expliquons ce diagramme. Les deux premi` eres lignes reproduisent le diagramme commutatif
+de la proposition 1.20 appliqu´ ee ` a Λ et ΛM, avec ´ echange des lignes et des colonnes. LesRepr´ esentation de Weil et β-extensions 23
+deux derni` eres lignes reproduisent le mˆ eme diagramme commutat if dans le groupe G+
+int, avec
+les paires ( S1,SM
+1) et (S2,SM
+2), et invers´ e ; noter que η(S2,SM
+2) =ηP|M(S2,SM
+2).
+Soit de nouveau WMla repr´ esentation de P+(ΛM
+oE) dans l’espace de ηM
+extd´ efinie dans la
+proposition 2.30, soit Int WMla bijection correspondante et soit Φ l’isomorphisme associ´ e par
+la proposition 2.26. Alors Φ entrelace des repr´ esentations de J+
+int,P(ΛM), en particulier de
+J+
+int,P(ΛM)∩M=J+
+int(ΛM)∩M=J+(ΛM)∩M= (J+(SM
+1)∩M)×J+(SM
+2) (cf (1.11)), et
+Φ est le produit tensoriel par ( wM
+U)−1=wM
+U. La compos´ ee de haut en bas, soit r−1
+P◦Φ◦rP,
+des applications verticales de droite du diagramme est la bijection Int WM.
+Partonsalorsde κ(Λ), prolongement de η(Λ)` aJ+(Λ). C’estune β-extensionrelativement
+` a ΛMsi et seulement si son image B(κ(Λ)), prolongement de η(Λ,ΛM) ` aP+(ΛoE)J1(ΛM),
+se prolonge ` a J+(ΛM) en une β-extension de η(ΛM). Puisque WMest une repr´ esentation de
+P+(ΛM
+oE) tout entier, cette condition ´ equivaut ` a : Int WM(B(κ(Λ))) se prolonge ` a J+
+int(ΛM)
+en uneβ-extension de ηint(ΛM) (proposition 2.33). Ceci signifie exactement que :
+B−1[IntWM(B(κ(Λ)))]est uneβ-extension de η(S1)⊗η(S2)relativement ` a ΛM.
+Il reste ` a faire le lien avec les applications rPde la colonne de gauche. L’application
+faisant commuter le diagramme du milieu est encore le produit tensorie l parwM
+Upuisque
+la bijection canonique est compatible ` a la torsion par les caract` ere s deP+(ΛoE)/P1(ΛoE)
+(lemme 1.5). Autrement dit :
+rP(κ(Λ)) =wM
+UrP(B−1[IntWM(B(κ(Λ)))]) = rP/parenleftbig
+wM
+UB−1[IntWM(B(κ(Λ)))]/parenrightbig
+./squaresolid
+Exemple 2.36. Pla¸ cons-nous dans un groupe symplectique avec dim W(1)= 1, de sorte que
+G≃Sp(2N+2,F)etGint≃Sp(2,F)×Sp(2N,F). Prenonsdans Z1lastratenulle[ S1,0,0,0]
+attach´ ee ` a une suite de r´ eseaux dont le fixateur est le sous-gr oupe d’Iwahori standard Ide
+Sp(2,F), dansZ2une stratesemi-simple sans composante nulle [ S2,n2,0,β2]. Alorsη(S1) est
+la repr´ esentation triviale de J1(S1), radical pro-unipotent de J(S1) =I. Un prolongement
+κ(S1) deη(S1) ` aIest un caract` ere Ξ de Ide la forme Ξ(a b
+c d) =ξ(a) pour un caract` ere ξ
+deo×
+Ftrivial sur 1+ pF. Un tel caract` ere est une β-extension relativement ` a une des deux
+suites de r´ eseaux minimales S+
+1etS−
+1extraites de S1si et seulement si κ(S1) est le caract` ere
+trivial de I(Corollaire 1.9, Proposition 1.6).
+Soientξet Ξ comme ci-dessus, soit κ(S2) un prolongement de η(S2) ` aJ(S2) et soit
+κP(Λ) le prolongement de ηP(Λ) ` aJP(Λ) dont la restriction ` a JP(Λ)∩Mestξ⊗κ(S2). Le
+th´ eor` eme 2.35 dit que IndJ(Λ)
+JP(Λ)κP(Λ) est une β-extension relativement ` a la suite de r´ eseaux
+ΛM=S+
+1⊥S2si et seulement si wM
+U(Ξ⊗κ(S2)) est une β-extension de 1 J1(S1)⊗η(S2)
+relativement ` a ΛM. D’apr` es nos hypoth` eses b0(S2) est un ordre autodual maximal, de sorte
+quewM
+Uκ(S2) est une β-extension si et seulement si κ(S2) en est une (D´ efinition 1.4 ; wM
+U
+est trivial sur les sous-groupes unipotents). Il reste :
+IndJ(Λ)
+JP(Λ)κP(Λ)est uneβ-extension relativement ` a ΛMsi et seulement si Ξ =wM
+U.
+La d´ etermination de wM
+Un´ ecessite d’expliciter le quotient J1
+ext(ΛM)∩U−/H1(ΛM)∩U−.
+A titre d’exemple, supposons que l’´ el´ ement β2engendre une extension totalement ramifi´ ee de
+degr´ e 2Ndans laquelle il a pour valuation −(2Nk+1),k∈N; il est en particulier minimal.
+On obtient J1
+ext(ΛM)∩U−sous la forme de matrices par blocs/parenleftBig1 0 0
+Y I0
+z X1/parenrightBig
+appartenant ` a
+Sp(2N+ 2,F) avecz∈pFetX∈/parenleftbig
+pk+1
+F,···,pk+1
+F,pk
+F,···,pk
+F/parenrightbig
+(Nfoispk+1
+FetNfois24 Corinne Blondel
+pk
+F). La description de H1(ΛM)∩U−est analogue avec cette fois N+1 foispk+1
+FetN−1
+foispk
+F, de sorte que le quotient cherch´ e est isomorphe ` a oF/pF. Pourx∈o×
+F, l’action
+deι(x)∈P+(ΛoE)∩Mest la multiplication par x−1dansoF/pF, la signature de cette
+permutation est l’unique caract` ere d’ordre 2 de k×
+F. En particulier, IndJ(Λ)
+JP(Λ)κP(Λ)est une
+β-extension relativement ` a ΛMsi et seulement si ξ2= 1etξ/ne}ationslash= 1.
+Le cas ΛM=S−
+1⊥S2est semblable.
+3 Application ` a la r´ eductibilit´ e
+3.1 Une paire couvrante et son alg` ebre de Hecke
+Soit [Λ,n,0,β]une strate semi-simple gauche et V=W(−1)⊕W(0)⊕W(1)une d´ ecomposition
+autoduale de Vtelle que W(0)contienne tous les Visauf un ; quitte ` a r´ eordonner les Vion
+peutdoncsupposer W(0)=V1∩W(0)⊥V2⊥···⊥Vlet/parenleftbig
+W(−1)⊕W(1)/parenrightbig
+⊥V1∩W(0)=V1.
+On demande que cette d´ ecomposition soit exactement subordonn´ ee ` a la strate [Λ ,n,0,β],
+c’est-` a-dire proprement subordonn´ ee et telle que b0(Λ(0)) soit un oE-ordre autodual maximal
+deEnd F(W(0))∩Betb0(Λ(1))unoE1-ordremaximaldeEnd E1(W(1))[21, Definition6.5]. Soit
+e1la p´ eriode sur E1de la suite de r´ eseaux Λ1; quitte ` a ´ echanger W(−1)etW(1), on suppose
+comme dans [21,§6.2] que l’unique entier q1v´ erifiant−e1
+2< q1≤e1
+2et Λ(1)(q1+1)/subsetn⋊teqlΛ(1)(q1)
+est positif ou nul ; comme la d´ ecomposition est proprement subord onn´ ee on a de fait q1>0.
+SoitPlesous-groupeparaboliquede Gstabilisantledrapeau W(−1)⊂W(−1)⊕W(0)⊂V,
+Uson radical unipotent, Mle fixateur de la d´ ecomposition autoduale, P−etU−les oppos´ es
+dePetUpar rapport ` a M. La d´ ecomposition V=W(−1)⊕W(0)⊕W(1)d´ etermine des injec-
+tions des alg` ebres End F(W(i)),i= 0,1,−1, dans End F(V), obtenues par un prolongement
+nul sur les deux autres sous-espaces. L’involution adjointe g/mapsto→¯gsur End F(V) se restreint ` a
+EndF(W(0))enuneinvolutionnot´ eedelamˆ eme fa¸ con, et induitunisomorphism e not´ ex/mapsto→ˆx
+de End F(W(1)) sur End F(W(−1)). On identifie le groupe GL F(W(1)) =/tildewideG(1)` a un sous-groupe
+deGau moyen du monomorphisme ιqui ` ag∈/tildewideG(1)associeι(g) = (ˆg−1,IW(0),g)∈G∩M.
+On note G(0)=G∩GLF(W(0)) ; ainsiM≃G(0)×ι(/tildewideG(1)).
+SoitE0
+1le corps des points fixes de l’involution adjointe dans E1ete0l’indice de ramifi-
+cation de E1surE0
+1; fixons une uniformisante ̟1deE1telle que ¯ ̟1= (−1)e0−1̟1. Notons
+enfinf1laE1/E0
+1-formeǫ-hermitienne non d´ eg´ en´ er´ ee sur V1associ´ ee ` a la restriction de h.
+Suivant toujours [21, §6.2] on choisit une base ordonn´ ee B(1)={v1,···,vd}du r´ eseau Λ(1)(0)
+suroE1. Les ´ el´ ements v−i, 1≤i≤d, d´ efinis par f1(vj,v−i) =̟1δij(1≤j≤d) constituent
+une base ordonn´ ee B(−1)={v−1,···,v−d}du r´ eseau Λ(−1)(1) suroE1. Ond´ efinit les´ el´ ements
+I1,−1,I−1,1deB1ets1ets̟
+1deG+
+Ecomme suit :
+I−1,1(vi) =v−ietI−1,1(v−i) = 0 pour 1≤i≤d;
+I1,−1(v−i) =vietI1,−1(vi) = 0 pour 1≤i≤d;
+s1=I−1,1+ǫ(−1)e0−1I1,−1+IW(0);s̟
+1=̟−1
+1I−1,1+ǫ̟1I1,−1+IW(0).
+Aucun des ´ el´ ements s1ets̟
+1n’appartient ` a P+(ΛoE) car la d´ ecomposition est exactement
+subordonn´ ee [21, Definition 6.8, Lemma 6.7], tandis que pour toute s uite autoduale ΛMdeRepr´ esentation de Weil et β-extensions 25
+oE-r´ eseaux telle que b0(ΛM) soit un oE-ordre autodual maximal contenant b0(Λ), l’un de ces
+deux ´ el´ ements appartient ` a P+(ΛM
+oE) [21, fin du§6.2]. Remarquons que
+s2
+1=ι/parenleftBig
+ǫ(−1)e0−1IW(1)/parenrightBig
+,(s̟
+1)2=ι(ǫIW(1)), s̟
+1s1=ι(ǫ̟1), s1s̟
+1=ι/parenleftBig
+ǫ(−1)e0−1̟−1
+1/parenrightBig
+.
+On obtient un automorphisme involutif σ1de/tildewideG(1)en posant : ι(σ1(g)) =s1ι(g)s−1
+1.
+Lemme 3.1 ([21] Lemma 7.11, Corollary 7.12). SoitJP=JP(Λ) =H1(Λ)(J(Λ)∩P).
+Posons ζ=ι/parenleftbig
+̟−1
+1/parenrightbig
+etζ′=ι/parenleftbig
+̟−1
+F/parenrightbig
+o` u̟Fest une uniformisante de F. Alorsζ′est un
+´ el´ ement fortement positif ou fortement n´ egatif du centr e deMet on a
+JPs1JPs̟
+1JP=JPζJPet(JPζJP)e(E1/F)=JPζe(E1/F)JP=JPζ′JP.
+Les mˆ emes r´ esultats sont valides en rempla¸ cant JP(Λ)parJP−(Λ).
+D´ emonstration. On reprend la d´ emonstration de [21, Lemma 7.11] en notant J1=J1(Λ)
+etc. Le changement de J0
+PenJPouJ+
+Pn’affecte que l’intersection avec M, normalis´ ee par s1
+ets̟
+1, la conjugaison par s1ous̟
+1´ echangeUetU−, et il suffit en fait de travailler dans /tildewideG.
+Notonse(i)la projection de VsurW(i)de noyau la somme des W(j)pourj/ne}ationslash=iet d´ etaillons
+les calculs de loc. cit.. On ´ ecrit ˜J1∩U−= 1+e(1)Je(0)+e(1)Je(−1)+e(0)Je(−1)d’o` u
+s1(˜J1∩U−)s−1
+1= 1+I−1,1e(1)Je(0)+I−1,1e(1)JI−1,1+e(0)JI−1,1e(1).
+Commee(i)appartient ` a b0(Λ),I−1,1e(1)appartient ` a b0(Λ) etb1(Λ)J⊂H, on en d´ eduit
+s1(˜J1∩U−)s−1
+1⊂˜H1∩U
+assertion un peu plus forte que loc. cit.(ii). On a de mˆ eme
+(s̟
+1)−1(˜J1∩U)s̟
+1⊂˜H1∩U−,
+qui est l’assertion (iii) de loc. cit.. Ces propri´ et´ es suffisent ` a conclure puisque les groupes
+JPetJP−sont les groupes ( H1∩U−)(J∩M)(J1∩U) et (J1∩U−)(J∩M)(H1∩U)./squaresolid
+Partons d’un caract` ere semi-simple θdeH1(β,Λ) et r´ esumons, d’apr` es [21, §5.3], les
+propri´ et´ es des objets associ´ es que voici :
+•son prolongement θP` aH1
+P(Λ) =H1(Λ)(J1(Λ)∩U) trivial sur J1(Λ)∩U;
+•l’unique repr´ esentation irr´ eductible ηPdeJ1
+P(Λ) =H1(Λ)(J1(Λ)∩P) contenant θP;
+•l’unique repr´ esentation irr´ eductible ηdeJ1(Λ) contenant θ, qui v´ erifie η≃IndJ1(Λ)
+J1
+P(Λ)ηP;
+•un prolongement κdeη` aJ+(Λ) ;
+•la repr´ esentation κPdeJ+
+P(Λ) =H1(Λ)(J+(Λ)∩P) obtenue par restriction de κ` a
+l’espace des J1(Λ)∩U-invariants de η, qui prolonge ηPet v´ erifie κ≃IndJ+(Λ)
+J+
+P(Λ)κP.26 Corinne Blondel
+Les groupes consid´ er´ es ont tous une d´ ecomposition d’Iwahori relative ` a ( M,P), les repr´ esen-
+tationsθP,ηPetκPsont triviales sur H1(Λ)∩U−etJ1(Λ)∩U=J+(Λ)∩Uet leur
+composante en Mv´ erifie :
+θP|H1(Λ)∩M=θ|H1(Λ)∩M=θ(0)⊗˜θ(1);
+ηP|J1(Λ)∩M≃η(0)⊗˜η(1);
+κP|J+(Λ)∩M≃κ(0)⊗˜κ(1);
+o` uθ(0)est un caract` ere semi-simple gauche de H1(β(0),Λ(0)),η(0)est l’unique repr´ esentation
+irr´ eductible de J1(β(0),Λ(0)) contenant θ(0),κ(0)est un prolongement de η(0)` aJ+(β(0),Λ(0)) ;
+˜θ(1)est un caract` ere semi-simple de /tildewideH1(β(1),Λ(1)) =/tildewideH1(2β(1),Λ(1)) attach´ e ` a la strate
+[Λ(1),n(1),0,2β(1)], ˜η(1)est l’unique repr´ esentation irr´ eductible de /tildewideJ1(β(1),Λ(1)) contenant
+˜θ(1)et ˜κ(1)est un prolongement de ˜ η(1)` a/tildewideJ(β(1),Λ(1)). En outre :
+Lemme 3.2 ([21] Proposition 6.3, Lemma 6.9, Corollary 6.10). L’involution σ1sta-
+bilise le groupe /tildewideJ(β(1),Λ(1)). De plus, si κest uneβ-extension de η, alorsκ(0)est une
+β(0)-extension, ˜κ(1)est une2β(1)-extension ´ equivalente ` a ˜κ(1)◦σ1ets1ets̟
+1entrelacent κP.
+Les groupes J(Λ)/J1(Λ) etJP(Λ)/J1
+P(Λ) sont isomorphes ` a P(ΛoE)/P1(ΛoE), lui-mˆ eme
+isomorphe ` a P(Λ(0)
+oE)/P1(Λ(0)
+oE)×ι(/tildewideP(L(1)
+oE)//tildewideP1(Λ(1)
+oE)) [21, Lemma 5.3]. Donnons-nous une
+repr´ esentation cuspidale irr´ eductible ρde ces groupes, que l’on peut ´ ecrire sous la forme
+ρ(0)⊗ι(/tildewideρ(1)) moyennant les identifications ci-dessus, et consid´ erons les repr ´ esentations λPde
+JP(Λ) etλdeJ(Λ) d´ efinies par
+λP=κP⊗ρ, λ=κ⊗ρ.
+D’apr` es [21, Lemma 6.1] – dont la d´ emonstration n’utilise pas le fait qu eκsoit une β-
+extension – elles sont irr´ eductibles, on a λ=c-IndJ
+JPλPet l’isomorphisme d’alg` ebres naturel
+deH(G,λP) surH(G,λ) envoie une fonction de support JPyJP,y∈GE, sur une fonction
+de support JyJ.
+Fixons, pour i= 2,···,l, uneoEi-base de la suite de r´ eseaux Λi, fixons une oE1-base de
+la suite de r´ eseaux Λ1∩W(0), et prenons la base B(−1)∪B(1)deW(−1)⊕W(1). Ceci d´ efinit
+un tore d´ eploy´ e maximal de G+
+Ede normalisateur N+. SoitN+
+Λl’intersection de N+avec le
+normalisateur dans G+
+EdeP0(ΛoE)∩M. On aN+
+Λ=N+
+Λ(0)×N+
+Λ/negationslash=0o` uN+
+Λ(0)etN+
+Λ/negationslash=0sont
+les objets analogues d´ efinis dans G+
+E∩GLF(W(0)) =G+
+E1∩GLF(W(0)∩V1)×G+
+E2×···G+
+El
+etG+
+E1∩GLF(W(−1)⊕W(1)) respectivement. Soit τune composante irr´ eductible de la
+restriction de ρ` aP0(ΛoE) ; on peut l’´ ecrire comme plus haut sous la forme τ(0)⊗ι(/tildewideρ(1)), de
+sorte que
+N+
+Λ(τ) :={n∈N+
+Λ/nτ≃τ}=N+
+Λ(0)(τ(0))×N+
+Λ/negationslash=0(ι(/tildewideρ(1)))
+etNΛ(τ) :=N+
+Λ(τ)∩G=/parenleftbig
+N+
+Λ(0)(τ(0))×N+
+Λ/negationslash=0(ι(/tildewideρ(1)))/parenrightbig
+∩G.
+Rappelons que s1ets̟
+1appartiennent ` a GEdans tous les cas except´ e celui du groupe
+sp´ ecial orthogonal avec β1= 0 et dim FW(1)impaire [21,§6.2]. Pla¸ cons-nous dans cette
+situation d’exception et supposons V1∩W(0)/ne}ationslash={0}. Le groupe G+
+E1∩GLF(W(0)∩V1) estRepr´ esentation de Weil et β-extensions 27
+alors un groupe orthogonal et l’on peut choisir p∈P+(Λ(0)
+oE∩V1) de d´ eterminant−1 tel que
+p2= 1 [21, Case (i) p.350] ; si V1∩W(0)est de dimension impaire on choisit p=−1. Alors
+ps1etps̟
+1v´ erifient toutes les propri´ et´ es de s1ets̟
+1utilis´ ees pr´ ec´ edemment et appartiennent
+` aGE. Dans tous les autres cas posons p= 1. Soit enfin Wle groupe de Weyl affine ` a
+deux g´ en´ erateurs engendr´ e par ps1etps̟
+1(par abus de langage - plus exactement c’est le
+quotient du groupe engendr´ e par ps1,ps̟
+1etι(o×
+E1) parι(o×
+E1)). Alors NΛ/negationslash=0/NΛ/negationslash=0∩P(Λ/negationslash=0
+oE)
+est isomorphe ` a Wlorsqueps1appartient ` a GE, c’est-` a-dire dans tous les cas except´ e celui
+dugroupe sp´ ecial orthogonal avec β1= 0,dimFW(1)impaire et V1∩W(0)={0}.
+Proposition 3.3 ([21]§6.4).PosonsJP=JP(Λ)et soitκun prolongement de η` aJ+(Λ).
+(i)Dans les trois cas suivants :
+(a)la repr´ esentation /tildewideλ(1)n’est pas ´ equivalente ` a /tildewideλ(1)◦σ1,
+(b)pn’entrelace pas τ(0),
+(c)ps1n’appartient pas ` a GE,
+le support de l’alg` ebre H(G,λP)estJP(IG(0)(λ(0))×ι(E×
+1))JP.
+(ii)Si/tildewideλ(1)est ´ equivalente ` a /tildewideλ(1)◦σ1etpentrelace τ(0)etps1appartient ` a GE, le support
+deH(G,λP)estJP(IG(0)(λ(0))⋊W)JP.
+D´ emonstration. Supposons tout d’abord que κest uneβ-extension standard de η; en
+particulier ˜ κ(1)est ´ equivalente ` a ˜ κ(1)◦σ1et la condition “ /tildewideλ(1)est ´ equivalente ` a /tildewideλ(1)◦σ1” est
+v´ erifi´ ee si et seulement si /tildewideρ(1)est ´ equivalente ` a /tildewideρ(1)◦σ1(variante de [5, d´ emonstration de la
+Proposition 5.3.2]). L’application de [21] (Proposition 6.15, Corollary 6.1 6 et d´ emonstration
+de la Proposition 6.18) montre que l’entrelacement de λPest contenu dans JPNΛ(τ)JP.
+D’autre part l’entrelacement de λPdansMest exactement l’entrelacement de λ(0)⊗ι(/tildewideλ(1))
+[6, Proposition 6.3 (ii)]. Si /tildewideρ(1)n’est pas ´ equivalente ` a /tildewideρ(1)◦σ1ou sips1n’appartient pas
+` aGE,NΛ/negationslash=0(ι(/tildewideρ(1))) se r´ eduit ` a ι(E×
+1) qui entrelace effectivement la repr´ esentation, d’o` u le
+r´ esultat. Il en est de mˆ eme si pn’entrelace pas τ(0), carps1alors n’entrelace pas τ. Dans le
+second cas au contraire, ps1entrelace τ, doncρ, doncλP.
+Un prolongement κarbitraire de η` aJ+(Λ) peut s’´ ecrire sous la forme κs⊗χo` uκs
+est uneβ-extension standard de ηetχest un caract` ere de P+(ΛoE)/P1(ΛoE) (par unicit´ e
+des op´ erateurs d’entrelacement de η` a scalaire pr` es [21, Proposition 3.5]). Ecrivons χ=
+χ(0)⊗ι(/tildewideχ(1)). Alors l’entrelacement de λP= (κs⊗χ)P⊗ρ=κs
+P⊗χρest d´ etermin´ e par
+l’action de σ1sur/tildewideχ(1)⊗/tildewideρ(1). Comme /tildewideκs(1)est ´ equivalente ` a /tildewideκs(1)◦σ1, la premi` ere partie
+donne le r´ esultat. Noter que si κest uneβ-extension le lemme 3.2 s’applique aussi bien ` a κ
+qu’` aκs, le caract` ere /tildewideχ(1)est donc σ1-invariant (voir aussi [21, Corollary 6.13]), de sorte que
+/tildewideρ(1)estσ1-invariante si et seulement si /tildewideρ(1)⊗/tildewideχ(1)l’est. /squaresolid
+Corollaire 3.4. Soitκun prolongement de η` aJ+(Λ)etλ=κ⊗ρcomme ci-dessus.
+NotonsJ(0)=J(β(0),Λ(0))et/tildewideJ(1)=/tildewideJ(β(1),Λ(1)). La paire (JP,λP)est une paire couvrante
+de(J(0)×ι(/tildewideJ(1)), λ(0)⊗ι(˜λ(1)))dansG. Il en est de mˆ eme de la paire analogue (JP−,λP−).28 Corinne Blondel
+Dans le cas (i) de la proposition, l’alg` ebre H(G,λP)est isomorphe ` aH(M,λP|J∩M). En
+particulier, la repr´ esentation induite parabolique d’un e repr´ esentation irr´ eductible de Mde
+typeλP|J∩Mest toujours irr´ eductible.
+Dans le cas (ii) de la proposition, si IG(0)(λ(0)) =J(0), l’alg` ebreH(G,λP)est une alg` ebre
+de convolution sur W. Sa structure est d´ etermin´ ee par les relations quadratiq ues v´ erifi´ ees
+par deux g´ en´ erateurs T1etT̟
+1de supports respectifs JPps1JPetJPps̟
+1JP.
+D´ emonstration. Danslecas(i) delaproposition, l’entrelacement de λPest form´ ededoubles
+classes d’´ el´ ements de M, la paireest donccouvrante par [6, Comments 8.2] ; lescons´ equen ces
+sur l’irr´ eductibilit´ e sont prouv´ ees dans loc. cit. et rappel´ ees au paragraphe suivant.
+Dans l’autre cas, l’entrelacement de λPestJPWJP. Grˆ ace au lemme 3.1, qui reste valide
+pourps1etps̟
+1, et aux r´ esultats de loc. cit., il suffit de montrer que les´ el´ ements de H(G,λP)
+de support JPps1JPetJPps̟
+1JPsont inversibles, ce que l’on peut faire comme dans [10] §2.2
+(en v´ erifiant que les d´ emonstrations des lemmes 2.10 et 2.13 s’appliq uent mot pour mot) ou
+en passant directement aux ´ enonc´ es plus pr´ ecis ci-apr` es.
+Pour passer de P` aP−il suffit de remarquer que l’action de κ|JP−sur lesJ1(Λ)∩U−-
+invariants de ηest aussi ´ egale ` a κ(0)⊗˜κ(1)surJ∩M. C’est ´ el´ ementaire en prenant le
+mod` ele de η≃c-IndJ1
+J1
+PηPform´ e des fonctions de J1∩U−/H1∩U−dans l’espace de ηP: les
+J1(Λ)∩U−-invariants sont les fonctions constantes. /squaresolid
+Pour obtenir des pr´ ecisions sur les relations quadratiques du seco nd cas ci-dessus nous
+suivons encore [21], §7.1 et 7.2.2. Commen¸ cons par construire deux suites de oE-r´ eseaux
+autoduales M0=M1
+0⊥Λ2···⊥ΛletM1=M1
+1⊥Λ2···⊥Λlcomme dans loc. cit.§7.2.2,
+` a partir des suites de oE1-r´ eseaux M1
+0etM1
+1dansV1, de p´ eriode 2 sur E1, caract´ eris´ ees par :
+M1
+0(0) = Λ1(1−q1),M1
+0(1) = Λ1(q1),M1
+1(0) = Λ1(1−e1
+2),M1
+1(1) = Λ1(e1
+2).
+Les ordres b0(M0) etb0(M1) sont des oE-ordres autoduaux maximaux contenant b0(Λ) et la
+d´ ecomposition V=W(−1)⊕W(0)⊕W(1)estsubordonn´ ee auxstrates[ Mi,nMi,0,β](i= 0,1).
+Enfins1appartient ` a P+(M0,oE),s̟
+1appartient ` a P+(M1,oE) et l’on a :
+P+(ΛoE) =/parenleftbig
+P+(M1,oE)∩P−/parenrightbig
+P1(M1,oE) =/parenleftbig
+P+(M0,oE)∩P/parenrightbig
+P1(M0,oE).(3.5)
+En particulier, la repr´ esentation ρ=ρ(0)⊗ι(/tildewideρ(1)) deJ(Λ)/J1(Λ)≃P(ΛoE)/P1(ΛoE) s’´ etend
+en une repr´ esentation toujours not´ ee ρdu sous-groupe parabolique P(ΛoE)/P1(M1,oE) de
+P(M1,oE) triviale sur son radical unipotent P1(ΛoE)/P1(M1,oE), et de mˆ eme avec M0,oE.
+Avec ces conventions on a :
+Proposition 3.6 ([21](7.3)). Siκest uneβ-extension de ηrelative ` a M0, il y a un homo-
+morphisme d’alg` ebres injectif
+j0:H(P(M0,oE),ρ)֒→ H(G,λP)
+pr´ eservant le support : Supp(j0(φ)) =JPSupp(φ)JP. Il en est de mˆ eme en rempla¸ cant M0
+parM1, avec un homomorphisme j1.Repr´ esentation de Weil et β-extensions 29
+D´ emonstration. Il faut justifier l’usage de [21, §7.1], qui s’occupe de repr´ esentations de
+groupes de la forme J0(Λ), pour nos repr´ esentations de groupes J(Λ). Par hypoth` ese, il exis-
+te uneβ-extension κ(M0), repr´ esentation de J+(M0), dont la restriction ` a P+(ΛoE)J1(M0)
+correspond ` a la repr´ esentation κdeJ+(Λ) par la bijection canonique B. Reprenant la
+d´ emonstration de [21] Proposition 7.1 pas ` a pas, on obtient un isom orphisme d’alg` ebres
+de Hecke, pr´ eservant le support, de H(G,κ(M0)|P(ΛoE)J1(M0)⊗ρ) surH(G,κ|J(Λ)⊗ρ).
+L’isomorphisme de la proposition 7.2 de loc. cit. reste valide puisque κ(M0)|P(ΛoE)J1(M0)
+se prolonge ` a J(M0). Le dernier isomorphisme de la composition de [21, (7.3)] provient de
+[21, Lemma 6.1]. /squaresolid
+Corollaire 3.7. Soitκun prolongement de η` aJ+(Λ)etλ=κ⊗ρcomme ci-dessus.
+Supposons que la repr´ esentation /tildewideλ(1)est ´ equivalente ` a /tildewideλ(1)◦σ1, quepentrelace τ(0)et que
+ps1appartient ` a GE. Soient χ0etχ1des caract` eres de P+(ΛoE)/P1(ΛoE)tels queκ⊗χ−1
+0
+soit une β-extension relative ` a M0etκ⊗χ−1
+1uneβ-extension relative ` a M1. Moyennant
+une normalisation convenable, les relations quadratiques v´ erifi´ ees par les g´ en´ erateurs T1et
+T̟
+1deH(G,λP)sont respectivement les relations quadratiques v´ erifi´ ee s par
+•le g´ en´ erateur deH(P(M0,oE),ρ⊗χ0), de support P(ΛoE)ps1P(ΛoE),
+•le g´ en´ erateur deH(P(M1,oE),ρ⊗χ1), de support P(ΛoE)ps̟
+1P(ΛoE).
+Ce corollaire fournit le moyen de calculer les relations quadratiques ch erch´ ees, ` a ho-
+moth´ etie pr` es, dans un groupe r´ eductif fini. Nous indiquons dan s le paragraphe suivant les
+conclusions qu’une telle connaissance permet de tirer en termes de r ´ eductibilit´ e d’induites.
+3.2 Relations quadratiques et points de r´ eductibilit´ e
+Nous expliquons dans cette section comment les coefficients des rela tions quadratiques satis-
+faites par les deux g´ en´ erateurs de l’alg` ebre de Hecke ci-dessus , sous les hypoth` eses communes
+aux corollaires 3.4 et 3.7, d´ eterminent enti` erement les parties r´ eelles des quatre points de
+r´ eductibilit´ e de l’induite correspondante.
+Commen¸ cons par rappeler la propri´ et´ e caract´ eristique des p aires couvrantes dans le
+groupe classique G, pour des repr´ esentations supercuspidales d’un sous-groupe d e Levi maxi-
+malM≃GL(n,F)×L. On se donne une repr´ esentation supercuspidale πde GL(n,F), un
+type (Jn,λn) pourπ, une repr´ esentation supercuspidale σdeL, un type ( J0,λ0) pourσ, un
+sous-groupe parabolique PdeGde facteur de Levi M, et on suppose construite une paire
+couvrante ( J,λ) de (Jn×J0,λn⊗λ0) dansG. Par d´ efinition [6, §7] il existe un morphisme
+d’alg` ebres injectif t:H(M,λn⊗λ0)−→H(G,λ) rendant le diagramme suivant commutatif :
+R[π⊗σ,M](G)−→ Mod-H(G,λ)
+indG
+P↑ t∗↑
+R[π⊗σ,M](M)h−→Mod-H(M,λn⊗λ0)(3.8)
+Noustravaillonsiciavecl’induitenormalis´ eeindG
+Petavecdesmodules` adroitesurlesalg` ebres
+de Hecke. Le foncteur t∗associe ` a un module Xle module Hom H(M,λn⊗λ0)(H(G,λ),X),30 Corinne Blondel
+dans lequel td´ etermine la structure de module de H(G,λ), et le foncteur hassocie ` a une
+repr´ esentation τle module Hom Jn×J0(λn⊗λ0,τ).
+Les fl` eches horizontales de (3.8) sont des ´ equivalences de cat´ e gories, ce diagramme im-
+plique donc que pour tout s∈C, la repr´ esentation indG
+Pπ|det|s⊗σest r´ eductible si et
+seulement si leH(G,λ)-module t∗h(π|det|s⊗σ) est r´ eductible.
+Choisissons pour σun type ( J0,λ0) construit par Stevens [21, Theorem 7.14] de sorte que
+σ≃IndL
+J0λ0. Alors l’alg` ebreH(M,λn⊗λ0) est isomorphe ` a H(GL(n,F),λn).
+Choisissons pour πun type simple maximal de Bushnell-Kutzko ( Jn,λn), attach´ e ` a une
+strate simple [Λ n,tn,0,βn]. Posons En=F[βn] et notons ̟Enune uniformisante de En.
+D’apr` es [5,§6], l’entrelacement de λnest ´ egal ` a /hatwiderJn=E×
+nJn=̟Z
+EnJnet on a une bijection
+Λτ/mapsto−→τ= IndGL(n,F)
+/hatwideJnΛτentre l’ensemble des prolongements de λn` a/hatwideJnet l’ensemble des
+classes d’´ equivalence de repr´ esentations irr´ eductibles de la cla sse d’inertie de π. SoitZun
+´ el´ ement deH(GL(n,F),λn) de support ̟EnJn; on a par [5,§5.5] :
+H(GL(n,F),λn)≃C[Z,Z−1].
+En particulier, les repr´ esentations irr´ eductibles de cette alg` eb re sont des caract` eres, d´ etermi-
+n´ es par leur valeur en Z. Le groupe X0des caract` eres non ramifi´ es de GL( n,F) agit sur
+H(GL(n,F),λn) parχ(f)(x) =χ(x)f(x) (χ∈X0,f∈H(GL(n,F),λn),x∈GL(n,F)), et
+siτest une repr´ esentation irr´ eductible comme ci-dessus on a : h(τ⊗χ−1) =h(τ)◦χsoit :
+h(τ⊗χ−1)(Z) =χ(̟En)h(τ)(Z). (3.9)
+Le plongement de GL( n,F) dansM⊂Gd´ etermine une involution g/mapsto→ˆg−1sur GL(n,F)
+d´ eduite de l’involution adjointe (voir le d´ ebut du paragraphe pr´ ec ´ edent) ; on dira que π
+estautoduale si sa classe d’isomorphisme est fix´ ee par cette involution. (Cela sign ifie que
+πest ´ equivalente ` a sa contragr´ ediente dans les cas symplectique et orthogonal, ` a la con-
+tragr´ ediente de sa compos´ ee avec la conjugaison de FsurF0dans le cas unitaire.) Si aucune
+des repr´ esentations π|det|sn’est autoduale on sait que la repr´ esentation induite est toujours
+irr´ eductible ([18, Corollary 1.8] ; voir aussi [23, Lemmas 4.1 and 5.1]).
+Supposons πautoduale et faisons les hypoth` eses suivantes :
+(i) L’alg` ebreH(G,λ) a deux g´ en´ erateurs T0etT1v´ erifiant des relations quadratiques de
+la forme ( Ti−ω1
+i)(Ti−ω2
+i) = 0 (i= 0,1) et on a t(Z) =T0T1.
+(ii) Les quotients des valeurs propres sont des nombres r´ eels str ictement n´ egatifs. On
+ordonne alors les valeurs propres de fa¸ con que : ω1
+i/ω2
+i=−qriavecri≥0.
+Soitξun caract` ere deH(M,λn⊗λ0). La repr´ esentation t∗(ξ) est de dimension 2, elle est
+r´ eductible si et seulement si elle contient un caract` ere, c’est-` a-dire si et seulement si il existe
+un caract` ere αdeH(G,λ) tel que ξ=α◦t[1, Proposition 1.13]. Ainsi la r´ eductibilit´ e de
+indG
+Pπ|det|s⊗σestobtenueexactement pourquatrevaleursde s(compt´ eesavecmultiplicit´ e),
+donn´ ees par h(π|det|s)(Z) =ωj
+0ωk
+1, j,k= 1,2, c’est-` a-dire
+h(π|det|s)(Z)∈ {α0=ω2
+0ω2
+1,−qr0α0,−qr1α0, qr0+r1α0}. (3.10)Repr´ esentation de Weil et β-extensions 31
+On sait par ailleurs que R[π,GL(n,F)](GL(n,F)) contient exactement deux repr´ esentations
+irr´ eductibles autoduales non ´ equivalentes : πetπ|det|e
+2n2iπ
+logq, pour un unique entier edi-
+visantn, qui est l’indice deramificationde EnsurF[5, Lemma 6.2.5]. Onsait enfin[18, The-
+orem1.6]quepourchaquerepr´ esentationsupercuspidaleautod ualeπ′deGL(n,F)l’ensemble
+des points de r´ eductibilit´ e r´ eelsde indG
+Pπ′|det|s⊗σest soit vide, soit de la forme {a,−a}
+pour un r´ eel positif ou nul a. Ainsi il existe au maximum quatre repr´ esentations donnant
+lieu ` a r´ eductibilit´ e (compt´ ees avec multiplicit´ e), ` a savoir : π◦|det|±s1etπ◦|det|±s2+e
+2n2iπ
+logq,
+pour uns1∈Ret uns2∈R, dont les images par hsont, via (3.9), les caract` eres ayant pour
+valeur en Zun ´ el´ ement de l’ensemble :
+{qs1n
+eh(π)(Z),q−s1n
+eh(π)(Z),−qs2n
+eh(π)(Z),−qs2n
+eh(π)(Z)}.(3.11)
+Sousleshypoth` eses(i)et(ii), unsous-ensemble Xconvenablede(3.11)doitco¨ ıncideravec
+(3.10). Comme (3.10) contient des paires d’´ el´ ements de quotien t n´ egatif, le sous-ensemble X
+doit avoir la mˆ eme propri´ et´ e : c’est (3.11) tout entier. Les hypo th` eses faites entraˆ ınent donc
+que les induites consid´ er´ ees ont effectivement des points de r´ ed uctibilit´ e. En comparant les
+quotients r´ eels positifs de valeurs de (3.10) et (3.11) on obtient la f ormule suivante :
+Proposition 3.12. Supposons que l’alg` ebre H(G,λ)a deux g´ en´ erateurs v´ erifiant des rela-
+tions quadratiques de la forme (Ti−ω1
+i)(Ti−ω2
+i) = 0 (i= 0,1)dont les quotients des valeurs
+propres s’´ ecrivent ω1
+i/ω2
+i=−qriavecri≥0, et tels que t(Z) =T0T1. Les param` etres r0,r1
+d´ eterminent alors les parties r´ eelles des points de r´ edu ctibilit´ e de indG
+Pπ|det|s⊗σ: ce sont
+les ´ el´ ements de l’ensemble
+{±e
+2n(r0+r1),±e
+2n(r0−r1)}.
+Remarque 3.13. Cettepropositionnepermet pasded´ eterminer s´ epar´ ement les deuxpoints
+de r´ eductibilit´ e r´ eels correspondant ` a chacune des deux repr ´ esentations autoduales πet
+π|det|e
+2n2iπ
+logq. Elle ne fournit que l’ensemble total des parties r´ eelles des points de r´ eductibi-
+lit´ e pour une repr´ esentation autoduale de la classe d’inertie de π.
+Remarque 3.14. Lorsque la structure de H(G,λ) est donn´ ee par un ´ enonc´ e semblable au
+corollaire 3.7 ci-dessus, l’hypoth` ese (i) est automatiquement v´ er ifi´ ee. L’hypoth` ese (ii) l’est
+aussi par [15,§6.7], apr` es [11] : le quotient des valeurs propres est −pc, i.e. l’oppos´ e du
+quotient des degr´ es des deux repr´ esentations irr´ eductibles f ormant l’induite de la cuspidale
+dans le groupe r´ eductif fini.
+Remarque 3.15. Pour des repr´ esentations σetπdont les types v´ erifient les hypoth` eses
+du paragraphe 3.1 (voir ` a ce sujet la remarque 3.18 plus loin) on retr ouve les r´ esultats de
+r´ eductibilit´ e connus par ailleurs. En effet, si Gest un groupe symplectique ou unitaire les
+´ enonc´ es de ce paragraphe se simplifient (car p= 1 ets1appartient ` a GE) et, compl´ et´ es par la
+proposition3.12, fournissent l’irr´ eductibilit´ e de indG
+Pπ|det|s⊗σsiaucunedesrepr´ esentations
+π|det|sn’est autoduale, et sa r´ eductibilit´ e pour un unique s≥0 siπest autoduale. Il en est
+de mˆ eme si Gest sp´ ecial orthogonal en dimension impaire : le seul cas o` u s1n’appartient pas
+` aGEest celui o` u β1= 0, les espaces V2,···,Vlsont alors de dimension paire, car chacun
+porte une strate simple gauche non nulle, donc W(0)∩V1est de dimension impaire d’o` u
+p=−1 etps1appartient ` a GE.32 Corinne Blondel
+Le cas du groupe sp´ ecial orthogonal en dimension paire est plus d´ elicat. L’irr´ eductibilit´ e
+de indG
+Pπ|det|s⊗σsi aucune des repr´ esentations π|det|sn’est autoduale est donn´ ee par le
+corollaire3.4. Via les lemmes deZhang cit´ es plus haut, onest assur´ e s de l’existence depoints
+de r´ eductibilit´ e r´ eels, pour πautoduale, dans tous les cas sauf le suivant : Gest un groupe
+sp´ ecial orthogonal et nest impair. De fait, si nest pair, s1appartient ` a GE,p= 1 et l’on
+retrouve les situations d´ ecrites plus haut. Supposons nimpair ; l’autodualit´ e de πentraˆ ıne
+n= 1 par [2, Theorem 1]. Le cas d’exception est donc celui de GL 1(F)×SO(2t,F), Levi de
+SO(2t+2,F). D’apr` es la proposition 3.3 il existera des points de r´ eductibilit´ e s i et seulement
+sipentrelace τ(0)etps1appartient ` a GE– en particulier il n’en existe pas si W(0)∩V1={0}.
+On pourrait comparer cette condition avec celle de Jantzen [12] selo n laquelle l’induite a des
+points de r´ eductibilit´ e si et seulement si σest ´ equivalente ` a sa conjugu´ ee par un ´ el´ ement
+du groupe orthogonal de d´ eterminant −1 (siW(0)∩V1={0}il est facile de voir que cette
+condition n’est jamais satisfaite). Ce n’est pas notre propos ici.
+Ces questions de r´ eductibilit´ e ont ´ et´ e beaucoup ´ etudi´ ees, en premier lieu par Shahidi [17,
+Theorem 8.1] ; pour le cas d’exception citons par exemple [14], [23], [12]...
+3.3 Conclusion
+Passons enfin ` a l’´ etude de la r´ eductibilit´ e sous les hypoth` eses communes aux paragraphes
+3.1 et 2.4, c’est-` a-dire une d´ ecomposition autoduale V=W(−1)⊕W(0)⊕W(1)exactement
+subordonn´ ee ` a [Λ ,n,0,β], avecW(0)=V2⊥···⊥ VletW(−1)⊕W(1)=V1. On pose
+Z1=W(−1)⊕W(1),Z2=W(0), comme au paragraphe 2.4, et on note G+
+1(resp.G+
+(0)) le
+groupe d’automorphismes de la restriction de la forme h` aZ1(resp.Z2),G1(resp.G(0)) sa
+composante neutre, de sorte que G+
+int=G+
+(0)×G+
+1.
+SoientPetMcomme en 3.1 et P1,M1leurs intersections avec G1. Le corollaire 3.4
+nous d´ ecrit une paire couvrante ( JP,λP), dansG, de la paire ( J(0)×ι(/tildewideJ(1)), λ(0)⊗ι(˜λ(1)))
+dansM=G(0)×ι(GLF(W(1))). Supposons que IG(0)(λ(0)) =J(0); la paire ( J(0), λ(0)) est
+alors un type pour la classe d’inertie de la repr´ esentation supercus pidaleσ=c-IndG(0)
+J(0)λ(0)de
+G(0)([21, Corollary 6.19] ; κ(0)est tordue d’une β-extension par un caract` ere, comme dans
+la d´ emonstration de la proposition 3.3). D’autre part b0(Λ(1)) est un oE1-ordre maximal
+de End E1(W(1)) donc ( /tildewideJ(1),˜λ(1)) est un type pour une classe d’inertie de repr´ esentations
+supercuspidales de GL F(W(1)) [5, Theorem 6.2.2] dont on note πun ´ el´ ement. Le diagramme
+3.8reliealorsl’´ etude der´ eductibilit´ e des induites IndG
+Pπ|det|s⊗σ` alastructure deH(G,λP).
+Notre but est de comparer les parties r´ eelles des points de r´ educ tibilit´ e des induites
+paraboliques IndG
+Pπ|det|s⊗σdansGaux parties r´ eelles des points de r´ eductibilit´ e des
+induites paraboliques IndG1
+P1π|det|sdansG1. DansG1nous avons une situation tout ` a fait
+analogue c’est-` a-dire une paire couvrante ( JP1(Λ1),λP1) de la paire ( ι(/tildewideJ(1)), ι(˜λ(1))) dansM1
+(ce qui d´ efinit λP1).
+D’apr` es le corollaire 3.4, si la repr´ esentation /tildewideλ(1)n’est pas ´ equivalente ` a /tildewideλ(1)◦σ1ou sis1
+n’appartient pas ` a GE, les alg` ebresH(G,λP) etH(G1,λP1) sont isomorphes ` a H(M,λP|J∩M)
+etH(M1,λP1|JP1(Λ1)∩M1) respectivement et les induites IndG
+Pπ|det|s⊗σet IndG1
+P1π|det|ssont
+toujours irr´ eductibles. Le premier cas est celui o` u la classe d’iner tie deπne contient pasRepr´ esentation de Weil et β-extensions 33
+de repr´ esentation autoduale et l’on retrouve le fait qu’une repr´ e sentation supercuspidale
+autoduale poss` ede un type autodual.
+Pla¸ cons-nous d´ esormais dans le cas autodual o` u /tildewideλ(1)est ´ equivalente ` a /tildewideλ(1)◦σ1, et sup-
+posons que s1appartient ` a GE. L’´ enonc´ e du corollaire 3.7, dont on reprend les notations,
+peut ˆ etre pr´ ecis´ e en ne faisant intervenir que les intersections avecG1. En effet P(M0,oE) est
+´ egal au produit P(M(0)
+0,oE)×P(M1
+0,oE), de mˆ eme pour P(M1,oE) etP(ΛoE). Comme b0(Λ(0))
+est unoE-ordre autodual maximal de End F(W(0))∩Bon aP(M(0)
+0,oE) =P(M(0)
+1,oE) =P(Λ(0)
+oE),
+de sorte que (cf. (3.5)) les alg` ebres :
+H(P(M0,oE),ρ⊗χ0)≃ H(P(M1
+0,oE),ι(ρ(1)⊗χ(1)
+0))
+etH(P(M1,oE),ρ⊗χ1)≃ H(P(M1
+1,oE),ι(ρ(1)⊗χ(1)
+1))
+d´ eterminent les relations quadratiques satisfaites par les g´ en´ e rateurs deH(G,λP). De la
+mˆ eme fa¸ con, les relations quadratiques satisfaites par les g´ en´ erateurs deH(G1,λP1) sont
+d´ etermin´ ees par les alg` ebres
+H(P(M1
+0,oE),ι(ρ(1)⊗χ(1)
+1,0)) etH(P(M1
+1,oE),ι(ρ(1)⊗χ(1)
+1,1))
+o` uχ1,0etχ1,1sont des caract` eres de P+(Λ1
+oE)/P1(Λ1
+oE) tels que κ1⊗χ−1
+1,0soit uneβ-extension
+relative ` a M1
+0etκ1⊗χ−1
+1,1uneβ-extension relative ` a M1
+1.
+Danslesdeuxcasils’agitd’alg` ebres H(G,γ)relatives` aunerepr´ esentationsupercuspidale
+autoduale γdu sous-groupe de Levi de Siegel d’un groupe r´ eductif fini G, soit
+P(Λ1
+oE)/P1(Λ1
+oE)≃/tildewideP(Λ(1)
+oE)//tildewideP1(Λ(1)
+oE)≃GL(f,kE1)
+o` ukE1est le corps r´ esiduel de E1etfla dimension de W(1)surE1. (On rappelle comme au
+paragraphe 3.2 que la notion d’autodualit´ e correspond ici ` a γ◦σ1≃γ, et que l’action de σ1
+sur GL(f,kE1) est ´ equivalente ` a g/mapsto→tg−1siβ1= 0 ou si E1est ramifi´ ee sur E0
+1, ` ag/mapsto→t¯g−1
+siE1est non ramifi´ ee sur E0
+1, la barre d´ esignant alors l’action de l’´ el´ ement non trivial du
+groupe de Galois de kE1surkE0
+1.) Les relations quadratiques correspondantes peuvent ˆ etre
+calcul´ ees au cas par cas ` a partir des travaux de Lusztig. Quoi q u’il en soit, il arrive que ces
+relations soient ind´ ependantes du choix de la repr´ esentation cus pidale autoduale γ. Lorsque
+c’est le cas, la proposition 3.12 implique que les parties r´ eelles des point s de r´ eductibilit´ e de
+IndG
+Pπ|det|s⊗σet IndG1
+P1π|det|ssont les mˆ emes.
+Exemple 3.16. SiE1est non ramifi´ ee sur E0
+1et de degr´ e maximal [ E1:F] = dim FW(1), le
+groupeGest isomorphe dans les deux cas ` a U(1,1)(kE1/kE0
+1). Les relationsquadratiques sont
+toujours homoth´ etiques ` a X2= (qE1−1)X+qE1, avec quotient des valeurs propres −qE1.
+L’application de la proposition 3.12 donne pour parties r´ eelles des poin ts de r´ eductibilit´ e 0
+(double) et±1
+2. SiE1est ramifi´ ee sur E0
+1et de degr´ e maximal, le groupe Gest isomorphe ` a
+SL(2,kE1)) ouO(1,1)(kE1)). Dans SL(2 ,kE1)) la relation quadratique est soit X2= (qE1−
+1)X+qE1, soitX2= 1, avec quotient des valeurs propres −qE1ou−1, elle est sensible ` a la
+torsion ; dans O(1,1)(kE1)) c’estX2= 1. Les parties r´ eelles des points de r´ eductibilit´ e sont
+soit 0 (double) et ±1
+2, soit 0 (quadruple).
+Dans ces deux cas, pour toute repr´ esentation σv´ erifiant les hypoth` eses du pr´ esent para-
+graphe (voir la remarque 3.18), les points de r´ eductibilit´ e de IndG
+Pπ|det|s⊗σsont de partie
+r´ eelle strictement inf´ erieure ` a 1.34 Corinne Blondel
+Si les relations ne sont pas ind´ ependantes du choix de γ, la comparaison cherch´ ee impose
+de comparer les caract` eres χ0etχ1d’une part, χ1,0etχ1,1d’autre part ; c’est l` a qu’intervient
+le th´ eor` eme 2.35. Soit donc w0
+U(resp.w1
+U−) le caract` ere de P+(ΛoE), trivial sur P1(ΛoE),
+qui ` ax∈P+(ΛoE)∩Massocie la signature de la permutation v/mapsto→xvx−1deJ1
+ext(M0)∩
+U−/ H1(M0)∩U−(resp.J1
+ext(M1)∩U/H1(M1)∩U). Rappelons que ces caract` eres ne
+d´ ependent que de la strate [Λ ,n,0,β] (Lemme 2.22).
+Les hypoth` eses du paragraphe 2.4 sont v´ erifi´ ees par le sous-g roupe parabolique Pet la
+suite de r´ eseaux M0, ainsi que par P−etM1(3.5) et les repr´ esentations κetκ1sont reli´ ees
+par la condition :
+rP(κ) =rP/parenleftbig
+κ(0)⊗κ1/parenrightbig
+=κ(0)⊗ι(/tildewideκ(1)) =rP−(κ) =rP−/parenleftbig
+κ(0)⊗κ1/parenrightbig
+.
+D’apr` es le th´ eor` eme 2.35, κ⊗χ−1
+0est uneβ-extension relative ` a M0si et seulement si
+w0
+Uχ−1
+0(κ(0)⊗κ1) est une β-extension relative ` a M0; de mˆ eme κ⊗χ−1
+1est uneβ-extension
+relative ` a M1si et seulement si w1
+U−χ−1
+1(κ(0)⊗κ1) est une β-extension relative ` a M1. Sur la
+composante en W(0)la torsion par w0
+Uouw1
+U−ne change pas le fait d’ˆ etre une β-extension
+car ces caract` eres sont triviaux sur tous les sous-groupes unip otents [21, Theorem 4.1]. La
+condition se simplifie donc comme suit :
+κ⊗χ−1
+0est uneβ-extension relative ` a M0si et seulement si (χ(0)
+0)−1κ(0)est uneβ-
+extension et w0
+Uχ−1
+0κ1est uneβ-extension relative ` a M1
+0;
+et de mˆ eme en rempla¸ cant M0parM1etχ0parχ1. Notons au passage que χ0etχ1
+diff` erent d’un caract` ere autodual [21, Remark 6.12] tandis que w0
+Uetw1
+U−sont quadratiques
+ou triviaux. Finalement :
+Proposition 3.17. Notons γ0=ι(ρ(1)⊗χ(1)
+0)etγ1=ι(ρ(1)⊗χ(1)
+1). Les g´ en´ erateurs de
+H(G,λP)se calculent dans
+H(P(M1
+0,oE)/P1(M1
+0,oE), γ0)etH(P(M1
+1,oE)/P1(M1
+1,oE), γ1)
+tandis que ceux de H(G1,λP1)se calculent dans
+H(P(M1
+0,oE)/P1(M1
+0,oE), γ0w0
+U)etH(P(M1
+1,oE)/P1(M1
+1,oE), γ1w1
+U−),
+o` u le caract` ere w0
+U(resp.w1
+U−) deP(Λ1
+oE)/P1(Λ1
+oE)associe ` a x∈P(Λ1
+oE)la signature de la
+permutation v/mapsto→xvx−1deJ1
+ext(M1
+0)∩U−/ H1(M1
+0)∩U−(resp.J1
+ext(M1
+1)∩U/ H1(M1
+1)∩U).
+Remarque 3.18. Bien entendu ce r´ esultat ne permet pas de traiter la r´ eductibilit´ e de
+IndG
+Pπ|det|s⊗σpour n’importe quelle paire ( π,σ), puisque les types attach´ es ` a πetσont
+´ et´ e construits ` a partir des ´ el´ ements suivants :
+•une strate semi-simple gauche [Λ ,n,0,β] et une d´ ecomposition autoduale exactement
+subordonn´ ee ` a cette strate V=W(−1)⊕W(0)⊕W(1)telle que V1=W(−1)⊕W(1);
+•uncaract` eresemi-simplegauche θdontlarestriction θ(0)⊗ι(˜θ(1))` aH1(Λ)∩Md´ etermine
+un caract` ere semi-simple gauche θ(0)intervenant dans σet un caract` ere simple ˜θ(1)
+intervenant dans π.Repr´ esentation de Weil et β-extensions 35
+On ne peut l’appliquer qu’` a des paires de repr´ esentations poss´ ed ant des strates et caract` eres
+semi-simples sous-jacents au-dessus desquels il existe une strat e semi-simple et un caract` ere
+semi-simpleconvenablement d´ ecompos´ escommeci-dessus, cequ ipeutimposerdesconditions
+de compatibilit´ e entre les caract` eres θ(0)etι(˜θ(1)) (voir la d´ efinition des caract` eres semi-
+simples [20, Definition 3.13] et la notion de “common approximation” de [7 ,§8]).
+Pr´ ecisons ceci, avec les notations du paragraphe 3.1. On choisit un type pour la repr´ esen-
+tation supercuspidale autoduale πavec strate simple sous-jacente [Λ(1),n(1),0,2β(1)] et ca-
+ract` ere simple sous-jacent ˜θ(1). On peut sommer la strate [Λ(1),n(1),0,β(1)] et la strate duale
+dansW(−1)de fa¸ con ` a obtenir une strate simple gauche [Λ1,n1,0,β1] dansV1` a laquelle
+V1=W(−1)⊕W(1)soit exactement subordonn´ ee [2] [10], avec β1=β(−1)⊕β(1). On choisit
+par ailleurs un type pour σavec strate semi-simple gauche sous-jacente [Λ(0),n(0),0,β(0)] et
+caract` ere semi-simple gauche θ(0). On se ram` ene au cas o` u Λ1et Λ(0)ont mˆ eme p´ eriode. La
+proposition 3.17 s’applique alors dans les cas suivants : n1=n(0)et les polynˆ omes carac-
+t´ eristiques des strates [Λ1,n1,0,β1] et [Λ(0),n(0),0,β(0)] sont premiers entre eux ; n1= 0 et
+β(0)est inversible ; n(0)= 0 etβ1est inversible. Si les strates sont moins “disjointes”, un
+examen plus approfondi est n´ ecessaire pour d´ eterminer si ˜θ(1)etθ(0)v´ erifient les conditions
+d’existence d’un caract` ere θcomme ci-dessus.
+En particulier ce r´ esultat s’applique toujours si la strate sous-ja cente ` aπest nulle et β(0)
+inversible. En revanche le cas d’un caract` ere autodual πdeGL(1,F) et d’une repr´ esentation
+σattach´ ee ` a une strate semi-simple dont une composante simple es t nulle (ce qui est au-
+tomatique dans un groupe sp´ ecial orthogonal impair) ne rel` eve pas de la proposition 3.17.
+3.4 Exemples
+Commen¸ cons par illustrer la signification de la proposition 3.17 ` a part ir d’exemples bas´ es
+sur les calculs de paires couvrantes et d’alg` ebres de Hecke d´ ej` a faits dans Sp(4 ,F) [1] ; le
+groupeGest ici symplectique.
+Le cas des repr´ esentations supercuspidales autoduales πde GL(2,F) vu comme sous-
+groupe de Levi de Siegel de Sp(4 ,F) [1, Tableau (3.17)] est conforme ` a l’exemple 3.16
+ci-dessus. Passons au cas le plus int´ eressant, celui d’un caract` ere quadratique ou trivial
+˜χde GL(1,F). Notons χla restriction de ˜ χ` ao×
+F. La paire couvrante ( JP1(Λ1),λP1) dans
+G1= Sp(2,F) correspondante est tout simplement la paire ( I,χ0) o` uI=/parenleftBig
+o×
+FoF
+pFo×
+F/parenrightBig
+∩G1
+est un sous-groupe d’Iwahori et χ0((a b
+c d)) =χ(a),(a b
+c d)∈I. Les deux sous-groupes para-
+horiques maximaux contenant IsontP0= SL(2,oF) etP1=/parenleftBig
+o×
+Fp−1
+F
+pFo×
+F/parenrightBig
+∩G1. La seule
+β-extension du caract` ere trivial de I1=/parenleftbig1+pFoF
+pF1+pF/parenrightbig
+∩G1` aIrelativement ` a l’un ou l’autre
+de ces parahoriques est le caract` ere trivial. Notant encore χ0le caract` ere du sous-groupe
+parabolique image de Idans le quotient r´ eductif de P0ouP1, on voit par le corollaire 3.7
+que les relations quadratiques d´ eterminant la r´ eductibilit´ e sont d eux fois celle satisfaite par
+un g´ en´ erateur deH(SL(2,kF),χ0), soit (X+1)(X−q) = 0 (A) (r0=r1= 1) siχest trivial
+suro×
+F, (X+ 1)(X−1) = 0 (B) (r0=r1= 0) sinon, d’o` u la structure bien connue pour
+H(SL(2,F),χ0) et les points de r´ eductibilit´ e de parties r´ eelles 0 et 1 dans le premie r cas, 0
+quatre fois dans le second, pour la repr´ esentation IndSp(2,F)
+P1˜χ||s.36 Corinne Blondel
+Occupons-nous maintenant des repr´ esentations induites IndSp(4,F)
+P˜χ||s⊗σo` uσest une
+repr´ esentation supercuspidale de Sp(2 ,F). L’examen des tableaux (2.17) p. 676 et 1 p. 677
+de [1] appelle quelques remarques.
+(i) Lorsque σest une repr´ esentation de la s´ erie non ramifi´ ee “exceptionnelle” , i.e. induite
+` a partir de l’inflation ` a SL(2 ,oF) d’une repr´ esentation cuspidale de SL(2 ,kF) qui ne
+se prolonge pas ` a GL(2 ,kF), et siχn’est pas trivial, on trouve une relation dont le
+quotient des valeurs propres est −q2. De fait, dans ce cas la strate sous-jacente ` a σest
+une strate nulle, la strate sous-jacente ` a π= ˜χaussi, et nous ne sommes pas dans les
+conditions d’application de la proposition 3.17 (voir la remarque 3.18). C ’est pourquoi
+cette proposition ne permet de pr´ evoir ni ce cas, ni les autres ca s dits “de niveau 0” o` u
+σest induite de l’inflation d’une repr´ esentation cuspidale de SL(2 ,kF) qui se prolonge
+` a GL(2,kF).
+(ii) Siχest trivial, on trouve deux relations de type (A) si σest de la s´ erie non ramifi´ ee et
+de niveau strictement positif, deux relations de type (B) si σest de la s´ erie ramifi´ ee ;
+siχest non trivial, c’est exactement le contraire. Ces cas rel` event d e la proposition
+3.17, les caract` eres w0
+Uetw1
+U−sont n´ ecessairement triviaux dans le cas non ramifi´ e et
+non triviaux dans le cas ramifi´ e.
+Dans un groupe symplectique on a calcul´ e dans l’exemple 2.36 les cara ct` eresw0
+Uetw1
+U−
+dans le cas d’un caract` ere autodual ˜ χde GL(1,F) et d’une repr´ esentation σde Sp(2N,F)
+attach´ ee ` a un type simple maximal o` u l’´ el´ ement β(0)engendre une extension totalement
+ramifi´ ee et y est de valuationcongrue` a −1 modulo dim W(0). Ces deux caract` eres sont´ egaux
+au caract` ere quadratique non trivial de o×
+F(comme dans (ii) pour N= 1). La proposition
+3.17 nous permet de comparer comme ci-dessus la r´ eductibilit´ e de I ndSp(2N+2,F)
+P ˜χ||s⊗σ` a
+celle de IndSp(2,F)
+P1˜χ||s. On en d´ eduit qu’il y a un unique caract` ere autodual ˜ χdeF×tel
+que IndSp(2N+2,F)
+P ˜χ||s⊗σsoit r´ eductible en s= 1, et que la restriction de ce caract` ere ` a o×
+F
+est non triviale. Il s’agit donc, comme on s’y attend par ailleurs, d’un c aract` ere quadratique
+ramifi´ e. Cependant (voir remarque 3.13), la d´ etermination exact e de ce caract` ere parmi les
+deux caract` eres quadratiques ramifi´ es n´ ecessite des calculs p lus pouss´ es.
+R´ ef´ erences
+[1] L. Blasco et C. Blondel, Alg` ebres de Hecke et s´ eries principales g´ en´ eralis´ ees deSp4(F),
+Proc. London Math. Soc. 85(3), 2002, 659–685.
+[2] C.Blondel, Sp(2N)-covers for self-contragredient supercuspidal rep resentations of GL(N) ,
+Ann. scient. Ec. Norm. Sup. 37, 2004, 533–558.
+[3] C. Blondel, Covers and propagation in symplectic groups , Functional analysis IX, 16–31,
+Various Publ. Ser. (Aarhus), 48, Univ. Aarhus, Aarhus, 2007.
+[4] C. Blondel and S. Stevens, Genericity of supercuspidal representations of p-adicSp4,
+Compositio Math. 145(1), 2009, 213-246.Repr´ esentation de Weil et β-extensions 37
+[5] C.J.Bushnell and P.C.Kutzko, The admissible dual of GL( N) via compact open sub-
+groups, Annals of Mathematics Studies 129, Princeton, 1993.
+[6] C.J.Bushnell and P.C.Kutzko, Smooth representations of reductive p-adic groups: struc-
+ture theory via types , Proc. London Math. Soc. 77, 1998, 582–634.
+[7] C.J.Bushnell and P.C.Kutzko, Semisimple types in GLn, Compositio Math. 119, 1999,
+53–97.
+[8] W.T. Gan and S. Takeda, The Local Langlands Conjecture for Sp(4), Int. Math. Res.
+Not., 2010.
+[9] P. G´ erardin, Weil representations associated to finite fields , J. of Algebra 46, 1977, 54–
+101.
+[10] D. Goldberg, P. Kutzko and S. Stevens, Covers for self-dual supercuspidal representa-
+tions of the Siegel Levi subgroup of classical p-adic groups , Int. Math. Res. Not., 2007.
+[11] R. Howlett andG. Lehrer, Induced cuspidal representations and generalised Hecke ri ngs,
+Invent. Math. 58, 1980, 37–64.
+[12] C. Jantzen, Discrete series for p-adicSO(2n)and restrictions of representations of
+O(2n), to appear.
+[13] P. Kutzko and L. Morris, Level zero Hecke algebras and parabolic induction : the Sieg el
+case for split classical groups , Int. Math. Res. Not., 2006.
+[14] C. Mœglin, Normalisation des op´ erateurs d’entrelacement et r´ educt ibilit´ e des induites
+de cuspidales ; le cas des groupes classiques p-adiques , Ann. of Math. 151, 2000, 817–847.
+[15] L. Morris, Tamely ramified intertwining algebras , Invent. Math. 114, 1993, 1-?54.
+[16] M. Neuhauser, An explicit construction of the metaplectic representatio n over a finite
+field, J. of Lie Theory 12, 2002, 15–30.
+[17] F. Shahidi, A proof of Langlands conjecture on Plancherel measure; comp lementary
+series for p-adic groups , Ann. of Math. 132, 1990, 273–330.
+[18] A. Silberger, Special representations of reductive p-adic groups are not integrable , Ann.
+of Math. 111, 1980, 571–587.
+[19] S. Stevens, Double coset decomposition and intertwining , manuscripta math. 106, 2001,
+349–364.
+[20] S.Stevens, Semisimple characters for p-adic classical groups , Duke Math. J. 127(1),
+2005, 123–173.
+[21] S.Stevens, The supercuspidal representations of p-adic classical groups , Invent. Math.
+172, 2008, 289–352.38 Corinne Blondel
+[22] F. Szechtman, Weil representations of the symplectic group , J. of Algebra 208, 1998,
+662–686.
+[23] Y. Zhang, Discrete series of classical groups , Canad. J. Math. 52(5), 2000, 1101–1120.
+Corinne Blondel
+Groupes, repr´ esentations et g´ eom´ etrie - Case 7012
+C.N.R.S. - Institut de Math´ ematiques de Jussieu - UMR 7586
+Universit´ e Paris 7
+F-75205 Paris Cedex 13
+France
+Corinne.Blondel@math.jussieu.fr
\ No newline at end of file
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+arXiv:1001.0130v2  [astro-ph.CO]  8 Jan 2010Mon. Not.R. Astron. Soc. 000, 1–15 (2002) Printed 8 September 2018 (MN L ATEX style file v2.2)
+Near-IRdustandline emissionfromthe centralregionof Mrk 1066:
+ConstraintsfromGeminiNIFS
+Rogemar.A. Riffel1,2⋆, Thaisa Storchi-Bergmann2and Neil M. Nagar3
+1Universidade Federal deSanta Maria, Departamento de F´ ısi ca, Centro deCiˆ encias Naturais eExatas, 97105-900, Santa Maria, RS,Brazil
+2Universidade Federal do Rio Grande do Sul, Departamento deA stronomia, Instituto deF´ ısica, CP 15051, 91501-970, Port o Alegre, RS,Brazil
+3AstronomyGroup, Departamento deF´ ısica, Universidad deC oncepci´ on, Casilla 160-C, Concepci´ on, Chile
+Accepted 1988 December 15. Received 1988 December 14;in ori ginal form 1988 October 11
+ABSTRACT
+We present integral field spectroscopy of the inner 700 ×700pc2of the Seyfert galaxy
+Mrk1066obtainedwith Gemini’sNear-InfraredIntegralFie ld Spectrograph(NIFS)at a spa-
+tial resolution of≈35pc. This high spatial resolutionallowed us to observe,fo r the first time
+in thisgalaxy,an unresolveddust concentrationwith mass ∼1.4×10−2M⊙. Thisunresolved
+concentration, with emission well reproduced by a blackbod y with temperature ∼830K, is
+possiblypartofthenucleardustytorus.Wecomparedmapsof emission-linefluxdistributions
+and ratios with a 3.6cm radio-continuumimage and [O iii] image in order to investigate the
+originofthenear-infraredemission.Theemission-lineflu xesareelongatedinPA =135/315◦
+in agreement with the [O iii] and radio images and, except for the H lines, are brighter to the
+north-westthantothesouth-east.Thiscloseassociationw iththeradiohotspotimpliesthatat
+least part of the emitting gas is co-spatial with the radio ou tflow. The H emission is stronger
+to the south-east, where we find a large region of star-format ion. The strong correlation be-
+tweentheradioemissionandthehighestemission-linefluxe sindicatesthattheradiojetplays
+afundamentalroleattheseintensitylevels.Atloweremiss ion-linefluxesthiscorrelationdis-
+appearssuggestinga contributionfromthe planeofthegala xyto the observedemission.The
+H2flux is more uniformlydistributedand has an excitationtemp eratureof≈2100K. Its ori-
+gin appears to be circum-nuclear gas heated by X-rays from th e central active nucleus. The
+[Feii] emission also is consistent with X-ray heating, but its spa tial correlation with the ra-
+dio jet and [Oiii] emission indicates additional emission due to excitation and/or abundance
+changescausedbyshocksintheradiojet.Thecoronal-linee missionof[Caviii]and[Six]are
+unresolved by our observations indicating a distribution w ithin 18pc from the nucleus. The
+reddeningmapobtainedviathePa β/Brγlineratiorangesfrom E(B−V)≈0toE(B−V)≈1.7
+with the highest values defining a S-shaped structure along P A≈135/315◦. The emission-line
+ratios are Seyfert-likewithin the ionization cone indicat ing that the line emission is powered
+by the central active nucleus in these locations. Low ioniza tion regions are observed away
+from the ionization cone, and may be powered by the di ffuse radiation field which filters
+throughthe ionization cone walls. Two regionsat 0 .′′5 south-east and at 1′′north-west of the
+nucleusshowstarburst-likeline ratios, co-spatialwith a n enhancementin the emissionof the
+H lines. We attribute this changeto additionalemission fro m star formingregions.The mass
+ofionizedgasis MHII≈1.7×107M⊙andthatofhotmoleculargasis MH2≈3.3×103M⊙.
+Key words: galaxies: individual (Mrk1066) – galaxies: Seyfert – galax ies: ISM – infrared:
+galaxies
+1 INTRODUCTION
+The studyof the gas distributionand excitationinthe Narro w Line
+Region (NLR) of active galaxies has a fundamental importanc e in
+the understanding of the physical phenomena in the vicinity of
+⋆E-mail: rogemar@smail.ufsm.brthe super-massive black holes (SMBH) in their centers. The e x-
+citation of the inner NLR can reveal how the radiation and mas s
+outflows from the active galactic nucleus (AGN) interact wit h the
+surrounding inter-stellarmedium (ISM). Several studies h ave been
+aimed to investigate the line emission from the NLR gas (e.g.
+Rodr´ ıguez-Ardila et al.2004;Rodr´ ıguez-Ardila, Ri ffel&Pastoriza
+2005a; Riffeletal. 2006, 2008, 2009a; Stoklasov´ a etal 2009;
+c∝circlecopyrt2002 RAS2Riffel,Storchi-Bergmann &Nagar
+Figure 1. Top left: Large scale image of Mrk1066 obtained at the R-band with the 1.22m Oschin Telescope on Palomar Mountain (Lasker et al. 1996). The
+central box shows NIFS field of view. Top right: 2 .22µm continuum map from NIFS observations. Bottom: Spectra for the nuclear position (N at top-right
+panel) and for the position A (0 .′′5north-west from the nucleus) for the J and K l-bands for an aperture of 0 .′′25×.′′25. The emission lines are identified at the
+nuclear spectrum and in the extra-nuclear spectrum wemark t he K-band CO absorption band heads.
+Storchi-Bergmannet al. 2009; Barbosa et al. 2009). Neverth eless,
+itsphysics isstillnot completely understood.
+Besidesthelineemission,thenatureofthenuclearcontinu um
+emission is an important issue in the study of AGNs. In partic ular,
+the study of near-infrared (hereafter near-IR) continuum c an be
+used to probe the emission from hot dust located in the putati ve
+torus postulated by the Unified Model (Antonucci 1993). Seve ral
+studies revealed the presence of hot dust with temperatures of
+T∼900−1500K and masses of MHD∼10−6−10−2M⊙(e.g.
+Rieke & Lebofsky 1981; Rodr´ ıguez-Ardila, Contini & Viegas
+2005b; Rodr´ ıguez-Ardila & Mazzalay 2006; Ri ffelet al.
+2009a,b,c). The high spatial resolution provided by adapti ve
+optics spectroscopy in large telescopes can be used to const raint
+the locationof the near-IR emittingdust (e.g.Ri ffelet al.2009b).
+This work is aimed to investigate the near-IR emission-
+line flux distributions, gas excitation and continuum emiss ion
+within the inner kiloparsec of Mrk1066 from spectroscopy ob -
+tained with the Gemini’s Near-Infrared Integral Field Spec tro-
+graph (NIFS, McGregor et al. 2003). In a following paper we
+discuss the kinematics of both the gas and stars. Mrk1066 wasselected for this study because it presents extended radio ( e.g
+Ulvestad& Wilson1989;Nagar et al.1999)andnear-IRlineem is-
+sion (e.g. Rodr´ ıguez-Ardila, Ri ffel& Pastoriza 2005a; Knop et al.
+2001)allowingustoinvestigatetheroleoftheradiojetint heemis-
+sion of the NLR gas. In addition, its central kpc has a high ext inc-
+tion in the optical (e.g. Stoklasov´ a et al 2009) and thus a ne ar-IR
+study is better suited for the investigation of the gas excit ation and
+kinematics.
+Mrk1066isanSBO +galaxyharboringaSeyfert2nucleus; it
+is located at a distance of 48.6Mpc1, for which 1′′corresponds to
+235pc at the galaxy. Hubble Space Telescope (HST) narrow-band
+images show a “jetlike” feature in the [O iii]+Hβemission extend-
+ing up to 1.′′4 north-west from the nucleus along the position an-
+gle PA=315◦, while the Hα+[Nii] image is extended to both sides
+of the nucleus (Bower etal. 1995). Radio continuum images of
+Mrk1066,at3.6,6and20cm,showextendedemissionupto1 .′′5to
+bothsidesofthenucleusorientedapproximatelyalongthes amePA
+1Distance quoted in NASA /IPAC Extragalactic Database (NED –
+http://nedwww.ipac.caltech.edu)
+c∝circlecopyrt2002 RAS, MNRAS 000,1–15Near-IR dustand lineemissionfromMrk1066 3
+Figure 2. Continuum images of the central 3′′×3′′of Mrk1066. Top left: Optical continuum image obtained with the HST-WFPC2 using the filter F606W
+(Malkan, Gorjian &Tam1998).Topright: H-band continuum im age obtained with HST-NICMOSwith filter F160W.Bottom: Cont inuum images centered at
+1.23µm(left) and 2.22µm(right) fromNIFSobservations. TheHSTimages arein arbit rary units and NIFSimages arein logarithmic flux units (ergs−1cm−2
+Å−1).Thestraight line showstheorientation ofthe major axis o fthegalaxy and thethick contours overlaid to theK-band map arefromtheoptical continuum
+image.
+of the optical line emission (Ulvestad& Wilson 1989; Nagar e t al.
+1999).Longslitspectroscopyshowsthatthenear-IRemissi onlines
+areextendedupto5′′fromthenucleusalongthePA =135/315◦with
+distinct flux distributions for [Fe ii], H2and H recombination lines
+suggesting different emission processes (Knopet al.2001).
+This paper is organized as follows: In Section 2 we present
+thedescriptionoftheobservations anddatareduction.InS ec.3we
+present the results for the continuum and line emission, as w ell as
+line-ratiomaps. The discussion of the results ispresented inSec.4
+and theconclusions are presented inSec.5.
+2 OBSERVATIONSAND DATA REDUCTION
+2.1 TheGemini NIFSdata
+TheIntegralFieldUnit(IFU)spectroscopic dataofMrk1066 were
+obtained withGeminiNIFS(McGregor et al.2003)operating w ith
+the Gemini North Adaptive Optics system ALTAIR in Septem-
+ber 2008 under the programme GN-2008B-Q-30. The NIFS hasa square field of view of ≈3.′′0×3.′′0, divided into 29 slices with
+an angular sampling of 0 .′′103×0.′′042, optimized for use with AL-
+TAIR.
+The observing procedure followed the standard Object-Sky-
+Sky-Objectdithersequence,witho ff-sourceskypositionssincethe
+target is extended, and individual exposure times of 600s. T wo set
+of observations were obtained at di fferent spectral ranges: the first
+at the J-band, centered at 1.25 µm and covering the spectral region
+from 1.15µm to 1.36µm, and the second at the K l-band, centered
+at 2.3µm covering the spectral range from 2.10 µm to 2.26µm. At
+the J-band, the selected instrument configuration was the J G5603
+grating and ZJ G0601 filter resulting in a spectral resolution of ≈
+1.7Å, as obtained from the measurement of the full width at half
+maximum (FWHM) of arc lamp lines. The K l-band observations
+were done using the Kl G5607 grating and HK G0603 filter and
+resultedina spectral resolutionof FWHM ≈3.3Å.
+Thetotalexposure timeateachbandwas4800s,consistingof
+8 individual on-source exposures. A small spatial ditherin g of 0.′′1
+was applied between on-source exposures in order to correct the
+c∝circlecopyrt2002 RAS, MNRAS 000, 1–154Riffel,Storchi-Bergmann &Nagar
+data from bad pixels.The FWHM of the spatial profile of the sta r
+was 0.′′13±0.′′02for the J-band and 0 .′′15±0.′′03for the K l-band.
+Thedatareductionwasaccomplishedusingtaskscontainedi n
+thenifspackage which is part of geminiiraf package as well as
+genericiraftasks. The data reduction followed the standard pro-
+cedure, which includes the trimming of the images, flat-field ing,
+cosmic ray rejection, sky subtraction, wavelength and s-di stortion
+calibrations.Inordertoremovetelluricabsorptionsfrom thegalaxy
+spectra we observed the standard star HIP15925 just after th e J-
+band observations of the galaxy and HIP10559 just before the Kl-
+band observations. The galaxy spectra was divided by the nor mal-
+ized spectrum of the tellurc standard star using the nftelluric task
+of thenifs.gemini.irafpackage. The galaxy spectra were flux cal-
+ibrated by interpolating a blackbody function to the spectr um of
+the telluric standard and the J and K l-bands data cubes were con-
+structedwithanangularsamplingof0 .′′05×0.′′05foreachindividual
+exposure. Finallythe individual data cubes were combined u sing a
+sigma clipping algorithm in order to eliminate bad pixels an d re-
+mainding cosmic rays by mosaicing the dithered spatial posi tions.
+ThefinalJandK ldatacubes containabout4200individual spectra
+and cover the central 3′′×3′′, corresponding to ≈700×700 square
+parsecs atthe galaxy.
+2.2 The3.6cm radioimage
+The 3.6 cm Very Large Array (VLA) radio image was made using
+datatakenon31December1992,initiallypublishedinNagar et al.
+(1999),and reprocessed as described inMundell et al.(2009 ).
+3 RESULTS
+In the top-left panel of Fig.1 we present a large scale R-band
+image of Mrk1066 from the Palomar Observatory Sky Survey
+(Lasker et al.1996).Inthetop-rightpanel wepresent anima geob-
+tainedfromtheNIFSdata cube forthecontinuum around2.22 µm.
+In the bottom panels we present two characteristic IFU spect ra:
+the nuclear one and a spectrum from a location at 0 .′′5 north-
+west of the nucleus. In the J-band we identified the [P ii] emis-
+sion lines at 1.14713 and 1.18861 µm, the [Feii] emission lines at
+1.25702, 1.27912, 1.29462, 1.29812, 1.32092 and 1.32814 µm, the
+HiPaβλ1.28216µm, the Heiiline at 1.16296µm and the [Six]
+coronal lineat1.25235 µmatthenucleus. TheH 2emissionlines at
+2.12183, 2.15420, 2.22344, 2.24776, 2.40847, 2.41367, 2.4 2180,
+2.43697 and 2.45485 µm are identified in the K-band spectra, as
+well as the HiBrγλ2.16612µm, the Heiλ2.14999µm and the
+[Caviii]λ2.32204µm coronal line at the nucleus. In the K-band
+spectrawehavealsoidentifiedtheCOstellarabsorptionban dheads
+around 2.3µm.
+3.1 Thecontinuumemission
+Figure2 presents optical and near-IR continuum images of th e
+central 3′′×3′′of Mrk1066. The top left panel shows an opti-
+cal image obtained with the HST Wide Field Planetary Cam-
+era 2 (WFPC2) through the filter F606W (Malkan, Gorjian&Tam
+1998).Thisimagepresentstwospiralarmsorientedapproxi mately
+along PA=128◦(approximately coincident with the major axis of
+the galaxy), which extend up to 1 .′′2 to both sides of the nucleus.
+Along the spiral arms there are several knots with enhanced e mis-
+sion. This image is in good agreement with the optical contin uum
+images presented by Bower etal. (1995).
+Figure 3. K/J continuum ratio obtained by averaging continuum regions
+(in units of ergs−1cm−2Å−1) centered on 2.22µm and 1.23µm.Thethick
+contours are fromthe reddening map for Fig. 5.
+In the top right panel of Fig.2 we present an HST H-band
+image obtained with the Near Infrared Camera and Multi-Obje ct
+Spectrometer (NICMOS) through the filter F160W (proposal ID
+7330 by J. S. Mulchaey).In the bottom panels of Fig.2 we prese nt
+the J and K l-band continuum images obtained from the NIFS data
+cubes by averaging continuum regions centered at 1.23 µm and
+2.22µm.The near-IR continuum is more extended along PA =128◦,
+whichissimilartotheorientationofthespiralarmsobserv edinthe
+optical continuum and approximately the orientation of the major
+axis of the galaxy. The spiral arms can also be seen in the near -IR,
+although with “less contrast” than in the optical, probably due to
+the stronger effect of the dust and contribution of line emission in
+the optical. A comparison between the J and K continuum shows
+that the extra-nuclear emission is bluer than the nuclear on e, indi-
+catingthe presence of dust nucleus. This behavior canbe obs erved
+inFig.3, which presents a K /J continuum ratiomap obtained from
+the NIFS data cubes by averaging the flux within spectral wind ow
+of50Åinregions freeofemissionandabsorptionlinescente redat
+2.22µm and1.23µm.
+3.2 Emission-linefluxdistributions
+As shown in Fig. 1, several emission lines can be observed in t he
+spectra.Table1presents the measuredfluxes fortheemissio nlines
+within an aperture of 0 .′′25×0.′′25 at the four positions identified in
+the top-right panel of Fig. 1: the nucleus, positionA at 0 .′′5north-
+west from the nucleus, positionB at 0 .′′5south-east and positionC
+at 0.′′5north-east from the nucleus. Most emission lines peak at th e
+nucleus,exceptfor[Fe ii]and[Pii]whichpeakat0.′′5north-westof
+the nucleus.
+In order to map the gas emission we have integrated the flux
+under theemission-lineprofilesandsubtractedtheunderly ingcon-
+tinuum obtained from two spectral windows, one from each sid e
+of the profile. Figure4 presents the resulting flux maps. For m ost
+emission lines, the highest intensity levels most extended along
+PA≈135/315◦, which is the orientation of the radio structure.
+A detailed inspection of each panel reveals distinct struct ures for
+each ionization level. The [P ii], [Feii] and H 2emission lines are
+c∝circlecopyrt2002 RAS, MNRAS 000,1–15Near-IR dustand lineemissionfromMrk1066 5
+Figure 4. Integrated flux maps for the [P ii]λ1.8861µm, [Feii]λ1.2570µm, Paβ, [Feii]λ1.3109µm, H2λ2.1218µm, Brγ, [Caviii]λ2.3220µm and
+H2λ2.1218µm emission lines in units of 10−17ergs−1cm−2. The central cross marks the position of the continuum peak. The thick contours overlaid on
+the Paβ,[Feii]λ1.2570µmand H 2λ2.1218µmmaps are from the3.6cm radio continuum image.
+c∝circlecopyrt2002 RAS, MNRAS 000, 1–156Riffel,Storchi-Bergmann &Nagar
+Table 1.Measured emission-line fluxes for thefour positions marked in Fig.1 within 0.′′25×0.′′25 aperture in units of 10−16ergs−1cm−2.
+λvac(Å) ID Nucleus A B C
+11471.3 [Pii]1D3−3P1 5.07±2.23 2.75±0.71 1.50±0.47 –
+11629.6 Heii7−5 2.41 ±1.52 0.99±0.32 0.67±0.13 –
+11886.1 [Pii]1D2−3P2 6.21±0.69 6.54±0.87 2.80±0.38 0.41±0.28
+12523.5 [Six]3P1−3P2 0.60±0.37 – – –
+12570.2 [Feii]a4D7/2−a6D9/218.88±0.80 21.15±1.82 13.06±0.95 5.76±1.21
+12706.9 [Feii]a4D1/2−a6D1/20.52±0.41 0.96±0.38 0.43±0.17 0.14±0.11
+12712.1 [Feiv]2G5
+7/2−2D5
+5/2? 0.21±0.19 – – –
+12791.2 [Feii]a4D3/2−a6D3/20.94±0.35 1.34±0.65 1.27±0.32 0.36±0.25
+12821.6 HiPaβ 41.18±4.26 25.17±1.98 26.41±1.98 6.79±0.55
+12946.2 [Feii]a4D5/2−a6D5/21.91±0.87 1.71±0.16 0.82±0.29 0.70±0.13
+12981.2 [Feii]a4D3/2−a6D1/21.72±0.55 0.53±0.22 0.94±0.23 0.46±0.29
+13209.2 [Feii]a4D7/2−a6D7/24.99±0.73 6.32±0.64 4.03±0.52 2.21±0.26
+13281.4 [Feii]a4D5/2−a6D3/21.54±1.05 1.31±0.59 0.54±0.40 –
+21218.3 H 22 1-0S(1) 13.41 ±1.02 11.74±0.49 5.27±0.48 2.52±0.11
+21499.9 Hei3S1−3P0
+10.40±0.21 0.11±0.21 0.27±0.19 0.18±0.07
+21542.0 H 21-0S(2) 0.51 ±0.23 0.45±0.50 0.25±0.18 0.16±0.04
+21661.2 HiBrγ 17.14±1.16 10.59±0.60 9.81±0.53 1.28±0.08
+22234.4 H 21-0S(0) 2.99 ±0.92 2.63±0.64 1.96±0.72 0.65±0.06
+22477.6 H 22-1S(1) 1.47 ±0.62 1.17±0.38 0.67±0.33 0.33±0.02
+23220.4 [Caviii]2P0
+3/2−2P0
+1/24.64±1.42 – – –
+24084.7 H 21-0Q(1) 10.04 ±1.33 8.14±0.45 4.82±0.86 1.42±0.11
+24136.7 H 21-0Q(2) 3.03 ±0.47 2.47±0.42 1.47±0.41 0.49±0.10
+24218.0 H 21-0Q(3) 10.61 ±1.05 8.32±0.57 5.23±0.56 1.50±0.21
+24369.7 H 21-0Q(4) 2.73 ±1.45 2.26±0.38 1.30±0.48 0.47±0.09
+24548.5 H 21-0Q(5) 7.08 ±2.18 5.41±0.43 2.82±0.77 4.36±2.24
+∼2 times brighter to the north-west than to the south-east, wh ile
+Paβand Brγare approximately equally bright to both sides of the
+nucleus. The peak fluxes of the [P ii] and [Feii] emission lines
+occur at≈0.′′85north-west of the nucleus, approximately at the
+same position where there is a hot spot in the radio map. The H i
+recombination-line emission peaks at the nucleus and prese nts a
+secondary peak at ≈0.′′5south-east from it. Another di fference be-
+tween the Hiand [Feii] emission lines is that the flux distribution
+of the latter traces the radio structure better than the form er. The
+H2emission also presents a peak at the nucleus and a secondary
+peak at≈0.′′65north-west of the nucleus, in a region between the
+nucleus andtheradiohot spot.Theemissioninthecoronal li nesof
+[Caviii]λ2.3220µm and [Six]λ1.2524µm are unresolved by our
+observations.
+3.3 Line-ratiomaps
+In order to investigate the excitation mechanisms of the [Fe ii] and
+H2emission lines we constructed flux ratio maps, shown in Fig-
+ure5.Inthisfigurewealsopresenta reddeningmapobtainedf rom
+the Paβ/Brγline ratioas
+E(B−V)=4.74log/parenleftBigg5.88
+FPaβ/FBrγ/parenrightBigg
+, (1)
+whereFPaβandFBrγare the fluxes of PaβandBrγemis-
+sion lines, respectively. We have used the reddening law of
+Cardelli,Clayton& Mathis (1989) and adopted the intrinsic ra-
+tioFPaβ/FBrγ=5.88 corresponding to case B recombination
+(Osterbrock& Ferland2006).Theresulting E(B−V)mapisshown
+in the top left panel of Fig.5. The highest values are observe d in
+a S-shaped structure, resembling the spiral arms observed i n the
+continuum image overlaidas thickcontours onthe reddening map.Typicalvalueswithinthisstructureare E(B−V)≈1.0,withseveral
+knots of higher values reaching E(B−V)≈1.7. In regions away
+fromthecenteroftheS-shapedstructurethereddeningdecr easesto
+E(B−V)≈0.4orbellow.Theerrorsin E(B−V)areapproximately
+0.15.
+A comparison between the emission-line reddening and the
+continuum ratio map is shown in Fig. 3. The thick contours, re p-
+resenting the reddening map, show that the S-shaped structu re is
+also present in the continuum ratio map as a region of higher K /J
+ratio. In order to estimate the reddening in the continuum, w e note
+that, from Fig. 3, a continuum ratio(K /J)1≈0.4 corresponds to the
+locations with E(B−V)1≈0.4inthe gas.Ifthe higher (K /J)2ratio
+alongtheS-shapedstructureisduetoreddening,thecorres ponding
+E(B−V)2canbe calculatedfrom
+χ[E(B−V)1−E(B−V)2]=log/bracketleftBigg(K/J)1
+(K/J)2/bracketrightBigg
+, (2)
+whereχ=−0.4(AK/AV−AJ/AV)RV=0.2083, where ( AK/AV)=
+0.114, (AJ/AV)=0.282 andRV=3.1 for the extinction law of
+Cardelli,Clayton& Mathis (1989). Using (K /J)2=0.5 measured
+in the S-shaped structure (from Fig. 3) we obtain reddening f or
+the continuum of E(B−V)2≈0.9, which is somewhat smaller
+thantheoneobservedfortheemittinggas(top-leftpanelof Fig.5).
+Nevertheless, right atthe nucleus the K /J ratiois much higher than
+intherestoftheS-shapedstructure,whatindicatesthatit isnotdue
+toreddening.Thenatureofthisreddercontinuumwillbedis cussed
+inSec.4.1.
+In the top-right panel of Fig.5 we present the
+[Feii]λ1.2570µm/Paβintensity ratio, which can be used
+to investigate the excitation mechanism of the [Fe ii]
+emission. Seyfert galaxies present typical values for
+this ratio between 0.6 and 2.0. (e.g. Ri ffelet al. 2006;
+c∝circlecopyrt2002 RAS, MNRAS 000,1–15Near-IR dustand lineemissionfromMrk1066 7
+Figure 5. Top: Reddening map obtained from the Pa β/Brγline ratio (left) and [Fe ii]λ1.2570µm/Paβline ratio map (right). Bottom:
+[Feii]λ1.2570µm/[Pii]λ1.8861µm line-ratio map (left) and H 2λ2.1218µm/Brγratio (right). The thick contours overlaid to the [Fe ii]/Paβand H2/Brγmaps
+are fromthe radio image.
+Rodr´ ıguez-Ardila etal.2004;Rodr´ ıguez-Ardila, Ri ffel& Pastoriza
+2005a; Storchi-Bergmannet al. 2009). The smallest value ob -
+served in Mrk1066 is [Fe ii]λ1.2570µm/Paβ=0.33±0.03 at
+0.′′3south-east from the nucleus, while the highest values are
+observed in regions located at distances larger than 1′′, reaching
+values of upto1.5.
+Another line ratio that can be used to investigate the [Fe ii]
+excitation mechanism is [Fe ii]λ1.2570µm/[Pii]λ1.8861µm (e.g.
+Oliva et al. 2001; Storchi-Bergmannet al. 2009). We present this
+ratio in the bottom-left panel of Fig.5. The lowest values of ≈3
+are observed at the nucleus and in regions to north-west at sm all
+distances from the nucleus, while the highest values up to 9. 5 are
+observed at largerdistances from the nucleus.
+In the bottom-right panel of Fig.5 we present the
+H2λ2.1218µm/Brγratio map, which is useful to investigate the
+excitation of the H 2emitting gas. This ratio presents values rang-
+ing from 0.6 to 2 for Seyfert galaxies (e.g. Ri ffelet al. 2006;
+Rodr´ ıguez-Ardila etal.2004;Rodr´ ıguez-Ardila, Ri ffel& Pastoriza
+2005a; Storchi-Bergmannet al. 2009). In Mrk 1066, the lowes t
+values of 0.29±0.06 are observed at 1′′north-west from the nu-
+cleus and the highest values of up to 2.6 are observed at dista ncesof>0.′′7inregionsawayfromtheradiostructure.Anotherregionof
+small values (≈0.4) is observed at≈0.′′5south-east of the nucleus,
+approximately coincident with the secondary peak observed in the
+Hirecombination-lines fluxmaps (see Fig.4),whileatthe nucl eus
+H2λ2.1218µm/Brγ=0.75±0.07.
+4 DISCUSSION
+4.1 Dustemission
+AsobservedinFig.3(seealsoFigs.1and2)theextra-nuclea rcon-
+tinuum is bluer than the nuclear continuum. At the nucleus th e
+ratio between the continuum at 2.22 µm and at 1.23µm is≈1,
+whileatlargerdistancesfromthenucleus,typicalvaluesa re≈0.45
+along PA≈128◦and even smaller at locations away from this PA
+(/lessorsimilar0.38). In order to to investigate the nature of the K-band exces s
+atthenucleus,wehaveextractedanuclearspectrumwithina naper-
+ture of 0.′′25 radius, which includes the nuclear flux down to 20%
+of the peak continuum flux in the K band. An extra-nuclear spec -
+trum for a circular ring within radii 0 .′′25/lessorequalslantr/lessorequalslant0.′′35 was extracted
+to represent the stellar population. This spectrum was norm alized
+c∝circlecopyrt2002 RAS, MNRAS 000, 1–158Riffel,Storchi-Bergmann &Nagar
+according to the ratio of the nuclear /extra-nuclear extraction aper-
+tures. Both spectra were corrected for extinction using the redden-
+ing values obtained in Sec. 3.3: E(B−V)≈1.03 for the nucleus
+andE(B−V)≈0.69 forthe stellar population.
+Undertheassumptionthattheunderlyingstellarpopulatio nin
+the nuclear spectrum is the same as in the extra-nuclear spec trum,
+wehavesubtractedthelatterfromtheformer.Theresulting nuclear
+continuum was then modeled by the sum of two components: a
+blackbody function, which dominates the emission in the K ba nd
+and a power-law, which is important for the J-band emission. The
+fit was performed using the nfit1dtask of thestsdasiraf package
+selecting only spectral regions not a ffected by line emission. The
+bestfitwasobtainedforapower-law Fλ∝λ−2.4±0.3andablackbody
+function withtemperature T=863±30K.
+We thenconsidered the possibilitythat the contribution of the
+stellar population to the nuclear spectrum is higher than th e one
+derived only from the normalization of the fluxes to the same e x-
+traction aperture. This possibility is suggested by the fac t that the
+stellar CO absorption bands in the nuclear spectrum do not di sap-
+pear after the subtractionof the extra-nuclear spectrum. I norder to
+completelyeliminatethestellarabsorptions theextra-nu clear spec-
+trum had to be multiplied by a factor of 1.45. After this corre ction
+we obtain Fλ∝λ−4.7±0.4andT=829±30K. In the top panel of
+Fig. 6 we show the nuclear and extra-nuclear spectra correct ed by
+extinction with the latter properly normalized as describe d above.
+Thebottom panel shows the subtractedspectrum, aswellasth e re-
+sultingfitasacontinuous line.Theblackbody functionissh ownas
+a dashed line andthe power-law as a dottedline.
+Recently, Riffelet al. (2009c)2presented stellar population
+synthesis fora sample ofSeyfertgalaxies,including Mrk10 66, us-
+ing near-IR spectra extracted within an aperture of 1 .′′6×0.′′8. Be-
+sides synthetic Simple Stellar Population (SSP) models, th e au-
+thors considered the contribution of a power-law to represe nt the
+featureless continuum from the AGN and five blackbody func-
+tions for temperatures ranging from 800K to 1400K in order
+to account for possible dust emission. They concluded that t he
+dust emission is not necessary in the case of Mrk1066. Never-
+theless, as shown in Fig.6, a K-band excess is clearly presen t in
+our nuclear spectrum. Ri ffeletal. (2009c) did not identified this
+component probably due to their larger aperture, which cove rs
+an area more than 6 times ours. Unresolved dust emission from
+the nucleus of Seyfert galaxies is commonly observed and is u su-
+ally attributed to the dusty torus postulated by the Unified M odel
+(e.g. Riffeletal. 2009a,b,c; Rodr´ ıguez-Ardila & Mazzalay 2006;
+Rodr´ ıguez-Ardila, Contini & Viegas 2005b). Previous work s have
+shown that the dust emission in the near-IR for Seyfert1 gala x-
+ies give blackbody temperatures T/greaterorsimilar1200K, while for Seyfert2
+galaxiestheblackbody temperature isusually T/lessorsimilar900K,inagree-
+ment with our result for Mrk1066. This di fference can be inter-
+preted as due to the fact that in Seyfert 1 galaxies one can see
+the inner wall of the torus, where the temperatures are close to the
+evaporation temperature of graphite and silicate grains (B arvainis
+1987; Granato& Danese 1994). In Seyfert 2 galaxies we do not
+see the inner wall, only dust located at larger distances fro m the
+nucleus, heated by a more diluted radiation field, part of whi ch is
+also blocked by clouds in the inner region of the torus (e.g. E litzur
+2008). Since the K-band continuum of Mrk1066 is unresolved b y
+ourobservations,wecanconstraintthelocationoftheemit tingdust
+2This is notthe present author, buthis brother.to within the inner ≈18 pc, although, as discussed in Ri ffelet al.
+(2009b), the dust isprobably locatedmuch closer in.
+We can derive the mass of the emittingdust from the parame-
+ters obtained from the fit of the blackbody function, togethe r with
+assumptionsaboutthephysicalpropertiesofthedustgrain s.Wees-
+timate the infrared luminosity of each grain assuming that t he dust
+composition isgraphite, by
+Lgr
+ν,ir=4π2a2QνBν(Tgr) [ergs−1Hz−1] (3)
+whereais the grain radius, Qν=qirνγis the absorption e ffi-
+ciency and Bν(Tgr) is its spectral energy distributionassumed to be
+a Planckfunction withtemperature Tgr(Barvainis 1987). The ratio
+between the total luminosity of the hot dust ( LHD
+ir) and that of one
+grain (Lgr
+ir) gives the number of dust grains NHD=LHD
+ir
+Lgr
+ir.LHD
+irwas
+obtained by integrating the flux under the Planck function fit ted to
+the nuclear spectrum and Lgr
+irwas obtained by the integration of
+equation3foratemperatureof Tgr=829Kassuming a=0.05µm,
+qir=1.4×10−24andγ=1.6 (Barvainis 1987; Kishimotoet al.
+2007).Finally, the mass of the emittinghot dust isobtained as:
+MHD≈4π
+3a3NHDρgr. (4)
+Assuming a graphite density of ρgr=2.26gcm−3
+(Granato& Danese1994),weobtain MHD=(1.4±0.5)×10−2M⊙.
+ThederivedmassiscomparabletovaluesobtainedforotherS eyfert
+galaxies, which are in the range 10−6−10−2M⊙(e.g. Riffelet al.
+2009b,c).
+4.2 Fluxdistributions
+The flux distributions shown in Fig.4, are more more extended
+along the PA≈135/315◦and resemble those in optical lines
+(Bower etal. 1995; Stoklasov´ a et al 2009). Using HST imagin g,
+Bower etal. (1995) observed a narrow “jetlike” feature in th e
+[Oiii]+Hβemission, extending to1 .′′4 north-west from the nucleus
+along PA=315◦. They concluded that this image is dominated by
+[Oiii] emission and thus that the high excitation gas is strongly
+concentrated in the jet. To the south-east the [O iii] emission is
+much fainter than to the north-west, similarly to what we obs erve
+in the [Feii] and [Pii] emission. Bower etal. (1995) present also
+a Hα+[Nii] narrow-band image, which shows the same “jetlike”
+structure, but with a flux distribution broader then in [O iii], with
+emission to both sides of the nucleus, similar to our Br γand Paβ
+flux distributions. The 3.6cm radio map presents a bipolar je t at
+approximatelythesameorientationoftheemission-line flu xdistri-
+butions (see Fig.4). By comparing the radio and optical line emis-
+sion, Bower etal. (1995) suggest that the low- and high-exci tation
+gas present distinct spatial distributions, with the forme r preferen-
+tially located in the galaxy plane and the latter in the jet re gion.
+They explain the lower intensity of [O iii] to south-west of the nu-
+cleus as being due toobscuration of the jet bythe galactic pl ane.
+A detailed analysis of the line flux distributions reveal som e
+differences between lines from distinct ionization levels. In o r-
+der to do this comparison, we extracted one-dimensional cut s
+along PA=135◦for a pseudo-slit with 0 .′′25 width. The resulting
+one-dimensional flux distributions for the [Fe ii], Paβ, Brγand
+H2λ2.12µm emission lines are shown in Figure7. As observed in
+this figure (see also the two-dimensional maps from Fig.4), t he
+[Feii] flux distribution peaks at ≈0.′′85 north-west of the nucleus,
+approximately coincident with a hot spot observed in the rad io
+map, and is fainter to the south-east, in good agreement with the
+c∝circlecopyrt2002 RAS, MNRAS 000,1–15Near-IR dustand lineemissionfromMrk1066 9
+Figure 6. Top: Nuclear and extra-nuclear spectra of Mrk1066. The nucl ear spectrum was extracted within a circular aperture with r adius 0.′′25, while the
+extra-nuclear spectrum was integrated within an aperture w ith inner radius 0.′′25 and outer radius of 0 .′′35. Both spectra were corrected by reddening. The
+extra-nuclear spectrum was multiplied by 1.45, after norma lization to the same aperture of the nuclear one, in order to e liminate the stellar absorption in
+the subtracted spectrum. Bottom: Di fference between the nuclear and extra-nuclear spectra, toge ther with a fit (solid line) considering the contribution of a
+power-law (dotted line) plus ablackbody function (dashed l ine).
+[Oiii]image.TheHrecombinationlinesandH 2lineemissionpeak
+at the nucleus. The H lines have a secondary peak at 0 .′′5 south-
+east of the nucleus, which is not observed in the H 2and [Feii] flux
+distributions. The H 2emission presents a secondary peak to north-
+west,betweenthenucleusandtheradiohotspot.Knopet al.( 2001)
+obtained J and K long-slit spectroscopy of Mrk1066 with a see ing
+of 0.′′7. They present fluxmeasurements for near-IR emissionlines
+along PA=45◦andPA=135◦,which canbe directlycompared with
+our one-dimensional cutsandfluxmaps(Figs.7and4).Ingene ral,
+our flux distributions are in good agreement with theirs, alt hough
+much more details are revealed by our measurements. In parti cu-
+lar, the secondary peaks observed in the H emission lines and the
+H2nuclear peak are not present in the flux distributions presen ted
+by Knop et al. (2001). We attribute these di fferences to our higher
+spatial resolution.
+Our [Feii] and H recombination lines present similar flux
+distributions to those of the [O iii] and Hα+[Nii] images of
+Bower et al.(1995),respectively.Weconclude thatthe[Fe ii]emit-tinggas extends tohigh galactic latitudes and isapproxima tely co-
+spatial with the radio jet, while the Br γand Paβemission origi-
+natemostlyfromthegalacticplane. TheH 2fluxismoreuniformly
+distributed over the whole IFU field, suggesting also an impo rtant
+contributionfrom gas inthe galactic plane, although part o f the H 2
+emitting gas close to the nucleus may be also outflowing to hig h
+latitudes, as judged from the highest flux to the north-west t han to
+the south-east (Fig. 4). These conclusions are supported by the ob-
+servation of distinct kinematics for these emission lines, with the
+[Feii] originating in a much more disturbed gas than the H and H 2
+emitting (Riffel et al., in preparation). This result is also supported
+bynear-IRIFUobservations ofother Seyfertgalaxies, whic hshow
+that the H 2and ionized gas have distinct distributions and kine-
+matics, with the former usually restricted to the galactic p lane and
+thelatterextendingtohigherlatitudes(e.g.Ri ffel etal.2006,2008,
+2009a; Storchi-Bergmannet al.2009).
+c∝circlecopyrt2002 RAS, MNRAS 000, 1–1510Riffel,Storchi-Bergmann &Nagar
+Figure 7. One-dimensional cuts from the flux distributions in [Fe ii], Paβ,
+Brγand H2λ2.12µm emission-lines extracted of a pseudo-slit with 0 .′′25
+widthoriented alongPA =135◦.Thefluxesarenormalized tothepeakvalue.
+4.3 Diagnostic diagram
+In order to investigate the nature of the circum-nuclear lin e-
+emitting region we constructed the spatially resolved spec tral-
+diagnostic diagram [Fe ii]λ1.25µm/PaβvsH2λ2.12µm/Brγ,
+originally proposed for non-resolved spectra (Larkinetal . 1998;
+Rodr´ ıguez-Ardila etal.2004;Rodr´ ıguez-Ardila, Ri ffel& Pastoriza
+2005a). Typical values for the nucleus of Seyfert galax-
+ies are 0.6/lessorsimilar[Feii]/Paβ/lessorsimilar2.0 and 0.6/lessorsimilarH2/Brγ/lessorsimilar2.0 (e.g.
+Rodr´ ıguez-Ardila, Ri ffel& Pastoriza 2005a). The diagram is
+shown in the top panel of Fig.8, in which black filled circles
+represent typical values for Seyfert galaxies, blue open ci rcles
+for Starbursts and red crosses for LINERs. Most ratios prese nt
+Seyfert-like, although some points are observed in the LINE R and
+Starburstregionsofthediagnosticdiagram.Stoklasov´ a e t al(2009)
+analyzed optical IFU data for Mrk1066 and presented several
+opticaldiagnosticdiagrams.Acomparisonbetweentheopti caland
+the present near-IR diagram shows that they are consistent w ith
+each other.
+The locations of the distinct line ratios are shown in the bot -
+tom panel of Fig. 8. Seyfert-like line ratios are observed ap prox-
+imately along the radio jet and ionization cone (PA ≈135/315◦,
+Bower et al.1995;Nagar et al.1999)suggestingthattheline emis-
+sion in these regions is driven by the AGN. The Starburst rati os
+are mainly observed in a region at 0 .′′5 south-east of the nucleus at
+the position of the secondary emission peak of the H recombin a-
+tion lines, as described in the Sec.4.2. Approximately at th is same
+position there is also a higher flux in the continuum (see Fig. 2).
+Fig.8shows alsosome Starburstratiosat1′′north-west inaregion
+approximatelycoincidentwithasmallenhancement oftheHe mis-
+sion (see Fig.7) and with a knot in the optical continuum imag e
+(see Fig. 2). We interpret the Starburst ratios as originati ng in star
+formation regions (SFRs) located at these positions. This c onclu-
+sion is supported by the observations of knots in the optical con-
+tinuum image and its absence in the near-IR continuum, as you ng
+starscontributemostlytotheopticalemission,whileinth enear-IR
+the continuum isdominatedbythe emissionfrom oldstars.At dis-
+tances/greaterorsimilar0.′′5 perpendicular tothe orientation of the ionization coneFigure8. Top:[Feii]λ1.25µm/PaβvsH2λ2.12µm/Brγline-ratio diagnostic
+diagram. The dashed lines delimit regions with ratios typic al of Starbursts
+(blue-opencircles),Seyferts(black-filledcircles)andL INERs(redcrosses).
+Bottom: Spatial position of each point from the diagnostic d iagram.
+the H2λ2.12µm/Brγratio reaches the highest values, being in the
+region of LINERs in the diagnostic diagram. We interpret thi s low
+ionizationregionasbeingoriginatedbyadi ffuseradiationfieldes-
+capingthroughthewallsoftheionizationcone,whichhavee nough
+energytoexcitetheH 2.Asimilarscenarioisobservedforthewell-
+studied Seyfert galaxy NGC4151 (Kraemer, Schmitt& Crensha w
+2008) and attributed to the origin of the near-IR lines in reg ions
+awayfrom itsionization cone (Storchi-Bergmannet al.2009 ).
+4.4 H 2excitation
+The molecular hydrogen flux distribution and kinematics in
+the central regions of active galaxies have been the subject
+c∝circlecopyrt2002 RAS, MNRAS 000,1–15Near-IR dustand lineemissionfromMrk1066 11
+Figure9. Correlation between emission-line fluxes (top left: [Fe ii]λ1.2570µm,topright: Paβ,bottom left: [Pii]λ1.1886µmand bottom left: H 2λ2.1218µm)
+and 3.6 cm radio emission.
+of several recent studies. Nevertheless, its excitation me cha-
+nisms is still in debate (e.g. Reunanen, Kotilainen &Prieto 2002;
+Rodr´ ıguez-Ardila etal.2004;Rodr´ ıguez-Ardila, Ri ffel& Pastoriza
+2005a; Davieset al. 2005; Ri ffeletal. 2006, 2008; S´ anchez et al.
+2009; Ramos Almeida,P´ erez Garc´ ıa & Acosta-Pulido 2009;
+Hicks et al.2009; Storchi-Bergmannet al.2009).The H 2emission
+lines can be excited by two mechanisms: (i) fluorescent excit ation
+through absorption of soft-UV photons (912–1108 Å) in the
+Lyman and Werner bands – present both in star-forming region s
+and surrounding AGNs (Black& vanDishoeck 1987) and (ii)
+collisional excitation (usually referred to as thermal pro cesses)
+due to heating of the gas by shocks, due to interaction of a rad io
+jet with the interstellar medium (Hollenbach & McKee 1989) o r
+by X-rays from the central AGN (Maloney, Hollenbach & Tielen s
+1996).
+We can use the H 2λ2.1218/Brγemission-line ratio to in-
+vestigate the origin of the H 2emission. Typical values for Star-
+burst galaxies, where the main heating agent is the UV radia-tion, are in the range H 2/Brγ/lessorsimilar0.6 (Rodr´ ıguez-Ardila et al. 2004;
+Rodr´ ıguez-Ardila, Ri ffel& Pastoriza 2005a), while for Seyferts
+this ratio is larger because of additional thermal excitati on by
+shocks or X-rays from the AGN. As observed in Figs. 5 and 8,
+Mrk1066 presents two regions with H 2λ2.1218/Brγ<0.6. The
+first located at∼1′′north-west from the nucleus and the second
+at≈0.′′5south-east. As discussed in Sec.4.3, we interpret this re-
+sultasdue toanenhancement ofBr γdue toionizationfromyoung
+stars in SFRs at these locations. In regions away from these S FRs,
+H2λ2.1218/Brγis larger than 0.6 reaching ∼2.6 in regions more
+than 0.′′7 from the nucleus in the direction perpendicular to the ra-
+diojet,suggestingadditionalH 2emissionduetothermalprocesses.
+Another line ratio which can be used to study the H 2
+excitation is H 2λ2.2477/λ2.1218. For fluorescent excitation,
+typical values are ∼0.55, while for thermal processes this ratio
+is∼0.1−0.2 (e.g. Mouri 1994; Reunanen, Kotilainen & Prieto
+2002; Rodr´ ıguez-Ardila etal. 2004; Storchi-Bergmannet a l.
+2009). In the case of active galaxies, the fluorescent proces s
+c∝circlecopyrt2002 RAS, MNRAS 000, 1–1512Riffel,Storchi-Bergmann &Nagar
+Figure10. Relationbetween Nupp=Fiλi
+AigiandEupp=TifortheH 2emission
+lines for thermal excitation at the nuclear position and at 0 .′′5north-west
+from the nucleus. Orthotransitions are shown as filled circles and para
+transitions as open circles. From left to right the transiti ons shown are: 1–
+0Q(1), 1–0S(0), 1–0Q(2), 1–0S(1), 1–0Q(4), 1–0Q(5), 2–1S( 1) and 1–
+0S(2)
+seems not to be important (e.g. Rodr´ ıguez-Ardila et al. 200 4;
+Rodr´ ıguez-Ardila, Ri ffel& Pastoriza 2005a; Ri ffeletal. 2006).
+As observed in Tab.1, H 2λ2.2477/λ2.1218≈0.1 in all positions,
+indicating that the H 2emission in most of the circum-nuclear
+regionof Mrk1066 isalsodue toexcitationbythermal proces ses.
+WecanalsousetheobservedfluxesforallH 2emissionlinesin
+theKbandtocalculatethethermalexcitationtemperature. Follow-
+ing Wilman,Edge & Juhnstone (2005), we have investigated th e
+expression (see alsoStorchi-Bergmannet al.2009):
+log/parenleftBiggFiλi
+Aigi/parenrightBigg
+=constant−Ti
+Texc, (5)
+whereFiis the flux of the ithH2line,λiis its wavelength, Aiis the
+spontaneous emission coe fficient,giis the statistical weight of the
+upper level of the transition, Tiis the energy of the level expressed
+as a temperature and Texcis the excitation temperature. This rela-
+tion is valid for thermal excitation, under the assumption o f anor-
+tho:paraabundance ratioof3:1.InFig.10wepresenttheobserved
+values for Nupp=Fiλi
+Aigi(plus an arbitrary constant) vs Eupp=Ti
+for the nuclear spectrum and for anextra-nuclear one at 0 .′′5 north-
+westfromthenucleus (positionAinFig.1).Theresultingfit ofthe
+above relation isshown inFig.10 as a continuum line and resu lted
+in an excitation temperature of Texc=2131±40K for the nucleus
+andTexc=2097±40K for the position A. The fact that the or-
+tholines (1-0 S(1), 2-1 S(1), 1-0 Q(1), etc) give the same result as
+theparalines (1-0 S(0), Q(2), Q(4), etc) confirm that the H 2is in
+thermal equilibrium at Texcand that the orthotopararatio is 3 as
+assumed. Thisis alsowhat isexpected for thermal equilibri um.
+The heating of the H 2can be provided by nuclear X-rays
+and/or by shocks due to the interaction of the radio jet withthe in -
+terstellarmedium.Neverthelessdistinguishingbetweenb othmech-
+anisms is usuallya hard task. Correlations between the radi oor X-
+ray emission with the H 2emission could be useful to investigate
+the importance of these mechanisms. Quillenet al. (1999), u sing
+HST, imaged a sample of 10 Seyfert galaxies in H 2and looked forcorrelations with radio 6 cm and hard X-ray flux. They found no
+correlation with X-rays and a weak correlation with radio 6- cm,
+suggesting that no single mechanism is likely tobe responsi ble for
+the H2excitation inSeyfert galaxies.
+In Fig.9 we present plots for the [Fe ii]λ1.2570µm (top
+left panel), [Pii]λ1.1886µm (bottom left), Pa β(top right) and
+H2λ2.1218µm (bottom right) emission-line fluxes versusthe radio
+flux at 3.6cm for each spatial pixel. As observed in this figure the
+flux distributions correlate very well with the radio emissi on. We
+usedtheroutinercorrelate inidl3programinglanguage toobtain
+the Sepearman correlation coe fficient (R). The best correlation is
+found for [Feii] withR=0.77, followed by Pa βwithR=0.70,
+[Pii]withR=0.64andH 2withR=0.62.Thesecorrelationcanbe
+understoodintwoways:(i)theinteractionbetweentheradi ojetand
+the ISM produces the heating necessary to collisionally exc itation
+the emitting gas and /or (ii) the radio jet only compresses the gas
+enhancing its emission, excited by other mechanism. The fac t that
+Paβemission shows a better correlation with the radio than the H 2
+emission suggests that the radio jet only compresses the gas , since
+theH recombination linesare not sensible tocollisional ex citation.
+Nevertheless, shocks due tothe radiojet cannot be complete ly dis-
+carded since the ionized and molecular hydrogen emission co uld
+arisefrom distinct locations alongthe line-of-sight.
+Following Rodr´ ıguez-Ardila et al. (2004) and Ri ffelet al.
+(2008) we evaluate the emergent flux for the H 2λ2.1218µm of a
+gascloudilluminatedbyasourceofhardX-raysusingthemod elof
+Maloney, Hollenbach & Tielens (1996). Shuetal. (2007) obta ined
+a2–10keVX-rayfluxof FX∼2.3×10−13ergs−1cm−2andaattenu-
+atingcolumndensityof NH>1024cm−2forMrk1066usingChan-
+dra observations. The NHvalue ismeasured along the line-of-sight
+and according to the unified model NHmay be smaller in other di-
+rections,sinceMrk1066hasaSeyfert2nucleus andthusthed usty
+torus is seen edge-on, enhancing the column density only alo ng
+the line-of-sight (Antonucci 1993). The column density in t he H2
+emitting region can be estimated using NH≈5.2×1021E(B−V)
+[cm−2] (Shull & vanSteenberg 1985). Using the E(B−V) values
+from Fig. 5 we obtain NH≈1021cm−2, which is in good agree-
+mentwiththevalueobtainedbyGuainazzi, Matt& Perola(200 5)–
+NH=1.21×1021cm−2–usingXMM-Newton X-rayobservations.
+Thelower value of NHobtained withthe XMM-Newton relativeto
+the one withChandra data maybe due tothe fact that former has a
+largeraperturethanlatter,dilutingthee ffectofthenucleardust.Us-
+ingNH≈1021cm−2as the appropriate value for the emitting gas,
+theemergentfluxescalculatedfromtheX-rayexcitationmod elcan
+account forthe H 2emissioninmost locations of the observed field
+of Mrk1066.
+ThelineH 22–1S(3)(λ2.0649µm) canalsobeusedtoinvesti-
+gate the H 2excitation. In the case of X-ray irradiated gas this line
+is expected to be absent (Davies etal. 2005). Our spectral ra nge
+doesnotinclude thisline,butRi ffel,Rodr´ ıguez-Ardila &Pastoriza
+(2006) obtained a near-IR nuclear spectrum for an aperture o f
+1.′′6×0.′′6 in which the H 22–1S(3) emission line is not detected,
+supporting the above conclusion that X-ray heating contrib ute to
+the observed H 2emission.
+From the discussion above, we conclude that heating by X-
+rays fromthe centralAGNmaybe the dominant excitationmech a-
+nismoftheH 2.Thisconclusionisingoodagreementwiththoseob-
+tainedfor other active galaxies (e.g. Storchi-Bergmannet al.1999;
+Riffeletal. 2008; Storchi-Bergmann etal. 2009). Nevertheless , as
+3http://ittvis.com/idl
+c∝circlecopyrt2002 RAS, MNRAS 000,1–15Near-IR dustand lineemissionfromMrk1066 13
+discussedabove,somecontributionoftheradiojettotheH 2excita-
+tioncannotbediscarded.Infact,theinteractionoftherad iojetwith
+the circum-nuclear gas seems to be important in the excitati on of
+the H2insome galaxies, suchas ESO428-G14 (Ri ffelet al.2006).
+4.5 [Feii] excitation
+The [Feii]λ1.2570µm/Paβline ratio is controlled by the ratio be-
+tween the volumes of partially to fully ionized regions, as t he
+[Feii]emissionisexcitedinpartiallyionizedgasregions.InAG Ns,
+such regions can be created by X-ray (e.g. Simpsonet al. 1996 )
+and/or shock (e.g. Forbes &Ward1993) heating of the gas.
+The [Feii]λ1.2570µm/Paβline ratio can be used to in-
+vestigate the excitation mechanism of the [Fe ii] emission.
+For Starburst galaxies, [Fe ii]/Paβ/lessorsimilar0.6 and for supernovae
+for which shocks are the main excitation mechanism, this
+ratio has values larger than 2 (Rodr´ ıguez-Ardila et al. 200 4;
+Rodr´ ıguez-Ardila, Ri ffel& Pastoriza 2005a). Thus this ratio can
+be used to measure the relative contribution of photoioniza tion
+and shocks, with values close to 0.6 meaning that photoioniz ation
+dominates, while values close to 2 that shock excitation is d om-
+inant. As observed in Figs.5 and 8, for Mrk1066 typical value s
+for this ratio are 0.6–1.5, although in the regions co-spati al with
+the SFRs [Feii]/Paβ/lessorsimilar0.6 due to an enhancement of the Pa βemis-
+sion.Thehighest values areobservedinlocations closetot heedge
+of the radio structure suggesting that the excitation due to shocks
+by the radio jet is important in these regions. The stronger c or-
+relation between the [Fe ii] and the radio emission than those ob-
+served for the other emission lines (see Fig.9) support a hig her
+contribution of the radio jet for the excitation of [Fe ii] than for
+H2. Radio shocks have been appointed as the main [Fe ii] exci-
+tation mechanism for some Seyfert galaxies such as ESO428-
+G14 (Riffelet al. 2006) and being not negligible in others, as in
+NGC4151 (Storchi-Bergmannet al.2009).
+Another tracer of the origin of the [Fe ii] emission is the
+lineratio[Feii]λ1.2570µm/[Pii]λ1.8861µm.These twolineshave
+similar excitation temperatures, and their parent ions hav e similar
+ionization potentials and radiative recombination coe fficients. As
+pointedoutinOliva et al.(2001)–seealsoStorchi-Bergman net al.
+(2009) – values larger than 2 indicate that shocks have passe d
+through the gas destroying the dust grains, releasing the Fe and
+thus enhancing its observed abundance. For supernova remna nts
+whereshocksarethedominantexcitationmechanism[Fe ii]/[Pii]is
+typically higher than 20 (Oliva etal. 2001). As observed in F ig.5,
+Mrk1066presents[Fe ii]/[Pii]∼3atmostlocations,butinsome re-
+gions closetothebordersoftheradiostructure[Fe ii]/[Pii]reaches
+values up to 9.5, indicating that shocks due to the radio jet a re im-
+portant in these locations in agreement with the highest val ues ob-
+tainedfor the [Feii]/Paβat the same locations.
+From the discussion above, we conclude that X-ray heat-
+ing may be the dominant excitation mechanism of the [Fe ii] in
+Mrk1066, exceptions are the regions 0 .′′85 north-west of the nu-
+cleus where there isa radiohot spot andinthe borders of the r adio
+structure, where shocks seem toplaya more important role. I nany
+case, the contribution of shocks due to the radio jet seems to be
+moreimportantforthe[Fe ii]emissionthanfortheH 2emission,as
+indicatedbythebettercorrelationobservedforwiththera dioemis-
+sion and the [Feii] than the one observed for the H 2. This result is
+in good agreement with the ones obtained for other galaxies ( e.g.
+Rodr´ ıguez-Ardila etal.2004;Rodr´ ıguez-Ardila, Ri ffel& Pastoriza
+2005a;Riffelet al.2006; Storchi-Bergmannet al.1999,2009).4.6 Mass ofionizedand molecular gas
+Following Riffeletal. (2008) and Storchi-Bergmannet al. (2009)
+we estimatethe mass of ionized gas inthe inner 700 ×700pc2by
+MHII≈3×1017/parenleftBiggFBrγ
+ergs−1cm−2/parenrightBigg/parenleftBiggd
+Mpc/parenrightBigg2
+[M⊙], (6)
+whereFBrγis the integrated flux for the Br γemission line and dis
+the distance to Mrk1066. We have assumed an electron tempera -
+tureT=104K andelectron density Ne=100cm−3.
+The mass of molecular gascan be obtained as
+MH2≈5.0776×1013/parenleftBiggFH2λ2.1218
+ergs−1cm−2/parenrightBigg/parenleftBiggd
+Mpc/parenrightBigg2
+[M⊙], (7)
+whereFH2λ2.1218is the integrated flux for the H 2λ2.1218µm emis-
+sion line and we have used the vibrational temperature T =2000K
+we have obtained inSec. 4.4.
+Integrating over the whole IFU field we obtain FBrγ≈2.5×
+10−14ergs−1cm−2andFH2λ2.1218≈2.8×10−14ergs−1cm−2, result-
+ing inMHII≈1.7×107M⊙andMH2≈3.3×103M⊙. The mass
+of molecular gas is 104times smaller than of ionized gas but, as
+discussed in Storchi-Bergmann etal. (2009), this H 2mass repre-
+sents onlythat of hot gas emittinginthe near-IR,heated byX -rays
+and maybe also by shocks from the central AGN. The total mass
+inmoleculargas(includingthecoldgaswhichdoesnot emiti nthe
+near-IR) is usually 105−107times that in hot H 2(Daleet al. 2005)
+suggesting that the totalmolecular gas mass isat least 108M⊙.
+5 CONCLUSIONS
+We used integral field J- and K l-Band spectroscopy of the inner
+700×700pc2of the Seyfert galaxy Mrk1066, obtained with the
+Gemini NIFS at spatial resolution ∼35pc and spectral resolution
+∼3.3Å, to map the near-IR continuum and emission-line flux dis -
+tributions, as well as the properties of the nuclear spectru m. Our
+mainconclusions are:
+•The nucleus contains an unresolved infrared source whose
+continuum is well reproduced by the emission of a dust struct ure
+withtemperature∼830Kandmass∼1.4×10−2M⊙whichweiden-
+tifyas the dusty torus of the UnifiedModel;
+•Emission-line fluxes (except for the coronal lines) are elon -
+gated in PA=135/315◦in agreement with previous optical [O iii]
+imagingand alsofollowingthe radio jetextending beyond th e IFU
+field. Except for the H lines, the emission is stronger to the n orth-
+west than tothe south-EAST inassociation withthe radio hot spot
+implying that at least part of the emittinggas is co-spatial with the
+radio outflow. The H emission is stronger to the south-east, w here
+we finda large regionof star-formation.
+•There is a strong correlation between the radio emission and
+the highest emission-line fluxes suggesting that the radio j et plays
+animportantrolefortheseintensitylevels.Atlowerlinefl uxvalues
+thiscorrelationdisappears indicatingacontributionfro mgasinthe
+galacticplane.
+•The coronal line emission in [Ca viii]λ2.3220µm and
+[Six]λ1.2524µm are unresolved by our observations indicating
+thatthey originate withina radius of 18pc from the nucleus.
+•The [Feii]/PaβvsH2/Brγline-ratio diagnostic diagram
+presents typical values for Seyfert nuclei along the ioniza tion cone
+indicating that the line emission is powered by the AGN in the se
+regions.Someradiationescapesthroughthewallsoftheion ization
+cone and powers a low ionizationemission-line region inloc ations
+c∝circlecopyrt2002 RAS, MNRAS 000, 1–1514Riffel,Storchi-Bergmann &Nagar
+perpendicular tothe cone and radiojet.At the positions 0 .′′5south-
+east and at 1′′north-west of the nucleus Starburst-like values are
+observed due to additional emission of the H lines from star f orm-
+ingregions.
+•The reddening map obtained via the Pa β/Brγline ratio
+present a S-shaped structure with high values oriented alon g the
+PA≈135/315◦, reaching up to E(B−V)=1.7. Lower values are
+observed inregions away from this structure withtypical va lues of
+E(B−V)=0.5. The reddening in the continuum is smaller with
+typical values of E(B−V)=0.9along the S-shaped structure.
+•The H 2gas has an excitation temperature Texc≈2100K and
+its emission is mainly due to X-ray excitation from the centr al
+AGN, which is also the main excitation mechanism of the [Fe ii]
+emission. Emission due to shocks produced by the radio jet ma y
+contribute a small fraction to the emission of both lines, wi th a
+larger contributionfor the [Fe ii] line.
+•Themassofionizedgasintheinner700 ×700pc2ofMrk1066
+isMHII≈1.7×107M⊙, while that of hot molecular gas is MH2≈
+3.3×103M⊙.
+ACKNOWLEDGMENTS
+Based on observations obtained at the Gemini Observatory, w hich
+is operated by the Association of Universities for Research in As-
+tronomy, Inc., under a cooperative agreement with the NSF on
+behalf of the Gemini partnership: the National Science Foun da-
+tion(UnitedStates),theScienceandTechnologyFacilitie sCouncil
+(UnitedKingdom),theNationalResearchCouncil(Canada), CON-
+ICYT (Chile), the Australian Research Council (Australia) , Min-
+ist´ erio da Ciˆ encia e Tecnologia (Brazil) and south-eastC YT (Ar-
+gentina). This research has made use of the NASA /IPAC Extra-
+galactic Database (NED) which is operated by the Jet Propuls ion
+Laboratory, CaliforniaInstituteof Technology, under con tract with
+theNationalAeronauticsandSpaceAdministration.Thiswo rkhas
+been partiallysupported bythe BrazilianinstitutionCNPq .
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+arXiv:1001.0132v1  [math.GT]  31 Dec 2009TWISTED ALEXANDER POLYNOMIALS AND FIBERED
+3–MANIFOLDS
+STEFAN FRIEDL AND STEFANO VIDUSSI
+Abstract. In a series of papers the authors proved that twisted Alexander p oly-
+nomials detect fibered 3-manifolds, and they showed that this implies that a closed
+3–manifold Nis fibered if and only if S1×Nis symplectic. In this note we summa-
+rize some of the key ideas of the proofs. We also give new evidence to the conjecture
+that ifMis a symplectic 4–manifold with a free S1–action, then the orbit space is
+fibered.
+1.Introduction
+1.1.Definitions and previous results. Amanifold pair is a pair (N,φ) where
+N/\e}atio\slash=S1×D2,N/\e}atio\slash=S1×S2is an orientable connected 3–manifold with toroidal or
+empty boundary and φ∈H1(N;Z) = Hom(π1(N),Z) is non–trivial. We say that a
+manifold pair ( N,φ)fibers over S1if there exists a fibration p:N→S1such that
+the induced map p∗:π1(N)→π1(S1) =Zcoincides with φ. Given a manifold pair
+(N,φ) theThurston norm ofφis defined as
+||φ||T= min{χ−(Σ)|Σ⊂Nproperly embedded surface dual to φ}.
+Here, given a compact surface Σ with connected components Σ 1∪ ···∪Σk, we de-
+fineχ−(Σ) =/summationtextk
+i=1max{−χ(Σi),0}. Thurston [Th86] showed that this defines a
+seminorm on H1(N;Z) which can be extended to a seminorm on H1(N;R).
+Given a manifold pair ( N,φ) and a homomorphism α:π1(N)→Gto a finite
+groupGwe can consider the corresponding twisted Alexander polynomial ∆α
+N,φ∈
+Z[t±1]. This invariant was initially introduced by Lin [Lin01], Wada [Wa94] and
+Kirk–Livingston [KL99]. We will recall the definition in Section 2 and we re fer to
+[FV09] for a survey of the theory of twisted Alexander polynomials.
+We say that ∆α
+N,φ∈Z[t±1] ismonicif its top coefficient equals ±1. Note that ∆α
+N,φ
+is palindromic, in particular if its top coefficient equals ±1, then its bottomcoefficient
+also equals ±1. Given a polynomial p(t)∈Z[t±1] withp=/summationtextl
+i=kaiti,ak/\e}atio\slash= 0,al/\e}atio\slash= 0
+we define deg( p) =l−k.
+We have the following theorem, that gives a characterization of fibe red 3-manifolds
+in terms of twisted Alexander polynomials.
+Date: November 13, 2018.
+The second author was partially supported by NSF grant #0906281 .
+12 STEFAN FRIEDL AND STEFANO VIDUSSI
+Theorem 1.1. Let(N,φ)be a manifold pair. Then (N,φ)is fibered if and only if for
+any epimorphism α:π1(N)→Gonto a finite group the twisted Alexander polynomial
+∆α
+N,φ∈Z[t±1]is monic and
+deg(∆α
+N,φ) =|G|/bardblφ/bardblT+(1+b3(N))divφα,
+whereφαdenotes the restriction of φ:π1(N)→Zto Ker(α), and where we denote
+by divφα∈Nthe divisibility of φα, i.e.
+divφα= max{n∈N|φα=nψfor someψ:Ker(α)→Z}.
+The ‘only if’ direction has been shown at various levels of generality by Cha [Ch03],
+Kitano and Morifuji [KM05], Goda, Kitano and Morifuji [GKM05], Pajitn ov [Pa07],
+Kitayama [Kiy08], [FK06] and [FV09, Theorem 6.2]. We will also outline the p roof
+in Section 2. The ‘if’ direction is the main result of [FV08c].
+The main goal of this paper is to provide a summary of the proof, and to show a
+few ways that the approach can be generalized to prove new result s.
+Revisiting the proof of Theorem 1.1 will also show that in fact the follow ing refine-
+ment of Theorem 1.1 holds:
+Theorem 1.2. Let(N,φ)be a manifold pair such that π1(N)is residually finite
+solvable (we refer to Section 3.4 for the definition). Then (N,φ)is fibered if and
+only if for any epimorphism α:π1(N)→Gonto a finite solvable group the twisted
+Alexander polynomial ∆α
+N,φ∈Z[t±1]is monic and if the following equality holds
+deg(∆α
+N,φ) =|G|/bardblφ/bardblT+(1+b3(N))divφα.
+Note that 3-manifolds with residually finite solvable fundamental gro up are fairly
+frequent. For example, as we will see in Theorem 3.4, any 3-manifold h as a finite
+cover such that its fundamental group is residually finite solvable. A lso note that the
+fundamental group of a fibered 3-manifold is always residually finite s olvable.
+1.2.Fibered manifolds and symplectic 4–manifolds. The main application of
+the “if” direction of Theorem 1.1 is in the proof of the following Theore m.
+Theorem 1.3. LetNbe a closed 3–manifold. Then S1×Nis symplectic if and only
+ifNis fibered.
+Proof. (outline) We first consider the ‘if’ direction. This direction was first proved
+by Thurston [Th76], and we present here a proof that is well-known t o the experts.
+Letp:N→S1be a fibration. We write ψ=p∗(dt) wheredtis the canonical
+non-degenerate closed 1–form on S1=R/Z. By [Ca69] we can find a metric on N
+such thatψis harmonic. Denote by ∗ψthe dual closed 2–form. We now consider
+the 1–form dsonS1as a 1–form on S1×Nby the pull back operation, similarly we
+considerψand∗ψas forms on S1×N. With this convention we now define:
+ω=ds∧ψ+∗ψ.TWISTED ALEXANDER POLYNOMIALS AND FIBERED 3–MANIFOLDS 3
+Clearlyωis a closed 2–form on S1×N. We furthermore calculate that
+ω∧ω= 2ds∧ψ∧∗ψ.
+But sinceψis non-zero everywhere, it follows that ψ∧∗ψis a 3-form on Nwhich is
+non-zero everywhere. Hence ω∧ωisa 4-formon S1×Nwhich is non-zero everywhere.
+This shows that ωis a symplectic form on S1×N.
+The ‘only if’ direction follows by showing that if S1×Nis symplectic, then there
+exists a class φ∈H1(N,Z) determined by the symplectic form that satisfies the
+constraints described in Theorem 1.1. This is achieved building on an ide a of Kro-
+nheimer in [Kr99]. Suppose that Nis a closed 3-manifold such that S1×Nadmits
+a symplectic form ω. Without loss of generality we can assume that ωrepresents an
+integral class. Taubes proved in [Ta94, Ta95] that the Seiberg–Wit ten invariants of a
+symplectic 4–manifoldsatisfyvery stringent constraints, thatca nbeviewed asakinto
+a condition of “monicness”. This, together with the relation betwee n Seiberg–Witten
+invariants of S1×Nand the Alexander polynomial of N, due to Meng and Taubes,
+translates in the condition that ∆ N,φis monic, where φ∈H1(N,Z) is the K¨ unneth
+component of [ ω]∈H2(S1×N;Z)∼=H1(N;Z)⊕H2(N;Z). Next, thanks to a the-
+orem of Donaldson ([Do96]), there exists a symplectic surface dual to (a sufficiently
+large multiple of) the symplectic form. Such surface satisfies the us ual adjunction
+formula for symplectic surfaces. This formula, played against Kron heimer’s adjunc-
+tion inequality for manifolds of type S1×N, gives a constraint on the top degree of
+the Alexander polynomial ∆ N,φin terms of the Thurston norm, more precisely
+deg(∆ N,φ) =/bardblφ/bardblT+2divφ.
+The constraints above hold for all finite covers of N, as all finite covers of S1×Nare
+symplectic as well. The connection between the twisted Alexander po lynomials of N
+and the ordinary Alexander polynomials of the finite covers of Nentails at this point
+that for any epimorphism α:π1(N)→Gto a finite group the twisted Alexander
+polynomial ∆α
+N,φis monic and
+deg(∆α
+N,φ) =|G|/bardblφ/bardblT+2divφα.
+(We refer to [FV08a] for the details of the argument). The theore m follows at this
+point from Theorem 1.1. /square
+We would like to mention that an alternative proof of the “only if” direc tion of
+Theorem1.3inthecasethat b1(N) = 1, andunder atechnical conditioninthegeneral
+case, follows from combining the work of Kutluhan–Taubes [KT09], Kr onheimer–
+Mrowka [KM08] and Ni [Ni08]. This proof requires a more sophisticated study of the
+Seiberg–Witten theory of S1×Nin the symplectic case.
+1.3.Non–fibered manifolds and vanishing twisted Alexander poly nomials.
+It is natural to ask whether the conditions in Theorem 1.1 can be wea kened. In4 STEFAN FRIEDL AND STEFANO VIDUSSI
+particular, in light of some partial results discussed below, we propo se the following
+conjecture.
+Conjecture 1.4. Let(N,φ)be a manifold pair. If (N,φ)is not fibered, then there
+exists an epimorphism α:π1(N)→Gonto a finite group Gsuch that ∆α
+N,φ= 0∈
+Z[t±1].
+Besides the interest per sein sharpening the results of Theorem 1.1 there are other
+reasons to investigate Conjecture 1.4. First, the proof of Theor em 1.3 would be sig-
+nificantly simplified, bypassing the use of Kronheimer’s refined adjun ction inequality:
+Taubes’ nonvanishing result for Seiberg–Witten invariants of symp lectic manifolds
+would suffice to carry the argument. But more importantly, Conjec ture 1.4 would
+imply a result akin to Theorem 1.3 for all symplectic 4–manifolds that ca rry a free
+circle action. For those manifolds, in fact, a refined adjunction ineq uality in the
+spirit of [Kr99] does not seem available, and Taubes’ constraints tr anslate to a mere
+monicness of the twisted Alexander polynomials of the orbit space. W e state these
+observations in the following form, referring to [FV07b] for details and for the extent
+to which the converse holds.
+Theorem 1.5. LetMbe a4–manifold which carries a free circle action with orbit
+spaceN. If Conjecture 1.4 holds for N, thenMadmits a symplectic structure only
+ifNfibers over the circle.
+We also refer to [Ba01] and [Bo09] for related work on the problem of determining
+which 4–manifolds with a free circle action admit a symplectic structur e. We refer
+also to the work by Silver and Williams [SW09a, SW09b] and by Pajitnov [P a08] for
+several interesting connections of Conjecture 1.4 to other prob lems in 3–dimensional
+topology.
+Conjecture 1.4 can be proven to hold for various classes of manifold s. In order to
+describe them in detail, we must introduce some definitions.
+Letπbe a group and Γ ⊂πa subgroup. We say Γ is separable if for anyg∈π\Γ
+there exists an epimorphism α:π→Gonto a finite group Gsuch thatα(g)/\e}atio\slash∈α(Γ).
+Put differently, we can tell that gis not in Γ by going to a finite quotient. We say π
+islocally extended residually finite (LERF) if any finitely generated subgroup of πis
+separable.
+The following theorem proves Conjecture 1.4 in various special case s:
+Theorem 1.6. Let(N,φ)be a manifold pair. Suppose that ∆α
+N,φ/\e}atio\slash= 0∈Z[t±1]for
+any epimorphism α:π1(N)→Gonto a finite group G. Furthermore suppose that
+one of the following holds:
+(1)N=S3\νKandKis a genus one knot,
+(2)||φ||T= 0,
+(3)Nis a graph manifold,
+(4)π1(N)is LERF.TWISTED ALEXANDER POLYNOMIALS AND FIBERED 3–MANIFOLDS 5
+Then(N,φ)fibers overS1.
+We refer to [FV07a, Theorem 1.3] and [FV08b, Theorem 1, Proposit ion 4.6, Corol-
+lary 5.6] for details and proofs. Note that it is conjectured (cf. [Th 82]) thatπ1(N) is
+LERF for any hyperbolic 3–manifold N.
+In Section 4 we will give new conditions under which Conjecture 1.4 hold s.
+Acknowledgments. The first author would like to express his gratitude to the
+organizers of the Georgia International TopologyConference 20 09 for the opportunity
+to speak and for organizing a most enjoyable and interesting meetin g.
+2.Twisted invariants of 3–manifolds
+We recall the definition of twisted homology and cohomology and their basic prop-
+erties. LetXbeatopologicalspaceandlet ρ:π1(X)→GL(n,R)bearepresentation.
+Denote by /tildewideXthe universal cover of X. Lettingπ=π1(X), we use the representation
+ρto regardRnas a left Z[π]–module. The chain complex C∗(/tildewideX) is also a left Z[π]–
+module via deck transformations. Using the natural involution g/mapsto→g−1on the group
+ringZ[π], we can view C∗(/tildewideX) as a right Z[π]–module and form the twisted homology
+groups
+Hρ
+∗(X;Rn) =H∗(C∗(/tildewideX)⊗Z[π]Rn).
+For most of the paper we will be interested in a particular type of rep resentation.
+Letφ∈H1(X;Z) and letα:π1(X)→GL(n,Z) be a representation. We can now
+define a left Z[π1(X)]–module structure on Zn⊗ZZ[t±1] =:Zn[t±1] as follows:
+g·(v⊗p) := (α(g)·v)⊗(φ(g)·p) = (α(g)·v)⊗(tφ(g)p),
+whereg∈π1(X),v⊗p∈Zn⊗ZZ[t±1] =Zn[t±1]. Put differently, we get a represen-
+tationα⊗φ:π1(X)→GL(n,Z[t±1]).
+We call the resulting twisted module Hα⊗φ
+1(X;Zn[t±1]) thetwisted Alexander mod-
+ule of(X,φ,α). Whenφandαare understood, then we just write H∗(X;Zn[t±1]).
+Now suppose Xhas finitely many cells in all dimensions. Using that Z[t±1] is a
+Noetherian UFD it follows that Hα⊗φ
+i(X;Zn[t±1]) is a finitely generated module over
+Z[t±1]. We now denote by ∆α
+X,φ,i∈Z[t±1] the order of Hα⊗φ
+1(X;Zn[t±1]) and refer
+to it as the twisted Alexander polynomial of (X,φ,α). We refer to [Tu01] or [FV09,
+Section 2] for the precise definitions. Note that the twisted Alexan der polynomials
+are well–defined up to multiplication by an element of the form ±tk,k∈Z.
+We adopt the convention that we drop αfrom the notation if αis the trivial
+representation to GL(1 ,Z). Ifα:π1(N)→Gis a homomorphism to a finite group
+G, then we get the regular representation π1(N)→G→Aut(Z[G]) where the second
+map is given by left multiplication. We can identify Aut( Z[G]) with GL( |G|,Z) and
+we obtain the corresponding twisted Alexander polynomial ∆α
+N,φ.6 STEFAN FRIEDL AND STEFANO VIDUSSI
+As an example we give an outline of the proof of the ‘only if’ direction in T heorem
+1.1.
+Lemma 2.1. Let(N,φ)be a fibered manifold pair. Then for any epimorphism α:
+π1(N)→Gonto a finite group the twisted Alexander polynomial ∆α
+N,φ∈Z[t±1]is
+monic and the following equality holds
+deg(∆α
+N,φ) =|G|/bardblφ/bardblT+(1+b3(N))divφα.
+Proof.First note that exists a short exact Mayer–Vietoris sequence
+0→H1(Σ;Z[G])⊗Z[t±1]tι+−ι−− −−− →H1(N\Σ;Z[G])⊗Z[t±1]→H1(N;Z[G][t±1])→0,
+where Σisafiber of( N,φ). Notethat inthecase ofaknot complement anduntwisted
+coefficients this is just the usual exact sequence relating the homo logy of a Seifert
+surface to the Alexander module of a knot (cf. e.g. [Lic97, Theorem 6.5]). We write
+r= rankH1(Σ;Z[G]), where the rank is taken as a Z–module. Picking a basis (over
+Z) forH1(Σ;Z[G]) andH1(N\Σ;Z[G]) =H1(Σ×[0,1];Z[G]) we can represent ι±by
+r×r-matricesA±. It follows from the definition of the Alexander polynomial that
+∆α
+N,φ= det(tA+−A−) =trdet(A+)+···+det(−A−).
+Sinceι±are homotopy equivalences it follows that det( A±) =±1. In particular ∆α
+N,φ
+is monic and of degree r. It remains to determine r. First note that we have
+2/summationdisplay
+i=0(−1)irankHi(Σ;Z[G]) =|G|·2/summationdisplay
+i=0(−1)irankHi(Σ;Z) =|G|·(−χ(Σ)) =|G|||φ||T.
+Furthermore recall that Σ is closed if and only if Nis closed. The formula for rnow
+follows from a direct calculation of the rank of H0(Σ;Z[G]) and from duality in the
+case that Σ is closed. /square
+3.Summary of the proof of Theorem 1.1
+Our goal is to give an outline of the proof of Theorem 1.1 given in [FV08c ] without
+descending into the many technical details required in a rigorous writ e up. We are
+acutely aware of the fact that the proof in [FV08c] hides the fores t behind a wall of
+trees.
+3.1.Step A: First observations. Let (N,φ) be a manifold pair and k∈N. Note
+that (N,φ) fibers if and only if ( N,kφ) fibers and note that ||kφ||T=k||φ||T. It
+follows now easily that it suffices to prove Theorem 1.1 for primitive φ∈H1(N;Z).
+Theorem A. Let(N,φ)be a manifold pair with φ∈H1(N;Z)primitive. Assume
+that∆N,φ/\e}atio\slash= 0. Then the following hold:
+(1)There exists a connected Thurston norm minimizing surface Σdual toφ.
+(2)Any connected surface Σdual toφintersects any boundary torus, in particular
+Σis closed if and only if Nis closed.TWISTED ALEXANDER POLYNOMIALS AND FIBERED 3–MANIFOLDS 7
+(3)If∆α
+N,φ/\e}atio\slash= 0for any epimorphism α:π1(N)→Gonto a finite group, then N
+is irreducible.
+(4)The pair (N,φ)fibers overS1if and only if the maps ι±:π1(Σ)→π1(N\νΣ)
+are isomorphisms.
+Proof.If ∆N,φ/\e}atio\slash= 0, then it follows from [McM02, Section 4 and Proposition 6.1] that
+there exists a connected Thurston norm minimizing surface Σ dual t oφ.
+NowletΣbeanyconnectedsurfacedualto φ. Supposethatthereexistsaboundary
+torusTofNwhich Σ does not intersect. Then Tlifts to the infinite cyclic cover ˆN
+ofNdetermined by φ:π1(N)→Z, in particular ˆNcontains infinitely many tori in
+its boundary. A standard argument now shows that b1(ˆN) =∞, but it is well-known
+(cf. [Tu01]) that b1(ˆN) = deg∆ N,φ.
+Statement (3) follows from an argument of McCarthy [McC01] (se e also [FV08c,
+Lemma 7.1] and [Bo09]). Note that the proof of (3) relies on the fact that 3-manifold
+groups are residually finite, which is a consequence of the proof of t he Geometrization
+Conjecture (cf. [Th82] and [He87]). The final statement is a conse quence of Stallings’
+fibering theorem ([St62] and [He76]). /square
+Throughout this section Σ will always denote a connected Thurston norm minimiz-
+ing surface dual to φ. We write M=N\νΣ and denote the two canonical inclusion
+maps of Σ into ∂Mbyι±. Since Σ ⊂Nis Thurston norm minimizing it follows from
+Dehn’s lemma that the inclusion induced maps π1(Σ)→π1(N) andπ1(M)→π1(N)
+are injective. In particular we can view π1(Σ) andπ1(M) as subgroups of π1(N).
+Given an epimorphism α:π1(N)→Gonto a finite group Gwe say ∆α
+N,φhas
+Property (M) if ∆α
+N,φ∈Z[t±1] is monic and if
+deg(∆α
+N,φ) =|G|/bardblφ/bardblT+(1+b3(N))divφα
+holds.
+3.2.Step B: Extracting information from twisted Alexander poly nomials.
+In view of Theorem A our strategy is now to translate the informatio n coming from
+twisted Alexander polynomials into information on the maps ι±:π1(Σ)→π1(M).
+We start with considering the untwisted polynomial:
+Lemma 3.1. If∆N,φhas Property (M), then the maps ι±:H1(Σ;Z)→H1(M;Z)
+are isomorphisms.
+This lemma is well-known in the case of the untwisted Alexander polynom ial for
+knots. An early reference is given by [CT63] but see also [Ni07, Prop osition 3.1] or
+[GS08].
+Proof.We translate the information on twisted Alexander polynomials into inf orma-
+tion on the maps ι±:π1(Σ)→π1(M) by considering, as in Lemma 2.1, the following8 STEFAN FRIEDL AND STEFANO VIDUSSI
+long exact Mayer–Vietoris sequence:
+(1)... →H2(N;Z[t±1])→
+→H1(Σ;Z)⊗Z[t±1]tι+−ι−− −−− →H1(M;Z)⊗Z[t±1]→H1(N;Z[t±1])→
+→H0(Σ;Z)⊗Z[t±1]tι+−ι−− −−− →H0(M;Z)⊗Z[t±1]→H0(N;Z[t±1])→0.
+Now suppose that ∆ N,φ∈Z[t±1] is monic and that
+deg∆N,φ=/bardblφ/bardblT+(1+b3(N))
+holds. Note that this in particular implies that H1(N;Z[t±1]) isZ[t±1]-torsion, since
+H0(Σ;Z)⊗Z[t±1]isafree Z[t±1]-moduleitfollowsimmediatelythatthemap H1(N;Z[t±1])→
+H0(Σ;Z)⊗Z[t±1]isthetrivialmap. NowrecallthatΣisconnectedandthatΣisclosed
+if and only if Nis closed. In our context this implies that /bardblφ/bardblT+(1+b3(N)) equals
+twice the genus gof Σ. Using elementary arguments one can show that H1(M;Z)
+is a free abelian group of the same rank as H1(Σ;Z), namely 2 g. Picking bases for
+H1(Σ;Z) andH1(M;Z) we denote the corresponding 2 g×2g–matrices for ι±byA±.
+It now follows from the definition of the Alexander polynomial that
+∆N,φ= det(tA+−A−) = det(A+)t2g+···+det(−A−).
+Recall that we assumed that ∆ N,φis monic and that deg∆ N,φ= 2g. Since ∆ N,φis
+palindromic it now follows that A−andA+are invertible matrices, in particular the
+mapsι±:H1(Σ;Z)→H1(M;Z) are isomorphisms. /square
+Clearly the conclusion of the claim is not enough to deduce that π1(Σ)→π1(M)
+is an isomorphism. In fact there exist many non-fibered knots whos e Alexander
+polynomial has Property (M). We will therefore use the information coming from all
+twisted Alexander polynomials.
+Using the idea of the proof the previous claim we can show the following (we refer
+to [FV08c, Theorem 3.2] for details). If α:π1(N)→Gis an epimorphism onto a
+finite group such that ∆α
+N,φhas Property (M), then the maps ι±:H1(Σ;Z[G])→
+H1(M;Z[G]) are isomorphisms. In fact, considering the ‘ H0-part’ of the Mayer–
+Vietoris sequence (1) we see that the assumption ∆α
+N,φ/\e}atio\slash= 0 implies that the maps
+ι±:H0(Σ;Z[G])→H0(M;Z[G]) are isomorphisms. Using well-known properties of
+0-th homology groups (cf. e.g. [HS97, Section VI]) this condition is eq uivalent to
+Im{π1(Σ)ι±− →π1(M)α− →G}= Im{π1(M)α− →G}.
+For future reference we now summarize the results of the above d iscussion in the
+following theorem.
+Theorem B. Letα:π1(N)→Gbe an epimorphism onto a finite group such that
+∆α
+N,φ/\e}atio\slash= 0, then
+(2) Im{π1(Σ)ι±− →π1(M)α− →G}=Im{π1(M)α− →G}.TWISTED ALEXANDER POLYNOMIALS AND FIBERED 3–MANIFOLDS 9
+If furthermore ∆α
+N,φhas Property (M), then
+(3) ι±:H1(Σ;Z[G])→H1(M;Z[G])
+are isomorphisms
+Ourgoalnowistoshow thattheinformationwejust obtainedfromt wisted Alexan-
+der polynomials is in fact enough to deduce that ι±:π1(Σ)→π1(M) are isomor-
+phisms.
+3.3.Step C: Finite solvable quotients. First recall that in the untwisted case
+we obtained the following conclusion: if ∆ N,φhas Property (M), then the maps
+ι±:H1(Σ;Z)→H1(M;Z) are isomorphisms. Another way of saying this is that
+the mapsι±:π1(Σ)→π1(M) ‘look like an isomorphism on the abelian level’. Our
+goal is now to show that if all twisted Alexander polynomials correspo nding to finite
+solvable groups have Property (M), then the maps ι±:π1(Σ)→π1(M) ‘look like
+isomorphisms on the finite solvable level’. More precisely, we will prove t he following
+theorem.
+Theorem C. Let(N,φ)be a manifold pair such that for any epimorphism π1(N)→S
+onto a finite solvable group the polynomial ∆α
+N,φhas Property (M). Then for any finite
+solvable group Sthe induced maps
+ι∗
+±:Hom(π1(M),S)→Hom(π1(Σ),S)
+are bijections.
+The outline of the proof of Theorem C will require the remainder of th is section.
+We will now need to introduce a couple of definitions. Given a solvable gr oupSwe
+denote byℓ(S) its derived length, i.e. the length of the shortest decomposition int o
+abelian groups. Note that ℓ(S) = 0 if and only if S={e}.
+Givenn∈N∪ {0}we denote by S(n) the statement that for any finite solvable
+groupSwithℓ(S)≤nthe maps
+ι∗
+±: Hom(π1(M),S)→Hom(π1(Σ),S)
+are bijections. It is a straightforward exercise to see that ι±:H1(Σ;Z)→H1(M;Z)
+are isomorphisms if and only if S(1) holds.
+Note that Theorem C says that S(n) holds for all nif all twisted Alexander polyno-
+mials corresponding to finite solvable groups have Property (M). We will show that
+this does indeed hold by induction on n. For the induction argument we use the
+following auxiliary statement: Given n∈N∪{0}we denote by H(n) the statement
+that for any epimorphism β:π1(M)→TwhereTis finite solvable with ℓ(T)≤n
+the maps
+ι±:H1(Σ;Z[T])→H1(M;Z[T])
+are isomorphisms.10 STEFAN FRIEDL AND STEFANO VIDUSSI
+Proposition 3.2. [FV08c, Proposition 3.3] IfH(n)andS(n)hold, then S(n+ 1)
+holds as well.
+Proof.LetGbe a group and α:G→San epimorphism onto a solvable group of
+derived length n. Then we obtain the following short exact sequence
+0→H1(G;Z[S])→G/[Ker(α),Ker(α)]α− →S→1.
+In particular if we can control solvable quotients of derived length a t mostnand the
+corresponding first homology groups, then we can control solvab le information on G
+up to length n+1. The proposition now follows from elaborating this principle. We
+refer to [FV08c, Section 3.3] for the full details. /square
+Proposition 3.3. [FV08c, Proposition 3.4] Assume that ∆α
+N,φhas Property (M) for
+any epimorphism α:π1(N)→Sonto a finite solvable group Swithℓ(S)≤n+1. If
+S(n)holds, then H(n)holds as well.
+Proof.In this proof we find it convenient to introduce the notation π=π1(N),
+A=π1(Σ) andB=π1(M). Recall that we can view AandBas subgroups of π.
+In fact we can view πas an HNN extension of BbyA, more precisely we have a
+canonical isomorphism
+π=/a\}bracketle{tB,t|tι−(g)t−1=ι+(g),g∈A/a\}bracketri}ht.
+As usual we will normally just write π=/a\}bracketle{tB,t|,ti−(A)t−1=ι+(A)/a\}bracketri}ht. Letβ:B→T
+beanepimorphismwhere Tisfinitesolvablewith ℓ(T)≤n. Givenafinitelygenerated
+groupCwe now define
+C(T) =/intersectiondisplay
+γ∈Hom(C,T)Ker(γ).
+It isstraightforwardtoseethat C/C(T)isafinitesolvablegroupwith ℓ(C/C(T))≤n
+(see [FV08c, Lemma 3.6]). It is a consequence of S(n) that the homomorphisms
+(4) ι±:A/A(T)→B/B(T)
+are in fact isomorphisms (see [FV08c, Lemma 3.6]). In particular we c an define an
+epimorphism
+π=/a\}bracketle{tB,t|,tι−(A)t−1=ι+(A)/a\}bracketri}ht → /a\}bracketle{tB/B(T),t|tι−(A/A(T))t−1=ι+(A/A(T))/a\}bracketri}ht.
+It is a consequence of (4) that the above group is in fact a semidirec t product, i.e. we
+have an isomorphism
+/a\}bracketle{tB/B(T),t|tι−(A/A(T))t−1=ι+(A/A(T))/a\}bracketri}ht∼=Z⋉B/B(T),
+where 1∈Zacts onB/B(T) viaι−◦ι−1
++. SinceB/B(T) is finite this automorphism
+has finite order, say k, therefore there exists an epimorphism
+α:π→Z⋉B/B(T)→Z/k⋉B/B(T) =:S.
+Note thatSis a finite solvable group of length n+1. It now follows from our assump-
+tion that the twisted Alexander polynomial ∆α
+N,φhas Property (M). The informationTWISTED ALEXANDER POLYNOMIALS AND FIBERED 3–MANIFOLDS 11
+coming from Theorem B is not quite what we wanted, since we replaced β:B→T
+byα:B→S. But since Ker( α)⊂Ker(β), the latter homomorphism contains in fact
+the information coming from β, using a few technical arguments we can now deduce
+that
+ι±:H1(A;Z[T])→H1(B;Z[T])
+is an isomorphism as well. We refer to [FV08c, Section 3.4] for the full d etails. /square
+Theorem C is now an immediate consequence of Propositions 3.2 and 3.3 and of
+the fact, observed above, that ∆ N,φhaving Property (M) implies that S(1) holds.
+3.4.Step D: Residually finite solvable fundamental groups. The conclusion
+of Theorem C loosely says that the maps 鱑look like isomorphisms on the finite
+solvable level’ if all ∆α
+N,φhave Property (M). With the methods from the previous
+section Theorem C is the maximum information on the map ι±:π1(Σ)→π1(M) we
+can obtain from twisted Alexander polynomials.
+InordertoanalyzethecontentoftheconclusionofTheoremCwen eedthefollowing
+definition. Let Pbe a property of groups(e.g. finite, finite solvable), then we say th at
+a groupπisresidually Pif for any non-trivial g∈ Pthere exists a homomorphism
+α:π→Gto a group Gwith Property Psuch thatα(g) is non-trivial. For example
+it is well-known that surface groups are residually finite solvable, and that 3-manifold
+groups are residually finite (cf. [Th82] and [He87]).
+On the other hand 3-manifold groups are in general not residually fin ite solvable.
+For example if Kis a non-trivial knot with Alexander polynomial equal to one, then
+standard arguments show that any homomorphism π1(S3\νK)→Sto a solvable
+groupSnecessarily factors throughthe abelianization π1(S3\νK)→Z. In particular
+π1(S3\νK) is not residually finite solvable. One can use such a knot to construc t a
+manifold pair ( N,φ) whereι±: Hom(π1(Σ),S)→Hom(π1(M),S) is a bijection for
+any solvable S, but such that π1(M) is not residually solvable. In particular Mis not
+a product.
+This discussion shows that the conclusion of Theorem C is not strong enough to
+ensure that ι±:π1(Σ)→π1(M) areisomorphisms, theproblem being that3-manifold
+groups are in general not residually finite solvable.
+Before we continue we need to introduce a few more notions. We say a group has
+a property virtually, if there exists a finite index subgroup which has this property.
+Also recall, that given a prime pap–groupis a group whose order is a power of p.
+IfGis a group which is residually a p–group, then we will normally just say Gis
+residuallyp.
+Wecan now formulatethe following recent theorem ofMatthias Asch enbrenner and
+the first author.
+Theorem 3.4. [AF10]LetNbe a 3-manifold. Then for almost all primes pthe group
+π1(N)is virtually residually p.12 STEFAN FRIEDL AND STEFANO VIDUSSI
+Recall that p-groups are finite solvable, in particular Theorem 3.4 says that 3-
+manifold groups are virtually residually finite solvable.
+IfNis hyperbolic then the theorem is a consequence of the fact that line ar groups
+are virtually residually p(cf. e.g. [We73, Theorem 4.7]). The proof of that fact is so
+short and elegant that we think it is worthwhile mentioning.
+Proof of Theorem 3.4 for hyperbolic N.We writeπ=π1(N). Since we assume that
+Nis hyperbolic we can assume that πis a subgroup of SL(2,C). Sinceπis finitely
+generated there exists a finitely generated subring RofCsuch thatπ⊂GL(n,R). It
+is well–known that for almost all primes pthere exists a maximal ideal mofRwith
+char(R/m) =p(see [LS03, p. 376f]).
+Nowletpbeaprimeforwhichthereexistsamaximalideal mofRwithchar(R/m) =
+p. We will show that πis virtually residually p. Before we continue note that R/mkis
+a finite ring forany k≥1 and that/intersectiontext∞
+k=1mk={0}by the Krull Intersection Theorem.
+Fork≥1 we let
+πk= Ker/parenleftbig
+π→GL(n,R)→GL(n,R/mk)/parenrightbig
+.
+Eachπkis a normal subgroup of π, of finite index, and clearly πk+1⊂πkfor every
+k≥1. Moreover/intersectiontext∞
+k=1πk={1}since/intersectiontext∞
+k=1mk={0}.
+We claim that π1is residually p. We will prove this by showing that π1/πkis a
+p-group for any k. This in turn follows from showing that any non–trivial element in
+πk/πk+1has orderp. In order to show this pick A∈πk. By definition we can write
+A= id+Cfor somen×n-matrixCwith entries in mk.
+Fromp∈mandk≥1 we get that
+Ap= (id+C)p= id+pC+p(p−1)
+2C2+···+Cp
+= id+(some n×n-matrix with entries in mk+1).
+HenceAp∈πk+1. /square
+The combination of Theorem C and 3.4 shows that proving Theorem 1.1 becomes
+much easier, if we can go to finite covers. Fortunately the following le mma tells us
+that we can indeed do so:
+Lemma 3.5. Letp:N′→Nbe a finite cover and let φ′=p∗(φ). Then the following
+hold:
+(1) (N,φ)fibers if and only if (N′,φ′)fibers,
+(2)if∆α
+N,φhas Property (M) for any epimorphism αfromπ1(N)onto a finite
+group, then ∆α
+N,φhas Property (M) for any epimorphism αfromπ1(N′)onto
+a finite group.
+Proof.Thefirst statement canforexample beproved using Stallings’ fiber ingtheorem
+[St62]: Indeed, ( N,φ) fibers if and only if Ker( φ) is finitely generated and ( N′,φ′)TWISTED ALEXANDER POLYNOMIALS AND FIBERED 3–MANIFOLDS 13
+fibers if and only if Ker( φ′) is finitely generated. But Ker( φ′) is subgroup of Ker( φ)
+of finite index. In particular if one is finitely generated, then so is the other.
+The second statement is fundamentally just an application of Shapir o’s lemma,
+which says that the homology of a finite cover of a space Nis nothing but the twisted
+homology of N. Making this principle work in this context is a little delicate though,
+and we refer the reader to [FV08c, Lemma 7.6] for the details. /square
+The following is now an immediate corollary to Theorem 3.4 and Lemma 3.5.
+Theorem D. Suppose the conclusion of Theorem 1.1 holds for all 3-manifo lds such
+thatπ1(N)is residually finite solvable, then Theorem 1.1 holds for all 3-manifolds.
+Note that the original proof of Theorem 1.1 (that appeared befor e the first author
+and M. Aschenbrenner completed the proof of Theorem 3.4) requir ed in [FV08c,
+Section 6] a rather convoluted argument based on the study of re sidual properties
+of each piece of the JSJ decomposition of N. The result of Theorem 3.4 therefore
+greatly simplifies the argument.
+3.5.Step E. Reformulation in terms of sutured manifolds and Agol ’s the-
+orem.In our final step we find it convenient to switch to the language of su tured
+manifolds. A sutured manifold is a triple ( M,Σ−,Σ+) whereMis an oriented 3-
+manifold, Σ ±are (possibly disconnected) disjoint oriented subsurfaces of ∂Mwith
+the following properties:
+(1) the orientation of Σ +agrees with the orientation of ∂M,
+(2) the orientation of Σ −is the opposite orientation of ∂M,
+(3) the closure of ∂M\Σ−∪Σ+consists of a union of annuli A1,...,A nsuch
+that for any ithe boundary of Aiconsists of a boundary curve of Σ −and of a
+boundary curve of Σ +. Furthermore the boundary curves have to be oriented
+the same way.
+A sutured manifold ( M,Σ−,Σ+) is called tautifMis irreducible and if Σ ±are
+Thurston norm minimizing in their homology class in H2(M,∂Σ±;Z). We refer to
+[Ju06], [Ga83, Definition 2.6] or [CC03, p. 364] for more on sutured ma nifolds.
+The following are the two most important types of examples for us:
+(1) If Σ is a oriented surface, then (Σ ×[−1,1],−Σ×−1,Σ×1) is a taut sutured
+manifold. We will refer to it as a product sutured manifold .
+(2) LetNbe an irreducible 3–manifold with empty or toroidal boundary. Let Σ
+be a Thurston norm minimizing surface which intersects all boundary tori of
+N. Denote by Mthe result of cutting Nalong Σ and denote by Σ ±the two
+copies of Σ in M. Then (M,Σ−,Σ+) is a taut sutured manifold.
+Theorem E. Let(M,Σ−,Σ+)be a taut sutured manifold. Suppose that π1(M)is
+residually finite solvable and suppose that for any finite sol vable group Sthe induced14 STEFAN FRIEDL AND STEFANO VIDUSSI
+maps
+ι∗
+±:Hom(π1(M),S)→Hom(π1(Σ±),S)
+are bijections. Then Mis a product on Σ±.
+Theorem 1.1 is an immediate consequence of Theorems A, C, D and E, a nd Theo-
+rem 1.2 is an immediate consequence of Theorems A, C and E.
+Note that the statement of Theorem E can be generalized to a ques tion about
+groups in general: Let Pbe a property of groups, let ϕ:A→Bbe a homomorphism
+of finitely presented groups which are residually Psuch that for any group Gwith
+Property Pthe map Hom( B,G)→Hom(A,G) is a bijection. Does this imply that ϕ
+is an isomorphism? For P={finite}this question goes back to Grothendieck [Gr70]
+and was answered in the negative by Bridson and Grunewald [BG04]. We refer to
+[AHKS07] for more on the case P={finite solvable }.
+This excursion into group theory shows that in order to prove Theo rem E we can
+not rely on a miracle in group theory, but we need a miracle which comes from our
+3-dimensional setting. This miracle is provided by a stunning theorem of Agol [Ag08].
+To explain it we need one more definition.
+A groupπis calledresidually finite Q–solvable orRFRSif there exists a filtration
+of groupsπ=π0⊃π1⊃π2...such that the following hold:
+(1)∩iπi={1},
+(2)πiis a normal, finite index subgroup of πfor anyi,
+(3) for any ithe mapπi→πi/πi+1factors through πi→H1(πi;Z),
+(4) for any ithe mapπi→πi/πi+1factors through πi→H1(πi;Z)/torsion.
+Note that conditions (1), (2) and (3) are equivalent to saying that πis residually
+finite solvable. But condition (4) means that the RFRS condition is con siderably
+more restrictive. The notion of an RFRS group was introduced by Ag ol [Ag08], we
+refer to Agol’s paper for more information on RFRS groups. For our context it is
+important to note that free groups and surface groups are RFRS . Indeed, it is well-
+known that these groups are residually finite solvable, in particular t here exists a
+sequenceπiwith Properties (1), (2) and (3). But the extra condition (4) is now
+always satisfied since the first homology of any finite index subgroup of a free group
+or a surface group is always torsion free.
+Given a sutured manifold M= (M,Σ−,Σ+) the double DMis defined to be the
+double ofMalong Σ −and Σ +. Note that the annuli ∂M\(Σ−∪Σ+) give rise to
+toroidal boundary components of DM.
+The following theorem is implicit in the proof of [Ag08, Theorem 6.1].
+Theorem 3.6 (Agol).LetM= (M,Σ−,Σ+)be a connected, taut sutured manifold
+which is not a product sutured manifold. Suppose that π1(M)is RFRS. Then there
+exists an epimorphism α:π1(M)→Sonto a finite solvable group, such that the
+corresponding cover ˜M= (˜M,˜Σ−,˜Σ+)ofM= (M,Σ−,Σ+)has the property that theTWISTED ALEXANDER POLYNOMIALS AND FIBERED 3–MANIFOLDS 15
+class[˜Σ−]∈H2(D˜M,∂D˜M;Z)lies on the closure of the cone over a fibered face of the
+Thurston norm ball of D˜M.
+Before we delve into the details of the proof of Theorem E, let us tak e a step
+back and think about what Theorem 3.6 does for us. Agol’s theorem h as as input
+information on finite solvable quotients of π1(M) and as output it gives us a strong
+topological conclusion. This is exactly the type of statement we wan t to make in
+Theorem E. It is now just a matter of time till bending and twisting tur ns Theorem
+3.6 into a proof of Theorem E.
+Proof of Theorem E. LetM= (M,Σ−,Σ+) be a taut sutured manifold such that
+π1(M) is residually finite solvable and such that for any finite solvable group Sthe
+induced maps
+(5) ι∗: Hom(π1(M),S)→Hom(π1(Σ±),S)
+are bijections. We have to show that Mis a product sutured manifold.
+Let us now suppose that Mis not a product sutured manifold. We write Σ =
+Σ−. Recall that we pointed out above that the surface group π1(Σ) is RFRS. By
+assumption π1(M) is residually finite solvable and by (5) the finite solvable quotients
+ofπ1(M) andπ1(Σ) ‘look the same’. It is now fairly elementary to show that π1(M)
+is also RFRS (see [FV08c, Section 4.2] for details).
+We can thus apply Theorem 3.6 to the taut sutured manifold ( M,Σ−,Σ+). In fact
+applying arguments similar to the ones used in Lemma 3.5 we can without loss of
+generality assume that already the class [Σ −]∈H2(DM,∂DM;Z) lies on the closure
+of the cone over a fibered face Fof the Thurston norm ball of DM. Moreover, as
+by hypothesis Mis not a product, we can assume that [Σ −] lies in the cone over
+the boundary of F, as otherwise ( N,φ) would fiber already. We refer to [FV08c,
+Lemma 4.3] for details.
+Note thatDMhas an obvious involution rgiven by ‘reflection’, i.e. interchanging
+the two copies of M. Also recall that (5) implies in particular that the inclusion
+induced maps H1(Σ±;Z)→H1(M;Z) are isomorphisms. This means that homologi-
+callyMlooks like a product, and hence homologically DMlooks likeS1×Σ. More
+precisely, there exists a canonical isomorphism Z·t⊕H1(Σ−;Z)∼=− →H1(DM;Z), where
+tis an oriented curve with r(t) =−twhich intersects each of Σ −and Σ +once. Note
+that the map r:H1(DM;Z)→H1(DM;Z) restricts to the identity on H1(Σ;Z) and
+sendstto−t.
+Applying duality we now obtain a dual isomorphism H2(DM,∂DM;Z) =Z·[Σ]⊕V
+whereracts as−id onV. Recall that r([Σ]) = [Σ] and that [Σ] sits on the boundary
+of the face F. It follows that [Σ] also sits on the boundary of the face r(F). Clearly
+r(F) is also a fibered face, and since racts as−id onVwe see that Fandr(F) are
+distinct faces. Also note that by the convexity of the Thurston no rm ballFandr(F)
+can not sit on the same plane. Schematically we now have the situation presented in
+Figure 1.16 STEFAN FRIEDL AND STEFANO VIDUSSI
+r(F)V
+F
+Figure 1. Thurston norm ball on H2(DM,∂DM;R) =R·Σ⊕V.
+We now consider the information contained in (5) coming from finite me tabelian
+groups. We see that Mlooks like a product on the ‘metabelian level’, and hence
+DMlooks likeS1×Σ on the ‘metabelian level’. Now recall that the multivariable
+Alexander polynomial of a 3-manifold is a metabelian invariant, we conc lude that
+the multivariable Alexander polynomial of DMequals the multivariable Alexander
+polynomial of S1×Σ which is well-known to be given by (1 −t)−χ(Σ), wheret∈
+H1(S1×M;Z) =H1(DM;Z) is the same generator introduced above (we refer to
+[FV08c, Lemma 4.9] for details).
+The norm ball dual to the Newton polygon of the multivariable Alexand er poly-
+nomial is called the Alexander norm ball ([McM02]). Note that the Alexa nder norm
+ball ofDMis a convex subset of Hom( H1(DM;Z);R) =H2(DM,∂DM;R). Since the
+Alexander polynomial is given by (1 −t)−χ(Σ)and since Σ is dual to twe see that the
+Alexander norm ball in our case is given by Figure 2. Note that for fibe red classes
+V
+Figure 2. Alexander norm ball on H2(DM,∂DM;R) =R·Σ⊕V.
+the Thurston norm and the Alexander norm agree (see [McM02]). In particular the
+distinct fibered faces Fandr(F) of the Thurston norm ball have to lie on distinct
+faces of the Alexander norm ball. But the Alexander norm ball only ha s two faces,
+both preserved under reflection. This leads to a contradiction, wh ich is schematically
+indicated by the mismatch of Figure 1 and Figure 2. This shows that th e assumptionTWISTED ALEXANDER POLYNOMIALS AND FIBERED 3–MANIFOLDS 17
+thatMis not a product leads to a contradiction. We refer to [FV08c, Sectio n 4] for
+a formal and completely rigorous version of the above argument. /square
+4.Vanishing twisted Alexander polynomials for non–fibered
+manifolds
+Throughoutthissectionweusethenotationfromtheprevioussec tion. Inparticular
+given a manifold pair ( N,φ) whereφ∈H1(N,Z) is a primitive class with ∆ N,φ/\e}atio\slash= 0,
+we will denote by Σ a connected Thurston norm minimizing surface dua l toφand we
+will writeM=N\νΣ. Furthermore we will denote the two natural inclusion maps of
+Σ intoMbyι±. We recall the following theorem, whose proof we had outlined above :
+Theorem B. Let(N,φ)be a manifoldpair and let α:π1(N)→Gbe an epimorphism
+onto a finite group such that ∆α
+N,φ/\e}atio\slash= 0, then
+Im{π1(Σ)ι±− →π1(M)α− →G}=Im{π1(M)α− →G}.
+We will see in this section that Theorem B can be used in many situations to show
+that a non-fibered manifold pair has zero twisted Alexander polynom ials.
+Theorem 4.1. [FV08b, Theorem 4.2] Let(N,φ)be a non-fibered manifold pair such
+thatφis dual to a connected incompressible surface Σ. Ifπ1(N)is LERF, then there
+exists an epimorphism α:π1(N)→Gonto a finite group Gsuch that ∆α
+N,φ= 0.
+Proof.We write Σ = Σ −. By Theorem A we know that the monomorphism π1(Σ)→
+π1(M) is not an isomorphism, in particular the set π1(M)\π1(Σ) is nonempty. By
+the separability of π1(Σ)⊂π1(N) we can now find for any g∈π1(M)\π1(Σ) an
+epimorphism α:π1(N)→Gonto a finite group Gsuch thatα(g)/\e}atio\slash∈α(π1(Σ)). In
+particular we have
+Im{π1(Σ)ι− →π1(M)α− →G}/subset⋉oteqlIm{π1(M)α− →G}.
+The theorem now follows from Theorem B. /square
+It is an important open question whether fundamental groups of h yperbolic 3–
+manifolds are LERF (see [Th82, Question 15]), and various partial re sults of separa-
+bility are known. In particular, Long and Niblo [LN91, Theorem 2] show ed that the
+subgroup carried by an embedded torus is separable. It follows tha t the separability
+condition required in the proof of Theorem 4.1 is always satisfied if Σ is a torus. We
+now easily obtain the following.
+Theorem 4.2. [FV08b, Proposition 4.6] LetNbe a closed 3-manifold and φ∈
+H1(N;Z)a non-trivial class with ||φ||T= 0. Then(N,φ)is fibered if and only if
+for any epimorphism α:π1(N)→Gonto a finite group Gwe have∆α
+N,φ/\e}atio\slash= 0.18 STEFAN FRIEDL AND STEFANO VIDUSSI
+Expanding on the ideas of Theorem 4.1 and 4.2 one can then continue t o prove
+Theorem 1.6. We refer to [FV08b] for details.
+Unfortunately not all 3-manifoldgroups areLERF (see [NW01]) and little is known
+evenconjecturallyabouttheseparabilitypropertiesofnon-geom etric3-manifoldgroups.
+In the remainder of this section we will therefore give two examples o f types of non-
+fibered manifold pairs where Theorem 4.1 cannot be applied, but wher e the weaker
+assumptions of Theorem B allow us to show that these pairs have twis ted Alexander
+polynomials which are zero.
+In order to prove our theorems we recall the following result of Lon g and Niblo
+[LN91]: Let Σ be an incompressible subsurface of the boundary of a 3 –manifoldM.
+Thenπ1(Σ)⊂π1(M) is separable. This is often referred to as peripheral subgroup
+separability . We will exploit this result in two cases. The first is the case of the
+double of the complement of a nonfibered surface Σ ⊂N. The second, perhaps of
+more conceptual breadth, is an application of ‘virtual retractibility ’.
+We start with the first case. Let Wbe a 3–manifold with empty or toroidal bound-
+ary and let Σ ⊂Wbe an incompressible nonseparating connected properly embed-
+ded surface. Consider the manifold with boundary M=W\νΣ. This manifold
+has two copies Σ ±sitting in the boundary. Consider the double DMofMalong
+Σ−∪Σ+⊂∂M. The images Σ ±⊂DMare nonseparating incompressible surfaces
+which are homologous in H2(DM,∂DM;Z).
+Theorem 4.3. LetDMbe defined as above, and let φ∈H1(DM,Z)be the primitive
+class Poincar´ e dual to [Σ±]. If(DM,φ)is a non-fibered pair, then there exists an
+epimorphism α:π1(DM)→Gonto a finite group Gsuch that ∆α
+DM,φ= 0.
+Proof.Suppose that ( DM,φ) is a non–fibered pair. First note that it is well-known
+that Σ +⊂DMis a fiber if and only if Mis a product on Σ +.
+Note that we have a folding map r:DM→Mthat is a retraction. In particular
+the induced map in homotopy r∗:π1(DM)→π1(M) is an epimorphism, and has
+as right inverse the inclusion–induced map i∗:π1(M)→π1(DM). Note that both
+r∗andi∗restrict to an isomorphism on the proper subgroups of the domain a nd
+image determined by a copy of π1(Σ+). Consider now the proper subgroup π1(Σ+)⊂
+π1(M); by peripheral subgroup separability, there is an epimorphism β:π1(M)→G
+to a finite group such that β(π1(Σ+))/subset⋉oteqlβ(π1(M)). The surface Σ +⊂DMis an
+incompressible surface dual to φ; defineZ:=DM\νΣ+. By the usual argument
+based on the incompressibility of Σ +,π1(Σ+) can be viewed as subgroup of π1(Z)
+and the latter is a subgroup of π1(DM).
+The inclusion–induced map i∗:π1(M)→π1(DM) has image in π1(Z). It follows
+that if we let α=β◦r∗:π1(DM)→G, then we have
+Im{π1(Σ+)ι− →π1(Z)α− →G}/subset⋉oteqlIm{π1(Z)α− →G}.
+It now follows from Theorem B that ∆α
+DM,φ= 0.
+/squareTWISTED ALEXANDER POLYNOMIALS AND FIBERED 3–MANIFOLDS 19
+The second application of peripheral subgroup separability is in the c ontext of
+virtual retractions . In [LR08] (see also [LR05]), Darren Long and Alan Reid define
+and explore the notion of virtual retraction of a group to one of its finitely generated
+subgroup, aswell asvariousrelatedproperties. Astheauthorso f[LR08]discuss, these
+notions are closely connected with subgroup separability propertie s of the group.
+Westartby giving theproperdefinitions, from[LR08], using anotatio nthat adapts
+to the case we have in mind.
+Definition. Letπbe a group and Ba subgroup. Then a homomorphism θ:B→
+Gextends over the finite index subgroup ˆπ⊂πifB⊂ˆπand if there exists a
+homomorphism Θ : ˆ π→Gsuch that Θ |B=θ.
+Definition. Letπbe a group and Ba subgroup. Then πvirtually retracts ontoBif
+the identity homomorphism θ= idBextends over some finite index subgroup of π.
+Quite clearly, if πvirtually retracts onto B,anyhomomorphism θ:B→Gex-
+tends over some finite index subgroup of π. A less trivial fact, observed in [LR08,
+Theorem 2.1], is that if πis LERF and if Bis finitely generated, then any homomor-
+phismθonto a finite group extends over a finite index subgroup. In light of t hat the
+following theorem can be viewed as a generalization of Theorem 4.1.
+Theorem 4.4. Let(N,φ)be a non-fibered manifold pair. Suppose that there exists
+a connected Thurston norm minimizing surface Σdual toφsuch that any homomor-
+phism ofπ1(N\νΣ)to a finite group extends to a finite index subgroup of π1(N).
+Then there exists an epimorphism α:π1(N)→Gonto a finite group Gsuch that
+∆α
+N,φ= 0.
+Proof.We writeM:=N\νΣ and we write π=π1(N),A=π1(Σ) andB=π1(M).
+The manifold Mhas, as boundary, two copies of Σ; the incompressibility of Σ entails
+in particular the existence of inclusion–induced injective morphisms ι±:A֒→B⊂π.
+As (a copy of) Σ occurs as boundary component of M, the image under say i+
+ofAinB(that we will denote by Aas well) is separable by peripheral subgroup
+separability. This means that for any element γ∈B\Athere exist an epimorphism
+θ:B→Honto some finite group such that θ(γ)/∈θ(A). Pick such an element γ.
+By assumption, θextends to an epimorphism Θ : ˆ π→Hwhere ˆπ⊂πis a finite
+index subgroup. The kernel ker Θ ⊂ˆπis a normal finite index subgroup of π. This
+subgroup may fail to be normal in π; define Γ = ∩g∈πg(ker Θ)g−1to be its normal
+core inπ, a normal finite index subgroup of both ˆ πandπ. Denote by G:=π/Γ, a
+finite group, and let α:π→Gbe the quotient map. As ˆ π/Γ surjects on ˆ π/ker Θ, it is
+not difficult to verify that the condition Θ( A)/subset⋉oteqlΘ(B) entails that α(A)/subset⋉oteqlα(B)⊂G.
+The theorem is now an immediate consequence of Theorem B. /square
+Clearly, Theorem 4.4 applies when π1(N) virtually retracts to π1(M). Examples
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+hyperbolic Coxeter subgroup of Isom( Hn) (see [LR08, Theorem 2.6]): these groups20 STEFAN FRIEDL AND STEFANO VIDUSSI
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+generated geometrically infinite subgroups are virtual fiber group s, hence excluded in
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+[Ta94] C. H. Taubes, The Seiberg-Witten invariants and symplectic forms , Math. Res. Lett. 1: 809–
+822 (1994)
+[Ta95] C. H. Taubes, More constraints on symplectic forms from Seiberg-Witten i nvariants , Math.
+Res. Lett. 2: 9–13 (1995)
+[Th76] W. P. Thurston, Some simple examples of symplectic manifolds , Proc. Amer. Math. Soc. 55
+(1976), no. 2, 467–468.
+[Th82] W. P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbol ic geometry ,
+Bull. Amer. Math. Soc. 6 (1982)
+[Th86] W. P. Thurston, A norm for the homology of 3–manifolds , Mem. Amer. Math. Soc. 339:
+99–130 (1986)
+[Tu01] V. Turaev, Introduction to combinatorial torsions , Birkh¨ auser, Basel, (2001)
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+241–256 (1994)
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+University of Warwick, Coventry, UK
+E-mail address :s.k.friedl@warwick.ac.uk
+Department of Mathematics, University of California, Rive rside, CA 92521, USA
+E-mail address :svidussi@math.ucr.edu
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+arXiv:1001.0133v2  [nlin.SI]  11 Jun 2010Solutions of matrix NLS systems and their
+discretisations: A unified treatment
+Aristophanes Dimakis
+Department of Financial and Management Engineering,
+University of the Aegean, 41, Kountourioti Str., GR-82100 Chios, Greece
+E-mail:dimakis@aegean.gr
+Folkert M ¨uller-Hoissen
+Max-Planck-Institute for Dynamics and Self-Organization
+Bunsenstrasse 10, D-37073 G¨ ottingen, Germany
+E-mail:folkert.mueller-hoissen@ds.mpg.de
+Abstract
+Usingabidifferentialgradedalgebraapproachto“integrable”part ialdifferentialordifference
+equations, a unified treatment of continuous, semi-discrete (Ablo witz-Ladik) and fully discrete
+matrix NLS systems is presented. These equations originate from a universal equation within
+this framework, by specifying a representation of the bidifferentia l graded algebra and imposing
+a reduction. By application of a general result, corresponding fam ilies of exact solutions are
+obtained that in particular comprise the matrix soliton solutions in the focusing NLS case. The
+solutions are parametrised in terms of constant matrix data subje ct to a Sylvester equation
+(which previously appeared as a rank condition in the integrable systems literature). These
+data exhibit a certain redundancy, which we diminish to a large extent .
+More precisely, we first consider more general AKNS-type system s from which two different
+matrix NLS systems emerge via reductions. In the continuous case , the familiar Hermitian
+conjugation reduction leads to a continuous matrix (including vecto r) NLS equation, but it is
+well-known that this does not work as well in the discrete cases. On t he other hand there is a
+complex conjugation reduction, which apparently has not been stu died previously. It leads to
+squarematrixNLSsystems, but worksinallthreecases(continuous, sem i-andfully-discrete). A
+large part of this work is devoted to an exploration of the correspo nding solutions, in particular
+regularity and asymptotic behaviour of matrix soliton solutions.
+Contents
+1 Introduction 3
+2 The framework 4
+3 NLS systems 8
+3.1 A class of bidifferential calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
+3.2 Continuous NLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
+3.3 Semi-discrete NLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
+3.4 Fully discrete NLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
+3.5 Normal form of the matrix J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
+3.6 Complex conjugation reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
+3.7 Hermitian conjugation reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
+14 Solutions of the matrix NLS systems 11
+4.1 Proof of proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
+4.2 Transformations of the matrix data determining solutions . . . . . . . . . . . . . . . . . . . . 15
+4.2.1 Similarity transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
+4.2.2 Reflection symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
+4.2.3 Reparametrisation transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
+4.3 Superposition of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
+4.4 Imposing reduction conditions on the solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 18
+4.4.1 Complex conjugation reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
+4.4.2 Hermitian conjugation reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
+5 Block-decomposition of the NLS systems and their solution s 20
+5.1 Transformations of the matrix data determining solutions . . . . . . . . . . . . . . . . . . . . 21
+6 The complex conjugation reduction 22
+6.1 On the structure of the solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
+6.1.1 Solving the Sylvester equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
+6.1.2 Superpositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
+6.1.3 The use of reflection symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
+6.2 Solutions of the scalarcontinuous and discrete NLS equations . . . . . . . . . . . . . . . . . . 27
+6.3 Solutions of the matrix NLS equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
+6.3.1 Rank one single solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
+6.3.2 Asymptotics of superpositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
+6.3.3 Superposition of rank one solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
+6.3.4 2-soliton solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
+6.3.5 Regularity condition for matrix solitons in the focusing NLS case . . . . . . . . . . . . 37
+6.3.6 Cases in which the Sylvester equation has not a unique solution . . . . . . . . . . . . . 38
+7 Hermitian conjugation reduction of the continuous NLS sys tem 40
+7.1 Solitons of the focusing matrix NLS equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
+7.1.1 Rank one single solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
+7.1.2 Superposing solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
+7.1.3 Superposition of rank one solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
+7.1.4 2-soliton solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
+8 Conclusions 47
+Appendix A: Solutions of the matrix KdV equation 48
+Appendix B: A relation between the semi-discrete NLS and mod ified KdV equations 49
+Appendix C: Equivalence of the fully discrete NLS equation w ith the corresponding
+equation of Ablowitz and Ladik in the scalar case 50
+Appendix D: A non-local Hermitian reduction of the semi-dic rete NLS system 50
+Appendix E: Lax pairs for the NLS systems 51
+Appendix F: Continuum limit 53
+References 53
+21 Introduction
+Using the framework of bidifferential calculi (or bidifferenti al graded algebras), a unification of
+integrability aspects and solution generating techniques for a wide class of integrable models has
+been achieved [1] (see also the references cited therein). I n particular, there is a fairly simple and
+universal method to construct large families of exact solut ions from solutions of a linear system of
+equations. The formalism treats continuous and discrete sy stems on an equal level. This suggests
+a corresponding study of the well-known nonlinear Schr¨ odi nger (NLS) equation (see e.g. [2,3]), its
+semi-discretisation, the integrable discrete NLS or Ablow itz-Ladik (AL) equation [4–7], and a full
+discretisation [5]. The literature on mathematical aspect s of the continuous NLS equation and its
+applications in science is meanwhile impossible to summari se. A rather incomplete list of references
+dealing with mathematical aspects of the AL equation is [3,8 –46], and [47–69] more generally for
+its hierarchy. Physical applications of the AL equation can be found e.g. in [70–76]. In this work,
+we consider more generally matrixversions of the (continuous and discrete) NLS equations and
+present corresponding solutions. Matrix generalisations , including vector versions, of the scalar
+continuous NLS equation have been studied in particular in [ 3,77–101]. A brief summary of the
+physical relevance can be found in [3]. In particular, a (sym metric) 2 ×2 matrix NLS equation
+turned out to be of relevance for the description of a special Bose-Einstein condensate (with atoms
+in a spin 1 state) [102–115]. Semi-discrete matrix NLS equat ions appeared in [3,116–129], a full
+discretisation has been elaborated in [130] and a dispersio nless limit studied in [127].
+In section 2 we introduce the general framework. Section 3 th en presents bidifferential calculi
+for the NLS system and its discrete versions. Section 4 deriv es exact solutions, by application
+of the method of section 2. The solutions are parametrised by matrix data that have to solve a
+Sylvester equation.1In section 5, these results are translated into a decomposed form, where the
+NLS systems attain a more familiar form.
+Section 6 deals with a complex conjugation reduction which a pplies to the continuous as well as
+the discrete NLS systems. Quite surprisingly, this has appa rently not been studied previously for
+matrix NLS equations. Our analysis in particular addresses regularity and asymptotic behaviour
+of matrix soliton solutions. Section 7 treats the more famil iar Hermitian conjugation reduction
+which, however, only works for the continuous NLS system, at least without severe restrictions or
+introduction of non-locality (cf. appendix D). Section 8 co ntains some concluding remarks.
+Some supplementary material has been shifted to appendices . Appendix A briefly treats the
+considerably simpler example of the matrix KdV equation, in particular to provide the reader with
+a quick access to the methods used in this work, but also in ord er to stress the similarities even
+in details of the calculation. Appendix B establishes a rela tion between the semi-discrete matrix
+NLS (AL) system and the matrix modified KdV equation. Appendi x C demonstrates that our
+fully discrete matrix NLS system reduces in the scalar case t o the corresponding equation in [5].
+Appendix D presents a non-local reduction of the semi-discr ete NLS system. Appendix E recovers
+Lax pairs for the three NLS systems, starting with the respec tive bidifferential calculus. Finally,
+Appendix F contains some remarks on the continuum limit of th e fully discrete matrix NLS system
+and the solutions obtained in this work.
+1See e.g. [131–133] for results on the Sylvester equation. A v ast literature deals with its solutions and applications.
+32 The framework
+Definition 2.1. A (complex) graded algebra is an algebra2Ω that has a direct sum decomposition
+Ω =/circleplusdisplay
+r≥0Ωr(2.1)
+into complex vector spaces Ωr, with the property
+ΩrΩs⊆Ωr+s. (2.2)
+Furthermore, A:= Ω0is a subalgebra and, for r>0, Ωris anA-bimodule.
+Definition 2.2. Abidifferential graded algebra orbidifferential calculus is a graded algebra Ω
+equipped with two graded derivations d ,¯d : Ω→Ω of degree one (hence dΩr⊆Ωr+1,¯dΩr⊆Ωr+1),
+with the properties
+d◦d = 0,¯d◦¯d = 0,d◦¯d+¯d◦d = 0, (2.3)
+and the graded Leibniz rule
+d(χχ′) = (dχ)χ′+(−1)rχdχ′,¯d(χχ′) = (¯dχ)χ+(−1)rχ¯dχ′, (2.4)
+for allχ∈Ωrandχ′∈Ω.
+For an algebra A, a corresponding graded algebra is given by
+Ω =A⊗/logicalanddisplay
+(C2), (2.5)
+where/logicalandtext(C2) denotes the exterior algebra of C2.3In this case it is sufficient to define the maps d ,¯d
+onA. They extend in an obvious way to Ω such that the Leibniz rule h olds. Elements of/logicalandtext(C2) are
+treated as “constants”. This structure underlies all our ex amples. In this work, ζ1,ζ2will denote
+a basis of/logicalandtext1(C2).
+Given a unital algebra Band a bidifferential graded algebra (Ω ,d,¯d) with Ω0=B, the actions
+of d and ¯d extend componentwise to matrices over B. A corresponding example is provided in
+Appendix A, the matrix KdV equation. By application of resul ts stated below we recover in a
+strikingly simple way a class of KdV solutions derived previ ously via inverse scattering theory (see
+in particular [134]). A treatment of NLS-type equations req uires a generalisation, however. Here d
+and¯d involve a structure that depends on the size of the matrix on which they act.
+The space of all matrices over B, with size greater or equal to that of n0×n0matrices,4
+Matn0(B) :=/circleplusdisplay
+n′,n≥n0Mat(n′,n,B), (2.6)
+has the structure of a complex algebra with the usual matrix p roduct extended trivially by setting
+AB= 0wheneverthedimensionsdonotmatch. Extending A= Mat n0(B)toagradedalgebra(2.5),
+weshallrequirethatdand ¯drespectthesizeofmatricesinthesensethatd(Mat( n′,n,B)⊗/logicalandtext(C2))⊆
+Mat(n′,n,B)⊗/logicalandtext(C2)), and correspondingly for ¯d. The following result originated in [135] and
+appeared in [1] in a more general form.
+2All algebras in this work will be (assumed to be) associative .
+3More generally, one may consider the exterior algebra of Ck,k >1.
+4In some cases it may turn out to be necessary to further restri ct the values of nandn′. In typical examples we
+can setn0= 1, but the NLS case requires n0= 2 (see section 3).
+4Theorem 2.3. Let(Ω,d,¯d)be a bidifferential graded algebra with5Ω =A ⊗/logicalandtext(C2)andA=
+Matn0(B), for some n0∈N. For fixed n,n′≥n0, letX∈Mat(n,n,B)andY∈Mat(n′,n,B)be
+solutions of
+¯dX= (dX)P,¯dY= (dY)P, (2.7)
+and6
+RX−XP=−QY, (2.8)
+withd- and¯d-constant matrices P,R∈Mat(n,n,B), and
+Q=˜VQ˜U, (2.9)
+where˜U∈Mat(m′,n′,B),˜V∈Mat(n,m,B)andQ∈Mat(m,m′,B)are also constants of the
+calculus. If Xis invertible, then
+φ:=˜UYX−1˜V∈Mat(m′,m,B) (2.10)
+satisfies
+¯dφ= (dφ)Qφ+dθwithθ=˜UYX−1R˜V. (2.11)
+Furthermore, (by application of d)φsolves
+¯ddφ= dφQdφ. (2.12)
+Proof:As a consequence of (2.7) and (2.8), we find (cf. [1]) that
+Φ:=YX−1∈Mat(n′,n,B)
+solves
+¯dΦ= (dΦ)QΦ+d(ΦR).
+Multiplying from the left by ˜Uand from the right by ˜V, resolving Qaccording to (2.9), and using
+the graded Leibniz rule for d and ¯d, we obtain (2.11). /square
+The theorem reveals an integrability feature of the nonline ar equation (2.12). From solutions of
+a system of n×n, respectively n′×n, matrix linear equations, it generates a large class of solu tions
+of them′×mmatrix equation (2.12). Since n,n′can be arbitrarily large, in this way we obtain
+infinite families of solutions of (2.14), where the members d epend on a number of parameters that
+increases with nandn′. In this work we will restrict our considerations to the case where
+m′=m, Q =Im. (2.13)
+Accordingly, our central equation will be
+¯ddφ= dφdφwhereφ∈Mat(m,m,B). (2.14)
+5More general choices of Ω are actually allowed.
+6(2.8) has a form analogous to that of a Sylvester equation. Th e ingredients of the latter are just matrices over
+C, however. Nevertheless, we shall see that in our examples (2 .8) actually results in an ordinary Sylvester equation.
+5Remark 2.4. Concerning the general case, if Qadmits a factorisation
+Q=A/parenleftbiggIr0r×(m′−r)
+0(m−r)×r0(m−r)×(m′−r)/parenrightbigg
+B
+into d- and ¯d-constant matrices, where AandBare invertible, then (2.12) is equivalent to a
+(inhomogeneous) linear extension (cf. [136]) of (2.14) (where mhas to be replaced by r), after the
+redefinition BφA/ma√sto→φ. This holds in particular if Qis a constant complex matrix (Smith form
+decomposition, see e.g. [137]), and if all such matrices are d- and¯d-constant. Our methods can
+also be applied to such a linear extension. In case of the bidi fferential calculi in section 3, d- and
+¯d-constancy of a matrix places additional restrictions on a constant matrix, so that (2.12) is not
+in general equivalent to a linear extension of (2.14). All th is opens additional possibilities that
+will be disregarded in this work, however. A matrix Qdifferent from an identity matrix effects a
+deformation of the ordinary matrix product, which indeed offe rs interesting applications [138]. /square
+Many (e.g. in the sense of the inverse scattering method) int egrable partial differential or
+difference equations can be recovered from the universal equa tion (2.14), by choosing a suitable
+bidifferential graded algebra. At first sight it seems that (2. 14) can only produce equations with
+quadratic nonlinearity. But φneed not directly correspond to the independent variable of the
+equation of final interest. Indeed, the main examples in this work, the NLS systems, possess a
+cubic nonlinearity. The (anti-) self-dual Yang-Mills equa tion (more precisely, a gauge reduced
+version of it), which is well-known as a source of many integr able partial differential equations, is
+a special case of (2.14) [1]. Whereas the KP equation and its h ierarchy do not quite fit into the
+self-dual Yang-Mills framework, they do fit well into our gen eralised framework. The generalisation
+(2.14) covers much more, however, and in particular integra ble partial difference equations, i.e.
+discretised versions of continuous integrable equations.
+In all examples that will be considered in this work, the gene ral solution of the linear equations
+(2.7) can be expressed in the form
+X=A0+A1ˆΞ,Y=B0+B1ˆΞ, (2.15)
+where the matrices A0,A1,B0,B1are d- and ¯d-constant, and ˆΞis not. Assuming that A0is
+invertible, we can rewrite the solution φdetermined by the theorem as
+φ=˜U(B0+B1ˆΞ)(I−ˆKˆΞ)−1V=ˆUˆΞ(I−ˆKˆΞ)−1V+˜UB0V (2.16)
+in terms of the d- and ¯d-constant matrices
+ˆK=−A−1
+0A1,ˆU=˜U(B1+B0ˆK),V=A−1
+0˜V. (2.17)
+The last summand in (2.16) is d- and ¯d-constant. Since such a term can always be added to any
+solution of (2.14), we shall neglect it. Next we insert (2.15 ) in (2.8) and separate the d- and ¯d-
+constant summands from those that depend on ˆΞ. Assuming that both parts have to be satisfied
+separately (which is true in our examples), the first part bec omesR= (A0P−˜V˜UB0)A−1
+0.
+Eliminating Rfrom the second, then yields
+[P,ˆKˆΞ] =VˆUˆΞ. (2.18)
+We summarise our results.
+6Corollary 2.5. Let(Ω =A ⊗/logicalandtext(C2),d,¯d)be a bidifferential graded algebra with A= Mat n0(B)
+for somen0∈N. LetˆΞ∈Mat(n,n,B)be any solution of
+¯dˆΞ= (dˆΞ)P (2.19)
+and (2.18), with d- and¯d-constant matrices P,ˆK∈Mat(n,n,B),V∈Mat(n,m,B)andˆU∈
+Mat(m,n,B). Then
+φ=ˆUˆΞ(I−ˆKˆΞ)−1V (2.20)
+solves (2.14), provided that I−ˆKˆΞis invertible. /square
+Having determined ˆΞ(which should not be d- and ¯d-constant) in a concrete example, the
+remaining task is thus to find, for given d- and ¯d-constant matrices ˆUandV, a d- and ¯d-constant
+matrixˆKsuch that (2.18) holds.
+Remark 2.6. The construction of exact solutions according to theorem 2. 3 hardly exhausts the
+set of all solutions of (2.14) (which is (2.12) with Q=Im). In examples like matrix KdV (see
+appendix A) and the matrix NLS systems treated in the main par t of this work, the latter equation
+is equivalent to ¯dφ= (dφ)φ+dθwith an arbitrary matrix function θ. But (2.11) chooses θfrom
+a certain family of matrix functions, which clearly means a r estriction to a subclass of solutions.
+On the other hand, it should be noted that the expression for θin (2.11) involves matrices of
+arbitrarily large size and, with suitable technical assump tions, we may allow them to be infinitely
+large, corresponding to operators on a Banach space (see als o [139–142]). It is by far not evident
+what kind of solutions can be reached in this way. /square
+Remark 2.7. Theorem 2.3 can be generalised by replacing (2.7) with the we aker equations
+¯dX= (dX)P+XΨ,¯dY= (dY)P+YΨ, (2.21)
+with any Ψ∈Ω1of sizen×n, and dropping the requirement that the n×nmatrixPis d- and
+¯d-constant. PandΨare constrained as a consequence of the above system, howeve r. Indeed,
+acting with ¯d on (2.8), using the Leibniz rule, (2.21) and (2.8) again, we obtain
+¯dP= (dP)P+PΨ−ΨP. (2.22)
+Acting with ¯d on (2.21), using the Leibniz rule and ¯dd =−d¯d, again (2.21) and then (2.22), leads
+to
+¯dΨ= (dΨ)P−Ψ2. (2.23)
+Setting
+Ψ=G−1/parenleftbig¯dG−(dG)P/parenrightbig
+, (2.24)
+with an invertible n×nmatrix function G, then 0 = ¯d2G=¯d(GΨ+(dG)P) =G(¯dΨ−(dΨ)P+
+Ψ2), by use of (2.22), shows that (2.23) is satisfied. In terms of P′=GPG−1, (2.22) then reads
+¯dP′= (dP′)P′. (2.25)
+If for any solution ( P,Ψ) of (2.22) and (2.23) the equation (2.24) has a solution G, it follows that
+Ψcan be set to zero without restriction of generality. A restr iction seems to remain nevertheless
+in theorem 2.3, since there we solved the non-linear equatio n (2.25) trivially by requiring that P′
+(renamed to P) is d- and ¯d-constant. In case of other solutions of (2.25), the linear equations (2.7)
+involve variable coefficients and also the constraint (2.8) b ecomes considerably more complicated.
+We will therefore not consider this generalisation further in this work. /square
+73 NLS systems
+3.1 A class of bidifferential calculi
+LetA= Mat 2(B) and Ω = A⊗/logicalandtext(C2). For every n≥2 we choose a matrix Jnwith the properties
+J2
+n=In, J n/ne}ationslash=±In, (3.1)
+whereIndenotes the n×nidentity matrix. Then
+e2(f) :=1
+2[J,f] :=1
+2(Jn′f−fJn) forf∈Mat(n′,n,B) (3.2)
+defines a derivation of A. Setting
+df=e1(f)ζ1+e2(f)ζ2,¯df= ¯e1(f)ζ1+ ¯e2(f)ζ2, (3.3)
+then determines a bidifferential calculus on Ω if e1,¯e1,¯e2are commuting derivations of Aand
+also commute with e2. In the following, e1,¯e1,¯e2will be related to partial derivative or partial
+difference operators. In order to be d- and ¯d-constant, a matrix must then not depend on the
+respective variables. We shall see that, with appropriate c hoices, (2.14) reproduces a continuous,
+semi-discrete, respectively fully discrete NLS system.
+3.2 Continuous NLS
+LetBbe the space of smooth complex functions on R2. In (3.3) we choose7
+e1=∂x,¯e1=−i∂t,¯e2=∂x, (3.4)
+where∂xis the partial derivative with respect to x, and i =√−1. (2.14) takes the form
+−i
+2[J,φt] =φxx+1
+2[φx,[J,φ]], (3.5)
+whereJ=Jm. Writing
+φ=Jϕ, (3.6)
+and decomposing the m×mmatrixϕas follows into the sum of an “even” and an “odd” part,
+ϕ=D+U,D=1
+2(ϕ+JϕJ),U=1
+2(ϕ−JϕJ), (3.7)
+we have
+JD=DJ, J U=−UJ . (3.8)
+Hence (3.5) splits into
+iJUt+Uxx+DxU+UDx= 0 (3.9)
+andDxx=−(U2)x. The latter can be integrated to
+Dx=−U2+C, (3.10)
+7Despite of our notation, in some important applications (no tably in optics) xplays the role of a time coordinate
+andtthat of a physical space coordinate.
+8whereCis an even matrix independent of x. This can be used to eliminate Din (3.9). We conclude
+that, for the chosen bidifferential calculus, (2.14) is equiv alent to
+iJUt+Uxx−2U3+CU+UC= 0,Dx=−U2+C. (3.11)
+Without restriction of generality, we can set8C= 0, so that
+iJUt+Uxx−2U3= 0,Dx=−U2. (3.12)
+This will be called ( m×mmatrix) continuous NLS system .
+Theζ2-part of (2.11) (with Q=Im) is
+ϕx=1
+2[J,ϕ]Jϕ+1
+2J[J,θ], (3.13)
+which implies Dx=−U2. As a consequence, the solutions generated via theorem 2.3 a ctually have
+vanishing C.9
+3.3 Semi-discrete NLS
+LetB0be the algebra of functions on Z×R, smooth in the second variable. We extend it to an
+algebraBby adjoining the shift operator Swith respect to the discrete variable (the discretised
+coordinate x). In (3.3) we choose
+e1(f) = [S,f],¯e1(f) =−i˙f,¯e2(f) =−[S−1,f], (3.14)
+where˙f=df/dt. In the following we use the notation f+=SfS−1,f−=S−1fS. Setting
+φ=JϕS−1, (3.15)
+(2.14) becomes
+i
+2[J,˙ϕ]+ϕ+−2ϕ+ϕ−+1
+2(ϕ+−ϕ)J[J,ϕ]−1
+2[J,ϕ]J(ϕ−ϕ−) = 0, (3.16)
+whereϕcan be restricted to be an element of Mat( m,m,B0). With the decomposition ϕ=D+U
+as used above, we find
+D+−D=−U+U+C, (3.17)
+whereCis even and does not depend on the discrete variable, and
+iJ˙U+U+−2U+U−−U+U2−U2U−+{C,U}= 0. (3.18)
+Withtheabovechoiceof bidifferential calculus, (2.14) isth usequivalent to(3.17) and(3.18). Again,
+Ccan be eliminated by redefinitions, and we have
+iJ˙U+U+−2U+U−−U+U2−U2U−= 0,D+−D=−U+U. (3.19)
+These two equations constitute a semi-discretisation of (3 .12) and will be called ( m×mmatrix)
+semi-discrete NLS system. By inspection of (2.11), the solutions generated vi a theorem 2.3 have
+vanishing C.
+8ForU′=T UT−1, where Tt=−iJCT, andD′=D − Cx, (3.12) holds without the terms involving C. The
+modification of the NLS equations in sections 6 and 7, caused b y a nonvanishing C, in the scalar case reproduces a
+“background term” which turned out to be convenient in order to study finite density type solutions (see e.g. (1.5)
+in [2]).
+9We actually lost the freedom of the choice of an even matrix Cin the step from (2.11) to (2.12), which would
+still allow the addition of a d-closed 1-form that is not d-ex act, namely JCζ2in the case under consideration.
+93.4 Fully discrete NLS
+Here we make the following choice in (3.3),
+e1(f) = [TS,f],¯e1(f) =−i[T,f],¯e2(f) =−[S−1,f], (3.20)
+whereTis the shift operator in discrete “time”. Here Bis the algebra of functions B0onZ×Z,
+extended by adjoining SandT. In the following, we also set f+=TfT−1andf−=T−1fT.
+Writing
+φ=JϕS−1, (3.21)
+(2.14) reads
+i
+2[J,ϕ+−ϕ]+(ϕ+−ϕ)+−(ϕ−ϕ−)+1
+2(ϕ+
++−ϕ)J[J,ϕ+]−1
+2[J,ϕ]J(ϕ+−ϕ−) = 0,(3.22)
+andϕcan be restricted to be an element of Mat( m,m,B0). With the decomposition ϕ=D+U,
+we obtain
+iJ(U+−U)+(U+−U)+−(U −U−)−U+
++U2
++−U2U−+(D+−D)U++U(D+−D) = 0,
+D+−D=−U+U, (3.23)
+which will be called ( m×mmatrix)fully discrete NLS system . We again dropped terms involving
+an even matrix Cthat does not depend on the discrete variable x. Again, inspection of (2.11) shows
+that solutions generated via theorem 2.3 have in fact vanish ingCand are therefore solutions of
+(3.23).
+3.5 Normal form of the matrix J
+By an application of the Jordan decomposition theorem, Jis related by a similarity transformation
+to a diagonal matrix with diagonal entries ±1. Since we can apply this similarity transformation
+to the above equations, without restriction of generality w e can choose
+J=/parenleftbiggIm10
+0−Im2/parenrightbigg
+(3.24)
+with any choice m1,m2∈Nsuch thatm1+m2=m. Note that, for m >2, the bidifferential
+calculus depends on the splitting of minto two summands. This means that there are actually
+different bidifferential calculi on Mat( m,m,B), and different choices correspond to different NLS
+systems.
+3.6 Complex conjugation reduction
+LetJsatisfy
+J∗=J, (3.25)
+where∗denotes complex conjugation, and let Γ be an invertible cons tantm×mmatrix with the
+properties
+JΓ =−ΓJ,Γ∗= Γ−1. (3.26)
+10Then the transformation
+U /ma√sto→ǫΓ−1U∗Γ withǫ=±1,D /ma√sto→Γ−1D∗Γ, (3.27)
+leaves all three NLS systems invariant, including (3.8) (wh ich applies to all three cases). Hence a
+consistent reduction is given by
+U∗=ǫΓUΓ−1,D∗= ΓDΓ−1. (3.28)
+Transforming Jto its normal form (3.24), the first of (3.26) restricts Γ to off- block-diagonal form.
+The second of conditions (3.26) then requires that mis even and
+m1=m2. (3.29)
+We will actually choose
+Γ =/parenleftbigg0Im1
+Im10/parenrightbigg
+. (3.30)
+3.7 Hermitian conjugation reduction
+Thecontinuous NLS system with
+J†=J, (3.31)
+where†denotes Hermitian conjugation (i.e. transposition combin ed with complex conjugation), is
+invariant under
+U /ma√sto→ǫU†withǫ=±1,D /ma√sto→ D†. (3.32)
+Hence
+U†=ǫU,D†=D (3.33)
+is a consistent reduction. This does notwork as well for the matrix discrete NLS systems, since
+†is an anti-involution (whereas∗is an involution). Applying it to the nonlinear term in the fir st
+of equations (3.19), we have ( U+U2+U2U−)†=U†2U†++U†−U†2. Clearly the matrices in this
+expression will not commute in general. In order to achieve t hat the equations obtained from (3.19)
+by Hermitian conjugation become equivalent to the original equations, we can either
+(1) require that U2commutes with U±, while keeping (3.33), or
+(2) replace (3.33) by U(x,t)†=ǫU(−x,t) andD(x,t)†=−D(−x,t) (see also [116]).
+We will not discuss the first case further in this work, but onl y refer to [3,92,118,123,124,130], see
+also remark 7.5. The second case leads (in a decomposed form i n the sense of section 5) in general
+to anon-local equation, see appendix D.
+4 Solutions of the matrix NLS systems
+A central result of this section is expressed in the next prop osition. In the following, I=Indenotes
+then×nidentity matrix.
+11Proposition 4.1.
+U=U(Ξ−1−KΞK)−1V,D=UΞK(Ξ−1−KΞK)−1V (4.1)
+solves10the continuous, semi-discrete, respectively fully discre tem×mmatrix NLS system (involv-
+ing them×mmatrixJwithJ2=Im) if
+J/ne}ationslash=±ISU VK
+sizen×nn×nm×nn×mn×n
+are constant matrices such that
+J2=I,JS=SJ, JU=−UJ,JV=VJ,JK=−KJ, (4.2)
+and
+NLS Sylvester equation Ξ
+continuous SK+KS=VU e−xS−itS2J
+semi-discrete S−1K−KS=VU Sxe−it(S+S−1−2I)J
+fully discrete S−1K−KS=VUSx[(I+iJ(I−S−1))(I−iJ(I−S))−1]t
+wherex∈Zin the semi- and fully discrete NLS case, t∈Zin the fully discrete NLS case, and
+x,t∈Rotherwise. Of course, we need Sinvertible in the discrete cases. /square
+The matrices S,U,V,Kare subject to a Sylvester equation . Given matrix data ( J,S,U,V),
+the problem is to find a solution Kof the Sylvester equation. Let σ(S) denote the spectrum of S,
+i.e. the set of its eigenvalues. A sufficient condition for the existence of a solution of the respective
+Sylvester equation is
+σ(S)∩σ(−S) =∅
+σ(S)∩σ(S−1) =∅forcontinuous NLS
+discrete NLS(4.3)
+(see e.g. theorem 4.4.6 in [133]). Moreover, the solution is then unique, and as a consequence K
+automatically satisfies JK=−KJ.
+There is a certain redundancy in the matrix data, since differe nt matrix data can determine the
+same solution, but we will narrow this down to a considerable extent (section 4.2, see also [143]).
+We note that the NLS equations can determine Donly up to addition of an arbitrary m×m
+matrixD0that does not depend on x(andD0has to commute with J). We shall therefore
+understand the expression for Din (4.1) always modulo addition of an arbitrary constant mat rix
+commuting with J. To put it another way, we simply drop any constant summand th at arises in
+an expression for D, and we actually did this in order to arrive at the expression in (4.1).
+In section 4.1 we derive the results formulated in propositi on 4.1. Some symmetries of the
+solution formulae in proposition 4.1 are revealed in subsec tion 4.2. Subsection 4.3 formulates a
+superposition rule on the level of matrix data. Furthermore , section 4.4 specialises to the complex
+conjugation, respectively Hermitian conjugation reducti ons.
+10More precisely, here we assume that the inverses appearing i n these formulae exist. This will also be done in the
+formulation of results that originate from this propositio n. A considerable part of this work deals with this regularit y
+problem, however.
+12Remark 4.2. The results formulated in proposition 4.1 will be derived be low by application of
+corollary 2.5. The latter actually allows to replace Ξin proposition 4.1 by CΞwith any constant
+matrixCthat commutes with JandS. But the line of arguments that led us from theorem 2.3
+to corollary 2.4 in fact shows that such a matrix can be elimin ated by redefinitions of the matrix
+variables entering the solution formula. Conversely, in sp ecial cases (see e.g. section 6.2) it will be
+possible and convenient to shift all the freedom in the choic e ofUandVinto such a matrix C. In
+other cases it will be convenient to move at least part of the f reedom from UandVto a matrix
+C. Although the corresponding reparametrisation transform ation (see section 4.2.3) requires some
+restrictions that result in an invertible C(commuting with S), the aforementioned ensures us that
+this restriction for Cis not really required in order to obtain an NLS solution. /square
+4.1 Proof of proposition 4.1
+In the following we start with the bidifferential calculi on Ω = Mat2(B)⊗/logicalandtext(C2) associated in
+section 3 with the continuous NLS, semi-discrete NLS, respe ctively fully discrete NLS system.
+Then we apply theorem 2.3 in the form specified in corollary 2. 5. For fixed nwe write JforJn.
+The requirement that the matrices P,ˆU,Vhave to be d- and ¯d-constant now means that they
+must not depend on the independent (continuous, respective ly discrete) variables, and they have
+to satisfy
+[J,P] = 0, JˆU=ˆUJ,JV=VJ . (4.4)
+Anyn×nmatrixAdecomposes into the sum of an “even” and an “odd” part,
+A=Ae+AowhereJAe=AeJ,JAo=−AoJ. (4.5)
+Such a matrix is d- and ¯d-constant iff it is even and constant (i.e. independent of xandt). A
+constantn×nmatrixΣthat satisfies
+Σ2=I,JΣ=−ΣJ, (4.6)
+can be used to convert odd into even matrices and vice versa.
+Continuous NLS. We write
+P=JS, (4.7)
+with a new object S. Then¯dX= (dX)P= (dX)JSbecomes
+−iXt=XxSJ,Xx=1
+2[J,X]JS. (4.8)
+The general solution is
+X=A0+A1ΣΞ with Ξ=e−xS−itS2J. (4.9)
+HereA0,A1are constant even matrices (hence d- and ¯d-constant). Assuming that A0is invertible,
+application of theorem 2.3 reduces to an application of coro llary 2.5. In terms of the redefined
+matrices
+K=ˆKΣ,U=ˆUΣ, (4.10)
+13and with ˆΞ=ΣΞ, (2.18) takes the form
+SK+KS=VU, (4.11)
+Note that JV=VJ, butJU=−UJ. (2.20) and (3.6) lead to
+ϕ=UΞ(I−KΞ)−1V=UΞ(I+KΞ)(I−(KΞ)2)−1V. (4.12)
+The last expression is convenient in order to split ϕinto an even and an odd part. In this way we
+obtain (4.1).
+Semi-discrete NLS. Writing
+X=S˜X,P=JS−1˜P, (4.13)
+we find that ¯dX= (dX)P= (dX)JS−1˜Pis equivalent to
+−i˙˜X= (˜X+−˜X)J˜P,˜X−˜X−=1
+2[J,˜X]J˜P, (4.14)
+with the general solution
+˜X=A0+A1ΣΞ, (4.15)
+where
+Ξ=Sxe−itω(S)J, ω(S) =S+S−1−2I,S= (I+˜P)−1, (4.16)
+andx∈Z. Using the same redefinitions (4.10) as in the continuous NLS case, (2.18) becomes
+S−1K−KS=VU, (4.17)
+and (4.1) is obtained in precisely the same way as in the conti nuous NLS case.
+Fully discrete NLS. Using again (4.13), we find that ¯dX= (dX)Pis equivalent to
+−i(˜X+−˜X) = (˜X+
++−˜X)J˜P,˜X−˜X−=1
+2[J,˜X]J˜P. (4.18)
+Assuming that I−iJ˜Pis invertible, it follows that the even part of ˜Xis constant. If furthermore
+S= (I+˜P)−1and [I−iJ(I−S)]−1exist, the above system yields the following equations for t he
+odd part of ˜X,
+˜X+
+o=˜XoS,(˜Xo)+=˜Xo[I+iJ(I−S−1)][I−iJ(I−S)]−1. (4.19)
+Hence we obtain the solution
+˜X=A0+A1ΣΞ, (4.20)
+with constant even coefficient matrices and
+Ξ=Sx[(I+iJ(I−S−1)][I−iJ(I−S))−1]tx,t∈Z. (4.21)
+Again, assuming A0invertible and adopting the definitions (4.10), we arrive at (4.1) and (4.17).
+144.2 Transformations of the matrix data determining solutio ns
+4.2.1 Similarity transformations
+A transformation
+J/ma√sto→MJM−1,K/ma√sto→MKM−1,S/ma√sto→MSM−1,U/ma√sto→UM−1,V/ma√sto→MV,(4.22)
+whereMis any constant invertible n×nmatrix, obviously leaves UandDin proposition 4.1
+invariant, and also the corresponding Sylvester equation. It can thus be used to achieve that J
+takes the form
+J=/parenleftbiggIn10
+0−In2/parenrightbigg
+, (4.23)
+withn1,n2∈Nsuch thatn1+n2=n. The remaining similarity transformations such that
+[J,M] = 0 (4.24)
+can then be exploited to diminish the redundancy in the “para metrisation” of the solutions by the
+set of matrices S,U,V. In particular, it can be used to transform Sto Jordan normal form.
+4.2.2 Reflection symmetries
+IfKis invertible and Πa constantn×nmatrix with
+JΠ=−ΠJ, (4.25)
+then the transformation
+S/ma√sto→S′,U/ma√sto→ −UK−1Π−1,V/ma√sto→ΠK−1V,K/ma√sto→ΠK−1Π−1, (4.26)
+with
+S′=/braceleftbigg−ΠSΠ−1continuous NLS
+ΠS−1Π−1semi- and fully discrete NLS(4.27)
+implies
+Ξ/ma√sto→ΠΞ−1Π−1(4.28)
+and leaves Uin proposition 4.1 and also the corresponding Sylvester equ ation invariant. We refer
+to such a transformation as a reflection . Since
+D /ma√sto→U(ΞK)−1(Ξ−1−KΞK)−1V=D+UK−1V, (4.29)
+Donly changes by a constant matrix, which can be dropped as poi nted out above. Hence a
+reflection is a symmetry of the solutions given in propositio n 4.1 for all the NLS systems. A
+symmetry transformation of this kind for continuous NLS sol utions already appeared in [143].
+4.2.3 Reparametrisation transformations
+LetA,Bbe any constant invertible n×nmatrices that commute with JandS, i.e.
+[J,A] = [J,B] = [S,A] = [S,B] = 0. (4.30)
+Then
+S/ma√sto→S,U/ma√sto→UB−1,V/ma√sto→A−1V,K/ma√sto→A−1KB−1,Ξ/ma√sto→BAΞ, (4.31)
+is areparametrisation of the solution (4.1). This is helpful in order to reduce the n umber of
+parameters of UandV(see in particular section 6.2).
+154.3 Superposition of solutions
+Let (Ji,Si,Ui,Vi),i= 1,2, be data that determine two solutions ( Ui,Di) of one of the three
+systems (3.12), (3.19) or (3.23). Let Ki,i= 1,2, be corresponding solutions of the Sylvester
+equations. Then
+J=/parenleftbiggJ10
+0J2/parenrightbigg
+,S=/parenleftbiggS10
+0S2/parenrightbigg
+,U=/parenleftbig
+U1U2/parenrightbig
+,V=/parenleftbiggV1
+V2/parenrightbigg
+,K=/parenleftbiggK1K12
+K21K2/parenrightbigg
+(4.32)
+determine new solutions if K12andK21satisfy
+J1K12=−K12J2,J2K21=−K21J1, (4.33)
+and solve the respective equations below.
+Continuous NLS :
+S1K12+K12S2=V1U2,S2K21+K21S1=V2U1. (4.34)
+These equations possess unique solutions if σ(S1)∩σ(−S2) =∅.
+Semi-discrete and fully discrete NLS :
+S−1
+1K12−K12S2=V1U2,S−1
+2K21−K21S1=V2U1. (4.35)
+Theseequationshaveuniquesolutionsif σ(S1)∩σ(S−1
+2) =∅. (4.3) appliedto S= block-diag( S1,S2)
+contains the abovementioned spectrum conditions and is the refore a sufficient (but not necessary)
+condition for the above superposition principle to generat e again a solution. Let us list some fairly
+obvious properties of superpositions.
+1. The superposition of two solutions is commutative. Note t hat
+M=/parenleftbigg0In2
+In10/parenrightbigg
+(4.36)
+is a similarity transformation that exchanges the matrix bl ocks corresponding to the two
+solutions in the superposed data.
+2. Superpositions are associative.
+3. Asimilarity transformationofthematrixdatacorrespon dingtoaconstituent solution extends
+to a similarity transformation of the superposition, hence remains an equivalence transfor-
+mation. This is simply based on the fact that a similarity tra nsformation of the superposed
+data, with an invertible matrix of the form
+M=/parenleftbiggM10
+0M2/parenrightbigg
+, (4.37)
+covers arbitrary similarity transformations of the consti tuent data.
+4. A reparametrisation transformation of a constituent ext ends to a reparametrisation of the
+superposition, hence remains an equivalence transformati on. (Choose A,Bblock-diagonal in
+(4.30) and (4.31).)
+16Recall that, for any data ( J,S,U,V) that determine a solution, via a similarity transformatio n
+Jcan be brought to standard form (4.23) and Ssimultaneously to Jordan normal form. We call
+the data ( J,S,U,V)simpleif at least one of the blocks of S, corresponding to a block of J, is a
+single Jordan block. For any solution given by data ( J,S,U,V) that are not simple, we have the
+block structure in (4.32). These data have to solve the Sylve ster equation, hence in particular there
+are matrices K1,K2solving its diagonal blocks. It follows that ( J,S,U,V) decomposes into data
+(Ji,Si,Ui,Vi),i= 1,2, that determine solutions. As a consequence, for any solut ion obtained
+from non-simple data, the simple subdata also determine sol utions. In this sense, the solution can
+always be regarded as a superposition of solutions.
+Though reflection symmetries are not quite compatible with t he structure of a superposition,
+we still have the following result (based on arguments in [14 3]).
+Proposition 4.3. Let (4.3) hold for S= block-diag( S1,S2). A reflection symmetry of (only) one
+of the two matrix data sets can then be extended in such a way th at it is also a symmetry of the
+superposition solution.
+Proof:We consider a reflection symmetry on the second data set. Henc e we have to assume that
+K2is invertible. Writing Zi=Ξ−1
+i−KiΞiKi, so that Ui=UiZ−1
+iVi, we have U=UZ−1Vwith
+Z=/parenleftbiggZ1−K12Ξ2K21−K1Ξ1K12−K12Ξ2K2
+−K2Ξ2K21−K21Ξ1K1Z2−K21Ξ1K12/parenrightbigg
+.
+According to the reflection transformation rules (4.26), we have
+S′=/parenleftbiggS10
+0S′
+2/parenrightbigg
+,
+whereS′
+2is defined for the three NLS cases in (4.27). The other transfo rmed quantities U′
+2,V′
+2,K′
+2
+are given by (4.26). Corresponding transformation rules fo rK12andK21should then be read off
+from (4.34), respectively (4.35). It turns out that also U1andV1have to be transformed. In fact,
+setting
+U′=UA,V′=BVwithA=/parenleftbiggI 0
+−K−1
+2K21−K−1
+2Π−1/parenrightbigg
+,B=/parenleftbiggI−K12K−1
+2
+0ΠK−1
+2/parenrightbigg
+,
+and
+K′=/parenleftbiggK1−K12K−1
+2K21−K12K−1
+2Π−1
+ΠK−1
+2K21 ΠK−1
+2Π−1/parenrightbigg
+=/parenleftbiggK1K12
+0−Π/parenrightbigg
+A,
+one obtains Z′=BZAand hence U′=U′Z′−1V′=U. Furthermore, we find that
+D′=D+U/parenleftbigg0 0
+0K−1
+2/parenrightbigg
+V,
+i.e.Donly changes by an irrelevant constant summand. /square
+If we superpose the same data, we obtain nothing new. More gen erally, we have the following
+result.11
+11One can still refine this proposition in order to weaken the as sumptions.
+17Proposition 4.4. Let(Ji,Si,Ui,Vi),i= 1,2, be two data sets that determine solutions of an NLS
+system according to proposition 4.1 and let (4.3) hold for S= block-diag( S1,S2). Suppose there
+are matrices A,Bsuch that U2=U1B,V2=AV1,AS1=S2AandS1B=BS2. Let further
+C=I+BAbe invertible. Then the superposed data (4.32) determine a so lution that can also be
+obtained from the data (J1,S1,U1C,V1)(i.e. simply by a redefinition of U1).
+Proof:We consider the continuous NLS case only. The discrete cases are treated in a similar way.
+Theassumptions together with the respective Sylvester equ ations for thedata ( Ji,Si,Ui,Vi) imply
+S2(K2−AK1B)+(K2−AK1B)S2= 0.
+Since the assumption (4.3) in particular implies σ(S2)∩σ(−S2) =∅, the last equation implies
+K2=AK1B. For (4.32) to determine a solution, (4.34) has to be satisfie d, and thus
+S1(K12−K1B)+(K12−K1B)S2= 0,S2(K21−AK1)+(K21−AK1)S1= 0.
+Since (4.3) also implies σ(S1)∩σ(−S2) =∅, these equations have the unique solutions K12=K1B
+andK21=AK1, hence
+K=/parenleftbiggI
+A/parenrightbigg
+K1/parenleftbig
+I B/parenrightbig
+.
+This in turn implies
+W:=I−ΞKΞK =I−/parenleftbiggI
+A/parenrightbigg
+Ξ1K1CΞ1K1/parenleftbig
+I B/parenrightbig
+,
+and thus, assuming that C=I+BAis invertible and introducing ˜Ξ1=CΞ1,
+W−1=I+/parenleftbiggI
+A/parenrightbigg
+C−1(˜W−1
+1−I)/parenleftbig
+I B/parenrightbig
+,
+where˜W1=I−˜Ξ1K1˜Ξ1K1. With straightforward computations (4.1) now leads to
+U=U1/parenleftig
+˜Ξ−1
+1−K1˜Ξ1K1/parenrightig−1
+V1,D=U1˜Ξ1K1/parenleftig
+˜Ξ−1
+1−K1˜Ξ1K1/parenrightig−1
+V1,
+from which our assertion is easily deduced. /square
+4.4 Imposing reduction conditions on the solutions
+4.4.1 Complex conjugation reduction
+LetΓbe ann×nmatrix with the properties
+JΓ=−ΓJ,Γ∗=Γ−1. (4.38)
+Choosing the normal form (4.23) of J, this implies that Γis off-block-diagonal. The last condition
+in (4.38) requires that nis even and
+n1=n2. (4.39)
+Imposing the following constraints on the matrices S,U,V,
+S∗=ΓSΓ−1,U∗=ǫǫ′ΓUΓ−1,V∗=ǫ′ΓVΓ−1, (4.40)
+18withǫ′=±1 and Γ as in section 3, the Sylvester equations demand
+K∗=ǫΓKΓ−1, (4.41)
+andUandDgiven by (4.1) indeed satisfy the reduction conditions (3.2 8).
+The transformations considered in section 4.2 should now be restricted in such a way that the
+reduction conditions are preserved. The matrix Mof a similarity transformation (preserving the
+form ofJ), now also has to satisfy M∗=ΓMΓ−1. For a reflection symmetry, the matrix Πhas
+to satisfy Π∗=ǫΓΠΓ−1. In case of a reparametrisation transformation, we have to a dd the
+conditions A∗=ΓAΓ−1andB∗=ΓBΓ−1.
+4.4.2 Hermitian conjugation reduction
+This only applies to the continuous NLS case. From (4.1) we ob tain
+U†=V†(Ξ†−1−K†Ξ†K†)−1U†, (4.42)
+In order to achieve the reduction condition U†=ǫUwithǫ=±1, we set
+S†=TST−1,U=V†T, (4.43)
+with an invertible n×nmatrixTwith the properties
+JT=−TJ,T†=ǫT. (4.44)
+These conditions again restrict us to the case where
+n1=n2. (4.45)
+The Sylvester equation now reads
+SK+KS=VV†T, (4.46)
+and is preserved by Hermitian conjugation if
+K†=ǫTKT−1. (4.47)
+Then the expression for Uin (4.1) indeed satisfies U†=ǫU. Furthermore, as a consequence of the
+identityZ−1KΞ=ΞKZ−1whereZ=Ξ−1−KΞK, the expression for Din (4.1) satisfies D†=D.
+For a similarity transformation with matrix M, which commutes with J, we have to require
+T/ma√sto→(M†)−1TM†. WithJgiven by (4.23) (where n1=n2), this can be used to achieve that
+T=/parenleftbigg0In1
+ǫIn10/parenrightbigg
+. (4.48)
+After having fixed T, the matrix Mof a similarity transformation preserving the Hermitian re -
+duction conditions has to satisfy M†=TM−1T−1. For a reflection symmetry, Π†=−ǫTΠ−1T−1.
+In case of a reparametrisation, A†=TBT−1.
+195 Block-decomposition of the NLS systems and their solution s
+In this section we express the NLS systems in a more familiar f orm12and translate proposition 4.1
+correspondingly. We choose for Jthe normal form (3.24) and write
+ϕ=/parenleftbiggp q
+¯q¯p/parenrightbigg
+, (5.1)
+wherep,¯p,q,¯qare, respectively, m1×m1,m2×m2,m1×m2andm2×m1matrices. Asaconsequence
+of (3.8),
+U=/parenleftbigg0q
+¯q0/parenrightbigg
+,D=/parenleftbiggp0
+0 ¯p/parenrightbigg
+. (5.2)
+Continuous NLS system. (3.12) becomes the matrix NLS system (see e.g. [3,79,80,144])
+iqt+qxx−2q¯qq= 0,−i¯qt+ ¯qxx−2¯qq¯q= 0, (5.3)
+together with
+px=−q¯q,¯px=−¯qq. (5.4)
+Semi-discrete NLS system. (3.19) leads to
+p+−p=−q+¯q,¯p+−¯p=−¯q+q, (5.5)
+and the matrix Ablowitz-Ladik (AL) system
+i ˙q+q+−2q+q−−q+¯qq−q¯qq−= 0,
+−i˙¯q+ ¯q+−2¯q+ ¯q−−¯q+q¯q−¯qq¯q−= 0. (5.6)
+Fully discrete NLS system. From (3.23) we obtain
+p+−p=−q+¯q,¯p+−¯p=−¯q+q, (5.7)
+(as in the semi-discrete case) and
+i(q+−q)+(q+−q)+−(q−q−)−(q+¯qq)+−q¯qq−+(p+−p)q++q(¯p+−¯p) = 0,
+−i(¯q+−¯q)+(¯q+−¯q)+−(¯q−¯q−)−(¯q+q¯q)+−¯qq¯q−+(¯p+−¯p)¯q++ ¯q(p+−p) = 0.(5.8)
+Unlike the situation in the continuous NLS and semi-discret e NLS (AL) cases, here the variables
+p,¯pcannot be eliminated from the equations for qand ¯qwithout introduction of non-local terms.
+A reduction of this system appeared in [5] (see also appendix C). Integrable full discretisations of
+matrix NLS systems appeared in [130].
+12See also appendix E for a derivation of Lax pairs for these NLS systems within the bidifferential calculus frame-
+work.
+20Solutions. ForJwechoosethenormalform(4.23). Fromthepropertiesofthem atricesappearing
+in proposition 4.1, we find the following block structures,
+S=/parenleftbiggS0
+0¯S/parenrightbigg
+,U=/parenleftbigg0U
+¯U0/parenrightbigg
+,V=/parenleftbiggV0
+0¯V/parenrightbigg
+,K=/parenleftbigg0K
+¯K0/parenrightbigg
+,(5.9)
+andΞis block-diagonal with diagonal blocks Ξ ,¯Ξ. Now proposition 4.1 translates into the following
+statement.
+Proposition 5.1. Let
+U ¯UV ¯VS ¯SK ¯K
+sizem1×n2m2×n1n1×m1n2×m2n1×n1n2×n2n1×n2n2×n1
+be constant complex matrices, with Sand¯Sinvertible in the discrete NLS cases, and
+NLS Sylvester equations Ξ ¯Ξ
+continuousSK+K¯S=VU
+¯S¯K+¯KS=¯V¯Ue−xS−itS2e−x¯S+it¯S2x,t∈R
+semi-discreteS−1K−K¯S=VU
+¯S−1¯K−¯KS=¯V¯USxe−itω(S)¯Sxeitω(¯S)x∈Z,t∈R
+fully discreteS−1K−K¯S=VU
+¯S−1¯K−¯KS=¯V¯USxΩ(S)t¯Sx¯Ω(¯S)tx,t∈Z
+where
+ω(S) =S+S−1−2In1, (5.10)
+Ω(S) = [In1+i(In1−S−1)][In1−i(In1−S)]−1,
+¯Ω(¯S) = [In2−i(In2−¯S−1)][In2+i(In2−¯S)]−1. (5.11)
+Then
+q=U/parenleftbig¯Ξ−1−¯KΞK/parenrightbig−1¯V , ¯q=¯U/parenleftbig
+Ξ−1−K¯Ξ¯K/parenrightbig−1V ,
+p=U¯Ξ¯K/parenleftbig
+Ξ−1−K¯Ξ¯K/parenrightbig−1V ,¯p=¯UΞK/parenleftbig¯Ξ−1−¯KΞK/parenrightbig−1¯V ,(5.12)
+solves the respective NLS system. /square
+5.1 Transformations of the matrix data determining solutio ns
+The transformations described in section 4.2 decompose cor respondingly.
+1.A similarity transformation satisfies (4.24), with Jgiven by (4.23), iff the transformation matrix
+Mis block-diagonal, i.e. M= block-diag( M,¯M). Then we obtain
+S/ma√sto→MSM−1,¯S/ma√sto→¯M¯S¯M−1, U/ma√sto→U¯M−1,¯U/ma√sto→¯UM−1,
+K/ma√sto→MK¯M−1,¯K/ma√sto→¯M¯KM−1, V/ma√sto→MV , ¯V/ma√sto→¯M¯V . (5.13)
+21As a consequence, withoutrestriction of generality wecan a ssumethat Sand¯Shave Jordan normal
+form.
+2.The matrix Πof a reflection symmetry has to be off-block-diagonal. For late r use we consider
+the example
+Π=/parenleftbigg0βIn1¯βIn10/parenrightbigg
+, (5.14)
+assumingn1=n2(and thusneven). The transformation then acts as follows,
+K/ma√sto→β¯β−1K−1, U /ma√sto→ −βUK−1, V /ma√sto→¯β−1K−1V ,
+¯K/ma√sto→β−1¯β¯K−1, ¯U/ma√sto→ −¯β¯U¯K−1, ¯V/ma√sto→β−1¯K−1¯V ,(5.15)
+together with
+S/ma√sto→ −¯S,¯S/ma√sto→ −S, (5.16)
+for continuous NLS, and
+S/ma√sto→¯S−1,¯S/ma√sto→S−1(5.17)
+for semi-discrete and fully discrete NLS. This implies
+Ξ/ma√sto→¯Ξ−1,¯Ξ/ma√sto→Ξ−1. (5.18)
+3.Incaseofareparametrisationtransformation, wehave A= block-diag( A,¯A),B= block-diag( B,¯B),
+and [S,A] = [S,B] = [¯S,¯A] = [¯S,¯B] = 0. The transformation is then given by
+S/ma√sto→S,¯S/ma√sto→¯S, U/ma√sto→U¯B−1,¯U/ma√sto→¯UB−1, V/ma√sto→A−1V ,¯V/ma√sto→¯A−1¯V
+K/ma√sto→A−1K¯B−1,¯K/ma√sto→¯A−1¯KB−1,Ξ/ma√sto→BAΞ,¯Ξ/ma√sto→¯B¯A¯Ξ. (5.19)
+4.A transformation
+U/ma√sto→ SU,¯U/ma√sto→¯S¯U, V/ma√sto→VS−1,¯V/ma√sto→¯V¯S−1, (5.20)
+with invertible constant matrices S,¯S, has the effect
+q/ma√sto→ Sq¯S−1,¯q/ma√sto→¯S¯qS−1, p/ma√sto→ SpS−1,¯p/ma√sto→¯S¯p¯S−1. (5.21)
+In contrast to the previous transformations, this is not a sy mmetry of the solutions. But it may be
+regarded as an equivalence transformation of solutions.
+6 The complex conjugation reduction
+We recall that, for
+m1=m2, (6.1)
+all three NLS systems admit the complex conjugation reducti on introduced in section 3. In the
+following we rename m1tom. Choosing Γ as in (3.30), the reduction conditions (3.28) ta ke the
+form
+¯q=ǫq∗withǫ=±1,¯p=p∗, (6.2)
+Hereq,¯q,p,¯pare allsquarematrices of size m×m.
+22Continuous NLS. (6.2) reduces the (continuous) NLS system to the following ( square) matrix
+NLS equations,
+iqt+qxx−2ǫqq∗q= 0, p x=−ǫqq∗. (6.3)
+The casesǫ=−1 andǫ= 1 are sometimes referred to as focusing anddefocusing , respectively.
+Form= 2 and symmetricq, i.e.q⊺=q(whereq⊺is the transpose of q), so thatqhas the form
+q=/parenleftbiggq1q0
+q0q−1/parenrightbigg
+, (6.4)
+the matrix NLS equation with ǫ=−1 describes a special case of a spin-1 Bose-Einstein condens ate
+[102–115]. (6.3) is then equivalent to the system
+iq1,t+q1,xx+2(|q1|2+2|q0|2)q1+2q2
+0q∗
+−1= 0,
+iq−1,t+q−1,xx+2(|q−1|2+2|q0|2)q−1+2q2
+0q∗
+1= 0,
+iq0,t+q0,xx+2(|q1|2+|q0|2+|q−1|2)q0+2q1q∗
+0q−1= 0. (6.5)
+Semi-discrete NLS. In this case (6.2) leads to the (square) matrix AL equations
+i ˙q+q+−2q+q−−ǫ(q+q∗q+qq∗q−) = 0, p+−p=−ǫq+q∗. (6.6)
+The scalar version of the first equation first appeared in [4–6 ] and has since been the subject of
+many publications (see the introduction).
+Fully discrete NLS. Now (6.2) yields
+i(q+−q)+(q+−q)+−(q−q−)−ǫ(q+q∗q)+−ǫqq∗q−+(p+−p)q++q(p+−p)∗= 0,
+p+−p=−ǫq+q∗. (6.7)
+Herepdoes not decouple from the equation for q. Appendix C shows that, by elimination of pat
+the priceof introducingnon-local terms in the firstequatio n, one recovers an equation firstobtained
+by Ablowitz and Ladik [5] in the scalar case.
+Solutions. We setn1=n2and rename it to n. Furthermore, we choose
+Γ=/parenleftbigg0In
+In0/parenrightbigg
+. (6.8)
+Using (5.9), the reduction conditions (4.40) and (4.41) bec ome
+¯S=S∗,¯U=ǫǫ′U∗,¯V=ǫ′V∗,¯K=ǫK∗. (6.9)
+Proposition 5.1 now implies the following result.
+Proposition 6.1. LetS,U,Vbe constant complex matrices of size n×n,m×nandn×m,
+respectively. In the discrete NLS cases, we assume invertibil ity ofS. Then
+q=±U/parenleftbig
+Ξ∗−1−ǫK∗ΞK/parenrightbig−1V∗, (6.10)
+p=ǫUΞ∗K∗(Ξ−1−ǫKΞ∗K∗)−1V (6.11)
+solves them×mmatrix continuous NLS equations (6.3), the semi-discrete NLS equations (6.6),
+respectively the fully discrete NLS equations (6.7), if ther e is a matrix Ksolving the following
+Sylvester equation and if Ξis given by the following expression,
+23NLS Sylvester equation Ξ
+continuous SK+KS∗=VUe−xS−itS2x,t∈R
+semi-discrete S−1K−KS∗=VUSxe−itω(S)x∈Z, t∈R
+fully discrete S−1K−KS∗=VUSxΩ(S)tx,t∈Z
+where
+ω(S) =S+S−1−2In,Ω(S) = [In+i(In−S−1)][In−i(In−S)]−1. (6.12)
+/square
+With reference e.g. to theorem 4.4.6 in [133], a sufficient con dition for the above Sylvester
+equations to possess a solution K, irrespective of the form of UandV, is
+σ(S)∩σ(−S∗) =∅
+σ(S∗)∩σ(S−1) =∅forcontinuous NLS
+discrete NLS(6.13)
+(whereσ(S) denotes the spectrum of the matrix S). The solution matrix Kis then unique.
+ChoosingSdiagonal, inthescalar focusingcase thesolutions determi ned by thelast proposition
+contain the familiar solitons. They will be described in a sl ightly different form in section 6.2.
+Remark 6.2. For given data ( S,U,V) with ann×nmatrixS, letKbe a solution of the Sylvester
+equation in the continuous NLS case. Then it follows that Kalso solves the Sylvester equation
+withS′=S+iβIn,β∈R. Hence there is a new NLS solution obtained by replacing Ξ in ( 6.10)
+and (6.11) by
+Ξ(x,t;S′) =e−xS′−it(S′)2=e−iβ(x−βt)Ξ(x−2βt,t;S). (6.14)
+This induces the following Galilean transformation of the N LS solution,
+q′(x,t) =eiβ(x−βt)q(x−2βt,t), (6.15)
+which is a symmetry of the NLS equation. /square
+Remark 6.3. IfSis symmetric, i.e. S⊺=S, and ifU=V†, in the continuous NLS case the
+Sylvester equation becomes the Lyapunov equation13
+SK+KS†=VV†, (6.16)
+whereKcan be chosen Hermitian. Furthermore, the solution qgiven by (6.10) is symmetric, i.e.
+q⊺=q. In particular, this yields solutions of the abovementione d spin-1 Bose-Einstein condensate
+model. For later reference (see remark 6.7), we mention the f ollowing result in [148].
+(6.16) possesses a Hermitian positive definite solution Kif and only if the following three conditions
+hold,
+(i) all eigenvalues of Shave non-negative real parts,
+(ii) the subspace of Cnspanned by all eigenvectors of Scorresponding to eigenvalues with positive
+real part coincides with the controllability subspace asso ciated with the pair ( S,V) (which is the
+subspace spanned by the columns of the matrices V,SV,S2V,...,Sn−1V),
+(iii) eigenvalues of Swith zero real parts are not degenerate (i.e. they are simple roots of the
+minimal polynomial). /square
+13More generally, the Lyapunov equation has the form SK+KS†=Qwith a complex matrix Q(see e.g. [132,
+133,145,146]). It plays an important role in control theory [147].
+246.1 On the structure of the solutions
+6.1.1 Solving the Sylvester equation
+Without restriction of generality we can assume that Shas Jordan normal form (since this can be
+achieved by a similarity transformation). The Sylvester eq uation then has to be satisfied for each
+Jordan block (and in addition we have to solve equations for t he off-diagonal blocks of K). Let us
+therefore now consider a matrix Sconsisting of an r×rJordan block only, hence
+S=sI+N, (6.17)
+whereIis ther×ridentity matrix and the components of Nare given by Nij=δi,j−1. Writing
+K= (kij) and
+U=/parenleftbig
+u1···ur/parenrightbig
+, V=
+v⊺
+1...
+v⊺
+r
+, (6.18)
+withm-component column vectors ui,vi,i= 1,...,r, in thecontinuous NLS case the Sylvester
+equation takes the form
+(s+s∗)kij+ki+1,j+ki,j−1=v⊺
+iuji,j= 1,...,r, (6.19)
+wherekr+1,j=ki,0= 0. Assuming s/ne}ationslash=−s∗(in which case we know that the Sylvester equation
+has a unique solution), this implies
+kr1=v⊺
+ru1
+s+s∗, k rj=1
+s+s∗(v⊺
+ruj−kr,j−1)j= 2,...,r,
+ki1=1
+s+s∗(v⊺
+iu1−ki+1,1)i=r−1,...,1,(6.20)
+which recursively determines the entries ki1,kri,i= 1,...,r. In fact, (6.19) also determines the
+remaining entries of Krecursively. For example, for r= 2 we obtain
+K=
+v⊺
+1u1
+s+s∗−v⊺
+2u1
+(s+s∗)2v⊺
+1u2
+s+s∗−v⊺
+1u1+v⊺
+2u2
+(s+s∗)2+2v⊺
+2u1
+(s+s∗)3
+v⊺
+2u1
+s+s∗v⊺
+2u2
+s+s∗−v⊺
+2u1
+(s+s∗)2
+. (6.21)
+Returning to arbitrary r, we observe that the first column of Kvanishes if u1= 0, and the
+last row of Kvanishes if vr= 0. Inspection of (6.10) and (6.11) shows that in these cases the
+solution is completely determined by the correspondingly r educed matrix data. In the case where
+u1= 0, these reduced data are obtained by deleting the first row a nd first column of S, and also
+deleting the first column of Uand the first row of V. This also holds in case of the discrete NLS
+versions. We conclude that, for any solution determined by m atrix data ( S,U,V), whereUandV
+are different from zero and Sis a Jordan block subject to the condition (6.13), there are e quivalent
+data (S′,U′,V′) whereS′is a Jordan block (with the same eigenvalue and of size smalle r or equal
+to that ofS) such that the first column of U′as well as the last row of V′are different from zero.
+More generally, for any non-vanishing solution determined by matrix data ( S,U,V), withS
+given in Jordan normal form subject to the condition (6.13), we can assume without restriction of
+generality that for any Jordan block of Sthe first column of the corresponding part of Uas well
+as the last row of the corresponding part of Vare different from zero.
+256.1.2 Superpositions
+Given twodatasets ( Si,Ui,Vi),i= 1,2, thatdetermineNLSsolutions, withcorrespondingsoluti ons
+Kiof the respective Sylvester equations, we can superpose the m,
+S=/parenleftbiggS10
+0S2/parenrightbigg
+, U=/parenleftbig
+U1U2/parenrightbig
+, V=/parenleftbiggV1
+V2/parenrightbigg
+, K=/parenleftbiggK1K12
+K21K2/parenrightbigg
+,(6.22)
+whereK12andK21have to solve
+S1K12+K12S∗
+2=V1U2, S2K21+K21S∗
+1=V2U1 continuous NLS (6.23)
+K12−S1K12S∗
+2=V1U2, K21−S2K21S∗
+1=V2U1 discrete NLS .(6.24)
+(6.13) is a sufficient condition for such a solution to exist (w hich is then unique). Then we have
+Z=/parenleftbiggZ1−ǫK∗
+12Ξ2K21−ǫK∗
+1Ξ1K12−ǫK∗
+12Ξ2K2
+−ǫK∗
+2Ξ2K21−ǫK∗
+21Ξ1K1Z2−ǫK∗
+21Ξ1K12/parenrightbigg
+, (6.25)
+whereZi= Ξ∗−1
+i−ǫK∗
+iΞiKi. Its inverse, needed to evaluate qandpmore explicitly, can be
+expressed in terms of Schur complements.
+Conversely, given solution data ( S,U,V), whereSis block-diagonal with at least two blocks,
+then these data can be understood as a superposition (see als o section 4.3).
+The following result is simply proposition 4.4 adapted to th e case under consideration. It will
+be helpful in the sequel.
+Proposition 6.4. Let(Si,Ui,Vi),i= 1,2, be two data sets that determine solutions of an NLS
+system according to proposition 6.1 and let (6.13) hold for S= block-diag( S1,S2). Suppose there
+are matrices A,Bsuch thatU2=U1B,V2=AV1,AS1=S2AandS∗
+1B=BS∗
+2. Let further C=
+I+B∗Abe invertible. Then the superposed data (6.22) determine a so lution that can equivalently
+be obtained from the data (S1,U1C,V1)(i.e. simply by a redefinition of U1).
+6.1.3 The use of reflection symmetries
+The following result has its origin in [143], where only the ( focusing) continuous NLS equation was
+considered, however. We show that for a solution determined by matrix data via proposition 6.1,
+in general equivalent matrix data ( S,U,V) exist such that all eigenvalues sofSwith Re(s)/ne}ationslash= 0 in
+the continuous NLS case, respectively |s| /ne}ationslash= 1 in the discrete cases, satisfy
+(i) Re(s)>0 in the continuous NLS case,
+(ii)|s|<1 in the semi- and fully discrete NLS case.
+In case (i) the matrix Sis called positive stable [133]. In case (ii) it is sometimes called stable with
+respect to the unit circle [145].
+SupposeS2is a Jordan block that does not satisfy condition (i), respec tively (ii). Writing
+S=/parenleftbiggS10
+0S2/parenrightbigg
+, U=/parenleftbig
+U1U2/parenrightbig
+, V=/parenleftbiggV1
+V2/parenrightbigg
+, (6.26)
+presents the solution data in the form of a superposition, an d one has to solve (6.23), respectively
+(6.24), in order to construct the solution. The idea is now to apply a reflection symmetry and to
+use the analogue of proposition 4.3. A reflection symmetry of the form considered in section 4.2.2
+preserves the reduction conditions if
+¯β=ǫβ∗. (6.27)
+26Let us choose β=ǫ(so that ¯β= 1). The reflection symmetry is then given by
+U2/ma√sto→ −ǫU2K−1
+2, V2/ma√sto→K−1
+2V2, K2/ma√sto→ −ǫU2K−1
+2, (6.28)
+together with
+S2/ma√sto→S′
+2=/braceleftbigg−S∗
+2continuous NLS
+S∗−1
+2semi- and fully discrete NLS. (6.29)
+This is a reflection in an obvious sense. As in the proof of prop osition 4.3, we can construct an
+extension to matrix data that determine the same superposit ion solution. For the above reflection,
+this is achieved by
+S/ma√sto→/parenleftbiggS10
+0S′
+2/parenrightbigg
+, (6.30)
+U/ma√sto→U/parenleftbiggI 0
+−K−1
+2K21−ǫK−1
+2/parenrightbigg
+, V/ma√sto→/parenleftbiggI−K12K−1
+2
+0K−1
+2/parenrightbigg
+V , (6.31)
+and
+K/ma√sto→/parenleftbiggK1−K12K−1
+2K21−ǫK12K−1
+2
+K−1
+2K21 ǫK−1
+2/parenrightbigg
+. (6.32)
+The only assumption needed for this to work is that the Sylves ter equation with data ( S2,U2,V2)
+has aninvertible solutionK2.
+In case of a Jordan block, the result of the reflection (6.29) h as no longer the standard Jordan
+form. But with the help of a similarity transformation one ca n restore the usual Jordan structure
+(without changing the eigenvalues).
+6.2 Solutions of the scalarcontinuous and discrete NLS equations
+In this subsection we concentrate on the scalarNLS case, i.e. m= 1. Without restriction of
+generality, we can assume that the n×nmatrixShas Jordan normal form.
+Lemma 6.5.
+(1) LetSbe a single k×kJordan block. For any k-component column vector V, there is a k×k
+matrixAthat commutes with Sand satisfies
+V=AV0whereV⊺
+0= (0,...,0,1). (6.33)
+Acan be chosen invertible if the last component of Vis different from zero.
+Furthermore, for any k-component row vector U, there is a k×kmatrixBthat commutes with
+Sand satisfies
+U=U0B∗whereU0= (1,0,...,0). (6.34)
+Bcan be chosen invertible if the first component of Uis different from zero.
+(2) LetS=block-diag (S1,...,S r)with Jordan blocks Si. For any correspondingly decomposed
+n-component vector V⊺= (V⊺
+1,...,V⊺
+r), there is an n×nmatrixAthat commutes with Sand
+satisfies
+V=AV0withV⊺
+0= (V⊺
+0,1,...,V⊺
+0,r), (6.35)
+27where each vector V0,i(of same size as Vi) has a1in the last entry and zeros otherwise. Acan be
+chosen invertible if the last component of each Viis different from zero.
+For any correspondingly decomposed n-component row vector U= (U1,...,U r)there is ann×n
+matrixBthat commutes with Sand satisfies
+U=U0B∗withU0= (U0,1,...,U 0,r), (6.36)
+where each row vector U0,i(of same size as Ui) has a1in the first entry. Bcan be chosen invertible
+if the first component of each Uiis different from zero.
+Proof:(1)Acommutes with the Jordan block Sif it is an upper-triangular Toeplitz matrix. Hence
+S=
+s1 0 ···0
+0............
+............0
+.........1
+0··· ··· 0s
+, A=
+a1a2a3···ak
+0............
+............a3
+.........a2
+0··· ··· 0a1
+,
+with arbitrary constants a1,...,a k. It is now immediately verified that, for any given vector V,
+there is a matrix Aof the above form such that AV0=Vholds.Ais invertible iff a1/ne}ationslash= 0, and
+a1coincides with the last component of V. Letu1,...,u kbe the components of UandBthe
+Toeplitz matrix built with the complex conjugates of these c onstants. Then U=U0B∗. Clearly,B
+is invertible iff u1/ne}ationslash= 0. (2) immediately follows from (1). /square
+As a consequence of the lemma, we can always achieve that
+U=U0B∗, V=AV0, (6.37)
+with matrices A,Bcommuting with S. As shown in section 6.1.1, the condition in the above lemma
+achieving that AandBare invertible is no restriction of generality, provided th atSsatisfies
+the spectrum condition (6.13). The latter implies that Sis invertible. By a reparametrisation
+transformation14(and redefinitions of KandVin the discrete cases) the above family of solutions
+can then be expressed as
+q=U0(I−ǫΞ∗K∗ΞK)−1Ξ∗S∗−ηV0, p=ǫU0Ξ∗K∗(I−ǫΞKΞ∗K∗)−1ΞS−ηV0,(6.38)
+with
+NLSηSylvester equation Ξ
+continuous 0SK+KS∗=V0U0 Ce−xS−itS2
+semi-discrete 1K−SKS∗=V0U0 CSx+1e−it(S+S−1−2I)
+fully discrete 1K−SKS∗=V0U0CSx+1[(I+i(I−S−1))(I−i(I−S))−1]t
+whereI=Inand the matrices A,Bonly appear through the combination
+C=BA. (6.39)
+14The complex conjugation reduction requires ¯A=A∗and¯B=B∗in (5.19).
+28With the restriction to the focusing case, i.e.ǫ=−1, we now consider the special case where
+Sisdiagonal, hence
+S= diag(s1,...,sn), U 0=V⊺
+0= (1,...,1). (6.40)
+In the following we assume the spectrum condition (6.13), i. e.si+s∗
+j/ne}ationslash= 0 in the continuous case,
+andsis∗
+j/ne}ationslash= 1 in the discrete cases, for all i,j= 1,...,n. According to proposition 6.4, with no
+restriction of generality we can then assume that the eigenv aluessiare pairwise different. As a
+consequence, AandBare diagonal, hence Cis diagonal. Writing
+Ξ = diag(ξ1,...,ξn), C= diag(c1,...,cn), K= (kij), (6.41)
+we obtain
+NLS ξi kij
+continuous cie−xsi−its2
+i 1/(si+s∗
+j)
+semi-discrete (AL) cisx+1
+ie−it(si+s−1
+i−2)1/(1−sis∗
+j)
+fully discrete cisx+1
+i[(1+i(1 −s−1
+i))(1−i(1−si))−1]t1/(1−sis∗
+j)
+Kis a Cauchy-like matrix and Hermitian.15
+Lemma 6.6. LetSbe diagonal with pairwise different eigenvalues si, subject to (6.13).
+(i) The matrix Kof the continuous NLS case is positive definite if Re(si)>0fori= 1,...,n
+(which means that Sis positive stable).
+(ii) The matrix Kof the discrete NLS cases is positive definite if all silie in the open unit disk
+(which means that Sis stable with respect to the unit circle).
+Proof:In case (i), the matrix Kis a positive semidefinite Cauchy matrix, see e.g. [146,149] . If the
+siare pairwise different, Cauchy’s determinant formula (see e. g. [146]) shows that Kis invertible.
+HenceKis positive definite. In case (ii), we can write K=DCD†with the Cauchy matrix
+C= (1/(λi+λ∗
+j)) andD= diag((λ1+1)/√
+2,...,(λn+1)/√
+2), wheresi= (λi−1)/(λi+1) defines
+a bijection between the open unit disk and the open right half plane. Since the siare assumed to
+be pairwise different, it follows that also the λiare pairwise different. Hence Cis positive definite,
+which then also holds for K. /square
+Remark 6.7. Since a diagonal matrix Sissymmetric , the Sylvester equation in the continuous
+NLS case is the Lyapunov equation (6.16). Theresult formula ted in part (i) of the preceding lemma
+can thus be deduced from a more general result in [148], which we recalled in remark 6.3. /square
+Proposition 6.8. LetS= diag(s1,...,sn)with pairwise different si, andRe(si)>0in the
+continuous NLS case, |si|<1in the discrete NLS cases. Then the associated solution (6.38) of
+the (continuous, semi-discrete, respectively fully discr ete) focusing NLS equations is regular (for all
+x,t∈R).
+Proof:Using a diagonal square root Ξ1/2of the diagonal matrix Ξ, we introduce
+L= Ξ1/2K(Ξ1/2)†.
+15We actually arranged the form of the Sylvester equations in o rder to achieve this (at the price of introducing η).
+29SinceKis Hermitian andpositive definiteby theprecedinglemma, al soLis Hermitian, i.e. L†=L,
+and positive definite. Hence Lpossesses a positive definite Hermitian square root L1/2= (L1/2)†,
+and we can write
+(ΞK)∗(ΞK) = (Ξ1/2)∗L∗L(Ξ1/2)∗−1
+= (Ξ1/2L1/2)∗/parenleftig
+[(L1/2)∗L1/2][(L1/2)∗L1/2]†/parenrightig
+(Ξ1/2L1/2)∗−1.
+For the expression in (6.38) that has to be inverted, this imp lies
+I+(ΞK)∗(ΞK) = (Ξ1/2L1/2)∗/parenleftig
+I+[(L1/2)∗L1/2][(L1/2)∗L1/2]†/parenrightig
+(Ξ1/2L1/2)∗−1.(6.42)
+Since any matrix of the form I+AA†is invertible, this proves our assertion. /square
+Because of the following reason we can relax the assumptions in proposition 6.8 to the require-
+ments that the eigenvalues siof the diagonal matrix Sare pairwise different, and si+s∗
+j/ne}ationslash= 0 in
+the continuous case, sis∗
+j/ne}ationslash= 1 in the discrete cases. If we have matrix data ( S,C,U 0,V0), where
+Shas an eigenvalue with negative real part (respectively whi ch lies outside the unit circle), then
+we can apply a reflection symmetry and obtain new matrix data ( S′,U′,V′) that determine the
+same solution, and S′is given by Swhere the respective eigenvalue now has positive real part,
+respectively lies inside the unit circle. Furthermore, we c an choose the data such that U′=U0B′∗
+andV′=A′V0with invertible matrices A′,B′that commute with S′. A reparametrisation trans-
+formation then achieves the form (6.38) with matrix data ( S′,C′,U0,V0). Hence the assumption in
+the proposition that the eigenvalues of Shave positive real parts, respectively lie in the open unit
+disk, is no restriction.
+6.3 Solutions of the matrix NLS equations
+By redefinitions of KandVin the discrete NLS cases, the solutions determined by propo sition 6.1
+can be expressed as16
+q=UZ−1S∗−ηV∗, p=ǫUΞ∗K∗Z∗−1S−ηV , (6.43)
+where
+Z= Ξ∗−1−ǫK∗ΞK (6.44)
+and
+NLSηSylvester equation Ξ
+continuous 0SK+KS∗=VU e−xS−itS2
+semi-discrete 1K−SKS∗=VU Sx+1e−it(S+S−1−2I)
+fully discrete 1K−SKS∗=VUSx+1[(I+i(I−S−1))(I−i(I−S))−1]t
+This and what follows includes the case m= 1, i.e. solutions of the scalar NLS equations. For
+m>1, the results are new.
+16Ifqandpsolves the respective NLS system, then also −qandp.
+306.3.1 Rank one single solitons
+Choosingn= 1,Uis a column and Va row vector (with mcomponents). Writing
+S=s,Ξ =ξ, U=u, V=v⊺, K=κv⊺u, (6.45)
+with (m-component) column vectors u,v, we obtain17
+q=κ−1˜ξ∗
+1−ǫ|˜ξ|2s∗−ηuv†
+(v⊺u)∗, p=κ−1
+1−ǫ|˜ξ|2s−ηuv⊺
+v⊺u+p0, (6.46)
+wherep0=−s−ηκ−1uv⊺/(v⊺u) and
+NLSηκ ˜ξ
+continuous 01
+s+s∗κv⊺ue−sx−is2t
+semi-discrete 11
+1−ss∗κv⊺usx+1e−i(s+s−1−2)t
+fully discrete 11
+1−ss∗κv⊺usx+1/parenleftig
+1+i(1−s−1)
+1−i(1−s)/parenrightigt
+Of course, we have to assume Re( s)/ne}ationslash= 0, respectively |s| /ne}ationslash= 1. In the focusing case, i.e.ǫ=−1,
+these solutions are regular and describe a single soliton wi th an attached “polarisation” determined
+by the vectors uandv. The solutions can be expressed as follows,
+q=csech(α(x−ρt)−δ)ei(βx−νt−θ)uv†
+(v⊺u)∗,
+p=2c
+1+e−2(α(x−ρt)−δ)uv⊺
+v⊺u+p0, (6.47)
+where
+s=α+iβ, c=α,v⊺u=κ−1eδ+iθ, ρ= 2β, ν=β2−α2(6.48)
+in the continuous NLS case, with α,β,δ,θ ∈R. In the discrete NLS cases,
+s=e−α−iβ, c= sinh(α)eiβ,v⊺u=κ−1eα+δ+i(θ−β), (6.49)
+and
+ρ=2
+αsinh(α) sin(β), ν= 2(1−cosh(α) cos(β)) (6.50)
+in the semi-discrete case, whereas in the fully discrete cas eρandνare given in terms of α,βvia
+eαρ+iν=1+i(1−eα+iβ)
+1−i(1−e−α−iβ). (6.51)
+Since hereqandpare rank one matrices, these are the most elementary matrix s oliton solutions.
+Such a soliton has velocity ρ. Whereas a scalar (i.e. m= 1) NLS soliton carries a phase factor
+that can change via scattering with other solitons, here the phase is extended to the polarisation
+matrix. We shall see below how the latter changes in a scatter ing process.
+17Note that P=uv⊺/(v⊺u) is a projector. Furthermore, ˜P=uv†/(v⊺u)∗satisfies ˜P˜P∗=P.
+316.3.2 Asymptotics of superpositions
+The following is a preparation for the elaboration of the asy mptotics of multi-soliton solutions
+in section 6.3.3. We note that (6.25) can be written as Z=A∗
+2ˇΞ∗−1B2+A∗
+1ˇΞB1withˇΞ =
+block-diagonal(Ξ 1,Ξ∗−1
+2) and
+A1=−/parenleftbiggǫK10
+ǫK21I/parenrightbigg
+, B1=/parenleftbiggK1K12
+0−I/parenrightbigg
+, A2=/parenleftbiggI K12
+0K2/parenrightbigg
+, B2=/parenleftbiggI0
+−ǫK21−ǫK2/parenrightbigg
+(6.52)
+(whereIstands for the identity matrix of the respective size). Assu ming that K1andK2are
+invertible, this implies
+Z−1=B−1
+1ˇΞ−1/parenleftig
+I−ǫ˜K∗−1ˇΞ∗−1˜K−1ˇΞ−1/parenrightig−1
+A∗−1
+1, (6.53)
+where
+˜K=−ǫA−1
+2A1=B1B−1
+2=/parenleftbiggL1−ǫK12K−1
+2
+K−1
+2K21ǫK−1
+2/parenrightbigg
+(6.54)
+(cf. (6.32)), with the Schur complement of K2(with respect to K),
+L1=K1−K12K−1
+2K21. (6.55)
+In a limit in which Ξ−1
+1→0 and Ξ 2∼ˆΞ2(asymptotic form), we obtain
+Z−1∼B−1
+1/parenleftigg0 0
+0ˆΞ∗
+2/parenleftig
+I−ǫL∗
+2ˆΞ2L2ˆΞ∗
+2/parenrightig−1/parenrightigg
+A∗−1
+1, (6.56)
+with the Schur complement of K1(with respect to K),
+L2=K2−K21K−1
+1K12. (6.57)
+This allows us to compute the asymptotic form of q, given by (6.43), in this limit. In order to
+compute the asymptotic form of p, we write
+Ξ∗K∗Z∗−1=B−1
+1(B1Ξ∗K∗B∗−1
+1ˇΞ∗−1)/parenleftig
+I−ǫ˜K−1ˇΞ−1˜K∗−1ˇΞ∗−1/parenrightig−1
+A−1
+1. (6.58)
+Since
+B1Ξ∗K∗B∗−1
+1ˇΞ∗−1=/parenleftbiggK1+K12Ξ∗
+2K∗
+21K∗−1
+1Ξ∗−1
+1−K12Ξ∗
+2L∗
+2Ξ2
+Ξ∗
+2K∗
+21K∗−1
+1Ξ∗−1
+1 −Ξ∗
+2L∗
+2Ξ2/parenrightbigg
+, (6.59)
+we find
+Ξ∗K∗Z∗−1∼B−1
+1/parenleftigg0 0
+0ǫL−1
+2/parenleftig
+I−ǫL2ˆΞ∗
+2L∗
+2ˆΞ2/parenrightig−1/parenrightigg
+A−1
+1. (6.60)
+In the same way, in a limit in which Ξ−1
+2→0 and Ξ 1∼ˆΞ1, we obtain
+Z−1∼B−1
+2/parenleftigg
+ˆΞ∗
+1/parenleftig
+I−ǫL∗
+1ˆΞ1L1ˆΞ∗
+1/parenrightig−1
+0
+0 0/parenrightigg
+A∗−1
+2,
+Ξ∗K∗Z∗−1∼B−1
+2/parenleftigg
+ǫL−1
+1/parenleftig
+I−ǫL1ˆΞ∗
+1L∗
+1ˆΞ1/parenrightig−1
+0
+0 0/parenrightigg
+A−1
+2. (6.61)
+326.3.3 Superposition of rank one solitons
+Now we consider the focusing case, i.e.ǫ=−1. We choose S= diag(s1,...,sn) and write
+U=/parenleftbig
+u1···un/parenrightbig
+, V=
+v⊺
+1...
+v⊺
+n
+, (6.62)
+withm-component columnvectors ui,vi,i= 1,...,n, andΞ = diag( ξ1,...,ξn). Thecorresponding
+Sylvester equation is solved by
+K= (kij) where kij=
+
+v⊺
+iuj
+si+s∗
+jcontinuous NLS
+v⊺
+iuj
+1−sis∗
+j(semi- and fully) discrete NLS.(6.63)
+The NLS solution can now be expressed as
+q=n/summationdisplay
+i,j=1ξ∗
+i[(ZΞ∗)−1]ijs∗−η
+juiv†
+j, p=n/summationdisplay
+i,k,j=1(K−1)ik[(Z∗Ξ)−1]kjs−η
+juiv⊺
+j+p0,(6.64)
+with the constant term p0=−UK−1S−ηV, dropped in the following, and
+(ZΞ∗)ij=δij+n/summationdisplay
+k=1k∗
+ikkkjξkξ∗
+j. (6.65)
+For eachsjwe introduce real constants αj,βj,ρj,νjas in (6.48), (6.49), (6.50), (6.51), and
+κi=/braceleftbigg(si+s∗
+i)−1continuous NLS
+(1−sis∗
+i)−1discrete NLS. (6.66)
+Furthermore, we choose18αi>0,i= 1,...,n, and19ρ1< ρ2<···< ρn. Asymptotically, as
+t→ −∞, the corresponding incoming solitons thus appear in decrea sing numerical order along
+thex-axis. The outgoing solitons (as t→ ∞) then appear in increasing numerical order. To
+extract these asymptotic solitons from the solution in the l imitst→ ±∞, we introduce a comoving
+coordinate ¯xi=x−ρit, with respect to which the i-th soliton is stationary up to a phase factor. We
+considerthelimits t→ ±∞whilekeeping ¯ xiconstant, forsome fixedi. Then|ξj|=e−αj(x−ρjt)+δj=
+e−αj[¯xi−(ρj−ρi)t]+δj. It follows that
+lim
+t→−∞ξj= 0 if j >i, lim
+t→−∞ξ−1
+j= 0 if j <i, (6.67)
+and
+lim
+t→+∞ξ−1
+j= 0 if j >i, lim
+t→+∞ξj= 0 if j <i. (6.68)
+In order to find the corresponding asymptotic forms of qandp, we express the solution as a
+superposition (6.22) with S1= diag(s1,...,si−1) andS2= diag(si,...,sn). LetU1,U2andV1,V2
+18This is not really a restriction since for αi/negationslash= 0 it can be achieved generically by a reflection symmetry.
+19For simplicity, we exclude the case where some of the ρiare equal.
+33be the correspondingparts of UandV, respectively. Let K1,K2andK12,K21bethe corresponding
+diagonal and off-diagonal blocks of K. Using (6.56), (6.60), (6.61), with ( ˆΞ2)kl=ξiδikδil, we obtain
+q∼κ−1
+i(γ(−)
+i˜ξi)∗
+1+|γ(−)
+i˜ξi|2s∗−η
+iu(−)
+iv(−)†
+i
+(γ(−)
+iv⊺
+iui)∗, p∼κ−1
+i
+1+|γ(−)
+i˜ξi|2s−η
+iu(−)
+iv(−)⊺
+i
+γ(−)
+iv⊺
+iuiast→ −∞
+¯xi= const.(6.69)
+where we introduced
+˜ξi=kiiξi=κiv⊺
+iuiξi (6.70)
+and
+γ(−)
+i=(L2)ii
+kii,u(−)
+i= (U2−U1K−1
+1K12)i,v(−)
+i=sη
+i[(S−η
+2V2−K21K−1
+1S−η
+1V1)i]⊺.(6.71)
+The latter vectors can also be expressed as follows,
+u(−)
+i=ui−i−1/summationdisplay
+j,k=1uj(K−1
+1)jkkki=1
+det(K1)det
+k11···k1i
+.........
+ki−1,1···ki−1,i
+u1···ui
+,
+v(−)
+i=sη
+i/parenleftig
+s−η
+ivi−i−1/summationdisplay
+j,k=1kij(K−1
+1)jks−η
+kvk/parenrightig
+=sη
+i
+det(K1)det
+k11···k1,i−1s−η
+1v1
+............
+ki1···ki,i−1s−η
+ivi
+.(6.72)
+The first equation e.g. can be verified by Laplace expansion of the determinant of the big matrix on
+the right hand side with respect to the last row, and subseque nt Laplace expansion of the cofactors
+with respect to the last column. The expressions in (6.72) ar e examples of quasideterminants (see
+e.g. [150]). Note also that
+(L2)ii= (K2−K21K−1
+1K12)ii=1
+det(K1)det
+k11···k1i
+.........
+ki1···kii
+, (6.73)
+so that all constituents of the formulae in (6.69) have nice e xpressions in terms of the vectors ui
+andvi, andkjlwithj,l= 1,...,i. The expressions in (6.69) are arranged in the form of the sin gle
+soliton solution (6.46). This is more clearly seen by noting the following result.
+Proposition 6.9.
+v(−)⊺
+iu(−)
+i=γ(−)
+iv⊺
+iui. (6.74)
+Proof:In the continuous NLS case, we have v⊺
+iuj= (si+s∗
+j)kij. Hence
+v(−)⊺
+iu(−)
+i=/parenleftig
+vi−i−1/summationdisplay
+j,k=1kij(K−1
+1)jkvk/parenrightig⊺/parenleftig
+ui−i−1/summationdisplay
+l,r=1ul(K−1
+1)lrkri/parenrightig
+=v⊺
+iui−i−1/summationdisplay
+l,r=1v⊺
+iul(K−1
+1)lrkri−i−1/summationdisplay
+j,k=1v⊺
+kuikij(K−1
+1)jk+i−1/summationdisplay
+j,k,l,r=1v⊺
+kulkij(K−1
+1)jk(K−1
+1)lrkri
+= (si+s∗
+i)/parenleftig
+kii−i−1/summationdisplay
+j,k=1kij(K−1
+1)jkkki/parenrightig
+−i−1/summationdisplay
+j,k=1kij(K−1
+1)jksk/parenleftig
+kki−i−1/summationdisplay
+l,r=1kkl(K−1
+1)lrkri/parenrightig
+−i−1/summationdisplay
+l,r=1/parenleftig
+kil−i−1/summationdisplay
+j,k=1kij(K−1
+1)jkkkl/parenrightig
+s∗
+l(K−1
+1)lrkri.
+34Now we observe that the last two sums vanish as a consequence o f the identities
+kri−i−1/summationdisplay
+j,k=1krj(K−1
+1)jkkki= 0, kir−i−1/summationdisplay
+j,k=1kij(K−1
+1)jkkkr= 0 forr<i.
+Thefirstidentity is e.g. obtainedfromthefirstidentity in( 6.72) byreplacing ujbykrj,j= 1,...,i.
+Hence
+v(−)⊺
+iu(−)
+i=(L2)ii
+kiiv⊺
+iui=γ(−)
+iv⊺
+iui.
+In the discrete NLS case, using v⊺
+iuj= (1−sis∗
+j)kij, we obtain
+v(−)⊺
+iu(−)
+i= (1−sis∗
+i)(L2)ii−sii−1/summationdisplay
+j,k=1kij(K−1
+1)jks−1
+k/parenleftig
+kki−i−1/summationdisplay
+l,r=1kkl(K−1
+1)lrkri/parenrightig
+−sii−1/summationdisplay
+l,r=1/parenleftig
+kil−i−1/summationdisplay
+j,k=1kij(K−1
+1)jkkkl/parenrightig
+s∗
+l(K−1
+1)lrkri=γ(−)
+iv⊺
+iui.
+/square
+Now we can rewrite (6.69) as follows,
+q∼κ−1
+i(γ(−)
+i˜ξi)∗
+1+|γ(−)
+i˜ξi|2s∗−η
+iu(−)
+iv(−)†
+i
+(v(−)⊺
+iu(−)
+i)∗, p∼κ−1
+i
+1+|γ(−)
+i˜ξi|2s−η
+iu(−)
+iv(−)⊺
+i
+v(−)⊺
+iu(−)
+iast→ −∞
+¯xi= const.(6.75)
+In the same way we obtain
+q∼κ−1
+i(γ(+)
+i˜ξi)∗
+1+|γ(+)
+i˜ξi|2s∗−η
+iu(+)
+iv(+)†
+i
+(v(+)⊺
+iu(+)
+i)∗, p∼κ−1
+i
+1+|γ(+)
+i˜ξi|2s−η
+iu(+)
+iv(+)⊺
+i
+v(+)⊺
+iu(+)
+iast→+∞
+¯xi= const.,(6.76)
+where
+γ(+)
+i=(L1)ii
+kii,u(+)
+i= (U1−U2K−1
+2K21)i,v(+)
+i=sη
+i[(S−η
+1V1−K12K−1
+2S−η
+2V2)i]⊺.(6.77)
+The vectors can also be expressed as follows,
+u(+)
+i=1
+det(K2)det
+ui···un
+ki+1,i···ki+1,n
+.........
+kni···knn
+,
+v(+)
+i=sη
+i
+det(K2)det
+s−η
+iviki,i+1···kin
+............
+s−η
+nvnkn,i+1···knn
+, (6.78)
+and satisfy
+v(+)⊺
+iu(+)
+i=γ(+)
+iv⊺
+iui, (6.79)
+35which is proved in the same way as proposition 6.9. Furthermo re, we have
+(L1)ii= (K1−K12K−1
+2K21)ii=1
+det(K2)det
+kii···kin
+.........
+kn1···knn
+. (6.80)
+All this nicely expresses the constituents of the formulae ( 6.76) in terms of the vectors uiandvi,
+andkjlwithj,l=i,...,n.
+Clearly, ifi= 1 respectively i=n, we have
+q∼κ−1
+1˜ξ∗
+1
+1+|˜ξ1|2s∗−η
+1u1v†
+1
+(v⊺
+1u1)∗, p∼κ−1
+1
+1+|˜ξ1|2s−η
+1u1v⊺
+1
+v⊺
+1u1ast→ −∞
+¯x1= const.
+q∼κ−1
+n˜ξ∗
+n
+1+|˜ξn|2s−η
+iunv†
+n
+(v⊺
+nun)∗, p∼κ−1
+n
+1+|˜ξn|2s−η
+nunv⊺
+n
+v⊺
+nunast→+∞
+¯xn= const.(6.81)
+Bycomparisonoftheasymptoticexpressions(6.75) and(6.7 6), thesolitonsexperienceaposition
+shift and a change of polarisation through the scattering pr ocess.
+6.3.4 2-soliton solutions
+Ifn= 2, we have
+S=/parenleftbiggs10
+0s2/parenrightbigg
+, U=/parenleftbig
+u1u2/parenrightbig
+, V=/parenleftbiggv⊺
+1
+v⊺
+2/parenrightbigg
+. (6.82)
+Using a reparametrisation transformation (with diagonal t ransformation matrices) and a transfor-
+mation (5.20), we can achieve that
+(v⊺
+iuj)∗=v⊺
+jui, (6.83)
+so thatKis Hermitian. But now we have to admit in the expression for ξja factorcj∈C. The
+matrixK, given by (6.63) in terms of the eigenvalues sjand the vectors uj,vj,j= 1,2, has the
+form
+K=/parenleftbiggk1k
+k∗k2/parenrightbigg
+(6.84)
+(where the kiare real), and we obtain the following explicit solution,
+q=1
+D/bracketleftig/parenleftbig
+1+k2
+2|ξ2|2+k2ξ1ξ∗
+2/parenrightbig
+ξ∗
+1s∗−η
+1u1v†
+1+/parenleftbig
+1+k2
+1|ξ1|2+k∗2ξ∗
+1ξ2/parenrightbig
+ξ∗
+2s∗−η
+2u2v†
+2
+−(kk1ξ1+k∗k2ξ2)ξ∗
+1ξ∗
+2(s∗−η
+2u1v†
+2+s∗−η
+1u2v†
+1)/bracketrightig
+,
+p=−1
+D/bracketleftig/parenleftbig
+k1+det(K)k2|ξ2|2/parenrightbig
+k1|ξ1|2s−η
+1u1v⊺
+1+/parenleftbig
+k2+det(K)k1|ξ1|2/parenrightbig
+k2|ξ2|2s−η
+2u2v⊺
+2
++(k∗−kdet(K)ξ1ξ∗
+2)ξ∗
+1ξ2s−η
+2u1v⊺
+2+(k−k∗det(K)ξ∗
+1ξ2)ξ1ξ∗
+2s−η
+1u2v⊺
+1/bracketrightig
+,(6.85)
+where
+D=/vextendsingle/vextendsingle/vextendsingle1−k
+k∗det(K)ξ1ξ∗
+2/vextendsingle/vextendsingle/vextendsingle2
++/vextendsingle/vextendsingle/vextendsinglek1ξ1+k∗
+kk2ξ2/vextendsingle/vextendsingle/vextendsingle2
+. (6.86)
+36Remark 6.10. IfS=sInwithn≤m, we have a “rank nsoliton”. It follows from proposition 6.4
+that we can choose all m-component vectors composing U, respectively V, linearly independent.
+In particular, this means that there are at most m2different such solitons. If n=m, then with
+a reparametrisation transformation we can achieve that U=V=Im, while replacing Ξ by CΞ
+with a constant m×mmatrixC. Imposing the condition (6.83) on the vectors ui,vj, so thatK
+is Hermitian, and choosing Csymmetric, the resulting NLS solution is regular. In the con tinuous
+NLS case,qis then symmetric. Superposing such rank msolitons (with symmetric Ci), we obtain
+again continuous m×mmatrix NLS solutions with symmetric q. This may be of interest for the
+aforementioned spin 1 Bose-Einstein condensate model. /square
+6.3.5 Regularity condition for matrix solitons in the focusing NLS case
+We address the question whether a superposition (in the abov e sense of “superposing” the cor-
+responding matrix data) of regular solutions again leads to a regular solution. Hence we should
+check the determinant of (6.25) for the occurence of zeros. A s an intermediate step, we compute
+the Schur complement of the first block-diagonal entry of Z, which is
+D:=Z2+K∗
+21Ξ1K12−(K∗
+2Ξ2K21+K∗
+21Ξ1K1)(Z1+K∗
+12Ξ2K21)−1(K∗
+1Ξ1K12+K∗
+12Ξ2K2).(6.87)
+Here we assume that, for the superposed data, a matrix Kas in (6.22) exists such that (6.23),
+respectively (6.24) holds. A sufficient condition is (6.13). We further assume that Z1andZ2are
+(for allx,t) invertible, so that the constituents of the superposition are regular, and also that
+Ξ−1
+2+K21Z−1
+1K∗
+12is invertible. By the matrix inversion lemma (see e.g. [137] , Corollary 2.8.8), the
+latter condition guarantees the existence of
+(Z1+K∗
+12Ξ2K21)−1=Z−1
+1−Z−1
+1K∗
+12(Ξ−1
+2+K21Z−1
+1K∗
+12)−1K21Z−1
+1. (6.88)
+Using this formula in the preceding equation, and noting tha t
+(Ξ−1
+2+K21Z−1
+1K∗
+12)−1K21Z−1
+1K∗
+12Ξ2= Ξ2−(Ξ−1
+2+K21Z−1
+1K∗
+12)−1
+= Ξ2K21Z−1
+1K∗
+12(Ξ−1
+2+K21Z−1
+1K∗
+12)−1,(6.89)
+we obtain
+D=Z2+K∗
+21Ξ1K12−K∗
+21Ξ1K1Z−1
+1K∗
+1Ξ1K12
++K∗
+21Ξ1K1Z−1
+1K∗
+12(Ξ−1
+2+K21Z−1
+1K∗
+12)−1K21Z−1
+1K∗
+1Ξ1K12
+−K∗
+2(Ξ−1
+2+K∗
+21Z−1
+1K∗
+12)−1K21Z−1
+1K∗
+1Ξ1K12
+−K∗
+21Ξ1K1Z−1
+1K∗
+12(Ξ−1
+2+K21Z−1
+1K∗
+12)−1K2
+−K∗
+2Ξ2K2+K∗
+2(Ξ−1
+2+K21Z−1
+1K∗
+12)−1K2. (6.90)
+Using the identity Ξ 1K1Z−1
+1K∗
+1=I−Z∗−1
+1Ξ−1
+1in the third summand, this can rewritten as follows,
+D= (K∗
+21Ξ1K1Z−1
+1K∗
+12−K∗
+2)(Ξ−1
+2+K21Z−1
+1K∗
+12)−1(K21Z−1
+1K∗
+1Ξ1K12−K2)
++Ξ∗−1
+2+K∗
+21Z∗−1
+1K12. (6.91)
+With the help of the identity Ξ 1K1Z−1
+1= (Z−1
+1K∗
+1Ξ1)∗, we obtain
+D= (K21Z−1
+1K∗
+1Ξ1K12−K2)∗(Ξ−1
+2+K21Z−1
+1K∗
+12)−1(K21Z−1
+1K∗
+1Ξ1K12−K2)
++(Ξ−1
+2+K21Z−1
+1K∗
+12)∗. (6.92)
+37If the data ( S2,U2,V2) belong to a 1-soliton solution, writing S2=s, Ξ2=ξ,U2=u,V2=v⊺,
+K2=k,K12=k1, andK21=k⊺
+2, we have
+Z=/parenleftbiggZ1+ξk∗
+1k⊺
+2K∗
+1Ξ1k1+ξkk∗
+1
+ξk∗k⊺
+2+k⊺∗
+2Ξ1K1Z2+k⊺∗
+2Ξ1k1/parenrightbigg
+, (6.93)
+whereZ2=ξ∗−1+|k|2ξ. Using determinant identities (see e.g. [137], Fact 2.14.2 ), we find
+det(Z) = det(Z1+ξk∗
+1k⊺
+2)D= det(Z1)(1+ξk⊺
+2Z−1
+1k∗
+1)D
+=ξdet(Z1)(ξ−1+k⊺
+2Z−1
+1k∗
+1)/parenleftig
+(ξ−1+k⊺
+2Z−1
+1k∗
+1)∗
++(k⊺
+2Z−1
+1K∗
+1Ξ1k1−k)∗(ξ−1+k⊺
+2Z−1
+1k∗
+1)−1(k⊺
+2Z−1
+1K∗
+1Ξ1k1−k)/parenrightig
+=ξdet(Z1)/parenleftig
+|ξ−1+k⊺
+2Z−1
+1k∗
+1|2+|k⊺
+2Z−1
+1K∗
+1Ξ1k1−k|2/parenrightig
+. (6.94)
+This can only vanish if k=k⊺
+2Z−1
+1K∗
+1Ξ1k1and simultaneously ξ−1=−k⊺
+2Z−1
+1k∗
+1at some value of
+xandt. Generically it is not possible to satisfy both equations at any value of x,t. (Note that
+these equations consist of fourreal equations.) In particular, this shows that the matrix n-soliton
+solutions are generically regular.
+6.3.6 Cases in which the Sylvester equation has not a unique solution
+Several of our results so far in this section rest upon the ass umption that the condition (6.13) is
+satisfied, in which case the Sylvester equation in propositi on 6.1 possesses a unique solution. Now
+we address those cases in which this condition is violated, c onfining ourselves to the continuous
+matrix NLS equation (6.3). The simplest such cases occur if t he matrixSconsists of a single
+Jordan block with a purely imaginary eigenvalue. By applica tion of a Galilean transformation (see
+remark 6.2), this case can be reduced to that where the eigenv alue is zero.
+Example 6.11. (1) Letn= 1 andS= 0. Writing U=uandV=v⊺withm-component vectors
+u,v, the Sylvester equation requires v⊺u= 0 and leaves K=kcompletely arbitrary. If m= 1 (the
+scalar NLS case), the corresponding NLS solution vanishes i dentically. But if m >1 (the matrix
+NLS case), this is not so20and we obtain the constant rank one matrix NLS solution
+q=1
+1−ǫ|k|2uv†wherev⊺u= 0. (6.95)
+A Galilean transformation generalises it to
+q=1
+1−ǫ|k|2eiβ(x−βt)uv†, (6.96)
+withβ∈R. Thisiswhatwewouldhaveobtained hadwestartedabovewith S= iβ, seeremark6.2.
+As a consequence of v⊺u= 0, we have qq∗= 0, so that qsolves the linear and the nonlinear parts
+of the NLS equation separately.
+(2) Letn= 2 andSa Jordan block with zero eigenvalue, hence
+S=/parenleftbigg0 1
+0 0/parenrightbigg
+, U=/parenleftbig
+a1a2/parenrightbig
+, V=/parenleftbiggb1
+b2/parenrightbigg
+. (6.97)
+20This is in contrast to the Hermitian reduction case treated i n section 7, where the corresponding Sylvester
+equation also leads to the trivial solution if m >1. See remark 7.2.
+38Figure 1: The norm /bardblq/bardblof the 2×2 matrix focusing NLS solution, determined by the data (6.99 ),
+att=−4,−2,0,2,4. This is a rational “matrix breather”.
+The Sylvester equation only has a solution if b⊺
+2a1= 0 and b⊺
+1a1=b⊺
+2a2. Let us more concretely
+considerthecase m= 2. Thesolution KoftheSylvesterequationdependsontwoarbitrarycomplex
+constants (expressing the non-uniqueness of K). The resulting components of qare quotients of
+a linear polynomial in xby a quadratic polynomial, hence qvanishes as |x| → ∞. Applying a
+Galilean transformation (see remark 6.2), these solutions becomet-dependent.21A special regular
+member from this family is
+q=eiβ(x−βt)
+1+(x−2βt)2/parenleftbigg1−(x−2βt)
+x−2βt 1/parenrightbigg
+β∈R. (6.98)
+(3) Forn>2, withSstill consisting of aJordan block with eigenvalue zero, one obtainst-dependent
+rational solutions. This includes regular solutions with a breather -like behaviour. For n= 4, a
+special solution of the 2 ×2 matrix focusing NLS equation is determined by
+S=
+0 1 0 0
+0 0 1 0
+0 0 0 1
+0 0 0 0
+, U=/parenleftbigg0 0 1 −1
+1 0 0 0/parenrightbigg
+, V=
+0−1
+0−1
+0 0
+1 0
+, K=
+0 0 0 0
+−1 0 0 0
+−1 1 0 0
+0 1−1 0
+.
+(6.99)
+This rational solution is regular, vanishes as |x| → ∞, and shows a breather-like behaviour, see
+Fig. 1. /square
+Of course, we can superpose any number of solutions of the kin d described in the last example,
+see the next example, and also compose such data with (e.g. so litonic) data satisfying (6.13), as
+in the next but one example. In doing so, we have to take the par ametersβiinto account, since
+a Galilean transformation applied to the composite solutio n can only eliminate or create one of
+these parameters. In the special case where S= diag(iβ1,...,iβn), with real βi/ne}ationslash= 0, the resulting
+solutions are periodic in xandt, since they are rational expressions in trigonometric func tions.
+Example 6.12. The following superposition of data of the kind considered i n the preceding ex-
+ample determines a regular solution that vanishes as |x| → ∞, see Fig. 2.
+S=
+iβ1 0 0
+0 iβ0 0
+0 0 0 1
+0 0 0 0
+, U=/parenleftbigg1 0 1 0
+0 1 0 1/parenrightbigg
+, V=
+1 0
+0 1
+1−1
+0 1
+, K=
+0 0 −i
+21
+2
+1 0 0 −i
+2i
+21−i
+20 0
+0i
+21−1
+.
+/square
+21The absolute values of the components are still rational fun ctions, hence these solutions resemble the “rational”
+soliton solutions known in the scalar NLS case [151].
+39Figure 2: Thereal part of q11for the matrix NLS solution qdetermined by the data in example 6.12
+att=−4,...,4.
+Example 6.13. LetS= diag(s,0),s/ne}ationslash∈iR, andU,Vas in (6.97) with m= 2. The Sylvester
+equation then admits a solution Kiffb⊺
+2a2= 0. Writing a⊺
+i= (ai1,ai2),b⊺
+i= (bi1,bi2), we solve
+this constraint by setting b21=ca22andb22=−ca21withc∈C. ThenKtakes the form
+K=/parenleftbigga11b11+a12b12
+s+s∗a21b11+a22b12
+s
+ca11a22−a12a21
+s∗ k/parenrightbigg
+, (6.100)
+wherek∈Cis another arbitrary constant. The resulting solution fami ly of the 2 ×2 matrix
+focusing NLS equation depends on nine complex parameters an d one real. Fig. 3 shows a regular
+solution from this family. /square
+Recall that any NLS solution correspondingto matrix data ( S,U,V) consisting of several blocks
+can be regarded as a (nonlinear) superposition of solutions corresponding to these blocks, say
+(S1,U1,V1) and (S2,U2,V2). Ifσ(S1)∩σ(−S∗
+2) =∅in the continuous NLS case (respectively
+σ(S∗
+1)∩σ(S−1
+2) =∅in the discrete cases), the solution is completely determin ed by the “constituent
+solutions”. Soliton solutions are the examples par excelle nce. Ifσ(S1)∩σ(−S∗
+2)/ne}ationslash=∅(respectively
+σ(S∗
+1)∩σ(S−1
+2)/ne}ationslash=∅), there is no unique solution for the off-diagonal blocks of th e matrixKsolving
+the Sylvester equation for the superposition. The parts the n no longer determine the composite
+solution completely, since additional freedom enters thro ugh arbitrary parameters in K. But if
+the eigenvalue of S1has a non-zero real part in the continuous NLS case, respecti vely a modulus
+different from 1 in the discrete NLS cases, then we can apply a re flection symmetry to S2(provided
+that the associated matrix K2is invertible). This leads to equivalent data involving Jor dan blocks
+S1andS′
+2, where now σ(S1)∩σ(−S′∗
+2) =∅. This means that such a NLS solution can be realised
+in different ways as a superposition, i.e. with different consti tuents.
+7 Hermitian conjugation reduction of the continuous NLS sys tem
+By decomposition, the Hermitian conjugation reduction con dition (3.33) for the continuous matrix
+NLS system becomes
+¯q=ǫq†, p=p†,¯p= ¯p†, (7.1)
+40Figure 3: A sequence of snapshots of a solution from the famil y in example 6.13. The plots
+show the absolute values of the components of the 2 ×2 matrix focusing NLS solution qat times
+t=−.2,0,.2,.4,.6. Here we chose the parameter values s= 2(1 +i), c=k=a11=a22= 1 and
+a21=−1,b12= i,a12=b11= 0.
+whereǫ=±1. It reduces the NLS system to the m1×m2matrixNLS equation (see e.g. [3])
+iqt+qxx−2ǫqq†q= 0, (7.2)
+supplemented by px=−ǫqq†.
+Ifm2= 1 andm1>1, (7.2) is the vector NLS equation (orcoupled NLS equations )
+i/vector qt+/vector qxx−2ǫ|/vector q|2/vector q= 0, (7.3)
+first treated in [77].
+Ifm1=m2and ifqissymmetric orskew-symmetric , i.e.q⊺=±q, then (7.2) coincides with
+(6.3). This includes the special case of a spin-1 Bose-Einst ein condensate mentioned earlier. For
+the skew-symmetric case, see e.g. [152].
+Proposition 7.1. LetS,V,¯Vbe constant matrices of size n×n,n×m1andn×m2, respectively,
+andK,¯Kconstant Hermitian solutions of
+SK+KS†=VV†, S†¯K+¯KS=¯V¯V†. (7.4)
+Then
+q=V†/parenleftig
+Ξ†−1−ǫ¯KΞK/parenrightig−1¯VwhereΞ =e−xS−itS2(7.5)
+solves them1×m2matrix NLS equation (7.2). /square
+Proof:Settingn1=n2, renamed to n, we recall from section 4.4 the reduction conditions to be
+imposed on the matrices entering the solution formulae. Wit h the block structure expressed in
+(4.23), (4.48) and (5.9), the statement then follows direct ly from proposition 5.1. /square
+41Theclass of matrix NLS solutions provided by proposition 7. 1 has also beenobtained in [98,153]
+via inverse scattering, including conditions under which t he solutions are everywhere regular and
+exponentially localised (see also [143]). Multiple solito n solutions of the matrix (including vector)
+NLS equation appeared previously also in [3,77,81,82,87–9 4,117], for example.
+IfKand¯Ksolve the Lyapunov equations (7.4), then also K†and¯K†. As a consequence, there
+are then also Hermitian solutions of these equations. We fur ther note that the equations (7.4) have
+unique solutions if
+σ(S)∩σ(−S†) =∅, (7.6)
+in which case Kand¯Kare then necessarily Hermitian.
+Remark 7.2. This concerns cases in which (7.6) does not hold. If S= 0, then (7.4) implies
+V=¯V= 0, leading to q= 0. IfSis anr×rJordan block with eigenvalue zero, i.e. Sij=δi,j−1, a
+more involved inspection of (7.4) shows that Kis left-upper-triangular, ¯Kright-lower-triangular,
+V†= (v1,...,vk,0,...,0) and¯V†= (0,...,0,¯vl,...,¯vr), wherek=r/2,l=k+ 1 ifris even,
+andk= (r−1)/2,l=k+2 ifris odd. (7.5) again leads to q= 0. Corresponding solutions of the
+complex conjugation invariant matrix NLS equation, obtain ed in section 6.3.6, have therefore no
+analogue in the Hermitian invariant NLS case. /square
+Remark 7.3. The first result noted in remark 6.3 for symmetric Scan also be obtained from
+proposition 7.1 by setting ¯V=±V∗and¯K=K⊺. /square
+Example 7.4. Letm1=m2= 1, hence we consider solutions of the scalarNLS equation. Let
+S= diag(s1,...,sn) with pairwise different eigenvalues si. According to lemma 6.5, without
+restriction of generality we can set
+V=A
+1
+...
+1
+,¯V=¯A
+1
+...
+1
+, (7.7)
+with matrices A,¯Asatisfying [A,S] = 0 = [ ¯A,S†]. Using an argument similar to that in sec-
+tion6.1.1,Aand¯Acanbechoseninvertible. Byapplication ofareparametrisa tion transformation22
+the solutions can be expressed as
+q=/parenleftbig
+1···1/parenrightbig/parenleftig
+˜Ξ†−1−ǫ˜Ξ†¯K˜ΞK/parenrightig−1
+1
+...
+1
+, (7.8)
+where
+˜Ξ =Ce−xS−itS2withC=¯A†A, (7.9)
+andK,¯Khave to solve
+SK+KS†=
+1···1
+.........
+1···1
+=S†¯K+¯KS (7.10)
+SinceSissymmetric (i.e.S⊺=S) andK=¯K⊺Hermitian, the solutions coincide with those ob-
+tained via the complex conjugation reduction in section 6, y ielding the well-known soliton solutions
+of the scalar focusing NLS equation. /square
+22The Hermitian conjugation reduction requires B=¯A†and¯B=A†in (5.19).
+42Remark 7.5. The Hermitian conjugation reduction does notwork for the discrete matrix NLS
+systems, except when m1=m2andqq†=q†q=|χ|2Iwith a scalar χ[3,92,118,123,124,130]
+(see also [119,120]). One is then led to the semi-discrete vector NLS equation (or coupled NLS
+equations)
+i˙/vector q+/vector q+−2/vector q+/vector q−−ǫ|/vector q|2(/vector q++/vector q−) = 0, (7.11)
+where the vector /vector qis obtained via expansion of the matrix qin terms of a basis of an appropriate
+Clifford algebra. A corresponding analysis in our framework w ill be postponed to a separate work.
+In appendix D we present a non-local modification of the Hermitian conjugation reduction which
+does work for the semi-discrete NLS system. /square
+7.1 Solitons of the focusing matrix NLS equation
+7.1.1 Rank one single solitons
+Settingn= 1, proposition 7.1 determines rank one single soliton solu tions of the focusing m1×m2
+matrix NLS equation. Writing V=v†and¯V=¯v†, with a reparametrisation we can achieve that
+them1-component columnvector vandthem2-componentcolumnvector ¯vsatisfy/bardblv/bardbl2:=v†v= 1
+and/bardbl¯v/bardbl2= 1, at the price of having to replace Ξ in (7.5) by
+ξ=ce−xs−its2= 2α˜ξwhere˜ξ=e−i(βx+(α2−β2)t)e−α(x−2βt)+δ, (7.12)
+withc>0 ands=α+iβwithα,β∈R,α>0.23Furthermore, we wrote c= 2αeδwithδ∈R.
+The solutions of the two equations (7.4) are then given by
+K=¯K=κwhereκ=1
+s+s∗=1
+2α, (7.13)
+and the matrix NLS solution (7.5), in the focusing case ǫ=−1, takes the form
+q=κ−1˜ξ∗
+1+|˜ξ|2v¯v†=αsech(α(x−2βt)+δ)ei(βx+(α2−β2)t)v¯v†. (7.14)
+This describes a soliton with amplitude given by α, with velocity 2 βand polarisation matrix v¯v†.
+7.1.2 Superposing solutions
+Given two data sets ( Si,Vi,¯Vi),i= 1,2, that determine solutions according to proposition 7.1,
+with corresponding Hermitian solutions Ki,¯Kiof the corresponding Sylvester equations, we can
+superpose them,
+S=/parenleftbiggS10
+0S2/parenrightbigg
+, V=/parenleftbiggV1
+V2/parenrightbigg
+,¯V=/parenleftbigg¯V1
+¯V2/parenrightbigg
+, K=/parenleftbiggK1K12
+K21K2/parenrightbigg
+,¯K=/parenleftbigg¯K1¯K12
+¯K21¯K2/parenrightbigg
+,(7.15)
+whereK12,K21and¯K12,¯K21have to solve
+S1K12+K12S†
+2=V1V†
+2, S2K21+K21S†
+1=V2V†
+1,
+S†
+1¯K12+¯K12S2=¯V1¯V†
+2, S†
+2¯K21+¯K21S1=¯V2¯V†
+1. (7.16)
+23α/negationslash= 0 is required in order to have a solution of the Sylvester equ ations (7.4). α >0 can then always be achieved
+by a reflection symmetry.
+43This system reduces to only two equations by setting
+K21=K†
+12,¯K21=¯K†
+12. (7.17)
+We have
+Z= Ξ†−1−ǫ¯KΞK=/parenleftbiggZ1−ǫ¯K12Ξ2K21−ǫ¯K1Ξ1K12−ǫ¯K12Ξ2K2
+−ǫ¯K2Ξ2K21−ǫ¯K21Ξ1K1Z2−ǫ¯K21Ξ1K12/parenrightbigg
+,(7.18)
+whereZi= Ξ†−1
+i−ǫ¯KiΞiKi.
+The following paves the way towards the asymptotic structur eof multi-soliton solutions, worked
+out in section 7.1.3, in analogy to the analysis in the case of the complex conjugation reduction in
+section 6.3. First we note that Z=¯AˇΞ†−1A+¯BˇΞBwithˇΞ = block-diag(Ξ 1,Ξ†−1
+2) and
+¯A=/parenleftbiggI¯K12
+0¯K2/parenrightbigg
+, A=/parenleftbiggI0
+−ǫK21−ǫK2/parenrightbigg
+,¯B=−/parenleftbiggǫ¯K10
+ǫ¯K21I/parenrightbigg
+, B=/parenleftbiggK1K12
+0−I/parenrightbigg
+.(7.19)
+Assuming that K1and¯K1are invertible, in a limit in which Ξ−1
+1→0 and Ξ 2∼ˆΞ2, we obtain
+Z−1∼B−1/parenleftigg0 0
+0ˆΞ†
+2/parenleftig
+I−ǫ¯L2ˆΞ2L2ˆΞ†
+2/parenrightig−1/parenrightigg
+¯B−1, (7.20)
+with the Schur complements of K1(with respect to K) and¯K1(with respect to ¯K),
+L2=K2−K21K−1
+1K12,¯L2=¯K2−¯K21¯K−1
+1¯K12, (7.21)
+which are also Hermitian.
+7.1.3 Superposition of rank one solitons
+A solution of the m1×m2matrix NLS equation composed of nrank one solitons is obtained from
+proposition 7.1 by choosing S= diag(s1,...,sn) withsi=αi+iβi. We assume αi>0 (which can
+be achieved generically by a reflection symmetry, see also [1 43]). Let us write
+V=
+v†
+1...
+v†
+n
+,¯V=
+¯v†
+1...
+¯v†
+n
+, (7.22)
+whereviand¯viarem1- respectively m2-component column vectors. Conditions (7.4) are then
+solved by
+K= (kij),¯K= (¯kij) where kij=v†
+ivj
+si+s∗
+j,¯kij=¯v†
+i¯vj
+s∗
+i+sj. (7.23)
+Kand¯Kare Hermitian. Using a reparametrisation transformation, we can achieve that the vectors
+vi,¯viare normalised, i.e.
+/bardblvi/bardbl= 1 =/bardbl¯vi/bardbl. (7.24)
+Hencekii=¯kii= 1/(2αi). Now Ξ in (7.5) has to be replaced by Ξ = diag( ξ1,...,ξn) with
+ξi=cie−xsi−is2
+it= 2αi˜ξi,˜ξi=e−i(βix+(α2
+i−β2
+i)t)e−αi(x−2βit)+δi, (7.25)
+44where we set ci= 2αieδi.
+Next we choose β1< β2<···< βn.24Ast→ −∞the solitons are then arranged along
+thex-axis in decreasing numerical order (recall that the veloci ty of thei-th soliton, when isolated
+from its partners, is 2 βi). After the interaction, as t→+∞, the order is reversed. In terms of
+thei-th comoving coordinate ¯ xi=x−2βit, we have |ξj|= 2αjexp(−αj¯xi+ 2(βj−βi)t] +δj).
+Hence, ast→ −∞(keeping ¯xiconstant), we have ξj→0 forj > iandξ−1
+j→0 forj < i.
+Furthermore, as t→ ∞, we haveξ−1
+j→0 forj > iandξj→0 forj < i. In order to find
+the corresponding limit of q, forfixedi, we express the solution as a superposition (7.15) with
+S1= diag(s1,...,si−1) andS2= diag(si,...,sn). LetV1,V2and¯V1,¯V2be the corresponding parts
+ofVand¯V, respectively. Let K1,K2andK12,K21be the corresponding diagonal and off-diagonal
+blocks ofK, and correspondingly for ¯K. Then we can use the results of section 7.1.2, in particular
+(7.20) with ( ˆΞ2)kl=ξiδikδil, in order to work out the limits of qgiven by (7.5) as t→ −∞,
+respectively t→+∞. To formulate the results, we introduce new vectors via
+η(−)
+iv(−)
+i= (V†
+2−V†
+1K−1
+1K12)i=1
+det(K1)det
+k11···k1i
+.........
+ki−1,1···ki−1,i
+v1···vi
+,
+¯η(−)
+i¯v(−)
+i= (¯V2−¯K21¯K−1
+1¯V1)†
+i=1
+det(¯K1)∗det
+¯k∗
+11···¯k∗
+1,i−1¯v1
+............
+¯k∗
+i,1···¯k∗
+i,i−1¯vi
+,(7.26)
+and
+η(+)
+iv(+)
+i= (V†
+1−K−1
+2K21V†
+2)i=1
+det(K2)det
+vi···vn
+ki+1,1···ki+1,n
+.........
+kn1···knn
+,
+¯η(+)
+i¯v(+)
+i= (¯V1−¯K12¯K−1
+2¯V2)†
+i=1
+det(¯K2)∗det
+¯vi¯k∗
+i,i+1···¯k∗
+in............
+¯vn¯k∗
+n,i+1···¯k∗
+nn
+,(7.27)
+where the constants η(−)
+i,¯η(−)
+i,η(+)
+i,¯η(+)
+iare chosen such that the following normalisation condition s
+hold,
+/bardblv(−)
+i/bardbl=/bardbl¯v(−)
+i/bardbl=/bardblv(+)
+i/bardbl=/bardbl¯v(+)
+i/bardbl= 1. (7.28)
+Proposition 7.6.
+|η(−)
+i|2=k−1
+ii(L2)ii,|η(+)
+i|2=k−1
+ii(L1)ii,
+|¯η(−)
+i|2=k−1
+ii(¯L2)ii,|¯η(+)
+i|2=k−1
+ii(¯L1)ii, (7.29)
+with the Schur complements L1=K1−K12K−1
+2K21,¯L1=¯K1−¯K12¯K−1
+2¯K21, andL2,¯L2given by
+(7.21).25
+24Here we leave aside the cases where some of the βicoincide.
+25Recall that Ki,¯Kiare Hermitian and (7.17) holds, hence Li,¯Liare Hermitian and the diagonal entries are
+therefore real.
+45Proof:Based on (7.23) and (7.24), the proof is similar to that of pro position 6.9. /square
+In particular, since we chose αi>0 and thuskii>0, this implies ( La)ii>0,(¯La)ii>0,a= 1,2.
+Let
+γ(−)
+i=k−1
+ii/radicalig
+(L2)ii(¯L2)ii, γ(+)
+i=k−1
+ii/radicalig
+(L1)ii(¯L1)ii. (7.30)
+Then the following asymptotic relations hold,26
+q∼k−1
+ii(γ(−)
+i˜ξi)∗
+1+|γ(−)
+i˜ξi|2v(−)
+i¯v(−)†
+iast→ −∞,¯xi= const., (7.31)
+q∼k−1
+ii(γ(+)
+i˜ξi)∗
+1+|γ(+)
+i˜ξi|2v(+)
+i¯v(+)†
+iast→+∞,¯xi= const.. (7.32)
+These asymptotic expressions determine the changes in rela tive positions and polarisations of the
+solitons due to the collision. For the vector NLS equation, s uch results have already been obtained
+in [3,91,93] in a different way.
+7.1.4 2-soliton solutions
+Forn= 2 we can write
+K=/parenleftbiggκ1k
+k∗κ2/parenrightbigg
+,¯K=/parenleftbiggκ1¯k∗
+¯k κ2/parenrightbigg
+, (7.33)
+where
+κi=1
+si+s∗
+i=1
+2αi, k=v†
+1v2
+s1+s∗
+2,¯k=¯v†
+2¯v1
+s1+s∗
+2. (7.34)
+The corresponding m1×m2matrix focusing NLS solution is then explicitly given by
+q=1
+D/bracketleftig
+(ξ∗
+1+κ2
+2|ξ2|2ξ∗
+1+k¯k|ξ1|2ξ∗
+2)v1¯v†
+1+(ξ∗
+2+κ2
+1|ξ1|2ξ∗
+2+k∗¯k∗|ξ2|2ξ∗
+1)v2¯v†
+2
+−(kκ1|ξ1|2ξ∗
+2+¯k∗κ2|ξ2|2ξ∗
+1)v1¯v†
+2−/parenleftbig¯kκ1|ξ1|2ξ∗
+2+k∗κ2|ξ2|2ξ∗
+1/parenrightbig
+v2¯v†
+1/bracketrightig
+, (7.35)
+where
+D=/parenleftig
+1+κ1
+κ2(κ1κ2−|k|2)|ξ1|2/parenrightig/parenleftig
+1+κ2
+κ1(κ1κ2−|¯k|2)|ξ2|2/parenrightig
++1
+κ1κ2/vextendsingle/vextendsingleκ1kξ1+κ2¯k∗ξ2/vextendsingle/vextendsingle2.(7.36)
+As aconsequenceof thedefinitions (7.34) and αi>0, we haveκ1κ2−|k|2≥0andκ1κ2−|¯k|2≥0, so
+that the solution is regular. If the vectors v1andv2are orthogonal (with respect to the Hermitian
+inner product), and also ¯v1and¯v2, thenKand¯Kare diagonal and the superposition is simply
+the sumq=q1+q2of the single soliton solutions.
+26Clearly, if i= 1 respectively i=n, keeping ¯ x1respectively ¯ xnconstant, then we have
+q∼k−1
+11˜ξ∗
+1
+1+|˜ξ1|2v1¯v1ast→ −∞, q∼k−1
+nn˜ξ∗
+n
+1+|˜ξn|2vn¯v†
+nast→+∞.
+46For the above solution we have the following asymptotic rela tions,
+q∼κ−1
+1˜ξ∗
+1
+1+|˜ξ1|2v1¯v†
+1+κ−1
+2γ(−)
+2˜ξ∗
+2
+1+|γ(−)
+2˜ξ2|2v(−)
+2¯v(−)†
+2ast→ −∞, (7.37)
+q∼κ−1
+1γ(+)
+1˜ξ∗
+1
+1+|γ(+)
+1˜ξ1|2v(+)
+1¯v(+)†
+1+κ−1
+2˜ξ∗
+2
+1+|˜ξ2|2v2¯v†
+2ast→+∞, (7.38)
+where˜ξi=κiξi,γ(−)
+2=γ(+)
+1=/radicalbig
+det(K)det(¯K)/(κ1κ2), and
+v(−)
+2=/radicalbiggκ1κ2
+det(K)(v2−k
+κ1v1),¯v(−)
+2=/radicalbiggκ1κ2
+det(¯K)(¯v2−¯k
+κ1¯v1),
+v(+)
+1=/radicalbiggκ1κ2
+det(K)(v1−k∗
+κ2v2),¯v(+)
+2=/radicalbiggκ1κ2
+det(¯K)(¯v1−¯k∗
+κ2¯v2). (7.39)
+8 Conclusions
+An appealing aspect of the bidifferential calculus approach i s the simplicity of the basic structure
+one starts with, as compared with the typical complexity of a Lax pair. Our work demonstrates
+this in case of NLS systems and their discretisations (see al so appendix E). The step from the
+continuum NLS bidifferential calculus to its discretisation s is quite evident and the general solution
+generating method of section 2 is easily adapted. In this wor k we concentrated on the application
+of this method, which seems to be the most direct and simplest method to generate a large class
+of relevant exact solutions. In [1] we have shown how other me thods, like B¨ acklund and Dar-
+boux transformations, universally arise in the bidifferenti al calculus framework. The basic general
+result formulated in theorem 2.3 may actually be regarded as a shortcut to the binary Darboux
+transformation method [1].
+In this work we presented infinite families of exact solution s of matrix continuous, semi- and
+fully discrete NLS systems and some of their reductions. The solutions are parametrised by con-
+stant matrices, and the only task left is to solve a Sylvester equation (which appeared as a “rank
+condition” in the literature). In the focusing case, they in clude the soliton solutions. But there
+are more and a complete understanding of the relations betwe en matrix data and the behaviour of
+solutions has not yet been reached. But the expression of sol utions in terms of matrix data turned
+out to be very convenient in this respect. The regularity pro ofs and the asymptotic analysis for
+some classes of solutions in sections 6 and 7 provide good exa mples.
+Besides a comparatively short treatment of the previously ( with different methods) studied
+matrix continuous NLS equation with a Hermitian conjugatio n invariance, we considered in great
+detail another matrix NLS equation with a complex conjugati on invariance. They coincide in
+the scalar case. But in the squarematrix case, the latter has discrete counterparts, whereas
+the Hermitian conjugation invariant matrix NLS equation se ems to admit a discretisation only if
+additional constraints aresatisfied [3,118,123,124,130] (but see also appendixD for a corresponding
+non-local version). We therefore concentrated on an explor ation of continuous and discrete matrix
+NLS solutions of the complex conjugation invariant NLS equa tions. We would also like to generate
+solutions of vector discrete NLS equations [3,118,123,124,130], usingthe framework pr esented here,
+and plan to elaborate on this in more detail in a separate work . Of further interest is the possibility
+to extend the present work in order to obtain solutions of (2 + 1)-dimensional generalisations of
+matrix NLS equations.
+In the defocusing NLS case we have not yet reached, via the met hod of section 2, the important
+class of dark soliton solutions (see in particular [39,81,8 7,154–160]) and also rational solutions as
+e.g. obtained in [161]. A generalisation of this method seem s to be required.
+47Acknowledgment. F M-H thanks Nils Kanning for discussions and comments. The a uthors are
+also grateful to Takayuki Tsuchida for some helpful comment s on the original manuscript.
+Appendix A: Solutions of the matrix KdV equation
+LetB0be the algebra of smooth complex functions on R2, andBits extension by adjoining an
+operator∂xthat satisfies ∂xf=fx+f∂x(wherefxdenotes the partial derivative of fwith respect
+tox). We define linear maps d ,¯d :B → B ⊗/logicalandtext1(C2) via
+df= [∂x,f]ζ1+1
+2[∂2
+x,f]ζ2,¯df=−1
+2[∂2
+x,f]ζ1+1
+3[∂t−∂3
+x,f]ζ2, (A.1)
+whereζ1,ζ2is a basis of/logicalandtext1(C2). In particular, we have d( ∂x) =¯d(∂x) = 0. The maps d ,¯d extend
+to linear maps B ⊗/logicalandtext(C2)→ B ⊗/logicalandtext(C2) satisfying (2.3) and the graded Leibniz rule. Moreover,
+they extend to Mat 1(B)⊗/logicalandtext(C2). Forφ∈Mat(m,m,B0), we have
+dφ=φxζ1+1
+2(φxx+2φx∂x)ζ2, (A.2)
+and thus
+dφ∧dφ=−1
+2(φx2)xζ1∧ζ2,¯ddφ=/parenleftbigg
+−1
+3φxt+1
+12φxxxx/parenrightbigg
+ζ1∧ζ2, (A.3)
+so that (2.14) turns out to be equivalent to the m×mmatrix KdV equation
+ut−1
+4uxxx−3
+2(u2)x= 0 where u=φx. (A.4)
+In the following we apply results of section 2 in order to gene rate exact solutions of (A.4).
+Inspection of the linear equation ¯dX= (dX)Psuggests setting P=S−I∂x, where d- and ¯d-
+constancy requires that Sdoes not depend on xandt. The linear equation is then equivalent
+to
+Xxx=−2XxS,Xt=XxS2. (A.5)
+These equations make sense for S∈Mat(n,n,C) andX∈Mat(n,n,B0). They are solved by
+X=A0+A1ˆΞ,ˆΞ=e−2(xS+tS3), (A.6)
+whereAiare constant complex matrices. Assuming A0invertible, application of theorem 2.3
+reduces to an application of corollary 2.5. (2.18) takes the form
+SˆK+ˆKS=VˆU, (A.7)
+and (2.20) becomes
+φ=ˆUˆΞ(I−ˆKˆΞ)−1V. (A.8)
+This solves the m×mmatrix KdV equation for any matrix data ( S,ˆU,V) for which (A.7) admits a
+solution ˆK. Under the additional assumption that Sis positive stable, so that e−yS→0 asy→ ∞,
+the condition27(A.7) is solved by [131,162]
+ˆK=/integraldisplay∞
+0e−ySVˆUe−ySdy. (A.9)
+27Form= 1, (A.7) is a well-known rank one condition in the context of the scalar KdV equation. See [141,142]
+for a corresponding general analysis in the framework of bou nded linear operators on a Banach space.
+48In the scalar case, i.e. m= 1, we can apply more directly the formulae of theorem 2.3 to o btain
+φ=˜UYX−1˜V= tr(˜V˜UYX−1) = tr((XS+Xx−˜RX)X−1)
+= tr(S−˜R)+tr(XxX−1), (A.10)
+where˜R=R+∂x. The second step requires that the matrices do not contain op erator terms
+(involving∂x). Dropping the irrelevant constant summand, this becomes
+φ= (logτ)x, (A.11)
+where
+τ= det/parenleftig
+I−e−xSˆKe−xS−2tS3/parenrightig
+= det/parenleftbigg
+I−/integraldisplay∞
+xe−ySVˆUe−ySe−2tS3dy/parenrightbigg
+. (A.12)
+In the last step we inserted (A.9), assuming positive stable S. This determines the family of (here
+in general complex) KdV solutions derived in [134] (see (4.1 ) therein).
+Alternatively, using lemma 6.5, (A.8) can be expressed in th e scalar case ( m= 1) asφ=
+V⊺
+0CˆΞ(I−ˆKCˆΞ)−1V0, withV⊺
+0= (1,...,1). Here ˆKnow has to solve SˆK+ˆKS=V0V⊺
+0, and
+Cis anyn×nmatrix over Cthat commutes with S. IfSis diagonal and if the eigenvalues satisfy
+si+sj/ne}ationslash= 0 for alli,j= 1,...,n, thenˆKis the Cauchy matrix with entries 1 /(si+sj).
+Appendix B: A relation between the semi-discrete NLS and mod i-
+fied KdV equations
+Let us recall from (3.19) the m×mmatrix equation for Uin the semi-discrete NLS (AL) case,
+iJ˙U −2U+U+(I−U2)+(I−U2)U−= 0. (B.1)
+Substituting
+U= i(iJ)xe−2iJ tVx∈Z, (B.2)
+with a new matrix variable V, we find28
+˙V+V+(I+V2)−(I+V2)V−= 0, (B.3)
+which is the m×mmatrix (semi-) discrete mKdV equation. As a complex matrix equation, the
+latter is thus an equivalent form of (B.1).
+Proposition B.1. LetS,U,Vbe complex matrices of size n×n,m×nandn×m, respectively,
+andKa solution of
+S−1K−KS=VU. (B.4)
+Then
+V=−U(I+(ΞK)2)−1ΞVwithΞ=Sxe−t(S−S−1)(B.5)
+solves them×mdiscrete modified KdV equation (B.3).
+28Note that Vanticommutes with J, sinceUdoes, and we have U2=−V2.
+49Proof:From proposition 4.1 we recall that
+U=U(Ξ−1−KΞK)−1V,withΞ=Sxe−it(S+S−1−2I)J
+solves (B.1) if Kis a solution of S−1K−KS=VU. Via (B.2), with the redefinitions
+S/ma√sto→ −iSJ,V/ma√sto→iV,K/ma√sto→JK,
+(and keeping U) and using (4.2), this is translated into (B.4) and (B.5). /square
+The proof moreover gives a concrete map between a class of sol utions of (B.1) and a class of
+solutions of (B.3).
+Appendix C: Equivalence of the fully discrete NLS equation w ith
+the corresponding equation of Ablowitz and Ladik in the scal ar case
+After elimination of the terms cubic in qin favour of p, (6.7) reads
+i(q+−q)+(q+−q)+−(q−q−)+(p+
++−p)q++q(p+−p−)∗= 0,
+p+−p=−ǫq+q∗. (C.1)
+We can solve the second equation with
+p(x,t) =−ǫx/summationdisplay
+k=−∞q(k,t)q(k−1,t)∗x,t∈Z, (C.2)
+assuming the existence of this infinite series. Inserting th is in the first equation, leads to
+i[q(x,t+1)−q(x,t)]+q(x+1,t+1)−q(x,t+1)−[q(x,t)−q(x−1,t)]
+−ǫq(x,t)x/summationdisplay
+k=−∞/parenleftig
+q(k,t+1)∗q(k−1,t+1)−q(k−1,t)∗q(k−2,t)/parenrightig
+−ǫx+1/summationdisplay
+k=−∞/parenleftig
+q(k,t+1)q(k−1,t+1)∗−q(k−1,t)q(k−2,t)∗/parenrightig
+q(x,t+1) = 0.(C.3)
+Expressing this in terms of a new variable rrelated toqbyq(x,t) =r(x,−t+1), and substituting
+tby−tin the resulting equation, in the scalar case we end up with eq uation (2.17) in [5] (with
+an obvious change in notation). The infinite sum in the last eq uation appears to be an awkward
+non-locality. But see [46] for how to achieve locality with t he help of conservation laws.
+Appendix D: A non-local Hermitian reduction of the semi-dic rete
+NLS system
+Imposing the condition (see also [116])
+¯q(x,t) =ǫq(−x,t)†withǫ=±1 (D.1)
+reduces the two equations (5.6) to the single equation
+i ˙q(x,t)+q(x+1,t)−2q(x,t)+q(x−1,t)−ǫq(x+1,t)q(−x,t)†q(x,t)
+−ǫq(x,t)q(−x,t)†q(x−1,t) = 0. (D.2)
+50This is a non-local equation unless one imposes the restrict ion to functions q(x,t) that are even in
+x. Setting
+U=V†,¯U=ǫ¯V†,¯S= (S−1)†, K†=−K,¯K†=−¯K, (D.3)
+then¯Ξ is given by Ξ†withxreplaced by −x, and proposition 5.1 implies that
+q=V†/parenleftig
+(S†)xe−itω(S†)−ǫ¯KSxe−itω(S)K/parenrightig−1¯V , (D.4)
+withω(S) defined in (5.10), solves (D.2) if K,¯Ksolve
+S−1K−K(S−1)†=VV†, S†¯K−¯KS=¯V¯V†. (D.5)
+Appendix E: Lax pairs for the NLS systems
+(2.14) is the integrability condition of the linear equatio n
+¯dΨ = (dφ)Ψ+(dΨ)∆ , (E.1)
+if ∆ solves (cf. [1])
+¯d∆ = (d∆)∆ . (E.2)
+In the following we show how Lax pairs for the NLS systems orig inate from these equations, by
+using the respective bidifferential calculus. (E.2) is obvio usly satisfied if d∆ = ¯d∆ = 0. These
+linear equations allow ∆ to carry an arbitrary constant fact or which turns out to play the role of
+a “spectral parameter” in the Lax pair. Although (E.1) is lin ear in this parameter, the resulting
+Lax pair exhibits a nonlinear parameter dependence.
+Continuous NLS system. (E.2) is trivially satisfied if ¯d∆ = d∆ = 0. Using the bidifferential
+calculus that led to the continuous NLS system, this means ∆ x= ∆t= [J,∆] = 0. A particular
+solution is ∆ = 2 λIwith a constant λ. Then (E.1) becomes
+Ψx=1
+2J[J,ϕ]Ψ+λ[J,Ψ],−iΨt=JϕxΨ+2λΨx (E.3)
+(whereφ=Jϕ). Writing
+Ψ =ψe−xλJ−2itλ2J, (E.4)
+after some straight computations we obtain the linear syste m in the canonical form
+ψx=Lψ, ψ t=Mψ, (E.5)
+where
+L=λJ+U=/parenleftbiggλIm1q
+¯q−λIm2/parenrightbigg
+,
+M= iJ(Ux−U2)+2iλL= i/parenleftbigg2λ2Im1−q¯q2λq+qx
+2λ¯q−¯qx−2λ2Im2+ ¯qq/parenrightbigg
+. (E.6)
+Thisisafamilar Laxpair forthematrix NLSsystem [3]. Thein tegrability condition (zero curvature
+condition)
+Lt−Mx+[L,M] = 0 (E.7)
+indeed reproduces the NLS system (5.3). Whereas our linear s ystem (E.1) is linearin the (spectral)
+parameterλ, the above Lax pair has a nonlinear λ-dependence.
+51Semi-discrete NLS system. (E.2) is trivially satisfied with ∆ = 2 αS−1, whereαis a constant.
+Then (E.1) becomes
+Ψ−Ψ−=UΨ−+α[J,Ψ], (E.8)
+−i˙Ψ =J(ϕ+−ϕ)Ψ+2α(Ψ+−Ψ)
+=J(U+−U −U+U)Ψ+2α(Ψ+−Ψ), (E.9)
+where we used the second of equations (3.19) in the last step. Multiplying from the right by
+(1/2)(I+J), and introducing ψ=1
+2Ψ−(I+J), we obtain
+ψ+=Lψ,˙ψ=Mψ, (E.10)
+where
+L= [(1+α)I−αJ]−1(I+U),M= iJ(U −U−−U U−)+2iα(L−I).(E.11)
+The integrability condition of this linear system, i.e. ˙L=M+L − LM , is invariant under the
+transformations L′=S+LS−1,M′=SMS−1+˙SS−1. Setting
+S=λxeit(λ2−1)/parenleftbiggIm10
+0λIm2/parenrightbigg
+, α=1
+2(λ2−1) (E.12)
+(wherexis the discrete variable), we recover the usual Lax pair of th e AL system,
+L′=/parenleftbiggλIm1q
+¯q λ−1Im2/parenrightbigg
+,M′= i/parenleftbigg(λ2−1)Im1−q¯q−λq−λ−1q−
+λ¯q−−λ−1¯q(1−λ−2)Im2+ ¯qq−/parenrightbigg
+.(E.13)
+Fully discrete NLS system. Choosing again ∆ = 2 αS−1, (E.1) yields
+Ψ−Ψ−=UΨ−+α[J,Ψ],−i(Ψ+−Ψ) =J(ϕ+
++−ϕ)Ψ++2α(Ψ+
++−Ψ).(E.14)
+Introducing again ψ=1
+2Ψ−(I+J) leads to
+ψ+=Lψ, ψ −ψ−=Mψ, (E.15)
+with
+L= [(1+α)I−αJ]−1(I+U),M=−1
+i+2α[J(ϕ−ϕ−
+−)+2α(L−I)].(E.16)
+The integrability condition L − L−=M+L − L−Mis invariant under the transformation L′=
+S+LS−1,M′=I−S−(I−M)S−1. Choosing
+S=λx(1−i−λ2)−t/parenleftbiggIm10
+0λIm2/parenrightbigg
+, α=1
+2(λ2−1) (E.17)
+(wherex,t∈Z), we obtain a Lax pair close to (E.13),
+L′=/parenleftbiggλIm1q
+¯q λ−1Im2/parenrightbigg
+,M′=/parenleftbigg(λ2+i)Im1+p−p−
+−λq−λ−1q−
+−
+λ¯q−
+−−λ−1¯q(2+i−λ−2)Im2−¯p+ ¯p−
+−/parenrightbigg
+.(E.18)
+52Appendix F: Continuum limit
+Let us introduce lattice spacing parameters a,bin the basic relations that determine the bidiffer-
+ential calculus for the fully discrete NLS system,
+df=1
+a[TS,f]ζ1+1
+2[J,f]ζ2,¯df=−i
+b[T,f]ζ1−1
+a[S−1,f]ζ2, (F.1)
+where now x∈aZandt∈bZ.1
+a[TS,f] taken at a lattice point ( x,t) reads1
+a[f(x+a,t+b)−
+f(x,t)]TS, which shows that an approach to the continuum can only work i f we let first b→0 and
+only afterwards a→0. The first limit yields the bidifferential calculus for the se mi-discrete NLS
+system and then the second that for the continuum NLS system ( provided that fis differentiable).
+In proposition 4.1 we only have to replace the expression for Ξby
+Ξ=Sx/a/parenleftig
+I−ib
+a2(S−1−I)J/parenrightigt/b/parenleftig
+I−ib
+a2(I−S)J]−1/parenrightig−t/b
+(F.2)
+b→0−→Sx/ae−it(S−1−I)J/a2eit(I−S)J/a2=Sx/ae−it(S+S−1−2I)J/a2
+a→0−→e−x˜Pe−it˜P2.
+In the last step we expressed Sin terms of ˜PviaS= (I+a˜P)−1, which replaces (4.16). Renaming
+˜PtoS, yields the expression for Ξin the continuous NLS case in proposition 4.1.
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+arXiv:1001.0134v2  [astro-ph.HE]  22 Sep 2010Draft version September 8, 2018
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+THE SPECTRUM OF COSMIC RAYS ESCAPING FROM RELATIVISTIC SHOC KS
+Boaz Katz1, Peter M ´esz´aros2and Eli Waxman1
+Draft version September 8, 2018
+ABSTRACT
+We derive expressions for the time integrated spectrum of Cosmic R ays (CRs) that are accelerated
+in a decelerating relativistic shockwaveand escape ahead of the sho ck. It is assumed that at any given
+time the CRs have a power law form, carry a constant fraction of th e energy Eof the shocked plasma,
+andescapecontinuouslyatthemaximalenergyattainable. Thespe ctrumofescapingparticlesishighly
+sensitive to the instantaneous spectral index due to the fact tha t the minimal energy, εmin∼Γ2mpc2
+where Γ is the shock Lorentz factor, changes with time. In particu lar, the escaping spectrum may be
+considerably harder than the canonical N(ε)∝ε−2spectrum. For a shock expanding into a plasma
+of density n, a spectral break is expected at the maximal energy attainable at the transition to non
+relativistic velocities, ε∼1019(ǫB/0.1)(n/1cm−3)1/6(E/1051erg)1/3eV where ǫBis the fraction of the
+energy flux carried by the magnetic field. If ultra-high energy CRs a re generated in decelerating
+relativistic blast waves arising from the explosion of stellar mass obje cts, their generation spectrum
+may therefore be different than the canonical N(ε)∝ε−2.
+Subject headings: cosmic rays
+1.INTRODUCTION
+Cosmic rays (CRs) are widely believed to be accel-
+erated in collisionless shock waves in various systems
+(see e.g. Blandford & Eichler 1987; Axford 1994). In
+particular, supernova remnant (SNR) shocks propagat-
+ing subrelativistically are thought to accelerate pro-
+tons up to ∼1015eV (see e.g. Ginzburg & Syrovatskii
+1969), while non or trans-relativistic internal shocks
+or ultra-relativistic external shocks of the jets of ac-
+tive galactic nuclei (AGNs, see e.g. Berezinsky 2008) or
+gamma-ray bursts (GRBs, Waxman 1995; Vietri 1995;
+Milgrom & Usov 1995) may be the sources of Ultra High
+Energy CRs (UHECRs) up to the GZK energies of ∼
+few×1020eV.
+In cases where the shocked material expands consider-
+ably before releasing the CRs, most of the cosmic rays
+loose most of their energy by adiabatic losses before es-
+caping. Thus, the instantaneous spectrum of CRs at a
+given time and the integrated spectrum of the CRs es-
+caping from the system may be different.
+In this paper, we write down simple analytic expres-
+sions forthe spectrum ofescapingCRs from a relativistic
+decelerating shock wave assuming that at any given time
+(a) CRs escape at the maximal energy attainable at that
+time, (b) the CRs have a power law energy spectrum,
+and (c) the CRs carry a constant fraction of the shocked
+thermal plasma. We show that even under these simple,
+widely acceptable assumptions, the resulting spectrum
+may be non trivial and in particular significantly harder
+than the canonical N(ε)∝ε−2spectrum.
+In section §2 we formulate our basic assumptions and
+give general expressions for the resulting flux. In section
+§3 we focus on energy conserving blast waves expanding
+in a uniform medium. The implications of the results
+are discussed in §4. Expressions for the escaping CR
+1Weizmann Institute of Science, Rehovot, Israel
+2Dpt. of Astronomy & Astrophysics; Dpt. of Physics; Center
+for Particle Astrophysics,Pennsylvania State University , Univer-
+sity Park, PA 16802, USAspectrum expected in other scenarios, including power
+law density profiles, radiative shocks, and a jet injection
+with constant luminosity, are derived in §A.
+2.BASIC MODEL AND ASSUMPTIONS
+Consider CRs that are accelerated by an expanding
+shock wave which is decelerating due to interaction with
+an external medium.
+2.1.Assumptions
+(i) The basic assumption that we make is that at any
+given time (or shock radius R), particles escape at the
+maximal energy εmaxattainable at that time. Specifi-
+cally, it is assumed that εNesc(ε), the number of CRs
+ejected within a logarithmic energy interval around ε,
+is similar to the number of particles in the system,
+εN(ε,R), at the radius Rfor which the maximal energy
+εmaxis equal to ε,
+εNesc(ε)∼εN(ε,R|εmax=ε). (1)
+Here and everywhere else in this letter, all quantities
+are measured in the observer frame. This assumption
+is expected to be true whenever the acceleration is lim-
+ited by the finite size of the system (for a recent dis-
+cussion see e.g. Caprioli et al. 2009). Note that for
+a relativistic shock with Lorenz factor Γ, only parti-
+cles that move in the shock propagation direction to
+within an angular deviation of 1 /Γ, can outrun the shock
+and escape. This does not introduce a further correc-
+tion however, due to the fact that all particles in the
+shocked region, including the thermal and accelerated
+particles, are beamed in the observer frame to an an-
+gular separation of 1 /Γ (for examples of the expected
+angular distribution of CRs in relativistic shocks see
+e.g. Kirk & Schneider 1987; Bednarz & Ostrowski 1998;
+Kirk et al. 2000; Keshet & Waxman 2005).
+For concreteness we further make the following as-
+sumptions.
+(ii) At any given radius R, the energy spectrum of CRs2 Katz et al.
+is a power law, N(ε)∝ε−2−xforεmin< ε < ε max, with
+x >0 and with total energy ECR. The spectrum can be
+expressed as
+ε2N(ε) =xECR/parenleftbiggε
+εmin/parenrightbigg−x
+, (2)
+where we assumed that ( εmax/εmin)−x≪1.
+(iii) The minimal, maximal and total cosmic ray energies
+are power law functions of the radius,
+εmin∝R−αmin, εmax∝R−αmax, ECR∝R−αE.(3)
+2.2.Resulting spectra
+Under the aboveassumptions, the spectrum of escaped
+particles is given by
+ε2Nesc(ε)∝ε−xesc(4)
+with
+xesc=x−(αminx+αE)/αmax. (5)
+For the case αmin= 0, this reduces to equation (28) of
+Ohira et al. (2009).
+Equation (5) is valid for x >0. We note that for the
+limiting value, x= 0, a logarithmic correction is intro-
+duced to the spectrum of escaped CRs, due to the loga-
+rithmic dependence of the total CR energy on εmin,max,
+ECR= log(εmax/εmin)ε2N(ε). Forx <0, the energy
+carried by the CRs is dominated by the particles with
+largest energies, ε=εmax. The resulting escaped spec-
+trum satisfies xesc=αE/αmaxand is not sensitive to the
+form of the instantaneous spectrum.
+3.ENERGY CONSERVING BLAST WAVES
+We next apply the abovegeneralresult to simple cases,
+focusing on a blast wave of fixed energy which starts off
+ultra-relativistic, and then decelerates to non relativis-
+tic velocity. The non relativistic and relativistic stages
+are analyzed in §3.1 and§3.2 respectively. The spec-
+trum resulting from the combination of the two stages is
+addressed in §4.
+3.1.Non-relativistic expansion
+Perhaps the simplest case to consider is the case of
+constant CR minimal energy εminand total energy ECR
+(αE,αmin= 0). This is expected in the Sedov-Taylor
+phase of non relativistic blast waves. The energy in CRs
+is dominated by the relativistic CRs, ε > m pc2, and is
+commonly assumed to be a constant fraction fof the
+total energy Ein the system. Hence εmin∼mpc2and
+ECR=fE, both independent of the shock radius. Eq.
+(5) implies that the escaped spectrum is the same as the
+instantaneous spectrum,
+xesc=x,i.e. Nesc(ε)∝ε−2−x. (6)
+This result can be obtained directly from Eq. (1) by not-
+ing that the entire spectrum, up to the maximal energy
+εmax(R), is independent of radius.
+3.2.Ultra-relativistic expansion
+We next consider the case of ultra-relativistic expan-
+sion. Below and in §A we make the following assump-
+tions in addition to assumptions (i)-(iii) above:(iv) The minimal CR energy is the energy of the thermal
+particles,
+εmin∼εth∼Γ2mpc2; (7)
+(v) TheCR pressureisa radiusindependent fraction fCR
+of the momentum flux in the shock frame, pCR∝Γ2ρ,
+implying
+ECR∝R3Γ2ρ; (8)
+(vi) The maximal CR energy is the maximal energy of
+CRsthat areconfined by afluid rest framemagnetic field
+Brest(equivalent to Diffusive Shock Acceleration in the
+Bohm limit),
+εmax∼eEobsδR∼eBrestR∝e(Γρ1/2)R,(9)
+whereEobs∼ΓBrestis the electric field in the ob-
+server frame corresponding to a magnetic field Brest∼
+(8πΓ2ρc2ǫB)1/2in the rest frame of the shocked plasma,
+assumed to carry a constant fraction ǫBof the momen-
+tum flux, and δR∼R/Γ is the maximal distance that a
+cosmic ray can propagate along the electric field in the
+observer frame.
+3.2.1.Ultra-relativistic impulsive expansion
+WenextconsideraBlandford-McKee(energyconserv-
+ing; Blandford & McKee 1976) shock expanding into a
+uniform medium. By assumption αE= 0. Using equa-
+tions (7)-(9), αmax= 1/2 andαmin= 3 are obtained.
+Using (5) we find xesc=−5x, or
+Nesc(ε)∝ε5x−2. (10)
+4.DISCUSSION
+The spectral index of escaped CRs, given by Eq. (10),
+may be surprisingly hard if the instantaneous spectrum
+(equ.[2]) is softer than a flat spectrum, i.e. x >0. The
+basic reason for this is the fact that the minimal CR
+energyεmin∼Γ2mpc2changes with radius much faster
+than the maximal energy, implying that at later times
+(corresponding to lower escaped energies) there is much
+lessCR energyat the escaping, highend ofthe spectrum.
+Forexample,considerthecommonlyassumedDiffusive
+Shock Acceleration (DSA) mechanism (Krymskii 1977;
+Axford et al. 1977; Bell 1978; Blandford & Ostriker
+1978), which for ultra-relativistic shocks with isotropic,
+small-angle scattering, leads to an instantaneous
+spectrum N∝ε−20/9(Keshet & Waxman 2005;
+Bednarz & Ostrowski 1998; Kirk et al. 2000). Using Eq.
+(10), this implies a very hard escaping spectrum, Nesc∝
+ε−8/9. We emphasize that DSA has not been shown to
+work based on first principles and even if it does, the
+resulting spectrum is sensitive to the scattering mecha-
+nism (e.g. Keshet & Waxman 2005; Keshet 2006) which
+is poorly understood. In fact, the instantaneous spec-
+trum does not necessarily need to have a power-law form
+(e.g. Caprioli et al. 2009; Ohira et al. 2009) or to have a
+constant spectral index(e.g. Ellison & Double 2004).
+Note that an instantaneous spectrum that is flat N∝
+ε−2(e.g. Katz et al. 2007) or harder (e.g. Keshet 2006;
+Stecker et al. 2007, and others) can be obtained in spe-
+cific particle acceleration models. These models, how-
+ever, are different from the possible hard spectrum of
+escaping particles discussed above. In fact, such hardCR spectrum 3
+instantaneous spectra lead to a flat escaped spectrum
+Nesc(ε)∝ε−2[see discussion following eq. (5)]. The
+important point is that the escaping spectrum is highly
+uncertain, very sensitive to the acceleration mechanism,
+and may be considerably harder than Nesc(ε)∝ε−2.
+Once the shock decelerates to non relativistic speeds,
+the integrated escaping spectrum changes its form. In
+the subsequent Sedov-Taylor phase, the minimal CR en-
+ergy does not change any more, εmin∼mpc2, and the
+escaping spectrum will be similar to the instantaneous
+spectrum [see Eq. (6)]. Note that in the extreme case
+in which CRs carry most of the energy, and assuming
+they are accelerated by non-linear DSA (for a recent re-
+view see Malkov & Drury 2001), the instantaneous spec-
+trum may not be a power law and correspondingly the
+escaping spectrum is different than that given here (e.g.
+Caprioli et al. 2009; Ohira et al. 2009). In any case, a
+significant break is expected in the escaping spectrum at
+an energy ε∗equal to the maximal energy achievable at
+the transition from relativistic to non-relativistic shock
+velocities. Using Eq. (9) for Γ β∼1, the break is ex-
+pected at
+ε∗∼eBrestR∼(8πρc2ǫB)1/2[3E/(4πρc2)]1/3
+∼1019ǫB,−1n1/6
+0E1/3
+51eV, (11)where we have taken, as an example, a gamma-ray burst
+source with E= 1051E51erg,ρ=n mpis the ambient
+densitywith n=n0cm−3, andǫB= 0.1ǫB,−1. Thespec-
+tral index above this energy will be −2−xesc=−2+5x,
+as given by Eq. (10). We note that the energy range
+wherexesc=−5xis limited between ε∗and max( ε)∼
+εmax(Γ0) whereΓ 0isthe initialLorentzfactor. The max-
+imal energy scales with Γ as εmax∝Γ1/3resulting in a
+narrow energy range ε∗/lessorsimilarε/lessorsimilarΓ1/3
+0ε∗. While this range
+depends on model parameters, it does point to a possible
+mechanism for achieving a hardening of the spectrum at
+the high end of the CR spectrum.
+We conclude that the spectrum of UHECRs, may be
+different than ε−2, and possibly considerably harder, e.g.
+if they originate from a relativistic decelerating blast-
+waveresultingfromtheexplosionofastellarmassobject.
+ACKNOWLEDGMENTS
+BK & EW acknowledge support by ISF, AEC and
+Minerva grants. PM acknowledges support from NSF
+PHY-0757155 grant.
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+Spectral indexes of escaping CRs for different relativisti c scenarios
+Scenario Conserved quantity αΓ xesc[ρ∝R0]xesc[ρ∝R−2]
+Impulsive, Energy conserving E∼Γ2ρR3(3−αρ)/2−5x −x
+Impulsive, Radiative MΓ∼ΓρR33−αρ−3/2−2x −1/2−x
+Continuous injection L=const. L∼4πρΓ4c3R2(2−αρ)/4-4+3x 2-x
+The value of the spectral index xescis given by Eqs. (A5) and (A6) for the various scenarios. N(ε)∝ε−2−xesc.
+APPENDIX
+OTHER PARTICULAR CASES
+Consider an ultra-relativistic shock, which expands into a medium with a density profile
+ρ∝R−αρ(A1)
+with a bulk Lorentz factor
+Γ∝R−αΓ. (A2)
+Using Eqs. (7)-(9) and (A1) we find:
+αmin= 2αΓ, αmax=αΓ+1
+2αρ−1,andαE= 2αΓ+αρ−3. (A3)
+Substituting this in equation (5) we find
+xesc= 3−2αΓ−αρ+(2−αρ)+2αΓ
+(2−αρ)−2αΓx. (A4)
+For a uniform density distribution, αρ= 0, and for a wind-like density distribution, αρ= 2, this reduces to
+xesc=3−2αΓ
+αΓ−1+1+αΓ
+1−αΓx, (A5)
+and
+xesc=1−2αΓ
+αΓ+1−x (A6)
+respectively. The escaping spectrum is then
+Nesc(ε)∝ε−2−xesc, (A7)
+which is flatter then −2 forxesc<0. The expected spectral indices xesc, for various physical scenarios, are provided
+in table 1.
\ No newline at end of file
diff --git a/1001.0135.txt b/1001.0135.txt
new file mode 100644
index 0000000000000000000000000000000000000000..953964c3c9f2d739f55bc9efb1411445104d7046
--- /dev/null
+++ b/1001.0135.txt
@@ -0,0 +1,1332 @@
+arXiv:1001.0135v1  [astro-ph.GA]  31 Dec 2009ACCEPTED FOR PUBLICATION IN THE ASTROPHYSICAL JOURNAL
+Preprint typesetusingL ATEX styleemulateapjv. 11/10/09
+ASSESSMENTOF STELLARSTRATIFICATIONINTHREE YOUNGSTARCL USTERS INTHELARGE MAGELLANIC
+CLOUD.
+DIMITRIOS A. GOULIERMIS
+Max Planck Institute for Astronomy, Königstuhl 17, 69117 He idelberg, Germany
+DOUGALMACKEY
+Institute for Astronomy, University of Edinburgh, Royal Ob servatory,
+Blackford Hill, Edinburgh, EH93HJ,UK
+YUXIN
+Argelander-Institut für Astronomie, Rheinische Friedric h-Wilhelms-Universität Bonn,
+Aufdem Hügel 71, 53121 Bonn, Germany
+AND
+BOYKEROCHAU
+Max Planck Institute for Astronomy, Königstuhl 17, 69117 He idelberg, Germany
+Accepted for Publication in theAstrophysical Journal
+ABSTRACT
+Wepresentacomprehensivestudyofstellarstratificationi nyoungstarclustersintheLargeMagellanicCloud
+(LMC).We applyourrecentlydeveloped effectiveradiusmethod forthe assessment ofstellar stratification on
+imaging data obtained with the Advanced Camera for Surveys o f three young LMC clusters to characterize
+the phenomenon and develop a comparative scheme for its asse ssment in such clusters. The clusters of our
+sample, NGC 1983, NGC 2002 and NGC 2010, are selected on the ba sis of their youthfulness, and their
+variety in appearance,structure, stellar content, and sur roundingstellar ambient. Our photometryis complete
+for magnitudes down to m814≃23 mag, allowing the calculation of the structural paramete rs of the clusters,
+the estimation of their ages and the determination of their s tellar content. Our study shows that each cluster
+in our sample demonstrates stellar stratification in a quite different manner and at different degree from the
+others. Specifically, NGC 1983 shows to be partiallysegregated with the effective radius increasing with
+fainter magnitudes only for the faintest stars of the cluste r. Our method on NGC 2002 provides evidence of
+strongstellar stratification for both bright and faint stars; the c luster demonstrates the phenomenon with the
+highest degreein the sample. Finally, NGC 2010is not segregated ,as its brightstellar content is not centrally
+concentrated, the relation of effective radius to magnitud e for stars of intermediate brightness is rather flat,
+and we find no evidence of stratification for its faintest star s. For the parameterization of the phenomenonof
+stellar stratification and its quantitative comparison amo ng these clusters, we propose the slope derived from
+thechangeintheeffectiveradiusoverthecorrespondingma gnituderangeasindicativeparameterofthe degree
+of stratification in the clusters. A positive value of this slope indicates mas s segregationin the cluster, while a
+negativeorzerovaluesignifiesthe lackofthephenomenon.
+Subject headings: Magellanic Clouds – galaxies: star clusters – globular clus ters: individual: NGC 1983,
+NGC 2002, NGC 2010 – Hertzsprung-Russell diagram – Methods: statistical – stellar dy-
+namics
+1.INTRODUCTION
+Stellar stratification is a characteristic phenomenon of star
+clusters, which has been well documented for more than 50
+years. It is directly linked to the dynamical evolution of st ar
+clusters(Chandrasekhar 1942;Spitzer1987),andspecifica lly
+to the central concentration of the massive stars as the clus -
+ter relaxes dynamically, a phenomenon known as mass seg-
+regation(for detailed reviews see Lightman&Shapiro 1978;
+Meylan&Heggie 1997). This behavior in massive Galactic
+globularclusters(GGCs)iswellunderstoodandtheobserve d
+mass segregation is explained as a result of their dynamical
+relaxation in a timescale, trelax, much longer than their ages,
+dgoulier@mpia-hd.mpg.de
+dmy@roe.ac.uk
+yxin@astro.uni-bonn.de
+rochau@mpia-hd.mpg.deτ, as derivedfrom stellar evolutionarymodels. This dynami-
+calmass segregation, thus, forces more massive stars to sink
+inwards to the center of a cluster through weak two-body in-
+teractions.
+However, segregation of massive stars is observed also
+in young star clusters, such as the ∼30 Myr-old clus-
+ter NGC 330 in the Small Magellanic Cloud (Sirianniet al.
+2002), oreventhe ∼1 Myr-oldGalactic starburst NGC 3603
+(Stolteetal.2006). Suchsystems,supposedlynotsufficien tly
+oldtobedynamicallyrelaxed,cannotbeincludedinthe “tra -
+ditional”pictureofdynamicalmasssegregationasitoccur sin
+GGCs. Forthesecasesthephenomenonhasbeen–ratherten-
+tatively–characterizedas primordial masssegregation,based
+on the suggestion that this stellar stratification is due to t he
+formationof the stars in, or near, the central part of the clu s-
+tersratherthanbeingdynamicallydriven
+Mass segregation at early stages of clustered star forma-2 Gouliermis,et al.
+tion is indeed predicted by theoretical studies, which sug-
+gest that the positions of massive stars for rich young clus-
+ters ofτ <<trelaxcannot be the result of dynamical evolu-
+tion (Bonnell& Davies 1998). Moreover, the appearance of
+gas in such clusters reduces further the efficiency of any dy-
+namicallydrivenmasssegregation. Accordingtotheory,ma s-
+sive protostars can be formed at the central part of the pro-
+tocluster through cohesive collisions and dissipative mer ging
+of cloudlets, which lead to extensive mass segregation and
+energy equipartition among them (Murray& Lin 1996), or
+through accretion from a distributed gaseous component in
+the cluster, which leads to mass segregation since stars lo-
+cated near the gravitational center of the cluster benefit fr om
+the attraction of the full potential and accrete at higher ra tes
+thanotherstars(Bonnell& Bate2006).
+Both of the aforementionedmassive star formationmecha-
+nisms predict primordial mass segregation through early dy -
+namical processes at the central part of the protocluster. T he
+significant difference of these processes from the subseque nt
+energy equipartition of the stars in the formed cluster is th at
+the latter, and consequently dynamical mass segregation, i s
+independentoftheinitialclusterconditions,whilethefo rmer,
+and therefore primordial mass segregation, is directly con -
+nected to these conditions. In order to achieve a complete
+understanding of the phenomenon of primordial mass segre-
+gation, many related studies are focused on clusters at very
+earlystagesoftheirformation,includingavarietyoftype sof
+stellar systems from loose stellar associations and open cl us-
+tersto compactmassivestarbursts.
+The Magellanic Clouds (MCs), being extremely rich in
+young star clusters (e.g., Bica &Schmitt 1995; Bica etal.
+1999), offer an outstanding sample of intermediate type
+of stellar systems, specifically compact young star clus-
+ters, which are not available in the Galaxy. In such clus-
+ters primordialmass segregationcan be sufficiently observ ed
+from ground-based observations (e.g., Kontizaset al. 1998 ;
+Santiagoet al. 2001; Kerberet al. 2002; Kumaret al. 2008),
+and they are also suitable for more detailed studies of this
+phenomenon from space with the high-resolving efficiency
+of theHubble Space Telescope (e.g. Fischer etal. 1998;
+deGrijsetal. 2002; Sirianniet al. 2002; Gouliermiset al.
+2004; Kerber&Santiago 2006). Such studies contribute sig-
+nificantly to the debate about the phenomenon of primordial
+mass segregation and its origin which which is still ongoing
+(see, e.g., Kissler-Patiget al. 2007). It is interesting to note
+that while the existence of the phenomenon itself is being
+questioned(Ascensoetal.2009),newmethodsarestill bein g
+designed for its comprehensive identification and quantific a-
+tion(e.g.,Allisonet al.2009a).
+In the first part of our study of stellar stratification we de-
+veloped and tested a new robust diagnostic method aimed at
+establishing the presence (or not) of this phenomenon in a
+cluster (Gouliermisetal. 2009, from here on Paper I). This
+methodisbasedonthecalculationofthemean-squareradius ,
+theso-called effectiveradius ,ofthestarsinaclusterfordiffer-
+entmagnituderanges,andtheinvestigationofthedependen ce
+of the effective radii of each stellar species on magnitude a s
+indicationofstellarstratificationinthe cluster.
+1.1.The EffectiveRadiusMethod
+Stellar stratification affects the presently observable ph ysi-
+cal propertiesof the star cluster, and thus diagnostictool sfor
+the identification of the phenomenonare based on the detec-
+tion of mass- or brightness-dependentchanges of such prop-
+FIG. 1.— Distribution of the uncertainties of our photometry fo r all stars
+detected in both the F555W and F814W bands in the areas of the c lusters
+of our sample. Uncertainties in F555W are plotted with filled black and in
+F814Win open grey circular symbols.
+erties. Previously established methods involve (i) the stu dy
+of the projected stellar density distribution of stars of di f-
+ferent magnitudes; here, the differences in the exponents o f
+the slopes fitted are seen as proof of mass segregation (e.g.,
+Subramaniametal. 1993), (ii) the radial dependence of the
+Luminosity or Mass Function (LF, MF) slope; a gradient of
+this slope outwards from the center of the cluster indicates
+stratification (e.g., Kingetal. 1995), and (iii) the varian ce of
+the core radius of the cluster as estimated for stars in spe-
+cificmagnitude(ormass)groups;atrendofthisvariancewit h
+brightness(or mass) signifies stratification (e.g., Brandl et al.
+1996). While these methods can provide useful information
+about the segregation in a cluster, all of them are sensitive toStellarStratificationinYoungStar Clustersin theLargeMa gellanicCloud 3
+modelassumptions,sincetheystronglydependondatafittin g
+for parametrizingstellar stratification; i.e., functiona lexpres-
+sions for the surface density profiles, or the MFs and LFs are
+requiredforthederivationoftheirslopesandcoreradii.
+Our method, developedin Paper I for assessment of stellar
+stratification in star clusters, is based on the notionof the dy-
+namicallystable Spitzerradius ofastarcluster,definedasthe
+mean-squaredistance of the stars fromthe center of the clus -
+ter (e.g., Spitzer 1958). The observable counterpart of thi s
+radius,theeffectiveradius,isgivenbythe expression:
+reff=/radicalBigg
+ΣN
+i=1r2
+i
+N, (1)
+whereriistheprojectedradialdistanceofthe ithstellarmem-
+ber of the cluster in a specific brightness range and Nthe
+corresponding total number of stars in the same brightness
+range. Different calculations of the effective radius are p er-
+formed,eachforstarsindifferentbrightnessrangesthatc over
+the whole observed stellar luminosity function of the clust er.
+The simple application of this method can yield direct infor -
+mation about the spatial extent (radial distribution) of st ellar
+groups in different magnitude (mass) ranges. In a star clus-
+ter, where stratification occurs, the segregated brighter s tars
+are expected to be more centrally concentrated and the cor-
+respondingeffective radius should be shorter than that of t he
+non-segregated fainter stars. As a consequence, stellar st rat-
+ification can be observed from the dependence of the effec-
+tiveradiusofstarsinspecificmagnitude(mass)rangesonth e
+corresponding mean magnitude (mass). This method is suf-
+ficiently described and tested in Paper I, where we show that
+it performsefficiently in the detection of stellar stratific ation,
+providedthat the incompletenessof the photometryhas been
+accurately measured and the contamination by the field pop-
+ulation has been thoroughlyremoved. Our diagnosis method
+is also independentof any model or theoretical prediction, in
+contrasttothoseusedthusfarforthedetectionofmasssegr e-
+gation.
+In this second part of our study of stellar stratification we
+apply the effective radius method on deep imaging data ob-
+tained with the Wide-Field Channel (WFC) of the Advanced
+Camera for Surveys (ACS) onboard the Hubble Space Tele-
+scope (HST) of three young star clusters in the Large Mag-
+ellanicCloud(LMC) fortheinvestigationofprimordialmas s
+segregationinthem. YoungclustersintheMagellanicCloud s
+are quite different to each other, so that their study requir es
+the detailed treatment of each cluster individually (see, e .g.,
+§§ 3, 4), and therefore a statistical investigation of the ph e-
+nomenonof mass segregationin a large sample of such clus-
+tersisnotyetmeaningful. Thepurposeofthepresentstudyi s
+the detailed investigationwith the use of a consistent meth od
+ofindividualselectedclusters,inordertoestablishacom pre-
+hensive comparative framework for the assessment of mass
+segregation in young LMC clusters. The objects of interest,
+clusters NGC 1983, NGC 2002 and NGC 2010, are selected
+among others also observed with ACS due to their youthful-
+ness and the variety in their characteristics. This selecti on is
+made in order to include in this comparative study clusters,
+which are very different from each other. In § 2 we describe
+theconsideredsampleofLMCclustersandthecorresponding
+datasets,aswellastheirreductionandphotometry. Westud y
+thedynamicalstatusoftheclustersanddiscusstheirstruc tural
+parameters in § 3, and in § 4 we present the observed stellar
+populations, the decontamination of the stellar samples fr om
+FIG. 2.— Completeness functions for NGC 1983 (top), NGC 2002 (mi d-
+dle), and NGC 2010 (bottom) for both filters F555W (solid line ) and F814W
+(dashedline). Averagecompletenessfunctionsoverthewho leobservedfields
+of view are shown. It should be noted that completeness is a fu nction of
+cluster-centric distance, as stars closer to the center of a cluster suffer from
+high crowding than those farther away.
+thecontributionofthelocalbackgroundfieldoftheLMCand
+thestellarcontentoftheclusters. Theapplicationofthed iag-
+nosticmethodisperformedforallthreeclustersin§5,wher e
+we also develop a comparative scheme for the quantification
+of stellar stratification among different clusters, and we d is-
+cussourresults. We concludeonthemin §6.
+2.THESTARCLUSTERSAMPLE
+The present study deals with stellar stratification in rich
+LMC star clusters. Such clusters are known to differ from4 Gouliermis,et al.
+TABLE1
+ACS/WFC OBSERVATIONS OF THE CLUSTERS OF OUR SAMPLE (HST P ROGRAM 9891).
+Cluster R.A. DEC Filter Dataset Exposure Date
+(J2000.0) (J2000.0) filename time (s)
+NGC 1983 05h27m44.s92−68◦59′07.′′0 F555W j8ne68req 20 October 7, 2003
+F814W j8ne68riq 20 October 7, 2003
+NGC 2002 05h30m20.s80−66◦55′02.′′3 F555W j8ne70b5q 20 August 23,2003
+F814W j8ne70b7s 20 August 23,2003
+NGC 2010 05h30m34.s89−70◦49′08.′′3 F555W j8ne71r5q 20 October 7, 2003
+F814W j8ne71r9q 20 October 7, 2003
+each other in terms of, e.g., their structural parameters, a ges,
+luminosities and masses (Mackey& Gilmore 2003). Since
+we focus on the phenomenon of primordial mass segrega-
+tion we limit our sample to young LMC clusters, and in or-
+der to use the most complete available data we select clus-
+ters observed with ACS/WFC. The advantages introduced to
+crowded-field photometry of MCs star clusters by the use
+of ACS, and specifically its Wide-Field Channel, have been
+documented extensively by studies based on such observa-
+tions (e.g., Mackeyetal. 2006; Rochauetal. 2007; Xin etal.
+2008). Consequently, we select three young LMC clus-
+ters, NGC 1983, NGC 2002 and NGC 2010, observed with
+ACS/WFC.Theseclustersaredifferentfromeachother,asfa r
+astheirappearance,structureandstellarcontentisconce rned.
+Consideringthatwealsowishtoinvestigatetheidentificat ion
+and quantification of the phenomenonof stratification in var -
+ious cluster-hosting environments, we based the selection of
+these specific clusterson the onthe differencesbetweenthe ir
+local surrounding field star populations. We discuss in de-
+tailthedifferencesoftheclustersthemselvesintheirstr ucture
+and dynamical behavior in § 3, and in their stellar content in
+§4. Thedecontaminationofthetrueclusterpopulationsfro m
+the contributionof the general backgroundfield of the LMC,
+whichdeterminestheactualstellarcontentoftheclusters and
+the stellar sample to be used in our methodfor assessment of
+stellarstratificationin theclustersisperformedin§ 4.3.
+2.1.DataReductionandPhotometry
+Our observations come from HST program 9891, a snap-
+shot survey of ∼50 rich star clusters in the Magellanic
+Clouds. Imagingwas obtainedin Cycle 12 using ACS/WFC.
+As snapshot targets, each cluster was observed for one orbit
+only, resulting in a single exposure through the F555W filter
+andanotherthroughthe F814Wfilter. Detailsof theobserva-
+tionsfortheclustersconsideredinthispapermaybefoundi n
+Table1.
+The ACS WFC consists of a mosaic of two 4096 ×
+2048 pixel CCDs covering an area of approximately 202′′×
+202′′at a plate scale of 0 .05 arcsec per pixel. The two CCDs
+are separated by a gap roughly 50 pixels wide. The core of
+each cluster was placed at the center of WFC chip 1. This
+ensuredthattheinter-chipgapdidnotinterferewiththema in
+partofthecluster,andmeantthat eachimagereachedamax-
+imum radius of approximately ∼150′′from the cluster cen-
+ter. A small dither was made between the two exposuresof a
+giventarget,to help facilitate the removalofcosmic raysa nd
+hot pixels; however with only two images per cluster it was
+not possible to eliminate the gap in spatial coverage due to
+theinter-chipseparation.
+The data products produced by the standard STScI reduc-
+tion pipeline, which we retrieved from the public archive,have had bias and dark-currentframes subtracted and are di-
+videdbya flat-fieldimage. Inaddition,knownhot-pixelsand
+other defects are masked, and the photometric keywords in
+theimageheadersarecalculated. Wealsoobtaineddistorti on-
+corrected(drizzled)imagesfrom the archive, producedusi ng
+thePYRAFtaskMULTIDRIZZLE .
+We used the DOLPHOT photometrysoftware (e.g., Dolphin
+2000), specifically the ACS module1, to photometer the
+flatfielded (but not drizzled) F555W and F814W images.
+DOLPHOT performspoint-spreadfunction(PSF) fitting using
+PSFsespecially tailoredto theACS camera. Beforeperform-
+ing the photometry, we first prepared the images using the
+DOLPHOT packages ACSMASK andSPLITGROUPS . Respec-
+tively, these two packages apply the image defect mask and
+thensplitthemulti-imageSTScIFITSfilesintoasingleFITS
+file per chip. We then used the main DOLPHOT routine to
+simultaneously make photometric measurements on the pre-
+processed images, relative to the coordinate system of the
+drizzledF814Wimage. Wechosetofittheskylocallyaround
+eachdetectedsource (importantdueto the crowdednatureof
+the targets), and keep only objects with a signal greater tha n
+10 times the standard deviation of the background. The out-
+put photometry from DOLPHOT is on the calibrated VEGA-
+MAGscaleofSirianniet al.(2005),andcorrectedforcharge -
+transferefficiency(CTE)degradation.
+To obtain a clean list of stellar detectionswith highqualit y
+photometry, we applied a filter employing the sharpness and
+“crowding” parameters calculated by DOLPHOT . The sharp-
+ness is a measure of the broadness of a detected object rela-
+tivetothe PSF –fora perfectly-fitstar thisparameteriszer o,
+while it is negative for an object which is too sharp (perhaps
+a cosmic-ray) and positive for an object which is too broad
+(e.g., a background galaxy). The crowding parameter mea-
+sures how much brighter a detected object would have been
+measuredhad nearbyobjectsnotbeenfit simultaneously. We
+selected only objects with −0.15≤sharpness ≤0.15 in both
+frames, and crowding ≤0.25 mag in both frames. We also
+only kept objects classified by DOLPHOT as good stars2with
+formal errors from the PSF fitting less than 0 .2 mag in both
+F555W and F814W. In Figure 1 typical uncertainties in the
+derived magnitudes of the stars detected in all three cluste rs
+areshownforbothF555WandF814Wfilters.
+Inordertoassignadetectioncompletenessfractiontoeach
+star, we used DOLPHOT to perform a large number of arti-
+ficial star tests. For each star in our final cleaned catalogue
+we first generateda list of150 associated fake stars usingth e
+1The ACS module of DOLPHOT is an adaptation of the photometry
+package HST PHOT(Dolphin 2000). The package can be retrieved from
+http://purcell.as.arizona.edu/dolphot/ .
+2Which have object type 1, as opposed to elongated or extended objects
+which haveobject types >1.StellarStratificationinYoungStar Clustersin theLargeMa gellanicCloud 5
+FIG. 3.— The observed ACS/WFC fields-of-view of the clusters (a) NGC 1983, (b) NGC 2002 and (c) NGC 2010 (left panel), with the c orresponding stellar
+density maps constructed with star counts on the photometri c catalogs (right panel). Symbol sizes correspond to the bri ghtness of the marked stars. These maps
+demonstrate that the selected clusters show avariety in siz e, structure, stellar density and ambient local field (see te xt in § 3).
+DOLPHOT utilityACSFAKELIST . We constrained these fake
+starstolieclosetotherealstaronthe(F555W,F814W)color -
+magnitude diagram (CMD) – the fake stars were randomly
+and uniformly distributed in a box of size 2 ∆V×2∆Ccen-
+teredontherealstar,where ∆Vwasthemaximumof0 .1mag
+and 3 times the uncertainty in the m555magnitude of the real
+star, and ∆Cwas the maximum of 0 .1 mag and 3 times the
+uncertaintyinthe ( m555−m814) colorofthereal star.
+Wewerealsocarefultoconstrainthespatialpositionsofth e
+fake stars. Often this is doneso that the fake stars lie at sim i-larcluster-centricradiitotherealstarunderconsiderat ion,but
+withunconstrainedpositionangle. Thissavesoncomputati on
+timebecausethecalculationsarethenvalidforgroupsofre al
+starsatatime,ratherthansimplyonanindividualbasis. Ho w-
+ever, because we are dealing with very young clusters in the
+presentwork,this implicitassumptionofsphericalsymmet ry
+is not necessarily valid. These objects could potentially s till
+be embedded in gas or dust and thus exhibit a strongly vary-
+ingbackground. Theyarealso notverydynamicallyevolved,
+so may yet possess azimuthal asymmetries in stellar density6 Gouliermis,et al.
+TABLE2
+CHARACTERISTICS OF THE CLUSTERS IN OUR
+SAMPLE DERIVED FROM OUR PHOTOMETRY .
+Cluster Size f3σ Age AV
+(pc) (stars arcsec−2) (Myr) (mag)
+NGC 1983 7.3 0.72 28 0.17
+NGC 2002 9.4 0.65 18 0.19
+NGC 2010 12.5 0.54 159 0.17
+NOTE. — Sizes are derived from circular annuli en-
+compassing the 3 σisopleth in the iso-density contour
+mapsoftheclusters inthe observed regions. Stellar den-
+sityf3σis measured within these annuli and are given
+for comparison with the field stellar density of Table 3.
+Age and interstellar extinction for the clusters are esti-
+mated from evolutionary model fitting on the CMDs of
+starswithinthe rcoftheclusters. Sizesaregiveninphys-
+ical units after a conversion 1′≃15 pc assuming a dis-
+tancemodulusof( m−M)0≃18.5mag(Clementini et al.
+2003;Alves 2004;Schaefer 2008)for all clusters.
+aswell asnumerousrandomlyplacedbrightstars. Therefore ,
+we adopted the more computationally expensive method of
+generating the list of fake stars locally around each real st ar
+on the CCD – in this case randomly and uniformly within a
+radiusof 5′′. In doingthis we were carefulto accountfor the
+edgesofthechips,includingtheinter-chipgap.
+We then ran DOLPHOT in artificial stars mode on each list
+offakestars. Whenoperatingin thisway DOLPHOT takesthe
+next fake star from the list, adds it to the input images, and
+then solvesfor the position andphotometricpropertiesof t he
+starusingexactlythesameparametersasfortheoriginal(r eal
+star)analysis. Oncecomplete,wetooktheoutputphotometr y
+andranit throughourqualityfilters,justaswedidfortheli st
+of realstars. Finally, we calculatedthe completenessfrac tion
+foragivenrealstarbydetermininghowmanyofitsassociate d
+fake stars had been successfully ‘found’. We considered a
+fake star to be found if it was recovered in both F555W and
+F814Wandpassedsuccessfullythroughourqualityfilter,an d
+provided its output position lay within 2 pixels of its input
+position. Figure 2 presents the completeness function of ou r
+photometry for all considered clusters in both F555W (solid
+line)andF814W(dashedline)filters.
+3.CLUSTERMORPHOLOGIESANDSTRUCTURES
+3.1.Morphologyofthe Clusters
+The stellar charts of the observed ACS/WFC fields of the
+clusters are shown in Figure 3 (left panel). Our observation s
+showcleardifferencesintheappearanceofthethreecluste rs.
+In order to reveal the spatial distribution of the detected s tars
+andthemorphologyoftheclusters,we performedstarcounts
+on our photometric catalogs. The constructed ‘iso-density ’
+contourmapsforeachregionarealsoshowninFigure3(right
+panel). The coordinate system used for these maps is that
+of the original ACS/WFC pixel coordinates. The star counts
+were performed in a quadrilateral grid divided in elements
+withsizes ∼190×190WFCpix2each,whichcorrespondto
+about 10′′×10′′(or≈2.5 pc×2.5 pc at the distance of the
+LMC).Thefirstisoplethinthecontourmapindicatestheleve l
+of 1σabove the mean background density, with σbeing the
+standarddeviationofthebackgrounddensity. Thesubseque nt
+levels are drawn in steps of 1 σ. All contours with density
+equal or higher than the 3 σlevel, considered as the statisti-
+callysignificantdensitythreshold,aredrawnwiththickli nes.
+All three star clusters are revealed in these maps as the dom-
+FIG. 4.— Radial surface density profiles of NGC 1983 (top) NGC 200 2
+(middle), and NGC 2010 (bottom) with the best-fitting King an d EFF mod-
+els superimposed. The error bars represent Poisson statist ics.Left: Density
+profiles of all detected stars, including the local field of th e LMC. The best-
+fitting EFFmodelisover-plotted withasolidblueline. Right: Stellar surface
+densityprofilesaftersubtracting thebackground densityl evelestimated from
+thebest-fitting EFFmodelshown intheplots on theleft. Both EFFand King
+models are overlaid. Since these clusters are not tidally tr uncated, the King
+models areused only tentatively for an estimation ofalimit ing radius forthe
+clusters. Both models, however, seem to fit very well the dens ity profiles at
+the inner parts of the clusters deriving comparable core rad ii. The derived
+structural parameters based on thebest-fitting EFF(solid b lueline) and King
+(dashed green line) models aregiven in Table 3.
+inant stellar concentrations in the observed regions. Thes e
+mapsdemonstratethatwhileNGC1983isacentrallyconcen-StellarStratificationinYoungStar Clustersin theLargeMa gellanicCloud 7
+trated compact cluster, NGC 2010 is a quite large and loose
+cluster, with NGC 2002beingan intermediatecase of a large
+compact concentration. Moreover, NGC 2002 appears to be
+the most spherical cluster in the sample, since NGC 1983 is
+moreellipticalandNGC2010ismoreamorphous,forwhich,
+as indicated by the isopleths of density ≥3σ, its most com-
+pact part is off-center. The sizes of the clusters as derived
+fromcircularannuliconsideredtoencirclethe3 σisoplethfor
+each cluster are given in Table 2. The correspondingsurface
+stellardensityofeachclusterisalso givenin thisTable.
+3.2.StellarSurfaceDensity Profiles
+Theobservedfieldscoverquitelargeareasaroundtheclus-
+tersinoursample,allowingtheconstructionofthestellar sur-
+facedensityprofilesoftheclustersatrelativelylargedis tances
+fromtheircenters. Wedividedtheareaofeachclusterincon -
+centric annuli of increasing steps outwards and we counted
+the number of stars within each annulus. We corrected the
+counted stellar numbers for incompleteness according to th e
+corresponding completeness factors on a star-by-star basi s.
+Completenessisafunctionofbothdistancefromthecentero f
+the cluster and magnitude. Let Ni,cbe the completeness cor-
+rectedstellar numberwithin the ith annulus. We obtainedthe
+stellar surface density, fi, by normalizing this number to the
+area of the correspondingannulusas fi=Ni,c/Ai, withAibe-
+ingtheareaoftheannulus. Theobservedfieldsdonotalways
+fullycoverthecompleteextentoftheclusters,and,theref ore,
+forthetruncatedannuli,weconsideredonlytheavailablea rea
+forthe estimationofthecorrespondingsurfacedensity.
+The “raw” radial stellar density profiles, meaning the sur-
+face stellar density as a function of distance from the cen-
+ter of each cluster, f(r), show a smooth drop outwards away
+fromthecentersoftheclusters. Theydroptoauniformlevel ,
+which represents the stellar density of the LMC field in the
+vicinityoftheclusters,anditismeasuredbyfittingthemod els
+ofElson,Fall &Freeman (1987)tothestellarsurfacedensit y
+profiles,log f(logr),(Figure4,left)asdescribedbelow. Inthe
+case ofNGC2002thereisa small increasein theoutskirtsof
+theprofileduetoanotherneighboringsmallstellarconcent ra-
+tion (see Figure 3). We do not consider this increase in our
+subsequentanalysisonthestructureofthiscluster. Theer rors
+in the profiles reflect the counting uncertainties, represen ting
+thePoissonstatistics.
+3.3.StructuralParameters
+We applied both the empirical model by King (1962) and
+the model of Elson,Fall& Freeman (1987) (from here-on
+EFF) to the stellar surface density profiles of the clusters i n
+orderto obtain theirstructural parameters. The EFF model i s
+suited for clusters which are not tidally limited, while Kin g’s
+empirical model represents tidally truncated clusters. Bo th
+models provide the opportunity to derive accurate characte r-
+istic radii for the clusters. The core radius ( rc) describes the
+distancefromthecenteroftheclusterwherethestellarden sity
+dropstothe halfofitscentralvalueandthetidalradius( rt) is
+thelimitwherethestellardensityoftheclusterdropstoze ro.
+The best-fittingEFF and King profilesare foundby perform-
+ing the Levenberg-Marquardt least-squares fit to the consid -
+eredfunctions. Thisfit was performedwith IDL3(Interactive
+Data Language) with the use of the specially developed pro-
+cedureMPFITFUN(Markwardt2008). Fortheapplicationof
+3http://www.ittvis.com/ProductServices/IDL.aspxKing’s model the stellar surface density of the cluster alon e
+(withnocontaminationfromthefield)isnecessary,whileth e
+EFF model does not require any field subtraction. We first
+apply the latter in order to estimate the core radius and the
+uniformbackgroundlevelforeachcluster. Thesubtraction of
+this density level from the measured surface density at each
+annulus gives the surface density profile of the cluster alon e,
+fromwhichwe derivethecoreandtidal radii.
+3.3.1.BestFittingEFFProfiles
+FromstudiesofyoungLMCclustersbyEFFitappearsthat
+these clusters are not tidally truncated. These authors dev el-
+opedamodelmoresuitabletodescribethestellarsurfacede n-
+sityprofileofsuchclusters:
+f(r)=f0(1+r2/α2)−γ/2+ffield, (2)
+wheref0isthecentralstellarsurfacedensity, aisameasureof
+the core radius and γis the power-law slope which describes
+the decrease of surface density of the cluster at large radii ;
+f(r)∝r−γ/2forr≫a. Theuniformbackgrounddensitylevel
+isrepresentedinEq.(2)by ffield. Themeasuredparameters α,
+γandffield, derivedfromthefitting procedureonourdataare
+givenforeachclusterinTable3. AccordingtoEFFmodel,the
+coreradius, rc,isgivenfromEq.(2)assumingnocontribution
+fromthefieldas:
+rc=α(22/γ−1)1/2. (3)
+The estimated core radii of the clusters derived from Eq. (3)
+are also given in Table 3. The best-fitting EFF models are
+shownsuperimposed(bluelines)onthestellarsurfacedens ity
+profilesofFigure4.
+3.3.2.BestfittingKingProfiles
+In order to construct the density profiles of the clusters
+alonewithnocontributionfromthefieldwesubtracttheback -
+ground density level, ffield, from the raw profiles of Figure 4
+(left). We then use the field-subtracted profile to derive an
+indicative tidal radius of the clusters as described by King
+(1962). Accordingtothismodelthedensityprofileofatidal ly
+truncatedclusterisgivenas:
+f(r)∝/parenleftBig1
+[1+(r
+rc)2]1
+2−1
+[1+(rt
+rc)2]1
+2/parenrightBig2
+, (4)
+wherefis the stellar surface density, rcandrtthe core and
+tidalradius,respectivelyand risthedistancefromthecenter.
+The tidal radius, which representsthe outskirtsof the clus ter,
+isfoundfromthe formula
+f(r)=f1(1/r−1/rt)2, (5)
+and the core radius, which describes the inner region of the
+cluster,isgivenfrom
+f(r)=f0
+1+(r/rc)2. (6)
+f0describesagainthecentralsurfacedensityoftheclustera nd
+f1isaconstant. Thebest-fittingKingprofilesapartfromcore
+and tidal radii for the clusters, deliver also the concentra tion
+parameters, c, definedas the logarithmicratio of tidal to core
+radiusc= log(rt/rc), which refers to the compactness of the
+cluster. The derived rtandcfor our clusters are given in Ta-
+ble 3, while rcderived from the best-fitting King profiles are8 Gouliermis,et al.
+TABLE3
+RESULTS OF THE APPLICATION OF BOTH EFFANDKING MODELS FIT TO THE OBSERVED RADIAL SURFACE DENSITY PROFIL ES OF THE CLUSTERS IN
+OUR SAMPLE .
+EFFProfiles King Profiles
+Cluster α γ ffield rc rt c
+(pc) (stars arcsec−2) (pc) (pc) log rt/rc
+NGC 1983 1.68 ±0.28 1.69 ±0.32 0.23 ±0.02 1.89 ±0.43 21.69 0.8
+NGC 2002 3.92 ±0.68 3.91 ±1.00 0.09 ±0.01 2.56 ±0.61 22.33 0.9
+NGC 2010 13.32 ±6.03 9.81 ±8.04 0.21 ±0.01 5.19 ±3.34 28.13 0.7
+NOTE. — Parameters αandγare derived from the best-fitting EFF profile according to Eq. (2). Core radii, rc, are derived from the same models with
+Eq. (3). Tidal radii, rt, are measured with Eq. (5) from the best-fitting King models. Concentration parameters, c, are estimated from the measured rt
+and core radii derived from King models and Eq. (6). All radii are given in physical units after a conversion 1′≃15 pc assuming a distance modulus of
+(m−M)0≃18.5 mag (Clementini etal. 2003;Alves 2004;Schaefer 2008)for all clusters.
+omitted, as they are found in excellent agreement with those
+derived above from the EFF profiles. The best-fitting King
+modelsof our clusters are shown with green dashed lines su-
+perimposed on the radial surface density profiles of Figure 4
+(right).
+In general, both King and EFF models are in good agree-
+ment to each other in the inner and intermediate regions of
+the clusters, which are equally well fitted by both models.
+However,basedonthefactthatyoungcompactclustersinthe
+LMCarenottidallytruncated,theEFF modelshouldbe con-
+sideredasthebestrepresentativeforthem. Ontheotherhan d,
+the decontaminationof the observedstellar samples from th e
+populationsofthelocalLMCfieldrequiresthedeterminatio n
+of a limiting radius, Rlim, for the clusters (§ 4.3). Since EFF
+models do not predict such a radius, we treat the tidal radii
+derived from the application of King’s theory as this radius
+for each cluster. It should be noted, though, that our cluste rs
+are not expectedto be dynamicallyrelaxedand therefore,th e
+results on their tidal radii based on the application of King
+profilefittingshould notbetakenliterallybutonlytentatively
+forthedeterminationofanindicative limitradiusfortheclus-
+ters, and for reasons of comparison. For example, as seen in
+the charts of Figure 3, NGC 1983 cannot be directly distin-
+guishedfromthe rich stellar field in whichit isembedded,as
+opposed, e.g., to NGC 2002, which shows up as a compact
+concentrationeasily separatedfromits surroundings.
+4.STELLARCONTENTOF THECLUSTERS
+In the stellar charts of the three observed regions, shown
+in Figure 3 (left), the star clusters immediately appear as t he
+most prominent stellar concentrations. These maps are con-
+structedonlyfromstars detectedwith photometricuncerta in-
+tiesσ≤0.1 mag in both F555W and F814W filters. In our
+subsequentanalysiswe consideronlythesestars foreachob -
+servedfield. The ACS/WFC field-of-viewofNGC 1983cov-
+ers 9 641 stars with good photometry, the one of NGC 2002
+4340,andthatofNGC20109006suchstars.
+4.1.Color-MagnitudeDiagrams
+Them555−m814,m814Color-MagnitudeDiagrams(CMDs)
+ofthetotal observedsamplesofstarsforall threeclusters are
+shown in Figure 5 (top panel). These CMDs of the Whole
+Areasdemonstrate the existence of a significant mixture of
+differentstellar populationsin each region;fromyoungst ars,
+which cover the main sequence (MS) to the turn-off,belong-
+ing to the clusters and their surrounding field, to the evolve d
+Red Giant Branch (RGB) and Red Clump (RC) stars, repre-
+senting mostly the local background LMC field. The CMDs
+of Figure 5 indicate that our photometry provides accuratemagnitudes for stars with m814∼<22 mag. It is interesting
+to notethat asseen fromthe CMDs of Figure5 (top),as well
+as from the stellar charts of Figure 3, the general field of the
+LMCinthe observedregionsshowsto varyin stellar density.
+Specifically, both NGC 1983 and NGC 2010 are located in
+muchdenserstellarfieldsthanNGC2002,withthelocalfield
+of NGC 1983 being particularly rich in red field stars. This
+is already demonstrated by the measurements of the surface
+stellardensity, ffield,ofthegeneralfieldofallthreeclustersin
+§3.3(Table3; column4).
+The tidal radii, rt, of the clusters, as they are derived in
+§ 3.3, basically define the spatial extent of each cluster, an d
+thereforewe treat the stars that are located at radial dista nces
+from the center of the clusters larger than rtas the best rep-
+resentativestellar populationsofthesurroundingfieldso fthe
+clusters. ThecorrespondingCMDs,forwhichweusetheterm
+Field CMDs , are shown in Figure 5 (middle panel). In these
+CMDs the features of both RGB and RC of the general field
+of the LMC are well apparent in all three observed regions.
+These populations are well documented as members of the
+LMCfieldinvariousHSTstudiesofthestarformationhistory
+(SFH) of this galaxy (e.g., Gehaet al. 1998; Holtzmanet al.
+1999; Castroet al. 2001; Smecker-Haneetal. 2002). Specif-
+ically, the SFH of the LMC is found to be continuous with
+almost constant star formationrate (SFR) for over ∼10 Gyr,
+withanincreaseoftheSFRataround1-4Gyrago(forasum-
+mary see Gouliermiset al. 2006, their § 3.1). This old LMC
+field populationis present in all the Field CMDs of Figure 5.
+Since it covers the same parts in all three of them, we define
+the age-limits of these stars only on the CMD of the field of
+NGC 2010. With the use of the Padova grid of evolutionary
+models (Girardiet al. 2002) for insignificant extinction an d
+metallicity and distance typical for the LMC these limits ar e
+foundtobe 1 ∼<τ/Gyr∼<10.
+Moreover,the Field CMDs of Figure 5 include also young
+MS stars, clearly suggesting that the general regions, wher e
+all three clusters are located, include young stellar popul a-
+tions that are not members of the clusters. This suggestion
+doesnotcontradictpreviousstudiesonsuchclustersandth eir
+surroundings. Indeed, young compact clusters in the LMC
+areknownto formtogetherin largerstructuresof youngstel -
+larpopulations,the StellarAggregates (e.g.,Oeyet al.2008),
+the size of which seems to correlate with the duration of star
+formation in them (e.g., Efremov&Elmegreen 1999). Such
+clusters are not isolated as they may be formed in binary or
+multiple cluster systems (Dieballetal. 2002). An inspecti on
+ofthegeneralregionswheretheclustersofoursamplearelo -
+catedverifiesthatnoneofthemisanexceptiontothisgenera l
+rule. Wide-fieldimagesoftheregionsaroundtheclustersan dStellarStratificationinYoungStar Clustersin theLargeMa gellanicCloud 9
+FIG. 5.— The m 555−m814vs. m814CMDs of the stars detected with the best photometry (uncerta intiesσ≤0.1 mag in both filters) in the observed HST
+ACS/WFCareas oftheLMCclusters ofoursample: NGC1983(lef t), NGC2002(middle) andNGC2010(right). Top: Thecomplete CMDsofallstarsdetected
+in the whole observed areas, which demonstrate the mixture o f the young stellar populations of the clusters (MS) with tho se of their surrounding field, and
+the evolved ones of the general LMC field (RGB, RC). Middle: The CMDs of the stars located outside the rtof the clusters, representing the most probable
+populations of the local ambient of each cluster. Bottom: TheCMDs of the stars comprised within the rcof each cluster, which represent best the true CMDs of
+the clusters. Isochrones from the grid of evolutionary mode ls by Girardi etal. (2002) are overlaid for an estimate of the age of the clusters and the old field (see
+§ 4.1).
+thewholeextentoftheLMCin B,RandIwereretrievedfrom
+theSuperCOSMOS Sky Survey (Hamblyet al. 2001), avail-
+able at the Aladin Sky Atlas (Bonnareletal. 2000). These
+images show that indeed all three clusters are members of
+larger young structures. Specifically, NGC 1983 is located
+at the central part of the star forming Shapley Constellation
+II(Shapley1951;McKibbenNail &Shapley1953)thatcoin-
+cideswiththesupergiantshell(SGS) LMC3(Meaburn1980),
+NGC2002belongstothenorthernpartof ConstellationIII or
+SGSLMC 4, and NGC 2010 is located in one of the eastern
+nebularfilamentsofSGS LMC9inConstellationIX .
+In order to compare these MS field populations with the
+actual stellar populations of the clusters we select the sta rs
+contained in the core radii of the clusters (see § 3.3; Table 3 ,
+column 5), as the most probable stellar members, and we
+construct the correspondingCMDs, which are shown in Fig-
+ure 5 (bottom panel). A comparison of the Field CMDs withthe CMDs of the clusters within their core radii shows that
+there is a confusion between the young field populationsand
+thoseoftheclusters,especiallyforNGC2002andNGC1983.
+The case of NGC 2010 is more straightforward, as it can be
+seen that while the cluster is somewhat evolved (see discus-
+sion below), it seems that it is ‘embedded’in a youngerfield
+with brighter MS stars. On the other hand, NGC 1983 and
+NGC2002aremorecomplicatedcasesasbothfieldandclus-
+tersseem to comprisethe same typeof MS stars. Takinginto
+accountthesimilaritiesoffieldstarstoclustermembersin the
+CMDs, theuse ofourmethodforthe statistical decontamina-
+tionoftheobservedstellarsamplesfromthefieldpopulatio ns,
+discussed in § 4.3, becomes quite important. As we show in
+§ 4.4 this method allows us an accurate statistical determin a-
+tionofthe mostprobablestellar membersofeachcluster.
+4.2.Agesof theClusters10 Gouliermis,etal.
+WeusetheCMDswithinthecoreradiioftheclusterstode-
+rive an estimation of their ages. In the correspondingCMDs
+of Figure 5 (bottom panel), the best-fitting isochrones from
+the Padova grid of evolutionary models (Girardiet al. 2002)
+are also plotted. The process of isochrone fitting of the ob-
+served CMD results in an accurate measurement of the age
+of the system, providingthat there are distinguished featu res
+in the CMD to be fitted. Moreover, a good estimate of the
+distance and the interstellar reddening of the cluster is re -
+quired. For clusters in the LMC, visual extinction is known
+to vary around very low values (e.g., deGrijsetal. 2002;
+Gouliermiset al. 2004; Kerber&Santiago 2006), while the
+distance of the galaxy is also well defined (e.g., Alves
+2004; Schaefer 2008). Indeed, for our age estimation we
+had to vary little both of these parameters around typi-
+cal values. Since the ACS photometric system is slightly
+different from the standard Johnson-Cousins system, we
+estimate the mean extinction in the F555W band, A555,
+which we arbitrarily symbolize as AV, using the compu-
+tations by DaRio, Gouliermis&Henning (2009). These
+authors derive a conversion R555=A555/E(m555−m814)≃
+2.18 andA555/A814≃1.85 assuming the extinction law of
+Cardelliet al. (1989) with RV≃3.1. In all cases we assume
+the typical metallicity of the LMC of Z≃0.3 - 0.5Z⊙(e.g.,
+Westerlund 1997), and thereforewe use the models designed
+forZ=0.008. The derived ages and indicative visual extinc-
+tionforallthreeclustersaregiveninTable2. Letusconsid er
+theresultsforeachclusterindividually:
+(i) NGC 1983 : The CMD of the most prominent member
+stars of the cluster consists only of the MS, indicating that
+the cluster is quite young, but making the accurate determi-
+nationofits agedifficult. Indicatively,we applytheyoung est
+isochronethat fits the brighterMS stars with m814∼14 mag,
+which shows that the cluster has an age about 28 Myr. The
+selected isochrone fits best the blue part of the MS for a
+nominal distance modulus ( m−M)0≃18.45 mag and red-
+dening of E(m555−m814)∼<0.05 mag, which corresponds to
+AV≃0.17mag.
+(ii) NGC 2002 : The CMD of NGC 2002 within the rcof
+the cluster includes the significant number of five Red Su-
+pergiants (RSG) located at 2 .2∼<(m555−m814)∼<2.6 mag
+and 11∼<m814∼<12 mag, which helps to constrain ac-
+curately the age of the system. However, the isochrone for
+metallicity Z= 0.008 that seems to fit best this cluster with
+an age of τ≃18 Myr, cannot reach the RSG stars, which
+thus require the consideration of models of higher metallic -
+ity. Indeed the next available models from the Padova grid
+are those of almost solar metallicity of Z= 0.019. The cor-
+responding isochrone for the same age (plotted with a green
+line in Figure 5) does fit the RSG, implying that either there
+is a significant changein the metallicity of the cluster, or t hat
+the evolutionary models need to be fine-tuned for RSGs in
+low metallicities. In any case, our data give a fixed age for
+NGC 2002 of around 18 Myr. It should be noted that while
+the high metallicity model requires a modest AV≃0.12 mag
+to fit the CMD, the low metallicity model predicts a higher
+extinctionof AV≃0.19mag.
+(iii) NGC 2010 : The CMD of Figure 5 (bottom) for
+NGC 2010 clearly suggests that this is the oldest cluster
+in our sample. The existence within rcof bright stars of
+m814≃15 mag, evolved away from the MS allows us an ac-
+curate determination of the cluster age of τ≃159 Myr for
+(m−M)0≃18.5 mag and AV≃0.17 mag. However, the ap-pearance in this spatially constrained CMD of a prominent
+RGB population, clearly suggests that the cluster may com-
+priseamixtureofdifferentstellarpopulations.
+4.3.FieldDecontamination
+As we discuss earlier and seen from the CMDs of Fig-
+ure 5, all three observed regions suffer from significant con -
+tamination by field stars. With no information about cluster -
+membership probabilities for the observed stars, obtained ,
+e.g., from radial velocities and/or proper motions, the qua n-
+titativedecontaminationoftheclustersfromthefieldstar son
+astatisticalbasisbecomesafundamentalmethodtoobtaint he
+CMD ofthecompletesampleoftruestellar membersofeach
+cluster. We decontaminate, thus, the observed CMDs from
+thecontributionofthe stellar membersofthesurroundingl o-
+cal field of the LMC with the use of a sophisticated random-
+subtractiontechnique.
+This field-subtraction algorithm, originally designed by
+Bonatto& Bica(2007),isalreadydevelopedinPaperI,where
+it is also thoroughly described. According to this techniqu e,
+the successful decontamination of the clusters from the fiel d
+isbasedonthedefinitionofthe limitingradius, Rlim, foreach
+cluster. For the clusters in our sample, Rlimis given by the
+tidal radii, rt, estimated in § 3.3.2with the use of King’s the-
+ory. All stars located at distances r>Rlimfrom the center
+of the cluster are treated as field stars, while the rest (with
+r≤Rlim)areconsideredasthemostprobablecluster-member
+stars. Assuming a homogeneous field-star distribution, the
+number density of field stars is calculated in the Field CMD
+(Figure5;middle)constructedforstarswith r>Rlim,consid-
+eringalsothe observationaluncertaintiesinthephotomet ry.
+The decontamination process is then applied in three steps
+for each cluster: (i) Division of the CMDs of the stars with
+r≤Rlim(clusterregion)andof those with r>Rlim(offsetre-
+gion), respectively, in 2D cells with the same axes along the
+m814and (m 555−m814) directions. (ii) Calculation of the ex-
+pected number density of field stars in each cell in the CMD
+of the offset region. (iii) Random subtractionof the expect ed
+number of field stars from each cell from the CMD of the
+cluster region. The algorithm is applied several times usin g
+different cell sizes ( ∆m814,∆(m555−m814)) so that many dif-
+ferentdecontaminationresultsarederived. Fromtheseres ults
+we calculate the final probability that a star is identified as a
+‘true’ cluster member, and the final decontaminatedCMD of
+theclustercontainsonlythestarswiththehighestprobabi lity
+ofbeingclustermembers. Thisprocessallowsustominimize
+any artificial effects intrinsic to the method itself. A deta iled
+descriptionofthemathematicalformulationofourmethodf or
+field-subtractionisgiveninPaperI.
+4.4.StellarPopulationsin theClusters
+We performed the field-subtraction process, as described
+above,15timesforeachclusterconsideringcombinationsf or
+the CMD cell sizes of ∆m814= 0.5, 1.0,1.5, 2.5and3.0 mag
+and∆(m555−m814)=1.0,1.5and2.0mag. Weobtained,thus,
+the probability for each star in each sample of being a mem-
+ber of cluster population. Since our method for diagnosis
+of stellar stratification is completeness-sensitive in the sub-
+sequent analysis we limit the sample of stellar members of
+all three clusters to those detected with our photometry wit h
+completeness ≥70%. The derived stellar samples comprise
+424 stars in NGC 1983, 1 102 stars in NGC 2002 and 2 265
+stars in NGC 2010 with probability larger than 70% of be-Stellar Stratificationin YoungStar ClustersintheLargeMa gellanicCloud 11
+FIG. 6.— Them 555−m814vs. m814CMDsof the stars retained as possible members of the cluster s after the statistical removal of thecontribution of theLM C
+field from the originally observed CMDs of Figure 5 (top panel ). The Field CMDs of the offset regions around the clusters (F igure 5, middle panel) are used as
+representative of the contaminating populations. Stars wi th probabilities >70% of being true members of the clusters (see § 4.3) are consi dered for these plots.
+Stars identified with 100% membership are plotted with thick points.
+ing true cluster members. Figure 6 shows the correspond-
+ingfield-star-decontaminatedCMDsoftheclusterregionsf or
+all clusters. Stars found by the field-subtractionmethod wi th
+probability 100% of being cluster members are plotted with
+thick points. From these CMDs it can be seen that our statis-
+tical subtractionof the CMD contaminationby the field stars
+was efficient in removingbothyoungfield populationsof the
+immediate surroundings of the clusters, andold populations
+ofthegeneralLMCfield. Specifically,forNGC1983theMS
+stars remaining in the CMD of Figure 6, after the applica-
+tionofthefield-subtraction,formasharpnarrowMSdownto
+m814≃21 mag. This MS resembles that of the CMD within
+thercoftheclustershowninFigure5(bottom),withnoindi-
+cation of the broad lower MS stars of the Field CMD of Fig-
+ure 5 (middle). The CMD of NGC 2002 also demonstrates
+a very sharp MS similar to the rcCMD of the cluster. The
+RSGslocatedwithinthe rcoftheclusteralsoremainedin the
+CMD of Figure 6 with 100% of membership probability. As
+farasNGC2010isconcerned,thestatistical fieldsubtracti on
+removed completely the bright MS young stars identified in
+§ 4.1 as young field stars, while the older MS and evolved
+stars remained with high probabilities of being true cluste r
+members.
+It should be noted that the lower main sequence of the
+CMDs of the clusters may be also sensitive to any potential
+pre-mainsequence(PMS)populationthatmightexist insuch
+young clusters. However, due to observational constrains,
+we would be able to detect only the turn-on of these PMS
+stars, which is known to coincide with the MSTO (see, e.g.,
+Notaet al. 2006; Gouliermiset al. 2007), very close to our
+detection limit. As a consequence, any PMS population in
+the clusters would be seriously affected by the incomplete-
+ness in our photometry and therefore does not have any sig-
+nificance in the measured stellar numbers. Another bias that
+may affect the final numbers of low main sequence cluster-
+membersis differentialreddeningthat could displace star s to
+the red-faint part of the CMD. However, differential redden -
+ing in young LMC clusters is known to account for no more
+than∆E(B−V)≃0.1 (e.g., Dirschet al. 2000; Baumeetal.
+2007),whichiswell coveredbythecoloruncertaintiesinou r
+photometryatthelowmainsequence. Asaconsequence,dif-
+ferentialreddeningisnotconsideredinourtreatmentforfi eld
+subtraction. It is interesting to note that while the Cluste r
+CMDs of Figure 6 for both NGC 2002 and NGC 2010 show
+thesamefeatureswiththecorresponding rcCMDsoftheclus-ters (Figure 5, bottom), that of NGC 1983 shows clear indi-
+cations of multiple young populations with m814∼<15 mag.
+Indeed, as we discuss in § 4.1, multiplicity is a typical char -
+acteristicofyoungLMCclusters,andnaturallyitmayleadt o
+a mergingprocessand mixingof differentstellar populatio ns
+within one cluster (see e.g., PortegiesZwart &Rusli 2007).
+As we cannot verify or reject this scenario for NGC 1983,
+we treat all 424 stars that meet our membership criteria as
+truemembersofthecluster. Ourdiagnosticmethodforstell ar
+stratification is applied in the following section on the fina l
+catalogs of stellar members of the clusters, with 70% mem-
+bership,theCMDsofwhichare showninFigure6.
+5.DIAGNOSIS OF STELLARSTRATIFICATIONIN THECLUSTERS
+Theeffective radius method for the diagnosis of stellar
+stratification is based on the assumption that the observabl e
+counterpartof the Spitzer radius , theeffectiveradius , of stars
+inaspecificmagnitude(ormass)rangewillbeauniquefunc-
+tion of this range if the system is segregated. In Paper I the
+comparison of the results of this method with those from the
+‘classical’ methods of the radial dependence of the cluster s
+LFs and MFs, showed that the effective radius method be-
+haves more efficiently in the detection of the phenomenon,
+providing direct proof of stellar stratification in star clu sters
+without considering any model or theoretical prediction, a nd
+withnoapplicationofanyfunctionalfit. Inthissectionwea p-
+ply this method for the detection and quantificationof stell ar
+stratificationto theyoungLMCclustersofoursample.
+5.1.EffectiveRadiiof theStarClusters
+We calculated the effective radii, reff, of the stars in differ-
+ent magnitude ranges, according to Eq. (1), using the com-
+pletenesscorrectednumbersofstarsforallthreeclusters . We
+performed this calculation only for the star-members of the
+clusters, as they are established in § 4.4 (CMDs of Figure 6),
+and we used their brightness measured in the F814W filter,
+sincethisfilterprovidedthemostcompleteandaccuratepho -
+tometry (see § 2.1). In order to identify any functional rela -
+tion between reffand the correspondingbrightness range, we
+binnedthestarsaccordingtotheirmagnitudes. Sincetheme a-
+surement of reffdepends strongly to the number of stars per
+bin, this processnaturallyinherits systematic uncertain tiesto
+thederivedrelations,becauseofthefactthatsomestarsco uld
+belong to neighboring bins due to photometric uncertaintie s.
+Therefore,weperformedthesamebinningprocesspercluste r12 Gouliermis,etal.
+FIG. 7.— Estimated effective radii of stars within different ma gnitude
+ranges vs. the corresponding mean magnitude for all three cl usters.
+several times by changingthe bin sizes and/orthe magnitude
+limits,withinwhichthestarsarebinned. Wefoundthatarea -
+sonablebinsize,whichprovidesgoodstatisticsandallows for
+the detailed observation of the dependence of reffon magni-
+tude, is that of 0.5 ∼<∆m814∼<1.0 mag. For this range of
+binsizeswedidnotfindanysignificantchangeinthegeneral
+trend ofreff(m814), which is also found to be independent of
+theselectedmagnitudelimitsforbinningthe stars.
+In Figure 7 we indicatively show the relations between
+the computed effective radii and the corresponding magni-
+tude bins derived from two different binning processes with
+bin sizes of 1 mag and magnitude ranges shifted by 0.5 mag
+from each other. The combination of both distributions with
+thesespecificbinningcharacteristics(eachplottedwithd iffer-
+entsymbols)providesmeasurementsof reffforevery0.5magrange. The derived trends of reff(m814), shown in these plots,
+arenotdifferentfromthecorrespondingdistributionsder ived
+for∆m814=0.5mag,exceptofbeingsomewhatsmoother. In
+general the relations of Figure 7 are representative of thos e
+found from any applied binning, and therefore we consider
+them in the followingsection for the derivationof our resul ts
+concerningstellar stratificationintheclusters.
+5.2.StellarStratificationintheStarClusters
+Thereff(m814) graphs of Figure 7 are plotted on the same
+scaleforreasonsofcomparisonbetweentheclusters. Inthe se
+plots it is shown that the effective radius does behave as a
+function of magnitude, but not necessarily providing proof
+of stellar stratification for all clusters. A definite indica tion
+of stellar stratification would require the effective radii of
+the faintest stars to be systematically larger than those of the
+brightest ones for the whole observed brightness range. Thi s
+is not the behavior we observe at least for two of the three
+clustersofoursample. Indeed,aswediscussinPaperI, reffis
+notamonotonicfunctionofbrightness,andmasssegregatio n,
+if any, is not expectedto behavein the same mannerin every
+cluster, but rather to dependon the intrinsic stellar chara cter-
+isticsofeachofthem. Forexample,whileinNGC2002there
+is a trend of fainter stars having systematically larger eff ec-
+tive radii for the whole luminosity range, this is not the cas e
+forNGC1983thatshowsthistrendforonlya limitedmagni-
+tude range, and certainly not for NGC 2010, which one may
+argueisnotevensegregated.
+Specifically, for NGC 1983, the brightest magnitude bins,
+downtom814≃16mag,seemtoincludestarsthatarelocated
+awayfromthecentralpartofthecluster. Indeed,thebright est
+stars in the Cluster CMD (Figure 6) with m814∼<15.5 mag
+are found to be distributedwithin the largest reffof about 1 .′2
+away from the center of the cluster and not closer than about
+0.′6. On the other hand, taking into account the remaining
+magnitude bins of m814≥15.5 mag, the graph of reff(m814)
+for NGC 1983 provides clear indications that the cluster is
+segregated with an almost monotonic dependency of larger
+refftofainterstars.
+ThegraphforNGC2002isquitedifferent,givingevidence
+that the cluster is segregatedin the whole extend of observe d
+stellar luminosities. There is, however, a dependency of th e
+appearance of a relation between reffand brightness on the
+consideredmagnituderange. Indeed,whilethethreebright est
+magnitude bins, comprising six stars (the five RSG and the
+brightest MS star), correspond to the smallest effective ra dii,
+the next fainter bins, which include six more MS stars, show
+that these stars are located well away from the center of the
+cluster. Moreover, from the plot of Figure 7 for the fainter
+stars, it is almost self-evident that they are segregated as the
+correspondingrelation reff(m814) demonstrates an increase of
+reffwithbrightness.
+The behavior of this relation for NGC 2010 is quite differ-
+ent from both the previouscases, as there is no general trend
+for larger effective radii with fainter magnitudes observa ble,
+and thus the cluster behavesas if there is no significant stra t-
+ification occurring. Specifically, apart from the fact that t he
+threebrightestmagnitudebinsareinconclusive,formoref aint
+stars there is a “plateau” in the function reff(m814), which ap-
+pearsratherflatformagnitudesdownto m814≃19mag,while
+for even fainter stars, down to the faintest observed magni-
+tude, the dependence of reffoverm814appears again, giving
+evidenceof mild mass segregationfor the faintest stars at t he
+outskirtsofthecluster.Stellar Stratificationin YoungStar ClustersintheLargeMa gellanicCloud 13
+TABLE4
+MAGNITUDE LIMITS ,THE CORRESPONDING EFFECTIVE RADII LIMITS ,
+AND THE DERIVED SLOPES SstratINDICATIVE OF THE DEGREE OF
+STRATIFICATION FOR SELECTED BRIGHTNESS RANGES FOR ALL THRE E
+CLUSTERS OF OUR SAMPLE .
+Cluster m814∆m814 reff ∆reff Sstrat
+(mag) (mag) (arcmin) (arcmin) (mag/arcmin)
+NGC 1983 12.0 -15.5 3.5 1.20 - 0.58 −0.62 −0.177
+15.5 -22.0 6.5 0.58 - 0.94 +0.36 +0.055
+12.0 -22.0 10.0 1.20 - 0.94 −0.26 −0.026
+NGC 2002 11.0 -14.5 3.5 0.11 - 0.99 +0.88 +0.251
+15.0 -23.0 8.0 0.49 - 1.23 +0.74 +0.093
+11.0 -23.0 12.0 0.11 - 1.23 +1.12 +0.093
+NGC 2010 13.0 -14.0 1.0 1.24 - 0.10 −1.14 −1.140
+14.5 -18.5 4.0 1.02 - 1.07 +0.05 +0.013
+19.0 -23.5 4.5 0.89 - 1.51 +0.62 +0.138
+13.0 -23.5 10.5 1.24 - 1.51 +0.27 +0.026
+NOTE. — Negative values of Sstrator values close to zero indicate non-
+segregated clusters atleast within thespecificmagnitude l imits (see§5.2). The
+third row for every cluster refers to the whole observed magn itude range.
+5.3.DegreeofStellarStratification
+In order to parametrize the phenomenon of stellar stratifi-
+cation as it exhibits (or not) itself in the considered clust ers,
+we assume as null hypothesisthat all clusters are segregate d,
+but to a different degree. This is evident, as discussed above,
+from thereffvs. magnitude plots of Figure 7. This degree of
+stratification in each cluster may be, thus, expressed in terms
+of the brightness range of its stars, and the effective radiu s
+withinwhichtheyareconfined. Asaconsequence,inorderto
+quantifystellar stratificationasobservedin everycluste r,one
+may use the two primary output parameters of the effective
+radius method, i.e., the magnitude range and thecorrespond-
+ing effective radii of segregation. These parameters actually
+represent the ranges in magnitudes and radial distances tha t
+definethe slope ofthe observedrelation reff(m814). Naturally,
+for a non-segregated cluster this slope, Sstrat, should be ≤0,
+as it is derived from the graphsof Figure 7. Considering that
+asteeperpositiveslopeofthisrelationrepresents ahigherde-
+gree of stratification , we parametrize stellar stratification in
+the clusters by measuring Sstrat≡∆reff/∆m814and quantify
+thephenomenonthroughthis degree.
+Asmentionedabove, reff(m814)doesnotrepresentasystem-
+atic relation throughoutthe whole observedmagnitude rang e
+in everycluster, but, as seen in Figure 7, it shows differenc es
+depending on the selected magnitudes. For every cluster we
+select one indicative magnitude limit to separate the brigh t
+from the faint stars and measure the correspondingdegree of
+stratification. This selection is based on the magnitude bin ,
+where we observe dramatic changes in the relation reff(m814)
+in each cluster. As a consequence, we select two magnitude
+ranges, one spanning from the brightest magnitudebin down
+totheselectedlimit,andtheotherstartingatthislimitun tilthe
+faintest magnitude bin, and we estimate their correspondin g
+Sstratforallthreeclusters. Thesemagnituderanges,thecorre-
+sponding ∆m814, and∆reff, and the derived Sstratare givenin
+Table 4, along with the derived Sstratfor the whole observed
+magnitude range for every cluster. The values of Sstrat, given
+inthistable,correspondtothe degreeofstratification foreach
+clusterineverygivenbrightnessrange.
+5.4.ResultsandDiscussion
+From the Sstratmeasurements of Table 4 one can see that
+while indeed the bright stars of NGC 1983 are not segre-gated, stars with m814∼>15.5 mag show a definite trend
+in their relation reff(m814), indicative of stratification with a
+degree of +0.06. On average, however, the cluster does not
+appear to demonstrate stellar stratification. We character ize,
+thus, NGC 1983 as partiallysegregated cluster. On the other
+hand,NGC2002showsadefinitetrendoflarger reffforfainter
+stars within both selected magnitude ranges, as well as for
+the whole extend of observed magnitudes. It is a definite
+case of stellar stratification with a degree of stratificatio n of
++0.09. Finally, NGC 2010 is the most peculiar case, as the
+results for the few brightest stars are quite inconclusive, and
+forthe fainter stars with 14.5 ∼<m814∼<18.5magthe clus-
+terappearsslightlysegregated,witharelation reff(m814)thatis
+ratherflat. Ontheotherhand,themostprominenttrendinthi s
+relationappearsfor evenfainterstars with m814∼>18.5mag
+and a degree of stratification Sstrat≃+0.14. However, if in-
+deed the cluster was segregatedfor this magnituderange, th e
+observed trend should appear as a continuation of the (non
+stratified)relation reff(m814)ofthebrighterstars. Morespecif-
+icallythe reff,wherethetrendstartsfor m814≃19mag(∼0.′9)
+issmallerthan that of the faintest non segregated stars of
+m814≃18.5 mag with reff∼1.′1. This behavior of the rela-
+tionreff(m814)correspondsto fluctuationsin thespatial distri-
+butionofthe faintstarsaroundthecenterofthe clusterrat her
+than to true segregation. As a consequence, we characterize
+NGC2010as notsegregated cluster.
+While the aforementioned results on three clusters cannot
+be used for a statistically complete interpretation of the p he-
+nomenon of stellar stratification in young LMC clusters in
+connectiontotheirindividualcharacteristics,it is wort hwhile
+to search for any dependence of the degree of stratification
+to the structural and evolutionary parameters we derived fo r
+eachclusterinoursample. Concerningthestructuralparam e-
+ters shown in Table 3, all clusters are foundwith comparable
+concentration parameters, and tidal radii, rt, that are not far
+from each other. The most important difference we find in
+these parametersisthe coreradius, rc, of NGC 2010,the non
+(or less) segregatedcluster, which is at least two times lar ger
+thanthoseoftheothertwoclusters. Inaddition,fromthech ar-
+acteristics of Table 2 it is shown that this cluster is the mos t
+extendedand least dense in the sample. Moreover, the CMD
+ofthecluster(Figure6)showsindicationsofmulti-ageste llar
+populations,suggestingthat NGC2010isa clusterwith mul-14 Gouliermis,etal.
+tiplestellargenerations. Insuchclustersalargefractio nofthe
+firstgenerationofstarsislostearlyintheclusterevoluti ondue
+toitsexpansionandstrippingofitsouterlayersresulting from
+earlymasslossassociated(e.g.,D’Ercoleetal.2008). Ing en-
+eral, the loose appearance of a low-density cluster with lar ge
+rc, comes to the support of this speculation. While this can-
+notbeverifiedwiththepresentdata,itwouldbereasonablet o
+suggestthatthelackofstellarstratificationinthisclust erwill
+maketheeffectofearlymasslossduetostellarevolutionle ss
+destructiveleadingtolongerlifetimes(Vesperiniet al.2 008).
+As far as NGC 2002 is concerned, it is interesting to note
+that this cluster, being the most segregated in the sample, i s
+actuallytheyoungest(Table2)andlocatedinaregion,whic h
+appears to correspond to the emptiest general LMC field of
+the three (Table 3). Naturally, these observations allow us to
+suggest that detection of stellar stratification may have be en
+benefitted by the loose contaminating stellar field, and that
+the observed stellar stratification in NGC 2002 may be pri-
+mordial in nature due to the star formationprocess, or due to
+very early dynamical evolution. Indeed recent studies show
+that cool, fractal clusters can dynamically mass segregate on
+timescalescomparabletotheircrossing-times,farshorte rthan
+usuallyexpected(e.g.,Allisonet al.2009b). Theexactnat ure,
+though, of the observed segregation cannot be fully under-
+stood without detailed kinematic information about the clu s-
+termembers,ordeeperphotometrythatwillprovidethecom-
+plete IMF of the cluster. Finally, the observed partialstel-
+lar stratification in NGC 1983 is driven by the fact that the
+brighteststarsof theclusterare distributedina wide rang eof
+distances from the cluster center. In general, the behavior of
+reffas function of stellar magnitude cannot be connected to
+any specific differenceof its characteristicsfrom those of the
+other clusters, apart from the fact that the cluster belongs to
+the richest region in field populations, and it is a rather ell ip-
+tical cluster.
+6.CONCLUSIONS
+We present a coherent comparative investigation of stellar
+stratificationinthreeyoungLMCclusterswiththeapplicat ion
+ofarobustmethodfortheassessment ofthisphenomenonon
+deep HST imaging. In a cluster where stellar stratification
+occurs, the segregated brighter stars are expected to be mor e
+centrally concentrated than the non-segregated fainter st ars.
+Stellar stratification can, thus, be investigated from the d e-
+pendence of the radial extent of stars in specific magnitude
+(mass) ranges on the correspondingmean magnitude (mass).
+According to this notion, in Paper I (Gouliermisetal. 2009)
+we developed and verified the efficiency of a robust method
+fortheassessmentofstellarstratificationinstarcluster s. This
+methodisbasedonthecalculationofthemean-squareradius ,
+theeffectiveradius reff,ofstarsindifferentmagnituderanges,
+andthe investigationof itsdependenceonmagnitudeasindi -
+cationofstellar stratificationinthecluster.
+In the present study we apply the effective radius method
+for the detection and quantification of stellar stratificati on in
+young star clusters in the LMC. We select three clusters ob-
+servedwiththehighresolvingefficiencyofHST/ACS,specif -
+ically NGC 1983, NGC 2002 and NGC 2010, on the basis
+of the differences in their appearance, structure, stellar con-
+tent, and surroundingstellar field. Our photometrydeliver ed
+complete stellar catalogs down to m814≃22 - 23 mag for all
+threeobservedfields(§2.1). Thestellarsurfacedensityma ps
+of the observed regions, constructed from star counts on the
+photometric catalogs, demonstrate that all three clusters aremorphologically quite different from each other. NGC 1983
+appears to be a centrally concentrated, rather elliptical c om-
+pact cluster, NGC 2002 a compact, spherical stellar concen-
+tration, and NGC 2010 a large, loose and rather amorphous
+cluster(§3.1).
+TheapplicationofbothEFF(Elson,Fall &Freeman 1987)
+and King’s (King 1962) models on the stellar surface den-
+sity profiles of the clusters allowed us the measurement of
+the background stellar field density of the regions, and the
+estimation of the core radii, rc, of the clusters. While these
+clusters do not seem to be tidally truncated, a tidal radius, rt
+is estimated, indicative of the limiting radius of each clus ter
+(§ 3). The latter is essential for the successful applicatio n
+of a random field subtraction technique for the decontamina-
+tionofthestellarsamplesoftheclustersfromtheirsurrou nd-
+ing young stellar ambient and the general LMC field (§ 4.3).
+Wederivetheagesoftheclustersfromthestellarpopulatio ns
+comprised within the rcof the clusters and we find ages of
+τ≃28Myr,18Myrand159MyrforNGC1983,NGC2002,
+andNGC2010respectively(§4.2).
+We finally apply the effective radius method for assess-
+ing stellar stratification in the clusters and for its quanti tative
+study (§ 5). We bin the stars according to their magnitudes
+intheF814Wfilter,andweestimate thecorrespondingeffec-
+tive radii, reff, for every cluster in our sample. We then plot
+the derived radii versus the corresponding mean magnitude
+andwe investigatethefunctionalrelation, reff(m814), between
+these parameters. With our method it is shown that stellar
+stratification behaves differently in every cluster. NGC 19 83
+appears to be partially segregated , since its brightest stellar
+content does not appear to be centrally concentrated, while
+its fainterstars show a mild dependencyof refftowardslower
+values for fainter magnitudes. The results on the bright sta rs
+of NGC 2010 show evidence of lack of stratification, since
+the brightest stars are located quite far away from its cente r.
+As far as the faint stars are concerned they show a rather flat
+reff(m814) relation, with no evidence of segregationin this re-
+lationforthefaintest stars. Consequently,thiscluster i schar-
+acterized as not segregated . Finally, NGC 2002 is found to
+be well segregatedfor both its brighter and fainter stars. I t is
+a clear case of proofof stellar stratification with the effec tive
+radiusmethod.
+We propose the slopeSstrat≡∆reff/∆m814, measured for
+stars in selected magnitude ranges, as well as in the whole
+magnitude range, as the most appropriate parameter for the
+quantification of the degree of stratification , and we use it to
+characterizethe phenomenonas it is observedin the cluster s.
+Fromthederivedvaluesof Sstrat(§5.3)wefindthatNGC2002
+is the most strongly segregated cluster in the sample with a
+degreeofstratification Sstrat≃+0.09,NGC1983isapartially
+segregatedclusterwith Sstrat≃+0.06forthefainteststarswith
+m814∼>15.5 mag, and NGC 2010 is not segregatedwith in-
+dicativeSstrat≃+0.03(§ 5.4). Finally, it should be notedthat
+the extension of this study to a larger sample of LMC clus-
+tersobservedwith HST will providea morecompletepicture
+of the phenomenonof stellar stratification. Naturally, the de-
+velopmentof such a comparativescheme, whichwill include
+starclustersinthewholeextentoftheevolutionarysequen ce,
+requires a comprehensiveset of ACS and, in the near future,
+WFC3 observations.
+D. A. G. kindly acknowledges the support of the Ger-
+man Research Foundation (Deutsche Forschungsgemein-Stellar Stratificationin YoungStar ClustersintheLargeMa gellanicCloud 15
+schaft, DFG) through grant GO 1659/1-2, and the German
+Aerospace Center (Deutsche Zentrum für Luft- und Raum-
+fahrt,DLR)throughgrant50OR0908. Sincereacknowledge-
+ments go also to M. Kontizas and E. Kontizas the collabora-
+tionwithwhomoriginallyplantedthe seedoftheideafortheEffective Radius method. Based on observations made with
+the NASA/ESA Hubble Space Telescope , obtained from the
+data archive at the Space Telescope Science Institute. STSc I
+isoperatedbytheAssociationofUniversitiesforResearch in
+Astronomy,Inc. underNASAcontractNAS5-26555.
+Facilities: HST
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+arXiv:1001.0136v1  [math.CO]  31 Dec 2009Spectral Properties of the Threshold Network Model
+Yusuke Ide∗
+Faculty of Engineering, Kanagawa University,
+Yokohama, 221-8686, Japan
+Norio Konno†
+Department of Applied Mathematics, Yokohama National Univ ersity,
+Yokohama, 240-8501, Japan
+Nobuaki Obata‡
+Graduate School of Information Sciences, Tohoku University ,
+Sendai, 980-8579, Japan
+Abstract
+We study the spectral distribution of the threshold network model. The results
+contain an explicit description and its asymptotic behavio ur.
+1 Introduction
+Thethreshold network model Gn(X,θ), where Xis a random variable, n≥2 is an integer
+andθ∈Ris a constant called a threshold, is a random graph on the vertex set V=
+{1,2,...,n}obtained as follows: let X1,X2,...,X nbe independent copies of Xand draw
+an edge between two distinct vertices i,j∈VifXi+Xj> θ. In other words, Gn(X,θ) is
+specified by the random adjacency matrix A= (Aij) defined by
+Aij=/braceleftBigg
+I(θ,∞)(Xi+Xj),ifi/ne}ationslash=j,
+0, otherwise ,
+whereIBdenotes the indicator function of a set B. As a small variant one may allow
+self-loops, see e.g., [4]. In this case the threshold network model is d enoted by ˜Gn(X,θ),
+∗E-mail: ide@kanagawa-u.ac.jp
+†E-mail: konno@ynu.ac.jp
+‡E-mail: obata@math.is.tohoku.ac.jp
+Abbr. title: Spectral Properties of Threshold Network Model
+Key words and phrases: spectral distribution, threshold network model, random graphs, complex
+networks.
+1where two vertices i,j∈V(possibly i=j) are connected if Xi+Xj> θ. The adjacency
+matrix˜A= (˜Aij) is given by
+˜Aij=I(θ,∞)(Xi+Xj), i,j ∈V.
+The threshold network model has been extensively studied as a rea sonable candidate
+model of real world complex graphs (networks), which are often c haracterized by small
+diameters, high clustering, and power-law (scale-free) degree dis tributions [1,2,20]. In
+fact, the threshold network model belongs to the so-called hidden variable models [5,22]
+and is known for being capable of generating scale-free networks. Their mean behavior
+[3,5,8,9,15,21,22] and limit theorems [7,11–13] for the degree, the clustering coefficients,
+the number of subgraphs, and the average distance have been an alyzed. See also [6,11–
+14,16,17] for related works.
+Spectral properties of the threshold network model are also of in terest. As a simple
+case, the binary threshold model appears in [23]. The strong law of la rge numbers and
+central limit theorem for the rank of the adjacency matrix of the m odel with self-loops
+are given by [4]. Eigenvalues and eigenvectors of the Laplacian matrix of the model have
+been studied [18,19]. For general results of spectral analysis of g raphs see e.g. Hora–
+Obata [10]. The main purpose of this paper is to study the spectral d istribution of
+the threshold network model. Our result covers the preceding stu dy of the rank of the
+adjacency matrix.
+This paper is organized as follows: In Section 2 we recall the hierarch ical structure of
+the threshold network model and derive the spectral distribution of each sample graph
+(threshold graph). In Section 3 we obtain similar results for the thr eshold network model
+which admits self-loops. In Section 4 we derive some asymptotic beha viors of the spectral
+distributions and in Section 5 we give a simple example called the binary th reshold model.
+2 Spectra of threshold graphs
+Each sample graph G∈ Gn(X,θ) has a hierarchical structure described by the so-called
+creation sequence, introduced by Hagberg–Schult–Swart [9]. He re we adopt a variant by
+Diaconis–Holmes–Janson [6]. Each Gbeing determined by the values of random variables
+X1,X2,...,X n, we arrange them in increasing order: X(1)≤X(2)≤ ··· ≤ X(n). If
+X(1)+X(n)> θ, we have
+θ < X (1)+X(n)≤X(2)+X(n)≤ ··· ≤X(n−1)+X(n),
+which means that the vertex corresponding to X(n)is connected with the n−1 other
+vertices. Otherwise, we have
+θ≥X(1)+X(n)≥ ··· ≥X(1)+X(3)≥X(1)+X(2),
+which means that the vertex corresponding to X(1)is isolatedD We set sn= 1 orsn= 0
+according as the former case or the latter occurs. Then, accord ing to the case we remove
+the random variable X(n)orX(1), we continue similar procedure to define sn−1,...,s 2.
+Finally, we set s1=s2and obtain a {0,1}-sequence {s1,s2,...,s n}, which is called the
+creation sequence ofGand is denoted by SG.
+2Given a creation sequence SGletkiandlidenote the number of consecutive bits of
+1’s and 0’s, respectively, as follows:
+SG={k1/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
+1,...,1,l1/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
+0,...,0,k2/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
+1,...,1,l2/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
+0,...,0,...,km/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
+1,...,1,lm/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
+0,...,0}. (1)
+It may happen that k1= 0 orlm= 0, but we have k2,...,k m,l1,...,lm−1≥1 andm≥1.
+Moreover, by definition we have two cases: (a) k1= 0 (equivalently s1= 0) and l1≥2;
+(b)k1≥2 (equivalently, s1= 1).
+For example, if SG={1,1,0,0,1,0,1,0}thenk1= 2, l1= 2, k2= 1, l2= 1, k3=
+1, l3= 1 and Fig. 1 shows the shape of G.
+1 0
+Figure 1: A threshold graph Gcorresponding to SG={1,1,0,0,1,0,1,0}
+The creation sequence SGgives rise to a partition of the vertex set:
+V=m/uniondisplay
+i=1V(1)
+i∪m/uniondisplay
+i=1V(0)
+i|V(1)
+i|=ki,|V(0)
+i|=li.
+The subgraph induced by V(1)
+iis the complete graph of kivertices, and that induced
+byV(0)
+iis the null graph of livertices. Moreover, every vertex in V(1)
+i(resp.V(0)
+i) is
+connected (resp. disconnected) with all vertices in
+V(1)
+1∪···∪V(1)
+i∪V(0)
+1∪···∪V(0)
+i−1.
+In general, a graph possessing the above hierarchical structure is called a threshold graph
+[14].
+Theorem 1. LetGbe atheresholdgraph with acreation sequence SG={s1=s2,s3,...,s n}.
+Definekiandlias in(1)and set
+Cn(−1) =m/summationdisplay
+i=1ki−(m−1)−I{1}(s1), C n(0) =m/summationdisplay
+i=1li−(m−1).(2)
+Then the spectral distribution of Gis given by
+µn(G) =Cn(−1)
+nδ−1+Cn(0)
+nδ0+1
+nJ/summationdisplay
+j=1δλj, J= 2(m−1)+I{1}(s1),(3)
+3where{λj}exhausts the eigenvalues of the matrix:
+
+km−1lm−1km−1lm−2... l1k1
+km0 0 0 ...0 0
+km0km−1−1lm−2... l1k1
+km0km−10...0 0
+.....................
+km0km−10...0 0
+km0km−10...0k1−1
+(4)
+fors1= 1(equivalently, k1≥2), or
+
+km−1lm−1km−1lm−2... k 2l1
+km0 0 0 ...0 0
+km0km−1−1lm−2... k 2l1
+km0km−10...0 0
+.....................
+km0km−10... k 2−1l1
+km0km−10... k 20
+(5)
+fors1= 0(equivalently, k1= 0). Moreover, any λjin(3)differs from 0and−1.
+Proof.Let1i,jdenote the i×jmatrix consisting of only 1, 0i,jthei×jzero matrix,
+Iithei×iidentity matrix, and ¯1i,i=1i,i−Ii. By the hierarchical structure mentioned
+above, the adjacency matrix of Gis represented in the form:
+AG=
+0lm,lm0lm,km0lm,lm−10lm,km−10lm,lm−2...0lm,l10lm,k1
+0km,lm¯1km,km1km,lm−11km,km−11lm,lm−2...1km,l11km,k1
+0lm−1,lm1lm−1,km0lm−1,lm−10lm−1,km−10lm−1,lm−2...0lm−1,l10lm−1,k1
+0km−1,lm1km−1,km0km−1,lm−1¯1km−1,km−11km−1,lm−2...1km−1,l11km−1,k1
+0lm−2,lm1lm−2,lm0lm−2,lm−11lm−2,km−10lm−2,lm−2...0lm−2,l10lm−2,k1........................
+0l1,lm1l1,km0l1,lm−11l1,km−10l1,lm−2...0l1,l10l1,k1
+0k1,lm1k1,km0k1,lm−11k1,km−10k1,lm−2...0k1,l1¯1k1,k1
+.
+The adjacency matrix Aacts onCnfrom the left. We define subspaces of Cnby
+Vi(−1) =
+
+
+0ui+li
+ξki
+0di
+:ξ1+ξ2+···+ξki= 0
+
+,1≤i≤m,
+Vi(0) =
+
+
+0ui
+ηli
+0ki+di
+:η1+η2+···+ηli= 0
+
+,1≤i≤m−1,
+Vm(0) =/braceleftbigg/bracketleftbiggηlm
+0km+dm/bracketrightbigg/bracerightbigg
+,
+4where
+ξk=
+ξ1
+ξ2
+...
+ξk
+,ηl=
+η1
+η2
+...
+ηl
+,1j=
+1
+1
+...
+1
+,0j=
+0
+0
+...
+0
+
+and
+ui=m/summationdisplay
+j=i+1(lj+kj), d i=i−1/summationdisplay
+j=1(lj+kj).
+SinceAGacts onVi(−1) as the scalar operator with −1, it possesses the eigenvalues −1
+with multiplicity at least
+m/summationdisplay
+i=1dimVi(−1) =m/summationdisplay
+i=1(ki−1) =m/summationdisplay
+i=1ki−m
+ifk1≥2 (i.e.,s1= 1), and
+m/summationdisplay
+i=2dimVi(−1) =m/summationdisplay
+i=2(ki−1) =m/summationdisplay
+i=2ki−(m−1)
+ifk1= 0 (i.e., s1= 0). In any case, the multiplicity is at least Cn(−1) defined in (2).
+Similarly, acting on Vi(0) as a scalar operator with 0, AGpossesses the eigenvalues 0 with
+multiplicity at least Cn(0).
+LetWbe the orthogonal complement to/circleplustextm
+i=1(Vi(−1)⊕Vi(0)). The matrix represen-
+tation of AGonWwith respect to the basis
+vi=
+0ui+li
+1ki
+0di
+,1≤i≤m,andwi=
+0ui
+1li
+0ki+di
+,1≤i≤m−1.
+is given by (4) or by (5) according as k1≥2 ork1= 0. Then, one may verify easily the
+eigenvalues of the matrices (4) and (5) are different from −1 nor 0.
+Remark After simple calculation we see that the eigenvalues λ1,...,λ Jin (3) are ob-
+tained from the characteristic equations
+M(λ) = 0,
+where
+M(λ) = det
+km−1−λ lm−1km−1lm−2... l 1k1
+km−λ 0 0 ...0 0
+km 0km−1−1−λ lm−2... l 1k1
+km 0 km−1−λ ... 0 0
+.....................
+km 0 km−10...−λ0
+km 0 km−10...0k1−1−λ
+
+5ifk1≥2 (i.e.,s1= 1), and
+M(λ) = det
+km−1−λ lm−1km−1lm−2... k 2l1
+km−λ 0 0 ... 0 0
+km 0km−1−1−λ lm−2... k 2l1
+km 0 km−1−λ ... 0 0
+.....................
+km 0 km−10... k2−1−λ l1
+km 0 km−10... k 2−λ
+
+ifk1= 0 (i.e., s1= 0). Simple calculation shows that
+M(−1) =/braceleftBigg
+k1···km·l1···lm−1, ifk1≥2 (i.e.,s1= 1),
+k2···km·(l1−1)·l2···lm−1,otherwise ,
+and
+M(0) =/braceleftBigg
+(k1−1)·k2···km·l1···lm−1,ifk1≥2 (i.e.,s1= 1),
+k2···km·l1···lm−1, otherwise ,
+from which we see also that {λj}do not contain −1 or 0.
+3 Spectra of threshold graphs with self-loops
+The idea of a creation sequence in Section 2 can be applied to the thre shold network
+model which allows self-loops. With each G∈/tildewideGn(X,θ) we associate a creation sequence
+/tildewideSG={˜s1,˜s2,...,˜sn}as follows: if X(1)+X(n)> θ, we have
+θ < X (1)+X(n)≤X(2)+X(n)≤ ··· ≤X(n−1)+X(n)≤X(n)+X(n),
+which implies that the vertex corresponding to X(n)is connected with the n−1 other
+vertices and has a self-loop. Otherwise,
+θ≥X(1)+X(n)≥ ··· ≥X(1)+X(3)≥X(1)+X(2)≥X(1)+X(1),
+which means that the vertex corresponding to X(1)is isolated and has no self-loopsD We
+set ˜sn= 1 or ˜sn= 0 according as the former case or the latter occurs. Then, acco rding
+to the case we remove the random variable X(n)orX(1), we continue similar procedure to
+define ˜sn−1,...,˜s2. Finally, letting X(∗)be the last remained random variable, set ˜ s1= 1
+ifX∗> θ/2 and ˜s1= 0 otherwise. In this case Gis called a threshold graph with self-loops
+associated with a creation sequence ˜S={˜s1,˜s2,...,˜sn}. We note that if ˜ sj= 1 the
+corresponding vertex has a self-loop and otherwise no self-loop.
+Given a creation sequence ˜S={˜s1,˜s2,...,˜sn}we define kjandljas in (1). It may
+happenthat k1= 0andlm= 0, but k2,...,k m,l1,...,lm−1≥1andm≥1. Theadjacency
+6matrix of Gis of the form:
+˜AG=
+0lm,lm0lm,km0lm,lm−10lm,km−10lm,lm−2...0lm,l10lm,k1
+0km,lm1km,km1km,lm−11km,km−11lm,lm−2...1km,l11km,k1
+0lm−1,lm1lm−1,km0lm−1,lm−10lm−1,km−10lm−1,lm−2...0lm−1,l10lm−1,k1
+0km−1,lm1km−1,km0km−1,lm−11km−1,km−11km−1,lm−2...1km−1,l11km−1,k1
+0lm−2,lm1lm−2,lm0lm−2,lm−11lm−2,km−10lm−2,lm−2...0lm−2,l10lm−2,k1........................
+0l1,lm1l1,km0l1,lm−11l1,km−10l1,lm−2...0l1,l10l1,k1
+0k1,lm1k1,km0k1,lm−11k1,km−10k1,lm−2...0k1,l11k1,k1
+.
+(6)
+Repeating a similar argument as in Theorem 1, we come to the following
+Theorem 2. LetGbe a threshold graph with self-loops associated with a creat ion sequence
+˜S={˜s1,˜s2,...,˜sn}and its adjacency matrix given as in (6). Set
+/tildewideCn(0) =n−2(m−1)−I{1}(˜s1). (7)
+Then the spectral distribution of Gis given by
+/tildewideµn(G) =/tildewideCn(0)
+nδ0+1
+nJ/summationdisplay
+j=1δλj, J= 2(m−1)+I{1}(˜s1) (8)
+where{λj}exhaust the eigenvalues of
+
+kmlm−1km−1lm−2... l1k1
+km0 0 0 ...0 0
+km0km−1lm−2... l1k1
+km0km−10...0 0
+.....................
+km0km−10...0 0
+km0km−10...0k1
+
+for˜s1= 1(i.e.,k1≥1), or
+
+kmlm−1km−1lm−2... k 2l1
+km0 0 0 ...0 0
+km0km−1lm−2... k 2l1
+km0km−10...0 0
+.....................
+km0km−10... k 2l1
+km0km−10... k 20
+
+for˜s1= 0(i.e.,k1= 0). Moreover, any λjin(8)differs from 0.
+7Remark The eigenvalues λ1,...,λ Jin(8)areobtainedfromthecharacteristic equations:
+det
+−λ0...0 0 0 ...0km
+λ−λ ... 0 0 0 ... km−10
+...........................
+0 0...−λ0k2...0 0
+0 0 ... λ k 1−λ0...0 0
+0 0 ... l 1λ−λ ... 0 0
+...........................
+0lm−2...0 0 0...−λ0
+lm−10...0 0 0 ... λ −λ
+= 0
+fors1= 1 (i.e., k1≥1), or
+det
+−λ0...0 0 0 0 ...0km
+λ−λ ... 0 0 0 0 ... km−10
+0λ...0 0 0 0 ...0 0
+..............................
+0 0 ... λ −λ k 20...0 0
+0 0 ...0l1+λ−λ0...0 0
+0 0 ... l 20λ−λ ... 0 0
+..............................
+0lm−2...0 0 0 0...−λ0
+lm−10...0 0 0 0 ... λ −λ
+= 0
+fors1= 0 (i.e., k1= 0).
+4 Limit theorems
+In this section we discuss asymptotic behaviors of the spectral dis tributions obtained in
+the previous sections.
+We first consider the case where the distribution of Xis discrete and given by
+P(X=i) =pi, i= 0,1,...,∞/summationdisplay
+i=0pi= 1.
+Letm≥1 be a fixed integer. Take a particlar threshold θ= 2m−1 and assume that
+pi>0 fori= 0,1,...,2m−1. It follows from the strong law of large numbers that
+li=♯{j:Xj=m−i}, i= 1,...,m,
+ki=♯{j:Xj=m−1+i}, i= 1,...,m−1,
+km=♯{j:Xj≥2m−1},
+li=ki= 0, i≥m+1,
+8for large nalmost surely. Moreover, denoting by Fthe distribution function of X, we
+have
+lim
+n→∞1
+nm/summationdisplay
+i=1li=F(m−1) a.s. lim
+n→∞1
+nm/summationdisplay
+i=1ki= 1−F(m−1) a.s.
+With these observation we easily obtain the following
+Theorem 3. Notations and assumptions being as above, the spectral dist ributions of
+Gn(X,2m−1)verifies
+lim
+n→∞µn(G) = (1−F(m−1))·δ−1+F(m−1)·δ0a.s.
+Similarly, the spectral distributions of ˜Gn(X,2m−1)verifies
+lim
+n→∞/tildewideµn(G) =δ0a.s.
+Remark Similar results hold when the distribution of Xis discrete and Fhas only finite
+number of jumps in ( −∞,θ/2] or (θ/2,∞). But no simple description is known for a
+general case.
+Next we consider the case where the distribution of Xis continuous. As is stated by
+Bose–Sen [4] implicitly, if the distribution of Xis continuous and symmetric around 0,
+then the distribution of zero and one entries in the creation sequen ce/tildewideSof each graph
+generated by /tildewideGn(X,0) is the same as the distribution of a sequence of i.i.d. Bernoulli
+random variables {/tildewideYi}i=1,2,...,nwith success probability 1 /2, that is,
+P(/tildewideYi= 0) =P(/tildewideYi= 1) = 1/2,fori= 1,2,...,n.
+This means that /tildewideS={˜s1,˜s2,...,˜sn}d={/tildewideY1,/tildewideY2,...,/tildewideYn}.Recall that the creation sequence
+Sof each graph generated by Gn(X,0) is always satisfied with s1=s2. Then we observe
+thatS={s1=s2,s3,...,s n}d={Y2,Y2,Y3,...,Y n},where{Yi}i=2,3,...,nbe the sequence
+of i.i.d. Bernoulli random variables with success probability 1 /2, similarly.
+Taking the above consideration into account, we obtain asymptotic behaviors of coef-
+ficients of point measures on −1 and 0 appearing in µn(G) and/tildewideµn(G).
+Theorem 4. Assume that the distribution of Xis continuous and symmetric around 0.
+DefineCn(−1),Cn(0)and/tildewideCn(0)as in(2)and(7). Then we have
+(1) lim
+n→∞Cn(−1)/n= lim
+n→∞Cn(0)/n= 1/4a.s.
+(2)√n(Cn(−1)/n−1/4)⇒N(0,1/4)and√n(Cn(0)/n−1/4)⇒N(0,1/4)asn→
+∞.
+(3) lim
+n→∞/tildewideCn(0)/n= 1/2a.s.
+(4)√n(/tildewideCn(0)/n−1/2)⇒N(0,1/4)asn→ ∞.
+9Proof.Note the following relations:
+Cn(−1) =m/summationdisplay
+i=1ki−(m−1)−I{1}(s1)
+d=/parenleftBigg
+Y2+n/summationdisplay
+i=2Yi/parenrightBigg
+−n−1/summationdisplay
+i=2(1−Yi)Yi+1−Y2=Y2+n−1/summationdisplay
+i=2YiYi+1,
+Cn(0) =m/summationdisplay
+i=1li−(m−1)
+d=/braceleftBigg
+(1−Y2)+n/summationdisplay
+i=2(1−Yi)/bracerightBigg
+−n−1/summationdisplay
+i=2(1−Yi)Yi+1
+= 2−Y2−Yn+n−1/summationdisplay
+i=2(1−Yi)(1−Yi+1),
+/tildewideCn(0) =n−2(m−1)−I{1}(˜s1)
+d=n−2n−1/summationdisplay
+i=1(1−/tildewideYi)/tildewideYi+1−/tildewideY1= 1−/tildewideYn+n−1/summationdisplay
+i=1(1−/tildewideYi+/tildewideYi+1)(1+/tildewideYi−/tildewideYi+1).
+We then easily check that
+E[Cn(−1)] =E[Cn(0)]−1
+2=n
+4
+and
+E[/tildewideCn(0)] =n
+2.
+Applying a similar argument as in [4, Theorem 1], we have the assertion.
+When the distribution of Xis continuous and symmetric around θ/2, we can obtain
+similar results for Gn(X,θ) and/tildewideGn(X,θ) by straightforward modification. Study covering
+a more general situation is now in progress.
+5 Binary threshold model
+In this section we give a simple example. The threshold network model defined by
+Bernoulli trials X1,X2,...,X nwith success probability p, i.e., 0< P(Xi= 1) =p <1,
+and a threshold 0 ≤θ <1 is called the binary threshold model and is denoted by Gn(p).
+ForG∈ Gn(p) the partition of the vertex set Vis given by
+V=V(1)∪V(0), V(1)={i;Xi= 1}, V(0)={i;Xi= 0}.
+Theorem 5. ForG∈ Gn(p)we set|V(1)|=kand|V(0)|=l. Then the spectral distribu-
+tion ofGis given by
+µk,l=k−1
+nδ−1+l−1
+nδ0+1
+nδλ++1
+nδλ−,
+10where
+λ±=k−1±/radicalbig
+(k−1)2+4kl
+2. (9)
+Proof.We need only to apply Theorem 1 with l1=l,l2=k1= 0,k2=kandm= 2. In
+this case, (5) becomes/bracketleftbiggk−1l
+k0/bracketrightbigg
+,
+of which the eigenvalues are λ±in (9).
+Corollary 1. Letµn(G)be the spectral distribution of G∈ Gn(p). Then we have
+lim
+n→∞µn(G) =p·δ−1+(1−p)·δ0a.s.
+Proof.By the strong law of large numbers, see also Theorem 3.
+As for the the mean spectral distribution we have
+Theorem 6. The mean spectral distribution of the binary threshold mode lGn(p)is given
+by
+µn=/parenleftbigg
+p−1
+n/parenrightbigg
+δ−1+/parenleftbigg
+1−p−1
+n/parenrightbigg
+δ0
++1
+nn/summationdisplay
+k=0/parenleftbiggn
+k/parenrightbigg
+pk(1−p)n−k/parenleftbig
+δλ−(k)+δλ+(k)/parenrightbig
+, (10)
+where
+λ±(k) =k−1±/radicalbig
+(k−1)2+4k(n−k)
+2, k= 0,1,...,n.
+Proof.Since
+P(|V(1)|=k,|V(0)|=l) =/parenleftbiggn
+k/parenrightbigg
+pk(1−p)l, k+l=n,
+the mean spectral distribution is given by
+µ=n/summationdisplay
+k=0/parenleftbiggn
+k/parenrightbigg
+pk(1−p)lµk,l.
+Then (10) follows from Theorem 5 by direct computation.
+Corollary 2. Letµnbe mean spectral distribution of the binary threshold model Gn(p).
+Then we have
+lim
+n→∞µn=p·δ−1+(1−p)·δ0.
+Acknowledgment. NK is funded by the Grant-in-Aid for Scientific Research (C) of
+Japan Society for the Promotion of Science (Grant No. 21540118) . NO is funded by the
+Grant-in-Aid for Challenging Exploratory Research of Japan Societ y for the Promotion
+of Science (Grant No. 19654024).
+11References
+[1] Albert, R., and Barab´ asi, A. -L. (2002). “Statistical mechanic s of complex networks,” Rev. Mod.
+Phys.74, 47–97.
+[2] Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., and Hwang, D. -U. (2006). “Complex networks:
+structure and dynamics,” Phys. Rep. 424, 175–308.
+[3] Bogu˜ n´ a, M., and Pastor-Satorras, R. (2003). “Class of cor related random networks with hidden
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+[4] Bose, A., and Sen, A. (2007). “On asymptotic properties of the rank of a special random adjacency
+matrix,” Elect. Comm. in Probab. 12, 200–205.
+[5] Caldarelli, G., Capocci, A., De Los Rios, P., and Mu˜ noz, M. A. (2002) . “Scale-free networks from
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+[6] Diaconis, P., Holmes, S., and Janson, S. (2009). “Threshold grap h limits and random threshold
+graphs,” Internet Mathematics 5, no. 3, 267–318.
+[7] Fujihara, A., Ide, Y., Konno, N., Masuda, N., Miwa, H., and Uchida, M . (2009).“Limit theoremsfor
+the average distance and the degree distribution of the threshold network model,” Interdisciplinary
+Information Sciences 15, no.3, 361–366.
+[8] Fujihara, A., Uchida, M., and Miwa, H. (2009). “Universal power la ws in threshold network model:
+theoretical analysis based on extreme value theory,” Physica A 389, 1124–1130.
+[9] Hagberg, A., Schult, D. A., and Swart, P. J. (2006). “Designing t hreshold networks with given
+structural and dynamical properties,” Phys. Rev. E 74, 056116.
+[10] Hora, A. and Obata, N. (2007). Quantum Probability and Spectral Analysis of Graphs , Springer.
+[11] Ide, Y., Konno, N., and Masuda, N. (2007). “Limit theorems for some statistics of a generalized
+threshold network model,” RIMS Kokyuroku , no.1551, Theory of Biomathematics and its Applica-
+tions III, 81-86.
+[12] Ide, Y., Konno,N., andMasuda,N.(2009).“Statisticalprope rtiesofageneralizedthresholdnetwork
+model,” to appear in Methodol. Comput. Appl. Probab.
+[13] Konno, N., Masuda, N., Roy, R., and Sarkar, A. (2005). “Rigoro us results on the threshold network
+model,”J. Phys. A: Math. Gen. 38, 6277–6291.
+[14] Mahadev, N.V.R., Peled, U.N. (1995). Threshold Graphs and Related Topics, Elsevier.
+[15] Masuda, N., Miwa, H., and Konno, N. (2004). “Analysis of scale-f ree networks based on a threshold
+graph with intrinsic vertex weights,” Phys. Rev. E 70, 036124.
+[16] Masuda, N., Miwa, H., and Konno, N. (2005). “Geographical thr eshold graphs with small-world
+and scale-free properties,” Phys. Rev. E 71, 036108.
+[17] Masuda, N., and Konno, N. (2006). “VIP-club phenomenon: Em ergence of elites and masterminds
+in social networks,” Social Networks 28, 297–309.
+[18] Merris, R. (1994). “Degree maximal graphs are Laplacian integ ral,”Linear Algebr. Appl. 199,
+381–389.
+[19] Merris, R. (1998). “Laplacian graph eigenvectors,” Linear Algebr. Appl. 278, 221–236.
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+167–256.
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+13
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+arXiv:1001.0137v1  [math.DG]  31 Dec 2009ONE-PARAMETER HOMOTHETIC MOTION
+IN THE HYPERBOLIC PLANE AND
+EULER-SAVARY FORMULA
+Soley ERSOY, Mahmut AKYIGIT
+sersoy@sakarya.edu.tr , makyigit@sakarya.edu.tr
+Department of Mathematics, Faculty of Arts and Sciences
+Sakarya University, 54187 Sakarya/TURKEY
+March 17, 2009
+Abstract
+In [10] one-parameter planar motion was first introduced and the re-
+lations between absolute, relative, sliding velocities (a nd accelerations) in
+the Euclidean plane E2were obtained. Moreover, the relations between
+the Complex velocities one-parameter motion in the Complex plane were
+provided by [10]. One-parameter planar homothetic motion w as defined
+in the Complex plane, [9]. In this paper, analogous to homoth etic mo-
+tion in the Complex plane given by [9], one-parameter planar homothetic
+motion is defined in the Hyperbolic plane. Some characterist ic properties
+about the velocity vectors, the acceleration vectors and th e pole curves
+are given. Moreover, in the case of homothetic scale hidentically equal
+to 1, the results given in [15] are obtained as a special case. In addi-
+tion, three hyperbolic planes, of which two are moving and th e other one
+is fixed, are taken into consideration and a canonical relati ve system for
+one-parameter planar hyperbolic homothetic motion is defin ed. Euler-
+Savary formula, which gives the relationship between the cu rvatures of
+trajectory curves, is obtained with the help of this relativ e system.
+Mathematics Subject Classification (2000). : 53A17, 11E88.
+Keywords : Kinematics, homotheticmotion, hyperbolicnumbers,Eule r-
+Savary Formula.
+1 Preliminaries
+Before proceeding any further, we require a definition for the set of hyperbolic
+number and assume the existence of any number jwhich has the property
+j/ne}ationslash=±1 . In terms of the standard basis {1,j}, the hyperbolic number can be
+written as
+z=x+jy
+1wherej/parenleftbig
+j2= 1/parenrightbig
+is the unipotent (hyperbolic) imaginary unit and the reel num-
+bersxandyare called the real and unipotent (or hallucinatory) parts of the
+hyperbolic number z, respectively, [2]-[4],[6]-[8],[11], [12]. The hyperbolic num-
+bers
+H=R[j] =/braceleftbig
+z=x+jy|x,y∈,j2= 1/bracerightbig
+are the real numbers extended to include the unipotent jin the same manner
+thatC=R[i] are the complex numbers extended to include the imaginary i,/parenleftbig
+i2=−1/parenrightbig
+, [13].
+The hyperbolic numbers are also called perplex numbers [6], split-com plex num-
+bers [1] or double numbers [1],[7],[11]. The hyperbolic number systems se rve as
+the coordinatesin the Lorentzianplanein the same wayasthe comple x numbers
+serve as coordinates in the Euclidean plane. The role played by the co mplex
+numbers in Euclidean space is played by the hyperbolic number system s in the
+pseudo-Euclidean space, [12].
+Addition and multiplication of the hyperbolic numbers are
+(x+jy)+(u+jv) = (x+u)+j(y+v),
+(x+jy)(u+jv) = (xu+yv)+j(xv+yu).
+respectively. This multiplication is commutative, associative and distr ibutes
+over addition. The hyperbolic conjugate of z=x+jyis defined by z=x−jy.
+The hyperbolic inner product is
+/an}bracketle{tz,w/an}bracketri}ht= Re(zw) = Re(zw) =xu−yv
+where;z=x+jyandw=u+jv. Hyperbolic numbers zandware hyperbolic
+(Lorentzian) orthogonal if /an}bracketle{tz,w/an}bracketri}ht= 0, [12].
+The hyperbolic modulus of z=x+jyis
+/bardblz/bardblh=/radicalbig
+|/an}bracketle{tz,z/an}bracketri}ht|=/radicalbig
+|zz|=/radicalbig
+|x2−y2|
+and it is the hyperbolic distance of the point zfrom the origin. This is the
+Lorentz invariant of two-dimensional special relativity and their un imodular
+multiplicative group (the group composed of quadratic matrices det erminant
+of which equals to 1) is the special relativity Lorentz group, [14]. The se rela-
+tions have been used to extend special relativity. Furthermore, b y using the
+functions of the hyperbolic variable, two-dimensional special relat ivity has been
+generalized, [3]. These applications make the hyperbolic numbers app ropriate
+for physics and the application of hyperbolic numbers is similar to the a pplica-
+tion of complex numbers to the Euclidean plane geometry, [14].
+Note that the points z/ne}ationslash= 0 on the lines y=xare isotropic in the sense that
+they are nonzero vectors with /bardblz/bardblh= 0. By this way, the hyperbolic distance
+creates Lorentzian geometry in R2. This is different from the usual Euclidean
+geometry of the complex plane, where /bardblz/bardblh= 0 only if z= 0 in the complex
+plane. The set of all points in the hyperbolic plane that satisfy the eq uation
+/bardblz/bardblh=r>0 is a four-branched hyperbola of hyperbolic radius r, [12].
+2The hyperbolic number z=x+jycan be written as follows:
+While the hyperbolic number zis on H-I or H-III plane, then
+z=±r(coshϕ+jsinhϕ) =±rejϕ,
+While the hyperbolic number zis on H-II or H-IV plane, then
+z=±r(sinhϕ+jcoshϕ) =±rjejϕ,
+[See Figure 1.]
+Figure 1. Hyperbolic Plane
+This formula can be derived by a power series expansion due to the fa ct that
+cosh has only even powers whereas sinh has odd powers. For all rea l values of
+the hyperbolic angle ϕ, the hyperbolic number ejϕhas norm 1 and lies on the
+right branch of the unit hyperbola, [12].
+A hyperbolic rotation defined by ejϕcorresponds to multiplication by the ma-
+trix, [12];/bracketleftbiggcoshϕsinhϕ
+sinhϕcoshϕ/bracketrightbigg
+.
+Another property of the hyperbolic inner product is
+/angbracketleftbig
+zejϕ,wejϕ/angbracketrightbig
+=/an}bracketle{tz,w/an}bracketri}ht.
+In addition, a vector multiplied by jis a hyperbolic orthogonal vector. This is
+similar to the role played by the multiplication i=ei(π/2)in the complex plane,
+[12].
+32 One-Parameter Homothetic Motion in theHy-
+perbolic Plane
+In this section, we will define one-parameter homothetic motion in th e hyper-
+bolic plane and obtain the relation between the velocities and the acce lerations
+of a point under one-parameter homothetic planar motions.
+Homothetic motion of a moving hyperbolic plane Hwith respect to a fixed
+hyperbolic plane H′will be considered, in that, the orthonormal coordinate sys-
+tems{O;h1,h2}and{O′;h′1,h′2}being on the moving and fixed hyperbolic
+planesHandH′, respectively, will be analyzed with respect to each other. As
+the vector− − →OO′represented by the hyperbolic number udetermines the distance
+between the origin point of the moving system and the origin point of t he fixed
+system, the vectorial representation is as follows
+x′=hx−u (1)
+where,h=h(t)/ne}ationslash= constant is the homothetic scale (Figure 2. and Figure 3.).
+Figure 2. H/H′hyperbolic homothetic motion that rotates with central ang leϕ
+4Figure 3. H/H′hyperbolic homothetic motion that rotates with hyperbolic angleϕ
+Let a fixed point Xchosen on the plane Hbe represented by the hyperbolic
+numbers x=x1+jx2andx′=x′
+1+jx′
+2on the planes HandH′, respectively.
+Thus, one-parameter homothetic hyperbolic motion of the moving c oordinate
+system{O;h1,h2}with respect to the fixed coordinate system {O′;h′1,h′2}
+represented by H/H′is defined as the following transformation;
+x′=u′+hxejϕ. (2)
+Here,ϕis the Lorentzian (either central or hyperbolic) rotation angle of t he
+motionH/H′, and the hyperbolic number u′represents the origin point of the
+moving system in the fixed system. The rotation angle ϕ, the homothetic scale
+handx,x′,uwill be regardedas the differentiable functions of a real parameter
+5tfrom the class C∞. Generally, this parameter twill be used as time and at
+the moment t= 0, the coordinate systems will be accepted as coincident.
+The hyperbolic number u=u1+ju2represents the origin point O′of the fixed
+system in the moving system. At this case, if X′=O′, thenx′=0andx=u.
+Thus, from the equation (2)
+u′=−uejϕ(3)
+is found. Using (2) and (3) the following is obtained.
+x′= (hx−u)ejϕ. (4)
+As ˙ϕ(t) would give only the translation, we will assumedϕ
+dt= ˙ϕ(t)/ne}ationslash= 0 during
+the motion H/H′and call it as the angular velocity of the motion.
+2.1 Velocities and the Composition of Velocities
+Let the point Xon the moving plane Hchange its location depending on a
+parameter twhile undergoing one-parameter homothetic motion of plane H
+with regards to the plane H′. At this case, two motions belonging to the point
+Xoccur. Let’s explore what kind of relation exists between these hom othetic
+motions and their velocities.
+The velocity vector of the point Xwith respect to the plane H, that is, the
+vectorial velocity which the point has while drawing the trajectory c urve on is
+called the relative velocity of the point and is represented as Vr. The relative
+velocityVrofXis
+Vr=h˙ xejϕ. (5)
+The velocity of the point Xwith respect to the fixed plane H′is called the
+absolutevelocityof the point Xand is representedby Va. The absolute velocity
+ofXis
+Va=/parenleftBig
+˙h+jh˙ϕ/parenrightBig
+xejϕ−(˙ u+ju˙ϕ)ejϕ+h˙ xejϕ(6)
+If the differential of equation (3) is substituted in the last equation
+Va=˙ u′+/parenleftBig
+˙h+jh˙ϕ/parenrightBig
+xejϕ+h˙ xejϕ(7)
+can be written. If
+Vf=/parenleftBig
+˙h+jh˙ϕ/parenrightBig
+xejϕ−(˙ u+ju˙ϕ)ejϕ(8)
+the following equation is obtained
+Va=Vf+Vr. (9)
+Here,Vfis the sliding velocity of the one-parameter planar homothetic motion
+H/H′. If the point Xis a fixed point on the moving plane H, thenVf=0. So
+it is easily seen that
+Va=Vr. (10)
+62.2 The Rotation Pole and the Pole Trajectories
+Studying the points of the one-parameter homothetic hyperbolic m otionH/H′
+where the Vfsliding velocity equals to zero in every moment twill reveal the
+rotation pole term. Thus, by taking Vf=0in the equation (8), the pole point
+P= (p1,p2)∈His the hyperbolic number as
+p=˙ u+j˙ϕu
+˙h+jh˙ϕ(11)
+or
+p=p1+jp2=˙h˙ u−h˙ϕ2u
+˙h2−h2˙ϕ2+j˙h˙ϕu−h˙ϕ˙ u
+˙h2−h2˙ϕ2.
+In the special case of h(t) = 1, the following equation exists:
+p=p1+jp2=u+j˙ u
+˙ϕ
+which was given in [15].
+Let the rotation pole of the homothetic hyperbolic motion H/H′bePand a
+moving point on H′beX. Given this condition, the pole ray− − →PXfrom the pole
+Pto the point Xis expressed by the following equation
+− − →PX= (hx−p)ejϕ. (12)
+In addition, if the equations (8) and (11) are considered together
+Vf=/parenleftBig
+˙h+jh˙ϕ/parenrightBig
+(x−p)ejϕ(13)
+is found. As in [15], Vfand− − →PXare seen to be orthogonal to each other given
+condition of h(t) = 1.
+The length of the vector obtained from equation (13) is
+/bardblVf/bardblh=/radicalbigg/parenleftBig
+˙h2−h2˙ϕ2/parenrightBig/bracketleftBig
+(x1−p1)2+(x1−p1)2/bracketrightBig
+.
+In special case of h(t) = 1 as given in [15],
+/bardblVf/bardblh=|˙ϕ|/vextenddouble/vextenddouble/vextenddouble− − →PX/vextenddouble/vextenddouble/vextenddouble
+h
+is found.
+During one-parameter homothetic hyperbolic motion H/H′the geometric locus
+of the pole points Pin eachtmoment is the moving pole curve ( P) on the plane
+Hand the fixed pole curve ( P′) on the plane H′, respectively. Due to equation
+(1), the following is written:
+p′= (hp−u)ejϕ
+7and from the differentiation of this last equation with respect to t
+˙ p′=/bracketleftBig/parenleftBig
+˙h+jh˙ϕ/parenrightBig
+p−(˙ u+ju˙ϕ)+h˙ p/bracketrightBig
+ejϕ
+is obtained. Here, if the equation of the pole point given by equation ( 11) is
+substituted in the last equation,
+˙ p′=h˙ pejϕ(14)
+is found.
+Thus, the tangent vectors at the contact points of the pole curv es coincide with
+each other after the Lorentzian rotation ϕand the translation h.
+Let the arc elements of the moving and the fixed pole curves be dsandds′,
+respectively. In this case
+ds=/bardbl˙ p/bardblhdtandds′=/bardbl˙ p′/bardblhdt
+can be written. With the help of this last equation and equation (14), we get
+ds′=|h|ds.
+Thus, the following theorem can be given.
+Theorem 1 In one-parameter planar homothetic hyperbolic motion H/H′, the
+moving pole curve (P)in the plane Hrolls by sliding on the fixed pole curve (P′)
+on the plane H′. The coefficient of this sliding, rolling motion is the homoth etic
+scaleh.
+Special Case In the special case of h= 1, we get ds′=ds, that is, the pole
+curves roll on each other without sliding, which is given in [15].
+2.3 Accelerations and the Composition of Accelerations
+LetXbe a moving point on the moving hyperbolic plane H. In this case, the
+acceleration of the point Xwith respect to His called relative acceleration and
+is defined byd2x
+dt2=¨ x. In addition, the relative acceleration brwith respect to
+fixed hyperbolic plane H′can be written as
+br=h¨ xejϕ. (15)
+The acceleration of X′with respect to H′is called the absolute acceleration ba,
+and with the help of the differentiation of Vawith respect to t,
+ba=dVa
+dt=˙Va= (x−p)/bracketleftBig
+¨h+h˙ϕ2+j/parenleftBig
+2˙h˙ϕ+h¨ϕ/parenrightBig/bracketrightBig
+ejϕ−˙ p/parenleftBig
+˙h+jh˙ϕ/parenrightBig
+ejϕ+2˙ x/parenleftBig
+˙h+jh˙ϕ/parenrightBig
+ejϕ+h¨ xejϕ
+(16)
+is obtained and in this last equation, the expression
+bf= (x−p)/bracketleftBig
+¨h+h˙ϕ2+j/parenleftBig
+2˙h˙ϕ+h¨ϕ/parenrightBig/bracketrightBig
+ejϕ−˙ p/parenleftBig
+˙h+jh˙ϕ/parenrightBig
+ejϕ(17)
+8is called the sliding acceleration and
+bc= 2˙ x/parenleftBig
+˙h+jh˙ϕ/parenrightBig
+ejϕ(18)
+is called the Coriolis acceleration.
+Thus, the following theorem can be given.
+Theorem 2 InH/H′one-parameter planar homothetic hyperbolic motion, there
+is the following relation between the accelerations
+ba=bf+bc+br.
+The accelerationpole is known bythe vanishing ofthe sliding accelerat ionunder
+one-parameter planar motion. Thus, the following theorem is obtain ed, given
+the condition of bf=0.
+Theorem 3 Let the pole point be Pof the one-parameter homothetic hyperbolic
+motionH/H′. During this motion the acceleration pole point Q= (q1,q2)∈H
+is the hyperbolic number as
+q=p+˙ p/parenleftBig
+˙h+jh˙ϕ/parenrightBig
+¨h+h˙ϕ2+j/parenleftBig
+2˙h˙ϕ+h¨ϕ/parenrightBig (19)
+where¨h+h˙ϕ2/ne}ationslash=∓/parenleftBig
+2˙h˙ϕ+h¨ϕ/parenrightBig
+.
+Special Case In the special case of h= 1, given the condition that ¨ ϕ2−˙ϕ4/ne}ationslash= 0,
+the acceleration pole point of one-parameter hyperbolic motion is
+q=p+˙ p/parenleftbig
+˙ϕ¨ϕ−j˙ϕ3/parenrightbig
+¨ϕ2−˙ϕ4
+which was given in [15].
+3 Canonical Relative System for Homothetic Mo-
+tion in the Hyperbolic Plane
+Let’s consider an Aplane which moves with regard to HandH′hyperbolic
+planes, first one moving and the second one fixed. Let’s examine the motion
+of the coordinate system {B;a1,a2}which defines hyperbolic plane A, and
+hyperbolic planes HandH′with regard to the coordinate systems {O;h1,h2}
+and{O′;h′1,h′2}. [See Figure 4. and 5.] If the vector− − →OBis defined by
+the hyperbolic number b=b1+jb2, by applying the hyperbolic inner product,
+b2
+1−b2
+2>0 orb2
+1−b2
+2<0 can be obtained. As seen in Figure 3.1. and Figure 3.2.
+9respectively,thevector− − →OBcanbeontheplaneH-IorH-IIinhyperbolicmotion.
+Figure 4.− − →OBvector is on H-I plane
+Figure 5.− − →OBvector is on H-II plane
+10The rotation angles of the one-parameter planar hyperbolic motion A/Hand
+A/H′areϕandψ, respectively. If the origin points of O, BandO′, Bare
+coincident, then there exists following relations;
+a1= coshϕh1+sinhϕh2
+a2= sinhϕh1+coshϕh2
+and
+a1= coshψh′1+sinhψh′2
+a2= sinhψh′1+coshψh′2
+respectively [See Figure 6.].
+Figure 6.
+LetXbe a point with coordinates x1, x2on the moving plane A. If we denote
+the vectors− − →BX,− − →OBand− − →OB′with the hyperbolic numbers ˜X=x1+jx2,
+b=b1+jb2andb′=b′
+1+jb′
+2, respectively; then we can write
+x= (b+h˜ x)ejϕ(20)
+and
+x′= (b′+h˜ x′)ejϕ(21)
+where,h=h(t)/ne}ationslash= constant is the homothetic scale of the motion and the
+hyperbolic numbers xandx′denote the point Xwith respect to the coordinate
+systems of HandH′, respectively.
+The velocities of the motion with the help of the differentiation of the e quations
+(20) and (21) can be found. By differentiating the equation (20)
+dx= (σ+(dh+jhτ)˜ x+hd˜ x)ejϕ(22)
+is obtained, in which
+σ=σ1+jσ2=db+jbdϕ , τ =dϕ (23)
+and the relative velocity vector of X(with respect to H) isVr=dx
+dt.
+If we assume the differentiation of the equation (21),
+d′x= (σ′+(dh+jhτ′)˜ x+d˜ x)ejψ(24)
+11can be obtained along with the equation
+σ′=σ′
+1+jσ′
+2=d′b+jb′dψ , τ′=dψ. (25)
+Also, the absolute velocity vector, that is, the velocity vector of Xwith respect
+toH′, isVa=d′x
+dt.
+Here,σi, σ′
+i,(i= 1,2), τ, τ′are linear differential forms of tand are called
+Lorentzian Pfaffian forms of one-parameter homothetic hyperbo lic motion. The
+real parameter trepresents time.
+IfVr=0andVa=0, the pointXis fixed on the hyperbolic planes HandH′,
+respectively. Thus, the conditions of Xbeing fixed on the HandH′planes are
+d˜ x=−1
+h(σ+(dh+jhτ)˜ x) (26)
+and
+d˜ x=−1
+h(σ′+(dh+jhτ′)˜ x) (27)
+respectively. If the equation (26) is substituted into equation (24 ),
+dfx= [(σ′−σ)+jh(τ′−τ)˜ x]ejψ(28)
+can be obtained, where the sliding velocity vector of the point XisVf=dfx
+dt.
+Thus, following can be easily obtained:
+d′x=dfx+dx (29)
+This satisfies the relation between velocities which is given in equation ( 9).
+Just to avoid translation, it is assumed that ˙ ϕ/ne}ationslash= 0 and ˙ψ/ne}ationslash= 0. The rotation
+pole of the motion H/H′is characterized by the sliding velocity Pbeing zero.
+For that reason, if dfx=0, from the equation (28), the pole point Pof the
+one-parameter planar hyperbolic homothetic motion is obtained as
+p=jσ′−σ
+τ−τ′(30)
+and if Lorentzian coordinates are preferred on the condition that− − →BP=p=
+p1+jp2, it can be written
+p1=σ′
+2−σ2
+τ−τ′, p2=σ′
+1−σ1
+τ−τ′(31)
+which is given in [5].
+In theH/H′one-parameter planar hyperbolic homothetic motion, moving and
+fixed pole curves determine the geometric locus of the point PinHandH′
+planes, respectively. In other words; ( P) and (P′) are the representation of the
+moving and fixed pole curves, respectively. Also, the pole tangents can be either
+on the plane H-I or H-II [See Figure 7.].
+12Figure 7.
+Let’s first choose the pole tangents of the pole curves ( P) and (P′) on the plane
+H-II because the same results would be obtained by following similar op erations
+on the plane H-I.
+3.1 The Euler-Savary Formula for One-Parameter Planar
+Hyperbolic Homothetic Motion
+Let’schoosethemovingplane A, representedbythecoordinatesystem {B;a1,a2},
+in such way to meet the following conditions:
+i)The origin of the system Bcoincides with the instantaneous rotation pole P
+ii)The axis {B;a2}is the pole tangent, that is, it coincides with the common
+tangent of the pole curves ( P) and (P′) (on the plane H-II) [See Figure 8.].
+Figure 8.
+When the condition (i) is considered: by using the equation (31),
+σ1=σ′
+1, σ2=σ′
+2 (32)
+13are obtained. From the equations (23) and (25),
+db= (db+jbdϕ)ejϕ=σejϕ
+d′b= (db′+jb′dϕ)ejψ=σ′ejψ (33)
+arefound. Ifthe equation(32) and the lastequation aretook into consideration:
+dp=d′p=db=d′b (34)
+is found.
+Thus, the moving pole curve ( P), the pole tangent of which is given, and the
+fixed pole curves ( P′) are rolling on each other without sliding.
+The second condition, that is, the condition that the pole tangent c oincides
+witha2, requires the coefficient of a1to be zero. Here, σ1=σ′
+1= 0 and
+σ=jσ2=jσ′
+2can be written. Consequently, the derivative equations of the
+canonical relative system {P;a1,a2}are
+da1=τa2=jτejϕ, da2=τa1=−τejϕ, dp=jσ2a1=σejϕ(35)
+and
+d′a1=τ′a2=jτ′ejψ, d′a2=τ′a1=jτ′ejψ, d′p=jσ′
+2a1=σejψ.
+(36)
+Hereσ=dsis the scalar arc element of the pole curves ( P) and (P′).τis
+the hyperbolic cotangent angle, that is, two neighboring tangent a ngles of (P).
+Thus, the curvature of ( P) on the point Pis represented byτ
+σ=dϕ
+ds.
+Similarly, the curvature of the fixed pole curve ( P′) on the point Pisτ′
+σ=dψ
+ds
+whereτ′is the hyperbolic cotangent angle.
+The inverse values of these ratios
+r=σ
+τ(37)
+and
+r′=σ
+τ′(38)
+give the curvature radius of the pole curves ( P) and (P′), respectively.
+Whendν=τ′−τis the infinitesimal small hyperbolic instantaneous rotation
+angle, the movinghyperbolic plane H, with respect to the fixed plane H′, rotates
+around the rotation pole Pas much as this hyperbolic angle in the dttime scale.
+Thus, the hyperbolic angular velocity of the rotational motion of Hwith respect
+toH′is
+τ′−τ
+dt=dν
+dt=•ν. (39)
+From the equations (37), (38), and the last equation, the following can be writ-
+ten:
+τ′−τ
+dt=dν
+dt=1
+r′−1
+r. (40)
+14Let the direction of the unit tangent vector a2be in the direction determined
+by time-based pole curves ( P) and (P′). Let’s choose the vector a2in such
+way to ensure thatds
+dt>0. In this case, r >0 as the curvature center of the
+moving pole ( P) curve is at the right side of the directed pole tangent {P;a2}.
+Similarly,r′>0.
+Accordingtothecanonicalrelativesystem, thedifferentiation x-thecoordinates
+of which are x1, x2- with respect to the planes HandH′are
+dx= (σ+(dh+jhτ)˜ x+hd˜ x)ejϕ(41)
+and
+d′x= (σ′+(dh+jhτ′˜ x)+hd˜ x)ejψ(42)
+respectively. If
+hd˜ x=−σ−(dh+jhτ)˜ x (43)
+then the point Xis fixed on the hyperbolic plane H. Similarly, if
+hd˜ x=−σ−(dh+jhτ′)˜ x (44)
+then the point Xis fixed on the hyperbolic plane H′. Also, the sliding velocity
+Vfof the movement H/H′corresponds to the differentiation
+dfx=jh(τ′−τ)˜ xejϕ. (45)
+Now, let’s examine the curvature centers of the trajectory curv es drawn on
+their fixed plane by the points of moving planes in the motion of H/H′. In the
+canonical relative system, the points X,X′having the coordinates x1,x2and
+x′
+1,x′
+2, respectively, are situated, -together with the instantaneous r otation pole
+Pin everytmoment on the instantaneous trajectory normal, which belongs to
+X. Moreover,this curvaturecenter can be consideredas the limit of the meeting
+point of the normals of the two neighboring points on the curve. Thu s,
+− − →PX=x1+jx2=x− − →PX′=x′
+1+jx′
+2=x′(46)
+vectors have the same direction which passes through P. Then, for the points
+XandX′, the equation is
+x
+x′=x1+jx2
+x′
+1+jx′
+2=λ∈R. (47)
+If the differential of this last equation is taken, then we get
+dxx′−xdx′= 0. (48)
+If the conditions that the point Xbe fixed on the plane Hand the point X′be
+fixed on the plane H′are provided, then
+σ[x−x′]+jhxx′(τ′−τ) = 0 (49)
+15can be obtained. As the vectors− − →PX,− − →PX′are on the plane H-II ,
+x=ajejα(50)
+and
+x′=a′jejα. (51)
+That is,aanda′, respectively, represent the distance of the points XandX′
+on the plane H-II from the rotation pole P. Also, the angle αis bounded by
+the pole curves− − →PX=− − →PX′, [See Figure 9.]
+Figure 9.
+If the equations (50) and (51) are substituted into equation (49) , then
+jσ(a−a′)+jhaa′ejα(τ′−τ) = 0 (52)
+can be obtained, and if the equation (40) is considered together wit h this last
+equation,/parenleftbigg1
+a−1
+a′/parenrightbigg
+e−jα=h/parenleftbigg1
+r′−1
+r/parenrightbigg
+=hdν
+ds. (53)
+is found. Here, randr′are the radii of curvature of the pole curves Pand
+P′, respectively. dsrepresents the scalar arc element and dνrepresents the
+infinitesimal hyperbolic angle of the motion of the pole curves.
+The equation (53) is called the Euler-Savary formula for one-param eter plane
+hyperbolic homothetic motion.
+Consequently, the following theorem can be given.
+16Theorem 4 LetHandH′be the moving and fixed hyperbolic planes, respec-
+tively. A point X, assumed on H, draws a trajectory whose instantaneous center
+of curvature is X′on the plane H′in one-parameter planar homothetic motion
+H/H′. In the inverse homothetic motion of H/H′, a pointX′assumed on H′
+draws a trajectory whose center of curvature is Xon the plane H. The relation
+between the points XandX′is given by the Euler-Savary formula given in the
+equation (53).
+Remark Let’s choose the moving plane Arepresented by the coordinate system
+{B;a1,a2}in such way to meet following conditions:
+i)The origin of the system Band the instantaneous rotation pole Pcoincide
+with each other, i.e. B=P, [See Figure 10.]
+ii)The axis {B;a1}is the pole tangent, that is, it coincides with the common
+tangent of the pole curves ( P) and (P′) (on the plane H-I)
+Figure 10.
+Thus, if the operations in section3.1. are performed considering th e conditions
+i) and ii), the Euler-Savary formula for one-parameter planar hype rbolic ho-
+mothetic motion remains unchanged, that is, it is the same as in the eq uation
+(53)[See Figure 11.]
+17Figure 11.
+References
+[1] D. Alfsmann, On families of 2Ndimensional Hypercomplex Algebras suit-
+able for digital signal Processing. Proc. EURASIP 14thEuropean Signal
+Processing Conference (EUSIPCO 2006), Florence, Italy, 2006.
+[2] F. Catoni, R. Cannata, V. Catoni and P. Zampetti, Hyperbolic Tr igonome-
+try in two-dimensional space-time geometry. N. Cim. B 118 B (2003) , 475-
+491.
+[3] F. Catoni and P. Zampetti, Two-dimensional space-time symmet ry in hy-
+perbolic functions. N. Cim. B 115 B no. 12 (2000), 1433-1440.
+[4] J. Cockle, On a new imaginary in algebra. Phil. Mag. 3 no. 34 (1847), 37-47.
+[5] M. Erg¨ ut, A. P. Aydin, N. Bildik, The Geometry of the Canonical Relative
+System and One-Parameter Motions in 2-Lorentzian Space, The Jo urnal of
+Firat University, 1988 3(1) 113-122.
+[6] P. Fjelstad, Extending special relativity via the perplex numbers . Amer. J.
+Phys. 54 no. 5 (1986), 416-422.
+[7] P. Fjelstad and S.G. Gal, n-dimensional hyperbolic complex number s. Adv.
+Appl. Clifford Algebr. 8 no. 1 (1998), 47-68.
+18[8] P. Fjelstad and S.G. Gal, Two-dimensional geometries, topologies ,
+trigonometries and physics generated by complex-type numbers. Adv. Appl.
+Clifford Algebr. 11 no.1 (2001), 81-107.
+[9] N. Kuruo˜ glu, A. Tutar and M. D¨ uld¨ul, On the 1-parameter Homothetic Mo-
+tions on the Complex Plane, International Journal of Applied Mathe matics,
+6 No 4, 439-447, 2001.
+[10] H.R. M¨ uller, Kinematik, Sammlung G¨ oschen, Walter de Gruyter, Berlin
+(1963).
+[11] D. Rochon and M. Shapiro, On algebraic properties of bicomplex a nd hy-
+perbolic numbers. Anal. Univ. Oradea, fasc. math. 11 (2004), 71- 110.
+[12] G. Sobczyk, The Hyperbolic Number Plane, The College Math. J. 2 6 no.
+4 (1995), 268-280.
+[13] I.M. Yaglom, Complex numbers in geometry. Academic Press, New York,
+1968.
+[14] I.M. Yaglom, A simple non-Euclidean geometry and its Physical Bas is.
+Springer-Verlag, New York, 1979.
+[15] S. Y¨ uce and N. Kuruo˜ glu, One-parameter plane hyperbolic Motions. Adv.
+Appl. Clifford Algebras, 18 no. 2 (2008), 279-285.
+19
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+arXiv:1001.0138v1  [math.DG]  31 Dec 2009The Euler-Savary Formula for One-Parameter
+Planar Hyperbolic Motion
+Soley ERSOY, Murat TOSUN
+Department of Mathematics, Faculty of Arts and Sciences
+Sakarya University, 54187 Sakarya/TURKEY
+March 17, 2009
+Abstract
+One-parameter hyperbolic planar motion was first studied by S. Y¨uce
+and N. Kuruo˜ glu. Moreover, they analyzed the relationships between the
+absolute, relative and sliding velocities of one-paramete r hyperbolic pla-
+nar motion as well as the related pole curves, [14]. One-para meter planar
+motions in the Euclidean plane E2and the Euler-Savary formula in one-
+parameter planar motions were given by M¨ uller, [9]. In the present article,
+one hyperbolic plane moving relative to two other hyperboli c planes, one
+moving and the other fixed, was taken into consideration and t he relation
+between the absolute, relative and sliding velocities of th is movement was
+obtained. In addition, a canonical relative system for one- parameter hy-
+perbolic planar motion was defined. Euler-Savary formula, w hich gives
+the relationship between the curvature of trajectory curve s, was obtained
+with the help of this relative system.
+Mathematics Subject Classification (2000). : 53A17, 11E88.
+Keywords : Kinematics, hyperbolicmotion, hyperbolicnumbers, Eule r-
+Savary formula.
+1 Preliminaries
+Before proceeding any further, we require a definition for the set of hyperbolic
+number and assume the existence of any number jwhich has the property
+j/ne}ationslash=±1 . In terms of the standard basis {1,j}, the hyperbolic number can be
+written as
+z=x+jy
+wherej/parenleftbig
+j2= 1/parenrightbig
+is the unipotent (hyperbolic) imaginary unit and the reel num-
+bersxandyare called the real and unipotent (or hallucinatory) parts of the
+hyperbolic number z, respectively, [2]-[4],[6]-[8],[10], [11]. The set of the hyper-
+bolic numbers is
+H=R[j] =/braceleftbig
+z=x+jy|x,y∈,j2= 1/bracerightbig
+1In just the same way C=R[i] are the complex numbers extended to include
+the imaginary i/parenleftbig
+i2=−1/parenrightbig
+number [12], the hyperbolic numbers are the real
+numbers extended to include the unipotent jnumber.
+The hyperbolic numbers are also called perplex numbers [6], split-com plex num-
+bers [1] or double numbers [1],[7],[10]. The hyperbolic number systems se rve as
+the coordinatesin the Lorentzianplanein the same wayasthe comple x numbers
+serve as coordinates in the Euclidean plane. The role played by the co mplex
+numbers in Euclidean space is played by the hyperbolic number system s in the
+pseudo-Euclidean space, [11].
+Addition and multiplication of the hyperbolic numbers are as follows:
+(x+jy)+(u+jv) = (x+u)+j(y+v),
+(x+jy)(u+jv) = (xu+yv)+j(xv+yu).
+This multiplication is commutative, associative and distributes over ad dition.
+The hyperbolic conjugate of z=x+jyis defined by z=x−jy. The hyperbolic
+inner product is
+/an}bracketle{tz,w/an}bracketri}ht= Re(zw) = Re(zw) =xu−yv
+where;z=x+jyandw=u+jv. Hyperbolic numbers zandware hyperbolic
+(Lorentzian) orthogonal if /an}bracketle{tz,w/an}bracketri}ht= 0. Hyperbolic modulus of z=x+jyis
+/bardblz/bardblh=/radicalbig
+|/an}bracketle{tz,z/an}bracketri}ht|=/radicalbig
+|zz|=/radicalbig
+|x2−y2|
+and it is the hyperbolic distance of the point zfrom the origin. This is the
+Lorentz invariant of two-dimensional special relativity and their un imodular
+multiplicative group (the group composed of quadratic matrices det erminant
+of which equals to 1) is the special relativity Lorentz group, [13]. The se rela-
+tions have been used to extend special relativity. Furthermore, b y using the
+functions of the hyperbolic variable, two-dimensional special relat ivity has been
+generalized, [3]. These applications make the hyperbolic numbers app ropriate
+for physics and the application of hyperbolic numbers is similar to the a pplica-
+tion of complex numbers to the Euclidean plane geometry, [13]. Note t hat the
+pointsz/ne}ationslash= 0 on the lines y=xare isotropic in the sense that they are nonzero
+vectors with /bardblz/bardblh= 0. By this way, the hyperbolic distance creates Lorentzian
+geometry in R2. This is different from the usual Euclidean geometry of the
+complex plane, where /bardblz/bardblh= 0 only if z= 0 in the complex plane. The set of
+all points in the hyperbolic plane that satisfy the equation /bardblz/bardblh=r >0 is a
+four-branched hyperbola of hyperbolic radius r, [11].
+The hyperbolic number z=x+jycan be written as follows:
+While the hyperbolic number zis on H-I or H-III plane, then
+z=±r(coshϕ+jsinhϕ) =±rejϕ,
+While the hyperbolic number zis on H-II or H-IV plane, then
+z=±r(sinhϕ+jcoshϕ) =±rjejϕ,
+[See Figure1.1.]
+2Figure 1.1. Hyperbolic Plane
+This formula can be derived by a power series expansion due to the fa ct that
+cosh has only even powers whereas sinh has odd powers. For all rea l values of
+the hyperbolic angle ϕ, the hyperbolic number ejϕhas norm 1 and lies on the
+right branch of the unit hyperbola, [11].
+A hyperbolic rotation defined by ejϕcorresponds to multiplication by the ma-
+trix, [11];/bracketleftbiggcoshϕsinhϕ
+sinhϕcoshϕ/bracketrightbigg
+.
+Another property of the hyperbolic inner product is
+/angbracketleftbig
+zejϕ,wejϕ/angbracketrightbig
+=/an}bracketle{tz,w/an}bracketri}ht.
+In addition, a vector multiplied by jis a hyperbolic orthogonal vector, [11].
+This is similar to the role played by the multiplication i=ei(π/2)in the complex
+plane.
+2 Planar Hyperbolic Motion
+Let’s consider an Aplane which moves with regard to HandH′hyperbolic
+planes, first one moving and the second one fixed. Let’s examine the motion
+of the coordinate system {B;a1,a2}which defines hyperbolic plane A, and
+hyperbolic planes HandH′with regard to the coordinate systems {O;h1,h2}
+and{O′;h′1,h′2}. [See Figure 2.1. and 2.2.] If the vector− − →OBis defined by
+the hyperbolic number b=b1+jb2, by applying the hyperbolic inner product,
+b2
+1−b2
+2>0 orb2
+1−b2
+2<0 can be obtained. As seen in Figure 2.1. and Figure 2.2.
+3respectively,thevector− − →OBcanbeontheplaneH-IorH-IIinhyperbolicmotion.
+Figure2.1.− − →OBvector is on H-I plane
+Figure2.2.− − →OBvector is on H-II plane
+4The rotation angles of the one-parameter planar hyperbolic motion A/Hand
+A/H′areϕandψ, respectively. If the origin points of O, BandO′, Bare
+coincident, then there exists following relations;
+a1= coshϕh1+sinhϕh2
+a2= sinhϕh1+coshϕh2
+and
+a1= coshψh′1+sinhψh′2
+a2= sinhψh′1+coshψh′2
+respectively [See Figure 2.3. and 2.4.].
+If we denote the vectors− − →BX,− − →OBand− − →OB′with the hyperbolic numbers ˜ x=
+x1+jx2,b=b1+jb2andb′=b′
+1+jb′
+2on the moving coordinate system of A,
+respectively; then we have
+x= (b+˜ x)ejϕ(2.1)
+and
+x′= (b+˜ x)ejψ(2.2)
+where, the hyperbolic numbers xandx′denote the point Xwith respect to the
+coordinate systems of HandH′, respectively.
+Let’s find the velocities of the one-parameter motion with the help of the dif-
+ferentiation of the equations (2.1) and (2.2). By differentiating equ ation (2.1),
+we get
+dx= (σ+jτ˜ x+d˜ x)ejϕ(2.3)
+in which
+σ=σ1+jσ2=db+jbdϕ , τ =dϕ (2.4)
+and the relative velocity vector of X(with respect to H) isVr=dx
+dt.
+If we assume the differentiation of the equation (2.2),
+d′x= (σ′+jτ′˜ x+d˜ x)ejψ(2.5)
+can be obtained along with the equation
+σ′=σ′
+1+jσ′
+2=d′b+jb′dψ , τ′=dψ (2.6)
+5Also, the absolute velocity vector, that is, the velocity vector of Xwith respect
+toH′, isVa=d′x
+dt.
+Here,σi, σ′
+i,(i= 1,2), τ, τ′are linear differential forms of tand are
+called Lorentzian Pfaffian forms of one-parameter hyperbolic motio n. The real
+parametertrepresents time.
+IfVr=0andVa=0, the pointXis fixed on the hyperbolic planes HandH′,
+respectively. Thus, the conditions of Xbeing fixed on the HandH′planes are
+d˜ x=−σ−jτ˜ x (2.7)
+and
+d˜ x=−σ′−jτ′˜ x (2.8)
+respectively. If the equation (2.7) is substituted into equation (2.5 ),
+dfx= [(σ′−σ)+j(τ′−τ)˜ x]ejψ(2.9)
+can be obtained, where the sliding velocity vector of the point XisVf=dfx
+dt.
+Thus, following can be easily obtained:
+d′x=dfx+dx (2.10)
+Just to avoid translation, it is assumed that ˙ ϕ/ne}ationslash= 0 and ˙ψ/ne}ationslash= 0. The rotation
+pole of the motion H/H′is characterized by the sliding velocity Pbeing 0.
+For that reason, if dfx=0, from the equation (2.9), the pole point Pof the
+one-parameter planar hyperbolic motion is obtained as
+p=jσ′−σ
+τ−τ′(2.11)
+and if Lorentzian coordinates are preferred on the condition that− − →BP=p=
+p1+jp2, it can be written
+p1=σ′
+2−σ2
+τ−τ′, p2=σ′
+1−σ1
+τ−τ′(2.12)
+which is given in [5].
+In theH/H′one-parameter planar hyperbolic motion, moving and fixed pole
+curves determine the geometric locus of the point PinHandH′planes, re-
+spectively. In other words; ( P) and (P′) are the representation of the moving
+and fixed pole curves, respectively. Also, the pole tangents can be either on the
+plane H-I or H-II [See Figure 2.5. and 2.6.].
+6Let’s first choose the pole tangents of the pole curves ( P) and (P′) on the plane
+H-II because the same results would be obtained by following similar op erations
+on the plane H-I.
+3 The Euler-Savary Formula for One-Parameter
+Planar Hyperbolic Motion
+Let’schoosethemovingplane A, representedbythecoordinatesystem {B;a1,a2},
+in such way to meet the following conditions:
+i)The origin of the system Bcoincides with the instantaneous rotation pole P
+ii)The axis {B;a2}is the pole tangent, that is, it coincides with the common
+tangent of the pole curves ( P) and (P′) (on the plane H-II) [See Figure 3.1.].
+Figure 3.1.
+When the condition (i) is considered: by using the equation (2.12),
+σ1=σ′
+1, σ2=σ′
+2 (3.1)
+7are obtained. From the equations (2.4) and (2.6),
+db= (db+jbdϕ)ejϕ=σejϕ
+d′b= (db′+jb′dψ)ejψ=σ′ejψ (3.2)
+arefound. Iftheequation(3.1)andthelastequationaretookinto consideration:
+dp=d′p=db=d′b (3.3)
+is found.
+Thus, the moving pole curve ( P), the pole tangent of which is given, and the
+fixed pole curves ( P′) are rolling on each other without sliding.
+The second condition, that is, the condition that the pole tangent c oincides
+witha2, requires the coefficient of a1to be zero. Here, σ1=σ′
+1= 0 and
+σ=jσ2=jσ′
+2can be written. Consequently, the derivative equations of the
+canonical relative system {P;a1,a2}are
+da1=τa2=jτejϕ, da2=τa1=−τejϕ, dp=jσ2a1=σejϕ(3.4)
+and
+d′a1=τ′a2=jτ′ejψ, d′a2=τ′a1=jτ′ejψ, d′p=jσ′
+2a1=σejψ
+(3.5)
+Hereσ=dsis the scalar arc element of the pole curves ( P) and (P′).τis
+the hyperbolic cotangent angle, that is, two neighboring tangent a ngles of (P).
+Thus, the curvature of ( P) on the point Pis represented byτ
+σ=dϕ
+ds.
+Similarly, the curvature of the hyperbolic cotangent angle τ′-that is, the fixed
+pole curve ( P′) on the point P- isτ′
+σ=dψ
+ds.
+The inverse values of these ratios
+r=σ
+τ(3.6)
+and
+r′=σ
+τ′(3.7)
+give the curvature radius of the pole curves ( P) and (P′), respectively.
+Whendν=τ′−τis the infinitesimal small hyperbolic instantaneous rotation
+angle, the movinghyperbolic plane H, with respect to the fixed plane H′, rotates
+around the rotation pole Pas much as this hyperbolic angle in the dttime scale.
+Thus, the hyperbolic angular velocity of the rotational motion of Hwith respect
+toH′is
+τ′−τ
+dt=dν
+dt=•ν. (3.8)
+From the equations (3.6), (3.7), and the last equation, the following can be
+written:
+τ′−τ
+dt=dν
+dt=1
+r′−1
+r. (3.9)
+Let the direction of the unit tangent vector a2be in the direction determined
+by time-based pole curves ( P) and (P′). Let’s choose the vector a2in such
+8way to ensure thatds
+dt>0. In this case, r >0 as the curvature center of the
+moving pole ( P) curve is at the right side of the directed pole tangent {P;a2}.
+Similarly,r′>0.
+Accordingtothecanonicalrelativesystem, thedifferentiation x-thecoordinates
+of which are x1, x2- with respect to the planes HandH′are
+dx= [(τx2+dx1)+j(σ2+τx1+dx2)]ejϕ= (σ+jτx+dx)ejϕ(3.10)
+and
+d′x= [(τ′x2+dx1)+j(σ′
+2+τ′x1+dx2)]ejϕ= (σ+jτ′x+dx)ejϕ(3.11)
+respectively. If
+dx1=τx2anddx2=−σ2−τx1 (3.12)
+then the point Xis fixed on the hyperbolic plane H. Similarly, if
+dx1=τ′x2anddx2=−σ′
+2−τ′x1 (3.13)
+then the point Xis fixed on the hyperbolic plane H′. Also, the sliding velocity
+Vfof the movement H/H′corresponds to the differentiation
+dfx=j(τ′−τ)(x1+jx2)ejϕ=j(τ′−τ)xejϕ(3.14)
+Now, let’s examine the curvature centers of the trajectory curv es drawn on
+their fixed plane by the points of moving planes in the motion of H/H′. In the
+canonical relative system, the points X,X′having the coordinates x1,x2and
+x′
+1,x′
+2, respectively, are situated, -together with the instantaneous r otation pole
+Pin everytmoment on the instantaneous trajectory normal, which belongs to
+X. Moreover,this curvaturecenter can be consideredas the limit of the meeting
+point of the normals of the two neighboring points on the curve. Thu s,
+− − →PX=x1+jx2=x− − →PX′=x′
+1+jx′
+2=x′(3.15)
+vectors have the same direction which passes through P. Then, for the points
+XandX′, the equation is
+x
+x′=x1+jx2
+x′
+1+jx′
+2=λ∈R. (3.16)
+If the differential of this last equation is taken,
+(x′
+1dx1+x′
+2dx2−x1dx′
+1−x2dx′
+2)+j(x′
+1dx2+x′
+2dx1−x1dx′
+2−x2dx′
+1) = 0.
+(3.17)
+If the conditions that the point Xbe fixed on the plane Hand the point X′be
+fixed on the plane H′are provided, then
+jσ2[(x1+jx2)−(x′
+1+jx′
+2)]+j(x1+jx2)(x′
+1+jx′
+2)(τ′−τ) = 0
+9can be obtained, that is,
+σ[x−x′]+jxx′(τ′−τ) = 0 (3.18)
+As the vectors− − →PX,− − →PX′are on the plane H-II ,
+x=ajejα(3.19)
+and
+x′=a′jejα. (3.20)
+That is,aanda′, respectively, represent the distance of the points XandX′
+on the plane H-II from the rotation pole P. Also, the angle αis bounded by
+the pole curves− − →PX=− − →PX′, [See Figure 3.2.]
+Figure 3.2.
+If the equations (3.19) and (3.20) are substituted into equation (3 .18), then
+jσ(a−a′)+jaa′ejα(τ′−τ) = 0 (3.21)
+can be obtained, and if the equation (3.9) is considered together wit h this last
+equation,
+dν
+ds=1
+r′−1
+r= (1
+a−1
+a′)e−jα(3.22)
+is found. Here, randr′are the radii of curvature of the pole curves Pand
+P′, respectively. dsrepresents the scalar arc element and dνrepresents the
+infinitesimal hyperbolic angle of the motion of the pole curves.
+10The equation (3.22) is called the Euler-Savary formula for one-para meter plane
+hyperbolic motion.
+Consequently, the following theorem can be given.
+Theorem 3.1 LetHandH′be the moving and fixed hyperbolic planes, respec-
+tively. A point X, assumed on H, draws a trajectory whose instantaneous center
+of curvature is X′on the plane H′in one-parameter planar motion H/H′. In
+the inverse motion of H/H′, a pointX′assumed on H′draws a trajectory whose
+center of curvature is Xon the plane H. The relation between the points Xand
+X′is given by the Euler-Savary formula given in the equation (3 .22).
+Remark Let’s choose the moving plane Arepresented by the coordinate system
+{B;a1,a2}in such way to meet following conditions:
+i)The origin of the system Band the instantaneous rotation pole Pcoincide
+with each other, i.e. B=P, [See Figure 4.1.]
+ii)The axis {B;a1}is the pole tangent, that is, it coincides with the common
+tangent of the pole curves ( P) and (P′) (on the plane H-I)
+Figure 4.1.
+Thus, if the operations in III. section are performed considering t he conditions
+i) and ii), the Euler-Savaryformula for one-parameter planar hype rbolic motion
+remains unchanged, that is, it is the same as in the equation (3.22)[Se e Figure
+4.2.]
+11Figure 4.2.
+References
+[1] D. Alfsmann, On families of 2Ndimensional Hypercomplex Algebras suit-
+able for digital signal Processing. Proc. EURASIP 14thEuropean Signal
+Processing Conference (EUSIPCO 2006), Florence, Italy, 2006.
+[2] F. Catoni, R. Cannata, V. Catoni and P. Zampetti, Hyperbolic Tr igonome-
+try in two-dimensional space-time geometry. N. Cim. B 118 B (2003) , 475-
+491.
+[3] F. Catoni and P. Zampetti, Two-dimensional space-time symmet ry in hy-
+perbolic functions. N. Cim. B 115 B no. 12 (2000), 1433-1440.
+[4] J. Cockle, On a new imaginary in algebra. Phil. Mag. 3 no. 34 (1847), 37-47.
+[5] M. Erg¨ ut, A. P. Aydin, N. Bildik, The Geometry of the Canonical Relative
+System and One-Parameter Motions in 2-Lorentzian Space, The Jo urnal of
+Firat University, 1988 3(1) 113-122.
+[6] P. Fjelstad, Extending special relativity via the perplex numbers . Amer. J.
+Phys. 54 no. 5 (1986), 416-422.
+[7] P. Fjelstad and S.G. Gal, n-dimensional hyperbolic complex number s. Adv.
+Appl. Clifford Algebr. 8 no. 1 (1998), 47-68.
+12[8] P. Fjelstad and S.G. Gal, Two-dimensional geometries, topologies ,
+trigonometries and physics generated by complex-type numbers. Adv. Appl.
+Clifford Algebr. 11 no.1 (2001), 81-107.
+[9] H.R. M¨ uller, Kinematik, Sammlung G¨ oschen, Walter de Gruyter, Berlin
+(1963).
+[10] D. Rochon and M. Shapiro, On algebraic properties of bicomplex a nd hy-
+perbolic numbers. Anal. Univ. Oradea, fasc. math. 11 (2004), 71- 110.
+[11] G. Sobczyk, The Hyperbolic Number Plane, The College Math. J. 2 6 no.
+4 (1995), 268-280.
+[12] I.M. Yaglom, Complex numbers in geometry. Academic Press, New York,
+1968.
+[13] I.M. Yaglom, A simple non-Euclidean geometry and its Physical Bas is.
+Springer-Verlag, New York, 1979.
+[14] S. Y¨ uce and N. Kuruo˜ glu, One-parameter plane hyperbolic Motions. Adv.
+Appl. Clifford Algebras, 18 no. 2 (2008), 279-285.
+13
\ No newline at end of file
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+arXiv:1001.0139v1  [astro-ph.SR]  31 Dec 2009Publications of the Astronomical Society of the Pacific
+KeplerAsteroseismology Program:
+Introduction and First Results
+Ronald L. Gilliland,1Timothy M. Brown,2Jørgen Christensen-Dalsgaard,3Hans Kjeldsen,3
+Conny Aerts,4Thierry Appourchaux,5Sarbani Basu,6Timothy R. Bedding,7William
+J. Chaplin,8Margarida S. Cunha,9Peter De Cat,10Joris De Ridder,4Joyce A. Guzik,11
+Gerald Handler,12Steven Kawaler,13L´ aszl´ o Kiss,7,14Katrien Kolenberg,12Donald
+W. Kurtz,15Travis S. Metcalfe,16Mario J.P.F.G. Monteiro,9Robert Szab´ o,14Torben
+Arentoft,3Luis Balona,17Jonas Debosscher,4Yvonne P. Elsworth,8Pierre-Olivier
+Quirion,3,18Dennis Stello,7Juan Carlos Su´ arez,19William J. Borucki,20Jon M. Jenkins,21
+David Koch,20Yoji Kondo,22David W. Latham,23Jason F. Rowe,20and Jason H. Steffen24– 2 –
+ABSTRACT
+Asteroseismology involves probing the interiors of stars and quant ifying their
+global properties, such as radius and age, through observations of normal modes
+of oscillation. The technical requirements for conducting asteros eismology in-
+clude ultra-high precision measured in photometry in parts per million, as well
+as nearly continuous time series over weeks to years, and cadence s rapid enough
+1Space Telescope Science Institute, 3700 San Martin Drive, Baltimor e, MD 21218, USA; gillil@stsci.edu
+2Las Cumbres Observatory Global Telescope, Goleta, CA 93117, US A
+3Department of Physics and Astronomy, Aarhus University, DK-80 00 Aarhus C, Denmark
+4Instituut voor Sterrenkunde, K.U.Leuven, Celestijnenlaan 200 D, 3001, Leuven, Belgium
+5Institut d’Astrophysique Spatiale, Universit´ e Paris XI, Bˆ atiment 121, 91405 Orsay Cedex, France
+6Astronomy Department, Yale University, P.O. Box 208101, New Hav en CT06520, USA
+7Sydney Institute for Astronomy, School of Physics, University o f Sydney, NSW 2006, Australia
+8School of Physics and Astronomy, University of Birmingham, Birming ham B15 2TT, England UK
+9Centro de Astrof´ ısica da Universidade do Porto, Rua das Estrelas , 4150-762 Porto, Portugal
+10Royal Observatory of Belgium, Ringlaan 3, B-1180 Brussels, Belgium
+11Applied Physics Division, LANL, X-2 MS T086, Los Alamos, NM 87545, US A
+12Institut f¨ ur Astronomie, Universit¨ at Wien, T¨ urkenschanzstr asse 17, A-1180 Wien, Austria
+13Department of Physics and Astronomy, Iowa State University, Am es, IA 50011, USA
+14Konkoly Observatory, H-1525, P.O. Box 67. Budapest, Hungary
+15Jeremiah Horrocks Institute for Astrophysics, University of Cen tral Lancashire, Preston PR1 2HE, UK
+16High Altitude Observatory and SCD, NCAR, P.O. Box 3000, Boulder, C O 80307, USA
+17South African Astronomical Observatory, Observatory, 7935 C ape Town, South Africa
+18CSA, 6767 Boulevard de l’A´ eroport, Saint-Hubert, QC J3Y 8Y9, Ca nada
+19Instituto de Astrof´ ısica de Andaluc´ ıa, C.S.I.C., Apdo. 3004, 18080 Granada, Spain
+20NASA Ames Research Center, Moffett Field, CA 94035, USA
+21SETI Institute/NASA Ames Research Center, Moffett Field, CA 940 35, USA
+22NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
+23Harvard-Smithsonian Astrophysical Observatory, Cambridge, M A 02138, USA
+24Fermilab Center for Particle Astrophysics, Batavia, IL 60510, USA– 3 –
+to sample oscillations with periods as short as a few minutes. We repor t on re-
+sults from the first 43 days of observations in which the unique capa bilities of
+Keplerin providing a revolutionary advance in asteroseismology are already well
+in evidence. The Keplerasteroseismology program holds intrinsic importance in
+supporting the core planetary search program through greatly e nhanced knowl-
+edge of host star properties, and extends well beyond this to rich applications in
+stellar astrophysics.
+Subject headings: Review (regular)
+1. INTRODUCTION
+TheKepler Mission science goals and initial results in the core planet-detection and
+characterization area, as well as the mission design and overall per formance, are reviewed
+by Borucki et al. (2010), and Koch et al. (2010).
+Asteroseismologyissometimesconsideredasthestellaranalogofhe lioseismology(Gough et al.
+1996), being the study of very low amplitude sound waves that are e xcited by near-surface,
+turbulent convection, leading to normal-mode oscillations in a natura l acoustical cavity. The
+Sun, when observed as a star without benefit of spatial resolution on its surface, shows ∼
+30 independent modes with white light amplitude of a few parts per million (ppm) and
+periods of 4 – 8 minutes. In many cases stars with stochastically driv en oscillations may
+deviate strongly from solar size, as in the case of red giants detailed below. For our purposes,
+however, we will broaden the definition of asteroseismology to also in clude the many types
+of classical variable stars, e.g. Cepheids, famous for helping to est ablish the scale of the
+universe (Hubble & Humason 1931).
+There are two primary motivations for performing asteroseismolog y withKepler, which
+is primarily a planet-detection and characterization mission (Boruck i et al. 2010). First,
+knowledge of planet properties is usually limited to first order by know ledge of the host star,
+e.g.Keplereasily measures the ratio of planet to star size through transit dep ths. Turning
+this into an absolute size for the planet requires knowledge of the ho st star size which, in
+favorable cases, asteroseismology is able to provide better than a ny other approach – in
+many cases radii can be determined to accuracies near 1%. Second , the instrumental charac-
+teristics already required for the exquisitely demanding prime mission (Borucki et al. 2010;
+Koch et al. 2010) can readily support the needs of seismology at ess entially no additional
+cost or modification, thus promising strong science returns by alloc ating a few percent of the
+observations to this exciting area of astrophysics.– 4 –
+Oscillations in stars similar to the Sun, which comprise the primary set f or the planet
+detections, have periods of only a few minutes and require use of th e Short Cadence (SC)
+mode with 58.8-second effective integrations. Many of the classical variables may be studied
+well with the more standard Long Cadence (LC) mode, which has 29.4 -minute effective
+integrations.
+Keplercomes at a propitious time for asteroseismology. The forerunner m issions MOST
+(Matthews2007), andespeciallyCoRoT(Baglin et al.2009; Michel et al.2008), havestarted
+our journey taking space-based asteroseismology from decades of promise to the transfor-
+mative reality we realistically expect with Kepler. The most salient features of the Kepler
+Missionfor asteroseismology are: (a) a stable platform from which nearly c ontinuous obser-
+vations can be made for months to years, (b) cadences of 1 and 30 minutes which support the
+vast majority of asteroseismology cases, (c) a large 100 square d egree field of view providing
+many stars of great intrinsic interest, (d) a huge dynamic range of over a factor of 10,000 in
+apparent stellar brightness over which useful asteroseismology ( not always of the same type
+of variable) can be conducted, and (e) exquisite precision that in ma ny cases is well under
+one ppm for asteroseismology purposes. Initial data characteris tics for SC (Gilliland et al.
+2010) and LC (Jenkins et al. 2010) have been shown to support res ults that nearly reach the
+limit of Poisson statistics.
+2. SUPPORT OF EXOPLANETARY SCIENCE
+TheKepler Input Catalog (KIC) (Koch et al. 2010) provides knowledge (30 – 50% errors
+on stellar radii) of likely stellar properties for ∼4.5 million stars in the Keplerfield of view
+at a level of accuracy necessary to specify the targets to be obs erved. One likely application
+of asteroseismology will follow from quantification of stellar radii to m ore than an order
+of magnitude better than this for a few thousand giant stars, and several hundred dwarfs,
+which can then be used to test the KIC entries, and quite possibly pr ovide the foundation
+for deriving generally applicable improvements to the calibrations ena bling redefined entries
+for the full catalog.
+Some 15% of the KIC entries were not classified, thus no radius estim ates were available
+to support selection of stars most optimal for small-planet transit searches. In Q0 (May
+2009) and Q1 (May – June 2009) a total of about 10,000 such unclas sified stars brighter
+than Kepler-mag = 13.8 were observed for either 10 and/or 33 days respectively. An early
+application ofasteroseismology was to identify stars inthis unclassifi ed set that areobviously
+red giants, a well-posed exercise given the quality of Keplerdata (Koch et al. 2010), and
+thus allow these to be dropped from further observation in favor o f bringing in smaller, and– 5 –
+photometrically quieter stars.
+At a more basic level asteroseismology can play an important role in qu antifying knowl-
+edge of individual planet host candidates. In particular, by showing that the stellar radius
+is significantly greater than the catalogued value, we can rule out a p lanetary candidate
+without the need to devote precious ground-based resources to measuring radial velocities.
+One such case is the Kepler Object of Interest1(KOI) identified as KOI-145, whose light
+curve shows a transit. The host star KIC-9904059 has a tabulate d radius of just under 4
+R⊙, adjusted to 3.6 R ⊙with addition of ground-based classification spectroscopy2. This
+radius allows a transit light curve solution with a transiting body below 1 .6 RJ, hence the
+KOI designation.
+Figure 1 shows that KIC-9904059 displays oscillations characterist ic of a red giant star.
+Thepatternofpeaksinthepowerspectrumfollowsthespacingexp ectedfromtheasymptotic
+relation for low-degree p modes (restoring force is pressure for s ound waves) which may be
+given as (Tassoul 1980; Gough 1986):
+νnl≈∆ν(n+l/2 +ǫ)−D0l(l+1). (1)
+Here ∆ν= (2/integraltextR
+0dr/c)−1is the so-called large separation, which corresponds to the inverse
+of the sound travel time across the stellar diameter. In this notat ionnrepresents the number
+of nodes in a radial direction in the star for the standing sound wave s, andlis the number
+of nodes around the circumference. For stars observed without spatial resolution only l= 0,
+1, and 2 modes are typically visible in photometry, since the higher-de gree modes that are
+easily visible on the Sun average out in disk-integrated measurement s. It can be shown that
+∆νscales as the square root of the stellar mean density, and more pre cise determinations of
+/angbracketleftρ∗/angbracketrightfollow from using stellar evolution models (Christensen-Dalsgaard et al. 2010). The ǫ
+term captures near-surface effects, while D0represents the so-called small separation, which
+between l= 0 and 2 we will refer to as: δ02. In main-sequence stars this small separation
+is sensitive to the sound speed in the stellar core, hence the evolving hydrogen content, and
+helps constrain stellar age.
+The asteroseismic solution for KOI-145 yields ∆ ν= 11.77±0.07µHz, with maximum
+power at a frequency νmaxof 143±3µHz, and an amplitude of 50 ppm for the radial modes,
+1The project uses these numbers for the internal tracking of can didates (Borucki et al. 2010), which are
+not synonymous with claimed detections of planets.
+2Based on Spectroscopy Made Easy – Valenti & Piskunov 1996 – analy sis by Debra Fischer using a
+spectrum of KOI-145 obtained by Geoff Marcy at the W. M. Keck Obs ervatory which gives T eff= 4980±
+60, log(g) = 3.47 ±0.1, and [Fe/H] = -0.02 ±0.06.– 6 –
+all in excellent agreement with canonical scaling relations. The ∆ νcorresponds to /angbracketleftρ∗/angbracketright=
+0.0113±0.0001 g cm−3(Kjeldsen, Bedding & Christensen-Dalsgaard 2008) and, using the
+spectroscopic constraints on stellar temperature and metallicity, yields solutions for mass
+and radius of 1.72 ±0.13 M ⊙and 5.99 ±0.15 R ⊙following the procedure of Brown (2009).
+Adopting the larger asteroseismic radius of 5.99 R ⊙, Figure 1 shows the light curve
+for KOI-145 (only one eclipse has been seen so far) and an eclipse so lution indicating that
+the transiting body has a radius of 0.45+0.11
+−0.07R⊙assuming an orbital period of 90 days (the
+minimum currently allowed). Assuming a longer orbital period allows yet larger values for
+the radius of the eclipsing body. In this case, asteroseismology sec urely sets the radius of
+the host star and forces the solution for the eclipsing object firmly into the stellar domain.
+The 3-σlower-limit to the transiting body radius is 3.4 R J. More detailed analyses await
+fixing the orbital period from repeated eclipses.
+In Figure 2a the filled symbols show the observed frequencies in KOI- 145 in a so-called
+´ echelle diagram, in which frequencies are plotted against the frequ encies modulo the average
+large frequency spacing (but note that the abscissa is scaled in unit s of ∆ν). The frequencies
+were extracted by iterative sine-wave fitting in the interval νmax±3∆νdown to a level
+of 19 ppm, corresponding to 3.3 times the mean noise level in the amplit ude spectrum
+at high frequency. The open symbols in Figure 2a show frequencies f rom a model in the
+grid calculated by Stello et al. (2009), after multiplying by a scaling fac tor ofr= 0.9331
+(Kjeldsen, Bedding & Christensen-Dalsgaard 2008).
+An interesting feature of Figure2a is the absence of a significant off set between observed
+and model frequencies for the radial modes. This contrasts to th e Sun, for which there is a
+long-standing discrepancy that increases with frequency betwee n observations and standard
+solar models, as shown in Figure 2b. This offset is known to arise from t he inability to ad-
+equately model the near-surface effects (Christensen-Dalsgaa rd, D¨ appen & Lebreton 1988)
+andhas been animpediment to progress in asteroseismology for sola r-type stars, althoughan
+effectiveempiricalcorrectionhasrecentlybeenfound(Kjeldsen, Bedding & Christensen-Dalsgaard
+2008) that allows accurate determination of the mean stellar densit y from ∆ ν. As shown in
+Figure 2 a, the offset between the measured and theoretically comp uted frequencies for l= 0
+modes in KOI-145 is very small, suggesting some simplification in interpr etations for red
+giants if this surface term can be generally ignored. Other factors for red giant oscillations,
+however, are already significantly more complex than in the Sun, as a rgued in the recent
+theoretical analysis of Dupret et al. (2009). In particular, less effi cient trapping of the modes
+in the envelope for red giants on the lower part of the red giant bran ch can lead to multiple
+non-radial modes, especially for l= 1 (Dupret et al. 2009). We see this in KOI-145 and also
+in other red giants observed by Kepler(Bedding et al. 2010).– 7 –
+The asteroseismic solution for HAT-P-7 by Christensen-Dalsgaard et al. (2010) illus-
+trated in Figure 3 demonstrates the great potential for refining h ost star properties. The
+stellar radius is determined to be 1.99 ±0.02 R ⊙, an order of magnitude gain in confidence
+interval from solutions based on ground-based transit light curve solutions by P´ al (2008),
+with an age of 1.9 ±0.5 Gyr. Such improvements to host star knowledge, thus ultimately
+planet size and density (when radial velocities provide mass) are crit ically important for
+informing studies of planet structure and formation.
+3. INDEPENDENT STELLAR ASTROPHYSICS
+The mission as a whole will benefit from enhanced knowledge of stellar s tructure and
+evolution theory. This canbest beadvancedbychallenging theoryw ithdetailed observations
+of stellar oscillations across a wide range of stellar types. Supportin g this goal has led to
+devoting 0.8% from the available LC target allocations (still a very hea lthy number of 1,320
+stars) to a broad array of classical variables. These LC targets a re usually large stars with
+characteristic periods of variation of hours, to in extreme cases m onths or even years, and
+hence are either massive and/or evolved stars. In addition, a set o f 1,000 red giants selected
+to serve as distant reference stars for astrometry (Monet 201 0) provide enticing targets for
+LC-based asteroseismology. A number of stellar variables, including , of course, close analogs
+of the Sun can only be probed with use of the SC (1-minute) observa tions, on which a
+cap of 512 targets exists. Most of these SC slots are now being app lied to a survey of
+asteroseismology targets rotated through in one-month periods . After the first year a subset
+of these surveyed targets will be selected for more extended obs ervations. As the mission
+progresses, a growing fraction of the SC targets will be used to fo llow up planet detections
+and candidates, for which better sampling will support transit timing searches for additional
+planets in the system (Holman & Murray 2005) and oscillation studies o f the host stars.
+Asteroseismology with Kepleris being conducted through the Kepler Asteroseismic Sci-
+ence Consortium (KASC)3, whose ∼250 members are organized into working groups by
+type of variable star. So far data have been available from the first 43 days of the mission
+3The Kepler Asteroseismic Investigation (KAI) is managed at a top lev el by the first 4 authors of this
+paper. The next level of authorship comprises the KASC working gr oup chairs, and members of the KASC
+Steering Committee. Data for KASC use first passes through the S TScI archive for Kepler, then if SC is
+filtered to remove evidence of any transits, and then is made availab le to the KASC community from the
+Kepler Asteroseismic Science Operations Centre (KASOC) at the De partment of Physics and Astronomy,
+Aarhus University, Denmark. Astronomers wishing to join KASC are welcome to do so by following the
+instructions at: http://astro.phys.au.dk/KASC/.– 8 –
+for∼2300 LC targets, some selected by KASC and some being the astrom etric red giants.
+The SC data have only been made available for a small number of targe ts (Chaplin et al.
+2010). The remainder of this paper reviews the science goals for Keplerasteroseismology
+and summarizes some of the first results.
+3.1. Solar-like Oscillations
+Stars like the Sun, which have sub-surface convection zones, disp lay a rich spectrum of
+oscillations that are predominantly acoustic in nature4. The fact that the numerous excited
+modes sample different interior volumes within the stars means that t he internal structures
+can be probed, and the fundamental stellar parameters constra ined, to levels of detail and
+precision thatwould nototherwise bepossible (see, forexample, Go ugh1987). Asteroseismic
+observations of many stars will allow multiple-point tests to be made o f stellar evolution
+theory and dynamo theory. They will also allow important constraint s to be placed on the
+ages and chemical compositions of stars, key information for cons training the evolution of
+the galaxy. Furthermore, the observations permit tests of phys ics under the exotic conditions
+found in stellar interiors, such as those underpinning radiative opac ities, equations of state,
+and theories of convection.
+3.1.1. Main Sequence and Near Main Sequence Stars
+Keplerwill observe more than 1500 solar-like stars during the initial survey phase of
+the asteroseismology program. This will allow the first extensive “se ismic survey” to be per-
+formed on this region of the color-magnitude diagram. On completion of the survey, a subset
+of 50 to 75 solar-like targets will be selected for longer-term, multi- year observations. These
+longer datasets will allow tight constraints to be placed on the intern al angular momenta
+of the stars, and also enable “sounding” of stellar cycles via measur ement of changes to the
+mode parameters over time (Karoff et al. 2009).
+Figure 4 showcases the potential of the Keplerdata for performing high-quality astero-
+seismology of solar-like stars from SC data (Chaplin et al. 2010). The left-hand panels show
+frequency-power spectra of three 9th-magnitude, solar tempe rature targets observed during
+4The solar-like oscillations are driven stochastically and damped by the vigorous turbulence in the near-
+surfacelayersofthe convectionzones, meaningthe modesareint rinsicallystablesee, e.g. Goldreich & Keeley
+(1977), and Houdek et al. (1999).– 9 –
+Q1. All three stars have a prominent excess of power showing a rich spectrum of acoustic
+(p) modes. The insets show near-regular spacings characteristic of the solar-like mode spec-
+tra, and highlight the excellent S/N observed in the individual mode pe aks. The sharpness
+of the mode peaks indicates that the intrinsic damping from the near -surface convection is
+comparable to that seen in solar p modes.
+The p modes sit on top of a smoothly varying background that rises in power towards
+lower frequencies. This background carries signatures of convec tion and magnetic activity
+in the stars. We see a component that is most likely due to faculae – br ight spots on the
+surface ofthestars formedfromsmall-scale, rapidlyevolving magn eticfield. This component
+is manifest in the spectra of the top two stars as a change in the slop e of the observed
+background just to the low-frequency side of the p-mode envelop e (see arrows). All three
+stars also show higher-amplitude components due to granulation, w hich is the characteristic
+surface pattern of convection.
+The near-regularity of the oscillation frequencies allows us to display them in so-called
+´ echelle diagrams, in Figure 4. Here, the individual oscillation frequen cies have been plotted
+against their values modulo ∆ ν(the average large frequency spacing – see Eq. 1). The
+frequencies align in three vertical ridges that correspond to radia l, dipole, and quadrupole
+modes. By making use of the individual frequencies and the mean spa cings we are able to
+constrain the masses and radii of the stars to within a few percent . The top two stars are
+both slightly more massive than the Sun (by about 5%), and also have larger radii (larger
+by about 20% and 30% respectively). The bottom star is again slightly more massive than
+the Sun (10%), and about twice the radius. It has evolved off the ma in sequence, having
+exhausted the hydrogen in its core. The ragged appearance of its dipole-mode ridge labeled
+“Avoided Crossing5” in Figure 4, is a tell-tale indicator of the advanced evolutionary stat e;
+the frequencies are displaced from a near-vertical alignment beca use of evolutionary changes
+to the deep interior structure of the star.
+5The effects are analogous to avoided crossings of electronic energ y levels in atoms e.g., see Osaki (1975)
+and Aizenman, Smeyers & Weigert (1977). Here, evolutionary chan ges to the structure of the deep interior
+of the star mean that the characteristic frequencies of modes wh ere buoyancy is the restoring force have
+moved into the frequency range occupied by the high-order acous tic modes. Interactions between acoustic
+and buoyancy modes give rise to the avoided crossings, displacing th e frequencies of the dipole modes so
+that they no longer lie on a smooth ridge.– 10 –
+3.1.2. Red Giants
+Red giants have outer convective regions and are expected to exh ibit stochastic oscil-
+lations that are ‘solar-like’ in their general properties but occur at much lower frequencies
+(requiring longer time series). The first firm discovery of solar-like o scillations in a giant was
+made using radial velocities by Frandsen et al. (2002). However, it w as only recently that
+the first unambiguous proof of nonradial oscillations in G and K giants was obtained, using
+spaced-based photometry from the CoRoT satellite (De Ridder et a l. 2009). This opened
+up the field of red giant seismology, which is particularly interesting be cause important un-
+certainties in internal stellar physics, such as convective oversho oting and rotational mixing,
+are more pronounced in evolved stars because they accumulate wit h age.
+The extremely highS/Nphotometry ofthe Keplerobservations brings redgiant seismol-
+ogy to the next level. With the first 43 days of LC data we were able to detect oscillations
+withνmaxranging from 10 µHz up to the Nyquist frequency around 280 µHz, as shown
+in Figure 5. The results include the first detection of oscillations in low- luminosity giants
+withνmax>100µHz (Bedding et al. 2010). These giants are important for constrain ing the
+star-formation rate in the local disk (Miglio et al. 2009). In addition, Keplerpower spectra
+have such a low noise level that it is possible to detect l=3 modes (Bedding et al. 2010) –
+significantly increasing the available asteroseismic information. The la rge number of giants
+thatKeplercontinuously monitors for astrometric purposes during the entire mission will
+allow pioneering research on the long-term interaction between osc illations and granulation.
+It is also expected that the frequency resolution provided by Keplerwill ultimately be suf-
+ficient to detect rotational splitting in the fastest rotating giants , and possibly allow the
+measurement of frequency variations due to stellar evolution on th e red giant branch.
+3.2. Classical Pulsating Stars
+Several working groups within KASC are devoted to the various clas ses of classical
+pulsators. The goals and first results are discussed below, ordere d roughly from smaller to
+larger stars (i.e., from shorter to longer oscillation timescales), givin g most emphasis to those
+areas in which scientific results have already been possible.
+3.2.1. Compact Pulsators
+The KASC Working Group on compact pulsators will explore the intern al structure
+and evolution of stars in the late stages of their nuclear evolution an d beyond. The two– 11 –
+main classes of targets are white dwarf stars (the ultimate fate of solar-mass stars) and hot
+subdwarf B stars – less-evolved stars that are undergoing (or ju st completing) core helium
+burning.
+While compact stars are among the faintest asteroseismic targets , the science payoff
+enabled by continuous short-cadence photometry will be profoun d. Pulsations in these stars
+are far more coherent than for solar-like pulsators, so extended coverage should provide
+unprecedented frequency resolution, and may also reveal very lo w amplitude modes.
+We will address several important questions about the helium-burn ing stage including
+the relative thickness of the surface hydrogen-rich layer (i.e. Cha rpinet et al. 2006), the role
+andextent ofradiativelevitationanddiffusionofheavyelements(Ch arpinet et al.2008), and
+the degree ofinternal differential rotation(Kawaler, Sekii & Goug h1999; Kawaler & Hostler
+2005; andCharpinet, Fontaine & Brassard2009). Evenifthesest arsrotateextremelyslowly,
+thehighfrequencyresolutionof Keplerdatashouldrevealrotationallysplitmultiplets. Aster-
+oseismology can also probe the role of binary star evolution in produc ing these hot subdwarfs
+(Hu et al. 2008).
+Similar questions about white dwarfs will be addressed, along with pro bing proper-
+ties of the core such as crystallization (e.g. Metcalfe, Montgomery & Winget 2004; and
+Brassard & Fontaine2005)andreliccompositionchangesfromnucle arevolution(seeWinget & Kepler
+2008, and Fontaine & Brassard 2008 for recent reviews of white dw arf asteroseismology).
+3.2.2. Rapidly Oscillating Ap Stars
+Rapidly oscillating Ap (or roAp) stars, with masses around twice solar , are strongly
+magnetic and chemically peculiar. They oscillate in high-overtone p mod es, similar to those
+seen in the Sun (Kurtz 1982; Cunha 2007). Their abnormal surfac e chemical composition
+results from atomic diffusion, a physical process common in stars bu t most evident in Ap
+stars due to their extremely stable atmospheres. Since ground-b ased photometry only re-
+veals a small number of oscillation modes in roAp stars, long-term con tinuous observations
+from space are needed to detect the large numbers of modes desir able for asteroseismology.
+This has been achieved for a few bright roAp stars using the MOST sa tellite, which found
+oscillation modes down to the level of 40 µmag (Huber et al. 2008). Since no roAp stars
+are present in the CoRoT fields, Kepler, with its 1 µmag precision should dramatically in-
+crease the number of detected oscillation modes – providing strong constraints on models of
+diffusion, magnetic fields, and the internal structure of roAp star s.– 12 –
+3.2.3.δSct andγDor Variables
+Stars that exhibit both p modes (pressure-driven) and g modes (b uoyancy-driven) are
+valuable for asteroseismology because they pulsate with many simult aneous frequencies.
+Theoretical calculations predict a small overlap in the H-R diagram of the instability region
+occupied by the γDor stars, which display high-order g modes driven by convective blo cking
+at the bottom of the envelope convection zone (Guzik et al. 2000), and the δSct stars,
+in which low-order g and p modes are excited by the κmechanism in the He iiionization
+zone6. Among the hundreds of known δSct andγDor variables, ground- and spaced-based
+observations have so far detected only a handful of hybrids (Han dler 2009). The g modes
+inγDor stars have low amplitudes and periods of order one day, making t hem particularly
+difficult to detect from the ground. The KeplerQ1 data in LC mode have revealed about
+40δSct candidates and over 100 γDor candidates, among which we find several hybrid δ
+Sct –γDor stars (Grigahc` ene et al. 2010). These are young hydrogen- burning stars with
+temperatures of 6500–8000K and masses of 1.5–2M ⊙. Figure 6 shows an example.
+TheKeplerdata, in conjunction with spectroscopic follow-up observations, w ill help to
+unravel theoretical puzzles for the hybrid stars. For example, Keplershould help us identify
+and explain the frequency of hybrid stars, discover previously unk nown driving mechanisms,
+test theoretical predictions of the δSct andγDor oscillation frequencies, find possible higher-
+degree modes that fill in frequency gaps, and determine whether a ll hybrids show abundance
+peculiarities similar to Am stars as were noted in previously known hybr ids. We should
+also learn about the interior or differential rotation of these objec ts via rotational frequency
+splittings. Perhaps just as interesting are those stars observed byKeplerthat reside in the
+γDor orδSct instability regions that show no significant frequencies. Indeed , the discovery
+of photometrically constant stars at µmag precision would be very interesting. Studying
+Kepler’s large sample of stars and looking for trends in the data will also help us better
+understand amplitude variation and mode selection.
+Another unique possibility arising from the long-baseline, high-precis ionKeplerdata is
+the ultimate frequency resolution it provides. The CoRoT mission has recently shown δSct
+stars to have hundreds and even thousands of detectable frequ encies, with degrees up to
+l≤14 (Poretti et al. 2009). However, it has long been known that mod e frequencies in δSct
+stars can be so closely spaced that years of data—such as that pr ovided by Kepler—may be
+needed to resolve them.
+6The mechanism operates through perturbations to the opacity κthat blocks radiation at the time of
+compression in a critical layer in the star, heating the layer and cont ributing to driving the oscillation.– 13 –
+3.2.4. RR Lyr Stars
+RRLyrstarsareevolvedlow-massstarsthathaveleftthemainseq uenceandareburning
+helium in their cores. Their “classical” radial oscillations with large amplit udes make them
+useful tracers of galactic history and touchstones for theoret ical modelling (Kolenberg et al.
+2010). Moreover, like Cepheids, they obey a period-luminosity-colo r relation which allows
+them to play a crucial role as distance indicators. Most RR Lyr stars pulsate in the radial
+fundamental mode, the radial first overtone, or in both modes sim ultaneously.
+The phenomenon of amplitude and phase modulation of RR Lyr stars – the so-called
+Blazhko effect (Blazhko 1907) – is one of the most stubborn problem s of the theory of radial
+stellar pulsations. With Keplerphotometry, we will be able to resolve important issues,
+such as period variation, the stability of the pulsation and modulation , the incidence rate
+of the Blazhko effect, multiple modulation periods, and the existence of ultra-low amplitude
+modulation. Our findings should constrain existing (Dziembowski & Miz erski 2004) and
+future models. Moreover, Keplermay find second and higher radial overtones and possibly
+nonradial modes (Gruberbauer et al. 2007) in the long uninterrupt ed time series. Figure 7
+shows two striking RR Lyrae light curves from the early Keplerdata contrasting examples
+of a constant wave form and strongly modulated one.
+3.2.5. Cepheids
+Classical Cepheids are the most important distance indicators of th e nearby Universe.
+Several candidates (Blomme et al. 2009) have been found in the Keplerfield. Being long-
+periodradialpulsators, Cepheidsmayshowinstabilitiesthathavebe enhiddeninthesparsely
+sampled, less accurate ground-based observations. Kepleralso provides an excellent oppor-
+tunity to follow period change and link it to internal structure variat ions due to stellar
+evolution. Period variation and eclipses will uncover binary Cepheids; faint companions may
+directly affect their role as distance calibrators (Szabados 2003).
+The discovery of non-canonical Cepheid light variations is highly prob able. Ultra-low
+amplitude (Szab´ o, Buchler & Bartee 2007) cases are in an evolution ary stage entering or
+leaving the instability strip (Buchler & Koll´ ath 2002), thus providing in formation on the
+driving mechanism and scanning the previously unexplored domains of the period-amplitude
+relation. Nonradial modes (Mulet-Marquis et al. 2007; Moskalik & Kola czkowski 2009), and
+solar-like oscillations driven by convection, may provide a valuable too l to sound inner stellar
+structure, adding yet another dimension to asteroseismic investig ations of Cepheids.– 14 –
+3.2.6. Slowly Pulsating B and βCep Stars
+Slowly pulsating B (SPB) stars are mid-to-late B-type stars oscillatin g in high-order g
+modes with periodsfrom0.3 to 3 days. These oscillations aredriven by theκmechanism act-
+ing in the iron opacity bump at around 200000 K (Dziembowski, Moskalik & Pamyatnykh
+1993; Gautschy & Saio 1993). Since most SPB stars are multi-period ic, the observed vari-
+ations have long beat periods and are generally complex. The large ob servational efforts
+required for in-depth asteroseismic studies are hard to achieve wit h ground-based observa-
+tions.
+With the ultra-precise Keplerphotometry of SPB stars spanning 3.5 years, it will be
+possible to search for signatures of different types of low-amplitud e oscillations to probe
+additional internal regions. Keplercan resolve the recent suggestion that hybrid SPB/ δSct
+stars exist (Degroote et al. 2009), thus filling the gap between the classical instability strip
+and the one for B stars with nonradial pulsations – which theory doe s not predict. The
+nature of the excess power recently found in βCep and SPB stars (Belkacem et al. 2009;
+Degroote et al. 2009) can be quantified. Keplershould also allow us to investigate the stabil-
+ity of the periods and amplitudes, and to search for evolutionary eff ects in the more evolved
+B stars.Kepler’s photometric capabilities will help us detect andexplore both the devia tions
+from regular period spacings predicted for g modes in the asymptot ic regime (Miglio et al.
+2008) and frequency multiplets induced by stellar rotation and/or m agnetic fields. Finally,
+we will test the developing models that include the effects of rotation on g mode pulsations.
+So far, some βCep stars have been studied asteroseismically with data from inten-
+sive ground-based observational campaigns (Aerts 2008). Thes e studies placed limits on
+convective core overshooting and demonstrated the presence o f non-rigid internal rotation
+(Aerts et al. 2003), in addition to identifying constraints on the inte rnal structure, opacities,
+andabundances(see, e.g., Pamyatnykh et al.2004; Briquet et al.2 007; Daszy´ nska-Daszkiewicz & Walczak
+2009). With the new level of precision provided by Kepler, we can go further, such as exam-
+ining the suspected presence of solar-like oscillations in βCep stars (Belkacem et al. 2009).
+The detection of more pulsation modes is key to detailed analyses of in ternal rotation. New
+methods using rotational mode splittings and their asymmetries hav e been developed by
+Su´ arez et al. (2009), and might help to test theories describing an gular momentum redistri-
+butionandchemical mixing dueto rotationallyinduced turbulence. Th eanalysis of“hybrid”
+p and g mode pulsators will result in tighter constraints on stellar str ucture, particularly
+on opacities (Handler et al. 2009). This increased understanding of βCep stars through as-
+teroseismology can provide useful information about the chemical evolution of the Universe
+becauseβCep stars, pulsating in modes of low radial order and periods betwee n about 2 −7
+hours, are ideal for determining the interior structure and compo sition of stars around 10– 15 –
+M⊙—precursors of type II supernovae.
+3.2.7. Miras and Semiregular Variables
+Miras and semiregular variables (M giants) are the coolest and most lu minous KASC
+targets, representing advanced evolutionary stages of low- and intermediate-mass stars such
+as the Sun. Ground-based photometry has shown many of these s tars to have complicated
+multi-mode oscillations on timescales ranging from several hundred d ays down to ten days
+and probably shorter (Wood 2000; Tabur et al. 2009). The oscillatio ns are strongly coupled
+to important, but still poorly understood mechanisms, such as con vection and mass loss.
+The uninterrupted coverage and unprecedented precision of Keplerphotometry will al-
+low the first application of asteroseismology to M giants. We anticipat e the first exciting
+results in this field after the first year of observations, while the mo st intriguing theoretical
+questions will require the full time span of the project. By measurin g the oscillation fre-
+quencies, amplitudes and mode lifetimes we will challenge stellar models a nd examine the
+interplay between convection and the κmechanism, and the roles of both in exciting and
+damping the oscillations (Xiong & Deng 2007). We also hope to shed light on the myste-
+rious Long Secondary Periods phenomenon (Nicholls et al. 2009) and to search for chaotic
+behavior and other non-linear effects that arise naturally in these v ery luminous objects
+(Buchler, Koll´ ath & Cadmus 2004).
+3.3. Oscillating Stars in Binaries and Clusters
+One of the goals of the asteroseismology program is to model the os cillation frequencies
+of stars in eclipsing binaries. In order to find such stars in the Keplerfield of view, a global
+variability classification treating all KASC stars observed in Q0 and Q1 was performed with
+the methodology by Debosscher et al. (2009). This revealed hundr eds of periodic variables
+all over the H-R Diagram, among which occur stars with activity and r otational modu-
+lation in the low-frequency regime, Cepheids, RRLyr stars, and mult iperiodic nonradial
+pulsators (Blomme et al. 2009). More than 100 new binaries were also identified. About
+half of those are likely to be ellipsoidal variables while the other half are eclipsing binaries,
+several ofwhichhave pulsatingcomponents. Oneexample foracas eunknown priortolaunch
+is shown in Figure 8, where we see pulsational behavior which is typical for gravity modes
+in a slowly pulsating B star (Degroote et al. 2009) in an eclipsing system with a period of
+about 11 days with both primary and secondary eclipses. Examples o f pulsating red giants– 16 –
+in eclipsing binaries are clear in early Keplerdata (Hekker et al. 2009), and discussed for
+the above case of KOI-145. Eclipsing binaries with pulsating compone nts not only provide
+stringent tests of stellar structure and evolution models, they als o offer the opportunity to
+discover planets through changes of the pulsation frequencies du e to the light travel time
+in the binary orbit (Silvotti et al. 2007). For these reasons, pulsat ing eclipsing binaries are
+prime targets within KASC.
+Star clusters are Rosetta Stones in stellar structure and evolutio n. Stars in clusters are
+believed to have been formed from the same cloud of gas at roughly t he same time, leaving
+fewer free parameters when analyzed as a uniform ensemble, which allows stringent tests of
+stellar evolution theory. Seismic data enable us to probe the interior of models, and thus
+should allow us to test aspects of stellar evolution that cannot be ad dressed otherwise, such
+as whether or not the sizes of convective cores in models are corre ct. Asteroseismology of
+star clusters has been a long sought goal that holds promise of rew arding scientific return. In
+particular, the advantages of asteroseismology for clusters are that, unlike estimates of colors
+and magnitudes, seismic data do not suffer from uncertainties in dist ance or extinction and
+reddening.
+Therearefouropenclustersinthe Keplerfieldofview, NGC6791,NGC6811,NGC6866
+and NGC 6819. In Figure 9 we show a color-magnitude diagram of star s in NGC 6819 and
+point out the relative flux variation of four stars along the giant bra nch. Initial data from
+Keplerhave allowed the first clear measurements of solar-like oscillations in c luster stars by
+Stello et al. (2010). The oscillations of the stars are clear even witho ut any further analysis.
+The power in the oscillations changes with luminosity and effective temp erature as expected
+fromground-based andearlyspace-based observations of near by field stars. The data further
+enable us to determine cluster membership by lining up the power spec tra of the observed
+stars in order of their apparent magnitude. We have detected sev eral possible non-members
+in this manner, which previously all had high membership probabilities ( P >80%) from
+radial-velocity measurements.
+While the currently available data from Keplerare for stars on the giant branch of
+the clusters, future observations will also provide data for the su bgiants and main-sequence
+stars. Having seismic data for stars at various stages of evolution will allow us to have
+independent constraints on the cluster age. Since stars in a cluste r are coeval and have the
+same metallicity, the process of modelling detailed seismic data of thes e stars will allow us
+to test the various physical processes that govern stellar evolut ion.– 17 –
+4. Summary
+Thephenomenalpromiseofthe Kepler Mission toinvigoratestellarastrophysicsthrough
+the study of stellar oscillations is clearly being realized. The unpreced ented combination of
+temporal coverage and precision is sure to provide new insights into many classical variable
+stars, to increase by more than two orders of magnitude the numb er of cool main-sequence
+and subgiant stars observed for asteroseismology and to complet ely revolutionize asteroseis-
+mology of solar-like stars.
+Kepleris the tenth Discovery mission. Funding for this mission is provided by N ASA’s
+Science Mission Directorate. CA, JDR and JD received funding from t he European Research
+Council under the European Community’s Seventh Framework Prog ramme (FP7/2007–
+2013)/ERC grant agreement n◦227224 (PROSPERITY), as well as from the Research Coun-
+cilofK.U.Leuven (GOA/2008/04)andfromtheBelgianFederalScien ce PolicyOfficeBelspo.
+We are grateful to the legions of highly skilled individuals at many privat e businesses, uni-
+versities and research centers through whose efforts the marve lous data being returned by
+Keplerhave been made possible.
+Facilities: The Kepler Mission
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+This preprint was prepared with the AAS L ATEX macros v5.2.– 22 –
+Fig. 1.— Panel A shows the power spectrum for KOI-145 with the sign ature of solar-like
+oscillations in a red giant. This shows the power between 80 and 180 µHz with an indication
+of the largeseparation parameter, ∆ νbetween two consecutive l= 1 modes. The inset shows
+the nearly perfect spectral window provided by these Keplerdata. Panel B shows the result
+of folding the power spectrum by ∆ νwhich clearly shows the resulting distribution of l=
+2, 0, and 1 (from left) modes corresponding to the relation given in E q. 1. Panel C shows
+part of the time series for KOI-145 and a superposed ‘transit’ light curve solution. Assuming
+the stellar radius as implied by the asteroseismology of R = 5.99 ±0.15 R ⊙results in a
+transiting object size of ∼0.4 R ⊙, well removed from the planetary realm. The eclipsing
+object is likely an M-dwarf star.– 23 –
+Fig. 2.— The distribution of low-angular-degree p modes in a so-called ´ echelle di-
+agram, in which individual oscillation frequencies are plotted against t he frequencies
+modulo the average large frequency spacing, and with the abscissa scaled to the ∆ ν
+units. Panel a shows results for KOI-145. The filled symbols are the observed fre-
+quencies and the open symbols show frequencies from a model in the grid calculated by
+Stello et al. (2009), with symbol sizes proportional to a simple estima te of the amplitude
+(see Christensen-Dalsgaard, Bedding & Kjeldsen 1996). Panel b s hows frequencies for the
+Sun, with filled symbols indicating observed frequencies (Broomhall, A .-M., et al. 2009),
+while open symbols show the frequencies of Model S (Christensen-D alsgaard et al. 1996).– 24 –
+      12       13       14       15       16       17       18       19       20       21       22       23       24
+       0        1        2
+Fig. 3.— The upper panel shows the power spectrum for the previou sly known HAT-P-
+7 host star based on KeplerQ0 and Q1 data. The marked modes are the radial, l= 0
+frequencies labelled with inferred radial order n. The lower panel shows the power spectrum
+folded by ∆ νwith the l= 1, 2, and 0 ridges (from left) as indicated.– 25 –
+Fig. 4.— Left-handpanels: Frequency-powerspectraof Keplerphotometryofthreesolar-like
+stars (grey) over 200 – 8000 µHz. The thick black lines show the result of heavily smoothing
+the spectra. Fitted estimates of the underlying power spectral d ensity contribution of p
+modes, bright faculae and granulation as labelled in the top left panel are also shown; these
+are color coded red, blue and green respectively in the on-line versio n. These components
+sit on top of a flat contribution from photon shot noise. The arrows mark a kink in the
+background power that is caused by the flattening toward lower fr equencies of the facular
+component. The insets show the frequency ranges of the most pr ominent modes. Right-
+hand panels: So-called ´ echelle plots of individual mode frequencies. Individual oscillation
+frequencies have been plotted against the frequencies modulo the average large frequency
+spacings (with the abscissa scaled to units of the large spacing of ea ch star). The frequencies
+align in three vertical ridges that correspond to radial modes ( l= 0, diamonds), dipole
+modes (l= 1, triangles) and quadrupole modes ( l= 2, crosses).– 26 –
+0 20 40 60 80 100 120 140 160 180 200 0100 200 300 400 500 600 700 800 900 1000 
+frequency (μHz) sequential star # 
+Fig. 5.— Keplerred giant power spectra have been stacked into an image after sor ting
+onνmaxfor the oscillation peak. Larger stars, with lower frequency variat ions are at the
+top. The ever-present slow variations due to stellar granulation ar e visible as the band at
+the left, generally in the 2 - 10 µHz range. The curve from top left to bottom right traces
+the oscillations. Low-mass, low-luminosity giants are clearly visible her e withνmax>100
+µHz (Bedding et al. 2010). The slope of the curve is a measure of how f ast the evolutionary
+stage is, the more horizontal the faster, although selection effec ts for the sample also need
+to be taken into account. The bulk of the giants are He-burning sta rs with a νmaxaround 40
+µHz. The fact that the top right part of the figure is so uniformly dar k illustrates how little
+instrumental noise there is for Kepler. By contrast the low-Earth orbit for CoRoT imposes
+extra systematic noise in the general domain of 160 µHz contributing to its lack of results
+on these smaller, more rapidly varying and lower-amplitude red giants .– 27 –
+Fig. 6.— Amplitude spectrum of the δSct –γDor star hybrid KIC 09775454 where the
+g-mode pulsations are evident in the 0 −70µHz range and the p-mode pulsation is evident
+at 173µHz (P= 96min). There are further frequencies in the δSct range that are not seen
+at this scale. The amplitude spectrum has a high frequency limit just b elow the Nyquist
+frequency of 283 µHz for LC data. In the range 70 −140µHz there are no significant peaks.
+The highest noise peaks have amplitudes of about 20 µmag; the noise level in this amplitude
+spectrum is about 5 µmag.– 28 –
+Fig. 7.— The upper panel shows direct and folded Q1 light curves for t he non-modulated
+RR Lyrae star NR Lyrae, while the lower panel illustrates the rapid ev olution of oscillation
+wave form present in RR Lyrae itself – characteristic of many RR Lyr ae stars monitored
+withKepler.– 29 –
+0 5 10 15 20 25 30 35
+HJD-0.1
+-0.05
+0
+0.05
+0.1Delta mag
+Fig. 8.— Time series over Q1 of the slowly pulsating B-star, KIC-11285 625, which shows
+both large amplitude multi-periodic pulsations, and primary and secon dary eclipses. With
+the addition of ground-based radial-velocity studies and the exquis iteKeplerlight curve
+elucidating the broad range of phenomena visible here, this is likely to b ecome one of the
+more important asteroseismic targets.– 30 –
+Fig. 9.— Color-magnitude diagram for NGC 6819. The grey points show stars that have
+high radial-velocity probabilities from Hole et al. (2009). The curve (r ed in on-line version)
+is a solar metallicity, 2.5 Gyr isochrone of Marigo et al. (2008). The larg er, dark-grey (blue)
+points are four of the Keplertargets; we show the time series of their brightness fluctuation
+in the insets. As can be seen from the inset, the timescale of the osc illations decreases with
+increasing apparent magnitude of the stars.
\ No newline at end of file
diff --git a/1001.0140.txt b/1001.0140.txt
new file mode 100644
index 0000000000000000000000000000000000000000..49a0d6d1207acc8e79f53049351d74f6b539f7ab
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@@ -0,0 +1,365 @@
+arXiv:1001.0140v1  [physics.ins-det]  31 Dec 20091
+ThermalControl ofaDualMode
+Parametric SapphireTransducer
+JacopoBelfi,Nicol `oBeverini,AndreaDe Michele,GianlucaGabbriellini,
+FrancescoMangoandRobertoPassaquieti
+Abstract—We propose a method to control the thermal stability of a sap phire dielectric transducer made with two
+dielectricdisksseparatedbyathingapandresonatinginth ewhisperinggallery(WG)modesoftheelectromagneticfield .
+The simultaneousmeasurementof the frequenciesof botha WG H mode and a WGE mode allowsone to discriminate
+the frequency shifts due to gap variations from those due to t emperature instability. A simple model, valid in quasi
+equilibriumconditions,describesthe frequencyshift of t he two modes in terms of four tuningparameters.A procedure
+forthedirectmeasurementof themispresented.
+✦
+1 INTRODUCTION
+ADIELECTRIC whispering gallery resonator,
+made with two dielectric disks separated
+by a thin gap is a very sensitive displacement
+sensor [1]. Indeed, parametric microwave sap-
+phire oscillators, operating at cryogenic tem-
+perature, have been proposed as readout sys-
+tems for bar-type gravitational antennae [2]
+and also as core sensors for space-gravity tests
+and geodesy research [3]. It has been shown
+that such devices provide high sensitive vi-
+bration measurements even when operating at
+room-temperature [4].
+The efficiency of these transducers is conve-
+niently described by the merit factor
+M=Q·∂zf
+f,
+whereQis the resonator quality factor, fits res-
+onance frequency and ∂zfis the tuning coeffi-
+cient, the derivative of the resonance frequency
+w.r.t. the gap spacing z.
+The modes are classified as WGH modes
+(characterized by Ez,Hφ,Hρ)andWGE modes
+(withHz,Eφ,Eρ), wherezdenotes the compo-
+nent of the field vector along the disks rota-
+•The authors are with the Department of Physics “Enrico Fermi ”
+and CNISM unit` a di Pisa , Universit` a di Pisa, 56127 Pisa, It aly.
+E-mail: belfi@df.unipi.ittional axis, ρthe radial component, and φthe
+azimuthal one.
+TheWGH modes with high azimuthal mode
+number present the highest merit factors [2].
+Even at room temperature, X-band reso-
+nances in a sapphire resonator made with disks
+4 cm in diameter and 0.5 cm in thickness, ex-
+hibit aQfactor of ∼105and a tuning coefficient
+of6 MHzµm−1so that M∼60µm−1. The
+frequency instability is dominated by thermal
+effects, which contribute with some tenths of
+MHz K−1. This strong temperature dependence
+of the dielectric tensor compromises the long
+term stability. On the other side, ultra-low
+frequency (i.e. diurnal timescale) displacement
+measurements have a key role in many appli-
+cations such as gravimetric exploration, envi-
+ronmental monitoring and materials testing.
+In the research field of ultra stable oscillators,
+several techniques for precision temperature
+stabilization have been proposed. At low tem-
+perature (below 100 K) the cancellation (to first
+order) of the temperature coefficient of fre-
+quency can be achieved by employing doped
+dielectric disks [5], composite sapphire-rutile
+resonators [6] and mechanically compensated
+structures [7]. High thermal stability at room
+temperature can be obtained with high pre-
+cision control stages, by the optimization of
+temperature sensors [8] and actuators [9], and
+also by accurately modelling and simulating2
+the thermodynamic system under test [10].
+Due to the anisotropy of the sapphire crystal,
+the temperature coefficient of frequency is in
+general different for WGH and WGE modes.
+Exciting two electromagnetic modes with dif-
+ferent polarization in the same resonator per-
+mit one to measure and stabilize the resonator
+temperature in oscillators [11], [12] even at
+room temperature [13]. In the case of a dis-
+placement transducer, WGH and WGE modes
+have to satisfy different boundary conditions at
+the gap spacing thus exhibiting also different
+tuning coefficients ∂zf.
+The measured resonance frequency varia-
+tions in a parametric sapphire transducer are
+given by the mixing of pure-displacement sig-
+nals, pure-temperature signals and temperature-
+induced displacement signals. These last are due
+to the thermal expansion of the material com-
+prising the enclosing chamber.
+In this paper we propose a calibration tech-
+nique providing an estimate of pure displace-
+ment signals in a dual mode parametric sap-
+phire resonator transducer operating at room
+temperature.
+Fig. 1. A generic displacement sensor based
+on a transducer converting the physical quan-
+tity “x”, in a vertical displacement δlx. A servo
+actuator controls the separation set point. The
+sapphire disks are cut with their c-axis parallel
+tothecylinder axis.
+2 THERMAL EFFECTS IN A SAPPHIRE
+PARAMETRIC TRANSDUCER
+The frequency dependence on temperature
+for the parametric displacement transducer ofFig. 1 is described by the following equa-
+tion [12]:
+1
+f∂f
+∂T=−pǫ⊥αǫ⊥−pǫ||αǫ|| (1)
+−pDαD−phαh−pz(2αls+αl2+αlx−αl1),
+whereαθ=1
+θ∂θ
+∂Tare coefficients depending on
+the materials and pθ=θ
+f∂f
+∂θare the filling factors
+of the considered WG mode and depend on
+the field distribution of the mode inside the
+resonating volume. The label θrefers respec-
+tively to: ǫ⊥andǫ||the dielectric constants per-
+pendicular and parallel to the c-axis, Dandh
+the resonator spatial dimensions perpendicular
+and parallel to the c-axis, ls,l1,l2,lx,zthe verti-
+cal dimensions of the single sapphire disk, the
+chamber length, the lower support (actuator),
+the upper support (moving part) and the gap
+spacing. Under stationary conditions, one can
+assume to be valid the following linear and
+time-independent relation between a given pair
+of WG modes, the temperature Tand the gap
+spacingz:
+/parenleftBigg
+δfWGH
+δfWGE/parenrightBigg
+=C/parenleftBigg
+δT
+δz/parenrightBigg
+(2)
+with
+C=/parenleftBigg
+CWGH
+TCWGH
+z
+CWGE
+TCWGE
+z/parenrightBigg
+. (3)
+The anisotropy of both the material and the
+field distribution assures that it is possible to
+invert Cand to obtain the following estimate
+for the effective temperature fluctuations δT∗
+and for the pure (not thermally-induced) dis-
+placement δz∗:
+/parenleftBigg
+δT∗
+δz∗/parenrightBigg
+=C−1/parenleftBigg
+δfWGH
+δfWGE/parenrightBigg
+. (4)
+3 CALIBRATION
+The basis of the method is to determine the
+coefficients of the matrix Cby means of a
+calibration procedure. A schematics of the ex-
+perimental apparatus for the sensor calibration
+is shown in Fig. 2. The sapphire transducer is
+placed inside a metallic chamber which is tem-
+perature stabilized at about 34◦C. Each cham-
+ber wall is electrically isolated in order to avoid3
+any influence from cavity modes on the chosen
+WG modes in the dielectric. A 4 cm thick insu-
+lation material covers the whole chamber to re-
+duce the heat losses. The temperature stabiliza-
+tion system consists of a standard PI controller
+driving a set of thermoresistances in thermal
+contact with the metallic cavity. The set point
+temperature is determined by a voltage input
+for calibration purposes. The sapphire disks
+composing the resonator transducer are cou-
+pled to four microwaves antennae (see Fig. 2).
+Two electric probes, coupled to the azimuthal
+electric field are used for sustaining the WGE
+oscillations. One electric probe, coupled to the
+axial electric field, and one magnetic probe,
+coupled to the azimuthal magnetic field, excite
+the WGH oscillations. We chose to test the
+system using the following modes (see Fig. 3):
+fWGE11,1,1∼11.38 GHz, (5)
+fWGH 10,1,1∼11.20 GHz,
+forz∼300µm. In this choice there is very little
+coupling [1] between the selected modes and
+furthermore they are close enough in frequency
+so that it is possible to measure their beat
+note by means of an RF frequency counter.
+The upper resonator disk is rigidly attached to
+the aluminium top plate and the lower disk is
+mounted on a piezoelectric actuator.
+A PC based data acquisition system moni-
+tors three physical quantities: internal temper-
+ature (via the resistance of a Pt100 temperature
+probe), the WGE mode frequency (with a mi-
+crowave frequency counter) and the WGE−
+WGH beat frequency (with an RF frequency
+counter).
+4 CHECK OF THE METHOD
+Once the two microwave oscillations are imple-
+mented, we can evaluate the four elements of
+theCmatrix by means of the temperature and
+position controllers. CWGE
+z andCWGH
+z are given
+by the ratio between the frequency variation
+offWGEandfWGHand the calibrated gap
+variation induced by the piezoelectric actuator
+moving the lower disk.
+CWGE
+T andCWGH
+T are instead the proportion-
+ality constants between the frequency varia-
+tion extrapolated at thermal equilibrium and
+Fig. 2. Microwave circuitry for the dual mode
+self oscillating transducer employed for the
+calibration. TC: Temperature Controller, PS:
+Phase Shifter, MA: Microwave Amplifier (ALC-
+ALN060029),Circ:Circulator,PD:PowerDetec-
+tor.
+Fig. 3. Resonance frequencies of WGH and
+WGEmodes versus gap spacing.4
+the temperature variation induced by thermal
+actuators. For the present setup we obtained:
+CWGE
+T= 0.155MHz/K,
+CWGH
+T= 3.64MHz/K,
+CWGE
+z= 0.567MHz/µm,
+CWGH
+z= 3.43MHz/µm.
+In Fig. 4 we show a comparison between
+the frequency-based temperature variation es-
+timateδT∗and the temperature variation mea-
+sured by a Pt100 thermometer placed inside the
+chamber. The measurement is referred to stabi-
+lized temperature conditions and rigid materi-
+als. The agreement between the two traces con-
+firms the validity of the model especially for
+very slow variations and makes it possible to
+measure the sensor temperature sensitivity by
+means of frequency measurements. In Fig. 5 we
+show a comparison between the displacement
+estimate δz∗, obtained from the calibrated dual
+frequency measurement, and δfWGH/CWGH
+z i.e.
+the displacement estimate one would get from
+theWGH mode alone (the most sensitive mode
+to the gap spacing variation). It is neatly visible
+that the trace δz∗displays a net reduction of the
+fluctuations over time scales typical of thermal
+phenomena.
+It is worth remarking that the assumption of
+thermal equilibrium for the system is essential
+to effectively filter out the temperature noise
+from the displacement signal. In Fig. 6 we show
+the Allan deviation sz(τ) =/radicalBig
+/angbracketleft(¯zi−¯zi+1)2/angbracketright/2
+(where¯ziis the average of the displacement
+zover the i–th sample–period of duration τ) of
+the two displacement traces of Fig. 5.
+It can be seen that for integration times be-
+low about 200 sec the dual mode based dis-
+placement measurements are noisier than the
+single frequency measurements. This is due to
+the fact that over these time scales the dif-
+ferent sensor components are not in thermal
+equilibrium and the two frequencies are al-
+most uncorrelated. For integration times above
+about 500 sec, a net reduction of noise can be
+observed for the trace due to δz∗. This time
+scale corresponds to the longest time constant
+(sapphire thermalization) in the sensor. Here
+thermal equilibrium approximation is almost
+Fig. 4. Comparison between frequency-based
+temperature estimate and temperature mea-
+sured by aPt100 probe.
+Fig.5. Comparisonbetweenthedualfrequency
+based displacement estimate and the estimate
+obtained from thesingle WGHmode.
+valid and then the method provides the ex-
+pected noise cancellation.
+5 CONCLUSIONS
+We proposed and tested an experimental tech-
+nique for reducing the instabilities induced by
+thermal fluctuations in a sapphire parametric
+displacement transducer. The effective temper-
+ature of the sensor and the pure displace-
+ment signal can be obtained from a double
+frequency measurement. The very simple time-
+independent model of the system has been5
+Fig. 6. Allan deviation of the displacement
+fluctuations showninFig. 5.
+tested on a trial setup in order to outline the
+potential and the limitations of the method. In
+the future we intend to improve the efficiency
+of the thermal stabilization and isolation of the
+system in order to improve the bandwidth of
+the noise filtering. Finally by implementing a
+double locking of the two reconstructed quan-
+tities (δT∗andδz∗) it will be possible to im-
+plement the experiment under static conditions
+and improve on the modelled assumptions.
+REFERENCES
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\ No newline at end of file
diff --git a/1001.0141.txt b/1001.0141.txt
new file mode 100644
index 0000000000000000000000000000000000000000..45a343f8c6652ad5624aa00939a341578e1be78d
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+++ b/1001.0141.txt
@@ -0,0 +1,213 @@
+1 See http://bifrost.crwu.edu/nstars/ LINE DENSITY INDICE AS AN ALTERNATIVE TO MK PROCESS  
+ 
+M. Kuassivi 
+170374 Cotonou, Benin 
+ 
+ 
+ABSTRACT 
+ 73 broadband optical spectra of dwarf stars later than F0 have been obtained from the Nearby 
+Stars Project website 1.  The number of absorption lines is computed for e ach spectrum 
+between 6000 and 6200 Angstrom. A correlation is fo und between the density of lines Κλ and 
+the spectral type. This method is independent of ca libration process, does not require high 
+resolution or  high signal—to—noise data and does n ot make use of a large library of standard 
+spectra. 
+ 
+Keywords : stars : late type – stars : fundamental parameters 
+ 
+1. INTRODUCTION 
+ 
+  Accurate spectral types provide basic 
+physical parameters (effective temperature 
+and surface gravity). They can also be used 
+to pick out peculiar or astrophysically 
+interesting stars.  
+  In 1943, “An Atlas of Stellar Spectra” 
+was produced at the Yerkes observatory. 
+The Atlas represented a new approach to 
+the classification of stars, one that depends 
+on a set of standard “specimens”. The 
+Morgan—Keenan (MK) System of spectral 
+classification is a classical application of 
+morphological techniques (Keenan, 1984). 
+Such techniques have a fundamental 
+characteristics: stability over time 
+(Garrison, 1994a). 
+  Usually, the stars are classified  on the 
+MK system by direct visual inspection on a 
+computer screen. To avoid this time 
+consuming process, softwares are available 
+that match a target spectrum to one from a 
+standard dataset (see the ALLSTAR 
+program described in Henry et al. 2002). 
+This is not an easy task: calibrated fluxes 
+are interpolated, spectra are normalized in 
+a region free of opacity, spurious features 
+must be rejected (telluric lines and cosmic 
+rays). Although these routines can be tuned 
+to work well with a particular dataset and a 
+well—defined interval of spectral types, an 
+universal procedure is yet to be written.     In Fig.1 are shown two normalized spectra 
+in a limited wavelength region with 
+different spectral types. The difference 
+between the two spectra on the basis of the 
+number of lines is obvious. 
+ Such visual inspection cannot give an 
+accurate spectral type for a given star: 
+many lines can show up that are not seen in 
+all spectra due to metallicity effect, 
+spurious absorption lines may also appear 
+in poor signal—to—noise data.  
+ 
+ 
+FIG 1.  Comparizon of spectral type 
+  However, the need for large computer 
+routines comparing precisely flux 
+calibrated spectra with a complete set of 
+standard spectra over a large wavelength 
+domain seems avoidable. Using a simple 
+method, Gray et al. (2002) have shown that 2 KUASSIVI 
+basic empirical indicators can be well 
+suited to characterize  giant stars. 
+ 
+  In this paper, I propose an alternative to a 
+global morphological comparizon of 
+standard spectra by focusing on one 
+empirical parameter, independant of 
+calibration : the density of lines detected in 
+a given spectral range (hereafter Κλ ). 
+ 
+Spectral 
+type HD names Spectral 
+type HD names 
+F0V HD29391 G4V HD52711 
+F2V HD128167 G4V HD179958 
+F3V HD32715 G5V HD4208 
+F5V HD114378 G5V HD25680 
+F5V HD126141 G5V HD72946 
+F5V HD197692 G5V HD134987 
+F6V HD30652 G5V HD140538 
+F6V HD69897 G5V HD172051 
+F6V HD142860 G6V HD202206 
+F7V HD25998 G7V HD111395 
+F7V HD88595 G8V HD144579 
+F7V HD126660 K0V HD166 
+F7V HD143333 K0V HD10780 
+F7V HD215648 K0V HDD69830  
+F8V HD5015 K0V HD112758 
+F8V HD9826 K0V HD158633 
+F8V HD85380 K0V HD185144 
+F8V HD179949 K1V HD170657 
+F9V HD78366 K1V HD26965 
+F9V HD81858 K1V HD52698 
+G0V HD1461 K2V HD208801 
+G0V HD4614 K2V HD61606 
+G0V HD39587 K2V HD23356 
+G0V HD48682 K2V HD22049 
+G0V HD84117 K2V HD4628 
+G0V HD101177 K2V HD18445 
+G0V HD101563 K3V HD29697 
+G0V HD140913 K3V HD223778 
+G0V HD157214 K3V HD219134 
+G0V HD160269 K3V HD192310 
+G0V HD206860 K3V HD128165 
+G1V HD126053 K3V HD82106 
+G1V HD130948 K3V HD110833 
+G1V HD146233 K3V HD50281 
+G2V HD14802 K4V HD131977 
+G3V HD30495 K4V HD190007 
+G4V HD38529   
+Table 1. List of targets as function of their 
+spectral type 
+ 
+2. OBSERVATIONS AND ANALYSIS 
+ 
+  The observations reported in this paper 
+are public data obtained through the Nearby Star Project web site  (http://bifrost- 
+-.crwu.edu/nstars). The goal of the project 
+is to seek information on the population of 
+stars in the solar neighborhood. All the 
+data have been observed with a single 
+instrument at Mc Donald Observatory 
+which makes the data set highly consistent 
+(Heiter & Luck, 2003).   Tables 1. lists the 
+73 stars used in this work as a function of 
+their assigned spectral type. 
+  The data analysis is straihgtforward. 
+Spectra of many types were first carefully 
+observed and the spectral domain between 
+6000 and 6200 Angstrom was chosen : 
+smooth continuum and large variability of 
+the number of lines with spectral types. 
+Then I wrote a short I.D.L. routine to sort 
+the number of lines in that domain that was 
+robust enough to detect artifacts. The 
+detection treshold was automatically 
+adjusted  for each spectrum depending on 
+the signal—to—noise ratio (line depth 
+between 80 and 92 per cent). 
+ 
+3. RESULTS AND DISCUSSION 
+ 
+   
+Fig 2. Line density versus effective temperature. 
+The dashed line is the linear regression. 
+ 
+In Fig.2, I plot the line density indice Κλ  
+as a function of the effective temperature. 
+A clear linear correlation is observed: 
+ 
+Κλ = -7.7 10 -4 (0.5 10 -4) × Teff  + 5.5 (0.3) 
+ No 1, 2009                                     LINE  DENSITY AND MK PROCESS 3 
+  Numbers in parenthesis are 1—sigma 
+uncertainties on the parameters. Those 
+uncertainties are relatively small in respect 
+to the many uncertainties that can affect 
+the data analysis:  errors on the derived 
+spectral type, lack of sophistication of the 
+detection procedure, and intrinsic 
+variability of the number of lines in various 
+physical conditions. 
+ 
+  Variation in the number of absorption 
+lines is a trivial observation (Fig 1.) 
+consistent with the increase of opacity with 
+decreasing Effective Temperature. One 
+that might have been neglected when 
+assigning spectral type. This work shows 
+that for late-type dwarf the number of lines 
+gives a fair guess of the spectral type. The 
+method can be improved in many ways: 
+multi wavelength analysis, extension to 
+earlier spectra types, to giant stars, more 
+sophisticated routines for the detection of 
+weak absorption lines and additional use of 
+line ratios. 
+ 
+This work is an attempt to underline that a 
+complete MK process reaches far beyond 
+the scope of assigning spectral types 
+(Garrison, 1994a; Garrison, 1994b; 
+Walborn, 2008). The speed and capacity of 
+current computers have led many to use 
+big tools for small jobs. Short routines 
+based on empirical parameters should be 
+properly fitted for spectral classification. 
+ 
+ 
+REFERENCES 
+ 
+Garrison, R.F. 1994a, ASP conference    
+  Series, Vol. 60 
+Garrison, R.F. 1994b, RMxAA, 29, 111 
+Gray, D.F., Scott, H.R., & Postma, J.E.      
+  2002, PASP, 114, 536 
+Gray, R.O., Corbally, C.J., & Robinson, 
+  P.E. 2003, AJ, 126, 2048 (Paper I) 
+Heiter, U. & Luck, R.E. 2003, AJ, 126,  
+  2015 
+Keenan, P.C. 1984, in The MK Process  
+  and Stellar Classification, ed.R.F.  
+  Garrison (Toronto: DDO), 20 Henry, T.J., Walcowicz, L.M., Barto, T.C.,  
+  & Golimowski, D.A. 2002, AJ, 123,  
+  2002 
+Walborn, N.R. 2008, RMxAA, 33,5 
\ No newline at end of file
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+arXiv:1001.0142v1  [astro-ph.SR]  31 Dec 2009Submitted to The Astrophysical Journal Letters
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+INITIAL CHARACTERISTICS OF KEPLER SHORT CADENCE DATA
+Ronald L. Gilliland1, Jon M. Jenkins2, William J. Borucki3, Steven T. Bryson3, Douglas A. Caldwell2, Bruce
+D. Clarke2, Jessie L. Dotson3, Michael R. Haas3, Jennifer Hall4, Todd Klaus4, David Koch3, Sean McCauliff4,
+Elisa V. Quintana2, Joseph D. Twicken2, and Jeffrey E. van Cleve2
+Submitted to The Astrophysical Journal Letters
+ABSTRACT
+TheKepler Mission offerstwooptions for observations– either Long Cadence(LC) us ed for the bulk
+of core mission science, or Short Cadence (SC) which is used for app lications such as asteroseismology
+of solar-like stars and transit timing measurements of exoplanets w here the 1-minute sampling is
+critical. We discuss the characteristics of SC data obtained in the 33 .5-day long Quarter 1 (Q1)
+observations with Keplerwhich completed on 15 June 2009. The truly excellent time series prec isions
+are nearly Poisson limited at 11th magnitude providing per-point meas urement errors of 200 parts-
+per-million per minute. For extremely saturated stars near 7th mag nitude precisions of 40 ppm are
+reached, while for background limited measurements at 17th magnit ude precisions of 7 mmag are
+maintained. We note the presence of two additive artifacts, one th at generates regularly spaced
+peaks in frequency, and one that involves additive offsets in the time domain inversely proportional to
+stellar brightness. The difference between LC and SC sampling is illustr ated for transit observations
+of TrES-2.
+Subject headings: planetary systems — stars: oscillations — techniques: photometric
+1.INTRODUCTION
+TheKepler Mission has a primary science goal of de-
+tecting analogs of the Earth orbiting stars in the ex-
+tended solar neighborhood as reviewed in Borucki et al.
+(2009). Overall mission design and performance are re-
+viewed by Koch et al. (2009). In this paper we establish
+the characteristics of data from Keplercollected at 1-
+minute cadence. Asteroseismology of solar-type stars re-
+quiresdetectionofoscillationswithperiodsof3–10min-
+utes, which thus cannot be studied at the LC of nearly
+30 minutes. Early science returns and goals of astero-
+seismology are reviewed by Gilliland et al. (2010). For
+exoplanets, precise determination of the transit times of
+many successive events to search for possible variations
+in those times due to the gravitational influence of other
+planets (Holman & Murray 2005) benefits from SC.
+2.TARGET, DATA, AND CADENCE SPECIFICS
+TheKeplertargets in Q1 consisted of 156097 stars ob-
+served at LC. A mere 512 targets, or 0.3% of the total
+may be carried at the roughly one minute SC. A new SC
+target list may be used each month. Early in the mis-
+sion nearly all the SC targets have been assigned to the
+Kepler Asteroseismic Consortium (KASC); some 4000
+targets will be surveyed, each for one month from which
+the best set will be selected for extended observations.
+As planetary candidates are discovered many of these
+are switched to short cadence in order to provide precise
+transit timing variation (TTV) measures, and for accu-
+rate determinations of the host star mass and radius if
+gillil@stsci.edu
+1Space Telescope Science Institute, 3700 San Martin Drive,
+Baltimore, MD 21218
+2SETI Institute/NASA Ames Research Center, MS 244-30,
+Moffett Field, CA 94035
+3NASA Ames Research Center, MS 244-30, Moffett Field, CA
+94035
+4Orbital Sciences Corporation/NASA Ames Research Center,
+MS 244-30, Moffett Field, CA 94035stellaroscillationscanbedetected. Theasteroseismology
+program through KASC is guaranteed access to 140 SC
+target slots throughout the mission. At the start of Q2
+(19 June 2009) 25 SC targets are reserved for the Guest
+Observer (GO) program.
+There is also a restriction that the total number of
+pixels for SC not exceed 512 ×85 = 43,520. Very bright
+stars require more than 85 pixels each, thus requiring an
+SCtargetsetwith abalanceddistributionofmagnitudes.
+Data is time-stamped so that the mid-time of each ca-
+dence is known with an accuracy of ±0.050 seconds.
+Details of how the data acquisition process is managed
+on-board the spacecraft may be found in the “ Kepler
+Instrument Handbook” (van Cleve & Caldwell 2009), as
+wellas awealth ofgeneralinformation that is beyond the
+scope of a Letter. Caldwell et al. (2009) discuss charac-
+teristics of the instrument and early calibration.
+At the most basic level it is important to understand
+that the integrations underlying both SC and LC data
+are the same 6.02 s, followed by 0.52 s readout, be-
+fore successive integrations are summed into memory.
+There is no difference at which saturation occurs for
+the 58.84876 s cadence data, versus the 29.4244 min LC,
+since both are based on 6 s exposures. Within the 58.85
+s SC periods, 54.18 s is spent usefully collecting pho-
+tons while the remainder of the time is divided between
+9 readouts, which generates “smear” along the columns
+asKeplerhas no shutter. This is removed (Jenkins et al.
+2009a) as an early pipeline calibration step. In Q1 138
+out of 49,170 cadences ( <0.3%) were rejected, being as-
+sociated with periods of excess spacecraft jitter yielding
+a 99.7% duty cycle (!) for these 33.49 days, which span
+HJD (-2454900) = 64.00 to 97.49.
+LC data involve collection of collateral data contin-
+uously in a spatial sense along overscan (trailing black)
+columns, plus maskedand virtual (smear)rows, while for
+SC only those collateral pixels that project onto the SC
+aperture are retained. Likewise, for LC there are 44642 Gilliland, et al.
+pixels on each channel assigned to apparently blank re-
+gions of the field to provide real time monitoring of sky
+brightness (which changes more than expected), SC does
+not have dedicated sky pixels, therefore interpolation be-
+tween LC measures is used by the current pipeline pro-
+cessing. Sky was not expected to change on a time scale
+of minutes, and inclusion of direct background monitor-
+ing at SC would have increased telemetry needs for SC
+dramatically leading to an early decision to forgo this.
+3.ERROR BUDGET, AND REALIZED NOISE LEVELS
+In pre-launch simulations dozens of factors were con-
+sidered in combination to arrive at predictions of noise
+levels in the Keplerdata. In this Letterwe consider
+the minimum number of error budget terms that to-
+gether provide a good empirical representation of the
+realized noise level. With Kepler’s exquisite precision
+most stars are variable at least on time scales of days.
+Detrending of photometric time series against variations
+ofx,ypointing, thermal and focus changes is discussed
+by Jenkins et al. (2009b) for LC data. For SC applica-
+tions wherethe time scalesareusually shortcomparedto
+thermal and focus forcing functions this is less of an is-
+sue, but still one that will remain a major analysis factor
+going forward.
+The pipeline (Jenkins et al. 2009a) provides “raw”
+time series which are the simple sums (after cosmic ray
+and background removal) over the pixels chosen in an
+aperture to provide optimal signal-to-noise (S/N), the
+units of these are detected electrons (e-) per 58.8 s SC.
+To turn these time series into relative photometry with a
+mean of zero, and remove slow trends the following steps
+for each star have been taken: (1) A 5th order polyno-
+mial is fit to the full 33.5 days of data to remove drifts on
+timescales of a few days to a month resulting from minor
+pointing drift, focus changes etc. Future releases from
+the pipeline are expected to have more effective detrend-
+ing, perhaps obviating any need for this step. (2) The
+polynomial fit is subtracted and then this value divided
+by the zero point term of the polynomial yielding zero
+mean, empiricallydetrended data. (3) Arunning median
+filter 361 cadences wide (6 hours) is evaluated and point-
+by-point subtracted from the data. SC data processed
+before November 2009 had an underlying software error
+that resulted in blocks of 3,000 successive pixels having
+photometric values too high by ∼0.3 mmag at 12th mag-
+nitude (scaling inversely with brightness). These ran-
+domly positioned blocks averaged about one per star,
+are effectively suppressed in the current data by the me-
+dian filter, and the software error has now been fixed in
+the pipeline. (4) The §4.1 correction is applied. (5) Any
+individual points more than 5- σdeviant from the mean
+of zero have values and weights set to zero. Typically
+this final step clips values for <0.05% of the data points.
+Fig. 1 shows the result of forming power spectra for all
+512 SC targets up to 8 mHz with a spacing of 0.05 µHz
+that over-samples the frequency resolution by a factor of
+aboutseven,aftertransformingthistoamplitudewithan
+appropriate scale factor to have units of ppm. We then
+measure the noise level (standard deviation of the ampli-
+tude spectrum) over ν= 1 – 7.5 mHz. Many stars have
+strong variations below 1 mHz, relatively fewer show sig-
+nificant power at higher frequencies, although a healthy
+subset ofintrinsically variable stars show excess powerathighν. (10 stars show high noise due to poorly designed
+apertures early in the mission.) Evident in Fig. 1 is a
+lower envelope of data points representing the empirical
+limit to precision as a function of magnitude.
+Fig. 1.— The noise level for each of 512 stars in Q1 Short Ca-
+dence data expressed as the standard deviation in ppm over 1. 0 –
+7.5 mHz in an amplitude spectrum (square root of power). The
+curve passing through the lower envelope consists of simple Pois-
+son, readout noise plus sky background terms. The boxes near
+magnitudes of 9.3, 11.4 and 14.9 respectively are used to del ineate
+samples of quiet stars as discussed in the text and Fig. 2. A mi -
+nor spread in values is expected from differences in sensitiv ity and
+readout noise over the 84 amplifiers as well as errors in the Kepler
+Input Catalog magnitudes.
+The Fig. 1 lower envelope is fit “by-eye” with
+106(c+9.5×105(14.0/Kp)5)1/2/(cN1/2) wherec= 1.28
+×10(0.4(12.−Kp)+7)is the number of detected e- per ca-
+denceand N=49032(the numberofSCsusedinQ1). In
+this empirical formulation aimed at achieving a minimal
+termmodelforthenoise, cprovidesthePoissontermand
+is within the small channel-to-channel calibrated value
+for the count rate, Kpis theKeplermagnitude (close
+toR-band in center wavelength, but much broader, e.g.
+see Koch et al. 2009), and the (14 .0/Kp)5term captures
+to first order the number of pixels used in the aperture
+extraction as a function of magnitude. We have assumed
+8-pixel apertures at Kp= 14, increasing to 256 at Kp=
+7. This in turn implies a per pixel variance of 11,875 per
+cadence, or a readout plus Poisson statistics on the sky
+of 109 e- per pixel per underlying 6 second exposure (9
+per cadence). Since this is approximately the expected
+value (Caldwell et al. 2009) we conclude that this mini-
+mal model successfully represents the noise (as measured
+at high frequencies from amplitude spectra) in the data.
+The set of 115 quiet stars at Kp= 11.44 has a mean
+time series rmsof 255 ppm, and a mean amplitude noise
+level of 1.03 ppm. (The mean rmsvalue divided by N1/2
+is 1.15 ppm, which is larger than the quoted amplitude
+spectrum noise level, since only the rmsincludes contri-
+butions from below 1 mHz.)
+As a simple rule of thumb reaching a noise level
+of 1 ppm is required to make the most basic astero-KeplerShort Cadence Data Characteristics 3
+seismic measurement, that of determining the so-called
+large splitting ∆ ν0which in turn can be used to ac-
+curately constrain the mean stellar density (see, e.g.
+Gilliland et al. 2010) in solar-type stars where severalin-
+dividual modes may be expected to have amplitudes of
+a few ppm. We reach this level with one month of data
+atKp∼11.3. Coincidentally, Kp∼11.3 is also the
+rough dividing line below which stars saturate the detec-
+tor during the 6 s integrations. It had long been a goal
+withKeplerto achievephotometrynearthe Poissonlimit
+on strongly saturated stars as had been demonstrated
+for multiple HSTCCD-based instruments without this
+having been a design goal (see, e.g. Gilliland (2004) for
+ACS). Nearly Poisson limited photometry is maintained
+toKp= 7 (factor of 50 over-saturated), and beyond as
+long as the bleed does not extend into columns beyond
+the edge of the detector, nor blend with other bright
+stars. Photometry of highly saturated stars merely re-
+quires using an aperture that consistently captures the
+super-set of pixels that are bled into at any time over the
+time series. The Kepler short cadence data have demon-
+strated a large dynamic range of 10 magnitudes, and for
+special applications can probably be extended usefully by
+one magnitude in each direction beyond the 7 – 17 range.
+A major factor contributing to the stability of Kepler
+photometry is the operating environment of the Earth-
+trailing orbit. The 2-axis, point-to-point jitter in SC
+centroids after removing a 5th order polynomial in x
+andyseparately is 4.2 ×10−4pixels, or only 1.7 milli-
+arcseconds. This remarkable result holds despite the use
+of two variable guide stars (removed for Q2) known to
+have degraded guiding in Q1 (Jenkins et al. 2009b).
+4.KNOWN ANOMALIES IN CURRENT SC TIME SERIES
+A number of anomalies, any specific one of which is
+a surprise, appear in the early KeplerSC data. The
+fact that some such unanticipated anomalies have arisen
+should not of course come as a surprise. We discuss the
+two most significant examples of these.
+4.1.Power at 1/LC-period and Overtones
+Fig. 2 illustrates the most serious artifact appearing
+in power spectra of SC time series. A number of peaks
+corresponding to the fundamental and all harmonics of
+the inverse of the LC-period, 29.4244 minutes, contami-
+nate the spectra. Comparison of the amplitude of these
+modes shows that the effect is additive – as can be seen
+in Fig. 2 the relative amplitudes scale inversely with
+the stellar intensity. The causal factor may be varia-
+tion in the detector electronics that “rings” in response
+to changes associated with initiation of each LC. For a
+given star these sinusoids appear to be coherent over
+the full 33 days, but different phases apply to each star.
+These peaks also appear in power spectra of the overscan
+(black) pixels. Attempts to amelioratethis effect, e.g. by
+usinglocalskyvariationsfromtheperipheryofSCtarget
+apertures, have failed to improve the situation. Further
+studywilldetermineifchangestopipelinecalibrationcan
+suppress this contamination which is introduced on the
+spacecraft. The SC photometry folded on the LC period
+(not shown) has significant ringing that is low amplitude
+near the boundaries, and larger peaks near the center at
+a time scale of 3 – 4 short cadences thus explaining the
+larger strength of intermediate harmonics,Fig. 2.— Mean amplitude spectra over samples of quiet stars
+spanning more than a factor of 100 in brightness are shown. Th e
+1/LC-cadence artifacts at the fundamental of 0.566391 mHz a nd
+all harmonics are visible for the faint star set in the bottom panel.
+Even at 9th magnitude in the upper panel this artifact remain s a
+dominant spectral feature from the 7th and 8th harmonics.
+whether this robust empirical behavior is significant in
+understanding the origin remains under investigation.
+For now, at least, asteroseismic investigations will ei-
+ther need to attempt correctionsin the time orfrequency
+domainsonastar-by-starbasis, orataminimumflagfre-
+quencies at multiples of 0.566391 mHz as suspect. Since
+the fraction of “phase space” contaminated by these
+peaks is roughly the number of harmonics (14) times
+the frequency resolution (1/33-days), divided by 8 mHz,
+which equals 0.06%, the ultimate impact of these spuri-
+ous peaks should be minimal.
+With the exception of this artifact the Keplerspectra
+are remarkably clean and free of problematic sidelobes.
+The low frequency turn up in power for 9th magnitude
+stars below ∼2 mHz in Fig. 2 likely follows from stellar
+oscillationsand granulationnoisewhich areeasilyseen in
+many individual amplitude spectra for bright stars. The
+peak at∼0.1 mHz likely corresponds to an artifact near
+3.2 hours discussed in Jenkins et al. (2009b); the wavy
+structure at low frequency in the middle panel is likely
+due to harmonics of this.
+4.2.Additive Corrections in the Time Domain
+A somewhat more problematic artifact is illustrated in
+Fig. 3, but unlike that from the previous section, this
+can likely be addressed with pipeline modifications.
+The primary offsets visible in Fig. 3 are vaguely
+transit-like, and clearly scale inversely with the intrin-
+sic stellar intensity. These appear at the times of the
+two major brightening events seen in Q1 LC sky back-
+ground (see §3 of Jenkins et al. 2009a for a discussion
+of these enigmatic and unexpected events) for which im-
+perfect background correction is applied in the pipeline
+for these SC data. The correction illustrated adopts the
+mean from 115 stars near Kp= 11.44, and for points
+deviating by more than 8 ×10−5from the mean of zero
+subtractsascaledversionofthis. Whilenotaperfectres-
+olution, the primaryfeatureslargelydisappear, aswell as
+several positive deviations appearing in the time series.
+5.SCIENCE APPLICATIONS OF SC TIME SERIES4 Gilliland, et al.
+Fig. 3.— The upper three panels show means, after sigma clip-
+ping, over the same sets of stars shown in Fig. 2; this time how ever
+for 5 days of the direct time series. Events in the pipeline pr ovided
+time series at HJD = 84.7 and 88.6 scale inversely with stella r
+brightness. The lower two panels show corrected mean time se ries
+for the 9th and 15th magnitude bins as discussed in the text.
+5.1.Detailed Study of Transits
+The previously known exoplanet hosts, TrES-2
+(O’Donovan et al.2006), HAT-P-7(P´ al2008), andHAT-
+P-11 (Bakos et al. 2009) have all been observed at SC
+from the start. TrES-2b has the narrowest transits of
+these and is used to illustrate differences between LC
+and SC data in Fig. 4.
+Fig. 4.— The upper panel shows the 33.5 days of Q1 LC data
+folded on the ∼2.47 day period of TrES-2 with a Keplermagnitude
+of 11.34. The lower panel shows folded data at SC for the same
+interval. The curve passing through the points in the upper p anel
+was based on fitting a transit light curve model integrated over the
+LC of 29.4 minutes, while for comparison the narrower curve ( at
+top) shows the fit to SC data. The horizontal bar in the upper
+panel shows the length of an LC interval in phase for TrES-2.
+LC observations effectively integrate over 29.4 minute
+intervals, which arenotshortcomparedto featuresin the
+TrES-2b transit, and therefore exhibit easily discerneddifferences with respect to SC data which much more
+finely sample the light curve. The two fits shown in Fig.
+4 use the Mandel & Agol (2002) transit model with iden-
+tical values assumed for stellar mass, radius and limb-
+darkening. Parameters solved for are: ( Rp/R∗)2, the
+orbital period and phase, a zero point outside of tran-
+sit, and a transit width related to the impact parameter.
+The SC model fit ( rmsresiduals are 237 ppm) evaluates
+the functional representation at each central time, while
+the LC model fit ( rmsof 66 ppm) is integrated over the
+LCintervals(usinganumericalsumcorrespondingtothe
+30 contributing SC intervals). Although the blurring in-
+troduced by the long LC intervals is very noticeable for
+TrES-2, the formal parameters returned from a simple
+non-linear, least-squares fit are nearly identical in the
+two cases. The solutions for planetary radius differ by
+only 0.012% for the LC and SC solutions, while the in-
+ferred orbital inclinations differ by only 0.001 degrees.
+Since systematic uncertainties on the stellar radius will
+likely remain >>0.01%, even if asteroseismology solu-
+tions for the mean stellar density are available, LC data
+seem to equally well support determination of the basic
+parameters associated with transit modeling.
+SC data are expected to provide a significant relative
+advantage for transit timings where LC data fail to re-
+solve the critical ingress and egress periods. Quantifi-
+cation of this awaits applications to longer time series
+generally needed for TTV analyses.
+5.2.Asteroseismology
+The case for short cadence data for asteroseismic ap-
+plications is clear. Indeed, the value of slightly less than
+1-minute, rather than, say 2-minutes for SC data which
+would have been more convenient in terms of on-board
+data storage and telemetry bandwidth was selected to
+support robust asteroseismology of solar-like stars some
+of which will have modes with periods as short as 2
+minutes. The Nyquist frequency for LC sampling of
+283.2µHz permits asteroseismology on relatively large
+stars, e.g. the R/R⊙∼6 star KOI-145 analyzed by
+Gilliland et al. (2010) has frequencies of maximum mode
+power (which scales with the surface acoustic cutoff fre-
+quency∝g/T1/2
+eff– Brown et al. 1991) at 143 µHz and
+can be studied as effectively at LC as SC. Thus near so-
+lar temperatures, stars with radii down to R/R⊙about
+4 can be studied well at LC, for stars with radii below
+this SC data are needed.
+6.DISCUSSION AND SUMMARY
+We have shown that the 58.8 s short cadence data
+reachesunparalleled noise levels near fundamental limits
+imposed by Poisson statistics on source and background
+plus readout noise. An annoying artifact imposes fre-
+quencies at harmonics of the LC period in power spec-
+tra, but these are restricted to a few known frequencies
+and will not generally degrade science applications. SC
+data provide excellent returns in both transit and aster-
+oseismic analyses. The SC data acquisition option is a
+limited resourcethat will be in greatdemand throughout
+theKeplermission.
+Funding for this Discovery Mission is provided by
+NASA’s Science Mission Directorate. We gratefully ac-KeplerShort Cadence Data Characteristics 5
+knowledge the many individuals through whose dedica-
+tion and excellence the remarkable capabilities of Keplerhave been made possible.
+Facilities: The Kepler Mission.
+REFERENCES
+Bakos, G. ´A., et al. 2009, ApJ, submitted, arXiv:0901.0282
+Borucki, W.J., et al. 2009, Science, submitted
+Brown, T.M., Gilliland, R.L., Noyes, R.W., & Ramsey, L.W.
+1991, ApJ, 368, 599
+Caldwell, D., et al. 2009, ApJ, submitted
+Gilliland, R.L. 2004, INS ACS 2004-01, (STScI)
+Gilliland, R.L., et al. 2010, PASP, in press.
+Holman, M.J., & Murray, N.M. 2005, Science, 307, 1288
+Jenkins, J.M., et al. 2009a, ApJ, submitted.Jenkins, J.M., et al. 2009b, ApJ, submitted.
+Koch, D., et al. 2009, ApJ, submitted.
+Mandel, K., & Agol, E. 2002, ApJ, 580, L171
+O’Donovan, F.T., et al. 2006, ApJ, 651, L61
+P´ al, A., et al. 2008, ApJ, 680, 1450
+van Cleve. J., & Caldwell, D. 2009, KeplerInstrument Handbook
+(KSCI-19033), Version 1 (NASA-Ames)
\ No newline at end of file
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+arXiv:1001.0143v1  [cs.FL]  31 Dec 2009ROMANIAN JOURNAL OF INFORMATION
+SCIENCE AND TECHNOLOGY
+Volume12, Number 2, 2009, 265279
+Undecidability Results for Finite
+Interactive Systems
+Alexandru Sofroniaaand Alexandru Popaband Gheorghe Stefanescuc,1
+aFaculty of Mathematics and Computer Science, University of Bucha rest
+E-mail:alexandrusofronia@yahoo.com
+bDepartment of Computer Science, University of Bristol
+E-mail:popa@cs.bris.ac.uk
+cDepartment of Computer Science, University of Illinois at Urbana-C hampaign
+E-mail:ghstef@yahoo.com
+Abstract. A new approach to the design of massively parallel and intera ctive
+programming languages has been recently proposed using rv- systems (interac-
+tive systems with registers and voices) and Agapia programm ing. In this paper
+we present a few theoretical results on FISs (finite interact ive systems), the un-
+derlying mechanism used for specifying control and interac tion in these systems.
+First, we give a proof for the undecidability of the emptines s problem for FISs,
+by reduction to the Post Correspondence Problem. Next, we us e the construc-
+tion in this proof to get other undecidability results, e.g. , for the accessibility of
+a transition in a FIS, or for the finiteness of the language rec ognized by a FIS.
+Finally, we present a simple proof of the equivalence betwee n FISs and tile sys-
+tems, making explicit that they precisely capture recogniz able two-dimensional
+languages.
+1. Introduction
+A new approach to the design of massively parallel and interactive pr ogramming
+languages has been recently proposed. The approach focuses on Agapia [2, 13], a
+1On leave from the Faculty of Mathematics and Computer Scienc e, University of Bucharest.2 Sofronia, Popa, Stefanescu
+programming language paradigm based on classical register machine s and space-time
+duality [16, 18]. Agapia extends usual programming languages with co ordination fea-
+tures, being a step forward to the integration of coordination fea tures (as in Klaim
+[4], Reo [1], Orc [12], etc.) into practical programming languages. A few distinc-
+tive features of Agapia are: high-level modularity, including a struc tured approach
+to interaction based on name-free processes; simple operational and relational seman-
+tics based on grids and scenarios (enriched two-dimensional words ); invariance with
+respect to space-time duality.
+Agapia language uses (i) complex spatial and temporal data for iner faces, (ii)
+modules over usual programming languages, and (iii) temporal, spat ial, or spatio-
+temporal while-programsfor coordination. In Agapia v0.1 one can w rite programsfor
+open processeslocatedatvarioussitesand havingtheir temporal windowsofadequate
+reaction to the environment. It naturally supports process migra tion, structured
+interaction, and deployment of modules on heterogeneous machine s. Agapia can be
+seen as an extension of usual procedural or functional program ming languages (used
+in the basic modules), for instance may be developed on top of langua ges as C, Java,
+Scheme, etc.
+The theoretical foundation of Agapia is strongly related to the the ory of two-
+dimensionallanguages[6,7,10,11]. ItisbasedonFISs(finite intera ctivesystems)[17,
+18], abstract mechanisms for specifying control and interaction in interactive systems.
+FISs can be used to recognize two-dimensional languages, and, in t his respect, FISs
+are equivalent to tile systems [6], existential monadic second order lo gic [7], or other
+equivalentpresentationsofregular(orrecognizable)two-dimens ionallanguages[6,10].
+However, they come equipped with abstract “states”and “intera ction classes,” which,
+by instantiation, may be used to design interactive programs.
+The present paper contains a few results on FISs. First, it presen ts a proof for the
+undecidability of the emptiness problem for FISs by reduction to the Post Correspon-
+dence Problem. Next, the construction in this proof is used to show the accessibility
+problem for FISs (i.e., whether, for a specific transition, there is an accepting scenario
+for a two-dimensional word using that transition) is undecidable. Fin ally, the paper
+includes a simple and direct proof of the equivalence between FISs an d tile systems,
+emphasizing the conceptual differences between these two equiva lent presentations of
+recognizable two-dimensional language; as a byproduct, this gives another (this time,
+indirect) proof of the undecidability of the emptiness problem for FI Ss via the unde-
+cidability of a similar problem for tile systems. A conference version of the paper has
+appeared in [14].
+2. Preliminaries
+Grids and scenarios Agrid(also called a two-dimensional word ) is arectangular
+two-dimensional area filled in with letters of a given alphabet. The colu mns in a
+grid correspond to processes, the top-to-bottom order descr ibing their progress in
+time. The left-to-right order corresponds to process interactio n in anonblockingUndecidability Results for Finite Interactive Systems 3
+message passing discipline : a process sends a message to the right, then it resumes
+its execution.
+Ascenario is a grid enriched with data around each letter. The data may have
+various interpretations: they either represent control/interac tion information, or cur-
+rent values of the variables, or both. In this paper, we only use abs tract scenarios
+of the first type resulting from accepting runs in finite interactive s ystems. A grid is
+presented in Fig. 1(a) and an abstract scenario in Fig. 1(b).
+aabbabb
+abbcdbb
+bbabbca
+ccccaaa1 1 1AaBbBbB2 1 1
+AcAaBbB2 2 1AcAcAaB2 2 2F1=A B1
+ab
+c 2
+(a) (b) (c)
+Figure 1: A grid (a), an abstract scenario (b), and a FIS (c).
+We use the following notation for operations on grids: ·and⋆denote vertical com-
+position and iteration, while ⊲and†denote the horizontal composition and iteration.
+Finite interactive systems Afinite interactive system (shortly FIS) is a finite
+hyper-graph with two types of vertices and one type of (hyper) e dges: the first type
+of vertices is for states(labeled by numbers), the second is for classes(labeled by
+capital letters) and the edges/transitions are labeled by letters d enoting the atoms of
+the grids; each transition has two incoming arrows (one from a class and the other
+from a state), and two outgoing arrows(one to a class and the oth er to a state). Some
+classes/states may be initial(indicated by small incoming arrows) or final(indicated
+by double circles); see, e.g., [17, 18].
+For theparsing procedure , given a FIS Fand a gridw, insert initial states/classes
+at the north/west border of wand parse the grid completing the scenario according
+to the FIS transitions; if the grid is fully parsed and the south/east border contains
+final states/classes only, then the grid wis recognized by F. Thelanguage ofFis the
+set of its recognized grids.
+LetF1be the FIS graphically represented as in Fig. 1(c). It is equivalently r epre-
+sented specifying its transitions1
+AaB
+2,1
+BbB
+1,2
+AcA
+2and pointing out that
+A,1 are initial and B,2 final. A parsing forabbcabccais
+1 1 1Aa b b
+Ac a b
+Ac c a1 1 1AaBb b2Ac a b
+Ac c a...1 1 1AaBbBbB2 1 1AcAaBbB2 2 1AcAcAaB2 2 24 Sofronia, Popa, Stefanescu
+Post Correspondence Problem Letx= (x1,...,x n) andy= (y1,...,y n) be
+two lists of nonempty words from an alphabet Σ, with at least two lett ers. The
+Post Correspondence Problem (PCP) is to decide whether or not there exist i1,...,ik
+wherek≥1 and21≤ij≤n,∀j∈1,ksuch thatxi1...xik=yi1...yik.It is known
+that PCP is undecidable [15] (if |Σ| ≥2).
+We use this result to prove the emptiness problem for FISs is undecid abile.
+3. The emptiness problem and the finiteness problem for FISs
+Letx= (x1,...,x n) andy= (y1,...,y n) be an instance of the PCP, labeled
+PCP(x,y). We construct a finite interactive system Swhich accepts a language
+L=L(S) such that: Lis finite iff it is empty iff PCP(x,y) has no solution. Let |w|
+denote the length of the string w∈Σ.
+The idea of the construction is as follows. The FIS associated to the PCP instance
+allowsto parse gridswhere the first tworowscontain a proposed so lution for the PCP.
+More precisely, it contains sequences of xi’s andyj’s (with a single letter in a cell)
+such that their product is equal. The next rows check if the chosen indices are equal,
+hence whether or not one gets a solution for the PCP.
+The states and the classes of S are:
+States
+•s
+•a(i,j)∀i∈1,n∀j∈1,|xi|
+•c(i,j)∀i∈0,n∀j∈0,n
+Classes
+•A
+•B(i,j)∀i∈1,n∀j∈1,|xi|−1
+•C(i,k)∀i∈1,n∀k∈1,|yi|−1
+•M(i,j,k)∀i∈1,n∀j∈0,|xi| ∀k∈0,|yi|
+Shas a unique initial state - s, a unique initial class - A, a unique final state -
+c(0,0) andn+ 1 final classes - AandM(i,0,0) for each i∈1,n. To simplify the
+definition of the transitions we use an extended notation A=B(i,0) =B(i,|xi|) =
+C(i,0) =C(i,|yi|)∀i∈1,n.
+Letxi=x1
+i·x2
+i···x|xi|
+i, wherexk
+i’s are the letters of the word xi, 1≤i≤n
+and use a similar notation for yi’s. The alphabet of the FIS Salso contains a special
+symbol $ such that $ /ne}ationslash∈Σ.
+We define the transitions ofSas follows:
+2p,qdenotes the set {p,p+1,...,q}Undecidability Results for Finite Interactive Systems 5
+(I)s
+B(i,j−1) xj
+i B(i,j)
+a(i,j)∀i∈1,n∀j∈1,|xi|
+(II)a(j,g)
+C(i,k−1) yk
+i C(i,k)
+c(j,i)∀i∈1,n∀k∈1,|yi|iff
+xg
+j=yk
+i, wherej∈1,nandg∈1,|xj|
+(III)c(0,0)
+A $ A
+c(0,0)
+(IV)c(i,i)
+A $M(i,|xi|−1,|yi|−1)
+c(0,0)∀i∈1,n
+(V)c(i,0)
+A $M(i,|xi|−1,|yi|)
+c(0,0)∀i∈1,n
+(VI)c(0,i)
+A $M(i,|xi|,|yi|−1)
+c(0,0)∀i∈1,n
+(VII)c(j1,j2)
+M(i,k1,k2)$ M(i,r1,r2)
+c(m1,m2)
+wherei∈1,n,j1,j2,m1,m2∈0,n,k1,r1∈0,|xi|,k2,r2∈0,|yi|andk1satisfies
+exactly one of the following conditions
+•k1= 0, m1=j1, r1=k1.
+•k1>0, j1=i, m1= 0, r1=k1−1.
+•k1>0, j1= 0, m1= 0, r1=k1.
+andk2satisfies exactly one of the following conditions
+•k2= 0, m2=j2, r2=k2.
+•k2>0, j2=i, m2= 0, r2=k2−1.
+•k2>0, j2= 0, m2= 0, r2=k2.
+ThefollowinglemmasrevealthebehaviorofthisFISandtheroleofth etransitions.
+Recall that the grids below are parsed from left to right and from to p to bottom.
+Note that equalities of indexes in the following lemmas must be interpre ted as
+equalities of lists over {1,...,n}rather than weaker string equalities. We use a
+simplified notation ininstead of ( i,...,i), whereiappearsntimes in a row.6 Sofronia, Popa, Stefanescu
+Lemma 1 Letwbe a grid with mlines andqcolumns. If there exists a successful
+running for wwith respect to the border conditions3bn,bw,bs,be, then:
+(i)m≥3,q≥1,bn∈ {s}⋆,bw∈ {A}⋆,bs∈ {c(0,0)}⋆.
+(ii) There exist k≥1,r≥1,i1,...,ikandj1,...,j rwhere 1≤il≤n∀l∈1,kand
+1≤jl≤n∀l∈1,rsuch thatxi1···xikandyj1···yjrare the first two lines of
+w(as strings) and xi1···xik=yj1···yjr.
+(iii) The southern border of the second line of wisc(a1,b1),...,c(aq,bq) where
+(a1,...,a q) = (i|xi1|
+1,...,i|xik|
+k) and (b1,...,b q) = (j|yj1|
+1,...j|yjr|
+r) (as lists over
+{1,...,n})
+(iv) The first two lines of ware parsed using type ( I) and type ( II) transitions,
+and no other lines can be parsed using type ( I) or type (II) transitions. In
+particular, all the lines of w, except the first two, are composed of $.
+Proof:
+(i)Sincesis not a final state, it follows that q≥1 andm≥1. The conditions for
+the border are trivial from uniqueness of initial and final states an d initial classes.
+Recall that the only final state of Sisc(0,0). A successful run of whas at least
+three transitions from a northern border sto a southern border c(0,0): a type ( I)
+transition, followed by a type ( II) transition and, as i,j>0 in type (II) transitions,
+a type (IV) or (VII) transition. Therefore, in order to be accepted, whas to have
+at least three lines.
+(ii)First, since Ais the only initial class and sthe only initial state, only type
+(I) transitions can be used to parse the first letter of w. Therefore this letter must
+be the first letter of some word xi. Note that this argument can be applied whenever
+we haveAon the western border and son the northern border.
+All the transitionson the firstline of whavesasthe northernborderand therefore
+aretype(I)transitions. Suchtransitionscontainonlylettersfromthewords inthe list
+x. Note that these transitions can be connected horizontally only if t he corresponding
+letters are adjacent in some xi(this is the role played by the B(i,j) classes) or are
+the final letter of some xiand the first letter of some xk(in which case, the class in
+the middle is A=B(i,|xi|) =B(k,0)).
+It follows that the words from the list xthat appear in the first line of wcannot be
+truncated (except for possibly the rightmost one). Since type ( I) transitions cannot
+have anyM(i,0,0) as a eastern border, the eastern border of the first line of wmust
+beA, the only final class remaining. Therefore, the first line of wis somexi1···xik
+wherek≥1 (asq≥1).
+The northern border of the second line of wis made ofa(i,j) states, therefore only
+type (II) transitions can be used for parsing. Since C(i,k) classes prevent truncation,
+3I.e., using on the borders the specified sequences bn,bw,bs,beof initial/final states/classesUndecidability Results for Finite Interactive Systems 7
+similar to the B(i,j) classes, a similar argument shows that the second line of wis
+yj1···yjrfor somer≥1, and it is parsed using only type ( II) transitions.
+A type (I) transition accepting a letter xk
+ihas on the southern border the state
+a(i,k). The first index encodes the position of the word xiin the listxand the second
+index encodes the position kin which the letter xk
+iappears in xi. The condition
+xg
+j=yk
+iin the definition of type ( II) transitions forces the letters in the first and
+second line to be identical. Therefore, xi1···xik=yj1···yjr.
+(iii)From(ii)it follows that the southern border of the first line has the following
+formata(i1,1)a(i1,2)...a(i1,|xi1|)...a(ik,1)...a(ik,|xik|). As type ( II) transitions
+copy the first index of the northern border a(j,g) to the first index of the southern
+borderc(j,i), the southern border of line 2 (or northern border of line 3) can b e
+writtenc(a1,b1),...,c(aq,bq), and, by extracting the first indexes, ( a1,...,a q) =
+(i|xi1|
+1,...,i|xik|
+k). The second index of the state c(j,i) in a type ( II) transition is
+equal to the position of the word yiin the listy, so from (ii)we get (b1,...,b q) =
+(j|yj1|
+1,...,j|yjr|
+r).
+(iv)Recall that the first line of wis parsed using only type ( I) transitions and
+the second line using only type ( II) transitions. To conclude the proof, note that
+the northern border of line 3 is made of c(∗,∗) states, type ( III)−(VII) transitions
+also produce c(∗,∗) states on the southern border and type ( I) and (II) transitions
+cannotbe used for any c(∗,∗) on the northern border.
+In particular, since only type ( III)−(VII) transitions can be used, all the lines
+ofwexcept the first two, are composed of $. ✷
+In order to obtain a successful “encoding” in wof a solution for PCP(x,y) we
+must prove that k=rand indexes il=jl∀l∈1,k.
+Using type ( III)−(VII) transitions, a p+ 2 line marks the letters of a corre-
+sponding pair of tiles ( xip,yip) in the PCP solution. Final states and a final class are
+obtained only after successful reduction of these tiles. Additiona l $ lines parsed with
+type (III) transitions can be added to such final positions. This behavior is ca ptured
+by the following lemma.
+Lemma 2 Letwbe a grid with mlines andqcolumns parsed by Sand let us use
+the notation xi1,...xik,yj1,...yjras in Lemma 1. Then:
+(i) For all pwith 0≤p≤k, the southern border of the p+ 2 line of wcan be
+writtenc(ap,1,bp,1),...,c(ap,q,bp,q) and satisfies the following equalities (as lists
+over{0,1,...,n}):
+(ap,1,...,a p,q) = (0α,i|xip+1|
+p+1,...,i|xik|
+k)
+(bp,1,...,b p,q) = (0β,j|yjp+1|
+p+1,...,j|yjr|
+r)
+whereα=/summationtextp
+z=1|xiz|andβ=/summationtextp
+z=1|yjz|.
+Furthermore, k=randip=jp∀p∈1,k.8 Sofronia, Popa, Stefanescu
+(ii) Any line of wafter thek+2 line, if any, is parsed using type ( III) transitions
+only.
+(iii) Theeasternborder beisofthefollowingtype be∈A·A·M(i1,0,0)···M(ik,0,0)·
+{A}⋆
+Proof:
+(i)Recall from the previous lemma that the two equalities of lists hold for p= 0.
+We will prove the equalities by induction over p.
+We refer to the index sequences ( ap,1,...,a p,q) and (bp,1,...,b p,q), as the first
+stream and the second stream of specifying the southern border of thep+2 line ofw
+(or the northern border of the p+3 line, if such a line exists).
+First, some remarks on type ( VII) transitions. These transitions can be com-
+posed horizontally only with transitions of the same type. These tra nsitions are only
+defined if the first index of the classes Mon the western and eastern border are equal.
+Therefore, whenever a class M(i,k1,k2) appears on a line, only M(i,,) classes can
+appear when parsing the rest of the line. Furthermore, whenever k1= 0, the second
+index inM(i,,) classes remains 0 until the end of the line and the rest of the first
+northern stream is copied to the first southern stream. Similarly, w heneverk2= 0
+the third index in M(i,,) classes remains 0 until the end of the line and the rest of
+the second northern stream is copied to the second southern str eam.
+Letp= 1. From the previous lemma, the northern border of the first lett er in
+the third line is c(i1,j1) withi1,j1≥1 and the western border is A. Theni1=j1
+since only a type ( IV) transition can be used to parse this letter and the transitions
+used to parse the rest of line 3 are only type ( VII) transitions, with M(i1,,) on the
+western and eastern borders. The processing of the following lett ers on the third line
+is deterministic, since depending on the northern and western bord er, at most one
+type (VII) transition can be chosen at each step.
+Letα=|xi1| ≥1 andβ=|yi1| ≥1 (recall that both lists in PCP have nonempty
+words). The first letter of this line is processed using the following ty pe (IV) transi-
+tion:
+c(i1,i1)
+A $M(i1,α−1,β−1)
+c(0,0)
+For the sake of simplicity, assume α<β. Then the following α−1 letters in this
+line are parsed using type ( VII) transitions, each creating a southern border of c(0,0)
+and decreasing the second and third indexes in M(i,,) by 1 until k1reaches 0 using
+c(i1,i1)
+M(i1,k1,k2)$M(i1,k1−1,k2−1)
+c(0,0).
+The first and second southern streams have αleading zeros and k1= 0 is carried over
+when parsing the rest of the line. The first equality is thus proven sin ce the rest of
+the first northern stream is copied to the first southern stream.Undecidability Results for Finite Interactive Systems 9
+After parsing αi1’s from the first northern stream there are still at least β−α=
+|yi1|−|xi1|i1’s on the second northern stream. For the next β−αletters only type
+(VII) transitions can be used:
+c(j,i1)
+M(i1,0,k2)$M(i1,0,k2−1)
+c(j,0)
+therefore the second southern stream has a total of βleading zeros. After that,
+k2reaches zero and the rest of the second northern stream is copie d to the second
+southern stream. Thus we have obtained the second required equ ality andM(i1,0,0)
+on the easternmost border of line 3. A symmetrical argument holds whenα≥β.
+Note that if k > p≥1 orr > p≥1, then the southern border of this line still
+contains non-final states - c(ik,jr) sowhas at least another line, which motivates the
+use of induction.
+Letp∈2,k. Let
+α=p−1/summationdisplay
+z=1|xiz|andβ=p−1/summationdisplay
+z=1|yjz|.
+We distinguish only two non-similar cases: α=βorα<β, due to the symmetry in
+type (VII) transitions.
+Ifα=β, then by the induction hypothesis, both northern streams of line p+2
+have exactly αleading zeros followed by q−αnonzero numbers. This allows us to
+useαtype (III) transitions to parse the first αletters on this line. The next letter
+has the northern border c(ip,jp) which forces a type ( IV) transition and therefore
+ip=jp. The two equalities are proved similarly with the case p= 1 above. Note that
+this case can be avoided entirely if we consider only atom solutions of PCP(x,y) (i.e.
+no non-trivial prefix of the solution forms a valid solution).
+Ifα < β, then after the first αleading zeros, the first northern stream contains
+only nonzero numbers, while the second one has another β−αleading zeros followed
+byq−βnonzero numbers. Then after αtype (III) transitions, the α+1 atom has
+c(ip,0) on the northern border and Aon the western border. The type ( V) transition
+used to parse the first letter is:
+c(ip,0)
+A $M(ip,ϕ−1,ψ)
+c(0,0).
+whereϕ=|xip|andψ=|yip|.
+The restofthe firstnorthernstreamcontainsatleast ϕ−1ip’sthat requireparsing
+by type (VII) transitions with k1>0. Thus, the corresponding first southern stream
+will contain an equal number of zeros in that part of the stream. Af ter that, only type
+(VII) transitions with k1= 0 can be used which copy the rest of the first northern
+stream to the first southern stream.
+The rest of the second northern stream has another β−α−1 leading zeros which
+force type ( VII) transitions with k2>0, j2= 0, that copy the zeros to the second10 Sofronia, Popa, Stefanescu
+southern stream. Afterwards, the second northern stream re aches theψ jp’s created
+in line 2 by the word yjpand carried over to this line.
+Since the case k2>0, jp>0jp/ne}ationslash=ipis undefined for type ( VII) transitions, the
+only remaining possibility is that jp=ip. Thus the next ψnumbers from the second
+northern stream are replaced by zeros in the second southern st ream, decreasing k2
+until it becomes 0. Afterwards the rest of the second stream is co pied from north to
+south, yielding the eastern border M(ip,0,0).
+This proves the required equalities. Moreover, wmust have an additional line and
+the process continues if p<min(k,r) becausek>porr>pimply that the southern
+border of the current line contains the non-final state c(ik,jr).
+Assumek<r, then the southern border of line k+2 has a first stream containing
+only zeros, and a second one with βleading zeros followed by 0 <q−βnonzero num-
+bers. In this case, the southern border contains the non-final s tatec(0,jr), therefore
+wmust have at least another line. However, in this case, it is impossible t o obtain a
+final class on the additional line. Indeed, the first βpairsof zerosaredeterministically
+parsed by type ( III) transitions. The following β+ 1 letter, with c(0,jk+1) on the
+northern border and Aon the eastern border, can only be parsed by a type ( VI)
+transition with M(jk+1,|xjk+1|,|yjk+1|−1) on the eastern border.
+However, since the rest of the first northern border contains on ly zeros, the rest
+of the line can be parsed only using type ( VII) transitions with k1=|xjk+1|>0 and
+j1= 0. Therefore r1=k1>0 and the eastern border of this line is M(jk+1,|xjk+1|,0)
+which is not a final class! A similar argument for k>rproves that k=r.
+(ii & iii) Ifm≥k+ 3, the northern border of the k+3 - th line of whas both
+streams full of 0’s. Therefore, only type ( III) transitions are used to parse this line,
+copying the northern border to the next line and resulting in an east ern border of A.
+Similarly, any line of wafter thek+2 line has only type ( III) transitions yielding
+Aon the eastern border. From Lemma 1 the first two lines have Aon the eastern
+border. From (i)the eastern border of the next klines isM(i1,0,0)...M(ik,0,0)
+which concludes the proof. ✷
+Using these two lemmas, one can prove the following result:
+Theorem 3 The finiteness and the emptiness problems for finite interactive sys tems
+are undecidable.
+Proof: Assume that the emptiness problem is decidable. Then for each PCP in -
+stance,PCP(x,y), we can use the above construction in order to obtain the FIS
+S=S(x,y). For this FIS, we can decide whether or not L(S) is empty. The previous
+lemmas show that for every w∈L(S), the first two lines of wdetermine a solution for
+PCP(x,y) and that from any solution of PCP(x,y), a gridwconstructed from the
+strings of the solution and composed vertically with enough $’s ( klines where kis the
+length of the solution), can be parsed according to the lemmas (the reforew∈L(S)).
+Therefore the assumption contradicts the fact that PCP is undec idable.
+For the undecidability of the finiteness problem, it suffices to observ e that ifL(S)Undecidability Results for Finite Interactive Systems 11
+is not empty, then it is infinite. That follows from the fact that if w∈L(S) then
+w·$†∈L(S) (parsing the additional lines with type ( III) transitions). Also, the
+iteration (any number of times) of a solution for PCP(x,y) gives another PCP(x,y)
+solution forwhich a w′∈L(S) can be constructedby horizontaliterationof w. There-
+foreL(S) is empty iff it is finite which concludes the proof. ✷
+4. The accessibility of a given FIS transition
+For any instance of PCP, we construct a FIS S1 which has all the states, classes,
+and transitions of the FIS Sin the previous section, an additional state q, and two
+additional classes Q,T. The initial states and classes of S1 are those of S, the only
+final state is qand the only final class is T. In addition to S,S1 has the following
+new transitions:
+(i)s
+A $T
+s
+(ii)s
+M(i,0,0)$T
+s∀i∈1,n
+(iii)c(0,0)
+A $Q
+q
+(iv)c(0,0)
+Q $Q
+q
+(v)s
+Q $T
+q
+This construction may be used to get the following result:
+Theorem 4 The accessibility of a transition in a FIS is undecidable.
+Proof: It suffices to prove that transition of type ( v) is accessible iff PCP(x,y) has
+a solution.
+IfPCP(x,y) has a solution then there is a wwithmlines andqcolumns accepted
+by the FIS in the previous section. Because of the border condition s established by
+Lemma 1 and 2, we can add one column to the right of wand an additional line,
+obtainingu∈L(S1). Indeed if we take a successful running of win the previous FIS,
+and parsethe last line of uusingqtype (iii) and (iv) transitions, the last column with12 Sofronia, Popa, Stefanescu
+type (i) and (ii) transitions, and finally a type ( v) transition for the southeasternmost
+letter ofu, we obtain a successful running of uonS1, which uses the transition ( v).
+If transition( v) is accessible, let ube agridthat canbe parsedsuchthat transition
+(v) can be applied to the southeasternmost letter. We shall prove th at there exists w
+such thatu= (w⊲$⋆)·$†andwis recognized by the FIS S in the previous section.
+Firstly, the letter parsedwith this type ( v) transitioncannotbe on the firstcolumn
+sinceQis not aninitial class. The onlywayofobtaining class Qon the westernborder
+on this line is by using a type ( iii) transition and some (if any) type ( iv) transitions.
+This implies that c(0,0) is on the northern border of all the atoms in the last line of
+uand sincec(0,0) is not an initial state, uhas at least 2 lines. Similarly, the state s
+cannot appear on the northern border of the last atom, unless on ly type (i) and some
+(if any) type ( ii) transitions were used for processing the atoms on the last column
+ofu. Then there exists wsuch thatu= (w⊲$⋆)·$†. The southern border of w
+is composed only of c(0,0)’s and the eastern border of A’s andM(i,0,0)’s for some
+i∈1,n.
+Since type ( i)-(v) transitions cannot lead to A’s orM(i,0,0)’s on the eastern bor-
+der, it follows that wwas parsed using transitions of the FIS in the previous section.
+Furthermore, due to the southern and eastern border condition s forw, this parsing is
+a successful running of win the FIS in the previous section and therefore PCP(x,y)
+has a solution. ✷
+5. The equivalence of FISs and tile systems
+In this section we present a simple, direct proof of the equivalence b etween FISs
+and tile systems. Tile systems [6, 10] are but one of many equivalent m echanisms
+specifying the class of recognizable two-dimensional languages.
+Tile systems Foragridwofsizem×n, let/hatwidewdenotethe gridofsize( m+2)×(n+2)
+obtained bordering wwith a special symbol ♯. A tile system is defined as follows:
+•for a bordered grid /hatwidew, letBr,s(/hatwidew) be the set of its sub-grids of size r×s;
+•a two-dimensional language LoverVislocalif there is a set ∆ of 2 ×2 grids
+overV∪{♯}such that
+L={w|wgrid overV∧B2,2(/hatwidew)⊆∆}
+•a two-dimensional language Lover an alphabet Vis recognized by a tile system
+if there is an alphabet V′, a local language L′overV′and a letter-to-letter
+homomorphism h:V′→Vsuch thath(L′) =L.
+From FISs to tile systems LetLbeagridlanguageoveranalphabet Vrecognized
+by a FISF. LetV1denote the extended alphabet consisting of tuples ( N,W,a,E,S )Undecidability Results for Finite Interactive Systems 13
+(N/W/E/S stands for north/west/east/south, respectively), with W,Eclasses inF,
+withN,Sstates inF, withainV, and such thatN
+WaE
+Sis a transition in F.
+Consider the local language L1overV1generated by the following set of 2 ×2 tiles
+overV1∪{♯}:
+•(tiles with letters in V1which agree on the connecting cells)
+w1w2
+w3w4, withwk= (Nk,Wk,ak,Ek,Sk),k∈1,4 and such that E1=W2,S1=
+N3,S2=N4,E3=W4.
+•(tiles with♯for handling the borders) - these are tiles with ♯and such that the
+next elements in V1have appropriate initial/final states/classes;
+– for instance, the north-east corner tile is♯ ♯
+♯w4withN4,W4initial;
+– the middle-north border tile is♯ ♯
+w3w4withN3,N4initial;
+– and so on.
+One can easily see that the grids in L1correspond to the scenarios in F. By
+dropping the information around the transition symbols with the hom omorphism h:
+(N,W,a,E,S )/mapsto→aone gets a tile systems specifying L. This proves one implication,
+namely:
+Lemma 5 A language recognized by a FIS can be specified with a tile system.
+From tile systems to FISs It is obvious that FIS languages are closed to letter-
+to-letter homomorphism, so we can restrict ourself to local langua ges.
+For a local language Lover an alphabet Vand specified by a set ∆ of tiles, we
+construct an equivalent FIS Fas follows:
+•the set of states is the same as the set of classes and consists of 2 ×2 tiles over
+V∪{♯}from ∆;
+•the transitions areN1N2
+N3N4
+W1W2
+W3W4aE1E2
+E3E4
+S1S2
+S3S4with:
+– (N1,N2,N3,N4) = (W1,W2,W3,W4),
+–N4=W4=a,
+– (N3,N4) = (S1,S2), and
+– (W2,W4) = (E1,E3);
+•initial states are tiles♯ ♯
+N3N4∈∆;
+initial classes are tiles♯ W2
+♯ W4∈∆;14 Sofronia, Popa, Stefanescu
+•final states are tilesN1N2
+♯ ♯∈∆;
+final classes are tilesW2♯
+W4♯∈∆;
+A gridwis inLif and only if it is recognized by F. In this way, the following
+reverse implication result is proved:
+Lemma 6 A language specified by a tile system can be recognized with a FIS.
+The two lemmas above prove the main result of this section.
+Theorem 7 A language is recognized by a FIS if and only if it can be specified with
+a tile system.
+Theorem 7 and a result in [11], stating that by extracting the first line from the
+gridsspecified bytile systemsonegetspreciselythe context-sens itivestringlanguages,
+yield an alternative, indirect proof of the undecidability of the emptin ess problem for
+FISs.
+6. Conclusions and future work
+We have proved that a few simple and easily decidable properties on fin ite au-
+tomata (like accessibility of a transition, finiteness, etc.) become un decidable when
+extended to finite interactive systems. It may be worthwhile to find interesting re-
+stricted classes of FIS languages for which these properties are d ecidable.
+One can also look at possible extensions of the technique in this paper for cover-
+ing different sets of restricted grids such as: non-rectangular gr ids, connected grids,
+bounded grids, etc. For these classes, one may use the bounded Post Correspondence
+Problem (which bounds the number of pairs used in a PCP solution to no more th an
+k, including repeated tiles) which is also known to be NP-complete [9].
+Other open area is to develop an algebraic theory for representing FIS languages,
+similar to the regular algebra used for regular languages and finite au tomata.
+Acknowledgment
+This research was partially supported by the GlobalComp Grant (PNC DI-II,
+Project 11052/18.09.2007, Romania).
+References
+[1] ARBAB, F.: Reo: a channel-based coordination model for component comp osi-
+tion, Mathematical Structures in Computer Science 14(3), 2004, pp. 329-366.Undecidability Results for Finite Interactive Systems 15
+[2] DRAGOI, C., STEFANESCU, G.: Agapia v0.1: A programming language for
+interactive systems and its typing systems. In: Proc. FINCO/ETAPS 2007,
+ENTCS, 203(3)(2008): pp. 69-94.
+[3] DRAGOI, C., STEFANESCU, G.: On compiling structured interactive programs
+with registers and voices. In: Proc. SOFSEM 2008. LNCS 4910, Springer, 2008:
+pp. 259-270
+[4] DE NICOLA, R., FERRARI, G., PUGLIESE, R.: Klaim: a Kernel Language
+for Agents Interaction and Mobility. IEEETransactionsonSoftwareEngineering,
+24(5): pp. 315-330, IEEE Computer Society, 1998.
+[5] GADDUCCI, F., MONTANARI, U.: The tile model. In: Proof, language, and
+interaction: Essays in honor of Robin Milner,pp. 133-168. MIT Press , 1999.
+[6] GIAMMARRESI,D., RESTIVO,A.: Two-dimensional languages. In: Handbook
+of formal languages. Vol. 3: Beyond words (Rozenberg, G., Saloma a, A., eds.),
+pp. 215-265. Springer, 1997.
+[7] GIAMMARRESI, D., RESTIVO, A., SEIBERT, S., THOMAS, W.: Monadic
+second order logic over rectangular pictures and recogniza bility by tiling systems.
+Information and Computation, 125(1996): pp. 32-45.
+[8] GOLDIN, D., SMOLKA, S., WEGNER, P.(Eds.): Interactive computation: The
+new paradigm. Springer, 2006.
+[9] GAREY, M.R. and JOHNSON, D.S.: Computers and Intractability: A Guide to
+the Theory of NP-Completeness. W. H. Freeman, 1979.
+[10] LINDGREN, K., MOORE, C., NORDAHL, M.: Complexity of two-dimensional
+patterns. Journal of Statistical Physics, 91(1998): pp. 909-951.
+[11] LATTEUX, M., SIMPLOT, D.: Context-sensitive string languages and recogniz-
+able picture languages. Information and Computation, 138(1997): pp. 160-169.
+[12] MISRA, J., COOK, W.R. : Computation Orchestration. Software and System
+Modeling 6(1)(2007): pp. 83-110
+[13] POPA, A., SOFRONIA, A., STEFANESCU, G.: High-level structured interac-
+tive programs with registers and voices. Journal of Universal Computer Science,
+13(11)(2007): pp. 1498-1500.
+[14] SOFRONIA, A., POPA, A., STEFANESCU, G.: Undecidability Results for Fi-
+nite Interactive Systems , Proc. 10th International Symposium on Symbolic and
+Numeric Algorithms for Scientific Computing - SYNASC 2008, IEEE, 20 08, pp.
+xxx-xxx.
+[15] POST, E.L.: Recursive unsolvability of a problem of Thue. Journal of Symbolic
+Logic, 12(1947): pp. 1-11.16 Sofronia, Popa, Stefanescu
+[16] STEFANESCU, G.: Network algebra. Springer, 2000.
+[17] STEFANESCU, G.: Algebra of networks: modeling simple networks as well as
+complex interactive systems. In: Proof and System-Reliability, Proc. Marktober-
+dorf Summer School 2001, pp. 49-78. Kluwer, 2002.
+[18] STEFANESCU, G.: Interactive systems with registers and voices. Fundamenta
+Informaticae 73(2006), pp. 285-306.
\ No newline at end of file
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+arXiv:1001.0144v1  [math-ph]  31 Dec 2009ON HIGHER ORDER ESTIMATES IN QUANTUM
+ELECTRODYNAMICS
+OLIVER MATTE
+Abstract. We propose a new method to derive certain higher order esti-
+mates in quantum electrodynamics. Our method is particularly conve nient
+in the application to the non-local semi-relativistic models of quantum elec-
+trodynamics as it avoids the use of iterated commutator expansion s. We
+re-derivehigher orderestimates obtained earlierby Fr¨ ohlich, Grie semer, and
+Schlein and prove new estimates for a non-local molecular no-pair op erator.
+1.Introduction
+The main objective of this paper is to present a new method to derive higher
+order estimates in quantum electrodynamics (QED) of the form
+/vextenddouble/vextenddoubleHn/2
+f(H+C)−n/2/vextenddouble/vextenddouble/lessorequalslantconst<∞, (1.1)
+/vextenddouble/vextenddouble[Hn/2
+f, H](H+C)−n/2/vextenddouble/vextenddouble/lessorequalslantconst<∞, (1.2)
+for alln∈ /C6, whereC >0 is sufficiently large. In these bounds Hfdenotes the
+radiation field energy of the quantized photon field and His the full Hamil-
+tonian generating the time evolution of an interacting electron-pho ton system.
+For instance, estimates of this type serve as one of the main techn ical ingredi-
+ents in the mathematical analysis of Rayleigh scattering. In this con text, (1.1)
+has been proven by Fr¨ ohlich et al. in the case where His the non- or semi-
+relativistic Pauli-Fierz Hamiltonian [4]; a slightly weaker version of (1.2) h as
+been obtained in [4] for all even values of n. Higher order estimates of the form
+(1.1) also turn out to be useful in the study of the existence of gro und states
+in a no-pair model of QED [8]. In fact, they imply that every eigenvect or of
+the Hamiltonian Hor spectral subspaces of Hcorresponding to some bounded
+interval are contained in the domains of higher powers of Hf. This information
+is very helpful in order to overcome numerous technical difficulties w hich are
+caused by the non-locality of the no-pair operator. In these applic ations it is
+actually necessary to have some control on the norms in (1.1) and ( 1.2) when
+Date: November 4, 2018.
+1991Mathematics Subject Classification. Primary 81Q10; Secondary 47B25.
+Key words and phrases. (Semi-relativistic) Pauli-Fierz operator, no-pair Hamiltonian,
+higher order estimates, quantum electrodynamics.
+1the operator Hgets modified. To this end we shall give rough bounds on the
+right hand sides of (1.1) and (1.2) in terms of the ground state ener gy and
+integrals involving the form factor and the dispersion relation.
+Various types of higher order estimates have actually been employe d in the
+mathematical analysis ofquantumfield theoriessince averylong time . Herewe
+only mention the classical works [5, 11] on P(φ)2models and the more recent
+articles [2] again on a P(φ)2model and [1] on the Nelson model.
+In what follows we briefly describe the organization and the content of the
+present article. In Section 2 we develop the main idea behind our tech niques in
+ageneralsetting. Bythecriterionestablishedtheretheproofof thehigherorder
+estimates is essentially boiled down to the verification of certain form bounds
+on the commutator between Hand a regularized version of Hn/2
+f. After that,
+in Section 3, we introduce some of the most important operators ap pearing in
+QED and establish some useful norm bounds on certain commutator s involving
+them. These commutator estimates provide the main ingredients ne cessary to
+apply the general criterion of Section 2 to the QED models treated in this
+article. Their derivation is essentially based on the pull-through form ula which
+is always employed either way to derive higher order estimates in quan tum
+field theories [1, 2, 4, 5, 11]; compare Lemma 3.2 below. In Sections 4, 5, and 6
+the general strategy from Section 2 is applied to the non- and semi- relativistic
+Pauli-Fierz operators and to the no-pair operator, respectively. The latter
+operatorsareintroducedindetailinthesesections. Apartfromt hefactthatour
+estimate (1.2) is slightly stronger than the corresponding one of [4] the results
+of Sections 4 and 5 are not new and have been obtained earlier in [4]. Ho wever,
+in order to prove the higher order estimate (1.1) for the no-pair op erator we
+virtually have to re-derive it for the semi-relativistic Pauli-Fierz oper ator by
+our own method anyway. Moreover, we think that the arguments e mployed
+in Sections 4 and 5 are more convenient and less involved than the pro cedure
+carried throughin[4]. The maintext isfollowed byanappendix where we show
+thatthesemi-relativisticPauli-Fierzoperatorforamolecularsyste mwithstatic
+nuclei issemi-boundedbelow, providedthatallCoulombcoupling cons tantsare
+less than or equal to 2 /π. Moreover, we prove the same result for a molecular
+no-pair operator assuming that all Coulomb coupling constants are strictly less
+than the critical coupling constant of the Brown-Ravenhall model [3]. The
+results of the appendix are based on corresponding estimates for hydrogen-like
+atoms obtained in [10]. (We remark that the considerably stronger s tability of
+matter of the second kind has been proven for a molecular no-pair o perator in
+[9] under more restrictive assumptions on the involved physical par ameters.)
+No restrictions onthevaluesof thefine-structure constant oro ntheultra-violet
+cut-off are imposed in the present article.
+2The main new results of this paper are Theorem 2.1 and its corollaries w hich
+provide general criteria for the validity of higher order estimates a nd Theo-
+rem 6.1 where higher order estimates for the no-pair operator are established.
+Some frequently used notation. Fora,b∈ /CA, we write a∧b:= min{a,b}
+anda∨b:= max{a,b}.D(T) denotes the domain of some operator Tacting
+in some Hilbert space and Q(T) its form domain, when Tis semi-bounded
+below.C(a,b,...),C′(a,b,...), etc. denote constants that depend only on the
+quantitiesa,b,...and whose value might change from one estimate to another.
+2.Higher order estimates: a general criterion
+Thefollowingtheoremanditssucceedingcorollariespresentthekey ideabehind
+of our method. They essentially reduce the derivation of the higher order
+estimates to the verification of a certain sequence of form bounds . These form
+bounds can be verified easily without any further induction argumen t in the
+QED models treated in this paper.
+Theorem 2.1. LetHandFε,ε>0, be self-adjoint operators in some Hilbert
+space Ksuch thatH/greaterorequalslant1,Fε/greaterorequalslant0, and each Fεis bounded. Let m∈ /C6∪
+{∞}, letDbe a form core for H, and assume that the following conditions are
+fulfilled:
+(a) For every ε>0,Fεmaps DintoQ(H)and there is some cε∈(0,∞)
+such that/angbracketleftbig
+Fεψ/vextendsingle/vextendsingleHFεψ/angbracketrightbig
+/lessorequalslantcε/a\}b∇acketle{tψ|Hψ/a\}b∇acket∇i}ht, ψ∈D.
+(b) There is some c∈[1,∞)such that, for all ε>0,
+/a\}b∇acketle{tψ|F2
+εψ/a\}b∇acket∇i}ht/lessorequalslantc2/a\}b∇acketle{tψ|Hψ/a\}b∇acket∇i}ht, ψ∈D.
+(c) For every n∈ /C6,n < m, there is some cn∈[1,∞)such that, for all
+ε>0,/vextendsingle/vextendsingle/a\}b∇acketle{tHϕ1|Fn
+εϕ2/a\}b∇acket∇i}ht−/a\}b∇acketle{tFn
+εϕ1|Hϕ2/a\}b∇acket∇i}ht/vextendsingle/vextendsingle
+/lessorequalslantcn/braceleftbig
+/a\}b∇acketle{tϕ1|Hϕ1/a\}b∇acket∇i}ht+/a\}b∇acketle{tFn−1
+εϕ2|HFn−1
+εϕ2/a\}b∇acket∇i}ht/bracerightbig
+, ϕ 1,ϕ2∈D.
+Then it follows that, for every n∈ /C6,n<m+1,
+(2.1) /ba∇dblFn
+εH−n/2/ba∇dbl/lessorequalslantCn:= 4n−1cnn−1/productdisplay
+ℓ=1cℓ.
+(An empty product equals 1by definition.)
+Proof.We define
+Tε(n) :=H1/2[Fn−1
+ε, H−1]H−(n−2)/2, n∈ {2,3,4,...}.
+Tε(n) is well-defined and bounded because of the closed graph theorem a nd
+Condition (a), which implies that Fε∈L(Q(H)), where Q(H) =D(H1/2)
+3is equipped with the form norm. We shall prove the following sequence of
+assertions by induction on n∈ /C6,n<m+1.
+A(n) :⇔The bound (2.1) holds true and, if n>3,we have
+∀ε>0 :/ba∇dblTε(n)/ba∇dbl/lessorequalslantCn/4c2. (2.2)
+Forn= 1, the bound (2.1) is fulfilled with C1=con account of Condition (b).
+Next, assume that n∈ /C6,n < m, and that A(1),...,A(n) hold true. To
+find a bound on /ba∇dblFn+1
+εH−(n+1)/2/ba∇dblwe write
+Fn+1
+εH−(n+1)/2=Q1+Q2 (2.3)
+with
+Q1:=FεH−1Fn
+εH−(n−1)/2, Q 2:=Fε/bracketleftbig
+Fn
+ε, H−1/bracketrightbig
+H−(n−1)/2.
+By the induction hypothesis we have
+(2.4) /ba∇dblQ1/ba∇dbl/lessorequalslant/ba∇dblFεH−1/2/ba∇dbl/ba∇dblH−1/2Fε/ba∇dbl/ba∇dblFn−1
+εH−(n−1)/2/ba∇dbl/lessorequalslantc2Cn−1,
+whereC0:= 1. Moreover, we observe that
+(2.5) /ba∇dblQ2/ba∇dbl=/ba∇dblFεH−1/2Tε(n+1)/ba∇dbl/lessorequalslantc/ba∇dblTε(n+1)/ba∇dbl.
+To find a bound on /ba∇dblTε(n+ 1)/ba∇dblwe recall that Fεmaps the form domain of
+Hcontinuously into itself. In particular, since Dis a form core for Hthe
+form bound appearing in Condition (c) is available, for all ϕ1,ϕ2∈ Q(H). Let
+φ,ψ∈D. Applying Condition (c), extended in this way, with
+ϕ1=δ1/2H−1/2φ∈ Q(H), ϕ 2=δ−1/2H−(n+1)/2ψ∈ Q(H),
+for someδ>0, we obtain
+|/a\}b∇acketle{tφ|Tε(n+1)ψ/a\}b∇acket∇i}ht|
+=/vextendsingle/vextendsingle/angbracketleftbig
+HH−1/2φ/vextendsingle/vextendsingleFn
+εH−(n+1)/2ψ/angbracketrightbig
+−/angbracketleftbig
+Fn
+εH−1/2φ/vextendsingle/vextendsingleHH−(n+1)/2ψ/angbracketrightbig/vextendsingle/vextendsingle
+/lessorequalslantcninf
+δ>0/braceleftbig
+δ/ba∇dblφ/ba∇dbl2+δ−1/ba∇dbl{H1/2Fn−1
+εH−n/2}H−1/2ψ/ba∇dbl2/bracerightbig
+/lessorequalslant2cn/ba∇dbl{H1/2Fn−1
+εH−n/2}/ba∇dbl/ba∇dblφ/ba∇dbl/ba∇dblψ/ba∇dbl.
+The operator {···}is just the identity when n= 1. Forn >1, it can be
+written as
+(2.6)H1/2Fn−1
+εH−n/2={H−1/2Fε}Fn−2
+εH−(n−2)/2+Tε(n).
+Applying the induction hypothesis and c,cℓ/greaterorequalslant1, we thus get /ba∇dblTε(2)/ba∇dbl/lessorequalslant2c1,
+/ba∇dblTε(3)/ba∇dbl/lessorequalslant6cc1c2,/ba∇dblTε(4)/ba∇dbl/lessorequalslant14c2c1c2c3<C4/4c2, and
+c/ba∇dblTε(n+1)/ba∇dbl=csup/braceleftbig
+|/a\}b∇acketle{tφ|Tε(n+1)ψ/a\}b∇acket∇i}ht|:φ,ψ∈D,/ba∇dblφ/ba∇dbl=/ba∇dblψ/ba∇dbl= 1/bracerightbig
+/lessorequalslant2cn(c2Cn−2+Cn/4c)< cnCn=Cn+1/4c, n> 3,
+4sincec2Cn−2/lessorequalslantCn/16, forn>3. Taking (2.3)–(2.5) into account we arrive at
+/ba∇dblF2
+εH−1/ba∇dbl/lessorequalslantc2+2cc1<C2,/ba∇dblF3
+εH−3/2/ba∇dbl/lessorequalslantc3+6c2c1c2<C3, and
+/ba∇dblFn+1
+εH−(n+1)/2/ba∇dbl< c2Cn−2+Cn+1/4c < C n+1, n> 3,
+which concludes the induction step. /square
+Corollary 2.2. Assume that HandFε,ε >0, are self-adjoint operators in
+some Hilbert space Kthat fulfill the assumptions of Theorem 2.1 with (c)
+replaced by the stronger condition
+(c’) For every n∈ /C6,n < m, there is some cn∈[1,∞)such that, for all
+ε>0,/vextendsingle/vextendsingle/a\}b∇acketle{tHϕ1|Fn
+εϕ2/a\}b∇acket∇i}ht−/a\}b∇acketle{tFn
+εϕ1|Hϕ2/a\}b∇acket∇i}ht/vextendsingle/vextendsingle
+/lessorequalslantcn/braceleftbig
+/ba∇dblϕ1/ba∇dbl2+/a\}b∇acketle{tFn−1
+εϕ2|HFn−1
+εϕ2/a\}b∇acket∇i}ht/bracerightbig
+, ϕ 1,ϕ2∈D.
+Then, in addition to (2.1), it follows that, for n∈ /C6,n < m,[Fn
+ε, H]H−n/2
+defines a bounded sesquilinear form with domain Q(H)×Q(H)and
+(2.7)/vextenddouble/vextenddouble[Fn
+ε, H]H−n/2/vextenddouble/vextenddouble/lessorequalslantC′
+n:= 4ncn−1n/productdisplay
+ℓ=1cℓ.
+Proof.Again, the form bound in (c’) is available, for all ϕ1,ϕ2∈ Q(H), whence
+/vextendsingle/vextendsingle/a\}b∇acketle{tHφ|Fn
+εH−n/2ψ/a\}b∇acket∇i}ht−/a\}b∇acketle{tFn
+εφ|HH−n/2ψ/a\}b∇acket∇i}ht/vextendsingle/vextendsingle
+/lessorequalslantcninf
+δ>0/braceleftbig
+δ/ba∇dblφ/ba∇dbl2+δ−1/vextenddouble/vextenddoubleH1/2Fn−1
+εH−n/2ψ/vextenddouble/vextenddouble2/bracerightbig
+/lessorequalslant2cn/ba∇dblH1/2Fn−1
+εH−n/2/ba∇dbl,
+for all normalized φ,ψ∈ Q(H). The assertion now follows from (2.1), (2.6),
+and the bounds on /ba∇dblTε(n)/ba∇dblgiven in the proof of Theorem 2.1. /square
+Corollary 2.3. LetH/greaterorequalslant1andA/greaterorequalslant0be two self-adjoint operators in some
+Hilbert space K. Letκ>0, define
+fε(t) :=t/(1+εt), t/greaterorequalslant0, F ε:=fκ
+ε(A),
+for allε >0, and assume that HandFε,ε >0, fulfill the hypotheses of
+Theorem 2.1, for some m∈ /C6∪{∞}. ThenRan(H−n/2)⊂ D(Aκn), for every
+n∈ /C6,n<m+1, and
+/vextenddouble/vextenddoubleAκnH−n/2/vextenddouble/vextenddouble/lessorequalslant4n−1cnn−1/productdisplay
+ℓ=1cℓ.
+IfHandFε,ε >0, fulfill the hypotheses of Corollary 2.2, then, for every
+n∈ /C6,n<m, it additionally follows that AκnH−n/2mapsD(H)into itself so
+that[Aκn, H]H−n/2is well-defined on D(H), and
+/vextenddouble/vextenddouble[Aκn, H]H−n/2/vextenddouble/vextenddouble/lessorequalslant4ncn−1n/productdisplay
+ℓ=1cℓ.
+5Proof.LetU:K→L2(Ω,µ) be a unitary transformation such that a=
+UAU∗is a maximal operator of multiplication with some non-negative mea-
+surable function – again called a– on some measure space (Ω ,A,µ). We pick
+someψ∈K, setφn:=UH−n/2ψ, and apply the monotone convergence
+theorem to conclude that/integraldisplay
+Ωa(ω)2κn|φn(ω)|2dµ(ω) = lim
+εց0/integraldisplay
+Ωfκ
+ε(a(ω))2n|φn(ω)|2dµ(ω)
+= lim
+εց0/ba∇dblFn
+εH−n/2ψ/ba∇dbl2/lessorequalslantCn/ba∇dblψ/ba∇dbl2,
+for everyn∈ /C6,n<m+1, which implies the first assertion. Now, assume that
+HandFε,ε>0, fulfill Condition(c’)of Corollary2.2. Applying thedominated
+convergence theorem in the spectral representation introduce d above we see
+thatFn
+εψ→Aκnψ, for every ψ∈ D(Aκn). Hence, (2.7) and Ran( H−n/2)⊂
+D(Aκn) imply, for n<mandφ,ψ∈ D(H),
+/vextendsingle/vextendsingle/angbracketleftbig
+φ/vextendsingle/vextendsingleAκnH−n/2Hψ/angbracketrightbig
+−/angbracketleftbig
+Hφ/vextendsingle/vextendsingleAκnH−n/2ψ/angbracketrightbig/vextendsingle/vextendsingle
+= lim
+εց0/vextendsingle/vextendsingle/angbracketleftbig
+Fn
+εφ/vextendsingle/vextendsingleHH−n/2ψ/angbracketrightbig
+−/angbracketleftbig
+Hφ/vextendsingle/vextendsingleFn
+εH−n/2ψ/angbracketrightbig/vextendsingle/vextendsingle
+/lessorequalslantlimsup
+εց0/vextenddouble/vextenddouble[Fn
+ε, H]H−n/2/vextenddouble/vextenddouble/ba∇dblφ/ba∇dbl/ba∇dblψ/ba∇dbl/lessorequalslantC′
+n/ba∇dblφ/ba∇dbl/ba∇dblψ/ba∇dbl.
+Thus,|/a\}b∇acketle{tHφ|AκnH−n/2ψ/a\}b∇acket∇i}ht|/lessorequalslant/ba∇dblφ/ba∇dbl/ba∇dblAκnH−n/2/ba∇dbl/ba∇dblHψ/ba∇dbl+C′
+n/ba∇dblφ/ba∇dbl/ba∇dblψ/ba∇dbl, for all
+φ,ψ∈ D(H). In particular, AκnH−n/2ψ∈ D(H∗) =D(H), for allψ∈ D(H),
+and the second asserted bound holds true. /square
+3.Commutator estimates
+In this section we derive operator norm bounds on commutators inv olving the
+quantized vector potential, A, the radiation field energy, Hf, and the Dirac
+operator,DA. The underlying Hilbert space is
+H:=L2( /CA3
+x× /CI4)⊗Fb=/integraldisplay⊕/CA3
+/BV4⊗Fbd3x,
+where the bosonic Fock space, Fb, is modeled over the one-photon Hilbert
+space
+F(1)
+b:=L2(A× /CI2,dk),/integraldisplay
+dk:=/summationdisplay
+λ∈ /CI2/integraldisplay
+Ad3k.
+With regards to the applications in [8] we define A:={k∈ /CA3:|k|/greaterorequalslantm}, for
+somem/greaterorequalslant0. We thus have
+Fb=∞/circleplusdisplay
+n=0F(n)
+b,F(0)
+b:= /BV,F(n)
+b:=SnL2/parenleftbig
+(A× /CI2)n/parenrightbig
+, n∈ /C6,
+6whereSn=S2
+n=S∗
+nis given by
+(Snψ(n))(k1,...,k n) :=1
+n!/summationdisplay
+π∈Snψ(n)(kπ(1),...,k π(n)), ψ(n)∈L2/parenleftbig
+(A× /CI2)n/parenrightbig
+,
+Sndenoting the group of permutations of {1,...,n}. The vector potential is
+determined by a certain vector-valued function, G, called the form factor.
+Hypothesis 3.1. The dispersion relation, ω:A →[0,∞), is a measurable
+function such that 0<ω(k) :=ω(k)/lessorequalslant|k|, fork= (k,λ)∈ A× /CI2withk/\e}atio\slash= 0.
+For everyk∈(A\{0})× /CI2andj∈ {1,2,3},G(j)(k)is a bounded continuously
+differentiable function, /CA3
+x∋x/ma√sto→G(j)
+x(k), such that the map (x,k)/ma√sto→G(j)
+x(k)
+is measurable and G(j)
+x(−k,λ) =G(j)
+x(k,λ), for almost every kand allx∈ /CA3
+andλ∈ /CI2. Finally, there exist d−1,d0,d1,...∈(0,∞)such that
+2/integraldisplay
+ω(k)ℓ/ba∇dblG(k)/ba∇dbl2
+∞dk/lessorequalslantd2
+ℓ, ℓ∈ {−1,0,1,2,...}, (3.1)
+2/integraldisplay
+ω(k)−1/ba∇dbl∇x∧G(k)/ba∇dbl2
+∞dk/lessorequalslantd2
+1, (3.2)
+whereG= (G(1),G(2),G(3))and/ba∇dblG(k)/ba∇dbl∞:= supx|Gx(k)|, etc.
+Example. In the physical applications the form factor is often given as
+(3.3) Ge,Λ
+x(k) :=−e
+/BD{|k|/lessorequalslantΛ}
+2π/radicalbig
+|k|e−ik·xε(k),
+for (x,k)∈ /CA3×( /CA3× /CI2) withk/\e}atio\slash= 0. Here the physical units are chosen such
+thatenergiesaremeasuredinunitsoftherestenergyoftheelect ron. Lengthare
+measured in units of one Compton wave length divided by 2 π. The parameter
+Λ>0 is an ultraviolet cut-off and the square of the elementary charge, e>0,
+equals Sommerfeld’s fine-structure constant in these units; we ha vee2≈1/137
+in nature. The polarization vectors, ε(k,λ),λ∈ /CI2, are homogeneous of degree
+zero inksuch that {˚k,ε(˚k,0),ε(˚k,1)}is an orthonormal basis of /CA3, for every
+˚k∈S2. This corresponds to the Coulomb gauge for ∇x·Ge,Λ= 0. We remark
+that the vector fields S2∋˚k/ma√sto→ε(˚k,λ) are necessarily discontinuous. ⋄
+It is useful to work with more general form factors fulfilling Hypoth esis 3.1
+since in the study of the existence of ground states in QED one usua lly en-
+counters truncated and discretized versions of the physical cho iceGe,Λ. For
+the applications in [8] it is necessary to know that the higher order e stimates
+established here hold true uniformly in the involved parameters and H ypothe-
+sis 3.1 is convenient way to handle this.
+7We recall the definition of the creation and the annihilation operator s of a
+photon state f∈F(1)
+b,
+(a†(f)ψ)(n)(k1,...,k n) =n−1/2n/summationdisplay
+j=1f(kj)ψ(n−1)(...,kj−1,kj+1,...), n∈ /C6,
+(a(f)ψ)(n)(k1,...,k n) = (n+1)1/2/integraldisplay
+f(k)ψ(n+1)(k,k1,...,k n)dk, n∈ /C60,
+and (a†(f)ψ)(0)= 0,a(f)(ψ(0),0,0,...) = 0, for all ψ= (ψ(n))∞
+n=0∈Fbsuch
+that the right hand sides again define elements of Fb.a†(f) anda(f) are
+formal adjoints of each other on the dense domain
+C0:= /BV⊕∞/circleplusdisplay
+n=1SnL∞
+comp/parenleftbig
+(A× /CI2)n/parenrightbig
+.(Algebraic direct sum.)
+For a three-vector of functions f= (f(1),f(2),f(3))∈(F(1)
+b)3, we writea♯(f) :=
+(a♯(f(1)),a♯(f(2)),a♯(f(3))), wherea♯isa†ora. Then the quantized vector
+potential is the triplet of operators given by
+A≡A(G) :=a†(G)+a(G), a♯(G) :=/integraldisplay⊕/CA3
+/BD/BV4⊗a♯(Gx)d3x.
+The radiation field energy is the direct sum Hf=/circleplustext∞
+n=0dΓ(n)(ω) :D(Hf)⊂
+Fb→Fb, wheredΓ(0)(ω) := 0, and dΓ(n)(ω) denotes the maximal multiplica-
+tion operator in F(n)
+bassociated with the symmetric function ( k1,...,k n)/ma√sto→
+ω(k1)+···+ω(kn). By the permutation symmetry and Fubini’s theorem we
+thus have
+(3.4)/angbracketleftbig
+H1/2
+fφ/vextendsingle/vextendsingleH1/2
+fψ/angbracketrightbig
+=/integraldisplay
+ω(k)/a\}b∇acketle{ta(k)φ|a(k)ψ/a\}b∇acket∇i}htdk, φ,ψ ∈ D(H1/2
+f),
+where we use the notation
+(a(k)ψ)(n)(k1,...,k n) = (n+1)1/2ψ(n+1)(k,k1,...,k n), n∈ /C60,
+almost everywhere, and a(k)(ψ(0),0,0,...) = 0. For a measurable function
+f: /CA→ /CAandψ∈ D(f(Hf)), the following identity in F(n)
+b,
+(a(k)f(Hf)ψ)(n)=f/parenleftbig
+ω(k)+dΓ(n)(ω)/parenrightbig
+(a(k)ψ)(n), n∈ /C60,
+valid for almost every k, is called the pull-through formula. Finally, we let
+α1,α2,α3, andβ:=α0denote hermitian four times four matrices that fulfill
+the Clifford algebra relations
+(3.5) αiαj+αjαi= 2δij
+/BD, i,j ∈ {0,1,2,3}.
+8They act on the second tensor factor in L2( /CA3
+x× /CI4) =L2( /CA3
+x)⊗ /BV4. As a
+consequence of (3.5) and the C∗-equality we have
+(3.6)/ba∇dblα·v/ba∇dblL( /BV4)=|v|,v∈ /CA3,/ba∇dblα·z/ba∇dblL( /BV4)/lessorequalslant√
+2|z|,z∈ /BV3,
+whereα·z:=α1z(1)+α2z(2)+α3z(3), forz= (z(1),z(2),z(3))∈ /BV3. A standard
+exercise using the inequality in (3.6), the Cauchy-Schwarz inequality , and the
+canonical commutation relations,
+[a♯(f), a♯(g)] = 0,[a(f), a†(g)] =/a\}b∇acketle{tf|g/a\}b∇acket∇i}ht /BD, f,g ∈F(1)
+b,
+reveals that every ψ∈ D(H1/2
+f) belongs to the domain of α·a♯(G) and
+(3.7)
+/ba∇dblα·a(G)ψ/ba∇dbl/lessorequalslantd−1/ba∇dblH1/2
+fψ/ba∇dbl,/ba∇dblα·a†(G)ψ/ba∇dbl2/lessorequalslantd2
+−1/ba∇dblH1/2
+fψ/ba∇dbl2+d2
+0/ba∇dblψ/ba∇dbl2.
+(Here and in the following we identify Hf≡ /BD⊗Hf, etc.) These relative bounds
+imply that α·Ais symmetric on the domain D(H1/2
+f).
+The operators whose norms areestimated in (3.9) and the following le mmata
+are always well-defined ` a priori on the following dense subspace of H,
+D:=C∞
+0( /CA3× /CI4)⊗C0.(Algebraic tensor product.)
+Given some E/greaterorequalslant1 we set
+(3.8) ˇHf:=Hf+E
+in the sequel. We already know from [10] that, for every ν/greaterorequalslant0, there is some
+constant,Cν∈(0,∞), such that
+(3.9)/vextenddouble/vextenddouble[α·A,ˇH−ν
+f]ˇHν
+f/vextenddouble/vextenddouble/lessorequalslantCν/E1/2, E/greaterorequalslant1.
+In our first lemma we derive a generalization of (3.9). Its proof rese mbles
+the one of (3.9) given in [10]. Since we shall encounter many similar but
+slightly different commutators in the applications it makes sense to int roduce
+the numerous parameters that obscure its statement (but simplif y its proof).
+Lemma 3.2. Assume that ωandGfulfill Hypothesis 3.1. Let ε/greaterorequalslant0,E/greaterorequalslant1,
+κ,ν∈ /CA,γ,δ,σ,τ/greaterorequalslant0, such that γ+δ+σ+τ/lessorequalslant1/2, and define
+(3.10) fε(t) :=t+E
+1+εt+εE, t∈[0,∞).
+Then the operator ˇHν+γ
+ffσ
+ε(Hf)[α·A, fκ
+ε(Hf)]ˇH−ν+δ
+ff−κ+τ
+ε(Hf), defined ` a pri-
+ori on D, extends to a bounded operator on Hand
+/vextenddouble/vextenddoubleˇHν+γ
+ffσ
+ε(Hf)[α·A, fκ
+ε(Hf)]ˇH−ν+δ
+ff−κ+τ
+ε(Hf)/vextenddouble/vextenddouble
+/lessorequalslant|κ|2(ρ+1)/2(d1+dρ)Eγ+δ+σ+τ−1/2, (3.11)
+whereρis the smallest integer greater or equal to 3+2|κ|+2|ν|.
+9Proof.We notice that all operators ˇHs
+fandfs
+ε(Hf) leave the dense subspace D
+invariant and that α·a♯(G) maps DintoD(ˇHs
+f), for every s∈ /CA. Now, let
+ϕ,ψ∈D. Then
+/angbracketleftbig
+ϕ/vextendsingle/vextendsingleˇHν+γ
+ffσ
+ε(Hf)[α·A, fκ
+ε(Hf)]ˇH−ν+δ
+ff−κ+τ
+ε(Hf)ψ/angbracketrightbig
+=/angbracketleftbig
+ϕ/vextendsingle/vextendsingleˇHν+γ
+ffσ
+ε(Hf)[α·a(G), fκ
+ε(Hf)]ˇH−ν+δ
+ff−κ+τ
+ε(Hf)ψ/angbracketrightbig
+(3.12)
+−/angbracketleftbig
+f−κ+τ
+ε(Hf)ˇH−ν+δ
+f[α·a(G), fκ
+ε(Hf)]fσ
+ε(Hf)ˇHν+γ
+fϕ/vextendsingle/vextendsingleψ/angbracketrightbig
+. (3.13)
+Foralmostevery k,thepull-throughformulayieldsthefollowingrepresentation,
+ˇHν+γ
+ffσ
+ε(Hf)[a(k), fκ
+ε(Hf)]ˇH−ν+δ
+ff−κ+τ
+ε(Hf)ψ=F(k;Hf)a(k)ˇH−1/2
+fψ,
+where
+F(k;t) := (t+E)ν+γfσ
+ε(t)/parenleftbig
+fκ
+ε(t+ω(k))−fκ
+ε(t)/parenrightbig
+·(t+E+ω(k))−ν+δ+1/2f−κ+τ
+ε(t+ω(k))
+=/parenleftBigt+E
+t+E+ω(k)/parenrightBigν
+(t+E)γ(t+E+ω(k))δ+1/2
+·/integraldisplay1
+0d
+dsfκ
+ε(t+sω(k))dsfσ
+ε(t)fτ
+ε(t+ω(k))
+fκε(t+ω(k)),
+fort/greaterorequalslant0. We compute
+(3.14)d
+dsfκ
+ε(t+sω(k)) =κω(k)fκ
+ε(t+sω(k))
+(t+sω(k)+E)(1+εt+εsω(k)+εE).
+Using that fεis increasing in t/greaterorequalslant0 and that
+(t+ω(k)+E)/(t+sω(k)+E)/lessorequalslant1+ω(k), s∈[0,1],
+thus
+fκ
+ε(t+sω(k))/fκ
+ε(t+ω(k))/lessorequalslant(1+ω(k))−(0∧κ), s∈[0,1],
+it is elementary to verify that
+|Fε(k;t)|/lessorequalslant|κ|ω(k)(1+ω(k))δ+τ−(0∧κ)−(0∧ν)+1/2Eγ+δ+σ+τ−1/2,
+10for allt/greaterorequalslant0 andk. We deduce that the term in (3.12) can be estimated as
+/vextendsingle/vextendsingle/angbracketleftbig
+ϕ/vextendsingle/vextendsingleˇHν+γ
+ffσ
+ε(Hf)[α·a(G), fκ
+ε(Hf)]ˇH−ν+δ
+ff−κ+τ
+ε(Hf)ψ/angbracketrightbig/vextendsingle/vextendsingle
+/lessorequalslant/integraldisplay
+/ba∇dblϕ/ba∇dbl/vextenddouble/vextenddoubleα·G(k)ˇHν+γ
+ffσ
+ε(Hf)[a(k), fκ
+ε(Hf)]ˇH−ν+δ
+ff−κ+τ
+ε(Hf)ψ/vextenddouble/vextenddoubledk
+/lessorequalslant√
+2/integraldisplay
+/ba∇dblϕ/ba∇dbl/ba∇dblG(k)/ba∇dbl∞/ba∇dblFε(k;Hf)/ba∇dbl/ba∇dbla(k)ˇH−1/2
+fψ/ba∇dbldk
+/lessorequalslant|κ|√
+2/parenleftBig/integraldisplay
+ω(k)(1+ω(k))2(δ+τ)−(0∧2κ)−(0∧2ν)+1/ba∇dblG(k)/ba∇dbl2
+∞dk/parenrightBig1/2
+·/parenleftBig/integraldisplay
+ω(k)/vextenddouble/vextenddoublea(k)ˇH−1/2
+fψ/vextenddouble/vextenddouble2dk/parenrightBig1/2
+/ba∇dblϕ/ba∇dblEγ+δ+σ+τ−1/2
+/lessorequalslant|κ|2(ρ−1)/2(d1+dρ)/ba∇dblϕ/ba∇dbl/vextenddouble/vextenddoubleH1/2
+fˇH−1/2
+fψ/vextenddouble/vextenddoubleEγ+δ+σ+τ−1/2.(3.15)
+In the last step we used δ+τ/lessorequalslant1/2 and applied (3.4). (3.15) immediately
+gives a bound on the term in (3.13), too. For we have
+f−κ+τ
+ε(Hf)ˇH−ν+δ
+f[α·a(G), fκ
+ε(Hf)]fσ
+ε(Hf)ˇHν+γ
+fϕ
+=ˇH−ν+δ
+ffτ
+ε(Hf)[f−κ
+ε(Hf),α·a(G)]ˇHν+γ
+ffκ+σ
+ε(Hf)ϕ,
+which after the replacements ( ν,κ,γ,δ,σ,τ )/ma√sto→(−ν,−κ,δ,γ,τ,σ ) andϕ/ma√sto→ −ψ
+is precisely the term we just have treated. /square
+Lemma 3.2 provides all the information needed to apply Corollary 2.3 to non-
+relativistic QED. For the application of Corollary 2.3 to the non-local s emi-
+relativistic models of QED it is necessary to study commutators that involve
+resolvents and sign functions of the Dirac operator,
+DA:=α·(−i∇x+A)+β.
+AnapplicationofNelson’s commutator theoremwithtest operator −∆+Hf+1
+shows that DAis essentially self-adjoint on D. The spectrum of its unique
+closed extension, again denoted by the same symbol, is contained in t he union
+of two half-lines, σ[DA]⊂(−∞,−1]∪[1,∞). In particular, it makes sense to
+define
+RA(iy) := (DA−iy)−1, y∈ /CA,
+and the spectral calculus yields
+/ba∇dblRA(iy)/ba∇dbl/lessorequalslant(1+y2)−1/2,/integraldisplay/CA/vextenddouble/vextenddouble|DA|1/2RA(iy)ψ/vextenddouble/vextenddouble2dy
+π=/ba∇dblψ/ba∇dbl2, ψ∈H.
+The next lemma is a straightforward extension of [10, Corollary 3.1] w here it
+is also shown that RA(iy) mapsD(Hν
+f) into itself, for every ν >0.
+11Lemma 3.3. Assume that ωandGfulfill Hypothesis 3.1. Then, for all κ,ν∈/CA, we findki≡ki(κ,ν,d1,dρ)∈[1,∞),i= 1,2, such that, for all y∈ /CA,ε/greaterorequalslant0,
+andE/greaterorequalslantk1, there exist Υκ,ν(iy),/tildewideΥκ,ν(iy)∈L(H)satisfying
+RA(iy)ˇH−ν
+ff−κ
+ε(Hf) =ˇH−ν
+ff−κ
+ε(Hf)RA(iy)Υκ,ν(iy) (3.16)
+=ˇH−ν
+ff−κ
+ε(Hf)/tildewideΥκ,ν(iy)RA(iy), (3.17)
+onD(ˇH−ν
+f), and/ba∇dblΥκ,ν(iy)/ba∇dbl,/ba∇dbl/tildewideΥκ,ν(iy)/ba∇dbl/lessorequalslantk2, whereρis defined in Lemma 3.2.
+Proof.Without loss of generality we may assume that ε >0 for otherwise we
+could simply replace νbyν+κandfκ
+0byf0
+0= 1. First, we assume in addition
+thatν/greaterorequalslant0. We observe that
+T0:=/bracketleftbigˇH−ν
+ff−κ
+ε(Hf),α·A/bracketrightbigˇHν
+ffκ
+ε(Hf) =T1+T2
+onD, where
+T1:= [ˇH−ν
+f,α·A]ˇHν
+f, T 2:=ˇH−ν
+f[f−κ
+ε(Hf),α·A]fκ
+ε(Hf)ˇHν
+f.
+Due to (3.9) (or (3.11) with ε= 0) the operator T1extends to a bounded
+operator on Hand/ba∇dblT1/ba∇dbl/lessorequalslantCν/E1/2. According to (3.11) we further have
+/ba∇dblT2/ba∇dbl/lessorequalslantCκ,ν(d1+dρ)/E1/2. We pick some φ∈Dand compute
+/bracketleftbig
+RA(iy),ˇH−ν
+ff−κ
+ε(Hf)/bracketrightbig
+(DA−iy)φ=RA(iy)/bracketleftbigˇH−ν
+ff−κ
+ε(Hf), DA/bracketrightbig
+φ
+=RA(iy)T0ˇH−ν
+ff−κ
+ε(Hf)φ
+=RA(iy)T0ˇH−ν
+ff−κ
+ε(Hf)RA(iy)(DA−iy)φ. (3.18)
+Since (DA−iy)Dis dense in Hand since ˇH−ν
+fandfκ
+ε(Hf) are bounded (here
+we use that ν/greaterorequalslant0 andε>0), this identity implies
+RA(iy)ˇH−ν
+ff−κ
+ε(Hf) =/parenleftbig/BD+RA(iy)T0/parenrightbigˇH−ν
+ff−κ
+ε(Hf)RA(iy).
+Taking the adjoint of the previous identity and replacing yby−ywe obtain
+ˇH−ν
+ff−κ
+ε(Hf)RA(iy) =RA(iy)ˇH−ν
+ff−κ
+ε(Hf)( /BD+T∗
+0RA(iy)).
+In view of the norm bounds on T1andT2we see that (3.16) and (3.17) are valid
+with Υ κ,ν(iy) :=/summationtext∞
+ℓ=0{−T∗
+0RA(iy)}ℓand/tildewideΥκ,ν(iy) :=/summationtext∞
+ℓ=0{−RA(iy)T∗
+0}ℓ,
+provided that Eis sufficiently large, depending only on κ,ν,d1, anddρ, such
+that the Neumann series converge.
+Now, letν <0. Then we write T0on the domain Das
+T0=ˇH−ν
+ff−κ
+ε(Hf)/bracketleftbig
+α·A,ˇHν
+ffκ
+ε(Hf)/bracketrightbig
+,
+and deduce that
+RA(iy)ˇHν
+ffκ
+ε(Hf)( /BD+T0RA(iy)) =ˇHν
+ffκ
+ε(Hf)RA(iy)
+12by a computation analogous to (3.18). Taking the adjoint of this iden tity with
+yreplaced by −ywe get
+/parenleftbig/BD+RA(iy)T∗
+0/parenrightbigˇHν
+ffκ
+εRA(iy) =RA(iy)ˇHν
+ffκ
+ε(Hf).
+Next, we invert /BD+RA(iy)T∗
+0by means of the same Neumann series as above.
+As a result we obtain
+ˇHν
+ffκ
+ε(Hf)RA(iy) =RA(iy)Υκ,ν(iy)ˇHν
+ffκ
+ε(Hf) =/tildewideΥκ,ν(iy)RA(iy)ˇHν
+ffκ
+ε(Hf),
+wherethedefinitionofΥ κ,νand/tildewideΥκ,νhasbeenextended tonegative ν. Itfollows
+thatRA(iy)Υκ,ν(iy) =/tildewideΥκ,ν(iy)RA(iy) mapsD(ˇH−ν
+f) =D(ˇH−ν
+ff−κ
+ε(Hf)) =
+Ran(ˇHν
+ffκ
+ε(Hf)) into itself and that (3.16) and (3.17) still hold true when νis
+negative. /square
+Inorder to control theCoulomb singularity 1 /|x|interms of |DA|andHfinthe
+proof of the following corollary, we shall employ the bound [10, Theor em 2.3]
+(3.19)2
+π1
+|x|/lessorequalslant|DA|+Hf+kd2
+1,
+which holds true in sense of quadratic forms on Q(|DA|)∩ Q(Hf). Herek∈
+(0,∞) is some universal constant. We abbreviate the sign function of th e Dirac
+operator, which can be represented as a strongly convergent pr incipal value [6,
+Lemma VI.5.6], by
+(3.20) SAψ:=DA|DA|−1ψ= lim
+τ→∞/integraldisplayτ
+−τRA(iy)ψdy
+π.
+Werecall from[10, Lemma3.3]that SAmapsD(Hν
+f)intoitself, forevery ν >0.
+This can also be read off from the proof of the next corollary.
+Corollary 3.4. Assume that ωandGfulfill Hypothesis 3.1. Let κ,ν∈ /CA.
+Then we find some C≡C(κ,ν,d1,dρ)∈(0,∞)such that, for all γ,δ,σ,τ/greaterorequalslant0
+withγ+δ+σ+τ/lessorequalslant1/2and allε/greaterorequalslant0,E/greaterorequalslantk1,
+/vextenddouble/vextenddoubleˇHν
+ffκ
+ε(Hf)SAˇH−ν
+ff−κ
+ε(Hf)/vextenddouble/vextenddouble/lessorequalslantC, (3.21)
+/vextenddouble/vextenddouble|DA|1/2ˇHν+γ
+ffσ
+ε(Hf)[SA, fκ
+ε(Hf)]ˇH−ν+δ
+ff−κ+τ
+ε(Hf)/vextenddouble/vextenddouble/lessorequalslantC, (3.22)
+/vextenddouble/vextenddouble|x|−1/2ˇHν
+ffσ
+ε(Hf)[SA, fκ
+ε(Hf)]ˇH−ν−σ−τ
+ff−κ+τ
+ε(Hf)/vextenddouble/vextenddouble/lessorequalslantC. (3.23)
+(k1is the constant appearing in Lemma 3.3, ˇHfis given by (3.8),fεby(3.10).)
+Proof.First, we prove (3.22). Using (3.20), writing
+[RA(iy), fκ
+ε(Hf)] =RA(iy)[fκ
+ε(Hf),α·A]RA(iy)
+13onDandemploying (3.16), (3.17), and(3.11) we obtainthe following estima te,
+for allϕ,ψ∈D, andE/greaterorequalslantk1,/vextendsingle/vextendsingle/angbracketleftbig
+|DA|1/2ϕ/vextendsingle/vextendsingleˇHν+γ
+ffσ
+ε(Hf)[SA, fκ
+ε(Hf)]ˇH−ν+δ
+ffε(Hf)−κ+τψ/angbracketrightbig/vextendsingle/vextendsingle
+/lessorequalslant/integraldisplay/CA/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig
+ˇHν+γ
+f|DA|1/2ϕ/vextendsingle/vextendsingle/vextendsinglefσ
+ε(Hf)[fκ
+ε(Hf), RA(iy)]×
+סH−ν+δ
+ff−κ+τ
+ε(Hf)ψ/angbracketrightBig/vextendsingle/vextendsingle/vextendsingledy
+π
+=/integraldisplay/CA/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig
+ϕ/vextendsingle/vextendsingle/vextendsingle|DA|1/2RA(iy)Υσ,ν+γ(iy)ˇHν+γ
+ffσ
+ε(Hf)[fκ
+ε(Hf),α·A]×
+סH−ν+δ
+ff−κ+τ
+ε(Hf)/tildewideΥκ−τ,ν−δ(iy)RA(iy)ψ/angbracketrightBig/vextendsingle/vextendsingle/vextendsingledy
+π
+/lessorequalslantCκ,ν(d1+dρ)Eγ+δ+σ+τ−1/2sup
+y∈ /CA{/ba∇dblΥσ,ν+γ(iy)/ba∇dbl/ba∇dbl/tildewideΥκ−τ,ν−δ(iy)/ba∇dbl}
+·/parenleftBig/integraldisplay/CA/vextenddouble/vextenddouble|DA|1/2RA(iy)ϕ/vextenddouble/vextenddouble2dy
+π/parenrightBig1/2/parenleftBig/integraldisplay/CA/vextenddouble/vextenddoubleRA(iy)ψ/vextenddouble/vextenddouble2dy
+π/parenrightBig1/2
+/lessorequalslantCκ,ν,d1,dρEγ+δ+σ+τ−1/2/ba∇dblϕ/ba∇dbl/ba∇dblψ/ba∇dbl.
+This estimate shows that the vector in the right entry of the scalar product in
+the first line belongs to D((|DA|1/2)∗) =D(|DA|1/2) and that (3.22) holds true.
+Next, we observe that (3.23) follows from (3.22) and (3.19). Finally, (3.21) fol-
+lows from /ba∇dblX/ba∇dbl/lessorequalslantconst(ν,κ,d1,dρ), whereX:=ˇHν
+ffκ
+ε(Hf)[SA,ˇH−ν
+ff−κ
+ε(Hf)].
+Such a bound on /ba∇dblX/ba∇dblis, however, an immediate consequence of (3.22) (where
+we can choose ε= 0) because
+X= [ˇHν
+f, SA]ˇH−ν
+f+ˇHν
+f[fκ
+ε(Hf), SA]f−κ
+ε(Hf)ˇH−ν
+f
+on the domain D. /square
+4.Non-relativistic QED
+The Pauli-Fierz operator for a molecular system with static nuclei an dN∈ /C6
+electrons interacting with the quantized radiation field is acting in the Hilbert
+space
+(4.1) HN:=ANL2/parenleftbig
+( /CA3× /CI4)N/parenrightbig
+⊗Fb,
+whereAN=A2
+N=A∗
+Ndenotes anti-symmetrization,
+(ANΨ)(X) :=1
+N!/summationdisplay
+π∈SN(−1)πΨ(xπ(1),ςπ(1),...,xπ(N),ςπ(N)),
+for Ψ∈L2(( /CA3× /CI4)N) and a.e.X= (xi,ςi)N
+i=1∈( /CA3× /CI4)N.`A priori it is
+defined on the dense domain
+DN:=ANC∞
+0/parenleftbig
+( /CA3× /CI4)N/parenrightbig
+⊗C0,
+14the tensor product understood in the algebraic sense, by
+(4.2) HV
+nr≡HV
+nr(G) :=N/summationdisplay
+i=1(D(i)
+A)2+V+Hf.
+A superscript ( i) indicates that the operator below is acting on the pair of
+variables ( xi,ςi). In fact, the operator defined in (4.2) is a two-fold copy of the
+usual Pauli-Fierz operator which acts on two-spinors and the ener gy has been
+shifted byNin (4.2). For (3.5) implies
+(4.3) D2
+A=TA⊕TA,TA:=/parenleftbig
+σ·(−i∇x+A)/parenrightbig2+1.
+Hereσ= (σ1,σ2,σ3) is a vector containing the Pauli matrices (when αj,j∈
+{0,1,2,3}, are given in Dirac’s standard representation). We write HV
+nrin the
+form (4.2) to maintain a unified notation throughout this paper.
+We shall only make use of the following properties of the potential V.
+Hypothesis 4.1. Vcan be written as V=V+−V−, whereV±/greaterorequalslant0is a
+symmetric operator acting in ANL2/parenleftbig
+( /CA3× /CI4)4/parenrightbig
+such that DN⊂ D(V±). There
+exista∈(0,1)andb∈(0,∞)such thatV−/lessorequalslantaH0
+nr+bin the sense of quadratic
+forms on DN.
+Example. The Coulomb potential generated by K∈ /C6fixed nuclei located at
+the positions {R1,...,RK} ⊂ /CA3is given as
+(4.4) VC(X) :=−N/summationdisplay
+i=1K/summationdisplay
+k=1e2Zk
+|xi−Rk|+N/summationdisplay
+i,j=1
+i<je2
+|xi−xj|,
+for somee,Z1,...,Z K>0 and a.e. X= (xi,ςi)N
+i=1∈( /CA3× /CI4)N. It is
+well-known that VCis infinitesimally H0
+nr-bounded and that VCfulfills Hypoth-
+esis 4.1. ⋄
+ItfollowsimmediatelyfromHypothesis4.1that HV
+nrhasaself-adjointFriedrichs
+extension – henceforth denoted by the same symbol HV
+nr– and that DNis a
+form core for HV
+nr. Moreover, we have
+(4.5) (D(1)
+A)2,...,(D(N)
+A)2, V+, Hf/lessorequalslantHV+
+nr/lessorequalslant(1−a)−1(HV
+nr+b)
+onDN. In [4] it is shown that D((HV
+nr)n/2)⊂ D(Hn/2
+f), for every n∈ /C6. We
+re-derive this result by means of Corollary 2.3 in the next theorem wh ere
+Enr:= infσ[HV
+nr], H′
+nr:=HV
+nr−Enr+1.
+15Theorem 4.2. Assume that ωandGfulfill Hypothesis 3.1 and that Vfulfills
+Hypothesis 4.1. Assume in addition that
+2/integraldisplay
+ω(k)ℓ/ba∇dbl∇x∧G(k)/ba∇dbl2
+∞dk/lessorequalslantd2
+ℓ+2, (4.6)
+/integraldisplay
+ω(k)ℓ/ba∇dbl∇x·G(k)/ba∇dbl2
+∞dk/lessorequalslantd2
+ℓ+2, (4.7)
+for allℓ∈ {−1,0,1,2,...}. Then, for every n∈ /C6, we have D((HV
+nr)n/2)⊂
+D(Hn/2
+f),Hn/2
+f(H′
+nr)−n/2mapsD(HV
+nr)into itself, and
+/vextenddouble/vextenddoubleHn/2
+f(H′
+nr)−n/2/vextenddouble/vextenddouble/lessorequalslantC(N,n,a,b,d −1,d1,d5+n)(|Enr|+1)(3n−2)/2,
+/vextenddouble/vextenddouble[Hn/2
+f, HV
+nr](H′
+nr)−n/2/vextenddouble/vextenddouble/lessorequalslantC′(N,n,a,b,d −1,d1,d5+n)(|Enr|+1)(3n−1)/2.
+Proof.We pick the function fεdefined in (3.10) with E= 1 and verify that the
+operatorsFn
+ε:=fn/2
+ε(Hf),ε>0,n∈ /C6, andH′
+nrfulfill the conditions (a), (b),
+and (c’) of Theorem 2.1 and Corollary 2.2 with m=∞. Then the assertion
+follows from Corollary 2.3. We set ˇHf:=Hf+Ein what follows. By means of
+(4.5) we find
+(4.8) /a\}b∇acketle{tΨ|F2
+εΨ/a\}b∇acket∇i}ht/lessorequalslant/a\}b∇acketle{tΨ|ˇHfΨ/a\}b∇acket∇i}ht/lessorequalslantEnr+b+E
+1−a/a\}b∇acketle{tΨ|H′
+nrΨ/a\}b∇acket∇i}ht,
+for all Ψ ∈DN, which is Condition (b). Next, we observe that Fεmaps DN
+into itself. Employing (4.5) once more and using −V−/lessorequalslant0 and the fact that
+V+/greaterorequalslant0 andFεact on different tensor factors we deduce that
+/angbracketleftbig
+FεΨ/vextendsingle/vextendsingle(V+Hf)FεΨ/angbracketrightbig
+/lessorequalslant/ba∇dblfε/ba∇dbl∞/angbracketleftbig
+Ψ/vextendsingle/vextendsingle(V++Hf)Ψ/angbracketrightbig
+/lessorequalslant/ba∇dblfε/ba∇dbl∞Enr+b+E
+1−a/a\}b∇acketle{tΨ|H′
+nrΨ/a\}b∇acket∇i}ht, (4.9)
+for every Ψ ∈DN. Thanks to (3.11) with κ= 1/2,ν=γ=δ=σ=τ= 0,
+and (4.5) we further find some C∈(0,∞) such that
+/vextenddouble/vextenddoubleD(i)
+AFεΨ/vextenddouble/vextenddouble2/lessorequalslant2/ba∇dblfε/ba∇dbl∞/ba∇dblD(i)
+AΨ/ba∇dbl2+2/ba∇dblfε/ba∇dbl∞/vextenddouble/vextenddoubleF−1
+ε[α·A, Fε]/vextenddouble/vextenddouble2/ba∇dblΨ/ba∇dbl2
+/lessorequalslantC/ba∇dblfε/ba∇dbl∞/a\}b∇acketle{tΨ|H′
+nrΨ/a\}b∇acket∇i}ht, (4.10)
+for all Ψ ∈DN. (4.9) and (4.10) together show that Condition (a) is fulfilled,
+too. Finally, we verify the bound in (c’). We use
+[α·(−i∇x),α·A] =Σ·B−i(∇x·A),
+whereB:=a†(∇x∧G)+a(∇x∧G) is the magnetic field and the j-th entry
+of the formal vector Σis−iǫjkℓαkαℓ,j,k,ℓ∈ {1,2,3}, to write the square of
+the Dirac operator on the domain Das
+D2
+A=D2
+0+Σ·B−i(∇x·A)+(α·A)2+2α·Aα·(−i∇x).
+16This yields
+[H′
+nr,Fn
+ε] =N/summationdisplay
+i=1/bracketleftbig
+(D(i)
+A)2,Fn
+ε/bracketrightbig
+=N/summationdisplay
+i=1/braceleftbig
+[Σ·B(i), Fn
+ε]−i[(∇x·A(i)), Fn
+ε]
++α·A(i)[α·A(i), Fn
+ε]+[α·A(i), Fn
+ε](2D(i)
+A−α·A(i)−2β)/bracerightbig
+onDN. For every i∈ {1,...,N}, we further write
+[α·A(i), Fn
+ε]D(i)
+A=Q(i)
+ε,n/parenleftbig
+D(i)
+AFn−1
+ε−Q(i)
+ε,n−1Fn−2
+ε/parenrightbig
+onDN, where
+Q(i)
+ε,n:= [α·A(i), Fn
+ε]F1−n
+ε, n∈ /C6, Q(i)
+ε,0:= 0. (4.11)
+Accordingto(3.11)wehave /ba∇dblQ(i)
+ε,n/ba∇dbl/lessorequalslantn2(n+2)/2(d1+d3+n),/ba∇dblˇH1/2
+fQ(i)
+ε,nˇH−1/2
+f/ba∇dbl/lessorequalslant
+n2(n+3)/2(d1+d4+n). Likewise, we write
+[α·A(i), Fn
+ε]α·A(i)=Q(i)
+ε,n/parenleftbig
+{α·A(i)ˇH−1/2
+f}ˇH1/2
+fFn−1
+ε−Q(i)
+ε,n−1Fn−2
+ε/parenrightbig
+onDN, where/ba∇dblα·AˇH−1/2
+f/ba∇dbl2/lessorequalslant2d2
+0+4d2
+−1by (3.7). Furthermore, we observe
+that Lemma 3.2 is applicable to Σ·Bas well instead of α·A; we simply have
+to replace the form factor Gby∇x∧Gand to notice that /ba∇dblΣ·v/ba∇dblL( /BV4)=|v|,
+v∈ /CA3, in analogy to (3.6). Note that the indices of dℓare shifted by 2 because
+of (4.6). Finally, we observe that Lemma 3.2 is applicable to ∇x·A, too.
+To this end we have to replace Gby (∇x·G,0,0) anddℓby some universal
+constant times d2+ℓbecause of (4.7). Taking all these remarks into account we
+arrive at
+/vextendsingle/vextendsingle/angbracketleftbig
+Ψ1/vextendsingle/vextendsingle[H′
+nr, Fn
+ε]Ψ2/angbracketrightbig/vextendsingle/vextendsingle/lessorequalslantN/summationdisplay
+i=1/braceleftBig
+/ba∇dblΨ1/ba∇dbl/vextenddouble/vextenddouble[Σ·B(i), Fn
+ε]F1−n
+ε/vextenddouble/vextenddouble/ba∇dblFn−1
+εΨ2/ba∇dbl
++/ba∇dblΨ1/ba∇dbl/vextenddouble/vextenddouble[divA(i), Fn
+ε]F1−n
+ε/vextenddouble/vextenddouble/ba∇dblFn−1
+εΨ2/ba∇dbl
++/ba∇dblΨ1/ba∇dbl/ba∇dblα·AˇH−1/2
+f/ba∇dbl/vextenddouble/vextenddoubleˇH1/2
+fQ(i)
+ε,nˇH−1/2
+f/vextenddouble/vextenddouble/ba∇dblˇH1/2
+fFn−1
+εΨ2/ba∇dbl
++/ba∇dblΨ1/ba∇dbl/ba∇dblQ(i)
+ε,n/ba∇dbl/parenleftbig
+2/ba∇dblD(i)
+AFn−1
+εΨ2/ba∇dbl+/ba∇dblα·AˇH−1/2
+f/ba∇dbl/ba∇dblˇH1/2
+fFn−1
+εΨ2/ba∇dbl/parenrightbig
++3/ba∇dblΨ1/ba∇dbl/ba∇dblQ(i)
+ε,n/ba∇dbl/ba∇dblQ(i)
+ε,n−1/ba∇dbl/ba∇dblFn−2
+εΨ2/ba∇dbl+2/ba∇dblΨ1/ba∇dbl/ba∇dblQ(i)
+ε,n/ba∇dbl/ba∇dblβ/ba∇dbl/ba∇dblFn−1
+εΨ2/ba∇dbl/bracerightBig
+,
+for all Ψ 1,Ψ2∈DN. From this estimate, Lemma 3.2, and (4.5) we readily infer
+thatCondition(c’)isvalidwith cn= (|Enr|+1)C′′(N,n,a,b,d −1,...,d 5+n)./square
+5.The semi-relativistic Pauli-Fierz operator
+The semi-relativistic Pauli-Fierz operator is also acting in the Hilbert sp ace
+HNintroduced in (4.1). It is obtained by substituting the non-local ope rator
+17|DA|forD2
+AinHV
+nr. We thus define, ` a priori on the dense domain DN,
+HV
+sr≡HV
+sr(G) :=N/summationdisplay
+i=1|D(i)
+A|+V+Hf,
+whereVis assumed to fulfill Hypothesis 4.1 with H0
+nrreplaced by H0
+sr. To
+ensure that in the case of the Coulomb potential VCdefined in (4.4) this yields
+a well-defined self-adjoint operator we have to impose appropriate restrictions
+on the nuclear charges.
+Example. In Proposition A.1 we show that HVCsris semi-bounded below on DN
+provided that Zk∈(0,2/πe2], for allk∈ {1,...,K}. Its proof is actually a
+straightforward consequence of (3.19) and a commutator estima te obtained in
+[10]. If all atomicnumbers Zkarestrictly less than2 /πe2we thus find a∈(0,1)
+andb∈(0,∞) such that
+(5.1)N/summationdisplay
+i=1K/summationdisplay
+k=1e2Zk
+|xi−Rk|/lessorequalslantaH0
+sr+b
+in the sense of quadratic forms on DN. In particular, VCfulfills Hypothesis 4.1
+withH0
+nrreplaced by H0
+sras long asZk∈(0,2/πe2), fork∈ {1,...,K}.⋄
+For potentials Vas aboveHV
+srhas a self-adjoint Friedrichs extension which we
+denote again by the same symbol HV
+sr. Moreover, DNis a form core for HV
+sr
+and we have the following analogue of (4.5),
+(5.2) |D(1)
+A|,...,|D(N)
+A|, V+, Hf/lessorequalslantHV+
+sr/lessorequalslant(1−a)−1(HV
+sr+b)
+onDN. In order to apply Corollary 2.3 to the semi-relativistic Pauli-Fierz
+operator we recall the following special case of [7, Corollary 3.7]:
+Lemma 5.1. Assume that ωandGfulfill Hypothesis 3.1. Let τ∈(0,1]. Then
+there existδ>0andC≡C(δ,τ,d1)∈(0,∞)such that
+(5.3)C+|DA|+τHf/greaterorequalslantδ(|D0|+Hf)/greaterorequalslantδ(|D0|+τHf)/greaterorequalslantδ2|DA|−δC
+in the sense of quadratic forms on D.
+In the next theorem we re-derive the higher order estimates obta ined in [4] for
+the semi-relativistic Pauli-Fierz operator by means of Corollary 2.3. ( The sec-
+ondestimateofTheorem5.2isactuallyslightlystrongerthanthecor responding
+one stated in [4].) The estimates of the following proof are also employe d in
+Section 6 where we treat the no-pair operator. We set
+Esr:= infσ[Hsr], H′
+sr:=HV
+sr−Esr+1.
+18Theorem 5.2. Assume that ωandGfulfill Hypothesis 3.1 and that Vfulfills
+Hypothesis 4.1 with H0
+nrreplaced by H0
+sr. Then, for every m∈ /C6, it follows that
+D((HV
+sr)m/2)⊂ D(Hm/2
+f),Hm/2
+f(H′
+sr)−m/2mapsD(HV
+sr)into itself, and
+/vextenddouble/vextenddoubleHm/2
+f(H′
+sr)−m/2/vextenddouble/vextenddouble/lessorequalslantC(N,m,a,b,d 1,d3+m)(|Esr|+1)(3m−2)/2,
+/vextenddouble/vextenddouble[Hm/2
+f, HV
+sr](H′
+sr)−m/2/vextenddouble/vextenddouble/lessorequalslantC′(N,m,a,b,d 1,d3+m)(|Esr|+1)(3m−1)/2.
+Proof.Letm∈ /C6. We pick the function fεdefined in (3.10) with E=k1∨C.
+(k1is the constant appearing in Lemma 3.3 with κ=m/2,ν= 0, and depends
+onm,d1, andd3+m;Cis the one in (5.3).) We fix some n∈ /C6,n/lessorequalslantm,
+and verify Conditions (a), (b), and (c’) of Theorem 2.1 and Corollary 2.2 with
+Fε=f1/2
+ε(Hf),ε>0. The estimates (4.8) and (4.9) are still valid without any
+further change when the subscript nr is replaced by sr. Employing ( 5.3) twice
+and using (5.2) we obtain the following substitute of (4.10),
+/angbracketleftbig
+FεΨ/vextendsingle/vextendsingle|DA|FεΨ/angbracketrightbig
+/lessorequalslantδ−1/ba∇dbl|D0|1/2FεΨ/ba∇dbl2+δ−1/ba∇dblˇH1/2
+fFεΨ/ba∇dbl2
+/lessorequalslantδ−1/ba∇dblfε/ba∇dbl∞/parenleftbig
+/ba∇dbl|D0|1/2Ψ/ba∇dbl2+/ba∇dblˇH1/2
+fΨ/ba∇dbl2/parenrightbig
+/lessorequalslantC′/ba∇dblfε/ba∇dbl∞/angbracketleftbig
+Ψ/vextendsingle/vextendsingleH′
+srΨ/angbracketrightbig
+,
+for all Ψ ∈DN. Altogether we see that Conditions (a) and (b) are satisfied. In
+order to verify (c’) we set
+(5.4)U(i)
+ε,n:= [S(i)
+A, Fn
+ε]F1−n
+ε=Fn
+ε[F−n
+ε, S(i)
+A]Fε, i∈ {1,...,N}.
+Byvirtueof (3.22)weknowthatthenormsof U(i)
+ε,nandU(i)
+ε,n|D(i)
+A|1/2arebounded
+uniformly in ε>0 by some constant, C∈(0,∞), that depends only on n,d1,
+andd3+n. We employ the notation (4.11) and (5.4) to write
+[H′
+sr, Fn
+ε] =N/summationdisplay
+i=1/bracketleftbig
+|D(i)
+A|, Fn
+ε/bracketrightbig
+=N/summationdisplay
+i=1/bracketleftbig
+S(i)
+AD(i)
+A, Fn
+ε/bracketrightbig
+=N/summationdisplay
+i=1/braceleftBig
+{U(i)
+ε,n|D(i)
+A|1/2}S(i)
+A|D(i)
+A|1/2Fn−1
+ε−U(i)
+ε,nQ(i)
+ε,n−1Fn−2
+ε+S(i)
+AQ(i)
+ε,nFn−1
+ε/bracerightBig
+.
+The previous identity, (5.2), and |DA|/greaterorequalslant1 permit to get
+/vextendsingle/vextendsingle/angbracketleftbig
+Ψ1/vextendsingle/vextendsingle[H′
+sr, Fn
+ε]Ψ2/angbracketrightbig/vextendsingle/vextendsingle/lessorequalslantN/summationdisplay
+i=1/ba∇dblΨ1/ba∇dbl/braceleftbig
+C/vextenddouble/vextenddouble|D(i)
+A|1/2Fn−1
+εΨ2/vextenddouble/vextenddouble
++C/ba∇dblQ(i)
+ε,n/ba∇dbl/ba∇dblFn−2
+εΨ2/ba∇dbl+/ba∇dblQ(i)
+ε,n/ba∇dbl/ba∇dblFn−1
+εΨ2/ba∇dbl/bracerightbig
+/lessorequalslantcn/braceleftbig
+/ba∇dblΨ1/ba∇dbl2+/angbracketleftbig
+Fn−1
+εΨ2/vextendsingle/vextendsingleH′
+srFn−1
+εΨ2/angbracketrightbig/bracerightbig
+,
+for all Ψ 1,Ψ2∈DNand some constant cn=C′′(n,a,b,d 1,d3+n)(|Esr|+1). So
+(c’) is fulfilled also and the assertion follows from Corollary 2.3. /square
+196.The no-pair operator
+We introduce the spectral projections
+(6.1)P+
+A:=E[0,∞)(DA) =1
+2
+/BD+1
+2SA, P−
+A:= /BD−P+
+A.
+The no-pair operator acts in the projected Hilbert space
+H+
+N≡H+
+N(G) :=P+
+A,NHN, P+
+A,N:=N/productdisplay
+i=1P+,(i)
+A,
+and is ` a priori defined on the dense domain P+
+A,NDNby
+HV
+np≡HV
+np(G) :=P+
+A,N/braceleftBigN/summationdisplay
+i=1D(i)
+A+V+Hf/bracerightBig
+P+
+A,N.
+Noticethatalloperators D(1)
+A,...,D(N)
+AandP+,(1)
+A,...,P+,(N)
+Acommuteinpairs
+owing to the fact that the components of the vector potential A(i)(x),A(j)(y),
+x,y∈ /CA3,i,j∈ {1,2,3}, commute in the sense that all their spectral pro-
+jections commute; see the appendix to [9] for more details. (Here w e use the
+assumption that Gx(−k,λ) =Gx(k,λ).) So the order of the application of
+the projections P+,(i)
+Ais immaterial. In this section we restrict the discussion
+to the case where Vis given by the Coulomb potential VCdefined in (4.4). To
+have a handy notation we set
+vi:=−K/summationdisplay
+k=1e2Zk
+|xi−Rk|, w ij:=e2
+|xi−xj|,
+for alli∈ {1,...,N}and 1/lessorequalslanti < j/lessorequalslantN, respectively. Thanks to [10,
+Lemma 3.4(ii)], which implies that P+
+Amaps DintoD(|D0|)∩D(Hν
+f), for ev-
+eryν >0, and Hardy’s inequality, we actually know that HVCnpis well-defined
+onDN. In order to apply Corollary 2.3 to HVCnpwe extendHVCnpto a continu-
+ously invertible operator on the whole space HN: We pick the complementary
+projection,
+P⊥
+A,N:= /BD−P+
+A,N,
+abbreviate
+P+,(i,j)
+A:=P+,(i)
+AP+,(j)
+A=P+,(j)
+AP+,(i)
+A,1/lessorequalslanti<j/lessorequalslantN,
+and define the operator /tildewideHnp` a priori on the domain DNby
+/tildewideHnp:=N/summationdisplay
+i=1/braceleftbig
+|D(i)
+A|+P+,(i)
+AviP+,(i)
+A/bracerightbig
++N/summationdisplay
+i,j=1
+i<jP+,(i,j)
+AwijP+,(i,j)
+A
++P+
+A,NHfP+
+A,N+P⊥
+A,NHfP⊥
+A,N. (6.2)
+20Evidently, we have [ /tildewideHnp, P+
+A,N] = 0 and /tildewideHnpP+
+A,N=HVCnpP+
+A,NonDN. In
+Proposition A.2 we show that the quadratic forms of the no-pair ope ratorHVCnp
+and of/tildewideHnpare semi-bounded below on DNprovided that the atomic numbers
+Z1,...,Z K/greaterorequalslant0 are less than the critical one of the Brown-Ravenhall model
+determined in [3],
+(6.3) Znp:= (2/e2)/(2/π+π/2).
+Therefore, both HVCnpand/tildewideHnppossess self-adjoint Friedrichs extensions which
+are again denoted by the same symbols in the sequel. DNis a form core for
+/tildewideHnpand we have the bound
+(6.4)/tildewideHnp−N/summationdisplay
+i=1P+,(i)
+AviP+,(i)
+A/lessorequalslantZnp+|Z|
+Znp−|Z|/parenleftbig/tildewideHnp+C(N,Z,R,d−1,d1,d5)/parenrightbig
+onDN, where|Z|:= max{Z1,...,Z K}< Znp. Moreover, it makes sense to
+define
+Enp:= infσ[HVC
+np],
+so that
+H′
+np:=/tildewideHnp−EnpP+
+A,N+ /BD/greaterorequalslant /BD.
+Theorem 6.1. Assume that ωandGfulfill Hypothesis 3.1 and let N,K∈ /C6,
+e >0,Z= (Z1,...,Z K)∈[0,Znp)K, and R={R1,...,RK} ⊂ /CA3, where
+Znpis defined in (6.3). ThenD((H′
+np)m/2)⊂ D(Hm/2
+f), for every m∈ /C6, and
+/vextenddouble/vextenddoubleHm/2
+f↾H+
+N(Hnp−(Enp−1) /BDH+
+N)−m/2/vextenddouble/vextenddouble
+L(H+
+N,HN)/lessorequalslant/vextenddouble/vextenddoubleHm/2
+f(H′
+np)−m/2/vextenddouble/vextenddouble
+/lessorequalslantC(N,m, Z,R,e,d−1,d1,d5+m)(1+|Enp|)(3m−2)/2<∞.
+Proof.Letm∈ /C6. Again we pick the function fεdefined in (3.10) and set
+Fε:=f1/2
+ε(Hf),ε >0. This time we choose E= max{kd2
+1,k1,C}wherekis
+the constant appearing in (3.19), C≡C(d1) is the one in (5.3), and k1the one
+appearing in Lemma 3.3 with |κ|= (m+ 1)/2,|ν|= 1/2. Thusk1depends
+only onm,d1, andd5+m. On account of Corollary 2.3 it suffices to show that
+the conditions (a)–(c) of Theorem 2.1 are fulfilled. To this end we obs erve that
+onDNthe extended no-pair operator can be written as H′
+np=H0
+sr+ /BD+W,
+where
+W:=N/summationdisplay
+i=1P+,(i)
+AviP+,(i)
+A+N/summationdisplay
+i,j=1
+i<jP+,(i,j)
+AwijP+,(i,j)
+A
+−EnpP+
+A,N−2Re/bracketleftbig
+P+
+A,NHfP⊥
+A,N/bracketrightbig
+.
+21The semi-relativistic Pauli-Fierz operator H0
+srhas already been treated in the
+previous section and the bound
+(6.5) Hf/lessorequalslant2P+
+A,NHfP+
+A,N+2P⊥
+A,NHfP⊥
+A,N
+together with (6.4) implies
+H0
+sr/lessorequalslant2/tildewideHnp−2N/summationdisplay
+i=1P+,(i)
+AviP+,(i)
+A/lessorequalslantC′(1+|Enp|)H′
+np (6.6)
+onDN, forsomeC′≡C′(N,Z,R,d−1,d1,d5)∈(0,∞). Hence, it onlyremains
+to consider the operator W.
+We fix some n∈ /C6,n/lessorequalslantm. When we verify (a) we can ignore the potentials
+visince they are negative. Using [ Fn
+ε,P⊥
+A,N] = [P+
+A,N,Fn
+ε] we obtain
+/vextendsingle/vextendsingle2Re/angbracketleftbig
+P+
+A,NFεΨ/vextendsingle/vextendsingleHfP⊥
+A,NFεΨ/angbracketrightbig/vextendsingle/vextendsingle
+/lessorequalslant/vextenddouble/vextenddoubleH1/2
+fP+
+A,NFεΨ/vextenddouble/vextenddouble2+/vextenddouble/vextenddoubleH1/2
+fP⊥
+A,NFεΨ/vextenddouble/vextenddouble2
+/lessorequalslant2/ba∇dblfε/ba∇dbl∞/vextenddouble/vextenddoubleH1/2
+fP+
+A,NΨ/vextenddouble/vextenddouble2+2/ba∇dblfε/ba∇dbl∞/vextenddouble/vextenddoubleH1/2
+fP⊥
+A,NΨ/vextenddouble/vextenddouble2
++ 4/vextenddouble/vextenddoubleH1/2
+f[P+
+A,N, Fε]ˇH−1/2
+f/vextenddouble/vextenddouble/ba∇dblˇH1/2
+fΨ/ba∇dbl2,
+for every Ψ ∈DN, where, for all n∈ /C6andν∈ /CA,
+ˇHν
+f[P+
+A,N, Fn
+ε]ˇH−ν
+fF1−n
+ε=N/summationdisplay
+i=1/braceleftBigi−1/productdisplay
+j=1ˇHν
+fP+,(j)
+AˇH−ν
+f/bracerightBig
+×
+×/braceleftbigˇHν
+f[P+,(i)
+A,Fn
+ε]ˇH−ν
+fF1−n
+ε/bracerightbig/braceleftBigN/productdisplay
+k=i+1ˇHν
+fFn−1
+εP+,(k)
+AˇH−ν
+fF1−n
+ε/bracerightBig
+onDN. On account of Corollary 3.4 we thus have, for |ν|/lessorequalslant1/2,
+(6.7) sup
+ε>0/vextenddouble/vextenddoubleHν
+f[P+
+A,N, Fn
+ε]ˇH−ν
+fF1−n
+ε/vextenddouble/vextenddouble/lessorequalslantC(N,n,d 1,d4+n).
+Likewise we have
+/vextendsingle/vextendsingle/angbracketleftbig
+FεΨ/vextendsingle/vextendsingleP+,(i,j)
+AwijP+,(i,j)
+AFεΨ/angbracketrightbig/vextendsingle/vextendsingle/lessorequalslant2/ba∇dblfε/ba∇dbl/vextenddouble/vextenddoublew1/2
+ijP+,(i,j)
+AΨ/vextenddouble/vextenddouble2
++4/vextenddouble/vextenddoublew1/2
+ij[P+,(i,j)
+A, Fε]ˇH−1/2
+f/vextenddouble/vextenddouble2/ba∇dblˇH1/2
+fΨ/ba∇dbl2, (6.8)
+where the first norm in the second line of (6.8) is bounded (uniformly in ε>0)
+due to Lemma 6.2. Taking these remarks, vi/lessorequalslant0, (6.4), and (6.5) into account
+we infer that
+/angbracketleftbig
+FεΨ/vextendsingle/vextendsingleH′
+npFεΨ/angbracketrightbig
+/lessorequalslantcε/angbracketleftbig
+Ψ/vextendsingle/vextendsingleH′
+npΨ/angbracketrightbig
+,Ψ∈DN,
+showingthat(a)isfulfilled. Condition(b)with c2=C(N,Z,R,d−1,d1,d5)(1+
+|Enp|) follows immediately from F2
+ε/lessorequalslantˇHf/lessorequalslantH0
+sr+EonDNand (6.6). Finally,
+22we turn to Condition (c). To this end let P♯
+A,NandP♭
+A,NbeP+
+A,NorP⊥
+A,N. On
+DNwe clearly have
+/bracketleftbig
+P♯
+A,NHfP♭
+A,N, Fn
+ε/bracketrightbig
+=±[P+
+A,N, Fn
+ε]HfP♭
+A,N±P♯
+A,NHf[P+
+A,N, Fn
+ε]. (6.9)
+For Ψ 1,Ψ2∈DN, we thus obtain
+/vextendsingle/vextendsingle/angbracketleftbig
+Ψ1/vextendsingle/vextendsingle/bracketleftbig
+P♯
+A,NHfP♭
+A,N, Fn
+ε/bracketrightbig
+Ψ2/angbracketrightbig/vextendsingle/vextendsingle
+/lessorequalslant/ba∇dblˇH1/2
+fΨ1/ba∇dbl/vextenddouble/vextenddoubleˇH−1/2
+f[P+
+A,N, Fn
+ε]H1/2
+fF1−n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleH1/2
+fFn−1
+εP♭
+A,NΨ2/vextenddouble/vextenddouble
++/ba∇dblH1/2
+fP♯
+A,NΨ1/ba∇dbl/vextenddouble/vextenddoubleH1/2
+f[P+
+A,N, Fn
+ε]ˇH−1/2
+fF1−n
+ε/vextenddouble/vextenddouble/ba∇dblˇH1/2
+fFn−1
+εΨ2/ba∇dbl, (6.10)
+where we can further estimate
+/vextenddouble/vextenddoubleH1/2
+fFn−1
+εP♭
+A,NΨ2/vextenddouble/vextenddouble
+/lessorequalslant/braceleftbig
+1+/vextenddouble/vextenddoubleH1/2
+fFn−1
+εP+
+A,NˇH−1/2
+fF1−n
+ε/vextenddouble/vextenddouble/bracerightbig
+/ba∇dblˇH1/2
+fFn−1
+εΨ2/ba∇dbl
+/lessorequalslant/braceleftbig
+1+/ba∇dblH1/2
+fFn−1
+εP+
+AˇH−1/2
+fF1−n
+ε/ba∇dblN/bracerightbig
+/ba∇dblˇH1/2
+fFn−1
+εΨ2/ba∇dbl, (6.11)
+and, of course,
+/ba∇dblˇH1/2
+fFn−1
+εΨ2/ba∇dbl/lessorequalslant/ba∇dblˇH1/2
+fP+
+A,NFn−1
+εΨ2/ba∇dbl+/ba∇dblˇH1/2
+fP⊥
+A,NFn−1
+εΨ2/ba∇dbl. (6.12)
+The operator norms in (6.10) can be estimated by means of (6.7) with ν=
+±1/2, the one in the last line of (6.11) is bounded by some C(n,d1,d3+n)∈
+(0,∞) due to (3.21). In a similar fashion we obtain, for all i,j∈ {1,...,N},
+i<j, and Ψ 1,Ψ2∈DN,
+/vextendsingle/vextendsingle/angbracketleftbig
+Ψ1/vextendsingle/vextendsingle[P+,(i,j)
+AwijP+,(i,j)
+A, Fn
+ε]Ψ2/angbracketrightbig/vextendsingle/vextendsingle
+/lessorequalslant/vextenddouble/vextenddoubleF1−n
+εw1/2
+ij[Fn
+ε, P+,(i,j)
+A]ˇH−1/2
+f/vextenddouble/vextenddouble/ba∇dblˇH1/2
+fΨ1/ba∇dbl/vextenddouble/vextenddoubleFn−1
+εw1/2
+ijP+,(i,j)
+AΨ2/vextenddouble/vextenddouble
++/vextenddouble/vextenddoublew1/2
+ijP+,(i,j)
+AΨ1/vextenddouble/vextenddouble/vextenddouble/vextenddoublew1/2
+ij[P+,(i,j)
+A, Fn
+ε]F1−n
+εˇH−1/2
+f/vextenddouble/vextenddouble/ba∇dblˇH1/2
+fFn−1
+εΨ2/ba∇dbl. (6.13)
+Here we can further estimate
+/vextenddouble/vextenddoublew1/2
+ijFn−1
+εP+,(i,j)
+AΨ2/vextenddouble/vextenddouble/lessorequalslant/vextenddouble/vextenddoublew1/2
+ijP+,(i,j)
+AFn−1
+εΨ2/vextenddouble/vextenddouble
++/vextenddouble/vextenddoublew1/2
+ij[Fn−1
+ε, P+,(i,j)
+A]ˇH−1/2
+fF1−n
+ε/vextenddouble/vextenddouble/ba∇dblˇH1/2
+fFn−1
+εΨ2/ba∇dbl. (6.14)
+Lemma 6.2 below ensures that all operator norms in (6.13) and (6.14) that
+involvew1/2
+ijare bounded uniformly in ε >0 by constants depending only on
+e,n,d1, andd5+n. Furthermore, it is now clear how to treat the terms involving
+viorEnp. (In order to treat vijust replace P+,(i,j)
+AbyP+,(i)
+A,wijbyvi, andw1/2
+ij
+by|vi|1/2in (6.13) and (6.14).) Combining (6.9)–(6.14) and their analogues for
+23the remaining operators in Wwe arrive at
+/vextendsingle/vextendsingle/angbracketleftbig
+Ψ1/vextendsingle/vextendsingle[W, Fn
+ε]Ψ2/angbracketrightbig/vextendsingle/vextendsingle
+/lessorequalslantC/summationdisplay
+♯∈{+,⊥}/braceleftbig/angbracketleftbig
+Ψ1/vextendsingle/vextendsingleP♯
+A,NHfP♯
+A,NΨ1/angbracketrightbig
++/angbracketleftbig
+Fn−1
+εΨ2/vextendsingle/vextendsingleP♯
+A,NHfP♯
+A,NFn−1
+εΨ2/angbracketrightbig/bracerightbig
++CN/summationdisplay
+i,j=1
+i<j/braceleftbig/angbracketleftbig
+Ψ1/vextendsingle/vextendsingleP+,(i,j)
+AwijP+,(i,j)
+AΨ1/angbracketrightbig
++/angbracketleftbig
+Fn−1
+εΨ2/vextendsingle/vextendsingleP+,(i,j)
+AwijP+,(i,j)
+AFn−1
+εΨ2/angbracketrightbig/bracerightbig
++CN/summationdisplay
+i=1/braceleftbig/angbracketleftbig
+Ψ1/vextendsingle/vextendsingleP+,(i)
+A|vi|P+,(i)
+AΨ1/angbracketrightbig
++/angbracketleftbig
+Fn−1
+εΨ2/vextendsingle/vextendsingleP+,(i)
+A|vi|P+,(i)
+AFn−1
+εΨ2/angbracketrightbig/bracerightbig
++C(1+|Enp|)/braceleftbig
+/ba∇dblΨ1/ba∇dbl2+/ba∇dblFn−1
+εΨ2/ba∇dbl2/bracerightbig
+,
+for all Ψ 1,Ψ2∈DNand someε-independent C≡C(N,n,e,d 1,d5+n)∈(0,∞).
+Employing successively (3.19), which implies |vi|/lessorequalslant(πe2|Z|/2)(|D(i)
+A|+ˇHf),
+after that (3.21), which yields /ba∇dblˇH1/2
+fP+,(i)
+AΨ/ba∇dbl2/lessorequalslantC(d1,d4)(/ba∇dblˇH1/2
+fP+
+A,NΨ/ba∇dbl2+
+/ba∇dblˇH1/2
+fP⊥
+A,NΨ/ba∇dbl2), and finally (6.4) we conclude that Condition (c) is fulfilled
+withcn=C(N,n,Z,R,e,d−1,d1,d5+n)(1+|Enp|). /square
+Lemma 6.2. For alli,j∈ {1,...,N},i < j,n∈ /CI, andσ,τ/greaterorequalslant0with
+σ+τ/lessorequalslant1,
+sup
+ε>0/vextenddouble/vextenddoubleFσ−n
+εw1/2
+ij[Fn
+ε, P+,(i,j)
+A]ˇH−1/2
+fFτ
+ε/vextenddouble/vextenddouble
+= sup
+ε>0/vextenddouble/vextenddoublew1/2
+ijFσ
+ε[P+,(i,j)
+A, F−n
+ε]ˇH−1/2
+fFn+τ
+ε/vextenddouble/vextenddouble/lessorequalslanteC(n,d1,d5+n)<∞.
+Proof.We write
+w1/2
+ijFσ
+ε[P+,(i)
+AP+,(j)
+A, F−n
+ε]ˇH−1/2
+fFn+τ
+ε=Y1+w1/2
+ijY2+Y3,
+where
+Y1:={w1/2
+ijFσ
+ε[P+,(i)
+A, F−n
+ε]ˇH−1/2
+fFn+τ
+ε}{ˇH1/2
+fF−n−τ
+εP+,(j)
+AˇH−1/2
+fFn+τ
+ε},
+Y2:=P+,(i)
+AFσ
+ε[P+,(j)
+A, F−n
+ε]ˇH−1/2
+fFn+τ
+ε,
+Y3:=w1/2
+ij[Fσ
+ε, P+,(i)
+A][P+,(j)
+A, F−n
+ε]ˇH−1/2
+fFn+τ
+ε.
+Applying Corollary 3.4 we immediately see that /ba∇dblY1/ba∇dbl/lessorequalslanteC(n,d1,d5+n) and
+that
+/ba∇dblY3/ba∇dbl/lessorequalslant/vextenddouble/vextenddoublew1/2
+ij[Fσ
+ε, P+,(i)
+A]F−σ
+ε/vextenddouble/vextenddouble/vextenddouble/vextenddoubleFσ
+ε[P+,(j)
+A, F−n
+ε]Fn+τ
+ε/vextenddouble/vextenddouble/lessorequalslanteC(n,d1,d3+n)
+24uniformly in ε>0. Employing (3.19) (with respect to the variable xjfor each
+fixedxi) and using [ |D(j)
+A|1/2,P+,(i)
+A] = 0, we further get
+/vextenddouble/vextenddoublew1/2
+ijY2Ψ/vextenddouble/vextenddouble2/lessorequalslant(πe2/2)/ba∇dblP+,(i)
+A/ba∇dbl2/vextenddouble/vextenddouble|D(j)
+A|1/2Fσ
+ε[P+,(j)
+A, F−n
+ε]Fn+τ
+ε/vextenddouble/vextenddouble2/ba∇dblˇH−1/2
+f/ba∇dbl2
++(πe2/2)/vextenddouble/vextenddoubleˇH1/2
+fP+,(i)
+AˇH−1/2
+f/vextenddouble/vextenddouble2/vextenddouble/vextenddoubleˇH1/2
+fFσ
+ε[P+,(i)
+A, F−n
+ε]ˇH−1/2
+fFn+τ
+ε/vextenddouble/vextenddouble2.
+By Corollary 3.4 all norms on the right hand side are bounded uniformly in
+ε>0 by constants depending only on n,d1, andd4+n. /square
+Appendix A.Semi-boundedness of HVCsrandHVCnp
+In this appendix we verify that the semi-relativistic Pauli-Fierz and no -pair op-
+erators with Coulomb potential are semi-bounded below for all nucle ar charges
+less than the critical charges without radiation fields. We do not att empt to
+give good lower bounds on their spectra since this is not the topic add ressed
+in this paper. Our aim here is essentially only to ensure that these ope rators
+possess self-adjoint Friedrichs extensions. We recall that the st ability of matter
+of the second kind has been proven for the no-pair operator in [9] under cer-
+tain restrictions on the fine-structure constant, the ultra-viole t cut-off, and the
+nuclear charges. The stability of matter of the second kind is a much stronger
+property than mere semi-boundedness. It says that the operat or is bounded
+below by some constant which is proportional to the total number o f nuclei
+and electrons and uniform in the nuclear positions. The restrictions imposed
+on the physical parameters in [9] do, however, not allow for all atom ic numbers
+less thanZnp.
+First, we consider the semi-relativistic Pauli-Fierz operator. The fo llowing
+proposition is a simple generalization of the bound (3.19) proven in [10] to the
+case ofN∈ /C6electrons and K∈ /C6nuclei.
+Proposition A.1. Assume that ωandGfulfill Hypothesis 3.1 and let N,K∈/C6,e >0,Z= (Z1,...,Z K)∈(0,2/πe2]K, and R={R1,...,RK} ⊂ /CA3.
+Then
+(A.1)N/summationdisplay
+i=1|D(i)
+A|+VC+δHf/greaterorequalslant−C(δ,N,Z,R,d1)>−∞,
+for everyδ>0in the sense of quadratic forms on DN.
+Proof.In view of (3.19) we only have to explain how to localize the non-local
+kinetic energy terms. To begin with we recall the following bounds pro ven in
+[10, Lemmata 3.5 and 3.6]: For every χ∈C∞( /CA3
+x,[0,1]),
+/ba∇dbl[χ,SA]/ba∇dbl/lessorequalslant/ba∇dbl∇χ/ba∇dbl∞,/vextenddouble/vextenddoubleDA/bracketleftbig
+χ,[χ,SA]/bracketrightbig/vextenddouble/vextenddouble/lessorequalslant2/ba∇dbl∇χ/ba∇dbl2
+∞. (A.2)
+Now, let Br(z) denote the open ball of radius r >0 centered at z∈ /CA3in /CA3.
+We set̺:= min{|Rk−Rℓ|:k/\e}atio\slash=ℓ}/2 and pick a smooth partition of unity
+25on /CA3,{χk}K
+k=0, such that χk≡1 onB̺/2(Rk) and supp( χk)⊂ B̺(Rk), for
+k= 1,...,K, and such that/summationtextK
+k=0χ2
+k= 1. Then we have the following IMS
+type localization formula,
+(A.3) |DA|=K/summationdisplay
+k=0/braceleftBig
+χk|DA|χk+1
+2/bracketleftbig
+χk,[χk,|DA|]/bracketrightbig/bracerightBig
+onD, for everyi∈ {1,...,N}. A direct calculation shows that
+/bracketleftbig
+χk,[χk,|DA|]/bracketrightbig
+= 2iα·(∇χk)[χk,SA]+DA/bracketleftbig
+χk,[χk,SA]/bracketrightbig
+(A.4)
+onD. By virtue of (3.6) and (A.2) we thus get
+(A.5)/vextenddouble/vextenddouble/bracketleftbig
+χk,[χk,|DA|]/bracketrightbig/vextenddouble/vextenddouble/lessorequalslant4/ba∇dbl∇χ/ba∇dbl2
+∞,
+for allk∈ {0,...,K}. Since we are able to localize the kinetic energy terms
+and since, by the choice of the partition of unity, the functions /CA3∋x/ma√sto→
+|x−Rk|−1χ2
+ℓ(x) are bounded, for k∈ {1,...,K},ℓ∈ {0,...,K},k/\e}atio\slash=ℓ,
+the bound (A.1) is now an immediate consequence of (3.19) (with δreplaced
+byδ/N). (Here we also make use of the fact that the hypotheses on Gare
+translation invariant.) /square
+Next, we turn to the no-pair operator discussed in Section 6. The s emi-
+boundedness of the molecular N-electron no-pair operator is essentially a con-
+sequence of the following inequality [10, Equation (2.14)], valid for all ωand
+Gfulfilling Hypothesis 3.1, γ∈(0,2/(2/π+π/2)), andδ>0,
+(A.6) P+
+A(D(i)
+A−γ/|x|+δHf)P+
+A/greaterorequalslantP+
+A(c(γ)|D0|−C)P+
+A,
+in the sense of quadratic forms on P+
+AD. HereC≡C(δ,γ,d−1,d0,d1)∈(0,∞)
+andc(γ)∈(0,∞) depends only on γ.
+Proposition A.2. Assume that ωandGfulfill Hypothesis 3.1 and let N,K∈/C6,e>0,Z= (Z1,...,Z K)∈(0,Znp)K, and R={R1,...,RK} ⊂ /CA3, where
+Znpis defined in (6.3). Then the quadratic form associated with the operator
+/tildewideHnpdefined in (6.2)is semi-bounded below,
+/tildewideHnp/greaterorequalslant−C(N,K, Z,R,d−1,d1,d5)>−∞,
+in the sense of quadratic forms on DN.
+Proof.We again employ the parameter ̺ >0 and the partition of unity in-
+troduced in the paragraph succeeding (A.2). Thanks to [10, Lemma 3.4(ii)]
+we know that P+
+AmapsD(D0⊗Hν
+f) into itself, for every ν >0. The IMS
+localization formula thus yields
+P+,(i)
+AviP+,(i)
+A=K/summationdisplay
+k=0/braceleftBig
+χ(i)
+kP+,(i)
+AviP+,(i)
+Aχ(i)
+k+1
+2/bracketleftbig
+χ(i)
+k,[χ(i)
+k, P+,(i)
+AviP+,(i)
+A]/bracketrightbig/bracerightBig
+26onD(D0⊗ /BD), where a superscript ( i) indicates that χk=χ(i)
+kdepends on the
+variablexi. Usingvi/lessorequalslant0, we observe that
+/bracketleftbig
+χ(i)
+k,[χ(i)
+k, P+,(i)
+AviP+,(i)
+A]/bracketrightbig
+=−2[χ(i)
+k, P+,(i)
+A]vi[P+,(i)
+A, χ(i)
+k]+2Re/braceleftbig
+P+,(i)
+Avi/bracketleftbig
+χ(i)
+k,[χ(i)
+k, P+,(i)
+A]/bracketrightbig/bracerightbig
+/greaterorequalslant2Re/braceleftbig
+P+,(i)
+Avi/bracketleftbig
+χ(i)
+k,[χ(i)
+k, P+,(i)
+A]/bracketrightbig/bracerightbig
+.(A.7)
+We recall the following estimate proven in [10, Lemma 3.6], for every χ∈
+C∞( /CA3
+x,[0,1]),
+/vextenddouble/vextenddouble1
+|x|/bracketleftbig
+χ,[χ, P+
+A]/bracketrightbigˇH−1/2
+f/vextenddouble/vextenddouble/lessorequalslant83/2/ba∇dbl∇χ/ba∇dbl2
+∞,
+whereˇHf=Hf+EwithE/greaterorequalslant1∨(4d1)2. Together with (A.7) it implies
+/angbracketleftbig
+Ψ/vextendsingle/vextendsingle/bracketleftbig
+χ(i)
+k,[χ(i)
+k, P+,(i)
+AviP+,(i)
+A]/bracketrightbig
+Ψ/angbracketrightbig
+/greaterorequalslant−δ/a\}b∇acketle{tΨ|ˇHfΨ/a\}b∇acket∇i}ht−(83/ba∇dbl∇χk/ba∇dbl4
+∞/δ)/ba∇dblΨ/ba∇dbl2,
+for allk∈ {0,...,K},i∈ {1,...,N},δ >0, and Ψ ∈ D(D0⊗ /BD). Next, we
+pick cut-off functions, ζ1,...,ζ K∈C∞
+0( /CA3
+x,[0,1]), such that ζk= 1 in a neigh-
+borhood of Rkand supp(ζk)⊂ B̺/4(Rk), fork∈ {1,...,K}. By construction,
+supp(ζk)∩supp(χℓ) =∅, for allk∈ {1,...,K}andℓ∈ {0,...,K}withk/\e}atio\slash=ℓ.
+Denotingζk:= 1−ζkand using the superscript ( i) to indicate that ζk=ζ(i)
+kis
+a function of the variable xi, we obtain
+/angbracketleftbig
+Ψ/vextendsingle/vextendsingleχ(i)
+kP+,(i)
+AviP+,(i)
+Aχ(i)
+kΨ/angbracketrightbig
+=−/angbracketleftBig
+Ψ/vextendsingle/vextendsingle/vextendsingleχ(i)
+kP+,(i)
+Ae2Zk
+|xi−Rk|P+,(i)
+Aχ(i)
+kΨ/angbracketrightBig
+−K/summationdisplay
+ℓ=1
+ℓ/negationslash=k/angbracketleftBig
+Ψ/vextendsingle/vextendsingle/vextendsingleχ(i)
+kP+,(i)
+Ae2Zℓζ(i)
+ℓ
+|xi−Rℓ|P+,(i)
+Aχ(i)
+kΨ/angbracketrightBig
+(A.8)
+−K/summationdisplay
+ℓ=1
+ℓ/negationslash=k/angbracketleftBig
+Ψ/vextendsingle/vextendsingle/vextendsingleχ(i)
+kP+,(i)
+Ae2Zℓζ(i)
+ℓ
+|xi−Rℓ|P+,(i)
+Aχ(i)
+kΨ/angbracketrightBig
+, (A.9)
+for all Ψ ∈ D(D0⊗ /BD). The operators appearing in the scalar products in
+(A.9) are bounded by definition of ζℓ. Their norms depend only on Rsince
+e2Zℓ<1. Furthermore, by virtue of Lemma A.3 below the term in (A.8)
+is bounded from below by −δ/a\}b∇acketle{tΨ|HfΨ/a\}b∇acket∇i}ht −Cδ/ba∇dblΨ/ba∇dbl2, for allδ >0 and some
+Cδ≡Cδ(R,d1,d4)∈(0,∞); see (A.11).
+Taking all the previous remarks into account, using (A.3)–(A.5), wij/greaterorequalslant0,
+|D(i)
+A|/greaterorequalslantP+,(i)
+AD(i)
+AP+,(i)
+A, and writing
+Hf=1
+NN/summationdisplay
+i=1K/summationdisplay
+k=0χ(i)
+k(P+,(i)
+A+P−,(i)
+A)Hfχ(i)
+k,
+27we deduce that
+/tildewideHnp/greaterorequalslant(1−3δ)P+
+A,NHfP+
+A,N+ (1−3δ)P⊥
+A,NHfP⊥
+A,N
++/summationdisplay
+♯∈{+,⊥}K/summationdisplay
+k=0P♯
+A,N/braceleftBigN/summationdisplay
+i=1χ(i)
+kP+,(i)
+A/parenleftBig
+D(i)
+A−e2Zk
+|xi−Rk|+δ
+NHf/parenrightBig
+P+,(i)
+Aχ(i)
+k
++δ
+NN/summationdisplay
+i=1/parenleftBig
+χ(i)
+kP−,(i)
+AHfP−,(i)
+Aχ(i)
+k+/summationdisplay
+♭=±χ(i)
+kP♭,(i)
+A[P♭,(i)
+A,Hf]χ(i)
+k/parenrightBig/bracerightBig
+P♯
+A,N
+−const(N,R,d1,d4)
+onDN, for every δ >0. Thanks to Corollary 3.4 (with ε= 0) we know that
+[P♭,(i)
+A,Hf]ˇH−1/2
+fextends to an element of L(HN) whose norm is bounded by
+some constant depending only on d1andd5, whence
+δ
+NN/summationdisplay
+i=1K/summationdisplay
+k=0/angbracketleftbig
+χ(i)
+kP♯
+A,NΨ/vextendsingle/vextendsingleP♭,(i)
+A[P♭,(i)
+A,Hf]χ(i)
+kP♯
+A,NΨ/angbracketrightbig
+/greaterorequalslant−(δ/2)/ba∇dblˇH1/2
+fP♯
+A,NΨ/ba∇dbl2−(δ/2)/vextenddouble/vextenddouble[P♭,(i)
+A,Hf]ˇH−1/2
+f/vextenddouble/vextenddouble2/ba∇dblΨ/ba∇dbl2,
+for every Ψ ∈DN,♯∈ {+,⊥}, and♭=±. For a sufficiently small choice of
+δ >0, the assertion now follows from the semi-boundedness of P+,(i)
+A(D(i)
+A−
+e2Zk/|xi−Rk|+ (δ/N)Hf)P+,(i)
+Aensured by (A.6) and the condition Zk<
+Znp. /square
+Lemma A.3. Letζ∈C∞
+0( /CA3,[0,1]),χ∈C∞( /CA3,[0,1]), such that 0∈supp(ζ)
+andsupp(ζ)∩supp(χ) =∅. SetˇHf:=Hf+E, whereE/greaterorequalslantk1∨d2
+1. Then
+/vextenddouble/vextenddoubleDAH1/2
+fζP+
+AχˇH−1/2
+f/vextenddouble/vextenddouble/lessorequalslantC(ζ,χ,d 1,d4), (A.10)
+/vextenddouble/vextenddoubleζ
+|x|P+
+AχˇH−1/2
+f/vextenddouble/vextenddouble/lessorequalslantC′(ζ,χ,d 1,d4). (A.11)
+Proof.We pick some /tildewideχ∈C∞( /CA3,[0,1]) such that supp( /tildewideχ)∩supp(ζ) =∅and
+/tildewideχ≡1 on supp( ∇χ). Usingζχ= 0 =ζ/tildewideχwe infer that, for all ϕ,ψ∈D,
+/vextendsingle/vextendsingle/angbracketleftbig
+DAϕ/vextendsingle/vextendsingleH1/2
+fζP+
+AχˇH−1/2
+fψ/angbracketrightbig/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/angbracketleftbig
+DAϕ/vextendsingle/vextendsingleH1/2
+fζ[P+
+A, χ]ˇH−1/2
+fψ/angbracketrightbig/vextendsingle/vextendsingle
+/lessorequalslant/integraldisplay/CA/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig
+DAϕ/vextendsingle/vextendsingle/vextendsingleH1/2
+fζ[RA(iy),/tildewideχ]iα·∇χRA(iy)ˇH−1/2
+fψ/angbracketrightBig/vextendsingle/vextendsingle/vextendsingledy
+2π
+=/integraldisplay/CA/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig
+DAϕ/vextendsingle/vextendsingle/vextendsingleH1/2
+fζRA(iy)iα·∇/tildewideχRA(iy)iα·∇χRA(iy)ˇH−1/2
+fψ/angbracketrightBig/vextendsingle/vextendsingle/vextendsingledy
+2π
+=/integraldisplay/CA/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig
+ζDAϕ/vextendsingle/vextendsingle/vextendsingleRA(iy)Υ0,1/2(iy)iα·∇/tildewideχRA(iy)Υ0,1/2(iy)×
+×iα·∇χRA(iy)Υ0,1/2(iy)ψ/angbracketrightBig/vextendsingle/vextendsingle/vextendsingledy
+2π.
+28In the last step we repeatedly applied (3.16). Commuting ζandDAand using
+/ba∇dblDARA(iy)/ba∇dbl/lessorequalslant1,/ba∇dblRA(iy)/ba∇dbl2/lessorequalslant(1 +y2)−1, and the fact that /ba∇dblΥ0,1/2(iy)/ba∇dblis
+uniformly bounded in y∈ /CA, we readily deduce that
+/vextendsingle/vextendsingle/angbracketleftbig
+DAϕ/vextendsingle/vextendsingleH1/2
+fζP+
+AχˇH−1/2
+fψ/angbracketrightbig/vextendsingle/vextendsingle/lessorequalslantC(ζ,χ,/tildewideχ,d1,d4)/ba∇dblϕ/ba∇dbl/ba∇dblψ/ba∇dbl,
+which implies (A.10). The bound (A.11) follows from (A.10) and the inequ ality
+/ba∇dbl|x|−1ϕ/ba∇dbl2/lessorequalslant4/ba∇dblDAϕ/ba∇dbl2+4/ba∇dblˇH1/2
+fϕ/ba∇dbl, ϕ∈ D(D0⊗H1/2
+f),
+which is a simple consequence of standard arguments (see, e.g., [10, Equa-
+tion (4.7)]). /square
+Acknowledgement. It is a pleasure to thank Martin K¨ onenberg and Edgardo
+Stockmeyer for interesting discussions and helpful remarks.
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+[6] Tosio Kato. Perturbation theory for linear operators . Classics in Mathematics. Springer-
+Verlag, Berlin, 1995. Reprint of the 1980 edition.
+[7] Martin K¨ onenberg, Oliver Matte, and Edgardo Stockmeyer. Ex istence of ground states
+of hydrogen-like atoms in relativistic quantum electrodynamics I: Th e semi-relativistic
+Pauli-Fierz operator. Preprint, arXiv:0912.4223, 2009.
+[8] Martin K¨ onenberg, Oliver Matte, and Edgardo Stockmeyer. Ex istence of ground states
+of hydrogen-like atoms in relativistic quantum electrodynamics II: T he no-pair operator.
+In preparation .
+[9] Elliott H. Lieb and Michael Loss. Stability of a model of relativistic qu antum electrody-
+namics.Comm. Math. Phys. ,228: 561–588, 2002.
+[10] Oliver Matte and Edgardo Stockmeyer. Exponential localization of hydrogen-like
+atoms in relativistic quantum electrodynamics. Comm. Math. Phys. , Online First,
+doi:10.1007/s00220-009-0946-6, 2009.
+[11] Lon Rosen.The ( φ2n)2quantum field theory: higherorderestimates. Comm. Pure Appl.
+Math.24: 417–457, 1971.
+29Oliver Matte Institut f ¨ur Mathematik, TU Clausthal, Erzstraße 1, D-
+38678 Clausthal-Zellerfeld, Germany, On leave from: Mathematisches Institut,
+Ludwig-Maximilians-Universit ¨at, Theresienstraße 39, D-80333 M ¨unchen, Ger-
+many.
+E-mail address :matte@math.lmu.de
+30
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+arXiv:1001.0145v3  [astro-ph.SR]  17 Nov 2010Accepted for publication in the Astrophysical Journal
+Preprint typeset using L ATEX style emulateapj v. 08/22/09
+PRODUCTION OF THE p-PROCESS NUCLEI IN THE CARBON-DEFLAGRATION MODEL FOR TYPE I a
+SUPERNOVAE
+Motohiko Kusakabe1, Nobuyuki Iwamoto2, and Ken’ichi Nomoto3
+Accepted for publication in the Astrophysical Journal
+ABSTRACT
+We calculate nucleosynthesis of proton-rich isotopes in the carbon -deflagration model for Type Ia
+supernovae (SNe Ia). The seed abundances are obtained by calcu lating the s-process nucleosynthesis
+that is expected to occur in the repeating helium shell flashes on the carbon-oxygen (CO) white dwarf
+(WD) during mass accretion from a binary companion. When the defla gration wave passes through
+the outer layer of the CO WD, p-nuclei are produced by photodisintegration reactions on s-nuclei in
+a region with the peak temperature ranging from 1.9 to 3.6 ×109K. We confirm the sensitivity of the
+p-processon the initial distribution of s-nuclei. We showthat the initial C/O ratioin the WD does not
+much affect the yield of p-nuclei. On the other hand, the abundance of22Ne left after the s-processing
+has a large influence on the p-process via22Ne(α,n) reaction. We find that about 50% of p-nuclides
+are co-produced when normalized to their solar abundances in all ad opted cases of seed distribution.
+Mo and Ru, which are largely underproduced in Type II supernovae ( SNe II), are produced more than
+in SNe II although they are underproduced with respect to the yield levels of other p-nuclides. The
+ratios between p-nuclei and iron in the ejecta are larger than the solar ratios by a fa ctor of 1.2. We
+also compare the yields of oxygen, iron, and p-nuclides in SNe Ia and SNe II, and suggest that SNe Ia
+could make a larger contribution than SNe II to the solar system con tent ofp-nuclei.
+Subject headings: Galaxy: abundances – nuclear reactions, nucleosynthesis, abund ances – supernovae:
+general
+1.INTRODUCTION
+Stable nuclides of atomic number Z≥34 located
+at the neutron deficient side of the β-stability line
+are classified as p-nuclei (e.g. Lambert 1992; Meyer
+1994; Arnould & Goriely 2003). They consist of 35
+nuclides. The production process of these p-nuclei is
+commonly called the p-process. The process through
+the photodisintegration reactions on the pre-existing
+heavy nuclides is, especially, referred to as the γ-
+process (Woosley & Howard 1978; Howard et al. 1991).
+These nuclides are observed only in the solar system,
+since the abundances are very small, typically 1% or less
+in the isotopes of element (by number). Some primitive
+meteorites which were not in equilibrium with the bulk
+of the solar system materials are also found to contain
+very poor p-nuclei (Anders & Grevesse 1989).
+Althoughnucleosyntheticprocessestoproduce p-nuclei
+have been studied for many astrophysical sites, the core-
+collapse supernova (SN) of massive stars with an H-rich
+envelope, i.e., a Type II supernova (SN II), has been con-
+sidered as a plausible site (e.g., Woosley & Howard1978;
+Rayet et al.1990;Prantzos et al.1990;Rayet et al.1995;
+Costa et al. 2000; Rauscher et al. 2002; Iwamoto et al.
+2005; Hayakawa et al. 2004, 2006, 2008). During the SN
+explosion, the γ-process plays an important role in pro-
+ducingp-nuclei in the O- and Ne-rich layers.
+1Institute for Cosmic Ray Research, University of Tokyo,
+Kashiwa, Chiba 277-8582, Japan
+kusakabe@icrr.u-tokyo.ac.jp
+2Nuclear Data Center, Japan Atomic Energy Agency, Tokai,
+Ibaraki 319-1195, Japan
+iwamoto.nobuyuki@jaea.go.jp
+3Institute for the Physics and Mathematics of the Universe, U ni-
+versity of Tokyo, Kashiwa, Chiba 277-8568, Japan
+nomoto@astron.s.u-tokyo.ac.jpMany studies of SNe II have shown that about a half
+ofp-nuclides are reproduced in the proportion of the so-
+larp-abundance. However, there remain two unresolved
+problems. First,92,94Mo,96,98Ru,115Sn, and138La are
+largelyunderproduced compared with the distribution of
+the solar p-abundances. Second, the contribution of p-
+nuclei from SNe II to their galactic evolution has been
+foundtobe smallerthan thatof16O,themain productof
+SNe II (Prantzos et al. 1990; Rayet et al. 1995). These
+results led them to conclude that SNe II could not be
+responsible for the whole content of solar p-nuclei.
+Thep-process nucleosynthesis in a supercritical ac-
+cretion disk around a compact object (Fujimoto et al.
+2003) and in a jet-like explosion (Nishimura et al. 2006)
+has been studied. Hoffman et al. (1996) suggested that
+nucleosynthesis in the neutrino-driven wind following
+the delayed explosion can produce light p-nuclei, i.e.,
+74Se,78Kr,84Sr, and92Mo. Recently, neutrinos emit-
+ted from the collapsed core has been found to largely
+contribute to the production of92,94Mo,96,98Ru, and
+other light p-nuclei (Pruet et al. 2005; Fr¨ ohlich et al.
+2006a,b; Pruet et al. 2006; Wanajo 2006). This contri-
+bution might resolvethe underproduction of92,94Mo and
+96,98Ru in the O- and Ne-rich layers in SNe II.
+Costa et al. (2000) have shown that the second prob-
+lem of underproduction of p-nuclei could be solved if
+the22Ne(α,n)25Mg reaction rate was larger by a fac-
+tor of∼10–50 in the temperature range of the s-process
+during core He burning. However, the uncertainty they
+expected is too large and now questioned (Jaeger et al.
+2001; Koehler 2002; Karakas et al. 2006). On the other
+hand, the second problem might be reconciled by con-
+sidering the contribution of more energetic SNe (hy-
+pernovae) to the production of p-nuclei (Iwamoto et al.2
+2005; Hayakawa et al. 2008).
+It is less likely that uncertainties of the nuclear reac-
+tion rates such as ( γ,n), (γ,α), and (γ,p) photodisinte-
+grations (and their inverse reactions) for nuclei heavier
+than iron give a significant impact on a p-process yield
+as seen from the sensitivity of the p-nuclei yields on re-
+action rates (Rapp et al. 2006). Dillmann et al. (2008)
+have performed p-process calculations for the same SN II
+model as in Rayet et al. (1995) by utilizing most recent
+stellar (n,γ) cross sections. They found that overpro-
+duction factors for almost all p-nuclides decreased by, on
+average, 7% and that the largest deviation is the reduc-
+tion in156Dy by 39.2% in comparison with the result
+in Rapp et al. (2006).
+Thep-process has also been suggested to occur in the
+outermost layer of the exploding carbon-oxygen (CO)
+white dwarf (WD) which is presumed to be a Type Ia
+supernova (SN Ia; Howard et al. 1991; Howard & Meyer
+1992; Goriely et al. 2002; Kusakabe et al. 2005;
+Goriely et al. 2005; Arnould & Goriely 2006).
+Howard et al. (1991) calculated the p-process in a
+simple parametric model. In this model, it is assumed
+that the s-process produces seed nuclei with the mass
+number higher than 90 prior to the explosion. The high
+density environment during the passage of the shock
+wave makes proton capture reactions efficient in produc-
+ing light p-nuclides, while the γ-process makes heavier
+p-nuclides. They reproduced the solar-like distribution
+ofp-nuclei. Howard & Meyer (1992) calculated the p-
+process nucleosynthesis in the delayed-detonation (DD)
+model and obtained an abundance pattern where the
+abundances of lighter p-nuclei are relatively large. When
+the solar abundance is used for the initial composition,
+the result implies that SN Ia has very small contribution
+to the galactic content of p-nuclei. However, when the
+s-process is assumed to occur as in the solar metallicity
+asymptotic giant branch (AGB) star, a sufficient yield of
+p-nuclei is obtained. In this case, the underproductions
+of Mo and Ru are reduced, compared to those in SNe II.
+Those authors concluded that contribution of SN Ia on
+Galactic p-nuclei was still uncertain. It is, therefore,
+worth further studies.
+Goriely et al. (2002) studied the p-process nucleosyn-
+thesis in the one-dimensional He-detonation model for a
+sub-Chandrasekhar mass CO WD, where the p-process
+would proceed in the accreting He layer. For the initial
+seed with solar abundances in accreting He-rich materi-
+als, the overproductionofCa-to-Fenuclei hasbeen found
+to be a factor of ∼100 with respect to p-nuclei. This re-
+sult means that the He detonating sub-Chandrasekhar
+mass CO WD model is not an efficient site for the syn-
+thesis of p-nuclei. Meanwhile, they have shown that
+by increasing the abundances of the initial heavy seeds
+with the solar composition by a factor of 100 the p-
+nuclei yields are comparable to those of Ca-to-Fe nu-
+clides. The resulting yields of p-nuclei in their one-
+dimensional model have been confirmed to be very sim-
+ilar to those calculated by using a three-dimensional ex-
+plosion model (Goriely et al. 2005).
+In light of the theoretical underproduction of92,94Mo
+and96,98Ru, a precise measurement of terrestrial abun-
+dances of those isotopes is needed. Recently, de Laeter
+(2008) has shown that the abundances of p-nuclei of Moand Ru were, on average, 4.3% lower than presently
+known ones. Unfortunately, the large underproduction
+problem of the Mo and Ru isotopes still remains.
+In this paper, we have adopted a carbon-deflagration
+model(W7) forSNIa(Nomoto et al.1984) and analyzed
+thep-process nucleosynthesis. The used trajectories of
+temperatureanddensityinexplodingWDarelargelydif-
+ferent from those in the DD model by Howard & Meyer
+(1992) and in the parametric model by Howard et al.
+(1991). It is, thus, important to investigate the C-
+deflagration model as one of the possible sites for the
+p-process. The structure of this paper is as follows. The
+adopted SN model, nuclear reaction network, and initial
+compositions are described in Section 2. The results for
+some cases of initial compositions are analyzed in Sec-
+tion 3. Finally, we present conclusions in Section 4.
+2.INPUT PHYSICS
+2.1.The Supernova Model
+The adopted SN model is W7 in Nomoto et al. (1984).
+The accreting WD with the initial mass of 1 .0M⊙
+has been cooled down for 5 .8×108yr before the on-
+set of mass accretion. This WD has the composition of
+X(12C) = 0.475,X(16O) = 0.5, andX(22Ne) = 0.025.
+The WD mass increases at the accretion rate of ˙M=
+4×10−8M⊙yr−1. When the WD mass approaches
+MWD= 1.378M⊙, carbon burning is ignited at the
+center. This forms a C-deflagration wave which prop-
+agates outward. The released nuclear energy of about
+1051erg exceeds the binding energy of the WD so that
+the whole star is exploded. We use the time variations
+(trajectories) of temperature and density in the explod-
+ing layers (Nomoto et al. 1984) in order to calculate the
+production of p-nuclei.
+We calculate the p-process nucleosynthesis in the
+heated layers where a peak temperature Tmhas the
+range of Tm,9= 1.86–3.60 (in units of 109K). The
+corresponding peak densities are ρm= 1.28×107to
+2.24×107g cm−3. These layers are located at mass
+coordinates of 1 .143< Mr/M⊙<1.280 and undergo
+explosive carbon and neon burning at a passage of the
+deflagration wave. The representative temperature and
+density trajectories are shown in Figures 1(a) and 1(b),
+respectively. Itshouldbenotedthattheyieldsof p-nuclei
+are very sensitive to the temperature and density tra-
+jectories of exploding WD. The DD and C-deflagration
+(W7) models show completely different characteristics of
+trajectories. In the W7 model, the decreasing timescale
+of temperature in regions of T/greaterorsimilar2×109K relevant to
+thep-process is 0 .15–0.3 s, depending on mass shells.
+This timescale is ∼1.5–2 times longer than in the DD
+model by Howard & Meyer (1992) and 2–4 times shorter
+than in the parametric model with the assumption of
+e-folding time (0.6 s) by Howard et al. (1991). During
+thep-process nucleosynthesis, the density in W7 model
+is higher than in the DD model and in the assumption
+in Howard et al. (1991). The difference in density is also
+importantandaffectsthe yieldsoflight p-nucleiresulting
+from proton capture reactions.
+2.2.Nuclear Reaction Network and Initial Composition
+Thep-process nucleosynthesis is calculated by using
+the nuclear reaction network, in which 2565 nuclei from3
+neutron and proton to Polonium ( Z= 84) are combined
+with neutron, proton, α-induced reactions, and their in-
+verses. We use the nuclear reaction rates based on the
+experiments and the Hauser-Feshbach statistical model,
+NON-SMOKER (Rauscher & Thielemann 2000). The
+theoretical and experimental β-decay rates are adopted
+from the REACLIB database (F.-K. Thielemann 1995,
+private communication), which are supplemented by the
+theoretical rates of M¨ oller et al. (1997). We take the so-
+lar abundances from Anders & Grevesse (1989) to calcu-
+late the overproduction factor, X/X⊙, of the produced
+p-nuclei, where Xis the mass fraction.
+The seed abundances are important to consider the
+nucleosynthesis by the p-process in SNe Ia. In the W7
+model, materials with solar compositions are accreted
+onto the surface of the CO WD. The accreted H-rich
+materials are transformed into He through the CNO cy-
+cle of H-burning. The main product of the CNO cycle,
+14N, is left in the He-rich region and is processed to22Ne
+through the αcapture reactions. The main nuclei in the
+CO WD result in12C,16O, and22Ne.
+The accretion rate for the W7 model is high enough
+to avoid the occurrence of an He detonation in the sub-
+Chandrasekhar mass stage (Nomoto 1982a,b). Instead,
+He burning is ignited in a very thin shell at the base of
+the He-rich layer where electrons are partially degener-
+ate. The He shell burning becomes thermally unstable
+in these conditions, which are similar to those in the
+He-burning shell of an AGB star. This results in the re-
+peated occurrence of thermal runawayof He burning (He
+shell flash).
+For enhanced p-process to occur in W7 model, s-
+process must occur when the WD mass is 1.143–1.280
+M⊙. For such high WD mass (as well as the high-mass
+C+O core of AGB stars), the temperature of the He-
+burning layer is high enough for the22Ne(α,n)25Mg re-
+action to take place (e.g. Fujimoto 1977; Truran & Iben
+1977; Straniero et al. 2000). There is an important dif-
+ference between the accreting WD and the AGB star,
+i.e., the mass of the H-rich layer in the accreting WD is
+much smaller than the AGB star and thus the entropy
+of the H-layer is much smaller in the WD. Therefore,
+the mixing of H-rich material into the He layer is more
+easily to occur (Sugimoto & Fujimoto 1978). Therefore,
+it is naturally expected that the s-process nucleosyn-
+thesis proceeds through the neutron source reactions of
+22Ne(α,n)25Mgaswellasof13C(α,n)16Oandfinallycre-
+ates a large amount of heavy nuclei.
+The efficiency of the s-process in the accreting CO
+WD remains uncertain, and thus, we calculate the initial
+seed abundances for the p-process by using the canonical
+s-process model (Howard et al. 1986; Aoki et al. 2003;
+Terada et al.2006). Thenuclearreactionnetworkforthe
+s-process includes 602 nuclei connected with the neutron
+capture reactions, whose rates are taken from Bao et al.
+(2000) and Rauscher & Thielemann (2000), and the β-
+decays, whoseratesareadoptedfromTakahashi & Yokoi
+(1987) and REACLIB. Nuclei above32S are included in
+the seed composition since the s-process nucleosynthesis
+is dominant. In this investigation, we assume two sets
+of initial seed abundances with mean neutron exposure
+τ0= 0.15 (Case A) and 0.33 mb−1(Case B) for the
+p-process. Figure 2 shows the seed abundance distribu-tions normalized to the solar abundances, and Tables 1
+and 2 show the seed abundances (by mass) in A and B,
+respectively. The seed abundance of B is fitted to the so-
+lars-only nuclei, and thus, the distribution is similar to
+the solar s-distribution. The abundances of B are higher
+than those of A because the larger production efficiency
+is needed to get the constant s-distribution normalized
+to solar.
+The He-exhausted core is composed of mainly12C
+and16O. The C/O abundance ratio (by mass) is as-
+sumed to be 0.95 (Case A1 in Table 3) in the W7
+model. However, the C/O ratio should be changed by,
+e.g., the adopted reaction rate of12C(α,γ)16O which
+has a large uncertainty in the current experiments at
+a low energy (e.g., Makii et al. 2007). We investigate
+the influence of the different C/O ratio on the p-process
+nucleosynthesis by changing the C/O ratio from 0.56
+(Case A2) to 2.55 (Case A3). The22Ne abundance left
+after the s-processing is also uncertain. The high abun-
+dance of22Ne may affect the p-process flows by the pro-
+duction ofneutrons through22Ne(α,n)25Mg reaction. In
+order to investigate the influence, we calculate the case
+in which no22Ne abundance is left after the He shell
+flashes as an extreme case (Case A4). The12C,16O, and
+22Ne abundances for the adopted cases are summarized
+in Table 3.
+3.RESULTS
+3.1.Case A1
+Here, we show the result of Case A1, which is consid-
+ered as the standard case in this investigation.
+3.1.1.Abundance Variations of Light Particles, p,n,α
+Figure3representstheabundancevariationsofproton,
+neutron,4He,12C,16O, and22Ne in each trajectory as
+a function of time after the onset of explosion. In layer
+1 where the peak temperature attains to Tm,9= 3.6,
+16O+16O fusion reaction starts to destroy16O as the
+temperature increases rapidly. Layers 2–4 are the most
+interesting region as the main site of p-process, where
+explosive C and Ne burning successively occurs.20Ne
+first produced is photodisintegrated, and thus,16O is
+left (Figure 3(e)). The explosive C-burning partially op-
+erates in layer 5, but the layer is not important for the
+p-process.22Ne is burnt by ( α,γ) and (α,n) reactions
+in the layers 1–4. The peak abundances of α-particle are
+Xα∼10−5in all layers, which are larger than those in
+thep-process in SNe II (e.g., Prantzos et al. 1990). In
+the SN II model of Prantzos et al. (1990), Xα∼10−5
+is realized by the efficient photodisintegration of20Ne
+atT9∼3, but it decreases to Xα∼10−7as the peak
+temperatures decrease. In the SNe Ia, αparticle is also
+produced by the12C(12C,α)20Ne(γ,α) in spite of no
+initial20Ne abundance. The peak proton and neutron
+abundances are Xp∼10−7andXn∼10−10, respec-
+tively, which are also larger than those in SNe II.
+In the present model, neutron is produced mainly by
+the22Ne(α,n) reaction. The peak neutron number den-
+sity isNn∼1022–1023cm−3, which is almost equiva-
+lent to that in the He-detonation of sub-Chandrasekhar-
+mass model (Goriely et al. 2002). On the other hand,
+the peak proton mass fraction is about five orders of4
+magnitude smaller than that in the He-detonation model
+(Xp∼6×10−3in the He-rich layer with Tm,9/greaterorsimilar3). In
+the He-detonation model, the plenty of α-particles are
+present in the region where the p-processoccurs and lead
+to the production of α-elements such as40Ca and44Ti.
+Further radiative α-captures bring the nuclear flow to
+the proton-rich side of the β-stability line, and then are
+followed by ( α,p) reactions. This results in the very high
+proton mass fraction. In the C-deflagration model, on
+the contrary, the overproduction of proton does not take
+place, since the p-process occurs in outer CO-rich layers
+as explained below.
+3.1.2.Nuclear Flow
+The general production trends of various p-nuclei are
+understood through the analysis of nuclear flows. First,
+we see a nuclear flow in the layers with a high peak tem-
+perature. After a passage of the deflagration wave the
+high neutron density produced brings almost all the seed
+nuclei to the neutron-rich side through ( n,γ) reactions.
+In this C-deflagration model, the abundances move to
+more neutron-rich region than in SNe II.4
+(1) When the temperature exceeds T9∼2, (γ,n) reac-
+tions bring back the nuclear flow to the β-stability line
+and further to the neutron-deficient region. There is a
+little flow which results in a leakage to lower Zthrough
+(γ,p) and (γ,α) photodisintegrations.
+(2) Furthermore, when the temperature exceeds T9∼
+3, (γ,p) and (γ,α) reactions become dominant, which
+drives nuclear materials from the neutron-deficient, high
+Aregiondowntowardtheiron-peakelementregion. This
+flow to the iron-peak elements is terminated, since the
+photodisintegrationsfreeze out as the temperature of the
+heated layers decreases.
+(3) In the layer with Tm,9≃2.6, the (γ,n) reactions
+are predominant to the seed nuclei with neutron number
+N >82 and create the production peak for most of the
+p-nuclei with N >82. This result is seen for196Hg in
+Figure 4.180Ta, which is one of the rare nuclei in nature,
+is produced directly through the photodisintegration of
+181Ta in Figure 4. We note that the production of180Ta
+might become efficient even in layers with lower peak
+temperatures ( Tm,9= 1.8–2.6; Prantzos et al. 1990).
+Unfortunately, we cannot investigate the p-process nu-
+cleosynthesis in such layers.5
+(4) In a somewhat higher peak temperature region
+(Tm,9∼2.7–2.9), (γ,n) reactions drive the nuclear flow
+to a sufficiently neutron-deficient region, and then, ( γ,p)
+and (γ,α) reactions get effective to destroy seed nuclei
+withN >82. Large amounts of heavy p-nuclei with
+N >82 are produced in this peak temperature range
+fromβ+-decays of unstable isobars after freezing-out of
+the nuclear reactions. For example, the second produc-
+tion peak is found for196Hg in Figure 4. The production
+of intermediate-mass p-nuclei with 50 < N < 82 (es-
+4We checked the p-process calculation in the following approx-
+imate model of SNe II. The temperature and density are given b y
+T(t) =Tmexp(−t/3τex) andρ(t) =ρmexp(−t/τex), respectively,
+whereτexis the expansion timescale taken to be 0.446 s and 1 s,
+andρm= 106g cm−3. This setup is the same as in Rayet et al.
+(1990).
+5A mesh zoning in the outer layer of the W7 model was sparse.
+This is because this region is not so important for main nucle osyn-
+thesis in SN Ia.pecially,113In,115Sn,120Te,126Xe,132Ba,138La, and
+136,138Ce) already proceeds by ( γ,n) photodisintegra-
+tions in this region.
+(5) In the layers where the peak temperatures reach
+Tm,9∼2.9–3.2, most of the intermediate-mass seed
+nuclei experience photodisintegrations. Then, the
+intermediate-mass p-nuclei are abundantly produced
+mainly through the β+-decays of unstable neutron-
+deficient isobars.
+(6) In the layers with Tm,9= 3.2–3.392,94Mo,96,98Ru
+and102Pd show production peaks, but the abundances
+rapidly decrease when the peak temperature exceeds
+Tm,9= 3.3. Finally, all the N >50 seed nuclei
+are photodisintegrated to the iron-peak elements. Only
+N <50p-nuclides (74Se,78Kr, and84Sr) are produced
+in the layers with Tm,9= 3.3–3.6. In these hottest lay-
+ers, the16O(γ,α)12C reaction produces12C, and follow-
+ing12C+12C fusion reaction produces protons and α-
+particles which are identified as the second peak of their
+abundances in Figures 3(a) and 3(c). The synthesis of
+74Se,78Kr, and84Sr through proton captures, therefore,
+becomes effective. This results in the increasesin the rel-
+ative yields ofthe three p-nuclides and the slight increase
+in the mean average overproduction factor. However,
+we find that even at those high temperatures and even
+for the lightest three p-nuclei, the contribution of proton
+capture reactions to the production does not exceed that
+of photodisintegration.
+3.1.3.Yields of p-nuclei
+We calculate the overproduction factor F=X/X⊙
+which is the ratio between the produced abundances and
+the corresponding solar abundances for 35 p-nuclides in
+the 21 trajectories. The overproduction factors for five
+p-nuclides are plotted as a function of Tmin Figure 4.
+The selected nuclides are the same as those in Figure 3
+of Prantzos et al. (1990). It is seen from Figure 4 that
+each nucleus is produced in a narrow range of the peak
+temperature.
+There is one clear difference between SNe Ia and
+SNe II. The temperature ranges for productions of p-
+nucleiinSNe Iaareshifted tohotterlayersthan inSNe II
+(Prantzos et al. 1990). In addition, in the W7 model the
+density of the p-process site ( ∼107g cm−3) before the
+explosionis higher than in SNe II ( ∼105g cm−3). Thus,
+the number of particles (baryons) is larger in SNe Ia and
+the particle-induced reactions are more dominant than
+in SNe II. It is also the reason why neutron capture reac-
+tions, which become efficient at first in this calculation,
+drive the nuclear flow into the neutron-rich region far
+fromβ-stability line (Section 3.1.1.). The peak temper-
+ature, at which photodisintegration gets predominant, is
+thus higher in SNe Ia so that the peak temperature re-
+gion,inwhich p-nucleiarecreated,totallyshiftstohigher
+one than in SNe II.
+The overproduction factor averaged over the consid-
+eredp-process layers is calculated for a p-nucleus by the
+prescription
+/angbracketleftF/angbracketright=N/summationdisplay
+j=21
+2(Fj+Fj−1)Mj−Mj−1
+Mp,(1)
+whereFjis the overproduction factor in a trajectory5
+j,Mjis the Lagrangian mass coordinate of jth trajec-
+tory, and Mpis the total mass (0 .137M⊙) of the p-
+process layer. We define the mean overproduction factor
+asF0=/summationtext
+i/angbracketleftF/angbracketrighti/35, where iis a species of p-nucleus
+and the summation is taken for /angbracketleftF/angbracketrightvalues of 35 p-nuclei.
+Figure 5 shows the /angbracketleftF/angbracketrightvalues normalized to F0, in which
+the filled circles represent the result for Case A1. The
+numerical values for these quantities are listed in Table
+4.
+In Case A1, 19 of 35 p-nuclides are produced in
+amounts within a factor of three around the mean of
+F0= 4657. As mentioned above, the peak tempera-
+tures of the used trajectories do not involve the range
+of 1.9< T m,9<2.6. Hence, some degrees of increase in
+yields are expected in a C-deflagration model with finer
+mesh zoning for nuclei of which their production yields
+increase in the peak temperature region of Tm,9≤2.6
+(i.e.,138La,152Gd,156,158Dy,162,164Er,168Yb,174Hf,
+180Ta,184Os, and196Hg). Taking this fact into account,
+115Sn might be the only p-nuclide which is markedly un-
+derproduced.
+We comparethe resultofCaseA1with previousworks.
+The pattern of normalized average overproduction fac-
+tor/angbracketleftF/angbracketright/F0forp-nuclides in this calculation is very sim-
+ilar to that in the SNe II models (Prantzos et al. 1990;
+Rayet et al. 1995). For instance, the lightest three nu-
+clides (74Se,78Kr, and84Sr) are produced more than the
+underproducedMoandRuisotopes. Intheintermediate-
+massp-nuclides,113In and115Sn are also underabundant
+while the other nuclei show their nearly equal overpro-
+ductions. The most remarkable difference is that in the
+C-deflagrationmodel, severeunderproductionsoftheMo
+and Ru p-isotopes, which are the most puzzling prob-
+lem of the p-process in SNe II (Prantzos et al. 1990;
+Rayet et al. 1995), are reduced by a factor of ∼6–12.
+However, they still remain underproduced with respect
+to the mean production level for 35 p-nuclei.
+3.2.Effect of the Initial Composition of the WD Core
+on Yields of p-nuclides
+3.2.1.Effect of C/O Ratio
+We investigate the effect of the change in the abun-
+dance ratio between12C and16O on the p-process nucle-
+osynthesis, keeping the initial abundances of22Ne and
+s-nuclei (Case A) fixed. Figure 6 shows the normalized
+average overproduction factors calculated for Cases A2
+(stars), A1 (circles), and A3 (triangles) in the increasing
+order of12C/16O ratio. The mean value F0for each case
+is listed in the fifth column of Table 3. For the larger
+C/O ratio, the relative yields of74Se,78Kr, and84Sr are
+larger, and those of115Sn,138La,180Ta, and184Os are
+smaller, although all the differences are slight. There
+is little variation in the overproductions F0(Table 3).
+We, thus, conclude that the nucleosynthetic results of
+p-process are not so sensitive to the abundance ratio be-
+tween12C and16O if the ratio does not affect much the
+explosion timescale. It should be noted, however, that
+a variation of the C/O ratio could have an influence on
+the produced amount and pattern of p-nuclei by strongly
+affecting the explosion timescale.
+From a comparison of the nuclear flows in Cases A1–
+A3, we note the following results. In layer 5 of Fig-
+ure 3, partial C-burning triggers the production of pro-tons and α-particles. The abundance of α-particle is,
+therefore, higher when the C/O ratio is larger. Then,
+the22Ne+αreaction proceeds more efficiently, so that
+the22Ne(α,n)25Mg reaction supplies more neutrons for
+larger C/O ratio.
+3.2.2.Effect of22Ne Abundance
+Second, we investigate the effect of22Ne abundance on
+thep-process yields by comparing the result of Case A1
+with A4. In Case A4, the initial abundance of22Ne is
+assumed to be zeroin orderto examine the extreme case.
+We compare the abundances of light particles in A1
+with A4. The neutron abundances in A4 are 10–100
+times smaller than those in A1 in the whole range of
+p-process layer. This is because the neutron supply
+through22Ne(α,n)25Mg reactiondecreases due to no ini-
+tial abundance of22Ne. Therefore, the nuclear flow does
+not reach so neutron-rich region in A4 at an early phase
+of the deflagration wave passage, while a larger amount
+of neutrons in A1 drive it to a more neutron-rich region.
+Theratiosofthe /angbracketleftF/angbracketrightofeachp-nuclideinA1andA4are
+plotted in Figure 7. It is found that74Se,78Kr,84Sr, and
+92Mo are produced more abundantly in A4. Since the
+values of F0except for those four nuclei are almost the
+same(∼4930),their efficient productionsareresponsible
+for the increase of F0in A4.
+Next, we investigate the differences of nuclear flow be-
+tween A1 and A4. In the case of A1,74Se is created by
+75Se(γ,n) reaction in layer 1, but the ( γ,n), (γ,p), and
+(γ,α) reactions photodisintegrate it. Also in layers 2–4,
+the75Se(γ,n) reaction produces74Se, but supplemented
+by73As(p,γ) reaction with a small contribution. For the
+production of78Kr,79Kr(γ,n) reaction is important in
+layer1, but78Kr is photodisintegrated by ( γ,p) reaction.
+Inlayers2and3,78Krisproducedby79Kr(γ,n)reaction,
+togetherwith asmall contributionby77Br(p,γ)reaction.
+However, in layer 477Br(p,γ) is a main production re-
+action of78Kr. For84Sr,85Sr(γ,n) reaction produces
+it in layers 1–3, but in layer 1 ( γ,n) and (γ,p) reac-
+tions destruct84Sr. In layer 4,83Rb(p,γ) and85Sr(γ,n)
+reactions enhance the abundance of84Sr.92Mo is cre-
+ated by the nuclear flow from93Mo(γ,n),93Tc(γ,p), and
+96Ru(γ,α) reactions, but the produced92Mo is soon de-
+structed by ( γ,p) reaction. In layers 2–4,93Mo(γ,n) re-
+action dominates to produce92Mo.
+In contrast, the proton abundances in A4 are by a
+factor of ∼10 larger than those in A1. This implies
+that the larger amount of protons involves the produc-
+tionofthe four p-nuclei. Inthe caseofA4,73As(p,γ)and
+75Se(γ,n) reactions first create74Se in layer 1. After the
+peaktemperatureexceeds T9= 3, (γ,p), (γ,α)and(γ,n)
+photodisintegrationsovercometheproductionreactionof
+75Se(γ,n) and destruct74Se. In layers 2–4, the contribu-
+tion of73As(p,γ) reactionto the production of74Se dom-
+inates that of75Se(γ,n) and75Br(γ,p) reactions. The
+production of78Kr comes from79Kr(γ,n) reaction, to-
+gether with minor contribution of77Br(p,γ) reaction in
+layer 1, but the produced78Kr is destructed by ( γ,p)
+reaction. In layers 2–4, the77Br(p,γ) is the dominant
+reaction to produce78Kr, compared to79Kr(γ,n) reac-
+tion. For84Sr, the production and destruction processes6
+in layer 1 of A4 are almost the same as in A1. However,
+in layers 2–483Rb(p,γ) reaction has also some contribu-
+tions to the production of84Sr, together with85Sr(γ,n)
+reaction. For92Mo, the production and destruction pro-
+cesses in layer 1 of A4 are the same as in A1. In layers
+2–4,91Nb(p,γ) reaction is important to produce92Mo
+and there is a small contribution of93Mo(γ,n). This
+result is largely different from the case of A1. Thus,
+proton capture reactions have an important role in the
+production of lightest four p-nuclides. Figure 8 shows
+the overproduction factor of74Se as a function of peak
+temperature for A1 and A4. The important contribution
+of proton captures to the production of74Se is found in
+the layer with the peak temperature T9≤3.3. The pro-
+duction region of almost all p-nuclides extends to cooler
+layers in A4 than in A1.
+Figure 7 reveals that180Ta and184Os are underpro-
+duced in A4, compared to A1. The abundance peak
+of these two p-nuclei in A1 is found to be in the layer
+withTm,9= 2.6.6In the layer,180Ta is first cre-
+ated by181Ta(γ,n) reaction, which is later destructed
+by180Ta(γ,n) reaction as the temperature increases to
+its peak. The production and destruction timescales of
+180Ta depend on the inverse reactions. This is because
+the nuclear flow from180Ta to179Ta depends on the
+rates of180Ta(γ,n) and179Ta(n,γ) reactions. The reac-
+tion rate of photodisintegration is a function of tempera-
+ture only. However, the reaction rate of neutron capture
+is dependent on the temperature, density, and neutron
+abundance which is larger in A1 than in A4. Thus, the
+contribution of neutron capture reactions in A1 is more
+important than in A4, and the destruction timescale of
+180Ta in A1 is longer than in A4. These facts lead to a
+largerabundanceof180TainA1leftafterthetermination
+of nuclear processing.
+The situation for the production of184Os is simi-
+lar to that for180Ta. The production and destruction
+processes of184Os are the competition between photo-
+disintegrations on184Os and185Os and neutron cap-
+ture reactions on183Os and184Os. The nuclear flux of
+185Os(γ,n)184Os(γ,n)183Os in A1 is relatively small due
+to a larger amount of neutrons than in A4. In addition,
+183Os(n,γ) creates184Os in the later phase of explosion
+in A1. As a result, the final abundance of184Os is by a
+factor of 15 larger in A1.
+3.3.Effect of the Initial Abundances of s-nuclei on
+Yields of p-nuclei
+Here, we examine the impact of different seed distri-
+butions of Cases A (A1–A4) and B with the enhance-
+ment from the solar abundance of heavy nuclei. In this
+investigation, we compare A1 with B, which has similar
+abundances to A1 for12C,16O, and22Ne in the CO core.
+Figure 2 shows that the overabundances of s-nuclei in
+Case B are by factors of 10–100 larger than those in A.
+Especially, in B heavy seed s-nuclei with A >140 are
+about 100 times more abundant than in A. This leads to
+the increase of F0by a factor of 50 in B, relative to A.
+This increase is seen for each p-nuclide in Figure 5, in
+6The zoning of W7 model in the outermost layer is relatively
+coarse. If the finer zones were used, the abundance peak of180Ta
+and184Os would appear in the outer layer where Tm,9would be
+lower than 2.6.which the normalized /angbracketleftF/angbracketrightis presented for Cases A1 and
+B. The values of normalized /angbracketleftF/angbracketrightfor heavy p-nuclei with
+A >140, especially168Yb,174Hf,180W, and196Hg, are
+larger than those in A. In contrast, lighter p-nuclei have
+smaller values in B than those in A. In particular,74Se–
+98Ru shows significant reductions, which come from the
+decrease in relative abundances of seed nuclei near the
+p-nuclei, compared to heavy seed nuclei with A >140.
+The resulting distribution of p-nuclei shows important
+decreases in74Se–98Ru as seen in B, if the distribution of
+heavy seed s-nuclei is similar to that of the solar distri-
+bution. This result is analogous to that obtained by the
+p-process in SNe II, but with the even worse reduction
+of74Se–84Sr. Thus, we find that the distribution simi-
+lar to that of the solar p-nuclei is obtained by the seed
+distributionattributedtothe s-processunderanenviron-
+ment with metallicity close to solar (Gallino et al. 1998),
+although main neutron source might be22Ne(α,n) reac-
+tion.
+The presupernova abundance of s-nuclei in the WD
+thus has a large impact on the p-process nucleosynthesis
+and is very crucial for the resulting yields of p-nuclei.
+3.4.Contribution of C-deflagration SNe to the Galactic
+Yields of p-nuclei
+The biggest question concerning the p-process nucle-
+osynthesis is where the main production site is. In order
+to investigate this question, we estimate the contribu-
+tion of SN Ia to galactic chemical evolution of p-nuclei
+by comparing the ejected mass of56Fe, a main product
+of SN Ia, with those of p-nuclei in Case A1.
+Here, weintroducea net yield ofeachnuclideinSNeIa,
+defined as a difference between the mass of the nuclide
+returned to the interstellar space at the SN explosion
+and the mass engulfed into a star at its birth. The p-
+nuclei present at the star formation remain inside the p-
+processlayerbeforethe explosion, but they aredestroyed
+by photodisintegrations. In addition, we assume that
+thep-nuclei present in an accreted matter are burned by
+neutron capture during the s-process developing in the
+He intershell (see, e.g., Tables 1 and 2). Therefore, the
+p-nuclei ejected from the SN Ia explosion are only those
+ultimately produced in the p-process layer.
+Assuming that the mass of CO core in the WD is ap-
+proximatelyequal to that of the whole WD, the net yield
+forp-nuclideiis given by
+yi=Xi,⊙(/angbracketleftF/angbracketrightiMp−MWD), (2)
+whereMWD= 1.378M⊙is the mass of the WD
+(Nomoto et al. 1984) and Xi,⊙is the solar mass frac-
+tion ofp-nuclidei. The above prescription could also be
+applied to56Fe, so that the net yield of56Fe is
+y56Fe=X56Fe,⊙(M56Fe/X56Fe,⊙−MWD),(3)
+whereM56Feis the mass of56Fe ejected by an SN Ia and
+X56Fe,⊙= 1.17×10−3is the solar mass fraction of56Fe.
+By taking a ratio between these quantities normalized
+to the corresponding solar mass fraction for respective
+nuclides, we can evaluate how many SN Ia events con-
+tribute to the galactic chemical evolution of one nuclide.
+The calculated ratio of yields between56Fe and a repre-7
+sentative p-nucleus is as follows:
+56Fe/p≡y56Fe/X56Fe,⊙
+(1/35)/summationtext35
+i=1yi/Xi,⊙
+=M56Fe/X56Fe,⊙−MWD
+F0Mp−MWD(4)
+where the yield of the p-nucleus is derived by summing
+the net yields of each p-nuclide and averaging it over 35
+p-nuclei.
+The yield ratio (16O/p) between16O and the repre-
+sentative p-nucleus is also defined in the same way, in
+which the solar mass fraction of16O is assumed to be
+X16O,⊙= 9.59×10−3. In this C-deflagration model for
+an SN Ia, we obtain56Fe/p= 0.82 and16O/p= 0.023
+for Case A1 (Tsujimoto et al. 1995). These results indi-
+cate that p-nuclei are produced more than56Fe and16O
+when normalized to their solar abundances. We, there-
+fore, conclude that SNe Ia can account for the galac-
+tic content of p-nuclei assuming that the seed s-nuclei
+are produced efficiently up to the overproduction level of
+∼103in He-rich layer accreting onto the degenerate CO
+WD.
+3.5.Comparison with Previous p-process Calculations
+Goriely et al. (2002) calculated p-process nucleosyn-
+thesis in the sub-Chandrasekhar mass He-detonation
+model. The stable nuclei from Ca to Fe are overabun-
+dant with respect to p-nuclei by a factor of ∼100. They,
+therefore, concluded that the He-detonation model is not
+an efficient site for production of p-nuclei. Nevertheless,
+Goriely et al. claimed that if the initial abundances of
+s-nuclei are enhanced over their solar values by a factor
+of 100,p-nuclei are produced at the same level as Ca–
+Fe, while such enhancement of the s-nuclei is not trivial
+in the He-detonation model. On the other hand, in the
+present C-deflagration model the overproduction factors
+for ejected p-nuclides and stable nuclides lighter than Fe-
+group elements are plotted as a function of mass number
+in Figure 9 and are listed in Table 5. From this figure, it
+is found that FCa/suppressFe∼Fpis realized with some enhance-
+ments of heavy seed nuclei, which are a natural conse-
+quence from the accretion of H-rich matter onto a CO
+WDwithanappropriaterateofmassaccretion, although
+the enhancement level of heavy seed nuclei remains am-
+biguous. This means that the C-deflagration model is
+∼100 times more effective in enriching a galaxy with p-
+nuclei, if the assumed level of enhancement is valid.
+As mentioned in Section 1, the DD model for SNe Ia
+may also produce an important amount of p-nuclei if the
+prior enhancement of s-nuclei in a Chandrasekhar mass
+WDmodelistakenintoaccount(Howard & Meyer1992)
+as in our present assumptions. However, the enhance-
+ment ofs-nuclei is not trivial for the double degenerate
+scenario of SN Ia in a Chandrasekhar mass model.
+Arnould & Goriely (2003) have calculated the p-
+process nucleosynthesis in the W7 model for SNe Ia
+adopting two cases as an initial abundance of heavy seed
+nuclei. The resulting pattern of yields of p-nuclei in
+Arnould & Goriely (2003) under the assumption of the
+solar abundance for initial seed shows the following dif-
+ference from our Case A. Normalized abundances of p-
+nuclei for Mo–Ce in Arnould & Goriely (2003) are muchsmaller, but those for heavier p-nuclei are larger. This
+trend represents the solar abundance of p-nuclei them-
+selves. On the other hand, their result for the initial
+seed distribution representative of the s-process in AGB
+stars of the solar metallicity is similar to that of Case
+A. It is, however, found that there exists a difference of
+the pattern between our Case A and Arnould & Goriely
+(2003) for p-nuclei with N >82, while the pattern for p-
+nuclei with N≤82 is similar. This result represents that
+the abundance pattern for heavier p-nuclei is very sensi-
+tive to initial seed distributions. It should be noted that
+the differences in yields of113In,158Dy, and190Pt are
+larger than those within nuclear uncertainties evaluated
+by Arnould & Goriely (2003).
+Prantzos et al. (1990) studied the p-process in SNe II
+using the model for SN1987A, and obtained FO/Fp
+(≈16O/p)∼12, where FOstands for the overproduc-
+tion factor of oxygen. They claimed that the solar sys-
+temcontentof p-nucleidoesnotcomefromSN1987A-like
+SNe. One reason may be attributed to the abundances
+of seed nuclei which reflect low metallicity in the Large
+Magellanic Cloud. Rayet et al. (1995) derived the ratios
+of yields for oxygen and p-nuclei from star models with
+masses of 13, 15, 20, and 25 M⊙and averaged them over
+the initial mass function (IMF). They found that the
+resulting ratio was FO/Fp= 4.2. These results, thus,
+imply that p-nuclei are not sufficiently produced, com-
+pared to the main product in SNe II and suggest that
+the other production sites have an important contribu-
+tion to the galactic evolution of p-nuclei. Accordingly,
+the C-deflagration model for SNe Ia is one of the promis-
+ing sites for the p-process because sufficient amounts of
+p-nuclei are produced relative to16O within our assump-
+tions.
+Using the results of this study and previous works,
+we estimate how many SNe Ia and SNe II contribute
+to the galactic evolution of p-nuclei. We assume that
+all SNe Ia would typically experience the p-process as in
+Case A1. First, we adopt the ratios of overproduction
+factors56Fe/p= 0.82 for our SN Ia model and16O/p
+= 4.2 for SNe II (Rayet et al. 1995). We also take the
+ejected masses of16O and56Fe from the W7 model for
+SN Ia and the IMF-averaged masses of these nuclides
+from SNe II (Tsujimoto et al. 1995). These masses nor-
+malized to the corresponding solar abundances are sum-
+marized in Table 6. Values for p-nuclei are estimated
+fromtheyieldratiosbetween p-nucleiandthemainprod-
+ucts (56Fe for SN Ia and16O for SN II). In order to com-
+pare these yields between SN Ia and SN II, we adopt the
+ratio of the occurrence frequency of SNe Ia to that of
+SNe II,∼0.15 (Tsujimoto et al. 1995). From the val-
+ues in Table 7, we find that SNe Ia contribute about
+70% of the p-nuclei in the solar system. It should be
+noted that there is still a small underproduction of p-
+nuclei with respect to16O, i.e.,16O/p= 2/1.46 = 1.4,
+while the ratio of the yields between56Fe andp-nuclei al-
+mostagreeswith that ofthe solarabundance, i.e.,56Fe/p
+= 1.57/1.46= 1.1.
+4.CONCLUSIONS
+We calculate the p-process nucleosynthesis in the
+carbon-deflagration model for SNe Ia (W7 model in
+Nomoto et al. 1984) with realistic initial abundances
+ofs-nuclei. The temperature and density trajecto-8
+ries in W7 are different from those of the DD model
+adopted in Howard & Meyer (1992) and the parameter
+study (Howard et al. 1991). The adopted s-process pat-
+terns are also different from the previous study. We in-
+vestigate the effects on productions of p-nuclei of (1) the
+different initial12C abundances which affect the explo-
+sive C-burning, (2) the uncertain initial abundance of
+22Ne at the explosion, whose ( α,n) reaction is an impor-
+tant neutron source for the s-process occurring during
+the He shell flashes, and (3) the different distributions in
+thes-process which provides initial seed abundances for
+thep-process. Our findings are summarized as follows.
+1. In all cases we considered, more than 50% of p-
+nuclides are co-produced at almost the same degree of
+enhancements with respect to their solar abundances.
+We find that SNe Ia can produce Mo and Ru p-isotopes
+∼6–12 times more in Case A1 than SNe II on the
+basis of the mean overproduction factor of p-nuclei,
+although the problem of the relative underproduction
+still remains. The patterns of p-nuclei obtained in this
+studyaredifferentfromthosein Howard & Meyer(1992)
+and Howard et al. (1991), reflecting the differences in
+temperature and density profiles and initial distribution
+of seed nuclei. In addition, it is confirmed that the p-
+process layer in this C-deflagration model shifts to a
+higher temperature region than that in SNe II.
+2. The effect of variable C/O ratio in the initial com-
+position of the CO WD on the p-nuclei yields is small.
+On the other hand, the effect of the initial22Ne abun-
+dances is large especially for74Se,78Kr,84Sr, and92Mo.
+If the initial22Ne is less abundant, the light p-nuclides
+are enhanced. In order to produce enough Mo and Ru
+p-isotopes, the abundance of22Ne smaller than the as-
+sumption in W7 model is expected. Initial abundances
+of various nuclides other than12C,16O, and22Ne consid-
+ered in this study also affect the p-process through sup-
+plies of neutron, proton, and α-particle emerging during
+the explosion.
+3. The effect of initial abundances of s-nuclei on the
+p-process is large. If the s-process efficiently contributes
+to the production of seed nuclei, yields of p-nuclei in-
+crease and relative yields of heavy p-nuclei are enhanced
+more than those of lighter ones. Such enhancement of
+s-nuclei is expected for the single degenerate scenario of
+the Chandrasekhar mass model, but not trivial in the
+double degenerate scenario.
+4. This result leads to a possibility that SNe Ia play an
+important role in the galactic evolution of p-nuclei. The
+p-nuclei are produced by a factor of ∼1.2 more than
+56Fe when normalized to the solar abundances. SNe Ia,
+therefore, may have contributed to the enrichment of
+p-nuclei more effectively than SNe II (about twice in
+Case A1). Our calculation involves an uncertainty in
+the initial abundances of nuclides (e.g.,22Ne) in the C-
+deflagration model of the exploding CO WD. This has a
+great influence on abundances of background particles,
+i.e., protons, neutrons, and α-particles in the p-process
+nucleosynthesis. There is also an important uncertainty
+in thes-process in presupernova stars which affects ini-
+tial distributions of the s-nuclei and the remaining22Ne
+abundance. Studies of the s-process nucleosynthesis dur-
+ingthe presupernovaevolutionofaccretingWDs arenec-
+essary to clarify if the p-process in SNe Ia could reallyproduce enough abundances of p-nuclei. Despite these
+uncertainties, the present study strongly suggests that
+SNe Ia, as well as SNe II, could be a probable produc-
+tion site of the solar p-nuclei.
+This work has been supported by Grant-in-Aid for
+the Japan Society for the Promotion of Science (JSPS)
+Fellows (21.6817), and for Scientific Research of JSPS
+(18104003, 18540231, 20244035, 20540226) and the Min-
+istry of Education, Culture, Sports, Science, and Tech-
+nology(MEXT;19047004,20040004). Thisworkhasalso
+beensupportedbyWorldPremierInternationalResearch
+Center Initiative, MEXT, of Japan.
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+Figure 1. Trajectories of temperature (a) and density (b) as a functio n of time in the W7 model. The labels refer to the trajectories in
+the layers with the peak temperatures in units of 109K, i.e.,Tm,9=Tm/(109K) = 3.60 (1), 3.17 (2), 3.01 (3), 2.85 (4), and 1.86 (5).
+(a) (b)
+Figure 2. Distributions of initial seed abundances relative to solar for Case A (a) and B (b) (Table 3), respectively. Filled circl es indicate
+s-only nuclides.11
+Figure 3. Time evolution of mass fractions of proton (a), neutron (b),4He (c),12C (d),16O (e), and22Ne (f) during the explosion in
+the five layers shown in Figure 1.12
+Figure 4. Dependence of the production of representative p-nuclei (74Se,112Sn,130Ba,180Ta, and196Hg) on the peak temperature Tm.
+Figure 5. Average overproduction factors /angbracketleftF/angbracketrightnormalized to F0for thep-nuclides as a function of mass number. The results of Cases A 1
+and B (Table 3) are shown by filled circles and open squares, re spectively. Horizontal solid and dashed lines correspond t o/angbracketleftF/angbracketright/F0= 1 and
+(3, 1/3), respectively.13
+Figure 6. Average overproduction factors normalized to F0for three initial compositions: Cases A1 (circles), A2 (sta rs), A3 (triangles;
+Table 3).
+Figure 7. Average overproduction factors in Case A4 with respect to th ose in Case A1 (Table 3), i.e., /angbracketleftF/angbracketrightA4//angbracketleftF/angbracketrightA1forp-nucleus.14
+Figure 8. Overproduction factors of74Se as a function of peak temperature for Cases A1 and A4.
+Figure 9. Overproduction of p-nuclei (filled circles) and nuclei lighter than Fe-group el ements (open diamonds) in the ejecta of SNe Ia.15
+Table 1
+Initial Mass Fractions
+Xof Seed Nuclei for
+Case A
+Nuclide Abundance
+32S 1.92E-04
+33S 1.03E-04
+34S 7.35E-05
+36S 4.99E-05
+35Cl 3.79E-05
+Note. — (This ta-
+ble is available in its
+entirety in a machine-
+readable form in the on-
+line journal. A portion
+is shown here for guid-
+ance regarding its form
+and content.)
+Table 2
+Initial Mass Fractions
+Xof Seed Nuclei for
+Case B
+Nuclide Abundance
+32S 1.35E-04
+33S 8.90E-05
+34S 9.04E-05
+36S 1.85E-04
+35Cl 6.22E-05
+Note. — (This ta-
+ble is available in its
+entirety in a machine-
+readable form in the on-
+line journal. A portion
+is shown here for guid-
+ance regarding its form
+and content.)
+Table 3
+Initial Compositions and Average
+Overproduction Factors of p-nuclei
+CaseX(12C)X(16O)X(22Ne) F0
+A1 0.475 0.500 0.023 4657
+A2 0.350 0.625 0.023 4606
+A3 0.700 0.275 0.023 4777
+A4 0.498 0.500 0.000 7621
+B 0.475 0.500 0.022 25073416
+Table 4
+Average Overproduction Factors /angbracketleftF/angbracketrightNormalized to F0forp-nuclei
+Nuclide Case A1 Case A2 Case A3 Case A4 Case B
+74Se 1.05E+00 6.87E-01 1.74E+00 2.81E+00 4.94E-02
+78Kr 3.67E-01 2.96E-01 5.01E-01 4.98E+00 3.07E-02
+84Sr 5.78E-01 5.19E-01 7.68E-01 6.59E+00 6.94E-02
+92Mo 1.81E-01 1.74E-01 1.84E-01 5.46E-01 5.40E-02
+94Mo 1.32E-01 1.35E-01 1.27E-01 7.66E-02 3.20E-02
+96Ru 1.80E-01 1.73E-01 1.83E-01 1.39E-01 6.74E-02
+98Ru 5.12E-01 5.21E-01 4.95E-01 3.11E-01 1.55E-01
+102Pd 2.09E+00 2.15E+00 2.00E+00 1.26E+00 7.89E-01
+106Cd 1.82E+00 1.81E+00 1.79E+00 1.15E+00 8.15E-01
+108Cd 1.47E+00 1.51E+00 1.42E+00 8.83E-01 5.20E-01
+113In 2.04E-01 2.34E-01 1.71E-01 1.06E-01 7.23E-02
+112Sn 1.68E+00 1.76E+00 1.59E+00 9.94E-01 8.51E-01
+114Sn 1.64E+00 1.68E+00 1.58E+00 1.02E+00 6.84E-01
+115Sn 1.06E-02 1.43E-02 7.36E-03 2.70E-03 3.93E-03
+120Te 1.98E+00 1.99E+00 1.92E+00 1.23E+00 9.32E-01
+124Xe 2.74E+00 2.88E+00 2.59E+00 1.58E+00 1.76E+00
+126Xe 3.41E+00 3.16E+00 3.58E+00 2.51E+00 2.38E+00
+130Ba 4.23E+00 4.41E+00 4.01E+00 2.43E+00 3.21E+00
+132Ba 3.07E+00 3.17E+00 2.90E+00 1.77E+00 2.43E+00
+138La 9.54E-02 1.21E-01 6.30E-02 9.24E-03 8.31E-02
+136Ce 7.69E-01 7.75E-01 7.53E-01 4.59E-01 9.18E-01
+138Ce 8.03E-01 7.87E-01 7.96E-01 5.03E-01 1.04E+00
+144Sm 6.36E-01 6.59E-01 6.06E-01 3.63E-01 1.68E+00
+152Gd 1.40E-02 1.55E-02 1.29E-02 8.20E-03 2.25E-02
+156Dy 2.81E-01 2.55E-01 3.18E-01 2.58E-01 6.70E-01
+158Dy 9.64E-02 9.79E-02 8.55E-02 5.05E-02 1.75E-01
+162Er 3.81E-01 3.40E-01 4.21E-01 3.22E-01 1.00E+00
+164Er 9.33E-02 9.14E-02 9.05E-02 4.65E-02 2.63E-01
+168Yb 1.14E+00 1.07E+00 1.15E+00 7.23E-01 3.38E+00
+174Hf 8.10E-01 8.46E-01 7.63E-01 4.69E-01 2.52E+00
+180Ta 4.36E-03 7.04E-03 2.32E-03 2.06E-04 7.29E-03
+180W 1.49E+00 1.51E+00 1.49E+00 8.67E-01 4.85E+00
+184Os 2.61E-01 4.12E-01 9.82E-02 1.19E-02 4.83E-01
+190Pt 1.23E-01 1.16E-01 1.23E-01 8.92E-02 4.11E-01
+196Hg 6.54E-01 6.39E-01 6.51E-01 4.36E-01 2.57E+00
+Note. — All values in respective columns are normalized to differe nt
+F0values, which are listed in Table 3.17
+Table 5
+Overproduction Factors of Ejecta Xejecta/X⊙for Stable Nuclei from C
+to Fe and p-nuclei
+Nuclide Abundance Nuclide Abundance Nuclide Abundance
+12C 7.66E+0046Ca 1.79E-0374Se 3.53E+02
+13C 1.51E-0848Ca 1.15E-0678Kr 1.23E+02
+14N 3.15E-0645Sc 1.06E+0184Sr 1.94E+02
+15N 3.49E-0346Ti 1.56E+0292Mo 6.07E+01
+16O 1.06E+0147Ti 3.45E+0094Mo 4.44E+01
+17O 5.42E-0548Ti 1.05E+0096Ru 6.06E+01
+18O 1.51E-0749Ti 7.10E+0198Ru 1.72E+02
+19F 5.20E-0550Ti 4.86E+00102Pd 7.03E+02
+20Ne 4.93E+0050V 6.59E+00106Cd 6.13E+02
+21Ne 7.21E-0351V 5.01E+01108Cd 4.94E+02
+22Ne 8.36E+0050Cr 3.23E+02113In 6.85E+01
+23Na 3.91E-0152Cr 3.76E+02112Sn 5.65E+02
+24Mg 3.24E+0153Cr 1.18E+03114Sn 5.52E+02
+25Mg 8.37E-0254Cr 5.00E+01115Sn 3.57E+00
+26Mg 2.24E-0155Mn 4.37E+02120Te 6.64E+02
+27Al 8.26E+0054Fe 1.53E+03124Xe 9.21E+02
+28Si 1.78E+0256Fe 3.79E+02126Xe 1.15E+03
+29Si 9.75E+0057Fe 2.80E+02130Ba 1.42E+03
+30Si 1.39E+0258Fe 4.12E+01132Ba 1.03E+03
+31P 1.51E+0159Co 1.02E+02138La 3.21E+01
+32S 1.50E+0258Ni 8.96E+02136Ce 2.59E+02
+33S 1.76E+0260Ni 4.08E+02138Ce 2.70E+02
+34S 6.61E+0161Ni 5.41E+00144Sm 2.14E+02
+36S 1.78E-0162Ni 7.06E+01152Gd 4.70E+00
+35Cl 3.44E+0164Ni 2.30E-01156Dy 9.44E+01
+37Cl 2.21E+0163Cu 5.43E-01158Dy 3.24E+01
+36Ar 2.06E+0265Cu 3.02E-02162Er 1.28E+02
+38Ar 4.63E+01164Er 3.14E+01
+40Ar 3.59E-02168Yb 3.84E+02
+39K 1.51E+01174Hf 2.72E+02
+41K 1.90E+01180Ta 1.47E+00
+40Ca 4.97E+02180W 5.01E+02
+42Ca 3.81E+01184Os 8.78E+01
+43Ca 1.38E-01190Pt 4.12E+01
+44Ca 4.19E+01196Hg 2.20E+0218
+Table 6
+Relative Ejected Mass in One SN
+Event
+Type p-nucleia 16Oa 56Fea
+SN II 45 188 72
+SN Ia 638 15 524
+Note. — As for values of
+SNe II, the IMF averages in
+Tsujimoto et al. 1995 are used.
+aValues of Mi/Xi,⊙fornuclei iand
+average over p-nuclei.
+Table 7
+Contributions to the Galactic
+Contents of p-nuclei
+Type p-nuclei16O56Fe
+SN II 0.46 2.0 0.75
+SN Ia 1 0 0.82
+Note. — Ratios of masses
+that have ever been ejected in
+the Galaxy by two types of SN
+events to the corresponding so-
+lar mass fractions. Values are
+normalized so that the p-nuclei
+for SNe Ia are unity.
\ No newline at end of file
diff --git a/1001.0147.txt b/1001.0147.txt
new file mode 100644
index 0000000000000000000000000000000000000000..dd5753e2cfff70fb1cc9be8acaa272e3b0581cc5
--- /dev/null
+++ b/1001.0147.txt
@@ -0,0 +1,1711 @@
+arXiv:1001.0147v1  [math.GR]  31 Dec 2009Large scale geometry of negatively curved Rn⋊R
+Xiangdong Xie
+Abstract
+We classify all negatively curved Rn⋊Rup to quasiisometry. We show that
+all quasiisometries between such manifolds (except when th ey are biLipschitz
+to the real hyperbolic spaces) are almost similarities. We p rove these results by
+studying the quasisymmetric maps on the ideal boundary of th ese manifolds.
+Keywords. quasiisometry, quasisymmetric map, negatively curved solvable Lie
+groups.
+Mathematics Subject Classification (2000). 20F65, 30C65, 53C20.
+Contents
+1 Introduction 1
+2 Some basic definitions 4
+3 Negatively curved Rn⋊R 4
+4Q-variation on the ideal boundary 10
+5 Proof of the main theorems 15
+6 Proof of the corollaries 23
+7 QS maps in the Jordan block case 27
+8 A Liouville type theorem 32
+References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4
+1 Introduction
+In this paper we study quasiisometries between negatively c urved solvable Lie groups of the
+formRn⋊Rand quasisymmetric maps between their ideal boundaries.
+1Given an n×nmatrixA, we letGAbe the semi-direct product Rn⋊AR, whereRacts
+onRnby (t,x)/mapsto→etAx. ThenGAis a solvable Lie group.
+LetGAbeequippedwith any left invariant Riemannian metric such t hat theRdirection
+is perpendicular to the Rnfactor. When A=In,GAis isometric to Hn+1. More generally,
+if the eigenvalues of Aall have positive real parts, then it follows from Heintze’s results [H]
+thatGAis Gromov hyperbolic. Hence GAhas a well defined ideal boundary ∂GA. The
+ideal boundary ∂GAcan benaturally identified with (the one-point compactifica tion of)Rn.
+On the ideal boundary Rn, there is a parabolic visual (quasi)metric DA, which is invariant
+under Euclidean translations and admits a family of dilatio ns{λt=etA}. See Section 3 for
+more details.
+Given an n×nmatrixA, thereal part Jordan form ofAis obtained from the Jordan
+form ofAby replacing each diagonal entry with its real part and reord ering to make it
+canonical. Notice that the real part Jordan form is different f rom the real Jordan form
+and the absolute Jordan form. It is related to the absolute Jo rdan form through matrix
+exponential.
+Here are the main results of the paper. See Theorem 5.12 for a m ore precise statement
+of Theorem 1.2. Also see Section 2 for basic definitions.
+Theorem 1.1. LetAandBben×nmatrices whose eigenvalues all have positive real
+parts. Then (Rn,DA)and(Rn,DB)are quasisymmetric if and only if there is some s >0
+such that AandsBhave the same real part Jordan form.
+Theorem 1.2. LetAandBben×nmatrices whose eigenvalues all have positive real parts.
+Denote by λ1andµ1the smallest real parts of the eigenvalues of AandBrespectively, and
+setǫ=λ1/µ1. If the real part Jordan form of Ais not a multiple of the identity matrix
+In, then for every quasisymmetric map F: (Rn,DA)→(Rn,DB), the map F: (Rn,Dǫ
+A)→
+(Rn,DB)is biLipschitz.
+WhenA=In, the manifold GAis isometric to the real hyperbolic space Hn+1. In this
+case, the ideal boundary is Rnwith the Euclidean metric, and hence the claim in Theorem
+1.2 fails: there are non-biLipschitz quasiconformal maps i n the Euclidean space Rn. More
+generally, if the real part Jordan form of Ais a multiple of In, then it follows from the
+result of Farb-Mosher (see Section 2) that ( Rn,DA) is biLipschitz to ( Rn,|·|ǫ), where |·|
+denotes the Euclidean metric and ǫ >0 is some constant. Hence the claim in Theorem 1.2
+also fails.
+There are several consequences of the main results.
+Recall that two geodesic Gromov hyperbolic spaces admittin g cocompact isometric
+group actions are quasiisometric if and only if their ideal b oundaries are quasisymmet-
+ric with respect to the visual metrics, see [Pa] or [BS]. Henc e Theorem 1.1 yields the
+quasiisometric classification of all negatively curved Rn⋊R.
+Corollary 1.3. LetAandBben×nmatrices whose eigenvalues all have positive real
+parts. Then GAandGBare quasiisometric if and only if there is some s >0such that A
+andsBhave the same real part Jordan form.
+The next three results are consequences of Theorem 1.2.
+2A mapf:X→Ybetween two metric spaces is called an almost similarity if there are
+constants L >0 andC≥0 such that Ld(x1,x2)−C≤d(f(x1),f(x2))≤Ld(x1,x2)+C
+for allx1,x2∈Xandd(y,f(X))≤Cfor ally∈Y.
+Corollary 1.4. LetAandBben×nmatrices whose eigenvalues all have positive real
+parts. Suppose the real part Jordan form of Ais not a multiple of the identity matrix In.
+Then every quasiisometry f:GA→GBis an almost similarity.
+We view the canonical projection hA:GA=Rn×R→Ras the height function for GA.
+LetAandBbe twon×nmatrices. A quasiisometry f:GA→GBisheight-respecting if
+it maps the fibers of hAto within uniformly bounded Hausdorff distance from the fiber s of
+hB.
+Corollary 1.5. LetAandBben×nmatrices whose eigenvalues all have positive real
+parts. Suppose the real part Jordan form of Ais not a multiple of the identity matrix In.
+Then every quasiisometry f:GA→GBis height-respecting.
+Corollary 1.6. LetAbe a square matrix whose eigenvalues all have positive real p arts.
+If the real part Jordan form of Ais not a multiple of the identity matrix, then GAis not
+quasiisometric to any finitely generated group.
+A group Gof bijections g:X→Xof aquasimetric space is a unform quasim¨ obius group
+if there is some homeomorphism η: [0,∞)→[0,∞) such that every element gofGisη-
+quasim¨ obius. Thefollowing result follows from Theorem 1. 2 and a theorem of Dymarz-Peng
+[DP].
+Corollary 1.7. LetAbe a square matrix whose eigenvalues all have positive real p arts.
+Suppose that the real part Jordan form of Ais not a multiple of the identity matrix. Let
+Gbe a uniform quasim¨ obius group of ∂GA(equipped with a visual metric). If the induced
+action of Gon the space of distinct triples of ∂GAis cocompact, then Gcan be conjugated
+by a biLipschitz map of (Rn,DA)into the group of almost homotheties of (Rn,DA).
+WhenAis a Jordan block, we describe all the quasisymmetric maps on (Rn,DA).
+Consequently, we are able to prove a Liouville type theorem. See Section 7 and Section 8.
+Theorem 1.2 was established in the diagonal case in [SX] and i n the 2×2 Jordan block
+case in [X]. We believe that Theorem 1.2 holds true for most ho mogeneous manifolds
+with negative curvature (HMNs), with only a few exceptions. Recall that HMNs were
+characterized by Heintze in [H]: each such manifold is isome tric to a solvable Lie group S
+with a left invariant Riemannian metric, and the group Shas the form S=N⋊R, where
+Nis a simply connected nilpotent Lie group, and the action of RonNis generated by
+a derivation whose eigenvalues all have positive real parts . An open problem now is to
+establish Theorem 1.2 for most HMNs, and to construct non-bi Lipschitz quasisymmetric
+maps (of the ideal boundary) for the few exceptions. The only exceptions known to the
+author are (those HMNs that are biLipschitz to) the real and c omplex hyperbolic spaces:
+there are quasisymmetric maps in the Euclidean spaces [GV] a nd the Heisenberg groups [B]
+that change Hausdorff dimensions (of certain subsets), so th ey can not be biLipschitz.
+Acknowledgment . I would like to thank Bruce Kleiner for suggestions and stim ulating
+discussions. I also would like to thank Tullia Dymarz for tel ling me about her joint paper
+with Irine Peng [DP]. Finally, I am grateful for the generous travel support offered by the
+Department of Mathematical Sciences at Georgia Southern Un iversity.
+32 Some basic definitions
+In this section we recall some basic definitions.
+Aquasimetric ρon a set Xis a function ρ:X×X→Rsatisfying the following three
+conditions:
+(1)ρ(x,y) =ρ(y,x) for allx,y∈X;
+(2)ρ(x,y)≥0 for all x,y∈X, andρ(x,y) = 0 if and only if x=y;
+(3) there is some M≥1 such that ρ(x,z)≤M(ρ(x,y)+ρ(y,z)) for all x,y,z∈X.
+For each M≥1, there is a constant ǫ0>0 such that ρǫis biLipschitz equivalent to a metric
+for all quasimetric ρwith constant Mand all 0 < ǫ≤ǫ0, see Proposition 14.5. in [Hn].
+For any quadruple Q= (x,y,z,w) of distinct points in a quasimetric space X, thecross
+ratiocr(Q) ofQis:
+cr(Q) =ρ(x,w)ρ(y,z)
+ρ(x,z)ρ(y,w).
+Letη: [0,∞)→[0,∞) be a homeomorphism. A bijection F:X→Ybetween two quasi-
+metric spaces is η-quasim¨ obius if cr(F(Q))≤η(cr(Q)) for all quadruples Q= (x,y,z,w)
+of distinct points in X, whereF(Q) = (F(x),F(y),F(z),F(w)). A bijection F:X→Y
+between two quasimetric spaces is η-quasisymmetric if for all distinct triples x,y,z∈X, we
+have
+ρ(F(x),F(y))
+ρ(F(x),F(z))≤η/parenleftbiggρ(x,y)
+ρ(x,z)/parenrightbigg
+.
+A mapF:X→Yis quasisymmetric if it is η-quasisymmetric for some η.
+LetK≥1 andC >0. A bijection F:X→Ybetween two quasimetric spaces is called
+aK-quasisimilarity (with constant C) if
+C
+Kρ(x,y)≤ρ(F(x),F(y))≤CKρ(x,y)
+for allx,y∈X. When K= 1, we say Fis asimilarity . It is clear that a map is a
+quasisimilarity if and only if it is a biLipschitz map. The po int of using the notion of
+quasisimilarity is that sometimes there is control on Kbut not on C.
+3 Negatively curved Rn⋊R
+In this section we first review some basics about negatively c urvedRn×R, then define the
+parabolic visual (quasi)metric on their ideal boundary and study its properties. We also
+recall a result of Farb-Mosher and the main results of [X] and [SX].
+LetAbe ann×nmatrix. Let Ract onRnby
+R×Rn→Rn
+(t,x)→etAx.
+We denote the corresponding semi-direct product by GA=Rn⋊AR. ThenGAis a solvable
+Lie group. Recall that the group operation in GAis given by:
+(x1,t1)·(x2,t2) = (x1+et1Ax2,t1+t2).
+4We will always assume that the eigenvalues of Ahave positive real parts. An admissible
+metriconGAisaleftinvariantRiemannianmetricsuchthatthe Rdirectionisperpendicular
+to theRnfactor. The standard metric onGAis the left invariant Riemannian metric
+determined by the standard inner product on the tangent spac e of the identity element
+(0,0)∈Rn×R=GA. We remark that GAwith the standard metric does not always have
+negative sectional curvature. However, Heintze’s result( [H]) says that GAhas an admissible
+metric with negative sectional curvature. Since any two lef t invariant Riemannian distances
+on a Lie group are biLipschitz equivalent, GAwith any left invariant Riemannian metric is
+Gromov hyperbolic. From now on, unless indicated otherwise ,GAwill always be equipped
+with the standard metric.
+At a point ( x,t)∈Rn×R≈GA, the tangent space is identified with Rn×R, and the
+standard metric is given by the symmetric matrix
+/parenleftbiggQA(t)0
+01/parenrightbigg
+,
+whereQA(t) =e−tATe−tA. HereTdenotes matrix transpose.
+For each x∈Rn, the map γx:R→GA,γx(t) = (x,t) is a geodesic. We call such a
+geodesic a vertical geodesic. It can be checked that all vert ical geodesics are asymptotic
+ast→+∞. Hence they define a point ξ0in the ideal boundary ∂GA. The sets Rn×{t}
+(t∈R) arehorospherescentered at ξ0. For each t∈R, theinducedmetricon thehorosphere
+Rn×{t} ⊂GAis determined by thequadratic form QA(t). Thismetric has distance formula
+dA,t((x,t),(y,t)) =|e−tA(x−y)|. Here|·|denotes the Euclidean norm.
+Each geodesic rayin GAis asymptotictoeither anupwardoriented vertical geodesi cor a
+downward oriented vertical geodesic. Theupward oriented v ertical geodesics are asymptotic
+toξ0and the downward oriented vertical geodesics are in 1-to-1 c orrespondence with Rn.
+Hence∂GA\{ξ0}can be naturally identified with Rn.
+We next define a parabolic visual quasimetric on ∂GA\{ξ0}=Rn. Given x,y∈Rn=
+∂GA\{ξ0}, the parabolic visual quasimetric DA(x,y) is defined as follows: DA(x,y) =et,
+wheretis the smallest real number such that at height tthe two vertical geodesics γxand
+γyare at distance one apart in the horosphere; that is,
+dA,t((x,t),(y,t)) =|e−tA(x−y)|= 1.
+For each g= (x,t)∈GA, the Lie group left translation Lgis an isometry of GAand
+fixes the point ξ0. It shifts all the horospheres centered at ξ0in the vertical direction by the
+same amount. It follows that the boundary map of Lgis a similarity of ( Rn,DA). When
+g= (z,0),Lgleaves invariant all the horospheres centered at ξ0, and the boundary map is
+the Euclidean translation by z. Hence Euclidean translations are isometries with respect to
+DA:
+DA(x+z,y+z) =DA(x,y) for all x,y,z∈Rn.
+Wheng= (0,t),Lgshifts all the horospheres centered at ξ0byt, and the boundary map is
+the linear transformation etA. HenceetAis a similarity with similarity constant et:
+DA(etAx,etAy) =etDA(x,y) for all x,y∈Rnand all t∈R.
+5We remark that DAin general is not a metric, but merely a quasimetric. See the r emark
+after the proof of Corollary 3.2.
+For any integer n≥2, let
+Jn=
+1 1 0 ···0 0
+0 1 1 ···0 0
+0 0 1 ···0 0
+··· ··· ··· ···
+0 0 0 ···1 1
+0 0 0 ···0 1
+
+be then×nJordan matrix with eigenvalue 1. We write Jn=In+N. Here we omit
+the subscript nforNto simplify the notation. Notice that e−tJn=e−tIne−tN=e−te−tN.
+HenceDJn(x,y) =etif andonly if tis thesmallest real numbersatisfying et=|e−tN(y−x)|.
+For later use, we notice here that
+etN=
+1tt2
+2!···tn−2
+(n−2)!tn−1
+(n−1)!
+0 1 t···tn−3
+(n−3)!tn−2
+(n−2)!
+0 0 1 ···tn−4
+(n−4)!tn−3
+(n−3)!
+··· ··· ··· ···
+0 0 0 ···1t
+0 0 0 ···0 1
+. (3.1)
+LetPbe a nonsingular n×nmatrix. Define a map f:GA→GPAP−1byf(x,t) =
+(Px,t). Then it is easy to check that fis a Lie group isomorphism. Hence fis an isometry
+ifGPAP−1is equipped with the standard metric and GAhas the admissible metric in which
+P−1e1,···,P−1en, en+1is orthonormal at the identity element of GA. Heree1,···,en
+denote the standard basis of Rn, anden+1is the standard basis for R. Hence, GAwith any
+admissible metric is isometric to GPAP−1with the standard metric for some nonsingular
+matrixP. By Heintze’s result [H], there is a nonsingular matrix Psuch that GPAP−1with
+the standard metric has negative sectional curvature.
+Now we suppose both GAandGPAP−1are equipped with the standard metric. Then it
+is easy to check that for each t∈R, the restricted map
+f|Rn×{t}: (Rn×{t}, dA,t)→(Rn×{t}, dPAP−1,t)
+isK-biLipschitz, where K:= max{||P||,||P−1||}. Here||M||= sup{|Mx|:x∈Rn,|x|= 1}
+denotes the operator norm of an n×nmatrixM. We next recall a more general result by
+Farb-Mosher [FM].
+Proposition 3.1. (Proposition 4.1 in [FM]) LetAandBbe twon×nmatrices. Suppose
+there are constants r,s >0such that rAandsBhave the same real part Jordan form. Then
+there is a height-respecting quasiisometry f:GA→GB. To be more precise, there exist an
+n×nmatrixMandK≥1such that for each t∈R, the map v→Mvis aK-biLipschitz
+homeomorphism from (Rn,dA,t)to(Rn,dB,s
+rt); it follows that the map f:GA→GBgiven
+by
+(x,t)/mapsto−→/parenleftig
+Mx,s
+r·t/parenrightig
+is biLipschitz with biLipschitz constant sup{K,s
+r,r
+s}.
+6Corollary 3.2. Suppose we are in the setting of Proposition 3.1. Assume furt her thatr= 1
+andGAhas negative sectional curvature. Then:
+(1) the boundary map ∂f: (Rn,Ds
+A)→(Rn,DB)is biLipschitz;
+(2)fis an almost similarity.
+Proof.(1) We observe that the boundary map is given by ∂f(x) =Mx. Letx,y∈Rn
+and assume Ds
+A(x,y) =et. Then DA(x,y) =et/s. By the definition of DA, we have
+dA,t/s((x,t/s),(y,t/s)) = 1. Since GAhas pinched negative sectional curvature, there is a
+constant adepending only on the curvature bounds of GA, such that dA,t′((x,t′),(y,t′))<
+1/Kfort′> t/s+aanddA,t′((x,t′),(y,t′))> Kfort′< t/s−a. It now follows from Propo-
+sition 3.1 that dB,t′′((Mx,t′′),(My,t′′))<1 fort′′> t+saanddB,t′′((Mx,t′′),(My,t′′))>1
+fort′′< t−sa. By the definition of DBwe have e−saet≤DB(Mx,My)≤esaet. Hence
+∂f: (Rn,Ds
+A)→(Rn,DB) is biLipschitz with biLipschitz constant esa.
+(2) Letp= (x1,t1),q= (x2,t2)∈GAbe arbitrary. We may assume t1≤t2. If
+x1=x2, then it is clear from the definition of fthatd(f(p),f(q)) =s·d(p,q). So we
+assumex1/\e}atio\slash=x2and that dA,t0((x1,t0),(x2,t0)) = 1 for some t0. First assume t0≤t2. Then
+d((x1,t2),q)< dA,t2((x1,t2),q)≤1 asGAhas negative sectional curvature. By the triangle
+inequality, we have |d(p,q)−(t2−t1)| ≤1. By Proposition 3.1, d((Mx1,st2),f(q))≤
+dB,st2((Mx1,st2),f(q))≤K. By the triangle inequality again we have |d(f(p),f(q))−
+(st2−st1)| ≤K. Hence|d(f(p),f(q))−s·d(p,q)| ≤s+K.
+Next we assume t0> t2. By Lemma 6.3 (1) of [SX] we have |d(p,q)−(t0−t1)−
+(t0−t2)| ≤C1for some constant C1depending only on the curvature bounds of GA. By
+Lemma 6.2 of [SX], the point ( x1,t0) is aC2-quasicenter of x1,x2,ξ0∈∂GAfor some
+constant C2depending only on the curvature bounds of GA. Sincefis a quasiisometry,
+f(x1,t0) = (Mx1,st0)isaC3-quasicenter of Mx1,Mx2,η0∈∂GB(hereη0denotesthepoint
+in∂GBcorresponding to upward oriented vertical geodesics), whe reC3depends only on C2,
+the quasiisometry constants of fand the Gromov hyperbolicity constant of GB. Similarly,
+the point ( Mx2,st0) is also a C3-quasicenter of Mx1,Mx2,η0∈∂GB. Now consider the
+geodesic triangle consisting of {Mx1}×R,{Mx2}×Rand a geodesic joining Mx1,Mx2.
+Notice that f(p)∈ {Mx1}×Rlies between Mx1and (Mx1,st0) andf(q)∈ {Mx2}×R
+lies between Mx2and (Mx2,st0). It follows that
+|d(f(p),f(q))−(st0−st1)−(st0−st2)|
+=|d(f(p),f(q))−d(f(p),(Mx1,st0))−d(f(q),(Mx2,st0))| ≤C4
+for some constant C4depending only on C3and the Gromov hyperbolicity constant of GB.
+This combined with |d(p,q)−(t0−t1)−(t0−t2)| ≤C1implies|d(f(p),f(q))−s·d(p,q)| ≤
+C4+sC1.
+We notice that Corollary 3.2 (1) implies that DAis indeed a quasimetric: by Heintze’s
+result, there is some nonsingular Psuch that GPAP−1has pinched negative sectional cur-
+vature and hence DPAP−1is a quasimetric (this can be proved by the arguments in [CDP,
+p.124] or by using the relation between parabolic visual qua simetric and visual quasimetric
+[SX, section 5]); since ( Rn,DA) and (Rn,DPAP−1) are biLipschitz, DAis also a quasimetric.
+7LetAbe ann×nmatrix in real part Jordan form with positive eigenvalues
+λ1<···< λkA.
+LetVi⊂Rnbe the generalized eigenspace of λi, and let di= dimVi.
+Ifk:=kA≥2, we write Ain the block diagonal form A= [A1,···,Ak], where
+Aiis the block corresponding to the eigenvalue λi; we also denote A′= [A1,···,Ak−1].
+Correspondingly, Rnadmits the decomposition Rn=V1× ··· ×Vk. Hence each point
+x∈Rncan be written x= (x1,···,xk), where xi∈Vi. Observe that, for each xk∈Vk,
+if we identify V1× ··· ×Vk−1× {xk}withV1× ··· ×Vk−1, then the restriction of DAto
+V1×···×Vk−1×{xk}agrees with DA′. It is not hard to check that for all xk,yk∈Vk, the
+following holds for the Hausdorff distance with respect to th e quasimetric DA:
+HD(V1×···×Vk−1×{xk},V1×···×Vk−1×{yk}) =DAk(xk,yk).(3.2)
+Also, for any x= (x1,···,xk)∈Rnand any yk∈Vk,
+DA(x,V1×···×Vk−1×{yk}) =DAk(xk,yk). (3.3)
+Whenk= 1, that is, when Ahas only one eigenvalue λ:=λ1>0, the matrix A
+also has a block diagonal form A= [λIn0,λIn1+N,···,λInr+N], where n0≥0 and
+λIni+Nis a Jordan block. We allow the case A=λIn. We write a point p∈Rnas
+p= (z,(x1,y1),···,(xr,yr))T, whereTdenotes matrix transpose, z∈Rn0corresponds to
+λIn0and (xi,yi)T∈Rni(xi∈Rni−1,yi∈R) corresponds to λIni+N. Set
+Rn0+r={p= (z,(x1,y1),···,(xr,yr))T∈Rn:x1=···=xr=0},
+and letπA:Rn→Rn0+rbe the projection given by:
+πA(p) = (z,(0,y1),···,(0,yr))Tforp= (z,(x1,y1),···,(xr,yr))T∈Rn.
+Set
+A(1) = [λIn1−1+N,···,λInr−1+N],
+whereλI1+Nis understood to be λI1.
+Lemma 3.3. The restriction of DAto the fibers of πAagrees with DA(1). To be more
+precise, for all p= (z,(x1,y1),···,(xr,yr))T,p′= (z,(x′
+1,y1),···,(x′
+r,yr))Twe have
+DA(p,p′) =DA(1)(x,x′),
+wherex= (x1,···,xr)Tandx′= (x′
+1,···,x′
+r)T.
+Proof.Assume DA(p,p′) =etandDA(1)(x,x′) =es. By the definition, sis the smallest
+real number such that |e−sA(1)(x′−x)|= 1. We calculate
+e−sA(1)(x′−x) =e−λs(e−sNn1−1(x′
+1−x1),···,e−sNnr−1(x′
+r−xr))T.
+Similarly, tis the smallest real number such that |e−tA(p′−p)|= 1. We calculate
+e−tA(p′−p) =e−λt(0,(e−tNn1−1(x′
+1−x1),0),···,(e−tNnr−1(x′
+r−xr),0))T.
+It follows that the two equations |e−sA(1)(x′−x)|= 1 and |e−tA(p′−p)|= 1 are the same.
+Hences=t.
+8Lemma 3.4. The following hold for all y,y′∈Rn0+r:
+(1) the Hausdorff distance HDDA(π−1
+A(y),π−1
+A(y′)) =|y−y′|1
+λ;
+(2) for any p∈π−1
+A(y), we have DA(p,π−1
+A(y′)) =|y−y′|1
+λ.
+Proof.Letp= (z,(x1,y1),···,(xr,yr))T∈π−1
+A(y),p′= (z′,(x′
+1,y′
+1),···,(x′
+r,y′
+r))T∈
+π−1
+A(y′), where yandy′are written y= (z,y1,···,yr),y′= (z′,y′
+1,···,y′
+r). Assume
+DA(p,p′) =et. Thentis the smallest real number such that
+/vextendsingle/vextendsingle/parenleftbig
+z′−z,e−tNn1(x′
+1−x1,y′
+1−y1)T,···,e−tNnr(x′
+r−xr,y′
+r−yr)T/parenrightbig/vextendsingle/vextendsingle=eλt.
+Notice that the last entry of e−tNni(x′
+i−xi,y′
+i−yi)Tisy′
+i−yi, which is independent of t. It
+follows that eλt≥ |(z′−z,y′
+1−y1,···,y′
+r−yr)|=|y′−y|, andhence DA(p,p′) =et≥ |y′−y|1
+λ.
+Sett0= ln|y′−y|/λ. Theneλt0=|y′−y|. Now let p= (z,(x1,y1),···,(xr,yr))T∈
+π−1
+A(y) be arbitrary. Since the matrix e−t0Nniis nonsingular, the equation
+e−t0Nni(ui,vi)T= (0,···,0,y′
+i−yi)T
+has a unique solution ( ui,vi)T, whereui∈Rni−1andvi∈R. Notice that vi=y′
+i−yi. Set
+x′
+i=ui+xiandp′= (z′,(x′
+1,y′
+1),···,(x′
+r,y′
+r))T. Thenp′∈π−1
+A(y′) and
+e−t0A(p′−p) =e−t0λ/parenleftbig
+z′−z,(0,y′
+1−y1),···,(0,y′
+r−yr)/parenrightbigT.
+It follows that t0is a solution of |e−tA(p′−p)|= 1 and so DA(p,p′)≤et0=|y−y′|1
+λ. This
+together with the first paragraph implies DA(p,p′) =|y−y′|1
+λ. So each point p∈π−1
+A(y)
+is within |y−y′|1
+λofπ−1
+A(y′). Similarly, every point p′∈π−1
+A(y′) is also within |y−y′|1
+λof
+π−1
+A(y). Therefore, (1) holds.
+(2) also follows from the above two paragraphs.
+The following two results will be used in the proof of Theorem s 1.1 and 1.2. They are
+the basic steps in the induction.
+Theorem 3.5. (Theorem 4.1 in [SX]) Suppose Ais diagonal with positive eigenvalues
+α1< α2<···< αr(r≥2). Then every η-quasisymmetry F: (Rn,DA)→(Rn,DA)is a
+K-quasisimilarity, where Kdepends only on ηandr.
+Theorem 3.6. (Theorems4.1 and5.1in [X]) Everyη-quasisymmetric map F: (R2,DJ2)→
+(R2,DJ2)is aK-quasisimilarity, where Kdepends only on η. Furthermore, a bijection
+F: (R2,DJ2)→(R2,DJ2)is a quasisymmetric map if and only of it has the following for m:
+F(x,y) = (ax+c(y),ay+b)for all(x,y)∈R2, wherea/\e}atio\slash= 0,bare constants and c:R→R
+is a Lipschitz map.
+94Q-variation on the ideal boundary
+In this section we introduce the main tool in the proof of the m ain results: Q-variation for
+maps between quasimetric spaces. It is a discrete version of the notion of capacity. The
+advantage of this notion is that it makes sense for quasimetr ic spaces and does not require
+the existence of rectifiable curves. We remark that, while de aling with ideal boundary
+of negatively curved spaces, very often either one has to wor k with quasimetric spaces in
+which the triangle inequality is not available, or one needs to work with metric spaces
+that contain no rectifiable curves. Both scenarios are unple asant from the point of view of
+classical quasiconformal analysis.
+The notion of Q-variation is due to Bruce Kleiner [K].
+Let (X,ρ) be a quasimetric space and L≥1. A subset A⊂Xis called an L-quasi-
+ballif there is some x∈Xand some r >0 such that B(x,r)⊂A⊂B(x,Lr). Here
+B(x,r) :={y∈X:ρ(y,x)< r}.
+For any ball B:=B(x,r) and any κ >0, we sometimes denote B(x,κr) byκB.
+For a subset Eof a quasimetric space ( Y,ρ), theρ-diameter of Eis
+diamρ(E) := sup{ρ(e1,e2) :e1,e2∈E}.
+Letu: (X,ρ1)→(Y,ρ2) be a map between two quasimetric spaces. For any subset A⊂X,
+the oscillation of uoverAis
+osc(u|A) = diam ρ2(u(A)).
+LetQ≥1. For a collection of disjoint subsets A={Ai}ofX, theQ-variation of uover
+A, denoted by VQ(u,A), is the quantity
+/summationdisplay
+i[osc(u|Ai)]Q.
+Forδ >0 andK≥1, set
+Vδ
+Q,K(u) = sup{VQ(u,A)},
+whereAranges over all disjoint collections of K-quasi-balls in ( X,ρ1) withρ1-diameter at
+mostδ. Finally, the ( Q,K)-variation VQ,K(u)ofuis
+VQ,K(u) = lim
+δ→0Vδ
+Q,K(u).
+We notice that VQ,K(u|E1)≤VQ,K(u|E2) whenever E1⊂E2⊂X.
+There are useful variants of this definition, for instance on e can look at the infimum
+over all coverings. Or one can take the infimum over all coveri ngs followed by the sup as the
+mesh size tends to zero. As a tool, Q-variation could be compared with Pansu’s modulus
+[P], but seems slightly easier to work with in our context.
+Since quasisymmetric maps send quasi-balls to quasi-balls quantitatively, it is easy to
+see that Q-variation is a quasisymmetric invariant. To be more precis e, we recall
+Lemma 4.1. (Lemma 3.1 in [X]) LetXbe a bounded quasimetric space and F:X→Zan
+η-quasisymmetric map. Then for every map u:X→Ywe haveVQ,K(u)≤VQ,η(K)(u◦F−1).
+10We next calculate the Q-variation of certain functions defined on the ideal boundar y of
+negatively curved Rn⋊R. These calculations will be used in the next section to show t hat
+certain foliations on the ideal boundary are preserved by qu asisymmetric maps.
+For later use we recall that, for any Q >1, any integer k≥1 and any nonnegative
+numbers a1,···,ak, Jensen’s inequality states
+/summationtextk
+i=1aQ
+i
+k≥/parenleftigg/summationtextk
+i=1ai
+k/parenrightiggQ
+,
+and equality holds if and only if all the ai’s are equal. In our applications, the a′
+iswill be
+the oscillations of a function ualong a “stack” of quasi-balls.
+LetAbe ann×nmatrix in real part Jordan form with positive eigenvalues
+λ1< λ2< ... < λ k,
+letVi⊂Rnbe the generalized eigenspace of λi, and let di= dimVi. ThenRnadmits the
+decomposition: Rn=V1× ··· ×Vk. SinceetAis a linear transformation with det( etA) =
+et(/summationtext
+idiλi), for any subset U⊂Rn, we have Vol( etA(U)) =et(/summationtext
+idiλi)Vol(U).
+There are constants C1,C2,C3depending only on the dimension nwith the following
+properties. If B:=B(o,1)⊂Rnis the unit ball (in the Euclidean metric), and t≤ −1,
+then
+B(o,C1etλk|t|−n+1)⊂etAB⊂B(o,C2etλ1|t|n−1), (4.1)
+while
+Vol(etAB) =C3et(/summationtext
+idiλi). (4.2)
+LetS=/producttextn
+i=1[0,1]⊂Rnbe the unit cube. We notice that both SandBareK0-quasi-
+balls with respect to DAfor some K0depending only on A. Hence there is some r >0 such
+thatBA(o,r)⊂B⊂BA(o,K0r). Here the subscript Arefers to DA. Also recall that DA
+is a quasimetric: there is a constant M≥1 such that DA(x,z)≤M(DA(x,y)+DA(y,z))
+for allx,y,z∈Rn.
+In the following, when we say a subset E⊂Rnis convex, we mean it is convex with
+respect to the Euclidean metric. The continuity of a functio nu:E→Ris with respect to
+the topology induced from the usual topology on Rn.
+Lemma 4.2. LetE⊂Rnbe a convex open subset. If u: (E,DA)→Ris a nonconstant
+continuous function, then VQ,K(u) =∞for allQ </summationtext
+idiλi
+λkand allK≥K0.
+Proof.Letp,q∈Ewithu(p)/\e}atio\slash=u(q). LetC⊂Ebe a fixed cylinder with axis pq, such
+that the minimum of uon one cap of Cis strictly greater than its maximum on the other
+cap. We pack Cwith translates of etAB, fort≪0, as follows. First pick a maximal set of
+linesL={Lj}inRnsatisfying the following conditions:
+(1) each line is parallel to pq;
+(2) each line intersects C;
+(3) the Hausdorff distance (with respect to DA) between any two of the lines is at least
+2MK0ret.
+11The maximality implies that for each x∈C, we have DA(x,Lj)≤2MK0retfor some j.
+For each j, consider a translate BjofetABcentered at some point on Lj. Then we move
+BjalongLj(in both directions) by translations until the translates j ust touch Bj. Repeat
+this and we obtain a “stack” of K0-quasi-balls centered on Lj. Do this for each jand we
+obtain a packing P={P}ofCby translates of etAB, after removing those that are disjoint
+fromC.
+We claim that the collection Pcovers a fixed fraction of the volume of C. To see this,
+first notice that the DA-distance between the centers x1,x2of two consecutive K0-quasi-
+balls along Ljis at most M(K0ret+K0ret) = 2MK0ret, due to the generalized triangle
+inequality for DA. Assume DA(x1,x2) =es. Then
+e(lnr−s)A(x2−x1)∈e(lnr−s)ABA(o,es) =BA(o,r)⊂B⊂BA(o,K0r).
+SinceBis convex, the line segment joining oande(lnr−s)A(x2−x1) is contained in B⊂
+BA(o,K0r). It follows that the segment joining oandx2−x1lies ine(s−lnr)ABA(o,K0r) =
+BA(o,K0es). Hence x1x2⊂BA(x1,K0es). This shows that every point on Lj∩Cis within
+K0es≤K1:= 2MK2
+0retof the center of some P∈ P. Now the choice of the lines {Lj}and
+the generalized triangle inequality for DAimply that Cis covered by DA-balls with radius
+K2:=M(2MK0ret+K1) and centers the centers of {P}. Since the volumes of etABand
+BA(o,K2) are comparable, the claim follows.
+The number of K0-quasi-balls in Palong each line Ljis/lessorsimilare−tλk|t|n−1in view of the
+estimate (4.1). By Jensen’s inequality, the Q-variation of ufor theK0-quasi-balls along Lj
+is at least as large as the Q-variation when the oscillations of uon these quasi-balls are
+equal. This common oscillation is /greaterorsimilaretλk|t|−n+1. SincePcovers a fixed fraction of C, the
+cardinality of Pis/greaterorsimilare−t(/summationtext
+idiλi). Hence the Q-variation of uonPis
+/greaterorsimilare−t(/summationtext
+idiλi)/parenleftig
+etλk|t|−n+1/parenrightigQ
+=et(Qλk−/summationtext
+idiλi)|t|(−n+1)Q,
+which→ ∞ast→ −∞forQ </summationtext
+idiλi
+λk. HenceVQ,K(u) =∞.
+Notice that/summationtext
+idiλi
+λk< nifk≥2 and/summationtext
+idiλi
+λk=nifk= 1. Hence we have the following:
+Corollary 4.3. Suppose k= 1. LetE⊂Rnbe a convex open subset. If u: (E,DA)→R
+is a nonconstant continuous function, then VQ,K(u) =∞for allQ < nand allK≥K0.
+Lemma 4.4. LetE⊂Rnbe a convex open subset. Let u: (E,DA)→Rbe a continuous
+function. Suppose there is an affine subspace Wparallel to the subspace/producttext
+i≤lVisuch that
+u|W∩Eis not constant. Then VQ,K(u) =∞for allQ </summationtext
+idiλi
+λland allK≥K0.
+Proof.Note that in the proof of Lemma 4.2, if pqis parallel to the subspace/producttext
+i≤lVi, then
+the number of quasi-balls in Palong a line Ljis/lessorsimilare−tλl|t|n−1, so the lower bound on
+Q-variation becomes
+Cet(Qλl−/summationtext
+idiλi)|t|(−n+1)Q,
+which tends to ∞ast→ −∞ifQ </summationtext
+idiλi
+λl.
+12Letπ:Rn=V1×···×Vk→Vkbe the natural projection.
+Lemma 4.5. Letπ′:Vk→Rbe a coordinate function on Vk, and set u=π′◦π. Then
+VQ,K(u|E) = 0for allQ >/summationtext
+idiλi
+λk, allK≥K0and all bounded subsets E⊂Rn.
+Proof.LetEbe a bounded open subset. Let 0 < δ <<1 and{Bj}j∈Ibe a packing of Eby
+K-quasi-balls with size < δ. Then for each jthere is some xj∈Rnand some tjsuch that
+BA(xj,etj)⊂Bj⊂BA(xj,Ketj).
+SinceBA(o,r)⊂B⊂BA(o,K0r), we have et′
+jAB⊂BA(o,etj) andBA(o,Ketj)⊂et′′
+jAB,
+wheret′
+j=tj−lnr−lnK0andt′′
+j=tj−lnr+lnK. SetB′
+j=xj+et′
+jABandB′′
+j=xj+et′′
+jAB.
+ThenB′
+j⊂Bj⊂B′′
+j. It follows that
+osc(u|Bj)≤osc(u|B′′
+j)/lessorsimilaret′′
+jλk|t′′
+j|dk−1,
+and
+(osc(u|Bj))Q/lessorsimilaret′′
+j(Qλk)|t′′
+j|Q(dk−1)/lessorsimilaret′
+j(Qλk)|t′
+j|Q(dk−1).
+IfQ >/summationtext
+idiλi
+λk, then this will be /lessorsimilar(Vol(B′
+j))s≤(Vol(Bj))sfors=Qλk+/summationtext
+idiλi
+2/summationtext
+idiλi>1, which
+implies that the Q-variation is zero.
+For the rest of this section, we will assume k= 1 and use the notation introduced before
+Lemma 3.3.
+Lemma 4.6. Suppose Ahas only one eigenvalue λ >0. Letπ′:Rn0+r→Rbe a coordinate
+function and u=π′◦πA. Then for any bounded open subset E:
+(1)VQ,K(u|E) = 0for allQ > nand allK≥K0;
+(2)0< Vn,K(u|E)<∞for allK≥K0.
+Proof.LetPbe aK-quasi-ball. Then there is a DA-ballUwithU⊂P⊂KU. Let
+t0= ln(KK0). For some t∈Rthere is a translate S(t) ofetASand a translate S(t+t0) of
+e(t+t0)ASsuch that1
+K0U⊂S(t)⊂UandKU⊂S(t+t0)⊂KK0U. Observe that for any
+translate S′ofetAS, we have osc( u|S′) =eλt. It follows that
+osc(u|P)≥osc(u|S(t)) =1
+(KK0)λosc(u|S(t+t0))≥1
+(KK0)λosc(u|P).
+Also notice that osc( u|S(t)) = (Vol( S(t)))1
+n≤(Vol(P))1
+n.
+Now let Ebe a bounded open subset and {Pi}a packing of Eby a disjoint collection
+ofK-quasi-balls with size < δ. For each Pi, letUibe aDA-ball with Ui⊂Pi⊂KUiand
+letSibe a translate of some etiASwith1
+K0Ui⊂Si⊂Ui. Then the preceding paragraph
+implies/summationdisplay
+iosc(u|Pi)Q≤(KK0)Qλ/summationdisplay
+iosc(u|Si)Q≤(KK0)Qλ/summationdisplay
+iVol(Pi)Q
+n.
+13From this it is clear that VQ,K(u|E) = 0 ifQ > nandVn,K(u|E)<∞since{Pi}is a disjoint
+collection in E.
+Now consider a particular packing {Pi}ofEby the images of the integral unit cubes
+inRnunderetA. Then osc( u|Pi) =eλt. The cardinality of {Pi}is approximatelyVol(E)
+enλt.
+HenceVδ
+n,K(u|E)≥/summationtext
+iosc(u|Pi)n≈Vol(E). Hence we have 0 < Vn,K(u|E)<∞.
+Lemma 4.7. Suppose Ahas only one eigenvalue λ >0. LetE⊂Rnbe a rectangular
+box whose edges are parallel to the coordinate axes. Let u: (E,DA)→Rbe a continuous
+function. Suppose there is some fiber HofπA:Rn→Rn0+rsuch that u|H∩Eis not
+constant. Then VQ,K(u) =∞for allQ≤nand allK≥K0.
+Proof.Suppose there is some fiber HofπAsuch that u|H∩Eis not constant. Then there is
+some Jordan block JinAwith the following property: if we denote by x= (x1,···,xm) the
+coordinates corresponding to J, then there is some index k, 1≤k≤m−1 such that uis
+constant along every line parallel to the xj-axis for j≤k−1, but is not constant along some
+lineLparallel to the xk-axis. We write Rn=Rk−1×R×Rn−k, where the Rcorresponds
+to thexk-axis and the Rk−1is spanned by the xj-axes (j≤k−1). After composing uwith
+an affine function, we may assume that for some rectangular box C=/producttextn
+i=1[ai,bi]⊂E, we
+haveu≤0 on the codimension 1 face F0:={x∈C:xk=ak}ofCandu≥1 on the
+codimension1 face F1:={x∈C:xk=bk}ofC. We will induct on k.
+Recall that for a Jordan block J=λIm+N, we have etJ=eλtetN. See (3.1) for an
+expression of etN.
+We first assume k= 1. For t <<0, consider the images of the integral unit cubes
+underetA. Let{Bi}be the collection of all those images that intersect the box C. Notice
+that a vertical stack (i.e. parallel to the xm-axis) of integral cubes is mapped by etAto
+a sequences of K0-quasi-balls which is almost parallel to the x1-axis. We divide {Bi}into
+such sequences which join F0andF1. Note that the projection of each Bito thex1-axis has
+length comparable to eλt|t|m−1. Hence the cardinality of each sequence is comparable to
+e−λt|t|1−m. TheQ-variation of ualong each sequence is at least the Q-variation of uwhen
+oscillations of uon the members of the sequence are equal. Since u≤0 onF0andu≥1 on
+F1, this common oscillation is at least comparable to eλt|t|m−1. Since each Bihas volume
+enλt, the cardinality of {Bi}is comparable to e−nλt. It follows that the Q-variation of u|C
+is at least comparable to
+e−nλt·/parenleftig
+eλt|t|m−1/parenrightigQ
+=eλt(Q−n)|t|Q(m−1),
+which→ ∞ast→ −∞ifQ≤n. HenceVQ,K(u|C) =∞forQ≤n.
+Now we assume m−1≥k≥2. Then uis constant along affine subspaces parallel to
+Rk−1×{0}×{0} ⊂Rn. Let
+U=/braceleftbig
+x∈F0: (3ai+bi)/4≤xi≤(ai+3bi)/4 for alli/\e}atio\slash=k/bracerightbig
+⊂F0.
+Fort <<0, denote by v(t) = (−1)m−ke−λtetAem. Notice that the components of v(t)
+corresponding to the Jordan block Jis
+(−1)m−k/parenleftbiggtm−1
+(m−1)!,tm−2
+(m−2)!,···, t,1/parenrightbigg
+14and all other components are 0. Hence for t <<0, lines parallel to v(t) travel much faster
+in thexi(1≤i≤m−1) direction than in the xi+1direction. Let Z⊂Rnbe the subset
+given by:
+Z=/braceleftig
+f+sv(t) :f∈U,0≤s≤(m−k)!
+|t|m−k(bk−ak)/bracerightig
+.
+Notice that for each fixed f, the segment {f+sv(t) : 0≤s≤(m−k)!
+|t|m−k(bk−ak)}joins the
+two hyperplanes xk=akandxk=bk. Also notice that these segments are parallel to the
+images of vertical stacks (i.e., parallel to the xm-axis) of integral cubes under etA. HenceZ
+has a packing Pthat can be divided into sequences such that each sequence jo insxk=ak
+andxk=bkand is the image (under etA) of a vertical stack of integral cubes.
+Forp= (x,y,z),q= (x′,y′,z′)∈Rk−1×R×Rn−k, definep∼qify′=y,z′=zand
+x′
+i−xiis an integral multiple of bi−aifor 1≤i≤k−1. SetY=Rn/∼and letπ:Rn→Y
+be the natural projection. Also let πC:C→Ybe the composition of the inclusion C⊂Rn
+andπ. It is clear that πCis injective on the interior of C. It is also easy to check that π|Zis
+injective. Now the packing PofZprojects onto a packing of Y, which can then be pulled
+back through πCto obtain a packing P′ofC(sinceπ(Z)⊂πC(C)). A sequence in Pgives
+rise to a broken sequence in P′: the broken sequence will first hit the boundary of Cat a
+point of ∂(/producttextk−1
+i=1[ai,bi])×/producttextn
+k[ai,bi]⊂∂C, it continues after a translation by an element
+of the form (/summationtextk−1
+i=1mi(bi−ai),0,0)∈Rn, wheremi∈Z; this can be repeated until the
+sequence hits xk=bk. Note that we can apply Jensen’s inequality to each broken se quence
+while considering Q-variations of usince by assumption uis constant along affine spaces
+parallel to Rk−1×{0}×{0}.
+Each broken sequence joins F0toF1. Since the projection of etASto thexk-axis has
+lengthcomparableto eλt|t|m−k, thecardinality ofeachsequenceiscomparableto e−λt|t|k−m.
+TheQ-variation of ualong the sequence is at least the Q-variation whenthe oscillations of u
+are the same on all members of the sequence. Thecommon oscill ation is at least comparable
+toeλt|t|m−k. Hence the Q-variation of uis at least comparable to
+1
+enλt·(eλt|t|m−k)Q=|t|Q(m−k)e(Q−n)λt,
+which→ ∞whent→ −∞ifQ≤n. HenceVQ,K(u|C) =∞forQ≤n.
+5 Proof of the main theorems
+In this section we prove the main results of the paper. The mai n tools are the notion of
+Q-variation (Section 4) and the arguments from Section 4 of [X ] and [SX]. The main results
+of [SX] and [X] are the basic steps in the induction.
+We first fix the notation. Let Abe ann×nmatrix in real part Jordan form with
+positive eigenvalues
+λ1<···< λkA.
+LetVi⊂Rnbe the generalized eigenspace of λi, and set di= dimVi. IfkA≥2, we write
+Ain the block diagonal form A= [A1,···,AkA], where Aiis the block corresponding to
+15the eigenvalue λi; we also denote A′= [A1,···,AkA−1]. IfkA= 1, that is, if Ahas only
+one eigenvalue λ=λ1, then we also write A= [λIn0,λIn1+N,···,λInr+N] in the block
+diagonal form, and we let πA:Rn→Rn0+rbe the projection defined before Lemma 3.3. If
+kA= 1 and r≥1, we set lA= max{n1,···,nr}.
+Similarly, let Bbe ann×nmatrix in real part Jordan form with positive eigenvalues
+µ1<···< µkB.
+LetWj⊂Rnbe the generalized eigenspace of µj, and set ej= dimWj. IfkB≥2, we write
+Bin the block diagonal form B= [B1,···,BkB], where Bjis the block corresponding to
+the eigenvalue µj; we also denote B′= [B1,···,BkB−1]. IfkB= 1, that is, if Bhas only
+one eigenvalue µ=µ1, we also write B= [µIm0,µIm1+N,···,µIms+N] in the block
+diagonal form, and we let πB:Rn→Rm0+sbe the projection defined before Lemma 3.3.
+IfkB= 1 and s≥1, we set lB= max{m1,···,ms}.
+Suppose there is an η-quasisymmetric map F: (Rn,DA)→(Rn,DB).
+Lemma 5.1. kA= 1if and only if kB= 1.
+Proof.Suppose kA= 1 and kB≥2. Fix any Qwith/summationtext
+jµjej
+µkB< Q < n . Letπ: (Rn,DB)→
+WkBbe the projection onto WkB, andπ′:WkB→Ra coordinate function on WkB. Set
+u=π′◦π. Then Lemma 4.5 implies VQ,η(K)(u|F(E)) = 0 for all sufficiently large Kand all
+bounded subsets E⊂(Rn,DA). By Lemma 4.1 VQ,K(u◦F|E) = 0. But this contradicts
+Corollary 4.3.
+Lemma 5.2. Suppose kA= 1. ThenA=λ1Inif and only if B=µ1In.
+Proof.Suppose B=µ1In. Letπi(i= 1,2,···,n) be the coordinate functions on ( Rn,DB).
+Then by Lemma 4.6 we have Vn,η(K)(πi|F(E))<∞for alli, all sufficiently large Kand all
+rectangular boxes E⊂(Rn,DA). Hence Vn,K(πi◦F|E)<∞by Lemma 4.1. Now Lemma
+4.7 implies that πi◦Fis constant on the fibers of πA. Since this is true for all 1 ≤i≤n,
+the fibers of πAmust have dimension 0. Hence Amust also be a multiple of In.
+Lemma 5.3. Suppose kA= 1andr≥1. ThenFmaps each fiber of πAonto some fiber of
+πB.
+Proof.Lemmas 5.1 and 5.2 imply that kB= 1 and s≥1. Notice that it suffices to show
+that each fiber of πAis mapped by Finto some fiber of πB: by symmetry each fiber of
+πBis mapped by F−1into some fiber of πAand hence the lemma follows. We shall prove
+this by contradiction and so assume that there is some fiber HofπAsuch that F(H) is
+not contained in any fiber of πB. Then there is some coordinate function π′:Rm0+s→R
+such that u◦Fis not constant on H, whereu:=π′◦πB. Now Lemma 4.6 implies that
+Vn,η(K)(u|F(E))<∞for all sufficiently large Kand all rectangular boxes E⊂(Rn,DA). By
+Lemma 4.1 we have Vn,K(u◦F|E)<∞. This contradicts Lemma 4.7 since we can choose
+Esuch that u◦Fis not constant on H∩E.
+16It follows from Lemma 5.3 that Finduces a map G:Rn0+r→Rm0+ssuch that
+F(π−1
+A(y)) =π−1
+B(G(y)) for all y∈Rn0+r. Define
+τA:Rn=Rn0×Rn1×···×Rnr−→Rn=Rn−n0−r×Rn0+r
+by
+τA(z,(x1,y1),···,(xr,yr)) = ((x1,···,xr),(z,y1,···,yr)),
+where (xi,yi)∈Rni=Rni−1×R. Similarly, there is an identification
+τB:Rn=Rm0×Rm1×···×Rms−→Rn=Rn−m0−s×Rm0+s.
+With the identifications τAandτB, we have π−1
+A(y) =Rn−n0−r× {y},π−1
+B(G(y)) =
+Rn−m0−s×{G(y)}, andF(Rn−n0−r×{y}) =Rn−m0−s×{G(y)}. Hence for each y∈Rn0+r,
+there is a map
+H(·,y) :Rn−n0−r→Rn−m0−s
+such that F(x,y) = (H(x,y),G(y)) for all x∈Rn−n0−r.
+Lemma 5.4. Suppose kA= 1andr≥1. Then:
+(1) The map G: (Rn0+r,|·|1
+λ)→(Rm0+s,|·|1
+µ)isη-quasisymmetric;
+(2) for each y∈Rn0+r, the map H(·,y) : (Rn−n0−r,DA(1))→(Rn−m0−s,DB(1))isη-
+quasisymmetric.
+Proof.(1) follows from Lemma 3.4 and the arguments on page 10 of [X]. The statement
+(2) follows from Lemma 3.3.
+Suppose kA= 1. Set ǫ=λ/µandη1(t) =η(t1
+ǫ). We notice that all the following maps
+areη1-quasisymmetric:
+(1)F: (Rn,Dǫ
+A)→(Rn,DB);
+(2)G: (Rn0+r,|·|1
+µ)→(Rm0+s,|·|1
+µ);
+(3)H(·,y) : (Rn−n0−r,Dǫ
+A(1))→(Rn−m0−s,DB(1)), for each y∈Rn0+r.
+Letg: (X1,ρ1)→(X2,ρ2) be a bijection between two quasimetric spaces. Suppose
+gsatisfies the following condition: for any fixed x∈X1,ρ1(y,x)→0 if and only if
+ρ2(g(y),g(x))→0. We define for every x∈X1andr >0,
+Lg(x,r) = sup{ρ2(g(x),g(x′)) :ρ1(x,x′)≤r},
+lg(x,r) = inf{ρ2(g(x),g(x′)) :ρ1(x,x′)≥r},
+and set
+Lg(x) = limsup
+r→0Lg(x,r)
+r, lg(x) = liminf
+r→0lg(x,r)
+r.
+Lemma 5.5. Consider the maps G: (Rn0+r,|·|1
+µ)→(Rm0+s,|·|1
+µ)andH(·,y) : (Rn−n0−r,Dǫ
+A(1))→
+(Rn−m0−s,DB(1)). The following hold for all y∈Rn0+r,x∈Rn−n0−r:
+(1)LG(y,r)≤η1(1)lH(·,y)(x,r)for anyr >0;
+(2)η−1
+1(1)lH(·,y)(x)≤lG(y)≤η1(1)lH(·,y)(x);
+(3)η−1
+1(1)LH(·,y)(x)≤LG(y)≤η1(1)LH(·,y)(x).
+17Proof.The proof is very similar to that of Lemma 4.3 in [X]. Let y∈Rn0+r,x∈Rn−n0−r
+andr >0. Lety′∈Rn0+rwith|y−y′|1
+µ≤randx′∈Rn−n0−rwithDǫ
+A(1)(x,x′)≥r.
+Sett0= ln|y′−y|/λ. Let (ui,vi) (ui∈Rni−1,vi∈R, 1≤i≤r) be the unique solution
+ofe−t0Nni(ui,vi)T= (0,···,0,y′
+i−yi)T. Letx′′
+i=ui+xiandx′′= (x′′
+1,···,x′′
+r). Then
+Dǫ
+A((x,y),(x′′,y′)) =|y−y′|1
+µ≤r≤Dǫ
+A((x,y),(x′,y)). Since F: (Rn,Dǫ
+A)→(Rn,DB) is
+η1-quasisymmetric, we have
+|G(y)−G(y′)|1
+µ≤DB(F(x′′,y′),F(x,y))≤η1(1)DB(F(x,y),F(x′,y))
+=η1(1)DB(1)(H(x,y),H(x′,y)).
+Sincey′andx′are chosen arbitrarily, (1) follows.
+The proofs of (2) and (3) are exactly the same as those for Lemm a 4.3 in [X].
+Lemma 5.6. Suppose kA= 1andlA= 2. ThenlB= 2and forǫ=λ/µ
+(1)AandǫBhave the same real part Jordan form;
+(2) The map F: (Rn,Dǫ
+A)→(Rn,DB)is aK-quasisimilarity, where Kdepends only on
+A,Bandη.
+Proof.(1) By Lemma 5.4 (2), for each y∈Rn0+r, the map H(·,y) : (Rn−n0−r,DA(1))→
+(Rn−m0−s,DB(1)) isη-quasisymmetric. Since lA= 2, all Jordan blocks of Ahave size 2 and
+A(1) =λIr. Now Lemma 5.2 applied to H(·,y) implies that B(1) =µIr. It follows that all
+Jordan blocks of Balso have size 2, and hence lB= 2 and B(1) =µIs. So we have r=s.
+That is, AandBhave the same number of 2 ×2 Jordan blocks. Now (1) follows.
+(2) The proof of (2) is very similar to the arguments in Sectio n 4 of [SX] and [X]. We
+will only indicate the differences here. First we notice that G: (Rn0+r,|·|)→(Rm0+s,|·|)
+is also quasisymmetric, and hence is differentiable a.e. Sinc eF: (Rn,Dǫ
+A)→(Rn,DB) is
+η1-quasisymmetric, the arguments in Section 4of [SX] and [X] i mply that there is a constant
+K1dependingonlyon η1, suchthat forevery y∈Rn0+rwhereG: (Rn0+r,|·|)→(Rm0+s,|·|)
+is differentiable, we have 0 < lG(y)<∞and the map
+H(·,y) : (Rn−n0−r,Dǫ
+A(1))→(Rn−m0−s,DB(1))
+is aK1-quasisimilarity with constant lG(y).
+Now let y,y′∈Rn0+rbe two points where Gis differentiable. We will show that
+lG(y) andlG(y′) are comparable. Let x∈Rn−n0−rand choose x′∈Rn−n0−rso that
+DA(1)(x,x′)>>|y′−y|1
+λ. Let (ui,vi) be as in the proof of Lemma 5.5. Let x′′
+i=xi+ui,
+x′′′
+i=x′
+i+ui(1≤i≤r), and set x′′= (x′′
+1,···,x′′
+r),x′′′= (x′′′
+1,···,x′′′
+r). Then
+DA((x,y),(x′′,y′)) =DA((x′,y),(x′′′,y′)) =|y′−y|1
+λ.
+Now the generalized triangle inequality implies
+DA((x′′,y′),(x′,y))≤M/braceleftig
+DA((x′′,y′),(x,y)) +DA((x,y),(x′,y))/bracerightig
+≤2MDA((x,y),(x′,y)).
+18By the quasisymmetry condition we have
+DB(F(x′′,y′),F(x′,y))≤η(2M)DB(F(x,y),F(x′,y)).
+Similarly, DB(F(x′′,y′),F(x′′′,y′))≤η(2M)DB(F(x′′,y′),F(x′,y)). So we have
+DB(F(x′′,y′),F(x′′′,y′))≤(η(2M))2DB(F(x,y),F(x′,y)).
+This together with the quasisimilarity properties of H(·,y) andH(·,y′) mentioned above
+implies that
+lG(y′)Dǫ
+A(1)(x′′,x′′′)≤K2
+1(η(2M))2lG(y)Dǫ
+A(1)(x,x′).
+SinceDA(1)(x′′,x′′′) =DA(1)(x,x′), we have lG(y′)≤K2
+1(η(2M))2lG(y). By symmetry, we
+alsohave lG(y)≤K2
+1(η(2M))2lG(y′). Nowfix yandsetC=lG(y). Thenatevery y′whereG
+is differentiable, H(·,y′) is aK2-quasisimilarity with constant C, whereK2=K3
+1(η(2M))2.
+Now a limiting argument shows that this is true for every y′∈Rn0+r. The arguments in
+Section 4 of [X] (using Lemma 5.5 from above instead of Lemma 4 .3 in [X]) then show that
+there is a constant K3=K3(K2,η1) such that G: (Rn0+r,| ·|1
+µ)→(Rm0+s,| · |1
+µ) and all
+H(·,y) areK3-quasisimilarities with constant C.
+Thefinaldifferenceisinfindingalowerboundfor DB(F(x,y),F(x′,y′)). IfDǫ
+A((x,y),(x′,y′))≤
+(2M)ǫ|y′−y|1
+µ, then
+DB(F(x,y),F(x′,y′))≥ |G(y′)−G(y)|1
+µ≥C
+K3|y′−y|1
+µ
+≥C
+(2M)ǫK3Dǫ
+A((x,y),(x′,y′)).
+Now assume Dǫ
+A((x,y),(x′,y′))≥(2M)ǫ|y′−y|1
+µ. Let (ui,vi) be as in the above paragraph.
+Letx′′
+i=x′
+i−uiand setx′′= (x′′
+1,···,x′′
+r). Then Dǫ
+A((x′′,y),(x′,y′)) =|y′−y|1
+µ. The
+generalized triangle inequality implies
+1
+2M≤DA((x,y),(x′′,y))
+DA((x,y),(x′,y′))≤2M.
+Now the quasisymmetric condition implies
+DB(F(x,y),F(x′,y′))≥1
+η(2M)DB(F(x,y),F(x′′,y))
+≥C
+K3η(2M)Dǫ
+A((x,y),(x′′,y))
+≥C
+(2M)ǫK3η(2M)Dǫ
+A((x,y),(x′,y′)).
+So we have found a lower boundfor DB(F(x,y),F(x′,y′)). The rest of the proof is the same
+as in Section 4 of [X]. We notice that the constant Mdepends only on A, andǫdepends
+only onAandB. HenceFis aK-quasisimilarity with Kdepending only on A,Bandη.
+19Lemma 5.7. Suppose kA= 1andlA≥2. Then for ǫ=λ/µ:
+(1)AandǫBhave the same real part Jordan form;
+(2) The map F: (Rn,Dǫ
+A)→(Rn,DB)is aK-quasisimilarity, where Kdepends only on
+A,Bandη.
+Proof.We induct on lA. The basic step lA= 2 is Lemma 5.6. Now assume lA=l≥3 and
+that the lemma holds for lA=l−1. For any y∈Rn0+r, the induction hypothesis applied
+to theη-quasisymmetric map H(·,y) : (Rn−n0−r,DA(1))→(Rn−m0−s,DB(1)) implies that
+forǫ=λ/µ:
+(a)A(1) andǫB(1) have the same real part Jordan form;
+(b)H(·,y) : (Rn−n0−r,Dǫ
+A(1))→(Rn−m0−s,DB(1)) is aK-quasisimilarity with Kdepending
+only onA(1),B(1) andη.
+Now (1) follows from (a), and (2) follows from (b), Lemma 5.5 a nd the arguments in Section
+4 of [X] (see the proof of Lemma 5.6(2)).
+Lemma 5.8. Suppose kA≥2. ThenkB≥2and/summationtext
+idiλi
+λkA=/summationtext
+jejµj
+µkB.
+Proof.Lemma 5.1 implies kB≥2. Suppose/summationtext
+idiλi
+λkA>/summationtext
+jejµj
+µkB. Pick any Qwith/summationtext
+idiλi
+λkA>
+Q >/summationtext
+jejµj
+µkB. Letπ: (Rn,DB)→WkBbe the projection onto WkB, andπ′:WkB→Ra
+coordinate function on WkB. Setu=π′◦π. By Lemma 4.5 we have VQ,η(K)(u|F(E)) = 0
+for all sufficiently large Kand all Euclidean balls E⊂(Rn,DA). Lemma 4.1 implies
+VQ,K(u◦F|E) = 0. This contradicts Lemma 4.2 since Q </summationtext
+idiλi
+λkAand the function u◦F
+is nonconstant. Similarly there is a contradiction if/summationtext
+idiλi
+λkA</summationtext
+jejµj
+µkB. The lemma follows.
+Recall that (see Section 3), if kA≥2, then the restriction of DAto each affine subspace
+Hparallel to/producttext
+i<kAViagrees with DA′, whereA′= [A1,···,AkA−1].
+Lemma 5.9. Denotek=kAandk′=kB. Suppose k≥2. Then each affine subspace H
+ofRnparallel to/producttext
+i<kViis mapped by Fonto an affine subspace parallel to/producttext
+j<k′Wj.
+Furthermore, F|H: (H,DA′)→(F(H),DB′)isη-quasisymmetric, and Finduces an
+η-quasisymmetric map G: (Vk,DAk)→(Wk′,DBk′)such that F((/producttext
+i<kVi)× {y}) =
+(/producttext
+j<k′Wj)×{G(y)}.
+Proof.As in the proof of Lemma 5.3, to establish the first claim it suffi ces to show that each
+affine subspace parallel to/producttext
+i<kViis mapped into an affine subspace parallel to/producttext
+j<k′Wj.
+By Lemma 5.8 we have/summationtext
+idiλi
+λk=/summationtext
+jejµj
+µk′. Pick any Qwith
+/summationtext
+idiλi
+λk< Q <min/braceleftbigg/summationtext
+idiλi
+λk−1,/summationtext
+jejµj
+µk′−1/bracerightbigg
+.
+Supposethere is an affinesubspace Hparallel to/producttext
+i<kVisuch that F(H) is not contained in
+any affine subspace parallel to/producttext
+j<k′Wj. Letπ:/producttext
+jWj→Wk′be the canonical projection.
+20Then there is some coordinate function π′:Wk′→Rsuch that u◦Fis not constant on H,
+whereu=π′◦π. AsQ >/summationtext
+jejµj
+µk′, Lemma 4.5 implies VQ,η(K)(u|F(E)) = 0 for all sufficiently
+largeKand all rectangular boxes E⊂(Rn,DA). By Lemma 4.1 VQ,K(u◦F|E) = 0. This
+contradicts Lemma 4.4 since Q </summationtext
+idiλi
+λk−1and we can choose a rectangular box Esuch that
+u◦Fis not constant on H∩E.
+Since by assumption Fisη-quasisymmetric, it follows from the remark preceding the
+lemma that F|H: (H,DA′)→(F(H),DB′) isη-quasisymmetric.
+Thefirst claim implies that there is a map G:Vk→Wk′such that F((/producttext
+i<kVi)×{y}) =
+(/producttext
+j<k′Wj)×{G(y)}for anyy∈Vk. ThatG: (Vk,DAk)→(Wk′,DBk′) isη-quasisymmetric
+follows from (3.2), (3.3) and the arguments on page 10 of [X].
+Lemma 5.10. Suppose kA= 2. ThenkB= 2and forǫ=λ1/µ1:
+(1)AandǫBhave the same real part Jordan form;
+(2) The map F: (Rn,Dǫ
+A)→(Rn,DB)is aK-quasisimilarity, where Kdepends only on
+A,Bandη.
+Proof.LetHbe an affine subspace of Rnparallel to/producttext
+i<kAVi. By Lemma 5.9 F(H)
+is an affine subspace parallel to/producttext
+j<kBWj, andF|H: (H,DA′)→(F(H),DB′) isη-
+quasisymmetric. Since kA= 2, we have kA′= 1. Now Lemma 5.1 applied to F|Him-
+plieskB′= 1, sokB=kB′+ 1 = 2. Now the η-quasisymmetric map F|H: (H,DA′)→
+(F(H),DB′) becomes ( V1,DA1)→(W1,DB1), and Lemmas 5.7 and 5.2 imply that A1and
+ǫ1B1have the same real part Jordan form, where ǫ1=λ1/µ1. By Lemma 5.9 Finduces an
+η-quasisymmetric map G: (V2,DA2)→(W2,DB2), and hence Lemmas 5.7 and 5.2 again
+imply that A2andǫ2B2have the same real part Jordan form, where ǫ2=λ2/µ2. Lemma
+5.9 also implies d1=e1andd2=e2. Now Lemma 5.8 implies λ1/µ1=λ2/µ2. Hence (1)
+holds.
+To prove (2), we consider two cases. First assume that A1=λ1IandA2=λ2I. In
+this case, (2) follows from (1) and Theorem 3.5. Next we assum e that either A1/\e}atio\slash=λ1Ior
+A2/\e}atio\slash=λ2Iholds. Then Lemma 5.7 implies that either F|H: (H,Dǫ
+A1)→(F(H),DB1) is a
+K1-quasisimilarity with K1depending only on A1,B1andη, orG: (V2,Dǫ
+A2)→(W2,DB2)
+is aK2-quasisimilarity with K2depending only on A2,B2andη. Then the arguments
+similar to those in the proof of Lemma 5.6(2) (also compare wi th Section 4 of [SX]) show
+thatF: (Rn,Dǫ
+A)→(Rn,DB) is aK-quasisimilarity with Kdepending only on A,Band
+η.
+Lemma 5.11. Suppose kA≥2. Then for ǫ=λ1/µ1:
+(1)AandǫBhave the same real part Jordan form;
+(2) The map F: (Rn,Dǫ
+A)→(Rn,DB)is aK-quasisimilarity, where Kdepends only on
+A,Bandη.
+Proof.We induct on kA. The basic step kA= 2 is Lemma 5.10. Now we assume kA=k≥3
+and that the lemma holds for kA=k−1. For each affine subspace HofRnparallel to/producttext
+i<kAVi, the induction hypothesis applied to F|H: (H,DA′)→(F(H),DB′) implies that
+21forǫ=λ1/µ1:
+(a)A′andǫB′have the same real part Jordan form;
+(b) The map F|H: (H,Dǫ
+A′)→(F(H),DB′) is aK-quasisimilarity, where Kdepends only
+onA′,B′andη.
+The statement (a) implies in particular kA−1 =kB−1 (hence kA=kB),λi=ǫµiand
+ei=difori < kA. Now it follows from Lemma 5.8 that λkA=ǫµkA. IfAkAis a multiple of
+I, then (1) follows from Lemmas 5.9 and 5.2. If AkAis not a multiple of I, then Lemma 5.9
+and Lemma 5.7 (1) imply that AkAandǫBkBhave the same real part Jordan form. Hence
+(1) holds as well in this case.
+IfAkAisamultipleof I, then(2)followsfromthestatement(b)aboveandtheargume nts
+in the proof of Lemma 5.6(2). If AkAis not a multiple of I, then Lemma 5.7 (2) implies
+thatG: (Vk,Dǫ
+Ak)→(Wk′,DBk′) is aK1-quasisimilarity with K1depending only on AkA,
+BkBandη. In this case, (2) follows from this, (b) and the arguments in the proof of Lemma
+5.6(2).
+Next we will finish the proofs of the main theorems. So let A,Bbe two arbitrary n×n
+matrices whoseeigenvalues have positive real parts. Let GA,GBbeequipped with arbitrary
+admissible metrics. Then there are nonsingular matrices P,Qsuch that GAis isometric to
+GPAP−1(equipped with the standard metric) and GBis isometric to GQBQ−1(equipped
+with thestandardmetric). Hencebelow in theproofswewill r eplace (Rn,DA) and(Rn,DB)
+with (Rn,DPAP−1) and (Rn,DQBQ−1) respectively. There also exist nonsingular matrices
+P0,Q0such that GP0AP−1
+0andGQ0BQ−1
+0have pinched negative sectional curvature. We
+may choose the same P0AP−1
+0for all conjugate matrices A. Denote by JandJ′the real
+part Jordan forms of AandBrespectively. By Proposition 3.1, there are biLipschitz
+mapsfJ:GP0AP−1
+0→GJandfP:GP0AP−1
+0→GPAP−1. Then Corollary 3.2 implies their
+boundarymaps ∂fJ: (Rn,DP0AP−1
+0)→(Rn,DJ) and∂fP: (Rn,DP0AP−1
+0)→(Rn,DPAP−1)
+are also biLipschitz. Similarly, there are biLipschitz map sfJ′:GQ0BQ−1
+0→GJ′and
+fQ:GQ0BQ−1
+0→GQBQ−1, whose boundary maps ∂fJ′: (Rn,DQ0BQ−1
+0)→(Rn,DJ′) and
+∂fQ: (Rn,DQ0BQ−1
+0)→(Rn,DQBQ−1) are also biLipschitz.
+Completing the proof of Theorem 1.1 . The“if” part follows from Proposition 3.1 since
+theboundarymap of aquasiisometry between Gromov hyperbol icspaces is quasisymmetric.
+We will prove the “only if” part. So we suppose ( Rn,DPAP−1) and (Rn,DQBQ−1) are
+quasisymmetric. Since the four maps ∂fP,∂fJ,∂fQand∂fJ′are biLipschitz, we see that
+(Rn,DJ) and (Rn,DJ′) are quasisymmetric. Now it follows from Lemma 5.2, Lemma 5. 7(1)
+and Lemma 5.11(1) that JandǫJ′have the same real part Jordan form, where ǫ=λ1/µ1.
+HenceAandǫBalso have the same real part Jordan form.
+Theorem 5.12. LetAandBben×nmatrices whose eigenvalues all have positive real
+parts, and let GAandGBbe equipped with arbitrary admissible metrics. Denote by λ1and
+µ1the smallest real parts of the eigenvalues of AandBrespectively, and set ǫ=λ1/µ1. If
+the real part Jordan form of Ais not a multiple of the identity matrix In, then for every
+η-quasisymmetric map F: (Rn,DA)→(Rn,DB), the map F: (Rn,Dǫ
+A)→(Rn,DB)is a
+K-quasisimilarity, where Kdepends only on η,A,Band the metrics on GA,GB.
+22Completing the proof of Theorem 5.12 . LetF: (Rn,DPAP−1)→(Rn,DQBQ−1) be
+anη-quasisymmetric map. Notice that the biLipschitz constant of the map ∂fJdepends
+only onA(actually the conjugacy class of A) as the same P0AP−1
+0is chosen for all matrices
+Ain the same conjugate class. However, the biLipschitz const ant of∂fPdepends on P
+and hence on the admissible metric on GA. Hence ∂fJ◦∂f−1
+PisL1-biLipschitz for some
+constant L1depending only on Aand the admissible metric on GA. Similarly, ∂fJ′◦∂f−1
+Q
+isL2-biLipschitz for some constant L2depending only on Band the admissible metric on
+GB. It follows that
+G:= (∂fJ′◦∂f−1
+Q)◦F◦(∂fJ◦∂f−1
+P)−1: (Rn,DJ)→(Rn,DJ′)
+isη1-quasisymmetric, where η1depends only on L1,L2andη. Now Lemma 5.7(2) and
+Lemma 5.11(2) imply that Gis aK-quasisimilarity, where Kdepends only on J,J′and
+η1. Consequently, Fis aKL1L2-quasisimilarity.
+6 Proof of the corollaries
+In this section we prove the corollaries from the introducti on and also derive a local version
+of Theorem 1.1.
+LetMbe a Hadamard manifold with pinched negative sectional curv ature,ξ0∈∂M,
+andx0∈Ma base point. Let γbe the geodesic with γ(0) =x0andγ(∞) =ξ0. LethM=
+−Bγ:M→R, whereBγis the Busemann function associated with γ. SetHt=h−1
+M(t).
+A parabolic visual quasimetric Dξ0on∂M\{ξ0}is defined as follows. For ξ,η∈∂M\{ξ0},
+Dξ0(ξ,η) =etif and only if ξξ0∩Htandηξ0∩Hthave distance 1 in the horosphere Ht.
+LetNbe another Hadamard manifold with pinched negative section al curvature, and
+f:M→Na quasiisometry. For any ξ∈∂Mandx∈M, we setξ′=∂f(ξ) andx′=f(x).
+Letγ′be the geodesic with γ′(0) =x′
+0andγ′(∞) =ξ′
+0. SethN=−Bγ′, whereBγ′is
+the Busemann function associated with γ′. Denote H′
+t=h−1
+N(t). LetDξ′
+0be the parabolic
+visual quasimetric on ∂N\{ξ′
+0}with respect to the base point x′
+0.
+Lemma 6.1. Lets >0. Then the following three conditions are equivalent:
+(1) there is a constant C≥0such that the Hausdorff distance HD(f(Ht),H′
+st)≤Cfor all
+t;
+(2) the boundary map ∂f: (∂M\{ξ0},Ds
+ξ0)→(∂N\{ξ′
+0},Dξ′
+0)is biLipschitz;
+(3) there exists a constant C≥0such that s·d(x,y)−C≤d(f(x),f(y))≤s·d(x,y)+C
+for allx,y∈M.
+Proof.The arguments in the proof of Lemma 6.4 in [SX] shows (2) = ⇒(1), while the
+arguments at the end of [SX] (proof of Corollary 1.2) yields ( 1) =⇒(3). We shall prove
+(3) =⇒(1) and (1) = ⇒(2).
+(1) =⇒(2): Suppose (1) holds. Let ξ/\e}atio\slash=η∈∂M\{ξ0}. Assume Dξ0(ξ,η) =etand
+Dξ′
+0(ξ′,η′) =et′. Letγξbethegeodesicjoining ξandξ0withγξ(0)∈H0andγξ(∞) =ξ0. By
+Lemma6.2of[SX], γξ(t)isaC1-quasicenter of ξ,η,ξ0, andγξ′(t′)isaC1-quasicenter of ξ′,η′,
+ξ′
+0, whereC1dependsonly on the curvature boundsof MandN. Sincefis a quasiisometry,
+23f(γξ(t)) is aC2-quasicenter of ξ′,η′,ξ′
+0, whereC2depends only on C1, the quasiisometry
+constants of fand the curvature bounds of N. It follows that d(f(γξ(t)),γξ′(t′))≤C3,
+whereC3depends only on C1,C2and the curvature bounds of N. By condition (1), the
+pointf(γξ(t)) is within CofH′
+st. It follows that γξ′(t′)∈H′
+t′is within C+C3ofH′
+stand
+so|t′−st| ≤C+C3. Therefore, e−(C+C3)est≤Dξ′
+0(ξ′,η′) =et′≤eC+C3est.
+(3) =⇒(1): Suppose (3) holds. Let ω:R→Mbe any geodesic with ω(0)∈H0and
+ω(∞) =ξ0. Thenf◦ωis a (L1,C1)-quasigeodesic in N, whereL1andC1depend only on s
+andC. By the stability of quasigeodesics in a Gromov hyperbolic s pace, there is a constant
+C2depending only on L1,C1and the Gromov hyperbolicity constant of N, and a complete
+geodesic ω′inNwith one endpoint ξ′
+0such that the Hausdorff distance between ω′(R) and
+f◦ω(R) is at most C2. Lett1< t2. Then it follows from condition (3) and the triangle
+inequality that
+|hN(f(ω(t2)))−hN(f(ω(t1)))−s(t2−t1)| ≤C3,
+whereC3dependsonlyon C,C2andtheGromov hyperbolicity constant of N. Inparticular,
+this applied to ω=γ,t2=tandt1= 0 (ort2= 0 andt1=tift <0) implies |hN(f(γ(t)))−
+st| ≤C3.
+Letx∈Htbe arbitrary. Let ω1be the geodesic with ω1(t) =xandω1(∞) =ξ0. Pick
+anyt2≥twithd(γ(t2),ω1(t2))≤1. By condition (3),
+|hN(f(γ(t2)))−hN(f(ω1(t2)))| ≤d(f(γ(t2)),f(ω1(t2)))≤s+C.
+The discussion from the preceding paragraph implies
+|hN(f(ω1(t2)))−hN(f(ω1(t)))−s(t2−t)| ≤C3
+and
+|hN(f(γ(t2)))−hN(f(γ(t)))−s(t2−t)| ≤C3.
+These inequalities together with the one at the end of last pa ragraph imply
+|hN(f(ω1(t)))−st| ≤C4:= 3C3+s+C.
+Hencef(x) =f(ω1(t)) is within C4ofH′
+st. This shows f(Ht) lies in the C4-neighborhood of
+H′
+st. Byconsideringaquasi-inverseof f,weseethattheHausdorffdistance HD(f(Ht),H′
+st)≤
+C5, whereC5depends only on s,Cand the Gromov hyperbolicity constants of MandN.
+A local version of Theorem 1.1 also holds:
+Theorem 6.2. LetA,Bben×nmatrices whose eigenvalues have positive real parts, and le t
+GAandGBbe equipped with arbitrary admissible metrics. Let U⊂(Rn,DA),V⊂(Rn,DB)
+be open subsets, and F: (U,DA)→(V,DB)anη-quasisymmetric map. Then AandsB
+have the same real part Jordan form for some s >0.
+Proof.By Corollary 3.2 and the discussion before the proof of Theor em 1.1 we may assume
+AandBare in real part Jordan form. Fix a base point x∈U. We may assume both xand
+24F(x) are the origin o. Then there is some constant a >1 and a sequence of distinct triples
+(xk,yk,zk) fromUsatisfying xk=o,DA(xk,yk)→0 and
+DA(xk,yk)
+DA(xk,zk),DA(yk,xk)
+DA(yk,zk),DA(zk,xk)
+DA(zk,yk)∈(1/a,a).
+Such a triple can be chosen from the eigenspace of λ1(the smallest eigenvalue of A) so that
+xk=ois the middle point of the segment ykzk. SinceFisη-quasisymmetric, there is a
+constant b >0 depending only on aandηsuch that:
+DB(F(xk),F(yk))
+DB(F(xk),F(zk)),DB(F(yk),F(xk))
+DB(F(yk),F(zk)),DB(F(zk),F(xk))
+DB(F(zk),F(yk))∈(1/b,b).
+Assume DA(xk,yk) =e−tkandDB(F(xk),F(yk)) =e−t′
+k. Then etkA: (U,etkDA)→
+(etkAU,DA) is an isometry. Hence the sequence of pointed metric spaces (U,etkDA,o)
+converges (as k→ ∞) in the pointed Gromov-Hausdorff topology towards ( Rn,DA). Sim-
+ilarly, the sequence of pointed metric spaces ( V,et′
+kDB,o) converges (as k→ ∞) in the
+pointed Gromov-Hausdorff topology towards ( Rn,DB). On the other hand, the sequence
+of maps Fk=F: (U,etkDA)→(V,et′
+kDB) are all η-quasisymmetric, and the triples
+(xk,yk,zk)∈(U,etkDA) and (F(xk),F(yk),F(zk))∈(V,et′
+kDB) are uniformly separated
+and uniformly bounded. Now the compactness property of quas isymmetric maps implies
+that a subsequenceof {Fk}converges in thepointed Gromov-Hausdorff topology towards an
+η-quasisymmetric map F′: (Rn,DA)→(Rn,DB). Now the theorem follows from Theorem
+1.1.
+Lemma 6.3. LetF:∂GA→∂GBbe a quasisymmetric map, where ∂GAand∂GBare
+equipped with visual metrics. Let ξ0∈∂GA,ξ′
+0∈∂GBbe the points corresponding to upward
+oriented vertical geodesic rays. If the real part Jordan form ofAis not a multiple of the
+identity matrix, then F(ξ0) =ξ′
+0.
+Proof.The proof is similar to that of Proposition 3.5, [X]. Suppose F(ξ0)/\e}atio\slash=ξ′
+0. By the
+relation between visual metrics and parabolic visual metri cs (see Section 5 of [SX]), the
+map
+F: (Rn\{F−1(ξ′
+0)},DA)→(Rn\{F(ξ0),DB)
+is locally quasisymmetric. By Theorem 6.2, AandsBhave the same real part Jordan form
+for some s >0. In particular, we have kB=kA; the fibers of πAandπBhave the same
+dimension if kA= 1, and the subspaces/producttext
+i<kAViand/producttext
+j<kBWjhave the same dimension
+ifkA≥2. IfkA= 1, let Hbe a fiber of πAnot containing F−1(ξ′
+0); ifkA≥2, then let
+Hbe an affine subspace parallel to/producttext
+i<kAViand not containing F−1(ξ′
+0). Letmbe the
+topological dimension of H. ThenH∪{ξ0} ⊂∂GAis anm-dimensional topological sphere.
+SinceF(ξ0)/\e}atio\slash=ξ′
+0andF−1(ξ′
+0)/∈H, the image F(H∪{ξ0}) is am-dimensional topological
+sphere in Rn=∂GB\{ξ′
+0}. In particular, F(H∪{ξ0}) (and hence F(H)) is not contained
+in any fiber of πB(ifkA= 1) or any affine subspace parallel to/producttext
+j<kBWj(ifkA≥2). Now
+the arguments of Lemma 5.3 and Lemma 5.9 yield a contradictio n. Hence F(ξ0) =ξ′
+0.
+25Now Corollary 1.3 follows from Proposition 3.1, Lemma 6.3, T heorem 1.1 and the fact
+that a quasiisometry between Gromov hyperbolic spaces indu ces a quasisymmetric map
+between the ideal boundaries.
+Proofs of Corollary 1.4 and Corollary 1.5 . We use the notation introduced before the
+proof of Theorem 1.1. Let f:GPAP−1→GQBQ−1be a quasiisometry. By Lemma 6.3,
+finduces a boundary map ∂f: (Rn,DPAP−1)→(Rn,DQBQ−1), which is quasisymmetric.
+By Theorem 1.2, there is some s >0 such that ∂f: (Rn,Ds
+PAP−1)→(Rn,DQBQ−1) is
+biLipschitz. Since ∂fPand∂fQare also biLipschitz, the boundary map ∂(f−1
+Q◦f◦fP) :
+(Rn,Ds
+P0AP−1
+0)→(Rn,DQ0BQ−1
+0) off−1
+Q◦f◦fP:GP0AP−1
+0→GQ0BQ−1
+0is biLipschitz. Since
+GP0AP−1
+0andGQ0BQ−1
+0have pinched negative sectional curvature, Lemma 6.1 impli es the
+mapf−1
+Q◦f◦fPis height-respecting and is an almost similarity. By Propos ition 3.1 and
+Corollary 3.2 the two maps fPandfQare height-respecting and are almost similarities.
+Hencefis height-respecting and is an almost similarity.
+The proof of Corollary 1.6 is the same as in [SX] (Corollary 1. 3).
+Next we give a proof of Corollary 1.7. Recall that a group Gof quasisimilarity maps
+of (Rn,DA) is a uniform group if there is some K≥1 such that every element of Gis
+aK-quasisimilarity. Dymarz and Peng have established the fol lowing (see [DP] for the
+definition of almost homotheties):
+Theorem 6.4. ([DP])LetAbe a square matrix whose eigenvalues all have positive real
+parts, and Gbe a uniform group of quasisimilarity maps of (Rn,DA). If the induced action
+ofGon the space of distinct couples of Rnis cocompact, then Gcan be conjugated by a
+biLipschitz map into the group of almost homotheties.
+Proof of Corollary 1.7. LetGbe a group of quasim¨ obius maps of ( ∂GA,d) such that
+every element of Gisη-quasim¨ obius, where dis a fixed visual metric on ∂GA. Letξ0∈∂GA
+be the point corresponding to vertical geodesic rays. Since the real part Jordan form of A
+is not a multiple of the identity matrix, Lemma 6.3 implies th at the point ξ0is fixed by all
+quasisymmetric maps ∂GA→∂GA. HenceGrestricts to a group of quasisymmetric maps
+of (Rn,DA). For any three distinct points ξ1,ξ2,ξ3∈Rn=∂GA\{ξ0}, the quasim¨ obius
+condition applied to the quadruple Q= (ξ1,ξ2,ξ3,ξ0) implies that every element of Gis an
+η-quasisymmetric map of ( Rn,DA). Now Theorem 1.2 implies that there is some K≥1
+such that every element of Gis aK-quasisimilarity. In other words, Gis a uniform group
+of quasisimilarities of ( Rn,DA).
+Since the induced action of Gon the space of distinct triples of ( ∂GA,d) is cocompact,
+there is some δ >0 such that for any distinct triple ( ξ1,ξ2,ξ3), there is some g∈Gsuch
+thatd(g(ξi),g(ξj))≥δfor all 1 ≤i/\e}atio\slash=j≤3. Now let ξ/\e}atio\slash=ξ2∈Rn=∂GA\{ξ0}be
+any distinct couple. Then there is an element g∈Gas above corresponding to the triple
+(ξ0,ξ1,ξ2). Since g(ξ0) =ξ0, there are two constants a,b >0 depending only on δsuch
+thatDA(g(ξ1),o)≤b,DA(g(ξ2),o)≤bandDA(g(ξ1),g(ξ2))≥a. This shows that Gacts
+cocompactly on the space of distinct couples of ( Rn,DA).
+Now the corollary follows from the theorem of Dymarz-Peng.
+267 QS maps in the Jordan block case
+In this section we describe all the quasisymmetric maps on th e ideal boundary in the case
+whenAis a Jordan block.
+Theorem 7.1. LetJn=In+Nbe then×n(n≥2) Jordan block with eigenvalue 1. Then
+a bijection F: (Rn,DJn)→(Rn,DJn)is a quasisymmetric map if and only of there are
+constants a0/\e}atio\slash= 0,a1,···,an−2∈R, a vector v∈Rnand a Lipschitz map C:R→Rsuch
+that
+F(x) = (a0In+a1N+···+an−2Nn−2)x+v+˜C(x)
+for allx= (x1,···,xn)T∈Rn, where˜C(x) = (C(xn),0,···,0)T. HereTindicates matrix
+transpose.
+We first prove that every map of the indicated form is actually biLipschitz. Notice that
+the map Fdescribed in the theorem decomposes as F=F1◦F2◦F3, withF1(x) =x+v,
+F2(x) =x+˜C1(x) andF3(x) = (a0In+a1N+···+an−2Nn−2)x, whereC1:R→Ris
+defined by C1(t) =C(t/a0). Since DJnis invariant under Euclidean translations, F1is an
+isometry. We shall prove that F2andF3are biLipschitz in the next two lemmas.
+For ann×nmatrixM= (mij), setQ(M) =/summationtext
+i,jm2
+ij. We will use the fact ||M|| ≤
+Q(M)1
+2, where||M||denotes the operator norm of M.
+Lemma 7.2. Suppose C:R→RisL-Lipschitz for some L >0. ThenF2: (Rn,DJn)→
+(Rn,DJn),F2(x) =x+˜C(x)isL′-biLipschitz, where L′depends only on Land the dimension
+n.
+Proof.Letx= (x1,···,xn)Tandx′= (x′
+1,···,x′
+n)Tbe two arbitrary points in Rn.
+ThenF2(x) = (x1+C(xn),x2,···,xn)TandF2(x′) = (x′
+1+C(x′
+n),x′
+2,···,x′
+n)T. Assume
+DJn(x,x′) =etandDJn(F2(x),F2(x′)) =es. We need to show that there is some constant
+adepending only on Landnsuch that |t−s| ≤a.
+SinceDJn(x,x′) =et, wehave et=|e−tN(x′−x)|, seeSection3. Similarly, DJn(F2(x),F2(x′)) =
+esimplieses=|e−sN(F2(x′)−F2(x))|. Notice F2(x′)−F2(x) = (x′−x) +w, where
+w= (C(x′
+n)−C(xn),0,···,0)T. The only nonzero entry in e−tNwisC(x′
+n)−C(xn). So
+we have
+|e−tNw|=|C(x′
+n)−C(xn)| ≤L|x′
+n−xn|.
+On the other hand, the last entry in e−tN(x′−x) is (x′
+n−xn), hence
+|e−tNw| ≤L|x′
+n−xn| ≤L|e−tN(x′−x)|=Let.
+We write
+e−sN(F2(x′)−F2(x)) =e(t−s)Ne−tN[(x′−x)+w] =e(t−s)N[e−tN(x′−x)+e−tNw].
+Now
+es=|e−sN(F2(x′)−F2(x))|=|e(t−s)N[e−tN(x′−x)+e−tNw]|
+≤ ||e(t−s)N||·|e−tN(x′−x)+e−tNw| ≤ ||e(t−s)N||·/braceleftbig
+|e−tN(x′−x)|+|e−tNw|/bracerightbig
+≤ ||e(t−s)N||·/braceleftbig
+et+Let/bracerightbig
+≤et(1+L)/radicalig
+Q(e(t−s)N).
+27From this we derive es−t≤(1 +L)/radicalbig
+Q(e(t−s)N). Notice that Q(e(t−s)N) is a polynomial
+of degree 2( n−1) int−sthat depends only on n. It follows that there is a constant
+adepending only on nandLsuch that s−t≤a. Since the inverse of F2isF−1
+2(x) =
+x+ (−C(xn),0,···,0)T, the above argument applied to F−1
+2yieldst−s≤a. Hence
+|s−t| ≤a, and we are done.
+Lemma 7.3. LetF3: (Rn,DJn)→(Rn,DJn)be given by
+F3(x) = (a0In+a1N+···+an−1Nn−1)x,
+wherea0/\e}atio\slash= 0,a1,···,an−1∈Rare constants. Then F3isL-biLipschitz for some Ldepend-
+ing only on nanda0,a1,···,an−1.
+Proof.The proof is similar to that of Lemma 7.2. Let x,x′∈Rnbe arbitrary. Assume
+DJn(x,x′) =etandDJn(F3(x),F3(x′)) =es. Then we have et=|e−tN(x′−x)|and
+es=|e−sN(F3(x′)−F3(x))|. We need to find a constant athat depends only on nand the
+numbers a0,···,an−1such that |s−t| ≤a.
+SetB1=e(t−s)NandB2=a0In+a1N+···+an−1Nn−1. Notice that B2commutes
+withN. We have
+es=|e−sN(F3(x′)−F3(x))|=|e(t−s)Ne−tNB2(x′−x)|
+=|B1B2e−tN(x′−x)| ≤ ||B1||·||B2||·|e−tN(x′−x)|
+≤/radicalbig
+Q(B1)/radicalbig
+Q(B2)et.
+Hencees−t≤/radicalbig
+Q(B1)Q(B2). SinceQ(B1)Q(B2) is a polynomial in t−sthat depends only
+onnand the numbers a0,···,an−1, there is some constant a >0 depending only on nand
+a0,···,an−1such that s−t≤a.
+Notice that F−1
+3(x) =B−1
+2x. Set
+β=−/parenleftbigga1
+a0N+···+an−1
+a0Nn−1/parenrightbigg
+.
+Thenβn= 0. We have B2=a0(I−β) andB−1
+2=a−1
+0(I+β+β2+···+βn−1). It follows
+thatB−1
+2has the expression B−1
+2=a−1
+0I+b1N+···+bn−2Nn−2+bn−1Nn−1, where
+b1,···,bn−1are constants depending only on a0,···,an−1. Now the preceding paragraph
+implies that t−s≤a′for some constant a′depending only on nanda−1
+0,b1,···,bn−1,
+hence only on nanda0,···,an−1. Therefore |s−t| ≤max{a,a′}, and the proof of Lemma
+7.3 is complete.
+To prove that every quasisymmetric map has the described typ e, we induct on n. The
+basic step n= 2 is given by Theorem 3.6. Now we assume n≥3 and that Theorem 7.1
+holds for Jn−1.
+LetF: (Rn,DJn)→(Rn,DJn) be a quasisymmetric map. Let Fi(i= 1,···,n−1) be
+the foliation of Rnconsisting of affine subspaces parallel to the linear subspac e
+Hi:={x= (x1,···,xn)T∈Rn:xi+1=···=xn= 0}.
+28Then the proof of Theorem 1.2 shows that the foliation Fiis preserved by F. To be more
+precise, if His an affine subspace parallel to Hi, thenF(H) is also an affine subspace
+parallel to Hi. In particular, Fmaps every line parallel to the x1-axis (that is, parallel to
+H1) to a line parallel to the x1-axis, and maps every horizontal hyperplane (that is, paral lel
+toHn−1) to a horizontal hyperplane. It follows that there is a map G:Rn−1→Rn−1such
+that for any y∈Rn−1,F(R× {y}) =R× {G(y)}. For each y∈Rn−1, there is a map
+H(·,y) :R→Rsuch that F(x1,y) = (H(x1,y),G(y)).
+Arguments similar to the proofs of Lemmas 3.3 and 3.4 show the following:
+(1) for each y∈Rn−1, the restriction of DJntoR×{y}agrees with the Euclidean distance
+onR;
+(2) forany two y1,y2∈Rn−1, theHausdorffdistance withrespect to DJn:HD(R×{y1},R×
+{y2}) =DJn−1(y1,y2);
+(3) for any p= (x1,y1)∈R×Rn−1and any y2∈Rn−1, we have DJn(p,R× {y2}) =
+DJn−1(y1,y2).
+Hence each H(·,y) : (R,|· |)→(R,| ·|) is quasisymmetric, and the arguments on page 10
+of [X] shows that G: (Rn−1,DJn−1)→(Rn−1,DJn−1) is also quasisymmetric.
+Now the induction hypothesis applied to Gshows that there are constants a0/\e}atio\slash= 0,a1,
+···,an−3,bi(2≤i≤n) and a Lipschitz map g:R→Rsuch that
+G
+x2
+x3
+···
+xn
+=
+a0x2+a1x3+···+an−3xn−1+b2+g(xn)
+a0x3+a1x4+···+an−3xn+b3
+···
+a0xn−1+a1xn+bn−1
+a0xn+bn
+.
+Notice that the horizontal hyperplane Rn−1× {xn}at height xnis mapped by Fto the
+horizontal hyperplane Rn−1×{a0xn+bn}at height a0xn+bn. Since the restriction of DJn
+to a horizontal hyperplane agrees with DJn−1(Lemma 3.3), the map
+F: (Rn−1×{xn},DJn−1)→(Rn−1×{a0xn+bn},DJn−1)
+is quasisymmetric. Now the induction hypothesis, the fact F(x1,y) = (H(x1,y),G(y)) and
+the expression of Gimply that
+H(x1,y) =a0x1+a1x2+···+an−3xn−2+c1(xn)+c2(xn−1,xn),
+wherec1:R→Randc2:R2→Rare two maps and for each fixed v,c2(u,v) is Lipschitz
+inu. Since Fis a homeomorphism, c1andc2are continuous. Define c3:R2→Rby
+c3(u,v) =c1(v) +c2(u,v). After composing Fwith a map of the described type, we may
+assumeFhas the following form
+F(x1,x2,···,xn) = (x1+c3(xn−1,xn),x2+g(xn),x3,···,xn).
+We need to show that there are constants an−2,d2and a Lipschitz map C:R→Rsuch
+thatg(xn) =an−2xn+d2andc3(xn−1,xn) =an−2xn−1+C(xn).
+Lemma 7.4. There is a constant Lsuch that the following holds for all u,v,v′∈R:
+/vextendsingle/vextendsingle/vextendsingle/braceleftbig
+c3(u+(v′−v)ln|v′−v|,v′)−c3(u,v)/bracerightbig
+−ln|v′−v|/braceleftbig
+g(v′)−g(v)/bracerightbig/vextendsingle/vextendsingle/vextendsingle≤L|v′−v|.
+29Proof.Letu,v,v′∈R. Letx∈Rnwithxn−1=u,xn=v. Sett= ln|v′−v|and let
+y= (y1,···,yn)Tbe the unique solution of e−tNy= (0,···,0,v′−v)T. Letx′=x+y.
+Noticeyn=v′−v,yn−1= (v′−v)ln|v′−v|,x′
+n=v′and
+x′
+n−1=xn−1+yn−1=u+(v′−v)ln|v′−v|.
+Notice also that tis the smallest solution for et=|e−tN(x′−x)|and soDJn(x,x′) =et.
+Suppose DJn(F(x),F(x′)) =es. Thenes=/vextendsingle/vextendsinglee−sN(F(x′)−F(x))/vextendsingle/vextendsingle. By Theorem 1.2, Fis
+L1-biLipschitz for some L1≥1. Hence et/L1≤es≤L1et. It follows that |t−s| ≤lnL1.
+Now we write
+e−sN(F(x′)−F(x)) =e−sN(x′−x)+e−sN
+c3(x′
+n−1,x′
+n)−c3(xn−1,xn)
+g(x′
+n)−g(xn)
+0
+·
+·
+·
+0
+
+=e(t−s)Ne−tN(x′−x)+e(t−s)Ne−tN
+c3(x′
+n−1,x′
+n)−c3(xn−1,xn)
+g(x′
+n)−g(xn)
+0
+·
+·
+·
+0
+
+=e(t−s)N
+/braceleftbig
+c3(x′
+n−1,x′
+n)−c3(xn−1,xn)/bracerightbig
+−t/braceleftbig
+g(x′
+n)−g(xn)/bracerightbig
+g(x′
+n)−g(xn)
+0
+·
+·
+·
+0
+x′
+n−xn
+.
+Set
+τ=/braceleftbig
+c3(x′
+n−1,x′
+n)−c3(xn−1,xn)/bracerightbig
+−t/braceleftbig
+g(x′
+n)−g(xn)/bracerightbig
+.
+The first entry of e−sN(F(x′)−F(x)) is
+q:=τ+(t−s)/braceleftbig
+g(x′
+n)−g(xn)/bracerightbig
++(t−s)n−1
+(n−1)!(x′
+n−xn).
+We have
+|q| ≤ |e−sN(F(x′)−F(x))|=es≤L1et=L1|v′−v|.
+Recall that gisL2-Lipschitz for some L2≥0. Hence,
+|g(x′
+n)−g(xn)| ≤L2|x′
+n−xn|=L2|v′−v|.
+30Now it follows from |t−s| ≤lnL1and the triangle inequality that
+|τ| ≤/parenleftbigg
+L1+L2lnL1+(lnL1)n−1
+(n−1)!/parenrightbigg
+|v′−v|.
+Recall that the map gis Lipschitz and for each fixed v,c3(u,v) is Lipschitz in u. Hence
+gis differentiable a.e., and for each fixed v, the partial derivative∂c3
+∂uexists for a.e. u.
+Lemma 7.5. Letvbe any point such that g′(v)exists. Then c3(u,v) =c3(0,v)+g′(v)ufor
+allu.
+Proof.Fix an arbitrary u∈R. Leta >0. For any positive integer n, define ( y0,z0) = (u,v)
+and (yi,zi) = (u+ia
+nlna
+n,v+ia
+n) (1≤i≤n). Applying Lemma 7.4 to yi−1,zi−1,ziwe
+obtain: /vextendsingle/vextendsingle/vextendsingle/braceleftbig
+c3(yi,zi)−c3(yi−1,zi−1)/bracerightbig
+−lna
+n/braceleftbig
+g(zi)−g(zi−1)/bracerightbig/vextendsingle/vextendsingle/vextendsingle≤La
+n.
+Now let k=k(n) be the integer part of n/lnn
+a. Thenk
+nlna
+n→ −1 asn→ ∞. Combining
+the above inequalities for 1 ≤i≤kand using the triangle inequality, we obtain
+/vextendsingle/vextendsingle/vextendsingle/braceleftbig
+c3(yk,zk)−c3(u,v)/bracerightbig
+−lna
+n/braceleftbig
+g(zk)−g(v)/bracerightbig/vextendsingle/vextendsingle/vextendsingle≤Lak
+n.
+Now divide both sides byak
+nlnn
+a(which converges to aasn→ ∞), we get:
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/braceleftbig
+c3(yk,zk)−c3(u,v)/bracerightbig
+ak
+nlnn
+a+/braceleftbig
+g(zk)−g(v)/bracerightbig
+ak
+n/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤L
+lnn
+a.
+Asn→ ∞, we have zk=v+ak
+n→v,yk→u−a. Also, since g′(v) exists, we have
+/braceleftbig
+g(zk)−g(v)/bracerightbig
+ak
+n→g′(v).
+Consequently,
+c3(u−a,v)−c3(u,v)
+a+g′(v) = 0.
+Hencec3(u−a,v)−c3(u,v) =−ag′(v) for all u∈Rand alla >0. It follows that
+c3(u,v) =c3(0,v)+g′(v)ufor allu.
+Lemma 7.6. Suppose gis differentiable at v1andv2. Theng′(v1) =g′(v2).
+Proof.By Lemma 7.5, we have c3(u,v1) =c3(0,v1)+ug′(v1) and
+c3(u+[v2−v1]ln|v2−v1|,v2) =c3(0,v2)+(u+[v2−v1]ln|v2−v1|)g′(v2)
+for allu. Now Lemma 7.4 applied to u,v1,v2implies that/vextendsingle/vextendsingleu(g′(v2)−g′(v1))/vextendsingle/vextendsingle≤Cholds
+for allu, whereCis a quantity independent of u. It follows that g′(v2)−g′(v1) = 0.
+31Completing the proof of Theorem 7.1 . Lemma 7.6 implies that gis an affine function
+and hence there are constants a,bsuch that g(v) =av+b. It now follows from Lemma
+7.5 that for any vwe have c3(u,v) =c3(0,v) +au. To finish the proof of Theorem 7.1, it
+remains to show that c3(0,v) is Lipschitz in v. This follows immediately from Lemma 7.4
+after plugging in the formulas for gandc3.
+Now the proof of Theorem 7.1 is complete.
+8 A Liouville type theorem
+In this section we prove a Liouville type theorem for GAin the case when Ais a Jordan
+block: every conformal map of the ideal boundary of GAextends to an isometry of GA.
+LetXandYbe quasimetric spaces with finite Hausdorff dimension. Denot e byHX
+andHYtheir Hausdorff dimensions and by HXandHYtheir Hausdorff measures. We say
+a quasisymmetric map f:X→Yis conformal if:
+(1)Lf(x) =lf(x)∈(0,∞) forHX-almost every x∈X;
+(2)Lf−1(y) =lf−1(y)∈(0,∞) forHY-almost every y∈Y.
+Wenowdescribesomeisometriesof GA. Forany g= (x,t)∈GA=Rn⋊R, theLiegroup
+left translation Lgis an isometry. If g= (x,0), then the boundary map ∂Lg:Rn→Rnof
+Lgis translation by x. Ifg= (0,t), then the boundary map of Lgis the similarity etA. Let
+τ′:GA→GAbe given by τ′(x,t) = (−x,t). Thenτ′is an isometry, and its boundary map
+isτ:Rn→Rn,τ(x) =−x.
+Theorem 8.1. LetJnbe then×n(n≥2) Jordan matrix with eigenvalue 1. Then every
+conformal map F: (Rn,DJn)→(Rn,DJn)is the boundary map of an isometry GJn→GJn.
+Proof.LetF: (Rn,DJn)→(Rn,DJn) be a quasisymmetric map. After composing with the
+boundary maps of isometries described above, we may assume Fhas the following form
+F(x) = (I+a1N+···+an−2Nn−2)x+(C(xn),0,···,0)T,
+whereC:R→Ris Lipschitz. We will prove the following statement by induc ting onn:
+IfFas above is conformal, then a1=···=an−2= 0andCis constant.
+The basic step n= 2 is Theorem 6.3 in [X]. Now we assume n≥3 and that the
+statement holds for Jordan matrices with sizes ≤n−1. Notice that Fmaps every horizontal
+hyperplane H(xn) :=Rn−1×{xn}to itself. By Lemma 3.3 the restriction of DJnonH(xn)
+agrees with the metric DJn−1. It now follows from Fubini’s theorem that for a.e. xn∈R,
+the restricted map
+F|H(xn): (H(xn),DJn−1)→(H(xn),DJn−1)
+is also conformal. Now the induction hypothesis applied to F|H(xn)implies that ai= 0 for
+1≤i≤n−2. It remains to show Cis constant.
+Suppose Cis not constant. Then there is some u∈Rsuch that C′(u)/\e}atio\slash= 0 and
+LF(p) =lF(p) for some p∈H(u). After pre-composing and post-composing with Euclidean
+32translations, we may assume u= 0,C(0) = 0 and pis the origin o. Notice that the
+restriction of Fto thex1-axis is the identity, so LF(o) =lF(o) = 1. Now for any xn>0,
+choosex1,···,xn−1such that x= (x1,···xn)Tsatisfies e−tNx= (0,···,0,xn)T, where
+t= lnxn. It follows that DJn(o,x) =et=xn. Suppose DJn(F(o),F(x)) =es. Then
+es=|e−sNF(x)|. We calculate as before that
+e−sNF(x) =/parenleftbigg
+C(xn)+(t−s)n−1
+(n−1)!xn,(t−s)n−2
+(n−2)!xn,···,(t−s)xn, xn/parenrightbiggT
+.
+SinceLF(o) =lF(o) = 1, we must havees
+et=DJn(F(x),F(o))
+DJn(x,o)→1 asxn→0 and hence
+t−s→0. Now
+es=/vextendsingle/vextendsinglee−sNF(x)/vextendsingle/vextendsingle=xn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggC(xn)
+xn+(t−s)n−1
+(n−1)!,(t−s)n−2
+(n−2)!,···,(t−s),1/parenrightbiggT/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
+Sincexn=et, we have
+es−t=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggC(xn)
+xn+(t−s)n−1
+(n−1)!,(t−s)n−2
+(n−2)!,···,(t−s),1/parenrightbiggT/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
+Now asxn→0, the right hand side converges to
+/vextendsingle/vextendsingle/parenleftbig
+C′(0),0,···,0,1/parenrightbigT/vextendsingle/vextendsingle=/radicalbig
+1+(C′(0))2,
+which is >1 sinceC′(0)/\e}atio\slash= 0. However, the left hand side converges to 1. The contradic tion
+showsCmust be a constant function.
+References
+[B] Z. Balogh, Hausdorff dimension distribution of quasiconformal mappin gs on the
+Heisenberg group, J. Anal. Math. 83(2001), 289–312.
+[BS] M. Bonk, O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal.
+10(2000), no. 2, 266–306.
+[CDP] M. Coornaert, T. Delzant, A. Papadopoulos, G´ eom´ etrie et th´ eorie des groupes, Lec-
+ture Notes in Mathematics 1441(1990).
+[DP] T. Dymarz, I. Peng, Bilipschitz maps of boundaries of certain negatively curve d ho-
+mogeneous spaces , http://www.math.yale.edu/ ∼td252/DymarzPeng-BilipNCHS.pdf
+[EFW1] A. Eskin, D. Fisher, K. Whyte, Coarse differentiation of quasi-isometries I: spaces
+not quasi-isometric to Cayley graphs, http://arxiv.org/pdf/math/0607207.pdf
+[EFW2] A. Eskin, D. Fisher, K.Whyte, Coarse differentiation of quasi-isometries II: Rigid-
+ity for Sol and Lamplighter groups, http://arxiv.org/pdf/0706.0940.pdf
+33[FM] B. Farb, L. Mosher, On the asymptotic geometry of abelian-by-cyclic groups, Acta
+Math.184(2000), no. 2, 145–202.
+[FM2] B. Farb, L. Mosher, A rigidity theorem for the solvable Baumslag-Solitar group s,
+Invent. Math. 131 (1998), no. 2, 419–451.
+[FM3] B. Farb, L. Mosher, Quasi-isometric rigidity for the solvable Baumslag-Solit ar
+groups. II, Invent. Math. 137 (1999), no. 3, 613–649.
+[GV] F. Gehring, J. Vaisala, Hausdorff dimension and quasiconformal mappings, J. London
+Math. Soc. (2) 6(1973), 504–512.
+[H] E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211(1974),
+23–34.
+[Hn] J. Heinonen, Lectures on analysis on metric spaces, Universitext. Springer-Verlag,
+New York, 2001.
+[K] B. Kleiner, Unpublished notes.
+[Pa] F. Paulin, Un groupe hyperbolique est d´ etermin´ e par son bord, J. London Math. Soc.
+(2) 54 (1996), no. 1, 50–74.
+[P] P. Pansu, Dimension conforme et sphere a l’infini des varietes a courbu re negative,
+Ann. Acad. Sci. Fenn. Ser. A I Math. 14(1989), no. 2, 177–212.
+[SX] N. Shanmugalingam, X. Xie, A Rigidity Property of Some Negatively Curved Solvable
+Lie Groups, preprint.
+[X] X. Xie, Quasisymmetric Maps on the Boundary of a Negatively Curved So lvable Lie
+Group, preprint.
+Address:
+Xiangdong Xie: Dept. of Mathematical Sciences, Georgia Sou thern University, Statesboro,
+GA 30460, U.S.A. E-mail: xxie@georgiasouthern.edu
+34
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+arXiv:1001.0148v1  [math.GR]  31 Dec 2009Quasisymmetric Maps on the Boundary of a
+Negatively Curved Solvable Lie Group
+Xiangdong Xie
+Abstract
+We describe all the self quasisymmetric maps on the ideal bou ndary of a
+particular negatively curved solvable Lie group. As applic ations, we prove a
+Liouville type theorem, and derive some rigidity propertie s for quasiisometries
+of the solvable Lie group.
+Keywords. quasiisometry, quasisymmetric map, negatively curved solvable Lie
+groups.
+Mathematics Subject Classification (2000). 20F65, 30C65, 53C20.
+1 Introduction
+Inthispaperwestudyquasisymmetricmapsontheidealbound aryofaparticularnegatively
+curved solvable Lie group.
+Let
+A=/parenleftbigg1 1
+0 1/parenrightbigg
+.
+LetRact onR2by (t,v)→etAv(t∈R,v∈R2). We denote the corresponding semi-direct
+product by GA=R2⋊AR. That is, GA=R2×Ras a smooth manifold, and the group
+operation is given by:
+(v,t)·(w,s) = (v+etAw,t+s)
+for all (v,t),(w,s)∈R2×R. The group GAis a simply connected solvable Lie group.
+We endow GAwith the left invariant Riemannian metric determined by tak ing the
+standard Euclidean metric at the identity of GA=R2×R=R3. With this metric GAhas
+pinched negative sectional curvature (and so is Gromov hype rbolic). Hence GAhas a well
+defined ideal boundary ∂GA. There is a so-called cone topology on GA=GA∪∂GA, in
+which∂GAis homeomorphic to the 2-dimensional sphere and GAis homeomorphic to the
+closed 3-ball in the Euclidean space. For each v∈R2, the map γv:R→GA,γv(t) = (v,t)
+is a geodesic. We call such a geodesic a vertical geodesic. It can be checked that all vertical
+1geodesics are asymptotic as t→+∞. Hence they define a point ξ0in the ideal boundary
+∂GA.
+Each geodesic ray in GAis asymptotic to either an upward oriented vertical geodesi c or
+a downward oriented vertical geodesic. The upward oriented geodesics are asymptotic to ξ0
+and the downward oriented vertical geodesics are in 1-to-1 c orrespondence with R2. Hence
+∂GA\{ξ0}can be naturally identified with R2.
+For any properGromov hyperbolicgeodesicspace Xandanyξ∈∂X, thereareso-called
+parabolic visual (quasi)metrics on ∂X\{ξ}. See [SX], Section 5. In our case, a parabolic
+visual quasimetric Don∂GA\{ξ0}is given by:
+D((x1,y1),(x2,y2)) = max/braceleftbig/vextendsingle/vextendsingley2−y1/vextendsingle/vextendsingle,/vextendsingle/vextendsingle(x2−x1)−(y2−y1)ln|y2−y1|/vextendsingle/vextendsingle/bracerightbig
+for all (x1,y1),(x2,y2)∈R2=∂GA\{ξ0}, where 0ln0 is understood to be 0.
+We remark that Dis not a metric on R2, but merely a quasimetric. Recall that a
+quasimetric ρonasetAisafunction ρ:A×A→Rsatisfyingthefollowing threeconditions:
+(1)ρ(x,y) =ρ(y,x)forallx,y∈A; (2)ρ(x,y)≥0forallx,y∈Aandρ(x,y) = 0ifandonly
+ifx=y; (3) there is some M≥1 such that ρ(x,z)≤M(ρ(x,y)+ρ(y,z)) for all x,y,z∈A.
+For each M≥1, there is a constant ǫ0>0 such that ρǫis biLipschitz equivalent to a metric
+for all quasimetric ρwith constant Mand all 0 < ǫ≤ǫ0, see Proposition 14.5. in [Hn].
+Letη: [0,∞)→[0,∞) be a homeomorphism. A bijection F:X→Ybetween two
+quasimetric spaces is η-quasisymmetric if for all distinct triples x,y,z∈X, we have
+d(F(x),F(y))
+d(F(x),F(z))≤η/parenleftbiggd(x,y)
+d(x,z)/parenrightbigg
+.
+A mapF:X→Yis quasisymmetric if it is η-quasisymmetric for some η.
+The following is the main result of the paper.
+Theorem 1.1. Every quasisymmetric map F: (R2,D)→(R2,D)is a biLipschitz map.
+Furthermore, a bijection F: (R2,D)→(R2,D)is a quasisymmetric map if and only of it
+has the following form: F(x,y) = (ax+c(y),ay+b)for all(x,y)∈R2, wherea/\e}atio\slash= 0,bare
+constants and c:R→Ris a Lipschitz map.
+One should compare this with quasiconformal maps on the sphe re or the Euclidean
+space, where there are plenty of non-biLipschitz quasiconf ormal maps. On the other hand,
+the conclusion of Theorem 1.1 is not as strong as in the cases o f quarternionic hyperbolic
+spaces, Cayley plane ([P2]) and Fuchsian buildings ([BP], [ X]), where every quasisymmetric
+map of the ideal boundary is actually a conformal map. In our c ase, there are many non-
+conformal quasisymmetric maps of the ideal boundary of GA.
+As applications, we describe all the isometries and all the s imilarities of ( R2,D), see
+Proposition 6.1. We also prove a Liouville type theorem for ( R2,D).
+Theorem 1.2. Every conformal map f: (R2,D)→(R2,D)is the boundary map of an
+isometry GA→GA.
+Theorem 1.1 also has geometric consequences. Let L≥1 andC≥0. A (not necessarily
+continuous ) map f:X→Ybetween two metric spaces is an ( L,A)-quasiisometry if:
+2(1)d(x1,x2)/L−C≤d(f(x1),f(x2))≤Ld(x1,x2)+Cfor allx1,x2∈X;
+(2) for any y∈Y, there is some x∈Xwithd(f(x),y)≤C.
+In the case L= 1, we call fanalmost isometry .
+Corollary 1.3. Every self quasiisometry of GAis an almost isometry.
+Notice thatanalmost isometryisnotnecessarily afinitedis tanceaway fromanisometry.
+Acknowledgment . IwouldliketothankBruceKleinerforstimulatingdiscuss ions. Ithank
+him for allowing me to include his proof of the fact that quasi symmetric maps preserve the
+horizontal foliation (Section 3). I also would like to thank the Department of Mathematical
+Sciences at Georgia Southern University for generous trave l support.
+2 Quasimetrics on the ideal boundary
+In this section, we will define three different parabolic visua l quasimetrics on the ideal
+boundary, and find an explicit formula for one of them. The thr ee quasimetrics are biLips-
+chitz equivalent with each other.
+LetAandGAbe as in the Introduction. We endow GAwith the left invariant metric
+determined by taking the standard Euclidean metric at the id entity of GA≈R2×R=R3.
+At a point ( x,t)∈R2×R≈GA, the tangent space is identified with R2×R, and the
+Riemannian metric is given by the symmetric matrix
+/parenleftbiggQA(t) 0
+0 1/parenrightbigg
+,
+whereQA(t) =e−tATe−tA. HereATdenotes the transpose of A. With this metric GAhas
+sectional curvature −(6+√
+29)/4 =−b2≤K≤ −a2=−(6−√
+29)/4. Hence GAhas a well
+defined ideal boundary ∂GA. All vertical geodesics γv(v∈R2) are asymptotic as t→+∞.
+Hence they define a point ξ0in the ideal boundary ∂GA.
+The sets R2×{t}(t∈R) are horospheres centered at ξ0. For each t∈R, the induced
+metric on the horosphere R2×{t} ⊂GAis determined by the quadratic form QA(t). This
+metric has distance formula dR2×{t}((v,t),(w,t)) =|e−tA(v−w)|. Here| · |denotes the
+Euclidean norm.
+Each geodesic ray in GAis asymptotic to either an upward oriented vertical geodesi c or
+a downward oriented vertical geodesic. The upward oriented geodesics are asymptotic to ξ0
+and the downward oriented vertical geodesics are in 1-to-1 c orrespondence with R2. Hence
+∂GA\{ξ0}can be naturally identified with R2.
+We next define three parabolic visual quasimetrics on ∂GA\{ξ0}=R2. Given v,w∈
+R2≈∂GA\{ξ0}, the parabolic visual quasimetric De(v,w) is defined as follows: De(v,w) =
+et, where tis the unique real number such that at height tthe two vertical geodesics
+γvandγware at distance one apart in the horosphere; that is, dRn×{t}((v,t),(w,t)) =
+|e−tA(v−w)|= 1.Here the subscript einDemeans it corresponds to the Euclidean norm.
+Recall that the supernorm on R2is given by: |(x,y)|s= max{|x|,|y|}for all (x,y)∈R2.
+Theparabolicvisualquasimetric Dson∂GA\{ξ0}isdefinedasfollows: Ds(v,w) =et, where
+3tis the smallest real number such that at height tthe two vertical geodesics γvandγware
+at distance one apart with respect to the norm |·|s; that is, |e−tA(v−w)|s= 1.Here the
+subscript sinDsmeans it corresponds to the super norm |·|s.
+Notice that |v|s≤ |v| ≤√
+2|v|sfor allv∈R2. Using this, one can verify the following
+lemma, whose proof is left to the reader.
+Lemma 2.1. For allv,w∈R2we have Ds(v,w)≤De(v,w)≤21/2aDs(v,w), where
+a=/radicalbig
+6−√
+29/2.
+The following result provides a parabolic visual quasimetr icDwhich admits an explicit
+formula and is also biLipschitz equivalent with DeandDs.
+Proposition 2.2. For allv= (x1,y1),w= (x2,y2)∈R2,
+D(v,w)/3≤Ds(v,w)≤3D(v,w),
+whereD(v,w) = max/braceleftbig/vextendsingle/vextendsingley2−y1/vextendsingle/vextendsingle,/vextendsingle/vextendsingle(x2−x1)−(y2−y1)ln|y2−y1|/vextendsingle/vextendsingle/bracerightbig
+and0ln0is understood
+to be0.
+Letg= ((x,y),t)∈R2×Rand denote by Lg:GA→GAthe left translation by g. We
+calculate
+Lg((x′,y′),t′) = ((x+et(x′+ty′),y+ety′),t′+t).
+We see that Lgmaps vertical geodesics to vertical geodesics. It follows t hatLginduces a
+mapTg:R2→R2,
+Tg(x′,y′) = (x+et(x′+ty′),y+ety′).
+SinceLgis an isometry of GAand it translates by tin the vertical direction, the definition
+of the quasimetric Deshows that
+De(Tg(x1,y1),Tg(x2,y2)) =etDe((x1,y1),(x2,y2))
+for all (x1,y1),(x2,y2)∈R2. In other words, Tgis a similarity of ( R2,De) with similarity
+constant et. When t= 0,Tgis simply a Euclidean translation and it is an isometry with
+respect to De. Similar statements also hold for the quasimetric Ds.
+Notice that Euclidean translations are also isometries wit h respect to the function D.
+Thistogether withthesamestatement about Dsimpliesthat wecanassume( x1,y1) = (0,0)
+in order to prove Proposition 2.2.
+Proof of Proposition 2.2. By the preceding remark, we may assume ( x1,y1) = (0,0),
+and write ( x,y) for (x2,y2). Recall Ds((x,y),(0,0)) =etiftis the smallest real number
+such that |e−tA(x,y)|s= 1.We calculate |e−tA(x,y)|s= max{e−t|x−ty|,e−t|y|}. We
+consider several cases:
+Case 1:y= 0. In this case, |e−tA(x,0)|s=e−t|x|. HenceDs((x,0),(0,0)) =et=|x|=
+D((x,0),(0,0)).
+Wheny/\e}atio\slash= 0, we let t0= ln|y|anda=x/y−ln|y|.
+4Case 2: y/\e}atio\slash= 0 and |x−t0y| ≤ |y|. In this case, |e−t0A(x,y)|s= max{e−t0|x−
+t0y|,e−t0|y|}=e−t0|y|= 1. Notice also |e−tA(x,y)|s≥e−t|y|>1 ift < t0. Hence
+Ds((x,y),(0,0)) =et0=|y|=D((x,y),(0,0)).
+Wheny/\e}atio\slash= 0 and |x−t0y|>|y|, we have |a|>1.
+Case 3:y/\e}atio\slash= 0,|x−t0y|>|y|anda >1. In this case, D((x,y),(0,0)) =|x−yln|y||.
+Lett1> t0be the smallest real number tsatisfying e−t|x−ty|= 1. Notice that e−t1|y|<1
+and soDs((x,y),(0,0)) =et1. Setu=t1−t0>0. The equality e−t1|x−t1y|= 1 implies
+eu=a−u. Clearly eu=a−u≤a. We claim eu=a−u≥a/3. Otherwise, u >2a/3.
+This contradicts a=u+eu> u+(1+u). Hence a/3≤eu≤aand
+|x−yln|y||/3 =a|y|/3≤ |y|eu=et1=Ds((x,y),(0,0))≤a|y|=|x−yln|y||.
+Case 4:y/\e}atio\slash= 0,|x−t0y|>|y|anda <−1. In this case, D((x,y),(0,0)) =|x−yln|y||.
+Lett1> t0be the smallest real number tsatisfying e−t|x−ty|= 1. Again we have
+Ds((x,y),(0,0)) =et1. Setu=t1−t0>0. Theequality e−t1|x−t1y|= 1 implies eu=u−a.
+Clearlyeu=u−a >−a. We claim eu=u−a≤ −3a. Otherwise, u >−2aand hence
+−a=eu−u >1+u2/2> u >−2a, a contradiction. Hence |a|=−a≤eu≤ −3a= 3|a|
+and
+|x−yln|y||=|ay| ≤ |y|eu=et1=Ds((x,y),(0,0))≤3|ay|= 3|x−yln|y||.
+We describe some isometries and similarities of the space ( R2,D). The following propo-
+sition can be easily proved by using the formula for D.
+Proposition 2.3. Let(R2,D)be as above.
+(1) Then Euclidean translations of R2are isometries with respect to D;
+(2) Letπ:R2→R2be defined by π(x,y) = (−x,−y). Thenπis an isometry with respect
+toD;
+(3) For any real number t, letλt:R2→R2be defined by λt(x,y) = (et(x+ty),ety). Then
+D(λt(x1,y1),λt(x2,y2)) =et·D((x1,y1),(x2,y2))for all(x1,y1),(x2,y2)∈R2.
+We notice that the three classes of maps in Proposition 2.3 ar e boundary maps of
+isometries of GA. Euclidean translations of R2=∂GA\{ξ0}are boundary maps of left
+translations LgofGAfor elements gof typeg= ((x,y),0)∈R2×R=GA. The map λtis
+the boundary map of left translation Lgforg= ((0,0),t). Finally, τis the boundary map
+of the automorphism τ′:GA→GA,τ′((x,y),t) = ((−x,−y),t). Notice that τ′is indeed an
+automorphism of GAand the tangential map of τ′at the identity is an isometry. It follows
+thatτ′is an isometry of GA.
+3 Quasisymmetric maps preserve horizontal folia-
+tion
+In this section we prove that every self quasisymmetric map o f (R2,D) maps horizontal
+lines to horizontal lines. The proof belongs to Bruce Kleine r. Here I am trying to provide
+5more details and I am responsible for the inaccuracies that m ight result from this. I would
+like to express my gratitude towards Bruce for allowing me to include his argument.
+Definition 3.1. Let (X,ρ) be a quasimetric space and L≥1. A subset A⊂Xis called
+anL-quasi-ball if there is some x∈Xand some r >0 such that B(x,r)⊂A⊂B(x,Lr).
+HereB(x,r) ={y∈X:ρ(y,x)< r}.
+The following notion is key to the proof.
+Definition 3.2. (Kleiner) Pick Q≥1. Letu:X→Rbe a function (not necessarily
+continuous) defined on a quasimetric space, and let Pbe a collection of subsets of X. The
+Q-variation of uoverP– denoted VQ(u,P) – is the quantity
+ΣP∈P[osc(u|P)]Q,
+whereosc(u|p) denotes the oscillation (sup minus inf) of the restriction ofuto the subset
+P⊂X. TheQ-variation VQ(u)ofuis sup{VQ(u,P)}wherePranges over all disjoint
+collections of balls in X. The (Q,K)-variation VQ,K(u)ofuis sup{VQ(u,P)}whereP
+ranges over all disjoint collections of K-quasi-balls in X.
+There are useful variants of this definition, for instance on e can look at the infimum
+over all coverings. Or one can take the infimum over all coveri ngs followed by the sup as
+the mesh size tends to zero. The definition preforms the same f unction as Pansu’s modulus
+[P1], but it seems easier to digest.
+Lemma 3.1. LetF:X→Ybe anη-quasisymmetric map between two quasimetric spaces.
+Then for every function u:X→Rwe have VQ,K(u)≤VQ,η(K)(u◦F−1).
+Proof.For any subset A⊂X, the oscillation of uonAequals the oscillation of u◦F−1on
+F(A). LetPbe a disjoint collection of K-quasi-balls in X. ThenF(P) ={F(P) :P∈ P}
+is a disjoint collection of η(K)-quasi-balls in Y, andVQ(u,P) =VQ(u◦F−1,F(P)). Hence
+VQ,K(u) = sup
+PVQ(u,P) = sup
+PVQ(u◦F−1,F(P))≤VQ,η(K)(u◦F−1).
+By Proposition 2.3 and the discussion preceding the proof of Proposition 2.2, for each
+g∈GA, the map Tg:R2→R2is a similarity with respect to the quasimetrics De,Dsand
+D. Hence, in particular, the images of the unit square Sunder the action of GAonR2are
+K-quasi-balls in these quasimetrics for some fixed K. In Lemmas 3.2 through 3.4, R2is
+equipped with one of the three quasimetrics.
+Lemma 3.2. The coordinate function y:R2→Rhas locally finite (2,L)-variation for any
+L.
+Proof.LetU⊂R2beany boundedopensubset. First observethat if two L-quasi-balls have
+comparable size, then the oscillation of yover the two quasi-balls will be comparable. Hence
+6when we calculate the 2-variation, it suffices to consider onl y packings of Uby quasi-balls
+of the form Tg(S) whereg∈GA. For each such square, we clearly have
+[osc(y|B)]2=area(B)
+wherearea(B) denotes the Euclidean area. It follows that the 2-variatio n ofy|Uis bounded
+by the area of U.
+Lemma 3.3. LetU⊂R2be an open subset. If u:U→Ris a continuous function which
+is not constant along some horizontal line segment in U, thenV2,K(u) =∞.
+Proof.Sinceuis continuous and is not constant along a horizontal line seg ment in U,
+after composing uwith an affine function, we may assume that there is a rectangle C=
+[a,b]×[c,d]⊂Usuch that u≤0 onF0:={(x,y)∈C:x=a}andu≥1 onF1:={(x,y)∈
+C:x=b}. LetGbe the standard unit coordinate grid. Pick t∈R,t <<0. The image of
+Gunder
+λt=/bracketleftbiggettet
+0et/bracketrightbigg
+is a “sheared grid”, whose tiles have area e2t. Organize these into nearly horizontal chains
+(which correspond to the image of vertical strips under λt). Notice that these chains have
+slope 1/tand intersect vertical lines in segments with Euclidean len gthet/|t|. It follows
+that there are at least
+(d−c)−(b−a)/|t|
+et/|t|=(d−c)|t|−(b−a)
+et
+such chains connecting the left edge F0ofCto the right edge F1ofC.
+Now consider a chain as above that connects F0andF1. Orient the chain from left
+to right. Let Tbe the last tile in the chain that intersects F0andT′the first tile in the
+chain that intersects F1. Order the tiles in the chain between TandT′from left to right
+and denote them by T1,···,Tk. SetT0=T,Tk+1=T′. Letpi(i= 1,···,k+1) be the
+upper left vertex of Ti. Also choose any p0∈T0∩F0andpk+2∈Tk+1∩F1. Notice that the
+difference between the x-coordinates of pi+1andpi(i= 1,···,k) is|t|et. It follows that
+k <b−a
+|t|et.
+Letaibe the oscillation of uonTi∩S. Thenai≥ |u(pi+1)−u(pi)|. By the triangle
+inequality, we have
+Σi=k+1
+i=0ai≥Σi=k+1
+i=0|u(pi+1)−u(pi)| ≥ |u(pk+2)−u(p0)| ≥1.
+In the last inequality we used the facts that u≤0 onF0andu≥1 onF1. Hence
+Σi=k+1
+i=0a2
+i≥1
+k+2(Σi=k+1
+i=0ai)2≥1
+k+2≥|t|et
+(b−a)+2|t|et.
+Since there are at least(d−c)|t|−(b−a)
+etchains connecting F0andF1, the (2,K) – variation of
+uover this particular packing is at least
+|t|et
+(b−a)+2|t|et×(d−c)|t|−(b−a)
+et=|t|/braceleftbig
+(d−c)|t|−(b−a)/bracerightbig
+(b−a)+2|t|et.
+7Ast→ −∞, we see that V2,K(u) =∞.
+Lemma 3.4. LetU,V⊂R2be two open subsets, and F:U→Vbe a quasisymmetric
+map. Then Fmaps each horizontal line segment in Uto a horizontal line segment in V.
+Proof.Assume F:U→Visη-quasisymmetric. Suppose that the claim in the lemma is
+false. Then there are two points p,qon the same horizontal line segment in Usuch that
+F(p) andF(q) are not on the same horizontal line. Then F(p) andF(q) have different y
+coordinates. Hence y◦Fis notconstant along horizontal lines. By Lemma3.3, V2,K(y◦F) =
+∞. On the other hand, applying Lemma 3.1 to the function y◦F:U→RandF:U→V,
+we obtain V2,η(K)(y)≥V2,K(y◦F) =∞. This contradicts Lemma 3.2.
+Proposition 3.5. LetF:∂GA→∂GAbe a quasisymmetric homeomorphism, where ∂GA
+is equipped with a visual metric. Then Ffixes the point ξ0and maps horizontal lines to
+horizontal lines.
+Proof.Suppose F(ξ0)/\e}atio\slash=ξ0. ThenFinduces a homeomorphism
+F1:∂GA\{ξ0,F−1(ξ0)} →∂GA\{F(ξ0),ξ0}
+between two open subsets of R2. Since a visual metric (away from ξ0) is locally quasisym-
+metrically equivalent with a parabolic visual metric (say a metric of the form Dǫ
+ewithǫ
+sufficiently small) (see [SX] Section 5), F1is locally quasisymmetric with respect to any
+one ofDe,DsandD. Now Lemma 3.4 implies that F1maps horizontal line segments to
+horizontal line segments. Let Lbe a complete horizontal line in R2which does not contain
+F−1(ξ0). ThenL∪{ξ0}is a circle in ∂GAand hence F(L∪{ξ0}) is a circle in R2. By the
+above argument, F(L) is horizontal and is dense in the circle F(L∪ {ξ0})⊂R2. This is
+clearly impossible. Hence Ffixesξ0. Now Lemma 3.4 implies Fmaps horizonal lines to
+horizontal lines.
+We omit the proof of the following consequence of Propositio n 3.5 since the proof is
+more or less routine and is already contained in [SX], Sectio n 6.
+Corollary 3.6. The group GAis not quasiisometric to any finitely generated group.
+4 Quasisymmetric maps are D-biLipschitz
+In this section we show that every quasisymmetric map of ∂GAis biLipschitz with respect
+toD. One should contrast this with the round sphere or the Euclid ean space, where there
+are plenty of non-biLipschitz quasisymmetric maps. On the o ther hand, ( R2,D) is not as
+rigid as the ideal boundary of a quarternionic hyperbolic sp ace or a Cayley plane ([P2]) or
+a Fuchsian building ([BP], [X]), where each self quasisymme try is a conformal map.
+8LetK≥1 andC >0. A bijection F:X1→X2between two quasimetric spaces is
+called aK-quasisimilarity (with constant C) if
+C
+Kd(x,y)≤d(F(x),F(y))≤CKd(x,y)
+for allx,y∈X1. When K= 1, we say Fis asimilarity . It is clear that a map is a
+quasisimilarity if and only if it is a biLipschitz map. The po int of using the notion of
+quasisimilarity is that sometimes there is control on Kbut not on C.
+Theorem 4.1. LetF: (R2,D)→(R2,D)be anη-quasisymmetry. Then Fis aK-
+quasisimilarity, where K= (η(1)/η−1(1))6.
+We first recall some definitions.
+Letg: (X1,ρ1)→(X2,ρ2) be a bijection between two quasimetric spaces. Suppose
+gsatisfies the following condition: for any fixed x∈X1,ρ1(y,x)→0 if and only if
+ρ2(g(y),g(x))→0. We define for every x∈X1andr >0,
+Lg(x,r) = sup{ρ2(g(x),g(x′)) :ρ1(x,x′)≤r},
+lg(x,r) = inf{ρ2(g(x),g(x′)) :ρ1(x,x′)≥r},
+and set
+Lg(x) = limsup
+r→0Lg(x,r)
+r, lg(x) = liminf
+r→0lg(x,r)
+r.
+Then
+Lg−1(g(x)) =1
+lg(x)andlg−1(g(x)) =1
+Lg(x)
+for anyx∈X1. Ifgis anη-quasisymmetry, then Lg(x,r)≤η(1)lg(x,r) for allx∈X1and
+r >0. Hence if in addition
+lim
+r→0Lg(x,r)
+ror lim
+r→0lg(x,r)
+r
+exists, then
+0≤lg(x)≤Lg(x)≤η(1)lg(x)≤ ∞.
+We notice that for every y1,y2∈R, the Hausdorff distance with respect to D,
+HD(R×{y1},R×{y2}) =|y1−y2|. (4.1)
+Also, for any p= (x1,y1)∈R2and any y2∈R,
+D(p,R×{y2}) =|y1−y2|. (4.2)
+LetF: (R2,D)→(R2,D) be an η-quasisymmetry. By Lemma 3.4 Fpreserves the
+horizontal foliation on R2. Hence it induces a map G:R→Rsuch that for any y∈R,
+F(R× {y}) =R× {G(y)}. For each y∈R, letH(·,y) :R→Rbe the map such that
+F(x,y) = (H(x,y),G(y)) for all x∈R. Notice that the restriction of Dto a horizontal line
+agrees with the Euclidean distance. Because F: (R2,D)→(R2,D) is anη-quasisymmetry,
+for each fixed y∈R, the map H(·,y) : (R,|·|)→(R,|·|) is also an η-quasisymmetry. The
+following lemma together with equations (4.1) and (4.2) imp ly thatG:R→Ris also an
+η-quasisymmetry with respect to the Euclidean metric on R.
+9Lemma 4.2. ([T, Lemma15.9]) Letg:X1→X2be anη-quasisymmetry and A,B,C⊂X1.
+IfHD(A,B)≤tHD(A,C)for some t≥0, then there is some a∈Asuch that
+HD(g(A),g(B))≤η(t)d(g(a),g(C)).
+We recall that if g:X1→X2is anη-quasisymmetry, then g−1:X2→X1is an
+η1-quasisymmetry, where η1(t) = (η−1(t−1))−1. See [V], Theorem 6.3.
+Theorem 4.1 is proved in Lemmas 4.3 through 4.7. In these proo fs, the quantities
+lG,LG,lG−1,LG−1andlH(·,y),LH(·,y),lIandLIare all defined with respect to the Euclidean
+metric on R, whereI:=H(·,y)−1:R→R.
+Lemma 4.3. The following hold for all y∈R,x∈R:
+(1)LG(y,r)≤η(1)lH(·,y)(x,r)for anyr >0;
+(2)η−1(1)lH(·,y)(x)≤lG(y)≤η(1)lH(·,y)(x);
+(3)η−1(1)LH(·,y)(x)≤LG(y)≤η(1)LH(·,y)(x).
+Proof.(1) Lety∈R,x∈Randr >0. Lety′∈Rwith|y−y′| ≤randx′∈Rwith|x−x′| ≥
+r. Denote x′′=x+(y′−y)ln|y′−y|. ThenD((x,y),(x′′,y′))≤r≤D((x,y),(x′,y)). Since
+Fisη-quasisymmetric, we have
+|G(y)−G(y′)| ≤D(F(x′′,y′),F(x,y))≤η(1)D(F(x,y),F(x′,y))
+=η(1)|H(x,y)−H(x′,y)|.
+Sincey′andx′are arbitrary, (1) follows.
+(2) and (3). It follows from lG(y,r)≤LG(y,r),lH(·,y)(x,r)≤LH(·,y)(x,r) and (1)
+thatLG(y,r)≤η(1)LH(·,y)(x,r) andlG(y,r)≤η(1)lH(·,y)(x,r) for any r >0. Hence
+LG(y)≤η(1)LH(·,y)(x) andlG(y)≤η(1)lH(·,y)(x). Notice that the inverse map F−1:
+(R2,D)→(R2,D) is anη1-quasisymmetry. Applying the inequality lG(y)≤η(1)lH(·,y)(x)
+toI:=H(·,y)−1andG−1we obtain:
+1
+LG(y)=lG−1(G(y))≤η1(1)·lI(H(x,y)) =1
+η−1(1)·1
+LH(·,y)(x),
+henceLG(y)≥η−1(1)LH(·,y)(x). Similarly we prove lG(y)≥η−1(1)lH(·,y)(x).
+Because G:R→Ris a quasisymmetry, it is differentiable a.e. (with respect to the
+Lebesgue measure).
+Lemma 4.4. Lety∈Rbe such that G′(y)exists. Then 0< lG(y) =LG(y) =G′(y)<∞.
+Proof.Lety∈Ybe such that G′(y) exists. Then 0 ≤lG(y) =LG(y) =G′(y)<∞.
+Suppose G′(y) = 0. Then Lemma 4.3 (3) implies LH(·,y)(x) = 0 for all x∈R. It follows
+thatH(·,y) :R→Ris a constant function, contradicting the fact that H(·,y) is a
+homeomorphism. Hence G′(y)/\e}atio\slash= 0.
+10Lemma 4.5. Lety∈Rbe such that G′(y)exists. Then the map H(·,y) :R→Ris an
+η(1)/η−1(1)-quasisimilarity with constant G′(y).
+Proof.By Lemma 4.3 (2) we have lH(·,y)(x)≥lG(y)/η(1) for all x∈R. Lemma 4.3 (3)
+and Lemma 4.4 imply LH(·,y)(x)≤LG(y)/η−1(1) =lG(y)/η−1(1) for all x∈R. Because
+Ris a geodesic space, the map H(·,y) is anη(1)/η−1(1)-quasisimilarity with constant
+lG(y) =G′(y).
+Lemma 4.6. There exists a constant C >0with the following properties:
+(1) For each y∈R,H(·,y)is an(η(1)/η−1(1))4-quasisimilarity with constant C;
+(2)G:R→Ris an(η(1)/η−1(1))5-quasisimilarity with constant C.
+Proof.(1) Fix any y0∈Rsuch that G′(y0) exists and set C=G′(y0). Lety∈Rbe any
+point such that G′(y) exists. By Lemma 4.5, the map H(·,y) :R→Ris anη(1)/η−1(1)-
+quasisimilarity with constant G′(y). Letx0∈Rand choose x∈Rsuch that |x−x0| ≥
+|y−y0|. Letx′=x+(y0−y)ln|y0−y|. Then
+D((x′,y0),(x0,y)) =D((x,y),(x0,y)) =|x−x0|.
+By picking xso that in addition
+κ:=/vextendsingle/vextendsingleH(x′,y0)−H(x0,y)−(G(y0)−G(y))ln|G(y0)−G(y)|/vextendsingle/vextendsingle>|G(y0)−G(y)|,
+by theη-quasisymmetry of Fwe have
+κ=D(F(x′,y0),F(x0,y))≤η(1)D(F(x,y),F(x0,y)) =η(1)|H(x,y)−H(x0,y)|.
+By Lemma 4.5 and the choice of y, we have
+|H(x,y)−H(x0,y)| ≤(η(1)/η−1(1))lG(y)|x−x0|.
+On the other hand, letting τ= (G(y0)−G(y))ln|G(y0)−G(y)|, we have
+κ≥ |H(x′,y0)−H(x0,y0)|−|H(x0,y0)−H(x0,y)|−|τ|
+≥G′(y0)
+η(1)/η−1(1)|x′−x0|−|H(x0,y0)−H(x0,y)|−|τ|.
+Combining the above inequalities and letting |x−x0| → ∞, we obtain
+G′(y) =lG(y)≥1
+(η(1))3(η−1(1))−2G′(y0) =C
+(η(1))3(η−1(1))−2.
+Switching the roles of yandy0we obtain
+G′(y0)≥1
+(η(1))3(η−1(1))−2G′(y).
+11HenceC
+(η(1))3(η−1(1))−2≤G′(y)≤C(η(1))3(η−1(1))−2. By Lemma 4.3, for all x∈R,
+LH(·,y)(x)≤LG(y)/η−1(1)≤C(η(1))3(η−1(1))−3
+and
+lH(·,y)(x)≥1
+η(1)lG(y)≥C
+(η(1))4(η−1(1))−2.
+Hence for a.e. y∈R, the map H(·,y) is an (η(1)/η−1(1))4-quasisimilarity with constant C.
+A limiting argument shows that this is true for all y.
+(2) Statement (1) implies the following for all x,y∈R,
+C
+(η(1)/η−1(1))4≤lH(·,y)(x)≤LH(·,y)(x)≤C(η(1)/η−1(1))4.
+Now Lemma 4.3 implies
+C
+(η(1)/η−1(1))5≤lG(y)≤LG(y)≤C(η(1)/η−1(1))5
+for ally∈R. Hence (2) holds.
+Lemma 4.7. Fis an(η(1)/η−1(1))6-quasisimilarity with constant C, whereCis the con-
+stant in Lemma 4.6.
+Proof.SetK= (η(1)/η−1(1))5. Let (x1,y1),(x2,y2)∈R2. We shall first establish a lower
+bound for D(F(x1,y1),F(x2,y2)). Setτ=x1−x2−(y1−y2)ln|y1−y2|. If|τ| ≤ |y1−y2|,
+thenD((x1,y1),(x2,y2)) =|y1−y2|and by Lemma 4.6 (2),
+D(F(x1,y1),F(x2,y2))≥ |G(y1)−G(y2)| ≥C
+K|y1−y2|=C
+KD((x1,y1),(x2,y2)).
+If|τ|>|y1−y2|, then
+D((x1,y1),(x2,y2)) =|τ|=D((x1−(y1−y2)ln|y1−y2|,y2),(x2,y2)),
+and since Fis anη-quasisymmetry, we have
+D(F(x1,y1),F(x2,y2))≥1
+η(1)D(F(x1−(y1−y2)ln|y1−y2|,y2),F(x2,y2))
+=1
+η(1)/vextendsingle/vextendsingleH(x1−(y1−y2)ln|y1−y2|,y2)−H(x2,y2)/vextendsingle/vextendsingle
+≥C
+η(1)K/vextendsingle/vextendsinglex1−x2−(y1−y2)ln|y1−y2|/vextendsingle/vextendsingle
+=C
+η(1)KD((x1,y1),(x2,y2)),
+with the second inequality following from Lemma 4.6 (1). Hen ce we have a lower bound for
+D(F(x1,y1),F(x2,y2)).
+12By Lemma 4.6 (2), G−1:R→Ris aK-quasisimilarity with constant C−1. Similarly,
+Lemma 4.6 (1) implies that for each y∈R, (H(·,y))−1is aK-quasisimilarity with constant
+C−1. Also recall that F−1is anη1-quasisymmetry and Fis anη-quasisymmetry. Now the
+argument in the previous paragraph applied to F−1implies
+D(F−1(x1,y1),F−1(x2,y2))≥1
+CKη1(1)D((x1,y1),(x2,y2)).
+It follows that
+D(F(x1,y1),F(x2,y2))≤CKη1(1)D((x1,y1),(x2,y2)) =CK/η−1(1)D((x1,y1),(x2,y2))
+for all (x1,y1),(x2,y2)∈R2. Hence we also obtain an upper bound for the quantity
+D(F(x1,y1),F(x2,y2)).
+5 Characterization of quasisymmetric maps
+In this section we give a complete description of all self qua sisymmetric maps of ∂GA.
+Theorem 5.1. A mapF: (R2,D)→(R2,D)is a quasisymmetric map if and only if it
+has the following form: F(x,y) = (ax+c(y),ay+b)for all(x,y)∈R2, wherea/\e}atio\slash= 0,bare
+constants and c:R→Ris a Lipschitz map.
+LetF: (R2,D)→(R2,D) be a quasisymmetric map. From Section 4, we know there
+is a quasisymmetric map G:R→R, and for each y∈Rthere is a quasisymmetric map
+H(·,y) :R→Rsuch that F(x,y) = (H(x,y),G(y)) for all ( x,y)∈R2. ThenG′(y) exists
+almost everywhere. Similarly, for each y∈R, the map H(·,y) has derivative Hx(x,y) for
+a.e.x∈R.
+Lemma 5.2. LetF: (R2,D)→(R2,D)be a quasisymmetric map. Let y∈Rbe such that
+G′(y)exists, and x∈Rsuch that Hx(x,y)exists at x. ThenG′(y) =Hx(x,y).
+Proof.LetF: (R2,D)→(R2,D) be an η-quasisymmetric map. By replacing Fwith
+T(−H(x,y),−G(y))◦F◦T(x,y), we may assume ( x,y) = (H(x,y),G(y)) = 0. Here T(x,y)denotes
+the Euclidean translation by ( x,y). Lemma 4.4 implies G′(0)/\e}atio\slash= 0. By composing Fwith
+a dilation λtfor a suitable twe may assume G′(0) = 1 or −1. IfG′(0) =−1, we further
+compose Fwith the rotation π:R2→R2,π(x,y) = (−x,−y). Hence we may assume
+G′(0) = 1. Denote λ=Hx(0,0). By Lemma 4.6 (2) we have λ/\e}atio\slash= 0. We shall prove that
+λ= 1.
+Sinceλtis a similarity, the family of maps {Ft:=λt◦F◦λ−t|t∈R}consists of
+η-quasisymmetric maps. Write Ft(x,y) = (Ht(x,y),Gt(y)). We notice that Ht(x,0) =
+etH(e−tx,0) andGt(y) =etG(e−ty). Sincethederivative Hx(0,0) exists, themaps Ht(·,0) :
+R→Rconverge (as t→ ∞) in the pointed Gromov-Hausdorff distance towards the map
+x→λx. Similarly, the maps Gt:R→Rconverge (as t→ ∞) in the pointed Gromov-
+Hausdorff distance towards the map y→y. The compactness property of quasisymmetric
+maps implies that there is a sequences ti→ ∞such that Fticonverges in the pointed
+13Gromov-Hausdorff distance towards an η-quasisymmetric map ˜F: (R2,D)→(R2,D). If
+we write ˜F(x,y) = (˜H(x,y),˜G(y)), then ˜G(y) =yand˜H(x,0) =λx.
+By Theorem 4.1, the map ˜FisL-biLipschitz for some L≥1. Fix some x∈R
+and a positive integer n≥1. Fori= 0,···,n, let (xi,yi) = (x−i
+nlnn,i
+n). Then
+D((xi,yi),(xi+1,yi+1)) = 1/n. Hence
+|˜H(xi,yi)−˜H(xi+1,yi+1)−1
+nlnn| ≤D(˜F(xi,yi),˜F(xi+1,yi+1))≤L·1
+n.
+Adding up all these inequalities for i= 0,···,n−1 and using the triangle inequality we
+obtain
+|˜H(x0,y0)−˜H(xn,yn)−lnn| ≤L. (5.1)
+On the other hand, D((xn,yn),(x−lnn,0)) = 1 and hence
+|˜H(xn,yn)−˜H(x−lnn,0)| ≤D(˜F(xn,yn),˜F(x−lnn,0))≤L. (5.2)
+It follows from (5.1) and (5.2) that |˜H(x0,y0)−˜H(x−lnn,0)−lnn| ≤2L. Notice that
+˜H(x0,y0) =˜H(x,0) =λxand˜H(x−lnn,0) =λ(x−lnn). So we have |(λ−1)lnn| ≤2L.
+Since this is true for all n≥1, we must have λ= 1.
+Lemma 5.3. There exist constants a/\e}atio\slash= 0andband also a function c:R→Rsuch that
+(1)G(y) =ay+b;
+(2)H(x,y) =ax+c(y)for all(x,y)∈R2.
+Proof.Lety∈Rbeany point where Gis differentiable. By Lemma5.2, the quasisymmetric
+mapH(·,y) :R→Ra.e. has derivative G′(y). It follows that H(·,y) is an affine map; to be
+more precise, there is a constant c(y) dependingonly on ysuch that H(x,y) =G′(y)x+c(y)
+for allx∈R.
+We claim that G′(y1) =G′(y2) holds for any two points y1,y2∈Rat which Gis
+differentiable. Set τ= (y2−y1)ln|y2−y1|. Letx >0 and denote p= (0,y1),q=
+(x,y1),p′= (τ,y2) andq′= (x+τ,y2). One checks that D(p,q) =D(p′,q′) =xand
+D(p,p′) =D(q,q′) =|y2−y1|. By Theorem 4.1 FisL-biLipschitz for some L≥1. We
+haveD(F(p),F(p′))≤L|y2−y1|andD(F(q),F(q′))≤L|y2−y1|. On the other hand, by
+the preceding paragraph, we have F(p) = (c(y1),G(y1)),F(q) = (G′(y1)x+c(y1),G(y1))
+andF(p′) = (G′(y2)τ+c(y2),G(y2)),F(q′) = (G′(y2)(x+τ) +c(y2),G(y2)). Setτ′=
+(G(y2)−G(y1))ln|G(y2)−G(y1)|. Since
+|[G′(y2)(x+τ)+c(y2)]−[G′(y1)x+c(y1)]−τ′| ≤D(F(q),F(q′))≤L|y2−y1|
+for allx >0, we must have G′(y1) =G′(y2).
+SinceGis differentiable a.e., it follows from the above claim that Ga.e.has constant
+derivative, hence must be an affine map. That is, there are cons tantsa/\e}atio\slash= 0,bsuch that
+G(y) =ay+bfor ally∈R. This proves (1). Now (2) follows from (1) and the first
+paragraph.
+14Completing the proof of Theorem 5.1. First suppose F: (R2,D)→(R2,D) is a qua-
+sisymmetric map. Then by Lemma 5.3 Fhas the form F(x,y) = (ax+c(y),ay+b),
+wherea/\e}atio\slash= 0,bare constants, and c:R→Ris a function. Now fix y1,y2∈R.
+Letτ= (y2−y1)ln|y2−y1|and denote p= (0,y1),q= (τ,y2). One checks that
+D(p,q) =|y2−y1|. By Theorem 4.1 Fis aL-biLipschitz map for some L≥1. Hence
+D(F(p),F(q))≤LD(p,q) =L|y2−y1|. On the other hand, F(p) = (c(y1),ay1+b) and
+F(q) = (aτ+c(y2),ay2+b). We have
+D(F(p),F(q))≥/vextendsingle/vextendsingle(aτ+c(y2)−c(y1))−a(y2−y1)ln|a(y2−y1)|/vextendsingle/vextendsingle
+=/vextendsingle/vextendsinglec(y2)−c(y1)−(aln|a|)(y2−y1)/vextendsingle/vextendsingle.
+Now the triangle inequality implies |c(y2)−c(y1)| ≤/parenleftbig
+L+/vextendsingle/vextendsinglealn|a|/vextendsingle/vextendsingle/parenrightbig
+|y2−y1|, that is, cis/parenleftbig
+L+/vextendsingle/vextendsinglealn|a|/vextendsingle/vextendsingle/parenrightbig
+-Lipschitz.
+Conversely, suppose Fhas the form F(x,y) = (ax+c(y),ay+b), where a/\e}atio\slash= 0,b
+are constants, and c:R→RisL-Lipschitz. One checks by direct calculation that F
+is Lipschitz, as follows. Let p= (x,y),q= (x′,y′)∈R2be two arbitrary points. Then
+F(p) = (ax+c(y),ay+b) andF(q) = (ax′+c(y′),ay′+b). Setτ= (x′−x)−(y′−y)ln|y′−y|.
+We have D(p,q) = max{|y′−y|,|τ|}and
+D(F(p),F(q)) = max/braceleftbig/vextendsingle/vextendsinglea(y′−y)/vextendsingle/vextendsingle,/vextendsingle/vextendsingleaτ+[c(y′)−c(y)]−(aln|a|)(y′−y)/vextendsingle/vextendsingle/bracerightbig
+.
+Now|a(y′−y)|=|a|·|y′−y| ≤ |a|D(p,q) and
+/vextendsingle/vextendsingleaτ+[c(y′)−c(y)]−(aln|a|)(y′−y)/vextendsingle/vextendsingle≤/vextendsingle/vextendsingleaτ/vextendsingle/vextendsingle+/vextendsingle/vextendsinglec(y′)−c(y)/vextendsingle/vextendsingle+/vextendsingle/vextendsingle(aln|a|)(y′−y)/vextendsingle/vextendsingle
+≤ |a|D(p,q)+L|y′−y|+/vextendsingle/vextendsinglealn|a|/vextendsingle/vextendsingle·|y′−y|
+≤/parenleftbig
+|a|+L+/vextendsingle/vextendsinglealn|a|/vextendsingle/vextendsingle/parenrightbig
+D(p,q).
+It follows that Fis Lipschitz with Lipschitz constant |a|+L+/vextendsingle/vextendsinglealn|a|/vextendsingle/vextendsingle.On the other hand,
+F−1has the form
+F−1(x,y) =/parenleftbigg1
+a·x−1
+a·c/parenleftbigg1
+ay−b
+a/parenrightbigg
+,1
+a·y−b
+a/parenrightbigg
+.
+As a composition of Lipschitz maps, the map c′:R→R,c′(y) =−1
+ac(1
+ay−b
+a) is also
+Lipschitz. Hence the above calculation shows that F−1is also Lipschitz.
+6 A Liouville type theorem for (R2,D)
+In this section we prove a Liouville type theorem for ( R2,D), which says that all conformal
+maps of ( R2,D) are boundary maps of isometries of GA. We first identify all the conformal
+maps of ( R2,D).
+Using Theorem 5.1, we can identify all the isometries and sim ilarities of ( R2,D). Recall
+that the map πand similarities λtare defined in Proposition 2.3.
+15Proposition 6.1. (1) The group of all isometries of (R2,D)is generated by Euclidean
+translations and π;
+(2) The group of all similarities of (R2,D)is generated by Euclidean translations, πand
+the similarities λt(t∈R).
+Proof.We only prove (2), the proof of (1) being similar. Let F: (R2,D)→(R2,D) be
+a similarity. By composing Fwith a suitable λt, we may assume Fis an isometry. By
+Theorem 5.1, Fhas the form F(x,y) = (ax+c(y),ay+b), where a/\e}atio\slash= 0,bare constants
+andc:R→Ris a Lipschitz map. By considering the restriction of Fon a horizontal line
+R× {y}, we see a= 1 or−1. By composing with πif necessary (when a=−1), we may
+assumea= 1. By further composing Fwith an Euclidean translation, we may assume
+b= 0 and c(0) = 0. Now Fhas the form F(x,y) = (x+c(y),y) for all ( x,y)∈R2, where
+c(0) = 0. We claim c(y) = 0 for all y∈R. Suppose c(y)/\e}atio\slash= 0 for some y/\e}atio\slash= 0. Let ǫ= 1 or
+−1 be such that ǫyandc(y) are either both positive or both negative. Let p= (0,0) and
+q= (ǫy+yln|y|,y). Then F(p) =pandF(q) = (ǫy+yln|y|+c(y),y). One calculates
+D(F(p),F(q)) =|ǫy+c(y)|>|y|=D(p,q), contradicting the fact that Fis an isometry.
+Hencec(y) = 0 for all yandFis the identity map.
+LetXandYbe quasimetric spaces with finite Hausdorff dimension. Denot e byHX
+andHYtheir Hausdorff dimensions and by HXandHYtheir Hausdorff measures (see [F]
+for definitions). We say a quasisymmetric map f:X→Yis conformal if:
+(1)Lf(x) =lf(x)∈(0,∞) forHX-almost every x∈X;
+(2)Lf−1(y) =lf−1(y)∈(0,∞) forHY-almost every y∈Y.
+Lemma 6.2. Every conformal map F: (R2,D)→(R2,D)is a similarity.
+Proof.SinceFis conformal, it is quasisymmetric in particular. By Theore m 1.1,Fhas the
+following form: F(x,y) = (ax+c(y),ay+b), where a/\e}atio\slash= 0,bare constants and c:R→Ris
+a Lipschitz map. By composing Fwith a similarity, we may assume a= 1 and b= 0; that
+is,Fhas the form F(x,y) = (x+c(y),y). We shall prove that c(y) is a constant function.
+Sincec:R→Ris a Lipschitz function, it is differentiable a.e. We shall sho w that
+c′(y) = 0 for a.e. y∈R. By the definition of a conformal map, LF(x,y) =lF(x,y) for a.e.
+(x,y)∈R2with respect to the Lebesgue measure in R2. It follows from Fubini’s theorem
+that for a.e. y∈R, the derivative c′(y) exists and LF(x,y) =lF(x,y) for a.e. x∈R. Lety0
+be an arbitrary such point and x0∈Rbe such that LF(x0,y0) =lF(x0,y0). We will show
+c′(y0) = 0.
+By pre-composing and post-composing with Euclidean transl ations if necessary, we may
+assume that ( x0,y0) = (0,0) andc(y0) = 0. We need to show c′(0) = 0. We will suppose
+c′(0)/\e}atio\slash= 0 and get a contradiction. Notice that F(x,0) = (x,0) for all x∈R. It follows that
+LF(0,0)≥1 andlF(0,0)≤1. Combining this with the assumption LF(0,0) =lF(0,0),
+we obtain LF(0,0) =lF(0,0) = 1. First suppose c′(0)>0. Then c(y)>0 for sufficiently
+smally >0. Letp= (0,0) andq= (r+rlnr,r) withr >0. Then F(p) =pand
+F(q) = (r+rlnr+c(r),r). One calculates D(p,q) =randD(F(p),F(q)) =r+c(r). It
+follows that LF(p,r)≥r+c(r) and hence LF(p)≥1+c′(0)>1, contradicting LF(0,0) = 1.
+Ifc′(0)<0, then letting q= (−r+rlnr,r) one similarly obtains a contradiction.
+16Theorem 6.3. LetF: (R2,D)→(R2,D)be a conformal map. Then Fis the boundary
+map of some isometry f:GA→GA.
+Proof.By Lemma 6.2, Fis a similarity. By Proposition 6.1 (2), Fis the composition of
+Euclidean translations, τand similarities λt. Now the theorem follows from the following
+facts (see the end of Section 2): (1) Euclidean translations ofR2are boundary maps of the
+Lie group left translations Lgfor elements of the form g= ((x,y),0)∈GA; (2)τis the
+boundary map of the isometry τ′:GA→GA; (3)λtis the boundary map of the Lie group
+left translation Lgforg= ((0,0),t).
+7 Quasiisometries of GA
+In this section we calculate the quasiisometry group of GAand identify all the quasiisome-
+tries ofGAup to bounded distance. From this it is easy to see that all qua siisometries of
+GAare almost isometries and are height-respecting.
+We first discuss the structure of the group QS(R2,D) of all quasisymmetric maps of
+(R2,D). We identify three subgroups of QS(R2,D). LetH1={λt:t∈R}∼=R. Let
+H2=< τ >∼=Z2={¯0,¯1}be the order 2 cyclic group generated by τ. LetH3be the group
+of homeomorphisms of R2of the form FC,b(x,y) = (x+C(y),y+b), where b∈Rand
+C:R→Ris a Lipschitz function. Direct calculations show that H1andH2commute, both
+H1andH2normalize H3, andH3∩< H1,H2>is trivial. On the other hand, Theorem
+5.1 implies that QS(R2,D) is generated by H1,H2andH3. It follows that we have the
+following isomorphism:
+QS(R2,D)∼=H3⋊(H1⊕H2).
+LetLbe the additive group consisting of Lipschitz functions C:R→R. LetRact on
+Lbyb∗C=C◦Tb, forb∈RandC∈L, whereTbis the translation on Rbyb. Then
+it is easy to check that the map given by FC,b/ma√sto→(C,b) defines an isomorphism from the
+groupH3to the opposite group L⋊RofL⋊R. It now follows that we have the following
+isomorphism:
+QS(R2,D)∼=(L⋊R)⋊(R×Z2).
+Here the action of R×{¯0}onL⋊Ris given by ( t,¯0)∗(C,b) = (C′,b′) for (t,¯0)∈R×{¯0}
+and (C,b)∈L⋊R, where
+C′(y) =et·C(e−ty)+btetandb′=etb;
+and the action of {0} ×Z2onL⋊Ris given by (0 ,¯1)∗(C,b) = (C′′,−b), where C′′(y) =
+−C(−y).
+Two quasiisometries f,g:X→Ybetween two metric spaces are said to be equivalent if
+sup{d(f(x),g(x)) :x∈X}<∞. For any metric space X, the quasiisometry group QI(X)
+consists of equivalence classes of quasiisometries X→Xand has group operation given by
+composition.
+For each quasiisometry f:GA→GA, let∂f:∂GA→∂GAbe its boundary map.
+17Theorem 7.1. We have QI(GA)∼=QS(R2,D)∼=(L⋊R)⋊(R×Z2), whereLand the
+various actions are as described above.
+Proof.By Proposition 3.5, each quasiisometry f:GA→GAinduces a quasisymmetric
+map∂f: (R2,D)→(R2,D). Notice that an quasiisometry fofGAis at finite distance
+from the identity map of GAif and only if the boundary map of fis the identity map on
+(R2,D). Hence the map ∂1:QI(GA)→QS(R2,D),∂1([f]) =∂fis well-defined and is
+injective. Here [ f] denotes the equivalence of f. On the other hand, by [BS] each element
+inQS(R2,D) is the boundary map of an quasiisometry. Hence ∂1:QI(GA)→QS(R2,d)
+is also surjective.
+We now identify a group of quasiisometries of GAthat is isomorphic to QI(GA). Let
+H′
+1={Lg:g= ((0,0),t)∈GA}. LetH′
+2=< τ′>, whereτ′((x,y),t) = ((−x,−y),t) is the
+automorphism of GAdefined in Section 2. The two groups H1andH2consist of isometries
+ofGA. LetH′
+3be the group of homeomorphisms of GAof the following form:
+fC,b:GA→GA, fC,b((x,y),t) = ((x+C(y),y+b),t),
+whereb∈RandC∈L. It is clear that H′
+3is isomorphic to H3∼=L⋊R. Using Lemma
+6.3 in [SX] and the fact that FC,b: (R2,De)→(R2,De) is biLipschitz, it is easy to check
+that each fC,b(b∈R,C∈L) is an almost isometry of GA. In particular, H3consists of
+quasiisometries of GA.
+LetQI′(GA) be the group of homeomorphisms of GAgenerated by H′
+1,H′
+2andH′
+3. A
+similar discussion as above shows that
+QI′(GA)∼=H′
+3⋊(H′
+1×H′
+2)∼=L⋊R⋊(R×Z2),
+where the various actions are as described above. Let ∂:QI′(GA)→QS(R2,D) be
+the map that assigns to each f∈QI′(GA) its boundary map. It is easy to see that ∂
+mapsH′
+i(i= 1,2,3) isomorphically onto Hi. It follows that ∂is an isomorphism. Let
+p:QI′(GA)→QI(GA) be the group homomorphism that assigns to each f∈QI′(GA) its
+equivalence class. Since ∂=∂1◦p(where∂1is defined in the proof of Theorem 7.1), it
+follows from Theorem 7.1 that pis an isomorphism. We obtain:
+Theorem 7.2. Every quasiisometry f:GA→GAis at a finite distance from exactly one
+element of QI′(GA).
+Now we can provide a proof of Corollary 1.3.
+Proof of Corollary 1.3. Sinceeach element in H′
+1andH′
+2is anisometry of GA, andevery
+element of H′
+3is an almost isometry, we see that every element of QI′(GA) is an almost
+isometry. Now Corollary 1.3 follows from this and Theorem 7. 2.
+Under the identification of GAwithR2×R, we view the map h:R2×R→R,
+h((x,y),t) =tas the height function. A quasiisometry f:GA→GAis height-respecting if
+|h(f((x,y),t))−t|is bounded independent of (( x,y),t)∈GA. Since every element of H′
+1,
+H′
+2andH′
+3is height-respecting, we have
+Corollary 7.3. All self quasiisometries of GAare height-respecting.
+18References
+[BP] M. Bourdon, H. Pajot, Rigidity of quasi-isometries for some hyperbolic building s.
+Comment. Math. Helv. 75(2000), no. 4, 701–736.
+[BS] M. Bonk, O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal.
+10(2000), no. 2, 266–306.
+[F] H. Federer, Geometric measure theory, Grundlehren der Mathematik 153. Springer
+Verlag, Berlin-New York-Heidelberg, 1969.
+[Hn] J. Heinonen, Lectures on analysis on metric spaces, Universitext. Springer-Verlag,
+New York, 2001.
+[K] B. Kleiner, Unpublished notes.
+[P1] P. Pansu, Dimension conforme et sphere a l’infini des varietes a courbu re negative,
+Ann. Acad. Sci. Fenn. Ser. A I Math. 14(1989), no. 2, 177–212.
+[P2] P. Pansu, Metriques de Carnot-Caratheodory et quasiisometries des esp aces
+symetriques de rang un, Ann. of Math. (2) 129(1989), no. 1, 1–60.
+[SX] N. Shanmugalingam, X. Xie, A Rigidity Property of Some Negatively Curved Solvable
+Lie Groups, preprint.
+[T] J. Tyson, Metric and geometric quasiconformality in Ahlfors regular Loewner spaces,
+Conform. Geom. Dyn. 5(2001), 21–73.
+[V] J. V¨ ais¨ al¨ a, The free quasiworld. Freely quasiconformal and related maps in Banach
+spaces,Quasiconformal geometry and dynamics, Banach Center Publ. (Lublin) 48
+(1996), 55–118.
+[X] X. Xie, Quasi-isometric rigidity of Fuchsian buildings . Topology 45(2006), no. 1, 101–
+169.
+Address:
+Xiangdong Xie: Dept. of Mathematical Sciences, Georgia Sou thern University, Statesboro,
+GA 30460, U.S.A. E-mail: xxie@georgiasouthern.edu
+19
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+arXiv:1001.0149v2  [math.NA]  21 Aug 2010Fast construction of hierarchical matrix representation
+from matrix-vector multiplication
+Lin Lina, Jianfeng Lub, Lexing Yingc
+aProgram in Applied and Computational Mathematics, Princet on University, Princeton,
+NJ 08544.
+bDepartment of Mathematics, Courant Institute of Mathemati cal Sciences, New York
+University, 251 Mercer St., New York, NY 10012.
+cDepartment of Mathematics and ICES, University of Texas at A ustin, 1 University
+Station, C1200, Austin, TX 78712.
+Abstract
+We develop a hierarchical matrix construction algorithm using matrix -
+vectormultiplications, basedontherandomizedsingularvaluedecom position
+of low-rank matrices. The algorithm uses O(logn) applications of the matrix
+on structured random test vectors and O(nlogn) extra computational cost,
+wherenis the dimension of the unknown matrix. Numerical examples on
+constructing Green’s functions for elliptic operators in two dimensio ns show
+efficiency and accuracy of the proposed algorithm.
+Keywords: fast algorithm, hierarchical matrix construction, randomized
+singular value decomposition, matrix-vector multiplication, elliptic
+operator, Green’s function
+1. Introduction
+Inthiswork, weconsiderthefollowingproblem: Assumethatanunkn own
+symmetric matrix Ghas the structure of a hierarchical matrix ( H-matrix) [1,
+2, 3], that is, certain off-diagonal blocks of Gare low-rank or approximately
+low-rank(see thedefinitions in Sections 1.3and2.2). The task istoco nstruct
+Gefficiently only froma “black box” matrix-vector multiplication subrou tine
+(which shall be referred to as matvecin the following). In a slightly more
+Email addresses: linlin@math.princeton.edu (Lin Lin), jianfeng@cims.nyu.edu
+(Jianfeng Lu), lexing@math.utexas.edu (Lexing Ying)
+Preprint submitted to Journal of Computational Physics Oct ober 23, 2018general setting when Gis not symmetric, the task is to construct Gfrom
+“black box” matrix-vector multiplication subroutines of both GandGT. In
+this paper, we focus on the case of a symmetric matrix G. The proposed
+algorithm can be extended to the non-symmetric case in a straightf orward
+way.
+1.1. Motivation and applications
+Ourmotivationismainlythesituationthat GisgivenastheGreen’sfunc-
+tion of an elliptic equation. In this case, it is proved that Gis anH-matrix
+under mild regularity assumptions [4]. For elliptic equations, methods lik e
+preconditioned conjugate gradient, geometric and algebraic multig rid meth-
+ods, sparse direct methods provide application of the matrix Gon vectors.
+The algorithm proposed in this work then provides an efficient way to c on-
+struct the matrix Gexplicitly in the H-matrix form.
+Once we obtain the matrix Gas anH-matrix, it is possible to apply G
+on vectors efficiently, since the application of an H-matrix on a vector is
+linear scaling. Of course, for elliptic equations, it might be more efficien t
+to use available fast solvers directly to solve the equation, especially if only
+a few right hand sides are to be solved. However, sometimes, it would be
+advantageous to obtain Gsince it is then possible to further compress G
+according to the structure of the data (the vectors that Gwill be acting on),
+for example as in numerical homogenization [5]. Another scenario is th at the
+data has special structure like sparsity in the choice of basis, the a pplication
+of the resulting compressed matrix will be more efficient than the “bla ck
+box” elliptic solver.
+Let us remark that, in the case of elliptic equations, it is also possible t o
+use theH-matrixalgebrato invert thedirect matrix(which is an H-matrixin
+e.g.finite element discretization). Our method, on the other hand, pro vides
+an efficient alternative algorithm when a fast matrix-vector multiplica tion
+is readily available. From a computational point of view, what is probab ly
+more attractive is that our algorithm facilitates a parallelized constr uction
+of theH-matrix, while the direct inversion has a sequential nature [2].
+As another motivation, the purpose of the algorithm is to recover t he
+matrix via a “black box” matrix-vector multiplication subroutine. A ge neral
+question of this kind will be that under which assumptions of the matr ix,
+one can recover the matrix efficiently by matrix-vector multiplication s. If
+the unknown matrix is low-rank, the recently developed randomized singular
+value decomposition algorithms [6, 7, 8] provide an efficient way to obt ain
+2the low-rank approximation through application of the matrix on ran dom
+vectors. Low-rank matrices play an important role in many applicatio ns.
+However, the assumption is too strong in many cases that the whole matrix
+is low-rank. Since the class of H-matrices is a natural generalization of the
+one of low-rank matrices, the proposed algorithm can be viewed as a further
+step in this direction.
+1.2. Randomized singular value decomposition algorithm
+A repeatedly leveraged tool in the proposed algorithm is the random ized
+singular value decomposition algorithm for computing a low rank appro x-
+imation of a given numerically low-rank matrix. This has been an active
+research topic in the past several years with vast literature. For the purpose
+of this work, we have adopted the algorithm developed in [7], although other
+variants of this algorithm with similar ideas can also be used here. For a
+given matrix Athat is numerically low-rank, this algorithm goes as following
+to compute a rank- rfactorization.
+Algorithm 1 Construct a low-rank approximation A≈U1MUT
+2for rank r
+1:Choose a Gaussian random matrix R1∈Rn×(r+c)wherecis a small
+constant;
+2:FormAR1and apply SVD to AR1. The first rleft singular vectors
+giveU1;
+3:Choose a Gaussian random matrix R2∈Rn×(r+c);
+4:FormRT
+2Aand apply SVD to ATR2. The first rleft singular vectors
+giveU2;
+5:M= (RT
+2U1)†[RT
+2(AR1)](UT
+2R1)†, where B†denotes the Moore-
+Penrose pseudoinverse of matrix B[9, pp. 257–258].
+The accuracy of this algorithm and its variants has been studied tho r-
+oughly by several groups. If the matrix 2-norm is used to measure the error,
+it is well-known that the best rank- rapproximation is provided by the singu-
+lar value decomposition (SVD). When the singular values of Adecay rapidly,
+it has been shown that Algorithm 1 results in almost optimal factoriza tions
+withanoverwhelming probability[6]. AsAlgorithm1istobeusedfreque ntly
+in our algorithm, we analyze briefly its complexity step by step. The ge n-
+eration of random numbers is quite efficient, therefore in practice o ne may
+ignore the cost of steps 1 and 3. Step 2 takes ( r+c)matvecof matrix A
+3andO(n(r+c)2) steps for applying the SVD algorithms on an n×(r+c)
+matrix. The cost of step 4 is the same as the one of step 2. Step 5 inv olves
+the computation of RT
+2(AR1), which takes O(n(r+c)2) steps as we have
+already computed AR1in step 2. Once RT
+2(AR1) is ready, the computation
+ofMtakes additional O((r+c)3) steps. Therefore, the total complexity of
+Algorithm 1 is O(r+c)matvecs plusO(n(r+c)2) extra steps.
+1.3. Top-down construction of H-matrix
+Weillustratethecoreideaofouralgorithmusingasimpleone-dimension al
+example. Thealgorithmofconstructingahierarchicalmatrix Gisatop-down
+pass. We assume throughout the article that Gis symmetric.
+For clarity, we will first consider a one dimension example. The details o f
+the algorithm in two dimensions will be given in Section 2. We assume that
+a symmetric matrix Ghas a hierarchical low-rank structure corresponding
+to a hierarchical dyadic decomposition of the domain. The matrix Gis of
+dimension n×nwithn= 2LMfor an integer LM. Denote the set for all
+indices as I0;1, where the former subscript indicates the level and the latter
+is the index for blocks in each level. At the first level, the set is partitio ned
+intoI1;1andI1;2, with the assumption that G(I1;1,I1;2) andG(I1;2,I1;1)
+are numerically low-rank, say of rank rfor a prescribed error tolerance ε.
+At level l, each block Il−1;ion the above level is dyadically decomposed
+into two blocks Il;2i−1andIl;2iwith the assumption that G(Il;2i−1,Il;2i) and
+G(Il;2i,Il;2i−1) are also numerically low-rank (with the same rank rfor the
+tolerance ε). Clearly, at level l, we have in total 2loff-diagonal low-rank
+blocks. We stop at level LM, for which the block ILM,ionly has one index
+{i}. For simplicity of notation, we will abbreviate G(Il;i,Il;j) byGl;ij. We
+remark that the assumption that off-diagonal blocks are low-rank matrices
+may not hold for general elliptic operators in higher dimensions. Howe ver,
+this assumption simplifies the introduction of the concept of our algo rithm.
+More realistic case will be discussed in detail in Sections 2.3 and 2.4.
+The overarching strategy of our approach is to peel off the off-dia gonal
+blocks level by level and simultaneously construct their low-rank ap proxima-
+tions. On the first level, G1;12is numerically low-rank. In order to use the
+randomized SVD algorithm for G1;12, we need to know the product of G1;12
+and also GT
+1;12=G1;21with a collection of random vectors. This can be done
+by observing that
+/parenleftbiggG1;11G1;12
+G1;21G1;22/parenrightbigg/parenleftbiggR1;1
+0/parenrightbigg
+=/parenleftbiggG1;11R1;1
+G1;21R1;1/parenrightbigg
+, (1)
+4/parenleftbigg
+G1;11G1;12
+G1;21G1;22/parenrightbigg/parenleftbigg
+0
+R1;2/parenrightbigg
+=/parenleftbigg
+G1;12R1;2
+G1;22R1;2/parenrightbigg
+, (2)
+whereR1;1andR1;2are random matrices of dimension n/2×(r+c). We
+obtain (G1;21R1;1)T=RT
+1;1G1;12by restricting the right hand side of Eq. (1)
+toI1;2and obtain G1;12R1;2by restricting the right hand side of Eq. (2) to
+I1;1, respectively. The low-rank approximation using Algorithm 1 results in
+G1;12≈/hatwideG1;12=U1;12M1;12UT
+1;21. (3)
+U1;12andU1;21aren/2×rmatrices and M1;12is anr×rmatrix. Due to the
+fact that Gis symmetric, a low-rank approximation of G1;21is obtained as
+the transpose of G1;12.
+Now on the second level, the matrix Ghas the form
+
+G2;11G2;12
+G2;21G2;22G1;12
+G1;21G2;33G2;34
+G2;43G2;44
+.
+The submatrices G2;12,G2;21,G2;34, andG2;43are numerically low-rank, to
+obtain their low-rank approximations by the randomized SVD algorith m.
+Similar to the first level, we could apply Gon random matrices of the form
+like(R2;1,0,0,0)T. Thiswillrequire4( r+c)numberofmatrix-vectormultipli-
+cations. However, this is not optimal: Since we already know the inter action
+between I1;1andI1;2, we could combine the calculations together to reduce
+the number of matrix-vector multiplications needed. Observe that
+
+G2;11G2;12
+G2;21G2;22G1;12
+G1;21G2;33G2;34
+G2;43G2;44
+
+R2;1
+0
+R2;3
+0
+=
+/parenleftbiggG2;11R2;1
+G2;21R2;1/parenrightbigg
++G1;12/parenleftbiggR2;3
+0/parenrightbigg
+/parenleftbigg
+G2;33R2;3
+G2;43R2;3/parenrightbigg
++G1;21/parenleftbigg
+R2;1
+0/parenrightbigg
+.
+(4)
+Denote
+/hatwideG(1)=/parenleftigg
+0/hatwideG1;12
+/hatwideG1;210/parenrightigg
+(5)
+with/hatwideG1;12and/hatwideG1;21the low-rank approximations we constructed on the first
+5level, then
+/hatwideG(1)
+R2;1
+0
+R2;3
+0
+=
+/hatwideG1;12/parenleftbigg
+R2;3
+0/parenrightbigg
+/hatwideG1;21/parenleftbigg
+R2;1
+0/parenrightbigg
+. (6)
+Therefore,
+(G−/hatwideG(1))
+R2;1
+0
+R2;3
+0
+≈
+G2;11R2;1
+G2;21R2;1
+G2;33R2;3
+G2;43R2;3
+, (7)
+so that we simultaneously obtain ( G2;21R2;1)T=RT
+2;1G2;12and (G2;43R2;3)T=
+RT
+2;3G2;34. Similarly, applying Gon (0,R2;2,0,R2;4)Tprovides G2;12R2;2and
+G2;34R2;4. We can then obtain the following low-rank approximations by
+invoking Algorithm 1.
+G2;12≈/hatwideG2;12=U2;12M2;12UT
+2;21,
+G2;34≈/hatwideG2;34=U2;34M2;34UT
+2;43.(8)
+The low-rank approximations of G2;21andG2;43are again given by the trans-
+poses of the above formulas.
+Similarly, on the third level, the matrix Ghas the form
+
+G3;11G3;12
+G3;21G3;22G2;12
+G2;21G3;33G3;34
+G3;43G3;44G1;12
+G1;21G3;55G3;56
+G3;65G3;66G2;34
+G2;43G3;77G3;78
+G3;87G3;88
+,(9)
+and define
+/hatwideG(2)=
+0/hatwideG2;12
+/hatwideG2;2100
+00/hatwideG2;34
+/hatwideG2;430
+. (10)
+6We could simultaneously obtain the product of G3;12,G3;34,G3;56andG3;78
+with random vectors by applying the matrix Gwith random vectors of the
+form
+(RT
+3;1,0,RT
+3;3,0,RT
+3;5,0,RT
+3;7,0)T,
+thensubtracttheproductof /hatwideG(1)+/hatwideG(2)withthesamevectors. Againinvoking
+Algorithm 1 provides us the low-rank approximations of these off-dia gonal
+blocks.
+The algorithm continues in the same fashion for higher levels. The com -
+bined random tests lead to a constant number of matvecat each level. As
+there are log( n) levels in total, the total number of matrix-vector multiplica-
+tions scales logarithmically.
+Whentheblocksize onalevel becomes smaller thanthegivencriteria ( for
+example, the numerical rank rused in the construction), one could switch to
+a deterministic way to get the off-diagonal blocks. In particular, we stop at
+a levelL(L < L M) such that each IL;icontains about rentries. Now only
+the elements in the diagonal blocks GL,iineed to be determined. This can
+be completed by applying Gto the matrix
+(I,I,...,I )T,
+whereIis the identity matrix whose dimension is equal to the number of
+indices in IL;i.
+Letussummarizethestructureofouralgorithm. Fromthetopleve ltothe
+bottom level, we peel off the numerically low-rank off-diagonal blocks using
+the randomized SVD algorithm. The matrix-vector multiplications req uired
+by the randomized SVD algorithms are computed effectively by combining
+several random tests into one using the zero pattern of the remaining matrix.
+In this way, we get an efficient algorithm for constructing the hierar chical
+representation for the matrix G.
+1.4. Related works
+Our algorithm is built on top of the framework of the H-matrices pro-
+posed by Hackbusch and his collaborators [1, 2, 4]. The definitions o f the
+H-matrices will be summarized in Section 2. In a nutshell, the H-matrix
+framework is an operational matrix algebra for efficiently represen ting, ap-
+plying, and manipulating discretizations of operators from elliptic par tial
+differential equations. Though we have known how to represent an d ap-
+ply these matrices for quite some time [10], it is the contribution of the
+7H-matrix framework that enables one to manipulate them in a general and
+coherent way. A closely related matrix algebra is also developed in a mo re
+numerical-linear-algebraic viewpoint under the name hierarchical semisepa-
+rable matrices by Chandrasekaran, Gu, and others [11, 12]. Here, we will
+follow the notations of the H-matrices as our main motivations are from
+numerical solutions of elliptic PDEs.
+A basic assumption of our algorithm is the existence of a fast matrix-
+vector multiplication subroutine. The most common case is when Gis the
+inverse of the stiffness matrix Hof a general elliptic operator. Since His
+often sparse, much effort has been devoted to computing u=Gfby solving
+the linear system Hu=f. Many ingenious algorithms have been developed
+for this purpose in the past forty years. Commonly-seen examples include
+multifrontal algorithms [13, 14], geometric multigrids [15, 16, 2], algeb raic
+multigrids (AMG) [17], domain decompositions methods [18, 19], wavelet -
+based fast algorithms [20] and preconditioned conjugate gradient algorithms
+(PCG) [21], to name a few. Very recently, both Chandrasekaran et al[22]
+and Martinsson [23] have combined the idea of the multifrontal algor ithms
+with the H-matrices to obtain highly efficiently direct solvers for Hu=f.
+Another common case for which a fast matrix-vector multiplication s ubrou-
+tine is available comes from the boundary integral equations where Gis often
+a discretization of a Green’s function restricted to a domain bounda ry. Fast
+algorithms developed for this case include the famous fast multipole m ethod
+[10], the panel clustering method [24], and others. All these fast a lgorithms
+mentioned above can be used as the “black box” algorithm for our me thod.
+As shown in the previous section, our algorithm relies heavily on the
+randomized singular value decomposition algorithm for constructing the fac-
+torizations of the off-diagonal blocks. This topic has been a highly ac tive
+research area in the past several years and many different algorit hms have
+been proposed in the literature. Here, for our purpose, we have a dopted
+the algorithm described in [7, 8]. In a related but slightly different prob lem,
+the goal is to find low-rank approximations A=CURwhereCcontains a
+subset of columns of AandRcontains a subset of rows. Papers devoted to
+this task include [25, 26, 27, 28]. In our setting, since we assume no d irect
+access of entries of the matrix Abut only its impact through matrix-vector
+multiplications, the algorithm proposed by [7] is the most relevant ch oice.
+An excellent recent review of this fast growing field can be found in [6].
+In a recent paper [29], Martinsson considered also the problem of co n-
+structing the H-matrix representation of a matrix, but he assumed that one
+8can access arbitrary entries of the matrix besides the fast matrix -vector mul-
+tiplication subroutine. Under this extra assumption, he showed tha t one
+can construct the H2representation of the matrix with O(1) matrix-vector
+multiplications and accesses of O(n) matrix entries. However, in many situ-
+ations including the case of Gbeing the inverse of the stiffness matrix of an
+elliptic differential operator, accessing entries of Gis by no means a trivial
+task. Comparing with Martinsson’s work, our algorithm only assumes the
+existence of a fast matrix-vector multiplication subroutine, andhe nce ismore
+general.
+As we mentioned earlier, one motivation for computing Gexplicitly is to
+further compress the matrix G. The most common example in the literature
+of numerical analysis is the process of numerical homogenization or upscaling
+[5]. Herethematrix Gisoftenagaintheinverseofthestiffnessmatrix Hofan
+elliptic partial differential operator. When Hcontains information from all
+scales, the standard homogenization techniques fail. Recently, Ow hadi and
+Zhang [30] proposed an elegant method that, under the assumptio n that the
+Cordes condition is satisfied, upscales a general Hin divergence form using
+metric transformation. Computationally, their approach involves dsolves of
+formHu=fwithdbeing the dimension of the problem. On the other hand,
+ifGis computed using our algorithm, one can obtain the upscaled operat or
+by inverting a low-passed and down-sampled version of G. Complexity-wise,
+our algorithm is more costly since it requires O(logn) solves of Hu=f.
+However, since our approach makes no analytic assumptions about H, it is
+expected to be more general.
+2. Algorithm
+We now present the details of our algorithm in two dimensions. In ad-
+dition to a top-down construction using the peeling idea presented in the
+introduction, the complexity will be further reduced using the H2property
+of the matrix [1, 3]. The extension to three dimensions is straightfor ward.
+In two dimensions, a more conservative partition of the domain is re-
+quired to guarantee the low-rankness of the matrix blocks. We will s tart
+with discussion of this new geometric setup. Then we will recall the no tion
+of hierarchical matrices and related algorithms in Section 2.2. The alg orithm
+to construct an H2representation for a matrix using matrix-vector multi-
+plications will be presented in Sections 2.3 and 2.4. Finally, variants of t he
+9algorithm for constructing the H1and uniform H1representations will be
+described in Section 2.5.
+2.1. Geometric setup and notations
+Letusconsideranoperator Gdefinedona2Ddomain[0 ,1)2withperiodic
+boundary condition. We discretize the problem using an n=N×Nuniform
+grid with Nbeing a power of 2: N= 2LM. Denote the set of all grid points
+as
+I0={(k1/N,k2/N)|k1,k2∈N,0≤k1,k2< N} (11)
+and partition the domain hierarchically into L+1 levels ( L < L M). On each
+levell(0≤l≤L), we have 2l×2lboxes denoted by Il;ij= [(i−1)/2l,i/2l)×
+[(j−1)/2l,j/2l) for 1≤i,j≤2l. The symbol Il;ijwill also be used to denote
+the grid points that lies in the box Il;ij. The meaning should be clear from
+the context. We will also use Il(orJl) to denote a general box on certain
+levell. The subscript lwill be omitted, when the level is clear from the
+context. For a given box Ilforl≥1, we call a box Jl−1on levell−1 its
+parent if Il⊂ Jl−1. Naturally, Ilis called a child of Jl−1. It is clear that
+each box except those on level Lwill have four children boxes.
+For any box Ion levell, it covers N/2l×N/2lgrid points. The last level
+Lcan be chosen so that the leaf box has a constant number of points in it
+(i.e.the difference LM−Lis kept to be a constant when Nincreases).
+For simplicity of presentation, we will start the method from level 3. It
+is also possible to start from level 2. Level 2 needs to be treated sp ecially, as
+for level 3. We define the following notations for a box Ion levell(l≥3):
+NL(I) Neighbor list of box I. This list contains the boxes on level lthat are
+adjacent to Iand also Iitself. There are 9 boxes in the list for each
+I.
+IL(I) Interaction list of box I. Whenl= 3, this list contains all the boxes
+on level 3 minus the set of boxes in NL(I). There are 55 boxes in total.
+Whenl >3, this list contains all the boxes on level lthat are children
+of boxes in NL(P) withPbeingI’s parent minus the set of boxes in
+NL(I). There are 27 such boxes.
+Noticethat these two lists determine two symmetric relationship: J ∈NL(I)
+if and only if I ∈NL(J) andJ ∈IL(I) if and only if I ∈IL(J). Figs. 1
+and 2 illustrate the computational domain and the lists for l= 3 and l= 4,
+respectively.
+10/BU/D3 /DCI3;3,3/BT /CS/CY/CP
/CT/D2 /D8/C1/D2 /D8/CT/D6/CP
/D8/CX/D3/D2
+Figure 1: Illustration of the computational domain at level 3. I3;3,3is the black box. The
+neighbor list NL(I3;3,3) consists of 8 adjacent light gray boxes and the black box itself,
+and the interaction list IL(I3;3,3) consists of the 55 dark gray boxes./BU/D3 /DCI4;5,5/BT /CS/CY/CP
/CT/D2 /D8/C1/D2 /D8/CT/D6/CP
/D8/CX/D3/D2
+Figure 2: Illustration of the computational domain at level 4. I4;5,5is the black box. The
+neighbor list NL(I4;5,5) consists of 8 adjacent light gray boxes and the black box itself,
+and the interaction list IL(I4;5,5) consists of the 27 dark gray boxes.
+11For a vector fdefined on the N×NgridI0, we define f(I) to be the
+restriction of fto grid points I. For a matrix G∈RN2×N2that represents a
+linear map from I0to itself, we define G(I,J) to be the restriction of Gon
+I ×J.
+A matrix G∈RN2×N2has the following decomposition
+G=G(3)+G(4)+···+G(L)+D(L). (12)
+Here, for each l,G(l)incorporates the interaction on level lbetween a box
+with its interaction list. More precisely, G(l)has a 22l×22lblock structure:
+G(l)(I,J) =/braceleftigg
+G(I,J),I ∈IL(J) (eq.J ∈IL(I));
+0, otherwise
+withIandJboth on level l. The matrix D(L)includes the interactions
+between adjacent boxes at level L:
+D(L)(I,J) =/braceleftigg
+G(I,J),I ∈NL(J) (eq.J ∈NL(I));
+0, otherwise
+withIandJboth on level L. To show that (12) is true, it suffices to prove
+that for any two boxes IandJon levelL, the right hand side gives G(I,J).
+Inthe case that I ∈NL(J), this is obvious. Otherwise, it is clear that we can
+find a level l, and boxes I′andJ′on levell, such that I′∈IL(J′),I ⊂ I′
+andJ ⊂ J′, and hence G(I,J) is given through G(I′,J′). Throughout the
+text, we will use /ba∇dblA/ba∇dbl2to denote the matrix 2-norm of matrix A.
+2.2. Hierarchical matrix
+Our algorithm works with the so-called hierarchical matrices. We rec all
+in this subsection some basic properties of this type of matrices and also
+some related algorithms. For simplicity of notations and representa tion, we
+will only work with symmetric matrices. For a more detailed introductio n of
+the hierarchical matrices and their applications in fast algorithms, w e refer
+the readers to [2, 3].
+2.2.1.H1matrices
+Definition 1. Gis a (symmetric) H1-matrix if for any ε >0, there exists
+r(ε)/lessorsimilarlog(ε−1) such that for any pair ( I,J) withI ∈IL(J), there exist
+12orthogonal matrices UIJandUJIwithr(ε) columns and matrix MIJ∈
+Rr(ε)×r(ε)such that
+/ba∇dblG(I,J)−UIJMIJUT
+JI/ba∇dbl2≤ε/ba∇dblG(I,J)/ba∇dbl2. (13)
+The main advantage of the H1matrix is that the application of such
+matrix on a vector can be efficiently evaluated: Within error O(ε), one
+can use/hatwideG(I,J) =UIJMIJUT
+JI, which is low-rank, instead of the origi-
+nal block G(I,J). The algorithm is described in Algorithm 2. It is standard
+that the complexity of the matrix-vector multiplication for an H1matrix is
+O(N2logN) [2].
+Algorithm 2 Application of a H1-matrixGon a vector f.
+1:u= 0;
+2:forl= 3 toLdo
+3:forIon levelldo
+4:forJ ∈IL(I)do
+5: u(I) =u(I)+UIJ(MIJ(UT
+JIf(J)));
+6:end for
+7:end for
+8:end for
+9:forIon levelLdo
+10:forJ ∈NL(I)do
+11: u(I) =u(I)+G(I,J)f(J);
+12:end for
+13:end for
+2.2.2. Uniform H1matrix
+Definition 2. Gis a (symmetric) uniform H1-matrix if for any ε >0, there
+existsrU(ε)/lessorsimilarlog(ε−1) such that for each box I, there exists an orthogonal
+matrixUIwithrU(ε) columns such that for any pair ( I,J) withI ∈IL(J)
+/ba∇dblG(I,J)−UINIJUT
+J/ba∇dbl2≤ε/ba∇dblG(I,J)/ba∇dbl2 (14)
+withNIJ∈RrU(ε)×rU(ε).
+The application of a uniform H1matrix to a vector is described in Algo-
+rithm 3. The complexity of the algorithm is still O(N2logN). However, the
+prefactor is much better as each UIis applied only once. The speedup over
+Algorithm 2 is roughly 27 r(ε)/rU(ε) [2].
+13Algorithm 3 Application of a uniform H1-matrixGon a vector f
+1:u= 0;
+2:forl= 3 toLdo
+3:forJon levelldo
+4:/tildewidefJ=UT
+Jf(J);
+5:end for
+6:end for
+7:forl= 3 toLdo
+8:forIon levelldo
+9:/tildewideuI= 0;
+10: forJ ∈IL(I)do
+11:/tildewideuI=/tildewideuI+NIJ/tildewidefJ;
+12: end for
+13:end for
+14:end for15:forl= 3 toLdo
+16:forIon levelldo
+17: u(I) =u(I)+UI/tildewideuI;
+18:end for
+19:end for
+20:forIon levelLdo
+21:forJ ∈NL(I)do
+22: u(I) = u(I) +
+G(I,J)f(J);
+23:end for
+24:end for
+2.2.3.H2matrices
+Definition 3. Gis anH2matrix if
+•it is a uniform H1matrix;
+•Suppose that Cis any child of a box I, then
+/ba∇dblUI(C,:)−UCTCI/ba∇dbl2/lessorsimilarε, (15)
+for some matrix TCI∈RrU(ε)×rU(ε).
+The application of an H2matrix to a vector is described in Algorithm 4
+and it has a complexity of O(N2), Notice that, compared with H1matrix,
+the logarithmic factor is reduced [3].
+Remark. Applying an H2matrix to a vector can indeed be viewed as the
+matrix form of the fast multipole method (FMM) [10]. One recognizes in
+Algorithm 4 that the second top-level forloop corresponds to the M2M
+(multipole expansion to multipole expansion) translations of the FMM; the
+third top-level forloop is the M2L (multipole expansion to local expansion)
+translations; and the fourth top-level forloop is the L2L (local expansion to
+local expansion) translations.
+In the algorithm to be introduced, we will also need to apply a partial
+matrixG(3)+G(4)+···+G(L′)for some L′≤Lto a vector f. This amounts
+to a variant of Algorithm 4, described in Algorithm 5.
+14Algorithm 4 Application of a H2-matrixGon a vector f
+1:u= 0;
+2:forJon levelLdo
+3:/tildewidefJ=UT
+Jf(J);
+4:end for
+5:forl=L−1 down to 3 do
+6:forJon levelldo
+7:/tildewidefJ= 0;
+8:foreach child CofJdo
+9:/tildewidefJ=/tildewidefJ+TT
+CJ/tildewidefC;
+10: end for
+11:end for
+12:end for
+13:forl= 3 toLdo
+14:forIon levelldo
+15:/tildewideuI= 0;
+16: forJ ∈IL(I)do
+17:/tildewideuI=/tildewideuI+NIJ/tildewidefJ;
+18: end for
+19:end for
+20:end for18:forl= 3 toL−1do
+19:forIon levelldo
+20: foreach child CofIdo
+21:/tildewideuC=/tildewideuC+TCI/tildewideuI;
+22: end for
+23:end for
+24:end for
+25:forIon levelLdo
+26:u(I) =UI/tildewideuI;
+27:end for
+28:forIon levelLdo
+29:forJ ∈NL(I)do
+30: u(I) = u(I) +
+G(I,J)f(J);
+31:end for
+32:end for
+2.3. Peeling algorithm: outline and preparation
+We assume that Gis a symmetric H2matrix and that there exists a fast
+matrix-vector subroutine for applying Gto any vector fas a “black box”.
+The goal is to construct an H2representation of the matrix Gusing only a
+small number of test vectors.
+The basicstrategyisatop-downconstruction: Foreachlevel l= 3,...,L,
+assume that an H2representation for G(3)+···+G(l−1)is given, we construct
+G(l)by the following three steps:
+1.Peeling. Construct an H1representation for G(l)using the peeling idea
+and theH2representation for G(3)+···+G(l−1).
+2.Uniformization. Construct a uniform H1representation for G(l)from
+itsH1representation.
+3.Projection. Construct an H2representation for G(3)+···+G(l).
+The names of these steps will be made clear in the following discussion.
+Variantsofthealgorithmthatonlyconstructan H1representation(auniform
+15Algorithm 5 Applicationofa partial H2-matrixG(3)+···+G(L′)ona vector
+f
+1:u= 0;
+2:forJon levelL′do
+3:/tildewidefJ=UT
+Jf(J);
+4:end for
+5:forl=L′−1 down to 3 do
+6:forJon levelldo
+7:/tildewidefJ= 0;
+8:foreach child CofJdo
+9:/tildewidefJ=/tildewidefJ+TT
+CJ/tildewidefC;
+10: end for
+11:end for
+12:end for
+13:forl= 3 toL′do
+14:forIon levelldo
+15:/tildewideuI= 0;
+16: forJ ∈IL(I)do
+17:/tildewideuI=/tildewideuI+NIJ/tildewidefJ;
+18: end for
+19:end for
+20:end for18:forl= 3 toL′−1do
+19:forIon levelldo
+20: foreach child CofIdo
+21:/tildewideuC=/tildewideuC+TCI/tildewideuI;
+22: end for
+23:end for
+24:end for
+25:forIon levelL′do
+26:u(I) =UI/tildewideuI;
+27:end for
+H1representation, respectively) of the matrix Gcan be obtained by only
+doing the peeling step (the peeling and uniformization steps, respec tively).
+These variants will be discussed in Section 2.5.
+After we have the H2representation for G(3)+···+G(L), we use the
+peeling idea again to extract the diagonal part D(L). We call this whole
+process the peelingalgorithm.
+Before detailing the peeling algorithm, we mention two procedures th at
+serve as essential components of our algorithm. The first proced ure con-
+cerns with the uniformization step, in which one needs to get a unifor m
+H1representation for G(l)from its H1representation, i.e., from/hatwideG(I,J) =
+UIJMIJUT
+JIto/hatwideG(I,J) =UINIJUT
+J, for all pairs of boxes ( I,J) with
+I ∈IL(J). To this end, what we need to do is to find the column space of
+[UIJ1MIJ1|UIJ2MIJ2| ··· |UIJtMIJt], (16)
+whereJjare the boxes in IL(I) andt=|IL(I)|. Notice that we weight the
+16singular vectors UbyM, so that the singular vectors corresponding to larger
+singular values will be more significant. This column space can be found by
+the usual SVD algorithm or a more effective randomized version pres ented in
+Algorithm 6. The important left singular vectors are denoted by UI, and the
+diagonal matrix formed by the singular values associated with UIis denoted
+bySI.
+Algorithm 6 Construct a uniform H1representation of Gfrom the H1
+representation at a level l
+1:foreach box Ion levelldo
+2:Generate a Gaussian random matrix R∈R(r(ε)×t)×(rU(ε)+c);
+3:Form product [ UIJ1MIJ1| ··· |UIJtMIJt]Rand apply SVD to it.
+The first rU(ε) left singular vectors give UI, and the corresponding
+singular values give a diagonal matrix SI;
+4:forJj∈IL(I)do
+5:IIJj=UT
+IUIJj;
+6:end for
+7:end for
+8:foreach pair ( I,J) on level lwithI ∈IL(J)do
+9:NIJ=IIJMIJIT
+JI;
+10:end for
+Complexity analysis: For a box Ion levell, the number of grid points
+inIis (N/2l)2. Therefore, UIJjare all of size ( N/2l)2×r(ε) andMIJ
+are of size r(ε)×r(ε). Forming the product [ UIJ1MIJ1| ··· |UIJtMIJt]R
+takesO((N/2l)2r(ε)(rU(ε)+c)) steps and SVD takes O((N/2l)2(rU(ε)+c)2)
+steps. As there are 22lboxes on level l, the overall cost of Algorithm 6
+isO(N2(rU(ε) +c)2) =O(N2). One may also apply to [ UIJ1MIJ1| ··· |
+UIJtMIJt] the deterministic SVD algorithm, which has the same order of
+complexity but with a prefactor about 27 r(ε)/(rU(ε)+c) times larger.
+The second procedure is concerned with the projection step of th e above
+list, in which one constructs an H2representation for G(3)+···G(l). Here, we
+are given the H2representation for G(3)+···+G(l−1)along with the uniform
+H1representation for G(l)and the goal is to compute the transfer matrix TCI
+for a box Ion levell−1 and its child Con levellsuch that
+/ba∇dblUI(C,:)−UCTCI/ba∇dbl2/lessorsimilarε.
+17In fact, the existing UCmatrix of the uniform H1representation may not be
+rich enough to contain the columns of UI(C,:) in its span. Therefore, one
+needs to update the content of UCas well. To do that, we perform a singular
+value decomposition for the combined matrix
+[UI(C,:)SI|UCSC]
+and define a matrix VCto contain rU(ε) left singular vectors. Again UI,UC
+shouldbeweightedbythecorrespondingsingularvalues. Thetrans fermatrix
+TCIis then given by
+TCI=VT
+CUI(C,:)
+and the new UCis set to be equal to VC. SinceUChas been changed, the
+matrices NCDforD ∈IL(C) and also the corresponding singular values SC
+need to be updated as well. The details are listed in Algorithm 7.
+Algorithm 7 Construct an H2representation of Gfrom the uniform H1
+representation at level l
+1:foreach box Ion levell−1do
+2:foreach child CofIdo
+3:Form matrix [ UI(C,:)SI|UCSC] and apply SVD to it. The
+firstrU(ε) left singular vectors give VC, and the corresponding
+singular values give a diagonal matrix WC;
+4:KC=VT
+CUC;
+5:TCI=VT
+CUI(C,:);
+6:UC=VC;
+7:SC=WC;
+8:end for
+9:end for
+10:foreach pair ( C,D) on level lwithC ∈IL(D)do
+11:NCD=KCNCDKT
+D;
+12:end for
+Complexity analysis: The main computational task of Algorithm 7 is
+again the SVD part. For a box Con levell, the number of grid points in
+Iis (N/2l)2. Therefore, the combined matrix [ UI(C,:)SI|UCSC] is of size
+(N/2l)2×2rU(ε). The SVD computation clearly takes O((N/2l)2rU(ε)2) =
+O((N/2l)2) steps. Taking into the consideration that there are 22lboxes on
+levellgives rise to an O(N2) estimate for the cost of Algorithm 7.
+182.4. Peeling algorithm: details
+With the above preparation, we are now ready to describe the peelin g
+algorithm in detail at different levels, starting from level 3. At each le vel, we
+follow exactly the three steps listed at the beginning of Section 2.3.
+2.4.1. Level 3
+First in the peeling step, we construct the H1representation for G(3). For
+each pair ( I,J) on level 3 such that I ∈IL(J), we will invoke randomized
+SVD Algorithm 1 to construct the low rank approximation of GI,J. How-
+ever, in order to apply the algorithm we need to compute G(I,J)RJand
+RT
+IG(I,J), where RIandRJare random matrices with r(ε)+ccolumns.
+For each box Jon level 3, we construct a matrix Rof sizeN2×(r(ε)+c)
+such that
+R(J,:) =RJandR(I0\J,:) = 0.
+Computing GRusingr(ε)+cmatvecs andrestricting the result to gridpoints
+I ∈IL(J) givesG(I,J)RJfor eachI ∈IL(J).
+After repeating these steps for all boxes on level 3, we hold for an y pair
+(I,J) withI ∈IL(J) the following data:
+G(I,J)RJandRT
+IG(I,J) = (G(J,I)RI)T.
+Now, applying Algorithm 1 to them gives the low-rank approximation
+/hatwideG(I,J) =UIJMIJUT
+JI. (17)
+In the uniformization step, in order to get the uniform H1representation
+forG(3), we simply apply Algorithm 6 to the boxes on level 3 to get the
+approximations
+/hatwideG(I,J) =UINIJUT
+J. (18)
+Finally in the projection step, since we only have 1 level now (level 3),
+we have already the H2representation for G(3).
+Complexity analysis: The dominant computation is the construction o f
+theH1representation for G(3). This requires r(ε)+cmatvecs for each box
+Ion level 3. Since there are in total 64 boxes at this level, the total c ost
+is 64(r(ε) +c)matvecs. From the complexity analysis in Section 2.3, the
+computation for the second and third steps cost an extra O(N2) steps.
+192.4.2. Level 4
+First in the peeling step, in order to construct the H1representation for
+G(4), we need to compute the matrices G(I,J)RJandRT
+IG(I,J) for each
+pair (I,J) on level 4 with I ∈IL(J). HereRIandRJare again random
+matrices with r(ε)+ccolumns.
+One approach is to follow exactly what we did for level 3: Fix a box J
+at this level, construct Rof sizeN2×(r(ε)+c) such that
+R(J,:) =RJandR(I0\J,:) = 0.
+Next apply G−G(3)toR, by subtracting GRandG(3)R. The former is
+computed using r(ε) +cmatvecs and the latter is done by Algorithm 5.
+Finally, restrict the result to grid points I ∈IL(J) givesG(I,J)RJfor each
+I ∈IL(J).
+However, we have observed in the simple one-dimensional example in
+Section 1.3 that random tests can be combined together as in Eq. (6 ) and
+(7). We shall detail this observation in the more general situation h ere as
+following. Observe that G−G(3)=G(4)+D(4), andG(4)(J,I)andD(4)(J,I)
+for boxes IandJon level 4 is only nonzero if I ∈NL(J)∪IL(J). Therefore,
+(G−G(3))RforRcoming from Jcan only be nonzero in NL(P) withP
+beingJ’s parent. The rest is automatically zero (up to error εasG(3)
+is approximated by its H2representation). Therefore, we can combine the
+calculation of different boxes as long as their non-zero regions do no t overlap.
+More precisely, we introduce the following sets Spqfor 1≤p,q≤8 with
+Spq={J4;ij|i≡p(mod 8), j≡q(mod 8)}. (19)
+There are 64 sets in total, each consisting of four boxes. Fig. 3 illust rates
+one such set at level 4. For each set Spq, first construct Rwith
+R(J,:) =/braceleftigg
+RJ,J ∈ S pq;
+0,otherwise .
+Then, we apply G−G(3)toR, by subtracting GRandG(3)R. The former
+is computed using r(ε) +cmatvecs and the latter is done by Algorithm 5.
+For each J ∈ S pq, restricting the result to I ∈IL(J) givesG(I,J)RJ.
+Repeatingthiscomputationforallsets Spqthenprovidesuswiththefollowing
+data:
+G(I,J)RJandRT
+IG(I,J) = (G(J,I)RI)T,
+20for each pair ( I,J) withI ∈IL(J). Applying Algorithm 1 to them gives
+the required low-rank approximations
+/hatwideG(I,J) =UIJMIJUT
+JI (20)
+withUIJorthogonal./CB/CT/D8S5,5/BT /CS/CY/CP
/CT/D2 /D8/C1/D2 /D8/CT/D6/CP
/D8/CX/D3/D2
+Figure 3: Illustration of the set S55at level 4. This set consists of four black boxes
+{I4;5,5,I4;13,5,I4;5,13,I4;13,13}. The light gray boxes around each black box are in the
+neighbor list and the dark gray boxes in the interaction list.
+Next in the uniformization step, the task is to construct the unifor mH1
+representation of G(4). Similar to thecomputationat level 3, wesimply apply
+Algorithm 6 to the boxes on level 4 to get
+/hatwideG(I,J) =UINIJUT
+J. (21)
+Finally in the projection step, to get H2representation for G(3)+G(4), we
+invoke Algorithm 7 at level 4. Once it is done, we hold the transfer mat rices
+TCIbetween any Ion level 3 and each of its children C, along with the
+updated uniform H1-matrix representation of G(4).
+Complexity analysis: The dominant computation is again the construc -
+tion ofH1representation for G(4). For each group Spq, we apply Gtor(ε)+c
+vectors and apply G(3)tor(ε)+cvectors. The latter takes O(N2) steps for
+each application. Since there are 64 sets in total, this computation t akes
+64(r(ε)+c)matvecs andO(N2) extra steps.
+212.4.3. Level l
+First in the peeling step, to construct the H1representation for G(l), we
+follow the discussion of level 4. Define 64 sets Spqfor 1≤p,q≤8 with
+Spq={Jl;ij|i≡p(mod 8), j≡q(mod 8)}. (22)
+Each set contains exactly 2l/8×2l/8 boxes. For each set Spq, construct R
+with
+R(J,:) =/braceleftigg
+RJ,J ∈ S pq;
+0,otherwise .
+Next, apply G−[G(3)+···+G(l−1)] toR, by subtracting GRand [G(3)+···+
+G(l−1)]R. The former is again computed using r(ε)+cmatvecs and the latter
+is done by Algorithm 5 using the H2representation of G(3)+···+G(l−1).
+For each J ∈ S pq, restricting the result to I ∈IL(J) givesG(I,J)RJ.
+Repeating this computation for all sets Spqgives the following data for any
+pair (I,J) withI ∈IL(J)
+G(I,J)RJandRT
+IG(I,J) = (G(J,I)RI)T.
+Now applying Algorithm 1 to them gives the low-rank approximation
+/hatwideG(I,J) =UIJMIJUT
+JI (23)
+withUIJorthogonal.
+Similar to the computation at level 4, the uniformization step that co n-
+structs the uniform H1representation of G(l)simply by Algorithm 6 to the
+boxes on level l. The result gives the approximation
+/hatwideG(I,J) =UINIJUT
+J. (24)
+Finally in the projection step, one needs to compute an H2representation
+forG(3)+···+G(l). To this end, we apply Algorithm 7 to level l.
+The complexity analysis at level lfollows exactly the one of level 4. Since
+we still have exactly 64 sets Spq, the computation again takes 64( r(ε) +c)
+matvecs along with O(N2) extra steps.
+These three steps (peeling, uniformization, and projection) are r epeated
+for each level until we reach level L. At this point, we hold the H2represen-
+tation for G(3)+···G(L).
+222.4.4. Computation of D(L)
+Finally we construct of the diagonal part
+D(L)=G−(G(3)+···+G(L)). (25)
+More specifically, for each box Jon levelL, we need to compute G(I,J)
+forI ∈NL(J).
+Define a matrix Eof sizeN2×(N/2L)2(recall that the box Jon level
+Lcovers (N/2L)2grid points) by
+E(J,:) =IandE(I0\J,:) = 0,
+whereIis the (N/2L)2×(N/2L)2identity matrix. Applying G−(G(3)+
+···+G(L)) toEand restricting the results to I ∈NL(J) givesG(I,J) for
+I ∈NL(J). However, we can do better as ( G−(G(3)+···+G(L)))Eis only
+non-zero in NL(J). Hence, one can combine computation of different boxes
+Jas long as NL(J) do not overlap.
+To do this, define the following 4 ×4 = 16 sets Spq, 1≤p,q≤4
+Spq={JL,ij|i≡p(mod 4), j≡q(mod 4)}.
+For each set Spq, construct matrix Eby
+E(J,:) =/braceleftigg
+I,J ∈ S pq;
+0,otherwise .
+Next, apply G−(G(3)+···+G(L)) toE. For each J ∈ S pq, restricting the
+result to I ∈NL(J) givesG(I,J)I=G(I,J). Repeating this computation
+for all 16 sets Spqgives the diagonal part D(L).
+Complexity analysis: The dominant computation is for each group Spq
+applyGandG(3)+···+G(L)toE, the former takes ( N/2L)2matvecs and the
+latter takes O((N/2L)2N2) extra steps. Recall by the choice of L,N/2Lis a
+constant. Therefore, the total cost for 16 sets is 16( N/2L)2=O(1)matvecs
+andO(N2) extra steps.
+Let us now summarize the complexity of the whole peeling algorithm.
+From the above discussion, it is clear that at each level the algorithm spends
+64(r(ε)+c) =O(1)matvecs andO(N2) extra steps. As there are O(logN)
+levels, the overall cost of the peeling algorithm is equal to O(logN)matvecs
+plusO(N2logN) steps.
+23It is a natural concern that whether the error from low-rank dec ompo-
+sitions on top levels accumulates in the peeling steps. As observed fr om
+numerical examples in Section 3, it does not seem to be a problem at lea st
+for the examples considered. We do not have a proof for this thoug h.
+2.5. Peeling algorithm: variants
+In this section, we discuss two variants of the peeling algorithm. Let us
+recall that the above algorithm performs the following three steps at each
+levell.
+1.Peeling. Construct an H1representation for G(l)using the peeling idea
+and theH2representation for G(3)+···+G(l−1).
+2.Uniformization. Construct a uniform H1representation for G(l)from
+itsH1representation.
+3.Projection. Construct an H2representation for G(3)+···+G(l).
+As this algorithm constructs the H2representation of the matrix G, we also
+refer to it more specifically as the H2version of the peeling algorithm. In
+what follows, we list two simpler versions that are useful in practice
+•theH1version, and
+•the uniform H1version.
+In theH1version, we only perform the peeling step at each level. Since
+this version constructs only the H1representation, it will use the H1repre-
+sentation of G(3)+···+G(l)in the computation of ( G(3)+···+G(l))Rwithin
+the peeling step at level l+1.
+In the uniform H1version, we perform the peeling step and the uni-
+formization step at each level. This will give us instead the uniform H1
+version of the matrix. Accordingly, one needs to use the uniform H1repre-
+sentation of G(3)+···+G(l)in the computation of ( G(3)+···+G(l))Rwithin
+the peeling step at level l+1.
+These two simplified versions are of practical value since there are m a-
+trices that are in the H1or the uniform H1class but not the H2class. A
+simplecalculationshows thatthesetwo simplified versions still take O(logN)
+matvecs but requires O(N2log2N) extra steps. Clearly, the number of extra
+steps is log Ntimes more expensive than the one of the H2version. However,
+if the fast matrix-vector multiplication subroutine itself takes O(N2logN)
+24stepsperapplication, usingthe H1ortheuniform H1versiondoesnotchange
+the overall asymptotic complexity.
+Between thesetwosimplifiedversions, theuniform H1versionrequiresthe
+uniformization step, while the H1version does not. This seems to suggest
+that the uniform H1version is more expensive. However, because (1) our
+algorithm also utilizes the partially constructed representations fo r the cal-
+culation at future levels and (2) the uniform H1representation is much faster
+to apply, the construction of the uniform H1version turns out to be much
+faster. Moreover, since the uniform H1representation stores one UImatrix
+for each box Iwhile the H1version stores about 27 of them, the uniform
+H1is much more memory-efficient, which is very important for problems in
+higher dimensions.
+3. Numerical results
+We study the performance of the hierarchical matrix constructio n algo-
+rithm for the inverse of a discretized elliptic operator. The computa tional
+domain is a two dimensional square [0 ,1)2with periodic boundary condition,
+discretized as an N×Nequispaced grid. We first consider the operator
+H=−∆+Vwith ∆ being the discretized Laplacian operator and the po-
+tential being V(i,j) = 1+W(i,j), i,j= 1,...,N. For all ( i,j),W(i,j) are
+independent random numbers uniformly distributed in [0 ,1]. The potential
+function Vischosen tohave this strong randomness inorder toshow that the
+existence of H-matrix representation of the Green’s function depends weakly
+on the smoothness of the potential. The inverse matrix of His denoted by
+G. The algorithms are implemented using MATLAB . All numerical tests are
+carried out on a single-CPU machine.
+We analyze the performance statistics by examining both the cost a nd
+the accuracy of our algorithm. The cost factors include the time co st and
+the memory cost. While the memory cost is mainly determined by how
+the matrix Gis compressed and does not depend much on the particular
+implementation, the time cost depends heavily on the performance o fmatvec
+subroutine. Therefore, we report both the wall clock time consum ption of
+the algorithm and the number of calls to the matvecsubroutine. The matvec
+subroutine used here is a nested dissection reordered block Gauss elimination
+method [14]. For an N×Ndiscretization of the computational domain, this
+matvecsubroutine has a computational cost of O(N2logN) steps.
+25Table 1 summarizes the matvecnumber, and the time cost per degree of
+freedom (DOF) for the H1, the uniform H1and theH2representations of
+the peeling algorithm. The time cost per DOF is defined by the total tim e
+cost divided by the number of grid points N2. For the H1and the uniform
+H1versions, the error criterion εin Eq. (13), Eq. (14) and Eq. (15) are all
+set to be 10−6.
+The number of callsto the matvecsubroutine isthe same inall threecases
+(as the peeling step is the same for all cases) and is reported in the t hird
+columnofTable1. Itisconfirmedthatthenumberofcallsto matvecincreases
+logarithmically with respect to Nif the domain size at level L, i.e. 2LM−L,
+is fixed as a constant. For a fixed N, the time cost is not monotonic with
+respect to L. WhenLis too small the computational cost of D(L)becomes
+dominant. When Lis too large, the application of the partial representation
+G(3)+...+G(L)to a vector becomes expensive. From the perspective of
+time cost, there is an optimal Loptfor a fixed N. We find that this optimal
+level number is the same for H1, uniform H1andH2algorithms. Table 1
+shows that Lopt= 4 forN= 32,64,128,Lopt= 5 forN= 256, and Lopt= 6
+forN= 512. This suggests that for large N, the optimal performance is
+achieved when the size of boxes on the final level Lis 8×8. In other words,
+L=LM−3.
+The memory cost per DOF for the H1, the uniform H1and theH2al-
+gorithms is reported in Table 2. The memory cost is estimated by summ ing
+the sizes of low-rank approximations as well as the size of D(L). For a fixed
+N, the memory cost generally decreases as Lincreases. This is because as L
+increases, an increasing part of the original dense matrix is repres ented using
+low-rank approximations.
+Both Table 1 and Table 2 indicate that uniform H1algorithm is signifi-
+cantly more advantageous than H1algorithm, while the H2algorithm leads
+to a further improvement over the uniform H1algorithm especially for large
+N. This fact can be better seen from Fig. 4 where the time and memory
+cost per DOF for N= 32,64,128,256,512 with optimal level number Lopt
+are shown. We remark that since the number of calls to the matvecsubrou-
+tine are the same in all cases, the time cost difference comes solely fr om the
+efficiency difference of the low rank matrix-vector multiplication subr outines.
+We measure the accuracy for the H1, the uniform H1and theH2rep-
+resentations of Gwith its actual value using the operator norm (2-norm) of
+the error matrix. Here, the 2-norm of a matrix is numerically estimat ed by
+power method [9] using several random initial guesses. We report b oth abso-
+26lute and relative errors. According to Table 3, the errors are well c ontrolled
+with respect to both increasing NandL, in spite of the more aggressive
+matrix compression strategy in the uniform H1and theH2representations.
+Moreover, for each box I, the rank rU(ε) of the uniform H1representation
+is only slightly larger than the rank r(ε) of theH1representation. This can
+be seen from Table 4. Here the average rank for a level lis estimated by
+averaging the values of rU(ε) (orr(ε)) for all low-rank approximations at
+levell. Note that the rank of the H2representation is comparable to or even
+lower than the rank in the uniform H1representation. This is partially due
+to different weighting choices in the uniformization step and H2construction
+step.
+NLmatvec H1time Uniform H1time H2time
+number per DOF (s) per DOF (s) per DOF (s)
+3243161 0.0106 0.0080 0.0084
+6443376 0.0051 0.0033 0.0033
+6454471 0.0150 0.0102 0.0106
+12844116 0.0050 0.0025 0.0024
+12854639 0.0080 0.0045 0.0045
+12865730 0.0189 0.0122 0.0125
+25647169 0.015 0.0054 0.0050
+25655407 0.010 0.0035 0.0033
+25665952 0.013 0.0058 0.0057
+25677021 0.025 0.0152 0.0154
+51258439 0.025 0.0070 0.0063
+51266708 0.018 0.0050 0.0044
+51277201 0.022 0.0079 0.0072
+Table 1: matvecnumbers and time cost per degree of freedom (DOF) for the H1, the
+uniformH1and theH2representations with different grid point per dimension Nand low
+rank compression level L. Thematvecnumbers are by definition the same in the three
+algorithms.
+27NLH1memory Uniform H1memory H2memory
+per DOF (MB) per DOF (MB) per DOF (MB)
+3240.0038 0.0024 0.0024
+6440.0043 0.0027 0.0026
+6450.0051 0.0027 0.0026
+12840.0075 0.0051 0.0049
+12850.0056 0.0029 0.0027
+12860.0063 0.0029 0.0027
+25640.0206 0.0180 0.0177
+25650.0087 0.0052 0.0049
+25660.0067 0.0030 0.0027
+25670.0074 0.0030 0.0027
+51250.0218 0.0181 0.0177
+51260.0099 0.0053 0.0049
+51270.0079 0.0031 0.0027
+Table 2: Memory cost per degree of freedom (DOF) for the H1, the uniform H1and the
+H2versions with different grid point per dimension Nand low rank compression level L.
+32 64 128 256 5120.0040.0060.0080.010.0120.0140.0160.018
+NTime per DOF (sec)
+  
+H1
+Uniform H1
+H2
+32 64 128 256 5123456789x 10−3
+NMemory per DOF (MB)
+  
+H1
+Uniform H1
+H2
+Figure 4: Comparison of the time and memory costs for the H1, the uniform H1and the
+H2versions with optimal level LoptforN= 32,64,128,256,512. The x-axis (N) is set to
+be in logarithmic scale.
+28NL H1Uniform H1H2
+Absolute Relative Absolute Relative Absolute Relative
+error error error error error error
+3242.16e-07 3.22e-07 2.22e-07 3.31e-07 2.20e-07 3.28e-07
+6442.10e-07 3.15e-07 2.31e-07 3.47e-07 2.31e-07 3.46e-07
+6451.96e-07 2.95e-07 2.07e-07 3.12e-07 2.07e-07 3.11e-07
+12842.16e-07 3.25e-07 2.26e-07 3.39e-07 2.24e-07 3.37e-07
+12852.60e-07 3.90e-07 2.68e-07 4.03e-07 2.67e-07 4.02e-07
+12862.01e-07 3.01e-07 2.09e-07 3.13e-07 2.08e-07 3.11e-07
+25641.78e-07 2.66e-07 1.95e-07 2.92e-07 2.31e-07 3.46e-07
+25652.11e-07 3.16e-07 2.26e-07 3.39e-07 2.27e-07 3.40e-07
+25662.75e-07 4.12e-07 2.78e-07 4.18e-07 2.30e-07 3.45e-07
+25671.93e-07 2.89e-07 2.05e-07 3.08e-07 2.24e-07 3.36e-07
+51252.23e-07 3.35e-07 2.33e-07 3.50e-07 1.42e-07 2.13e-07
+51262.06e-07 3.09e-07 2.17e-07 3.26e-07 2.03e-07 3.05e-07
+51272.67e-07 4.01e-07 2.74e-07 4.11e-07 2.43e-07 3.65e-07
+Table 3: Absolute and relative 2-norm errors for the H1, the uniform H1and the H2
+algorithms with different grid point per dimension Nand low rank compression level L.
+The 2-norm is estimated using power method.
+lH1Uniform H1H2
+average rank average rank average rank
+4 6 13 13
+5 6 13 11
+6 6 12 9
+Table 4: Comparison of the average rank at different levels between theH1, the uniform
+H1, and the H2algorithms, for N= 256.
+29The peeling algorithm for the construction of hierarchical matrix ca n
+be applied as well to general elliptic operators in divergence form H=
+−∇·(a(r)∇)+V(r). The computational domain, the grids are the same as
+theexampleabove, andfive-pointdiscretizationisusedforthediffe rentialop-
+erator. The media is assumed to be high contrast: a(i,j) = 1+U(i,j), with
+U(i,j) being independent random numbers uniformly distributed in [0 ,1].
+The potential functions under consideration are (1) V(i,j) = 10−3W(i,j);
+(2)V(i,j) = 10−6W(i,j).W(i,j) are independent random numbers uni-
+formly distributed in [0 ,1] and are independent of U(i,j). We test the H2
+version for N= 64,L= 4, with the compression criterion ε= 10−6. The
+resulting L2absolute and relative error of the Green’s function are reported
+in Table 5. The results indicate that the algorithms work well in these c ases,
+despite the fact that the off-diagonal elements of the Green’s fun ction have a
+slower decay than the first example. We also remark that the small r elative
+error for case (2) is due to the large 2-norm of H−1whenVis small.
+Potential Absolute error Relative error
+V(i,j) = 10−3W(i,j)5.91e-04 2.97e-07
+V(i,j) = 10−6W(i,j)3.60e-03 1.81e-09
+Table 5: Absolute and relative 2-norm errors for the H2representation of the matrix
+(−∇·(a∇)+V)−1withN= 64,L= 4 and two choice of potential function V. The
+2-norm is estimated using power method.
+4. Conclusions and future work
+In this work, we present a novel algorithm for constructing a hiera rchical
+matrix from its matrix-vector multiplication. One of the main motivatio ns
+is the construction of the inverse matrix of the stiffness matrix of a n elliptic
+differential operator. The proposed algorithm utilizes randomized s ingular
+value decomposition of low-rank matrices. The off-diagonal blocks o f the
+hierarchical matrix are computed through a top-down peeling proc ess. This
+algorithm is efficient. For an n×nmatrix, it uses only O(logn) matrix-
+vector multiplications plus O(nlogn) additional steps. The algorithm is also
+friendly to parallelization. The resulting hierarchical matrix represe ntation
+can be used as a faster algorithm for matrix-vector multiplications, as well
+as for numerical homogenization or upscaling.
+30The performance of our algorithm is tested using two 2D elliptic opera -
+tors. The H1, theuniform H1andtheH2versions oftheproposedalgorithms
+are implemented. Numerical results show that our implementations a re ef-
+ficient and accurate and that the uniform H1representation is significantly
+more advantageous over H1representation in terms of both the time cost
+and the memory cost, and H2representation leads to further improvement
+for large N.
+Although the algorithms presented require only O(logn)matvecs, the ac-
+tualnumber ofmatvecscanbequite large(forexample, several t housands for
+the example in Section 3). Therefore, the algorithms presented he re might
+not be the right choice for many applications. However, for comput ational
+problems in which one needs to invert the same system with a huge of u n-
+knowns or for homogenization problems where analytic approaches do not
+apply, our algorithm does provide an effective alternative.
+Thecurrent implementation depends explicitly onthegeometricpart ition
+ofthe rectangular domain. However, the idea of ouralgorithmcan b eapplied
+to general settings. For problems with unstructured grid, the on ly modifica-
+tion is to partition the unstructured grid with a quadtree structur e and the
+algorithms essentially require no change. For discretizations of the boundary
+integral operators, the size of an interaction list is typically much sm aller as
+many boxes contain no boundary points. Therefore, it is possible to design
+a more effective combination strategy with small number of matvecs. These
+algorithms can also be extended to the 3D cases in a straightforwar d way,
+however, we expect the constant to grow significantly. All these c ases will be
+considered in the future.
+Acknowledgement:
+L. L. is partially supported by DOE under Contract No. DE-FG02-
+03ER25587 and by ONR under Contract No. N00014-01-1-0674. L . Y. is
+partially supported by an Alfred P. Sloan Research Fellowship and an N SF
+CAREER award DMS-0846501. The authors thank Laurent Demane t for
+providing computing facility and Ming Gu and Gunnar Martinsson for he lp-
+ful discussions. L. L. and J. L. also thank Weinan E for support and encour-
+agement.
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+34
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+arXiv:1001.0150v1  [math.GR]  31 Dec 2009A Rigidity Property of Some Negatively Curved
+Solvable Lie Groups
+Nageswari Shanmugalingam, Xiangdong Xie
+Abstract
+We show that for some negatively curved solvable Lie groups, all self quasi-
+isometries are almost isometries. We prove this by showing t hat all self qua-
+sisymmetric maps of the ideal boundary (of the solvable Lie g roups) are bilips-
+chitz with respect to the visual metric. We also define parabo lic visual metrics
+on the ideal boundary of Gromov hyperbolic spaces and relate them to visual
+metrics.
+Keywords. quasiisometry, quasisymmetric map, negatively curved solvable Lie
+groups.
+Mathematics Subject Classification (2000). 20F65, 30C65, 53C20.
+1 Introduction
+In recent years, there have been a lot of interest in the large scale geometry of solvable Lie
+groups and finitely generated solvable groups ([D], [EFW1], [EFW2], [FM1], [FM2], [FM3],
+[Pe]). In particular, Eskin, Fisher and Whyte ([EFW1], [EFW 2]) proved the quasiisometric
+rigidity of the 3-dimensional solvable Lie group Sol. In thi s paper, we use quasiconformal
+analysis to prove a rigidity property of some negatively cur ved solvable Lie groups.
+LetAbe ann×ndiagonal matrix with real eigenvalues αiwithαi+1> αi>0:
+A=
+α1In10···0
+0α2In2···0
+··· ··· ··· ···
+0 0 ···αrInr
+,
+whereIniis theni×niidentity matrix and the 0’s are zero matrices (of various sizes).
+LetRact onRnby the linear transformations etA(t∈R) and we can form the semidirect
+product GA=Rn⋊AR. That is, GA=Rn×Ras a smooth manifold, and the group
+operation is given for all ( x,t),(y,s)∈Rn×Rby:
+(x,t)·(y,s) = (x+etAy,t+s).
+1The group GAis a simply connected solvable Lie group and is the subject of study in this
+paper.
+We endow GAwith the left invariant metric determined by taking the stan dard Eu-
+clidean metric at the identity of GA≈Rn×R=Rn+1. With this metric GAhas sectional
+curvature −α2
+r≤K≤ −α2
+1(and so is Gromov hyperbolic). Hence GAhas a well defined
+ideal boundary ∂GA. There is a so-called cone topology on GA=GA∪∂GA, in which
+∂GAis homeomorphic to the n-dimensional sphere and GAis homeomorphic to the closed
+(n+1)-ball in the Euclidean space. For each x∈Rn, the map γx:R→GA,γx(t) = (x,t)
+is a geodesic. We call such a geodesic a vertical geodesic. It can be checked that all vertical
+geodesics are asymptotic as t→+∞. Hence they define a point ξ0in the ideal boundary
+∂GA.
+SinceGAis Gromov hyperbolic, there is a family of visual metrics on ∂GA. For each
+ξ∈∂GA, there is also the so-called parabolic visual metric on ∂GA\{ξ}. The relation
+between visual metrics and parabolic visual metrics is anal ogous to the relation between
+spherical metric (on the sphere) and the Euclidean metric (o n the one point complement of
+the sphere). See Section 5 for a discussion of all these in the setting of Gromov hyperbolic
+spaces. We next recall the parabolic visual metric Don∂GAviewed from ξ0.
+The set ∂GA\{ξ0}can be naturally identified with Rn(see Section 2). Write Rn=
+Rn1× ··· ×Rnr, whereRniis the eigenspace associated to the eigenvalue αiofA. Each
+pointx∈Rncan be written as x= (x1,···,xr) withxi∈Rni. The parabolic visual metric
+Don∂GA\{ξ0} ≈Rnis defined by:
+D(x,y) = max{|x1−y1|,|x2−y2|α1/α2,···,|xr−yr|α1/αr},
+for allx= (x1,···,xr),y= (y1,···,yr)∈Rn.
+Letη: [0,∞)→[0,∞) be a homeomorphism. An embedding of metric spaces f:X→
+Yis anη-quasisymmetric embedding if for all distinct triples x,y,z∈X, we have
+d(f(x),f(y))
+d(f(x),f(z))≤η/parenleftbiggd(x,y)
+d(x,z)/parenrightbigg
+.
+Iffis further assumed to be a homeomorphism, we say it is η-quasisymmetric . A map
+f:X→Yis quasisymmetric if it is η-quasisymmetric for some η.
+Whenr≥2, Bruce Kleiner has proved that ([K]) every self quasisymme try of∂GA
+(equippedwithavisualmetric)preservesthehorizontalfo liation (seeSection3)andfixesthe
+pointξ0. This is one of the main ingredients in the proof of our main re sult. Since Kleiner’s
+proof is unpublished, we include a proof here for completene ss. Notice that Kleiner’s result
+implies that a self quasisymmetry of ∂GAinduces a self map of ( Rn,D).
+The following is the main result of this paper.
+Theorem 1.1. LetGAandξ0∈∂GAbe as above. If r≥2, then every self quasisymmetry
+of∂GA(equipped with a visual metric) is bilipschitz on ∂GA\ {ξ0}with respect to the
+parabolic visual metric D.
+One should compare this with quasiconformal maps on Euclide an spaces ([GV]) and
+Heisenberg groups ([B]), where there are non-bilipschitz q uasiconformal maps. On the
+2other hand, the conclusion of Theorem 1.1 is not as strong as i n the cases of quaternionic
+hyperbolic spaces, Cayley plane ([P2]) and Fuchsian buildi ngs ([BP], [X]), where every
+quasisymmetric map of the ideal boundary is actually a confo rmal map. In our case, there
+are many non-conformal quasisymmetricmaps of theideal bou ndaryof GA. We also remark
+that in [T2, Section 15] Tyson has previously classified (qua si)metric spaces of the form
+(Rn,D) up to quasisymmetry.
+We list three consequences of Theorem 1.1.
+LetL≥1 andC≥0. A (not necessarily continuous ) map f:X→Ybetween two
+metric spaces is an ( L,A)-quasiisometry if:
+(1)d(x1,x2)/L−C≤d(f(x1),f(x2))≤Ld(x1,x2)+Cfor allx1,x2∈X;
+(2) for any y∈Y, there is some x∈Xwithd(f(x),y)≤C.
+In the case L= 1, we call fanalmost isometry .
+Corollary 1.2. Assume that r≥2. Then every self quasiisometry of GAis an almost
+isometry.
+Notice thatanalmost isometryisnotnecessarily afinitedis tanceaway fromanisometry.
+The following result was previously obtained by B. Kleiner [ K].
+Corollary 1.3. Ifr≥2, thenGAis not quasiisometric to any finitely generated group.
+In the identification of GAwithRn×R, we view the map h:Rn×R,h(x,t) =tas the
+height function. A quasiisometry ϕofGAis height-respecting if |h(ϕ(x,t))−t|is bounded
+independent of x,t.
+Corollary 1.4. Assume that r≥2. Then all self quasiisometries of GAare height-
+respecting.
+The question of whether a quasiisometry of GAis height-respecting is important for
+the following three reasons. First, Mosher and Farb ([FM1]) have classified a large class of
+solvable Lie groups (including groups of type GA) up to height-respecting quasiisometries.
+Second, there is no known examples of non-height-respectin g quasiisometries except for
+rank one symmetric spaces of noncompact type. Finally, show ing a quasiisometry is height-
+respecting is a key step in the proof of the quasiisometric ri gidity of Sol ([EFW1], [EFW2]).
+Whenr= 1, the group GAis isometric to a rescaling of the real hyperbolic space. In
+this case, all the above results fail.
+This paper is structured as follows. In Section 2 we review so me basics about the group
+GA. In Section 3 we prove that quasisymmetric self-maps of ∂GA\ {ξ0}equipped with
+the parabolic visual metric preserve horizontal foliation s, and in Section 4 we will prove
+that such maps are bilipschitz with respect to this metric. T he main result of this paper,
+Theorem 1.1, is proven in Section 5, where a discussion of par abolic visual metrics on the
+ideal boundary and their connection to the visual metrics ca n also be found. In Section 6
+we provide the proofs of the Corollaries stated in Section 1.
+Acknowledgment . We would like to thank Bruce Kleiner for helpful discussion s. The
+second author would also like to thank the Department of Math ematical Sciences at Georgia
+Southern University for generous travel support.
+32 The Solvable Lie Groups GA
+In this section we review some basic facts about the group GAand define several parabolic
+visual (quasi)metrics on the ideal boundary.
+LetAandGAbe as in the Introduction. We endow GAwith the left invariant metric
+determinedby takingthestandardEuclideanmetric at theid entity of GA≈Rn×R=Rn+1.
+At a point ( x,t)∈Rn×R≈GA, the tangent space is identified with Rn×R, and the
+Riemannian metric is given by the symmetric matrix
+/parenleftbigge−2tA0
+0 1/parenrightbigg
+.
+With this metric GAhas sectional curvature −α2
+r≤K≤ −α2
+1. HenceGAhas a well defined
+ideal boundary ∂GA. All vertical geodesics γx(x∈Rn) are asymptotic as t→+∞. Hence
+they define a point ξ0in the ideal boundary ∂GA.
+The sets Rn×{t}(t∈R) are horospheres centered at ξ0. For each t∈R, the induced
+metric on the horosphere Rn×{t} ⊂GAis determined by the quadratic form e−2tA. This
+metric has distance formula dRn×{t}((x,t),(y,t)) =|e−tA(x−y)|. Here| · |denotes the
+Euclidean norm. The distance between two horospheres, corr esponding to t=t1and
+t=t2, is|t1−t2|. It follows that for ( x1,t1),(x2,t2)∈GA=Rn×R,
+d((x1,t1),(x2,t2))≥ |t1−t2|. (2.1)
+Each geodesic ray in GAis asymptotic to either an upward oriented vertical geodesi c or
+a downward oriented vertical geodesic. The upward oriented geodesics are asymptotic to ξ0
+and the downward oriented vertical geodesics are in 1-to-1 c orrespondence with Rn. Hence
+∂GA\{ξ0}can be naturally identified with Rn.
+Givenx,y∈Rn≈∂GA\{ξ0}, the parabolic visual quasimetric De(x,y) is defined
+as follows: De(x,y) =et, where tis the unique real number such that at height tthe
+two vertical geodesics γxandγyare at distance one apart in the horosphere; that is,
+dRn×{t}((x,t),(y,t)) =|e−tA(x−y)|= 1.Here the subscript einDemeans it corresponds
+to the Euclidean norm. This definition of parabolic visual qu asimetric is very natural, but
+Dedoes not have a simple formula. Next we describe another para bolic visual quasimetric
+which is bilipschitz equivalent with Deand admits a simple formula. Recall that a quasi-
+metric on a set Ais a function ρ:A×A→[0,∞) satisfying: (1) ρ(x,y) =ρ(y,x) for
+allx,y∈A; (2)ρ(x,y) = 0 only when x=y; (3) there is a constant L≥1 such that
+ρ(x,z)≤L(ρ(x,y)+ρ(y,z)) for all x,y,z∈A.
+In addition to the Euclidean norm, there is another norm on Rnthat is naturally
+associated to GA. WriteRn=Rn1×··· ×Rnr, whereRniis the eigenspace associated to
+the eigenvalue αiofA. Each point x∈Rncan be written as x= (x1,···,xr) withxi∈Rni.
+The block supernorm is given by: |x|s= max{|x1|,···,|xr|}forx= (x1,···,xr). Using
+this norm one can define another parabolic visual quasimetri c on∂GA\{ξ0}as follows:
+Ds(x,y) =et, wheretis the unique real number such that at height tthe two vertical
+geodesics γxandγyare at distance one apart with respect to the norm | · |s; that is,
+|e−tA(x−y)|s= 1.Here the subscript sinDsmeans it corresponds to the block supernorm
+4|·|s. ThenDsis given by [D, Lemma 7]:
+Ds(x,y) = max{|x1−y1|1
+α1,···,|xr−yr|1
+αr},
+for allx= (x1,···,xr),y= (y1,···,yr)∈Rn.
+Notice that |x|s≤ |x| ≤√r|x|sfor allx∈Rn. Using this, one can verify the following
+elementary lemma, whose proof is left to the reader.
+Lemma 2.1. For allx,y∈Rnwe have Ds(x,y)≤De(x,y)≤r1/2α1Ds(x,y).
+In general, Dsdoes not satisfy the triangle inequality. However, for each 0< ǫ≤α1, the
+function Dǫ
+sis always a metric, called a parabolic visual metric. In this paper we consider
+the following parabolic visual metric
+D(x,y) =Dα1s(x,y) = max{|x1−y1|,|x2−y2|α1/α2,···,|xr−yr|α1/αr}.
+With respect to this metric the rectifiable curves in Rn≈∂GA\{ξ0}are necessarily curves
+of theform γ:I→Rnwithγ(t) = (γ1(t),c2,···,cr) whereci∈Rni, 2≤i≤r, are constant
+vectors. This follows from the fact that the directions corr esponding to Rni,i≥2, have
+their Euclidean distance components “snowflaked” by the pow erα1/αi<1.
+3 Quasisymmetric maps preserve horizontal folia-
+tions
+In this section we show that every self-quasisymmetry of ∂GAfixes the point ξ0∈∂GAand
+preserves a natural foliation on ∂GA\{ξ0}.
+Recall that a metric space Xendowed with a Borel measure µis anAhlfors Regular
+space of dimension Q (for short, a Q-regular space) if there exists a constant C0≥1 so
+that
+C−1
+0rQ≤µ(Br)≤C0rQ
+for every ball Brwith radius r <diam(X).
+We need the following result; see [T1] for the definition of th e modulusMod Qof a family
+of curves.
+Theorem 3.1 ([T1, Theorem 1.4]) .LetXandYbe locally compact, connected, Q-regular
+metric spaces ( Q >1) and let f:X→Ybe anη-quasisymmetric homeomorphism. Then
+there is a constant Cdepending only on η,Qand the regularity constants of XandYso
+that1
+CModQΓ≤ModQf(Γ)≤CModQΓ
+for every curve family ΓinX.
+Recall that we write RnasRn=Rn1×···×Rnr. SetY=Rn2×··· ×Rnrand write
+Rn=Rn1×Y. Since we assume r≥2, the set Yis nontrivial. The subsets {Rn1×{y}:
+y∈Y}form a foliation of Rn. We call this foliation the horizontal foliation and each le af
+5Rn1×{y}a horizontal leaf. Sinceα1
+αi<1 for all 2 ≤i≤r, we notice that a curve in ( Rn,D)
+is not rectifiable if it is not contained in a horizontal leaf.
+Observe that ( Rni,|·|α1/αi) with the Hausdorff measure (which is comparable to the ni-
+dimensional Lebesgue measure) is niαi/α1-regular. Let µbe the product of the Hausdorff
+measures on the factors ( Rni,|·|α1/αi). Then it is easy to see that ( Rn,D) with the measure
+µisQ-regular with Q= Σr
+i=1niαi
+α1. It follows that Theorem 3.1 applies to the metric
+space (Rn,D). We also point out here that the Hausdorff measure µis comparable to the
+canonical n-dimensional Lebesgue measure on Rn.
+Theorem 3.2. Ifr≥2, then every quasisymmetry F: (Rn,D)→(Rn,D)preserves the
+horizontal foliation on Rn.
+Proof.Suppose Fdoes not preserve the horizontal foliation. Then there are t wo points p
+andqin someRn1×{y}such that f(p) andf(q) are not in the same horizontal leaf. Let γ
+be the Euclidean line segment from ptoqand Γ be the family of straight segments parallel
+toγinRnwhose union is an n-dimensional circular cylinder with γas the central axis.
+The curves in Γ are rectifiable with respect to the metric D. Sincefis a homeomorphism,
+by choosing the radius of the circular cylinder to be sufficien tly small (by a compactness
+argument) wemayassumethat nocurveinΓismappedintoahori zontal leaf. Itfollows that
+f(Γ) has no locally rectifiable curve and so Mod Qf(Γ) = 0. On the other hand, [V1], 7.2
+(page 21) shows that Mod QΓ>0 (the Euclidean length element on each β∈Γ is the same
+as the length element on βobtained from the metric D). Since Q= Σr
+i=1niαi
+α1>1, this
+contradicts Theorem 3.1. Hence each horizontal leaf is mapp ed to a horizontal leaf.
+4 Quasisymmetry implies Bilipschitz
+Inthis section weshowthat each selfquasisymmetryof ( Rn,D)isactually abilipschitzmap.
+One should contrast this with the case of Euclidean spaces an d Heisenberg groups, where
+there are non-bilipschitz quasisymmetric maps ([GV], [B]) . On the other hand, ( Rn,D) is
+not as rigid as the ideal boundary of a quaternionic hyperbol ic space or a Cayley plane
+([P2]) or a Fuchsian building ([BP], [X]), where each self qu asisymmetry is a conformal
+map.
+LetK≥1 andC >0. A bijection F:X1→X2between two metric spaces is called a
+K-quasisimilarity (with constant C) if
+C
+Kd(x,y)≤d(F(x),F(y))≤CKd(x,y)
+for allx,y∈X1. It is clear that a map is a quasisimilarity if and only if it is a bilipschitz
+map. The point of using the notion of quasisimilarity is that sometimes there is control on
+Kbut not on C.
+Theorem 4.1. LetF: (Rn,D)→(Rn,D)be anη-quasisymmetry. Then Fis aK-
+quasisimilarity with K= (η(1)/η−1(1))2r+2.
+6In this section, we first develop some intermediate results, and then use these results to
+provide a proof of this theorem. We first recall some definitio ns.
+Letg:X1→X2be a homeomorphism between two metric spaces. We define for ev ery
+x∈X1andr >0,
+Lg(x,r) = sup{d(g(x),g(x′)) :d(x,x′)≤r},
+lg(x,r) = inf{d(g(x),g(x′)) :d(x,x′)≥r}.
+Notice that if X1is connected and X1\B(x,r) is non-empty, then lg(x,r)≤Lg(x,r). In
+this paper, we only consider connected metric spaces. Set
+Lg(x) = limsup
+r→0Lg(x,r)
+r, lg(x) = liminf
+r→0lg(x,r)
+r.
+Then
+Lg−1(g(x)) =1
+lg(x)andlg−1(g(x)) =1
+Lg(x)
+for anyx∈X1. Ifgis anη-quasisymmetry, then Lg(x,r)≤η(1)lg(x,r) for allx∈X1and
+r >0. Hence if in addition
+lim
+r→0Lg(x,r)
+ror lim
+r→0lg(x,r)
+r
+exists, then
+0≤lg(x)≤Lg(x)≤η(1)lg(x)≤ ∞.
+Recall the decomposition Rn=Rn1×Y. Given points y= (x2,···,xr) andy′=
+(x′
+2,···,x′
+r)∈Ywithxi,x′
+i∈Rni, set
+DY(y,y′) = max{|x2−x′
+2|α1
+α2,|x3−x′
+3|α1
+α3,···,|xr−x′
+r|α1
+αr}.
+Forp= (x1,y),p′= (x′
+1,y′)∈Rn1×Y, we have D(p,p′) = max{|x1−x′
+1|,DY(y,y′)}. We
+notice that for every y1,y2∈Y, theHausdorff distance in the metric Dof the two horizontal
+leaves,
+HD(Rn1×{y1},Rn1×{y2}) =DY(y1,y2). (4.1)
+Also, for any p= (x1,y1)∈Rn1×Yand any y2∈Y,
+D(p,Rn1×{y2}) =DY(y1,y2). (4.2)
+By Theorem 3.2 the quasisymmetry Fpreserves the horizontal foliation. Hence it
+induces a map G:Y→Ysuch that for any y∈Y,F(Rn1× {y}) =Rn1× {G(y)}. For
+eachy∈Y, letH(·,y) :Rn1→Rn1be the map such that F(x,y) = (H(x,y),G(y)) for
+allx∈Rn1. Because F: (Rn,D)→(Rn,D) is anη-quasisymmetry, it follows that for
+each fixed y∈Y, the map H(·,y) :Rn1→Rn1is anη-quasisymmetry with respect to
+the Euclidean metric on Rn1. The following lemma together with equations (4.1) and (4.2 )
+imply that G: (Y,DY)→(Y,DY) is also an η-quasisymmetry.
+7Lemma 4.2. ([T2, Lemma 15.9]) Letg:X1→X2be anη-quasisymmetry and A,B,C⊂
+X1. IfHD(A,B)≤tHD(A,C)for some t≥0, then there is some a∈Asuch that
+HD(g(A),g(B))≤η(t)d(g(a),g(C)).
+We recall that if g:X1→X2is anη-quasisymmetry, then g−1:X2→X1is an
+η2-quasisymmetry, where η2(t) = (η−1(t−1))−1, see [V2, Theorem 6.3]. Note that η2(1) =
+1/η−1(1) andη−1
+2(1) = 1/η(1).
+In the proofs of the following lemmas, the quantities lG,LG,lG−1,LG−1are defined with
+respect to the metric DY. Similarly, lH(·,y),LH(·,y),lIyandLIyare defined with respect
+to the Euclidean metric on Rn1, where Iy:=H(·,y)−1:Rn1→Rn1. Lemmas 4.6 and
+4.7 together verify Theorem 4.1 for the case r= 2. At the end of this section we will use
+induction to then complete the proof of Theorem 4.1 for the ge neral case r≥2.
+Lemma 4.3. The following holds for all y∈Yandx∈Rn1:
+(1)LG(y,r)≤η(1)lH(·,y)(x,r)forr >0;
+(2)η−1(1)lH(·,y)(x)≤lG(y)≤η(1)lH(·,y)(x);
+(3)η−1(1)LH(·,y)(x)≤LG(y)≤η(1)LH(·,y)(x).
+Proof.To prove (1), let y∈Y,x∈Rn1andr >0. Lety′∈Ybe an arbitrary point with
+DY(y,y′)≤randx′∈Rn1an arbitrary point with |x−x′| ≥r. ThenD((x,y),(x,y′))≤
+r≤D((x,y),(x′,y)). Since Fisη-quasisymmetric, we have
+DY(G(y),G(y′))≤D(F(x,y),F(x,y′))≤η(1)D(F(x,y),F(x′,y))
+=η(1)|H(x,y)−H(x′,y)|.
+Sincey′andx′are chosen arbitrarily and are independent of each other, th e inequality
+follows.
+Next we prove (2) and (3). Since Yis connected, we have lG(y,r)≤LG(y,r). Now
+the second inequality of (2) follows from (1). Similarly the second inequality of (3) follows
+from (1) and the fact that lH(·,y)(x,r)≤LH(·,y)(x,r).
+To prove the first inequalities in (2) and (3), observe that th e inverse map F−1:
+(Rn,D)→(Rn,D) is anη2-quasisymmetry, with
+F−1(x,y) = (H(·,G−1(y))−1(x),G−1(y)) = (IG−1(y)(x),G−1(y)).
+Applying the second inequality of (2) proven above to IyandG−1, we obtain:
+1
+LG(y)=lG−1(G(y))≤η2(1)·lIy(H(x,y)) =1
+η−1(1)·1
+LH(·,y)(x),
+henceLG(y)≥η−1(1)LH(·,y)(x), which is the first inequality of (3). Similarly, using the
+second inequality of (3) we obtain the first inequality of (2) .
+Whenr= 2, we have Y=Rn2andDY=|·|α1
+α2.
+Lemma 4.4. Assume that r= 2. Then0< lG(y)≤LG(y)≤η(1)lG(y)<∞for a.e.
+y∈Ywith respect to the Lebesgue measure on Y=Rn2.
+8Proof.Observe in this case that DY(y,y′) =|y−y′|α1/α2fory,y′∈Y=Rn2. Because Gis
+anη-quasisymmetry with respect to the metric DY, it isη1-quasisymmetric with respect to
+theEuclideanmetric, where η1(t) = (η(tα1/α2))α2/α1. Hencethemap G: (Rn2,|·|)→(Rn2,|·
+|) is differentiable a.e. with respect to the Lebesgue measure. WithLe
+G,le
+Gthe distortion
+quantities of the map Gwith respect to the Euclidean metric, the differentiability p roperty
+ofGshows that lim r→0Le
+G(y,r)
+rand lim r→0le
+G(y,r)
+rexist. Since LG(y,r) =Le
+G(y,rα2/α1)α1/α2
+andlG(y,r) =le
+G(y,rα2/α1)α1/α2, this implies that both lim r→0LG(y,r)
+rand lim r→0lG(y,r)
+r
+exist for a.e. y∈Y. It follows that
+0≤lG(y)≤LG(y)≤η(1)lG(y)≤ ∞.
+Fixy∈Ysuch that both lim r→0LG(y,r)
+rand lim r→0lG(y,r)
+rexist. We next prove that
+LG(y)/\e}atio\slash= 0,∞. Suppose that LG(y) =∞. ThenlG(y) =∞and so by Lemma 4.3 (2),
+lH(·,y)(x) =∞for allx∈Rn1. Hence Iy=H(·,y)−1:Rn1→Rn1has the property that
+LIy(x) = 0 for all x∈Rn1. This implies that Iyis a constant map, contradicting the fact
+that it is a homeomorphism. Similarly we use Lemma 4.3 (3) to s how that LG(y)/\e}atio\slash= 0.
+In the next two lemmas we use the fact that η(1)≥1 and 0< η−1(1)≤1.
+Lemma 4.5. Suppose that r= 2. Then, for a.e. y∈Y, the map H(·,y) :Rn1→Rn1is
+anη(1)/η−1(1)-quasisimilarity with constant lG(y)>0.
+Proof.By Lemma 4.3 (2) we have lH(·,y)(x)≥lG(y)/η(1). Lemma 4.3 (3) and Lemma 4.4
+imply that, for a.e. y∈Y, we have lG(y)>0 and
+LH(·,y)(x)≤LG(y)/η−1(1)≤(η(1)/η−1(1))lG(y)
+for allx∈Rn1. Because Rn1is a geodesic space, for a.e. y∈Ythe map H(·,y) is an
+η(1)/η−1(1)-quasisimilarity with constant lG(y).
+Lemma 4.6. Ifr= 2, then there exists a constant C >0with the following properties:
+(1) For each y∈Y,H(·,y)is an(η(1)/η−1(1))4-quasisimilarity with constant C;
+(2)G: (Y,DY)→(Y,DY)is an(η(1)/η−1(1))5-quasisimilarity with constant C.
+Proof.(1) Fix any y0∈Ythat satisfies both Lemma 4.4 and Lemma 4.5. Set C=lG(y0).
+Lety∈Ybe an arbitrary point satisfying both Lemma 4.4 and Lemma 4.5 . Fixx0∈Rn1
+and choose x∈Rn1such that |x−x0| ≥DY(y,y0). Then
+D((x,y0),(x0,y)) =D((x,y),(x0,y)) =|x−x0|.
+By choosing xso that in addition |H(x,y0)−H(x0,y)|> DY(G(y0),G(y)), by the η-
+quasisymmetry of Fwe have
+|H(x,y0)−H(x0,y)|=D(F(x,y0),F(x0,y))
+≤η(1)D(F(x,y),F(x0,y)) =η(1)|H(x,y)−H(x0,y)|.
+9By the choice of yand Lemma 4.5, we have
+|H(x,y)−H(x0,y)| ≤(η(1)/η−1(1))lG(y)|x−x0|.
+On the other hand,
+|H(x,y0)−H(x0,y)| ≥ |H(x,y0)−H(x0,y0)|−|H(x0,y0)−H(x0,y)|
+≥lG(y0)
+η(1)/η−1(1)|x−x0|−|H(x0,y0)−H(x0,y)|.
+Combining the above inequalities and letting |x−x0| → ∞, we obtain
+lG(y)≥1
+(η(1))3(η−1(1))−2lG(y0) =C
+(η(1))3(η−1(1))−2. (4.3)
+Switching the roles of y0andy, we obtain lG(y)≤(η(1))3(η−1(1))−2lG(y0). By Lemma 4.4,
+we have
+LG(y)≤η(1)lG(y)≤(η(1))4(η−1(1))−2C. (4.4)
+Because Rn1is a geodesic space, to show that H(·,y) is a quasisimilarity it suffices to
+gain control over lH(·,y)andLH(·,y). By (4.4) and Lemma 4.3 (3),
+LH(·,y)(x)≤LG(y)/η−1(1)≤C(η(1))4(η−1(1))−3
+for allx∈Rn1, and by (4.3) and Lemma 4.3 (2),
+lH(·,y)(x)≥1
+η(1)lG(y)≥C
+(η(1))4(η−1(1))−2.
+for allx∈Rn1. Hence for a.e. y,H(·,y) is an (η(1)/η−1(1))4-quasisimilarity with constant
+C. A limiting argument shows this is true for all y. Hence (1) holds.
+(2) Recall that when r= 2 we have Y=Rn2andDY=| · |α1/α2. Hence to prove
+(2) it suffices to show that G: (Rn2,| · |)→(Rn2,| · |) is aK-quasisimilarity with K=
+(η(1)/η−1(1))5α2/α1. As observed before, Gisη1-quasisymmetric with respect to the Eu-
+clidean metric, where η1(t) = (η(tα1/α2))α2/α1. Because Rn2is a geodesic space, it suf-
+fices to gain control over le
+GandLe
+G, where le
+GandLe
+Gare similar to lGandLG, but
+with the Euclidean metric instead of the metric DY. Because le
+G(p) =lG(p)α2/α1and
+Le
+G(p) =LG(p)α2/α1, it suffices to gain control over the quantities lGandLGin terms of
+(η(1)/η−1(1))5.
+Notice that (1) implies
+C
+(η(1)/η−1(1))4≤lH(·,y)(x)≤LH(·,y)(x)≤C(η(1)/η−1(1))4
+for allx∈Rn1and ally∈Y. By Lemma 4.3, for all y∈Ywe have
+C
+(η(1)/η−1(1))5≤lG(y)≤LG(y)≤C(η(1)/η−1(1))5.
+Hence (2) holds.
+10Lemma 4.7. Suppose that r≥2and there are constants K≥1andC >0with the
+following properties:
+(1)G: (Y,DY)→(Y,DY)is aK-quasisimilarity with constant C;
+(2) For each y∈Y,H(·,y)is aK-quasisimilarity with constant C.
+ThenFis an(η(1)/η−1(1))K-quasisimilarity with constant C.
+Proof.Let(x1,y1),(x2,y2)∈Rn1×Y. Weshallfirstestablishalowerboundfor D(F(x1,y1),F(x2,y2)).
+If|x1−x2| ≤DY(y1,y2), thenD((x1,y1),(x2,y2)) =DY(y1,y2) and by (1),
+D(F(x1,y1),F(x2,y2))≥DY(G(y1),G(y2))≥C
+KDY(y1,y2) =C
+KD((x1,y1),(x2,y2)).
+If|x1−x2|> DY(y1,y2), then
+D((x1,y1),(x2,y2)) =D((x1,y2),(x2,y2)) =|x1−x2|,
+and since Fis anη-quasisymmetry, by using (2),
+D(F(x1,y1),F(x2,y2))≥1
+η(1)D(F(x1,y2),F(x2,y2))
+=1
+η(1)|H(x1,y2)−H(x2,y2)|
+≥C
+η(1)K|x1−x2|
+=C
+η(1)KD((x1,y1),(x2,y2)).
+Hence we have a lower bound for D(F(x1,y1),F(x2,y2)).
+By (1),G−1: (Y,DY)→(Y,DY) is aK-quasisimilarity with constant C−1. Similarly,
+(2) implies that for each y∈Y, (H(·,y))−1is aK-quasisimilarity with constant C−1. Also
+recall that F−1is anη2-quasisymmetry and Fis anη-quasisymmetry. Now the argument
+in the previous paragraph applied to F−1implies
+D(F−1(x1,y1),F−1(x2,y2))≥1
+CKη2(1)D((x1,y1),(x2,y2)).
+It follows that
+D(F(x1,y1),F(x2,y2))≤CKη2(1)D((x1,y1),(x2,y2)) =CK
+η−1(1)D((x1,y1),(x2,y2))
+for all (x1,y1),(x2,y2)∈Rn, completing the proof.
+Proof of Theorem 4.1. We induct on r. Lemmas 4.6 and 4.7 yield the desired result
+in the case r= 2. Now we assume that r≥3 and that the Theorem is true for r−
+1. By Lemma 4.2, Finduces an η-quasisymmetry G: (Y,DY)→(Y,DY). It follows
+thatGisη1-quasisymmetric with respect to the metric Dα2/α1
+Y(and it is easy to verify
+11that this is indeed a metric), where η1(t) = [η(tα1/α2)]α2/α1. We point out here that for
+(x2,···,xr),(x′
+2,···,x′
+r)∈Y,
+DY((x2,···,xr),(x′
+2,···,x′
+r))α2/α1= max{|x2−x′
+2|,|x3−x′
+3|α2/α3,···,|xr−x′
+r|α2/αr}.
+Hence the induction hypothesis applied to G: (Y,Dα2/α1
+Y)→(Y,Dα2/α1
+Y) shows that Gis
+an (η1(1)/η−1
+1(1))2r-quasisimilarity with constant C. Therefore G: (Y,DY)→(Y,DY) is a
+K1-quasisimilarity with constant Cα1/α2, where
+K1=/parenleftbiggη1(1)
+η−1
+1(1)/parenrightbigg2rα1
+α2=/parenleftbiggη(1)
+η−1(1)/parenrightbigg2r
+. (4.5)
+This implies that Cα1/α2/K1≤lG(y)≤LG(y)≤Cα1/α2K1for ally∈Y. Now Lemma 4.3
+yields
+Cα1/α21
+K1η(1)≤lH(·,y)(x)≤LH(·,y)(x)≤Cα1/α2K1
+η−1(1)
+for ally∈Yand allx∈Rn1. SinceRn1is a geodesic space, for each y∈Ythe map
+H(·,y) is aK1η(1)
+η−1(1)-quasisimilarity with constant Cα1/α2. By Lemma 4.7, the map Fis a
+K1(η(1)
+η−1(1))2-quasisimilarity with constant Cα1/α2. HereK1is as in (4.5).
+5 Parabolic Visual Metrics
+In this section we introduce parabolic visual metrics, disc uss their relation with the visual
+metrics and give a sufficient condition for them to be doubling . We then use these results
+to complete the proof of Theorem 1.1.
+Parabolic visual metrics have been defined by Hersonsky-Pau lin ([HP], see also [BK])
+for CAT( −1) spaces. Here we formally construct parabolic visual metr ics in the setting of
+Gromov hyperbolic spaces. Since GAis Gromov hyperbolic, the theory developed here is
+applicable to ∂GAas well. The metric D(onRn=∂GA\{ξ0}) used in the previous sections
+is bilipschitz equivalent with a parabolic visual metric co nstructed in this section, see the
+discussion after Proposition 5.1.
+Parabolic visual metric is defined on the one-point compleme nt of the ideal boundary.
+The relationship between visual metric and parabolic visua l metric is similar to the rela-
+tionship between the spherical metric (on the sphere) and th e Euclidean metric (on the one
+point complement of the sphere). See Proposition 5.4 for the precise statement.
+LetXbe aδ-hyperbolic proper geodesic metric space for some δ≥0. Letξ∈∂Xand
+p∈X. Then there exists a ray from ptoξ. Letγ: [0,∞)→Xbe such a ray. Define
+Bγ:X→RbyBγ(x) = lim t→+∞(d(γ(t),x)−t). The triangle inequality implies that the
+limit exists and that |Bγ(x)−Bγ(y)| ≤d(x,y) for allx,y∈X. Note that Bγ(γ(t0)) =−t0
+for allt0≥0. Since any two rays γ1andγ2fromptoξare at Hausdorff distance at most
+δfrom each other, we have |Bγ1(x)−Bγ2(x)| ≤δfor allx∈X.
+TheBuseman function Bξ,p:X→Rcentered at ξwith base point pis:
+Bξ,p(x) = sup{Bγ(x) :γis a geodesic ray from ptoξ}.
+12Because Bγis 1-Lipschitz, Bξ,pis 1-Lipschitz. The above discussion shows that Bγ(x)≤
+Bξ,p(x)≤Bγ(x)+δfor allx∈Xand every ray γfromptoξ. By Proposition 8.2 of [GdlH],
+there exists a constant c=c(δ) such that for any two points p1,p2∈X, anyξ∈∂Xand
+allx∈Xwe have
+|Bξ,p1(x)−Bξ,p2(x)−Bξ,p1(p2)| ≤c. (5.1)
+Letǫ >0,p∈X,ξ∈∂X, andη1/\e}atio\slash=η2∈∂X\{ξ}. Given a complete geodesic σfrom
+η1toη2, letHξ,p(σ) = inf{Bξ,p(x) :x∈σ}. Define
+Dξ,p,ǫ(η1,η2) =e−ǫHξ,p(η1,η2),
+where
+Hξ,p(η1,η2) = inf{Hξ,p(σ) :σis a complete geodesic from η1toη2}.
+Since any two complete geodesics from η1toη2are at most Hausdorff distance 2 δapart,
+we have Hξ,p(σ)−2δ≤Hξ,p(η1,η2)≤Hξ,p(σ) for any complete geodesic σfromη1toη2.
+An argument similar to that found in [CDP, p.124] shows the fo llowing:
+Proposition 5.1. There exists a constant ǫ0, depending only on δ, with the following
+property. If Xis aδ-hyperbolic proper geodesic metric space, for each 0< ǫ≤ǫ0, each
+p∈Xand each ξ∈∂Xthere exists a metric dξ,p,ǫon∂X\{ξ}such that1
+2Dξ,p,ǫ(η1,η2)≤
+dξ,p,ǫ(η1,η2)≤Dξ,p,ǫ(η1,η2)for allη1,η2∈∂X\{ξ}.
+The metric dξ,p,ǫis called a parabolic visual metric . WithX=GA,p= (0,0), by using
+Lemmas 6.1 and 6.2 one can see that Dξ0,p,1is bilipschitz equivalent with De. It follows
+from Lemma 2.1 and Proposition 5.1 that dξ0,p,α1is bilipschitz equivalent with the metric
+Dconsidered in the previous sections.
+We next discuss how dξ,p,ǫvaries with pandǫ.
+Proposition 5.2. Suppose Xis aδ-hyperbolic proper geodesic metric space. Then
+(1) For any p1,p2∈X, the identity map id : (∂X\{ξ},dξ,p1,ǫ)→(∂X\{ξ},dξ,p2,ǫ)is a
+K-quasisimilarity, where Kdepends only on δ;
+(2) For0< ǫ1,ǫ2≤ǫ0, the identity map id : (∂X\{ξ},dξ,p,ǫ1)→(∂X\{ξ},dξ,p,ǫ2)isη-
+quasisymmetric with η(t) = 21+ǫ2
+ǫ1tǫ2
+ǫ1;
+(3) For any p1,p2∈Xand any 0< ǫ1,ǫ2≤ǫ0, the identity map id : (∂X\{ξ},dξ,p1,ǫ1)→
+(∂X\{ξ},dξ,p2,ǫ2)is quasisymmetric.
+Proof.To prove (1) let η1,η2∈∂X\{ξ}. Then Proposition 5.1 and inequality (5.1) imply
+dξ,p2,ǫ(η1,η2)≤Dξ,p2,ǫ(η1,η2) =e−ǫHξ,p2(η1,η2)≤e−ǫHξ,p1(η1,η2)+ǫBξ,p1(p2)+cǫ
+≤2ecǫ·eǫBξ,p1(p2)·dξ,p1,ǫ(η1,η2).
+Similarly, we obtain dξ,p2,ǫ(η1,η2)≥1
+2ecǫ·eǫBξ,p1(p2)·dξ,p1,ǫ(η1,η2). The statement holds
+withK= 2ecǫ0and constant C=eǫBξ,p1(p2).
+The claim (2) follows from Proposition 5.1, and (3) follows f rom (1) and (2).
+13We next discuss the relation between the parabolic visual me tric and the visual metric.
+Recall that there is a constant ǫ1depending only on δsuch that for any p∈Xand any
+0< ǫ≤ǫ1, there is a visual metric dp,ǫon∂Xsatisfying
+1
+2e−ǫ(η1|η2)p≤dp,ǫ(η1,η2)≤e−ǫ(η1|η2)p(5.2)
+for allη1,η2∈∂X. Here (ξ|η)pdenotes the Gromov product of ξandηbased at p, and is
+defined by
+(ξ|η)p=1
+2sup liminf
+i,j→∞(d(p,xi)+d(p,yj)−d(xi,yj))
+where the supremum is taken over all sequences {xi} →ξ,{yi} →η. By the δ-hyperbolicity
+ofX,
+(ξ|η)p−2δ≤liminf
+i,j→∞(xi|yj)p≤(ξ|η)p (5.3)
+for allp∈X, allξ,η∈∂Xand all sequences {xi} →ξ,{yi} →η; we refer the interested
+reader to Chapter 7 of [GdlH].
+To formulate the relation between visual metric and parabol ic visual metric, we need to
+recall the notion of metric inversion and sphericalization . The reader is referred to [BHX]
+for more details.
+Given a metric space ( X,d) andp∈X, there is a metric dponX\{p}satisfying
+d(x,y)
+4d(x,p)d(y,p)≤dp(x,y)≤d(x,y)
+d(x,p)d(y,p)
+for allx,y∈X\{p}. Furthermore, the identity map ( X\{p},d)→(X\{p},dp) isη-
+quasim¨ obius with η(t) = 16t. We call dpthemetric inversion of (X,d) atp.
+LetXbe an unbounded metric space and p∈X. LetSp(X) =X∪{∞}, where∞is a
+point not in X. We define a function sp:Sp(X)×Sp(X)→[0,∞) as follows:
+sp(x,y) =sp(y,x) =
+
+d(x,y)
+[1+d(x,p)][1+d(y,p)]ifx,y∈X,
+1
+1+d(x,p)ifx∈Xandy=∞,
+0 if x=∞=y.
+It was shown in [BK] that there is a metric /hatwidedponSp(X) satisfying
+1
+4sp(x,y)≤/hatwidedp(x,y)≤sp(x,y) for all x,y∈Sp(X). (5.4)
+Furthermore, the identity map ( X,d)→(X,/hatwidedp) isη-quasim¨ obius with η(t) = 16t. We call
+/hatwidedpthesphericalization of (X,d) atp.
+If (Y,d) is a bounded metric space, and if a metric inversion is appli ed toY, fol-
+lowed by an application of sphericalization, the resulting space is bilipschitz equivalent
+to (Y,d). To be more precise, let p/\e}atio\slash=q∈Y; assume pis non-isolated in Yand let
+f: (Y,d)→(Sq(Y\{p}),/hatwidest(dp)q) be the map that is identity on Y\{p}withf(p) =∞. Then
+fis bilipschitz (see for example [BHX, Proposition 3.9]).
+We need the following result for the proof of Proposition 5.4 .
+14Theorem 5.3 ([CDP, Chapter 8]) .Let(Y,h)be aδ-hyperbolic space, y0∈Y, andY0=
+{y0,y1,···,yn}be a set of n+1points in Y∪∂Y. For each 1≤i≤n, let[y0,yi]be a fixed
+geodesic connecting y0andyi. LetXdenote the union of the geodesics [y0,yi], and choose a
+positive integer ksuch that 2n≤2k+1. Then there exists a simplicial tree, denoted T(X),
+and a continuous map u:X→T(X)which satisfies the following properties:
+(i) For each i, the restriction of uto the geodesic [y0,yi]is an isometry;
+(ii) For every xandyinXwe have h(x,y)−2kδ≤d(u(x),u(y))≤h(x,y), wheredis
+the metric on T(X).
+Proposition 5.4. LetXbe aδ-hyperbolic proper geodesic metric space, ξ∈∂X,p∈X
+and0< ǫ≤min{ǫ0,ǫ1}.
+(1) The identity map
+id: (∂X\{ξ},dξ,p,ǫ)→(∂X\{ξ},(dp,ǫ)ξ)
+isL-bilipschitz, where Lis a constant depending only on δ. In particular, the parabolic
+visual metric and the metric inversion of the visual metric a bout the point ξare bilipschitz
+equivalent;
+(2) Assume ξis non-isolated in ∂X. Letη∈∂X\{ξ}and
+f: (∂X,dp,ǫ)→(Sη(∂X\{ξ}),/hatwider(dξ,p,ǫ)η)
+be the bijection that is identity on ∂X\{ξ}and maps ξto∞. Thenfis bilipschitz. In
+particular, the visual metric and the sphericalization of t he parabolic visual metric are bilip-
+schitz equivalent.
+Proof.LetD=dp,ǫdenote the visual metric. Then ( dp,ǫ)ξ=Dξ.
+We first prove (1). Let η1,η2∈∂X\{ξ}. By Proposition 5.1 and inequality (5.2),
+Dξ(η1,η2)
+dξ,p,ǫ(η1,η2)≤dp,ǫ(η1,η2)
+dp,ǫ(ξ,η1)dp,ǫ(ξ,η2)2eǫHξ,p(η1,η2)
+≤e−ǫ(η1|η2)p2eǫ(ξ|η1)p2eǫ(ξ|η2)p2eǫHξ,p(η1,η2)
+= 8eǫ{Hξ,p(η1,η2)+(ξ|η1)p+(ξ|η2)p−(η1|η2)p}.
+Similarly,
+Dξ(η1,η2)
+dξ,p,ǫ(η1,η2)≥1
+8eǫ{Hξ,p(η1,η2)+(ξ|η1)p+(ξ|η2)p−(η1|η2)p}.
+Now (1) follows from the following claim.
+Claim: There is a constant Cdepending only on δsuch that if η1,η2,ξ∈∂Xare pairwise
+distinct, then |Hξ,p(η1,η2)+(ξ|η1)p+(ξ|η2)p−(η1|η2)p| ≤C.
+We now prove the claim. Let γbe a ray from ptoξ. Pick a point y0∈γthat is far away
+from any complete geodesic joining η1andη2. Letγi(i= 1,2) be a ray from y0toηi. Set
+X=γ∪γ1∪γ2. By Theorem 5.3 (with the choice k= 3) there is a tree T:=T(X) and a
+mapu:X→Twith the properties stated in Theorem 5.3. Let y′
+0,p′∈Tandξ′,η′
+1,η′
+2∈∂T
+15bethe points correspondingto y0,p,ξ η1andη2respectively. Also let x′bethe branchpoint
+ofξ′η′
+1andξ′η′
+2, and let y′be the projection of p′onto the tripod Y:=x′ξ′∪x′η′
+1∪x′η′
+2. Let
+y∈γbe the point on γthat is mapped to y′byu(by choosing y0far away from pwe may
+assume that ylies between pandy0). Similarly let xi∈γibe the point mapped to x′byu.
+Letσbe a complete geodesic from η1toη2. Because Xisδ-hyperbolic, geodesic triangles
+inX∪∂Xare 24δ-thin. Also notice that the union x′η′
+1∪x′η′
+2is a complete geodesic in T.
+Now the properties of the map ugiven by Theorem 5.3 imply that the Hausdorff distance
+between σandx1η1∪x2η2is bounded above by a constant c1=c1(δ).
+Choosezj∈γ1andwj∈γ2withzj→η1andwj→η2. Then the property of the map
+uand inequality (5.3) imply that |(η1|η2)p−(η′
+1|η′
+2)p′| ≤11δ. Notice that on the tree Twe
+have (η′
+1|η′
+2)p′=d(p′,η′
+1η′
+2). Hence |(η1|η2)p−d(p′,η′
+1η′
+2)| ≤11δ. Similar inequalities hold
+for (ξ|η1)pand (ξ|η2)p.
+Since the Hausdorff distance between σandx1η1∪x2η2is at most c1, the definition
+ofHξ,p(σ) and the property of the map uimply that |Hξ,p(σ)−Hξ′,p′(η′
+1,η′
+2)| ≤c1+13δ.
+The discussion about Hξ,p(η1,η2) shows that |Hξ,p(η1,η2)−Hξ,p(σ)| ≤2δ. It follows that
+|Hξ,p(η1,η2)−Hξ′,p′(η′
+1,η′
+2)| ≤c1+ 15δ. Now on the tree T, by considering three cases
+depending on whether y′∈x′ξ′,y′∈x′η′
+1ory′∈x′η′
+2, we can verify that
+Hξ′,p′(η′
+1,η′
+2)+d(p′,ξ′η′
+1)+d(p′,ξ′η′
+2)−d(p′,η′
+1η′
+2) = 0.
+Now the claim follows by combining the above estimates.
+We now prove (2). By (1), the identity map
+id: (∂X\{ξ},dξ,p,ǫ)→(∂X\{ξ},Dξ)
+is bilipschitz. Pick η∈∂X\{ξ}. Then the map idextends to a map Fbetween their
+sphericalizations
+F: (Sη(∂X\{ξ}),/hatwider(dξ,p,ǫ)η)→(Sη(∂X\{ξ}),/hatwide(Dξ)η).
+Sinceidis bilipschitz, inequality (5.4) can be used on /hatwider(dξ,p,ǫ)ηand/hatwide(Dξ)ηto verify that F
+is also bilipschitz. On the other hand, the natural identific ation between ( ∂X,dp,ǫ) and
+(Sη(∂X\{ξ}),/hatwide(Dξ)η) is bilipschitz. The statement now follows.
+We next give a sufficient condition for the parabolic visual me tric to be doubling. Recall
+that a metric space is doubling if there is a constant Nsuch that every open ball with radius
+R >0 can be covered by at most Nopen balls with radius R/2. By a theorem of Assouad
+([A]), a metric space is doubling if and only if the metric spa ce admits a quasisymmetric
+embedding into some Euclidean space.
+A metric space X has bounded growth at some scale , if there are constants r,Rwith
+R > r > 0, and an integer N≥1 such that every open ball of radius RinXcan be covered
+byNopen balls of radius r.
+The following is a consequence of Proposition 5.4, a result o f Bonk-Schramm and As-
+souad’s theorem.
+16Theorem 5.5. LetXbe aδ-hyperbolic geodesic metric space with bounded growth at so me
+scale. Then for any ξ∈∂X,p∈Xand any 0< ǫ≤ǫ0, the metric space (∂X\{ξ},dξ,p,ǫ)
+is doubling.
+Proof.Under the assumption of the Theorem, Bonk-Schramm has prove d that the ideal
+boundary with the visual metric is doubling ([BS, Theorem 9. 2]). Hence there is a qua-
+sisymmetric embedding f: (∂X,dp,ǫ)→Rnfor some n≥1. By Lemma 5.6 below,
+f: (∂X\{ξ},(dp,ǫ)ξ)→(Rn\{f(ξ)},| · |f(ξ)) is also a quasisymmetric embedding, where
+| · |denotes the Euclidean metric. However, the metric inversio n of the Euclidean space
+is still a Euclidean space (with one point removed). Hence ( ∂X\{ξ},(dp,ǫ)ξ) admits a
+quasisymmetric embedding into a Euclidean space, and so is d oubling. Since doubling is
+invariant under bilipschitz map, the theorem now follows fr om Proposition 5.4 (1).
+Recall that a homeomorphism f:X→Ybetween two metric spaces is η-quasim¨ obius
+for somehomeomorphism η: [0,∞)→[0,∞), if for every fourdistinct points x1,x2,x3,x4∈
+X, we have
+d(f(x1),f(x3))d(f(x2),f(x4))
+d(f(x1),f(x4))d(f(x2),f(x3))≤η/parenleftbiggd(x1,x3)d(x2,x4)
+d(x1,x4)d(x2,x3)/parenrightbigg
+.
+Lemma 5.6. Suppose that f: (X,d)→(Y,d)is a quasisymmetric embedding. Then for
+anyp∈X,f: (X\{p},dp)→(Y\{f(p)},df(p))is also a quasisymmetric embedding.
+Proof.Suppose fisanη-quasisymmetricembeddingforsome η. Thenfisan ˜η-quasim¨ obius
+embeddingfor some ˜ ηdependingonly on η, see [V2, Theorem 6.25]. Now let x,y,z∈X\{p}
+be three distinct points. Set q=f(p). We calculate
+dq(f(x),f(z))
+dq(f(y),f(z))≤d(f(x),f(z))
+d(f(x),f(p))d(f(z),f(p))·4d(f(y),f(p))d(f(z),f(p))
+d(f(y),f(z))
+= 4d(f(x),f(z))d(f(y),f(p))
+d(f(x),f(p))d(f(y),f(z)).
+Similarly,
+dp(x,z)
+dp(y,z)≥1
+4d(x,z)d(y,p)
+d(x,p)d(y,z).
+It follows that
+dq(f(x),f(z))
+dq(f(y),f(z))≤4 ˜η/parenleftbigg
+4dp(x,z)
+dp(y,z)/parenrightbigg
+.
+Hencef: (X\{p},dp)→(Y\{f(p)},df(p)) isη′-quasisymmetric with η′(t) = 4˜η(4t).
+Let (X,d) be an unbounded complete metric space with an Ahlfors Q-regular ( Q >1)
+Borel measure µ, andp∈X. On the sphericalization ( Sp(X),/hatwidedp) we define a measure µ′as
+follows:µ′({∞}) = 0, and on X=Sp(X)\{∞},µ′is absolutely continuous with respect to
+µwith Radon-Nikodym derivative
+dµ′
+dµ(x) =1
+(1+d(p,x))2Q
+forx∈X. It can be shown that ( Sp(X),/hatwidedp) withµ′is alsoQ-regular.
+17Proof of Theorem 1.1 .LetF: (∂GA,dp,ǫ)→(∂GA,dp,ǫ) be a quasisymmetric map,
+wheredp,ǫis a visual metric ( p∈GAandǫ >0 is sufficiently small). We first prove that
+F(ξ0) =ξ0. LetDbethe metric on ∂GA\{ξ0}=Rnconsidered in theprevious sections. We
+have observed that Dis bilipschitz equivalent with a parabolic visual metric on ∂GA\{ξ0}.
+Letθ∈∂GA\{ξ0}=Rn. Proposition 5.4 implies that the natural identification
+(Sθ(Rn),/hatwideDθ) = (Sθ(∂GA\{ξ0}),/hatwideDθ)→(∂GA,dp,ǫ)
+is bilipschitz. It follows that (after the above natural ide ntification)
+F: (Sθ(Rn),/hatwideDθ)→(Sθ(Rn),/hatwideDθ)
+isquasisymmetric. Let µbetheproductoftheHausdorffmeasuresonthefactors( Rni,|·|α1
+αi)
+ofRn. Since the metric measure space ( Rn,D,µ) isQ-regular with Q= Σr
+i=1niαi
+α1, the
+remark preceding the proof shows that the metric measure spa ce (Sθ(Rn),/hatwideDθ,µ′) is alsoQ-
+regular. Here µ′is obtained from µas described in the remark preceding the proof. Hence
+Theorem 3.1 applies to the map F: (Sθ(Rn),/hatwideDθ)→(Sθ(Rn),/hatwideDθ) and the measure µ′.
+Suppose F(ξ0)/\e}atio\slash=ξ0. Under the above natural identification, this means that F(∞)/\e}atio\slash=
+∞. ThenF−1(∞) lies in exactly one horizontal leaf. Fix some y∈Ysuch that Rn1×{y}
+does not contain F−1(∞). Notice that the subset ( Rn1× {y})∪{∞}ofSθ(Rn) is ann1-
+dimensional topological sphere. So F(Rn1× {y} ∪ {∞}) is ann1-dimensional topological
+sphere in Rn. Since each horizontal leaf is an n1-dimensional Euclidean space, the set
+F(Rn1×{y}∪{∞}) is not contained in any horizontal leaf. It follows that as a dense subset
+ofF(Rn1×{y}∪{∞}), the set F(Rn1×{y}) is also not contained in any horizontal leaf.
+Hence there are two points pandqinRn1× {y}such that F(p) andF(q) are not in the
+same horizontal leaf.
+Letγbethe Euclidean line segment from ptoqand Γ bethe family of straight segments
+parallel to γinRnwhose union is an n-dimensional circular cylinder Cwithγas the
+central axis. The curves in Γ are rectifiable with respect to t he metric D. SinceFis a
+homeomorphism, by choosing the radius of the circular cylin der to be sufficiently small (by
+a compactness argument) we may assume that no curve in Γ is map ped into a horizontal
+leaf and that F−1(∞) is not in this cylinder. It follows that F(Γ) has no locally rectifiable
+curve with respect to D. Now notice that both CandF(C) are compact subsets of Rn.
+Hence the two metrics Dand/hatwideDθare bilipschitz equivalent on C, as well as on F(C). It
+follows that F(Γ) has no locally rectifiable curve with respect to /hatwideDθ. Hence Mod QF(Γ) = 0
+in the metric measure space ( Sθ(Rn),/hatwideDθ,µ′). Theorem 3.1 then implies that Mod QΓ = 0
+in the metric measure space ( Sθ(Rn),/hatwideDθ,µ′). On the other hand, Mod QΓ>0 in the metric
+measure space ( Rn,D,µ) (see the proof of Theorem 3.2). Since Dand/hatwideDθare bilipschitz
+equivalent on C, andµandµ′are also comparable on C, we have Mod QΓ>0 in the metric
+measure space ( Sθ(Rn),/hatwideDθ,µ′), a contradiction. Hence F(ξ0) =ξ0.
+Next we prove that Fis bilipschitz with respect to the metric D. Since the map
+F: (∂GA,dp,ǫ)→(∂GA,dp,ǫ) is quasisymmetric, Lemma 5.6 implies that
+F: (∂GA\{ξ0},(dp,ǫ)ξ0)→(∂GA\{ξ0},(dp,ǫ)ξ0)
+18isalsoaquasisymmetricmap. ByProposition5.4, id: (∂GA\{ξ0},(dp,ǫ)ξ0)→(∂GA\{ξ0},dξ0,p,ǫ)
+is bilipschitz, where dξ0,p,ǫis a parabolic visual metric. It follows that
+F: (∂GA\{ξ0},dξ0,p,ǫ)→(∂GA\{ξ0},dξ0,p,ǫ)
+is quasisymmetric. By Proposition 5.2, any two parabolic vi sual metrics are quasisym-
+metrically equivalent. By the discussion following Propos ition 5.1, it follows that F:
+(∂GA\{ξ0},D)→(∂GA\{ξ0},D) is quasisymmetric. Now the result follows from Theo-
+rem 4.1.
+6 Consequences
+In this section we will prove the corollaries from the introd uction.
+We note that because GAhas sectional curvature −α2
+r≤K≤ −α2
+1,GAis a proper
+geodesic δ-hyperbolic space with δdepending only on α1.
+Proof of Corollary 1.3. .Suppose there is a quasiisometry f:GA→GfromGAto
+a finitely generated group G, whereGis equipped with a fixed word metric. Since GAis
+Gromov hyperbolic, it follows that Gis Gromov hyperbolic and finduces a quasisymmetric
+map∂f:∂GA→∂G. The left translation of Gon itself induces an action of Gon the
+Gromov boundary ∂Gby quasisymmetric maps. By conjugating this action with ∂fwe
+obtain an action of Gon∂GAby quasisymmetric maps. By Theorem 1.1, this action has
+a global fixed point. It follows that the action of Gon∂Ghas a global fixed point. This
+can happen only when Gis virtually infinite cyclic, in which case the Gromov bounda ry
+∂Gconsists of only two points. This contradicts the fact that ∂GAis a sphere of dimension
+n≥2 (sincer≥2).
+The proofs of Corollaries 1.4 and 1.2 require some preparati on.
+LetXbe a proper geodesic δ-hyperbolic space and ξ1,ξ2,ξ3∈∂Xbe three distinct
+points in the Gromov boundary. For any constant C≥0, a point x∈Xis called a C-
+quasicenter of the three points ξ1,ξ2,ξ3if for each i= 1,2,3, there is a geodesic σijoining
+ξiandξi+1(ξ4:=ξ1) such that the distance from xtoσiis at most C. For any C≥0,
+there is a constant C′that depends only on δandCsuch that the distance between any
+twoC-quasicenters of ξ1,ξ2,ξ3is at most C′.
+The following three lemmas hold in all Hadamard manifolds wi th pinched negative
+sectional curvature.
+Lemma 6.1. Let(x,t)∈GA=Rn×Rbe an arbitrary point, and σa geodesic through (x,t)
+and tangent to the horosphere Rn×{t}. Letp,q∈Rn≈∂GA\{ξ0}be the two endpoints of
+σ. Then(x,t)is a12δ-quasicenter for p,q,ξ0.
+Proof.As an ideal geodesic triangle in a δ-hyperbolic space, σ∪γp∪γqis 4δ-thin. Hence
+there is some point m∈γp∪γqwithd((x,t),m)≤4δ. We may assume m= (p,t′)∈γpfor
+somet′∈R. We may further assume that ( p,t′) is the point on γpnearest to ( x,t). Then
+the geodesic segment from ( x,t) to (p,t′) must be perpendicular to the geodesic γp. This
+19impliest′> t. Since(x,t) is thehighestpointon σandismorethan4 δbelowthehorosphere
+through ( p,t′+4δ), we have d((p,t′+4δ),σ)>4δ. Now the thin triangle condition applied
+to the point ( p,t′+4δ) and the triangle σ∪γp∪γqimplies there is some ( q,t′′)∈γqwith
+d((p,t′+4δ),(q,t′′))≤4δ. The triangle inequality together with d((p,t′),(p,t′+4δ)) = 4δ
+impliesd((x,t),(q,t′′))≤12δ. Hence ( x,t) is a 12δ-quasicenter for p,q,ξ0.
+LetMbe a simply connected Riemannian manifold with sectional cu rvature−b2≤
+K≤ −a2, whereb > a > 0. For any ξ∈∂M, any horosphere Hcentered at ξ, and any
+two points x,y∈ H, the distance dH(x,y) between xandyin the horosphere is related to
+d(x,y) by (see [HI]):
+2
+asinh/parenleftiga
+2d(x,y)/parenrightig
+≤dH(x,y)≤2
+bsinh/parenleftbiggb
+2d(x,y)/parenrightbigg
+. (6.1)
+For anys >0, letHsbe the horosphere centered at ξthat is closer to ξthanHand is at
+distance sfromH. Letφs:H → H sbe the map which sends each x∈ Hto the unique
+intersection point of xξwithHs. Then for each tangent vector v∈TxHofHatxwe have
+(see [HI]): e−bs/bardblv/bardbl ≤ /bardbldφs(v)/bardbl ≤e−as/bardblv/bardbl.It follows that for any rectifiable curve cinH,
+the lengths of candφ(c) are related by e−bsℓ(c)≤ℓ(φ(c))≤e−asℓ(c).
+Lemma 6.2. Letp,q∈Rn≈∂GA\{ξ0}and suppose that De(p,q) =et0. Then(p,t0)is a
+C-quasicenter for p,q,ξ0, whereCdepends only on α1andαr.
+Proof.Letσbe the geodesic in GAjoiningp,q∈∂GA\{ξ0}, and (x,t) the highest point on
+σ. We may assume d((x,t),γp)≤4δ. Let (p,t1)∈γpbe the point nearest to ( x,t). Then
+t1> tand the argument in the proof of Lemma 6.1 gives a point ( q,t2)∈γqsuch that
+d((p,t1),(q,t2))≤8δ. It follows that |t1−t2| ≤8δ. The triangle inequality then implies
+d((p,t1),(q,t1))≤16δ.
+By the definition of De, we have dRn×{t0}(p,q) = 1. Hence d((p,t0),(q,t0))≤1. If
+t0≤t1, then the convexity of the distance function f(t) :=d(γp(t),σ) =d((p,t),σ) implies
+d((p,t0),σ)≤d((p,t1),σ)≤4δ. In this case, ( p,t0) is a max {1,4δ}-quasicenter of p,q,ξ0.
+Now we suppose t0> t1. Join (p,t1) and (q,t1) by a shortest path cin the horosphere
+H:=Rn×{t1}. By (6.1) we have ℓ(c)≤2
+αrsinh(8αrδ). The projection φt0−t1(c) is a path
+in the horosphere Rn×{t0}joining (p,t0) and (q,t0). Hence
+1 =dRn×{t0}(p,q)≤ℓ(φt0−t1(c))≤e−(t0−t1)α1ℓ(c)≤2
+αre−(t0−t1)α1sinh(8αrδ).
+It follows that t0−t1≤C1, where
+C1=ln[2
+αrsinh(8αrδ)]
+α1.
+The triangle inequality then implies d((p,t0),σ)≤C1+4δ. Hence ( p,t0) is aC-quasicenter
+forp,q,ξ0, whereC= max{1,C1+4δ}.
+20Lemma 6.3. Letp,q∈Rn≈∂GA\{ξ0}and suppose that De(p,q) =et0.
+(1) Ift1,t2< t0, then|d((p,t1),(q,t2))−(t0−t1)−(t0−t2)| ≤C, whereCdepends only
+onα1andαr;
+(2) Ift1≥t0ort2≥t0, then|t1−t2| ≤d((p,t1),(q,t2))≤ |t1−t2|+1.
+Proof.(1) Letσbe the geodesic in GAjoiningpandq, and (x,t) the highest point on σ. By
+Lemmas 6.1 and 6.2, the three points ( p,t0), (q,t0) and (x,t) are allc1-quasicenters of ξ0,p,
+q, wherec1depends only on α1andαr. Henced((p,t0),(x,t))≤c2andd((q,t0),(x,t))≤c2
+for some c2=c2(c1,δ) =c2(α1,αr). Sincet1< t0, the convexity of distance function implies
+thatd((p,t1),m1)≤d((p,t0),(x,t))≤c2for some point m1∈σlying between ( x,t) and
+p. Similarly, there is some point m2∈σbetween ( x,t) andqwithd((q,t2),m2)≤c2. By
+triangle inequality we have |d((p,t1),(q,t2))−d(m1,m2)| ≤2c2. Sinced((p,t0),(x,t))≤c2
+andd((p,t1),m1)≤c2, thetriangle inequality alsoimplies |d((p,t0),(p,t1))−d(m1,(x,t))| ≤
+2c2. Similarly, |d((q,t0),(q,t2))−d(m2,(x,t))| ≤2c2. Sinced(m1,m2) =d(m1,(x,t)) +
+d((x,t),m2) andd((p,t0),(p,t1)) =t0−t1,d((q,t0),(q,t2)) =t0−t2, the above estimates
+together yield |d((p,t1),(q,t2))−(t0−t1)−(t0−t2)| ≤6c2.
+(2) We may assume t1≥t0. Then the convexity of distance function and the definition
+ofDeimply
+d((p,t1),(q,t1))≤d((p,t0),(q,t0))≤dRn×{t0}((p,t0),(q,t0)) = 1.
+Now (2) follows from the triangle inequality and (2.1).
+Corollary 1.4 follows from Theorem 1.1 and the following lem ma. Notice that, by
+Theorem 1.1, for any quasiisometry f:GA→GA, the boundary map ∂ffixesξ0and
+restricts to a homeomorphism of ∂GA\{ξ0}, which we still denote by ∂f.
+Lemma 6.4. Letf:GA→GAbe a quasiisometry. Then fis height-respecting if and only
+if∂f: (∂GA\{ξ0},D)→(∂GA\{ξ0},D)is a bilipschitz map.
+Proof.Dymarz ([D, Lemma 7]) proved that the boundary map of a height -respecting quasi-
+isometry is a bilipschitz map with respect to the quasimetri cDs. It follows that the bound-
+ary map is also bilipschitz with respect to the metric D. Hence we only prove the “if” part.
+So we assume ∂fis bilipschitz w.r.t. D. Notice that it is also bilipschitz w.r.t. De. Hence
+there is a constant L≥1 such that for all p,q∈∂GA\{ξ0}=Rn,
+De(p,q)/L≤De(∂f(p),∂f(q))≤LDe(p,q).
+Let (x,t)∈GA=Rn×R. Pick any geodesic σthrough ( x,t) that is tangent to the
+horosphere Rn× {t}. Then the two endpoints p,qofσare in∂GA\{ξ0}=Rn. Ift0is
+the real number such that dRn×{t0}((p,t0),(q,t0)) = 1, then by the definition of Dewe
+haveDe(p,q) =et0. By Lemmas 6.1 and 6.2 both ( x,t) and (p,t0) arec1-quasicenters of
+the three points p,q,ξ0∈∂GA, wherec1depends only on α1andαr. Hence there is a
+constant c2depending only on c1andδsuch that d((x,t),(p,t0))≤c2. By (2.1), we have
+|t−t0| ≤c2. Lett′
+0be the real number such that dRn×{t′
+0}((∂f(p),t′
+0),(∂f(q),t′
+0)) = 1.
+ThenDe(∂f(p),∂f(q)) =et′
+0. Since fis a quasiisometry between δ-hyperbolic spaces,
+21f(x,t) is ac3-quasicenter of ∂f(p),∂f(q),∂f(ξ0) =ξ0, wherec3depends only on δ,c1and
+the quasiisometry constants of f. As (∂f(p),t′
+0) is ac1-quasicenter of these three points,
+we have d((∂f(p),t′
+0),f(x,t))≤c4, withc4depending only on c1,c3andδ. Lett′be the
+height of f(x,t). Then by (2.1) again, |t′−t′
+0| ≤c4.
+Thebilipschitzassumptionof ∂fandtheformulas De(∂f(p),∂f(q)) =et′
+0andDe(p,q) =
+et0imply that |t0−t′
+0| ≤lnL. Combining this with |t−t0| ≤c2and|t′−t′
+0| ≤c4, we obtain
+|t−t′| ≤lnL+c2+c4. Hence the heights of any point ( x,t) and its image f(x,t) differ by
+at most a constant that is independent of ( x,t). The corollary follows.
+Proof of Corollary 1.2 .Letf:GA→GAbe an (L,A)-quasiisometry. By Theorem 1.1,
+the boundary map ∂f:∂GA→∂GAfixes the point ξ0. Let (x1,t1),(x2,t2)∈Rn×R=GA.
+Suppose De(x1,x2) =et0. We only consider the case t0> t1,t2, the other cases being
+similar. By Lemma 6.3 there is a constant c1=c1(α1,αr) such that
+|d((x1,t1),(x2,t2))−(t0−t1)−(t0−t2)| ≤c1. (6.2)
+Lett′
+i(i= 1,2) be the height of f(xi,ti). By Corollary 1.4, there is a constant c2≥0
+such that |ti−t′
+i| ≤c2. Sincef(γxi) is an (L,A)-quasigeodesic joining ξ0and∂f(xi), there is
+a constant c3dependingonly on L,Aandδsuch that the Hausdorff distance between f(γxi)
+andγ∂f(xi)is at most c3. Hence there is some t′′
+isuch that d((∂f(xi),t′′
+i),f(xi,ti))≤c3. It
+follows that |t′
+i−t′′
+i| ≤c3and hence |ti−t′′
+i| ≤c2+c3.
+Suppose De(∂f(x1),∂f(x2)) =et′
+0. By Lemma 6.2 ( ∂f(x1),t′
+0) is ac4-quasicenter for
+ξ0,∂f(x1),∂f(x2), where c4=c4(α1,αr). Similarly, ( x1,t0) is ac4-quasicenter for ξ0,x1,
+x2. On the other hand, since fis an (L,A) quasiisometry, f(x1,t0) is ac5-quasicenter of ξ0,
+∂f(x1) and∂f(x2), where c5=c5(L,A,c4,δ). It follows that d((∂f(x1),t′
+0),f(x1,t0))≤c6
+for some constant c6=c6(c4,c5,δ). Lett′′
+0be the height of f(x1,t0). Then |t′
+0−t′′
+0| ≤c6.
+By Corollary 1.4 we have |t0−t′′
+0| ≤c2. Hence|t0−t′
+0| ≤c6+c2.
+Next we consider two cases:
+Case 1. Botht′′
+1,t′′
+2< t′
+0.
+In this case, by Lemma 6.3 (1) again we have
+|d((∂f(x1),t′′
+1),(∂f(x2),t′′
+2))−(t′
+0−t′′
+1)−(t′
+0−t′′
+2)| ≤c1.
+Combining this with (6.2) and the estimates |ti−t′′
+i| ≤c2+c3,|t0−t′
+0| ≤c6+c2, and
+d((∂f(xi),t′′
+i),f(xi,ti))≤c3, we obtain
+|d((x1,t1),(x2,t2))−d(f(x1,t1),f(x2,t2))| ≤2c1+4c2+4c3+2c6.
+Case 2. Either t′′
+1≥t′
+0ort′′
+2≥t′
+0.
+Without loss of generality, we may assume t′′
+1≥t′
+0andt′′
+1≥t′′
+2. Then Lemma 6.3 (2) implies
+|d((∂f(x1),t′′
+1),(∂f(x2),t′′
+2))−(t′′
+1−t′′
+2)| ≤1. (6.3)
+On the other hand, t′′
+1≥t′
+0and the assumption t0> t1together with |t0−t′
+0| ≤c6+c2and
+|ti−t′′
+i| ≤c2+c3imply that |t0−t1| ≤2c2+c3+c6. Now it follows from (6.2) and the
+triangle inequality that
+|d((x1,t1),(x2,t2))−(t1−t2)| ≤c1+4c2+2c3+2c6. (6.4)
+22Now (6.3), (6.4), |ti−t′′
+i| ≤c2+c3, andd((∂f(xi),t′′
+i),f(xi,ti))≤c3imply
+|d((x1,t1),(x2,t2))−d(f(x1,t1),f(x2,t2))| ≤1+c1+6c2+6c3+2c6.
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+24
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+arXiv:1001.0151v1  [physics.flu-dyn]  31 Dec 2009Physical aspects of the field-theoretical
+description
+of two-dimensional ideal fluids
+Florin Spineanu and Madalina Vlad
+National Institute of Laser, Plasma and Radiation Physics
+Magurele, 077125 Bucharest, Romania
+November 2, 2018
+Abstract
+The two-dimensional ideal (Euler) fluids can be described by the
+classical fields of streamfunction, velocity and vorticity and, in an
+equivalent manner, by a model of discrete point-like vortic es inter-
+acting in plane by a self-generated long-range potential. T his latter
+model can be formalized, in the continuum limit, as a field the ory
+of scalar matter in interaction with a gauge field, in the su(2) alge-
+bra. This description has already offered the analytical deri vation of
+thesinh-Poisson equation, which was known to govern the stationary
+coherent structures reached by the Euler fluid at relaxation . In or-
+der this formalism to become a familiar theoretical instrum ent it is
+necessary to have a better understanding of the physical mea ning of
+the variables and of the operations used by the field theory. S everal
+problems will be investigated below in this respect.
+1 Introduction
+Thetwo-dimensional incompressible idealfluidisgovernedbytheEule requa-
+tiondω
+dt= 0 (1)
+whereωis the vorticity ω=∇×v, a vector perpendicular on the plane
+where the flow with the velocity vis contained. It is useful to define the
+streamfunction ψfrom which the velocity field is derived v=/hatwideez×∇ψand
+1the vorticity is ω=/hatwideez∆ψ. Most of the results in the theory of the ideal 2 D
+Euler fluid have been obtained using these three quantities, which ha ve clear
+physical meaning and are measurable experimentally.
+On the other hand the striking result that the asymptotic stationa ry
+states obtained at relaxation exhibit a strong coherency of the flo w cannot
+be easily described in the framework defined by ( ψ,v,ω). The stationary
+flows from Eq.(1) obey the equation [( −∇ψ×/hatwideez)·∇]∇2ψ= 0 which has
+a very large space of possible solutions. Clearly the selection of the fi nal
+states is dictated by an additional constraint that may have the fo rm of a
+functional extremal condition. Finding a functional defined on the space of
+flow configuration, as for example a density of a Lagrangian and an a ction
+functional, has not been possible in the traditional approach.
+A model (Hamilton, Kirchhoff, Onsager) which is equivalent with the 2 D
+Euler fluid consists of a set of discrete point-like vortices interactin g in plane
+by a potential which is created by themselves and has long range, i.e.it is
+Coulombian
+dxi
+dt=−∂
+∂yN/summationdisplay
+j/negationslash=iω0ln/parenleftbigg|x−xj|
+L/parenrightbigg
+(2)
+dyi
+dt=∂
+∂xN/summationdisplay
+j/negationslash=iω0ln/parenleftbigg|x−xj|
+L/parenrightbigg
+The following observation will have an important consequence later in the
+theory. In any further development it is not sufficient to only work w ith the
+system (2). The system (2) as it is still needs to specify which are th eobjects
+whose positions ( xi,yi) evolve in plane.
+The elementary objects can be charges. For large Nthe system can be
+treated as a statistical ensemble. In particular it has been shown t hat it
+has negative temperature (S.F. Edwards and J. B. Taylor) which su ggests an
+intrinsic tendency to self-organization into large structures.
+On the other hand the discrete elementary objects can be vorticesas ac-
+tually is requested by the equivalence with the Euler fluid, where the n atural
+logarithm appears as the propagator inverting the equation relatin g the vor-
+ticity and the streamfunction, ∆ ψ=ω(Kraichnan, Montgomery). Again in
+a statistical approach, the stationary states corresponding to the maximum
+entropy under the conservation of the mass, momentum and ener gy have
+been found to be described by the differential equation sinh-Poisson
+∆ψ+sinhψ= 0 (3)
+whereψis the streamfunction.
+2The problem is how to differentiate between the system consisting of
+Eqs.(2)plusthe information that the objects are charges and the system
+consisting of Eqs.(2) plusthe information that the objects are vortices. S.F.
+Edwards and J. B. Taylor observe that the two-dimensional plasma (i.e.
+charges) and the vortex fluid ( i.e.vortices) are described by the same sys-
+tem of equations. The system of chargesand the system of vortices are
+formally identical if we replace the charge by the circulation. Howeve r for
+statistical considerations it is not sufficient. The field theory clearly shows
+this, obtaining the distinct results that the system of charges is de scribed
+by the Liouville equation and the system of vortices by the sinh-Poisson
+equation. The field theory describes the charges by an Abelian mode l and a
+vortex fluid by a non-Abelian one.
+Thesinh-Poisson equation has been confirmed by numerical simulations
+of relaxation of Euler fluid from states of turbulence to the cohere nt and
+cuasi-stationary asymptotic states (Montgomery).
+In consequence of the discussion above we note that the derivatio n of
+thesinh-Poisson equation from purely statistical consideration still requir es
+some care. In addition the statistical approach could not be exten ded to
+other type of fluids like the 2 Datmosphere or the plasma in strong magnetic
+field. However the fact that the streamfunction ψverifies the sinh-Poisson
+equation has been confirmed after careful numerical verification . Any new
+theory oftheEuler fluidwill haveto beconfronted tothischallange, to derive
+thesinh-Poisson equation.
+The field theoretical (FT) model for the Euler fluid (Spineanu and Vla d,
+2003) is able to provide a purely analytical derivation of the sinh-Poisson
+equation. Moreover it can be extended to the more complicated pro blem of
+the 2Dplasma in strong transversal magnetic field and 2 Dplanetary atmo-
+sphere. TheFTmodelsforfluid, plasma, atmospherehaveclearlysh ownthat
+they are able to derive new results which are inaccessible to the trad itional
+approaches based on ( ψ,v,ω). However these FT models would be more
+easily adopted as an instrument of theoretical investigation if ther e would
+be a physical understanding of the meaning of the field theoretical concepts,
+operations, etc. in terms of the more familiar ( ψ,v,ω).
+32 The field theoretical representation of the
+elementary vortex as a global string
+The definition of the elementary object in the discrete model of poin t-like
+vortices consists of two characteristics:
+1. the elementary object in plane is strictly point-like, there is no spa tial
+extension.We can represent it however as a line extending in the z
+direction, i.e.perpendicular on the plane
+2. the elementary object carry a vorticity “content” which, altho ugh it is
+not a scalar charge, it is not introduced as the result of a fluid rotat ing
+around the vortex
+For the reason resulting from the second characteristic, the con venient
+representation of the elementary vortex in field theory is the globa l string.
+This is a time-independent solution of the equation of motion of a spon -
+taneously broken global U(1) Higgs model (Davies and Shellard). By this
+representation we dispose of a field theory for the elementary vor tex as a
+Nambu-Goldstone boson α(x) with the complex scalar field
+/hatwideφ=ηexp(iα) (4)
+whereηis the vacuum value of /hatwideφ. Later this model will prove to be essen-
+tial in the investigation of the fermionic zero-modes propagating alo ng the
+string and strictly confined to it. The axial anomaly related to these modes
+interacting with a gauge field will allow us to get an understanding of th e
+vorticity concentration, a process of high importance in the tropic al cyclone
+and tornadoes.
+For the global string there is no Magnus force acting on the string w hen
+it is moving with velocity vin the fluid. This is not a problem since the lows
+of motion of the point-like objects is given by the purely kinematic equ ations
+of motion and the motion resulting from these equation is not regard ed as a
+physical interaction with a fluid environment.
+However we can not be satisfied with the definition of the elementary
+vortex as a global string plusthe statement that it carries a fixed vorticity.
+As shown by Davies and Shellard the global string gets a fixed vorticit y when
+it is assumed that there is interaction between the string and a back ground
+field which breaks the Lorentz invariance and is equivalent with inducin g
+rotation of the global string around its axis, or a linear time variation of the
+bosonic field α=θ+const×twithθthe azimuthal angle. Taking the energy
+4density of the pure background field pthe fixed vorticity carried by a global
+string in such background is
+ω0= (4πη/√p)/hatwideez (5)
+This is a mechanism by which we can attach a fixed vorticity to the globa l
+string but it is not less arbitrary than defining the elementary objec t as
+carrying a fixed vorticity ω0.
+3 The physical meaning of the concepts in-
+volved in the field theoretical model of the
+Euler fluid
+3.1 Review of field theoretical model for the Euler
+fluid
+Jackiw and Pi have developed a field-theoretical for the continuum limit of a
+system of charges in plane. The equations of motion can be derived f rom the
+density of a Lagrangian and the theory leads to a differential equat ion de-
+scribing the asymptotic stationary states of the system. This is th e Liouville
+equation.
+We have found that the continuum limit of the discrete model of vor-
+ticesis given by a field theory with the Lagrangian that is characterized
+by: non-relativistic (Schrodinger), Chern-Simons, 4thorder scalar field self-
+interaction. The Lagrangian is
+L=−κεµνρtr/parenleftbigg
+(∂µAν)Aρ+2
+3AµAνAρ/parenrightbigg
+(6)
++itr/parenleftbig
+φ†(D0φ)/parenrightbig
+−1
+2tr/parenleftBig
+(Dkφ)†/parenleftbig
+Dkφ/parenrightbig/parenrightBig
++1
+4κtr/parenleftBig/bracketleftbig
+φ†,φ/bracketrightbig2/parenrightBig
+whereφ,φ†,Aµ,A†
+µareSU(2) elements. The covariant derivatives are
+Dµ=∂µ+[Aµ,] (7)
+with the metric g00=−1,g11=g22= 1. The equations of motion are
+iD0φ=−1
+2mDkDkφ−1
+2mκ/bracketleftbig/bracketleftbig
+φ,φ†/bracketrightbig
+,φ/bracketrightbig
+(8)
+−κεµνρFνρ=iJµ
+5where
+J0=/bracketleftbig
+φ†,φ/bracketrightbig
+(9)
+Jk=−i
+2m/parenleftBig/bracketleftbig
+φ†,/parenleftbig
+Dkφ/parenrightbig/bracketrightbig
+−/bracketleftBig/parenleftbig
+Dkφ/parenrightbig†,φ/bracketrightBig/parenrightBig
+which is covariantly conserved DµJµ= 0. The action functional calculated
+with this Lagrangian has the fundamental property that can be wr itten in
+the Bogomolnyi form, i.e.as a sum of squares, which clearly identifies the
+absolute extrema in the space of states by simply taking these squa re terms
+to zero. The energy is
+E=1
+2tr/parenleftBig
+(Dkφ)†/parenleftbig
+Dkφ/parenrightbig/parenrightBig
+−1
+2κtr/parenleftBig/bracketleftbig
+φ†,φ/bracketrightbig2/parenrightBig
+(10)
+The Gauss law is the zero component of the second equation of motio n
+−2κF12=iJ0(11)
+Introducing the notations D±=Dx±iDyand similarly for other variables,
+the first term in Eq.(10) is
+tr/parenleftBig
+(Dkφ)†/parenleftbig
+Dkφ/parenrightbig/parenrightBig
+= tr/parenleftBig
+(D−φ)†(D−φ)/parenrightBig
+−itr/parenleftbig
+φ†[F12,φ]/parenrightbig
+(12)
+Replacing in the expression of the energy, the last term, coming fro m the
+potential energy or the scalar field self-interaction is canceled and we obtain
+E=1
+2tr/parenleftBig
+(D−φ)†(D−φ)/parenrightBig
+(13)
+which leads to the equation for the states which correspond to the lowest
+energy
+D−φ= 0 (14)
+Then the equations of motion are replaced by
+D−φ= 0 (15)
+∂+A−−∂−A++[A+,A−] =1
+κ/bracketleftbig
+φ,φ†/bracketrightbig
+The solutions are stationary flow configurations obeying the set (1 5) of two
+first order partial differential equations from which, under a reas onable alge-
+braicansatzone can derive the sinh-Poisson equation, a unique differential
+6equation for the streamfunction. The algebraic ansatz uses the g enerator of
+the Cartan sub-algebra and the two ladder generators
+φ=φ1E++φ2E− (16)
+φ†=φ∗
+1E−+φ∗
+2E+
+A+=Ax+iAy=aH (17)
+A−=Ax−iAy=−a∗H
+from which the first SD equation leads to
+a=∂
+∂z∗ln(φ∗
+1) (18)
+a∗=∂
+∂zln(φ1) (19)
+and
+tr/parenleftbig
+φφ†/parenrightbig
+=ρ1+ρ2 (20)/bracketleftbig
+φ,φ†/bracketrightbig
+= (ρ1−ρ2)H
+with the notation
+ρ1≡ |φ1|2(21)
+ρ2≡ |φ2|2
+Using the algebraic ansatz in Eq.(15) we obtain the sinh-Poisson equation.
+The particular property of the system allowing the Bogomolnyi form of
+the action is called Self-Duality. It is worth to mention that the field th eoret-
+ical formalism for the Euler fluid has been able to show that the asymp totic
+coherent structures of the Euler fluid belong to the same family as a ll the
+other structures known: the solitons, the instantons, all are so lutions of
+equations derived at Self-Duality. It would be desirable to have the p hysical
+meaning of this concept in terms of classical fluid variables, i.e.(ψ,v,ω).
+3.2 Connections derived from the fermionic nature of
+the elementary point-like vortex
+The elementary object of the discrete model is a point-like vortex. The
+magnitude is the same for all elementary vortices, say ω0. It is not admitted
+7to work with multiple vortices as single objects, for example consistin g of
+two elementary objects superposed into a unique vortex of magnit ude 2ω0.
+This can be expressed by saying that two elementary vortices cann ot be in
+the same point. There are only two possible orientations, given by ±ω0.
+We conclude that there are similarities between the elementary point -like
+vortex and a spin-1 /2 object. Further this means that if one looks for a field
+theoretical formulation of the model of point-like vortices the mos t natural
+way would involve fermionic fields.
+Onthe other hand, Jackiw and Pi have shown that a system of point elec-
+tric charges moving in plane according to the same Eqs.(2) can be des cribed
+by a classical Abelian field theory of a bosonic scalar and gauge fields w ith
+Chern-Simons term and with ϕ4scalar field self-interaction. This model has
+self-dual states that are described by the Liouville equation, ∆ ψ+exp(ψ) =
+0. Since however the Euler fluid is described by point-like vorticesinstead of
+electric charges , thefield theory must be re-formulatedto reflect the spinorial
+nature of the elementary objects. Since the spinors are the lowes t represen-
+tations of the Lorentz group whose complex covering is SL(2,C) we expect
+to represent the spinorial nature of the elementary vortices by t aking all
+bosonic variables of the Jackiw-Pi model as elements of the sl(2,C) alge-
+bra. This is the model of Eq.(6) from which the sinh-Poisson equation has
+been derived. This places the non-Abelian, bosonic scalar field model, with
+Chern-Simons term for the gauge field and scalar potential nonlinea rity of
+order four Eq.(6) as the main framework where we should look for mo re
+conventional physical significance. However we also note the poss ibility that
+neighboring “fermionic” models can be useful. The connection should be
+realised via bosonization procedures, where it is possible.
+There are several two-dimensional models that present similarities with
+themodelofpoint-likevorticesandthathavereceivedconsiderab le attention,
+withvariouspurposes: theThirringmodel, theSchwinger model, the Nambu-
+Jona-Lasinio model, etc.
+We note that the scalar-field self-interaction of order four, which is in
+Non-Abelian form1
+2tr/parenleftBig/bracketleftbig
+φ†,φ/bracketrightbig2/parenrightBig
+(22)
+or, in Abelian case
+1
+2(φ∗φ)2(23)
+is the same as the fourth order term in the expansion
+cosφ−1≈ −1
+2(φ∗φ)+1
+24(φ∗φ)2(24)
+8The term ( φ∗φ) should be considered separately together with the mass
+term. If the fourth order scalar-field non-linearity ∼(φ∗φ)2comes from
+cos(φ)−1 then the nonlinearity is the same as in the sine-Gordon model.
+Then the fermionic model to which we should look is the Thirring model
+with the Lagrangian
+LTh=−ψ/parenleftBig
+i/∂µ/parenrightBig
+ψ (25)
+−1
+2/parenleftbig
+ψγµψ/parenrightbig/parenleftbig
+ψγµψ/parenrightbig
+which is characterized by a current-current ( JJ) interaction. Writting the
+nonlinearity as
+/parenleftbig
+ψγµψ/parenrightbig/parenleftbig
+ψγµψ/parenrightbig
+=1
+2/bracketleftBig/parenleftbig
+ψψ/parenrightbig2−/parenleftbig
+ψγ5ψ/parenrightbig2/bracketrightBig
+(26)
+We propose the following identification
+/parenleftbig
+ψψ/parenrightbig2= (ρ1+ρ2)2(27)
+/parenleftbig
+ψγ5ψ/parenrightbig2= (ρ1−ρ2)2
+such that
+1
+2/bracketleftBig/parenleftbig
+ψψ/parenrightbig2−/parenleftbig
+ψγ5ψ/parenrightbig2/bracketrightBig
+(28)
+=1
+2/bracketleftbig
+(ρ1+ρ2)2−(ρ1−ρ2)2/bracketrightbig
+= 2ρ1ρ2
+By this identification JJis connected with the product
+ρ1ρ2 (29)
+which plays a major role in the field theory for Euler. The extremum of the
+action for the Thirring model is
+JJ= const (30)
+and this means, after normalizations,
+ρ1ρ2= 1 (31)
+This equation plays a major role in the field-theoretical derivation of
+sinh-Poisson equation for Euler and demands an interpretation in physic al
+9terms. The identification proposed above can open the way to a phy sical
+interpretation.
+The two components are
+/parenleftbig
+ψψ/parenrightbig
+=ψ†γ0ψ=ψ†σ3ψthe density of spin (32)/parenleftbig
+ψγ5ψ/parenrightbig
+chiral current
+We recall ( Coleman, Faber and Ivanov ) that the two models: sine-
+Gordon (bosonic) and Thirring (fermionic) are equivalent if 4 π/β2= 1+
+g/πand the fermionic field ψ(x) respectively the bosonic field θ(x) satisfy
+the Abelian bosonization relation
+mψ(x)/parenleftbigg1∓γ5
+2/parenrightbigg
+ψ(x) =−α
+β2exp[±iβθ(x)] (33)
+(here the constants g,β,α,m, are constants). Witten uses the notations
+ψ(x)/parenleftbigg1±γ5
+2/parenrightbigg
+ψ(x)≡ O± (34)
+which are called chiral densities . According to the identification our ρ1,2
+ρ1,ρ2∼ O± (35)
+the coefficients of the ladder generators. Then we should compare
+ψ(x)/parenleftbigg1+γ5
+2/parenrightbigg
+ψ(x)⇐⇒φ∗
+1φ1(fromφ1E+) (36)
+ψ(x)/parenleftbigg1−γ5
+2/parenrightbigg
+ψ(x)⇐⇒φ∗
+2φ2(fromφ2E−)
+We have
+tr/parenleftbig
+φφ†/parenrightbig
+=ρ1+ρ2 (37)/bracketleftbig
+φ,φ†/bracketrightbig
+= (ρ1−ρ2)H
+ρ1−ρ2=/bracketleftbig
+φ†,φ/bracketrightbig
+/H= vorticity Euler ω (38)
+↓
+ψ(x)γ5ψ(axial current)
+ρ1+ρ2= tr/parenleftbig
+φφ†/parenrightbig
+(39)
+↓
+ψ(x)ψ(x) density of spin
+10In conclusion a possible identification is
+O+=ψ/parenleftbigg1+γ5
+2/parenrightbigg
+ψ=φ∗
+1φ1=ρ1= density of positive-chirality modes (40)
+O−=ψ/parenleftbigg1−γ5
+2/parenrightbigg
+ψ=φ∗
+2φ2=ρ2= density of negative-chirality modes
+ψψ=ρ1+ρ2= tr/parenleftbig
+φ†φ/parenrightbig
+density of spin (Boyanovsky) (41)
+ψγ5ψ=ρ1−ρ2=/bracketleftbig
+φ†,φ/bracketrightbig
+/H=ωaxial current
+From Eq.(33) we have
+ψψ−ψγ5ψ=−2α
+mβ2exp(iβθ) (42)
+ψψ+ψγ5ψ=−2α
+mβ2exp(−iβθ)
+ψψ=−2α
+mβ2cos(βθ) (43)
+ψγ5ψ=i2α
+mβ2sin(βθ)
+/parenleftbig
+ψψ/parenrightbig2−/parenleftbig
+ψγ5ψ/parenrightbig2=/parenleftbigg2α
+mβ2/parenrightbigg2/bracketleftbig
+(cos(βθ))2+(sin(βθ))2/bracketrightbig
+(44)
+=/parenleftbigg2α
+mβ2/parenrightbigg2
+which could correspond to the equation
+(ρ1+ρ2)2−(ρ1−ρ2)2=/parenleftbigg2α
+mβ2/parenrightbigg2
+(45)
+or
+ρ1ρ2=1
+4/parenleftbigg2α
+mβ2/parenrightbigg2
+(46)
+The fact that none of ρ1orρ2can be zero results from the fact that each
+represents a chiral density
+O+=ρ1 (47)
+O−=ρ2
+11The vanishing of any of the two densities ρ1orρ2would lead to the
+equation
+exp(iθ) = 0 (48)
+with no solution. The fact that none of these two functions can van ish is
+essential for the structure of the differential equation describin g the self-dual
+states: the sinh-Poisson equation contains both exp(+ ψ) and exp( −ψ). If
+we think to the derivation of this equation, reviewed above, it leaves the
+impression that the value of the vorticity in a particular point is always a
+result of two opposite actions, and none of them can be absent: cr eation of
+vorticity in a small spatial patch by the effect of densification of poin t-like
+vortices by the action of the ladder generator E+followed by decrease of
+the local vorticity by rarefaction of elementary vortices in the sam e patch,
+realised by the second generator E−. Actually we see that the FT operations
+mean the substraction of the two chiral densities O+−O−such that the spin
+density is eliminated leaving only the chiral current, i.e.vorticity.
+3.3 Review of the connections and comments on the
+interpretation
+The connection goes through these steps:
+•the Euler equation
+•the point-like vortices
+•the Jackiw-Pi model, constructed for chargedpoint-like objects
+•the non-relativistic Non-Abelian SU(2) CS 4th; this is for point-like
+vortices.
+•observation that the scalar-field part is close at low amplitudes to th e
+sine-Gordon model with the nonlinearity cos φ−1 expanded to second
+order,i.e.toφ4term.
+•exploitingby“anti-bosonization”theconnectionwiththeThirring fermionic
+non-linearity, JJ.
+•identification of the chiral densities defined within the Thirring model
+as the amplitudes of the algebraic operations of the ladder generat ors
+in theansatzfor thesu(2) scalar field φof the FT model for Euler.
+•JJconstant, leading to SD states
+12If this identifications are confirmed then it formulates the field theo retical
+model in a more accessible way, preparing for physical discussions in terms
+of classical fluid approaches.
+Weneedanon-Abelianfieldtheory(FT)andTWOfunctions, ρ1andρ2to
+describe the Euler fluid. This comes from the nature of the point-like object,
+which is a vortexis not a charge like in Jackiw-Pi. For Jackiw-Pi model an
+Abelian charge was sufficient and the result was the Liouville equation. The
+intermediate Thirring-sine-Gordon mapping (bosonization) is fully Abelian.
+Since for the Euler fluid the point-like objects carry vorticity the mo del must
+benon-Abelian and the final equation at self-duality is sinh-Poisson.
+Taking the physical streamfunction as Ψ, at self-duality in FT we hav e
+/parenleftbig
+ψγ5ψ/parenrightbig
+=ω (49)
+=ρ1−ρ2
+= sinhΨ
+and
+/parenleftbig
+ψψ/parenrightbig
+=ρ1+ρ2 (50)
+= coshΨ
+and
+JJ=/parenleftbig
+ψψ/parenrightbig2−/parenleftbig
+ψγ5ψ/parenrightbig2(51)
+= (coshΨ)2−(sinhΨ)2
+= 1
+which means that SD is equivalent with constant interaction JJ.
+4 Discussion of the meaning of ρ1ρ2= 1in the
+non-Abelian model
+We have seen the meaning of the relation ρ1ρ2= 1 in the case where we
+use the fermionic counterpart of the theory. Now we return to th e bosonic,
+non-Abelian model.
+For the Euler fluid the condition ρ1ρ2= 1 arises from the equality of
+two distinct expression for the magnetic field F+−calculated first with only
+φ1and alternatively with only φ2. This is possible because the algebraic
+operation involved in the first self-duality equation
+D−φ= 0 (52)
+13preserves without mixing the generators of the algebra in the origin al places
+where they exist according to the algebraic ansatz. The commutators repro-
+duce the same generator (with change of sign for E−) and the equality with
+zero gives two equation in which the potentials A±are expressed either by
+φ1or byφ2. It is now essential that the algebraic ansatz connects formally
+the two potentials A+=aHandA−=−a∗H. The operations that are
+performed in the equation D−φ= 0 involve commutators of A−(from the
+covariant operator D−) with the function φ, or
+[H,E±] =±2E± (53)
+It is essential that the commutators do NOT mix the algebra genera tors. If
+A−werenotalong the Cartan generator Hbut it contained E±then there
+would have been mixing of generators.They remain separate and the two
+equations derived from D−φ= 0 involve φ1and respectively φ2, separately.
+This looks like an equation of eigenfunctions where the generator H(which is
+thepotential A−)leaves invarianttheladder E+, asifitwasitseigenfunction.
+This means that the potential which is the fluid physical velocity does not
+affect the ladder generator when it is compatible with it.
+It is equally important that the function aand its complex conjugate a∗
+are the potentials A±. This means physically to consider the same velocity
+in the transversal plane, acting to determine φ1and respectively φ2. The two
+functionsA+andA−represent, in this algebraic ansatzthe same physical
+velocity in the plane but seen fromtwo opposite directions, from up a ndfrom
+down but along the same zdirection. This means that φ1andφ2must be
+connected and finally the relationship
+ρ1ρ2= 1 (54)
+is obtained.
+We can express the magnetic field F+−in two ways, using each time only
+onevariableφ1orφ2.
+In conclusion, we have two ways to think the Eq.(54): JJ=const, in
+every point. The other way is:
+∆ln|φ1|2+∆ln|φ2|2= 0
+coming from the fact that we can express the field F+−in two different ways,
+using either φ1(which means increase by E+) orφ2(usingE−).
+This is the reason for ρ1ρ2= 1.
+145 Conclusion
+Less than a physical interpretation, the present discussion may b e useful for
+further investigation of the connection between the traditional a pproach to
+the fluid physics and the field theoretical approach.
+Acknowledgement 1 This work has been supported by the Romanian Min-
+istry of Education and Research under Ideas Exploratory Res earch Project
+No.557
+References
+[1] D. Fyfe, D. Montgomery and G. Joyce, J. Plasma Phys. 17, 369 (1976).
+[2] D. Montgomery, W.H. Mathaeus, W.T. Stribling, D. Martinez and S.
+Oughton, Phys. Fluids A4(1992) 3.
+[3] R. H. Kraichnan and D. Montgomery, Rep. Prog. Phys. 43, 547 (1980)
+[4] D. Montgomery and G. Joyce, Phys. Fluids 17, 1139 (1974)
+[5] D. Montgomery, L. Turner and G. Vahala, J. Plasma Phys. 21, 239
+(1979)
+[6] G. Joyce and D. Montgomery, J. Plasma Phys. 10, 107 (1973)
+[7] F. Spineanu and M. Vlad, Phys. Rev. E 67 (2003) 046309.
+[8] F. Spineanu and M. Vlad, arXiv.org/physics/0501020.
+[9] F. Spineanu and M. Vlad, arXiv.org/physics/0503155.
+[10] R.L. Davies and E.P.S. Shellard, Phys.Rev.Letters 63 (1989) 2021 .
+[11] R. Jackiw and So-Young Pi, Phys. Rev. D 42, 3500 (1990).
+[12] R. Jackiw and So-Young Pi, Phys. Rev. Lett. 64, 2969 (1990).
+[13] G. Dunne, Self-dual Chern-Simons theories , hep-th/9410065.
+[14] M. faber and A.N. Ivanov, hep-th/0206034.
+[15] S. Coleman, Comm. Math. Phys. 31 (1973) 259.
+[16] E. Witten, preprint HUTP 78/A027.
+15
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+arXiv:1001.0152v2  [gr-qc]  7 Jul 2010Gravitational instability of the inner static region of a Re issner-Nordstr¨ om black hole
+Gustavo Dotti and Reinaldo J. Gleiser
+Facultad de Matem´ atica, Astronom´ ıa y F´ ısica (FaMAF),
+Universidad Nacional de C´ ordoba and
+Instituto de F´ ısica Enrique Gaviola, CONICET.
+Ciudad Universitaria, (5000) C´ ordoba, Argentina
+Reissner–Nordstr¨ om black holes have two static regions: r > r oand 0< r < r i, whereri
+androare the inner and outer horizon radii. The stability of the ex terior static region has been
+established long time ago. In this work we prove that the inte rior static region is unstable under
+linear gravitational perturbations, by showing that field p erturbations compactly supported within
+this region will generically excite a mode that grows expone ntially in time. This result gives an
+alternative reason to mass inflation to consider the space ti me extension beyond the Cauchy horizon
+as physically irrelevant, and thus provides support to the s trong cosmic censorship conjecture, which
+is also backed by recent evidence of a linear gravitational i nstability in the interior region of Kerr
+black holes found by the authors. The use of intertwiners to s olve for the evolution of initial data
+plays a key role, and adapts without change to the case of supe r-extremal Reissner-Nordstr¨ om black
+holes, allowing to complete the proof of the linear instabil ity of this naked singularity. A particular
+intertwiner is found such that the intertwined Zerilli field has a geometrical meaning -it is the first
+order variation of a particular Riemann tensor invariant-. Using this, calculations can be carried
+out explicitely for every harmonic number.
+PACS numbers: 04.50.+h,04.20.-q,04.70.-s, 04.30.-w
+I. INTRODUCTION
+In the course of a program [1–4] to study the stability under linear g ravitational perturbations of the most notable
+nakedly singular solutions of Einstein’s field equations, namely, negat ive mass Schwarzschild’s solution [1, 4, 6],
+|Q|>M >0 Reissner–Nordstr¨ om space-time [2] and |J|>M2Kerr space-time [2, 3] (see also [7]), we noticed that
+the stationary interior region beyond the inner horizon of a Kerr black hole is unstable [3, 5]. The existence of an
+initially bounded and exponentially growing solution of Teukolsky equat ions in the “super-extreme” case |J|>M2, of
+which some numerical evidence had been given earlier in [2], was establis hed in [3], where it was shown that there are
+actually infinitely many unstable modes. It was also shown in [3], that th e stationary region beyond the inner horizon
+of a Kerr black hole (i.e.|J| ≤M2) is linearly unstable under gravitational perturbations. These res ults show that
+linear perturbation theory is a valuable tool to study not only weak c osmic censorship (impossibility of formation of
+naked singularities), but also strong cosmic censorship (impossibility of formation of Cauchy horizons).
+For the Kerr spacetime, an explicit expression for the unstable mod es is not given in [3], since they involve solutions
+of complicated second order ordinary differential equations, which can at best be written in terms of Heun functions,
+providing little extra information. This, added to the complexity of th e reconstruction of the perturbed metric from
+a solution of Teukolski’s equations, makes it extremely difficult to evalu ate the physical meaningfulness of these
+unstable modes.
+The situation is different for the negative mass Schwarzschild and th e super-extremal Reissner–Nordstr¨ om space-
+times, where explicit expressions for some unstable modes, which inv olve only elementary functions, are given in [1, 4]
+and [2] respectively. Since the metric reconstruction process in th e spherically symmetric case is much simpler, it is
+possibletostudy the effectofperturbationsonthe singularity, by calculatingthe perturbed Riemanntensorinvariants.
+We use these to select appropriate boundary conditions at the sing ularity that ensure the self-consistency of the linear
+perturbation scheme, by requiring that curvature scalars do not get corrections that diverge faster at the singularity
+than the zeroth order term. As an example, in the case of the Schw arzschild negative mass naked singularity there are
+infinitely many possible boundary conditions at the singularity, param eterized by S1[1, 6], only one of which satisfies
+the above requirement. Thus, besides assuring the self-consiste ncy of the perturbative treatment, the above procedure
+solvesthe problem ofhavinga unique, well defined evolutionofpertu rbations in a non globallyhyperbolic background.
+The unstable modes in [1] were recognized by Cardoso and Cavaglia [7] to correspond to Chandrasekar’s “algebraic
+special” (AS) solutions of the linearized Einstein’s equations [9, 10]. Th is observation hinted in the right direction
+where to look for unstable modes of super-extremal Reissner-No rdstr¨ om black holes [2, 7]. As shown in [2], some of
+the Reissner–Nordstr¨ om AS modes grow exponentially in time while ke eping appropriate spatial boundary conditions2
+in the super-extremal case, as happens in the negative mass Schw arzschild case. With no exception, the AS solutions
+are irrelevant to the stability problem of the exterior region of black holes , since they do not satisfy suitable boundary
+conditions, a probable reason why they remained unnoticed for suc h a long time. For the Kerr solution, the AS
+modes do not satisfy appropriate boundary conditions, neither fo r the black hole stationary regions, nor for the
+naked singularity. However, it was proved in [3] that unstable modes exist for every harmonic (i.e., spin weighted
+spheroidal harmonic) of the Teukolski equations in the super-ext reme case. Moreover, in [3] a connection was
+spot between the unstable modes of Kerr naked singularities, and u nstable modes for the interior stationary region
+(r<ri) of a Kerr black hole . Given the similarities in the structures of the maximal analytic exten sions of Kerr and
+Reissner-Nordstr¨ om black holes, and the fact that both are affe cted by Cauchy horizon issues, one is naturally led to
+ask whether the interior region of a Reissner-Nordstr¨ om black ho le is also unstable. In this paper we show that this
+is the case. We give a detailed proof of the instability of the inner regio n of a Reissner-Nordstr¨ om black hole under
+linear gravitational perturbations initially restricted to a compact s ubset of the inner region.
+This provides an alternative reason to the mass inflation mechanism t o disregard the extension of the space time
+manifold beyond the Cauchy horizon: since the inner region is static b ut unstable, it cannot be the endpoint of an
+evolving space time. A similar result for the Kerr black hole would imply cu tting off the inner region of this space
+time, which has closed time like curves and other pathologies.
+We will concentrate on type one (also called “gravitational”, as oppo sed to type two or “electromagnetic”) scalar
+(also called “polar”) linear perturbations of the metric and electrom agnetic fields around the Reissner-Nordstr¨ om
+solution, since it is in this sector that we have found explicit unstable m odes. The linearized Einstein’s equations for
+these type of perturbations can be reduced to a 1+1 wave equatio n on a field Φ+
+1in a semi-infinite domain bounded
+by the singularity world-line, with a time independent potential (Zerilli’s equation [11–13]). This formalism was used
+to prove the stability of the exterior static region [11] of the Reissner-Nordstr¨ om black hole. In this case, Zerilli’s
+equation can be written as a wave equation in a complete 1+1 Minkowsk i spacetime, with a nonsingular potential
+which, being positive definite, guarantees the stability under this kin d of gravitational perturbations [11].
+When applied to the static black hole inner region r <ri, instead, one gets a wave equation on a half 1+1 fiducial
+Minkowski spacetime bounded by the singularity worldline, and the po tential has an unexpected second order pole at
+an inner point in the domain (that we call “kinematic singularity”), bes ides the expected divergence at the singularity.
+This makes the initial value problem for the inner region far more difficu lt than that for r > ro. These technical
+difficulties, however, are entirely analogous to those that arise in th e linear perturbation problem of a negative mass
+Schwarzschild spacetime, a problem which was solved recently in [4]. As in the Schwarzschild case, the second order
+pole in the potential can be traced back to the fact that the Zerilli fi eld Φ+
+1, as defined, is a singular function of the
+perturbed metric and electromagnetic fields at the kinematic singula rity (from where the name “kinematic” comes).
+Thus, an alternative field ˆΦ has to be introduced to properly analyze perturbations [4]. This is r elated to Φ+
+1by an
+intertwiner operator: ˆΦ =IΦ+
+1, whereI=∂/∂x+g, andxis a tortoise radial coordinate. In terms of ˆΦ, the type
+one scalar gravitational perturbation equation is a 1 + 1 wave equat ion∂2ˆΦ/∂t2+ˆHˆΦ = 0, ˆH=∂2/∂x2+ˆV(x),
+with a potential ˆVthat is regular everywhere. Once an appropriate self-adjoint ext ension of ˆHis chosen -and, as
+explained above, there is a unique physically motivated choice-, the e volution of initial data is unambiguously defined
+by means of an ˆHmode expansion of the data. This gives a dynamics in spite of the fact that the background is non
+globally hyperbolic (see [15] for a similar approach). The intertwining t echnique and choice of self-adjoint extension
+is explained in detail in Section III. Previously, in Section II, we review of the basics of the Reissner-Nordstr¨ om so-
+lution, itslinearperturbations,thefactorizationofZerilli’sHamiltonia n, andChandrasekhar’salgebraicspecialmodes.
+Several technical aspects of the problem are dealt with in the Appe ndixes. In particular, we include an algebraic
+procedure for the explicit construction of the vector and scalar z ero modes considered in the paper.
+II. LINEAR PERTURBATIONS OF A REISSNER-NORDSTR ¨OM BLACK HOLE
+This section contains all the material required for the proof of inst ability in Section III. We first review some
+basic facts on the maximal analytic extension of the Reissner-Nord str¨ om solution to the Einstein-Maxwell equations
+(Section IIA), and on the reduction of the linearized field equations around this solution to decoupled 1+1 wave
+equations (Section IIB). Then in Section IIC we calculate the pertu rbed Riemann tensor invariants, to determine
+the appropriate boundary conditions at the singularity for the self -consistency of the perturbation method. In Section
+IID we review from [9, 10] the factorization of the Regge-Wheeler a nd Zerilli hamiltonians, and its connection to
+Chandrasekhar’s “algebraic special” modes, which are central in th e proof of instability that follows.3
+A. The Reissner-Nordstr¨ om spacetime and its maximal analy tic extension
+The Reissner–Nordstr¨ om space-time metric
+ds2=−fdt2+dr2
+f+r2(dθ2+sin2θdφ2) =:gabdyadyb+r2ˆgijdxidxj, (1)
+is a warped product N ×r2S2of a two dimensional Lorentzian “orbit” manifold times a unit two sphe re. The Maxwell
+field on this space-time is
+F=Q
+r2dt∧dr. (2)
+In (1),fis the norm of the Killing vector ka=∂/∂t,
+f= 1−2M
+r+Q2
+r2=(r−ro)(r−ri)
+r2, (3)
+the latter form being useful when |Q|< M, in which case the roots of fare positive real numbers, 0 < ri< ro,
+and correspond to the horizon radii. It is useful to keep in mind the r elation between the alternative two-parameter
+descriptions of (1)
+ri=M−/radicalbig
+M2−Q2, ro=M+/radicalbig
+M2−Q2 (4)
+M=1
+2(ri+ro), Q2=riro. (5)
+kais timelike in the exterior( r>ro) and interior0 <r<riregions. As longaswe restrictto aregionwhere r/negationslash=ri,ro,
+the coordinates in (1) are appropriate. These coordinates becom e singular at riandro, yet the Reissner-Nordstr¨ om
+spacetime can be extended through the horizons, and new regions isometric to I: r > ro, II:ri< r < roand III:
+0<r<riarise ad infinitum, giving rise to the Penrose diagram displayed in Figure 1. The stability of those regions
+isometric to III is the subject of this paper.
+Given a complete spacelike surface such as Sin Figure 1, a Cauchy horizon (thicker rihorizon in the figure)
+develops at ri. This is the boundary of the maximal domain of development of the da ta, and it is connected to
+Sby timelike curves of finiteproper time. The solution of the Einstein-Maxwell equations is unique only up to
+the Cauchy horizon, and, although the spacetime is C∞extensible beyond it -as shown in Figure 1- the extension
+is not determined by the data in S, and is non unique. This lack of predictability in a classical theory of fie lds
+moved Penrose [16] to postulate what is known as the strong cosmic censorship conjecture , according to which, for
+generic initial data in an appropriate class, the maximal domain of dev elopment is inextensible (thus guaranteeing
+the preservation of predictability). The idea that under more realis tic assumptions than perfect spherical symmetry
+a Cauchy horizon would not develop, is supported by the finding that certain natural derivatives of a perturbation
+field diverge as the Cauchy horizon is approached from region II [18], and by Israel and Poisson “mass inflation”
+model [19]. An alternative, non perturbative approach was carried out by Dafermos [20]. In [20], spherical symmetric
+solutions to the Einstein-Maxwell-scalar field system are studied. Th e (uncharged) scalar field was added to get
+around the uniqueness Birchoff theorems in the spherically symmetr ic case. A characteristic problem is solved
+combining Reissner-Nordstr¨ om data at the event horizon with gen eric matching data at the other null edge coming
+out the bifurcation sphere at ro. It is shown that, generically, the Hawking mass diverges at the Cau chy horizon, and
+thus the spacetime fails to be C1extendible beyond it.
+The results in this paper contribute to the idea that the extended s pacetime depicted in Figure 1 is an irrelevant
+solution of the Einstein-Maxwell equations. We show that a linear per turbation of the metric and Maxwell fields in
+region III, compactly supported away from the Cauchy horizon an d the singularity, will grow exponentially in time,
+showing that region III is in fact an unstable static solution of the Ein stein-Maxwell equations.
+B. Linearized gravity around the Reissner-Nordstr¨ om solu tion
+ThelinearizedEinstein-Maxwellequationsaround(2)havebeenana lyzedbymanyauthors, startingwiththe papers
+by Regge, Wheeler and Zerilli [12], generalized to higher dimensional ch arged black holes with constant curvature4
+SI II I
+IIIIIIII
+III IIIIII III
+III IIIr=0r=0
+r=0 r=0r=0r=0
+rorir
+roi
+ri
+r=
+8
+FIG. 1: Penrose diagram for the maximal analytic extension o f the Reissner-Nordstr¨ om space-time. All regions labeled I are
+isometric, and so are those labeled II, and III. Regions I and III are stationary, and region I is known to be linearly stabl e.
+The function ris globally defined, with ro< rin I ,ri< r < r oin II , and 0 < r < r iin III. Also r→ ∞at the conformal
+boundaries J±. Ther=rihorizon drawn thicker is a Cauchy horizon for the initial dat a surfaceS, theC∞extension beyond
+it (in particular, the two copies of region III just above it b eing non unique, unless analyticity of the metric is require d.
+horizons by Kodama and Ishibashi [13, 14]. For polar (scalar in [13, 14 ], here denoted (+) following [9, 10]) and axial
+(vector in [13, 14], here denoted ( −) following [9, 10]) modes with harmonic number ℓ(ℓ= 2,3,...), the metric and
+electromagneticperturbations of a Reissner–Nordstr¨ omspace -time can be encoded in two functions, Φ±
+α(t,r),α= 1,2,
+that satisfy wave equations [9, 14]
+0 =∂Φ±
+α
+∂t2−∂Φ±
+α
+∂x2+V±
+αΦ±
+α=:∂Φ±
+α
+∂t2+H±
+αΦ±
+α (6)
+with potentials
+V±
+α=±βαdfα
+dx+βα2fα2+κfα (7)
+whereκ= (ℓ−1)ℓ(ℓ+1)(ℓ+2),βα= 3M+(−1)α/radicalbig
+9M2+4Q2(ℓ−1)(ℓ+2),
+fα=f
+rβα+(ℓ−1)(ℓ+2)r2(8)
+withfgiven in (3) (equation (8) corrects a typo in [2]) and xis a “tortoise” coordinate, defined by dx/dr= 1/f. Eq.
+(6) admits separation of variables Φ±
+α(t,r) = exp(−iωt)ψ±
+α(r), leading to the Schr¨ odinger like equation
+ω2ψ±
+α=H±
+αψ±
+α (9)
+Unstable modes correspond to purely imaginary ω, and thus to negative eigenvalues of the “hamiltonian” H.
+We are interested in the case M >|Q|and 0<r<ri, then
+x=r+r2
+o
+ro−riln/parenleftbiggro−r
+ro/parenrightbigg
++r2
+i
+ri−roln/parenleftbiggri−r
+ri/parenrightbigg
+(10)5
+where the integration constant was chosen so that xranges from zero to infinity as rgoes from zero to ri. In these
+limits
+x≃/braceleftBigg1
+3riror3+ri+ro
+(2riro)2r4+O(r5),r→0+
+r2
+i
+ri−roln/parenleftBig
+ri−r
+ri/parenrightBig
++... ,r →ri−(11)
+Note form (8) that f1has a singularity at
+rc=/radicalbig
+9M2+4Q2(ℓ−1)(ℓ+2)−3M
+(ℓ−1)(ℓ+2), (12)
+then, from (7), we expect a singularity at rcinV±
+1. However, the divergences from the different terms cancel out a nd
+the vector potential V−
+1is smooth at rc. This is not the case for the scalar mode V+
+1, which has a quadratic pole
+atrc, with a positive coefficient. The singularities at r= 0 andr=rchave very different origins. The first one is
+due to the spacetime singularity at this point, whereas the second o ne can be traced back to the definition of Φ+
+1,
+which happens to be a singular function of the metric and Maxwell field first order variations at this point (see, e.g.,
+(63)-(57)), this being the reasonwhy we referto it asa “kinematic ” singularity. We should stress herethat the wayΦ+
+1
+is defined in terms of the perturbed metric and Maxwell fields is crucia l to disentangle the linearized Einstein-Maxwell
+equations, and that rchappens to fall outside the domain of interest r>rowhen the stability of the exteriorregion
+is studied. Note that rcis a decreasing function of ℓand an increasing function of Q. Thus, for large enough ℓwe
+have 0< rc< ri, andV+
+1is singular in the inner region, the one that we study in this paper. The “safest” mode
+isℓ= 2, for which rc> ri, and therefore the potential regular for 0 < r < ri, as long as Q/M <√
+7/4≃0.66. For
+largerQ/Mrate,rc<rifor every harmonic mode.
+Fig. 2 depicts the ℓ= 2 scalar potentials V+
+αfor some particular values of the parameters. The behaviour of th e
+±4000±20000200040006000
+V1+
+0.05 0.1 0.15 0.2 0.25 0.3 0.35
+x
+±0.1±0.08±0.06±0.04±0.0200.020.04
+V2+1 2 3 4 5 6 7x
+FIG. 2: Scalar potentials V+
+1(left) andV+
+2(right) forℓ= 2,ri= 1,ro= 2 plotted against x. Thexrange was chosen in each
+case to exhibit the relevant details, beyond these ranges th e behavior is that captured in equations (13)
+potentials near the spacetime singularity and the inner horizon is
+V+
+α≃/braceleftBigg
+−2
+9x2+... x ≃0
+C+
+αexp/parenleftBig
+−(ro−ri)x
+r2
+i/parenrightBig
+x→ ∞(13)
+where
+C+
+α=/parenleftbiggro−ri
+ri4/parenrightbigg/bracketleftbiggκri2−βα(ro−ri)
+βα+(ℓ−1)(ℓ+2)ri/bracketrightbigg
+(14)
+The local solutions of the equation
+H+
+αψ+
+α=−k2ψ+
+α, (15)
+near the inner horizon and the singularity are (for both α= 1 andα= 2) of the form
+ψ+
+α≃/braceleftBigg
+Acos(θ)[x1/3/summationtext
+n≥0a(1)
+nxn/3]+Asin(θ)[x2/3/summationtext
+n≥0a(2)
+nxn/3] forx≃0
+b1[ exp(−kx)+...]+b2[ exp(kx)+...] for x→ ∞(16)6
+where we have set a(1)
+0= 1 =a(2)
+0. The differential equation (15), rewritten using ras the independent variable, has
+a regular singular point at r= 0, whose indicial equation has roots 1 and 2. The terms between sq uare brackets
+above are just the Frobenius series solution for this equation, writ ten in terms of xby inverting (11) (thus the powers
+ofx1/3). The two arbitrary constants in front of them where parameter ized withA >0 andθ∈[0,2π) for later
+convenience. Similar expansions are made for local solutions of differ ential equations near the singularity at different
+points below.
+Fig. 3 depicts the ℓ= 2 vector potentials V−
+αfor the same parameter values as those in Fig. 2.
+024681012
+V1-
+1 2 3 4
+x012345678
+V2-
+1 2 3 4
+x
+FIG. 3: Vector potentials V−
+1(left) andV−
+2(right) forℓ= 2,ri= 1,ro= 2 plotted against x.
+The behaviour of the potentials near the spacetime singularity and t he inner horizon is
+V−
+α≃/braceleftBigg4
+9x2+... x ≃0
+C−
+αexp/parenleftBig
+−(ro−ri)x
+r2
+i/parenrightBig
+x→ ∞(17)
+where
+C−
+α=/parenleftbiggro−ri
+ri4/parenrightbigg/bracketleftbiggκri2+βα(ro−ri)
+βα+(ℓ−1)(ℓ+2)ri/bracketrightbigg
+(18)
+The local solutions of the equation
+H−
+αψ−
+α=−k2ψ−
+α, (19)
+which correspond to a mode ω=±ik, are, for both α= 1,2, of the form
+ψ−
+α≃/braceleftBigg
+Acos(θ)[x−1/3/summationtext
+n≥0a(1)
+nxn/3]+Asin(θ)[x4/3/summationtext
+n≥0a(2)
+nxn/3] forx≃0
+b1[exp(−kx)+...]+b2[exp(kx)+...] for x→ ∞(20)
+where we have set a(1)
+0= 1 =a(2)
+0.
+C. Consistency of the linearized analysis
+In the linearized approach, a solution gab,Aaof the Einstein-Maxwell equations is replaced with a “perturbed”
+metric and electromagnetic field potential gab+ǫhab,Aa+ǫBa, and the field equations are then required to hold
+to first order in ǫ. Given that the background solution we are interested in, region II I of the Reissner-Nordstr¨ om
+spacetime, has a curvature singularity as r→0+, the perturbation treatment will certainly be inconsistent if we find
+that the first order correction to a perturbed divergent curvat ure scalar diverges faster than the unperturbed piece in
+ther→0+limit, since in this case the notion of a “uniformly small metric perturba tion” is lost. This is why the first
+order piece of the Kretschmann or some other invariant is usually co mputed. Here we take a systematic approach
+to make sure that noneof the algebraic invariant made out of the Riemman tensor acquires a correction diverging
+faster than the corresponding background metric invariant.7
+Any algebraic polynomial invariant of the Riemann tensor Rabcdcan be written as a polynomial on a set of basic
+invariants, the basic invariants being generically subject to syzygie s (constraints). The basic invariants are more
+compactly written in terms of the Ricci tensor Rab:=Rcacb, the Ricci scalar R=Raa, the trace free Ricci tensor
+Sab=Rab−gabR/4, the Weyl tensor
+Cabcd:=Rabcd−2
+n−2(ga[cSd]b−gb[cSd]a)−2
+n(n−1)Rga[cgd]b
+and its dual C∗
+abcd:=1
+2ǫabefCefcd, or just using the Ricci and Weyl spinors ΦAB˙A˙B,ΨABCD. In the case of spacetimes
+with a perfect fluid or a Maxwell field, the basic invariants are those g iven in Table I (from [17].)
+Table I: Basic Riemann tensor invariants for perfect fluid or Maxwell field spacetimes
+R:=Raa
+R1:= ΦAB˙A˙BΦAB˙A˙B=1
+4SabSba
+R2:= ΦAB˙A˙BΦBC˙B˙CΦCA˙C˙A=−1
+8SabSbcSca
+R3:= ΦAB˙A˙BΦBC˙B˙CΦCD˙C˙DΦDA˙D˙A=1
+16SabSbcScdSda
+w1:= ΨABCDΨABCD=1
+8(Cabcd+iC∗
+abcd)Cabcd
+w2:= ΨABCDΨCDEFΨEFAB=−1
+16(Cabcd+iC∗
+abcd)CcdefCefab
+m1:= ΨABCDΦAB˙A˙BΦCD˙A˙B=1
+8SabScd(Cacdb+iC∗
+acdb)
+m2:= ΨABCDΨABEFΦCD˙A˙BΦEF˙A˙B
+m3:= ΨABCD¯Ψ˙A˙B˙C˙DΦAB˙A˙BΦCD˙C˙D
+m4:= ΨABCD¯Ψ˙A˙B˙C˙DΦAB˙C˙EΦCE˙A˙BΦDE˙D˙E
+m5:= ΨABCDΨCDEF¯Ψ˙A˙B˙E˙FΦAB˙A˙BΦEF˙E˙F
+In the Maxwell case, R=R2= 0, and the syzygies among the remaining invariants are [17]
+R12= 4R3, m4= 0, m1¯m2=R1¯m5, m2¯m2m3=R1m5¯m5 (21)
+For Reissner-Nordstr¨ om R1andm2do not vanish, then the syzygies imply that, as long as R1,w1,w2,m1andm2
+behave properly (correction does not diverge faster than unper turbed term), the same will happen to any curvature
+invariant, of any degree. Using references [12, 13], we have recon structed the perturbed metric for each mode, and
+then calculated the perturbed invariants with the help of the grten sor symbolic manipulation package [21]. For the
+vector (axial) modes we could reduce all expressions, by repeated ly applying (6), to
+R1=Q4
+r8
+w1=6(Q2−Mr)2
+r8+iǫ6(Q2−Mr)Z
+r8Yℓm(θ,φ)
+w2=6(Q2−Mr)3
+r12+iǫ9(Q2−Mr)2Z
+r12Yℓm(θ,φ) (22)
+m1=2(Q2−Mr)Q4
+r12+iǫQ4Z
+r12Yℓm(θ,φ)
+m2=4(Q2−Mr)2Q4
+r16−iǫ4(Q2−Mr)Q4Z
+r16Yℓm(θ,φ)
+where
+Z:=/bracketleftbiggκ(β2r−4Q2)ψ−
+2
+2(β2−β1)+2Q(β2+(ℓ+2)(ℓ−1)r)ℓ(ℓ+1)ψ−
+1
+β2−β1/bracketrightbigg
+(23)8
+From (20), (22) and (23), it is clear that an unstable vector mode g ives an inconsistent perturbation unless
+a(1)
+0=b2= 0 in (20). It is only in this case that the perturbation can be uniform ly bound in the whole of region III.
+For the scalar modes, the invariants can be expressed entirely in te rms of the Zerilli field and its first r−derivative
+but we were not able to reduce the resulting expressions to a reaso nably compact form, except for R1, for which we
+found that
+R1=Q4
+r8/bracketleftbigg
+1−4ǫ/parenleftbigg
+f∂ψ+
+1
+∂r−f∂χ+
+1
+∂rψ+
+1
+χ1/parenrightbigg
+Yℓm(θ,φ)−4ǫ/parenleftbigg
+f∂ψ+
+2
+∂r−f∂χ+
+2
+∂rψ+
+2
+χ2/parenrightbigg
+Yℓm(θ,φ)/bracketrightbigg
+, (24)
+where theχ+
+αis one of Chandrasekhar’s algebraic special modes, introduced in th e following Section (see equation
+(26)). The above formula will turn out to be very useful in the follow ing sections.
+D. Factorization of Zerilli’s Hamiltonian and algebraic sp ecial modes
+Chandrasekhar’salgebraicspecialmodes(ASM) aresolutionsofeq uations(15), (19)withreal k, i.e., unstablemodes
+of the perturbation equations. These modes do not satisfy appro priate boundary conditions as linear perturbations of
+region I of the Reissner-Nordstr¨ om black hole, in agreement with t he fact that this region is linearly stable. However,
+some ASM wereshown in [2, 7] to satisfy appropriateboundary cond itions as perturbations of the Reissner-Nordstr¨ om
+naked singularity, showing that this spacetime is unstable. In this Se ction we consider algebraic special modes as
+perturbations of region III of a Reissner-Nordstr¨ om black hole, and analyze their behaviour near the singularity and
+the inner horizon. Following [9], we introduce χ±
+αdefined by
+d
+dxlnχ±
+α=±/parenleftbigg
+βαfα+κ
+2βα/parenrightbigg
+, (25)
+the general solution of this equation being an irrelevant constant t imes
+χ+
+α=Mrexp/parenleftBig
+κx
+2βα/parenrightBig
+βα+(ℓ−1)(ℓ+2)rχ−
+α=Mexp/parenleftbigg−κx
+2βα/parenrightbigg(βα+(ℓ−1)(ℓ+2)r)
+r. (26)
+In terms of the functions χ±
+α, equation (9) reads
+1
+ψ±αd2ψ±
+α
+dx2+/bracketleftbigg
+ω2+κ2
+4(βα)2/bracketrightbigg
+=1
+χ±αd2χ±
+α
+dx2, (27)
+which can easily be integrated if
+ω=±ikα, kα:=κ
+2|βα|>0 (28)
+Equation (28) defines the algebraic special modes , (ASM) which give unstable solutions Φ±
+α(t,r) = exp(kαt)ζ±
+α(r) of
+the perturbation equations, with ζ±
+αa solution of (27) when the term between brackets vanishes:
+ζ±
+α=A±
+αχ±
+α+B±
+ατ±
+α,R∗, τ±
+α,R∗:=χ±
+α/integraldisplayx
+R∗Mdx
+[χ±α(x)]2. (29)
+Note that a change of choice of R∗amounts to adding a constant times χtoτ, and thus the constants AandBin
+(29) are unambiguously defined only after R∗has been chosen.
+Alternatively, ASM can be obtained from the factorization propert y of the Hamiltonians H±
+α
+H+
+α=AαBα−/parenleftbiggκ
+2βα/parenrightbigg2
+(30)
+H−
+α=BαAα−/parenleftbiggκ
+2βα/parenrightbigg2
+, (31)9
+where
+Aα:=∂
+∂x+/parenleftbigg
+βαfα+κ
+2βα/parenrightbigg
+(32)
+Bα:=−∂
+∂x+/parenleftbigg
+βαfα+κ
+2βα/parenrightbigg
+. (33)
+ζ+
+αspan de kernel of AαBα, withχ+
+αin kerBα, whereasζ−
+αspan de kernel of BαAα, withχ−
+αin kerAα.
+1. Algebraic special vector modes
+The asymptotic behaviour of χ−
+αnear the spacetime singularity and the inner horizon is
+χ−
+α≃/braceleftBiggMβα
+(3riro)1/3x−1/3+... x ≃0
+M
+ri[βα+(ℓ−1)(ℓ+2)ri]exp/parenleftBig
+−κx
+2βα/parenrightBig
++... x→ ∞(34)
+The vector τmodes are
+τ−
+α,R∗:=
+βα+(ℓ−1)(ℓ+2)r
+rexp/parenleftBig
+κx
+2βα/parenrightBig
+/integraldisplayx
+R∗r2exp/parenleftBig
+κx
+βα/parenrightBig
+dx
+[βα+(ℓ−1)(ℓ+2)r]2. (35)
+Sinceβ1<0 and 0<β2, forα= 2 the integral in (35) near x=∞(r=ri) diverges, we can give R∗anyfinitevalue,
+and the asymptotic behaviour will depend on whether R∗= 0 or not:
+τ−
+α=2,R∗=0≃/braceleftBigg
+x4/3+... x ≃0
+exp/parenleftBig
+κx
+2β2/parenrightBig
++... x→ ∞(36)
+τ−
+α=2,R∗/negationslash=0≃/braceleftBigg
+x−1/3+... x ≃0
+exp/parenleftBig
+κx
+2β2/parenrightBig
++... x→ ∞(37)
+Since those type-2 vector ASM that grow slower than r0asr→0+blow up at the inner horizon, it follows from
+the analysis of invariants in the previous subsection (equations (22 )-(23)), that a pure mode Φ−
+2=ek2tζ−
+2=
+ek2t[A−
+2χ−
+2+B−
+2τ−
+2,R∗] cannot be consistently treated as a first order perturbation on the entire Reissner-Nordstr¨ om
+region III.
+Ifα= 1 the integral in (35) near infinity converges, thus, we can give R∗anyfinitevalue, or take R∗=∞. The
+asymptotic behaviour will be
+τ−
+α=1,R∗=0≃/braceleftBigg
+x4/3+... x ≃0
+exp/parenleftBig
+−κx
+2β1/parenrightBig
++... x→ ∞(38)
+τ−
+α=1,R∗/negationslash=0≃/braceleftBigg
+x−1/3+... x ≃0
+exp/parenleftBig
+−κx
+2β1/parenrightBig
++... x→ ∞(39)
+τ−
+α=1,R∗=∞≃/braceleftBigg
+x−1/3+... x ≃0
+exp/parenleftBig
+κx
+2β1/parenrightBig
++... x→ ∞(40)
+It follows again from equations (22)-(23) that a pure AS mode Φ−
+1=ek1tζ−
+1=ek1t[A−
+1χ−
+1+B−
+1τ−
+1,R∗] cannot be
+consistently treated as a first order perturbation in the entire Re issner-Nordstr¨ om region III.10
+Note that some care is required when interpreting equation (35) fo r theα= 1 vector mode, due to the singularity
+of the integrand at rc. Take, e.g., the case R∗=∞. The one form under the integral sign in
+τ−
+1,∞:=
+(rc−r)
+(ℓ−1)(ℓ+2)rexp/parenleftBig
+κx
+2β1/parenrightBig
+/integraldisplay∞
+xr2
+(r−rc)2exp/parenleftbiggκx
+β1/parenrightbigg
+dx. (41)
+can be written as/bracketleftBig
+A
+(r−rc)2+B
+r−rc+Z(r)/bracketrightBig
+dr,Z(r) the regular function obtained by subtracting the second and firs t
+order poles. The integration constants at both sides of rccan then be adjusted such that
+τ−
+1,∞=A−B(r−rc)ln|r−rc|+/parenleftBig
+Bln|ri−rc|−A
+ri−rc/parenrightBig
+(r−rc)+(r−rc)/integraltextri
+rZ(r)dr
+(ℓ−1)(ℓ+2)rexp/parenleftBig
+κx
+2β1/parenrightBig .
+This is well defined across rc.
+2. Algebraic special scalar modes
+The asymptotic behaviour of χ+
+αnear the spacetime singularity and the inner horizon is
+χ+
+α≃/braceleftBiggM
+βα(3riro)1/3x1/3+... x ≃0
+Mri
+βα+(ℓ−1)(ℓ+2)riexp/parenleftBig
+κx
+2βα/parenrightBig
++... x→ ∞(42)
+The asymptotic behaviour of τ±
+α,R∗depends on the choice of R∗in
+τ+
+α,R∗:=
+rexp/parenleftBig
+κx
+2βα/parenrightBig
+βα+(ℓ−1)(ℓ+2)r
+/integraldisplayx
+R∗[βα+(ℓ−1)(ℓ+2)r]2dx
+r2exp/parenleftBig
+κx
+βα/parenrightBig. (43)
+Ifα= 1 the integral (43) diverges near infinity, then R∗is restricted to finitevalues, and
+τ+
+α=1,R∗=0≃/braceleftBigg
+x2/3+... x ≃0
+exp/parenleftBig
+−κx
+2β1/parenrightBig
++... x→ ∞(44)
+τ+
+α=1,R∗/negationslash=0≃/braceleftBigg
+x1/3+... x ≃0
+exp/parenleftBig
+−κx
+2β1/parenrightBig
++... x→ ∞(45)
+Ifα= 2 there are three possibilities:
+τ+
+α=2,R∗=0≃/braceleftBigg
+x2/3+... x ≃0
+exp/parenleftBig
+κx
+2β2/parenrightBig
++... x→ ∞(46)
+τ+
+α=2,R∗/negationslash=0≃/braceleftBigg
+x1/3+... x ≃0
+exp/parenleftBig
+κx
+2β2/parenrightBig
++... x→ ∞(47)
+τ+
+α=2,R∗=∞≃/braceleftBigg
+x1/3+... x ≃0
+exp/parenleftBig
+−κx
+2β2/parenrightBig
++... x→ ∞(48)
+The only scalar AS modes that behave appropriately near the inner h orizon areχ+
+α=1andτ+
+α=2,R∗=∞. Since the
+integrals defining the latter one are non elementary, we proceed wit hχ+
+α=1, for which an explicit reconstruction of the11
+perturbed metric and invariants is relatively simple. No invariant that is trivial at order zeroth develops a first order
+correction. The non zero invariants to first order for the χ+
+1perturbation are
+R1=Q4
+r8(49)
+w1=6(Mr−Q2)2
+r8−ǫ/bracketleftbigg3Mβ2ℓ(ℓ+1) (Mr−Q2)
+2β1r7/bracketrightbigg
+Yℓm(θ,φ)e−κ
+2β1(t−x)(50)
+w2=−6(Mr−Q2)3
+r12+ǫ/bracketleftbigg9Mβ2ℓ(ℓ+1) (Mr−Q2)2
+4β1r11/bracketrightbigg
+Yℓm(θ,φ)e−κ
+2β1(t−x)(51)
+m1=−2(Mr−Q2)Q4
+r12+ǫ/bracketleftbiggMQ4β2ℓ(ℓ+1)
+4β1r11/bracketrightbigg
+Yℓm(θ,φ)e−κ
+2β1(t−x)(52)
+m2=m3=4(Mr−Q2)2Q4
+r16−ǫ/bracketleftbiggMQ4β2ℓ(ℓ+1) (Mr−Q2)
+β1r15/bracketrightbigg
+Yℓm(θ,φ)e−κ
+2β1(t−x)(53)
+This algebraic special mode satisfies all our requirements for a self c onsistent first order formalism in this singular
+spacetime. Thus, we will restrict to type one scalar perturbations from now on. χ+
+1is an example of a spatially
+uniformly bounded perturbation that grows exponentially in time, an d thus a signal of a gravitational instability. In
+the following Section, we will show that this mode can be excited by a ge neric perturbation that is initially compactly
+supported within region III. The treatment will follow closely the cas e of the Schwarzschild naked singularity treated
+in [4].
+III. PROOF OF THE LINEAR INSTABILITY OF THE INNER STATIC REGI ON
+The linear instability of a static spacetime is established once an unstable mode is found. W e have shown in the
+previous section that, out of the four modes existing for every ha rmonic pair ( ℓ,m), the type=1 scalarmode admits an
+unstable solution to the linearized Einstein-Maxwell equations -Chan drasekhar’s AS mode- that can consistently be
+treated to first order in the whole domain of region III, 0 <r<ri. The purpose of this section is to show how generic
+initial data with compact support in region III excites this mode. Alth ough this problem is trivial for perturbations
+in region I, it exhibits a number of unexpected difficulties when dealing w ith perturbations of region III. Zerilli’s wave
+equation for this mode:
+0 =∂Φ+
+1
+∂t2−∂Φ+
+1
+∂x2+V+
+1Φ+
+1=:∂Φ+
+1
+∂t2+H+
+1Φ+
+1 (54)
+has a potential with a singularity at rcgiven in equation (12), that generically falls in region III (see left pan el in
+Figure 2). The origin of this singularity (a second order pole) in V+
+1can be traced back to the definition of Φ+
+1
+[12, 13]. The first order variation of the electromagnetic field has
+δFrθ=∂A
+∂t∂Yℓm
+∂θf−1, (55)
+δFtθ=∂A
+∂r∂Yℓm
+∂θf, (56)
+thenAmust be smooth for δFto be smooth. Since
+A=β2(r−rc)
+8QrΦ+
+1(t,r), (57)
+we conclude that, for generic smooth perturbations, Φ+
+1has a first order pole at r=rc, and can be Laurent expanded
+aroundrcas
+Φ+
+1=/summationdisplay
+k≥−1ck(r−rc)k. (58)
+The variation of gtrcan be simplified to the form [12, 13]
+δgtr=Yℓm∂
+∂t/bracketleftbigg
+r∂Φ+
+1
+∂r+BΦ+
+1/bracketrightbigg
+, (59)12
+where
+B=/parenleftbig
+(ℓ−1)(ℓ+2)r4−3M/parenleftbig
+−3+ℓ2+ℓ/parenrightbig
+r3+/parenleftbig
+2 (ℓ−1)(ℓ+2)Q2−12M2/parenrightbig
+r2+13Q2Mr−4Q4/parenrightbig
+(r2−2Mr+Q2)((ℓ−1)(ℓ+2)r2+6Mr−4Q2)
++r/radicalbig
+9M2+4Q2(ℓ−1)(ℓ+2)
+((ℓ−1)(ℓ+2)r2+6Mr−4Q2)(60)
+It can be checked that the ( r−rc)−2coefficient of the series expansion of (59) vanishes. The ( r−rc)−1coefficient
+will vanish if
+c0
+c−1=−(ℓ−1)2(ℓ+2)2(2M−β1)
+β1(2(ℓ−1)(ℓ+2)M+(ℓ2+ℓ+2)β1)(61)
+It turns out that (58) and (61) are not only necessary but also su fficient conditions for the perturbations of the metric
+and electromagnetic fields to be smooth. Thus, we have proved tha t the Zerilli functions Φ+
+1corresponding to smooth
+type-1 scalar perturbations are those admitting, at any fixed time , a Laurent expansion (58) around rcsatisfying the
+condition (61).
+We also need to check that the perturbation does not change the c haracter of the singularity. As explained in the
+previous section, this guarantees the self-consistency of the line arized theory. It then follows from equation (24) that
+we must demand that, for some positive δandN
+/vextendsingle/vextendsingle/vextendsinglef∂Φ+
+1
+∂r−f∂χ1
+∂rΦ+
+1
+χ1/vextendsingle/vextendsingle/vextendsingle≤N,if 0<r<δ (62)
+We will show in the following Section that this condition is also sufficient to assure that all the invariants behave
+properly. Thus, we arrive at:
+Lemma 1 In order that the metric and electromagnetic scalar type-1 pertu rbations be smooth and the linearized
+approach be self consistent, the Zerilli function Φ+
+1has to satisfy (58) and (61) whenever rc<ri. It also has to satisfy
+condition (62), and decay properly as r→ri. In particular, both initial data functions Φ+
+1(t= 0,r),˙Φ+
+1(t= 0,r)
+must satisfy all these conditions.
+The rather odd initial value problem that these conditions pose is in fa ct very similar to that of the propagation
+of scalar gravitational perturbations on a negative mass Schwarz schild background, which also has a “kinematic”
+singularity. This latter problem was worked out in [4] using an intertwin er operator [24]. We will apply the same
+technique in what follows.
+A. Basics of the intertwining technique
+Consider a wave equation with a time independent potential:
+∂Φ
+∂t2−∂Φ
+∂x2+VΦ =:∂Φ
+∂t2+HΦ = 0 (63)
+on a domain t∈R,x∈(a,b) wherea=−∞and/orb=∞is a possibility. An intertwinner for this equation has the
+form [4, 24]
+I=∂
+∂x−g, g =1
+ψIdψI
+dx(64)
+withψIsatisfying
+HψI=EIψI. (65)
+The above equations neither assume that His self adjoint in L2((a,b),dx) nor thatψIis an eigenfunction of such an
+operator.ψIin equation (65) is just anysolution of this differential equation, without any consideration on b oundary
+conditions, boundedness or finiteness of some L2norm.
+In terms of
+ˆΦ :=IΦ (66)13
+equation (63) reads
+0 =∂ˆΦ
+∂t2+ˆHˆΦ
+ˆH:=−∂ˆΦ
+∂x2+ˆV (67)
+ˆV:=V−2dg
+dx
+That is, if Φ is a solution of (63) with initial data
+(Φ(t= 0,x),˙Φ(t= 0,x)), (68)
+thenIΦ =:ˆΦ is a solution of (67) with initial data
+(ˆΦ(t= 0,x),˙ˆΦ(t= 0,x)) = (IΦ(t= 0,x),I˙Φ(t= 0,x)). (69)
+The general idea of the intertwiner technique is to use this fact to s earch for an appropriate intertwiner such that ˆV
+is simpler than the potential in the original problem.
+The operator Ihas a nontrivial kernel spanned by ψI, so information is lost when switching from Φ to ˆΦ :=IΦ.
+The operator
+ˆI=∂
+∂x+g, (70)
+can easily be seen to map solutions of (67) onto solutions of (63). A s traightforward computation shows that
+ˆII=EI−H (71)
+SinceIandˆIhave non trivial kernels, information is lost when applying these oper ators. However, in the case of a
+solution of the wave equation (71) implies that
+ˆIˆΦ =ˆIIΦ = (EI−H)Φ =EIΦ+∂2Φ
+∂t2, (72)
+Thus the information lost is precisely the Φ initial data. The above equ ation can be regarded as a t−ODE on ˆΦ for
+everyx, and can easily be integrated to give Φ back:
+IfEI>0,
+Φ(t,x) =1√EI/integraldisplayt
+0sin/parenleftBig/radicalbig
+EI(t−t′)/parenrightBig
+ˆIˆΦ(t′,x)dt′+cos(/radicalbig
+EIt)Φ(0,x)+sin(√EIt)√EI˙Φ(0,x); (73)
+ifEI<0,
+Φ(t,x) =1√−EI/integraldisplayt
+0sinh/parenleftBig/radicalbig
+−EI(t−t′)/parenrightBig
+ˆIˆΦ(t′,x)dt′+cosh(/radicalbig
+−EIt)Φ(0,x)+sinh(√−EIt)√−EI˙Φ(0,x); (74)
+ifEI= 0,
+Φ(t,x) =/integraldisplayt
+0(t−t′)ˆIˆΦ(t′,x)dt′+Φ(0,x)+t˙Φ(0,x) =/integraldisplayt
+0/parenleftBigg/integraldisplayt′
+0ˆIˆΦ(t′′,x)dt′′/parenrightBigg
+dt′+t˙Φ(0,x)+Φ(0,x).(75)
+We conclude that we can solve equation (63) subject to (68) by mea ns of the following procedure:
+1. From the initial Φ data (68) construct initial ˆΦ data using (69).
+2. Find the solution ˆΦ of equation (67) with initial data (69).
+3. Apply (73)-(75) to obtain the solution Φ to the original equation.
+Note that the evolution problem is solved in step 2, and that the initial Φ data is used twice: in steps 1 an d 3.14
+B. The initial value problem for perturbations of region III
+Why would be one be interested in solving (63) using the complicated int ertwiner method? The intertwiner is
+certainly useful if ˆVis simpler than V. Our motivation, however, comes from a deeper problem: there is n o available
+theory to deal with the initial value problem possed in Lemma 1. Unless rc>ri, which may only happen for a finite
+number of harmonic numbers ℓ, the function space in Lemma 1 is unrelated to any recognizable Hilber t space, and the
+Zerilli wave equation has singular coefficients in region III (on top of t his, there is the issue of non global hyperbolicity
+of region III, even when rc> ri.) A similar situation is found when studying perturbations of a negativ e mass
+Schwarzschild spacetime. In this case, the problem was solved using intertwiners [4]. Less sophisticated approaches,
+such as using algebraic redefinitions of the variables, can be shown t o fail.
+As we show in Appendix A, for any(even complex) EI, the intertwiner (64)-(65) produces a ˆVthat is smooth at
+r=rc, while sending functions satisfying (58) and (61) onto functions wh ich are smooth at r=rc. Also, there are
+options for ψIfor which Isends initial data (as characterized in Lemma 1) onto a subspace D ⊂L2((0,∞),dx) where
+ˆHis a self-adjoint operator. This allows us to use the resolution of the identity for ˆHto solve the hat wave equation
+by separation of variables, by expanding the initial data using norma lized eigenfunctions of ˆH,ˆHˆψE=EˆψE:
+ˆΦ(t,x) =/summationdisplay
+EaE(t,x)ˆψE(x) (76)
+ˆΦ(0,x) =/summationdisplay
+Ea0
+EˆψE(x) (77)
+˙ˆΦ(t,x) =/summationdisplay
+E˙a0
+EˆψE(x). (78)
+Here the coefficients are obtained by integrating against the comple x conjugate of ˆψE(x), and the wave equation then
+reduces to an infinite set of ODEs:
+¨aE=−EaE
+˙aE(0) = ˙a0
+E:=/integraldisplay
+ˆψE˙ˆΦ(t= 0,x)dx (79)
+aE(0) =a0
+E:=/integraldisplay
+ˆψEˆΦ(t= 0,x)dx.
+whose solution is
+aE(t) =
+
+a0
+Ecos(√
+Et)+ ˙a0
+EE−1/2sin(√
+Et) ,E >0
+a0
+E+t˙a0
+E ,E= 0
+a0
+Ecosh(√
+−Et)+ ˙a0
+E(−E)−1/2sinh(√
+−Et),E <0(80)
+The above equations definethe evolution of the fields outside the domain of dependence of the in itial data. This
+same technique was applied, e.g., in [4, 15], in similar contexts. A subtle is sue is that of defining the domain
+D ⊂L2((0,∞),dx) where ˆHis self-adjoint. This problem is identical to that of quantum mechanic s on a half axis,
+treated in detail in the first reference in [22]. Consider the two dimen sional vector space of local (Frobenius) solutions
+of the eigenvalue equation ˆHˆψ=Eˆψ,ˆψ/negationslash= 0, nearx= 0. Given that an overall factor on ˆψis irrelevant, the space of
+local solutions can be regarded as a circle (this is why we used Acos(θ) andAsin(θ) for the two arbitrary constants
+in equations (16), (20), etc. θ∈[0,2π) labels points in this circle.)
+If any eigenfunction is square integrable near x= 0 we say, following [22], that ˆHbelongs to the “limit circle case”.
+In this case, ˆHwill be a self-adjoint operator only after restricting to a subspace Dθo⊂L2((0,∞),dx) of functions
+behaving near x= 0 as local eigenfunctions with a fixed θ=θo. Equations (76)-(80) will then hold in Dθo, for initial
+data in this space. Note, however, that initial data of compact sup port belongs to Dθforanyθ, and evolve in a
+different way if some θ′
+o/negationslash=θois chosen in (76)-(80). Of course, the solution will be different only outside the domain
+of dependence of the initial data , but still there is an ambiguity, which must be resolved. Physical inpu t must then
+dictate what the right choice of θis in order to get rid of this ambiguity.
+The other possibility is that the hamiltonian piece of the wave equation belongs to the “limit point case”, i.e., that
+there is a single θvalue giving local solutions which are square integrable near x= 0. In this case we say that ˆHis
+“essentially self-adjoint” (since it is only self-adjoint in the domain de fined by this particular θvalue) and there is no
+ambiguity in the dynamics. This would be the case if one of the roots of the indicial equation of of the Frobenius
+local solution of the hamiltonian eigenfunction were less than −1/2.15
+For the scalar gravitational perturbation problem, the situation is that of a limit circle hamiltonian. The self
+consistency condition (62), however, singles out a unique θ. With this choice, the degree of divergency as x→0+
+not only of R1, but also of allthe remaining algebraic invariants of the Riemann tensor, gets cont rolled, and, since
+the evolution (76)-(80) preserves the local behaviour at x= 0, the invariants will stay properly bounded near the
+singularity at later times. The dynamics is thus unambiguous once we e nforce the self consistency condition (62).
+In [4], a “zero mode” ( EI= 0) was used to construct the intertwiner and produce a self-adj ointˆH. The resulting
+hamiltonian has a negative energy eigenvalue, and thus exhibits the in stability of the spacetime. The Zerilli field
+is then recovered using equation (74), from where it is clear that th e exponential growing in time of ˆΦ shows up
+in the metric and electromagnetic field perturbations. We have tried this same approach here, and found that an
+appropriate zero mode can be explicitly constructed for ℓ= 2 and, as happens in the Schwarzschild naked singularity
+case.ˆHhas a smooth potential and contains a negative energy eigenvalue, at least for some Q/Mvalues for which
+rc< ri(see Section IIIC). Since generic perturbation initial data with com pact support in region III will have a
+nonzero projection onto the ℓ= 2 type-one scalar mode, this is certainly enough to show that such initial data will
+excite unstable mode in these cases.
+However, we were not able to prove that for arbitraryℓandQ/Msuch thatrc<rithere is a zero mode intertwiner
+which does not introduce a newsingularity in ˆV(although it is still trivial to show that any intertwiner washes out th e
+singularity at r=rc, see Appendix A.) A new singularity would be introduced if ψIhad a zero for r∈(0,rc)∪(rc,ri).
+For this reason, in Section IIID we exhibit an alternative intertwiner for which computations can be carried out
+explicitly for any ℓ. This uses ψI=χ+
+1, Chandrasekhar’s mode, and gives, for any Q/Mandℓ,ˆH=H−
+1(the
+Hamiltonian for type-1 vector perturbations !). Since ˆHis positive definite in this case, the hat wave equation is
+stable, and the scalar instability shows up only when reconstructing the Zerilli field using (74). This is so because
+those factors inside the integral which are exponential in tdo not cancel the exponential factors outside the integral
+(as it would happen if the original wave equation were stable). These factors will then show up in the metric and
+electromagnetic field perturbations, the Riemann tensor, and its in variants.
+C. Theℓ= 2zero mode intertwiner
+In [4], a“zeromode”(solution of Hψ= 0 that isnot necessarilynormalizableorwellbehavedat the bounda ries)was
+used to construct an intertwiner to deal with the initial value proble m for the scalarmode negativemass Schwarzschild
+perturbations, which has difficulties similar to those found in the pres ent case.
+We may try the same approach here, however, given the complexity of the potential in H+
+1, it is rather difficult to
+obtain the solutions of
+H+
+1ψ+
+o= 0 (81)
+forψI=ψ+
+orequired to construct the EI= 0 intertwiner I(64). One possibility is to use the relations (30)-(31) to
+obtain scalar zero modes from vector ones, since, for ψ−
+oavectorzero mode,
+H−
+1ψ−
+o= 0, (82)
+it follows from (30)-(31) that A1ψ−
+ois ascalarzero mode. Since the vector potential V−
+1is much simpler than V+
+1,
+there is some hope that we could carry on calculations in a more explicit way using this idea. This is indeed the case
+forℓ= 2, for which the general solution of equation (82) (see Appendix B ) can be shown to be
+ψ−
+o=Acos(α)/braceleftbigg
+r3+β2
+4(Q2−r2)−Q4
+r/bracerightbigg
++Asin(α)/braceleftBigg
+3/parenleftbigg
+4r−β2+4Q2
+r/parenrightbigg/parenleftBigg
+r2−Q2
+2/radicalbig
+M2−Q2/parenrightBigg/bracketleftBigg
+ln/parenleftBigg
+−r+M/radicalbig
+M2−Q2−1/parenrightBigg
+−ln/parenleftBigg
+−r+M/radicalbig
+M2−Q2+1/parenrightBigg/bracketrightBigg
+−/parenleftBigg
+12r2+3 (β1−2M)r+/parenleftbig
+3Mβ1−2(M2+2Q2)/parenrightbig
++12M3−2(β2−β1)/parenleftbig
+M2−Q2/parenrightbig
+r/parenrightBigg/bracerightBigg
+.(83)
+whereAandαare arbitrary constants. Note that the ℓ= 2 intertwiner operator constructed using ψI=ψ+
+o=A1ψ−
+o
+in (64) will depend on αbut certainly not on A. It can be easily shown, however, that ˆVis smooth at r=rc
+irrespective of the choice of α, the first and second order poles in V+
+1being canceled by the poles in dg/dx(Figure 4).16
+This, of course, is to be expected from the more general consider ations in Appendix A. The asymptotic behaviour of
+ˆVnear the singularity and the inner horizon is also independent of α:
+ˆV≃/braceleftBigg4
+9x2+... ,x ≃0
+Cexp/parenleftBig
+−(ro−ri)x
+r2
+i/parenrightBig
+,x→ ∞(84)
+Near the singularity the eigenfunctions of ˆHbehave as
+ˆψ=Acos(θ)[x−1/3/summationdisplay
+n≥0a(1)
+nxn/3]+Asin(θ)[x4/3/summationdisplay
+n≥0a(2)
+nxn/3] (85)
+ThusˆHbelongs to the limit circle case, and only restricting to a subspace Dθoof functions behaving as (85) with a
+fixedθ=θovalue does ˆHbecomes self-adjoint.
+We will make the choice θ=π/2 of slowest decaying functions. This condition is certainly preserve d by the hat wave
+equation (see (76)), and implies
+ˆIˆΦ =/summationdisplay
+k≥1akrk, a2=β2
+4Q2a1, (86)
+nearr= 0. If Φ+
+1(0,r),˙Φ+
+1(0,r) also admit expansions like those in (86), then, using (74) follows tha t
+Φ+
+1(t,r) =/summationdisplay
+k≥1ak(t)rk, a2(t) =β2
+4Q2a1(t), (87)
+for allt, a condition that can be shown to guarantee that allalgebraic invariants of the Riemann tensor behave
+properly near the singularity.
+Regarding the choice of αin (83), although the results do not depend on the intertwiner that we use, it is certainly
+easier to understand how the instability is excited by an initially compac tly bounded perturbation if we use
+tan(α) =2Q2β2/radicalbig
+M2−Q2
+3Q2β2ln/parenleftbigg
+M−√
+M2−Q2
+M+√
+M2+Q2/parenrightbigg
++2(3Mβ2−16(M2−Q2))/radicalbig
+M2−Q2, (88)
+since in this case the resulting intertwiner will send χ+
+1onto the the Hilbert space Dπ/2selected by the self consistency
+argument, and thus Iχ+
+1will be one of the eigenfunctions of ˆH(it will actually be the only negative energy ˆH
+eigenfunction). The transformed potential ˆV, together with Iχ+
+1for this choice are given in Figure 4 below for some
+specificQandMvalues. An explicit expression for ˆVcan be readily obtained using equations (64), (67), ψo=A1ψ−
+o,
+(83) and (88).
+Now suppose some perturbation data (Φ+
+1(t= 0,x),˙Φ+
+1(t= 0,x)) of compact support is given. The hat wave
+equation data ( IΦ+
+1(t= 0,x),I˙Φ+
+1(t= 0,x)) will be of compact support and then it will belong to Dπ/2. Expanding
+it using (77)-(78) will generically give a nonzero projection onto the fundamental, unstable ˆHeigenfunction Iχ+
+1,
+and thus, from (80), an exponentially growing term in (76), which su rvives when Φ+
+1is reconstructed using (75) and
+shows up in the metric and electromagnetic field perturbations.
+The use of a zero mode intertwiner has some drawbacks: although w e can show that the kinematic singularity is
+absent from ˆV(see Appendix B), we do not have a complete proof, even for ℓ= 2, that the zero mode ψI=ψ+
+ohas
+no zeroes in (0 ,rc)∪(rc,ri), which would introduce new singularities in ˆV. However, for ℓ= 2, we have numerically
+verified that this is the case for a range of values of Q/M. A particular example of a smooth ˆVforℓ= 2 is that given
+in Figure 4.
+In the following Section, we show that all these issues can be avoided by using an alternative intertwiner that allows
+explicit calculations for every harmonic number and charge and mass values.17
+±100±50050100
+ 
+0.2 0.4 0.6 0.8 1
+x
+FIG. 4:ℓ= 2 potential ˆV-continuous line- for the transformed Zerilli equation (67 ), and the transformed of the unstable mode
+χ+
+1. In this example ri= 1 andro= 2.
+D. Intertwining using Chandrasekhar’s algebraic special m ode
+We can apply the intertwining technique using Chandrasekhar’s algeb raic special mode χ+
+1in (64), for which
+EI=−/parenleftbiggκ
+2β1/parenrightbigg2
+. (89)
+Using this intertwiner has a number of advantages. Contrary to wh at happens for the zero mode, we have an explicit
+expression for χ+
+1for every harmonic number ℓ, equation (26). It is also simple to construct ˆVusing (7) and (25) (a
+prime denotes derivative with respect to x):
+ˆV=V+
+1−2/parenleftBigg
+χ+
+1′
+χ+
+1/parenrightBigg′
+=V+
+1−/parenleftbigg
+β1f1+κ
+2β1/parenrightbigg′
+=V−
+1=f
+r4/parenleftbig
+ℓ(ℓ+1)r2−β1r+4Q2/parenrightbig
+(90)
+(This relation between the scalar and vector modes was first notice d by Chandrasekhar [9, 10].) The fact that
+ˆV=V−
+1, the type-1 vector potential, simplifies the analysis considerably, s ince it is clear that V−
+1is smooth in (0 ,ri)
+foranyvalue ofQ/M.
+From the first line in (20) follows that ˆH=H−
+1belongs to the limit circle case. As explained above, a choice θohas
+to be made to fix the domain Dθo⊂L2((0,∞),dx) where ˆHis self adjoint. However, for type-1 scalar perturbation
+the consistency requirement (62) (see also Lemma 1) reduces to:
+/vextendsingle/vextendsingle/vextendsingleIΦ+
+1/vextendsingle/vextendsingle/vextendsingle≤N,if 0<r<δ (91)
+which rules the x−1/3piece of (20), and forces θ=π/2. Once again, this condition is preserved by the hat wave
+equation (see (76)), and imply (86) and (87) for initial data satisfy ing this condition (where now Ihas to be
+understood as the intertwiner made using Chandrasekhar’s mode) Condition (87) guarantees that allalgebraic
+invariants of the Riemann tensor behave properly near the singular ity at later times.
+SinceˆV(and thus ˆH) is positive definite, the hat wave equation is stable, and its modes os cillate in time. The insta-
+bility of Φ+
+1arises as a consequence of the fact that, generically, the expone ntial terms in the integrand of (74) do not
+cancel out with those outside the integral. As a trivial example, tak e Φ+
+1(t= 0,x) and˙Φ+
+1(t= 0,x) both proportional
+toχ+
+1(this certainly passes all the requirements in Lemma 1). Since Iχ+
+1= 0, the initial data in hat space is trivial,
+thenˆΦ(t,x) = 0forall t, and(74) reducesto the lasttwoterms, whicharegenericallyexpo nentiallygrowingforlarge t.
+A final observation is the nontrivial fact that, unlike the Zerilli field, the intertwined variable has a geometrical
+significance, as, from (24), it gives the first order variation of the Riemann invariant R1:
+IΦ+
+1=ˆΦ =−r8
+4Q2δR1 (92)18
+This gives further significance to the dual relation between vector and scalar modes first found by Chandrasekhar,
+which was limited to some observations on their mode spectra. Equat ions (24) and (90) prove that the field giving
+the first order variation of R1(timesQ4/r8) associated to a scalarmode perturbation is a solution of the vectormode
+perturbation equation of the same harmonic number.
+IV. CONCLUSIONS
+We have proved that the inner static region 0 < r < riof a Reissner-Nordstr¨ om black hole is unstable under
+linear perturbations of the metric and electromagnetic field. More p recisely, we have shown that a perturbation with
+compact support within this region will excite unstable type-1 polar m odes, of which there is one for every harmonic
+number (ℓ,m). This instability is relevant to the strongcosmic censorship conjec ture, accordingto which this regionof
+the black hole, lying beyond the Cauchy horizon (of a Cauchy surfac e like the one in Figure 1), should be disregarded,
+as it could not arise as a result of the collapse of ordinary matter dep arting from spherical symmetry.
+This result has implications on some simple models of halted collapse of a p ressure-less charged perfect fluid star [25],
+according to which the world tube of the surface of the star trace s a path going from the (right copy of) region I in
+Figure 1, through region II, into the leftcopy of region III, upper copy of region II, then upper right copy of region
+I. To the right of this curve, this spacetime agrees with the extend ed Reissner-Nordstr¨ om spacetime, which contains
+an entire copy of region III, thus being unstable. We are currently studying these models in more detail.
+The difficulty in establishing in a rigorouswaythe Reissner-Nordstr¨ o minstability lies in the fact that the field variable
+which succeeds in disentangling the linearized equations, the Zerilli fie ld Φ+
+1, happens to be a singular function of
+the perturbed fields in region III. This cannot be cured by any simple field redefinition, but requires the use of an
+intertwiner operator I=∂/∂x+gthat maps to a smooth field ˆΦ :=IΦ+
+1. The information lost due to the nontrivial
+kernel of Iis entirely contained in the initial data, and thus is available. The evolut ion of perturbations on the non
+globally hyperbolic backgroundis well defined by using the spectral t heorem and a unique self-adjoint extension of the
+spatial piece of the wave operator that gives the dynamics of the ˆΦ field. We should comment here that intertwiners
+in the context of linear perturbations were first considered in [24], w hile they were first used to deal with the issue of
+the singularities of the Zerilli field in the proof of the instability of the S chwarzschild naked singularity in [4]. The
+idea of defining dynamics on non globally hyperbolic backgrounds by us ing a suitable self adjoint extension of the
+spatial piece of the wave equation together with the spectral the orem was first suggested in [15]. The main difference
+between the cases considered in [15], and the Reissner-Nordstr¨ o m and negative mass Schwarzschild cases, lies in the
+fact that the spatial operators (“Hamiltonian”) in the last two cas es are not positive definite. A difference between
+the negative mass Schwarzschild and the Reissner-Nordstr¨ om ca ses, is that the intertwiner used in the first case gives
+a Hamiltonian with a single self-adjoint extension (limit point case in [22]), whereas the one for Reissner-Nordstr¨ om
+corresponds to the limit circle case in [22].
+Two different intertwiners were used: one constructed out of a ze ro mode, the other using one of Chandrasekhar’s
+algebraic special modes. The first intertwiner has the advantage o f exhibiting the instability in a rather obvious
+way, and the drawback that we lack explicit expressions for the inte rwtined potential, or a proof of its smoothness
+within the relevant parameterrange. The intertwiner that uses Ch andrasekhar’s“hides” the instability, which is made
+explicit in the metric reconstruction process through the original Z erilli field. This mode allows explicit calculations
+for every harmonic number and QandMvalues. It also exhibits a very interesting connection between vect or and
+scalar modes: the intertwined field ˆΦ gives the first order variation of the Riemann invariant R1(see equation (92).
+An alternative way of stating this is that the first order variation δR1ofR1underscalarperturbations is a solution
+of the Zerilli vectorperturbation equation.
+The results presented here adapt easily to the case of a super-ex treme (|Q|>M) Reissner-Nordstr¨ om spacetime, and
+thus can be used to fill in the details left untreated in [2] to show that a perturbation of an overcharged Reissner-
+Nordstr¨ om spacetime, compactly supported away from the singu larity, will excite modes that grow exponentially in
+time. This, of course, is relevant to weak cosmic censorship, this be ing the original motivation for our work.
+Acknowledgements
+ThisworkwassupportedbygrantsPIP112-200801-02479fromC ONICET(Argentina), Secyt05/B384and05/B253
+from Universidad Nacional de C´ ordoba (Argentina), and a Partne r Group grant from the Max Planck Institute for
+Gravitational Physics, Albert-Einstein- Institute (Germany). RJ G and GD are supported by CONICET. GD wishes
+to thank the AEI, where part of this work was done, for hospitality and support.19
+Appendix A: Intertwined potential
+In this section we analyze the local behaviour of the intertwined pot ential and ˆψgiven in equations (64),(65) and
+(67) as the singularity, inner horizon, and r=rcare approached. This is done by studying the behaviour of generic
+solutions of the equation
+H+
+1ψI=−fd
+dr/parenleftbigg
+fdψI
+dr/parenrightbigg
++V+
+1ψI=EIψ+
+1 (A1)
+forr≃0,r≃rc, andr≃ri.
+Note thatV+
+1may be written as,
+V+
+1=f
+r(β1+(ℓ−1)(ℓ+2)r)2/bracketleftbigg/parenleftbigg
+κ+β1df
+dr/parenrightbigg
+(β1+(ℓ−1)(ℓ+2)r)−2f(r)β1(ℓ−1)(ℓ+2)/bracketrightbigg
+(A2)
+showing explicitly the double pole at,
+r=rc=−β1
+(ℓ−1)(ℓ+2). (A3)
+1. Behaviour of ˆV
+Nearr=rc, the general solution of (A1) is of the form,
+ψI=1
+r−rc/bracketleftbigg
+c0−(ℓ−1)2(ℓ+2)2(2M−β1)c0
+β1(2(ℓ−1)(ℓ+2)M+(ℓ2+ℓ+2)β1)(r−rc)
++8EIβ2
+1c0
+(2(ℓ−1)(ℓ+2)M+(ℓ2+ℓ+2)β1)2(r−rc)2+c3(r−rc)3+c4(r−rc)4+.../bracketrightBigg
+(A4)
+wherec0andc3are arbitrary constants, and c4and higher coefficients in the series are linear combinations of c0
+andc3with coefficients that depend on EI,Q,Mandℓ. Note that the generic local eigenfunction above satisfies the
+requirement (61).
+If we use the generic ψIgiven above to construct the potential /hatwideV,
+/hatwideV=V−2fd
+dr/parenleftbiggf
+ψId
+drψI/parenrightbigg
+(A5)
+a straightforward computation shows that, provided a0/negationslash= 0, nearr=rc,
+/hatwideV=8k2β4
+1+(ℓ+2)3(ℓ−1)3[(ℓ2+ℓ+4)β2
+1−20Mβ1−12(ℓ+2)(ℓ−1)M2]
+4β4
+1+O(r−rc). (A6)
+This means that, for anyEI, anarbitrary solution of (A1) with a0/negationslash= 0 gives an intertwined potential /hatwideVthat is
+smooth atr=rc. We notice also that, as can be checked, if a0= 0, the second term in the R.H.S. of (A5) does not
+compensate the double pole in V, so that/hatwideVis also singular.
+We consider next the local behaviour near r= 0. The general solution of (A1) admits an expansion in powers of r
+of the form,
+ψI(r) =a1r+a2r2+/bracketleftbigg(ℓ−1)(ℓ+2)((ℓ−1)(ℓ+2)Q2+Mβ1)
+Q2β2
+1a1+M
+Q2a2/bracketrightbigg
+r3+a4r4+... (A7)
+wherea1, anda2are arbitrary constants and the higher order coefficients depend linearly on them. A dependence on
+EIappears first at order r7. This result implies that, near r= 0, assuming a1/negationslash= 0, we have,
+/hatwideV=4
+9x−2−2
+35/3/parenleftbiggM
+Q4/3−2a2Q2/3
+a1/parenrightbigg
+x−5/3+O(x−4/3) (A8)20
+while, ifa1= 0,
+/hatwideV=10
+9x−2+32/3
+108/parenleftbigg10ℓ(ℓ+1)−12)
+Q2/3+M(28M−5β1)
+Q8/3/parenrightbigg
+x−4/3+O(x−1) (A9)
+Finally we consider the behaviour near r=ri. The cases EI/negationslash= 0 andEI= 0 require separate treatment. For
+EI/negationslash= 0, since the potential vanishes for r=ri, the leading terms of the two linearly independent parts of the solut ion
+for realEIare of the form,
+ψI=C1(ri−r)
+r2
+i√EI
+ro−ri
+
++C2(ri−r)
+−r2
+i√EI
+ro−ri
+
+=˜C1exp/parenleftBig/radicalbig
+EIx/parenrightBig
++˜C2exp/parenleftBig
+−/radicalbig
+EIx/parenrightBig
+(A10)
+whereC1,C2,˜C1, and˜C2are constants. For EI= 0, on the other hand, the general solution admits an expansion of
+the form,
+ψ+
+1(r) =a0+/parenleftbig
+(ri−ro)β1+κr2
+i/parenrightbig
+a0−ri/parenleftbig
+β1+(ℓ+2)(ℓ−1)((2ℓ2+2ℓ−1)ri+2ro)/parenrightbig
+b0
+ri(ri−ro)(β1+ri(ℓ+2)(ℓ−1))(r−ri)
++a2(r−ri)2+... (A11)
++ln(ri−r)/bracketleftBigg
+b0+/parenleftbig
+(ri−ro)β1+κr2
+i/parenrightbig
+b0
+ri(ri−ro)(β1+ri(ℓ+2)(ℓ−1))(r−ri)+b2(r−ri)2+.../bracketrightBigg
+.
+wherea2,b2and higher order coefficients depend linearly on a0andb0.
+TheEI= 0 transformed potential behaves as
+/hatwideV(r) =2(ro−ri)2b2
+0
+(a0+b0ln(ri−r))2r4
+i+... (A12)
+where the dots indicate terms that vanish as a ( ri−r). In terms of x, forb0/negationslash= 0, this implies,
+/hatwideV(x) =2
+x2+... (A13)
+2. Behaviour of ˆψ
+We consider now the behaviour of the intertwined field /hatwideψ(64), which can be written as
+/hatwideψ=Iψ=fψId
+dr/parenleftbiggψ
+ψI/parenrightbigg
+. (A14)
+We will use the following result, whose proof is straightforward:
+Lemma 2: Assume that ψandψIadmit a Laurent expansion
+ψ= (r−ro)p/summationdisplay
+k≥0ak(r−ro)k, ψI= (r−ro)p/summationdisplay
+k≥0aI
+k(r−ro)k(A15)
+wherea0= 1 =aI
+0, andpis any integer number. If sis the highest number for which ak=aI
+kfor everyk≤s(s
+measures the degree of contact of these functions at ro, and, generically, s= 0), then
+ψId
+dr/parenleftbiggψ
+ψI/parenrightbigg
+= (r−ro)p/summationdisplay
+k≥sdk(r−ro)k, ds=s(as+1−aI
+s+1). (A16)
+Consider first the action of the intertwiner on a function ψsatisfying (58) and (61). Since, as follows from (A4),
+ψIsatisfies this same condition, generically ψandψIhave (as functions of r) degree of contact s= 2, in the notation
+of Lemma 2, and thus ˆψ=fψId
+dr/parenleftBig
+ψ
+ψI/parenrightBig
+will be smooth at r=rc.
+The local solutions of H+
+1ψ=Eψ, are of the form ψ=/summationtext
+k≥1akrkwitha2anda1arbitrary,akindependent of Eup
+tok= 7. Consider the action of Ion a function like this further subject to the condition (86). If the intertwiner also
+satisfies (86), the degree of contact will be s= 5, thenψId
+dr/parenleftBig
+ψ
+ψI/parenrightBig
+will beO(r6), and thus ˆψwill beO(r4) =O(x4/3).21
+Appendix B: The scalar and vector zero modes.
+In this Appendix we describe a procedure that allows the construct ion of the zero mode solutions for both vector
+and scalar modes. We start with the vector zero modes by first con sidering the differential equation they satisfy. In
+accordance with (6, 7), and (9) with ω= 0, this is given by,
+−f2d2ψ−
+0
+dr2−fdf
+drdψ−
+0
+dr+f
+r4/parenleftbig
+ℓ(ℓ+1)r2−β1r+4Q2/parenrightbig
+ψ−
+0= 0 (B1)
+We notice that (B1) has regular singular points for r= 0,r=riandr=ro, and no other singularity. From now
+on we will write simply ψforψ−
+0.
+A simple analysis of the indicial equation shows that near r= 0 this equation has two independent solutions, one
+behaving as r−1, and the other as r4, both admitting a power series expansion. We consider therefore a n expansion
+forψ−
+0(r) of the form,
+ψ−
+0(r) =1
+r∞/summationdisplay
+i=0airi(B2)
+Replacing in (B1) we find that we must set a2= 0, and,
+a1=−β2
+4Q2a0;a3=ℓ(ℓ+1)β2
+24Q4a0;a4=ℓ(ℓ2−1)(ℓ+2)
+24Q4a0 (B3)
+The coefficient a5can be chosen arbitrarily, in accordance with the indicial equation, a nda6is given by,
+a6=−ℓ(ℓ2−1)(ℓ2−4)(ℓ+3)
+144Q6a0−β1−16M
+6Q2a5 (B4)
+For the remaining coefficients, including a6, we find a three term recursion relation of the form,
+−(ℓ+j−2)(ℓ−j+3)aj−1+(β1−2(j−1)(j−3)M)aj+(j+1)(j−4)Q2aj+1= 0 (B5)
+and, therefore, all the coefficients are determined once a0, anda5are given. But this implies that for any given ℓ, and
+a0/negationslash= 0, we may choose a5such thataℓ+3= 0, and then all coefficients for j≥ℓ+3 vanish. Calling ψathis solution
+we have,
+ψa(r) =1
+rPℓ(r) (B6)
+wherePℓ(r) is a polynomial of order ℓ+2. The lowest order polynomials are,
+P2(r) = 1−β2
+4Q2r+β2
+4Q4r3−1
+Q4r4
+P3(r) = 1−β2
+4Q2r+β2
+2Q4r3−5
+Q4r4+30
+(β2+10M)Q4r5(B7)
+P4(r) = 1−β2
+4Q2r+5β2
+6Q4r3−15
+Q4r4+21(24M+β2)
+4(3Q2+Mβ2+6M2)Q4r5−42
+(3Q2+6M2+Mβ2)Q4r6
+(B8)
+where we have fixed, for simplicity, Pℓ(r= 0) = 1. With this normalization, the polynomials are positive and
+decreasing functions of rnearr= 0. We shall now prove that they have no zeros in the interval 0 ≤r≤ri. We write
+(B1) in the form,
+d2ψa
+dr2=(2riro−(ri+ro)r)
+r(ro−r)(ri−r)dψa
+dr+(ℓ(ℓ+1)r2−β1r+4riro)
+r2(ro−r)(ri−r)ψa (B9)
+and notice that ψacan have only simple zeros in 0 <r<ribecause the coefficients in (B9) are regular functions of r
+in 0<r<ri. Since sufficiently near r= 0 we have ψa>0 anddψa/dr<0, and the coefficients on the RHS in (B9)
+are both positive, the sign of d2ψa/dr2is not fixed. We notice however that at the first zero of ψaforr>0 we must
+havedψa/dr<0, and therefore, we also have d2ψa/dr2<0. Since to the right of such a zero, and as long as r<ri,22
+(since the coefficients are still positive) we must have both dψa/dr <0, andd2ψa/dr2<0, and therefore ψa<0,
+there can be no other zero for r<ri. This proves that there is at most one zero for r∈(0,ri).
+But now we notice that near r=ri, equation (B9) has a singular solution (diverging as ln( ri−r)) and a unique
+regular solution of the form,
+ψ(r) =ψ(ri)/bracketleftbigg
+1−(4ro+ℓ(ℓ+1)ri−β1)
+ri(ro−ri)(r−ri)+O(r−ri)2/bracketrightbigg
+(B10)
+Sinceψais regular, it has the form (B10), and this implies that close to r=rithe regular solution and its first
+derivative have opposite signs. This contradicts the result obtaine d under the assumption that there is a zero in
+0<r<ri. We conclude that ψadoes not vanish for r∈(0,ri).
+Similarly, we find that near r=ro, equation (B9) has a singular solution (diverging as ln( ro−r)) and a unique non
+vanishing regular solution. This implies that Pℓ(r) cannot vanish for r=ro.
+We note in passing that for the extreme case Q=M, whereri=ro, we have the exact (regular at the horizon)
+solutions,
+ψa(r) =C(ℓr+2M)(r−M)(ℓ+1)
+r(B11)
+whereCis a constant.
+Going back to (B1), for any fixed ℓ, given the solution (B6), a linearly independent solution is given by,
+ψb(r) =C1
+rPℓ(r)/integraldisplayr
+0y4
+(y2−2My+Q2)(Pℓ(y))2dy (B12)
+whereCis a constant. It is easy to check that ψbis regular and non vanishing in 0 <r<ri, and,
+ψb(r)∼C1r4;r→0+
+ψb(r)∼C2ln(ri−r) ;r→ri−(B13)
+whereC1andC2are constants. We may obtain and expansion of this solution in power s ofrusing (B12), or directly
+from (B1),
+ψb(r) =r4+(16M−β1)r5
+6Q2+5((ℓ2+ℓ−8)Q2+4M(12M−β1))r6
+42Q4
++((ℓ(ℓ+1)(36M−β1)+15β1−360M)Q2+30M2(32M−3β1))r7
+84Q6+... (B14)
+where we have set an arbitrary multiplicative constant so that the c oefficient of r4is equal to one.
+Forℓ >2 the integrals in (B12) cannot be computed directly, because that would require explicit expressions for
+the zeros of polynomials of degree larger that 4. We may, neverthe less, infer their general form as follows. We first
+notice that Pℓ(r) may be written in the form,
+Pℓ(r) =/producttextℓ+2
+k=1(r−rk)
+/producttextℓ+2
+k=1(−rk)(B15)
+whererkare the zeros of Pℓ(r), which, as indicated are simple. Therefore, since y2−2My+Q2= (y−ri)(y−ro),
+we should have,
+/integraldisplayr
+0y4
+(y2−2My+Q2)(Pℓ(y))2dy=Aln(ri−r)+Bln(ro−r)+C
++ℓ+2/summationdisplay
+k=1ailn(r−rk)+ℓ+2/summationdisplay
+k=1bi
+(r−rk)(B16)
+whereA,B,C,akandbkare constants that depend on M,Q, andrk. The last term in (B16) may be written in the
+form,
+ℓ+2/summationdisplay
+k=1bi
+(r−rk)=Uℓ(r)
+Pℓ(r)(B17)23
+whereUℓ(r) is a polynomial of order ℓ−1, or lower. Replacing in (B12),
+ψb(r) =1
+rPℓ(r)(Aln(ri−r)+Bln(ro−r)+C)+1
+rUℓ(r)
++1
+rPℓ(r)ℓ+2/summationdisplay
+k=1ailn(r−rk) (B18)
+But we notice that ψb(r) is a solution of (B1), which can be singular only at the regular singular pointsr= 0,r=ri,
+andr=ro, and that the zeros rkdo not coincide with these points. Therefore, we must have ak= 0 for allk, and the
+last term in (B18) vanishes identically. This result implies that, for any ℓ, we may construct algebraically the solution
+(B12) as follows. We first compute the coefficients of Pℓ(r) as indicated above, and then replace in (B18) leaving A,
+B,Cand the coefficients of Uℓarbitrary. Next we replace in (B1) and impose the condition that ψis a solution of
+that equation, and that ψ∼r4nearr= 0. It can be checked that this procedure determines all the coeffi cients, up
+to an arbitrary multiplicative constant, a simple example being (83) fo rℓ= 2. Since by construction these solutions
+satisfy the appropriate boundary condition at r= 0, the construction of the corresponding scalarzero modes, and the
+associated intertwining potential, is now a simple algebraic procedure . The resulting expressions are, unfortunately,
+very long and rather difficult to analyze in detail. In particular, we hav e not been able to show explicitly that for
+0<rc<rithe scalar zero modes that satisfy the required boundary conditio n atr= 0 are non vanishing everywhere
+in the interval 0 <r <ri, as required for the regularity of the intertwining potential. We rem ark, nevertheless, that
+this appears to be the case in all the particularsolutions analyzed nu merically after assigning definite numerical values
+for the parameters, as in the example described in Section III - C.
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+arXiv:1001.0153v1  [math.AG]  31 Dec 2009Effective descent for differential operators
+Elie Compoint∗, Marius van der Put†, Jacques-Arthur Weil‡
+Dedicated to the memory of Jerry Kovacic.
+September 2009
+Abstract
+A theorem of N. Katz [Ka] p.45, states that an irreducible dif -
+ferential operator Lover a suitable differential field k, which has an
+isotypical decomposition over the algebraic closure of k, is a tensor
+product L=M⊗kNof an absolutely irreducible operator Moverk
+and an irreducible operator Noverkhaving a finite differential Galois
+group. Using the existence of the tensor decomposition L=M⊗N,
+an algorithm is given in [C-W], which computes an absolutely irre-
+ducible factor FofLover a finite extension of k. Here, an algorithmic
+approach to finding MandNis given, based on the knowledge of F.
+This involves a subtle descent problem for differential opera tors which
+can be solved for explicit differential fields kwhich are C1-fields.
+1 Introduction
+Cdenotes an algebraically closed field of characteristic 0 and the differ ential
+fieldkis a finite extension of ( C(z),∂=d
+dz). The algebraic closure of k
+will be written as k. LetL∈k[∂] be a (monic) differential operator. The
+∗D´ epartement de math´ ematiques, Universit´ e de Lille I, 59655 , Ville neuve d’Ascq
+Cedex, France. compoint@math.univ-lille1.fr
+†University of Groningen, Department of Mathematics P.O. Box 407, 9700 AK Gronin-
+gen, The Netherlands. mvdput@math.rug.nl
+‡XLIM, D´ epartement de Math´ ematiques et Informatique, Univer sit´ e de
+Limoges, 123 avenue Albert Thomas, 87060 Limoges Cedex, France .
+jacques-arthur.weil@unilim.fr
+1operatorLis called irreducible if it does not factor over kand absolutely ir-
+reducible if it does not factor over k. Here we are interested in the following
+special situation:
+Lis irreducible and Lfactors over kas a product F1...Fsofs>1equivalent
+(monic) absolutely irreducible operators.
+There are algorithms for factoring Loverk, i.e., as element of k[∂] ([H1,
+H2, vdP-S]). Algorithms for finding factors of order 1 in k[∂] are proposed in
+[S-U, H-R-U-W, vdP-S]. An algorithm for finding factors of arbitrar y order
+ink[∂] is given in [C-W].
+According to N. Katz [Ka], Proposition 2.7.2, p.45, the assumption on L
+implies that Lis a tensor product M⊗Nof monic operators in k[∂] such
+thatMis absolutely irreducible and the irreducible operator Nhas a finite
+differentialGaloisgroup(orequivalentlyallitssolutionsarealgebraic overk).
+We will present a quick, down-to-earth proof of this in terms of diffe rential
+modules over k. Further we note that the converse statement is immediate
+becauseNdecomposes over kas a “direct sum” (least common left multiple)
+of operators ∂−f′
+fwithf∈k∗.
+The absolute factorization algorithm in [C-W] uses the existence of t his
+tensor decomposition to ensure its correctness but, in full gener ality, the
+problem of computing MandNis yet left open in there. For MorNof
+small order, methods for detecting and computing MandNare given in
+[C-W, N-vdP] and particularly in [H3] where additional references ca n be
+found. We will illustrate this (and propose another method) at the e nd of
+the paper.
+The problem which we address to produce the operators M,N∈k[∂] by
+some decision procedure for MandNof arbitrary order. It follows from
+L=M⊗NthatF1∈k[∂] is equivalent to M(i.e.,F1descends to k). Let
+K1⊃kbe the smallest Galois extension such that F1∈K1[∂] (or equiva-
+lentlyK1is the field extension of kgenerated by all the coefficients of all Fi).
+One might think that the equivalence between F1andM, seen as elements of
+K1[∂], takes place over K1. However, in general, a (finite) extension K′⊃K1
+is needed for this equivalence. The title of this paper refers to this d escent
+problem1.
+1Techniques for arithmetic descent were proposed in [H-P], where the case of a differ-
+ential field kwith non-algebraically closed constant field is handled
+2Our method for finding K′is as follows. First the smallest extension
+K⊃K1is computed which guarantees that the factors F1,...,F sare equiv-
+alent over the field K. Using these equivalences, a certain 2-cocyle cfor
+Gal(K/k) with values in C∗, i.e. the obstruction for the descent of F1, is
+computed. Since kis aC1-field, the 2-cocycle cbecomes trivial over a finite
+(cyclic) computable extension K′⊃K: we give a construction of K′in
+section 3.1.2, particularly part (4) of remark 3.4 which produces firs t order
+operators having the same obstruction to descent and for which t he problem
+can be solved. Finally, once K′is found, the computation of M,Nis easily
+completed.
+In the sequel we will use differential modules because these are mor e
+natural for the problem. A translation in terms of differential oper ators is
+presented at the moment that actual algorithms are involved since the latter
+are frequently phrased in terms of differential operators.
+Thefollowingnotationisused. Thetrivial1-dimensionaldifferentialm od-
+ule over a field KisKewith∂e= 0. This module will be denoted by 1or
+1K. For a differential module AoverKof dimension aone writes det Afor
+the 1-dimensional module ΛaA.
+2 A version of Katz’ theorem
+In the proof we will use the following notion of twist of a differential module .
+LetGal(K/k) be the Galois group of any Galois extension Kofk(finite or
+infinite). Let Abeadifferential moduleover Kandσ∈Gal(K/k). Thetwist
+σAis equal toAas additive group, has the same ∂asA, but its structure as
+K-vector space is given by λ∗a:= (σ−1λ)·aforλ∈K, a∈A.
+The elements σ∈Gal(K/k) act in a natural way on K[∂] by the for-
+mulaσ(/summationtext
+nan∂n) =/summationtext
+nσ(an)∂n. If one presents AasK[∂]/K[∂]F(withF
+monic), thenσA=K[∂]/K[∂]σ(F).
+An isomorphism φ(σ) :σA→Acan also be interpreted as a C-linear
+bijection Φ( σ) :A→A, commuting with ∂and such that Φ( σ)(λa) =
+σ(λ)·Φ(a). In other words Φ( σ) isσ-linear isomorphism.
+Proposition 2.1 LetLbe a differential module over k. Suppose that:
+(1). The field of constants Cofkis algebraically closed, khas characteristic
+3zero andkis aC1-field.
+(2).Lis irreducible and L:=k⊗kLdecomposes as a direct sum ⊕s
+i=1Aiof
+isomorphic irreducible differential modules over k.
+Then there are modules M,Noverksuch thatL∼=M⊗kN,Mis
+absolutely irreducible and the irreducible module Nhas a finite differential
+Galois group. The pair (M,N)is unique up to a change (M⊗kD,N⊗kD−1)
+whereDhas dimension 1 and D⊗tis the trivial module for some t≥1.
+Proof.WriteA=A1and letGaldenote the Galois group of k/k. For any
+σ∈Gal, the twisted moduleσAis a submodule ofσ(k⊗kL). As the latter
+moduleisisomorphicto k⊗kL, thereisaσ-linearisomorphismΦ( σ) :A→A.
+This induces a 2-cocycle cforGalwith values in C∗, defined by Φ( στ) =
+c(σ,τ)·Φ(σ)·Φ(τ). Sincekis aC1-field, the 2-cocycle is trivial ([Se], II-9,
+§3.2). Aftermultiplying theisomorphisms {Φ(σ)}bysuitableelements in C∗,
+one obtains descent data {Φ(σ)|σ∈Gal}satisfying the descent condition
+Φ(στ) = Φ(σ)·Φ(τ) for allσ,τ∈Gal(k/k). DefineM:={a∈A|Φ(σ)a=
+afor allσ∈Gal}. It is easily verified that Mis a differential module over k
+and that the canonical morphism M:=k⊗kM→Ais an isomorphism.
+Consider now Hom ∂(M,L). This is a vector space, isomorphic to Cs
+and provided with an action of Gal. Thenk⊗CHom∂(M,L) is a trivial
+differential module over kprovided with an action of Gal. It is a submodule
+of the differential module Hom( M,L). Taking invariants under Galone
+obtains a differential module
+N:= (k⊗CHom∂(M,L))Galoverkwhich is a submodule of Hom( M,L).
+The canonical morphism k⊗N→k⊗CHom∂(M,L) is an isomorphism and
+thus the differential Galois group of Nis finite.
+The canonical morphism of differential modules M⊗kHom(M,L)→L,
+namelym⊗ℓ/ma√sto→ℓ(m), can be restricted to a morphism f:M⊗kN→L.
+By construction the induced morphism M⊗kN→Lis an isomorphism and
+thus so isf.
+The descent data {Φ(σ)|σ∈Gal}forAare not unique. They can be
+changed into {h(σ)Φ(σ)|σ∈Gal}whereh:Gal→C∗is any continuous
+homomorphism. We note that the image of his a subgroup µtof thet-th
+roots of unity for some t. Consider the trivial differential module kewith
+∂e= 0 and with Galaction given by σe=h(σ)efor allσ∈Gal. By taking
+the invariants under Galone obtains a 1-dimensional module Doverksuch
+4that the canonical morphism k⊗D→keis an isomorphism and respects
+the actions of Gal. FurtherD⊗tis the trivial differential module 1.
+Thedifferential moduleobtainedwiththenewdescent datacanbese ento
+beM⊗kDand thusNwill be changed into N⊗D−1. From this observation
+the last statement of the proposition follows. ✷
+Remarks 2.2 The non unicity of the pair ( M,N) can be restricted by the
+condition that det N=1. Then only a change ( M⊗D,N⊗D−1) is possible
+withD⊗s=1.
+3 Algorithmic approach
+First we make the relation between differential modules and different ial oper-
+ators explicit. Let Abe a differential module and a∈Aa cyclic vector. One
+associates to this the monic differential operator op(A,a)∈k[∂] of minimal
+degree satisfying op(A,a)a= 0. The morphism k[∂]→A, which maps 1 to
+a, induces an isomorphism k[∂]/k[∂]op(A,a)→A.
+LetB⊂Abe a submodule. This yields a factorization op(A,a) =LR
+withR ∈k[∂] is the monic operator of minimal degree such that Ra∈B.
+One observes that R=op(A/B,a+B).
+FurtherL ∈k[∂] is the operator of minimal degree satisfying Lb= 0,
+whereb:=Ra. Clearlybis a cyclic vector for BandL=op(B,b).
+Moreover, any factorization op(A,a) =LRwith monic L,Rcorresponds
+in this way to a unique submodule B⊂A, namelyB=k[∂]Ra.
+Letk′be an algebraic extension of k. Then the above bijection extends
+to a bijection between the (monic) factorizations of op(A,a) ink′[∂] and the
+submodules of k′⊗kA.
+As before,Ldenotes an irreducible differential module over ksuch thatk⊗L
+is a direct sum of s>1copies of an absolutely irreducible differential module .
+Chooseℓ∈L,ℓ/ne}ationslash= 0. SinceLisirreducible, ℓisacyclicvector. We assume
+the knowledge of a factorization op(L,ℓ) =F.Rwith monic F,R∈k[∂]andF
+absolutely irreducible , given by [C-W].Using this information we will describe
+the computation leading to a tensor product decomposition L=M⊗N.
+53.1 The special case dimM= 1
+Assume that the irreducible Lis equal toM⊗kNwith dimM= 1,dimN=
+s>1 andk⊗kNis trivial. Thus the Picard-Vessiot extension K+ofNis a
+finite extension of kand can be considered as a subfield of k. The (covariant)
+solution space VofNis equal to ker( ∂,K+⊗kN). The differential Galois
+groupG+=Gal(K+/k) acts onVand there is a canonical isomorphism
+K+⊗CV→K+⊗kN. Moreover, M:=k⊗kMis not a trivial module
+(equivalently, the differential Galois group of Mis infinite and then equal to
+the multiplicative group Gm).
+There is a trivial way to produce a decomposition L=M⊗kN. Indeed,
+writeop(L,ℓ) = (∂s+as−1∂s−1+···+a0). Then the tensor product decom-
+positionop(L,ℓ) = (∂+as−1
+s)⊗(∂s+bs−2∂s−2+···+b0), for suitable elements
+bi∈k, solves already the problem, since (as one easily sees) all the solutio ns
+of∂s−1+bs−2∂s−2+···+b0are ink. However, the aim of this subsection is
+to describe in this easy situation an algorithm for obtaining the pair ( M,N),
+up to a change ( M⊗D,D−1⊗N), which works with small modifications for
+the general case.
+After fixing a non zero element ℓ∈L, the module is represented by the
+monic operator op(L,ℓ).The first step is to produce the smallest subfield
+K⊂k(containing k) such that op(L,ℓ) decomposes in K[∂] as a product
+F1···Fsof (monic) equivalent operators of degree 1.
+The (monic) left hand factors F=∂+u∈k[∂] ofop(L,ℓ) correspond to
+the 1-dimensional submodules of
+k⊗L=M⊗k(k⊗kN) =M⊗k(k⊗CV)
+andthesearethe M⊗k(k⊗CW)whereWrunsinthesetofthe1-dimensional
+subspaces of V. The same can be done with kreplaced by K+. Therefore
+u∈K+andK0:=k(u)⊂K+. More precisely, let St(W)⊂G+be the
+stabilizer of W. This is the subgroup of G+leavingFinvariant and thus
+K0= (K+)St(W)⊂K+.
+Now we suppose that a (monic) left hand factor F=∂+uofop(L,ℓ)is
+known and explain how to obtain Kfrom this . LetK1⊂kbe the normal
+closure ofK0. ThenK1⊗kLcontains a 1-dimensional submodule that we will
+call againD. The submodule/summationtext
+σ∈Gal(K1/k)σ(D) ofK1⊗kLis invariant under
+6theaction of Gal(K1/k). SinceLis irreducible,/summationtext
+σ∈Gal(K1/k)σ(D) =K1⊗kL
+and it follows that K1⊗kLis a direct sum of 1-dimensional submodules. As
+a consequence, op(L,ℓ) factors as F1···FsinK1[∂].
+A priori, the factors Fi=∂+ui∈K1[∂] need not be equivalent. For
+i < jwe consider a non zero element fij∈ksatisfyingf′
+ij
+fij=ui−uj. Put
+K=K1({fij}). ThenK⊃kis the smallest field such that op(L,ℓ) factors
+asF1···Fs∈K[∂] where the monic degree one factors Fiare equivalent.
+ClearlyK⊂K+. Fromtheconditionthat Kisminimalsuchthat K⊗kN
+is a direct sum of isomorphic 1-dimensional submodules and the irredu ciblity
+ofN, it follows easily that the center ZofG+⊂GL(V) is the finite cyclic
+group (C∗·idV)∩G+and thatK= (K+)Z.
+Remarks 3.1 (1) The field Kand the above algorithm for Kdo not change
+ifLis replaced by D⊗kL, whereDis a 1-dimensional module satisfying
+D⊗t=1for somet≥1.
+(2) The fields K0andK1depend on the given left hand factor of degree 1 of
+op(L,ℓ). We illustrate this by an example where the differential Galois group
+G+⊂GL(C3) is generated by the matrices
+0 0 1
+1 0 0
+0 1 0
+and
+a0 0
+0b0
+0 0c
+
+witha6=b6=c6= 1. If this left hand factor corresponds to Ce1(orCe2
+orCe3), then its stabilizer is the subgroup of the diagonal matrices in G+.
+Otherwise it is just the center Z. In the first case K1/ne}ationslash=Kand in the last
+oneK1=K. ✷
+3.1.1 The 2-cocycle cand descent fields
+Now we have arrived at the situation where K⊗kLis a direct sum of scopies
+of a known 1-dimensional differential module D=M⊗kEoverK, whereE
+is an, a priori, unknown 1-dimensional submodule of K⊗kN. We want to
+produce a field extension K′⊃Ksuch thatK′⊗KDdescends to k.
+LetDcorrespond to the operator ∂−u. Then, for σ∈Gal(K/k), the
+operator∂−σ(u) corresponds toσD. There are elements fσ∈K∗such that
+σ(u)−u=f′
+σ
+fσ. One obtains a 2-cocycle cforGal(K/k) with values in C∗by
+fστ=c(σ,τ)·fσ·σfτfor allσ,τ∈Gal(K/k). The class of the 2-cocycle cin
+7H2(Gal(K/k),C∗) is theobstruction for the descent ofD(or, equivalently,
+for the descent of E).
+Indeed, ifDdescends to k, then one can represent Dby∂−uwithu∈k.
+On the other hand, if the class of cis trivial, then after changing the {fσ}
+one hasfστ=fσ·σfτ. By Hilbert 90, there exists F∈K∗withfσ=σF
+Ffor
+allσ∈Gal(K/k). Then∂−uis equivalent to ∂−u+F′
+F. Furtheru−F′
+F
+lies ink, since it is invariant under Gal(K/k).
+The exact sequence 1 →Z→G+pr→Gal(K/k)→1 induces a 2-
+cocycle with values in Z, in the following way. Let φ:Gal(K/k)→G+
+be a section, i.e., pr◦φ(g) =gfor allg∈Gal(K/k). Thend, defined by
+φ(g1g2) =d(g1,g2)φ(g1)φ(g2), is a 2-cocycle with values in Z. The class of
+dinH2(Gal(K/k),Z) is independent of the choice of the section φ. As be-
+fore,Zis identified with a subgroup of C∗. Thusdinduces an element of
+H2(Gal(K/k),C∗). We note that the homomorphism H2(Gal(K/k),Z)→
+H2(Gal(K/k),C∗) is, in general, not injective.
+Lemma 3.2 Thecocycles canddhavethe sameimagein H2(Gal(K/k),C∗).
+This image does not depend of the choice of N. Lets= dimN. Then the
+image ofcsinH2(Gal(K/k),C∗)is trivial.
+Proof.LetD, as before, be given by ∂−uwithu∈K. Writeu=F′
+Fwith
+F∈(K+)∗. Letφ:G→G+be a section. Then σ(u)−u=φ(σ)F′
+φ(σ)F−F′
+F=f′
+σ
+fσ
+wherefσ:=φ(σ)F
+F. One easily sees that fσis invariant under Zand thusfσ∈
+K∗. The equality fστ=c(σ,τ)fσσfτimpliesφ(στ)F=c(σ,τ)φ(σ)φ(τ)F.
+Henced(σ,τ) =c(σ,τ).
+ReplacingNby (∂−v)⊗kNwithv∈k, induces the change of ∂−u
+into∂−u−v. Sinceσ(u+v)−(u+v) =σ(u)−u, the element cand its
+image inH2(Gal(K/k),C∗) are unchanged.
+Suppose that Nis chosen such that det N=1. The cocycle dhas values
+inµs, since dim N=s. Thus the image of dsinH2(Gal(K/k),C∗) is trivial
+and the same holds for cs. ✷
+Definition 3.3 Let (K,c) be a Galois extension K/kandca 2-cocycle for
+Gal(K/k) with values in C∗and such that csis a trivial 2-cocycle. A descent
+fieldfor (K,c) is a Galois extension K′⊃kcontaining K, such that the in-
+duced 2-cocycle c′forGal(K′/k), defined by c′(g1,g2) =c(prg1,prg2), yields
+a trivial element in H2(Gal(K′/k),C∗). Herepr:Gal(K′/k)→Gal(K/k)
+denotes the natural map. ✷
+8The assumption that kis aC1-field implies the existence of a descent field
+for every pair (K,c). Indeed, by [Se], II §3, the cohomological dimension of
+Gal(k/k) is 1 and therefore H2(Gal(k/k),µ∞) = 1, where µ∞denotes the
+torsion subgroup of C∗. One has H2(Gal(K/k),C∗) =H2(Gal(K/k),µ∞)
+and lim
+→H2(Gal(K′/k),µ∞) =H2(Gal(k/k),µ∞) ={1}, where the direct
+limit is taken over all Galois extensions K′⊃k, containing K. Hence for
+every class c∈H2(Gal(K/k),C∗) there exists a Galois extension K′⊃k,
+containingK, such that the image of cinH2(Gal(K′/k),C∗) is 1.
+Our contribution is now to produce a descent field for (K,c)by some algo-
+rithm.
+3.1.2 A decision procedure constructing a descent field for (K,c)
+Sincekis aC1-field,H2(Gal(K/k),K∗) is trivial ([Se], II-9, §3.2) and this
+implies the existence of elements {fσ|σ∈Gal(K/k)} ⊂K∗satisfyingfστ=
+c(σ,τ)·fσ·σfτfor allσ,τ∈Gal(K/k).
+Suppose that cis given in the form fστ=c(σ,τ)fσσfτwith{fσ} ⊂K∗.
+(This is trivially true for the present case dim M= 1. For the general case,
+Remarks 3.4 part (4) describes a decision procedure producing suit able{fσ},
+a key step in the construction). Then the following algorithm produces a de-
+scent field .
+One hasf′
+στ
+fστ=f′
+σ
+fσ+σ(f′
+τ
+fτ) and since H1(Gal(K/k),K) = 0 there is an
+elementv∈Ksuch thatf′
+σ
+fσ=σ(v)−vfor allσ∈Gal(K/k); explicitly
+v=−1
+[K:k]/summationdisplay
+τ∈Gal(K/k)f′
+τ
+fτ.
+One observes that cis the obstruction for descent of the operator ∂−v.
+Further−mv=G′
+GwithG=/producttext
+τ∈Gal(K/k)fτandm= [K:k]. Hence the
+fieldK(m√
+G) contains the Picard-Vessiot field of ∂−vand is a descent field.
+A less brutal way to compute a descent field is as follows. Since the co cy-
+clecsis trivial, there are computable elements {d(σ)|σ∈Gal(K/k)} ⊂C∗
+satisfyingd(στ) =c(σ,τ)sd(σ)d(τ). The elements {fs
+σ
+d(σ)}form a 1-cocycle.
+SinceH1(Gal(K/k),K∗) ={1}, one can effectively compute F∈K∗such
+thatfs
+σ
+d(σ)=σF
+Ffor allσ∈Gal(K/k) (see [Se2], Chapitre X, §1, Prop. 2).
+One observes that v−1
+sF′
+F∈ksince it is invariant under Gal(K/k). The
+9field extension K′=K(s√
+F) has the property that ( ∂−v) is equivalent to
+∂−v+1
+sF′
+FoverK′. HenceK′is a descent field. ✷
+Remarks 3.4
+(1) We note that the above algorithm proves, by considering 1-dime nsional
+differential modules over K, the existence of a descent field only using that
+H2(Gal(K/k),K∗) ={1}(see Lemma 4.1 for the general statement).
+(2). Instead of assuming that the 2-cocycle csis trivial, we may consider a
+classc∈H2(Gal(K/k),µ∞), whereµ∞⊂C∗denotes, as before, the group
+of the roots of unity. Any finite group Goccurs as some Gal(K/k). There-
+fore the group H2(Gal(K/k),µ∞) is in general not trivial and the descent
+problem, i.e., finding an extension K′⊃Ksuch that the image of cin
+H2(Gal(K′/k),µ∞) is 1, is non trivial. However for a cyclic Gal(K/k) one
+hasH2(Gal(K/k),µ∞) ={1}([Se2], VIII, §4). In particular, non trivial
+examples for the descent problem tend to be complicated.
+(3). We ignore how to compute or characterize all minimal descent fi elds for
+a given pair ( K,c).
+(4). Computing elements fσ∈K∗satisfyingfστ=c(σ,τ)·fσ·σfτappears to
+be far from trivial. A possible method, which uses explicitly the C1-property
+ofk, is the following. Starting with the 2-cocyle c, there is a well known
+construction (see [G-S]) of an algebra A=⊕σ∈GK[σ], whereG=Gal(K/k),
+of dimension m= #G= [K:k] overK, defined by the rules:
+[σ]·λ=σ(λ)·[σ] forλ∈K, σ∈G,
+[σ1σ2] =c(σ1,σ2)[σ1]·[σ2].
+ThenAis a central simple algebra with center k. Sincekis aC1-field
+there exists an isomorphism I:A→Matr(m,k). The latter algebra can be
+identified (by standardGaloistheory) withthe groupalgebra K[G] =⊕K·σ.
+Suppose that we knew already elements fσ∈K∗satisfyingfστ=c(σ,τ)·
+fσ·σfτ. ThenI0:A/ma√sto→K[G], given by I0(/summationtextλσ[σ]) =/summationtextλσfσ.σis an
+isomorphism and is also K-linear. By the Skolem-Noether theorem, any
+isomorphism Iofk-algebras has the form I(/summationtextλσ[σ]) =x−1{/summationtextλσfσ.σ}x,
+wherexis an invertible element of Matr(m,k). Our aim is to compute a
+K-linear isomorphism and in that case xcommutes with Kand therefore
+belongs to K∗. NowIhas the form I(/summationtextλσ[σ]) =/summationtextλσfσ.σ(x)
+x.σand thus
+anyK-linear isomorphism A→K[G] has the form [ σ]/ma√sto→gσσfor suitable
+elementsgσ∈K∗. It follows that gστ=c(σ,τ)·gσ·σgτ.
+10The computation of the isomorphism Iuses the reduced norm Normof
+A(see e.g. [F, Pi] or [R] section 4, which adapts to C1fields). With respect
+to a basis of Aoverk, the reduced norm is a homogeneous form of degree m
+inm2variables. Again the C1property of kasserts that there are non trivial
+solutionsa∈A, a/ne}ationslash= 0 forNorm(a) = 0. An explicit calculation of such ais
+possible (but rather expensive). Applying this several times one ob tains the
+isomorphism I:A→Matr(m,k).
+Example 3.5 Consider the case where K⊃khas degree 2 and cis a 2-
+cocycle for G={1,σ}with values in K∗.
+One easily sees that the 2-cocyle can be given by c(1,1) =c(1,σ) =
+c(σ,1) = 1andc(σ,σ) =α−1∈k∗. TheK-linear morphism φ:A:=
+K[1]⊕K[σ]→K[G] =K⊕Kσshould have the form φ([1]) = 1, φ([σ]) =fσ
+withf∈K∗. We have to find f. Now the condition is α=fσ(f). Write
+K=k⊕kwwithw2∈k∗and writef=a+bw. Then we have to solve
+a2−b2w2=α. Consider the equation X2
+1−X2
+2w2−X2
+3α= 0. By theC1-
+property of kthere is a solution (x1,x2,x3)/ne}ationslash= 0. Nowx3/ne}ationslash= 0, sincew2∈k∗
+is not a square. Then we can normalize to x3= 1and the problem is solved.
+(5) For dim N= 2 andMof any dimension we will give in the Appendix an
+easier algorithm for descent fields, not using the 2-cocycle cexplicitly (and
+recall the former algorithms of [H3, C-W, N-vdP]).
+(6) Let again a Galois extension k⊂Kwith Galois group Gand 2-cocycle
+c∈H2(G,C∗) be given. The 2-cocycle class has finite order (dividing s) and
+corresponds to a short exact sequence 1 →Z→G+→G→1, whereZ
+is a finite cyclic group, lying in the center of G+. Suppose that the Galois
+extensionk⊂K+with group G+is such that ( K+)Z=K. Then the image
+ofcinH2(G+,C∗) is trivial and the descent condition holds for the field K+.
+TheC1-property of the field kguarantees the existence of K+, however there
+seems to be no explicit algorithm, based on the C1-property, producing an
+K+. Examples 4.4 are based on this remark.
+(7). Finally, we note that H2(Gal(K/k),µs) = 1 ifg.c.d.([K:k],s) = 1. In
+that case there is no field extension needed for the descent. ✷
+3.2 Description of the algorithm for the general case
+We will search for a decomposition L=M⊗Nwith detN= 1. The module
+k⊗Lcan be written as M⊗k(k⊗kN) =M⊗k(k⊗CV) whereVis the
+11solutionspace of N. The absolutely irreducible left handfactors Fofop(L,ℓ)
+correspond to the 1-dimensional subspaces WofV.
+We suppose that at least one Fis given. Let K0⊃kdenote the field
+extension generated by the coefficients of F. LetK1be the normal closure
+ofK0. For eachσ∈Gal(K1/k) one considers the absolutely irreducible left
+hand factor σ(F). This factor is over kequivalent to F. We have to compute
+the field extension of K1needed for this equivalence.
+An algorithm in terms of differential modules (which easily translates in
+terms of differential operators) is based upon the following lemma.
+Lemma 3.6 LetAbe an irreducible differential module over k. Then the
+differential module Hom(A,A)overkhas only one 1-dimensional submodule,
+namelyk·idA.
+Proof.It is possible to prove this by using [N-vdP] and irreducible represen-
+tations of semi-simple Lie algebras.
+However, a more down-to-earth proof is the following. Let Vbe the
+solution space of A. This is a C-vector space of dimension equal to a:=
+dimkA, equipped with a faithful irreducible action of the differential Galois
+groupGofA. The group Gis connected since kis algebraically closed.
+A 1-dimensional submodule of Hom( A,A) corresponds to a 1-dimensional
+subspaceCfof Hom C(V,V), invariant under the action of G. There is a
+homomorphism c:G→C∗suchthatgfg−1=c(g)·fholdsforall g∈G. The
+kernel offisG-invariant and is {0}since the representation is irreducible.
+The action of Gon Hom(ΛaV,ΛaV) is trivial. In particular, the isomorphism
+Λa(f) : ΛaV→ΛaVis invariant under G. Alsog(Λa(f))g−1=c(g)a·Λa(f)
+andc(g)a= 1. Since Gis connected, c(g) = 1 for all g∈G. Thusfis
+G-invariant and is a multiple of idVsince the representation is irreducible.
+✷
+Corollary 3.7 The differential module
+T(σ) := Hom(K1[∂]/K1[∂]σ(F),K1[∂]/K1[∂]F)
+overK1has a single 1-dimensional submodule A(σ). Moreover, the Picard–
+Vessiot field of A(σ)is a finite cyclic extension K′
+1⊂kofK1.
+Proof.S:=k⊗K1T(σ) is isomorphic to Hom( M,M), whereM:=k⊗kM.
+By Lemma 3.5, Shas a unique 1-dimensional submodule, say, B. By unique-
+ness,Bis invariant under the action of Gal(k/K1) onSand has therefore
+12the formk⊗K1A(σ) for some submodule A(σ) ofT(σ). The uniqueness of
+A(σ) is clear.
+Furtherk⊗A(σ) is isomorphic to the trivial differential module k·idM.
+Thus thePicard–Vessiot field K′
+1isafiniteextension of K1andthisextension
+is cyclic since A(σ) has dimension 1. ✷
+Byfactorizationthe1-dimensional submodule A(σ) canbeobtained. The
+Picard–Vessiot field of A(σ) is a finite cyclic extension K′
+1⊂kofK1. Then
+ker(∂,K′
+1⊗T(σ)) has dimension 1 over Cand a generator φ(σ) of this kernel
+is an isomorphism φ(σ) :K′
+1[∂]/K′
+1[∂]σ(F)→K′
+1[∂]/K′
+1[∂]F.
+The fieldK⊂kis the compositum of the Picard–Vessiot fields of all
+A(σ). We note that Kis the field called “ stabilisateur ” in [C-W]. The
+isomorphisms φ(σ) :K[∂]/K[∂]σ(F)→K[∂]/K[∂]Fare now also known,
+they areK-rational solutions of the modules K⊗K1A(σ). The 2-cocycle c
+forGal(K/k) with values in C∗has the property that csis trivial by the
+assumption that det N=1. Then, as in Subsection 3.1, one can construct a
+cyclic extension K′⊃Ksuch that the module K′[∂]/K′[∂]Fdescends to k.
+The result is called M.
+The module Nis obtained by computing the unique irreducible direct
+summand of M∗⊗kLhaving dimension s. Indeed, this direct summand of
+M∗⊗kL=M∗⊗kM⊗kN= Hom(M,M)⊗kNis (k·idM)⊗kN∼=N.
+3.2.1 An exemple for the construction of a descent field
+Consider the irreducible operator
+L=∂4+(−4+8z)
+z2−1∂3+(53z2−40z−1)
+4(z2−1)2∂2+(5z3−z2−13z−3)
+2(z2−1)3∂+61z2+64z+67
+16(z2−1)4.
+The algorithm of [C-W] produces the following absolutely irreducible rig ht-
+hand factor
+L1=∂2+(3u+4z2−26z+24)
+2(z2−1)(4z−5)∂+(−3z−6)u+45z2−40z−13
+4(z2−1)2(4z−5),
+whereu2=z2−1. It is isomorphic to its conjugate L2overK=k(Φ) with
+Φ4−2zΦ2+1 = 0 (or Φ =√z−u). Explicitely, there exists S∈K[∂] such
+thatL2.R=S.L1with
+R=(1−2z+2u)
+Φ(4z−5)/parenleftbig
+2/parenleftbig
+z2−1/parenrightbig
+∂−z−2/parenrightbig
+,
+13i.e.Rmaps a solution of L1to a solution of L2.
+LetG=Gal(K/k), acting via σ1(Φ) = Φ,σ2(Φ) =−Φ,σ3(Φ) = 1/Φ,
+σ4(Φ) =−1/Φ. The 2-cocycle cisgivenbyc(σ2,σ3) =c(σ2,σ4) =c(σ4,σ3) =
+c(σ4,σ4) =−1 (andc(σi,σj) = 1 otherwise).
+Remark 3.4 part (4) (see also Example 4.3 in the Appendix) produce th e
+elementsfσwhich are, respectively, 1 ,1,Φ,Φ ; hence the construction from
+section 3.2.1 shows that K′=K(√
+Φ) is a descent field. An anihilating
+operator for√
+Φ is
+N=∂2+z
+z2−1∂−1/16/parenleftbig
+z2−1/parenrightbig−1
+(one could find it by writing K′as ak[∂]-module and decomposing it).
+Decomposing L⊗N⋆(overk), we then obtain
+M=∂2−2
+z−1∂+35z+37
+16(z+1)(z−1)2
+andL1is isomorphic over K′toM. At the end of the appendix, we give
+alternative (easier) methods to handle such small order examples.
+4 Appendix
+The following lemma makes the relation between 2-cocycles and desce nt for
+1-dimensional differential modules more explicit. We present some ex amples
+and present an algorithm producing descent fields for the case dim N= 2.
+Lemma 4.1 Letkbe a differential field having the properties: the field of
+constantsCis algebraically closed and has characteristic 0; kis aC1-field.
+LetK/kbe a Galois extension (finite or infinite). The collection H(K)of
+the (isomorphy classes of the) 1-dimensional differential m odulesAoverK,
+satisfyingσA∼=Afor allσ∈Gal(K/k), forms a group with respect to the
+operation tensor product. Let h(K)⊂H(K)denote the subgroup consisting
+of the modules of the form K⊗kB, whereBis a 1-dimensional differential
+module over k.
+There is a canonical isomorphism H(K)/h(K)→H2(Gal(K/k),C∗).
+Proof.The first statement is obvious. Let the differential module Kewith
+∂e=uelieinH(K). Thenforany σ∈Gal(K/k)thereisanelement fσ∈K∗
+14such thatσ(u)−u=f′
+σ
+fσ. Define the 2-cocycle cbyfστ=c(σ,τ)·fσ·σfτfor
+allσ,τ∈Gal(K/k). Replacing the fσbyd(σ)fσ, withd(σ)∈C∗, changes
+the 2-cocycle into an equivalent one. Tensoring Kewith an element of h(K)
+changesuintou+vwithv∈kandthisdoesnoteffectthe fσ. Thustheabove
+construction defines a homomorphism H(K)/h(K)→H2(Gal(K/k),C∗).
+This homomorphism is injective since the triviality of the 2-cocycle clas s
+cimplies that fστ=fσ·σfτ. SinceH1(Gal(K/k),K∗) ={1}there is an
+F∈K∗withfσ=σF
+Ffor allσ. Thusσ(u)−u=σ(F′
+F)−F′
+Fandu:=u−F′
+Fis
+invariant under Gal(K/k) and belongs to k. NowKe=K·ewithe:=F−1e
+and∂e=ue. ThusKebelongs toh(K).
+The homomorphism is surjective. Indeed, consider a 2-cocycle cfor
+Gal(K/k) with values in C∗. SinceH2(Gal(K/k),K∗) ={1}there are el-
+ementsfσ∈K∗such thatfστ=c(σ,τ)·fσ·σfτfor allσ,τ∈Gal(K/k).
+Thenf′
+στ
+fστ=f′
+σ
+fσ+σ(f′
+τ
+fτ) and since H1(Gal(K/k),K) ={0}there is an element
+u∈Ksuch thatσ(u)−u=f′
+σ
+fσfor allσ∈Gal(K/k). Thus the class of cis
+the image of the module Kewith∂e=ue, belonging to H(K). ✷
+Example 4.2 Letk=C(z) andK=C(t) witht2=z. Letσbe the non
+trivial element in Gal(K/k).
+The module Kewith∂e=uebelongs toh(K) if and only if
+u=w+1
+2t/summationdisplay
+α/negationslash=0dα√α
+z−αwith alldα∈Zandw∈k.
+(We note that√αdenotes an arbitrary choice of a square root for α∈C∗).
+This follows from the computation: if K∗∋F=tn0/producttext
+β/negationslash=0(t−β)nβ, then
+F′
+F=n0
+2t2+/summationdisplay
+β2/negationslash=0nβ+n−β
+2(t2−β2)+1
+2t/summationdisplay
+β2/negationslash=0nββ−n−ββ
+t2−β2.
+Consider now Kewith∂e=ue, belonging to H(K). Writeu=a+1
+2tb
+witha,b∈k. By assumption−1
+tb=σ(u)−uhas the formG′
+Gfor some
+G∈K∗. WriteG=tm0/producttext
+β/negationslash=0(t−β)mβ, then, using the above formula, one
+finds thatm0= 0 andm−β=−mβ. Thusb=/summationtext
+α/negationslash=0m√α√α
+z−α, whereα=β2
+(and some choice of√αis made). According to the above result, one has
+thatKelies inh(K). Therefore H(K)/h(K) ={0}. This is in accordance
+withH2(Gal(K/k),C∗) ={1}(sinceGal(K/k) is cyclic). ✷
+15Example 4.3 The group D2∼=(Z/2Z)2has the property that H2(D2,C∗)
+contains an element of order 2. In order to obtain this element for a n ob-
+struction to descent we consider the following differential fields
+k=C(z)⊂K=C(t)⊂K′=C(s) withz=t2+t−2, t=s2.
+The group Gal(K/k) ={1,a,b,ab}∼=D2where the elements a, bare given
+bya(t) =−tandb(t) =t−1. One considers the differential module Ke
+with∂e=ueandu=1
+4(t2−t−2)=s′
+s. One observes that a(u)−u= 0 and
+b(u)−u=(t−1)′
+t−1. Thus we may take f1= 1, fa= 1, fb=t−1, fab=t−1. The
+corresponding 2-cocycle chas values in {±1}andfab=c(a,b)·fa·afbholds
+withc(a,b) =−1. It is easily verified that c∈H2(D2,C∗) is not trivial.
+The module K′⊗Keis trivial since u=s′
+sand therefore descends to k. In
+particularK′is a descent field for ( K,c). ✷
+We note that for a finite group G, acting trivially on C∗, the cohomology
+groupH2(G,C∗) is called the Schur multiplier of G. This group is well
+studied, see [Su].
+Examples 4.4 A construction of many examples of the type L=M⊗kN
+under consideration, involving a non trivial descent problem, is the f ollowing.
+Suppose that G+is given as a finite irreducible subgroup of GL( V) where
+dimCV=n>1 and that the center ZofG+is non trivial. Further, assume
+that a Galois extension K+⊃k=C(z) with Galoisgroup G+is given. Then
+K:= (K+)Zis a Galois extension of kwith group G:=G+/Z.
+Consider the differential module K+⊗CVoverK+, defined by ∂(f⊗v) =
+f′⊗vforf∈K+, v∈V. This is a trivial differential module. The action of
+G+onK+⊗CVis defined by σ(f⊗v) =σ(f)⊗σ(v). This action commutes
+with∂.
+DefineN:= (K+⊗CV)G+. This is an irreducible differential module
+overkwith Picard–Vessiot field K+. The subfield Kis the smallest field
+such thatK⊗kNis a direct sum of isomorphic copies of a 1-dimensional
+differential module DoverK. In particularσD∼=Dfor allσ∈Gal(K/k).
+The 2-cocycle attached to Dis non trivial if there is no subgroup H⊂G+
+mapping bijectively to G. More precisely, K+is a smallest field over which
+the 2-cocycle becomes trivial if and onlyif no proper subgroup HofG+maps
+surjectively to G.
+Take now any absolutely irreducible differential module Mof dimension
+m>1overk. ThenL:=M⊗kNhastherequired properties. Fordim N= 2
+16thereisarichchoiceofexamplesandtherearesimilarexplicit casesfo rn= 3,
+see [vdP-U]. ✷
+Example 4.5 Algorithms for the descent field for the case dimN= 2.
+LetL=M⊗Nbe given with Mabsolutely irreducible and an irreducible N
+with dimN= 2,detN=1and finite differential Galois group G+. For this
+case, methods for finding N(hence the descent field) and Mare proposed,
+e.g, in [H3, C-W, N-vdP] (and references therein).
+Below is another nice method, adapted to our case. We have G+∈
+{DSL2
+k, ASL2
+4, SSL2
+4, ASL2
+5} ⊂SL(2,C) with center Z={±/parenleftbig1 0
+0 1/parenrightbig
+}and there
+is no proper subgroup of G+mapping onto G:=G+/Z∈ {Dk, A4, S4, A5}.
+According to Lemma 3.2, the 2-cocycle class c=d∈H2(G,C∗) is non
+trivial. The method of Section 3 provides the field KwithGal(K/k) =G,
+from the data op(L,ℓ) and an absolutely irreducible monic left hand factor
+Fofop(L,ℓ).
+The (unknown) group G+is a subgroup of SL( V) with dim CV= 2.
+WriteW=sym2Vand letS∈sym2(W) be a generator of the kernel
+ofsym2(W)→sym4(V). ThenSis a non degenerate symmetric form of
+degree two. The homorphism ψ: SL(V)→SL(W), defined by A/ma√sto→A⊗
+sA,
+has kernel {±/parenleftbig1 0
+0 1/parenrightbig
+}and its image is {B∈SL(W)|Sis invariant under B}.
+One observes that ψ(G+) =G. Conversely, for any subgroup G⊂SL(W)
+preserving the form S, one has that ψ−1(G) =G+.
+Leta∈Kbe a general element, then the orbit Gais a basis of K/kand
+theC-vector space with basis Gais the regular representation of G. This
+vector space contains an irreducible representation WofGof dimension
+three. In each of the cases for G, there exists a (unique) non degenerated
+symmetric form SonWwhich is invariant under G.
+The unique monic differential operator T3∈K[∂] of degree 3 which is 0
+onWbelongs to k[∂] becauseWis invariant under G. This operator (or
+the corresponding differential module) is equivalent to the second s ymmetric
+power of an operator T2∈k[∂] (which can be found using [H3]). Let ˜K
+denote the Picard-Vessiot field of T2. Then [ ˜K:K] = 2. LetV⊂˜Kdenote
+the space of solutions of T2. ThenW={v1v2|v2,v2∈V}=sym2V. The
+differential Galois group of T2isψ−1(G) and thus isomorphic to G+. Hence
+˜Kis a descent field for ( K,d) and then also for ( K,c). Using this descent
+field one computes MandN.
+17Yet another observation (though less practical) is that J.J. Kovac ic’s
+fundamental algorithm for order 2 equations, [Ko], could also be app lied
+toL=M⊗N. For example, the symmetric power symm+1(L) contains
+an irreducible factor over k, which is projectively isomorphic to M, for
+m= 2,4,6,12 for the cases G+=DSL2
+k,ASL2
+4,SSL2
+4,ASL2
+5. More refined
+factorisation patterns may be established for each of these case s.
+Example 4.6 The referee’s example. The operator
+L4=∂4+6z
+z2−1∂3+1971z2−947
+288(z2−1)2∂2+27z
+32(z2−1)2∂+9
+4096(z2−1)2has the
+absolutely irreducible righthand factor L2=∂2+3z−α+1
+6(z2−1)∂+3
+64(z2−1),
+whereαis a root of T4+12(z−1)T2−32(z−1)T−12(z−1)2= 0.
+There are the following methods:
+(1). By [C-W, H3]. L4=M⊗Nwith detM= detN=1. The two factors
+of Λ2L4=sym2(M)⊕sym2(N) are easily computed and, using [H3], one
+findsMandN.
+(2). By [N-vdP], Theorem 6.2. One computes F∈sym2(L4) with∂F=
+0, F/ne}ationslash= 0 and a 2-dimensional isotropic subspace for F. From the last part
+of the proof of [N-vdP], Theorem 6.2 one reads off MandN.
+(3). Example 4.5 works here as follows. K0=k(α) and its normal closure
+K1has Galois group A4andK=K1. TheC-vector space Wspanned by
+A4αhas dimension 3. The operator T3=∂3+a2∂2+a1∂+a0∈k[∂] with
+solution space Wis determined by the equation T3(α) = 0. This yields
+T3=∂3+3z−1
+(z+1)(z−1)∂2+27z+5
+36(z+1)2(z−1)2∂−9z+23
+36(z+1)2(z−1)3.
+The operator T2=∂2+3z−1
+3(z+1)(z−1)∂−3z−11
+48(z+1)(z−1)2
+satisfiessym2(T2) =T3and its Picard-Vessiot field (an extension of kof
+degree 24) is a descent field. A minimum polynomial of an algebraic solut ion
+ofT2is
+P=Y8+1
+3(z−1)Y4+4
+27(z−1)Y2−1
+108(z−1)2.
+18It factors over k(α) as
+/parenleftBig
+Y2+α
+6/parenrightBig/parenleftbigg
+Y6−α
+6Y4+1
+3/parenleftbigg
+z−1+α2
+12/parenrightbigg
+Y2−α3
+216−1
+18(z−1)α+4
+27(z−1)/parenrightbigg
+,
+which illustrates the fact that K+is obtained from K1by adjunction of a
+square root, here/radicalbig−α
+6(in fact, adjoining any solution of T2would do).
+Continuation of our method (decomposing L4⊗T∗
+2overk) then yields
+M=∂2−1
+3(−1+3x)
+x2−1∂+1
+192189x2−96x+227
+(x2−1)2
+and we may check that
+(M⊗T2)./parenleftbig
+(x2−1)∂/parenrightbig
+=/parenleftbig/parenleftbig
+x2−1/parenrightbig
+∂+4x/parenrightbig
+.L4.
+As solutions of both MandT2can be expressed in terms of special functions
+(e.g using the methods of van Hoeij), this allows to solve L4in terms of
+algebraic and special functions.
+Acknowledgments
+The authors are grateful to Mark van Hoeij and Lajos Ronyai and to the
+referee for interesting suggestions and for providing Example 4.6 le ading to
+an improvement of our paper.
+References
+[C-W]´E. Compoint and J.-A. Weil – Absolute reducibility of differential
+operators and Galois groups , J. Algebra 275 (2004), no. 1, 77–105.
+[F] T.Fisher – How to trivialise a central simple algebra,
+http://www.faculty.jacobs-university.de/mstoll/work shop2007/fisher2.pdf ,
+see also
+http://www.warwick.ac.uk/staff/J.E.Cremona/descent/ second/
+[G-S] Ph. Gille and T. Szamuely – Central simple algebras and Galois
+cohomology , Cambridge Studies in Advanced Mathematics, Vol 101,
+2006.
+19[H-R-U-W] M. van Hoeij, J-F. Ragot, F. Ulmer and J-A. Weil – Liouvillian
+solutions of linear differential equations of order three an d higher, J.
+Symbolic Comput. Vol 28, p. 589-609,1999.
+[H-P] M. van Hoeij and M. van der Put – Descent for differential modules
+and skew fields , J. Algebra 296 (2006), no. 1, 18–55.
+[H1] M. van Hoeij – Rational Solutions of the Mixed Differential Equation
+and its Application to Factorization of Differential Operat ors,ISSAC
+1996, p. 219-225.
+[H2] M. van Hoeij – Factorization of differential operators with rational
+functions coefficients , J. Symbolic Comput., Vol 24, p.537-561, 1997.
+[H3] M. van Hoeij – Solving third order linear differential equations in
+terms of second order equations , ISSAC 2007, 355–360, ACM, 2007.
+http://www.math.fsu.edu/ ~hoeij/files/ReduceOrder/
+[Ka] N. Katz – Exponential Sums and Differential Equations , Annals of
+Mathematical Studies 124, Princeton University Press 1990.
+[Ko] J. J. Kovacic – An algorithm for solving second order linear homo-
+geneous differential equations , Journal of Symbolic Computation 2,
+3-43, 1986.
+[N-vdP] K.A. Nguyen and M. van der Put – Solving differential equa-
+tions, to be published in the Tate volume in PAMQ. Also
+arXiv.org/abs/0810.4039.
+[Pi] J. Pilnikova – Trivializing a central simple algebra of degree 4 over
+the rational numbers , Journal of Symbolic Computation, vol. 42
+(2007) 587-620
+[vdP-S] M. van der Put and M.F. Singer – Galois theory of linear differential
+equations , Grundlehren Volume 328, Springer Verlag 2003.
+[vdP-U] M. van der Put and F. Ulmer – Differential equations and finite
+groups, J. Algebra 226, 920-966, 2000.
+[R] L. Ronyai – Computing the structure of finite algebras , Journal of
+Symbolic Computation, vol. 9 (1990), no. 3, 355–373.
+20[Se] J.-P. Serre – Cohomologie Galoisienne , Lecture Notes in Mathemat-
+ics 5, Springer Verlag 1973.
+[Se2] J.-P. Serre – Corps locaux , Hermann Paris, 1968.
+[S-U] M.F. Singer and F. Ulmer – Linear differential equations and prod-
+ucts of linear forms , J. Pure Appl. Algebra 117/118(1997), 549–563.
+[Su] M. Suzuki – Group Theory I, Comprehensive Studies in Mathemat-
+ics, vol. 248, Springer-Verlag, Berlin, 1986
+21
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+arXiv:1001.0155v2  [hep-th]  21 Feb 2010Singularities in Horava-Lifshitz theory
+Rong-Gen Caia∗and Anzhong Wangb†
+aKey Laboratory of Frontiers in Theoretical Physics, Instit ute of Theoretical Physics,
+Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China
+bGCAP-CASPER, Physics Department, Baylor University, Waco , TX 76798-7316, USA
+(Dated: November 7, 2018)
+Singularities in (3 + 1)-dimensional Horava-Lifshitz (HL) theory of gravity are studied. These
+singularities canbedividedintoscalar, non-scalar curva ture,andcoordinatesingularities. Because of
+the foliation-preserving diffeomorphisms of the theory, th e number of scalars that can be constructed
+from the extrinsic curvature tensor Kij, the 3-dimensional Riemann tensor and their derivatives is
+much large than that constructed from the 4-dimesnional Rie mann tensor and its derivatives in
+general relativity (GR). As a result, even for the same space time, it may be singular in the HL
+theory but not in GR. Two representative families of solutio ns with projectability condition are
+studied, one is the (anti-) de Sitter Schwarzschild solutio ns, and the other is the Lu-Mei-Pope
+(LMP) solutions written in a form satisfying the projectabi lity condition - the generalized LMP
+solutions. The (anti-) de Sitter Schwarzschild solutions a re vacuum solutions of both HL theory and
+GR, while the LMP solutions with projectability condition s atisfy the HL equations coupled with an
+anisotropic fluid with heat flow. It is found that the scalars KandKijKijare singular only at the
+center for the de Sitter Schwarzschild solution, but singul ar at both the center and r= (3M/|Λ|)1/3
+for the anti-de Sitter Schwarzschild solution. The singula rity atr= (3M/|Λ|)1/3is absent in GR.
+In addition, all the generalized LMP solutions have two scal ar curvature singularities, located at
+eitherr= 0 and r=rs>0, orr=r1andr=r2withr2> r1>0, orr=rs>0 andr=∞,
+depending on the choice of the free parameter λ.
+PACS numbers: 04.60.-m; 98.80.Cq; 98.80.-k; 98.80.Bp
+I. INTRODUCTION
+There has been considerable interest recently on a the-
+ory of quantum gravity proposed by Horava [1], moti-
+vated by Lifshitz theory in solid state physics [2], for
+which the theory is usually referred to as the Horava-
+Lifshitz (HL) theory. The HL theory is based on the
+perspective that Lorentz symmetry should appear as an
+emergent symmetry at long distances, but can be fun-
+damentally absent at high energies [3]. With such a
+perspective, Horava considered systems whose scaling
+at short distances exhibits a strong anisotropy between
+space and time,
+x→ℓx, t→ℓzt. (1.1)
+In (3 + 1)-dimensional spacetimes, in order for the the-
+ory to be power-counting renormalizable, it needs z≥
+3. At low energies, the theory is expected to flow to
+z= 1, wherebythe Lorentzinvarianceis“accidentallyre-
+stored.” So, the HL theory is non-relativistic, ultra-violet
+(UV) complete, explicitly breaks Lorentz invariance at
+short distances, but is expected to reduce to general rel-
+ativity (GR) in the infrared (IR) limit.
+The effective speed of light in this theory diverges in
+the UV regime, which could potentially resolve the hori-
+zon problem without invoking inflation [4]. The spatial
+∗Electronic address: cairg@itp.ac.cn
+†Electronic address: anzhong˙wang@baylor.educurvature is enhanced by higher-order curvature terms
+[5–7], andthis opens anew approachto the flatnessprob-
+lem and to a bouncing universe [5, 8, 9]. In addition, in
+the super-horizon region scale-invariant curvature per-
+turbations can be produced without inflation [10, 11],
+and the perturbations become adiabatic during slow-roll
+inflation driven by a single scalar field and the comov-
+ing curvature perturbation is constant [11]. Due to all
+these remarkable features, the theory has attracted lot of
+attention lately [12–14].
+To formulate the theory, Horava assumed two condi-
+tions –detailed balance and projectability (He also con-
+sidered the case where the detailed balance condition
+was softly broken) [1]. The detailed balance condition
+restricts the form of a general potential in a ( D+ 1)-
+dimensional Lorentz action to a specific form that can
+be expressed in terms of a D-dimensional action of a rel-
+ativistic theory with Euclidean signature, whereby the
+number of independent-couplings is considerablylimited.
+The projectability condition, on the other hand, orig-
+inates from the fundamental symmetry of the theory –
+the foliation-preservingdiffeomorphismsofthe Arnowitt-
+Deser-Misner (ADM) form,
+ds2=−N2c2dt2+gij/parenleftbig
+dxi+Nidt/parenrightbig/parenleftbig
+dxj+Njdt/parenrightbig
+,
+(i, j= 1,2,3),(1.2)
+which require coordinate transformations be only of the
+types,
+t→f(t), xi→ζi(t,x), (1.3)
+that is, space-dependent time reparameterizationsare no
+longer allowed, although spatial diffeomorphisms are still2
+a symmetry. Then, it is natural, but not necessary, to
+restrict the lapse function Nto be space-independent,
+while the shift vector Niand the 3-dimensional metric
+gijin general depend on both time and space,
+N=N(t), Ni=Ni(t,x), gij=gij(t,x).(1.4)
+This is the projectability condition, and clearly is pre-
+served by the foliation-preserving diffeomorphisms (1.3).
+However, due to these restricted diffeomorphisms, one
+more degree of freedom appears in the gravitational sec-
+tor-aspin-0graviton. Thisispotentiallydangerous,and
+needs to be highly suppressed in the IR regime, in order
+to be consistent with observations. Similar problems also
+raise in other modified theories, such as massive gravity
+[15].
+Under the rescaling (1.1), the dynamical variables
+N, Niandgijscale as,
+N→N, Ni→ℓ−2Ni, gij→gij.(1.5)
+Note that in [7, 11], the constant cin the metric (1.2) was
+absorbed into N, so that there the lapse function scaled
+asℓ−2.
+So far most of the work on the HL theory has aban-
+doned the projectability condition but kept the detailed
+balance [4, 6, 12–14]. One of the main reasons is that
+the detailed balance condition leads to a very simple ac-
+tion, and the resulted theory is much easier to deal with,
+while abandoningprojectabilityconditiongivesrisetolo-
+cal rather than globalHamiltonian constraintand energy
+conservation. However, with detailed balance a scalar
+field is not UV stable [5], and gravitational perturba-
+tions in the scalar section have ghosts [1] and are not
+stable for any given value of the dynamical coupling con-
+stantλ[16]. In addition, detailed balance also requires
+a non-zero (negative) cosmological constant, breaks the
+parity in the purely gravitational sector [17], and makes
+the perturbations not scale-invariant [18]. Breaking the
+projectability condition, on the other hand, can cause
+strong couplings [19] and gives rise to an inconsistency
+theory [20].
+To resolve these problems, various modifications have
+been proposed [21]. In particular, Blas, Pujolas and
+Sibiryakov (BPS) [22] showed that the strong coupling
+problem can be solved without projectability condition
+(in which the lapse function becomes dependent on both
+tandxi), when terms constructed from the 3-vector
+ai≡∂iN
+N, (1.6)
+are included. Contrary claims can be found in [23]. In
+addition, it is not clear how the inconsistency problem
+[20] is resolved in such a generalization.
+On the other hand, Sotiriou, Visser and Weifurtner
+(SVW) formulated the most general HL theory with pro-
+jectability but without detailed balance conditions [17].
+The total action consists of three parts, kinetic, potentialand matter,
+S=ζ2/integraldisplay
+dtd3xN√g/parenleftbig
+LK−LV+ζ−2LM/parenrightbig
+,(1.7)
+whereg= detgij, and
+LK=1
+c2/bracketleftBig
+KijKij−(1−ξ)K2/bracketrightBig
+,
+LV= 2Λ−R+1
+ζ2/parenleftbig
+g2R2+g3RijRij/parenrightbig
++1
+ζ4/parenleftBig
+g4R3+g5R RijRij+g6Ri
+jRj
+kRk
+i/parenrightBig
++1
+ζ4/bracketleftbig
+g7R∇2R+g8(∇iRjk)/parenleftbig
+∇iRjk/parenrightbig/bracketrightbig
+.(1.8)
+Hereζ2= 1/16πG, andcdenotes the speed of light.
+In the “physical” units, one can set c= 1 [17]. The
+covariant derivatives and Ricci and Riemann terms are
+all constructed from the three-metric gij, whileKijis the
+extrinsic curvature,
+Kij=1
+2N(−˙gij+∇iNj+∇jNi),(1.9)
+whereNi=gijNj. The constants ξ,gI(I= 2,...8) are
+coupling constants, and Λ is the cosmological constant.
+In the IR limit, all the high order curvature terms (with
+coefficients gI) drop out, and the total action reduces
+whenξ= 0 to the Einstein-Hilbert action.
+TheSVWgeneralizationseemstohavethepotentialto
+solve the above mentioned problems [24], although it was
+found that gravitational scalar perturbations either have
+ghosts (0 ≤ξ≤2/3) or are not stable ( ξ <0) [7, 25].
+In order to avoid ghost instability, one needs to assume
+ξ≤0. Then, the sound speed c2
+ψ=ξ/(2−3ξ) becomes
+imaginary, which leads to an IR instability. Izumi and
+Mukohyama showed that this type of instability does not
+show up if |cψ|is less than a critical value [26].
+It is fair to say, in order to have a viable HL theory,
+much work needs to be done, and various aspects of the
+theory ought to be explored, including the renormaliza-
+tion group flows [27], Vainshtein mechanism [15, 28], so-
+lar system tests [29], Lorentz violations [30], and its ap-
+plications to cosmology [7, 11].
+In this paper, we shall study another important issue
+in the HL theory - the problem of singularities, which is
+closely related to the issue of black holes in this the-
+ory [12]. Although we are initially interested in the
+case with projectability condition, our conclusions can
+be equally applied to the HL theory without projectabil-
+ity condition. The extrinsic curvature Kijand the 3-
+dimensional Riemann tensor Ri
+jklare not tensors under
+the 4-dimensional Lorentz transformations,
+xµ→x′µ=ζµ/parenleftbig
+t,xi/parenrightbig
+. (1.10)
+As a result, in GR one usually does not use them to
+construct gauge-invariant quantities. However, in the3
+HL theory, due to the restricted diffeomorphisms, these
+quantities become tensors, and can be easily used to con-
+struct various scalars. If any of such scalars is singular,
+such a singularity cannot be limited by the restricted co-
+ordinate transformations (1.3). Then, we may say that
+the spacetime is singular. It is exactly in this vein that
+we study singularities in the HL theory. In particular,
+we first generalize the definitions of scalar, non-scalar
+and coordinate singularities in GR to the HL theory in
+Sec. II, and then in Sec. III we study two representa-
+tivefamilies ofsphericalstatic solutionsoftheHL theory,
+and identify scalar curvature singularities using the three
+quantities K, KijKijandR. In Sec. IV, we present
+our main conclusions and remarks. There is also an Ap-
+pendix, in which we show explicitly that the second class
+oftheLMPsolutionswrittenintheADMframewithpro-
+jectability condition in general satisfy the HL equations
+coupled with an anisotropic fluid with heat flow.
+Before proceeding further, we would like to note that
+black holes in GR for asymptotically-flat spacetimes are
+well-defined [31]. However, how to generalize such defini-
+tions to more general spacetimes is still an open question
+[32, 33]. The problem in the HL theory becomes more
+complicated [26, 34], partially because of the fact that
+particles in the HL theory can have non-standard disper-
+sion relations, and therefore no uniform maximal speed
+exists. As a result, the notion of a horizon is observer-
+depedent.
+II. SINGULARITIES IN HL THEORY
+In GR, there are powerful Hawking-Penrose theorems
+[31], from which one can see that spacetimes with quite
+“physically reasonable” conditions are singular. Al-
+though the theorems did not tell the nature of the sin-
+gularities, Penrose’s cosmic censorship conjecture states
+that those formed from gravitational collapse in a “phys-
+ically reasonable” situation are always covered by hori-
+zons [35].
+To study further the nature of singularities in GR, El-
+lis and Schmidt divided them into two different kinds,
+spacetime curvature singularities andcoordinate singu-
+larities[36]. The former is real and cannot be made dis-
+appear by any Lorentz transformations (1.10), while the
+latteriscoordinate-depedent,andcanbe madedisappear
+by proper Lorentz transformations. Spacetime curva-
+ture singularitiesare further divided into two sub-classes,
+scalar curvature singularities andnon-scalar curvature
+singularities . If any of the scalars constructed from the
+4-dimensional Riemann tensor Rσ
+µνλand its derivatives
+is singular, then the spacetime is said singular, and the
+corresponding singularity is a scalar one. If none of these
+scalars is singular, spacetimes can be still singular. In
+particular, tidal forces and/or distortions (which are the
+double integrals of the tidal forces), experienced by an
+observer, may become infinitely large [37]. These kinds
+of singularities are usually referred to as non-scalar cur-vature singularities.
+To generalize these definitions to the HL theory, as
+mentioned in the Introduction, both the extrinsic curva-
+tureKijand the 3-dimensional Riemann tensor Ri
+jklcan
+be used to construct gauge-invariant quantities, as now
+they are all tensors under the restricted transformations
+(1.3). Then, we can see that now there are three kinds of
+scalars: one is constructed from Kijand its derivatives;
+one is from the 3-dimensional Riemann tensor Ri
+jkland
+its derivatives; and the other is the mixture of Kij, Ri
+jkl
+and their derivatives. Therefore, we may define a scalar
+curvature singularity in the HL theory as the one where
+any of the scalars constructed from Kij, Ri
+jkland their
+derivatives is singular. A non-scalar curvature singular-
+ity is the one where none of these scalars is singular, but
+some other physical quantities, such as tidal forces and
+distortionsexperiencedbyobservers,becomeunbounded.
+Coordinate singularities are the ones that can be limited
+by the restricted coordinate transformations (1.3).
+Several comments now are in order. The scalars con-
+structedfromthe3-dimensionaltensors KijandRi
+jkland
+their derivatives include all scalars constructed from the
+4-dimensional Rσ
+µνλand its derivatives. Thus, according
+to the above definitions, all scalar singularities under the
+generalLorentztransformations(1.10)arealsoscalarsin-
+gularities under the restricted transformations (1.3), but
+not the other way around. In this sense, scalar singulari-
+ties in the HL theory are more general than those in GR.
+One simple example is the anti-de Sitter Schwarzschild
+solutions, which are also solutions of the SVW general-
+ization with ξ= 0, as in this case the 3-dimensional Ricci
+tensorRijvanishes identically, and the contributions of
+high order derivatives of curvature to the potential LV
+are zero, as can be seen from Eq. (1.8). However, as
+shown in the next section, the corresponding two scalars
+KandKijKijall become singular at r= (3M/|Λ|)1/3.
+This singularity is absent in GR [31].
+In 3-dimensional space, the Weyl tensor vanishes iden-
+tically, and the Riemann tensor is determined alge-
+braically by the curvature scalar and the Ricci tensor:
+Rijkl=gikRjl+gjlRik−gjkRil−gilRjk
+−1
+2/parenleftbig
+gikgjl−gilgjk/parenrightbig
+R. (2.1)
+Therefore, the singular behavior of the scalars made of
+the 3-dimensional Riemann tensor Ri
+jklmay well be rep-
+resented by the 3-dimensional curvature scalar R.
+III. SINGULARITIES IN SPHERICAL STATIC
+SPACETIMES
+The metric of general spherically symmetric static
+spacetimes that preserve the ADM form of Eq. (1.2)
+with the projectability condition can be cast in the form
+[38],
+ds2=−dt2+e2ν/parenleftbig
+dr+eµ−νdt/parenrightbig2+r2dΩ2,(3.1)4
+whereµ=µ(r), ν=ν(r). Then, we have
+R=2e−2ν
+r2/bracketleftBig
+2rν′−/parenleftbig
+1−e2ν/parenrightbig/bracketrightBig
+,
+K=eµ−ν/parenleftbigg
+µ′+2
+r2/parenrightbigg
+,
+KijKij=e2(µ−ν)/parenleftbigg
+µ′2+2
+r2/parenrightbigg
+, (3.2)
+whereν′≡dν/dr, etc. It is interesting to note that for
+the metric (3.1) we have
+Cij= 0 =ǫijkRil∇jRl
+k, (3.3)
+whereCijis the Cotton tensor, defined as
+Cij=ǫikl∇k/parenleftbigg
+Rj
+l−1
+4Rδj
+l/parenrightbigg
+. (3.4)
+As a result, the HL theory with detailed balance [1]
+SHLd=/integraldisplay
+dtdx3N√g/braceleftBigg
+2
+κ2/bracketleftBig
+KijKij−λK2/bracketrightBig
++κ2µ2/parenleftbig
+ΛWR−3Λ2
+W/parenrightbig
+8(1−3λ)+κ2µ2/parenleftbig
+1−4λ)R2
+32(1−3λ)
+−κ2µ2
+8RijRij+κ2µ
+2w2ǫijkRil∇jRl
+k
+−κ2
+2w2CijCij/bracerightBigg
+, (3.5)
+can be effectively considered as a particular case of the
+general SVW action (1.7) with
+G=κ2
+32πc2, c2=κ4µ2ΛW
+16(1−3λ),Λ =3ΛW
+2,
+g2=4λ−1
+4ΛWζ2, g3=1−3λ
+ΛWζ2,
+ξ= 1−λ, g4=g5=...=g8= 0. (3.6)
+It should be noted that these relations are valid only
+when the conditions of Eq. (3.3) hold. In general, these
+two terms do not vanish and violate parity, while the
+SVW action always preserves it. It must not be confused
+with the parameter µused in the action (3.5) and the
+metric coefficient used in (3.1).
+To study singularities in the HL theory, in the rest of
+thispaperweshallrestrictourselvestotworepresentative
+cases, the (anti-) de Sitter Schwarzschild solutions and
+the solutions found by Lu, Mei and Pope (LMP) [6].
+A. (Anti-) de Sitter Schwarzschild Solutions
+The (anti-) de Sitter Schwarzschild solutions are given
+by [38]
+µ=1
+2ln/parenleftbiggM
+r+Λ
+3r2/parenrightbigg
+, ν= 0.(3.7)When Λ >0, it represents the de Sitter Schwarzschild
+solutions, and when Λ <0 it represents the anti-de Sitter
+Schwarzschild solutions. As mentioned previously, they
+are also solutions of the SVW generalization with ξ= 0.
+Inserting the above into Eq.(3.2), we find that
+R= 0,
+K=/parenleftbigg3M+Λr3
+12r3/parenrightbigg1/2/parenleftbigg4
+r−3M−2Λr3
+3M+Λr3/parenrightbigg
+,
+KijKij=3M+Λr3
+12r3/bracketleftBigg
+8+/parenleftbigg3M−2Λr3
+3M+Λr3/parenrightbigg2/bracketrightBigg
+.(3.8)
+Clearly, when Λ ≥0,KandKijKijare singular only at
+the center r= 0. However, when Λ <0, they are also
+singular at r=rΛ≡(3M/|Λ|)1/3. In contrast to GR,
+this singularity is a scalar one, and cannot be removed
+by any coordinate transformations given by Eq. (1.3).
+In GR, the (anti-) de Sitter Schwarzschildsolutionsare
+usually given in the orthogonal gauge,
+ds2=−e2Ψ(r)dτ2+e2Φ(r)dr2+r2dΩ2,(3.9)
+with
+Ψ =−Φ =1
+2ln/parenleftbigg
+1−2M
+r+1
+3Λr2/parenrightbigg
+.(3.10)
+Clearly, the metric (3.9) does not satisfy the pro-
+jectability condition, and its coefficient grris singular
+ate2Φ(rEH)= 0. But, in GR this is a coordinate singu-
+larity, and all scalarsmade ofthe 4-dimensionalRiemann
+tensor and its derivatives are finite at r=rEH.
+It is interesting to note that in the orthogonal gauge
+(3.9),KandKijKijall vanish, as can be seen from Eqs.
+(1.9), while the 3-dimensional Ricci scalar is given by
+Rorth=−2Λ, where Rorthdenotes the quantity calcu-
+lated in the orthogonal gauge (3.9).
+In [38], it was showed explicitly that metric (3.9) is
+relatedtometric(3.1)bythecoordinatetransformations,
+τ=t−/integraldisplayr/radicalbig
+e−2Ψ−1eΦdr, (3.11)
+under which we have
+Φ(r) =ν(r)−1
+2ln/parenleftBig
+1−e2µ/parenrightBig
+,
+Ψ(r) =1
+2ln/parenleftBig
+1−e2µ/parenrightBig
+, (3.12)
+or inversely,
+µ=1
+2ln/parenleftBig
+1−e2Ψ/parenrightBig
+, ν= Φ(r)+Ψ(r).(3.13)
+Note that the coordinate transformations (3.11) are
+not allowed by the foliation-preserving diffeomorphisms
+(1.3). In addition, since K, KijKijandRare not scalars
+under these transformations, it explains why we have
+completely different physical interpretations in the two5
+different gauges, defined, respectively, by Eqs. (3.1) and
+(3.11). Infact, intheframeworkoftheHLtheorythetwo
+different gauges represent two different theories - metric
+(3.1) represents a HL theory with projectability condi-
+tion, while metric (3.11) represents a HL theory without
+projectability condition.
+B. The LMP Solutions
+The LMP solutions were originally found for the HL
+theory with detailed balance but without projectability
+conditions in the form (3.9). There are two classes of
+solutions, given, respectively, by
+Φ =−1
+2ln/parenleftbig
+1+x2/parenrightbig
+, (3.14)
+for any Ψ( r), and
+Φ =−1
+2ln/parenleftBig
+1+x2−αxα±/parenrightBig
+,
+Ψ =−β±ln(x)+1
+2ln/parenleftBig
+1+x2−αxα±/parenrightBig
+,(3.15)
+wherex≡√−ΛWr, and
+α−≡2λ−√
+6λ−2
+λ−1=2(2λ−1)
+2λ+√
+6λ−2,
+α+≡2λ+√
+6λ−2
+λ−1, β±≡2α±−1,(3.16)
+withαbeing an arbitraryreal constant. Fig. 1 schemati-
+cally showsthe curves of α±vsλ, which shows that there
+is no solution for the α+branch when λ= 1.
+Solutions given by Eqs. (3.14) and (3.15) shall be re-
+ferred to, respectively, as Class A and B solutions. Par-
+ticular values of α±are
+α−(λ) =
+
+−1, λ= 1/3,
+0, λ= 1/2,
+1/2, λ= 1,
+2/3, λ≃1.375,
+1, λ= 3,
+2, λ=∞,
+α+(λ) =
+
+−1, λ= 1/3,
+−∞, λ= 1−0,
++∞, λ= 1+0,
+2, λ=∞,(3.17)
+which are useful in the following discussions.
+Inthe orthogonalgauge(3.9), asmentioned previously,
+the extrinsic curvature Kijvanishes identically, while the
+3-dimensional curvature for the above two classes of so-
+lutions can be written as
+Rorth=2
+r2/braceleftBig
+α/parenleftbig
+1+α±/parenrightbig
+xα±−3x2/bracerightBig
+.(3.18)
+Whenα= 0, the corresponding Rorthis for the solution
+(3.14), which reduces to a constant. That is, in this caseα
+α
+α+
++−
+0 1λ
+1/21/32
+−1
+FIG. 1: The functions α±(λ) defined by Eq. (3.16).
+all the three scalars K, KijKijandRare finite. When
+α/negationslash= 0, it is for the solution (3.15), which is singular only
+whenα±≤2 atr= 0. From Fig.1 we can see that
+α−is always less than two for a finite λ. Therefore, this
+branch of solutions is always singular at the center. α+
+is always less than two for 1 /3≤λ <1 and greater
+than two for λ >1. Therefore, Rorthis finite for the α+
+solutionswith λ >1forany r, including the center r= 0,
+while it is singular at the center for the α+solutions with
+1/3≤λ <1.
+1. Class A Solutions
+Transforming the above solutions into the canonical
+ADM form (3.1), we find that for the solution (3.14), we
+have
+µ=−∞, ν=−1
+2ln/parenleftBig
+1−ΛWr2/parenrightBig
+,(3.19)
+where in writing the above expressions, we had chosen
+Ψ = 0. Then, the corresponding three scalars are given
+by
+K=KijKij= 0, R= 6ΛW,(3.20)
+which are all finite. As shown in [38], this solution is also
+a vacuum solution of the SVW generalization (1.7) with
+a non-vanishing constant curvature k= 6ΛW. Since it
+does not depend explicitly on the coupling constant ξ, it
+is a vacuum solution for any given ξ, including ξ= 0.
+For detail, we refer readers to Sec. V of [38].6
+2. Class B Solutions
+For Class B solutions (3.15), they can be written in the
+canonical ADM form (3.1) with
+µ=1
+2ln∆, ν=/parenleftbig
+1−2α±/parenrightbig
+ln(x), (3.21)
+where
+∆≡1−x2(1−2α±)−x4(1−α±)+αx2−3α±.(3.22)
+Clearly, to have real solutions, we must assume ∆ ≥0.
+It should be noted that unlike Class A solutions, this
+class of solutions do not satisfy the vacuum equations of
+the HL theory with projectability condition, due to the
+fact that the HL actions are not invariant under the co-
+ordinate transformations (3.11). As shown in Appendix,
+they can be interpreted as solutions of the HL theory
+with projectability condition coupled with a spherical
+anisotropic fluid with heat flow. The properties of sin-
+gularities of these quantities can be well represented by
+the three scalars K, KijKijandR, as can be seen from
+Appendix, so in the following we shall not consider them
+specifically.
+Inserting the above solutions into Eq. (3.2) we find
+that
+R=2
+r2/braceleftBig
+1−/parenleftbig
+1+2β±/parenrightbig
+x2β±/bracerightBig
+,
+K=√−ΛW
+2∆1/2x3−2α±/parenleftBig
+4/radicalbig
+−ΛW∆−x2δ/parenrightBig
+,
+KijKij=−ΛW
+4∆x4(1−α±)/parenleftbig
+8∆−x2δ2/parenrightbig
+,(3.23)
+where
+δ≡2/parenleftbig
+1−2α±/parenrightbig
+x1−4α±+4/parenleftbig
+1−α±/parenrightbig
+x3−4α±
+−α/parenleftbig
+2−3α±/parenrightbig
+x1−3α±. (3.24)
+To study these solutions further, it is found convenient
+to consider the two branches α±separately. We shall use
+β=β±to denote the α±branches.
+Case i) β=β+,1/3≤λ <1:In this case we have
+α+≤ −1andβ+≤ −3. Then, from the expression(3.22)
+we find that
+∆ =/braceleftBig1, x= 0,
+−∞, x→ ∞.(3.25)
+Thus, there must exist a point x=xsat which we have
+∆(xs) = 0. Since ∆ ≥0, we must restrict the solutions
+to the range 0 ≤x≤xs(or equivalently, 0 ≤r≤rs,
+wherers≡xs/√−ΛW.). Fig. 2 shows this case. From
+the expressions (3.23) we find that all these three scalars
+diverge at the center r= 0, while KandKijKijdi-
+verge at r=rs. Therefore, in the present case there are
+two scalar curvature singularities, located, respectively,
+atr= 0 and r=rs.1
+0∆
+r(r)
+rsα > 0
+α < 0
+FIG. 2: The function ∆( x) defined by Eq. (3.22) for the α+
+branch with 1 /3≤λ <1. All three scalars R, KandKijKij,
+definedbyEq. (3.23), become unboundedat r= 0, while only
+KandKijKijdiverge at r=rs.
+Case ii) β=β+, λ >1:In this case we have α+≥2
+andβ+≥3, where the equality holds only when λ=∞.
+Then, from the expression (3.22) we find that
+∆ =/braceleftBig−∞, x= 0,
++1, x→ ∞.(3.26)
+Fig.3 shows this case, from which we find that for any
+givenλ >1, there always exists a minimum rs>0 so
+that ∆(r)≥0 forr≥rs, where rsis the solution of
+∆(r) = 0. Then, from the expressions (3.23) we find
+thatKandKijKijdiverge at r=rs, whileRdiverges
+asr→ ∞. Therefore, in the present case there are also
+two scalar curvature singularities, located, respectively,
+atr=rsandr=∞.
+Case iii) β=β−,1/3≤λ <1:In this case we have
+−1≤α−<1/2 and−3≤β−<0, where the equality
+holds only for λ= 1/3. Then, from the expression (3.22)
+we find that
+∆ =/braceleftBig+1, x= 0,
+−∞, x→ ∞.(3.27)
+from Fig.4, we find that for any given λin this range,
+there always exists a maximum rs>0 so that ∆( r)≥0
+for 0≤r≤rs. Then, From the expressions (3.23) we
+find that all three scalars R, KandKijKijbecome
+unbounded at the center, while only KandKijKijdi-
+verge at r=rs. Therefore, in the present case there are
+two scalar curvature singularities, located, respectively,
+atr= 0 and r=rs.
+Case iv) β=β−= 0, λ= 1:In this case we have
+α−= 1/2, and
+∆ =αx1/2−x2. (3.28)
+Thus, to have ∆ non-negative, we must assume α >0.
+Then, ∆ ≥0 for 0≤r≤rs≡α2/3, where ∆( x) = 0 at7
+1
+0∆
+r(r)
+α < αα > α c
+c
+rs
+FIG. 3: The function ∆( x) defined by Eq. (3.22) for the α+
+branch with λ >1, where αcis the value of α, for which
+∆′(x)>0 forα < α c.KandKijKijdiverge at r=rs, while
+Rdiverges as r→ ∞.
+1
+0∆
+r(r)α > α c
+α < α
+crs
+FIG. 4: The function ∆( x) defined by Eq. (3.22) for the α−
+branch with 1 /3≤λ <1, where αcis the value of α, for
+which ∆′(x)<0 forα < α c.R, KandKijKijdiverge at the
+centerr= 0, while only KandKijKijdiverges at r=rs.
+bothr= 0 and r=rs. At the center, all three scalars
+R, KandKijKijbecome unbounded, while only Kand
+KijKijdiverge at r=rs.
+Case v) β=β−,1< λ <3:In this case we have
+1/2≤α−<1,0≤β−<1, and
+∆ =/braceleftBig−∞, x= 0,
+−∞, x→ ∞.(3.29)
+Fig.5 shows the general properties of ∆( r), from which
+we can see that for anygiven λ, there alwaysexists a crit-
+ical value αc, for which ∆( r)≥0 is possible only when
+α > αc. In the latter case, ∆( r) = 0 always has two
+positive roots, say, r1andr2. Without loss of general-
+ity, we assume r2> r1>0. When r1≤r≤r2, ∆(r)∆(r)
+r 0α > α 
+α < αα = α
+cc
+cr r1 2
+FIG. 5: The function ∆( x) defined by Eq. (3.22) for the
+α−branch with 1 < λ <3, where αcis the value of α, for
+which ∆( x)≥0 is possible only when α > α c.KandKijKij
+diverge at the two points r=r1,2, whileRremains finite
+there.
+is non-negative. At the two points r=r1andr2we
+have ∆(ri) = 0, and Eq. (3.23) shows that both Kand
+KijKijbecome unbounded at these points, while Rre-
+mains finite. Therefore, in the present case there are also
+two scalar curvature singularities, located, respectively,
+atr=r1andr=r2forα > αc. Solutions with α≤αc
+are not physical.
+Case vi) β=β−= 1, λ= 3:In this case we have
+α−=β−= 1, and
+∆ =α
+x−1
+x2=/braceleftBig−∞, x= 0,
+0, x→ ∞.(3.30)
+Clearly, to have ∆ ≥0, we must assume α >0. Then,
+forr≥rs≡1/(√−ΛWα), ∆ is non-negative [See Fig.6,
+Curve (a)]. At r=rs, ∆(r) = 0, and Eq. (3.23)
+shows that both KandKijKijbecome unbounded at
+this point, while Rremains finite. As r→ ∞, ∆(r)→0,
+andKandKijKijbecome unbounded again, while R
+still remains finite. Therefore, in the present case there
+are also two scalar curvature singularities, located, re-
+spectively, at r=rsandr=∞.
+Case vii) β=β−, λ >3:In this case we have α−>
+1, β−>1, and
+∆ =/braceleftBig−∞, x= 0,
++1, x→ ∞.(3.31)
+Fig.6(b)showsthegeneralpropertiesof∆( r), fromwhich
+we can see that for any given λ, there always exists a
+pointr=rswhere ∆( rs) = 0. When r > rs, ∆ is
+always positive. Eq. (3.23) shows that now both Kand
+KijKijdiverge at r=rs, while all three scalars become
+unboundedas r→ ∞. Thus, inthe presentcasethereare
+two scalar curvature singularities, located, respectively,
+atr=rsandr=∞.8
+1
+0∆
+r(r)
+(a)(b)
+rs
+FIG. 6: The function ∆( x) defined by Eq. (3.22) for the α−
+branch with (a) λ= 3, α >0; and (b) λ >3 for any α.
+IV. CONCLUSIONS
+In this paper, we have studied singularities in the HL
+theory, and classified them into three different kinds, the
+scalar, non-scalar, and coordinate singularities, following
+the classification given in GR [36]. Due to the restricted
+diffeomorphisms (1.3), the number of the scalars that
+can be constructed from the extrinsic curvature tensor
+Kij, the 3-dimensional Riemann tensor Ri
+jkland their
+derivatives is much larger than that constructed from
+the 4-dimensional Riemann tensor Rσ
+µνλand its deriva-
+tives. The latter is invariant under the general Lorentz
+transformations (1.10). As a result, even for the same
+spacetime, it may be singular in the HL theory, but not
+singular in GR. One simple example is the anti-de Sitter
+Schwarzschild solution written in the ADM form (3.1).
+This solution is a solution of both HL theory and GR.
+However, in the HL theory, there are two scalar singu-
+larities located, respectively, at the origin r= 0 and
+r= (3M/|Λ|)1/3, as the two scalars KandKijKijbe-
+come singular at these points. It is well-known that the
+scalar singularity at r= (3M/|Λ|)1/3is absent in GR.
+This is because both KandKijKijare not scalars un-
+der the general Lorentz transformations (1.10). Thus,
+even they are singular at this point, it only represents a
+coordinate singularity, and all scalars constructed from
+the 4-dimensional Riemann tensor and its derivatives are
+finite. On the other hand, due to the restricted transfor-
+mations (1.3), KandKijKijare scalars in the HL the-
+ory, and once they are singular, the resulting singularity
+cannot be transferred away by the restricted transforma-
+tions. As a result, in the HL theory it represents a real
+spacetime singularity.
+With the above in mind, we have studied the LMP so-
+lutions [6], and found that their singularity behavior in
+the orthogonal frame defined by (3.9) is different from
+that in the ADM frame defined by (3.1) for the secondclass of the LMP solutions. In particular, in the orthog-
+onal frame, Kijvanishes, so do KandKijKij, while the
+3-dimensional Ricci scalar R[cf. Eq. (3.18)] can be sin-
+gular at the origin or infinity, depending on the choice of
+the parameter λ. However, in the ADM frame at least
+one of the three scalars K, KijKijandRis always sin-
+gular at two different points, either r= 0 andr=rs>0,
+orr=r1andr=r2withr2> r1>0, orr=rs>0 and
+r=∞, depending on the choice of the free parameter λ,
+wherersis a finite non-zero positive constant. This dif-
+ferent singular behavior originates from the fact that the
+two frames are related by the coordinate transformations
+(3.11), which is not allowed by the foliation-preserving
+diffeomorphisms (1.3), or in other words, K, KijKijand
+Rare not scalars under such transformations. In fact,
+in the framework of the HL theory, the two frames actu-
+ally represent two different HL theories, one is with the
+projectability condition, while the other is without. In
+particular, the second class of the LMP solutions in the
+orthogonalframe satisfy the vacuum HL equations, while
+in the ADM frame they satisfy the HL equations coupled
+with an anisotropic fluid with heat flow, as shown explic-
+itly in the Appendix.
+Our above results show clearly that the problem of
+singularities in the HL theory is a very delicate prob-
+lem, due to the restricted diffeomorphisms (1.3), which
+preserve the ADM foliations (1.2). Further investiga-
+tions are needed, in particular, in terms of the strength
+of these singularities. In the examples studied in this
+paper, all singularities indicated by the two scalars K
+andKijKijatr=rs>0, including that of the anti-de
+Sitter Schwarzschild solution, seems weak in the sense
+of tidal forces and distortions experiencing by observers.
+Therefore, it is not clear whether or not the spacetime is
+extendable across such a singularity [37].
+Finally, we would like to note that the ADM form (3.1)
+can be considered as a particular case of the HL theory
+without projectability condition. Therefore, restricting
+ourselves only to the HL theory without projectability
+condition does not solve the singularity problem occur-
+ring atr=rs.
+Acknowledgements:
+We would like to express our gratitude to Jared Green-
+wald and Antonios Papazoglou for valuable discussions
+and suggestions. Part of the work was supported by
+NNSFC under Grant 10535060, 10821504 and 10975168
+(RGC); and No. 10703005 and No. 10775119 (AZ).
+Appendix: The Generalized LMP Solutions with
+Projectability Condition
+The second class of the LMP solutions with pro-
+jectability condition takes the form of Eq. (3.1) with
+µandνgiven by Eq. (3.21). Unlike the first class,
+this one does not satisfy the HL vacuum equations with
+projectability condition, due to the fact that HL actions9
+are not invariant under the coordinate transformations
+(3.11). In particular, under these transformations, we
+have
+Rij→Rij+δRij. (A.1)
+Forsphericallysymmetricstaticsolutions, the extraterm
+δRijin generalgivesrise to an anisotropicfluid with heat
+flow [38], possibly subjected to some energy conditions
+[31]. In particular, for the LMP solutions of Eqs. (3.1)
+and (3.21), the energy density Jt, defined by
+Jt= 2/parenleftbigg
+NδLM
+δN+LM/parenrightbigg
+, (A.2)
+is given by the Hamiltonian equation,
+/integraldisplay
+d3x√g(LK+LV) = 8πG/integraldisplay
+d3x√gJt,(A.3)
+where
+LK=−ΛWe2(µ−ν)
+4x2∆/braceleftBig
+ξx2∆′2−8(1−ξ)x∆∆′
+−8(1−2ξ)x∆2/bracerightBig
+,
+LV=ΛW
+x2/bracketleftBig
+2+3x2+2/parenleftbig
+1−4α±/parenrightbig
+x2β±/bracketrightBig
++ΛW
+x8(1−α±)/braceleftBig/bracketleftbig
+8ξα2
+±−8(1−ξ)/parenleftbig
+1−α±/parenrightbig
++3/bracketrightbig
++2/bracketleftbig
+4(1−ξ)(1−α±)−1/bracketrightbig
+x2(1−2α±)
++(1−2ξ)x4(1−2α±)/bracerightBig
+,
+(A.4)
+where ∆ is given by Eq. (3.22), and ∆′≡d∆/dx. The
+quantity v, defined by
+Ji≡ −NδLM
+δNi=e−(µ+ν)/parenleftbig
+v,0,0/parenrightbig
+,(A.5)
+which is related to heat flow [38], is given by
+v=ΛWe2(µ−ν)
+32πGx2∆2/braceleftBigg
+ξx/bracketleftBig
+2x∆∆′′+x∆′2−2β±∆∆′/bracketrightBig
++4∆/bracketleftBig
+ξx∆′−2(1−ξ)β±∆/bracketrightBig
+−8ξ∆2/bracerightBigg
+.(A.6)
+On the other hand, the corresponding stress part τij
+defined by,
+τij=2√gδ/parenleftbig√gLM/parenrightbig
+δgij, (A.7)can be written in the form,
+τij=e2νprδi
+rδj
+r+r2pθΩij, (A.8)
+where Ωij≡δθ
+iδθ
+j+sin2θδφ
+iδφ
+j, and
+pr=ΛWe2(µ−ν)
+64πGx2∆2/braceleftBigg
+ξx/bracketleftBig
+4x∆∆′′−3x∆′2+4β±∆∆′/bracketrightBig
++8∆/bracketleftBig
+x∆′−2(1−ξ)β±∆/bracketrightBig
++8(1−4ξ)∆2/bracerightBigg
+−e−2ν
+8πGFrr,
+pθ=ΛWe2(µ−ν)
+16πGx2∆2/braceleftBigg
+(1−ξ)x2∆∆′′+1
+4ξx2∆′2
++(1−ξ)β±x∆∆′+2(1−2ξ)/parenleftbig
+x∆′+β±∆/parenrightbig
+∆/bracerightBigg
++ΛW
+8πGx2Fθθ, (A.9)
+with
+Frr=ΛW
+2x2/bracketleftBig
+2−2x−2β±−3x2(1−β±)/bracketrightBig
+−ΛW
+4x4−2β±/bracketleftBigg
+/parenleftbig
+19−32ξ/parenrightbig
++4(3−5ξ)/parenleftbig
+2−3β±/parenrightbig
+β±
+−2/parenleftbig
+7−12ξ/parenrightbig
+x−2β±−/parenleftbig
+5−8ξ/parenrightbig
+x−4β±/bracketrightBigg
+,
+Fθθ=3
+2x2−β±x2β±+1
+2x2−4β±/braceleftBigg
+(1+2ξ)
++2(11−16ξ)β±−2ξ/parenleftbig
+7−6β±/parenrightbig
+β2
+±
+−2/parenleftbig
+1−β±/parenrightbig
+x−2β±+(1−2ξ)x−4β±/bracerightBigg
+.(A.10)
+Clearly, to interpret the above as an anisotropic fluid
+with heat flow, some energyconditions [31] mightneed to
+be imposed. Since in this paper we are mainly concerned
+with the nature of singularities, we shall not study these
+conditions here.
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+arXiv:1001.0156v6  [quant-ph]  11 Jun 2010Entanglement distribution maximization over one-side Gau ssian
+noisy channel
+Xiang-Bin Wang,∗Zong-Wen Yu, and Jia-Zhong Hu
+Department of Physics and the Key Laboratory of Atomic and Na nosciences,
+Ministry of Education, Tsinghua University, Beijing 10008 4, China
+Abstract
+The optimization of entanglement evolution for two-mode Ga ussian pure states under one-side
+Gaussian map is studied. Even there isn’t complete informat ion about the one-side Gaussian noisy
+channel, one can still maximize the entanglement distribut ion by testing the channel with only two
+specific states.
+∗Electronic address: xbwang@mail.tsinghua.edu.cn
+1Introduction . The study of properties about quantum entanglement has drawn much
+interest for a long time[1–4]. Although initially quantum information pro cessing(QIP) was
+studied with discrete quantum states, it was then extended to the continuous variable (CV)
+quantum states[5]. So far, many concepts and results with 2-level quantum systems have
+been extended to the continuous variable case with parallel results , such as the quantum
+teleportation[6], the inseparability criteion[7], the degree of entang lement[8, 9], the entangle-
+ment purification[10–12], the entanglement sudden death[13], the c haracterization of Gaus-
+sian maps[14], and so on. However, this does not mean allresults with 2-level quantum
+systems can have parallel results for Gaussian states.
+Entanglement distribution is the first step towards many novel tas ks in quantum commu-
+nication and QIP[1]. In practice, there is no perfect channel for en tanglement distribution.
+Naturally, how to maximize the entanglement after distribution is an im portant question
+in practical QIP. If we distribute the quantum entanglement by sen ding one part of the
+entangled state to a remote place through noisy channel, we can us e the model of one-side
+noisy channel, or one-side map.
+Given the factorization law presented by Konrad et al[15], such a ma ximization problem
+for entanglement distribution over one-side map does not exist for the 2×2 system because
+any one-side map will produce the same entanglement on the output states provided that
+the entanglement of the input pure states are same. The result ha s been experimentally
+tested[16] and also been extended [17] recently. However, such a factorization does not hold
+for the continuous variable state as shown below. In this work, we c onsider the following
+problem: Initially we have a bipartite Gaussian pure state. Given a one -side Gaussian map
+(or a one-side Gaussian noisy channel), how to maximize the entangle ment of the output
+state by taking a Gaussian unitary transformation on the input mod e before it is sent to
+the noisy channel. We find that by testing the channel with only two d ifferent states, if a
+certain result is verified, then we can find the right Gaussian unitary transformation which
+optimizes the entanglement evolution for any input Gaussian pure st ate. That is to say, we
+canmaximize theoutput entanglement even thoughwe don’t have th efull informationofthe
+one-side map. In what follows we shall first show by specific example t hat the factorization
+law for 2 ×2 system presented by Konrad et al[15] does not hold for Gaussian s tates. We
+then present an upper bound of the entanglement evolution for init ial Gaussian pure states.
+Based on this, we study how to optimize the entanglement evolution o ver one-side Gaussian
+2map by taking a local Gaussian unitary transformation to the mode b efore sent to the noisy
+channel.
+Output entanglement of one-side Gaussian map and single-mo de squeezing. Most gener-
+ally, a two-mode Gaussian pure state is
+|g(U,V,q)/an}bracketri}ht=U⊗V|χ(q)/an}bracketri}ht (1)
+and|χ(q)/an}bracketri}ht=/radicalbig
+1−q2eqa†
+1a†
+2|00/an}bracketri}ht(−1≤q≤1) is a two-mode squeezed state (TMSS). We
+define map $ as a Gaussian map which acts on one mode of the state on ly. A Gaussian map
+changes a Gaussian state to a Gaussian state only. In whatever re asonable entanglement
+measure, the entanglement of a Gaussian pure state in the form of Eq.(1) is uniquely deter-
+mined byq. Therefore, we define the characteristic value of entanglement of the Gaussian
+pure stateρ(q) =|g(U,V,q)/an}bracketri}ht/an}bracketle{tg(U,V,q)|as
+E[ρ(q)] =|q|2. (2)
+On the other hand, any bipartite Gaussian pure state is fully charac terized by its covariance
+matrix (CM). Suppose the CM of state U⊗V|χ(q)/an}bracketri}htis
+Λ =
+A C
+CTB
+, (3)
+|q|2is uniquely determined by |A|(the determinant of the matrix A). So, to compare
+the entanglement of two Gaussian pure state, we only need to comp are|A|value of their
+covariance matrices.
+We start with the projection operator ˆTk(qα) which acts on mode konly:
+ˆTk(qα) =∞/summationdisplay
+n=0qn
+α|n/an}bracketri}ht/an}bracketle{tn|=qa†
+kak
+α. (4)
+This operator has an important mathematical property
+ˆTk(qα)(a†
+k,ak)ˆT−1
+k(qα) = (qαa†
+k,ak/qα) (5)
+which shall be used latter in this paper. For simplicity, we sometimes om it the subscripts of
+states and/or operators provided that the omission does not affe ct the clarity.
+Define the one-mode squeezed operator S(r) =er(a†2−a2)whereris a real number and
+bipartite state |ψr(q0)/an}bracketri}ht=I⊗S(r)|χ(q0)/an}bracketri}ht. We have
+3Theorem 1. Consider the one-side map I⊗ˆT(q1) acting on the initial state |ψr(q0)/an}bracketri}ht. The
+entanglement for the outcome state I⊗ˆT(q1)|ψr(q0)/an}bracketri}htis a descending function of |r|. Math-
+ematically, it is to say that if |r1|>|r2|then
+E[I⊗ˆT(q1)|ψr1(q0)/an}bracketri}ht]<E[I⊗ˆT(q1)|ψr2(q0)/an}bracketri}ht]. (6)
+This theorem actually shows that there isn’t a factorization law similar to that in 2 ×2
+states for the continuous variable states, in whatever good enta nglement measure. Using
+Backer-Compbell-Horsdorff (BCH) formula, up to a normalization fa ctor, we have
+|ψr(q0)/an}bracketri}ht=e−1
+2a†
+12q2
+0tanh(2r)+1
+2a†
+22tanh(2r)+q0a†
+1a†
+2
+cosh(2r)|00/an}bracketri}ht. (7)
+Detailed derivation of this identity is given in the appendix. Based on Eq .(4), the one-side
+mapI⊗ˆT(q1) changes state |ψr(q0)/an}bracketri}htinto
+|ψ′/an}bracketri}ht=ef1a†
+12+f2a†
+22+f3a†
+1a†
+2|00/an}bracketri}ht (8)
+wheref1=−1
+2q2
+0tanh(2r),f2=1
+2q2
+1tanh(2r), andf3=q0q1
+cosh(2r). Here we have omitted the
+normalization factor. Since we only need the covariance matrix of st ate|ψ′/an}bracketri}ht, the normaliza-
+tion can be disregarded because it does not change the covariance matrix. The characteristic
+function of state ρ′=|ψ′/an}bracketri}ht/an}bracketle{tψ′|has the form
+C(α1,α2) = tr[ρ′ˆD1(α1)ˆD2(α2)] =e−1
+2¯αΛ¯αT(9)
+whereˆDk(αk) =eαka†
+k−α∗
+kakand ¯α= (x1,y1,x2,y2) withαk=1√
+2(xk+iyk). Writing Λ here
+in the form of Eq.(3), we find A= diag[b1,b2],C= diag[c1,c2] andB= diag[d1,d2] with
+b1=−1
+2+1+2f2
+1+2f1+2f2+4f1f2−f2
+3,b2=−1
+2+1−2f2
+1−2f1−2f2+4f1f2−f2
+3,d1=−1
+2+1+2f1
+1+2f1+2f2+4f1f2−f2
+3,d2=
+−1
+2+1−2f1
+1−2f1−2f2+4f1f2−f2
+3,c1=−f3
+1+2f1+2f2+4f1f2−f2
+3,c2=f3
+1−2f1−2f2+4f1f2−f2
+3. The entanglement
+in whatever measure of state |ψ′/an}bracketri}htis a rising functional of |A|and
+|A|=1
+4+2q2
+0q2
+1
+1−4q2
+0q2
+1+q4
+1+q4
+0(1+q4
+1)+(1−q4
+0)(1−q4
+1)cosh(4r). (10)
+This is obviously a descending functional of |r|.
+Upper bound of entanglement evolution. SinceU⊗IandI⊗$ commute, the unitary
+operatorUplaces no role in the entanglement evolution under one-side map I⊗$, and hence
+we only need consider the initial state |g(I,V,q)/an}bracketri}ht=I⊗V|χ(q)/an}bracketri}ht=|ϕ(q)/an}bracketri}ht. We also define
+ρG(qα) =I⊗$(|ϕ(qα)/an}bracketri}ht/an}bracketle{tϕ(qα)|).
+4Using Eq.(5), one easily finds |ϕ(q=qaqb)/an}bracketri}ht=ˆT(qa)⊗I|ϕ(qb)/an}bracketri}ht. Since the operator
+ˆT(qa)⊗Iand the map I⊗$ commute, there is:
+ρG(q=qaqb) =ˆT(qa)⊗IρG(qb)ˆT†(qa)⊗I. (11)
+Using entanglement of formation[9, 18], we can calculate the entang lement of the state of a
+Gaussian state through its optimal decomposition form[9]. Suppose ρG(qb) has the following
+optimal decomposition[9]:
+ρG(qb) =U1⊗U2ρs(q0)U†
+1⊗U†
+2 (12)
+HereU1,U2are two local Gaussian unitaries and ρsis in the form
+ρs(q0) =/integraldisplay
+d2β1d2β2P(β1,β2)
+ˆD(β1,β2)|χ(q0)/an}bracketri}ht/an}bracketle{tχ(q0)|ˆD†(β1,β2),(13)
+whereP(β1,β2) is positive definite, ˆD(β1,β2) =ˆD1(β1)⊗ˆD2(β2) is a displacement operator
+defined as ˆDk(βk) =eβka†
+k−β∗
+kak. According to the definition of optimal decomposition[9,
+18], there don’t exist any other U1,U2and positive definite functional P(β1,β2) which can
+decompose ρG(qb) in the form of Eq.(12) with a smaller |q0|. The entanglement of ρG(qb)
+is equal to that of a TMSS |χ(q0)/an}bracketri}ht, i.e.q2
+0. For the Gaussian state ρG(qb) with its optimal
+decomposition of Eq.(12), we define the characteristic value of ent anglement of ρG(qb) as
+E[ρG(qb)] =|q0|2.
+Lemma 1. For any local Gaussian unitary Uand operator ˆT(qa), we can find θ,θ′andβ′′
+satisfying
+ˆT(qa)U1⊗U2·ˆD(β1,β2)|χ(q0)/an}bracketri}ht
+=R(θ′)⊗R(θ)·ˆD(β′
+1,β′
+2)·ˆT(qa)S(r)⊗U2|χ(q0)/an}bracketri}ht,(14)
+where,S(r) is a squeezing operator defined earlier, R(θ) is a rotation operator defined by
+R(θ)(a†,a)R†(θ) = (e−iθa†,eiθa),β′
+1,β′
+2andβ1,β2are related by a certain linear transfor-
+mation.
+Proof: Any local Gaussian unitary operator U1can be decomposed into the product form
+ofR(θ′)S(r)R(θ). Also, S(r)R(θ)⊗U2·ˆD(β1,β2) =ˆD(β′′
+1,β′′
+2)· S(r)R(θ)⊗U2. Define
+5ˆd=ˆT(qa)⊗I·ˆD(β′′
+1,β′′
+2)·ˆT−1(qa)⊗I, we have
+ˆT(qa)U⊗I·ˆD(β1,β2)|χ(q0)/an}bracketri}ht
+=ˆT(qa)R(θ′)S(r)R(θ)⊗I·ˆD(β1,β2)|χ(q0)/an}bracketri}ht
+=R(θ′)⊗I·ˆd·ˆT(qa)S(r)R(θ)⊗I|χ(q0)/an}bracketri}ht
+=R(θ′)⊗R(θ)·ˆD(β′
+1,β′
+2)·ˆT(qa)S(r)⊗I|χ(q0)/an}bracketri}ht.
+This completes the proof of Eq.(14). In the second equality above, we have used the fact
+ˆT(qa) andR(θ′) commute. Also, ˆdthere isnotunitary. However, using BCH formula and
+the vacuum state property ak|00/an}bracketri}ht= 0, we can always construct a unitary operator ˆD(β′
+1,β′
+2)
+so that the final equality above holds. Here β′
+1, β′
+2are certain linear functions of β1, β2.
+Using Eq.(11) and Eq.(12) with Eq.(14) we have
+E[ρG(q=qaqb)]
+=E[I⊗U2·ˆT(qa)U1⊗IρsU†
+1ˆT†(qa)⊗I·I⊗U†
+2]
+=E/bracketleftbigg
+R(θ′
+1)⊗U2R(θ1)/parenleftbigg/integraldisplay
+d2β1d2β2P(β1,β2)
+ˆD(β′
+1,β′
+2)·ˆT(qa)S(r1)⊗I|χ(q0)/an}bracketri}ht/an}bracketle{tχ(q0)|S†(r1)ˆT†(qa)
+⊗I·ˆD†(β′
+1,β′
+2)/parenrightBig
+R†(θ′
+1)⊗R†(θ1)U†
+2/bracketrightBig
+≤E/bracketleftbigg/integraldisplay
+d2β1d2β2P(β1,β2)ˆD(β′
+1,β′
+2)·ˆT(qa)⊗I
+|χ(q0)/an}bracketri}ht/an}bracketle{tχ(q0)|ˆT†(qa)⊗I·ˆD†(β′
+1,β′
+2)/bracketrightBig
+≤|qaq0|2=E[|χ(qa)/an}bracketri}ht/an}bracketle{tχ(qa)|]·E[ρG(qb)].(15)
+In the third step above we have used theorem 1 for the inequality sig n. This gives rise to
+the second theorem:
+Theorem 2. Using the entanglement formation as the entanglement measure, if the entan-
+glement of ρG(qb) is equal to that of TMSS |χ(q0)/an}bracketri}ht, the entanglement of ρG(q=qaqb) must
+be not larger than that of TMSS |χ(qaq0)/an}bracketri}ht. Mathematically, it is to say that if |q| ≤ |qb| ≤1
+we have
+E[I⊗$(|ϕ(q)/an}bracketri}ht/an}bracketle{tϕ(q)|)]
+E[I⊗$(|ϕ(qb)/an}bracketri}ht/an}bracketle{tϕ(qb)|)]≤E[|ϕ(q)/an}bracketri}ht/an}bracketle{tϕ(q)|]
+E[|ϕ(qb)/an}bracketri}ht/an}bracketle{tϕ(qb)|]. (16)
+Here|ϕ(q)/an}bracketri}ht=I⊗V|χ(q)/an}bracketri}htas defined earlier, Vcan be any Gaussian unitary operator.
+Definitely, the inequality also holds if we replace |ϕ(q)/an}bracketri}htby|g(U,V,q)/an}bracketri}htand replace |ϕ(qb)/an}bracketri}ht
+6by|g(U′,V,qb)/an}bracketri}ht, andU, U′can be arbitrary unitary operators. Theorem 2 also gives rise to
+the following corollary.
+Corollary 1. Given the one-side Gaussian map I⊗$, if the equality sign holds in formula
+(16) for two specific values q, qband 0<|q|<|qb| ≤1, then the equality sign there holds
+evenq,qbthere are replaced by any q′,q′′, respectively, as long as |q′|,|q′′| ∈[|q|,1].
+Proof. For simplicity, we first consider the case where qis replaced by any q′. (1) suppose
+|q′| ∈[|q|,|qb|]. The left side of formula (16) is equivalent to w′·z′, andw′=E[I⊗$(|ϕ(q)/an}bracketri}ht/an}bracketle{tϕ(q)|)]
+E[I⊗$(|ϕ(q′)/an}bracketri}ht/an}bracketle{tϕ(q′)|)]
+andz′=E[I⊗$(|ϕ(q′)/an}bracketri}ht/an}bracketle{tϕ(q′)|)]
+E[I⊗$(|ϕ(qb)/an}bracketri}ht/an}bracketle{tϕ(qb)|)]. The right side of formula (16) is equivalent to w·zandw=
+E[|ϕ(q)/an}bracketri}ht/an}bracketle{tϕ(q)|]
+E[|ϕ(q′)/an}bracketri}ht/an}bracketle{tϕ(q′)|]andz=E[|ϕ(q′)/an}bracketri}ht/an}bracketle{tϕ(q′)|]
+E[|ϕ(qb)/an}bracketri}ht/an}bracketle{tϕ(qb)|]. Theorem 2 itself says that w′≤wandz′≤z. If the
+equality sign holds in formula (16), we have w′·z′=w·zhence we must have w=w′and
+z=z′which is just corollary 1 in the case qis replaced by q′. (2) Suppose |q′|>|qb|. As we
+have already known, ρG(q) =ˆT(qa)⊗IρG(qb). Consider Eq.(14). Unitary U1in the optimal
+decomposition of Eq.(12) must be a rotation operator only, i.e., it con tains no squeezing,
+for, otherwise, according to theorem 1, E(ρG(q′)) is strictly less than q2
+0q2
+awhich means the
+equality in formula (16) does not hold.
+We denote q′=qb/qcand|qc|<1. We have
+ρG(q′=qb/qc)
+=ˆT−1(qc)⊗IρG(qb)/parenleftBig
+ˆT−1(qc)⊗I/parenrightBig†
+=ˆT−1(qc)⊗I·R1⊗U2ρsR†
+1⊗U†
+2·ˆT−1(qc)⊗I
+=R1⊗U2·/integraldisplay
+d2β1d2β2P(β1,β2)ˆD(β′
+1,β′
+2)
+|χ(q0/qc)/an}bracketri}ht/an}bracketle{tχ(q0/qc)|ˆD†(β′
+1,β′
+2)·R†
+1⊗U†
+2. (17)
+Here we have used ˆT−1(qc)⊗I|χ(qb=q′qc)/an}bracketri}ht=|χ(q′)/an}bracketri}ht. We have used the optimal decompo-
+sition forρG(qb) in the second equality, and lemma 1 in the last equality above. Eq.(17) is
+one possible decomposition of the state ρG(q′), but not necessarily the optimized decompo-
+sition. Therefore, E[ρG(q′=qb/qc)]≤ |q0|2/|qc|2=|q′|2/|qb|2·E[ρG(qb)]. On the other hand,
+according to theorem 2, we further obtain that E[ρG(qb=q′qc)]≤ |qb|2/|q′|2·E[ρG(q′)].
+Remark: Since here |q′| ≥qb, sign≤should be replaced by sign ≥in formula (16), when q
+is replaced by q′. These two inequalities and result of (1) lead to
+E[ρG(q′)]
+E[ρG(qb)]=E[|χ(q′)/an}bracketri}ht/an}bracketle{tχ(q′)|]
+E[|χ(qb)/an}bracketri}ht/an}bracketle{tχ(qb)|]. (18)
+7for anyq′provided that |q| ≤ |q′| ≤1. Replacing symbol q′above by symbol q′′, we have
+another equation. Comparing these two equations we conclude cor ollary 1.
+Lemma 2 : Given any Gaussian unitaries U, V, we have
+E[I⊗$(U⊗V|φ+/an}bracketri}ht/an}bracketle{tφ+|U†⊗V†)] =E[I⊗$(|φ+/an}bracketri}ht)]. (19)
+Here|φ+/an}bracketri}htis themaximally entangled statedefined as thesimultaneous eigensta te ofposition
+difference ˆx1−ˆx2and momentum sum ˆ p1+ ˆp2, with both eigenvalues being 0. Also, when
+q= 1, the state |χ(q)/an}bracketri}ht=|φ+/an}bracketri}ht. We shall use the following fact.
+Fact 1:For any local Gaussian unitary operators UandV, we can always find another
+Gaussian unitary operator Vso that
+U⊗V|φ+/an}bracketri}ht=V ⊗I|φ+/an}bracketri}ht. (20)
+Proof: Any local Gaussian unitary operator can be decomposed int o the product form of
+R(θ′)S(r)R(θ). For any TMSS |χ(q)/an}bracketri}htwe haveR(θ1)⊗R(θ2)|χ(q)/an}bracketri}ht=I⊗R(θ1+θ2)|χ(q)/an}bracketri}ht.
+For the maximally TMSS |φ+/an}bracketri}htwe haveS(r)⊗S(r)|φ+/an}bracketri}ht=|φ+/an}bracketri}ht, for, the both sides are the
+simultaneous eigenstates of position difference and momentum sum, with both eigenvalues
+being 0. This also means S(r)⊗I|φ+/an}bracketri}ht=I⊗S†(r)|φ+/an}bracketri}ht. SupposeV=R(θ′
+B)S(rB)R(θB),
+then
+U⊗V|φ+/an}bracketri}ht=V ⊗I|φ+/an}bracketri}ht (21)
+whereV=UR(θB)S†(rB)R(θ′
+B). This completes the proof of Eq.(20). If the equality sign
+in formula (16) holds, we can apply corollary 1 of theorem 2 through r eplacingqbby 1 and
+we obtain that E[ρG(q′)] =|q′|2·E[I⊗$(|φ+/an}bracketri}ht)]. On the other hand, by using theorem 2 and
+lemma 2 we have E[ρG(q′)]≤ |q′|2·E[I⊗$(|φ+/an}bracketri}ht)]. This means
+E[ρG(q′)] = max
+{V′}{E[I⊗$(|g(I,V′,q′)/an}bracketri}ht)]} (22)
+whereρG(q′) =I⊗$(|g(I,V,q′)/an}bracketri}ht/an}bracketle{tg(I,V,q′)|) as defined earlier, {V′}is the set containing all
+single-mode Gaussian unitary transformations. The equality holds f oranyq′provided that
+the equality of formula(16) holds for two specific values q, qband|q′| ≥ |q|. We arrive at
+the following major conclusion of this Letter:
+Major conclusion : Suppose that we have a TMSS |χ(q′)/an}bracketri}ht. We want to maximize the entan-
+glement distribution over a one-side Gaussian map I⊗$ by taking local Gaussian unitary
+8operationI⊗V′before entanglement distribution. Although we don’t have complete in-
+formation of the map I⊗$, it’s still possible for us to find out a specific Gaussian unitary
+operationVso that the entanglement distribution is maximized over all V′, for an initial
+state|χ(q′)/an}bracketri}htwithany|q′| ≥ |q|, as long as we can find two specific values |qb|>|q|, such
+that the equality sign in formula (16) holds. Obviously, the conclusion is also correct for
+any initial state which is a Gaussian pure state.
+The conclusion actually says that, in verifying that Vcan maximize the entanglement
+distribution for all initial states {|χ(q′)/an}bracketri}ht||q′| ≥ |q|}, we only need to verify the equality sign
+of formula (16) for two specific values.
+Experimental proposal . To experimentally test our major conclusion, we can consider the
+following beamsplitter channel: Initially, beams 1 and 2 are in a TMSS, wh ich is the initial
+bipartite Gaussian pure state. Beam 3 is in a squeezed thermal stat eρ3=˜S(u3)ρth˜S†(u3)
+here˜S(u) is a squeezing operator defined by ˜S(u)(ˆx,ˆp)˜S†(u) = (uˆx,ˆp/u) andρthis a ther-
+mal state whose CM is diag[ b3,b3]. Beam 3 together with the beamsplitter makes the
+one-side Gaussian channel. A beamsplitter will transform ˆ x2,ˆx3byUB(ˆx2,ˆx3)U−1
+B−→
+(ˆx2,ˆx3)
+cosθsinθ
+−sinθcosθ
+.In an experiment, we can take, e.g., q= 0.02 andqb= 0.5, test-
+ing with many different Vwe should find that the equality sign in formula (16) can hold
+withV=˜S(u2=u3) . Our major conclusion is verified if we can find that the same
+V=˜S(u3) always maximizes the output entanglement for any input state |χ(q′)/an}bracketri}htprovided
+that|q′| ≥0.02. Numerical calculation is shown in the following figure.
+123456789100.20.210.220.230.240.250.260.27The numerical example with q=2/3, θ=π/6, b3=1, u3=3.
+The squeezing factor u2.The entanglement← M(3,0.26451)
+FIG.1: Theentanglement withdifferentsqueezingfactor u2. Themaximumentanglement obtained
+whenu2=u3= 3. Here we set u3= 3 and q′= 2/3,θ=π/6,b3= 1.
+9In summary, we present an upper bound of the entanglement evolu tion of a 2-mode
+Gaussian pure state under one-side Gaussian map. We show that on e can maximize the
+entanglement distribution over an unknown one-side Gaussian noisy channel by testing the
+channel with only two specific states. An experimental scheme is pr oposed.
+Acknowledgement. This work was supported in part by the National Basic Research Pro -
+gram of China grant nos 2007CB907900 and 2007CB807901, NSFC g rant number 60725416,
+and China Hi-Tech program grant no. 2006AA01Z420.
+Appendix. Details of the proof of Eq.(7). We will use the following lemma.
+Lemma 2. IfAandBare two noncommuting operators that satisfy the conditions
+[A,[A,B]] = [B,[A,B]] = 0, (23)
+then
+eA+B=eAeBe−1
+2[A,B]. (24)
+This is a special case of the Baker-Hausdorff theorem of group the ory[19].
+The squeezing operator S(r) =er(a†2−a2)can be normally ordered as[20]
+S(r) =1/radicalbig
+cosh(2r)exp/bracketleftBigg
+a†2
+2tanh(2r)/bracketrightBigg
+·exp/bracketleftbig
+−a†a(ln(cosh(2r)))/bracketrightbig
+exp/bracketleftbigg
+−1
+2a2tanh(2r)/bracketrightbigg
+. (25)
+We neglect the constant of normalization in all the following calculation .
+I⊗S(r)|χ(q0)/an}bracketri}ht=er(a†
+22−a2
+2)eq0a†
+1a†
+2|00/an}bracketri}ht
+=eq0a†
+1(a†
+2cosh(2r)−a2sinh(2r))er(a†2
+2−a2
+2)|00/an}bracketri}ht
+=eq0a†
+1(a†
+2cosh(2r)−a2sinh(2r))e1
+2a†
+22tanh(2r)|00/an}bracketri}ht
+=e1
+2a†
+22tanh(2r)eq0a†
+1{a†
+2cosh(2r)−[a2+a†
+2tanh(2r)]sinh(2r)}|00/an}bracketri}ht
+=e1
+2a†
+22tanh(2r)eq0a†
+1(a†
+2
+cosh(2r)−a2sinh(2r))|00/an}bracketri}ht
+=e1
+2a†
+22tanh(2r)eq0a†
+1a†
+2
+cosh(2r)e−1
+2a†
+12q2
+0tanh(2r)|00/an}bracketri}ht
+(26)
+This is just Eq.(7). In the last equality we have used lemma 2. This comp letes the proof of
+10Eq.(7).
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+arXiv:1001.0157v3  [math.RA]  16 Dec 2011EVERY CENTRAL SIMPLE ALGEBRA IS HOPF
+SCHUR
+EHUD MEIR
+Abstract. We show that every central simple algebra Aover a
+fieldkis Brauer equivalent to a quotient of a finite dimensional
+Hopf algebra over the same field (that is- Ais Hopf Schur). If
+the characteristic of the field is zero, or if the algebra has a Galois
+splitting field ofdegreeprime to the characteristicof k, we can take
+this Hopf algebra to be semisimple. We also show that if Fis any
+finite extension of k, thenFis a quotient of a finite dimensional
+Hopf algebra over k. We use it in order to show why the algebric
+closeness assumption is necessary in a weak form of Kaplansky’s
+tenth conjecture, due to Stefan.
+1.Introduction
+Letkbeafield. In[1]weaskedwhatcentralsimple k-algebrascanwe
+get (up to Brauer equivalence) as quotients of finite dimensional Ho pf
+algebras over k. We called such algebras Hopf Schur algerbas, and we
+defined the Hopf Schur group of k,HS(k), as the subgroup of Br(k)
+which contains all classes of Hopf Schur algebras. This is analogue
+to the definition of the Schur group, S(k), which contains all classes
+of central simple algebras which are quotients of finite dimensional
+group algebrs, and also to the projective Schur group, PS(k), which
+contains all classes of central simple algebras which are quotients o f
+finite dimensional twisted group algebrs. Since any group algebra is
+a Hopf algebra, clearly S(k)< HS(k). The Schur group is a “small”
+subgroup of Br(k). It is known by a theorem of Brauer (see [8]) that
+any element in S(k) has a cyclotomic splitting field, and therefore if k
+contains all roots of unity then S(k) = 0, whereas Br(k) may be large
+(e.g.k=C(x1,x2,...,xn),n≤2, see [5]). We refer the reader to [9]
+andto [12] for acomprehensive acount ontheSchur group. In[1] it was
+shown that HS(k) might be much bigger than S(k). The authors have
+proved that PS(k)< HS(k). The projective Schur group is already
+a much bigger subgroup of the Brauer group. It was conjectured that
+PS(k) =Br(k) and this is indeed the case for many interesting fields
+(e.g. number fields). It was disproved, however, by Aljadeff and So nn
+in [2]. In [1] it was also proved that any product of cyclic algebras is
+Date: December 31, 2009.
+12 EHUD MEIR
+inHS(k), and there is a conjecture which says that this is all of the
+Brauer group. In this paper we will prove the following:
+Theorem 1.1. Anyk-central simple algebra is Bruaer equivalent to a
+quotient of a finite dimensional Hopf algebra (that is- Br(k) =HS(k)).
+If ak-central simple algebra Ahas a Galois splitting field Fsuch
+thatchar(k)∤|F:k|, thenAis Brauer equiavelnt to a quotient of
+a semisimple finite dimensional Hopf algebra.
+Since division algebras arise naturally as Endomorphism rings of
+simple representations, we have the following:
+Corollary 1.2. LetDbe ak-central division algebra. Then there is
+a finite dimensional Hopf algebra H, and a simple representation Vof
+Hsuch thatEndH(V)∼=D. If in addition Dhas a splitting field F
+such thatchar(k)∤|F:k|, we can take Hto be semisimple.
+Proof.LetDbe ak-central division algebra. By the above theorem,
+we have a Hopf algebra H(seimsimple in case char(k)∤|F:k|, where
+Fis some splitting field of D), and an algebra map π:H։Mn(Dop)
+for somen, whereDopis the algebra Dwith opposite multiplication.
+ThenV= (Dop)nis a representation of Hviaπ, and it is easy to see
+thatEndH(V)∼=D. /square
+In the course of the proof of Theorem 1.1 we will consider, in section
+3, forms of the function algebra of a finite group (i.e. the dual of a
+finite group algebra). As a result, we will see that there might be an
+infinite number of non-isomorphic semisimple and cosemisimple Hopf
+algebras over kof a given dimension, if kis not algebraically closed.
+In [11] Stefan have proved that for a given number n, there are only
+finitelymanyisomorphismclasses ofsemisimple andcosemisimple Hopf
+algebras of dimension nover an algebraically closed field. We there-
+fore conclude that the algebraic closeness assumption in Stefan’s T he-
+orem is necessary. Stefan’s Theorem is a weaker form of Kaplansky ’s
+tenth conjecture, which states that for a given number nthere are only
+finitely many isomorphism classes of (not necessarily semisimple and
+cosemisimple) Hopfalgebrasofdimension noveranalgebraicallyclosed
+field. Kaplansky’s tenth conjecture was disproved by Andruskiewit sch
+and Schneider (see [3]) by Beattie, Dascalescu and Grunenfelder (s ee
+[4]), and by Gelaki (see [6]) (and so, also the semisimplicity and the
+cosemisimplicity of the Hopf algebra is necessary in Stefan’s Theorem ).
+In order to prove Theorem 1.1 we will revise the construction from
+[1]. We will first review the relevant notions from Galois descent in
+Section 2. The main idea we shall use from Galois descent is that a
+Hopf algebra over kis the same thing as a Hopf algebra over a Galois
+extension of ktogether with a certain “semilinear” action. In section
+3 we use Galois descent in order to understand the forms of functio n
+algebras on finite groups. We use this in order to prove that any finit eTHE HOPF SCHUR GROUP 3
+field extension of kis a quotient of a Hopf algebra over k. Besides
+Galois descent, we shall also need to consider semidirect products o f
+Hopf algebras. The relevant details will be discussed in Section 4.
+Finally, in Section 5 we will make a reduction and show that it is
+enough to prove that any crossed product algebra for which the v alues
+of the cocycle are all roots of unity is a quotient of a Hopf algebra.
+We then show how we can construct such a Hopf algebra, using Galois
+descent and semidirect products.
+2.Galois descent
+We shall need to use a very small portion of the descent theory in
+here. The reader is referred to [7] for more comprehensive treat ment.
+LetL/kbe a Galois extension with a galois group G.
+Definition 2.1. LetVbea vector space over L. AG-semilinear action
+onVis an action of GonVas ak-vector space such that g(x·v) =
+g(x)·g(v), for every g∈G,x∈Landv∈V.
+We shall simply say “semilinear action” instead of “ G-semilinear
+action” if the group is clear from the context. The typical example w e
+should have in mind for a semilinear action is the following: suppose
+thatˆVis a vector space over k. ThenV=L⊗kˆVis a vector space over
+L, and we have a semilinear action given by g·(x⊗v) =g(x)⊗v. In
+descent theory it is proved that every semilinear action is of this for m.
+We shall need to use the following lemma, whose proof is rather easy:
+Lemma 2.2. LetHbe a Hopf algebra over L. Suppose that Hhas a
+semilinear action which respects the Hopf structure (i.e. i t commutes
+with all the Hopf algebra structure maps). Then HGis a Hopf algebra
+overk(the structure maps are just the restrictions of the structu re
+maps ofH).
+So in order to construct Hopf algebras over k, we can construct Hopf
+algebras over Ltogether with a semilinear action. In descent theory, all
+thesemilinear actionsareclassifiedbyacertainnonabeliancohomolog y
+group. Inourcaseitwouldbeeasiertofindsemilinear actionsdirectly .
+3.Semilinear actions on function algebras
+LetL,kandGbe as before, and let Tbe any finite group. We con-
+sider the function algebra (which is also the dual of the group algebr a
+ofT,L[T] = (LT)∗). This is the L-algebra of all the functions from
+TtoL. This algebra has a basis of simple idempotents {et}t∈T, where
+et(s) =δt,sfort,s∈T. This algebras also has a Hopf structure given
+by ∆(et) =/summationtext
+rs=ter⊗es. In this section we shall describe the semilin-
+ear actions of GonL[T] which respects the Hopf structure, and how
+the corresponding invariant algebras look like.4 EHUD MEIR
+Lemma 3.1. There is a one to one correspondence between semilinear
+actions on L[T]and homomorphism G→Aut(T)(i.e. actions of G
+onT).
+Remark 3.2. The reader who is familiar with descent theory will no-
+tice that this correspondence is exactly the correspondence bet ween
+k-forms ofk[T] andH1(G,Aut(T)), whereGacts trivially on Aut(T).
+Proof.The correspondence is given in the following way: for φ:G→
+Aut(T) we have the semilinear action gφ·xet=g(x)eφ(g)(t)forg∈G,
+x∈Landt∈T. Any semilinear action is of this form due to the
+following reason: since the action is by algebra automorphisms, ever y
+g∈Gpermutes the set of simple idempotents {et}t∈T, and so acts on
+T. The fact that the action preserves the coalgebra structure me ans
+that this permutation is an automorphism of Tas a group. /square
+Remark 3.3. In Section 3 of [1] we have described explicitly a spe-
+cific form of a specific function algebra. The situation there was the
+following: we had an abelian Galois extension L/kwith an (abelian)
+Galois group G. We have constructed a form for the function algebra
+k[Z2⋉G], whereZ2acts onGby inversions. If we denote the generator
+ofZ2byσ, then the map G→Aut(T) which corresponds to this form,
+is the map which sends g∈Gto the automorphism φgofZ2⋉Gwhich
+fixesGpointwise, and sends σtoσg.
+Let us describe, for a given φ:G→Aut(T), what would be the
+algebra of invariants ( L[T])G. We will describe only the algebra struc-
+ture and not the coalgebra structure, since it will be enough for ou r
+purposes. Let a=/summationtext
+t∈Tatet∈L[T]G. It is easy to see that the fact
+thatais invariant is equivalent to the fact that g(at) =aφ(g)(t)for
+everyg∈Gandt∈T. In particular at∈Lstab(t), where by stab(t)
+we denote the stabilizer of tinGwith respect to the action φ. If we
+fix representatives of the different orbits t1,...,tm, then we have an
+isomorphism of algebras
+Lstab(t1)⊕Lstab(t2)⊕...⊕Lstab(tm)→(L[T])G
+given by
+x/mapsto→/summationdisplay
+g∈G/stab(ti)g(x)eφ(g)ti
+forx∈Lstab(ti). Notice in particular that all the fields Lstab(ti)are
+quotient of the k-Hopf algebra ( L[T])G. Can we get any subfield of L
+that way? the answer is yes. Using Galoiscorrespondence, any sub field
+ofLis of the form LH, for someH < G. Therefore we need to prove
+that for every H < Gwe have a group Tand an action of GonT
+such thatTcontains an element tfor whichstab(t) =H. Let us take
+T=Z2G/H, the vector space over Z2with the coset space G/Has a
+basis. the group Gacts from the left on G/Hand therefore also on T.THE HOPF SCHUR GROUP 5
+It is clear that if we take t=H(the trivial coset), then stab(t) =H
+as required. Since Lwas an arbitrary Galois extension of k, and any
+finite extension of kis contained in its Galois closure, we have proved
+the following:
+Theorem 3.4. LetF/kbe any finite extension. Then there is a
+semisimple commutative Hopf algebra Hoverksuch thatFis a quo-
+tient ofH.
+Now assume that we have an infinite number of nonisomorphic Ga-
+lois extensions of kwith the same Galois group Gwhich satisfies the
+conditionchar(k)∤|G|(e.g.k=QandG=Z2). Then any Galois
+extensionLofkwith a Galois group Gis a quotient of a twisted form
+ofk[Z2G]. The algebra k[Z2G] (and each of its forms) is semisimple
+and cosemisimple (by the assumption on the order of G). It is easy to
+see that by considering all these forms, we get an infinite number of
+nonisomorpic commutative semisimple andcosemisimple Hopfalgebras
+of dimension 2|G|, as was claimed in Section 1.
+4.A semidirect product
+Inthis section we will construct semidirect products of Hopf algebr as
+overk. LetNandTbe groups such that Nacts onTby group
+automorphisms, i.e. wehaveahomomorphism ψ:N→Aut(T). Given
+ψ, we will construct a Hopf algebra Xψwhich would be the semidirect
+product of kNandk[T]. As a coalgebra, Xψwould bek[T]⊗kkN.
+The product in Xψis given by the rule
+et1⊗h1·et2⊗h2=δt1,ψ(h1)(t2)et1⊗h1h2.
+In other words- k[T] andkNare subalgebras of Xψ, andn∈Nacts
+by conjugation on etviaψ. The algebra Xψis a Hopf algebra. It is the
+bicrossed product of the Hopf algebras kNandk[T]. For the definition
+of bicrossed products in general, see [10]. We have an algebra map
+Xψ։kN. Therefore, if kNis not semisimple, the same is true for Xψ.
+On the other hand, a brief calculation shows that if kNis semisimple,
+then so isXψ. Using Maschke’s Theorem, we have the following
+Lemma 4.1. The algebra Xψis semisimpleif and only if char(k)∤|N|.
+5.A proof of Theorem 1.1
+In this section we will show that HS(k) =Br(k). As before, let L/k
+be a Galois extension with a Galois group G. Let [α]∈H2(G,L∗),
+where the action of GonL∗is the Galois action.
+Definition 5.1. We say that the cocycle αisfiniteif all its values are
+roots of unity.6 EHUD MEIR
+Note that this definition depends on the particular cocycle α, and
+not just on its cohomology class [ α]. We would like to prove that any
+central simple algebra is Brauer equivalent to a quotient of a finite
+dimensional Hopf algebra. We shall do this in the following way: Since
+anycentralsimplealgebraisknowntobeBrauerequivalent toacros sed
+product algebra, we will prove first that any crossed product alge bra
+Lα
+tGis Brauer equivalent to a crossed product algebra Kβ
+tNin which
+the cocycle βis finite. We then prove that any such crossed product is
+a quotient of a finite dimensional Hopf algebra.
+Lemma 5.2. Thecrossedproduct Lα
+tGis Brauerequivalentto a crossed
+product algebra Kβ
+tNwithβfinite.
+Proof.Letmbe the order of αinH2(G,L∗). Suppose that f:G→L∗
+satisfies
+αm(g1,g2) =∂f(g1,g2) =f(g1)f(g2)f−1(g1g2),
+forg1,g2∈G. LetKbe a Galois extension of Lwhich contains ele-
+mentsrg, for everyg∈G, for which the equations rm
+g=f(g) hold. If
+we denote by Nthe Galois group of Koverk, we have an onto map
+π:N։G. Define a two cocycle β∈H2(N,K∗) by
+β(h1,h2) =α(π(h1),π(h2))r−1
+π(h1)r−1
+π(h2)rπ(h1h2).
+A direct calculation shows that all the values of βarem-th roots of
+unity, and so βis finite. The cocycle βis cohomologous to infN
+G(α).
+By Brauer theory we thus know that the central simple algebras Lα
+tG
+andKβ
+tNare Brauer equivalent, as required. /square
+We will therefore assume from now, without loss of generality, that
+all the values of αare roots of unity. Recall that Lα
+tGhas anL-
+basis{Ug}g∈G, and the multiplication in Lα
+tGis given by xUg·yUh=
+xg(y)α(g,h)Ugh. By considering the subgroup of Lα
+tGgenerated by the
+Ug’s, we get an extension of finitegroups
+1→µ→/hatwideG→G→1,
+whereµis thefinitesubgroup of L∗generated by all elements of the
+formα(g1,g2), forg1,g2∈G. LetTbe the group Z2G, the vector space
+overZ2with basisG(multiplication in Tis just addition of vectors).
+We have a natural action of G(and thus of /hatwideG, using the map /hatwideG→G)
+onTby left multiplication. If we denote the action of /hatwideGonTbyψ, we
+can construct the semidirect product Xψ, which would be k[T]⊗kk/hatwideG
+as a vector space. Notice that by Lemma 4.1 Xψis semisimple if and
+only ifchar(k)∤|G|(it is easy to see, by the proof of Lemma 5.2 and
+by the fact that the order of αinH2(G,L∗) divides the order of G, that
+the prime divisors of |µ|are also prime divisors of |G|). Now consider
+the induced L-Hopf algebra XL=L⊗kXψ. We have an action of GTHE HOPF SCHUR GROUP 7
+onTnot only from the left but also from the right by multiplication.
+We define an action of GonXLvia
+g⋆(l⊗et⊗h) =g(l)⊗et·g−1⊗h.
+We claim the following
+Lemma 5.3. The action ⋆is a semilinear action of GonXL.
+Proof.This is a straightforward verification. The crux of the proof is
+the fact that the two actions of GonTfrom the left and from the right
+commute with each other. /square
+Consider now the k-Hopf algebra H= (XL)G, where we are taking
+invariants with respect to the ⋆action. We claim that we have an onto
+mapH։Lα
+tG. To see why this is true, notice first that we have a
+decomposition of Has a direct sum of two sided ideals H=H1⊕H2.
+The idealH1is the intersection of ( XL)Gwith theLsubspace spanned
+by all 1⊗et⊗ˆg, where ˆg∈/hatwideG, andt∈G(We can consider Gas the
+subset ofTwhich contains the basis elements). The ideal H2is the
+interesection of ( XL)Gwith theLsubspace spanned by all 1 ⊗et⊗ˆg,
+where ˆg∈/hatwideGandt /∈G. So it is enough to prove that we have an onto
+mapH1։Lα
+tGof algebras. It is easy to see that H1is spanned by
+elements of the form/summationtext
+g∈Gg(l)⊗eg−1⊗ˆu, wherel∈Land ˆu∈/hatwideG. So
+H1hask/hatwideGandLas subalgebras. The inclusion of Las a subalgebra
+ofH1is given by l/mapsto→/summationtext
+g∈Gg(l)⊗eg−1⊗1, and the inclusion of k/hatwideG
+as a subalgebra of H1is given by x/mapsto→1⊗1⊗x. It is easy to see
+thatH1is the semidirect product of k/hatwideGwithL, where the action of
+/hatwideGonLis given by the Galois action of GonL(recall that Gis a
+quotient of /hatwideG). We would like to define now an algebra map from H1
+ontoLα
+tG. To do this, recall that the group /hatwideGwas constructed as a
+subgroup of the group of invertible elements in Lα
+tG. We therefore
+have a natural inclusion map /hatwideGi→Lα
+tG. We can now define an algebra
+homomorphism Φ : H1→Lα
+tGin the following way: since H1is the
+semidirect product of k/hatwideGandL, it is enough to define Φ on products
+of the form lˆgwherel∈Land ˆg∈/hatwideG. We define Φ( lˆg) =li(ˆg). It
+is easy to see that Φ is a well defined onto algebra map. This proves
+Theorem 1.1.
+References
+[1] E. Aljadeff, J. Cuadra, S. Gelaki and E. Meir, On the Hopf Schur g roup of a
+field, Journal of Algebra 319, 12 (2008), 5165-5177. arXiv:0708.1 943v1.
+[2] E. Aljadeff and J. Sonn, On the projective Schur group of a field, Journal of
+Algebra Volume 178, Issue 2, 1 December 1995, Pages 530-540
+[3] N. Andruskiewitsch and H. J. Schneider, Lifting of Quantum Linea r Spaces
+and Pointed Hopf Algebras of Order p3, Journal of Algebra Volume 209, Issue
+2, (1998), Pages 658-691.8 EHUD MEIR
+[4] M. Beattie, S. Dascalescu and L. Grunenfelder, On the number o f types of
+finite dimensional Hopf algebras, Inventiones Mathematicae 136 (1 999), p.1-7.
+[5] B. Fein and M. Schacher, Brauer groups of rational function fie lds, in ”Groupe
+de Brauer”, Lecture Notes in Mathematics, vol 844, Springer-Ve rlag, New
+York/Berlin, (1981).
+[6] S. Gelaki, Pointed Hopf Algebras and Kaplansky’s 10th Conjectur e, Journal
+of Algebra Volume 209, Issue 2, 15 November (1998), Pages 635-6 57
+[7] P. Gille and T. Szamuely- Central Simple Algebras and Galois Cohomolo gy,
+Cambridge Studies in Advanced Mathematics, Cambridge University P ress,
+(2006).
+[8] I. M. Issacs, Character Theory of Finite Groups, Dover Publica tions, (1994).
+[9] G. J. Janusz, The Schur group of an algebraic number field. Ann. of Math. (2)
+103 (1976), no. 2, 253–281.
+[10] C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer-
+Verlag, 1995.
+[11] D. Stefan, The Set of Types of n-Dimensional Semisimple and Cos emisimple
+Hopf Algebras Is Finite, Journal of Algebra 193, 571-580 (1997).
+[12] T. Yamada, The Schur Subgroup of the Brauer Group, Lectur e Notes in Math-
+ematics397, Springer-Verlag, 1970.arXiv:1001.0157v3  [math.RA]  16 Dec 2011EVERY CENTRAL SIMPLE ALGEBRA IS BRAUER
+EQUIVALENT TO A HOPF SCHUR ALGEBRA
+EHUD MEIR
+Abstract. We show that every central simple algebra Aover a field k
+is Brauer equivalent to a quotient of a finite dimensional Hop f algebra
+over the same field. This shows that the natural generalizati on of the
+Schur group for Hopf algebras (which we call the Hopf Schur gr oup) is
+in fact the entire Brauer group of k. If the characteristic of the field is
+zero, or if the algebra has a Galois splitting field with certa in properties,
+we can take this Hopf algebra to be semisimple. We also show th at if
+Fis any finite separable extension of k, thenFis a quotient of a finite
+dimensional commutative semisimple and cosemisimple Hopf algebra
+overk.
+1.Introduction
+1Letkbe a field. In [1] we asked what central simple k-algebrascan we get
+(up to Brauer equivalence) as quotients of finite dimensional Hopf a lgebras
+overk. We called such algebras Hopf Schur algebras, and we defined the Ho pf
+Schur group of k,HS(k), as the subgroup of Br(k) which contains all classes
+of Hopf Schur algebras. This is analogue to the definition of the Schu r group,
+S(k), which contains all classes of central simple algebras which are quo tients
+of finite group algebras, and also to the projective Schur group, PS(k), which
+contains all classes of central simple algebras which are quotients o f finite
+dimensional twisted group algebras.
+Since any group algebra is a Hopf algebra, clearly S(k)< HS(k). The
+Schur group is a “small” subgroup of Br(k). It is known by a theorem of
+Brauer (see [9]) that any element in S(k) has a cyclotomic splitting field, and
+therefore if kcontains all roots of unity then S(k) = 0, whereas Br(k) may
+be large (e.g. k=C(x1,x2,...,xn),n≥2, see [6]). We refer the reader to
+[10] and to [16] for a comprehensive account on the Schur group. I n [1] it was
+shown that HS(k) might be much bigger than S(k). The authors have proved
+thatPS(k)< HS(k). The projective Schur group is already a much bigger
+subgroup of the Brauer group. It was conjectured that PS(k) =Br(k) and
+this is indeed the case for many interesting fields (e.g. number fields) . It was
+Date: 23 November, 2010.
+1AMS subject classification: 16T20, 16K50
+12 EHUD MEIR
+disproved, however, by Aljadeff and Sonn in [2]. In [1] it was also prove d that
+any product of cyclic algebras is in HS(k), and there is a conjecture which
+says that this is all of the Brauer group.
+In this paper we will prove that HS(k) =Br(k). In addition, we will give
+sufficient conditions for a central simple k-algebra to be Brauer equivalent to
+a quotient of a semisimple Hopf algebra. For that reason, we introduce the
+following definition:
+Definition 1.1. LetAbe a central simple k-algebra. Denote by mthe order
+of [A] inBr(k). We will say that the algebra Aisgoodifchar(k) = 0,
+or ifchar(k) =p,p∤m, andAhas a Galois splitting field Lsuch that
+p∤|L(ξm) :k|, whereξmis a primitive mth root of unity.
+The main result of this paper is the following:
+Theorem 1.2. Anyk-central simple algebra Ais Bruaer equivalent to a
+quotient of a finite dimensional Hopf algebra H. (that is-Br(k) =HS(k)).
+IfAis a good algebra, then Ais Brauer equivalent to a quotient of a finite
+dimensional semisimple Hopf algebra.
+Since division algebras arise naturally as Endomorphism rings of simple
+representations, we have the following:
+Corollary 1.3. LetDbe ak-central division algebra. Then there is a finite
+dimensional Hopf algebra H, and a simple representation VofHsuch that
+EndH(V)∼=D. If in addition Dis good, then we can take Hto be semisimple.
+Proof.LetDbe ak-central division algebra. By the above theorem, we
+have a Hopf algebra H(seimsimple in case Dis good), and an algebra map
+π:H։Mn(Dop) for some n, whereDopis the algebra Dwith opposite
+multiplication. Then V= (Dop)nis a representation of Hviaπ, and it is
+easy to see that EndH(V)∼=D. /square
+Remark 1.4. In the proof above we have used the fact that Dis good if and
+only ifDopis good.
+In the course of the proof of Theorem 1.2 we will consider, in Section 3,
+forms of the function algebra of a finite group (i.e. the dual of a finit e group
+algebra). As a result, we will see that there might be an infinite numbe r of
+non-isomorphic semisimple and cosemisimple Hopf algebras over kof a given
+dimension,if kisnotalgebraicallyclosed. In[15]Stefanprovedthatforagiven
+numbern, there are only finitely many isomorphism classes of semisimple and
+cosemisimple Hopf algebras of dimension nover an algebraically closed field.
+We therefore conclude that the algebraic closeness assumption in S tefan’s
+Theorem is necessary (this was observed also by Caenepeel Dasca lescu and
+Le Bruyn in [5]). Stefan’s Theorem is a weaker form of Kaplansky’s ten th
+conjecture, which states that for a given number nthere are only finitelyTHE HOPF SCHUR GROUP 3
+many isomorphism classes of (not necessarily semisimple and cosemisim ple)
+Hopf algebras of dimension nover an algebraically closed field. Kaplansky’s
+tenth conjecture was disproved by Andruskiewitsch and Schneide r (see [3]) by
+Beattie, Dascalescu and Grunenfelder (see [4]), and by Gelaki (see [7]) (and
+so, also the semisimplicity and the cosemisimplicity of the Hopf algebra is
+necessary in Stefan’s Theorem).
+The main idea behind the proof of Theorem 1.2 will be the followiong
+observation: if L/kis aGaloisextension offields, then a Hopfalgebraover kis
+equivalent to a Hopf algebra over Ltogether with a certain “Hopf-semilinear”
+action. This generalizes the idea from descent theory, that an alge bra overk
+is the same thing as an algebra over Ltogether with a certain “semilinear”
+action. In descent theory, this idea gives a description of the relat ive Brauer
+groupBr(L/k) (that is- of all Brauer classes in Br(k) which split by L). In
+our setting, this idea gives us a tool to construct a variety of nonis omorphic
+Hopf algebras over k(which become isomorphic over L). We will be able to
+show in Section 5 that by choosing the “right” Hopf algebra and the “ right”
+Hopf-semilinear action, we will be able to “catch” all the Brauer equiv alence
+classes inBr(L/k). SinceLis arbitrary, this will prove that HS(k) =Br(k).
+We will begin by defining what are Hopf-semilinear actions in Section 2,
+where we will also describe the relevant parts from descent theory which will
+be in use. In Section 3 we use descent theory in order to describe all the
+k-forms of the Hopf algebra L[T] of functions on some finite group T(by this
+we mean all the k-Hopf algebras which become isomorphic to L[T] overL).
+Another tool which we will need will be semidirect products of Hopf alg ebras.
+This will be described in Section 4. In Section 5 we will prove Theorem 1.2 .
+2.Galois descent
+We shall need to use a very small portion of the descent theory in he re.
+The reader is referred to the second chapter of [8] for more com prehensive
+treatment. Let L/kbe a Galois extension with a Galois group G.
+Definition 2.1. LetVbe a vector space over L. AG-semilinear action on
+Vis an action of GonVas ak-vector space such that g(x·v) =g(x)·g(v),
+for everyg∈G,x∈Landv∈V.
+We shall simply say “semilinear action” instead of “ G-semilinear action” if
+the group is clear from the context. The typical example we should h ave in
+mind for a semilinear action is the following: suppose that ˆVis a vector space
+overk. ThenV=L⊗kˆVis a vector space over L, and we have a semilinear
+action given by g·(x⊗v) =g(x)⊗v. In descent theory it is proved that
+every semilinear action is of this form (this is Speiser Lemma, see [8], pp . 27).
+In this paper, we will be interested in semilinear actions which also comm ute
+with the Hopf structure.4 EHUD MEIR
+Definition 2.2. LetHbe a Hopf algebra over L. AG-semilinear action on
+His said to be Hopf-semilinear if it commutes with the Hopf structure of H.
+That is- for every g∈Ganda,b∈Hwe haveg(ab) =g(a)g(b), and similar
+equations hold for the coproduct, counit, unit and antipode.
+We will need to use the following lemma, whose proof is based on Speiser
+Lemma together with general arguments.
+Lemma 2.3. LetHbe a Hopf algebra over L. Suppose that Hhas aG-Hopf-
+semilinear action. Then the subspace of invariant elements HGis a Hopf
+algebra over k(the structure maps are just the restrictions of the structu re
+maps ofH), andHG⊗kL∼=HasL-Hopf algebras.
+Remark 2.4. The algebra HGis called ak-form of the L-Hopf algebra H.
+We thus see that in order to construct Hopf algebras over k, we can con-
+struct Hopf algebras over Ltogether with semilinear actions. In descent the-
+ory, allthe semilinearactionsofagivenHopfalgebraareclassifiedby acertain
+nonabeliancohomologygroup. In ourcaseitwouldbeeasiertofind se milinear
+actionsdirectly. FormsofHopfalgebraswerealsoconsideredbyRa dford, Taft
+and Wilson in [14], by Pareigis in [12], by Parker in [13], and by Caenepeel,
+Dascalescu and Le Bruyn in [5].
+3.Hopf-semilinear actions on function algebras
+LetL,kandGbe as before, and let Tbe any finite group. We consider
+the function algebra (which is also the dual of the group algebra of T,L[T] =
+(LT)∗). This is the L-algebra of all the functions from TtoL. This algebra
+has a basis of simple idempotents {et}t∈T, whereet(s) =δt,sfort,s∈T.
+This algebras also has a Hopf structure given by ∆( et) =/summationtext
+rs=ter⊗es. In
+this section we will describe the Hopf-semilinear actions of GonL[T] and how
+the corresponding invariant Hopf algebras look like.
+Lemma 3.1. There is a one to one correspondence between Hopf-semilinea r
+actions onL[T]and homomorphism G→Aut(T)(i.e. actions of GonT).
+Remark 3.2. The reader who is familiar with descent theory will notice that
+this correspondence is exactly the correspondence between k-forms ofL[T]
+andH1(G,Aut(T)), whereGacts trivially on Aut(T).
+Proof.The correspondence is given in the following way: for φ:G→Aut(T)
+we have the semilinear action gφ·xet=g(x)eφ(g)(t)forg∈G,x∈Land
+t∈T. Any Hopf-semilinear action is of this form due to the following rea-
+son: since the action is by algebra automorphisms, every g∈Gpermutes
+the set of simple idempotents {et}t∈T, and so acts on T. The fact that the
+action preserves the coalgebra structure means that this permu tation is an
+automorphism of Tas a group. /squareTHE HOPF SCHUR GROUP 5
+Remark 3.3. In Section3 of[1]we havedescribedexplicitly aspecific formof
+a specific function algebra. Let us describe the corresponding Hop f-semilinear
+action. We have an abelian Galois extension L/kwith an (abelian) Galois
+groupG. We have constructed a k-formHof theL-Hopf algebra L[Z2⋉G]
+(the action of Z2=/an}bracketle{tσ/an}bracketri}htis by inversion) which is isomorphic as an algebra to
+L⊕k[G]. The form Hcorresponds to a semilinear action, which corresponds,
+by Lemma 3.1, to a homomorphism φ:G→Aut(Z2⋉G). Forg∈G, the
+homomorphism φ(g) is the homomorphism which sends σtoσgand fixesG
+pointwise.
+Let us describe, for a given φ:G→Aut(T), the structure of the algebra
+of invariants ( L[T])G. Leta=/summationtext
+t∈Tatet∈L[T]G. It is easy to see that the
+fact thatais invariant is equivalent to the fact that g(at) =aφ(g)(t)for every
+g∈Gandt∈T. In particular at∈Lstab(t), where by stab(t) we denote the
+stabilizer of tinGwith respect to the action φ. If we fix representatives of
+the different orbits t1,...,tm, then we have an isomorphism of algebras
+Lstab(t1)⊕Lstab(t2)⊕...⊕Lstab(tm)→(L[T])G
+given by
+x/mapsto→/summationdisplay
+g∈G/stab(ti)g(x)eφ(g)ti
+forx∈Lstab(ti). Notice in particular that all the fields Lstab(ti)are quotient
+of thek-Hopf algebra ( L[T])G. Can we get any subfield of Lin this way? the
+answer is yes. Using Galois correspondence, any subfield of Lis of the form
+LH, for someH < G. Therefore we need to prove that for every H < Gwe
+have a group Tand an action of GonTsuch thatTcontains an element t
+such thatstab(t) =H. Let us take T=Z2G/H, the vector space over Z2
+with the coset space G/Has a basis. The group Gacts from the left on G/H
+and therefore also on T. It is clear that if we take t=H(the trivial coset),
+thenstab(t) =Has required. Since Lwas an arbitrary Galois extension of k,
+and any finite separable extension of kis contained in its Galois closure, we
+have (almost) proved the following:
+Theorem 3.4. LetF/kbe any finite separable extension. Then there is a
+semisimple cosemisimple commutative Hopf algebra Hoverksuch thatFis
+a quotient of H.
+Proof.The only thing that requires a proof is the cosemisimplicity. The func-
+tion algebra k[T] is cosemisimple if and only if char(k)∤|T|(this is Maschke’s
+Theorem applied to the dual Hopf algebra). If char(k)/ne}ationslash= 2 we can take
+T=Z2G/Has above, and if char(k) = 2, we can take T=Z3G/H./square
+Now assume that we have an infinite number of nonisomorphic Galois ex -
+tensions of kwith the same Galois group Gwhich satisfies the condition
+char(k)∤|G|(e.g.k=QandG=Z2). Then any Galois extension Lofk6 EHUD MEIR
+with a Galois group Gis a quotient of a twisted form of k[Z2G]. The algebra
+k[Z2G] (and each of its forms) is semisimple and cosemisimple (if char(k) = 2
+we can take k[Z3G] as in the proof of the theorem above). It is easy to see
+that by considering all these forms, we get an infinite number of non isomorpic
+commutative semisimple and cosemisimple Hopf algebras of dimension 2|G|,
+as was claimed in Section 1. This generalizes the example given in Section 2
+of [5] which shows that over a non algebraically closed field a group alge bra
+of an abelian group can have infinitely many nonisomorphic forms.
+4.A semidirect product
+In this section we will construct some specific semidirect products o f Hopf
+algebras over k. LetNandTbe groups such that Nacts onTby group
+automorphisms (that is- we have a homomorphism ψ:N→Aut(T)). We
+will construct a Hopf algebra k[T]⋊kNwhich we call the semidirect product
+ofkNandk[T]. As a coalgebra, k[T]⋊kNisk[T]⊗kkN. The product in
+k[T]⋊kNis given by the rule
+et1⊗n1·et2⊗n2=δt1,ψ(n1)(t2)et1⊗n1n2.
+In other words- k[T] andkNare subalgebras of k[T]⋊kN, andn∈Nacts
+by conjugation on etviaψ. The algebra k[T]⋊kNis a Hopf algebra. It is
+the bicrossed product of the Hopf algebras kNandk[T]. For the definition of
+bicrossed products in general, see Chapter IX.2 of [11].
+Remark 4.1. It can be seen that k[T]⋊kNis isomorphic as an algebra to a
+directsum oftheform/circleplustext
+iMni(kHi), whereHiaresubgroupsof Nwhicharise
+as stabilizers of element in T. Therefore, if char(k)∤|N|, then the semidirect
+productk[T]⋊kNis semisimple, and thus also every form of it. We will need
+this observation later, in order to deal with semisimplicity questions.
+5.A proof of Theorem 1.2
+In this section we will show that HS(k) =Br(k). LetAbe ak-central
+simple algebra. The algebra Asplits by some Galois extension L/k. Let
+G=Gal(L/k). By Galois descent, we know that Ais equivalent to a crossed
+product algebra Lα
+tG, whereα∈H2(G,L∗), and the action on L∗is the
+Galois action. This algebra has an L-basis{Uσ}σ∈G, and the multiplication
+is given by the rule:
+xUgyUh=xσ(y)α(g,h)Ugh
+whereg,h∈Gandx,y∈L. Wewillshowthat Ais(uptoBrauerequivalence)
+a quotient of a Hopf algebra. We begin with the following definition:
+Definition 5.1. We say that the cocycle αisfiniteif all its values are roots
+of unity.THE HOPF SCHUR GROUP 7
+Note that this definition depends on the particular cocycle α, and not just
+on its cohomology class [ α]. We will prove that Ais Brauer equivalent to a
+quotient of a Hopf algebra in the following way: we will first show that A
+is Brauer equivalent to a product of cyclic algebras with a crossed pr oduct
+algebra in which the cocycle is finite, and then we will prove that a cros sed
+product algebra with a finite cocycle is a quotient of a Hopf algebra. I n [1]
+we have proved that any cyclic algebra is a quotient of a Hopf algebra , and
+therefore [A] (the Brauer class of A) is inHS(k).
+Remark 5.2. In casechar(k) = 0,Ais equivalent to a crossed product
+algebra with a finite cocycle, and we do not need to use the result fro m [1].
+The following lemma seems to be well known. We have included it here
+nevertheless, as it makes our construction more explicit.
+Lemma 5.3. Letα∈H2(G,L∗). Denote the order of αbym. Ifchar(k) = 0
+or ifchar(k) =pandp∤m, then the crossed product algebra Lα
+tGis Brauer
+equivalent to a crossed product algebra Kβ
+tNwhereβis a finite cocycle.
+Proof.Since the order of αism, there is a function f:G→L∗which satisfies
+αm(g1,g2) =∂f(g1,g2) =f(g1)g1(f(g2))f−1(g1g2),
+forg1,g2∈G. LetKbe a Galois extension of Lwhich contains, for every
+g∈G, an element rgwhich satisfies rm
+g=f(g) (the fact that we have such
+a Galois extension follows from the assumption on mandchar(k)). If we
+denote byNthe Galois group of Koverk, we have an onto map π:N։G.
+Define a two cocycle β∈H2(N,K∗) by
+β(h1,h2) =α(π(h1),π(h2))r−1
+π(h1)h1(r−1
+π(h2))rπ(h1h2).
+A direct calculation shows that all the values of βarem-th roots of unity, and
+soβis finite. The cocycle βis cohomologous to infN
+G(α). By Brauer theory
+we thus know that the central simple algebras Lα
+tGandKβ
+tNare Brauer
+equivalent, as required. /square
+Remark 5.4. The fieldKin the lemma can be taken to be L(ξm,{rg}g∈G).
+Notice that if ord([A]) =mis prime to p, and ifp∤|L(ξm) :k|, then also
+p∤|L(ξm,{rg}g∈G) :k|. Thus, the lemma implies that if Ais a good algebra,
+thenAis Brauer equivalent to a crossed product algebra Kβ
+tGwith a finite
+cocycle, such that p∤|K:k|. By the next proposition, this implies that A
+is Brauer equivalent to a quotient of a finite dimensional semisimplie Hop f
+algebra.
+Suppose now that kis a field of characteristic p, and that the order of α,m,
+is not prime to p. Writem=rpe, whereris prime to p. ThenLα
+tGis Brauer
+equivalent to a tensor product of the form Lα1
+tG⊗kLα2
+tGwhereord(α1) =r
+andord(α2) =pe. By a theorem of Teichmueller, if char(k) =pthen any8 EHUD MEIR
+central simple algebra whose order is peis Brauer equivalent to a product of
+cyclic algebras (see Chapter 9.1 of [8]), and Brauer classes of cyclic a lgebras
+are inHS(k) (see [1]). By the lemma above, Lα1
+tGis Brauer equivalent to
+a crossed product algebra in which the cocycle is finite. The proof of the
+following proposition together with the above remark therefore fin ishes the
+proof of Theorem 1.2:
+Proposition 5.5. LetA=Lα
+tGbe a crossed product algebra such that αis
+finite. Then Ais a quotient of a Hopf algebra. If char(k)∤|G|, thenAis a
+quotient of a semisimple Hopf algebra.
+Proof.Consider the subgroup of Lα
+tGgenerated by the Ug’s. Sinceαis finite,
+we get an extension of finitegroups
+1→µ→/hatwideG→G→1,
+whereµis thefinitesubgroup of L∗generated by all elements of the form
+α(g1,g2), forg1,g2∈G(by assumption, they are all roots of unity). We thus
+know how to get the field Las a quotient of a Hopf algbera, and how to get
+the subalgebra generated by the Ug’s as a quotient of a Hopf algebra (the
+group algebra k/hatwideG). We now use the semidirect product construction in order
+to combine these two constructions into one Hopf algebra.
+LetTbe the group Z2G, the vector space over Z2with basis G(multi-
+plication in Tis just addition of vectors). We have a natural action of G
+(and thus of /hatwideG, using the map /hatwideG→G) onTby left multiplication. If we
+denote the action of /hatwideGonTbyψ, we can construct the semidirect product
+k[T]⋊k/hatwideGas explained in Section 4. Notice that by Remark 4.1 k[T]⋊k/hatwideGis
+semisimple if char(k)∤|G|(it is easy to see by the fact that the order of αin
+H2(G,L∗) divides |G|, that the prime divisors of |µ|are also prime divisors
+of|G|). Now consider the induced L-Hopf algebra XL=L⊗k(k[T]⋊k/hatwideG).
+We have an action of GonTnot only from the left but also from the right
+by multiplication. We define an action of GonXLvia
+g⋆(l⊗et⊗h) =g(l)⊗et·g−1⊗h.
+We claim the following
+Lemma 5.6. The action ⋆is a Hopf-semilinear action of GonXL.
+Proof.This is a straightforward verification. The crux of the proof is the f act
+that the two actions of GonTfrom the left and from the right commute with
+each other. /square
+Consider now the k-Hopf algebra H= (XL)G(which is semisimple in case
+char(k)∤|G|), where we take invariants with respect to the ⋆action. We
+claim that we have an onto map H։Lα
+tG. To see why this is true, we
+first decompose Has an algebra (we do not care any more about the Hopf
+structure at this stage).THE HOPF SCHUR GROUP 9
+Denote by H1the intersection of ( XL)Gwith theLsubspace spanned by
+all 1⊗et⊗ˆg, where ˆg∈/hatwideG, andt∈G(we can consider Gas the subset of T
+which contains the basis elements). Denote by H2the intersection of ( XL)G
+with theLsubspace spanned by all 1 ⊗et⊗ˆg, where ˆg∈/hatwideGandt /∈G. Since
+G, as a subset of T, is stable under the action of /hatwideGfrom the right and under
+the action of Gfrom the left, we see that H1andH2are two sided ideals, and
+we have a decomposition of algebras H=H1⊕H2.
+SinceH1is a quotient of H, it will be enough to prove that Lα
+tGis a
+quotient of the algbera H1. In order to provethis, we give a neater description
+ofH1. The group /hatwideGacts onLviaπ:/hatwideG→G. We define the algebra Bto be
+L⊗kk/hatwideGas a vector space, with the product
+(l1⊗ˆg1)·(l2⊗ˆg2) =l1ˆg1(l2)⊗ˆg1ˆg2.
+Thus,Bis the semidirect product (of algebras) of Landk/hatwideG. We define a
+linear map φ:B→H1by
+l⊗ˆg/mapsto→/summationdisplay
+g∈Gg(l)⊗eg−1⊗ˆg.
+The following lemma is quite easy to prove
+Lemma 5.7. The mapφis an isomorphism of algebras.
+By the lemma, it is enough to prove that we have an onto map B→Lα
+tG.
+To do this, recall that the group /hatwideGwas constructed as a subgroup of the
+group of invertible elements in Lα
+tG, and thus we have a natural algebra map
+k/hatwideG→Lα
+tG. We define the following linear map
+Ψ :B→Lα
+tG
+l⊗ˆg/mapsto→lˆg
+We claim the following:
+Lemma 5.8. The map Ψis an onto algebra map.
+Proof.The fact that Ψ is onto is easily seen by the fact that Lα
+tGis spanned
+overLby theUg’s. The fact that Ψ is an algebra map follows from a direct
+calculation. /square
+We thus see that Lα
+tGis a quotient of a Hopf algebra. This finished the
+proof of Theorem 1.2. /square
+Remark 5.9. The Hopf algebra His actually a form of a group algebra (i.e.
+H⊗kLis a group algebra). This is due to the following fact: the group Tis
+abelian, and therefore k[T]∼=kT♯, whereT♯is the character group of T. The
+semidirect product k[T]⋊k/hatwideGcan therefore be seen to be the group algebra
+of the semidirect product /hatwideG⋉T♯. SinceL⊗k(k[T]⋊k/hatwideG) andL⊗kHare
+isomorphic, we see that His a form of a group algebra.10 EHUD MEIR
+References
+[1] E. Aljadeff, J. Cuadra, S. Gelaki and E. Meir, On the Hopf Sc hur group of a field,
+Journal of Algebra 319, 12 (2008), 5165-5177. arXiv:0708.1 943v1.
+[2] E. Aljadeff and J. Sonn, On the projective Schur group of a fi eld, Journal of Algebra
+Volume 178, Issue 2, 1 December 1995, Pages 530-540
+[3] N. Andruskiewitsch and H.J. Schneider, Lifting of Quant um Linear Spaces and Pointed
+Hopf Algebras of Order p3, Journal of Algebra Volume 209, Issue 2 (1998), Pages 658-
+691.
+[4] M. Beattie, S. Dascalescu and L. Grunenfelder, On the num ber of types of finite di-
+mensional Hopf algebras, Inventiones Mathematicae 136 (19 99), p.1-7.
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+Manuscripta Math. 100 (1999), 35-53.
+[6] B. Fein and M. Schacher, Brauer groups of rational functi on fields, in ”Groupe de
+Brauer”, Lecture Notes in Mathematics, vol 844, Springer-V erlag, New York/Berlin
+(1981).
+[7] S. Gelaki, Pointed Hopf Algebras and Kaplansky’s 10th Co njecture, Journal of Algebra
+Volume 209, Issue 2, 15 November (1998), Pages 635-657
+[8] P. Gille and T. Szamuely- Central Simple Algebras and Gal ois Cohomology, Cambridge
+Studies in Advanced Mathematics, Cambridge University Pre ss (2006).
+[9] I. M. Issacs, Character Theory of Finite Groups, Dover Pu blications (1994).
+[10] G. J. Janusz, The Schur group of an algebraic number field . Ann. of Math. (2) 103
+(1976), no. 2, 253–281.
+[11] C. Kassel, Quantum Groups, Graduate Texts in Mathemati cs155, Springer-Verlag,
+1995.
+[12] B. Pareigis, Forms of Hopf Algebras and Galois Theory, B anach Center Publications,
+Vol. 26, pp. 75-93 (1990)
+[13] D. Parker, Forms of Coalgebras and Hopf Algebras, Journ al of Algebra, Volume 239,
+Issue 1, (2001) Pages 1-34
+[14] D. Radford, E. Taft and R. Wilson, Forms of certain Hopf a lgebras, manuscripta math.
+17, 333 - 338 (1975)
+[15] D. Stefan, The Set of Types of n-Dimensional Semisimple and Cosemisimple Hopf
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+[16] T. Yamada, The Schur Subgroup of the Brauer Group, Lectu re Notes in Mathematics
+397, Springer-Verlag, 1970.
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+arXiv:1001.0159v2  [cond-mat.soft]  5 May 2010New analytical progress in the theory of vesicles under line ar flow
+Alexander Farutin,1Thierry Biben,2and Chaouqi Misbah1
+1Laboratoire de Spectrométrie Physique, UMR5588,
+140 avenue de la physique, Université Joseph Fourier Grenob le,
+and CNRS, 38402 Saint Martin d’Hères, France
+2Université de Lyon, F-69000, France; Univ. Lyon 1,
+Laboratoire PMCN; CNRS, UMR 5586; F-69622 Villeurbanne Ced ex
+Vesicles are becoming a quite popular model for the study of r ed blood cells (RBCs). This is a
+free boundary problem which is rather difficult to handle theo retically. Quantitative computational
+approaches constitute also a challenge. In addition, with n umerical studies, it is not easy to scan
+within a reasonable time the whole parameter space. Therefo re, having quantitative analytical
+results is an essential advance that provides deeper unders tanding of observed features and can
+be used to accompany and possibly guide further numerical de velopment. In this paper shape
+evolution equations for a vesicle in a shear flow are derived a nalytically with precision being cubic
+(which is quadratic in previous theories) with regard to the deformation of the vesicle relative to
+a spherical shape. The phase diagram distinguishing region s of parameters where different types
+of motion (tank-treading, tumbling and vacillating-breat hing) are manifested is presented. This
+theory reveals unsuspected features: including higher ord er terms and harmonics (even if they
+are not directly excited by the shear flow) is necessary, what ever the shape is close to a sphere.
+Not only does this theory cure a quite large quantitative dis crepancy between previous theories
+and recent experiments and numerical studies, but also it re veals a new phenomenon: the VB
+mode band in parameter space, which is believed to saturate a fter a moderate shear rate, exhibits
+a striking widening beyond a critical shear rate. The wideni ng results from excitation of fourth
+order harmonic. The obtained phase diagram is in a remarkabl y good agreement with recent three
+dimensional numerical simulations based on the boundary in tegral formulation. Comparison of our
+results with experiments is systematically made.
+PACS numbers: 87.16.D- 83.50.Ha 87.17.Jj 83.80.Lz 87.19.r h
+I. INTRODUCTION
+A vesicle is a closed simply connected membrane sepa-
+rating two liquids. Its dynamics in an ambient flow is an
+important research target due to various medical appli-
+cations. For instance, they are employed as biochemical
+reactors [1, 2], as vectors for targeted drug and gene de-
+livery [3, 4], and as artificial cells for hemoglobin encap-
+sulation and oxygen transport [5].
+One can also regard a vesicle, from mechanical point
+of view, as a simplified model of a living cell. The most
+prominent biological counterpart for a vesicle is the red
+blood cell (RBC). The vesicle system allows to iden-
+tify the elementary processes of visco-elastic properties
+of cells moving passively in a fluid. The study of a single
+vesicle under a shear flow is one of the simplest nonequi-
+librium example. Nevertheless, this problem gives rise to
+a very complicated dynamics with different types of vesi-
+cle motion because of the nontrivial competition between
+the interactions of straining and rotational parts of shear
+flow with the vesicle. Three main types of motion for a
+vesicle in a shear flow have been identified so far: (i) tank-
+treading (TT), when shape and orientation of the vesicle
+is a steady-state, (ii) tumbling (TB), when the vesicle
+makes full rotations, and (iii) vacillating-breathing (VB ,
+aka trembling or swinging), when the longest axis of the
+vesicle oscillates around certain direction with the shape
+strongly changing during these oscillation cycles. Since
+many properties of a vesicle (e.g. the effective viscosity)strongly depend on its type of motion, it is important
+to define the phase diagram of a vesicle in a shear flow,
+predicting which type of motion will be realized for each
+set of parameters defining the vesicle dynamics.
+A widely accepted model for a vesicle proposes the fol-
+lowing assumptions: zero Reynolds number limit, mem-
+brane impermeability and local inextensibility, and the
+continuity of velocity field and mechanical stress across
+the membrane. The force exerted on the membrane by
+the liquids is balanced by the membrane rigidity force
+that is calculated from the energy of membrane bending
+taken as[6]
+E=κ
+2/integraldisplay
+(2H)2dA+/integraldisplay
+ZdA, (1)
+whereκis the bending rigidity coefficient of the mem-
+brane,His the mean curvature and Zis a Lagrange
+multiplier, ensuring the local membrane inextensibility.
+Under these assumptions, and after a certain choice of
+units for all variables, the vesicle dynamics can be com-
+pletely described by three dimensionless numbers (see for
+example [7]): the viscosity contrast
+λ=ηint
+ηext, (2)
+the excess area relative to a sphere
+∆ =Ar−2
+0−4π, (3)2
+and the capillary number
+Ca=ηext˙γr3
+0
+κ. (4)
+Hereηint,ext are the viscosities of fluids inside and outside
+the membrane respectively, Ais the surface area of the
+vesicle,r0=/parenleftbig3V
+4π/parenrightbig1/3is the radius of a sphere containing
+the same volume as the vesicle, and ˙γis the shear rate.
+It is convenient to choose r0as the unit of distance so
+that the volume of the vesicle is
+V=4π
+3. (5)
+We also use ηextas the unit of viscosity and ˙γ−1as the
+unit of time.
+Several studies have been recently dedicated to the
+determination of the phase diagram regarding different
+types of motion of an almost spherical vesicle in a shear
+flow. These studies were theoretical[8–11], experimental
+[12–15], and numerical [16, 17]. Those theoretical works
+treat the membrane deviation from a spherical shape as
+a perturbation ( ǫ=√
+∆is the small expansion parame-
+ter), represented as a series of spherical harmonics. Then
+it is suggested that one or two first orders in the expan-
+sion of the shape evolution equations must be retained.
+Generally, the resulting equations differ only because the
+authors propose distinct rules of neglection. The Leading
+Order Theory[8] (LOT) keeps only the terms of order of
+the imposed flow and the effects of the vorticity in the
+final equations. This theory is precise up to O(ǫ)in the
+expansion scheme. Later studies have included higher or-
+der terms in the expansion (up to O(ǫ2)). Lebedev et al.
+[11] included the next order terms in the membrane Hel-
+frich force (as compared to LOT), while another study
+(hereafter called Higher Order Theory[10] (HOT)) ac-
+counted for the next order terms not only in the Helfrich
+force but also in the hydrodynamic field as well.
+In the theories of Ref. [10] and Ref. [11], as well as in
+numerical studies based on dissipative particles dynamics
+[17], a saturation of the VB/TB phase border for large
+enoughCawas suggested– i.e. the value of λat which the
+transition from VB to tumbling occurs becomes almost
+independent on the capillary number (or equivalently on
+shear rate).
+Recent advances in three dimensional numerical com-
+putations based on the boundary integral formulation
+[16] have made it possible to study vesicle dynamics
+quantitatively. The results of this study show, that the
+loss of stability of the TT solution occurs at values of
+λsignificantly higher than those predicted by analytical
+calculations[10, 11]. Furthermore, no saturation of the
+VB/TB phase border upon increasing shear rate (or Ca)
+was found. These results provide a new input that is
+worth understanding from the analytical point view.
+The main objective of this work is to investigate the
+reasons for the discrepancies between the results provided
+by the recent numerical simulations [16] and analyticaltheories. It was pointed out in Ref.[11] that the critical
+values of λat which the steady-state solution loses its
+stability, scales as O(∆−1/2)for a fixed shear rate. How-
+ever, this behaviour, which also agrees with the result
+of Keller and Skalak [18], does not provide the correct
+information about the next order terms, as will be seen
+here. The next correction turns out to cause a rather big
+shift of the phase borders.
+This peculiar behavior (i.e. that the critical λdiverges
+at vanishing ∆) confers to the vesicle problem a special
+status: in the small ∆limit, one further order and only
+onein the expansion scheme in powers of ∆is needed
+as compared to previous studies. As a consequence, it
+will be shown, in particular, that the next order term,
+previously unaccounted for, survives whatever small the
+deviation from a sphere is. This is the source of devia-
+tion between the recent numerical work and the existing
+analytical theories. As will be seen inclusion of one more
+order in the expansion will allow us to extract a phase
+diagram that is in quantitative agreement with the re-
+sults of numerical simulations[16]. It will also be shown
+that fourth order spherical harmonics (in previous theo-
+ries only second order harmonics were included) can not
+be neglected, and especially beyond a certain shear rate.
+Indeed, it will be revealed that these harmonics give rise
+to a new important feature recently revealed in numerical
+simulations [16]. More precisely, the VB/TB phase bor-
+der does not saturate for fixed ∆when shear rate exceeds
+a certain value, rather a significant widening is revealed,
+in contrast to previous theoretical [10, 11], numerical [17 ]
+and experimental investigations [14].
+II. SHAPE EVOLUTION EQUATIONS
+A. Basic definitions and the expansion scheme
+Here we perform the expansion of the shape evolution
+equations one order higher as compared to the most ad-
+vanced of previous papers[10]. Since a slightly deflated
+vesicle in the absence of flow takes a centro-symmetric
+shape, and the shear flow has a symmetry center in ev-
+ery point, we consider only symmetric shapes.
+Naturally, we take the origin of coordinates at the cen-
+ter of the vesicle moving with the velocity of the undis-
+turbed shear flow. That the vesicle moves with the same
+velocity as the applied shear flow is a consequence of
+the symmetry of the problem. We use the conventional
+parametrization of the vesicle membrane with reference
+points belonging to a sphere of radius unity
+R=x[1+f(x)]. (6)
+The shape function f(x)is then expanded in spherical
+harmonics of x.Note that the shape depends also on
+time, but the temporal variable will be specified only
+when need be. Following Ref.[9, 10] we introduce a for-
+mal expansion parameter ε=√
+∆which helps classifying3
+different orders. Only second order harmonics are present
+to the order of O(ε)in the equilibrium shape function of
+an almost spherical vesicle in the absence of flow. They
+are also the only harmonics that can interact with linear
+flow directly. For consistency considerations, pushing the
+expansion to next order implies that the fourth and ze-
+roth order harmonics cannot be neglected. They enter
+dynamics as a result of interactions between the vesicle
+shape and the Helfrich force and the flow; they are of or-
+derO(ε2).The shape function does not contain spherical
+harmonics of odd orders owing to the centro-symmetry.
+The spherical harmonics of even orders higher than 4
+have an amplitude O(ε3)and thus are neglected. As fol-
+lows from the inequality
+1 =3V
+4π=/integraldisplay
+R(x)3d2x
+4π≥/bracketleftbigg/integraldisplay
+R(x)d2x
+4π/bracketrightbigg3
+=R3
+0
+the average radius of the vesicle R0is not exactly 1.
+Therefore a negative correction must be added to it in
+order to satisfy (5). This correction is of order O(ε2)
+and leads to a non-zero coefficient for zero-order spher-
+ical harmonic in the expansion of f(x),which will be
+denoted f0below.
+The expansion takes formally the form:
+f(x) =ε2f0+εf2+ε2f4+O(ε3), (7)
+fk=k/summationdisplay
+l=−kfk,l(t)Yk,l(x).
+There are various definitions of spherical harmonics,
+here we use those that satisfy the following normalization
+conditions:
+/integraldisplay
+Yk,l(x)Y∗
+k,l(x)d2x=4π(k!)2
+(2k+1)(k+l)!(k−l)!
+However, all equations are written in universal form and
+are valid for any rescaling of spherical harmonics.
+The condition (5) together with the definition ∆ =
+ε2provide two additional constraints on the coefficients
+of the expansion (7). We use the former to express f0
+through f2,l
+/integraldisplay
+f0(x)d2x=−/integraldisplay
+f2(x)2d2x−ε
+3/integraldisplay
+f2(x)3d2x+O(ε2)
+(8)
+and the latter to norm the coefficients f2,l
+2/integraldisplay
+f2(x)2d2x−2ε
+3/integraldisplay
+f2(x)3d2x= 1+O(ε2).(9)
+We retain one more order in these expressions compared
+to previous works for the sake of consistency. Note that
+the volume and the surface have been expanded to the
+orderO(ǫ3)in order to obtain the equalities (8) and (9).B. Derivation of the evolution equations of the fk
+from the boundary integral formulation
+We shall adopt here a different spirit from that of pre-
+vious theories [8–11]. There the Lamb solution is used
+(a solution of the Stokes equations inside and outside
+the vesicle). We find it convenient to take the bound-
+ary integral formulation as a starting point. The present
+spirit is equivalent to using the Lamb solution, but it has
+the advantage that the boundary conditions at the mem-
+brane are already implemented in the boundary integral
+formulation. Indeed, the Stokes equations together with
+the boundary conditions on the membrane, and far away
+from it, can be converted into boundary integral formu-
+lation [19]. This leads to the following integral equation
+vl(x)(1+λ) =
+= 2ul(x)+2/integraldisplay
+Gjl(R(x)−R(x′))Fj(x′)d2R(x′)+
++(1−λ)/integraldisplay
+Kjlm(R(x)−R(x′))vj(x′)Nm(x′)d2R(x′).
+(10)
+Herev(x)is the actual velocity of point R(x),u(x)is
+the velocity of point R(x)in the imposed flow, F(x)
+is the membrane rigidity force at point R(x),N(x)is
+the outward pointing normal to the vesicle surface at
+pointR(x).The integration is taken over the surface of
+the vesicle and d2R(x′)is the surface area element. The
+integration kernels have the following form:
+Gij(R) =1
+8π/parenleftbiggδij
+R+RiRj
+R3/parenrightbigg
+, Kijk(R) =3
+4πRiRjRk
+R5.
+Using the expression for the Helfrich force, we can find
+from (10) the velocity field on the surface of the vesicle.
+The Helfrich force is given in terms of the shape as
+Fi=−2/parenleftbig
+κ[2H(H2−K)+∆SH]−ZH/parenrightbig
+Ni+
++∂Z(x)
+∂Rj(x)(δij−NiNj),(11)
+whereKis the Gaussian curvature and ∆Sis the
+Laplace-Beltrami operator. It can easily be checked that
+expression (11) vanishes for a spherical shape, and thus
+is of order O(ε).In order to balance (10) at order O(1)
+we need to assume the imposed flow to be of the same
+order as (11). This requirement can equivalently be ful-
+filled, following Refs. [8, 10], by the demand that κscale
+asε−1.We then set κ=ε−1¯κ,with¯κof order O(1).
+The precise technical details will be given elsewhere,
+while here we shall present only the spirit and some in-
+termediate steps. We expand v(x),u(x),andF(x)into
+vector spherical harmonics of x.For convenience we de-
+fine them as
+Yi
+1,k,l(x) =eimnxm∂nYk,l(x),
+Yi
+2,k,l(x) = (2k+1)xiYk,l(x)−∂iYk,l(x),4
+Yi
+3,k,l(x) =∂iYk,l(x).
+Here the differentiation with respect to xiis taken for-
+mally as if xwere a regular 3D vector not bound to a
+sphere of radius unity, and eimnis the Levi-Civita (unit
+anti-symmetric) tensor. The advantage of such a defini-
+tion is that not only do these spherical harmonics consti-
+tute an orthogonal set with respect to the integration of
+dot product over x,i.e. the quantity
+/integraldisplay
+Yi
+j,k,l(x)Yi
+j′,k′,l′(x)∗d2x
+is zero unless j=j′, k=k′, l=l′,but also operators
+GandKare diagonal in the chosen basis for a spherical
+vesicle, so that the integrals
+/integraldisplay /integraldisplay
+Yi
+j,k,l(x)Gim(x−x′)Ym
+j′,k′,l′(x′)∗d2xd2x′,
+/integraldisplay /integraldisplay
+Yi
+j,k,l(x)Kimn(x−x′)Ym
+j′,k′,l′(x′)∗xnd2xd2x′
+are zero unless j=j′, k=k′, l=l′.Given the centro-
+symmetry we impose, in our expansion of v(x),u(x),and
+F(x),the absence of Yi
+1,k,l(x),for even k,andYi
+2,k,l(x),
+andYi
+3,k,l(x),for oddk.In order to find the evolution
+equations we need to expand the velocity field to the
+order of O(ε2).Coefficients for Yi
+j,k,l(x)fork >6are
+O(ε3)and thus can be neglected. It will be shown later
+that coefficients of 5th and 6th orders of vector spherical
+harmonics make corrections to the evolution equations
+forfk,lwithk≤4which are O(ε3).So only five first
+orders (starting from the zeroth one) of vector spherical
+harmonics should be taken into account in the expansion
+of any space-dependent quantity under consideration.
+In order to fulfil the condition of local inextensibility
+of the membrane we impose zero surface divergence of
+the velocity field[8].
+dA(x)
+dt=∂vi(x)
+∂Rj(x)[δij−Ni(x)Nj(x)]dA(x) = 0.(12)
+Here
+∂vi(x)
+∂Rj(x)=∂vi(x)
+∂xk∂xk
+∂Rj(x)
+is the Jacobian matrix. We can take the derivatives as
+ifxwere a regular 3D vector, because the surface di-
+vergence is fully determined by the distribution of the
+velocity field on the membrane and does not depend on
+the continuation of the flow into the liquids.
+We expand equations (10,12) to order O(ε2)and the
+resulting integrals could be performed analytically. Pro-
+jecting the results of the integration on the space of vector
+spherical harmonics up to the fourth order, we obtain a
+set of equations satisfied by the coefficients entering the
+expansions of v(x)andZ(x).Since the surface area isevaluated up to order O(ε3),while the velocity field in
+(12) is expanded only up to order O(ε2),it is not appro-
+priate to use (12) in order to ensure the conservation of
+the whole surface area. Therefore, we only use the projec-
+tions of (12) on Yk,l(x)forkequal to2or4.Accordingly,
+we leave the isotropic part of Z(x)(denoted here as Z0)
+undetermined. Once the final shape evolution equations
+are obtained we shall use the constraint that the time
+derivative of (9) is equal to zero in order to determine
+Z0.This way of reasoning was used in previous studies
+[8–11]. Note that unlike other theories [8–11], since we
+expand the equations to higher order, it is not legitimate
+to replace dA(x)in (12) with d2xprior to projection
+on the spherical harmonics sub-space. Such a substitu-
+tion would imply neglecting terms of order O(ε2)in final
+equations, and would be inconsistent with the spirit of
+the present theory.
+Having determined the velocity field thanks to the
+above expansions, we are in a position to obtain the fi-
+nal evolution equation by making use of the kinematic
+equation expressing the fact that the membrane velocity
+is equal to the fluid velocity at the membrane
+∂f(x)
+∂t=vi(x)xi−∂if(x)vj(x)(δij−xixj)
+R(x)(13)
+Then the task is to substitute the expanded velocity
+field into this equation (in terms of coefficients of the
+velocity field) and project the resulting expression onto
+the space of spherical harmonics of interest. This then
+leads to the determination of the evolution equations that
+must be satisfied by the shape coefficients fk,l.It can be
+shown that the following identities hold Yi
+1,5,l(x)xi= 0,
+Yi
+2,6,l(x)xi= 7Y6,l(x),andYi
+2,6,l(x)xi= 6Y6,l(x).There-
+fore the projection of the first part of the right hand
+side of (13) on the subspace of spherical harmonics of
+orders up to four does not depend on the coefficients of
+the vector spherical harmonics of orders five and six in
+the velocity field. Regarding the the second part, ∂if(x)
+is of order O(ε)and the coefficients for fifth and sixth
+orders of vector spherical harmonics in the velocity field
+expansion are of order O(ε2)so that their contribution
+in the shape evolution equation is of order O(ε3),that is
+beyond our accuracy.
+It is convenient to decompose the applied shear flow
+into its elongational and rotational parts to simplify
+the final equations. The same decomposition can be
+applied to general linear flow: the quantity E2(x) =
+Ui(x)xi/2defines the straining part, while the vector
+Ωi=eijk∂jUk(x)/2represents the vorticity. Note, that
+u(x)in (10) is the velocity of the imposed flow evaluated
+at the point R(x),which takes the following form
+ul(x) = (∂lE2+eljkxjΩk)(1+f(x))
+The shape evolution equations for the second and
+fourth order harmonics can be written in a compact form:
+εDf2,l
+Dt=/integraltextF2(x)Y2,l(x)∗d2x/integraltext
+Y2,l(x)Y2,l(x)∗d2x, (14)5
+ε2Df4,l
+Dt=/integraltext
+F4(x)Y4,l(x)∗d2x/integraltext
+Y4,l(x)Y4,l(x)∗d2x, (15)
+The left hand side term of the equation is a special
+derivative of fk,lthat naturally arises in a non-rotating
+coordinate system, and its definition is
+Dfk,l
+Dt=∂fk,l
+∂t+/integraltext
+eijm∂ifkxjωmYk,l(x)∗d2x/integraltext
+Yk,l(x)Yk,l(x)∗d2x.(16)
+To the order at which our expansion is performed, the
+rotational velocity of the vesicle is not equal to the vor-
+ticity of the imposed flow (unlike lower order calculations
+[8–11]), but rather it has a new contribution originating
+from the shape function
+ωj= Ωj+εejkl∂ikE2∂ilf2
+2+O(ε2). (17)
+Note that if only the first term Ωiis retained [8–11], then
+(16) coincides with the Jaumann derivative.
+The functions Fkcan be written as
+F2=a1f2+εa2f2
+2+ε2(a3f3
+2+a4f2f4)+b1E2+εb2E2f2+
++ε2(b3f2∂if2∂iE2+b4∂if2∂ijf2∂jE2+
++b5∂if2∂ijE2∂jf2+b6E2f4)+O(ε3),
+F4=ε(c1f4+c2f2
+2)+ε2(c3f3
+2+c4f2f4)+εd1E2f2+
++ε2(d2f4E2+d3f2
+2E2+d5f2∂iE2∂if2)+O(ε3).
+The coefficients ai, bi, ci,anddiare rational functions of
+λ,¯κ=ε/Ca,andZ0.The exact expressions are listed in
+the appendix.
+III. THE PHASE DIAGRAM
+A. The analytical phase diagram and comparison
+with previous works
+The phase diagram for ∆ = 0.43is presented on the
+Fig.1. The value ∆ = 0.43is chosen for the sake of com-
+parison with available numerical data [16]. Also shown
+on that Figure are the results obtained in two previous
+theories[10, 11]. As can be seen the basic structure of
+the phase diagram bears similarity with previous ones.
+There are however some important distinctions. First,
+the phase borders are significantly shifted towards the
+region of higher viscosity contrasts. Second, the VB/TB
+transition curve does not saturate upon increasing shear
+rate (or Ca), rather it exhibits a striking widening. We
+have also compared the results with those obtained re-
+cently by full three dimensional simulations. A remark-
+able agreement between the two approaches is found (see
+comparison in Ref. [16]).0 2 4 6 8 10
+Ca051015
+λPresent Theory
+Higher Order Theory
+Lebedev et al.  TheoryTB
+VB
+TTλc0λc∞
+Figure 1: Continuous lines represent the phase diagram for
+∆ = 0.43,the results of the higher order theory[10] and Lebe-
+devet al. theory[11] are added in dashed and dotted lines for
+comparison. The TT phase borders are almost indistinguish-
+able for last two theories.
+B. Basic reasons for necessity of higher order
+expansion
+Our first concern is to understand why the inclusion of
+the terms of order O(ε2)provides such a dramatic shift of
+the phase borders even though the shape is almost spher-
+ical (the relative excess area is only 0.43/(4π) = 0.034).
+We have studied the evolution of the phase diagram as a
+function of ∆in order to gain further insight. We have
+tracked the viscosity contrast λat which the loss of sta-
+bility of TT motion occurs in the two asymptotic limits
+Ca→0(the corresponding value is denoted as λc0) and
+Ca=∞(denoted as λc∞). It was reported in [11] that
+λc∝ε−1.Since we expect the expansions of λcto be an-
+alytic in ε,we have thus attempted the following ansatz:
+λc=λ(−1)
+cε−1+λ(0)
+c+O(ε). (18)
+To check the validity of this expansion we plot these crit-
+ical values as a function of ∆−1/2= 1/ε.We expect then
+a linear behavior. This is presented on Fig.2 where we
+also compare our results with those of previous theories.
+A key point to be discussed further below is the fact that
+previous theories provide correct values for the dominant
+termλ(−1)
+cin the expansions (18), but not for λ(0)
+cwhich
+does not vanish even in the limit of almost spherical vesi-
+cles (note that the next order terms in the expansion (18)
+tend to0with the excess area). Because λ(0)
+cremains fi-
+nite whatever small is the deviation from a sphere, the
+discrepancy between the TT phase borders obtained by
+different theories (with one lower order below the present
+one, namely of the order of O(ε)) persists for any excess
+area. In contrast the present theory valid to order O(ǫ2),
+has the property that even if one wishes to make an ex-
+pansion to the next order (i.e. order O(ǫ3)), then the
+shift in the borders in the phase diagram would be negli-6
+0 2 4 68 10
+∆-1/201020304050
+λ0Higher Order Theory
+Keller-Skalak Theory
+Present Theory
+0 2 4 68 10
+∆-1/201020304050
+λ∞Present Theory
+Higher Order Theory
+Figure 2: λ0andλ∞as a function of ∆−1/2.
+gibly small, i.e. the shift would vanish in the small excess
+area limit. In other words, the present theory shows that
+there is a convergence of the small deformation scheme at
+orderO(ǫ2).This implies that there is no need to continue
+the expansion of the shape evolution equation beyond the
+orderO(ε2)since this would neither notably modify the
+results for ∆<0.43nor significantly extend the applica-
+bility to higher values of ∆.Comparison of the present
+theory with numerical results (see comparison provided
+in Ref. [16]) shows a good agreement for ∆ = 0.43.(the
+only value explored to date by the numerical scheme).
+C. Discussion of the tank-treading phase borders
+As can be seen the dependencies shown on Fig.2 are
+fairly linear for ∆<1and are almost indistinguishable
+from straight lines for ∆<0.5.The region of 1/√
+∆
+where the curves noticeably deviate from straight lines
+is wider for present theory than for HOT. This is most
+likely caused by the introduction of fourth order spherical
+harmonics, their amplitude is no more small compared to
+that of the second order harmonics for ∆>1.The ex-
+pression (9) becomes inapplicable, and as a result f4,l(t)
+grow uncontrollably with time in solutions of the differ-
+ential equations. This problem can be fixed by expanding
+the excess area to the order O(ε4),but the results of such
+patching are not trustworthy because they go beyond the
+precision of the differential equations.
+Note also that the second term in (17) tends to align
+the vesicle to the elongating direction of the straining
+part of the shear flow which makes the angle π/4with
+the direction of the flow. In the TT phase close to the
+loss of the stability of the steady-state solution the vesi-
+cle is almost parallel to the flow. The effective vorticity
+(17) turns out to be less than Ωin the TT phase, and
+it leads to the increased stability of the steady-state so-
+lution. The correction to the vorticity is proportional to
+εand thus increases with ǫand dominates over Ωin theTheory λ(−1)
+c0 λ(0)
+c0
+LOT[8]8
+23√
+30π(3.38)−32
+23(−1.39)
+HOT[10] 3.91 −1.22
+Lebedev et al.[11]16
+23√
+10π(3.90)−32
+23(−1.39)
+Keller and Skalak [18]2
+3√
+10π(3.74)73
+63(1.17)
+Present 3.89 1.54
+Table I: λ(−1)
+c0andλ(0)
+c0extracted from different theories. It
+should be noted that Lebedev et al.[11] theory provided only
+values for λ(−1)
+c0and we have formally extracted from their
+theory the value of λ(0)
+c0for comparison purpose.
+Theory λ(−1)
+c∞ λ(0)
+c∞
+LOT[8]8
+23√
+30π(3.38)−32
+23(−1.39)
+HOT[10] 4.79 −1.38
+Lebedev et al.[11]16
+23√
+15π(4.78)−32
+23(−1.39)
+Present 4.76 1.13
+Table II: λ(−1)
+c∞andλ(0)
+c∞extracted from different theories. It
+should be noted that Lebedev et al.[11] theory provided only
+values for λ(−1)
+c∞and we have formally extracted from their
+theory the value of λ(0)
+c∞for comparison purpose.
+large excess area limit. As a consequence, the truncation
+(17) becomes illegitimate. Generally, we may assert that
+∆<1is a good estimate for the applicability limits of
+the small deformation approximation.
+Some theories[8, 11, 18] allow for analytical extraction
+of the coefficients λ(−1)
+candλ(0)
+c,and for HOT and the
+present theory the extraction is numerical by using the
+slope and offset of the lines on the Fig.2. The combined
+data are presented in the tables 1 and 2.
+HOT and Lebedev et al. theories show a good agree-
+ment for the coefficients λ(−1)
+cwith the present calcula-
+tion. This means, that expansion of the shape evolution
+equations to the order O(ε)was indeed sufficient in or-
+der to capture the correct value of λ(−1)
+ceven for large
+values of Ca,where Lebedev et al. theory loses its ap-
+plicability. The present theory shows that the correct
+value of λ(0)
+cis captured thanks to the fact that the ex-
+pansion scheme is pushed one step further in the excess
+area. Keller-Skalak theory gives a slightly different resul t
+forλ(−1)
+c,the difference is caused by the fact they used a
+velocity field having a surface divergence of order O(ε),
+whereas the membrane local inextensibility requires zero
+surface divergence instead. On the other hand Keller-
+Skalak theory provides a better estimate for the value of
+λ(0)
+cthan the one proposed by HOT and this explains why
+its results are relatively close to the results of numerical
+simulations for ∆∼1.7
+D. Discussion of TB/VB transition
+We plan to dedicate a separate research to the proper-
+ties of different types of vesicle motion, so here we only
+briefly discuss further implication of the new theory on
+the TB/VB transition. Neglecting harmonics of orders
+higher than two, as done in previous theories, provides
+two major simplifications. First, the vesicle shape has
+three symmetry planes, which make the definition of the
+orientation angle obvious, and second the in-plane mo-
+tion is defined by only two independent variables. As a
+consequence, the dynamics relaxes with time to a cyclic
+motion (which degenerates into a point in the TT phase).
+Thus only a simple limit cycle can exist, and this shows
+that each of the VB and TB modes has its own region of
+existence in parameter space. The above assertions can
+not be made in general, and especially when the fourth
+order harmonics are included. It also turns out that near
+the transition region the vesicle tends to finish its TB
+quasi-cycles by assuming an oblate almost axisymmetric
+shape in the shear plane rather than an elongated shape
+perpendicular to the flow. For such shape the error in
+the definition of the orientation angle is very large.
+Let us now discuss how each type of motion is de-
+termined from our evolution equations. We start with
+some initial values for fkl(t)and wait a ceratin time in-
+terval until the initial data are irrelevant. Then it was
+checked whether during one oscillation quasi-cycle f(x)
+has a maximum for xlying in the shear plane and perpen-
+dicular to the shear velocity. Despite some noise due to
+the aforementioned complications it is clearly seen that
+unlike with previous theories the VB/TB phase border
+does not saturate even for Ca= 10,and the VB region
+broadens with the increase of the capillary number (Fig.
+1).
+Note that it was suggested recently [20] that higher
+orders of spherical harmonics can be excited close to the
+VB/TB phase border and this may cause some widen-
+ing of the VB phase region (but still saturation at large
+Cais found, unlike our theory). It can be checked that
+whenZ0+ 20ǫ/Ca<0, the fourth order harmonics are
+excited (i.e. the corresponding decay time becomes infi-
+nite to leading order). Here, we found that close to the
+VB/TB transition at some times during the oscillation
+the quantity
+−Z0Ca
+ε(19)
+exceeds 20. This is probably the reason why the VB/TB
+phase border remains unsaturated for much larger values
+than in previous analytical studies. It was proposed in
+[20], however, that excitation of higher harmonics was
+due to thermal fluctuations. This contrasts in spirit with
+our theory where the fourth order harmonic is excited as
+non-linear interaction of second harmonics and no refer-
+ence to temperature is required.0 2 4 68 10 12 14 1618 20
+S123
+Λ∆=0.16
+∆=0.43
+∆=0.8
+Lebedev et al.  Theory
+Figure 3: Phase diagrams for various values of ∆inS−Λ
+coordinates.
+IV. COMPARISON WITH EXPERIMENTS
+Experiments have been performed recently regarding
+determination of the phase diagram of vesicle motions
+under shear flow[14]. They referred to a previous the-
+oretical work [11] which had suggested that the phase
+diagram should depend on only two independent dimen-
+sionless control parameters (and not three), namely
+Λ =4√
+30π/parenleftbigg
+1+23
+32λ/parenrightbigg√
+∆, S=7π
+3√
+3Ca
+∆(20)
+It has been reported [14] (with a certain degree of uncer-
+tainty) that the experimental data were consistent with
+the fact that only the above two parameters determine
+the phase diagram. Our results do not comply with this
+report, as shown on Figure 3. Indeed, besides SandΛ,
+the excess area ∆plays an important role. Experiments
+mixed data for different ∆’s in the plane (S,Λ). The
+band of the VB mode in experiments looks quite wide,
+and we believe that this reflects the sensitivity of the lo-
+cation of the VB band to excess area (in other words
+the experimental data may be viewed as juxtaposition of
+bands each representing a value of ∆; see the example
+of Figure 3). It is hoped that a study representing each
+value of ∆will be performed in the future with the aim
+of making comparison with theory clearer.
+Furthermore, it must be noted that our estimates for
+the TT-VB phase borders are higher than those observed
+in experiments. This is not very surprising since tran-
+sients are found to be very long close to TT-VB transi-
+tion (as also discussed in our recent full numerical simu-
+lations [16]), and therefore a firm conclusion on the type
+of motion can not be made on the basis of the current
+experimental data, which are limited to only few periods
+of oscillation (while transient can exhibit up to hundreds
+of cycles as discussed recently [16]). In other words, this
+kind of long relaxation would convey the impression that
+dynamics is of VB type, whereas in reality it is a TT one.8
+0 5 10 15 20
+S11,522,5
+ΛPresent Theory, ∆=0.43
+Present Theory, ∆=1
+Lebedev et al.  Theory
+~
+~
+Figure 4: Phase diagrams of vesicle motions under general
+planar linear flows. The viscosity contrast λis fixed to 1and
+the ratio of rotating and straining parts of the flow ω0/s0is
+varied. The theory of Lebedev et al.[11] is plotted in dotted
+lines for comparison.
+0 5 10 15 20
+S11,522,53
+ΛPresent Theory, ω0=1/2 (simple shear flow)
+Present Theory, ω0=3/4
+Present Theory, ω0=5/4
+Lebedev et al.  Theory, arbitrary ω0∆=0.43, s0=1/2
+~
+~
+Figure 5: Phase diagrams of vesicles under general planar
+linear flows. The straining and rotational parts of the flow
+are denoted as s0andω0respectively.
+Other experiments[15] were performed under linear
+flow with applied velocity field ux= (s0+ω0)y, uy=
+(s0−ω0)x, uz= 0, which is different from a simple shear
+flow. The experiments were performed with no viscos-
+ity contrast and the ratio of the rotational ( ω0) and the
+straining ( s0) parts of the flow was varied instead. The
+results of the experiments were plotted in coordinates
+˜Λ =4ω0√
+30πs0/parenleftbigg
+1+23
+32λ/parenrightbigg√
+∆,˜S=14π
+3√
+3s0
+κ∆(21)
+representing the generalization of (20). It was then
+claimed [15] that the phase diagram depends on these
+two parameters only (at least with no viscosity contrast).
+The Fig.4 simulates this experiment using the present
+theory. The discrepancy between resulting phase dia-grams for various values of ∆is indeed not as striking as
+for the simple shear flow (Fig.3): the TT phase borders
+fall close to each other and VB bands overlap for a wide
+range of ∆.At the same time, the VB/TB phase borders
+still vary significantly with the excess area. However, by
+fixing∆and varying λwe can produce series of phases
+diagrams in the ˜Sand˜Λplane corresponding to different
+values of ω0/s0,(see Fig.5). Here we find the same kind
+of discrepancy as on Fig.3 .
+Finally, let us compare our results with other set of ex-
+periments in the TT regime[13]. We represent the vesicle
+orientation angle φ0under strong shear flow(Fig. 3) (for
+the sake of comparison with experiments which were per-
+formed at high shear rates). It can be checked that the
+exact form of the bending energy becomes insignificant
+under strong flows, because the dominant contribution to
+the membrane force comes form the tension part which
+enforces incompressibility of the membrane. Unlike for
+the VB/TB phase border, the dependence of φ0onλ
+quickly saturates with Ca.We take Ca= 100 in our cal-
+culation. We use the artificial parameter Λ(20) instead
+ofλin order to show the effect of O(ǫ2)terms taken into
+account by the present calculation. Note that the results
+of Lebedev it et al. theory [11] do not depend on the
+excess area ∆in this representation, unlike our results
+and those reported by experiments (see Fig. 5). The re-
+sults of HOT depend also on ∆, but that dependence is
+weak so that they hardly differ from theoretical results
+of Ref.[11] (we do not plot them here in order to avoid
+encumbering of the figure). Experimental results[13] are
+provided for four different values of ∆ : 0.15,0.24,0.42,
+and1.43.We exclude ∆ = 1.43, since we do not expect
+our theory to be applicable to a such large value of excess
+area. As can be seen, theoretical results and experimen-
+tal ones show a rather good agreement provided that one
+is not close to the TT/VB border (i.e. if the angle is
+not too close to zero). The discrepancy for small incli-
+nation angles may be attributed to thermal fluctuations
+or interactions with walls. However, a systematic theo-
+retical study of these factors should be performed before
+drawing conclusive answers.
+V. DISCUSSION AND CONCLUSION
+Here we have presented a small deformation theory of a
+vesicle in shear flow keeping one more order in the expan-
+sion of the shape evolution equations than prior studies.
+We have confirmed the leading term in the asymptotic
+expansion of critical viscosity ratios determined by previ -
+ous analytical studies, but we also were able to determine
+accurately the next term in the expansion. This term is
+constant and survives no matter how small the excess
+area is. Moreover, this theory provides a correction to
+the phase borders that is significant in a wide range of
+excess areas. Unlike previous analytical works, but in
+agreement with the results of the numerical simulations
+[16], we observe an unsaturated growth of the VB/TB9
+0 0,5 1 1,5 2
+Λ00,10,20,30,40,50,6φ0Experiment, ∆=0.15
+Experiment, ∆=0.24
+Experiment, ∆=0.42
+Present Theory, ∆=0.15
+Present Theory, ∆=0.24
+Present Theory, ∆=0.42
+Lebedev et al. TheoryTT inclination angle for Ca=100
+Figure 6: inclination angle of vesicles in TT phase for large
+Ca.
+phase border in a quite large range of the capillary num-
+ber.
+We hope that this work will incite future experimental
+research. It will also be interesting to investigate ex-
+perimentally the presence of widening of the VB band
+as a function of shear rate. Finally, we did not in-
+clude third order spherical harmonics in the expansion
+of the shape function, restricting our calculations only
+to centro-symmetric vesicles. In the full numerical simu-
+lation [16], no symmetry restriction is imposed and so
+far no manifestation of the third order harmonic has
+been observed. This confers to our assumption of centro-
+symmetry a certain legitimacy.
+Acknowledgements: We would like to thank
+G. Danker, S. S. Vergeles, and P. M. Vlahovska for helpful
+discussions. A.F. and C.M. acknowledge financial sup-
+port from CNES (Centre National D’études Spatiales)
+and ANR (Agence Nationale pour la Recherche); "MOSI-
+COB project".
+Appendix A: The shape evolution equations and the
+expression of various coefficients in terms of physical
+parameters
+We remind that
+F2=a1f2+εa2f2
+2+ε2(a3f3
+2+a4f2f4)+b1E2+εb2E2f2+
++ε2(b3f2∂if2∂iE2+b4∂if2∂ijf2∂jE2++b5∂if2∂ijE2∂jf2+b6E2f4)+O(ε3),
+F4=ε(c1f4+c2f2
+2)+ε2(c3f3
+2+c4f2f4)+
++εd1E2f2+ε2(d2f4E2+d3f2
+2E2+d5f2∂iE2∂if2)+O(ε3).
+The coefficients are then written as
+a1=−24Z0+6¯κ
+23λ+32
+a2= 24(49λ+136)Z0+(432λ+1008)¯κ
+(23λ+32)2
+b1=120
+23λ+32
+b2= 2400λ−2
+(23λ+32)2
+b6=−40241λ+344
+(23λ+32)2
+c1=−40Z0+20¯κ
+19λ+20
+c2=16
+3(92−3λ)Z0+(1822λ+3112)¯κ
+(19λ+20)(23 λ+32)
+d1=20
+31047λ+1072
+(23λ+32)(19 λ+20)
+d2= 10257λ−32
+(23λ+32)(19 λ+20)
+d3=1
+6−125055λ3+594716 λ2+1107168 λ+392320
+(2λ+5)(23λ+32)(19 λ+20)2
+a3=−8
+175(2479595 λ3+6703156 λ2+18601472 λ+16622592
+(23λ+32)3(19λ+20)Z0−
+−8
+17578181390 λ3+390845256 λ2+713624832 λ+429094912
+(23λ+32)3(19λ+20)¯κ10
+a4= 4(18335λ2+57376λ+44224) Z0+(318896 λ2+1026064 λ+809600)¯ κ
+(19λ+20)(23 λ+32)2
+b3=1
+7078420885 λ4+815632786 λ3+2193572112 λ2+1954954752 λ+481607680
+(2λ+5)(19λ+20)(23 λ+32)3
+b4=−1
+630490295475 λ4+3878418742 λ3+9801602064 λ2+9403966464 λ+2954997760
+(2λ+5)(19λ+20)(23 λ+32)3
+b5=1
+315136559745 λ4+385825454 λ3+141580368 λ2−26564352 λ+98232320
+(2λ+5)(19λ+20)(23 λ+32)3
+c3=−4
+632552583λ3+1777744 λ2−913584λ+315392
+(23λ+32)2(19λ+20)2Z0−
+−16
+6332203871 λ3+118114286 λ2+154157080 λ+69803008
+(23λ+32)2(19λ+20)2¯κ
+c4= 2(21383λ2+63844λ+44928) Z0+48(13411 λ2+35444λ+23040)¯ κ
+(23λ+32)(19 λ+20)2
+d4=1
+4241587815 λ4+95846332 λ3−88522016 λ2−295841536 λ−152842240
+(2λ+5)(23λ+32)2(19λ+20)2
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+C.Misbah, Phys. Rev. E 76, 041905 (2007).
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\ No newline at end of file
diff --git a/1001.0160.txt b/1001.0160.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3502a3e470a7d27b3970f17e2c1a8bf1a695bda7
--- /dev/null
+++ b/1001.0160.txt
@@ -0,0 +1,813 @@
+arXiv:1001.0160v2  [stat.ML]  19 Aug 2010LEARNING THE STRUCTURE OF DEEP SPARSE
+GRAPHICAL MODELS
+By Ryan P. Adams∗, Hanna M. Wallach and Zoubin Ghahramani
+University of Toronto, University of Massachusetts
+and University of Cambridge
+Deep belief networks are a powerful way to model complex prob -
+ability distributions. However, learning the structure of a belief net-
+work, particularly one with hidden units, is difficult. The In dian buf-
+fet process has been used as a nonparametric Bayesian prior o n the
+directed structure of a belief network with a single infinite ly wide
+hidden layer. In this paper, we introduce the cascading Indi an buffet
+process (CIBP), which provides a nonparametric prior on the struc-
+ture of a layered, directed belief network that is unbounded in both
+depth and width, yet allows tractable inference. We use the C IBP
+prior with the nonlinear Gaussian belief network so each uni t can
+additionally vary its behavior between discrete and contin uous rep-
+resentations. We provide Markov chain Monte Carlo algorith ms for
+inference in these belief networks and explore the structur es learned
+on several image data sets.
+1. Introduction. Thebeliefnetwork ordirectedprobabilisticgraphical
+model [Pearl,1988] is a popular and useful way to represent complex prob-
+ability distributions. Methods for learning the parameter s of such networks
+are well-established. Learning network structure, howeve r, is more difficult,
+particularly when the network includes unobserved hidden u nits. Then, not
+only must the structure (edges) be determined, but the numbe r of hidden
+units must also be inferred. This paper contributes a novel n onparametric
+Bayesian perspective on the general problem of learning gra phical models
+with hidden variables. Nonparametric Bayesian approaches to this problem
+are appealing because they can avoid the difficult computatio ns required
+for selecting the appropriate a posteriori dimensionality of the model. In-
+stead, they introduce an infinite number of parameters into t he model a pri-
+oriand inference determines the subset of these that actually c ontributed
+to the observations. The Indian buffet process (IBP) [ Ghahramani et al. ,
+2007,Griffiths and Ghahramani ,2006] is one example of a nonparamet-
+ric Bayesian prior and it has previously been used to introdu ce an infi-
+nite number of hidden units into a belief network with a singl e hidden
+layer [Wood et al. ,2006].
+∗http://www.cs.toronto.edu/ ~rpa
+12 R.P. ADAMS ET AL.
+This paper unites two important areas of research: nonparam etric Baye-
+sian methods and deep belief networks. To date, work on deep b elief net-
+works has not addressed the general structure-learning pro blem. We there-
+fore present a unifying framework for solving this problem u sing nonpara-
+metric Bayesian methods. We first propose a novel extension t o the Indian
+buffet process — the cascading Indian buffet process (CIBP) — and use
+the Foster-Lyapunov criterion to prove convergence proper ties that make it
+tractable with finite computation. We then use the CIBP to gen eralize the
+single-layered, IBP-based, directed belief network to con struct multi-layered
+networks that are both infinitely wide and infinitely deep, an d discuss use-
+ful properties of such networks including expected in-degr ee and out-degree
+for individual units. Finally, we combine this framework wi th the powerful
+continuous sigmoidal belief network framework [ Frey,1997]. This allows us
+to infer the type (i.e., discrete or continuous) of individu al hidden units—an
+important property that is not widely discussed in previous work. To sum-
+marize, we present a flexible, nonparametric framework for d irected deep
+belief networks that permits inference of the number of hidd en units, the
+directed edge structure between units, the depth of the netw ork and the
+most appropriate type for each unit.
+2. Finite Belief Networks. We consider belief networks that are lay-
+ered directed acyclic graphs with both visible and hidden un its. Hidden
+units are random variables that appear in the joint distribu tion described
+by the belief network but are not observed. We index layers by m, increasing
+with depth up to M, and allow visible units (i.e., observed variables) only in
+layerm=0. Werequirethat unitsinlayer mhaveparents onlyinlayer m+1.
+Within layer m, we denote the number of units as K(m)and index the units
+withkso that the kth unit in layer mis denoted u(m)
+k. We use the no-
+tationu(m)to refer to the vector of all K(m)units for layer mtogether.
+A binary K(m−1)×K(m)matrixZ(m)specifies the edges from layer mto
+layerm−1, so that element Z(m)
+k,k′=1 iff there is an edge from unit u(m)
+k′to
+unitu(m−1)
+k.
+A unit’s activation is determined by a weighted sum of its par ent units.
+The weights for layer mare denoted by a K(m−1)×K(m)real-valued ma-
+trixW(m), so that the activations for the units in layer mcan be writ-
+ten asy(m)=(W(m+1)⊙Z(m+1))u(m+1)+γ(m), whereγ(m)is aK(m)-dimen-
+sional vector of bias weights and the binary operator ⊙indicates the Hada-
+mard (elementwise) product.
+To achieve a wide range of possible behaviors for the units, w e use the
+nonlinear Gaussian belief network (NLGBN) [ Frey,1997,Frey and Hinton ,LEARNING THE STRUCTURE OF DEEP SPARSE GRAPHICAL MODELS 3
+1999] framework.IntheNLGBN, thedistributionon u(m)
+karises fromadding
+zero mean Gaussian noise with precision ν(m)
+kto the activation sum y(m)
+k.
+This noisy sum is then transformed with a sigmoid function σ(·) to ar-
+rive at the value of the unit. We modify the NLGBN slightly so t hat the
+sigmoid function is from the real line to ( −1,1), i.e.σ:R→(−1,1), via
+σ(x) = 2/(1+exp{x})−1. Thedistributionof u(m)
+kgiven its parents is then
+p(u(m)
+k|y(m)
+k,ν(m)
+k) =exp/braceleftbigg
+−ν(m)
+k
+2/bracketleftBig
+σ−1(u(m)
+k)−y(m)
+k/bracketrightBig2/bracerightbigg
+σ′(σ−1(u(m)
+k))/radicalBig
+2π/ν(m)
+k
+whereσ′(x)=d
+dxσ(x). As discussed in Frey[1997] and shown in Figure 1,
+different choices of ν(m)
+kyield different belief unit behaviors from effectively
+discrete binary units to nonlinear continuous units. In the multilayered con-
+struction we have described here, the joint distribution ov er the units in a
+NLGBN is
+(1)p({u(m)}M
+m=0|{Z(m),W(m)}M
+m=1,{γ(m),{ν(m)
+k}K(m)
+k=1}M
+m=0,) =
+K(M)/productdisplay
+k=1p(u(M)|γ(M)
+k,ν(M)
+k)
+M−1/productdisplay
+m=0K(m)/productdisplay
+k=1p(u(m)
+k|y(m)
+k,ν(m)
+k).
+3. Infinite Belief Networks. Conditioned onthe numberof layers M,
+the layer widths K(m)and the network structures Z(m), inference in be-
+lief networks can be straightforwardly implemented using M arkov chain
+Monte Carlo[ Neal,1992]. Learningthedepth,widthandstructure,however,
+presents significant computational challenges. In this sec tion, we present a
+novel nonparametric prior, the cascading Indian buffet process , for multi-
+layered belief networks that are both infinitely wide and infi nitely deep. By
+using an infinite prior we avoid the need for the complex dimen sionality-
+altering proposals that would otherwise be required during inference.
+3.1.The Indian buffet process. Section2used the binary matrix Z(m)as
+aconvenient waytorepresenttheedgesconnectinglayer mtolayerm−1.We
+stated that Z(m)was a finite K(m−1)×K(m)matrix. We can use the Indian
+buffet process (IBP) [Griffiths and Ghahramani ,2006] to allow this matrix
+to have an infinite number of columns. We assume the two-param eter IBP
+[Ghahramani et al. ,2007], and use Z(m)∼IBP(α,β) toindicate that the ma-
+trixZ(m)∈{0,1}K(m−1)×∞is drawn from an IBP with parameters α,β >0.4 R.P. ADAMS ET AL.
+−1−0.5 00.5 1
+(a)ν=1
+2−1−0.5 00.5 1
+(b)ν= 5−1−0.5 00.5 1
+(c)ν= 1000
+Fig 1:Three modes of operation for the NLGBN unit. The black solid line show s
+the zero mean distribution (i.e. y= 0), the red dashed line shows a pre-sigmoid
+mean of +1 and the blue dash-dot line shows a pre-sigmoid mean of −1. (a) Bi-
+nary behavior from small precision. (b) Roughly Gaussian behavior f rom medium
+precision. (c) Deterministic behavior from large precision.
+The eponymous metaphor for the IBP is a restaurant with an infi nite num-
+ber of dishes available. Each customer chooses a finite set of dishes to taste.
+The rows of the binary matrix correspond to customers and the columns
+correspond to dishes. If the jth customer tastes the kth dish, then Zj,k=1,
+otherwise Zj,k=0. The first customer into the restaurant samples a number
+of dishes that is Poisson distributed with parameter α. After that, when
+thejth customer enters the restaurant, she selects dish kwith probabil-
+ityηk/(j+β−1), where ηkis the number of previous customers that have
+tried the kth dish. She then chooses a number of additional dishes to tas te
+that is Poisson distributed with parameter αβ/(j+β−1). Even though each
+customer chooses dishes based on their popularity with prev ious customers,
+the rows and columns of the resulting matrix Z(m)are infinitely exchange-
+able.
+AsinWood et al. [2006], ifthemodelofSection 2hadonlyasinglehidden
+layer, i.e. M=1, then the IBP could be used to make that layer infinitely
+wide. While a belief network with an infinitely-wide hidden l ayer can rep-
+resent any probability distribution arbitrarily closely [ Le Roux and Bengio ,
+2008], it is not necessarily a useful prior on such distributions . Without
+intra-layer connections, the the hidden units are independ enta priori. This
+“shallowness” is a strong assumption that weakens the model in practice
+and the explosion of recent literature on deep belief networks (see, e.g.
+Hinton and Salakhutdinov [2006],Hinton et al. [2006]) speaks to the em-
+pirical success of belief networks with more hidden structu re.
+3.2.The cascading Indian buffet process. To build a prior on belief net-
+works that are unbounded in both width and depth, we use an IBP -like ob-
+ject that providesaninfinitesequenceof binarymatrices Z(0),Z(1),Z(2),···.LEARNING THE STRUCTURE OF DEEP SPARSE GRAPHICAL MODELS 5
+We require the matrices in this sequence to inherit the usefu l sparsity prop-
+erties of the IBP, with the constraint that the columns from Z(m−1)corre-
+spond to the rows in Z(m). We interpret each matrix Z(m)as specifying the
+directed edge structure from layer mto layerm−1, where both layers have
+a potentially-unbounded width.
+We propose the cascading Indian buffet process to provide a pri or with
+these properties. The CIBP extends the vanilla IBP in the fol lowing way:
+each of the “dishes” in the restaurant are also “customers” i n another Indian
+buffet process. The columns in one binary matrix correspond to the rows in
+another binary matrix. The CIBP is infinitely exchangeable i n the rows of
+matrixZ(0). Each of the IBPs in the recursion is exchangeable in its rows
+and columns, so it does not change the probability of the data to propagate
+a permutation back through the matrices.
+If there are K(0)customers in the first restaurant, a surprising result is
+that, for finite K(0),α, andβ, the CIBP recursion terminates with proba-
+bility one. By “terminate” we mean that at some point the cust omers do
+not taste any dishes and all deeper restaurants have neither dishes nor cus-
+tomers. Here we only sketch the intuition behind this result . A proof is
+provided in Appendix A.
+The matrices in the CIBP are constructed in a sequence, start ing with
+m=0. The number of nonzero columns in matrix Z(m+1),K(m+1), is de-
+termined entirely by K(m), the number of active nonzero columns in Z(m).
+We require that for some matrix Z(m), there are no nonzero columns. For
+this purpose, we can disregard the fact that it is a matrix-va lued stochastic
+process and instead consider the Markov chain that results o n the number
+of nonzero columns. Figure 2ashows three traces of such a Markov chain
+onK(m). If we define λ(K;α,β) =α/summationtextK
+k′=1β
+k′+β−1, then the Markov chain
+has the transition distribution
+p(K(m+1)=k|K(m),α,β) =1
+k!exp/braceleftBig
+−λ(K(m);α,β)/bracerightBig
+λ(K(m);α,β)k, (2)
+which is simply a Poisson distribution with mean λ(K(m);α,β). Clearly,
+K(m)= 0 is an absorbing state, however, the state space of the Mark ov
+chain is countably-infinite and to know that it will reach the absorbing state
+with probability one, we must know that K(m)does not blow up to infinity.
+In such a Markov chain, this requirement is equivalent to the statement
+that the chain has an equilibrium distribution when conditi oned on nonab-
+sorption (has a quasi-stationary distribution ) [Seneta and Vere-Jones ,1966].
+For countably-infinite state spaces, a Markov chain has a (qu asi-) station-
+ary distribution if it is positive-recurrent, which is the p roperty that there
+is a finite expected time between consecutive visits to any st ate. Positive6 R.P. ADAMS ET AL.
+102030405060708002040
+Depth mWidth K(m)
+(a)Example traces with K(0)= 50
+0510152005101520
+Current Width K(m)Expected K(m+1)
+(b)Expected K(m+1)05101520−10−505
+Current Width K(m)Drift
+(c)Drift
+Fig 2:Properties of the Markov chain on layer width for the CIBP, with α= 3,
+β= 1. Note that these values are illustrative and are not necessarily a ppropriatefor
+a network structure. a) Example traces of a Markov chain on layer width, indexed
+by depth m. b) Expected K(m+1)as a function of K(m)is shown in blue. The
+Lyapunov function L(·) is shown in green. c) The drift as a function of the current
+widthK(m). This corresponds to the difference between the two lines in (a). No te
+that it goes negative when the layer width is greater than eight.
+recurrency can be shown by proving the Foster–Lyapunov stability crite-
+rion(FLSC) [ Fayolle et al. ,2008]. Taken together, satisfying the FLSC for
+the Markov chain with transition probabilities given by Eqn 2demonstrates
+that eventually the CIBP will reach a restaurant in which the customers
+try no new dishes. We do this by showing that if K(m)is large enough, the
+expected K(m+1)is smaller than K(m).
+The FLSC requires a Lyapunov function L(k) :N+→R≥0, with whichLEARNING THE STRUCTURE OF DEEP SPARSE GRAPHICAL MODELS 7
+(a)α= 1,β= 1
+(b)α= 1,β=1
+2
+(c)α=1
+2,β= 1(d)α= 1,β= 2
+(e)α=3
+2,β= 1
+Fig 3:Samples from the CIBP-based prior on network structures, with fi ve visible
+units.
+we define the drift function :
+Ek|K(m)[L(k)−L(K(m))] =∞/summationdisplay
+k=1p(K(m+1)=k|K(m))(L(k)−L(K(m))).
+The drift is the expected change in L(k). If there is a K(m)above which
+all drifts are negative, then the Markov chain satisfies the F LSC and is
+positive-recurrent. In the CIBP, this is satisfied for L(k) =k. That the drift
+eventually becomes negative can be seen by the fact that
+Ek|K(m)[L(k)] =λ(K(m);α,β)
+isO(lnK(m)) andEk|K(m)[L(K(m))] =K(m)isO(K(m)). Figures 2band2c
+show a schematic of this idea.
+3.3.The CIBP as a prior on the structure of an infinite belief network.
+The CIBP can be used as a prior on the sequence Z(0),Z(1),Z(2),···from
+Section2, to allow an infinite sequence of infinitely-wide hidden laye rs. As
+before, there are K(0)visible units. The edges between the first hidden layer
+and the visible layer are drawn according to the restaurant m etaphor. This
+yields a finite number of units in the first hidden layer, denot edK(1)as8 R.P. ADAMS ET AL.
+before. These units are now treated as the visible units in an other IBP-
+based network. While this recurses infinitely deep, only a fin ite number of
+units are ancestors of the visible units. Figure 3shows several samples from
+the prior for different parameterizations. Only connected un its are shown in
+the figure.
+The parameters αandβgovern the expected width and sparsity of the
+network at each level. The expected in-degree of each unit (n umber of par-
+ents)isαandtheexpectedout-degree(numberofchildren)is K//summationtextK
+k=1β
+β+k−1,
+forKunits used in the layer below. For clarity, we have presented the CIBP
+results with αandβfixed at all depths; however, this may be overly restric-
+tive. For example, in an image recognition problem we would n ot expect the
+sparsity of edges mapping low-level features to pixels to be the same as that
+for high-level features tolow-level features. Toaddresst his, weallow αandβ
+to vary with depth, writing α(m)andβ(m). The CIBP terminates with prob-
+ability one as long as there exists some finite upper bound for α(m)andβ(m)
+for allm.
+3.4.Priors on other parameters. Other parameters in the model also re-
+quire prior distributions and we use these priors to tie para meters together
+accordingtolayer.Weassumethattheweightsinlayer maredrawnindepen-
+dently from Gaussian distributions with mean µ(m)
+wand precision ρ(m)
+m. We
+assume a similar layer-wise prior for biases with parameter sµ(m)
+γandρ(m)
+γ.
+We use layer-wise gamma priors on the ν(m)
+k, with parameters a(m)andb(m).
+We tie these prior parameters together with global normal-g amma hyper-
+priors for the weight and bias parameters, and gamma hyperpr iors for the
+precision parameters.
+4. Inference. We have so far described a prior on belief network struc-
+ture and parameters, along with likelihood functions for un it activation.
+The inference task in this model is to find the posterior distr ibution over
+the structure and the parameters of the network, having seen N K(0)-
+dimensional vectors {xn∈(−1,1)K(0)}N
+n=1. This posterior distribution is
+complex, so we use Markov chain Monte Carlo (MCMC) to draw sam ples
+fromp({Z(m),W(m)}∞
+m,{γ(m),ν(m)}∞
+m,{xn}N
+n), which, for fixed {xn}N
+n, is
+proportional to the posterior distribution. This joint dis tribution requires
+marginalizing over the states of the hidden units that led to each of the N
+observations. The values of these hidden units are denoted {{u(m)
+n}∞
+m=1}N
+n=1,
+and we augment the Markov chain to include these as well.
+In general, one would not expect that a distribution on infini te networks
+would yield tractable inference. However, in our construct ion, conditionedLEARNING THE STRUCTURE OF DEEP SPARSE GRAPHICAL MODELS 9
+on the sequence Z(1),Z(2),···, almost all of the infinite number of units
+are independent. Due to this independence, they trivially m arginalize out
+of the model’s joint distribution and we can restrict infere nce only to those
+units that are ancestors of the visible units. Of course, sin ce this trivial
+marginalization only arises from the Z(m)matrices, we must also have a
+distribution on infinite binary matrices that allows exact m arginalization of
+all the uninstantiated edges. The row-wise and column-wise exchangeabil-
+ity properties of the IBP are what allows the use of infinite ma trices. The
+bottom-up conditional structure of the CIBP allows an infini te number of
+these matrices.
+To simplify notation, we will use Ωfor the aggregated state of the model
+variables,i.e. Ω=({Z(m),W(m),{u(m)
+n}N
+n=1}∞
+m=1,{γ(m),ν(m)}∞
+m=0,{xn}N
+n=1).
+Given the hyperparameters, we can then write the joint distr ibution as
+(3)p(Ω) =
+p(γ(0))p(ν(0))K(0)/productdisplay
+k=1N/productdisplay
+n=1p(xk,n|y(0)
+k,n,ν(0)
+k)
+
+×
+∞/productdisplay
+m=1p(W(m))p(γ(m))p(ν(m))K(m)/productdisplay
+k=1N/productdisplay
+n=1p(u(m)
+k,n|y(m)
+k,n,ν(m)
+k)
+.
+Although this distribution involves several infinite sets, it is possible to sam-
+ple from the relevant parts of the posterior. We do this by MCM C, updating
+part of the model, while conditioning on the rest. In particu lar, condition-
+ing on the binary matrices {Z(m)}∞
+m=1, which define the structure of the
+network, inference becomes exactly as it would be in a finite b elief network.
+4.1.Sampling from the hidden unit states. Since we cannot easily inte-
+grate out the hidden units, it is necessary to explicitly rep resent them and
+sample from them as part of the Markov chain. As we are conditi oning on
+the network structure, it is only necessary to sample the uni ts that are an-
+cestors of the visible units. Frey[1997] proposed a slice sampling scheme for
+the hidden unit states but we have been more successful with a specialized
+independence-chain variant of multiple-try Metropolis–H astings [Liu et al. ,
+2000]. Our method proposes several ( ≈5) possible new unit states from the
+activation distribution and selects from among them (or rej ects them all)
+according to the likelihood imposed by its children. As this operation can
+be executed in parallel by tools such as Matlab, we have seen s ignificantly
+better mixing performance by wall-clock time than the slice sampler.
+4.2.Sampling from the weights and biases. Given that a directed edge
+exists, we sample the posterior distribution over its weigh t. Conditioning on10 R.P. ADAMS ET AL.
+the rest of the model, the NLGBN results in a convenient Gauss ian form
+for the distribution on weights so that we can Gibbs update th em using a
+Gaussian with parameters
+µw−post
+m,k,k′=ρ(m)
+wµ(m)
+w+ν(m−1)
+k/summationtext
+nu(m)
+n,k′(σ−1(u(m−1)
+k)−ξ(m)
+n,k,k′)
+ρ(m)
+w+ν(m−1)
+k/summationtext
+n(u(m)
+n,k′)2(4)
+ρw−post
+m,k,k′=ρ(m)
+w+ν(m−1)
+k/summationdisplay
+n(u(m)
+n,k′)2, (5)
+where
+ξ(m)
+n,k,k′=γ(m−1)
+k+/summationdisplay
+k′′/negationslash=k′Z(m)
+k,k′′W(m)
+k,k′′u(m)
+n,k′′. (6)
+The bias γ(m)
+kcan be similarly sampled from a Gaussian distribution with
+parameters
+µγ−post
+m,k=ρ(m)
+γµ(m)
+γ+ν(m)
+k/summationtextN
+n=1(σ−1(u(m)
+n,k)−χ(m)
+n,k)
+ρ(m)
+γ+Nν(m)
+k(7)
+ργ−post
+m,k=ρ(m)
+γ+Nν(m)
+k(8)
+where
+χ(m)
+n,k=K(m+1)/summationdisplay
+k′=1Z(m+1)
+k,k′W(m+1)
+k,k′u(m+1)
+n,k′. (9)
+4.3.Sampling from the activation variances. We use the NLGBN model
+togain theability tochangethemodeofunitbehaviorsbetwe en discreteand
+continuous representations. This corresponds to sampling from the posterior
+distributions over the ν(m)
+k. With a conjugate prior, the new value can be
+sampled from a gamma distribution with parameters
+aν−post
+m,k=a(m)
+ν+N/2 (10)
+bν−post
+m,k=b(m)
+ν+1
+2N/summationdisplay
+n=1(σ−1(u(m)
+n,k)−y(m)
+k)2. (11)
+4.4.Sampling from the structure. A model for infinite belief networks is
+only useful if it is possible to perform inference. The appea l of the CIBP
+prioristhat itenablesconstruction ofatractableMarkov c hainforinference.LEARNING THE STRUCTURE OF DEEP SPARSE GRAPHICAL MODELS 11
+To do this sampling, we must add and remove edges from the netw ork,
+consistent with the posterior equilibrium distribution. W hen adding a layer,
+we must sample additional layerwise model components. When introducing
+an edge, we must draw a weight for it from the prior. If this new edge
+introduces a previously-unseen hidden unit, we must draw a b ias for it and
+also draw its deeper cascading connections from the prior. F inally, we must
+sample the Nnew hidden unit states from any new unit we introduce.
+We iterate over each layer that connects to the visible units . Within each
+layerm≥0, we iterate over the connected units. Sampling the edges in ci-
+dent to the kth unit in layer mhas two phases. First, we iterate over each
+connected unit in layer m+1, indexed by k′. We calculate η(m)
+−k,k′, which is
+the number of nonzero entries in the k′th column of Z(m+1), excluding any
+entry in the kth row. If η(m)
+−k,k′is zero, we call the unit k′asingleton parent,
+to be dealt with in the second phase. If η(m)
+−k,k′is nonzero, we introduce (or
+keep) the edge from unit u(m+1)
+k′tou(m)
+kwith Bernoulli probability
+p(Z(m+1)
+k,k′= 1|Ω\Z(m+1)
+k,k′) =1
+Z
+η(m)
+−k,k′
+K(m)+β(m)−1
+
+×N/productdisplay
+n=1p(u(m)
+n,k|Z(m+1)
+k,k′= 1,Ω\Z(m)
+k,k′)
+p(Z(m+1)
+k,k′= 0|Ω\Z(m+1)
+k,k′) =1
+Z
+1−η(m)
+−k,k′
+K(m)+β(m)−1
+
+×N/productdisplay
+n=1p(u(m)
+n,k|Z(m+1)
+k,k′= 0,Ω\Z(m+1)
+k,k′),
+whereZis the appropriate normalization constant.
+In the second phase, we consider deleting connections to sin gleton parents
+of unitk, or adding new singleton parents. We do this via a Metropolis –
+Hastings operator using a birth/death process. If there are currently K◦
+singleton parents, then with probability 1 /2 we proposeaddinga new one by
+drawing it recursively from deeper layers, as above. We acce pt the proposal
+to insert a connection to this new parent unit with M–H accept ance ratio
+amh−insert=α(m)β(m)
+(K◦+1)2(β(m)+K(m)−1)N/productdisplay
+n=1p(u(m)
+n,k|Z(m+1)
+k,j=1,Ω\Z(m+1)
+k,j)
+p(u(m)
+n,k|Z(m+1)
+k,j=0,Ω\Z(m+1)
+k,j).12 R.P. ADAMS ET AL.
+If we do not propose to insert a unit and K◦≥0, then with probability 1 /2
+we select uniformly from among the singleton parents of unit kand propose
+removing the connection to it. We accept the proposal to remo ve thejth
+one with M–H acceptance ratio
+amh−remove=K2
+◦(β(m)+K(m)−1)
+α(m)β(m)N/productdisplay
+n=1p(u(m)
+n,k|Z(m+1)
+k,j=0,Ω\Z(m+1)
+k,j)
+p(u(m)
+n,k|Z(m+1)
+k,j=1,Ω\Z(m+1)
+k,j).
+After these phases, chains of units that are not ancestors of the visible
+units can be discarded. Notably, this birth/death operator samples from the
+IBP posterior with a non-truncated equilibrium distributi on, even without
+conjugacy. Unlike the stick-breaking approach of Teh et al. [2007], it allows
+use of the two-parameter IBP, which is important to this mode l.
+4.5.Sampling From CIBP Hyperparameters. When applying this model
+to data, it is infrequently the case that we would have a good a priori idea
+of what the appropriate IBP parameters should be. These cont rol the width
+and sparsity of the network and while we might have good initi al guesses
+for the lowest layer, in general we would like to infer {α(m),β(m)}as part of
+the larger inference procedure. This is straightforward in the fully-Bayesian
+MCMC procedure we have constructed, and it does not differ mark edly
+from hyperparameter inference in standard IBP models when c onditioning
+onZ(m). As in some other nonparametric models (e.g. Tokdar[2006] and
+Rasmussen and Williams [2006]), we have found that light-tailed priors on
+the hyperparameters helps ensure that the model stays in rea sonable states.
+5. Reconstructing Images. We applied the model and MCMC-based
+inference procedure to three image data sets: the Olivetti f aces, the MNIST
+digits and the Frey faces. We used these data to analyze the st ructures and
+sparsity that arise in the model posterior. To get a sense of t he utility of
+the model, we constructed a missing-data problem using held -out images
+from each set. We removed the bottom halves of the test images and asked
+the model to reconstruct the missing data, conditioned on th e top half. The
+prediction itself was done by integrating out the parameter s and structure
+via MCMC.
+Olivetti Faces. The Olivetti faces data [ Samaria and Harter ,1994] consists
+of 400 64 ×64 grayscale images of the faces of 40 distinct subjects. We d i-
+vided these into 350 test data and 50 training data, selected randomly. This
+data set is an appealing test because it has few examples, but many dimen-
+sions. Figure 4ashows six bottom-half test set reconstructions on the right ,LEARNING THE STRUCTURE OF DEEP SPARSE GRAPHICAL MODELS 13
+(a)
+ (b)
+(c)
+ (d)
+Fig 4:Olivettifacesa) Test imageson the left, with reconstructedbott om halveson
+the right. b) Sixty features learned in the bottom layer, where blac k shows absence
+of an edge. Note the learning of sparse features corresponding t o specific facial
+structures such as mouth shapes, noses and eyebrows. c) Raw p redictive fantasies.
+d) Feature activations from individual units in the second hidden laye r.
+compared to the ground truth on the left. Figure 4bshows a subset of sixty
+weight patterns from a posterior sample of the structure, wi th black indicat-
+ing that no edge is present from that hidden unit to the visibl e unit (pixel).
+The algorithm is clearly assigning hidden units to specific a nd interpretable
+features, such as mouth shapes, the presence of glasses or fa cial hair, and
+skin tone, while largely ignoring the rest of the image. Figu re4cshows ten
+pure fantasies from the model, easily generated in a directe d acyclic belief
+network. Figure 4dshows the result of activating individual units in the sec-
+ond hidden layer, while keeping the rest unactivated, and pr opagating the14 R.P. ADAMS ET AL.
+(a) (b) (c)
+(d)
+Fig 5:MNIST Digits a) Eight pairs of test reconstructions, with the botto m half of
+eachdigitmissing.Thetruth istheleft imageineachpair.b) 120featu reslearnedin
+the bottom layer, where black indicates that no edge exists. c) Act ivations in pixel
+space resulting from activating individual units in the deepest layer. d) Samples
+from the posterior of Z(0),Z(1)andZ(2)(transposed).
+activations down to the visible pixels. This provides an ide a of the image
+space spanned by the principal components of these deeper un its. A typical
+posterior network had three hidden layers, with about 70 uni ts in each layer.
+MNIST Digit Data. We used a subset of the MNIST handwritten digit
+data [LeCun et al. ,1998] for training, consisting of 50 28 ×28 examples of
+each of the ten digits. We used an additional ten examples of e ach digit for
+test data. Inthiscase, thelower-level featuresareextrem ely sparse,as shown
+in Figure 5b, and the deeper units are simply activating sets of blobs at t he
+pixel level. This is shown also by activating individual uni ts at the deepest
+layer, as shown in Figure 5c. Test reconstructions are shown in Figure 5a.
+A typical network had three hidden layers, with approximate ly 120 in the
+first, 100 in the second and 70 in the third. The binary matrice sZ(0),Z(1),
+andZ(2)are shown in Figure 5d.LEARNING THE STRUCTURE OF DEEP SPARSE GRAPHICAL MODELS 15
+(a) (b)
+Fig 6:Frey faces a) Eight pairs of test reconstructions, with the botto m half of
+each face missing. The truth is the left image in each pair. b) 260 feat ures learned
+in the bottom layer, where black indicates that no edge exists.
+Frey Faces. TheFreyfaces data1are1965 20 ×28grayscale videoframesof
+a single face with different expressions. We divided these int o 1865 training
+data and 100 test data, selected randomly. While typical pos terior samples
+of the network again typically used three hidden layers, the networks for
+these data tended to be much wider and more densely connected . In the
+bottom layer, as shown in Figure 6b, a typical hidden unit would connect to
+many pixels. We attribute this to global correlation effects f rom every image
+only coming from a single person. Typical widths were 260 uni ts, 120 units
+in the second hidden layer, and 35 units in the deepest layer.
+In all three experiments, our MCMC sampler appeared to mix we ll and
+begins to find reasonable reconstructions after a few hours o f CPU time.
+It is interesting to note that the learned sparse connection patterns in Z(0)
+varied fromlocal (MNIST), throughintermediate (Olivetti ) to global (Frey),
+despite identical hyperpriors on the IBP parameters. This s trongly suggests
+that flexiblepriorsonstructures areneededto adequately c apturethestatis-
+tics of different data sets.
+1http://www.cs.toronto.edu/ ~roweis/data.html16 R.P. ADAMS ET AL.
+6. Discussion. Thispaperunitestwoareasofresearch—nonparametric
+Bayesian methodsanddeepbeliefnetworks—toprovideanove l nonparamet-
+ric perspective on the general problem of learning the struc ture of directed
+deep belief networks with hidden units.
+Weaddressedthreeoutstandingissuesthatsurrounddeepbe liefnetworks.
+First, we allowed the units to have different operating regime s and infer
+appropriate local representations that range from discret e binary behavior
+to nonlinear continuous behavior. Second, we provided a way for a deep
+belief network to contain an arbitrary number of hidden unit s arranged
+in an arbitrary number of layers. This structure enables the hidden units
+to have nontrivial joint distributions. Third, we presente d a method for
+inferring the appropriate directed graph structure of deep belief network. To
+addressthese issues, weintroduceda novel cascading extension to theIndian
+buffet process—the cascading Indian buffet process (CIBP)—and proved
+convergence properties that make it useful as a Bayesian pri or distribution
+for a sequence of infinite binary matrices.
+This work can be viewed as an infinite multilayer generalizat ion of the
+density network [ MacKay,1995], and also as part of a more general litera-
+ture of learning structure in probabilistic networks. With a few exceptions
+(e.g.,Beal and Ghahramani [2006],Elidan et al. [2000],Friedman [1998],
+Ramachandran and Mooney [1998]), most previous work on learning the
+structure of belief networks has focused on the case where al l units are ob-
+served [Buntine,1991,Friedman and Koller ,2003,Heckerman et al. ,1995,
+Koivisto and Sood ,2004]. The framework presented in this paper not only
+allows for an unboundednumber of hidden units, but fundamen tally couples
+the model for the number and behavior of the units with the non parametric
+model for the structure of the infinite directed graph. Rathe r than compar-
+ing structures by evaluating marginal likelihoods of differe nt models, our
+nonparametric approach makes it possible to do inference in a single model
+with an unbounded number of units and layers, thereby learni ng effective
+model complexity. This approach is more appealing both comp utationally
+and philosophically.
+There are a variety of future research paths that can potenti ally stem
+from the model we have presented here. As we have presented it , we do
+not expect that our MCMC-based unsupervised inference sche me will be
+competitive on supervised tasks with extensively-tuned di scriminative mod-
+els based on variants of maximum-likelihood learning. Howe ver, we believe
+that this model can inform choices for network depth, layer s ize and edge
+structure in such networks and will inspire further researc h into flexible
+nonparametric network models.LEARNING THE STRUCTURE OF DEEP SPARSE GRAPHICAL MODELS 17
+Acknowledgements. The authors wish to thank Brendan Frey, David
+MacKay, Iain Murray and Radford Neal for valuable discussio ns. RPA is
+funded by the Canadian Institute for Advanced Research.
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+sity Estimation, Regression and Sufficient Dimension Reduct ion. PhD thesis, Purdue
+University, August 2006.
+F. Wood, T. L. Griffiths, and Z. Ghahramani. A non-parametric B ayesian method for in-
+ferring hiddencauses. In Proceedings of the 22nd Conference in Uncertainty in Artific ial
+Intelligence , 2006.
+APPENDIX A: PROOF OF GENERAL CIBP TERMINATION
+In the main paper, we discussed that the cascading Indian buffe t process
+for fixed and finite αandβeventually reaches a restaurant in which the
+customers choose no dishes. Every deeper restaurant also ha s no dishes.
+Here we show a more general result, for IBP parameters that va ry with
+depth, written α(m)andβ(m).
+Let there be an inhomogeneous Markov chain Mwith state space N.
+Letmindex time and let the state at time mbe denoted K(m). The initial
+stateK(0)is finite. The probability mass function describing the tran sition
+distribution for Mat timemis given by
+(12)p(K(m+1)=k|K(m),α(m),β(m)) =
+1
+k!exp
+
+−α(m)K(m)/summationdisplay
+k′=1β(m)
+k′+β(m)−1
+
+
+α(m)K(m)/summationdisplay
+k′=1β(m)
+k′+β(m)−1
+.
+Theorem A.1.If there exists some ¯α <∞and¯β <∞such that ∀m,
+α(m)<¯αandβ(m)<¯β, thenlimm→∞p(K(m)= 0) = 1 .
+Proof. LetN+be the positive integers. The N+are a communicating
+class for the Markov chain (it is possible to eventually reac h any mem-
+ber of the class from any other member) and each K(m)∈N+has a
+nonzero probability of transitioning to the absorbing stat eK(m+1)= 0,LEARNING THE STRUCTURE OF DEEP SPARSE GRAPHICAL MODELS 19
+i.e.p(K(m+1)= 0|K(m))>0,∀K(m). If, conditioned on nonabsorption, the
+Markov chain has a stationary distribution (is quasi-stationary ), then it
+reaches absorption in finite time with probability one. Heur istically, this is
+the requirement that, conditioned on having not yet reached a restaurant
+with no dishes, the number of dishes in deeper restaurants wi ll not explode.
+The quasi-stationary condition can be met by showing that N+are posi-
+tive recurrent states. We usetheFoster–Lyapunov stabilit y criterion (FLSC)
+to show positive-recurrency of N+. The FLSC is met if there exists some
+function L(·) :N+→R+such that for some ǫ >0 and some finite B∈N+,
+∞/summationdisplay
+k=1p(K(m+1)=k|K(m))/parenleftBig
+L(k)−L(K(m))/parenrightBig
+<−ǫforK(m)> B (13)
+∞/summationdisplay
+k=1p(K(m+1)=k|K(m))L(k)<∞forK(m)≤B. (14)
+For Lyapunov function L(k) =k, the first condition is equivalent to
+
+α(m)K(m)/summationdisplay
+k=1β(m)
+k+β(m)−1
+−K(m)<−ǫ. (15)
+We observe that
+α(m)K(m)/summationdisplay
+k=1β(m)
+k+β(m)−1<¯αK(m)/summationdisplay
+k=1¯β
+k+¯β−1, (16)
+for allK(m)>0. Thus, the first condition is satisfied for any Bthat satisfies
+the condition for ¯ αand¯β. That such a Bexists for any finite ¯ αand¯βcan
+be seen by the equivalent condition
+
+¯αK(m)/summationdisplay
+k=1¯β
+k+¯β−1
+−K(m)<−ǫforK(m)> B. (17)
+As the first term is roughly logarithmic in K(m), there exists some finite B
+that satisfies this inequality. The second FLSC condition is trivially satisfied
+by the observation that Poisson distributions have a finite m ean.
+Ryan P. Adams
+Department of Computer Science
+University of Toronto
+10 King’s College Road
+Toronto, Ontario M5S 3G4, CA
+E-mail: rpa@cs.toronto.eduHanna M. Wallach
+Department of Computer Science
+University of Massachusetts Amherst
+140 Governors Drive
+Amherst, MA 01003, USA
+E-mail: wallach@cs.umass.edu20 R.P. ADAMS ET AL.
+Zoubin Ghahramani
+Department of Engineering
+University of Cambridge
+Trumpington Street
+Cambridge CB2 1PZ, UK
+zoubin@eng.cam.ac.uk
\ No newline at end of file
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@@ -0,0 +1,824 @@
+arXiv:1001.0161v1  [gr-qc]  31 Dec 2009Phenomenology of loop quantum cosmology
+Mairi Sakellariadou
+Department of Physics, King’s College London, University o f London, Strand WC2R 2LS, UK
+E-mail:Mairi.Sakellariadou@kcl.ac.uk
+Abstract. After introducing the basic ingredients of Loop Quantum Cos mology, I will briefly
+discuss some of its phenomenological aspects. Those can giv e some useful insight about the full
+Loop Quantum Gravity theory and provide an answer to some lon g-standing questions in early
+universe cosmology.
+1. Introduction
+The variety of precise astrophysical and cosmological data , available at present, combined with
+the large number of high energy physics experiments, are exp ected to give us the necessary
+ingredients to test fundamental theories and understand th e very early phases of the evolution
+of our universe.
+Cosmological inflation [1] remains the most promising candi date to solve the shortcomings
+of the standard hot big bang model, while it offers a causal expl anation for the primordial
+fluctuations with the correct Cosmic Microwave Background ( CMB) features. Despite this
+success, inflation suffers from a number of drawbacks [2]. In pa rticular, compatibility between
+theory and measurements often necessitates fine-tuning of t he inflationary parameters, inflation
+remains still a paradigm in search of a model, while it must pr ove itself generic [3, 4, 5, 6, 7, 8, 9].
+There is undoubtfully an additional list of fundamental cos mological questions, still lacking
+a satisfying answer. One does not know, for instance, how clo se to the big bang a smooth
+space-time can be considered as the correct framework. Quan tum gravity, a full theory which is
+supposed to resolve the big bang singularity, is still missi ng, while it is not known whether a new
+boundary condition is needed at the big bang, or whether quan tum dynamical equations remain
+well-behaved at singularities. It is nevertheless clear th at a smooth space-time background
+cannot be assumed as the correct description close to the big bang.
+In a Hamiltonian formulation to a quantum theory, the absenc e of background metric
+indicates that Hamiltonian dynamics is generated by constr aints. Physical states are solutions
+to quantum constraints implying that all physical laws are o btained from these solutions, while
+thereisnoexternaltimeaccordingwhichevolutioncanbest udied. Onehastodefineamonotonic
+variable to play the rˆ ole of emergent, or internal, time, an d then build a framework within which
+the short-distance drawbacks of General Relativity near th e big bang can be cured, maintaining
+however the agreement with General Relativity at large scal es.
+Loop Quantum Gravity (LQG) [10, 11] is a non-perturbative an d background independent
+canonical quantisation of General Relativity in four space -time dimensions. Loop Quantum
+Cosmology (LQC) [12] is a cosmological mini-superspace mod el quantised with methods of
+LQG. The discreteness of spatial geometry and the simplicit y of the setting allow for a completestudy of dynamics. The difference between LQC and other approa ches of quantum cosmology,
+is that the input is motivated by a full quantum gravity theor y. The simplicity of the setting,
+combined with the discreteness of spatial geometry provide d by LQG, render feasible the overall
+study of LQC dynamics.
+In what follows, I will briefly discuss some of the phenomenol ogical consequences of LQC,
+a subject which gains constantly an increasing interest fro m the scientific community, due in
+particular to its successes.
+2. Elements of LQG/LQC
+Loop quantum cosmology is formulated in terms of SU(2) holon omies of the connection and
+triads. The canonically conjugated variables are defined by the densitised triad Ea
+i, and an
+SU(2) valued connection Ai
+a, whereirefers to the Lie algebra index, and ais a spatial index
+witha,i= 1,2,3. The densitised triad gives information about spatial geo metry (i.e., the three
+metric), while the connection gives information about curv ature (spatial and extrinsic one).
+Let me explain this: Quantum gravity introduces a discreten ess to space-time. To quantise
+quantumgravity, thisdiscreteness manifests itself asqua nta ofspace. Sincethequantaarethree-
+dimensional, they can be characterised by a triad of numbers . The connection between adjacent
+pairs of these quantaform a two-dimensional surface, through which the geometri c connection is
+defined. To ensure that local rotations do not induce different geometries, this connection must
+beSO(3) invariant. By using connection-triad variables, a rising from a canonical transformation
+of Arnowitt-Desner-Misner ( variables, one can make an anal ogy with gauge theories, which is
+particularly useful when dealing with quantisation issues .
+For any quantisation scheme based on a Hamiltonian framewor k (e.g., LQC) or an action
+principle ( e.g., path integral) for a homogeneous and flat model, divergence s appear which need
+to be regularised. To remove the divergences that occur in no n-compact topologies, the spatial
+homogeneity andHamiltonian arerestricted toafinitefiduci alcell, ofscale µ0, withfinitevolume
+V0=/integraltext
+d3x/radicalbig
+|0q|, where0qis the determinant of the fiducial background metric.
+In the case of a spatially flat background, derived from the Bi anchi I model, the isotropic
+connection can be expressed in terms of the dynamical compon ent of the connection ˜ c(t) as
+Ai
+a= ˜c(t)ωi
+a, (1)
+withωi
+aa basis of left-invariant one-forms ωi
+a= dxi. The densitised triad can be decomposed
+using the Bianchi I basis vector fields Xa
+i=δa
+ias
+Ea
+i=/radicalbig
+0q˜p(t)Xa
+i, (2)
+where0qstands for the determinant of the fiducial background metric0qab=ωi
+aωbi, and ˜p(t)
+denotes the remaining dynamical quantity after symmetry re duction.
+In terms of the metric variables with three-metric qab=a2ωi
+aωbi, the dynamical quantity is
+just the scale factor a(t). Given that the Bianchi I basis vectors are Xa
+i=δa
+i,
+|˜p|=a2, (3)
+where the absolute value is taken because the triad has an ori entation. Since the basis vector
+fields are spatially constant in the spatially flat model, the connection component is
+˜c= sgn(˜p)γ˙a
+N; (4)
+Nis the lapse function and γthe Barbero-Immirzi parameter, a quantum ambiguity parame ter
+approximately equal to 0.2375.The canonical variables ˜ c,˜pare related through the Poisson bracket
+{˜c,˜p}=κγ
+3V0, (5)
+whereV0the volume of the elementary cell adapted to the fiducial tria d andκ≡8πG.
+Defining the triad component p, determining the physical volume of the fiducial cell, and
+the connection component c, determining the rate of change of the physical edge length o f the
+fiducial cell, as
+p=V2/3
+0˜p , c =V1/3
+0˜c , (6)
+respectively, we obtain
+{c,p}=κγ
+3, (7)
+independent of the volume V0of the fiducial cell.
+To quantise the theory one faces the usual difficulty of quantu m cosmology. Namely, the
+metric itself has to be considered as a physical field, thus it must be quantised; it is not a
+fixed background. Any standard quantisation scheme fails: Fock quantisation fails, while even
+free Hamiltonians are non-quadratic in metric dependence, so there is no simple perturbation
+analysis. The way out is to use gauge theory variables to defin e holonomies of the connection
+along a given edge
+he(A) =Pexp/integraldisplay
+ds˙γµ(s)Ai
+µ(γ(s))τi, (8)
+wherePindicates a path ordering of the exponential, γµis a vector tangent to the edge and
+τi=−iσi/2, withσithe Pauli spin matrices, and fluxes of a triad along an Ssurface
+E(S,f) =/integraldisplay
+SǫabcEcifidxadxb, (9)
+withfian SU(2) valued test function. Certainly, at the level of LQC these variables may seem
+ad-hoc and unnatural, however their motivation follows nat urally from the full LQG theory.
+Thus, the basic configuration variables in LQC are holonomie s of the connection
+h(µ0)
+i(A) = cos/parenleftigµ0c
+2/parenrightig
+1+2sin/parenleftigµ0c
+2/parenrightig
+τi, (10)
+along a line segment µ00ea
+iand the flux of the triad
+FS(E,f)∝p ,
+where the basic momentum variable is the triad component p,1is the identity 2 ×2 matrix and
+τi=−iσi/2 is a basis in the Lie algebra SU(2) satisfying the relation
+τiτj= (1/2)ǫijkτk−(1/4)δij.
+In the classical theory, curvature can be expressed as a limi t of the holonomies around a loop
+as the area enclosed by the loop shrinks to zero. In quantum ge ometry however, the loop cannot
+be continuously shrunk to zero area and the eigenvalues of th e area operator are discrete. Thus,
+there is a smallest non-zero eigenvalue, the area gap ∆ [13]. As a result, the Wheeler-de Witt
+differential equation gets replaced by a difference equation wh ose step size is controlled by ∆.
+Let me start with the oldquantisation procedure. Along the lines of LQG, one conside rs
+eiµ0c/2(withµ0an arbitrary real number) and p, as the elementary classical variables, whichhave well-defined analogues. Using the Dirac bra-ket notati on and setting eiµ0c/2=∝an}bracketle{tc|µ∝an}bracketri}ht, the
+action of the operator ˆ pacting on the basis states |µ∝an}bracketri}htreads
+ˆp|µ∝an}bracketri}ht=κγ/planckover2pi1|µ|
+6|µ∝an}bracketri}ht, (11)
+whereµ(a real number) stands for the eigenstates of ˆ p, satisfying the orthonormality relation
+∝an}bracketle{tµ1|µ2∝an}bracketri}ht=δµ1,µ2. The action of the/hatwiderexp/bracketleftig
+iµ0
+2c/bracketrightig
+operator acting on basis states |µ∝an}bracketri}htis
+/hatwider
+exp/bracketleftbiggiµ0
+2c/bracketrightbigg
+|µ∝an}bracketri}ht= exp/bracketleftbigg
+µ0d
+dµ/bracketrightbigg
+|µ∝an}bracketri}ht=|µ+µ0∝an}bracketri}ht, (12)
+whereµ0is any real number. Thus, in the oldquantisation, the operator eiµ0c/2acts as a simple
+shift operator. As in the full LQG theory, there is no operato r corresponding to the connection,
+however the action of its holonomy is well defined. The action of the holonomies, ˆh(µ0)
+i, of the
+gravitational connection on the basis states is given by [14 ]
+ˆh(µ0)
+i|µ∝an}bracketri}ht= (/hatwidecs1+2/hatwidesnτi)|µ∝an}bracketri}ht, (13)
+where,
+/hatwidecs|µ∝an}bracketri}ht ≡/hatwidercos(µ0c/2)|µ∝an}bracketri}ht= [|µ+µ0∝an}bracketri}ht+|µ−µ0∝an}bracketri}ht]/2,
+/hatwidesn|µ∝an}bracketri}ht ≡/hatwidersin(µ0c/2)|µ∝an}bracketri}ht= [|µ+µ0∝an}bracketri}ht−|µ−µ0∝an}bracketri}ht]/(2i). (14)
+The gravitational part of the Hamiltonian operator in terms of SU(2) holonomies and the
+triad component, in the irreducible J= 1/2 representation, reads [14]
+ˆCg=2i
+κ2/planckover2pi1γ3µ3
+0tr/summationdisplay
+ijkǫijk/parenleftig
+ˆh(µ0)
+iˆh(µ0)
+jˆhi(µ0)−1ˆh(µ0)−1
+jˆh(µ0)
+k/bracketleftig
+ˆh(µ0)−1
+k,ˆV/bracketrightig/parenrightig
+sgn(ˆp).(15)
+The action of the self-adjoint Hamiltonian constraint oper ator,ˆHg= (ˆCg+ˆC†
+g)/2 on the basis
+states|µ∝an}bracketri}htis
+ˆHg|µ∝an}bracketri}ht=3
+4κ2γ3/planckover2pi1µ3
+0/braceleftig/bracketleftbig
+R(µ)+R(µ+4µ0)/bracketrightbig
+|µ+4µ0∝an}bracketri}ht−4R(µ)|µ∝an}bracketri}ht+/bracketleftbig
+R(µ)+R(µ−4µ0)/bracketrightbig
+|µ−4µ0∝an}bracketri}ht/bracerightig
+,
+(16)
+where
+R(µ) = (κγ/planckover2pi1/6)3/2/vextendsingle/vextendsingle/vextendsingle|µ+µ0|3/2−|µ−µ0|3/2/vextendsingle/vextendsingle/vextendsingle. (17)
+The dynamics are then determined by the Hamiltonian constra int
+(ˆHg+ˆHφ)|Ψ∝an}bracketri}ht= 0, (18)
+whereˆHφstands for the matter Hamiltonian. In the full LQG theory the re is an infinite number
+of constraints, while in LQC there is only one integrated Ham iltonian constraint. Matter is
+introduced by just adding the actions of matter components t o the gravitational action. We can
+finally obtain difference equations analogous to the differenti al Wheeler-de Witt equations.More precisely, the constraint equation on the physical wav e-functions |Ψ∝an}bracketri}ht, which can be
+expanded using the basis states as |Ψ∝an}bracketri}ht=/summationtext
+µΨµ(φ)|µ∝an}bracketri}htwith summation over values of µand
+wherethedependenceof thecoefficients on φrepresents thematter degrees of freedom, reads [15]
+/bracketleftigg/vextendsingle/vextendsingle/vextendsingleVµ+5µ0−Vµ+3µ0/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingleVµ+µ0−Vµ−µ0/vextendsingle/vextendsingle/vextendsingle/bracketrightigg
+Ψµ+4µ0(φ)−4/vextendsingle/vextendsingle/vextendsingleVµ+µ0Vµ−µ0/vextendsingle/vextendsingle/vextendsingleΨµ(φ)
++/bracketleftigg/vextendsingle/vextendsingle/vextendsingleVµ−3µ0−Vµ−5µ0/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingleVµ+µ0−Vµ−µ0/vextendsingle/vextendsingle/vextendsingle/bracketrightigg
+Ψµ−4µ0(φ) =−4κ2γ3/planckover2pi1µ3
+0
+3Hφ(µ)Ψµ(φ) ; (19)
+the matter Hamiltonian ˆHφis assumed to act diagonally on the basis states with eigenva lues
+Hφ(µ). Equation (19) is the quantum evolution (in internal time µ) equation; there is no
+continuous variable (the scale factor in classical cosmolo gy), but a label µwith discrete steps.
+The wave-function Ψ µ(φ), depending on internal time µand matter fields φ, determines the
+dependence of matter fields on the evolution of the universe, with a massless scalar field playing
+the rˆ ole of emergent time. Thus, in LQC the quantum evolutio n is governed by a second order
+difference equation, rather than the second order differential equation of the Wheeler-de Witt
+quantum cosmology. As the universe becomes large and enters the semi-classical regime, its
+evolution can be approximated by the differential Wheeler-de Witt equation.
+3. Lattice refinement
+The quantised holonomies were at first assumed to be shift ope rators with a fixed magnitude,
+leadingtoaquantisedHamiltonianconstraintbeingadiffere nceequationwithaconstantinterval
+between points on the lattice. While these models can be used to study certain aspects of the
+quantum regime, they were found to lead to serious instabili ties in the continuum, semi-classical
+limit [16, 17]. Considering the continuum limit of the Hamil tonian constraint operator, we
+have so far assumed that Ψ does not vary much on scales of the or der of 4µ0(known as pre-
+classicality ), so that one can smoothly interpolate between the points on the discrete function
+Ψµ(φ) and approximate them by the continuous function Ψ( µ,φ). Under this assumption, the
+difference equation can be very well approximated by a different ial equation for a continuous
+wave-function. However, the form of the wave-function indi cates that the period of oscillations
+can decrease as the scale increases. Thus, the assumption of pre-classicality can break down at
+large scales, leading to deviations from the classical beha viour.
+Let me explain this point a bit further: In the underlying ful l LQG theory, the contributions
+to the (discrete) Hamiltonian operator depend on the state w hich describes the universe. As
+the volume grows (ı.e., the universe expands), the number of contributions increases. Thus, the
+Hamiltonian constraint operator is expected to create new v ertices of a lattice state (in addition
+to changing their edge labels), which in LQC results in a refin ement of the discrete lattice [18].
+Latticerefinementisalsorequiredfromphenomenological r easons[17,19]; forinstance, itrenders
+a successful inflationary era more natural [19]. Theeffect of l attice refinement has been modelled
+and the elimination of the instabilities in the continuum er a has been explicitly shown.
+The appropriate lattice refinement model should be obtained from the full LQG theory. In
+principle, oneshouldusethefullHamiltonian constraint a ndfindtheway thatitsaction balances
+the creation of new vertices as the volume increases. Instea d, phenomenological arguments have
+been used, where the choice of the lattice refinement model is constrained by the form of the
+matter Hamiltonian [20]. In particular, LQC can genericall y support inflation, and other matter
+fields, withouttheonset of large scale quantumgravity corr ections, only for aparticular model of
+lattice refinement [20]. This choice is the only one for which physical quantities are independent
+oftheelementary cell adopted toregulatespatial integrat ions [21], andmoreover, it isexactly the
+choice required for the uniqueness of the factor ordering of the Wheeler-de Witt equation [22].Allowing the length scale of the holonomies to vary dynamica lly, the form of the difference
+equation, describing the evolution, changes. Since the par ameterµ0determines the step-size
+of the difference equation, assuming the lattice size is growi ng, the step-size of the difference
+equation is not constant in the original triad variables. Le t us consider the particular case of
+µ0→˜µ(µ) =µ0µ−1/2, (20)
+suggested by certain intuitive heuristic approaches, such as noting that the minimum area used
+to regulate the holonomies should be a physical area [23], or that the discrete step size of
+the difference equation should always be of the order of the Pla nck volume. Moreover, this
+choice also results in a significant simplification of the diffe rence equation, compared to more
+general lattice refinement schemes. The basic operators are given by replacing µ0with ˜µ. Upon
+quantisation [23]
+/hatwiderei˜µc/2|µ∝an}bracketri}ht=e−i˜µd
+dµ|µ∝an}bracketri}ht, (21)
+which is no longer a simple shift operator since ˜ µis a function of µ. Changing the basis to
+ν=µ0/integraldisplaydµ
+˜µ=2
+3µ3/2, (22)
+one gets
+e−i˜µd
+dµ|ν∝an}bracketri}ht=e−iµ0d
+dν|ν∝an}bracketri}ht=|ν+µ0∝an}bracketri}ht. (23)
+Thus, in the new variables the holonomies act as simple shift operators, with parameter length
+µ0. In this sense, the basis |ν∝an}bracketri}htis a much more natural choice than |µ∝an}bracketri}ht. One can then proceed
+as in the previous case of a fixed spatial lattice and write dow n the Hamiltonian constraint.
+3.1. Constraints on inflation
+Let me consider first a fixed and then a dynamically varying lat tice and solve the second order
+difference equation in the continuum limit. Two constraints c an be imposed on the inflaton
+potential: the first one so that the continuum approximation is valid, and the second one so
+that there is agreement with the CMB measurements on large an gular scales. A combination
+of the two constraints in the context of a particular inflatio nary model, will give us [20] the
+conditions for natural and successful inflation within LQC.
+More precisely, let us separate the wave-function Ψ( p,φ) into Ψ(p,φ) = Υ(p)Φ(φ) and
+approximate the dynamics of the inflaton field, φ, by setting V(φ) =Vφpδ−3/2, whereVφis
+a constant and δ= 3/2 in the case of slow-roll, to get [20]
+p−1/2d
+dp/bracketleftbigg
+p−1/2d
+dp/parenleftig
+p3/2Υ(p)/parenrightig/bracketrightbigg
++βVφpδΥ(p) = 0, (24)
+with solutions [20]
+Υ(p)≈p−(9+2δ)/8/radicaligg
+2δ+3
+2/radicalbig
+βVφπ/bracketleftigg
+C1cos/parenleftbigg
+x−3π
+2(2δ+3)−π
+4/parenrightbigg
++C2sin/parenleftbigg
+x−3π
+2(2δ+3)−π
+4/parenrightbigg/bracketrightigg
+,
+(25)
+wherex= 4/radicalbigβVφ(2δ+3)−1p(2δ+3)/4andβ= 96/(κ/planckover2pi12).
+Without lattice refinement, the discrete nature of the under lying lattice would eventually
+be unable to support the oscillations and the assumption of pre-classicality will break down,
+implying that the discrete nature of space-time becomes sig nificant on large scales. For the end
+of inflation to be describable using classical General Relat ivity, it must end before a scale, atwhich the assumption of pre-classicality breaks down and the semi-classical description is no
+longer valid, is reached. Let me quantify this constraint: T he separation between two successive
+zeros of Υ( p) is
+∆p=π/radicalbigβVφp(1−2δ)/4. (26)
+For the continuum limit to be valid, the wave-function must v ary slowly on scales of the order
+ofµc= 4˜µ, leading to [20]
+∆p>4µ0/parenleftbiggκγ/planckover2pi1
+6/parenrightbigg3/2
+p−1/2, (27)
+which implies [20]
+Vφ<27π2
+192µ2
+0γ3κ2/planckover2pi1p(3−2δ)/2. (28)
+Assuming slow-roll inflation, we set δ≈3/2. Setting µ0= 3√
+3/2, the constraint on the
+inflationary potential, in units of /planckover2pi1= 1, reads
+V(φ)<∼2.35×10−2l−4
+Pl, (29)
+which is a weaker constraint than the one imposed for fixed lat tices, namely [20]
+Vφ≪10−28l−4
+pl, (30)
+assuming that half of the inflationary era takes place during the classical era.
+One can further constrain the inflationary potential so that the fractional over-density in
+Fourier space and at horizon crossing is consistent with the COBE-DMR measurements, namely
+[V(φ)]3/2
+V′(φ)≈5.2×10−4M3
+Pl. (31)
+To do so, one must however adopt a particular inflationary mod el. As such, let us select
+V(φ) =m2φ2/2. Combining the two constraints we obtain [20]
+m<∼70(e−2Ncl)MPlandm<∼10MPl, (32)
+for the fixed and varying lattices, respectively.
+In conclusion, the requirement that a significant proportio n of a successful (in particular
+with respect to the CMB measurements) inflationary regime ta kes place during the classical era,
+imposes a strong constraint on the inflaton mass. Such a const raint turns out however to be
+much softer, and therefore rather natural, once lattice refi nement is considered. In this sense,
+we argue [20] that lattice refinement is essential to achieve a successful inflationary era, provided
+inflation can be generically ( i.e., with generic initial conditions) realised.
+3.2. Constraints on the matter Hamiltonian
+Given a lattice refinement model, we will show [19] that only c ertain types of matter can be
+allowed. To do so, we parametrise the lattice refinement by Aand the matter Hamiltonian
+byδand solve the Hamiltonian constraint. The restrictions on t he two-dimensional parameter
+space will become apparent once physical restrictions to th e solutions of the wave-functions are
+imposed [19].
+To be more precise, consider
+˜µ=µ0µA, (33)-0.5-0.4-0.3-0.2-0.10
+-1.5 -1 -0.5 0 0.5 1 1.5 2A
+δ
+Break down of
+preclassicalityNon-
+normalisableLimit of
+Bessel Expansion
+Figure 1. The allowed types of matter content are significantly restri cted. For a varying
+lattice (A∝ne}ationslash= 0) it is not always possible to treat the large scale behavio ur of the wave-functions
+perturbatively (dashed line with crosses) [19].
+leading to
+ν=˜µ0µ1−A
+µ0(1−A). (34)
+Being only interested in the large scale limit, we approxima te [19] the matter Hamiltonian by
+ˆHφ= ˆνδˆǫ(φ), implying
+ˆǫ(φ)Ψ≡ǫ(φ)Ψ =−ν−δˆHgΨ. (35)
+A necessary condition so that the wave-functions are physic al, is that the finite norm of the
+physical wave-functions, defined by/integraltext
+φ=φ0dν|ν|δΨ1Ψ2, is independent of the choice of φ=φ0.
+The solutions of the constraint are renormalisable provide d they decay, on large scales, faster
+thanν−1/(2δ).
+To solve the constraint equation, we need to specify the from ofHφ, which has in general
+two terms with different scale dependence. Being interested i n the large scale limit, there is one
+dominant term, allowing to write [19]
+βHφ=ǫν(φ)νδν, (36)
+where the function ǫνis constant with respect to ν.
+Solving [19] the constraint equation, we only consider the p hysical solutions. The large
+scale behaviour of the wave-functions must be normalisable , which is a necessary condition for
+havingphysical wave-functions, while thewave-functions should preserve pre-classicality at large
+scales, which is a necessary condition for the validity of th e continuum limit. We thus obtain [19]
+constraints to the two-dimensional parameter space ( A,δ), shown in Fig. 1 for Ain the range
+0<A<−1/2, imposed from full LQG theory considerations [17].
+In conclusion, the continuum limit of the Hamiltonian const raint equation is sensitive to the
+choice of the lattice refinement model and only a limited rang e of matter components can be
+supported within a particular choice.3.3. Uniqueness of the factor ordering in the Wheeler-de Wit t equation
+I will now show that the lattice refinement model ˜ µ=µ0µ−1/2, argued [22] to be the only
+one achieved by both physical considerations of large scale physics and consistency of the
+quantisation structure, is also the only model which makes t he factor ordering ambiguities of
+LQC to disappear in the continuum limit [22].
+Let me be more specific. Writing the gravitational part of the Hamiltonian constraint in
+terms of the triad and the holonomies of the connection, one r ealises that there are many ways
+of doing so. Considering for example,
+ˆCg=2i
+κ2/planckover2pi1γ3k3tr/summationdisplay
+ijkǫijk/parenleftig
+ˆhiˆhjˆh−1
+iˆh−1
+jˆhk/bracketleftig
+ˆh−1
+k,ˆV/bracketrightig/parenrightig
+, (37)
+there are many possible choices of factor ordering that coul d have been made at this point;
+classically, the actions of the holonomies commute. Howeve r, each of these factor ordering
+choices leads to a different factor ordering of the Wheeler-de Witt equation in the continuum
+limit. The action of the factor ordering chosen in Eq. (37) le ads [22] to
+ǫijktr/parenleftig
+ˆhiˆhjˆh−1
+iˆh−1
+jˆhk/bracketleftig
+ˆh−1
+k,ˆV/bracketrightig/parenrightig
+=−24/hatwidesn2/hatwidecs2/parenleftig
+/hatwidecsˆV/hatwidesn−/hatwidesnˆV/hatwidecs/parenrightig
+, (38)
+while other choices have certainly a different action.
+Defining ˆV|ν∝an}bracketri}ht=Vν|ν∝an}bracketri}ht, withˆVthe volume operator with eigenvalues Vν, the action of the
+above factor ordering on a general state |Ψ∝an}bracketri}ht=/summationtext
+νψν|ν∝an}bracketri}htin the Hilbert space, reads [22]
+ǫijktr/parenleftig
+ˆhiˆhjˆh−1
+iˆh−1
+jˆhk/bracketleftig
+ˆh−1
+k,ˆV/bracketrightig/parenrightig
+|Ψ∝an}bracketri}ht=−3i
+4/summationdisplay
+ν/bracketleftigg/parenleftig
+Vν−3k−Vν−5k/parenrightig
+ψν−4k−2/parenleftig
+Vν+k−Vν−k/parenrightig
+ψν
++/parenleftig
+Vν+5k−Vν+3k/parenrightig
+ψν+4k/bracketrightigg
+|ν∝an}bracketri}ht; (39)
+the action of any other factor ordering choice can be obtaine d in a similar manner. By noting
+that the volume is given by
+Vν|ν∝an}bracketri}ht ∼[µ(ν)]3/2|ν∝an}bracketri}ht, (40)
+whereµ(ν) is obtained by
+ν=kµ1−A
+µ0(1−A), (41)
+we find [22]
+Vν±nk∼/bracketleftig
+(ν±nk)α/bracketrightig3/[2(1−A)]
+, (42)
+whereα=µ0(1−A)/k.
+We then take the continuum limit of these expressions by expa ndingψν≈ψ(ν) as a
+Taylor expansion in small k/ν. For the particular factor ordering chosen above, the large scale
+continuum limit of the Hamiltonian constraint reads [22]:
+lim
+k/ν→0ǫijktr/parenleftig
+ˆhiˆhjˆh−1
+iˆh−1
+jˆhk/bracketleftig
+ˆh−1
+k,ˆV/bracketrightig/parenrightig
+|Ψ∝an}bracketri}ht ∼
+−36i
+1−Aα3/[2(1−A)]k3/summationdisplay
+νν(1+2A)/[2(1−A)]/bracketleftigg
+d2ψ
+dν2+1+2A
+1−A1
+νdψ
+dν+(1+2A)(4A−1)
+(1−A)21
+4ν2ψ(ν)/bracketrightigg
+|ν∝an}bracketri}ht.
+(43)One can easily confirm that one obtains the same continuum lim it for the Wheeler-de Witt
+equation, only for the choice A=−1/2 [22], in which case the Wheeler-de Witt equation
+reads [22]
+lim
+k/ν→0Cg|Ψ∝an}bracketri}ht=72
+κ2/planckover2pi1γ3/parenleftbiggκγ/planckover2pi1
+6/parenrightbigg3/2/summationdisplay
+νd2ψ
+dν2|ν∝an}bracketri}ht. (44)
+Thus, there is only one lattice refinement model, namely ˜ µ=µ0µ−1/2, with a non-ambiguous
+continuum limit.
+In conclusion, phenomenological and consistency requirem ents lead to a particular lattice
+refinement model, implying that LQC predicts a unique factor ordering of the Wheeler-de Witt
+equation in its continuum limit. Alternatively, demanding that factor ordering ambiguities
+disappear in LQC at the level of Wheeler-de Witt equation lea ds to a unique choice for the
+lattice refinement model.
+3.4. Numerical techniques in solving the Hamiltonian const raint
+Lattice refinement leads to new dynamical difference equation s, which being of a non-uniform
+step-size, imply technical complications. More precisely , the information needed to calculate the
+wave-function at a given lattice point is not provided by pre vious iterations. This becomes clear
+in the case of two-dimensional wave-functions, as for insta nce in the study of Bianchi models or
+black hole interiors. I will present below a method [24] base d on Taylor expansions that can be
+used to perform the necessary interpolations with a well-de fined and predictable accuracy.
+For a one-dimensional difference equation defined on a varying lattice, the Hamiltonian
+constraint can be mapped onto a fixed lattice simply by a chang e of basis [24]. This method is
+however not usefulfor the two-dimensional case, wheretheH amiltonian constraint is a difference
+equation on a varying lattice [17]:
+C+(µ,τ)/bracketleftbig
+Ψµ+2δµ,τ+2δτ−Ψµ−2δµ,τ+2δτ/bracketrightbig
++C0(µ,τ)/bracketleftbig
+(µ+2δµ)Ψµ+4δµ,τ−2/parenleftbig
+1+2γ2δ2
+µ/parenrightbig
+µΨµ,τ+(µ−2δµ)Ψµ−4δµ,τ/bracketrightbig
++C−(µ,τ)/bracketleftbig
+Ψµ−2δµ,τ−2δτ−Ψµ+2δµ,τ−2δτ/bracketrightbig
+=δτδ2
+µ
+δ3HφΨµ,τ, (45)
+with
+C±≡2δµ/parenleftig/radicalbig
+|τ±2δτ|+/radicalbig
+|τ|/parenrightig
+, (46)
+C0≡/radicalbig
+|τ+δτ|−/radicalbig
+|τ−δτ|, (47)
+whereδµandδτdenote the step-sizes along the µandτdirections, respectively.
+In the case of lattice refinement, δµandδτare decreasing functions of µandτ, respectively,
+and the data needed to calculate the value of the wave-functi on at a particular lattice site are
+not given by previous iterations, as it is illustrated in Fig .2. We propose [24] to use Taylor
+expansions to calculate the necessary data points. Let us as sume that the matter Hamiltonian
+acts diagonally on the basis states of the wave-function, na mely
+ˆHφ|Ψ∝an}bracketri}ht ≡ˆHφ/summationdisplay
+µ,τΨµ,τ|µ,τ∝an}bracketri}ht=/summationdisplay
+µ,τHφΨµ,τ|µ,τ∝an}bracketri}ht. (48)
+Given a function evaluated at three (non-collinear) coordi nates, the Taylor approximation to
+the value at a fourth position is
+f(x4,y4) =f(x2,y2)+δx
+42∂f
+∂x/vextendsingle/vextendsingle/vextendsingle
+x2,y2+δy
+42∂f
+∂y/vextendsingle/vextendsingle/vextendsingle
+x2,y2
++O/parenleftbigg
+(δx
+42)2∂2f
+∂x2/vextendsingle/vextendsingle/vextendsingle
+x2,y2/parenrightbigg
++O/parenleftbigg
+(δy
+42)2∂2f
+∂y2/vextendsingle/vextendsingle/vextendsingle
+x2,y2/parenrightbigg
+, (49)(a)µ−4δµ µ µ +4δµ µ+8δµ µ+12δµ
+µ−2δµ µ+2δµ µ+6δµ µ+10δµτ−2δτττ+2δττ+4δττ+6δτ
+(b)µ µ−4δµ µ+4δµ
+µ−2δµ µ+2δµ¯µ−4δ¯µ
+¯µ−2δ¯µ¯µ
+¯µ+2δ¯µ¯µ+4δ¯µ
+¯τ
+¯τ−2δ¯τ¯τ+2δ¯τ
+Figure 2. (a) For the fixed lattice case the two-dimensional wave-func tion can be calculated,
+given suitable initial conditions (solid circles). (b) In t he case of a refining lattice, the data
+needed to calculate the value of the wave-function at a parti cular lattice site (open square) is
+not given by previous iterations (solid squares) [24].
+where the Taylor expansion is taken about the position ( x2,y2), we have defined δx
+ij≡xi−xj
+andδy
+ij≡yi−yj, and the differentials can be approximated using
+f(x1,y1) =f(x2,y2)+δx
+12∂f
+∂x/vextendsingle/vextendsingle/vextendsingle
+x2,y2+δy
+12∂f
+∂y/vextendsingle/vextendsingle/vextendsingle
+x2,y2+···, (50)
+f(x3,y3) =f(x2,y2)+δx
+32∂f
+∂x/vextendsingle/vextendsingle/vextendsingle
+x2,y2+δy
+32∂f
+∂y/vextendsingle/vextendsingle/vextendsingle
+x2,y2+···, (51)
+where the dots indicate higher order terms.
+For slowly varying wave-functions, linear approximation i s very accurate and higher order
+terms in Taylor expansion can only improve the accuracy by 10−2% [24]. This method can be
+applied in any lattice refinement model, while its accuracy c an be estimated.
+By using this Taylor expansions method, we confirmed [24] num erically the stability
+criterion of the Schwarzchild interior, found [17] earlier using a von Neumann analysis, and
+investigated [24] the way that lattice refinement modifies th e stability properties of the system.
+Let me now show how the underlying discreteness of space-tim e leads to a twist [24] in the
+wave-functions, for both a fixed and a varying lattice model. In the case of a constant lattice,
+an initially centred Gaussian will move to larger µ, asτis increased [24]. However, for regions
+where the lattice discreteness is important, this no longer holds, and the value at one lattice
+point introduces a non-zero component to the value at a latti ce point with larger µcoordinate,
+i.e., the wave-function moves to larger µ. This implies the existence of some induced rotation
+on the wave-function due to the underlying discreteness of t he space-time. Including lattice
+refinement, this effect persists [24]. In other words, there is no motion for τ≫δτ, while in the
+case of lattice refinement this requirement is reached for lo werτ, sinceδτreduces asτincreases.
+As the wave-function moves into a region in which the discret eness of the lattice is important,
+a motion will be induced, as in the constant lattice case.4. Anisotropic LQC
+Various aspects of anisotropic cosmologies have been studi ed within LQC in the past [17, 25],
+however the first full and consistent quantisation of a Bianc hi I cosmology (the simplest of
+anisotropic cosmological models) was achieved recently in Ref. [26]. Moreover, the link back
+to the underlying full LQG theory has been strengthened, by c onsidering the flux of the triads
+through surfaces consistent with the Bianchi I anisotropic case. With the quantisation of the
+Bianchi I model under control, it is possible to ask whether L QC features, obtained within the
+context of isotropic Friedmann-Lemaˆ ıtre-Roberston-Wal ker cosmology, are robust, at least with
+respect to this limited extension of the symmetries of the sy stem. Bianchi type I models, apart
+their simplicity, they present a particular interest for sp ace-like singularities within the full LQG
+theory. Following the same vein as for the isotropic case, a m assless scalar field plays the rˆ ole of
+an internal time parameter. In the absence of such a field, phy sical evolution can be found by
+constructing families of unitarily related partial observ ables, parametrised by geometry degrees
+of freedom [27].
+4.1. Unstable Bianchi I LQC
+Studyingstability conditions of thefull Hamiltonian cons traint equation describingthe quantum
+dynamics of the diagonal Bianchi I model, I will show [28] tha t there is robust evidence of an
+instability in the explicit implementation of the difference equation. On the one hand, such a
+result may question the choice of the quantisation approach , the model of lattice refinement,
+and/or the rˆ ole of ambiguity parameters. On the other hand, one may argue that such an
+instability may not be necessarily a problem since it might b e that unstable trajectories are
+explicitly removed by the physical inner product.
+Considerthedifferenceequationarisingfromtheloopquanti sation oftheBianchiImodel[26]:
+∂2
+TΨ(λ1,λ2,ν;T) =πG
+2√ν/bracketleftig
+(ν+2)√
+ν+4Ψ+
+4(λ1,λ2,ν;T)−(ν+2)√νΨ+
+0(λ1,λ2,ν;T)
+−(ν−2)√νΨ−
+0(λ1,λ2,ν;T)+(ν−2)/radicalbig
+|ν−4|Ψ−
+4(λ1,λ2,ν;T)/bracketrightig
+,(52)
+where
+Ψ+
+4(λ1,λ2,ν;T) =/summationdisplay
+i/negationslash=j=(0,1,2)Ψ(aiλ1,ajλ2,ν+4;T)
+Ψ−
+4(λ1,λ2,ν;T) =/summationdisplay
+i/negationslash=j=(−3,−2,0)Ψ(aiλ1,ajλ2,ν−4;T)
+Ψ+
+0(λ1,λ2,ν;T) =/summationdisplay
+i/negationslash=j=(−1,0,1)Ψ(aiλ1,ajλ2,ν;T)
+Ψ−
+0(λ1,λ2,ν;T) =/summationdisplay
+i/negationslash=j=(−2,0,3)Ψ(aiλ1,ajλ2,ν;T), (53)
+a−3≡/parenleftbiggν−4
+ν−2/parenrightbigg
+, a−2≡/parenleftbiggν−2
+ν/parenrightbigg
+, a−1≡/parenleftbiggν
+ν+2/parenrightbigg
+,
+a0≡1, a1≡/parenleftbiggν+4
+ν+2/parenrightbigg
+, a2≡/parenleftbiggν+2
+ν/parenrightbigg
+, a3≡/parenleftbiggν
+ν−2/parenrightbigg
+. (54)
+Note thatλ1,λ2,λ3are related to the volume of the fiducial cell Vby
+ˆVΨ(λ1,λ2,ν) = 2π|γ|√
+∆|ν|l3
+plΨ(λ1,λ2,ν), (55)ν
+λ1λ2
+λ1a−2λ1a−3λ1λ2 a−2λ2 a−3λ2
+ν−4λ1a−2λ1
+a−1λ1a−2λ2a−1λ2λ2
+λ1
+a2λ1
+a3λ1
+λ2a2λ2a3λ2νλ1
+a1λ1
+a2λ1
+λ2a1λ2a2λ2ν+ 4
+Figure 3. The geometry of the points used in the difference equation that results from the
+Hamiltonian constraint, for the Bianchi I model [28].
+withγ= sgn(p1p2p3)|γ|and
+ν= 2λ1λ2λ3, (56)
+We will investigate the stability of the vacuum solutions, i n which case the solution is static,
+namely Ψ( λ1,λ2,ν;T) = Ψ(λ1,λ2,ν), simplifying considerably Eq. (52), whose geometry is
+drawn in Fig. 3. To search for growing mode solutions we will p erform a von Neumann analysis.
+In addition to specifying the boundary conditions on the νandν−4 planes, we are also required
+to specify the value at five of the points given in Ψ+
+4(λ1,λ2,ν). There are in total 23 values
+that are required and with such initial data the difference equ ation, Eq. (52), can be used to
+evaluate the 24thpoint. Once this point has been evaluated, it can be used to movethe central
+point and evaluate the wave-function at subsequent positio ns in theν+4 plane.
+Following the standard von Neumann stability analysis, we d ecompose the solutions of the
+difference equation into Fourier modes and look for growing mo des. Considering the ansatz [28]
+Ψ(λ1,λ2,ν) =T(λ1)exp(i(ωλ2+χν)), (57)
+where we have chosen the λ1direction to be the direction in which the ν+4 plane is evolved,the difference equation becomes
+e4χi/summationdisplay
+i/negationslash=j=(0,1,2)T(aiλ1)ei(ωajλ2+χν)=/radicalbiggν
+ν+4/summationdisplay
+i/negationslash=j=(−1,0,1)T(aiλ1)ei(ωajλ2+χν)
++/parenleftbiggν−2
+ν+2/parenrightbigg/radicalbiggν
+ν+4/summationdisplay
+i/negationslash=j=(−2,0,3)T(aiλ1)ei(ωajλ2+χν)
+−/parenleftbiggν−2
+ν+2/parenrightbigg/radicalbigg
+|ν−4|
+ν+4e−4χi/summationdisplay
+i/negationslash=j=(−3,−2,0)T(aiλ1)ei(ωajλ2+χν).
+(58)
+As it has been explicitly shown in Ref. [28], expanding in ter ms of small 1 /ν, the difference
+equation can be written in the form of a vector equation as
+M1T3=M2T2, (59)
+where we have defined the vectors
+Ti=
+T(aiλ1)
+T(ai−1λ1)
+T(ai−2λ1)
+T(ai−3λ1)
+T(ai−4λ1)
+T(ai−5λ1)
+fori= 2,3 (60)
+and the matrices
+M1=
+A0 0 0 0 0
+0 1 0 0 0 0
+0 0 1 0 0 0
+0 0 0 1 0 0
+0 0 0 0 1 0
+0 0 0 0 0 1
+, M 2=
+B C D E F G
+1 0 0 0 0 0
+0 1 0 0 0 0
+0 0 1 0 0 0
+0 0 0 1 0 0
+0 0 0 0 1 0
+.(61)
+The condition for stability can be written as follows: If
+max|˜λ| ≤1∀ωandχ , (62)
+where˜λare the eigenvalues of the matrix ( M1)−1M2, the amplitude T(a3λ1) is less than that
+of previous points, in other words, the difference equation is stable. Let us define the parameter
+Λ =ωλ2/ν. The presence of an instability in the difference equation has been demonstrated [28]
+in several ways: (i) there is a particular set of critical mod es, Λ = (2n−1)π/2, withn∈Z,
+for which the system is unstable; (ii) in the large νlimit, the system is unstable for the modes
+Λ =π/4 andχ= 0; (iii) the system is unstable for a general ν, for modes that approach the
+critical value.
+In conclusion, the difference equation, Eq. (52), is uncondit ionally unstable, in the sense that
+there is no region of ( λ1,λ2,ν) in which Eq. (52) is stable.4.2. Lattice refinement from isotropic embedding of anisotr opic cosmology
+Given a consistent quantum anisotropic model, one can find [2 9] isotropic states for which the
+discrete step-size of the isotropically embedded Hamilton ian constraint is not necessarily that
+of thep−1/2(new quantisation ) step-size. The choice of different embeddings has important
+consequences for the precise form of discretisation in the i sotropic sub-system. In this sense,
+lattice refinement could be interpreted as being due to the de grees of freedom that are absent
+in the isotropic model [29].
+In the standard approach, isotropic states are taken to be th ose in which the three scale
+factors are λ1=λ2=λ3, or more precisely, defining the volume of the state as ν= 2λ1λ2λ3to
+eliminate one of the directions ( λ3), the map
+|Ψ(λ1,λ2,ν)∝an}bracketri}ht →/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+λ1,λ2Ψ(λ1,λ2,ν)∝an}bracketri}ht ≡ |Ψ(ν)∝an}bracketri}ht, (63)
+produces isotropic states. Working with the three scale fac torsλi, one can show [29] that there
+is an ambiguity in exactly what the volume of such an isotropi c state is. Consider the state
+|˜Ψ(λ1,λ2,λ3)∝an}bracketri}ht ≡1
+A/bracketleftig
+|Ψ(λ1,λ2,λ3)∝an}bracketri}ht+|Ψ(λ3,λ1,λ2)∝an}bracketri}ht+|Ψ(λ2,λ3,λ1)∝an}bracketri}ht/bracketrightig
+,(64)
+with the restriction
+∝an}bracketle{tΨ(λ1,λ2,λ3)|Ψ(λ3,λ1,λ2)∝an}bracketri}ht=∝an}bracketle{tΨ(λ1,λ2,λ3)|Ψ(λ2,λ3,λ1)∝an}bracketri}ht=∝an}bracketle{tΨ(λ3,λ1,λ2)|Ψ(λ2,λ3,λ1)∝an}bracketri}ht,
+on the anisotropic states. The expectation values of the sca le factors along each direction of
+such a state are
+∝an}bracketle{tˆλi∝an}bracketri}ht=λ1+λ2+λ3
+3. (65)
+The measured scale factor is equal in each direction and is gi ven by the average of the scale
+factorsoftheunderlying, anisotropicstates. However, th emeasuredvolumeofsuchastateisjust
+ν= 2λ1λ2λ3, which is notthe cube of the measured scale factor. Thus, while it is the ei genvalue
+oftheanisotropically definedvolumeoperator, itisnot nec essarilywhatwewouldmeasureasthe
+volume. Essentially, the reason for this is that while the sc ale factorsλiare measured to beequal
+in each direction, they arenot eigenvalues of the state, i.e.,ˆλi|˜Ψ(λ1,λ2,λ3)∝an}bracketri}ht ∝ne}ationslash=λi|˜Ψ(λ1,λ2,λ3)∝an}bracketri}ht.
+However, both the average and the product ( i.e., the volume) of the scale factors are eigenvalues.
+It is this ambiguity, that leads to the possibility of deviat ions from the standard isotropic case.
+In conclusion, the difference between the two procedures is es sentially due to what one
+considers to be more fundamental, the volume of the underlyi ng states (ν) or the measured
+volume of the symmetric state ( ∝an}bracketle{tλ∝an}bracketri}ht3), which are not necessarily equal. Choosing νleads to the
+newquantised Hamiltonian of isotropic cosmology, while choos ing∝an}bracketle{tλ∝an}bracketri}ht3results in some kind of
+different lattice refinement . Thislattice refinement is significantly more complicated that the
+single power law behaviour, pA, usually considered.
+5. Conclusions
+Loop Quantum Gravity proposes a method of quantising gravit y in a background independent,
+non-perturbative way. Quantum gravity is essential when cu rvature becomes large, as for
+example in the early stages of the evolution of the universe. Applying LQG in a cosmological
+context leads to Loop Quantum Cosmology which is a symmetry r eduction of the infinite
+dimensional phase space of the full theory, allowing us to st udy certain aspects of the theory
+analytically. The discreteness of spatial geometry, a key e lement of the full theory, leads to
+successes in LQC which do not hold in the Wheeler-de Witt quan tum cosmology.
+By studyingphenomenological consequences of LQC we can get some useful insight about the
+full LQG theory and get an answer to some long-standing quest ions in early universe cosmology.
+Here, I have briefly described some of the phenomenological a spects of LQC.Acknowledgments
+It is a pleasure to thank the organisers of the First Mediterr anean Conference on Classical and
+Quantum Gravity (MCCQG) (http://www.phy.olemiss.edu/mc cqg/), for inviting me to give
+this talk, in the beautiful island of Crete in Greece. This wo rk is partially supported by the
+European Union through the Marie Curie Research and Trainin g Network UniverseNet (MRTN-
+CT-2006-035863).
+References
+[1] Guth A 1981 Phys. Rev. D 23347
+[2] Sakellariadou M 2008 Lect. Notes Phys. 738359
+[3] Piran T 1986 Phys. Lett. B 181238
+[4] Goldwirth D 1991 Phys. Rev. D 433204
+[5] Calzetta E and Sakellariadou M 1992 Phys. Rev. D 452802
+[6] Calzetta E and Sakellariadou M 1993 Phys. Rev. D 473184
+[7] Germani C, Nelson W and Sakellariadou M 2007 Phys. Rev. D 76043529
+[8] Gibbons G and Turok N 2008 Phys. Rev. D 77063516
+[9] Ashtekar A and Sloan D 2009 Loop quantum cosmology and slo w roll inflation ( Preprint 0912.4093)
+[10] Rovelli C 2004 Quantum Gravity (Cambridge: Cambridge University Press)
+[11] Thiemann T 2007 Modern Canonical Quantum General Relativity (Cambridge: Cambridge University Press)
+[12] Bojowald M 2005 Living Rev. Rel. 811
+[13] Ashtekar A and Lewandowski J 1997 Class. & Quant. Grav. 14A55
+[14] Ashtekar A, Pawlowski T and Singh P 2006 Phys. Rev. D 74084003
+[15] Vandersloot K 2005 Phys. Rev. D 71103506
+[16] Rosen J, Jung J H and Khanna G 2006 Class. Quant. Grav. 237075
+[17] Bojowald M, Cartin D and Khanna G 2007 Phys. Rev. D 76064018
+[18] Sakellariadou M 2009 J. Phys. Conf. Ser. 189012035
+[19] Nelson W and Sakellariadou M (2007) Phys. Rev. D 76104003
+[20] Nelson W and Sakellariadou M 2007 Phys. Rev. D 76044015
+[21] Corichi A and Singh P 2008 Phys. Rev. D 78024034
+[22] Nelson W and Sakellariadou M 2008 Phys. Rev. D 78024006
+[23] Ashtekar A, Pawlowski T and Singh P 2006 Phys. Rev. D 74, 084003
+[24] Nelson W and Sakellariadou M 2008 Phys. Rev. D 78024030
+[25] Chiou D W and Vandersloot K 2007 Phys. Rev. D 76084015
+[26] Ashtekar A and Wilson-Ewing E 2009 Phys. Rev. D 79083535
+[27] Martin-Benito M, Mena Marugan G A and Pawlowski T 2009 Phys. Rev. D 80084038
+[28] Nelson W and Sakellariadou M 2009 Phys. Rev. D 80063521
+[29] NelsonWandSakellariadou M2009LatticeRefiningLoopQ uantumCosmology fromanIsotropicEmbedding
+of Anisotropic Cosmology ( Preprint 0906.0292)
\ No newline at end of file
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+arXiv:1001.0162v2  [math.AG]  18 Aug 2011FAMILIES OF DETERMINANTAL SCHEMES
+JAN O. KLEPPE, ROSA M. MIR ´O-ROIG∗
+Abstract. Given integers a0≤a1≤ ··· ≤ at+c−2andb1≤ ··· ≤ bt, we denote by
+W(b;a)⊂Hilbp(Pn) the locus of good determinantal schemes X⊂Pnof codimension
+cdefined by the maximal minors of a t×(t+c−1) homogeneous matrix with entries
+homogeneous polynomials of degree aj−bi. The goal of this paper is to extend and
+completetheresultsgivenbytheauthorsin[12]anddetermineunde rweakenednumerical
+assumptions the dimension of W(b;a) as well as whether the closure of W(b;a) is a
+generically smooth irreducible component of Hilbp(Pn).
+1.Introduction
+In this paper, we will deal with good and standard determinantal sc hemes. A scheme
+X⊂Pnof codimension cis called standard determinantal if its homogeneous saturated
+ideal can be generated by the maximal minors of a homogeneous t×(t+c−1) matrix, and
+Xis said to be good determinantal if it is standard determinantal and a generic complete
+intersection. We denote the Hilbert scheme by Hilbp(Pn). Given integers a0≤a1≤ ··· ≤
+at+c−2andb1≤ ··· ≤ bt, we denote by W(b;a)⊂Hilbp(Pn) (resp. Ws(b;a)) the locus
+of good (resp. standard) determinantal schemes X⊂Pnof codimension cdefined by
+the maximal minors of a t×(t+c−1) homogeneous matrix with entries homogeneous
+polynomials of degree aj−bi.
+In [11] and [12], we addressed the following 3 crucial problems:
+(1) To determine the dimension of W(b;a) in terms of ajandbi,
+(2) Is the closure of W(b;a) an irreducible component of Hilbp(Pn)? and
+(3) Is Hilbp(Pn) generically smooth along W(b;a)?
+In [12] we obtained an upper bound for dim W(b;a) in terms of ajandbiwhich was
+achieved in the cases 2 ≤c≤5 andn−c >0 (assuming char(k) = 0 ifc= 5), and in
+codimension c >5 provided certain numerical conditions are satisfied (See [12], Theo rems
+3.5 and 4.5; and Corollaries 4.7, 4.10 and 4.14). Concerning problems (2 ) and (3), we
+gave in [12] an affirmative answer to both questions in the range 2 ≤c≤4 andn−c≥2,
+and in the cases c≥5 andn−c≥1 provided certain numerical assumptions are verified
+(See [12], Corollaries 5.3, 5.7, 5.9 and 5.10. See also [7], [11] for the case s 2≤c≤3).
+Note that since every element of W(b;a) has the same Hilbert function, the assumption
+n > cis close to being necessary for problem (2). Indeed if n=cthe problems (2) and (3)
+become more natural provided we replace Hilbp(Pn) by the postulation Hilbert scheme,
+see [10].
+Date: November 16, 2018.
+1991Mathematics Subject Classification. Primary 14M12, 14C05, 14H10, 14J10.
+∗Partially supported by MTM2010-15256 .
+12 JAN O. KLEPPE, ROSA M. MIR ´O-ROIG
+In this work we attempt to extend and complete the results of [11] a nd [12]. Indeed if
+at+3> at−2we almost solve problem (1) in Theorem 3.2 while Theorem 3.4 and Corol-
+lary 3.8, for c >4, generalize results of [12] for the problems (2) and (3) substan tially.
+To prove these results we use induction on the codimension by succe ssively deleting the
+columns of the highest degree and the Eagon-Northcott complex a ssociated to a standard
+determinantal scheme. We also use the theory of Hilbert flag schem es and the depth of
+certain mixed determinantal schemes (see Theorem 2.7). We end th e paper with two con-
+jectures which are supported by our results and by a huge number of examples computed
+using Macaulay 2.
+Notation: Throughout this paper Pnis then-dimensional projective space over an alge-
+braically closed field k,R=k[x0,x1,...,x n] andm= (x0,...,x n). ByHomOX(F,G) we
+denote the sheaf of local morphisms between coherent OX-modules while Hom( F,G) de-
+notesthegroupofmorphismsfrom FtoG. Moreoverwesethom( F,G) = dim kHom(F,G)
+and we correspondingly use small letters for the dimension, as a k-vector space, of similar
+groups. For any quotient AofRof codimension c, we letKA= Extc
+R(A,R)(−n−1).
+Inthe sequel, µHomR(M,N) denotes homomorphisms of degree µof graded R-modules.
+Moreover, we denote the Hilbert scheme by Hilbp(Pn),pthe Hilbert polynomial, and
+(X)∈Hilbp(Pn) the point which corresponds to the subscheme X⊂Pnwith Hilbert
+polynomial p. We denote by IXthe saturated homogeneous ideal of X⊂Pn. We say that
+Xisgeneralin some irreducible subset W⊂Hilbp(Pn) if (X) belongs to a sufficiently
+small open subset UofW(small enough so that any ( X)∈Uhas all the openness
+properties that we want to require).
+2.Preliminaries
+This section provides the background and basic results on standar d determinantal
+ideals, good determinantal ideals and mixed determinantal ideals nee ded in the sequel.
+We refer to [3], [6] [12] and [15] for the details.
+LetA= (fij)j=0,...,t+c−2
+i=1,...t, degfij=aj−bi, be at×(t+c−1) homogeneous matrix and
+let
+(2.1) ϕ:F=t/circleplusdisplay
+i=1R(bi)−→G:=t+c−2/circleplusdisplay
+j=0R(aj)
+be the graded morphism of free R-modules represented by the transpose, At, ofA. Let
+I(A) =It(A) be the ideal of Rgenerated by the maximal minors of A. A codimension
+csubscheme X⊂Pnis said to be standard determinantal ifIX=I(A) for some homo-
+geneoust×(t+c−1) matrix Aas above. Moreover Xisgood determinantal ifXis
+standard determinantal and a generic complete intersection in Pn([13], Theorem 3.4). In
+this paper we suppose c≥2,t≥2,b1≤...≤btanda0≤a1≤...≤at+c−2. (Note
+that the case t= 1 for determinantal schemes corresponds to the well-known com plete
+intersections).
+LetW(b;a) (resp. Ws(b;a)) be the stratum in Hilbp(Pn) consisting of good (resp.
+standard) determinantal schemes as above. Since our definition d oes not assume Ato be
+minimal (i.e. fij= 0 when bi=aj) forX= Proj(R/It(A))∈W(b;a) (orWs(b;a)), weFAMILIES OF DETERMINANTAL SCHEMES 3
+must reconsider Corollary 2.6 of [12] where Awas supposed minimal in the proof (a slight
+correction to [11] and [12]!). We may, however, use that proof to se e that
+(2.2)W(b;a)/ne}ationslash=∅ ⇔Ws(b;a)/ne}ationslash=∅ ⇔ai−1≥bifor alliandai−1> bifor some i.
+Indeed if we assume the converse of the condition on the right hand side, then either
+It(A)∋1 or one of the maximal minors vanishes, i.e. Proj( R/It(A))/∈Ws(b;a). Con-
+versely assuming the right hand side condition (to simplify notations, assumeai−1≥bi
+for 1≤i≤sandai−1> bifors < i≤tfor some integer s < t), then we may take
+A= (I O
+OA′) whereIis thes×sidentity matrix, Oare matrices of zero’s and A′, for
+t−s >1, the (t−s)×(t−s+c−1) matrix used in [12], Corollary 2.6 to define a good
+determinantal scheme (if t−s= 1 we take the entries of A′to be a regular sequence).
+We get Proj( R/It(A))∈W(b;a) and we easily deduce (2.2). Note that by [12], end of
+p. 2877 and [12], Remark 3.7 we still have that the closures of W(b;a) andWs(b;a) in
+Hilbp(Pn) are equal and irreducible.
+LetA=R/IXbethehomogeneouscoordinateringofastandarddeterminantal scheme.
+By [3], Theorem 2.20 and [6], Corollaries A2.12 and A2.13 the Eagon-Northcott complex
+yields a minimal free resolution of A
+(2.3) 0 −→ ∧t+c−1G∗⊗Sc−1(F)⊗∧tF−→ ∧t+c−2G∗⊗Sc−2(F)⊗∧tF−→...
+−→ ∧tG∗⊗S0(F)⊗∧tF−→R−→A−→0
+which allows us to deduce that any standard determinantal scheme is arithmetically
+Cohen-Macaulay (ACM). Moreover if MA:= coker( ϕ∗) thenKA(n+ 1)∼=Sc−1MA(ℓc)
+where
+(2.4) ℓi:=t+i−2/summationdisplay
+j=0aj−t/summationdisplay
+k=1bkfor 2≤i≤c.
+LetBbe the matrix obtained by deleting the last column of A, letB=R/IBbe thek-
+algebra given by the maximal minors of Band letMBbe the cokernel of φ∗= Hom R(φ,R)
+whereφ:F=⊕t
+i=1R(bi)→G′:=⊕t+c−3
+j=0R(aj) is the graded morphism induced by Bt.
+Recall that if c >2 there is an exact sequence
+(2.5) 0 −→B−→MB(at+c−2)−→MA(at+c−2)−→0
+in which B−→MB(at+c−2) is a regular section given by the last column of A. Moreover,
+(2.6) 0 −→MB(at+c−2)∗:= Hom B(MB(at+c−2),B)−→B−→A−→0
+is exact by [13] or [11] (e.g. see the text after (3.1) of [11]). Note th at the proofs of (2.5)-
+(2.6) rely heavily on the equality Ann( MB) =IBestablished in [4]. If c= 2 (codim RB=
+1) we have at least Ann( MB)It−1(B)⊂IB⊂Ann(MB) by [4]; thus the kernel, IA/B,
+ofB→Asatisfies IA/B=MB(at+c−2)∗(resp.˜IA/B|U=˜MB(at+c−2)∗|UwhereU=
+Y−V(It−1(B))) forc >2 (resp. c= 2). Due to the Buchsbaum-Rim resolution of
+M:=MA,Mis a maximal Cohen-Macaulay A-module, and so is IA/Bforc >2 by (2.6).
+By successively deleting columns from the right hand side of A, and taking maximal
+minors, one gets a flag of standard determinantal subschemes
+(2.7) ( X.) :X=Xc⊂Xc−1⊂...⊂X2⊂X1⊂Pn4 JAN O. KLEPPE, ROSA M. MIR ´O-ROIG
+where each Xi+1⊂Xi(with ideal sheaf IXi+1/Xi=Ii) is of codimension 1, Xi⊂Pnis
+of codimension i(i= 1,...,c) and there exist OXi-modules Mifitting into short exact
+sequences
+0→ OXi(−at+i−1)→ Mi→ Mi+1→0 for 2≤i≤c−1,
+such that Ii(at+i−1) is theOXi-dual of Mifor 2≤i≤c(all this holds also for i= 1
+provided werestrict thesheaves to X1−V(It−1(ϕ1))where ϕ1isgivenby X1=V(It(ϕ1))).
+In this context, we let Di:=R/IXi,IDi:=IXiandIi:=IDi+1/IDi.
+Remark 2.1. Assumet≥2,b1≤...≤bt,a0≤a1≤...≤at+c−2and letα≥1 be an
+integer. If Xis general in W(b;a) andai−min(α,t)−bi≥0 for min( α,t)≤i≤t, then
+(2.8) codim XjSing(Xj)≥min{2α−1,j+2}for allj= 2,···,c.
+This follows from the theorem of [5], arguing as in [5], Example 2.1. In pa rticular, if
+α≥3, weget foreach i >0thatXi֒→PnandXi+1֒→Xiarelocalcomplete intersections
+(l.c.i.’s) outside some set Ziof codimension at least min(4 ,i+1) inXi+1, cf. theparagraph
+below. Finally note that it is easy to show (2.8) for ( j,α) = (1,2) provided ai−2> bifor
+2≤i≤t, cf. [1], (1.10).
+Now letZ⊂X(resp.Zi⊂Xi) be some closed subset such that U:=X−Z ֒→Pn
+(resp.Ui:=Xi−Zi֒→Pn) is an l.c.i. By the fact that the 1stFitting ideal of Mis equal
+toIt−1(ϕ), we get that ˜Mis locally free of rank 1 precisely on X−V(It−1(ϕ)) [2], Lemma
+1.4.8. Since the set of non-l.c.i points of X ֒→Pnis precisely V(It−1(ϕ)) by e.g. [17],
+Lemma 1.8, we get that U⊂X−V(It−1(ϕ)) and that ˜Mis locally free on U. Indeed
+MiandIXi/I2
+Xiare locally free on Ui, as well as on Ui−1∩Xi. Note also that since
+V(It−1(B))⊂V(It(A)), we may suppose Zi−1⊂Xi.
+Let us recall the following useful comparison of cohomology groups . IfLandNare
+finitely generated A-modules such that depthI(Z)L≥r+ 1 and ˜Nis locally free on
+U:=X−Z, then the natural map
+(2.9) Exti
+A(N,L)−→Hi
+∗(U,HomOX(˜N,˜L))
+is an isomorphism, (resp. an injection) for i < r(resp.i=r) cf. [9], exp. VI. Note that
+we interpret I(Z) asmifZ=∅.
+In [12], Conjecture 6.1, we conjectured the dimension of (a non-em pty)W(b;a) in terms
+of the invariant
+(2.10)λc:=/summationdisplay
+i,j/parenleftbiggai−bj+n
+n/parenrightbigg
++/summationdisplay
+i,j/parenleftbiggbj−ai+n
+n/parenrightbigg
+−/summationdisplay
+i,j/parenleftbiggai−aj+n
+n/parenrightbigg
+−/summationdisplay
+i,j/parenleftbiggbi−bj+n
+n/parenrightbigg
++1.
+Here the indices belonging to aj(resp.bi) range over 0 ≤j≤t+c−2 (resp. 1 ≤i≤t)
+and/parenleftbiga
+b/parenrightbig
+= 0 whenever a < b.
+Conjecture 2.2. Given integers a0≤a1≤...≤at+c−2andb1≤...≤bt, we set
+ℓi:=/summationtextt+i−2
+j=0aj−/summationtextt
+k=1bkandhi−3:= 2at+i−2−ℓi+n, fori= 3,4,...,c. Assume
+ai−min([c/2]+1,t)≥biformin([c/2]+1,t)≤i≤t. Then we have
+dimW(b;a) =λc+K3+K4+...+KcFAMILIES OF DETERMINANTAL SCHEMES 5
+whereK3=/parenleftbigh0
+n/parenrightbig
+andK4=/summationtextt+1
+j=0/parenleftbigh1+aj
+n/parenrightbig
+−/summationtextt
+i=1/parenleftbigh1+bi
+n/parenrightbig
+and in general
+Ki+3=/summationdisplay
+r+s=i
+r,s≥0/summationdisplay
+0≤i1<...<ir≤t+i
+1≤j1≤...≤js≤t(−1)i−r/parenleftbigghi+ai1+···+air+bj1+···+bjs
+n/parenrightbigg
+for 0≤i≤c−3.
+In [12] we proved that the right hand side in the formula for dim W(b;a) in Conjec-
+ture 2.2 is always an upper bound for dim W(b;a) ([12], Theorem 3.5) and, moreover,
+that Conjecture 2.2 holds in the range
+(2.11) 2 ≤c≤5 andn−c >0 (supposing char(k) = 0 ifc= 5),
+cf. [12], Theorems 3.5 and 4.5; and Corollaries 4.7, 4.10 and 4.14. See als o [7], [11] for the
+cases 2≤c≤3. In[10], however, thefirst authorgaveacounterexample toCon jecture 2.2
+for zero-dimensional schemes (see the section of conjectures in this paper).
+Remark 2.3. Conjecture 2.2 holds in codimension c≥6 provided the numerical condi-
+tions of [12], Corollary 4.15 are satisfied. By [12], Remarks 4.16, 4.17 an d Corollary 4.18
+Conjecture 2.2 holds (without assuming n > corchar(k) = 0) for any W(b;a) satisfying
+at+3> at−1+at+at+1−a0−a1 in codimension c= 5,
+at+2> at−1+at−a0 forn=c= 4,
+at+1> at−1andai−2> bifor 2≤i≤t forn=c= 3.
+Note that in the case n=c= 3 we need ai−2> bi(notai−2≥bias assumed in [12]) for
+2≤i≤tfor the proof of [12], Corollary4.18 to hold, see the last sentence of Remark 2.1.
+A goal of this paper is to extend the results summarized in (2.11) and in the above
+remark. To do this we will need the following result where a=a0,a1,...,at+c−2and
+a′=a0,a1,...,at+c−3.
+Proposition 2.4. Letc≥3, let(X)∈W(b;a)and suppose dimW(b;a′)≥λc−1+K3+
+K4+...+Kc−1anddepthI(Z)B≥2for a general Y= Proj(B)∈W(b;a′). If
+(2.12) 0homR(IY,IX/Y)≤t+c−3/summationdisplay
+j=0/parenleftbiggaj−at+c−2+n
+n/parenrightbigg
+,
+thendimW(b;a) =λc+K3+K4+...+Kcand(2.12)turns out to be an equality.
+Proof.See [10], Proposition 3.4. /square
+Let us recall how we proved Remark 2.3 since we want to generalize th at approach.
+Lettinga=at+i−2−at+i−1we showed in [12] that Hom Di−1(Ii−1,Ii)∼=Di(a) fori < c
+using depthI(Zi−1)Di≥2. Indeed this follows from (2.9), i.e. from
+H0
+∗(Ui−1,Hom(Ii−1,Ii))∼=H0
+∗(Ui−1,HomOXi(Ii−1⊗OXi−1OXi⊗I∗
+i,OXi))
+because ˜Mi|Ui−1∼=I∗
+i(−at+i−1)|Ui−1is locally free, ˜Mi|Ui−1∼=˜Mi−1⊗OXi|Ui−1and hence
+Ii−1⊗OXi−1OXi|Ui−1∼=Ii(−a)|Ui−1, and note that Ui−1∩Xi⊂Ui. Then since
+(2.13) 0 →HomR(Ii−1,Ii)→HomR(IDi,Ii)→HomR(IDi−1,Ii)6 JAN O. KLEPPE, ROSA M. MIR ´O-ROIG
+isexactwewereabletoshow(2.12)for X=Xc⊂Y=Xc−1byputting 0HomR(IDc−2,Ic−1) =
+0 (through making the minimal generators of IDc−2andIc−1explicit via (2.3)). Using
+Proposition 2.4 we obtained Remark 2.3 if the conjecture holds for W(b;a′)∋Y:=Xc−1.
+Letusfinishthissectionbygatheringalltheresultsonthedepthof mixeddeterminantal
+idealsandcogeneratedidealsneeded inthenext section. Westartb yfixing somenotation.
+Let
+(2.14) A=
+f1
+1f1
+2···f1
+q
+f2
+1f2
+2···f2
+q............
+fp
+1fp
+2···fp
+q
+
+be a homogeneous p×qmatrix with entries homogeneous polynomials fj
+i∈k[x0,···,xn]
+ofdegree aj−bi. Foranychoice ( α;β) = (α1,···,αm;β1,···,βm) ofrowindexes 1 ≤α1<
+···< αm≤pand of column indexes 1 ≤β1<···< βm≤q, we denote by I(α;β)(A) the
+ideal cogenerated by (α;β), i.e.I(α,β)is the homogeneous ideal of k[x0,···,xn] generated
+by all (m+ 1)×(m+ 1) minors of A, alli×iminors of the rows 1 ,···,αi−1 for
+i= 1,···,mand alli×iminors of the columns 1 ,···,βi−1 fori= 1,···,m.
+Example 2.5. (1) If (α;β) = (1,···,t−1;1,···,t−1), then I(α;β)(A) is the ideal
+generated by the t×tminors of A, i.e.,I(α;β)(A) =It(A).
+(2) LetAbe a homogeneous p×qmatrix and let Aibe the matrix obtained by deleting
+the last i−1 columns of A. We assume p≤qand we fix an integer m < pand
+(α;β) = (1,···,m;β1,···,βm). Setj:= min{i|βi> i},c1= 1 and cs=q+2−βm+2−s
+for 2≤s≤m+2−j. The ideal cogenerated by ( α;β) = (1,···,m;β1,···,βm) can be
+identified with the following mixed determinantal ideal
+I(α;β)(A) =Im+1(Ac1)+Im(Ac2)+...+Ij(Acm+2−j),
+whereIλ(A̺) denotes the ideal generated by all λ×λminors of A̺.
+Theorem 2.6. LetA= (xi,j)be ap×qmatrix of indeterminates and let (α;β) =
+(α1,···,αm;β1,···,βm)be a choice of mrow indexes and of mcolumn indexes. Let
+I(α;β)be the ideal cogenerated by (α;β). Then,I(α;β)(A)is a Cohen-Macaulay ideal and
+(2.15) ht(I(α;β)(A)) =pq−(p+q+1)m+m/summationdisplay
+i=1(αi+βi).
+Proof.See [3], Corollary 5.12. /square
+We are now ready to state the main result of this section and to prov e that the above
+formula for the height of a cogenerated ideal associated to a matr ix with entries that
+are indeterminates also works for a general homogeneous matrix w ith entries thar are
+homogeneous polynomials of positive degree. Indeed, we have
+Theorem 2.7. Fix integers b1,···,bqanda1,···,ap. LetA= (fj
+i)j=1,···,p
+i=1,···,qbe a general
+homogeneous p×qmatrixAwith entries that are homogeneous forms of degree aj−bi, and
+assume that aj> bifor allj,i. Let(α;β) = (α1,···,αm;β1,···,βm)be any choice of mFAMILIES OF DETERMINANTAL SCHEMES 7
+row indexes and of mcolumn indexes, and suppose n+1≥pq−(p+q+1)m+/summationtextm
+i=1(αi+βi).
+ThenI(α;β)(A)is a Cohen-Macaulay ideal and
+ht(I(α;β)(A)) =pq−(p+q+1)m+m/summationdisplay
+i=1(αi+βi).
+Proof.We clearly have ht(I(α;β)(A))≤pq−(p+q+1)m+/summationtextm
+i=1(αi+βi) and it will be
+enoughtoconstructanexampleofhomogeneous p×qmatrixMwithentrieshomogeneous
+formsfj
+i∈k[x0,···,xn] such that the ideal I(α;β)(M) cogenerated by ( α;β) is Cohen-
+Macaulay and ht(I(α;β)(M)) =pq−(p+q+1)m+/summationtextm
+i=1(αi+βi).Therefore, let n+1≥
+pq−(p+q+1)m+/summationtextm
+i=1(αi+βi).We will distinguish 2 cases:
+Case 1.Assumeaj−bi= 1 for all i,j(i.e. the entries of the matrix Aare linear forms).
+By Theorem 2.6 for any choice of p,qand (α;β), we have the matrix A= (xi,j) of
+indeterminates and the ideal I(α;β)(A)⊂S:=k[x1,1···,xp,q] cogenerated by ( α;β) is
+Cohen-Macaulay of height
+ht(I(α;β)(A)) =pq−(p+q+1)m+m/summationdisplay
+i=1(αi+βi).
+We choose pq−n−1 general linear forms ℓ1,···,ℓpq−n−1∈S=k[x1,1···,xp,q] and we set
+S/(ℓ1,···,ℓpq−n−1)∼=k[x0,x1,···,xn] =:R. Let us call I⊂Rthe ideal of Risomorphic
+to the ideal I(α;β)(A)/(ℓ1,···,ℓpq−n−1) ofS/(ℓ1,···,ℓpq−n−1). Note that Iis a Cohen-
+Macaulay ideal of height ht(I) =pq−(p+q+1)m+/summationtextm
+i=1(αi+βi) and, in addition, I
+is nothing but the ideal I(α;β)(M) whereM= (mj
+i) is ap×qhomogeneous matrix with
+entries linear forms in k[x0,x1,···,xn] obtained from A= (xi,j) by substituting using the
+equations ℓ1,···,ℓpq−n−1, which proves what we want.
+Case 2.Assumeaj−bi≥1 for alli,j. In this case, it is enough to raise the entry mj
+iof
+the above matrix Mto the power aj−bi. /square
+Remark 2.8. Fix integers a0≤a1≤...≤at+c−2andb1≤...≤bt. LetX⊂Pn,
+(X)∈W(b;a) be a general determinantal ideal associated to a t×(t+c−1) matrix A
+represented by a graded morphism as in (2.1). Let
+(X.) :X=Xc⊂Xc−1⊂...⊂X2⊂X1⊂Pn
+be the flag of standard determinantal subschemes that we obtain by successively deleting
+columns from the right hand side of Aand letϕibe the graded morphism associated to
+the matrix which defines Xi. Assume a0> btandc≥3. Applying Theorem 2.7, we get
+the following formula
+(2.16) dim R/(It−1(ϕ1)+It(ϕc−1)) = dim Dc−1−2.
+Even more, if dim Dc−1≥3 andc≥4 we have the equalities
+(2.17) dim R/(It−1(ϕi)+It(ϕc−1)) = dim Dc−1−i−1
+for 1≤i≤2 which will play an important role in the next section.8 JAN O. KLEPPE, ROSA M. MIR ´O-ROIG
+Remark 2.9. Ift= 2 we can prove (2.16) directly as follows. Let A= [C,D,v] whereCis
+a2by2generalenoughmatrixinthevariables x0,x1,x2,x3(e.g. withrows( xa0−b1
+0,xa1−b1
+1)
+and (xa0−b2
+2,xa1−b2
+3) ),vis some column and Dis a 2 by c−2 matrix whose first row is
+(xa2−b1
+4,xa3−b1
+5,...,xac−2−b1c,0) and whose second row is (0 ,xa3−b2
+4,xa4−b2
+5,...,xac−1−b2c). Note
+thatϕ1correspondsto Candϕc−1to[C,D]. SinceCisgeneral, wegetcodim RR/It−1(ϕ1) =
+4, i.e. the radical of It−1(ϕ1) is (x0,x1,x2,x3). Since it is clear that the radical of the
+ideal generated by the maximal minors of Dis (x4,x5,...,xc), it follows that the scheme
+Xc−1= Proj(Dc−1)∈W(b;a′) given by the maximal minors of [ C,D] satisfies (2.16). The
+general element XofW(b;a) will therefore have a flag where Xc−1satisfies (2.16).
+3.Smoothness and dimension of the determinantal locus
+This section is the heart of the paper and contains the results which generalize quite a
+lot of our previous contributions to problems (1)-(3) stated in the introduction.
+Proposition 3.1. With notation as in Remark 2.8, let c≥3and suppose (2.16)(this
+holds ifa0> bt). Ifat+c−2> at−2then(2.12)holds for X:=Xc⊂Y:=Xc−1, i.e.
+0homR(IDc−1,IDc/Dc−1)≤t+c−3/summationdisplay
+j=0/parenleftbiggaj−at+c−2+n
+n/parenrightbigg
+.
+In particular we get dimW(b;a) =λc+K3+K4+...+Kcprovided dimW(b;a′) =
+λc−1+K3+K4+...+Kc−1wherea′=a0,a1,...,at+c−3.
+Proof.Weclaimthat depthI(Zj)Dc−1≥2 for all jsatisfying 0 < j < c −1. Indeed
+by the discussion right after Remark 2.1 we see that we may take I(Zj) to be the ideal
+It−1(ϕj)⊂Rof submaximal minors of the matrix which defines Xj, andI(Zj)Dc−1
+to be (It−1(ϕj) +It(ϕc−1))Dc−1. It follows that depthIt−1(ϕj)Dc−1≥2 forj= 1, i.e.
+dimDc−1−dimDc−1/(It−1(ϕ1)+It(ϕc−1))Dc−1≥2, implies the claim. Hence we conclude
+the proof of the claim by (2.16).
+For every j, 0< j < c−1, puta:=at+j−1−at+c−2. Weclaimthat
+HomDj(Ij,Ic−1)∼=Dc−1(a).
+To prove this claim we remark that depthI(Zj)Ic−1≥2 sinceIc−1is maximally CM. Using
+thatIjis locally free on Ujand the arguments in the text before (2.13), see the text
+accompanying (2.7) for j= 1, we get
+Ij⊗OXjOXc−1|Uj∼=Ij⊗OXjOXj+1⊗...⊗OXc−2OXc−1|Uj∼=Ic−1(−a)|Uj.
+It follows that HomOXj(Ij,Ic−1)(−a)∼=HomOXc−1(Ic−1,Ic−1)∼=OXc−1are isomorphic
+as sheaves on Uj∩Xc−1, i.e. we get that H0
+∗(Uj,Hom(Ij,Ic−1))∼=H0
+∗(Uj,OXc−1)(a) and
+hence the claim from depthI(Zj)Dc−1= depthI(Zj)Ic−1≥2 and (2.9).
+Now we repeatedly use the exact sequence
+(3.1) 0 →Dc−1(a)∼=HomDj(Ij,Ic−1)→HomR(IDj+1,Ic−1)→HomR(IDj,Ic−1)→FAMILIES OF DETERMINANTAL SCHEMES 9
+forj=c−2,c−3,...,1. Sinceat+j−1≤at+c−2, we have dim Dc−1(a)0=/parenleftbiga+n
+n/parenrightbig
+. It follows
+that
+0hom(IDc−1,Ic−1)≤0hom(ID1,Ic−1)+t+c−3/summationdisplay
+i=t/parenleftbiggai−at+c−2+n
+n/parenrightbigg
+where we have replaced at+j−1byaiin which case 1 ≤j≤c−2 corresponds to t≤i≤
+t+c−3.
+It remains to prove that 0hom(ID1,Ic−1)≤/summationtextt−1
+i=0/parenleftbigai−at+c−2+n
+n/parenrightbig
+since we then by Propo-
+sition 2.4 get the dimension formula. Using that X1= Proj(D1) is a hypersurface of
+degreeℓ1, we find 0hom(ID1,Ic−1)∼=dimIc−1(ℓ1)0whereℓk=/summationtextt+k−2
+j=0aj−/summationtextt
+i=1bi. Now
+we have to make the degrees of the minimal generators of Ic−1explicit. Taking a close
+look at (2.3), we see that a minimal generator fofIc−1∼=IDc/IDc−1of the smallest pos-
+sible degree has degree s(Ic−1) :=ℓc−/summationtextt+c−3
+j=t−1ajbecausea0≤a1≤ ··· ≤at+c−2. Since
+ℓ1−s(Ic−1) =at−1−at+c−2≤0 by the definition of ℓk, we get either dim Ic−1(ℓ1)0= 0
+orat−1=at+c−2. In the latter case the assumption at+c−2> at−2implies that the de-
+grees of all minimal generators, except for f, are strictly greater than s(Ic−1), i.e. we get
+dimIc−1(ℓ1)0=/parenleftbigat−1−at+c−2+n
+n/parenrightbig
+and we are done. /square
+By repeatedly using Proposition 3.1 we get the
+Theorem 3.2. LetX⊂Pn,(X)∈W(b;a), be a general determinantal scheme and
+supposea0> bt. Moreover if c≥6we suppose at+3> at−2(orat+4> at−2provided
+chark= 0) and if3≤c≤5we suppose at+c−2> at−2. Then we have
+dimW(b;a) =λc+K3+K4+···+Kc.
+Proof.If 3≤c≤6 (chark= 0 ifc= 6) and at+c−2> at−2we use (2.11) to find
+dimW(b;a′) and we conclude the proof by Proposition 3.1.
+Ifc≥6andat+3> at−2(resp.at+4> at−2ifc≥7) werepeatedly use Proposition3.1to
+reduce to the case c= 5 (resp. c= 6) and we conclude by the first part of the proof (note
+that the assumption at+c−2> at−2of Proposition 3.1 is satisfied in this induction). /square
+Remark 3.3. We expect that the assumption a0> btcan be weakened in Theorem 3.2,
+as well as in (2.16). At least it does for c= 3 provided we assume ai−2> bifor 2≤i≤t.
+Indeed since It(ϕ2)⊂It−1(ϕ1) we first show (2.16) using Remark 2.1. Then the proof
+above applies to conclude as in Theorem 3.2 provided at+1> at−2, cf. Remark 2.3.
+If the condition (2.17) is satisfied, then we can prove the following re sult forW(b;a) to
+be a generically smooth irreducible component.
+Theorem 3.4. LetX⊂Pn,(X)∈W(b;a), be a general determinantal scheme of di-
+mension n−c≥1, letc >2and letX=Xc⊂Xc−1⊂...⊂X2⊂Pn,Xi= Proj(Di), be
+the flag obtained by successively deleting columns from the r ight hand side. If a0> bt,
+0Ext1
+D2(ID2/I2
+D2,I2) = 0and0Ext1
+D3(ID3/I2
+D3,Ii) = 0fori= 3,...,c−1,
+thenW(b;a)isa genericallysmoothirreduciblecomponentof the Hilber tscheme Hilbp(Pn).
+Remark 3.5. Ifn−c≥2 then we have 0Ext1
+D2(ID2/I2
+D2,I2) = 0 provided c= 3, and
+0Ext1
+D2(ID2/I2
+D2,I2) =0Ext1
+D3(ID3/I2
+D3,I3) = 0 provided c= 4 by [12], (5.4)-(5.8). Hence10 JAN O. KLEPPE, ROSA M. MIR ´O-ROIG
+the conclusion of Theorem 3.4 holds provided n−c≥2 and 3 ≤c≤4. Moreover if
+n−c≥1 andc >4 then both Ext-groups above still vanish by [12] and we may in this
+case replace the assumption of Theorem 3.4 given by the displayed fo rmula with
+0Ext1
+D3(ID3/I2
+D3,Ii) = 0 for i= 4,...,c−1.
+Note that in the case n−c≥1 andc= 2, the conclusion of Theorem 3.4 holds and
+moreover, Hilbp(Pn) is smooth at any ( X)∈W(b;a) by [7].
+Remark 3.6. Forc >2 one knows that Hilbp(Pn) is not always smooth at any ( X)∈
+W(b;a) [14]. Indeed, since W(b;a) is irreducible, it is not difficult to find singular points
+of Hilbp(Pn) by first computing its tangent space dimension, h0(NX), at a general ( X)∈
+W(b;a), using Macaulay 2. Then by experimenting with special choices of ( X0)∈W(b;a)
+one may find h0(NX)< h0(NX0) which means that Hilbp(Pn) is singular at ( X0), see [16]
+which even computes the obstructions of deformations, using Sing ular, in a related case.
+Proof.Due to Theorem 5.1 of [12] we must show that
+(3.2) 0Ext1
+Di(IDi/I2
+Di,Ii) = 0 for i= 2,...,c−1.
+By assumption we need to prove the vanishing (3.2) for i= 4,...,c−1 andc >4. By
+induction on cit suffices to show it for i=c−1,c≥5. Hence it suffices to see that there
+exist injections
+(3.3) 0Ext1
+Dj+1(IDj+1/I2
+Dj+1,Ic−1)֒→0Ext1
+Dj(IDj/I2
+Dj,Ic−1) forj= 2,...,c−2.
+By (2.17) and the arguments in the first paragraph of the proof of Proposition 3.1 we may
+suppose depthI(Zj)Dc−1≥3 for alljsatisfying 1 < j < c−1.
+Weclaimthat the left-exact sequence (3.1) is also right-exact, i.e. that th e rightmost
+map of the Hom-groups is surjective for 1 < j < c−1. To show this, it certainly suffices
+to prove Ext1
+R(Ij,Ic−1) = 0. However, by paying closer attention to the modules of (3.1)
+we shall see that also Ext1
+Dj(Ij,Ic−1) = 0 for 1 < j < c−1 suffices for proving the claim.
+Indeed if we apply ( −)⊗RDjto 0→IDj→IDj+1→Ij→0 we get the right-exact
+sequence IDj⊗RDj→IDj+1⊗RDj→Ij→0 whereIDj+1⊗RDj∼=IDj+1/IDj·IDj+1and
+sinceIDj⊗RDj+1∼=IDj/IDj·IDj+1= ker(IDj+1/IDj·IDj+1→Ij) we obtain the exact
+sequence
+0−→IDj⊗RDj+1−→IDj+1⊗RDj−→Ij−→0
+to which we apply Hom Dj(−,Ic−1). Then we get exactly (3.1) continued to the right by
+Ext1
+Dj(Ij,Ic−1). Thus the vanishing of Ext1
+Dj(Ij,Ic−1) implies that (3.1) is right-exact.
+Toseethat Ext1
+Dj(Ij,Ic−1) = 0weusetheisomorphism HomOXj(Ij,Ic−1)(−a)|Uj∩Xc−1∼=
+OXc−1|Uj∩Xc−1which we obtained in the proof of Proposition 3.1. By (2.9) it follows tha t
+Ext1
+Dj(Ij,Ic−1)∼=H1
+∗(Uj,Hom(Ij,Ic−1))∼=H1
+∗(Uj,OXc−1(a)) = 0
+because we have depthI(Zj)Dc−1≥3 and hence depthI(Zj)Ic−1≥3 (Ic−1is maximally
+CM). This proves the claim.
+Now we can rewrite the exact sequence (3.1) as
+(3.4) 0 →Dc−1(a)→HomDj+1(IDj+1/I2
+Dj+1,Ic−1)→HomDj(IDj/I2
+Dj,Ic−1)→0.FAMILIES OF DETERMINANTAL SCHEMES 11
+Sheafifying, restricting to Uj∩Xc−1(note that IDj+1is also locally free on Uj∩Xc−1)
+and taking cohomology, we get
+→H1(Uj,OXc−1(a))→H1(Uj,Hom(IXj+1/I2
+Xj+1,Ic−1))→H1(Uj,Hom(IXj/I2
+Xj,Ic−1))→
+Since depthI(Zj)Dc−1≥3, the two latter H1-groups are by (2.9) isomorphic to the 0Ext1-
+groups quoted in (3.3) and since H1(Uj,OXc−1(a)) = 0 we are done. /square
+Remark 3.7. Since (2.17) holds also for i= 3 provided dim Dc−1≥4,c≥5 anda0> bt
+by Theorem 2.7, we may continue the proof above to see that the inj ections (3.3) are
+isomorphisms for j≥3. Hence if Xis general and n−c≥2, then
+0Ext1
+D3(ID3/I2
+D3,Ic−1)∼=0Ext1
+Dc−1(IDc−1/I2
+Dc−1,Ic−1).
+Here the leftmost Ext1-group is computed much faster by Macaulay 2 than the right-
+most one. We also get an injection in (3.3) for j= 2, but now it is not necessarily an
+isomorphism.
+Corollary 3.8. Letn−c≥1,c≥5and suppose a0> btandat+3> at−1+at−b1.
+ThenW(b;a)is a generically smooth irreducible component of Hilbp(Pn)of dimension
+λc+K3+...+Kc.
+Proof.We get dim W(b;a) from Theorem 3.2. Hence by (3.3) and Remark 3.5 it suffices
+to show that 0Ext1
+R(ID2,Ii) = 0 for 4 ≤i≤c−1 since 0Ext1
+D2(ID2/I2
+D2,Ii) is a subgroup
+of0Ext1
+R(ID2,Ii). First let i=c−1. By the Eagon-Northcott resolution (2.3) we see
+that the largest possible degree of a relation for ID2isℓ2−b1and the smallest possible
+degree of a generator of Ic−1∼=IDc/IDc−1isℓc−/summationtextt+c−3
+j=t−1aj. Sinceℓc=ℓ2+/summationtextt+c−2
+j=t+1aj, we
+get0Ext1
+R(ID2,Ic−1) = 0 from
+ℓ2−b1< ℓ2+t+c−2/summationdisplay
+j=t+1aj−t+c−3/summationdisplay
+j=t−1aj=ℓ2−at−1−at+at+c−2,
+i.e. from at+c−2> at−1+at−b1.Since we need the vanishing of 0Ext1
+R(ID2,Ii) for any
+i= 4,5,...,c−1, we must suppose at+3> at−1+at−b1and hence we get the corollary. /square
+Remark 3.9. Note that if c= 3 (resp. c= 4) we can argue as above to see that the
+conclusions ofCorollary3.8 holdprovided at+1> at−1+at−b1(resp.at+2> at−1+at−b1).
+This is, however, proved in [12], Corollary 5.10. For c≥5, Corollary 3.8 generalizes the
+corresponding result [12], Corollary 5.9 quite a lot.
+4.Conjectures
+In [10] the first author discovered a counterexample to Conjectu re 2.2 for every cin the
+rangen=c≥3. Indeed the vanishing all 2 ×2 minors of a general 2 ×(c+ 1) matrix
+of linear entries defines a reduced scheme of c+1 different points in Pc. The conjectured
+dimension of W(0,0;1,1,...,1) isc(c+ 1) +c−2 while its actual dimension is at most
+c(c+1).12 JAN O. KLEPPE, ROSA M. MIR ´O-ROIG
+On the other hand Theorem 3.2 is quite close to proving Conjecture 2 .2. The crucial
+assumption in Theorem 3.2 is the inequality at+c−2> at−2(orat+3> at−2ifc >5). Since
+we, in addition to proving Theorem 3.2, have computed quite a lot of ex amples where we
+haveat+c−2=at−2andai−min([c/2]+1,t)> bi, and each time, except for the counterexample,
+obtained (2.12) and hence the conjecture, we now want to slightly c hange Conjecture 2.2
+to
+Conjecture 4.1. Given integers a0≤a1≤...≤at+c−2andb1≤...≤bt, we assume
+ai−min([c/2]+1,t)≥biprovided n > candai−min([c/2]+1,t)> biprovided n=cformin([c/2]+
+1,t)≤i≤t. Except for the family W(0,0;1,1,...,1)ofzero-dimensional schemes above
+we have, for W(b;a)/ne}ationslash=∅, that
+dimW(b;a) =λc+K3+K4+...+Kc.
+Indeed in the situation of Proposition 2.4 we even expect(2.12) to hold! This will imply
+Conjecture 4.1 provided the conjecture holds for W(b;a′). Note that the conclusion of the
+conjecture is true provided n−c≥1 and 2≤c≤5 (char(k) = 0 ifc= 5) by [12].
+Finallywewillstateaconjecturerelatedtotheproblems(2)and(3) oftheIntroduction:
+Conjecture 4.2. Given integers a0≤a1≤...≤at+c−2andb1≤...≤bt, we suppose
+n−c≥2,c≥5anda0> bt. ThenW(b;a)is a generically smooth irreducible component
+of the Hilbert scheme Hilbp(Pn).
+Indeed due to the results of this paper and many examples compute d by Macaulay 2
+in the range a0> btwe evenexpectthe groups 0Ext1
+D3(ID3/I2
+D3,Ii) fori= 4,...,c−1 of
+Theorem 3.4 to vanish! This will imply Conjecture 4.2. The conclusion of the conjecture
+may even be true for 0 ≤n−c≤1 (we have no counterexample), but in this range we
+have verified that the Ext1-groups above do not always vanish. Note that the conclusion
+of Conjecture 4.2 is true provided n−c≥2 and 2≤c≤4 ([7], [11], [12]).
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+sheaves, J. Pure Appl. Algebra, 150(2000), 155-174.
+[14] M. Martin-Deschamps and R. Piene, Arithmetically Cohen-Macaulay curves in P4of degree 4and
+genus0, Manuscripta Math. 93(1997), 391-408.
+[15] R.M. Mir´ o-Roig, Determinantal ideals . Progress in Mathematics, 264(2008) Birkhuser Verlag,
+Basel.
+[16] A. Siqveland, Generalized Matric Massey Products for Graded Modules . J. Gen. Lie Theory Appl.
+(in press), arXiv:math/0603425
+[17] B. Ulrich, Ring of invariants and linkage of determinantal ideals , Math. Ann. 274(1986), 1-17.
+Faculty of Engineering, Oslo University College, Pb. 4 St. O lavs plass, N-0130 Oslo,
+Norway
+E-mail address :JanOddvar.Kleppe@iu.hio.no
+Facultat de Matem `atiques, Departament d’Algebra i Geometria, Gran Via de les Corts
+Catalanes 585, 08007 Barcelona, SPAIN
+E-mail address :miro@ub.edu
\ No newline at end of file
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+arXiv:1001.0163v1  [cond-mat.stat-mech]  31 Dec 2009Short-time dynamic in the Majority vote model: The ordered a nd disordered initial
+cases
+Francisco Sastre1
+1Departamento de Ingenier´ ıa F´ ısica, Divisi´ on de Ciencia s e Ingenier´ ıa,
+Campus Le´ on de la Universidad de Guanajuato,
+AP E-143, CP 37150, Le´ on, Guanajuato, M´ exico∗
+This work presents short-time Monte Carlo simulations for t he two dimensional Majority-vote
+model starting from ordered and disordered states. It has be en found that there are two pseudo-
+critical points, each one within the error-bar range of prev ious reported values performed using
+fourth order cumulant crossing method. The results show tha t the short-time dynamic for this
+model has a dependence on the initial conditions. Based on th is dependence a method is proposed
+for the evaluation of the pseudo critical points and the extr action of the dynamical critical exponent
+zand the static critical exponent β/νfor this model.
+PACS numbers: 64.60.Ht,75.40.-s,05.70.Ln
+Keywords:
+I. INTRODUCTION
+Critical phenomena in equilibrium statistical systems
+is one of the most important topics in physics. Much
+of the attention has been focused on the universality of
+thecriticalexponents, withseveraluniversalityclassesal-
+readycharacterizedin equilibrium systems. On the other
+hand the critical behavior of non-equilibrium statistical
+systems has been receiving a lot of attention in recent
+years, but the characterization of the different universal-
+ity classes is far from be complete. One of the simplest
+non-equilibrium models is the two dimensional majority
+votemodel, an Ising-likesystem (up-down symmetry and
+spin-flip dynamic) with a continuous order-disordertran-
+sition with the same critical exponents that the two di-
+mensionalIsing model [1–3], as expected from the predic-
+tionofGrinstein et al[4]: everyspinsystemwithspin-flip
+dynamic and up-down symmetry falls in the Ising model
+universality class. However, there is some controversy
+about the universality class for higher dimensions. A re-
+cent work claims [5] that the upper critical dimension for
+the majority vote model is 6 instead of 4, based on nu-
+merical calculations. Another discrepancy in the critical
+exponents have been found in simulations on non-regular
+lattices [6, 7]. It must be mentioned that all ofthe results
+mentioned above were performed using standard ”Monte
+Carlo” simulations and Finite Size Scaling approaches
+for the evaluation of the static critical exponents. On
+the other hand the time evolution can gives important
+information about the universality of a given system. It
+has been shown by Janssen et al.[8] that when systems
+with relaxation dynamics are quenched from high tem-
+peratures to the critical temperature there is a short crit-
+ical universal behavior. Numerical simulations have con-
+firmed this behaviorin the Isingand the Pottmodels (see
+∗Electronic address: sastre@fisica.ugto.mxreference [9] for a review of these results). Concerning
+the critical dynamic in systems without detailed balance,
+there are some works that evaluate the critical dynamic
+exponent z[11, 12] and the fluctuation-dissipation ra-
+tioX∞[13] for the Majority model, but always starting
+from a disordered state. As expected the results were
+compatibles with the Ising ones. Given all these results
+one should expect that the basic assumption for the dy-
+namic relaxation of the k-th moment of the order param-
+eter starting from an completely ordered state will be the
+same as in the Ising model, but this has not been proved
+yet.
+The aims of this work are: a) to evaluate the critical
+pointusingshorttimedynamicstartingfromorderedand
+disorderedinitial conditions and b) evaluate the dynamic
+critical exponent zand the static critical exponent β/ν.
+This will test if the Grinstein prediction holds for the
+short time dynamic in the majority vote model.
+II. MODELS AND DEFINITIONS
+In the Majority vote model each lattice site has a spin
+whose values are σ=±1 and its dynamic can be grouped
+by the spin flip rule
+Wi=1
+2[1−σif(Hi)], (1)
+hereHiis the local field/summationtext
+nnσj=,0,±2,±4 produced
+by its nearest neighbors and fis a function with up
+down symmetry that depends on two control parameters
+f(0) = 0, f(2) =−f(−2) =xandf(4) =−f(4) =y.
+The parameters xandycan be associated with inter-
+face and bulk temperatures respectively [15] using the
+relations
+x= tanh2β2, y= tanh4β4. (2)
+The Majority model is obtained setting x=y(β4<
+β2)and the critical point is at xc= 0.850(2) [2, 3].2
+The equilibrium case can be obtained along the line
+y= 2x/(1+x2) (Glauber dynamic), where the temper-
+atures are equal ( β2=β4=β) and the critical point is
+βc=1
+2log(1+√
+2).
+The order parameter is the standard for Ising-like sys-
+tems, defined by
+m=1
+N/angbracketleft/summationdisplay
+iσi/angbracketright (3)
+whereN=L2is the total number of lattice sites ( Lis
+the lateral size.
+Starting from a disorderedstate the dynamic for the k-
+thmomentoftheorderparameterwasderivedbyJanssen
+et al[8], the mathematical expression is given by
+m(k)(t,τ,L,m 0) =b−kβ/νm(k)(b−zt,b1/ντ,b−1L,bx0m0),
+(4)
+hereτ= (T−Tc)/Tcis the reduced temperature, or the
+reduced control parameter for non-equilibrium systems,
+tis the dynamic time variable, bis the re-scaling factor,
+βandνare static critical exponents, zis the dynamic
+critical exponent, x0is the scaling dimension of the ini-
+tial (small) order parameter m0(see [9] and reference
+therein).
+At the critical point for sufficiently small m0and large
+systems ( L→ ∞) the order parameter follows a power
+law dynamic
+m(t)∼m0tθ, (5)
+whereθis defined by ( x0−β/ν)/z. One must remark
+that there is a strong dependence on the initial value of
+theorderparameter. Thisdependencedoesnotaffectthe
+power law behavior at the critical point, what is affected
+is the exponent θthat tends to the real value in the limit
+m0→0
+For the dynamic starting, from the ordered state we
+have the assumption that the scaling dynamic form is
+given by
+m(k)(t,τ,L) =b−kβ/νm(k)(b−zt,b1/ντ,b−1L),(6)
+again it can be obtained the scaling form at the critical
+point taking the limit L→ ∞.
+M(t)∼t−β/νz. (7)
+This has been proved in equilibrium systems, like the
+Ising or the Potts model.
+For the evaluation of the critical point it has been used
+the fact that theoretically the order parameter evolves
+as a power law at the critical point If it is evaluated the
+differencebetween apowerlawandthe time evolutionfor
+different values of the control parameter xwe will expect
+a minimum in those differences at the critical point.
+The simulations were performed choosing the lattice
+site randomly starting from both, a completely ordered
+state (m= 1.0) and a carefully prepared disordered state0.20.3m(t)
+10 50 100 200t0.650.700.75a)
+b)
+FIG. 1: (color online) a) Dynamic starting from an ordered
+state. b) Dynamic starting from a disordered state ( m0=
+0.0875). In both cases the top continuous line corresponds to
+axabove the critical point and the lower one to a xbelow
+the critical point. The dashed line shows the expected power
+law behavior.
+with small magnetization values m0<0.1. The or-
+der parameter is evaluated as a function of the time t,
+with ∆t= 1 corresponding to a Monte Carlo time step
+(MCTS). In order to avoid finite size effects we used lat-
+ticeswithlateralsize L= 29(N=L2). Theaveragewere
+taken with at least 105independent simulations and 250
+and 1000 MCTS for the evaluation of the critical point
+and dynamical exponents respectively.
+III. RESULTS
+Given that the power law behavior of the order pa-
+rameter as function of t(decay from an ordered state,
+increase for a disordered one) one can performed simula-
+tions for different values of xaround the expected critical
+point. In Fig. 1 we can observe that above and below a
+certain value of xwe have dynamic that differ from a
+power law (illustrated by the dashed line).
+From here one must define the criterion that will mea-
+sure which is the best power law curve, in this work it
+was used the measurement of the χ2for each curve with
+respect to a power law behavior. I must remark that pre-
+vious results for equilibrium systems give the same result
+for the critical point for both initial states. However, two
+different values were found for the majority vote model,
+see Fig. 2.
+The critical point values obtained were xc=
+0.85007(6)and xc= 0.85147(2)fortheorderanddisorder
+state respectively, both results are in perfect agreement
+with the obtained in the static case (references [1–3]) and
+are quite close.
+We have a clearly dependence on the initial condition
+in short time dynamic of this model, which is not the3
+0.849 0.850 0.851 0.852 0.853
+x0.01×10−52×10−5χ2 
+FIG. 2: (color online) Deviation from the power line behavio r
+for the decay ( m0= 1, left curve) and the increase ( m0=
+0.0875, right curve) cases.
+TABLE I: Critical point for initial disordered states.
+m00.0375 0.0500 0.0625 0.0750 0.0875
+xc0.84929(10) 0.84972(9) 0.85019(6) 0.85076(5) 0.85147(2)
+case for equilibrium systems. There is no doubt about
+the critical point obtained from the ordered phase, since
+m0= 1 is one of the fixed point under renormalization
+group transformations. However, one must remember
+that the power law is valid only in the limit m0→0 (the
+other fixed point) for the disorder case, assuming that
+the ”real” critical point is located at this limit, one can
+proceed to evaluate the critical point for another values
+ofm0and with this values extrapolate the critical point
+forthe disorderedphase. TheresultsareshowedintableI
+Once that each value xc(m0) has been evaluated, the
+dynamical exponent θcan be obtained. For the evalua-
+tionofthisexponentthesimulationswereperformedwith
+1000 MCTS discarding the first time steps, since there is
+an initial time scale tmicwhere the power law stabilizes
+tmic∼20 (see Fig. 3). The results for the exponent θare
+showed in table II.
+With a extrapolation of these values to m0= 0 the
+value of the critical point and the θexponent were eval-
+uated (see Fig. 4). The result for the critical point was
+xc= 0.84860(10), which is clearly different from the or-
+dered one, however both values are within the error bar
+from the obtained in the static case 0 .848≤xc≤0.852.
+From now on the pseudo critical point evaluated with
+TABLE II: thetaexponents for growing process.
+m00.0375 0.0500 0.0625 0.0750 0.0875
+θ0.1769(8) 0.1774(4) 0.1782(3) 0.1788(4) 0.1792(3)1 10 100 1000t0.1m(t)
+FIG. 3: (color online) Growing of the order param-
+eter at xc(m0), the continuous curves are for m0=
+0.0875,0.075,0.0625,0.5,0.0375 from top to bottom. The
+dashed line represents the power law growing with θ= 0.1751
+(the result in this work for the majority vote model).
+the decay process will be denoted as xo
+cand the evalu-
+ated with the growing process with xd
+c. This surprising
+result seems similar to the obtained for weak first-order
+phase transitions [10], where two pseudo-critical points
+exits due to the metastable states above and below the
+critical point. However, there is an important difference
+in this case: for weak order phase transitions the smaller
+critical point corresponds to the decay process, and the
+bigger corresponds to the growing process, contrary to
+the majority vote model case.
+In the evaluation of the exponent θa linear extrapola-
+tion gives the result of 0 .175(3), which is lower from the
+values of the two dimensional Ising model, θ= 0.191(1),
+and from the previously evaluated in references [11, 12]
+for the majorityvote model, θ= 0.192(2). The difference
+with respect to the Ising model could be understood con-
+sidering that the results obtained here seems to indicate
+a hole new dynamic. The differences with previous re-
+sults for the majority model can be explained observing
+the simulations details used previously: first the critical
+point used was x=0.850, which is above the result for
+xd
+c. Second the systems sizes used previously were really
+small (L= 32), at this size the growing process is not
+very long and is really hard to see the power law behav-
+ior.
+One can obtain the dynamical exponent zevaluating
+the second moment of the magnetization at the critical
+pointxd
+c
+m(2)∼ty, y= (d−2β/ν)/z, (8)
+and the autocorrelation
+A(t) =/summationtext
+iσi(t= 0)σ(t),
+A(t)∼t−λ, λ=d
+z−θ.(9)4
+0.8480.850xc
+0 0.02 0.04 0.06 0.08m00.1750.18θa)
+b)
+FIG. 4: a) Evaluation of the critical point xcand b) the θ
+exponent.
+0.0010.0100.1001.000A(t)
+1 10 100 1000t1×10−61×10−51×10−4m(2)(t)a)
+b)
+FIG. 5: (color online) a) Evaluation of λ, the continuous line
+shows the autocorrelation time evolution and the dashed lin e
+shows the power law behavior with λ= 0.758. b) Evaluation
+ofy, the continuous line shows the second moment order
+parameter and the dashed line shows the power law behavior
+withy= 0.799.
+Both starting from m0= 0 and using 1000 MCTS. Again
+there is a tmicin each case (around 20 for the autocor-
+relation and 75 for the second moment, see Fig. 5). The
+results obtained are y= 0.799(17) and λ= 0.758(2).
+Combining both results it can be obtained the values
+z= 2.143(9) and β/ν= 0.143(18). A summary of the
+results are showed in table III, where it can be observed
+discrepancies between most of the values for the major-
+ity model and the Ising ones. It must be remark that all
+these results were obtained using just the growing pro-
+cess.
+Finally the decay exponent β/νzwas evaluated start-
+ing from an ordered phase (at xo
+c), using 1000 MCTSTABLE III: Summary of the results in this work and of the
+Ising model.
+Majority vote model Ising
+θ 0.175(3) 0.191(1)
+λ 0.758(2) 0.737(1)
+z 2.143(9) 2.155(3)
+y 0.799(17) 0.817(7)
+β/ν 0.143(18) 1/4
+1 10 100 1000t0.60.70.80.9m(t)
+FIG. 6: (color online) Order parameter relaxation ( m0= 1),
+herethedashed lineshows thepower lawbehavior with β/ν=
+0.0526.
+(Fig. 6), the result was β/ν= 0.0526(5), that is lower
+compared to the Ising one, 0 .0580(5). Again the first
+time steps were discarded for the evaluation of the expo-
+nent (Fig. 6). Theoretically it is possible to obtain the z
+exponent using the known value of β/ν, or knowing the z
+value one can obtain the β/νvalue, but in both cases the
+results depend on values obtained with a growingprocess
+(z) or the static simulations ( β/nu). The approachtaken
+in this work is that the growing and the decay process
+are different and it could be possible that the dynamic
+exponent zis different in each case, so for the decay case
+the reporting value is z= 2.37(2).
+The fact that we have two pseudo-critical points (that
+not corresponds to a weak phase transition) and that the
+decay and growing process are slower that in the Ising
+model must be related to the absence of detailed balance
+condition. One of the consequences of this absence is
+that we do not have a unique thermodynamic temper-
+ature, in this case we have two, so looking at the snap-
+shots for different initial conditions at the pseudo-critical
+points we can speculate about the competition between
+thetwo”temperatures”thatgovernsthedynamicin non-
+equilibrium Ising systems. Figure 7 shows the time evo-
+lution with m0= 0 at the two pseudo-critical points, a)
+forxd
+cand b) for xo
+c. Initially the number of sites with
+spin-flip probability depending on β2are very similar to
+the ones depending on β4, the time increases from left to
+right and it can be observed that at xo
+cthe coarsening5
+FIG. 7: Snapshots at the pseudo-critical points starting wi th
+m0= 0, a) xd
+cand b)xo
+c. Times are (from left to right) 1,
+100, 1000, 10000 and 20000.
+FIG. 8: Snapshots at the pseudo-critical points starting wi th
+m0= 1, a) xd
+cand b)xo
+c. Times are (from left to right) 1,
+100, 1000, 10000 and 20000.
+seems to appear faster that at xd
+c.
+Figure 8 shows the same time evolution for m0= 1.
+In this case at the beginning of the evolution all sites
+have spin-flip probability that depends only on β4and
+the decay is slightly lower at xo
+c.
+It seems that the coarsening differs at the two pseudo-
+critical points, so as a final test the time evolution for
+a special initial condition were performed setting m0re-
+ally close to zero putting almost half of the spins in one
+state and the other half in the other state with a circular
+border, in this way we have a large number of sites with
+β4while the number of possible sites with b2increases
+(Fig. 9). We can observe that the circular shape last
+longer at xo
+c. In order to corroborate the effect of the
+difference between temperatures it should be performed
+simulationsin spin-like systemsfor different ratios β4/β2,
+except for the equilibrium case β4/β2= 1.
+FIG. 9: Snapshots at the pseudo-critical points starting wi th
+m0almost zero and a circular border, a) xd
+cand b)xo
+c. Times
+are (from left to right) 1, 100, 1000, 10000 and 20000.IV. CONCLUSION
+In this work it has been shown that the short time
+dynamic in the majority vote model presents power law
+behavior at different control parameters for the grow-
+ing,xd
+c= 0.84860(10), and the decay processes, xo
+c=
+0.85007(6). These pseudo-critical points are compati-
+bleswithresultsforthecriticalpointreportedpreviously,
+xc= 0.850(2). It has been show also that the dynamic
+in both cases is slower that in Ising model for all the
+quantities calculated ( m, m(2)andA). These results
+seems to be related to the competing dynamic between
+the interface ( β2) and bulk ( β4) temperatures associated
+to the dynamic, and as consequence to the absence of
+detailed balance in the system. In order to corroborate
+these results additional simulations must be carry on in
+systemswithout detailedbalance. Thedynamicalcritical
+exponent ( z) and the static critical exponent ( β/ν) has
+been evaluated independently using a growingprocess, in
+both cases the results were close to the Ising ones. For
+the decay process the zexponent was evaluated using re-
+sults from static simulations founding that the value is
+different from the obtained in the growing process.
+V. ACKNOWLEDGMENTS
+I wish to thank G. P´ erezfor his useful comments. This
+work was supported by Conacyt M´ exico through Grant
+No. 61418/2007.
+VI. REFERENCES6
+[1] M. J. de Oliveira, J. of Stat. Phys. ,66, 273 (1992).
+[2] M. J. de Oliveira, J. F. F. Mendes and M. A. Santos, J.
+Phys. A: Math. and Gen. ,262317 (1993).
+[3] KwakWooseop, Jae-SukYang, Jang-il SohnandIn-mook
+Kim,Phys. Rev. E 75, 061110 (2007).
+[4] G. Grinstein, C. Jayaprakash and Y.He, Phys. Rev. Lett.
+55,2527 (1985).
+[5] Jae-SukYangandIn-mookKim, Phys. Rev. E 77, 051122
+(2008).
+[6] F. W. S. Lima, U. L. Fulco and R. N. Costa Filho, Phys.
+Rev. E71, 036105 (2005).
+[7] F. W. S. Lima and K. Malarz, Int. J. of Moderm Phys.
+C17, 1273 (2006).
+[8] H. K. Janssen, B. Schaub and B. Schmittmann, Z. Phys.
+B73, 539 (1989).[9] B. Zheng, 1998 Int. J. of Mod. Phys. ,12(1419).
+[10] L. Sch¨ ulke and B. Zheng, Phys. Rev. E 62, 7482 (2000)
+[11] J. F. F. Mendes and M. A. Santos, Phys. Rev. E 57108,
+(1998).
+[12] T. Tom´ e and M. J. de Oliveira, Phys. Rev. E 58, 4242
+(1998).
+[13] F. Sastre, I. Dornic and H. Chat´ e Phys. Rev. Lett. 91,
+267205 (2003).
+[14] M. P. Nightingale and H. W. J. Bl¨ ote, Phys. Rev. B 62,
+1089 (2000).
+[15] J. M. Drouffe and C. Godr` eche, J. Phys. A: Math. and
+Gen.32, 249 (1999).
+[16] P. Calabrese, A. Gambassi and F. Krzakala, J. of Stat.
+Mec.: Th. Exp. , 06016 (2006).
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+arXiv:1001.0164v1  [nlin.SI]  31 Dec 2009Manuscript submitted to Website: http://AIMsciences.org
+AIMS’ Journals
+Volume X, Number 0X, XX200X pp.X–XX
+BOSE-EINSTEIN CONDENSATES AND SPECTRAL PROPERTIES
+OF MULTICOMPONENT NONLINEAR SCHR ¨ODINGER
+EQUATIONS
+Vladimir S. Gerdjikov
+Institute for Nuclear Research and Nuclear Energy,
+Bulgarian academy of sciences
+72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria
+Abstract. We analyze the properties of the soliton solutions of a class of mod-
+els describing one-dimensional BEC with spin F. We describe the minimal sets
+of scattering data which determine uniquely both the corres ponding potential
+of the Lax operator and its scattering matrix. Next we give se veral reduc-
+tions of these MNLS, derive their N-soliton solutions and analyze the soliton
+interactions. Finally we prove an important theorem provin g that if the initial
+conditions satisfy the reduction then one gets a solution of the reduced MNLS.
+1.INTRODUCTION. It is well known that Bose-Einstein condensate (BEC)
+of alkali atoms in the F= 1 hyperfine state, elongated in xdirection and con-
+fined in the transverse directions y,zby purely optical means are described by a
+3-component normalized spinor wave vector Φ(x,t) = (Φ 1,Φ0,Φ−1)T(x,t). Con-
+sidering dimensionless units and using special choices for the scatte ring lengths one
+can show that Φ(x,t) satisfies the multicomponent nonlinear Schr¨ odinger (MNLS)
+equation [ 14], see also [ 15,19,26,2,22]:
+i∂tΦ1+∂2
+xΦ1+2(|Φ1|2+2|Φ0|2)Φ1+2Φ∗
+−1Φ2
+0= 0,
+i∂tΦ0+∂2
+xΦ0+2(|Φ−1|2+|Φ0|2+|Φ1|2)Φ0+2Φ∗
+0Φ1Φ−1= 0,(1)
+i∂tΦ−1+∂2
+xΦ−1+2(|Φ−1|2+2|Φ0|2)Φ−1+2Φ∗
+1Φ2
+0= 0.
+Similarly spinor BEC with F= 2 is described by a 5-component normalized spinor
+wave vector Φ(x,t) = (Φ 2,Φ1,Φ0,Φ−1,Φ−2)T(x,t). For specific choices of the
+scattering lengths in dimensionless coordinates the corresponding set of equations
+forΦ(x,t) take the form [ 27]:
+i∂tΦ±2+∂xxΦ±2+2(/vectorΦ,/vectorΦ∗)Φ±2−(2Φ2Φ−2−2Φ1Φ−1+Φ2
+0)Φ∗
+∓2= 0,
+i∂tΦ±1+∂xxΦ±1+2(/vectorΦ,/vectorΦ∗)Φ±1+(2Φ2Φ−2−2Φ1Φ−1+Φ2
+0)Φ∗
+∓1= 0,(2)
+i∂tΦ0+∂xxΦ0+2(/vectorΦ,/vectorΦ∗)Φ0−(2Φ2Φ−2−2Φ1Φ−1+Φ2
+0)Φ∗
+0= 0.
+Both models havenaturalLie algebraicinterpretationandarerelat edto the sym-
+metricspaces BD.I≃SO(n+2)/SO(n)×SO(2)with n= 3andn= 5respectively.
+They are integrable by means of inverse scattering transform met hod [4,25,12].
+2000Mathematics Subject Classification. Primary: 35Q51, 37K40; Secondary: 34K17 .
+Key words and phrases. Bose-Einstein condensates, Multicomponent nonlinear Sch r¨ odinger
+equations, Soliton solutions, Soliton interactions, Redu ctions of MNLS.
+12 VLADIMIR S. GERDJIKOV
+Using a modification of the Zakharov-Shabat ‘dressing method’ we d escribe the
+soliton solutions [ 14,17] and the effects of the reductions on them.
+Sections 2 contains the basicdetails on the direct and inversescatt eringproblems
+for the Lax operator. Section 3 is devoted to the construction of their soliton
+solutions. In Section 4 we formulate the minimal sets of scattering d ataLwhich
+determine uniquely both the scattering matrix and the potential Q(x,t). Section
+5 gives a few important examples of algebraic reductions of the MNLS . In Section
+6 we analyze the soliton interactions of the MNLS. To this end we evalu ate the
+limits of the generic two-soliton solution for t→ ±∞. As a result we establish that
+the effect of the interactions on the soliton parameters is analogou s to the one for
+the scalar NLS equation and consists in shifts of the ‘center of mass ’ and shift in
+the phase. In Section 7 we prove an important theorem proving tha t if the initial
+conditions satisfy the reduction then one gets a solution of the red uced MNLS.
+2.The method for solving MNLS for any F.
+2.1.The Lax representation. The above MNLS equations ( 1) and (2) are the
+firsttwomembersofaseriesofMNLSequationsrelatedtothe BD.I-typesymmetric
+spaces. They allow Lax representation as follows [ 4,10,12]
+Lψ(x,t,λ)≡i∂xψ+U(x,t,λ)ψ(x,t,λ) = 0,
+Mψ(x,t,λ)≡i∂tψ+V(x,t,λ)ψ(x,t,λ) = 0,(3)
+where
+U(x,t,λ) =Q(x,t)−λJ,
+V(x,t,λ) =V0(x,t)+λV1(x,t)−λ2J,
+V1(x,t) =Q(x,t), V 0(x,t) =iad−1
+JdQ
+dx+1
+2/bracketleftbig
+ad−1
+JQ,Q(x,t)/bracketrightbig
+.(4)
+For those familiar with Lie algebras I remind that, as usual, Q(x,t) andJare
+elements of the corresponding Lie algebra, which in our case is g≃so(n+2). The
+choice of the Cartan subalgebra element Jdetermines the co-adjoint orbit of g; in
+our caseJis dual toe1, see [13]. It introduces grading in g=g(0)⊕g(1)where
+g(0)≃so(n). The root system of g(0)consists of all roots of so(n+ 2) which are
+orthogonalto e1; the linear subspace g(1)is spanned by the Weyl generators Eαand
+E−αfor which the roots α∈∆+
+1are such that their scalar products ( α,e1) = 1.
+Thus the potential
+Q(x,t) =/summationdisplay
+α∈∆+
+1(qα(x,t)Eα+pα(x,t)E−α) (5)
+may be viewed as local coordinate of the above mentioned symmetric space. The
+linear operator ad JX= [J,X] and ad−1
+Jis well defined on the image of ad Jing.
+In what follows we will use the typical representation of so(n+2) withn= 2r−1
+in whichQandJtake the following block-matrix structure:
+Q(x,t) =
+0/vector qT0
+/vector p0s0/vector q
+0/vector pTs00
+, J= diag(1,0,...0,−1). (6)
+Forphysical applications one uses mostly potentials satisfying the t ypical reduction,
+i.e./vector p(x,t) =/vector q∗(x,t). The vector /vector q(x,t) for integer F=rhas 2r+1 components
+/vector q(x,t) = (Φ r−1,...,Φ0,...,Φ−r+1)T(x,t), (7)BEC AND SPECTRAL PROPERTIES OF MNLS 3
+and the corresponding matrices s0enter in the definition of the orthogonal algebras
+so(2r−1); namely X∈so(2r+1) if
+X+S0XTS0= 0, S0=2r+1/summationdisplay
+s=1(−1)s+1Es,n+1−s, S0=
+0 0 1
+0−s00
+1 0 0
+.(8)
+ByEspabove we mean 2 r+1×2r+1 matrix with matrix elements ( Esp)ij=δsiδpj.
+With the definition of orthogonality used in ( 8) the Cartan generators Hk=Ek,k−
+E2r+2−k,2r+2−kare represented by diagonal matrices.
+If we make use of the typical reduction Q=Q†(or/vector p∗=/vector q) the generic MNLS
+type equations related to BD.I.acquire the form:
+i/vector qt+/vector qxx+2(/vector q†,/vector q)/vector q−(/vector q,s0/vector q)s0/vector q∗= 0. (9)
+and forr= 2 (resp. r= 3) coincides with the MNLS eq. ( 1) (resp. with eq. ( 3)).
+The Hamiltonians for the MNLS equations ( 9) are given by
+HMNLS=/integraldisplay∞
+−∞dx/parenleftbigg
+(∂x/vector q†,∂x/vector q)−(/vector q†,/vector q)2+1
+2|(/vector qT,s0/vector q)|2/parenrightbigg
+. (10)
+2.2.The Direct and the Inverse scattering problem for L.We remind some
+basic features of the scattering theory for the Lax operators L, see [10,12]. There
+we have made use of the general theory developed in [ 32,33,29,3,6] and the
+references therein. The Jost solutions of Lare defined by:
+lim
+x→−∞φ(x,t,λ)eiλJx=11,lim
+x→∞ψ(x,t,λ)eiλJx=11 (11)
+and the scattering matrix T(λ,t)≡ψ−1φ(x,t,λ). The special choice of Jand
+the fact that the Jost solutions and the scattering matrix take va lues in the group
+SO(2r+1) we can use the following block-matrix structure of T(λ,t)
+T(λ,t) =
+m+
+1−/vectorb−Tc−
+1
+/vectorb+T22−s0/vectorB−
+c+
+1/vectorB+Ts0m−
+1
+,ˆT(λ,t) =
+m−
+1/vectorB−Tc−
+1
+−/vectorB+ˆT22s0/vectorb−
+c+
+1−/vectorb+Ts0m+
+1
+,
+(12)
+where/vectorb±(λ,t) and/vectorB±(λ,t) are 2r−1-component vectors, T22(λ) is 2r−1×2r−1
+block matrix, and m±
+1(λ), andc±
+1(λ) are scalar functions. Below we often use ˆXto
+denote the matrix inverse to X.
+Remark 1. The typical reduction /vector p(x,t) =/vector q∗(x,t) mentioned above imposes on
+T(λ,) the constraint T†(λ,t) =ˆT(λ,t) for real values of λ∈R, i.e.
+m+
+1(λ) =m−,∗
+1(λ),/vectorB1−(λ) =/vectorb1+,∗(λ),
+c+
+1(λ) =c−,∗
+1(λ),/vectorB1+(λ) =/vectorb1−,∗(λ).(13)
+The Lax representation ( 1) allows one to prove that if /vector q(x,t) satisfies the MNLS
+(9) then the scattering matrix T(λ,t) satisfies the linear evolution equation [ 12]:
+idT
+dt−λ2[J,T(λ,t)] = 0, (14)4 VLADIMIR S. GERDJIKOV
+or in components:
+id/vectorb±
+dt±λ2/vectorb±(t,λ) = 0, id/vectorB±
+dt±λ2/vectorB±(t,λ) = 0,
+idm±
+1
+dt= 0, idm±
+2
+dt= 0.(15)
+Thustheblock-diagonalmatrices D±(λ) canbeconsideredasgeneratingfunctionals
+of the integrals of motion. Thus the problem of solving the MNLS eq. is based on
+the effective analysis of the mapping between the potential Q(x,t) ofLand the
+scattering matrix T(λ,t).
+2.3.The fundamental analytic solution and the Riemann-Hilbert prob-
+lem.The most effective method for the above mentioned analysis consist s in con-
+structing the fundamental analytic solution (FAS) of L-operators of type ( 3) and
+reducing the inverse scattering problem to an equivalent Riemann-H ilbert problem
+(RHP). Skipping the details (see [ 11]) we just outline the construction of FAS for
+L. Obviously the FAS, like any other fundamental solutions of Lmust be linearly
+related to the Jost solutions. For the class of potentials Q(x,t) with vanishing
+boundary conditions there exist two FAS χ±(x,t,λ) which allow analytic extension
+forλ∈C±respectively. For real λthey are related to the Jost solutions by
+χ±(x,t,λ) =φ(x,t,λ)S±
+J(t,λ) =ψ(x,t,λ)T∓
+J(t,λ)D±
+J(λ), (16)
+whereT∓
+J(t,λ),D±
+J(λ) andT∓
+J(t,λ) are the generalized Gauss factors of T(λ,t),
+see [29,5,7]:
+T(λ,t) =T−
+JD+
+JˆS+
+J, T (λ,t) =T+
+JD−
+JˆS−
+J,
+T∓
+J(λ,t) =e±(/vector ρ±,/vectorE∓
+1), S±
+J(λ,t) =e±(/vector τ±,/vectorE±
+1),
+D±
+J(λ) = diag/parenleftbig
+(m±
+1)±1,m±
+2,(m±
+1)∓1/parenrightbig
+,(17)
+Here
+/vector τ±(λ,t) =/parenleftbig
+τ±
+r−1,...,τ±
+0,...,τ±
+−r+1/parenrightbigT(λ,t),
+/parenleftig
+/vector τ+,/vectorE+
+1/parenrightig
+=r−1/summationdisplay
+k=1(τ+
+kEe1−ek+1+τ+
+−kEe1+ek+1)+τ+
+0Ee1,
+/parenleftig
+/vector τ−,/vectorE−
+1/parenrightig
+=r−1/summationdisplay
+k=1(τ−
+kE−e1+ek+1+τ−
+−kE−e1−ek+1)+τ−
+0E−e1,(18)
+and similar expressions for/parenleftig
+/vector ρ±,/vectorE∓
+1/parenrightig
+. Above we have made use of the fact that ∆+
+1
+consists of the roots {e1−ek,e1,e1+ek}r−1
+k=1. The functions m±
+1andn×nmatrix-
+valued functions m±
+2are analytic for λ∈C±. One can check, that the analogs of
+the reflection coefficients /vector ρ±and/vector τ±are expressed by:
+/vector ρ−=/vectorB−
+m−
+1, /vector τ−=/vectorB+
+m−
+1, /vector ρ+=/vectorb+
+m+
+1, /vector τ+=/vectorb−
+m+
+1.
+Remark 2. The typical reduction means that for λ∈Rthe reflection coefficients
+are constrained by (see remark 1above):
+/vector ρ+(λ) =/vector ρ−,∗(λ), /vector τ+(λ) =/vector τ−,∗(λ), λ∈R. (19)BEC AND SPECTRAL PROPERTIES OF MNLS 5
+There are some additional relations which ensure that both T(λ) and its inverse
+ˆT(λ) belong to the orthogonal group SO(2r+1) and that T(λ)ˆT(λ) =11.
+The FASχ±(x,t,λ) are related by:
+χ+(x,t,λ) =χ−(x,t,λ)G0,J(λ,t), G 0,J(λ,t) =ˆS−
+J(λ,t)S+
+J(λ,t) (20)
+Below for convenience we introduce ξ±(x,λ) =χ±(x,λ)eiλJxwhich satisfy the
+equation:
+idξ±
+dx+Q(x)ξ±(x,λ)−λ[J,ξ±(x,λ)] = 0, (21)
+and the relation
+lim
+λ→∞ξ±(x,t,λ) =11, (22)
+Thenξ±(x,λ) satisfy the RHP’s
+ξ+(x,t,λ) =ξ−(x,t,λ)GJ(x,t,λ),
+GJ(x,t,λ) =e−iλJ(x+λt)G0,J(λ)eiλJ(x+λt),(23)
+with sewing function GJ(x,t,λ) uniquely determined by the Gauss factors S±
+J(t,λ)
+taken fort= 0:
+G0,J(λ) =ˆS−
+J(0,λ)S+
+J(0,λ).
+The analyticity properties of these FAS follow from the equivalent se t of integral
+equations:
+ξ+
+1j(x,λ) =δ1j+i/integraldisplayx
+∞dye−iλ(x−y)2r−1/summationdisplay
+p=1qr−p(y)ξ+
+p+1,j(y,λ),
+ξ+
+kj(x,λ) =δkj+i/integraldisplayx
+−∞dy(q∗
+r−k+1(y)ξ+
+1,j(y,λ)
+−(−1)r+kq−r+k−1(y)ξ+
+2r+1,j(y,λ)),2≤k,j≤2r;
+ξ+
+2r+1,j(x,λ) =δ2r+1,j+i/integraldisplayx
+−∞dyeiλ(x−y)2r−1/summationdisplay
+p=1(−1)p+1q∗
+−r+p(y)ξ+
+p+1,j(y,λ),
+(24)
+and a similar set of integral equations for
+ξ−
+1j(x,λ) =δ1j+i/integraldisplayx
+−∞dye−iλ(x−y)2r−1/summationdisplay
+p=1qr−p(y)ξ−
+p+1,j(y,λ),
+ξ−
+kj(x,λ) =δkj+i/integraldisplayx
+−∞dy(q∗
+r−k+1(y)ξ−
+1,j(y,λ)
+−(−1)r+kq−r+k−1(y)ξ−
+2r+1,j(y,λ)),2≤k,j≤2r;
+ξ−
+2r+1,j(x,λ) =δ2r+1,j+i/integraldisplayx
+∞dyeiλ(x−y)2r−1/summationdisplay
+p=1(−1)p+1q∗
+−r+p(y)ξ−
+p+1,j(y,λ),(25)
+The RHP ( 23) with the additional condition ( 22) is known as an RHP with
+canonical normalization.
+Remark 3. An immediate consequence of the analyticity of ξ±(x,t,λ) is that
+D±(λ) are analytic functions for λ∈C±. This fact follows from the relation
+limx→∞ξ±(x,λ) =D±(λ).6 VLADIMIR S. GERDJIKOV
+Zakharov and Shabat proved a theorem [ 32,33] which states that if GJ(x,λ,t)
+satisfies:
+idG
+dx−λ[J,G(x,λ,t)] = 0,
+idG
+dt−λ2[J,G(x,λ,t)] = 0,(26)
+then the corresponding solutions of the RHP allow one to construct χ±(x,λ) =
+ξ±(x,λ)e−iλJxas a fundamental solution of the Lax pair eq. ( 1).
+We will say that ξ±
+0(x,λ) is a regular solution to the RHP ( 23) if the block-
+diagonal part of it has neither zeroes nor poles in its whole region of a nalyticity.
+If we have solved the RHP’s and know the FAS ξ+(x,t,λ) then the formula
+Q(x,t) = lim
+λ→∞λ/parenleftig
+J−ξ+(x,t,λ)Jˆξ+(x,t,λ)/parenrightig
+, (27)
+allows us to recover the corresponding potential of L.
+3.Singular solutions of RHP and soliton solutions of MNLS. Zakharov-
+Shabat’s theorem ensures that if a given RHP allows regular solution, then this
+solution is unique. However the RHP may have many singular solutions. The
+construction of such singular solutions starting from a given regula r one is known
+as the dressing Zakharov-Shabat method [ 32,33]. Indeed, if ξ±
+0(x,t,λ) are regular
+solutions to the RHP, then
+ξ±(x,t,λ) =u(x,t,λ)ξ±
+0(x,t,λ) (28)
+with conveniently chosen dressing factor u(x,t,λ) may again be a solution of the
+RHP [32,33]. Obviously this factor must be analytic (with the exception of finite
+number of singular points) in the whole complex λ-plane and can explicitly be
+constructed using only the solution of the regular RHP.
+In order to obtain N-soliton solutions one has to apply the dressing procedure
+to the trivial solution of the RHP ξ0(x,t,λ) =11. We choose a dressing factor with
+2N-poles [11]:
+u(x,t,λ) =11+N/summationdisplay
+k=1/parenleftbiggAk(x,t)
+λ−λ+
+k+Bk(x,t)
+λ−λ−
+k/parenrightbigg
+. (29)
+TheN-soliton solution itself can be generated via the following formula
+QN,s(x,t) =N/summationdisplay
+k=1[J,Ak(x,t)+Bk(x,t)]. (30)
+The dressing factor u(x,λ) must satisfy the equation
+i∂u
+∂x+QN,s(x,t)u(x,t,λ)−λ[J,u(x,t,λ)] = 0 (31)
+and the normalization condition lim λ→∞u(x,λ) =11.
+The residues of uadmit the following decomposition
+Ak(x,t) =Xk(x,t)FT
+k(x,t), B k(x,t) =Yk(x,t)GT
+k(x,t),
+where all matrices involved are supposed to be rectangular and of m aximal rank s
+[30,9]. By comparing the coefficients before the same powers of λ−λ±
+kin (31) we
+convince ourselves that the factors FkandGkcan be expressed by the fundamental
+analytic solutions χ±
+0(x,t,λ) =e−iλ(x+λt)Jas follows
+FT
+k(x,t) =FT
+k,0[χ+
+0(x,t,λ+
+k)]−1, GT
+k(x,t) =GT
+k,0[χ−
+0(x,t,λ−
+k)]−1.BEC AND SPECTRAL PROPERTIES OF MNLS 7
+The constant rectangular matrices Fk,0andGk,0obey the algebraic relations
+FT
+k,0S0Fk,0= 0, GT
+k,0S0Gk,0= 0.
+The other two types of factors Xk(x,t) andYk(x,t) are solutions to the algebraic
+system
+S0Fk=Xkαk+/summationdisplay
+l/negationslash=kXlFT
+lS0Fk
+λ+
+l−λ+
+k+/summationdisplay
+lYlGT
+lS0Fk
+λ−
+l−λ+
+k,
+S0Gk=/summationdisplay
+lXlFT
+lS0Gk
+λ+
+l−λ−
+k+Ykβk+/summationdisplay
+l/negationslash=kYlGT
+lS0Gk
+λ−
+l−λ−
+k.(32)
+The square s×smatricesαk(x,t) andβk(x,t) introduced above depend on χ+
+0and
+χ−
+0and their derivatives by λas follows
+αk(x,t) =−FT
+0,k[χ+
+0(x,t,λ+
+k)]−1∂λχ+
+0(x,t,λ+
+k)S0F0,k+α0,k,
+βk(x,t) =−GT
+0,k[χ−
+0(x,t,λ−
+k)]−1∂λχ−
+0(x,t,λ−
+k)S0G0,k+β0,k.(33)
+Below for simplicity we will choose FkandGkto be 2r+1-component vectors.
+Then one can show that αk=βk= 0 which simplifies the system ( 32). We also
+introduce the following more convenient parametrization for FkandGk, namely
+(see eq. ( 35)):
+Fk(x,t) =S0|nk(x,t)/an}bracketri}ht=
+e−zk+iφk
+−√
+2s0/vector ν0k
+ezk−iφk
+,
+Gk(x,t) =|n∗
+k(x,t)/an}bracketri}ht=
+ezk+iφk√
+2/vector ν∗
+0k
+e−zk−iφk
+,(34)
+where/vector ν0kare constant 2 r−1-component polarization vectors and
+zj=νj(x+2µjt)+ξ0j, φ j=µjx+(µ2
+j−ν2
+j)t+δ0j,
+/an}bracketle{tnT
+j(x,t)|S0|nj(x,t)/an}bracketri}ht= 0,or (/vector ν0,js0/vector ν0,j) = 1.(35)
+With this notations the polarization vectors automatically satisfy th e condition
+/an}bracketle{tnj(x,t)|S0|nj(x,t)/an}bracketri}ht= 0. Thus for N= 1 we get the system:
+|Y1/an}bracketri}ht=−(λ+
+1−λ−
+1)|n1/an}bracketri}ht
+/an}bracketle{tn†
+1|n1/an}bracketri}ht,|X1/an}bracketri}ht=(λ+
+1−λ−
+1)S0|n∗
+1/an}bracketri}ht
+/an}bracketle{tn†
+1|n1/an}bracketri}ht, (36)
+which is easily solved. As a result for the one-soliton solution we get:
+/vector q1s=−i√
+2(λ+
+1−λ−
+1)e−iφ1
+∆1/parenleftbig
+e−z1s0|/vector ν01/an}bracketri}ht+ez1|/vector ν∗
+01/an}bracketri}ht/parenrightbig
+,∆1= cosh(2z1)+/an}bracketle{t/vector ν†
+01|/vector ν01/an}bracketri}ht.
+(37)8 VLADIMIR S. GERDJIKOV
+Forn= 3 we put ν0k=|ν0k|eα0kand get:
+Φ1s;±1=−/radicalbig
+2|ν01;1ν01;3|(λ+
+1−λ−
+1)
+∆1e−iφ1±iβ13
+×(cosh(z1∓ζ01)cos(α13)−isinh(z1∓ζ01)sin(α13)),
+Φ1s;0=−√
+2|ν01;2|(λ+
+1−λ−
+1)
+∆1e−iφ1(sinhz1cos(α02)+icoshz1sin(α02)),
+β13=1
+2(α03−α01), ζ 01=1
+2ln|ν01;3|
+|ν01;1|, α 13=1
+2(α03+α01),(38)
+Note that the ‘center of mass‘ of Φ 1s;1(resp. of Φ 1s;−1) is shifted with respect to
+the one of Φ 1s;0byζ01to the right (resp to the left); besides |Φ1s;1|=|Φ1s;−1|, i.e.
+they have the same amplitudes.
+Forn= 5 we put ν0k=|ν0k|eα0kand get analogously:
+Φ1s;±2=−/radicalbig
+2|ν01;1ν01;5|(λ+
+1−λ−
+1)
+∆1e−iφ1±iβ15
+×(cosh(z1∓ζ01)cos(α15)−isinh(z1∓ζ01)sin(α15)),
+Φ1s;±1=/radicalbig
+2|ν01;2ν01;4|(λ+
+1−λ−
+1)
+∆1e−iφ1±iβ24
+×(cosh(z1∓ζ02)cos(α24)−isinh(z1∓ζ01)sin(α24)),
+Φ1s;0=−√
+2|ν01;3|(λ+
+1−λ−
+1)
+∆1e−iφ1(coshz1cos(α03)−isinhz1sin(α03)),
+β15=1
+2(α05−α01), ζ 01=1
+2ln|ν01;5|
+|ν01;1|, α 15=1
+2(α05+α01),
+β24=1
+2(α04−α02), ζ 02=1
+2ln|ν01;4|
+|ν01;2|, α 24=1
+2(α04+α02).(39)
+Similarly the ‘center of mass‘ of Φ 1s;2and Φ 1s;1(resp. of Φ 1s;−2and Φ 1s;−1) are
+shifted with respect to the one of Φ 1s;0byζ01andζ02to the right (resp to the left);
+besides|Φ1s;2|=|Φ1s;−2|and|Φ1s;1|=|Φ1s;−1|.
+ForN= 2 we get:
+|n1(x,t)/an}bracketri}ht=X2(x,t)f21
+λ+
+2−λ+
+1+Y1(x,t)κ11
+λ−
+1−λ+
+1+Y2(x,t)κ21
+λ−
+2−λ+
+1,
+|n2(x,t)/an}bracketri}ht=X1(x,t)f12
+λ+
+1−λ+
+2+Y1(x,t)κ12
+λ−
+1−λ+
+2+Y2(x,t)κ22
+λ−
+2−λ+
+2,
+S0|n∗
+1(x,t)/an}bracketri}ht=X1(x,t)κ11
+λ+
+2−λ+
+1+X2(x,t)κ11
+λ+
+2−λ−
+1+Y2(x,t)f∗
+21
+λ−
+2−λ−
+1,
+S0|n∗
+2(x,t)/an}bracketri}ht=X1(x,t)κ21
+λ+
+1−λ−
+2+X2(x,t)κ22
+λ+
+2−λ−
+2+Y1(x,t)f∗
+12
+λ−
+1−λ−
+2,(40)
+where
+κkj(x,t) =ezk+zj+i(φk−φj)+e−zk−zj−i(φk−φj)+2/parenleftig
+/vector ν†
+0k,/vector ν0j/parenrightig
+,
+fkj(x,t) =ezk−zj−i(φk−φj)+ezj−zk+i(φk−φj)−2/parenleftbig
+/vector νT
+0ks0/vector ν0j/parenrightbig
+,(41)BEC AND SPECTRAL PROPERTIES OF MNLS 9
+In other words:
+M/vectorX≡
+0f21
+λ+
+2−λ+
+1κ11
+λ−
+1−λ+
+1κ21
+λ−
+2−λ+
+1f12
+λ+
+1−λ+
+20κ12
+λ−
+1−λ+
+2κ22
+λ−
+2−λ+
+2
+κ11
+λ+
+1−λ−
+1κ12
+λ+
+2−λ−
+10f∗
+21
+λ−
+2−λ−
+1
+κ21
+λ+
+1−λ−
+2κ22
+λ+
+2−λ−
+2f∗
+12
+λ−
+1−λ−
+20
+
+X1
+X2
+Y1
+Y2
+=
+|n1/an}bracketri}ht
+|n2/an}bracketri}ht
+S0|n∗
+1/an}bracketri}ht
+S0|n∗
+2/an}bracketri}ht
+.(42)
+We can rewrite Min block-matrix form:
+M=/parenleftbigg
+M11M12
+M21M22/parenrightbigg
+,M22=M∗
+11,M21=−MT
+12,
+M11=f12
+λ+
+2−λ+
+1/parenleftbigg0 1
+−1 0/parenrightbigg
+,M12=/parenleftiggκ11
+λ−
+1−λ+
+1κ21
+λ−
+2−λ+
+1κ12
+λ−
+1−λ+
+2κ22
+λ−
+2−λ+
+2/parenrightigg
+.(43)
+The inverse of Mis given by:
+M−1=/parenleftbigg
+N−1
+1 −N−1
+1M12ˆM∗
+11
+−N−1
+2M21ˆM11 N−1
+2/parenrightbigg
+,
+N1=M11−M12ˆM∗
+11M21,N2=M∗
+11−M21ˆM11M12(44)
+From eqs. ( 42) and (44) we obtain [ 12]:
+|X1/an}bracketri}ht=1
+Z/parenleftbiggf∗
+12
+λ−
+1−λ−
+2|n2/an}bracketri}ht−κ22
+λ+
+2−λ−
+2S0|n∗
+1/an}bracketri}ht+κ12
+λ+
+2−λ−
+1S0|n∗
+2/an}bracketri}ht/parenrightbigg
+,
+|X2/an}bracketri}ht=1
+Z/parenleftbigg
+−f∗
+12
+λ−
+1−λ−
+2|n1/an}bracketri}ht+κ21
+λ+
+1−λ−
+2S0|n∗
+1/an}bracketri}ht−κ11
+λ+
+1−λ−
+1S0|n∗
+2/an}bracketri}ht/parenrightbigg
+,
+|Y1/an}bracketri}ht=1
+Z/parenleftbiggκ22
+λ+
+2−λ−
+2|n1/an}bracketri}ht−κ21
+λ+
+1−λ−
+2|n2/an}bracketri}ht−f12
+λ+
+1−λ+
+2S0|n∗
+2/an}bracketri}ht/parenrightbigg
+,
+|Y2/an}bracketri}ht=1
+Z/parenleftbigg
+−κ12
+λ+
+2−λ−
+1|n1/an}bracketri}ht+κ11
+λ+
+1−λ−
+1|n2/an}bracketri}ht+f12
+λ+
+2−λ+
+1S0|n∗
+1/an}bracketri}ht/parenrightbigg
+,(45)
+where
+Z=/parenleftbigg|f12|2
+|λ+
+2−λ+
+1|2−κ12κ21
+|λ+
+2−λ−
+1|2+κ11κ22
+4ν1ν2/parenrightbigg
+. (46)
+Inserting this result into eq. ( 30) we obtain the following expression for the
+2-soliton solution of the MNLS:
+Q2s(x,t) = [J,A1+B1+A2+B2] =1
+Z[J,C(x,t)−S0CT(x,t)S0],
+C(x,t) =κ22
+λ+
+2−λ−
+2|n1/an}bracketri}ht/an}bracketle{tn†
+1|−κ12
+λ+
+2−λ−
+1|n1/an}bracketri}ht/an}bracketle{tn†
+2|−κ21
+λ+
+1−λ−
+2|n2/an}bracketri}ht/an}bracketle{tn†
+1|
++κ11
+λ+
+1−λ−
+1|n2/an}bracketri}ht/an}bracketle{tn†
+2|−f∗
+12
+λ−
+1−λ−
+2|n1/an}bracketri}ht/an}bracketle{tn2|S0−f12
+λ+
+1−λ+
+2S0|n∗
+2/an}bracketri}ht/an}bracketle{tn†
+1|.(47)
+4.The minimal sets of scattering data. It is well known that the locations of
+the singularitiesofthe RHP λ±
+k∈C±arezeroesofthe functions m±
+1(λ) anddiscrete
+eigenvalues of the Lax operator L. We will say that these eigenvalues are simple
+if the corresponding eigensubspaces are one dimensional. This corr esponds to our
+choice ofFk(x,t) andGk(x,t) as vectors. Eigensubspaces of higher multiplicities
+s >1 can be obtained choosing Fk(x,t) andGk(x,t) ass×(2r+ 1) matrices of
+ranks.10 VLADIMIR S. GERDJIKOV
+Theorem 4.1. Let the potential Q(x,t)be such that the corresponding Lax operator
+Lhas finite number of simple discrete eigenvalues located at t he pointsλ±
+k∈C±
+respectively, k= 1,...,N. Then as minimal sets of scattering data uniquely de-
+termining both the scattering matrix T(λ,t)and the corresponding potential Q(x,t)
+one can consider the sets
+T1≡ {/vector τ+(λ,0), λ∈R;/vector τ+
+k, λ+
+k∈C+, k= 1,...N},
+T2≡ {/vector ρ±(λ,0), λ∈R;/vector ρ+
+k, λ+
+k∈C+, k= 1,...N},(48)
+where the constant vectors /vector τ+
+0kand/vector ρ±
+0k
+/vector τ+
+0k=
+eξ0k−iδ0k√
+2/vector ν0k
+e−ξ0k+iδ0k
+, /vector ρ+
+0k=
+eη0k−iθ0k√
+2/vector µ0k
+e−η0k+iθ0k
+, (49)
+and the vectors /vector ν0kand/vector µ0ksatisfy the normalization condition (/vector νT
+0ks0/vector ν0k) = 1and
+(/vector µT
+0ks0/vector µ0k) = 1.
+Remark 4. The dataλ+
+kandλ−
+k= (λ+
+k)∗characterize the discrete eigenvalues of
+L. The vectors /vector τ+
+0kand/vector τ−
+0k= (/vector τ+
+0k)∗(resp./vector ρ+
+0kand/vector ρ−
+0k= (/vector ρ+
+0k)∗) determine the
+corresponding eigenfunction of L. Note also that by definition these vectors satisfy
+(/vector τ+,T
+0ks0/vector τ+
+0k) = 0 and ( /vector ρ+,T
+0ks0/vector ρ+
+0k) = 0.
+Outline of the proof. Let us be given T1. Using/vector τ+(λ,t) and/vector τ−(λ,t) = (/vector τ+(λ,t))∗
+we construct S+
+0J(λ,t) andS−
+0J(λ,t) and therefore obtain also the sewing function
+G0(λ,t) =ˆS−
+0J(λ,t)S+
+0J(λ,t) for a regular RHP. According to the Zakharov-Shabat
+theorem it has unique solution ξ±
+0(x,t,λ). The corresponding regular potential is
+obtained by:
+Q0(x,t) = lim
+λ→∞λ/parenleftig
+J−ξ±
+0(x,t,λ)Jˆξ±
+0(x,t,λ)/parenrightig
+= [J,ξ+
+01(x,t)],(50)
+whereξ+
+01(x,t) = lim λ→∞λ(ξ+
+0(x,t,λ)−11).
+Next we use the dressing method to dress the regular solution ξ±
+0(x,t,λ) with
+the dressing factor u(x,t,λ) of the form ( 29). In order to do it we make use of the
+set of eigenvalues λ+
+kandλ−
+k= (λ+
+k)∗and instead of the polarization vectors ( 34)
+we use:
+|nk(x,t)/an}bracketri}ht=ξ+
+0(x,t,λ+
+k)e−iλ+
+k(x+λ+
+kt)J/vector τ0k+,
+|n∗
+k(x,t)/an}bracketri}ht=ξ−
+0(x,t,λ−
+k)e−iλ−
+k(x+λ−
+kt)J/vector τ0k−.(51)
+After solving the algebraic equations for |Xk(x,t)/an}bracketri}htand|Yk(x,t)/an}bracketri}htwe find explicitly
+the dressed potential
+Q(x,t) =Q0(x,t)+N/summationdisplay
+k=1[J,Ak(x,t)+Bk(x,t)], (52)
+which proves the first part of the theorem.
+Let us now show how one can recover T(λ,t) fromT1. Given the regular solution
+ξ±
+0(x,t,λ) we can find
+D±
+0,J(λ) = lim
+x→∞ξ±
+0(x,t,λ), (53)
+and also
+T∓
+0,J(λ)D±
+0,J(λ) = lim
+x→∞ei(λx+λ2t)Jξ±
+0(x,t,λ)e−i(λx+λ2t)J. (54)BEC AND SPECTRAL PROPERTIES OF MNLS 11
+Thus we have recovered all Gauss factors T∓
+0,J(λ),D±
+0,J(λ) andS±
+0,J(λ) of the ‘un-
+dressed’ scattering matrix T0(λ,t), so
+T0,J(λ,t) =T∓
+0,J(λ,t)D±
+0,J(λ)ˆS±
+0,J(λ,t). (55)
+In order to take into account the effect of dressing we make use of the relations
+between the dressed and undressed Jost solutions:
+ψ(x,t,λ) =u(x,t,λ)ψ0(x,t,λ)ˆu+(λ),
+φ(x,t,λ) =u(x,t,λ)φ0(x,t,λ)ˆu−(λ),(56)
+whereu±(λ) = lim x→±∞u(x,t,λ). As a result we get:
+T(λ,t) =ˆψ(x,t,λ)φ(x,t,λ)
+=u+(λ)ˆψ0(x,t,λ)φ0(x,t,λ)ˆu−(λ)
+=u+(λ)T0(λ,t)ˆu−(λ).(57)
+Skipping the details we state the result:
+u+(λ) =
+c(λ) 0 0
+011 0
+0 0 1/c(λ)
+, u −(λ) =
+1/c(λ) 0 0
+011 0
+0 0c(λ)
+,(58)
+wherec(λ) =/producttextN
+j=1λ−λ+
+j
+λ−λ−
+j.
+The fact that the set T2is also a minimal set of scattering data is proved analo-
+gously.
+5.Reductions of MNLS. Along with the typical reduction Q=Q†mentioned
+above one can impose additional reductions using the reduction gro up proposed by
+Mikhailov [ 21]. They are automatically compatible with the Lax representation of
+the corresponding MNLS eq. Below we make use of two types of Z2-reductions[ 8]:
+1)C1U†(x,t,λ∗)C−1
+1=U(x,t,λ), C 1V†(x,t,λ∗)C−1
+1=V(x,t,λ),
+2)C2UT(x,t,λ)C−1
+2=−U(x,t,λ), C 2VT(x,t,λ)C−1
+2=−V(x,t,λ),
+(59)
+whereC1andC2are involutions of the Lie algebra so(2r+1, i.e.C2
+i=11. They can
+be chosen to be either diagonal(i.e., elements ofthe Cartansubgro upofSO(2r+1))
+or elements of the Weyl group.
+The typical reductions of the MNLS eqs. is a class 1) reduction obta ined by
+specifyingC1to be the identity automorphism of g; below we list several choices
+forC1leading to inequivalent reductions:
+1a)C1=11, /vector p(x) =/vector q∗(x), 1b)C1=K1, /vector p(x) =K01/vector q∗(x),
+1c)C1=Se2, /vector p(x) =K02/vector q∗(x),1d)C1=Se2Se3, /vector p(x) =K03/vector q∗(x).
+(60)
+We also make use of type 2) reductions:
+2e)C2=K4, /vector q(x) =−K04s0/vector q(x), /vector p (x) =−K04s0/vector p(x),
+2f)C2=K5, /vector q(x) =K05/vector q(x), /vector p (x) =K05/vector p(x),(61)
+where
+Kj= block-diag(1 ,K0j,1), K 01= diag(ǫ1,...,ǫ r−1,1,ǫr−1,...,ǫ 1),(62)12 VLADIMIR S. GERDJIKOV
+forj= 1,2,3,5 andǫj=±1. The matrices K02,K03andK4are not diagonal and
+may take the form:
+K02=
+0 0 1
+0−1 0
+1 0 0
+, K 4=
+0 0 1
+0K040
+1 0 0
+,
+K02=
+0 0 0 0 −1
+0 1 0 0 0
+0 0−1 0 0
+0 0 0 1 0
+−1 0 0 0 0
+, K 03=
+0 0 0 0 −1
+0 0 0 1 0
+0 0−1 0 0
+0 1 0 0 0
+−1 0 0 0 0
+.(63)
+Each of the above reductions impose constraints on the FAS, on th e scattering
+matrixT(λ) and on its Gauss factors S±
+J(λ),T±
+J(λ) andD±
+J(λ). For the type 1
+reductions (cases 1a) – 1d)) these have the form:
+(S+(λ∗))†=K−1
+jˆS−(λ)Kj(T+(λ∗))†=K−1
+jˆT−(λ)Kj
+(D+(λ∗))†=K−1
+jˆD−(λ)Kj
+/vector τ+=K0j/vector τ−,∗, /vector ρ+=K0j/vector ρ−,∗, j = 1,2,3(64)
+where the matrices Kjare specific for each choice of the automorphisms C1, see
+eqs. (60). In particular, from the last line of ( 64) and (61) we get:
+(m+
+1(λ∗))∗=m−
+1(λ), (65)
+and consequently, if m+
+1(λ) has zeroes at the points λ+
+k, thenm−
+1(λ) has zeroes at:
+λ−
+k= (λ+
+k)∗, k= 1,...,N. (66)
+For the type 2) reductions we obtain:
+2e) (S±(λ))T=K−1
+4ˆS±(λ)K4(T±(λ))T=K−1
+4ˆT±(λ)K4
+(D±(λ))T=K−1
+4ˆD±(λ)K4
+/vector τ±=−K04s0/vector τ±, /vector ρ±=−K04s0/vector ρ±,(67)
+and
+2f) (S+(λ))T=K−1
+5ˆS−(−λ)K5(T+(λ))T=K−1
+5ˆT−(−λ)K5
+(D+(λ))T=K−1
+5ˆD−(−λ)K5
+/vector τ+(λ) =K05/vector τ−(−λ), /vector ρ+(λ) =−K05/vector ρ−(−λ),(68)
+For the 2e) reduction with n= 3 we may choose K4to corresponds to the Weyl
+group element Se1, soK04=11. As a result we get:
+Φ1=−Φ−1 (69)
+and Φ 0arbitrary. This reduction of eq. ( 1) is also important for the BEC [ 22].
+From (67) we findν01=ν03. The effect of this constraint is that for the one-soliton
+solution we get Φ 1s;1=−Φ1s;−1.
+Our next remark following [ 23] is that this reduction applied to the F= 1 MNLS
+(1) leads to a 2-component MNLS which after the change of variables
+Φ1=1
+2(w1+iw2),Φ0=i√
+2(w1−iw2), (70)
+leads to two disjoint NLS equations for w1andw2respectively.BEC AND SPECTRAL PROPERTIES OF MNLS 13
+It is only logical that applying the constraint ν01=ν03the explicit expression for
+the one-soliton solution ( 38) simplifies and reduces to the standard soliton solutions
+of the scalar NLS.
+For the other two examples of type 2) reductions we choose n= 5 andK4, and
+K5correspond to the Weyl group elements Se2Se3andSe2−e3respectively. Then
+K04=−s0and
+K05=
+0 1 0 0 0
+1 0 0 0 0
+0 0 1 0 0
+0 0 0 0 −1
+0 0 0 −1 0
+, (71)
+For these choices of K4,K5we obtain:
+2e) Φ 2= Φ−2, Φ1= Φ−1,
+2f) Φ ±2=±c√
+1+c2Φ′
+±1,Φ±1=1√
+1+c2Φ′
+±1,(72)
+It reduces the F= 2 spin BEC model into the F= 1 model.
+The correspondingrelationsfor the Gaussfactorsand forthe po larizationvectors
+are given by:
+Φ±2=±c√
+1+c2Φ′
+±1,Φ±1=1√
+1+c2Φ′
+±1, (73)
+6.Two Soliton interactions. In this section we generalize the classical results of
+Zakharovand Shabat about soliton interactions [ 31] to the class of MNLS equations
+related to BD.I symmetric spaces. For detailed exposition see the mo nographs
+[29,3]. These results were generalized for the vector nonlinear Schr¨ od ingerequation
+by Manakov [ 20], see also [ 1,16,28]. The Zakharov Shabat approach consisted
+in calculating the asymptotics of generic N-soliton solution of NLS for t→ ±∞
+and establishing the pure elastic character of the generic soliton int eractions. By
+generic here we mean N-soliton solution whose parameters λ±
+k=µk±iνkare such
+thatµk/ne}ationslash=µjfork/ne}ationslash=j. The pure elastic character of the soliton interactions is
+demonstrated by the fact that for t→ ±∞the generic N-soliton solution splits
+into sum of None soliton solutions each preserving its amplitude 2 νkand velocity
+µk. The only effect of the interaction consists in shifting the center of mass and the
+initial phase of the solitons. These shifts can be expressed in terms ofλ±
+konly; for
+detailed exposition see [ 3].
+Let us apply these ideas to the MNLS equations studied above. Name ly we use
+the 2-soliton solution ( 47) derived above and calculate its asymptotics along the
+trajectory of the first soliton. To this end we keep z1(x,t) fixed and let τ=z2−z1
+tend to±∞. Therefore it will be enough to insert the asymptotic values of the
+matrix elements of Mforτ→ ±∞and keep only the leading terms. For τ→ ∞
+that gives:
+κ22≃e2τexp(ν2z1/ν1)+2C1,
+κ12=eτexp((1+ν2/ν1)z1+i(φ1−φ2))+O(1),
+κ21=eτexp((1+ν2/ν1)z1−i(φ1−φ2))+O(1),
+f12=eτexp(−(1−ν2/ν1)z1+i(φ1−φ2))+O(1),(74)14 VLADIMIR S. GERDJIKOV
+while forτ→ −∞we get:
+κ22≃e−2τexp(−ν2z1/ν1)+2C1,
+κ12=e−τexp(−(1+ν2/ν1)z1−i(φ1−φ2))+O(1),
+κ21=e−τexp(−(1+ν2/ν1)z1+i(φ1−φ2))+O(1),
+f12=e−τexp((1−ν2/ν1)z1−i(φ1−φ2))+O(1),(75)
+After somewhat lengthy calculations we get:
+lim
+τ→∞/vector q2s(x,t) =−i√
+2ν1e−i(φ1−α+)(e−z1−r+s0|/vector ν01/an}bracketri}ht+ez1+r+|/vector ν∗
+01/an}bracketri}ht)
+cosh(2(z1+r+))+(/vector ν†
+01,/vector ν01),
+lim
+τ→−∞/vector q2s(x,t) =i√
+2ν1e−i(φ1+α+)(e−z1+r+s0|/vector ν01/an}bracketri}ht+ez1−r+|/vector ν∗
+01/an}bracketri}ht)
+cosh(2(z1−r+))+(/vector ν†
+01,/vector ν01),(76)
+where
+r+= ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleλ+
+1−λ+
+2
+λ+
+1−λ−
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, α += argλ+
+1−λ+
+2
+λ+
+1−λ−
+2.
+Forn= 3 andn= 5 the right hand sides of ( 76) coincide with the one-soliton
+solutions ( 38) and (39) respectively. This means that the 2-soliton interaction for
+the above MNLS eqs. is purely elastic. The solitons preserve their sh apes and
+velocities and the only effect of the interaction consist in shifts of th e center of mass
+and the phase. From this point of view the interaction is the same like f or the scalar
+NLS eq.
+It is important to check whether the N-soliton interactions consist of sequence
+of elementary 2-soliton interactions and the shifts are additive.
+7.Effects of reductions and initial conditions on MNLS.
+Theorem 7.1. Let the minimal set of scattering data Tj,j= 1,2fort= 0satisfy
+the reduction conditions ( 67). Then the solution /vector q(x,t)of the MNLS with such
+initial data will satisfy the corresponding reduction 2e) ( 61).
+Proof.Let the minimal sets of scattering data, say T1satisfy the reduction condi-
+tions (67) fort= 0. It is easy to check that their evolution law ( 15) is compatible
+with the reduction, so ( 67) will hold for all t >0. As a result the corresponding
+Gauss factors S±,T±andD±, and consequently, the sewing function in the RHP
+G(x,t,λ) will satisfy
+G(x,t,λ) =K−1
+4ˆGT(x,t,λ)K4. (77)
+Thenextconsequenceisthatboth ξ±andK−1
+4ˆξ±,TK4aresolutionsoftheRHP( 23)
+with the same sewing function and the same canonical normalization. Therefore
+from the uniqueness of the solution of RHP we get that the regular s olutions of this
+RHP satisfy:
+ξ±
+0(x,t,λ) =K−1
+4ˆξ±,T
+0(x,t,λ)K4. (78)
+Next we note that the scattering data related to the discrete spe ctrum also satisfy
+the reduction conditions. This means that the dressing factor u(x,t,λ) and the
+singular solutions ξ±(x,t,λ) =u(x,t,λ)ξ±
+0(x,t,λ)ˆu−(λ) also satisfy:
+u(x,t,λ) =K−1
+4ˆuT(x,t,λ)K4, ξ±(x,t,λ) =K−1
+4ˆξ±,T(x,t,λ)K4.(79)
+It remains to check that from equations ( 27) and (79) there follows:
+Q(x,t) =−K−1
+4QT(x,t)K4. (80)BEC AND SPECTRAL PROPERTIES OF MNLS 15
+Remark 5. Note that the above arguments are not specific for the choice of K4.
+The above theorem can be proved along the same lines for any reduc tion of type 1
+and type 2.
+A simple consequence of the above theorem is the following. Consider n= 3 and
+chooseτ+
+1=τ+
+3fort= 0. Then the corresponding solution of F= 1 BEC ( 1) will
+also satisfy Φ 1=−Φ−1for allt>0, i.e. will be a solution to
+i∂tΦ1+∂2
+xΦ1+2(|Φ1|2+2|Φ0|2)Φ1−2Φ∗
+1Φ2
+0= 0,
+i∂tΦ0+∂2
+xΦ0+2(2|Φ1|2+|Φ0|2)Φ0−2Φ∗
+0Φ2
+1= 0,(81)
+If we insert eq. ( 70) into (81) we obtain
+i∂tw1+∂2
+xw1+2|w1|2w1= 0,
+i∂tw2+∂2
+xw2+2|w2|2w2= 0,(82)
+Therefore, if we want to analyze the specific features of F= 1 BEC we have to
+avoid such initial conditions.
+Similarly, if for n= 5 we choose in ( 39)ν01;1=ν01,5,ν01;2=−ν01,4we will
+obtain in fact a solution to F= 1 BEC.
+8.Conclusions and discussion. Using the Zakharov-Shabat dressing method
+we have obtained the two-soliton solution and have used it to analyze the soliton
+interactionsof the MNLS equation. The conclusion is that after the interactionsthe
+solitons recover their polarization vectors ν0k, velocities and frequency velocities.
+The effect of the interaction is, like in for the scalar NLS equation, sh ift of the
+center of mass z1→z1+r+and shift of the phase φ1→φ1+α+. Both shifts are
+expressed through the related eigenvalues λ±
+jonly.
+The next step would be to analyze multi-soliton interactions. Our hyp othesis is
+that each soliton will acquire a total shift of the center of mass tha t is sum of all
+elementary shifts from each two soliton interactions. Similar result is expected for
+the total phase shift of the soliton.
+Finally we have proved a theorem, stating that a symmetry imposed o n the
+minimal set of scattering data leads to a symmetry of the correspo nding solution.
+So if we want to analyze the specific features of a given MNLS we have to avoid
+such initial conditions.
+Acknowledgements. I am grateful to Professors J. C. Maraver, P. G. Kevrekidis
+andR. Carretero-Gonzalesfor givingme the chance totake partin this proceedings.
+I also wish to thank Professor N. Kostov and Dr. T. Valchev for use ful discussions
+and the referee for useful remarks.
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+E-mail address :gerjikov@inrne.bas.bg
\ No newline at end of file
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+DRAFT VERSION 24 O CTOBER 2018
+Preprint typeset using L ATEX style emulateapj v. 08/22/09
+SEARCH FOR GRA VITATIONAL-WA VE INSPIRAL SIGNALS ASSOCIATED WITH SHORT GAMMA-RAY BURSTS
+DURING LIGO’S FIFTH AND VIRGO’S FIRST SCIENCE RUN
+J. A BADIE29, B. P. A BBOTT29, R. A BBOTT29, T. A CCADIA27, F. A CERNESE19AC, R. A DHIKARI29, P. A JITH29, B. A LLEN2,77, G. A LLEN52,
+E. A MADOR CERON77, R. S. A MIN34, S. B. A NDERSON29, W. G. A NDERSON77, F. A NTONUCCI22A, S. A OUDIA43A, M. A. A RAIN64,
+M. A RAYA29, K. G. A RUN26, Y. A SO29, S. A STON63, P. A STONE22A, P. A UFMUTH28, C. A ULBERT2, S. B ABAK1, P. B AKER37,
+G. B ALLARDIN12, S. B ALLMER29, D. B ARKER30, F. B ARONE19AC, B. B ARR65, P. B ARRIGA76, L. B ARSOTTI32, M. B ARSUGLIA4, M.A.
+BARTON30, I. B ARTOS11, R. B ASSIRI65, M. B ASTARRIKA65, TH. S. B AUER41A, B. B EHNKE1, M.G. B EKER41A, A. B ELLETOILE27,
+M. B ENACQUISTA59, J. B ETZWIESER29, P. T. B EYERSDORF48, S. B IGOTTA21AB, I. A. B ILENKO38, G. B ILLINGSLEY29, S. B IRINDELLI43A,
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+S. B OSE78, L. B OSI20A, S. B RACCINI21A, C. B RADASCHIA21A, P. R. B RADY77, V. B. B RAGINSKY38, J. E. B RAU70, J. B REYER2,
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+E. E. D OOMES51, M. D RAGO44CD, R. W. P. D REVER6, J. D RIGGERS29, J. D UECK2, I. D UKE32, J.-C. D UMAS76, M. E DGAR65,
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+R. S CHOFIELD70, B. S CHULZ2, B. F. S CHUTZ1,8, P. S CHWINBERG30, J. S COTT65, S. M. S COTT5, A. C. S EARLE29, F. S EIFERT2,29,
+D. S ELLERS31, A. S. S ENGUPTA29, D. S ENTENAC12, A. S ERGEEV24, B. S HAPIRO32, P. S HAWHAN66, D. H. S HOEMAKER32, A. S IBLEY31,
+X. S IEMENS77, D. S IGG30, A. M. S INTES61, G. S KELTON77, B. J. J. S LAGMOLEN5, J. S LUTSKY34, J. R. S MITH53, M. R. S MITH29,
+N. D. S MITH32, K. S OMIYA7, B. S ORAZU65, A. J. S TEIN32, L. C. S TEIN32, S. S TEPLEWSKI78, A. S TOCHINO29, R. S TONE59,
+K. A. S TRAIN65, S. S TRIGIN38, A. S TROEER39, R. S TURANI17AB, A. L. S TUVER31, T. Z. S UMMERSCALES3, M. S UNG34, S. S USMITHAN76,
+P. J. S UTTON8, B. S WINKELS12, G. P. S ZOKOLY14, D. T ALUKDER78, D. B. T ANNER64, S. P. T ARABRIN38, J. R. T AYLOR2, R. T AYLOR29,
+K. A. T HORNE31, K. S. T HORNE7, A. T H¨URING28, C. T ITSLER54, K. V. T OKMAKOV65,75, A. T ONCELLI21AB, M. T ONELLI21AB,
+C. T ORRES31, C. I. T ORRIE29,65, E. T OURNEFIER27, F. T RAVASSO20AB, G. T RAYLOR31, M. T RIAS61, J. T RUMMER27, L. T URNER29,
+D. U GOLINI60, K. U RBANEK52, H. V AHLBRUCH28, G. V AJENTE21AB, M. V ALLISNERI7, J. F. J. VAN DEN BRAND41AB,
+C. V ANDENBROECK8, S. VAN DER PUTTEN41A, M. V. VAN DER SLUYS42, S. V ASS29, R. V AULIN77, M. V AVOULIDIS26, A. V ECCHIO63,
+G. V EDOVATO44C, A. A. VAN VEGGEL65, J. V EITCH63, P. J. V EITCH62, C. V ELTKAMP2, D. V ERKINDT27, F. V ETRANO17AB,
+A. V ICER ´E17AB, A. V ILLAR29, J.-Y. V INET43A, H. V OCCA20A, C. V ORVICK30, S. P. V YACHANIN38, S. J. W ALDMAN32, L. W ALLACE29,
+A. W ANNER2, R. L. W ARD29, M. W AS26, P. W EI53, M. W EINERT2, A. J. W EINSTEIN29, R. W EISS32, L. W EN7,76, S. W EN34, P. W ESSELS2,
+M. W EST53, T. W ESTPHAL2, K. W ETTE5, J. T. W HELAN46, S. E. W HITCOMB29, B. F. W HITING64, C. W ILKINSON30, P. A. W ILLEMS29,
+H. R. W ILLIAMS54, L. W ILLIAMS64, B. W ILLKE2,28, I. W ILMUT47, L. W INKELMANN2, W. W INKLER2, C. C. W IPF32, A. G. W ISEMAN77,
+G. W OAN65, R. W OOLEY31, J. W ORDEN30, I. Y AKUSHIN31, H. Y AMAMOTO29, K. Y AMAMOTO2, D. Y EATON -MASSEY29, S. Y OSHIDA50,
+P. P. Y U77, M. Y VERT27, M. Z ANOLIN13, L. Z HANG29, Z. Z HANG76, C. Z HAO76, N. Z OTOV35, M. E. Z UCKER32, J. Z WEIZIG29
+The LIGO Scientific Collaboration & The Virgo Collaboration
+Draft version 24 October 2018
+ABSTRACT
+Progenitor scenarios for short gamma-ray bursts (short GRBs) include coalescenses of two neutron stars or a
+neutron star and black hole, which would necessarily be accompanied by the emission of strong gravitational
+waves. We present a search for these known gravitational-wave signatures in temporal and directional coincidence
+with 22 GRBs that had sufficient gravitational-wave data available in multiple instruments during LIGO’s fifth
+science run, S5, and Virgo’s first science run, VSR1. We find no statistically significant gravitational-wave
+candidates within a [5;+1)swindow around the trigger time of any GRB. Using the Wilcoxon–Mann–
+Whitney U test, we find no evidence for an excess of weak gravitational-wave signals in our sample of GRBs.
+We exclude neutron star–black hole progenitors to a median 90% confidence exclusion distance of 6.7 Mpc.
+Subject headings: compact object mergers – gamma-ray bursts – gravitational waves
+1Albert-Einstein-Institut, Max-Planck-Institut f ¨ur Gravitationsphysik, D-
+14476 Golm, Germany
+2Albert-Einstein-Institut, Max-Planck-Institut f ¨ur Gravitationsphysik, D-
+30167 Hannover, Germany
+3Andrews University, Berrien Springs, MI 49104 USA
+4AstroParticule et Cosmologie (APC), CNRS: UMR7164-IN2P3-
+Observatoire de Paris-Universit ´e Denis Diderot-Paris 7 - CEA : DSM/IRFU
+5Australian National University, Canberra, 0200, Australia
+6California Institute of Technology, Pasadena, CA 91125, USA
+7Caltech-CaRT, Pasadena, CA 91125, USA
+8Cardiff University, Cardiff, CF24 3AA, United Kingdom
+9Carleton College, Northfield, MN 55057, USA
+10Charles Sturt University, Wagga Wagga, NSW 2678, Australia
+11Columbia University, New York, NY 10027, USA
+12European Gravitational Observatory (EGO), I-56021 Cascina (Pi), Italy
+13Embry-Riddle Aeronautical University, Prescott, AZ 86301 USA
+14E¨otv¨os University, ELTE 1053 Budapest, Hungary
+15ESPCI, CNRS, F-75005 Paris, France
+16Hobart and William Smith Colleges, Geneva, NY 14456, USA
+17INFN, Sezione di Firenze, I-50019 Sesto Fiorentinoa; Universit `a degli
+Studi di Urbino ’Carlo Bo’, I-61029 Urbinob, Italy
+18INFN, Sezione di Genova; I-16146 Genova, Italy
+19INFN, sezione di Napolia; Universit `a di Napoli ’Federico II’bComp-
+lesso Universitario di Monte S.Angelo, I-80126 Napoli; Universit `a di Salerno,
+Fisciano, I-84084 Salernoc, Italy
+20INFN, Sezione di Perugiaa; Universit `a di Perugiab, I-6123 Perugia,Italy
+21INFN, Sezione di Pisaa; Universit `a di Pisab; I-56127 Pisa; Universit `a di
+Siena, I-53100 Sienac, Italy
+22INFN, Sezione di Romaa; Universit `a ’La Sapienza’b, I-00185 Roma,
+Italy
+23INFN, Sezione di Roma Tor Vergataa; Universit `a di Roma Tor Vergatab,
+Istituto di Fisica dello Spazio Interplanetario (IFSI) INAFc, I-00133 Roma;Universit `a dell’Aquila, I-67100 L’Aquilad, Italy
+24Institute of Applied Physics, Nizhny Novgorod, 603950, Russia
+25Inter-University Centre for Astronomy and Astrophysics, Pune - 411007,
+India
+26LAL, Universit ´e Paris-Sud, IN2P3/CNRS, F-91898 Orsay, France
+27Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP),
+IN2P3/CNRS, Universit ´e de Savoie, F-74941 Annecy-le-Vieux, France
+28Leibniz Universit ¨at Hannover, D-30167 Hannover, Germany
+29LIGO - California Institute of Technology, Pasadena, CA 91125, USA
+30LIGO - Hanford Observatory, Richland, WA 99352, USA
+31LIGO - Livingston Observatory, Livingston, LA 70754, USA
+32LIGO - Massachusetts Institute of Technology, Cambridge, MA 02139,
+USA
+33Laboratoire des Mat ´eriaux Avanc ´es (LMA), IN2P3/CNRS, F-69622
+Villeurbanne, Lyon, France
+34Louisiana State University, Baton Rouge, LA 70803, USA
+35Louisiana Tech University, Ruston, LA 71272, USA
+36McNeese State University, Lake Charles, LA 70609 USA
+37Montana State University, Bozeman, MT 59717, USA
+38Moscow State University, Moscow, 119992, Russia
+39NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
+40National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
+41Nikhef, National Institute for Subatomic Physics, P.O. Box 41882, 1009
+DBa, VU University Amsterdam, De Boelelaan 1081, 1081 HVbAmsterdam,
+The Netherlands
+42Northwestern University, Evanston, IL 60208, USA
+43Universit ´e Nice-Sophia-Antipolis, CNRS, Observatoire de la C ˆote
+d’Azur, F-06304 Nicea; Institut de Physique de Rennes, CNRS, Univer-
+sit´e de Rennes 1, 35042 Rennesb; France
+44INFN, Gruppo Collegato di Trentoaand Universit `a di Trentob, I-38050
+Povo, Trento, Italy; INFN, Sezione di Padovacand Universit `a di Padovad,
+I-35131 Padova, Italy3
+1.INTRODUCTION
+The past decade has seen dramatic progress in the under-
+standing of gamma-ray bursts (GRBs), intense flashes of 
-
+rays that are observed to be isotropically distributed over the
+sky (see e.g. Klebesadel et al. 1973; M ´esz´aros 2006, and refer-
+ences therein). The short-time variability of the bursts indicates
+that the sources are very compact. They are observed directly
+by
-ray and X-ray satellites in the Interplanetary Network
+(IPN) such as HETE, Swift, Konus–Wind, INTEGRAL, and
+Fermi (see Ricker et al. 2003; Gehrels et al. 2004; Aptekar et al.
+1995; Winkler et al. 2003; Atwood et al. 2009, and references
+therein).
+GRBs are usually divided into two types (see Kouveliotou
+et al. 1993; Gehrels et al. 2006), distinguished primarily by the
+duration of the prompt burst. Long-duration bursts, with a du-
+ration of &2 s, are generally associated with hypernova explo-
+sions in star-forming galaxies. Several nearby long GRBs have
+been spatially and temporally coincident with core-collapse
+supernovae as observed in the optical (Campana et al. 2006;
+Galama et al. 1998; Hjorth et al. 2003; Malesani et al. 2004).
+Follow-up observations by X-ray, optical, and radio telescopes
+of the sky near GRBs have yielded detailed measurements
+of afterglows from more than 500 GRBs to date; some of
+these observations resulted in strong host galaxy candidates,
+which allowed redshift determination for more than 200 bursts
+(Greiner 2009).
+Short GRBs, with a duration .2 s, are thought to originate
+45IM-PAN 00-956 Warsawa; Warsaw Univ. 00-681 Warsawb; Astro. Obs.
+Warsaw Univ. 00-478 Warsawc; CAMK-PAN 00-716 Warsawd; Białystok
+Univ. 15-424 Bial ystoke; IPJ 05-400 ´Swierk-Otwockf; Inst. of Astronomy
+65-265 Zielona G ´orag, Poland
+46Rochester Institute of Technology, Rochester, NY 14623, USA
+47Rutherford Appleton Laboratory, HSIC, Chilton, Didcot, Oxon OX11
+0QX United Kingdom
+48San Jose State University, San Jose, CA 95192, USA
+49Sonoma State University, Rohnert Park, CA 94928, USA
+50Southeastern Louisiana University, Hammond, LA 70402, USA
+51Southern University and A&M College, Baton Rouge, LA 70813, USA
+52Stanford University, Stanford, CA 94305, USA
+53Syracuse University, Syracuse, NY 13244, USA
+54The Pennsylvania State University, University Park, PA 16802, USA
+55The University of Melbourne, Parkville VIC 3010, Australia
+56The University of Mississippi, University, MS 38677, USA
+57The University of Sheffield, Sheffield S10 2TN, United Kingdom
+58The University of Texas at Austin, Austin, TX 78712, USA
+59The University of Texas at Brownsville and Texas Southmost College,
+Brownsville, TX 78520, USA
+60Trinity University, San Antonio, TX 78212, USA
+61Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
+62University of Adelaide, Adelaide, SA 5005, Australia
+63University of Birmingham, Birmingham, B15 2TT, United Kingdom
+64University of Florida, Gainesville, FL 32611, USA
+65University of Glasgow, Glasgow, G12 8QQ, United Kingdom
+66University of Maryland, College Park, MD 20742 USA
+67University of Massachusetts - Amherst, Amherst, MA 01003, USA
+68University of Michigan, Ann Arbor, MI 48109, USA
+69University of Minnesota, Minneapolis, MN 55455, USA
+70University of Oregon, Eugene, OR 97403, USA
+71University of Rochester, Rochester, NY 14627, USA
+72University of Salerno, 84084 Fisciano (Salerno), Italy
+73University of Sannio at Benevento, I-82100 Benevento, Italy
+74University of Southampton, Southampton, SO17 1BJ, United Kingdom
+75University of Strathclyde, Glasgow, G1 1XQ, United Kingdom
+76University of Western Australia, Crawley, WA 6009, Australia
+77University of Wisconsin–Milwaukee, Milwaukee, WI 53201, USA
+78Washington State University, Pullman, WA 99164, USAprimarily in the coalescence of a neutron star (NS) with another
+compact object (see, e.g., Nakar 2007, and references therein),
+such as a neutron star or black hole (BH). There is growing
+evidence that finer distinctions may be drawn between bursts
+(Zhang et al. 2007; Bloom et al. 2008); for example, it is
+estimated that up to 15% of short GRBs could be associated
+with soft gamma repeaters (Nakar et al. 2006; Chapman et al.
+2009), which emit bursts of X-rays and gamma rays at irregular
+intervals with lower fluence than compact binary coalescence
+engines (Hurley et al. 2005; Palmer et al. 2005).
+In the compact binary coalescence model of short GRBs, a
+neutron star and compact companion in otherwise stable orbit
+lose energy to gravitational waves and inspiral. The neutron
+star(s) tidally disrupt shortly before coalescence, providing
+matter, some of which is ejected in relativistic jets. The prompt
+
-ray emission is widely thought to be created by internal
+shocks, the interaction of outgoing matter shells at different
+velocities, while the afterglow is thought to be created by
+external shocks, the interaction of the outflowing matter with
+the interstellar medium (M ´esz´aros 2006; Nakar 2007). If the
+speed of gravitational radiation equals the speed of light as
+we expect, then for an observer in the cone of the collimated
+outflow, the gravitational-wave inspiral signal will arrive a few
+seconds before the electromagnetic signal from internal shocks.
+Several semi-analytical calculations of the final stages of a NS–
+BH inspiral show that the majority of matter plunges onto the
+BH within 1 s(Davies et al. 2005). Numerical simulations on
+the mass transfer suggest a timescales of milliseconds (Shibata
+& Taniguchi 2008) or some seconds at maximum (Faber et al.
+2006). Also, it has been found in simulations that the vast
+majority of the NS matter is accreted onto the BH directly and
+promptly (within hundreds of ms) without a torus that gets
+accreted later (Rosswog 2006; Etienne et al. 2008).
+Compact binary coalescence is anticipated to generate strong
+gravitational waves in the sensitive frequency band of Earth-
+based gravitational-wave detectors (Thorne 1987). The direct
+detection of gravitational waves associated with a short GRB
+would provide direct evidence that the progenitor is indeed
+a compact binary; with such a detection it would be possi-
+ble to measure component masses (Cutler & Flanagan 1994;
+Finn & Chernoff 1993), measure component spins (Poisson
+& Will 1995), constrain NS equations of state (Flanagan &
+Hinderer 2008; Read et al. 2009), test general relativity in
+the strong-field regime (Will 2005), and measure calibration-
+free luminosity distance (Nissanke et al. 2009), which is a
+measurement of the Hubble expansion and dark energy.
+In this paper, we report on a search for gravitational-wave
+inspiral signals associated with the short GRBs that occurred
+during the fifth science run (S5) of LIGO (Abbott et al. 2009a),
+from 4 November 2005 to 30 September 2007, and the first
+science run (VSR1) of Virgo (Acernese et al. 2008), from 18
+May 2007 to 30 September 2007. S5 represents the combined
+operation of the three LIGO detectors, one Michelson inter-
+ferometer with 4 km long orthogonal arms at Livingston, LA,
+USA, named L1, and two interferometers located at Hanford,
+WA, USA, named H1 and H2, with lengths of 4 km and2 km
+respectively. VSR1 represents the operation of the Virgo in-
+terferometer located at Cascina, Italy, named V1, which has a
+length of 3 km . During the S5/VSR1 joint run, 212 GRBs were
+discovered by different satellite missions (39 of them during
+VSR1 times), 33 of which we classified as search targets (8 of
+them in VSR1 times). See section 2.2 for more details on the
+selected GRBs.
+A similar search in the same LIGO/Virgo data-set was per-4
+formed in Abbott et al. (2009c), looking for short-duration
+gravitational-wave bursts in association with 137 GRBs
+recorded during S5/VSR1, both long and short. The anal-
+ysis reported upper limits on the strain of a generic burst of
+circularly polarized gravitational radiation, predominantly at
+the detectors’ most sensitive frequencies. These were trans-
+lated into lower limits in distance by assuming that 0:01M
+is converted into isotropically emitted gravitational waves. In
+contrast, the search described in this paper does not make any
+assumption on the polarization of the gravitational waves and
+searches for the specific signals expected from binary coa-
+lescenses. Importantly, the present search can distinguish a
+coalescence signal from other models and estimate the progen-
+itor parameters.
+The remainder of the paper is organized as follows. In Sec-
+tion 2, we discuss the set of GRBs we chose for this analysis
+and outline our analysis methods. In Section 3, we present
+the results and astrophysical implications for the GRBs in our
+sample.
+2.SEARCH METHODS
+2.1.Experimental setup
+The binary coalescence model suggests that the time delay
+between the arrival of a gravitational wave and the arrival of
+the subsequent electromagnetic burst, referred to as trigger
+time, is a few seconds. We assessed uncertainties in reported
+trigger times and quantization in our own analysis along integer
+second boundaries, finding that these each contribute less than
+one second. For example, when the Swift BAT instrument
+determines that the count rate has risen above a threshold,
+it waits for the maximum to pass, checking with a 320 ms
+cadence (Gehrels et al. 2008); it reports the start time of the
+block containing the maximum, rather than making any attempt
+to identify the start of the burst, and does so with a 320 ms
+granularity. As another example, there have been reports of
+sub-threshold precursors to many GRBs (Burlon et al. 2009).
+For each GRB in our sample, we checked tens of seconds
+of light-curve by eye to look for both excessive difference
+between the trigger time and the apparent rise time, and also for
+precursors, but found nothing to suggest that we should correct
+the published trigger times. The largest timing uncertainty
+we identified is the delay between the compact merger and
+the prompt emission of the internal shocks. We search for
+gravitational-wave signals within an on-source segment of
+[5;+1)saround each trigger time for each GRB of interest,
+feeling that this window captures the physical model with some
+tolerance for its uncertainties.
+Because we believe that a gravitational wave associated with
+a GRB only occurs in the on-source segment, we use 324
+off-source trials, each 6 slong, to estimate the distribution of
+background due to the accidental coincidences of noise triggers.
+We also re-analyze the off-source trials with simulated signals
+added to the data to test the response of our search to signals;
+these we call injection trials. The actual number of off-source
+trials included in the analysis varied by GRB, as the trials that
+overlapped with data-quality vetoes were discarded (Abbott
+et al. 2009d). To prevent biasing our background estimation
+due to a potential loud signal in the on-source trial, the off-
+source segments do not use data within 48 sof the on-source
+segment, reflecting the longest duration of templates in our
+bank. Finally, we discard 72 sof data subject to filter transients
+on both ends of the off-source region. Taking all of these
+requirements into account, the minimum analyzable time isGRB Redshift Duration (s) References
+051114 . . . 2:2 G4272, G4275
+051210 . . . 1:2 G4315, G4321
+051211 . . . 4:8 G4324, G4359
+060121 . . . 2:0 G4550
+060313 <1:7 0 :7 G4867, G4873, G4877
+060427B . . . 2:0 G5030
+060429 . . . 0:25 G5039
+061006 . . . 0:50 G5699, G5704
+061201 . . . 0:80 G5881, G5882
+061217 0.827 0:30 G5926, G5930, G5965
+070201 . . . 0:15 G6088, G6103
+070209 . . . 0:10 G6086
+070429B . . . 0:50 G6358, G6365
+070512 . . . 2:0 G6408
+070707 . . . 1:1 G6605, G6607
+070714 . . . 2:0 G6622
+070714B 0.92 64:0 G6620, G6623, G6836
+070724 0.46 0:40 G6654, G6656, G6665
+070729 . . . 0:90 G6678, G6681
+070809 . . . 1:3 G6728, G6732
+070810B . . . 0:08 G6742, G6753
+070923 . . . 0:05 G6818, G6821
+Table 1
+Parameters of the 22 GRBs selected for this search. The values in the
+References column give the number of the GRB Coordinates Network (GCN)
+notice from which we took the preceding information (Barthelmy 2009).
+2190 s.
+2.2.Sample selection
+X-ray and
-ray instruments identified a total of 212 GRBs
+during the S5 run, 211 with measured durations; 30 of them
+have aT90duration smaller than 2 seconds .T90is the time in-
+terval over which 90% of all counts from a GRB are recorded.
+While theT90classifies a burst as long or short, it is not a
+definitive descriminator of progenitor systems. In addition
+to the short GRBs, GRB 051211 (Kawai et al. 2005) and
+GRB 070714B (Barbier et al. 2007) are formally long GRBs,
+but they have spectral features hinting at an underlying coales-
+cence progenitor. GRB 061210 is another long-duration burst,
+but it exhibits the typical short spikes of a short GRB (Can-
+nizzo et al. 2006). This gives a list of 33 interesting GRBs with
+which to search for an association with gravitational waves
+from compact binary coalescence.
+Around the trigger time of each interesting GRB, we re-
+quired 2190 s of multiply-coincident data. The detectors oper-
+ated with individual duty cycles of 67% to 81% over the span
+of the S5 and VSR1 runs. Where more than two detectors had
+sufficient data, we selected the most sensitive pair based on the
+average inspiral range, because including a third, less sensi-
+tive detector does not enhance the sensitivity greatly. The one
+exception was GRB 070923, described below. In descending
+order of sensitivity, the detectors are H1, L1, H2, and V1. This
+procedure yielded 11 GRBs searched for in H1–L1 coincident
+data, 9 GRBs in H1–H2, and 1 in H2–L1.
+In addition to these 21, we analyze GRB 070923 because of
+its sky location relative to the detectors’ antenna patterns. The
+antenna pattern changes with the location of a source relative to
+a detector and can be expressed by the responseq
+F2
+++F2
+,
+in whichF+andFdenote the antenna-pattern functions
+(Allen et al. 2005). A value of 1 corresponds to an optimal
+location of the putative gravitational-wave source relative to
+the observatory, while a value of 0 corresponds to a source
+location that will not induce any strain in the detector. For
+this particular GRB, the optimal antenna response for Virgo5
+is around 0.7, while those for the two LIGO sites are about
+half that (see Table 2), yielding a comparable sensitivity in
+the direction of GRB 070923 for all three of them. Data from
+H1, L1, and V1 were analyzed, making this the only GRB
+involving triple coincidences.
+Table 1 lists all 22 target GRBs after applying the selec-
+tion criteria described in this section. Plausible redshifts have
+been published for only three of these GRBs, placing them
+well outside of our detectors’ range, but short GRB redshift
+determinations are in general sufficiently tentative to warrant
+searching for all of these GRBs.
+GRB 070201 is also worth special mention. It was already
+analyzed in a high-priority search because of the striking
+spatial coincidence of this GRB with M31, a galaxy only
+780 kpc from Earth. No gravitational-wave signal was found
+and a coalescence scenario could be ruled out with >99%
+confidence at that distance (Abbott et al. 2008a), lending ad-
+ditional support for a soft gamma repeater hypothesis (Ofek
+et al. 2008). However, because of improvements in the analysis
+pipeline, we reanalyzed this GRB and report the results in this
+paper. See Section 3.1.1 for details.
+2.3.Candidate generation
+We generated candidates using the standard, untriggered
+compact binary coalescence search pipeline described in de-
+tail in Abbott et al. (2009e). The core of the inspiral search
+involves correlating the measured data against the theoreti-
+cal waveforms expected from compact binary coalescence, a
+technique called matched filtering (Helmstrom 1968). The
+gravitational waves from the inspiral phase, when the binary
+orbit decays under gravitational-wave emission prior to merger,
+are accurately modeled by post-Newtonian approximants in
+the band of the detector’s sensitivity for a wide range of binary
+masses (Blanchet 2006). The expected gravitational-wave sig-
+nal, as measured by LIGO and Virgo, depends on the masses
+(mNS;mcomp ) and spins ( ~ sNS;~ scomp ) of the neutron star and
+its companion (either NS or BH), as well as the spatial loca-
+tion (;), inclination angle , and polarization angle  of the
+orbital axis. In general, the power of matched filtering depends
+most sensitively on accurately tracking the phase evolution
+of the signal. The phasing of compact binary inspiral signals
+depends on the masses and spins, the time of merger, and an
+overall phase.
+We adopted a discrete bank of template waveforms that
+span a two-dimensional parameter space (one for each com-
+ponent mass) such that the maximum loss in signal to noise
+ratio (SNR) for a binary with negligible spins would be 3%
+(Cokelaer 2007). While the spin is ignored in the template
+waveforms, we verify that the search can still detect binaries
+with most physically reasonable spin orientations and magni-
+tudes with only moderate loss in sensitivity. For simplicity, the
+template bank is symmetric in component masses, spanning
+the range [2;40)Min total mass. The number of template
+waveforms required to achieve this coverage depends on the
+detector noise spectrum; for the data analyzed in this paper the
+number of templates was around 7000 for each detector.
+We filtered the data from each of the detectors through each
+template in the bank. If the matched filter SNR exceeds a
+threshold, the template masses and the time of the maximum
+SNR are recorded. For a given template, threshold crossings
+are clustered in time using a sliding window equal to the du-
+ration of the template (Allen et al. 2005). For each trigger
+identified in this way, the coalescence phase and the effective
+distance — the distance at which an optimally oriented andoptimally located binary, with masses corresponding to those
+of the template, would give the observed SNR — are also com-
+puted. Triggers identified in each detector are further required
+to be coincident with their time and mass parameters with a
+trigger from at least one other detector, taking into account
+the correlations between those parameters (Robinson et al.
+2008). This significantly reduces the number of background
+triggers that arise from matched filtering in each detector inde-
+pendently.
+The SNR threshold for the matched filtering step was chosen
+differently depending on which detectors’ data are available
+for a given GRB. If data from H1 and L1 were analyzed, the
+threshold for each detector was set to 4.25, reflecting their
+comparable sensitivity. If data from H1 and H2 were analyzed,
+the threshold of the latter detector — the less sensitive of the
+two — was set to 3.5 to gain maximum network sensitivity,
+while the threshold of the more sensitive detector, H1, was set
+to 5.5 since any signal seen in H2 would be twice as loud in
+H1, with some uncertainty. In the single case of analyzing only
+H2–L1 data (GRB 070707) the threshold was 4.25 for L1 and
+3.5 for H2, and for the single case of analyzing data with Virgo
+(GRB 070923), the threshold was set to 4.25 for all involved
+detectors (H1, L1, and V1). For comparison, a uniform SNR
+threshold of 5.5 was used in the untriggered S5 search (Abbott
+et al. 2009b).
+We applied two signal-based tests to reduce and refine our
+trigger sets. First, we computed a 2statistic (Allen 2005) to
+measure how different a trigger’s SNR integrand looks from
+that of a real signal; triggers with large 2were discarded. Sec-
+ond, we applied the r2veto (Rodr ´ıguez 2007) which discards
+triggers depending on the duration that the 2statistic stays
+above a threshold. The SNR and 2from a single detector
+were combined into an effective SNR (Abbott et al. 2008b).
+The effective SNRs from the analyzed detectors were then
+added in quadrature to form a single quantity 2
+ewhich pro-
+vided better separation between signal candidate events and
+background than SNR alone. The list of coincident triggers at
+this stage are then called candidate events .
+2.4.Ranking candidates
+The distribution of effective SNRs from background and
+from signals can vary strongly across the template bank, de-
+pending most strongly on the chirp mass, a combination of the
+two component masses that appears in the leading term of the
+signal amplitude and phase (Thorne 1987). For this reason, we
+refine our candidate ranking with a likelihood-ratio statistic,
+which we compute for every candidate in the on-source, off-
+source, and injection trials. In short, we define the likelihood
+ratioLfor a candidate cto be the efficiency divided by the
+false-alarm probability. The efficiency here is the probability
+of obtaining a candidate as loud or louder than c(by effec-
+tive SNR) within the same region of template space given a
+signal in the data. The efficiency is a function of the signal
+parametersmcomp andDand is marginalized over all other
+parameters; it is obtained by simply counting across injection
+trials. The false-alarm probability here is the probability of
+obtaining a candidate as loud or louder than cin the same re-
+gion of chirp mass from noise alone; it is obtained by counting
+across off-source trials.
+At the end of the search (i.e., Table 2 and Figure 1), we
+report a different false-alarm probability. It is the fraction of
+off-source likelihood ratios larger than the largest on-source
+likelihood ratio.
+There is another noteworthy difference with respect to un-6
+Antenna response Excluded distance (Mpc)
+GRB H1 H2 L1 V1 F.A.P. NS–NS NS–BH
+051114 0.56 0.56 . . . . . . 1 2:3 6 :2
+051210 0.61 0.61 . . . . . . 0.10 3:3 4 :3
+051211 0.53 . . . 0.62 . . . 0.66 2:3 8 :9
+060121 0.11 . . . 0.09 . . . 0.58 0:4 1 :3
+060313 0.59 0.59 . . . . . . 0.16 1:4 4 :3
+060427B 0.91 . . . 0.92 . . . 1 7:0 12 :7
+060429 0.92 0.92 . . . . . . 0.21 4:3 6 :2
+061006 0.61 0.61 . . . . . . 1 2:3 8 :2
+061201 0.85 0.85 . . . . . . 1 4:3 10 :1
+061217 0.77 . . . 0.52 . . . 0.23 3:2 11 :8
+070201 0.43 0.43 . . . . . . 0.07 3:3 5 :3
+070209 0.19 . . . 0.12 . . . 0.76 2:3 4 :2
+070429B 0.99 . . . 0.93 . . . 0.31 8:9 14 :6
+070512 0.38 . . . 0.51 . . . 0.97 6:1 8 :9
+070707 . . . 0.87 0.79 . . . 0.87 4:2 7 :1
+070714 0.28 . . . 0.40 . . . 0.72 4:2 2 :3
+070714B 0.25 . . . 0.38 . . . 0.54 3:2 5 :1
+070724 0.53 . . . 0.70 . . . 0.84 5:1 11 :8
+070729 0.85 0.85 . . . . . . 0.40 7:0 10 :8
+070809 0.30 0.30 . . . . . . 1 2:3 4 :3
+070810B 0.55 . . . 0.34 . . . 0.50 2:3 6 :1
+070923 0.32 . . . 0.40 0.69 0.74 5:1 7 :9
+Table 2
+Summary of the results for the search for gravitational waves from each GRB.
+The Antenna Response column contains the response for each detector as
+explained in Section 2.2; an ellipsis (. . . ) denotes that a detector’s data were
+not used. F.A.P. is the false-alarm probability of the most significant
+on-source candidate for a GRB as measured against its off-source trials, as
+explained in Section 2.4. On-source trials with no candidates above threshold
+are assigned a F.A.P. of 1. The last two columns show the lower limits at 90%
+CL on distances, explained in Section 3.2.
+triggered inspiral searches. For background estimation, untrig-
+gered searches use coincidences found between triggers from
+different detectors, to which unphysical time-shifts greater
+than the light-travel time between detector sites have been
+applied. Unfortunately, H1 and H2, being co-located, share a
+common environmental noise that is absent from the time-shift
+background measurement. Being unable to estimate the signif-
+icance of H1–H2 candidates reliably, the untriggered search
+examines them with significantly greater reservation and does
+not consider them at all in upper limit statements on rates.
+The present search performs its background estimation with
+unshifted coincidences under the assumption that any gravita-
+tional wave signal will appear only in the on-source trial. Thus,
+we regain the unconditional use of H1–H2 candidates.
+3.RESULTS
+3.1.Individual GRB results
+We found no evidence for a gravitational-wave signal in
+coincidence with any GRB in our sample. We ran the search
+as described in the previous section and found that the loud-
+est observed candidates in each GRB’s on-source segment is
+consistent with the expectation from its off-source trials. The
+results are summarized in Table 2, with brief highlights in the
+following subsections. A graphical comparison of on-source
+to off-source false-alarm probability is shown in Figure 1.
+3.1.1. GRB 070201
+The reanalysis of GRB 070201 yielded candidates in the
+on-source segment, despite having no coincident candidate
+at all in the previous analysis (Abbott et al. 2008a). This is
+consistent because the threshold for H2 has been lowered from
+4.0 to 3.5 and the coincident trigger found in this reanalysis
+happened to lie very close to the larger threshold in the previous
+search. The reanalysis yields a false-alarm probability of 6.8%,
+0.03 0.1 0.3 1
+false-alarm probability1310cumulative numberoff-source (normalized)
+on-sourceFigure 1. Cumulative false-alarm probabilities for the most significant candi-
+date in each on- and off-source trial, as described in Section 2.4.
+the smallest in the set of analyzed GRBs1. This value is
+completely within our expectations when we consider that we
+examined 22 GRBs.
+3.1.2. GRB 070923
+GRB 070923 was the GRB for which H1, L1, and V1 had
+comparable sensitivity and we accepted triggers from all three
+detectors. There were no triply-coincident candidates in the
+on-source trial, but there were surviving doubly-coincident
+candidates, the loudest of which had a false-alarm probability
+of 74.5%.
+3.2.Distance exclusions
+With our null observations and a large number of simula-
+tions, we can constrain the distance to each GRB assuming it
+was caused by a compact binary coalescence with a neutron
+star (with a mass in the range [1;3)M) and a companion
+of massmcomp . For a given mcomp range, we used the ap-
+proach of Feldman & Cousins (1998) to compute regions in
+distance where gravitational-wave events would, with a given
+confidence, have produced results inconsistent with our ob-
+servations. Figure 2 shows the lower Feldman–Cousins dis-
+tances for the 22 analyzed GRBs at 90% confidence for two
+illustrative choices for the companion mass range. The val-
+ues are also listed in Table 2. Because the companion mass
+range has been divided into equally spaced bins, we report
+on a ‘NS–NS’ system in which the companion mass is in the
+range [1;4)Mand a ‘NS–BH’ system in which the BH has a
+mass in the range [7;10)M. The median exclusion distance
+for a NS–BH system is 6.7 Mpc and for a NS–NS system is
+3.3 Mpc . These distances were derived assuming no beaming
+1In public presentations of preliminary results, GRB 061006 was erro-
+neously highlighted as having the loudest candidate due to a 22.8 s offset
+in the GRB time. Swift’s initial GCN alert (Schady et al. 2006a) was later
+corrected (Schady et al. 2006b), but we initially overlooked this correction.7
+0 5 10 15 20
+distance (Mpc)070923070810B070809070729070724070714B070714070707070512070429B070209070201061217061201061006060429060427B060313060121051211051210051114NS–NS
+NS–BH
+Figure 2. Lower limits on distances at 90% CL to putative NS–NS and
+NS–BH progenitor systems, as listed in Table 2 and explained in Section 3.2.
+(uniform prior on cos). NS–BH distances are typically higher
+than NS–NS because more massive systems radiate more total
+gravitational-wave energy. The excluded distance depends on
+various parameters: the location of the GRB on the sky, the
+detectors used for the GRB, the noise floor of the data itself,
+and the likelihood ratio of the loudest on-source candidate
+event for the GRB.
+We drew the simulations from a distribution in which
+our marginalized parameters roughly reflect our pri-
+ors on these astrophysical compact binary systems.
+In our models, a signal is completely specified by
+(mNS;mcomp;~ sNS;~ scomp;; ;t 0;D;; ). Of these,
+we wish to constrain mcomp andD, marginalizing over
+everything else. We drew the NS mass mNSuniformly
+from [1;3)M; the magnitudes of the NS spins j~ sNSjwere
+half 0 and half uniform in [0;0:75)(Cook et al. 1994); the
+magnitude of the companion’s spin j~ scompjwere half 0 and
+half uniform in [0;0:98)(Mandel & O’Shaughnessy 2009);
+the orientations of the spins were uniform in solid angle;
+the inclination of the normal to the binary’s orbital plane
+relative to our line of sight was conservatively chosen to be
+uniform in cosinstead of making an assumption about the
+GRB beaming angle; the polarization angle was uniform
+in[0;2); the coalescence time t0was uniform over the
+off-source region; the declination was set to that of the GRB;
+the right ascension was also set to that of the GRB, but
+was adjusted based on t0to keep each simulation at the same
+location relative to the detector as the GRB.
+A number of systematic uncertainties enter into this analy-
+sis, but amplitude calibration error and Monte–Carlo counting
+statistics from the injection trials have the largest effects. We
+multiplied exclusion distances by 1:28(1 +cal), wherecal
+is the fractional uncertainty (10% for H1 and H2; 13% for L1;
+6% for V1 (Marion et al. 2008)). The factor of 1.28 corre-
+sponds to a 90% pessimistic fluctuation, assuming Gaussianity.
+To take the counting statistics into account, we stretched theFeldman–Cousins confidence belts to cover the probability
+CL + 1:28p
+CL(1CL)=n, where CL is the desired confi-
+dence limit and nis the number of simulations contained in
+the(mcomp; D)bin for which we are constructing the belt.
+3.3.Population statement
+In addition to the individual detection searches above, we
+would like to assess the presence of gravitational-wave signals
+that are too weak to stand out above background separately,
+but that are significant when the entire population of analyzed
+GRBs is taken together. We compare the cumulative distribu-
+tion of the false alarm probabilities of the on-source sample
+with the off-source sample. The on-source sample consist of
+the results of all 22 individual searches, including those for
+GRBs with known redshifts, and the off-source sample con-
+sists of 6801 results from the off-source trials. This number is
+lower than 22324because for some GRBs, some off-source
+trials were discarded due to known data quality issues.
+These two distributions are compared in Figure 1. To deter-
+mine if they are consistent with being drawn from the same
+parent distribution, we employ the non-parametric Wilcoxon–
+Mann–Whitney U statistic (Mann & Whitney 1947), which is
+a measure of how different two populations are. Applying the
+U test, we find that the two distributions are consistent with
+each other; if the on-source and off-source significances were
+drawn from the same distribution, they would yield a U statis-
+tic greater than what we observed 53% of the time. Therefore
+we find no evidence for an excess of weak gravitational-wave
+signals associated with GRBs.
+4.DISCUSSION
+We searched data taken with the three LIGO detectors and
+the Virgo detector for gravitational-wave signatures of compact
+binary coalescences associated with 22 GRBs but found none.
+We were sensitive to systems with total masses 2M<m<
+40M. We also searched for a population of signals too weak
+to be individually detected, but again found no evidence. While
+there are few redshift determinations for short GRBs, it appears
+that the distribution is peaked around hzi0:25(Nakar 2007),
+far outside initial detector sensitivity, so it is not surprising that
+the S5/VSR1 run yielded no detections associated with short
+GRBs.
+We are indebted to the observers of the electromagnetic
+events and the GCN for providing us with valuable data. The
+authors gratefully acknowledge the support of the United States
+National Science Foundation for the construction and operation
+of the LIGO Laboratory, the Science and Technology Facilities
+Council of the United Kingdom, the Max-Planck-Society, and
+the State of Niedersachsen/Germany for support of the con-
+struction and operation of the GEO600 detector, and the Italian
+Istituto Nazionale di Fisica Nucleare and the French Centre
+National de la Recherche Scientifique for the construction and
+operation of the Virgo detector. The authors also gratefully
+acknowledge the support of the research by these agencies
+and by the Australian Research Council, the Council of Scien-
+tific and Industrial Research of India, the Istituto Nazionale di
+Fisica Nucleare of Italy, the Spanish Ministerio de Educaci ´on
+y Ciencia, the Conselleria d’Economia Hisenda i Innovaci ´o
+of the Govern de les Illes Balears, the Foundation for Fun-
+damental Research on Matter supported by the Netherlands
+Organisation for Scientific Research, the Royal Society, the
+Scottish Funding Council, the Scottish Universities Physics8
+Alliance, The National Aeronautics and Space Administration,
+the Carnegie Trust, the Leverhulme Trust, the David and Lu-
+cile Packard Foundation, the Research Corporation, and the
+Alfred P. Sloan Foundation.
+This document bears the LIGO document number P0900074.
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\ No newline at end of file
diff --git a/1001.0166.txt b/1001.0166.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d75792e5d2b5b4ee5d4ec72925fea35a2c001526
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+++ b/1001.0166.txt
@@ -0,0 +1,720 @@
+arXiv:1001.0166v1  [nlin.SI]  31 Dec 2009On Reductions ofSoliton Solutions of
+multi-component NLSmodels and Spinor
+Bose-Einsteincondensates
+V.S.Gerdjikov
+InstituteforNuclearResearchandNuclearEnergy,Bulgari anAcademyofSciences,72
+Tsarigradskochaussee1784Sofia,Bulgaria
+Abstract. WeconsideraclassofmulticomponentnonlinearSchrödinge requations(MNLS)related
+to the symmetric BD.I-type symmetric spaces. As important particular case of the se MNLS we
+obtain the Kulish-Sklyanin model. Some new reductions and t heir effects on the soliton solutions
+areobtainedbypropermodifyingtheZakahrov-Shabatdress ingmethod.
+Keywords: MulticomponentnonlinearSchrödingerequations,dressin g method,soliton solutions,
+reductiongroup
+PACS:35Q51,37K40
+INTRODUCTION
+Consider Bose-Einstein condensate (BEC) of alkali atoms in theF=1 hyperfine state,
+elongated in xdirection and confined in the transverse directions y,zby purely optical
+means.Thedynamicsofthisassemblyofatomsisdescribedby a3-componentnormal-
+ized spinor wave vector /vectorΦ= (Φ1,Φ0,Φ−1)T(x,t)satisfying the multicomponent non-
+linear Schrödinger (MNLS) equation [1, 2, 3], which in dimen sionless coordinates can
+bewritten downas:
+i∂tΦ±1+∂2
+xΦ±1+2(|Φ±1|2+|Φ0|2)Φ±1+Φ∗
+∓1Φ2
+0=0,
+i∂tΦ0+∂2
+xΦ0+(2|Φ+1|2+|Φ0|2+2|Φ−1|2)Φ0+2Φ∗
+0Φ+1Φ−1=0,(1)
+ThesecondmodelwhichdescribesBECwith F=2hyperfinestructureisa5-component
+MNLSsystem:
+i∂tΦ±2+∂2
+xΦ±2+2/parenleftig
+(/vectorΦ†,/vectorΦ)−|Φ±2|2/parenrightig
+Φ±2+2Φ∗
+∓2Φ+1Φ−1−Φ∗
+∓2Φ2
+0=0,
+i∂tΦ±1+∂2
+xΦ±1+2/parenleftig
+(/vectorΦ†,/vectorΦ)−|Φ±1|2/parenrightig
+Φ±1+2Φ∗
+∓1Φ−1Φ+1+Φ∗
+∓1Φ2
+0=0,
+i∂tΦ0+∂2
+xΦ0+2/parenleftbigg
+(/vectorΦ†,/vectorΦ)−1
+2|Φ0|2/parenrightbigg
+Φ0+2Φ∗
+0Φ+1Φ−1−2Φ∗
+0Φ+2Φ−2=0,(2)
+where(/vectorΦ†,/vectorΦ) =∑2
+k=−2|Φk|2. Both models allow Lax representations and therefore
+are integrable by the inverse scattering transform method [ 4, 2, 3]. The Lax pairs have
+natural Lie algebraic structure which relates them to the sy mmetric spaces BD.I≃
+SO(n+2)/SO(n)×SO(2)withn=3 andn=5 respectively. From algebraic pointof view this means that the potential Q(x,t)ofLtakes the form Q(x,t) = [J,X(x,t)]
+whereX(x,t)is agenericelementoftheLiealgebra so(n+2)and theconstantelement
+J=diag(1,0,...,0−1)is a specially chosen element of the Cartan subalgebra h⊂g.
+Formoredetailssee[5, 4].
+Thepresentpaperextendstheresultsof[2, 3]fortheclasso fMNLSrelatedto BD.I-
+type symmetric spaces, i.e. for any n. We briefly outline how the direct and inverse
+scatteringproblemfortheLaxoperatorarereducedtoaRiem ann-Hilbertproblem.Next
+we find that a simple change of variables can cast the above-me ntioned MNLS into the
+Kulish-Sklyanin model (KSM) [6]. We also apply Mikhailov re duction group method
+[7] and derive several new types of MNLS interactions. We der ive also the constraints
+on the polarization vectors in the dressing factors that are imposed by the reductions.
+Finally we apply a proper modification (see [2, 3]) of the Zakh arov-Shabat dressing
+method [8, 9] and derive the soliton solutions of the MNLS and of KSM in particular.
+ThusweobtainseveralnewtypesofintegrablevectorMNLSan dtheirsolitonsolutions.
+The majority of papers devoted to soliton equations analyze and solve the inverse
+scattering problem (ISP) for the relevant Lax operators usi ng the typical (lowest di-
+mensional)representation of the correspondingLie algebr a. At the end of ourpaper we
+briefly compare the properties of the dressing factors in two of the fundamental repre-
+sentationsoftheLiealgebra so(2r).Wealsoelucidatesomeadditionalissuesconsidered
+in [3, 10] such as the structure of the soliton solutions and t he effect of additional Z2-
+reductions.
+MNLSEQUATIONSFOR BD.ISERIESOFSYMMETRICSPACES
+MNLSequationsforthe BD.I.seriesofsymmetricspaces(algebrasofthetype so(n+2)
+andJdual toe1)havetheLax representation [L,M]=0 as follows
+Lψ(x,t,λ)≡i∂xψ+(Q(x,t)−λJ)ψ(x,t,λ)=0. (3)
+Mψ(x,t,λ)≡i∂tψ+(V0(x,t)+λV1(x,t)−λ2J)ψ(x,t,λ)=0, (4)
+V1(x,t) =Q(x,t),V0(x,t)=iad−1
+JdQ
+dx+1
+2/bracketleftbig
+ad−1
+JQ,Q(x,t)/bracketrightbig
+.(5)
+wheread JX=[J,X]and ad−1
+Jis welldefined ontheimageofad Jing;
+Q=
+0/vectorqT0
+/vectorp0s0/vectorq
+0/vectorpTs00
+,J=diag(1,0,...0,−1). (6)
+Then-componentvectors /vectorqand/vectorphavetheform
+/vectorq=(q1,...,qn)T, /vectorp=(p1,...,pn)T, (7)
+whilethematrix s0=S(n)enters in thedefinitionof so(n):
+X∈so(n),X+S(n)XTS(n)=0,S(n)=n
+∑
+s=1(−1)s+1E(n)
+s,n+1−s,(8)forn=2r+1 and
+S(n)=r
+∑
+s=1(−1)s+1(E(n)
+s,n+1−s+E(n)
+n+1−s,s) (9)
+forn=2r. ByE(n)
+spabove we mean n×nmatrix whose matrix elements are (E(n)
+sp)ij=
+δsiδpj. With the definition of orthogonality used in (8) the Cartan g enerators Hk=
+E(n)
+k,k−E(n)
+n+1−k,n+1−kare represented by diagonalmatrices.
+The Lax pairs, related to the symmetric spaces SO(n+2)/(SO(n)×SO(2))have
+special algebraic properties. They are determined by choos ingJ=H1to be dual to
+e1∈Er. It allowsoneto introduceagradingin g, i.e.g=g0⊕g1so that:
+[X1,X2]∈g0,[X1,Y1]∈g1,[Y1,Y2]∈g0, (10)
+for any choice of the elements X1,X2∈g0andY1,Y2∈g1. The grading splits the set
+of positive roots of so(n)into two subsets Δ+=Δ+
+0∪Δ+
+1whereΔ+
+0contains all the
+positive roots of gwhich are orthogonal to e1, i.e.(α,e1) =0; the roots in β∈Δ+
+1
+satisfy(β,e1)=1. Formoredetailssee[5].
+In writing down the Lax pair (3) we made use of the typical n×nrepresentation of
+so(n).The Lax pair can be considered in any representation of so(n), then the potential
+Qwilltaketheform:
+Q(x,t)=∑
+α∈Δ+
+1(qα(x,t)Eα+pα(x,t)E−α). (11)
+Nextweintroduce n-component‘vectors’formedbytheWeylgeneratorsof so(n+2)
+correspondingto theroots in Δ+
+1:
+/vectorE±
+1=(E±(e1−e2),...,E±(e1−er),E±e1,E±(e1+er),...,E±(e1+e2)),(12)
+forn=2r+1 and
+/vectorE±
+1=(E±(e1−e2),...,E±(e1−er),E±(e1+er),...,E±(e1+e2)), (13)
+forn=2r. Then the generic form of the potentials Q(x,t)related to these type of
+symmetricspaces can bewrittenas sumoftwo"scalar" produc ts
+Q(x,t)=(/vectorq(x,t)·/vectorE+
+1)+(/vectorp(x,t)·/vectorE−
+1). (14)
+In terms of these notations the generic MNLS type equations c onnected to BD.I.
+acquiretheform
+i/vectorqt+/vectorqxx+2(/vectorq,/vectorp)/vectorq−(/vectorq,s0/vectorq)s0/vectorp=0,
+i/vectorpt−/vectorpxx−2(/vectorq,/vectorp)/vectorp+(/vectorp,s0/vectorp)s0/vectorq=0,(15)
+Withthetypicalreduction pk=q∗
+kit gives:
+i/vectorqt+/vectorqxx+2(/vectorq†,/vectorq)/vectorq−(/vectorq,s0/vectorq)s0/vectorq∗=0, (16)If we put n=3 and introduce the new variables Φ±1=q1,3,Φ0=q2we recover
+equations (1). Likewise with n=5 andΦ±2=q1,5,Φ±1=q2,4,Φ0=q3we find eq.
+(2). TheHamiltoniansfortheMNLSequations(15)are givenb y
+HMNLS=/integraldisplay∞
+−∞dx/parenleftbigg
+(∂x/vectorpT,∂x/vectorq)−(/vectorpT,/vectorq)2+1
+2(/vectorpT,s0/vectorp)(/vectorqT,s0/vectorq)/parenrightbigg
+,(17)
+THE DIRECTAND THEINVERSESCATTERINGPROBLEM
+Thefundamentalanalytic solution
+Hereinweremindsomebasicfeaturesoftheinversescatteri ngtheoryfortheoperator
+L(4),see[2,3].Therewehavemadeuseofthegeneraltheoryde velopedin[21,11,12,
+13]and thereferences therein.TheJostsolutionsof Lare defined by:
+lim
+x→−∞φ(x,t,λ)eiλJx=11,lim
+x→∞ψ(x,t,λ)eiλJx=11 (18)
+and the scattering matrix T(λ,t)≡ψ−1φ(x,t,λ). The special choice of Jand the fact
+that the Jost solutions and the scattering matrix take value s in the group SO(n+2)we
+can usethefollowingblock-matrixstructureof T(λ,t)
+T(λ,t)=
+m+
+1−/vectorb−Tc−
+1/vectorb+T22−s0/vectorB−
+c+
+1/vectorB+Ts0m−
+1
+,ˆT(λ,t)=
+m−
+1/vectorB−Tc−
+1
+−/vectorB+ˆT22s0/vectorb−
+c+
+1−/vectorb+Ts0m+
+1
+,
+(19)
+where/vectorb±(λ,t)and/vectorB±(λ,t)aren-component vectors, T22(λ)isn×nblock matrix,
+andm±
+1(λ),andc±
+1(λ)arescalarfunctions.Suchparametrizationiscompatiblew iththe
+generalized Gaussdecompositionsof T(λ)which read as follows:
+T(λ,t)=T−
+JD+
+JˆS+
+J,T(λ,t)=T+
+JD−
+JˆS−
+J,
+T∓
+J=e±(/vectorρ±,/vectorE∓
+1), S±
+J=e±(/vectorτ±,/vectorE±
+1),D±
+J=diag/parenleftbig
+(m±
+1)±1,m±
+2,(m±
+1)∓1/parenrightbig
+.
+The functions m±
+1andn×nmatrix-valued) functions m±
+2are are analytic for λ∈C±.
+Wehaveintroducedalso thenotations:
+/vectorρ−=/vectorB−
+m−
+1, /vectorτ−=/vectorB+
+m−
+1, /vectorρ+=/vectorb+
+m+
+1, /vectorτ+=/vectorb−
+m+
+1,
+c±
+1=m±
+1(/vectorρ±Ts0/vectorρ±)
+2=m∓
+1(/vectorτ∓Ts0/vectorτ∓)
+2,
+/vectorb−=µ−,T
+2
+m−
+1/vectorB−,/vectorb+=µ−
+2
+m−
+1/vectorB+,/vectorB+=s0µ+,T
+2s0
+m+
+1/vectorb+,/vectorB−=s0µ+
+2s0
+m+
+1/vectorb−,
+whereµ+=m+
+2−/vectorb+/vectorb−,T/(2m+
+1),µ−=m−
+2−s0/vectorB+/vectorB−,Ts0/(2m−
+1). There are some
+additional relations which ensure that both T(λ)and its inverse ˆT(λ)belong to the
+orthogonalgroup SO(n+2)and thatT(λ)ˆT(λ)=11.Important tools for reducing the ISP to a Riemann-Hilbert pr oblem (RHP) are the
+fundamental analytic solution (FAS) χ±(x,t,λ). We will introduce two pairs of FAS
+usingthegeneralized Gaussdecompositionof T(λ,t),see[11,13, 14]:
+χ±(x,t,λ)=φ(x,t,λ)S±
+J(t,λ)=ψ(x,t,λ)T∓
+J(t,λ)D±
+J(λ),
+χ′,±(x,t,λ)=φ(x,t,λ)S±
+J(t,λ)ˆD±
+J(λ)=ψ(x,t,λ)T∓
+J(t,λ).(20)
+Moreprecisely,thisconstructionensuresthat ξ±(x,λ)=χ±(x,λ)eiλJxandξ′,±(x,λ)=
+χ′,±(x,λ)eiλJxare analytic functions of λforλ∈C±. IfQ(x,t)is a solution of the
+MNLS eq. (15) then the matrix elements of T(λ)satisfy the linear evolution equations
+[2,3]
+id/vectorb±
+dt±λ2/vectorb±(t,λ)=0,id/vectorB±
+dt±λ2/vectorB±(t,λ)=0,
+idm±
+1
+dt=0, idm±
+2
+dt=0.(21)
+Thustheblock-diagonalmatrices D±(λ)canbeconsideredasgeneratingfunctionalsof
+theintegralsofmotion.Thefactthatall (2r−1)2matrixelementsof m±
+2(λ)forλ∈C±
+generateintegralsofmotionreflectthesuperintegrabilit yofthemodelandareduetothe
+degeneracyofthedispersionlawof(15).Weremindthat D±
+J(λ)allowanalyticextension
+forλ∈C±and thattheirzeroes and polesdeterminethediscreteeigen valuesof L.
+TheRiemann-HilbertProblem
+TheFASforreal λare linearlyrelated [2, 3]
+χ+(x,t,λ)=χ−(x,t,λ)G0,J(λ,t),G0,J(λ,t)=ˆS−
+J(λ,t)S+
+J(λ,t)
+χ′,+(x,t,λ)=χ′,−(x,t,λ)G′
+0,J(λ,t),G′
+0,J(λ,t)=ˆT+
+J(λ,t)T−
+J(λ,t).(22)
+Onecanrewriteeq.(22)inanequivalentformfortheFAS ξ±(x,t,λ)=χ±(x,t,λ)eiλJx
+andξ′,±(x,t,λ)=χ′,±(x,t,λ)eiλJxwhichsatisfytheequation:
+idξ±
+dx+Q(x)ξ±(x,λ)−λ[J,ξ±(x,λ)]=0,
+idξ′,±
+dx+Q(x)ξ′,±(x,λ)−λ[J,ξ′,±(x,λ)]=0(23)
+andtherelations
+lim
+λ→∞ξ±(x,t,λ)=11,lim
+λ→∞ξ′,±(x,t,λ)=11. (24)
+ThentheseFASsatisfytheRHP’s
+ξ+(x,t,λ)=ξ−(x,t,λ)GJ(x,λ,t),GJ(x,λ,t)=e−iλJ(x+λt)G−
+J(λ,t)eiλJ(x+λt),
+ξ′,+(x,t,λ)=ξ′,−(x,t,λ)GJ(x,λ,t),G′
+J(x,λ,t)=e−iλJ(x+λt)G′
+J(λ,t)eiλJ(x+λt).
+(25)Obviously the sewing function GJ(x,λ,t)(resp.G′
+J(x,λ,t)) is uniquely determined by
+the Gauss factors S±
+J(λ,t)(resp.T±
+J(λ,t)). In addition Zakharov-Shabat’s theorem [8]
+states that if sewing functions GJ(x,λ,t)andG′
+J(x,λ,t)depend on xandtin the way
+prescribedaboveensures thatthecorrespondingFASsatisf ythelinearsystems(23).
+Assume we have solved the RHP’s above and know the FAS ξ+(x,t,λ). Then the
+correspondingpotentialof Lis recoveredby
+Q(x,t)=lim
+λ→∞λ/parenleftig
+J−ξ+(x,t,λ)Jˆξ+(x,t,λ)/parenrightig
+. (26)
+REDUCTIONSOFMNLS
+The reduction group proposed by Mikhailov [7] provides four classes of reductions
+which are automatically compatible with the Lax representa tion of the corresponding
+MNLSeq.
+The reduction group GRis a finite group which preserves the Lax representation
+[L,M]=0,i.e.itensuresthatthereductionconstraintsareautoma ticallycompatiblewith
+theevolution. GRmust havetwo realizations: i) GR⊂Autgand ii)GR⊂ConfC, i.e. as
+conformal mappings of the complex λ-plane. To each gk∈GRwe relate a reduction
+conditionfortheLaxpairas follows[7]:
+Ck(L(Γk(λ)))=ηkL(λ),Ck(M(Γk(λ)))=ηkM(λ), (27)
+whereCk∈AutgandΓk(λ)∈ConfCaretheimagesof gkandηk=1or−1depending
+on the choice of Ck. SinceGRis a finitegroup then for each gkthere exist an integer Nk
+such that gNk
+k=11. In all the cases below Nk=2 and the reduction group is isomorphic
+toZ2.Morespecificallytheautomorphisms Ck,k=1,...,4listedaboveleadtothefour
+possibleclasses ofreductionsforthematrix-valuedfunct ions
+U(x,t,λ)=Q(x,t)−λJ,V(x,t,λ)=V0(x,t)+λV1(x,t)−λ2J,(28)
+oftheLax representation:
+1) C1(U†(κ1(λ)))=U(λ), C1(V†(κ1(λ)))=V(λ),
+2)C2(UT(κ2(λ)))=−U(λ),C2(VT(κ2(λ)))=−V(λ),
+3) C3(U∗(κ1(λ)))=−U(λ),C3(V∗(κ1(λ)))=−V(λ),
+4) C4(U(κ2(λ)))=U(λ), C4(V(κ2(λ)))=V(λ),(29)
+In what follows we will examine the typical reductions of MNL S eqs. of the class 1)
+obtained by specifying κ1(λ) =λ∗andC1to be aZ2-automorphism of gsuch that
+C1(J)=J. Belowwelistseveralchoices for C1leadingto inequivalentreductions:
+a)C1=11, /vectorp(x)=/vectorq∗(x),b)C1=K1, /vectorp(x)=K01/vectorq∗(x),
+c)C1=Se2, /vectorp(x)=K02/vectorq∗(x),d)C1=Se2Se3, /vectorp(x)=K03/vectorq∗(x),(30)
+where
+Kj=block-diag (1,K0j,1),K01=diag(ε1,...,εr−1,1,εr−1,...,ε1),(31)andεj=±1.Thematrices K02andK03correspondingtotheWeylreflections Se2,Se2Se3
+etc. arenotdiagonal;theyhavedimension n×nand forn=3,4 and5 are givenby:
+n=3,K02=
+0 0 1
+0−1 0
+1 0 0
+,
+n=4,K02=
+0 0 0 −1
+0 1 0 0
+0 0 1 0
+−1 0 0 0
+,K03=
+0 0 0 1
+0 0−1 0
+0−1 0 0
+1 0 0 0
+,
+n=5,K02=
+0 0 0 0 −1
+0 1 0 0 0
+0 0−1 0 0
+0 0 0 1 0
+−1 0 0 0 0
+,K03=
+0 0 0 0 −1
+0 0 0 1 0
+0 0 1 0 0
+0 1 0 0 0
+−1 0 0 0 0
+,(32)
+Each oftheabovereductionsimposeconstraintsontheFAS, o nthescatteringmatrix
+T(λ)and onitsGaussfactors S±
+J(λ),T±
+J(λ)andD±
+J(λ).Thesehavetheform:
+(φ(x,λ∗))†=K−1
+jˆφ(x,λ)Kj,(ψ(x,λ∗))†=K−1
+jˆψ(x,λ)Kj,
+(χ+(x,λ∗))†=K−1
+jˆχ−(x,λ)Kj(T(λ∗))†=K−1
+jˆT(λ)Kj,
+(S+(λ∗))†=K−1
+jˆS−(λ)Kj(T+(λ∗))†=K−1
+jˆT−(λ)Kj
+(D+(λ∗))†=K−1
+jˆD−(λ)Kj(33)
+wherethematrices Kjarespecificforeachchoiceoftheautomorphism C1,seeeq.(31).
+Inparticular, from thelast lineof(33)and(31)weget:
+(m+
+1(λ∗))∗=m−
+1(λ), (34)
+andconsequently,if m+
+1(λ)has zeroes at thepoints λ+
+k,thenm−
+1(λ)has zeroes at:
+λ−
+k=(λ+
+k)∗,k=1,...,N. (35)
+Below we will write down the effects of these reductions on th e corresponding
+Hamiltonians.Forthetypicalreduction /vectorp=/vectorq∗weget:
+HMNLS=/integraldisplay∞
+−∞dx/braceleftbigg
+((∂x/vectorq,∂x/vectorq∗)−(/vectorq,/vectorq∗)2+1
+2(/vectorq,s0/vectorq)(/vectorq∗,s0/vectorq∗)/bracerightbigg
+, (36)
+H(j)
+MNLS=/integraldisplay∞
+−∞dx/parenleftbigg
+(∂x/vectorqKj∂x/vectorq∗)−(/vectorq,Kj/vectorq∗)2+1
+2(/vectorq,s0/vectorq)(/vectorq∗,s0/vectorq∗)/parenrightbigg
+,(37)
+TheHamiltonian H(1)
+MNLSwithK01(31)hasindefinitekineticterm.Asaconsequencethe
+correspondingMNLShas singularsolitonsolutionswhich‘b low-up’infinitetime.The above Hamiltonians, after the change of variables can be written in more ‘aes-
+thetic’form.Indeed, forodd n=2r−1 wecan put:
+q2k−1,2r−2k+1=v2k−1±iv2r−2k+1√
+2,q2k,2r−2k=iv2k∓v2r−2k√
+2,qr=c0,rvr;
+(38)
+withk=1,2,...,r−1 andc0,r=e(r−1)πi/2; forn=2rweput:
+q2k−1,2r−2k+2=v2k−1±iv2r−2k+2√
+2,q2k,2r−2k+1=iv2k∓v2r−2k+1√
+2(39)
+withk=1,2,...,r.
+Insertingtheabovechanges ofvariablesintotheHamiltoni an(36)weget
+HKS=/integraldisplay∞
+−∞dx
+
+n
+∑
+j=1|∂xvj|2−/parenleftigg
+n
+∑
+j=1|vj|2/parenrightigg2
++1
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen
+∑
+j=1v2
+j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+
+, (40)
+which is the Hamiltonian of the n-component Kulish-Sklyanin model (KSM) [6]. Thus
+we have demonstrated that the Lax pairs (3), (4) can be used al so for integrating
+the MNLS (40). In their original paper [6] Kulish and Sklyani n have used Lax pair
+whosepotential is an element of a Clifford algebra. Later So kolov and Svinolupov[15]
+discovered another class of Lax pairs for these models whose potentials take values in
+Jordan algebras. The above Lax pairs allowed to prove integr ability of the KSM but
+were not convenient for solving the inverse scattering prob lem and constructing exact
+solutions.Anotherimportantpropertyofthesemodelsisth attheypossessbothclassical
+[4]andquantum R-matrices[6].
+Another way to obtain KSM is to apply the reduction of type 4) w ith
+K0=block-diag (1,εs0,1), whereε=±1. For odd values of n=2r−1 this reduction
+meansthat:
+qk=(−1)k+1εq2r−k=wk,k=1,...,r, (41)
+whilefor n=2ronegets:
+qk=(−1)k+1εq2r−k+1=wk,k=1,...,r. (42)
+Thisreductionleadsto r-componentKSM.
+Let us write down the Hamiltonians for the different reducti ons. Below for conve-
+niencewe willsplit HMNLSintokineticand interactionterms: HMNLS=H(j)
+kin−H(j)
+int.
+Reduction b):
+H(1)
+kin=/integraldisplay∞
+−∞dx/braceleftigg
+r−1
+∑
+j=1εj(|∂xqj|2+|∂xq2r−j|2)+|∂xqr|2/bracerightigg
+,
+H(1)
+int=/integraldisplay∞
+−∞dx/braceleftigg
+r−1
+∑
+j=1εj(|qj|2+|q2r−j|2)+|qr|2)2
+−1
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingler−1
+∑
+j=1(−1)j+12qjq2r−j+(−1)rq2
+r/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+
+,(43)One can construct other reductions, e.g. ones of type c) with reduction matrix Kj.
+Then
+H(j)
+MNLS=/integraldisplay∞
+−∞dx/braceleftbigg
+(∂x/vectorq†Kj∂x/vectorq)−(/vectorq†Kj/vectorq)2+1
+2/vextendsingle/vextendsingle(/vectorqTs0/vectorq)/vextendsingle/vextendsingle2/bracerightbigg
+,(44)
+Characteristic feature of the reductions involving Weyl gr oup elements is that they
+leadto‘non-diagonal’formofthekineticterms[16].Makin gsimplechangeofvariables
+diagonalizing Kjwecanrecoverthediagonalformofthekinetictermsbutunfo rtunately
+wecan not makeit positivedefinite. Thisis related to thefac t thatK2
+j=1 and sohas as
+eigenvaluesboth +1 and−1 withcertain multiplicities.
+Letusgivealsoanimportantexampleofclass2)reductions( 28).Theconstraintsthat
+these class of reductions impose on the FAS and on the scatter ing matrix T(λ)and on
+itsGaussfactors S±
+J(λ),T±
+J(λ)andD±
+J(λ)taketheform:
+(χ+(x,λ))T=K′,−1
+jˆχ−(x,λ)K′
+j(T(λ))T=K′,−1
+jˆT(λ)K′
+j,
+(S±(λ))T=K′,−1
+jˆS±(λ)K′
+j(T±(λ))T=K′,−1
+jˆT±(λ)K′
+j(45)
+and(D±(λ))T=K′,−1
+jˆD±(λ)K′
+j. The explicit form of the matrices K′
+jis determined
+by the particular realization of the automorphism C2. Choosing n=3 andC2=Se1we
+obtaintheconstraint q1=q3and thereduced Hamiltoniantakes theform:
+H=/integraldisplay∞
+−∞dx/braceleftbigg
+|∂xv1|2+|∂xv2|2−(|v1|2+|v2|2)2+1
+2/vextendsingle/vextendsinglev2
+1−v2
+2/vextendsingle/vextendsingle2/bracerightbigg
+,(46)
+wherewehaveput q1=q2=1√
+2v1andq2=v2.Thismodelalsobeenderivedasrelevant
+forF=1BEC [17].
+DRESSINGMETHOD AND SOLITON SOLUTIONS
+ThedressingZakharov-Shabatmethod[8,9]forconstructin gsolitonsolutionsofMNLS
+has been modified in [3] for the BD.I-type symmetric spaces. T here we also analyzed
+the different types of soliton solutions. Below we briefly di scuss the properties of the
+genericone-solitonsolutions
+It is obtained by dressing the regular FAS χ±
+0(x,λ)of the RHP (25). Using them we
+constructthesingularsolutions χ±
+0(x,λ)oftheRHP
+χ±(x,λ)=u(x,λ)χ±
+0(x,λ)ˆu−,χ′,±(x,λ)=u(x,λ)χ±
+0(x,λ)ˆu+,
+u(x,λ)=11+(c1(λ)−1)P1(x)+(c−1
+1(λ)−1)¯P1(x),u±=lim
+x→±∞u(x,λ).(47)
+Fortheabovechoiceof Jitisenoughtoconsiderrank1projectors P1(x,t)and¯P1(x,t)=
+S0PT
+1S0. Together with the constraint P1¯P1=0, the last condition ensures that u(x,t)∈
+SO(n+2). It remains to only to give the explicit form of P1(x,t). Generically it is
+determinedbytwopolarizationvectors |n0,1/angbracketrightand/angbracketleftm0,1|,andtheinitialregularsolutions:
+P1(x,t)=|n1(x,t)/angbracketright/angbracketleftm1(x,t)|
+/angbracketleftm1(x,t)|n1(x,t)/angbracketright,¯P1(x,t)=|m1(x,t)/angbracketright/angbracketleftn1(x,t)|
+/angbracketleftn1(x,t)|m1(x,t)/angbracketright,(48)|n1(x,t)/angbracketright=χ+
+0(x,t,λ+
+1)|n0,1/angbracketright,/angbracketleftm1(x,t)|=/angbracketleftm0,1|ˆχ−
+0(x,t,λ−
+1),
+|m1(x,t)/angbracketright=χ−
+0(x,t,λ−
+1)|m0,1/angbracketright,/angbracketleftm1(x,t)|=/angbracketleftn0,1|ˆχ−
+0(x,t,λ−
+1).(49)
+The one soliton solution is parametrized by the two eigenval uesλ±
+1and by the polar-
+izationvectors |n0,1/angbracketrightand/angbracketleftm0,1|.Thelatterafterrenormalizationhave n−1independent
+componentseach:
+|n0,1/angbracketright=
+√A0,
+/vectorν0,1/√A0
+1/√A0
+,/angbracketleftm0,1|=/parenleftig/radicalbig
+B0,/vectorµ0,1//radicalbig
+B0,1//radicalbig
+B0/parenrightig
+,
+whereA0=1
+2(/vectorνT
+0,1s0/vectorν0,1)andB0=1
+2(/vectorµT
+0,1s0/vectorµ0,1). The constraint P1¯P1=0 means that
+the vectors /vectorµ0,1and/vectorν0,1must satisfy /vectorµT
+0,1s0/vectorν0,1=0. Therefore the one-soliton solution
+can be viewed as a dynamical system with 2 n−1 degrees of freedom. After some
+simplificationsit takestheform:
+qk(x,t)=−4iν1
+Δe−iµ1˜zke˜ξ0,k[cos(δ0k)cosh(z0k)+isin(δ0k)sinh(z0k)],(50)
+Δ=2cosh(2z0)+C,C=(/vectorν0†/vectorν0)
+|A0|, ˜zk=x+w1t−˜δ0,k/µ1,
+z0=ν1(x−u1t)+ξ0,z0k=ν1x+ξ0,k, ˜z0k=ν1(x−v1t)+˜ξ0,k,
+ξ0=1
+2ln|A0|, ξ0,k=1
+2ln|/vectorν0,2r−k|
+|A0||/vectorν0,k|,˜ξ0,k=1
+2ln|/vectorν0,k||/vectorν0,2r−k|
+|A0|,(51)
+δ0,k=(α0,2r−k+α0,k−α0−πk)/2,˜δ0,k=(α0,2r−k−α0,k+α0+πk)/2,
+whereα0=argA0andα0,k=arg/vectorν0,k.
+Each of the reductions of the type (29) imposesconstraints n ot only on λ+
+1=(λ−
+1)∗,
+butalsoon thepolarizationvectors:
+/vectorµ0,1=K0,j/vectorν0,1∗,/vectorνT
+0,1K0,js0/vectorν0,1=0. (52)
+As a result, after the reduction the number of independent pa rameters of the soliton
+solution becomes n−1. The velocities u1andw1are given by u1=−2µ1andw1=
+(ν2
+1−µ2
+1)/µ1.
+Special attentiondeserves thefact that generically all z0,kare different and as aresult
+each component qk(x,t)has itscenterofmassshifted withrespect totheothers.
+Let us now consider a Z2×Z2reduction by applying simultaneously two reduction:
+thefirstisthetypicaloneandthesecondistheclass2)reduc tionsasforthemodel(46).
+Thefirst reductionimposestherelation (52) between thetwo polarizationvectors |n0,1/angbracketright
+and/angbracketleftn0,1|. Thesecondreductionimposesconstraintonthevector |n0,1/angbracketright, namely:
+|n0,1/angbracketright=K′
+0j|n0,1/angbracketright.
+In particular, for n=3 andC2=Se1the vector |n0,1/angbracketrighthas 3 components and
+K′
+0j=diag(1,−1,1). Thus only two independent complex coefficients are enough
+to parametrize thecorresponding polarization vector, and thecorresponding solitoncan
+beviewed as dynamicalsystemwiththreedegrees offreedom.DISCUSSIONAND CONCLUSIONS
+One of the important consequences of the FAS is that with thei r help one can construct
+the kernel of the resolvent of L(see [12, 18]) and prove the completeness relation
+for its eigenfunctions. From these expressions it becomes o bvious that the resolvent
+developspoles at all points λ±
+k∈C±for which m±
+1(λ±
+k)=0. Combining this fact with
+the equivalence between the solutions of the RHP and the FAS o f the Lax operator we
+concludethatthesingularitiesoftheRHPcorrespond tothe discreteeigenvaluesof L.
+QuiteoftenthegeneralanalysisoftheMNLS(1)isfollowedb ysimplificationswhich
+oftenreducetheMNLStoasingle-componentNLS. Oneway doto thiswas mentioned
+above: it is to impose the reduction Φ+1=Φ−1. Another less obvious way to this is
+to impose this reduction on the initial conditions. Indeed, one can show that impos-
+ingΦ+1(x,t=0) =Φ−1(x,t=0)ensures that Φ+1(x,t) =Φ−1(x,t)for allt>0. At
+the same time there is a substantial difference between the s olitons of the scalar NLS
+or Manakov model and the solitons of MNLS (1). Unlike the soli tons of the Man-
+akov model, all three components of the one-soliton solutio n of (1) have different x-
+dependence;genericallyeach componenthasdifferent‘cen terofmass’position.There-
+fore,ifonewantstodemonstratenewnontrivialaspects ofs olitondynamicsoneshould
+usegenericinitialvaluesfor Φ±1(x,t=0)andΦ−1(x,t=0).
+Another still open problem is the interrelation between the solutions of the direct
+and inverse scattering problem for L, considered in different irreduciblerepresentations
+(IRREP) of the corresponding Lie algebra g. From the point of view of the relevant
+NLEE,theirLaxrepresentationshavepurelyalgebraicnatu reandtherefore,theformof
+theNLEE doesnotdepend on thechoiceoftheIRREP of g.
+From the point of view of the spectral theory, the different I RREP have different
+dimensions; therefore changing the IRREP we change the order of the corresponding
+operator. Since we are dealing with simple Lie algebras whos e IRREP are well known
+[5].Inparticular,itiswellknownthatthefinitedimension alrepresentationscanbereal-
+ized as invariant subspaces of the tensor products of the typ ical one. Let us assumethat
+weareabletoconstructtheFASandtherelevantRHPanddress ingfactorsinthetypical
+representation.Obviously,takingthetensorsproductsof theFAStheiranalyticityprop-
+ertieswillpersistand wewillgetthecorrespondingFASand RHP inthecorresponding
+IRREP. Howevernontrivialthingsmay takeplace whenonecon sidersthemultiplicities
+ofthecorrespondingdiscreteeigenvalues.
+AsanexampleIwilljustmentionthatthedressingfactorcan beevaluatedalsoforthe
+otherfundamentalrepresentationsof g[19].Ifinthetypicalrepresentationof g≃so(2r)
+u(x,λ)isgivenby (47)thenin thespinorrepresentation itwilltak etheform [20]:
+u(x,λ)=/radicalbig
+c1(λ)π1(x,t)+1/radicalbig
+c1(λ)¯π1(x,t),¯π1(x,t)=˜s0π1(x,t)˜s−1
+0,
+and the projectors satisfy π1(x,t)¯π1(x,t)=0 andπ1(x,t)+¯π1(x,t)=11. Note the sub-
+stantial change in the λ-dependence of u(x,λ), as well as the fact that now instead of
+havingrank oneprojectors P1(x,t)weget projectors π1(x,t)and¯π1(x,t)ofrankr.
+Wewilldiscusstheseproblemsin moredetailselsewhere.ACKNOWLEDGMENTS
+ItismypleasuretothanktheorganizersoftheAMITANSconfe renceforkindinvitation.
+IamgratefultoprofessorN.Kostovandtoananonymousrefer eeforusefulsuggestions.
+REFERENCES
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+3. V. S. Gerdjikov, D. J. Kaup, N. A. Kostov, T. I. Valchev. On classification of soliton solutions of
+multicomponentnonlinearevolutionequations.J.Phys.A: Math.Theor. 41315213(2008)(36pp).
+4. A. P. Fordy, and P. P. Kulish. Nonlinear Schrödinger equat ions and simple Lie algebras. Commun.
+Math.Phys. 89,427–443(1983).
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+integrablesystem. Phys.Lett. 84A,349-352(1981).
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+73–117(1981).
+8. V. E. Zakharov, A. B. Shabat. A scheme for integrating the n onlinear equations of mathematical
+physicsby the methodof the inverse scattering problem.I. FunctionalAnalysis and Its Applications ,
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+V. E. Zakharov, A. B. Shabat. Integration of nonlinear equat ions of mathematical physics by the
+methodofinversescattering.II. FunctionalAnalysisandItsApplications ,13,166–174(1979).
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+dimensionalSpace-time Comm.Math.Phys. 74,21–40(1980).
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+method,Plenum,N.Y.(1984).
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+13. V. S. Gerdjikov. Generalised Fourier transformsfor the soliton equations. Gauge covariant formula-
+tion.InverseProblems 2,no.1,51–74,(1986).
+14. V. S. Gerdjikov. The Generalized Zakharov–Shabat Syste m and the Soliton Perturbations. Theor.
+Math.Phys. 99,No.2,292–299(1994).
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+Theor.Math.Phys. 100,214-218,(1994).
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+Structures. EuropeanJ. Phys. 29B,177-182(2002).
+17. H. E. Nistazakis, D.J. Frantzeskakis, P.G. Kevrekidis, B.A. Malomed, and R. Carretero-Gonzt’alez.
+Bright-DarkSolitonComplexesinSpinorBose-EinsteinCon densates. Phys.Rev.A 77,033612(2008).
+18. V. S. Gerdjikov.Algebraic and Analytic Aspects of N-wave Type Equations. nlin.SI/0206014 ;Con-
+temporaryMathematics 301,35-68(2002).
+19. V.S.Gerdjikov.TheZakharov–Shabatdressingmethodam dtherepresentationtheoryofthesemisim-
+pleLiealgebras. Phys.Lett. A, 126A,n.3,184–188,(1987).
+20. R. Ivanov.On the dressing method for the generalised Zak harov˝UShabatsystem. Nuclear Physics B
+694[PM] 509˝U524(2004).
+21. A. B. Shabat. Inverse-scatteringproblem for a system of differential equations. Functional Analysis
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\ No newline at end of file
diff --git a/1001.0167.txt b/1001.0167.txt
new file mode 100644
index 0000000000000000000000000000000000000000..210a6c37c5550561bec19ea99ad7b0d07b8277b3
--- /dev/null
+++ b/1001.0167.txt
@@ -0,0 +1,671 @@
+arXiv:1001.0167v1  [cs.IT]  31 Dec 20091
+Position Modulation Code
+for Rewriting Write-Once Memories
+Yunnan Wu and Anxiao (Andrew) Jiang
+Abstract—A write-once memory (wom) is a storage medium
+formed by a number of “write-once” bit positions (wits), whe re
+each wit initiallyis in a ‘0’ state and can be changed to a ‘1’ s tate
+irreversibly. Examples of write-once memories include SLC flash
+memories andoptical disks.Thispaper presents a low comple xity
+coding scheme for rewriting such write-once memories, whic h
+is applicable to general problem configurations. The propos ed
+scheme is called the position modulation code , as it uses the
+positions of the zero symbols to encode some information. Th e
+proposed technique can achieve code rates higher than state -of-
+the-art practical solutions for some configurations. For in stance,
+there is a position modulation code that can write 56 bits 10
+times on 278 wits, achieving rate 2.01. In addition, the posi tion
+modulation code is shown to achieve a rate at least half of the
+optimal rate.
+Index Terms —Write-once memories, flash memories, position
+modulation.
+I. INTRODUCTION
+In their pioneeringwork [18], Rivest and Shamir considered
+the problem of rewriting a “write-once memory” (wom). A
+write-oncememoryconsistsofa numberof“write-once”cell s,
+where each wit initially is in a ‘0’ state and can be changed
+to a ‘1’ state irreversibly. Several types of storage media
+follow such a write-once memory model. Examples include
+SLC flash memories and optical disks. Rivest and Shamir [18]
+demonstrated that it is possible to rewrite such a write-onc e
+memorymultipletimes,usingcodingtechniques.Forexampl e,
+using 3 wits, we can write a 2-bit variable twice via a wom-
+code. The rateof this wom-code (i.e., capacity per wit) is
+2×2/3 = 1.33.
+In [18], Rivest and Shamir presented a wom-code that has
+the best asymptotic rate (as the cardinality of the variable ap-
+proaches infinity), for any given number of writes. This resu lt
+is done via a counting argument and the scheme is not con-
+structive. Rivest and Shamir and others have proposed vario us
+other wom-codes, which have low complexity. However, the
+coding rates of existing wom-codes with low complexity are
+still far from the theoretically achievable coding rates. T hus,
+the problem of finding good practical wom-codes remains an
+open challenge.
+In this paper, we present a low complexity coding scheme,
+which we call the position modulation code , for rewriting a
+write-once memory. The scheme is built upon a simple yet
+fundamental observation: If we flip kcells out of nbinary
+cells that are initially zero, there are n-choose-kways of
+Yunnan Wu is with Microsoft Research, One Microsoft Way, Red mond,
+WA, 98052. yunnanwu@microsoft.com .Anxiao (Andrew) Jiang is with
+the Computer Science and Engineering Dept., Texas A&MUnive rsity, College
+Station, TX 77843. ajiang@cse.tamu.edu11 
+01 10 
+00 
+0a be
+Fig. 1. Two wits are used to implement a symbol that can take fo ur values
+{0,a,b,e}. The all-one value is used to represent the “erased” state.
+doing so, which can represent n-choose-kdistinct messages.
+We call this encoding method position modulation . Clearly,
+position modulationcan be directly applied to the first writ e in
+write-once memories, because we start from the all-zero sta te.
+Several wom-codes can be viewed as performing position
+modulationin the first write.However,it isnot straightfor ward
+to apply position modulation to subsequent writes, because
+each write flips some wits, and the decoder only sees the
+overall set of written wits but not when they are written.
+The scheme proposed in this paper performs position mod-
+ulation to all writes except the last one. Instead of directl y
+applying position modulation to the wits, we apply position
+modulation to symbols that are formed by multiple wits each.
+For example, as illustrated in Figure 1, using every two wits
+in a group, we obtain a symbol that can take four values.
+The all-one value is used to represent the “erased” state. At
+the beginning of each write, we erase all nonzero symbols
+by setting them to the all-one value. After this operation, t he
+remainingsymbols are all zero and thus we can apply position
+modulation.
+The rest of the paper is organized as follows. In Section II,
+we present a brief overview of existing research on wom
+codingand related topics. In Section III, we highlighta sim ple
+observation that encoding and decoding for position modula -
+tion can be implemented with polynomial time complexity.
+The proposed position modulation code is then described in
+Section IV. In Section V we compare the performance of
+the position modulation code with existing wom-codes. In
+Section VI we present the conclusions.
+II. OVERVIEW OF RELATEDSTUDIES
+We briefly overview the existing studies on wom codes.
+They will be referred to with more details later in the paper
+when we compare the performance of different codes. Rivest
+and Shamir defined wom codes in their pioneering paper [18],
+in which they presented a number of individual wom codes2
+(the most well-known of which is the code for writing two
+bits twice in three wits) and some families of wom codes
+(including the linear wom codes and the tabular codes). The
+wom code model was generalized in [4] by allowing the
+cell state transitions to be any acyclic directed graph; in
+addition, two families of wom codes for data with alphabet
+sizethreeandfourwereshown.In[16]severalindividualwo m
+codes were constructed using projective geometries. In [3] a
+wom code construction based on error-correcting codes was
+presented. As an example, based on the C[23,12,7]Golay
+code, a wom code for writing 11 bits three times in 23 wits
+was shown. More works on wom include [6], [9], [21], [22],
+which studied the capacity and error correction of write-on ce
+memories.
+WOM is related to the study on defective memory [8], [10],
+[14], write unidirectional memory [17], [19], [20], and wri te
+efficient memory [1], [7]. It is also related to the study on
+codingforflashmemories,wheremanyoftheproposedcoding
+schemes are based on the monotonic transitions of flash cell
+states [11], [13]. In particular, the works on rewriting cod es
+for flash memories extend wom codes [2], [5], [12], [15].
+Themotivationforthisstudyis to lookfora generalmethod
+for constructing wom codes that can achieve low encod-
+ing/decoding complexity and high rates, which can potentia lly
+be used in practice (e.g., for flash memories). As a result, ou r
+focus is on cases where the number of rewrites is reasonably
+small and the data size is moderate, instead of asymptotic
+settings.
+III. POSITION MODULATION HASPOLYNOMIAL
+COMPLEXITY
+We use the following notations for wom codes in [18].
+A/an}brack⌉tl⌉{tv/an}brack⌉tri}htt/nwomcode is a code that can write a variable
+of cardinality v ttimes, using nwits. More generally, a
+/an}brack⌉tl⌉{tv1,...,v t/an}brack⌉tri}ht/nwomcode is a code that can write a variable
+of cardinality v1the first time, a variable of cardinality v2the
+second time, and so on, using nwits. The i-th write is also
+called the i-th generation. The rate of the code is:
+log2(v1...vt)
+n. (1)
+In position modulation, we use a length- nbinary vector
+withkones to represent a variable of cardinality/parenleftbiggn
+k/parenrightbigg
+. Let
+Udenote the set of all length- nbinary vectors with kones.
+The most natural approach is to associate each vector u∈ U
+with its index in the lexical sorted list of U. Let
+ℓ:U /mapsto→/braceleftbigg
+0,1,...,/parenleftbigg
+n
+k/parenrightbigg
+−1/bracerightbigg
+(2)
+be a function that computes the lexical order of a vector
+u∈ U. We now show a simple observation, that the encoding
+and decoding functions ℓ(·)andℓ−1(·)can be implemented
+efficiently.
+First, consider an example where n= 7andk= 3.
+Figure 2 illustrates the process of computing the lexical or der
+of the sequence 0101100. The idea is to count the number of
+sequences that have a lower index. We start with the leftmost00***** 
+0100*** 
+01010** 
+Fig. 2. Illustration of the process of computing the lexical order of the
+sequence 0101100.
+1 and move to the right. First, if we flip the leftmost 1 to
+0, then clearly vectors in Uof the form 00∗ ∗ ∗ ∗∗ are all
+lexically before 0101100; there are/parenleftbigg
+5
+3/parenrightbigg
+of such vectors. If
+we keep the second bit as 0, the next group of vectors with a
+lower index than 0101100 is of the form 0100∗∗∗. There are /parenleftbigg3
+2/parenrightbigg
+of such vectors. If we keep the first two 1’s, the next
+group of vectors with a lower index than 0101100 is of the
+form01010∗∗. There are/parenleftbigg2
+1/parenrightbigg
+of such vectors. Therefore,
+the index of 0101100 is
+g(0101100)=/parenleftbigg
+5
+3/parenrightbigg
++/parenleftbigg
+3
+2/parenrightbigg
++/parenleftbigg
+2
+1/parenrightbigg
+= 15.
+More generally, consider a vector u∈Uthat haskone’s at
+positions n > i1> ... > i k≥0(here the bit positions are
+labeled as 0 to n−1from right to left). We have:
+ℓ(u) =/parenleftbigg
+i1
+k/parenrightbigg
++/parenleftbigg
+i2
+k−1/parenrightbigg
++...+/parenleftbigg
+ik
+1/parenrightbigg
+.(3)
+Conversely, given ℓ(u), to find u, we determine the bits
+from left to right. First, i1is determined as the largest integer
+suchthat/parenleftbigg
+i1
+k/parenrightbigg
+≤ℓ(u). Next,i2isdeterminedasthe largest
+integer such that/parenleftbiggi2
+k−1/parenrightbigg
+≤ℓ(u)−/parenleftbiggi1
+k/parenrightbigg
+. This process
+can be continued until all ones have been determined. All the
+above computation has time complexity polynomial in n.
+IV. POSITION MODULATION CODE
+Theorem 1: Given integers v1,...,v tandm≥2, lethi,
+1≤i≤tbe integers satisfying
+(a)h1> h2> ... > h t>0.
+(b)/summationtexth1−h2
+k=0/parenleftbigg
+h1
+k/parenrightbigg
+(2m−1)k≥v1,
+(c)/summationtexthi−hi+1
+k=1/parenleftbigghi
+k/parenrightbigg
+(2m−2)k≥vi, i= 2,...,t−1.
+(d)(2m−1)ht−1≥vt.
+Then, there exists a /an}brack⌉tl⌉{tv1,...,v t/an}brack⌉tri}ht/(m×h1)wom-code.
+Proof:We organize the n=mh1wits ash1groups, where
+eachgroupconsistsof mwitsandrepresentsa symbolthatcan
+take2mvalues. The state of the storage cells is then described
+bya vector x= [x1,...,x h1], whereeachcomponentcantake
+values from {0,1,...,2m−1}(each value corresponds to its
+binary representation). Two of the states are special, the z ero
+state implemented by the all-zero codeword and the erased
+state implemented by the all-one codeword.3
+At the beginning of each write, we erase all nonzero
+symbols by setting them to the all-one value. The remaining
+symbols are thus all zeros. Using these remaining symbols,
+we encode the message by the number of zero symbols, the
+positions of the zero symbols, and the values of the nonzero
+symbols.
+We now describe the encoding process in detail. For the
+first write, we select 0 up to h1−h2symbols and write a
+variablewithvaluefrom {1,...,2m−1}toeachofthem.Ifwe
+write into ksymbols, then the positions of the ksymbols can
+represent a variable of cardinality/parenleftbigg
+h1
+k/parenrightbigg
+, and the values of
+theseksymbols can represent a variable of cardinality (2m−
+1)k. Since we can write 0,...,h 1−h2symbols, in total we
+can represent a variable of cardinality
+h1−h2/summationdisplay
+k=0/parenleftbiggh1
+k/parenrightbigg
+(2m−1)k, (4)
+which is at least v1according to condition (b). Hence the first
+write can be done.
+For thei-th write with 1< i < t , first we erase all
+nonzero symbols by changing them into the all-one state.
+If there are more than hiremaining zero symbols, then we
+arbitrarily erase some additional zero symbols so that ther e
+are exactly hiremaining zeros. Then among the hiremaining
+zero symbols, we pick 1 up to hi−hi+1entries and write a
+value from {1,...,2m−2}to each of them. Since we can
+write1,...,h i−hi+1symbols, in total we can represent a
+variable of cardinality
+hi−hi+1/summationdisplay
+k=1/parenleftbigg
+hi
+k/parenrightbigg
+(2m−2)k(5)
+which is at least viaccording to condition (c). Hence the i-th
+write can be done.
+The last write is donedifferentlyin that positionmodulati on
+is not used. First we erase all nonzero symbols and possibly
+some additional zero symbols so that there are exactly htzero
+symbols. We simply use these htpositions to represent the
+variable by setting each position from {0,1,...,2m−2},
+except that the all zero codeword is not used. Since (2m−
+1)ht−1≥vt, the last write can be done.
+Decoding is done accordingly. First, the generation number
+(i.e.,whichwrite)isdecodedfromthenumberofzerosymbol s
+k0. Ifk0≥h2, then it is the first write. If h2> k0≥h3, then
+it is the second write; and so on. If ht> k0, then it is the
+last write. Next, the information message is decoded from th e
+number of zeros, the positions of the zeros, and the values of
+the nonzero entries (with erased symbols discarded in all bu t
+the first generation). .
+Since the encoding and decoding for position modulation
+can be done with polynomial complexity (in logv1,...,
+logvt), the resulting position modulation code has polynomial
+encoding and decoding complexity.
+A. Determining the Code Parameters
+To find the coding parameters, we determine the tnumbers
+h1,...,h tin reverse order. First, we choose htas the smallestnumber that satisfies condition (d). More specifically, we
+choosehtto be:
+ht=/ceilingleftbigglog2(vt+1)
+log2(2m−1)/ceilingrightbigg
+. (6)
+Next, we choose ht−1as the smallest number greater than ht
+that satisfies condition (c); and so on. More specifically, fo r
+1< i < twe choose hito be:
+hi=hi+1+δi, (7)
+δi= min/braceleftBigg
+δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleδ/summationdisplay
+k=1/parenleftbigg
+hi+1+δ
+k/parenrightbigg
+(2m−2)k≥vi/bracerightBigg
+(8)
+Wechoose h1tobethesmallestnumberthatsatisfiescondition
+(b), i.e.,
+h1=h2+δ1, (9)
+δ1= min/braceleftBigg
+δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleδ/summationdisplay
+k=0/parenleftbiggh2+δ
+k/parenrightbigg
+(2m−1)k≥v1/bracerightBigg
+(10)
+Lemma 1: Considergivenparameters m,t,andv1=...=
+vt=v. Letδi=hi−hi+1. The above process (namely (6)-
+(10)) will output an increasing sequence ht,ht−1,...,h 1with
+nondecreasing increments, i.e.,
+δt−1≥δt−2≥...≥δ1. (11)
+Proof:From (8), we have
+δi/summationdisplay
+k=1/parenleftbigghi+1+δi
+k/parenrightbigg
+(2m−2)k≥v. (12)
+Sincehi> hi+1, this implies that
+δi/summationdisplay
+k=1/parenleftbigghi+δi
+k/parenrightbigg
+(2m−2)k≥v. (13)
+and
+δi/summationdisplay
+k=0/parenleftbigg
+hi+δi
+k/parenrightbigg
+(2m−1)k≥v. (14)
+From (8) and (10) we know that δi−1≤δi.
+Example 1: Consider a code that can write 56 bits of data
+10 times using 278 wits, where a symbol is formed by m= 2
+wits. The rate of this code is:
+56×10
+139×2= 2.01...
+In this case, the code paramters are:
+h10= 36,
+h9= 51,
+h8= 64,
+h7= 76,
+h6= 88,
+h5= 99,
+h4= 110,
+h3= 120,
+h2= 130,
+h1= 139.4
+Note from (6)–(10) that if we want to obtain a code that
+writes56bitsofdatafor t′>10times,thenthefirst9numbers
+in the above list will remain the same.
+B. Performance Characterization
+We next present a performance characterization of the
+position modulationcode, by comparingit with a performanc e
+bound on womcodes in [18]. The original bound in [18] is for
+the/an}brack⌉tl⌉{tv/an}brack⌉tri}htt/ncase; however,the argumentcan easily be extended
+tothe/an}brack⌉tl⌉{tv1,...,v t/an}brack⌉tri}ht/ncase,asmentionedin[16].Thefollowing
+lemma presents the bound for the /an}brack⌉tl⌉{tv1,...,v t/an}brack⌉tri}ht/ncase.
+Lemma 2 (Bound on womcodes [18]):
+Letw(/an}brack⌉tl⌉{tv1,...,v t/an}brack⌉tri}ht)denote the least nfor which a
+/an}brack⌉tl⌉{tv1,...,v t/an}brack⌉tri}ht/n-womcode exists. Then
+w(/an}brack⌉tl⌉{tv1,...,v t/an}brack⌉tri}ht)≥Zt(v1,...,v t). (15)
+HereZt(...)is at-variable function defined recursively as
+follows:
+Zt(v1,...,v t) =Zt−1(v2,...,v t)+δ(v1,Zt−1(v2,...,v t)),
+t≥1, (16)
+Z0= 0, (17)
+δ(v,m)∆= min/braceleftBigg
+δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleδ/summationdisplay
+i=0/parenleftbiggm+δ
+i/parenrightbigg
+≥v/bracerightBigg
+(18)
+Proof:The proof is by induction on t. The case t= 0is
+trivial. Now consider any /an}brack⌉tl⌉{tv1,...,v t/an}brack⌉tri}ht/nwomcode for t >0.
+For this code, after the first write, there must be at least
+Zt−1(v2,...,v t)zeros.(Ifthereareless than Zt−1(v2,...,v t)
+zeros after the first write, it is not possible to accommo-
+date the remaining t−1writes since w(/an}brack⌉tl⌉{tv2,...,v t/an}brack⌉tri}ht)≥
+Zt−1(v2,...,v t).)
+Therefore, the first write can only use codewords with at
+mostκ∆=n−Zt−1(v2,...,v t)ones. Since the first generation
+needs to represent a variable of size v1, we must have:
+κ/summationdisplay
+i=0/parenleftbiggn
+i/parenrightbigg
+≥v1 (19)
+From the definition of δ(v,m), we see that
+κ≥δ(v1,Zt−1(v2,...,v t)). (20)
+Thus
+n≥Zt−1(v2,...,v t)+δ(v1,Zt−1(v2,...,v t)).(21)
+Since (21) holds for any /an}brack⌉tl⌉{tv1,...,v t/an}brack⌉tri}ht/nwomcode,
+w(/an}brack⌉tl⌉{tv1,...,v t/an}brack⌉tri}ht)≥Zt(v1,...,v t).
+Theorem 2: Givenv1,...,v twithvi≥2, consider
+h1,...,h tdetermined based on (6)–(10) for m= 2. Then
+h1≤Zt(v1,...,v t)≤w(/an}brack⌉tl⌉{tv1,...,v t/an}brack⌉tri}ht).(22)
+Therefore, the position modulation code achieves a rate tha t
+is at least half the optimal rate.Proof:Withm= 2, we have
+ht=/ceilingleftbigglog2(vt+1)
+log23/ceilingrightbigg
+(23)
+ht−1=ht+δt−1, (24)
+...
+h1=h2+δ1, (25)
+where
+δt−1= min/braceleftBigg
+δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleδ/summationdisplay
+k=1/parenleftbigght+δ
+k/parenrightbigg
+2k≥vt−1/bracerightBigg
+,(26)
+δt−2= min/braceleftBigg
+δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleδ/summationdisplay
+k=1/parenleftbigg
+ht−1+δ
+k/parenrightbigg
+2k≥vt−2/bracerightBigg
+,(27)
+...
+δ1= min/braceleftBigg
+δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleδ/summationdisplay
+k=0/parenleftbigg
+h2+δ
+k/parenrightbigg
+3k≥v1/bracerightBigg
+(28)
+By expanding the expression of Zt(v1,...,v t), we have
+Z1(vt) =δ(vt,0) (29)
+Z2(vt−1,vt) =Z1(vt)+δ′
+t−1 (30)
+... (31)
+Zt(v1,...,v t) =Zt−1(v2,...,v t)+δ′
+1,(32)
+where
+δ′
+t−1∆=δ(vt−1,Z1(vt)) (33)
+δ′
+t−2∆=δ(vt−2,Z2(vt−1,vt)) (34)
+...
+δ′
+1∆=δ(v1,Zt−1(v2,...,v t)). (35)
+To simplify notations, we use the shorthand notation Zkto
+refer toZk(vt−k+1,...,v t). We now show via induction over
+kthat
+ht−k+1≤Zk. (36)
+The case k= 1follows from the fact that for vt>1,
+δ(vt,0) =⌈log2vt⌉ ≥/ceilingleftbigglog2(vt+1)
+log23/ceilingrightbigg
+=ht.(37)
+Supposeht−k+1≤Zk. We now show that ht−k≤Zk+1.
+Note that δ′
+t−k≥1and
+δ′
+t−k/summationdisplay
+k=1/parenleftbiggZk+1
+k/parenrightbigg
+2k≥δ′
+t−k/summationdisplay
+k=0/parenleftbiggZk+1
+k/parenrightbigg
+≥vt−k(38)
+δ′
+t−k/summationdisplay
+k=0/parenleftbiggZk+1
+k/parenrightbigg
+3k≥δ′
+t−k/summationdisplay
+k=0/parenleftbiggZk+1
+k/parenrightbigg
+≥vt−k.(39)5
+SinceZk≥ht−k+1,Zk+δ′
+t−k−ht−k+1≥δ′
+t−k. Thus
+Zk+δ′
+t−k−ht−k+1/summationdisplay
+k=1/parenleftbigg
+Zk+1
+k/parenrightbigg
+2k≥vt−k(40)
+Zk+δ′
+t−k−ht−k+1/summationdisplay
+k=0/parenleftbiggZk+1
+k/parenrightbigg
+3k≥vt−k.(41)
+SinceZk+1=Zk+δ′
+t−k−ht−k+1+ht−k+1, from (26)–(28)
+we see that
+δt−k≤Zk+δ′
+t−k−ht−k+1, (42)
+or equivalently,
+ht−k≤Zk+1. (43)
+Thus, using induction we can show that h1≤Zt(v1,...,v t).
+V. PERFORMANCE COMPARISON
+In this section we compare the performance of the position
+modulation codes with existing wom-codes. For the position
+modulation code, there are some cases where setting m=
+3gives slightly better performance than setting m= 2. For
+example, to write 56 bits twice, the position modulation cod e
+needs 98 wits with m= 2, and 96 wits with m= 3. However,
+settingm= 2generally gives good performance among all
+choices of m. In the following, we use m= 2for the position
+modulation code.
+Table I gives the performance comparison of position mod-
+ulation code (we use v= 256) and known low complexity
+/an}brack⌉tl⌉{tv/an}brack⌉tri}htt/ncodeswith the best rates, for t= 2,...,10. Theknown
+low complexity /an}brack⌉tl⌉{tv/an}brack⌉tri}htt/ncodes with the best rates are given in
+the top row of the table. The position modulation codes and
+their rates are given in the bottom row of the table.
+We now explain the codes in the top row from left to right.
+The/an}brack⌉tl⌉{t26/an}brack⌉tri}ht2/7code is presented in [18] and this code is found
+via computer search, according to [18]. The /an}brack⌉tl⌉{t63/an}brack⌉tri}ht3/12code is
+from James B. Saxe’s construction of a /an}brack⌉tl⌉{t65,81,63/an}brack⌉tri}ht/12code,
+which was mentioned in [18]. There are two constructions
+of/an}brack⌉tl⌉{t7/an}brack⌉tri}ht4/7codes. One is a cyclic womcode constructed by
+David Leavitt (mentioned in [18]) and the other is a womcode
+constructed from projective geometry by Frans Merkx [16].
+The/an}brack⌉tl⌉{t11/an}brack⌉tri}ht5/11womcode, designed by M. Beveraggi, is based
+on Steiner pentagonal systems, according to [3]. There are
+two constructions of the /an}brack⌉tl⌉{t16/an}brack⌉tri}ht6/15code. One is the linear
+coset code given in [3] (Proposition 5 in [3]). The other is
+the linear code given in [18]. The /an}brack⌉tl⌉{t15/an}brack⌉tri}ht7/15womcode is also
+constructed from projective geometry [16]. The next three
+codes are all obtained by combining the /an}brack⌉tl⌉{t15/an}brack⌉tri}ht7/15womcode
+with some other small womcodes, since there are no known
+specific designs for t= 8,9,10with rates higher than 1.62.
+This is done by using two observations given in [18]. First,
+by concatenating a /an}brack⌉tl⌉{tv/an}brack⌉tri}htt1/n1code and a /an}brack⌉tl⌉{tv/an}brack⌉tri}htt2/n2code side
+by side, we can have a /an}brack⌉tl⌉{tv/an}brack⌉tri}htt1+t2/(n1+n2)code (Information
+is represented by the modulo- vsum of the values of the two
+subcodes). Second, by concatenating a /an}brack⌉tl⌉{tv1/an}brack⌉tri}htt/n1code and a/an}brack⌉tl⌉{tv2/an}brack⌉tri}htt/n2code side by side, we obtain a /an}brack⌉tl⌉{tv1·v2/an}brack⌉tri}htt/(n1+n2)
+wom-code (Information is represented as the ordered pair of
+thetwovalues).Weobtainthe /an}brack⌉tl⌉{t15/an}brack⌉tri}ht8/19codebyconcatenating
+the/an}brack⌉tl⌉{t15/an}brack⌉tri}ht7/15code with a /an}brack⌉tl⌉{t15/an}brack⌉tri}ht1/4code (simply using the 4
+bits to represent a 15-ary variable). We obtain the /an}brack⌉tl⌉{t15/an}brack⌉tri}ht9/21
+code by concatenating the /an}brack⌉tl⌉{t15/an}brack⌉tri}ht7/15code with a /an}brack⌉tl⌉{t15/an}brack⌉tri}ht2/6code
+(based on the /an}brack⌉tl⌉{t16/an}brack⌉tri}ht2/6code given in [18]). We obtain the
+/an}brack⌉tl⌉{t15/an}brack⌉tri}ht10/24code by concatenating the /an}brack⌉tl⌉{t15/an}brack⌉tri}ht7/15code with a
+/an}brack⌉tl⌉{t15/an}brack⌉tri}ht3/9code, which is obtained by concatenating a /an}brack⌉tl⌉{t5/an}brack⌉tri}ht3/5
+code and a /an}brack⌉tl⌉{t3/an}brack⌉tri}ht3/4code. The /an}brack⌉tl⌉{t5/an}brack⌉tri}ht3/5code is a cyclic code
+designed by David Klarner (see [18]) and the /an}brack⌉tl⌉{t3/an}brack⌉tri}ht3/4code is
+given in [4].
+For eacht∈ {2,...,10},the best rate is shown in boldface.
+It is seen that the position modulation code offers higher ra tes
+than state-of-the-art solutions for t= 5,6,8,9,10.
+For larger values of t, we compare the position modulation
+code with three existing general code designs that produce
+classes of codes. In [18], Rivest and Shamir presented a
+/an}brack⌉tl⌉{tv/an}brack⌉tri}ht1+v/4/(v−1)linear code. The rate of this code is less
+than 2 for t≤50. In [4], Fiat and Shamir presented a
+/an}brack⌉tl⌉{t3/an}brack⌉tri}htt/(t+1)code that works for arbitrary t. The rate of this
+code is always less than 1.59. In [3], Cohen et al.presented
+a/an}brack⌉tl⌉{t2r/an}brack⌉tri}ht2r−2+2/(2r−1)code that works for r≥4; according
+to the paper, the encoding process is NP-hard. Figure 3 shows
+the rates of these wom-coding schemes, where the position
+modulation code is for m= 2andv= 232. It is seen that the
+position modulation code achieves higher rates.
+VI. CONCLUSION
+We presented a method for constructing wom codes with
+low encoding and decoding complexity and high rates, which
+works for general problem configurations. The proposed
+method is called position modulation code , as it is built upon
+a simple and yet fundamental observation, that positions of
+zeros in a binary array can be used to encode information.
+The position modulation code applies position modulation
+to symbols formed by multiple cells and performs position
+modulation in all writes except the last one, by using a “soft ”
+erasing operation to reset the state between writes.
+It is proven that the position modulation code achieves a
+rate that is at least half of the optimal rate. Furthermore, t he
+proposed position modulation codes is seen to have superior
+rates compared to existing wom codes. The low complexity
+and high rate of the position modulation code make it a
+promising candidate for practical applications.
+In addition, as a family of codes that accommodate general
+parameter configurations, the position modulation code can
+be readily used in related rewriting schemes. For example,
+the trajectory code [12] for a generalized rewriting model
+can use the position modulation code as a subcode for better
+performance.
+REFERENCES
+[1] R. Ahlswede and Z. Zhang, “On multiuser write-efficient m em-
+ories,”IEEE Trans. on Inform. Theory , vol. 40, no. 3, pp. 674–
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+TABLE I
+PERFORMANCE COMPARISON OF POSITION MODULATION CODE AND KNOW N/an}bracketle{tv/an}bracketri}htt/nCODES
+t= 2 t= 3 t= 4 t= 5 t= 6 t= 7 t= 8 t= 9 t= 10
+Known /an}bracketle{t26/an}bracketri}ht2/7/an}bracketle{t63/an}bracketri}ht3/12 /an}bracketle{t7/an}bracketri}ht4/7/an}bracketle{t11/an}bracketri}ht5/11/an}bracketle{t16/an}bracketri}ht6/15/an}bracketle{t15/an}bracketri}ht7/15/an}bracketle{t15/an}bracketri}ht8/19/an}bracketle{t15/an}bracketri}ht9/21/an}bracketle{t15/an}bracketri}ht10/24
+/an}bracketle{tv/an}bracketri}htt/n
+codesR=1.34R=1.49R=1.60R= 1.57R= 1.60R=1.82R= 1.65R= 1.67R= 1.63
+Position /an}bracketle{t256/an}bracketri}ht2/98/an}bracketle{t256/an}bracketri}ht3/124/an}bracketle{t256/an}bracketri}ht4/150/an}bracketle{t256/an}bracketri}ht5/172/an}bracketle{t256/an}bracketri}ht6/196/an}bracketle{t256/an}bracketri}ht7/216/an}bracketle{t256/an}bracketri}ht8/238/an}bracketle{t256/an}bracketri}ht9/258/an}bracketle{t256/an}bracketri}ht10/278
+modulation
+codeR= 1.14R= 1.35R= 1.49R=1.63R=1.71R= 1.81R=1.88R=1.95R=2.01
+0510152025303540455011.21.41.61.822.22.42.62.83
+t (number of writes)rateRates of various wom−coding schemes
+  
+Position modulation with m=2 and v=232
+Linear code by Rivest and Shamir
+<3>t/(t+1) code by Fiat and Shamir
+Coset code by Cohen et al.
+Fig. 3. Rates for various wom-coding schemes.
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+[22] G. Z´ emor and G. Cohen, “Error-correcting WOM-codes,” IEEE
+Trans.onInform.Theory ,vol.37,no.3,pp.730–734, May1991.
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+arXiv:1001.0168v1  [nlin.SI]  31 Dec 2009Bose-Einstein condensates with F= 1andF= 2. Reductions and
+soliton interactions of multi-component NLS models.
+V. S. Gerdjikov, N. A. Kostov and T. I. Valchev
+Institute for Nuclear Research and Nuclear Energy,
+Bulgarian academy of sciences
+72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria
+ABSTRACT
+We analyze a class of multicomponent nonlinear Schr¨ odinger equatio ns (MNLS) related to the symmetric BD.I-
+type symmetric spaces and their reductions. We briefly outline the d irect and the inverse scattering method for
+the relevant Lax operators and the soliton solutions. We use the Za kharov-Shabatdressing method to obtain the
+two-soliton solution and analyze the soliton interactions of the MNLS equations and some of their reductions.
+Keywords: Bose-Einstein condensates, Multicomponent nonlinear Schr¨ oding er equations, Soliton solutions,
+Soliton interactions
+1. INTRODUCTION
+Bose-Einstein condensate (BEC) of alkali atoms in the F= 1 hyperfine state, elongated in xdirection and
+confined in the transverse directions y,zby purely optical means are described by a 3-component normalized
+spinor wave vector Φ(x,t) = (Φ 1,Φ0,Φ−1)T(x,t) satisfying the nonlinear Schr¨ odinger (MNLS) equation1see
+also:2–6
+i∂tΦ1+∂2
+xΦ1+2(|Φ1|2+2|Φ0|2)Φ1+2Φ∗
+−1Φ2
+0= 0,
+i∂tΦ0+∂2
+xΦ0+2(|Φ−1|2+|Φ0|2+|Φ1|2)Φ0+2Φ∗
+0Φ1Φ−1= 0, (1)
+i∂tΦ−1+∂2
+xΦ−1+2(|Φ−1|2+2|Φ0|2)Φ−1+2Φ∗
+1Φ2
+0= 0.
+spinor BEC with F= 2 for rather specific choices of the scattering lengths in dimension less coordinates takes
+the form:7
+i∂tΦ±2+∂xxΦ±2+2(/vectorΦ,/vectorΦ∗)Φ±2−(2Φ2Φ−2−2Φ1Φ−1+Φ2
+0)Φ∗
+∓2,
+i∂tΦ±1+∂xxΦ±1+2(/vectorΦ,/vectorΦ∗)Φ±1−(2Φ2Φ−2−2Φ1Φ−1+Φ2
+0)Φ∗
+∓1, (2)
+i∂tΦ0+∂xxΦ0+2(/vectorΦ,/vectorΦ∗)Φ0−(2Φ2Φ−2−2Φ1Φ−1+Φ2
+0)Φ∗
+0.
+Both models have natural Lie algebraic interpretation and are relat ed to the symmetric spaces BD.I≃
+SO(n+2)/SO(n)×SO(2) with n= 3 andn= 5 respectively. They are integrable by means of inverse scatterin g
+transform method.8,9,12Using a modification of the Zakharov-Shabat ‘dressing method’ we d escribe the soliton
+solutions1,10and the effects of the reductions on them.
+Sections 2 and 3 contain the basic details on the direct and inverse sc attering problems for the Lax operator.
+Section 4 outlines the effects of the algebraic reductions of the MNL S. In Section 5 using the Zakharov-Shabat
+dressing method we derive the one- and two-soliton solutions of the MNLS and discuss their properties. Section
+6 is dedicated to the analysis of the soliton interactions of the MNLS. To this end we evaluate the limits of the
+generic two-soliton solution for t→ ±∞. As a result we establish that the effect of the interactions on the s oliton
+parameters is analogous to the one for the scalar NLS equation and consists in shifts of the ‘center of mass’ and
+shift in the phase.
+Further author information: (Send correspondence to V. S. G erdjikov)
+V. S. Gerdjikov: E-mail: gerjikov@inrne.bas.bg, Telephon e: +3592 979 56382. THE METHOD FOR SOLVING MNLS WITH F= 1ANDF= 2
+MNLS equations for the BD.I.series of symmetric spaces (algebras of the type so(n+2) andJdual toe1) have
+the Lax representation [ L,M] = 0 as follows8,11,12
+Lψ(x,t,λ)≡i∂xψ+U(x,t,λ)ψ(x,t,λ) = 0, Mψ (x,t,λ)≡i∂tψ+V(x,t,λ)ψ(x,t,λ) = 0,
+U(x,t,λ) =Q(x,t)−λJ, V (x,t,λ) =V0(x,t)+λV1(x,t)−λ2J,
+V1(x,t) =Q(x,t), V 0(x,t) =iad−1
+JdQ
+dx+1
+2/bracketleftbig
+ad−1
+JQ,Q(x,t)/bracketrightbig
+.(3)
+where ad JX= [J,X] and ad−1
+Jis well defined on the image of ad Jing;
+Q=
+0/vector qT0
+/vector p∗0s0/vector q
+0/vector p†s00
+, J= diag(1,0,...0,−1). (4)
+The vector /vector qforF= 1 (resp.F= 2) is 3- (resp. 5-) component and has the form
+/vector q= (Φ1,Φ0,Φ−1)T, /vector q= (Φ2,Φ1,Φ0,Φ−1,Φ−2)T, (5)
+and the corresponding matrices s0enter in the definition of so(2r+1) withr= 2 andr= 3:
+X∈so(2r+1), X+S0XTS0= 0, S 0=2r+1/summationdisplay
+s=1(−1)s+1Es,n+1−s, S 0=
+0 0 1
+0−s00
+1 0 0
+.(6)
+ByEspabove we mean 2 r+ 1×2r+ 1 matrix with matrix elements ( Esp)ij=δsiδpj. With the definition of
+orthogonalityused in (6) the Cartan generators Hk=Ek,k−En+3−k,n+3−kare represented by diagonal matrices.
+If we make use of the typical reduction Q=Q†(or/vector p∗=/vector q) the generic MNLS type equations related to
+BD.I.acquire the form:
+i/vector qt+/vector qxx+2(/vector q†,/vector q)/vector q−(/vector q,s0/vector q)s0/vector q∗= 0. (7)
+The Hamiltonians for the MNLS equations (7) are given by
+HMNLS=/integraldisplay∞
+−∞dx/parenleftbigg
+(∂x/vector q†,∂x/vector q)−(/vector q†,/vector q)2+1
+2|(/vector qT,s0/vector q)|2/parenrightbigg
+. (8)
+3. THE DIRECT AND THE INVERSE SCATTERING PROBLEM
+3.1 The fundamental analytic solution
+We remind some basic features of the inverse scattering theory fo r the Lax operators L, see.11,12There we
+have made use of the general theory developed in13–16,18and the references therein. The Jost solutions of Lare
+defined by:
+lim
+x→−∞φ(x,t,λ)eiλJx=11,lim
+x→∞ψ(x,t,λ)eiλJx=11 (9)
+and the scattering matrix T(λ,t)≡ψ−1φ(x,t,λ). The special choice of Jand the fact that the Jost solutions
+and the scattering matrix take values in the group SO(2r+1) we can use the following block-matrix structure
+ofT(λ,t)
+T(λ,t) =
+m+
+1−/vectorb−Tc−
+1
+/vectorb+T22−s0/vectorB−
+c+
+1/vectorB+Ts0m−
+1
+,ˆT(λ,t) =
+m−
+1/vectorB−Tc−
+1
+−/vectorB+ˆT22s0/vectorb−
+c+
+1−/vectorb+Ts0m+
+1
+, (10)where/vectorb±(λ,t) and/vectorB±(λ,t) are 2r−1-component vectors, T22(λ) is 2r−1×2r−1 block matrix, and m±
+1(λ),
+andc±
+1(λ) are scalar functions. Such parametrization is compatible with the g eneralized Gauss decompositions
+ofT(λ) which read as follows:
+T(λ,t) =T−
+JD+
+JˆS+
+J, T(λ,t) =T+
+JD−
+JˆS−
+J,
+T∓
+J=e±(/vector ρ±,/vectorE∓
+1), S±
+J=e±(/vector τ±,/vectorE±
+1), D±
+J= diag/parenleftbig
+(m±
+1)±1,m±
+2,(m±
+1)∓1/parenrightbig
+,(11)
+where/parenleftig
+/vector ρ+,/vectorE+
+1/parenrightig
+=r−1/summationdisplay
+k=1(ρ+
+kEe1−ek+1+ρ+
+¯kEe1+ek+1)+ρ+
+rEe1,
+/parenleftig
+/vector ρ−,/vectorE−
+1/parenrightig
+=r−1/summationdisplay
+k=1(ρ−
+kE−e1+ek+1+ρ−
+¯kE−e1−ek+1)+ρ−
+rE−e1,(12)
+and similar expressions for/parenleftig
+/vector τ±,/vectorE±
+1/parenrightig
+. The functions m±
+1andn×nmatrix-valued functions m±
+2are analytic for
+λ∈C±. We have introduced also the notations:
+/vector ρ−=/vectorB−
+m−
+1, /vector τ−=/vectorB+
+m−
+1, /vector ρ+=/vectorb+
+m+
+1, /vector τ+=/vectorb−
+m+
+1.
+There are some additional relations which ensure that both T(λ) and its inverse ˆT(λ) belong to the orthogonal
+groupSO(2r+1) and that T(λ)ˆT(λ) =11.
+Next we introduce the fundamental analytic solution (FAS) χ±(x,t,λ) using the generalized Gauss decom-
+position of T(λ,t), see:15,19,20
+χ±(x,t,λ) =φ(x,t,λ)S±
+J(t,λ) =ψ(x,t,λ)T∓
+J(t,λ)D±
+J(λ), (13)
+This construction ensures that ξ±(x,λ) =χ±(x,λ)eiλJxare analytic functions of λforλ∈C±. IfQ(x,t) is a
+solution of the MNLS eq. (7) then the matrix elements of T(λ) satisfy the linear evolution equations12
+id/vectorb±
+dt±λ2/vectorb±(t,λ) = 0, id/vectorB±
+dt±λ2/vectorB±(t,λ) = 0,
+idm±
+1
+dt= 0, idm±
+2
+dt= 0.(14)
+Thus the block-diagonal matrices D±(λ) can be considered as generating functionals of the integrals of mo tion.
+The fact that all (2 r−1)2matrix elements of m±
+2(λ) forλ∈C±generate integrals of motion reflect the
+superintegrability of the model and are due to the degeneracy of t he dispersion law of (7). Note that D±
+J(λ)
+allow analytic extension for λ∈C±and that their zeroes and poles determine the discrete eigenvalues11ofL.
+3.2 The Riemann-Hilbert Problem
+The FAS for real λare linearly related12
+χ+(x,t,λ) =χ−(x,t,λ)G0,J(λ,t), G 0,J(λ,t) =ˆS−
+J(λ,t)S+
+J(λ,t) (15)
+Eq. (15) can be rewriten in an equivalent form for the FAS ξ±(x,t,λ) =χ±(x,t,λ)eiλJx:
+idξ±
+dx+Q(x)ξ±(x,λ)−λ[J,ξ±(x,λ)] = 0, (16)
+and the relation
+lim
+λ→∞ξ±(x,t,λ) =11, (17)Then these FAS satisfy the RHP’s
+ξ+(x,t,λ) =ξ−(x,t,λ)GJ(x,λ,t), G J(x,λ,t) =e−iλJ(x+λt)G−
+J(λ,t)eiλJ(x+λt), (18)
+Obviously the sewing function GJ(x,λ,t) is uniquely determined by the Gauss factors S±
+J(λ,t). In addition
+Zakharov-Shabat’s theorem13,14states that GJ(x,λ,t) depends on xandtin the way prescribed above then the
+corresponding FAS satisfy the linear systems (16).
+If we have solved the RHP’s and know the FAS ξ+(x,t,λ) then the formula
+Q(x,t) = lim
+λ→∞λ/parenleftig
+J−ξ+(x,t,λ)Jˆξ+(x,t,λ)/parenrightig
+, (19)
+allows us to recover the corresponding potential of L.
+4. REDUCTIONS OF MNLS
+Along with the typical reduction Q=Q†mentioned above one can impose additional reductions using the
+reduction group proposed by Mikhailov.21They are automatically compatible with the Lax representation of
+the corresponding MNLS eq. Below we make use of two Z2-reductions:22
+1)C1U†(x,t,λ∗)C−1
+1=U(x,t,λ), C 1V†(x,t,λ∗)C−1
+1=V(x,t,λ),
+2)C2UT(x,t,λ)C−1
+2=−U(x,t,λ), C 2VT(x,t,λ)C−1
+2=−V(x,t,λ),(20)
+whereC1andC2are involutions of the Lie algebra so(2r+ 1, i.e.C2
+i=11. They can be chosen to be either
+diagonal (i.e., elements of the Cartan subgroup of SO(2r+1)) or elements of the Weyl group.
+The typical reductions of the MNLS eqs. is a class 1) reduction obta ined by specifying C1to be the identity
+automorphism of g; below we list several choices for C1leading to inequivalent reductions:
+1a)C1=11, /vector p(x) =/vector q∗(x), 1b)C1=K1, /vector p (x) =K01/vector q∗(x),
+1c)C1=Se2, /vector p(x) =K02/vector q∗(x),1d)C1=Se2Se3, /vector p (x) =K03/vector q∗(x),
+2e)C2=K4, /vector q(x) =−K04s0/vector q(x), /vector p (x) =−K04s0/vector p(x),(21)
+where
+Kj= block-diag(1 ,K0j,1), K 01= diag(ǫ1,...,ǫ r−1,1,ǫr−1,...,ǫ 1), j= 1,2,3, (22)
+andǫj=±1. The matrices K02,K03andK4are not diagonal and may take the form:
+K02=
+0 0 1
+0−1 0
+1 0 0
+, K 4=
+0 0 1
+0K040
+1 0 0
+,
+K02=
+0 0 0 0 −1
+0 1 0 0 0
+0 0−1 0 0
+0 0 0 1 0
+−1 0 0 0 0
+, K 03=
+0 0 0 0 −1
+0 0 0 1 0
+0 0−1 0 0
+0 1 0 0 0
+−1 0 0 0 0
+,(23)
+Each of the above reductions impose constraints on the FAS, on th e scattering matrix T(λ) and on its Gauss
+factorsS±
+J(λ),T±
+J(λ) andD±
+J(λ). These have the form:
+(S+(λ∗))†=K−1
+jˆS−(λ)Kj(T+(λ∗))†=K−1
+jˆT−(λ)Kj(D+(λ∗))†=K−1
+jˆD−(λ)Kj
+/vector τ+=K0j/vector τ−,∗, /vector ρ+=K0j/vector ρ−,∗,(24)
+where the matrices Kjare specific for each choice of the automorphisms C1, see eqs. (21), (22).
+In particular, from the last line of (24) and (22) we get:
+(m+
+1(λ∗))∗=m−
+1(λ), (25)
+and consequently, if m+
+1(λ) has zeroes at the points λ+
+k, thenm−
+1(λ) has zeroes at:
+λ−
+k= (λ+
+k)∗, k= 1,...,N. (26)5. SOLITON SOLUTIONS
+Let us now make use of one of the versions of the dressing method13,14which allows one to construct singular
+solutions of the RHP. In order to obtain N-soliton solutions one has to apply dressing procedure with a 2 N-poles
+dressing factor of the form
+u(x,λ) =11+N/summationdisplay
+k=1/parenleftbiggAk(x)
+λ−λ+
+k+Bk(x)
+λ−λ−
+k/parenrightbigg
+. (27)
+TheN-soliton solution itself can be generated via the following formula
+QN,s(x) =N/summationdisplay
+k=1[J,Ak(x)+Bk(x)]. (28)
+The dressing factor u(x,λ) must satisfy the equation
+i∂xu+QN,su−λ[J,u] = 0 (29)
+and the normalization condition lim λ→∞u(x,λ) =11. The construction of u(x,λ)∈SO(n+2) is based on an
+appropriate anzatz specifying the form of its λ-dependence23,24
+The residues of uadmit the following decomposition
+Ak(x) =Xk(x)FT
+k(x), B k(x) =Yk(x)GT
+k(x).
+where all matrices involved are supposed to be rectangular and of m aximal rank s. By comparing the coefficients
+before the same powers of λ−λ±
+kin (29) we convince ourselves that the factors FkandGkcan be expressed by
+the fundamental analytic solutions χ±
+0(x,λ) as follows
+FT
+k(x) =FT
+k,0[χ+
+0(x,λ+
+k)]−1, GT
+k(x) =GT
+k,0[χ−
+0(x,λ−
+k)]−1.
+The constant rectangular matrices Fk,0andGk,0obey the algebraic relations
+FT
+k,0S0Fk,0= 0, GT
+k,0S0Gk,0= 0.
+The other two types of factors XkandYkare solutions to the algebraic system
+S0Fk=Xkαk+/summationdisplay
+l/negationslash=kXlFT
+lS0Fk
+λ+
+l−λ+
+k+/summationdisplay
+lYlGT
+lS0Fk
+λ−
+l−λ+
+k,
+S0Gk=/summationdisplay
+lXlFT
+lS0Gk
+λ+
+l−λ−
+k+Ykβk+/summationdisplay
+l/negationslash=kYlGT
+lS0Gk
+λ−
+l−λ−
+k.(30)
+The square s×smatricesαk(x) andβk(x) introduced above depend on χ+
+0andχ−
+0and their derivatives by λ
+as follows
+αk(x) =−FT
+0,k[χ+
+0(x,λ+
+k)]−1∂λχ+
+0(x,λ+
+k)S0F0,k+α0,k,
+βk(x) =−GT
+0,k[χ−
+0(x,λ−
+k)]−1∂λχ−
+0(x,λ−
+k)S0G0,k+β0,k.(31)
+Below for simplicity we will choose FkandGkto be 2r+ 1-component vectors. Then one can show that
+αk=βk= 0 which simplifies the system (30). We also introduce the following mor e convenient parametrization
+forFkandGk, namely (see eq. (33)):
+Fk(x,t) =S0|nk(x,t)/angbracketright=
+e−zk+iφk
+−√
+2s0/vector ν0k
+ezk−iφk
+, G k(x,t) =|n∗
+k(x,t)/angbracketright=
+ezk+iφk√
+2/vector ν0k∗
+e−zk−iφk
+, (32)where/vector ν0kare constant 2 r−1-component polarization vectors and
+zj=νj(x+2µjt)+ξ00, φ j=µjx+(µ2
+j−ν2
+j)t+δ00,
+/angbracketleftnT
+j(x,t)|S0|nj(x,t)/angbracketright= 0,or (/vector ν0,js0/vector ν0,j) = 1.(33)
+With this notations the polarization vectors automatically satisfy /angbracketleftnj(x,t)|S0|nj(x,t)/angbracketright= 0.
+Thus forN= 1 we get the system:
+|Y1/angbracketright=−(λ+
+1−λ−
+1)|n1/angbracketright
+/angbracketleftn†
+1|n1/angbracketright,|X1/angbracketright=(λ+
+1−λ−
+1)S0|n∗
+1/angbracketright
+/angbracketleftn†
+1|n1/angbracketright, (34)
+which is easily solved. As a result for the one-soliton solution we get:
+/vector q1s=−i√
+2(λ+
+1−λ−
+1)e−iφ1
+∆1/parenleftbig
+e−z1s0|/vector ν01/angbracketright+ez1|/vector ν∗
+01/angbracketright/parenrightbig
+,∆1= cosh(2z1)+/angbracketleft/vector ν†
+01|/vector ν01/angbracketright. (35)
+Forn= 3 we put ν0k=|ν0k|eα0kget:
+Φ1s;±1=−/radicalbig
+2|ν01;1ν01;3|(λ+
+1−λ−
+1)
+∆1e−iφ1±iβ13(cosh(z1∓ζ01)cos(α13)−isinh(z1∓ζ01)sin(α13)),
+Φ1s;0=−√
+2|ν01;2|(λ+
+1−λ−
+1)
+∆1e−iφ1(sinhz1cos(α02)+icoshz1sin(α02)),
+β13=1
+2(α03−α01), ζ 01=1
+2ln|ν01;3|
+|ν01;1|, α 13=1
+2(α03+α01),(36)
+Note that the ‘center of mass‘ of Φ 1s;1(resp. of Φ 1s;−1) is shifted with respect to the one of Φ 1s;0byζ01to the
+right (resp to the left); besides |Φ1s;1|=|Φ1s;−1|, i.e. they have the same amplitudes.
+Forn= 5 we put ν0k=|ν0k|eα0kand get analogously:
+Φ1s;±2=−/radicalbig
+2|ν01;1ν01;5|(λ+
+1−λ−
+1)
+∆1e−iφ1±iβ15(cosh(z1∓ζ01)cos(α15)−isinh(z1∓ζ01)sin(α15)),
+Φ1s;±1=/radicalbig
+2|ν01;2ν01;4|(λ+
+1−λ−
+1)
+∆1e−iφ1±iβ24(cosh(z1∓ζ02)cos(α24)−isinh(z1∓ζ01)sin(α24)),
+Φ1s;0=−√
+2|ν01;3|(λ+
+1−λ−
+1)
+∆1e−iφ1(coshz1cos(α03)−isinhz1sin(α03)),
+β15=1
+2(α05−α01), ζ 01=1
+2ln|ν01;5|
+|ν01;1|, α 15=1
+2(α05+α01),
+β24=1
+2(α04−α02), ζ 02=1
+2ln|ν01;4|
+|ν01;2|, α 24=1
+2(α04+α02),(37)
+Similarly the ‘center of mass‘ of Φ 1s;2and Φ 1s;1(resp. of Φ 1s;−2and Φ 1s;−1) are shifted with respect to the one
+of Φ1s;0byζ01andζ02to the right (resp to the left); besides |Φ1s;2|=|Φ1s;−2|and|Φ1s;1|=|Φ1s;−1|.
+ForN= 2 we get:
+|n1(x,t)/angbracketright=X2(x,t)f21
+λ+
+2−λ+
+1+Y1(x,t)κ11
+λ−
+1−λ+
+1+Y2(x,t)κ21
+λ−
+2−λ+
+1,
+|n2(x,t)/angbracketright=X1(x,t)f12
+λ+
+1−λ+
+2+Y1(x,t)κ12
+λ−
+1−λ+
+2+Y2(x,t)κ22
+λ−
+2−λ+
+2,
+S0|n∗
+1(x,t)/angbracketright=X1(x,t)κ11
+λ+
+2−λ+
+1+X2(x,t)κ11
+λ+
+2−λ−
+1+Y2(x,t)f∗
+21
+λ−
+2−λ−
+1,
+S0|n∗
+2(x,t)/angbracketright=X1(x,t)κ21
+λ+
+1−λ−
+2+X2(x,t)κ22
+λ+
+2−λ−
+2+Y1(x,t)f∗
+12
+λ−
+1−λ−
+2,(38)where
+κkj(x,t) =ezk+zj+i(φk−φj)+e−zk−zj−i(φk−φj)+2/parenleftig
+/vector ν†
+0k,/vector ν0j/parenrightig
+,
+fkj(x,t) =ezk−zj−i(φk−φj)+ezj−zk+i(φk−φj)−2/parenleftbig
+/vector νT
+0ks0/vector ν0j/parenrightbig
+,(39)
+In other words:
+M/vectorX≡
+0f21
+λ+
+2−λ+
+1κ11
+λ−
+1−λ+
+1κ21
+λ−
+2−λ+
+1f12
+λ+
+1−λ+
+20κ12
+λ−
+1−λ+
+2κ22
+λ−
+2−λ+
+2
+κ11
+λ+
+1−λ−
+1κ12
+λ+
+2−λ−
+10f∗
+21
+λ−
+2−λ−
+1
+κ21
+λ+
+1−λ−
+2κ22
+λ+
+2−λ−
+2f∗
+12
+λ−
+1−λ−
+20
+
+X1
+X2
+Y1
+Y2
+=
+|n1/angbracketright
+|n2/angbracketright
+S0|n∗
+1/angbracketright
+S0|n∗
+2/angbracketright
+. (40)
+We can rewrite Min block-matrix form:
+M=/parenleftbigg
+M11M12
+M21M22/parenrightbigg
+,M22=M∗
+11,M21=−MT
+12,
+M11=f12
+λ+
+2−λ+
+1/parenleftbigg0 1
+−1 0/parenrightbigg
+,M12=/parenleftiggκ11
+λ−
+1−λ+
+1κ21
+λ−
+2−λ+
+1κ12
+λ−
+1−λ+
+2κ22
+λ−
+2−λ+
+2/parenrightigg
+.(41)
+The inverse of Mis given by:
+M−1=/parenleftbigg(M11−M12ˆM∗
+11M21)−1−(M11−M12ˆM∗
+11M21)−1M12ˆM∗
+11
+−(M∗
+11−M21ˆM11M12)−1M21ˆM11 (M∗
+11−M21ˆM11M12)−1/parenrightbigg
+, (42)
+One can check by direct calculation that:
+M11−M12ˆM∗
+11M21=f∗
+12
+λ−
+2−λ−
+1Z/parenleftbigg0 1
+−1 0/parenrightbigg
+,
+M∗
+11−M21ˆM11M12=f12
+λ+
+2−λ+
+1Z/parenleftbigg
+0 1
+−1 0/parenrightbigg
+,
+Z=/parenleftbigg|f12|2
+|λ+
+2−λ+
+1|2−κ12κ21
+|λ+
+2−λ−
+1|2+κ11κ22
+4ν1ν2/parenrightbigg
+,(43)
+Finally we get:
+M−1=1
+Z
+0f∗
+12
+λ−
+1−λ−
+2−κ22
+λ+
+2−λ−
+2κ12
+λ+
+2−λ−
+1
+−f∗
+12
+λ−
+1−λ−
+20κ21
+λ+
+1−λ−
+2−κ11
+λ+
+1−λ−
+1κ22
+λ+
+2−λ−
+2−κ21
+λ+
+1−λ−
+20−f12
+λ+
+1−λ+
+2
+−κ12
+λ+
+2−λ−
+1κ11
+λ+
+1−λ−
+1f12
+λ+
+2−λ+
+10
+, (44)
+From eqs. (40) and (44) we obtain:
+|X1/angbracketright=1
+Z/parenleftbiggf∗
+12
+λ−
+1−λ−
+2|n2/angbracketright−κ22
+λ+
+2−λ−
+2S0|n∗
+1/angbracketright+κ12
+λ+
+2−λ−
+1S0|n∗
+2/angbracketright/parenrightbigg
+,
+|X2/angbracketright=1
+Z/parenleftbigg
+−f∗
+12
+λ−
+1−λ−
+2|n1/angbracketright+κ21
+λ+
+1−λ−
+2S0|n∗
+1/angbracketright−κ11
+λ+
+1−λ−
+1S0|n∗
+2/angbracketright/parenrightbigg
+,
+|Y1/angbracketright=1
+Z/parenleftbiggκ22
+λ+
+2−λ−
+2|n1/angbracketright−κ21
+λ+
+1−λ−
+2|n2/angbracketright−f12
+λ+
+1−λ+
+2S0|n∗
+2/angbracketright/parenrightbigg
+,
+|Y2/angbracketright=1
+Z/parenleftbigg
+−κ12
+λ+
+2−λ−
+1|n1/angbracketright+κ11
+λ+
+1−λ−
+1|n2/angbracketright+f12
+λ+
+2−λ+
+1S0|n∗
+1/angbracketright/parenrightbigg
+,(45)Inserting this result into eq. (28) we obtain the following expression for the 2-soliton solution of the MNLS:
+Q2s(x,t) = [J,A1+B1+A2+B2] =1
+Z[J,C(x,t)−S0CT(x,t)S0],
+C(x,t) =κ22
+λ+
+2−λ−
+2|n1/angbracketright/angbracketleftn†
+1|−κ12
+λ+
+2−λ−
+1|n1/angbracketright/angbracketleftn†
+2|−κ21
+λ+
+1−λ−
+2|n2/angbracketright/angbracketleftn†
+1|+κ11
+λ+
+1−λ−
+1|n2/angbracketright/angbracketleftn†
+2|
+−f∗
+12
+λ−
+1−λ−
+2|n1/angbracketright/angbracketleftn2|S0−f12
+λ+
+1−λ+
+2S0|n∗
+2/angbracketright/angbracketleftn†
+1|.(46)
+At the end of this section we note that the effect of the reductions (20)–(21) consists in constraining the
+polarization vectors. For the reduction 2e) we get
+/vector ν0k=K04/vector ν0k (47)
+In particular, for n= 3 and for K04=−11 we haveq1=q3, andq2arbitrary. This reduction of eq. (1) is also
+important for the BEC.6From (47) we find ν01=ν03. The effect of this constraint is that for the one-soliton
+solution we get Φ 1s;1= Φ1s;3.
+Our next remark following25is that this reduction applied to the F= 1 MNLS (1) leads to a 2-component
+MNLS which after the change of variables
+q1=1
+2(w1+iw2), q 2=1√
+2(w1−iw2), (48)
+leads to two disjoint NLS equations for w1andw2respectively.
+It is only logical that applying the constraint ν01=ν03the explicit expression for the one-soliton solution
+(36) simplifies and reduces to the standard soliton solutions of the s calar NLS.
+6. TWO SOLITON INTERACTIONS
+In this section we generalize the classical results of Zakharov and S habat about soliton interactions26to the class
+of MNLS equations related to BD.I symmetric spaces. For detailed ex position see the monographs.15,16These
+resultsweregeneralizedforthe vectornonlinearSchr¨ odingereq uationbyManakov,17seealso.27–29The Zakharov
+Shabat approach consisted in calculating the asymptotics of gener icN-soliton solution of NLS for t→ ±∞and
+establishing the pure elastic character of the generic soliton intera ctions. By generic here we mean N-soliton
+solution whose parameters λ±
+k=µk±iνkare such that µk/negationslash=µjfork/negationslash=j. The pure elastic character of the
+soliton interactions is demonstrated by the fact that for t→ ±∞the generic N-soliton solution splits into sum
+ofNone soliton solutions each preserving its amplitude 2 νkand velocity µk. The only effect of the interaction
+consists in shifting the center of mass and the initial phase of the so litons. These shifts can be expressed in terms
+ofλ±
+konly; for detailed exposition see.16
+We start with the simplest non-trivial case. Namely we use the 2-solit on solution derived above and calculate
+its asymptotics along the trajectory of the first soliton. To this en d we keepz1(x,t) fixed and let τ=z2−z1tend
+to±∞. Therefore it will be enough to insert the asymptotic values of the m atrix elements of Mforτ→ ±∞
+and keep only the leading terms. That gives:
+κ22=/braceleftbigge2τexp(ν2z1/ν1)+2C1,forτ→ ∞,
+e−2τexp(−ν2z1/ν1)+2C1,forτ→ −∞,
+κ12=/braceleftbigg
+eτexp((1+ν2/ν1)z1+i(φ1−φ2))+O(1),forτ→ ∞,
+e−τexp(−(1+ν2/ν1)z1−i(φ1−φ2))+O(1),forτ→ −∞,
+κ21=/braceleftbiggeτexp((1+ν2/ν1)z1−i(φ1−φ2))+O(1),forτ→ ∞,
+e−τexp(−(1+ν2/ν1)z1+i(φ1−φ2))+O(1),forτ→ −∞,
+f12=/braceleftbiggeτexp(−(1−ν2/ν1)z1+i(φ1−φ2))+O(1),forτ→ ∞,
+e−τexp((1−ν2/ν1)z1−i(φ1−φ2))+O(1),forτ→ −∞,(49)After somewhat lengthy calculations we get:
+lim
+τ→∞/vector q2s(x,t) =−i√
+2ν1e−i(φ1−α+)(e−z1−r+s0|/vector ν01/angbracketright+ez1+r+|/vector ν∗
+01/angbracketright)
+cosh(2(z1+r+))+(/vector ν†
+01,/vector ν01),
+lim
+τ→−∞/vector q2s(x,t) =i√
+2ν1e−i(φ1+α+)(e−z1+r+s0|/vector ν01/angbracketright+ez1−r+|/vector ν∗
+01/angbracketright)
+cosh(2(z1−r+))+(/vector ν†
+01,/vector ν01),(50)
+where
+r+= ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleλ+
+1−λ+
+2
+λ+
+1−λ−
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, α += argλ+
+1−λ+
+2
+λ+
+1−λ−
+2.
+In other words the 2-soliton interaction for the MNLS eqs. related to the BD.I symmetric spaces is the same
+as the one of the scalar NLS. Again we have that for large times the 2 -soliton solution splits into sum of 1-soliton
+solutions with shifted center of masses and phases and the value of these shifts r+andα+are independent on
+the number of components of MNLS. It will be interesting to check w hether the N-soliton interactions consist
+of sequence of elementary 2-soliton interactions and the shifts ar e additive.
+7. CONCLUSIONS AND DISCUSSION
+Using the Zakharov-Shabat dressing method we have obtained the two-soliton solution and have used it to
+analyze the soliton interactions of the MNLS equation. The conclusio n is that after the interactions the solitons
+recover their polarization vectors ν0k, velocities and frequency velocities. The effect of the interaction is , like in
+for the scalar NLS equation, shift of the center of mass z1→z1+r+and shift of the phase φ1→φ1+α+. Both
+shifts are expressed through the related eigenvalues λ±
+jonly.
+The next step would be to analyze multi-soliton interactions. Our hyp othesis is that each soliton will acquire
+a total shift of the center of mass that is sum of all elementary shif ts from each two soliton interactions. Similar
+result is expected for the total phase shift of the soliton. Proofs of these facts will be published elsewhere.
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\ No newline at end of file
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+arXiv:1001.0169v3  [math.OA]  27 Jul 2011Mixing subalgebras of finite von Neumann algebras
+Jan Cameron †, Junsheng Fang ‡and Kunal Mukherjee∗§
+†Department of Mathematics, Vassar College, Poughkeepsie, N Y12604, USA
+(email: jacameron@vassar.edu )
+‡School of Mathematical Sciences, Dalian University of Tech nology, Dalian, 116024, China
+(email: jfang@math.tamu.edu )
+§The Institute of Mathematical Sciences, Taramani,Chennai 600113, India.
+(email: kunal@neo.tamu.edu )
+Abstract
+Jolissaint and Stalder introduced definitions of mixing and weak mixing for
+von Neumann subalgebras of finite von Neumann algebras. In th is note, we
+study various algebraic and analytical properties of subal gebras with these mix-
+ing properties. We prove some basic results about mixing inc lusions of von
+Neumann algebras and establish a connection between mixing properties and
+normalizers of von Neumann subalgebras. The special case of mixing subal-
+gebras arising from inclusions of countable discrete group s finds applications
+to ergodic theory, in particular, a new generalization of a c lassical theorem of
+Halmos on the automorphisms of a compact abelian group. For a finite von
+Neumann algebra Mand von Neumann subalgebras A,BofM, we introduce
+a notion of weak mixing of B⊆Mrelative to A. We show that weak mixing of
+B⊂Mrelative to Ais equivalent to the following property: if x∈Mand there
+exist a finite number of elements x1,...,x n∈Msuch that Ax⊂/summationtextn
+i=1xiB,
+thenx∈B. We conclude the paper with an assortment of further example s
+of mixing subalgebras arising from the amalgamated free pro duct and crossed
+product constructions.
+∗The third author was supported in part by NSF grant DMS-0600814 during graduate studies at
+Texas A&M University.
+12
+1 Introduction
+In [6], Jolissaint and Stalder defined weak mixing and mixing for abelian vo n Neu-
+mann subalgebras of finite von Neumann algebras. These propertie s arose as natural
+extensions of corresponding notions in ergodic theory in the followin g sense: If σis a
+measure preserving action of a countable discrete abelian group Γ 0on a finite measure
+space (X,µ), then the action is (weakly) mixing in the sense of [1] if and only if the
+abelian von Neumann subalgebra L(Γ0) is (weakly) mixing in the crossed product
+finite von Neumann algebra L∞(X,µ)⋊Γ0.
+In this note, we extend the definitions of weak mixing and mixing to gen eral von
+Neumann subalgebras of finite von Neumann algebras, and study va rious algebraic
+and analytical properties of these subalgebras. In a forthcoming note, the authors
+will specialize to the study of mixing properties of maximal abelian von N eumann
+subalgebras. If Bis a von Neumann subalgebra of a finite von Neumann algebra M,
+andEBdenotes the usual trace-preserving conditional expectation on toB, we call B
+aweakly mixing subalgebra of Mif there exists a sequence of unitary operators {un}
+inBsuch that
+lim
+n→∞/bardblEB(xuny)−EB(x)unEB(y)/bardbl2= 0,∀x,y∈M.
+We callBamixing subalgebra of Mif the above limit is satisfied for all elements x,y
+inMand all sequences of unitary operators {un}inBsuch that lim n→∞un= 0 in the
+weak operator topology. When Bis an abelian algebra, our definition of weak mixing
+is precisely the weak asymptotic homomorphism property introduced by Robertson,
+Sinclair and Smith [15]. Although our definitions of weak mixing and mixing a re
+slightly different from those of Jolissaint and Stalder, our definitions coincide with
+theirs in the setting of the action of a countable discrete group on a probability space.
+Using arguments similar to those in the proofs of Proposition 2.2 and P roposition 3.6
+of [6], one can show
+Proposition 1.1. Ifσis a measure preserving action of a countable discrete group Γ0
+on a finite measure space (X,µ), then the action is (weakly )mixing in the sense of [1]
+if and only if the von Neumann subalgebra L(Γ0)is(weakly )mixing in the crossed
+product finite von Neumann algebra L∞(X,µ)⋊Γ0.
+For an inclusion of finite von Neumann algebras B⊆M, we call a unitary operator
+u∈Manormalizer ofBinMifuBu∗=B[3]. Clearly, every unitary in Bsatisfies
+this condition; the subalgebra Bis said to be singular inMif the only normalizers of
+BinMare elements of B. There is a close relationship between the concepts of weak
+mixing and singularity. Sinclair and Smith [17] noted one connection in p roving that
+weakly mixing von Neumann subalgebras are singular in their containing algebras. The
+converse was proved by Sinclair, Smith, White and Wiggins [20] under the assumption
+that the subalgebra is also masa (maximal abelian self-adjoint subalg ebra) in the3
+ambient von Neumann algebra. In other words, the measure prese rving action of a
+countable discrete abelian group Γ 0on a finite measure space ( X,µ) is weakly mixing
+if and only if the associated von Neumann algebra L(Γ0) is singular in L∞(X,µ)⋊Γ0.
+This provides an operator algebraic characterization of weak mixing in the abelian
+setting, which is the main motivation for the study undertaken here . In contrast to the
+abelian case, Grossman and Wiggins [4] showed that for general finit e von Neumann
+algebras, weakly mixing is not equivalent to singularity, so techniques beyond those
+known for singular subalgebras are required. In what follows, we de velop basic theory
+for mixing properties of general subalgebras of finite von Neumann algebras. This
+leads to a number of new observations about mixing properties of su balgebras and
+group actions, a characterization of weakly mixing subalgebras in te rms of their finite
+bimodules, and a variety of new examples of inclusions of von Neumann algebras
+satisfying mixing conditions. The paper is organized as follows.
+Section 2 contains some preliminary technical results. We show that ifBis a
+diffuse finite von Neumann algebra, then Bω⊖B={x∈Bω:τω(x∗b) = 0,∀b∈B}is
+the weak operator closure of the linear span of unitary operators inBω⊖B, whereBω
+is the ultra-power algebra of B. This result plays an important role in the subsequent
+sections.
+In Section 3, we prove that if Bis a mixing von Neumann subalgebra of a finite
+von Neumann algebra M, one has
+lim
+n→∞/bardblEB(xbny)−EB(x)bnEB(y)/bardbl2= 0,∀x,y∈M,
+if{bn}is a bounded sequence of operators in Bsuch that lim n→∞bn= 0 in the weak
+operator topology. As applications, we show that if Bis mixing in M,kis a positive
+integer, and e∈Bis a projection, then Mk(C)⊗Bis mixing in Mk(C)⊗MandeBe
+is mixing in eMe. We also show that, in contrast to weakly mixing masas, one cannot
+distinguish mixing masas by the presence or absence of centralizing s equences in the
+masa for the containing II 1factor.
+Section 4 concerns the special case of inclusions of group von Neum ann algebras.
+We extend some results of [6] for abelian subgroups to the case of a general inclusion
+of countable, discrete groups Γ 0⊂Γ in showing that L(Γ0) is mixing in L(Γ) if and
+only ifgΓ0g−1∩Γ0is a finite group for every g∈Γ\Γ0. These two conditions are
+seen to be equivalent the property that for every diffuse von Neum ann subalgebra A
+ofBand every y∈M,yAy∗⊂Bimpliesy∈B. Some applications to ergodic theory
+are given. In particular, Theorem 4.3 generalizes results of Kitchen s and Schmidt [9]
+and Halmos [5].
+In Section 5, we introduce and study the concept of relative weak m ixing for a
+triple of finite von Neumann algebras, and obtain several characte rizations of weakly
+mixing triples. It turns out that relative weak mixing of one subalgebr a with respect
+to another is closely related to the bimodule structure between the two subalgebras. In
+particular, we show that B⊂Mis weakly mixing relative to A(Ais not necessarily4
+contained in B) if and only the following property holds: if x∈MsatisfiesAx⊂/summationtextn
+i=1xiBfor a finite number of elements x1,...,x ninM, thenx∈B.
+The results in Section 6 show that mixing von Neumann subalgebras ha ve heredi-
+tary properties which are notably different from those of general singular subalgebras.
+We also consider the relationship between mixing and normalizers; in pa rticular, we
+show that subalgebras of mixing algebras inherit a strong singularity property from
+the containing algebra. Finally, we provide an assortment of new exa mples of mixing
+von Neumann subalgebras which arise from the amalgamated free pr oduct and crossed
+product constructions.
+We collect here some basic facts about finite von Neumann algebras, which will
+be used in the sequel. Throughout this paper, Mis a finite von Neumann algebra
+with a given faithful normal trace τ. Denote by L2(M) =L2(M,τ) the Hilbert space
+obtained by the GNS-construction of Mwith respect to τ. The image of x∈Mvia
+the GNS-construction is denoted by ˆ x, and the image of a subset LofMis denoted
+byˆL. The trace norm of x∈Mis defined by /bardblx/bardbl2=/bardblx/bardbl2,τ=τ(x∗x)1/2. Suppose
+thatBis a von Neumann subalgebra of M. Then there exists a unique faithful normal
+conditional expectation EBfromMontoBpreserving τ. LeteBbe the projection
+ofL2(N) ontoL2(B). Then the von Neumann algebra /a\}bracketle{tM,eB/a\}bracketri}htgenerated by Mand
+eBis called the basic construction ofM, which plays a crucial role in the study of
+von Neumann subalgebras of finite von Neumann algebras. There is a unique faithful
+tracial weight Tr on /a\}bracketle{tM,eB/a\}bracketri}htsuch that
+Tr(xeBy) =τ(xy),∀x,y∈M.
+Forξ∈L2(/a\}bracketle{tM,eB/a\}bracketri}ht,Tr), define /bardblξ/bardbl2,Tr= Tr(ξ∗ξ)1/2. For more details of the basic
+construction, we refer to [2, 7, 11, 18]. For a detailed account of fi nite von Neumann
+algebras and the theory of masas, we refer the reader to [18].
+Acknowledgements : The authors thank Ken Dykema, David Kerr, Roger Smith,
+and Stuart White for valuable discussions throughout the completio n of this work.
+2 Unitary operators in M⊖B
+LetMbe a finite von Neumann algebra with a faithful normal trace τ, and let Bbe
+a von Neumann subalgebra of M. We denote by M⊖Bthe orthogonal complement
+ofBinMwith respect to the standard inner product on M, that is,
+M⊖B={x∈M:τ(x∗b) = 0 for all b∈B}.
+Thenx∈M⊖Bif and only if EB(x) = 0, where EBis the trace-preserving conditional
+expectation of MontoB. Note that if x∈M⊖B, thenτ(x) =τ(EB(x)) = 0, so
+the unique positive element in M⊖Bis 0. On the other hand, it is easy to see that
+M⊖Bis the linear span of self-adjoint elements in M⊖B.5
+In the following section, we will use the fact that a bounded sequenc e (bn) in a
+finite von Neumann algebra Bconverges to 0 in the weak operator topology if and
+only if it defines an element of the ultrapower Bωwhich is orthogonal to Bin the
+above sense. A key step in the proof of Theorem 3.3 will then be to ap proximate an
+arbitrary z∈Bω⊖Bby linear combinations of unitary operators in Bω⊖B. That
+such an approximation is possible is the main technical result of this se ction.
+WhenB⊂Mcomes from an inclusion of countable discrete groups, there is an
+obvious dense linear subspace of M⊖B: ifGis a subgroup of a discrete group Γ, then
+/suppress L(Γ)⊖/suppress L(G) is the weak closure of the linear span of unitary operators corres ponding
+to elements in Γ \G. Although in the case of a general inclusion B⊆M, such a
+canonical set of unitaries is not available, we nevertheless obtain a p artial answer to
+the following question: If Bis a subalgebra of a diffuse finite von Neumann algebra M
+such that eMe/\e}atio\slash=eBefor every nonzero projection e∈B, isM⊖Bthe weak closure
+of the linear span of unitaries in M⊖B?
+The assumption that eMe/\e}atio\slash=eBefor every nonzero projection e∈Bis necessary,
+as is the assumption that Mis diffuse. For instance, if M=C⊕CandB=Cand
+τ(1⊕0)/\e}atio\slash=τ(0⊕1), then there are no unitary operators in M⊖B.
+Let (M)1be the closed unit ball of M, and let Λ = {x∈(M)1:x=x∗,EB(x) = 0}.
+Then Λ is a convex set which is closed, hence also compact, in the weak operator
+topology. By the Krein-Milman Theorem, Λ is the weak operator closu re of the convex
+hull of its extreme points. Thus, we need only characterize the ext reme points of Λ.
+Lemma 2.1. Suppose that for every nonzero projection p∈M, there exists a nonzero
+elementxp∈pMpsatisfying EB(xp) = 0. Then the extreme points of Λare
+/braceleftbigg
+2e−1 :e∈MandEB(e) =1
+2/bracerightbigg
+.
+Proof. Suppose a∈Λ is an extreme point of Λ and a/\e}atio\slash= 2e−1. By the spectral
+decomposition theorem, there exists an ǫ >0 and a nonzero spectral projection eofa
+such that
+(−1 +ǫ)e≤ae≤(1−ǫ)e.
+By assumption, there is a nonzero self-adjoint element x∈eMe such that EB(x) = 0.
+By multiplying a scalar, we may assume that −ǫe≤x≤ǫe. Thenx+a,x−a∈Λ
+andx=1
+2(x+a) +1
+2(x−a). Soxis not an extreme point of Λ, contradicting our
+assumption. Therefore, x= 2e−1 for some projection e∈M. SinceEB(x) = 0,
+EB(e) =1
+2. Note that 2 e−1 is an extreme point of ( M)1. This completes the
+proof.
+The following example shows that the assumptions of the above Lemm a are essen-
+tial.6
+Example: In the inclusion C⊆M3(C), there is no projection e∈M3(C) satisfing
+τ(e) =1
+2. In this case, the partial isometry
+/parenleftBigg1 0 0
+0 0 0
+0 0−1/parenrightBigg
+is an extreme point of Λ.
+Corollary 2.2. LetMbe a diffuse finite von Neumann algebra with a faithful normal
+traceτ. ThenM⊖C1is the weak operator closure of the linear span of self-adjoi nt
+unitary operators in M⊖C1.
+Proof. For every nonzero projection p∈M,pMp is diffuse and hence pMp/\e}atio\slash=Cp. So
+there is an operator xp∈pMp withτ(xp) = 0. By Lemma 2.1, M⊖C1 is the weak
+operator closure of linear span of self-adjoint unitary operators inM⊖C1.
+Lemma 2.3. SupposeBis a diffuse finite von Neumann algebra with a faithful normal
+traceτ. Forǫ >0andx1,...,x n∈B, there exists a Haar unitary operator u∈B
+such that
+|τ(xiu∗)|< ǫ, 1≤i≤n.
+Proof. SinceBis diffuse, Bcontains a Haar unitary operator v. Note that vn→0 in
+the weak operator topology. So there exists an Nsuch that
+|τ(xi(vN)∗)|< ǫ, 1≤i≤n.
+Letu=vN. Thenuis a Haar unitary operator and the lemma follows.
+LetBbe a separable diffuse von Neumann algebra with a faithful normal tr aceτ,
+and letBωbe the ultrapower algebra of Bwith a faithful normal trace τω(see [16]).
+Let (Bω⊖B)1be the closed unit ball of Bω⊖B. The following proposition is the
+main result of this section.
+Proposition 2.4. SupposeBis a separable diffuse finite von Neumann algebra with
+a faithful normal trace τ. Then (Bω⊖B)1is the trace norm closure of the convex hull
+of self-adjoint unitary operators in Bω⊖B.
+Proof. We claim that for every nonzero projection p∈Bω, there exists a nonzero
+elementxpinpBωpsuch that EB(xp) = 0, where EBis the conditional expectation of
+BωontoBpreserving τω. Letp= (pn), where pn∈Bis a projection with τ(pn) =
+τω(p)>0. SinceBis separable, there is a dense sequence {yn}ofBin the trace norm.
+We may assume that y1= 1. By Lemma 2.3, for any finite subset {y1,...,y n}of the
+dense sequence, there is a Haar unitary operator un∈pnBpnsuch that
+|τ(pnyipnu∗
+n)|<1
+n,∀1≤i≤n.7
+Letxp= (un). Then/bardblxp/bardbl2
+2= limn→ω/bardblun/bardbl2
+2=τ(p)>0. Hence, xp/\e}atio\slash= 0 andxp∈pBωp.
+Note that for each yk,
+τω(yk(xp)∗) =τω((pykp)(xp)∗) = lim
+n→ωτ(pnykpnu∗
+n) = 0.
+Since{yk}is dense in Bin the trace norm topology, τω(y(xp)∗) = 0 for all y∈B.
+This implies EB(xp) = 0. By Lemma 2.1, ( Bω⊖B)1is the weak operator closure
+of the convex hull of self-adjoint unitary operators in Bω⊖B. Note that ( Bω⊖B)1
+is a convex set, so its weak operator closure coincides with its closur e in the strong
+operator and trace norm topologies. This proves the result.
+Corollary 2.5. Suppose Bis a separable diffuse finite von Neumann algebra with a
+faithful normal trace τ. ThenBω⊖Bis the weak operator closure of the linear span
+of self-adjoint unitary operators in Bω⊖B.
+Using a similar approach, we can also prove the following result.
+Proposition 2.6. IfMis a separable type II1factor and Bis an abelian von Neumann
+subalgebra of M, thenM⊖Bis the weak operator closure of the linear span of unitary
+operators in M⊖B.
+It is not clear whether Proposition 2.6 holds for nonabelian subalgebr as. We are
+unable, for instance, to establish the conclusion of the result when Bis a hyperfinite
+subfactor of a nonhyperfinite type II 1factorM, e.g.LF2.
+3 Mixing von Neumann subalgebras
+LetMbe a finite von Neumann algebra with a faithful normal trace τ, and let Bbe
+a von Neumann subalgebra of M.
+Definition 3.1. An algebra Bis amixing von Neumann subalgebra ofMif
+lim
+n→∞/bardblEB(xuny)−EB(x)unEB(y)/bardbl2= 0
+holds for all x,y∈Mand every sequence of unitary operators {un}inBsuch that
+lim
+n→∞un= 0 in the weak operator topology. If Bis a mixing von Neumann subalgebra
+ofM, then we say B⊆Mamixing inclusion of finite von Neumann algebras.
+It is easy to see that Bis a mixing von Neumann subalgebra of Mif and only if
+for all elements x,yinMwithEB(x) =EB(y) = 0, one has
+lim
+n→∞/bardblEB(xuny)/bardbl2= 0
+if{un}is a sequence of unitary operators in Bsuch that lim
+n→∞un= 0 in the weak
+operator topology.8
+Remark 3.2. By the Kaplansky density theorem, we may assume that xandyare
+in a subset FofMsuch that Mis the von Neumann algebra generated by Fin
+Definition 3.1.
+The following theorem provides a useful equivalent condition for mixin g inclusions
+of finite von Neumann algebras.
+Theorem 3.3. IfBis a mixing von Neumann subalgebra of Mandx,y∈Mwith
+EB(x) =EB(y) = 0, then
+lim
+n→∞/bardblEB(xbny)/bardbl2= 0
+if{bn}is a bounded sequence of operators in Bsuch that lim
+n→∞bn= 0in the weak
+operator topology.
+Proof. Letωbe a free ultrafilter of the set of natural numbers and let Mωbe the
+ultrapower algebra of M. ThenMωis a finite von Neumann algebra with a faithful
+normal trace τω. We can identify Bωwith a von Neumann subalgebra of Mωin the
+natural way. Every bounded sequence ( bn) ofBdefines an element zinBω. We
+may assume that /bardblz/bardbl ≤1. It is easy to see that lim n→ωbn= 0 in the weak operator
+topology if and only if
+τω(zb) = 0,∀b∈B.
+Recall that M⊖B={x∈M:τ(x∗b) = 0 for all b∈B}. It is easy to see Definition 3.1
+is equivalent to the following: For any elements x,yinM⊖B, andu∈Bω⊖B, one
+hasEBω(xuy) = 0. By Proposition 2.4, ( Bω⊖B)1is the trace norm closure of the
+convex hull of unitary operators in Bω⊖B. Letǫ > 0. Then there exist unitary
+operators u1,...,u ninBω⊖Band positive numbers α1,...,α nwithα1+···+αn= 1
+such that /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublez−n/summationdisplay
+k=1αkuk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+2,τω< ǫ.
+For any elements xandyinM⊖B,
+/bardblEBω(xzy)/bardbl2,τω=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleEBω/parenleftBigg
+x/parenleftBigg
+z−n/summationdisplay
+k=1αkuk/parenrightBigg
+y/parenrightBigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+2,τω≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublex/parenleftBigg
+z−n/summationdisplay
+k=1αkuk/parenrightBigg
+y/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+2,τω
+≤ /bardblx/bardbl·/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublez−n/summationdisplay
+k=1αkuk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+2,τω·/bardbly/bardbl ≤ǫ/bardblx/bardbl/bardbly/bardbl.
+Sinceǫ >0 is arbitrary, EBω(xzy) = 0, which is equivalent to
+lim
+n→∞/bardblEB(xbny)/bardbl2= 0.9
+Two applications of the above theorem are the following.
+Corollary 3.4. IfBis a mixing von Neumann subalgebra of Mandkis a positive
+integer, then Mk(C)⊗Bis mixing in Mk(C)⊗M.
+Proof. Note that x= (xij)∈(Mk(C)⊗M)⊖(Mk(C)⊗B) if and only if xij∈M⊖B
+for all 1≤i,j≤k, andbn= (bn
+ij)∈Mk(C)⊗Bconverges to 0 in the weak operator
+topology if and only if bn
+ijconverges to 0 in the weak operator topology for all 1 ≤
+i,j≤k. Now the corollary follows from Theorem 3.3.
+Corollary 3.5. IfBis a mixing von Neumann subalgebras of Mandeis a projection
+ofB, theneBeis mixing in eMe.
+Proof. Let (bn) be a bounded sequence of eBe which converges to 0 in the weak
+operator topology. For x,y∈eMe⊖eBe, we have x,y∈M⊖B. By Theorem 3.3,
+lim
+n→∞/bardblEeBe(xbny)/bardbl2= lim
+n→∞/bardblEB(xbny)/bardbl2= 0.
+It is well-known that the presence of centralizing sequences in a mas a for its con-
+taining II 1factor is a conjugacy invariant for the masa. More generally, it is po ssible
+to build non-conjugate masas of a II 1factor by controlling the existence of centraliz-
+ing sequences in various cutdowns of each masa. Sinclair and White [19 ] developed
+this technique to produce uncountably many non-conjugate weak ly mixing masas in
+the hyperfinite II 1factor with the same Puk´ anszky invariant. The final result of this
+section implies that, in contrast to the larger class of weakly mixing ma sas, there is no
+hope of distinguishing mixing masas along these lines. Following the nota tion of [19],
+for a von Neumann subalgebra Bof a II 1factorM, we denote by Γ( B) the maximal
+trace of a projection e∈Bfor which eBecontains a non-trivial centralizing sequences
+foreMe.
+Proposition 3.6. IfBis a mixing subalgebra of a type II1factorMandeBe/\e}atio\slash=eMe
+for each nonzero projection e∈B, then Γ(B) = 0.
+Proof. By Corollary 3.5, we need only show that there is no nontrivial centra l sequence
+{bn}inBofM. Suppose {bn} ⊂Bis a central sequence of M. We may assume that
+τ(bn) = 0 for each n. Suppose lim n→ωbn=z∈Bin the weak operator topology, then
+for allx∈M,
+zx= lim
+n→ωbnx= lim
+n→ωxbn=xz.
+SinceMis a type II 1factor,z=τ(z)1 = 0. Hence lim n→ωbn= 0 in the weak
+operator topology. Choose a nonzero element x∈Msuch that τ(xb) = 0 for all b∈B.
+Note that
+/bardblxbn−bnx/bardbl2
+2=/bardblxbn/bardbl2
+2+/bardblbnx/bardbl2
+2−2Reτ(b∗
+nx∗bnx)10
+≥τ(b∗
+nx∗xbn)−2Reτ(b∗
+nEB(x∗bnx))
+=τ(x∗xbnb∗
+n)−2Reτ(b∗
+nEB(x∗bnx)).
+Since{bn}is a central sequence of M,{bnb∗
+n}is also a central sequence of M. The
+uniqueness of the trace on Mimplies that
+lim
+n→ωτ(x∗xbnb∗
+n) = lim
+n→ωτ(x∗x)τ(bnb∗
+n) = lim
+n→ω/bardblx/bardbl2
+2·/bardblbn/bardbl2
+2.
+By Theorem 3.3,
+0 = lim
+n→∞/bardblxbn−bnx/bardbl2≥ /bardblx/bardbl2lim
+n→∞/bardblbn/bardbl2,
+which implies that lim n→ω/bardblbn/bardbl2= 0. This completes the proof.
+Corollary 3.7. IfBis a mixing masa of a type II1factorM, then Γ(B) = 0.
+4 Mixing inclusions of group von Neumann alge-
+bras
+In this section, we apply our operator-algebraic machinery to the s pecial case of mixing
+inclusions of von Neumann algebras that arise from actions of count able, discrete
+groups. This direction was taken up in [6], where it was shown that, fo r an infinite
+abelian subgroup Γ 0of a countable group Γ ,the inclusion L(Γ0)⊂L(Γ) is mixing if
+and only if the following condition (called (ST)) is satisfied:
+For every finite subset Cof Γ\Γ0,there exists a finite exceptional set E⊂Γ0such
+thatgγh /∈Γ0for allg0∈Γ0\Eandg,h∈C.
+Theorem 4.3 of this section supplies a similar characterization for the case in which
+Γ0is not abelian, and also establishes a connection between the group n ormalizer of
+the subgroup Γ 0and the “analytic” normalizer of its associated group von Neumann
+algebra. The key observation required is the following, which shows t hat mixing
+subalgebras satisfy a much stronger form of singularity.
+Theorem 4.1. LetBbe a mixing von Neumann subalgebra of M, and suppose that A
+is a diffuse von Neumann subalgebra of B. Ify∈MsatisfiesyAy∗⊆B, theny∈B.
+Proof. We may assume that Ais a diffuse abelian von Neumann algebra. Then Ais
+generated by a Haar unitary operator w. In particular, lim n→∞wn= 0 in the weak
+operator topology. Let x∈MandEB(x) = 0. Then
+|τ(xy)|2≤ /bardblEA′∩M(xy)/bardbl2
+2.11
+Note that
+EA′∩M(xy) = lim
+n→ω/summationtextn
+k=1wk(xy)(w∗)k
+n
+in the weak operator topology. Hence,
+|τ(xy)|2≤ /bardblEA′∩M(xy)/bardbl2
+2
+≤lim
+n→ω/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationtextn
+k=1wk(xy)(w∗)k
+n/vextenddouble/vextenddouble/vextenddouble/vextenddouble2
+2
+= lim
+n→ω1
+n2n/summationdisplay
+i,j=1τ(wi(xy)(w∗)iwj(y∗x∗)(w∗)j)
+≤lim
+n→ω1
+n2n/summationdisplay
+i,j=1|τ(x(ywj−iy∗)x∗(w∗)j−i)|
+≤lim
+n→ω1
+n2n/summationdisplay
+i,j=1/bardblEB(x(ywj−iy∗)x∗(w∗)j−i)/bardbl2
+= lim
+n→ω1
+n2n/summationdisplay
+i,j=1/bardblEB(x(ywj−iy∗)x∗)/bardbl2.
+By hypothesis, ywny∗∈B. Note that lim n→∞ywny∗= 0 in the weak operator
+topology. By Theorem 3.3,
+lim
+n→∞/bardblEB(x(ywny∗)x∗)/bardbl2= 0.
+So
+|τ(xy)|2≤lim
+n→ω1
+n2n/summationdisplay
+i,j=1/bardblEB(x(ywj−iy∗)x∗)/bardbl2= 0.
+Therefore, τ(xy) = 0 for all y∈M⊖B. This implies that y∈B.
+Remark 4.2. In Theorem 4.1, it is not necessary that the unit of Abe the same as
+the unit of B.
+Theorem 4.3. LetM=L(Γ)andB=L(Γ0). Then the following conditions are
+equivalent:
+1.B=L(Γ0)is mixing in M=L(Γ);
+2.gΓ0g−1∩Γ0is a finite group for every g∈Γ\Γ0;12
+3. for every diffuse von Neumann subalgebra AofBand every unitary operator
+v∈M, ifvAv∗⊆B, thenv∈B;
+4. for every diffuse von Neumann subalgebra AofBand every operator y∈M, if
+yAy∗⊆B, theny∈B.
+Proof. “1⇒4” follows from Theorem 4.1 and “4 ⇒3” is trivial.
+“3⇒2”. Suppose M=L(Γ) and B=L(Γ0). Suppose for some g∈Γ\Γ0,
+gΓ0g−1∩Γ0is an infinite group. Let Γ 1= Γ0∩g−1Γ0g=g−1(gΓ0g−1∩Γ0)g. Then
+Γ1is an infinite group, and gΓ1g−1⊆Γ0. Soλ(g)L(Γ1)λ(g−1)⊆L(Γ0). By the third
+statement, λ(g)∈L(Γ0) andg∈Γ0. This is a contradiction.
+“2⇒1”. First, we show that if g1,g2∈Γ\Γ0, theng1Γ0g2∩Γ0is a finite set.
+Suppose h1,h2∈Γ0andg1h1g2,g1h2g2∈Γ0. Then
+g1h1h−1
+2g−1
+1=g1h1g2(g1h2g2)−1∈Γ0∩g1Γ0g−1
+1.
+Since Γ 0∩g1Γ0g−1
+1is a finite group, {h1h−1
+2:h1,h2∈Γ0andg1h1g2,g1h2g2∈Γ0}is a
+finite set. Hence, g1Γ0g2∩Γ0is a finite set.
+Let{vn}be a sequence of unitary operators in Bsuch that lim
+n→∞vn= 0 in the weak
+operator topology. Write vn=/summationtext∞
+k=1αn,kλ(hk). Then for each k, limn→∞αn,k= 0.
+Suppose g1,g2∈Γ\Γ0. There exists an Nsuch that for all m≥N,g1hmg2/∈Γ0.
+Hence,
+/bardblEB(g1vng2)/bardbl2=N/summationdisplay
+i=1/bardblαn,iEB(g1λ(hi)g2)/bardbl2≤N/summationdisplay
+i=1|αn,i| →0
+whenn→ ∞ . By Remark 3.2, Mis mixing relative to B.
+We now apply Theorem 4.3 to the group-theoretic situation arising fr om a semidi-
+rect product Γ = G⋊Γ0, where Γ 0is an infinite group. Let σh(g) =hgh−1forh∈Γ0
+andg∈G. Thenσhis an automorphism of G. Note that hg=hgh−1h=σh(g)hfor
+h∈Γ0andg∈G.
+Proposition 4.4. LetM=L(G⋊Γ0)andB=L(Γ0). ThenBis mixing in Mif
+and only if for each g∈G,g/\e}atio\slash=e, the group
+{h∈Γ0:σh(g) =g}
+is finite.
+Proof. Letg∈Gandh∈Γ0. Suppose h∈gΓ0g−1∩Γ0. Thenghg−1∈Γ0. Note
+thatghg−1=hh−1ghg−1=h(σh−1(g)g−1). Soghg−1∈Γ0implies that σh−1(g)g−1∈13
+Γ0∩G={e}, i.e.,σh−1(g) =gand hence σh(g) =g. Conversely, suppose σh(g) =g.
+Thenσh−1(g) =gand hence ghg−1=hσh−1(g)g−1=h∈Γ0∩gΓ0g−1. This proves
+{h∈Γ0:σh(g) =g}={h∈Γ0:h∈gΓ0g−1∩Γ0}
+Suppose Bis mixing in M. By 2 of Theorem 4.3, gΓ0g−1∩Γ0is a finite group
+for every g∈Gwithg/\e}atio\slash=e. So the group {h∈H:σh(g) =g}is finite. Conversely,
+suppose for each g∈G,g/\e}atio\slash=e, the group {h∈Γ0:σh(g) =g}is finite, which implies
+thatgΓ0g−1∩Γ0is finite. A group element of Γ \Γ0can be written as gh,g∈G,
+g/\e}atio\slash=e,h∈Γ0. Note that
+ghΓ0h−1g−1∩Γ0=gΓ0g−1∩Γ0
+is finite. So Bis mixing in Mby 2 of Theorem 4.3.
+Recall that the action σof a group Hon a finite von Neumann algebra Nis called
+ergodic ifσh(x) =xfor allh∈Himplies that x=λ1. The following result extends
+Theorem 2.4 of [9] to the noncommutative setting.
+Corollary 4.5. LetM=L(G⋊Γ0)andB=L(Γ0). Suppose Γ0is a finitely generated,
+infinite, abelian group or Γ0is a torsion free group. Then Bis mixing in Mif and
+only if every element h∈Γ0of infinite order is ergodic on L(G).
+Proof. IfBis mixing in M, then clearly every element h∈Γ0of infinite order is ergodic
+onL(G). Now suppose every element h∈Γ0of infinite order is ergodic on L(G). If
+Bis not mixing in M, then there is a g∈G,g/\e}atio\slash=e, such that {h∈Γ0:σh(g) =g}
+is an infinite group. Under the above hypotheses on Γ 0, there exits an element h0of
+infinite order such that σh0(g) =g. This implies that the action of h0onL(G) is not
+ergodic, which is a contradiction.
+Corollary 4.6. LetM=L(G⋊Z)andB=L(Z). Then the following conditions are
+equivalent:
+1. the action of ZonL(G)is mixing, i.e., Bis mixing in M;
+2. the action of ZonL(G)is weakly mixing, i.e., Bis weakly mixing in M;
+3. the action of ZonL(G)is ergodic;
+4. for every g∈G,g/\e}atio\slash=e, the orbit {σh(g)}is infinite;
+5. for every g∈G,g/\e}atio\slash=e,{h∈Z:σh(g) =g}={e}.14
+Proof. Letγbe a generator of Z. Clearly “1 ⇒2⇒3”.
+“3⇒4”. Suppose σγn(g) =gandnis the minimal positive integer satisfies this
+condition. Let x=Lg+Lσγ(g)+...+Lσγn−1(g). Thenx∈L(G),x/\e}atio\slash=λ1, andσh(x) =x
+for allh∈Z. This implies that the action of ZonL(G) is not ergodic.
+“4⇒5”. Suppose σγn(g) =gfor some positive integer n. Then the orbit {σh(g)}
+has at most nelements.
+“5⇒1” follows from Proposition 4.4.
+A special case of Corollary 4.6 implies the following classical result of Ha lmos [5].
+Corollary 4.7 (Halmos’s Theorem) .LetXbe a compact abelian group, and T:X→
+Xa continuous automorphism. Then Tis mixing if and only if Tis ergodic.
+Proof. By the Pontryagin duality theorem, the dual group GofXis a discrete abelian
+group. Furthermore, there is an induced action of ZonG, and the action is unitarily
+conjugate to the action of TonX. Now the corollary follows from Corollary 4.6.
+5 Relative weak mixing
+Suppose Mis a finite von Neumann algebra with a faithful normal trace τ, andA,B
+are von Neumann subalgebras of M. We say B⊂Misweakly mixing relative to Aif
+there exits a sequence of unitary operators un∈Asuch that
+lim
+n→∞/bardblEB(xuny)−EB(x)unEB(y)/bardbl2= 0,∀x,y∈M.
+SoBis weakly mixing in Mif and only if B⊂Mis weakly mixing relative to B.
+Since every diffuse von Neumann algebra contains a sequence of unit ary operators
+converging to 0 in the weak operator topology, Bis mixing in Mimplies that B⊂M
+is weakly mixing relative to Afor all diffuse von Neumann subalgebras AofB.
+It is easy to see that B⊂Mis weakly mixing relative to Aif and only if there
+exits a sequence of unitary operators un∈Asuch that for all elements x,yinMwith
+EB(x) =EB(y) = 0, one has
+lim
+n→∞/bardblEB(xuny)/bardbl2= 0.
+The main result of this section is the following, which is inspired by [14].
+Theorem 5.1. LetMbe a finite von Neumann algebra with a faithful normal trace
+τ, and let A,Bbe von Neumann subalgebras of M. Then the following conditions are
+equivalent:15
+1.B⊂Mis weakly mixing relative to A, i.e., there exists a sequence of unitary
+operators {uk}inAsuch that
+lim
+k→∞/bardblEB(xuky)/bardbl2= 0,∀x,y∈N⊖B;
+2. ifz∈A′∩/a\}bracketle{tM,eB/a\}bracketri}htsatisfies Tr(z∗z)<∞, theneBzeB=z;
+3. ifp∈A′∩/a\}bracketle{tM,eB/a\}bracketri}htsatisfies Tr(p)<∞, theneBpeB=p;
+4. ifx∈MsatisfiesAx⊂/summationtextn
+i=1xiBfor a finite number of elements x1,...,x n∈
+M, thenx∈B.
+Before we prove Theorem 5.1, we state some corollaries of the theo rem.
+Corollary 5.2. LetMbe a finite von Neumann algebra with a faithful normal trace
+τ, and let Bbe a von Neumann subalgebra of M. Then the following conditions are
+equivalent:
+1.Bis a weakly mixing von Neumann subalgebra of M;
+2. ifx∈MsatisfiesBx⊂/summationtextn
+i=1xiBfor a finite number of elements x1,...,x n∈
+M, thenx∈B.
+The following corollary gives an operator algebraic characterization of weak mixing
+actions of countable discrete groups.
+Corollary 5.3. Ifσis a measure preserving action of a countable discrete group Γ0
+on a finite measure space (X,µ), then weak mixing of σis equivalent to the following
+property: if x∈L∞(X,µ)⋊Γ0andxL(Γ0)⊂/summationtextn
+i=1xiL(Γ0)for a finite number of
+elements x1,...,x ninL∞(X,µ)⋊Γ0, thenx∈L(Γ0).
+Corollary 5.4. LetMbe a finite von Neumann algebra with a faithful normal trace
+τ, and let Bbe a mixing von Neumann subalgebra of M. IfA⊂Bis a diffuse
+von Neumann subalgebra and x∈MsatisfiesAx⊂/summationtextn
+i=1xiBfor a finite number of
+elements x1,...,x n∈M, thenx∈B.
+To prove Theorem 5.1, we need the following lemmas.
+Lemma 5.5. Letp∈ /a\}bracketle{tM,eB/a\}bracketri}htbe a finite projection, p≤1−eB, andǫ >0. Then
+there exist x1,...,x n∈M⊖Bsuch that EB(x∗
+jxi) =δijfi, wherefiis a projection in
+B, and /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublep−n/summationdisplay
+i=1xieBx∗
+i/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+2,Tr< ǫ.16
+Proof. Letq=eB+p. Thenqis a finite projection in /a\}bracketle{tM,eB/a\}bracketri}ht. By Lemma 1.8 of [12],
+there are x0,x1,...,x n∈M,x0= 1, such that EB(x∗
+jxi) =δijfifor 0≤i,j≤nand
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleq−n/summationdisplay
+i=0xieBx∗
+i/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+2,Tr< ǫ.
+Clearly, /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublep−n/summationdisplay
+i=1xieBx∗
+i/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+2,Tr< ǫ.
+Suppose that H ⊂L2(M) is a right B-module. We denote by /suppress L B(L2(B),H) the
+set of bounded right B-modular operators from L2(B) intoH. The dimension of H
+overBis defined as
+dimB(H) = Tr(1) ,
+where Tr is the unique tracial weight on B′satisfying the following condition
+Tr(x∗x) =τ(xx∗),∀x∈/suppress LB(L2(B),H).
+We sayHis afinite right Bmodule if Tr(1)<∞. For details on finite modules, we
+refer the reader to appendix A of [21].
+Suppose that H ⊂L2(M) is a right B-module. We say that Hisfinitely generated
+if there exist finitely many elements ξ1,...,ξ n∈ H such that His the closure of/summationtextn
+i=1ξiB. A set{ξi}n
+i=1is called an orthonormal basis of HifEB(ξ∗
+iξj) =δijpi∈B,
+p2
+i=pi, and for every ξ∈ H we have
+ξ=/summationdisplay
+iξiEB(ξ∗
+iξ).
+Letpbe the orthogonal projection of L2(M) ontoH. Thenp=/summationtextn
+i=1ξieBξi, where
+ξi∈L2(M) is viewed as an unbounded operator affilated with M. Every finitely
+generated right Bmodule has an orthonormal basis. For finitely generated right B
+modules, we refer to 1.4.1 of [13].
+The following lemma is proved by Vaes in [21] (see Lemma A.1).
+Lemma 5.6. Suppose His a finite right B-module. Then there exists a sequence
+of projections znofZ(B) =B′∩Bsuch that limn→∞zn= 1in the strong operator
+topology and Hznis unitarily equivalent to the left pnMkn(B)pnrightBbimodule
+pn(L2(B)(n))for each n. In particular, Hznis a finitely generated right Bmodule.
+The following lemma is motivated by Lemma 1.4.1 of [13].17
+Lemma 5.7. SupposeH ⊂L2(M)is a leftAand finitely generated right Bbimodule.
+Letpdenote the orthogonal projection of L2(M)ontoH. Then there exists a sequence
+of projections zninA′∩Msuch that limn→∞zn= 1in the strong operator topology
+and for each n, there exist a finite number of elements xn,1,...,x n,k∈Msuch that
+znpzn(ˆx) =k/summationdisplay
+i=1/hatwiderxn,iEB(x∗
+n,ix),∀x∈M.
+Proof. Let{ξi}k
+i=1⊂ H ⊂L2(M,τ) be an orthonormal basis for H, i.e.,H=⊕k
+i=1[ξiB].
+Thenp=/summationtextk
+i=1ξieBξ∗
+i∈A′∩/a\}bracketle{tM,eB/a\}bracketri}ht, whereξi∈L2(M) is viewed as an unbounded
+operator affilated with M. Fora∈A, we have
+a/parenleftBiggn/summationdisplay
+i=1ξieBξ∗
+i/parenrightBigg
+=/parenleftBiggn/summationdisplay
+i=1ξieBξ∗
+i/parenrightBigg
+a.
+Applying the pull down map to both sides, we obtain
+a/parenleftBiggn/summationdisplay
+i=1ξiξ∗
+i/parenrightBigg
+=/parenleftBiggn/summationdisplay
+i=1ξiξ∗
+i/parenrightBigg
+a.
+Henceaq=qafor all spectral projections qofξiξ∗
+i. Since/summationtextn
+i=1ξiξ∗
+iis a densely defined
+operator affilated with M,q∈A′∩M. We thus obtain a sequence of projections zn∈
+A′∩Msuch that lim n→∞zn= 1 in the strong operator topology and/summationtextk
+i=1znξiξ∗
+iznis
+a bounded operator for each n. Letxn,i=znξi, 1≤i≤k. Thenxn,i∈Mand
+znpzn(ˆx) =k/summationdisplay
+i=1znξieBξ∗
+izn(ˆx) =k/summationdisplay
+i=1xn,ieBx∗
+n,i(ˆx) =k/summationdisplay
+i=1/hatwiderxn,iEB(x∗
+n,ix),∀x∈M.
+Proof of Theorem 5.1. “1⇒2”. Suppose eBzeB=zis not true. We may assume
+that (1−eB)z/\e}atio\slash= 0 (otherwise, consider z(1−eB)). Replacing zby a nonzero spectral
+projection of (1 −eB)zz∗(1−eB) corresponding to an interval [ c,1] withc >0, we
+may assume that z=p/\e}atio\slash= 0 is a subprojection of 1 −eB.
+Letǫ >0. By Lemma 5.5, there is a natural number nandx1,...,x n∈M⊖B
+such that EB(x∗
+jxi) =δijfi, wherefiis a projection in B, and
+/bardblp−n/summationdisplay
+i=1xieBx∗
+i/bardbl2,Tr< ǫ/2.18
+Letp0=/summationtextn
+i=1xieBx∗
+i. Thenp0is a projection. Note that ukpu∗
+k=p. So
+/bardblukp0u∗
+k−p0/bardbl2,Tr≤ /bardbluk(p0−p)u∗
+k/bardbl2,Tr+/bardblp0−p/bardbl2,Tr< ǫ.
+Therefore,
+2/bardblp0/bardbl2
+2,Tr=/bardblukp0u∗
+k−p0/bardbl2
+2,Tr+ 2Tr(ukp0u∗
+kp0)
+=/bardblukp0u∗
+k−p0/bardbl2
+2,Tr+ 2/summationdisplay
+1≤i,j≤nTr(ukxieBx∗
+iu∗
+kxjeBx∗
+j)
+≤ǫ2+ 2/summationdisplay
+1≤i,j≤nτ(EB(x∗
+iu∗
+kxj)x∗
+jukxi)
+≤ǫ2+ 2/summationdisplay
+1≤i,j≤n/bardblEB(x∗
+jukxi)/bardbl2
+2,τ.
+By the assumption of the lemma, 2/summationtext
+1≤i,j≤n/bardblEB(x∗
+jukxi)/bardbl2
+2,τ→0 whenk→ ∞ .
+Hence,/bardblp0/bardbl2,Tr≤ǫ. Sinceǫ >0 was arbitrary, this says p= 0. This is a contradiction.
+“2⇒1”. Suppose 1 is false. Then there exists an ǫ0>0 andx1,...,x n∈N⊖B
+such that/summationtext
+1≤i,j≤n/bardblEB(xiux∗
+j)/bardbl2
+2,τ≥ǫ0for allu∈ U(A). Letz=/summationtextn
+i=1x∗
+ieBxi. Then
+z⊥eB, Tr(z)<∞, and
+Tr(zuzu∗) =n/summationdisplay
+i,j=1Tr(x∗
+ieBxiux∗
+jeBxju∗) =n/summationdisplay
+i,j=1Tr(EB(xiux∗
+j)eBxju∗x∗
+i)
+=n/summationdisplay
+i,j=1τ(EB(xiux∗
+j)xju∗x∗
+i) =n/summationdisplay
+i,j=1/bardblEB(xiux∗
+j)/bardbl2
+2≥ǫ,∀u∈ U(A).
+Consider Γ z, the weak operator closure of the convex hull of {uzu∗:u∈ U(A)}. Then
+there exists a unique element y∈Γzsuch that /bardbly/bardbl2,Tr= min{/bardblx/bardbl2,Tr:x∈Γz}. The
+uniqueness implies that uyu∗=yfor allu∈ U(A) and hence y∈A′∩/a\}bracketle{tN,eB/a\}bracketri}ht. Since
+Tr(zuzu∗)≥ǫ0, Tr(zy)≥ǫ0>0. Soy >0 andy⊥eB. Note that
+Tr(y2)≤ /bardbly/bardblTr(y)≤ /bardbly/bardblTr(z)<∞.
+This contradicts the assumption of 2.
+“2⇔3” is easy to see.
+“3⇒4”. Suppose Ax⊂/summationtextn
+i=1xiB. LetHbe the closure of /hatwideAxB inL2(N,τ).
+ThenHis a leftAfinitely generated right Bbimodule. Let pbe the projection of
+L2(N,τ) ontoH. Thenp∈A′∩ /a\}bracketle{tN,eB/a\}bracketri}htis a finite projection of /a\}bracketle{tN,eB/a\}bracketri}ht. By the
+assumption of 3, p≤eB. So ˆx=p(ˆx) =eB(ˆx)∈ˆBandx∈B.19
+“4⇒3”. Suppose p∈A′∩ /a\}bracketle{tM,eB/a\}bracketri}htsatisfies Tr( z∗z)<∞. ThenH=pL2(M)
+is a leftAfinite right Bbimodule. By Lemma 5.6, we may assume that His a left
+Afinitely generated right Bbimodule. By Lemma 5.7, there exists a sequence of
+projections zninA′∩Msuch that lim n→∞zn= 1 in the strong operator topology and
+for eachn, there exist xn,1,...,x n,k∈Msuch that
+znpzn(ˆx) =k/summationdisplay
+i=1/hatwiderxn,iEB(x∗
+n,ix),for allx∈M.
+Note that znpzn∈A′∩/a\}bracketle{tM,eN/a\}bracketri}ht, and for every x∈M,
+A(znpzn(ˆx)) = (znpzn)(/hatwiderAx)⊂n/summationdisplay
+i=1/hatwidexn,iB.
+By the assumption of 4, znpzn(ˆx)∈ˆB⊂L2(B) for every x∈M. Hence, for each
+ξ∈L2(M),znpzn(ξ)∈L2(B). Since lim n→∞zn= 1 in the strong operator topology,
+p(ξ) = lim n→∞znpzn(ξ)∈L2(B), i.e.,p≤eB.
+6 Further results and examples
+In this section, we explore the hereditary properties of mixing suba lgebras of finite
+von Neumann algebras; that is, we show that if B⊂Mis a mixing inclusion, then the
+properties of an inclusion B1⊂Bcan force certain mixing properties on the inclusion
+B1⊂M. In particular, Proposition 6.1 below allows us to construct examples of
+weakly mixing subalgebras which are not mixing. We also use the crosse d product
+and amalgamated free product constructions to produce furthe r examples of mixing
+inclusions.
+6.1 Hereditary properties of mixing algebras
+Proposition 6.1. LetBbe a mixing von Neumann subalgebra of M, and let B1be a
+diffuse von Neumann subalgebra of B. We have the following
+1.B′
+1∩M=B′
+1∩B;
+2. ifB1is singular in B, thenB1is singular in M;
+3.NM(B1)′′⊆B, whereNM(B1) ={u∈ U(M) :uB1u∗=B1};
+4. ifB1is weakly mixing in B, thenB1is weakly mixing in M;20
+5. ifB1is mixing in B, thenB1is mixing in M;
+Proof. 1,2,3 follow from Theorem 4.1.
+4. By Corollary 5.2, we need to show that if x∈MsatisfiesB1x⊂/summationtextn
+i=1xiB1for
+a finite number of elements x1,...,x n∈M, thenx∈B1. Note that Bis mixing in
+M. By Corollary 5.4, x∈B. Letbi=EB(xi) for 1≤i≤n. Applying EBto both
+sides of the inclusion B1x⊂/summationtextn
+i=1xiB1, we have B1x⊂/summationtextn
+i=1biB1. SinceB1is weakly
+mixing in B,x∈B1by Corollary 5.2.
+5. Suppose B1is mixing in Bandunis a sequence of unitary operators in B1with
+limn→∞un= 0 in the weak operator topology. For x,y∈M, we have
+lim
+n→∞/bardblEB(xuny)−EB(x)unEB(y)/bardbl2= 0
+sinceBis mixing in M. Applying EB1toEB(xuny)−EB(x)unEB(y), we have
+lim
+n→∞/bardblEB1(xuny)−EB1(EB(x)unEB(y))/bardbl2= 0 (1)
+SinceB1is mixing in B,
+lim
+n→∞/bardblEB1(EB(x)unEB(y))−EB1(x)unEB1(u)/bardbl2= 0. (2)
+Combining (1) and (2), we have
+lim
+n→∞/bardblEB1(xuny)−EB1(x)unEB1(u)/bardbl2= 0,
+which implies that B1is mixing in M.
+Remark 6.2. Suppose Biis a diffuse von Neumann subalgebra of Mifori= 1,2.
+IfB1/\e}atio\slash=M1orB2/\e}atio\slash=M2, thenB1¯⊗B2is not a mixing von Neumann subalgebra of
+M1¯⊗M2by Proposition 6.1. On the other hand, it is easy to check that B1¯⊗B2is
+weakly mixing in M1¯⊗M2ifB1andB2are weakly mixing in M1andM2, respectively.
+This gives examples of weakly mixing but not mixing subalgebras.
+Note that in the proof of statement 4 of Proposition 6.1, we use an e quivalent con-
+dition of weak mixing (Corollary 5.2) instead of the definition. The esse ntial difficulty
+is that in the definition of weak mixing, we do not assume that lim n→∞un= 0 in the
+weak operator topology. However, we have the following result.
+Proposition 6.3. LetMbe a type II1factor with the faithful normal trace τ, and let
+Bbe a proper subfactor of M. If{un}is a sequence of unitary operators in Bsuch
+that for all elements x,yinMwithEB(x) =EB(y) = 0, one has
+lim
+n→∞/bardblEB(xuny)/bardbl2= 0,
+then lim
+n→∞un= 0in the weak operator topology.21
+Proof. Note that Bis weakly mixing in Mand hence singular in M. In particular
+B′∩M=C1. Letωbe a non principal ultrafilter of Nand suppose lim n→ωun=bin
+the weak operator topology. For x,yinMwithEB(x) =EB(y) = 0,
+EB(xby) = lim
+n→ωEB(xuny) = 0.
+Letb=u|b|be the polar decomposition of b. Note that EB(xu∗) =EB(x)u∗= 0.
+Hence,
+EB(x|b|y) =EB(xu∗u|b|y) =EB(xu∗by) = 0.
+Letx=y∗. ThenEB(y∗|b|y) = 0 and hence y∗|b|y= 0. This implies that |b|y= 0
+for ally∈MwithEB(y) = 0. For b′∈B,EB(b′y) =b′EB(y) = 0. Hence, |b|b′y= 0.
+This implies that |b|R(b′y) = 0, where R(b′y) is the range projection of b′y. Let
+p=∨b′∈BR(b′y). Then|b|p= 0. On the other hand, 0 /\e}atio\slash=p∈B′∩M. Sop= 1. So
+|b|= 0 and b= 0. Therefore, lim n→ωun= 0 in the weak operator topology. Since
+ωis an arbitrary non principal ultrafilter of N, limn→∞un= 0 in the weak operator
+topology.
+6.2 Further examples of mixing subalgebras
+Lemma 6.4. LetBbe a von Neumann subalgebra of M. Then the following conditions
+are equivalent:
+1.Bis atomic type I;
+2. for every bounded sequence {xn}inMwith limn→∞xn= 0in the weak operator
+topology, limn→∞/bardblEB(xn)/bardbl2= 0.
+Proof. 1⇒2: Since Bis a finite atomic type I von Neumann algebra, B=⊕N
+k=1Mnk(C),
+where 1≤N≤ ∞ . So there exists a sequence of finite rank central projections pn∈B
+such that pn→1 in the strong operator topology. Therefore, τ(pn)→1. Let{xn}’ be
+a bounded sequence in Mwithxn→0 in the weak operator topology, and let ǫ >0.
+We may assume that /bardblxn/bardbl ≤1. Choose pksuch that τ(1−pk)< ǫ2/4. Note that the
+mapx∈M→pkEB(x) is a finite rank operator. There is an m > 0 such that for all
+n≥m,/bardblpkEB(xn)/bardbl2< ǫ/2. Then
+/bardblEB(xn)/bardbl2≤ /bardblpkEB(xn)/bardbl2+/bardbl(1−pk)EB(xn)/bardbl2≤ǫ/2 +ǫ/2 =ǫ.
+This proves that /bardblEB(xn)/bardbl2→0.
+2⇒1: IfMis not atomic type I. Then there is a nonzero central projection p∈M
+such that pMis diffuse. So there is a Haar unitary operator v∈pM. Note that
+vn→0 in the weak operator topology. But /bardblEB(vn)/bardbl2=/bardblvn/bardbl2=τ(p)1/2does not
+converge to 0. This contradicts to 2.22
+Proposition 6.5. LetM=M1∗AM2be the amalgamated free product of diffuse
+finite von Neumann algebras (M1,τ1)and(M2,τ2)over an atomic finite von Neumann
+algebraA. ThenM1is a mixing von Neumann subalgebra of M.
+Proof. The following spaces are mutually orthogonal with respect to the un ique trace τ
+onM:M2⊖A, (M1⊖A)⊗(M2⊖A), (M2⊖A)⊗(M1⊖A), (M1⊖A)⊗(M2⊖A)⊗(M1⊖A),
+···. Furthermore, the trace-norm closure of the linear span of the a bove spaces is
+L2(M,τ)⊖L2(M1,τ). Suppose {un}is a sequence of unitary operators in M1satisfying
+lim
+n→∞un= 0 in the weak operator topology. To prove M1is a mixing von Neumann
+subalgebra of M, we need only to show for xin each of the above space, we have
+lim
+n→∞/bardblEM1(xunx∗)/bardbl2= 0.
+We will give the proof for xin one of the following spaces: ( M1⊖A)⊗(M2⊖A),
+(M2⊖A)⊗(M1⊖A). The other cases can be proved similarly.
+Suppose x=x1y1, wherex1∈M1⊖Aandy1∈M2⊖A. Then
+xunx∗=x1y1(un−EA(un))y∗
+1x1+x1y1EA(un)y∗
+1x∗
+1.
+Note that EM1(x1y1(un−EA(un))y∗
+1x1) = 0 and lim n→∞/bardblEA(un)/bardbl2= 0 by Lemma 6.4.
+So
+lim
+n→∞/bardblEM1(xunx∗)/bardbl2= 0.
+Suppose x=y1x1, wherex1∈M1⊖Aandy1∈M2⊖A. Then
+xunx∗=y1x1unx∗
+1y1=y1(x1unx∗
+1−EA(x1unx∗
+1))y∗
+1−y1EA(x1unx∗
+1)y∗
+1.
+Note that EM1(y1(x1unx∗
+1−EA(x1unx∗
+1))y∗
+1) = 0 and lim n→∞/bardblEA(x1unx∗
+1)/bardbl2= 0 by
+Lemma 6.4. So
+lim
+n→∞/bardblEM1(xunx∗)/bardbl2= 0.
+Note, in particular, that Proposition 6.5 implies that if Ais a diffuse mixing masa
+in a finite von Neumann algebra M1, andM2is also diffuse, then Ais mixing in the
+free product M1∗M2.
+Now let Bbe a diffuse finite von Neumann algebra with a faithful normal trace
+τ, and let Gbe a countable discrete group. Let ∗g∈GBgbe the free product von
+Neumann algebra, where Bgis a copy of Bfor each g. The shift transformation
+σ(g)((xh)) = (xg−1h) defines an action of Gon∗g∈GBg. LetM=∗g∈GBg⋊G. Then
+Mis a type II 1factor and we can identify BwithBe.
+Proposition 6.6. The above algebra Bis a mixing von Neumann subalgebra of M.23
+Proof. Suppose vgis the classical unitary operator corresponding to the action gin
+M. Then for every ( xh) in∗g∈GBg,
+vg(xh)v−1
+g= (σg(xh)) = (xg−1h).
+Suppose bn∈B=Be,bn→0 in the weak operator topology, g/\e}atio\slash=e, andxh∈Bh. We
+may assume τ(bn) = 0 for each n. Note that
+xhvgvnv∗
+gx∗
+h=xhσg(bn)x∗
+h.
+Ifh/\e}atio\slash=e, it is clear that xhσg(bn)x∗
+his free with B=Beand hence orthogonal to B.
+Ifh=e, direct computations show that xeσg(bn)x∗
+eis orthogonal to B=Be. So we
+have
+EB(xhvgbnv∗
+gx∗
+h) =EB(xhσg(bn)x∗
+h) =τ(xhσg(bn)x∗
+h) =τ(σg(bn)x∗
+hxh) =τ(bnσg−1(x∗
+hxh)),
+and this last expression above converges to zero. Note that the lin ear span of the
+above elements xhvgis dense in M⊖Bin the weak operator topology. This proves
+thatBis mixing in M.
+References
+[1] V. Bergelson and J. Rosenblatt, Mixing actions of groups. Illinois J. Math ,
+32(1):65–80, 1988.
+[2] E. Christensen, Subalgebras of a finite algebra. Math. Ann. , 243(1):17–29, 1979.
+[3] J. Dixmier, Sous-anneaux ab´ eliens maximaux dans les facteurs d e type fini. Ann.
+of Math. (2) , 59:279–286, 1954.
+[4] P. Grossman and A. Wiggins, Strong singularity for subfactors, preprint ,
+arXiv:math/0703673 2008.
+[5] P. R. Halmos, On automorphisms of compact groups. Bull. Amer. Math. Soc. ,
+49:619–624, 1943.
+[6] P. Jolissaint and Y. Stalder, Strongly singular MASAs and mixing act ions in
+finite von Neumann algebras. Ergodic Theory Dynam. Systems , 28(6):1861–1878,
+2008.
+[7] V. F. R. Jones, Index for subfactors. Invent. Math. , 72(1):1–25, 1983.
+[8] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator alge-
+bras. Vol. II , volume 16. American Mathematical Society, Providence, RI, 1997 .24
+[9] B. Kitchens and K. Schmidt, Automorphisms of compact groups. Ergodic Theory
+Dynam. Systems , 9 (1989), no. 4, 691–735.
+[10] M. G. Nadkarni, Spectral theory of dynamical systems . Birkh¨ auser Advanced
+Texts: Basler Lehrb¨ ucher. Birkh¨ auser Verlag, Basel, 1998.
+[11] M. Pimsner and S. Popa, Entropy and index for subfactors. Ann. Sci. ´Ecole
+Norm. Sup. (4) , 19(1):57–106, 1986.
+[12] S. Popa, A short proof of “injectivity implies hyperfiniteness” f or finite von
+Neumann algebras. J. Operator Theory , 16(2):261–272, 1986.
+[13] S. Popa, On a class of type II 1factors with Betti numbers invariants. Ann. of
+Math. (2) , 163(3):809–899, 2006.
+[14] S. Popa, Strong rigidity of II 1factors arising from malleable actions of w-rigid
+groups. I. Invent. Math. , 165(2):369–408, 2006.
+[15] G. Robertson, A. M. Sinclair, and R. R. Smith, Strong singularity for subalgebras
+of finite factors. Internat. J. Math. , 14(3):235–258, 2003.
+[16] S. Sakai, “The Theory of W* Algebras”, Lecture notes, Yale Un iversity, 1962.
+[17] A. M. Sinclair and R. R. Smith, Strongly singular masas in type II 1factors.
+Geom. Funct. Anal. , 12(1):199–216, 2002.
+[18] A. M. Sinclair and R R. Smith, Finite von Neumann algebras and masas , volume
+351 ofLondon Mathematical Society Lecture Note Series . Cambridge University
+Press, Cambridge, 2008.
+[19] A. M. Sinclair and S. A. White, A continuous path of singular masas in the
+hyperfinite II 1factor,J. Lond. Math. Soc . (2)75(2007), no. 1, 243–254.
+[20] A. M. Sinclair, R R. Smith, S A. White, and A. Wiggins, Strong singula rity of
+singular masas in II 1factors.Illinois J. Math. , 51(4):1077–1084, 2007.
+[21] S. Vaes, Rigidity results for Bernoulli actions and their von Neum ann algebras
+(after Sorin Popa). Ast´ erisque , (311):Exp. No. 961, viii, 237–294, 2007. S´ eminaire
+Bourbaki. Vol. 2005/2006.
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+arXiv:1001.0170v2  [q-bio.PE]  31 Jan 2010Speeding up disease extinction with a limited amount of vacc ine
+M. Khasin1, M.I. Dykman1and B. Meerson2
+1Department of Physics and Astronomy, Michigan State Univer sity, East Lansing, MI 48824 USA and
+2Racah Institute of Physics, Hebrew University of Jerusalem , Jerusalem 91904, Israel
+We consider optimal vaccination protocol where the vaccine is in short supply. In this case, disease
+extinction results from a large and rare fluctuation. We show that the probability of such fluctuation
+can be exponentially increased by vaccination. For periodi c vaccination with fixed average rate, the
+optimal vaccination protocol is model independent and pres ents a sequence of short pulses. The
+effect of vaccination can be resonantly enhanced if the pulse period coincides with the characteristic
+period of the disease dynamics or its multiples. This resona nt effect is illustrated using a simple
+epidemic model. If the system is periodically modulated, th e pulses must be synchronized with the
+modulation, whereas in the case of a wrong phase the vaccinat ion can lead to a negative result.
+The analysis is based on the theory of fluctuation-induced po pulation extinction in periodically
+modulated systems that we develop.
+PACS numbers: 87.23.Cc, 02.50.Ga, 05.40.-a
+I. INTRODUCTION
+Spreading of an infectious disease is a random pro-
+cess. An important source of the randomness is demo-
+graphic noise associated with the stochastic character of
+the events of infection, recovery, birth, death, etc. In
+a large population demographic noise is small on aver-
+age, and the infection spread leads to an endemic state
+where a certain fraction of the population stays infected
+for a long time. The disease, however, can disappear as
+a result of a large rare fluctuation, an unlikely chain of
+events where, for example, susceptible individuals hap-
+pen to avoid getting infected while infected ones recover
+[1–3]. Then, if there is no influx of infected individu-
+als from the outside, the population will be disease-free.
+Such spontaneous disappearance of a disease is an exam-
+ple of population extinction studied in stochastic popu-
+lation dynamics.
+Spontaneous extinction is also important for various
+physical and chemical reaction systems. This is a conse-
+quence of the underlying similarity of the dynamics that
+result from short random events, like collisions between
+molecules that lead to chemical reactions and interac-
+tions between individuals that lead to the disease spread.
+Extinction can be understood for different types of sys-
+tems within the same general formalism, which provides
+a broader scope for the present paper. Moreover, the
+method of optimal control of extinction that we propose
+can be applied to systems of various types.
+A conventional way of fighting epidemics is via vac-
+cination. If there is enough vaccine, the infection can
+be eradicated “deterministically” by eliminating the en-
+demic state [4]. The amount of available vaccine, how-
+ever, is often insufficient. The vaccine may be expensive,
+or it may be dangerous to store in large amounts, as in
+the case of anthrax, or it may be effectively short-lived
+due to mutations of the infection agent, as for HIV [5]
+and influenza [6].
+Even where the endemic state may not be eliminated
+deterministically, vaccination can dramatically affect thestochastic dynamics of the epidemics. The underlying
+mechanism is the change of the rate of large fluctuations
+leading to disease extinction. For a well mixed popu-
+lation, this rate Weis usually exponentially small for
+a large total population size N≫1,We∝exp(−Q)
+withQ ∝N, [3, 7–14]. We call Qthe disease extinc-
+tion barrier. Vaccination changes the value of Q/N. In
+turn, this changes the disease extinction rate exponen-
+tially strongly. This effect was previously discussed for
+vaccination applied at random [13].
+The goal of this paper is to find an optimal way of
+administering a limited amount of vaccine which would
+maximally increase the disease extinction rate. We find
+a vaccination protocol that applies for a broad class of
+epidemic models. Our approach is based on the obser-
+vation that, in a large fluctuation that leads to disease
+extinction, the population is most likely to evolve in a
+well-defined way. It moves along the most probable path
+inthespaceofthedynamicalvariableswhichcharacterize
+different sub-populations [13–15]. Vaccination perturbs
+the system as it moves along the optimal path. One can
+think of vaccination as “force” and its effect as “work”
+done on the system. This work reduces the barrier Q.
+The problem then is to maximize the work for given con-
+straints on the vaccine.
+Optimization of the effect of vaccination resembles an-
+other problem of optimal control of random systems:
+controlling large fluctuations in noise-driven dynamical
+systems by applying an external field with a given av-
+erage intensity [16, 17]. There are, however, important
+differences, which come from the very nature of the con-
+trol field. Indeed, vaccination only reducesthe number
+of susceptible individuals. In other words, as a control
+field, vaccination never changes sign. Then, as we find, if
+the available amount of vaccine is constrained by a given
+mean vaccination rate, the optimal vaccination protocol
+turns out to be model-independent. This applies also to
+using a limited amount of medications and other situa-
+tions where the control field drives the system only in
+one direction.2
+We assume that vaccination is applied periodically in
+time. In this case, the optimal protocol is to apply the
+vaccine as a sequence of δ-like pulses. The disease ex-
+tinction rate can strongly depend on the period of this
+sequence. Furthermore, the extinction rate can display
+exponentially sharp peaks when the pulse sequence pe-
+riod is close to the characteristic period of oscillations
+of the system in the absence of fluctuations, or to its
+multiples. We illustrate this resonant phenomenon for
+the Susceptible-Vaccinated-Infected-Recovered (SVIR)
+model.
+Epidemics often display seasonal modulation [18]. It
+is natural to apply a vaccine with period equal to the
+modulation period. As we show, there is a qualitative
+difference between the effect of a periodic vaccination in
+this case and in the case where seasonal modulation is
+absent. For a system with seasonal modulation, an im-
+properly applied pulsed vaccination can actually reduce
+the disease extinction rate and therefore prolong the du-
+ration of the epidemic. The overall effect of the pulsed
+vaccination critically depends here on the phaseat which
+the periodic pulses are applied.
+The analysis of periodic vaccination, with and with-
+out seasonal variations, necessitates a general formula-
+tion of the extinction problem in periodically modulated
+stochastic populations. We extend the previous results
+for single-population systems [19, 20] to provide a com-
+plete extinction theory for multi-population systems in
+theeikonalapproximation,andemphasizethedistinction
+from the well-understood problem of switching between
+metastablestatesin periodicallymodulated systemswith
+noise [21].
+Section II describes the class of epidemic models we
+consider in this work and develops an eikonal theory
+of disease extinction rate in periodically modulated sys-
+tems. SectionIIIformulatestheoptimizationproblemfor
+vaccination and presents its solution for a time-periodic
+vaccination in the limit of a small average vaccination
+rate. In Section IV we discuss the vaccination-induced
+reduction of the disease extinction barrier for two types
+of constraints on the vaccination period, a limited life-
+time of the vaccine and a limited vaccine accumulation.
+Section V illustrates, on the example of the stochastic
+SVIR model, the phenomenon of resonant response to
+vaccination. Section V contains concluding remarks.
+II. THE DISEASE EXTINCTION RATE
+A. The model of population dynamics
+We consider stochastic disease dynamics in a well-
+mixed population which includes infected ( I) and sus-
+ceptible ( S) individuals and possibly other population
+groups such as recovered or vaccinated. The system
+state is described by a vector X= (S,I,...) with in-
+teger components equal to the sizes of different popula-
+tion groups. Along with Xit is convenient to consider aquasi-continuous vector x=X/N, whereNis the char-
+acteristic total population size, N≫1. We assume that
+the population dynamics is Markovian. It is quite gener-
+ally described by the master equation for the probability
+distribution P(X,t),
+˙P(X,t) =/summationdisplay
+r[W(X−r,r,t)P(X−r,t)
+−W(X,r,t)P(X,t)]. (1)
+HereW(X,r,t) is the rate of an elementary transition
+X→X+rin which the population size changes by r=
+(r1,r2,...). Examples of such transitions are infection
+of a susceptible individual as a result of contacting an
+infected individual, recovery of an infected individual or
+arrival of a susceptible individual.
+We assume that there is no influx of infected individ-
+uals into the population. Therefore, there are no tran-
+sitions from states where there are no infected to states
+where infected are present,
+W(X,r,t) = 0 for XE= 0, rE∝ne}ationslash= 0,(2)
+where subscript Eis used for the component of Xwhich
+enumerates infected, XE≡I.
+In the neglect of demographic noise the population dy-
+namicscanbedescribedbythedeterministic(mean-field)
+equation for the population size ¯x,
+˙¯x=/summationdisplay
+rrw(¯x,r,t), w(x,r,t) =W(X,r,t)/N.(3)
+It immediately follows from Eq. (1) if the width of the
+probability distribution P(X,t) is set equal to zero.
+1. Stationary systems
+We start with the case where the transition rates
+W(X,r) are independent of time. An endemic state,
+where a finite fraction of population is infected for a long
+time, corresponds to an attracting fixed point xAof the
+dynamical system, Eq. (3). We will assume throughout
+this work that there is only one such point. We will also
+assume that Eq. (3) has one fixed point xSin the hy-
+perplane xE= 0. The state xSis stable with respect to
+all variables except xE. We call it the disease extinction
+state. If xE>0 (there is a nonzero number of infected),
+the deterministic trajectory leaves the vicinity of xSand
+approaches the endemic state xA.
+Due to demographic noise the endemic state is ac-
+tuallymetastable . A rare large fluctuation ultimately
+drives the population into a disease-free state. The most
+probable fluctuation brings the system to the fixed point
+xS[13, 14]. The probability of such a fluctuation per
+unit time, i.e., the disease extinction rate We, is given
+by the probability current to xS, similarly to the prob-
+lem of escape from a metastable state [22]. For time-
+independent W(X,r) this current is quasistationary for3
+timestr≪t≪W−1
+e, wheretris the characteristicrelax-
+ation time for the noise-free motion described by Eq. (3).
+We note that, even though the state xSis a saddle
+point in the mean-field approximation, it differs from
+the saddle-point states encountered in the problem of
+switching between metastable states of reaction systems.
+In the case of interstate switching, the rates of elemen-
+tary transitions W(X,r) in the unstable direction are
+nonzero, and ultimately fluctuations drive the system
+away from the saddle point. In the case of extinction,
+fluctuations around xSoccur only in the extinction hy-
+perplane, whereas the probability of exiting this hyper-
+plane is zero.
+If the system has, in the mean-field approximation,
+more than one steady state away from the extinction hy-
+perplane, extinction can go in steps: from the endemic
+state to another steady state and, ultimately to the ex-
+tinction hyperplane. In particular, if the only additional
+steady state is a saddle point at the boundary between
+the basins of attraction of xAandxS, the problem of
+extinction can be reduced to the problem of escape over
+this saddle point [23].
+2. Periodically modulated systems
+The above picture can be extended to the case where
+the transition rates are periodic functions of time,
+W(X,r,t+T) =W(X,r,t). Periodicity of some of
+the rates in time is a natural way of modeling sea-
+sonal variations of epidemics [18]. The attracting so-
+lution of Eqs. (3), which describes the endemic state,
+is no longer stationary. We will assume that this solu-
+tion,xA(t), is periodic in time with the same period T,
+xA(t+T) =xA(t). The asymptotic disease extinction
+statexS(t) is also periodic in time; it lies in the hyper-
+planexE= 0.
+The most probable fluctuation which causes extinction
+of the disease corresponds to a transition from xA(t) to
+xS(t) [20]. An important characteristic of such a tran-
+sition is the period-averaged disease extinction rate We.
+It can be introduced if the modulation period T≪W−1
+e
+and, in addition, tr≪W−1
+e. In this case, for time tsuch
+thattr,T≪t≪W−1
+e, a quasi-stationary time-periodic
+probabilitydistributionisformed, centeredat xA(t). The
+probabilitycurrent from xA(t) toxS(t) is also periodic in
+time, and the period-averaged value of this current gives
+We[20], in a direct analogy with the problem of switch-
+ing between metastable states in noise-driven dynamical
+systems [24–27].
+B. Eikonal approximation
+1. Equations of motion
+We will be interested in evaluating the disease extinc-
+tion barrier Qwhich gives the exponent in the diseaseextinction rate, We∝exp(−Q), atN≫1. This bar-
+rier is entropic in nature, as it results from an unlikely
+sequence of elementary transitions. It can be found by
+either solving the mean first passage time problem for
+reaching xS(t)[7–9]orbycalculatingthetailofthequasi-
+stationary probability distribution P(X,t) forxclose to
+xS(t) [11, 13, 14]. Here we choose the latter strategy and
+determine, to the leading order in N, the logarithm of
+the distribution tail. We seek the solution of Eq. (1) in
+eikonal form, P(X,t) = exp[ −Ns(x,t)] [28–30]. In the
+limit of large N, from Eq. (1) we obtain the following
+equation for s(x,t):
+∂ts=−H(x,∂xs,t), (4)
+H(x,p,t) =/summationdisplay
+rw(x,r,t)[exp(pr)−1].
+Here,wehavetakenintoaccountthat, typically, |r| ≪N,
+andW(X,r,t) depends on Xpolynomially, whereas Pis
+exponential in X. Therefore we expanded P(X+r,t)≈
+P(X,t)exp(−r∂xS) and replaced, to the leading order in
+1/N,w(x−N−1r,r,t) byw(x,r,t).
+Equation (4) has the form of the Hamilton-Jacobi
+equation for an auxiliary Hamiltonian system with
+Hamiltonian H(x,p,t);s(x,t) is the action of this sys-
+tem. The Hamilton equations of motion are
+˙x=/summationdisplay
+rrw(x,r,t)epr,
+˙p=−/summationdisplay
+r∂xw(x,r,t)(epr−1).(5)
+These trajectories determine, in turn, the most probable,
+or optimal, trajectories that the system follows in a fluc-
+tuation to a given state xat timet. We will calculate
+actions(x,t) using these trajectories and thus find the
+exponent in the distribution P(X,t).
+2. Boundary conditions for the optimal extinction
+trajectory
+To find the boundary conditions for Hamiltonian tra-
+jectories (5), we note that the quasi-stationary distribu-
+tionP(X,t) has a Gaussian maximum at XA(t). This
+means that, close to attractor xA(t), action s(x,t) is
+quadratic in |x−xA)|for stationary systems, whereas
+for periodically modulated systems s(x,t) =s(x,t+T)
+is quadratic in the distance from trajectory xA(t) [31].
+On the Hamiltonian trajectories that give such action,
+the momentum p≡∂xs→0 forx→xA(t), and since
+x=xA(t),p= 0 is a fixed point (a periodic trajectory)
+of the Hamiltonian dynamics, the trajectories of interest
+start att→ −∞,
+s(x,t) =/integraldisplayt
+−∞dtL(˙x,x,t), (6)
+L(˙x,x,t) =/summationdisplay
+rw(x,r,t)[(pr−1)epr+1].
+In the Lagrangian L, Eq. (6), pshould be expressed in
+terms of ˙x,xusing Eq. (5). Since w≥0, we have L≥0.4
+Extinction barrier Qis determined by Ns(x,t) forx
+in the extinction hyperplane, xE= 0. In the spirit of the
+eikonal approximation, we have to find such ( x,t) in this
+hyperplane that s(x,t) be minimal. The minimum de-
+termines the boundary conditions for the optimal Hamil-
+tonian trajectory of extinction, ( xopt(t),popt(t)). The
+condition that s(x,t) is minimal with respect to xi/negationslash=E
+on the extinction hyperplane means that pi=∂xis→0
+fori∝ne}ationslash=Eas the trajectory ( xopt(t),popt(t)) approaches
+the hyperplane. The minimum of s(x,t) with respect to t
+withintheperiodofmodulationisreachedif H(x,p,t)→
+0 as the trajectory approaches the hyperplane.
+A consequence of conditions H(x,p,t)→0 and
+pi/negationslash=E→0 is that the momentum component pEremains
+bounded on trajectory ( xopt(t),popt(t)). Indeed, near
+the extinction hyperplane, xE≪1, we have
+˙xE=/summationdisplay
+rrEw(x,r,t)epr
+≈xE/summationdisplay
+rrE[∂w(x,r,t)/∂xE]xE=0epErE.(7)
+Here,weassumedthat w(x,r,t) isnonsingularat xE→0
+and, since w= 0 forxE= 0 and rE∝ne}ationslash= 0 [cf. Eq. (2)],
+we expanded winxEto the lowest order. Let us assume
+nowthat |pE| → ∞forxE→0. Then onlythe term with
+maximal −rE≡ −rEmshould be kept in the sum over
+rEin Eq. (7); it is also clear that pEshould be negative,
+otherwise the trajectory would not approach xE= 0.
+From the Hamilton equation for pEand Eq. (7) it follows
+thatdpE/dxE≈ −1/xErEm. This relation, along with
+the explicit form of the Hamiltonian H, show that, if pE
+were diverging for xE→0, the Hamiltonian would not
+become equal to zero but would remain ≈∂w/∂x Ewith
+the derivative calculated for xE= 0 andrE=rEm. This
+contradiction shows that the assumption |pE| → ∞is
+wrong,pEremains limited for xE→0.
+Equation(7) shows that xE→0 exponentially as
+t→ ∞. AsxEapproaches zero, variables xi/negationslash=Eare
+approaching the equilibrium position in the hyperplane
+xE= 0. This happens because pi/negationslash=E→0 and the dy-
+namics of xi/negationslash=Ein the hyperplane is described by the
+mean-field equations, Eq. (3). Therefore,
+Q=Nsext, s ext=/integraldisplay∞
+−∞dtL(˙x,x,t),(8)
+x(t)→xS(t),p(t)→pS(t) fort→ ∞.
+Function pS(t) is periodic in time, with pi/negationslash=E= 0 and
+with hitherto unknown component pE(t), which is dis-
+cussed below.
+The optimal trajectory ( xopt(t),popt(t)) is a hetero-
+clinic Hamiltonian trajectory that goes from periodic or-
+bit (xA(t),p=0) to periodic orbit ( xS(t),pS), and the
+action for extinction sextis calculated along this trajec-
+tory. Thetrajectory xopt(t)istheoptimalpathtodisease
+extinction: it describes the most probable sequence of el-
+ementarytransitionsleadingto extinction. We notethat,
+in periodically modulated systems, there is one optimal
+path per period, whereas in stationary systems trajecto-
+ries (xopt(t),popt(t)) are time-translation invariant.3. The t→ ∞value of the momentum on the Hamilton
+trajectory
+The momentum component pEis generically nonzero,
+as found for stationary systems [9, 13, 14, 32]. For peri-
+odically modulated systems, one can show that pE∝ne}ationslash= 0
+by extending the arguments presented in Ref. [13, 32].
+This amounts to showing that the stable manifold of the
+periodic orbit ( xS(t),p=0) lies entirely in the invariant
+hyperplane xE= 0,pi/negationslash=E= 0, and, as a consequence,
+does not intersect the unstable manifold of the periodic
+orbit (xA(t),p=0) . Such intersection is necessary in
+ordertohaveaheteroclinictrajectorythatwouldgofrom
+(xA(t),p=0) to (xS(t),p=0).
+The hyperplane xE= 0,pi/negationslash=E= 0 is formed by trajec-
+tories
+˙xi/negationslash=E=/summationdisplay
+r[w(x,r,t)]xE=0ri, (9)
+˙pE=−/summationdisplay
+r[∂xEw(x,r,t)]xE=0(epErE−1).
+The invariance of this hyperplane is a consequence of
+Eq.(2), which leadsto ˙ pi/negationslash=E= 0 and ˙xE= 0forpi/negationslash=E= 0
+andxE= 0.
+To prove that the stable manifold of ( xS(t),p=0) lies
+entirelyin theinvarianthyperplane xE= 0,pi/negationslash=E= 0, we
+first show that the trajectories, which are described by
+Eq. (9) and which start close to the state ( xS(t),p=0),
+approach this state for t→ ∞. Then, since the dimen-
+sion of the hyperplane xE= 0,pi/negationslash=E= 0 is equal to
+the dimension of the stable manifold of ( xS(t),p=0),
+we conclude that the stable manifold indeed lies in the
+hyperplane.
+Equations (9) for xi/negationslash=Eare the mean-field equations
+in the extinction hyperplane xE= 0, cf. Eqs. (3), and
+therefore xi→(xS(t))ifort→ ∞. Linearization of
+Eq. (9) for pEabout (xS(t),p=0) gives
+˙pE=−/summationdisplay
+r[∂xEw(x,r,t)]xS(t)pErE.(10)
+We compare this equation with the mean-field equa-
+tion for xEnearxS(t). The latter has the form ˙ xE=
+xE/summationtext
+rrE[∂xEw(x,r,t)]xS(t). Since the state xS(t) is
+unstable in xEdirection in the mean-field approxima-
+tion, from Eq. (10) ˙ pE/pE<0. Therefore, all trajecto-
+ries on the hyperplane xE= 0,pi/negationslash=E= 0 close to the
+state (xS(t),p=0) approach this state asymptotically
+ast→ ∞, and thus the stable manifold of ( xS(t),p=0)
+lies in this hyperplane.
+From the above analysis one concludes that there
+are no Hamiltonian trajectories that would go from
+(xA(t),p=0) to/parenleftbig
+xS(t),p=0/parenrightbig
+. Therefore the opti-
+mal trajectory leading to extinction should go to a state/parenleftbig
+xS(t),[pS(t)]E/parenrightbig
+with [pS(t)]E∝ne}ationslash= 0.5
+III. OPTIMAL VACCINATION
+A. Constraint on the vaccination protocol
+Vaccination increases the number of individuals who
+are at least temporarily immune to the disease. It thus
+reducesthepoolofsusceptibleindividualsandultimately
+leads to a reduction of the number of infected. When the
+available amount of vaccine is small, so that the disease
+extinction still requires a large fluctuation, the goal of
+vaccination is to reduce the disease extinction barrier Q.
+An outcome of vaccination is often modeled as the cre-
+ation of a sub-population of vaccinated individuals out
+of susceptibles. The corresponding elementary transi-
+tion rate is W(X,r) =ξ0(t)XSforrS=−1,rV= 1
+andri/negationslash=S,V= 0, where subscripts VandSrefer to
+vaccinated and susceptible individuals, respectively, and
+ξ0(t) is the control field that characterizes the vaccina-
+tion (subscript Sshould not be confused with subscript
+Sused to indicate the extinction state). Another model
+is vaccination of newly arrived susceptibles, which leads
+to an effective reduction of the arrival rate µN. In this
+model, the elementary transition rate for the arrival is
+W(X,r) =N[µ−ξ0(t)] forrS= 1 and ri/negationslash=S= 0, with
+ξ0(t)Nbeing the change in the arrival rate due to vacci-
+nation.
+We will consider a general model where vaccination
+modifies the rate of an elementary transition of a certain
+type; the change of the population in the corresponding
+transition is rξ. The field ξ0(t) characterizing the vacci-
+nation is assumed to be weak. The affected rate has the
+formW(X,rξ,t) =W(0)(X,rξ,t) +ξ0(t)W(1)(X,rξ,t),
+withW(0)being the rate without vaccination. The vac-
+cinationeitherincreasesordecreasesthe rate,asfortran-
+sitions from susceptibles to vaccinated or for vaccination
+of newly arrived susceptibles, respectively. Therefore, we
+will assume without loss of generality that ξ0(t)≥0 and
+thatW(1)(X,rξ,t) is either positive or negative. We con-
+sider models in which the number of susceptibles changes
+by 1 in an elementary transition associated with vacci-
+nation, ( rξ)S=±1. We note that the analysis can be
+immediately extended to describe other processes, like
+modification of the infection rates [3] or recovery accel-
+eration by administrating medicine.
+It should be noted that the vaccination model adopted
+in this work is probabilistic by nature. An alternative
+is where vaccination is done in a pre-determined fashion,
+when a certain number of individuals are vaccinated per
+unit time at a given time. The analysis of such determin-
+istic vaccination lies beyond the scope of this paper.
+We will assume that vaccination is periodic, ξ0(t) =
+ξ0(t+T), and that the amount of vaccine available per
+periodTis limited. We model this limitation as a con-
+straint on the ensemble-averaged number of individuals
+vaccinated per period T. The constraint can be writtenas
+T−1/integraldisplayT
+0dtξ0(t)/summationdisplay
+X/vextendsingle/vextendsingle/vextendsingleW(1)(X,rξ,t)/vextendsingle/vextendsingle/vextendsingleP(X,r,t)
+=NΞ. (11)
+Here, Ξ is the average vaccination rate rescaled by the
+characteristic population size N. The constraint is well-
+defined for tr≪t≪W−1
+e, where P(X,r,t+T)≈
+P(X,r,t). Since for N≫1 the population distribution
+sharply peaks at the endemic state XA(t), the sum over
+Xin Eq. (11) can be replaced by/vextendsingle/vextendsingleW(1)(XA(t),rξ,t)/vextendsingle/vextendsingle, in
+the leading order in 1 /N.
+In the presence of vaccination, one can still seek a so-
+lution of the master equation in the eikonal form. The
+exponent Qin the extinction rate is again given by the
+action of an auxiliary Hamiltonian system, Eq. (8). The
+Hamiltonian now has the form
+H(x,p) =H(0)(x,p)+ξ0(t)H(1)(x,p),
+H(0)(x,p) =/summationdisplay
+rw(0)(x,r,t)(epr−1),(12)
+H(1)(x,p) =w(1)(x,rξ,t)(eprξ−1).
+Our goal is to find the optimal form of ξ0(t) which would
+minimize the disease extinction barrier Qsubject to con-
+straint (11). Since w(1)(xA(t),rξ,t) is a known periodic
+function of time, we can equivalently search for the op-
+timalvaccination rate ξ(t)≡ξ0(t)/vextendsingle/vextendsinglew(1)(xA(t),rξ,t)/vextendsingle/vextendsingle. It
+minimizes the functional
+˜sext[ξ(t)] =sext[ξ(t)]+λT−1/integraldisplayT
+0[ξ(t)−Ξ]dt,(13)
+ξ(t) =ξ(t+T) =ξ0(t)/vextendsingle/vextendsingle/vextendsinglew(1)(xA(t),rξ,t)/vextendsingle/vextendsingle/vextendsingle≥0,
+whereλisthe Lagrangemultiplier. Thefunctional sext[ξ]
+is given by Eq. (8), where the Lagrangian corresponds
+to the Hamiltonian (12) and depends on the vaccination
+rateξ(t).
+B. Vaccination protocol for stationary systems
+1. Double optimization problem
+For a low average vaccination rate Ξ it suffices to keep
+intheaction sext[ξ(t)]onlytheleading-ordertermin ξ(t).
+Since Hamiltonian (12) is linear in ξ, this term is linear
+inξ, too. In the spirit of the standard perturbation the-
+ory for Hamiltonian systems [33], the change in the ac-
+tion, caused by the small perturbation, can be calculated
+along the unperturbed trajectory ( x(0)
+opt(t),p(0)
+opt(t)) of the
+Hamiltonian H(0), which describes the optimal path of
+disease extinction in the absence of vaccination.
+We startwith the caseofsystems, which arestationary6
+in the absence of vaccination. For such systems
+sext[ξ(t)] =s(0)
+ext+s(1)
+ext[ξ(t)], (14)
+s(1)
+ext[ξ(t)] = min
+t0/integraldisplay∞
+−∞dtχ(t−t0)ξ(t),
+χ(t) =−H(1)/parenleftbig
+x(0)
+opt(t),p(0)
+opt(t)/parenrightbig/vextendsingle/vextendsingle/vextendsinglew(1)(xA,rξ)/vextendsingle/vextendsingle/vextendsingle−1
+.
+The quantity χ(t) is called logarithmic susceptibility [13,
+20, 34, 35]; it gives the change of the logarithm of the
+extinction rate, which is linear in the vaccination rate for
+moderately low vaccination rate.
+The minimization over t0in Eq. (14) accounts for lift-
+ing the time-translational invariance of the optimal ex-
+tinction paths mentioned earlier. For ξ(t)≡0, extinc-
+tion can occur at any time ( tr≪t≪W−1) with rate
+We. Periodic vaccination synchronizes extinction events;
+it periodically modulates the extinction rate, and the
+modulation is exponentially strong for N|s(1)
+ext| ≫1 (see
+below). Formally, in a modulated system there is only
+one optimal extinction path per period, as explained in
+Sec. II, which is here reflected in the minimization over
+t0. This optimal path minimizes the disease extinction
+barrierQ=Nsext[19, 20, 34, 35]. Equation (14) is
+closely related to the Mel’nikov theorem for dynamical
+systems [20, 36].
+The constraint for minimizing the action over ξ(t) in
+Eq. (13) has a form of an integral over the vaccination
+periodT. It is therefore convenient to write action s(1)
+ext
+also in the form of such an integral,
+s(1)
+ext[ξ(t)] = min
+t0/integraldisplayT
+0dtξ(t)χT(t−t0),
+χT(t) =∞/summationdisplay
+n=−∞χ(t+nT). (15)
+The function χT(t) is obtained by superimposing the
+parts of χ(t) which differ by T. By construction, χT(t)
+is periodic in time t.
+2. Temporal shape of optimal vaccination
+To find the optimal shape of vaccination rate ξ(t) we
+first minimize the time integral in the variational prob-
+lem Eqs. (13) – (15) with respect to ξ(t) for a given t0.
+Sinceξ(t)≥0, it is convenient to perform the minimiza-
+tion with respect to ξ1/2(t) rather than ξ(t). The min-
+imization shows that ξ1/2(t)∝ne}ationslash= 0 only for t=tλ, where
+tλis given by equation χT(tλ−t0) =−λ/T. From the
+constraint on the period-averaged ξ(t) we then have
+ξ(t) = ΞT/summationdisplay
+nδ(t−tλ+nT). (16)
+Substituting this expression into the functional ˜ sextand
+minimizing with respect to t0, we obtain the action in asimple explicit form
+sext= min˜sext=s(0)
+ext+s(1)
+ext,
+s(1)
+ext= ΞTmin
+0≤t<TχT(t). (17)
+Alternatively, this expression can be rewritten in terms
+of the Fourier transform of the logarithmic susceptibility:
+s(1)
+ext= Ξ min
+t/summationdisplay
+n˜χ(nΩ)exp[inΩt],(18)
+˜χ(ω) =/integraldisplay∞
+−∞dtχ(t)exp(iωt),
+where Ω = 2 π/Tis the cyclic frequency of vaccination.
+We areinterested in the solution for which s(1)
+extis nega-
+tive, which requires min χT(t)<0. Only in this case will
+vaccination reduce the barrier for disease extinction and
+increase the disease extinction rate. The barrier reduc-
+tion due to vaccination, Q(1)=Ns(1)
+ext∝NΞ, becomes
+large for N≫1 even if the average vaccination rate Ξ is
+small. Thus, for not too small vaccination rates, where
+the eikonal approximation is valid [20, 34], the effect of
+vaccinationonthediseaseextinctionrateisexponentially
+strong.
+Theexpressionfortheactionchange s(1)
+ext, Eq.(17), can
+be also obtained in a more intuitive way. Indeed, since
+ξ(t) is non-negative, it follows from Eq. (15) that
+s(1)
+ext[ξ(t)]≥min
+tχT(t)/integraldisplayT
+0dtξ(t) = ΞTmin
+tχT(t).(19)
+The minimum is provided by ξ(t) = ΞT/summationtext
+nδ(t−tmin+
+nT). Formally, tminis the instance of time where χT(t)
+is minimal. In fact, it is the optimal path that adjusts to
+the vaccination pulses so as to increase the probability of
+disease extinction. This provides the mechanism of syn-
+chronization by vaccination. Equation (19) immediately
+leads to Eqs. (16) and (17) with tλreplaced by tmin.
+In addition to the constraint on the average vacci-
+nation rate, there may be an upper limit on the in-
+stantaneous vaccination rate, which is imposed by con-
+ditionw(x,r) =w(0)(x,rξ) +ξ(t)w(1)(x,rξ)≥0. In
+the case where w(1)(x,rξ,t)<0, as for vaccination of
+newly arrivedsusceptibles, this condition is met provided
+ξ0(t)≤ξ0m≡min{w(0)(x,rξ)//vextendsingle/vextendsinglew(1)(x,rξ)/vextendsingle/vextendsingle}. In this
+case the optimal vaccination protocol changes.
+The new protocol can be found from the variational
+problem (13) by changingfrom ξ(t)≡ξ0(t)/vextendsingle/vextendsinglew(1)(xA,rξ)/vextendsingle/vextendsingle
+to an auxiliary function η(t) such that ξ0(t) =ξ0m[1 +
+η2(t)]−1, and then finding the minimum of ˜ sextwith
+respect to η(t). This choice satisfies the constraints
+0≤ξ0(t)≤ξ0m. Variation with respect to η(t) shows
+that ˜sexthas an extremum for η(t) = 0 or η(t) =∞for
+t∝ne}ationslash=tλ, wheretλisgivenbyequation χT(tλ−t0) =−λ/T.
+The value of η(t) at the isolated instances t=tλis arbi-
+trary. Only regions where η(t) = 0, so that ξ0(t) =ξ0m,
+contribute to ˜ sext. Obviously, ˜ sextis minimal when
+ξ0(t) =ξ0mfor|t−tmin| ≤∆t/2, where tminis the7
+time when χT(t) is minimal and ∆ tis determined by the
+average vaccination rate Ξ. In other words, the vaccina-
+tion rate ξ(t) has the form of periodic rectangular pulses
+of width ∆ t, centered at tmin+nT,n= 0,±1,±2,....
+The pulse width is
+∆t=ΞT
+ξ0m/vextendsingle/vextendsinglew(1)(xA,rξ)/vextendsingle/vextendsingle. (20)
+Since the vaccination rate is limited by the rate
+of elementary transitions without vaccine, we have
+ξ0m/vextendsingle/vextendsinglew(1)(xA,r)/vextendsingle/vextendsingle/lessorsimilart−1
+r. Then for weak vaccination,
+ΞT≪1, from Eq. (20) ∆ t≪tr. Therefore, χT(t) =
+χT(tmin) during the pulse of ξ(t), to the leading order in
+ΞT[we note that χT(t) may vary on a time scale shorter
+thantr, see below; however, this time scale is alwayslong
+compared to ∆ tfor sufficiently weak modulation]. The
+resulting change of the action is again given by Eq.(17).
+C. Vaccination protocol for periodically modulated
+systems
+Optimalvaccinationinperiodicallymodulatedsystems
+requires a separate consideration. Here, there is only one
+optimal extinction path per period Tin the absence of
+vaccination. When the average vaccination rate Ξ is low,
+vaccination with the same period Twill only weakly per-
+turb this path. To first order in Ξ, the linear in ξterm in
+the action still has the formofEq. (15), but without min-
+imization over t0. Sinceξ(t)≥0, the minimum of action
+is still achieved for ξ(t) = ΞT/summationtext
+nδ(t−tmin+nT), but
+nowtminis uniquely determined by the strong modula-
+tion. Inotherwords,themodulationuniquelydetermines
+the phase of the optimal vaccination pulses. The result-
+ing expression for s(1)
+extfor the optimal vaccination pro-
+tocol has the form of Eq. (17). If the vaccination pulses
+are applied at a wrong time, i.e.if the phase difference
+between the vaccination and the modulation differs from
+the optimal one, the vaccination will be not as efficient
+and may even be harmful: it may prolong the lifetime
+of the endemic state by increasing the disease extinction
+barrierQ.
+IV. DISEASE EXTINCTION BARRIER AS A
+FUNCTION OF VACCINATION PERIOD
+The vaccination-induced reduction of the disease ex-
+tinction barrier Q(1)=Ns(1)
+ext, as given by Eqs. (17), de-
+pendsontheinterrelationbetweenthevaccinationperiod
+Tand the characteristic time scales of the logarithmic
+susceptibility χ(t). Function χ(t) may or may not oscil-
+late in time, but generally χ(t) is relatively large within
+a time interval of the order of the relaxation time of the
+systemtr[20, 34]. To reveal some qualitative features of
+the effect ofvaccination and in particular, its dependence
+on the vaccination period, we will consider s(1)
+extfor two
+types of constraint on this period.1. Limited lifetime of the vaccine
+The vaccination period Tis naturally limited by the
+effectivelifetime τvofthevaccine. Thislifetime is usually
+determined by the maximum storage time of the vaccine
+and/or by the mutation rates of the infectious agent. If
+τvis long,τv≫tr, vaccination can be made most ef-
+ficient by increasing the vaccination period up to ∼τv.
+Indeed, as it follows from Eq. (17), s(1)
+ext∝Tin this case.
+This result is easy to understand. Even though a de-
+crease of the vaccine pulse frequency Ω = 2 π/Tcauses
+a decrease of the prefactor in the disease extinction rate
+We, the exponential factor exp( −Ns(1)
+ext) inWeincreases
+sharply. Indeed, it can be seen from Eqs. (5), (12) and
+(14) that, as the system moves along the optimal path
+to extinction, χ(t) is significant when the system is far
+from the stationary states xAandxS. The characteristic
+time scale of this motion is ∼tr. ForT≫trwe have
+min0≤t≤TχT(t)≈mintχ(t) and
+s(1)
+ext= ΞTmin
+tχ(t), τ v/greaterorsimilarT≫tr.(21)
+In the opposite limit of τv≪tr, and thus T≪tr, we
+have from Eq. (18)
+s(1)
+ext= Ξ˜χ(0), t r≫τv/greaterorsimilarT, (22)
+In this case the vaccination-induced reduction of the ex-
+tinction barrier is independent of the vaccination period
+and is determined by the zero-frequency component of
+the logarithmic susceptibility.
+An interesting situation may occur in the intermediate
+rangeτv∼trif, in the mean-field description, the system
+approaches the endemic state in an oscillatory manner.
+In this case function χ(t) is also expected to oscillate.
+The oscillations are well-pronounced if their typical fre-
+quencyis ω0≫t−1
+r. ItisclearfromEq.(18)thatastrong
+effect on disease extinction can be achievedby tuning the
+vaccination frequency Ω = 2 π/Tor its overtones in res-
+onance with ω0. An example of such a resonance will be
+discussed in Sec. V.
+2. Limited vaccine accumulation
+A different situation occurs if the total amount of vac-
+cine that can be accumulated is limited. This limitation
+implies that Ξ T≤M(note that Mis the limit on the
+ensemble-averaged amount of the accumulated vaccine).
+Such a constraint is typical for live vaccines, as it may
+be dangerous to store too much vaccine in this case. The
+actual average vaccination rate in this case is now T-
+dependent. We use the notation Ξ afor this rate, with
+Ξa= min(Ξ ,M/T). This is Ξ athat should be used now
+in Eqs. (21) and (22) for s(1)
+extin the limits T≫trand
+T≪tr, respectively.8
+The behavior of s(1)
+extwith varying vaccination period
+Tdepends on the form of the logarithmic susceptibil-
+ityχ(t). Let us first consider the case where χ(t) has
+a single local minimum (at t=t∗), and|χ(t)|mono-
+tonically decays to zero with increasing |t−t∗|. Here,
+once the vaccine accumulation has reached saturation
+with increasing T(which happens for Ξ T=M), func-
+tion|s(1)
+ext|=M|minχT(t)|monotonically decreases with
+further increase in T. Indeed,
+d
+dTmin
+0<t<TχT(t) =∂
+∂T/summationdisplay
+nχ(t∗−a∗T+nT)
+=/summationdisplay
+n/parenleftbiggdχ(t−a∗T+nT)
+dt/parenrightbigg
+t=t∗(n−a∗)>0,(23)
+wherea∗gives the position of the minimum of χT(t)
+overtand is given by equation ∂χT(t∗−a∗T)/∂a∗= 0;
+we choose 0 < a∗<1 and take into account that
+min0<t<TχT(t)<0. In Eq. (23) we have used that,
+ifχ(t) is minimal for t=t∗, thendχ/dt > 0 fort > t∗
+anddχ/dt < 0 fort < t∗. It follows from Eq. (23) that,
+once the vaccine accumulation has reached saturation,
+further increase in Twill only reduce the effect of the
+vaccine. This result is understandable because, if Tin-
+creases beyond M/Ξ, the actual average vaccination rate
+Ξadecreases.
+A counterintuitive situation may occur if χ(t) is os-
+cillating. Here the inequality (23) may be violated. As
+a result, the dependence of the effect of the vaccine on
+Tand, consequently, on the actual vaccination rate Ξ a,
+may be nonmonotonic. An example of this behavior is
+discussed in the next section.
+V. RESONANCES IN THE STOCHASTIC SVIR
+MODEL
+We now apply some of our results to an important and
+widely used stochastic epidemic model, the Susceptibles-
+Vaccinated-Infected-Recovered (SVIR) model. The
+modelis sketchedin Fig.1. In theabsenceofvaccination,
+ξ0(t) = 0, the SVIR model reduces to the stochastic SIR
+model with population turnover, which was originally in-
+troduced to describe the spread of measles, mumps, and
+rubella, see [1, 3]. In the SIR model, susceptible in-
+dividuals are brought in, individuals in all population
+groups leave (for example, die), a susceptible individual
+can become infected upon contacting an infected indi-
+vidual, and an infected individual can recover. If we set
+X1=S,X2=I, andX3=R, the rates of the cor-
+responding processes are: (i) influx of the susceptibles,
+W(X,r) =µNforr1= 1,ri/negationslash=1= 0, (ii) leaving, with
+the same rate for all populations, W(X,r) =µXifor
+ri=−1,rj/negationslash=i= 0, (iii) infection, W(X,r) =βX1X2/N
+forr1=−1,r2= 1,ri/negationslash=1,2= 0, and (iv) recovery of the
+infected, W(X,r) =γX2forr2=−1,r3= 1,ri/negationslash=2,3= 0,
+Fig. (1).
+
 
+ 
+
+ 	 
+ 
+ 

+   
+
+
+FIG. 1: The SVIR epidemic model with susceptible, vacci-
+nated, infected and recovered sub-populations. The arrows
+indicate processes leading to changes of sub-population si zes;
+the corresponding rates are scaled per individual.
+Forβ >Γ≡γ+µthe SIR model possesses a sin-
+gle endemic state. This state corresponds, in the mean-
+field theory, to an attracting fixed point on the two-
+dimensional phase plane of susceptibles and infected. At
+µ <4(β−Γ)(Γ/β)2this attracting point is a focus.
+The populations of susceptibles, infected and recovered
+exhibit decaying oscillations in time as the system ap-
+proaches the endemic state. It was found in Ref. [14]
+that, in this parameter range, the populations oscillate
+also on the optimal disease extinction path. These oscil-
+lations are illustrated in Fig. 2.
+We will now incorporate vaccination and introduce a
+sub-population of vaccinated X4=V. The vaccination
+is described by the transition rate W(X,r) =ξ0(t)x1for
+r1=−1,r4= 1,ri/negationslash=1,4= 0. The corresponding term in
+the Hamiltonian Eq. (12) has the form ξ0(t)H(1)with
+H(1)(x,p) =x1/parenleftbig
+ep4−p1−1/parenrightbig
+. (24)
+Vaccinated individuals leave at the same rate µas in-
+dividuals in other populations. For simplicity, we as-
+sume that the immunity from the vaccination is never
+lost. In this case fluctuations of the vaccinated pop-
+ulation do not affect fluctuations of other populations,
+andp4≡0 along the optimal extinction path. Then
+from Eq. (14), the logarithmic susceptibility is χ(t) =
+x(0)
+1opt(t)x−1
+1A/parenleftBig
+1−exp[−p(0)
+1opt(t)]/parenrightBig
+, wherex(0)
+1opt(t),p(0)
+1opt(t)
+andx1Aare calculated for the SIR model.
+The Fourier spectrum of the logarithmic susceptibility
+˜χ(ω)isplottedinFig.3(a). Itcorrespondstotheoptimal
+extinction path shown in Fig. 2. As one can see, the
+spectrum has a peak at the characteristic frequency of
+oscillations of the system in the absence of vaccination
+ω0(for the chosen parameter values ω0≈5.2µ).
+Wenowconsidertheeffectoftheresonantpeakin ˜ χ(ω)
+on vaccination. The dependence of the scaled change of
+the disease extinction barrier s(1)
+ext=Q(1)/Non vaccina-
+tion period Tis shown in Fig. 3 (b). The solid line in
+Fig. 3 (b) shows the behavior of s(1)
+extwhere there is no
+limit on vaccine accumulation or, equivalently, for such
+periods where the limitation does not come into play and9
+the actual vaccinationrate Ξ ais independent of T. Func-
+tion|s(1)
+ext| ≡ −s(1)
+extis seen to be strongly nonmonotonic,
+it displays pronounced maxima (which correspond to the
+minima of s(1)
+ext). They occur where the vaccination pe-
+riodTcoincides with the multiples of the characteristic
+period of the system motion without vaccination 2 π/ω0.
+For limited vaccine accumulation M, the actual aver-
+age vaccination rate depends on the vaccination period,
+Ξa= min(Ξ ,M/T). Beyond a certain value of T, the
+increase of Tis accompanied by the decrease Ξ a. This
+leads to a change of the dependence of s(1)
+extonT. Re-
+markably, |s(1)
+ext|still displays resonant peaks at 2 πn/ω0
+with integer n. Their amplitude decreases with increas-
+ingn. The occurrence of the peaks shows that, by tuning
+the vaccination period, the effect of the vaccination can
+be resonantly enhanced; the resonance in this case is in
+the exponent of the disease extinction rate, and there-
+fore it is extremely strong. Counter-intuitively, since the
+actual average vaccination rate decreases with increasing
+T, a strong enhancement of the vaccine can be achieved
+where this rate is decreased. For example, in Fig. 3 (b)
+the maxima of |s(1)
+ext|forµM/Ξ = 1 and µM/Ξ = 3 are
+achieved for Tin the range where Ξ a<Ξ.
+0.40.50.60.70.80.9 100.020.040.060.08
+x1x2
+FIG. 2: The most probable trajectories in the stochastic SIR
+model on the plane of the scaled numbers of susceptibles and
+infected, x1=X1/Nandx2=X2/N, respectively. The
+dashed line shows a mean-field trajectory toward the endemic
+state, and the solid line shows the most probable trajec-
+tory followed during the fluctuation-induced disease extin c-
+tion [14]. The plot refers to β/µ= 80, and γ/µ= 50.
+VI. CONCLUSIONS
+We have developed a theory of optimal periodic vacci-
+nation against an endemic disease for low average vacci-
+nation rate, as in the case where the vaccine is in short
+supply, or short lived, or cannot be stored in the suf-
+ficient amount. We assume that the vaccination rate is
+insufficient foreliminating the endemicstateand thusex-
+terminating the disease by “brute force”. However, vac-
+cination can change the rate of disease extinction, which
+occurs spontaneously as a result of a comparatively rare02468−0.015−0.01−0.0050
+s′ext 
+ T′0 5 1000.511.522.5x 10−3
+ω′χ(ω′)(b)
+4 π2 π~
+0ω′(a)
+0ω′
+0M′=3M′=1
+ω′
+FIG. 3: (a) The Fourier transform of the logarithmic suscep-
+tibility in the SIR model. The parameters are the same as in
+Fig. 2; the rescaled frequency is ω′=ω/µ. The susceptibility
+spectrum displays a sharp peak at the characteristic vibrat ion
+frequency ω0. (b) The change of the scaled extinction barrier
+s′
+ext=µs(1)
+ext/Ξ with vaccination period T. The solid line
+showss′
+extwhere there is no limit on vaccine accumulation,
+whereas the dashed lines refer to the case of limited accumu-
+lated vaccine amount. The accumulation limit Mis scaled
+by the small- Taverage vaccination rate Ξ, M′=µM/Ξ, and
+T′=Tµ. The locations of the resonances of s′
+extare indepen-
+dent ofM.
+fluctuation. We show that the optimal vaccination leads
+to an exponentially strong increase of the disease extinc-
+tion rate. This happens because the vaccine changes the
+effective entropic barrier that needs to be overcome for
+spontaneous extinction.
+We find that the optimal vaccinationprotocolis a peri-
+odic sequence of δ-like pulses. This protocol is essentially
+model-independent, it only requires that the population
+be spatially uniform. In stationary systems, the phase
+of the pulses is irrelevant. In contrast, in periodically
+modulated systems, like in the case of seasonally vary-
+ing infection, it is necessary to appropriately synchronize
+vaccination pulses with the modulation. Moreover, if the
+pulse phase is wrong, vaccination may hamper disease
+extinction.
+For fixed average vaccination rate, the effect of vacci-
+nation in stationary systems increases with the increas-
+ing vaccination period. However, this increase is gener-
+ally nonmonotonic and the disease extinction rate can
+display exponentially strong resonances. They occur if
+the vaccination period coincides with the period of de-
+caying oscillations of the population, which character-
+ize the approach to the endemic state in the mean-field
+(fluctuation-free) approximation. The resonances occur
+also where the vaccination period coincides with a mul-
+tiple of the dynamical period. They are illustrated using
+the well-known SVIR model of population dynamics.
+It turns out that, counterintuitively, the effect of vacci-
+nation can be sometimes enhanced by reducing the aver-
+age vaccination rate. This happens where the mean-field
+dynamics is characterized by decaying oscillations and
+there is a constraint on the amount of vaccine that can
+be stored. In this case lowering the average vaccination10
+rate can allow one to tune the vaccination period in res-
+onance with the system dynamics.
+The analysis is based on the master equation for the
+population dynamics. We solve it in the eikonal approx-
+imation and reduce the problems of the tail of the dis-
+tribution and of the extinction to Hamiltonian dynamics
+of an auxiliary system. A general formulation of the cor-
+responding Hamiltonian problem is obtained for period-
+ically modulated systems. The optimal vaccination pro-
+tocol is found using this formulation with account taken
+of the constraint on the average vaccination rate. The
+feature of the problem that makes it different from other
+problems ofoptimal controlofrareevents is that the vac-cinationratecannotbe negativeandit istheaveragevac-
+cination rate that is given. The analysis can be extended
+tootherproblemsofoptimalcontroloffluctuation-driven
+extinction with similar constraints.
+Acknowledgments
+The work at MSU was supported in part by the Army
+Research Office and by NSF Grant PHY-0555346. B. M.
+was supported by the US-Israel Binational Science Foun-
+dation Grant 2008075.
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\ No newline at end of file
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+arXiv:1001.0171v1  [cond-mat.mtrl-sci]  31 Dec 2009Novel Structural Motifs in Oxidized Graphene
+H. J. Xiang,1Su-Huai Wei,2and X. G. Gong1
+1Key Laboratory for Computational Physical Sciences,
+Ministry of Education, P. R. China, and Department of Physic s,
+Fudan University, Shanghai 200433, P. R. China
+2National Renewable Energy Laboratory, Golden, Colorado 80 401, USA
+(Dated: November 8, 2018)
+Abstract
+The structural and electronic properties of oxidized graph ene are investigated on the basis of
+the genetic algorithm and density functional theory calcul ations. We find two new low energy
+semiconducting phases of the fully oxidized graphene (C 1O). In one phase, there is parallel epoxy
+pair chains running along the zigzag direction. In contrast , the ground state phase with a slightly
+lower energy and a much larger band gap contains epoxy groups in three different ways: normal
+epoxy, unzippedepoxy, andepoxypair. Forpartiallyoxidiz edgraphene,aphaseseparationbetween
+bare graphene and fully oxidized graphene is predicted.
+PACS numbers: 61.48.Gh,73.61.Wp,71.20.-b,73.22.-f
+1Graphene, a single layer of honeycomb carbon lattice, exhibits many exotic behaviors,
+ranging from the anomalous quantum Hall effect [1–3] and Klein parad ox [4] to coherent
+transport [5]. Because of its exceptional electrical, mechanical, an d thermal properties,
+graphene holds great promise for potential applications in many tec hnological fields such as
+nanoelectronics, sensors, nanocomposites, batteries, superc apacitors and hydrogen storage
+[6]. Nevertheless, several of these applications are still not feasib le because the large-scale
+production of pure graphene sheets remains challenging. Current ly, graphene oxide (GO) is
+of particular interest since chemical reduction of GO has been demo nstrated as a promising
+solution based route for mass production of graphene [7–11]. In a ddition, GO shows promise
+for use in several technological applications such as polymer compo site [12], dielectric layers
+in nanoscale electronic devices, and the active region of chemical se nsors.
+GO, which was first prepared by Brodie [13] in 1859, consists of oxidiz ed graphene layers
+with the hexagonal graphene topology [14]. Although it is now widely a ccepted that GO
+bears hydroxyl and epoxide functional groups on its basal plane [ 15], as confirmed by a
+recent high-resolution solid-state13C-NMR measurement [16], the complete structure of GO
+has remained elusive because of the pseudo-random chemical func tionalization of each layer,
+as well as variations in exact composition.
+Several first-principles studies [17–20] have been performed to in vestigate the structure
+and energetics of epoxide and hydroxyl groups on single-layer gra phene. In those calcu-
+lations, some possible structures were manually selected and examin ed in order to obtain
+a good structural model for GO. In this Letter, we adopt the gen etic algorithm (GA) in
+combination with density functional theory (DFT) to search the glo bal minimum structures
+of oxidized graphene with different oxidation levels. As a first step, w e consider only the
+arrangement of epoxide groups on single-layer graphene. We find t wo new graphene based
+low energy structures of C 1O. In both structures, there are epoxy pairs and some carbon-
+carbonσbonds are broken. However, the two structures of C 1O have dramatically different
+electronic properties: one has a much larger band gap than the oth er, and the low band gap
+structure has a conduction band minimum (CBM) which is even lower th an the Dirac point
+of graphene. We also show that for C xO with low oxygen (high x) concentration, a phase
+separation between bare graphene and the low energy structure s of C 1O will take place in
+the ground state. In addition, the electronic structure of C xO will depend on which low
+energy structure is present in the system. Our results are direct ly relevant to the oxidation
+2of graphene in O 2atmosphere [21] or oxygen plasma [22]. They also have important impli-
+cations for understanding the structural and electronic proper ties of GO made by mineral
+acid attack.
+We perform global structure searches based on the GA. The GA ha s been successfully
+applied to find the ground state (GS) structure of clusters [23], allo ys [24], and crystals [25].
+In this work, we consider all possible inequivalent (6 in total) graphen e supercells up to 8
+carbon per cell. For each graphene supercell, we perform several GA simulations to confirm
+the obtained GS structure. Here, the oxygen atoms can only occu py the bridge sites of
+both sides of the graphene plane. It can be easily seen that the num ber of possible oxygen
+positions on both sides of the graphene is thrice the number of carb on atoms. However,
+two oxygen atoms at nearest bridge sites is very unstable due to th e short O-O distances
+(about 1.2 ˚A), thus the lowest C/O ratio for a possible stable structure is 1. Diff erent
+oxygen concentrations are simulated: C/O= 1, C/O= 2, C/O= 3 and C /O= 4. For a given
+graphene supercell and C/O ratio, we first randomly generate ten s (e.g., 16) of structures of
+CxO. Then we fully optimize the internal coordinations and cell of struc ture. To generate
+new population, we perform the cut and splice crossover operator [23, 25] on parents chosen
+through the tournament selection. In an attempt to avoid stagna tion and to maintain
+population diversity, a mutation operation in which some of the oxyge n atoms can hop to
+other empty bridge sides is introduced. In this study, we use DFT to calculate the energies
+and relax the structures. Our first principles DFT calculations are p erformed on the basis
+of the projector augmented wave method [26] encoded in the Vienn a ab initio simulation
+package [27] using the local density approximation (LDA) and the pla ne-wave cutoff energy
+of 500 eV.
+To characterize the stability of oxidized graphene structures, we define the formation
+energy or oxygen absorption energy as:
+Ef=E(CxO)−xµC−µO (1)
+whereµCis set to be the energy of graphene per carbon atom, and µO= 1/2E(O2) [E(O2)
+is the energy of an isolated triplet oxygen molecule]. It should be note d that the relative
+stability between various structures of oxidized graphene does no t depend on the particular
+choice of the oxygen chemical potential ( µO). Our GA simulations reveal two low energy
+structures of C 1O, as shown in Fig. 1. In both structures, each of the carbon atom s bonds
+3withtwoneighborcarbonatomsandtwoOatoms, thusthereisno sp2carbon. Thestructure
+shown in Fig. 1(b) is rather simple: There are epoxy pair chains along t he zigzag direction
+which are parallel to each other. We will refer to this model as the ep oxy pair model ( D2h
+symmetry) hereafter. The formation of an isolated epoxy pair was suggested in a theoretical
+study on the graphene oxidative process [28]. It was also shown tha t an isolated epoxy pair
+is less stable than a carbonyl pair [28]. However, in the fully oxidized gr aphene case, we find
+that the epoxy pair chain is more stable than the carbonyl pair chain by about 0.84 eV per
+pair. The lowest energy structure ( C2vsymmetry) [Fig. 1(a)] of C 1O has a lower energy than
+the epoxy pair model by only 60 meV/O. The small energy difference s uggests that both
+phases might coexist at finite temperature. In the lowest energy s tructure, there are isolated
+six-membered carbon rings, which are connected to the neighborin g six-membered carbon
+rings through two epoxy pairs and four unzipped epoxy groups. Fo r each six-membered
+carbon ring, there are two normal epoxy groups. Therefore, we term this complex structure
+as the mix model. One can obtain the epoxy pair model from the mix mod el by moving the
+unzipped epoxy groups on top of the normal epoxy groups.
+The low energy structures of oxidized graphene with C/O= 2 and C/O = 4 are displayed
+in Fig. 2. Among all C 2O structures with no more than eight C atoms per cell, the lowest
+energy structure has two neighboring epoxy pair chains and half of the carbon atoms remain
+sp2hybridized. For comparison, we also show the C 2O structure predicted by Yan et al.
+[19] in which no carbon-carbon bond is broken and all carbon atoms a resp3hybridized.
+Test calculations show that our identified new structure is more sta ble by almost 0.40 eV/O
+than Yan’s structure with normal epoxy groups. For C 4O, we find that oxygen atoms tend
+to form isolated chains. In the lowest energy structure, the grap hene is unzipped by an
+oxygen chain and the normal epoxy groups are connected with the unzipped epoxy groups.
+In another metastable structure, there is an isolated epoxy pair c hain. The isolated epoxy
+pair chain structure is less stable than the lowest energy structur e by only 30 meV/O. The
+isolated epoxy pair chain structure is found to be the lowest energy structure of C 3O with
+no more than six carbon atoms.
+The calculated formation energies of the above discussed lowest en ergy structures are
+about−1.2 eV/O, −0.9 eV/O, and −0.5 eV/O for C 1O, C2O, and C 4O, respectively. We
+can clearly see that C 1O has the lowest formation energy and the formation energy increa ses
+with the decrease of the oxygen concentration. We note that the above results are based
+4on the GA simulations with finite supercell size. We now discuss the pos sible structure of
+oxidized graphene with an infinite large graphene supercell. The two lo w energy structures
+of C1O should remain the same because of the perfect ordering and low fo rmation energy.
+For oxidized graphene with less oxygen atoms than carbon atoms, t he phase separation
+between the C 1O phase and bare graphene will occur. The phase separation is caus ed by
+the tendency to minimize the number of broken carbon-carbon π-πbonds, as in the case of
+partially hydrogenated graphene [29]. The tendency toward phase separation is manifested
+in the lowest energy structure of C 2O [Fig. 2(a)] and is confirmed in the case of C 4O: If
+we double the cell of Fig. 2(d) along the direction perpendicular to th e isolated epoxy pair
+chain and move the two epoxy pair chains close to each other, the re sulting C 4O structure
+will have a formation energy of −0.87 eV/O. In contrast, increasing the cell of Fig. 2(c) does
+not lead to a significant decrease of the formation energy.
+Our calculations indicate that the two low energy phases of C 1O will probably exist
+in any oxidized graphene. Therefore, the electronic properties of the C 1O phases is of
+paramount importance. Because the LDA is well known to underest imate the band gaps
+of semiconductors, we calculate the band structure of the C 1O phases by employing the
+screened Heyd-Scuseria-Ernzerhof 06 (HSE06) hybrid function al [30], which was shown to
+give a good band gap for many semiconductors including graphene ba sed systems [31]. Our
+calculations show that the C 1O phase with the mix model is a semiconductor with a large
+band gap of 5.90 eV [Fig. 3(a)]. The valence band maximum (VBM) locate s at Γ, but the
+CBM is almost non-dispersive, which is consistent with the fact that t he CBM state at Γ
+is an antibonding C-O orbital mostly localized in the six-membered carb on rings [see the
+inset of Fig. 3(a)]. Fig. 3(b) shows that the C 1O phase with the epoxy pair model is also
+a semiconductor but with a smaller indirect band gap of 2.14 eV. The VB M and CBM
+locate at (0.5, 0, 0) and Γ, respectively. An important difference be tween the two phases
+is that the C 1O phase with the epoxy pair model has a much smaller electron and hole
+effective mass. This is because the band overlap is stronger in the ep oxy pair model with
+a higher symmetry, as can be seen from the CBM wavefunction displa yed in the inset of
+Fig. 3(b). The larger band gap and lower energy in the mix model is a co nsequence of
+its lower symmetry ( C2v): Level repulsions between the occupied and unoccupied bands
+which are symmetrically forbidden in the epoxy pair model ( D2h) become possible in the
+mix model.
+5We also calculate the absolute values of the VBM and CBM levels using th e HSE06
+hybrid functional, which was found [32, 33] to give improved results compared to the LDA
+functional. The vacuum level is defined as the average electrostat ic potential in the vacuum
+region where it approaches a constant [32]. For comparison, we als o calculate the work
+function of graphene. Our results are shown in Fig. 3(c). As expec ted, the VBM of both
+C1O phases are below the Fermi level of graphene due to the oxygen 2 porbitals with low
+energies. And the CBM level of the C 1O phase with the mix model is higher than the Fermi
+level of graphene by 1.6 eV. Interestingly and surprisingly, the CBM level of the C 1O phase
+with the epoxy pair model is lower than the Dirac point of graphene. T his is because the
+low oxygen 2s and 2p orbitals and the formation of one dimensional ziz ag chain by carbon
+2pzorbitals dramatically lowers the eigenvalue of the CBM state [see the in set of Fig. 3(b)].
+It is noted that LDA gives qualitatively similar band alignment. This will ha ve interesting
+consequences as will be discussed below.
+The experimentally synthesized graphene oxide usually has a much low er oxygen con-
+centration than the C 1O phases. Here, we will discuss the electronic structures of partia lly
+oxidized graphene. As shown above, there is a phase separation be tween the graphene part
+and the C 1O phase in the partially oxidized graphene under thermodynamic equilib rium.
+The C 1O phase with the epoxy pair model could match graphene well via a zigz ag-like edge
+with a lattice mismatch of 6.0 %[Fig. 4(b)]. However, the latticemismatc h between the C 1O
+phase with the mix model and graphene is much larger: The smallest mis match (21.9%)
+occurs when the C 1O phase connects with graphene via an armchair edge [Fig. 4(a)]. As
+expected, there is a band gap openning [See Fig. 4(c)] in the partially oxidized graphene
+with a phase separation between graphene and the C 1O phase with the mix model. This is
+due to quantum confinement effect, similar to the armchair graphen e nanoribbon case [34].
+In contrast, the partially oxidized graphene with a zigzag-like edge b etween graphene and
+the C1O phase with the epoxy pair model is metallic [See Fig. 4(d)]. And there is no flat
+band and inclusion of spin degree of freedom does not open a gap, diff erent from the zigzag
+graphene nanoribbon case [34]. The metallicity of the system is not du e to the peculiar
+zigzag-like interface between graphene and the C 1O phase because a test calculation on a
+system similar to that shown in Fig. 4(a) but with an armchair interfac e between graphene
+and the C 1O phase with the epoxy pair model is also metallic. This is due to the fact that
+the CBM of the C 1O phase with the epoxy pair model is lower than graphene, thus ther e is
+6almost no quantum confinement effect and no flat edge states.
+In summary, we have performed global search of the lowest energ y structures of oxidized
+grapheneusingthegeneticalgorithmapproachcombinedwithdensit yfunctionaltheory. Our
+calculations unravel two novel low energy semiconducting phases o f fully oxidized graphene
+C1O. The C 1O phase with the epoxy pair model has an indirect band gap of about 2 .14 eV,
+and an extremely low CBM that is below the Fermi level of graphene. T he ground state of
+the C1O phase with three mixed epoxy groups has a lower energy by only 60 m eV/O than
+the C1O phase within the epoxy pair model and a much larger band gap. Our c alculations
+predict that the phase separation between bare graphene and fu lly oxidized graphene is
+thermodynamically favorable in partially oxidized graphene. The C 1O phase with the epoxy
+pair model has a much smaller lattice mismatch with graphene than the case of the mix
+model. In the partially oxidized graphene with a zigzag-like edge betwe en graphene and the
+C1O phase with the epoxy pair model, there is no band gap openning and n o local spin
+moment as a result of the unusually low CBM of the C 1O phase.
+Work at Fudan was supported by the National Science Foundation o f China. Work at
+NREL was supported by the U.S. Department of Energy, under Con tract No. DE-AC36-
+08GO28308. We thank Dr. Gus Hart for useful discussion at the ea rly stage on this project.
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+(2007).
+9mix model
+−1.22 eV/OC2vepoxy pair
+−1.16 eV/OD2h
+C
+O
+a b(a) (b)
+a
+bI
+IIIII
+FIG. 1: (Color online) (a) The structure of the mix model ( C2vsymmetry) which is predicted
+to be the lowest energy C 1O phase. (b) The low energy C 1O phase with the epoxy pair model
+(D2hsymmetry), which has a higher energy by only 0.06 eV/O than th e mix model. “I”, “II”, and
+“III” in (a) indicate normal epoxy, unzipped epoxy, and epox y pair, respectively. The unit cells
+are enclosed by dashed lines. The numbers give the formation energy of the oxidized graphene
+structures.
+(c)(a) (b)
+(d) OC4C2O −0.92 eV/O −0.52 eV/O
+−0.54 eV/O −0.51 eV/O
+FIG. 2: (Color online) (a) shows the newly predicted lowest e nergy structure of the C 2O phase
+among all configurations with no more than eight carbon atoms per cell. (b) is the model of the
+C2O phase predicted by Yan et al.[19]. (c) shows the lowest energy structure of the C 4O phase
+among all configurations with no more than eight carbon atoms per cell. (d) is a low energy
+structure of the C 4O phase which has a higher energy by only 0.03 eV/O than the low est energy
+structure.
+10-505Energy (eV)
+Γ (0.5,0.5,0) (0,0.5,0) Γ (0,0.5,0) (0.5,0,0)
+−4.27 eV −4.43 eVCBM
+VBM
+VBMDirac point−2.67 eVCBM
+−8.57 eV −6.57 eVmix model graphene epoxy pair(a) (b)
+(c)
+FIG. 3: (Color online) (a) and (b) show the band structures fr om the hybrid HSE06 calculations
+of the C 1O phases with the mix model and epoxy pair model, respectivel y. In the insets, we show
+the wavefunction plots of the LUMO states at Γ. The k-points a re given in terms of the reciprocal
+lattice. (c) Schematic illustration of the band alignment b etween the two C 1O phases and graphene
+from the hybrid HSE06 calculations.
+-4-2024Energy (eV)
+Γ X
+-4-2024Energy (eV)
+Γ X(c)
+(d)(a)
+(b)
+FIG. 4: (Color online) (a) shows a partially oxidized graphe ne with an armchair edge between
+bare graphene and the C 1O phase with the mix model. (b) displays a partially oxidized graphene
+with a zigzag-like edge between bare graphene and the C 1O phase with the epoxy pair model. (c)
+and (d) show the corresponding LDA band structures.
+11
\ No newline at end of file
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new file mode 100644
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+++ b/1001.0172.txt
@@ -0,0 +1,843 @@
+arXiv:1001.0172v3  [math.MG]  16 Jun 2014CHARACTERIZING THE UNIVERSAL RIGIDITY OF GENERIC
+FRAMEWORKS
+STEVEN J. GORTLER AND DYLAN P. THURSTON
+Abstract. Aframework is a graph and a map from its vertices to Ed(for some d). A
+framework is universally rigid if any framework in any dimension with the same graph
+and edge lengths is a Euclidean image of it. We show that a gene ric universally rigid
+framework has a positive semi-definite stress matrix of maxi mal rank. Connelly showed
+that the existence of such a positive semi-definite stress ma trix is sufficient for universal
+rigidity, so this provides a characterization of universal rigidity for generic frameworks. We
+also extend our argument to give a new result on the genericit y of strict complementarity
+in semidefinite programming.
+1.Introduction
+In this paper we characterize generic frameworks which are u niversally rigid in d-dimen-
+sional Euclidean space. A framework is universally rigid if , modulo Euclidean transforms,
+there is no other framework of the same graph in anydimension that has the same edge
+lengths. A series of papers by Connelly [ Con82 ,Con99 ,Con01 ] and later Alfakih [ Alf07 ,Alf10 ]
+described a sufficient condition for a generic framework to be universally rigid. In this paper
+we show that this is also a necessary condition.
+Universally rigid frameworks are especially relevant when applying semidefinite program-
+ming techniques to graph embedding problems. Suppose some v ertices are embedded in Ed
+and we are told the distances between some of the pairs of vert ices. The graph embedding
+problem is to compute the embedding (up to an unknown Euclide an transformation) from
+the data. This problem is computationally difficult as the gra ph embeddablity question is in
+general NP-HARD [ Sax79 ], but because of its utility, many heuristics have been atte mpted.
+One approach is to use semidefinite programming to find an embe dding [ LLR95 ], but such
+methods are not able to find the solutions that are specificall yd-dimensional, rather than
+embedded in some larger dimensional space. However, when th e underlying framework is, in
+fact, universally rigid, then the distance data itself auto matically constrains the dimension,
+and therefore the correct answer will be (approximately) fo und by the semidefinite program.
+The connection between rigidity and semi-definite programm ing was first explored by So and
+Ye [SY07 ].
+Many rigidity questions are easier to answer in the generic c ase (see Definition 1.2below),
+where various singular cases can simply be ignored. It is als o a reasonable assumption to
+make in many graph embedding problems that arise in practice . For example, in the problem
+of sensor network localization [ EGW+04], we assume that there are a set of robotic sensors
+moving independently in space, and that some inter-sensor d istances are measured physically.
+For such a setting, we only expect to observe generic behavio r.
+Date: September 14, 2018.
+12 GORTLER AND THURSTON
+We also discuss the more general topic of strict complementa rity in semidefinite pro-
+gramming. Strict complementarity is a strong form of dualit y that is needed for the fast
+convergence of many interior point optimization algorithm s. Using our arguments from uni-
+versal rigidity, we show that if the semidefinite program has a sufficiently generic primal
+solution, then it must satisfy strict complementarity. Thi s is in distinction with previous
+results on strict complementarity [ AHO97 ,PT01 ] that require the actual parameters of the
+program to be generic. In particular, our result applies to p rograms where the solution is of
+a lower rank than would be found generically.
+1.1.Rigidity definitions.
+Definition 1.1. Agraph Γ is a set of vvertices V(Γ) and eedges E(Γ), where E(Γ) is a set
+of two-element subsets of V(Γ). We will typically drop the graph Γ from this notation. A
+configuration pis a mapping from VtoEv. Let C(V) be the space of configurations. For
+p∈C(V) and u∈ V, letp(u) denote the image of uunder p. Let Cd(V) denote the space
+of configurations that lie entirely in the Ed, contained as the first ddimensions of Ev. A
+framework (p,Γ) is the pair of a graph and a configuration of its vertices. Fo r a given graph Γ
+thelength-squared function ℓΓ:C(V)→Reis the function assigning to each edge of Γ its
+squared edge length in the framework. That is, the component ofℓΓ(p) in the direction of
+an edge {u, w}is|p(u)−p(w)|2.
+Definition 1.2. A configuration in Cd(V) isproper if it does not lie in any affine subspace
+ofEdof dimension less than d. It is generic if its first dcoordinates (i.e., the coordinates
+not constrained to be 0) do not satisfy any algebraic equatio n with rational coefficients.
+Remark 1.3.A generic configuration in Cd(V) with at least d+ 1 vertices is proper.
+Definition 1.4. The configurations p, qinC(V) are congruent if they are related by an
+element of the group of Eucl( v) of rigid motions of Ev.
+A framework ( p,Γ) with p∈Cd(V) isuniversally rigid if any other configuration in C(V)
+with the same edge lengths under ℓΓis a configuration congruent to p.
+A graph Γ is generically universally rigid inEdif any generic framework ( p,Γ) with
+p∈Cd(V) is universally rigid.
+Essentially, universal rigidity means that the lengths of t he edges of pare consistent with
+essentially only one embedding of Γ in anydimension, up to v. (In general, a configuration
+in any higher dimension can be related by a rigid motion to a co nfiguration in Ev.) This is
+stronger than global rigidity , where the lengths fully determine the embedding in the smal ler
+spaceEd. And global rigidity is, in turn, stronger than (local) rigidity , which only rules out
+continuous flexes in Edthat preserve edge lengths.
+Remark 1.5.Universal rigidity or closely related notions have also bee n called dimensional
+rigidity [Alf07 ],uniquely localizable [SY07 ], and super stability [Con99 ].
+Remark 1.6.In Definition 1.4, it would be equivalent to require that ( p,Γ) be (locally) rigid
+inEvsince by a result of Bezdek and Connelly [ BC04 ] any two frameworks with the same
+edge lengths can be connected by a smooth path in a sufficiently large dimension.
+Remark 1.7.Universal rigidity, unlike local and global rigidity, is no t a generic property of
+a graph: for many graphs, some generic frameworks are univer sally rigid and others are not.
+For instance, an embedding of a 4-cycle in the line E1is universally rigid iff one side is longCHARACTERIZING THE UNIVERSAL RIGIDITY OF GENERIC FRAMEWOR KS 3
+Figure 1. Planar graph embeddings with increasing types of rigidity. From
+left to right, generically locally flexible graph, generica lly locally rigid graph,
+generically globally rigid graph, generic universally rig id framework, generi-
+cally universally rigid graph. (More precisely, the fourth framework is not
+itself generic, but it and any generic framework close to it a re universally
+rigid.) This paper focuses on frameworks of the type of the fo urth one. The
+fifth graph is generically universally rigid as it is a 2-tril ateration graph.
+in the sense that its length is equal to the sum of the lengths o f the others. On the other
+hand, some graphs, such as a simplex, and any trilateration g raph [ EGW+04] are generically
+universally rigid in Ed. (A d-trilateration graph has the property that one can order its
+vertices such that the following holds: the first d+ 2 vertices are part of a simplex in Γ
+and each subsequent vertex is adjacent in Γ to d+ 1 previous vertices.) We do not know a
+characterization of generically universally rigid graphs .
+See Figure 1for examples of embedded graphs with increasing types of rig idity.
+1.2.Equilibrium stresses.
+Definition 1.8. Anequilibrium stress matrix of a framework ( p,Γ) is a matrix Ω indexed
+byV × V so that
+(1) for all u, w∈ V, Ω(u, w) = Ω( w, u);
+(2) for all u, w∈ Vwith u/\e}atio\slash=wand{u, w} /\e}atio\slash∈ E , Ω(u, w) = 0;
+(3) for all u∈ V,/summationtext
+w∈VΩ(u, w) = 0; and
+(4) for all u∈ V,/summationtext
+w∈VΩ(u, w)p(w) = 0.
+(The last condition is the equilibrium condition.) Let SΓ(p) be the linear space of equilibrium
+stress matrices for ( p,Γ).
+Conditions ( 3) and ( 4) give us:
+(1.9) ∀u∈ V/summationdisplay
+{w∈V|{ u,w}∈E}Ω(u, w)(p(w)−p(u)) = 0 .
+The kernel of an equilibrium stress matrix Ω of a framework ( p,Γ) always contains the
+subspace of Rvspanned by the coordinates of palong each axis and the vector /vector1 of all
+1’s. This corresponds to the fact that any affine image of psatisfies all of the equilibrium
+stresses in SΓ(p). If pis a proper d-dimensional configuration, these kernel vectors span a
+(d+ 1)-dimensional space, so for such frameworks rank Ω ≤v−d−1.
+Definition 1.10. We say that the edge directions of ( p,Γ) with p∈Cd(V) are on a conic
+at infinity if there exists a symmetric d-by-dmatrix Qsuch that for all edges ( u, v) of Γ, we
+have
+[p(u)−p(v)]tQ[p(u)−p(v)] = 0 ,
+where the square brackets mean the projection of a vector in EvtoEd(i.e., dropping the 0’s
+at the end of p(u)).4 GORTLER AND THURSTON
+Remark 1.11.If the edges of ( p,Γ) are not on a conic at infinity in Cd(V), then in particular
+pis proper.
+A framework has edges on a conic at infinity iff there is a contin uous family of d-dimensional
+non-orthonormal affine transforms that preserve all of the ed ge lengths. However, if the
+configuration is not proper, this family of affine transforms m ight all be congruent to the
+original one.
+1.3.Results. Connelly in a series of papers has studied the relationship b etween various
+forms of rigidity and equilibrium stress matrices. In parti cular he proved the following
+theorem that gives a sufficient condition for a framework to be universally rigid.
+Theorem 1.12 (Connelly) .Suppose that Γis a graph with d+ 2or more vertices and pis a
+(generic or not) configuration in Cd(V). Suppose that there is an equilibrium stress matrix
+Ω∈SΓ(p)that is positive semidefinite (PSD) and that rank Ω = v−d−1. Also suppose that
+the edge directions of pdo not lie on a conic at infinity. Then (p,Γ)is universally rigid.
+Connelly also proved a lemma that allows him to ignore the con ic at infinity issue when p
+is generic.
+Lemma 1.13 (Connelly) .Suppose that Γis a graph with d+ 2or more vertices and pis a
+generic configuration in Cd(V). Suppose that there is an equilibrium stress matrix Ω∈SΓ(p)
+with rank v−d−1. Then the edge directions of pdo not lie on a conic at infinity.
+Putting these together, we can summarize this as
+Corollary 1.14. Suppose that Γis a graph with d+ 2or more vertices and pis a generic
+configuration in Cd(V). Suppose that there is a PSD equilibrium stress matrix Ω∈SΓ(p)
+with rank Ω = v−d−1. Then (p,Γ)is universally rigid.
+The basic ideas for the proof of Theorem 1.12appear in [ Con82 ] where they were applied
+to show the universal rigidity of Cauchy polygons. It is also described in [ Con99 ] and stated
+precisely in [ Con01 , Theorem 2.6]. Lemma 1.13can be derived from the proof of Theorem 1.3
+in [Con05 ]. Corollary 1.14in two dimensions is summarized in Jordán and Szabadka [ JS09].
+A different set of sufficient conditions for the related concep t of dimensional rigidity were
+described by Alfakih [ Alf07 ]. In [ Alf10 ] he showed that these conditions were equivalent to
+the existence of a PSD equilibrium stress matrix of maximal r ank. In [ Alf10 ] he also showed
+that a generic dimensionally rigid framework (with v≥d+ 2) must be universally rigid;
+this has the same effect as Lemma 1.13, and thus results in Corollary 1.14. In these papers,
+he also conjectured that for a generic universally rigid fra mework, a maximal rank PSD
+equilibrium stress matrix must exist.
+In the related context of frameworks with pinned anchor vert ices, So and Ye [ SY07 ] showed
+the appropriate analogue of Theorem 1.12 follows from complementarity in semidefinite
+programming (see section 5below for more on this).
+In this paper our main result is the converse to Corollary 1.14:
+Theorem 1. A universally rigid framework (p,Γ), with pgeneric in Cd(V)and having d+2
+or more vertices, has a PSD equilibrium stress matrix with ra nkv−d−1.
+Remark 1.15.Alfakih has given an example [ Alf07 , Example 3.1] showing that Theorem 1
+is false if we drop the assumption that pis generic. For any universally rigid framework
+(generic or not), it’s not hard to see that there is a non-zero PSD equilibrium stress matrix.CHARACTERIZING THE UNIVERSAL RIGIDITY OF GENERIC FRAMEWOR KS 5
+(See [ Alf11 , Theorem 5.1].) The difficulty in Theorem 1is finding a stress matrix of high
+rank.
+Theorems 1.12and1compare nicely with the situation for global rigidity, wher e a frame-
+work is generically globally rigid iff it has an equilibrium s tress matrix of rank v−d−1 (with
+no PSD constraint). Sufficiency was proved by Connelly[ Con05 ], and necessity was proved
+by the authors together with Alex Healy [ GHT10 ].
+Question 1.16. For a graph Γ that is generic globally rigid Ed, is there always a generic
+framework in Cd(V) that is universally rigid?
+In this paper, we also prove more general versions of Theorem 1in the general context
+of convex optimization and strict complementarity in semid efinite programming. See Theo-
+rem2in Section 4and Theorem 3in Section 5.
+2.Algebraic geometry preliminaries
+We start with some preliminaries about semi-algebraic sets from real algebraic geometry,
+somewhat specialized to our particular case. For a general r eference, see, for instance, the
+book by Bochnak, Coste, and Roy [ BCR98 ].
+Definition 2.1. An affine real algebraic set orvariety Vcontained in Rnis a subset of Rn
+that is defined by a set of algebraic equations. It is defined over Qif the equations can be
+taken to have rational coefficients. An algebraic set has a dimension dim(V), which we will
+define as the largest tfor which there is an open subset of V(in the Euclidean topology)
+homeomorphic to Rt.
+Definition 2.2. Asemi-algebraic set Sis a subset of Rndefined by algebraic equalities
+and inequalities; alternatively (by a non-trivial theorem ), it is the image of an algebraic set
+(defined only by equalities) under an algebraic map. It is defined over Qif the equalities
+and inequalities have rational coefficients. Like algebraic sets, a semi-algebraic set has a well
+defined (maximal) dimension t.
+We next define genericity in larger generality and give some b asic properties.
+Definition 2.3. A point in a (semi-)algebraic set Vdefined over Qisgeneric if its coor-
+dinates do not satisfy any algebraic equation with coefficien ts inQbesides those that are
+satisfied by every point on V.
+A point xon a semi-algebraic set Sislocally generic if for small enough ǫ,xis generic in
+S∩Bǫ(x).
+Remark 2.4.A semi-algebraic set Swill have no generic points if its Zariski closure over Q
+is reducible over Q. In this case all points in each irreducible component will s atisfy some
+specific equations not satisfied everywhere over S. But even in this case, almost every point
+in the set will be generic within its own component. Such poin ts will still be locally generic
+inS.
+We will need a few elementary lemmas on generic points in semi -algebraic sets.
+Lemma 2.5. IfXandYare both semi-algebraic sets defined over Q, with X⊂Yand
+Xdense in Y(in the Euclidean topology), then Xcontains all of the locally generic points
+ofY.6 GORTLER AND THURSTON
+Proof. Due to the density assumption, Y\Xmust be a semi-algebraic set defined over Q
+with dimension less than that of Y. The Zariski closure of a semi-algebraic set maintains
+its dimension, thus all points in Y\Xmust satisfy some algebraic equation that is non-zero
+over Yand thus these points must be non-generic. To see that a point y∈Y\Xis not
+locally generic either, apply this argument in an ǫ-neighborhood of y. (That is, apply the
+argument to Y∩Bǫ(y) and X∩Bǫ(y).) /square
+Lemma 2.6. LetVbe a semi-algebraic set, fbe an algebraic map from Vto a range space X,
+andWbe an semi-algebraic set contained in the image of f, with V,W, and fall defined
+overQ. Ifx0∈Vis generic and f(x0)∈W, then f(x0)is generic in W.
+Proof. Letφbe any algebraic function on Xwith rational coefficients so that φ(f(x0)) = 0.
+Then φ◦fvanishes at x0, which is generic in V, soφ◦fvanishes identically on V. This
+implies that φvanishes on W. Thus any algebraic function defined over Qthat vanishes at
+f(x0) must vanish on all of W. This proves that f(x0) is generic in W. /square
+3.The geometry of PSD stresses
+We now turn to the main construction in our proof.
+3.1.The measurement set.
+Definition 3.1. Thed-dimensional measurement set Md(Γ) of a graph Γ is defined to be the
+image of Cd(V) under the map ℓΓ. These are nested by Md(Γ)⊂Md+1(Γ) and eventually
+stabilize at Mv−1(Γ), also called the absolute measurement set M(Γ).
+Since Md(Γ) is the image of Cd(V) under an algebraic map, by Lemma 2.6, ifpis generic
+inCd(V) then ℓΓ(p) is generic in Md(Γ) (which must be irreducible).
+Lemma 3.2. The set M(Γ)is convex.
+Proof. The squared edge lengths of a framework pare computed by summing the squared
+edge lengths of each coordinate projection of p, soMd(Γ) is the d-fold Minkowski sum of
+M1(Γ) with itself. Since M1(Γ) is invariant under scaling by positive reals, Md(Γ) can also
+be described as an iterated chord variety, and in particular M(Γ) is the convex hull. /square
+A particular case of interest is M(∆v), the absolute measurement set of the complete
+graph on the vertices. It is a cone in R(v
+2).
+Lemma 3.3. The measurement set Md(∆v)is isomorphic (as a subset of the linear space
+R(v
+2)) to the set of PSD (v−1)×(v−1)matrices of rank at most d.
+Proof. There is a standard linear map from M(∆v) to the convex cone of v−1 by v−1
+positive semidefinite matrices, which we recall for the read er. (See, e.g., [ Sch35 ,Gow85 ].) For
+a point x∈Md(∆v), by the universal rigidity of the simplex, up to d-dimensional Euclidean
+isometry there is a unique p∈Cd(V) with x=ℓ∆(p). If we additionally constrain the last
+vertex to be at the origin, then pis unique up to an orthonormal transform. Think of such
+a constrained pas a ( v−1)×dmatrix, denoted ̺. The map sends xto the matrix ̺̺t,
+which has rank at most d. (The rank of ̺̺tis less than difphas an affine span of dimension
+less than d). It is easy to check that ̺̺tdoes not change if we change ̺by an orthonormal
+transform, and that in fact the entries of ̺̺tare linear combinations of the squared lengths
+of edges, so the map is well-defined and linear. /squareCHARACTERIZING THE UNIVERSAL RIGIDITY OF GENERIC FRAMEWOR KS 7
+Definition 3.4. LetKbe a non-empty convex set in an affine space. The dimension ofKis
+the dimension of the smallest affine subspace containing K. A point x∈Kis in the relative
+interior Int(K) ofKif there is a neighborhood UofxinEeso that U∩K∼=RdimK. (It is
+easy to see that the relative interior of Kis always non-empty.)
+ThefaceF(x) of a point x∈Kis the (unique) largest convex set contained in Kso that
+x∈Int(F(x)). The faces of Kare the sets F(x) asxranges over K. (In Section 4.1we will
+see more characterizations of F(x).) Let f(x) be the dimension of F(x).
+Lemma 3.5. Letx=ℓ∆(p)with p∈Cd(V). Then F(x) =ℓ∆(Aff d(p)) = ℓ∆(GLd(p)).
+When pis proper, f(x) =/parenleftBigd+1
+2/parenrightBig
+.
+Here GLdis the group of d-dimensional linear transforms, and Aff dis the corresponding
+affine group.
+Proof. By Lemma 3.3, this reduces to understanding the faces of the cone of v−1 by v−1
+PSD matrices, which is well understood (see, e.g., [ Pat00 , Theorem A.2]). In particular, for
+̺a (v−1)×dmatrix, the points in F(̺̺t) in the semidefinite cone are those matrices of
+the form ̺LtL̺tfor some d×dmatrix L. But each such ̺LtL̺tmaps under our isometry to
+ℓ∆(σ) where σ∈Cd(V) is obtained from pby a d-dimensional affine transform, with linear
+part described by L. When phas an affine span of dimension d,̺is of rank d. This implies
+thatF(̺̺t) has dimension/parenleftBigd+1
+2/parenrightBig
+, as we can see by computing the dimension of GLd/Od, the
+linear transforms modulo the orthonormal transforms. /square
+3.2.Dual vectors and PSD stresses.
+Definition 3.6. Let us index each of the coordinates of R(v
+2)with an integer pair ijwith
+1≤i < j ≤v. Given φ, a functional in the dual space/parenleftBig
+R(v
+2)/parenrightBig∗, define the v-by-vmatrix
+M(φ) as follows: for i/\e}atio\slash=j,M(φ)ij=M(φ)ji:=−φijandM(φ)ii:=−/summationtext
+j/ne}ationslash=iM(φ)ij
+Lemma 3.7. Letφbe a dual vector in/parenleftBig
+R(v
+2)/parenrightBig∗and let Ω:=M(φ)be the corresponding
+matrix. Let p∈Cd(V). Then
+(3.8) /a\}bracketle{tφ, ℓ ∆(p)/a\}bracketri}ht=1
+2d/summationdisplay
+k=1(pk)tΩpk.
+Here we use the notation pkfor vector in Rvdescribing the component of pin the k’th
+coordinate direction of Ed.
+Proof. Both sides are equal to/summationtext
+k/summationtext
+i,j|i<j(pk
+i−pk
+j)2φij. /square
+Definition 3.9. ForK⊂Rna convex set and φa functional in the dual space ( Rn)∗, we
+say that φistangent to Katx∈Kif for all y∈K,/a\}bracketle{tφ, y/a\}bracketri}ht ≥0 while /a\}bracketle{tφ, x/a\}bracketri}ht= 0.
+Lemma 3.10. Letφbe a dual vector in/parenleftBig
+R(v
+2)/parenrightBig∗that is tangent to M(∆v)atℓ∆(p)for some
+p∈C(V). Let Ω:=M(φ)be the corresponding matrix. Then Ωis a PSD equilibrium stress
+matrix for (p,∆v).
+Proof. Conditions (1) and (3) in Definition 1.8are automatic by definition of Ω, and condi-
+tion (2) is vacuous for the complete graph ∆ v. It remains to check condition (4) and that Ω
+is PSD.8 GORTLER AND THURSTON
+By Lemma 3.7,/a\}bracketle{tφ,·/a\}bracketri}htcan be evaluated as a quadratic form using the matrix Ω. Since /a\}bracketle{tφ,·/a\}bracketri}ht
+is not negative anywhere on M(∆v), Ω must be PSD. Because Ω is positive semidefinite and/summationtext
+k=1(pk)tΩpk= 0, it must also be true that for each k, (pk)tΩpk= 0 and so Ω pk= 0, which
+is the last necessary condition to show Ω is an equilibrium st ress matrix. /square
+Lemma 3.11. In the setting of Lemma 3.10, suppose furthermore that φis only tangent to
+points in F(ℓ∆(p)), and suppose the affine span of phas dimension d.
+Then Ωis a PSD equilibrium stress matrix for (p,∆v)with rank v−d−1.
+Proof. From Lemma 3.10, Ω is a PSD equilibrium stress matrix for ( p,∆v). By assumption,
+its kernel is spanned by the coordinates of frameworks that m ap under ℓ∆toF(ℓ∆(p)). From
+Lemma 3.5, such frameworks are d-dimensional affine transforms of p. Thus the kernel is
+spanned by the dcoordinates of pand the all-ones vector. Since the kernel has dimension
+d+ 1, Ω has rank v−d−1. /square
+Lemma 3.12. In the setting of Lemma 3.11, suppose furthermore that φij= 0for{i, j} /\e}atio\slash∈
+E(Γ)for a graph Γ.
+Then Ωis a positive semidefinite equilibrium stress matrix for (p,Γ)with rank v−d−1.
+Proof. Ω must have zeros in all coordinates corresponding to non-ed ges in Γ, thus it will be
+an equilibrium stress matrix for Γ as well as for ∆ v. The rest follows from Lemma 3.11./square
+To be able to use Lemma 3.12, we want to find a dual vector φthat is only tangent to
+points in F(ℓ∆(p))⊂M(∆v). Additionally we need that φij= 0 for {i, j} /\e}atio\slash∈ E . As we will
+see, this will correspond to finding a dual vector in ( Re)∗that is only tangent to points in
+F(ℓΓ(p))⊂M(Γ). The proper language for this kind of condition is the not ion of convex
+extreme and exposed points, which we turn to next.
+4.Convex sets and projection
+We will prove our main theorems in the more general setting of closed convex cones. Our
+goal is the following general theorem about projections of c onvex sets, which will quickly
+imply Theorem 1. (See Definitions 4.1,4.6, and 4.11for the terms used.)
+Theorem 2. LetKbe a closed line-free convex semi-algebraic set in Rm, and π:Rm→Rn
+a projection, both defined over Q. Suppose xis locally generic in extk(K)and universally
+rigid under π. Then π(x)isk-exposed.
+4.1.Extreme points.
+Definition 4.1. LetKbe a non-empty, convex set. A point x∈Kisk-extreme iff(x)≤k.
+(Recall from Definition 3.4that f(x) is the dimension of the face of Kcontaining x.) Let
+extk(K) be the set of k-extreme points in K.
+We will also use the following elementary propositions (see for example the exercises in
+chapter 2.4 of [ Grü03 ].)
+Proposition 4.2. ForKa convex set and x∈K, the following statements are equivalent:
+•f(x)≤k, and
+•xis not in the relative interior of any non-degenerate (k+1)-simplex contained in K.
+Proposition 4.3. ForKa convex set and x∈K, the face F(x)is the set of points z∈K
+so that there is an z′∈Kwith xin the relative interior of the segment [z′, z].CHARACTERIZING THE UNIVERSAL RIGIDITY OF GENERIC FRAMEWOR KS 9
+ab
+cd
+Figure 2. Examples of extreme and exposed points. For the racetrack sh ape
+illustrated (bounded by semi-circles and line segments), ais 0-extreme and
+0-exposed, bis 1-extreme and 1-exposed, cis 2-extreme and 2-exposed, and d
+is 0-extreme and 1-exposed.
+Remark 4.4.One special case of Proposition 4.3is when z=x, in which case we also take
+z′=xand the segment consists of a single point.
+Corollary 4.5. ForKa convex set, the faces of Kare the convex subsets FofKsuch that
+every line segment in Kwith a relative interior point in Fhas both endpoints in F.
+4.2.Exposed points.
+Definition 4.6. A point x∈Kisk-exposed if there is a closed half-space Hcontaining x
+so that dim H∩K≤k. Let expk(K) be the set of k-exposed points in K.
+See Figure 2for some examples of k-extreme and k-exposed points, including a case where
+they differ. If xisk-exposed, it is also k-extreme, as any simplex containing xin its relative
+interior is contained in any supporting hyperplane.
+The following theorem is a crucial tool in our proof.
+Theorem 4.7 (Asplund [ Asp63 ]).ForKa closed convex set, the k-exposed points are dense
+in the k-extreme points.
+Remark 4.8.Asplund only states and proves Theorem 4.7for compact convex sets. The
+result follows for any closed convex set Kas follows. For x∈extk(K), fix R > 0 and
+consider the compact convex set K′:=K∩BR(x). Then Asplund’s theorem says we can
+findk-exposed points zinK′that are arbitrarily close to x. Ifzis sufficiently close to x
+(say, within R/2 ofx), it will also be k-exposed in K, as desired.
+Theorem 4.7is an extension of a theorem of Straszewicz [ Str35 ], who proved it in the
+casek= 0. There are several improvements of Theorem 4.7, replacing “dense” with various
+stronger assertions. In our applications we will consider s emi-algebraic sets Kdefined over Q.
+Note that for any such set K, ext k(K) and expk(K) are also semi-algebraic over Q, as the
+k-extreme and k-exposed conditions can be phrased algebraically.
+Corollary 4.9. LetKbe a closed, convex, semi-algebraic set. If xis locally generic in
+extk(K), then xisk-exposed.
+Proof. This follows from an application of Lemma 2.5to the inclusion expk(K)⊂extk(K)
+in a small neighborhood around x. /square
+4.3.Projection. The two convex sets M(∆v) and M(Γ) are related by a projection from
+R(n
+2)toRe. We will consider this situation more generally.10 GORTLER AND THURSTON
+Throughout this section, fix two Euclidean spaces, EmandEn, with a surjective projection
+mapπ:Em→Enand a closed convex set K⊂Em. We will work with the image π(K)⊂En.
+LetπKbe the restriction of πtoK.
+Remark 4.10.In general, π(K) need not be closed, even if Kis. We continue to work with
+π(K) and its “faces” as defined in Definition 3.4, even if it is not closed.
+Definition 4.11. We say that x∈Kisuniversally rigid (UR) under πifπ−1
+K(π(x)) is the
+single point x.
+Note that for y∈π(K),π−1
+K(y) is always a convex set in its own right.
+Lemma 4.12. LetKbe a convex set and πa projection. For any x∈K,π(F(x))⊂
+F(π(x)).
+Proof. The set π(F(x)) is a convex set containing π(x) in its relative interior, so by definition
+is contained in F(π(x)). /square
+A point that is 0-extreme in a convex set is called a vertex of the set. A convex set is line
+freeif it contains no complete affine line. Recall that a non-empty , closed, line-free convex
+set has a vertex (see e.g., [ Grü03 , 2.4.6]). Our convex sets of interest, M(K∆) and M(Γ),
+are closed and also automatically line free, as the length-s quared takes only positive values
+and any line contains both positive and negative values in at least one coordinate.
+Proposition 4.13. LetKbe a convex set and πa projection. For any y∈π(K)andxa
+vertex of π−1
+K(y),F(x)maps injectively under πKintoF(y). In particular, f(x)≤f(y). If
+Kis closed and line free, then for every y∈π(K)there is a x∈π−1(y)so that f(x)≤f(y).
+Proof. LetA(x) be the smallest affine subspace containing F(x). Suppose A(x) contained
+a direction vector vin the kernel of π. Then, since xis in the relative interior of F(x) for
+small enough ǫ, we would have the segment [ x+ǫv, x −ǫv] contained in F(x) and also in
+π−1
+K(y). This would contradict the assumption that xis a vertex of π−1
+K(y). Thus F(x) maps
+injectively under πK.
+From Lemma 4.12we see that f(x)≤f(y). For the last part of the lemma, observe that
+ifKis closed and line free, so is π−1
+K(y), and so π−1
+K(y) has a vertex, which is the desired
+point xby the first part. /square
+Proposition 4.14. LetKbe a convex set and πa projection. For any y∈π(K)andxin
+the relative interior of π−1
+K(y),F(x) =π−1
+K(F(y)). In particular, f(x)≥f(y).
+Proof. By Lemma 4.12, we already know that F(x)⊂π−1
+K(F(y)), so we just need to show
+the other inclusion. Pick any y1∈F(y) and x1∈π−1
+K(y1). We must show that x1∈F(x).
+Since y1∈F(y), by Proposition 4.3there is an y2∈π(K) so that y∈Int([y1, y2]). Let
+x2∈π−1
+K(y2). Let x′∈Kbe the (unique) point of intersection of [ x1, x2] with π−1
+K(y). Since
+x∈Int(π−1
+K(y)), there is a point x′′∈π−1
+K(y) with x∈Int([x′, x′′]). But then xis in the
+interior of the simplex [ x1, x2, x′′], which implies that x1∈F(x). /square
+In order to be able to apply Asplund’s theorem at π(x) we need π(x) to be locally generic
+in ext k(π(K)). The following lemma will assist us.
+Lemma 4.15. LetKbe a closed, convex set in Rmand let πK:K→Rnbe a projection. Let
+x∈Kbe universally rigid under π. Then for any ǫthere is a δ >0so that π−1
+K(Bδ(π(x)))⊂
+Bǫ(x)∩K.CHARACTERIZING THE UNIVERSAL RIGIDITY OF GENERIC FRAMEWOR KS 11
+Intuitively, the only points in Kthat map close to π(x) are close to x.
+Proof. Suppose not. Then there is an ǫand a sequence of points xiwith the following
+property: π(xi) approach π(x), while d(x, x i)> ǫ. Let x′
+ibe the point on the interval [ x, x i]
+that is a distance exactly ǫfrom x. By convexity x′
+i∈K. Then the x′
+iare in a bounded and
+closed set and therefore have an accumulation point x′∈K, which will also have distance ǫ
+from x. By the linearity of πtheπ(x′
+i) also approach π(x). So by continuity π(x′) =π(x).
+This contradicts the universal rigidity of x. /square
+And now we can prove the following.
+Lemma 4.16. LetKbe a closed line-free convex semi-algebraic set in Rmandπ:Rm→Rn
+a projection, both defined over Q. Suppose xis locally generic in extk(K)and universally
+rigid under π. Then π(x)is locally generic in extk(π(K)).
+Proof. By Proposition 4.13, ext k(π(K))⊂π(ext k(K)). By Proposition 4.14and the universal
+rigidity of x, we have that π(x) is in ext k(π(K)).
+LetV:= ext k(K)∩Bǫ(x) and W:= ext k(π(K))∩Bδ(π(x)). For sufficiently small ǫ, by
+local genericity, xis generic in V. Meanwhile, W⊂π(ext k(K)), and, from Lemma 4.15, for
+small enough δ, we have π−1
+K(W)⊂Bǫ(x) and thus W⊂π(V).
+Thus from Lemma 2.6, we have π(x) generic in W. Thus π(x) is locally generic in
+extk(π(K)). /square
+Remark 4.17.In our special case where Kis the cone of positive semidefinite matrices, xis
+in fact generic in ext k(K) (which is irreducible), and thus the second paragraph in th e above
+proof (and in turn Lemma 4.15) are not needed.
+Finally, in order to be able to apply Asplund’s theorem to π(K) we need π(K) to be a
+closed set. In the graph embedding case, we can use the fact th atπis a proper map whenever
+Γ is connected, and thus π(K) must be closed. In the setting of a general Kandπwe can
+argue closedness using standard techniques.
+Definition 4.18. A direction of recession vfor a convex set Kis a vector such that for every
+(equivalently, any) x∈Kand every λ≥0 we have x+λv∈K(see e.g. [ Roc70 , Chapter
+8]).
+A sufficient condition for closedness is given by the followin g theorem [ Roc70 , Theorem
+9.1].
+Theorem 4.19. LetKbe a line free closed convex set and πa projection. If Kdoes not
+have a direction of recession vwith π(v) = 0 , then then π(K)is closed.
+Corollary 4.20. LetKbe a line free closed convex set and πa projection. If there is a
+point x∈Kthat is universally rigid under π, then π(K)is closed.
+Proof. Suppose there was a direction of recession vforKwith π(v) = 0. Then π(x+v) =π(v)
+with x+vinK. This would contradict the universal rigidity of x. /square
+Putting this all together, we can deduce Theorem 2.
+Proof of Theorem 2.From Corollary 4.20,π(K) is closed. From Lemma 4.16,π(x) is locally
+generic in ext k(π(K)). Thus by Corollary 4.9,π(x) isk-exposed. /square12 GORTLER AND THURSTON
+4.4.Proof of main theorem. We are now in position to complete the proof of Theorem 1.
+Recall that we have a universally rigid framework ( p,Γ) with pgeneric in Cd(V). Let Kbe
+M(∆v), and πbe its projection to M(Γ),kbe/parenleftBigd+1
+2/parenrightBig
+. and x∈Md(∆v)⊂Kbeℓ∆(p). The
+assumption that pis generic in Cd(V) implies that xis generic in Md(∆v) = ext k(K).
+From Theorem 2,π(x) isk-exposed. So there must be a closed halfspace HinRewhose
+intersection with π(K) is exactly F(π(x)).
+The preimage π−1(H) is a closed halfspace in R(v
+2)whose intersection with Kis exactly
+π−1
+K(F(π(x))), which by Proposition 4.14 isF(x). Since Kis a cone, π−1(H) must also
+include the origin. When v > d + 1, then F(x) is not the entire measurement cone M(∆v),
+and the halfspace π−1(H) must not include any interior points of M(∆v).
+Thus π−1(H) is represented by a dual vector φ∈/parenleftBig
+R(v
+2)/parenrightBig∗, in the sense that π−1(H) =
+/braceleftBig
+x∈R(v
+2)/vextendsingle/vextendsingle/vextendsingleφ(x)≤0/bracerightBig
+. Moreover, this φis tangent to M(∆v) exactly at F(x).
+Since we started with a halfspace in Re,φij= 0 for {i, j} /\e}atio\slash∈ E . From Lemma 3.12, then,
+Ω:=M(φ) must be a positive semi-definite equilibrium stress matrix of (p,Γ) with rank
+v−d−1.
+5.Relation to semidefinite programming
+Theorem 1can be interpreted as a strict complementarity statement ab out a particular
+semidefinite program (SDP) and its associated dual program, as we will now explain. For
+a general survey on semidefinite programming see Vandenberg he and Boyd [ VB96 ]. Our
+notation is similar to that of Pataki and Tunçel [ PT01 ]. For a related discussion on universal
+rigidity, semidefinite programming and complementarity, s ee, for instance, the paper by So
+and Ye [ SY07 ].
+5.1.Semidefinite programming. We first recall the basic definitions of semidefinite pro-
+gramming. Let Snbe the linear space of all symmetric n-by-nmatrices and Sn
++⊂Sn
+be the cone of positive semidefinite matrices. A semidefinite program is given by a triple
+(L, b, β ) where L⊂Snis a linear subspace, b∈Snis a matrix, and β∈(Sn)∗is a
+dual to a matrix (which we can also view as a matrix, using the n atural inner product
+/a\}bracketle{tA, B/a\}bracketri}ht= tr( AtB) =/summationtext
+i,jAijBij). This describes a primal constrained optimization proble m:
+infx{/a\}bracketle{tβ, x/a\}bracketri}ht |x∈(L+b)∩Sn
++}
+That is, we optimize a linear functional over all symmetric p ositive semidefinite matrices x,
+subject to the constraint that xlies in a chosen affine space.
+Associated with every primal SDP is the (Lagrangian) dual pr ogram
+inf
+Ω{/a\}bracketle{tΩ, b/a\}bracketri}ht |Ω∈(L⊥+β)∩(Sn
++)∗}
+Since the cone of PSD matrices is self-dual, elements of ( Sn
++)∗also correspond to PSD
+matrices, so this dual program can be thought of as a semidefin ite program itself.
+Definition 5.1. We say a pair of primal/dual points ( x,Ω) with x∈Sn
++and Ω ∈(Sn
++)∗are
+acomplementary pair for the program ( L, b, β ) if both are feasible points for the respective
+programs and /a\}bracketle{tΩ, x/a\}bracketri}ht= 0.
+A calculation shows that for any feasible primal/dual pair o f points ( x,Ω), we have /a\}bracketle{tβ, b/a\}bracketri}ht−
+/a\}bracketle{tΩ, b/a\}bracketri}ht ≤ /a\}bracketle{t β, x/a\}bracketri}htand that complementarity implies /a\}bracketle{tβ, b/a\}bracketri}ht − /a\}bracketle{tΩ, b/a\}bracketri}ht=/a\}bracketle{tβ, x/a\}bracketri}ht.CHARACTERIZING THE UNIVERSAL RIGIDITY OF GENERIC FRAMEWOR KS 13
+Thus, writing the dual problem in the alternative form
+sup
+Ω{/a\}bracketle{tβ, b/a\}bracketri}ht − /a\}bracketle{tΩ, b/a\}bracketri}ht |Ω∈(L⊥+β)∩(Sn
++)∗}
+we see that a complementary pair must represent optimal solu tions to the primal and dual
+SDPs respectively.
+Definition 5.2. We say that an SDP problem ( L, b, β ) isgap free if it has complementary
+pairs of solutions.
+Gap freedom is typically seen as a fairly mild constraint on a n SDP problem.
+For every complementary pair ( x,Ω) we have [ AHO97 ]
+(5.3) rank( x) + rank(Ω) ≤n.
+Definition 5.4. A complementary pair ( x,Ω) is said to be strictly complementary if we have
+rank( x) + rank(Ω) = n.
+There is a rich theory on when SDPs have strictly complementa ry solution pairs. (See
+[AHO97 ,PT01 ,NRS10 ]; see also Section 5.9.) In particular, certain SDP algorithms converge
+more quickly on problems that satisfy strict complementari ty [LSZ98 ]. The linear program-
+ming counterpart of strict complementarity is strict compl ementary slackness [ DT03 ]. In
+the linear case, strict complementarity is always achieved . By contrast, for SDP problems it
+depends on the particular problem.
+5.2.Graph embedding as an SDP. In the context of graph embedding, suppose we are
+looking for a configuration of unconstrained dimension, p∈Cv(V)/Eucl( v), such that the
+framework of Γ is constrained to have a set of squared edge len gthsd2
+ijforij∈ E(Γ). It is well
+known that we can set up this graph embedding problem as a semi definite program [ LLR95 ],
+as we will now review.
+To a configuration pwe can associate a v×vGram matrix x, with xij=p(i)·p(j). The
+Gram matrix of pis unchanged by elements of O(v) (although it does change when pis
+translated). The matrix rank, r, of such an xis the dimension of the linear span of the
+associated configuration p. This will be typically be d+ 1; one greater than the affine span,
+d, of the framework.
+In the graph embedding SDP, we set n:=vand take the Gram matrix xas our unknown.
+The distance constraint at an edge ijcan be expressed as the linear constraint xii+xjj−2xij=
+(dij)2. The collection of matrices satisfying these constraints f or all edges forms an affine
+space; we choose Landbsuch that L+bis this affine space. In our context, we are only
+interested in feasibility, and thus have no objective funct ion, so we set β:= 0. Note that
+semidefinite programs do not allow us to explicitly constrai n the rank of the solution (which
+is related to the dimension of the configuration).
+Let us now look at the dual program to our graph embedding SDP p roblem. In the primal
+problem, the linear space Lcorresponds to symmetric matrices xwith xii+xjj−2xij= 0
+for all {i, j} ∈ E (Γ). Thus the space L⊥, when represented as matrices, is spanned by a basis
+elements Bij=−eii+eij+eji−ejjfor{i, j} ∈ E (Γ), where eijis the elementary matrix
+with a 1 in the ijentry and 0 elsewhere. The space L⊥is therefore those matrices Ω with
+row-sums of zero and with Ω ij= 0 for all i/\e}atio\slash=j,{i, j}/∈ E(Γ). That is, L⊥is the space of all
+(not necessarily equilibrium) stress matrices. We also hav eβ= 0, so we optimize over these
+matrices.14 GORTLER AND THURSTON
+Minimizing /a\}bracketle{tΩ, b/a\}bracketri}htimposes the further constraint that the solution be an equil ibrium stress
+matrix. This can be seen as follows. Since β= 0, the all-zeros matrix is dual feasible.
+Whenever there is some primal feasible xcorresponding to a configuration p, the SDP must
+be gap free as /a\}bracketle{t0, x/a\}bracketri}ht= 0, and the optimal Ω are the feasible Ω with /a\}bracketle{tΩ, x/a\}bracketri}ht= 0. For any
+Ψ∈(Sn
++)∗we have /a\}bracketle{tΨ, x/a\}bracketri}ht=1
+2/summationtextr
+k=1(pk)tΨpk. (Here we use the notation pkfor vector in
+Rndescribing the component of pin the k’th coordinate direction of Rr.) And thus, as in
+Lemma 3.10, the feasible Ω with /a\}bracketle{tΩ, x/a\}bracketri}ht= 0 are the PSD equilibrium stress matrices for
+(p,Γ).
+In this language we can restate Theorem 1as follows.
+Proposition 5.5. Letpbe generic in Cd(V), let (p,Γ)be universally rigid in Ed, and let
+(L, b,0)be the associated SDP using the graph Γand the distances of the edges in (p,Γ).
+Then (L, b,0)has a strictly complementary solution pair (x,Ω).
+Proof. Simply let xbe the Gram matrix corresponding to a translation of psuch that its affine
+span does not include the origin. Thus xwill have rank d+ 1. Let Ω be a PSD equilibrium
+stress matrix of rank v−d−1, which exists by Theorem 1. Then rank x+ rank Ω = v./square
+5.3.SDP feasibility. Our discussion of universal rigidity and equilibrium stres s matrices
+carries over directly to any SDP feasibility problem ( L, b,0) (i.e., where β= 0). To see
+this we first set up some notation. Pick an integer r >0 and let k:=/parenleftBigr+1
+2/parenrightBig
+. Then, as in
+Lemma 3.5, ext k(Sn
++) is the set of the PSD matrices of rank less than or equal to r, which
+we will denote Sn
++r. Let πbe projection from SntoSn/L, and π+its restriction to Sn
++. A
+point x∈Sn
++is a solution to the feasibility problem iff π(x) =π(b), or x∈π−1
++(π(b)).
+We say that a point x∈Sn
++is universally rigid under πiffπ−1
++(π(x)) is a single point: A
+universally rigid feasible point is the unique feasible sol ution to ( L, b,0). As we vary bwith
+fixed L, we will see as solutions all points in Sn
++; it may happen that for some bthere is a
+unique, generic feasible solution.
+Proposition 5.6. Let(L, b,0)be an SDP feasibility problem, where Lhas rational coeffi-
+cients. Suppose there is a unique feasible solution xwhich is generic in Sn
++r. Then there
+exists an optimal dual solution Ωsuch that (x,Ω)is a strictly complementary pair.
+Proof. Since xis generic in Sn
++r= ext k(Sn
++), by Theorem 2π(x) isk-exposed. Thus there is
+a PSD matrix Ω in ( Sn/L)∗=L⊥that is tangent to π(Sn
++) atπ(x), with contact F(π(x)).
+Since Ω ∈L⊥, it is a feasible point of the dual SDP. Since Ω is tangent to π(Sn
++) atπ(x), it
+is also tangent to Sn
++atx; thus /a\}bracketle{tΩ, x/a\}bracketri}ht= 0 and Ω is a complementary and therefore optimal
+dual solution.
+The tangency of Ω to Sn
++atxhas contact π−1
++(F(π(x)) which, by Proposition 4.14, isF(x).
+Letpbe an r-dimensional configuration of npoints with Gram matrix x. From the facial
+stucture of the PSD cone we see that F(x) consists only of Gram matrices of r-dimensional
+linear transforms of p. We also have the relation /a\}bracketle{tΩ, x/a\}bracketri}ht=1
+2/summationtextr
+k=1(pk)tΩpk= 0. Thus we see
+that Ω must have a kernel of dimension rand have rank n−r. /square
+5.4.SDP optimization. We can apply the approach above to a more general SDP problem
+(L, b, β ), where β/\e}atio\slash= 0. Specifically, we prove the following.
+Theorem 3. Let(L, b, β )be a gap-free SDP problem, with Landβrational. Suppose
+there is a unique optimal solution xwhich is generic in Sn
++r. Then there exists a strictly
+complementary pair (x,Ω).CHARACTERIZING THE UNIVERSAL RIGIDITY OF GENERIC FRAMEWOR KS 15
+We will prove the theorem by reducing to the β= 0 case. For any SDP optimization
+problem ( L, b, β ) with an optimal solution x, there is an associated SDP feasibility problem
+(L′, b′,0), where L′+b′is{y∈L+b| /a\}bracketle{tβ, y/a\}bracketri}ht=/a\}bracketle{tβ, x/a\}bracketri}ht}. In particular, L′isL∩kerβ(which
+is still rational) and the dual space L′⊥isL⊥+/a\}bracketle{tβ/a\}bracketri}ht. Then feasible solutions to ( L′, b′,0)
+correspond to optimal solutions to ( L, b, β ).
+Lemma 5.7. Let(L, b, β )be an SDP problem, with Landβrational. Suppose there is a
+unique optimal solution xwhich is generic in Sn
++r. Then there exists a PSD Ωsuch that
+rank Ω = n−r, with Ω∈L⊥+/a\}bracketle{tβ/a\}bracketri}htand/a\}bracketle{tΩ, x/a\}bracketri}ht= 0.
+Proof. Apply Proposition 5.6to the feasibility problem ( L′, b′,0) constructed above. This
+gives a Ω ∈L′⊥=L⊥+/a\}bracketle{tβ/a\}bracketri}htthat is strictly complementary to x. /square
+Note that we have not yet proved Theorem 3, as Lemma 5.7gives Ω in the linear space
+L⊥+/a\}bracketle{tβ/a\}bracketri}htrather than the desired affine space L⊥+β.
+Proof of Theorem 3.Let Ω 1be the PSD matrix given by Lemma 5.7. From the assumption
+of gap freedom there exists a PSD Ω 2with Ω 2∈L⊥+βand/a\}bracketle{tΩ2, x/a\}bracketri}ht= 0. Thus for any
+positive scalars λ1andλ2, the matrix Ω :=λ1Ω1+λ2Ω2is PSD, has rank no less than n−r,
+and has /a\}bracketle{tΩ, x/a\}bracketri}ht= 0. By adjusting λ1andλ2we can achieve Ω ∈L⊥+β. By Equation ( 5.3),
+we in fact have rank Ω = n−r. /square
+We note though that for a given ( L, β) and rank r, as we vary b, there may be no unique
+solutions with rank r. Even if there are such x, they may all be non-generic. In such cases,
+Theorem 3is vacuous, as there are no choices of ( L, b, β ) satisfying the hypotheses.
+5.5.Genericity. In the context of SDP optimization, the hypothesis in Theore m3that L
+andβbe rational may seem a little unnatural. This hypothesis can be relaxed if we work
+with points that are generic over a field that is larger than Q.
+Definition 5.8. Letkbe a field containing Qand contained in R. A semi-algebraic set S⊂
+Rnisdefined over kif there is a set of equalities and inequalities defining Swith coefficients
+ink. IfSis defined over k, a point x∈Sisgeneric over kif the coordinates of xdo
+not satisfy any algebraic equation with coefficients in kbeyond those that are satisfied by
+every point in S. Similarly, xislocally generic over kif for small enough ǫ,xis generic in
+S∩Bǫ(x).
+Adefining field of a semi-algebraic set S, written Q[S], is any field over which it is defined.
+(There is a unique smallest field for algebraic sets by a resul t of Weil [ Wei46 , Corollary IV.3].
+For our purposes, we can use any field over which Sis defined.) Similarly, if f:X→Yis a
+map between semi-algebraic sets, then Q[f] is a field over which it is defined (or, equivalently,
+a field over which the graph of fis defined).
+For instance, if Sis a single point x, we can take Q[S] to be the same as Q[x], the smallest
+field containing all the coordinates of x. Also, if xis generic over Q, then yis generic over
+Q[x] iff the pair ( x, y) is generic over Q.
+With this definition, Theorems 2and3can be improved to allow non-rational sets and
+projections.
+Theorem 2′.LetKbe a closed line-free convex semi-algebraic set in Rm, and π:Rm→Rn
+a projection. Suppose xis locally generic over Q[K, π]inextk(K)and universally rigid
+under π. Then π(x)isk-exposed.16 GORTLER AND THURSTON
+Theorem 3′.Let(L, b, β )be a gap-free SDP problem. Suppose there is a unique optimal
+solution xwhich is generic over Q[L, β]inSn
++r. Then there exists a strictly complementary
+pair(x,Ω).
+The proofs follow exactly the proofs of the versions given ea rlier. For instance, in Theo-
+rem 3′, we apply Theorem 2′to the cone Sn
++(which is defined over Q) and the projection
+onto Sn/(L+/a\}bracketle{tβ/a\}bracketri}ht), which is defined over whatever field is needed to define Landβ.
+5.6.Relation to previous results. A fundamental result of [ AHO97 ,PT01 ] on strict com-
+plementarity can be summarized as follows.
+Theorem 5.9. Suppose (L, b, β )is a generic SDP program that is gap free. Then (L, b, β )
+admits a strictly complementary pair of solutions.
+This result is neither stronger nor weaker than our Theorem 3. In particular Theorem 5.9
+requires that all parameters be generic. In contrast, Theor em3does not assume genericity of
+any of the parameters but rather assumes genericity of the so lution within its rank. Indeed, in
+our application to rigidity of graphs, β= 0, which is obviously not generic. (The parameter
+bis also not usually generic.) In fact, there are very few situ ations where both theorems can
+apply, as the following proposition shows.
+Proposition 5.10. In an SDP problem (L, b, β )with dimL=Dand primal solution xof
+rank r, if Theorem 5.9applies, then
+/parenleftBiggn−r+ 1
+2/parenrightBigg
+≤D.
+On the other hand, if Theorem 3applies, then
+/parenleftBiggn−r+ 1
+2/parenrightBigg
+≥D.
+Proof. For a given ( L, b, β ), ifbis generic over Q[L], then rsatisfies the first inequality, as
+described in [ AHO97 , Theorem 12] and [ NRS10 , Proposition 5]. In particular, for generic b,
+the intersection of L+bwith Sn
++rmust be transversal, which from a dimension count gives
+the first inequality. This takes care of the first part of the pr oposition. (If βis generic over
+Q[L], as in Theorem 5.9, there is also an upper bound on r:/parenleftBigr+1
+2/parenrightBig
+≤/parenleftBign+1
+2/parenrightBig
+−D. But we do
+not need this.)
+For the second part, recall that if Theorem 3applies, there is a point x, generic over
+Q[L, β] inSn
++rwhich is the unique solution to the SDP problem ( L, b, β ). Recall that there is
+an associated feasibility problem ( L′, b′,0). Let π:Sn→Sn/L′be the associated projection
+andπ+its restriction to Sn
++. Since xis unique (in both ( L, b, β ) and ( L′, b′,0)),xis the
+only point in Sn
++mapping to π(x), and π(x)∈∂π(Sn
++). Moreover, since xis generic in Sn
++r,
+there is an open neighborhood UofxinSn
++rwith these properties. In particular, Umaps
+injectively by πto∂π(Sn
++)⊂Sn/L′.
+Sn
++rhas dimension/parenleftBign+1
+2/parenrightBig
+−/parenleftBign−r+1
+2/parenrightBig
+. On the other hand, dim( Sn/L′)≤/parenleftBign+1
+2/parenrightBig
+−D+ 1
+(with equality iff β/\e}atio\slash∈L⊥or equivalently L′/\e}atio\slash=L), so dim( ∂π(Sn
++))≤/parenleftBign+1
+2/parenrightBig
+−D. If there is
+a smooth injection from Sn
++rto∂π(Sn
++), we must have/parenleftBign−r+1
+2/parenrightBig
+≥D, as desired. /squareCHARACTERIZING THE UNIVERSAL RIGIDITY OF GENERIC FRAMEWOR KS 17
+5.7.Complexity of universal rigidity. Assuming ( p,Γ) is not at a conic at infinity and
+is translated so that its affine span does not include 0, ( p,Γ) is UR iff there is no higher rank
+solution to the SDP than p[Alf07 ]. Thus, to test for universal rigidity, the main step is to te st
+if there is a feasible solution with rank higher than that of a known input feasible solution p
+(given say as integers). Numerically speaking, SDP optimiz ation algorithms that use interior
+point methods produce approximate solutions of the highest possible rank [ GY93 ,DKRT97 ]
+and so in practice one could try such a method to produce a gues s about the universal
+rigidity of ( p,Γ).
+The complexity of getting a definitive answer is a trickier qu estion, even assuming one
+could reduce the UR question to one of “yes-no” SDP feasibili ty. In particular, an approx-
+imate solution to an SDP optimization problem can be found in polynomial time but the
+complexity of the SDP feasibility problem remains unknown, even with strict complemen-
+tarity. See [ Ram97 ] for formal details.
+Theorem 1tells us that for generic inputs, the UR question can be answe red by finding
+the highest rank dual optimal solution. (A framework with in teger coordinates will not be
+generic; however, if the integers are large enough we are lik ely to avoid all special behavior, as
+in [GHT10 , Section 5].) This does not appear to be any help in determini ng the complexity
+of testing algorithmically for universal rigidity.
+References
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+semidefinite programming , Mathematical Programming 77(1997), no. 1, 111–128.
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+B5(2011), 113–128.
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+Discrete Comput. Geom. 32(2004), no. 1, 101–106.
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+[Con99] ,Tensegrity structures: Why are they stable? , Rigidity Theory and Applications (M. F.
+Thorpe and P. M. Duxbury, eds.), Klewer Academic/Plenum Pub lishers, New York, 1999, pp. 47–
+54.
+[Con01] ,Stress and stability , Chapter from an unpublished book, available from
+http://www.math.cornell.edu/~connelly/Tensegrity.Ch apter.2.pdf , 2001.
+[Con05] ,Generic global rigidity , Discrete Comput. Geom 33(2005), no. 4, 549–563.
+[DKRT97] E. De Klerk, C. Roos, and T. Terlaky, Initialization in semidefinite programming via a self-dual
+skew-symmetric embedding , Operations Research Letters 20(1997), no. 5, 213–221.
+[DT03] G. B. Dantzig and M. N. Thapa, Linear Programming: Theory and extensions , Springer Verlag,
+2003.
+[EGW+04] T. Eren, O. K. Goldenberg, W. Whiteley, Y. R. Yang, A. S. Mo rse, B. D. O. Anderson, and P. N.
+Belhumeur, Rigidity, computation, and randomization in network local ization , INFOCOM 2004.
+Twenty-third Annual Joint Conference of the IEEE Computer a nd Communications Societies,
+vol. 4, 2004, pp. 2673–2684.
+[GHT10] Steven Gortler, Alex Healy, and Dylan Thurston, Characterizing generic global rigidity , American
+Journal of Mathematics 132(2010), no. 4, 897–939, arXiv:0710.0926.18 GORTLER AND THURSTON
+[Gow85] J. C. Gower, Properties of Euclidean and non-Euclidean distance matric es, Linear Algebra and
+its Applications 67(1985), 81–97.
+[Grü03] B. Grünbaum, Convex polytopes , Springer Verlag, 2003.
+[GY93] O. Güler and Y. Ye, Convergence behavior of interior-point algorithms , Mathematical Program-
+ming 60(1993), no. 1, 215–228.
+[JS09] T. Jordán and Z. Szabadka, Operations preserving the global rigidity of graphs and fra meworks
+in the plane , Computational Geometry: Theory and Applications 42(2009), no. 6-7, 511–521.
+[LLR95] N. Linial, E. London, and Y. Rabinovich, The geometry of graphs and some of its algorithmic
+applications , Combinatorica 15(1995), no. 2, 215–245.
+[LSZ98] Z. Q. Luo, J. F. Sturm, and S. Zhang, Superlinear convergence of a symmetric primal-dual
+path following algorithm for semidefinite programming , SIAM Journal on Optimization 8(1998),
+no. 1, 59–81.
+[NRS10] J. Nie, K. Ranestad, and B. Sturmfels, The algebraic degree of semidefinite programming , Math-
+ematical Programming 122(2010), no. 2, 379–405.
+[Pat00] G. Pataki, The geometry of semidefinite programming , Internat. Ser. Oper. Res. Management
+Sci., vol. 27, pp. 29–65, Kluwer Acad. Publ., Boston, MA, 200 0.
+[PT01] G. Pataki and L. Tunçel, On the generic properties of convex optimization problems i n conic
+form, Mathematical Programming 89(2001), no. 3, 449–457.
+[Ram97] M. V. Ramana, An exact duality theory for semidefinite programming and its complexity impli-
+cations , Mathematical Programming 77(1997), no. 1, 129–162.
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+[Sax79] J. B. Saxe, Embeddability of weighted graphs in k-space is strongly NP-hard , Proc. 17th Allerton
+Conf. in Communications, Control, and Computing, 1979, pp. 480–489.
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+d’espace distanciés vectoriellement applicable sur l’esp ace de Hilbert” , Annals of Mathematics
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+Mathematical Society, Providence, Rhode Island, 1946.
+School of Engineering and Applied Sciences, Harvard Universit y, Cambridge, MA 02138
+E-mail address :sjg@cs.harvard.edu
+Department of Mathematics, Indiana University, Bloomington IN 47405
+E-mail address :dpthurst@indiana.edu
\ No newline at end of file
diff --git a/1001.0173.txt b/1001.0173.txt
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+++ b/1001.0173.txt
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+Accepted for publication in the Astrophysical Journal
+The Radio Properties and Magnetic Field Conguration in the Crab-like
+Pulsar Wind Nebula G54.1+0.3
+Cornelia C. Lang
+Department of Physics & Astronomy, 703 Van Allen Hall, University of Iowa, Iowa City, IA
+52242
+cornelia-lang@uiowa.edu
+Q. Daniel Wang
+Department of Astronomy, University of Massachusetts, Amherst, MA 01002
+Fangjun Lu1
+Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
+Sciences, Beijing 100049, China
+and
+Kelsey I. Clubb2
+Department of Physics & Astronomy, Van Allen Hall, University of Iowa, Iowa City, IA 52242
+ABSTRACT
+We present a multifrequency radio investigation of the Crab-like pulsar wind nebula
+(PWN) G54.1+0.3 using the Very Large Array. The high resolution of the observations
+reveals that G54.1+0.3 has a complex radio structure which includes lamentary and
+loop-like structures that are magnetized, a diuse extent similar to the associated diuse
+X-ray emission. But the radio and X-ray structures in the central region dier strikingly,
+indicating that they trace very dierent forms of particle injection from the pulsar
+and/or particle acceleration in the nebula. No spectral index gradient is detected in
+the radio emission across the PWN, whereas the X-ray emission softens outward in
+the nebula. The extensive radio polarization allows us to image in detail the intrinsic
+magnetic eld, which is well-ordered and reveals that a number of loop-like laments are
+1Department of Astronomy, University of Massachusetts, Amherst, MA 01002
+2Now at: Department of Physics and Astronomy, San Francisco State University 1600 Holloway Avenue, San
+Francisco, CA 94132arXiv:1001.0173v1  [astro-ph.GA]  31 Dec 2009{ 2 {
+strongly magnetized. In addition, we determine that there are both radial and toroidal
+components to the magnetic eld structure of the pulsar wind nebula. Strong mid-IR
+emission detected in Spitzer Space Telescope data is closely correlated with the radio
+emission arising from the southern edge of G54.1+0.3. In particular, the distributions
+of radio and X-ray emission compared with the mid-IR emission suggest that the PWN
+may be interacting with this interstellar cloud. This may be the rst PWN where we are
+directly detecting its interplay with an interstellar cloud that has survived the impact
+of the supernova explosion associated with the pulsar's progenitor.
+Subject headings: ISM: Individual: Alphanumeric: G54.1+0.3, ISM: Supernova Rem-
+nants
+1. Introduction
+Crab-like supernova remnants, also known as pulsar wind nebulae (PWNe), provide unique
+laboratories for studying the interaction between high-energy pulsar wind materials and supernova
+ejecta. The synchrotron emission (normally observed at radio and X-ray wavelengths) arises from
+ultra-relativistic particles in reverse-shocked pulsar wind materials (Rees & Gunn 1974; Reynolds
+& Chevalier 1984). SNR G54.1+0.3 (or hereafter, G54.1+0.3) is considered to be a Crab-like
+source and we will refer to it and other similar objects as PWNe. Recent high-resolution Chandra
+observations (100or better) have revealed physical features similar to the Crab: a bright X-ray ring
+surrounding a central point source (corresponding to a 136 ms pulsar; Camilo et al. 2002), a western
+jet perpendicular to the ring, and an eastern protrusion, all in addition to the known low-brightness,
+diuse component (Lu et al. 2002). The X-ray spectra of the above features are all nonthermal
+and steepen with increasing distance from the central pulsar, indicating signicant particle energy
+evolution. The distance to G54.1+0.3 has been recently revisited by an HI absorption and13CO
+emission study (Leahy et al. 2008), which suggests that the distance to the PWN is approximately
+d=6.2 kpc, based on the morphological association with a molecular cloud. This is slightly farther
+than the previous distance, estimated to be 5 kpc (Lu et al. 2002). Here, we adopt the 6.2 kpc
+distance, which implies that 10=1.8 pc.
+However, unlike the Crab, the diuse X-ray emission in G54.1+0.3 extends well beyond the
+central X-ray torus/jet structures. This extensive, low surface brightness X-ray component corre-
+sponds well to the radio emission associated with this object. Reich et al. (1985) rst identied
+G54.1+0.3 as a small-diameter (1.50) radio source and suggested its nonthermal nature at 4.8 GHz.
+Velusamy & Becker (1988) present Very Large Array (VLA) of the National Radio Astronomy Ob-
+servatory (NRAO)1observations of G54.1+0.3 at 4.8 and 1.4 GHz with resolutions of 500and 1400,
+1The NRAO is a facility of the National Science Foundation operated under cooperative agreement by Associated
+Universities, Inc.{ 3 {
+respectively. They conrm the nonthermal nature of the radio source, its center-lled morphology
+and linear polarization ( 10%). The similarity of the extent of diuse X-ray and radio emission
+also diers from that of the Crab, where the radio emission is much more extensive than the X-ray
+component. Similarities in the size of radio and X-ray PWNe have been observed in sources such
+as G292.0+1.8, the PWN powered by B1509-58, and 3C58 (Gaensler & Wallace 2003; Gaensler et
+al. 2002; Slane et al. 2004), and have interesting implications for the particle energetics and source
+structure.
+Although detailed, multi-wavelength studies have been carried out for a growing number of
+PWNe (see recent review by Gaensler & Slane 2006), many issues relevant to Crab-like sources
+remain unresolved: (1) determining which acceleration mechanism can produce both the radio-
+emitting particles in addition the to the X-ray emitting particles is dicult (Atoyan 1999; Arons
+2002; Bientenholz et al. 2004); (2) the absence of outer shells in many of these PWNe (e.g., 3C58
+and the Crab). This question is related to the ages of the SNRs as well as the the properties of
+both the SN progenitors and the surrounding interstellar medium; and (3) the role and geometry of
+the magnetic eld in the inner and outer regions of PWNe are not well-studied. Radio polarization
+studies have been carried out for a few PWNe (e.g., Crab, Vela, 3C58, G292.0+1.8, Boomerang) and
+show that the magnetic eld structures in PWNe are highly-organized and often oriented radially
+toward the outer parts of the nebula. However, the transition from the predicted torodial magnetic
+eld orientation at the very center of the PWN near the location of the pulsar (Bucciantini et al.
+2005), to the outer, diuse parts of the PWN has not been fully studied.
+Because of its similarities (and important dierences) to the Crab nebula, a multi-wavelength
+study of G54.1+0.3 can therefore provide insight as to many of the above unresolved issues. Under-
+standing the nature of the diuse X-ray emission and its relationship to the diuse radio emission
+in G54.1+0.3 is crucial for understanding the energetic history of the PWN and its expansion into
+the surrounding interstellar medium. Archival VLA radio observations of G54.1+0.3 (Velusamy
+& Becker 1988) have relatively short integration times ( 45 minutes at each frequency in the C
+array conguration) and low resolution (5-1400). Therefore, we present new multi-frequency, multi-
+conguration VLA observations of G54.1+0.3. These observations, which have resolutions of 3-500,
+allow us to (1) characterize the radio emission on scales more comparable to the Chandra X-ray ob-
+servations, (2) search for large-scale diuse emission surrounding the central radio source, (3) study
+the radio spectral index variations across G54.1+0.3, (4) investigate the distribution and strength
+of polarized emission in G54.1+0.3, and (5) determine the Faraday rotation toward G54.1+0.3 and,
+ultimately, the intrinsic magnetic eld structure of the PWN. The observations and data reduction
+are presented inx2, the results inx3, and the discussion and interpretation of results follow in x4.
+2. Observations and Imaging
+VLA multifrequency observations of G54.1+0.3 were made during 2002-2004 using the B, C,
+and D array congurations at 1.4, 4.7 and 8.5 GHz. Table 1 summarizes the observational details.{ 4 {
+At all frequencies, J1331+305 and J1924+334 were used as 
ux and phase calibrators. Standard
+procedures for calibration, editing and imaging were carried out using the Astronomical Image
+Processing Software (AIPS) of the NRAO. Data from dierent arrays were combined to produce
+nal images at each frequency. Table 2 summarizes the image parameters. In order to measure
+
ux densities, the images were corrected for primary beam attenuation, but the images shown in
+the paper have not been corrected for this attenuation as the correction causes the rms noise level
+to increase near the edges of the image, particularly at 8.5 GHz.
+Polarization calibration was done for all three frequencies using the frequent observations of
+the phase calibrator J1924+334 in the B and C array observations. In the D array observations,
+polarization calibration was carried out only at 4.7 and 8.2 GHz; the 1.4 GHz observations did
+not have enough parallactic angle coverage for adequate polarization calibration. Stokes' Q and
+U images were made at 4.7 and 8.2 GHz and images of the polarized intensity (I p=p
+Q2+U2)
+were created. In order to study the Faraday rotation toward G54.1+0.3, additional Stokes' Q and
+U images were made at 4.585, 4.885, 8.085, and 8.465 GHz. For each of these IFs, images of the
+polarization angle (PA=1
+2Arctan(U
+Q)) and the error in polarization angle were created. Using the
+polarization angle and error images at the four observed frequencies, the rotation angle as a function
+of wavelength was tted to a 2law with the AIPS algorithm RM. This task produces images of
+the distribution of the rotation measure and also the distribution of the orientation of the magnetic
+eld intrinsic to the source by correcting the observed position angles for Faraday rotation.
+3. Results
+3.1. Radio Continuum Morphology
+3.1.1. Structure of G54.1+0.3
+Figures 1 and 2 show the detailed structure of the G54.1+0.3 radio source at 4.7 and 8.2 GHz.
+The radio continuum morphology at all frequencies (including 1.4 GHz) is nearly identical. The
+gures show that the source extends for approximately 2.502.00(4.53.6 pc at d=6.2 kpc), with
+diuse radio emission elongated in the NE-SW direction. The brightest emission is concentrated in
+the central 1-1.50area (1.8 to 2.7 pc), with intensities in this area of 1.5 mJy beam1at 4.7 GHz.
+The radio emission gets considerably weaker ( 0.1 mJy beam1at 4.7 GHz) moving outwards from
+the center of the nebula. Velusamy & Becker (1988) also note these same central features of the
+brightness distribution in their lower resolution observations.
+The high resolution ( 300) observations presented here reveal several new features in the mor-
+phology of G54.1+0.3. The brightness peaks near the center of G54.1+0.3, in several very pro-
+nounced ridges which appear to be tracing a ring or torus of 203000diameter. In addition, there
+is a central cavity in the distribution of radio emission near RA, DEC (J2000): 19 30 30, 18 52
+15. Slices of intensity across of the source indicate that the intensity in the central cavity is just{ 5 {
+a fraction of that contained in the surrounding ridges: 0.7 mJy/beam compared to 1.3 mJy/beam
+at 8.5 GHz, (this trend in brightness is also observed at 4.7 GHz and 1.4 GHz). Figure 3 shows
+a slice of intensity across the center of G54.1+0.3. Velusamy & Becker (1988) also report on this
+decrease of intensity at the very center of G54.1+0.3, but did not have the resolution or sensitivity
+to distinguish the ring.
+Figure 1 also reveals a number of lamentary structures in the NE and S portions of the
+nebula (labeled in Figure 1). In the NE part of G54.1+0.3, several elongated radio structures have
+sizes of30-4500(0.7-1 pc) near RA, DEC (J2000): 19 30 32, 18 52 15. Two of these features are
+mainly aligned with the NE-SW orientation of the radio nebula and suggestive of a 
ow. The third
+feature is oriented perpendicular to the NE-SW axis. In the southern part of the nebula, a number
+of loop-like structures are detected, ranging in size from 15-3000(0.45-0.9 pc). Such lamentary
+and loop-like features are observed in the diuse emission of several other PWNe, including both
+the Crab and 3c58, with similar physical sizes in all cases of 0.6 to 1 pc (Reynolds & Aller 1988;
+Velusamy 1985). Finally, G54.1+0.3 shows several \bays" to the NE and SW sides of the central
+ring or torus of radio emission which are similar to the optical bays in the Crab and 3C58 and the
+radio structure of PWN G21.5-0.9 (Furst et al. 1988; Velusamy 1985). These are discussed further
+inx4.6.
+3.1.2. Diuse Emission Surrounding G54.1+0.3
+Since G54.1+0.3 lies in the Galactic plane, it is surrounded by diuse large-scale radio emission.
+These larger structures are apparent in single-dish radio surveys of the Galactic plane (Caswell et al.
+1985; Velusamy et al. 1986) but with resolutions of 1-30, such images can not distinguish compact
+and shell-like features as in interferometric studies. Similar to the Crab, no shell surrounding
+G54.1+0.3 has ever been detected. Figure 4 shows the region surrounding G54.1+0.3 at 1.4 GHz;
+several shell-like radio sources and other large lamentary features are apparent in addition to the
+PWN which lies at the center of the image. In particular, a new radio shell-like structure has
+been detected (labeled in red in the gure) that appears to be centered around G54.1+0.3, with a
+diameter of80(14.4 pc at 6.2 kpc). The radio shell is brightest on three sides of G54.1+0.3, with
+the faintest emission located in the SW portion of the shell. If the radio emission is nonthermal then
+this shell may be interpreted as the supernova remnant from the the G54.1+0.3 supernova explosion.
+However, determining the spectrum of the radio emission using the current interferometric data
+is dicult. At 4.7 GHz (image not shown; a larger eld of view than Figures 1 and 2), extended
+emission is present to the NW and SE at approximately 80in radius from G54.1+0.3. At 1.4 GHz,
+there is considerably more diuse emission in the eld of view than at 4.7 GHz, presumably because
+of the better sensitivity to extended features. In fact, the largest angular size detectable with the
+VLA at 4.7 GHz is only 30000(50), so emission associated with this large ( 80) shell will not be
+observed with the VLA at this frequency. The brightest of the extended emission has an intensity
+of 100-200 Jy beam1, which corresponds to a signal-to-noise of 10 or less at best, and is not{ 6 {
+well represented by the (u,v) coverage of the 4.7 GHz observation. Therefore we are unable to
+provide any reliable measure of the spectral index of the shell-like source surrounding G54.1+0.3.
+This shell is visible partly in the VLA Galactic Plane Survey image (D-array survey, 10resolution)
+of Leahy et al. (2008), but it is on a much smaller scale than their large image (eld of view 1
+). The nature of this shell is further discussed in x4.6.
+3.2. Spectral Index of G54.1+0.3
+Since we have detected G54.1+0.3 at three frequencies (8.2, 4.7 and 1.4 GHz), it is possible
+to derive the spectral index of the radio emission. The radio emission can be integrated over
+the entire source (a region of 201.50). In all cases, the integrated 
ux density measurements
+have come from identical regions of G54.1+0.3. However, G54.1+0.3 lies in a crowded part of the
+Galactic plane (see above), many large scale extended features are present in 1.4 GHz image, which
+has a much larger eld of view (300) compared with the 4.7 or 8.2 GHz images (FOV 5-90).
+The background 
ux levels in the 1.4 GHz image are much higher and variable over the image
+size than at other frequencies, and these variations contaminate the 
ux density measurements at
+1.4 GHz. Therefore, we have corrected the 1.4 GHz measurement to yield S 1:4=433.030.0 mJy,
+S4:7=327.025 mJy, and S 8:5=252.020.0 mJy. The average value of the integrated spectral index
+for G54.1+0.3 is =0.30.1. The spectral index between 1.4 GHz and 4.7 GHz has a value of
+=0.20.1 and between 4.7 and 8.5 GHz, =0.50.2. In addition, Figure 5 shows the spectrum
+of G54.1+0.3 based on the integrated 
ux density values and the best t to this plot gives a spectral
+index value of =0.28.
+In order to search for variations of the spectral index across the source, the spectral index can
+be calculated pixel by pixel across G54.1+0.3 using the AIPS task COMB (opcode=SPIX). Figure
+6 shows the distribution of spectral index between 1.4 and 4.7 GHz in greyscale and overlaid by
+contours of 4.7 GHz continuum emission. This image was created by using data from matched arrays
+so that the spatial frequency sampling (i.e., (u,v) coverage) is consistent between both frequencies.
+In practice, this means that the 1.4 GHz image is constructed from B-array data only, the 4.7 GHz
+from C-array data only and the 8.2 GHz from D-array data only. Further, the spectral index images
+were obtained by convolving both images to the same resolution (500) and determining the spectral
+index value at every pixel. The spectral index between 4.7 and 1.4 GHz in G54.1+0.3 does not
+show any variation across the nebula, with a fairly constant value of =0.2. The spectral index
+between 4.7 and 8.2 GHz also does not show any signicant variation across G54.1+0.3, but has a
+slightly steeper average value, =0.3.{ 7 {
+3.3. Polarization
+There is strong linear polarization detected in G54.1+0.3 at both 4.7 and 8.5 GHz. The
+polarization at 1.4 GHz was signicantly weaker ( 10% fractional polarization) presumably because
+of bandwidth depolarization across the 50 MHz bandwidth at 1.4 GHz. Figure 7 shows polarized
+intensity at both 4.7 and 8.5 GHz. These gures illustrate that there is signicant polarized
+intensity arising from G54.1+0.3 that closely follows the total intensity structure. The polarization
+is concentrated along the two central radio ridges, and the polarized intensity peak is oset by
+10-1500from the center of the PWN. Diuse polarized intensity extends over much of the nebula.
+Figure 7 (top) shows that several of the radio loop-like features along the NW-SE extensions of
+the nebula are polarized. There are two striking linear features that extend for up to 3000(0.9 pc)
+in extent; they are labeled in Figure 7 (top). The fractional polarization was sampled across the
+nebula and varies between 20 50% at 8.2 and 4.7 GHz. Typical values of the fractional polarization
+are 27% at 8.2 GHz, 23% at 4.7 GHz. The highest values of fractional polarization are labeled in
+Figure 7 (bottom) and have values close to 40% at 8.5 GHz near RA, DEC (J2000): 19 30 30, 18
+52 30.0 and 50% at 8.5 GHz near RA, DEC (J2000): 19 30 31, 18 52 50.0.
+3.4. Rotation Measure and Intrinsic Magnetic Field Orientation
+Figure 8 shows the distribution of rotation measure (RM) in rad m2toward G54.1+0.3, based
+on tting the polarization angle as a function of wavelength squared (see x2) for interstellar Faraday
+rotation. Across much of the G54.1+0.3 nebula, the values for RM are in the range of 400-650
+rad m2. In two regions (in the NE near RA, DEC (J2000): 19 30 33, 18 52 45, and in the SW
+near RA, DEC (J2000): 19 30 28, 18 52 15) the RM values appear to be lower by several hundred
+rad m2. Figure 9 shows a sample of the ts of polarization angle as a function of wavelength
+squared, where each panel represents the data averaged over a few pixels from two small regions of
+the source. The errors in the RM ts are on the order of 25-75 rad m2.
+The vectors shown in Figure 10 represent the orientation of the magnetic eld intrinsic to
+G54.1+0.3, obtained by correcting the observed polarization angles for the calculated RM. The mag-
+netic eld is very well ordered over the majority of the region which is polarized in G54.1+0.3 (see
+Figure 9, left). In particular, the magnetic eld appears to follow closely the curvature of the
+central radio \ring" region. At the edges of the nebula, the magnetic eld appears to be oriented
+along the linear extensions which are present in both total and polarized intensity and have a ra-
+dial orientation. In the SE and SW portions of G54.1+0.3, the eld becomes less-ordered as the
+polarized intensity fades.{ 8 {
+4. Discussion
+4.1. Comparison of Radio Emission and X-Ray Emission
+Figure 11 shows a comparison of the 4.7 GHz radio emission in contours and the Chandra X-ray
+emission in colorscale (from Lu et al. 2002). The left panel has a colorscale stretch to emphasize
+the extended X-ray emission, and the right panel is set to emphasize the concentrated central X-
+ray emission. This gure, especially the left panel, illustrates that the distribution of large-scale
+and diuse X-ray emission in G54.1+0.3 is strikingly similar to the extent of the radio emission.
+Figure 12 shows radial proles of brightness taken along the NE (left) and SW (right) portions of
+G54.1+0.3. The dotted line illustrates the \edge" of the PWN in radio emission, corresponding
+approximately to the lowest radio contour level in Figure 11 (right panel). Although the X-ray
+emission drops more signicantly than the radio emission, there is still diuse and low-level X-ray
+emission arising from the same volume as the radio emission out to the lowest radio contour level.
+However, in the center of the PWN, the radio and X-ray emission are not as well-correlated.
+The X-ray emission peaks at RA, DEC (J2000)= 19 30 30.1, 18 52 14.10, whereas the maximum in
+radio emission occurs at RA, DEC (J2000)= 19 30 30.5, 18 52 11.7. These positions are 600apart.
+Figure 11 (top) shows that the peak in X-ray emission lies in a slight depression of radio emission,
+but with ridges of radio emission on either side. In addition, the projected orientation of the
+extended radio emission diers from the orientation of jet-like X-ray features (e.g., Figure 4 from
+Lu et al. 2002). These compact X-ray features in the center of G54.1+0.3 are oriented at angles of
+45with respect to the extended radio nebula (which runs along a NE to SW line). Apparently,
+the X-ray and radio emissions trace two rather distinct components of relativistic particles in the
+inner region of the PWN. While both emissions are apparently due to synchrotron radiation, their
+corresponding particle energies are very dierent ( 100 TeV vs.1 GeV) and clearly represent
+dierent forms of pulsar injections and/or accelerations in the region.
+Several PWN have been studied recently at high-resolution in the X-ray and radio: G0.9+0.1
+(Dubner et al. 2008), 3C58 (Slane et al. 2004), G292.0+1.8 (Gaensler & Wallace 2003), and G21.5-
+0.9 (Furst et al. 1988; Bietenholz & Bartel 2008). G54.1+0.3 bears resemblance to several of these
+PWN, but is most similar to 3C58 in the following ways: (1) the substructure in G54.1+0.3 is la-
+mentary and magnetized and (2) the extents of the diuse radio and X-ray emission are comparable.
+Both 3C58 and G54.1+0.3 show evidence for lamentary and loop-like structures in X-ray and radio
+emission on size scales of 0.7-1 pc (e.g., in 3C 58 these structures are on size scales of 10-1500for a
+distance of3 kpc and those scale directly to loop-like features of size 20-3000in G54.1+0.3 at a
+distance of6 kpc). Slane et al. (2004) propose that the lamentary loops in 3C58 are magnetic
+loops torn from the toroidal eld (associated with the pulsar) by kink instabilities. Figures 7 and
+10 show that in G54.1+0.3 these lamentary loops are strongly polarized and have magnetic eld
+orientations that align along their lengths. In fact, the curvature of magnetic structures that extend
+for as much as 15 3000are one of the more remarkable features of G54.1+0.3.{ 9 {
+The overall diuse structure of the radio and X-ray emission (that extends along the NE to
+SW line) in G54.1+0.3 closely resembles that of 3C58, in which the X-ray and radio emission ll
+essentially the same volume. Slane et al. (2004) report that the orientation of the elongation
+could be related to the toroidal magnetic eld of the pulsar and that the elongation should line up
+with the orientation of the projected pulsar spin axis, as is roughly the case in 3C58. However,
+as described earlier there is a discontinuity between the orientation of the X-ray jet out
ows and
+the extended radio emission in G54.1+0.3. Assuming that the pulsar spin axis is aligned with the
+X-ray jet-like feature, Ng and Romani (2004) estimate that the pulsar spin axis to be 90with
+respect to N, or45with respect to the diuse radio emission.
+4.2. Source Energetics and Nebular Magnetic Field Strength
+Figure 13 shows the spectrum of G54.1+0.3 over the range of 108to1018Hz. The 
ux
+in the diuse X-ray emission is 5.2 1012ergs cm2s1, which corresponds to 1.73 Jy. The
+radio spectrum (as discussed in x3.2 and shown in Figure 5) is relatively 
at, with R=0.28,
+corresponding to a photon index ()= 1.28. The diuse X-ray nebula (not including the pulsar
+emission) has = 1.960.08, orX0.96. A spectrum was made and t for this emission,
+dened by the \outer nebula" region from Lu et al. (2002). There is a trend in the diuse X-ray
+emission, though, to soften outwards in the nebula: the inner X-ray ring has = 1.6, and the
+photon index gradually decreases outwards in the nebula to nearly 2. This indicates that there is
+cooling in the X-ray particles as they extend to ll a similar volume as the radio particles. However,
+the X-ray index of uncooled particles, = 1.6, is signicantly steeper than the radio index. This
+shows that the responsible particles cannot be described by a single power law energy distribution.
+According to Gaensler et al. (2002), the X-ray spectrum in PWNe are typically steeper than
+at radio wavelengths in part due to synchrotron losses which generate a break in the spectrum at
+the \break frequency". In this case, one can use Figure 13 to derive the break frequency. Figure
+13 illustrates that the break frequency is the position where the two solid curves meet and occurs
+nearb=50 GHz for the X-ray spectrum of 0.96. However, the errors on the X-ray spectral t
+give a large range of possible break frequencies, from b=3 to 200 GHz. A variety of PWNe have
+been shown to have a steep spectral break at relatively low frequencies ( break<100 GHz; Woltjer
+et al. 1997); examples include 3C58 (Slane et al. 2004), G292.0+1.8 (Gaensler & Wallace 2003),
+and G21.5-0.9 (Bock et al. 2001) and G54.1+0.3 may be consistent with this class of PWNe. Here,
+we assume a break frequency from our radio and X-ray observations b=50 GHz.
+With a radio 
ux density of 327 mJy at 4.7 GHz, the =0.28 spectral index and a distance
+of 6.2 kpc, the radio luminosity over the range of 107Hz to 1011Hz is 8.71032erg s1. Over a
+similar range in frequency, the Crab has a radio luminosity of 1.8 1035erg s1. The value for
+radio luminosity of G54.1+0.3 is several orders of magnitude below what has been found for many
+other PWNe, such as the Crab. What may be a more relevant quantity to compare is the ratio of
+radio luminosity to the spin-down power of the pulsar. For G54.1+0.3 this ratio (L R/_E) is 8.7 x{ 10 {
+1032/3.01036= 0.0003. For the Crab, we calculcate almost exactly the same number: L R/_E=
+1.8 x 1035/4.61038= 0.0004.
+The magnetic eld strength in the PWN may play a role in regulating its expansion and in
+the energetics of the out
owing particles. If the break frequency is known or estimated, and the
+age of the PWN is known, then the nebular magnetic eld can be calculated using the expression
+in Frail et al. (1996). The characteristic age (P/2 _P) of the pulsar associated with G54.1+0.3 was
+determined to be 2900 years (Camilo et al. 2002). Using this age and the break frequency of 50
+GHz, we estimate a magnetic eld strength of 1250G. However, since the break frequency is
+not very well known, this estimate is more uncertain.
+Alternatively, we can use the radio luminosity above and assume that there is equipartition
+between the particles and the magnetic eld in the nebula. If we approximate the nebula as a
+sphere with diameter of 3.1 pc, then we derive an equipartition magnetic eld strength of 38 G.
+Another way to estimate the nebular magnetic eld is by using the lifetime of the X-ray emitting
+particles. The particle 
ow velocity just down stream from the termination shock is derived by Lu
+et al. (2002) to be about v=0.4 c. Assuming a simple situation in which the 
ow is radial, we
+estimate a lifetime of 12.5 years (e.g., radius of PWN/the 
ow velocity). The lower limit on the
+mean 
ow velocity is unknown, but if we assume v=0.1 c, the lifetime of the particles will be 50
+years. Then, assuming that the synchrotron lifetime at X-ray energies (i.e., 1.4 keV) depends on
+the magnetic eld strength, we can derive a magnetic eld in the nebula between 80-200 G using
+the above velocities. Here, we understand that the magnetic eld will be an upper limit as the
+particles may diuse out to the edge of the nebula instead.
+However, this magnetic eld substantially higher than the equipartition magnetic eld 
+suggests that G54.1+0.3 is lled by a magnetically-dominated plasma, consistent with the presence
+of the polarization (and hence the magnetic eld) organized on large scales of the nebula. Therefore,
+the magnetic eld strength in G54.1+0.3 is probably stronger than the equipartition magnetic eld,
+although the original wind from the pulsar is particle dominated (Lu et al. 2002). What might be
+the origin of this magnetic eld? According to the convetional model for the Crab nebula (e.g.,
+Kennel and Coroniti 1984), the pulsar wind is particle-dominated. But when it is terminated, the
+magnetic eld is expected to be amplied in the down stream. The magnetic eld can eventually
+become higher than the equipartition value. A detailed modeling, which accounts for the detailed
+magnetic eld structure as reported here (see below), could potentially shed new light into the
+evolution and dynamics of the PWN.
+4.3. Rotation Measure Distribution Toward G54.1+0.3
+The fairly constant RM distribution across G54.1+0.3 indicates that we are detecting the
+Faraday rotation by the interstellar medium in the direction of G54.1+0.3 rather than an internal
+rotation process. There is a slight variation, however, of the RM, with larger values (near 500-600{ 11 {
+rad m2) being concentrated toward the central 10of the source and lower values (200-400 rad
+m2) concentrated outwards in the nebula. In particular, there are gradients in the RM in the
+NE and SW extents of the nebula where it is thought that the PWN may be interacting with a
+component of the interstellar medium (see below).
+4.4. Nature of the Magnetic Field Conguration in G54.1+0.3
+Figure 7 shows that strong linear polarization is evident across much of G54.1+0.3 at both
+8.5 and 4.7 GHz and that little change in polarized intensity morpholoy or fractional polarization
+occurs between frequencies. Because the polarization extends over much of the PWN, it is possible
+to determine the intrinsic magnetic eld structure over this region. Studies that map out the
+intrinsic magnetic eld have only been carried out for a handful of PWN sources (e.g., G106.6+2.9
+(Kothes et al. 2006); Vela (Dotson et al. 2003); G21.5-0.9 (Furst et al. 1988), and G292.0+1.8
+(Gaensler & Wallace 2003)). A large sample is needed to fully understand the magnetic eld
+structure in these objects.
+Kothes et al. (2006) present an interpretation of radio polarization measurements for G106.6+2.9,
+but generalized for many PWNe. They suggest that a toroidal eld will appear to be tangential
+to the spin axis of the pulsar, whereas the radial eld will appear along the elongation axis of the
+PWN. In many cases, the geometry will not be straightforward because of projection eects and
+the orientation of the observer. However, if the spin axis of the pulsar is known, then it is possible
+to imagine how the transition between the magnetic eld in the inner region of the pulsar may be
+projected into something observable in the outer parts of the PWN.
+In G54.1+0.3, the spin axis of the pulsar is assumed to be aligned roughly with the X-ray jets
+(i.e., Ng & Romani 2004), as in the Crab nebula and 3C58. Therefore, a toroidal eld component
+would appear to be oriented perpendicular to this direction, and a radial magnetic eld would align
+along this axis. Figure 14 shows the orientation of the intrinsic magnetic eld vectors (in the plane
+of the sky) overlaid on greyscale representing the X-ray emission in G54.1+0.3. It is apparent that
+the majority of the magnetic eld vectors are aligned in the central part of the nebula (where the
+X-ray emission is strongest) essentially tangential to the spin axis, but in the outer parts of the
+nebula, the eld is radial and extends for the most part along the elongation. The exception is in
+the linear part of the loop-like feature in the NE part of the PWN. There appears to be a large
+magnetized loop emerging from this part of the PWN, perhaps an expanding part of the inner eld
+structure. The diculty with this is that the elongation of the diuse nebula and the spin-axis
+from the X-ray are tilted with respect to each other, so this distribution of magnetic structure is
+complex. G54.1+0.3 appears to exhibit a complex radial magnetic eld pattern, according to the
+classication of Kothes et al. (2006) by showing both radial and toroidal eld components.
+Finally, the role of the magnetic eld in G54.1+0.3 may be similar to that in 3C 58. As
+described in the previous section, Slane et al. (2004) conclude that in 3C 58 the possible disruption{ 12 {
+of a toroidal magnetic eld conguration as the pulsar wind expands is responsible for producing
+the radio and X-ray "loops" and lamentary radio polarization structures. This may be the same
+mechanism at work in G54.1+0.3 where a toroidal magnetic eld is present and may be disrupted
+in the outer parts of the nebula.
+4.5. New Radio Shell: Large-scale Emission Around G54.1+0.3
+A clue to the nature of the new radio shell surrounding G54.1+0.3 can be gained by comparing
+the radio emission to the Spitzer Space Telescope 24m image made as part of the MIPS Legacy
+Project (MIPSGAL; Carey et al. 2009). The 24 m emission traces stocastically-heated, small
+dust grains which trace massive stellar activity in the Galaxy. The radio continuum and 24 m
+emission turns out to align very closely as evidenced in Figure 15. In addition to the large radio
+shell, Figure 15 also illustrates that there is a signicant source of 24 m emission associated with
+G54.1+0.3 itself that has a smaller, \loop-like" geometry (8000or 2.4 pc across). This correspon-
+dance has been noted before by Koo et al. (2008) and Slane et al. (2008). Figure 16 shows archival
+Spitzer data (ID:3647) taken at 24 m pointed at G54.1+0.3 with radio continuum contours at 4.7
+GHz overlaid. However, as pointed out in x3.1.2, the nature of the radio continuum emission is
+unclear and could be either free-free brehmstralung from ionized gas or nonthermal synchrotron
+emission.
+Koo et al. (2008) indicate there may be some young stellar object (YSOs) sources and early
+B-stars embedded in the bright mid-infrared (mid-IR) loop that could be responsible for heating
+this large-scale radio shell (i.e., ionized gas). However, these ionizing sources may not be strong
+enough to account for the extensive radio emission in the radio shell if the total content of YSOs
+is on the order of 100M as suggested by Koo et al. (2008). In this case, the correspondance
+might represent the coincidence of a mid-IR shell with nonthermal radio emission. More sensitive
+multi-frequency radio observations which include the single dish 
ux are necessary to measure the
+spectral index accurately.
+4.6. Interplay of G54.1+0.3 with a Mid-infrared Cloud
+As mentioned above, a more detailed view of the correlation between G54.1+0.3 and the mid-IR
+loop is shown in Figure 16 and reveals several morphological details: the bright radio ridges in the
+center of the PWN exactly ll a depression in the mid-IR loop, and the contours of radio emission,
+especially at the SW edge of the PWN very closely track the morphology of the mid-IR emission.
+In addition, an overlay between the X-ray emission in G54.1+0.3 and the 24 m emission made
+by Slane et al. (2008) also reveals that the brightest X-ray emission concentrated in the central
+region of G54.1+0.3 is closely correlated with the edge of the mid-IR loop, indicating they may be
+interacting (see their Figure 3). Koo et al. (2008) have suggested that a burst of star-formation{ 13 {
+evidenced by the presence of various near-to mid-IR sources (proposed to be massive YSOs), the
+formation of which may be triggered by the interplay of the stellar wind from the progenitor star
+of PWN G54.1+0.3. The large 24 m luminosity and the presence of massive young stellar objects
+(Koo et al. 2008) indicate that the cloud is a pre-existing interstellar cloud, instead of part of the
+SN ejecta. It would also be dicult to imagine how the ejecta could remain so dense and cool in
+such a young SNR.
+Several other features observed in G54.1+0.3 can be naturally explained with an interaction
+scenario: (1) the anti-correlation between peaks of the mid-IR emission and the distribution of RM
+(or similarly, polarized emission) shown in Figure 18. The polarization disappears abruptly at the
+edges of G54.1+0.3, before the radio continuum edge of the PWN, right at the positions where the
+thermal IR emission from the cloud appears to peak. In fact, closer inspection shows this change
+quite clearly: Figure 17 presents slices in fractional polarization at 4.7 GHz in (left) the SE part
+of G54.1+0.3, and (right) the SW part of G54.1+0.3 (see caption for exact locations of slices). In
+both cases, abrupt decreases in fractional polarization are detected before the edge of the radio
+continuum emission has been reached and coinciding with the peak of the mid-IR emission. In
+the case of the slice along the W side of G54.1+0.3, the fractional polarization falls o and then
+increases brie
y again near the peak of the mid-IR emission. The sudden decreases in polarization
+can naturally be explained if the pulsar wind 
ow (hence, the magnetic eld) changes direction
+because of the connement of the mid-IR shell. (2) The change in direction of the pulsar wind 
ow
+will also have the eect of changing the orientation of the intrinsic magnetic eld. Figures 10 and 14
+show that the magnetic eld becomes much less regular in the SW part of the PWN compared to the
+well-ordered and primarily radially-oriented elds in the N of the PWN. (3) Overall, the expansion
+of the PWN may be conned by the cloud which can explain the dierences in the radio and X-ray
+emission. The brightest X-ray features are concentrated in the center of the PWN (see Figure 11
+(top panel) and are dominated by the jets and central X-ray torus (e.g., Lu et al. 2002)). However,
+the diuse X-ray emission and radio emission are well-aligned and their morphology is likely to be
+shaped by the ram-pressure connement of thermalized particles. (3) The radio intensity contours
+(see Figure 16) appear to be compressed SW to the pulsar, relatively to the NE, which again can be
+a natural consequence of the ram-pressure connement provided by the IR shell to the SW. There
+is no clear detection of the mid-IR cloud in existing13CO (1-0) data. One possibility is that the
+cloud could have been heated substantially and may be bright in higher order transitions. Sensitive
+observations of various molecular tracers are needed to further the study of the interplay between
+the G54.1+0.3 and the cloud.
+The apparent connement of G54.1+0.3 by the surrounding mid-IR cloud is somewhat similar
+to that of another PWN N157B in the Large Magellanic Cloud (Wang et al. 2001). In the case of
+PWN N157B, however, the connement is probably due to the re
ection of the supernova ejecta
+from a large-scale dense cloud, in contrast to the direct interaction of G54.1+0.3 with the mid-IR
+cloud. In both cases, the X-ray emission is largely extended, comparable to the radio emission
+in size, which is apparently caused by the bulk motion of pulsar wind materials due the direction{ 14 {
+asymmetry in the ram-pressure connement. The consideration of such bulk motion is important
+in understanding the evolution of the relativistic particles in the pulsar wind of G54.1+0.3 and the
+resultant X-ray and radio emission. Figure 19 shows a sketch of a possible interaction geometry
+between G54.1+0.3 and an interstellar cloud. The morphological \bays" mentioned earlier in the
+text and illustrated in Figure 1 are therefore likely to be more complicated that simple analogs to
+the bays in the Crab or in G21.5-0.9. The nature of the bays is far from clear and may be related to
+how the relativistic particles are injected from the pulsar and/or further accelerated in the nebula.
+The issue may also be complicated by the apparent connement of the nebula by the mid-IR cloud.
+In summary, this may be the rst PWN where we are directly detecting its interplay with an
+interstellar cloud that has survived the impact of a supernova explosion. This interplay seems to
+give a natural explanation of the morphology and polarization change of the radio emission as well
+as its similarity and dissimilarity to the X-ray intensity distribution of the G54.1+0.3.
+5. Conclusions
+Here we summarize our radio observations of G54.1+0.3 and comparision with multi-wavelength
+observations of this source and draw the following conclusions:
+(1) The higher resolution radio observations presented here reveal that G54.1+0.3 has a com-
+plex structure which closely resembles other PWNe (e.g., in particular, the Crab, 3C 58, and
+G.21.5-0.9). G54.1+0.3 shows wispy and lamentary radio structures in the outer parts of the
+nebula that are loop-like and are polarized. On large scales, the radio emission in G54.1+0.3 is ex-
+tended along the NE SW direction. The radio emission in the center of the nebula is concentrated
+into a ring or torus-like object with a larger radius that the X-ray torus.
+(2) Comparisons between G54.1+0.3 in the radio and X-ray are striking as there is much simi-
+larity at these two dierent frqeuencies. Both the radio and X-ray emission have comparable diuse
+extents with the presence of lamentary and loop-like structures as mentioned above. However,
+the inner part of G54.1+0.3 is dominated by bright X-ray emission associated with the jet-like
+features. These structures are oset from the central radio features, which peak further outwards
+in the nebula than the X-ray emission. The radio emission also tends to dominate the nebula on
+larger scales.
+(3) No spectral index gradient is detected in the radio emission across the PWN, whereas
+the X-ray emission softens outward in the nebula, consistent with the synchrotron cooling of the
+X-ray-emitting particles in a magentic eld of about 100 G.
+(4) The strong radio polarization present in G54.1+0.3 allows us to image in detail the in-
+trinsic magnetic eld across the PWN. Clear radial and toroidal components of the magnetic eld
+are detected in G54.1+0.3, which is an intermediate morphology for the eld, according to the
+classication scheme of Kothes et al. (2006).{ 15 {
+(5) Strong mid-IR emission from a bright loop is closely correlated with the southern edge of
+G54.1+0.3. In particular, the distribution radio and X-ray emission compared with the mid-IR
+emission suggest that the PWN may be interacting with this interstellar cloud. Figure 19 shows a
+sketch of such a possible interaction.
+(6) An interaction between the PWN G54.1+0.3 and the interstellar cloud (traced in mid-IR
+emission) may naturally explain the dierences in the orientation of X-ray and radio emission in
+G54.1+0.3, the change of magnetic eld orientation and the RM distribution and the apparent
+compression of the radio emission along the southern part of G54.1+0.3. This may be the rst
+PWN where we are directly detecting its interplay with an interstellar cloud that has survived the
+impact of the supernova explosion associated with the pulsar's progenitor.
+The authors thank an anonymous referee for helpful comments. We also thank Dr. Wenwu
+Tian for helpful discussions regarding the assocation of G54.1+0.3 with surrounding interstellar
+media. F.J. Lu is supported by the Nature Science Foundation of China through grants 10533020
+and by National Basic Research Program of China (973 Program 2009CB824800).
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+This preprint was prepared with the AAS L ATEX macros v5.2.{ 17 {
+Fig. 1.| VLA 4.7 GHz continuum image of G54.1+0.3 shown in greyscale representing 0 to 1.25
+mJy beam1(see axis on top of gure for greyscale range in mJy beam1). The resolution of this
+image is 3.28003.2100, PA=41.6and the rms noise is 15 Jy beam1.{ 18 {
+Fig. 2.| VLA 8.2 GHz continuum image of G54.1+0.3 shown in greyscale representing 0 to 1.25
+mJy beam1(see axis on top of gure for greyscale range in mJy beam1) and contour levels
+representing -5, 5, 10, 15, 15, 20, 25, 30, 40, 45, 55, 65, 70, 75, 85, 95, 100 and 104 times the rms
+level of 10Jy beam1. The resolution of this image is 3.22003.0100, PA=20.0.{ 19 {
+Fig. 3.| VLA 8.5 GHz radio continuum intensity as function of position (in arcseconds) across
+G54.1+0.3. This slice was taken through the broadest part of the radio \ring/torus" on a line that
+is at a 45angle (E of N). The beginning point is RA, DEC (J2000): 19 30 34, 18 51 45, and the
+end point is RA, DEC (J2000): 19 30 26, 18 53 00. This gure illustrates that the central radio
+emission is concentrated in a ring-like distribution.{ 20 {
+Fig. 4.| VLA 1.4 GHz continuum image of the region surrounding the compact source G54.1+0.3,
+with a resolution of 6.82006.6000, PA=5.5. The new radio shell is marked in the gure.{ 21 {
+Fig. 5.| Radio spectrum of G54.1+0.3, based on 
ux densities measured at 1.4, 4.7 and 8.2 GHz
+and shown with errors. The dotted lines represent the best t to the spectral index.{ 22 {
+Fig. 6.| VLA 4.7 GHz continuum continuum contours of G54.1+0.3 shown overlaid on greyscale
+representing the 1.4 to 4.7 GHz spectral index, with greyscale that ranges from =+0.5 to1.0.{ 23 {
+Fig. 7.| Polarized intensity images shown in greyscale, which ranges from 0 to 200 Jy beam1,
+and have resoltions of 3.5003.500at(top) : 4.9 GHz and (bottom): 8.2 GHz overlaid with
+contours of total intensity at levels of 0.25, 0.5, 0.75, 1.0, and 1.25 mJy beam1.{ 24 {
+Fig. 8.| Colorscale representing the distribution of rotation measure (RM) toward G54.1+0.3 in
+units of 100-500 rad m2, with a spatial resolution of 3.5003.500.{ 25 {
+Fig. 9.| Polarization rotation angle versus wavelength squared for two regions in G54.1+0.3.{ 26 {
+Fig. 10.| Vectors of the intrinsic magnetic eld orientation in G54.1+0.3 overlaid on greyscale
+representing 8.5 GHz emission between 0 and 1.0 mJy beam1(see top axis).{ 27 {
+Fig. 11.| Comparison of radio emission (4.7 GHz data from this study; see Figure 1) and X-ray
+emission from the Chandra X-ray observatory (Lu et al. 2002). The top image has a color stretch
+selected to show the centrally-concentrated X-ray emission for comparison with the radio emission
+in the nebula, and the bottom image has a color stretch set to highlight the diuse X-ray emission
+which lls the same volume as the radio nebula.{ 28 {
+Fig. 12.| Comparison of the radio and X-ray extent of G54.1+0.3. The upper panels are radio
+frequencies (4.7 and 8.5 GHz) and the bottom panels are X-ray emission. The brightness proles
+show a radial slice from the center of the nebula outwards at two dierent locations. The dashed
+line marks the edge of the radio nebula.{ 29 {
+Fig. 13.| Spectrum of the G54.1+0.3 PWN over the radio to X-ray regime, with the radio spectrum
+(R=0.28) shown as a solid line, and the X-ray spectrum ( X=0.96) shown with a solid line.
+The position where these two curves meet is known as the break frequency and occurs near b=50
+GHz. However, the errors on the X-ray spectral t give a large range of possible break frequencies,
+fromb=3 to 200 GHz.{ 30 {
+Fig. 14.| Vectors of the intrinsic magnetic eld orientation in G54.1+0.3 from these data (pre-
+sented in Figure 10) overlaid on greyscale showing the 0.2-10 keV X-ray emission from Lu et al.
+(2002).{ 31 {
+Fig. 15.| Comparison of the large-scale structure surrounding G54.1+0.3 showing 1.4 GHz radio
+emisson (white contours) and the Spitzer 24m data from the MIPSGAL survey (Carey et al.
+2009). G54.1+0.3 is the compact object in the middle of the image in green contours and the
+colorscale shows the new shell of radio emission in addition to the local galactic background. This
+image is shown in Galactic coordinates.{ 32 {
+Fig. 16.| Zoomed-in view of the comparison of radio continuum emisson (contours of 4.7 GHz
+emission) and colorscale showing Spitzer 24m data from the archive (program=3647).{ 33 {
+Fig. 17.| Slices in fractional polarization at 4.7 GHz (polarized intensity/total intensity) across
+G54.1+0.3. (left) Slice in the SE part of the PWN beginning near the center of the source near
+RA, DEC (J2000): 18 52 15, 19 30 31, and extending essentially south to RA, DEC (J2000): 18 52
+45, 19 30 30. (right) Slice originating at the center of the PWN near RA, DEC (J2000): 18 52 30,
+19 30 30 and extending essentially eastward to the edge of the nebula at RA, DEC (J2000): 18 52
+15, 19 30 28. In this slice, it is clear that the polarization falls o, then increases again before the
+edge of the PWN.{ 34 {
+Fig. 18.| Distribution of rotation measure in rad m2(as shown in Figure 8) with contours of
+24m emission from the Spitzer archive (program=3647) overlaid. What is notable here is the
+anti-correlation between the rotation measure and the intensity of 24 m emission.{ 35 {
+Fig. 19.| Sketch of a possible scenario for the interaction of G54.1+0.3 with an interstellar cloud.
\ No newline at end of file
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+arXiv:1001.0174v3  [math.GT]  1 Feb 2011AN INTRINSIC APPROACH TO INVARIANTS OF
+FRAMED LINKS IN 3-MANIFOLDS
+EFSTRATIA KALFAGIANNI
+Abstract. We study framed links in irreducible 3-manifolds that are
+Z-homology 3-spheres or atoroidal Q-homology 3-spheres. We calculate
+the dual of the Kauffman skein module over the ring of two varia ble
+power series with complex coefficients. For links in S3we give a new
+construction of the classical Kauffman polynomial.
+Keywords. characteristic submanifold, framed links, finite type inva ri-
+ants, Kauffman skein module, loop space, Seifert fibered 3-ma nifolds,
+toroidal decompositions.
+Mathematics Subject Classication (2010). 57N10, 57M2, 57R42,
+57R56.
+1.Introduction
+The Kauffman polynomial is a 2-variable Laurent polynomial in variant
+for links in S3[17] that has interesting applications and connections wit h
+contact geometry. The degree in one of the variables of the Ka uffman poly-
+nomial provides an upper bound for the Thurston-Bennequin n orm of Leg-
+endrian links [8, 26]. The inequality is known to be sharp for several classes
+of links (e.g. alternating links) and the proof of this sharp ness has led to
+deeper connections between knot polynomials and contact ge ometry [22].
+In this paper we study framed links in oriented, irreducible 3-manifolds
+that are Z-homology 3-spheres or atoroidal Q-homology 3-spheres. We give
+conditions under which an invariant that is defined on framed singular links
+with one double point gives rise to an invariant of framed lin ks (Theorem
+2.2). This allows us to construct formal power series framed link invariants
+obeying the Kauffman polynomial skein relations. The coefficie nts of these
+series are finite type framed link invariants and are perturb ative versions
+of the Reshetikhin-Turaev, Witten SO(n)-invariants [25, 29] in the sense
+of Le-Murakami-Ohtsuki [20]. Using weight systems corresp onding to ap-
+propriate representations of the Lie algebras so(n) and the naturallity of
+the LMO invariant, one obtains a Kauffman type power series inv ariant for
+framed links in all Q-homology 3-spheres. Our approach in this paper is
+quite different from this line and allows us to solve the subtle r problem of
+Supported in part by NSF grant DMS-0805942.
+12 E. KALFAGIANNI
+constructingpowerseries invariants withgiven valueson a set of initial links.
+Our approach here, that exhibits the interplay between skei n framed link
+theory and the topology of 3-manifolds, is inspired by the st udy of Vassiliev
+invariants (a.k.a. finite type invariants) [27] using 3-dim ensional topology
+techniques [12]. The precise relation of the power series co nstructed here to
+the one obtained via the LMO invariant is not clear to us at thi s point.
+Definition 1.1. LetMbe an oriented Q-homology 3-sphere. A framedm-
+component link is a collection of munordered (unoriented) circles smoothly
+and disjointly embedded in Mand such that each component is equipped
+with a continuous unit normal vector field. Two framed links a re equivalent
+if they are isotopic by an ambient isotopy that preserves the homotopy class
+of the vector field on each component. Let ¯L:=¯L(M) denote the set of
+isotopy classes of framed links in M.
+Figure 1. The parts of L+,L−andLoandL∞inB.
+LLrLl
+Figure 2. LrandLlare obtained by a full twist from L.
+To state the main result of the paper we need some notation and conven-
+tions: LetL+,L−,LoandL∞denote four framed links that are identical
+everywhere except in a 3-ball BinM. There under a suitable projection
+of the parts in B,L+,L−,LoandL∞look as shown in Figure 1. Also for
+every framed link we denote by Lr,Llthe framed links that are identical
+toLeverywhere except in a 3-ball where they differ as shown in Figu re 2.
+Here we suppose that the orientation of Magrees with the right-handed
+orientation of the 3-balls containing the link parts in Figu res 1 and 2 and
+that the framing vector for link parts in these figures is perp endicular to
+the page. The framings of the links coincide everywhere outs ide the parts
+shown in Figures 1 and 2.
+LetˆΛ :=C[[x, y]] denote the ring of formal power series in x,yoverC
+and lett:=ex= 1+x+x2
+2+.... Let us set a:=iey=i+iy+iy2
+2+...
+and setz:=it−(it)−1=iex+ie−x= 2i+ix2+.... Note that aandzare
+invertible in ˆΛ.INVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 3
+Definition 1.2. TheKauffman skein module ofMoverˆΛ, denotedby F(M),
+is the quotient of the free ˆΛ-module with basis ¯Lby its ideal generated by
+all the relations of the following two types:
+L+−L−=z/bracketleftbig
+Lo−L∞/bracketrightbig
+,
+Lr=aLandLl=a−1L.
+We will use F∗(M) := Hom ˆΛ/parenleftbig
+F(M),ˆΛ/parenrightbig
+to denote the ˆΛ-dual of F(M).
+Remark 1.1. The usual convention in skein module theory is to allow an
+empty link as part of the set ¯L. In contrast to that, in this paper, we find
+it convenient to work with non-empty links (Definition 1.1).
+Remark 1.2. Since the links are unoriented the declarations L+andL−,
+when considering a crossing, are arbitrary. However this do esn’t matter for
+our purposes since the first skein relation in Definition 1.2 i s invariant under
+simultaneously interchanging L+withL−andLowithL∞.
+To continue let π:=π(M) denote the set of non-trivial conjugacy classes
+ofπ1(M)and ˆπdenotetheset obtainedfrom πbyindentifyingtheconjugacy
+class of every element 1 /\e}a⊔io\slash=x∈π1(M) with that of x−1. Also letS(ˆπ) denote
+the symmetric algebra of the free ˆΛ-module, say ˆΛˆπ, with basis ˆ π. Finally,
+letS∗(ˆπ) := Hom ˆΛ/parenleftbig
+S(ˆπ),ˆΛ/parenrightbig
+denote the ˆΛ-dual ofS(ˆπ).
+Theorem 1.3. LetMbe an oriented Q-homology 3-sphere with π2(M) = 0
+and such that if H1(M)/\e}a⊔io\slash= 0thenMis atoroidal. Then there is a ˆΛ-module
+isomorphism
+F∗(M)∼=S∗(ˆπ).
+For components that are homologically trivial in Mthe homotopy class of
+the framing vector field is determined by an integer: the alge braic intersec-
+tion number of a push-out of the component in the direction of the framing
+vector field with a Seifert surface boundedby thecomponent. This algebraic
+intersection number is the self-linking number of the compo nent. There is
+a canonical framing defined by the Seifert surface that corre sponds to the
+integer zero. This implies that in a Z-homology sphere, for every underly-
+ing (unframed) isotopy class of knots the framed knot types c orrespond to
+integers. The self-linking number can also be defined in term s of Vassiliev-
+Gusarov axioms; it is a finite type framed link invariant of or der one. As
+shown by Chernov [3] this point of view generalizes to all fra med knots in
+3-manifolds; inparticular for knots inirreducible Q-homology 3-spheres that
+we study here. For Mas above, given a conjugacy class cinπ1(M) and a
+fixed framed knot CKrepresenting c, Chernov shows that there is a unique
+Z-valued invariant for all framed knots representing cwith given value on
+CK(Theorem 2.2 of [3]). His work implies that, with a chosen set of initial
+knots, for every underlying (unframed) isotopy class of kno ts the framed
+knot types correspond to integers. This point will be useful to us in the
+next sections.4 E. KALFAGIANNI
+The isomorphism in Theorem 1.3 also depends on a choice of ini tial links
+which we now discuss: For every unordered sequence of elemen ts in ˆπ∪{1}
+we choose a framed link CLthat realizes it and call it an initial link . For
+elementsin ˆ π∪{1}thataretrivialin H1(M)wechoosethecanonicalframing.
+This means that the integer describing the framing on each co mponent of
+an initial link is zero. For an initial link CLwithkhomotopically trivial
+components we choose CL=CL∗⊔Uk, whereCL∗is an initial link with no
+homotopically trivial components and Ukis the standard unlink in a 3-ball
+disjoint from CL∗. The one component unlink U1will be abbreviated to U.
+In general we will assume that each component of an initial li nkCLis the
+chosen initial knot for the corresponding element in ˆ π∪ {1}. We will also
+assume that each component is the initial knot required to de fine Chernov’s
+self-linking invariant. We will denote by CL∗the set of all initial links with
+no homotopically trivial components.
+The elements in the set CL∗∪{U}are in one-to-one correspondence with
+a basis ofS(ˆπ). An element RM∈F∗(M) gives rise to one in S∗(ˆπ) by
+restriction on the set CL∗∪ {U}. Theorem 1.3 will follow easily once we
+have proven the following result (see Section 4 for details) .
+Theorem 1.4. LetMbe an oriented Q-homology 3-sphere with π2(M) = 0,
+and such that if H1(M)/\e}a⊔io\slash= 0thenMis atoroidal. Given a map RM:
+CL∗∪{U} →ˆΛthere exists a unique map RM:¯L →ˆΛsuch that:
+(1)The restriction of RMonCL∗∪{U}is equal to RM.
+(2)RMsatisfies the Kauffman skein relation
+RM(L+)−RM(L−) =z/bracketleftbig
+RM(Lo)−RM(L∞)/bracketrightbig
+,
+for every skein quadruple of links L+,L−,LoandL∞as in Figure
+1.
+(3)RM(Lr) =aRM(L)andRM(Ll) =a−1RM(L)for everyL∈¯L.
+Let Λ := C[a±1,z±1] denote the ring of Laurent polynomials in aandz.
+We can define the Kauffman skein module ofMover Λ, denoted by FΛ(M),
+and consider its Λ-dual, F∗
+Λ(M) := Hom Λ/parenleftbig
+FΛ(M),Λ/parenrightbig
+. As we will discuss in
+Section 4, for links in S3, if we choose the value RS3(U) to lie in Λ then
+RS3(L)∈Λ, for every L∈¯L. This implies that F∗
+Λ(S3)∼=Λ and leads to
+the following question:
+Question. LetMbe as in Theorem 1.3. Can we choose the initial links
+CL∗∈ CL∗so that we have a Λ-module isomorphism
+F∗
+Λ(M)∼=S∗
+Λ(ˆπ)?
+Here,S∗
+Λ(ˆπ) denotes the Λ-dual of the symmetric algebra of the free Λ-
+module with basis ˆ π.
+In [13] we constructed formal power series invariants that s atisfy the
+HOMFLY skein change formula for unframed oriented links in l arge classesINVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 5
+ofQ-homology 3-spheres. Cornwell [4, 5, 6] shows that for lens s paces both
+the question above and its analogue for the HOMFLY skein modu le of [14]
+have a positive answer. As a result he obtains analogues of th e aforemen-
+tioned results of [8, 26] for Legendrian links in contact len s spaces.
+Theorem 2.2 of this paper is the framed link analogue of the “i ntegrability
+of singular link invariants” results proved in [12, 13]. The orem 2.2 doesn’t
+follow from the results in these papers: In [12] we only treat knots while
+in [13] we treat links in some classes of irreducible Z-homology 3-spheres.
+In this paper we are able to remove those restrictions and dea l with all
+irreducible Z-homology 3-spheres; see Theorem 3.1 and Remark 3.2. If one
+forgets the framing, Theorem 3.1 generalizes the integrabi lity results and
+Theorem A of [13] for links in all irreducible Z-homology 3-spheres.
+Framed links in general 3-manifolds and their skein modules were stud-
+ied by several authors before; see [24] and references there in. In particular,
+Przytycki [23] introduced a two term homotopy skein module o f framed
+links in oriented 3-manifolds as quantum deformation of the fundamental
+group. In [16] Kaiser calculated this module over the ring of Laurent poly-
+nomials with Z-coefficients. He showed that if a 3-manifold contains no
+non-separating 2-spheres or tori then Przytycki’s module i s a symmetric al-
+gebra of the free module with basis the set of non-trivial con jugacy classes
+ofπ1(M). Kaiser also studied several variations of two term skein m odules
+and put the classical self-linking number for null homologo us knots as well
+as Chernov’s generalization of it in the skein module theory framework. For
+details the reader is referred to [16].
+The paper is organized as follows: In Section 2 we formulate t he prob-
+lem of integrating framed singular link invariants to invar iants of framed
+links. Then we state an integrability theorem and prove it fo r atoroidal
+Q-homology spheres. In Section 3 we treat manifolds containi ng essential
+tori and in Section 4 we construct the Kauffman power series inv ariants and
+prove Theorems 1.3 and 1.4.
+Throughout the paper we will work in the smooth category.
+Acknowledgment: I thank Chris Cornwell for his interest in this work
+and for several stimulating questions about link theory in 3 -manifolds that
+motivated me to go back and work on this project. I thank Vladi mir Turaev
+for suggesting that I formulate the main result of the paper i n terms of skein
+modules. I am grateful to the anonymous referees for reading the paper
+carefully and making thoughtful comments and suggestions t hat helped me
+improve the exposition.
+2.Framed oriented Singular Link Invariants
+Throughout this section we will work with oriented links in o riented 3-
+manifolds. Theorem 2.2, as well as its unframed counterpart s [12, 13, 21],6 E. KALFAGIANNI
+are proved for oriented links in oriented 3-manifolds. For e xample, the defi-
+nitions of the signs of resolutions of double points below us e the orientation
+of links as well as that of the ambient 3-manifolds.
+2.1.Framed oriented singular links and resolutions. LetMbean ori-
+entedQ-homology 3-sphere. An m-component oriented framed singular link
+of ordernis a collection of unordered oriented circles, smoothly imm ersed in
+Msuch that (i) the only singularities are exactly ntransverse double points;
+and (ii) the image of each component is equipped with a contin uous unit
+normal vector field. We consider framed singular links up to a mbient iso-
+topy that preserves the orientations, the transversality o f the double points
+and the homotopy class of the vector field on each component. F orn= 0
+we have an oriented framed link. We will denote by L(n):=L(n)(M) (resp.
+L:=L(M)) the set of isotopy classes of oriented framed singular lin ks of
+ordern(resp. links) in M.
+Convention: To simplify the exposition, for the remaining of the section
+and the next section, we will say a framed link (resp. singula r link) to mean
+anoriented framedlink(resp. singularlink). Alsowhenwes ay a3-manifold,
+we will mean an oriented 3-manifold.
+LetPdenote a disjoint union of oriented circles and consider a fr amed
+singular link represented by a smooth immersion L:P−→M. Letp∈M
+be a double point of L; the inverse image consists of two points p1,p2∈P.
+There are disjoint intervals σ1andσ2onPwithpi∈int(σi),i= 1,2,
+such that for a neighborhood Bofpwe haveL∩B=L(σ1)∪L(σ2).
+Moreover, there is a proper 2-disc DinBsuch thatL(σ1),L(σ2)⊂D
+intersect transversally at p. NowL(σ1)∪L(σ2) intersects ∂Dat four points
+and, sinceσiinherits an orientation from that of P, we can talk of the initial
+and terminal point of L(σi). Choose arcs a1,a2,b1,b2with disjoint interiors
+such that
+(1)a1anda2go from the initial point of L(σ1) to the terminal point of
+L(σ1) and lie in distinct components of ∂B\∂D; and
+(2)b1andb2lie on∂Dwithb1going from the initial point of L(σ1) to
+the terminal point of L(σ2) andb2from the initial point of L(σ2)
+to the terminal point of L(σ1). The complement of b1⊔b2in∂D
+consists of two arcs, say c1,c2.
+The orientation of Mand that of L(σ2) define an orientation of a1⊔a2; sup-
+pose that this induced orientation agrees with the one of a1and is opposite
+to that ofa2. Define the positive resolution of Latpto be
+L+=L\L(σ2)∪a1,
+and the negative resolution to be
+L−=L\L(σ2)∪a2.INVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 7
+In the case that n= 1 we also define
+Lo=L\(L(σ2)∪L(σ1))∪(b1⊔b2)
+L∞=L\(L(σ2)∪L(σ1))∪(c1⊔c2)
+Note thatL∞only makes sense as an unoriented link.
+Definition 2.1. A framed singular link Lis called inadmissible if there is a
+2-discD⊂Msuch thatL∩D=∂Dand exactly one double point of Llies
+on∂D. Otherwise the singular link is called admissible . A crossing change
+on a link that produces an inadmissible singular link as inte rmediate step
+will be called an inadmissible crossing change .
+In the proof of Theorem 2.2 it will be convenient for us to work with
+framed links with ordered components: Let ˜Ldenote the set of isotopy
+classes of such framed links in M. Similarly, let ˜L(n)denote the set of
+isotopy classes of ordered framed singular links with n-double points. There
+is an obvious map r:˜L −→ L that forgets the ordering of the components
+of links; similarly we have forgetful maps rn:˜L(n)−→ L(n), for alln∈N.
+Recall from the Introduction that the framing of a knot is det ermined by
+an integer, where in the case of not homologically trivial kn ots this integer
+is provided by Chernov’s work. Thus the framing of an m-component link
+in˜Lis determined by an ordered sequence {f1,...,fm}ofmintegers; one
+assigned to each component of the link. Every entry of the seq uence is the
+affine self-linking number of a link component and it changes b y 2 under an
+inadmissible crossing change while it remains unchanged un der admissible
+crossing changes (Theorems 2.2, [3]). Then, via r, an unordered link L∈ L
+inherits an unordered sequence of integers: More specifical ly, givenL∈ L,
+there is a set of ordered integer sequences, say f, corresponding to elements
+inr−1(L). We assign to Lthe map
+r−1(L)−→f,
+sending each element to its corresponding ordered sequence . We will often
+abuse the terminology and refer to fas theframingof the linkL.
+Definition 2.2. Thetotal framing of a linkL∈ Lis defined to be τ(L) :=/summationtextm
+i=1fiwhere{f1,...,fm}is the ordered sequence corresponding to an ap-
+propriate lift ˜L∈r−1(L) ofL.
+Definition 2.3. For an ordered, framed singular link ˜L×∈˜L(1)we define
+a sequence of integers {f1,...,fm}by
+fi(˜L×) :=
+
+fi(˜L+)−fi(˜L−),if× ∈˜Li
+fi(˜L+) =fi(˜L−),otherwise.
+Note that, in the first case, fi(˜L×) is non-zero only if L×is inadmissible, in
+which case it is equal to 2. For an unordered singular link L×∈ L(1)we8 E. KALFAGIANNI
+have a set of ordered integer sequences, say f, corresponding to elements
+inr−1(L×). The map r−1(L×)−→f, assigning to every ordered link in
+that preimage its corresponding sequence, gives an unorder ed sequence of
+integers for L×.
+2.2.Integration of singular link invariants. Given an abelian group A
+and a framed link invariant F:L −→A, we can extend it to an invariant
+of framed singular links by defining
+f(L×) :=F(L+)−F(L−), (1)
+for everyL×∈ L(1). Continuing inductively we can extend the invariant
+on singular links in L(n)for alln∈N. We are interested in reversing
+this process; the reverse process is usually referred to as i ntegration of the
+singular link invariant to an invariant of links [1, 12, 13, 2 1]. In this section
+we deal with the following question: Suppose that we are give n an invariant
+of framed singular links f:L(1)−→A. Under what conditions is there a
+framed link invariant F:L −→Aso that (1) holds for all singular links
+L×∈ L(1)? We will address this question for links in Q-homology 3-spheres
+with trivial π2.
+Definition 2.4. LetNbe an oriented compact 3-manifold with or without
+boundary. A map Φ : S1×S1−→Nis called essential if it induces an
+injection on π1and it cannot be homotoped to a map Φ′:S1×S1−→∂N.
+Otherwise Φ is called inessential . The manifold Nis calledatoroidal if there
+are noessential mapsS1×S1−→N.
+Remark 2.1. LetL××∈ L(2)be a framed singular link with two inadmis-
+sible singular points. By resolving the singular points, on e at a time, we
+obtain four singular links in L(1). These are shown in Figure 3, where the
+notation is consistent with that of Figure 2. We note that L×ris equivalent
+Figure 3. From left to right: L×r,Lr×,L×l,Ll×
+toLr×. Similarly, L×lis equivalent to Ll×. Thus iff:L(1)−→Ais an
+invariant of framed singular links then we have
+f(L×r) =f(Lr×) andf(L×l) =f(Ll×). (2)
+Now (2) implies that the signed sum of fon the four singular links in Figure
+3, where signs are determined by (1), is equal to zero. Next we will showINVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 9
+that if this holds true for all L××∈ L(2), thenfcan be integrated to a
+framed link invariant.
+Theorem 2.2. Suppose that Mis aQ-homology sphere with π2(M) = 0
+and such that that if H1(M)/\e}a⊔io\slash= 0thenMis atoroidal. Let f:L(1)−→Abe
+an invariant of framed singular links with one double point. Suppose that A
+is torsion free and that the invariant fsatisfies the relation
+f(L×+)−f(L×−) =f(L+×)−f(L−×), (3)
+for everyL××∈ L(2). Then there exists a framed link invariant Fsuch that
+fis derived from Fvia equation (1). Here, the four singular links appearing
+in (3) are obtained by resolving the singular points of L××one at a time.
+Theorem 2.2 is the framed link analogue of Theorem 3.16 of [12 ] and
+Theorem 3.1.2 of [13]. As explained in the Introduction, how ever, here we
+work in a more general class of manifolds. Also the presence o f framing
+requires an adaptation of the arguments: to formulate the co rrect “global
+integrability condition” (equation (6) below) we need a not ion of global
+framing around homotopies of links. The definition of such a n otion is
+facilitated by the works of Chernov and Kaiser [3, 16] (Defini tion 2.5). For
+arguments that are very similar to these in [12, 13] we will re fer the reader
+to these articles for details.
+2.3.Loop space and framing control. Because in this section we work
+withorientedlinksweneedtoslightlymodifythesetofinit iallinksCL∗∪{U}
+chosenintheIntroduction. Recall that L(resp.¯L)denotestheset ofisotopy
+classes of framed oriented (resp. unoriented) links in M. Consider the set
+of oriented links CL:=o−1(CL∗∪ {U}), where o:L −→¯Lis the obvious
+forgetful map. Also recall that ˜Ldenotes the set of isotopy classes of ordered
+framed links in Mand that we defined a forgetful map r:˜L −→ L. Given
+CL∈ CL, wepickL∈r−1(CL). Wewillalsouse Ltodenotearepresentative
+L:P−→MofL, wherePis a disjoint union of oriented circles. Let
+ML(P,M) denotethe spaceof ordered smoothframed immersions P−→M
+homotopicto L, equippedwiththecompact-opentopology. Forevery L′∈˜L
+and representative L′∈ ML(P,M), let Φ :P×[0,1]−→Mbe a homotopy
+with Φ(P×{0}) =L′and Φ(P×{1}) =L. After a small perturbation we
+can assume that for only finitely many points 0 < t1< t2<···< tn<1,
+φt:= Φ(P×{t}) is not anembeddingand it is asingular framedlink of order
+1. For different t′sin an interval of [0 ,1]\{t1, t2, ...,t n}the corresponding
+framed links are equivalent and when tpasses through ti,φtchanges from
+one resolution of φtito the other.
+ForCL∈ CL, letMCL(M) denote the space of unordered smooth framed
+immersions homotopic to CL, equipped with the compact-open topology.
+The projection q:ML(P,M)−→ MCL(M) is a covering map away from
+points that are fixed under permutation of components.10 E. KALFAGIANNI
+Definition 2.5. Let Φ be a homotopy between ordered links L1,L2∈
+ML(P,M) with points 0 < t1<···< tn<1 such that φtj∈˜L(1). For
+each singular link φtjwe have a sequence {fj
+i|i= 1,...,m}as in Defini-
+tion 2.3. We define the total framing of Φ to be the sequence of integers
+{∆fi|i= 1,...,m}, where
+∆fi:=n/summationdisplay
+j=1δi
+jǫjfj
+i(φtj). (4)
+Hereδi
+j= 1 if thei-th component of φtjcontains the double point and 0
+otherwise. Also ǫj= 1 ifφtj+δ, forδ >0 sufficiently small, is a positive
+resolution of φtjandǫj=−1 otherwise. We will say that the total framing
+is zero iff ∆ fi= 0, for all 1 ,...,m.
+Given a loop Φ ∈ MCL(M) we obtain a set of ordered sequences ∆ fΦ
+associated to theset of all lifts of Φ in ML(P,M). Themap q−1(Φ)−→∆fΦ
+defines an unordered sequence of integers for Φ. The homotopy Φ is called
+framing preserving iff the total framing of every element in q−1(Φ) is zero.
+We will write ∆ fΦ=0.
+2.4.Beginning the proof of Theorem 2.2. We want to define an invari-
+antF:L −→Athat is obtained from the given f:L(1)−→Avia (1). First
+we assign values of Fon the set of initial links CL. Now fixCL∈ CLand
+letL′∈ MCL(M) be a framed link. Choose a generic homotopy Φ from L′
+toCL. Let 0<t1<t2<···<tn<1 denote the points where φtis not an
+embedding. Recall that φti∈ L(1)such that for different t′sin an interval
+of [0,1]\ {t1, t2, ...,t n}, the corresponding framed links are equivalent.
+Whentpasses through ti,φtchanges from one resolution of φtito another.
+We define
+F(L′) =F(CL)+n/summationdisplay
+i=1ǫif(φti) (5)
+Hereǫi=±1 is determined as follows: If φti+δ, forδ >0 sufficiently small,
+is a positive resolution of φtithenǫi= 1. Otherwise ǫi=−1.
+To prove that Fis well defined we have to show that modulo “the integra-
+tion constant” F(CL), the definition of F(L′) is independent of the choice of
+the homotopy. For this we consider a closed homotopy Ψ from CLto itself.
+After a small perturbation, we can assume that there are only finitely many
+pointsx1,x2,...,x n∈S1, ordered cyclicly according to the orientation of
+S1, so thatψxi∈ L1andψxis equivalent to ψyfor allxi<x,y<x i+1. To
+prove thatFis well defined we need to show that
+XΨ:=n/summationdisplay
+i=1ǫif(ψti) = 0 (6)
+whereǫi=±1 is determined by the same rule as above.INVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 11
+Independence of link component orderings: To prove (6) we will turn our
+attention to ordered links: First we note that the invariant fpulls back
+to an invariant on ˜L(1)via the forgetful map r. After iterating Φ several
+times if necessary we can assume that it lifts to a loop in ML(P,M) based
+atL(compare, page 3874 of [16]). Given a self-homotopy Φ of CLand
+the associated quantity XΦ, lift Φ to a closed homotopy Ψ in ML(P,M)
+and letXΨdenote the lift of XΦ. Note that XΨ=aXΦ, for some integer
+a∈Z. Since Ais torsion free we have XΦ= 0 exactly when XΨ= 0.
+Thus, it is enough to check (6) for homotopies that preserve t he ordering of
+components.
+Restriction to framing preserving homotopies: Next we observe that it is
+enough to check (6) for homotopies that are framing preservi ng in the
+sense of Definition 2.5: To see that we recall that given a fram ed link
+L′∈ MCL(M) we need to check that (5) does not depend on the homo-
+topy from L′to the framed link CLused to define it. Thus the closed
+homotopies Φ that we need (6) to hold for, are those obtained b y composing
+two homotopies from L′toCL. Each component of CLis equipped with a
+vector field and going around Φ does not change the homotopy cl ass of this
+vector field (that is the equivalence class of CLas a framed link). We can
+think that the framing of CLtransports to a “new” framing around Φ. The
+two framings might differ by twists on the components of CLbut the total
+singed number of the twists must be zero. The total sum of such twists is
+captured exactly by the quantity ∆ fΦ(compare, Theorem 6 of [16]). The
+framing of CLlifts to one on Land going around the self-homotopy of L
+that lifts Φ also preserves the homotopy class of the framing vector field.
+The proof of (6), which occupies the remaining of Section 2 an d Section
+3, will be divided into several steps. In this section we will give the proof of
+(6) for closed homotopies in atoroidal 3-manifolds and in th e next section
+we deal with essential tori.
+To continue, suppose that Phasmcomponents P=⊔m
+i=1Pi, where each
+Piis an oriented circle. Let L:P−→Mbe a link. Pick a base point pi∈Pi
+and letaidenote the homotopy class of L(Pi) inπ1(M,L(pi)). We denote
+byZ(ai) the centralizer of aiinπ1(M,L(pi)). We begin with the following
+lemma (see, for example, the proof of Proposition 4.3 of [21] ).
+Lemma 2.3. Suppose that Mis an orientable 3-manifold with π2(M) = 0
+and let the notation be as above. Then
+π1(ML(P,M),L)∼=⊕m
+i=1Z(ai).
+2.5.Integrating around inessential tori. Here we show how to derive
+(6) in the case where the closed homotopy Φ represents a colle ction of
+inessential tori in M. Since∂M=∅this means that the induced map
+(Φi)∗:π1(Pi×S1)−→π1(M) has non-trivial kernel. Here Φ i:= Φ|Pi×S1,
+fori= 1,...,m.12 E. KALFAGIANNI
+Lemma 2.4. LetΦbe a loop in ML(P,M)representing a framing pre-
+serving self-homotopy of L. Suppose that Φcan be extended to a map
+ˆΦ :P×D2−→MwhereD2is a 2-disc with ∂D2={∗} ×S1. Then
+XΦ= 0.
+Proof.We perturb ˆΦ, relatively ∂D2, so that it is in general position in the
+sense of Proposition 1.1 of [12]. Then the set
+SˆΦ:={x∈D2|ˆφx:=ˆΦ(P×{x}) is not an embedding },
+is a graph in D2with properties (1)-(5) given in Proposition 1.1 of [12]. Th e
+vertices ofSˆΦin the interior of D2are of valence one or four (see Figure 4).
+Figure 4. The set of singularities SˆΦwith the types of dou-
+ble points they represent.
+The invariant fassigns an element of Ato every edge of SˆΦ. We observe
+that condition (3) in the statement of Theorem 2.2 implies th atXΦis inde-
+pendent on the order in which the crossing changes around Φ := ˆΦ|P×∂D2
+occur. Thus, without loss of generality, we may assume that t he valence
+one vertices of SˆΦin the interior of D2correspond to inadmissible cross-
+ing changes on ∂D2. With the notation as above, we will assume that the
+framed singular link φxi∈ L1is inadmissible for i= 1,...,sand admissi-
+ble fori=s,...,n. In particular, there are sedges ofSˆΦemanating from
+x1,...,x srespectively and ending at an interior vertex of valence one , and
+these are the only valence one vertices of SˆΦ.
+For every interior vertex of SˆΦwe draw a small circle Caround it so
+that the number of points in C∩SˆΦis equal to the valence of the vertex.
+See Figure 5. Let C1,...,C sdenote the circles surrounding the valence one
+vertices ofSˆΦand let Γ denote the disjoint union of the circles surroundin g
+the vertices of valence four. For a vertex of valence four the four points in
+C∩SˆΦcorrespond exactly to these appearing in equation (3). Thus by (3)
+we have /summationdisplay
+x∈Γ∩SˆΦǫxf(ˆφx) = 0, (7)INVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 13
+Cx
+x2
+1e
+e'
+Figure 5. The singular links φx1,φx2form a pair of type
+L×r,Lr×(orL×l,Ll×) shown in Figure 3. The framed links
+corresponding to the components e,e′of∂D2\{x1,x2,...}
+are isotopic.
+whereˆφx:=ˆΦ(P×{x}).Now observe that
+n/summationdisplay
+i=s+1ǫif(φxi) =/summationdisplay
+x∈Γ∩SˆΦǫxf(ˆφx) = 0.
+The last equation and (7) imply that
+XΦ=s/summationdisplay
+i=1ǫif(φxi). (8)
+Since Φ is framing preserving we have ∆ fΦ=0. By Definitions 2.3 and 2.5
+and the fact that fremains unchanged under admissible crossing changes
+we have ∆ fC=0, for every loop C∈Γ. This in turn implies that
+∆fΓ:=/summationdisplay
+C∈Γ∆fC=0
+Since we have
+∆fΦ=s/summationdisplay
+i=1ǫif(φxi)+∆fΓ=0
+we conclude that/summationtexts
+i=1ǫif(φxi) =0. This in turn implies that the inadmis-
+sible singular links φxican be partitioned into pairs of the forms shown in
+Figure 3. Relation (2) in Remark 2.1 shows that the right hand side of (8)
+is identically zero. Thus XΦ= 0, as desired. /square14 E. KALFAGIANNI
+Remark 2.5. Let¯XΦdenote the contribution of the admissible singular
+links around Φ to XΦ. The proof of Lemma 2.4 shows that regardless of
+whether Φ is framing preserving, relation (3) implies that ¯XΦ= 0.
+Remark 2.6. Proposition 1.1 of [12], referenced in the proof of Lemma
+2.4, is stated in there for the PL-category. However, as expl ained by Kaiser
+in Section 3 of [15], the statement is true in the smooth categ ory which is
+actually whatweneedhere. We shouldalsoremarkthat, as exp lainedbyLin
+in [21], the conclusion holds if the disc D2is replaced by any planar surface
+F. Furthermore, ifΦ |∂Fisalreadyingeneralpositionthenthemodifications
+that put Φ into general position on Fcan be performed relatively ∂F.
+A slight variation of the proof of Lemma 2.4 shows the followi ng:
+Lemma 2.7. LetΦbe a loop in ML(P,M)representing a framing preserv-
+ing self-homotopy of a framed link L. LetP′:=P\P1. Suppose that Φ|P′
+can be extended to a map ˆΦ :P′×D2−→MwhereD2is a 2-disc with
+∂D2={∗}×S1. Suppose moreover that Φ|(P1×S1)is an embedding. Then
+XΦ= 0.
+The proof of the next lemma is given in the proof of Lemma 3.3.4 of [13].
+Lemma 2.8. LetMbe aQ-homology 3-sphere with π2(M) = 0. Suppose
+thatπ1(M)is infinite and that Lhas no homotopically trivial components.
+LetΦ⊂ ML(P,M)be a framing preserving closed homotopy such that the
+restriction Φ|Pi×S1−→Mis inessential, for all i= 1,...,m. There exists
+a 2-discD2and a map ˜Φ :P×D2−→Msuch that
+X∂˜Φ=aXΦ, (9)
+for somea∈Z. Here∂˜Φ =˜Φ|P×∂D2.
+2.6.Theorem 2.2 for atoroidal manifolds. Before we can proceed with
+the proof of the theorem we need two additional lemmas.
+Lemma 2.9. Consider Φ,Φ′:S1−→ ML(P,M)two self-homotopies of
+L. Let¯XΦand¯XΦ′denote the contribution to XΦandXΦ′coming from
+admissible singular links around ΦandΦ′, respectively. Suppose that Φ,Φ′
+are freely homotopic as loops in ML(P,M). Then we have ¯XΦ′=¯XΦ.
+Furthermore, there is a group homomorphism ψ:π1(ML(P,M), L)−→A
+defined byψ([Φ]) := ¯XΦ.
+Proof.By a slight variation of the argument in the proof of Lemma 3.3 .2 of
+[13] we have the following: There exists a map ˆΨ :D2−→ ML(P,M) such
+that if we set Ψ := ˆΨ|∂D2then Ψ :S1−→ ML(P,M) is a self-homotopy of
+Lwith
+XΨ=XΦ−XΦ′.
+Lemma 2.4 and Remark 2.5 imply ¯XΨ= 0; thus ¯XΦ=¯XΦ′.INVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 15
+For the remaining of the claim define ψ:π1(ML(P,M), L)−→Aas
+follows: Given α∈π1(ML(P,M), L), let Φ is be a self-homotopy of L
+representing α. Defineψ(α) =¯XΦ. By our earlier arguments ψ(α) is inde-
+pendent on the representative Φ. The fact that ψis a group homomorphism
+follows easily. /square
+The next lemma is Lemma 3.2.5 in [13]. We point out that the pro of of
+this lemma uses the hypothesis that the group Ain which the invariants
+take values is torsion free.
+Lemma 2.10. Suppose that Mis aQ-homology 3-sphere with π2(M) = 0.
+LetL:P−→Mbe a framed link and let Φ :P×S1−→Mbe a framing
+preserving self-homotopy of L. Assume that, for some i= 1,...,m, we have
+ai= 1. SetP′:=P\PiandΦ′:= Φ|P′. IfXΦ′= 0thenXΦ= 0.
+We are now ready to give the proof of Theorem 2.2 in the case whe reM
+is an atoroidal Q-homology 3-sphere.
+Theorem 2.11. Suppose that Mis an atoroidal Q-homology 3-sphere with
+π2(M) = 0. Then the conclusion of Theorem 2.2 is true for M.
+Proof.Letf:L1−→Abe a framed singular link invariant satisfying (3)
+of the statement of Theorem 2.2 and let Φ : P×S1−→Mbe a framing
+preserving self-homotopy of a framed link L:P−→M. We have to show
+that
+XΦ= 0
+whereXΦis the signed sum of values of faround Φ defined in (6).
+Firstsupposethat π1(M)isfinite. Then,byLemma2.3, π1(ML(P,M), L)
+is finite. Since Ais torsion free the homomorphism ψof Lemma 2.9 must
+be the trivial one. Thus, in particular, XΦ= 0.
+Now suppose that π1(M) is infinite. If the link Lto begin with contains
+no homotopically trivial components, then since Mis atoroidal, Lemma 2.8
+applies to conclude that X∂˜Φ=aXΦ, for a map ˜Φ :P×D2−→M. By
+Lemma 2.4, X∂˜Φ=aXΦ= 0 and thus, since Ais torsion free, XΦ= 0.
+Next suppose that all the components of Lare homotopically trivial; that
+isai= 1, fori= 1,...,m. Then, by Lemma 2.3,
+π1(ML(P,M),L)∼=⊕m
+iπ1(M,L(pi)).
+SinceH1(M) is finite the above equality implies that the abelianizatio n of
+the groupπ1(ML(P,M),L) is a finite group. By Lemma 2.9 we have a
+homomorphism ψ:π1(ML(P,M),L)−→Awithψ([Φ]) =XΦ. SinceAis
+abelianψfactors through the abelianization of π1(ML(P,M),L); a finite
+group. But since Ais torsion free ψis the trivial homomorphism. Thus
+XΦ= 0.
+To handle the general case let h(L) denote the number of components of
+Lthat are homotopically trivial. The proof is by induction on h(L). In the
+light of our discussion above, the conclusion is true if h(L) = 0 orh(L) =m.16 E. KALFAGIANNI
+Thus we may assume that h(L)/\e}a⊔io\slash= 0,m. LetLi⊂Lbe a component that
+is homotopically trivial and let L′:=L\Li. Also let Φ be a self-homotopy
+ofLand let Φ′denote the restriction of Φ on P′, whereP′:=P\Pi. Since
+h(L′)<h(L), by induction, XΦ′= 0. Then, by Lemma 2.10, XΦ= 0./square
+3.Integration of invariants in toroidal 3-manifolds
+Tostudythequestion ofintegrability ofsingularlink inva riants intoroidal
+3-manifolds we need several results from the theory of the ch aracteristic
+submanifold of Jaco-Shalen [10] and Johannson [11]. The sta tements of the
+resultsfromthesetheories, intheformneededinoursettin g, aresummarized
+in Section 2 of [12] and in Section 2 of [13]. It will be conveni ent for us to
+recall the statements we need below from therein, instead fr om the original
+references. In particular we will need the Enclosing Theore m and the Torus
+Theorem both stated on pp. 679 of [12]. The later, in the form n eeded for
+our purposes, follows from work of Scott, Casson-Jungreis a nd Gabai.
+Theorem 3.1. LetMbe aZ-homology 3-sphere with π2(M) = 0and letA
+be a torsion free abelian group. Suppose that a map f:L(1)−→Asatisfies
+(3) of Theorem 2.2. Then there exists a framed link invariant Fsuch that
+fis derived from Fvia equation (1).
+Remark 3.2. The restriction to Z-homology 3-spheres in Theorem 3.1 is
+necessary. As explained in Remark 3.13 of [12] and the discus sion at the
+end of Section 3 in [13], in general, local conditions are not sufficient for the
+integration of singular link invariants. When the characte ristic submanifold
+contains Seifert fibered components over non-orientable su rfaces one needs
+to impose extra non-local conditions. Specific constructio ns demonstrating
+these phenomena are given by Kirk and Livingston in [19]. The necessity of
+working with irreducible 3-manifolds is demonstrated by [1 9] as well as the
+work of Eiserman [7].
+TheproofofTheorem2.2will becompleted oncewehaveproved Theorem
+3.1. For the proof of Theorem 3.1 we will need the following:
+Lemma 3.3. LetMbe aZ-homology 3-sphere with π2(M) = 0. Suppose
+thatΦ :T=S1×S1−→Mis an essential map. Then there exists a map
+Ψ :T−→Mhomotopic to Φand such that one of the following holds:
+(1) Ψ(T)lies on an essential embedded torus in M.
+(2)There exists an oriented surface Fwith∂F/\e}a⊔io\slash=∅, and a trivial fiber
+bundleY=S1×F, with the following property: Ψextends to a
+mapˆΨ :Y−→Mso that the image ˆΨ(∂Y\T)is contained on a
+collection of embedded tori in M.
+Proof.By the Torus Theorem and the discussion at the end of Section 2 of
+[13], either Mis Haken or it is a Seifert fibered 3-manifold that fibers over
+S2with three or less exceptional fibers.INVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 17
+First supposethat Mis Haken. Thenby theEnclosingTheorem thereis a
+Seifert fibered submanifold S⊂Mand a homotopy Φ′
+t:T−→Msuch that
+Φ′
+0= Φ and Φ′
+1(T)⊂S. If Φ′
+1(T) can be further homotoped in Sso that it
+lies on a component of ∂Sthen we have conclusion (1). Otherwise, by the
+classification of essential tori in Haken Seifert fibered spa ces (Proposition
+2.11 of [12]) we can homotope Φ′
+1inSto a map Ψ : T−→Swhich is
+vertical with respect to the fibration.
+Next suppose that Mis a Seifert fibered space. By Proposition 2.2.5 of
+[13], Φ is homotopic to a map Ψ : T−→Mwhich is vertical with respect
+to the fibration of M.
+Thus, in both cases, either (1) holds or we have a Seifert fiber ed manifold
+S⊆M, with orbit space Band fiber projection p, such that Φ is homotopic
+to a map Ψ : T−→Mthat is vertical with respect to the fibration of S.
+This means that Ψ is a composition Φ 1◦q, whereqis a covering map from
+the torusTto itself and Φ 1:T−→Sis an immersion without triple points.
+Then, there exists a decomposition T=S1×S1such that
+a) Φ1(S1×{∗}) maps onto a regular fiber hofS;
+b) we have p(Φ1({∗}×S1)) =p(Φ1(T)) on the orbit surface BofS.
+LetH(resp.Q) denote the curve S1×{∗}(resp.{∗}×S1) onT. Now
+α:=p(Φ1(T)) is an immersed closed curve on Bwith singularities finitely
+many transverse double points. A neighborhood N:=N(α)⊂BofαonB
+is an oriented planar surface. Choose Nsmall enough so that Y:=p−1(N)
+contains no exceptional fibers of S. Nowp:Y−→Nis anS1-bundle and
+sinceH2(N) = 0 this bundle is trivial. Choose a trivialization Y∼=S1×N
+so thatNis embedded as a cross-section. Pick a base point b∈Nand arcs
+frombto the components of ∂Nwhose homotopy classes freely generate
+π1(N); we pick one arc for each such component. Assume that these a rcs
+intersectαonly at its double points; let x1,...,x sdenote the resulting
+generators of π1(N,b). Writeαas a word in these generators; say
+[α] =xk1
+i1xk2
+i2···xkr
+ir.
+We can extend the restriction Ψ |{∗} ×S1to a map ˆΨ : (F, ∂F)−→
+(N, ∂N), whereFis a planar surface, such that: (i) the induced map
+ˆΨ∗:π1(F)−→π1(N) is onto; (ii) π1(F) is freely generated by elements
+a1
+1,···,a1
+k1,a2
+1,···,a2
+k2,ar
+1,···,ar
+kr; (iii)ˆΨ∗([Q]) =xk1
+i1xk2
+i2···xkr
+ir(see proof
+of Lemma 3.11 of [12]). We pull back the fiber bundle structure byˆΨ to
+obtain a fiber bundle ˆΨ∗(Y)−→F, overF. The pull-back of the cross-
+sectionαis a cross-section of ˆΨ∗(Y). Extending this cross-section over F,
+and conclusion. /square
+We now recall that the proof of Theorem 3.1 is reduced to showi ng (6)
+for every framing preserving self-homotopy of L. Using Lemmas 2.3, 2.9,
+and 2.10 we will see that the general case is essentially redu ced to the case18 E. KALFAGIANNI
+of knots. Before we continue with the proof Theorem 3.1 some r emarks are
+in order.
+Remark 3.4. Let Φ :P×S1−→Mdenote a framing preserving self-
+homotopy of a framed link Land let Φ′be obtained by a free homotopy of
+ΦinM. Considerthehomotopy fromΦtoΦ′asamap H:P×S1×[0,1]−→
+M. We can smoothly approximate Hby a homotopy in general position as
+intheproofof Lemma2.4 (see Remark2.6). Thenwecan view Has afamily
+of smooth framed immersions S1−→Mparametrized by an annulus. We
+note that the closed homotopy Φ′is not necessarily framing preserving.
+Remark 3.5. Suppose that we have a map Φ : Y:=S1×F−→M, such
+thatFis a planar surface so that there is a component α⊂∂Fsuch that
+the restriction Φ |S1×αis a loop in ML(P,M). We can view Φ as a family of
+framed immersions in M, parametrized by F. We can cut Y:=S1×F−→
+Malong a collection of properly embedded annuli (the project ion of which
+onFdecomposes Finto a disc) into a product S1×D2. By considering the
+pull back of Φ on S1×D2we obtain a family of framed immersions in M
+parametrized by D2.
+In the next lemma we treat homotopies that involve essential tori. The
+proof treats separately the case of knots and that of links. I n the case of
+knots (m= 1 below) the proof is very similar to that of Case 1 of Lemma
+3.3.3 in [13]. The starting ingredient in the proof of [13] is Lemma 3.3.2
+therein. Here we replace that ingredient with Lemma 3.3 and w e outline the
+argument below.
+Lemma 3.6. LetMbe aZ-homology 3-sphere with π2(M) = 0and let
+Φ :P×S1−→Mbe a framing preserving self-homotopy of a framed link
+L. Suppose that Φi:= Φ|Pi×S1is an essential map, for some i= 1,...,m.
+Suppose, moreover, that Φicannot be homotoped so that its image lies on
+an essential embedded torus in M. Then we have XΦ= 0.
+Proof.Letmbe the number of components of L. We distinguish two cases
+according to whether m= 1 orm>1.
+We havem= 1:Since Φ is framing preserving, relation (2) implies that
+thetotal contribution oftheinadmissiblesingular linksa long ΦtoXΦis zero
+(proof of Lemma 2.4). Thus, without loss of generality, we ca n assume that
+no inadmissible crossing changes occur along Φ. Now let Ψ : P×S1−→M
+bea map that is freely homotopic to Φ in M. By Lemma 2.9, and our earlier
+assumption on Φ, we have ¯XΨ=XΦ.
+SetT:=P×S1,l:=P× {∗}andm:={∗} ×S1. By assumption
+Φ|P×S1−→Mis an essential map and it cannot be homotoped so that
+its image lies on an essential embedded torus in M. By Lemma 3.3 we can
+homotope Φ to a map Ψ : P×S1−→Mso that: There is a trivial fiber
+bundleY=S1×F, over a planar surface F, such that Ψ extends to a mapINVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 19
+ˆΨ :S1×F−→Mand the image ˆΨ(∂Y\T) is contained on a collection
+of embedded tori in M. LetHdenote a simple closed curve Trepresenting
+a fiber ofYand letQdenote the component of ∂F(embedded as a cross-
+section of the bundle) on T. Inπ1(T) we have [ l] =a[H] +b[Q], for some
+a,b∈Z.
+First suppose that a= 0. Then Lemma 3.12 of [12] (or Lemma 3.3.1
+of [13]) applies to conclude that ¯XΨ= 0. By our discussion above, XΦ=
+¯XΨ= 0 and the conclusion in this case follows.
+Suppose now that a/\e}a⊔io\slash= 0. Letq:˜Y−→Ybe the covering of Ycorre-
+sponding to the subgroup aZ×π1(F) ofπ1(Y) =Z×π1(F). Liftl,H, and
+Qto curves ˜l,˜H,˜Q, respectively, on the torus ˜T:=q−1(T). Now ˜Yis a
+trivial fiber bundle over a surface ˜Fwith fiber ˜l; we will write ˜Y=˜lטF.
+Consider the composition ˜Ψ :=ˆΨ◦qand its restriction on ˜T∼=˜lטQ. Since
+˜Ψ(˜l×{x}) = Ψ(l×q({x})), for allx∈˜Q, the restriction ˜Ψ|˜lטQis a self-
+homotopy of a framed knot; the parameter space is ˜Q. As in Remark 3.5 we
+will think of ˜Ψ as a family of framed immersions parametrized by a disc D2.
+Then we can consider X∂˜Ψ. As in the proof of Lemma 3.14 of [12] we obtain
+thatX∂˜Ψ=cXΨ, for somec∈Z. Since, as discussed at the beginning of
+this proof we have ¯XΨ=XΦ, it follows that ¯X∂˜Ψ=cXΦ. By Remark 2.5,
+we have ¯X∂˜Ψ= 0. Hence we conclude that we have cXΦ= 0 for some c∈Z.
+SinceAis torsion free this implies that XΦ= 0; finishing thereby the proof
+of the Lemma in the case m= 1.
+We havem>1.By Lemma 2.3, π1(ML(P,M),L) is isomorphic to a
+direct productof thegroups π1(ML(Pi,M),Li) fori= 1,...,m. By Lemma
+2.9 it is enough to verify (6) only for homotopies Φ that are fix ed on all but
+one component of L. To that end, let Ψ be a homotopy in general position
+that only moves one component; say L1. Suppose, without loss of generality,
+that Ψ|P1×S1−→Mis an essential map that cannot be homotoped so
+that its image lies on an essential embedded torus in M. By (3), we may
+decompose Ψ into two homotopies Ψ 1and Ψ2such that during Ψ 1we only
+have self-crossing changes on L1, while during Ψ 2we only have crossing
+changes between L1and the rest of the components. The argument of Case
+1 applies to Ψ 1to conclude that XΨ1= 0. Since the restriction of Ψ 2on
+P′×S1, whereP′=P\P1, is constant; it extends to a map P′×D2−→M.
+Then by Lemma 2.7 we have XΨ2= 0. /square
+3.1.The completion of the proof of Theorem 3.1. Let Φ be a framing
+preserving loop in ML(P,M). Suppose that Φ |Pi×S1−→Mrepresents an
+essential torus for some i= 1,...,m. First suppose that some component,
+say Φi:= Φ|Pi×S1−→M, can be homotoped to lie on an embedded
+essential torus in M. Then a theorem of Nielsen ([9], theorem 13.1) implies
+that after further homotopy, we may assume that Φ iis a covering map of an
+embedded torus. It follows that the contribution of Φ itoXΦis zero. Thus,
+for our purposes, we can assume that if Φ iinduces an injection on π1then20 E. KALFAGIANNI
+it cannot be homotoped to lie on an embedded torus. Then by Lem ma 3.6
+we obtainXΦ= 0.
+As in the proof of Lemma 3.6 we may assume that Φ fixes all but one
+component of L; sayL1. If Φ :P1×S1−→Mis inessential the argument
+in the proof of Theorem 2.11 applies to conclude that XΦ= 0. Assume that
+Φ :P1×S1−→Mis essential. Then XΦ= 0 by Lemma 3.6.
+4.Kauffman power series
+4.1.Links in oriented Q-homology 3-spheres. For framed links in S3
+the Kauffman polynomial is equivalent to a sequence of 1-varia ble Laurent
+polynomials {Rn=Rn(t)}n∈Zdetermined by relations:
+Rn(U) = 1
+Rn(Lr) =t−(n+1)Rn(L)
+Rn(Ll) =t(n+1)Rn(L)
+Rn(L+)−Rn(L−) = (t−t−1)[Rn(Lo)−Rn(L∞)]
+whereL+,L−,Lo,L∞are as in Figure 1 and Lr,Llare as in Figure 2.
+Notice that the initial value Rn(U) = 1 is just a normalization. Any choice
+of the initial value together with the rest of the relations w ill determine a
+uniqueRn. Set
+un(t) :=tn+1−t−(n+1)
+t−t−1+1. (10)
+By the relations above one obtains Rn(L⊔U) =un(t)Rn(L), where the
+linkL⊔Uis obtained from Lby adding an unknotted and unlinked com-
+ponentU. The coefficients of the power series Rn(x) obtained from Rn(t)
+by substituting t=exare invariants of finite type [1, 2]. In the theorem
+below we reverse this procedure and guided by the axioms abov e we will
+construct power series invariants generalizing the Rn(x)’s: Suppose that M
+is aQ-homology sphere with π2(M) = 0 and such that if H1(M)/\e}a⊔io\slash= 0 then
+Mis atoroidal. For every n∈Zwe will construct a sequence of framed link
+invariants {vm
+n|m∈N}such that the formal power series
+R{M,n}=∞/summationdisplay
+m=0vm
+nxm
+satisfy the axioms above under the change of variable t=ex: We will
+construct our invariants inductively (induction on m) by using Theorem
+2.2. Each vm
+nis going to be obtained by integrating a suitable singular
+link invariant determined by the vj
+n’s withj < m. Although the resulting
+invariantswillbeinvariantsofunorientedframedlinks, f ortheirconstruction
+we need to work with oriented links. The reason is that Theore m 2.2 applies
+to oriented framed links. Recall that L(resp.¯L) denotes the set of isotopy
+classes of framed oriented (resp. unoriented) links in M. Also recall the set
+of oriented initial links CL:=o−1(CL∗∪{U}), defined in the beginning ofINVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 21
+subsection §2.3. By Theorem 2.2 and its proof the invariant vm
+nis unique
+once the values on the set CLare specified.
+Theorem 4.1. Assume that Mis aQ-homology 3-sphere with π2(M) = 0
+and such that if H1(M,Z)/\e}a⊔io\slash= 0thenMis atoroidal. Fix n∈Z. Given maps
+Vm
+n:CL∗∪ {U} −→C,m∈N, there exists a unique sequence of complex
+valued link invariants {vm
+n|m∈N}with the following properties:
+(1)vm
+n(CL) =Vm
+n(o(CL))for allCL∈ CLandm∈N.
+(2)vm
+n(L) =vm
+n(o(L))for allL∈ Landm∈N. Thus the values of the
+invariants are independent of the link orientation.
+(3)If we define a formal power series
+Rn:=R{M,n}(L) =∞/summationdisplay
+m=0vm
+n(L)xm
+then we have
+Rn(U) = 1 (11)
+Rn(Lr) =t−(n+1)Rn(L) (12)
+Rn(Ll) =t(n+1)Rn(L) (13)
+Rn(L+)−Rn(L−) = (t−t−1)[Rn(Lo)−Rn(L∞)] (14)
+wheret:=ex= 1+x+x2
+2+....
+Proof.Definevm
+n(CL) =Vm
+n(o(CL)) for allCL∈ CLandm∈N. Now we
+can form the power series Rn(CL). Guided by (12)-(13) we define
+Rn(CLr) =t−(n+1)Rn(CL) andRn(CLl) =t(n+1)Rn(CL).(15)
+Now guidedbythesewecandefinethevalues of Rnonall framedlinks whose
+underlying unframed isotopy class is CL. To explain this suppose that CL
+hasscomponents. Let CL(f) be a framed link in the same (unframed)
+isotopy class with CLwith framing unordered sequence f(see Definition 2.2
+and preceding discussion). Then define
+Rn(CL(f)) =t(n+1)τRn(CL),
+whereτ:=τ(CL(f)) is the total framing of CL(f). Using (14)-(15), and
+inducting on k, we can check that
+Rn(CL⊔Uk) = [un(t)]k−1Rn(CL) (16)
+whereun(t) is given by (10). Now Rnhas been defined on all framed links
+in the unframed isotopy classes of the links in CL.
+To continue for every L(f)∈ Lwith framing sequence fwe define
+v0
+n(L(f)) =v0
+n(CL(f)),22 E. KALFAGIANNI
+whereCLis the initial link homotopic to L. Inductively, suppose that the
+invariantsv0
+n,v1
+n,...,vm−1
+nhave been defined such that if we let
+R(m−1)
+n(L) :=m−1/summationdisplay
+i=1vi
+n(L)xi,
+then we have
+R(m−1)
+n(Lr) =t−(n+1)R(m−1)
+n(L) modxm(17)
+R(m−1)
+n(Ll) =t(n+1)R(m−1)
+n(L) modxm(18)
+R(m−1)
+n(L⊔U) =un(t)Rn(L) modxm(19)
+and
+R(m−1)
+n(L+)−R(m−1)
+n(L−) = (t−t−1)[R(m−1)
+n(Lo)−R(m−1)
+n(L∞)] modxm.
+Furthermore, supposethat theseinvariants donot dependon theorientation
+of the links. The last equation leads us to define
+R(m)
+n(L×) := (t−t−1)[R(m−1)
+n(Lo)−R(m−1)
+n(L∞)] modxm+1(20)
+We want to define the invariant vn
+m: Recall that it is already defined
+on the initial links. Next we examine the right hand side of (2 0). It is a
+polynomial of degree msuch that the coefficient of xmcomes from
+(t−t−1)[R(m−1)
+n(Lo)−R(m−1)
+n(L∞)].
+The expression above has no constant term and thus the coeffici ent ofxm
+depends on the inductively well defined invariants vi
+n,i= 1, 2,...,m−1.
+Thus the coefficient of xmin (20) is a “new” framed singular link invariant.
+We are going to prove that it is derived from aframed link inva riant by using
+Theorem 2.2. For that we need to check that condition (3) in th e statement
+2.2 is satisfied. It is enough to check it modulo xm+1. In what follows the
+symbol “ ≡” will denote calculation modulo xm+1.
+LetL×+andL×−∈ L(1)be two singular framed links as in the left hand
+side of (3) in the statement 2.2. From (20) we have
+R(m)
+n(L×+)−R(m)
+n(L×−)≡
+≡(t−t−1)/bracketleftbig
+R(m−1)
+n(Lo+)−R(m−1)
+n(L∞+)/bracketrightbig
+−
+−(t−t−1)/bracketleftbig
+R(m−1)
+n(Lo−)−R(m−1)
+n(L∞−)/bracketrightbig
+≡
+≡(t−t−1)/bracketleftbig
+R(m−1)
+n(Lo+)−R(m−1)
+n(Lo−)/bracketrightbig
+−
+−(t−t−1)/bracketleftbig
+R(m−1)
+n(L∞+)−R(m−1)
+n(L∞−)/bracketrightbig
+≡
+≡(t−t−1)2/bracketleftbig
+R(m−1)
+n(Loo)−R(m−1)
+n(Lo∞)/bracketrightbig
+−
+−(t−t−1)2/bracketleftbig
+R(m−1)
+n(L∞o)−R(m−1)
+n(L∞∞)/bracketrightbig
+≡
+≡(t−t−1)2/bracketleftbig
+R(m−1)
+n(Loo)+R(m−1)
+n(L∞∞)/bracketrightbig
+−
+−(t−t−1)2/bracketleftbig
+R(m−1)
+n(L∞o)+R(m−1)
+n(Lo∞)/bracketrightbig
+.INVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 23
+Since the result is symmetric with respect to the two double p oints we
+deduce that
+R(m)
+n(L×+)−R(m)
+n(L×−)≡R(m)
+n(L+×)−R(m)
+n(L−×).
+Thus the framed singular link invariant defined above is indu ced by a
+framed link invariant. Recall that we have already defined th e values ofvm
+n
+on all framed links with unframed underlying isotopy classe s inCL. Using
+this values we can define a link invariant vm
+nfor all links in Lsuch that if
+we let
+R(m)
+n(L) =m/summationdisplay
+i=1vm
+n(L)xi
+we have
+R(m)
+n(L+)−R(m)
+n(L−) =R(m)
+n(L×), (21)
+for everyL×∈ L(1). Now it is a straightforward calculation to check that
+the inductive hypotheses hold mod xm+1. For example let us check (18);
+the others are similar. Consider a framed link Lr. Keeping the kink intact
+in a small 3-ball, make a sequence of crossing changes to tran sformLlto an
+initial link say CLl. Over all such sequences of crossing changes, and initial
+linksCLl, choose one that minimizes the number of the required crossi ng
+changes. Suppose, without loss of generality, that the first crossing to be
+changed in that sequence is a positive crossing. By (20) and ( 21) we have
+R(m)
+n(Ll+)≡R(m)
+n(Ll−)+(t−t−1)[R(m−1)
+n(Llo)−R(m−1)
+n(Ll∞)] modxm+1.
+By (15) and induction on the number of crossing changes neede d to go from
+Ll+toCLlwe can assume that
+R(m)
+n(Ll−)≡t(n+1)R(m)
+n(L)
+By (18) we have
+R(m−1)
+n(Llo) =t(n+1)R(m−1)
+n(Lo) modxm,
+and
+R(m−1)
+n(Ll∞) =t(n+1)R(m−1)
+n(L∞) modxm.
+Combining the last four equations we have
+R(m)
+n(Ll+)≡t(n+1)R(m)
+n(L+),
+as desired. To finishthe proof we need to show that vm
+nis independentof the
+link orientation. Inductively, we assume that v0
+n,v1
+n,...,vm−1
+nare uniquely
+determined by their values on CLand independent of the (singular) link
+orientation. We have that
+vm
+n(L) =vm
+n(CL)+s/summationdisplay
+i=1±vm
+n(Li)
+whereL1,...,L s∈ L(1)are singular links appearing in a homotopy from L
+toCL, whereCLis the representative of LinCL(compare relation (5)).24 E. KALFAGIANNI
+Recall that we defined vm
+n(CL) to be independent of the orientation for
+CL. The proof of Theorem 2.2 establishes that vm
+n(L) doesn’t depend on
+the homotopy from LtoCLchosen. By induction v0
+n,v1
+n,...,vm−1
+ndo not
+depend on orientations. It follows that vm
+n(L) is unique once vm
+n(CL) is
+chosen and independent on link orientation. /square
+Theorem 1.4 stated in the Introduction is obtained from Theo rem 4.1 if
+we setz:=it−(it)−1=iex+ie−xanda:=iey, wherey= (n+1)x. Now
+we derive Theorem 1.3 stated in the Introduction.
+Proof.The elements in the set CL∗∪{U}are in one-to-one correspondence
+with a basis of S(ˆπ). An element R∈F∗(M) gives rise to one in S∗(ˆπ)
+by restriction on the set CL∗∪ {U}. Thus one direction of the theorem
+follows. For the other direction, we note that an element in S∗(ˆπ) defines
+a mapRM:CL∗∪{U} →ˆΛ. Then by Theorem 1.4 there is a unique map
+RM:¯L →ˆΛ with properties (1)-(3). These properties guarantee that RM
+factors through the Kauffman module F(M) to give an element in F∗(M)
+(see Definition 1.2). /square
+4.2.Links inS3.Links inS3are studied via projections on a sphere S2⊂
+S3. LetUmdenote the standard m-component unlink and Um(f) denote
+the one with framing f. Everym-component link projection L⊂S2is
+transformedtoaframedunlinkbyfinitelymanycrossingchan gesandregular
+isotopy moves on S2(i.e. isotopy using the Reidemeister moves of type II
+and III only). For a link projection L⊂S2, we define a complexity
+s(L) := (u(L), c(L)),
+as follows: c(L) is the number of crossings of Landu(L) is the number
+of admissible crossing changes required to transform Linto a diagram of
+a framed unlink that that admits a type I Reidemeister move th at reduces
+its crossing number. We order the complexities lexicograph ically. LetR:=
+RS3:L →ˆΛ be a map constructed as in Theorem 1.4 and recall that
+Λ :=C[a±1,z±1]. Note that the complexity s(L) defined above has the
+properties that s(Lr),s(Ll)>s(L).
+Proposition 4.2. DefineR(U(f)) =a−τ(a+a−1)z−1+ 1, where τ:=/summationtextm
+i=1fi. Then,R(L)∈Λ for every link. In fact, R(L) is the two variable
+Kauffman polynomial.
+Proof.Given a diagram Lfirst perform all type I Reidemeister moves that
+reduce the number of crossings of L. If there are no such moves and L
+is non-trivial then there is a crossing change such that thre e of the terms
+s(L−),s(Lo),s(L∞),s(L+) are strictly less that the remaining fourth one.
+Thus the skein relations
+R(L+)−R(L−) =z/bracketleftbig
+R(Lo)−R(L∞)/bracketrightbig
+,INVARIANTS OF FRAMED LINKS IN 3-MANIFOLDS 25
+R(Lr) =aR(L) andR(Ll) =a−1R(L)
+allow us to write the invariant R(L) of every link Las a finite sum of the
+invariants of links of strictly less complexity than s(L) and with coefficients
+in Λ. The result follows by induction on s(L) and the observation that
+R(U(f))∈Λ. The last claim in the statement of the proposition follows by
+the uniqueness properties of R. /square
+References
+[1] D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 2, 423–472.
+[2] J. S. Birman and X.-S. Lin,, Knot polynomials and Vassiliev’s invariants, Invent.
+Math. 111 (1993), no. 2, 225–270.
+[3] V. Chernov, Framed knots in 3-manifolds and affine self-linking numbers, J. Knot
+Theory Ramifications 14 (2005), no. 6, 791–818.
+[4] C. Cornwell, A polynomial invariants for links in lens spaces, arXiv:1002.1543.
+[5] C. Cornwell, Bennequin type inequalities in lens spaces, arXiv:1002.1546.
+[6] C. Cornwell, Ph.D thesis, MSU, in preparation.
+[7] M. Eisermann, A geometric characterization of Vassiliev invariants, Trans. Amer.
+Math. Soc. 355 (2003), no. 12, 4825–4846.
+[8] E. Ferrand, On Legendrian knots and polynomial invariants, Proc. Amer. Math.
+Soc. 130 (2002), no. 4, 1169–1176.
+[9] J. Hempel, 3 -Manifolds. Ann. of Math. Studies, No. 86. Princeton University
+Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1 976.
+[10] W. Jaco and P. Shalen, Seifert fibered spaces in 3-manifolds, Mem. Amer. Math.
+Soc. 21(220) (1979).
+[11] K. Johannson, Homotopy Equivalences of 3- Manifolds with Boundaries, LNM
+761, Springer, Berlin.
+[12] E. Kalfagianni, Finite type invariants for knots in 3-manifolds, Topology 37
+(1998), 3, 673–707.
+[13] E. Kalfagianni and X.-S. Lin, The HOMFLY polynomial for links in rational ho-
+mology 3-spheres, Topology Vol 38 (1999), 1, 95-115.
+[14] E. Kalfagianni, Power series link invariants and the Thurston norm, Topology
+Appl. 101 (2000), no. 2, 107–119.
+[15] U. Kaiser, Presentations of homotopy skein modules of oriented 3-mani folds,J.
+Knot Theory Ramifications 10 (2001), no. 3, 461–491.
+[16] U. Kaiser, Quantum deformations of fundamental groups of oriented 3-m anifolds,
+Trans. Amer. Math. Soc. 356 (2004), no. 10, 3869–3880.
+[17] L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318
+(1990), no. 2, 417–471.
+[18] L.H.Kauffman, Knotsandphysics. Thirdedition. SeriesonKnotsandEverything,
+1. World Scientific Publishing Co., Inc., River Edge, NJ, 200 1.
+[19] P. Kirk and C. Livingston, Knot invariants in 3-manifolds and essential tori,
+Pacific J. Math. 197(2001), no. 1, 73-96.
+[20] T. Le, J. Murakami and T. Ohtsuki On a universal perturbative invariant of 3-
+manifolds, Topology 37 (1998), 3, 539–574.
+[21] X.-S. Lin, Finite type invariants of 3-manifolds, Topology Vol 33 (1994) 1, 45-71.
+[22] D. Rutherford, Thurston-Bennequin number, Kauffman polynomial, and ruling
+invariants of a Legendrian link: the Fuchs conjecture and be yond,Int. Math. Res.
+Not. 2006, Art. ID 78591.26 E. KALFAGIANNI
+[23] J. Przytycki, A q-analogue of the first homology group of a 3-manifold , Contem-
+porary Mathematics Volume 214, 1998, 135–143.
+[24] J. Przytycki, Skein modules, math.GT/0602264.
+[25] N. Reshetikhin and V. Turaev, Invariants of 3-manifolds via link polynomials and
+quantum groups, Invent. Math. 103 (1991), no. 3, 547–597.
+[26] S. Tabachnikov, Estimates for the Bennequin number of Legendrian links from
+state models for knot polynomials, Math. Res. Lett. 4 (1997), no. 1, 143–156.
+[27] V. Vassiliev, Cohomology of knot spaces. Theory of singularities and its applica-
+tions,23–69, Adv. Soviet Math., 1, Amer. Math. Soc., Providence, R I, 1990.
+[28] V.Vassiliev, On invariants and homology of spaces of knots in arbitrary ma nifolds.
+Topics in quantum groups and finite-type invariants, 155–182, Amer. Math. Soc.
+Transl. Ser. 2, 185, Amer. Math. Soc., Providence, RI, 1998.
+[29] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys.
+121 (1989), 360–379.
+(kalfagia@math.msu.edu) E. Kalfagianni Mathematics Department, Michigan
+State University, E. Lansing, MI 48824, USA
\ No newline at end of file
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+arXiv:1001.0176v4  [math.RA]  19 Jun 2012A note on the Schur multiplier of a nilpotent Lie
+algebra
+Peyman Niroomanda, Francesco Russob
+aSchool of Mathematics and Computer Science Damghan University o f Basic Sciences,
+Damghan, Iran.
+bUniversity of Naples Federico II via Cinthia, 80126, Naples, Italy.
+Abstract
+ForanilpotentLiealgebra Lofdimension nanddim( L2) =m(m≥
+1), we find the upper bound dim( M(L))≤1
+2(n+m−2)(n−m−1)+1,
+whereM(L) denotes the Schur multiplier of L. In case m= 1 the
+equality holds if and only if L∼=H(1)⊕A, where Ais an abelian
+Lie algebra of dimension n−3 andH(1) is the Heisenberg algebra of
+dimension 3.
+Key Words : Schur multiplier, nilpotent Lie algebras.
+2000 Mathematics Subject Classification : Primary 17B30; Secondary
+17B60, 17B99.
+1 Introduction
+It is well known that restrictions on the Schur multiplier of finitep–groups
+(pa prime) are related to significant information on the Schur m ultiplier
+M(L) of a nilpotent Lie algebra Lof dimension n. Many times it is possible
+to get structural results on Lonly looking at the size of M(L). This fact
+was already noted by Batten and others in [1, Theorem 5]. Give nt(L) =
+1
+2n(n−1)−dim(M(L)), they prove in [1, Theorem 3] that t(L) = 1 if and
+only ifL∼=H(1), where H(1) is the Heisenberg algebra of dimension 3.
+Moreover [1, Theorem 5] shows that t(L) = 2 if and only if L∼=H(1)⊕A,
+whereAis an abelian algebra of dim( A) = 1.
+To convenience of the reader, we recall that a finite dimensio nal Lie
+algebraLis called Heisenberg provided that L2=Z(L) and dim( L2) = 1.
+Suchalgebrasareodddimensionalwithbasis v1,...,v2m,vandtheonlynon-
+zero multiplication between basis elements is [ v2i−1,v2i] =−[v2i,v2i−1] =v
+1fori= 1,...,m. The symbol H(m) denotes the Heisenberg algebra of
+dimension 2 m+1.
+There are successive contributions on the same line of inves tigation, once
+we prescribea value for t(L). In [3] the cases t(L) = 3,4,5,6 are studied and
+weaker characterizations are obtained (see [3, Theorems 2, 3, 4]). Under the
+same prospective we should read [2, 5], where homological ma chineries are
+involved.
+It is instructive to note that in [1, Section 1] Batten and oth ers declare
+explicitly that their contributions originated from the cl assification of Zhou
+in [9], where a corresponding situation for p–groups was analyzed. In a
+certain sense the same motivation allows us to write the pres ent paper.
+A classic restriction of Jones [4, Theorem 3.1.4] on the Schu r multiplier
+of a non-abelian p–group has been recently improved by the first author.
+More precisely it is proved in [6] that a non-abelian p–groupGof order pn
+with derived subgroup of order pkhas
+|M(G)| ≤p1
+2(n+k−2)(n−k−1)+1.
+In particular,
+|M(G)| ≤p1
+2(n−1)(n−2)+1
+and the equality holds in this last bound if and only if G=H×Z, whereH
+is extra special of order p3and exponent p, andZis an elementary abelian
+p–group.
+The present paper is devoted to obtain similar results for Li e algebras.
+2 Preliminaries
+The present section illustrates how the ideas of Zhou [9] hav e been adapted
+in [3] to the context of Lie algebras. This is an important fee dback for our
+main theorems. For instance, the following is analogous to [ 4, Theorem
+2.5.2].
+Proposition 2.1. (See[1, Lemma 4] ).LetLbe a finite dimensional Lie al-
+gebra,Kan ideal of LandH=L/K. Then there exists a finite dimensional
+Lie algebra Jand and ideal MofJsuch that
+(i)L2∩K∼=J/M;
+(ii)M(L)∼=M;
+(iii)M(H)is an epimorphic image of J.
+2The Schur multiplier of the direct product of two finite group s is equal to
+the direct product of the Schur multipliers of the two factor s plus the tensor
+product of the abelianization of the two groups (see [4, Theo rem 2.2.10]).
+This is a general fact of homology, which is known as the K¨ unn eth Formula
+(see [7]), and it is true also for two finite dimensional Lie al gebrasHandK.
+The K¨ unneth Formula was originally obtained by Schur in 190 4 (see [8]).
+We recall that the symbol ⊗denotes the usual tensor product of abelian Lie
+algebras. Then we have
+M(H⊕K) =M(H)⊕M(K)⊕(H/H2⊗K/K2).
+At this point we may give a short proof of [1, Theorem 1] as foll ows.
+Theorem 2.2. LetAandBbe finite dimensional Lie algebras. Then
+dim(M(A⊕B)) = dim( M(A))+dim( M(B))+dim( A/A2⊗B/B2).
+Proof.Use the K¨ unneth Formula above mentioned.
+The following result is proved in [4, Theorem 2.5.5 (ii)] for groups.
+Corollary 2.3. LetLbe a finite dimensional Lie algebra, Kan ideal of L
+andH=L/K. Then
+dim(M(L))+dim(L2∩K)≤dim(M(H))+dim(M(K))+dim(H/H2⊗K/K2).
+Proof.From Theorem 2.2 it is enough to prove that dim( M(L))+dim( L2∩
+K)≤dim(M(H⊕K)). From Proposition 2.1,
+dim(M(L))+dim(L2∩K) = dim( M)+dim(J/M) = dim( J) = dim( ǫ(M(H)))
+for a suitable epimorphism ǫof Lie algebras. Since ǫsends systems of
+generators in systems of generators, dim( ǫ(M(H))≤dim(M(H⊗K)), as
+claimed.
+Now we mention three results from [1, 2, 3, 5] to convenience o f the
+reader.
+Lemma 2.4. (See[2, Example 3] and[5, Theorem 24] ).
+(i)dim(M(H(1))) = 2 .
+(ii)dim(M(H(m))) = 2m2−m−1for allm≥2.
+3Lemma 2.5. (See[1, Lemma 3] ).A Lie algebra Lof dimension nis abelian
+if and only if dim(M(L)) =1
+2n(n−1).
+Proposition 2.6. (See[3, Proposition 1] ).A nilpotent Lie algebra Lof di-
+mension nhasdim(L2)+dim(M(L))≤1
+2n(n−1).
+Hardy and others declare in [3] that they were working on high er values
+oft(L) (namely, the case t(L)≥9 is still open). Their investigations come
+from the bound in Proposition 2.6.
+3 Main Theorem
+Thefollowing resultprovides aboundwhich is less than theb oundinPropo-
+sition 2.6 except for the case L∼=H(1).
+Theorem 3.1. LetLbe a nilpotent Lie algebra of dim(L) =nanddim(L2) =
+m(m≥1). Then
+dimM(L)≤1
+2(n+m−2)(n−m−1)+1.
+Moreover, if m= 1, then the equality holds if and only if L∼=H(1)⊕A,
+whereAis an abelian Lie algebra of dim(A) =n−3.
+Proof.Assume dim( L2) = 1. Then L/L2is an abelian Lie algebra of
+dim(L/L2) =n−1. Since L2⊆Z(L), we may consider a complement
+H/L2ofZ(L)/L2inL/L2. So we have L=H+Z(L) andL2=H2. On the
+other hand, H∩Z(L)⊆Hand soZ(H) =L2. SinceL2⊆Z(L),L2must
+have a complement KinZ(L). LetL2⊕K=Z(L) so we have L∼=K⊕H.
+By Theorem 2.2,
+dim(M(K⊕H)) = dim( M(K))+dim( M(H))+dim( K/K2⊗H/H2).
+SinceHis a Heisenberg algebra and Kis abelian, two cases should be
+considered.
+Case 1. Assume dim( H) = 2m+1 form≥2. By Lemmas 2.4 and 2.5,
+dim(M(H)) = 2m2−m−1,
+dim(M(K)) =1
+2(n−2m−1)(n−2m−2),
+dim(K⊗H/H2) = dim( K)·dim(H/H2) = (n−2m−1)(2m).
+4and we deduce that
+dim(M(K⊕H)) =1
+2(n−2m−1)(n−2m−2)+(2m2−m−1)+(n−2m−1)(2m)
+=1
+2n(n−3)<1
+2(n−1)(n−2)+1.
+Case 2. Assume m= 1 and H∼=H(1). By Lemmas 2.4 and 2.5,
+dim(M(H)) = 2,
+dim(M(K)) =1
+2(n−3)(n−4),
+dim(K⊗H/H2) = dim( K)·dim(H/H2) = (n−3)·(2)
+and we deduce that
+dim(M(K⊕H)) =1
+2(n−3)(n−4)+2+2( n−3)
+=1
+2n(n−3)+2.
+Now we proceed by induction on m.
+Letm >1. Since 0 /negationslash=L2∩Z(L), there exists an ideal Kof dimension 1
+contained in L2∩Z(L). By induction hypothesis and Lemma 2.3, we have
+1+dim( M(L))≤dim(M(L/K))+dim( M(K))+dim( L/L2⊗K)
+and so
+dim(M(L))≤1
+2(n+m−4)(n−m−1)+1+ n−m−1
+=1
+2(n+m−2)(n−m−1)+1,
+as claimed.
+References
+[1] P. Batten, K. Moneyhun and E. Stitzinger, On characteriz ing nilpotent
+Lie algebras by their multipliers, Comm. Algebra 24(1996), 4319–4330.
+[2] P. Batten and E. Stitzinger, On covers of Lie algebras, Comm. Algebra
+24, (1996), 4301–4317.
+5[3] P. Hardy and E. Stitzinger, On characterizing nilpotent Lie algebras by
+their multipliers t(L) = 3,4,5,6,Comm. Algebra 26(1998), 3527–3539.
+[4] G. Karpilovsky, The Schur multiplier , London Math. Soc. Monogr.
+(N.S.) 2, London, 1987.
+[5] K. Moneyhun, Isoclinisms in Lie algebras, Algebras Groups Geom. 11
+(1994), 9–22.
+[6] P. Niroomand, On the order of Schur multiplier of non-abe lianp–
+groups,J. Algebra 322(2009), 4479–4482.
+[7] J. Rotman, An Introduction to Homological Algebra , Academic Press,
+San Diego, 1979.
+[8] I. Schur, Untersuchungen ber die Darstellung der endlic hen Gruppen
+durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 132
+(1907), 85--137
+[9] X. Zhou, Ontheorder oftheSchurmultiplier offinite p–groups, Comm.
+Algebra22(1994), 1–8.
+6
\ No newline at end of file
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+arXiv:1001.0177v3  [math.AG]  17 Aug 2010ON THE SLOPE OF RELATIVELY MINIMAL FIBRATIONS ON
+RATIONAL COMPLEX SURFACES
+CLAUDIA R. ALC ´ANTARA, ABEL CASTORENA, ALEXIS G. ZAMORA
+Abstract. Given a relatively minimal fibration f:S→P1, defined on a ra-
+tional surface S, with a general fiber Cof genus g, we investigate under what
+conditions the inequality 6( g−1)≤K2
+foccurs, where Kfis the canonical
+relative sheaf of f. We give sufficient conditions for having such inequality,
+depending on the genus and gonality of Cand the number of certain excep-
+tional curves on S. We illustrate how these results can be used for constructin g
+fibrations with the desired property. For fibrations of genus 11≤g≤49 we
+prove the inequality:
+6(g−1)+4−4√g≤K2
+f.
+1.Introduction
+LetSbe a projective complex nonsingular surface. A fibration f:S→X
+onSis a morphism onto a projective curve Xwith connected fibers. Through-
+out this paper Xwill be equal to P1. We use the usual identification of divisors
+andtheir associatedsheavesand write, forexample, Hi(L) instead of Hi(S,OS(L)).
+Given a fibration f, the relative canonical sheaf of fis defined as Kf=KS⊗
+f∗K−1
+X, ifXis rational then Kf=KS(2C), withCa general fiber (which is as-
+sumed to havegenus g≥2). This sheafis known to be big and nef for non-isotrivial
+and relatively minimal fibrations, a result proved in the foundational papers by
+Arakelov and Parshin (see [1] and [9]). In the case when Sis rational and g >0,
+Kfis nef even if we drop the hypothesis on isotriviality. This is due to the f act
+that in this case Kfis an effective divisor, thus Kf·E <0 would imply that Eis
+a vertical ( −1)-curve which gives a contradiction with the relatively minimal hy-
+pothesis (see proof of 2.1).
+The basic numerical invariant associated to a non-isotrivial fibratio n is the so
+calledslopeoffdefined as:
+λf=K2
+f
+deg(f∗Kf).
+In the case of our interest, when Sis a rational surface, deg(f∗Kf) =g(see
+Lemma 2.2) for any fibration fandλf=K2
+f/g.
+2000Mathematics Subject Classification. Primary 14D06, 14J26.
+Key words and phrases. rational surface, fibration.
+The first author was partially supported by CONACyT Grant 058 486. The second author
+was partially supported by CONACyT Grant 48668-F and PAPIIT Grant IN100909-2. The third
+author was partially supported by CONACyT Grants 25811.
+12 CLAUDIA R. ALC ´ANTARA, ABEL CASTORENA, ALEXIS G. ZAMORA
+The study of the restrictions that the slope of a fibration must sat isfy in relation
+to the genus gis a central issue in the theory. As noted before, in the case of
+rational surfaces the study of λfis equivalent to that of K2
+f.
+This paper is devoted to the relation between K2
+fandgin the case of rational
+surfaces. Beside its importance for the study of the slope, this re lation is relevant
+for another problem, the bounding of the minimal number σof singular fibers that
+a semi-stable non isotrivial fibration must have.
+The strict canonical inequality (see [11] and [13]) states that:
+K2
+f<(σ−2)(2g−2),
+for any semi-stable, non isotrivial fibration f:S→P1of genus g≥2.
+In this way inequalities of the sort of n(g−1)≤K2
+f, for some integer n, lead to
+lower bounds for the number σ(see [11] and [12]). For instance, the inequality
+6(g−1)≤K2
+f,
+implies, for semistable and non isotrivial fibrations, that σ≥6.
+For the case of surfaces of non-negative Kodaira dimension it is kno wn that in
+fact the inequality 6( g−1)≤K2
+f(and in consequence σ≥6) holds for any semi-
+stable and non-isotrivial fibration (see [6] and [12]).
+However, it is a hard problem to determine for which fibrations on a ra tional or
+ruled surface the inequality 6( g−1)≤K2
+fis valid. Simple examples of rational
+surface admitting a fibration for which K2
+f<6(g−1) are shown after the statement
+of Theorem 3.7. In this paper we obtain several general conditions to guarantee
+the validity of this inequality for rational surfaces.
+Our method is based on the study of the linear systems |C+nKS|withn= 2,3.
+If any of these linear systems is non-empty, then it is possible to com pute its
+Zariski-Fujita decomposition P+N, withPa nef divisor. The resulting inequality
+P2≥0 gives inequalities involving K2
+f,gand some auto-intersection numbers of
+exceptional divisors on S. If moreover, Pis big then χ(P+KS)≥0 also gives
+useful inequalities. Extra hypotheses on the genus gand the gonality of Callows
+us to guarantee the non-emptiness of these linear systems. Thes e hypotheses are
+imposed in order to be able to apply Reider’s method to the study of th e linear
+systems. These results are summarized in Theorems 3.3 and 3.7.
+First of all we can compute the negative part N1of the Zariski-Fujita decompo-
+sition ofC+2KSand the auto-intersection N2
+1=−l.N1is given by the expression:
+N1=s/summationdisplay
+i=1/bracketleftbig
+(l(Γi)+1)Γ i+l(Γi)/summationdisplay
+j=1(l(Γi)−j+1)Eij/bracketrightbig
+,
+where{Γi}is the set of ( −1)−sections of fand/summationtextl(Γi)
+j=1Eijis a maximal chain
+(possibly empty) of vertical ( −2) curves satisfying:ON THE SLOPE OF FIBRATIONS 3
+Γi·Ei1= 1, Eik·Eim=/braceleftbigg
+1 if|k−m|= 1,
+0 otherwise.
+Thus, in this notation l(Γi) is the length of the chain of ( −2)−curves attached
+to Γi, andsthe number of −1 sections of f. The resulting auto-intersection num-
+berl=−N2
+1isl=/summationtext
+i(l(Γi)+1). Wecallsuchachainof( −2)-curvesa( −2)-divisor.
+Theorem 3.3 Letf:S→P1be a non-isotrivial relatively minimal fibration on a
+rational surface S, with general fiber Cof genus g. Then the following statements
+hold:
+i) IfC+2KSis effective, then C+2KS−N1is nef, and
+0≤4K2
+f−24(g−1)+l.
+ii) Ifg≥7and the gonality of Cis at least 4, thenC+2KSis effective.
+iii) Ifg≥11and the gonality of Cis at least 5, then C+2KS−N1is also big
+and
+0≤3K2
+f−19(g−1)+l+1.
+In particular if l+1≤g−1 then 6( g−1)≤K2
+f.
+We remark that if we contract the ( −1)−sections we obtain a new rational sur-
+faceand an associatedpencil havingits base locusjust on the image ofthe Γ i. After
+this contraction the images of Ei1are (−1) curves and can be contracted again to
+a non-singular rational surface if we continue this procedure we ca n finally contract
+the divisor N1and obtain a new rational nonsingular surface and a pencil on it
+with its base locus in the image of the connected components of N1. Conversely,
+resolvingthe base locus of the resulting pencil we obtain the original fibration on S.
+In order to explain the content of Theorem 3.3, we introduce a basic example.
+Start with a pencil generated by two irreducible, nonsingular plane c urves of de-
+greedintersecting each other transversally and consider the fibration f:S→P1
+obtained by blowing up the d2base points. The invariants associated are:
+g−1 =d(d−3)
+2, K2
+f=KS(2C)2= 3d2−12d+9,
+and finally, P2being a minimal surface, lcoincides in this case with the number of
+(−1) sections of f, therefore l=d2.
+In this way,
+K2
+f−6(g−1) = 9−3d,
+which is negative for d >3. On the other hand, C+2KSis effective for d≥6 (see
+the computations in Section 4, Example (1)).4 CLAUDIA R. ALC ´ANTARA, ABEL CASTORENA, ALEXIS G. ZAMORA
+Now, ifweaddtheprescribedterminTheorem3.3i), namely l/4(wearedividing
+the inequality by 4), then we obtain:
+d2
+4−3d+9,
+that is in fact positive for all values of d. Note also that in these examples the
+gonality of the general fiber is d−1.
+Returning to the general situation, a similar analysis can be made for C+3KS.
+The negative part of C+ 3KS−N1in the Zariski-Fujita decomposition is of the
+formN1+N′
+1+N2. Explicitly the divisors N′
+1andN2, are given by:
+N′
+1=l′(Γi)/summationdisplay
+j=1(l′(Γi)−j+1)E′
+ij,
+whereE′
+i=/summationtext
+iE′
+ijare maximal ( −2) divisors such that E′
+i·C= 1 and E′
+i·Γi= 1,
+and
+N2=t/summationdisplay
+i=1[(m(∆i)+1)∆ i+m(∆i)/summationdisplay
+j=1(m(∆i)−j+1)Fij],
+with{∆1,...,∆t}the set of ( −1) curves on Ssatisfying ∆ i·C= 2, and Fi=/summationtextFij
+are vertical maximal ( −2)−divisors such that Fi·∆i= 1.
+Denotel′=/summationtexts′
+i=1(l′(Γi)+1) and m=/summationtextt
+i=1(m(∆i)+1). Then N′
+12=−l′and
+N2
+2=−m.
+With this notation in mind we have:
+Theorem 3.7 Letf:S→P1a non-isotrivial relatively minimal fibration on a
+rational surface S, with general fiber Cof genus g. Then the following statements
+hold:
+i) IfC+3KS−N1is effective, then C+3KS−2N1−N′
+1−N2is nef, and
+0≤9K2
+f−60(g−1)+4l+l′+m.
+ii) If the gonality of Cis at least 6andg≥23, thenC+3KS−N1is effective.
+Once again, the divisors N′
+1andN2can be contracted to obtain a new non-
+singular surface with a pencil associated to f. IfN2is non-empty the generic
+element of this pencil will have in general singularities. These singular ities are of
+nodal type if the chains Fijare empty.
+Thus, in order to produce examples of fibrations satisfying the hyp othesis of
+Theorems 3.3 or 3.7 we must start with pencils of curves with a certain bounded
+kind of singularities. This is the task in section 4, where we explain how t he pre-
+vious theorems can be used in order to obtain fibrations satisfying 6 (g−1)≤K2
+f.
+The examples are given by blowing up the base locus of pencils of nodal curves onON THE SLOPE OF FIBRATIONS 5
+minimal rational surfaces.
+Before section4, asa part ofa preparatorydiscussionfor Theor em3.7, we obtain
+Theorem 3.5, which gives a general and uniform bound for the slope o f a relatively
+minimal fibration.
+Theorem 3.5 Letf:S→P1be a relatively minimal fibration on a rational
+surfaceS. If the genus gof the fiber is greater than 11and the gonality is at least
+5, then
+(5+1
+2)(g−1)−5≤K2
+f
+holds.
+Finally, returning to the example on pencils of non-singular plane curv es, note
+thatinordertoobtainapositivequantityitissufficienttoadd3 d−9toK2
+f−6(g−1).
+The number 3 d−9 is approximately equal to√g. In section 5 we find another class
+of fibrations satisfying an inequality of the type:
+6(g−1)≤K2
+f+O(√g),
+(denoting by O(√g) a quantity comparable to√g). More precisely we prove:
+Theorem 5.2 Iff:S→P1, is a relatively minimal fibration of genus 11≤g≤49
+on a rational surface Ssuch that the gonality of the general fiber Cis at least 5and
+the surface Tobtained by blowing-down the divisor N1satisfies that K2
+T<0, then:
+6(g−1)+4−4√g≤K2
+f.
+Theorem 5.2, together with the previous example gives some evidenc e in the
+sense that probably any relatively minimal fibration on a rational sur face satisfies
+an inequality of the type:
+6(g−1)≤K2
+f+O(√g).
+It should be noticed that Theorem 3.3 is valid on any algebraic surface satisfying
+h1(S,OS) =h2(S,OS) = 0, andallthe resultsin ourpaperarestillvalid ifmoreover
+weassumethat K2
+T≤9, withKTstandingforthesurfaceobtainedaftercontracting
+the support of the divisor N1. However, as remarked before, the inequality
+6(g−1)≤K2
+fistruefornon-isotrivialfibrationsonsurfacesofnon-negativeK odaira
+dimension. Obtaining analogous results in the case of non-rational s urfaces of
+negative Kodaira dimension seems to be of natural interest.
+The authors express their gratitude to anonymous referees for their valuable
+comments and critics.
+2.Preliminaries and notation
+We always denote by f:S→P1a relatively minimal fibration on a rational
+surfaceS. We will denote by KSa canonical divisor of S.6 CLAUDIA R. ALC ´ANTARA, ABEL CASTORENA, ALEXIS G. ZAMORA
+Cwill denote a general fibre of fwhich is assumed to have genus g, and
+Kf=KS(2C) will be the relative canonical sheaf of f.
+In order to simplify the notation we shall write:
+a=K2
+fandb=g−1.
+Some standard equalities are used systematically:
+C·Kf= 2b, K2
+S=a−8b.
+The following Lemma, a result taken from [10], will be invoked several t imes (S
+is here an arbitrary surface):
+Lemma 2.1. LetLbe a nef divisor on S.
+i) IfL2≥5and|L+KS|=∅then there exists a base point free pencil |E|onS
+such that either E·L= 0or1.
+ii) IfL2≥10and|L+KS|does not define a birational map then there exists a
+base point free pencil |E|onSsuch that either E·L= 1or2.
+Proof.ii) is just Corollary 2 in [10]. Even when i) is not explicitly stated in [10] it
+follows just by the same argument used in the proof of Corollary 2 in [ 10]. /square
+Now, in part for the sake of completeness, in part for illustrating th e kind of
+argument that will be used in the sequel, we state and prove a Lemma that is by
+now well known (see [6] and [12]).
+Lemma 2.2. Letfbe, as before, a relatively minimal fibration on a rational su rface
+S. Suppose that g >0, then|C+KS|is effective and nef.
+Proof.We can be more specific about the dimension of the space of sections of
+C+KS.
+Indeed, thesurface Sisrational,thus h0(KS) =h1(KS) = 0. ByLeray’sspectral
+sequence we have also h0(f∗KS) =h1(f∗KS) = 0. Thus in the decomposition of
+f∗KSlike a sum of invertible sheaves in P1we must have:
+f∗KS=g/circleplusdisplay
+OP1(−1),
+therefore,
+h0(C+KS) =h0(f∗KS⊗OP1(1)) =g.
+This proves the first assertion. Now, if ( C+KS)·E <0 for some irreducible
+curveE, then, being Cnef,Emust be a vertical ( −1)−curve (see [5], proof of
+Proposition 4.1) and in consequence C·E= 0. But this is impossible because fis
+relatively minimal. This proves the Lemma. /squareON THE SLOPE OF FIBRATIONS 7
+3.Adjoint systems and the slope of f
+In this section we investigate the properties of the linear systems |C+mKS|for
+m= 1,2,3. We apply their properties to the study of the slope of f.
+Lemma 3.1. Ifb≥6and the gonality of Cis at least 4, thenh0(C+2KS)>0
+anda≥5b.
+Proof.This follows easily from Corollary 4.4 of [7].
+/square
+IfC+2KSis effective, it admits a Zariski-Fujita decomposition as the sum of a
+nef divisor and a negative part, let us compute this negative part.
+Let
+N1=s/summationdisplay
+i=1/bracketleftbig
+(l(Γi)+1)Γ i+l(Γi)/summationdisplay
+j=1(l(Γi)−j+1)Eij/bracketrightbig
+,
+where{Γ1,...,Γs}is the set of ( −1)−sections of fand/summationtextEijis a maximal chain
+(possibly empty) of vertical ( −2) curves satisfying:
+Γi·Ei1= 1, Eik·Eim=/braceleftbigg1 if|k−m|= 1,
+0 otherwise,
+andl(Γi) will denote the length of the maximal chain/summationtextEij. If the chain is empty,
+then we convey that l(Γi) = 0. In the language of [3], this is expressed by saying
+that the divisor Ei=/summationtextEijis a (−2)−curve and Γ i·Ei= 1. We prefer to call
+such a chain a ( −2)−divisor, in order to distinguish it from the irreducible case. It
+should be noticed that there is only one maximal connected ( −2)−divisor (possibly
+empty) attached to each ( −1)−section.
+We will also use the notation Γ =/summationtexts
+i=1Γi.
+Lemma 3.2. The negative part of C+2KSin the Zariski-Fujita decomposition is
+N1andN2
+1=−l, wherel=/summationtexts
+i=1(l(Γi)+1).
+Proof.Although our calculus relies on the argument used in [5] Proposition 4 .1, it
+is slightly different and uses the Zariski-Fujita algorithm in order to ob tain a more
+explicit expression.
+If (C+2KS)·D <0 for some irreducible curve D, then being Cnef andD2<0
+we must have that Dis a (−1)-curve such that C·D≤1. Since fis relatively
+minimal we conclude that Dis a (−1)-section.
+Next we need to find the irreducible curves Dsuch that:
+(1) ( C+2KS−Γ)·D <0.
+SinceD/ne}ationslash= Γiwe have that it is not a ( −1)−curve, therefore D·KS≥0 and
+D·Γ>0.8 CLAUDIA R. ALC ´ANTARA, ABEL CASTORENA, ALEXIS G. ZAMORA
+LetTbe the surface obtained by contracting Γ. Then in T, (1) becomes:
+(C0+KT)·π∗D <0,
+withπstanding for the contraction and C0∈ |π(C)|.
+Just by the same argument as before we get that π∗Dis a (−1)−curve with
+0≤C0·π∗D≤1. In particular Γ ·D≤1. Thus we obtain that D·Γ = 1,
+KS·Γ = 0 and C·D= 0. In this way D=Ei1for some i.
+Now, the Zariski-Fujita algorithm commands solving the system of eq uation in
+αj,βk:
+(C+2KS−Γ)·Γi= (/summationdisplay
+αjΓj+/summationdisplay
+βkEk1)·Γi,
+(C+2KS−Γ)·Ei1= (/summationdisplay
+αjΓj+/summationdisplay
+βkEk1)·Ei1,
+fori= 1,...s. It is easy to see that the solution is αj=βk= 1 for all jandk. So
+in this step we need to subtract Γ+/summationtextEi1toC+2KS−Γ.
+The rest of the computation is iterative. Assume that on some step of the
+algorithm we have obtained the divisor:
+C+2KS−N,
+withN=/summationtext
+i[(li+1)Γi+/summationtextli
+j=1(li−j+1)Eij].
+LetTnow be the surface obtained by contracting the support of N. We obtain,
+with the analogous notation:
+(C0+KT)·π∗D <0,
+therefore, as before, C·D= 0 and N·D= 1. So, Dmust be equal to Ei,li+1for
+somei. The Zariski-Fujita algorithm leads again to subtract/summationtext
+i[Γi+/summationtextli+1
+j=1Eij].
+Therefore, the negative part is:
+N1=s/summationdisplay
+i=1/bracketleftbig
+(l(Γi)+1)Γ i+l(Γi)/summationdisplay
+j=1(l(Γi)−j+1)Eij/bracketrightbig
+.
+Now we will calculate N2
+1:ON THE SLOPE OF FIBRATIONS 9
+N2
+1=/parenleftBigs/summationdisplay
+i=1/bracketleftbig
+(l(Γi)+1)Γ i+l(Γi)/summationdisplay
+j=1(l(Γi)−j+1)Eij/bracketrightbig/parenrightBig2
+=s/summationdisplay
+i=1/bracketleftbig
+(l(Γi)+1)Γ i+l(Γi)/summationdisplay
+j=1(l(Γi)−j+1)Eij/bracketrightbig2
+(because Γ i·Γk= Γk·Eij=Ekj·Eij= 0 ifk/ne}ationslash=i)
+=l(Γi)/summationdisplay
+j=1/bracketleftbig
+−(l(Γi)+1)2+2(l(Γi)+1)l(Γi)+(l(Γi)/summationdisplay
+j=1(l(Γi)−j+1)Eij)2/bracketrightbig
+(because Γ i·Eij= 0 ifj >1)
+=l(Γi)/summationdisplay
+j=1/bracketleftbig
+−(l(Γi)+1)2+2(l(Γi)+1)l(Γi)−2l(Γi)−1/summationdisplay
+j=1(l(Γi)−j+1)−2/bracketrightbig
+(because (l(Γi)/summationdisplay
+j=1(l(Γi)−j+1)Eij)2=l(Γi)/summationdisplay
+j=1(−2(l(Γi)−j+1)2)+2l(Γi)−1/summationdisplay
+j=1(l(Γi)−j+1)(l(Γi)−j))
+=l(Γi)/summationdisplay
+j=1/bracketleftbig
+−(l(Γi)+1)(l(Γi)+1−2l(Γi)+l(Γi))/bracketrightbig
+=−l.
+/square
+After these computations we have:
+Theorem 3.3. Letf:S→P1be a relatively minimal fibration on a rational
+surfaceS, with general fiber Cof genus g. Then the following statements hold:
+i) IfC+2KSis effective, then C+2KS−N1is nef, and
+0≤4K2
+f−24(g−1)+l.
+ii) Ifg≥7and the gonality of Cis at least 4, thenC+2KSis effective.
+iii) Ifg≥11and the gonality of Cis at least 5, then C+2KS−N1is also big
+and
+0≤3K2
+f−19(g−1)+l+1.
+In particular if l+1≤g−1thena≥6b.
+Proof.Parts i) and ii) follow from Lemma 3.2, the inequality in i) merely expresse s
+the fact that ( C+2KS−N1)2≥0.
+We proceed with the proof of iii). We will prove that |C+ 2KS|defines a bi-
+rationalmap. Thiswouldimplythatthenefpartofthedivisorisbig(se e[2], 14.18).
+Assume, for contradiction, that |C+2KS|does not define a birational map.We
+know, by Lemma 3.1, that a≥5b, thus, (C+KS)2=a−4b≥b≥10, since g≥11.
+By Lemma 2.1 ii), Sadmits a base point free pencil |E|withE·(C+KS) = 1 or 2.
+If, for instance, E·(C+KS) = 1, then10 CLAUDIA R. ALC ´ANTARA, ABEL CASTORENA, ALEXIS G. ZAMORA
+1 =KS·E+C·E= 2gE−2+C·E.
+The only possibility for this equality holding is gE= 0 and C·E= 3, but then
+Cmust be trigonal. Similarly, E·(KS+C) = 2 implies that Cis tetragonal or
+hyperelliptic. The last assertion follows from Mumford’s vanishing the orem:
+0≤χ(C+3KS−N1) =h0(C+3KS−N1)
+=(C+2KS−N1)·(C+3KS−N1)
+2+1 = 3K2
+f−19(g−1)+l+1.
+/square
+Now, we can make a similar analysis for the linear system |C+3KS|. We start
+by proving the following bound for l:
+Lemma 3.4. Assumeg≥11and the gonality of Cis at least 5, then
+l≤5
+2b+14.
+Proof.Letπ:S→Tbe the contraction of the divisor ( N1)red. After contracting
+Γi,Ei1is contracted to a ( −1) curve, and the same is valid for Ei,j+1after con-
+tracting Ei,j, so we see that Tis a nonsingular surface.
+Moreover,
+KS=π∗KT+N1,
+and,
+K2
+S= (π∗KT+N1)2=a−8b.
+Thus,π∗K2
+T=a−8b+l. Combining this with part iii) of Theorem 3.3, we
+obtain the equality:
+h0(C+3KS−N1)−1+2l−3π∗K2
+T= 5b.
+UsingK2
+T≤9 we obtain the desired inequality. /square
+So far we have obtained, under the hypothesis of Theorem 3.3, a bo und for the
+slope off:
+Theorem 3.5. Letf:S→P1be a relatively minimal fibration on a rational
+surfaceS. If the genus gof the fiber is greater than or equal to 11and the gonality
+ofCis at least 5then
+(5+1
+2)(g−1)−5≤K2
+f.
+Proof.The statement is just a combination of Theorem 3.3 and Lemma 3.4. /square
+We return to the analysis of the linear system |C+ 3KS|. The following is
+analogous to Lemma 3.1.
+Proposition 3.6. Ifb≥22and the gonality of Cis at least 6, then |C+3KS−
+N1| /ne}ationslash=∅.ON THE SLOPE OF FIBRATIONS 11
+Proof.The hypotheses of Theorem 3.3, iii) are satisfied, therefore C+2KS−N1is
+nef. Note that, by Lemma 3.4 and part iii) of Theorem 3.3
+(C+2KS−N1)2= 4a−24b+l
+≥4
+3(19b−l−1)−24b+l
+=1
+3(4b−l−4)≥4
+3b−5
+6b−6 =1
+2b−6.
+Thus, ifb≥22then ( C+2KS−N1)2≥5. ByLemma2.1i)if |C+3KS−N1|=∅,
+then there exists a base point free pencil |E|such that:
+E·(C+2KS−N1) = 1 or 0 .
+We have either that Eis contracted by |C+2KS−N1|orEis a rational curve
+withE2= 0 and E·(C+2KS−N1) = 1.
+In the first case Eis contracted as well by C+ 2KS, butC+KSis nef and
+(C+KS)2=a−4b≥10. Thus, by part ii) of Lemma 2.1, there exists a pencil
+|E′|, such that:
+(C+KS)·E′= 2 or 3.
+But then:
+C·E′−2≤2 or 3,
+which gives a contradiction with the assumption on the gonality of C.
+On the other hand, if Eis rational then E·KS=−2 and:
+E·C−4−N1·E= 1.
+Consider the P1−fibre bundle associated with E(see [3], V 4.3):
+Sφ/d47/d47R/d47/d47P1.
+Ris a Hirzebruch surface Fnand, ifn/ne}ationslash= 1 then Fnis minimal and N1must be
+contracted by φ. We conclude that N1is a sum of vertical curves with respect to
+|E|, andN1·E= 0.
+IfR=F1then we could have N1·E= 1, but in this case C·E= 6 and the
+image of CinF1becomes equivalent to Γ 0+6E, with Γ 0denoting the (-1)-section,
+that is, the image of N1underφ. A simple calculus using the adjunction formula
+onF1shows that b= 17. Therefore, the only possibility is N1·E= 0 andE·C≤5.
+This final contradiction proves the Proposition. /square
+Next we need to obtain the negative part of C+3KS−N1, the computation is
+quite analogous to that of the negative part of C+ 2KS, using in this case that
+C+2KS−N1is nef. This negative part is N1+N′
+1+N2with12 CLAUDIA R. ALC ´ANTARA, ABEL CASTORENA, ALEXIS G. ZAMORA
+N′
+1=l′(Γi)/summationdisplay
+j=1(l′(Γi)−j+1)E′
+ij,
+whereE′
+i=/summationtext
+iE′
+ijare maximal ( −2) divisors such that E′
+i·C= 1 and E′
+i·Γi= 1,
+of length l′(Γi); and
+N2=t/summationdisplay
+i=1[(m(∆i)+1)∆ i+m(∆i)/summationdisplay
+j=1(m(∆i)−j+1)Fij],
+with{∆1,...,∆t}the set of ( −1) curves on Ssatisfying ∆ i·C= 2, and Fi=/summationtextFij
+are vertical maximal ( −2)−divisors such that Fi·∆i= 1 and of length m(∆i). To
+each (−1)−section Γ ithere is associated a unique (possible empty) divisor E′
+iand
+the same is true for divisors ∆ iwith respect to the chains Fi.
+Denotel′=/summationtexts′
+i=1(l′(Γi)+1) and m=/summationtextt
+i=1(m(∆i)+1).
+As in the case of N2
+1, a similar calculus shows that N′
+12=−l′andN2
+2=−m.
+We have obtained, in analogy with Theorem 3.3:
+Theorem 3.7. Letf:S→P1be a relatively minimal fibration on a rational
+surfaceS, with general fiber Cof genus g. Then the following statements hold:
+i) IfC+3KS−N1is effective, then C+3KS−2N1−N′
+1−N2is nef, and
+0≤9K2
+f−60(g−1)+4l+l′+m.
+ii) If the gonality of Cis at least 6andg≥23, thenC+3KS−N1is effective.
+The proof of i) follows, as in the analogous case in Theorem 3.2, from t he fact
+that (C+3KS−2N1−N′
+1−N2)2≥0, since the divisor is nef. Part ii) follows from
+Proposition 3.6.
+In particular, under the hypothesis of Theorem 3.7 i), if 4 l+l′+m≤6(g−1),
+then 6(g−1)≤K2
+f.
+4.Examples
+In the following examples we start with a surface Tand a pencil Wof nodal
+curves on T. The fibration will be obtained in a surface Sby blowing-up the base
+locus of the pencil. The general fiber of the fibration will be denoted byCand will
+correspond with the proper transform of the general element C0∈W.
+In general it is not true that blowing up the base locus of a pencil on a minimal
+surfaceTgives rise to a relatively minimal fibration. The simplest example comes
+from considering the pencil generated by an irreducible conic QinP2and the prod-
+uct of two lines L1,L2that intersects in a point not contained in Q. After blowing
+up the base locus of this pencil the proper transform of both, L1andL2become
+vertical ( −1)−curves. More complicated examples can be constructed. Fortuna tely
+if we limit the singularities of the general element of the pencil we can g ain control
+of the situation.ON THE SLOPE OF FIBRATIONS 13
+Thus, we start by computing the conditions to have in the following ex amples
+relatively minimal fibrations. Consider a curve C0=D0+D1in the pencil W
+and letting p1,...,plbe the nonsingular points in the base locus of Wand letting
+q1,...,qmbe the nodal points in the base locus of W, we will use the following
+notation:
+l01= #{pk:pk∈D0}
+l02= #{qk:qkis simple for D0}
+m0= #{qk:qkis node for D0}
+l0=l01+l02.
+Suppose that the proper transform /tildewideD0⊂Sis a vertical (-1)-curve. We have the
+following equations:
+(i)/tildewideD2
+0=−1 =D2
+0−ℓ01−ℓ02−4m0,
+(ii)g/tildewideD0= 0, which implies thatD0·(D0+KT)
+2−m0=−1,
+(iii)/tildewideD0·/tildewideC0= 0, that is, D0·C0−ℓ01−2ℓ02−4m0= 0.
+From (i) and (ii) we have that D2
+0−ℓ01−ℓ02+ 1 = 2 D2
+0+ 2D0·KT+ 4, and
+D2
+0+2D0·KT+3 =−(ℓ01+ℓ02), that is, if D2
+0+2D0·KT+3>0 curveD0cannot
+exist. On the other hand we have from (i) and (iii) that D0·C0−ℓ01−2ℓ02=
+D2
+0−ℓ01−ℓ02+1, then D0·C0−D2
+0=ℓ02+1>0. This implies that if there exists
+curveD0the following two conditions hold simultaneously:
+D2
+0+2D0·KT<0, D0·C0−D2
+0>0.
+For the case T=P2we have that:
+/tildewiderD02=d2
+0−l0−4m0=−1
+g/tildewideD0−1 =d0(d0−3)
+2−m0=−1
+C0·/tildewideD0=d0d−l01−2l02−4m0= 0.
+whered0is the degree of D0. From the above equations we can conclude that
+d0(d0−6)+3 = −l0andd0(2d0−d)>0. Ifd0≥6 then we have a contradiction
+with the first equation and if d0<6 thend <12. In conclusion a nodal pencil of
+degreed≥12 inP2gives place to a relatively minimal fibration.
+Similar computations show that if T=F0and the class of C0is (α,β) with
+α,β≥8 then the associated fibration is relatively minimal.
+(1) We want to use Theorem 3.7 i) in orderto construct a pencil of pla ne curves
+such that the associated fibration satisfies 6( g−1)≤K2
+f.
+Consider positive integers l,m,dsatisfying the conditions l+ 4m=d2
+andl= 2m, i.e., 6m=d2. Fixmpointsq1,...,qm∈P2, ifV(q1,...,qm)
+denotes the linear system of plane curves of degree dhaving nodes at qi,
+then:14 CLAUDIA R. ALC ´ANTARA, ABEL CASTORENA, ALEXIS G. ZAMORA
+dimV(q1,...,qm)≥d(d+3)
+2−3m=l−2m+3d
+2=3d
+2,
+which is positive. Taking a general pencil contained in V(q1,...,qk), and
+blowing up its base locus, we will obtain a rational fibered surface S.
+Ford >9 the condition |C+3KS| /ne}ationslash=∅is satisfied because:
+KS= [−3,1,1,1,...,1]
+C= [d,−1,...,−1,−2,...,−2].
+The above equalities mean equality of classes in the Picard group of S,
+whereweusethestandardorderedbasisfor Pic(S)≃Pic(P2)⊕iΓiZ⊕j∆jZ
+with Γ ithe exceptional divisors associated with piand ∆ jthe exceptional
+divisors associated with qj, and denote by Hthe hyperplane class in P2.
+Therefore
+C+3KS= [d−9,2,...,2,1,...,1].
+Note that the decomposition of the divisor C+3KS= (d−9)H+(/summationtext2Γi+/summationtext∆i) is the Zariski-Fujita decomposition of C+3KS. In this case, being
+P2a minimal surface the divisor N1defined in the previous section is just/summationtextΓiandN2=/summationtext∆i. The numbers landmcoincide with the previously
+defined in section 3. By the genus formula 6 b= 3d2−9d−6m, in order to
+be in the hypothesisof the remarkafter Theorem3.7 we need the ine quality
+4l+m≤6bwhich is equivalent to 3 d≤m. If we take d= 18,m= 54 we
+obtain a curve of genus 82 , so with this numerical conditions we have a
+fibration satisfying 6 b≤a. Moreover the gonality of Cisd−2 = 16 (see
+[4]).
+(2) Consider the Hirzebruch surface T=F0and letC0be an effective divisor
+onTof class ( α,β). The arithmetic genus of C0isga= 1+C2
+0+C0·KT
+2, i.e.,
+ga−1 =αβ−α−β. Suppose also that C0hasmnodes. Then its geometric
+genusgisαβ−α−β+1−m. The classes of KSandCinPic(S) are given
+by:
+KS= [(−2,−2),1,1,...,1]
+C= [(α,β),−1,...,−1,−2,...,−2].
+Here we use again the standard ordered basis for Pic(S)≃Pic(F0)⊕i
+ΓiZ⊕j∆jZwith Γ ithe exceptional divisors associated with piand ∆ jthe
+exceptional divisors associated with qj. The divisors N1andN2, as defined
+in the previous section are given again in this example by N1=/summationtextΓiand
+N2=/summationtext∆i.
+Sinceh0(T,OT(C0))−1≥C2
+0−C0·KT
+2, we have that dim |C0|>0 if and
+only ifαβ+α+β >3m. In order to apply Theorem 3.7 we need |C+
+3KS−N1| /ne}ationslash=∅, for this, it is enough to have α≥7 andβ≥7, because
+3KS+C= [(α−6,β−6),−2,−2,−1,...,−1]. In order to guarantee thatON THE SLOPE OF FIBRATIONS 15
+the resulting fibration is relatively minimal we must take both α,β≥8.
+We also need to have 4 l+m≤6b. Since 2 αβ=C2
+0=l+ 4m, then
+8αβ−15m≤6αβ−6α−6β−6m, therefore 3 m≥2
+3αβ+2α+2β. Then
+the condition 3 m∈[2
+3αβ+ 2α+ 2β,αβ+α+β] is satisfied taking for
+example α=β= 8 and m= 26. Then if we blow-up the nodes in the
+pencil given by C0we obtain the following diagram:
+S
+f/d32/d32/d64/d64/d64/d64/d64/d64/d64/d64/d47/d47T
+/d15/d15
+P1,
+wherefis a fibration of genus 23 that satisfies 6 b≤a.
+(3) In this example we exhibit a fibration for which the equality K2
+f= 6(g−1)
+holds. Following the notation of the above example consider the numb ers
+α,β,landmsatisfying 3 m < αβ +α+β,l+ 4m= 2αβ, sincea=
+8(α−1)(β−1)−l−9mandb=αβ−α−β−m, the condition a= 6bis
+equivalent to having 2 α+2β=m+8. For example the numbers α=β= 8,
+m= 24 satisfy the above conditions. In this case we obtain a fibration o f
+genus 25. We must observe that the minimal degree of a map C0→P1is
+αbut we can not guarantee that αis the gonality of its normalization C,
+it could be in fact lower (see [8]).
+5.Slope of fibration with 11≤g≤49
+Let as before f:S→P1be a relatively minimal fibration on a rational surface.
+Throughthis section assume moreverthat g≥11and the gonalityof Cis at least 5.
+Using Lemma 2.2 and Lemma 3.1 we have that Kfis big and nef. Thus:
+h0(nKf) =nKf((n−1)Kf+2C)
+2+1
+=n(n−1)a+2bn
+2+1.
+On the other hand, consider the direct image sheaves:
+f∗nKf=b(2n−1)/circleplusdisplay
+i=1O(ai).
+NotethatbyMumford’svanishing h1(nKf−(n+1)C) =h1((n−1)(C+KS)+KS) =
+0 forn≥2, asC+KSis nef. Thus, by the projection formula
+ai−(n+1)≥ −1,
+ai≥n.
+In this way 2( n+1)(2n−1)b≤2h0(f∗nKf) = 2h0(nKf) =n(n−1)a+4bn+2.
+Substraction of both terms leads to the conclusion that q(x) = (a−4b)x2−(a−16 CLAUDIA R. ALC ´ANTARA, ABEL CASTORENA, ALEXIS G. ZAMORA
+2b)x+2b+2 is positive when evaluated in any integer n≥2.
+Some properties of this polynomial are summarized in:
+Proposition 5.1. Letf:S→P1be a semistable, non isotrivial fibration on a
+rational surface S. Then:
+i)qis positive when evaluated in any integer n.
+ii) Ifqhas real roots then they are located in (0,1)∪(1,2).
+iii) The discriminant of qis∆q= (a−6b)2−8(a−4b).
+iv) If∆q≤0then:
+6(g−1)+4−4√g≤K2
+f.
+Proof.i) By construction q(n)>0 forn≥2. It is easy to verify that q(1) = 2 and
+q(0) = 2b+2. Moreover, the critical value of qis2(a−4b)
+a−2bwhich turns out to satisfy
+0<2(a−4b)
+a−2b<2.
+This proves i) and ii). Part iii) is just a direct computation.
+For iv), consider the discriminant ∆ qas a polynomial in a:
+∆q(a) =a2−(12b+8)a+36b2+32b.
+If ∆q(a)≤0 thenais in the “negative region” of this quadratic function. Thus,
+12b+8−/radicalbig
+(12b+8)2−4(36b2+32b)
+2≤a.
+This proves the Proposition. /square
+The next Theorem gives an inequality for the slope of fibrations of ge nus 11≤
+g≤49, with an extra assumption on the surface Tobtained after blowing down
+the negative part of C+2KS.
+Theorem 5.2. Iff:S→P1is a relatively minimal fibration of genus 11≤g≤49
+on a rational surface Ssuch that the gonality of the general fiber Cis at least 5and
+the surface Tobtained by blowing-down the divisor N1satisfies that K2
+T<0, then:
+6(g−1)+4−4√g≤K2
+f.
+Proof.We can assume a≤6b. Following the proof of Lemma 3.4 and by Theorem
+3.3 (iii) we have 2 ℓ≤5b+1+3π∗(K2
+T), the condition K2
+T<0 implies ℓ+1≤5
+2b,
+since 3a≥19b−ℓ−1 we obtain11
+2b≤a. Hence we have:
+∆q= (a−6b)2−8(a−4b)≤(1/2)2b2−12b=b(b−48)
+4.
+Combining with part iv) of Proposition 5.1 we get the theorem. /squareON THE SLOPE OF FIBRATIONS 17
+References
+[1] Arakelov S.J.: Families of algebraic curves with fixed degeneracies . Math. USSR-Izv. 5, 1971.
+[2] Badescu L.: Algebraic Surfaces . Springer-Verlag, 2001.
+[3] Barth W., Peters C., Van de Ven A.: Compact Complex Surfaces . Springer Verlag, 1984.
+[4] Coppens M., Kato T.: The gonality of smooth curves with plane models. Manuscripta math.
+70, 5-25, 1990.
+[5] Calabri A., Ciliberto C.: Birational classification of curves on rational surfaces. Arxiv
+0906.49603.
+[6] Kitagawa S., Konno K.: Fibred rational surfaces with extremal Mordell-Weill lati ces.Math-
+ematische Zeitschrift, 251, 179-204, 2005.
+[7] Konno K.: Clifford index and the slope of fibered surfaces. J. Alg. Geom. 8, 207-220, 1999.
+[8] Martens G.: The gonality of curves on a Hirzebruch surface. Arch. Math., Vol 67, 349-352,
+1996.
+[9] Parshin A.: Algebraic curves over functions fields . Izvest. Akad. Nauk., 32, 1968.
+[10] Reider I.: Vector Bundles of rank 2and linear systems on algebraic surfaces. Ann. Math.
+127, 309-316, 1988.
+[11] Tan S-L: The minimal number of singular fibers of a semi stable curve ov erP1.J. Algebraic
+Geom., 4, 591–596, 1995.
+[12] Tan S-L , Tu Y. , Zamora A. G. : On complex surfaces with 5or6semistable singular fibers
+overP1. Math. Zeitschrift, No. 249, 427-438, 2005.
+[13] Vojta P.: Diophantine Inequalities and Arakelov Theory . Appendix to introduction to
+Arakelov theory by S. Lan , Springer-Verlag, 155-178, 1988.
+Claudia R. Alc´ antara: Departamento de Matem´ aticas, Universidad de Gua-
+najuato; Jalisco s/n, Valenciana, C.P. 36240, Guanajuato, Guana juato, M´ exico.
+claudia@cimat.mx
+Abel Castorena: Instituto de Matem´ aticas, Unidad Morelia, Universidad Na-
+cional Aut´ onoma de M´ exico; Apartado Postal 61-3 (Xangari), C .P. 58089, Morelia,
+Michoac´ an, M´ exico. abel@matmor.unam.mx
+Alexis G. Zamora: Unidad Acad´ emica de Matem´ aticas, Universidad Aut´ onoma
+de Zacatecas; Camino a la Bufa y Calzada Solidaridad C.P. 98060, Zaca tecas, Zac.,
+M´ exico. alexiszamora06@gmail.com
\ No newline at end of file
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+arXiv:1001.0178v1  [gr-qc]  31 Dec 2009Small deformations of extreme Kerr black hole
+initial data
+Sergio Dain1,2and Mar´ ıa E. Gabach Cl´ ement1
+1Facultad de Matem´ atica, Astronom´ ıa y F´ ısica, FaMAF,
+Universidad Nacional de C´ ordoba,
+Instituto de F´ ısica Enrique Gaviola, IFEG, CONICET,
+Ciudad Universitaria, (5000) C´ ordoba, Argentina.
+2Max Planck Institute for Gravitational Physics,
+(Albert Einstein Institute), Am M¨ uhlenberg 1,
+D-14476 Potsdam Germany.
+October 31, 2018
+Abstract
+We prove the existence of a family of initial data for Einstei n equa-
+tions which represent small deformations of the extreme Ker r black
+hole initial data. The data in this family have the same asymp totic
+geometry as extreme Kerr. In particular, the deformations p reserve
+the angular momentum and the area of the cylindrical end.
+1 Introduction
+Black holes are one of the most spectacular predictions of Ge neral
+Relativity. Thereis growing experimental evidences that i ndicate that
+black holes do indeed exists in nature. Among the most impres sive
+ones are the evidences for the existence of a supermassive bl ack hole
+in the center of our galaxy (see the review article [31]).
+In vacuum, the only stationary black hole is expected to be th e
+Kerr black hole, characterized by the mass mand the angular mo-
+mentum J(see [11] and reference therein for updated results on this
+problem). The Kerr black hole satisfies the inequality m≥/radicalbig
+|J|. The
+limit case m=/radicalbig
+|J|is called the extreme black hole. It represents the
+1stationary black hole with maximum amount of angular moment um
+per mass unit. The extreme limit/radicalbig
+|J| →mis singular because the
+geometry of the spacetime changes at the limit. This is someh ow to
+be expected since the extreme case is the borderline between a black
+hole and a spacetime with a naked singularity (i.e. the Kerr s olution
+with 0< m </radicalbig
+|J|.)
+There exists relevant reasons to study extreme black holes. The
+first one is that there is good experimental evidences for the existence
+of nearly extreme black holes in the universe (see [29] for ex perimental
+evidences of a black hole with J/m2>0.98). Then, it is important to
+understand the dynamics of black holes near the extreme limi t. The
+second reason is less clear but, we believe, equally importa nt. As often
+happens in physical theories, solutions that arise as asymp totic limits
+are simpler than other solutions and they provide useful ins ight into
+the theory. In the set of solutions of Einstein equation, ext reme black
+holes represent a kind of barrier that divide black holes and naked sin-
+gularities. From the pure classical point of view, there are evidences
+that extreme black holes have some special properties that m ake them
+simpler than non-extreme ones (see the discussion in [18]). Also, from
+a completely different perspective, namely holographic dual ities, par-
+ticular features of extreme black holes play an important ro le (see [4]
+[24], see also the review article [3]). It appears that extre me black
+holes have a deep mathematical structure that it is still unc over.
+Finally, there is a third reason to study extreme black holes . In the
+study of extreme black hole initial conditions (which is the subject of
+this article), a particular kind of geometry appears: geome tries with
+cylindrical ends. This geometries have provided to be very u seful in
+the numerical computations of black holes collisions (they are called
+’trumpet’ initial conditions in this context, see [25] [26] [27]).
+As a first step to understand the dynamics near an extreme Kerr
+black hole, in this article we study small deformation of the extreme
+Kerr black hole initial conditions. We prove the existence o f a family
+of initial data that are close to extreme Kerr initial data. I n particu-
+lar, the asymptotic geometry of these initial data is the sam e as the
+extreme Kerr geometry. These data are, generically, non-st ationary.
+It is important to emphasize that the existence of these init ial condi-
+tions it is a priori by no means obvious due to the character of the
+extreme Kerr geometry.
+The paper is organized as follows. We begin in section 2 with a
+review of some of the main properties of the extreme Kerr blac k hole.
+Then we state our main result avoiding technical details. We also
+discuss how the cylindrical geometry is preserved along the evolution.
+In section 3 we state our main theorem in a precise form and pro ve
+2i0 i0SCauchy horizon
+Event horizonSingularity
+Bifurcation
+sphere
+Figure 1: Conformal diagram of the Kerr black hole in the non-extre me case.
+it. We conclude the article with a discussion of some relevan t open
+problems in section 4. Finally, we have included three appen dices. In
+appendix A we prove a decay property of the Sobolev spaces use d in
+our proof. In appendix B we prove a property of the extreme Ker r
+initial data that plays a central role in the proof. Appendix C is
+brief summary of the implicit function theorem, which is the central
+analytical tool used in the proof.
+2 Main Result
+Consider the Kerr black hole with mass mand angular momentum
+J. In the non-extreme case (i.e. m >/radicalbig
+|J|) the maximal analytical
+extension of the metric has the well known global structure s hown in
+figure 1 (see [6][7] and also [8]). Take the spacelike surface Sdrawn
+in this figure. This surface runs from one spacelike infinity ( denoted
+byi0) to the other. The topology of this surface is S=S2×R. The
+triple (S,hij,Kij), where hijis the induced intrinsic metric on Sand
+Kijis the second fundamental form of S, constitute an initial data set
+for Einstein equations. That is, they are solutions of the co nstraint
+equations
+DjKij−DiK= 0, (1)
+R−KijKij+K2= 0, (2)
+3Figure 2: The initial data for the non-extreme case. The dark circle in the
+middle represents the minimal surface.
+whereDandRare the Levi-Civita connection and the Ricci scalar
+associated with hij, andK=Kijhij. In these equations the indices
+are moved with the metric hijand its inverse hij.
+TheRiemannian manifold ( S,hij) has two asymptotically flat ends
+(see figure 2). This asymptotic geometry is identical to the a nalogous
+slice of Kruskal extension for Schwarzschild black hole. Th e surface S
+in figure 1 corresponds to a slice t= 0 of the Boyer-Lindquist coordi-
+nates (t,˜r,θ,φ) in Kerr metric (see the appendix B). It intersects the
+bifurcation sphere (denoted by a dark dot in figure 1 and by a da rk
+circle in figure 2). The slice is isometric across this sphere . The bifur-
+cation sphere on the slice is both a minimal surface and an app arent
+horizon. In these coordinates, spacelike infinity i0is represented by
+the limit ˜ r→ ∞. The intrinsic metric and the second fundamental
+form satisfy the standard asymptotically flat fall-off condi tions
+hij=δij+O(˜r−1), Kij=O(˜r−3),as ˜r→ ∞, (3)
+whereδijis the flat metric. The strong fall-off behavior of the second
+fundamentalformimpliesthat thelinearmomentumof theini tial data
+vanishes. The angular momentum is contained in the term O(r−3) of
+Kij.
+The maximal development of the initial data set ( S,hij,Kij) is
+shown in light gray in figure 1. This region does not cover the w hole
+analytical extension (asinthecaseofSchwarzschild’s), i t hasasmooth
+boundary in the spacetime. This boundary is known as Cauchy h ori-
+zon. Indarkgraythedomainofoutercommunications isshown , which
+is bounded by the black hole event horizon.
+In the extreme case m=/radicalbig
+|J|the global structure of the space-
+time changes. The maximal analytical extension is shown in fi gure 3.
+The spacelike surface Shas the same topology S2×Ras in the non-
+extreme case, however, the asymptotic geometry of the Riema nnian
+manifold ( S,hij) is different. It has one asymptotically flat end and
+4i0SEvent horizon
+icSingularity
+Figure3: ConformaldiagramfortheextremeKerrblackhole. Thec ylindrical
+end is denoted by ic.
+Figure 4: The initial data for the extreme Kerr black hole.
+one cylindrical end, see figure 4. The cylindrical end asympt otically
+approaches the event horizon. Contrary to the asymptotical ly flat
+case, this end is in the strong field region of the spacetime. N ote that
+(S,hij) is a complete Riemannian manifold without boundary which
+lies completely in the black hole exterior region. Let us tak e a closer
+look at the structure of the cylindrical end. In isotropic co ordinates
+(r,θ,φ), withr:= ˜r−m(see appendix B), the induced metric on S
+has the form
+h0
+ij= Φ4
+0˜h0
+ij,˜h0=e2q0(dr2+r2dθ2)+r2sin2θdφ2,(4)
+where Φ 0andq0are given by equation (95) in appendix B. The
+extrinsic curvature is given by
+K0
+ij=2
+ηS(iηj), Si=1
+ηǫijkηj∂kω0, (5)
+5whereηiis the axial Killing vector, ηthe square of its norm (see
+equation (91)), ǫijkdenotes the volume element with respect to the
+metrichijandω0is given by (96). The advantage of this particular
+form of writing K0
+ijis that it is easy to check from (5) that K0
+ijsatisfies
+the momentum constraint (1). We will discuss and use this fac t in
+section 3. In particular, we have that K0
+ijis trace-free
+K0= 0. (6)
+That is, these initial data are maximal surfaces.
+In isotropic coordinates, the asymptotically flat end is giv en by the
+limitr→ ∞and the cylindrical end by the limit r→0. The radial
+coordinate ris a good coordinate in the asymptotically flat end since
+the metric and the extrinsic curvature take the asymptotic f orm (3).
+On the other hand, in the limit r→0 the conformal factor Φ 0
+blows up. This is, however, just a coordinate problem. To see this, let
+s=−lnr, then the cylindrical end corresponds to s→ ∞, and the
+metric has the form
+h0= (√rΦ0)4/parenleftbig
+e2q0(ds2+dθ2)+sin2θdφ2/parenrightbig
+. (7)
+The functions√rΦ0andq0are smooth and uniformly bounded in
+the whole range −∞< s <∞(see lemma B.2). In particular, the
+Riemannian manifold ( S,h0
+ij) has bounded curvature.
+It is interesting to note (although we will not make use of it) that
+themetric(7)andthesecondfundamentalform(5)haveawell defined
+limits→ ∞as initial data. Namely
+h0=m2(1+cos2θ)/parenleftbig
+ds2+dθ2/parenrightbig
++4m2sin2θ
+(1+cos2θ)dφ2,ass→ ∞,(8)
+where we have used the limits (99)–(100). The extrinsic curv ature
+Kij
+0has the form (5) where ω0is replaced by its limiting value (101)
+and all the other quantities are computed with respect to the metric
+(8). These are in fact solutions of the constraint equations (1)–(2).
+They isolate the cylindrical geometry cutting off the asympt otically
+flat end. In particular, the metric (8) has non-negative Ricc i scalar,
+given by the limit (102) and it has another symmetry, namely t rans-
+lations in s. These limit initial data are slices t=constant of the
+four dimensional vacuum geometry described in [4], known as the the
+near-horizon extreme Kerr. This geometry has also been stud ied in
+[8] (see eq. (5.63) in that reference).
+A relevant parameter for extreme black hole data is the area o f the
+cylindrical end. Consider the area A(r) of the surfaces r=constant
+of the metric (4). In the limit r→0 we have
+A0= lim
+r→0A(r) = 8πm2. (9)
+6For extreme Kerr, this corresponds to the area of the black ho le event
+horizon. Finally, for completeness, let us mention that the ergoregion
+onSis given in these coordinates by
+0< r < m sinθ. (10)
+We have described a particular class of initial data sets for the
+extreme Kerr black hole which run from ictoi0. There exist similar
+initial data sets in Reissner-Nordstr¨ om and Kerr-Newman b lack hole.
+Remarkably enough, for a Schwarzschild black hole there als o exist
+initial data that having the same asymptotic geometry (see [ 26] and
+references therein). All these examples are stationary. Mo reover, all
+thesedataariseasasingularlimit inwhichthegeometry cha nges. The
+first numerical evidence for the existence of non-stationar y cylindrical
+data with a similar structure as the one described above was g iven in
+[21] and the first analytical proof was provided in [20], [23] . These
+data are also obtained as a singular limit from non-extreme d ata. The
+point we want to address to in this article is the following: g iven
+extreme Kerr initial data, does there exist a neighborhood o f similar
+data? The following theorem, which constitutes the main res ult of
+this article, gives an affirmative answer to this question.
+Theorem 2.1. Let(S,h0
+ij,K0
+ij)be the extreme Kerr data set described
+above with angular momentum Jand mass m=/radicalbig
+|J|. Then there is
+a smallλ0>0such that for −λ0< λ < λ 0there exists a family of
+initial data sets (S,hij(λ),Kij(λ))(i.e., solutions of the constraints
+onS) with the following properties:
+(i) We have hij(0) =h0
+ijandKij(0) =K0
+ij. The family is differ-
+entiable in λand it is close to extreme Kerr with respect to an
+appropriate norm which involves two derivatives of the metr ic.
+(ii) The data has the same asymptotic geometry as the extreme K err
+initial data set. The angular momentum and the area of the
+cylindrical end in the family do not depend on λ, they have the
+same value as in (S,h0
+ij,K0
+ij), namely Jand8π|J|respectively.
+(iii) The data are axially symmetric and maximal (i.e. K(λ) = 0).
+In section 3 we provide a more precise version of this theorem
+(theorem 3.1). Let us discuss here other relevant propertie s of the
+initial data family ( S,hij(λ),Kij(λ)).
+We mention that the angular momentum of the family remains
+constant, the total mass however is not. As a consequence of t he
+general theorems [19] [10] we have the following inequality for allλ
+m(λ)≥/radicalbig
+|J|, (11)
+7with equality only for λ= 0 (i.e. for extreme Kerr). This family
+realizes the local minimum behavior of extreme Kerr studied in [15].
+Inequality (11) allows us to define the following positive qu antity
+E(λ) =m(λ)−/radicalbig
+|J|. (12)
+The energy Eprovides (if we assume cosmic censorship) an upper
+bound for the total amount of radiation emitted by the system at null
+infinity for these initial data (see the discussion in [18]).
+Let us consider now some aspects of the evolution of these dat a.
+In the asymptotically flat case, it is well know that the asymp totic
+behavior (3) is preserved by evolution if we impose appropri ate fall
+off conditions for the lapse and shift. This is of course impor tant,
+since it is related to conservation of total mass in the space time. The
+natural question is whether this kind of persistence under e volution
+also holds for the cylindrical asymptote. To study this ques tion we
+need non-stationary data as the ones constructed here.
+Let us consider a member of the family for some λ/\e}atio\slash= 0 (we will
+suppress the λin the notation in the following). Take a short period
+of timet, then we have
+hij(t)≈hij(0)+˙hij(0)t. (13)
+Kij(t)≈Kij(0)+˙Kijt, (14)
+where dot denotes time derivative. The time derivatives ˙hij,˙Kijcan
+be computed using the evolution equations
+˙hij= 2αKij+Lβhij, (15)
+˙Kij=∇i∇jα+LβKij+α(2Kk
+iKjk−KKij−Rij),(16)
+whereαandβiare the lapse and shift of the foliation, Ldenotes the
+Lie derivative and Rijis the Ricci tensor of hij. If we want to preserve
+the cylindrical geometry under the evolution, we must have
+lim
+s→∞˙hij= 0,lim
+s→∞˙Kij= 0. (17)
+From equations (15)–(16) we deduce the following condition s for the
+lapse
+lim
+s→∞α= lim
+s→∞∂α= lim
+s→∞∂2α= 0. (18)
+and the shift
+lim
+s→∞βi= lim
+s→∞∂βi= 0, (19)
+where∂denotes partial derivatives with respect to the space coord i-
+nates. Note that for the particular Boyer-Lindquist foliat ion in ex-
+treme Kerr these requirements are satisfied (see equations ( 97)–(98)
+8in appendix B). Conditions (18) and (19) are analogous to the asymp-
+totically flat conditions for lapse and shift.
+In this article, we have assumed vacuum for simplicity. We ex pect
+that an analogous result as theorem (4) holds for the Kerr-Ne wman
+extreme black hole. In that case, inequality (11) should be r eplaced
+by it generalized charged version recently proved in [12], [ 13].
+3 Proof of Main Result
+A particular feature of axial symmetry is that it allows one t o reduce
+the constraint equations (1)–(2) to just one scalar equatio n for a con-
+formal factor (the so called Lichnerowicz equation). This p rocedure
+is well known (see, for example, [19] and reference therein) . Let us
+briefly review it. Consider the metric
+˜hij=e−2q(dr2+r2dθ2)+r2sin2θdϕ2, (20)
+whereq=q(r,θ) is an arbitrary function. This metric will be used as
+a conformal background for the physical metric hij. We first discuss
+how to construct solutions of the momentum constraint (1) fr om an
+arbitrary axially symmetric potential ω(r,θ). Consider the following
+tensor
+˜Kij=2
+ρ2˜S(iηj), (21)
+where
+˜Si=1
+2ρ2˜ǫijkηj∂kω, (22)
+and ˜ǫijkdenotes the volume element with respect to ˜hij,˜Dis the
+connexion with respect to ˜hijandρ=rsinθis the cylindrical radius.
+The indices on tilde quantities are moved with ˜hijand its inverse ˜hij.
+The tensor ˜Kijis symmetric, trace free, and satisfies the following
+equation (see, for example, the appendix in [14])
+˜Di˜Kij= 0, (23)
+for arbitrary qandω. Equation (23) essentially solves (up to a con-
+formal factor) the momentum constraint (1). Assume that we h ave a
+solution Φ of Lichnerowicz equation
+∆˜hΦ−˜R
+8Φ =−˜Kij˜Kij
+8Φ7, (24)
+where ∆ ˜his the Laplacian with respect to ˜hijand˜Ris the Ricci scalar
+of˜hij. Consider the rescaling
+hij= Φ4˜hij, Kij= Φ−2˜Kij. (25)
+9Then, as a consequence of (23) the pair ( hij,Kij) satisfies the con-
+straints (1)–(2). That is, the problem reduces to solving eq uation
+(24). This equation can be written in the following remarkab ly simple
+form in axial symmetry
+∆Φ =−(∂ω)2
+16ρ4Φ7−∆2q
+4Φ, (26)
+where, ∆ and ∆ 2are flat Laplace operators in three and two dimen-
+sions respectively (see (103)). In particular, extreme Ker r initial data
+satisfies this equation, namely
+∆Φ0=−(∂ω0)2
+16ρ4Φ7
+0−∆2q0
+4Φ0. (27)
+The idea is to perturbequation (26) around the extreme Kerr s olution
+by taking
+q0+λq, ω 0+λω, (28)
+for some fixed functions qandωand small λ, and then to find a
+solution udefined by
+Φ = Φ 0+u. (29)
+Inserting (28) and (29) in equation (26) and using (27) we obt ain our
+final equation
+G(λ,u) = 0, (30)
+where we have defined
+G(λ,u) = ∆u+(∂w0+λ∂w)2
+16ρ4(Φ0+u)7−∂w2
+0
+16ρ4Φ7
+0+λ∆2q
+4(Φ0+u)+∆2q0
+4u.
+(31)
+Then, theorem 2.1 is a direct consequence of the following ex istence
+theorem for equation (30).
+Theorem 3.1. Letw∈C∞
+0(R3\Γ)andq∈C∞
+0(R3\ {0}). Then,
+there isλ0>0such that for all λ∈(−λ0,λ0)there exists a solution
+u(λ)∈H′2
+−1/2of equation (30). The solution u(λ)is continuously
+differentiable in λand it satisfies Φ0+u(λ)>0. Moreover, for small
+λand small u(in the norm H′2
+−1/2) the solution u(λ)is the unique
+solution of equation (30).
+We have used the following notation: Γ denotes the axis ρ= 0,
+C∞
+0(Ω) are smooth functions with compact support in Ω and H′2
+−1/2
+denotes the Sobolev weighted spaces defined in appendix A.
+Thefact ωvanishesat theaxisimplies that theangularmomentum
+remains fixed for the whole family (see the discussion in [19] ). Also,
+using lemma A.1, from u∈H′2
+−1/2it follows that the perturbation u
+does not change the area of the cylindrical end at r= 0.
+10Proof.The proof uses the Implicit Function Theorem (see theorem
+C.1 in appendix C, in the rest of the proof we will follow the no tation
+introduced in that theorem) for the map Gdefined in equation (31).
+The proof is divided in two steps.
+In the first step, we find the appropriate Banach spaces X,Yand
+Zrequired by theorem C.1, together with the neighborhoods U⊂X
+andV⊂Y, such that G:V×U→Zdefines a C1map. The delicate
+part of this step is to take into account in the definition of th e Banach
+spaces the fall off behavior at infinity and the singular behav ior at
+the origin of the background functions Φ 0,q0andω0. In particular,
+it is clear from the equation that we can not expect the soluti onuto
+be regular at the origin, and hence standard Sobolev spaces a re not
+appropriate. Also, the presence of the singular background functions
+Φ0,q0andω0inthemap Gprevents onefromusingstandardtheorems
+(for example the chain rule in Sobolev spaces) to prove that Gis
+C1. We need to explicitly compute the functional partial deriv atives
+from their very definition as a limit. This makes this part of t he
+proof laborious. The asymptotic behavior of the background Kerr’s
+functions is typical of any data with one asymptotically flat end and
+one cylindrical end and that is the main ingredient needed in this step.
+In the second step, we prove that the derivative D2G(0,0) is an
+isomorphism between YandZ. In this part we use very specific
+properties of extreme Kerr initial data (namely, lemma B.1) which
+are not valid for generic cylindrical data. See the comment a fter the
+proof of lemma B.1. This step represents the key part of the pr oof.
+Step 1. To handle both the fall off behavior at infinity and the
+singular behavior at the origin of the functions Φ 0,q0andω0we
+will make use of weighted Sobolev spaces defined in appendix A . We
+chooseX=R,Y=H′2
+−1/2andZ=L′2
+−5/2. We also choose U=R. It
+is clear that the map Gis only defined when Φ 0+u >0. Hence, we
+need to find an appropriate neighborhood Vof 0 in the Banach space
+Ysuch that this condition is satisfied. Let us consider Vgiven by the
+open ball
+||u||H′2
+−1/2< ξ, (32)
+where the constant ξis computed as follows. From lemma A.1 we
+have that for u∈V√r|u| ≤C0ξ, (33)
+where the constant C0is a Sobolev constant independent of u. By
+lemma B.2 we have√rΦ0≥√m. (34)
+11Then, if we choose ξsuch that
+√m
+C0> ξ >0, (35)
+we have that for all u∈V
+√r(Φ0+u)≥√m−C0ξ >0. (36)
+The constant ξwill remain fixed for the rest of the proof.
+We first prove that G:R×V→L′2
+−5/2is well defined as a map.
+That is, we need to check that for λ∈Randu∈Vwe obtain
+G(λ,u)∈L′2
+−5/2. Let us compute the norm L′2
+−5/2ofG(λ,u). Using
+the definition (31) and the triangle inequality we get
+/ba∇dblG(λ,u)/ba∇dblL′2
+−5/2≤ /ba∇dbl∆u/ba∇dblL′2
+−5/2+/vextenddouble/vextenddouble/vextenddouble/vextenddoubleλ∂ω(2∂ω0+λ∂ω)
+16ρ4(Φ0+u)7/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+L′2
+−5/2+
++λ
+4/ba∇dbl(Φ0+u)∆2q/ba∇dblL′2
+−5/2+
++/vextenddouble/vextenddouble/vextenddouble/vextenddouble(∂ω0)2
+16ρ4/bracketleftbigg1
+(Φ0+u)7−1
+Φ7
+0/bracketrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+L′2
+−5/2+1
+4/ba∇dblu∆2q0/ba∇dblL′2
+−5/2.(37)
+From the definition of the H′2
+−1/2-norm it is clear that the first term in
+the right-hand side of (37) is bounded. For the second and thi rdterms
+weusethehypothesisthat ωhascompact supportoutsidetheaxis and
+qcompact support outside the origin together with the lower b ound
+(36) to conclude that these terms are also bounded. The delic ate
+terms are the last two.
+For the fourth term we proceed as follows. Using the followin g
+elementary identity for real numbers aandb
+1
+ap−1
+bp= (b−a)p−1/summationdisplay
+i=0ai−pb−1−i. (38)
+we find that
+r−4/parenleftbigg1
+Φ7
+0−1
+(Φ0+u)7/parenrightbigg
+=uH, (39)
+whereHis given by
+H=6/summationdisplay
+i=0(√r(Φ0+u))i−7(√rΦ0)−1−i. (40)
+Using inequalities (34) and (36) we obtain
+H≤C, (41)
+12where the constant Cdepends only on the mass parameter mof the
+background extreme Kerr solution. Inthefollowing we will g enerically
+denote by Cconstants depending at most on m. Then, we have
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble(∂ω0)2
+ρ4/bracketleftbigg1
+(Φ0+u)7−1
+Φ7
+0/bracketrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+L′2
+−5/2≤/vextenddouble/vextenddouble/vextenddouble/vextenddoubleC
+r6(r4uH)/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+L′2
+−5/2(42)
+=C/ba∇dblu/ba∇dblL′2
+−1/2≤C/ba∇dblu/ba∇dblH′2
+−1/2.(43)
+Where we have used the bound (109) in Lemma B.2 to bound the
+factor with ω0in the first inequality in (42). The last inequality in
+(42) comes from the definition of the weighted Sobolev space H′2
+−1/2.
+For the fifth term, which involves q0, we use the bound (110) in
+lemma B.2, to find
+/ba∇dblu∆2q0/ba∇dblL′2
+−5/2≤C/vextenddouble/vextenddouble/vextenddoubleu
+r2/vextenddouble/vextenddouble/vextenddouble
+L′2
+−5/2=C/ba∇dblu/ba∇dblL′2
+−1/2≤C/ba∇dblu/ba∇dblH′2
+−1/2.(44)
+These computations show that all norms involved in /ba∇dblG(λ,u)/ba∇dblL′2
+−5/2
+are finite, hence G:R×V→L′2
+−5/2is a well defined map.
+We will now prove that GisC1between the mentioned Sobolev
+spaces. Let us denote by D1G(λ,u) the partial Fr´ echet derivative
+ofGwith respect to the first argument evaluated at ( λ,u) and by
+D2G(λ,u) the partial derivative with respect to the second argument .
+By definition, the partial derivatives are linear operators between the
+following spaces
+D1G(λ,u) :R→L′2
+−5/2, (45)
+D2G(λ,u) :H′2
+−1/2→L′2
+−5/2. (46)
+We use the notation D1G(λ,u)[γ] to denote the operator D1G(λ,u)
+acting on γ∈R. That is, D1G(λ,u)[γ] defines a function on L′2
+−5/2.
+In the same way we denote by D2G(λ,u)[v] the operator acting on a
+function v∈H′2
+−1/2.
+Weproposeas candidates forthesepartial derivatives thef ollowing
+linear operators
+D1G(λ,u)[γ] =/parenleftbigg2(∂w0+λ∂w)·∂w
+16ρ4(Φ0+u)7+∆2q
+4(Φ0+u)/parenrightbigg
+γ, (47)
+D2G(λ,u)[v] = ∆v+/parenleftbigg
+−7(∂w0+λ∂w)2
+16ρ4(Φ0+u)8+λ∆2q
+4+∆2q0
+4/parenrightbigg
+v.(48)
+These operators arise by taking formally the following dire ctional
+13derivatives to the map G
+d
+dtG(λ+tγ,u)|t=0=D1G(λ,u)[γ], (49)
+d
+dtG(λ,u+tv)|t=0=D2G(λ,u)[v]. (50)
+To prove that the map G:R×V→ZisC1we need to prove the
+following items:
+(i) The linear operators (47) and (48) are bounded, namely
+/ba∇dblD1G(λ,u)[γ]/ba∇dblL′2
+−5/2≤C1|γ|, (51)
+/ba∇dblD2G(λ,u)[v]/ba∇dblL′2
+−5/2≤C2/ba∇dblv/ba∇dblH′2
+−1/2, (52)
+where the constants C1andC2do not depend on γandvre-
+spectively.
+(ii) The operators (47) and (48) are continuous in ( λ,u) with respect
+to the operator norms. That is, for every δ >0 there exists ǫ >0
+such that
+|λ1−λ2|< ǫ⇒ /ba∇dblD1G(λ1,u)−D1G(λ2,u)/ba∇dblL(X,Z)< δ,(53)
+and
+/ba∇dblu1−u2/ba∇dblH′2
+−1/2< ǫ⇒ /ba∇dblD1G(λ,u1)−D1G(λ2,u2)/ba∇dblL(Y,Z)< δ,
+(54)
+where the operator norms used in the right hand side of this
+inequalities are defined in appendix C.
+(iii) The operators (47) and (48) are the partial Fr´ echet de rivatives
+ofG(see the definition in appendix C). That is
+lim
+γ→0/ba∇dblG(λ+γ,u)−G(λ,u)−D1G(λ,u)[γ]/ba∇dblL′2
+−5/2
+|γ|= 0,(55)
+and
+lim
+v→0/ba∇dblG(λ,u+v)−G(λ,u)−D2G(λ,u)[v]/ba∇dblL′2
+−5/2
+/ba∇dblv/ba∇dblH′2
+−1/2= 0.(56)
+By performing similar computations as above it is straightf orward
+to prove (i) and also the following estimate
+/ba∇dblD1G(λ1,u)−D1G(λ2,u)/ba∇dblL′2
+−5/2≤C|λ1−λ2|,(57)
+14whereCdoes not depend on λ1andλ2. From inequality (57) the
+continuity with respect to λfollows, equation (53) of item (ii). In
+fact, estimate (57) is a bit stronger since it gives uniform c ontinuity.
+Continuity in the udirection is more delicate. Using again the
+identity (38) we have
+r−9/2/parenleftbigg1
+(Φ0+u1)−1
+(Φ0+u2)/parenrightbigg
+= (u2−u1)H, (58)
+where
+H=7/summationdisplay
+i=0(√r(Φ0+u1))i−8(√r(Φ0+u2))−1−i. (59)
+Using that u1,u2∈Vand the lower bound (36) we obtain
+H≤C. (60)
+We use the upper bound (109), together with (60) to find
+/ba∇dblD1G(λ,u1)−D1G(λ2,u2)/ba∇dblL′2
+−5/2≤C/vextenddouble/vextenddouble/vextenddouble/vextenddoublev(u1−u2)
+r3/2/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+L′2
+−5/2.(61)
+We bound the right hand side of (61) as follows
+/vextenddouble/vextenddouble/vextenddouble/vextenddoublev(u1−u2)
+r3/2/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+L′2
+−5/2=/parenleftbigg/integraldisplay
+R3v2(u1−u2)2
+rdx/parenrightbigg1/2
+, (62)
+=/parenleftbigg/integraldisplay
+R3(√rv)2(u1−u2)2
+r2dx/parenrightbigg1/2
+(63)
+≤C/ba∇dblv/ba∇dblH′2
+−1/2/parenleftbigg/integraldisplay
+R3(u1−u2)2
+r2dx/parenrightbigg1/2
+(64)
+≤C/ba∇dblv/ba∇dblH′2
+−1/2/ba∇dblu1−u2/ba∇dblH′2
+−1/2. (65)
+Equation (62) is just the definition of the L′2
+−5/2-norm and equation
+(63) is a trivial rearrangement of factors. Thecrucial ineq uality is (64)
+wherewehave usedlemmaA.1. Finally, line(65) trivially fo llows from
+the definition of H′2
+−1/2-norms. Hence, we obtain our final inequality
+/ba∇dblD1G(λ,u1)−D1G(λ2,u2)/ba∇dblL′2
+−5/2≤C/ba∇dblv/ba∇dblH′2
+−1/2/ba∇dblu1−u2/ba∇dblH′2
+−1/2.
+(66)
+From this inequality, the continuity (54) follows.
+We now prove (iii). The first limit (55) is straightforward. T he
+delicatepartisthesecondlimit(56). Wewillfollowasimil arargument
+15as in the previous calculation. We first compute
+G(λ,u+v)−G(λ,u)−D2G(λ,u)[v] =
+(∂ω0+λ∂ω)2
+16ρ4/parenleftbigg1
+(Φ0+u+v)7−1
+(Φ0+u)7+7v
+(Φ0+u)8/parenrightbigg
+(67)
+We have
+r−9/2/parenleftbigg1
+(Φ0+u+v)7−1
+(Φ0+u)7+7v
+(Φ0+u)8/parenrightbigg
+=v2H,(68)
+with
+H=1
+(√r(Φ0+u+v))7(√r(Φ0+u))8/summationdisplay
+i+j+k=6
+i,j,k≥0Cijk(√rΦ0)i(√ru)j(√rv)k,
+(69)
+whereCijkare numerical constants. To bound Hwe use the upper
+and lower bounds for Φ 0given by (108) and the fact that u,v∈V
+(and hence they satisfy the bound (33)). We obtain
+|H| ≤C(r+m)6/2
+(/radicalbig
+(r+m)−C0ξ)15≤C. (70)
+Then, we have
+/ba∇dblG(λ,u+v)−G(λ,u)−D2G(λ,u)[v]/ba∇dblL′2
+−5/2≤C/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubler9/2v2H
+r6/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+L′2
+−5/2,
+(71)
+=/vextenddouble/vextenddouble/vextenddouble/vextenddoublev2
+r3/2/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+L′2
+−5/2.(72)
+Using the same argument as we used in equation (62)–(65) we fin ally
+get the desired estimate
+/ba∇dblG(λ,u+v)−G(λ,u)−D2G(λ,u)[v]/ba∇dblL′2
+−5/2≤C/parenleftBig
+/ba∇dblv/ba∇dblH′2
+−1/2/parenrightBig2
+.
+(73)
+From (73) it follows (56).
+Step 2. We will prove that D2G(0,0) :H′2
+−1/2→L′2
+−5/2is an
+isomorphism. We write this linear operator in the following form
+D2G(0,0)[v] = ∆v−αv, (74)
+where
+α= 7(∂ω0)2
+16ρ4Φ8
+0−∆2q0
+4. (75)
+16By lemma B.1 we have that α=hr−2wherehis a positive and
+bounded function in R3. In [23] it has been proved that under such
+conditions for αthe map (74) is an isomorphism between H′2
+−1/2and
+L′2
+−5/2.
+We have satisfied all the hypothesis of the Implicit Function Theo-
+rem. Hence, there exists a neighborhood W= (−λ0,λ0) of the origin
+inRsuch that the conclusion of theorem 3.1 holds.
+Remarks: Wehaveimposedtheperturbationfunctions ωandqto
+have compact support. This can be relaxed by requiring appro priate
+fall off conditions at the axis and at the origin.
+The axially symmetric data considered here are not the most g en-
+eral one, since we are assuming in the form of the metric (20) t hat the
+axial Killing vector is hypersurface orthogonal on the surf aceS(but,
+of course, has a non zero twist in the spacetime). This simpli fication
+allows one to use the explicit expression’s (21) for the seco nd funda-
+mental form. We expect that this result can be generalized wi thout
+this assumption. However, it is important to emphasize that given a
+data as the one constructed in this theorem, the time evoluti on de-
+scribed in section 2, under the condition for lapse and shift (18)–(19),
+willdevelopeinitial datawiththesameasymptoticgeometr y forwhich
+the Killing vector is not surface orthogonal. And hence we ge t from
+our family also non-trivial initial data for which the Killi ng vector is
+not hypersurface orthogonal.
+4 Final Comments
+We have prove the existence of a initial data family close to e xtreme
+Kerr black hole initial data. This family represent the natu ral ini-
+tial data to study the evolution near an extreme black hole in axial
+symmetry, in the spirit of [17] [22].
+Thereexists also relevant openproblemsthat can beaddress at the
+level of theinitial data. As we have seen in section 2, the ext reme Kerr
+black data lies outside the black hole region and hence they c ontain no
+trapped surfaces. Does the family ( S,hij(λ),Kij(λ)) contains trapped
+surfaces for λ >0? If these data have no trapped surfaces, then there
+is a chance that they also lies outside the black hole region. This
+can, of course, only been answered after the whole evolution has been
+analyzed. On the other hand, if there are trapped surfaces, t hen the
+data necessarily penetrate the black hole. The formation of trapped
+surfaces for arbitrary small λ >0 will indicate that extreme Kerr
+data is a very special element in the family ( S,hij(λ),Kij(λ)). In
+that case, these kind of data could be very useful in the study of
+17geometric inequalities which relates angular momentum and area of
+trapped surfaces (see section 8 in the review article [28]).
+Acknowledgments
+S. D. thanks Piotr Chru´ sciel and Raffe Mazzeo for useful discu ssions.
+The discussions with P. Chru´ sciel took place at the Institu t Mittag-
+Leffler, during the program “Geometry, Analysis, and General Rela-
+tivity”, 2008 fall. The discussions with R. Mazzeo took plac e at the
+Mathematisches Forschungsinstitut Oberwolfach during th e workshop
+“Mathematical Aspects of General Relativity”, October 11t h – Octo-
+ber 17th, 2009. S. D. thanks the organizers of these events fo r the in-
+vitation and the hospitality and support of the Institut Mit tag-Leffler
+and Mathematisches Forschungsinstitut Oberwolfach.
+The authors want to thank Robert Beig for useful discussions that
+took place at FaMAF during his visit in 2009.
+S. D. is supported by CONICET (Argentina). M. E. G. C. is sup-
+ported by a fellowship of CONICET (Argentina). This work was sup-
+ported in part by grant PIP 6354/05 of CONICET (Argentina), g rant
+05/B415 Secyt-UNC (Argentina) and the Partner Group grant o f
+the Max Planck Institute for Gravitational Physics, Albert -Einstein-
+Institute (Germany).
+A Weighted Sobolev spaces
+The Bartnik’s weighted Sobolev spaces W′k,p
+δ[5] are appropriate for
+studying geometries with one cylindrical and one asymptoti cally flat
+end. These functional spaces have weights both at infinity an d at the
+origin.
+The weighted Lebesgue spaces L′p
+δare defined as the completion
+ofC∞
+0(Rn\{0}) functions under the norms
+/ba∇dblf/ba∇dbl′
+p,δ=/parenleftBigg/integraldisplay
+R3\{0}|f|pr−δp−ndx/parenrightBigg1/2
+. (76)
+The weighted Sobolev spaces W′k,p
+δare defined in the usual way
+/ba∇dblf/ba∇dbl′
+k,p,δ=m/summationdisplay
+0/ba∇dblDjf/ba∇dblp,δ−j. (77)
+In this article we only use the cases n= 3 and p= 2, we have denoted
+these spaces by H′k
+δ=W′k,2
+δand the norms by /ba∇dblf/ba∇dblL′2
+δ=/ba∇dblf/ba∇dbl′
+2,δand
+/ba∇dblf/ba∇dblH′k
+δ=/ba∇dblf/ba∇dbl′
+k,2,δ.
+18The next lemma plays a crucial role in the proof of theorem 3.1 .
+Lemma A.1. Assume u∈W′k,p
+δwithn−kp <0, then we have the
+following estimate
+r−δ|u| ≤C/ba∇dblu/ba∇dbl′
+k,p,δ. (78)
+Moreover, we have
+lim
+r→0r−δ|u|= lim
+r→∞r−δ|u|= 0. (79)
+We will use this lemma only for the particular case p= 2,n= 3,
+k= 2 and δ=−1/2, we state however the proof for the general case
+since it can have other applications.
+Proof.This proof is adapted from [5], Theorem 1.2, where the state-
+ment is proved for weighted spaces at infinity (namely, Wk,p
+δspaces in
+the notation of [5]).
+LetBRbe the ball of radius Rcentered at the origin, and let AR
+be the annulus AR=B2R\BR. We define the rescaled function
+uR(x) :=u(Rx). (80)
+Then, the fundamental scaling property of the spaces W′k,p
+δ(cf. equa-
+tion after equation (1.3) in [5]) is given by
+/ba∇dbluR/ba∇dblk,p,δ;A1=Rδ/ba∇dblu/ba∇dblk,p,δ;AR, (81)
+where we have used the same notation as in [5] for norms over su bsets
+ofRn.
+We have
+sup
+ARr−δ|u|= sup
+A1R−δr−δ|uR|, (82)
+≤CR−δ/ba∇dblr−δuR/ba∇dblk,p;A1, (83)
+≤CR−δ/ba∇dbluR/ba∇dbl′
+k,p,δ;A1, (84)
+=C/ba∇dblu/ba∇dbl′
+k,p,δ;AR. (85)
+The line (82) is a trivial change of coordinates. For the ineq uality (83)
+we have used the standard Sobolev estimate on the bounded dom ain
+A1, which is valid for n−kp <0. We have denoted the standard
+Sobolev norm on a domain Ω by /ba∇dbl·/ba∇dblk,p;Ω. It is important to note that
+the constant Cdoes not depend on R, since the domain A1does not
+either. The inequality in (84) is trivial because on the doma inA1the
+two norms (standard and weighted) are equivalent. Finally, in (85)
+we applied the scaling property (81).
+19Consider the set of annulus A2jand define uj=u|A2j. It is clear
+that
+u=∞/summationdisplay
+j=−∞uj. (86)
+Then, we use the estimate (82) on A2jand sum over all j
+(supr−δ|u|)p≤∞/summationdisplay
+j=−∞(supr−δ|uj|)p≤C∞/summationdisplay
+j=−∞/ba∇dbluj/ba∇dbl′p
+k,p,δ,(87)
+=C/ba∇dblu/ba∇dbl′p
+k,p,δ. (88)
+which proves (78).
+To prove (79) we observe that the sum/summationtext∞
+j=−∞(supr−δ|uj|)pis an
+infinite sum of positive real numbers which is bounded, hence in the
+limit we must have
+lim
+j→±∞(supr−δ|uj|) = 0, (89)
+which is equivalent to (79).
+B Properties of extreme Kerr initial
+data
+The spacetime metric for extreme Kerr black hole in Boyer-Li ndquist
+coordinates ( t,˜r,θ,φ), is given by
+g=−∆sin2θ
+ηdt2+η(dφ−Ωdt)2+Σ
+∆d˜r2+Σdθ2(90)
+whereηis the square norm of the axial Killing vector
+ηµ=/parenleftbigg∂
+∂φ/parenrightbiggµ
+, η=gνµηµηµ, (91)
+given by
+η=(˜r2+a2)2−a2∆sin2θ
+Σsin2θ. (92)
+The functions ∆ and Σ are given by
+∆ = (˜r−m)2,Σ = ˜r2+a2cos2θ, (93)
+and Ω is the angular velocity
+Ω =2a2˜rsin2θ
+ηΣ. (94)
+20Herea=J/mis the angular momentum per unit mass, and we con-
+sider the extreme case/radicalbig
+|J|=m. Note that for extreme Kerr we have
+two possible values for the angular momentum J=±m2(and hence
+a=±m).
+Take a surface t=constant , define the radius rasr= ˜r−m.
+From (90) we deduce that the intrinsic metric on this surface has the
+form (4) with
+e2q0=Σsin2θ
+η,Φ4
+0=η
+ρ2. (95)
+The twist potential of the Killing vector ηµis given by
+ω0= 2J(cos3θ−3cosθ)−2Jm2cosθsin4θ
+Σ. (96)
+The lapse function and shift vector for this foliation are gi ven by
+α=r/radicalbig
+Σ+a2(1+2a(r+a)/Σ)sin2θ, (97)
+βφ=−2a2sin2θ(r+a)
+Σ3r2. (98)
+The following asymptotic limit are interesting
+lim
+r→0√rΦ0=/parenleftbigg4m2
+1+cos2θ/parenrightbigg1/4
+, (99)
+lim
+r→0e2q0=/parenleftbigg1+cos2θ
+2/parenrightbigg2
+, (100)
+lim
+r→0ω0=−8Jcosθ
+1+cos2θ, (101)
+lim
+r→0R=2sin2θ
+m2(1+cos2θ)3. (102)
+We take the opportunity to correct a misprint in equation A.1 5 of [2].
+There is a missing exponent 3 in the denominator of this formu la, it
+should be the same as equation (102).
+In the following, we use ∆ to denote the flat Laplace operator i n
+three dimensions, the two dimensional Laplacian ∆ 2is given by
+∆2=1
+r∂r(r∂r)+1
+r2∂2
+θ. (103)
+The next lemma plays a crutial role in the proof of theorem 3.1 .
+Lemma B.1. Letq0andΦ0be given by (95)andω0by(96). Then
+the function αdefined in (75), has the form α=hr−2whereh≥0
+andhis bounded in R3.
+21Proof.From the Hamiltonian constraint
+−∆2q0
+4=∆Φ0
+Φ0+(∂w0)2
+16η2. (104)
+and the stationary equation satisfied by extreme Kerr’s init ial data
+(see [2])
+∆Φ0
+Φ0=−(∂ω0)2
+4η2+(∂Φ0)2
+Φ2
+0(105)
+we obtain
+−∆2q0
+4=−3
+16(∂w0)2
+η2+(∂Φ0)2
+Φ2
+0. (106)
+Therefore
+α=(∂ω0)2
+4η2+(∂Φ0)2
+Φ2
+0, (107)
+which is clearly a non negative quantity. By an explicit calc ulation it
+can be seen that αis in fact a strictly positive function. Since we do
+not need this property for our purposes, we omit the details. Also by
+explicit means, we note that αisO(r−2) at the origin, and O(r−4) at
+infinity, beingotherwisebounded. Thereby, theremustexis tapositive
+function hsuch that α=hr−2.
+Itis important to note that in theproof of lemma B.1 we have us ed
+the fact that extreme Kerr satisfies the stationary Einstein equations
+and also that the topology of extreme Kerr allows us to choose these
+coordinates. In particular, the proof fails for non-extrem e Kerr. See
+a similar discussion in [16] at the end of page 6868.
+Lemma B.2. LetΦ0,q0andω0be defined by (95)and(96), and
+assume that m >0. Then we have the following bounds:
+√m≤√
+r+m≤√rΦ0≤√
+2√
+r+m, (108)
+(∂ω0)2
+ρ4≤116m4
+r6, (109)
+|∆2q0| ≤90
+r2. (110)
+Proof.Inequality (108) has been proved in [2] (see equations (10) a nd
+(12) in this reference).
+We have
+(∂ω0)2=4m4ρ6F
+r8Σ4(111)
+where
+F= 4r2a4˜r2sin2(2θ)+/parenleftbig
+3˜r4+a2˜r2+a2(˜r2−a2)cos2θ/parenrightbig2(112)
+22and ˜r=r+m. Then
+F≤4r2a4˜r2+/parenleftbig
+3˜r4+a2˜r2+a2˜r2/parenrightbig2≤29(r+a)8.(113)
+We also find, bounding Σ ≥(r+a)2andρ≤r, that
+(∂ω0)2≤4a4ρ429(r+a)8
+r6(r+a)8= 116m4
+r2. (114)
+Finally, using the explicit expressions for Φ 0andω0one can check,
+after a laborious but straightforward calculation, the bou ndon|∆2q0|.
+C The implicit function theorem
+To facilitate the readability of the article and also to fix th e notation,
+we reproduce in this appendix well known results on differenti al cal-
+culus in Banach spaces (see, for example [1], [9], and also th e more
+introductory text books [30], [32]).
+LetXandZbe Banach spaces. Let A:X→Zbe a linear
+bounded operator. We denote by L(X,Z) the set of all linear and
+bounded operators from XtoZ. The set L(X,Z) is itself a Banach
+space with the operator norm defined by
+/ba∇dblA/ba∇dblL(X,Z)= sup
+/bardblx/bardbl/\e}atio\slash=0/ba∇dblA(x)/ba∇dblZ
+/ba∇dblx/ba∇dblX. (115)
+Letxbe a point in Xand letGbe a mapping from a neighborhood
+ofxintoZ. ThenGis called Fr´ echet differentiable at the point xif
+there exists a linear operator DG(x)∈ L(X,Z) such that
+lim
+v→0/ba∇dblG(x+v)−G(x)−DG(x)[v]/ba∇dbl
+/ba∇dblx/ba∇dblX= 0. (116)
+The map Gis called continuously differentiable (i.e. C1) if the deriva-
+tiveDG(x) as an element of L(X,Z) depends continuously on x.
+Namely, for every δ >0 there exists ǫ >0 such that
+/ba∇dblx1−x2/ba∇dblX< ǫ⇒ /ba∇dblDG(x1)−DG(x2)/ba∇dblL(X,Z)< δ. (117)
+LetX,YandZbe Banach spaces and let Gbea map G:X×Y→Z,
+in a similar way we define the partial derivatives with respec t to the
+first argument by D1G(x,y) and with respect to the second argument
+byD2G(x,y).
+23Theorem C.1 (Implicit Function Theorem) .Suppose Uis a neigh-
+borhood of 0inX,Vis a neighborhood of 0inY, andG:X×Y→Z
+isC1. Suppose G(0,0) = 0andD2G(0,0) :Y→Zdefines a bounded
+operator and it is an isomorphism. Then, there exists a neighb or-
+hoodWof the origin in Xand a continuously differentiable mapping
+f:W→Ysuch that G(x,f(x)) = 0. Moreover, for small xandy,
+f(x)is the only solution yof the equation G(x,y) = 0.
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+arXiv:1001.0179v1  [math.SG]  31 Dec 2009Deformations of Poisson structures
+by closed 3-forms1
+O. I. Mokhov
+Abstract
+We prove that an arbitrary Poisson structure ωij(u) and an arbitrary closed 3-
+formTijk(u) generate the local Poisson structure Aij(u,ux) =Mi
+s(u,ux)ωsj(u),where
+Mi
+s(u,ux)(δs
+j+ωsp(u)Tpjk(u)uk
+x) =δi
+j, on the corresponding loop space. We obtain also
+a special graded ε-deformation of an arbitrary Poisson structure ωij(u) by means of an
+arbitrary closed 3-form Tijk(u).
+In this paper we prove that an arbitrary Poisson structure ωij(u) and an arbitrary
+closed 3-form Tijk(u) generate the local Poisson structure
+Aij(u,ux) =Bi
+s(u,ux)ωsj(u), (1)
+where
+Bi
+s(u,ux)Ms
+j(u,ux) =δi
+j, Ms
+j(u,ux) =δs
+j+ωsp(u)Tpjk(u)uk
+x, (2)
+i.e., the matrix operator Aij(u,ux) gives the Poisson bracket
+{I,J}=/integraldisplayδI
+δui(x)Aij(u,ux)δJ
+δuj(x)dx (3)
+on the space of functionals on the corresponding loop space.
+LetMNbe an arbitrary smooth N-dimensional manifold with the local coordinates
+u= (u1,...,uN). By the loop space ΩMof the manifold MNwe mean, in this paper,
+the space of all smooth parametrized mappings of the circle S1intoMN,γ:S1→MN,
+γ(x) ={ui(x)}, x∈S1. The tangent space TγΩMof the loop space Ω Mat the point γ
+consist of all smooth vector fields ξ={ξi,1≤i≤N}, defined along the loop γwith
+ξ(γ(x))∈Tγ(x)M,∀x∈S1, whereTγ(x)Mis a tangent space of the manifold Mat the
+1The work was supported by the Max-Planck-Institut f¨ ur Mathem atik (Bonn, Germany), by the
+Russian Foundation for Basic Research (project no. 08-01-0046 4) and by the programme “Leading
+Scientific Schools” (project no. NSh-1824.2008.1).
+1pointγ(x). All closed 2-forms (presymplectic structures) on the loop spac e ΩMthat
+are given by matrix operators of the form ωij(u,ux,...,u (k)), i.e., all closed 2-forms of
+the form
+ω(ξ,η) =/integraldisplay
+S1ξiωij(u,ux,...,u (k))ηjdx, (4)
+whereξ,η∈TγΩM, were completely described in [1] (see also descriptions of various
+differential-geometric classes of symplectic (presymplectic) and Po isson structures in
+[2]–[9]).
+Theorem 1 [1]. A bilinear form (4)is a closed skew-symmetric 2-form (a presym-
+plectic structure )on the loop space ΩMif and only if
+ωij(u,ux,...,u (k)) =Tijk(u)uk
+x+Ωij(u), (5)
+whereTijk(u)is anarbitrary closed3-formonthe manifold MNandΩij(u)is anarbitrary
+closed 2-form on MN.
+If the matrix ωij(u,ux,...,u (k)) is nondegenerate, det( ωij(u,ux,...,u (k)))/ne}ationslash= 0, then
+the corresponding presymplectic form (4), (5) is symplectic and th e inverse matrix
+ωij(u,ux,...,u (k)),ωis(u,ux,...,u (k))ωsj(u,ux,...,u (k)) =δi
+j, gives the Poisson struc-
+ture
+{I,J}=/integraldisplayδI
+δui(x)ωij(u,ux,...,u (k))δJ
+δuj(x)dx (6)
+on the loop space Ω M, i.e., the bracket (6) is skew-symmetric and satisfy the Jacobi
+identity. Therefore Theorem 1 gives the complete description of all nondegenerate
+Poisson structures on the loop space Ω Mthat are given by matrix operators of the
+formωij(u,ux,...,u (k)), i.e., all the nondegenerate Poisson brackets of the form (6),
+det(ωij(u,ux,...,u (k)))/ne}ationslash= 0 (such nondegenerate Poisson structures were studied by As-
+tashov and Vinogradov in [9], see also [7]–[8] and [1]–[6]). We note that if t he closed
+2-form Ω ij(u) is nondegenerate, det(Ω ij(u))/ne}ationslash= 0, i.e, the form Ω ij(u) is symplectic on
+MN, then the 2-form (5) is a nondegenerate form on Ω Mfor any closed 3-form Tijk(u)
+on the manifold MNsince it is obvious that in this case det( Tijk(u)uk
+x+ Ωij(u))/ne}ationslash= 0.
+Thus we can define, on the loop space of an arbitrary symplectic man ifoldMN, the
+Poisson bracket
+{I,J}=/integraldisplayδI
+δui(x)ωij(u,ux)δJ
+δuj(x)dx, (7)
+where
+ωli(u,ux)(Tijk(u)uk
+x+Ωij(u)) =δl
+j, (8)
+IandJbeing arbitrary functionals on Ω M. The Poisson bracket (7), (8) is a par-
+tial case of the bracket (1)–(3), namely, the case when the Poiss on structure ωij(u) is
+nondegenerate, det( ωij(u))/ne}ationslash= 0,ωij(u) = Ωij(u),Ωis(u)Ωsj(u) =δi
+j,since
+ωij(u,ux) =Ci
+s(u,ux)Ωsj(u), (9)
+2where
+Cl
+s(u,ux)(δs
+j+Ωsi(u)Tijk(u)uk
+x) =δl
+j. (10)
+The case of degenerate Poisson structures ωij(u), det(ωij(u)) = 0, is much more com-
+plicated. We note that in contrast to the case of all closed 2-forms (presymplectic
+structures) of the form (4) (Theorem 1) the problem of descript ion of all degenerate
+Poisson structures of the form (6) is a very complicated and unsolv ed problem.
+Theorem 2. An arbitrary Poisson structure ωij(u)and an arbitrary closed 3-form
+Tijk(u)give the local Poisson bracket (1)–(3).
+First of all, we note that obviously the matrix operator Aij(u,ux) (1), (2) is skew-
+symmetric.
+Lemma. A skew-symmetric matrix operator Aij(u,ux) (1)gives a Poisson bracket
+(3)if and only if the following relations hold:
+ωij(u)ωrp(u)∂Ms
+r
+∂uix=ωis(u)ωrj(u)∂Mp
+r
+∂uix, (11)
+ωij(u)ωrp(u)∂Ms
+r
+∂ui−ωij(u)d
+dx/parenleftbigg∂Ms
+r
+∂ui
+xωrp(u)/parenrightbigg
++∂ωij
+∂urωrp(u)Ms
+i(u)+
++ωis(u)ωrj(u)∂Mp
+r
+∂ui+d
+dx/parenleftbig
+ωis(u)/parenrightbig∂Mp
+r
+∂uixωrj(u)+∂ωis
+∂urωrj(u)Mp
+i(u)+
++ωip(u)ωrs(u)∂Mj
+r
+∂ui+d
+dx/parenleftbig
+ωip(u)/parenrightbig∂Mj
+r
+∂ui
+xωrs(u)+∂ωip
+∂urωrs(u)Mj
+i(u) = 0.(12)
+IfMi
+s(u,ux) =δs
+j+ωsp(u)Tpjk(u)uk
+x, then relations (11), (12) hold for an arbitrary
+Poisson structure ωij(u) and an arbitrary closed 3-form Tijk(u).
+Let us add an arbitrary parameter εin the formula for our Poisson structure:
+Aij(ε,u,ux) =Bi
+s(ε,u,ux)ωsj(u), (13)
+where
+Bi
+s(ε,u,ux)(δs
+j+εωsp(u)Tpjk(u)uk
+x) =δi
+j. (14)
+We can now expand the Poisson structure Aij(ε,u,ux) in series in ε:
+Aij(ε,u,ux) =ωij(u)−εωis(u)Tsrk(u)ωrj(u)uk
+x+···. (15)
+This expansion give an ε-deformation of an arbitrary Poisson structure ωij(u) by means
+of an arbitrary closed 3-form Tijk(u).
+We note that this ε-deformation of an arbitrary Poisson structure ωij(u) belongs to a
+special class of graded ε-deformations of Poisson structures (see, for example, [10], [11]) .
+3Acknowledgements. The work was supported by the Max-Planck-Institut f¨ ur
+Mathematik (Bonn, Germany), by the Russian Foundation for Basic Research (project
+no. 08-01-00464) and by a grant of the President of the Russian F ederation (project no.
+NSh-1824.2008.1).
+References
+[1] O. I. Mokhov, “Symplectic and Poisson geometry on loop spaces o f manifolds and
+nonlinearequations”, In: TopicsinTopologyandMathematicalPhys ics, Ed. S.P.Novikov,
+Amer. Math. Soc., Providence, RI, 1995, pp. 121–151;
+http://arXiv.org/hep-th/9503076 (1995).
+[2] O. I. Mokhov, “Poisson and symplectic geometry on loop spaces o f smooth man-
+ifolds”, In: Geometry from the Pacific Rim, Proceedings of the Pacifi c Rim Geometry
+Conference held at National University of Singapore, Republic of Sin gapore, December
+12–17, 1994, Eds. A.J.Berrick, B.Loo, H.-Y.Wang, Walter de Gruyter , Berlin, 1997, pp.
+285–309.
+[3] O. I. Mokhov, “Differential geometry of symplectic and Poisson s tructures onloop
+spaces of smooth manifolds, and integrable systems”, Trudy Mate m. Inst. Akad. Nauk,
+Vol. 217, Moscow, Nauka, 1997, pp. 100–134; English translation in Proceedings of the
+Steklov Institute of Mathematics (Moscow), Vol. 217, 1997, pp. 9 1–125.
+[4] O. I. Mokhov, “Symplectic and Poisson structures on loop space s of smooth
+manifolds, and integrable systems”, Uspekhi Matematicheskikh Na uk, Vol. 53, No. 3,
+1998, pp. 85–192; English translation in Russian Mathematical Surv eys, Vol. 53, No.
+3, 1998, pp. 515–622.
+[5] O. I. Mokhov, “Symplectic and Poisson geometry on loop spaces o f smooth man-
+ifolds and integrable equations”, Moscow–Izhevsk, Institute of C omputer Studies, 2004
+(In Russian); English version: Reviews in Mathematics and Mathemat ical Physics, Vol.
+11, Part 2, Harwood Academic Publishers, 2001; Second Edition: Re views in Math-
+ematics and Mathematical Physics, Vol. 13, Part 2, Cambridge Scien tific Publishers,
+2009.
+[6] O. I. Mokhov, “Symplectic forms on loop space and Riemannian geo metry”,
+Funkts. Analiz i Ego Prilozh., Vol. 24, No. 3, 1990, pp. 86–87; English translation in
+Functional Analysis and its Applications, Vol. 24, No. 3, 1990, pp. 24 7–249.
+[7] A. M. Astashov, “Normal forms of Hamiltonian operators in field t heory”, Dokl.
+Akad. Nauk SSSR, Vol. 270, No. 5, 1983, pp. 363–368; English tran slation in Soviet
+Math. Dokl., Vol. 27, No. 3, 1983, pp. 685–689.
+[8] A. M. Vinogradov, “Hamiltonian structures in field theory”, Dokl. Akad. Nauk
+SSSR, Vol. 241, No. 1, 1978, pp. 18–21; English translation in Soviet Math. Dokl., Vol.
+19, No. 4, 1978, pp. 790–794.
+4[9] A. M. Astashov and A. M. Vinogradov, “On the structure of Ham iltonian op-
+erators in field theory”, J. Geom. Phys., Vol. 3, No. 2, 1986, pp. 78 1–785; English
+translation in Soviet Math. Dokl., Vol. 27, 1983, pp. 263–287.
+[10] B. Dubrovin and Y. Zhang, “Normal forms of hierarchies of inte grable PDEs,
+Frobenius manifolds and Gromov - Witten invariants”; arXiv:math/01 08160 (2001).
+[11] B. Dubrovin, S.-Q. Liu and Y. Zhang, “On Hamiltonian perturbatio ns of hyper-
+bolic systems of conservation laws, I: Quasi-triviality of bi- Hamiltonia n perturbations,”
+Comm. Pure Appl. Math., Vol. 59, 2006, 559 - 615; arXiv:math/04100 27 (2004).
+O. I. Mokhov
+Centre for Nonlinear Studies,
+L.D.Landau Institute for Theoretical Physics,
+Russian Academy of Sciences,
+Kosygina str., 2,
+Moscow, 117940, Russia;
+Department of Geometry and Topology,
+Faculty of Mechanics and Mathematics,
+M.V.Lomonosov Moscow State University,
+Moscow, 119992, Russia
+E-mail: mokhov@mi.ras.ru; mokhov@landau.ac.ru; mokhov@bk.ru
+5
\ No newline at end of file
diff --git a/1001.0180.txt b/1001.0180.txt
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+++ b/1001.0180.txt
@@ -0,0 +1,458 @@
+arXiv:1001.0180v4  [math.QA]  23 Aug 2010THE BIJECTIVITY OF THE ANTIPODE REVISITED
+M.C. IOVANOV AND S ¸. RAIANU
+Abstract. We provide a very short approach to several fundamental resu lts for Hopf
+algebras with nonzero integrals. Besides being short, our a pproach is the first to prove
+the bijectivity of the antipode without using the uniquenes s of the integrals of Hopf
+algebras and to obtain the uniqueness of integrals as a corol lary in a way similar to the
+classical theory of the Haar measure on compact groups.
+Introduction
+One of the fundamental notions of the theory of Hopf algebras is that of an integral,
+which is an analog of the Haar integral of a compact group and d raws its name from
+there. More precisely, if Gis a compact group and R(G) is the algebra of continuous
+representative functions on G, i.e. the space spanned by the coefficients ηijof all con-
+tinuous representations η:G→GLn(C), then the restriction of the Haar integral to
+R(G) becomes an integral in Hopf algebra sense. In this respect, a Hopf algebra having a
+nonzero integral is a generalization of the algebra of conti nuous representative functions
+on a compact group. Integrals for Hopf algebras were introdu ced by Sweedler in 1969 [14].
+In that paper he proves a series of fundamental results about Hopf algebras with nonzero
+integrals, including the fact that integrals are unique and the antipode is bijective when
+the Hopf algebra is finite dimensional. The questions about t he validity of these results
+for Hopf algebras with nonzero integrals of possibly infinit e dimension appear explicitly
+in Sweedler’s book [15]. These questions were given affirmati ve answers: the uniqueness
+of the integrals was proved by Sullivan in 1971 [13], then, us ing Sullivan’s result, Radford
+proved in 1977 that the antipode of a Hopf algebra with nonzer o integrals is bijective [10].
+Many other proofs for the uniqueness of integrals were found later. Some of these proofs
+have a strong homological flavor and use the fact that integra ls are just comodule maps
+[12], [2], [9], [6]. In contrast with the abundance of proofs for the uniqueness of integrals,
+Radford’s proof for the bijectivity of the antipode was virt ually the only one available
+(with a simplification due to [4]) until very recently, when a lternate proofs were obtained
+in [7] and [8] by using a purely coalgebraic approach, as a byp roduct of a general theory
+of algebraic “integrals” or infinite dimensional generaliz ed Frobenius algebras. All proofs
+for the bijectivity of the antipode used the uniqueness of in tegrals, and it was hard to say
+whether this happened by necessity or it was just an effect of th e order in which the two
+results were obtained. Moreover, the classical proof for th e uniqueness of Haar measures
+adapted for Hopf algebras requires the bijectivity of the an tipode (see [16] and [11]). In
+the classical case of compact groups, the Hopf algebra of rep resentative continuous func-
+tions clearly has a bijective antipode because it is commuta tive, and this probably made
+Date: May 15, 2010.
+12 M.C. IOVANOV AND S ¸. RAIANU
+the causative relationship between bijectivity and unique ness harder to understand. The
+fact that the antipode of a Hopf algebra with nonzero integra ls might not be necessarily
+bijective was the only obstacle in proving the uniquenessof the integrals by using the same
+technique as in the case of Haar measures.
+In this note we find a very short approach to explain the above m entioned results. We
+first prove the bijectivity of the antipode without using the uniqueness of the integrals.
+This is the first proof constructed in this manner, and it foll ows by using a technique
+from [7]. We can then just use the classical proof of the uniqu eness of the Haar measure
+from locally compact groups, as was done in [16] for multipli er Hopf algebras (see also
+the chapter on Haar measures in [3]). Thus, besides being sho rt, this proof also has the
+advantage that it shows oncemoreaneven stronger parallel t hannoted previouslybetween
+Hopf algebras and locally compact groups.
+The Proofs
+LetHbe a Hopf algebra over the field k. Recall that an integral λof the Hopf algebra
+His an element in H∗such that αλ=α(1)λfor allα∈H∗.
+For (M,ρ)∈ MH,
+ρ:M−→M⊗H, ρ(m) =m0⊗m1,
+we defineSM∈HMwith comodule structure given by
+m/mapsto−→m(−1)⊗m(0)=S(m1)⊗m0
+It is clear that we have a functor F:MH−→HM,F(M) =SM, andFis the identity
+on morphisms. Then we have:
+Proposition 1.SRat(H∗), with left H-module structure given by
+H⊗SRat(H∗)−→SRat(H∗), x⊗α−→x⇀α
+and leftH-comodule structure as above is a left H-Hopf module.
+Proof.We need to prove that
+(x⇀α)(−1)⊗(x⇀α)(0)=x1α(−1)⊗x2⇀α(0)
+which is
+S((x⇀α)1)⊗(x⇀α)0=x1S(α1)⊗x2⇀α0
+or
+< βS((x⇀α)1)(x⇀α)0,y >=< β(x1S(α1))(x2⇀α0),y >,∀β∈H∗, y∈HTHE BIJECTIVITY OF THE ANTIPODE REVISITED 3
+We have
+< βS((x⇀α)1)(x⇀α)0,y >=<(β◦S)∗(x⇀α),y >(rtH-com str of Rat(H∗))
+=βS(y1)(x⇀α)(y2)
+=βS(y1)α(y2x)
+=βS(y1)α(y2x2)ε(x1)
+=β(ε(x1)S(y1))α(y2x2)
+=β(x1S(x2)S(y1)α(y2x3)
+= (β↼x1)(S(y1x2))α(y2x3)
+=<(β↼x1)◦S,(yx2)1>< α,(yx2)2>
+=<((β↼x1)◦S)∗α,yx2>
+=<((β↼x1)◦S)(α1)α0,yx2>
+=β(x1S(α1))α0(yx2)
+=< β(x1S(α1))(x2⇀α0),y >,
+which ends the proof. /square
+LetCbe a coalgebra and M∈CM. The coalgebra CMassociated to Mis the smallest
+subcoalgebra CMofCsuch that ρ(M)⊆CM⊗M, i.e.CM=∩A⊆C,ρ(M)⊆A⊗MA(see [5,
+p. 102]). With this notation we have:
+Proposition 2. IfMf։Nis a surjective morphism of left C-comodules, then CN⊆CM.
+Proof.LetK=Ker(f). ThenρM(K) =ρK(K)⊆C⊗K. Also, since ρM(M)⊆CM⊗M,
+it follows that ρM(K)⊆CM⊗M. SoρM(K)⊆(CM⊗M)∩(C⊗K), i.e.ρK(K) =
+ρM(K)⊆CM⊗K. Therefore, Kis aCM-submodule, so ρM/K(M/K)⊆CM⊗M/K, i.e.
+ρN(N)⊆CM⊗N, henceCN⊆CMby definition. /square
+We are now ready to prove
+Theorem 3. IfHis co-Frobenius, Sis bijective.
+Proof.By Proposition 1 and the fundamental theorem of Hopf modules , we have that
+SRat(H∗)≃H⊗(SRat(H∗))co=H⊗/integraltext
+l, since it is easy to see that (SRat(H∗))co=/integraltext
+l.
+Also, since ( Rat(H∗)H≃/integraltext
+l⊗H=H(dim/integraltext
+l)inMH, we getSRat(H∗)≃(SH)(dim/integraltext
+l)=/circleplustext
+dim/integraltext
+lSH, using the fact that the functor Fclearly commutes with direct sums. Since
+Rat(H∗)/\e}atio\slash= 0 (equivalently/integraltext
+l/\e}atio\slash= 0) we can find a surjection of left H-comodules
+π: (SH)(dim/integraltext
+l)≃SRat(H∗)≃H⊗(SRat(H∗))co։HH
+ThenCH⊆/summationtextCSH=CSHby Proposition 2 and the obvious fact that C/circleplustext
+i∈IMi=/summationtext
+i∈ICMi. Obviously, CH=H(by the counit property), and also CSH=S(H), since
+∀h∈H,S(h) =S(h2)ε(h1)∈CSH, because ρSH(h) =S(h2)⊗h1∈H⊗SH. SoH⊆S(H),
+and the proof is complete. /square
+Corollary 4. Ift∈/integraltext
+l,t/\e}atio\slash= 0, thent◦S∈/integraltext
+r,t◦S/\e}atio\slash= 0.4 M.C. IOVANOV AND S ¸. RAIANU
+Proof.Obvious. /square
+Asaconsequenceofthisproof,theprooffortheuniquenesso fintegrals canbetranslated
+verbatim to Hopf algebras from the case of Haar measures, as w as done by Van Daele in
+[16] for regular multiplier Hopf algebras. This proof could not be used for Hopf algebras
+becauseit requiresthebijectivity of theantipode, andunt il now all proofsof thebijectivity
+of the antipode used the uniqueness of integrals. A modified v ersion of the proof below
+not requiring the bijectivity of the antipode was given in [1 1].
+Corollary 5. The dimension of/integraltext
+lis at most one.
+Proof.(Identical to the proof of [16, Theorem 3.7]) Let t1,t2∈/integraltext
+l,t2/\e}atio\slash= 0. By Corollary 4
+λ=t2◦S∈/integraltext
+r\{0}. Then for any h∈Hthere is a g∈Hsuch that
+t1(xh) =t2(xg)∀x∈H. (1)
+Indeed, let l,m∈Hsuch that λ(l) = 1 and t2(m) = 1. Then
+t1(xh) =λ(l)t(xh)
+=λ(x1h1l)t1(x2h2) (t1∈/integraltext
+l)
+=λ(x1h1l1)t1(x2h2l2S(l3))
+=λ(xhl1)t1(S(l2)) (λ∈/integraltext
+r)
+=λ(xe) (e=hl1t1(S(l2))
+=λ(xe)t2(m)
+=λ(x1e1)t2(x2e2m) (λ∈/integraltext
+r)
+=λ(x1e1m2S−1(m1))t2(x2e2m3)
+=λ(S−1(m1))t2(xem2) (t2∈/integraltext
+l)
+=t2(xg) (g=em2λ(S−1(m1)).
+We finish the proof by showing that t1is a scalar multiple of t2. Fory∈Hwe have:
+t1(y) =λ(l)t1(y)
+=λ(l1)t1(yl2) (λ∈/integraltext
+r)
+=λ(S(y1)y2l1)t1(y3l2)
+=λ(S(y1))t1(y2l) (t1∈/integraltext
+l)
+=λ(S(y1))t2(y2g) (1)
+=λ(e)t2(y),
+where the last equality follows from reversing the previous three equalities, and the proof
+is complete. /square
+Remarks 6. a) Note that aside from the bijectivity of the antipode, the p roof above uses
+only the definition of integrals.
+b) To compensate for not being able to use the inverse of S, a special left integral had to be
+chosen in [11]and it was shown to form a basis of/integraltext
+l. Once we are able to use the inverse
+ofS, the proof above shows that any non-zero left integral will d o.THE BIJECTIVITY OF THE ANTIPODE REVISITED 5
+Following the work of Lin, Larson, Sweedler, and Sullivan, t he existence of a nonzero
+integral is equivalent to various representation theoreti cal properties of the Hopf algebra,
+such as that of being co-Frobenius as a coalgebra, or having n onzero rational part. As a
+final application, we show how our approach may be used to simp lify the proof of some of
+these results (see [5, Theorem 5.3.2]):
+Corollary 7. For a Hopf algebra Hthe following assertions are equivalent:
+(1)His left co-Frobenius
+(2)His left quasi-co-Frobenius
+(3)His left semiperfect
+(4)Rat(H∗)/\e}atio\slash= 0(H∗as a left H∗-module)
+(5)/integraltext
+l/\e}atio\slash= 0
+(6) The right hand version of (1)-(5)
+Proof.Allimplications followdirectlyfromthedefinitionsorfro mSweedler’sisomorphism,
+with the exception of 5) ⇒6) which follows from Corollary 4. /square
+References
+[1] E. Abe, Hopf Algebras , Cambridge Univ. Press, 1977.
+[2] M. Beattie, S. D˘ asc˘ alescu, L. Gr¨ unenfelder, C. N˘ ast ˘ asescu, Finiteness Conditions, Co-Frobenius
+Hopf Algebras and Quantum Groups, J. Algebra 200 (1998), 312 -333.
+[3] Bourbaki, N. ´El´ ements de math´ ematique. Fascicule XXIX. Livre VI: Int´ egration. Chapitre 7:
+Mesure de Haar. Chapitre 8: Convolution et repr´ esentation s. (French) Actualit´ es Scientifiques et
+Industrielles, No. 1306 Hermann, Paris 1963.
+[4] C. C˘ alinescu, On the bijectivity of the antipode in a co- Frobenius Hopf algebra, Bull. Math. Soc.
+Sci. Math. Roumanie (N.S.) 44(92)(2001), no. 1, 59-62.
+[5] S. D˘ asc˘ alescu, C. N˘ ast˘ asescu, S ¸. Raianu, Hopf Algebras: an Introduction , Monographs and Text-
+books in Pure and Applied Mathematics 235, Marcel Dekker, Inc., New York, 2001.
+[6] S. D˘ asc˘ alescu, C. N˘ ast˘ asescu, B. Torrecillas, Co-F robenius Hopf Algebras: Integrals, Doi-
+Koppinen Modules and Injective Objects, J. Algebra 220 (199 9), 542-560.
+[7] M.C. Iovanov, Generalized Frobenius algebras and the th eory of Hopf algebras, preprint,
+arXiv:0803.0775.
+[8] M.C. Iovanov, Abstract Integrals in Algebra, preprint, arXiv:0810.3740.
+[9] C. Menini, B. Torrecillas, R. Wisbauer, Strongly regula r comodules and semiperfect Hopf algebras
+over QF rings, J. Pure Appl. Algebra 155(2001), no. 2-3, 237-255.
+[10] D.E. Radford, Finiteness conditions for a Hopf algebra with a nonzero integral, J. Algebra 46
+(1977), no. 1, 189-195.
+[11] S ¸. Raianu, An easy proof of the uniqueness of integrals , in Hopf Algebras and Quantum Groups,
+Proceedings of the Brussels Conference 1998, (eds. S. Caene peel and F. Van Oystaeyen), Marcel
+Dekker Lecture Notes in Pure and Appl. Math., vol 209, 2000, 2 37-240.
+[12] D. S ¸tefan, The uniqueness of integrals. A homological approach, Comm. Algebra, 23(1995),
+1657-1662.
+[13] J.B. Sullivan, The Uniqueness of Integrals for Hopf Alg ebras and Some Existence Theorems of
+Integrals for Commutative Hopf Algebras, J. Algebra, 19(1971), 426-440.
+[14] M.E. Sweedler, Integrals for Hopf algebras, Ann. of Mat h.,89(1969), 323-335.
+[15] M.E. Sweedler, Hopf Algebras , Benjamin, New York, 1969.6 M.C. IOVANOV AND S ¸. RAIANU
+[16] A. Van Daele, An algebraic framework for group duality, Adv. Math. 140(1998), 323-366.
+University of Southern California, 3600 Vermont Ave., Los A ngeles, CA 90089, USA,
+and, University of Bucharest, Fac. Matematica & Informatic a, Str. Academiei nr. 14,
+Bucharest 010014, Romania
+E-mail address :yovanov@gmail.com, iovanov@usc.edu
+Mathematics Department, California State University, Dom inguez Hills, 1000 E Vic-
+toria St, Carson CA 90747, USA
+E-mail address :sraianu@csudh.eduarXiv:1001.0180v4  [math.QA]  23 Aug 2010THE BIJECTIVITY OF THE ANTIPODE REVISITED
+M.C. IOVANOV AND S ¸. RAIANU
+Dedicated to Mia Cohen on the occasion of her retirement
+Abstract. We provide a very short approach to several fundamental resu lts for Hopf
+algebras with nonzero integrals. Besides being short, our a pproach is the first to prove
+the bijectivity of the antipode without using the uniquenes s of the integrals of Hopf
+algebras and to obtain the uniqueness of integrals as a corol lary in a way similar to the
+classical theory of the Haar measure on compact groups.
+Introduction
+One of the fundamental notions of the theory of Hopf algebras is that of an integral,
+which is an analog of the Haar integral of a compact group and d raws its name from
+there. More precisely, if Gis a compact group and R(G) is the algebra of continuous
+representativefunctionson G, i.e. thespacespannedbythecoefficients ηijofallcontinuous
+representations η:G→GLn(C), thentherestrictionoftheHaarintegral to R(G) becomes
+an integral in the Hopf algebra sense (see Abe (1977) or D˘ asc ˘ alescu et al. (2001)). In
+this respect, a Hopf algebra having a nonzero integral is a ge neralization of the algebra of
+continuous representative functions on a compact group. In tegrals for Hopf algebras were
+introduced by Sweedler (1969a). In that paper he proves a ser ies of fundamental results
+about Hopf algebras with nonzero integrals, including the f act that integrals are unique
+and the antipode is bijective when the Hopf algebra is finite d imensional. The questions
+about the validity of these results for Hopf algebras with no nzero integrals of possibly
+infinite dimension appear explicitly in Sweedler (1969b). T hese questions were given
+affirmative answers: the uniqueness of the integrals was prov ed by Sullivan (1971), then,
+using Sullivan’s result, Radford (1977) proved that the ant ipode of a Hopf algebra with
+nonzero integrals is bijective. Many other proofs for the un iqueness of integrals were found
+later. Some of these proofs have a strong homological flavor a nd use the fact that integrals
+are just comodule maps: S ¸tefan (1995), Beattie et al. (1998 ), Menini et al. (2001),
+D˘ asc˘ alescu et al. (1999). In contrast with the abundance o f proofs for the uniqueness
+of integrals, Radford’s proof for the bijectivity of the ant ipode was virtually the only
+one available (with a simplification due to C˘ alinescu (2001 )) until very recently, when
+alternate proofs were obtained by Iovanov (preprint1) and ( preprint2) by using a purely
+coalgebraic approach, as a byproduct of a general theory of a lgebraic “integrals” or infinite
+dimensional generalized Frobenius algebras. All proofs fo r the bijectivity of the antipode
+usedtheuniquenessof integrals, and it was hardto say wheth er thishappenedbynecessity
+or it was just an effect of the order in which the two results were obtained. Moreover, the
+The first author was partially supported by CNCSIS grant TE no . 45.
+12 M.C. IOVANOV AND S ¸. RAIANU
+classical proof for the uniqueness of Haar measures adapted for Hopf algebras requires the
+bijectivity of the antipode (see Van Daele (1998) and Raianu (2000)). In the classical
+case of compact groups, the Hopf algebra of representative c ontinuous functions clearly
+has a bijective antipode because it is commutative, and this probably made the causative
+relationship between bijectivity and uniqueness harder to understand. The fact that the
+antipode of a Hopf algebra with nonzero integrals might not b e necessarily bijective was
+the only obstacle in proving the uniqueness of the integrals by using the same technique
+as in the case of Haar measures.
+In this note we find a very short approach to explain the above m entioned results. We
+first prove the bijectivity of the antipode without using the uniqueness of the integrals.
+This is the first proof constructed in this manner, and it foll ows by using a technique
+from Iovanov (preprint1). We can then just use the classical proof of the uniqueness
+of the Haar measure from locally compact groups, as was done i n Van Daele (1998) for
+multiplier Hopf algebras (see also the chapter on Haar measu res in Bourbaki (1963)).
+Thus, besides being short, this proof also has the advantage that it shows once more an
+even stronger parallel than noted previously between Hopf a lgebras and locally compact
+groups.
+The Proofs
+LetHbe a Hopf algebra over the field k. Recall that a left integral λof the Hopf
+algebraHis an element in H∗such that αλ=α(1)λfor allα∈H∗. We also recall that
+whenever nonzero left integrals exist, Sweedler (1969b) pr oved that the antipode Sof
+the Hopf algebra His injective, and therefore it has a left inverse Sl. Sweedler proved
+the injectivity of the antipode after twisting by Sthe module structure in a Hopf module
+structure on the rational module Rat(H∗). Therefore, it makes sense that when trying to
+prove the surjectivity one should consider twisting by Sthe comodule structure in some
+natural Hopf module structure on the rational part. This is p recisely what we are going
+to do.
+For (M,ρ)∈ MH,
+ρ:M−→M⊗H, ρ(m) =m0⊗m1,
+we defineSM∈HMwith comodule structure given by
+m/mapsto−→m(−1)⊗m(0)=S(m1)⊗m0
+It is clear that we have a functor F:MH−→HM,F(M) =SM, andFis the identity
+on morphisms.
+Ifx,y∈Handα∈H∗, we denote ( x⇀α)(y) =α(yx) and (α↼x)(y) =α(xy). Then
+we have:
+Proposition 1.SRat(H∗), with left H-module structure given by
+H⊗SRat(H∗)−→SRat(H∗), x⊗α/mapsto−→x⇀α, x ∈H, α∈Rat(H∗)
+and leftH-comodule structure as above is a left H-Hopf module.
+Proof.The first problem is that it is not obvious whySRat(H∗) is a left H-module under
+the⇀action. To see this, let α∈Rat(H∗), which means that there exist xα
+i∈HandTHE BIJECTIVITY OF THE ANTIPODE REVISITED 3
+gα
+i∈H∗such that for all β∈H∗andh∈Hwe have
+βα(h) =β(h1)α(h2) =β(xα
+i)gα
+i(h) (1)
+Now letx∈H, denote as before the left inverse of SbySl, and let us compute
+β(x⇀α)(h) =β(h1)(x⇀α)(h2)
+=βSl(S(h1))α(h2x)
+=βSl(x1S(x2)S(h1))α(h2x3)
+= (βSl↼x1)(S((hx2)1))α((hx2)2)
+= (βSl↼x1)◦S(xα
+i)gα
+i(hx2)−by(1)
+=β(Sl(x1S(xα
+i)))(x2⇀gα
+i)(h)
+Therefore, we proved that x⇀α∈Rat(H∗).
+To finish the proof, we need to show that
+(x⇀α)(−1)⊗(x⇀α)(0)=x1α(−1)⊗x2⇀α(0)
+which is
+S((x⇀α)1)⊗(x⇀α)0=x1S(α1)⊗x2⇀α0
+or
+< βS((x⇀α)1)(x⇀α)0,y >=< β(x1S(α1))(x2⇀α0),y >,∀β∈H∗, y∈H
+We have
+< βS((x⇀α)1)(x⇀α)0,y >=<(β◦S)∗(x⇀α),y >(rtH-com str of Rat(H∗))
+=βS(y1)(x⇀α)(y2)
+=βS(y1)α(y2x)
+=βS(y1)α(y2x2)ε(x1)
+=β(ε(x1)S(y1))α(y2x2)
+=β(x1S(x2)S(y1)α(y2x3)
+= (β↼x1)(S(y1x2))α(y2x3)
+=<(β↼x1)◦S,(yx2)1>< α,(yx2)2>
+=<((β↼x1)◦S)∗α,yx2>
+=<((β↼x1)◦S)(α1)α0,yx2>
+=β(x1S(α1))α0(yx2)
+=< β(x1S(α1))(x2⇀α0),y >,
+which ends the proof. /square
+LetCbe a coalgebra and M∈CM. The coalgebra CMassociated to Mis the smallest
+subcoalgebra CMofCsuch that ρ(M)⊆CM⊗M, i.e.CM=∩A⊆C,ρ(M)⊆A⊗MA(see
+D˘ asc˘ alescu et al. (2001, p. 102)). With this notation we ha ve:
+Proposition 2. IfMf։Nis a surjective morphism of left C-comodules, then CN⊆CM.4 M.C. IOVANOV AND S ¸. RAIANU
+Proof.LetK=Ker(f). Then clearly CK⊆CM. Therefore, Kis aCM-subcomodule, so
+ρM/K(M/K)⊆CM⊗M/K, i.e.ρN(N)⊆CM⊗N, henceCN⊆CMby definition. /square
+We are now ready to prove
+Theorem 3. IfHis co-Frobenius, Sis bijective.
+Proof.By Proposition 1 and the fundamental theorem of Hopf modules , we have that
+SRat(H∗)≃H⊗(SRat(H∗))co=H⊗/integraltext
+l, since it is easy to see that (SRat(H∗))co=/integraltext
+l.
+Also, since ( Rat(H∗))H≃/integraltext
+l⊗H=H(dim/integraltext
+l)inMH, we getSRat(H∗)≃(SH)(dim/integraltext
+l)=/circleplustext
+dim/integraltext
+lSH, using the fact that the functor Fclearly commutes with direct sums. Since
+Rat(H∗)/\e}atio\slash= 0 (equivalently/integraltext
+l/\e}atio\slash= 0) we can find a surjection of left H-comodules
+π: (SH)(dim/integraltext
+l)≃SRat(H∗)≃H⊗(SRat(H∗))co։HH
+ThenCH⊆/summationtextCSH=CSHby Proposition 2 and the obvious fact that C/circleplustext
+i∈IMi=/summationtext
+i∈ICMi. Obviously, CH=H(by the counit property), and also CSH=S(H), since
+∀h∈H,S(h) =S(h2)ε(h1)∈CSH, because ρSH(h) =S(h2)⊗h1∈H⊗SH. SoH⊆S(H),
+and the proof is complete. /square
+Corollary 4. Ift∈/integraltext
+l,t/\e}atio\slash= 0, thent◦S∈/integraltext
+r,t◦S/\e}atio\slash= 0.
+Proof.Obvious. /square
+Asaconsequenceofthisproof,theprooffortheuniquenesso fintegrals canbetranslated
+verbatim to Hopf algebras from the case of Haar measures, as w as done by Van Daele
+(1998) for regular multiplier Hopf algebras. This proof cou ld not beused for Hopf algebras
+becauseit requiresthebijectivity of theantipode, andunt il now all proofsof thebijectivity
+of the antipode used the uniqueness of integrals. A modified v ersion of the proof below
+not requiring the bijectivity of the antipode was given by Ra ianu (2000).
+Corollary 5. The dimension of/integraltext
+lis at most one.
+Proof.(Identical to the proof of Van Daele (1998, Theorem 3.7)) Let t1,t2∈/integraltext
+l,t2/\e}atio\slash= 0.
+By Corollary 4 λ=t2◦S∈/integraltext
+r\{0}. Then for any h∈Hthere is a g∈Hsuch that
+t1(xh) =t2(xg)∀x∈H. (2)THE BIJECTIVITY OF THE ANTIPODE REVISITED 5
+Indeed, let l,m∈Hsuch that λ(l) = 1 and t2(m) = 1. Then
+t1(xh) =λ(l)t1(xh)
+=λ(x1h1l)t1(x2h2) (t1∈/integraltext
+l)
+=λ(x1h1l1)t1(x2h2l2S(l3))
+=λ(xhl1)t1(S(l2)) (λ∈/integraltext
+r)
+=λ(xe) (e=hl1t1(S(l2))
+=λ(xe)t2(m)
+=λ(x1e1)t2(x2e2m) (λ∈/integraltext
+r)
+=λ(x1e1m2S−1(m1))t2(x2e2m3)
+=λ(S−1(m1))t2(xem2) (t2∈/integraltext
+l)
+=t2(xg) (g=em2λ(S−1(m1)).
+We finish the proof by showing that t1is a scalar multiple of t2. Fory∈Hwe have:
+t1(y) =λ(l)t1(y)
+=λ(l1)t1(yl2) (λ∈/integraltext
+r)
+=λ(S(y1)y2l1)t1(y3l2)
+=λ(S(y1))t1(y2l) (t1∈/integraltext
+l)
+=λ(S(y1))t2(y2g)−by(2)
+=λ(g)t2(y),
+where the last equality follows from reversing the previous three equalities, and the proof
+is complete. /square
+Remarks 6. a) Note that aside from the bijectivity of the antipode, the p roof above uses
+only the definition of integrals.
+b) To compensate for not being able to use the inverse of S, a special left integral had to
+be chosen in Raianu (2000) and it was shown to form a basis of/integraltext
+l. Once we are able to
+use the inverse of S, the proof above shows that any non-zero left integral will d o.
+Following the work of Lin, Larson, Sweedler, and Sullivan, t he existence of a nonzero
+integral is equivalent to various representation theoreti cal properties of the Hopf algebra,
+such as that of being co-Frobenius as a coalgebra, or having n onzero rational part. As a
+final application, we show how our approach may be used to simp lify the proof of some of
+these results (see D˘ asc˘ alescu et al. (2001, Theorem 5.3.2 )):
+Corollary 7. For a Hopf algebra Hthe following assertions are equivalent:
+(1)His left co-Frobenius
+(2)His left quasi-co-Frobenius
+(3)His left semiperfect
+(4)Rat(H∗)/\e}atio\slash= 0(H∗as a left H∗-module)
+(5)/integraltext
+l/\e}atio\slash= 0
+(6) The right hand version of (1)-(5)6 M.C. IOVANOV AND S ¸. RAIANU
+Proof.Allimplications followdirectlyfromthedefinitionsorfro mSweedler’sisomorphism,
+with the exception of 5) ⇒6) which follows from Corollary 4. /square
+ACKNOWLEDGMENTS
+We thank the referee for some valuable suggestions.
+References
+Abe, E. (1977). Hopf Algebras , Cambridge Univ. Press.
+Beattie, M., D˘ asc˘ alescu, S., Gr¨ unenfelder, L., N˘ ast˘ a sescu, C. (1998). Finiteness Condi-
+tions, Co-Frobenius Hopf Algebras and Quantum Groups, J. Algebra 200:312-333.
+Bourbaki, N. (1963). ´El´ ements de math´ ematique . Fascicule XXIX. Livre VI: Int´ egration.
+Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et repr´ esentations. (French)
+Actualit´ es Scientifiques et Industrielles, No. 1306 Herma nn, Paris.
+C˘ alinescu, C. (2001). On the bijectivity of the antipode in a co-Frobenius Hopf algebra,
+Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 44(92):59-62.
+D˘ asc˘ alescu, S., N˘ ast˘ asescu, C., Raianu, S ¸. (2001). Hopf Algebras: an Introduction , Mono-
+graphs and Textbooks in Pure and Applied Mathematics 235, Ma rcel Dekker, Inc.,
+New York.
+D˘ asc˘ alescu, S., N˘ ast˘ asescu, C., Torrecillas, B. (1999 ). Co-Frobenius Hopf Algebras: Inte-
+grals, Doi-Koppinen Modules and Injective Objects, J. Algebra 220:542-560.
+Iovanov, M.C. Generalized Frobenius algebras and the theor y of Hopf algebras, preprint,
+arXiv:0803.0775.
+Iovanov, M.C. Abstract Integrals in Algebra, preprint, arX iv:0810.3740.
+Menini, C., Torrecillas, B,. Wisbauer, R. (2001). Strongly regular comodules and semiper-
+fect Hopf algebras over QF rings, J. Pure Appl. Algebra 155:237-255.
+Radford, D.E. (1977). Finiteness conditions for a Hopf alge bra with a nonzero integral, J.
+Algebra46:189-195.
+Raianu, S ¸. (2000). An easy proof of the uniqueness of integr als, inHopf Algebras and
+Quantum Groups , Proceedings of the Brussels Conference 1998; Caenepeel, S ., Van
+Oystaeyen, F., Eds.; Marcel Dekker Lecture Notes in Pure and Appl. Math. vol 209,
+pp:237-240.
+S ¸tefan, D. (1995). The uniqueness of integrals. A homologi cal approach, Comm. Algebra
+23:1657-1662.
+Sullivan, J.B. (1971). The Uniqueness of Integrals for Hopf Algebras and Some Existence
+Theorems of Integrals for Commutative Hopf Algebras, J. Algebra 19:426-440.
+Sweedler, M.E. (1069a). Integrals for Hopf algebras, Ann. of Math. 89:323-335.
+Sweedler, M.E. (1969b). Hopf Algebras , Benjamin, New York.
+Van Daele, (1998). An algebraic framework for group duality ,Adv. Math. 140:323-366.
+University of Southern California, 3620 South Vermont Ave. KAP 108, Los Angeles, CA
+90089, USA, and, University of Bucharest, Fac. Matematica & Informatica, Str. Academiei
+nr. 14, Bucharest 010014, Romania
+E-mail address :yovanov@gmail.com, iovanov@usc.edu
+Mathematics Department, California State University, Dom inguez Hills, 1000 E Victoria
+St, Carson CA 90747, USA
+E-mail address :sraianu@csudh.edu
\ No newline at end of file
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+arXiv:1001.0184v1  [math.SG]  31 Dec 2009DEGENERATE MAXIMA IN HAMILTONIAN SYSTEMS
+MIKE CHANCE
+Abstract. In this paper we explore loops of non-autonomous Hamiltonia n
+diffeomorphisms with degenerate fixed maxima. We show that su ch loops can
+not have totally degenerate fixed global maxima. This has app lications for the
+Hofer geometry of the group of Hamiltonians for certain symp lectic 4 manifolds
+and also gives criteria for certain 4 manifolds to be unirule d.
+1.Introduction and Main Results
+In this paper we examine Hamiltonian flows whose associated Hamiltonia ns have
+fixed maxima. By this wemean points wherea time dependent Hamiltonia n attains
+a maximum for all time. In [6] McDuff proves several results for loops of Hamil-
+tonian diffeomorphisms for which the fixed global maxima are nondege nerate. The
+aim of this paper is to extend these results to the degenerate case .
+Such Hamiltonian flows may be viewed as generalizations of the autono mous
+case, while loops of this form are natural generalizationsof Hamilton ianS1actions.
+The existence of such actions gives useful information about the g eometry of the
+underlying manifold, see e.g. [2, 9]. Furthermore, both types of flow s may be
+exploited to study the Hofer geometry of the Hamiltonian group.
+Throughout the paper ( M,ω) will be a closed, connected symplectic manifold.
+Ht:M→RwillbeasmoothfamilyofHamiltoniansparameterizedby t∈[0,1]and
+XH
+twill denote the associated Hamiltonian vector field defined by ω(XH
+t,·) =dHt
+andφH
+twill the corresponding flow.
+Definition 1.1. A point,x0∈M, satisfying φH
+t(x0) =x0,∀tis called nondegen-
+erate att0ifd
+dt|t=t0DφH
+t(x0)v/ne}ationslash= 0for all0/ne}ationslash=v∈Tx0M, nondegenerate if it is
+nondegenerate for all time. It is called totally degenerate ifDφH
+t(x0) =Idfor all
+values oft.
+Definition 1.2. A pointx0∈Mis called a fixed local maximum on U of Htif
+there exists a neighborhood U⊂Mofx0such thatHt(x)≥Ht(y)for all values of t
+and∀y∈U. Similarly, x0∈Mis called a fixed global maximum if Ht(x0)≥Ht(y)
+for all values of tand∀y∈M. We will denote the collection of fixed local maxima
+Dmaxand fixed global maxima Fmax.
+Our first result is the following theorem.
+Theorem 1.3. Given a loop of Hamiltonian diffeomorphisms, the collection of
+totally degenerate fixed local maximum points Dmax⊂Mis open.
+In the event that our maximum is global, we prove the following conseq uence.
+Date: November 18, 2018.
+1991Mathematics Subject Classification. 53D05, 65P10.
+12 MIKE CHANCE
+Corollary 1.4. Let{φH
+t},t∈[0,1], be a nonconstant loop in Ham(M,ω)based
+atId. Ifx0is a fixed global maximum, then we must have DφH
+t(x0)/ne}ationslash=Idfor some
+value oft.
+Ofcourseasimilarstatementholdsforminimabysimplyconsideringthe function
+−Ht. Our proofs use methods of holomorphic curves and the requireme nt that the
+maximum is global in Corollary 1.4 cannot be dropped. This result allows u s
+to then construct loops of Hamiltonian diffeomorphisms with fixed non degenerate
+global maxima. Combining these constructions with results of Slimowit z [11] we
+obtain:
+Theorem 1.5. LetMbe a symplectic manifold with dimM≤4. If{φH
+t}is
+any nonconstant loop of Hamiltonian diffeomorphisms with x0∈Ma fixed global
+maximum, then there is a nonconstant loop φK
+twithx0still a fixed global maximum,
+which is an effective S1action near x0.
+The dimensional restriction here is due to the fact that the homoto py results of
+Slimowitz have only been proved for dim ≤4, although in principle those results
+should hold in all dimensions.
+A symplectic manifold is called uniruled if some point class nonzero Gromo v-
+Witteninvariantdoesnotvanish. Morespecificallythismeanstheree xista2,...,a k∈
+H∗(M) so that
+/an}b∇acketle{tpt,a2,...,a k/an}b∇acket∇i}htM
+k,β/ne}ationslash= 0 for some 0 /ne}ationslash=β∈HS
+2(M),
+whereptis the point class in H0(M). We refer the reader to [7] for details on
+the Gromov-Witten invariants. In [6] McDuff uses the Seidel element and methods
+of relative Gromov-Witten invariants to show that manifolds admittin g a loop of
+Hamiltonian diffeomorphisms with a fixed nondegenerate global maximu m must be
+uniruled. Thus, combining the results of McDuff with Theorem 1.5 we ha ve:
+Theorem 1.6. IfdimM≤4and there exists a nonconstant loop of Hamiltonian
+diffeomorphisms with a fixed global maximum, then (M,ω)is uniruled.
+McDuff relies heavily on the algebraic structure of the quantum homo logy ofM
+as well as the invertibility of the Seidel element. While methods used in t his paper
+are largely inspired by these, we rely solely on the geometric structu res of a certain
+Hamiltonian bundle over S2, as opposed to the algebraic information the bundle
+gives rise to.
+Given a path φH
+t,t∈[0,1], the Hofer length is defined as
+L(φH
+t) =/integraldisplay1
+0/parenleftBig
+max
+xHt(x)−min
+xHt(x)/parenrightBig
+dt.
+This allows one to construct a nondegenerate Finsler metric on the g roup of
+Hamiltonian diffeomorphisms, Ham(M,ω), whose geometry has been studied ex-
+tensively, see e.g. [1, 3, 4, 10]. In particular, in [4] Lalonde and McDuff show that
+ifφH
+tis a Hofer length minimizing geodesic then its generating Hamiltonian has at
+least one fixed global minimum and one fixed global maximum. Thus Theo rem 1.6
+implies the following:
+Theorem 1.7. Let(M,ω)be a closed, connected symplectic 4-manifold, and sup-
+pose thatγ∈π1(Ham(M,ω))is nontrivial. If there exists a representative {φH
+t}
+ofγwhich is Hofer length minimizing, then (M,ω)is uniruled.DEGENERATE MAXIMA IN HAMILTONIAN SYSTEMS 3
+Of course this says nothing if the Hamiltonian group is simply connecte d. In
+[5] McDuff demonstrates that π1(Ham(M,ω))/ne}ationslash= 0 ifMis a suitable two point
+blow up of any symplectic 4-manifold. Thus, if Mis not uniruled (e.g. T4, a
+K3 surface, or a surface of general type), this two point blow up is a 4-manifold
+for which there are nontrivial elements of π1(Ham(M,ω)) having no Hofer length
+minimizing representatives.
+This paper is organizedas follows. Section 2 contains a discussionof p ositive and
+semipositive paths. It also contains the proofs needed for Theore m 1.5 assuming
+Corollary 1.4. Section 3 contains a discussion of the Hamiltonian fibrat ions used.
+As the machinery needed to prove Theorem 1.3 is discussed here, its proof is left
+to the end of this section.
+1.1.Acknowledgements. Much of this work is part of the author’s thesis under
+the advisement of Dusa McDuff. The author is deeply grateful to he r for many
+valuable discussions and advice. The author would also like to thank Ba ¸ sak G¨ urel
+and Aleksey Zinger for their helpful conversations and suggestion s.
+2.Positive and Semipositive Paths
+2.1.Positive and Semipositive Paths. Consider R2nwith the standard sym-
+plectic structure ωand almost complex structure J. Recall that Sp(2n) consists of
+all matrices satisfying ATJA=J, and its Lie algebra, sp(2n), consists of matrices
+which satisfy JAJ=AT. Throughout, when in R2n, we use these structures and
+the metric given by, g(·,·) =ω(·,J·).
+A differentiable path in At∈Sp(2n) is called positiveif it satisfies
+d
+dtAt=JQtAt
+whereQtis a positive definite symmetric matrix for each t. Such paths are natural
+generalizations of circle actions near maxima of the corresponding a utonomous
+Hamiltonian. The linearization of a Hamiltonian has the form
+Ht(x) =const−1
+2/an}b∇acketle{tx,Qtx/an}b∇acket∇i}ht,
+and if it is semipositive, it corresponds precisely with the linearized flow near a
+maximum of some Hamiltonian. The simplest example of such a path is the counter
+clockwise rotation At=e2πkJt, withk>0. HereQt= 2πkI. In the event that Qt
+is symmetric, but only positive semidefinite (i.e., Qtcould have eigenvalues of zero
+for certain values of t), the path is called semipositive . In [11] Slimowitz proves the
+following:
+Theorem 2.1. (Slimowitz) Let n= 1,2and letAt∈Sp(2n)be a positive loop.
+ThenAtcan be homotoped through positive loops to a circle action.
+As mentioned, in principle this should be true in all dimensions, but the d etails
+have only been worked out for these cases. Slimowitz shows furthe r that
+Lemma 2.2. (Slimowitz) In Sp(4), any two loops of matrices of the form
+/parenleftbigge2πbiJt0
+0e2πdiJt/parenrightbigg
+fori= 1,2andt∈[0,1]are homotopic through positive loops provided bi,di≥1
+andb1+d1=b2+d2.4 MIKE CHANCE
+2.2.Proof of Theorem 1.5. In our setting we wish to consider Hamiltonians on
+manifolds. Fixed maxima must be fixed points of the associated flow fo r all time
+(i.e.,φH
+t(x) =x,∀t). Choosing a Darboux chart around such a point x,Htmay be
+written as
+Ht(x) =const−1
+2/an}b∇acketle{tx,Qtx/an}b∇acket∇i}ht+O(/ba∇dblx/ba∇dbl3) (2.1)
+and as before we will call the path (semi)positive if Qtis positive (semi)definite.
+As the point xis only assumed to be a fixed maximum, it may be degenerate and
+we may only assume Qt≥0 and the flow of its linearization is a semipositive path.
+In [6] McDuff proved the following result:
+Lemma 2.3. (McDuff) Suppose the loop γinHam(M,ω)has a nondegenerate
+fixed maximum at x0. Suppose also that the linearized flow at x0is homotopic
+through positive paths to a linear circle action. Then γis homotopic through loops
+of Hamiltonian diffeomorphisms with fixed maximum at x0to a loopγ′that is the
+given circle action near x0.
+Thusgivenadegenerateglobalmaximum, wemustconstructanewlo opwithanon-
+degenerate global maximum, and then apply the results of McDuff an d Slimowitz
+to obtain Theorem 1.5.
+The first results deal with the case when Qt0>0 for some t0, and thus is
+a positive path for some ǫtime. We describe a method of “spreading out the
+positivity” to homotop our path to a new one which is positive on all of [0 ,1]. We
+do so in such a way that, if xis a maximum of Hton some set V, it will remain a
+maximum of the new Hamiltonian on V.
+Lemma 2.4. Let{φH
+t} ⊂Ham(M,ω)fort∈[0,1]be a path of Hamiltonian
+diffeomorphisms whose generating function, Ht, has a fixed local maximum at x0.
+Let−1
+2/an}b∇acketle{tx,Qtx/an}b∇acket∇i}htbe the quadratic part of Ht, and letI+={t∈[0,1]|Qt>0}. If
+∅ /ne}ationslash=I+/ne}ationslash= [0,1]chooset0∈I+andt1/∈I+. Then the path may be homotoped
+through semipositive paths with fixed endpoints to a new one, whose quadratic part
+is positive in a δ′>0neighborhood of t1and remains so in I+. Furthermore, if
+x0was a maximum of Hton a neighborhood Vofx0for allt, it will remain a
+maximum of FtonVfor allt, andδ′will depend only on the initial neighborhood
+oft0∈I+.
+Proof.For the purposes of this proof, consider tas a variable in R/Z. Letδbe
+such thatQt>0 for|t−t0|< δ. Note that we must have |t1−t0| ≥δ. We will
+show that we may take δ′=δ/3.
+Letabe smaller than any of the eigenvalues of Qtfor|t−t0|< δ/2 and let
+b >>1. Define a function α:R≥0→Rsatisfying: α′(r)≥0,α(r) =br−afor
+r < a/2b, andα(r) = 0 forr > a/b. Next consider the autonomous Hamiltonian
+Kdefined on R2ngiven byK(x) =α(/ba∇dblx/ba∇dbl)/ba∇dblx/ba∇dbl2. Also, let β: [0,1]→[0,1] be a
+smooth nonincreasing function which is 1 on [0 ,δ/3], and 0 on [ δ/2,1].
+For each value of 0 ≤s≤1 define a function:
+Ks,t=
+
+0 for t<t1−δ/2
+sβ(|t−t1|)Kfort1−δ/2≤t≤t1+δ/2
+0 for t1+δ/2≤t≤t0−δ/2
+−sβ(|t−t0|)Kfort0−δ/2≤t≤t0+δ/2
+0 for t0+δ/2≤tDEGENERATE MAXIMA IN HAMILTONIAN SYSTEMS 5
+For each value of s, this time-dependent function will generate a smooth path of
+Hamiltonian diffeomorphisms, {ψK
+s,t}. Since any perturbation from the identity
+map is eventually undone, the path will satisfy ψK
+s,0=ψK
+s,1=Id, regardless of the
+values oft0andt1, and thus will always be a loop.
+We now consider the composition φF
+s,t=ψK
+s,t◦φH
+t, and note that the correspond-
+ing time dependent family of functions Fs,tare given by the formula
+Fs,t=Ks,t#Ht=Ks,t+Ht◦(φK
+s,t)−1. (2.2)
+We now claim that for a suitable choice of b, our pathφF
+s,tis positive for t∈I+
+and|t−t1|< δ/3. To show positivity, we need only show that Hamiltonian has
+non-degenerate quadratic part at x0. Fixsandtwith|t−t1|<δ/3 andv∈R2n,
+and consider the limit
+lim
+r→0/parenleftBig
+Ks,t(rv)+Ht◦(ψK
+s,t)−1(rv)/parenrightBig
+/ba∇dblrv/ba∇dbl2=−aβ(|t−t1|)s+ lim
+r→0Ht◦(φK
+s,t)−1(rv)
+/ba∇dblrv/ba∇dbl2
+≤ −aβ(|t−t1|)s
+<0
+where the inequality and subsequent minus sign on the right are expla ined by our
+convention of the quadratic portion actually being negative semidefi nite.
+CallingQ′
+tthe quadratic portion of Ft,Q′
+t>0 for|t−t0|< δ/2, sinceawas
+chosen smaller than any of the eigenvalues of Qthere. To see that Q′
+t>0 on the
+rest ofI+, we note that β= 0 in this region, and
+lim
+r→0/parenleftBig
+Ks,t(rv)+Ht◦(ψK
+s,t)−1(rv)/parenrightBig
+/ba∇dblrv/ba∇dbl2= lim
+r→0Ht◦(φK
+s,t)−1(rv)
+/ba∇dblrv/ba∇dbl2
+<0
+Assandtare fixed while taking this limit, the inequality holds by the definition
+ofI+.
+Finally, our perturbed function will remain a maximum in Vfor|t−t0| ≥δ′,
+and by choosing blarge enough (the choice depends on the third order terms of the
+initialHt), it will remain a maximum for |t−t0|<δ′, as well. /square
+Proposition 2.5. Let{φH
+t}fort∈[0,1]be a path of Hamiltonian diffeomorphisms
+based atIdwith generating function Ht. Supposex0is a maximum of Hton some
+neighborhood Vofx0, for allt. LettingQtbe as in Lemma 2.4, if Qt0>0for
+some0≤t0≤1, then{φH
+t}may be homotoped through semipositive paths with
+fixed endpoints to a new path {φF
+t}whose associated quadratic portion is strictly
+positive for all t∈[0,1]. Furthermore x0will be a maximum of FtonVfor allt,
+as well.
+Proof.As theδ′>0 value from Lemma 2.4 depended only on the neighborhood
+oft0inI+, we may carry out the process a finite number of times to homotop o ur
+path through semipositive paths with fixed endpoints to one which is p ositive for
+allt. Furthermore, by construction, x0remains a maximum on Vthroughout. /square
+The next result deals with the case when Qt≯0 for anyt, butQt0/ne}ationslash= 0 for some
+t0. Thus while it is never a positive path, it is positive in at least one directio n at
+timet0.6 MIKE CHANCE
+Proposition 2.6. Let{φH
+t} ⊂Ham(M,ω)fort∈[0,1]be a path of Hamiltonian
+diffeomorphisms with generating function Ht. Letx0be a maximum of Hton
+some neighborhood Vofx0. IfDφH
+t/ne}ationslash=Idfor somet0, then there is a new path,
+{φK
+t}, whose associated Hamiltonian, Kt, is nondegenerate at x0and fort=t0.
+Furthermore, φK
+tcan be chosen to be homotopic to {(φH
+t)m}for somem≤1 +
+dim(ker(DφH
+t0−Id)). Furthermore, x0will remain a maximum of KtonV.
+Proof.Throughout, for convenience of notation, we explicitly work in R2nand the
+linearization of φH
+t. We refer to the linearization as the path At∈Sp(2n,R), and
+note that it satisfies A0=A1=Idandd
+dtAt(x) =JQt(x)At(x) withQt≥0 and
+symmetric. Let t0be such that Qt0/ne}ationslash= 0.
+Identify R2n=E0⊕E1withE0=ker(Qt0) andE1the sum of eigenspaces of
+Qt0with nonzero eigenvalues. We first consider the case when J(E0) =E0. Choose
+v∈E1to be an eigenvector for Qt0, and let 0 /ne}ationslash=w∈E0. SplitR2n=R4⊕R2n−4
+withR4spanned by {v,w,Jv,Jw }andR2n−4= (R4)ωits symplectic orthogonal.
+DefineB∈Sp(2n) by
+BA−1
+t0w=v, BA−1
+t0Jw=Jv
+BA−1
+t0v=−w, BA−1
+t0Jv=−Jw,
+BA−1
+t0|R2n−4=Id.
+LetBs∈Sp(R2n) fors∈[0,1] satisfyB0=IdandB1=B, and letfs:R2n→R2n
+be a family of symplectomorphisms fixing the origin and supported in a s mall
+neighborhood of it satisfying Dfs(0) =Bs, whereDfs(0) :R2n→R2nis the
+derivative at the origin. We may assume that x0is a maximum of Htthroughout
+the support of the family fs. We now wish to consider the family of paths given
+by:
+φK
+s,t=φH
+t(f−1
+sφH
+tfs)
+which will provide a homotopy from/parenleftbig
+φH
+t/parenrightbig2toφK
+1,t. For each sthis will remain a
+Hamiltonian flow, and will be generated by:
+Ks,t=Ht+Ht(fs◦(φH
+t)−1).
+By our choice of f,x0will remain a constant maximum of φK
+s,tfor all values of
+s. To determine the degeneracy of our maximum, we simply differentiat e:
+d
+dtDφH
+s,t(0) =d
+dtAtB−1
+sAtBs
+=˙AtB−1
+sAtBs+AtB−1
+s˙AtBs
+=JQt(AtB−1
+sAtBs)+AtB−1
+sJQtAtBs
+=J/parenleftBig
+Qt+(BsA−1
+t)TQt(BsA−1
+t)/parenrightBig
+(AtB−1
+sAtBs).
+Call Γ s,t= (B1A−1
+t)TQt(B1A−1
+t). Since both Qtand Γs,tremain symmetric and
+nonnegative for all values of sandt, so is their sum. Furthermore, if Qt+Γs,thas
+any kernel, it must be contained in the kernel of Qt. Thus we need only check that
+it is nondegenerate in the v,wplane when s= 1 andt=t0. Abusing notation, call
+Γ1,t0= Γ. We compute:DEGENERATE MAXIMA IN HAMILTONIAN SYSTEMS 7
+/an}b∇acketle{tv+aw,(Qt0+Γ)(v+aw)/an}b∇acket∇i}ht=/an}b∇acketle{tv,Qt0v/an}b∇acket∇i}ht+/an}b∇acketle{tv,Γv/an}b∇acket∇i}ht+a/an}b∇acketle{tv,Γw/an}b∇acket∇i}ht (2.3)
++a/an}b∇acketle{tw,Γv/an}b∇acket∇i}ht+a2/an}b∇acketle{tw,Γw/an}b∇acket∇i}ht
+= (1+a2)/an}b∇acketle{tv,Qt0v/an}b∇acket∇i}ht
+>0.
+To see that we created no new kernel, let u∈R2n. Then
+/an}b∇acketle{tu,(Qt0+Γ)u/an}b∇acket∇i}ht=/an}b∇acketle{tu,Qt0u/an}b∇acket∇i}ht+/an}b∇acketle{tu,Γu/an}b∇acket∇i}ht
+with both matricesbeingnonnegative. Thusthe sum canonlybe zero if/an}b∇acketle{tu,Qt0u/an}b∇acket∇i}ht=
+0.
+In the case when Jdoes not preserve E0, we may choose w∈E0so that
+/an}b∇acketle{tJw,Q t0Jw/an}b∇acket∇i}ht>0. In this case, setting v=Jwwe define
+BA−1
+t0w=v, BA−1
+t0v=−w,
+BA−1
+t0|R2n−2=Id.
+As (2.3) remains the same, the remainder of the proof is identical to the previous
+case. /square
+Corollary 2.7. Letx0∈Mbe a fixed local maximum on Uof a family of Hamil-
+toniansHt, such that DφH
+t(x)/ne}ationslash=Idfor somet0. Then we may construct a new
+pathφF
+twhich is positive for all t∈[0,1]and such that if Ht(x0)≥Ht(y)for some
+y∈U, thenFt(x0)≥Ft(y). Furthermore, φF
+tis an iterate of φH
+t.
+Assuming Corollary 1.4, combining the above results finishes the proo f of Theo-
+rem 1.5.
+Remark 2.8. One may note that there is a slight error in [6]. Proposition 1.4
+of that paper states that if dimM≤4, and the loop {φH
+t}has a nondegenerate
+fixed global maximum, then it can be homotoped so that it is an e ffectiveS1action
+near the maximum. This is actually only true if dimM= 4. IfdimM= 2, the
+existence of such a loop forces the manifold to be S2, and thus there certainly exists
+an effective S1action. However a two time rotation is not homotopic to an effe ctive
+action.
+3.Hamiltonian Fibrations
+This section contains proofs of Theorem 1.3 and Corollary 1.4. We beg in with
+a discussion of the Hamiltonian fibration that will be used. One may not e that
+the autonomous case is much more straightforward, even if not re stricted to global
+circle actions. It is a standard result that given an autonomous pat h of Hamilton-
+ian diffeomorphisms, around any totally degenerate fixed point, the re is an entire
+neighborhood containing no nontrivial periodic orbits (see e.g. Lemm a 12.27, [7]),
+and thus cannot be a loop.
+3.1.Hamiltonian Bundles Over S2.Given (M,ω) and any loop of Hamiltonian
+diffeomorphisms, {φH
+t}, there is an associated Hamiltonian fibration P→S2with
+fibersymplectomorphicto( M,ω). Throughoutweusethestandardalmostcomplex
+structure on S2, which we call j. Begin with two copies of M×D±, whereD±8 MIKE CHANCE
+denotes two different copies of the unit disk with opposite orientatio ns. Define the
+equivalence relation:
+P:=M×D+∪M×D−/∼,(φH
+t(x),e2πit)+= (x,e2πit)−. (3.1)
+SinceφH
+0(x) =φH
+1(x) =x,∀x∈M, the two copies of Dglue together along
+their boundaries to give a copy of S2, andP→S2will be a fibration with fiber
+diffeomorphic to M. Denote the projection map by π:P→S2. The vertical
+tangent bundle here is given by TVertP=ker(Dπ)⊂TP. Because the fibers are
+symplectic, they have Chern classes and we denote the vertical fir st Chern class by
+cVert
+1.
+Define a symplectic form Ω on Pby
+Ω−:=ω+δd(r2)∧dt,onM×D− (3.2)
+Ω+:=ω+/parenleftBig
+κ(r2,t)d(r2)−d(ρ(r2)Ht)/parenrightBig
+∧dtonM×D+
+where we have used normalized polar coordinates ( r,t) onDwitht:=θ/2π. Here
+ρ(r2) is a nondecreasing function that equals 0 near 0 and 1 near 1, and δ>0 is a
+small constant. As long as κ(r2,t) =δnearr= 1, these two forms will fit together
+to give a closed form on P. To be symplectic, Ω must be nondegenerate, but this
+can be seen to happen iff κ(r2,t)−ρ′(r2)Ht(x)>0,∀(r,t)∈D±andx∈M.
+Ω restricted to TVertPis nondegenerate. Thus Ω gives a connection 2-form on
+P, and we have a well defined horizontal distribution, which we will deno te by
+THorP. To be more precise:
+THor
+pP={v∈TpP|Ω(v,w) = 0,∀w∈kerDπ(p)} (3.3)
+Definition 3.1. ([8]§8.2) An almost complex structure, /tildewideJ:TP→TPwill be
+called compatible with the fibration if the following condit ions are met:
+(1)π:P→S2is holomorphic
+(2)/tildewideJ|TV erzPis tamed by ω,∀z∈S2
+(3)/tildewideJ(THorP)⊂THorP.
+Note here that varying κin (3.2) does not affect the horizontal distribution defined
+in (3.3).
+By our choice of Ω, THor(M×D−) is spanned by ∂rand∂t, andTHor(M×D+)
+is spanned by the vectors ∂rand∂t−XH
+tat each point, and conditions (1) and
+(3) completely determine /tildewideJonTHorP. Also, because /tildewideJis tamed by Ω, the bilinear
+form
+g/tildewideJ(v,w) =1
+2/parenleftbig
+Ω(v,/tildewideJw)+Ω(w,/tildewideJv)/parenrightbig
+defines a Riemannian metric on P, with associated Levi-Civita connection, ∇. To
+obtain a connection which will preserve /tildewideJwe use ([8] §3.1)
+/tildewide∇vX=∇vX−1
+2/tildewideJ(∇v/tildewideJ)X. (3.4)
+This bundle contains lots of sections. To see this explicitly, choose an yx∈M.
+As{φH
+t}is a loop, every point gives rise to a contractible 1-periodic orbit. A
+contraction of the orbit {φH
+t(x)}is a map from the unit disk f:D→MwithDEGENERATE MAXIMA IN HAMILTONIAN SYSTEMS 9
+f(e2πit) =φH
+t(x). Thus an explicit formula for a section s:S2→Pwould be
+s(r,t) =x×(r,t), onD− (3.5)
+s(r,t) =f(r,t)×(r,t), onD+.
+In the event that x0is fixed by φH
+tfor all time, we may choose f(r,t) to be
+constant. We will denote such a constant section by s0. These sections have
+particularly nice properties as they are holomorphic with respect to compatible
+almost complex structures on P. Another important property holds when x0is a
+fixed maximum of Ht. We now argue as in McDuff ([6], Proposition 2.11) as well
+as McDuff and Tolman ([9], Lemma 3.1).
+Lemma 3.2. Suppose that x0∈Mis a fixed maximum on the open set U⊂Mof a
+loop of Hamiltonian diffeomorphisms for all time, and consid er the constant section
+s0: (r,t)/mapsto→x0×(r,t). Then, given any /tildewideJcompatible with the fibration, the only
+nearby holomorphic sections in class [s0]are constant ones, and are parameterized
+by elements of a component of the fixed local maximum set, DmaxforHt.
+Proof.We use the symplectic form given by (3.2). At a point in the image of our
+sectionu:S2→P, splitTP=TVertP⊕THorP, and write elements of Tu(r,t)Pas
+v+h. Ifuissufficientlycloseto s0, thenitmustonlypassthroughourneighborhood
+U. We compute:
+Ω(v+h,/tildewideJ(v+h)) =ω(v,/tildewideJv)+Ω(h,/tildewideJh)≥Ω(h,/tildewideJh)
+≥2r/parenleftbig
+κ(r2,t)−ρ′(r2)max
+x∈UHt(x)/parenrightbig
+dr∧dt(h,/tildewideJh).
+The first inequality is an equality only if the curve is horizontal, and the second
+is an equality only if the section is contained in the same component of Dmax×S2
+asx0. Since another curve representing the same class as [ s0] must have the same
+symplectic area, it must be constant. /square
+Note that a slight adjustment to the previous argument gives the f ollowing.
+Lemma 3.3. Suppose that x0∈Mis a fixed global maximum on Mof a loop
+of Hamiltonian diffeomorphisms for all time, and consider th e constant section
+s0: (r,t)/mapsto→x0×(r,t). Then, given any /tildewideJcompatible with the fibration, the
+only holomorphic sections in class [s0]are constant ones, and are parameterized by
+elements of the fixed global maximum set, FmaxforHt.
+3.2.Totally Degenerate Maxima. LetφH
+tbe a loop in Ham(M,ω) based at
+Id, withDmaxthe fixed local maxima set. We show that this set must be open.
+The compatibility conditions of Definition 3.1 do not determine /tildewideJonTVertP, so
+we now construct one explicitly. In our case, we wish the almost comp lex structure
+we construct to be regular for a constant section through Dmax.
+First, choose a Darboux chart around x0∈Dmax, call itU, and identify it with
+a neighborhood of 0 ∈R2n. Using the standard {xi,yi}coordinates on R2nand
+the standard J, choose an almost complex structure on Mwhich is the pullback of
+JonU, and refer to it as J0.
+Take/tildewiderJ0VertonTVert(M×D−) to beJ0. This forces /tildewideJVert= (φH
+t)∗J0on
+M×∂D+, andwemustextendthistotherestof TVert(M×D+). Inourcoordinates10 MIKE CHANCE
+onU, (φH
+t)∗J0will be given by conjugation by DφH
+t(x), so that at a point x, we
+have
+(φH
+t)∗J0= (DφH
+t(x))−1◦J0◦DφH
+t(x)
+where we have realized DφH
+t(x) as a loop of maps DφH
+t:U→Sp(2n) based at the
+constant map U/mapsto→Id. SinceDφH
+t(0) =Idfor all time, we may also assume our
+initial neighborhood Uis small enough that there is a loop ofmaps Yt:U→sp(2n)
+based at the constant map U/mapsto→0 satisfying
+exp(Yt(x)) =DφH
+t(x)
+where exp is the standard exponential map from sp(2n)→Sp(2n). Letting β:
+[0,1]→[0,1] be a smooth, nondecreasing function which is 0 near 0 and 1 near 1 ,
+we may consider the family of maps Yr,t:U→sp(2n) given byYr,t(x) =β(r)Yt(x).
+By our choice of β,Yr,t(x) = 0 forrclose to 0 and we may now consider this as a
+family ofmapssmoothly parameterizedby D+. We nowextend our almostcomplex
+structure to all of U×D+by the formula:
+/tildewideJV ert(x×(r,t)) = exp(Yr,t(x))−1◦J0◦exp(Yr,t(x)), (3.6)
+wherex×(r,t)∈U×D+.
+Finally extend /tildewideJV ertto the rest of Tvert(M×D+) in a way compatible with
+the fibration (see [8], §8.2), and take /tildewideJ=/tildewideJV ert⊕/tildewideJHor. We now claim that the /tildewideJ
+just constructed is a regular almost complex structure for our co nstant maximum
+section.
+Lemma 3.4. Letξ∈Ω0(S2,s∗
+0(TP))be any vector field along s0. Then∇ξ/tildewideJ= 0.
+Proof.Given a section ξofTPdefined in a neighborhood of Im(s0), we may write
+it asvξ+hξwherevξis a section of TVertPandhξa section of THorP, both defined
+in a small neighborhood of Im(s0). We consider /tildewideJHorand/tildewideJV ertseparately.
+Itisclearthatif histangentto Im(s0),then∇h/tildewideJHor= 0. Ifv∈TVert(M×D−),
+one also has ∇v/tildewideJHor= 0 alongx0×D−. Ifx0×(r,t)∈x0×D+, then because ∇
+is Levi-Civita, we must have ∇v(∂t−X) =ar,t∂rand∇v∂r=−ar,t(∂t−X) with
+ar,t∈R. But then using the identity
+(∇v/tildewideJHor)(X) =∇v(/tildewideJHor(X))−/tildewideJHor(∇v(X))
+as well as the Leibniz rule, one can easily see that ∇v/tildewideJHor= 0 alongx0×D+, as
+well. Thus we have ∇ξ/tildewideJHor= 0 for any ξ∈Ω0(S2,u∗(TP)).
+Similar rationale holds to show ∇h/tildewideJV ert= 0 forhtangent to Im(s0), and
+∇v/tildewideJV ert= 0 forv∈TVertPalongx0×D−. Thus we need only concern ourselves
+with the value of ∇v/tildewideJV ertat points in U×D+withUa neighborhood of x0.
+Locally, we may expand φH
+taboutx0, so that
+φH
+t(x) =x+/summationdisplay
+i≤jAi,j(t)xixj+O(/ba∇dblx/ba∇dbl3) (3.7)
+withAi,j(t) a time-dependent loop of vectors in R2nand the higher order terms
+also depending on time. Since x0is a totally degenerate maximum, we may write
+|Ht(x)−Ht(0)| ≤C/ba∇dblx/ba∇dbl4in our neighborhood for some C, and thus /ba∇dblXH
+t(x)/ba∇dbl ≤DEGENERATE MAXIMA IN HAMILTONIAN SYSTEMS 11
+C′/ba∇dblx/ba∇dbl3in our neighborhood for some C′. We now use the fact that
+XH
+t(φH
+t0(x)) =d
+dtφH
+t(x)|t=t0
+=/summationdisplay
+i≤j/parenleftBigd
+dtAi,j(t)|t=t0/parenrightBig
+xixj+O(/ba∇dblx/ba∇dbl3)
+for every 0 ≤t0≤1. In order for /ba∇dbld
+dtφH
+t(x)/ba∇dbl=/ba∇dblXH
+t(φH
+t(x))/ba∇dbl ≤C′/ba∇dblx/ba∇dbl3, we must
+have eachAi,j(t) a constant function of t. AsφH
+0=Id,Ai,j(t) = 0 for all t. Thus,
+φH
+t(x) =x+O(/ba∇dblx/ba∇dbl3)
+DφH
+t(x) =Id+O(/ba∇dblx/ba∇dbl2).
+SinceDφH
+t(x) = exp(Yt(x)) we have ∇vexp(Yt(x)) = 0, and it is easy to see
+that∇vexp(Yr,t(x)) = 0 also. Finally, since exp( Yr,t(x)) =Idalong our section,
+we may say
+∇v/parenleftBig
+(exp(Yr,t(x))−1◦J0◦exp(Yr,t)/parenrightBig
+= 0
+along our section, as well. Thus ∇ξ/tildewideJVert= 0, for any ξ∈Ω0(S2,u∗(TP))./square
+Proposition 3.5. LetφH
+t,t∈[0,1]be a loop of Hamiltonian diffeomorphisms
+based atId. LetDmaxbe the set of fixed local maxima of Ht, and suppose that
+DφH
+t(x0)≡Idfor somex0∈Dmaxand for all values of t. Lets0denote the
+constant section through x0and let/tildewideJbe as constructed above. Then s0is a regular
+/tildewideJholomorphic map.
+Proof.For/tildewideJto be regular for s0, the differential,
+Ds0: Ω0(S2,s∗
+0(TP))→Ω0,1(S2,s∗
+0(TP))
+which maps smooth sections of s∗
+0(TP) to/tildewideJantiholomorphic s∗
+0(TP) valued 1-
+forms onS2, must be surjective. An explicit formula for Ds0evaluated at ξ∈
+Ω0(S2,s∗
+0(TP)) is given by:
+Ds0ξ=1
+2/parenleftbig/tildewide∇ξ+/tildewideJ(s0)/tildewide∇ξ◦j/parenrightbig
++1
+4N/tildewideJ(ξ,ds0) (3.8)
+where/tildewide∇is from (3.4) and N/tildewideJis the Nijenhuis tensor, see [8] Remark 3.1.2.
+As∇ξ/tildewideJ= 0 for all ξ∈Ω0(S2,s∗
+0(TP)), (3.4) becomes /tildewide∇=∇. A formula for
+N/tildewideJ(X,Y) (which can be found in [8] Lemma C.7.1) is given by
+N(X,Y) = (J∇YJ−∇JYJ)X−(J∇XJ−∇JXJ)Y.
+Thus∇ξ/tildewideJ= 0 also implies the Nijenhuis tensor vanishes, so that Ds0reduces to
+Ds0ξ=1
+2/parenleftbig
+∇ξ+/tildewideJ(s0)∇ξ◦j/parenrightbig
+. (3.9)
+The complex bundle s∗
+0(TP) splits as TS2⊕νs0withνs0=s∗
+0(TVertP). A
+trivialization for νs0is given by the path {DφH
+t}, and we are assuming DφH
+t(x0)≡
+Id. Furthermore, /tildewideJalong this section is the constant product J0×j, and so the
+complex bundle ( νs0,/tildewideJVert) is trivial, and the connection ∇on this bundle is also
+trivial. Thus we may split s∗
+0(TP) as a sum of complex line bundles ⊕n
+0=Li, with
+L0corresponding to TS2and we have c1(L0) = 2 andc1(Li) = 0 fori/ne}ationslash= 0.12 MIKE CHANCE
+Moreover by (3.9) Ds0preserves this splitting. This shows that (3.9) gives the
+formula for the standard Cauchy-Riemann operator. The vertica l portion of Ds0
+actson a trivialbundle, and we see that the verticalportionof Ds0is surjective. /square
+Proof of Theorem 1.3 :
+LetM1([s0],/tildewideJ) be the space of equivalence classes [ u,z] of simple holomorphic
+sections in class [ s0] with one marked point. Here two holomorphic section maps
+(u,z) and (u′,z′) are called equivalent if there is f∈PSL(2,C) so that
+u′=u◦fandf(z′) =z.
+Identifyx0with its image over0 ∈D+. There is only one /tildewideJholomorphiccurvein
+class [s0] passing through x0, and all other sections through x0have larger energy.
+Since all stable /tildewideJholomorphic maps through x0must involve a section, there can
+be no bubbling.
+We have the evaluation map
+ev:M1([s0],/tildewideJ)×S2→P, by
+ev([u,z]) =u(z).
+GivenM1(A,J), the moduli space of Jholomorphic curves u:S2→Mrep-
+resentingA∈H2(M) and a submanifold X⊂M, one may consider the space
+ev−1(X). This is referred to as the “cutdown” moduli space and consists o f ele-
+ments of M1(A,J) which send the marked point to X. Referring to this space as
+MCut
+1(A,J,X), in order to use such a cutdown moduli space, three conditions mu st
+be satisfied:
+• MCut(A,J,X) must be compact
+•Every curve in MCut(A,J,X) must be regular
+•The differential of the evaluation map must be transverse to X.
+We consider the cutdown moduli space given by ev−1((x0,0))⊂ M1([s0],/tildewideJ)
+with 0∈D+. Note that as M1([s0],/tildewideJ) had been quotiented out by PSL(2,C),
+MCut
+1([s0],/tildewideJ,(x0,0)) consists of a single map.
+Thetangentspaceto M([s0],/tildewideJ)canbeidentifiedwith kerDu⊂Ω0(S2,u∗(TVertP)),
+and the differential of the evaluation map at the point ( u,w) is given by
+devu,w(ξ) =ξ(w).
+This is surjective at MCut
+1([s0],/tildewideJ,(x0,0)) if, given any v∈TVert
+s0(w)P, there is
+ξ∈Ω0(S2,s∗
+0(TVertP)) satisfying
+ξ(0) =v, andDs0ξ= 0.
+But ass∗
+0(TVertP) has been shown to be a trivial holomorphic bundle, we may
+chooseξto be a constant section. One can see from (3.9) that Ds0(ξ) = 0 ifξis
+constant, and devs0,0must then be surjective.
+Asdim(M1([s0],/tildewideJ)) = 2n+2cVert
+1([s0]) = 2n, the fact that M1([s0],/tildewideJ) contains
+2ndimensions worth of constant sections allows us to apply Lemma 3.2 to see that
+Dmaxis open. This completes the proof of Theorem 1.3DEGENERATE MAXIMA IN HAMILTONIAN SYSTEMS 13
+Proof of Corollary 1.4 :
+Supposex0is a totally degenerate global maximum. Then as in Theorem 1.3,
+x0is an interior point of Fmax. AsFmaxconsists of global maxima, we must
+have∂Fmax⊂Fmax. SinceMis assumed to be connected, Fmaxmust equal M.
+As our loop φH
+twas assumed to be nonconstant, this contradiction completes the
+argument.
+Remark 3.6. One may note that elements of ∂Dmaxwill still be totally degener-
+ate fixed points. While Proposition 3.5 continues to hold at t otally degenerate fixed
+points which are not maxima, elements of ∂Dmaxwill not necessarily be local max-
+ima, so that Lemma 3.2 fails to hold. The difficulty for these bo undary sections is
+that nearby sections need not be constant. Thus one cannot co nclude that elements
+of∂Dmaxare interior points.
+References
+[1] Misha Bialy and Leonid Polterovich. Geodesics of Hofer’ s metric on the group of Hamiltonian
+diffeomorphisms. Duke Math. J. , 76(1):273–292, 1994.
+[2] Eduardo Gonzalez. Quantum cohomology and S1-actions with isolated fixed points. Trans.
+Amer. Math. Soc. , 358(7):2927–2948 (electronic), 2006.
+[3] E. Kerman and F. Lalonde. Length minimizing Hamiltonian paths for symplectically aspher-
+ical manifolds. Ann. Inst. Fourier (Grenoble) , 53(5):1503–1526, 2003.
+[4] Fran¸ cois Lalonde and Dusa McDuff. Hofer’s L∞-geometry: energy and stability of Hamilton-
+ian flows. I, II. Invent. Math. , 122(1):1–33, 35–69, 1995.
+[5] Dusa McDuff. The symplectomorphism group of a blow up. Geom. Dedicata , 132:1–29, 2008.
+[6] Dusa McDuff. Hamiltonian S1-manifolds are uniruled. Duke Math. J. , 146(3):449–507, 2009.
+[7] Dusa McDuff and Dietmar Salamon. Introduction to symplectic topology . Oxford Mathemat-
+ical Monographs. The Clarendon Press Oxford University Pre ss, New York, second edition,
+1998.
+[8] Dusa McDuff and Dietmar Salamon. J-holomorphic curves and symplectic topology , vol-
+ume 52 of American Mathematical Society Colloquium Publications . American Mathematical
+Society, Providence, RI, 2004.
+[9] Dusa McDuff and Susan Tolman. Topological properties of H amiltonian circle actions. IMRP
+Int. Math. Res. Pap. , pages 72826, 1–77, 2006.
+[10] Leonid Polterovich. The geometry of the group of symplectic diffeomorphisms . Lectures in
+Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel, 2001.
+[11] Jennifer Slimowitz. The positive fundamental group of Sp(2) and Sp(4). Topology Appl. ,
+109(2):211–235, 2001.
+Mathematics Department 1326 Stevenson Center Vanderbilt Un iversity Nashville,
+TN 37240
+E-mail address :michael.j.chance@vanderbilt.edu
\ No newline at end of file
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+arXiv:1001.0185v1  [cond-mat.str-el]  31 Dec 2009physica statussolidi, 9 March 2018
+Quantum Criticality and Novel
+Phases:Apanel discussion.
+PiersColeman
+Dept ofPhysics and Astronomy,
+Rutgers University,
+Piscataway,
+NJ08854, U.S.A.
+ReceivedXXXX, revisedXXXX,accepted XXXX
+Publishedonline XXXX
+PACS64.70.Tg, 71.27.+a, 71.10.Hf, 05.70.Jk, 05.60.Gg
+∗Corresponding author: e-mail coleman@physics.rutgers.edu , Phone: +1732 445 5500 ext 5082
+PhysicistsgatheredinaugustatDresdenforaconference
+about “Quantum Criticality and Novel Phases”. As one
+part of the meeting, nine panelists hosted an open and
+free-wheelingdiscussiononthetopicofthemeeting.This article outlines the discussions that took place dur-
+ingatthispanel-meetingontheafternoonofAugust3rd,
+2009.
+Copyrightlinewillbe provided by the publisher
+1 Introduction Eighty years ago, physicist Paul
+Dirac,reflectingonthenewquantumtheoryhehadplayed
+such an important part in developing, wistfully remarked
+that
+“theunderlyingphysicallawsnecessaryforthemathemat-
+ical theory of ...physics and the whole of chemistry
+are thus completely known, and the difficulty is only
+that...these laws lead to equationsmuch too compli-
+catedtobesoluble” P.A. M.Dirac
+∼1929.[1]
+Discoveries spanning eight decades have revealed Dirac’s
+remark to be one of the great physics understatements of
+all time, for understanding the link between the quantum
+micro-worldandouremergentmacroscopicworldprovesa
+singularchallenge.Today,we knowthat therulesofquan-
+tummechanicsendowmatterwith a propensityto develop
+unexpectedly simple, yet completely new kinds of collec-
+tive behavior. From the practical perspective of the mate-
+rial physicist,emergencemeansthat the periodictable is a
+forge of fabulous potential, from which wholly new kinds
+of material can be crafted, high temperature superconduc-
+tors,materialswithnewkindsofmulti-functionalbehavio r
+such as multiferrocityand materials with possibilities th at
+wehaveyettodiscoverorevenimagine.Butitseasytoget
+lost,andguidingprinciplesare invaluable.
+One such principle that has appears increasingly fruit-
+ful, is to seek materials that lie at the point of instabil-ity between one phase and another: this point is called a
+“quantumphasetransition”(QPT)[2].Conventionalphase
+transitions are driven by thermal motions at finite temper-
+ature. Quantum phase transitions occur at the point where
+the transition temperatureis tunedto absolutezero.At ab-
+solute zero, thermal motion vanishes, yet quantum phase
+transitionsarefarfromstatic,andcanbelikenedtoamelt-
+ing phenomenondrivenbythe zeropoint quantummotion
+that arises from Heisenberg’s uncertainty principle. Such
+zero point motion is particularlyimportant at a second or-
+derquantumphasetransition,wherefluctuationsinthe or-
+der parameter develop an infinite correlation length and
+aninfinitecorrelationtimethatengulfstheentiremateria l:
+such a singular point of transition is a “Quantum Critical
+Point”(QCP)[3,2,4].
+Experimentsshow that radically new kindsof metallic
+behaviordevelopwhenaquantumcriticalmetaliswarmed
+to a finite temperature[5,6] Such materials also have a
+marked propensity to nucleate new kinds of order. The
+“dome”of superconductivityin the phase diagramof high
+temperature cuprate superconductors is thought by many,
+to hide a quantumcritical point, but the superconductivity
+is so robust that huge magnetic fields are required to strip
+away the superconductivityto reveal the underlyingquan-
+tum critical point, and the idea is still controversial. For -
+tunately, similar situations occur in heavy fermionmateri -
+als, such as CeRhIn 5, where superconductivitynucleates
+Copyrightlinewillbe provided by the publisher2 P. Coleman: QCNP09: A panel discussion.
+around a pressure-tuned antiferromagnetic quantum criti-
+cal point and can be removed by more modest magnetic
+field[7].Theimportantpointisthatquantumcriticalityap -
+pears to provide a vital way of inducing high temperature
+superconductivity and other novel material behavior; it is
+thispotential,togetherwiththedramatictransformation sin
+metallic behavior that appear to accompany quantum crit-
+icality that motivate the research behind the the Dresden
+“QuantumCriticality and Novel Phases” conferences,dis-
+cussedinthepaneldiscussionreportedhere.
+Nine panelistsMeiganAronson(BrookhavenNational
+LaboratoryandStonyBrookUniversity,NewYork,USA),
+Piers Coleman (Rutgers University, New Jersey, USA),
+Philipp Gegenwart (University of G¨ ottingen, Germany),
+Hilbert von L¨ ohneysen (Karlsruhe University, Germany),
+Brian Maple (University of California, San Diego, USA),
+SuchitraSebastian(CambridgeUniversity,UK),T.Senthil
+(MIT,USA),KazuoUeda(InstituteforSolidStatePhysics,
+Tokyo, Japan) and Tomo Uemura (Columbia University,
+New York, USA). came together at the conference for a
+wide-rangingdiscussion on this topic. As a prequel to the
+discussions, each participant posted their questions and
+thoughtsona wikidiscussionsite[8].
+In reporting the discussions, I have not followed the
+originalorderingofspeakers,butinstead,togroupthedis -
+cussions by topic. Any mistakes in the rendition of the
+ideas that were presented at the discussion are most likely
+my own, for which I apologize in advance. I am particu-
+larlyindebtedtothosespeakerswhosentmenotesontheir
+discussion.
+2 f-electrons as a route to wider understanding
+One of the persistent threads throughout the discussion,
+wastheusefulnessoff-electronmaterialsasaresearchba-
+sis for studying quantum criticality and novel phases of
+strongly correlated materials. Meigan Aronson empha-
+sized how so much of our understanding about quantum
+criticality and zero temperature phase transitions comes
+from systematic studies of f-electron based compounds,
+where the low energy/temperature range of the f-bands
+meansthatthestabilityofmagneticordercanbetunedand
+fully suppressedby modest variationsin pressure, compo-
+sitionandmagneticfields.
+Kazuo Ueda agreed, and described how in Japan, the
+utility of f-electron research has recently been recognize d
+by the Japanese Ministry of Education through the es-
+tablishment of a distributed research network, consisting
+of sevean closely linked research consortia across Japan
+working on a wide-band of projects in f-electron physics.
+The network also has funds for several independent re-
+searchprojectsforsmaller researchgroupssome of anap-
+plied nature, to link up with the consortium. One of the
+vital reasons, he said, for a distributed consortium, is tha t
+itmakesitpossibletosharematerials,skills,resourcesa nd
+highqualityspectroscopywithoutthembeingconcentrated
+ata singleinstitution.Meigan Aronson referred to the mounting evidence
+from the phase diagrams of systems with very different
+microscopic physics - such as organic conductors, and
+the new Fe-based superconductors - that unconventional
+superconductivity may generically occur near magnetic
+quantum critical points, suggesting a universality to the
+overall behavior which was originally found in f-electron
+basedsystems.
+3 Strange Metalsand Quantum Criticality Oneof
+the key topics, was strange metal behavior and its possi-
+ble connection with quantumcriticality. There was a wide
+rangingdiscussion aboutthe meaningof quantumcritical-
+ity. Questions were also raised about whether the linkage
+between bulk quantum criticality, and the development of
+strangemetalbehaviorisindeedalwaysapparent?
+Hilbert von Lohneysen emphasized that even though
+a Quantum Phase Transition (QPT) is strictly speaking
+a zero-temperature instability, its experimental manifes ta-
+tionsareclearlyseenatfinitetemperatures.Inmetalsthes e
+anomalies give rise to departures from canonical Landau
+Fermi liquid behavior that are often refered to as “non-
+Fermi-liquid”(NFL)behavior.Ingeneralterms,heargued,
+onecanunderstandthisfinitetemperatureeffectofaquan-
+tumcriticalpointintermsofaquantum-classicalmapping
+in whichtemperaturesets the maximumtime scale for co-
+herent quantum processes[9,3,4,10] , introducing a finite
+system size Lτin thetimedirection.
+Lτ=¯h
+kBT.
+Accordingto thispicture,heemphasized,oneexpectsthat
+the fan-shaped catchment area of a QCP will extend at
+mosttotemperatureswhere ¯h/Lτremainssmallcompared
+with other relevantenergyscales, such as the Kondo scale
+kBTK.
+Puzzlingly,though,inseveralsystems,theNFLbehav-
+ior associated with the QCP extendsto temperaturesup to
+or even in excess of TK. For instance, in CeCu 5.9Au0.1
+withTK≈5K,the anomalousscaling exponent α= 0.75
+inthedcsusceptibility χ(T)−1= (A+BTα)extendsupn
+to 6 K [1].Thisresembles,he remarked,theanomalousT-
+linearresistivityinthecupratesuperconductors,extend ing
+up to≈1000Kin optimallydopedYBa 2Cu3O7. BothSu-
+chitraSebastian andHilbertvonLohneysen questioned
+whether the remarkably large temperature ranges
+over which strange metal behavior is observedis a
+signature of the asymptotic low temperature quan-
+tum criticality, or whether it is some other kind of
+separateprecursor?
+For example, in quantum critical YbRh 2Si2−xGexa log-
+arithmic specific heatC
+T∼1
+T0ln(T0
+T)is seen over two
+decadesof temperature,where T0= 24K,but at tempera-
+tures below T∗∼0.3K, the specific heat is seen to cross
+overtoa power-lawbehavior C/T∼T−1/3[11](SeeFig.
+1).
+Copyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 3
+Figure 1 The cross-over between
+“strange metal” behavior that occurs at
+high temperatures, to “quantum critical
+behavior”inthevicinityofthequantum
+criticalpoint.
+Suchitra Sebastian askedwhat do unconventional
+power laws really mean and with what theory should one
+compare them? It is well know that in the approach to a
+classical critical point, that larger temperature deviati ons
+T−Tcfrom the critical point are governed by Ginzburg
+Landau theory and true critical behavior only develops
+oncetheGinzburgcriterionisviolated.Couldacross-over
+between two different types of behavior be at work in
+quantumcriticalitytoo?
+Kazuo Ueda remarked that a key observation in f-
+electronsystems,isthatthemassofelectronquasiparticl es
+becomesveryheavynearaquantumcriticalpoint,andthat
+as the characteristic Fermi temperature collapses, strang e
+metal behavioris seen to develop.In addition to spin fluc-
+tuations,hesaid,therearea varietyofotherslow quantum
+fluctuations that may be important in driving up quasipar-
+ticle mass, including local orbital fluctuations and anhar-
+monic lattice vibrations. Two groups in the new consor-
+tium will explore such new mechanisms, he said. One of
+the guidingprinciples that researchersmay follow here, is
+to look for systems where the magnetic ion lies at a point
+ofhighsymmetry.
+Brian Maple expandedfurtheron the need for a more
+general exploration of quantum criticality. He argued that
+the view that the identification of strange metal physics
+with a bulk quantum critical point may be too restrictive.Indeed,keyanomalouscharacteristicsofstrangemetalbe-
+havior,suchasanomaloustemperaturedependencein
+–resistivity, ρ(T)∼ρ0±ATn(1≤n≤1.5with n
+usuallyclose to1).
+–specific heat C(T)/T∼ −lnT, T−n, (n∼0.2−0.4
+)and
+–magnetic susceptibility χ(T)∼ −lnT, T−n(n∼
+0.2−0.4),alongwith
+–the observation of ω/Tscaling in neutron scattering
+withχ′′(q,ω)∼f(ω/T).
+are found in widely different circumstance. Not only
+only are they found near bulk antiferromagnetic quan-
+tum critical points, as in YbRh 2Si2, CeRhIn 5under pres-
+sure, and CeRh 1−xCoxIn5, they are also observed at
+spin-glass quantum critical points in U 1−xYxPd3Al2,
+UCu5−xPdx, near ferromagnetic quantum critical points
+(in URu 2−xRexSi2and CePd 1−xRhx). Moreover, he
+noted, they are also observed far from any readily iden-
+tifiable bulk quantum critical point, as in (Y 1−xUxPd3,
+Sc1−xUxPd3and U1−xThxPd3Al2) promptinga specula-
+tionthattheremightbeasingleion,localquantum-critica l
+character.TheseobservationsledBrianMapletoask:
+Is there a more general scenario that encompasses
+these situations and presently proposed mechanisms (e.g.,
+Kondo disorder, quadrupolar Kondo, Griffiths phase, 2nd
+Copyrightlinewillbe provided by the publisher4 P. Coleman: QCNP09: A panel discussion.
+orderAFM,SG,orFMtransitionsuppressedto0K,etc.)?
+As an example, Maple noted Y 1−xUxPd3, the first f-
+electron material in which non-Fermi liquid behavior was
+observed), Sc 1−xUxPd3, and URu 2Si2, where there is
+evidence that the magnetic uranium ion is tetra-valent
+(U4+,5f2), with two localized f-electrons. In this situ-
+ation, there is the possibility of a Γ3orΓ5non-Kramers
+doublet ground state, setting the stage for a quadrupolar
+Kondo effect, originally proposed for Y 1−xUxPd3. Per-
+haps, he proposed, this scenario, modified to account for
+interactionsbetweenUions,can,afterall,accountforNFL
+behaviorin thesesystems.
+4 Themechanismofquantumcriticality. Thedis-
+cussion turned to the mechanism of quantum criticality as
+observedinf-electronmaterials.Oneofthesubjectsofpar -
+ticular interest, concerned the evolution of the Fermi sur-
+face through a quantum critical point. Meigan Aronson
+discussed how measurements have documentedthe extent
+towhichthecriticalfluctuationsassociatedwiththe T= 0
+K transition affect a wide range of measured quantities,
+magnetic,thermal,andtransport.Yetatthesametime,she
+noted,variousexperimentssuchasHall conductivity,neu-
+tron and de Haas van Alphen suggest the f-electron may
+belocalizingatamagneticheavyelectronquantumcritical
+point,rasingherto posethekeyquestion:
+how do the anomalous fluctuations at a quantum
+critical point impact the underlying electronic
+structure and give rise to the apparent f-electron
+delocalizationtransition that has been foundto ex-
+ist at or near the quantum critical point in some
+systems?
+The possibility of a class of quantumphase transitions
+where the Fermi surface jumps in area or volume was
+taken up in detail by Todadri Senthil . Senthil cited both
+heavy fermion Quantum Critical Points and Mott Metal-
+insulator transitions (which may occur in under and over-
+doped cuprate superconductors)as possible examples. He
+argued that since these transitions appear to be second-
+order, non-Fermi liquid physics follows very naturally at
+such a QCP, but that the intuition built up from “bosonic
+quantumcriticality” (electronscoupledto a fluctuatingor -
+derparameter)wouldnotberelevant.
+Senthil described how a model of “quantum critical
+Fermi surfaces” is appropriate to describe the paradoxi-
+cal combination of a first-order jump in the Fermi sur-
+face at a second-order phase transition[12]. According to
+this picture, the large and small Fermi surface co-exist at
+a QCP, while the jump Zin momentum-space occupancy
+is expected to vanish[5,13] at the QCP to be replaced by
+a Fermi surface with a power-law singularity in the elec-
+tron Green functions, similar to what is found in a one-
+dimensionalLuttingerLiquid(Fig.2).
+One of the interesting questions raised by Senthil, is
+whetherthe jump in the Fermi surface volume and the de-
+velopmentofmagneticorderarenecessarilylinked
+to one-another, or whether are they two different
+phenomenon?
+PiersColeman describedhow[14]insightsfromquan-
+tummagnetismandrecentexperimentstendtosupportthis
+point ofview. Imaginehe said, connectinga frustratedan-
+tiferromagnet to a conduction sea via a tunable Kondo
+interaction. Various groups[15,16,17,18] have considere d
+a two-dimensional phase diagram with x−ordinate de-
+scribing the tuning K=TK/JHof the ratio between the
+Kondo temperature and the nearest-neighbor RKKY in-
+teraction, and y−ordinate describing the intensity Q−of
+antiferromagnetic quantum zero-point fluctuations (which
+canbetunedforexample,byincreasingtheamountoffrus-
+tration). While there there is a common antiferromagnetic
+phaseatsmall KandQ,thetheparamagnetic“spinliquid”
+at largeQhas a small Fermi surface, while the the param-
+agnetic heavy Fermi liquid at large Khas a large Fermi,
+suggesting that the two are separated by zero-temperature
+phase transition (Fig. 3). The existence of this transition
+appearsto have been observedin field-tuningexperiments
+on Ir and Ge doped YbRh 2Si2(YbRh 2−xIrxSi2[19] and
+YbRh2Si2−xGex[18]. In these materials, there is a field-
+tuned temperature scale T∗(B) where various anomalies
+are seen in the Hall constant, susceptibility and Gr¨ uneise n
+parameter are seen to sharpen up at the critical field Bc
+where T∗(Bc)→0. This point has been interpreted as
+the point of field-tuned transition between a small and
+a large Fermi surface. Philipp Gegenwart pointed out
+that in YbRh 2Si2, the field-tuned T∗scale and the N´ eel
+temperature line and T N(B) convergeat a single quantum
+criticalpointbutthattheyseparateinIrorCoorGedoped
+systems[19,18]. Similar features, though less intensivel y
+studied, are seen at higher magnetic fields in YbGeSi[20].
+In both types of material, a strange metal phase appears
+to lie lie between the ordered antiferromagnet and the
+heavy electron state. Philipp Gegenwart raised various
+questionsabouttheseideas.He asked,
+–whatisthecoincidenceofscalesinundopedYbRh 2Si2
+thatbringsthe N´ eeland“T∗line”together?,
+–What is the nature of the “spin liquid phase” that is
+predictedto developinIrdopedYbRh 2Si2?
+–In YRS, why does pressure fail to influence the “T∗
+line”?[21]
+Senthilhad several points to make about this kind of
+phasediagram,whichheepitomizedbythephrase “Quan-
+tum is different” . These important differences can occur
+in manydifferentguises.Forexample,fromfromworkon
+frustrated two-dimensional antiferromagnets, there are i n-
+dications that the continuous QPT from antiferromagnet
+to spin liquid, or valence bond solid may involve a new
+kind of “deconfined criticality” with emergent fractional
+degreesof freedom[22]. He also mentionedthe possibility
+oftopologicalorder[23,24]and “selforganizedcriticality”
+Copyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 5
+Figure 2 (Color online) Schematic il-
+lustrating Senthil’s “quantum critical
+Fermi surface” scenario[12], whereby
+the jump in the occupancy Zkat the
+small Fermi surface vanishes at the
+quantum critical point[5], leaving be-
+hind a power-law singularity. As the
+tuningparameter passes beyond itscrit-
+ical value gc, a new “large” Fermi sur-
+face develops at k=kF2, describing
+the heavy Fermiliquid.
+where strange metal phases containing algebraic order in
+space,ortime,mightdevelopwithoutfine-tuningtoaQCP,
+suchasa criticalspin liquidphase[25,26].
+Philipp Gegenwart ,Hilbert von Lohneysen and
+Brian Maple turned the subject towards the practical, ex-
+perimental classification and characterization of differe nt
+typesofquantumcriticalpoint.
+Gegenwart described how antiferromagnetic quantum
+critical points in metals appear to divide into two classes-
+“conventional”,spin-densitywave transitionsand “uncon -
+ventional” QCP’s where the physics appears more local-
+ized.Oneofthemostusefulmethodstodelineatebetween
+different classes of quantum critical point, he said, is the
+Gr¨ uneisenparameter,whichmeasurestheratio
+Γ=V−1dV/dT
+CP.
+Under the assumption that the free energy contains a sin-
+gle energy scale, so that F(T)∼Tφ(T/T∗(P)),Γ∝
+dln(T∗)
+dP.Ithasbeenshownbyscalingarguments[27]that Γ
+divergesintheapproachtoanyQCP.ForQCPsoftheSDW
+type,the critical Gr¨ uneisenratio divergesas 1/T.By con-
+trast, a divergence with fractional exponent ( Γ∼T−0.7)
+hasbeenfoundinGe-dopedYbRh 2Si2[28],whichappears
+tobecompatiblewiththe predictionsofa locallyquantum
+critical senario. If Γhas no divergence, one can exclude
+a generic QCP as the origin of non-Fermi liquid behavior.
+Gegenwart noted that experimentally, both types of QCP
+(SDW and local) are observed in different heavy fermion
+metals. Unconventional transitions pose the greatest chal -
+lenge,withmanyopenquestions:
+–Whichtypesofunconventionalexist?
+–What aretheconditionstheyarise from?
+–Whichobservedfeaturesaregenericandwhicharema-
+terial specific?
+Gegenwart turned to discuss the field-tuned quan-
+tum criticality of of YbRh 2Si2, one where the mag-netic Gr¨ uneisen parameter ΓM=−dM/dT
+CHdiverges as
+ΓM=Gr/(H−Hc)withGr=−0.3[29]. Gegenwart
+asked whether this result might be consistent with the
+critical FermisurfacemodelofSenthil?
+HilbertvonLohneysen emphasizedthatthe difficulty
+withanykindofglobalphasediagram,isthatwedonotun-
+derstand the rolesof differenttuning parametersat a QPT.
+He pointed out that there are only few systems where dif-
+ferent parameters have been employed to tune a QCP. In
+the case of CeCu 6−xAux, three different tuning parame-
+tershavebeenemployed:chemicaldopingwithgold,con-
+centrationx,hydrostaticpressure P,andmagneticfield B.
+While P and x tuning lead to the same T dependencies in
+resistivity ρandspecific heat C, bothsuggestiveof a local
+QCP, field tuning shows dependencies in ρand C that are
+more indicative of a spin-density wave (SDW)-type QCP
+[2]. The different scaling behavior for x and B tuning has
+been corroborated by inelastic neutron scattering experi-
+ments, directly measuring the critical fluctuations [1, 3].
+Theseexperimentsputdefiniteconstraintsontheconstruc-
+tion of a multi-parameter zero-temperature (global) phase
+diagram.Moreexperimentsalongtheseslinesarecertainly
+needed.
+BrianMaple discussedhowlittleweunderstandabout
+ferromagnetic quantum critical points in f−electron sys-
+tems. Unfortunately, he said, most ferromagnetic sys-
+tems exhibit a first order QPT under pressure, but a sec-
+ond order QCP under chemical doping. He introduced
+URu2−xRexSi2as a fascinating new development in this
+respect.UndopedURu 2Si2exhibits“hiddenorder”,buton
+doping with rhenium, he said, the transition temperature
+of the hidden order is continuously suppressed to zero at
+xc≈0.15.TheQPTwherethehiddenorderdisappearsco-
+incideswitha suddenappearanceof ferromagnetism,with
+aCurietemperature Tcthatgrowslinearlywith x−xc.This
+ferromagnetic phase exhibits non-Fermi liquid properties :
+a logarithmic temperature dependence of the specific heat
+and an anomalous temperature dependence of the resis-
+Copyrightlinewillbe provided by the publisher6 P. Coleman: QCNP09: A panel discussion.
+Figure 3 (Color online) Schematic
+“global phase diagram” for heavy
+fermion materials, adapted from refer-
+ence[18],showing“Kondoaxis”tuning
+the ratio K=TK/JHof the Kondo
+temperature to Heisenberg interaction
+versus the “Quantum axis” tuning the
+strength of magnetic zero point fluctu-
+ationsQ.Redlinedenotes zerotemper-
+ature quantum phase transitionbetween
+smallandlargeFermisurfacestates.La-
+beled are the hypothetical locations of
+stoichiometric YbRh 2Si2(YRS)andthe
+same compound, doped with Ge or Ir
+(YRS(Ge,Ir)),showingpresumedeffect
+of magnetic fieldand doping.
+tivity. The logarithmic specific heat exhibits a remarkable
+parabolicdependenceondoping,
+γ(x,T)∼ −(x−xc)(x2−x)×1
+T0lnT,
+wherex=x2= 0.6,eventhoughferromagnetismcontin-
+ues to higher doping. Maple reported that a careful study
+of evolution of the temperature exponent of the suscepti-
+bilityχ∼t−γ, the field and temperature dependence of
+the magnetization M∼H1/δ, tβ(linked by the relation
+δ−1 =γ/β) shows that as a function of doping, γand
+δgrow linearly with x−xcwhileβ≈1is constant[30].
+Thisfascinatingphysicsawaitsa theory.
+Tomo Uemura andSuchitra Sebastian both returned
+thediscussionofthenatureofthesoftquantummodesnear
+aQPT.Bothraisedtheissueoftherelationshipofspinand
+chargedegreesoffreedomin materialsclose to a quantum
+critical point, and indeed, whether spin is the most impor-
+tantslowdegreeoffreedom.
+TomoUemura raisedtwointerestingpointsinconnec-
+tion with the soft-modesat a QCP. First, he asked, what is
+the role of importance of first-order quantum phase tran-
+sitions. Even though many quantum phase transitions are
+firstorder,theymaystillexhibitavarietyofimportantsof t
+modes that while strictly speaking, remain gapped at the
+transition,stillinfluencethephysicsoverawidefinitetem -
+perature range. Such soft modes are the analog of “roton
+modes”inHe4,buttheymightoccurinthespin,thecharge
+andeven the phase channel.In general,evenif the QCP is
+firstorder,theenergyofsuchsoftmodescanbeusedasan
+indicatorofthe closenesstoa competingstate[31,32].
+Uemura speculatively introduced the idea of “reso-
+nant spin-chargecoupling”- while most of us think of the
+charge and spin modes of strongly correlated systems as
+separatedegreesoffreedom,is it possible, he mused, that that slow spin and
+chargemodesbecomeresonantlycoupled?
+Uemura he showed how the superconducting transition
+temperature Tcin a wide range of unconventional super-
+conductorsscaleswiththeFermitemperature TF(obtained
+fromthelinearspecificheat)andthespin-fluctuationscale
+(obtainedfromthemagneticsusceptibility),namely
+Tc∝TF, TSF.
+Suchscalingrelationshipsseemtoholdovermanydecades
+of variation in Tc. Conventionally, these kinds of scaling
+relationshipsareinterpretedintermsofananisotropicpa ir-
+ing within a Fermi liquid, in which the a single renor-
+malized Fermi temperature of the pairing electrons also
+governs the characteristic spin fluctuation scale. Uemura
+argues that an alternative way to interpret tracking be-
+tweenTFandTSFillustrates a resonance between spin
+andchargefluctuationmodes.
+Suchitra Sebastian viewed the problem from another
+perspective.Sheasked:
+should we think of a separation of charge and spin
+Quantumcriticalpoints?
+Suchaseparationhas,shepointedout,beenusedtounder-
+stand the effect of pressure in CeCu 2Si2−xGex, where the
+superconducting transition temperature is seen to exhibit
+two separate maxima as a function of pressure - a lower
+pressure maximum in the vicinity of a magnetic plus a
+higherpressuremaximuminthevicinityofavalenceinsta-
+bility of the material[33,34]. But could this, she asked, be
+part of a muchmore generalphenomenon?Could one, for
+example, understand the two transitions seen in Ge and Ir
+doped YbRh 2Si2as a spin and a charge critical point?[19,
+18]. Sebastian also showed de Haas van Alphen measure-
+ments on CeIn 3, in where a field-induced quantum phase
+transition is observed within the antiferromagnetic phase
+Copyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 7
+at which the effective mass of the quasiparticles diverges
+while the orbit areas collapse to zero[35]. A similar diver-
+gence in quasiparticle effective masses has also been re-
+cently seen in the de Haas van Alphen measurements on
+under-doped YBa 2Cu3O6+x[36]. Could these, she asked,
+befurtherexamplesofthisgeneralphenomenon-QPTin-
+volvingcharge?
+5 New Phases, superconductivity and the d-f
+connection Throughout the discussion, physicists con-
+stantlyreturnedto thethemeof superconductivity,its con -
+nection with quantum criticality and the usefulness of f-
+electronresearch as a platformfor understandingordering
+phenomenonat higher energy scales, d-electron transition
+metals. The “d-f” connection was discussed by several of
+thepanelists.
+Tomo Uemura discussed how the broad trends in Tc
+with superfluiddensitywhich cut acrossanomalousf-and
+d-electronsuperconductorssupporttheideathatweshould
+endeavortounderstandthesephenomenonwithinaunified
+framework.Aspartofthisframework,heargued,thereare
+twoextremelimitstoconsider-oneextreme-thatofBCS
+pairing,whileonthe other- that ofBose-Einsteinconden-
+sation ofpre-formedpairs. Uemuraarguesthat the scaling
+ofTcwith superfluid density is an indication that anoma-
+loussuperconductivityliesattheBCS-BECcross-overbe-
+tweenthesetwoextremeregimes.
+Kazuo Ueda Discussed how a major part of the new
+Japanese f-electron consortium, is the discovery of new
+materials, with new types of broken symmetry ground-
+state will broadenourunderstandingof strongcorrelation ,
+helping the d-f connection. This thrust has two compo-
+nents,he said-
+–By extending the search for novel f-electron behavior
+along the rare earth series - at one end of the series,
+fromCerium to Paladiumcompounds,andat the other
+end of the series, from Ytterbium to Thullium (Tm)
+materials.
+–Bypushinguptohigherenergyscales,intermediatebe-
+tween the 4f-and 3d materialsthroughthe exploration
+of 5f correlated electron materials. By going from the
+Cerium 115 materials to related transuranic materials,
+PuCoGa 5andNpAl 2Pd5,ithasprovedpossibletosub-
+stantially raise Tc. Are there other examples of this
+trend?
+Meigan Aronson discussed the difficulties in making
+the d-f connection, noting that while an extensive body
+of measurements on itinerant ferromagnetssuch as ZrZn 2
+and MnSi, led the way in establishing magnetic quantum
+criticality in a d-electron context, [37,38], unlike their f-
+counterparts, these systems have ‘large’ Fermi surfaces in
+both the ordered and paramagnetic regimes, where the d-
+electronsareincludedintheFermisurface[39].Whilesys-
+tems such as V 2O3[40] and Ni(S 1−xSex)2[41] are con-
+sidered exemplars of Mott-Hubbard physics, with a firstorder transition from strongly correlated metal to local-
+ized moment insulator [42], she points out we still have
+notdiscoveredatransitionmetalcompoundthatistheana-
+log of magnetic field tuned YbRh 2Si2[43] or pressurized
+CeRhIn 5[7], where quantum critical points are the source
+ofbothstrongquantumcriticalfluctuationsandtransition s
+where an f-electron is delocalized, driving the Fermi sur-
+facefromsmallto large[44].
+6 Conclusion The panel discussion prompted lively
+debate throughout the QCNP09 meeting, and appears to
+providea useful model for future scientific conferencesof
+this time. Thestudy of quantumcriticalityand its intimate
+relationship with material physics and the emergent quan-
+tum mechanics of the periodic table, make it an area of
+burgeoning discovery. As a reporter on this event, I’d like
+end with a quote from Meigan Aronson at this event, on
+the prospectsofa futured-fconnectionsurroundingquan-
+tumcriticalityandhightemperaturesuperconductivity(s ee
+Fig. 4):
+“So far, our interest has focused on ferromagnetic
+systems, which are ultimately discontinuous, and
+not quantum critical, when the Curie temperature
+becomes sufficiently low [45,46]. Whether a sim-
+ilar behavior will be found in (d-electron) quan-
+tum critical antiferromagnets remains an interest-
+ing and controversial question, little explored due
+to a lack of suitable host systems. Still, these are
+thecompoundsinwhichunconventionalsupercon-
+ductivityispresumedtobemostlikely,particularly
+if it is possible to realize the strongly correlated
+state foundon the metallic side of a Mott-Hubbard
+transition. For these reasons, it seems more press-
+ingthaneverto refreshourinterestin intermetallic
+compoundswherethemagneticentityderivesfrom
+transitionmetalmoments.”
+Acknowledgements Iamindebtedtoalltheparticipantsin
+the QCNP09 panel discussion for providing their view-graph s in
+advance of the session, andsubsequently notes on theirpres enta-
+tions. Piers Coleman is supported by the Department of Energ y
+grant DE-FG02-99ER45790.
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+Copyrightlinewillbe provided by the publisher
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+arXiv:1001.0186v1  [math.MG]  31 Dec 2009ON THE SQUARE PEG PROBLEM AND SOME RELATIVES
+BENJAMIN MATSCHKE
+Abstract. The Square Peg Problem asks whether every continuous simple
+closed planar curve contains the four vertices of a square. T his paper proves
+this for the largest so far known class of curves.
+Furthermore we solve an analogous Triangular Peg Problem affi rmatively,
+state topological intuition why the Rectangular Peg Proble m should hold true,
+and give a fruitfulexistence lemma of edge-regular polygon s on curves. Finally,
+we show that the problem of finding a regular octahedron on emb edded spheres
+in /CA3has a “topological counter-example”, that is, a certain tes t map with
+boundary condition exists.
+1.Introduction
+The Square Peg Problem was first posed by O. Toeplitz in 1911:
+Conjecture 1.1 (Square Peg Problem, Toeplitz [Toe11]) .Every continuous em-
+beddingγ:S1→ /CA2contains four points that are the vertices of a square.
+The name Square Peg Problem might be a bit misleading: We do not requir e
+the square to lie inside the curve, otherwise there are easy counte r-examples:
+Toeplitz’ problem has been solved affirmatively for various restricte d classes of
+curves such as convex curves and curves that are “smooth enou gh”, by various
+authors; the strongest version so far was due to W. Stromquist [ Str89, Thm. 3]
+who established the Square Peg Problem for “locally monotone” curv es. All known
+proofs are based on the fact that “generically” the number of squ ares on a curve
+is odd, which can be measured in various topological ways. See [Pak08 ], [VrˇZi08],
+[CDM10], and[Mat08]forsurveys. Forgeneralembedded planecur ves, theproblem
+is still open.
+Date: Dec. 31, 2009.
+Supported by Studienstiftung des dt. Volkes and Deutsche Te lekom Stiftung.
+12 BENJAMIN MATSCHKE
+We start our discussion in Section 2 with a description of a convenient parameter
+space for the polygons on a given curve. We then present a proof id ea due to
+Shnirel’man [Shn44] (in a modern version, in terms of a bordism argume nt), which
+establishes the Square Peg Problem for the class of smooth curves .
+Then we prove it for a new class of curves, which includes W. Stromqu ist’s
+locally monotone curves. The first drawing above is an example that lie s in this
+new class, but not in Stromquist’s. See Definition 2.2 and Corollary 2.5 f or an
+interesting special case. The proof generalises to curves in metric spaces under a
+suitable definition of a square; see Remark 2.6.3).
+In Section 3 we ask the analogous question for equilateral triangles instead of
+squares and get a positive answer, even in a larger generality. Similar ly we look
+for edge-regular polygons on curves, that is, polygons whose edg es are all of the
+same length. In Section 4 we prove the existence of an interesting f amily of ε-close
+edge-regular polygons on smooth curves and deduce some immediat e corollaries.
+Section 5 deals with the existence of rectangles with a given aspect r atio on smooth
+curves. I have no proof for this, but we will see some intuition why th ose rectangles
+should exist.
+The last section, Section 6, treats higher-dimensional analogs. We ask ford-
+dimensional regular crosspolytopes on smoothly embedded ( d−1)-spheres in /CAd.
+The Square Peg Problem for smooth curves is the case d= 2. The problem is open
+for alld≥3, but we use Koschorke’s obstruction theory [Kos81] to derive th at for
+d= 3, a natural topological approach for a proof fails: The strong t est map in
+question exists.
+This paper is an extended extract of [Mat08, Chap. III]. Some of th e new results
+have been announced in [Mat09].
+2.Squares on Curves
+2.1.Notations and the parameter space of polygons on curves. For any
+spaceX, we denote by
+∆Xn:={(x,...,x)∈Xn}
+the thin diagonalof Xn. For an element xof the unit circle S1∼=
+/CA//CIandt∈ /CAwe
+definex+t∈S1as the counter-clockwise rotation of xby the angle 2 πtaround 0.
+Letσn={(t0,...,tn)∈ /CAn+1
+≥0|/summationtextti= 1}be the standard n-simplex.
+The natural parameter space of polygons is
+Pn:=S1×σn−1.
+It parametrises polygons on S1or on some given curve S1→ /CA∞by their vertices
+in the following way
+ϕ:Pn→(S1)n: (x;t0,...,tn−1)/mapsto→(x,x+t0,x+t0+t1,...,x+n−2/summationdisplay
+i=0ti).
+The so parametrised polygons are the ones that are lying counter- clockwise on S1.
+The map ϕis not injective, as all ( x;0,...,0,1,0,...,0) are mapped to the same
+point (x,...,x); but it is injective on Pn\(S1×vert(σn−1)), and onthis set ϕbijects
+to (S1)n\∆(S1)n. LetP◦
+n=S1×(σn−1)◦denote the interior of Pn. The map ϕ
+identifies P◦
+nwith the set of n-tuples of pairwise distinct points in counter-clockwise
+order on S1. We define the boundary as ∂P◦
+n:=Pn\P◦
+n.ON THE SQUARE PEG PROBLEM AND SOME RELATIVES 3
+We let /CIn:= /CI/n /CI=:/a\}bracketle{tε/a\}bracketri}htact onPnby
+ε·(x;t0,...,tn−1) = (x+t0;t1,...,tn−1,t0).
+This corresponds to a cyclic relabeling of the vertices of the parame trised polygon.
+2.1.1.A substitution. Thefollowingcoordinatetransformationmakesthe /CIn-action
+onPnlook nicer. We substitute ( x;t0,...,tn−1)∈Pnby (x∗;t0,...,tn−1), where
+x∗:=x+/summationtextn−1
+k=1n−k
+n·tk−1∈S1. In terms of the new coordinates,
+ε·(x∗;t0,...,tn−1) = (x∗+1
+n;t1,...,tn−1,t0).
+2.1.2.Further notations. When we talk about an arconS1from a point xtoy, we
+always mean the arc that goes counter-clockwise. For x,y∈S1, we denote by y−x
+the length of the arc from xtoy, normalised with the factor1
+2π. For an n-tuple
+(x1,...,x n)∈ϕ(Pn)⊂Snwe write
+[x1,...,x n] := (x1;x2−x1,x3−x2,...,x n−xn−1,1−n/summationdisplay
+k=2(xk−xk−1))∈Pn.
+The function [ ...] :ϕ(Pn)→(S1)nis right-inverse to ϕ, but not continuous.
+SmoothmeansC∞for us. An ε-close square is a quadrilateral whose ratios
+between the edges and diagonals are up to an ε-error the ones of a square. The
+precise definition will not matter. We will use “ ε-closeness” with other polygons
+analogously.
+2.2.Shnirel’man’s proof for the smooth Square Peg Problem. We start
+with L. G. Shnirel’man’s proof [Shn44], since it is in my point of view the mos t
+beautiful one. The following presentation uses transversality and a bordism argu-
+ment; in Shnirel’man’s days, these notions had not been formalised an d baptised
+yet, but his argument works like this.
+Proof.Suppose that γis smooth. P◦
+4parametrises quadrilaterals on γ. Letf:
+P4→ /CA6be the function that measures the four edges and the two diagona ls of
+the quadrilaterals,
+(1)f:P4 −→ /CA4× /CA2
+[x1,x2,x3,x4]/mapsto−→(||γ(x1)−γ(x2)||,||γ(x2)−γ(x3)||,||γ(x3)−γ(x4)||,
+||γ(x4)−γ(x1)||,||γ(x1)−γ(x3)||,||γ(x2)−γ(x4)||)
+We can compose fwith the quotient map /CA6→ /CA6/∆/CA4×∆/CA2∼=
+/CA4and get
+f:P4→ /CA4. The test-map f′measures squares, since Q:= (f′)−1(0)\∆(S1)4=
+(f′)−1(0)∩P◦
+4is the set of all squares that lie counter-clockwise on γ.f′is /CI4-
+equivariant with respect to the natural /CI4-actions. We can deform /tildewidefrelative to
+a small neighborhood of ∂P◦
+4equivariantly by a small ε-homotopy to make 0 a
+regular value of f′. SoQbecomes a zero-dimensional /CI4-manifold (note that Q
+lies inP◦
+4, which is free) of ε-close squares. If we deform the curve smoothly to
+another curve (e.g. the ellipse), which can also happen in /CA4to construct such
+a homotopy easily, then Qchanges by a /CI4-bordism. This bordism stays away
+from the boundary of P◦
+4, if the homotopy is chosen smoothly, since then no curve
+inscribes ε-close squares which have arbitrarily small edges (the angles get to o close
+toπ). Hence Qrepresents a unique class [ Q] in the zero-dimensional unoriented
+bordism group N0(P◦
+4//CI4)∼=H0(P◦
+4//CI4; /CI2)∼=
+/CI2. Ifγis an ellipse then 0 is a
+regular value of f′andQconsists of one point. Hence [ Q] is the generator of /CI2,4 BENJAMIN MATSCHKE
+soQis non-empty for any smooth curve γ. Taking a convergent subsequence of
+ε-close squares finishes the proof. /square
+Ifγis only continuous one might try to approximate it with smooth curves
+and then take a convergent subsequence of the squares that we get on them. The
+problem is to guarantee that this subsequence does not converge to a square that
+degenerates to a point. Natural candidates for which this works a re continuous
+curves with bounded total curvature without cusps, see Cantar ella, Denne & Mc-
+Cleary [CDM10]. So far, nobody managed to do this for all continuous curves.
+Shnirel’man’s proof can be refined to get a slightly stronger result.
+Corollary 2.1 (of the proof) .We may assume that γgoes counter-clockwise
+around its interior. Then one can find and order four vertices of a square on γ,
+such that they lie counter-clockwise on γand also label the square counter-clockwise.
+Proof.This can be achievedby restricting Qin the aboveproofto the set ofsquares
+[x1,x2,x3,x4]∈P4that are labeled by ( γ(x1),...,γ(x4)) in counter-clockwise or-
+der. Alonga path in the bordismthis cannot change(herewe takea b ordismthat is
+induced by a deformation of the curve in the plane). If γis an ellipse then it is clear
+that the restricted Qis equal to Q, so it represents the generator in N0(P◦
+4//CI4)./square
+2.3.New cases of the Square Peg Problem. First of all we will establish the
+main theorem of this section, which gives a larger class of curves for which in-
+scribed squares exist. Then we deduce two handy corollaries that a re more directly
+applicable.
+Letγ:S1→ /CA2be a simple closed curve (that is, injective and continuous).
+We need some preparation. Let f:P4→ /CA6be the corresponding test map that
+measure the four edges and two diagonals, which was defined in equa tion (1) in
+Section 2.2. For y1,y4∈S1,y1/\e}atio\slash=y4, let
+P4(y1,y4) :={[y1,x2,x3,y4]∈P◦
+4}
+the set of all quadrilaterals counter-clockwise on S1where the first and last vertex
+are given. For a path y:S1→(S1)2\∆(S1)2,y(t) = (y1(t),y2(t)), we define
+P4(y) :=/uniondisplay
+t∈S1P4(y(t)) ={[y1(t),x2,x3,y4(t)]∈P◦
+4|t∈S1}.
+Definition 2.2. We call a quadrilateral on γgiven by [ x1,x2,x3,x4]specialif
+f([x1,x2,x3,x4]) = (a,a,a,b,e,e ) witha≥b, for some reals a,b,e.
+Thesizeofaspecialquadrilateral[ x1,x2,x3,x4]isthenormalisedarclength x4−x1.
+LetSdenote the set of all special quadrilaterals in P4. The following figure
+shows a special quadrilateral of small size on γ.
+γ(x3)γ(x2)
+γ(x1) γ(x4)ON THE SQUARE PEG PROBLEM AND SOME RELATIVES 5
+Theorem 2.3. Suppose there is a path y:S1→(S1)2\∆(S1)2∼=P◦
+2,y(t) =
+(y1(t),y4(t)), that represents a generator in π1((S1)2\∆(S1)2)∼=π1(S1)∼=
+/CI. Ifγ
+does not inscribe a square then the mod- 2intersection number of P4(y)andSis1.
+The mod-2intersectionnumberwill be describedin the proof. The pr oofisbased
+on equivariant obstruction theory, which was first used in connect ion to the Square
+Peg Problem by Vre´ cica and ˇZivaljevi´ c [Vr ˇZi08]. The second part of our proof will
+be very close to what they did. One can of courseuse different topo logicalmethods,
+but their way is quite straight-forward and beautiful. Another poin t of view will
+be sketched in the remarks 2.6.
+Proof.P4(y) can be parametrised by g:S1×σ2→P4(y), where S1parametrises
+yandσ2the three arc lengths between the points y1(t),x2,x3andy4(t). The map
+gis injective if and only if yis.
+The mod-2 intersection number in the theorem is defined as the mod- 2 intersec-
+tion number of f(g(S1×σ2)) andV:={(a,a,a,b,e,e )∈ /CA6|a≥b}in /CA6. This
+is only well-defined if f(g(S1×∂σ2))∩V=∅and im(f◦g)∩∂V=∅. The former
+is trivially fulfilled, the latter if and only if no quadrilateral on γgiven by P4(y) is
+a square (this is interesting if one deformes y; compare with Remark 2.6.1.). The
+mapf◦gcould now be deformed by a homotopy rel S1×∂σ2, such that at no time
+it intersects the boundary of V, and such that it becomes transversal to V. The
+intersection number then counts the pre-images of Vunderf◦gmodulo 2.
+Suppose that γdoes not inscribe a square, but the described mod-2 intersection
+number is zero. We want to derive a contradiction.
+For some ε∈(0,1
+2) (later we might choose ε=1
+3), letT=Tε⊂σ3be a
+polytope obtained from a tetrahedron by cutting an open vertex fi gure of size ε
+from the vertices (we delete all points ( t0,...,t3)∈σ3that have an entry >1−ε).
+The four vertices of σ3are given by the standard basis vectors e0,...,e 3of /CA3.
+The four corresponding triangular facets of Tare denoted by T0,...,T 3, and their
+opposite hexagonal facets by H0,...,H 3.
+S1×T3⊂P4parametrises the 4-tuples ( x1,...,x 4)∈(S1)4withx4−x1=ε.
+Here is a sketch of Tin one dimension smaller where we draw S1×T⊂P4as a
+cylinder whose the bottom and top face are identified:
+S1×Ti S1×T:
+S1×Hj
+We will construct for some small δ >0 an /CI4-equivariant map
+h:S1×Tε−→/CI4S1×Tδ
+that satisfies the following conditions:
+(1)hmapsS1×HitoS1×Hi, 0≤i≤3,
+(2)his prescribed on S1×Tε
+3⊂P4as
+h(t;t0,t1,t2,t3= 1−ε) := (y1(t);λtt0,λtt1,λtt2,y1(t)−y4(t)),6 BENJAMIN MATSCHKE
+whereλt>0 is chosen uniquely such that the last four entries sum up to
+one, that is, we want h(t;,,,1−ε)∈P4(y1(t),y2(t)).
+The second condition prescribes hon allS1×Ti,i= 0,...,3, since his /CI4-
+equivariant.
+Nowweconstruct h. Ify= (y1,y4)isgivenby(id S1,idS1+ε), then wecanchoose
+δ=εandh= idS1×Tε. Otherwise there is a homotopy Ys:S1→(S1)2\∆(S1)2,
+s∈[0,1], from yto the previous one. For each time s∈[0,1] we can now ask
+how to find an hsas above for Ys. If we only require condition (2) then this is a
+homotopy extension problem. Since ( S1×Tε,S1×(T0∪...∪T3)) is a pair of free/CI4-CW-complexes, we can solve this. The standard proof for this give s a solution
+that automaticallysatisfiescondition(1) ateachtime, so especiallyf ory. Therefore
+hexists.
+Hence we get a test map
+t:=pr◦f◦h:S1×Tf◦h−→/CI4
+/CA6\(∆/CA4×∆/CA2)pr
+−→
+≃/CI4
+/CA4\{0}
+which is avoiding 0 ∈ /CA4, since we assumed that γinscribes no square.
+The range /CA4\{0}oftis a product of the standard /CI4-representation W4:=/CA4/∆/CA4andU:= /CA2/∆/CA2(ε·u=−u,u∈U), with 0 deleted. The corresponding
+components of taretWandtU. The images f0,...,f 3∈W4of the four standard
+basis vectors e0,...,e 3of /CA4span a tetrahedron which defines a fan with apex in
+0 and with four facets, which we label by F0,...,F 3, such that −fi∈Fi.V⊂ /CA6
+projects under prin /CA4=W4×UtoV′:= /CA≤0·(−f3)×{0}.
+We have enough information to disprove the existence of tusing an obstruction
+argument. Assume that only the restriction of tto∂(S1×T) =S1×∂Tis given,
+we look whether we can extend it.
+We are allowed to deform tby an arbitrary /CI4-homotopy. First of all we make t
+transversal to V′onS1×T3relative to its boundary (and extend this deformation/CI4-equivariantly). Let t−1(V′)∩(S1×T3) ={p1,...,p 2k}.
+From now on we write S1×T⊂Pnin the coordinates that were introduces in
+Section 2.1.1. We see that it has a simple /CI4-CW-complex structure with only one
+four-dimensional /CI4-cell orbit:
+One three-cell eshall be∗×T,∗ ∈S1. We may assume that t(∂(e)∩T3)∩V′=∅
+and analogously for the other Ti, since there are only finitely many points ∗ ∈S1
+which are forbidden in this way (namely the S1-coordinates of the piand their/CI4-translates).
+Notethat tW(S1×Hi)⊂Fi\{0}. Thisisbecauseonsuchpointsthe ti-coordinate
+is zero, hence the corresponding edge of the parametrised quadr ilateral is zero and
+thusminimalamongalledges. Thereforewecan /CI4-deformtUonasufficientlysmall
+neighborhoodof S1×(H0∪...∪H3)suchthat tUbecomeszeroon S1×(H0∪...∪H3)
+and such that during no time of this deformation change the new inte rsections of
+t(S1×T3) andV′.ON THE SQUARE PEG PROBLEM AND SOME RELATIVES 7
+By thedegreeof a map Sn−1→ /CAn\{0}we mean the degree of the normalised
+map toSn−1, or the scaling factor of the induced map on homology Hn−1().
+Sincet(∂(e)∩T3)∩V=∅, we can also deform ton a small neighborhood of
+∂(e)∩T3such that tW|∂(e)∩T3lies inF0∪F1∪F2and such that tU|∂(e)∩T3is zero,
+without changing the intersections of t(S1×T3) andV. Suppose we have extended
+tonesuch that tUis positive on the interior of e. ThentUis negative on the
+interior of ε·e( /CI4=/a\}bracketle{tε/a\}bracketri}ht). LetEbe the 4-cell of S1×Tthat has eandε·eas
+boundary faces. The degree of tWon∂eis one.
+Recallt−1(V)∩(S1×T3) ={p1,...,p 2k}. If 2k= 0, then one could also deform
+ton∂E∩∂(S1×T) as we did with ton∂e. In this case, t|∂Eis homotopic to
+the suspension of tW|∂e, hence it was of degree 1. However for every pi∈∂Ethe
+degree changes by one. This also happens at the other facets ∂E∩(S1×Ti) of∂E
+with the /CI4-translates of {p1,...,p 2k}. In total there are 2 ksuch points, hence the
+degree of t|∂Eis odd. If t|ewas chosen differently, the degree of t|Ewould change
+twice±the same number, once for eand once for ε·e. Hence one cannot extend t
+toE, contradiction. /square
+Corollary 2.4. Suppose there is a path y:S1→(S1)2\∆(S1)2=P◦
+2,y(t) =
+(y1(t),y4(t)), that represents a generator in π1((S1)2\∆(S1)2)∼=π1(S1)∼=
+/CI. If
+P4(y)∩S=∅, thenγcircumscribes a square.
+Proof.The mod-2 intersection number of Theorem 2.3 is here trivially zero. /square
+Corollary 2.5. Suppose there is an ε∈(0,1), such that γinscribes no (or gener-
+ically an even number of) special quadrilateral of size ε. Thenγcircumscribes a
+square.
+Proof.Use Theorem 2.3 with y1:= idS1andy4:= idS1+ε. /square
+Remarks 2.6.1.) An alternative view point is to look at Sas a 1-dimensional
+manifold, after one made ftransversal to Vby a small ε-homotopy, at first on
+P4(y) and then on P4. Here a technical trick is to choose εnot as a constant but
+as a function on P◦
+4that becomes arbitrarily small at the boundary, such that all
+technicalities work out. What Theorem 2.3 measures is the following.
+P4(y) can be seen as a “membrane”, which separates P4into two components if
+yis injective. If γcircumscribes no square then there is an oddnumber of paths
+inSthat pass through P4(y) and approach the boundary at S1×e3,e3being the
+one vertex of σ3. These paths might look very chaotic close to the boundary. On
+the other side of the membrane P4(y), this odd number of paths cannot all end in
+each other. One of them has to end somewhere else. It might end su ddenly in P◦
+4,
+which means that it found a square, or it might end somewhere else at ∂P◦
+4. My
+hope was that the latter is not possible, but it is:8 BENJAMIN MATSCHKE
+The drawn path of special quadrilaterals starts in the middle of the s piral at
+S1×e3with a quadrilateral that is degenerate to a point, and it stops when x1and
+x4moved together again, x4−x1= 1.
+2.) The corollaries are sometimes good for proving the existence of a square, if
+the curve is piecewise C1but has cusps (points in which the tangent vector changes
+the direction). This howeverworksnot in alargegeneralityas the pr eviousexample
+shows.
+3.) The whole Section 2.3 deals with the curve intrinsically , since the only
+datum of γwe used is the distances between points on γ. If we define a square
+in a metric space ( X,d) to be a 4-tuple ( x0,...,x 3)∈X4such that d(x0,x1) =
+d(x1,x2) =d(x2,x3) =d(x3,x0) andd(x0,x2) =d(x1,x3), then the whole section
+also works for curves γ:S1→X. More generally, Xdoes not need to fulfill
+the triangle inequality. In other words, we do not need an embedded curve but a
+distance defining function d:S1×S1→ /CAthat is continuous, positive definite,
+and symmetric.
+3.Equilateral Triangles on Curves
+For our first result, suppose we are given a symmetric distance fun ctiondon
+the circle. This might occur if we embed S1into a metric space and pull back the
+metric.
+Theorem 3.1 (“Triangular Peg Problem”) .Letd:S1×S1−→ /CAbe a continuous
+function satisfying d(x,y) =d(y,x). Then there are three points x,y,z∈S1, not
+all of them equal, forming an equilateral triangle, that is d(x,y) =d(y,z) =d(z,x).
+Proof.We use the configuration-space test-map scheme. Suppose ther e is a curve
+admitting no such triangle. This induces us an S3-equivariant map
+(S1)3\∆(S1)3S3−→ /CA3\∆/CA3≃S1.
+The configuration space ( S1)3\∆(S1)3deformation retracts equivariantly to the fol-
+lowing figure.
+It is a nice exercise in equivariant obstruction theory to show that s uch a map
+cannot exist. For more details see [Mat08, Chap. III.3] /squareON THE SQUARE PEG PROBLEM AND SOME RELATIVES 9
+If the distance function comes from a planar continuous embedding γ:S1→ /CA2
+then M. J. Nielsen [Nie92] has proven much more. Then there are ev en infinitely
+many triangles inscribed in the γwhich are similar to a given triangle T, and if
+one fixes a vertex of smallest angle in Tthen the set of the corresponding vertices
+onγis dense in γ. In the next section we show even a bit more if γis smooth. For
+example the latter holds true for any angle and the set of correspo nding vertices is
+all ofγ.
+4.Polygons on Curves
+This section is very similar to independently obtained results of V. V. M akeev
+[Mak05] and J. Cantarella, E. Denne and J. McCleary [CDM10].
+So far we asked about triangles and quadrilaterals that we can find u p to simi-
+larity on curves. There are several possibilities to generalise this pr oblem to other
+polygons, and the most natural one seems to consider fixed edge r atios. Suppose
+we are given an non-degenerate planar n-gon. If we take the quotient of the first
+n−1 edges by the last one, we get n−1edge ratios ρ1,...,ρ n−1∈ /CA>0. They are
+characterised by the property that any number of ρ1,...,ρ n−1,1 is smaller than
+the sum of the others.
+Letγ:S1→ /CA∞be a given smoothcurve. We could also let γmap into any
+Riemannian manifold, which would not make a difference by Nash’s embed ding
+theorem. We proceed as in Shnirel’man’s proof of Section 2.2. n-gons that are
+lying counter-clockwise on γare parametrised by Pn. One can measure their edges
+by a test map Pn→ /CAn, make this by an ε-homotopy relative to ∂Pntransversal
+to the vector subspace spanned by the edge length vector of the given polygon, and
+find the solution set Sof alln-gons inP◦
+nwith the given edge ratios as a pre-image,
+which then defines a unique element [ S]∈Ω1(Pn) in the one-dimensional oriented
+bordism group of Pn. The projection onto the first factor induces a homotopy
+equivalence Pn≃S1, hence [ S]∈ /CI. Forγa circle we deduce that [ S] =±1.
+HenceS/\e}atio\slash= 0. This can be interpreted in terms of winding numbers (by the winding
+numberof a component I mean its bordism class in Ω 1(Pn)∼=
+/CI, where fixing this
+isomorphism fixes orientation issues).
+Lemma 4.1. Sis a disjoint union of circles that wind around Pn≃S1and the
+winding numbers add with orientation up to ±1. /square
+If all edges ratios are one, so all edges are equal, then the test ma p is /CIn-
+equivariant. Pnis free, hence the ε-homotopy can also be equivariant, so Sis a/CIn-manifold. The generator of /CInpreserves the orientation of Pnif and only if it
+preserves the orientation of /CAn. The test-space ∆/CAnis the fixed point set of /CAn,
+so /CInacts on it orientation preserving. Hence /CInacts onSorientation preserving,
+which we use in the following lemma.
+Lemma 4.2. Letγ:S1−→ /CA∞be a smoothly embedded curve and let nbe a prime
+power≥3. Then there is a closed one-parameter family S1−→Pnof polygons such
+that10 BENJAMIN MATSCHKE
+(1)each of the polygons are ε-close edge-regular, that is, the edge ratios lie in
+[1−ε,1+ε], and
+(2)this one-parameter family (that is, its image) is /CIn-invariant.
+Proof. /CInacts by permutation on the set of components of S. The lemma just
+claims that there is a fixed point. If there was no fixed point, all orbit s had a
+cardinality divisible by p, wheren=pk. All components in one orbit have the
+same winding number, since /CInacts onSorientation preserving and the induced
+action of /CInon Ω1(Pn) is trivial. Thus the sum of all winding numbers would be
+divisible by p, but it is ±1, contradiction. /square
+Lemma 4.2 has some simple applications.
+Another proof of the smooth Square Peg Problem. Foragivencurve γ,choosea /CI4-
+invariant one-parameter family S1→Pnofε-close rhombi. Then go along this
+family from one of these rhombi to its translate by the generator o f /CI4. What
+happens is that the short diagonal becomes the long diagonal, henc e in the middle
+there was a square. Letting εgo to zero and taking a convergent subsequence of
+theε-close squares finishes the proof. /square
+Corollary 4.3 (A Conjecture of Hadwiger, [Mak05, Thm. 4], [Vr ˇZi08, Thm. 11]) .
+Each knot, that is, a smoothly embedded circle in /CA3, contains four points spanning
+a planar rhombus.
+Proof.As the proof before, but we look at the angle between the triangles of a
+triangulation of the rhombus instead of looking at a diagonal. Somewh ere in the
+middle it hasto be the straightangle(The anglehas to be preventedf rom becoming
+zero, which can be done by a compactness argument). /square
+Corollary 4.4 (Blagojevi´ c–M., [Mat08, Thm. III.6.1]) .Letd1andd2be two
+symmetric distance functions on S1, where d1is given by a smooth embedding
+ofS1into a Riemannian manifold. Then there are three pairwise di stinct points
+onS1forming an equilateral triangle with respect to d1and an isosceles triangle
+with respect to d2. /square
+Lemma 4.2 has the following generalisation to non-prime powers n, which will
+be useful in the next section.ON THE SQUARE PEG PROBLEM AND SOME RELATIVES 11
+Lemma 4.5. Letγ:S1−→ /CA∞be a smoothly embedded curve and let prbe
+a prime power dividing n≥3. We think of /CIpras the subgroup of /CInwithpr
+elements. Let Pbe a polygon whose edge lenths are /CIpr-invariant. Then there is a
+closed one-parameter family S1−→Pnof polygons such that
+(1)each of the polygons have up to a factor and an ε-error the same edge lengths
+asP, and
+(2)this one-parameter family is /CIpr-invariant. /square
+5.Rectangles on Curves
+H. B. Griffiths [Gri91] proved that every smooth planar embedded cir cle circum-
+scribes a rectangle with arbitrary aspect ratio. However there ar e unfortunately
+some errors in his computation concerning orientations (see [Mat08 , Chap. III.7]
+for details), which seem to invalidate the proof. Hence the problem is open:
+Conjecture 5.1 (“Rectangular Peg Problem”) .For all reals r >0, every smooth
+embedding γ:S1→ /CA2contains four points spanning a rectangle of aspect ratio r.
+Since there is not as much symmetry in the Rectangular Peg Problem a s in the
+Square Peg Problem, the symmetry group being /CI2instead of /CI4, the number of
+rectangles of a fixed aspect ratio on curves will be generically even. Hence purely
+topological arguments will not work. But they give some intuition, he re are two
+approaches. Assuming that Conjecture 5.1 admitted a counter-e xample ( γ,r), both
+lemmas derive conclusions that seem to be unintuitive, but more geom etric ideas
+are needed to yield a contradiction.
+Lemma 5.2. Suppose there was a counter-example (γ,r). Then for all ε >0,
+there is a /CI2-invariant one-parameter family S1→P4ofε-close parallelograms
+with aspect ratio in [r−ε,r+ε]and with an odd winding number, such that during
+the whole one-parameter family one of the diagonals stays la rger than the other one.
+Proof.We would like to use Lemma 4.5 with n= 4,pr= 2, and Pa rectangle
+with aspect ratio r. However the solution set Sof quadrilaterals on γthat have the
+desired edge ratios is too large. There exist skewquadrilaterals on γhaving the
+same edge ratios as P, which we do not want in our solution set Ssince they are
+not parallelograms. To solve this problem, we can simply ignore them an d argue
+that all arguments still go through. This turns out to be quite tech nical, but there
+is an easier proof:
+We define another test map,
+g:P4−→ /CA2× /CA
+that maps [ x1,x2,x3,x4] to
+/parenleftbig
+(γ(x1)+γ(x3))−(γ(x2)+γ(x4)),
+(||γ(x1)−γ(x2)||+||γ(x3)−γ(x4)||)−r·(||γ(x2)−γ(x3)||+||γ(x4)−γ(x1)||)/parenrightbig
+.12 BENJAMIN MATSCHKE
+The pre-image ( g|P◦
+4)−1(0) is exactly the set of parallelograms on γof aspect
+ratior. Using a bordism argument, the proof works now exactly as the one of
+Lemma 4.5. /square
+Remark 5.3.In Lemma 5.2, instead of looking at the set of parallelograms with
+aspect ratio r, we might look as well on the set of parallelograms whose diagonals
+intersect in an angle α, where αis the intersection angle of the diagonals in a
+rectangle of aspect ratio r. This gives an analogous lemma, which might be easier
+to deal with geometrically.
+Now we come to the second lemma, which gives similarly an intuition why th e
+Rectangular Peg Problem should hold true.
+Lemma 5.4. Suppose there was a counter-example (γ,r). Then for all ε >0, there
+is a /CI4-invariant one-parameter family S1→P4ofε-close rectangles.
+Proof.Letf:P◦
+4−→/CI4
+/CA4× /CA2be the restricted map (1) from Section 2.2,
+measuring the edges and diagonals.
+First of all we make f /CI4-equivariantly transversal to ∆/CA4×∆/CA2by a small
+δ-homotopy, and let Q:=f−1(∆/CA4×∆/CA2) be the set of all squares (up to an
+δ-error, where δis a function that decreases sufficiently fast near the boundary
+ofP◦
+4). Then we make f /CI4-equivariantly transversal to the /CI4-invariant subspace
+V:={(a,b,a,b,e,e )∈ /CA4× /CA2}by a small δ-homotopy which leaves Qfixed, and
+letR:=f−1(V) be the set of all rectangles on γ(up to an δ-error). If δwas chosen
+small enough, Rconsists only of ε-close rectangles.
+LetRQbe the set of all components of Rthat contain a square. We may assume
+that all these components are circles, otherwise a component wou ld come arbitrary
+close to the boundary of P4, so there would be an ε-closerectangle on it with aspect
+ratior. If we could do this for all ε, then a limit argument would give us a proper
+rectangle of aspect ratio r. So if need be, we choose a smaller εfor which this does
+not happen.
+Ris a one-dimensional /CI4-manifold, so /CI4acts onRQas well. We decompose
+RQ=R1⊎R2⊎R4, whereR1is the set of components with isotropy group /a\}bracketle{t0/a\}bracketri}ht,R2
+with isotropy group /CI2=/a\}bracketle{tε2/a\}bracketri}ht ⊂ /CI4andR4with /CI4. Now we only need to count
+the number of squares on each Ri.
+•♯Q= 4 mod 8, since modulo /CI4it is odd (see Section 2.2).
+•Every component C∈ /CAQcontains an even number of squares, since while
+passing a square the rectangle changes from fat to skinny or vice v ersa (this
+follows from the bijectivity of the differential d fat points in Q).
+•4 divides ♯R1, and every component in R1contains two squares. So the number
+of squares on components of R1is divisible by 8.
+•2 divides ♯R2, and if a component in R2contains a square S, then it contains
+alsoε2·S. When it goes through a square and changes from fat to skinny, th en
+so it does at ε2·S. Hence it has to go through 4 ksquares, k≥1. Thus the
+number of squares on components of R2is divisible by 8.
+•If a component CofR4goes through a square Sand changes from fat to skinny,
+then it alsogoes through ε·Sand changesfrom skinny to fat. That is, in between
+it had to go through an even number of squares, all of which of cour se belong to a
+different /CI4-orbit. Hence the number of square-orbits on Cis odd,♯(Q∩C) = 4
+mod 8.ON THE SQUARE PEG PROBLEM AND SOME RELATIVES 13
+Putting this modulo 8 together, we get ♯R4= 1 mod 2, which is even a bit
+stronger than what is stated in the lemma. /square
+6.Crosspolytopes on Spheres
+H. Guggenheimer [Gug65] proved that any smoothly embedded sphe reSd−1→/CAdcontains the vertices of a regular d-dimensional crosspolytope. However there
+is unfortunately an error in his main lemma (see [Mat08, Chap. III.9] f or details),
+which seems to invalidate the proof. Hence the problem is open:
+Conjecture 6.1 (“Crosspolytopal Peg Problem”) .Every smooth embedding γ:
+Sd−1→ /CAdcontains the vertices of a regular d-dimensional crosspolytope.
+Recently, R. N. Karasev [Kar09] has shown this conjecture to hold true for
+boundaries of non-angular (e.g. smooth) convex bodies, if dis an odd prime power.
+The topological counter-example. Theconjectureingeneralisprobablyverydifficult
+anda solutionwould involvedeeper geometricreasoning,since there is the following
+“topologicalcounter-example”for d= 3. Supposewearegivenasmoothembedding
+Γ :S2→ /CA3. LetG∼=( /CI2)3⋊S3be the symmetry group of the regular octahedron
+andGor⊂Gbethe subgroupoforientationpreservingsymmetries. Gactson(S2)6
+by permuting the coordinates in the same way as it permutes the ver tices of the
+regular octahedron. Let Gact on /CA12by permuting the coordinates in the same
+way as it permutes the edges of the regular octahedron. The subr epresentation
+(∆/CA12)⊥⊂ /CA12is denoted by Y. Let ∆fat
+(S2)6be the space of all 6-tuples in ( S2)6
+that contain at least two equal elements, that is, the fat diagonal. LetBbe a small
+ε-neighborhood of ∆fat
+(S2)6, whereεdepends only on an isotopy of Γ to some nice
+embedding, that we will describe later. Then the complement X:= (S2)6\Bis a
+free compact G-manifold with boundary and
+X≃G{(x1,...,x 6)∈(S2)6|xiare pairwise distinct }= (S2)6\∆fat
+(S2)6.
+Then Γ gives us a test map
+t:X−→GY,
+which measures the edges of the parametrised octahedra modulo /BD= (1,...,1).
+Sinceεwas chosen small, t|∂Xis mapping uniquely up G-homotopy to Y\{0},
+if we change Γ by an isotopy. The solution set Sof regular octahedra on Γ is
+S:=t−1(0). The subset Sor⊂Sof positively oriented octahedra is a part of the
+pre-image t−1(0), and tinduces an isomorphism of Gor-vector bundles over Sor,
+TSor⊕(iSor)∗(X×Y)∼=(iSor)∗(TX),
+whereiSordenotes the inclusion Sor֒→X. ThusSorgives us together with this
+normal data an element [ Sor] in the equivariant normal bordism group (see U.
+Koschorke [Kos81, Chap. 2])
+ΩGor
+1(X,X×Y−TX) = Ω1(X/Gor,X×GorY−T(X/Gor)),14 BENJAMIN MATSCHKE
+which is well-defined, since isotopies of Γ change Sonly by a normal bordism that
+stays away from the ∂Xofεwas chosen small enough, and components of octahe-
+dra of different orientation are always separated from each other . In Koschorke’s
+notation, [ Sor] is the obstruction
+/tildewideω1(
+/CA
+∼,X×GorY,(id∂X,t|∂X)/Gor),
+where
+/CA
+∼is the trivial line bundle.
+Theorem 6.2. The above defined [Sor]is zero. Hence
+[S]∈ΩG
+1(X,X×Y−TX)
+is zero as well.
+In particular, the test map t can be deformed G-equivariantly relative to ∂Xto
+a mapt′, such that 0/\e}atio\slash∈t′(X).
+The map t′is what I call a topological counter-example.
+Sketch of Proof. To construct a nice representative for [ Sor] we take the standard
+2-sphere and scale it down linearly along the z-axis of /CA3. This is our Γ and we let
+tandSbe the corresponding test map and solution set, respectively. Sis a disjoint
+union of 16 =1
+3·♯Gcircles. One octahedron on the scaled sphere looks as follows
+(one looks along the z-axis):
+If we rotate it around the z-axis then we get up to symmetry all octahedra on Γ.
+TheG-bundles X×YandTXareG-orientable, therefore the relevant part of
+Koschorke’s exact sequence [Kos81, Thm. 9.3] becomes
+H2(X/Gor; /CI)→ /CI2→Ω1(X/Gor,X×GorY−T(X/Gor))
+→H1(X/Gor; /CI)→0.
+It is not difficult to see that the image of [ Sor] inH1(X/Gor; /CI) =H1(Gor; /CI) is
+zero. Thisisbecausethe120degreerotationofaregularoctahed ronaroundtheline
+connecting the midpoints of two opposite triangles is an element of th e commutator
+ofGor. It requires much more visualisation to see that [ Sor] is in fact the image
+of the generator of /CI2. The hard part is to show that /CI2unfortunately lies in the
+image of H2(X/Gor; /CI), which I could manage to do only with a very long program.
+It finds that H2(X/Gor; /CI)∼=
+/CI4×( /CI2)3, where one can choose the generators such
+that the first three map to zero and the last one to the generator of /CI2.
+TheGor-null-bordism of Sorcan be extended to a G-null-bordism of S. By
+Theorem 3.1 of U. Koschorke [Kos81], we can extend the section as s tated. /squareON THE SQUARE PEG PROBLEM AND SOME RELATIVES 15
+Remarks to the algorithm. An economical S6-CW-complex structure on ( S2)6is
+based on an S6-cell decomposition of /CA2of V. A. Vassiliev [Vas94], which has few
+high dimensional cells. ∆fat
+(S2)6is a subcomplex, so one can compute H2(X/Gor)∼=
+H10((S2)6/Gor,(∆fat
+(S2)6)/Gor). The Smith normal form is used to compute this cel-
+lularcohomologyandtheLLL-algorithmtochoosenicegenerators. Theimagein /CI2
+is determined by computing second Stiefel-Whitney classes, which I im plemented
+as obstruction classes.
+Acknowledgements
+IamgratefulforveryfruitfuldiscussionswithPavleBlagojevi´ c ,JasonCantarella,
+Carsten Schultz, John Sullivan, Julia Ruscher, Rade ˇZivaljevi´ c, and in particular
+G¨ unter Ziegler.
+References
+[BlZi08b] P. V. M. Blagojevi´ c, G. M. Ziegler. Tetrahedra on deformed spheres and integral group
+cohomology , arXiv:0808.3841v1, (2008), 8 pages
+[CoFl64] P. E. Conner, E. E. Floyd. Differentiable Periodic Maps , Ergebnisse Series vol. 33,
+Springer-Verlag, (1964)
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+in preparation.
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+No 3, (1990), 647-672
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+proach, Lect. Notes Math. 847, Springer Verlag, (1981)
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+[Mat08] B. Matschke. Equivariant Topology and Applications , Diploma Thesis, Chap. III,
+www.math.tu-berlin.de/ ∼matschke/DiplomaThesis.pdf, (2008)
+[Mat09] B. Matschke. Extended abstract to Square Pegs and Beyond , Oberwolfach Reports No.
+2 (2009), 51-54
+[Nie92] M.J.Nielsen. Triangles inscribed in simple closed curves , Geometriae Dedicata 43, Kluwer
+Acad. Publisher, (1992), 291-297
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+[Shn44] L. G. Shnirel’man. On some geometric properties of closed curves (in Russian), Usp.
+Mat. Nauk 10, (1944), 34-44
+[Str89] W. Stromquist. Inscribed squares and square-like quadrilaterals in close d curves , Mathe-
+matika 36, (1989), 187-197
+[Toe11] O.Toeplitz. Ueber einige Aufgaben der Analysis situs , Verhandlungen der Schweizerischen
+Naturforschenden Gesellschaft in Solothurn 4 (1911), p. 19 7
+[Vas94] V. A. Vassiliev. Complements of discriminants of smooth maps: topology and a pplica-
+tions, Transl. Math. Monographs 98, Providence, RI: Amer. Math. S oc., (1994), 265 pages
+[VrˇZi08] S. Vre´ cica, R. T. ˇZivaljevi´ c. Fulton-MacPherson compactification, cyclohedra, and the
+polygonal pegs problem , arXiv:0810.1439, (2008), 23 pages
+Benjamin Matschke
+Institut f¨ ur Mathematik, MA 6–2
+Technische Universit¨ at Berlin, 10623 Berlin, Germany
+benjaminmatschke@googlemail.com
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+arXiv:1001.0188v5  [math.ST]  20 Mar 2013Bernoulli 19(2), 2013, 521–547
+DOI:10.3150/11-BEJ410
+Least squares after model selection in
+high-dimensional sparse models
+ALEXANDRE BELLONI1and VICTOR CHERNOZHUKOV2
+1100 Fuqua Drive, Durham, North Carolina 27708, USA. E-mail: abn5@duke.edu
+250 Memorial Drive, Cambridge, Massachusetts 02142, USA. E- mail:vchern@mit.edu
+In this article we study post-model selection estimators th at apply ordinary least squares (OLS)
+to the model selected by first-step penalized estimators, ty pically Lasso. It is well known that
+Lasso can estimate the nonparametric regression function a t nearly the oracle rate, and is thus
+hard to improve upon. We show that the OLS post-Lasso estimat or performs at least as well as
+Lasso in terms of the rate of convergence, and has the advanta ge of a smaller bias. Remarkably,
+this performance occurs even if the Lasso-based model selec tion “fails” in the sense of missing
+some components of the “true” regression model. By the “true ” model, we mean the best
+s-dimensional approximation to the nonparametric regressi on function chosen by the oracle.
+Furthermore, OLS post-Lasso estimator can perform strictl y better than Lasso, in the sense of
+a strictly faster rate of convergence, if the Lasso-based mo del selection correctly includes all
+components of the “true” model as a subset and also achieves s ufficient sparsity. In the extreme
+case, when Lasso perfectly selects the “true” model, the OLS post-Lasso estimator becomes
+the oracle estimator. An important ingredient in our analys is is a new sparsity bound on the
+dimension of the model selected by Lasso, which guarantees t hat this dimension is at most of
+the same order as the dimension of the “true” model. Our rate r esults are nonasymptotic and
+hold in both parametric and nonparametric models. Moreover , our analysis is not limited to the
+Lasso estimator acting as a selector in the first step, but als o applies to any other estimator,
+for example, various forms of thresholded Lasso, with good r ates and good sparsity properties.
+Our analysis covers both traditional thresholding and a new practical, data-driven thresholding
+scheme that induces additional sparsity subject to maintai ning a certain goodness of fit. The
+latter scheme has theoretical guarantees similar to those o f Lasso or OLS post-Lasso, but it
+dominates those procedures as well as traditional threshol ding in a wide variety of experiments.
+Keywords: Lasso; OLS post-Lasso; post-model selection estimators
+1. Introduction
+In this work, we study post-model selection estimators for linear r egression in high-
+dimensional sparse models (hdsms). In such models, the overall nu mber of regressors p
+is very large, possibly much larger than the sample size n. However, there are s=o(n)
+regressors that capture most of the impact of all covariates on t he response variable.
+This is an electronic reprint of the original article publis hed by the ISI/BSinBernoulli ,
+2013, Vol. 19, No. 2, 521–547 . This reprint differs from the original in pagination and
+typographic detail.
+1350-7265 c/circlecopyrt2013 ISI/BS2 A. Belloni and V. Chernozhukov
+hdsms [9,16] have emerged to deal with many new applications arising in biometrics ,
+signal processing, machine learning, econometrics, and other are as of data analysis where
+high-dimensional data sets have become widely available.
+Several authors have investigated estimation of hdsms, focusing primarily on mean
+regression with the ℓ1-norm acting as a penalty function [ 4,6–9,12,16,22,24,26]. The
+results of [ 4,6–8,12,16,24,26] demonstrate the fundamental result that ℓ1-penalized
+least squares estimators achieve the rate/radicalbig
+s/n√logp, which is very close to the oracle
+rate/radicalbig
+s/nachievable when the true model is known. [ 12,22] demonstrated a similar
+fundamental result on the excess forecasting error loss under b oth quadratic and non-
+quadratic loss functions. Thus the estimator can be consistent an d can have excellent
+forecasting performance even under very rapid, nearly exponen tial growth of the total
+number of regressors p. In addition, [ 3] investigated the ℓ1-penalized quantile regression
+processand obtained similar results. See [ 4,6–8,11,13,14,17] for many other interesting
+developments and a detailed review of the existing literature.
+In this article we derive theoretical properties of post-model sele ction estimators that
+apply ordinary least squares (OLS) to the model selected by first- step penalized esti-
+mators, typically Lasso. It is well known that Lasso can estimate th e mean regression
+function at nearly the oracle rate, and thus is hard to improve on. W e show that OLS
+post-Lasso can perform at least as well as Lasso in terms of the ra te of convergence, and
+has the advantage of a smaller bias. This nice performance occurs e ven if the Lasso-based
+model selection “fails” in the sense of missing some components of th e “true” regression
+model. (By the “true” model, we mean the best s-dimensional approximation to the re-
+gression function chosen by the oracle.) The intuition for this result is that Lasso-based
+model selection omits only those components with relatively small coe fficients. Further-
+more, OLS post-Lasso can perform better than Lasso in the sens e of a strictly faster rate
+of convergence, if the Lasso-based model correctly includes all c omponents of the “true”
+model as a subset and is sufficiently sparse. Of course, in the extre me case, when Lasso
+perfectly selects the “true” model, the OLS post-Lasso estimato r becomes the oracle
+estimator.
+Importantly,our rateanalysisis not limited to the Lassoestimatorin the first step, but
+appliestoawidevarietyofotherfirst-stepestimators,including,f orexample,thresholded
+Lasso, the Dantzig selector, and their various modifications. We pr ovide generic rate
+results that cover any first-step estimator for which a rate and a sparsity bound are
+available. We also present a generic result from using thresholded La sso as the first-step
+estimator, where thresholding can be performed by a traditional t hresholding scheme
+(t-Lasso) or by a new fitness-thresholding scheme that we introd uce here (fit-Lasso). The
+new thresholding scheme induces additional sparsity subject to ma intaining a certain
+goodness of fit in the sample and is completely data-driven. We show t hat OLS post-fit
+Lasso estimator performs at least as well as the Lasso estimator, but can be strictly
+better under good model selection properties.
+Finally, we conduct a series of computational experiments and find t hat the results
+confirm our theoretical findings. Figure 1provides a brief graphical summary of our
+theoretical results, showing how the empirical risk of various estim ators change with the
+signal strength C(coefficients of relevant covariates are set equal to C). For very lowLeast squares after model selection 3
+Figure 1. This figure plots the performance of the estimators listed in the text under the
+equicorrelated design for the covariates xi∼N(0,Σ), Σ jk=1/2 ifj/negationslash=k. The number of regres-
+sors isp= 500, and the sample size is n= 100 with 1000 simulations for each level of signal
+strength C. In each simulation, there are 5 relevant covariates whose c oefficients are set equal
+to the signal strength C, and the variance of the noise is set to 1.
+signal levels, all estimators perform similarly. When the signal stren gth is intermediate,
+OLS post-Lasso and OLS post-fit Lasso significantly outperform L asso and the OLS
+post-t Lasso estimators. However, we find that the OLS post-fit Lasso outperforms OLS
+post-Lasso whenever Lasso does not produce very sparse solut ions, which occurs if the
+signal strength level is not low. For large levels of signal, OLS post-fi t Lasso and OLS
+post-t Lasso perform very well, improving on Lasso and OLS post-L asso. Thus, the main
+message here is that OLS post-Lasso and OLS post-fit Lasso perf orm at least as well as
+Lasso and sometimes a lot better.
+To the best of our knowledge, this article is the first to establish the aforementioned
+rate results on OLS post-Lasso and the proposed OLS post-fitne ss-thresholded Lasso in
+the mean regression problem. Our analysis builds on the ideas of [ 3], who established the
+properties of postpenalized procedures for the related, but diffe rent problem of median
+regression. Our analysis also builds on the fundamental results of [ 4] and the other works
+cited above that established the properties of the first-step Las so-type estimators. An
+important ingredient in our analysis is a new sparsity bound on the dime nsion of the
+model selected by Lasso, which guarantees that this dimension is at most of the same
+order as the dimension of the “true” model. This result builds on some inequalities
+for sparse eigenvalues and reasoning previously given by [ 3] in the context of median4 A. Belloni and V. Chernozhukov
+regression. Our sparsity bounds for Lasso improve on the analogo us bounds of [ 4] and
+are comparable to the bounds of [ 26] obtained under a larger penalty level. We also rely
+on the maximal inequalities of [ 26] to provide primitive conditions for the sharp sparsity
+bounds to hold.
+The article is organized as follows. Section 2reviews the model and discusses the
+estimators. Section 3revisits some benchmark results of [ 4] for Lasso, allowing for a
+data-driven choice of penalty level, develops an extension of model selection results
+of [13] to the nonparametric case, and derives a new sparsity bound for Lasso. Sec-
+tion4presents a generic rate result on OLS post-model selection estima tors. Section 5
+applies the generic results to the OLS post-Lasso and the OLS post -thresholded Lasso
+estimators.The Appendix containsmain proofs,and the supplemental article[ 2] contains
+auxiliary proofs, as well as the results of our computational exper iments.
+Notation
+When making asymptotic statements, we assume that n→∞andp=pn→∞, and also
+allow for s=sn→∞. In what follows, all parameter values are indexed by the sample
+sizen, but we omit the index whenever this omission will not cause confusion . We use
+the notation ( a)+=max{a,0},a∨b=max{a,b}, anda∧b=min{a,b}. Theℓ2-norm is
+denoted by /bar⌈bl·/bar⌈bl, theℓ1-norm is denoted by /bar⌈bl·/bar⌈bl1, theℓ∞-norm is denoted by /bar⌈bl·/bar⌈bl∞, and
+theℓ0-norm/bar⌈bl·/bar⌈bl0denotes the number of nonzero components of a vector. Given a v ector
+δ∈Rpand a set of indices T⊂{1,...,p}, we denote by δTthe vector in Rpin which
+δTj=δjifj∈TandδTj= 0 ifj /∈T. The cardinality of Tis denoted by |T|. Given
+a covariate vector xi∈Rp, we letxi[T] denote the vector {xij,j∈T}. The symbol E[ ·]
+denotes the expectation. We also use standard empirical process notation
+En[f(z•)]:=n/summationdisplay
+i=1f(zi)/nandGn(f(z•)):=n/summationdisplay
+i=1(f(zi)−E[f(zi)])/√n.
+We denote the L2(Pn) norm by /bar⌈blf/bar⌈blPn,2=(En[f2
+•])1/2. Given covariate values x1,...,x n,
+we define the prediction norm of a vector δ∈Rpas/bar⌈blδ/bar⌈bl2,n={En[(x′
+•δ)2]}1/2=
+(δ′En[x•x′
+•]δ)1/2. We use the notation a/lessorsimilarbto denote a≤Cbfor some constant C >0
+that does not depend on n(and thus does not depend on quantities indexed by nlikep
+ors), anda/lessorsimilarPbto denote a=OP(b). For an event A, we say that Awp→1 whenAoc-
+curswithprobabilityapproaching1as nincreases.Inaddition,wewrite ¯ c=(c+1)/(c−1)
+for a chosen constant c>1.
+2. Setting, estimators, and conditions
+2.1. Setting
+Condition M. We have data {(yi,zi),i=1,...,n}such that for each n,
+yi=f(zi)+ǫi, ǫ i∼N(0,σ2),i=1,...,n, (2.1)Least squares after model selection 5
+whereyiare the outcomes, ziare vectors of fixed regressors, and ǫiare i.i.d. errors. Let
+P(zi)be a given p-dimensional dictionary of technical regressors with resp ectzi, that is,
+a p-vector of transformation of zi, with components
+xi:=P(zi)
+of the dictionary normalized so that
+En[x2
+•j]=1 forj=1,...,p.
+In making asymptotic statements, we assume that n→∞andp=pn→∞, and that all
+parameters of the model are implicitly indexed by n.
+Wewouldliketoestimate thenonparametricregressionfunction fatthe designpoints,
+namely the values fi=f(zi) fori= 1,...,n. To set up the estimation and define a
+performance benchmark, we consider the following oracle risk minimiz ation program:
+min
+0≤k≤p∧nc2
+k+σ2k
+n, (2.2)
+where
+c2
+k:= min
+/bardblβ/bardbl0≤kEn[(f•−x′
+•β)2]. (2.3)
+Note that c2
+k+σ2k/nis an upper bound on the risk of the best k-sparse least squares
+estimator, that is, the best estimator among all least squares est imators that use kout
+ofpcomponents of xito estimate fifori=1,...,n. The oracle program ( 2.2) chooses
+the optimal value of k. Letsbe the smallest integer among these optimal values, and let
+β0∈arg min
+/bardblβ/bardbl0≤sEn[(f•−x′
+•β)2]. (2.4)
+We callβ0the oracle target value, T:=support( β0) the oracle model, s:=|T|=/bar⌈blβ0/bar⌈bl0
+the dimension of the oracle model, and x′
+iβ0the oracle approximation to fi. The latter is
+our intermediary target, which is equal to the ultimate target fiup to the approximation
+error
+ri:=fi−x′
+iβ0.
+Ifweknew T,thenwecouldsimplyuse xi[T]asregressorsandestimate fi,fori=1,...,n,
+using the least squares estimator, achieving the risk of at most
+c2
+s+σ2s/n,
+which we call the oracle risk. Because Tis not known, we estimate Tusing Lasso-type
+methods and analyze the properties of post-model selection least squares estimators,
+accounting for possible model selection mistakes.6 A. Belloni and V. Chernozhukov
+Remark 2.1 (The oracle program). Note that if argmin is not unique in the problem
+(2.4), then it suffices to select one of the values in the set of argmins. Su pplemental
+article [2] provides a more detailed discussion of the oracle problem. The idea o f using
+oracle problems such as ( 2.2) for benchmarking the performance follows its previous uses
+in the literature (see, e.g., [ 4], Theorem 6.1, where an analogous problem appears in
+upper bounds on performance of Lasso).
+Remark 2.2 (A leading special case). When contrasting the performance of Lasso
+andOLSpost-LassoestimatorsinRemarks 5.1and5.2givenlater,wementionabalanced
+case where
+c2
+s/lessorsimilarσ2s/n, (2.5)
+which says that the oracle program ( 2.2) is able to balance the norm of the bias squared
+to be not much larger than the variance term σ2s/n. This corresponds to the case where
+the approximation error bias does not dominate the estimation erro r of the oracle least
+squaresestimator,sothatthe oraclerateofconvergencesimplifi esto/radicalbig
+s/n,asmentioned
+in theIntroduction .
+2.2. Model selectors based on Lasso
+Given the large number of regressors p>n, some regularization or covariate selection is
+needed to obtain consistency. The Lasso estimator [ 19], defined as follows, achieves both
+tasks by using the ℓ1penalization:
+/hatwideβ∈arg min
+β∈Rp/hatwideQ(β)+λ
+n/bar⌈blβ/bar⌈bl1,where/hatwideQ(β)=En[(y•−x′
+•β)2], (2.6)
+andλis the penalty level, the choice of which is described later. If the solut ion is not
+unique, then we pick any solution with minimum support. The Lasso is of ten used as an
+estimator, and most often only as a model selection device, with the model selected by
+Lasso given by
+/hatwideT:=support(/hatwideβ).
+Moreover, we let /hatwidem:=|/hatwideT\T|denote the number of components outside Tselected by
+Lasso and let /hatwidefi=x′
+i/hatwideβ,i=1,...,n, denote the Lasso estimate of fi,i=1,...,n.
+Often, additional thresholding is applied to remove regressors with small estimated
+coefficients, defining the so-called “thresholded” Lasso estimator ,
+/hatwideβ(t)=(/hatwideβj1{|/hatwideβj|>t},j=1,...,p), (2.7)
+wheret≥0 is the thresholding level. The corresponding selected model is then
+/hatwideT(t):=support(/hatwideβ(t)).
+Note that, when setting t=0, we have /hatwideT(t)=/hatwideT, so Lasso is a special case of thresholded
+Lasso.Least squares after model selection 7
+2.3. Post-model selection estimators
+Given the foregoing, all of our post-model selection estimators or OLS post-Lasso esti-
+mators will take the form
+/tildewideβt=arg min
+β∈Rp/hatwideQ(β):βj=0 for each j∈/hatwideTc(t). (2.8)
+Thatis, giventhat the model selectedathreshold Lasso /hatwideT(t), including the Lasso’smodel
+/hatwideT(0) as a special case, the post-model selection estimator applies O LS to the selected
+model.
+Along with the case of t=0, we also consider the following choices for the threshold
+level:
+traditional threshold (t): t>ζ= max
+1≤j≤p|/hatwideβj−β0j|,
+fitness-based threshold (fit): t=tγ:=max
+t≥0{t:/hatwideQ(/tildewideβt)−/hatwideQ(/hatwideβ)≤γ},(2.9)
+whereγ≤0, and|γ|is the gain of the in-sample fit allowed relative to Lasso.
+As discussed in Section 3.2, the standard thresholding method is particularly ap-
+pealing in models in which oracle coefficients β0are well separated from 0. How-
+ever, this scheme may perform poorly in models with oracle coefficient s not well sep-
+arated from 0 and in nonparametric models. Indeed, even in parame tric models with
+many small but nonzero true coefficients, thresholding the estimat es too aggressively
+may result in large goodness-of-fit losses and, consequently, slow rates of convergence
+and even inconsistency for the second-step estimators. This issu e directly motivates
+our new goodness-of-fit based thresholding method, which sets a s many small co-
+efficient estimates as possible to 0, subject to maintaining a certain g oodness-of-fit
+level.
+Depending on how we select the threshold, we consider three types of post-model
+selection estimators:
+OLS post-Lasso: /tildewideβ0(t=0),
+OLS post-t Lasso: /tildewideβt(t>ζ),
+OLS post-fit Lasso: /tildewideβtγ(t=tγ).(2.10)
+The first estimator is defined by OLS applied to the model selected by Lasso, also
+called Gauss-Lasso; the second, by OLS applied to the model select ed by the thresh-
+olded Lassol and the third, by OLS applied to the model selected by fi tness-thresholded
+Lasso.
+The main purpose of this article is to derive the properties of the pos t-model selection
+estimators ( 2.10). If model selection works perfectly, which is possible only under ra ther
+special circumstances, then the post-model selection estimator sare the oracleestimators,
+whose properties are well known. However, of much more general interest is the case
+when model selection does not work perfectly, as occurs for many designs of interest in
+applications.8 A. Belloni and V. Chernozhukov
+2.4. Choice and computation of penalty level for Lasso
+The key quantity in the analysis is the gradient of /hatwideQat the true value,
+S=2En[x•ǫ•].
+This gradientis the effective “noise”in the problem that should be dom inated by the reg-
+ularization. However, we would like to make the bias as small as possible . This reasoning
+suggests choosing the smallest penalty level λpossible to dominate the noise, namely
+λ≥cn/bar⌈blS/bar⌈bl∞with probability at least 1 −α, (2.11)
+where probability 1 −αneeds to be close to 1 and c>1. Therefore, we propose setting
+λ=c′/hatwideσΛ(1−α|X) for some fixed c′>c>1, (2.12)
+where Λ(1 −α|X) is the (1 −α) quantile of n/bar⌈blS/σ/bar⌈bl∞, and/hatwideσis a possibly data-driven
+estimate of σ. Note that the quantity Λ(1 −α|X) is independent of σand can be easily
+approximated by simulation. We refer to this choice of λas the data-driven choice,
+reflecting the dependence of the choice on the design matrix X= [x1,...,x n]′and a
+possibly data-driven /hatwideσ. Note that the proposed ( 2.12) is sharper than c′/hatwideσ2/radicalbig
+2nlog(p/α)
+typically used in the literature. We impose the following conditions on /hatwideσ.
+Condition V. The estimated /hatwideσobeys
+ℓ≤/hatwideσ/σ≤uwith probability at least 1−τ,
+where0<ℓ≤1and1≤uand0≤τ <1are constants possibly dependent on n.
+We can construct a /hatwideσthat satisfies this condition under mild assumptions, as follows.
+First, set/hatwideσ=/hatwideσ0, where/hatwideσ0is an upper bound on σthat is possibly data-driven, for
+example, the sample standard deviation of yi. Second, compute the Lasso estimator
+based on this estimate and set /hatwideσ2=/hatwideQ(/hatwideβ). We demonstrate that /hatwideσconstructed in this
+way satisfies Condition Vand characterize quantities uandℓandτin the supplemental
+article [2]. We can iterate on the last step a bounded number of times. We also c an use
+OLS post-Lasso for this purpose.
+2.5. Choices and computation of thresholding levels
+Our analysis covers a wide range of possible threshold levels. Here, h owever, we propose
+some basic options that give both good finite-sample and theoretica l results. In the
+traditional thresholding method, we can set
+t=˜cλ/n, (2.13)Least squares after model selection 9
+for some ˜ c≥1. This choice is theoretically motivated by Section 3.2, which presents the
+perfect model selection results, where under some conditions, ζ≤˜cλ/n. This choice also
+leads to near-oracleperformance of the resulting post-model se lection estimator. Regard-
+ing the choice of ˜ c, we note that setting ˜ c=1 and achieving ζ≤λ/nis possible based on
+the results of Section 3.2if the empirical Gram matrix is orthogonal and approximation
+errorcsvanishes. Thus, ˜ c=1 is the least aggressive traditional thresholding that can be
+performed under conditions of Section 3.2. (Also note that ˜ c=1 has performed better
+than ˜c>1 in our computational experiments.)
+Our fitness-based threshold tγrequires specification of the parameter γ. The simplest
+choice delivering near-oracle performance is γ=0; this choice leads to the sparsest post-
+model selection estimator that has the same in-sample fit as Lasso. However, we prefer
+to set
+γ=/hatwideQ(/tildewideβ0)−/hatwideQ(/hatwideβ)
+2<0, (2.14)
+where/tildewideβ0is the OLS post-Lasso estimator. The resulting estimator is sparse r and pro-
+ducesabetterin-samplefitthanLasso.Thischoicealsoresultsinne ar-oracleperformance
+and leads to the best performance in computational experiments. Note also that for any
+γ, we can compute tγby a binary search over t∈sort{|/hatwideβj|,j∈/hatwideT}, where sort is the sort-
+ing operator. This is the case because the final estimator depends only on the selected
+support, not on the specific value of tused. Therefore, because there are at most |/hatwideT|
+different values of tto be tested, using a binary search, we can compute tγexactly by
+running at most ⌈log2|/hatwideT|⌉OLS problems.
+2.6. Conditions on the design
+For the analysis of Lasso, we use the following restricted eigenvalue condition on the
+empirical Gram matrix:
+Condition (RE (¯ c)).For a given ¯c≥0,
+κ(¯c):= min
+/bardblδTc/bardbl1≤¯c/bardblδT/bardbl1,δ/\egatio\slash=0√s/bar⌈blδ/bar⌈bl2,n
+/bar⌈blδT/bar⌈bl1>0.
+This condition is a variant of the restricted eigenvalue condition intro duced by [ 4],
+which is known to be quite general and plausible (see [ 4] for related conditions).
+For the analysis of post-model selection estimators, we use the fo llowing restricted
+sparse eigenvalue condition on the empirical Gram matrix:
+Condition (RSE (m)).For a given m<n,
+/tildewideκ(m)2:= min
+/bardblδTc/bardbl0≤m,δ/\egatio\slash=0/bar⌈blδ/bar⌈bl2
+2,n
+/bar⌈blδ/bar⌈bl2>0, φ(m):= max
+/bardblδTc/bardbl0≤m,δ/\egatio\slash=0/bar⌈blδ/bar⌈bl2
+2,n
+/bar⌈blδ/bar⌈bl2.10 A. Belloni and V. Chernozhukov
+Condition RSE(m)depends on Tand can be viewed as an extension of the restricted
+isometry condition [ 9]. Heremdenotes the restriction on the number of nonzero compo-
+nentsoutsidethe support T.The standardconcept of(unrestricted) m-sparseeigenvalues
+corresponds to the restricted sparse eigenvalues when T=∅(see, e.g., [ 4]). It is con-
+venient to define the following condition number associated with the e mpirical Gram
+matrix:
+µ(m)=/radicalbig
+φ(m)
+/tildewideκ(m). (2.15)
+The following lemma demonstrates the plausibility of the foregoing con ditions for the
+case where the values xi,i=1,...,n, have been generated as a realization of the random
+sample; there are other primitive conditions as well. In this case, the empirical restricted
+sparse eigenvalues are bounded away from 0 and from above, so th at (2.15) is bounded
+from above with high probability. The lemma assumes as a primitive cond ition that
+the sparse eigenvalues of the population Gram matrix bounded away from zero and
+from above. The lemma allows for many standard bounded dictionarie s that arise in the
+nonparametric estimation, for example, regression splines, ortho gonal polynomials, and
+trigonometric series (see [ 10,20,21,25]). Similar results are known to hold for standard
+Gaussian regressors as well [ 26].
+Lemma 1 (Plausibility of REandRSE).Suppose that ˜xi,i=1,...,n, are i.i.d.
+mean-zero vectors, such that the population Gram matrix E[˜x˜x′]has all of the diagonal
+elements equal to 1, and
+0<κ2≤min
+1≤/bardblδ/bardbl0≤slognδ′E[˜x˜x′]δ
+/bar⌈blδ/bar⌈bl≤max
+1≤/bardblδ/bardbl0≤slognδ′E[˜x˜x′]δ
+/bar⌈blδ/bar⌈bl≤ϕ<∞.
+Definexias a normalized form of ˜xi, namely xij= ˜xij/(En[˜x2
+•j])1/2. Suppose that
+max
+1≤i≤n/bar⌈bl˜xi/bar⌈bl∞≤Kna.s. and K2
+nslog2(n)log2(slogn)log(p∨n)=o(nκ4/ϕ).
+Then, for any m+s≤slogn, the restricted sparse eigenvalues of the empirical Gram
+matrix obey the following bounds:
+φ(m)≤4ϕ,/tildewideκ(m)2≥κ2/4andµ(m)≤4√ϕ/κ,
+with probability approaching 1 as n→∞.
+3. Results on Lasso as an estimator and model selector
+The properties of the post-model selection estimators depend cr ucially on both the esti-
+mation and model selection properties of Lasso. In this section we d evelop the estimation
+properties of Lasso under the data-dependent penalty level, ext ending the results of [ 4],
+and also develop the model selection properties of Lasso for nonpa rametric models, gen-
+eralizing the results of [ 13] to the nonparametric case.Least squares after model selection 11
+3.1. Estimation properties of Lasso
+The following theorem describes the main estimation properties of La sso under the data-
+driven choice of the penalty level.
+Theorem 1 (Performance bounds for Lasso under data-driven p enalty). Sup-
+pose that Conditions MandRE(¯c)hold for ¯c=(c+1)/(c−1). Ifλ≥cn/bar⌈blS/bar⌈bl∞, then
+/bar⌈bl/hatwideβ−β0/bar⌈bl2,n≤/parenleftbigg
+1+1
+c/parenrightbiggλ√s
+nκ(¯c)+2cs.
+Moreover, suppose that Condition Vholds. Under the data-driven choice ( 2.12), for
+c′≥c/ℓ, we have λ≥cn/bar⌈blS/bar⌈bl∞with probability at least 1−α−τ, so that with at least the
+same probability,
+/bar⌈bl/hatwideβ−β0/bar⌈bl2,n≤(c′+c′/c)√s
+nκ(¯c)σuΛ(1−α|X)+2cs,whereΛ(1−α|X)≤/radicalbig
+2nlog(p/α).
+If in addition RE(2¯c)holds, then
+/bar⌈bl/hatwideβ−β0/bar⌈bl1≤/parenleftbigg(1+2¯c)√s
+κ(2¯c)/bar⌈bl/hatwideβ−β0/bar⌈bl2,n/parenrightbigg
+∨/parenleftbigg/parenleftbigg
+1+1
+2¯c/parenrightbigg2c
+c−1n
+λc2
+s/parenrightbigg
+.
+This theorem extends the result of [ 4] by allowing for a data-driven penalty level and
+deriving the rates in ℓ1-norm. These results may be of independent interest and are
+necessary for the subsequent results.
+Remark 3.1. Furthermore, a performance bound for the estimation of the reg ression
+function follows from the relation
+|/bar⌈bl/hatwidef−f/bar⌈blPn,2−/bar⌈bl/hatwideβ−β0/bar⌈bl2,n|≤cs, (3.1)
+where/hatwidefi=x′
+i/hatwideβis the Lasso estimate of the regression function fevaluated at zi. It is
+interesting to know some lower bounds on the rate, which follow from Karush–Kuhn–
+Tucker conditions for Lasso (see equation ( A.1) in theAppendix ):
+/bar⌈bl/hatwidef−f/bar⌈blPn,2≥(1−1/c)λ/radicalBig
+|/hatwideT|
+2n/radicalbig
+φ(/hatwidem),
+where/hatwidem=|/hatwideT\T|. We note that a similar lower bound was first derived by [ 15] with
+φ(p) instead of φ(/hatwidem).
+The preceding theorem and discussion imply the following useful asym ptotic bound on
+the performance of the estimators.12 A. Belloni and V. Chernozhukov
+Corollary1 (Asymptotic bounds on performance of Lasso). Under the conditions
+of Theorem 1, if
+φ(/hatwidem)/lessorsimilar1, κ(¯c)/greaterorsimilar1, µ(/hatwidem)/lessorsimilar1,log(1/α)/lessorsimilarlogp,
+(3.2)
+α=o(1), u/ℓ /lessorsimilar1andτ=o(1)
+hold asngrows, then we have
+/bar⌈bl/hatwidef−f/bar⌈blPn,2/lessorsimilarPσ/radicalbigg
+slogp
+n+cs.
+Moreover, if |/hatwideT|/greaterorsimilarPs– in particular, if T⊆/hatwideTwith probability going to 1– then we have
+/bar⌈bl/hatwidef−f/bar⌈blPn,2/greaterorsimilarPσ/radicalbigg
+slogp
+n.
+In Lemma 1we established fairly general sufficient conditions for the first thre e rela-
+tions in ( 3.2) to hold with high probability as ngrows, when the design points z1,...,zn
+are generated as a random sample. The remaining relations are mild co nditions on the
+choice of αand the estimation of σthat are used in the definition of the data-driven
+choice (2.12) of the penalty-level λ.
+It follows from the corollary that as long as κ(¯c) is bounded away from 0, Lasso with
+data-driven penalty estimates the regression function at a near- oracle rate. The second
+part of the corollary generalizes to the nonparametric case the low er bound obtained for
+Lassoby[ 15].Itshowsthattheratecannotbeimprovedingeneral.Weusethea symptotic
+rates of convergence to compare the performance of Lasso and the post-model selection
+estimators.
+3.2. Model selection properties of Lasso
+Our main results do not require that the first-step estimators like L asso perfectly select
+the “true” oracle model. In fact, we are specifically interested in th e most common cases,
+where these estimators do not perfectly select the true model. Fo r these cases, we prove
+thatpost-modelselectionestimatorssuchasOLSpost-Lassoac hievenear-oraclerateslike
+those of Lasso. However, in some special cases where perfect mo del selection is possible,
+these estimators can achieve the exact oracle rates, and thus ca n be even better than
+Lasso.In this section wedescribe these veryspecial casesin which perfect model selection
+is possible.
+Theorem 2 (Some conditions for perfect model selection in no nparametric
+settings). Suppose that Condition Mholds.
+(1) If the coefficients are well separated from 0, that is,
+min
+j∈T|β0j|>ζ+t,for some t≥ζ:= max
+j=1,...,p|/hatwideβj−β0j|,Least squares after model selection 13
+then the true model is a subset of the selected model, T:=support( β0)⊆/hatwideT:=support(/hatwideβ).
+Moreover, Tcan be perfectly selected by applying level tthresholding to /hatwideβ, that is,
+T=/hatwideT(t).
+(2) In particular, if λ≥cn/bar⌈blS/bar⌈bl∞and there is a constant U >5¯csuch that the empirical
+Gram matrix satisfies |En[x•jx•k]|≤1/(Us)for all1≤j <k≤p, then
+ζ≤λ
+n·U+¯c
+U−5¯c+σ√n∧cs+6¯c
+U−5¯ccs√s+4¯c
+Un
+λc2
+s
+s.
+These results substantively generalize the parametric results of [ 13] on model selection
+by thresholded Lasso. These results cover the more general non parametric case and may
+beofindependentinterest.Alsonotethatthestatedconditionsf orperfectmodelselection
+require a strong assumption on the separation of coefficients of th e oracle from 0, along
+with near-perfect orthogonality of the empirical Gram matrix. This is the sense in which
+the perfect model selection is a rather special, nongeneral pheno menon. Finally, we note
+that it is possible to perform perfect selection of the oracle model b y Lasso without
+applying any additional thresholding under additional technical con ditions and higher
+penalty levels [ 5,24,27]. In the supplement, we state the nonparametric extension of the
+parametric result due to [ 24].
+3.3. Sparsity properties of Lasso
+Here we derive new sharp sparsity bounds for Lasso, which may be o f independent inter-
+est.We begin with a preliminary sparsity bound for Lasso.
+Lemma 2 (Empirical presparsity for Lasso). Suppose that Conditions MandRE(¯c)
+hold and that λ≥cn/bar⌈blS/bar⌈bl∞, and let/hatwidem=|/hatwideT\T|. For¯c=(c+1)/(c−1), we have that
+√
+/hatwidem≤√s/radicalbig
+φ(/hatwidem)2¯c/κ(¯c)+3(¯c+1)/radicalbig
+φ(/hatwidem)ncs/λ.
+The foregoing lemma states that Lasso achieves the oracle sparsit y up to a factor of
+φ(/hatwidem). Under the conditions ( 2.5) andκ(¯c)/greaterorsimilar1, the lemma immediately yields the simple
+upper bound on the sparsity of the form
+/hatwidem/lessorsimilarPsφ(n), (3.3)
+as obtained for examples of [ 4] and [16]. Unfortunately, this bound is sharp only when
+φ(n)isbounded.When φ(n) diverges–forexample,when φ(n)/greaterorsimilarP√logpintheGaussian
+design with p≥2nby lemma 6 of [ 1] – the bound is not sharp. However, for this case
+we can construct a sharp sparsity bound by combining the precedin g presparsity result
+with the following sublinearity property of the restricted sparse eig envalues.
+Lemma 3 (Sublinearity of restricted sparse eigenvalues). For any integer k≥0
+and constant ℓ≥1, we have φ(⌈ℓk⌉)≤⌈ℓ⌉φ(k).14 A. Belloni and V. Chernozhukov
+A version of this lemma for (unrestricted) sparse eigenvalues has b een proven by [ 3].
+The combination of the preceding two lemmas gives the following spars ity theorem.
+Theorem 3 (Sparsity bound for Lasso under data-driven penal ty).Suppose that
+Conditions MandRE(¯c)hold, and let/hatwidem:=|/hatwideT\T|. The event λ≥cn/bar⌈blS/bar⌈bl∞implies that
+/hatwidem≤s·/bracketleftBig
+min
+m∈Mφ(m∧n)/bracketrightBig
+·Ln,
+whereM={m∈N:m>sφ(m∧n)·2Ln}andLn=[2¯c/κ(¯c)+3(¯c+1)ncs/(λ√s)]2.
+The main implication of Theorem 3is that under ( 2.5), if min m∈Mφ(m∧n)/lessorsimilar1 and
+λ≥cn/bar⌈blS/bar⌈bl∞hold with high probability, which is valid by Lemma 1for important designs
+and by the choice of penalty level ( 2.12), then, with high probability,
+/hatwidem/lessorsimilars. (3.4)
+Consequently, for these designs and penalty levels,the sparsity of Lasso is of the same
+order as that of the oracle, namely /hatwides:=|/hatwideT|≤s+/hatwidem/lessorsimilars, with high probability. This is
+because min m∈Mφ(m)≪φ(n) for these designs, which allows us to sharpen the previous
+sparsity bound ( 3.3) considered by [ 4] and [16]. Moreover, our new bound is comparable
+to the bounds of [ 26] in terms of order of sharpness, but it requires a smaller penalty
+levelλ, which also does not depend on the unknown sparse eigenvalues (as in [26]).
+4. Performance of post-model selection estimators
+with a generic model selector
+Here we present a general result on the performance of a post-m odel selection estimator
+with a generic model selector.
+Theorem 4 (Performance of post-model selection estimator w ith a generic
+model selector). Suppose that Condition Mholds, and let /hatwideβbe any first-step estimator
+acting as the model selector. Denote by /hatwideT:=support(/hatwideβ)the model that it selects, such
+that|/hatwideT|≤n. Let/tildewideβbe the post-model selection estimator defined by
+/tildewideβ∈arg min
+β∈Rp/hatwideQ(β):βj=0for each j∈/hatwideTc. (4.1)
+LetBn:=/hatwideQ(/hatwideβ)−/hatwideQ(β0)andCn:=/hatwideQ(β0/hatwideT)−/hatwideQ(β0)and/hatwidem=|/hatwideT\T|be the number of
+incorrect regressors selected. Then, if Condition RSE(/hatwidem)holds, for any ε>0, there is a
+constant Kεindependent of nsuch that with probability at least 1−ε, for/tildewidefi=x′
+i/tildewideβ, we
+have
+/bar⌈bl/tildewidef−f/bar⌈blPn,2≤Kεσ/radicalbigg
+/hatwidemlogp+(/hatwidem+s)log(eµ(/hatwidem))
+n+3cs+/radicalbig
+(Bn)+∧(Cn)+.Least squares after model selection 15
+Furthermore, for any ε >0, there is a constant Kεindependent of nsuch that with
+probability at least 1−ε,
+Bn≤/bar⌈bl/hatwideβ−β0/bar⌈bl2
+2,n+/bracketleftBigg
+Kεσ/radicalbigg
+/hatwidemlogp+(/hatwidem+s)log(eµ(/hatwidem))
+n+2cs/bracketrightBigg
+/bar⌈bl/hatwideβ−β0/bar⌈bl2,n,
+Cn≤1{T/\⌉}atio\slash⊆/hatwideT}/parenleftBigg
+/bar⌈blβ0/hatwideTc/bar⌈bl2
+2,n+/bracketleftBigg
+Kεσ/radicalBigg
+log/parenleftbigs
+/hatwidek/parenrightbig
++/hatwideklog(eµ(0))
+n+2cs/bracketrightBigg
+/bar⌈blβ0/hatwideTc/bar⌈bl2,n/parenrightBigg
+.
+Three implications of Theorem 4are worth noting. First, the bounds on the prediction
+norm stated in Theorem 4apply to the OLS estimator on the components selected
+by any first-step estimator /hatwideβ, provided that we can bound both /bar⌈bl/hatwideβ−β0/bar⌈bl2,n, the rate of
+convergenceofthe first-step estimator,and /hatwidem, the number ofincorrectregressorsselected
+by the model selector. Second, note that if the selected model co ntains the true model,
+T⊆/hatwideT, then we have ( Bn)+∧(Cn)+=Cn=0. In that case, Bnhas no affect on the rate,
+and the performance of the second-step estimator is determined by the sparsity /hatwidemof the
+first-step estimator, which controls the magnitude of the empirica l errors. Otherwise, if
+the selected model fails to contain the true model (i.e., T/\⌉}atio\slash⊆/hatwideT), then the performance
+of the second-step estimator is determined by both the sparsity /hatwidemand the minimum
+between BnandCn. The quantity Bnmeasures the in-sample loss of fit induced by
+the first-step estimator relative to the “true” parameter value β0, andCnmeasures the
+in-sample loss of fit induced by truncating the “true” parameter β0outside the selected
+model/hatwideT.
+The proof of Theorem 4relies on the sparsity-based control of the empirical error
+provided by the following lemma.
+Lemma 4 (Sparsity-based control of empirical error). Suppose that Condition M
+holds.
+(1) For any ε>0, there is a constant Kεindependent of nsuch that with probability
+at least1−ε,
+|/hatwideQ(β0+δ)−/hatwideQ(β0)−/bar⌈blδ/bar⌈bl2
+2,n|≤Kεσ/radicalbigg
+mlogp+(m+s)log(eµ(m))
+n/bar⌈blδ/bar⌈bl2,n+2cs/bar⌈blδ/bar⌈bl2,n,
+uniformly for all δ∈Rpsuch that /bar⌈blδTc/bar⌈bl0≤m, and uniformly over m≤n.
+(2) Furthermore, with at least the same probability,
+|/hatwideQ(β0/tildewideT)−/hatwideQ(β0)−/bar⌈blβ0/tildewideTc/bar⌈bl2
+2,n|≤Kεσ/radicalBigg
+log/parenleftbigs
+k/parenrightbig
++klog(eµ(0))
+n/bar⌈blβ0/tildewideTc/bar⌈bl2,n+2cs/bar⌈blβ0/tildewideTc/bar⌈bl2,n,
+uniformly for all /tildewideT⊂Tsuch that |T\/tildewideT|=k, and uniformly over k≤s.
+The proof of this lemma in turn relies on the following maximal inequality, the proof
+of which involves the use of a Samorodnitsky–Talagrand type of ineq uality.16 A. Belloni and V. Chernozhukov
+Lemma 5 (Maximal inequality for a collection of empirical pr ocesses). Let
+ǫi∼N(0,σ2)be independent for i=1,...,n, and for m=1,...,n, define
+en(m,η):=σ2√
+2/parenleftBigg/radicalBigg
+log/parenleftbigg
+p
+m/parenrightbigg
++/radicalbig
+(m+s)log(Dµ(m))+/radicalbig
+(m+s)log(1/η)/parenrightBigg
+for anyη∈(0,1)and some universal constant D. Then,
+sup
+/bardblδTc/bardbl0≤m,/bardblδ/bardbl2,n>0/vextendsingle/vextendsingle/vextendsingle/vextendsingleGn/parenleftbiggǫix′
+iδ
+/bar⌈blδ/bar⌈bl2,n/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤en(m,η)for allm≤n,
+with probability at least 1−ηe−s/(1−1/e).
+5. Performance of least squares after Lasso-based
+model selection
+In this section we apply our results on post-model selection estimat ors to the case where
+Lasso is the first-step estimator. Our previous generic results allo w us to use the sparsity
+bounds and rate of convergence of Lasso to derive the rate of co nvergence of post-model
+selection estimators in the parametric and nonparametric models.
+5.1. Performance of OLS post-Lasso
+Here we show that the OLS post-Lasso estimator has good theore tical performance de-
+spite (generally) imperfect selection of the model by Lasso.
+Theorem 5 (Performance of OLS post-Lasso). Suppose that Conditions M,RE(¯c),
+andRSE(/hatwidem)hold, where ¯c=(c+1)/(c−1)and/hatwidem=|/hatwideT\T|. Ifλ≥cn/bar⌈blS/bar⌈bl∞occurs with
+probability at least 1−α, then for any ε>0, there is a constant Kεindependent of n
+such that with probability at least 1−α−ε, for/tildewidefi=x′
+i/tildewideβ, we have
+/bar⌈bl/tildewidef−f/bar⌈blPn,2≤Kεσ/radicalbigg
+/hatwidemlogp+(/hatwidem+s)log(eµ(/hatwidem))
+n
++3cs+1{T/\⌉}atio\slash⊆/hatwideT}/radicalBigg
+λ√s
+nκ(1)/parenleftbigg(1+c)λ√s
+cnκ(1)+2cs/parenrightbigg
+.
+In particular, under Condition Vand the data-driven choice of λspecified in ( 2.12)
+withlog(1/α)/lessorsimilarlogp,u/ℓ/lessorsimilar1, for any ε>0there is a constant K′
+ε,αsuch that
+/bar⌈bl/tildewidef−f/bar⌈blPn,2≤3cs+K′
+ε,ασ/bracketleftBigg/radicalbigg
+/hatwidemlog(peµ(/hatwidem))
+n+/radicalbigg
+slog(eµ(/hatwidem))
+n/bracketrightBiggLeast squares after model selection 17
+(5.1)
++1{T/\⌉}atio\slash⊆/hatwideT}/bracketleftBigg
+K′
+ε,ασ/radicalbigg
+slogp
+n1
+κ(1)+cs/bracketrightBigg
+with probability at least 1−α−ε−τ.
+ThistheoremprovidesaperformanceboundforOLSpost-Lassoa safunctionofLasso’s
+sparsity (characterized by /hatwidem), rate of convergence, and model selection ability. For com-
+mon designs, this bound implies that OLS post-Lasso performs at lea st as well as Lasso
+andcan be strictlybetter in some cases,and hasasmallerregulariza tionbias.We provide
+further theoretical comparisons in what follows, and give computa tional examples sup-
+porting these comparisons in the supplemental article [ 2]. It is also worth repeating here
+that performance bounds in other norms of interest follow immediat ely by the triangle
+inequality and by the definition of /tildewideκ, as discussed in Remark 3.1.
+The following corollary summarizes the performance of OLS post-La sso under com-
+monly used designs.
+Corollary 2 (Asymptotic performance of OLS post-Lasso). Under the conditions
+of Theorem 5, (2.5), and (3.2), asngrows, we have that
+/bar⌈bl/tildewidef−f/bar⌈blPn,2/lessorsimilarP
+
+σ/radicalbigg
+slogp
+n+cs, in general,
+σ/radicalbigg
+o(s)logp
+n+σ/radicalbiggs
+n+cs,if/hatwidem=oP(s)andT⊆/hatwideTwp→1,
+σ/radicalbig
+s/n+cs, ifT=/hatwideTwp→1.
+Remark 5.1 (Comparison of the performance of OLS post-Lasso and Lasso).
+We now compare the upper bounds on the rates of convergence of Lasso and OLS post-
+Lassounder conditions ofthe corollary.In general,the ratescoin cide. Of note, this occurs
+despite the fact that Lasso generally may fail to correctly select t he oracle model Tas a
+subset, that is, T/\⌉}atio\slash⊆/hatwideT. However, if the oracle model has well-separated coefficients and
+conditions and the approximation error does not dominate the estim ation error, then the
+OLS post-Lasso rate improves on the rate of Lasso. Specifically, t his occurs if condition
+(2.5) holds and/hatwidem=oP(s) andT⊆/hatwideTwp→1, as under the conditions of Theorem 2Part
+1 or, in the case of perfect model selection, when T=/hatwideTwp→1, as under the conditions
+specified by [ 24]. In such cases, we know from Corollary 1that the rates for Lasso are
+sharp and cannot be faster than σ/radicalbig
+slogp/n. Thus the faster rate of convergence of OLS
+post-Lasso over Lasso is strict in such cases.
+5.2. Performance of OLS post-fit Lasso
+In what follows we provide performance bounds for OLS post-fit La sso/tildewideβdefined in
+equation ( 4.1) with threshold ( 2.9) for the case where the first-step estimator /hatwideβis Lasso.
+We let/tildewideTdenote the model selected.18 A. Belloni and V. Chernozhukov
+Theorem 6 (Performance of OLS post-fit Lasso). Suppose that Conditions M,
+RE(¯c),andRSE(/tildewidem)hold, where ¯c=(c+1)/(c−1)and/tildewidem=|/tildewideT\T|.Ifλ≥cn/bar⌈blS/bar⌈bl∞occurs
+with probability at least 1−α, then for any ε>0, there is a constant Kεindependent of
+nsuch that with probability at least 1−α−ε, for/tildewidefi=x′
+i/tildewideβ, we have
+/bar⌈bl/tildewidef−f/bar⌈blPn,2≤Kεσ/radicalbigg
+/tildewidemlogp+(/tildewidem+s)log(eµ(/tildewidem))
+n
++3cs+1{T/\⌉}atio\slash⊆/tildewideT}/radicalBigg
+λ√s
+nκ(1)/parenleftbigg(1+c)λ√s
+cnκ(1)+2cs/parenrightbigg
+.
+Under Condition Vand the data-driven choice of λspecified in ( 2.12) withlog(1/α)/lessorsimilar
+logp,u/ℓ/lessorsimilar1, for any ε>0there is a constant K′
+ε,αsuch that
+/bar⌈bl/tildewidef−f/bar⌈blPn,2≤3cs+K′
+ε,ασ/bracketleftBigg/radicalbigg
+/tildewidemlog(peµ(/tildewidem))
+n+/radicalbigg
+slog(eµ(/tildewidem))
+n/bracketrightBigg
+(5.2)
++1{T/\⌉}atio\slash⊆/tildewideT}/bracketleftBigg
+K′
+ε,ασ/radicalbigg
+slogp
+n1
+κ(1)+cs/bracketrightBigg
+,
+with probability at least 1−α−ε−τ.
+This theorem provides a performance bound for OLS post-fit Lass o as a function of
+its sparsity (characterized by /tildewidem), Lasso’s rate of convergence, and the model selection
+ability of the thresholding scheme. Generally, this bound is as good as the bound for
+OLS post-Lasso, because the OLS post-fitness-thresholded La sso thresholds as much as
+possible subject to maintaining a certain goodness of fit. Another a ppealing feature is
+that this estimator determines the thresholding level in a completely data-driven fashion.
+Moreover, by construction, the estimated model is sparser than the OLS post-Lasso
+model, which leads to an improved performance of OLS post-fitness -thresholded Lasso
+over OLS post-Lasso in some cases. We provide further theoretic al comparisons below
+and computational examples in the supplemental article [ 2].
+The following corollary summarizes the performance of OLS post-fit Lasso under com-
+monly used designs.
+Corollary 3 (Asymptotic performance of OLS post-fit Lasso). Under the con-
+ditions of Theorem 6, if conditions in ( 2.5) and (3.2) hold, then as ngrows, the OLS
+post-fitness-thresholded Lasso satisfies
+/bar⌈bl/tildewidef−f/bar⌈blPn,2/lessorsimilarP
+
+σ/radicalbigg
+slogp
+n+cs, in general,
+σ/radicalbigg
+o(s)logp
+n+σ/radicalbiggs
+n+cs,if/tildewidem=oP(s)andT⊆/tildewideTwp→1,
+σ/radicalbiggs
+n+cs, ifT=/tildewideTwp→1.Least squares after model selection 19
+Remark 5.2 (Comparison of the performance of OLS post-fit Las so, Lasso,
+and OLS post-Lasso). Under the conditions of the corollary, the OLS post-fitness-
+thresholded Lasso matches the near-oracle rate of convergenc e of Lasso and OLS post-
+Lasso:σ/radicalbig
+slogp/n+cs.If/tildewidem=oP(s)andT⊆/tildewideTwp→1and(2.5)hold,thenOLSpost-fit
+Lasso strictly improves on Lasso’s rate. That is, if the oracle model has coefficients well
+separated from 0 and the approximation error is not dominant, the n the improvement is
+strict. An interesting question is whether OLS post-fit Lasso can o utperform OLS post-
+Lasso in terms of the rates. We cannot rank these estimators in te rms of rates in general;
+however, this necessarily occurs when the Lasso does not achieve the sufficient sparsity
+but the model selection works well, namely when /tildewidem=oP(/hatwidem) andT⊆/tildewideTwp→1. Finally,
+under conditions ensuring perfect model selection – namely, the co ndition of Theorem 2
+holding for t=tγ– OLS post-fit Lasso achieves the oracle performance, σ/radicalbig
+s/n+cs.
+5.3. Performance of the OLS post-thresholded Lasso
+We next consider the traditional thresholding scheme, which trunc ates to 0 all com-
+ponents below a set threshold, t. This is arguably the most widely used thresholding
+scheme in the literature. To state the result, recall that /hatwideβtj=/hatwideβj1{|/hatwideβj|>t},/tildewidem:=|/tildewideT\T|,
+mt:=|/hatwideT\/tildewideT|andγt:=/bar⌈bl/hatwideβt−/hatwideβ/bar⌈bl2,n, where/hatwideβis the Lasso estimator.
+Theorem 7 (Performance of OLS post-t Lasso). Suppose that Conditions M,
+RE(¯c), andRSE(/tildewidem)hold, where ¯c= (c+1)/(c−1)and/tildewidem=|/tildewideT\T|. Ifλ≥cn/bar⌈blS/bar⌈bl∞
+occurs with probability at least 1−α, then for any ε>0, there is a constant Kεindepen-
+dent ofnsuch that with probability at least 1−α−ε, for/tildewidefi=x′
+i/tildewideβ, we have
+/bar⌈bl/tildewidef−f/bar⌈blPn,2≤Kεσ/radicalbigg
+/tildewidemlogp+(/tildewidem+s)log(eµ(/tildewidem))
+n+3cs
++1{T/\⌉}atio\slash⊆/tildewideT}/parenleftbigg
+γt+1+c
+cλ√s
+nκ(¯c)+2cs/parenrightbigg
++1{T/\⌉}atio\slash⊆/tildewideT}
+×/radicaltp/radicalvertex/radicalvertex/radicalbt/bracketleftBigg
+Kεσ/radicalbigg
+/tildewidemlogp+(/tildewidem+s)log(eµ(/tildewidem))
+n+2cs/bracketrightBigg/parenleftbigg
+γt+1+c
+cλ√s
+nκ(¯c)+2cs/parenrightbigg
+,
+whereγt≤t/radicalbig
+φ(mt)mt. Under Condition Vand the data-driven choice of λspecified in
+(2.12) forlog(1/α)/lessorsimilarlogp,u/ℓ/lessorsimilar1, for any ε>0, there is a constant K′
+ε,αsuch that
+with probability at least 1−α−ε−τ,
+/bar⌈bl/tildewidef−f/bar⌈blPn,2≤3cs+K′
+ε,α/bracketleftBigg
+σ/radicalbigg
+/tildewidemlog(peµ(/tildewidem))
+n+σ/radicalbigg
+slog(eµ(/tildewidem))
+n/bracketrightBigg
++1{T/\⌉}atio\slash⊆/tildewideT}/bracketleftBigg
+γt+K′
+ε,ασ/radicalbigg
+slogp
+n1
+κ(¯c)+4cs/bracketrightBigg
+.20 A. Belloni and V. Chernozhukov
+This theorem provides a performance bound for OLS post-thresh olded Lasso as a
+function of (1) its sparsity, characterized by /tildewidem, and improvements in sparsity over Lasso,
+characterized by mt; (2) Lasso’s rate of convergence; (3) the thresholding level tand
+resulting goodness-of-fit loss, γt, relative to Lasso induced by thresholding; and (4) the
+model selection ability of the thresholding scheme. Generally, this bo und may be worse
+than the bound for Lasso, because the OLS post-thresholded La sso potentially uses too
+much thresholding, resulting in large goodness-of-fit losses, γt. We provide further theo-
+retical comparisons below and computational examples in Section 4 o f the supplemental
+article [2].
+Remark 5.3 (Comparison of the performance of OLS post-thres holded Lasso,
+Lasso, and OLS post-Lasso). In this work, we also assume conditions in ( 2.5) and
+(3.2) presented in the foregoing formal comparisons. Under these co nditions, OLS post-
+thresholded Lasso obeys the bound
+/bar⌈bl/tildewidef−f/bar⌈blPn,2/lessorsimilarPσ/radicalbigg
+/tildewidemlogp
+n+σ/radicalbiggs
+n+cs+1{T/\⌉}atio\slash⊆/tildewideT}/parenleftBigg
+γt∨σ/radicalbigg
+slogp
+n/parenrightBigg
+.(5.3)
+In this case, we have /tildewidem∨mt≤s+/hatwidem/lessorsimilarPsby Theorem 3. In general, the foregoing rate
+cannot improve on Lasso’s rate of convergence given in Lemma 1.
+As expected, the choice of t, which controls γtvia the bound γt≤t/radicalbig
+φ(mt)mt, can
+have a significant effect on the performance bounds. If
+t/lessorsimilarσ/radicalbigg
+logp
+nthen/bar⌈bl/tildewidef−f/bar⌈blPn,2/lessorsimilarPσ/radicalbigg
+slogp
+n+cs. (5.4)
+The choice ( 5.4), suggested by [ 13] and Theorem 3, is theoretically sound, because it
+guarantees that OLS post-thresholded Lasso achieves the near -oracle rate of Lasso. Note
+that to implement the choice ( 5.4) in practice, we suggest setting t=λ/n, given that
+the separation of the coefficients from 0 is unknown in practice. Not e that using a much
+largertcan lead to inferior rates of convergence.
+Furthermore, there is a special class of models – a neighborhood of parametric models
+with well-separated coefficients – for which improvements in the rate of convergence
+of Lasso are possible. Specifically, if /tildewidem= oP(s) andT⊆/tildewideTwp→1, then OLS post-
+thresholded Lasso strictly improves on the Lasso’s rate. Further more, if/tildewidem=oP(/hatwidem) and
+T⊆/tildewideTwp→1, then OLS post-thresholded Lasso also outperforms OLS post- Lasso:
+/bar⌈bl/tildewidef−f/bar⌈blPn,2/lessorsimilarPσ/radicalbigg
+o(/hatwidem)logp
+n+σ/radicalbiggs
+n+cs.
+Finally, with the conditions of Theorem 2holding for given t, OLS post-thresholded
+Lasso achieves oracle performance, /bar⌈bl/tildewidef−f/bar⌈blPn,2/lessorsimilarPσ/radicalbig
+s/n+cs.Least squares after model selection 21
+Appendix: Proofs
+A.1. Proofs for Section 3
+Proof of Theorem 1.The bound in /bar⌈bl·/bar⌈bl2,nnorm follows by the same steps specified
+by [4], and thus we defer the derivation to the supplement.
+Under the data-driven choice ( 2.12) ofλand Condition V, we have c′/hatwideσ≥cσwith
+probability at least 1 −τ, because c′≥c/ℓ. Moreover, with the same probability, we also
+haveλ≤c′uσΛ(1−α|X). The result follows by invoking the /bar⌈bl·/bar⌈bl2,nbound.
+The bound in /bar⌈bl · /bar⌈bl1is proven as follows. First, assume that /bar⌈blδTc/bar⌈bl1≤2¯c/bar⌈blδT/bar⌈bl1.In
+this case, by the definition of the restricted eigenvalue, we have /bar⌈blδ/bar⌈bl1≤(1+2¯c)/bar⌈blδT/bar⌈bl1≤
+(1 + 2¯c)√s/bar⌈blδ/bar⌈bl2,n/κ(2¯c),and the result follows by applying the first bound to /bar⌈blδ/bar⌈bl2,n
+because ¯c>1. On the other hand, consider the case where /bar⌈blδTc/bar⌈bl1>2¯c/bar⌈blδT/bar⌈bl1. Here the
+relation
+−λ
+cn(/bar⌈blδT/bar⌈bl1+/bar⌈blδTc/bar⌈bl1)+/bar⌈blδ/bar⌈bl2
+2,n−2cs/bar⌈blδ/bar⌈bl2,n≤λ
+n(/bar⌈blδT/bar⌈bl1−/bar⌈blδTc/bar⌈bl1),
+which is established in (2.3) in the supplemental article [ 2], implies that /bar⌈blδ/bar⌈bl2,n≤2csand
+also
+/bar⌈blδTc/bar⌈bl1≤¯c/bar⌈blδT/bar⌈bl1+c
+c−1n
+λ/bar⌈blδ/bar⌈bl2,n(2cs−/bar⌈blδ/bar⌈bl2,n)≤/bar⌈blδT/bar⌈bl1+c
+c−1n
+λc2
+s≤1
+2/bar⌈blδTc/bar⌈bl1+c
+c−1n
+λc2
+s.
+Thus,
+/bar⌈blδ/bar⌈bl1≤/parenleftbigg
+1+1
+2¯c/parenrightbigg
+/bar⌈blδTc/bar⌈bl1≤/parenleftbigg
+1+1
+2¯c/parenrightbigg2c
+c−1n
+λc2
+s.
+The result follows by taking the maximum of the bounds on each case a nd invoking the
+bound on /bar⌈blδ/bar⌈bl2,n. /square
+Proof of Theorem 2.Part (1) follows immediately from the assumptions. To show
+part (2), let δ:=/hatwideβ−β0, and proceed in two steps:
+Step 1. By the first-order optimality conditions of /hatwideβand the assumption on λ,
+/bar⌈blEn[x•x′
+•δ]/bar⌈bl∞≤/bar⌈blEn[x•(y•−x′
+•/hatwideβ)]/bar⌈bl∞+/bar⌈blS/2/bar⌈bl∞+/bar⌈blEn[x•r•]/bar⌈bl∞
+≤λ
+2n+λ
+2cn+min/braceleftbiggσ√n,cs/bracerightbigg
+,
+because/bar⌈blEn[x•r•]/bar⌈bl∞≤min{σ√n,cs}by step 2 below.
+Next, let ejdenote the jth canonical direction. Thus, for every j=1,...,p, we have
+|En[e′
+jx•x′
+•δ]−δj|=|En[e′
+j(x•x′
+•−I)δ]|
+≤max
+1≤j,k≤p|(En[x•x′
+•−I])jk|/bar⌈blδ/bar⌈bl1
+≤/bar⌈blδ/bar⌈bl1/[Us].22 A. Belloni and V. Chernozhukov
+Then, combining the two bounds above and using the triangle inequalit y, we have
+/bar⌈blδ/bar⌈bl∞≤/bar⌈blEn[x•x′
+•δ]/bar⌈bl∞+/bar⌈blEn[x•x′
+•δ]−δ/bar⌈bl∞≤/parenleftbigg
+1+1
+c/parenrightbiggλ
+2n+min/braceleftbiggσ√n,cs/bracerightbigg
++/bar⌈blδ/bar⌈bl1
+Us.
+The result follows by Theorem 1to bound /bar⌈blδ/bar⌈bl1and the arguments of [ 4] and [13] to show
+that the bound on the correlations imply that for any C >0,
+κ(C)≥/radicalbig
+1−s(1+2C)/bar⌈blEn[x•x′•−I]/bar⌈bl∞,
+so thatκ(¯c)≥/radicalbig
+1−[(1+2¯c)/U] andκ(2¯c)≥/radicalbig
+1−[(1+4¯c)/U] under this particular
+design.
+Step 2. In this step, we show that /bar⌈blEn[x•r•]/bar⌈bl∞≤min{σ√n,cs}.First, note that for
+everyj= 1,...,p, we have |En[x•jr•]| ≤/radicalBig
+En[x2
+•j]En[r2•] =cs. Next, by the definition
+ofβ0in (2.2), forj∈T, we have En[x•j(f•−x′
+•β0)] =En[x•jr•] =0, because β0is a
+minimizer over the support of β0. Forj∈Tc, we have that for any t∈R,
+En[(f•−x′
+•β0)2]+σ2s
+n≤En[(f•−x′
+•β0−tx•j)2]+σ2s+1
+n.
+Therefore, for any t∈R, we have
+−σ2/n≤En[(f•−x′
+•β0−tx•j)2]−En[(f•−x′
+•β0)2]=−2tEn[x•j(f•−x′
+•β0)]+t2En[x2
+•j].
+Taking the minimum over ton the right-hand side at t∗=En[x•j(f•−x′
+•β0)], we obtain
+−σ2/n≤−(En[x•j(f•−x′
+•β0)])2or, equivalently, |En[x•j(f•−x′
+•β0)]|≤σ/√n./square
+Proof of Lemma 2.Let/hatwideT= support(/hatwideβ) and/hatwidem=|/hatwideT\T|. From the optimality
+conditions, we have that |2En[x•j(y•−x′
+•/hatwideβ)]|=λ/nfor allj∈/hatwideT.Therefore, for R=
+(r1,...,rn)′, we have
+/radicalBig
+|/hatwideT|λ≤2/bar⌈bl(X′(Y−X/hatwideβ))/hatwideT/bar⌈bl
+≤2/bar⌈bl(X′(Y−R−Xβ0))/hatwideT/bar⌈bl+2/bar⌈bl(X′(R+Xβ0−X/hatwideβ))/hatwideT/bar⌈bl
+≤/radicalBig
+|/hatwideT|·n/bar⌈blS/bar⌈bl∞+2n/radicalbig
+φ(/hatwidem)(En[(x′
+•/hatwideβ−f•)2])1/2,
+using the definition of φ(/hatwidem) and the Holder inequality,
+/bar⌈bl(X′(R+Xβ0−X/hatwideβ))/hatwideT/bar⌈bl≤ sup
+/bardblαTc/bardbl0≤/hatwidem,/bardblα/bardbl≤1|α′X′(R+Xβ0−X/hatwideβ)|
+≤sup
+/bardblαTc/bardbl0≤/hatwidem,/bardblα/bardbl≤1/bar⌈blα′X′/bar⌈bl/bar⌈blR+Xβ0−X/hatwideβ/bar⌈bl
+= sup
+/bardblαTc/bardbl0≤/hatwidem,/bardblα/bardbl≤1/radicalbig
+|α′X′Xα|/bar⌈blR+Xβ0−X/hatwideβ/bar⌈bl
+=n/radicalbig
+φ(/hatwidem)(En[(x′
+•/hatwideβ−f•)2])1/2.Least squares after model selection 23
+Because λ/c≥n/bar⌈blS/bar⌈bl∞, we have
+(1−1/c)/radicalBig
+|/hatwideT|λ≤2n/radicalbig
+φ(/hatwidem)(En[(x′
+•/hatwideβ−f•)2])1/2. (A.1)
+Moreover, because /hatwidem≤|/hatwideT|, and by Theorem 1and Remark 3.1, (En[(x′
+•/hatwideβ−f•)2])1/2≤
+/bar⌈bl/hatwideβ−β0/bar⌈bl2,n+cs≤(1+1
+c)λ√s
+nκ(¯c)+3cs, we have
+(1−1/c)√
+/hatwidem≤2/radicalbig
+φ(/hatwidem)(1+1/c)√s/κ(¯c)+6/radicalbig
+φ(/hatwidem)ncs/λ.
+The result follows by noting that (1 −1/c)=2/(¯c+1) by definition of ¯ c. /square
+Proof of Theorem 3.By Lemma 2,√
+/hatwidem≤/radicalbig
+φ(/hatwidem)·2¯c√s/κ(¯c)+3(¯c+1)/radicalbig
+φ(/hatwidem)·ncs/λ,
+which, by letting Ln=(2¯c
+κ(¯c)+3(¯c+1)ncs
+λ√s)2, can be rewritten as
+/hatwidem≤s·φ(/hatwidem)Ln. (A.2)
+Note that/hatwidem≤nby optimality conditions. Consider any M∈ M, and suppose that
+/hatwidem>M. Therefore, by Lemma 3on the sublinearity of restricted sparse eigenvalues,
+/hatwidem≤s·/ceilingleftbigg/hatwidem
+M/ceilingrightbigg
+φ(M)Ln.
+Thus, because ⌈k⌉<2kfor anyk≥1, we have M < s·2φ(M)Ln, which violates the
+condition of M∈ M. Therefore, we must have /hatwidem≤M. In turn, applying ( A.2) once
+more with/hatwidem≤(M∧n), we obtain/hatwidem≤s·φ(M∧n)Ln.The result follows by minimizing
+the bound over M∈M. /square
+A.2. Proofs for Section 4
+Proof of Theorem 4.Let/tildewideδ:=/tildewideβ−β0. By the definition of the second-step estimator,
+it follows that /hatwideQ(/tildewideβ)≤/hatwideQ(/hatwideβ) and/hatwideQ(/tildewideβ)≤/hatwideQ(β0/hatwideT). Thus,
+/hatwideQ(/tildewideβ)−/hatwideQ(β0)≤(/hatwideQ(/hatwideβ)−/hatwideQ(β0))∧(/hatwideQ(β0/hatwideT)−/hatwideQ(β0))≤Bn∧Cn.
+By Lemma 4part (1), for any ε>0 there exists a constant Kεsuch that with probability
+at least 1 −ε,|/hatwideQ(/tildewideβ)−/hatwideQ(β0)−/bar⌈bl/tildewideδ/bar⌈bl2
+2,n|≤Aε,n/bar⌈bl/tildewideδ/bar⌈bl2,n+2cs/bar⌈bl/tildewideδ/bar⌈bl2,n, where
+Aε,n:=Kεσ/radicalBig
+(/hatwidemlogp+(/hatwidem+s)log(eµ(/hatwidem)))/n.
+Combining these relations, we obtain the inequality /bar⌈bl/tildewideδ/bar⌈bl2
+2,n−Aε,n/bar⌈bl/tildewideδ/bar⌈bl2,n−2cs/bar⌈bl/tildewideδ/bar⌈bl2,n≤
+Bn∧Cn.Solvingthis,weobtainthestatedinequality, /bar⌈bl/tildewideδ/bar⌈bl2,n≤Aε,n+2cs+/radicalbig
+(Bn)+∧(Cn)+.
+Finally, the bound on Bnfollows from Lemma 4part (1). The bound on Cnfollows from
+Lemma4part (2). /square24 A. Belloni and V. Chernozhukov
+Proof of Lemma 4.The proof of part (1) follows from the relation
+|/hatwideQ(β0+δ)−/hatwideQ(β0)−/bar⌈blδ/bar⌈bl2
+2,n|=|2En[ǫ•x′
+•δ]+2En[r•x′
+•δ]|,
+and then bounding |2En[r•x′
+•δ]|by 2cs/bar⌈blδ/bar⌈bl2,nusing the Cauchy–Schwarz inequality, ap-
+plying Lemma 5on sparse control of noise to |2En[ǫ•x′
+•δ]|, where we bound/parenleftbigp
+m/parenrightbig
+bypm
+and setKε=6√
+2log1/2max{e,D,1/(esε[1−1/e])}. The proof part (2) also follows from
+Lemma5, but applying it with s= 0,p=s(because only the components in Tare
+modified), m=k, and noting that we can take µ(m) withm=0. /square
+Proof of Lemma 5.We divide the proof into steps.
+Step 0. Note that we can restrict the supremum over /bar⌈blδ/bar⌈bl=1 because the function is
+homogenous of degree 0.
+Step 1. For each nonnegative integer m≤nand each set/tildewideT⊂{1,...,p}, with|/tildewideT\T|≤
+m, define the class of functions
+G/tildewideT={ǫix′
+iδ//bar⌈blδ/bar⌈bl2,n:support( δ)⊆/tildewideT,/bar⌈blδ/bar⌈bl=1}. (A.3)
+Also define Fm={G/tildewideT:/tildewideT⊂{1,...,p}:|/tildewideT\T|≤m}.It follows that
+P/parenleftBig
+sup
+f∈Fm|Gn(f)|≥en(m,η)/parenrightBig
+≤/parenleftbigg
+p
+m/parenrightbigg
+max
+|/tildewideT\T|≤mP/parenleftBig
+sup
+f∈G/tildewideT|Gn(f)|≥en(m,η)/parenrightBig
+.(A.4)
+We apply the Samorodnitsky–Talagrandinequality(PropositionA.2.7o fvan der Vaart
+and Wellner [ 23]) to bound the right-hand side of ( A.4). Let
+ρ(f,g):=/radicalbig
+E[Gn(f)−Gn(g)]2=/radicalbig
+EEn[(f−g)2]
+forf,g∈G/tildewideT. By step 2 below, the covering number of G/tildewideTwith respect to ρobeys
+N(ε,G/tildewideT,ρ)≤(6σµ(m)/ε)m+sfor each 0 <ε≤σ, (A.5)
+andσ2(G/tildewideT) := max f∈G/tildewideTE[Gn(f)]2=σ2. Then, by the Samorodnitsky–Talagrand in-
+equality,
+P/parenleftBig
+sup
+f∈G/tildewideT|Gn(f)|≥en(m,η)/parenrightBig
+≤/parenleftbiggDσµ(m)en(m,η)√m+sσ2/parenrightbiggm+s
+¯Φ(en(m,η)/σ) (A.6)
+for some universal constant D≥1, where ¯Φ = 1−Φ and Φ is the cumulative proba-
+bility distribution function for a standardized Gaussian random varia ble. For en(m,η)
+defined in the statement of the theorem, it follows that P(supf∈G/tildewideT|Gn(f)|≥en(m,η))≤
+ηe−m−s//parenleftbigp
+m/parenrightbig
+by simple substitution into ( A.6).Then,
+P/parenleftBig
+sup
+f∈Fm|Gn(f)|>en(m,η),∃m≤n/parenrightBig
+≤n/summationdisplay
+m=0P/parenleftBig
+sup
+f∈Fm|Gn(f)|>en(m,η)/parenrightBigLeast squares after model selection 25
+≤n/summationdisplay
+m=0ηe−m−s≤ηe−s/(1−1/e),
+which proves the claim.
+Step 2. This step establishes ( A.5). Fort∈Rpand/tildewidet∈Rp, consider any two functions
+ǫi(x′
+it)
+/bar⌈blt/bar⌈bl2,nandǫi(x′
+i/tildewidet)
+/bar⌈bl/tildewidet/bar⌈bl2,ninG/tildewideT,for a given/tildewideT⊂{1,...,p}:|/tildewideT\T|≤m.
+We have that
+/radicalBigg
+EEn/bracketleftbigg
+ǫ2•/parenleftbigg(x′•t)
+/bar⌈blt/bar⌈bl2,n−(x′•/tildewidet)
+/bar⌈bl/tildewidet/bar⌈bl2,n/parenrightbigg2/bracketrightbigg
+≤/radicalBigg
+EEn/bracketleftbigg
+ǫ2•(x′•(t−/tildewidet))2
+/bar⌈blt/bar⌈bl2
+2,n/bracketrightbigg
++/radicalBigg
+EEn/bracketleftbigg
+ǫ2•/parenleftbigg(x′•/tildewidet)
+/bar⌈blt/bar⌈bl2,n−(x′•/tildewidet)
+/bar⌈bl/tildewidet/bar⌈bl2,n/parenrightbigg2/bracketrightbigg
+.
+By definition of G/tildewideTin (A.3), support( t)⊆/tildewideTand support(/tildewidet)⊆/tildewideT, so that support( t−
+/tildewidet)⊆/tildewideT,|/tildewideT\T|≤m, and/bar⌈blt/bar⌈bl=1 by (A.3). Thus, by the definition of RSE(m),
+EEn/bracketleftbigg
+ǫ2
+•(x′
+•(t−/tildewidet))2
+/bar⌈blt/bar⌈bl2
+2,n/bracketrightbigg
+≤σ2φ(m)/bar⌈blt−/tildewidet/bar⌈bl2//tildewideκ(m)2,and
+EEn/bracketleftbigg
+ǫ2
+•/parenleftbigg(x′
+•/tildewidet)
+/bar⌈blt/bar⌈bl2,n−(x′
+•/tildewidet)
+/bar⌈bl/tildewidet/bar⌈bl2,n/parenrightbigg2/bracketrightbigg
+=EEn/bracketleftbigg
+ǫ2
+•(x′
+•/tildewidet)2
+/bar⌈bl/tildewidet/bar⌈bl2
+2,n/parenleftbigg/bar⌈bl/tildewidet/bar⌈bl2,n−/bar⌈blt/bar⌈bl2,n
+/bar⌈blt/bar⌈bl2,n/parenrightbigg2/bracketrightbigg
+=σ2/parenleftbigg/bar⌈bl/tildewidet/bar⌈bl2,n−/bar⌈blt/bar⌈bl2,n
+/bar⌈blt/bar⌈bl2,n/parenrightbigg2
+≤σ2/bar⌈bl/tildewidet−t/bar⌈bl2
+2,n//bar⌈blt/bar⌈bl2
+2,n≤σ2φ(m)/bar⌈bl/tildewidet−t/bar⌈bl2//tildewideκ(m)2,
+so that
+/radicalBigg
+EEn/bracketleftbigg
+ǫ2•/parenleftbigg(x′•t)
+/bar⌈blt/bar⌈bl2,n−(x′•/tildewidet)
+/bar⌈bl/tildewidet/bar⌈bl2,n/parenrightbigg2/bracketrightbigg
+≤2σ/bar⌈blt−/tildewidet/bar⌈bl/radicalbig
+φ(m)//tildewideκ(m)=2σµ(m)/bar⌈blt−/tildewidet/bar⌈bl.
+Then the bound ( A.5) follows from the bound of [ 23], page 94, N(ε,G/tildewideT,ρ)≤N(ε/R,
+B(0,1),/bar⌈bl·/bar⌈bl)≤(3R/ε)m+s, withR=2σµ(m) for any ε≤σ. /square
+A.3. Proofs for Section 5
+Proof of Theorem 5.First, notethat if T⊆/hatwideT, we then have Cn=0,so that Bn∧Cn≤
+1{T/\⌉}atio\slash⊆/hatwideT}Bn.
+Next, we bound Bn. Note that by the optimality of /hatwideβin the Lasso problem, and letting
+/hatwideδ=/hatwideβ−β0,
+Bn:=/hatwideQ(/hatwideβ)−/hatwideQ(β0)≤λ
+n(/bar⌈blβ0/bar⌈bl1−/bar⌈bl/hatwideβ/bar⌈bl1)≤λ
+n(/bar⌈bl/hatwideδT/bar⌈bl1−/bar⌈bl/hatwideδTc/bar⌈bl1).(A.7)26 A. Belloni and V. Chernozhukov
+If/bar⌈bl/hatwideδTc/bar⌈bl1>/bar⌈bl/hatwideδT/bar⌈bl1, then we have /hatwideQ(/hatwideβ)−/hatwideQ(β0)≤0. Otherwise, if /bar⌈bl/hatwideδTc/bar⌈bl1≤/bar⌈bl/hatwideδT/bar⌈bl1, then,
+byRE(1), we have
+Bn:=/hatwideQ(/hatwideβ)−/hatwideQ(β0)≤λ
+n/bar⌈bl/hatwideδT/bar⌈bl1≤λ
+n√s/bar⌈bl/hatwideδ/bar⌈bl2,n
+κ(1). (A.8)
+The result follows by applying Theorem 1to bound /bar⌈bl/hatwideδ/bar⌈bl2,n, under the condition that
+RE(1)holds, along with Theorem 4.
+The second claim follows from the first by using λ/lessorsimilar√nlogpunder Condition V, the
+specified conditions on the penalty level. The final bound follows by ap plying the relation
+that for any nonnegative numbers a,b, we have√
+ab≤(a+b)/2. /square
+Acknowledgements
+WethankDonAndrews,WhitneyNewey,andAlexandreTsybakovas wellasparticipants
+of the Cowles Foundation Lecture at the 2009 Summer Econometric Society meeting and
+the joint Harvard–MIT seminar for useful comments. We also than k Denis Chetverikov,
+Brigham Fradsen, Joonhwan Lee, two refereers, and the associa te editor for numerous
+suggestions that helped improve the articler. We thank Kengo Kato for pointing out
+the usefulness of the approach of [ 18] for bounding sparse eigenvalues of the empirical
+Gram matrix. We gratefully acknowledgethe financial support from the National Science
+Foundation.
+Supplementary Material
+Supplementary material for Least squares after model selec tion in high-
+dimensional sparse models (DOI:10.3150/11-BEJ410SUPP ; .pdf). The online sup-
+plementalarticle[ 2]containsafinite sampleresultsfortheestimationof σ,detailsregard-
+ing the oracle problem, omitted proofs, uniform control of sparse eigenvalues, and Monte
+Carlo experiments to access the performance of the estimators p roposed in the paper.
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+arXiv:1001.0189v2  [cond-mat.other]  29 Jan 2010Non-linear Elastic Response in Solid Helium: critical velo city or strain?
+James Day, Oleksandr Syshchenko, and John Beamish
+Department of Physics, University of Alberta, Edmonton, Al berta, Canada, T6G 2G7
+(Dated: October 15, 2018)
+Torsional oscillator experiments show evidence of mass dec oupling in solid4He. This decoupling
+is amplitude dependent, suggesting a critical velocity for supersolidity. We observe similar behavior
+in the elastic shear modulus. By measuring the shear modulus over a wide frequency range, we can
+distinguish between an amplitude dependence which depends on velocity and one which depends
+on some other parameter like displacement. In contrast to th e torsional oscillator behavior, the
+modulus depends on the magnitude of stress, not velocity. We interpret our results in terms of the
+motion of dislocations which are weakly pinned by3He impurities but which break away when large
+stresses are applied.
+PACS numbers: 67.80.bd, 67.80.de, 67.80.dj
+Helium is a uniquely quantum solid and recent tor-
+sional oscillator (TO) measurements[1–8] provide evi-
+dence for a supersolid phase in hcp4He. At tempera-
+tures below about 200 mK the TO frequency increases,
+suggesting that some of the4He decouples from the os-
+cillator. This evidence of non-classical rotational iner-
+tia (NCRI) inspired searches for other unusual thermal
+or mechanical behavior in solid helium. Heat capacity
+measurements[9, 10] do show a small peak near the on-
+set temperature of decoupling, supporting the idea of a
+phase transition in solid4He. Mass flow is an obvious
+possiblesignatureofsupersoliditybutexperimentsinthis
+temperature range[11–13] show no pressure-induced flow
+through the solid (although recent measurements[14, 15]
+at higher temperatures showed intriguing behavior).
+We recently made low frequency measurements[16, 17]
+of the shear modulus of hcp4He and found a large stiff-
+ening, with the same temperature dependence as the fre-
+quency changes seen in TO experiments. This modulus
+increase also had the same dependence on measurement
+amplitude and on3He concentration and, like the TO
+decoupling, its onset was accompanied by a dissipation
+peak. It is clear that the shear stiffening and the TO de-
+coupling are closely related. Subsequent experiments[18]
+with3Heshowedsimilarelasticstiffeninginthehcpphase
+below 0.4 K, but not in the bcc phase (the bcc phase
+exists only over a temperature range in4He). TO mea-
+surements with hcp3He did not, however, show any sign
+of a transition in the temperature range where the stiff-
+ening occurred, nor was a transition seen with bcc3He.
+Thestiffeningappearstodependoncrystalstructure(ap-
+pearing in the hcp but not the bcc phase) while the TO
+frequencyand dissipationchangesoccuronlyfor the bose
+solid,4He.
+Although it is clear that solid4He shows unusual be-
+havior below 200 mK, the interpretation in terms of su-
+persolidity rests almost entirely upon torsional oscillator
+experiments. In addition to frequency changes which im-
+ply mass decoupling, two other features of the TO ex-
+periments are invoked as evidence of superflow. One isthe blocked annulus experiment[1, 19] in which NCRI is
+greatly reduced by inserting a barrier into the flow path,
+thus indicating that long-range coherent flow is involved.
+The other is the reduction of the NCRI fraction when the
+oscillation amplitude exceeds a critical value[1]. In anal-
+ogytosuperfluidity in liquidhelium, this isinterpretedin
+terms of a critical velocity v c(of order 10 µm/s), above
+which flow becomes dissipative. However, torsional oscil-
+lators are resonant devices and measurements made at a
+single frequency cannot distinguish an amplitude depen-
+dence which sets in at a critical velocity from one which
+begins at a critical displacement or a critical accelera-
+tion. A recent experiment[6] used a compound torsional
+oscillator which operated in two modes, allowing mea-
+surements to be made on the same solid4He sample at
+two different frequencies (496 and 1173 Hz). The ampli-
+tude dependence scaled somewhat better with velocity
+than with acceleration or displacement, supporting the
+superflow interpretation of TO experiments. However,
+recent measurements[20] in which one mode was driven
+at large amplitude while monitoring the low-amplitude
+response of the other mode gave unexpected results. The
+suppression of NCRI, as seen in the low amplitude mode,
+appeared to depend on the acceleration generated by the
+high amplitude mode, rather than on the velocity. To
+settle the question of whether the amplitude dependence
+seen in torsional oscillators reflects a critical velocity for
+superflow, measurements over a wider frequency range
+are needed.
+We have made direct measurements[16] of the ampli-
+tude dependence of the shear modulus, µ, of hcp4He in
+a narrow gap of thickness D. An AC voltage, V, with
+angular frequency ω= 2πf, is applied to a shear trans-
+ducer (with piezoelectric coefficient d 15) to generate a
+displacement δx=d15Vat its surface. This produces a
+quasi-static shear strain in the helium, ǫ=δx/D. The
+resulting stress, σ, generates a charge, q, and thus a
+current, I = ωq, in a second transducer on the oppo-
+site side of the gap. The shear modulus µ=σ/ǫis then
+proportional to I/fV. In contrast to torsional oscillator2
+measurements, this technique is non-resonant and so we
+could measure the shear modulus over a wide frequency
+range, from a few Hz (a limit set by preamplifier noise)
+to a maximum of 2500 Hz (due to interference from the
+first acoustic resonance in our cell, around 8 kHz). The
+technique is very sensitive, allowing us to make modulus
+measurements at strains as low as ǫ≈10−9, correspond-
+ing to stresses σ≈0.02 Pa. The drive voltage can be
+increased substantially without heating the sample, so
+measurements can be made at strains up to ǫ≈10−5, al-
+lowing us to study the amplitude dependence, including
+hysteretic effects, at low temperatures.
+Figure1showsthe temperaturedependence ofthe nor-
+malized shear modulus µ/µofor an hcp4He sample at a
+pressure of 38 bar and a frequency of 2000 Hz. The crys-
+tal was grown from standard isotopic purity4He (con-
+taining about 0.3 ppm3He) using the blocked capillary
+method. A piezoelectric shear stack[21] was used to gen-
+erate strains in a narrow gap (D = 500 µm) between
+it and a detecting transducer. Other experimental de-
+tails are the same as in refs. 16 to 18. The curves in
+Fig. 1 correspond to different transducer drive voltages,
+i.e. to different strains, and were measured during cool-
+ing from a temperature of 0.7 K. The shear modulus in-
+creases at low temperature, with ∆ µ/µo≈17% at the
+lowest amplitude. Similar stiffening was seen in all hcp
+4He crystals[16, 17], although the onset temperature was
+lower in crystals of higher isotopic purity and the magni-
+tude of the modulus change varied by a factor of about 2
+from sample to sample (the TO NCRI varies much more,
+by a factor of 1000, although its temperature dependence
+is always essentially the same[18]). The amplitude de-
+pendence of the shear modulus stiffening is essentially
+the same as that of the TO NCRI. At the lowest drive
+voltages and temperatures, both are independent of am-
+plitude but they decrease at high amplitudes. The onset
+of stiffening shifts to lower temperatures at high ampli-
+tudes, as does the onset of TO decoupling. However, in
+the case of elastic measurements, it is more natural to
+think of this behavior in terms of a critical stress (pro-
+portional to the strain, i.e. to transducer displacement),
+rather than a critical velocity.
+The amplitude dependence of the shear modulus is
+shown in more detail in Fig. 2. Open circles show the
+modulus at 48 mK, taken from the temperature sweeps
+(i.e. the points marked by open circles in Fig. 1). The
+solid circles are the modulus measured when the drive
+voltage was reduced at fixed temperature (48 mK), after
+cooling from high temperature at the highest drive am-
+plitude (3 V pp). Figure 2 also shows the corresponding
+amplitude dependence at temperatures well above the
+shear modulus anomaly (open squares are data at 700
+mK from the temperature sweeps of Fig. 1; solid squares
+are from amplitude sweeps at 800 mK). The data taken
+with the two protocols agree very well. The critical drive
+voltage (where the modulus becomes amplitude depen-FIG. 1: Temperature and amplitude dependence of the shear
+modulus inasolid4He sample at 38 bar, grown bytheblocked
+capillary method. The modulus was measured at 2000 Hz for
+transducer drive voltages (peak to peak) from 10 mV to 3 V.
+Values are normalized by µothe low temperature value at the
+lowest drive voltage.
+FIG. 2: Low and high temperature shear modulus for the
+crystal of Fig. 1, as a function of drive voltage (bottom axis )
+or strain (top axis). The open symbols are taken from tem-
+perature sweeps at different drive levels (the data of Fig. 1) .
+The solid symbols are taken while decreasing the drivevolta ge
+at fixed temperature.
+dent) is around 30 mV pp(corresponding to σ≈4x10−8,
+σ≈0.8 Pa) and at the highest drive levels (3 V ppcorre-
+spondingto σ≈80Pa)thestiffeningisalmostcompletely
+suppressed. The amplitude dependence of the modulus
+closely resembles that of the NCRI in TO experiments,
+behavior which is attributed to a superflow critical ve-
+locity (typically around 10 µm/s). However, even in TO
+measurements there are inertial stresses in the helium,
+produced by its acceleration, which could lead to the ob-
+served amplitude dependence.
+Anumberofexperiments[6,19,22,23]haveshownthat3
+the TO amplitude dependence is hysteretic at low tem-
+peratures. If a sample is cooled at high oscillation ampli-
+tude, the apparent NCRI is small. When the amplitude
+is reduced at low temperature, the TO frequency (NCRI)
+rises, becoming constant below some critical amplitude.
+When the drive is then increased at low temperature,
+the NCRI does not begin to decrease at the critical am-
+plitude - it remains essentially constant at substantially
+larger drives. This hysteresis between data taken while
+decreasingand increasing the drive amplitude disappears
+at temperatures above about 70 mK.
+Figure 3 shows the corresponding hysteresis in the
+shear modulus. At 120 mK the maximum stiffening is
+about half as large as at 36 mK and already depends
+on amplitude at the lowest strains shown. The modu-
+lus measured when the amplitude is reduced (open cir-
+cles) and when it is subsequently increased (solid circles)
+agree. Hysteresis appears when the sample is cooled be-
+low 60 mK and is nearly temperature independent below
+45 mK. Figure 3 shows this hysteresis at 36 mK. The
+sample was cooled from high temperature while driving
+at high amplitude (3 V pp). The amplitude was then low-
+ered at 36 mK (open circles) which resulted in a shear
+modulus increase ∆ µ/µoof about 15%. When the am-
+plitude was then raised (solid circles), the modulus re-
+mained constant to much higher amplitude, the same
+behavior seen for the NCRI in TO experiments. At
+very high amplitudes (above about 1 V pp, correspond-
+ing toǫ≈10−6,σ≈20 Pa) the modulus decreased,
+nearly closing the hysteresis loop. After each change
+we waited 2 minutes for the modulus to stabilize at the
+new amplitude. The only region where we observed fur-
+ther time dependence was while increasing the amplitude
+at drive levels above 1 V pp. The modulus decrease was
+sharper when we waited longer at each point. In acoustic
+resonance measurements[17], we found that even larger
+stresses( σ≈700Pa)produced irreversiblechangeswhich
+only disappeared after annealing above 0.5 K.
+The only obvious mechanism which can produce shear
+modulus changes as large as those shown in Figs. 1 to 3
+involves the motion of dislocations[24]. At low tempera-
+tures, dislocations are pinned by3He impurities and the
+intrinsic modulus is measured[17]. As the temperature is
+raised,3He impurities thermallyunbind fromthe disloca-
+tions, allowing them to move and reducing the modulus.
+At high amplitudes, elastic stresses can also tear the dis-
+locations away from the impurities. The hysteresis seen
+in Fig. 3 can be understood if, when a crystalis cooled at
+high strain amplitudes, the rapid motion of dislocations
+prevents3He atoms from attaching to them. When the
+driveisreducedatlowtemperatures,impuritiescanbind,
+thus pinning the dislocations and increasing the modu-
+lus. Once the3He impurities bind, the pinning length
+of dislocations is smaller and larger stresses are required
+to unpin them so the modulus retains its intrinsic value
+to much higher amplitudes. The critical amplitude forFIG.3: Hysteresis betweentheshear modulusmeasuredwhile
+decreasing and while increasing the drive amplitude at low
+temperature (36 mK). At 120 mK the amplitude dependence
+is reversible.
+this stress-induced breakaway can be estimated[25, 26] if
+the dislocation length and impurity binding energy are
+known. In single crystals of helium, a typical dislocation
+network pinning length[27] is (L ∼5µm), and disloca-
+tions would break away from a3He impurity at strain
+ǫ≈3x10−7. Our crystals are expected to have higher dis-
+location densities and smaller network lengths, so break-
+away would occur at higher strains as the amplitude is
+increased. Measurements with different3He concentra-
+tions would be useful to confirm that the amplitude de-
+pendence is due to this mechanism.
+This interpretation of the shear modulus behavior in-
+volves elastic stress (which is proportional to strain)
+rather than velocity, and so is at odds with the inter-
+pretation of the TO amplitude dependence in terms of a
+superfluid-like critical velocity. Since we can make mod-
+ulus measurements over a wide frequency range, we can
+unambiguously distinguish between an amplitude depen-
+dence which scales with stress (or strain) and one which
+depends on velocity. Figure 4 shows the modulus at 18
+mK, measured at three different frequencies (2000, 200
+and 20 Hz) as the drive amplitude was reduced from
+its maximum value. In Fig. 4a the modulus is plot-
+ted vs. shear strain ǫ(calibrated using the low tempera-
+ture piezoelectric coefficient of the shear stack, d 15=1.25
+nm/V) and in Fig. 4b the same data is plotted versusthe
+correspondingvelocities (v= ωǫD). The amplitude depen-
+dence scales much better with strain than with velocity
+(and the scaling with acceleration is even less satisfac-
+tory). The critical strain appears be slightly larger at
+lower frequency.
+It is clear that the amplitude dependence of the shear
+modulus is most closely associated with stress (or strain)
+amplitude, rather than with a superfluid-like critical ve-4
+FIG. 4: Scaling of the shear modulus amplitude dependence
+with (a) strain and (b) velocity for three different frequenc ies:
+20 Hz (triangles), 200 Hz (squares) and 2000 Hz (circles).
+locity. The many similarities to the TO behavior (e.g.
+the dependence on temperature,3He concentration and
+frequency, the amplitude dependence and its hysteresis)
+suggest that the apparent velocity dependence of the
+NCRI may have a similar origin, e.g. in inertial stresses
+which exceed the critical value for the shear modulus.
+However, estimates of the inertial stress corresponding
+to TO critical velocities give values significantly lower
+than the critical stress for the shear modulus. For an an-
+nular TO geometry, the maximum inertial stress can be
+estimated as σ=ρtωv/2, where ρis the helium density,
+t is the width of the annular channel, ωis the angular
+frequency of the TO and v is the oscillation velocity. For
+the oscillator of ref. 1, we estimate σ≈0.002 to 0.015
+Pa at the apparent critical velocities of 5 to 38 µm/s.
+Measurements in an open cylindrical TO[23] show am-
+plitude dependence at velocities corresponding to iner-
+tial stresses below 0.01 Pa, also much smaller than the
+stresses at which we observe amplitude dependence in
+the shear modulus. There is also other evidence that the
+TO frequency changes and dissipation are not just me-
+chanical consequences of the modulus changes. First, the
+apparent NCRI is too large to be explained simply by
+mechanical stiffening of the torsional oscillator[18, 23].
+Secondly, comparable modulus changes in hcp3He arenot reflected in the corresponding TO behavior[18]. The
+existence of a critical velocity for superflow in solid he-
+lium can only be definitively shown if TO measurements
+can be made over a wide range of frequency and/or TO
+geometries.
+This work was supported by the Natural Sciences and
+Engineering Research Council of Canada.
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\ No newline at end of file
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+arXiv:1001.0190v1  [astro-ph.EP]  31 Dec 2009Kepler-7b: A Transiting Planet with Unusually Low Density†
+David W. Latham1, William J. Borucki2, David G. Koch2, Timothy M. Brown3,
+Lars A. Buchhave1,4, Gibor Basri5, Natalie M. Batalha6, Douglas A. Caldwell7,
+William D. Cochran8, Edward W. Dunham9, Gabor F˝ ur´ esz1, Thomas N. Gautier III10,
+John C. Geary1, Ronald L. Gilliland11, Steve B. Howell12, Jon M. Jenkins7,
+Jack J. Lissauer2, Geoffrey W. Marcy5, David G. Monet13, Jason F. Rowe14,2,
+Dimitar D. Sasselov1
+Received ; accepted
+†BasedinpartonobservationsobtainedattheW.M.KeckObservat ory, whichisoperated
+by the University of California and the California Institute of Techno logy.
+1Harvard-Smithsonian Center for Astrophysics, 60 Garden Stree t, Cambridge, MA 02138
+2NASA Ames Research Center, Moffett Field, CA 94035
+3Las Cumbres Observatory Global Telescope, Goleta, CA 93117
+4Niels Bohr Institute, Copenhagen University, DK-2100 Copenhage n, Denmark
+5University of California, Berkeley, Berkeley, CA 94720
+6San Jose State University, San Jose, CA 95192
+7SETI Institute, Mountain View, CA 94043
+8University of Texas, Austin, TX 78712
+9Lowell Observatory, Flagstaff, AZ 86001
+10Jet Propulsion Laboratory/California Institute of Technology, Pa sadena, CA 91109
+11Space Telescope Science Institute, Baltimore, MD 21218
+12National Optical Astronomy Observatory, Tucson, AZ 85719
+13US Naval Observatory, Flagstaff Station, Flagstaff, AZ 86001
+14NASA Postdoctoral Program Fellow– 2 –
+Version resubmitted — 26 December 2009– 3 –
+ABSTRACT
+We report the discovery and confirmation of Kepler-7b, a transitin g planet
+with unusually low density. The mass is less than half that of Jupiter, MP=
+0.43MJ, but the radius is fifty percent larger, RP= 1.48RJ. The resulting
+density,ρP= 0.17gcm−3, is the second lowest reported so far for an extrasolar
+planet. The orbital period is fairly long, P= 4.886 days, and the host star is
+not much hotter than the Sun, Teff= 6000K. However, it is more massive and
+considerably larger than the sun, M⋆= 1.35M⊙andR⋆= 1.84R⊙, and must be
+near the end of its life on the Main Sequence.
+Subject headings: planetary systems — stars: individual (Kepler-7, KIC 57808 85,
+2MASS 19141956+4105233) — techniques: spectroscopic– 4 –
+1. INTRODUCTION
+The final test of the Keplerphotometer at the end of commissioning was a run of
+9.7 continuous days in science mode, to evaluate the noise performa nce of the instrument.
+TheKepler Input Catalog (KIC) was used to select fifty thousand isolated targets, all with
+magnitudes brighter than 13.8 in the Keplerpassband, and with no nearby companions
+that would contaminate the photometry. The preliminary light curve s from this test run
+were inspected by team members with great excitement, and a few d ozen obvious planet
+candidates were quickly identified and passed on to the team respon sible for ground-based
+follow-up observations. Kepler-7 was observed but was not identifi ed among the sample of
+initial candidates.
+After a gap of 1.3 days, normal science observations began for a f ull list of more than
+150,000 planet-search targets and continued for 33.5 days until in terrupted on 15 June
+2009, followed by a data download and roll of the spacecraft to the summer orientation. By
+the middle of July the preliminary light curves were available for inspect ion, and dozens
+of additional candidates were identified and passed on to the follow- up team. This time
+Kepler-7 was included. Along with the other candidates, Kepler-7 wa s scrutinized for
+evidence of astrophysical false positives involving eclipsing binaries. It survived this stage
+of the follow up and was then observed spectroscopically for very p recise radial velocities
+using the FIber-fed Echelle Spectrograph (FIES) on the Nordic Op tical Telescope (NOT)
+during a ten night run in early October. These observations yielded a spectroscopic orbit
+that confirmed that an unseen companion with a planetary mass was responsible for the
+dips in the light curve observed by Kepler.
+The KIC used ground-based multi-band photometry to assign an eff ective temperature
+and surface gravity of Teff= 5944 K and log g= 4.27(cgs) to Kepler-7, corresponding to a
+late F or early G dwarf. Stellar gravities in this part of the H-R Diagram are notoriously– 5 –
+difficult to determine from photometry alone, and one of the conclus ions of this paper is
+that the star is near the end of its Main Sequence lifetime, with a radiu s that has expanded
+toR⋆= 1.843+0.048
+−0.066R⊙and a surface gravity that has weakened to log g= 4.030+0.018
+−0.019(cgs).
+In turn this implies an inflated radius for the planet, resulting in an unu sually low density
+ofρP= 0.17gcm−3. This conclusion is hard to avoid, because the relatively long duration
+of the transit, more than 5 hours from first to last contact, dema nds a low density and
+expanded radius for the star.
+2. KEPLER PHOTOMETRY
+The light curve for Kepler-7 (= KIC 5780885, α= 19h14m19.s56,δ= +41◦05′23.′′3,
+J2000, KIC r= 12.815mag) is plotted in Figure 1. The numerical data are available
+electronically from the Multi Mission Archive at the Space Telescope S cience Institute
+(MAST) High Level Science Products (HLSP) website1. Only a modest amount of
+detrending has been applied (Koch et al. 2010) to this time series of lo ng cadence data
+(29.4-minute accumulations). There is no evidence for any systema tic difference between
+alternating events, which are plotted with + and ×symbols, supporting the interpretation
+that all the events are primary transits. Indeed, there isweak evidence for a secondary
+eclipse centered at phase 0.5, as would be expected for a circular or bit, but the significance
+is only about 2 .4σ. If this detection is real, it is not inconsistent with the thermal emiss ion
+expected from the planet for reasonable assumptions (Koch et al. 2010).
+1http://archive.stsci.edu/prepds/kepler hlsp– 6 –
+3. FOLLOW-UP OBSERVATIONS
+As described in more detail by Gautier et al. (2010), the initial follow- up observations
+ofKeplerplanet candidates involved reconnaissance spectroscopy to look f or evidence of
+a stellar companion or a nearby eclipsing binary responsible for the ob served transits.
+However, the follow-up team soon learned that the astrometry de rived from the Kepler
+images themselves, when combined with high-resolution images of the target neighborhood,
+could provide a very powerful tool for identifying background eclip sing binaries blended
+with and contaminating the target images (Batalha et al. 2010; Mone t et al. 2010). The
+astrometry of Kepler-7 indicated a very slight image centroid shift d uring transits of +0.1
+millipixels in its CCD row direction only.
+The only star listed in the KIC that is closer than 30′′to Kepler-7 and that can
+contribute significant light to the Kepler-7 photometry is KIC 57808 99, which is 4.4 mag
+fainter and lies at a separation of 15 .5′′. KIC 5780899 cannot be the source of the observed
+dips, because that would induce centroid shifts of about 25 millipixels. If KIC 5780899 is
+constant and Kepler-7 is the source of the transits, the predicte d shifts are in the right
+direction and have an amplitude of roughly 0.1 millipixels if a quarter of KI C 5780899’s light
+leaks into the Kepler-7 aperture. Thus KIC 5780899 provides a sat isfactory explanation for
+the observed shifts.
+To check for very close companions, a speckle observation of Keple r-7 was obtained by
+S. Howell with the WIYN 3.5-m telescope on Kitt Peak. It showed no co mpanions in a 2′′
+box centered on Kepler-7. Subsequently, images obtained by H. Is aacson with the HIRES
+guider on Keck 1, and independently by G. Mandushev with the 1.8-m P erkins telescope
+and PRISM camera at the Lowell Observatory and by N. Baliber with t he LCOGT Faulkes
+Telescope North on Haleakala, Maui, all detected a companion at a se paration of 1 .8′′(just
+outside the WIYN speckle window) and about 4.4 mag fainter in the red . This companion– 7 –
+cannot be the source of the observed centroid shifts. If it is the s ource of the dips in the
+light curve, the centroid shifts would have to be larger than 1 millipixel, and in the wrong
+direction. If it is constant, the shifts would be much too small to det ect. However, this
+companion does dilute the photometry of Kepler-7 with a contributio n of about 2.1%.
+Adding in a quarter of the light from the more distant companion gives a total dilution of
+about 2.5±0.4%. This dilution has been included in the analysis of the light curve.
+Reconnaissance spectra obtained by M. Endl and W. Cochran with t he coud´ e echelle
+spectrograph on the 2.7-m Harlan J. Smith Telescope at the McDona ld Observatory showed
+that there was no significant velocity variation at the level of 1 kms−1, and therefore that an
+orbiting stellar companion could not be responsible for the observed transits. Furthermore,
+there was no sign of a composite spectrum or contamination by the s pectrum of an eclipsing
+binary. The McDonald spectra were classified by L. Buchhave by find ing the best match
+between the observed spectra and a library of synthetic spectra calculated by J. Laird for
+an extensive grid of stellar models (Kurucz 1992) using a line list develo ped by J. Morse.
+This yielded Teff= 6000±125K, log g= 4.0±0.2(cgs), and vsini= 4kms−1, very close to
+the final values reported in Table 2.
+4. FIES SPECTROSCOPY
+The FIbre-fed Echelle Spectrograph (FIES) on the 2.5-m Nordic Op tical Telescope
+(NOT) at La Palma, was not originally designed with very precise radial velocities in mind.
+In particular, the fiber feed does not incorporate a scrambler, th ere is no attempt to control
+the atmospheric pressure (e.g. by housing the optics in a vacuum en closure), and there is no
+correction of the images for atmospheric dispersion. However, th e spectrograph does reside
+in its own well-insulated room with active control of the temperature to a few hundreths K,
+with the result that the optics are quite stable. Furthermore, FIE S has good throughput,– 8 –
+partly because the seeing is often excellent at the NOT site, and an a utomatic guider keeps
+the image well centered on a fiber 1 .3′′in diameter. These advantages encouraged us to
+develop specialized observing procedures and a new data reduction pipeline with the goal of
+measuring radial velocities to better than 10ms−1for the relatively faint planet candidates
+identified by Kepler.
+To establish a wavelength calibration that tracks slow drifts during a long exposure,
+we adopted the strategy of obtaining strong exposures of a Thor ium-Argon hollow cathode
+lamp through the science fiber immediately before and after each sc ience exposure. Long
+science exposures are divided into three or more sub-exposures, to allow detection of and
+correction for radiation events. Contamination by scattered moo nlight can be a serious
+problem for very precise velocities of faint targets. FIES does not yet have a separate fiber
+for monitoring the sky brightness, so care is needed to avoid the mo on, especially if there
+are thin clouds.
+A new reduction and analysis pipeline optimized for measuring precise r adial velocities
+was developed by L. Buchhave. After extraction of intensity- and wavelength-calibrated
+spectra, relative velocities are derived for each echelle order by cr oss correlation against
+a combined template created by shifting all the observed spectra o f the same star to a
+common velocity scale and co-adding them. The final velocity for eac h observation is the
+mean of the results for the individual orders, weighted by the numb er of detected photons
+but not by the velocity information content. Orders with very low sig nal levels and orders
+contaminated by telluric lines are not used. The internal error of th e mean is estimated
+from the scatter over the orders.
+We observed Kepler-7 with FIES for an hour on each of ten consecu tive nights in
+October 2009. On every night we observed a standard star, HD 18 2488, soon before
+Kepler-7 and also soon after on half the nights. HD 182488 is conven iently located close to– 9 –
+theKeplerfield of view and is known from HIRES observations over several yea rs to be
+stable to better than 3ms−1, and thus was adopted as the primary velocity standard by the
+follow-up team. Our 15 velocities for HD 182488 show an rms of 7ms−1, with a slow drift
+pattern with an amplitude of several ms−1. Therefore we interpolated a correction to our
+velocity zero point for each observation of Kepler-7 by assuming th at HD 182488 should
+not vary. One of the 10 observations was obtained through clouds and clearly showed a
+distortion of the correlation peak due to contamination by scatter ed moonlight for several
+of the blue orders. This observation was rejected. The results fo r the other 9 observations
+are reported in Table 1, including the variations in the line bisectors an d errors.
+We fit a circular orbit to the 9 velocities reported in Table 1, adopting t he photometric
+ephemeris, which leaves the orbital semi-amplitude, K, and center-of-mass velocity, γ, as
+the only free parameters. A plot of this orbital solution is shown in Fig ure 2, together with
+the velocity residuals and the line bisector variations. There is no evid ence of a correlation
+between the velocities and the bisectors, which supports the inter pretation that the velocity
+variations are due to a planetary companion. The orbital paramete rs are listed in Table 2.
+Allowing the eccentricity to be a free parameter reduced the velocit y residuals by only a
+small amount and yielded an eccentricity that was not significantly diff erent from circular.
+A solution for a circular orbit using the velocities uncorrected for th e drifts exhibited by
+the standard star gave similar velocity residuals, but a smaller value o fKby 7.6ms−1,
+corresponding to an 18% smaller mass.
+The combined template spectrum for Kepler-7 from FIES was analyz ed by A. Sozzetti
+using MOOG2, to provide the stellar parameters needed to estimate the mass an d radius
+of the host star using stellar evolution tracks. The critical input pa rameters to the models
+areTeffand [Fe/H], but the spectroscopic log gis also of interest for a consistency check.
+2http://verdi.as.utexas.edu/moog.html– 10 –
+A spectrum of Kepler-7 obtained by H. Isaacson and G. Marcy with H IRES on Keck
+1 was analyzed by D. Fischer using SME (Valenti & Piskunov 1996), wit h very similar
+results:Teff= 5933±44 vs. 6000 ±75 K, [Fe /H] = +0.11±0.03 vs. +0 .13±0.07, and
+logg= 3.98±0.10 vs. 4.00±0.10(cgs), for SME and MOOG, respectively. For the
+results reported in Table 2, we used the SME values. The mean absolu te velocity of
+Kepler-7, +0 .40±0.10kms−1, was determined from the FIES observations by adopting
+−21.508kms−1as the velocity for the standard star HD 182488.
+5. DISCUSSION
+The analysis of the Keplerphotometry and the determination of the stellar and
+planetary parameters for Kepler-7 followed exactly the procedur es reported in Koch et al.
+(2010) and Borucki et al. (2010). The results are reported in Tab le 2. These results were
+checked and confirmed by independent analyses carried out by C. B urke and G. Torres.
+The Kepler-7 host star is not much hotter than the Sun, Teff= 6000±75K. However,
+it is more massive and considerably larger than the sun, M⋆= 1.347+0.072
+−0.054M⊙and
+R⋆= 1.843+0.048
+−0.066R⊙, which puts it in a region of the H-R Diagram near the end of its
+Main Sequence lifetime. Indeed, the Yale-Yonsei evolutionary trac ks have hooks that cross
+at the position of Kepler-7, and the probability distribution for the s tellar mass has two
+peaks. The stronger peak is for an evolutionary state not long bef ore Hydrogen burning
+in the core is exhausted with M⋆= 1.362±0.040M⊙andR⋆= 1.857±0.047R⊙, while
+the weaker peak corresponds to a state soon after the star sta rts to evolve rapidly, with
+M⋆= 1.204±0.035M⊙andR⋆= 1.781±0.042R⊙. The mass for the evolved peak is 12%
+smaller, and the radius is 4% smaller (as it must be to yield the same stella r density). The
+corresponding planetary radius is also 4% smaller, while the planetary mass is 8% smaller
+(because of the dependence on the 2/3 power of the system mass ). As our best guess for– 11 –
+the mass and radius of the host star and for the mass, radius, and density of the planet, in
+Table 2 we report the mode and errors for the corresponding prob ability distributions. This
+takes into account all the possible evolutionary states for the hos t star that are consistent
+with the observations.
+The planetary radius is fifty percent larger than that of Jupiter, RP= 1.478+0.050
+−0.051RJ,
+but the mass is less than half, MP= 0.433+0.040
+−0.041MJ, which leads to an unusually low density
+ofρP= 0.166+0.019
+−0.020gcm−3. Among the known planets, only WASP-17b appears to have a
+lower density (Anderson et al. 2009), although the actual value fo r that planet is not yet
+well determined. The position of Kepler-7b on the mass/radius diagr am is illustrated in
+Figure 3, which plots all of the transiting planets with known paramet ers as of 5 November
+2009. Because of possible systematic errors in the radial velocities measured using FIES,
+the mass of Kepler-7b may be smaller than we report by as much as 20 % or even more.
+However, the systematic error in the mass on the high side is unlikely t o be this large,
+because a larger orbital amplitude is less vulnerable to systematic ve locity errors. For the
+planetary radius, it is hard to avoid the conclusion that the planet is s trongly inflated,
+because the relatively long duration of the transit demands a low den sity and expanded
+radius for the star. A robust measure of the transit duration is th e time between the
+moment when the center of the planet crosses the limb of the star d uring ingress and the
+corresponding moment during egress. A general formula for this d uration including the
+effect of orbital eccentricity is given by P´ al et al. (2010), leading t o a value of 4 .63±0.06
+hours for Kepler-7. We conclude that future observational refin ements to the characteristics
+of Kepler-7b are more likely to decrease the density than increase it , with a significant
+uncertainty remaining as long as the evolutionary state of the host star is uncertain.
+Many people have contributed to the success of the Kepler Mission , and it is
+impossible to acknowledge them all by name. We offer our special than ks to the team of– 12 –
+scientists and programmers working with J. M. Jenkins to create th e photometric pipeline
+- H. Chandrasekaran, S. T. Bryson, J. Twicken, E Quintana, B. Cla rke, C. Allen, J. Li,
+P. Tenenbaum, and H. Wu; to C. J. Burke and G. Torres for running independent checks of
+the analysis of the Kepler-7 light curve and system parameters; to J. Andersen for help with
+the FIES observations and unwavering moral support; to M. Endl, H. Isaacson, D. Ciardi,
+G. Mandushev, N. Baliber, and M. Crane for important contribution s to the follow-up
+work; to A. Sozzetti for his analysis of the FIES combined template spectrum and to
+D. Fischer for her analysis of the HIRES template spectrum; to M. E verett and G. Esquerdo
+for critical contributions to the KIC; to E. Bachtel and his team at Ball Aerospace for
+their work on the Keplerphotometer; to R. Duren and R. Thompson for key contributions
+to engineering; and to C. Botosh, M. Haas, and J. Fanson, for able management. DWL
+gratefully acknowledges partial support from NASA Cooperative A greement NCC2-1390
+and the help of S. Cahill and L. McArthur-Hines. Funding for this Disc overy mission is
+provided by NASA’s Science Mission Directorate.
+Facilities: The Kepler Mission, NOT (FIES), Keck:I (HIRES), WIYN (Speckle)– 13 –
+REFERENCES
+Anderson, D. R., et al. ApJ, submitted, arXiv:0908.1553
+Batalha, N. M., et al. 2010, ApJ, this issue
+Borucki, W. J., et al. 2010, ApJ, this issue
+Dunham, E. W., et al 2010, ApJ, this issue
+Gautier, T. N., et al. 2010, ApJ, this issue
+Koch, D. G., et al. 2010, ApJ, this issue
+Kurucz, R. L. 1992, in The Stellar Populations of Galaxies, IAU Symp. No. 149, ed. B.
+Barbuy and A. Renzini (Kluwer Acad. Publ.: Dordrecht), 225
+Monet, D. G., et al. 2010, ApJ, this issue
+P´ al, A., et al. 2008, ApJ, 680, 1450
+P´ al, A., et al. 2010, MNRAS in press (arXiv:0908.1705)
+Skrutskie, M. J., et al. 2006, AJ, 131, 1163
+Valenti, J. A., & Piskunov, N. 1996, A&AS, 118, 595
+Yi, S. K., Demarque, P., Kim, Y.-C., Lee, Y.-W., Ree, C. H., Lejeune, T., & Barnes, S.
+2001, ApJS, 136, 417
+This manuscript was prepared with the AAS L ATEX macros v5.2.– 14 –
+Table 1. Relative Radial-Velocity Measurements of Kepler-7
+HJD Phase RV σRV BS σBS
+(days) (cycles) (ms−1) (ms−1) (ms−1) (ms−1)
+2455107.37937 28.677 +43 .7±6.8 +19 .9±7.3
+2455108.36845 28.879 +32 .7±7.1 +1 .5±5.4
+2455110.50735 29.317 −34.2±9.8 +4 .8±17.9
+2455111.40251 29.500 −11.5±6.7−4.0±7.2
+2455112.41378 29.707 +33 .2±8.2−4.6±5.4
+2455113.40824 29.911 +27 .9±6.1−12.0±8.2
+2455114.44632 30.123 −31.1±8.1−5.5±8.9
+2455115.44411 30.328 −29.2±10.7−14.8±8.9
+2455116.37077 30.517 −0.1±9.4 +13 .8±10.6– 15 –
+Table 2. System Parameters for Kepler-7
+Parameter Value Notes
+Transit and orbital parameters
+Orbital period P(d) 4 .885525±0.000040 A
+Midtransit time E(HJD) 2454967 .27571±0.00014 A
+Scaled semimajor axis a/R⋆ 7.22+0.16
+−0.13A
+Scaled planet radius RP/R⋆ 0.08241+0.00030
+−0.00043A
+Impact parameter b≡acosi/R⋆ 0.445+0.032
+−0.044A
+Orbital inclination i(deg) 86 .◦5±0.4 A
+Orbital semi-amplitude K(ms−1) 42 .9±3.5 A,B
+Orbital eccentricity e 0 (adopted) A,B
+Center-of-mass velocity γ(ms−1) 0 A,B
+Observed stellar parameters
+Effective temperature Teff(K) 5933 ±44 C
+Spectroscopic gravity log g(cgs) 3 .98±0.10 C
+Metallicity [Fe/H] +0 .11±0.03 C
+Projected rotation vsini(kms−1) 4 .2±0.5 C
+Mean radial velocity (kms−1) +0 .40±0.10 B
+Derived stellar parameters
+MassM⋆(M⊙) 1 .347+0.072
+−0.054C,D
+RadiusR⋆(R⊙) 1 .843+0.048
+−0.066C,D
+Surface gravity log g⋆(cgs) 4 .030+0.018
+−0.019C,D
+Luminosity L⋆(L⊙) 4 .15+0.63
+−0.54C,D
+Age (Gyr) 3.5±1.0 C,D
+Planetary parameters
+MassMP(MJ) 0 .433+0.040
+−0.041A,B,C,D
+RadiusRP(RJ, equatorial) 1 .478+0.050
+−0.051A,B,C,D
+Density ρP(gcm−3) 0 .166+0.019
+−0.020A,B,C,D
+Surface gravity log gP(cgs) 2 .691+0.038
+−0.045A,B,C,D
+Orbital semimajor axis a(AU) 0 .06224+0.00109
+−0.00084E
+Equilibrium temperature Teq(K) 1540 ±200 F
+Note. —
+A: Based on the photometry.– 16 –
+B: Based on the radial velocities.
+C: Based on a MOOG analysis of the FIES spectra.
+D: Based on the Yale-Yonsei stellar evolution tracks.
+E: Based on Newton’s version of Kepler’s Third Law and total m ass.
+F: Assumes Bond albedo = 0.1 and complete redistribution.– 17 –
+Fig. 1.— The detrended light curve for Kepler-7. The time series for t he entire data set
+is plotted in the upper panel. The lower panel shows the photometry folded by the period
+P= 4.885525days. Themodel fit tothe primarytransit is plottedinred, a ndour attempt to
+fit a corresponding secondary eclipse for a circular orbit is shown in g reen with an expanded
+and offset scale.– 18 –
+      -40-2002040Radial velocity (m s-1)
+       -15015 Residual (m s-1)
+0.0 0.2 0.4 0.6 0.8 1.0
+Phase -15015 Bisector
+var. (m s-1)
+Fig. 2.— a) The orbital solution for Kepler-7. The observed radial ve locities obtained with
+FIES on the Nordic Optical Telescope are plotted together with the velocity curve for a
+circular orbit with the period and time of transit fixed by the photome tric ephemeris. The
+γvelocity has been subtracted from the relative velocities here and in Table 1, and thus the
+center-of-mass velocity for the orbital solution is 0 by definition. b ) The velocity residuals
+from the orbital solution. The rms of the velocity residuals is 7.4ms−1. c) The variation in
+the bisector spans for the 9 FIES spectra. The mean value has bee n subtracted.– 19 –
+Fig. 3.— The Mass/Radius diagram for all the transiting planets with kn own parameters
+as of 5 November 2009. The four new Keplerplanets are labeled and plotted as diamonds.
+Kepler-7 has an unusually low density.
\ No newline at end of file
diff --git a/1001.0191.txt b/1001.0191.txt
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+arXiv:1001.0191v2  [math.RA]  30 Nov 2012GROUP GRADINGS ON RESTRICTED CARTAN TYPE
+LIE ALGEBRAS
+YURI BAHTURIN AND MIKHAIL KOCHETOV
+Abstract. For a given abelian group G, we classify the isomorphismclasses of
+G-gradings on the simple restricted Lie algebras of types W(m;1) andS(m;1)
+(m≥2), in terms ofnumerical and group-theoretical invariants . Ourmain tool
+is automorphism group schemes, which we determine for the si mple restricted
+Lie algebras of types S(m;1) andH(m;1). The ground field is assumed to be
+algebraically closed of characteristic p >3.
+1.Introduction
+LetUbe an algebra (not necessarily associative) over a field Fand letGbe a
+group, written multiplicatively.
+Definition 1.1. AG-grading onUis a vector space decomposition
+U=/circleplusdisplay
+g∈GUg
+such that
+UgUh⊂Ughfor allg,h∈G.
+Ugis called the homogeneous component of degreeg. Thesupportof theG-grading
+is the set
+{g∈G|Ug/\e}atio\slash= 0}.
+IfUis finite-dimensional, then, replacing Gwith the subgroup generated by the
+support of the grading, we may assume without loss of generality th atGis finitely
+generated.
+Definition 1.2. We say that two G-gradings,U=/circleplustext
+g∈GUgandU=/circleplustext
+g∈GU′
+g,
+areisomorphic if there exists an algebra automorphism ψ:U→Usuch that
+ψ(Ug) =U′
+gfor allg∈G,
+i.e.,U=/circleplustext
+g∈GUgandU=/circleplustext
+g∈GU′
+gare isomorphic as G-graded algebras.
+Since the same vector space decomposition can be regarded as a G-grading for
+different groups G, one can also define equivalence of gradings, which is a weaker
+relation than isomorphism — see e.g. [6] for a discussion. Fine gradings (i.e., those
+that cannot be refined) as well as their universal groups are of pa rticular interest.
+We are interested in the problem of finding all possible group gradings on finite-
+dimensional simple Lie algebras. If Lis a simple Lie algebra, then it is known that
+Keywords: graded algebra, simple Lie algebra, grading, Cartan type Li e algebra.
+2000 Mathematics Subject Classification: Primary 17B70; Secondary 17B60.
+The first author acknowledges support by by NSERC grant # 2270 60-04. The second author
+acknowledges support by NSERC Discovery Grant # 341792-07.
+12 BAHTURIN AND KOCHETOV
+the support of any G-gradingon Lgenerates an abelian group. Hence, in this paper
+we will always assume that Gisabelian. We will also assume that the ground field
+Fisalgebraically closed .
+For an arbitrary abelian group G, the classification of G-gradings (up to isomor-
+phism) is known for almost all classical simple Lie algebras over an algeb raically
+closed field of characteristic 0 or p >2. The classification of fine gradings (up to
+equivalence) is also known. The reader may refer to [2, 5] and refer ences therein.
+Our goal is to carry out the same classification for simple restricted Lie algebras
+of Cartan type. In the present paper, we achieve this goal for Wit t algebras and
+for special algebras (see definitions in Section 2) in characterisitc p>3: Theorem
+4.13 with Corollary 4.14 and Theorem 4.17 with Corollary 4.18, respectiv ely.
+In a number of cases, a fruitful approach to the classification of g radings by
+abelian groups on an algebra Uis to use another algebra, R, that shares with Uthe
+automorphism group scheme (see Section 3) and whose gradingsar e easier to study.
+This approach often requires equipping Rwith some additional structure. For
+classicalsimpleLiealgebrasofseries A,B,CandD,Risthematrixalgebra Mn(F),
+possibly equipped with an antiautomorphism. For Lie algebrasof Cart an type,Ris
+the “coordinate algebra” O(m;n) (see Definition 2.1), which, in the restricted case,
+is just the truncated polynomial algebra O(m;1) =F[x1,...,x m]/(xp
+1,...,xp
+m). It
+is not difficult to classify G-gradings on the algebra O(m;1) — see Theorem 4.8
+and Corollary 4.11. Since it is known that O(m;1) has the same automorphism
+groupscheme as the Witt algebra W(m;1), this immediately gives the classification
+of gradings for the latter. In Section 3, we show that the automor phism group
+schemeofthespecialalgebra S(m;1)(1),m≥3—respectively,Hamiltonianalgebra
+H(m;1)(2),m= 2r— is isomorphic to the stabilizer of the differential form ωS—
+respectively, ωH—intheautomorphismgroupschemeof O(m;1); seeTheorems3.2
+and 3.5, respectively. As a consequence, all gradings on S(m;1)(1)andH(m;1)(2)
+come from gradings on O(m;1). However, this does not yet give a classification
+of gradings on S(m;1)(1)andH(m;1)(2)up to isomorphism, because one must
+make sure that the isomorphism preserves the appropriate differe ntial form — see
+Corollary 4.2. With some more work, we obtain a classification of gradin gs on
+S(m;1)(1),m≥3, andS(2;1)(2)=H(2;1)(2). The classification for H(2r;1)(2)
+withr>1 remains open.
+The paper is structured as follows. In Section 2, we recall the defin itions and
+basic facts regarding Lie algebras of Cartan type. In Section 3, we briefly recall
+background information on automorphism group schemes and dete rmine them for
+S(m;1)(1)andH(m;1)(2). In Section 4, we obtain the classification of gradings for
+Witt and special algebras (using the results on automorphism group schemes).
+2.Cartan type Lie algebras
+We start by briefly recalling the definitions and relevant properties o f Cartan
+type Lie algebras. We will use [11] as a standard reference. Fix m≥1 and
+n= (n1,...,n m) whereni≥1. Set
+Z(m;n):={α∈Zm|0≤αi<pnifori= 1,...,m}.
+The elements of Z(m;n)will be called multi-indices and denoted by Greek letters α,
+β,γ. Forα= (α1,...,α m), we set
+|α|=α1+···+αm.GRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 3
+We will denote by 1 the multi-index that has 1 in all positions and by εithe multi-
+index that has 1 in position iand zeros elsewhere.
+LetFbe a field of characteristic p>0.
+Definition 2.1. The algebra O=O(m;n) overFis a commutative associative
+algebra with a basis {x(α)|α∈Z(m,n)}where multiplication given by
+x(α)x(β)=/parenleftbiggα+β
+α/parenrightbigg
+x(α+β)where/parenleftbiggα+β
+α/parenrightbigg
+=m/productdisplay
+i=1/parenleftbiggαi+βi
+αi/parenrightbigg
+.
+Ifn= 1, thenO∼=F[x1,...,x m]/(xp
+1,...,xp
+m) by identifying xiwithx(εi):
+x(α)=1
+α1!···αm!xα1
+1···xαmm.
+The algebra Ohas a canonical Z-grading O=/circleplustext
+ℓ≥0Oℓdefined by declaring the
+degree ofx(α)to be|α|. The associated filtration will be denoted by
+O(ℓ):=/circleplusdisplay
+j≥ℓOj.
+Note that M:=O(1)is the unique maximal ideal of O.
+We will now define the following graded Cartan type Lie algebras: Witt, special
+and Hamiltonian. The contact algebras and generalized (i.e., non-gra ded) Cartan
+type Lie algebras will not be considered in this paper.
+Definition 2.2. Define a linear map ∂i:O → O by∂ix(α)=x(α−εi)where the
+right-hand side is understood to be zero if αi= 0. Then ∂iis a derivation of O.
+TheWitt algebra W=W(m;n) is the subalgebra of Der( O) that consists of all
+operators of the form
+f1∂1+···+fm∂mwherefi∈ O.
+The canonical Z-grading of Oinduces a Z-grading on End( O). SinceWis a
+graded subspace of End( O), it inherits the Z-grading:W=/circleplustext
+ℓ≥−1Wℓ. We will
+denote the associated filtration by W(ℓ).
+The de Rahm complex Ω0d−→Ω1d−→Ω2d−→...is defined as follows: Ω0=O,
+Ω1= Hom O(W,O), and Ωk= (Ω1)∧kfork≥2. The map d: Ω0→Ω1is defined
+by (df)(D) =D(f) for allf∈ OandD∈W. The remaining maps d: Ωk→Ωk+1
+are defined in the usual way: d(fdxi1∧···∧dxik) =df∧dxi1∧···∧dxik.
+Any element D∈Wacts on Ω1= Hom O(W,O) by setting
+D(ω)(E) =D(ω(E))−ω([D,E]) for allω∈Ω1andE∈W.
+This action turns all Ωk= (Ω1)∧kintoW-modules. Of course, all Ωkalso have
+canonical Z-gradings and associated filtrations.
+WewillneedthefollowingdifferentialformstodefinethespecialandH amiltonian
+algebras:
+ωS:=dx1∧dx2∧···∧dxm∈Ωm, (m≥2);
+ωH:=dx1∧dxr+1+dx2∧dxr+2+···+dxr∧dx2r∈Ω2(m= 2r).
+Definition 2.3. Thespecial algebra S=S(m;n) is the stabilizerof ωSinW(m;n):
+S={D∈W|D(ωS) = 0}.4 BAHTURIN AND KOCHETOV
+TheHamiltonian algebra H=H(m;n) is the stabilizer of ωHinW(m;n):
+H={D∈W|D(ωH) = 0}.
+Note that in the case m= 2, we have ωS=ωHand henceS=H. It is well-
+known that W(m;n) is simple unless p= 2 andm= 1. The algebras S(m;n) and
+H(m;n) are not simple, but the first derived algebra S(m;n)(1)(m≥3) and the
+second derived algebra H(m;n)(2)are simple.
+The Lie algebras W(m;n),S(m;n) andH(m;n) arerestrictable — i.e., admit
+p-maps making them restricted Lie algebras — if and only if n= 1. From now on,
+we will assume that this is the case. Since W,SandHhave trivial center, their
+p-maps are unique. Also, in this case W= Der(O) and hence the p-map is just the
+p-th power in the associative algebra End( O). The algebras SandHare restricted
+subalgebras of W.
+3.Automorphism group schemes
+LetO=O(m;1). Any automorphism µof the algebra Ogives rise to an auto-
+morphism Ad( µ) ofWgiven by Ad( µ)(D) =µ◦D◦µ−1. Then we can define the
+action ofµon Ω1= Hom O(W,O) by setting µ(ω)(D) =µ(ω(Ad(µ−1)(D))) for
+allω∈Ω1andD∈W. This action turns all Ωk= (Ω1)∧kinto Aut( O)-modules.
+Clearly, these actions can still be defined in the same way if we extend the scalars
+from the base field Fto any commutative associative F-algebraK, i.e., replace O
+withO(K) :=O ⊗K,WwithW(K) :=W⊗Kand Ωkwith Ωk(K) := Ωk⊗K.
+Recall the automorphism group scheme of a finite-dimensional F-algebraU(see
+e.g. [14] for background on affine group schemes). As a functor, t he (affine) group
+schemeAut(U)isdefinedbysetting Aut(U)(K) = Aut K(U⊗K)foranycommuta-
+tiveassociative F-algebraK. Fromthe abovediscussionitfollowsthatwehavemor-
+phisms of group schemes Ad : Aut(O)→Aut(W) and also Aut(O)→GL(Ωk).
+(We identify a smooth algebraic group scheme such as GL( V) with the correspond-
+ing algebraic group.) Note that, since the p-map ofW⊗Kis uniquely determined,
+the automorphism group scheme of Was a Lie algebra is the same as its au-
+tomorphism group scheme as a restricted Lie algebra. Note also tha t the maps
+d: Ωk→Ωk+1areAut(O)-equivariant.
+The algebraic group scheme Aut(U) contains the algebraic group Aut( U) as the
+largest smooth subgroupscheme. The tangent Lie algebra of Aut(U) is Der(U), so
+Aut(U) is smooth if and only if Der( U) equals the tangent Lie algebra of the group
+Aut(U). The automorphism group schemes of simple Cartan type Lie algebr as,
+unlikethoseoftheclassicalsimpleLiealgebras,arenotsmooth. Ind eed, thetangent
+Lie algebra of Aut( O) isW(1), which is a proper subalgebra of W= Der(O), so
+Aut(O) is not smooth. In view of the following theorem, we see that Aut(W) is
+not smooth.
+Theorem 3.1 ([13]).LetO=O(m;1)andW=W(m;1). Assumep >3. Then
+the morphism Ad :Aut(O)→Aut(W)is an isomorphism of group schemes. /square
+The automorphism group scheme of the general W(m;n) is also known — see
+[15] (p >2, with small exceptions in the case p= 3) and [9, 10] (any p, with
+small exceptions in the cases p= 2 andp= 3). In particular, Theorem 3.1 holds
+forp= 3 ifm≥2 and forp= 2 ifm≥3. In this section we will establish
+analogues of Theorem 3.1 for the simple algebras S(m;1)(1)andH(m;1)(2). WeGRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 5
+will follow the approach of [13]. Suppose Φ : G→His a morphism of algebraic
+group schemes. Let GredandHredbe the largest smooth subgroupschemes, which
+will be regarded as algebraic groups. In order for Φ to be an isomorp hism, the
+following two conditions are necessary:
+A) the restriction Φ : Gred→Hredis a bijection;
+B) the tangent map Lie(Φ) : Lie( G)→Lie(H) is a bijection.
+However, unless Gis known to be smooth (i.e., G=Gred), these two conditions
+arenotsufficient for Φ to be an isomorphism. In general, one has to show tha t
+the associated map of distribution algebras /tildewideΦ :G→His surjective. (The two
+conditions above imply that /tildewideΦ :G→His injective.)
+Recall that the distribution algebra Gof an algebraic group scheme Gis a con-
+nected cocommutative Hopf algebra (see e.g. [8] for background on Hopf algebras),
+with the space of primitive elements Prim( G) = Lie(G) of finite dimension. Hence
+Lie(G) has a descending chain of restricted Lie subalgebras (see e.g. [12] or [4, II,
+§3, No. 2]):
+Lie(G) = Lie0(G)⊃Lie1(G)⊃Lie2(G)⊃...,
+defined by Liek(G) := Liek(G) =Vk(G)∩Prim(G), where V:G→Gis the Ver-
+schiebung operator (see e.g. [12, Theorem 1] or [4, II, §2, No. 7]). The intersection
+of this chain is Lie( Gred), which can be identified with the tangent algebra of the
+algebraic group Gred.
+Recall that a sequence of elements 1 =0h,1h,...,nhin a connected cocom-
+mutative Hopf algebra Gis called a sequence of divided powers (lying over1h) if
+∆(jh) =/summationtextj
+i=0ih⊗j−ihfor allj= 1,...,n. It follows that1h∈Prim(G) and
+thatε(jh) = 0 for all j= 1,...,n. It is easy to see that Vk(pkh) =1hfor any
+pk≤n. Hence, if there exists a sequence of divided powers of length pklying over
+h∈Prim(G), thenh∈Liek(G). The converse is also true [12, Theorem 2].
+Now we come back to the problem of proving that a morphism Φ : G→Hof
+algebraic group schemes is an isomorphism. Assuming that Φ satisfies the above
+two conditions, we need to show that the Hopf subalgebra /tildewideΦ(G)⊂Hin fact equals
+H. Regarding Has a Hopf subalgebra in the distribution algebra of GL( V) for a
+suitable space V, we can apply [4, II, §3, No. 2, Corollary 1] to conclude that
+/tildewideΦ(G) =Hif and only if
+C) Lie(Φ) maps Liek(G) onto Liek(H) for allk.
+Here we are interested in the case G=Aut(O), where O=O(m;1), and its
+subgroupschemes AutS(O) := Stab G(/a\}bracketle{tωS/a\}bracketri}ht) andAutH(O) := Stab G(/a\}bracketle{tωH/a\}bracketri}ht). We
+have
+Aut(O)red= Aut(O), Lie(Aut(O)) = Der( O) =W;
+AutS(O)red= Stab Aut(O)(/a\}bracketle{tωS/a\}bracketri}ht),Lie(AutS(O)) = Stab W(/a\}bracketle{tωS/a\}bracketri}ht) =:CS;
+AutH(O)red= Stab Aut(O)(/a\}bracketle{tωH/a\}bracketri}ht),Lie(AutH(O)) = Stab W(/a\}bracketle{tωH/a\}bracketri}ht) =:CH.
+We will denote Stab Aut(O)(/a\}bracketle{tωS/a\}bracketri}ht) by Aut S(O) and Stab Aut(O)(/a\}bracketle{tωH/a\}bracketri}ht) by Aut H(O)
+for brevity.
+Assumep >3. It is known that the morphism Ad : Aut(O)→Aut(W)
+as well as its restrictions AutS(O)→Aut(S(1)) (m≥3) andAutH(O)→
+Aut(H(2)) (m= 2r) induce bijections Aut( O)→Aut(W), Aut S(O)→Aut(S(1))
+and Aut H(O)→Aut(H(2)) — see e.g. [11, Theorem 7.3.2]. Also, the tangent map
+ad :W→Der(W) as well asits restrictions CS→Der(S(1)) andCH→Der(H(2))6 BAHTURIN AND KOCHETOV
+are bijective — see e.g. [11, Theorem 7.1.2]. Thus, the above condition s A) and B)
+are satisfied for the morphisms Aut(O)→Aut(W),AutS(O)→Aut(S(1)) and
+AutH(O)→Aut(H(2)).
+By [1, Lemma 3.5, 2], Liek(Aut(O)) = Stab W(M) =W(0)for allk>0. Hence,
+fork>0, we have
+Liek(AutS(O))⊂Liek(Aut(O))∩CS=CS(0),
+Liek(AutH(O))⊂Liek(Aut(O))∩CH=CH(0).
+On the other hand, Lie(Aut S(O)) =CS(0)and Lie(Aut H(O)) =CH(0). It follows
+that, in fact,
+(1) Liek(AutS(O)) =CS(0)and Liek(AutH(O)) =CH(0)for allk>0.
+By [1, Lemma 3.5, 4], Liek(Aut(W)) = ad(W(0)) for allk >0, so condition
+C) is satisfied for the morphism Aut(O)→Aut(W). This is how Theorem 3.1 is
+proved in [13]. We are now ready to prove our analogues for SandH.
+Theorem 3.2. LetO=O(m;1),m≥3, andS(1)=S(m;1)(1). Let
+AutS(O) = Stab Aut(O)(/a\}bracketle{tωS/a\}bracketri}ht).
+Assumep>3. Then the morphism Ad :AutS(O)→Aut(S(1))is an isomorphism
+of group schemes.
+Proof.By the above discussion, we have to prove that condition C) is satisfi ed for
+Ad :AutS(O)→Aut(S(1)). LetGbe the distribution algebra of Aut(S(1)). In
+view of (1), it suffices to show that Lie1(G)⊂ad(CS(0)). In other words, we have
+to verify, for any D∈CS, that ifD /∈CS(0), then adD /∈Lie1(G). We can write
+D=λ1∂1+···+λm∂m+D0whereD0∈CS(0)and the scalars λ1,...λmare not
+all zero. Now, D0∈Lie1(AutS(O)) implies ad D0∈Lie1(G), so it suffices to prove
+that ad(λ1∂1+···+λm∂m)/∈Lie1(G). Applying an automorphism of Oinduced
+by a suitable linear transformation on the space Span {∂1,...,∂ m}, we may assume
+without loss of generality that D=∂1.
+By way of contradiction, assume that ad ∂1∈Lie1(G). Then there exists a
+sequence of divided powers 1 =0h,1h,...,phinGsuch that1h= ad∂1.
+As pointed out in the proof of [1, Lemma 3.5, 4], it follows from [12, Le mma 7]
+thatwemayassumewithoutlossofgeneralitythatkh=1
+k!(1h)kfork= 0,...,p−1.
+The distribution algebra Gacts canonically on S(1), so we have a homomorphism
+η:G→End(S(1)). The restriction of ηto Prim( G) = Der(S(1)) is the identity
+map. Letkδ=η(kh) fork= 0,...,p. Then we have
+(2)kδ=1
+k!(ad∂1)kfork= 0,...,p−1,
+and, sinceS(1)is aG-module algebra,
+(3)kδ([X,Y]) =k/summationdisplay
+j=0/bracketleftbigjδ(X),k−jδ(Y)/bracketrightbig
+for allk= 0,...,pandX,Y∈S(1).
+Note that the action of GonS(1)extends canonically to the universal enveloping
+algebraU(S(1)) and, since the p-map ofS(1)is uniquely determined, the G-action
+passes on to the restricted enveloping algebra u(S(1)). By abuse of notation, weGRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 7
+will usekδto denote the action ofkhonu(S(1)) as well as on S(1). We note for
+future reference that
+(4)kδ(XY) =k/summationdisplay
+j=0(jδ)(X)(k−jδ)(Y) for allk= 0,...,pandX,Y∈u(S(1)).
+Observe that if we replacepδbypδ+ξ, whereξis any derivation of S(1), then
+equations (3) and (4) will still hold (with the same0δ,...,p−1δ). We will use this
+observation to simplify the operatorpδ.
+Letzi= 1+xi,i= 1,...,m. For each multi-index α∈Z(m;1), we set
+zα=zα1
+1···zαm
+m.
+Sincezp
+i= 1 for alli, we may regard the components of αas elements of the cyclic
+groupZpwhen dealing with zα. It is this property that will make the basis {zα}
+ofO=O(m;1) more convenient for us than the standard basis {xα}.
+Recall [11, Section 4.2] that S(1)=S(m;1)(1)is spanned by the elements of the
+form
+Di,j(f) :=∂j(f)∂i−∂i(f)∂j
+wheref∈ Oand 1≤i<j≤m. Hence,S(1)is spanned by the elements
+Di,j(zα) =αjzα−εj∂i−αizα−εi∂j.
+For the calculations we are about to carry out, we will need the follow ing:
+Lemma 3.3. For any 1≤i<j≤mandα,β∈Z(m;1), the commutator
+/bracketleftbig
+D1,2(zα),Di,j(zβ)/bracketrightbig
+is given by the following expressions:
+(5) −(α1β2−α2β1)Di,j(zα+β−ε1−ε2)if2<i<j;
+(6) −(α1(β2−1)−α2β1)D2,j(zα+β−ε1−ε2)if2 =i<j;
+(7)−α1βjD1,2(zα+β−ε1−εj)+α2β1D1,j(zα+β−ε1−ε2)−α1β1D2,j(zα+β−2ε1)
+ifi= 1,j >2;
+(8) −(α1β2−α2β1)D1,2(zα+β−ε1−ε2)ifi= 1,j= 2.
+Proof.The verification of the above expressions is straightforward. Her e we will
+verify (7), which is somewhat special, and leave the rest to the read er. We have,
+forj >2,
+/bracketleftbig
+D1,2(zα),D1,j(zβ)/bracketrightbig
+=/bracketleftbig
+α2zα−ε2∂1−α1zα−ε1∂2,βjzβ−εj∂1−β1zβ−ε1∂j/bracketrightbig
+=α2βjβ1zα+β−ε1−ε2−εj∂1−α2βjα1zα+β−ε1−ε2−εj∂1
++α2β1αjzα+β−ε1−ε2−εj∂1−α2β1(β1−1)zα+β−2ε1−ε2∂j
++α1βj(α1−1)zα+β−2ε1−εj∂2−α1βjβ2zα+β−ε1−ε2−εj∂1
++α1β1β2zα+β−2ε1−ε2∂j−α1β1αjzα+β−2ε1−εj∂2
+=(α2βjβ1−α2βjα1+α2β1αj−α1βjβ2)zα+β−ε1−ε2−εj∂1
++α1(βj(α1−1)−β1αj)zα+β−2ε1−εj∂2
++β1(−α2(β1−1)+α1β2)zα+β−2ε1−ε2∂j.8 BAHTURIN AND KOCHETOV
+Comparing the above with
+D1,2(zα+β−ε1−εj) =(α2+β2)zα+β−ε1−ε2−εj∂1−(α1+β1−1)zα+β−2ε1−εj∂2,
+D1,j(zα+β−ε1−ε2) =(αj+βj)zα+β−ε1−ε2−εj∂1−(α1+β1−1)zα+β−2ε1−ε2∂j,
+D2,j(zα+β−2ε1) =(αj+βj)zα+β−2ε1−εj∂2−(α2+β2)zα+β−2ε1−ε2∂j,
+one readily sees that (7) holds. /square
+Another useful fact is the following:
+(9) [ ∂ℓ,Di,j(f)] =Di,j(∂ℓ(f)) for alli,j,ℓ= 1,...,mandf∈ O.
+The remaining part of the proof of Theorem 3.2 will be divided into four steps.
+Step 1: Without loss of generality, we may assume
+(10)pδ(D1,2(z1z2)) = 0.
+Substituting α=ε1+ε2into expressions (5) through (8), we see that each
+nonzero element Di,j(zβ) is an eigenvector for the operator ad D1,2(z1z2), with
+eigenvalueλ(i,j,β) =β1−β2−1,β1−β2orβ1−β2+1, depending on i,j. (For the
+casei= 1 andj >2, one has to combine the first and the third terms in expression
+(7), which gives −(β2+ 1)D1,j(zβ).) Sinceλ(i,j,β) is in the field GFp, we have
+D1,2(z1z2) =D1,2(z1z2)p. Applying the operatorpδto both sides and using (4),
+we obtain:
+pδ(D1,2(z1z2)) =/summationdisplay
+i1+···+ip=p(i1δ)(D1,2(z1z2))···(ipδ)(D1,2(z1z2))
+Taking into account (2) and (9), we see thatkδ(D1,2(z1z2)) = 0 for 1 < k < p .
+Hence,
+pδ(D1,2(z1z2)) =/parenleftbig1δ(D1,2(z1z2))/parenrightbigp(11)
++p−1/summationdisplay
+k=0D1,2(z1z2)k(pδ)(D1,2(z1z2))D1,2(z1z2)p−k−1.
+Since1δ(D1,2(z1z2)) = [∂1,D1,2(z1z2)] =D1,2(z2) =∂1and∂p
+1= 0, the first term
+on the right-hand side of (11) vanishes. The second term can be re written using
+the identity
+p−1/summationdisplay
+k=0XkYXp−k−1= (adX)p−1(Y),
+whereX=D1,2(z1z2) andY=pδ(X). Thus, (11) yields
+Y= (adD1,2(z1z2))p−1(Y).
+It follows that Ycan be written as a linear combination of those Di,j(zβ) for which
+the eigenvalue λ(i,j,β) is nonzero:
+pδ(D1,2(z1z2)) =/summationdisplay
+i,j,β:λ(i,j,β)/ne}ationslash=0σi,j
+βDi,j(zβ).
+It remains to replacepδwith
+pδ+/summationdisplay
+i,j,βσi,j
+β
+λ(i,j,β)adDi,j(zβ),GRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 9
+to complete Step 1.
+Step 2: Without loss of generality, we may assume that, in addition to (10), we
+also have
+(12)pδ(∂1) =/summationdisplay
+i,j,β:β1=p−1τi,j
+βDi,j(zβ)
+for some scalars τi,j
+β.
+By (9), we have [ ∂1,D1,2(z1z2)] =∂1. Applying the operatorpδto both sides
+and using (3), (2) and (10), we obtain:
+pδ(∂1) =p/summationdisplay
+k=0/bracketleftbigkδ(∂1),p−kδ(D1,2(z1z2))/bracketrightbig
+= [pδ(∂1),D1,2(z1z2)].
+Hence [D1,2(z1z2),Y] =−YwhereY=pδ(∂1). It follows that Ycan be written as
+a linear combination of those Di,j(zβ) for which the eigenvalue λ(i,j,β) is−1:
+(13)pδ(∂1) =/summationdisplay
+i,j,β:λ(i,j,β)=−1τi,j
+βDi,j(zβ).
+Now replacepδwith
+pδ+/summationdisplay
+i,j,β:β1/ne}ationslash=p−1τi,j
+β
+β1+1adDi,j(zβ+ε1).
+Using (9), one readily sees that, for the newpδ, we have (12), because all terms
+withβ1/\e}atio\slash=p−1 in the right-hand side of (13) will cancel out. It remains to check
+that we still have (10) for the newpδ. In other words, we have to check that
+[D1,2(z1z2),Di,j(zβ+ε1)] = 0 for all i,j,βwithτi,j
+β/\e}atio\slash= 0. But this is clear, because
+λ(i,j,β+ε1) =λ(i,j,β)+1 and thus we have λ(i,j,β+ε1) = 0 for all i,j,βthat
+occur in the right-hand side of (13). Step 2 is complete.
+Step 3: For any element X=f1∂1+···+fm∂m∈W, define
+pr1(X) :=f1∈ O.
+Assume (10) and (12). Then, for any k= 1,...,p−1, pr1/parenleftbigpδ(D1,2(zk
+1z2))/parenrightbig
+is a
+linear combination of zγwith 0≤γ1<k.
+We proceed by induction on k. The basis for k= 1 follows from (10). Now
+suppose the claim holds for some k≥1. By (9), we have
+[∂1,D1,2(zk+1
+1z2)] = (k+1)D1,2(zk
+1z2).
+Applying the operatorpδto both sides, we obtain
+(14)pδ([∂1,D1,2(zk+1
+1z2)]) = (k+1)(pδ)(D1,2(zk
+1z2)).
+Using (3),(2) and (12), the left-hand side of (14) becomes
+
+/summationdisplay
+i,j,β:β1=p−1τi,j
+βDi,j(zβ),D1,2(zk+1
+1z2)
++/bracketleftbig
+∂1,pδ(D1,2(zk+1
+1z2))/bracketrightbig
+.
+SettingY=pδ(D1,2(zk+1
+1z2)), we can rewrite (14) as follows:
+(15) [∂1,Y] = (k+1)(pδ)(D1,2(zk
+1z2))+/summationdisplay
+i,j,β:β1=p−1τi,j
+β/bracketleftbig
+D1,2(zk+1
+1z2),Di,j(zβ)/bracketrightbig
+.10 BAHTURIN AND KOCHETOV
+Our goal is to show that monomials zγwithγ1≥k+1 do not occur in f1:=
+pr1(Y). Since pr1[∂1,Y] =∂1f1, it suffices to show that elements zγ∂1withγ1≥k
+do not occur in the right-hand side of (15), when it is regarded as an element of
+W. The induction hypothesis tells us that such elements do not occur in the first
+term of the right-hand side of (15). We are going to prove the same for the second
+term.
+In the case 2 <i<j, we have by (5):
+/bracketleftbig
+D1,2(zk+1
+1z2),Di,j(zβ)/bracketrightbig
+= (β1−β2(k+1))Di,j(zβ+kε1).
+Hence, no elements zγ∂1occur here.
+In the case 2 = i<j, we have by (6):
+/bracketleftbig
+D1,2(zk+1
+1z2),Di,j(zβ)/bracketrightbig
+= (β1−(β2−1)(k+1))D2,j(zβ+kε1).
+Again, no elements zγ∂1occur.
+In the case i= 1 and 2<j, we can write/bracketleftbig
+D1,2(zk+1
+1z2),Di,j(zβ)/bracketrightbig
+, using (7) and
+β1=p−1, as follows:
+−(k+1)βjD1,2(zβ+kε1+ε2−εj)−D1,j(zβ+kε1)+(k+1)D2,j(zβ+(k−1)ε1+ε2).
+Elementszγ∂1occur only in the first two summands, and we have γ=β+kε1−εj
+in either case. Therefore, γ1=β1+k=k−1 modp(recall that we may take the
+exponents of zmodulop).
+In the case i= 1,j= 2, we have by β1=p−1 and (8):
+/bracketleftbig
+D1,2(zk+1
+1z2),D1,2(zβ)/bracketrightbig
+=−((k+1)β2+1)D1,2(zβ+kε1).
+Hence, elements zγ∂1occur with γ=β+kε1−ε2. Once again, γ1=k−1 modp.
+The inductive proof of Step 3 is complete.
+Step 4: We can finally obtain a contradiction. By (8), we have
+/bracketleftBig
+D1,2(zp−1
+1z2),D1,2(z2
+1z2)/bracketrightBig
+= 3D1,2(z2) = 3∂1.
+Applyingpδand taking into account (3), (2) and (12), we obtain:
+3/summationdisplay
+i,j,β:β1=p−1τi,j
+βDi,j(zβ) (16)
+=/bracketleftBig
+pδ(D1,2(zp−1
+1z2)),D1,2(z2
+1z2)/bracketrightBig
++/bracketleftBig
+D1,2(zp−1
+1z2),pδ(D1,2(z2
+1z2))/bracketrightBig
++/bracketleftbigg
+D1,2/parenleftbigg1
+(p−1)!∂p−1
+1(zp−1
+1z2)/parenrightbigg
+,D1,2/parenleftbigg1
+1!∂1(z2
+1z2)/parenrightbigg/bracketrightbigg
++/bracketleftbigg
+D1,2/parenleftbigg1
+(p−2)!∂p−2
+1(zp−1
+1z2)/parenrightbigg
+,D1,2/parenleftbigg1
+2!∂2
+1(z2
+1z2)/parenrightbigg/bracketrightbigg
+.
+One readily verifies that the sum of the third and fourth terms in the right-hand
+side of (16) is 3 ∂1. Let us consider the first and the second terms. Denote
+X:=pδ(D1,2(zp−1
+1z2)) andY:=pδ(D1,2(z2
+1z2)).
+By Step 3, we know that pr1(X) is a linear combination of zγwith 0≤γ1≤p−2
+and pr1(Y) is a linear combination of zγwith 0≤γ1≤1.GRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 11
+SinceD1,2(z2
+1z2) =z2
+1∂1−2z1z2∂2, we see that the coefficient of ∂1depends only
+onz1and hence all terms with ∂1in the commutator [ X,D1,2(z2
+1z2)] come from
+terms with ∂1inX. In other words,
+pr1[X,D1,2(z2
+1z2)] = pr1[pr1(X)∂1,D1,2(z2
+1z2)].
+Since
+[zγ∂1,D1,2(z2
+1z2)] = (2−γ1+2γ2)zγ+ε1∂1−2zγ+ε2∂2,
+we conclude that pr1[X,D1,2(z2
+1z2)] is a linear combination of monomials zγ+ε1
+with 0≤γ1≤p−2. Therefore, elements zα∂1withα1= 0 do not occur in the
+first term in the right-hand side of (16).
+SinceD1,2(zp−1
+1z2) =zp−1
+1∂1+zp−2
+1z2∂2, we also see that
+pr1[D1,2(zp−1
+1z2),Y] = pr1[D1,2(zp−1
+1z2),pr1(Y)∂1].
+Since
+[D1,2(zp−1
+1z2),zγ∂1] = (1+γ1+γ2)zγ+(p−2)ε1∂1+2zγ+(p−3)ε1+ε2∂2,
+weconcludethatpr1[D1,2(zp−1
+1z2),Y]isalinearcombinationofmonomials zγ+(p−2)ε1
+with 0≤γ1≤1. Therefore, elements zα∂1withα1= 0 do not occur in the second
+term in the right-hand side of (16).
+Finally, all elements zα∂1that occur in the left-hand side of (16) have α1=p−1.
+Summarizing our analysis, we obtain
+3∂1= 0,
+which is a contradiction, since p>3. The proof of Theorem 3.2 is complete. /square
+Corollary 3.4. Under the assumptions of Theorem 3.2,
+Liek(Aut(S(1))) = ad(CS(0))for allk>0.
+Theorem 3.5. LetO=O(m;1),m= 2r, andH(2)=H(m;1)(2). Let
+AutH(O) = Stab Aut(O)(/a\}bracketle{tωH/a\}bracketri}ht).
+Assumep >3. Then the morphism Ad :AutH(O)→Aut(H(2))is an isomor-
+phism of group schemes.
+Proof.LetGbe the distribution algebra of Aut(H(2)). As in the proof of Theorem
+3.2, we have to show that Lie1(G)⊂ad(CH(0)) — i.e., for any D∈CH, ifD /∈
+CH(0), thenadD /∈Lie1(G). Again, wecanwrite D=λ1∂1+···+λm∂m+D0where
+D0∈CH(0)and the scalars λ1,...λmare not all zero. Since D0∈Lie1(AutH(O))
+implies adD0∈Lie1(G), it suffices to prove that ad( λ1∂1+···+λm∂m)/∈Lie1(G).
+Applying an automorphism of Oinduced by a suitable symplectic transformation
+on the space Span {∂1,...,∂ m}, we may assume without loss of generality that
+D=∂1.
+By way of contradiction, assume that ad ∂1∈Lie1(G). Then there exists a
+sequence of divided powers 1 =0h,1h,...,phinGsuch that1h= ad∂1. We may
+assume without loss of generality thatkh=1
+k!(1h)kfork= 0,...,p−1.
+Thedistributionalgebra Gactscanonicallyon H(2), sowehaveahomomorphism
+η:G→End(H(2)). The restriction of ηto Prim( G) = Der(H(2)) is the identity
+map. Letkδ=η(kh) fork= 0,...,p. Then we have
+(17)kδ=1
+k!(ad∂1)kfork= 0,...,p−1,12 BAHTURIN AND KOCHETOV
+and, sinceH(2)is aG-module algebra,
+(18)kδ([X,Y]) =k/summationdisplay
+j=0[jδ(X),k−jδ(Y)] for allk= 0,...,pandX,Y∈H(2).
+For the extended action of Gon the restricted enveloping algebra u(H(2)), we have
+(19)kδ(XY) =k/summationdisplay
+j=0(jδ)(X)(k−jδ)(Y) for allk= 0,...,pandX,Y∈u(H(2)).
+We may replacepδbypδ+ξ, whereξis any derivation of H(2), without affecting
+equations (18) and (19). We will use this observation to simplify the o peratorpδ.
+As in [11, Section 4.2], we define
+σ(i) :=/braceleftbigg
+1 ifi= 1,...,r;
+−1 ifi=r+1,...,2r;andi′:=i+σ(i)r.
+Also we define a map DH:O →Has follows:
+DH(f) :=2r/summationdisplay
+i=1σ(i)∂i(f)∂i′.
+Note that the kernel of DHisF1.
+Letzi= 1+xi,i= 1,...,2r. For each multi-index α∈Z(2r;1), we set
+zα=zα1
+1···zα2r
+2r.
+Wemayregardthecomponentsof αaselements ofthe cyclicgroup Zpwhendealing
+withzα. For the calculations we are about to carry out, we will need the follo wing
+formula:
+(20) [ DH(zα),DH(zβ)] =DH/parenleftbig
+DH(zα)(zβ)/parenrightbig
+for allα,β∈Z(2r;1).
+In particular,
+(21) [∂ℓ,DH(zβ)] =DH/parenleftbig
+∂ℓ(zβ)/parenrightbig
+for allℓ= 1,...,2randβ∈Z(2r;1).
+Letτ= (p−1,...,p−1)∈Z(2r;1). It follows from [11, Section 4.2] that
+{DH(zα)|0<α<τ}is a basis of H(2).
+The remaining part of the proof of Theorem 3.5 will be divided into four steps,
+which are similar to the steps in the proof of Theorem 3.2.
+Step 1: Without loss of generality, we may assume
+(22)pδ(DH(z1zr+1)) = 0.
+Substituting α=ε1+ε2into equation (20), we see that DH(zβ) is an eigenvector
+for the operator ad DH(z1zr+1), with eigenvalue βr+1−β1. Sinceβr+1−β1is in
+the fieldGFp, we haveDH(z1zr+1) =DH(z1zr+1)p. Applying the operatorpδto
+both sides and using (19), we obtain:
+pδ(DH(z1zr+1)) =/summationdisplay
+i1+···+ip=p(i1δ)(DH(z1zr+1))···(ipδ)(DH(z1zr+1))GRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 13
+Taking into account (17) and (21), we see thatkδ(DH(z1zr+1)) = 0 for 1 <k<p.
+Hence,
+pδ(DH(z1zr+1)) =/parenleftbig1δ(DH(z1zr+1))/parenrightbigp(23)
++p−1/summationdisplay
+k=0DH(z1zr+1)k(pδ)(DH(z1zr+1))DH(z1zr+1)p−k−1.
+Since1δ(DH(z1zr+1)) = [∂1,DH(z1zr+1)] =DH(zr+1) =−∂1and∂p
+1= 0, the first
+term on the right-hand side of (23) vanishes. The second term can be rewritten
+using the identity
+p−1/summationdisplay
+k=0XkYXp−k−1= (adX)p−1(Y),
+whereX=DH(z1zr+1) andY=pδ(X). Thus, (23) yields
+Y= (adDH(z1zr+1))p−1(Y).
+It follows that Yis a linear combination of those DH(zβ) for which the eigenvalue
+βr+1−β1is nonzero:
+pδ(DH(z1zr+1)) =/summationdisplay
+β:βr+1/ne}ationslash=β1σβDH(zβ).
+It remains to replacepδwith
+pδ+/summationdisplay
+βσβ
+βr+1−β1adDH(zβ),
+to complete Step 1.
+Step 2: Without loss of generality, we may assume that, in addition to (22), we
+also have
+(24)pδ(∂1) =/summationdisplay
+β:β1=p−1τβDH(zβ)
+for some scalars τβ.
+By (21), we have [ ∂1,DH(z1zr+1)] =−∂1. Applying the operatorpδto both
+sides and using (18), (17) and (22), we obtain:
+−pδ(∂1) =p/summationdisplay
+k=0/bracketleftbigkδ(∂1),p−kδ(DH(z1zr+1))/bracketrightbig
+= [pδ(∂1),DH(z1zr+1)].
+Hence [DH(z1zr+1),Y] =YwhereY=pδ(∂1). It follows that Yis a linear combi-
+nation of those DH(zβ) for which the eigenvalue βr+1−β1is 1:
+(25)pδ(∂1) =/summationdisplay
+β:βr+1−β1=1τβDH(zβ).
+Now replacepδwith
+pδ+/summationdisplay
+β:β1/ne}ationslash=p−1τβ
+β1+1adDH(zβ+ε1).
+Using (21), one readily sees that, for the newpδ, we have (24), because all terms
+withβ1/\e}atio\slash=p−1 in the right-hand side of (25) will cancel out. It remains to check
+that we still have (22) for the newpδ. In other words, we have to check that
+[DH(z1zr+1),DH(zβ+ε1)] = 0 for all βwithτβ/\e}atio\slash= 0. But this is clear, because we14 BAHTURIN AND KOCHETOV
+haveβr+1−(β1+1) = 0 for all βthat occur in the right-hand side of (25). Step 2
+is complete.
+Step 3: Assume (22) and (24). Then, for any k= 1,...,p−1,pδ(DH(zk
+1zr+1))
+is a linear combination of DH(zγ) with 0 ≤γ1<k.
+We proceed by induction on k. The basis for k= 1 follows from (22). Now
+suppose the claim holds for some k≥1. By (21), we have
+[∂1,DH(zk+1
+1zr+1)] = (k+1)DH(zk
+1zr+1).
+Applying the operatorpδto both sides and taking into account (18) and (17), we
+obtain
+(26) [pδ(∂1),DH(zk+1
+1zr+1)]+[∂1,pδ(DH(zk+1
+1zr+1))] = (k+1)(pδ)(DH(zk
+1zr+1)).
+Writing
+(27)pδ(DH(zk+1
+1zr+1)) =/summationdisplay
+0<γ<τσγDH(zγ)
+and taking into account (24), (20) and (21), we can rewrite the lef t-hand side of
+(26) as follows:
+/summationdisplay
+β:β1=p−1τβ[DH(zβ),DH(zk+1
+1zr+1)]+/summationdisplay
+γσγ[∂1,DH(zγ)]
+=−/summationdisplay
+β:β1=p−1τβDH(((k+1)zk
+1zr+1∂r+1−zk+1
+1∂1)(zβ))+/summationdisplay
+γσγDH(∂1(zγ))
+=−/summationdisplay
+β:β1=p−1τβDH((k+1)βr+1zβ+kε1−β1zβ+kε1)+/summationdisplay
+γσγγ1DH(zγ−ε1)
+=−/summationdisplay
+β:β1=p−1τβ((k+1)βr+1+1)DH(zβ+kε1)+/summationdisplay
+γσγγ1DH(zγ−ε1).
+Settingτ′
+β:=τβ((k+1)βr+1+1), we can now rewrite (26) as follows:
+(28)/summationdisplay
+γσγγ1DH(zγ−ε1) = (k+1)(pδ)(DH(zk
+1zr+1))+/summationdisplay
+β:β1=p−1τ′
+βDH(zβ+kε1).
+Since the induction hypothesis applies to the first term in the right-h and side of
+(28), we see that all elements DH(zα) that occur in the right-hand side have 0 ≤
+α1≤k−1. Comparing this with the left-hand side, we conclude that for any γ
+withσγ/\e}atio\slash= 0 we have either γ1= 0 or 0 ≤γ1−1≤k−1. Hence the elements
+DH(zγ) that occur in the right-hand side of (27) have 0 ≤γ1≤k.
+The inductive proof of Step 3 is complete.
+Step 4: We can finally obtain a contradiction. By (20), we have
+[DH(z2
+1zr+1),DH(zp−1
+1zr+1)] = 3DH(zr+1) =−3∂1.GRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 15
+Applyingpδand taking into account (18) and (17), we obtain:
+−3(pδ)(∂1) =[pδ(DH(z2
+1zr+1)),DH(zp−1
+1zr+1)]+[DH(z2
+1zr+1),pδ(DH(zp−1
+1zr+1))]
++/bracketleftbigg
+DH/parenleftbigg1
+1!∂1(z2
+1zr+1)/parenrightbigg
+,DH/parenleftbigg1
+(p−1)!∂p−1
+1(zp−1
+1)zr+1/parenrightbigg/bracketrightbigg
++/bracketleftbigg
+DH/parenleftbigg1
+2!∂2
+1(z2
+1zr+1)/parenrightbigg
+,DH/parenleftbigg1
+(p−2)!∂p−2
+1(zp−1
+1)zr+1/parenrightbigg/bracketrightbigg
+=[pδ(DH(z2
+1zr+1)),DH(zp−1
+1zr+1)]+[DH(z2
+1zr+1),pδ(DH(zp−1
+1zr+1))]
++3[DH(z1zr+1),DH(zr+1)]
+=[pδ(DH(z2
+1zr+1)),DH(zp−1
+1zr+1)]+[DH(z2
+1zr+1),pδ(DH(zp−1
+1zr+1))]
+−3∂1.
+So we have
+3∂1=[pδ(DH(z2
+1zr+1)),DH(zp−1
+1zr+1)]+[DH(z2
+1zr+1),pδ(DH(zp−1
+1zr+1))] (29)
++3(pδ)(∂1).
+By Step 3, we know that
+pδ(DH(z2
+1zr+1)) =/summationdisplay
+α:0≤α1≤1σ(2)
+αDH(zα);
+pδ(DH(zp−1
+1zr+1)) =/summationdisplay
+α:0≤α1≤p−2σ(p−1)
+αDH(zα).
+Hence, by (20) we obtain:
+[pδ(DH(z2
+1zr+1)),DH(zp−1
+1zr+1)]
+=/summationdisplay
+α:0≤α1≤1σ(2)
+α[DH(zα),DH(zp−1
+1zr+1)]
+=−/summationdisplay
+α:0≤α1≤1σ(2)
+αDH((−zp−2
+1zr+1∂r+1−zp−1
+1∂1)(zα))
+=/summationdisplay
+α:0≤α1≤1σ(2)
+α(αr+1+α1)DH(zα−2ε1),
+and, similarly,
+[DH(z2
+1zr+1),pδ(DH(zp−1
+1zr+1))]
+=/summationdisplay
+α:0≤α1≤p−2σ(p−1)
+α[DH(z2
+1zr+1),DH(zα)]
+=/summationdisplay
+α:0≤α1≤p−2σ(p−1)
+αDH((2z1zr+1∂r+1−z2
+1∂1)(zα))
+=/summationdisplay
+α:0≤α1≤p−2σ(p−1)
+α(2αr+1−α1)DH(zα+ε1).16 BAHTURIN AND KOCHETOV
+Using the above calculations and (24), we can rewrite (29) as follows :
+3∂1=/summationdisplay
+α:0≤α1≤1σ(2)
+α(αr+1+α1)DH(zα−2ε1)
++/summationdisplay
+α:0≤α1≤p−2σ(p−1)
+α(2αr+1−α1)DH(zα+ε1)
++3/summationdisplay
+β:β1=p−1τβDH(zβ).
+This equation is impossible, because p>3 and none of the sums in the right-hand
+side involves DH(zr+1). The proof of Theorem 3.5 is complete. /square
+Corollary 3.6. Under the assumptions of Theorem 3.5,
+Liek(Aut(H(2))) = ad(CH(0))for allk>0.
+4.Group gradings
+LetGbe an abelian group. In this section we will give a classification of G-
+gradings on the algebra O=O(m;1) and on the simple Lie algebras W(m;1),
+S(m;1)(1)(m≥3) andS(2;1)(2)=H(2;1)(2).
+Given aG-grading Γ O:O=/circleplustext
+g∈GOg, we obtain an induced grading on
+End(O). It is easy to see that W= Der(O) is a graded subspace, so it inherits a
+G-grading, which will be denoted by Γ W:W=/circleplustext
+g∈GWg. The spaces Ωkalso re-
+ceiveG-gradings in a natural way, and one can verify that the maps d: Ωk→Ωk+1
+respect the G-gradings. (The canonical Z-gradings of Wand Ωkare induced by
+the canonical Z-grading of Oin this manner.) However, S=S(m;1) (respectively,
+H=H(m;1)) is not a graded subspace of W, in general. It is certainly a graded
+subspace if we assume that ωS(respectively, ωH) is a homogeneous element with
+respect to the G-grading on Ωm(respectively, Ω2).
+Definition 4.1. We will say that a G-grading Γ O:O=/circleplustext
+g∈GOgisS-admissible
+(respectively, H-admissible )of degreeg0∈Gif the form ωS(respectively, ωH) is a
+homogeneous element of degree g0.
+If ΓOisS-admissible (respectively, H-admissible), then we will denote the in-
+ducedG-gradings on S(respectively, H) and its derived algebra(s) by Γ S(respec-
+tively, Γ H).
+We will now recall the connection between group gradings on an algeb ra and
+certain subgroupschemes of its automorphism group scheme. Let Ube an algebra.
+For any group G, aG-grading on Uis equivalent to a structure of an FG-comodule
+algebra (see e.g. [8]). Assuming Ufinite-dimensional and Gfinitely generated
+abelian, we can regard this comodule structure as a morphism of alge braic group
+schemesGD→Aut(U) whereGDis the Cartier dual of G, i.e., the group scheme
+represented by the commutative Hopf algebra FG. TwoG-gradings are isomorphic
+if and only if the corresponding morphisms GD→Aut(U) are conjugate by an
+automorphism of U.
+If charF= 0, thenGD=/hatwideG, the algebraic group of multiplicative characters of
+G, andAut(U) = Aut(U), the algebraic group of automorphisms. The image of /hatwideG
+in Aut(U) is aquasitorus , i.e., a diagonalizable algebraic group. The G-grading on
+Uis, of course, the eigenspace decomposition of Uwith respect to this quasitorus.
+Hence, group gradings on Ucorrespond to quasitori in Aut( U).GRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 17
+Here we are interested in the case char F=p >0. Then we can write G=
+Gp′×GpwhereGp′has nop-torsion and Gpis ap-group. Hence GD=/hatwidestGp′×GD
+p,
+where/hatwidestGp′is smooth and GD
+pis finite and connected. The algebraic group /hatwidestGp′
+(which is equal to /hatwideG) is a quasitorus, and it acts by automorphisms of Uas follows:
+χ∗X=χ(g)Xfor allX∈Ugandg∈G.
+IfGpis an elementary p-group, then the distribution algebra of GD
+pis the restricted
+envelopingalgebra u(T) whereTis the group of additive charactersof Gp, regarded
+as an abelian restricted Lie algebra. If {a1,...,a s}is a basis of Gp(as a vector
+space over the field GFp), then the dual basis {t1,...,ts}ofThas the property
+(ti)p=tifor alli. Therefore, Tis atorusin the sense of restricted Lie algebras. It
+acts by derivations of Uas follows:
+t∗X=t(g)Xfor allX∈Ugandg∈G.
+IfGpis not elementary, then the distribution algebra of GD
+pis not generated by
+primitive elements and hence its action on Udoes not reduce to derivations. Re-
+gardless of what the case may be, the image of GDinAut(U) is a diagonalizable
+subgroupscheme. Insomesense, the G-gradingon Uisitseigenspacedecomposition
+(see e.g. [14]).
+Now, Theorems 3.1 (where, as was pointed out, one can include the c asesp= 2
+andp= 3), 3.2 and 3.5 give us the following corollary.
+Corollary 4.2. LetGbe an abelian group. Let Lbe one of the following simple
+Lie algebras: W=W(m;1)(m≥3ifp= 2andm≥2ifp= 3),S(1)=S(m;1)(1)
+(m≥3andp>3) orH(2)=H(m;1)(2)(m= 2randp>3). Then any G-grading
+onLis induced by a G-grading on O=O(m;1). More precisely,
+1)The correspondence ΓO/mapsto→ΓWis a bijection between the G-gradings on O
+and theG-gradings on W. It induces a bijection between the isomorphism
+classes of these gradings.
+2)The correspondence ΓO/mapsto→ΓSis a bijection between the S-admissible G-
+gradings on Oand theG-gradings on S(1). It induces a bijection between
+the isomorphism classes of G-gradings on S(1)and theAutS(O)-orbits of
+theS-admissible G-gradings on O.
+3)The correspondence ΓO/mapsto→ΓHis a bijection between the H-admissible G-
+gradings on Oand theG-gradings on H(2). It induces a bijection between
+the isomorphism classes of G-gradings on H(2)and theAutH(O)-orbits of
+theH-admissible G-gradings on O.
+Proof.Let Γ :L=/circleplustext
+g∈GLgbe aG-grading. Replacing Gwith the subgroup
+generated by the support, we may assume that Gis finitely generated. We also
+obtain that the corresponding morphism GD→Aut(L) is a closed imbedding.
+Consider the case L=W. Due to the isomorphism Ad : Aut(O)→Aut(L),
+we obtain a closed imbedding GD→Aut(O), which corresponds to a G-grading
+ΓO:O=/circleplustext
+g∈GOg(whose support also generates G, since otherwise we would
+not have a closed imbedding). The induced G-grading Γ WonWis obtained by
+inducing the FG-comodule structure from Oto End(O) and then restricting to L,
+which agrees with how Ad : Aut(O)→Aut(L) is defined. Therefore, Γ = Γ W.18 BAHTURIN AND KOCHETOV
+In the case L=S(1), we obtain a closed imbedding GD→AutS(O), so the
+subspace /a\}bracketle{tωS/a\}bracketri}htof ΩmisGD-invariant, i.e., /a\}bracketle{tωS/a\}bracketri}htis anFG-subcomodule. Hence ωS
+is a homogeneous element in the corresponding G-grading Γ O:O=/circleplustext
+g∈GOg.
+The proof in the case L=H(2)is similar. /square
+Remark 4.3. It follows from the proof that the supports of the gradings Γ O, ΓW,
+ΓSand ΓHgenerate the same subgroup in G.
+We will now describe all possible G-gradings on O=O(m;1).
+Proposition 4.4. LetO=O(m;1)and letMbe its unique maximal ideal. Let G
+be an abelian group and let O=/circleplustext
+g∈GOgbe aG-grading.
+1)There exist elements y1,...,y mofMand0≤s≤msuch that the elements
+1 +y1,...,1 +ys,ys+1,...,y mareG-homogeneous and {y1,...,y m}is a
+basis ofMmoduloM2.
+2)LetP={g∈G| Og/\e}atio\slash⊂M}. ThenPis an elementary p-subgroup of G.
+3)Let{b1,...,b s}be a basis of P. Then the elements y1,...,y mcan be chosen
+in such a way that the degree of 1+yiisbi, for alli= 1,...,s.
+Proof.1) Pick a basis for Oconsisiting of G-homogeneous elements and select a
+subset{f1,...,f m}ofthisbasisthatislinearlyindependentmodulo F1⊕M2. Order
+the elements fiso thatf1,...,f shave a nonzero constant term and fs+1,...,f m
+belong to M. Rescalef1,...,f sso that the constant term is 1. Let yi=fi−1
+fori= 1,...,sandyi=fifori=s+1,...,m. Theny1,...,y mis a basis of M
+moduloM2.
+2) Clearly, e∈P. Ifa,b∈P, then there exist elements u∈ Oaandv∈ Obthat
+are not in M. Then the element uv∈ Oabis not in M, soab∈P. Also, since upis
+a nonzero scalar, we have ap=e. It follows that Pis an elementary p-subgroup.
+3) Any element of Ocan be uniquely written as a (truncated) polynomial in the
+variables 1+ y1,...,1+ys,ys+1,...,y m. Hence, for any g∈G,
+(30)Og= Span{(1+y1)j1···(1+ys)jsyjs+1
+s+1···yjm
+m|0≤ji<p,aj1
+1···ajm
+m=g},
+wherea1,...,a m∈Gare the degreesof 1+ y1,...,1+ys,ys+1,...,y m, respectively.
+It follows that a1,...,a sgenerateP. Suppose they do not form a basis of P— say,
+as=aℓ1
+1···aℓs−1
+s−1. Set/tildewideyi=yifori/\e}atio\slash=sand
+/tildewideys:= 1+ys−(1+y1)ℓ1···(1+ys−1)ℓs−1.
+Then 1+/tildewidey1,...,1+/tildewideys−1,/tildewideys,...,/tildewideymare homogeneous of degrees a1,...,a m, respec-
+tively. Also,/tildewideys∈Mand
+/tildewideys=ys−(ℓ1y1+···+ℓs−1ys−1) (mod M2),
+so/tildewidey1,...,/tildewideymstill form a basis of MmoduloM2. We have decreased sby 1.
+Repeating this process as necessary, we may assume that {a1,...,a s}is a basis of
+P. Finally, if {b1,...,b s}is another basis of P, we can write bj=/producttexts
+i=1aℓij
+i, where
+(ℓij) is a non-degenerate matrix with entries in the field GFp. Set
+/tildewideyj:=s/productdisplay
+i=1(1+yi)ℓij−1 forj= 1,...,s,
+and/tildewideyj=yjforj=s+1,...,m. Then/tildewidey1,...,/tildewideymform a basis of MmoduloM2,
+and 1+/tildewideyjis homogeneous of degree bj,j= 1,...,s. /squareGRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 19
+Remark 4.5. Withoutlossofgenerality,assumethat Gisgeneratedbythesupport
+of the grading O=/circleplustext
+g∈GOg. LetQbe the image of GDunder the corresponding
+closed imbedding GD→Aut(O). LetH= Stab Aut(O)(M). (In fact, H= Aut(O),
+regarded as the largest smooth subgroupscheme of Aut(O).) LetQ0=Q∩H.
+ThenPis the subgroup of Gcorresponding to the Hopf ideal of FGdefining the
+subgroupscheme Q0ofQ.
+Proof.LetI0be the Hopf ideal defining the subgroupscheme Q0and letG0be the
+corresponding subgroup of G. Consider the coarsening O=/circleplustext
+g∈G/G0Ogof the
+G-grading induced by the natural homomorphism G→G/G0, i.e.,Og=/circleplustext
+g∈gOg.
+This coarsening corresponds to the subgroupscheme Q0⊂Q. SinceQ0stabilizes
+the subspace M⊂ O, we have M=/circleplustext
+g∈G/G0(Og∩M). Hence Og⊂Mforg/\e}atio\slash=e
+andOe=F1⊕(Oe∩M). Hence Og⊂Mfor allg /∈G0, which proves P⊂G0.
+To prove that P=G0, consider the Hopf ideal IofFGcorresponding to P. Then
+I⊂I0. The subgroupscheme /tildewideQofQdefined by Iacts trivially on each Ogwith
+g∈P. It follows that /tildewideQstabilizes M. Hence/tildewideQ⊂Q0andI⊃I0. /square
+Thedescriptionof G-gradingson O(m;1)resemblesthedescriptionof G-gradings
+on the matrix algebra Mn(F) — see e.g. [2, 5]. Namely, Proposition 4.4 shows
+that theG-graded algebra O(m;1) is isomorphic to the tensor product O(s;1)⊗
+O(m−s;1) where the first factor has a division grading (in the sense that each
+homogeneouscomponent is spanned by an invertible element) and th e secondfactor
+has anelementary grading (in the sense that it is induced by a grading of the
+underlyingvectorspace M/M2). The isomorphismin questionis, ofcourse, the one
+defined byy1/mapsto→x1⊗1,...,y s/mapsto→xs⊗1 andys+1/mapsto→1⊗x1,...,y m/mapsto→1⊗xm−s. The
+first factor, O(s;1), is isomorphic to the group algebra FPas aG-graded algebra
+(whereFPhas the standard P-grading, which is regarded as a G-grading).
+To state the classification of G-gradings on Oup to isomorphism, we introduce
+some notation.
+Definition 4.6. LetP⊂Gbe an elementary p-subgroup of rank s, 0≤s≤m.
+Lett=m−sand let
+γ= (g1,...,g t)∈Gt.
+We endow the algebra O=O(m;1) with aG-grading as follows. Select a basis
+{b1,...,b s}forPand declare the degrees of 1 + x1,...,1 +xsto beb1,...,b s,
+respectively. Declare the degrees of xs+1,...,x mto beg1,...,g t, respectively. We
+will denote the resulting G-grading on Oby ΓO(G,b1,...,b s,g1,...,g t). Since the
+gradings corresponding to different choices of basis for Pare isomorphic to each
+other, we will also denote this grading (abusing notation) by Γ O(G,P,γ).
+Definition 4.7. Letγ,/tildewideγ∈Gt. We will write γ∼/tildewideγif there exists a permutation
+πof the set {1,...,t}such that/tildewidegi≡gπ(i)(modP) for alli= 1,...,t.
+Theorem 4.8. LetFbe an algebraically closed field of characteristic p>0. LetG
+be an abelian group. Let O=/circleplustext
+g∈GOgbe a grading on the algebra O=O(m;1)
+overF. Then the grading is isomorphic to some ΓO(G,P,γ)as in Definition 4.6.
+TwoG-gradings, ΓO(G,P,γ)andΓO(G,/tildewideP,/tildewideγ), are isomorphic if and only if P=/tildewideP
+andγ∼/tildewideγ(Definition 4.7).20 BAHTURIN AND KOCHETOV
+Proof.Lety1,...,y mbe as in Proposition 4.4. Let g1,...,g t∈Gbe the degrees
+ofys+1,...,y m, respectively. Then the automorphism of Odefined by yi/mapsto→xi,
+i= 1,...,m, sends the grading O=/circleplustext
+g∈GOgto ΓO(G,b1,...,b s,g1,...,g t).
+If/tildewidegi=gπ(i),i= 1,...,t, for some permutation π, then the automorphism of O
+defined by
+(31) xi/mapsto→xifori= 1,...,sandxs+i/mapsto→xs+π(i)fori= 1,...,t,
+sends Γ O(G,P,/tildewideγ) to ΓO(G,P,γ).
+If/tildewidegi=gibℓi1
+1···bℓiss, then the automorphism of Odefined by
+(32)xi/mapsto→xifori= 1,...,sandxs+i/mapsto→xs+is/productdisplay
+j=1(1+xj)ℓijfori= 1,...,t,
+sends Γ O(G,P,/tildewideγ) to ΓO(G,P,γ).
+Hence, ifγ∼/tildewideγasinDefinition4.7, thenΓ O(G,P,/tildewideγ)isisomorphictoΓ O(G,P,γ).
+It remains to show that the subgroup Pand the equivalence class of γare
+invariants of the G-graded algebra O=/circleplustext
+g∈GOg. This is obvious for P, since
+P={g∈G| Og/\e}atio\slash⊂M}. LetG=G/Pand consider the coarsening of the G-
+grading, O=/circleplustext
+g∈GOg, induced by the natural homomorphism G→G. It follows
+from the definition of PthatMis aG-graded subspace of O. Consequently, M2is
+also aG-graded subspace, and the quotient V:=M/M2inherits aG-grading:
+V=Va1⊕···⊕Vaℓ.
+Letki= dimVaiifai/\e}atio\slash=eandki= dimVai−sifai=e. Clearly, a1,...,aℓ
+andk1,...,k ℓare invariants of the G-graded algebra O=/circleplustext
+g∈GOg. If theG-
+grading on Ois ΓO(G,P,γ), then, up to a permutation, g1P=...=gk1P=a1,
+gk1+1P=...=gk1+k2P=a2, and so on. /square
+Remark 4.9. Instead of using γ= (g1,...,g t) where some of the cosets giPmay
+be equal to each other, one can take multiplicities,
+κ= (k1,...,k ℓ) wherekiare positive integers ,
+with|κ|:=k1+···+kℓ=t,
+γ= (g1,...,g ℓ) wheregi∈Gare such that g−1
+igj/∈Pfor alli/\e}atio\slash=j,
+and write
+ΓO(G,P,κ,γ ) = ΓO(G,P,g 1,...,g 1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+k1times,...,g ℓ,...,g ℓ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+kℓtimes).
+Then Γ O(G,P,κ,γ ) is isomorphic to Γ O(G,/tildewideP,/tildewideκ,/tildewideγ) if and only if κand/tildewideκhave the
+same number of components ℓand there exists a permutation πof the set {1,...,ℓ}
+such that/tildewideki=kπ(i)and/tildewidegi≡gπ(i)(modP) for alli= 1,...,ℓ.
+Definition 4.10. Fix 0≤s≤m. For a multi-index α∈Zm, let
+α:= (α1+pZ,...,α s+pZ,αs+1,...,α m)∈Zs
+p×Zm−s.
+Define a Zs
+p×Zm−s-grading on O=O(m;1) by declaring the degree of 1 + xi,
+i= 1,...,s, and the degree of xi,i=s+ 1,...,m, to beεi. This is the
+grading Γ O(G,P,γ) whereG=Zs
+p×Zm−s(written additively), P=Zs
+p, and
+γ= (εs+1,...εm). We will denote this grading by Γ O(s).GRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 21
+Corollary 4.11. LetO=O(m;1). Then, up to equivalence, there are exactly
+m+ 1fine gradings of O. They are ΓO(s),s= 0,...,m. The universal group of
+ΓO(s)isZs
+p×Zm−s.
+Proof.All homogeneous components of Γ O(s) are 1-dimensional, so it is a fine
+grading. All relations in the grading group Zs
+p×Zm−scome from the fact that
+0/\e}atio\slash= (Og)p⊂ Oefor certain elements g. HenceZs
+p×Zm−sis the universal group of
+ΓO(s).
+For any abelian group Gand ap-subgroupP⊂Gwith a basis {b1,...,b s}, any
+G-grading Γ O(G,b1,...,b s,g1,...,g m−s) is induced from the Zs
+p×Zm−s-grading
+ΓO(s) by the homomorphism Zs
+p×Zm−s→Gdefined by
+εi/mapsto→bifori= 1,...,s,andεi/mapsto→gi−sfori=s+1,...,m.
+It follows that, up to equivalence, there are no other fine gradings . The grad-
+ings Γ O(s) are pair-wise non-equivalent, because their universal groups ar e non-
+isomorphic. /square
+Definition 4.12. TheG-grading induced by Γ O(G,b1,...,b s,g1,...,g t) (Defini-
+tion 4.6) on the Lie algebra Wwill be denoted by Γ W(G,b1,...,b s,g1,...,g t) or
+ΓW(G,P,γ). Explicitly, we declare the degree of each element
+(1+x1)α1···(1+xs)αsxαs+1
+s+1···xαmm∂iwhereα∈Z(m;1),1≤i≤m,
+to be
+bα1−δi,1
+1···bαs−δi,s
+sgαs+1−δi,s+1
+1 ···gαm−δi,m
+t,
+whereδi,jis the Kronecker delta. In particular, the gradings induced by Γ O(s)
+(Definition 4.10) will be denoted by Γ W(s).
+The following is a generalization of a result in [3] (see also [11, Corollary 7.5.2])
+on maximal tori of the restricted Lie algebra W, which corresponds to the case
+whenGis an elementary p-group.
+Theorem 4.13. LetFbe an algebraically closed field of characteristic p>0. LetG
+be an abelian group. Let W=W(m;1)overF. Assumem≥3ifp= 2andm≥2
+ifp= 3. Then any grading W=/circleplustext
+g∈GWgis isomorphic to some ΓW(G,P,γ)as
+in Definition 4.12. Two G-gradings, ΓW(G,P,γ)andΓW(G,/tildewideP,/tildewideγ), are isomorphic
+if and only if P=/tildewidePandγ∼/tildewideγ(Definition 4.7).
+Proof.A combination of Theorem 4.8 and Corollary 4.2. /square
+Corollary 4.14. LetW=W(m;1). Assumem≥3ifp= 2andm≥2ifp= 3.
+Then, up to equivalence, there are exactly m+ 1fine gradings of W. They are
+ΓW(s),s= 0,...,m. The universal group of ΓW(s)isZs
+p×Zm−s. /square
+We now turn to special algebras.
+Proposition 4.15. In the notation of Proposition 4.4, assume that O=/circleplustext
+g∈GOg
+is anS-admissible G-grading of degree g0. Then the elements y1,...,y mcan be cho-
+sen in such a way that the degrees a1,...,a m∈Gof1+y1,...,1+ys,ys+1,...,y m
+(respectively) satisfy the equation g0=a1···am.22 BAHTURIN AND KOCHETOV
+Proof.Choose elements y1,...,y mas in Proposition 4.4. Let a1,...,a m∈Gbe
+the degrees of the elements 1 + y1,...,1 +ys,ys+1,...,y m, respectively. We are
+going to adjust y1,...,y mto makea1,...,a msatisfy the above equation.
+The formdy1∧ ··· ∧dymisG-homogeneous of degree a0:=a1···am. On the
+other hand, we have
+dy1∧···∧dym=fωSwheref= det(∂jyi).
+SinceωSisG-homogeneous of degree g0, we conclude that fisG-homogeneous of
+degreea0g−1
+0. Sincef /∈M, we havea0g−1
+0∈P.
+First consider the case s=m. Thena0∈Pand thusg0∈P. Also, the G-
+grading in this case is the eigenspace decomposition of Owith respect to a torus
+T⊂Der(O) =W, whereTis isomorphic to the group of additive characters of P,
+soThas ranks=m. Ifg0=e, thenωSisT-invariant, so T⊂StabW(ωS) =S,
+which is a contradiction, because the toral rank of S=S(m;1) is less than m(in
+fact, it ism−1). Therefore, in this case we necessarily have g0/\e}atio\slash=e. It follows
+that there exists a basis {b1,...,b m}ofPsuch thatg0=b1···bm. By Proposition
+4.4, we can replace y1,...,y mwith/tildewidey1,...,/tildewideymso that 1+/tildewideyiisG-homogeneous of
+degreebi,i= 1,...,m. The proof in this case is complete.
+Now assume that s<m. Writea0g−1
+0=aℓ1
+1···aℓss. Set/tildewideyi=yifori<mand
+/tildewideym=ym(1+y1)−ℓ1···(1+ys)−ℓs.
+Then/tildewideymisG-homogeneous of degree /tildewideam=ama−ℓ1
+1···a−ℓssand hence/tildewidey1,...,/tildewideym
+are as desired. /square
+Recall that in Definition 4.6 of Γ O(G,P,γ), we had to choose a basis {b1,...,b s}
+forP. Theisomorphismclass,i.e., theAut( O)-orbit, ofthegradingdoesnotdepend
+on this choice. Clearly, the grading is S-admissible of degree g0=b1···bsg1...gt
+and hence it induces a G-grading on the Lie algebra Sand its derived subalgebras.
+LetL=S(m;1)(1)ifm≥3 andL=S(m;1)(2)ifm= 2. Since g0is Aut S(O)-
+invariant, the induced gradings on Lcorresponding to different values of g0are
+not isomorphic. Conversely, suppose {/tildewideb1,...,/tildewidebs}is another basis of Psuch that
+/tildewideb1···/tildewidebs=b1···bs(i.e., this basis leads to the same value of g0). Write/tildewidebj=/producttexts
+i=1bαij
+iwhere (αij) is a non-degenerate matrix with entries in the field GFp. Set
+(33) /tildewidexj:=s/productdisplay
+i=1(1+xi)αij−1 forj= 1,...,s,
+and/tildewidexj=xjforj=s+1,...,m. Then/tildewidex1,...,/tildewidexmform a basis of MmoduloM2,
+and 1+/tildewidexjis homogeneous of degree /tildewidebj,j= 1,...,s. One readily computes that
+(34) det( ∂j/tildewidexi) = det(αij)s/productdisplay
+i=1(1+xi)−1+/summationtexts
+j=1αij.
+Now/tildewideb1···/tildewidebs=b1···bsmeans that/summationtexts
+j=1αij= 1 for all i, so det(∂j/tildewidexj) is inF.
+Therefore, the automorphism of Odefined by xi/mapsto→/tildewidexi,i= 1,...,m, belongs to
+the subgroup Aut S(O). We have proved that two G-gradings on Larising from the
+same dataPandγ, but different choices of basis for P, are isomorphic if and only
+if they have the same value of g0. This justifies the following:
+Definition 4.16. LetPandγbe as in Definition 4.6. Let g0∈Gbe such that
+g0g−1
+1···g−1
+t∈P\{e}.GRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 23
+Select a basis {b1,...,b s}forPsuch thatg0=b1···bsg1···gt. TheG-grading in-
+duced by Γ O(G,b1,...,b t,g1,...,g s) on the Lie algebra Sand its derived subalge-
+bras will be denoted by Γ S(G,b1,...,b t,g1,...,g s) or ΓS(G,P,γ,g 0). In particular,
+theZs
+p×Zm−s-grading induced by Γ O(s) (Definition 4.10, with {ε1,...,εs}as a
+basis for Zs
+p), will be denoted by Γ S(s).
+The following is a generalization of a result in [3] (see also [11, Theorem 7 .5.5])
+on maximal tori of the restricted Lie algebra CS, which corresponds to the case
+whenGis an elementary p-group.
+Theorem 4.17. LetFbe an algebraically closed field of characteristic p>3. LetG
+be an abelian group. Let L=S(m;1)(1)ifm≥3andL=S(m;1)(2)=H(m;1)(2)if
+m= 2(a simple Lie algebra over F). Then any grading L=/circleplustext
+g∈GLgis isomorphic
+to someΓS(G,P,γ,g 0)as in Definition 4.16. Two G-gradings, ΓS(G,P,γ,g 0)and
+ΓS(G,/tildewideP,/tildewideγ,/tildewideg0), are isomorphic if and only if P=/tildewideP,γ∼/tildewideγ(Definition 4.7) and
+g0=/tildewideg0.
+Proof.First we show that the grading L=/circleplustext
+g∈GLgis isomorphic to some grading
+ΓS(G,P,γ,g 0). We can apply Corollary4.2 to translate this problem to the algebra
+O. LetΓ′:O=/circleplustext
+g∈GO′
+gbetheS-admissiblegradingon O, ofsomedegree g0∈G,
+that induces the grading L=/circleplustext
+g∈GLg. As usual, let P={g∈G| O′
+g/\e}atio\slash⊂M}and
+letsbe the rank of P. By Proposition 4.15, there exist elements y1,...,y m∈M
+that form a basis of MmodM2and such that 1 + yi,i≤s, andyi,i > s, are
+G-homogeneous of some degrees ai,i= 1,...,m, where {a1,...,a s}is a basis
+ofPandg0=a1···am. We want to show that there exists an automorphism
+in Aut S(O) that sends Γ′to the grading Γ O= ΓO(G,a1,...,a s,as+1,...,a m).
+Denote the latter grading by O=/circleplustext
+g∈GOg. Letµbe the automorphism of O
+defined byyi/mapsto→xi,i= 1,...,m. Thenµsends Γ′to ΓO, butµmay not belong to
+AutS(O). Writeµ(ωS) =fωSfor somef∈ O. Nowµ(ωS) has degree g0relative
+to the grading induced on Ωmby ΓO,ωShas degree a1···amrelative to the said
+grading, and g0=a1···am, so we conclude that fhas degree erelative to Γ O. If
+s=m, this implies that fis inFand henceµ∈AutS(O), completing the proof.
+So we assume s<m.
+Now we follow the idea of the proof of [11, Proposition 7.5.4], which is due to
+[7]. Observe that
+µ(ωS) =µ(d(x1dx2∧···∧dxm))
+=d(µ(x1)dµ(x2)∧···∧dµ(xm))
+=d/parenleftBiggm/summationdisplay
+i=1(−1)i−1hidx1∧···∧dxi−1∧dxi+1∧···∧dxm/parenrightBigg
+=/parenleftBiggm/summationdisplay
+i=1∂ihi/parenrightBigg
+ωS,
+whereh1,...,h m∈ O. SetE:=/summationtextm
+i=1hi∂i∈W. Sinceµ(ωS) =fωS, we have
+div(E) =/summationtextm
+i=1∂ihi=f. One can immediately verify that div( Wg)⊂ Ogfor all
+g∈G, where Γ W:W=/circleplustext
+g∈GWgis the grading induced on Wby ΓO. (Also,
+this is a consequence of the fact that div : W→ OisAut(O)-equivariant.) Since
+f∈ Oe, replacingEwith itsG-homogeneous component of degree ewill not affect
+the equation div( E) =f, so we will assume that E∈We.24 BAHTURIN AND KOCHETOV
+Define a Z-grading on Oby declaring the degree of x1,...,x s(or, equivalently,
+1+x1,...,1+xs) to be 0 and the degree of xs+1,...,x mto be 1. This Z-grading is
+compatiblewiththe G-gradingΓ Ointhesensethatthehomogeneouscomponentsof
+one grading are graded subspaces of Orelative to the other grading. We will denote
+the filtration associated to this Z-grading by O{ℓ},ℓ= 0,1,2,..., to distinguish it
+from the filtration O(ℓ)associated to the canonical Z-grading.
+Writef=/summationtext
+k≥0fk, wherefkhas degreekin theZ-grading and degree ein the
+G-grading. Observe that the constant term of fis equal to the constant term of
+f0, sof0is an invertible element of O. Letτ1be the automorphism of Odefined
+by
+τ1(xi) =xifori<mandτ1(xm) =f−1
+0xm.
+Sincef0has degree ein theG-grading,τ1preserves Γ O, i.e.,τ1(Og) =Ogfor all
+g∈G. We also have τ1(O{ℓ}) =O{ℓ}for allℓ. Sincexmhas degree 1 in the
+Z-grading, it does not occur in f0. Henceτ1(f0) =f0and we can compute:
+(τ1◦µ)(ωS) =τ1(fωS) =τ1(f)τ1(ωS)
+=(f0+τ1(/tildewideh))f−1
+0ωS= (1+h)ωS,
+where/tildewideh=/summationtext
+k≥1fkandh=f−1
+0τ1(/tildewideh). Note that h∈ O{1}.
+Claim: For anyℓ= 1,2,...there exists an automorphism τℓofOthat preserves
+theG-grading Γ Oand has the following property:
+(35) ( τℓ◦µ)(ωS) = (1+h)ωSwhereh∈ O{ℓ}.
+We proceed by induction on ℓ. The basis for ℓ= 1 was proved above. Assume
+(35) holds for some ℓ≥1 andτℓ. Sinceτℓpreserves Γ O, we have 1+ h∈ Oeand
+henceh∈ Oe. Writeh=/summationtext
+k≥ℓhkwherehkhas degree kin theZ-grading and
+degreeein theG-grading. As was shown above, there exists E∈Wesuch that
+div(E) = 1 +h. WriteE=/summationtext
+k≥−1EkwhereEkhas degree kin theZ-grading
+induced from our Z-grading of Oand degree ein theG-grading. Since div preserves
+theZ-grading, we have div Ek=hkfork≥1. Let/tildewideτbe the automorphism of O
+defined by
+/tildewideτ(xi) =xi−Eℓ(xi),i= 1,...,m.
+SinceEℓ∈We, the automorphism /tildewideτpreserves the G-grading Γ O. We also have
+/tildewideτ(f) =f(modO{k+1}) for allf∈ O{k}and
+/tildewideτ(ωS) = (1−div(Eℓ)+/tildewidef)ωS= (1−hℓ+/tildewidef)ωS
+for some/tildewidef∈ O{2ℓ}. Hence we can compute:
+(/tildewideτ◦τℓ◦µ)(ωS) =/tildewideτ((1+h)ωS) =/tildewideτ(1+h)/tildewideτ(ωS)
+=(1+hℓ+/hatwidef)(1−hℓ+/tildewidef)ωS= (1+/tildewideh)ωS,
+where/hatwidef∈ O{ℓ+1}and/tildewideh=−h2
+ℓ+/hatwidef(1−hℓ) +/tildewidef(1 +hℓ+/hatwidef)∈ O{ℓ+1}. Setting
+τℓ+1=/tildewideτ◦τℓ, we complete the induction step.
+Set/tildewideµ=τℓ◦µforℓ= (p−1)(m−s)+1. Then/tildewideµsends Γ′to ΓOand belongs to
+AutS(O), since/tildewideµ(ωS) =ωS. We have proved the first assertion of the theorem.
+Now, the subgroup Pand the equivalence class of γ= (g1,...,g t) are invari-
+ants of the G-graded algebra O, andg0is Aut S(O)-invariant. It remains to show
+that, ifγ∼/tildewideγandb1···bsg1···gt=g0=˜b1···˜bs˜g1···˜gtwhere{b1,...,b s}and
+{/tildewideb1,...,/tildewidebs}are bases of Pas in Definition 4.16, then Γ O(G,b1,...,b s,g1,...,g t)GRADINGS ON RESTRICTED CARTAN TYPE LIE ALGEBRAS 25
+and Γ O(G,˜b1,...,˜bs,˜g1,...,˜gt) are in the same Aut S(O)-orbit. Clearly, the au-
+tomorphism (31) of O, determined by a permutation πof{1,...,t}, belongs to
+AutS(O). So it suffices to consider the case /tildewidegi≡gi(modP). Write/tildewidebj=/producttexts
+i=1bαij
+i
+where(αij)isanon-degeneratematrixwithentriesin thefield GFp. Alsowrite/tildewidegi=
+gi/producttexts
+j=1bℓij
+j,i= 1,...,t. Then the composition µof the automorphism defined by
+xj/mapsto→/tildewidexj,j≤s, andxj/mapsto→xj,j >s, where/tildewidexjare as in (33), and the automorphism
+defined by (32), sends Γ O(G,/tildewideb1,...,/tildewidebs,/tildewideg1,...,/tildewidegt) to Γ O(G,b1,...,b s,g1,...gt).
+Now, (34) implies that
+µ(ωS) =det(αij)/parenleftBiggs/productdisplay
+i=1(1+xi)−1+/summationtexts
+j=1αij/parenrightBigg
+t/productdisplay
+i=1s/productdisplay
+j=1(1+xj)ℓij
+ωS
+=det(αij)/parenleftBiggs/productdisplay
+i=1(1+xi)−1+/summationtexts
+j=1αij+/summationtextt
+j=1ℓji/parenrightBigg
+ωS.
+On the other hand,
+/tildewideb1···/tildewidebs/tildewideg1···/tildewidegt=g1···gts/productdisplay
+i=1b/summationtexts
+j=1αij+/summationtextt
+j=1ℓji
+i ,
+so the equality /tildewideb1···/tildewidebs/tildewideg1···/tildewidegt=g0=b1···bsg1···gtimplies that
+s/summationdisplay
+j=1αij+t/summationdisplay
+j=1ℓji= 1 for alli
+and henceµ∈AutS(O). /square
+Corollary 4.18. Under the assumptions of Theorem 4.17, there are, up to equiv a-
+lence, exactly m+1fine gradings of L. They are ΓS(s),s= 0,...,m. The universal
+group of ΓS(s)isZs
+p×Zm−s. /square
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+Department of Mathematics and Statistics, Memorial Univers ity of Newfoundland,
+St. John’s, NL, A1C5S7, Canada
+E-mail address :bahturin@mun.ca
+Department of Mathematics and Statistics, Memorial Univers ity of Newfoundland,
+St. John’s, NL, A1C5S7, Canada
+E-mail address :mikhail@mun.ca
\ No newline at end of file
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+arXiv:1001.0192v1  [astro-ph.SR]  31 Dec 2009Finding the Instability Strip for Accreting Pulsating Whit e
+Dwarfs from HST and Optical Observations1
+Paula Szkody2, Anjum Mukadam2, Boris T. G¨ ansicke3, Arne Henden4, Matthew
+Templeton4, Jon Holtzman5, Michael H. Montgomery6, Steve B. Howell7, Atsuko Nitta8,
+Edward M. Sion9, Richard D. Schwartz10, William Dillon4
+ABSTRACT
+Time-resolved low resolution Hubble Space Telescope ultraviolet spec tra to-
+gether with ground-based optical photometry and spectra are u sed to constrain
+the temperatures and pulsation properties of six cataclysmic varia bles containing
+pulsating white dwarfs. Combining our temperature determinations for the five
+pulsating white dwarfs that are several years past outburst with past results on
+six other systems shows that the instability strip for accreting puls ating white
+dwarfs ranges from 10,500-15,000K, a wider range than evident fo r ZZ Ceti pul-
+sators. Analysis of the UV/optical pulsation properties reveals so me puzzling
+aspects. While half the systems show high pulsation amplitudes in the U V com-
+paredtotheiropticalcounterparts, othersshowUV/optical am plituderatiosthat
+are less than one or no pulsations at either wavelength region.
+Subjectheadings: cataclysmicvariables—stars: individual(PQAnd,REJ1255+266,
+SDSSJ074531.92+453829.6,SDSSJ091945.11+085710.0,SDSSJ13 3941.11+484727.5,
+SDSS J151413.72+454911.9) — stars: oscillations — ultraviolet:stars
+2Department of Astronomy, University of Washington, Box 351580 , Seattle, WA 98195
+3Dept. of Physics, University of Warwick, Coventry CV4 7AL, UK
+4American Association of Variable Star Observers, 25 Birch Street, Cambridge, MA 02138
+5Department of Astronomy, New Mexico State University, Box 3000 1, Las Cruces, NM 88003
+6Department of Astronomy, University of Texas, C1400, Austin, T X 78712
+7NOAO, 950 N. Cherry Ave., Tucson, AZ 85719
+8Gemini Observatory, 670 N A’ohoku Pl., Hilo, HI 96720 and Subaru Tele scope, 650 N A’ohoku Pl., Hilo,
+HI 96720
+9Dept. of Astronomy & Astrophysics, Villanova University, Villanova, PA 19085
+10Galaxy View Observatory, Sequim, WA 98382– 2 –
+1. Introduction
+In the decade since the white dwarf in the cataclysmic binary system GW Lib was found
+to show non-radial pulsations (Warner & van Zyl 1998), a dozen mo re systems of this type
+havebeendiscovered (Warner&Woudt 2004; Wouldt &Warner 2004 ; Araujo-Betancoretal.
+2005a; Vanlandingham et al. 2005; Patterson et al. 2005a,b; Mukad am et al. 2007; G¨ ansicke
+et al. 2006; Nilsson et al. 2006; Patterson et al. 2008; Pavlenko 200 9). For convenience,
+we will refer to the objects found from the Sloan Digital Sky Survey as SDSShhmm ±deg
+i.e. SDSS0745+45. The work on all the known objects has allowed som e progress toward
+understanding how accretion affects the instability zones. While nor mal non-interacting
+white dwarfs with hydrogen atmospheres (DAVs or ZZ Ceti stars) show pulsations if they
+have temperatures in the range of 10,800-12,300K with some depen dence of the temperature
+range on log g (Koester & Holberg 2001; Bergeron et al. 2004; Giann inas et al. 2006;
+Mukadam et al. 2006; Castanheira et al. 2007), the white dwarfs in c ataclysmic variables
+(CVs) are known to be heated by accretion (summaries in Sion 1991; 1999; Townsley &
+G¨ ansicke 2009). Moreover, CV white dwarfs typically rotate an or der of magnitude more
+rapidly than single DA pulsators, and have solar composition or subso lar metal composition
+accreted atmospheres. With very few exceptions, the temperat ures of non-magnetic white
+dwarfsinCVsareabove12,000Kandsotheywerepreviouslynotexp ectedtobeZZCetitype
+pulsators. However, HST ultraviolet observations of GW Lib clearly s howed high amplitude
+pulsations as well as a high temperature (Szkody et al. 2002a). In a ddition, the best fit to
+thedataoccurred witha2-temperaturemodel, with 63%ofthewhit e dwarf atatemperature
+of 13,300K and the rest at 17,100K. It was not known if the dual tem perature was related to
+the pulsation or to the presence of a hotter boundary layer where the accretion disk meets
+the white dwarf.
+The existence of the pulsations at such a high temperature led to sp eculation that
+the pulsations in GW Lib could be due to a higher mass white dwarf (Town sley, Arras &
+Bildsten 2004) since a shift in g of about a factor of 10 can shift the b lue edge of the H/HeI
+instability strip to hotter temperatures by about 2000K. They also stated that the effects
+of accretion i.e. heavier elements in the atmosphere of the white dwa rf, or a faster spin than
+present in ZZ Ceti pulsators could affect their models. Arras et al. ( 2006) accomplished a
+more detailed study that showed that the atmospheric composition of the accreting white
+dwarf is significant for determining the location of the instability strip . They discovered that
+1Based on observations made with the NASA/ESA Hubble Space Telesc ope, obtained at the Space
+Telescope Science Institute, which is operated by the Association o f Universities for Research in Astronomy,
+Inc., under NASA contract NAS 5-26555, and with the Apache Point Observatory 3.5m telescope which is
+owned and operated by the Astrophysical Research Consortium.– 3 –
+accreting model white dwarfs with a high He abundance ( >0.38) would form an additional
+hotter instability strip at ∼15,000K due to HeII ionization.
+However, the discovery of pulsations in V455 And (Araujo-Betanc or et al. 2005a) and
+a subsequent snapshot HST spectrum showed that this white dwar f was in the normal ZZ
+Ceti instability zone (within the uncertainties), with T wd∼10,500K and log g=8, thus very
+different from GW Lib. Arras et al. (2006) suggested that the differ ences in the two objects
+could be due to differences in the mass of the white dwarf and to the H e abundance in
+the driving zone. Temperatures determined for three more accre ting pulsators provided
+further confusion (Szkody et al. 2007), as all three systems (SD SS0131-09, SDSS1610-01,
+SDSS2205+11) showed white dwarfs with temperatures of 14,500- 15,000K. Thus, it was
+not clear if GW Lib or V455 And was the normal case for pulsating white dwarfs in CVs.
+Further complicating the issue was the fact that the other known c ool non-magnetic white
+dwarf in a CV (EG Cnc) shows no evidence of pulsations even though it s temperature is
+12,300K (Szkody et al. 2002b) and there are several CVs with T effnear 15,000K as well
+that don’t pulsate e.g. WZ Sge, BC UMa, SW UMa (Sion et al. 1995; G¨ an sicke et al. 2005).
+The situation for CVs containing magnetic white dwarfs is similar. Arau jo-Betancor et al.
+(2005b) determined temperatures for seven systems with magne tic white dwarfs and found
+a range of 10,800-14,200K but none are known to show any signs of p ulsations.
+Our past HST observations of GW Lib, SDSS0131-09, SDSS1610-01 and SDSS2205+11
+showed similar pulsation frequencies in the UV as the optical, with incre ased amplitudes
+(6-17) in the UV over the optical. This amplitude ratio is consistent wit h low order modes
+(Robinsonetal. 1995). Eachnon-radialpulsationmodecanbedesc ribed withauniquesetof
+indices. Mode identification is essential in asteroseismology to deter mine the inner structure
+ofthestars. Thenon-radialluminosityvariationsobservedinwhite dwarfsarealmostentirely
+due to temperature fluctuations (Robinson, Kepler & Nather 1982 ). The stellar surface is
+divided into zones of higher and lower T effdepending on the degree of spherical harmonic ℓ.
+Since the stellar disk can’t be resolved, geometric cancellation result s in smaller observable
+amplitudes for these modes. At UV wavelengths, the increased limb d arkening decreases the
+contribution of zones near the limb, so modes with ℓ=3 are canceled less effectively in the
+UV compared to the lower ℓ≤2 modes. But ℓ=4 modes do not show a significant change
+in amplitude as a function of wavelength. Robinson et al. (1995) sugg ested a new mode
+identification technique based on this effect. While the amplitude ratio s for SDSS1610-01
+were consistent with those expected for an ℓ=1 mode and T eff=12,500K white dwarf, the
+derived temperature from the spectral fit was 14,500K (Szkody e t al. 2007). In addition,
+long term optical coverage of GW Lib (van Zyl et al. 2004) and SDSSJ 0131-09 (Szkody et
+al. 2007) showed that there is a high degree of variablility in the amplitu des of the pulses so
+that at times, some of the periodsarenot visible. This is normal beha vior (termed amplitude– 4 –
+modulation) that is observed in thecool ZZCeti stars e.g. Kleinman e t al. (1998), Mukadam
+et al. (2007).
+In order to gain further insight into the location of the instability str ip for accreting
+white dwarf pulsators, and to constrain the mode identification of t he observed pulsation
+periods, we obtained HST and nearly simultaneous optical observat ions of six other CV
+systems known to contain pulsating white dwarfs (PQ And, REJ1255 +26, SDSS1514+45,
+SDSS1339+48, SDSS0745+45, SDSS0919+08 and SDSS1514+45). The basic properties
+known for these six objects, along with the full SDSS names, are giv en in Table 1. Some
+preliminary results appear in Mukadam et al. (2009).
+2. Observations
+2.1. HST Ultraviolet Data
+The Hubble Space Telescope (HST) Solar Blind Channel (SBC) on the A dvanced Cam-
+era for Surveys (ACS) was used to observe each of the six object s for 5 satellite orbits
+with either grating PR110L or PR130L. While both gratings provide sp ectra from ∼1200˚A
+to∼2000˚A, the PR110L extends slightly bluer while the PR130L has increased s ensitivity
+near 1300 ˚A. In both cases, the prism produces non-linear resolution, with ab out 2˚A pixel−1
+(PR110L) and 1 ˚A pixel−1(PR130L) at 1200 ˚A and about 40 ˚A pixel−1at 2000˚A. Since there
+is no time-tag mode forACS, 60 or61s integrations times were used t hroughout 5 HST orbits
+on each source. With the setup time during the first orbit, there we re 134 or 138 integrations
+on each object. The dead-time between integrations was 40s so th e time resolution is 100
+or 101s. The initial exposure and centering of the target was done with the F140LP filter
+with integration times of 25-60s depending on the brightness of the source. The observation
+times and gratings are summarized in Table 2.
+The HST data were analyzed with the reduction package aXe1.6 prov ided by STScI.
+The targets were extracted with different widths that were deter mined to optimize the spec-
+tral and light curve results. For the spectra, a wide extraction ( ±17 pixels correspond-
+ing to±0.5 arcsec) was used to maximize the flux level, whereas a narrower e xtraction of
+5 (PQ And, REJ1255+26), 7 (SDSS0745+45, SDSS1514+45), 11 (S DSS1339+48) and 13
+(SDSS0919+08) pixels was used for the light curves to optimize the b est S/N for each sys-
+tem. To obtain light curves that could be analyzed for periodicity, all the narrow extractions
+were summed over the useful wavelength range to obtain one UV flu x point per integration
+time interval.
+The entire set of wide-extraction spectra during the five HST orbit swere added together– 5 –
+to produce an average final spectrum for each object. These av erage spectra are shown in
+Figure 1 ordered by increasing far-UV flux from top to bottom. While the resolution is poor,
+the emission line of CIV (1550 ˚A) is apparent in all systems. It is clear from this figure that
+there is a large range in temperature as well as emission line flux for th e six objects.
+2.2. Optical Data
+Due to the remote but possible chance of an outburst during the HS T observations that
+couldproducemoreUVlightthanthelimitsoftheACSdetector, each system wasmonitored
+prior to and during each HST observation by amateurs (from the Am erican Association
+of Variable Star Observers) and professional astronomers world wide. These observations
+showed all six systems to be close to quiescent values. The outburs t history of the six
+objects is given in Table 1. While PQ And and REJ1255+26 had published o utbursts in
+the literature, the previous outburst of SDSS0745+45 was only fo und from the Catalina
+Real-time Transient Survey (Drake et al. 2009) which recorded an o utburst with a minimum
+amplitude of 5 mag in 2006 October, with the system brightness declin ing to its quiescent
+level over many months2.
+Time-resolved ground-based observations as close in time as possib le to the HST UV
+observations were also coordinated in order to determine the amplit ude and period of optical
+pulsations that would aid in mode identification. Seven observatories (Table 2) participated
+in providing observations. The Apache Point Observatory (APO) 3.5 m telescope was used
+with the time-series photometric system Agile which uses a frame-tr ansfer CCD and a BG40
+filter to provide broad-band blue light. The McDonald Observatory ( MO) 2.1m telescope
+with their time-series system Argos (Nather & Mukadam 2004) and a BG40 filter was used
+in a similar fashion. The 1m New Mexico State University (NMSU) telesco pe with a CCD
+and BG40 filter and the 1m US Naval Observatory Flagstaff Station ( NOFS) telescope with
+a BG38 filter were also used for several nights. The 0.35m Schmidt-C assegrain telescope
+equipped with an unfiltered SBIG CCD also provided data from the Son oita Research Tele-
+scope (SRO) in Arizona. In addition to the observations obtained clo se in time to the HST
+times, further data on SDSS0919+08 (APO) and on PQ And (using th e WIYN 3.5m tele-
+scopeequipped withOPTICandaBG39filter)weretakenayearafte rtheHSTobservations.
+All photometric points were made using differential photometry with respect to comparison
+stars on the frames and light curves were constructed using stan dard IRAF3programs for
+2click on object link in table at http://nesssi.cacr.caltech.edu/catalina /20050301/SDSSCV.html
+3IRAF (Image Reduction and Analysis Facility) is distributed by the Nat ional Optical Astronomy Ob-– 6 –
+sky-subtracted aperture photometry. To search for periodicit ies, a discrete Fourier transform
+up to the Nyquist frequency was computed for each object, afte r first converting the light
+curves to a fractional amplitude scale by dividing by the mean and the n subtracting one. A
+summary of the optical observations is also given in Table 2.
+3. Spectral Results
+Figure 1 shows that there is a large range in continuum flux and shape as well as in the
+emission line flux of CIV (1550 ˚A) for the six objects. Since the core of Ly αdoes not reach
+zero for any of the objects, there is an additional source of cont inuum light other than the
+white dwarf, although this contribution is small in most cases. This “s econd component”
+has been observed in UV observations of most quiescent dwarf nov ae, and has been modelled
+in terms of a hot boundary layer (e.g. Long et al. 1993) or an accret ion belt (e.g. G¨ ansicke
+& Beuermann 1996; Sion et al. 1996). Long et al. (2009) showed tha t the WD parameters
+inferred from a composite fit depend only very mildly on the details of t he model assumed
+for the second component.
+The CIV (and to a lesser extent CII at 1335 ˚A, SIV at 1400 ˚AandNV at 1240 ˚A) emission
+lines probablyoriginateinthedisk chromosphere. There mayalso bes ome contributionfrom
+Lαfrom the disk, although the large upturn at Ly αin Figure 1 for SDSS1514, SDSS1339
+and SDSS0919 is likely geocoronal. From our past work on CV systems (e.g. G¨ ansicke et
+al. 2005), we have derived a procedure to pull out the temperatur e of the white dwarf. We
+account for the disk continuum contribution by adding in a componen t whose flux raises the
+continuum in the core of Ly αto the value observed. As our objects were observed during
+(or close to) dwarf nova quiescence, our current understanding of the theory of dwarf nova
+outbursts suggests that the accretion disk should be very cool a nd optically thin. Since we
+do not know the spectral shape of an optically thin accretion disk, w e try three possibilities:
+a black body, a power law and a constant flux level in F λ. While these possibilities do
+not model line features from a disk, the disk contribution is generally small in all cases.
+Finally, we use simple broad Gaussians at the positions of the known UV lines to fill in
+their contribution. For each disk contribution type, we match the s pectrum with a grid of
+Hubeny white dwarf models (Hubeny & Lanz 1995). The ACS data do n ot allow us to infer
+the surface gravity and we hence fix it at log g= 8.0, as well as the metal abundances at
+0.01 times their solar value. Assuming log ghigher (lower) by 0.5dex would result in best-fit
+servatories, which are operated by AURA, Inc., under cooperativ e agreement with the National Science
+Foundation.– 7 –
+temperatures lower (higher) by ∼1000K, which should be considered as a systematic error
+of our method.
+The results of our fits are listed in Table 3. The final resulting temper ature of the white
+dwarf is indeed almost independent on the assumed spectral shape of the second component,
+but the exercise provides an estimate of the error of our fits (100 0K). If one expects a wildly
+varying white dwarf mass distribution among these objects (which is not suggested by the
+observations, e.g. Littlefair et al. 2008), the two uncertainties re lated to the unknown white
+dwarf mass and the nature of the second component should be add ed in quadrature.
+For completeness, we also include in the Table the results for the 3 ob jects observed
+with the SBC in the past (Szkody et al. 2007) that were re-analyzed with the new aXe1.6
+software and refit in the same manner as for our six new observatio ns. An example of the
+fits for the three spectral shapes of the disk are shown for SDSS 1339+48 in Figure 2 while
+the fitting results using a black-body for the disk contribution are s hown for all six of our
+new objects in Figure 3.
+As a check on the white dwarf temperature, the gmagnitude for the resulting white
+dwarf model is also listed and compared to the observed SDSS photo metric values in Table
+3. All the white dwarf values are fainter than observed, which is rea sonable given that the
+accretion disk will have some contribution to the optical light above t hat of the white dwarf.
+For a further exercise in the total spectral fit, we combined the H ST data with the
+available SDSS spectra for 5 of our objects. REJ1255+26 only has S DSS photometry as a
+spectrum was not obtained. For this object, the ugrizmagnitudes were converted to fluxes.
+The disk component (using the constant distribution) was then sub tracted from the SBC
+spectrum and the result plotted with the optical spectrum. Figure 4 shows the combined
+UV and optical data for each system along with the best fit white dwa rf temperature models
+within∼2000K. These plots show the goodness of our temperature fits as well as the amount
+of the disk contribution to the optical flux (the excess seen in the o bserved fluxes over the
+model white dwarf).
+4. Light Curves
+We computed DFTs of all the HST and available optical light curves, an d used linear
+and non-linear least squares analyses to determine periods, amplitu des, and phases of any
+coherentvariabilitypresentinthedata. Whitenoisewasdetermined byalight-curveshuffling
+technique where each time value of fractional intensity was random ly reassigned to another
+existing time value. This shuffling destroys coherent frequencies bu t keeps the same time– 8 –
+sampling and white noise as the original data. The DFT of the shuffled lig ht curve provides
+the noise at each frequency; the mean of the average noise from 1 0 shufflings is taken as the
+white noise of the light curve and 3 times this value is quoted in our pape r as a detection
+limit for pulsations. The amplitudes are given in millimodulation amplitudes ( mma) where
+1 mma is a 0.1% change in intensity. The DFTs for the HST data are show n in Figure 5
+along with the window functions. The window function is the DFT of a sin gle frequency
+noiseless sinusoid sampled exactly the same times as the actual light c urve. These functions
+look quite similar to each other as the HST data for all six systems is sa mpled in almost
+the same way (since we deleted data points when the background no ise was too high, the
+sampling is not completely identical for all six targets). A summary of the observed periods
+and limits from the HST and optical data is contained in Table 4, and the details for each
+system are discussed below.
+4.1. PQ And
+For this system, the ground and space observation time overlaps f or 4 hrs (Table 2),
+so periods and amplitudes can be optimally compared. Figures 5 and 6 s how the DFTs for
+the UV and optical data obtained on 2007 Sept. 13. The ground dat a were obtained with
+a smaller telescope but the observation interval is longer so that th e mean noise is lower
+(5 mma) than for the UV data (14 mma). While the optical data show t he presence of
+two periods (2337 and 1285s) with amplitudes near 20 mma, there is n o signifant pulsation
+evident in the UV. This lack of UV pulsations is surprising and without ex planation. The
+1285s optical period is the pulsation period reported by Vanlandingh am et al. (2005) (Table
+1) and the data on 2007 Sept. 15 and 17 show a similar period within the errors. Thus, we
+regard this as the non-radial pulsation. The 2337s period is not rep eated on the other dates
+and is likely due to flickering noise.
+Since the four previous systems observed with HST had shown pulsa tion amplitudes at
+least 6 times higher in the UV than the optical, consistent with ℓ=1 or 2 modes, we would
+expect a UV amplitude of at least 60 mma at the 1285speriodthat was evident in the optical
+within the same time interval. Thus, the lack of detection of any pulsa tion in the UV implies
+a highℓmode in this system, if the optical periodicity at 1285s is a non-radial pulsation.
+Highℓmodes (ℓ >2) have never been clearly and unambiguously identified in the ZZ Ceti
+stars. For example, Thompson et al. (2004) identified the 141.9s pe riod in PY Vul (G185-
+32) as an ℓ=4 mode, but Pech et al. (2006) suggest that it is an ℓ=2 mode. Furthermore,
+they explain that the low value of UV/optical amplitude could be attrib uted to a resonance
+between the ℓ=2 mode and nearby ℓ=3 andℓ=5 modes, which remain undetectable.– 9 –
+The high quality WIYN data obtained a year after the HST observatio n show an even
+higher amplitude new periodicity at 679s along with its sub-harmonic at 1355s (Figure 7).
+Harmonics and linear combinations in our data could arise as a result of non-linearities
+introduced by relatively thick convection zones (Brickhill 1992, Bra ssard et al. 1995, Wu
+2001, Montgomery 2005). However, since we observe a sub-harm onic at 2P=1355s and do
+not detect a harmonic at P/2=339.5s, we can rule out convection as the likely cause of
+the observed non-linearity. Since there is only one observation of t his new periodicity, it is
+possible that both the 679s period and its sub-harmonic at 1355s we re caused by flickering.
+Should frequent monitoring of the system fail to reveal the 679s p eriod again, then the
+flickering hypothesis is likely. If however the 679smode proves to be persistent, then it might
+be an unusual pulsation mode with an observed sub-harmonic instea d of a typical harmonic.
+As a pulsation mode, we could explain its appearance and the absence of previously observed
+modes as due to amplitude modulation. Amplitude modulation is apparen t in GW Lib
+(Van Zyl et al. 2004) and SDSS0131-09 (Szkody et al. 2007) and als o the ZZ Ceti G29-38
+(Kleinmanet al. 1998). Wecurrently favor theidea thatthe679smo deanditssub-harmonic
+at 1355s are caused by flickering.
+4.2. SDSS0745+45
+The HST observations of this system took place in the midst of a grou nd-based run of
+several nights at McDonald Observatory and SRO. Both observat ories overlapped with HST
+times for about 2.5 hrs (Table 2). While the light curve is highly modulate d in both the
+UV and optical, the modulation is at the long period of 86 min (and its 2nd harmonic at
+43 min and 3rd harmonic at 28 min), not within the previously observed range of pulsation
+periods between 1166-1290s that were clearly evident during 7 nigh ts from 2005 Oct through
+2006 Jan (Mukdadam et al. 2007). Figure 8 shows the intensity curv es and DFTs for the
+2007 data in comparison to that of 2006. While the pulsation is clearly e vident in the light
+curves in 2006, it disappeared at the time of the HST observation. C ombining the 4 nights
+of optical data from 2007 Oct 30-Nov 2, we obtain a 3 σlimit to the pulsation amplitude of
+8.5 mma. The same limit from the HST data is 12 mma (Figure 5).
+Further complicating the issue of the disappearance of the pulsatio n is the origin of the
+86 min modulation. This period (or its harmonics) was evident in 2 of the 7 nights in 2005-
+2006where itwas presumed tobetheorbital period(Mukadamet al. 2007). However, recent
+spectroscopy (John Taylor 2009, private communication) has rev ealed an orbital period of
+77.8±1.5 min. Combining the 4 nights of photometry listed in Table 4, we obtain periods of
+89.3±0.02 min and 43.23 ±0.01 min. Averaging these numbers (weighted inverseley as the– 10 –
+squares of the uncertainties), we obtain a photometric period of 8 7.4±1.3 min. Clearly, the
+spectroscopic period is significantly less than the photometric perio d. Photometric periods
+that are 2-4% longer than spectroscopic periods are usually ascrib ed to superhumps, a phe-
+nomenon usually observed following outbursts in short period syste ms due to precession of
+an eccentric disk caused by the heating from the outburst (Warne r, 1995; Patterson 2001).
+While it is unusual to have a superhump present at quiescence, one is also apparent in V455
+And (Araujo-Betancor et al. 2005a). It is perhaps even more unu sual that the periods are
+so different in SDSS0745+45 (12%). Even if the error bars are unde restimated by a factor of
+2, the difference between the spectroscopic and photometric per iods is still 5%. While this
+difference is 2.8% in V455 And, there is a one-day alias in the photometr ic period so the
+difference could possibly be as large as 9.2%, similar to what we find for S DSS0745+45.
+Aspreviouslynoted, thedetectionofanoutburstaroundmid-Oct 2006fromtheCatalina
+Sky Survey data means it is likely that the white dwarf in SDSS0745+45 had not completely
+cooled to its quiescent temperature by the time of the HST observa tion. Indeed, this object
+is the hottest one of all the objects we have observed, which is con sistent with it having
+been heated during the outburst. Past work on the cooling times of white dwarfs following
+outbursts e.g. WZ Sge (Slevinsky et al. 1999; Long et al. 2003; Sion e t al. 2003; Godon et
+al. 2006), and AL Com (Szkody et al. 2003) have shown that this coo ling can take more
+than 3 years. The outburst could have caused some increase in the eccentricity of the disk
+that causes the long period modulation to be more prominent than pr e-outburst. We expect
+the white dwarf to continue to cool during the next two years and t he pulsations to resume,
+while the long period modulation decreases.
+4.3. SDSS0919+08
+Whereas five out of six past optical light curves of this object from 2005 Dec to 2007
+Mar showed a period near 260s with amplitudes of 7-16 mma (Mukadam et al. 2007), Figure
+5 shows that the HST data in 2007 Nov only reveals a harmonic of the o rbital period at
+40.75 min, with a limit of 15 mma to any shorter term periodicity in the UV ( the two peaks
+on both sides of the 40 min period are aliases). Optical data obtained 2 days prior to the
+HST observations also reveal no period to a limit of 4 mma (Figure 10) a nd the data at the
+end of 2008 show no pulsation to a limit of 7 mmma (Table 4). Since there are no known
+outbursts of this system, and the white dwarf temperature is not high, it is not at all clear
+why the white dwarf in this system has stopped pulsating. Mukadam e t al. (2007) thought
+the 260s period was a close doublet and the one night it was not obser ved in their data could
+be a beating of the two frequencies. However, the lack of periods o n the three separated– 11 –
+days of observations in our data and especially the lack of pulsations in the UV indicates
+this system has actually stopped pulsating. It is possible that if this s ystem is close to the
+edge of its instability strip, accretion related heating could push it ou tside. However, the
+amount of accretion would need to be small enough that it would not n oticeably affect the
+visual magnitude of the system.
+4.4. SDSS1339+48
+From three nights of photometry in 2005, G¨ ansicke et al. (2006) id entified a prominent
+pulsation at a period of 641s, as well as a long period at either 320 or 3 44 min but no
+modulation at the spectroscopically determined period of 82.52 min. N either our HST data
+(Figure5)norourground-basedphotometryobtainedtwodaysp riortotheHSTdata(Figure
+10) show the pulsation period of 641s. The HST data do show one sign ificant long period at
+7.4 hrs and two short periods (210.3 and 229.6s) that are close to th e Nyquist frequency for
+the data resolution but just below the 3 σnoise level of 53 mma. The optical data reveal none
+of these periods but do show a period consistent with the orbital pe riod (within the error
+bars), as well as a period at 1539s. Due to the limited data, it is difficult to conclude that
+any of these periods result from anything other than flickering or a ccretion effects. However,
+it is clear that there is no large amplitude UV pulsation at the previously determined optical
+period of 641s .
+4.5. SDSS1514+45
+Our optical data from APO (Figure 11) have about 1.5 hrs of overlap with the HST
+UV data for this source. Neither wavelength shows the 559s period previously reported by
+Nilsson et al. (2006). As that period has not been evident since their data in 2005, further
+observations are needed to confirm if this really is a pulsator with a ch anging amplitude.
+The DFT from the UV data (Figure 5) does show a longer period at 88.8 min which
+could be the orbital period. The faintness of this system has preclu ded the determination
+of an orbital period from spectroscopy at the current time. Ther e is a period near 700s
+that is almost at the 3 σlimit, but as this is not evident in the optical, it is likely related to
+flickering.– 12 –
+4.6. REJ1255+26
+Our APO optical time-resolved data on this system only encompass o ne light curve on
+the night following the HST observations. The UV data show the 1.9 hr orbital period but
+a limit of 52 mma to any shorter periods. The optical data (Figure 13) reveals two short
+periods at 654.5 and 582.1. Patterson et al. (2005b) had previously found a period of 668s.
+As in PQ And, the presence of optical periods without detection in th e UV at a higher
+amplitude implies a high ℓmode of pulsation, if the periods are due to non-radial pulsations.
+5. Discussion
+Figures 1,3,4 and Table 3 show that the 9 systems with similar SBC data s how a
+range of temperatures from 10,000-17,000K. The hottest tempe rature white dwarf exists in
+SDSS0745+45. However, since that system underwent an outbur st only 1 year prior to the
+HST observation, it is likely that its white dwarf has not cooled to its qu iescent value. This
+is corroborated by the lack of pulsations evident in the HST data. Lo ng term studies of the
+white dwarf cooling in low accretion rate CVs with short orbital period s have shown that it
+takes more than 3 yrs for the white dwarf to cool following outburs t (Piro et al. 2005; Godon
+et al. 2006). Eliminating SDSS0745+45 and adding in the temperature s derived from STIS
+observations of GW Lib (15,400 for a black body disk contribution; Sz kody et al. 2002a)
+and V455 And (11,500; Araujo-Betancor et al. 2005a), and from t he eclipse modeling of
+SDSS1507+52 (11,000; Littlefair et al. 2008), gives an instability ran ge of 10,000-15,000K
+with the CV pulsators spread throughout this range. Thus, if the h igher temperatures of
+pulsating accretors relative to ZZ Ceti stars is due to mass or comp osition, then no unique
+parameter characterizes the accreting pulsators. Half of the 11 temperatures lie within the
+ZZ Cet instability strip (within the error bars and with a log g of 8) and h alf are hotter.
+Figure 13 shows our 11 quiescent temperatures along with the ZZ Ce ti empirical instability
+strip (Gianninas et al. 2007). While a few of the systems could fall with in the strip if they
+have massive white dwarfs (higher log g), it would be difficult to have ma ss be the primary
+cause of the width of the strip. Arras et al. (2006) can account fo r increased width with an
+increase in He in the driving zone.
+Besides the width of the instability strip for accreting pulsators, th e other oddity is why
+all CVs with white dwarfs in the temperature range of the known puls ators do not show
+pulsations. Table 6 lists all the white dwarfs with reliable temperature determinations (from
+the recent summary in Townsley & G¨ ansicke 2009, with the addition o f SDSS1507+52 from
+Littlefair et al. 2008) and they are also plotted in Figure 13. The two c losest systems to the
+ZZ Cet instability strip are EG Cnc (at the blue edge) and SDSS1035+0 5 (at the red edge).– 13 –
+Further monitoring of these systems is needed to determine if they never pulsate or if they
+were observed during a hiatus of their pulsations.
+GW Lib still stands out as the only system among the 11 with available UV spectra
+in which the best fit is obtained with a dual temperature white dwarf r ather than a white
+dwarf plus an accretion disk. It is the only system in which the core of Lyαdoes reach zero.
+If this dual temperature is due to a boundary layer ring that is hott er due to the accretion,
+it is not clear why this is not the case in the other 10 systems, especia lly for SDSS0745+45
+which is heated by the recent outburst.
+Within the limited numbers, there does not seem to be any correlation of white dwarf
+temperature with orbital period as would be expected if the white dw arfs are cooling due to
+decreasing mass accretion rate as they evolve to shorter orbital periods (Howell, Nelson &
+Rappaport 2001). Figures 2-4 give some indication of the accretion luminosity (hence white
+dwarf mass and accretion rate) from the contribution of the accr etion disk to the light.
+Excluding SDSS0745+45, whose white dwarf is likely not at its quiescen t temperature, the
+white dwarfs in our models contribute 75-89%of the UV light and 42-7 5%of the optical light
+fortheother8objectsinTable3. Thethreehottest whitedwarfs (SDSS2205+11,SDSS0131-
+09 and SDSS1610-01) do have the three highest disk contributions to the optical light (50,
+58 and 41%) which is consistent with a higher mass white dwarf or a high er accretion rate
+that would heat the white dwarf.
+Figure 14 shows the accreting pulsators as well as the non-pulsatin g white dwarfs (Table
+6) as a function of their orbital period. As noted in G¨ ansicke et al. ( 2009b), most of the
+pulsating white dwarfs lie in a period regime termed the ”period spike” b etween 80-86 min.
+The most egregious outlier is REJ1255+266 near 2 hrs. Since the per iod determination for
+REJ1255+266isaphotometriconeandwenowknowthatatleast2of ourpulsatingaccretors
+(V455 And and SDSS0745+45) show different photometric periods t han spectroscopic, this
+period may be suspect and awaits a spectroscopic determination. H owever, the pulsator
+SDSS1507+52 has a period of 66.6 min determined from eclipses (Patt erson et al. 2008) so
+it appears this one really is an outlier to the rest of the group. GW Lib a nd SDSS0745+45
+(not shown on the plot as its temperature may not be its quiescent o ne) are outside the
+spike as well. It is clear from Figure 14 that most of the accreting puls ators are concentrated
+in a narrow period range near 82 min. It is easy to explain why longer pe riod systems are
+not evident as pulsators, as those typically have increased mass ac cretion rates which can
+hide the white dwarf, and thus make pulsations harder to detect. B ut it is not obvious why
+the pulsations would not be present in systems with shorter orbital periods and why the
+transition is so abrupt.
+A pure instability strip with non-pulsators outside and only pulsators within, implies– 14 –
+that pulsations are an evolutionary phase along the cooling track (F ontaine et al. 1982,
+2003, Bergeron et al. 2004, Castanheira et al 2007). In other wo rds,allaccreting white
+dwarfs would undergo the pulsation phase. An impure instability strip with pulsators and
+non-pulsators mixed in implies that parameters other than the white dwarf temperature
+and mass are at play in deciding whether the star will pulsate or not (K epler & Nelan
+1993, Mukadam et al. 2004). Our preliminary results indicate an impur e strip with 13 non-
+pulsators intherangeof 10500–15400K;thislends credence to th etheorythat Heabundance
+is the third parameter that decides the shape of the instability strip for accretors (Arras et
+al. 2006).
+In addition to the temperature data, we now also have ten systems with pulsation
+properties determined from UV through optical wavlengths (Table 5). This table shows
+a perplexing mix of results. While several objects (GW Lib, SDSS0131 -09, SDSS1610-01
+and SDSS2205+11) behave similarly to ZZ Ceti stars in having high UV/ optical amplitude
+ratios, and similar periods evident in UV and optical, others (PQ And an d REJ1255+26)
+have very low UV/optical amplitude ratios or show no pulsations in eith er UV nor optical at
+the time of our observations (SDSS0745+45, SDSS0919+08, SDSS 1339+48, SDSS1514+45).
+The absence of pulsations in SDSS0745+45 can be explained by its rec ent outburst, which
+has likely heated its white dwarf and moved it out of the instability strip . This explanation is
+corroboratedbythelackofopticalandUVpulsationsofGWLibfollow ingitsrecentoutburst
+(Szkody et al. 2009; Copperwheat et al. 2009). SDSS1514+45 suff ers from insufficient
+data to confirm pulsations. However, there is no clear explanation f or SDSS0919+08 and
+SDSS1339+48. Recent observations of SDSS2205+11 (Southwor th et al. 2008) also show
+a lack of pulsation on two nights as compared to previous years (War ner & Woudt 2004;
+Szkody et al. 2007). While SDSS1339+48 is just outside the blue edge of the ZZ Cet
+instability strip for log g=8 (Figure 13), which might account for its lac k of pulsation if its
+white dwarf underwent a slight temperature increase, SDSS0919+ 08 and SDSS2205+11 are
+much further away from this edge. Southworth et al. (2008) discu ss several possible reasons
+forthe lack ofpulsations (destructive interference, changes int hethermal stateof thedriving
+region, or changing visibility of the modes over the surface of the wh ite dwarf) but conclude
+further observations over long timescales are needed to sort out the cause. While different
+frequencies are known to be dominant at different times during the c ourse of several years
+in GW Lib (van Zyl et al. 2004) and in SDSS0131-09 (Szkody et al. 2007 ), information as
+to the length of time that systems show no pulsations at all is not ava ilable.– 15 –
+6. Conclusions
+Our recent ultraviolet and optical data on six systems combined with past data on six
+others produces a dataset of 12 accreting pulsating white dwarfs with temperatures (Tables
+3, 6) and ten with pulsation properties determined from UV through optical wavelengths
+(Table 5). These datasets are beginning to contain enough object s to begin to see some
+trends as well as to reveal some abnormalities.
+•The temperature range for the 11 objects that have been in quies cence for years are in
+the range of 10,500-15,000K. This range includes the temperature s found in ZZ Ceti
+pulsators but extends to higher temperatures. The temperatur es appear to uniformly
+spread throughout the entire range, but we are still in the domain o f small number
+statistics. Since we do not have masses or He abundances for thes e 11 systems, it is
+difficult to test the predictions of (Arras et al. 2006) as to the caus e of the width of
+the strip.
+•Several CVs are known to have white dwarfs with temperature with in this range
+of 10,500-15,000K but they are not pulsating. Table 6 presents a list of the white
+dwarfs in disk-accreting CV that have reliable quiescent temperatu res (from Townsley
+& G¨ ansicke 2009, Littlefair et al. 2008, and this work) and whether they have shown
+pulsations. It is unclear why not all CVs in this temperature range ar e pulsating. In
+addition to the disk-accretors, there are an additional 11 highly ma gnetic white dwarfs
+in CVs (Polars) in the Townsley & G¨ ansicke (2009) list within this tempe rature range,
+none of which show pulsations.
+•GW Lib is the only system among the 11 with UV spectra in which the best fit is
+obtained with a dual temperature white dwarf rather than a white d warf plus a disk
+contribution.
+•The object that is one year past outburst (SDSS0745+45) has a h otter temperature
+(17,000K) and is no longer pulsating. This corroborates past work t hat shows the
+white dwarfs in CVs are heated for several years following outburs t (Piro et al. 2005;
+Godon et al. 2006) and it is likely that this object has moved out of the instability
+strip.
+•Four systems (GW Lib, SDSS0131-09, SDSS1610-01 and SDSS2205 +11) show iden-
+tical pulsation periods in the UV and optical with high amplitude ratios o f UV/opt.
+These four are similar to the low order modes evident in ZZ Cet stars; however, the
+temperatures of all 4 are 14,000-15,000, far outside the range fo r ZZ Ceti objects.– 16 –
+•Two objects (PQ And and REJ1255+26) have UV/optical amplitude r atios that are
+lessthan1. Bothoftheseobjectshavewhitedwarftemperature sof12,000K,justinside
+theblueedgeoftheZZCeti instabilitystrip(Figure13). Iftheoptic alperiodsobserved
+are non-radial pulsations, this implies a high ℓmode not unambiguosly observed in
+ZZ Ceti pulsators. It remains a problem for theorists to determine if the differences
+between accreting and non-accreting white dwarfs could cause th is effect.
+•Four objects (SDSS0745+45, SDSS0919+08, SDSS1339+48 and S DSS1514+45) did
+not show pulsations in either UV nor optical. The lack of pulsations in SD SS0745+45
+canbeexplainedbyitsoutburstonly1yearpriortotheHSTdata, wh ileSDSS1514+45
+has had only minimal observational data in the past (Nilsson et al. 200 6), so further
+data are needed to confirm it as an accreting pulsator. For the oth er 2 systems, there
+is no clear explanation for why the pulsations that were evident from several nights
+of ground-based data in the past have disappeared. Both of thes e objects are in the
+intermediate temperature zone (12,500-13,500K) between the no rmal ZZ Ceti instabil-
+ity edge and the hotter temperatures of the 4 pulsators that do s how UV pulsations
+of high amplitude. Since the temperatures are not high (as in SDSS07 45+45), it is
+unlikely that an outburst has occurred in these 2 systems. It is pos sible that these
+pulsators were close to the edge of the instability strip (SDSS0919+ 08 would have to
+be high mass to be near the edge), and changes in accretion heating take them in and
+out of the instability zone. We intend to keep observing these objec ts to determine if
+pulsations return at the previous periods.
+•Two objects (V455 And and SDSS0745+45) show photometric perio ds that are longer
+than the spectroscopic periods by 3-12%. The presence of such p eriods are typically
+due to superhumps caused by precessing, eccentric disks in short orbital period CVs
+during an outburst. If this is the cause in these two objects, it is un usual to find the
+superhumps during quiescence and even more unusual to have suc h a large percentage
+difference in the periods.
+To make progress toward understanding the edges of the instabilit y strip and the pul-
+sation modes that relate to the appearance/disappearance of pe riods at different times,
+further data are needed. Mass determinations of a few of the hot vs cool white dwarfs can
+determine how this parameter effects the width of the strip. Contin ued observations of the
+systems that have undergone outbursts in 2006-2007 (SDSS074 5+45, GW Lib, V455 And
+and SDSS0804+51) as they re-enter the instability strip following th e heating during the
+outburst will provide valuable information about the modes and dept h of heating. Long
+observation sequences for the objects in Table 6 that are not kno wn to pulsate are needed to
+place stringent limits on the amplitudes of possible pulsations. The iden tification of further– 17 –
+accreting pulsating systems will help to enlarge the database on whic h to draw conclusions
+about the stability of periods and the behavior as a function of the t emperature of the white
+dwarf.
+We gratefully acknowledge the many amateur and professional obs ervers who monitored
+our objects prior to and during the HST observations which allowed t he UV observations
+to proceed with the knowledge that the objects were at quiescenc e. Special thanks go to
+Agatha Raup, Joanne Hughes and Gary Walker. This research was s upported by NASA
+grant HST-GO-11163.01-A from the Space Telescope Science Inst itute which is operated by
+the Association of Universities for Research in Astronomy, Inc., fo r NASA, under contract
+NAS 5-26555. BTG was supported by a PPARC Advanced Fellowship.
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+Table 1. Objects Observed with HST
+Name Mag P orb(min) Opt Pulse P (sec) Amp (mma) Outbursts (yr) Refa
+PQ And 19.1(V) 80.6 1263, 1286 25 1938, 1967, 1988 1,2,3
+SDSS0745b19.0(g) 77.8 1166-1290 45-70 2006 4
+SDSS0919 18.2(g) 81.3 260 7-16 none 4
+SDSS1339 17.7(g) 82.5 641 25 none 5
+SDSS1514 19.7(g) ... 559 12 none 6
+REJ1255b19.1(g) 119.5 668, 1236, 1344 7-30 1994 7
+a1 Schwarz et al. (2004), 2 Patterson et al. (2005a), 3 Vanlandingh am et al. (2005), 4 Mukadam
+et al. (2007), 5 G¨ ansicke et al. (2006), 6 Nilsson et al. (2006), 7 Pa tterson et al. (2005b)
+bFull names of objects are: SDSS J074531.92+453829.6; SDSS J091 945.11+085710.0; SDSS
+J133941.11+484727.5; SDSS J151413.72+454911.9; RE J1255+266– 21 –
+Table 2. Summary of Observations
+Name Obs Instr Filter Int(s) UT Time
+PQ And HST SBC PR110L 60x138 2007 Sep 13 03:36-10:45
+PQ And NOFS CCD none 180 2007 Sep 13 06:43-12:18
+PQ And NMSU CCD BG38 180 2007 Sep 13 05:05-05:51
+PQ And NMSU CCD BG38 180 2007 Sep 15 04:33-09:41
+PQ And NMSU CCD BG38 180 2007 Sep 19 06:59-11:04
+PQ And WIYN OPTIC BG39 25 2008 Oct 21 08:14-10:57
+SDSS0745 HST SBC PR110L 61x138 2007 Nov 01 02:48-09:57
+SDSS0745 MO Argos BG40 30 2007 Oct 30 07:13-12:31
+SDSS0745 MO Argos BG40 30 2007 Oct 31 07:05-12:33
+SDSS0745 MO Argos BG40 30 2007 Nov 01 07:21-12:26
+SDSS0745 MO Argos BG40 30 2007 Nov 02 07:17-12:34
+SDSS0745 SRO CCD none 1200 2007 Oct 31 07:32-12:25
+SDSS0745 SRO CCD none 900 2007 Nov 01 07:27-12:27
+SDSS0919 HST SBC PR130L 61x134 2007 Nov 14/15 18:36-01:41
+SDSS0919 MO Argos BG40 15 2007 Nov 12 10:13-12:43
+SDSS0919 NMSU CCD BG39 180 2007 Nov 8-15 2 per night
+SDSS0919 SRO CCD none 300 2007 Nov 14 08:55-12:37
+SDSS0919 SRO CCD none 300 2007 Nov 15 09:05-10:18
+SDSS0919 APO Agile BG40 20 2008 Dec 30 10:10-13:28
+SDSS1339 HST SBC PR130L 61x138 2008 Jan 25 07:42-14:51
+SDSS1339 APO Agile BG40 15 2008 Jan 23 11:26-12:57
+SDSS1514 HST SBC PR130L 60x138 2008 May 08 05:24-12:29
+SDSS1514 APO Agile BG40 40 2008 May 08 03:00-07:00
+SDSS1514 APO Agile BG40 40,80 2008 May 09 04:53-07:04
+REJ1255 HST SBC PR110L 61x134 2008 May 14 01:50-08:58
+REJ1255 APO Agile BG40 40 2008 May 15 03:00-07:02– 22 –
+Table 3. Model Fits to SBC Spectra
+Obj+Model T wd(K) d(pc) T BB/PLagwd
+SDSS1514+const 10,500 408 ... 20.0
+SDSS1514+BB 10,000 358 9500 19.9
+SDSS1514+PL 10,500 416 1.0 20.1
+PQ And+const 12,000 340 ... 19.3
+PQ And+BB 12,000 361 16,500 19.4
+PQ And+PL 12,000 337 -0.16 19.2
+REJ1255+const 12,000 403 ... 19.6
+REJ1255+BB 12,000 543 11,500 20.3
+REJ1255+PL 12,000 420 0.90 19.7
+SDSS1339+const 12,500 191 ... 17.9
+SDSS1339+BB 12,500 187 25,000 17.9
+SDSS1339+PL 12,500 187 0.08 17.9
+SDSS0919+const 13,500 319 ... 18.9
+SDSS0919+BB 13,500 333 15,500 19.0
+SDSS0919+PL 13,500 330 1.0 19.0
+SDSS0131+const 14,000 388 ... 19.3
+SDSS0131+BB 14,500 432 14,000 19.4
+SDSS0131+PL 14,000 371 0.66 19.2
+SDSS1610+const 14,000 504 ... 19.8
+SDSS1610+BB 14,500 611 13,500 20.2
+SDSS1610+PL 14,500 562 1.0 20.0
+SDSS2205+const 14,000 859 ... 21.0
+SDSS2205+BB 15,000 919 11,000 21.0
+SDSS2205+PL 14,000 862 0.66 21.0
+SDSS0745+constb17,000 445 ... 19.2
+SDSS0745+BB 17,000 474 19,000 19.3
+SDSS0745+PL 17,000 463 -0.02 19.3
+aTBBis the black body temperature of the 2nd component
+and PL is the slope of the power law for the 2nd component.
+bAs this object underwent an outburst one year prior to the
+HST observation, the white dwarf is likely not yet at its quie s-
+cent temperature.– 23 –
+Table 4. Summary of Observed Periods and 3 σLimits
+Object Wavelength ( ˚A) Period Amp (mma) UT Date
+PQ And 1200-1820 ... <42 09-13-07
+PQ And optical 1285 ±10s, 2337 ±10s 22, 22 09-13-07
+PQ And optical 1309 ±13s 28 09-15-07
+PQ And optical 1301 ±13s 27 09-19-07
+PQ And optical 679.0 ±1.4s, 1355 ±16s 28, 10 10-21-08
+SDSS0745 1240-1910 ... <12 11-01-07
+SDSS0745 optical 84.4 ±0.4mina, 44.1±0.3minb68, 30 10-30-07
+SDSS0745 optical 86.4 ±0.4mina, 41.5±0.2minb, 27.8±0.2 68, 36, 17 10-31-07
+SDSS0745 optical 91.1 ±0.8mina70 11-01-07
+SDSS0745 optical 83.6 ±0.3mina, 42.2±0.3minb, 28.5±0.2min 62, 22, 17 11-02-07
+SDSS0919 1226-1955 40.72 ±0.09minb85 11-14/15-07
+SDSS0919 optical ... <4 11-12-07
+SDSS0919 optical ... <7 12-30-08
+SDSS1339 1245-1955 7.39 ±0.25hr 104 01-25-08
+SDSS1339 optical 83.2 ±3.0mina, 1539±32s 24, 14 01-23-08
+SDSS1514 1324-1935 88.8 ±minc108 05-08-08
+SDSS1514 optical ... <6 05-07-08
+REJ1255 1335-1885 1.91 ±0.04hra76 05-14-08
+REJ1255 optical 54.5 ±1.0min, 582.1 ±1.7s, 654.5 ±1.9s 12, 13 05-15-08
+aOrbital period or superhump period.
+bFirst harmonic of orbital or superhump period.
+cikely orbital period.– 24 –
+Table 5. Summary of UV and Optical Periods and Amplitude Ratios
+Object UV Periods (s) Opt Periods (s) UV/opt amplitude
+GW Lib 648, 376, 236 646, 376, 237 6-17
+SDSS0131 213 211 6
+SDSS1610 608, 304, 221 608, 304, 221 6
+SDSS2205 576 575 6
+PQ And ... 1285 <1
+REJ1255 ... 582, 655 <1
+SDSS0745 ... ... ...
+SDSS0919 ... ... ...
+SDSS1339 ... ... ...
+SDSS1514 ... ... ...– 25 –
+Table 6. All WDs in Disk CVs with Temperatures of 10,000-15,000K
+Name P orb(hr) T effK Pulsating
+SDSS1507+52 1.11 11,000 ±500 Y
+GW Lib 1.28 13,300+17,100 Y
+BW Scl 1.30 14,800 ±900 N
+LL And 1.32 14,300 ±1000 N
+PQ And 1.34 12,000 ±1000 Y
+SDSS1610-01 1.34 14,500 ±1500 Y
+V455 And 1.35 10,500 ±750 Y
+AL Com 1.36 16,300 ±1000 N
+SDSS0919+08 1.36 13,500 ±1000 Y
+WZ Sge 1.36 14,900 ±250 N
+SW UMa 1.36 13,900 ±900 N
+SDSS1035+05 1.37 10,500 ±1000 N
+HV Vir 1.37 13,300 ±800 N
+SDSS1339+48 1.38 12,500 ±1000 Y
+SDSS2205+11 1.38 15,000 ±1000 Y
+WX Cet 1.40 13,500 N
+EG Cnc 1.41 12,300 ±700 N
+XZ Eri 1.47 15,000 ±1500 N
+SDSS1514+45 1.48? 10,000 ±1000 Y?
+VY Aqr 1.51 14,500 N
+OY Car 1.52 15,000 ±2000 N
+SDSS0131-09 1.63 14,500 ±1000 Y
+HT Cas 1.77 14,000 ±1000 N
+REJ1255+26 1.99 12,000 ±1000 Y
+EF Peg 2.00 16,600 ±1000 N– 26 –
+Fig. 1.— HST SBC average spectra of all 6 of our objects using a 17 pix el extraction.– 27 –
+Fig. 2.— White dwarf model (mid curve with absorption lines) fit to the S BC data (dark
+curve) on SDSS1339+48 with the disk contribution component (lowe st lines) of a black
+body (top), a power law (mid) and a constant (bottom). In each ca se, the emission lines are
+approximated by broad Gaussians and the sum of the WD, disk and em ission lines is the
+dotted curve on top of the observed values.– 28 –
+Fig. 3.— White dwarf plus black-body + Gaussian lines model fits to the S BC data for our
+6 objects.– 29 –
+Fig. 4.— SBC data (with constant disk component removed) togethe r with SDSS spectra
+(ugriz fluxes for REJ1255+26) for our 6 objects shown with dark lin es. Grey lines are white
+dwarf models within 2000K of our best fit temperature white dwarfs . The difference between
+the WD model and the observed SDSS data is the disk contribution to the optical. Since
+SDSS0745+45 had an outburst since the SDSS spectra were obtain ed, the optical spectrum
+may not be representative of true quiescence.– 30 –
+05010088.8 min04080Amplitude (mma)050
+PQ And
+SDSS0745+4538
+SDSS0919+0857
+SDSS1339+4847
+SDSS1514+4549
+REJ1255+263σ = 42.1 mma
+3σ = 12.3 mma
+3σ = 34.2 mma
+3σ = 53.3 mma
+3σ = 88.7 mma
+3σ = 52.2 mma0255075 40.72 min
+0 0.001 0.002 0.003 0.004 0.005
+Frequency (Hz)0507.4 hr
+1.9 hr010
+Fig. 5.— DFTs of the HST data for our 6 objects along with a dashed line showing the
+3σvalues of the noise for each object. The window functions are show n in the right-hand
+column, with the same scaling as the DFT along the x-axis.– 31 –
+01020Amplitude (mma)036007200 10800 14400 18000 21600
+Time (s)-0.200.2Fractional Intensity
+0 0.0005 0.001 0.0015 0.002
+Frequency (Hz)DFT
+Window3σ = 23.1 mma
+Fig. 6.— Normalized intensity (top), DFT (middle) and window function ( bottom) of the
+NOFS data on PQ And taken simultaneously with the SBC on 2007 Septe mber 13. The two
+major peaks occur at 2337s and 1285s.– 32 –
+0102030Amplitude (mma)0 3600 7200
+Time (s)-0.0500.05Fractional Intensity
+0 0.002 0.004 0.0060.008 0.01 0.012 0.014
+Frequency (Hz)WindowDFTP = 679s
+3σ = 9.9 mma 2P
+Fig. 7.— Intensity, DFT and window function for the WIYN data on PQ A nd obtained one
+year after the SBC data.– 33 –
+Fig. 8.— Comparison of SDSS0745+45 data from 2006 with the 4 nights surrounding the
+2007 HST observations. Note the visible difference of the light curve as well as the changes
+in the period evident in the DFTs. Dashed lines show the 3 σvalues for the noise for each
+dataset.– 34 –
+00.0050.010.0150.020.0250.03051015Amplitude (mma)0 3600 7200
+Time (s)-0.100.1Fractional Intensity
+0 0.001 0.002 0.003 0.004 0.005
+Frequency (Hz)051015DFT
+DFT3σ = 4.9 mma
+Window
+260s
+Fig. 9.— Intensity, DFT with window function, and expanded view of low frequencies for
+McDonald Observatory data on SDSS0919+08 obtained two nights p rior to the HST data.
+The position of the 260s pulsation period that was previously observ ed is marked.– 35 –
+0 0.01 0.02 0.0301020Amplitude (mma)0 1800 3600 5400Time (s)-0.0500.05Fractional Intensity
+0 0.001 0.002 0.003 0.004 0.005
+Frequency (Hz)01020DFT
+DFTWindow
+1539s83.2 min3σ = 8.4mma
+3σ
+Fig. 10.— Intensity, DFT with noise level marked, and expanded view w ith window function
+for APO data on SDSS1339+48 obtained two nights prior to the HST d ata. The previously
+observed pulsation period at 641s (0.00156 Hz) is not evident.– 36 –
+0510Amplitude (mma)0 3600 7200 10800 14400Time (s)-0.100.1Fractional Intensity
+0 0.002 0.004 0.0060.008 0.01 0.012
+Frequency (Hz)WindowDFT 3σ = 6.2 mma
+Fig. 11.—Intensity, DFTandwindowfunctionforAPO dataonSDSS15 14+45thatoverlaps
+1.5 hrs with HST observations. No periods are evident.– 37 –
+051015Amplitude (mma)0 3600 7200 10800 14400Time (s)-0.100.1Fractional Intensity
+0 0.002 0.004 0.006
+Frequency (Hz)Window654.5s 582.1s
+DFT54.5min3σ = 9.5 mma
+Fig. 12.— Intensity, DFT and window function for APO data on REJ125 5+26 obtained one
+night after the HST data. Two significant short periods are evident .– 38 –
+8.48.287.87.6
+Temperature (K)
+Fig. 13.— Temperatures of accreting pulsators (solid dots) and of n on-pulsating CV white
+dwarfs (triangles) as a function of log g. The ZZ Ceti instability strip limits (Gianninas et
+al. (2007) are shown as dashed lines. The 3 pulsators within the strip are SDSS1507+52,
+REJ1255+26 and PQ And. The two objects just outside the blue edg e are the non-pulsator
+EG Cnc and the pulsator SDSS1339+48.– 39 –
+80 100 120
+Orbital P (min)
+Fig. 14.— Temperatures of accreting pulsators (solid dots) and of n on-pulsating CV white
+dwarfs (triangles) as a function of orbital period. The limits of the Z Z Ceti instability strip
+for log g=8 (Gianninas et al. 2007) are shown as dashed lines.
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+arXiv:1001.0193v1  [math.CO]  31 Dec 2009A NOTE ON MASSPARTITIONS BY HYPERPLANES
+BENJAMIN MATSCHKE
+Abstract. A triple of positive integers ( d,h,m) isadmissible if
+for anymgiven masses in /CAdthere exist hhyperplanes that cut
+eachofthesemassesinto2hequalpieces. Wepresentanelementary
+reduction which combined with results by Ramos (1996) yields
+all the admissible triples that were known up to now (with one
+exception) as well as new ones.
+1.Introduction
+Amassin our sense is a finite measure on the Borel σ-algebra on /CAd
+for which hyperplanes are zero-sets. It is cut byhhyperplanes into 2h
+equal pieces if all the 2horthants given by the hyperplanes have the
+same measure. A triple of positive integers ( d,h,m) isadmissible if for
+anymgiven masses in /CAdthere exist hhyperplanes that cut each of
+these masses into 2hequal pieces.
+The famous Ham Sandwich Theorem states that ( d,h= 1,m=d)
+is admissible, which is due to Banach [Ste38]. The problem of find-
+ing other admissible tripes arose in the sixties. Gr¨ unbaum [Gr¨ u60]
+asked which triples of the form ( d,h= 2d,m= 1) are admissible, and
+Hadwiger [Had66] showed the admissibility of the triples ( d= 3,h=
+1,m= 3) and ( d= 3,h= 2,m= 1). The general problem was posed
+by Ramos [Ram96]. It admits interesting topological approaches, se e
+e. g. [Had66], [Ram96], [ ˇZiv04], [MV ˇZ06], [Mat08, Chap. II]. It is seen
+as one of the prime model problems for equivariant algebraic topolog y;
+compare [Mat03, Sect. 3.1].
+If (d,h,m) is admissible then so is ( d+1,h,m): Project the masses
+from /CAd+1to /CAd, find cutting hyperplanes, and take their pre-image
+under the projection. Hence, let ∆( h,m) be the smallest dimension d
+such that ( d,h,m) is admissible. The famous Ham Sandwich Theorem
+states ∆( h= 1,m) =m.
+Here we present a new reduction, Lemma 2.2.
+Date: Dec. 31, 2009.
+Supported by Studienstiftung des dt. Volkes and Deutsche Teleko m Stiftung.
+This small note is an extract of [Mat08, Chap. II].
+12 BENJAMIN MATSCHKE
+2.Elementary Reductions
+The idea of the following Lemma 2.1 was already used by Hadwiger
+[Had66] and by Ramos [Ram96].
+Lemma 2.1. ∆(h,m)≤∆(h−1,2m), for allh≥2,m≥1. That is,
+if(d,h−1,2m)is admissible then so is (d,h,m).
+Proof.Suppose that ( d,h−1,2m) is admissible and we are given m
+masses in /CAd. We can first bisect them with one hyperplane using the
+Ham Sandwich Theorem, since d≥m. The resulting 2 mmasses can
+then be cut into equal parts using h−1 further hyperplanes, by the
+definition of ∆( h−1,2m).
+The next reduction will give new admissible triples.
+Lemma 2.2. ∆(h,m)≤∆(h,m+1)−1, for allh,m≥1. That is, if
+(d+1,h,m+1)is admissible then so is (d,h,m).
+Proof.Assume we are given mmasses in /CA∆(h,m+1)−1, which is embed-
+ded in /CA∆(h,m+1)with last coordinate equal to zero. Add a ball with
+radius1
+2at the point (0 ,...,0,1)∈ /CA∆(h,m+1)and view it as another
+mass. Thicken the first mmasses by an ε >0 into the direction of the
+last coordinate, such that they are now actually masses in /CA∆(h,m+1)
+(the thickened masses have in fact the property that hyperplane s are
+zero-sets)./CA∆(h,m+1)−1
+/CA∆(h,m+1)
+An example for m= 1 and h= 2.
+We then find an equipartition of the m+1 masses by hhyperplanes.
+All of the hyperplanes go through the point (0 ,...,0,1), therefore they
+intersect /CA∆(h,m+1)−1in hyperplanes of /CA∆(h,m+1)−1. These yield an
+equipartition of the given mmasses up to a small error which depends
+on the chosen ε. A limit argument finishes the proof (take a convergent
+subsequence).
+3.Result
+E. Ramos [Ram96] has shown among other things the following in-
+equalities ( x≥0): ∆(1,2x+1)≤2·2x, ∆(2,2x+1)≤3·2x, ∆(3,2x+1)≤NOTE ON MASSPARTITIONS BY HYPERPLANES 3
+5·2x, ∆(4,2x+1)≤9·2x, ∆(5,2x+1)≤15·2x. Note that the factors in
+front of 2xare of the form 2h−1+1, except for h= 5 it is 15 = 2h−1−1.
+Hisformula∆(5 ,2x+1)≤15·2xgivestogetherwiththeabovelemmas
+new admissible triples.
+Corollary 3.1. Leth≥5,m≥2and write m= 2q+r(0< r≤2q).
+Then
+∆(h,m)≤2h−5(14·2q+r).
+Proof.First verify the inequality for h= 5 with Lemma 2.2, then prove
+it for larger hwith Lemma 2.1.
+We note that Ramos’ mentioned results together with the above
+reduction lemmas give all known admissible triples, except for one,
+∆(2,5) = 8, which is due to Mani-Levitska, Vre´ cica, and ˇZivaljevi´ c
+[MVˇZ06].
+Acknowledgements
+I thank Julia Ruscher and G¨ unter Ziegler for their support.
+References
+[Gr¨ u60] B. Gr¨ unbaum. Partitions of mass-distributions and of convex bodies by
+hyperplanes , Pacific J. Math. 10, (1960), 1257-1261.
+[Had66] H. Hadwiger. Simultane Vierteilung zweier K¨ orper , Arch. Math. (Basel)
+17, (1966), 274-278.
+[MVˇZ06] P. Mani-Levitska, S. Vre´ cica, and R. ˇZivaljevi´ c. Topology and combina-
+torics of partitions of masses by hyperplanes , Adv. Math. 207, (2006), 266-296.
+[Mat03] J. Matouˇ sek. Using the Borsuk-Ulam Theorem , Universitext, Springer-
+Verlag, 2003.
+[Mat08] B. Matschke. Equivariant Topology and Applications , Diploma Thesis,
+Chap. II, www.math.tu-berlin.de/ ∼matschke/DiplomaThesis.pdf, (2008).
+[Ram96] E. Ramos. Equipartitions of mass distributions by hyperplanes , Discr.
+Comp. Geometry 10, (1996), 157-182.
+[Ste38] H. Steinhaus. A note on the ham sandwich theorem , Mathesis Polska 9,
+(1938), 2628.
+[ˇZiv04] R. T. ˇZivaljevi´ c. Equipartitions of measures in /CA4, arxiv:math/0412483v1,
+(2004), 19 pages.
+Benjamin Matschke
+Institut f¨ ur Mathematik, MA 6–2
+Technische Universit¨ at Berlin, 10623 Berlin, Germany
+benjaminmatschke@googlemail.com
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+arXiv:1001.0194v1  [math.LO]  31 Dec 2009A COLORING THEOREM FOR SUCCESSORS OF SINGULAR
+CARDINALS
+TODD EISWORTH
+Abstract. We formulate and prove (in ZFC) a strong coloring theorem which
+holds at successors of singular cardinals, and use it to answ er several questions
+concerning Shelah’s principle Pr 1(µ+,µ+,µ+,cf(µ)) for singular µ.
+1.Introduction
+Inearlierwork(see[5]), weobtainedthefollowingcoloringtheoremfo rsuccessors
+of singular cardinals:
+Letµbe singular. There is a function D: [µ+]2→[µ+]2×cf(µ)
+such that for any unbounded A⊆µ+, there is a stationary S⊆µ+
+with
+(1.1) [ S]2×cf(µ)⊆ran(D↾[A]2).
+This paperaroseoutofanattempt atfinding similarcoloringswitheve nstronger
+properties. This attempt was successful, with the happy consequ ence that we can
+use this new coloring theorem to settle a few questions which have ar isen in the
+author’s previous work.
+Beforeproceedinganyfurther, weneedtoalertthereadertoon eofournotational
+conventions.
+Definition 1.1. IfAandBare sets of ordinals, then we define
+(1.2) A⊛B:={/a\}bracketle{tα,β/a\}bracketri}ht ∈A×B:α<β}.
+IfCis also a set of ordinals, then we define
+(1.3) A⊛B×C:={/a\}bracketle{tα,β,γ/a\}bracketri}ht:/a\}bracketle{tα,β/a\}bracketri}ht ∈A⊛Bandγ∈C}
+This notation is a bit ad hoc(especially (1.3)), but it makes it much easier to
+state our main theorem. Note that if κis a cardinal, then it is quite common to
+identify [κ]2with those ordered pairs /a\}bracketle{tα,β/a\}bracketri}htfor whichα<β. In our notation, [ κ]2
+therefore corresponds exactly with κ⊛κ.
+Moving on, we can now state the principal result of this paper:
+Main Theorem. Letµbe a singular cardinal. There is a function
+(1.4) D: [µ+]2→µ+×µ+×cf(µ)
+such that whenever /a\}bracketle{ttα:α < µ+/a\}bracketri}htis a family of pairwise disjoint members of
+[µ+]<cf(µ), there are stationary subsets SandTofµ+such that whenever
+(1.5) /a\}bracketle{tα,β,δ/a\}bracketri}ht ∈S⊛T×cf(µ),
+Date: October 31, 2018.
+12 TODD EISWORTH
+there areα<β <µ+such that
+(1.6) D↾tα×tβis constant with value /a\}bracketle{tα∗,β∗,δ/a\}bracketri}ht.
+At this point, the reader may well be asking “So what?”, as this theo rem seems
+at first glance to be only a much more technical version of the earlier result, which
+was quite technical in its own right. This first impression is misleading, t hough. In
+the first place, we note that this theorem is a ZFCresult — a bit of a rarity given
+the mystery which still clouds the subject of partition relations at s uccessors of
+singular cardinals. More importantly, though, it turns that this res ult is powerful
+enough to have some important consequences for combinatorial s et theory. In
+particular, it puts us in a position to answer several questions conc erning a family
+of combinatorial principles studied extensively by Shelah in his book [1 0]:
+Definition 1.2. Supposeµis a singular cardinal and σ≤µ+. The principle
+Pr1(µ+,µ+,σ,cf(µ)) holds if there is a function f: [µ+]2→σsuch that whenever
+/a\}bracketle{ttα:α< λ/a\}bracketri}htis a sequence of pairwise disjoint elements of [ µ+]<cf(µ), then for any
+ǫ<σwe can find α<β <µ+such thatf↾tα×tβis constant with value ǫ.
+Theabovedefinitionisclearlyonlyaspecialcaseofsomethingmuchmo regeneral
+(after all, the notation involves five parameters), but we have foc used only on
+the cases of interest to us here. Note as well that Pr 1(µ+,µ+,θ,cf(µ)) is a very
+strong version of more standard negative square-brackets par tition relations, as the
+following implications hold:
+(1.7) Pr 1(µ+,µ+,θ,cf(µ)) =⇒µ+/notarrowright[µ+]2
+θ=⇒µ+/notarrowright[µ+]<ω
+θ.
+What sortsofconsequencescanwededucefromourmaintheorem ? Oneexample
+of importance to the author’s work is the equivalence of the two sta tements
+Pr1(µ+,µ+,µ+,cf(µ)) (1.8)
+and
+Pr1(µ+,µ+,µ,cf(µ)). (1.9)
+The equivalence of these two statements answers a question that has been around
+since Shelah’s original work on [7] in the mid 1990s. This is important bec ause
+in many situations it is the first statement which is sought, while the pr oof only
+establishes the second. In some situations, an ad hocargument depending on the
+nature of the function obtained to witness (1.9) has been given to a llow one to
+obtain (1.8). For example, this is what occurs in [7]. In other instance s, however,
+thefunction constructedtoverify(1.9)did notseemto admitsuch anupgrade, aset
+of circumstances which occurred in the author’s [4] and Chapter IV of Shelah’s [10].
+The equivalence of (1.8) and (1.9) removes these concerns, and allo ws us to
+resolve the associated questions. For example, we can show now th at
+(1.10) pp( µ) =µ+=⇒Pr1(µ+,µ+,µ+,cf(µ)),
+extending a chain of theorems containing results of Erd˝ os and Haj nal, Shelah, and
+Todorˇ cevi´ c. Another consequence is that the main theorem of [4] can be fully ex-
+tended to coversingular cardinals of countable cofinality, and this h as consequences
+for stationary reflection. We will discuss these matters in much mor e detail in the
+final section of the paper, as well as obtaining several other relat ed results. Readers
+willing to accept our main theorem as a “black box” can certainly the las t section
+with no problems.A COLORING THEOREM FOR SUCCESSORS OF SINGULAR CARDINALS 3
+2.Background material
+The background material we need is almost identical to that require d for the
+results of [5], although we do need to be a little more sophisticated in ou r use of
+scales. The background divides neatly into four separate areas, s o we consider each
+in turn.
+Minimal Walks
+We must be content with only a brief introduction to Todorˇ cevi´ c’s technique of
+minimal walks and its generalizations. We apology in advance for the no tation —
+we will be mixing methods of Todorˇ cevi´ c together with arguments of Shelah, and
+so our notation is a hybrid of notations they utilize.
+We start by recalling that ¯ e=/a\}bracketle{teα:α<λ/a\}bracketri}htis aC-sequence for the cardinal λif
+eαis closed unbounded in αfor eachα < λ. Givenα < β < λ theminimal walk
+fromβtoαalong¯eis defined to be the sequence β=β0>···>βn=αobtained
+by setting
+(2.1) βi+1= min(eβi\α).
+We will need to use several functions defined in terms of minimal walks . For
+example, we need the function ρ2: [λ]2→ωgiving the length of the walk from β
+toα, that is,
+(2.2) ρ2(α,β) = leastifor whichβi(α,β) =α.
+Next, fori≤ρ2(α,β), we set
+β−
+i(α,β) =/braceleftBigg
+0 if i= 0,
+sup(eβj(α,β)∩α) ifi=j+1 forj <ρ2(α,β).
+Note that if 0 <i<ρ 2(α,β) then
+•β−
+i(α,β) = max(eβi−1(α,β)∩α) (as opposed to “sup”),
+•β−
+i(α,β)<α<β i(α,β), and
+•βi(α,β) = min(eβi−1(α,β)\β−
+i(α,β)+1).
+Thus, for 0 <i<ρ 2(α,β), the ordinals β−
+i(α,β) andβi(α,β) are the two consecu-
+tive elements in eβi−1(α,β)which “bracket” α.
+The limiting case where i=ρ2(α,β) turns out to be quite important to us as
+well. In this situation, we have
+•β−
+ρ2(α,β)(α,β)≤α=βρ2(α,β)(α,β), and
+•β−
+ρ2(α,β)(α,β)<αif and only if α∈nacc(eβρ2(α,β)−1(α,β)).
+Notice that αmust be an element of eβρ2(α,β)−1(α,β)by definition, and β−
+ρ2(α,β)(α,β)
+is less than αprecisely when αfails to be an accumulation point of eβρ2(α,β)−1(α,β)
+(this should also explain the meaning of “nacc”).
+Continuing our discussion, we define
+γ(α,β) =βρ2(α,β)−1(α,β), (2.3)
+γ−(α,β) = max{β−
+i(α,β) :i<ρ2(α,β)}, (2.4)
+and
+η(α,β) = max{β−
+i(α,β) :i≤ρ2(α,β)}. (2.5)4 TODD EISWORTH
+The following proposition contains some standard facts about minima l walks
+couched in our notation. The proof is an easy induction.
+Proposition 2.1. Let ¯ebe aC-system, and suppose α<β.
+(1)γ−(α,β)<α, and ifγ−(α,β)<α∗≤αthen
+(2.6) βi(α,β) =βi(α∗,β) fori<ρ2(α,β).
+(2)η(α,β)≤α, and if it happens that η(α,β)<α∗≤α, then
+(2.7) βi(α,β) =βi(α∗,β) fori≤ρ2(α,β).
+In particular,
+(2.8) βρ2(α,β)(α∗,β) =α.
+Note that part (2) of the above proposition is of no interest unless we can guar-
+anteeη(α,β)<α(or equivalently, guarantee α∈nacc(eγ(α,β))); this will be one of
+our concerns in the sequel.
+Proposition 2.1 captures the only properties of minimal walks we need , and we
+refer the readerto to [12] or [13] for information on more sophistic ated applications.
+We do need to use a generalization of the minimal walks machinery in ord er to
+handle some issues that arise when dealing with successors of singula r cardinals of
+countable cofinality. These techniques were introduced by the aut hor and Shelah
+in [8], and they were further developed in [4].
+Definition 2.2. Letλbe a cardinal. A generalized C-sequence is a family
+¯e=/a\}bracketle{tem
+α:α<λ,m<ω /a\}bracketri}ht
+such that for each α<λandm<ω,
+•em
+αis closed unbounded in α, and
+•em
+α⊆em+1
+α.
+One can think of a generalized C-sequence as a countable family of C-sequences
+which are increasing in a sense. One can also utilize generalized C-sequences in
+the context of minimal walks. In this paper, we do this in the simplest f ashion
+— givenm < ωandα < β < λ , we let the m-walk from βtoαalong¯econsist
+of the minimal walk from βtoαusing theC-sequence /a\}bracketle{tem
+γ:γ < λ/a\}bracketri}ht. Such walks
+have their associated parameters, and we use the superscript mto indicate which
+part of the generalized C-sequence is being used in computations. So, for example,
+them-walk from βtoαalong ¯ewill have length ρm
+2(α,β), and consist of ordinals
+denotedβm
+i(α,β) fori≤ρm
+2(α,β).
+The requirement that em
+α⊆em+1
+αis relevant for the following reason. Given
+α < β, we note that the sequence /a\}bracketle{tmin(em
+β\α) :m < ω/a\}bracketri}htis non-increasing, and
+therefore eventually constant. From this it follows easily that the m-walk from β
+toαalong ¯eis exactly the same for all sufficiently large m, that is, there is an m∗
+such that
+(2.9) /a\}bracketle{tβm
+i(α,β) :i<ρm
+2(α,β)/a\}bracketri}ht=/a\}bracketle{tβm∗
+i(α,β) :i<ρm∗
+2(α,β)/a\}bracketri}ht
+wheneverm≥m∗.
+Club-guessingA COLORING THEOREM FOR SUCCESSORS OF SINGULAR CARDINALS 5
+Just as in [5], we are going to need generalized C-sequences that have been
+carefully selected to interact with certain club-guessingsequence s. The sort of club-
+guessing sequence we use depends on whether or not the cofinality of our singular
+cardinalµis uncountable, and we will handle each case separately. In either ca se,
+we will be defining a stationary set S⊆µ+, a club-guessing sequence ¯C, and a
+generalized C-sequence ¯e.
+If the cofinality of µis uncountable, then we define
+(2.10) S:=Sµ+
+cf(µ)={δ<µ+: cf(δ) = cf(µ)}.
+Claim 2.6 on page 127 of [10] (or see Theorem 2 of [8]) gives us a sequen ce/a\}bracketle{tCδ:
+δ∈S/a\}bracketri}htsuch that
+•Cδis club inδ,
+•otp(Cδ) = cf(µ),
+• /a\}bracketle{tcf(α) :α∈nacc(Cδ)/a\}bracketri}htincreases to µ, and
+•wheneverEis club inµ+, there are stationarily many δ∈Sfor which
+Cδ⊆E.
+(Recall “nacc( Cδ)” refers to the non-accumulation points of Cδ, that is, those ele-
+ments ofCδthat are not limits of points in Cδ.)
+We now use the “ladder swallowing” trick (see Lemma 13 of [6]) to build a
+C-sequence /a\}bracketle{teα:α<µ+/a\}bracketri}htsuch that for each α<µ+,
+|eα|<µ, (2.11)
+and
+δ∈S∩eα=⇒Cδ⊆eα. (2.12)
+We then construct a (admittedly somewhat trivial) generalized C-sequence ¯e=
+/a\}bracketle{tem
+α:m<ω,α<µ+/a\}bracketri}htby settingem
+α=eαfor allm<ω.
+In the case where µis of countable cofinality, our definition of S,¯C, and ¯eis a
+little more involved because of some open questions concerning club- guessing. A
+reader interested in these issues can find a more detailed discussion in [8], but we
+shall rely on technology developed in [4].
+Start by setting
+(2.13) S:=Sµ+
+ℵ1={δ<µ+: cf(δ) =ℵ1},
+and assume /a\}bracketle{tµm:m<ω/a\}bracketri}htis an increasing sequence of uncountable cardinals cofinal
+inµ.
+We are going to present a simplified version of the conclusion of Theor em 4
+of [4]; the reader can consult that paper for a detailed proof (Prop osition 5.8 is
+particularly relevant). In particular, the work in [4] provides us with a sequence
+/a\}bracketle{tCδ:δ∈S/a\}bracketri}htsuch that each Cδis club inδ, andCδ=/uniontext
+m<ωCδ[m] where
+Cδ[m] is closed and unbounded in δ, (2.14)
+|Cδ[m]| ≤µ+
+m, (2.15)
+and such that for every club E⊆µ+, there are stationarily many δ∈Ssuch that
+for eachm < ω, nacc(Cδ[m])∩Econtains unboundedly many ordinals of cofinal-
+ity greater than µ+
+m. (Note that the use of “nacc” is redundant as the cofinality
+assumption guarantees such an ordinal cannot be a limit point of Cδ[m].)6 TODD EISWORTH
+An applicationofLemma5.10from[4]providesus with ageneralized C-sequence
+¯e=/a\}bracketle{tem
+α:α<µ+,m<ω/a\}bracketri}htsatisfying
+|em
+α| ≤cf(α)+µ+
+m (2.16)
+and
+δ∈S∩em
+α=⇒Cδ[m]⊆em
+α. (2.17)
+No matter what the cofinality of µ, the reader should give the phrase “choose
+δ∈Ssuch thatCδguessesE” the obvious interpretation in light of the above
+discussion.
+Scales
+The next ingredient we need for our theorem is the concept of a sca le for a
+singular cardinal.
+Definition 2.3. Letµbe asingularcardinal. A scale forµisapair(/vector µ,/vectorf)satisfying
+(1)/vector µ=/a\}bracketle{tµi:i<cf(µ)/a\}bracketri}htis an increasing sequence of regular cardinals such that
+supi<cf(µ)µi=µand cf(µ)<µ0.
+(2)/vectorf=/a\}bracketle{tfα:α<µ+/a\}bracketri}htis a sequence of functions such that
+(a)fα∈/producttext
+i<cf(µ)µi.
+(b) Ifγ < δ < β thenfγ<∗fβ, where the notation f <∗gmeans that
+{i<cf(µ) :g(i)≤f(i)}is bounded in cf( µ).
+(c) Iff∈/producttext
+i<cf(µ)µithen there is an α<βsuch thatf <∗fα.
+Of course, we will be using the important theorem of Shelah (see Main Claim 1.3
+on page 46 of [10]) that scales exist for any singular µ. Readers seeking a gentler
+exposition of this and related topics can consult [2], or [3].
+We are going to need a result from [6] concerning scales. We remind th e reader
+that notation of the form “( ∃∗β < λ)ψ(β)” means {β < λ:ψ(β) holds}is un-
+bounded below λ, while “( ∀∗β < λ)ψ(β)” means that {β < λ:ψ(β) fails}is
+bounded below λ.
+Lemma 2.4. Letµbe singular, and suppose ( /vector µ,/vectorf) is a scale for µ. Then there is
+a closed unbounded C⊆µ+such that the following holds for every β∈C:
+(2.18) ( ∀∗i<cf(µ))(∀η <µi)(∀ν <µi+1)(∃∗α<β)[fα(i)>η∧fα(i+1)>ν].
+Proof.See Lemma 7 of [6]. /square
+Elementary Submodels
+We have the usual conventions when dealing with elementary submod els. For
+example, we always assume that χis regular cardinal much larger than anything
+relevant to discussion at hand, and we let Adenote the structure /a\}bracketle{tH(χ),∈,<χ/a\}bracketri}ht
+whereH(χ) is the collection of sets hereditarily of cardinality less than χ, and<χ
+is some suitable well-order of H(χ). We include <χin our structure so that we can
+talk about Skolem hulls, as the well-ordering gives us definable Skolem f unctions.
+In general, if B⊆H(χ), then we denote the Skolem hull of BinAby SkA(B).
+The following result of Baumgartner [1] is critical:A COLORING THEOREM FOR SUCCESSORS OF SINGULAR CARDINALS 7
+Lemma 2.5. Assume that M≺Aand letσ∈Mbe a cardinal. If we define
+N= SkA(M∪σ) then for all regular cardinals τ∈Mgreater than σ, we have
+sup(M∩τ) = sup(N∩τ).
+Proof.See the last section of [3], or Lemma 9 of [6]. /square
+The abovelemma gives us acrucial fact about characteristic functions of models,
+which we define next.
+Definition 2.6. Letµbe a singular cardinal, and let /vector µ=/a\}bracketle{tµi:i <cf(µ)/a\}bracketri}htbe
+an increasing sequence of regular cardinals cofinal in µ. IfMis an elementary
+submodel of Asuch that
+• |M|<µ,
+• /a\}bracketle{tµi:i<cf(µ)/a\}bracketri}ht ∈M, and
+•cf(µ)+1⊆M,
+then the characteristic function of Mon/vector µ(denoted Ch/vector µ
+M) is the function with
+domain cf(µ) defined by
+Ch/vector µ
+M(i) :=/braceleftBigg
+sup(M∩µi) if sup(M∩µi)<µi,
+0 otherwise.
+If/vector µis clear from context, then we suppress reference to it in the nota tion.
+In the above situation, it is clear that Ch/vector µ
+Mis an element of the product/producttext
+i<κµi,
+and furthermore, Ch/vector µ
+M(i) = sup(M∩µi) for all sufficiently large i <cf(µ). We
+obtain the following as an immediate consequence of Lemma 2.5.
+Corollary 2.7. Letµ,/vector µ, andMbe as in Definition 2.6. If i∗<cf(µ) and we
+defineNto be Sk A(M∪µi∗), then
+(2.19) Ch/vector µ
+M↾[i∗+1,cf(µ)) = Ch/vector µ
+N↾[i∗+1,cf(µ)).
+We need one final bit of notation concerning elementary submodels.
+Definition 2.8. Letλbe a regular cardinal. A λ-approximating sequence is a
+continuous ∈-chainM=/a\}bracketle{tMi:i<λ/a\}bracketri}htof elementary submodels of Asuch that
+(1)λ∈M0,
+(2)|Mi|<λ,
+(3)/a\}bracketle{tMj:j≤i/a\}bracketri}ht ∈Mi+1, and
+(4)Mi∩λis a proper initial segment of λ.
+Ifx∈H(χ), then we say that Mis aλ-approximating sequence over xifx∈M0.
+Note that if Mis aλ-approximating sequence and λ=µ+, thenµ+ 1⊆M0
+because of condition (4) and the fact that µis an element of each Mi.
+3.The minimal walk lemma
+This section begins with a technical ad hocdefinition. The definition actually
+depends on a generalized C-system ¯e, a scale (/vector µ,/vectorf), and a sequence /a\}bracketle{ttα:α<µ+/a\}bracketri}ht
+of pairwise disjoint elements of [ µ+]<cf(µ), but we suppress this dependence in the
+notation.8 TODD EISWORTH
+Definition 3.1. Supposek<cf(µ) andm<ω. The formula ψk,m(η,η+,β∗,γ,β)
+asserts
+η≤η+<β∗<γ≤min(tβ), (3.1)
+η= min{ηm(β∗,ǫ) :ǫ∈tβ}, (3.2)
+η+= sup{ηm(β∗,ǫ) :ǫ∈tβ}, (3.3)
+γ= min{γm(β∗,ǫ) :ǫ∈tβ}, (3.4)
+and
+(∀ǫ∈tβ)/bracketleftbig
+fη↾[k,cf(µ))≤fηm(β∗,ǫ)↾[k,cf(µ))≤fη+↾[k,cf(µ))/bracketrightbig
+. (3.5)
+The above definition probably seems quite mysterious, but it isolates a certain
+configurationwhose importancewill become clearas the proofprog resses. The next
+result is one of two technical lemmas crucial for our arguments.
+Lemma 3.2. Supposeµis a singular cardinal, and S,¯C, and ¯eare as in the
+preceding section. Further suppose that ( /vector µ,/vectorf) is a scale for µand/a\}bracketle{ttβ:β < µ+/a\}bracketri}ht
+is a sequence of pairwise disjoint elements of [ µ+]<cf(µ). Then there are an m<ω
+andk<cf(µ) such that
+(3.6) (∃∗η<µ+)(∃η+<µ+)(∃statβ∗<µ+)
+(∃∗γ <µ+)(∃β <µ+)[ψk,m(η,η+,β∗,γ,β)].
+Proof.We start by defining x:={µ,S,(/vector µ,/vectorf)),¯C,¯e,/a\}bracketle{ttβ:β < µ+/a\}bracketri}ht}, that is,x
+consists of the parameters needed to comprehend ψ. We then let /a\}bracketle{tMα:α < µ+/a\}bracketri}ht
+be aµ+-approximating sequence over x. It is clear that the set
+(3.7) E:={δ<µ+:Mδ∩µ+=δ}
+is closed and unbounded in µ+, so can fix δ∈Sfor whichCδguessesE.
+Chooseβsuch that
+(3.8) δ<min(tβ),
+and define
+(3.9) γ⊗= sup{γm,−(δ,ǫ) :ǫ∈tβ,m<ω}.
+We observe
+(3.10) max {γ⊗,M0∩µ+}<δ,
+asδ∈Eand|tβ|+ℵ0<cf(δ).
+Proposition 3.3. We can find β∗<δandm<ωsuch that
+β∗∈nacc(Cδ[m])∩E, (3.11)
+cf(β∗)>cf(µ)+sup{|em
+γm(δ,ǫ)|:ǫ∈tβ}, (3.12)
+and
+max{γ⊗,M0∩µ+}<max(Cδ[m]∩β∗). (3.13)A COLORING THEOREM FOR SUCCESSORS OF SINGULAR CARDINALS 9
+Proof.Our first observation is that for any m<ω, we have
+(3.14) cf( µ)+sup{|em
+γm(δ,ǫ)|:ǫ∈tβ}<µ
+as|tβ|<cf(µ).
+Ifµis of uncountable cofinality, then we set m= 1 (recall that ¯ eis somewhat
+trivialin this case). Ourchoice of ¯Candδtells us that all sufficiently largeelements
+of nacc(Cδ) satisfy (3.11) and (3.12), and so we can easily find β∗satisfying (3.13)
+as well.
+If the cofinality of µis countable, then we note that tβis finite. Thus, we can
+fix a single m0<ωsuch that
+(3.15) /a\}bracketle{tβm
+i(δ,ǫ) :i<ρm
+2(δ,ǫ)/a\}bracketri}ht=/a\}bracketle{tβm0
+i(δ,ǫ) :i<ρm0
+2(δ,ǫ)/a\}bracketri}ht
+wheneverǫ∈tβandm≥m0.
+Now choose m≥m0such that
+(3.16) µ+
+m≥max{cf(γm0(δ,ǫ)) :ǫ∈tβ}.
+In light of (3.15), we see
+(3.17) µ+
+m≥max{cf(γm(δ,ǫ)) :ǫ∈tβ}.
+Now nacc(Cδ[m])∩Econtains unboundedly many elements of cofinality greater
+thanµ+
+m, so we can find a β∗with the required properties. /square
+Proposition 3.4. For anyǫ∈tβ, we have
+(∀i<ρm
+2(δ,ǫ)[βm
+i(β∗,ǫ) =βm
+i(δ,ǫ)], (3.18)
+ρ2
+m(β∗,ǫ) =ρ2
+m(δ,ǫ), (3.19)
+γm(β∗,ǫ) =γm(γ,ǫ), (3.20)
+and
+β∗∈nacc(em
+γm(δ,ǫ)). (3.21)
+Proof.The first statement holds as γ⊗< β∗< δ. Givenǫ∈tβ, we note that
+δ∈em
+γm(δ,ǫ)by definition, and so Cδ[m]⊆em
+γm(δ,ǫ)by our choice of ¯ e. It follows that
+β∗∈em
+γm(δ,ǫ)and both (3.19) and (3.20) are immediate. Given that β∗∈em
+γm(δ,ǫ)
+was established earlier in the proof, the statement (3.21) follows fr om (3.12). /square
+Now that we have isolated β∗andm, we define
+η:=min{ηm(β∗,ǫ) :ǫ∈tβ}, (3.22)
+η+:=sup{ηm(β∗,ǫ) :ǫ∈tβ}, (3.23)
+and
+γ:=min{γm(β∗,ǫ) :ǫ∈tβ}. (3.24)
+Proposition 3.5. We have
+(3.25) M0∩µ+<η≤η+<β∗=Mβ∗∩µ+<δ=Mδ∩µ+<γ.10 TODD EISWORTH
+Proof.It is clear that η≤η+≤β∗<δ. By our choice of β∗we know
+M0∩µ+<max(Cδ[m]∩β∗).
+SinceCδ[m]⊆em
+γm(δ,ǫ)=em
+γm(β∗,ǫ)for eachǫ∈tβ, it follows that
+M0∩µ+<max(Cδ[m]∩β∗)≤η.
+By (3.20) and (3.21), we know that ηm(β∗,ǫ)<β∗for everyǫ∈tβ. Since cf(β∗)>
+|tβ|, it follows that η+<β∗.
+Finally, (3.20)andthe definitionof γimply thatδ<γ, andtheproofiscomplete.
+/square
+Now letA:={ηm(β∗,ǫ) :ǫ∈tβ}. It is clear that |A|<cf(µ) so there must exist
+a singlek<cf(µ) such that
+ξ0<ξ1inA=⇒fξ0↾[k,cf(µ))<fξ1↾[k,cf(µ)).
+Given this choice of ktogether with the preceding work, we see that
+(3.26) ψk,m(η,η+,β∗,γ,β)
+holds.
+We now finish the proof of the lemma by a standard elementary submo del argu-
+ment. Since x∪{η,η+,β∗,k,m} ∈Mδandγ >M δ∩µ+, we know
+(∃∗γ <µ+)(∃β <µ+)[ψk,m(η,η+,β∗,γ,β)].
+Sincex∪{η,η+,k,m} ∈Mβ∗andβ∗=Mβ∗∩µ+, we know
+(∃statβ∗<µ+)(∃∗γ <µ+)(∃β<µ+)[ψk,m(η,η+,β∗,γ,β)].
+Finally, since M0∩µ+<ηandx∪{k,m} ∈M0, we conclude
+(∃∗η<µ+)(∃η+<µ+)(∃statβ∗<µ+)(∃∗γ <µ+)(∃β<µ+)[ψk,m(η,η+,β∗,γ,β)],
+as required.
+/square
+4.TheΓlemma
+This sectionmakesheavyuse ofscalesand the associatedcombinat orics,solet us
+begin by assuming ( /vector µ,/vectorf) is a scale for some singular cardinal µ. There is a natural
+(and well-known) function Γ : [ µ+]2→cf(µ) associated with our scale, namely
+(4.1) Γ( α,β) := sup{i<cf(µ) :fβ(i)≤fα(i)}.
+Notice that Γ( α,β) is defined whenever α<β <µ+because (/vector µ,/vectorf) is a scale. We
+make the convention that Γ( α,α) is defined and equal to some symbol ∞as well.
+We also define a partial function Γ+: [µ+]2→cf(µ) by setting
+(4.2) Γ+(α,β) := max {i<cf(µ) :fβ(i)≤fα(i)},
+shouldthismaximumexist, andleavingΓ+undefinedinallothersituations. Clearly
+Γ and Γ+agree whenever the latter function is defined.
+Lemma 4.1. Assumeµis a singular cardinal and ( /vector µ,/vectorf) is a scale for µ. Further
+assume:
+•M0∈M1∈M2are elementary submodels of Aof cardinality µsuch that
+Mi∩µ+is an initial segment of µ+,
+•(/vector µ,/vectorf)∈M0,A COLORING THEOREM FOR SUCCESSORS OF SINGULAR CARDINALS 11
+•β∗=M0∩µ+,
+•¯s=/a\}bracketle{tsα:α < µ+/a\}bracketri}htis sequence of pairwise disjoint elements of [ µ+]<cf(µ)
+with ¯s∈M0, and
+•t∈[µ+]<cf(µ)withM2∩µ+≤min(t).
+Then for all sufficiently large i<cf(µ), there are unboundedly many α<β∗such
+that for all ǫa∈sαandǫb∈t, we have
+(4.3) Γ+(ǫa,ǫb) =i,
+but
+(4.4) fβ∗(i+1)<fǫa(i+1).
+Proof.Our first observation is that it suffices to prove the lemma under the follow-
+ing assumptions about ¯ s:
+•α<β=⇒sup(tα)<min(tβ),
+•α≤min(tα), and
+•there is an i0<cf(µ) such that for all α<µ+,
+(4.5) fmin(tα)↾[i0,cf(µ))≤fǫ↾[i0,cf(µ)) for allǫ∈tα.
+The point isthat given asequence ¯ sasin the assumptionsofthe lemma, we canfind
+an unbounded J⊆µ+such that /a\}bracketle{tsα:α∈J/a\}bracketri}htenjoys the three additional properties
+we want. (For the last property, we need to use that |sα|<cf(µ) for eachα.) Thus,
+there will be such a JinM0, and obtaining the conclusion for /a\}bracketle{tsα:α∈J/a\}bracketri}ht ∈M0
+is enough.
+Next, we observe that if we define /vector g=/a\}bracketle{tgα:α<µ+/a\}bracketri}htwhere
+(4.6) gα:=fmin(tα),
+then (/vector µ,/vector g) forms a scale for µ. This new scale is definable from parameters in M0,
+and hence ( /vector µ,/vector g) is an element of M0as well. In particular, we can apply Lemma 2.4
+to (/vector µ,/vector g) in the model M0. Taken in conjunction with the fact that β∗=M0∩µ+,
+we see that there is an i1<cf(µ) such that whenever i1≤i<cf(µ),ξ0<µi, and
+ξ1<µi+1,
+(4.7) ( ∃∗α<β∗)[fmin(tα)(i)>ξ0∧fmin(tα)(i+1)>ξ1].
+The main part of our argument relies on an application of Lemma 2.7. Le tCbe
+a closed unbounded subset of β∗of order-type cf( β∗), and define
+(4.8) x={µ,(/vector µ,/vectorf),¯s,β∗}∪cf(µ)∪C.
+We defineMto be the Skolem hull of xin the structure A.
+Note thatMis of cardinality max {cf(β∗),cf(µ)}, andM∈M2asMcan be
+computedin M2bytakingthehullof xusingtherestrictionsoftheSkolemfunctions
+toM1. Letgdenote the characteristic function of Min/vector µ, that is,
+(4.9) g= Ch/vector µ
+M.
+It is clear that
+(4.10) g∈M2∩/productdisplay
+i<cf(µ)µi.12 TODD EISWORTH
+Leti2<cf(µ) be least with |M|<µi2, so whenever i2≤i<cf(µ) we have
+(4.11) g(i) = sup(M∩µi).
+Note as well that C⊆Mand soM∩β∗is unbounded in β∗.
+Sinceg∈M2∩/producttext
+i<cf(µ)µi, it follows that g<∗fγwheneverM2∩µ+≤γ <µ+.
+Since|t|<cf(µ), we can fix i3<cf(µ) such that
+(4.12) g↾[i3,cf(µ))<fǫ↾[i3,cf(µ)) for allǫ∈t.
+Now define
+(4.13) i∗= max{i0,i1,i2,i3}.
+We claim that whenever i∗≤i <cf(µ), there are unboundedly many α < β∗for
+which the conclusion of the lemma holds.
+Let such an ibe given, and define
+(4.14) N:= SkA(M∪µi),
+and
+(4.15) h:= Ch/vector µ
+N.
+Recall that by Corollary 2.7, we know
+(4.16) g↾[i+1,cf(µ)) =h↾[i+1,cf(µ)).
+Now let us define
+(4.17) ξ0:= sup{fǫ(i) :ǫ∈t},
+and
+(4.18) ξ1:=fβ∗(i+1).
+Bothξ0andξ1are inN—ξ0gets in asµi⊆N, whileξ1is inNbecauseβ∗and
+the other parameters needed to define it are in N.
+The next two claims constitute the heart of the matter:
+Claim1.Supposeα∈N∩β∗satisfies
+•fmin(sα)(i)>ξ0, and
+•fmin(sα)(i+1)>ξ1.
+Then for all ǫa∈sα, we have
+•Γ+(ǫa,ǫb) =ifor allǫb∈t, and
+•fβ∗(i+1)<fǫa(i+1).
+Proof.Letα∈N∩β∗satisfy the assumptions of the claim, and fix ǫa∈sα. Since
+i0≤i, we know
+(4.19) fβ∗(i+1) =ξ1<fmin(sα)(i+1)≤fǫa(i+1),
+and therefore Γ( ǫa,β∗)/\e}atio\slash=i.
+Now fixǫb∈t. Sincei0≤i, we know
+(4.20) fǫb(i)≤ξ0<fmin(sα)(i)≤fǫa(i),
+and therefore
+(4.21) Γ+(ǫa,ǫb) =i.A COLORING THEOREM FOR SUCCESSORS OF SINGULAR CARDINALS 13
+Sincesα∈Nand|sα|<cf(|µ|)⊆N, we know that ǫa∈N. Sincei≥i2, it
+follows that
+(4.22) fǫa↾[i+1,cf(µ))≤h↾[i+1,cf(µ)).
+Sincei≥i3, a glance at (4.16) shows us that
+(4.23) fǫa↾[i+1,cf(µ))≤g↾[i+1,cf(µ))<fǫb↾[i+1,cf(µ)),
+and therefore
+(4.24) Γ( ǫa,ǫb)≤i.
+The conjunction of (4.21) and (4.24) finishes the proof. /square
+Claim2.The set ofα∈N∩β∗satisfying the hypotheses of the preceding claim is
+unbounded in β∗.
+Proof.The scale (/vector µ,/vector g) is inN, and so (4.7) holds in Nby elementarity. We have
+explicitly ensured that N∩β∗is unbounded in β∗(as we demanded C⊆M⊆N),
+and the result follows immediately after another application of elemen tarity. /square
+/square
+5.Main Theorem
+We come now to the proof of the main theorem stated in the introduc tion.
+Theorem 1 (Main Theorem) .Letµbe a singular cardinal. There is a function
+(5.1) D: [µ+]2→µ+×µ+×cf(µ)
+such that whenever /a\}bracketle{ttα:α < µ+/a\}bracketri}htis a family of pairwise disjoint members of
+[µ+]<cf(µ), there are stationary subsets SandTofµ+such that whenever
+(5.2) /a\}bracketle{tα,β,δ/a\}bracketri}ht ∈S⊛T×cf(µ),
+there areα<β <µ+such that
+(5.3) D↾tα×tβis constant with value /a\}bracketle{tα∗,β∗,δ/a\}bracketri}ht.
+Proof.Let(/vector µ,/vectorf)beascalefor µ,andfixageneralized C-sequence ¯easinLemma3.2.
+Letι: cf(µ)→ω×cf(µ) be a function such that for any m < ωandδ <cf(µ),
+there are unboundedly many i<cf(µ) withι(i) =/a\}bracketle{tm,δ/a\}bracketri}ht.
+Givenα<β, we letmandδdenote functions defined by the recipe
+(5.4) ι(Γ(α,β)) =/a\}bracketle{tm(α,β),δ(α,β)/a\}bracketri}ht.
+Following [6] and [8], given an natural number m, we letcm(α,β) be defined as
+βm
+i(α,β), whereiis the least number for which
+(5.5) Γ( α,β)/\e}atio\slash= Γ(α,βm
+i(α,β)).
+Note thatcm(α,β) is always defined when α < βbecause of our convention that
+Γ(α,α) is equal to ∞.
+We now define
+(5.6) β∗(α,β) =cm(α,β)(α,β)
+and
+(5.7) η∗(α,β) =ηm(α,β)(β∗(α,β),β),
+whereηmis as in (2.5), interpreted in the context of a generalized C-system.14 TODD EISWORTH
+Ifη∗(α,β)<α, we define
+(5.8) α∗(α,β) =cm(α,β)(η∗(α,β),α),
+and then set
+(5.9) D(α,β) :=/a\}bracketle{tα∗(α,β),β∗(α,β),δ(α,β)/a\}bracketri}ht.
+Ifη∗(α,β)≥αwe defineD(α,β) arbitrarily.
+The informal description associated with the preceding definition is m uch clearer
+than the notation required to write it down precisely. Given α < β, we compute
+D(α,β) in the following manner. First, we use Γ and ιto obtain a natural number
+mand an ordinal δ<cf(µ). The number mtells us which piece of the generalized
+C-system ¯ewe will be using for our minimal walks, while δappears in the final
+output ofD. The next step is to take the m-walk from βdown toαuntil we reach
+a spotβ∗where Γ(α,β∗) is different from Γ( α,β) — this is the ordinal β∗(α,β).
+Givenβ∗, we proceed as in the preceding section and isolate the ordinal ηm(β∗,β),
+whichwe name η∗=η∗(α,β)≤β∗. If it happensthat η∗<α, then wem-walkfrom
+αdown toη∗until we reach a point α∗where Γ(η∗,α∗) is different from Γ( η∗,α).
+The function Dnow returns the value /a\}bracketle{tα∗,β∗,δ/a\}bracketri}ht.
+The rest of the proof consists in showing that the function Dhas the required
+properties,soletusassume /a\}bracketle{ttα:α<µ+/a\}bracketri}htisapairwisedisjointcollectionofmembers
+of [µ+]<cf(µ). After a bit of culling and re-indexing, we may assume that
+α≤min(tα), (5.10)
+and
+α<β=⇒sup(tα)<min(tβ). (5.11)
+We now apply Lemma 3.2 to obtain m<ωandk<cf(µ) such that
+(5.12) ( ∃∗η<µ+)(∃η+<µ+)(∃statβ∗<µ+)
+(∃∗γ <µ+)(∃β <µ+)[ψk,m(η,η+,β∗,γ,β)],
+and then fix ordinals ηa≤η+
+asuch that
+(5.13) ( ∃statα∗<µ+)(∃∗γ <µ+)(∃α<µ+)[ψk,m(ηa,η+
+a,α∗,γ,α)].
+The next definition and claim are quite technical, but they are critical for our
+argument.
+Definition 5.1. We say that {ηb,η+
+b}is anη-candidate if
+•η+
+a<ηb≤η+
+b<µ+, and
+•(∃statβ∗<µ+)(∃∗γ <µ+)(∃β <µ+)[ψk,m(ηb,η+
+b,β∗,γ,β)].
+We say that {ηb,η+
+b}works for {α∗,α}atiaif
+•ηb≤η+
+b<α∗<min(tα),
+•ia<cf(µ),
+•Γ+(η,βm
+i(α∗,ǫ)) =iafor allǫ∈tα,i<ρm
+2(α∗,ǫ), andη∈ {ηb,η+
+b}, and
+•fηb(ia+1)>fα∗(ia+1).
+The next lemma says that there are many η-candidates that will be suitable for
+our construction.A COLORING THEOREM FOR SUCCESSORS OF SINGULAR CARDINALS 15
+Lemma 5.2. For all sufficiently large ia<cf(µ), we can find an η-candidate
+{ηb,η+
+b}such that
+(5.14) ( ∃statα∗<µ+)(∃∗γ <µ+)(∃α<µ+)
+[ψk,m(ηa,η+
+a,α∗,γ,α) and{ηb,η+
+b}works for {α∗,α}atia].
+Proof.Our choice of kandmtells us that there are unboundedly many ηfor
+which we can find an η+such that {η,η+}is anη-candidate, so we can construct
+a sequence /a\}bracketle{tsα:α<µ+/a\}bracketri}htof pairwise disjoint subsets of µ+such that each sαis an
+η-candidate.
+LetM0∈M1∈M2be elementary submodels of Aas in Lemma 4.1, chosen so
+that
+(5.15) {ηa,η+
+a,/a\}bracketle{tsα:α<µ+/a\}bracketri}ht} ∈M0,
+and, forα∗=M0∩µ+,
+(5.16) ( ∃∗γ <µ+)(∃α<µ+)[ψk,m(ηa,η+
+a,α∗,γ,α)].
+This can be done because there are stationarily α∗satisfying (5.16).
+Chooseγ≥M2∩µ+andα < µ+such thatψk,m(ηa,η+
+a,α∗,γ,α) holds, and
+define
+(5.17) t:={βm
+i(α∗,ǫa) :ǫa∈tαandi<ρm
+2(α∗,ǫa)}.
+We can now apply Lemma 4.1 to conclude that for all sufficiently large ia<cf(µ),
+there is an α<α∗such that
+(5.18) Γ+(η,ǫ) =iafor allη∈sαandǫ∈t,
+and
+(5.19) fα∗(ia+1)<fη(ia+1) for all η∈sα.
+It should be clear that {ηb,η+
+b}works for {α∗,α}atia. Since{ηb,η+
+b} ∈M0, the
+conclusion of the lemma now follows by a standard elementary submod el argument
+like that used to finish the proof of Lemma 3.2. /square
+In light of the preceding claim, we can fix ia> kand anη-candidate {ηb,η+
+b}
+such that
+(5.20) ( ∃statα∗<µ+)(∃∗γ <µ+)(∃α<µ+)
+[ψk,m(ηa,η+
+a,α∗,γ,α) and{ηb,η+
+b}works for {α∗,α}atia].
+Let us now define
+S:={α∗<µ+: (∃∗γ <µ+)(∃α<µ+)
+[ψk,m(ηa,η+
+a,α∗,γ,α) and{ηb,η+
+b}works for {α∗,α}atia]}
+and
+T∗:={β∗<µ+: (∃∗γ <µ+)(∃β <µ+)[ψk,m(ηb,η+
+b,β∗,γ,β)].
+Our choices make it clear that S∗andT∗are both stationary. We will thin out
+these sets a bit to obtain the promised stationary sets SandT. To do this, let x
+consist of those parameters needed to comprehend ψk,m(see the first line of the
+proof of Lemma 3.2), and let
+y:=x∪{S∗,T∗,ηa,η+
+a,ηb,η+
+b}.16 TODD EISWORTH
+Let/a\}bracketle{tMδ:δ<µ+/a\}bracketri}htbe aµ+-approximating sequence over y, and let
+E:={δ<µ+:Mδ∩µ+=δ}.
+We define
+S:=S∗∩E,
+and
+T:=T∗∩E.
+Now suppose /a\}bracketle{tα∗,β∗,δ/a\}bracketri}ht ∈S⊛T×cf(µ). We must produce α < βsuch that
+D↾tα×tβis constant with value /a\}bracketle{tα∗,β∗,δ/a\}bracketri}ht. To this point, we know
+(5.21) ηa≤η+
+a<ηb≤η+
+b<α∗<β∗.
+We choose now γbandβsuch that
+•ψk,m(ηb,η+
+b,β∗,γb,β), and
+•Mβ∗+2∩µ+≤γb.
+This can be done by the definition of T. Next, we define
+(5.22) t:={βm
+i(β∗,ǫb) :ǫb∈tβandi<ρm
+2(β∗,ǫb)}.
+By definition, γb= min(t) and soMβ∗+2∩µ+≤min(t).
+Finally, define Jto be the set of all α<µ+such that for some γ <µ+, we have
+•ψk,m(ηa,η+
+a,α∗,γ,α), and
+• {ηb,η+
+b}works for {α∗,α}atia.
+It is clear that Jis unbounded in µ+sinceα∗∈S. Furthermore, the set Jis
+definable in the model Mα∗+1, henceJ∈Mβ∗.
+Thus, the objects Mβ∗∈Mβ∗+1∈Mβ∗+2,β∗,/a\}bracketle{ttα:α∈J/a\}bracketri}ht, andtsatisfy the
+hypotheses of Lemma 4.1.
+We conclude that for all sufficiently large ib<cf(µ), there are unboundedly
+manyα∈J∩β∗such that for all ǫa∈tαand allǫ∈t,
+(5.23) Γ+(ǫa,ǫ) =ib,
+and
+(5.24) fβ∗(ib+1)<fǫa(ib+1).
+In particular, we can choose αandibin such a way that the above conditions are
+satisfied, and in addition such that
+(5.25) ι(ib) =/a\}bracketle{tm,δ/a\}bracketri}ht
+Notice that for any ǫa∈tαandǫb∈tβ, we have
+(5.26) ηa≤η+
+a<ηb≤η+
+b<α∗<ǫa<β∗<ǫb.
+The rest of the proof consists of show that D↾tα×tβis constant with value
+/a\}bracketle{tα∗,β∗,δ/a\}bracketri}ht, so assume now that ǫa∈tαandǫb∈tβ.
+Right away, we see that Γ( ǫa,ǫb) =ibsinceǫa∈tαandǫb∈t. As an immediate
+corollary, it follows that
+m(ǫa,ǫb) =m (5.27)
+and
+δ(ǫa,ǫb) =δ. (5.28)A COLORING THEOREM FOR SUCCESSORS OF SINGULAR CARDINALS 17
+Claim3.β∗(ǫa,ǫb) =β∗.
+Proof.Sinceψk,m(ǫb,ǫ+
+b,β∗,γb,β) holds, we know
+(5.29) ηb≤ηm(β∗,ǫb)≤η+
+b.
+Since
+(5.30) η+
+b<α∗<ǫa<β∗,
+it follows that
+(5.31) βm
+i(ǫa,ǫb) =βm
+i(β∗,ǫb) fori≤ρm
+2(β∗,ǫb).
+In particular,
+(5.32) βm
+i(ǫa,ǫb)∈tfori<ρm
+2(β∗,ǫb),
+and
+(5.33) βm
+ρm
+2(β∗,ǫb)(ǫa,ǫb) =β∗.
+Given our choice of α(see (5.23) and (5.24)), we obtain
+(5.34) β∗(ǫa,ǫb) =cm(ǫa,ǫb) =β∗,
+as required. /square
+Note that the preceding claim tells us
+(5.35) η∗(ǫa,ǫb) =ηm(β∗,ǫb)
+as well.
+Claim4.α∗(ǫa,ǫb) =α∗.
+Proof.Our choice of {ηb,η+
+b}tells us that
+ηa≤η+
+a<ηb≤η∗(ǫa,ǫb) =ηm(β∗,ǫb)≤η+
+b<α∗<ǫa,
+and therefore
+(5.36) βm
+i(η∗(ǫa,ǫb),ǫa) =βm
+i(α∗,ǫa) for alli≤ρm
+2(α∗,ǫa).
+Furthermore, we know
+(5.37) fηb↾[k,cf(µ))≤fη∗(ǫa,ǫb)↾[k,cf(µ))≤fη+↾[k,cf(µ)),
+and this implies
+fηb(ia)≤fη∗(ǫa,ǫb)(ia) (5.38)
+fηb(ia+1)≤fη∗(ǫa,ǫb)(ia+1) (5.39)
+and
+fη∗(ǫa,ǫb)↾[ia+1,cf(µ))≤fη+
+b↾[ia+1,cf(µ)). (5.40)
+Fori<ρm
+2(α∗,ǫa), we know that {ηb,η+
+b}works for {α∗,α}atia, and so
+(5.41) Γ+/parenleftbig
+ηb,βm
+i(α∗,ǫa)/parenrightbig
+= Γ+/parenleftbig
+η+
+b,βm
+i(α∗,ǫa)/parenrightbig
+=ia,
+and
+(5.42) fα∗(ia+1)<fηb(ia+1).
+From (5.36), (5.37), (5.38), (5.40), and (5.41) we conclude
+(5.43) Γ/parenleftbig
+η∗(ǫa,ǫb),βm
+i(η∗(ǫa,ǫb),ǫa)/parenrightbig
+=iafor alli<ρm
+2(α∗,ǫa).18 TODD EISWORTH
+Since
+(5.44) βm
+ρ2m(α∗,ǫa)(η∗(ǫa,ǫb),ǫa) =α∗,
+it follows from (5.39), (5.42), and (5.37) that
+(5.45) Γ/parenleftbig
+η∗(ǫa,ǫb),βm
+ρ2m(α∗,ǫa)(η∗(ǫa,ǫb),ǫa)/parenrightbig
+/\e}atio\slash=ia.
+From (5.43), (5.45), and the definition of cm(ǫa,ǫb), we conclude that
+(5.46)α∗(ǫa,ǫb) =cm(ǫa,ǫb)(η∗(ǫa,ǫb),ǫa) =βm
+ρ2m(α∗,ǫa)(η∗(ǫa,ǫb),ǫa) =α∗,
+as required. /square
+Putting Claim 3, Claim 4, and (5.28) together, we see
+(5.47) D(α,β) =/a\}bracketle{tα∗,β∗,δ/a\}bracketri}ht
+and the proof is complete. /square
+6.Consequences
+We turn our attention now to consequencesof Theorem 1 and pick u p the discus-
+sion of the introduction once more. Our first result gives the promis ed equivalence
+of (1.8) and (1.9), and also establishes an even stronger fact.
+Theorem 2. The following are equivalent for a singular cardinal µ:
+Pr1(µ+,µ+,µ+,cf(µ)) (6.1)
+Pr1(µ+,µ+,µ,cf(µ)) (6.2)
+Pr1(µ+,µ+,θ,cf(µ))for arbitrarily large θ<µ. (6.3)
+Proof.It is easy to see that each statement implies the one following, so we p rove
+that (6.3) implies (6.1). Fix an increasing sequence of cardinals /a\}bracketle{tθi:i <cf(µ)/a\}bracketri}ht
+cofinal inµsuch that Pr 1(µ+,µ+,θi,cf(µ)) holds for each i<cf(µ), and letcibe a
+coloring witnessing this for θi. Also, we fix for each β <µ+a functiongβmapping
+µontoβ.
+LetDbe a coloring as in our main theorem, and let α∗,β∗, andδdenote
+functions defined by the recipe
+(6.4) D(α,β) =/a\}bracketle{tα∗(α,β),β∗(α,β),δ(α,β)/a\}bracketri}ht.
+Finally, define the function c: [µ+]2→µ+as follows:
+(6.5) c(α,β) :=gβ∗(α,β)/parenleftbig
+cδ(α,β)(α∗(α,β),β∗(α,β))/parenrightbig
+.
+Now suppose /a\}bracketle{ttβ:β < µ+/a\}bracketri}htis a sequence of pairwise disjoint elements of
+[µ+]<cf(µ), and letς <µ+be arbitrary. We must find α<βsuch thatc↾tα×tβ
+is constant with value ς.
+Fix stationary sets SandTas in the conclusion of our main theorem. An
+application of Fodor’s Theorem allows us to find ǫ<µand stationary T∗⊆Tsuch
+thatgβ∗(ǫ) =ςfor allβ∗∈T∗.
+Next, we construct a sequence /a\}bracketle{tsγ:γ < µ+/a\}bracketri}htof pairwise disjoint elements of
+S⊛Tsuch that max( sζ)<min(sη) whenever ζ <η. This is easily done, as both
+SandT∗are unbounded in µ+. Now letδ<cf(µ) be chosen so that ǫ<θδ.
+Our choice of cδprovides us with ζ <η<µ+such that
+(6.6) cδ↾sζ×sηis constant with value ǫ.A COLORING THEOREM FOR SUCCESSORS OF SINGULAR CARDINALS 19
+Now supposing
+sζ={α∗
+ζ,β∗
+ζ} (6.7)
+and
+sη={α∗
+η,β∗
+η}, (6.8)
+we define
+α∗=α∗
+ζ (6.9)
+and
+β∗=β∗
+η (6.10)
+It should be clear that /a\}bracketle{tα∗,β∗/a\}bracketri}ht ∈S⊛T.
+Our assumptions about Dnow give us α<βsuch that
+(6.11) D↾tα×tβis constant with value /a\}bracketle{tα∗,β∗,δ/a\}bracketri}ht.
+Clearly we can also demand that sup( tα)<min(tβ), and now we show
+(6.12) c↾tα×tβis constant with value ς.
+Givenǫa∈tαandǫb∈tβ, we know
+α∗(ǫa,ǫb) =α∗, (6.13)
+β∗(ǫa,ǫb) =β∗, (6.14)
+and
+δ(ǫa,ǫb) =δ. (6.15)
+A glance at (6.5) tells us
+(6.16) c(ǫa,ǫβ) =gβ∗(cδ(α∗,β∗)),
+and now the result follows immediately as β∗∈T∗andcδ(α∗,β∗) =ǫ. /square
+The main theorem of our paper [4] established, among other things, that ifµ
+is singular of uncountable cofinality, then Pr 1(µ+,µ+,µ+,cf(µ)) holds unless the
+stationary subsets of µ+possess many instances of stationary reflection. In the
+case where the cofinality of µis countable, we were only able to get the analogous
+result with Pr 1(µ+,µ+,µ,cf(µ)), but this defect is repaired now by the equivalence
+of (6.2) and (6.1). This allows to state the following theorem without r estrictions
+on the cofinality of µ:
+Theorem 3. Ifµis singular and Pr1(µ+,µ+,µ+,cf(µ))fails then there is a θ<µ
+such that for any sequence /a\}bracketle{tSα:α < σ/a\}bracketri}htof stationary subsets of Sµ+
+≥θof length
+σ <cf(µ), there is an ordinal δ < µ+such thatSα∩δis stationary in δfor all
+α<σ.
+Proof.This is restatement the contrapositive of parts (2) and (3) of the main the-
+orem of [4], in light of the equivalence of (6.2) and (6.1). /square20 TODD EISWORTH
+It is still open whether Pr 1(µ+,µ+,µ+,cf(µ)) can fail for a singular cardinal.
+The above theorem tells us that obtaining such a consistency result will necessarily
+involve some considerable large cardinals. In light of the implications in ( 1.7), we
+see that this is true for the consistency of µ+being Jonsson, or the consistency of
+µ+→[µ+]2
+µ+as well.
+Our next result is a relative of one of the conclusions derived in [5].
+Theorem 4. The following are equivalent for a singular cardinal µand cardinal
+θ≤µ+:
+(1) Pr 1(µ+,µ+,θ,cf(µ)).
+(2)There is a function c: [µ+]2→θsuch that for any unbounded subsets A
+andBofµ+,
+(6.17) θ⊆ran(c↾A⊛B).
+(3)There is a function d: [µ+]2→θsuch that for any stationary subsets S
+andTofµ+,
+(6.18) θ⊆ran(d↾S⊛T).
+We are abusing notation a little bit in (6.17) and (6.18), as elements of S⊛T
+are technically ordered pairs and not pairs of ordinals, but the mean ing should be
+clear. Also note that (2) is relative of the relation
+(6.19) µ+/notarrowright[(µ+:µ+)]2
+θ
+(see [9]), which states that there is a function f: [µ+]2→θsuch that for any
+unbounded subsets AandBofµ+and anyς <θ, there areα∈Aandβ∈Bwith
+f(α,β) =ς. The difference between (2) and (6.19) is very slight — in (6.19), it is
+not required that αis less than β, while we need this in order to apply Theorem 1.
+Proof.The fact that (1) implies (2) is well-known (we did something similar in the
+proof of Theorem 2), but we give it for completeness. We show some thing a little
+bit stronger, namely that any function witnessing (1) also works fo r (2).
+Thus, letcwitness that Pr 1(µ+,µ+,θ,cf(µ)) holds, let ς <θbe given, and let S
+andTbe stationary subsets of µ+. Construct a sequence /a\}bracketle{ttα:α<µ+/a\}bracketri}htof pairwise
+disjoint elements of S⊛Tsuch that max( tα)<min(tβ) whenever α<β, and then
+fixα < βsuch thatc↾tα×tβis constant with value ς. Letα∗= min(tα) and
+β∗= max(tβ). Then/a\}bracketle{tα∗,β∗/a\}bracketri}ht ∈S⊛Tis as required.
+It is clear that (2) implies (3), so assume dbe as in (3), and let Dbe the function
+from Theorem 1, with
+(6.20) D(α,β) =/a\}bracketle{tα∗(α,β),β∗(α,β),δ(α,β)/a\}bracketri}ht.
+We define a function f: [µ+]2→θby
+(6.21) f(α,β) =c(α∗(α,β),β∗(α,β)),
+and the verification that fhas the required properties is straightforward. /square
+We will present only one application of the preceding theorem here, b ut we note
+that we can use Theorem 4 to greatly simplify the proof of the main re sult of [7].
+It also allows us to solve the main problem left open by [8]. We intend to pr esent
+this work elsewhere, as it is joint with Shelah.
+We start with a lemma, proved by a standard argumentA COLORING THEOREM FOR SUCCESSORS OF SINGULAR CARDINALS 21
+Lemma 6.1. Let (/vector µ,/vectorf) be a scale for the singular cardinal µ, and suppose A⊆µ+
+is unbounded. Then
+(6.22) ( ∀∗i<cf(µ))(∃∗ξ<µi)(∃∗α∈A)[fα(i) =ξ].
+Proof.The proof is by contradiction, so suppose the conclusion fails for so me un-
+boundedA⊆µ+. Parsing what this means, we find that there are unboundedly
+manyi<cf(µ), for all sufficiently large ξ <µi, the set of α∈Awithfα(i) =ξis
+bounded in µ+.
+LetIconsist of those i<cf(µ) for which the above is true, and for i∈Ichoose
+ξisuch that
+(6.23) ξi≤ξ<µi=⇒ |{α∈A:fα(i) =ξ}|<µ+.
+Giveni∈Iandξwithξi≤ξ<µi, we can fix an ordinal α(ξ,i)<µ+such that
+(6.24) {α∈A:fα(i) =ξ} ⊆α(ξ,i),
+and then define
+(6.25) α∗:= sup{α(ξ,i) :i<cf(µ),ξi≤ξ<µi}.
+It is clear that α∗<µ+as there are only µpossibilities for ξandi.
+After the dust has settled, we see that if α∈Ais greater than α∗, then
+(6.26) i∈I=⇒fα(i)<ξi,
+and this easily contradicts our assumption that ( /vector µ,/vectorf) is a scale. /square
+The theorem we prove below has many antecedents in the literature , but our
+result seems to be the first in which the partition relation holding at th e successor
+of the singular cardinal represents an upgrade over those assum ed to hold at the
+smaller cardinals.
+Theorem 5. Supposeµis singular, and there is a scale (/vector µ,/vectorf)forµsuch that
+(6.27) µi/notarrowright[(µi:µi)]2
+µi
+for alli<cf(µ). ThenPr1(µ+,µ+,µ+,cf(µ))holds.
+Proof.ByConclusion4.1Aonpage67of[10],weknowthatPr 1(µ+,µ+,cf(µ),cf(µ))
+holds, so we can fix a function c: [µ+]2→cf(µ) as in part (2) of Theorem 4 with
+cf(µ) standing in for θ.
+For eachi <cf(µ), letdi: [µi]2→µibe a witness for (6.27), and define a
+functiond: [µ+]2→µby
+(6.28) d(α,β) =dc(α,β)(fα(c(α,β)),fβ(c(α,β))).
+Givenς <µand unbounded subsets AandBofµ+, we will find /a\}bracketle{tα,β/a\}bracketri}htinA⊛B
+withd(α,β) =ς. This implies Pr 1(µ+,µ+,µ,cf(µ)) by Theorem 4, which in turn
+gives us Pr 1(µ+,µ+,µ+,cf(µ)) by Theorem 2.
+We choose i∗<cf(µ) so that
+ς <µi∗, (6.29)
+(∃∗ζ <µi∗)(∃∗α∈A)[fα(i∗) =ζ], (6.30)
+and
+(∃∗η<µi∗)(∃∗β∈B)[fα(i∗) =η]. (6.31)22 TODD EISWORTH
+Clearly this is possible by Lemma 6.1.
+Next, we define
+A∗:={ζ <µi∗: (∃∗α∈A)[fα(i∗) =ζ]} (6.32)
+and
+B∗:={η<µi∗: (∃∗β∈B)[fα(i∗) =η]}. (6.33)
+Both ofthese sets areunbounded in µi∗, andso wecan find α∗∈A∗andβ∗∈B∗
+with
+(6.34) di∗(α∗,β∗) =ς.
+Now define
+A†={α∈A:fα(i∗) =α∗} (6.35)
+and
+B†={β∈B:fβ(i∗) =β∗}. (6.36)
+Both of these sets are unbounded in µ+, and so we can find /a\}bracketle{tα,β/a\}bracketri}htinA⊛Bwith
+c(α,β) =i∗.
+We find now that
+(6.37)d(α,β) =dc(α,β)(fα(c(α,β)),fβ(c(α,β)))
+=di∗(fα(i∗),fβ(i∗)) =di∗(α∗,β∗) =ς,
+as required. /square
+We comenowtoaresultpromisedatthe end ofthe introduction. The hypothesis
+pp(µ) =µ+refers to Shelah’s pseudo-power function, but we will not elaborat e as
+we need only one easily understood consequence of this assumption . The reader
+can consult [10] for the definition of pp, and the author’s [3] contain s the proof of
+the relevant facts.
+Corollary 6.2. Pr1(µ+,µ+,µ+,cf(µ)) holds for any singular µwith pp(µ) =µ+.
+Proof.Since pp(µ) =µ+, there is a scale ( /vector µ,/vectorf) such that for each i <cf(µ),
+µi=κ++
+ifor some uncountable regular cardinal κi. By [11], it follows that
+(6.38) µi/notarrowright[(µi:µi)]2
+µi
+for eachi<cf(µ), and so we get what we need by way of Theorem 5. /square
+The proof of the preceding is deceptively short, for [11] is quite a diffi cult paper.
+Shelah shows there that
+(6.39) Pr 1(κ++,κ++,κ++,κ)
+holdsforregular κ, but theargumentforthe implication“(1) = ⇒(2)”inTheorem4
+tells us that
+(6.40) κ++/notarrowright[(κ++:κ++)]2
+κ++
+for every regular κas well.
+Corollary 6.2 can also be proved in the following manner. Claim 4.1E on pag e 70
+of [10] implies, when suitably interpreted and combined with the result quoted
+in (6.39), that Pr 1(µ+,µ+,µ,cf(µ)) holds. This fact was noted by Shelah in aA COLORING THEOREM FOR SUCCESSORS OF SINGULAR CARDINALS 23
+personal communication with the author, but Theorem 2 is still nece ssary to ob-
+tain a coloring with µ+colors. We chose our approach because Theorem 5 is of
+independent interest and of more general applicability.
+References
+[1] James E. Baumgartner. A new class of order types. Ann. Math. Logic , 9(3):187–222, 1976.
+[2] James Cummings. Notes on singular cardinal combinatori cs.Notre Dame J. Formal Logic ,
+46(3):251–282 (electronic), 2005.
+[3] Todd Eisworth. Successors of singular cardinals. Chapt er in the forthcoming Handbook of
+Set Theory.
+[4] Todd Eisworth. Club-guessing, stationary reflection, a nd coloring theorems. Submitted to
+APAL, July 2009.
+[5] Todd Eisworth. Getting more colors. submitted to JSL, De cember 2009.
+[6] Todd Eisworth. A note on strong negative partition relat ions.Fund. Math. , 202:97–123, 2009.
+[7] Todd Eisworth and Saharon Shelah. Successors of singula r cardinals and coloring theorems
+I.Arch. Math. Logic , 44:597–618, 2005.
+[8] Todd Eisworth and Saharon Shelah. Successors of singula r cardinals and coloring theorems
+II.J. Symbolic Logic , 74(4):1287–1309, Dec. 2009.
+[9] Paul Erd˝ os, Andr´ as Hajnal, Attila M´ at´ e, and Richard Rado.Combinatorial set theory: par-
+tition relations for cardinals , volume 106 of Studies in Logic and the Foundations of Mathe-
+matics. North-Holland Publishing Co., Amsterdam, 1984.
+[10] Saharon Shelah. Cardinal Arithmetic , volume 29 of Oxford Logic Guides . Oxford University
+Press, 1994.
+[11] Saharon Shelah. Colouring and non-productivity of ℵ2-c.c.Ann. Pure Appl. Logic , 84(2):153–
+174, 1997.
+[12] Stevo Todorˇ cevi´ c. Partitioning pairs of countable o rdinals. Acta Math. , 159(3-4):261–294,
+1987.
+[13] Stevo Todorcevic. Walks on ordinals and their characteristics , volume 263 of Progress in
+Mathematics . Birkh¨ auser Verlag, Basel, 2007.
+Department of Mathematics, Ohio University, Athens, OH 4570 1
+E-mail address :eisworth@math.ohiou.edu
\ No newline at end of file
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+ 
+A Distributed File System for a Wide-Area High Performance Computing 
+Infrastructure 
+ 
+Edward Walker 
+University of Texas at Austin 
+ 
+Abstract
+ 
+We describe our work in implementing a wide-area distributed file system for the NSF TeraGrid.  The system, called XUFS, 
+allows private distributed name spaces to be created for transparent access to personal files across over 9000 computer nodes. XUFS builds on many principles from prior distributed file systems research, but extends key design goals to support the workflow of computational science researchers.   Specifically, XUFS supports file a ccess from the desktop to the wide-area 
+network seamlessly, survives transient disconnected operations robustly, and demonstrates comparable or better throughput than some current high performance file systems on the wide-area network.   
+1. Introduction 
+How do computational scientists access files in a widely-
+distributed computing environment?  And what needs to be provided to enable them to access files easily and be productive in these environments?  We provide one data point from the NSF TeraGrid towards answering the first question and we propose a system that we believe provides a simple answer to the second question.   
+The TeraGrid is a multi-year, multi-million dollar, 
+National Science Foundation (NSF) project to deploy the world’s largest distributed infrastructure for open scientific research [1].  The project currently links over 9000 compute nodes, contributed from 19 systems, located at nine geographically distributed sites, through a dedicated 30 Gbps wide-area network (WAN).  In aggregate, the project provides over 100 TFLOP of compute power and four PB of disk capacity for computational science research.  Future system acquisitions for the project are expected to expand the compute capability of the infrastructure into the PFLOP range. 
+In 2005, TeraGrid conducted a survey amongst 412 of its 
+users.  One of the results from the survey highlighted the secure copy command, SCP,  as the most important data 
+management tool currently being used [2].   To address this 
+issue, the GPFS-WAN distributed parallel file system [3] went into production that same year at three sites on the TeraGrid: the San Diego Supercomputing Center (SDSC), the National Center for Supercomputing Applications (NCSA) and Argonne National Laboratory (ANL).  Users and jobs running on any compute node on systems at these three sites could now access their files from a common /gpfs-wan/  
+directory cross-mounted at all three sites.   GPFS-WAN was 
+selected over other wide-area distributed file systems, like OpenAFS [4] and NFSv4 [5], because of the high speed parallel file access capabilities provided by the underlying GPFS technology [6][7].  Some of the largest supercomputers in the world currently use GPFS as their primary work file system.   
+However, a number of issues persist with this solution.  
+First, files (data and code), residing on the user’s personal workstation, still needs to be manually copied to at least one of the TeraGrid sites, causing multiple versions of these files to be created.  Second, not all sites use GPFS as their primary work file system, precluding these sites from participating in the GPFS-WAN solution.  Other parallel file system technologies in use include Lustre [8] and IBRIX [9].    
+In this paper, we describe a distributed file system we have 
+implemented to address these concerns, and in the process extend some of the traditional design goals assumed by many past distributed file systems projects.   Many techniques proposed by past projects are valid today, and we have therefore incorporated these into our design.  But there is an opportunity to reexamine some of these past design goals, incorporating the current reality of personal mobility, hardware commoditization and infrastructure advancements, into a solution that not only builds on prior art, but introduces new, more productive, usage patterns.   
+Today, the commoditization of disk and personal 
+computers has resulted in many scientists owning personal systems with disk capacity approaching hundreds of GB.  These computational scientists have the ability to routinely run simulations and store/analyze data on a laptop computer or a desktop workstation.  Only the largest compute runs are scheduled on high-end systems at the NSF supercomputing centers.   
+Also, wide-area network bandwidth is no longer as scarce a 
+resource as it once was in the computational science community.  In addition to the TeraGrid network, other national projects like NLR [10] and Abilene-Internet2 [11] are providing multi-Gbps wide-area networks, with the goal of 100 Mbps a ccess to every desktop.  Furthermore, a lot of 
+this available bandwidth is not currently fully utilized [12], so there is an opportunity for software systems to better utilize this spare capacity, and provide for a richer computing experience.  
+In light of these conditions, this paper reconsiders the 
+following often cited distributed file system design goals [14][15][16][17][30]: 
+a. Location transparency :  All past distributed file 
+systems projects assumed this goal.  Although being able to walk down a corridor to physically access the same files from a different workstation may still be important, a scientist is equally mobile by physically carrying his (or hers) personal system in the form of a laptop, or a memory stick, into a different network.  Thus, the ability to extend a users personal work-space from a laptop, or a foreign desktop with an attached memory stick, is equally important.  
+b. Centralized file servers :  Many systems, such as AFS 
+[14], Decorum [15] and Coda [16], make a strict distinction between clients and servers.  Some treat clients primarily as memory caches, like Sprite [17].  However, personal systems now have large disk capacity, and this is changing the role of the “client”.  Files residing on multiple, distributed, personal systems need to be exported to a few central systems for access by users and their applications.  In other words, these large central systems are now the “clients” for files residing on the user’s personal workstation.  
+c. Minimize network load : Systems like Decorum and 
+Sprite strive to minimize network load as one of their fundamental design goals.  Although this is still an important design goal, it is no longer as critical as it once was.  Cost-benefit tradeoffs may be made to determine if the increased user productivity justifies the cost of the extra network bandwidth utilized.  
+d. File sharing between users : Previous studies [19][20] 
+have shown that file sharing between users on a file system is not common.  However, many distributed file systems still consider it a major benefit to provide the ability to share files between users [21].  Recently, many other non-file system mechanisms have been developed to facilitate wide-area data sharing: e.g. web portals, peer-to-peer systems, etc.  Therefore, developing a system to enable private , transparent, 
+access to personal files across remote sites not only addresses the most common use case, but also simplifies the design space considerably. 
+For the remainder of this paper, we describe the XUFS 
+distributed file system.  XUFS is a distributed file system that takes the assumption of personal mobility, with significant local disk resource and access to high-bandwidth research networks, as important considerations in its design criteria.   Section 2 further expounds on the system requirements, including motivating empirical observations made on the TeraGrid.   Section 3 describes the component architecture, the cache coherency protocol, the recovery mechanism, an example access client, and the security framework used in XUFS.  Section 4 looks at some comparison benchmarks between XUFS and GPFS-WAN, and section 5 examines related work.  Section 6 concludes this paper. 2. Requirements Overview 
+2.1. Computational Science Workflow 
+From our interviews with many scientists, the 
+computational science workflow shows many commonalities.  This workflow usually involves 1) developing and testing simulation code on a local resource (personal workstation or departmental cluster), 2) copying the source code to a supercomputing site (where it needs to be ported), 3) copying input data into the site’s parallel file system scratch space, 4) running the simulation on the supercomputer (where raw output data may be generated in the same scratch space), 5) analyzing the data in the scratch space, 6) copying the analysis results back to the local resource, and 7) moving the raw output to an archival facility such as a tape store.   
+In this scenario, files move from the local disk to a remote 
+disk (where they reside for the duration of the job run, and final analysis), with some new files moving back to the local disk, and the remainder of the new files moving to some other space (like a tape archive). 
+In this usage pattern, a traditional distributed file system 
+will not suffice.  Some simulations generate very large raw output files, and these files are never destined for the user’s local resource.  However, users still require transparent a ccess 
+to these large data files from their personal work-space in order to perform result analysis [24].   
+2.2. File-Sharing in Computational Science 
+Previous studies have shown that file-sharing between 
+users is not common [19][20].  However in computational science this appears to be even less prevalent.  We looked at the directory permissions for the 1,964 TeraGrid users in the parallel file system scratch space on the TeraGrid cluster at the Texas Advanced Computing Center (TACC).    
+We looked at this cluster because of the site’s policy of 
+setting the default umask  for every user to “077”.  Thus a 
+directory created by the user will only have user read-write 
+permission enabled by default.  Group read-write permission needs to be explicitly given by the user to permit file sharing.  Other sites have a more lenient umask  policy; hence no 
+conclusion can be derived by simply looking at the directory 
+permissions at those sites.   However, since the directories we examined at TACC belonged to TeraGrid-wide users, with many having the same accounts on the other systems on the TeraGrid, our conclusion represents a valid data point for the entire project as a whole. 
+Of the 1,964 directories examined  at the time of this paper 
+preparation, only one user explicitly enabled group read-write 
+permission.  This of course does not preclude file sharing in the future through a different work space, web site or central database.  However, we conclude that at least for the duration covering the period of input data preparation, the simulation run and the analysis of the simulation output files, little file-sharing between users is done.  
+2.3. Where are the Bytes? 
+This time we looked at a snapshot of the size distribution of 
+all files in the parallel file system scratch space on the TACC TeraGrid cluster.  TACC has a policy of purging any files in its scratch file system that has not been accessed for more then ten days.  We can therefore assume that the files were actively accessed by running jobs or users.  Again, the directories where these files were examined belonged to TeraGrid-wide users, so our conclusions represent one data point for the TeraGrid as well. 
+Table 1. Cumulative file size distribution for the parallel 
+file system scratch space on the TACC TeraGrid cluster. 
+Size Files Frequency  Total gigabytes Frequency 
+> 500M 130 0.09%  302.471 35% 
+> 400M 204 0.14%  335.945 38.87% 
+> 300M 271 0.19%  359.140 41.55% 
+> 200M 1413 0.99%  623.137 70.09% 
+> 100M 2523 1.76%  779.611 90.19% 
+> 1M 12856 9%  851.347 98.49% 
+> 0.5M 16077 11.23%  853.755 98.77% 
+> 0.25M 30962 21.62%  859.584 99.45% 
+Total 143190 100%  864.385 100% 
+Table 1 summarizes our findings.  We note that even 
+though only 9% of the files were greater then 1 megabyte in size, over 98.49% of the bytes are from files in this range. The file bytes represent the used/generated bytes by computational simulation runs in the TeraGrid cluster; hence they represent the majority of the I/O activity for the parallel file system.   Out study validates other studies [23] that have also shown this trend towards larger files in scientific computing environments.   
+2.4. Design Assumptions 
+From our empirical observations we make the following 
+design assumptions.  First, file access is more important then file-sharing between users.  This is additionally borne out in an internal 2004 TeraGrid survey [24].  If file sharing between users is required, this can be easily achieved by copying files from XUFS to some other space if needed. 
+Second, file access needs to originate from a user’s 
+personal work-space, from a desktop workstation or laptop.  A consequence of this is that the file server is now the user’s personal system, and we assume this to be unreliable, i.e. disconnects from the server is the norm. 
+Third, some files should never be copied back to the 
+personal work-space.  In our case, we allow the user to specify hints as to which directory any newly generated files are kept local to the client.  We call these localized  directories .   
+Fourth, client machines can have plenty of disk capacity.  
+User files residing on personal workstations need to be accessed by TeraGrid “client” sites where over 4 PB of combined disk capacity exists.     
+Fifth, the interface to the local parallel file system should 
+be exposed to the application at the client, enabling the use of any available advanced file system features.   3. Detailed Design 
+3.1. Component architecture 
+The current implementation of XUFS is provided in a 
+shared object, libxufs.so , and a user-space file server.  
+XUFS uses the library preloading feature in many UNIX 
+variants to load shared objects into a process image.  The XUFS shared object implement functions that interpose the libc  file system calls, allowing applications to transparently 
+access files and directories across the WAN by redirecting 
+operations to local cached copies.    
+Shell, Unix commands, user 
+application, etc.
+Interposition shared 
+object
+Meta-operations 
+queueFile server
+Cache spaceLocal name 
+spaceHome name 
+spaceSync managerNotification 
+callback manager
+Operation 
+LogExtended  file 
+name spaceLease managerExtended Work-
+space (TeraGrid site)Personal work-
+space (local 
+workstation)
+ 
+Figure 1.  XUFS component architecture 
+The component architecture of XUFS is shown in Figure 1.  
+When an application or user first mounts  a remote name 
+space in XUFS, a private cache space  is created on the client 
+host.   At TeraGrid sites, this cache space is expected to be on the parallel file system work partition.   
+When an opendir()  is first invoked on a directory that 
+resides in the remote  home name space , the interposed 
+version of the system call contacts a sync manager , 
+downloads the directory entries into the cache space, and redirects all directory operations to this local cache copy.  XUFS essentially recreates the entire remote directory in cache space, and stores the directory entry attributes in hidden files alongside the initial empty file entries.  Subsequent stat()  system calls on entries in this directory 
+return the attributes stored in the hidden file associated with 
+each entry.  Only when an open()  is first invoked on a file 
+in this directory will the interposed version of the open 
+system call contact the sync manager again to download the file into the cache space.    
+System calls that modify a file (or directory) in a XUFS 
+partition return when the local cache copy is updated, and the 
+operation is appended to a persisted meta-operation queue .  
+No file (or directory) operation blocks on a remote network call.  A write()  operations is treated differently from other 
+attribute modification operations however, in that the write 
+offsets and contents are stored into an internal shadow file   
+and the shadow file flush is only appended to the meta-
+operations queue on a close() .  Thus only the aggregated 
+change to the content of a file is sent back to the file server on 
+a close.   XUFS therefore implements the last-close-wins  
+semantics for files modified in XUFS mounted partitions.   
+Cache consistency with the home space is maintained by 
+the notification callback  manager .  This component registers 
+with the remote file server, via a TCP connection, for notification of any changes in the home name space.  Any change at the home space will result in the invalidation of the cached copy, requiring it to be re-fetched prior to being accessed again.  This guarantees the same semantics the user is expected to encounter when the file is similarly manipulated locally. 
+In the case of a client crash, we provide a command-line 
+tool for users to sync operations, which were in the meta-operations queue at the time of the crash, with the file server. In the case of a server crash, the file server is restarted with a crontab  job on recovery, with the client periodically 
+attempting to re-establish the notification callback channel 
+when it notices its termination. 
+File locking operations, except for files in localized 
+directories , are forwarded to the file server through the lease 
+manager .  The lease manager is responsible for periodically 
+renewing the lease on a remote lock to prevent orphaned locks.  Files in a localized directory can use the locking mechanisms provided by the cache space file system.     
+3.2. Security 
+XUFS provides multiple means to a ccess its distributed file 
+system functionality.  However, we provide an OpenSSH client, USSH, as one default mechanism.  With USSH, users can login from their personal system into a site on the TeraGrid and access personal files from within this login session by mounting entire directory trees from their personal work-space.  USSH provides XUFS with a framework for authenticating all connections between client and server.   
+When USSH is used to log into a remote site, the command 
+generates a short-lived secret <key,phrase>  pair, starts a 
+personal XUFS file server, authenticates with the remote site 
+using standard SSH mechanisms, and starts a remote login shell.   USSH then preloads the XUFS shared object in the remote login shell, and sets the <key, phrase>  pair in the 
+environment for use by the preloaded shared object.  
+Subsequent TCP connections between the client and file server is then authenticated with the <key, phrase>  pair 
+using an encrypted challenge string.  Communication 
+encryption can be further configured by enabling the use of port-forwarding through a SSH tunnel.  This option can also be used for tunneling through local firewalls if needed.  
+3.3. Striped Transfers and Parallel Pre-Fetches 
+For large latency wide-area networks, the caching of entire 
+files can take a non-negligible amount of time.  The situation is further exacerbated by the fact that there are a significant number of very large files in use in our environment.   In order to alleviate this potential usability issue, we take advantage of the large network bandwidth available on the 
+TeraGrid.   
+All data transfers in XUFS over 64 Kbytes are striped 
+across multiple TCP connections.  XUFS uses up to 12 stripes with a minimum 64 kilobytes block size each when performing file caching transfers.  XUFS also tries to maximize the use of the network bandwidth for caching smaller files by spawning multiple (12 by default) parallel threads for pre-fetching files smaller then 64 kilobytes in size.  It does this every time the user or application first changes  
+into a XUFS mounted directory.   
+4. Performance 
+We perform three experiments to examine the micro and 
+macro behavior of XUFS, compared to GPFS-WAN, on the TeraGrid WAN.  The experiments measure the comparative performance of the two file systems using 1) the IOzone micro-benchmarks [29], 2) the build time of a moderate size source code tree, and 3) the turnaround times for a ccessing a 
+large file located across the WAN. 
+For the GPFS-WAN scenarios, the benchmarks were all 
+run at NCSA.  For the XUFS scenarios, configured with no encryption, we import a directory from the SDSC GPFS scratch space into the NCSA GPFS scratch space, and we ran the benchmarks in the imported directory. Thus in both scenarios, the authoritative versions of the files were always at SDSC (the GPFS-WAN file servers are at SDSC).   The network bandwidth between SDSC and NCSA is 30 Gbps. 
+4.1. Micro-Benchmark 
+The IOzone benchmark measures the throughput of a 
+variety of file operations on a file system.  In this experiment we use the benchmark to measure the read  and write  
+performance.    
+110100100010000100000100000010000000
+1MB5MB100MB200MB300MB400MB500MB600MB
+700MB
+800MB900MB1000MB
+File s izeKilobytes/secGPFS-WAN
+XUFS
+ 
+Figure 2. Write performance of the WAN file systems 
+100010000100000100000010000000
+1MB5MB
+100MB
+200MB300MB
+400MB
+500MB600MB
+700MB800MB
+900MB
+1000MB
+File sizeKilobytes/secGPFS-WAN
+XUFS
+ 
+Figure 3. Read performance of the WAN file systems 
+We ran the benchmark for a range of file sizes from 1 MB 
+to 1 GB, and we also included the time of the close   
+operation in all our measurements to include the cost of cache 
+flushes.  The throughput for the write and read performance is shown in Figure 2 and Figure 3.   
+XUFS demonstrates better performance then GPFS-WAN 
+in the read case for files larger then 1 MB.  As we have seen, the majority of bytes on the TeraGrid are in files greater then 1 MB, so this is an important benefit.  XUFS does well because it directly accesses files from the local cache file system.   
+Also, XUFS performance is generally comparable to GPFS-
+WAN for the write case.  However, GPFS-WAN demonstrate far better write performance than XUFS for the 1 MB file.  This is probably demonstrates the benefit of memory caching in GPFS.
+ 
+4.2. Source Code Build Times  
+In this experiment we built a source code tree, containing 
+24 files of approximately 12000 lines of C source code distributed over 5 sub-directories.  A majority of the files in this scenario were less then 64 KB in size.  In our measurements we include the time to change to the source code tree directory and perform a clean “ make ” .    T h i s  
+experiment measures the amortized overhead of an expected 
+common use case on the WAN file systems. 
+The timings of consecutive runs are shown in Figure 4.  
+Surprisingly XUFS mostly out performs GPFS-WAN in this benchmark.  We speculate this is due to our aggressive parallel file pre-fetching strategy.  
+1101001000
+1 2 3 4 5 6 7 8 9 1 01 11 21 31 41 51 61 71 81 9
+Consecutive runsBuild time (secs)GPFS-WAN
+XUFS
+GPFS Scratch
+ 
+Figure 4. Build times on the WAN file systems and the 
+local GPFS partition. 
+4.3. Large File Access  
+In this experiment we measured the time of a shell 
+operation on a 1 GB file on the WAN file systems.  The command used was “ wc –l ”: the command opens an input 
+file, counts the number of new line characters in that file, and 
+prints this count.  The timings of consecutive runs of the command are shown in Figure 5.      
+The results show the command takes approximately 60 
+seconds to complete in the first instance when run in XUFS.  This delay is due to the system copying the file into its cache space for the first invocation of open() .  However, 
+subsequent invocation of the command performs considerably 
+better because XUFS is redirecting all file access to the local cache file.  GPFS-WAN show a consistent access time of 33 
+seconds in all 5 runs of the experiment.  110100
+123456
+Consecutive runsSecsGPFS-WAN
+XUFS
+GPFS Scratch
+ 
+Figure 5. Access timings for a 1GB file on the WAN file 
+systems and the local GPFS partition. 
+We also compared the XUFS timings, with the timing from 
+manually copying the 1GB file from SDSC to NCSA using the copy commands TGCP (a GridFTP client) and SCP.  The timings are summarized in Table 2.  The results show TGCP as having only a slight advantage over XUFS in terms of the turnaround time of a ccessing the large file.  The poor 
+performance of SCP is probably due to its use of encryption, and its lack of TCP stripping. 
+ 
+Table 2. Mean time of "wc -l" on a 1GB file in XUFS, 
+compared to copying the file across the WAN using the TGCP and SCP copy commands. 
+XUFS (secs) TGCP (secs) SCP (secs) 
+57 49 2100 
+5.  Related Work 
+The XUFS approach of using a shared object to interpose a 
+global name space for file systems located across a WAN is similar to the approach expounded by the Jade file system [22].  Like Jade, XUFS allows private name spaces to be created, and aggressively caches remote files to hide access latency on the WAN.  However, unlike Jade, XUFS employs a different cache consistency protocol, and uses striped block transfers and parallel file pre-fetching to better utilize large bandwidth networks.   
+XUFS caches entire files on disk instead of using smaller 
+memory block caches.  Most distributed file systems cache memory blocks for fast repeat access [5][6][15][17][18].  A few past systems have also used entire file caches; e.g. Cedar [28], AFS-1 [14], and Coda [16].  Our primary reason for caching entire files is because of our assumption that the server is unreliable.   Computation jobs need reliable access to input data files, and having the entire file in cache allows XUFS to provide access to files even during temporary server or network outages. 
+XUFS reliance on a callback notification mechanism for its 
+cache consistency protocol is similar to the approach used by AFS-2 and AFS-3 [30].  Cached copies are always assumed to be up-to-date unless otherwise notified by the remote server.  This protocol is different from the cache consistency protocol in NFS and Jade where clients are responsible for checking content versions with the remote server on every file open() .  This is also different from the token-based cache 
+consistency protocols used in GPFS [6], Decorum [15] and 
+Locus [27].   Token-based consistency protocols are particularly efficient for multiple-process write-sharing  
+scenarios.  However, XUFS allows the high performance 
+writing-sharing features in a parallel file system, configured as the cache space, to be made available to an application.  This satisfies the overwhelming majority of cases where an application is scheduled to run on multiple CPUs at one site.  Exposing the local file system features in the distributed file system interface is also supported by the Decorum file system [15] and extensions to the NFSv4 [25] protocol. 
+XUFS share similar motivation with other r ecent work on 
+wide-area network file systems, such as Shark [33].  However, XUFS differs with Shark in its fundamental design goals and detailed implementation.  Shark uses multiple cooperating replica proxies for optimizing network bandwidth utilization and alleviating the burden of serving many concurrent client requests on the central file server.  XUFS uses stripped file transfers to maximize network bandwidth utilization, and instantiates a private user-space file server for each user.  
+Finally, the semantics of our USSH access client is similar 
+in many respects to the import  command in Plan 9 [13], 
+and to the remote execution shell Rex [32].  However, USSH 
+differs from these other commands in the underlying 
+protocols used to access files and directories: 9P in the case of import , and SFS in the case of Rex. 
+6.  Conclusion 
+XUFS is motivated by real user requirements; hence we 
+expect the system to evolve over time based on feedback from scientists using it while engaged in distributed computational research.  However, future work on XUFS will also investigate integration with alternative interposition mechanisms such as SFS [31], FUSE [35] and Windows Detours  [34].  These will broaden the range of XUFS 
+supported platforms and usage scenarios, eliciting a wider community of users to instigate its evolution.    
+Acknowledgement 
+We would like to thank the anonymous reviewers for their 
+helpful and constructive comments.  In particular we would like to thank our shepherd, Mike Dahlin, for diligently reviewing and providing valuable feedback to the final manuscript. 
+References 
+[1] NSF TeraGrid, http://www.teragrid.org   
+[2] NSF Cyberinfrastructure User Survey 2005, http://www.ci-
+partnership.org/survey/report2005.html   
+[3] P. Andrews, P. Kovatch and C. Jordan, “Massive High-Performance 
+Global File Systems for Grid Computing”, in Proc of the ACM/IEEE 
+Conference on High Performance Networking and Computing (SC05) , 
+November 12-18, Seattle, WA, USA, 2005. 
+[4] OpenAFS, http://www.openafs.org   
+[5] B. Pawlowski, S. Shepler, C. Beame, B. Callagham, M. Eisler, D. Noveck, 
+D. Robinson, and R. Thurlow, “The NFS Version 4 Protocol”, in Proc. of 
+the 2nd Intl. System Admin. And Networking Conference (SANE2000),  
+2000 
+[6] F. Schmuck, and R. Haskin, “GPFS: A Shared-Disk File System for Large 
+Computing Clusters”, in Proc. of the 1st USENIX Conference on File and 
+Storage Technologies , Monterey, CA, 2002 [7] IBM Research, General Parallel File System (GPFS), 
+http://www.almaden.ibm.com/StorageSystems/file_systems/G PFS/   
+[8] P. Schwan, “Lustre: Building a File System for 1,000-node Clusters”, in 
+Proc. of Ottawa Linux Symposium , 2003. 
+[9] IBRIX, http://www.ibrix.com   
+[10] National LambdaRail, http://www.nlr.net   
+[11] Abilene, http://abilene.internet2.edu   
+[12] NLR Weather Map, http://weathermap.grnoc.iu.edu/nlrmaps/layer3.html   
+[13] Plan 9 from Bell Labs.  http://cm.bell-labs.com/plan9/  
+[14] J. H. Howard, M. L. Kazar, S. G. Menees, D. A. Nichols, M. 
+Satyanarayanan, R. N. Sidebotham, and M. J. West, “Scale and Performance in a Distributed File System”, ACM Trans. on Computer Systems, 6(1), Feb 1988, pp. 51-81. 
+[15] M. L. Kazar, et. al ., “Decorum File System Architectural Overview”, in 
+Proc. of the 1990 USENIX Technical Conf ., CA, June 1990, pp. 151-164. 
+[16] M. Satyanarayanan, “Coda: A Highly Available File System for 
+Distributed Workstation Environment”, IEEE Trans. on Computers, 39(4), 1990, pp. 447-459.  
+[17] M. N. Nelson, et. al ., “Caching in the Sprite Network File System”, ACM 
+Trans. on Computer Systems,  6(1), Feb 1988, pp. 134-154.  
+[18] T. Anderson, M. Dahlin, J. Neefe, D. Patterson, D. Roselli, and R. Wang, 
+“Serverless Network File Systems”, ACM Trans. on Computer Systems, 14(1), Feb. 1996, pp. 41—79.  
+[19] M. G. Baker, J. H. Hartman, M. D. Kupfer, K. W. Shirrif, and J. K. 
+Ousterhout, “Measurements of a Distributed File System”, in Proc. of the 
+13
+th ACM Symp. on Operating Systems Principles ,  1991. 
+[20] J. Ousterhout, H. Da Costa, D. Harrison, J. Kunze, M. Kupfer, and J. 
+Thomson, “A Trace-Driven Analysis of the 4.2BSD File System”, in Proc. 
+of the 10th ACM Symp. on Operating System Principles , 1985. 
+[21] M. Spasojevic, and M. Satyanarayanan, “An Empirical Study of a Wide-
+Area Distributed File System”, ACM Trans. on Computer Systems, 14(2), 1996, pp. 200-222. 
+[22] H. C. Rao and L. L. Peterson, “Accessing Files in an Internet: The Jade 
+File System”, IEEE Trans. on Software Engineering, 19.(6), June 1993 
+[23] F. Wang, Q. Xin, B. Hong, S. A. Brandt, E. L. Miller, D. D. E. Long, and 
+T. T. McLarty, “File System Workload Analysis for Large Scale Scientific 
+Computing Applications”, in Proc. 21
+st IEEE Conf. on Mass Storage 
+Systems and Technologies , MD, 2004, pp. 129-152.  
+[24] TeraGrid Director Presentation (slide 27): 
+http://www.tacc.utexas.edu/~ewalker/director.ppt  
+[25] A. Adamson, D. Hildebrand, P. Honeyman, S. McKee, and J. Zhang, 
+“Extending NFSv4 for Petascale Data Management”, in Proc. of 
+Workshop on Next-Generation Distributed Data Management , Paris, 
+June 2006. 
+[26] J. J. Kistler, and M. Satyanarayanan, “Disconnected Operation in the Coda 
+File System”, ACM Trans. On Computer Systems, 10(1), 1992. 
+[27] G. J. Popek and B. J. Walker, eds., The LOCUS Distributed System 
+Architecture, MIT Press, Cambridge MA, 1985.  
+[28] D. K. Gifford, R. M. Needham, an d M. D. Schroeder, “The Cedar File 
+System”, Comm. of the ACM, 31(3), 1988, pp. 288-298. 
+[29] IOzone filesystem benchmark, http://www.iozone.org   
+[30] M. Satyanarayanan, “Scalable, Secure and Highly Available File Access in 
+a Distributed Workstation Environment”, IEEE Computer, 23(5), May 1990, pp. 9-22.  
+[31] D. Mazieres, “A Toolkit for User-Level File Systems”, in Proc. of the 
+2001 USENIX Technical conf ., Boston, MA, June 2001. 
+[32] M. Kaminsky, E. Peterson, D. B. Giffin, K. Fu, D. Mazieres, and F. 
+Kaashoek, “Rex: Secure, Extensible Remote Execution”, in Proc. of 2004 
+USENIX Technical Conf.,  2004. 
+[33] S. Annapureddy, M. J. Freedman, D. Mazieres, “Shark: Scaling File 
+Servers via Cooperative Caching”, in Proc. of 2
+nd USENIX/ACM Symp. on 
+Networked System Design and Implementation , 2005. 
+[34] G. Hunt and D. Brubacher, “Detours: Binary Interception of Win32 
+Functions”, in Proc. of the 3rd USENIX Windows NT Symp osium, Seattle, 
+WA 1999, pp. 135-143. 
+[35] M. Szeredi, FUSE, http://sourceforge.net/projects/fuse 
\ No newline at end of file
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+arXiv:1001.0197v1  [astro-ph.IM]  1 Jan 2010Astronomical Data Analysis Software and Systems XIX P24
+ASP Conference Series, Vol. XXX, 2009
+Y. Mizumoto, K.-I. Morita, and M. Ohishi, eds.
+Pointing the SOFIA Telescope
+Michael A. K. Gross1, John J. Rasmussen2, Elizabeth M. Moore1
+Abstract. SOFIA is anairborne, gyroscopicallystabilized 2.5minfraredtele-
+scope, mounted to a spherical bearing. Unlike its predecessors, S OFIA will work
+in absolute coordinates, despite its continually changing position and attitude.
+In order to manage this, SOFIA must relate equatorial and telesco pe coordi-
+nates using a combination of avionics data and star identification, ma nage field
+rotation and track sky images. We describe the algorithms and syst ems required
+to acquire and maintain the equatorial reference frame, relate it t o tracking im-
+agers and the science instrument, set up the oscillating secondary mirror, and
+aggregate pointings into relocatable nods and dithers.
+1. SOFIA Telescope-Pointing Architecture
+MCCSAvionicsScience
+Instrument
+Telescope
+FFI FPIWFI
+Figure 1. The telescope-pointing systems on the SOFIA aircraft.
+The SOFIA pointing architecture is shown in Fig. 1. The Teles cope As-
+sembly (TA) keeps track of its attitude using gyroscopes cor rected by tracking
+imagers, but is unaware of sky coordinates. The Science Inst rument (SI) is
+responsible for science data acquisition. Avionics provid e aircraft location and
+attitude necessary to initialize sky coordinates. The Miss ion Communications &
+Control System (MCCS) correlates gyroscopes with sky coord inates and trans-
+lates pointing building blocks between coordinate systems .
+2. Coordinate Systems
+The TA describes its attitude as a 3 degree-of-freedom (DOF) rotation from the
+telescope reference frame (TARF) to various coordinate sys tems mechanically
+attached to the telescope. The MCCS translates these to exte rnal coordinates,
+and accounts for the optical path. The TA is far from rigid and orthogonal;
+thus many intermediate coordinate systems are established . Directly-observable
+1Universities Space Research Assoc., NASA/Ames Research Ce nter, Moffett Field, CA, USA
+2Critical Realm Corp., San Jose, CA, USA
+12 Gross, Rasmussen, and Moore
+MCCS coordinate systems are Equatorial (ERF) and grid coord inates for the
+three tracking imagers and the Science Instrument (SIRF). K ey internal coor-
+dinate systems are the telescope and inertial references (T ARF and IRF) with
+IRF derived from a set of gyroscopes. The MCCS distributes po sitions in all
+coordinates for real-time monitoring, FITS headers, and ar chiving.
+3. Establishing Equatorial Coordinates
+The MCCS uses a 2-3 stage approach to relate ERF to IRF, to∼<1′′:
+1. Rough-estimate ( ∼1◦) ERF coordinates from avionics-reported aircraft
+position and attitude and the TA’s IRF and airframe attitude s.
+2. Fine-tune using field recognition (PIXY; Yoshida 2000) on the Wide Field
+Imager (WFI; 6◦FOV).
+3. Optionally, refine further using a 2DOF correction, based upon two Focal
+Plane Imager (FPI; 0.57′′pixels) acquisitions several degrees apart.
+TheoperatormonitorsIRF /mapsto→ERFwithanoverlay ofcalculated positionsofstars
+on imagers against sky images (see Fig. 2), and executes the 2 DOF refinement
+when mismatch drifts too large.
+4. Simple Pointing and Tracking
+The TA can be pointed in any directly observable coordinate s ystem. Areas
+of Interest (AOIs) can be defined on imagers; the TA continual ly calculates
+AOI centroids. This provides stability feedback to the TA, n ecessary because
+gyroscopes are not sufficient to maintain pointing over hours . A centroiding
+AOI can be attached to a position of interest; when activated , the TA keeps
+the image fixed at the desired position, by modifying IRF to ca ncel its motions
+inferred from the AOI. The AOI need not coincide with the poin t of interest.
+5. Chopping
+A chop is a high-frequency square-wave oscillation of the se condary mirror
+(SMA),synchronizedelectronically withimageracquisiti on tosubtractskyback-
+ground. Only the FPI and SI observe chopped images; other ima gers do not
+share the optical path with the telescope. A given FPI or SI pi xel has two ERF
+coordinates since any images taken are a superpositionof tw o images with differ-
+ent centers; theMCCSpublishesbothforkeypositions. For p ointingcommands,
+positions can be modified by chop beam designation. The SMA is commanded
+by a 2D center offset, amplitude, and position angle, and a coor dinate system.
+SMA state is propagated through telescope motions by keepin g its specification
+constant in the coordinate system specified. A “3-point” cho p is also supported;
+this includes a pause at an intermediate point (labelled “0” ).
+6. Nodding and Dithering
+A nod is a low-frequency motion of the whole telescope, usual ly relating to an
+accompanying chop, to remove instrument backgrounds. Nods are defined inPointing the SOFIA Telescope 3
+Figure 2. Typical telescope setup for a single data acquisition. Two A OIs
+are defined on the WFI (left) to monitor rotation. A third is defined o n the
+FPI (right) to monitor transverse drift. Computed positions are s uperposed
+(for both chopped beams on the FPI), along with their names. Misma tch
+represents several arcseconds of drift in absolute pointing; dep ending on the
+precision required, a realignment may be necessary. Data shown is s ky simu-
+lation (Br¨ uggenwirth, Gross, Nelbach, & Shuping 2009) with a ther mal dark
+current model. The FPI is at 20◦C and the WFI is at −40◦C.
+φ
+XY
+Nod A
+Nod B(Nod X)b1
+b2A(ref)
+E
+DCB
+Figure 3. Graphical description of nodding (left) and dithering (righ t).
+Nods are specified in the same manner as chops, so they can be matc hed
+unconditionally, without direct connections. The center position (“ Nod X”)
+is optional, and can be used to match a 3-point chop, which includes th e
+center position as well. A formulation based upon circular lists of relat ive
+pointings is used for dithers – all positions are understood to be rela tive to
+the first.
+the same manner as chops, without direct coupling, and are pr opagated through
+telescope motions identically. Dithers are a generalizati on of nods to arbitrary-
+length sequences, intended to image the same star on several different imager
+pixels, in order to remove outliers such as cosmic ray hits or hot pixels. The first
+position in a dither is considered to be a “reference,” with a ll other positions
+definedrelative to it, for relocatability on thesky. Chops, nods, and dithers have
+no direct coupling and can therefore be mixed and matched in a ny combination.4 Gross, Rasmussen, and Moore
+7. Managing Field Rotation
+TA motion is significantly constrained by clearance to the ai rframe. Thus, az-
+imuth is mostly controlled by turning the aircraft (Gross & S huping 2009).
+Rotation against the sky can be held constant as the aircraft rotates up to 5 .6◦.
+As this rotation nears its limit, the telescope is periodica lly “rewound” manually
+to a new rotation angle, between data acquisitions. During r ewind, all building
+blocks are held constant in the coordinate systems in which t hey were defined.
+8. Key Design Points and Tests
+In order to minimize development risk for the MCCS, several c ritical design
+features have been included. Pointing building blocks do no t depend directly
+on each other and state derives strictly from the TA, allowin g for independent
+development, testing, and operational use. Much attention has been given to
+abstraction andsimplification, toaidindevelopment of tes tingprocedures. Each
+building block has a realistic operational “scenario,” defi ning detailed inputs,
+operator commands, user-visible outputs, and intermediat e calculations.
+Coordinate systems present special problems, and are notor iously difficult
+to orient correctly and keep that way; to that end, regressio n tests include:
+•Diurnal rotation rate measurements with zero aircraft spee d.
+•A “home position” at the equator on the vernal equinox, with n orthbound
+heading and zero speed (enables manual calculations).
+•Modifying each coordinate parameter individually.
+•Composite tests (verifies calculations are in the correct or der).
+•Corner cases (identifies quadrant problems).
+•Complete-system tests pointing to specific stars, using syn thetic images.
+Some tests violate TA physical limits; these are evaluated a gainst a simulator
+(Br¨ uggenwirth, Gross, Nelbach, & Shuping 2009)
+9. Conclusion
+Placing a telescope on a moving platform, whilestill desiri ngto point in absolute
+coordinates, presents a difficult challenge for software dev elopment. Eventual
+success of the SOFIA platform in this regard depends upon car eful abstraction
+and testing strategies for pointing the telescope.
+References
+Br¨ uggenwirth, S., Gross, M. A. K., Nelbach, F. J., & Shuping, R. Y. 2 009, in ASP
+Conf. Ser. XXX, ADASS XVIII, ed. D.A. Bohlender, D. Durand & P. D owler
+(San Francisco: ASP), 485
+Gross, M. A. K. & Shuping, R. Y. 2009, in ASP Conf. Ser. XXX, ADASS XVIII, ed.
+D.A. Bohlender, D. Durand & P. Dowler (San Francisco: ASP), 314
+Yoshida, S. 2000, PIXY System, http://www.aerith.net/misao/pixy 1/index.html
\ No newline at end of file
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+arXiv:1001.0198v2  [gr-qc]  13 Jan 2010Features of gravitational waves in higher dimensions
+Otakar Sv´ ıtek
+Institute of Theoretical Physics, Charles University in Pr ague, V Holesovickach 2, 180 00
+Praha 8 - Holesovice, Czech Republic
+E-mail:ota@matfyz.cz
+Abstract. There are several fundamental differences between four-dim ensional and higher-
+dimensional gravitational waves, namely in the so called br aneworld set-up. One of them is their
+asymptotic behavior within the Cauchy problem. This study i s connected with the so called
+Hadamard problem, which aims at the question of Huygens prin ciple validity. We investigate
+the effect of braneworld scenarios on the character of propag ation of gravitational waves on
+FRW background.
+1. Introduction
+Gravitational waves in higher dimensions have several feat ures that might distinguish them
+from the waves propagating on the four-dimensional manifol d only. Therefore, many features
+were already studied, including quasi-normal modes, modifi cation of late-time tails, and also the
+higher-dimensional exact radiative spacetimes.
+Aside from these studies in general higher-dimensional spa cetimes, special attention has been
+paid to specific braneworld models describing our universe a s a hypersurface (brane) in a higher-
+dimensional spacetime (bulk). Apart from the quickly disco vered short-distance modification
+of the Newtonian gravitational potential, a nontrivial cor rection to the quadrupole radiation
+formula has been derived [1]. Thereare also the notorious KK modes that are effectively massive
+on the brane.
+2. Huygens principle and tails
+Huygens principle (and related Hadamard problem) deals wit h the dependence of field values
+on the Cauchy data. Namely, whether it suffices to know the data only in an arbitrarily
+small neighborhood of the intersection of the past null cone with the Cauchy surface. In
+numerous mathematical studies of the problem (namely works by Hadamard [2], G¨ unther [3]
+and McLenaghan [4]) it was shown that the result depends on bo th the differential operator
+properties and underlying spacetime. This makes it a possib le candidate for studying deviations
+that might occur when we move to higher-dimensional spaceti mes.
+Related to the above described qualitative description the re is a problem of presence and
+characteristic decay of late-time tails. It quantifies the f all-of of the initially sharp signal from
+distant source as seen by the observer at fixed position. In th e context of physically interesting
+situations in four dimensions the study was started mainly b y Price [5], in higher dimensions it
+was recently studied for example in [6].
+The usual explanation of these late-time tails is the effect of backscaterring on the curvature
+(or the potential), but one should be careful to use this expl anation for any conclusions.3. Brane model
+Our concern here is to analyze how the specific brane model affec ts the validity of Huygens
+principle of gravitational waves on FRW spacetime. In brane world scenarios gravity feels the
+higher dimensional spacetime while other fields are confined only to four-dimensional brane.
+There are several different models and we will assume the Randa ll-Sundrum single brane set-up
+[7].
+For the description of the model we use the Gaussian embeddin g technique [8] with the
+following form of the metric
+ds2=dw2+iµνdxµdxν. (1)
+We assume Z2symmetry with bulk cosmological constant. The five-dimensi onal Einstein
+equations have the form
+GMN=κ2
+5[−Λ5gMN+τµνδµ
+Mδν
+Nδ(w)]. (2)
+The matter content on the brane is given by the Israel junctio n conditions
+K+
+µν=−K−
+µν=κ2
+5/bracketleftbigg
+τµν−1
+3iµν(0)ταα/bracketrightbigg
+, (3)
+where the brane stress energy tensor is decomposed in the fol lowing way
+τµν=−λiµν(0)+Sµν. (4)
+Four-dimensional Einstein equations then reduce to the for m
+4Gµν=−Λ4iµν(0)+κ2
+4Sµν+κ4
+5πµν+¯Eµν(0) (5)
+where Λ 4=κ2
+5
+12(κ2
+5λ2+6Λ5),κ2
+4=κ4
+5λ
+6,π∼S2, and¯Eµν(0) is a continuous part of certain Weyl
+tensor projection at the brane.
+Since we would like to study cosmological model we assume the perfect fluid on the brane
+Sµν= (p+ρ)uµuν+piµν. (6)
+Then even πµνcan be put to perfect fluid form with parameters quadratic in p,ρ.
+4. Modified Friedmann equation
+Restricting iµν(0) to the form of a standard FRW metric on the brane and assumi ng¯E00(0) =
+Λ4= 0 we arrive at the modified Friedmann equation correspondin g to the previously derived
+results [9]
+3(˙a2+k)
+a2=κ2
+4ρ+κ4
+5ρ2
+12. (7)
+For example the behavior of expansion function in the radiat ion era (ρ=ρ0a−4) withk= 0
+is the following
+a(t)∼t1/2/bracketleftbigg
+1+ℓ
+2t/bracketrightbigg1/4
+, (8)
+where the modification due to brane model is clearly visible a s the second term in the bracket.5. Perturbations of the brane metric
+We consider FRW metric in the following form
+ds2=a2(−dη2+dr2+σ2dΩ2), (9)
+with conformal time ηand radial function σ=r,sin(r),sinh(r) (depending on the spatial
+curvature k= 0,1,−1).
+Axial perturbations of gravitational field in the Regge-Whe eler gauge were derived in [10]
+and two polarizations might be parametrized as
+h0φ=h0(η,ρ)sin(θ)∂θYlm
+h1φ=h1(η,ρ)sin(θ)∂θYlm (10)
+If compatibility condition
+(σσ,r),r+2kσ2−1 = 0
+is satisfied, then equations for both modes can be reduced to o ne equivalent scalar equation.
+Defining F=h1
+σathis equation is
+∂2
+ηF−∂2
+rF+l(l+1)
+σ2F−kF−∂2
+ηa
+aF= 0 (11)
+From the modified Friedmann equation (7) with conformal time and assuming radiation era
+we get
+∂2
+ηa=−ka+κ4
+5ρ2a3
+18, (12)
+which after combining with (11) leads to
+∂2
+ηF−∂2
+rF+l(l+1)
+σ2F−κ4
+5ρ2a2
+18F= 0. (13)
+The behavior of the solutions to this equation will now be ana lyzed both in the case of standard
+and modified Friedmann evolution equations.
+6. Behavior analysis
+For standard cosmology the term ∼ρ2is missing in (13) and the solution is of the form
+Fl∼σl∂rσ−1···∂rσ−1
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+l−1∂r/parenleftbiggf+g
+σ/parenrightbigg
+, (14)
+with functions f,gdepending only on combinations r−ηandr+ηand so the gravitational
+radiation is bounded by null cones. Therefore, Huygens prin ciple is valid in this case.
+In our braneworld scenario the analysis is more difficult and t he approach from [10] will
+be used. We start with the decomposition of solution F=F0+f, where F0satisfies the
+reduced equation (without the brane-induced modification) with initial data of compact support
+σ(r)≤σ(r0) andffulfills
+∂2
+ηf−∂2
+rf+l(l+1)
+σ2f=κ4
+5ρ2a2
+18(f+F0) (15)
+with homogeneous initial conditions.Next, we introduce the energy functional
+E(F)≡1
+2/integraldisplay/bracketleftbigg
+(∂ηF)2−(∂rF)2+l(l+1)
+σ2F2/bracketrightbigg
+dr (16)
+which is conserved for F0but not for fbecause we have
+d
+dηE(f) =/integraldisplayκ4
+5ρ2a2
+18(F0+f)∂ηf dr . (17)
+Sinceσ(r0+η)≥σ(r) (assuming σ(r+η)≤π
+2fork= 1) and using the Schwartz inequality
+together with the bounds implied by E(f),E(F0) we arrive at the following inequality
+eαd
+dη(e−α/radicalbig
+E(f))≤dα
+dη/radicalbig
+E(F0), (18)
+whereα=/integraltextη
+η0κ4
+5ρ2a2
+18√
+l(l+1)σ(r0+ ¯η)d¯η.
+We continue by estimation of parameter αwith the help of the behavior of functions σ,H
+and the relationκ4
+5ρ2
+36/lessorsimilarH2(fork= 1 one has to assumek
+a2≪H2).
+Finally, we obtain
+α≤σ(r0+η)a(η)H(η)
+2/radicalbig
+l(l+1)≡β (19)
+After further manipulation we get the following estimate fo r the energy functional of f
+E(f)≤E(F0)(eβ−1)2(20)
+Similar estimates can be obtained for derivatives of f. Using Sobolev estimates the field f
+can be controlled pointwisely provided we have initial data of compact support and β≪1. The
+last condition is satisfied only for high enough mode number l.
+Conclusion
+We have shown that due to the modification of Friedmann equati on in brane model the Huygens
+principle for gravitational waves seems to be satisfied only for high enough multipoles. This is
+in contrast with the standard cosmology. However, one shoul d be aware that the presented
+analysis is something like a zero approximation of the more d ifficult analysis that would involve
+complete spectrum of brane perturbations. Here we have anal yzed just the so called massless
+mode and reduced the brane influence to the modification of Fri edmann equation.
+Acknowledgements
+This work was supported by grants GACR 202/07/P284, GACR 202 /09/0772 and the Czech
+Ministry of Education project Center of Theoretical Astrop hysics LC06014.
+References
+[1] Kinoshita S et al 2005 Class. Quantum Grav. 223911-3922
+[2] Hadamard J 1923 Lectures on Cauchys problem in linear partial differential eq uations(Yale, New Haven:
+Yale University Press)
+[3] G¨ unther F 1988 Huygens Principle and Hyperbolic Equations (Boston, MA: Academic)
+[4] McLenaghan R G 1982 Ann. Inst. H. Poincare 37211
+[5] Price R H 1972 Phys. Rev. D52419, 2439
+[6] Bizo´ n P, Chmaj T and Rostworowski A 2009 Class. Quantum Grav. 26175006
+[7] Randall L and Sundrum R 1999 Phys. Rev. Lett. 834690
+[8] Shiromizu T, Maeda K and Sasaki M 2000 Phys. Rev. D62024012
+[9] Binetruy P, Deffayet C and Langlois D 2000 Nucl. Phys. B565269-287
+[10] Malec E and Wyl¸ e˙ zek G 2005 Class. Quantum Grav. 223549-3553
\ No newline at end of file
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+arXiv:1001.0199v1  [astro-ph.IM]  4 Jan 2010Astronomical Data Analysis Software and Systems XVII C.10
+ASP Conference Series, Vol. XXX, 2008
+J. Lewis, R. Argyle, P. Bunclarck, D. Evans, and E. Gonzales- Solares, eds.
+Planning and Executing Airborne Astronomy Missions for
+SOFIA
+Michael A. K. Gross and Ralph Y. Shuping
+Universities Space Research Association, NASA/Ames Resea rch Center,
+MS 211-3, Moffett Field, CA 94035 USA
+Abstract. SOFIA is a 2.5 meter airborne infrared telescope, mounted in
+a Boeing 747SP aircraft. Due to the large size of the telescope, only a few
+degrees of azimuth are available at the telescope bearing. This mean s the head-
+ing of the aircraft is fundamentally associated with the telescope’s o bservation
+targets, and the ground track necessary to enable a given mission is highly com-
+plex and dependent on the coordinates, duration, and order of ob servations to
+be performed. We have designed and implemented a Flight Managemen t In-
+frastructure (FMI) product in order to plan and execute such mis sions in the
+presence of a large number of external constraints (e.g. restric ted airspace, in-
+ternational boundaries, elevation limits of the telescope, aircraft performance,
+winds at altitude, and ambient temperatures). We present an over view of the
+FMI, including the process, constraints and basic algorithms used t o plan and
+execute SOFIA missions.
+1. Introduction
+The Flight Management Infrastructure (FMI) product is inte nded to keep the
+aircraft from interfering with preplanned observations on the sky. It predicts
+the ground tracks necessary to execute its mission, and corr ects the plan for
+actual conditions while airborne. To support this, it conta ins both a planning
+component that can run on the ground and in the air, and an exec ution com-
+ponent that runs in the air. The planning component manages a set of ordered
+observations and optional aircraft repositioning request s. The execution com-
+ponent compares the plan to actual conditions in flight and re quests headings
+(indirectly) from the autopilot.
+SOFIA mission planning differs from satellite or ground based observatory
+planning in a few key areas. Most importantly, the observato ry position is a
+function of observation target history, which prevents obs ervations from being
+considered as time-slots alone. Assignment of flight dates i s also nontrivial
+— targets cannot be localized on the sky or the observatory wi ll always fly
+in about the same direction (requiring nearly equivalent de ad time to return);
+this suggests entire flight series should be considered at on ce, for greater target
+variety.
+Flight planning and execution differs from conventional as we ll. All con-
+ventional aircraft fly from point to point along specified pat hs on the ground,
+and “drift” the aircraft to compensate for winds. Typical dr ift angles (course
+- heading) exceed 3◦, and the worst possible case approaches 30◦, so SOFIA
+cannot necessarily observe in this manner. Expected winds c an be planned
+12 Gross and Shuping
+for using a weather forecast. Correcting course for unexpected winds can be
+accomplished by adjusting observation durations, by reloc ating the aircraft be-
+tween observations, or by “observation triage” as a last res ort. This requires
+an astronomically-aware airborne monitoring function to c ompare current con-
+ditions to plan. Aircraft capabilities are a strong functio n of fuel weight, which
+argues against simple parametrizing by time, in favor of fue l.
+2. External Constraints
+In addition to the geometrical and practical constraints de scribed above, the
+SOFIA flight planning problem also has a number of external co nstraints, all of
+which prevent a truly automated, or even rigorously sequent ial, flight planning
+process. Forinstance, specialuseairspace(SUA)incursio nsmayrequireexternal
+approval, and it cannot be known ahead of time whether such ap proval will be
+forthcoming in all cases. National airspace boundaries req uire international
+agreements. Over-ocean operations are prohibited for safe ty reasons for the first
+flights; forlaterflights, fairlycomplexfuelreserveconst raintsarerequired. Gross
+takeoff weight has a hard limit of 700,000 pounds, which limit s the duration of
+SOFIA missions.
+Some science-driven constraints require interaction with scientists or de-
+tailed knowledge of the observations; especially, trading off water vapor over-
+burden estimates with altitude and duration, and for tradin g off observations
+against each other.
+3. SOFIA FMI
+Prior to any particular mission, several iterations of fligh t planning are per-
+formed. This is expected to include fully integrated automa ted flight planning
+(Frank, Gross, & K¨ urkl¨ u 2004), routinely. Manual choice o f observation (in-
+cluding order) will occur subsequently. Upon execution, re planning might oc-
+cur if conditions are sufficiently different from assumptions. Figure 1 shows a
+color-processed screenshot of a simulated flight intended f or April, 2008, from
+Palmdale, CA. Actual conditions for the simulation shown di ffer from planned
+only by small timing errors of the order of several seconds be tween segments
+and at takeoff.
+As mentioned earlier, it is advantageous to consider entire flight series at
+once, in order to trade observations between flights. The dat a structure sup-
+porting this is shown in Fig. 2.
+FMI requires substantial input data in order to accurately p redict a flight
+track and its constraints. Weather forecast time-series ar e taken from the Na-
+tional Center for Environmental Prediction Global Forecas ting System, quadri-
+linearly interpolated. An alternative is required for date s more than several
+days in the future, since accurate forecasts are not availab le then; we use a
+set of stacked monthly means for 1997–2001 (the last years av ailable) from the
+European Center for Medium Ranger Weather Forcasting 40 Yea r Reanalysis
+(Uppala et. al. 2005) also quadrilinearly interpolated. Ai rcraft performance is
+interpolated from tables generated by Boeing INFLT runs, fo r cruising, thrust-
+limited climbs, and descents.Planning and Executing SOFIA Missions 3
+Figure 1. A sample flight in southern California airspace, executing in
+a simulated environment. Coastlines and Arizona and Nevada border s are
+shown as coarsely dotted lines. Planned and actual tracks are sho wn as solid
+lines. Since actual conditions never exactly match plans, the remain der of the
+flight is projected from the current actual location, shown as a da shed line.
+Offshore polygons are warning zones; others are restricted airsp ace. The cur-
+rent position and heading is shown as an aircraft glyph, near Santa B arbara,
+CA.
+Flight
+series
+Flight
+plan
+Flight
+leg
+WaypointFlight
+leg
+trackingNot Tracking
+fixed target moving targetTrackingFuel loadFlight Plan:Start date/time
+Start airport/rwyWaypoint:lat/lon/UT
+altitude
+heading
+airspeed
+water vapor estimate
+telescope elevation
+moon angle
+sun elevation
+pitch angle
+telescope azimuth biascourse
+ground speed
+field rotation
+remaining fuelFlight Leg:Obs ID or
+RA/dec/dur or
+hdg/dur or
+lat/lon or
+ending airport/rwy;
+end altitude0..n
+0..n
+3..n
+Figure 2. Structure of a flight series, corresponding to all flights b etween a
+given instrument’s installation and its removal.
+Planned flight track intersections with special use airspac e (SUA) bound-
+aries (including non-US zones, from the US National Imagery and Mapping
+Agency) are evaluated using a quadtree-based search on the 8 000+ SUA bound-
+aries for each flight segment. Observation segments are trea ted as initial value
+problems in Cartesian coordinates, others may be boundary o r initial problems,
+as appropriate. Desired headings are calculated during exe cution from planned
+(not actual) sky coordinates and actual position; direct st eering by the telescope
+cannot be allowed for safety reasons.4 Gross and Shuping
+In order to test FMI components and integrate other systems, as well as for
+training purposes, we use a simulation environment. This in cludes a medium-
+fidelity aircraft simulator, an automated pilot simulator, a method to set the
+time arbitrarily, and a telescope simulator (Br¨ uggenwirt h, Gross, Nelbach, &
+Shuping 2008).
+Leg constructor
+(API)
+Manual
+flight
+plannerSolved
+flight
+series(API)flight plannerAutomated
+Obs
+plans
+Flight
+executor
+(backend)Leg target
+order,
+duration
+Leg target
+order,
+duration
+Flight
+seriesWinds, water vapor,
+temperature,
+aircraft performance,
+airspace
+MCCS ProxyMD or
+operatorTweaks
+ResponsesSCL
+SCL
+subscribesMCCS (PIS)HousekeepingA/C Position, attitude, etc.
+Figure3. Dataflowwithin the airborneconfigurationforearlyflights . Later
+flights will have direct connections to the data acquisition system (M CCS),
+rather than through a proxy.
+While airborne, the FMI software components interact with t he airborne
+data acquisition software to acquire the aircraft’s positi on and attitude. While
+on the ground, the planner portion of FMI interacts with obse rvers’ planning
+software and must support multiple simultaneous planning s essions. The air-
+borneconfigurationisshowninFig. 3; thegroundconfigurati onissimilar, except
+thereis noflight executor norMCCS data(except in testing co nfigurations), and
+there is a connection to the Observation Planning Database.
+4. Conclusion
+SOFIA presents a unique flight planning problem due to the nat ure of astro-
+nomical observation. FMI provides a connection between sci entific needs of an
+observatory with the practical constraints of operating an aircraft, without in-
+troducing excessive safety considerations or pilot worklo ad, or planner effort.
+Acknowledgments. The authors wish to thank the European Center for
+Medium Range Weather Forecasting for access to their reanal ysis data.
+References
+Br¨ uggenwirth, S., Gross, M. A. K., Nelbach, F. J., & Shuping, R. Y. 2 009, in ASP
+Conf. Ser. XXX, ADASS XVII, ed. J. Lewis, R. Argyle, P. Bunclark, D. Evans,
+& E. Gonzalez-Solares (San Francisco: ASP), 485
+Frank, J., Gross, M. A. K., & K¨ urkl¨ u, E. 2004, in Proc. 16th Conf . on Innovative Appli-
+cations of Artificial Intelligence, ed. D. L. McGuinness & G. Ferguso n (Boston:
+MIT Press), 828
+Uppala, S. M. et al. 2005, Quart. J. R. Meterol. Soc.,131, 2961
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+arXiv:1001.0200v1  [math.ST]  1 Jan 2010How many Laplace transforms of probability
+measures are there?
+Fuchang Gao∗Wenbo V. Li†Jon A. Wellner‡
+May 29, 2018
+Abstract
+A bracketing metric entropy bound for the class of Laplace tr ansforms of prob-
+ability measures on [0 ,∞) is obtained through its connection with the small de-
+viation probability of a smooth Gaussian process. Our resul ts for the particular
+smooth Gaussian process seem to be of independent interest.
+Keywords : Laplace Transform; bracketing metric entropy; completely mono tone func-
+tions; smooth Gaussian process; small deviation probability
+∗Department of mathematics, University of Idaho, fuchang@uidaho.edu .
+†Department of Mathematical Sciences, University of Delaware, wli@math.udel.edu . Supported in
+part by NSF grant DMS-0805929.
+‡DepartmentofStatistics, UniversityofWashington, jaw@stat.washington.edu . Supportedin part
+by NSF Grant DMS-0804587
+11 Introduction
+Letµbe a finite measure on [0 ,∞). The Laplace transform of νis a function on (0 ,∞)
+defined by
+f(t) =/integraldisplay∞
+0e−tyµ(dy). (1)
+It is easy to check that such a function has the property that ( −1)nf(n)(t)≥0 for all
+non-negative integer nand allt >0. A function on (0 ,∞) with this property is called
+a completely monotone function on (0 ,∞). A characterization due to Bernstein (c.f.
+Williamson (1956)) says that fis completely monotone on (0 ,∞) if and only if there is
+a non-negative measure µ(not necessary finite) on [0 ,∞) such that (1) holds. Therefore,
+due to monotonicity, the class of Laplace transforms of finite meas ures on [0 ,∞) is the
+same as the class of bounded completely monotone functions on (0 ,∞). These functions
+can be extended to continuous functions on [0 ,∞), and we will call them completely
+monotone on [0 ,∞).
+Completely monotonic functions have remarkable applications in vario us fields, such
+as probability and statistics, physics and potential theory. The ma in properties of these
+functions are given in Widder (1941), Chapter IV. For example, the class of completely
+monotonic functions is closed under sums, products and pointwise c onvergence. We
+refer to Alzer and Berg (2002) for a detailed list of references on c ompletely monotonic
+functions. Closely related to the class of completely monotonic func tions are the so-
+calledk-monotone functions, where the non-negativity of ( −1)nf(n)is required for all
+integersn≤k. In fact, completely monotonic functions can be viewed as the limiting
+case of the k-monotone functions as k→ ∞. In this sense, the present work is a partial
+extension of Gao (2008) and Gao and Wellner (2009).
+LetM∞be the class of completely monotone functions on [0 ,∞) that are bounded
+by 1. Then
+M∞=/braceleftbigg
+f: [0,∞)→[0,∞)/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(t) =/integraldisplay∞
+0e−txµ(dx),/⌊ar⌈⌊lµ/⌊ar⌈⌊l ≤1/bracerightbigg
+.
+It is well known (see e.g. Feller (1971), Theorem 1, page 439) that t he sub-class of M∞
+withf(0) = 1 corresponds exactly to the Laplace transforms of the clas s of probability
+measures µon [0,∞). For a random variable with distribution function F(t) =P(X≤
+t), thesurvival function S(t) = 1−F(t) =P(X > t). Thus the class
+S∞=/braceleftbigg
+S: [0,∞)→[0,∞)/vextendsingle/vextendsingle/vextendsingle/vextendsingleS(t) =/integraldisplay∞
+0e−txµ(dx),/⌊ar⌈⌊lµ/⌊ar⌈⌊l= 1/bracerightbigg
+is exactly the class of survival functions of all scale mixtures of the standard exponential
+distribution (with survival function e−t), with corresponding densities
+p(t) =−S′(t) =/integraldisplay∞
+0xe−xtµ(dx), t≥0.
+2It is easily seen that the class P∞of such densities with p(0)<∞is also a class
+of completely monotone functions corresponding to probability mea suresµon [0,∞)
+with finite first moment. These classes have many applications in stat istics; see e.g.
+Jewell (1982) for a brief survey. Jewell (1982) considered nonpa rametric estimation of a
+completely monotone density and showed that the nonparametric m aximum likelihood
+estimator (or MLE) for this class is almost surely consistent. The br acketing entropy
+bounds derived below canbe considered asa first step toward globa l rates ofconvergence
+of the MLE.
+In probability and statistical applications, one way to understand t he complexity
+of a function class is by way of the metric entropy for the class unde r certain common
+distances. Recall thatthemetricentropyofafunctionclass Funder distance ρisdefined
+to be log N(ε,F,ρ) whereN(ε,F,ρ) is the minimum number of open balls of radius ε
+needed to cover F. In statistical applications, sometimes we also need bracketing met ric
+entropy which is defined as log N[](ε,F,ρ) where
+N[](ε,F,ρ) := min/braceleftBigg
+n:∃f1,f1,...,fn,fns.t.ρ(fk,fk)≤ε,F ⊂n/uniondisplay
+k=1[fk,fk]/bracerightBigg
+,
+and
+[fk,fk] =/braceleftBig
+g∈ F:fk≤g≤fk/bracerightBig
+.
+ClearlyN(ε,F,ρ)≤N[](ε,F,ρ) and they are close related in our setting below.
+In this paper, we study the metric entropy of M∞under the Lp(ν) norm given by
+/⌊ar⌈⌊lf/⌊ar⌈⌊lp
+Lp(ν)=/integraldisplay∞
+0|f(x)|pν(dx),1≤p≤ ∞,
+whereνis a probability measure on [0 ,∞). Our main result is the following
+Theorem 1.1. (i) Letνbe a probability measure on [0 ,∞). There exists a constant C
+depending only on p≥1 such that for any 0 < ε <1/4,
+logN[](ε,M∞,/⌊ar⌈⌊l·/⌊ar⌈⌊lLp(ν))≤Clog(Γ/γ)·|logε|2,
+for any 0 < γ <Γ<∞such that ν([γ,Γ])≥1−4−pεp. In particular, if there exists a
+constant K >1, such that ν([εK,ε−K])≥1−4−pεp, then
+logN[](ε,M∞,/⌊ar⌈⌊l·/⌊ar⌈⌊lLp(ν))≤CK|logε|3.
+(ii) Ifνis Lebesgue measure on [0 ,1], then
+logN[](ε,M∞,/⌊ar⌈⌊l·/⌊ar⌈⌊lL2(ν))≍logN(ε,M∞,/⌊ar⌈⌊l·/⌊ar⌈⌊lL2(ν))≍ |logε|3,
+whereA≍Bmeans there exist universal constants C1,C2>0 such that C1A≤B≤
+C2B.
+As an equivalent result for part (ii) of the above theorem, we have t he following
+important small deviation probability estimates for an associated sm ooth Gaussian pro-
+cess. In particular, it may be of interest to find a probabilistic proof for the lower bound
+directly.
+3Theorem 1.2. LetY(t),t >0, be a Gaussian process with covariance EY(t)Y(s) =
+(1−e−t−s)/(t+s), then for 0 < ε <1
+logP/parenleftbigg
+sup
+t>0|Y(t)|< ε/parenrightbigg
+≍ −|logε|3.
+The rest of the paper is organized as follows. In Section 2, we provid e the upper
+bound estimate in the main result by explicit construction. In Section 3, we summarize
+various connections between entropy numbers of a set (and its co nvex hull) and small
+ball probability for the associated Gaussian process. Some of our o bservations in a
+general setting are stated explicitly for the first time. Finally we iden tify the particular
+Gaussian process suitable for our entropy estimates. Then in Sect ion 4, we obtain the
+required upper bound small ball probability estimate (which implies the lower bound
+entropy estimates as discussed in section 3) by a simple determinant estimates. This
+method of small ball estimates is made explicit here for the first time a nd can be used
+in many more problems. The technical determinant estimates are als o of independent
+interests.
+2 Upper Bound Estimate
+In this section, we provide an upper bound for N[](ε,M∞,/⌊ar⌈⌊l · /⌊ar⌈⌊lLp(ν)), where νis a
+probability measure on [0 ,∞) and 1≤p≤ ∞. This is accomplished by an explicit
+construction of ε-brackets under Lp(ν) distance.
+For each 0 < ε <1/4, we choose γ >0 and Γ = 2mγwheremis a positive integer
+such that ν([γ,Γ])≥1−4−pεp. We use the notion I(a≤t < b) to denote the indicator
+function of the interval [ a,b). Now for each f∈ M∞, we first write in block form
+f(t) =I(0≤t < γ)f(t)+I(t≥Γ)f(t)+m/summationdisplay
+i=1I(2i−1γ≤t <2iγ)f(t).
+Then for each block 2i−1γ≤t <2iγ, we separate the integration limits at the level
+22−i|logε|/γand use the first Nterms of Taylor’s series expansion of e−uwith error
+terms associated with ξ=ξu,N, 0≤ξ≤1, to rewrite
+f(t) =I(0≤t < γ)f(t)+I(t≥Γ)f(t)+m/summationdisplay
+i=1(pi(t)+qi(t)+ri(t))
+where
+pi(t) =:I(2i−1γ≤t <2iγ)N/summationdisplay
+n=0(−1)ntn
+n!/integraldisplay22−i|logε|/γ
+0xnµ(dx)
+qi(t) =:I(2i−1γ≤t <2iγ)/integraldisplay22−i|logε|/γ
+0(−ξtx)N+1
+(N+1)!µ(dx)
+ri(t) =:I(2i−1γ≤t <2iγ)/integraldisplay∞
+22−i|logε|/γe−txµ(dx).
+4We choose the integer Nso that
+4e2|logε|−1≤N <4e2|logε|. (2)
+Then, by using the inequality k!≥(k/e)kand the fact that 0 < ξ <1, we have within
+the block 2i−1γ≤t <2iγ,
+|qi(t)| ≤/integraldisplay22−i|logε|/γ
+0(tx)N+1
+(N+1)!µ(dx)
+≤|4logε|N+1
+(N+1)!≤/parenleftbigg4e|logε|
+N+1/parenrightbiggN+1
+≤e−(N+1)≤ε4e2,
+where we used tx≤2iγ·22−i|logε|/γ= 4|logε|in the second inequality above. This
+implies, due to disjoint supports of qi(t),
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglem/summationdisplay
+i=1qi(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ε4e2. (3)
+Next, we notice that for t≥2i−1γandx≥22−iγ−1|logε|,e−tx≤ε2. Thus
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglem/summationdisplay
+i=1ri(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤m/summationdisplay
+i=1I(2i−1γ≤t <2iγ)/integraldisplay∞
+22−iγ−1|logε|ε2µ(dx)≤ε2. (4)
+Finally, because |f| ≤1 andν([0,γ))+ν([Γ,∞))≤4−pεp, we have
+/⌊ar⌈⌊l10≤t<γf(t)+1t≥Γf(t)/⌊ar⌈⌊lLp(ν)≤ε/4.
+Together with (3) and (4), we see that the set
+R=:/braceleftBiggm/summationdisplay
+i=1qi(t)+m/summationdisplay
+i=1ri(t)+I(t < γ)f(t)+I(t≥Γ)f(t) :f∈ M∞/bracerightBigg
+has diameter in Lp(ν)-distance at most ε2+ε4e2+ε/4< ε/2.
+Therefore, if we denote Pi={pi(t) :f∈ M∞}, then the expansion of fabove
+implies that M∞⊂/summationtextm
+i=1Pi+R, and consequently, we have
+N[](ε,M∞,/⌊ar⌈⌊l·/⌊ar⌈⌊lLp(ν))≤N[]/parenleftBigg
+ε/2,m/summationdisplay
+i=1Pi,/⌊ar⌈⌊l·/⌊ar⌈⌊lLp(ν)/parenrightBigg
+.
+For any 1 ≤i≤mand any pi∈ Pi, we can write
+pi(t) =I(2i−1γ≤t <2iγ)Ni/summationdisplay
+n=0(−1)nani(2−iγ−1t)n, (5)
+5where 0≤ani≤ |4logε|n/n!. Now we can construct
+pi=I(2i−1γ≤t <2iγ)N/summationdisplay
+n=0(−1)nbni(2−iγ−1t)n,
+pi=I(2i−1γ≤t <2iγ)N/summationdisplay
+n=0(−1)ncni(2−iγ−1t)n,
+where
+bni=
+
+ε
+2n+2⌈2n+2ani
+ε⌉nis even
+ε
+2n+2⌊2n+2ani
+ε⌋nis odd, cni=
+
+ε
+2n+2⌊2n+2ani
+ε⌋nis even
+ε
+2n+2⌈2n+2ani
+ε⌉nis odd.
+Clearly,pi(t)≤pi(t)≤pi(t), and
+|pi−pi| ≤I(2i−1γ≤t <2iγ)N/summationdisplay
+n=0|cni−bni|(2−iγ−1t)n
+≤I(2i−1γ≤t <2iγ)N/summationdisplay
+n=0ε
+2n+2(2−iγ−1t)n
+≤ε
+2I(2i−1γ≤t <2iγ).
+Hencem/summationdisplay
+i=1pi≤m/summationdisplay
+i=1pi≤m/summationdisplay
+i=1pi≤m/summationdisplay
+i=1pi+ε/2.
+That is, the sets
+P=:/braceleftBiggm/summationdisplay
+i=1pi:pi∈ Pi,1≤i≤m/bracerightBigg
+andP=:/braceleftBiggm/summationdisplay
+i=1pi:pi∈ Pi,1≤i≤m/bracerightBigg
+formε/2 brackets of/summationtextm
+i=1PiinL∞-norm, and thus in Lp(ν)-norm for all 1 ≤p <∞.
+Now we count the number of different realizations of PandP. Note that, due to
+the uniform bound on aniin (5) there are no more than
+2n+1
+ε·|4logε|n
+n!+1
+realizations for bni. So, the number of realizations of piis bounded by
+N/productdisplay
+n=0/parenleftbigg2n+1
+ε·|4logε|n
+n!+1/parenrightbigg
+.
+Becausen!>(n/e)n, for all 1 ≤n≤N, we have
+2n+1
+ε·|4logε|n
+n!+1≤3
+ε/parenleftbigg8e|logε|
+n/parenrightbiggn
+.
+6Thus, the number of realizations of piis bounded by
+/parenleftbigg3
+ε/parenrightbiggN+1
+·exp/parenleftBiggN/summationdisplay
+n=1(nlog|8elogε|−nlogn)/parenrightBigg
+≤/parenleftbigg3
+ε/parenrightbiggN+1
+·exp/parenleftbiggN(N+1)
+2log|8elogε|−/integraldisplayN
+1xlogxdx/parenrightbigg
+≤/parenleftbigg3
+ε/parenrightbiggN+1
+·exp/parenleftbiggN(N+1)
+2log|8elogε|−N2
+2logN+N2
+4/parenrightbigg
+≤exp/parenleftbig
+C|logε|2/parenrightbig
+for some absolute constant C, where in the last inequality we used the bounds on N
+given in (2).
+Hence the total number of realizations of Pis bounded by exp( Cm|logε|2). Similar
+estimate holds for the total number of realizations of P, and we finally obtain
+logN[](ε,M∞,/⌊ar⌈⌊l·/⌊ar⌈⌊lLp(ν))≤C′m|logε|2
+for some different constant C′. This finishes the proof since m= log2(Γ/γ).
+3 Entropy of Convex Hulls
+A lower bound estimate of metric entropy is typically difficult, because it often involves
+a construction of a well-separated set of maximal cardinality. Thus we introduce some
+soft analytic arguments to avoid this difficulty and change the proble m into a familiar
+one in this section. The hard estimates are given in the next section.
+First note that M∞is just the convex hull of the functions ks(·), 0< s <∞, where
+ks(t) =e−ts. We recall a general method about the entropy of convex hulls tha t was
+introduced in Gao (2004). Let Tbe a set in Rnor in a Hilbert space. The convex hull
+ofTcan be expressed as
+conv(T) =/braceleftBigg∞/summationdisplay
+n=1antn:tn∈T,an≥0,n∈N,∞/summationdisplay
+n=1an= 1/bracerightBigg
+;
+while the absolute convex hull of Tis defined by
+abconv(T) =/braceleftBigg∞/summationdisplay
+n=1antn:tn∈T,n∈N,∞/summationdisplay
+n=1|an| ≤1/bracerightBigg
+.
+Clearly, by using probability measures and signed measures, we can e xpress
+conv(T) =/braceleftbigg/integraldisplay
+Ttµ(dt) :µis a probability measure on T/bracerightbigg
+;
+abconv(T) =/braceleftbigg/integraldisplay
+Ttµ(dt) :µis a signed measure on T,/⌊ar⌈⌊lµ/⌊ar⌈⌊lTV≤1/bracerightbigg
+.
+7For any norm /⌊ar⌈⌊l·/⌊ar⌈⌊lthe following is clear:
+conv(T)⊂abconv(T)⊂conv(T)−conv(T).
+Therefore,
+N(ε,conv(T),/⌊ar⌈⌊l·/⌊ar⌈⌊l)≤N(ε,abconv(T),/⌊ar⌈⌊l·/⌊ar⌈⌊l)≤[N(ε/2,conv(T),/⌊ar⌈⌊l·/⌊ar⌈⌊l)]2.
+In particular, at the logarithmic level, the two entropy numbers are comparable, modulo
+constant factors on ε. The benefit of using absolute convex hull is that it is symmetric
+andcanbeviewed astheunit ballofa Banachspace, which allows usto use thefollowing
+duality lemma of metric entropy:
+logN(ε,abconv(T),/⌊ar⌈⌊l·/⌊ar⌈⌊l)≍logN(c2ε,B,/⌊ar⌈⌊l·/⌊ar⌈⌊lT)
+whereBis dual ball of the norm /⌊ar⌈⌊l·/⌊ar⌈⌊l, and/⌊ar⌈⌊l·/⌊ar⌈⌊lTis the norm introduced by T, that is,
+/⌊ar⌈⌊lx/⌊ar⌈⌊lT:= sup
+t∈T|/an}⌊ra⌋k⌉tl⌉{tt,x/an}⌊ra⌋k⌉tri}ht|= sup
+t∈abconv(T)|/an}⌊ra⌋k⌉tl⌉{tt,x/an}⌊ra⌋k⌉tri}ht|.
+Strictly speaking, the duality lemma remains as a conjecture in the ge neral case. How-
+ever, whenthenorm /⌊ar⌈⌊l·/⌊ar⌈⌊listheHilbertspacenorm,thishasbeenproved. SeeTomczak-Jaeg ermann
+(1987), Bourgain et al. (1989), and Artstein et al. (2004).
+A striking relation discovered by Kuelbs and Li (1993) says that the entropy number
+logN(ε,B,/⌊ar⌈⌊l·/⌊ar⌈⌊lT) is determined by the Gaussian measure of the set
+Dε=:{x∈H:/⌊ar⌈⌊lx/⌊ar⌈⌊lT≤ε}
+under some very weak regularity assumptions. For details, see Kue lbs and Li (1993),
+Li and Linde (1999), and also Corollary 2.2 of Aurzada et al. (2008). Using this rela-
+tion, we can now summarize the connection between metric entropy of convex hulls and
+Gaussian measure of Dεinto the following
+Proposition 3.1. LetTbe a precompact set in a Hilbert space. For α >0 andβ∈R,
+logP(Dε)≤ −C1ε−α|logε|β
+if and only if
+logN(ε,conv(T),/⌊ar⌈⌊l·/⌊ar⌈⌊l)≥C2ε−2α
+2+α|logε|2β
+2+α;
+and forβ >0 andγ∈R,
+logP(Dε)≤ −C1|logε|β(log|logε|)γ
+if and only if
+logN(ε,conv(T),/⌊ar⌈⌊l·/⌊ar⌈⌊l2)≥C2|logε|β(log|logε|)γ.
+Furthermore, the results also hold if the directions of the inequalitie s are switched.
+8The result of this proposition can be implicitly seen in Gao (2004), wher e an expla-
+nation of the relation between N(ε,B,/⌊ar⌈⌊l · /⌊ar⌈⌊lT) and the Gaussian measure of Dεis also
+given.
+Perhaps, the most useful case of Proposition 3.1 is when Tis a set of functions:
+K(t,·),t∈T, where for each fixed t∈T,K(t,·) is a function in L2(Ω), and where Ω is
+a bounded set in Rd,d≥1. For this special case, we have
+Corollary 3.2. LetX(t) =/integraltext
+ΩK(t,x)dB(x),t∈T, whereK(t,·) are square-integrable
+functions on a bounded set Ω in Rd,d≥1, andB(x) is thed-dimensional Brownian
+sheet on Ω. If Fis the convex hull of the functions K(·,ω),ω∈Ω, then
+logP/parenleftbigg
+sup
+t∈T|X(t)|< ε/parenrightbigg
+≍ε−α|logε|β
+forα >0 andβ∈Rif and only if
+logN(ε,F,/⌊ar⌈⌊l·/⌊ar⌈⌊l)≍ε−2α
+2+α|logε|2β
+2+α;
+and forβ >0 andγ∈R,
+logP/parenleftbigg
+sup
+t∈T|X(t)|< ε/parenrightbigg
+≍ −|logε|β(log|logε|)γ
+if and only if
+logN(ε,F,/⌊ar⌈⌊l·/⌊ar⌈⌊l2)≍ |logε|β(log|logε|)γ.
+The authors found this corollary especially useful. For example, it wa s used in
+Blei et al. (2007) and Gao (2008) to change a problem of metric entr opy to a problem
+of small deviation probability of a problem about a Gaussian process w hich is relatively
+easier. The proof is given in Gao (2008) for the case Ω = [0 ,1], and in Blei et al. (2007)
+for the case [0 ,1]d. For the general case, it can be proved as easily. Indeed, the only
+thing we need to prove is that P(Dε) can be expressed as the probability of the set
+supt∈T|X(t)|< ε. We outline a proof below. Let φnbe an orthonormal basis of L2(Ω),
+then
+X(t) =/integraldisplay
+ΩK(t,s)dB(s) =∞/summationdisplay
+n=1ξn/integraldisplay
+ΩK(t,s)φn(s)ds
+whereξnare i.i.d standard normal random variables. Thus,
+P(Dε) =P/braceleftbigg
+g∈L2(Ω) :/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Ωf(s)g(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle< ε,f∈ F/bracerightbigg
+=P/braceleftbigg
+g∈L2(Ω) :/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+T/integraldisplay
+ΩK(t,s)g(s)dsµ(dt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle< ε,/⌊ar⌈⌊lµ/⌊ar⌈⌊lTV≤1/bracerightbigg
+=P/braceleftBigg∞/summationdisplay
+n=1anφn(s) :∞/summationdisplay
+n=1a2
+n<∞,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+n=1an/integraldisplay
+T/integraldisplay
+ΩK(t,s)φn(s)dsµ(dt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle< ε,/⌊ar⌈⌊lµ/⌊ar⌈⌊lTV≤1/bracerightBigg
+=P/braceleftBigg∞/summationdisplay
+n=1anφn(s) :∞/summationdisplay
+n=1a2
+n<∞,sup
+t∈T/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+n=1an/integraldisplay
+ΩK(t,s)φn(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle< ε/bracerightBigg
+=P/parenleftbigg
+sup
+t∈T|X(t)|< ε/parenrightbigg
+.
+9Now back to our problem of estimate log N(ε,M∞,/⌊ar⌈⌊l · /⌊ar⌈⌊l2) in the statement of (ii)
+of the theorem, where /⌊ar⌈⌊l · /⌊ar⌈⌊l2is theL2norm under the Lebesgue measure on [0 ,1], we
+notice that M∞is the convex hull of the functions C(·,s),s∈[0,∞), on [0,1] with
+C(t,s) =e−ts. However, [0 ,∞) is not bounded. In order to use Corollary 3.2, we
+need to make a change of variables. Notice that by letting y=e−s, we can view M∞
+as convex hull of K(·,y),y∈(0,1], where K(t,y) =yt. Clearly, K(t,·) are square-
+integrable functions on the bounded set (0 ,1]. Now, for this K, the corresponding X(t)
+is a Gaussian process on [0 ,1] with covariance
+EX(t)X(s) =ts−1
+log(ts), s,t∈(0,1],(s,t)/n⌉}ationslash= (1,1),
+andEX(1)2= 1. Thus, the problem becomes how to estimate
+P/parenleftBigg
+sup
+t∈(0,1]|X(t)|< ε/parenrightBigg
+,or equivalently P/parenleftbigg
+sup
+t≥0|Y(t)|< ε/parenrightbigg
+,
+whereY(t) =X(e−t), which has covariance structure
+EY(t)Y(s) =1−e−t−s
+t+s, s,t≥0. (6)
+We now turn to the lower estimates of this probability.
+4 Lower Bound Estimate
+LetY(t),t≥0 be the centered Gaussian process defined in (6). Our goal in this s ection
+is to prove that
+logP(sup
+t≥0|Y(t)|< ε)≤ −C|logε|3,
+for some constant C >0.
+Note that for any sequence of positive numbers {δi}n
+i=1,
+P/parenleftbigg
+sup
+t≥0|Y(t)|< ε/parenrightbigg
+≤P(max
+1≤i≤n|Y(δi)|< ε)
+= (2π)−n/2(detΣ)−1/2/integraldisplay
+max1≤i≤n|yi|≤εexp/parenleftbig
+−/an}⌊ra⌋k⌉tl⌉{ty,Σ−1y/an}⌊ra⌋k⌉tri}ht/parenrightbig
+dy1···dyn
+≤(2π)−n/2(detΣ)−1/2(2ε)n
+≤εn(detΣ)−1/2. (7)
+where the covariance matrix
+Σ = (EY(δi)Y(δj))1≤i,j≤n=/parenleftbigg1−e−δi−δj
+δi+δj/parenrightbigg
+1≤i,j≤n.
+To find a lower bound for det(Σ), we need the following lemma:
+10Lemma 4.1. If 0< bij< aijfor all 1≤i,j≤nthen
+det(aij−bij)≥det(aij)−n/summationdisplay
+k=1max
+1≤l≤nbkl
+akl·per(aij).
+where per( aij) is the permanent of the matrix ( aij).
+Proof.For notational simplicity, we denote cij=aij−bij, then
+det(aij−bij)−det(aij)
+=/summationdisplay
+σ(−1)σc1,σ(1)c2,σ(2)cn,σ(n)−/summationdisplay
+σ(−1)σa1,σ(1)a2,σ(2)···an,σ(n)
+=/summationdisplay
+σ(−1)σn/summationdisplay
+k=1[c1,σ(1)···ck−1,σ(k−1)](ck,σ(k)−ak,σ(k))[ak+1,σ(k+1)···an,σ(n)]
+≥ −/summationdisplay
+σn/summationdisplay
+k=1[a1,σ(1)···ak−1,σ(k−1)](bk,σ(k))[ak+1,σ(k+1)···an,σ(n)]
+≥ −n/summationdisplay
+k=1max
+1≤l≤nbkl
+akl/summationdisplay
+σ[a1,σ(1)···ak−1,σ(k−1)](ak,σ(k))[ak+1,σ(k+1)···an,σ(n)]
+=−n/summationdisplay
+k=1max
+1≤l≤nbkl
+akl·per(aij).
+In order to use Lemma 4.1 to estimate det(Σ), we set
+aij=1
+δi+δj,andbij=e−δi−δjaij
+for a specific sequence {δi}n
+i=1defined by
+δmp+q= 4p+m(m+q),0≤p < m,1≤q≤m
+forn=m2.
+Clearly, we have
+0< bkl/akl≤e−2m4m,1≤k,l≤n=m2. (8)
+It remains to estimate det( aij) and per( aij), which is given in the following lemma.
+Lemma 4.2. For the matrix ( aij) defined above, we have per( aij)≤1, and det( aij)≥
+(240e)−2m3.
+Proof.It is easy to see
+per(aij)≤n!(max
+i,jaij)n≤(m2)!
+(2m4m)m2≤1
+11sinceaij≤(2m4m)−1for 1≤i,j≤n=m2.
+To estimate det( aij), we use the Cauchy’s determinant identity, see Krattenthaler
+(1999),
+det(aij) = det/parenleftbigg1
+δi+δj/parenrightbigg
+=/producttext
+1≤i<j≤n(δj−δi)2
+/producttext
+1≤i,j≤n(δj+δi)=1
+2n/producttextn
+i=1δi·/productdisplay
+1≤i<j≤n/parenleftbiggδj−δi
+δj+δi/parenrightbigg2
+.
+For 1≤i < j≤n=m2, writei=mp+qandj=mr+swith 1≤q,s≤m. Denote
+A={(i,j) :i=mp+q,j=mp+s,0≤p≤m−1,1≤q < s≤m},
+B={(i,j) :i=mp+q,j=m(p+1)+s,0≤p≤m−2,1≤q,s≤m},
+C={(i,j) :i=mp+q,j=mr+s,0≤p≤m−3,p+2≤r≤m−1,1≤q,s≤m}.
+ThenA,BandCform a partition of 1 ≤i < j≤n=m2.
+Next we estimate each part separately. First, for p=r,
+δj−δi
+δj+δi=s−q
+2m+s+q>s−q
+4m.
+Thus
+/productdisplay
+(i,j)∈A/parenleftbiggδj−δi
+δj+δi/parenrightbigg2
+≥m−1/productdisplay
+p=0/productdisplay
+1≤q<s≤m/parenleftbiggs−q
+4m/parenrightbigg2
+=m−1/productdisplay
+k=1m−k/productdisplay
+q=1/parenleftbiggk
+4m/parenrightbigg2m
+≥m−1/productdisplay
+k=1/parenleftbiggk
+4m/parenrightbigg2m2
+=/parenleftbigg(m−1)!
+(4m)m−1/parenrightbigg2m2
+≥(8e)−2m3.
+Second, for r−p= 1,
+δj−δi
+δj+δi=(4m+4s)−(m+q)
+(4m+4s)+(m+q)≥1
+5.
+Thus we have
+/productdisplay
+(i,j)∈B/parenleftbiggδj−δi
+δj+δi/parenrightbigg2
+≥m−2/productdisplay
+p=0/productdisplay
+1≤q,s≤m5−2≥5−2m3.
+Third, for r−p≥2,
+/parenleftbiggδj−δi
+δj+δi/parenrightbigg
+=4r(m+s)−4p(m+q)
+4r(m+s)+4p(m+q)= 1−2·4p(m+q)
+4r(m+s)+4p(m+q)>1−1
+4r−p−1.
+12Thus we have
+/productdisplay
+(i,j)∈C/parenleftbiggδj−δi
+δj+δi/parenrightbigg2
+≥m−3/productdisplay
+p=0m−1/productdisplay
+r=p+2/productdisplay
+1≤q,s≤m/parenleftbigg
+1−1
+4r−p−1/parenrightbigg2
+≥m−2/productdisplay
+k=1/parenleftbig
+1−4−k/parenrightbig2m3
+= exp/parenleftBigg
+−2m3m−2/summationdisplay
+k=1∞/summationdisplay
+l=14−kl
+l/parenrightBigg
+≥exp/parenleftBigg
+−2m3∞/summationdisplay
+l=11
+(4l−1)l/parenrightBigg
+≥exp/parenleftBigg
+−2m3∞/summationdisplay
+l=11
+3ll/parenrightBigg
+= (2/3)2m3.
+Therefore, we have
+/productdisplay
+1≤i<j≤n/parenleftbiggδj−δi
+δj+δi/parenrightbigg2
+=/productdisplay
+(i,j)∈A·/productdisplay
+(i,j)∈B·/productdisplay
+(i,j)∈C/parenleftbiggδj−δi
+δj+δi/parenrightbigg2
+≥(60e)−2m3.
+On the other hand, it is not difficult to see that
+2nn/productdisplay
+i=1δi= 2m2m/productdisplay
+q=1m−1/productdisplay
+p=04p+m(m+q)<2m2·4m2(m−1)/2+m3(2m)m2
+= 43m3/2+m2/2+m2log4m<42m3
+form >1. Therefore,
+det(aij) =/parenleftBigg
+2nn/productdisplay
+i=1δi/parenrightBigg−1
+·/productdisplay
+1≤i<j≤n/parenleftbiggδj−δi
+δj+δi/parenrightbigg2
+≥(240e)−2m3.
+Now combining the two lemmas above, and using the estimate in (8), we obtain
+det(Σ)≥(240e)−2m3−m2·e−2m4m≥e−16m3
+provided that mis large enough. Plugging into (7), we have
+P/parenleftbigg
+sup
+t≥0|Y(t)|< ε/parenrightbigg
+≤e8m3εm2.
+Minimizing the right-hand side by choosing m≈ |logε|/12,we obtain
+P/parenleftbigg
+sup
+t≥0|Y(t)|< ε/parenrightbigg
+/lessorsimilarexp/parenleftbig
+−(432)−1|logε|3/parenrightbig
+.
+Statement (ii) of Theorem 1.1 follows by applying Corollary 3.2. At the s ame time, we
+also finished the proof of Theorem 1.2.
+13References
+Alzer, H. andBerg, C. (2002). Some classes of completely monotonic functions.
+Ann. Acad. Sci. Fenn. Math. 27445–460.
+Artstein, S. ,Milman, V. ,Szarek, S. andTomczak-Jaegermann, N. (2004).
+On convexified packing and entropy duality. Geom. Funct. Anal. 141134–1141.
+Aurzada, F. ,Ibragimov, I. ,Lifshits, M. andvan Zanten J.H. (2008). Approx-
+imation, metric entropy and small ball estimates for Gaussian measu res.preprint
+.
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+521525–1538.
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+arXiv:1001.0202v1  [math.DG]  1 Jan 2010CAUSAL GEOMETRIES AND THIRD-ORDER ORDINARY
+DIFFERENTIAL EQUATIONS
+JONATHAN HOLLAND AND GEORGE SPARLING
+Abstract. We discuss contact invariantstructures on the space of solu tions of
+a third order ordinary differential equation. Associated to any third-order dif-
+ferential equation modulo contact transformations, Chern [Che40] introduced
+a degenerate conformal Lorentzian metric on the space J2of 2-jets of functions
+of one variable. When the scalar invariant of W¨ unschmann [W ¨ un05] vanishes,
+the degenerate metric descends to a proper conformal Lorent zian metric on
+the space of solutions. In the general case, when the W¨ unsch mann invariant is
+not zero, we define the notion of a causal geometry, and show th at the space
+of solutions supports one. The W¨ unschmann invariant is the n related to the
+projective curvature of the indicatrix curve cut out by the c ausal geometry
+in the projective tangent space. When the W¨ unschmann vanis hes, the causal
+geometry is then precisely the sheaf of null geodesics of the Chern conformal
+structure. We then introduce a Lagrangian and associated Ha miltonian from
+which the degenerate conformal Lorentzian metric are const ructed. Finally,
+necessary and sufficient conditions are given for a rank three degenerate con-
+formal Lorentzian metric in four dimensions to correspond t o a third-order
+differential equation.
+1.Introduction
+Thepurposeofthepresentpaperistostudythegeometryofthir d-orderordinary
+differential equations: equations of the form
+y′′′=F(x,y,y′,y′′).
+By settingp=y′andq=y′′, the solutions of this equation coincide with curves in
+the second jet space J2with coordinates ( x,y,p,q) that are everywhere tangent to
+the contact distribution annihilated by the one-forms
+θ1=dy−pdx, θ 2=dp−qdx
+and that are also annihilated by the one-form
+ω=dq−F(x,y,p,q)dx.
+A contact transformation Φ : J2→J2is a local diffeomorphism that preserves the
+contact filtration on J2, meaning that
+Φ∗θ1≡0 (modθ1),Φ∗θ2≡0 (modθ1,θ2).
+Contact transformations act on the set of differential equations by composition.
+The paper is concerned with structures that are invariant with res pect to trans-
+formations of this form. An alternative characterization of third- order equations
+and contact transformations relying only on the first jet space J1and the contact
+structure associated to θ1will be given in Section 2.
+12 JONATHAN HOLLAND AND GEORGE SPARLING
+The equivalence problem for third-order ordinary differential equa tions has a
+long history. It was first studied under the class of point transfor mations by ´Elie
+Cartan [Car55]. In 1940, S. S. Chern [Che40] focused on the equiva lence problem
+under contact transformations. A certain scalar invariant of the structure, called
+the W¨ unschmann invariant, divides the family of third-order equat ions into two
+classes. Those with vanishing W¨ unschmann invariant admit a natura l conformal
+Lorentzian structure on the space of solutions. For Chern, this c onformal struc-
+ture presented itself in the form of a normal SO(3,2) Cartan connection on the
+solution space or, what is the same, a conformal Lorentzian metric on that space.
+A geometrical description of this conformal structure was prese nted much later by
+Fritelli, Kozameh, and Newman [FKN01], who showed that the genera l third-order
+equation with vanishing W¨ unschmann invariant could be obtained by c onsidering
+one-parameter families of null hypersurfaces in a 3-dimensional sp ace with a con-
+formal Lorentzian metric. More directly, as discussed in Section 2.1 , this case can
+be understood also in terms of the null geodesic spray on the bundle of null rays:
+null hypersurfaces then being related by means of an envelope con struction.
+Chern also determined all of the contact invariants of the general third-order
+equation in which the W¨ unschmann is nonzero. This was presented in the modern
+languageofbundlesandconnectionsbySatoandYoshikawa[SY98], w hoinaddition
+clarifiedthe geometricalinterpretationofthe structureasa nor malSO(3,2)Cartan
+connection on the space J2. Nurowski and the second author later showed the
+existence of a conformal O(3,3) structure on a certain fiber bundle over J2(see
+[Nur05]). This O(3,3) structureis ofFefferman [Fef76] type, in the sense ofGraham
+[Gra87], if and only if the W¨ unschmann invariant vanishes. Godli´ nsk i [God08] then
+proved that the associated normal SO(4,4) Cartan connection included the Chern–
+Sato–Yoshikawa connection as its o(3,2) part.
+The structure of Chern–Sato–Yoshikawacan doubtless be under stood directly in
+termsofthegeometryofthespaceofsolutionsofthedifferential equation. However,
+whereas when the W¨ unschmann invariant vanishes, there is a stan dard geometry
+underlying the presence of certain connections—namely a conform al Lorentzian
+metric on the space of solutions—when the W¨ unschmann invariant is nonzero,
+such an underlying geometry appears to be missing. The main task of the present
+article is to supply the missing geometry and to examine its precise rela tionship
+with these constructions.
+Invariantlyassociatedtothestructureon J2isadegenerateconformalLorentzian
+metric (see Nurowski [Nur05]) whose degeneracy is in the direction o f the total de-
+rivative vector field coming from the differential equation. The seco nd task of the
+present article is to show that this degenerate conformal struct ure arises naturally
+from elementary geometrical constructions on a certain curve (t hepolar curve ) in
+the projective cotangent bundle of the space of solutions. The W¨ unschmann in-
+variant itself is precisely the projective curvature of this curve. F inally, the paper
+introduces a new conformal invariant of fourth order in the metric that gives a com-
+plete geometrical characterization of degenerate metrics that a rise in this manner
+from third-order differential equations.
+The first innovation of the paper is that of an incidence relation on the space
+of solutions, described in Section 3.1. An infinitesimal or linearized ver sion of this
+idea is implicit in W¨ unschmann’s [W¨ un05] investigations into Monge equa tions of
+the second degree (see also [Lie05] and [Che40]). When the W¨ unsch mann invariantCAUSAL GEOMETRIES AND THIRD-ORDER ORDINARY DIFFERENTIAL E QUATIONS 3
+vanishes, two solutions are incident if and only if they lie on the same nu ll geodesic.
+When the W¨ unschmann is nonzero, the incidence relation still define s a decent
+structure in a sense that is axiomatized in Section 3: roughly, the sh eaf of curves
+defining the incidence relation is envelope-forming. Such a family of en velope-
+formingcurvesisdubbeda causal geometry , theterminologysuggestedbyanaffinity
+with structures that typically arise in the study of hyperbolic partia l differential
+equations. Thespaceofincidencecurvesinthesolutionspacecorr espondsnaturally
+to the points of the 1-jet space J1. When the incidence curves through a point are
+linearized at that point, the resulting cone in the tangent space res embles the null
+cone associated to a Lorentzian structure. The null cone projec ts to a curve in the
+projective tangent space, called the indicatrix curve, borrowing terminology from
+optics [Arn97].
+The indicatrix gives rise to a Lagrangian in a natural manner that can be writ-
+ten down in terms of the general solution of the differential equatio n, as described
+in Section 3.2. The Lagrangian is a function on the tangent bundle whic h is ho-
+mogeneous of degree two with respect to the scalar homothety of the bundle. It
+is not fully contact-invariant, but its locus of zeros in the projectiv e tangent bun-
+dle is invariant, and coincides with the indicatrix. Null geodesics—extr emals of
+the Lagrangian along which the Lagrangian vanishes identically—are p recisely the
+incidence curves. The resulting structure is a Finsler [Fin18] analog of conformal
+Lorentzian geometry.
+The Lagrangianis in addition regular at every point of the indicatrix, a nd there-
+fore gives rise to a Hamiltonian on the cotangent bundle, which is desc ribed in
+Section 3.3. The zero locus of the Hamiltonian inside the projective co tangent
+space is the polar curve of the indicatrix. The total space of the ind icatrix or its
+polar curve defines a 4-dimensional bundle over the space of solutio ns, and the
+projective Hamiltonian spray defines a projective vector field on th is bundle. The
+3-dimensional quotient space under the flow of the vector field inhe rits a natural
+contact form from the cotangent bundle. The resulting space is co ntactomorphic
+toJ1, the space of 1-jets in the plane, and on it the polar curves descen d to a path
+geometry that defines a third-order differential equation in the ma nner described
+in Section 2.
+The entire procedure sketched here is summarized in the theorem.
+Theorem 1. There is a natural local isomorphism between the set of third -order
+equations under contact equivalence and the set of isomorph ism classes of causal
+geometries.
+The rest of the paper is devoted to studying the degenerate rank three conformal
+Lorentzian metrics gon a four-dimensional space N. The fundamental invariant is
+Γ =g∧LVg∧L2
+Vg∧L3
+Vg∧L4
+Vg∈ ∧5S2ker(V/rightanglese)∼=S2(TN/V)
+whereVis the degenerate direction. Here ker( V/rightanglese) is the space of one-forms an-
+nihilated by V, and each of the Lie derivatives Lk
+Vglies in the symmetric square
+S2ker(V/rightanglese). The space S2ker(V/rightanglese) is six-dimensional, and its fifth exterior power
+is naturally isomorphic to the symmetric square S2(TN/V) of the quotient of the
+tangent bundle of Nby the vertical direction V.
+Theorem 2. The degenerate Lorentzian metric gon the four-manifold Narises
+(locally) from a third-order differential equation if and on ly if either4 JONATHAN HOLLAND AND GEORGE SPARLING
+(1)Γis nonzero and the classical adjoint of Γvanishes identically (equivalently,
+Γhas rank 1). In this case, there exists a natural conformal isometry of
+NwithJ2equipped with its invariant degenerate metric coming from a
+third-order differential equation with non-zero W¨ unschma nn invariant.
+(2)LVgis proportional to g. In this case, the W¨ unschmann invariant van-
+ishes and there exists a conformal isometry of NwithJ2equipped with its
+invariant degenerate metric coming from a third-order diffe rential equation
+that is natural up to a gauge transformation of J2.
+2.Third-order differential equations
+Athird-orderdifferentialequationunder contactequivalencecan be conveniently
+regarded as the following data.
+(1) A three-dimensional contact manifold J1.
+(2) A generic three-parameter family of (unparameterized) cont act curves in
+J1.
+The contact structure on J1can represented as a one-form θwhich is only invari-
+antly defined up to scale. By Darboux’ theorem, there exist coord inates (x,y,p) on
+J1such thatθ=dy−pdx. A contact curve is a curve whose tangent annihilates
+θat each point. The family of curves is generic if at each given point of J1and
+tangent direction vatxannihilating θthere exists a unique curve through xwith
+tangent along v. These curves are identified with the (prolongation of) solutions
+of the differential equation. Two such structures are locally equiva lent if there is a
+local diffeomorphism of J1to itself that preserves the contact structure and sends
+one system of curves to the other.
+Given a third-order differential equation, it is clear how to generate such a struc-
+turebyprolongation(see, forinstance, [Olv95]), andtheresulting structuredepends
+only on the contact-equivalence class of the differential equation, by B¨ acklund’s
+theorem. Conversely, suppose we have chosen coordinates ( x,y,p) onJ1such that
+θ=dy−pdx. The distinguished class of curves is of the form
+x=χ(s;a,b,c)
+y=ψ(s;a,b,c)
+p=π(s;a,b,c)
+wherea,b,care the three parameters defining a curve in the class, and sis an
+evolution parameter of the curve. There is a gauge freedom in selec ting the pa-
+rameterization sof the curve, and so this freedom is eliminated this by imposing
+the condition dx/ds= 1 (that is, by effectively taking xitself to be the parame-
+ter). The contact relation takes the form p=dy/dx. Imposing this relation and
+differentiating ψthree times gives the system of equations
+y=ψ(x;a,b,c)
+y′=ψx(x;a,b,c)
+y′′=ψxx(x;a,b,c)
+y′′′=ψxxx(x;a,b,c).
+Solving the first three equations for a,b,cin terms of x,y′,y′′and substituting
+the result into the third equation gives a third-order differential eq uationy′′′=
+F(x,y,y′,y′′).CAUSAL GEOMETRIES AND THIRD-ORDER ORDINARY DIFFERENTIAL E QUATIONS 5
+Associated to the jet space J1there is naturally a fibration π:J2→J1. This is
+the subbundle of the projective tangent bundle PTJ1ofJ1given as the annihilator
+ofθ:J2=θ⊥. Specifically, J2is given fiberwise by
+J2
+x=/braceleftbig
+v∈PTxJ1|v/rightangleseθ= 0/bracerightbig
+.
+It is a four-dimensional space fibered over J1withS1fibers. The space J2defined
+here supports the following contact-invariant structure, indepe ndently of the differ-
+ential equation. This characterization is the four-dimensional ana log of structures
+studied in five dimensions by the authors in [DHS09].
+Lemma 1. (1) There exists a natural filtration
+T1⊂T2⊂T3⊂T4=TJ2
+of the tangent bundle of J2. HereT1is the vertical distribution for the
+fibrationJ2→J1,T2is a tautological bundle of 2-planes, and T3is the
+annihilator of the pullback of θ.
+(2) On sections, [Γ(T1),Γ(T2)] = [Γ(T1),Γ(T3)] = Γ(T3)and[Γ(T3),Γ(T3)] =
+Γ(T4).
+Specifically, T2is the tautological 2-plane bundle whose fiber at a point ( x,u)∈
+J2
+x⊂PTxJ1consists of all vectors vsuch thatπ∗vis in the direction of u. In terms
+of the (x,y,p) coordinates on J1, any vector field of the form ∂/∂x+p∂/∂y+q∂/∂p
+annihilatesthecontactform θ=dy−pdx. Thereforethis qdefinesafibercoordinate
+for the fibration J2→J1that allows the vector fields generating T2J2to be
+expressed as X=∂/∂q, the vertical vector field for the fibration, and ∂/∂x+
+p∂/∂y+q∂/∂p. The lemma follows by taking commutators.
+Lemma 2. Any two four-manifolds equipped with this structure are loc ally iso-
+morphic: a filtration T1⊂T2⊂T3⊂T4on the tangent bundle, such that
+[Γ(T1),Γ(T2)] = [Γ(T1),Γ(T3)] = Γ(T3)and[Γ(T3),Γ(T3)] = Γ(T4).
+Proof.LetMbe a manifold equipped with such a filtration and let θbe a nonva-
+nishing one-form annihilating T3M. LetNbe the 3-manifold obtained by passing
+to the (locally defined) quotient modulo the flow of T1M. The distribution T3M
+is Lie derived along T1M, and so descends to a distribution of 2-planes on N.
+Nowhere is this distribution Frobenius integrable, and so it defines a c ontact struc-
+ture onN. By Darboux’ theorem, Nis locally contactomorphic to J1with its
+standard contact structure and coordinates ( x,y,p). Letting qbe a fiber coordi-
+nate onM→N,X=∂/∂qgeneratesT1Mandθ=dy−pdx. Plane subbundles
+ofT3on which LXmaps surjectively onto T3are all related by a change in the
+fiber coordinate q. Indeed, as Xcommutes with ∂/∂p, the latter vector field does
+not lie inT2. The bundle T2must contain a solution YofLXY=∂/∂p.One
+such solution is Y=∂/∂x+p∂/∂y+q∂/∂p, and the ambiguity, modulo T1, in the
+solution is a transformation of the form Y/mapsto→Y+λ∂/∂pwhereX(λ) = 0. Noting
+thatλis independent of q, this ambiguity in the choice of Ycan be absorbed into
+a change of coordinates q/mapsto→q+µwhereµ(x,y,p) satisfies the differential equation
+µ−µx−µy−µp=λ. So we are free to choose the fiber coordinate qso thatT2is
+generated by XandY=∂/∂x+p∂/∂y+q∂/∂p. The coordinates ( x,y,p,q) now
+defined onMestablish a local diffeomorphism with J2that sendsT1M,T2M, and
+T3MtoT1J2,T2J2, andT3J2, respectively. /square6 JONATHAN HOLLAND AND GEORGE SPARLING
+The differential equation is specified in terms of a splitting of the first level of
+the filtration T1J2⊂T2J2. This is achieved by means of a vector field Vof
+the formV=∂/∂x+p∂/∂y+q∂/∂p+F(x,y,p,q)∂/∂qrepresenting the total
+derivative. While Vitself is not contact-invariant, the splitting direction consisting
+of all multiples of Vis. This splitting can invariantly be described in terms of
+the system of curves on J1that gives the differential equation. Lying over a point
+x∈J1, the point u∈J2
+xof the fiber is by definition a projective tangent vector
+atx. Passing through xin the direction defined by uis a distinguished curve of
+the differential equation. This curve determines a tangent directio n at every point
+which therefore specifies a lift to J2. The vector V(x,u)is the tangent direction to
+the lifted curve at the point ( x,u)∈J2.
+2.1.Conformal structure. It is possible to construct from these data a degener-
+ate conformal Lorentzian metric gonJ2. The degenerate direction for the metric
+isV, and the vector field Xis null with respect to the metric. In terms of the
+coordinates ( x,y,p,q) onJ2, this metric is given by
+(1)g= 2[dy−pdx][dq−1
+3Fqdp+Kdy+(1
+3qFq−F−pK)dx]−[dp−qdx]2
+whereK=1
+6V(Fq)−1
+9F2
+q−1
+2Fp. In Section 5, the present paper constructs this
+metric in a manifestly contact-invariant fashion. The (locally defined ) quotient
+spaceM=J2/Vis a three-manifold, but the metric gmay not be Lie derived up
+to scale along V, and so need not pass down to the quotient. The Lie derivative
+LVgis proportional to gif and only if the W¨ unschmann invariant of the original
+differential equation vanishes. The W¨ unschmann invariant is given b y
+(2) W=Fy+(V−2
+3Fq)K.
+Vanishing of this invariant is a necessary and sufficient condition for Mto possess
+an invariant Lorentzian conformal structure.
+Going the other way, let Mbe a conformal Lorentzian 3-manifold. Define Sto be
+the bundle over Mwith fiberS1that, at each point P, consists of all null directions
+inPT∗
+PM. The pullback metric is degenerate in the vertical direction, and so S
+supports the structure of a degenerate conformal Lorentzian 4-manifold for which
+the degenerate direction is a conformal Killing symmetry. There is a c anonical
+symplectic potential ψdefined on the total space of the cotangent bundle of M.
+The formψis annihilated by the scaling in the fiber, and is Lie derived up to scale,
+and so descends to give a form θup to scale on S. The condition θ∧dθ/\e}atio\slash= 0 follows
+sinceψ∧dψonT∗Mdoes not vanish when pulled back to nonzero sections M→
+T∗M. Indeed, in local coordinates x= (x1,x2,x3) onMwith fiber coordinates
+p= (p1,p2,p3) onT∗M,ψ∧dψ= (p·dx)(dp·dx) =−1
+2(p×dp)·(dx×dx). Since
+dx×dxhas three linearly independent components, and p×dpvanishes only on
+vectors parallel to the generator of scalings in T∗M,ψ∧dψdoes not vanish when
+pulled back along any section of the projective cotangent bundle, a nd soa fortiori
+it does not vanish when pulled back to S.
+The vector field Xis the given by the null geodesic spray in S. To describe this,
+fix a metric gin the conformal class on M. OnT∗Mthe geodesic Hamiltonian
+isH=π∗g−1(ψ,ψ), which gives rise to the Hamiltonian vector field /hatwideH, defined
+by/hatwideH/rightanglesedψ=dH. An integral curve µofXprojects to a geodesic of M, and the
+fiber component of µis the covelocity of the geodesic. The image of Sunder the
+mapS→T∗Mis the null cone at every point of M. This is everywhere tangentCAUSAL GEOMETRIES AND THIRD-ORDER ORDINARY DIFFERENTIAL E QUATIONS 7
+to the spray /hatwideH, because a geodesic being initially null will always remain null.
+Furthermore, /hatwideHscales quadratically in the cotangent bundle, and descends to a
+direction field on the projective cotangent bundle PT∗M. This direction field is
+everywhere tangent to S, and so restricts to a direction field XonS.
+Passing to the (locally defined) quotient by Xgives the space of null geodesics—
+the twistor space T. It follows from the definition of XandθthatX/rightangleseθ= 0 and
+X/rightanglesedθ= 0. The second assertion follows by pulling back X/rightanglesedψ=dHto the null
+cone, and using the fact that Hvanishes identically there. Thus θdescends to
+a contact structure on the twistor space. The fibers of S→Mproject to the
+distinguished curves of the twistor space.
+To prove that these distinguished curves are suitably generic, we v erify that S
+admits a structure satisfying the conditions of Lemma 2. Let T1be the bundle
+spanned by the null geodesic spray XandT3be annihilator of θ. Fromθ∧dθ/\e}atio\slash= 0,
+it follows that T3is not Frobenius integrable, and so [Γ( T3),Γ(T3)] = Γ(TS).
+Moreover, since θis annihilated by Xand Lie derived along it, [Γ( T1),Γ(T3)] =
+Γ(T3).
+It remains to identify T2and to show that [Γ( T1),Γ(T2)] = Γ(T3). LetVbe
+a nonvanishing vertical vector field for S→Mand letT2= span{X,V}. One
+such vector field Vcan be given in terms of the angular momentum operator
+L=p×∂/∂p. Then Lis tangent to the null cone, because it annihilates H.
+On homogeneous functions of degree 0, the angular momentum fac tors through
+a scalar operator L(f) =V(f)p, which defines the vector field V. Because θ
+is horizontal, any such vector field, being vertical, annihilates θ. Furthermore,
+[X,V]/rightangleseθ=LX(V/rightangleseθ) = 0 as well, so [ X,V]∈T3. It remains only to show that
+X,V,[X,V] are linearly independent. Each of these vector fields is at most firs t
+order in the metric, and therefore independence follows by a calcula tion in normal
+coordinates. In these coordinates, X=−p·∂/∂xand so on the one hand
+[X,L] =p×∂
+∂x
+and on the other hand, on the null cone this acts as p[X,V] on functions homoge-
+neous of degree zero. Finally,
+dθ([X,L],L) = (p·p)g−p⊗p
+which vanishes nowhere on the null cone. Thus [Γ( T1),Γ(T2)] = Γ(T3).
+So modulo the explicit construction of the degenerate metric and th e assertion
+that specifically it is the W¨ unschmann invariant that governs wheth er the degen-
+erate metric descends to the 3-manifold, we have proven the follow ing theorem
+(proven in Fritelli, Kozameh, Newman [FKN01] by entirely different me thods):
+Theorem 3. There is a natural local equivalence between third-order di fferential
+equations under contact transformations with vanishing W¨ unschmann invariant and
+conformal Lorentzian 3-manifolds.
+3.Causal geometries on the space of solutions
+This section develops the natural geometric structure associate d to the space of
+solutions Mto a third-order equation. This structure reduces to a conventio nal
+Lorentzian conformal structure if and only if the W¨ unschmann inv ariant vanishes.8 JONATHAN HOLLAND AND GEORGE SPARLING
+To motivate this discussion, the previous section establishes the ba sic properties of
+the (possibly locally defined) double fibration
+S
+/d127/d127/d0/d0/d0/d0/d0/d0/d0
+/d31/d31/d64/d64/d64/d64/d64/d64/d64/d64
+T M
+An individual fiber of the submersion S→Mprojects down to give a trajectory
+solving the differential equation in T: this is a re-expression of the notion that Mis
+aspace of solutions of the differential equation. The fibers for the other submersion
+S→Talsoprojectdowntothespaceofsolutions M, althoughtheirprecisemeaning
+has heretofore not been identified in general.
+WhentheW¨ unschmanninvariantvanishes, McarriesanaturalconformalLorentzian
+metric by Theorem 3, and Sis canonically identified with the null cone bundle asso-
+ciatedto this metric. The space Tis then the twistorspace: the quotient of Sby the
+null geodesic spray. The fibration S→Mcan be understood as the subbundle of
+the projective tangent bundle PTMof null directions. Under this correspondence,
+a null cone with vertex at P∈Mcorresponds to a one-parameter family of null
+geodesics, which in turn is identified with the trajectory defining the solutionP.
+This structure can be axiomatized in a manner that allows construct ion of the
+spacesSandT, along with their natural contact structure. The one parameter
+families of null geodesics starting at each point Pgive rise to a cone in the tangent
+spaceTPMwith vertex at the origin. Equivalently, such a cone is the affine cone
+over some curve in the projective space PTPM. Here locality considerations may
+mean that the cone may fail to close up completely, or the associate d curve may
+have one or more singular points. Henceforth, we shall work only ne ar regular
+points of the curve.1
+3.1.Incidence relation and the indicatrix. Two solutions y1andy2to the
+differential equation y′′′=F(x,y,y′,y′′) are said to be incident if, at some point x,
+one has
+y1(x) =y2(x), y′
+1(x) =y′
+2(x).
+That is to say, two curves are incident if and only if the associated so lution curves
+in theJ1intersect. In the latter interpretation, the incidence relation is ma nifestly
+contact-invariant.
+The set of all solutions incident with a given solution P∈Mcuts out a surface
+NPwith a conical type singularity at the vertex P. The generators of this cone
+can be described as follows. If Qis a solution incident with P, then the curve
+NPQconsisting of solutions Rincident with Pat the same point of J1asQ. (Our
+localization assumption implies that two distinct solutions are incident a t most at
+a single point of J1.) The surface NPis thus ruled by the pencil of curves NPQas
+Qvaries over solutions incident with P.
+The incidence relation described here is called a causal geometry . The following
+axioms are assumed to hold in a sufficiently small neighborhood of each point
+P∈M:
+1We can localize near a point of Tand a point of Msimultaneously. Thus all solutions can
+be assumed to be fully regular, but cannot necessarily be con tinued for all time. The geometrical
+implication is that we may isolate a smooth part of the null co ne at each point of M, but the
+“cone” need not then close up.CAUSAL GEOMETRIES AND THIRD-ORDER ORDINARY DIFFERENTIAL E QUATIONS 9
+Axiom 1. NP\{P}is a smooth embedded hypersurface in M
+Axiom 2. The singularvariety NPis ruled by a pencil of smooth curves NPQfrom
+PtoQ, asQvaries over NP\{P}.
+Axiom 3. For everyQ∈NP\{P}, the curveNPQonNPcoincides with the curve
+NQPonNQ:NPQ=NQP.
+Axiom 4. LetCPbe the set of tangents at Pto the curves NPQasQvaries
+overNP\{P}. ThenCPis a regular (smooth) curve in the projective
+tangent space PTPM, and does not make second order contact with any
+of its tangent lines. (The curve CPis called the indicatrix .)
+Axiom 5. LetQ,R∈NP\{P}be given distinct points which are mutually inci-
+dent. Then the tangent plane to NPatRis the same as the tangent
+plane toNQatR:TRNP=TRNQ. (Envelope condition )
+Theorem 5 establishes that these axioms hold in the case of the caus al geometry on
+the space of solutions of a third-order differential equation. The fi rst four axioms
+are fairly natural assumptions that make precise the notion that t heNPshould
+be a cone based at P. The envelope condition of axiom 5, in terms of the bundle
+J1→M, guarantees that the one-form θ, whose annihilator is the 3-plane bundle
+that lifts the tangent planes through the vertices of the cones NP, is Lie derived
+along the fibers of the fibration J2→J1.
+Causal geometries also appear naturally in connection with the follow ing varia-
+tional problem. Let L:TM\{0} →Rbe a homogeneous function of degree two
+(aLagrangian ), possibly only defined on an open conical subbundle of TM\ {0}.
+Suppose that foreach P∈M, the curveCP={v∈PTPM|L(P,v) = 0}is smooth.
+For a curve γinM, consider the energy functional
+E[γ] =1
+2/integraldisplayb
+aL(γ(t),γ′(t))dt.
+Anullgeodesicisacurve γthat isanextremalsfor Ealongwhich L(γ(t),γ′(t))≡0.
+The family of null geodesics for a Lagrangian defines a causal geome try if and only
+if the Lagrangian is regular at every point of the cone over CPinTPMfor eachP,
+in the sense that its vertical Hessian is nondegenerate in directions tangent toCP.
+In coordinates xiforMand fiber coordinates ˙ xiforTM, this is the condition that
+the 3×3matrix∂2L
+∂˙xi∂˙xjbe nonsingularin directions tangent to the null cone. This is
+alsoa sufficient condition for Lto be conservedalong an extremal, and thus the null
+geodesics are precisely those extremals of the energy for which L(γ(0),γ′(0)) = 0.
+The Euler–Lagrange system is a second order ordinary differential equation for
+the curveγ(t). By smooth dependence on initial conditions, for initial conditions
+lying on the hypersurface L(P,γ′(0)) = 0 sufficiently near the origin of the tan-
+gent space TPM, the null geodesics foliate a smooth hypersurface in M, giving
+axioms 1–2. Axiom 3 follows since the null geodesics are critical points of the en-
+ergy under compactly supported variations, and so in particular ar e characterized
+independently of the direction of their parameterization. Axiom 4 fo llows from the
+regularity of the Lagrangian. Finally, for axiom 5, it is sufficient to sho w that, for
+any pointP, and any variation γsof null geodesics through P,
+(3)∂L
+∂˙xidγi
+ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+s=0= 0.10 JONATHAN HOLLAND AND GEORGE SPARLING
+This then establishes that the null cone at Pis tangent to the indicatrix curve at
+every point, which is equivalent to axiom 5. By the Euler–Lagrange eq uations,
+0 =d
+dsL(γs,˙γs) =∂L
+∂xidγi
+ds+∂L
+∂˙xid˙γi
+ds
+=/parenleftbiggd
+dt∂L
+∂˙xi/parenrightbiggdγi
+ds+∂L
+∂˙xid˙γi
+ds
+=d
+dt/parenleftbigg∂L
+∂˙xidγi
+ds/parenrightbigg
+So the left-hand side of (3) is constant along the curve γ(t), as required.
+Conversely, a Lagrangian can be associated to any causal geomet ry, up to a
+certain ambiguity. Let L:TM→Rbe a function homogeneous of degree two that
+vanishes at every point of the cone over the indicatrix curve CPfor every point
+P∈M, and suppose further that the vertical Hessian of Lis nondegenerate. It
+is always possible to select such a function, at least locally, and the am biguity in
+selecting such a function is of the form
+L(x,˙x)→Ω(x,˙x)L(x,˙x)
+whereΩ is anonvanishingfunction. A causalgeometryis alreadyass ociatedto such
+a Lagrangian, by the preceding argument, and so it is sufficient to sh ow that the
+causal geometry induced by the Lagrangian agrees with the one alr eady given. To
+show that the curves NPcoincide with the null geodesics of L, it is enough to show
+that they are critical points for the energy with respect to variat ionsγstangent to
+the null cone, meaning that (3) holds. Indeed, the Euler–Lagrang e system for the
+constrained problem is
+∂L
+∂xi−d
+dt∂L
+∂˙xi=λ∂L
+∂˙xi
+whereλ=λ(t) is a free function of t. The freedom in λcan be absorbed by a
+change in parameterization of the curve γ(t).
+Now consider the curve γ=NPQ, with some parameterization. Let γsbe a
+variation of γobtained by varying the endpoint QalongNP(that is,γsis a vari-
+ation through null geodesics). In the statement of Axiom 5, taking the pointRto
+approachQ, a necessary condition for γsto be on the null cone is that (3) holds.
+Moreover, also by Axiom 5, γssatisfiesL(γs,˙γs)≡0. Therefore we have on the
+one hand
+0 =d
+dsL(γs,˙γs) =∂L
+∂xidγi
+ds+∂L
+∂˙xi˙γi
+ds.
+And on the other hand also
+0 =d
+dt/parenleftbigg∂L
+∂˙xidγi
+ds/parenrightbigg
+=/parenleftbiggd
+dt∂L
+∂˙xi/parenrightbiggdγi
+ds+∂L
+∂˙xid˙γi
+ds.
+Equating the two gives/parenleftbigg∂L
+∂xi−d
+dt∂L
+∂˙xi/parenrightbiggdγi
+ds= 0
+alongNPQ. ThusNPQis a null geodesic for the Lagrangian L.
+These considerations are summarized in the following theorem:
+Theorem 4. LetL:TM→Rbe a homogeneous function of degree two, such that
+for eachP∈M, the curveCP={L= 0} ⊂PTPMis smooth. Suppose also that the
+vertical Hessian of Lis nondegenerate at each point of CP. Then the null geodesicCAUSAL GEOMETRIES AND THIRD-ORDER ORDINARY DIFFERENTIAL E QUATIONS 11
+curves ofLform a causal geometry. Every causal geometry arises (local ly) in this
+manner.
+3.2.The causal geometry associated to a differential equation. At each
+pointPofM, denote by
+y=f(x;P)
+the solution to the differential equation represented by P. We break contact invari-
+ance in specifying the differential equation, but shall ultimately be co ncerned only
+with the contact-invariant information that can be extracted fro mf. Locally, this
+solution depends smoothly on xandP. The underlying assumption under which
+Mis a differentiable manifold smoothly parameterizing a space of solution s is that
+in local coordinates P= (p1,p2,p3) onM, the Wronskian determinant
+(4)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂f
+∂p1∂f
+∂p2∂f
+∂p3
+∂fx
+∂p1∂fx
+∂p2∂fx
+∂p3
+∂fxx
+∂p1∂fxx
+∂p2∂fxx
+∂p3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/\e}atio\slash= 0.
+To obtain a more concrete description of the cone NPthrough a particular
+solutionP∈Mand the associated curves NPQthat rule the cone defined in the
+previous section, suppose that γ(t) is a parameterization of the curve NPQsuch
+thatγ(0) =P. Since all points along γ(t) are incident with Pat the same point
+(x,y,y′), the following two equations must hold along γ:
+(5)f(x;γ(t)) =f(x;γ(0))
+fx(x;γ(t)) =fx(x;γ(0)).
+Under generic conditions, the second equation can be used to solve forxin terms of
+γ(0)andγ(t), and then substituted into the firstequation whichmaythen be so lved
+for the admissible values γ(t). Although ostensibly this system of equations is not
+invariant under contact transformations, by B¨ acklund’s theore m a contact trans-
+formation will preserve the space of solutions γ(t). This is geometrically evident
+because the causal geometry itself is contact-invariant.
+Solutions of these equations arise as first integrals of the different iated forms
+(6)d
+dtf(x;γ(t)) = 0
+d
+dtfx(x;γ(t)) = 0
+with initial conditions γ(0),γ′(0). The initial conditions are not completely arbi-
+trary. Rather if γ(0) is fixed, then γ′(0) is constrained by the requirement that
+(7)d
+dtf(x;γ(t))/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+t=0= 0
+d
+dtfx(x;γ(t))/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+t=0= 0
+The second equation of (7) can be solved to obtain the parameter xin terms of
+the initial conditions γ(0),γ′(0), by (4). The following Lagrangian is homogeneous12 JONATHAN HOLLAND AND GEORGE SPARLING
+of degree two in γ′(0):
+(8)L(γ(0),γ′(0)) =d
+dtf(x(γ(0),γ′(0));γ(t))d
+dtfxx(x(γ(0),γ′(0));γ(t))/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+t=0
+The second factor ensures that the Lagrangian is regular (Lemma 3). For fixed
+γ(0), the values of the tangent γ′(0) satisfying equation (7) cut out an affine cone
+over a curve in the projective tangent space at γ(0). This cone is the “null cone”
+for the Lagrangian L, and it coincides with the tangent cone to Nγ(0)at the vertex
+γ(0). As in Axiom 4, for P∈M, denote by CP⊂PTPMthe curve cut out in the
+projective tangent space by the equation L(P,v) = 0.
+Although the Lagrangian Lis not itself contact-invariant, by the argument al-
+ready given its locus of zeros is contact-invariant. Under contact transformations,
+the Lagrangian is determined up to rescaling by a nonvanishing funct ion ofγ(0)
+andγ′(0),
+L(γ(0),γ′(0))→Ω(γ(0),γ′(0))L(γ(0),γ′(0)).
+The curves NPQthat generate the causal cone NPare extremals for the energy
+functional
+E[γ] =1
+2/integraldisplayb
+aL(γ(t),γ′(t))dt.
+Theorem 5. A third-order differential equation determines a causal geo metry sat-
+isfying Axioms 1–5.
+Proof.Axiom 1 reflects our running localization assumption that the third-o rder
+differential equation under consideration is regular. In local coord inates on M, the
+system of equations
+(9)f(x;q1,q2,q3) =f(x;p1,p2,p3), fx(x;q1,q2,q3) =fx(x;p1,p2,p3)
+for unknowns x,Q= (q1,q2,q3) and fixed P= (p1,p2,p3) has Jacobian matrix
+
+fx(x;q1,q2,q3)−fx(x;p1,p2,p3)∂f
+∂q1∂f
+∂q2∂f
+∂q3
+fxx(x;q1,q2,q3)−fxx(x;p1,p2,p3)∂fx
+∂q1∂fx
+∂q2∂fx
+∂q3
+.
+The matrix always has rank two by (4). Moreover, at a solution Q/\e}atio\slash=Pof the orig-
+inal system, the lower left-hand corner cannot be zero, by unique ness of solutions.
+Thus the first column, along with one of the remaining three columns m ust yield
+an invertible 2 ×2 submatrix. The implicit function theorem then implies that the
+solution is a smooth submanifold away from the vertex Q=P.
+For axiom 2, NPRis the set of solutions Q= (q1,q2,q3) of (9) for a given value
+ofx,P= (p1,p2,p3). Because (9) has rank two in the qivariables, the space of
+solutions is a smoothly embedded curve. These curves clearly cover NP. Further-
+more, two distinct curves meet only at the vertex P, by uniqueness of solutions of
+differential equations.
+Axiom 3 is obvious. Axiom 4 is equivalent to the assertion that Lis a regular
+Lagrangian, which is proven in Lemma 3 of the next section.
+Finally, suppose without loss of generality that in the statement of a xiom 5,Ris
+a point between PandQonNPQ. Along the curve NPQthe value of the parameterCAUSAL GEOMETRIES AND THIRD-ORDER ORDINARY DIFFERENTIAL E QUATIONS 13
+xis fixed, and NPQitself consists of all points Rsuch that
+f(x;R) =f(x;P)
+fx(x;R) =fx(x;P).
+The tangent plane to NPat the point Ris the annihilator of dRf(x,R).2This is
+the same tangent plane as that obtained by interchanging the roles ofPandQ./square
+3.3.Associated Hamiltonian. As in the previous section, for P∈M, lety=
+f(x;P) denote the solution of the differential equation corresponding to P. Once
+again, contact-invariance is broken, but ultimately we will only be con cerned with
+contact-invariantinformationcontainedinthesolution. Let dPfdenotetheexterior
+derivative of fregardingxas constant. In local coordinates ( p1,p2,p3) onM,
+dPf(x;p1,p2,p3) =∂f
+∂p1dp1+∂f
+∂p2dp2+∂f
+∂p3dp3.
+Thenx/mapsto→dPf(x;P) defines a curve in the cotangent space T∗
+PM. Denote the
+associated projective curve by ˜CP⊂PT∗M.
+This curve is linked to the curve CPcut out by the Lagrangian via the following
+construction. Let Vbe a three-dimensional vector space and Ca smooth curve
+in the projective plane PV. The dual projective plane PV∗is naturally identified
+with the space of lines in PV. The dual curve C∗is the curve in PV∗defined by
+locus of lines tangent to C. Suppose that Cis a nondegenerate curve in PV, with
+parameterization t/mapsto→γ(t). At a point γ(t) ofC, the corresponding point of the
+dual curve is obtained by solving for γ∗(t)∈PV∗the equations
+/a\}bracketle{tγ∗(t),γ(t)/a\}bracketri}ht= 0
+/a\}bracketle{tγ∗(t),γ′(t)/a\}bracketri}ht= 0.
+Properly speaking, to make sense of the second equation, it is nece ssary to choose
+a lift ofγto a curve in V. Modulo the first equation, the second equation does
+not depend on the choice of lift, and so there is no ambiguity in speakin g of the
+solution of the system of equations.
+Proposition 1. ˜CP⊂PT∗
+PMandCP⊂PTPMare mutually dual.
+Proof.The curve ˜CPis characterized as the image of the map
+x/mapsto→dPf(x;P).
+The dual curve to ˜CPis defined by the equations
+(10)/a\}bracketle{tγ∗(x),dPf(x;P)/a\}bracketri}ht= 0
+/a\}bracketle{tγ∗(x),dPfx(x;P)/a\}bracketri}ht= 0.
+But these two equations are identical with the equations that char acterizeCP./square
+Alternatively, chooselocalcoordinatesat Pandlinearizethedifferentialequation
+at the solution defined by P. Then
+f(x;p1,p2,p3) =φ1(x)p1+φ2(x)p2+φ3(x)p3
+and so
+dPf=φ1(x)dp1+φ2(x)dp2+φ3(x)dp3.
+2Here and elsewhere, the notation dRf(x;R) is the exterior derivative of fwith respect to the
+variable Ronly. Equivalently, it is the exterior derivative of fmodulo the relation dx= 0.14 JONATHAN HOLLAND AND GEORGE SPARLING
+The incidence relation between f(x;p1,p2,p3) and a nearby solution f(x;p1+
+dp1,p2+dp2,p3+dp3) is then precisely
+φ1(x)dp1+φ2(x)dp2+φ3(x)dp3= 0
+φ′
+1(x)dp1+φ′
+2(x)dp2+φ′
+3(x)dp3= 0
+but these are the same equations that characterize the dual cur ve of˜CP.
+Lemma 3. LetL:TM→Rbe a Lagrangian (11). ThenLis regular in a
+neighborhood of each point of CP.
+Indeed, it is sufficient to showthat the Hessian matrix ∂2L/∂˙qi∂˙qjis nonsingular
+at each point of CP. The Lagrangian is defined by
+(11) L(q,˙q) =/parenleftbigg
+˙qi∂f
+∂qi(x(q,˙q),q)/parenrightbigg/parenleftbigg
+˙qj∂fxx
+∂qj(x(q,˙q),q)/parenrightbigg
+.
+The function x(q,˙q) is defined by
+(12) ˙ qi∂fx
+∂qi(x(q,˙q),q) = 0.
+In particular, by implicit differentiation,
+(13)∂x
+∂˙qi=−∂fx/∂qi
+˙qk∂fxx/∂qk.
+At a point of CP, in addition the following holds:
+(14) ˙ qi∂f
+∂qi(x(q,˙q),q) = 0.
+The Hessian at a point of CPis computed by differentiating (11), imposing
+(12) and (14) along the way, and then finally substituting (13). In d etail, denote
+fi=∂f/∂q i,fxi=∂fx/∂qi, etc., andxi=∂x/∂˙qi. Then
+L= ˙qkfk˙qℓfxxℓ
+Li=fi˙qℓfxxℓ+ ˙qkfkfxxi+ ˙qkfxkxi˙qℓfxxℓ+ ˙qkfk˙qℓfxxxℓxi
+Lij=fxixj˙qℓfxxℓ+fifxxj+fi˙qℓfxxxℓxj+fjfxxi+
+fxjxi˙qℓfxxℓ+ ˙qkfxxkxixj˙qℓfxxℓ+fj˙qℓfxxxℓxi(mod (12),(14))
+=−fxifxj+fifxxj+fxxifj−fifxj/parenleftbigg˙qℓfxxxℓ
+˙qkfxxk/parenrightbigg
+−fxifj/parenleftbigg˙qℓfxxxℓ
+˙qkfxxk/parenrightbigg
+by (13). Thus the Hessian matrix of Lhas the form
+HessL=−AAT+BCT+CBT
+where the column vectors A,B,Care defined by
+Ai=fxi
+Bi=fi
+Ci=fxxi−˙qℓfxxxℓ
+˙qkfxxkfxi.
+By the hypothesis (4), A,B,Care linearly independent, and so
+detHessL=−(det[A B C])3/\e}atio\slash= 0,
+which establishes the lemma.CAUSAL GEOMETRIES AND THIRD-ORDER ORDINARY DIFFERENTIAL E QUATIONS 15
+The Legendre transformation associated to the Lagrangian Lis a function L:
+TM→T∗Mcovering the projection onto M. Over a point P∈M,L:TPM→R,
+andL=d∨L, the vertical exterior derivative of L. The Hamiltonian associated
+to the degree 2 homogeneous Lagrangian Lis defined by H=L◦L−1. Here
+L−1is the inverse, possibly only locally defined near points of CP, of the function
+L:TPM→T∗
+PM.
+By Lemma 3, His well-defined in a neighborhood of the preimage of CPunder
+L. The Hamiltonian, where it is defined, vanishes precisely on the dual c urve˜CP.
+Indeed, in local coordinates,
+Li=∂L
+∂˙qi=fi˙qℓfxxℓ+ ˙qkfkfxxi+ ˙qkfxkxi˙qℓfxxℓ+ ˙qkfk˙qℓfxxxℓ
+=fi˙qℓfxxℓ
+when evaluated at any point of CP. Thus Lis proportional to dpfat each point
+ofCP. Inverting, we conclude that L◦L−1(dPf) = 0, soHvanishes along ˜CP.
+The following alternative construction of the Hamiltonian also applies, by lin-
+earizing the problem at P. In local coordinates at P,φi(x) =∂f
+∂pi,i= 1,2,3 define
+independent solutions of the linearized ordinary differential equatio n. In terms of
+these three solutions, the linearized equation itself can be recover ed by solving the
+3×3 system for the unknown coefficients hi:
+(15)φi(x)h0(x)+φ′
+i(x)h1(x)+φ′′
+i(x)h2(x) =φ′′′
+i(x), i= 1,2,3.
+Cramer’s rule gives
+h0(x) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′′′
+1φ′
+1φ′′
+1
+φ′′′
+2φ′
+2φ′′
+2
+φ′′′
+3φ′
+3φ′′
+3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ1φ′
+1φ′′
+1
+φ2φ′
+2φ′′
+2
+φ3φ′
+3φ′′
+3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, h 1(x) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ1φ′′′
+1φ′′
+1
+φ2φ′′′
+2φ′′
+2
+φ3φ′′′
+3φ′′
+3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ1φ′
+1φ′′
+1
+φ2φ′
+2φ′′
+2
+φ3φ′
+3φ′′
+3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+h2(x) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ1φ′
+1φ′′′
+1
+φ2φ′
+2φ′′′
+2
+φ3φ′
+3φ′′′
+3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ1φ′
+1φ′′
+1
+φ2φ′
+2φ′′
+2
+φ3φ′
+3φ′′
+3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
+The Lagrangian of the original equation localized at the point Pis equal to the
+Lagrangian of the linearized equation. It is given first by solving
+φ′
+1(x)q1+φ′
+2(x)q2+φ′
+3(x)q3= 0
+forxas a function of q1,q2,q3. In that case,
+L(q) = (φ1(x(q))q1+φ2(x(q))q2+φ3(x(q))q3)(φ′′
+1(x(q))q1+φ′′
+2(x(q))q2+φ′′
+3(x(q))q3).
+The associated Hamiltonian is obtained by the same construction, bu t applied
+to solutions ˜φ1,˜φ2,˜φ3of the adjoint equation to (15):
+y(x)h0(x)−(y(x)h1(x))′+(y(x)h2(x))′′=−y′′′(x).
+The following theorem is due to Wilczynski [Wil06]; cf. also Olver [Olv95]:16 JONATHAN HOLLAND AND GEORGE SPARLING
+Theorem 6. The projective curves x/mapsto→φ1(x)∂/∂q1+φ2(x)∂/∂q2+φ3(x)∂/∂q3
+inPTPMandx/mapsto→˜φ1(x)dq1+˜φ2(x)dq2+˜φ3(x)dq3are mutually dual.
+4.Hamiltonian spray
+As the point Pvaries, the dual curve ˜CPcut out by the Hamiltonian defines a
+subfibration of the projective cotangent bundle PT∗M. The four-manifold defined
+by the total space of this fibration is denoted here by S, and the cone over Sin
+the cotangent bundle T∗Mis denoted by ˜S. The canonical one-form θonT∗Mis
+homogeneous of degree one, and pulls back to a natural one-form defined up to
+scale on ˜S.
+For a fixed choice of Hamiltonian homogeneous of degree two, define the Hamil-
+tonian spray on T∗Mas the unique vector field Xsuch that
+X/rightanglesedθ=dH.
+The Hamiltonian spray is invariant up to scale under rescalings of H. Moreover, it
+is tangent to the variety ˜Scut out by H= 0 sinceX/rightanglesedH= 0.The vector field X
+is homogeneous of degree one: if µt:T∗M→T∗Mdenotes the dilation mapping
+in the fibers, then Xtα=t(µt)∗Xα.
+The Hamiltonian spray will now be used to define a filtration on TSin a manner
+analogous to the proof of Theorem 3. Let Vbe a nonvanishing vector field that
+is vertical for the fibration S→M. LetT1⊂TSbe the subbundle spanned
+byX,T2the subbundle spanned by X,V. LetT3⊂TSbe the annihilator of
+θ. SinceXandVboth annihilate θ,T2⊂T3. Moreover, since θis annihilated
+byXand is Lie derived along it, [Γ( T1),Γ(T2)] = Γ(T3). Because θ∧dθdoes
+not vanish when pulled back along on any section of the cotangent bu ndle, it also
+does not vanish on Sand soT3is not Frobenius integrable at any point, and thus
+[Γ(T3),Γ(T3)] = Γ(TS)
+It remains only to show that [Γ( T1),Γ(T2)] = Γ(T3). It is sufficient to prove
+thatX,V,[X,V] are linearly independent. As in the proof of Theorem 3, it is
+convenient to work with a particular choice of vector field V. Define the tensor
+h∈Sym2T(T∗M) to be the vertical Hessian of H. In coordinates,
+hij=∂2H
+∂pi∂pj,
+and lethijbe the inverse of hij. These two tensors can be used to raise and lower
+indices. Let ǫbe the associated volume tensor in the fiber. In coordinates
+ǫ=ǫijkdpi⊗dpj⊗dpk.
+The angular momentum
+Li=ǫijkpjhkℓ∂
+∂pℓ
+killsH, since∂H/∂p ℓ=pℓ, and so is tangent to the null cone bundle /tildewideS. Fur-
+thermore, on homogeneous functions of degree zero along the nu ll cone,Lifactors
+through a scalar operator Li=piV, becausepiand∂/∂piare an orthogonal basis
+for the orthogonalcomplement of pi. This defines a vertical vector V. To show that
+X,V,[X,V] are linearly independent, it is enough to show that dθ([X,Li],Lj)/\e}atio\slash= 0.CAUSAL GEOMETRIES AND THIRD-ORDER ORDINARY DIFFERENTIAL E QUATIONS 17
+The Lie bracket is given by
+[X,Li] =−ǫijkpj∂
+∂xk+Ti
+j∂
+∂pj
+for some tensor T. So
+dθ([X,Li],Lj) =ǫimnpmhknǫjknpk=pipj−pkpkhij
+which does not vanish when restricted to the null cone.
+Thus by Lemma 2, Sequipped with its geodesic spray and vertical vector field
+is locally isomorphic to the space J2. Because of the preferred direction V, the
+inclusion of bundles T1⊂T2splits, and thus gives rise to a third-order differential
+equation the distinguished curves of which are the fibers of S→M. The entire pro-
+cedure is reversible, by the construction of the preceding section , which establishes
+Theorem 1.
+5.Recovering the degenerate metric
+The degenerate metric on Sis defined as follows. A point of Sconsists of a point
+P∈Mandv∈CP. Now, through v∈CP, there is a uniquely defined osculating
+conictoCPatv. This osculating conic in turn defines a unique conformal metric
+hP,v:T∗M×T∗M→R. A degenerate conformal Lorentzian metric is defined by
+pullback on the subspace of the cotangent bundle of Sthat annihilates the vertical
+direction:
+gP,v(α,β) =hP,v(π∗α,π∗β).
+A proper degenerate conformal Lorentzian metric is obtained by d ualizing.3In
+order to derive the formula for the metric (1), it is necessary to ob tain explicit
+formulas for the osculating conic of a projective curve. The overa ll program is
+inspired by the work of Wilczynski [Wil06].
+Suppose that φ(t) = (φ1(t),φ2(t),φ3(t)) parametrically specifies the homoge-
+neous coordinates of a projective curve, with det( φ,φ′,φ′′)/\e}atio\slash= 0.. The projective
+curve is a conic provided that there exists a 3 ×3 symmetric non-singular matrix
+Asuch that
+φTAφ= 0.
+Supposing that φis given, the task is to determine a matrix Asuch that at a
+given point t=t0this holds to as many orders in the expansion in powers of
+t−t0as possible. Since Ais regarded projectively, it has 5 independent numerical
+components. These are obtained by solving the system of 5 equatio ns linear in the
+3IfVis a vector space and W⊂V, andBis a nondegenerate bilinear form on W, thenB
+gives rise to a linear isomorphism TB:W→W′. The dual (degenerate) form on V′is given by
+the mapping T˜B:V′→V′/W⊥∼=− →W′T−1
+B− −− →W⊂− →Vwhere the first is the quotient map, the
+second is the natural isomorphism, the third is the inverse o fTB, and the last is the inclusion
+map.18 JONATHAN HOLLAND AND GEORGE SPARLING
+entries ofA:
+(φTAφ)(t0) = 0
+(φTAφ)′(t0) = 0
+(φTAφ)′′(t0) = 0
+(φTAφ)′′′(t0) = 0
+(φTAφ)(4)(t0) = 0.
+Once such a matrix is found, the obstruction to continuing to the fif th order is the
+derivative ( φTAφ)(5)(t0), and is the projective curvature associated to the curve.
+By the first two equations, Aφ(t0) is proportional to the cross product φ(t0)×
+φ′(t0), and since Ais taken projectively, we can fix a scale by taking
+Aφ(t0) =φ(t0)×φ′(t0)
+det(φ(t0),φ′(t0),φ′′(t0))
+or, equivalently, φ(t0)TAφ′′(t0) = 1. The third and fourth equations then give,
+respectively
+φ′T(t0)Aφ′(t0) =−1
+φ′T(t0)Aφ′′(t0) =−1
+3det(φ,φ′,φ′′′)(t0)
+det(φ,φ′,φ′′)(t0).
+The final equation now gives
+3φ′′T(t0)Aφ′′(t0)+4φ′T(t0)Aφ′′′(t0) =−det(φ,φ′,φ(4))(t0)
+det(φ,φ′,φ′′)(t0).
+Now the coordinates of the curve φ(t) satisfy a third-order differential equation
+φ′′′(t) =h0(t)φ(t)+h1(t)φ′(t)+h2(t)φ′′(t)
+wherehiare given explicitly in terms of determinants of φand its first three deriva-
+tives as in (15). The above equations reduce to
+φT(t0)Aφ(t0) = 0
+φT(t0)Aφ′(t0) = 0
+φT(t0)Aφ′′(t0) = 1
+φ′T(t0)Aφ′(t0) =−1
+φ′T(t0)Aφ′′(t0) =−1
+3h2(t0)
+3φ′′T(t0)Aφ′′(t0)+4φ′T(t0)Aφ′′′(t0) =−h2
+2(t0)−h′
+2(t0)−h1(t0).
+The last equation simplifies by substituting (15) for φ′′′and then using the remain-
+ing equations to give
+3φ′′T(t0)Aφ′′(t0) =1
+3h2
+2(t0)−h′
+2(t0)+3h1(t0).
+The obstruction ( φTAφ)(5)(t0) can now be calculated by expanding any terms in-
+volvingφ′′′,φ(4),φ(5)in terms of lower order and then using the above equations.CAUSAL GEOMETRIES AND THIRD-ORDER ORDINARY DIFFERENTIAL E QUATIONS 19
+We find that
+(16)
+(φTAφ)(5)(t0) = 12h0(t0)+4h1(t0)+8
+9h2(t0)3−6h′
+1(t0)−4h2(t0)h′
+2(t0)+2h′′
+2(t0),
+which is precisely the W¨ unschmann invariant for the equation (15).
+The matrix Aobtainedfromthisprocedureisalsoofinterest, becauseit givesth e
+conformal Lorentzian structure. In the basis ( φ,φ′,φ′′),4the symmetric 2-tensor A
+is given by
+A=
+0 0 1
+0−1 −1
+3h2(t0)
+1−1
+3h2(t0)h2
+2(t0)−3h′
+2(t0)+9h1(t0)
+9
+.
+When, as in section 3.3, the curve φ(t)∈TPMis the linearization of the solution
+f(x;P) to the differential equation at a point P∈M, then (15) is the linearization
+of the differential equation at P:
+h0(x) =Fy(x,f(x;P),fx(x;P),fxx(x;P)), h1(x) =Fp(x,f(x;P),fx(x;P),fxx(x;P)),
+h2(x) =Fq(x,f(x;P),fx(x;P),fxx(x;P))
+Theprojectivecurvatureobtainedfrom(16) agreeswiththe W¨ u nschmanninvariant
+(2) of the original equation. Substituting in for the components of the 2-tensor A
+gives the metric
+g=−/parenleftbigg∂
+∂p/parenrightbigg2
++2∂
+∂q∂
+∂y−2
+3Fq∂
+∂p∂
+∂q+F2
+q−3V(Fq)+9Fp
+9/parenleftbigg∂
+∂q/parenrightbigg2
+.
+The dual degenerate conformal metric, defined on the full tange nt bundle of S,
+agrees with (1).
+6.Inverse problems
+By Theorem 1, every causal geometry gives rise to a third-order d ifferential
+equation. Amoresubtleinverseproblemis, givenarankthreedegen erateconformal
+Lorentzian metric on the tangent bundle of a four manifold, when is t here a causal
+geometry from which it arises? More precisely, given a one-paramet er family of
+plane conics, defined by 3 ×3 nondegenerate symmetric 2-tensors A(t), when is
+there a plane curve φ(t) such that, as t→t0,
+φ(t)TA(t0)φ(t) =O(t−t0)5?
+Interchanging tandt0, a necessary and sufficient condition is that
+φ(t0)TA(t)φ(t0) =O(t−t0)5.
+4That this basisis“nonholonomic” (i.e., nonconstant) is si gnificantforthe inverse construction,
+discussed presently.20 JONATHAN HOLLAND AND GEORGE SPARLING
+So at eacht0, the point φ(t0) must satisfy five equations
+φ(t0)TA(t0)φ(t0) = 0
+φ(t0)TA′(t0)φ(t0) = 0
+φ(t0)TA′′(t0)φ(t0) = 0
+φ(t0)TA′′′(t0)φ(t0) = 0
+φ(t0)TA(4)(t0)φ(t0) = 0
+In this system, the matrices A(t0),A′(t0),A′′(t0),A′′′(t0),A(4)(t0) should be re-
+garded as given, and the 3-vector φ(t0) as unknown homogeneous coordinates. The
+system is clearly overdetermined: the two (projective) degrees o f freedom in φ(t0)
+must satisfy five equations. Geometrically the point φ(t0) must simultaneously lie
+onfiveplaneconics, but the intersectionofmorethan twoplane con icsisgenerically
+empty.
+The overdetermined system gives rise to a consistency condition on Aand its
+first four derivatives, which we now describe. In the generic case, we can solve the
+linear equations for a 3 ×3 symmetric matrix X
+(17)trA(t0)X= 0
+trA′(t0)X= 0
+trA′′(t0)X= 0
+trA′′′(t0)X= 0
+trA(4)(t0)X= 0.
+This can be solved uniquely for X, up to scaling, provided the system has rank five.
+The consistency condition is then that the solution Xhas rank one and so splits as
+an outer product
+X=φ(t0)φ(t0)T.
+This happens if and only if every 2 ×2 minor ofXvanishes.5
+6.1.Intermediate cases. When (17) has rank one, the W¨ unschmann invariant
+vanishes. When it has full rank, then it gives rise to a causal curve p rovided the
+2×2 minors of the solution Xall vanish. A calculation done in Mathematica
+shows that, for structures coming from third-order equations, these are the only
+two possibilities: either the system has full rank (and thus nonzero W¨ unschmann)
+or it has rank one (and zero W¨ unschmann).
+However, a priorisuch a system, coming from an arbitrarydegenerate conformal
+Lorentzian structure in four dimensions, can have any rank betwe en 1 and 5. It is
+interestingto understand why these intermediate casesdo not lea d to causal curves.
+Rank 2. Suppose that (17) has rank two (in an interval around t0). Then
+A′′(t) =f(t)A(t) +g(t)A′(t) for some functions fandg. The initial matrices
+A(t0),A′(t0) can be brought simultaneously into diagonal form by a transforma tion
+of the form
+A(t0)/mapsto→MTA(t0)M, A′(t0)/mapsto→MTA′(t0)M.
+5The minors are not independent, however. The variety in P(Sym2(R3)) on which the 2 ×2
+minors of a symmetric 3 ×3 matrix vanish is the well-known Veronese surface, which is not a
+complete intersection. Locally it is the zero locus of any th ree minors coming from distinct rows
+and columns.CAUSAL GEOMETRIES AND THIRD-ORDER ORDINARY DIFFERENTIAL E QUATIONS 21
+Relative to this fixed initial basis, A(t) andA′(t) remain diagonal throughout the
+interval of existence. It is convenient to put
+/vectorA(t) =
+A11(t)
+A22(t)
+A33(t)
+,/vectorX=
+X11
+X22
+X33
+.
+The two equations of (17) reduce to
+trA(t)X=/vectorA(t)·/vectorX= 0
+trA′(t)X=A′
+11(t)X11+A′
+22(t)X22+A′
+33(t)X33= 0
+Solving:
+/vectorX=/vectorA(t)×/vectorA′(t)
+up to an overall scale. Now if φ(t) is a curve solving
+φ(t)TA(t)φ(t) = 0φ(t)TA′(t)φ(t) = 0
+then the entries of φmust square to the entries of /vectorX, so that
+φ(t) =
+±/radicalBig
+/vectorA×/vectorA′·e1
+±/radicalBig
+/vectorA×/vectorA′·e2
+±/radicalBig
+/vectorA×/vectorA′·e3
+.
+Differentiating gives
+φ′
+i(t) =±g(t)
+2/radicalBig
+/vectorA×/vectorA′·ei
+and soφ′is proportional to φ. Thus the range of φ(t) is a projective point (in the
+complex sense).6Therefore there are one or no solutions, depending on whether
+the square roots are all real.
+Rank 3 and 4. We argue indirectly that, if (17) has rank 3 or 4 throughout an
+interval and φ(t) is aC3solution in that interval, then φ(t) parameterizes either
+a projective point (as in the rank 2 case) or a line (which is degenerat e from our
+point of view). We were unable to devise a direct argument analogous to the
+rank 2 case. In general, because φ(t) is a three-vector, there must be a non-trivial
+linear relation between φ(t) and its first three derivatives. This must either give a
+proper third-order equation in an interval, or else there is a linear re lation between
+φ,φ′,φ′′. In the latter case, φ(t) does indeed parameterize a line (if it satisfies a
+second order equation) or a point (if the equation is first order). I n the former
+case, the coefficients of A(t) can be given in terms of φ(t),φ′(t),φ′′(t) and the
+coefficients of the third-order equation h1(t),h2(t),h3(t), as in the previous section.
+But, as already indicated, this implies that the system (17) has rank either 1 or 5,
+a contradiction.
+6Indeed, if φ(0) and φ′(0) are linearly independent, then φ(0)×φ′(0)·φ(t) satisfies a second
+order ode with both initial conditions zero, so φ(0)×φ′(0)·φ(t)≡0 which implies that φ(t) is
+constrained to a projective line. If φ(0) and φ′(0) are linearly dependent, then smoothness of
+dependence on initial conditions gives the result.22 JONATHAN HOLLAND AND GEORGE SPARLING
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+arXiv:1001.0203v2  [math.PR]  5 Jan 2010On the generalized Feynman-Kac
+transformation for nearly symmetric
+Markov processes
+Li Ma Wei Sun
+Department of Mathematics and Statistics
+Concordia University, Canada
+Abstract
+Suppose Xis a right process which is associated with a non-symmetric
+Dirichlet form ( E,D(E)) onL2(E;m). Foru∈D(E), we have Fukushima’s
+decomposition: ˜ u(Xt)−˜u(X0) =Mu
+t+Nu
+t. In this paper, we investigate
+the strong continuity of the generalized Feynman-Kac semig roup defined by
+Pu
+tf(x) =Ex[eNu
+tf(Xt)]. LetQu(f,g) =E(f,g)+E(u,fg) forf,g∈D(E)b.
+Denote by J1the dissymmetric part of the jumping measure Jof (E,D(E)).
+Under the assumption that J1is finite, we show that ( Qu,D(E)b) is lower
+semi-boundedifandonly ifthereexists aconstant α0≥0suchthat /ba∇dblPu
+t/ba∇dbl2≤
+eα0tfor every t >0. If one of these conditions holds, then ( Pu
+t)t≥0is strongly
+continuous on L2(E;m). IfXis equipped with a differential structure, then
+this result also holds without assuming that J1is finite.
+Keywords: Non-symmetricDirichlet form; generalized Feynman-Kac se mi-
+group; strong continuity; lower semi-bounded; Beurling-D eny formula; Le-
+Jan’s transformation rule
+1 Introduction
+LetEbe a metrizable Lusin space and X= ((Xt)t≥0,(Px)x∈E) be a right (contin-
+uous strong Markov) process on E(cf. [22, IV, Definition 1.8]). Suppose that Xis
+associated with a (non-symmetric) Dirichlet form ( E,D(E)) onL2(E;m), wherem
+is aσ-finite measure on the Borel σ-algebraB(E) ofE. Then, by [22, IV, Theorem
+6.7] (cf. also [15, Theorem 3.22]), ( E,D(E)) is quasi-regular. Moreover, ( E,D(E))
+is quasi-homeomorphic to a regular Dirichlet form (see [11]). We refe r the reader
+to [22] and [17] for the theory of Dirichlet forms. The notations and terminologies
+of this paper follow [22] and [17].
+Letu∈D(E). Then, we have Fukushima’s decomposition (cf. [22, VI, Theorem
+2.5])
+˜u(Xt)−˜u(X0) =Mu
+t+Nu
+t,
+1where ˜uis a quasi-continuous m-version ofu,Mu
+tis a square integrable martingale
+additive functional (MAF) and Nu
+tis a continuous additive functional (CAF) of
+zero energy. For x∈E, denote by Exthe expectation with respect to (w.r.t.) Px.
+Define the generalized Feynman-Kac transformation
+Pu
+tf(x) =Ex[eNu
+tf(Xt)], f≥0 andt≥0.
+In this paper, we will investigate the strong continuity of the semigr oup (Pu
+t)t≥0
+onL2(E;m).
+The strong continuity of generalized Feynman-Kac semigroups for symmetric
+Markov processes has been studied extensively by many people. No te that in gen-
+eral (Nu
+t)t≥0is not of finite variation (cf. [17, Example 5.5.2]). Hence the classical
+results of Albeverio and Ma given in [1] do not apply directly. Under the as-
+sumption that Xis the standard d-dimensional Brownian motion, uis a bounded
+continuousfunctionon Rdand|∇u|2belongstotheKatoclass, Gloveretal. proved
+in [18] that ( Pu
+t)t≥0is a strongly continuous semigroup on L2(Rd;dx). Moreover,
+they gave an explicit representation for the closed quadratic form corresponding to
+(Pu
+t)t≥0. In [25], Zhang generalized the results of [18] to symmetric L´ evy pr ocesses
+onRdand removed the assumption that uis bounded continuous. Furthermore,
+Z.Q. Chen and Zhang established in [13] the corresponding results fo r general sym-
+metric Markov processes via Girsanov transformation. They prov ed that ifµ/an}bracketle{tu/an}bracketri}ht,
+the energy measure of u, is a measure of the Kato class, then ( Pu
+t)t≥0is a strongly
+continuous semigroup on L2(E;m). Also, they characterized the closed quadratic
+form corresponding to ( Pu
+t)t≥0. In [16], Fitzsimmons and Kuwae established the
+strong continuity of ( Pu
+t)t≥0under the assumption that Xis a symmetric diffusion
+process and µ/an}bracketle{tu/an}bracketri}htis a measure of the Hardy class. Furthermore, Z.Q. Chen et al.
+established in [9] the strong continuity of ( Pu
+t)t≥0for general symmetric Markov
+processes under the assumption that µ/an}bracketle{tu/an}bracketri}htis a measure of the Hardy class.
+All the results mentioned above give sufficient conditions for ( Pu
+t)t≥0to be
+strongly continuous, where µ/an}bracketle{tu/an}bracketri}htis assumed to be of either the Kato class or the
+Hardy class. In [5], under the assumption that Xis a symmetric diffusion process,
+C.Z. Chen and Sun showed that the semigroup ( Pu
+t)t≥0is strongly continuous on
+L2(E;m) if and only if the bilinear form ( Qu,D(E)b) is lower semi-bounded. Here
+and henceforth
+Qu(f,g) :=E(f,g)+E(u,fg), f,g∈D(E)b:=D(E)∩L∞(E;m).(1.1)
+Furthermore, C.Z. Chen et al. generalized this result to general sy mmetric Markov
+processes in [4]. In [10], Z.Q. Chen et al. studied general perturbatio ns of sym-
+metric Markov processes and gave another proof for the equivale nce of the strong
+continuity of ( Pu
+t)t≥0and the lower semi-boundedness of ( Qu,D(E)b).
+The aim of this paper is to study the strong continuity problem of gen eralized
+Feynman-Kacsemigroups for nearly symmetric Markov processes . Note that many
+useful tools of symmetric Dirichlet forms, e.g. time reversal and Ly ons-Zheng de-
+composition, do not apply well to the non-symmetric setting. That m akes the
+2problem more difficult. Also, we would like to point out that the Girsanov trans-
+formed process of Xinduced by Mu
+tand the Girsanov transformed process of ˆX
+induced by ˆMu
+tare not in duality in general (cf. [6]), where ˆXis the dual process
+ofXandˆMu
+tis the martingale part of ˜ u(ˆXt)−˜u(ˆX0). The method of this paper
+is inspired by [4] and [10]. We will combine the h-transform method of [4] and
+the localization method used in [10]. It is worth to point out that the Be urling-
+Deny formula given in [20] and LeJan’s transformation rule developed in [21] play
+a crucial role in this paper.
+Denote byJandKthe jumping and killing measures of ( E,D(E)), respectively.
+WriteˆJ(dx,dy) =J(dy,dx). Denote by J1:= (J−ˆJ)+the positive part of the
+Jordan decomposition of J−ˆJ.J1is called the dissymmetric part of J. Note that
+J0:=J−J1is the largest symmetric σ-finite positive measure dominated by J.
+Denote by dthe diagonal of the product space E×E; and denote by /ba∇dbl · /ba∇dbl2and
+(·,·)mthe norm and inner product of L2(E;m), respectively.
+Now we can state the main results of the paper.
+Theorem 1.1. Suppose that Xis a right process which is associated with a (non-
+symmetric) Dirichlet form (E,D(E))onL2(E;m). Letu∈D(E). Assume that
+J1(E×E\d)<∞. Then the following two conditions are equivalent:
+(i) There exists a constant α0≥0such that
+Qu(f,f)≥ −α0(f,f)m,∀f∈D(E)b.
+(ii) There exists a constant α0≥0such that
+/ba∇dblPu
+t/ba∇dbl2≤eα0t,∀t>0.
+Furthermore, if one of these conditions holds, then the semi group(Pu
+t)t≥0is
+strongly continuous on L2(E;m).
+Theorem 1.2. LetUbe an open set of Rdandmbe a positive Radon measure on
+Uwithsupp[m] =U. Suppose that Xis a right process which is associated with a
+(non-symmetric) Dirichlet form (E,D(E))onL2(U;m)such thatC∞
+0(U)is dense
+inD(E). Then the conclusions of Theorem 1.1 remain valid without as suming that
+J1(E×E\d)<∞.
+The rest of this paper is organized as follows. In Section 2, we give th e proof of
+Theorems 1.1 and1.2. InSection 3, we give two examples, oneis finite- dimensional
+and the other one is infinite-dimensional.
+2 Proof of the Main Results
+In this section, we will prove Theorems 1.1 and 1.2. By quasi-homeomo rphism,
+we assume without loss of generality that Xis a Hunt process and ( E,D(E)) is a
+3regular (non-symmetric) Dirichlet form on L2(E;m), whereEis a locally compact
+separable metric space and mis a positive Radonmeasure on Ewith supp[m] =E.
+We denote by ∆ and ζthe cemetery and lifetime of X, respectively. It is known
+that everyf∈D(E) has a quasi-continuous m-version. To simplify notation, we
+still denote this version by f.
+Letu∈D(E). By [22, III, Proposition 1.5], there exists |u|E∈D(E) such that
+|u|E≥ |u|m−a.e.onEandE1(|u|E,w)≥0 for allw∈D(E) withw≥0m−a.e.
+onE. Similar to [17, Theorems 2.2.1 and 2.2.2], we can show that there exists a
+positive Radon measure ηuonEsuch that
+E1(|u|E,w) =/integraldisplay
+Ewdηu, w∈D(E). (2.1)
+Define
+u∗:=u+|u|E. (2.2)
+Then,u∗has a quasi-continuous m-version which is nonnegative q.e. on E.
+Moreover, there exists an E-nest{Fn}n∈Nconsisting of compact sets of Esuch
+thatu∗is continuous and hence bounded on Fnfor eachn∈N. Define
+τFn= inf{t >0|Xt∈Fc
+n}. By [22, IV, Proposition 5.30], lim n→∞τFn=ζ Px-
+a.s. for q.e. x∈E.
+Let (N,H) be a L´ evy system of Xandνbe the Revuz measure of H. Define
+Bt=/summationdisplay
+s≤t/bracketleftbig
+e(u∗(Xs−)−u∗(Xs))−1−(u∗(Xs−)−u∗(Xs))/bracketrightbig
+. (2.3)
+Note that for any M >0 there exists CM>0 such that ( ex−1−x)≤CMx2
+for allxsatisfyingx≤M. Since (u∗(Xt−))t≥0is locally bounded, ( u∗(Xt))t≥0is
+nonnegative and M−u∗is a square integrable martingale for q.e. x∈E, hence
+(Bt)t≥0is locally integrable on [0 ,ζ) for q.e.x∈E. Here and henceforth the
+phrase “on [0 ,ζ)” is understood as “on the optional set [[0 ,ζ)) of interval type”
+in the sense of [19, Chap. VIII, 3]. By [17, (A.3.23)], one finds that t he dual
+predictable projection of ( Bt)t≥0is given by
+Bp
+t=/integraldisplayt
+0/integraldisplay
+E△[e(u∗(Xs)−u∗(y))−1−(u∗(Xs)−u∗(y))]N(Xs,dy)dHs.
+We set
+Md
+t=Bt−Bp
+t (2.4)
+and denote
+Mt=M−u∗
+t+Md
+t. (2.5)
+Note that for any M >0 there exists DM>0 such that ( ex−1−x)2≤DMx2
+for allxsatisfyingx≤M. Since (u∗(Xt−))t≥0is locally bounded, ( u∗(Xt))t≥0
+is nonnegative and M−u∗is a square integrable martingale for q.e. x∈E, hence
+(Md
+t)t≥0isa locallysquare integrableMAF on[0 ,ζ)forq.e.x∈Eby[19, Theorem
+47.40]. Therefore ( Mt)t≥0is alocally squareintegrable MAF on[0 ,ζ)for q.e.x∈E.
+We denote the Revuz measure of ( <M > t)t≥0byµ<M>(cf. [8, Remark 2.2]).
+LetM−u∗,c
+tbe the continuous part of M−u∗
+t. Define
+A−u∗
+t=Bp
+t+1
+2<M−u∗,c>t. (2.6)
+Then (A−u∗
+t)t≥0is a positive CAF (PCAF). Denote by µ−u∗the Revuz measure of
+(A−u∗
+t)t≥0. Then
+µ−u∗(dx) =/integraldisplay
+E△[e(u∗(x)−u∗(y))−1−(u∗(x)−u∗(y))]N(x,dy)ν(dx)
++1
+2µ<M−u∗,c>(dx). (2.7)
+Define
+µ−u:=µ−u∗+ηu−|u|Em (2.8)
+and
+µ′
+−u:=µ−u∗+ηu+|u|Em.
+Recall that a smooth measure µis said to be of the Kato class if
+lim
+t→0inf
+Cap(N)=0sup
+x∈E−NEx[Aµ
+t] = 0,
+where (Aµ
+t)t≥0is the PCAF associated with µ. Denote by SKthe Kato class of
+smooth measures. Similar to [2, Theorem 2.4], we can show that there exists an
+E-nest{F′
+n}n∈Nconsisting of compact sets of Esuch thatIF′n(µ<M>+µ′
+−u)∈SK.
+To simplify notation, we still use Fnto denoteFn∩F′
+nforn∈N. LetEnbe
+the fine interior of Fnw.r.t.X. DefineD(E)n:={f∈D(E)|f= 0 q.e.onEc
+n},
+τEn= inf{t>0|Xt∈Ec
+n}and
+¯Pu,n
+tf(x) :=Ex[eM−u∗
+t−N|u|E
+tf(Xt);t<τEn].
+2.1 The bilinear form associated with (¯Pu,n
+t)t≥0onL2(En;m)
+Forn∈N, we define the bilinear form ( ¯Qu,n,D(E)n) by
+¯Qu,n(f,g) =E(f,g)−/integraldisplay
+Egdµ<Mf,M>−/integraldisplay
+Efgdµ−u, f,g∈D(E)n.(2.9)
+By [12, Lemma 4.3], for every ε>0, there exists a constant An
+ε>0 such that
+/integraldisplay
+Ew2d(µ<M>+µ′
+−u)≤εE(w,w)+An
+ε/ba∇dblw/ba∇dbl2
+2, w∈D(E)n.
+5Suppose that |E(f,g)| ≤k1E1(f,f)1
+2E1(g,g)1
+2for allf,g∈D(E) andsome constant
+k1>0. Then
+|¯Qu,n(f,g)| ≤k1E1(f,f)1
+2E1(g,g)1
+2+/parenleftbigg/integraldisplay
+Edµ<Mf>/parenrightbigg1
+2/parenleftbigg/integraldisplay
+Eg2dµ<M>/parenrightbigg1
+2
++/parenleftbigg/integraldisplay
+Ef2dµ′
+−u/parenrightbigg1
+2/parenleftbigg/integraldisplay
+Eg2dµ′
+−u/parenrightbigg1
+2
+≤k1E1(f,f)1
+2E1(g,g)1
+2+(max(ε,An
+ε))1
+2[2E(f,f)]1
+2E1(g,g)1
+2
++max(ε,An
+ε)·E1(f,f)1
+2E1(g,g)1
+2
+≤θnE1(f,f)1
+2E1(g,g)1
+2, (2.10)
+whereθn:= (k1+/radicalbig
+2max(ε,An
+ε)+max(ε,An
+ε)).
+Fix anε<(√
+2−1)/(√
+2+1) and set αn:= 2An
+ε. Then
+¯Qu,n
+αn(f,f) :=¯Qu,n(f,f)+αn(f,f)
+≥ E(f,f)−/parenleftbigg/integraldisplay
+Edµ<Mf>/parenrightbigg1
+2/parenleftbigg/integraldisplay
+Ef2dµ<M>/parenrightbigg1
+2
+−/integraldisplay
+Ef2dµ′
+−u+αn(f,f)
+≥ E(f,f)−(εE(f,f)+An
+ε/ba∇dblf/ba∇dbl2
+2)1
+2[2E(f,f)]1
+2
+−(εE(f,f)+An
+ε/ba∇dblf/ba∇dbl2
+2)+αn(f,f)
+≥ E(f,f)−1√
+2((1+ε)E(f,f)+An
+ε/ba∇dblf/ba∇dbl2
+2)
+−(εE(f,f)+An
+ε/ba∇dblf/ba∇dbl2
+2)+αn(f,f)
+≥√
+2−1−(√
+2+1)ε√
+2E(f,f)+(√
+2−1)An
+ε√
+2/ba∇dblf/ba∇dbl2
+2.(2.11)
+By (2.10), (2.11) and [22, I, Proposition 3.5], we know that ( ¯Qu,n
+αn,D(E)) is a
+coercive closed form on L2(En;m).
+Theorem 2.1. For eachn∈N,(¯Pu,n
+t)t≥0is a strongly continuous semigroup of
+bounded operators on L2(En;m)with/ba∇dbl¯Pu,n
+t/ba∇dbl2≤eβntfor everyt >0and some
+constantβn>0. Moreover, the coercive closed form associated with (e−βnt¯Pu,n
+t)t≥0
+is given by (¯Qu,n
+βn,D(E)n).
+Proof.The proof is much similar to that of [16, Theorem 1.1], which is based on
+a key lemma (see [16, Lemma 3.2]) and a remarkable localization method. In fact,
+the proof of our Theorem 2.1 is simpler since IFn(µ<M>+µ′
+−u) is of the Kato class
+instead of the Hardy class and there is no time reversal part in the s emigroup
+(¯Pu,n
+t)t≥0. We omit the details of the proof here and only give the following key
+lemma, which is the counterpart of [16, Lemma 3.2].
+6Lemma 2.2. Let(L¯Qu,n,D(L¯Qu.n))be the generator of (¯Qu,n,D(E)n). Then, for
+anyf∈D(L¯Qu,n), we have
+f(Xt)eM−u∗
+t−N|u|E
+t=f(X0)+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−dMf
+s
++/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)dMs
++/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−L¯Qu,nf(Xs)ds (2.12)
+Pm-a.s. on{t<τEn}.
+Proof.Letf∈D(L¯Qu,n) andg∈D(E)n. Then, by (2.9), we get
+E(f,g) =¯Qu,n(f,g)+/integraldisplay
+Egdµ<Mf,M>+/integraldisplay
+Efgdµ−u
+=−(L¯Qu,nf,g)+/integraldisplay
+Egdµ<Mf,M>+/integraldisplay
+Efgdµ−u.(2.13)
+By (2.1), (2.13) and [23, Theorem 5.2.7], we find that ( N|u|E
+t)t≥0is a CAF of
+bounded variation and
+Nf
+t=/integraldisplayt
+0L¯Qu,nf(Xs)ds−<Mf,M > t−/integraldisplayt
+0f(Xs)d(A−u∗
+s−N|u|E
+s)
+fort<τEn. Therefore, for t<τEn, we have
+f(Xt)−f(X0) =Mf
+t+Nf
+t
+=Mf
+t+/integraldisplayt
+0L¯Qu,nf(Xs)ds−<Mf,M > t
+−/integraldisplayt
+0f(Xs)d(A−u∗
+s−N|u|E
+s). (2.14)
+By Itˆ o’s formula (cf. [24, II, Theorem 33]), (2.14) and (2.4)-(2.6) , we obtain that
+fort<τEn
+f(Xt)eM−u∗
+t−N|u|E
+t
+=f(X0)+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−df(Xs)+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)d(M−u∗
+s−N|u|E
+s)
++1
+2/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)d<M−u∗,c>s+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−d<Mf,c,M−u∗,c>s
++/summationdisplay
+s≤t[f(Xs)eM−u∗
+s−N|u|Es−f(Xs−)eM−u∗
+s−−N|u|E
+s−
+−eM−u∗
+s−−N|u|E
+s−△f(Xs)−f(Xs−)eM−u∗
+s−−N|u|E
+s−△M−u∗
+s]
+7=f(X0)+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−dMf
+s+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−L¯Qu,nf(Xs)ds
+−/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−d<Mf,M > s−/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs)d(A−u∗
+s−N|u|E
+s)
++/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)d(M−u∗
+s−N|u|E
+s)
++1
+2/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)d<M−u∗,c>s+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−d<Mf,c,M−u∗,c>s
++/summationdisplay
+s≤t[f(Xs)eM−u∗
+s−N|u|Es−f(Xs−)eM−u∗
+s−−N|u|E
+s−
+−eM−u∗
+s−−N|u|E
+s−△f(Xs)−f(Xs−)eM−u∗
+s−−N|u|E
+s−△M−u∗
+s]
+=/braceleftbigg
+f(X0)+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−dMf
+s+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−L¯Qu,nf(Xs)ds
++/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)dM−u∗
+s/bracerightbigg
++/braceleftbigg
+−/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs)dA−u∗
+s+1
+2/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)d<M−u∗,c>s
+−/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−d<Mf,M > s+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−d<Mf,c,M−u∗,c>s
++/summationdisplay
+s≤t[f(Xs)eM−u∗
+s−N|u|Es−f(Xs−)eM−u∗
+s−−N|u|E
+s−
+−eM−u∗
+s−−N|u|E
+s−△f(Xs)−f(Xs−)eM−u∗
+s−−N|u|E
+s−△M−u∗
+s]/bracerightBig
+=/braceleftbigg
+f(X0)+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−dMf
+s+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−L¯Qu,nf(Xs)ds
++/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)dM−u∗
+s/bracerightbigg
++/braceleftbigg
+−/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)dBp
+s−/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−d<Mf,d,Md>s
+−/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−d<Mf,d,M−u∗,d>s
++/summationdisplay
+s≤t[f(Xs)eM−u∗
+s−N|u|Es−f(Xs−)eM−u∗
+s−−N|u|E
+s−
+−eM−u∗
+s−−N|u|E
+s−△f(Xs)−f(Xs−)eM−u∗
+s−−N|u|E
+s−△M−u∗
+s]/bracerightBig
+:=I+II. (2.15)
+Note that
+II=−/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)dBp
+s+/summationdisplay
+s≤t[−eM−u∗
+s−−N|u|E
+s−△f(Xs)△Bs
+8−eM−u∗
+s−−N|u|E
+s−△f(Xs)△M−u∗
+s]+/summationdisplay
+s≤t[f(Xs)eM−u
+s−N|u|Es
+−f(Xs−)eM−u∗
+s−−N|u|E
+s−−eM−u∗
+s−−N|u|E
+s−△f(Xs)−f(Xs−)eM−u∗
+s−−N|u|E
+s−△M−u∗
+s]
+=−/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)dBp
+s+/summationdisplay
+s≤t[−eM−u∗
+s−−N|u|E
+s−△f(Xs)(e△M−u∗
+s−1)
++f(Xs)eM−u∗
+s−N|u|Es−f(Xs−)eM−u∗
+s−−N|u|E
+s−−eM−u∗
+s−−N|u|E
+s−△f(Xs))
+−f(Xs−)eM−u∗
+s−−N|u|E
+s−△M−u∗
+s]
+=−/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)dBp
+s+/summationdisplay
+s≤t[eM−u∗
+s−−N|u|E
+s−f(Xs−)e△M−u∗
+s
+−f(Xs−)eM−u∗
+s−−N|u|E
+s−−f(Xs−)eM−u∗
+s−−N|u|E
+s−△M−u∗
+s]
+=−/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)dBp
+s+/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)dBs
+=/integraldisplayt
+0eM−u∗
+s−−N|u|E
+s−f(Xs−)dMd
+s. (2.16)
+Therefore (2.12) follows from (2.15) and (2.16).
+2.2 The bilinear form associated with (¯Pu,n
+t)t≥0on
+L2(En;e−2u∗m)
+Forn∈N, sinceu∗·IEnis bounded, ( ¯Pu,n
+t)t≥0is also a strongly continuous
+semigroup on L2(En;e−2u∗m) by Theorem 2.1. In this subsection, we will study
+the bilinear form associated with ( ¯Pu,n
+t)t≥0onL2(En;e−2u∗m).
+DefineD(E)n,b:=D(E)n∩L∞(E;m). Letf,g∈D(E)n,b. Note that e−2u∗g=
+(e−2u∗−1)g+g∈D(E)n,b. Define
+Eu,n(f,g) :=¯Qu,n(f,e−2u∗g), f,g∈D(E)n,b. (2.17)
+Then, by Theorem 2.1, we get
+Eu,n(f,g) = lim
+t→01
+t(f−¯Pu,n
+tf,e−2u∗g)m= lim
+t→01
+t(f−¯Pu.n
+tf,g)e−2u∗m.(2.18)
+(Eu,n,D(E)n,b) is called the bilinear from associated with ( ¯Pu,n
+t)t≥0on
+L2(En;e−2u∗m).
+Note that
+<Mf,Md>t
+= [Mf,Md]p
+t
+=/braceleftBigg/summationdisplay
+s≤t[f(Xs)−f(Xs−)][e(u∗(Xs−)−u∗(Xs))−1−(u∗(Xs−)−u∗(Xs))]/bracerightBiggp
+9=/integraldisplayt
+0/integraldisplay
+E△[f(y)−f(Xs)][e(u∗(Xs)−u∗(y))−1−(u∗(Xs)−u∗(y))]N(Xs,dy)dHs.
+Then
+/integraldisplay
+Egdµ<Mf,Md>
+=/integraldisplay
+E/integraldisplay
+E△g(x)[f(y)−f(x)][e(u∗(x)−u∗(y))−1−(u∗(x)−u∗(y))]N(x,dy)ν(dx).
+(2.19)
+By (2.7) and (2.8), we get
+/integraldisplay
+Efgdµ−u=/integraldisplay
+E/integraldisplay
+E△f(x)g(x)[e(u∗(x)−u∗(y))−1−(u∗(x)−u∗(y))]N(x,dy)ν(dx)
++1
+2/integraldisplay
+Efgdµ<M−u∗,c>+/integraldisplay
+Efgdηu−/integraldisplay
+Efg|u|Edm. (2.20)
+Similar to [17, Theorem 5.3.1] (cf. also [23, Chapter 5]), we can show th at
+J(dx,dy) =1
+2N(y,dx)ν(dy) andK(dx) =N(x,△)ν(dx). Therefore, we obtain by
+(2.17), (2.9), (2.19) and (2.20) that
+Eu,n(f,g) =¯Qu.n(f,e−2u∗g)
+=E(f,e−2u∗g)−/integraldisplay
+Ee−2u∗gdµ<Mf,M>−/integraldisplay
+Ee−2u∗fgdµ−u
+=E(f,e−2u∗g)−/integraldisplay
+Ee−2u∗gdµ<Mf,M−u∗>−/integraldisplay
+Ee−2u∗gdµ<Mf,Md>
+−/integraldisplay
+Ee−2u∗fgdµ−u
+=E(f,e−2u∗g)−/integraldisplay
+Ee−2u∗gdµ<Mf,M−u∗>
+−2/integraldisplay
+E×E−de−2u∗(y)g(y)f(x)[e(u∗(y)−u∗(x))−1−(u∗(y)−u∗(x))]J(dx,dy)
+−1
+2/integraldisplay
+Ee−2u∗fgdµ<M−u∗,c>−E(|u|E,e−2u∗fg).
+(2.21)
+Theorem 2.3. For eachn∈N, under either the assumption of Theorem 1.1 or
+Theorem 1.2, we have
+Eu,n(f,g) =Qu(fe−u∗,ge−u∗), f,g∈D(E)n,b. (2.22)
+Proof.We fix ann∈N. Define
+Ψu∗,n(f,g) :=E(f,e−2u∗g)−/integraldisplay
+Ee−2u∗gdµ<Mf,M−u∗>
+10−2/integraldisplay
+E×E−de−2u∗(y)g(y)f(x)[e(u∗(y)−u∗(x))−1−(u∗(y)−u∗(x))]J(dx,dy)
+−1
+2/integraldisplay
+Ee−2u∗fgdµ<M−u∗,c>, f,g∈D(E)n,b. (2.23)
+Then, by (2.21) and (1.1), we find that (2.22) is equivalent to
+Ψu∗,n(f,g) =E(fe−u∗,ge−u∗)+E(u∗,e−2u∗fg), f,g∈D(E)n,b.(2.24)
+Sinceu∗·IEnisbounded, thereexists l0∈Nsuch that |u∗(x)| ≤l0forallx∈En.
+Forl∈N, defineu∗
+l:= ((−l)∨u∗)∧l. Thenu∗
+l∈D(E)bandu∗=u∗
+lonEnfor
+l≥l0. Similarto[17,Lemma5.3.1],wecanshowthat µ<M−u∗,c>|En=µ<M−u∗
+l,c>|En
+forl≥l0. Forφ∈D(E)b, we define
+Ψφ,n(f,g) :=E(f,e−2φg)−/integraldisplay
+Ee−2φgdµ<Mf,M−φ>
+−2/integraldisplay
+E×E−de−2φ(y)g(y)f(x)[e(φ(y)−φ(x))−1−(φ(y)−φ(x))]J(dx,dy)
+−1
+2/integraldisplay
+Ee−2φfgdµ<M−φ,c>, f,g∈D(E)n,b. (2.25)
+Then, by (2.23) and (2.25), we find that for l≥l0
+Ψu∗,n(f,g) = Ψu∗
+l,n(f,g)+/integraldisplay
+Ee−2u∗gdµ<Mf,Mu∗−u∗
+l>, f,g∈D(E)n,b.
+Note that by [23, (5.1.3)]
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Ee−2u∗gdµ<Mf,Mu∗−u∗
+l>/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2e2l0/ba∇dblg/ba∇dbl∞E(f,f)1
+2E(u∗−u∗
+l,u∗−u∗
+l)1
+2
+→0 asl→ ∞,
+and
+E(u∗
+l,e−2u∗fg)→ E(u∗,e−2u∗fg) asl→ ∞.
+Hence, to establish (2.24), it is sufficient to show that for any φ∈D(E)band
+f,g∈D(E)n,b
+Ψφ,n(f,g) =E(fe−φ,ge−φ)+E(φ,e−2φfg). (2.26)
+Letφ∈D(E)b. By [23, (5.3.2)], we have
+/integraldisplay
+gdµ<Mf,M−φ>=−E(f,gφ)−E(φ,gf)+E(fφ,g). (2.27)
+By (2.25) and (2.27), we find that (2.26) is equivalent to
+E(f,e−2φg)+E(f,e−2φgφ)−E(fφ,e−2φg)
+−2/integraldisplay
+E×E−de−2φ(y)g(y)f(x)[e(φ(y)−φ(x))−1−(φ(y)−φ(x))]J(dx,dy)
+−1
+2/integraldisplay
+Ee−2φfgdµ<M−φ,c>
+=E(fe−φ,ge−φ). (2.28)
+11Denote byM−φ,j
+tandM−φ,k
+tthe jumping and killing parts of M−φ
+t, respectively.
+Then, similar to [17, (5.3.9) and (5.3.10)], we get
+µ<M−φ,j>(dx) = 2/integraldisplay
+E(φ(x)−φ(y))2J(dy,dx) andµ<M−φ,k>(dx) =φ2(x)K(dx).
+Thus, for any w∈D(E)b, we have
+/integraldisplay
+Ewdµ<M−φ,c>=/integraldisplay
+Ewd(µ<M−φ>−µ<M−φ,j>−µ<M−φ,k>)
+= 2E(φ,φw)−E(φ2,w)
+−2/integraldisplay
+E×E−d(φ(y)−φ(x))2w(y)J(dx,dy)−/integraldisplay
+Ewφ2dK.
+(2.29)
+By (2.29), we find that (2.28) is equivalent to
+E(f,e−2φg)+E(f,e−2φgφ)−E(fφ,e−2φg)−E(φ,e−2φφfg)+1
+2E(φ2,e−2φfg)
+−2/integraldisplay
+E×E−de−2φ(y)g(y)f(x)[e(φ(y)−φ(x))−1−(φ(y)−φ(x))]J(dx,dy)
++/integraldisplay
+E×E−d(φ(y)−φ(x))2e−2φ(y)f(y)g(y)J(dx,dy)+1
+2/integraldisplay
+Ee−2φfgφ2dK
+=E(fe−φ,ge−φ). (2.30)
+Proof of (2.30) under the assumption of Theorem 1.1.
+Denote by ˜Ethe symmetric part of E. Then ( ˜E,D(E)) is a regular symmetric
+Dirichlet form. Denote by ˜Jand˜Kthe jumping and killing measures of ( ˜E,D(E)),
+respectively. Then
+/integraldisplay
+E×E−d(φ(y)−φ(x))2J(dx,dy)+/integraldisplay
+Eφ2dK
+≤2/braceleftbigg/integraldisplay
+E×E−d(φ(y)−φ(x))2˜J(dx,dy)+/integraldisplay
+Eφ2d˜K/bracerightbigg
+≤2E(φ,φ) (2.31)
+and/integraldisplay
+E×E−d[e(φ(y)−φ(x))−1−(φ(y)−φ(x))]J(dx,dy)
+≤C/bardblφ/bardbl∞/integraldisplay
+E×E−d(φ(y)−φ(x))2J(dx,dy)
+≤C/bardblφ/bardbl∞E(φ,φ) (2.32)
+for some constant C/bardblφ/bardbl∞>0. Hence, to establish (2.30) for φ∈D(E)band
+f,g∈D(E)n,b, it is sufficient to establish (2.30) for φ,f,g∈D:=C0(E)∩D(E)
+by virtue of the density of DinD(E) and approximation.
+12By [20, Theorem 4.8 and Proposition 5.1], we have the following Beurling- Deny
+decomposition
+E(f,g) =Ec(f,g)+SPV/integraldisplay
+E×E−d2(f(y)−f(x))g(y)J(dx,dy)
++/integraldisplay
+EfgdK, f,g ∈D(E)b, (2.33)
+whereSPV/integraltext
+denotes the symmetric principle value integral (see [20, Definition
+2.5]) and Ec(f,g) satisfies the left strong local property in the sense that Ec(f,g) =
+0 iffis constant E-q.e. on a quasi-open set containing the quasi-support of g(see
+[20, Theorem 4.1]). By (2.33), we obtain that for any w∈D(E)b,
+2E(φ,φw)−E(φ2,w)
+−2/integraldisplay
+E×E−d(φ(y)−φ(x))2w(y)J(dx,dy)−/integraldisplay
+Ewφ2dK
+= 2Ec(φ,φw)−Ec(φ2,w).
+Hence (2.30) is equivalent to
+E(f,e−2φg)+E(f,e−2φgφ)−E(fφ,e−2φg)−Ec(φ,e−2φφfg)+1
+2Ec(φ2,e−2φfg)
+−2/integraldisplay
+E×E−de−2φ(y)g(y)f(x)[e(φ(y)−φ(x))−1−(φ(y)−φ(x))]J(dx,dy)
+=E(fe−φ,ge−φ). (2.34)
+In the following, we will establish (2.34) by showing that its left hand sid e and
+its right hand side have the same diffusion, jumping and killing parts. We assume
+without loss of generality that φ,f,g∈D.
+First, let us consider the diffusion parts of both sides of (2.34). Follo wing [21,
+(3.4)], we introduce a linear functional <L(w,v),·>forw,v∈Dby
+<L(w,v),f >:=ˇEc(w,vf) :=1
+2(Ec(w,vf)−ˆEc(w,vf)), f∈D,(2.35)
+whereˆEcis the left strong local part of the dual Dirichlet form ( ˆE,D(E)). Define
+Dloc:={w|for any relatively compact open set GofE,there
+exists a function v∈Dsuch thatw=vonG}.
+Then, the linear functional < L(w,v),·>can be extended and defined for any
+w,v∈Dloc(cf. [21, Definition 3.6]). Note that J1is assumed to be finite. Similar
+to [21, Theorem 3.8], we can prove the following lemma.
+Lemma 2.4. Letw1,...,w l,v∈Dlocandf∈D. Denotew:= (w1,...,w l). If
+ψ∈C2(Rl), thenψ(w)∈Dloc,ψxi(w)∈Dloc,1≤i≤l, and
+<L(ψ(w),v),f >=l/summationdisplay
+i=1<L(wi,v),ψxi(w)f >. (2.36)
+13By (2.35) and (2.36), we get
+ˇEc(f,e−2φg)+ˇEc(f,e−2φgφ)−ˇEc(fφ,e−2φg)
+−ˇEc(φ,e−2φφfg)+1
+2ˇEc(φ2,e−2φfg)
+=ˇEc(f,e−2φg)+ˇEc(f,e−2φgφ)−ˇEc(fφ,e−2φg)
+=ˇEc(f,e−2φg)−ˇEc(φ,e−2φfg)
+=ˇEc(f,e−2φg)+ˇEc(e−φ,e−φfg)
+=ˇEc(fe−φ,ge−φ). (2.37)
+By LeJan’s formula (cf. [17, Theorem 3.2.2 and Page 117], we can chec k that
+˜Ec(f,e−2φg)+˜Ec(f,e−2φgφ)−˜Ec(fφ,e−2φg)
+−˜Ec(φ,e−2φφfg)+1
+2˜Ec(φ2,e−2φfg)
+=1
+2/integraldisplay
+Ed˜µc
+<f,e−2φg>+1
+2/integraldisplay
+Ed˜µc
+<f,e−2φgφ>−1
+2/integraldisplay
+Ed˜µc
+<fφ,e−2φg>
+−1
+2/integraldisplay
+Ed˜µc
+<φ,e−2φφfg>+1
+4/integraldisplay
+Ed˜µc
+<φ2,e−2φfg>
+=1
+2/integraldisplay
+Ed˜µc
+<fe−φ,ge−φ>
+=˜Ec(fe−φ,ge−φ), (2.38)
+where˜Ecdenotes the strong local part of ( ˜E,D(E)) and ˜µcdenotes the local part
+of energy measure w.r.t. ( ˜E,D(E)). Then the diffusion parts of both sides of (2.34)
+are equal by (2.37) and (2.38).
+For the jumping parts of (2.34), we have
+Ej(f,e−2φg)+Ej(f,e−2φgφ)−Ej(fφ,e−2φg)−Ej(fe−φ,ge−φ)
+−2/integraldisplay
+E×E−de−2φ(y)g(y)f(x)[e(φ(y)−φ(x))−1−(φ(y)−φ(x))]J(dx,dy)
+= 2SPV/integraldisplay
+E×E−d{(f(y)−f(x))e−2φ(y)g(y)+(f(y)−f(x))φ(y)e−2φ(y)g(y)
+−(f(y)φ(y)−f(x)φ(x))e−2φ(y)g(y)−(f(y)e−φ(y)−f(x)e−φ(x))e−φ(y)g(y)
+−e−2φ(y)g(y)f(x)[e(φ(y)−φ(x))−1−(φ(y)−φ(x))]}J(dx,dy)
+= 0.
+For the killing parts of (2.34), we have
+Ek(f,e−2φg)+Ek(f,e−2φgφ)−Ek(fφ,e−2φg)−Ek(fe−φ,ge−φ)
+=/integraldisplay
+E(fe−2φg+fe−2φgφ−fφe−2φg−fe−2φg)dK
+= 0.
+14The proof is complete.
+Proof of (2.30) under the assumption of Theorem 1.2.
+LetGbe a relatively compact open subset of Usuch that the distance between the
+boundary of Gand that of Uis greater than some constant δ>0. Then, similar to
+[21, Theorem 4.8], we can show that ( E,C∞
+0(G)) has the following representation:
+E(w,v) =d/summationdisplay
+i,j=1/integraldisplay
+U∂w
+∂xi∂v
+∂xjdνG
+ij+d/summationdisplay
+i=1<FG
+i,∂w
+∂xiv >
++SPV/integraldisplay
+U×U−d2/parenleftBiggd/summationdisplay
+i=1(yi−xi)∂w
+∂yi(y)I{|x−y|≤δ
+2}(x,y)/parenrightBigg
+v(y)˜J(dx,dy)
++/integraldisplay
+U×U−d2/parenleftBigg
+w(y)−w(x)−d/summationdisplay
+i=1(yi−xi)∂w
+∂yi(y)I{|x−y|≤δ
+2}(x,y)/parenrightBigg
+v(y)J(dx,dy)
++/integraldisplay
+UwvdK, w,v ∈C∞
+0(G), (2.39)
+where{νG
+ij}1≤i,j≤daresignedRadonmeasureson Usuchthatforevery K⊂U,Kis
+compact,νG
+ij(K) =νG
+ji(K) and/summationtextd
+i,j=1ξiξjνG
+ij(K)≥0 for allξ= (ξ1,...,ξ d)∈Rd,
+{FG
+i}1≤i≤dare generalized functions on U.
+By (2.39), we can check that (2.30) holds for all φ,f,g∈C∞
+0(U). Therefore
+(2.30) holds for φ∈D(E)bandf,g∈D(E)n,bby (2.31), (2.32) and approximation.
+The proof is complete.
+2.3 Proof of Theorems 1.1 and 1.2 and some remarks
+Proof.By Theorem 2.1, for each n∈N, (¯Pu,n
+t)t≥0is a strongly continuous semi-
+group of bounded operators on L2(En;m) with/ba∇dbl¯Pu,n
+t/ba∇dbl2≤eβntfor everyt >0
+and some constant βn>0. Moreover, the coercive closed form associated with
+(e−βnt¯Pu,n
+t)t≥0is given by ( ¯Qu,n
+βn,D(E)n). Note that ( ¯Pu,n
+t)t≥0is also a strongly
+continuous semigroup of bounded operators on L2(En;e−2u∗m) and the bilinear
+from associated with ( ¯Pu,n
+t)t≥0onL2(En;e−2u∗m) is given by ( Eu,n,D(E)n,b) (see
+(2.18)).
+Define
+Pu,n
+tf(x) :=Ex[eNu
+tf(Xt);t<τEn].
+Then
+Pu,n
+tf(x) =Ex[eNu∗
+t−N|u|E
+tf(Xt);t<τEn]
+=Ex[eu∗(Xt)−u∗(X0)+M−u∗
+t−N|u|E
+tf(Xt);t<τEn]
+=e−u∗(x)¯Pu,n
+t(eu∗f)(x). (2.40)
+15Hence (Pu,n
+t)t≥0is a strongly continuous semigroup of bounded operators on
+L2(En;m). Let (Qu,n,D(E)n,b) be the restriction of QutoD(E)n,b. Then, by
+(2.40), (2.18) and Theorem 2.3, we know that the bilinear from assoc iated with
+(Pu,n
+t)t≥0onL2(En;m) is given by ( Qu,n,D(E)n,b). That is,
+Qu,n(f,g) = lim
+t→01
+t(f−Pu,n
+tf,g)m, f,g∈D(E)n,b. (2.41)
+(i) Suppose that there exists a constant α0≥0 such that
+Qu(f,f)≥ −α0(f,f)m,∀f∈D(E)b.
+Forn∈N, let (Ln,D(Ln)) be the generator of ( Pu,n
+t)t≥0onL2(En;m). Then
+D(Ln−α0) is dense in L2(En;m).
+Define
+¯Lnf(x) =eu∗(x)Ln(e−u∗f)(x), f∈D(¯Ln) :={eu∗g|g∈D(Ln)}.(2.42)
+Then, by (2.40), ( ¯Ln,D(¯Ln)) is the generator of ( ¯Pu,n
+t)t≥0onL2(En;e−2u∗m).
+(¯Ln,D(¯Ln)) is also the generator of ( ¯Pu,n
+t)t≥0onL2(En;m) due to the bound-
+edness ofu∗onEn. Since (e−βnt¯Pu,n
+t)t≥0is a strongly continuous contraction
+semigroup on L2(En;m), Range(λ−¯Ln) =L2(En;m) for allλ > β n. Hence
+Range(λ−(Ln−α0)) =L2(En;m) for allλ>βn−α0by (2.42).
+Letf∈L2(En;m). Then, for any α>0, we obtain by (2.41) that
+/ba∇dbl[α−(Ln−α0)]f/ba∇dbl2·/ba∇dblf/ba∇dbl2=/ba∇dbl[(α+α0)−Ln]f/ba∇dbl2·/ba∇dblf/ba∇dbl2
+≥([(α+α0)−Ln]f,f)m
+=Qu,n(f,f)+(α+α0)(f,f)m
+≥α(f,f)m.
+HenceLn−α0is dissipative on L2(En;m). Therefore ( e−α0tPu,n
+t)t≥0is a strongly
+continuous contraction semigroup on L2(En;m) by the Hille-Yosida theorem (cf.
+[14, Chapter 1, Theorem 2.6]).
+Letg∈L2(E;m) andt>0. Then
+/ba∇dblPu
+tg/ba∇dbl2≤ /ba∇dblPu
+t|g|/ba∇dbl2
+= lim
+l→∞/ba∇dblPu
+t|g·IEl|/ba∇dbl2
+≤liminf
+l→∞liminf
+n→∞/ba∇dblPu,n
+t|g·IEl|/ba∇dbl2
+≤eα0t/ba∇dblg/ba∇dbl2.
+Sinceg∈L2(E;m) is arbitrary, we get
+/ba∇dblPu
+t/ba∇dbl2≤eα0t,∀t>0.
+16(ii) Suppose that there exists a constant α0≥0 such that
+/ba∇dblPu
+t/ba∇dbl2≤eα0t,∀t>0. (2.43)
+Letn∈Nandf∈L2(En;m). Then
+/ba∇dblPu,n
+tf/ba∇dbl2≤ /ba∇dblPu,n
+t|f|/ba∇dbl2≤ /ba∇dblPu
+t|f|/ba∇dbl2≤eα0t/ba∇dblf/ba∇dbl2.
+Hence (e−α0tPu,n
+t)t≥0is a strongly continuous contraction semigroup on L2(En;m).
+By (2.41), we get
+Qu,n(f,f)+α0(f,f)m= lim
+t→01
+t(f−e−α0tPu,n
+tf,f)m≥0,∀f∈D(E)n,b.(2.44)
+By (2.44) and approximation, we find that
+Qu(f,f)≥ −α0(f,f)m,∀f∈D(E)b.
+Now we show that ( Pu
+t)t≥0is strongly continuous on L2(E;m). Letn∈N
+andf∈L2(En;m) satisfying f≥0. Then, we obtain by (2.43) and the strong
+continuity of ( Pu,n
+t)t≥0that
+limsup
+t→0/ba∇dblf−e−α0tPu
+tf/ba∇dbl2
+2
+= limsup
+t→0{2(f−e−α0tPu
+tf,f)m−[(f,f)m−/ba∇dble−α0tPu
+tf/ba∇dbl2
+2]}
+≤2limsup
+t→0(f−e−α0tPu
+tf,f)m
+≤2limsup
+t→0(f−e−α0tPu,n
+tf,f)m
+= 0.
+Sincefandnare arbitrary, ( Pu
+t)t≥0is strongly continuous on L2(E;m) by (2.43).
+The proof is complete.
+Remark 2.5. Letu∈D(E). Define
+Bu
+t=/summationdisplay
+s≤t/bracketleftbig
+e(u(Xs−)−u(Xs))−1−(u(Xs−)−u(Xs))/bracketrightbig
+. (2.45)
+Note that (Bu
+t)t≥0may not be locally integrable (cf. [4, Theorem 3.3]). To over come
+this difficulty, we introduced the nonnegative function u∗and the locally integrable
+increasing process (Bt)t≥0(see (2.2) and (2.3)). This technique has been used
+in [4] to show that if Xis symmetric and u∈D(E)e, then(Pu
+t)t≥0is strongly
+continuous if and only if (Qu,D(E)b)is lower semi-bounded. Here and henceforth
+D(E)edenotes the extended Dirichlet space of (E,D(E)).
+In fact, if we assume that (E,D(E))satisfies the strong sector condition instead
+of the weak sector condition (cf. [22, Pages 15 and 16] for the definitions), then
+similar to [4, Page 158] we can introduce a function |u|g
+Efor eachu∈D(E)e.
+17Defineu∗:=u+|u|g
+E. Using this defined u∗, similar to the above proof of this
+section, we can show that Theorems 1.1 and 1.2 hold for all u∈D(E)e.
+On the other hand, suppose we still assume that (E,D(E))satisfies the weak
+sector condition and u∈D(E)e. Define
+Fu
+t=/summationdisplay
+s≤t/bracketleftbig
+e(u(Xs−)−u(Xs))−1−(u(Xs−)−u(Xs))/bracketrightbig2.
+If(Fu
+t)t≥0is locally integrable on [0,ζ)for q.e.x∈E, then we can show that
+Theorems 1.1 and 1.2 still hold. The proof is similar to the ab ove proof of this
+section but we directly apply the (Bu
+t)t≥0defined in (2.45) instead of the (Bt)t≥0
+defined in (2.3). Note that if uis lower semi-bounded or eu∈D(E)e(cf. [4,
+Example 3.4 (iii)]), then (Fu
+t)t≥0is locally integrable on [0,ζ)for q.e.x∈E.
+Remark 2.6. If(E,D(E))is a symmetric Dirichlet form, then the assumption
+of Theorem 1.1 is automatically satisfied. Note that (Pu
+t)t≥0is symmetric on
+L2(E;m). If(Pu
+t)t≥0is strongly continuous, then (2.43) holds (cf. [10, Remark
+1.6(ii)]). Therefore, the following three assertions are e quivalent:
+(i)(Qu,D(E)b)is lower semi-bounded.
+(ii) There exists a constant α0≥0such that /ba∇dblPu
+t/ba∇dbl2≤eα0tfort>0.
+(iii)(Pu
+t)t≥0is strongly continuous on L2(E;m).
+Remark 2.7. Denote by Sthe set of all smooth measures on (E,B(E)). Let
+µ=µ1−µ2∈S−S,(A1
+t)t≥0and(A2
+t)t≥0be PCAFs with Revuz measures µ1and
+µ2, respectively. Define
+¯PA
+tf(x) =Ex[eA2
+t−A1
+tf(Xt)], f≥0 andt≥0,
+and
+/braceleftbiggEµ(f,g) :=E(f,g)+/integraltext
+Efgdµ,
+f,g∈D(Eµ) :={w∈D(E)|wis (µ1+µ2)−square integrable }.
+Then, by a localization argument similar to that used in the p roof of Theorems 1.1
+and 1.2 (cf. also [7]), we can show that the following two cond itions are equivalent:
+(i) There exists a constant α0≥0such that
+Eµ(f,f)≥ −α0(f,f)m,∀f∈D(Eµ).
+(ii) There exists a constant α0≥0such that
+/ba∇dbl¯PA
+t/ba∇dbl2≤eα0t,∀t>0.
+Furthermore, if one of these conditions holds, then the semi group(¯PA
+t)t≥0is
+strongly continuous on L2(E;m).
+This result generalizes the corresponding results of [1] an d [3]. Note that, similar
+to Theorems 1.1 and 1.2, it is not necessary to assume that the bilinear form
+(Eµ,D(Eµ))satisfies the sector condition.
+183 Examples
+Example 3.1. Inthisexample, we study thegeneralized Feynman-Kac semigroup
+for the non-symmetric Dirichlet form given in [22, II, 2 d)].
+Letd≥3,Ube an open set of Rdandm=dx, the Lebesgue measure on U. Let
+aij∈L1
+loc(U;dx), 1≤i,j≤d,bi,di∈Ld
+loc(U;dx),di−bi∈Ld(U;dx)∪L∞(U;dx),
+1≤i≤d,c∈Ld/2
+loc(U;dx). Define for f,g∈C∞
+0(U)
+E(f,g) =d/summationdisplay
+i,j=1/integraldisplay
+U∂f
+∂xi∂g
+∂xjaijdx+d/summationdisplay
+i=1/integraldisplay
+Uf∂g
+∂xididx
++d/summationdisplay
+i=1/integraldisplay
+U∂f
+∂xigbidx+/integraldisplay
+Ufgcdx.
+Denote ˜aij:=1
+2(aij+aji) and ˇaij:=1
+2(aij−aji), 1≤i,j≤d. Suppose that the
+following conditions hold
+(C1)There exists γ∈(0,∞) such that/summationtextd
+i,j=1˜aijξiξj≥γ/summationtextd
+i=1|ξi|2,∀ξ=
+(ξ1,...,ξ d)∈Rd.
+(C2)|ˇaij| ≤M∈(0,∞) for 1≤i,j≤d.
+(C3)cdx−/summationtextd
+i=1∂di
+∂xi≥0 andcdx−/summationtextd
+i=1∂bi
+∂xi≥0 (in the sense of Schwartz distri-
+butions, i.e.,/integraltext
+U(cf+/summationtextd
+i=1di∂f
+∂xi)dx,/integraltext
+U(cf+/summationtextd
+i=1bi∂f
+∂xi)dx≥0 for allf∈C∞
+0(U)
+withf≥0).
+Then (E,C∞
+0(U)) is closable and its closure ( E,D(E)) is a regular Dirichlet form
+onL2(U;dx) (see [22, II, Proposition 2.11]).
+Letu∈C∞
+0(U). Then, for f∈C∞
+0(U), we have
+Qu(f,f) =E(f,f)+E(u,f2)
+=d/summationdisplay
+i,j=1/integraldisplay
+U∂f
+∂xi∂f
+∂xjaijdx+/integraldisplay
+Uf2/parenleftBigg
+c(1+u)+d/summationdisplay
+i=1∂u
+∂xibi/parenrightBigg
+dx
++/integraldisplay
+Ud/summationdisplay
+i=1∂f2
+∂xi/parenleftBigg
+di+bi
+2+udi+d/summationdisplay
+j=1∂u
+∂xjaji/parenrightBigg
+dx.
+Suppose that the following condition holds
+(C4)There exists a constant α0≥0 such that
+/parenleftBigg
+α0+c(1+u)+d/summationdisplay
+i=1∂u
+∂xibi/parenrightBigg
+dx−d/summationdisplay
+i=1∂(di+bi
+2+udi+/summationtextd
+j=1∂u
+∂xjaji)
+∂xi≥0
+in the sense of Schwartz distribution.
+19ThenQu(f,f)≥ −α0(f,f) for anyf∈C∞
+0(U) and thus for any f∈D(E)bby
+approximation.
+LetXbe a Hunt process associated with ( E,D(E)) and (Pu
+t)t≥0be the gener-
+alized Feynman-Kac semigroup induced by u. Then, by Theorem 1.1 or Theorem
+1.2, (e−α0tPu
+t)t≥0is a strongly continuous contraction semigroup on L2(U;dx).
+Example 3.2. Inthisexample, we study thegeneralized Feynman-Kac semigroup
+for the non-symmetric Dirichlet form given in [22, II, 3 e)].
+LetEbe a locally convex topological real vector space which is a (topologic al)
+Souslin space. Let m:=µbe a finite positive measure on B(E) such that supp µ=
+E. LetE′denote the dual of EandE′/a\}b∇acketle{t,/a\}b∇acket∇i}htE:E′×E→Rthe corresponding
+dualization. Define
+FC∞
+b:={f(l1,...,lm)|m∈N,f∈C∞
+b(Rm),l1,...,lm∈E′}.
+Assume that there exists a separable real Hilbert space ( H,/a\}b∇acketle{t,/a\}b∇acket∇i}htH) densely and
+continuously embedded into E. Identifying Hwith its dual H′we have that
+E′⊂H⊂Edensely and continuously ,
+andE′/a\}b∇acketle{t,/a\}b∇acket∇i}htErestricted to E′×Hcoincides with /a\}b∇acketle{t,/a\}b∇acket∇i}htH. Forf∈ FC∞
+bandz∈E,
+define∇u(z)∈Hby
+/a\}b∇acketle{t∇u(z),h/a\}b∇acket∇i}htH=∂u
+∂h(z), h∈H.
+Let (Eµ,FC∞
+b), defined by
+Eµ(f,g) =/integraldisplay
+E/a\}b∇acketle{t∇f,∇g/a\}b∇acket∇i}htHdµ, f,g ∈ FC∞
+b,
+be closable on L2(E;µ) (cf. [22, II, Proposition 3.8 and Corollary 3.13]). Let
+L∞(H)denotetheset ofallboundedlinearoperatorson Hwithoperatornorm /ba∇dbl/ba∇dbl.
+Supposez→A(z),z∈E, is a map from EtoL∞(H) such that z→ /a\}b∇acketle{tA(z)h1,h2/a\}b∇acket∇i}htH
+isB(E)-measurable for all h1,h2∈H. Furthermore, suppose that the following
+conditions hold
+(C1)There exists γ∈(0,∞) such that /a\}b∇acketle{tA(z)h,h/a\}b∇acket∇i}htH≥γ/ba∇dblh/ba∇dbl2
+Hfor allh∈H.
+(C2)/ba∇dbl˜A/ba∇dbl∞∈L1(E;µ)and/ba∇dblˇA/ba∇dbl∞∈L∞(E;µ), where ˜A:=1
+2(A+ˆA),ˇA:=1
+2(A−ˆA)
+andˆA(z) denotes the adjoint of A(z),z∈E.
+(C3)Letc∈L∞(E,µ) andb,d∈L∞(E→H;µ) such that for u∈ FC∞
+bwith
+u≥0/integraldisplay
+E(/a\}b∇acketle{td,∇u/a\}b∇acket∇i}htH+cu)dµ≥0,/integraldisplay
+E(/a\}b∇acketle{tb,∇u/a\}b∇acket∇i}htH+cu)dµ≥0.
+Define forf,g∈ FC∞
+b
+E(f,g) =/integraldisplay
+E/a\}b∇acketle{tA∇f,∇g/a\}b∇acket∇i}htHdµ+/integraldisplay
+Ef/a\}b∇acketle{td,∇g/a\}b∇acket∇i}htHdµ
+20+/integraldisplay
+E/a\}b∇acketle{tb,∇f/a\}b∇acket∇i}htHgdµ+/integraldisplay
+Efgcdµ.
+Then (E,FC∞
+b) is closable and its closure ( E,D(E)) is a quasi-regular Dirichlet
+form onL2(E,µ) (see by [22, II, 3 e)].
+Letu∈ FC∞
+b. Then, for f∈ FC∞
+b, we have
+Qu(f,f) =E(f,f)+E(u,f2)
+=/integraldisplay
+E/a\}b∇acketle{tA∇f,∇f/a\}b∇acket∇i}htHdµ+/integraldisplay
+E(c(1+u)+/a\}b∇acketle{tb,∇u/a\}b∇acket∇i}htH)f2dx
++/integraldisplay
+E/angbracketleftbiggd+b
+2+ud+A∇u,∇f2/angbracketrightbigg
+Hdµ.
+Suppose that the following condition holds
+(C4)There exists a constant α0≥0 such that
+/integraldisplay
+E/braceleftbigg
+(α0+c(1+u)+/a\}b∇acketle{tb,∇u/a\}b∇acket∇i}htH)f+/angbracketleftbiggd+b
+2+ud+A∇u,∇f/angbracketrightbigg
+H/bracerightbigg
+dµ≥0
+for allf∈ FC∞
+bwithf≥0.
+ThenQu(f,f)≥ −α0(f,f) for anyf∈ FC∞
+band thus for any f∈D(E)bby
+approximation.
+LetXbe aµ-tight special standard diffusion process associated with ( E,D(E))
+and (Pu
+t)t≥0be the generalized Feynman-Kac semigroup induced by u. Then,
+by Theorem 1.1, ( e−α0tPu
+t)t≥0is a strongly continuous contraction semigroup on
+L2(E;µ).
+References
+[1] S. Albeverio and Z.M. Ma: Perturbation of Dirichlet forms-lower se mibound-
+edness, closablility, and form cores, J. Funct. Anal. 99 (1991) 332 -356.
+[2] S. Albeverio and Z.M. Ma: Additive functionals, nowhere Radon and Kato
+class smooth measures associated with Dirichlet forms, Osaka J. Ma th. 29
+(1992) 247-265.
+[3] C.Z. Chen: A note on perturbation of nonsymmetric Dirichlet form s by signed
+smooth measures, Acta Math. Scientia 27B (2007) 219-224.
+[4] C.Z. Chen, Z.M. Ma and W. Sun: On Girsanov and generalized Feynma n-
+Kac transfromations for symmetric Markov processes, Infin. Dim ens. Anal.
+Quantum Probab. Relat. Top. 10 (2007) 141-163.
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+groups: necessary and sufficient conditions, J. Funct. Anal. 237 ( 2006)446-465.
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+Canad. J. Math. 61 (2009) 534-547.
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+Proceedings of Functional Analysis IX, Dubrovnik, Croatia (2005) 39-43.
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+for symmetric Markov processes, Ann. Probab. 36 (2008) 931-9 70.
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+275.
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+forms, Nagoya Math. J. 136 (1994) 1-15.
+[12] Z.Q. Chen and R.M. Song: Conditional gauge therem for non-loca l Feynman-
+Kac transforms, Probab. Theory Relat. Fields 125 (2003) 45-72.
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+(2002) 475-505.
+[14] S.N. Ethier and T.G. Kurtz: Markov Processes Characterizatio n and Conver-
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+Markov Processes, Walter de Gruyrer, Berlin, 1994.
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+the generalized Schr¨ odinger semigroups, J. Funct. Anal. 125 (19 94) 358-378.
+[19] S.W. He, J.G. Wang and J.A. Yan: Semimartingale Theory and Stoch astic
+Calculus, Science Press, Beijing, 1992.
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+http://www.springerlink.com/content/h8427thw43263564/
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+Dirichlet Forms, Springer-Verlag, Berlin, 1992.
+[23] Y. Oshima: Lecture on Dirichlet Space, Univ. Erlangen-Nurnber g, 1988.
+[24] P.E. Protter: Stochastic Integration and Differential Equatio ns, Springer,
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+23
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+arXiv:1001.0204v1  [astro-ph.HE]  1 Jan 20102009 Fermi Symposium, Washington, D.C., Nov. 2-5 1
+Extensive near infrared monitoring of millimeter-wave / ga mma-ray bright blazars
+Alberto Carrami˜ nana, Luis Carrasco, Alicia Porras, Elsa Recillas
+INAOE, Luis Enrique Erro 1, Tonantzintla, Puebla 72840, M´ e xico
+We established a sample of millimeter-wave and γ-ray bright active galactic nuclei matching the
+WMAPcatalog with the EGRET catalog, highest energy photons and t heFermibright source list.
+We have monitored over 80 of these objects in the near infrare d, obtaining over 2000 JHKs data
+points directly comparable with Fermidata. We present examples of correlated near infrared and
+γ-ray activity of known blazars and recently identified sourc es.
+I. MM-WAVE γ-RAY BRIGHT BLAZARS
+CGRO-EGRET established the predominance of
+flat spectrum radio quasars (FSRQ) as extragalactic
+sources of high-energy γ-rays [1]. These are believed
+to be powered by supermassive black holes ejecting
+large amounts of material along relativistic jets per-
+pendicular to accretion disks and producing high en-
+ergy particles in these jets. Observations at the high-
+est photon energies shown a predominance of Bl Lac
+objects over FSRQ at E∼1 TeV. Although these
+observations may be biased by the horizon limita-
+tion due to pair absorption with extragalactic back-
+ground light (EBL), the Fermiγ-ray Space Telescope,
+more sensitive and more responsive to GeV photons
+than EGRET, reported a larger fraction of Bl Lac
+objects among blazars in its bright source list release
+(0FGL) [2, 3]. The wider spectral coverageof Fermiis
+allowing detailed studies of the intrinsic properties of
+various types of active galactic nuclei (AGN), cosmic
+evolution and their use as EBL tracers [4].
+The extrapolation of FSRQ to high radio frequen-
+cies makes the coincidence of foreground WMAP
+sources with γ-ray blazars expectable. Furthermore,
+it is clear than most of the 390 WMAP sources
+are AGNs [5]. These emitters are dominated in
+the mm-wave band by the same synchrotron com-
+ponent observed at lower frequencies and can be
+expected to emit photons of very high energy, re-
+lated to the hardest part of the synchrotron emission.
+WMAPsources have fluxes above 0.1 Jy at frequen-
+cies>∼40 GHz, making them suitable positional refer-
+ences for large millimeter telescopes. Known common
+WMAP/EGRET sources, found up to z>∼2.3, are
+relatively nearby analogs of more distant blazars de-
+tectable in principle by Fermiup toz>∼7 and by
+the Large Millimeter Telescope (LMT [6]) at z>∼30,
+if such an object could have existed at such an early
+phase of the Universe. The combination of Fermiand
+LMT data promises the availability of a sample of
+thousands of objects up to the highest redshifts ide-
+ally suited to study radio loud AGN evolution and the
+connection of the EBL with the star formation history
+of the Universe.II. MATCHING WMAP WITH γ-RAY
+SOURCES
+As a by product of the study of the cosmic
+microwave background, the Wilkinson Microwave
+Anisotropy Probe (WMAP) produced a catalog of 390
+bright sources detected at frequencies between 23 and
+94GHz, outsideamaskdefinedbythe mm-waveemis-
+sion from the Galactic plane [5]. These objects con-
+stitute a fairly homogeneous sample suitable for com-
+parison with all-sky γ-ray catalogs.
+We compared the WMAP foreground source cat-
+alog [5] with: (1) the Third EGRET (3EG) catalog
+of high-energy γ-ray sources [1]; (2) the E >10 GeV
+EGRET photons compiled by [7]; (3) the list of bright
+sources detected with signal-to-noise (√
+TS) larger
+than 10 by the Fermiγ-raySpace Telescope in its first
+three months of operations [2]. The positional uncer-
+tainty of WMAPequals 4′, similar to that of Fermi,
+but much better than that of EGRET. Preferring to
+accept false positives than to reject real coincidences,
+we used a relaxed matching criterion of 2.5 times
+the combined positional accuracy, ( σ2
+wmap+σ2
+γ)1/2,
+whereσγrefers to 3EG ( ∼1◦), EGRET-VHE (0 .5◦)
+or 0FGL (2 −10′). For each comparison we estimated
+the expected number of random coincidences quanti-
+fying the solid angle covered by the γ-ray sources (or
+events) outside the WMAPGalactic mask. For each
+WMAP/γpair we blindly listed potential radio, op-
+tical and X-ray counterparts from SIMBAD / Vizier.
+The results are listed in tables II and III.
+TABLE I: WMAPand EGRET source matches by type.
+(†= excludes P, G and S types [1]).
+Source 3EGWMAP Matches
+Typecatalog mask Observed Expected
+A 67 54 40 6.6
+a28 23 17 3.3
+U170 61 11 10.0
+Total 265†138 68 20
+eConf C0911222 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
+TABLE II: em WMAP sources in the 3EG catalog and 0FGL bright so urce list – Part I.
+Ddenotes the integer part of the distance in standard deviati ons.
+WMAP 3EG / 0FGL DCounterpart WMAP 3EG / 0FGL DCounterpart
+J0050–0649 0FGL J0051.1–0647 [0]PKS 0048–071 J0334–4007 0FGL J0334.1–4006 [0]PKS 0332–403
+J0051–0927 0FGL J0050.5–0928 [0]PKS 0048–097 J0339–0143 3EG J0340–0201 [0] CTA 026
+J0108+0135 3EG J0118+0248 [2]4C 01.02 J0407–3825 0FGL J0407.6–3829 [0]PKS 0405–385
+J0132–1653 3EG J0130–1758 [1]QSO B0130–171 J0416–2051 3EG J0412–1853 [1](QSO B0413–21)??
+J0137+4753 0FGL J0137.1+4751 [0]QSO B0133+47 J0423–0120 0FGL J0423.1–0112 [0]QSO B0420–015
+3EG J0422–0102 [0]
+J0204+1513 3EG J0204+1458 [0]4C +15.05 J0428–3757 0FGL J0428.7–3755 [0]PKS 0426–380
+J0205–1704 0FGL J0204.8–1704 [0]PKS 0202–17 J0442–0017 3EG J0442–0033 [0]QSO B0440–004
+J0210–5100 0FGL J0210.8–5100 [0]QSO B0208–5115 J0455–4617 3EG J0458–4635 [0] 0454–463
+3EG J0210–5055 [0]
+J0218+0138 0FGL J0217.8+0146 [1]PKS 0215+015 J0456–2322 0FGL J0457.1–2325 [0]QSO B0454–234
+3EG J0456–2338 [0]
+J0220+3558 0FGL J0220.9+3607 [0]B2 0218+35 J0501–0159 3EG J0500–0159 [0] 4C –02.19
+J0223+4303 0FGL J0222.6+4302 [1] 3C 66A J0506–6108 3EG J0512–6150 [1] 0506–612?
+3EG J0222+4253 [0]
+J0237+2848 0FGL J0238.4+2855 [0]4C +28.07 J0523–3627 3EG J0530–3626 [2]QSO J0522–3627
+3EG J0239+2815 [1]
+J0238+1637 0FGL J0238.6+1636 [0]AO 0235+164 J0538–4405 0FGL J0538.8–4403 [0]PKS 0537–441
+3EG J0237+1635 [1]QSO B0235+16 3EG J0540–4402 [0]
+J0319+4131 0FGL J0320.0+4131 [0]NGC 1275 J0539–2844 3EG J0531–2940 [2](QSO J0539–2839)??
+A. Matching WMAP with EGRET
+The comparison between the WMAPand 3EG cat-
+alogs produced 69 matches out of the 390 WMAP
+and 138 EGRET sources outside the WMAPGalactic
+mask. Thenumberexpectedrandomlyis20; theprob-
+ability of having 69 matches among the 390 WMAP
+sources is P<∼10−17, indicating that most -but not
+all- the matches are real.
+When accounting for the source type we count 40
+matches out of the 54 high confidence blazar associa-
+tion, labelled A in the 3EG catalog, compared to 6.6
+expected randomly. The low confidence ”a” associa-
+tions have 17 matches out of 23 tries and 3.3 expected
+chance coincidences. On the other hand we have 11
+matches among the 61 unidentified sources out of the
+WMAPmask, expecting 10.0 by chance. On statisti-
+cal grounds, we can confirm the physical association
+of foreground WMAPsources with 3EG blazars, ac-
+counted by both ”A” and ”a” classes, but not with
+unidentified EGRET sources (table I).
+We also compared the WMAP positions with the
+list of very high energy (VHE) photons, E >
+10GeV [7]: 510 out of the 1506 VHE photons are
+outside the WMAPGalactic mask. The combined po-
+sitional accuracyofVHE photonsand WMAPsources
+is 30.2′. We obtained coincidences between 33 VHE
+photons and 29 WMAPsources as follows:– 20WMAP sources coincide with a single isolated
+VHE photon;
+– 4WMAPsources coincide with a single VHE pho-
+ton and a 3EG, with no 0FGL counterpart;
+– 1WMAP source (WMAP J1408–0749) coincides
+with a 3EG source (3EG J1409–0745) and 4 VHE
+photons (VHE 494, 498, 1058 and 1061) – but with
+no 0FGL counterpart:
+– 3WMAPsources coincide with VHE photons, 3EG
+and 0FGL sources; of these WMAP J0210-5100 has
+two VHE photons;
+– 1WMAP source (WMAP J0137+4753) coin-
+cides with a VHE photon and a 0FGLsource
+(0FGL 0137.1+4751), with no 3EG counterpart.
+We note that given the number and error box sizes
+of the VHE photons, we expect 30 random matches
+under our 2.5 σcriterion. Most or all of the single
+WMAP-VHE coincidences are likely to be spurious.
+We note the case of WMAP J0137+4753, which be-
+longed to the WMAP& VHE only category prior to
+the publication of the 0FGL and turned out to be a
+realγ-ray source.
+eConf C0911222009 Fermi Symposium, Washington, D.C., Nov. 2-5 3
+B. Matching the WMAP catalog with the Fermi
+bright source list
+TheFermibright source list (0FGL), made public
+in February 2009, consists of 205 bright γ-ray sources
+detected with significance√
+TS >10 in the first three
+months of observations [2]. Of these, 121 are of the
+AGN class, mostly blazars [3]. The 0FGL list has
+122 objects outside the WMAPGalactic mask, which
+we compared with the respective WMAPandFermi
+positions. The improvedpositionalaccuracyof Fermi,
+in the 4′−10′range, results in only 0.82 spurious
+coincidences expected. We found 54 matches between
+WMAPand the 0FGL, 25 of which are common with
+EGRET sources and 29 are independent.
+C. Sample overview
+The sample is presented in tables II and III. The
+Dcolumn expresses the distance between the WMAP
+and the γ-ray event in terms of the integer part of
+the combined positional accuracy; associations with
+[0] have intersecting error countours and are more
+likely to be real than those with [2], separated by
+≥2σ. Most of the sources have a suitable radio, opti-
+caland/orX-raycounterpart, often in the intersection
+of theWMAP/γ-ray error boxes. In a few cases the
+candidate counterpart is not unique and the one dis-
+playedis a subjective election. Counterpartsin paren-
+thesis are tentative. We note the following:
+•WMAP J0051–0927 :: 0FGL J0050.5–0928 has
+a preferred association with the Bl Lac object
+PKS 0048–071 = PHL 856; the radiogalaxy
+FIRST J005051.9–092529 is an alternative.
+•WMAP J0237+2848 :: 0FGL J0238.4+2855 ::
+3EG J0239+2815. This is listed as a low confi-
+denceFermiassociation with 4C +28.07 in [2].
+TheWMAPandFermiboxes match, both con-
+taining 4C +28.07. All are somewhat outside
+the EGRET box.
+•WMAP J0319+4131 :: 0FGL J0320.0+4131 is
+identified with the radiogalaxy NGC 1275. We
+note that the radiogalaxy Cen A is excluded of
+this study by the WMAPGalactic mask.
+•WMAP J0423–0120 :: 0FGL J0423.1–0112 ::
+3EGJ0422–0102. Thisisalowconfidence Fermi
+association with PKS 0420–014, which is at the
+center of the WMAPcircle.
+•WMAP J0909+0119 :: 0FGL J0909.7+0145.
+The 0FGL error radius is 17′, theWMAP
+and 0FGL positions differing by 25′. PKS
+0907+022 is a low confidence Fermiassociation
+not compatible with WMAP. 4C+01.24(PKS =QSO B0906+015) is inside the WMAPbox and
+marginally compatible with the Fermisource.
+•WMAP J1517–2421 has two possible EGRET
+counterparts but only one (3EG J1517–2538) is
+compatible with the 0FGL source.
+•WMAP J1642+3948 is 19′away from
+0FGL J1641.4+3939, which has an posi-
+tional error of 9 .5′, making the association
+tentative. There are suitable candidates for
+both sources, with 3C345 being positionally
+the best common counterpart, as discussed in
+§VC. While there is no 3EG source associa-
+tion, we note the revised EGRET counterpart
+EGR J1642+3940 [8].
+III. NEAR INFRARED MONITORING
+On August 2007, we started a dedicated monitoring
+program with up to 60 nights per semester awarded
+on the 2.1m telescope of the Observatorio Astrof´ ısico
+GuillermoHaro(OAGH),inCananea,Sonora,Mexico
+(lat=+31 .05, long= −110.38). The program consists
+of optical photometry (BRVI), low resolution spec-
+troscopy (4000 - 7500 ˚A) and near infrared JHKs
+imagingofthe sampleabove,with the additionofhigh
+priority GLAST/Fermi sources and targets notified
+via the multi-wavelength Fermigroup. The current
+study is to lead to programs of follow-up and identi-
+fication of γ-ray sources using near infrared, optical
+and mm-wave facilities.
+Particularly successful has been the JHKs photo-
+metricsurveyusingthe CAnanea NearInfraredCAm-
+era (CANICA). CANICA is equiped with a Rockwell
+1024×1024 pixel Hawaii infrared detector working at
+75.4 K with standard near infrared filters. The scale
+plate is 0.32′′/pixel. Observations are usually carried
+out in series ofdithered frames in eachfilter. Datasets
+are coadded after correcting for bias and flat-fielding
+using IRAF based macros. Figure 1 shows the pho-
+tometric magnitudes measured with CANICA, with
+detection thresholds around magnitudes 19, 18 and
+17 for J, H and Ks respectively, i.e. about two mag-
+nitudes fainter than 2MASS.
+IV. CORRELATED INFRARED/ γ-RAY
+ACTIVITY
+We have found correlated infrared and γ-ray cor-
+relations with little or no evidence for time delays in
+our sample. We show here joint CANICA H band
+(1.6µm⇒0.76eV) and Fermi(1−300GeV) light
+curves, normalizing the maximum γ-ray flux to the
+maximum H-band flux and setting both zeros at the
+same level. We are currently studying the infrared
+eConf C0911224 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
+FIG. 1: CANICA photometric measurements of mm-
+wave/γ-ray sources. We show only detections.
+color behavior of our sample during flares and con-
+straining possible delays in the timing of near infrared
+flares relative to the γ-ray ones. Our current results
+on 3C 454.3 are presented in [9].
+A. PKS 0235+164
+PKS 0235+164 is positionally coincident with
+WMAPJ0238+1637,mm-wavesourcecoincidentwith
+3EG J0237+1635 = 0FGL J0238.6+1636. The
+WMAPsource is labelled as probable variable. Fig-
+ure 2 shows the joint H band and 1-300 GeV fluxes
+scaled. The near infrared flux at the end of 2008
+and beginning of 2009 matched the 2MASS values.
+On JD2454715 we found PKS 0235+164 almost three
+magnitudes brighter, prior to peaking 25 days later -
+in coincidence with the Fermiflare.
+B. PKS 0454–234
+TheWMAPandFermierrorcircles in this regionof
+the sky intersect closeto the center ofthe 3EG J0456–
+2338 error box. PKS 0454–234 is within the intersec-
+tion, in a neat positional match. We started monitor-
+ing on JD 2454748, when the near infrared flux was
+0.85 magnitudes above the 2MASS reference value.
+The source flared by a factor of nearly 3 in the next
+50 days to drop to a half 18 days later. The near in-
+frared peak seems delayed by about a week relative
+to the 1-300 GeV maximum. The AGN has been inFIG. 2: Joint CANICA - Fermilight curve of PKS
+0235+164. CANICA is given by the (black) points while
+Fermifluxesform the(red)histogram witherror bars. The
+discontinuous horizontal line marks the 2MASS flux. The
+dotted vertical lines indicate the change of year.
+relative quiescence since (fig. 3).
+C. PKS 1510–089
+PKS 1510–89 is the high confidence counter-
+part of 3EG J1512–0849, WMAP J1512–0904 and
+0FGL J1512.7–0905, with an excellent positional
+match between all data. We started monitoring PKS
+1510–89inearly2008,afewmonthspriortothe Fermi
+launch, when the flux fluctuated around the 2MASS
+reference value. We caught a simultaneous infrared -
+γ-ray flare on JD 2454924 and the subsequent decay
+a month later.
+V. SOURCE IDENTIFICATIONS
+A. The identification of QSO B0133+476
+QSO B0133+476 is a rather active quasar first
+catalogued originally as DA55 in the Dominion DA
+1420 MHz survey [10]. Also known as Mis V1436,
+archival optical data exists since at least 1953, when
+it was around magnitude R=18.7. This object has
+been monitored by other groups since 2007, when it
+was found to be 4.5 magnitudes brighter. Its optical
+variability is well documented in [11].
+This object has no EGRET counterpart, the only
+evidence for γ-ray emission prior to the Fermilaunch
+eConf C0911222009 Fermi Symposium, Washington, D.C., Nov. 2-5 5
+FIG. 3: Joint CANICA - Fermilight curve of PKS 0454–
+234. The discontinuous horizontal line marks the 2MASS
+flux. The dotted vertical lines indicate the change of year.
+FIG. 4: CANICA - Fermilight curve of PKS 1510–089.
+The discontinuous horizontal line marks the 2MASS flux.
+The dotted vertical lines indicate the change of year.
+being its marginal closeness to a 85( ±38)GeV photon.
+The unprovable association of this single photon with
+QSO B0133+476 is of interest as its energy is close to
+the pair absorption EBL limit for the redshift of the
+object,z= 0.859. If true, this association would in-
+dicate the potential of this source for testing the EBL
+horizon using Fermi, specially under phases of highFIG. 5: Positional circumstance of QSO B0133+476. Left:
+the VHE photon error circle is shown at α∼23◦,δ∼
+48.6◦, withthealmost concentric WMAPandFermicircles
+in the middle. Right:a zoom panel showing the 4′WMAP
+circle contained in the 7′Fermiposition. QSOB0133+476
+is indicated by the open hexagon and an asterix.
+FIG. 6: CANICA light curve of QSO B0133+47. The
+flaring at the end of 2008 is clearly visible, at a factor of
+10 above the 2MASS flux levels indicated by the dashed
+horizontal line.
+emission activity. Our report of near infrared flaring
+fromdatatakenbetweenJD2454788and2454795[12]
+was followed by the Fermidetection report, at a flux
+level above the EGRET limiting sensitivity [13]. The
+infrared light curve is shown in figure 6. The ob-
+ject was caught undergoing a rapid flare which has
+declined relatively slowly during 2009, but always at
+levels above 2MASS.
+eConf C0911226 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
+FIG. 7: CANICA - FermiQSO J0808–0751 light curve.
+The dots indicate the H band fluxes in mJy, while the
+histogram and upper limit arrows indicate the 1-300 GeV
+fluxes and limits in scaled units. The discontinuous hor-
+izontal line marks the 2MASS flux. The dotted vertical
+lines indicate the change of year.
+B. QSO J0808–0751
+The EGRETsource3EGJ0812–0646wasnot in the
+high priority GLAST/Fermi list, with the association
+with QSO J0808–0751 been only tentative ( D= 2).
+WMAP J0808–0750 is somewhat displaced from the
+3EG location, but contains this z= 1.84 QSO. The
+rapid flare observed simultaneously in the near in-
+frared and by Fermiof this source, not in the 0FGL,
+confirms the identity of the QSO as the γ-ray source.
+We missed a second flare occurring while the source
+was on the daylight. The flux increase in the near
+infrared reached a factor of ten within 50 days.
+C. The identification of 0FGL J1641.4+3939
+with 3C 345
+The angular distance between WMAP J1642+3948
+and 0FGL J1641.4+3939 is 19 .2′= 1.9σtimes the
+combined positional uncertainty, the association
+between both objects being tentative only. This is
+a rather populated region of the sky, where several
+potential counterparts can be found for each object:
+- candidate counertparts of 0FGL J1641.4+3939
+include 3C 345, QSO B1641.5+3956, QSOB1641.6+3949, QSO B1640+398, QSO
+B1641.8+3956, QSO B1641+3958, QSO
+B1640.5+3944, QSO B1640+396, three more QSOs,
+FIG. 8: Left:position of 0FGL J1641.4+3939 (larger
+circle) and WMAP J1642+3948 (smaller circle) with a
+few potential counterparts (AGN class - open hexagons).
+Right:CANICA H band light curve of 3C345 compared
+with the (1-300 GeV) fluxes from 0FGL J1641.4+3939
+byFermi. The discontinuous horizontal lines mark the
+2MASS flux and error. The dotted vertical line indicates
+the change of year.
+a Seyfert 1 and a radiogalaxy from SDSS.
+- potential counterparts for WMAP J1642+3948
+are GB6 J1642+3948 = 3C 345, FIRST
+J164304.3+394836, QSO B1641+3949, ...
+3C345hasalower χ2relativetothecombined WMAP
+andFermipositions (fig. 8), being the best candidate
+under the assumption of a common association. Even
+though more data are desirable, the simultaneous flux
+changes measured provide evidence for the physical
+association between 0FGL 1641.4+3939 and 3C345.
+VI. SUMMARY
+Wehaveselectedasampleofmm-wave/ γ-raybright
+blazarsfromthe WMAP, 3EGcatalogsand 0FGLlist.
+We have monitored these in the near infrared finding
+correlatedvariability, which has also allowed the iden-
+tification or confirmation of some of these objects.
+Acknowledgments
+We acknowledge the use of 2MASS, WMAP, Fermi,
+SIMBAD, Vizier and POSS databases. We appreciate
+the support of the technical staff at the Observatorio
+Astrof´ ısico Guillermo Haro.
+eConf C0911222009 Fermi Symposium, Washington, D.C., Nov. 2-5 7
+[1] Hartman, R.C., et al. 1999, ApJS 123, 79.
+[2] Abdo, A.A., et al. 2009a, ApJS 183, 46.
+[3] Abdo, A.A., et al. 2009b, ApJ 700, 597.
+[4] Dermer, C.D. 2007, AIP Conf. Proc. 921, 122.
+[5] Wright, E.L., et al. 2009, ApJS 180, 283.
+[6] Serrano P´ erez-Grovas, A., Schloerb, F.P., Hughes, D.,
+Yun, M. 2006, SPIE 6267, 1.
+[7] Thompson, D.J., Bertsch, D.L., O’Neal Jr., R.H.
+2005, ApJS 157, 324.
+[8] Casandjian, JM, Grenier, IA. 2008, A&A 489, 849.[9] Carrami˜ nana, A., Carrasco, L., Chavushyan, V. et al.
+2009, Fermi Symposium eConf proceedings.
+[10] Galt, J.A., Kennedy, J.E.D. 1968, AJ 73, 135.
+[11] http://www.aerith.net/misao/variable/MisV1436.h tml
+[12] Carrami˜ nana, A., Carrasco, L., Recillas, E.,
+Chavushyan, V. 2008, ATEL 1874.
+[13] Takahashi, H., Tosti, G. on behalf of the Fermi-LAT
+collaboration 2008, ATEL 1877.
+eConf C0911228 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
+TABLE III: WMAP sources in the 3EG catalog and 0FGL bright sou rce list.
+WMAP 3EG / 0FGL DCounterpart WMAP 3EG / 0FGL DCounterpart
+J0720–6222 3EG J0702–6212 [2] ... J1510–0546 0FGL J1511.2–0536 [0]PKS 1508–05
+J0721+7122 0FGL J0722.0+7120 [0]PKS 0716+714 J1512–0904 0FGL J1512.7–0905 [0]QSO B1510–089
+3EG J0721+7120 [0] 3EG J1512-0849 [0]
+J0738+1743 0FGL J0738.2+1738 [0]QSO B0735+178 J1517–2421 0FGL J1517.9–2423 [0]QSO B1514–241
+3EG J0737+1721 [0] 3EG J1517–2538a[1]
+J0808–0750 3EG J0812–0646 [2]QSO J0808–0751 J1608+1027 3EG J1608+1055 [0]4C +10.45
+J0813+4817 3EG J0808+4844 [1](4C+48.22) J1613+3412 3EG J1614+3424 [1]QSO B1611+343
+J0825+0311 3EG J0828+0508 [2]QSO B0823+033 J1642+3948 0FGL J1641.4+3939 [1](4C+39.48)?
+J0831+2411 3EG J0829+2413 [0]QSO J0830+2411 J1654+3939 0FGL J1653.9+3946 [1]Mrk 501
+J0841+7053 3EG J0845+7049 [0]0836+710 J1703–6214 3EG J1659–6251 [1]
+J0854+2006 0FGL J0855.4+2009 [0] OJ 287 J1736–7934 3EG J1720–7820 [2]PKS 1725–795
+3EG J0853+1941 [0]
+J0909+0119 0FGL J0909.7+0145 [1]PKS 0907+022 J1740+5212 3EG J1738+5203 [0]QSO B1739+522
+J0920+4441 0FGL J0921.2+4437 [0]RGB J0920+446 J1800+7827 0FGL J1802.2+7827 [0]S5 1803+78
+3EG J0917+4427 [1]
+J0957+5527 0FGL J0957.6+5522 [0]4C +55.17 J1820–6343 3EG J1813–6419 [1]
+3EG J0952+5501 [1]
+J0959+6530 3EG J0958+6533 [0]QSO B0954+65 J1848+3223 0FGL J1847.8+3223 [0]TXS 1846+322?
+J1058+0134 0FGL J1057.8+0138 [0]PKS 1055+018 J1849+6705 0FGL J1849.4+6706 [0]S4 1849+67
+J1059–8003 0FGL J1100.2–8000 [0]PKS 1057–79 J1923–2105 0FGL J1923.3–2101 [0]PMN J1923–2104
+3EG J1921–2015 [1]
+J1130–1451 0FGL J1129.8–1443 [0]PKS 1127–14 J1937–3957 3EG J1935–4022 [2]
+J1147–3811 0FGL J1146.7–3808 [0]PKS 1144–379 J1939–1525 3EG J1937–1529 [0]QSO B1939–155
+3EG J1134–1530 [2]
+J1159+2915 0FGL J1159.2+2912 [0]4C +29.45 J2011–1547 3EG J2020–1545 [2]QSO B2008–159
+3EG J1200+2847 [0]
+J1223–8306 3EG J1249-8330 [1] J2035+1055 3EG J2036+1132 [1]QSO B2032+107
+J1229+0203 0FGL J1229.1+0202 [0] 3C 273 J2056–4716 0FGL J2056.1–4715 [0]PMN J2056–4714
+3EG J1229+0210 [0] 3EG J2055–4716 [0]
+J1246–2547 0FGL J1246.6–2544 [0]PKS 1244–255 J2143+1741 0FGL J2143.2+1741 [0] OX 169
+J1256–0547 0FGL J1256.1–0547 [0] 3C 279 J2151–3027 3EG J2158–3023 [2]PKS 2155–304
+3EG J1255–0549 [1]
+J1310+3222 0FGL J1310.6+3220 [0]B2 1308+32 J2202+4217 0FGL J2202.4+4217 [0] Bl Lac
+3EG J2202+4217 [0]
+J1316–3337 3EG J1314–3431 [1]QSO B1313–333 J2203+1723 0FGL J2203.2+1731 [0]PKS 2201+171
+J1327+2213 (3EG J1323+2200)? [2]QSO B1324+224 J2207–5348 0FGL J2207.0–5347 [0]PKS 2204–54
+J1337–1257 3EG J1339–1419 [1]QSO B1335–127 J2211+2352 3EG J2209+2401 [0]QSO B2209+236
+J1354–1041 0FGL J1355.0–1044 [0]PKS 1352–104 J2229–0833 0FGL J2229.8–0829 [0]QSO B2227–088
+J1408–0749 3EG J1409–0745 [0]QSO B1406–074 J2232+1144 0FGL J2232.4+1141 [0]CTA 102
+3EG J2232+1147 [0]
+J1419+3822 3EG J1424+3734 [1]QSO B1417+385 J2254+1608 0FGL J2254.0+1609 [0]3C 454.3
+3EG J2254+1601 [0]
+J1427–4206 3EG J1429-4217 [0]QSO B1424–418 J2322+4448 3EG J2314+4426 [2]GB6 B2319+4429
+J1457–3536 0FGL J1457.6–3538 [0]PKS 1454–354 J2327+0937 0FGL J2327.3+0947 [0]GB6 B2325+0923
+3EG J1500–3509 [0]
+J1504+1030 0FGL J1504.4+1030 [0]PKS 1502+106 J2349+3846 3EG J2352+3752 [1]QSO B2346+385
+J1506–1644 3EG J1504–1537 [1](QSO B1504–1626) J2354+4550 3EG J2358+4604 [1]4C 45.51
+eConf C091122
\ No newline at end of file
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+arXiv:1001.0205v1  [cond-mat.str-el]  1 Jan 2010Dipole trap model for the metallic state in gated silicon-in version layers
+T. Hörmann
+Institute for Semiconductor Physics, Johannes Kepler Univ ersity, 4040 Linz, Austria; and
+Christian Doppler Labor for Surface Optics, Johannes Keple r University, 4040 Linz, Austria
+G. Brunthaler∗
+Institute for Semiconductor Physics, Johannes Kepler Univ ersity, 4040 Linz, Austria
+In order to investigate the metallic state in high-mobility Si-MOS structures, we have further
+developed and precised the dipole trap model which was origi nally proposed by B. L. Altshuler and
+D.L. Maslov [Phys. Rev. Lett. 82, 145 (1999)]. Our additiona l numerical treatment enables us to
+drop several approximations and to introduce a limited spat ial depth of the trap states inside the
+oxide as well as to include a distribution of trap energies. I t turns out that a pronounced metallic
+state can be caused by such trap states at appropriate energi es whose behavior is in good agreement
+with experimental observations.
+PACS numbers: 71.30.+h; 73.40.Qv; 72.10.Fk
+Keywords: Metal-insulator transition; Si-MOS structures ; Scattering model
+I. INTRODUCTION
+The discovery of the metal-insulator transition (MIT)
+in two-dimensional (2D) electron systems in 19941,2has
+attracted large attention, as it was in apparent con-
+tradiction to the scaling theory of localization3,4which
+states that in the limit of zero temperature, a metal-
+lic state should exist only in three dimensional systems,
+whereas in two dimensions disorder should always be
+strong enough to induce an insulating state. The MIT
+in high-mobility n-type silicon inversion layers shows a
+strong decrease of resistivity ρtowards low temperature
+Tfor high electron densities, manifesting the metallic re-
+gion, whereas a strong exponential increase of ρdemon-
+strated the insulting regime for low densities. A similar
+but weaker behavior was observed in many other semi-
+conductor systems at low densities and low temperatures
+(e. g.p-GaAs,5n-GaAs,6SiGe,7AlAs8)
+Several models were suggested in order to explain
+the unexpected finding of metallic behavior in 2D.
+The most important ones are i) temperature-dependent
+screening,9–14ii) quantum corrections in the diffusive
+regime,15–18and iii) quantum corrections in the ballistic
+regime.19,20Numerous argumentations for the different
+models are given in literature,21–25but a clear decision
+for one of them could not been drawn yet.
+As an alternative, Altshuler and Maslov (AM) intro-
+duced the dipol scattering scenario for Si-MOS struc-
+tures in which charged trap states in the oxide layer form
+dipols together with the image charge of the screening
+2D electrons.26The interplay between the gate voltage
+dependent energetic position of the trap states and the
+height of the chemical potential may lead as well to a
+metal-insulator transition in that system. It should not
+be assumed that the dipol scattering effect is active alone,
+as the temperature dependence of screening and quantum
+corrections will surely contribute at low temperatures,
+but the charging of trap states might be the generator
+of the particularly large effect in Si-MOS structures. Itis known that the misfit at the silicon/silicon-oxide in-
+terface produces charged defect states in the thermally
+grown oxide layer.27–29Arguments on the importance of
+trap states in Si-MOS structures were also given by Klap-
+wijk and Das Sarma.30
+AM could show within their analytical calculations
+that a trap level at energy ETwhich is either filled or
+empty, depending on its position relative to the Fermi
+energyEF, can lead to a critical behavior in electron
+scattering if ETandEFare degenerate. This dipole trap
+model is able to explain the main properties of the metal-
+insulator transition in gated Si-MOS structures.
+For the analytical calculations AM made a number of
+assumptions. These are:
+a1) the trap states possess a δ-like distribution in en-
+ergy (i. e. have all the same energy),
+a2) the spatial density distribution in the oxide is ho-
+mogeneous,
+a3) the states occupied with electrons behave neutral
+and cause no scattering of 2D electrons whereas the un-
+occupied states are positively charged and lead to scat-
+tering (AM work in the hole trap picture, but we describe
+occupation in terms of electrons),
+a4) a charged trap state is screened by the 2D electrons
+so that the resulting electrostatic potential can be de-
+scribed by the trap charge and an apparent mirror charge
+with opposite sign on the other side of the interface,
+a5) the scattering efficiency of the 2D electrons is de-
+scribed by a dipole field of the trap charge and its mirror
+charge,
+a6) a parabolic saddle-point approximation for the ef-
+fective potential between the Si/SiO2 interface and the
+metallic gate was used in order to perform analytical cal-
+culations,
+a7) the energy of the trap state ETis fixed relative to
+the quantization energy E0of the 2D ground state inside
+the inversion potential, and
+a8) the chemical potential µin the 2D layer has (A)
+either the same temperature dependence as in the bulk2
+substrate or (B) as in a 2D electron system with constant
+electron density.
+In this work, we precise and develop the AM trap
+model further in order to better understand the influ-
+ence of charged traps on the metallic state in Si-MOS
+structures. We present detailed numerical calculations of
+the temperature and density dependent resistivity ρdue
+to electronic scattering in the dipole trap model. Due to
+the numerical treatment we were able to drop the approx-
+imations a6), a7), and a8) of the analytic AM model. In
+addition, we have further extended our calculations for
+the more realistic case with energetic broadening and spa-
+tial distribution profile of the defect states, i. e. droppin g
+also approximations a1) and a2).
+As a result of our calculations, we find good agreement
+with the calculations of AM. There are mainly deviations
+in the overall behavior of the resistivity at low electron
+densities and at high temperatures. In order to under-
+stand the approximations of AM, we have also recalcu-
+lated the analytical model and were able to formulate it
+in a simplified way. Several mathematical terms are rear-
+ranged so that the scattering efficiency is expressed in the
+same form as in the usual Drude formulation. In addi-
+tion, their result contains two integrals, which we could
+replace by Fermi-Dirac integrals. Thus the known ap-
+proximations for the Fermi-Dirac integrals lead to simple
+equations for the resistivity ρat low temperatures.
+The paper is organized as follows. In Sec. II the an-
+alytical formulation of the dipole trap model is given in
+detail and we show that within the saddle-point approxi-
+mation the result can be written in terms of Fermi-Dirac
+integrals. Section III treats the chemical potential and
+Sec. IV the analytic approximations for low tempera-
+tures. From the numerical integration, first results are
+given in Sec. V whereas in Sec. VI the calculations are
+extended for the case that the conduction band is the ref-
+erence energy for the trap energy and not the electronic
+ground state E0. In Sec. VII the model and the calcula-
+tion are extended for a spatial density distribution of the
+trap states and in Sec. VIII energetic broadening of trap
+states is taken into account. Conclusions are drawn in
+Sec. IX. In two appendices the behavior of the chemical
+potential and the ground state energy of the inversion
+layer are described in detail. Please note that we use SI
+units throughout this work.
+II. TRAPMODEL
+In the AM model it is assumed that a large number
+of hole trap states exists in the oxide at a certain trap
+energyET. If the trap energy lies above the chemical
+potential µ, the trap is empty (in the electron picture,
+or has captured a hole in the equivalent description) and
+is positively charged, whereas if ETlies below µit is
+filled with an electron and thus is neutral, see Fig. 1.
+Please note that we use in this work the terminology
+µ(T)for the chemical potential, the Fermi energy EFFigure 1. Schematic representation of the trap states toget her
+with the 2D electron system in the Si inversion layer. For
+ET> µ(T)the trap state is positively charged and scatters
+electrons in the 2D layer whereas for ET< µ(T)the trap is
+neutral.
+denotesµ(T= 0).
+A potential gradient due to an applied gate voltage
+Vgcauses a decrease of the trap energy ET=ETs−
+eVinsZ/D, with the unscreened trap energy ETsat the
+oxide semiconductor (OS) interface, the voltage drop
+across the oxide (insulator) Vins, the distance from the
+OS interface Z, and the thickness of the oxide D. In
+their corresponding equation AM use the symbol Vgin-
+stead of Vins. . But the total gate voltage Vgis equal
+to the voltage drop across the oxide Vinsplus the voltage
+drop across the depletion layer Vdepl. Later in their paper
+AM write that the threshold voltage is incorporated into
+Vg, nevertheless they use practically an equation equiva-
+lent to
+Vins=ensD/ǫinsǫ0, (1)
+whereǫinsis the dielectric constant of the oxide and ǫ0
+is the electric field constant. But if the threshold volt-
+age is incorporated into Vg, the latter cannot be used to
+calculate the slope of the electrical potential within the
+oxide, as a part of Vgfalls off between the OS interface
+and the bulk layer. As the charges within the depletion
+layer (2D charge density −endepl) also contribute to the
+gradient of the potential, we use the equation
+Vins=e(ns+ndepl)D/ǫinsǫ0 (2)
+together with
+Vg=Vdepl+Vins. (3)
+According to AM another term has to be added to the
+trap energy ETwhich describes the interaction with the
+two dimensional electron gas (2DEG) because of the im-
+age force. In Ref. 26 this term is given as −e2/8πǫinsǫ0Z
+(translated to the SI unit system). We think there
+should be a factor 16in the denominator, see for instance
+Ref. 27 or Ref. 28. This factor does not change the re-
+sults of the trap model qualitatively therefore we write
+−e2/Cπǫinsǫ0Z.
+AM define an energy
+εD=e2
+Cπǫinsǫ0D(4)3
+ZO
+zS
+EC
+ECsµE0ETs
+Zm−eVinsZ/D
+εm
+−εDD/Zεe(Z) ϕ
+ET(Z)
+Figure 2. Oxide semiconductor interface. For simplicity we
+use different coordinate systems for the oxide ( Z) and the
+semiconductor side ( z) so that both, Zandz, are positive
+on their sides. Trap energy ETwhich reaches its maximum
+atZ=Zm, unscreened trap energy ETsat the OS inter-
+face, electrostatic energy εe, its components −eVinsZ/Dand
+−εDD/Z, and its maximum εm, chemical potential µ, ground
+state energy of the inversion layer E0and the corresponding
+wave function ϕ, conduction band edge ECand its value at
+the interface ECs.
+withC= 8(AM) or C= 16 (our assumption). (We
+use in this work the notation of capital Efor absolute
+energies and Greek εfor energy differences.) Finally the
+trap energy can be written as
+ET(Z) =ETs+εe(Z), (5)
+εe(Z) =−eVinsZ
+D−εDD
+Z, (6)
+where the subscript einεestands for ’electrostatic’. The
+shape of ET(Z)is shown in Fig. 2 together with the
+relevant energy notation.
+Without the presence of a magnetic field the probabil-
+ity of a trap to be charged is given by
+p+(Z) =1
+1
+2exp/parenleftBig
+−ET(Z)−µ
+kBT/parenrightBig
++1. (7)
+This formula is similar to the Fermi distribution function,
+differences are the minus sign in the exponent and the
+prefactor 1/2. A trap is charged when it is not occupied
+by an electron, thus the minus sign. If the trap is charged
+there are two possibilities for the spin orientation but
+only one if it is not charged, leading to the factor 1/2.
+As mentioned before p+is determined by ET−µ(the
+vertical lines in Fig. 2) and the temperature.
+AM assume that a positive charged trap is screened by
+the electrons in the 2DEG and the trap forms together
+with that image charge a dipole. For the transport scat-tering cross section σtof this dipole they found classically
+σt(ε,Z) = 2.74/parenleftbigge2Z2
+8πǫ∗ǫ0ε/parenrightbigg1/3
+≡cσε−1/3Z2/3,(8)
+→cσ= 2.74/parenleftbigge2
+8πǫ∗ǫ0/parenrightbigg1/3
+, (9)
+whereZis the distance between the trap and the oxide
+semiconductor interface, εis the energy of the scattered
+electron relative to the ground state energy of the inver-
+sion layer E0:
+ε=E−E0, (10)
+andǫ∗is an effective dielectric constant
+ǫ∗=ǫins+ǫsc
+2, (11)
+whereǫscis the dielectric constant of the semiconductor.
+We have recalculated Eq. (8) and got the same result as
+AM.
+The Drude formula together with the Boltzmann equa-
+tion in relaxation time approximation yields the resis-
+tivitydρ(Z)caused by the charged traps within the
+layer[Z,Z+dZ]. By integrating this contributions over
+the whole oxide [0,D]one gets
+ρ=√2mccσ¯ε1/6/integraltextD
+0N+
+TZ2/3dZ
+nse2, (12)
+¯ε=εF0/bracketleftBigg/integraldisplay∞
+01
+4kBT/parenleftbiggε
+εF0/parenrightbigg5/6
+cosh−2/parenleftbiggε−µE0
+2kBT/parenrightbigg
+dε/bracketrightBigg−6
+,
+(13)
+wheremcis the conductivity mass of the free electrons
+within the inversion layer and ¯εis an effective elec-
+tron energy as used by AM. In the corresponding AM
+equation the argument of the coshfunction contains the
+Fermi energy, but should be replaced by the chemical
+potential.31Furthermore εF0=EF−E0is the Fermi
+energy,µE0=µ−E0is the chemical potential, each rel-
+ative to the ground state energy of the inversion layer,
+andN+
+Tis the 3D density of charged traps. With the 3D
+trap density NT(Z)it is given by
+N+
+T(Z) =NT(Z)p+(Z). (14)
+Now we like to present Eq. (12) in a form similar to
+the Drude formula
+ρ=mc
+nse2∝an}bracketle{tτ∝an}bracketri}ht, (15)
+and further find a term for the scattering rate 1/∝an}bracketle{tτ∝an}bracketri}htin
+the usual form that scattering rate is equal to scattering
+cross section times density of scattering centers times ve-
+locity of scattered particles, which leads to one of the
+basic equations used throughout this work
+ρ=mcσt/parenleftbig
+¯ε,¯Z/parenrightbig
+n+
+Tv(¯ε)
+nse2. (16)4
+Hereσt/parenleftbig
+¯ε,¯Z/parenrightbig
+is the transport scattering cross section
+from Eq. (8) for the effective electron energy ¯εand an
+effective distance ¯Zbetween the traps and the OS inter-
+face,n+
+T=/integraltextD
+0N+
+T(Z)dZis the 2D density of charged
+traps, and v(¯ε)is the electron velocity which corresponds
+with¯ε=mcv2/2. Comparing Eq. (12) and (16) we find
+¯Z2/3=/integraltextD
+0N+
+T(Z)Z2/3dZ
+/integraltextD
+0N+
+T(Z)dZ=/integraltextD
+0N+
+T(Z)Z2/3dZ
+n+
+T(17)
+for this factor in σt/parenleftbig
+¯ε,¯Z/parenrightbig
+=cσ¯ε−1/3¯Z2/3.
+In order to calculate the resistivity ρthe knowledge
+ofn+
+Tis not necessary, n+
+Tcancels out with the denom-
+inator of ¯Z2/3withinσt/parenleftbig
+¯ε,¯Z/parenrightbig
+, AM do not use it. As
+mentioned in the introduction we are interested in the
+metal-insulator transition depending on the electron den-
+sityns, i. e. we want to know the temperature behavior
+ofρas a function of ns. In this context n+
+Tis very useful
+in order to understand on the basis of Eq. (16) that it
+contributes the main variations to the resistivity ρ(ns,T)
+whereas σt/parenleftbig
+¯ε,¯Z/parenrightbig
+andv(¯ε)show only weak dependence
+onnsandT. The benefit of Eq. (16) against (12) is, that
+the physical meaning of the terms is immediately clear.
+The integral in the numerator and that in the denom-
+inator of ¯Z2/3can be treated in quite the same way, so
+we define
+Ωj≡/integraldisplayD
+0N+
+T(Z)ZjdZ=
+=NT/integraldisplayD
+0ZjdZ
+1
+2exp/parenleftBig
+−ET(Z)−µ
+kBT/parenrightBig
++1.(18)
+In the last step we followed AM and assumed that the
+trap density is constant within the oxide, respectively in
+the region where p+(Z)does not vanish. Now we can
+write
+σt/parenleftbig
+¯ε,¯Z/parenrightbig
+=cσ¯ε−1/3¯Z2/3=cσ¯ε−1/3Ω2/3
+Ω0,(19)
+n+
+T= Ω0, (20)
+ρ=√2mccσ¯ε1/6Ω2/3
+nse2. (21)
+To be able to calculate the integral which corresponds
+withΩ2/3AM expanded the electrostatic energy εT(Z)
+into a Taylor series about the point Zmwhere it reaches
+its maximum εm. This procedure is called saddle-point
+approximation.
+Zm=D/radicalbiggεD
+eVins, (22)
+εm=−2/radicalbig
+eVinsεD, (23)
+εe(Z)≃εm−εDD
+Z3m(Z−Zm)2, (24)ZO
+zSEC
+ECsµE0ETs
+ϕ
+ET(Z)p+(Z)
+Figure 3. Oxide-semiconductor interface. Trap energy ET(Z)
+(full line), its Taylor approximation (dashed line), and th e
+resulting probabilities p+(Z)in arbitrary units with µas zero
+point.
+see Fig. 2 and 3. Now (18) can be written as
+Ωj≃NT/integraldisplayD
+0ZjdZ
+1
+2exp/parenleftbiggµE0−εTs0−εm+εDD
+Z3m(Z−Zm)2
+kBT/parenrightbigg
++1,
+(25)
+εTs0=ETs−E0=const. (26)
+AM assume that the energy ETsrelative to the ground
+state energy E0is constant, but we believe that rather
+the conduction band edge at the interface ECshas to be
+used as reference energy, i. e. εTsCs=ETs−ECs=const.
+This issue will be further treated in section VI.
+The integrand is a peak around Zmwhich drops off ex-
+ponentially on both sides. In order to come to the same
+result as AM, we further apply the following approxima-
+tions: (i) Because of the exponential decrease one can
+integrate from −∞to∞. (ii) The integrand is domi-
+nated by the denominator, so Zj≃Zj
+mcan be set in the
+numerator. Now the integrand is symmetric around Zm
+and with help of the substitution Z=εDD
+Z3mkBT(Z−Zm)2
+one gets
+Ωj=NTZj+3/2
+m/radicalbigg
+kBT
+εDD
+×/integraldisplay∞
+0Z−1/2dZ
+exp/parenleftBigµE0−εTs0−εm
+kBT−ln2+Z/parenrightBig
++1.(27)
+This corresponds to the integral in equation (9c) in
+Ref. 26 ( Zcorresponds to x2). We brought it into
+the form above as it corresponds now to a Fermi-Dirac
+integral32
+Fk(η) =1
+Γ(k+1)/integraldisplay∞
+0ZkdZ
+exp(Z −η)+1, (28)5
+whereΓis the Gamma function. A comparison yields
+Ωj=NTZj+3/2
+m/radicalbigg
+kBT
+εDD
+×√π/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+Γ(1
+2)F−1/2/parenleftbigg
+ln2+εTs0+εm−µE0
+kBT/parenrightbigg
+.(29)
+On the right hand side jappears only in the exponent of
+Zm, so within the saddle-point approximation Eq. (17)
+simplifies to
+¯Z2/3=Ω2/3
+Ω0≃Z2/3
+m. (30)
+III. CHEMICAL POTENTIAL
+AM described two scenarios for the temperature be-
+havior of the chemical potential: (A) The chemical po-
+tential of the 2DEG and of the Si substrate coincide. (B)
+The 2DEG is disconnected form the substrate. For the
+case (A) they assumed that the temperature behavior in
+the 2DEG is the same as in the bulk. However, they did
+not take into account that the chemical potential in the
+2DEG is measured against the ground state energy E0
+and in the bulk against the conduction or valence band
+edge, i. e. they assumed E0and the band bending to be
+fixed.
+For (B) AM used an equation analogous to
+µE0=kBTln/bracketleftbigg
+exp/parenleftbiggεF0
+kBT/parenrightbigg
+−1/bracketrightbigg
+. (31)
+If only one subband of the inversion layer is occupied
+(quantum limit), the Fermi energy relative to its ground
+state energy is given by28
+εF0=2π/planckover2pi12ns
+gsgv2Dmd2D, (32)
+wheregs,gv2D, andmd2Dare the spin degeneracy, the
+valley degeneracy (for the 2DEG), and the density-of-
+states mass (2D) respectively. The two equations above
+can be derived from
+ns=gsgv2Dmd2D
+2π/planckover2pi12
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+2Ddensityof states/integraldisplay∞
+E01
+exp/parenleftBig
+E−µ
+kBT/parenrightBig
++1
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+Fermi−DiracdistributiondE.(33)
+Our assumptions are shown schematically in Fig. 4
+(semiconductor side of the OS interface) and are as fol-
+lows. In thermal equilibrium there is a single chemical
+potential µthroughout the structure. For a certain tem-
+perature T,µis determined in the bulk by the (residual)
+doping density (giving µCb=µ−ECb, for details see
+App. B). In the inversion layer the position of µrelative
+toE0(i. e.µε0) follows directly from the 2D density ns.Figure 4. Band bending and notation on semiconductor side
+of the OS interface for thermal equilibrium between 2D layer
+and bulk. The electronic ground state energy of the inversio n
+layerE0, the conduction band edge EC, its values at the in-
+terfaceECsand in the bulk ECb, the valence band edge EV,
+and the chemical potential µare shown schematically, also
+the ground state energy relative to the conduction band edge
+ε0, the chemical potential relative to conduction band edge in
+the bulk µCband relative to the ground state energy µE0are
+noted.
+The band bending ECb−ECsadjusts so that
+ECb−ECs=ε0+µE0−µCb. (34)
+If the band bending is increased, the quantum well gets
+narrower and the ground state energy increases. Thus ε0
+itself is a function of the band bending, a self consistent
+calculation solves this problem (fix point iteration).
+Another possibility is that the 2D electron system is at
+low temperatures decoupled from the bulk substrate and
+no common chemical potential exists. We do not treat
+this case in this work, but expect a similar behavior.
+Fig. 5 shows the behavior of µ(T,ns)which is crucial
+for the understanding of the behavior of ρ(T,ns). The
+chemical potential relative to the ground state energy of
+the inversion layer µE0(T,ns)increases for decreasing T
+and for increasing ns.
+The parameters, which were used here and for the fol-
+lowing calculations are collected in Table I. We assumed
+a {001} plane for the silicon surface.
+IV. APPROXIMATION FOR LOW
+TEMPERATURES
+For given values Tandnswe calculate the resistivity
+ρwith help of the equations (21) and (29). Beside the
+explicit temperature dependence also the chemical poten-
+tialµE0and the effective electron energy ¯εare functions
+ofT.
+As a first step we replace the integral in (13) by a6
+(a)
+0 100 200 300−100−80−60−40−20020
+T (K)µE0 (meV)
+(b)
+0 5 10 15 20−200−150−100−50050
+ns (1011cm−2)µE0 (meV)
+Figure 5. Chemical potential µrelative to the ground state
+energyE0. a) Curves with constant ns, bottom up: ns=
+1,2,5,10,20×1011cm−11. b) Curves for constant T, top
+down:T= 0,50,100,150,200,250,300K.
+Table I. Values used for calculations.
+gs= 2
+gv2D= 2
+gv3D= 6
+me= 9.1094×10−31kg
+md2D= 0.1905me
+mz= 0.9163me
+mde3D= 0.322me
+mdh3D= 0.59me
+ǫsc= 11.9
+ǫins= 3.9
+→ǫ∗= 7.9
+D= 200nm
+C= 16
+→εD= 0.4615meV
+NA= 2×1015cm−3
+ND= 0
+εg0Si= 1.17eV
+αSi= 4.73×10−4eVK−1
+βSi= 636KFermi-Dirac integral. For the Fermi distribution function
+f=1
+exp/parenleftBig
+E−µ
+kBT/parenrightBig
++1=1
+exp/parenleftBigε−µE0
+kBT/parenrightBig
++1(35)
+the following identity holds,
+−∂f(ε)
+∂ε=1
+4kBTcosh−2/parenleftbiggε−µE0
+2kBT/parenrightbigg
+.(36)
+So Eq. (13) can be written as
+¯ε=εF0/bracketleftBigg/integraldisplay∞
+0/parenleftbiggε
+εF0/parenrightbigg5/6/parenleftbigg
+−∂f(ε)
+∂ε/parenrightbigg
+dε/bracketrightBigg−6
+.(37)
+An integration by parts yields
+¯ε=ε6
+F0/bracketleftbigg5
+6/integraldisplay∞
+0ε−1/6f(ε)dε/bracketrightbigg−6
+,
+¯ε=εF0/parenleftbiggεF0
+kBT/parenrightbigg5
+×
+5
+6/integraldisplay∞
+0/parenleftBig
+ε
+kBT/parenrightBig−1/6
+exp/parenleftBigε−µE0
+kBT/parenrightBig
++1d/parenleftbiggε
+kBT/parenrightbigg
+−6
+,
+¯ε=εF0/parenleftbiggεF0
+kBT/parenrightbigg5/bracketleftbigg
+Γ/parenleftbigg11
+6/parenrightbigg
+F−1/6/parenleftbiggµE0
+kBT/parenrightbigg/bracketrightbigg−6
+.(38)
+For low temperatures we can take advantage of the
+behavior of µE0(T)and¯ε(T)forT→0. It can easily be
+shown that for constant nsand therefore constant εF0
+all derivatives vanish at this point.
+dnµE0(T)
+dTn/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+T→0= 0 for n≥1, (39)
+dn¯ε(T)
+dTn/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+T→0= 0 for n≥1. (40)
+SoµE0(T)and¯ε(T)are very flat functions at T→0,
+forkBT≪εF0they can be approximated by
+µE0(T)≃εF0, (41)
+¯ε(T)≃εF0. (42)
+See appendix A for details.
+The Fermi-Dirac integral Fj(η)can be approximated
+by32
+Fj(η)≃ηj+1
+Γ(j+2)forη≫0, (43)
+which is the first term of an asymptotic series derived
+with help of a Sommerfeld expansion,33and by
+Fj(η)≃expηforη≪0. (44)
+ForT→0the argument of the Fermi-Dirac integral in
+Eq. (29) increases beyond any border. Which approxima-
+tion for the Fermi-Dirac integral is applicable depends7
+on the sign of εTs0+εm−εF0. (Here µE0is replaced
+byεF0as by definition µE0→εF0forT→0.) This
+is conform with AM’s definition of the transition point,
+(εTs0+εm−εF0)/kBT= 0. Accordingly we define
+εmF=εTs0+εm−εF0,
+εmF>0→insulating ,
+εmF<0→metallic,
+εmF= 0→transitionpoint
+and a critical density
+nsc=ns|εmF=0. (45)
+Applying the appropriate approximations results in
+Ωj≃
+
+2NTZj+3/2
+m/radicalBig
+kBTln2
+εDD+εmF
+εDDforεmF>0
+2NTZj+3/2
+m/radicalbigg
+kBTπ
+εDDexp/parenleftBig
+εmF
+kBT/parenrightBig
+forεmF<0
+F−1/2(ln2)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+≃0.891NTZj+3/2
+m/radicalBig
+kBTπ
+εDDforεmF= 0.
+(46)
+Please note an interesting behavior. When setting εmF=
+0in the first two equations for T >0they converge nei-
+ther into each other nor into the third one. This apparent
+discrepancy can be understood, as |εmF|gets smaller and
+smaller the maximum temperature where the approxima-
+tions for the Fermi-Dirac integral are just applicable also
+gets smaller and smaller and finally vanishes for εmF= 0.
+Indeed for εmF= 0andT→0the three cases yield the
+same result, i. e. Ωj= 0.
+V. COMPARISON OF ANALYTIC AND
+NUMERICAL RESULTS
+In order to get rid of the restrictions which came from
+the saddle-point approximation we have performed nu-
+merical calculations of the integrals Ωj(Eq. (25)). In
+Fig. 6 we show the resistivity ρdepending on the tem-
+perature Tand the 2D electron density in the inversion
+layernscalculated with the help of the saddle-point ap-
+proximation (full lines), the approximation for low tem-
+peratures (dashed lines), and the numerical integration
+(markers). We chose εTs0≃42.02meV in order to get
+nsc= 1011cm−11, for the 3D trap density we assumed
+NT= 1018cm−3. The value for NTseems to be quite
+high, but only in a very narrow layer the traps will in-
+deed be charged (where the trap states are above the
+chemical potential) and contribute to scattering. A fur-
+ther limitation of the available trap states into a narrow
+region besides the interface will follow later in this work.
+The critical behavior of ρ(T)versusnsfor the homo-
+geneous trap density NTcan be seen most clearly on the
+double logarithmic Figs. 6a) and c). We added 6b) as it
+is in this form directly comparable with Fig. 1b) in the
+work of AM.26We like to explain the shape of ρ(T,ns)starting from
+a valuens> nsc. The maximum of εT(Z)lies below
+µ, therefore the p+(Z)-peak is very small and narrow.
+Whennsis decreased µdrops off (see Fig. 5) and the
+resistivity ρrises fast because −(ET−µ)/kBTis the ex-
+ponent in the denominator of p+. It rises the faster the
+smaller the temperature is. When µreaches the max-
+imum of the trap energy ETthep+-peak has a height
+of2/3, therefore further decreasing of nsandµcannot
+increase the height of the peak appreciable (on a loga-
+rithmic scale, which covers some orders of magnitude),
+only the width.
+For increasing temperature Talso the chemical poten-
+tialµmeasured against E0decreases which results in
+increasing resistivity ρ. Additionally the p+(Z)-peak is
+broadened because the Fermi distribution function de-
+clines over several kBT.
+The saddle-point approximation works best for small
+temperatures Tand densities ns> nsc, in this domain
+thep+(Z)-peak is very narrow and therefore the Taylor
+approximation of the trap energy is accurate within the
+peak. For large Tand/orns< nscthe deviations of the
+saddle-point approximation from the numerical calcula-
+tions is visible in Fig. 6.
+It should be noted that according to the calculations,
+the resistivity ρdrops to arbitrarily low values in the
+metallic regime. This is caused by the fact that electron
+scattering is taken into account only by the trap states
+at a single trap energy. If this trap energy is below the
+Fermi energy, with decreasing Tthe number of charged
+scattering centers goes to zero. Only if other scattering
+effects, like residual impurities, surface roughness, acce p-
+tor states in the depletion layer, etc. are included, the
+low-Tresistivity would be limited. As will be seen later,
+an energetic broadening of the trap states will have a
+similar effect.
+VI. CONDUCTION BAND AS REFERENCE
+ENERGY
+If the bands in the semiconductor and in the oxide are
+bent due to an applied gate voltage all energies move
+up and down with the bands. The energetic position of
+the trap states ETsshould thus be fixed relative to the
+conduction band edge and not to the ground state energy
+E0of the inversion layer as assumed by AM.
+From Eq. (7) we see that ET−µ=ETs+εe(Z)−µ
+determines the probability p+of a trap to be charged.
+So if we measure ETsagainstECswe also have to know
+µCs=µ−ECs. We find
+µCs=µ−E0+E0−ECs=µE0+ε0,(47)
+whereε0=E0−ECsis the ground state energy relative
+to the conduction band edge at the OS interface.
+The chemical potential µE0follows from Eq. (31), but
+an accurate calculation of ε0is rather complex. For sim-
+plicity we follow the calculation method of Ando, Fowler8
+(a) (b)
+10−110010110210−310−210−1100101
+T (K)ρ (h/e2)
+0 5 10 15 200123456
+T (K)ρ (h/e2)
+(c) (d)
+10−110010110210−310−210−1100101
+T (K)ρ (h/e2)
+0.95 1 1.0510−310−210−1100101
+ns/nscρ (h/e2)
+Figure 6. Behavior of the resistivity ρdepending on the temperature Tand the 2D electron density in the inversion layer ns.
+Full lines: saddle-point approximation, point symbols: nu merical integration, dashed lines: low temperature approx imations.
+On the curves in a) - c) nsand in d) Tis constant. Critical density nsc= 1011cm−2. a) logarithmic and b) linear display
+ofρ(T)with parameters: ns= 0.75,...,1.25×nscin steps of 0.05×nscin top down order for the individual curves, c)
+logarithmic view of ρ(T)with top down parameters: ns= 0.95,...,1.05×nscin steps of 0.01×nsc, d)ρ(ns)with parameters
+T= 0,0.2,0.5,1,2,5,10,20Kin bottom up order.
+and Stern (AFS)28and neglect the exchange interaction
+and correlation effects and use the Ritz variational prin-
+ciple. (In the mentioned article also more sophisticated
+methods for the calculation of ε0are given.)
+For convenience we introduce a new coordinate system
+z=−Z, i. e. the z-axis is perpendicular to the OS inter-
+face, positive z-values correspond with the semiconduc-
+tor side. For the electrons in the inversion layer the bent
+conduction band of the semiconductor together with the
+step at the interface builds the quantum well. We use the
+Fang-Howard envelope wave function according to AFS34
+ϕ(z,b) =/braceleftBigg/radicalBig
+b3
+2zexp/parenleftbig
+−bz
+2/parenrightbig
+forz≥0
+0 for z <0.(48)
+The parameter bis varied in order to make the total
+energy per electron minimal. For the potential several
+approximations are taken, see App. B for details.
+In Fig. 7 the ground state energy ε0versus the elec-
+tron density nsis shown, ε0decreases with decreasing ns.
+Now we hold the difference between trap energy and con-02468100102030405060
+ns (1011cm−2)ε0 (meV)
+Figure 7. Ground state energy of the inversion layer ε0versus
+the density of electrons in the inversion layer nsfor three
+temperatures, top-down: T= 0,200,300K.
+duction band edge at the interface εTsCs=ETs−ECs=
+ETs−µ+µE0+ε0constant (instead of εTs0as before).9
+Whennsis decreased not only µE0but alsoε0decreases,
+thusµdrops off faster against ETsand the transition
+is more abrupt. This can be seen in Fig. 8a) where re-
+sults for εTsCs= const.(full lines) and εTs0= const.
+(dashed lines) are compared. In both cases Ωjwas cal-
+culated numerically. The critical curves ns=nsccoin-
+cide atT→0because the values εTs0≃42.0meV and
+εTsCs≃68.4meV were chosen in order to get the same
+nscand for higher temperatures because ε0(T)is nearly
+constant over a wide temperature range (Fig. 8c)). For
+the 3D trap density we assumed again NT= 1018cm−3.
+It has to be emphasized that we still use Eq. (1). But
+the 2D density of positive charges in the depletion layer
+ndeplis a byproduct of the calculation of ε0. So it is no
+longer a problem to use Eq. (2) Vins∝ns+ndeplinstead
+of (1)Vins∝ns. We will do so henceforward. As a result
+the slope of the line ETs−eVinsZ/Din Fig. 2 is increased,
+the maximum of ET(Z)falls off against µ, thep+(Z)-
+peak gets smaller and narrower. But it is still possible to
+let coincide the critical curves ρ(ns=nsc)by increasing
+the trap density NTin order to compensate the narrower
+p+(Z)-peak and by increasing εTsCsin order to get the
+same critical value nsc. Here it is also important that the
+2D charge carrier density of the depletion layer ndepl(T)
+has almost no nsdependence for low temperatures as
+can be seen in Fig. 8d). The resistivity ρ(T,ns), with
+NT= 2.98×1018cm−3andεTsCs≃95.7meV , for which
+we get the same nscas before, is shown in Fig. 8b). The
+above used depletion density ndeplwas calculated under
+the assumption of a background doping density of NA=
+2×1015cm−3, which is a typical value for high mobility
+Si-MOS samples.35Here the values for NTandεTsCs
+were chosen in order to get the requested nscfor the
+givenNA. In reality the value εTsCsis determined by
+the chemical nature of the defect and thus the critical
+density may be different from sample to sample if the
+background doping is different.
+If now the slope of the energy ETs−eVinsZ/D
+is higher due to the inclusion of ndepl, the variable
+max(ET(Z)−µ)which is crucial for the resistivity is
+less sensitive on Vinsandns, therefore the transition is
+less abrupt. The nsdependence of ndepldoes not play a
+role because it hardly exists.
+VII. SPATIAL TRAP PROFILE
+At higher temperatures the large value of kBTleads to
+charged trap states in regions where the chemical poten-
+tial is even somewhat below the chemical potential (i. e.
+thep+(Z)-peak broadens) and therefore the resistivity is
+increased to unrealistic high values, see curves at higher
+temperature in Fig. 8. But an appreciable density of
+traps should exist only within the strained region of the
+oxide,27and thus the broadening of the peak beyond the
+width of this region leads to an unrealistic description.
+We can resolve this problem by introducing a spatial trap
+density profile NT(Z).If now the trap density NTis a function of Zit has to
+remain inside the integral Ωj(compare Eq. 18)
+Ωj≡/integraldisplayD
+0N+
+T(Z)ZjdZ=/integraldisplayD
+0NT(Z)p+(Z)ZjdZ.
+(49)
+For simplicity we use here an rectangular spatial trap
+profile from the OS interface to an arbitrary depth Zmax,
+NT(Z) =/braceleftBigg
+nT
+Zmaxfor0≤Z≤Zmax
+0 for Z > Z max,(50)
+wherenTis the 2D trap density. Fig. 9 shows ρ(T,ns)
+forZmax= 4 nm , the conduction band edge ECswas
+used as reference energy, εTsCs≃95.7meV was chosen
+in order to get nsc= 1011cm−2. Where the 3D trap
+densityNTdoes not vanish its value is assumed to be
+2.98×1018cm−3as before (see Fig. 8b)), resulting in
+nT=NT·Zmax= 1.19×1012cm−3of which again only
+a part is charged.
+As can be seen in Fig. 9, the behavior for low tem-
+peratures has hardly changed, but for high temperatures
+ρ(T,ns)now saturates as a broadening of the p+(Z)-peak
+beyondZmaxdoes not lead to a further increase in the
+number of charged scattering centers. This saturation of
+ρ(T,ns)is in fairly good agreement with experiments,
+whereρfor high Tis limited as well.
+VIII. BROADENING OF THE TRAP ENERGY
+As the trap states will not all be identical and in ad-
+dition the stochastic position distribution will influence
+them mutually, their energetic position has to be broad-
+ened.
+We describe the broadening ∆ETwith the help of a
+normalized distribution function g/parenleftBig
+˜ETs,ETs,∆ET/parenrightBig
+for
+the trap energy ETswhich characterizes the trap, see
+Fig. 2. Now ETshas the meaning of a mean value. (Mean
+value should not be understood in a strict mathematical
+sense, e. g. for the Lorentz distribution the mean value
+does not exist, but in this case it is obvious to take the
+energyETswhere the distribution reaches its maximum.)
+Furthermore ˜ETsis the value for a particular trap. The
+probability of ˜ETsto lie within the interval [E,E+dE]
+is given by g(E,ETs,∆ET)dE. Therefore we replace the
+probability p+of a trap to be charged by
+P+(Z) =/integraldisplay∞
+−∞g/parenleftBig
+˜ETs,ETs,∆ET/parenrightBig
+1
+2exp/parenleftBig
+−˜ETs−eVinsZ
+D−εDD
+Z−µ
+kBT/parenrightBig
++1d˜ETs.
+(51)
+The denominator is that of p+, onlyETsis replaced by10
+(a) (b)
+10−110010110210310−210−1100101102
+T (K)ρ (h/e2)
+(c) (d)
+10−1100101102103232425262728
+T (K)ε0 (meV)
+10−11001011021031.41.51.61.71.8
+T (K)ndepl (1011cm−2)
+Figure 8. Resistivity ρ(T,ns)for the potential across the oxide layer a) Vins∝nswithNT= 1018cm−3andεTsCs≃68.4meV
+and b)Vins∝ns+ndeplwithNT= 2.98×1018cm−3andεTsCs≃95.7meV . The parameters are chosen to give a critical
+density of nsc= 1011cm−2for both cases. At the individual curves ns= 0.75,...,1.25×nscin steps of 0.05×nscfrom top to
+bottom, full lines represent εTsCs= const.(realistic case), dashed lines εTs0= const.(for comparison). In c) the ground state
+energy of the inversion layer ε0(T,ns)is shown for same nsvalues as before, but now assigned bottom up. In d) the deplet ion
+densityndepl(T,ns)is shown, but the nsdependence vanishes within the line width.
+10−110010110210310−310−210−1100101
+T (K)ρ (h/e2)
+Figure 9. Resistivity ρ(T,ns)for rectangular spatial trap
+profile with Zmax= 4 nm with curves of constant ns. The
+critical density is nsc= 1011cm−2which means that εTsCs
+has to be 95.7meV . Densities for full lines are ns=
+0.75,...,1.25×nscin steps of 0.05×nsc, and for dashed
+linesns= 0.96,...,1.04×nscin steps of 0.01×nsc(always
+top down), with NT=nT/Zmax= 2.98×1018cm−3.˜ETs. By introducing the dimensionless parameters
+α=∆ET
+kBT, (52)
+β=ETs−eVinsZ
+D−εDD
+Z−µ
+kBT(53)
+and a dimensionless distribution function h(η)defined
+by
+η=˜ETs−ETs
+∆ET, (54)
+g/parenleftBig
+˜ETs,ETs,∆ET/parenrightBig
+=1
+∆ETh/parenleftBigg˜ETs−ETs
+∆ET/parenrightBigg
+,(55)
+g/parenleftBig
+˜ETs,ETs,∆ET/parenrightBig
+=1
+∆ETh(η), (56)
+the probability P+can be written as
+P+(α,β) =/integraldisplay∞
+−∞h(η)
+1
+2exp(−αη−β)+1dη. (57)11
+As a rule this integral cannot be calculated analytically.
+An exception from this rule is the uniform distribution.
+If we define the width of the ’rectangle’ as 2∆ETwe get
+h(η) =/braceleftBigg
+1
+2for−1< η <1
+0 elsewhere(58)
+and
+P+(α,β) =1
+2αln1+2exp( β+α)
+1+2exp( β−α). (59)
+We also use the normal distribution
+h(η) =1√
+2πexp/parenleftbigg
+−η2
+2/parenrightbigg
+(60)
+with the standard deviation as ∆ETand the Lorentz
+distribution (natural line broadening)
+h(η) =1
+π1
+η2+1(61)
+with the half full width at half maximum (FWHM) as
+∆ET.
+Fig. 10 shows the numerical results for the three differ-
+ent broadening distributions and different width ∆ET=
+0.02,0.1and0.5meV . Again we took the conduction
+band edge at the interface ECsas reference energy, as-
+sumed that traps exist in the oxide only within 4nm
+from the OS interface (constant trap density within this
+region), and used Eq. (2) instead of (1). We chose
+εTsCs≃95.7meV in order to get nsc= 1011cm−2and
+for the 3D trap density NT=nT/Zmax= 1018cm−3.
+As one would expect, the transition is less abrupt
+for higher ∆ETand does not vanish even for ∆ET=
+0.5meV . If the broadening is indeed caused by potential
+fluctuations due to disorder or trap-trap interaction, the
+value of ∆ETcould even be much larger than kBTand
+the transition might be smeared out even stronger.
+In the metallic regime the mean trap energy is below
+the chemical potential. As the normal and the Lorentz
+distribution have tails, there always remain some charged
+traps from the upper tail when otherwise all traps would
+be filled with electrons and therefore would be neutral.
+On a logarithmic resistivity scale the ρ(T)-behavior is
+changed drastically by the few additional charged traps.
+In the insulating regime the mean trap energy is above
+the chemical potential. Here a large part of the traps is
+charged and by contrast the few uncharged traps due to
+the lower tail of the energy distribution hardly play any
+role.
+By means of analytical considerations we got the fol-
+lowing estimations for the temperature Tbbelow which
+the resistivity becomes almost constant. In general the
+resistivity ρvaries by some orders of magnitude, so as
+a criterion for being almost constant we took that re-
+gion where ρchanges finally by only a factor two down
+to zero temperature. The markers in Fig. 10 represent
+these temperatures Tb.According to this definition, for the insulating behavior
+ns< nscfor all three distributions we get
+Tb=maxET(Z)−EF
+(4−ln2)kB(62)
+and for the metallic behavior ns> nsc
+Tb=/braceleftBigg∆E2
+T
+(4+ln2) kB(EF−maxET(Z))...normaldistr .
+EF−maxET(Z)
+2(4+ln2) kB...Lorentzdistr .
+(63)
+The uniform distribution has no tails so in the metallic
+regime there is no temperature range where ρis almost
+constant.
+IX. CONCLUSIONS
+In this work we have performed numerical calcula-
+tions within the dipole trap model for Si-MOS structures.
+Originally this model was proposed by Altshuler and
+Maslov with several approximations, in order to be able
+to get analytical solutions. Due to our numerical treat-
+ment we could eliminate several approximations. We de-
+scribe the potential inside the insulator by its detailed
+spatial dependence instead of the saddle-point approxi-
+mation with the quadratic dependence around its max-
+imum, we fix the trap state energy relative to the con-
+duction band edge instead of relative to the electronic
+ground state inside the triangular potential well and we
+have taken into account the detailed change of the chem-
+ical potential in the two-dimensional electron layer with
+respect to the bulk material, which seems to be more
+realistic than the two cases in the original treatment.
+According to our calculations, the metallic regime at
+high electron densities ns, where the resistivity is de-
+creasing towards lower temperature, is strongly devel-
+oped . Also a critical density nsccan be identified with a
+characteristic temperature dependence different from the
+metallic and the ’insulating’ region. For electron densi-
+tiesns< nsc, the resistivity curves satturate towards
+lower temperature and remain constant when the tem-
+perature approaches zero. They do not show an insult-
+ing behavior in the sense that ρincreases towards zero
+temperature. Such an increase can in principle be caused
+e. g. by a further decrease of the chemical potential µwith
+temperature, as in the work of Altshuler and Maslov for
+case A, where it was assumed that the temperature de-
+pendence of µist the same for the two-dimensional elec-
+tron layer as it is in the Si-bulk material. In the dipol
+trap model a constant efficient screening is assumed in
+order to clarify the effects which are caused when the
+traps change their charge state. A realistic treatment
+of the temperatue dependence of the electronic screening
+could also cause an increase in ρtowards lower temper-
+ature in that it favors the formation of dipol trap states.12
+10−110010110210−410−310−210−1100101
+T (K)ρ (h/e2)
+10−110010110210−410−310−210−1100101
+T (K)ρ (h/e2)
+10−110010110210−410−310−210−1100101
+T (K)ρ (h/e2)
+10−110010110210−410−310−210−1100101
+T (K)ρ (h/e2)
+10−110010110210−410−310−210−1100101
+T (K)ρ (h/e2)
+10−110010110210−410−310−210−1100101
+T (K)ρ (h/e2)
+10−110010110210−410−310−210−1100101
+T (K)ρ (h/e2)
+10−110010110210−410−310−210−1100101
+T (K)ρ (h/e2)
+10−110010110210−410−310−210−1100101
+T (K)ρ (h/e2)
+Figure 10. Resistivity ρ(T,ns)for different types of trap energy broadening functions and d ifferent width ∆ET. The picture
+columns show uniform distribution, normal distribution, L orentz distribution from top to bottom. The rows show energy width
+of∆ET= 0.02,0.1,0.5meV from left to right. For the curves nsis constant. The critical density is nsc= 1011cm−2for
+εTsCs≃95.7meV . Full lines show ns= 0.75,...,1.25×nscin steps of 0.05×nsc, dashed lines: ns= 0.96,...,1.04×nscin
+steps of0.01×nsc(from top to bottom), for a 3D trap density of NT=nT/Zmax= 1018cm−3.
+Furthermore, the quantum corrections in the weak and
+strong localization regime are also neglected here, but
+would finally increase the resistance ρat lowT.
+In addition, we have further generalized the dipole
+trap model by dropping the assumptions that the trap
+states are homogeneously distributed inside the oxide
+layer and that the energy distribution is δ-like. A nar-
+row spatial distribution of the trap states near the oxide-
+semiconductor interface limits the number of charged
+states at high temperatures and thus gives an upper limit
+for the increase of the resistivity ρas well. This leads
+to a good agreement with experimental observations at
+higher temperatures. The energetic broadening of the
+trap states on the other hand leads to a finite amount
+of unoccupied and thus charged states in cases where
+otherwise all states would lie below the chemical poten-
+tialµand the number of charged trap states would goto zero for kBT→0. Thus for high electron densities
+with metallic behavior the resistivity will not further de-
+crease towards lower temperature, but saturate at a finite
+values, as has been observed in experiments on Si-MOS
+structures as well.
+The effect of a magnetic field can be taken into account
+by the Zeeman splitting of the trap states with spin ±1/2.
+As shown by Althuler and Maslov, the energetic splitting
+can turn a metallic behavior into an insulating one. We
+did not include magnetic field effect in our calculations,
+but an according energetic shift of the trap states has to
+lead to the same effects in our refined model as well.
+We also like to mention that for low electron densities
+care has to be taken for the dipol scattering model. It is
+assumed that the electrons in the two-dimensional layer
+shield the potential of the charged trap states and thus
+form together a dipol field which is responsible for the13
+scattering. At very low electron densities this screening
+becomes weaker and the scattering will finally increase so
+that the resistivity should be higher in this regime. These
+effects have not been taken into account in the frame of
+the current work, as we like to present the basic effects
+due to charging of trap states.
+Althogether, our detailed numerical calculations
+within the dipol trap model show that a pronounced
+metallic state can be caused by trap states at an appro-
+priate energy level inside the oxide of Si-MOS structures.
+For the realistic assumptions of energetic broadening and
+narrow spatial distribution near the oxide-semiconductor
+interface, the behavior is in close agreement with exper-
+imental observations.
+ACKNOWLEDGMENTS
+Work was supported by the Austrian Science Founda-
+tion (FWF) project no. P16160. We thank our former
+colleague A. Prinz to perform first calculations within
+the Altshuler-Maslov Trap model.
+Appendix A: Temperature behavior of the chemical
+potential and of the effective electron energy for low
+temperatures
+First we show that
+dnµE0(T)
+dTn/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+εF0=const.,T→0= 0 forn≥1(A1)
+holds for µE0(T)from Eq. (31). In our calculations we
+use the temperature Tand the electron density in the
+inversion layer nsas independent variables. So it is al-
+lowed to set ns=const. and therefore also εF0=const.
+while varying T, see Eq. (32).
+For simplicity we introduce the auxiliary variable
+x=kBT
+εF0(A2)
+and the function
+M(x) =µE0(T(x))−εF0
+εF0. (A3)
+With the chain rule we find
+dnM(x)
+dxn=εn−1
+F0
+kn
+BdnµE0
+dTnforn≥1. (A4)
+So Eq. (A1) is equivalent to
+dnM(x)
+dxn/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+x→0= 0 for n≥1. (A5)
+From Eq. (31) we get
+M(x) =xln/bracketleftbigg
+1−exp/parenleftbigg
+−1
+x/parenrightbigg/bracketrightbigg
+. (A6)The first two derivatives are
+dM
+dx= ln/bracketleftbigg
+1−exp/parenleftbigg
+−1
+x/parenrightbigg/bracketrightbigg
+−1/bracketleftbig
+exp/parenleftbig1
+x/parenrightbig
+−1/bracketrightbig
+x,(A7)
+d2M
+dx2=−1/bracketleftbig
+exp/parenleftbig1
+x/parenrightbig
+−1/bracketrightbig
+x3−1
+/bracketleftbig
+exp/parenleftbig1
+x/parenrightbig
+−1/bracketrightbig2x3.(A8)
+The second derivative contains only terms of the form
+ga,b=1/bracketleftbig
+exp/parenleftbig1
+x/parenrightbig
+−1/bracketrightbigaxb. (A9)
+Differentiating ga,byields
+dga,b
+dx=aga,b+2+aga+1,b+2−bga,b+1, (A10)
+therefore all higher derivativesdnM
+dxnalso contain
+only terms gj,kwithj∈ {1,2,...,n}andk∈
+{n+1,n+2,...,2n−1}. Now we can multiply the nu-
+merator and the denominator of gj,kwithexp(−j/x)and
+write
+lim
+x→0gj,k(x) = lim
+x→0exp/parenleftbig
+−j
+x/parenrightbig
+/bracketleftbig
+1−exp/parenleftbig
+−1
+x/parenrightbig/bracketrightbigjxk
+= lim
+x→0exp/parenleftbig
+−j
+x/parenrightbig
+xk
+= lim
+y→∞yk
+jkexp(y),
+withy=j/x. Applying the rule of L’Hospital ktimes
+we get
+lim
+x→0gj,k(x) = lim
+y→∞k!
+jkexp(y)= 0. (A11)
+The limit for the first derivative also vanishes
+lim
+x→0dM
+dx= ln1−lim
+x→0g1,1= 0, (A12)
+so Eq. (A5) holds and
+µE0≃EF0forkBT≪εF0. (A13)
+With help of Eq. (A1) we show now that
+dn¯ε(T)
+dTn/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+εF0=const.,T→0= 0 forn≥1 (A14)
+holds for Eq. (38).
+For not too small ηthe Fermi-Dirac integral Fj(η)can
+be approximated by32
+Fj(η)≃ηj+1
+Γ(j+2). (A15)
+For low temperatures µE0approaches εF0and therefore
+it is positive, so from Eq. (38) we get
+¯ε≃εF0/parenleftbiggεF0
+µE0/parenrightbigg5
+forkBT≪εF0, (A16)14
+and with the chain rule respectively Faá di Bruno’s for-
+mula
+d¯ε
+dT=d¯ε
+dµE0dµE0
+dT, (A17)
+dn¯ε
+dTn=/summationdisplayn!
+k1!...kn!dk¯ε
+dµk
+E0
+×/parenleftbigg1
+1!dµE0
+dT/parenrightbiggk1
+.../parenleftbigg1
+n!dnµE0
+dTn/parenrightbiggkn
+,(A18)
+k=k1+···+kn,
+where the sum runs over all integer numbers k1,...,k n≥
+0which fulfill
+k1+2k2+···+nkn=n. (A19)
+We do not have to find this numbers, we only need to
+know that for n >1at least one kj>0, therefore each
+term in (A18) contains a factordjµE0
+dTj, so forns= const.,
+T→0the sum vanishes and from Eq. (A16) we get
+¯ε≃εF0forkBT≪εF0. (A20)
+Appendix B: Ground state energy of the inversion
+layer
+As mentioned before we calculate the ground state en-
+ergy of the inversion layer with help of the Ritz vari-
+ational principle using the Fang-Howard test envelope
+wave function34
+ϕ(z,b) =/braceleftBigg/radicalBig
+b3
+2zexp/parenleftbig
+−bz
+2/parenrightbig
+forz≥0
+0 for z <0(B1)
+and as an approximation for the potential
+U(z) =/braceleftBigg
+Ud(z)+Us(z)+Ui(z) forz≥0
+∞ forz <0,(B2)
+Ud(z)≃
+
+e2ndepl
+ǫscǫ0z/parenleftBig
+1−z
+2zd/parenrightBig
+forz < zd
+e2ndepl
+2ǫscǫ0zd forz > zd,(B3)
+Us(z)≃e2ns
+2bǫscǫ0/bracketleftBig
+6−/parenleftBig
+(bz)2+4bz+6/parenrightBig
+exp(−bz)/bracketrightBig
+,
+(B4)
+Ui(z)≃be2
+32πǫscǫ0ǫsc−ǫins
+ǫsc+ǫins1
+4z. (B5)
+The term Udcomes from the charged acceptors within the
+depletion layer with thickness zd,Usdescribes the inter-
+action with all other electrons in the inversion layer, and
+Uithe interaction with image charges. To write Us(z)
+in this form we have to assume that only the first sub-
+band is occupied, this is the so called quantum limit.
+The conduction band edge is built by Ud(z) +Us(z),
+therefore the zero point of the energy scale was chosento getUd(0)+Us(0) = 0 which means that the resulting
+ground state energy is measured against the conduction
+band edge at the interface ECsas requested.
+The Hamiltonian is given by ˆH=ˆT+Uwith the oper-
+ator for the kinetic energy ˆT=−/planckover2pi12
+2mz∂2
+∂z2, wheremzis the
+z-component of the effective mass of the semiconductor
+in the bulk. The ground state energy is the expectation
+value of the Hamiltonian,
+ε0=/angbracketleftBig
+ˆT/angbracketrightBig
++/angbracketleftBig
+Ud/angbracketrightBig
++/angbracketleftBig
+Us/angbracketrightBig
++/angbracketleftBig
+Ui/angbracketrightBig
+, (B6)
+calculated with the value of bwhich makes the total en-
+ergy per electron
+˜ε=/angbracketleftBig
+ˆT/angbracketrightBig
++/angbracketleftBig
+Ud/angbracketrightBig
++1
+2/angbracketleftBig
+Us/angbracketrightBig
++/angbracketleftBig
+Ui/angbracketrightBig
+(B7)
+minimal. The factor 1/2in the third term prevents from
+double counting the electron-electron interaction. With
+the density n∗=ndepl+11
+32nsintroduced by AFS28one
+gets
+˜ε=α
+2b2+βb+γb−1−δ(b)
+2b−2, (B8)
+α=/planckover2pi12
+4mz, (B9)
+β=e2
+32πǫscǫ0ǫsc−ǫins
+ǫsc+ǫins, (B10)
+γ=3e2n∗
+ǫscǫ0, (B11)
+δ=12e2(NA−ND)
+ǫscǫ0(B12)
+×/braceleftBigg
+1−/bracketleftBigg
+(bzd)2
+12+bzd
+2+1/bracketrightBigg
+exp(−bzd)/bracerightBigg
+.(B13)
+NAandNDare the densities of the acceptors and donors
+respectively. The coefficients have been chosen in order
+to getα, β, γ, δ > 0and to get a most simple equation
+d˜ε
+db=αb+β−γb−2+δ(b)b−3= 0. (B14)
+As AFS usede2ndepl
+ǫscǫ0z/parenleftBig
+1−z
+2zd/parenrightBig
+for0< z <∞instead
+of Eq. (B3) they did not get the term proportional to
+exp(−bzd), under normal circumstances it is very small
+but formally it is necessary to see that b→0,˜ε→ −∞
+is not the global minimum. Furthermore they neglected
+βandδ(b)b−3and got
+b=/parenleftBigγ
+α/parenrightBig1/3
+=/parenleftbigg12e2n∗mz
+ǫscǫ0/planckover2pi12/parenrightbigg1/3
+. (B15)15
+We neglected only δ(b)b−3, this leads to
+b=F/parenleftbigg
+K+1
+K−1/parenrightbigg
+, (B16)
+F=β
+3α=e2mz
+24πǫscǫ0/planckover2pi12, (B17)
+K=/bracketleftbigg1
+2/parenleftBig
+3B−2+/radicalbig
+3B(3B−4)/parenrightBig/bracketrightbigg1/3
+, (B18)
+B=9α2γ
+β3=γ
+βF2=96πn∗
+F2ǫsc+ǫins
+ǫsc−ǫins. (B19)
+The density ndeplcan be calculated from the total band
+bending eφ0=ECb−ECs(b=bulk, s=surface),
+endepl=e(NA−ND)zd=/radicalbig
+2(NA−ND)ǫscǫ0eφ0d,
+(B20)
+eφ0d=eφ0−eφ0s−kBT. (B21)
+Hereeφ0dis the band bending caused by the charges in
+the depletion layer (acceptors and donors) and eφ0sthat
+caused by the free electrons in the inversion layer. To
+get this equations one has to assume that in the deple-
+tion layer all acceptors are charged and that in the bulk
+there is charge neutrality. The boundary between the de-
+pletion layer and the bulk is not sharp, this is described
+by the term −kBTineφ0d.36To calculate φ0sone has to
+solve the Poisson equation with the charge density which
+corresponds with the test wave function ϕ, the result is
+φ0s=3ens
+bǫscǫ0. (B22)As can be seen from Fig. 4 for the total band bending
+eφ0=ECb−ECs,
+eφ0=ECb−µ+µ−E0+E0−ECs,
+eφ0=−µCb+µE0+ε0 (B23)
+holds. The chemical potential relative to the conduction
+band edge in the bulk µCbis determined by the charge
+neutrality and can be calculated by solving
+NCF1/2/parenleftbiggµCb
+kBT/parenrightbigg
++NA
+gAexp/parenleftBig
+εA−εg−µCb
+kBT/parenrightBig
++1=
+=NVF1/2/parenleftbigg−εg−µCb
+kBT/parenrightbigg
+(B24)
+numerically. NCandNVare the effective densities of
+state in the conduction band and in the valence band, εA
+is the acceptor ionization energy, gAthe acceptor degen-
+eracy factor, and εgthe gap energy. According to Sze27
+this quantities are given by
+NC=gsgv3D/parenleftbiggmde3DkBT
+2π/planckover2pi12/parenrightbigg3/2
+, (B25)
+NV=gs/parenleftbiggmdh3DkBT
+2π/planckover2pi12/parenrightbigg3/2
+, (B26)
+εg=εg0Si−αSiT2
+T+βSi. (B27)
+The values we used can be found in Table I.
+The ground state energy ε0itself is a (small) part of
+the the total band bending, this problem is solved by a
+fix point iteration using ε0= 0as start value.
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\ No newline at end of file
diff --git a/1001.0206.txt b/1001.0206.txt
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+arXiv:1001.0206v1  [math.PR]  4 Jan 2010Stochastic control under progressive enlargement of filtra tions
+and applications to multiple defaults risk management∗
+Huyˆ en PHAM
+Laboratoire de Probabilit´ es et
+Mod` eles Al´ eatoires
+CNRS, UMR 7599
+Universit´ e Paris 7
+e-mail: pham@math.jussieu.fr
+and Institut Universitaire de France
+October 16, 2018
+Abstract
+We formulate and investigate a general stochastic control proble m under a progre-
+ssive enlargement of filtration. The global information is enlarged fr om a reference
+filtration and the knowledge of multiple random times together with as sociated marks
+when they occur. By working under a density hypothesis on the con ditional joint dis-
+tribution of the random times and marks, we prove a decomposition o f the original
+stochastic control problem under the global filtration into classica l stochastic control
+problems under the reference filtration, which are determined in a fi nite backward
+induction. Our method revisits and extends in particular stochastic control of diffu-
+sion processes with finite number of jumps. This study is motivated b y optimization
+problems arising in default risk management, and we provide applicatio ns of our de-
+composition result for the indifference pricing of defaultable claims, a nd the optimal
+investment under bilateral counterparty risk. The solutions are e xpressed in terms of
+BSDEs involving only Brownian filtration, and remarkably without jump terms coming
+from the default times and marks in the global filtration.
+Key words: stochastic control, progressive enlargement of filtration s, decomposition in
+the reference filtration, multiple default times, risk mana gement.
+MSC Classification (2000): 60J75, 93E20, 60H20.
+∗I would like to thank N. El Karoui, Y. Jiao and I. Kharroubi for useful comments.
+11 Introduction
+The field of stochastic control has known important developm ents over these last years,
+inspiredespeciallybyvariousproblemsineconomicsandfin ancearisinginriskmanagement,
+option hedging, optimal investment, portfolio selection o r real options valuation. A vast li-
+terature on this topic and its applications has grown with di fferent approaches ranging from
+dynamic programming method, Hamilton-Jacobi-Bellman Par tial Differential Equations
+(PDEs) and Backward Stochastic Differential Equations (BSDE s) to convex martingale
+duality methods. We refer to the monographs [7], [26], [18] o r [19] for recent updates on
+this subject. In particular, the theory of BSDEs has emerged as a major research topic
+with original and significant contributions related to stoc hastic control and its financial
+applications, see a recent overview in [5].
+On the other hand, the field of enlargement of filtrations is a t raditional subject in prob-
+ability theory initiated by fundamental works of the French school in the 80s, see e.g. [11],
+[9], [12], and the recent lecture notes [16]. It knows a renew ed interest due to its natural
+application in credit risk research where it appears as a pow erful tool for modelling default
+events. For an overview, we refer to the books [3], [4], [23] o r the lecture notes [2]. The
+standard approach of credit event is based on the enlargemen t of a reference filtration F
+(the default-free information structure) by the knowledge of a default time when it occurs,
+leading to the global filtration G, and called progressive enlargement of filtrations. More-
+over, it assumes that the credit event should arrive by surpr ise, i.e. it is a totally inacessible
+random time for the reference filtration. Hence, the main app roaches consist in modelling
+the intensity of the random time (usually referred to as the r educed-form approach), or
+more generally in the modelling of the conditional law of thi s random time, and referred to
+as density hypothesis, see [6]. The stability of the class of semimartingale, usually called
+(H’)hypothesis, and meaning that any F-semimartingale remains a G-semimartingale, is
+a fundamental property both in probability and finance where it is closely related to the
+absence of arbitrage. It holds true under the density hypoth esis, and the related canoni-
+cal decomposition in the enlarged filtration can be explicit ly expressed, as shown in [10].
+A stronger assumption than (H’)hypothesis is the so-called immersion property or (H)
+hypothesis, denoting the fact that F-martingales remain G-martingales.
+The purpose of this paper is to combine both features of stoch astic control and progre-
+ssive enlargement of filtrations in view of applications in fi nance, in particular for defaults
+risk management. We formulate and study the general structu re for such control problems
+by consideringaprogressiveenlargement with multipleran domtimes andassociated marks.
+These marks represent for example in credit events jump size s of asset values, which may
+arrive several times by surprise and cannot be predicted fro m the past observation of as-
+set processes. We work under the density hypothesis on the co nditional joint distribution
+of the random times and marks. Our new approach consist in dec omposing the initial
+control problem in the G-filtration into a finite sequence of control problems formul ated
+in theF-filtration, and which are determined recursively. This is b ased on an enlight-
+ening representation of any G-predictable or optional process that we split into indexed
+F-predictable or optional processes between each random tim e. This point of view allows
+2us to change of regimes in the state process, and to modify the control set and the gain
+functions between randomtimes. Thisflexibility intheform ulation of thestochastic control
+problem appears also quite useful and relevant for financial interpretation. Our method
+consist basically in projecting G-processes into the reference F-filtration between two ran-
+dom times, and features some similarities with filtering app roach. This contrasts with the
+standard approach in progressive enlargement of filtration focusing on the representation of
+controlled state process in the G-filtration wherethe control set has to befixed at the initial
+time. Moreover, in this global approach, one usually assume s that(H)hypothesis holds in
+order to get a martingale representation in the G-filtration. In this case, the solution is then
+characterized from dynamic programming method in the G-filtration via PDEs with inte-
+grodifferential terms or BSDEs with jumps. By means of our F-decomposition result under
+the density hypothesis (and without assuming (H)hypothesis), we can solve each stochas-
+tic control problem by dynamic programming in the F-filtration, which leads typically to
+PDEs or BSDEs related only to Brownian motion, thus simpler a priori than Integro-PDEs
+and BSDEs with jumps. Our decomposition method revisits and more importantly extends
+stochastic control of diffusion processes with finite number o f jumps, and gives some new
+insight for studying Integro-PDEs and BSDEs with jumps. We i llustrate our methodology
+with two financial applications in default risk management. The first one considers the
+problem of indifference pricing of defaultable claims, and th e second application deals with
+an optimal investment problem under bilateral contagion ri sk with two nonordered default
+times. The solutions are explicitly expressed in terms of BS DEs involving only Brownian
+motion.
+The paper is organized as follows. The next section presents the general framework of
+progressive enlargement of filtration with successive rand om times and marks. We state
+the decomposition result for a G-predictable and optional process, and as a consequence
+we derive under the density hypothesis the computation of ex pectation functionals of G-
+optional processes in terms of F-expectations. In Section 3, we formulate the abstract
+stochastic control problem in this context and connect it in particular to diffusion processes
+with jumps. Section 4 contains the main F-decomposition result of the initial stochastic
+control problem. The case of enlargement of filtration with m ultiple (and not necessarily
+successive) random times is considered in Section 5, and we s how how to derive the results
+from the case of successive random times with auxiliary mark s. Finally, Section 6is devoted
+to some applications in risk management, where we present th e results and postpone the
+detailed proofs and more examples in a forthcoming paper [13 ].
+2 Progressive enlargement of filtration with successive ran -
+dom times
+We fix a probability space (Ω ,G,P), and we start with a reference filtration F= (Ft)t≥0
+satisfying the usual conditions ( F0contains the null sets of PandFis right continuous: Ft
+=Ft+:=∩s>tFs). We consider a vector of nrandom times τ1,...,τn(i.e. nonnegative
+G-random variables) and a vector of nG-measurable random variables ζ1,...,ζnvalued in
+some Borel subset EofRm. The default information is the knowledge of these default
+3timesτkwhen they occur, together with the associated marks ζk. For eachk= 1,...,n, it
+is defined in mathematical terms as the smallest right-conti nuous filtration Dk= (Dk
+t)t≥0
+such thatτkis aDk-stopping time, and ζkisDk
+τk-measurable. In other words, Dk
+t=
+˜Dk
+t+, where ˜Dk
+t=σ(ζk1τk≤s,s≤t)∨σ(1τk≤s,s≤t). The global market information is
+then defined by the progressive enlargement of filtration G=F∨D1∨...∨Dn. The
+filtration G= (Gt)t≥0is the smallest filtration containing F, and such that for any k=
+1,...,n,τkis aG-stopping time, and ζkisGτk-measurable. With respect to the classical
+framework of progressive enlargement of filtration with a si ngle random time extensively
+studied in the literature, we consider here multiple random times together with marks. For
+simplicity of presentation, we first consider the case where the random times are ordered,
+i.e.τ1≤...≤τn, and so valued in ∆ non{τn<∞}, where
+∆k=/braceleftBig
+(θ1,...,θk)∈(R+)k:θ1≤...≤θk,/bracerightBig
+, k= 1,...,n.
+This means actually that the observations of interest are th e ranked default times (together
+with the marks). We shall indicate in Section 5 how to adapt th e results in the case of
+multiple random times not necessarily ordered.
+We introduce some notations used throughout the paper.
+-P(F) (resp. P(G)) is theσ-algebra of F(resp.G)-predictable measurable subsets on
+R+×Ω, i.e. the σ-algebra generated by the left-continuous F-adapted (resp. G-adapted)
+processes. We also let PF(resp.PG) denote the set of processes that are F-predictable
+(resp.G-predictable), i.e. P(F)-measurable (resp. P(G)-measurable).
+-O(F) (resp.O(G)) is theσ-algebra of F(resp.G)-optional measurable subsets on R+×Ω,
+i.e. theσ-algebra generated bytheright-continuous F-adapted(resp. G-adapted)processes.
+We alsolet OF(resp.OG)denotetheset ofprocessesthat are F-optional (resp. G-optional),
+i.e.O(F)-measurable (resp. O(G)-measurable).
+- Fork= 1,...,n, we denote by Pk
+F(∆k,Ek) (resp. Ok
+F(∆k,Ek)) the set of of indexed
+processesYk(.) such that the map ( t,ω,θ1,...,θk,e1,...,ek)→Yk
+t(ω,θ1,...,θk,e1,...,ek)
+isP(F)⊗B(∆k)⊗B(Ek)-measurable (resp. O(F)⊗B(∆k)⊗B(Ek)-measurable).
+- Forθ= (θ1,...,θn)∈∆n,e= (e1,...,en)∈En, we denote by
+θ(k)= (θ1,...,θk), e(k)= (e1,...,ek), k= 1,...,n.
+The following result provides the key decomposition of pred ictable and optional pro-
+cesses with respect to this progressive enlargement of filtr ation. This extends a classical
+result, see e.g. Lemma 4.4 in [11] or Chapter 6 in [21], stated for a progressive enlargement
+of filtration with a single random time.
+Lemma 2.1 AnyG-predictable process Y= (Yt)t≥0is represented as
+Yt=Y0
+t1t≤τ1+n−1/summationdisplay
+k=1Yk
+t(τ1,...,τk,ζ1,...,ζk)1τk<t≤τk+1
++Yn
+t(τ1,...,τn,ζ1,...,ζn)1τn<t, t≥0, (2.1)
+4whereY0∈ PF, andYk∈ Pk
+F(∆k,Ek), fork= 1,...,n. AnyG-optional process Y=
+(Yt)t≥0is represented as
+Yt=Y0
+t1t<τ1+n−1/summationdisplay
+k=1Yk
+t(τ1,...,τk,ζ1,...,ζk)1τk≤t<τk+1
++Yn
+t(τ1,...,τn,ζ1,...,ζn)1τn≤t, t≥0, (2.2)
+whereY0∈ OF, andYk∈ Ok
+F(∆k,Ek), fork= 1,...,n.
+Proof.We prove the decomposition result for predictable processe s by induction on n. We
+denote by Gn=F∨D1∨...∨Dn.
+Step 1. Suppose first that n= 1, so that G=F∨D1. Let us consider generators of P(G),
+which are processes in the form
+Yt=fsg(ζ11τ1≤s)h(τ1∧s)1t>s, t≥0,
+withs≥0,fsFs-measurable, gmeasurable defined on E∪{0}, andhmeasurable defined
+onR+. By taking
+Y0
+t=fsg(0)h(s)1t>s,andY1
+t(θ1,e) =fsg(e1θ1≤s)h(θ1∧s)1t>s,
+we see that the decomposition (2.1) holds for generators of P(G). We then extend this
+decomposition for any P(G)-measurable processes, by the monotone class theorem.
+Step 2.Suppose that the result holds for n, and consider the case with n+1 ranked default
+times, so that G=Gn∨Dn+1,Dn+1
+t=˜Dn+1
+t+, where ˜Dn+1
+t=σ(ζn+11τn+1≤s,s≤t)∨
+σ(1τn+1≤s,s≤t). By the same arguments of enlargement of filtration with one default time
+as in Step 1, we derive that any P(G)-measurable process Yis represented as
+Yt=Y0,(n)
+t1t≤τn+1+Y1,(n)
+t(τn+1,ζn+1)1τn+1<t, (2.3)
+whereY0,(n)isP(Gn)-measurable, and ( t,ω,θn+1,en+1)/mapsto→Y1,(n)
+t(ω,θn+1,en+1) isP(Gn)⊗
+B(R+)⊗B(E)-measurable. Now, from the induction hypothesis for Gn, we have
+Y0,(n)
+t=Y0,0,(n)
+t1t≤τ1+n−1/summationdisplay
+k=1Yk,0,(n)
+t(τ1,...,τk,ζ1,...,ζk)1τk<t≤τk+1
++Yn,0,(n)
+t(τ1,...,τn,ζ1,...,ζn)1τn<t, t≥0,
+whereY0,0,(n)∈ PF, andYk,0,(n)∈ Pk
+F(∆k,Ek), fork= 1,...,n. Similarly, we have
+Y1,(n)
+t(θn+1,en+1) =Y0,1,(n)
+t(θn+1,en+1)1t≤τ1
++n−1/summationdisplay
+k=1Yk,1,(n)
+t(τ1,...,τk,ζ1,...,ζk,θn+1,en+1)1τk<t≤τk+1
++Yn,1,(n)
+t(τ1,...,τn,ζ1,...,ζn,θn+1,en+1)1τn<t, t≥0,
+5whereY0,1,(n)∈P1
+F(R+,E),Yk,1,(n)∈Pk+1
+F(∆k×R+,Ek+1),k=1,...,n. Finally, plugging
+these two decompositions with respect to P(Gn) into relation (2.3), and recalling that
+τ1≤...≤τn≤τn+1, we get the required decomposition at level n+1 forG:
+Yt=Y0,0,(n)
+t1t≤τ1+n/summationdisplay
+k=1Yk,0,(n)
+t(τ1,...,τk,ζ1,...,ζk)1τk<t≤τk+1
++Yn+1
+t(τ1,...,τn+1,ζ1,...,ζn+1)1τn+1<t, t≥0,
+where we notice that the indexed process Yn+1defined byYn+1(θ1,...,θn+1,e1,...,en+1)
+:=Yn,1,(n)(θ1,...,θn,e1,...,en,θn+1,en+1), lies in Pn+1
+F(∆n+1,En+1).
+The decomposition result for G-optional processes is proved similarly by induction and
+considering generators of O(G1), which are processes in the form
+Yt=fsg(ζ11τ1≤s)h(τ1∧s)1t≥s, t≥0,
+withs≥0,fsFs-measurable, gmeasurable defined on E∪{0}, andhmeasurable defined
+onR+. ✷
+In view of the decomposition (2.1) or (2.2), we can then ident ify anyY∈ PG(resp.OG)
+with ann+1-tuple (Y0,...,Yn)∈ PF×...×Pn
+F(∆n,En) (resp.OF×...×On
+F(∆n,En)).
+We now require a density hypothesis on the random times and th eir associated jumps
+by assuming that for any t, the conditional distribution of ( τ1,...,τn,ζ1,...,ζn) given
+Ftis absolutely continuous with respect to a positive measure λ(dθ)η(de) onB(∆n)⊗
+B(En), withλthe Lebesgue measure λ(dθ) =dθ1...dθn, andηa product measure η(de)
+=η1(de1)...η1(den) onB(E)⊗...⊗B(E). More precisely, we assume that there exists γ
+∈ On
+F(∆n,En) such that
+(DH)P/bracketleftbig
+(τ1,...,τn,ζ1,...,ζn)∈dθde|Ft/bracketrightbig
+=γt(θ1,...,θn,e1,...,en)
+dθ1...dθnη1(de1)...η1(den), a.s.
+Remark 2.1 In the particular case where γis in the form γt(θ,e) =ϕt(θ)ψt(e), the con-
+dition(DH)means that the random times ( τ1,...,τn) and the jump sizes ( ζ1,...,ζn) are
+independent given Ft, for allt≥0, and
+P/bracketleftBig
+(τ1,...,τn)∈dθ|Ft/bracketrightbig
+=ϕt(θ)λ(dθ),P/bracketleftBig
+(ζ1,...,ζn)∈de|Ft/bracketrightbig
+=ψt(e)η(de), a.s.
+This condition extends the usual density hypothesis for a ra ndom time in the theory of
+initial or progressive enlargement of filtration, see [9] or [10]. An important result in
+the theory of enlargement of filtration under the density hyp othesis is the semimartingale
+invariance property, also called (H’)hypothesis, i.e. any F-semimartingale remains a G-
+semimartingale. This result is related in finance to no-arbi trage conditions, and is thus also
+a desirable property from a economical viewpoint. Random ti mes satisfying the density
+hypothesis are very well suitable for the analysis of credit risk events, as shown recently
+in [6]. We also refer to this paper for a discussion on the rela tion between the density
+hypothesis and the reduced-form (or intensity) approach in credit risk modelling.
+6In the sequel, it is useful to introduce the following notati ons. We denote by γ0the
+F-optional process defined by
+γ0
+t=P/bracketleftbig
+τ1>t/vextendsingle/vextendsingleFt/bracketrightbig
+(2.4)
+=/integraldisplay
+En/integraldisplay∞
+t/integraldisplay∞
+θ1.../integraldisplay∞
+θn−1γt(θ1,...,θn,e1,...,en)dθn...dθ1η1(de1)...ηn(den),
+and we denote by γk,k= 1,...,n−1, the indexed process in Ok
+F(∆k,Ek) defined by
+γk
+t(θ1,...,θk,e1,...,ek)
+=/integraldisplay
+En−k/integraldisplay∞
+t/integraldisplay∞
+θk+1.../integraldisplay∞
+θn−1γt(θ1,...,θn,e1,...,en)dθn...dθk+1η1(dek+1)...η1(den),
+so that for k= 1,...,n−1,
+P/bracketleftbig
+τk+1>t/vextendsingle/vextendsingleFt/bracketrightbig
+=/integraldisplay
+Ek/integraldisplay
+∆kγk
+t(θ1,...,θk,e1,...,ek)dθ1...dθkη1(de1)...η1(dek).(2.5)
+Notice that the family of measurable maps γk,k= 0,...,ncan be also written in backward
+induction by
+γk
+t(θ1,...,θk,e1,...,ek) =/integraldisplay
+E/integraldisplay∞
+tγk+1
+t(θ1,...,θk+1,e1,...,ek+1)dθk+1η1(dek+1),
+fork= 0,...,n−1, starting from γn=γ. In view of (2.4)-(2.5), the process γkmay be
+interpreted as the survival density process of τk+1,k= 0,...,n−1.
+Thenextresultprovidesthecomputationfortheoptionalpr ojectionofa O(G)-measurable
+process on the reference filtration F.
+Lemma 2.2 LetY= (Y0,...,Yn)be a nonnegative (or bounded) G-optional process.
+Then for any t≥0, we have
+ˆYF
+t:=E/bracketleftbig
+Yt/vextendsingle/vextendsingleFt/bracketrightbig
+=Y0
+tγ0
+t+n/summationdisplay
+k=1/integraldisplay
+Ek/integraldisplayt
+0.../integraldisplayt
+θk−1Yk
+t(θ1,...,θk,e1,...,ek)
+γk
+t(θ1,...,θk,e1,...,ek)dθk...dθ1η1(de1)...η1(dek),
+where we used the convention that θk−1= 0fork= 1in the above integral. Equivalently,
+we have the backward induction formula for ˆYF
+t=ˆY0,F
+t, where the ˆYk,F
+tare given for any
+t≥0, by
+ˆYn,F
+t(θ,e) =Yn
+t(θ,e)γt(θ,e)
+ˆYk,F
+t(θ(k),e(k)) =Yk
+t(θ(k),e(k))γk
+t(θ(k),e(k))
++/integraldisplayt
+θk/integraldisplay
+EˆYk+1,F
+t(θ(k),θk+1,e(k),ek+1)η1(dek+1)dθk+1,
+forθ= (θ1,...,θn)∈∆n∩[0,t]n,e= (e1,...,en)∈En.
+7Proof.LetY= (Y0,...,Yn) be a nonnegative (or bounded) G-optional process, decom-
+posed as in (2.2) so that:
+E/bracketleftbig
+Yt/vextendsingle/vextendsingleFt/bracketrightbig
+=E/bracketleftBig
+Y0
+t1t<τ1/vextendsingle/vextendsingle/vextendsingleFt/bracketrightBig
++n/summationdisplay
+k=1E/bracketleftBig
+Yk
+t(τ1,...,τk,ζ1,...,ζk)1τk≤t<τk+1/vextendsingle/vextendsingle/vextendsingleFt/bracketrightBig
+,(2.6)
+with the convention that τn+1=∞. Now, for any k= 1,...,n, we have under the density
+hypothesis (DH)
+E/bracketleftBig
+Yk
+t(τ1,...,τk,ζ1,...,ζk)1τk≤t<τk+1/vextendsingle/vextendsingle/vextendsingleFt/bracketrightBig
+=/integraldisplay
+∆n×EnYk
+t(θ1,...,θk,e1,...,ek)1θk≤t<θk+1γt(θ1,...,θn,e1,...,en)λ(dθ)η(de)
+=/integraldisplay
+Ek/integraldisplayt
+0.../integraldisplayt
+θk−1Yk
+t(θ1,...,θk,e1,...,ek)γk
+t(θ1,...,θk,e1,...,ek)dθk...dθ1
+η1(de1)...η1(dek),
+where the second inequality follows from Fubini’s theorem a nd the definition of γk. We
+also have
+E/bracketleftBig
+Y0
+t1t<τ1/vextendsingle/vextendsingle/vextendsingleFt/bracketrightBig
+=Y0
+tP[τ1>t|Ft] =Y0
+tγ0
+t.
+We then get the required result by plugging these two last rel ations into (2.6). Finally,
+the backward formula for the F-optional projection of Yis obtained by a straightforward
+induction. ✷
+As a consequence of the above backward induction formula for the optional projection,
+we derive a bakward formula for the computation of expectati on functionals of G-optional
+processes, which involves only F-expectations.
+Proposition 2.1 LetY= (Y0,...,Yn)andZ= (Z0,...,Zn)be two nonnegative (or
+bounded) G-optional processes, and fix T∈(0,∞).
+The expectation E[/integraltextT
+0Ytdt+ZT]can be computed in a backward induction as
+E/bracketleftBig/integraldisplayT
+0Ytdt+ZT/bracketrightBig
+=J0
+where theJk,k= 0,...,nare given by
+Jn(θ,e) =E/bracketleftBig/integraldisplayT
+θnYn
+tγt(θ,e)dt+Zn
+TγT(θ,e)/vextendsingle/vextendsingle/vextendsingleFθn/bracketrightBig
+Jk(θ(k),e(k)) =E/bracketleftBig/integraldisplayT
+θkYk
+tγk
+t(θ(k),e(k))dt+Zk
+Tγk
+T(θ(k),e(k))
++/integraldisplayT
+θk/integraldisplay
+EJk+1(θ(k),θk+1,e(k),ek+1)η1(dek+1)dθk+1/vextendsingle/vextendsingle/vextendsingleFθk/bracketrightBig
+,
+forθ= (θ1,...,θn)∈∆n∩[0,T]n,e= (e1,...,en)∈En, with the convention θ0= 0.
+8Proof.For anyθ= (θ1,...,θn)∈∆n∩[0,T]n,e= (e1,...,en)∈En, let us define
+Jk(θ(k),e(k)) =E/bracketleftBig/integraldisplayT
+θkˆYk,F
+t(θ(k),e(k))dt+ˆZk,F
+T(θ(k),e(k))/vextendsingle/vextendsingleFθk/bracketrightBig
+,
+where the ˆYk,FandˆZk,Fare defined in Lemma 2.2, associated respectively to YandZ.
+ThenJ0=E[/integraltextT
+0Ytdt+ZT], and we see from the backward induction for ˆYk,FandˆZk,F
+that theJk,k= 0,...,n, satisfy
+Jn(θ,e) =E/bracketleftBig/integraldisplayT
+θnYn
+tγt(θ,e)dt+Zn
+T(θ,e)γT(θ,e)/vextendsingle/vextendsingle/vextendsingleFθn/bracketrightBig
+Jk(θ(k),e(k)) =E/bracketleftBig/integraldisplayT
+θkYk
+tγk
+t(θ(k),e(k))dt+Zk
+Tγk
+T(θ(k),e(k))
++/integraldisplayT
+θk/integraldisplayt
+θk/integraldisplay
+EˆYk+1,F
+t(θ(k),θk+1,e(k),ek+1)η1(dek+1)dθk+1dt
++/integraldisplayT
+θk/integraldisplay
+EˆZk+1,F
+T(θ(k),θk+1,e(k),ek+1)η1(dek+1)dθk+1/vextendsingle/vextendsingle/vextendsingleFθk/bracketrightBig
+=E/bracketleftBig/integraldisplayT
+θkYk
+tγk
+t(θ(k),e(k))dt+Zk
+Tγk
+T(θ(k),e(k))
++/integraldisplayT
+θk/integraldisplay
+E/integraldisplayT
+θk+1ˆYk+1,F
+t(θ(k),θk+1,e(k),ek+1)dtη1(dek+1)dθk+1
++/integraldisplayT
+θk/integraldisplay
+EˆZk+1,F
+T(θ(k),θk+1,e(k),ek+1)η1(dek+1)dθk+1/vextendsingle/vextendsingle/vextendsingleFθk/bracketrightBig
+=E/bracketleftBig/integraldisplayT
+θkYk
+tγk
+t(θ(k),e(k))dt+Zk
+Tγk
+T(θ(k),e(k))
++/integraldisplayT
+θk/integraldisplay
+EJk+1(θ(k),θk+1,e(k),ek+1)η1(dek+1)dθk+1/vextendsingle/vextendsingle/vextendsingleFθk/bracketrightBig
+,
+where we used Fubini’s theorem in the second equality for Jk, and the law of iterated
+condional expectations for the last equality. This proves t he required induction formula for
+Jk,k= 0,...,n. ✷
+3 Abstract stochastic control problem
+In this section, we formulate the general stochastic contro l problem in the context of pro-
+gressively enlargement of filtration with successive rando m times and marks.
+3.1 Controls and state process
+A control is a G-predictable process α= (α0,...,αn)∈ PF×...×Pn
+F(∆n,En), where the
+αk,k= 0,...,n, are valued in some given Borel set Akof an Euclidian space. We denote by
+PF(A0) (resp.Pk
+F(∆k,Ek;Ak),k= 1,...,n), the set of elements in PF(resp.Pk
+F(∆,Ek),
+k= 1,...,n) valued in A0(resp.Ak,k= 1,...,n). We setA=A0×...×An, and denote
+byAGthe set of admissible controls as the product A0
+F×...×An
+F, whereA0
+F(resp.Ak
+F,k
+9= 1,...,n) is some separable metric space of PF(A0) (resp.Pk
+F(∆k,Ek;Ak),k= 1,...,n).
+The separability condition is required for measurability s election issue.
+The description of the controlled state process is formulat ed as follows:
+•Controlled state process between default times: we are given a collection of measurable
+mappings:
+(x,α0)∈Rd×A0
+F/mapsto−→X0,x,α0∈ OF (3.1)
+(x,αk)∈Rd×Ak
+F/mapsto−→Xk,x,αk∈ Ok
+F(∆k,Ek), k= 1,...,n, (3.2)
+such that we have the initial data:
+X0,x,α0
+0=x,∀x∈Rd,
+Xk,ξ,αk
+θk(θ1,...,θk,e1,...,ek) =ξ,∀ξFθk−measurable , k= 1,...,n.
+•Jumps of the controlled state process: we are given a collection of maps ΓkonR+×Ω×
+Rd×Ak−1×E, fork= 1,...,n, such that
+(t,ω,x,a,e )/mapsto→Γk
+t(ω,x,a,e) isP(F)⊗B(Rd)⊗B(Ak−1)⊗B(E)−measurable .
+•Global controlled state process: the controlled state process is then given by the mapping
+(x,α= (α0,...,αn))∈Rd×AG/mapsto−→Xx,α∈ OG,
+whereXx,αis the process equal to
+Xx,α
+t=¯X0
+t1t<τ1+n−1/summationdisplay
+k=1¯Xk
+t(τ1,...,τk,ζ1,...,ζk)1τk≤t<τk+1
++¯Xn
+t(τ1,...,τn,ζ1,...,ζn)1τn≤t, t≥0, (3.3)
+with (¯X0,...,¯Xn)∈ OF×...×On
+F(∆n,En) given by
+¯X0=X0,x,α0
+¯Xk(θ1,...,θk,e1,...,ek) =Xk,Γk
+θk(¯Xk−1
+θk,αk−1
+θk,ek),αk
+(θ1,...,θk,e1,...,ek),
+fork= 1,...,n.
+The interpretation is the following. Between the time inter valτk=θkandτk+1=θk+1,
+k= 0,...,n−1 (with the convention θ0= 0), the state process X=¯Xkis controlled
+byαk, which is based on the basic information F, and the knowledge of the past jump
+times and marks ( θ1,...,θk,e1,...,ek). Then, at time θk+1, there is a jump on the state
+process determined by the map Γk+1, which depends on the current state value, control
+and information, but also on a “nonpredictable” mark ζk+1=ek+1at timeθk+1:
+Xτk+1= Γk+1
+τk+1(Xτ−
+k+1,αk
+τk+1,ζk+1).
+103.2 Typical controlled state process
+In typical applications, the dynamics of X0=X0,x,α0,Xk=Xk,x,αk,k= 1,...,n, are
+governed by diffusion processes:
+dX0
+t=b0
+t(X0
+t,α0
+t)dt+σ0
+t(X0
+t,α0
+t)dWt, t≥0 (3.4)
+dXk
+t=bk
+t(Xk
+t,αk
+t,θ1,...,θk,e1,...,ek)dt (3.5)
++σk
+t(Xk
+t,αk
+t,θ1,...,θk,e1,...,ek)dWt, t≥θk,
+Here,Wis a standard m-dimensional ( P,F)-Brownian motion, and ( t,ω,x,a)→b0
+t(ω,x,a),
+σ0
+t(ω,x,a) areP(F)⊗B(Rd)⊗B(A0)-measurable maps valued respectively in RdandRd×m,
+fork=1,...,n,themaps( t,ω,x,a,θ 1,...,θk,e1,...,ek)→bk
+t(ω,x,a,θ 1,...,θk,e1,...,ek),
+σk
+t(ω,x,u,θ 1,...,θk,e1,...,ek) areP(F)⊗ B(Rd)⊗ B(Ak)⊗ B(∆k)⊗ B(Ek)-measurable
+valued respectively in RdandRd×m. To alleviate notations, we omitted in (3.5) the
+dependence of Xk,αkin (θ1,...,θk,e1,...,ek). We make the linear growth and Lips-
+chitz assumptions on the functions x→bk
+t(x,.),σk(x,.),k= 0,...,n, in order to en-
+sure for all ( θ1,...,θk,e1,...,ek)∈∆k×Ek, the existence and uniqueness of a solution
+Xk(θ1,...,θk,e1,...,ek) tothesde(3.4), (3.5), given thecontrols andtheinitial c onditions,
+and this indexed process Xklies inOk
+F(∆k,Ek). The dependence of the coefficients bk,σk
+on the past jump times θ1,...,θk, and marks e1,...,ek, corresponds to change of regimes
+after each jump time, and may be interpreted in finance as rati ng upgrades or downgrades.
+Also, typical choice for the set of admissible controls Ak
+Fis subset of indexed F-predictable
+processes in Lp,p∈[1,∞), and the separability of Ak
+Ffollows from the separability of Lp,
+see the discussion in [24].
+Connection with controlled jump-diffusion processes.
+Consider the particular case where the sets of controls Akare identical, equal to A, and let
+us define the mappings bandσonR+×Ω×Rd×Aby:
+bt(x,a) =b0
+t(x,a)1t≤τ1+n−1/summationdisplay
+k=1bk
+t(x,a,τ1,...,τk,ζ1,...,ζk)1τk<t≤τk+1
++bn
+t(x,a,τ1,...,τn,ζ1,...,ζn)1t>τn,
+σt(x,a) =σ0
+t(x,a)1t≤τ1+n−1/summationdisplay
+k=1σk
+t(x,a,τ1,...,τk,ζ1,...,ζk)1τk<t≤τk+1
++σn
+t(x,a,τ1,...,τn,ζ1,...,ζn)1t>τn,
+and notice that the maps ( t,ω,x,a)→bt(ω,x,a),σt(ω,x,a) areP(G)⊗ B(Rd)⊗ B(A)-
+measurable. Denote also by δthe mapping on R+×Ω×Rd×A×E:
+δt(x,a,e) =n−1/summationdisplay
+k=1/parenleftBig
+Γk
+t(x,a,e)−x/parenrightBig
+1τk<t≤τk+1+/parenleftBig
+Γn
+t(x,a,e)−x/parenrightBig
+1t>τn,
+which is P(G)⊗B(Rd)⊗B(A)⊗B(E)-measurable. Let us denote by µ(dt,de) the integer-
+valued random measure associated to the times τkand the marks ζk,k= 1,...,n, which
+11is then given by
+µ([0,t]×B) =/summationdisplay
+k≥11τn≤t1B(ζk),∀t≥0, B∈ B(E).
+The progressive enlarged filtration Gcan then be written also as: G=F∨Fµwhere
+Fµis the right-continuous filtration generated by the integer -valued random measure µ.
+Now, since the semimartingale property is preserved under t he density hypothesis for this
+progressive enlargement of filtration, (see [10]), the proc essWremains a semimartingale
+under (P,G) (with a canonical decomposition, which can be explicitly e xpressed in terms
+of the density). Then, we can write the dynamics of the state p rocessX=Xx,αin (3.3)
+as a controlled jump-diffusion process under ( P,G):
+dXt=bt(Xt,αt)dt+σt(Xt,αt)dWt+/integraldisplay
+Eδt(Xt−,αt,e)µ(dt,de).
+However, noticethatintheabove G-formulation, theprocess WisnotingeneralaBrownian
+motion under ( P,G), unless the so-called (H)immersion property is satisfied, i.e. the
+martingale property is preserved from FtoG, which corresponds to the particular case
+where the density satisfies: γt(θ,e) =γθ(θ,e) fort≥θ.
+In the classical formulation by controlled jump-diffusion pr ocesses, one has to fix a
+controlsetA, whichisinvariantduringthetimehorizon. Here, themoreg eneralformulation
+(3.3) allows us to consider different control sets Akbetween two default times, and this may
+be relevant in practical applications. Moreover, we have a s uitable decomposition of the
+coefficients and controlled state process between random tim es, which provides a natural
+interpretation in economics and finance.
+3.3 Stochastic control problem
+In the general framework for the controlled process in (3.3) , let us formulate the objective
+functionforthestochastic control problemonafinitehoriz onT. Theterminalgain function
+is given by a nonnegative map GTon Ω×Rdsuch that (ω,x)/mapsto→GT(ω,x) isGT⊗B(Rd)-
+measurable, and which may be represented as
+GT(x) =G0
+T(x)1T<τ1+n−1/summationdisplay
+k=1Gk
+T(x,τ1,...,τk,ζ1,...,ζk)1τk≤T<τk+1
++Gn
+T(x,τ1,...,τn,ζ1,...,ζn)1τn≤T,
+whereG0
+TisFT⊗B(Rd)-measurable, and Gk
+TisFT⊗B(Rd)⊗B(∆k)⊗B(Ek)-measurable,
+fork= 1,...,n. The running gain function is given by a nonnegative map fon Ω×Rd×A
+such that (t,ω,x,a)/mapsto→ft(ω,x,a) isO(G)⊗B(Rd)⊗B(A)-measurable, and which may be
+decomposed as
+ft(x,a) =f0
+t(x,a0)1t<τ1+n−1/summationdisplay
+k=1fk
+t(x,ak,τ1,...,τk,ζ1,...,ζk)1τk≤t<τk+1
++fn
+t(x,an,τ1,...,τn,ζ1,...,ζn)1τn≤t,
+12fora= (a0,...,a n)∈A=A0×...×An, wheref0isO(F)⊗B(Rd)⊗B(A0)-measurable,
+andfkisO(F)⊗B(Rd)⊗B(Ak)⊗B(∆k)⊗B(Ek)-measurable, for k= 1,...,n.
+The value function for the stochastic control problem is the n defined by:
+V0(x) = sup
+α∈AGE/bracketleftBig/integraldisplayT
+0ft(Xx,α
+t,αt)dt+GT(Xx,α
+T)/bracketrightBig
+, x∈Rd. (3.6)
+Remark 3.1 In the formulation (3.6) of our stochastic control problem, there is a change
+of regimes in the running and terminal gain after each defaul t time. This is in the spirit of
+the recent concept of forward or progressive utility functi ons introduced in [17].
+4F-decomposition of the stochastic control problem
+In this section, we provide a decomposition of the value func tion for the stochastic control
+problem in the G-filtration, defined in (3.6), that we formulate in a backward induction
+for value functions of stochastic control in the F-filtration. To alleviate notations, we shall
+often omit in (3.2) the dependence of Xk,xonαkand (θ1,...,θk,e1,...,ek) when there is
+no ambiguity.
+Theorem 4.1 The value function V0is obtained from the backward induction formula:
+Vn(x,θ,e) = esssup
+αn∈An
+FE/bracketleftBig/integraldisplayT
+θnfn
+t(Xn,x
+t,αn
+t,θ,e)γt(θ,e)dt
++Gn
+T(Xn,x
+T,θ,e)γT(θ,e)/vextendsingle/vextendsingle/vextendsingleFθn/bracketrightBig
+(4.1)
+Vk(x,θ(k),e(k)) = esssup
+αk∈Ak
+FE/bracketleftBig/integraldisplayT
+θkfk
+t(Xk,x
+t,αk
+t,θ(k),e(k))γk
+t(θ(k),e(k))dt
++Gk
+T(Xk,x
+T,θ(k),e(k))γk
+T(θ(k),e(k))
++/integraldisplayT
+θk/integraldisplay
+EVk+1/parenleftbig
+Γk+1
+θk+1(Xk,x
+θk+1,αk
+θk+1,ek+1),θ(k),θk+1,e(k),ek+1/parenrightbig
+η1(dek+1)dθk+1/vextendsingle/vextendsingle/vextendsingleFθk/bracketrightBig
+, k= 0,...,n−1, (4.2)
+for allθ= (θ1,...,θn)∈∆n∩[0,T]n,e= (e1,...,en)∈En,x∈Rd.
+Remark 4.1 Each step in the backward induction for the determination of the original
+value function V0leads to the formulation of a family of value functions assoc iated to
+standardstochastic control probleminthe F-filtration. Indeed, at step n,Vn(x,.) is afamily
+of value functions parametrized by ( θ1,...,θn)∈∆n, (e1,...,en)∈En, and corresponding
+to the stochastic control problem after the last default at t imeθn, with a running gain
+functionfn
+tand terminal gain function Gn
+Ton the controlled state process Xnin theF-
+filtration, and weighted by the O(F)-measurable process γ. Now, suppose that at step
+k+1, we have determined the family of value functions Vk+1(x,.), (θ1,...,θk+1)∈∆k+1,
+13(e1,...,ek+1)∈Ek+1, and denote by ˆVk+1the map on Ω ×Rd×Ak×∆k+1×Ek:
+ˆVk+1/parenleftbig
+x,ak,θ(k),θk+1,e(k)/parenrightbig
+=/integraldisplay
+EVk+1/parenleftbig
+Γk+1
+θk+1(x,ak,ek+1),θ(k),θk+1,e(k),ek+1/parenrightbig
+η1(dek+1).
+Then, the family of value functions at step k, representing the value for the stochastic
+control problem after kdefaults, is computed from the stochastic control problem i n the
+F-filtration with the running gain function fk
+tand terminal gain function Gk
+Tweighted by
+theO(F)-measurable random variable γk, and with the running gain function ˆVk+1:
+Vk(x) = esssup
+αk∈Ak
+FE/bracketleftBig/integraldisplayT
+θkfk
+t(Xk,x
+t,αk
+t)γk
+tdt+Gk
+T(Xk,x
+T)γk
+T
++/integraldisplayT
+θkˆVk+1(Xk,x
+θk+1,αk
+θk+1,θk+1)dθk+1/vextendsingle/vextendsingle/vextendsingleFθk/bracketrightBig
+. (4.3)
+Here, we omitted the dependence in θ(k)=(θ1,...,θk),e(k)= (e1,...,ek) to alleviate nota-
+tions. The two first terms in the rhs of (4.3) represent the gai n functional when there is no
+more default after the k-th one, while the last term represents the gain in the case wh en a
+k+ 1-th default would occur between the last one at time τk=θkand the finite horizon
+T. Finally, the decomposition in Theorem 4.1 also shows that a n optimal control for the
+global problem in the G-filtration is obtained by a concatenation of optimal contro ls for
+each subproblems Vkin theF-filtration.
+Proof of Theorem 4.1 .
+Fixx∈Rd,α= (α0,...,αn)∈ AG, and consider the controlled state process Xx,α. By
+definition of Xx,αin (3.3),GT(.) andft(.), observe that the GT-measurable random variable
+GT(Xx,α
+T) is decomposed according to the n+1-tuple (G0
+T(¯X0
+T),...,Gn
+T(¯Xn
+T)), and the G-
+optional process ft(Xx,α
+t,αt) is decomposed as ( f0
+t(¯X0
+t,α0
+t),...,fn
+t(¯Xn
+t,αn
+t)). Let us now
+define by backward induction the maps Jk,k= 0,...,nby
+Jn(x,θ,e,α) =E/bracketleftBig/integraldisplayT
+θnfn
+t(Xn,x
+t,αn
+t,θ,e)γt(θ,e)dt+Gn
+T(Xn,x
+T,θ,e)γT(θ,e)/vextendsingle/vextendsingle/vextendsingleFθn/bracketrightBig
+Jk(x,θ(k),e(k),α) =E/bracketleftBig/integraldisplayT
+θkfk
+t(Xk,x
+t,αk
+t,θ(k),e(k))γk
+t(θ(k),e(k))dt
++Gk
+T(Xk,x
+T,θ(k),e(k))γk
+T(θ(k),e(k))
++/integraldisplayT
+θk/integraldisplay
+EJk+1/parenleftbig
+Γk+1
+θk+1(Xk,x
+θk+1,αk
+θk+1,ek+1),θ(k),θk+1,e(k),ek+1,α/parenrightbig
+η1(dek+1)dθk+1/vextendsingle/vextendsingle/vextendsingleFθk/bracketrightBig
+, (4.4)
+for anyx∈Rd,θ= (θ1,...,θn)∈∆n∩[0,T]n,e= (e1,...,en)∈En, andα= (α0,...,αn)
+∈ A0
+F×...× An
+F. Let us denote by ¯Jk(θ(k),e(k)) =Jk(¯Xk
+θk,θ(k),e(k),α),k= 0,...,n,
+and observe by definition of Xx,αand¯Xkin (3.3) that ¯Jksatisfy the backward induction
+14formula:
+¯Jn(θ,e) =E/bracketleftBig/integraldisplayT
+θnfn
+t(¯Xn
+t,αn
+t,θ,e)γt(θ,e)dt+Gn
+T(¯Xn
+T,θ,e)γT(θ,e)/vextendsingle/vextendsingle/vextendsingleFθn/bracketrightBig
+¯Jk(θ(k),e(k)) =E/bracketleftBig/integraldisplayT
+θkfk
+t(¯Xk
+t,αk
+t,θ(k),e(k))γk
+t(θ(k),e(k))dt
++Gk
+T(¯Xk
+T,θ(k),e(k))γk
+T(θ(k),e(k))
++/integraldisplayT
+θk/integraldisplay
+E¯Jk+1(θ(k),θk+1,e(k),ek+1)η1(dek+1)dθk+1/vextendsingle/vextendsingle/vextendsingleFθk/bracketrightBig
+.
+Therefore, from Proposition 2.1, we have the equality:
+E/bracketleftBig/integraldisplayT
+0f(Xx,α
+t,αt)dt+GT(Xx,α
+T)/bracketrightBig
+=¯J0=J0(x,α). (4.5)
+Let us now define the value function processes:
+Vk(x,θ(k),e(k)) := esssup
+α∈AGJk(x,θ(k),e(k),α), (4.6)
+fork= 0,...,n,x∈Rd, andθ= (θ1,...,θn)∈∆n∩[0,T]n,e= (e1,...,en)∈En. First,
+observe that this definition for k= 0 is consistent with the definition of the value function
+V0of the stochastic control problem (3.6) from the relation (4 .5). Then, it remains to prove
+that the value functions Vkdefined in (4.6) satisfy the backward induction formula in th e
+assertion of the theorem. For k=n, and since Jn(x,θ,e,α) depends on αonly through its
+last component αn, the relation (4.1) holds true. Next, from the backward indu ction (4.4)
+forJk, and the definition of Vk+1, we have for all α= (α0,...,αn)∈ AG:
+Jk(x,θ(k),e(k),α)≤E/bracketleftBig/integraldisplayT
+θkfk
+t(Xk,x
+t,αk
+t,θ(k),e(k))γk
+t(θ(k),e(k))dt
++Gk
+T(Xk,x
+T,θ(k),e(k))γk
+T(θ(k),e(k))
++/integraldisplayT
+θk/integraldisplay
+EVk+1/parenleftbig
+Γk+1
+θk+1(Xk,x
+θk+1,αk
+θk+1,ek+1),θ(k),θk+1,e(k),ek+1/parenrightbig
+η1(dek+1)dθk+1/vextendsingle/vextendsingle/vextendsingleFθk/bracketrightBig
+≤¯Vk(x,θ(k),e(k)), (4.7)
+where¯Vkis defined by the rhs of (4.2). By taking the supremum over αin the inequality
+(4.7), this shows that Vk≤¯Vk. Conversely, fix x∈Rd,θ= (θ1,...,θn)∈∆n∩[0,T]n,e=
+(e1,...,en)∈En, and let us prove that Vk(x,θ(k),e(k))≥¯Vk(x,θ(k),e(k)). Fix an arbitrary
+αk∈ Ak
+F, and the associated controlled process Xk,x. By definition of Vk+1, for anyω∈
+Ω,ε >0, there exists αω,ε∈ AG, which is an ε-optimal control for Vk+1(.,θ(k),e(k)) at
+(ω,Γk+1
+θk+1(Xk,x
+θk+1,αk
+θk+1,ek+1)). Recalling that the set of admissible controls is a separa ble
+metric space, one can use a measurable selection result (see e.g. [25]) to find αε∈ AGs.t.
+αε
+t(ω) =αω,ε
+t(ω),dt⊗dPa.e., and so
+Vk+1/parenleftbig
+Γk+1
+θk+1(Xk,x
+θk+1,αk
+θk+1,ek+1),θ(k),θk+1,e(k),ek+1/parenrightbig
+−ε
+≤Jk+1/parenleftbig
+Γk+1
+θk+1(Xk,x
+θk+1,αk
+θk+1,ek+1),θ(k),θk+1,e(k),ek+1,αε/parenrightbig
+, a.s.
+15Denote by ( αε,0,...,αε,n) then+1-tuple associated to αε∈ AG, and let us consider the
+admissible control ˜ αε= (αε,0,...,αk,αε,k+1,...,αε,n)∈ AGconsisting in substituting the
+k-th component of αεbyαk∈ Ak
+F. SinceJk+1(x,θ,e,α) depends on αonly through its last
+components ( αk+1,...,αn), we have from (4.4)
+Vk(x,θ(k),e(k))≥Jk(x,θ(k),e(k),˜αε)
+=E/bracketleftBig/integraldisplayT
+θkfk
+t(Xk,x
+t,αk
+t,θ(k),e(k))γk
+t(θ(k),e(k))dt
++Gk
+T(Xk,x
+T,θ(k),e(k))γk
+T(θ(k),e(k))
++/integraldisplayT
+θk/integraldisplay
+EJk+1/parenleftbig
+Γk+1
+θk+1(Xk,x
+θk+1,αk
+θk+1,ek+1),θ(k),θk+1,e(k),ek+1,αε/parenrightbig
+η1(dek+1)dθk+1/vextendsingle/vextendsingle/vextendsingleFθk/bracketrightBig
+≥E/bracketleftBig/integraldisplayT
+θkfk
+t(Xk,x
+t,αk
+t,θ(k),e(k))γk
+t(θ(k),e(k))dt
++Gk
+T(Xk,x
+T,θ(k),e(k))γk
+T(θ(k),e(k))
++/integraldisplayT
+θk/integraldisplay
+EVk+1/parenleftbig
+Γk+1
+θk+1(Xk,x
+θk+1,αk
+θk+1,ek+1),θ(k),θk+1,e(k),ek+1/parenrightbig
+η1(dek+1)dθk+1/vextendsingle/vextendsingle/vextendsingleFθk/bracketrightBig
+−ε.
+Fromthearbitrarinessof αk∈Ak
+Fandε>0, weobtaintherequiredinequality: Vk(x,θ(k),e(k))
+≥¯Vk(x,θ(k),e(k)), and the proof is complete. ✷
+5 The case of enlarged filtration with multiple random times
+In this section, we consider the case wherethe random times a re not assumed to beordered.
+In other words, this means that one has access to the default t imes themselves with their
+indexes, and not only to the ranked default times. This gener al case can actually be
+derived from the case of successive random times associated with suitable auxiliary marks.
+Let us consider the progressive enlargement of filtration fr omFtoGwith multiple random
+times (τ1,...,τn) associated with the marks ( ζ1,...,ζn). Denote by ˆ τ1≤...≤ˆτnthe
+corresponding ranked times, and by ιithe index mark (valued in {1,...,n}) of thei-
+th order statistic of ( τ1,...,τn) fori= 1,...,n. Then, it is clear that the progressive
+enlargement of filtration of Fwith the successive random times (ˆ τ1,...,ˆτn) together with
+the marks ( ι1,ζι1,...,ιn,ζιn) leads to the filtration G, so that one can apply the results of
+the previous sections. For simplicity of notations, we shal l focus on the case of two random
+timesτ1andτ2, associated to the marks ζ1andζ2valued inEBorel space of Rm.
+The decomposition of optional and predictable process with respect to this progressive
+enlargement of filtration is given by the following lemma, wh ich is derived from Lemma
+2.1, with the specific feature that we have also to take into ac count the index of the order
+statistic in ( τ1,τ2).
+16Lemma 5.1 AnyG-optional (resp. predictable) process Y= (Yt)t≥0is represented as
+Yt=Y0
+t1t<ˆτ1+Y1,1
+t(τ1,ζ1)1τ1≤t<τ2+Y1,2
+t(τ2,ζ2)1τ2≤t<τ1+Y2
+t(τ1,τ2,ζ1,ζ2)1t≥ˆτ2,
+(resp.=Y0
+t1t≤ˆτ1+Y1,1
+t(τ1,ζ1)1τ1<t≤τ2+Y1,2
+t(τ2,ζ2)1τ2<t≤τ1+Y2
+t(τ1,τ2,ζ1,ζ2)1t>ˆτ2),
+for allt≥0, whereY0∈ OF(resp.PF),Y1,1,Y1,2∈ O1
+F(R+,E)(resp.P1
+F(R+,E)), and
+Y2∈ O2
+F(R2
++,E2)(resp.P2
+F(R2
++,E2)).
+AnyY∈ OG(resp.PG) can then be identified with a quadruple ( Y0,Y1,1,Y1,2,Y2)
+∈ OF× O1
+F(R+,E)× O1
+F(R+,E)× O2
+F(R2
++,E2) (resp. PF× P1
+F(R+,E)× P1
+F(R+,E)×
+P2
+F(R2
++,E2)).
+Similarly as in Section 1, we now make a density hypothesis on the conditional dis-
+tribution of ( τ1,τ2,ζ1,ζ2) given the reference information. We assume that there exis ts a
+O(F)⊗B(R2
++)⊗B(E2)-measurable map ( t,ω,θ1,θ2,e1,e2)→γt(ω,θ1,θ2,e1,e2) such that
+(DH)P/bracketleftbig
+(τ1,τ2,ζ1,ζ2)∈dθde|Ft/bracketrightbig
+=γt(θ1,θ2,e1,e2)dθ1dθ2η(de1)η(de2), a.s.
+whereηis a nonnegative measure on B(E).
+We next introduce some useful notations. We denote by γ0theF-optional process
+defined by
+γ0
+t=P[τ1>t|Ft] =/integraldisplay
+E2/integraldisplay
+[t,∞)2γt(θ1,θ2,e1,e2)dθ1dθ2η(de1)η(de2),
+and we denote by ( t,ω,θ1,e1)→γ1,1
+t(θ1,e1), and (t,ω,θ2,e2)→γ1,2
+t(θ2,e2),t≥0, the
+O(F)⊗B(R+)⊗B(E)-measurable maps defined by
+γ1,1
+t(θ1,e1) =/integraldisplay
+E/integraldisplay∞
+tγt(θ1,θ2,e1,e2)dθ2η(de2),
+γ1,2
+t(θ2,e2) =/integraldisplay
+E/integraldisplay∞
+tγt(θ1,θ2,e1,e2)dθ1η(de1),
+so that
+P[τ2>t|Ft] =/integraldisplay
+E/integraldisplay∞
+0γ1,1
+t(θ1,e1)dθ1η(de1),P[τ1>t|Ft] =/integraldisplay
+E/integraldisplay∞
+0γ1,2
+t(θ2,e2)dθ2η(de2).
+The next result, which is analog to Proposition 2.1, provide s a backward induction for-
+mula involving F-expectations for the computation of expectation function als ofG-optional
+processes.
+Proposition 5.1 LetY= (Y0,Y1,1,Y1,2,Y2)andZ= (Z0,Z1,1,Z1,2,Z2)be two nonneg-
+ative (or bounded) G-optional processes, and fix T∈(0,∞).
+The expectation E[/integraltextT
+0Ytdt+ZT]can be computed in a backward induction as
+E/bracketleftBig/integraldisplayT
+0Ytdt+ZT/bracketrightBig
+=J0
+17where the (J0,J1,1,J1,2,J2)are given by
+J2(θ1,θ2,e1,e2) =E/bracketleftBig/integraldisplayT
+θ1∨θ2Y2
+tγt(θ1,θ2,e1,e2)dt+Z2
+TγT(θ1,θ2,e1,e2)/vextendsingle/vextendsingle/vextendsingleFθ1∨θ2/bracketrightBig
+J1,1(θ1,e1) =E/bracketleftBig/integraldisplayT
+θ1Y1,1
+tγ1,1
+t(θ1,e1)dt+Z1,1
+Tγ1,1
+T(θ1,e1)
++/integraldisplay
+E/integraldisplayT
+θ1J2(θ1,θ2,e1,e2)dθ2η(de2)/vextendsingle/vextendsingle/vextendsingleFθ1/bracketrightBig
+J1,2(θ2,e2) =E/bracketleftBig/integraldisplayT
+θ2Y1,2
+tγ1,2
+t(θ2,e2)dt+Z1,2
+Tγ1,2
+T(θ2,e2)
++/integraldisplay
+E/integraldisplayT
+θ1J2(θ1,θ2,e1,e2)dθ2η(de2)/vextendsingle/vextendsingle/vextendsingleFθ2/bracketrightBig
+J0=E/bracketleftBig/integraldisplayT
+0Y0
+tγ0
+tdt+Z0
+Tγ0
+T
++/integraldisplay
+E/integraldisplayT
+0J1,1(θ1,e1)dθ1η(de1)+/integraldisplay
+E/integraldisplayT
+0J1,2(θ2,e2)dθ2η(de2)/bracketrightBig
+.
+Let us now formulate the general stochastic control problem in this framework.
+A control is a G-predictable process α= (α0,α1,1,α1,2,α2)∈ PF×P1
+F(R+,E)×P1
+F(R+,E)×
+P2
+F(R2
++,E2), whereα0,α1,1,α1,2andα2are valued respectively in A0,A1,1,A1,2andA2,
+Borel sets of some Euclidian space. We denote by A=A0×A1,1×A1,2×A2, and by AG
+the set of admissible control processes, which is a product s paceA0
+F× A1,1
+F× A1,2
+F× A2
+F,
+whereA0
+F,A1,1
+F,A1,2
+FandA2
+Fare some separable metric spaces respectively in PF(A0),
+P1
+F(R+,E;A1,1),P1
+F(R+,E;A1,2) andP2
+F(R2
++,E2;A2).
+We are next given a collection of measurable mappings:
+(x,α0)∈Rd×A0
+F/mapsto−→X0,x,α0∈ OF
+(x,α1,1)∈Rd×A1,1
+F/mapsto−→X1,1,x,α1,1∈ O1
+F(R+,E)
+(x,α1,2)∈Rd×A1,2
+F/mapsto−→X1,2,x,α1,2∈ O1
+F(R+,E)
+(x,α2)∈Rd×A2
+F/mapsto−→X2,x,α2∈ O2
+F(R2
++,E2),
+such that we have the initial data
+X0,x,α0
+0=x,∀x∈Rd,
+X1,1,ξ,α1,1
+θ1(θ1,e1) =ξ,∀ξFθ1−measurable ,
+X1,2,ξ,α1,2
+θ2(θ2,e2) =ξ,∀ξFθ2−measurable ,
+X2,ξ,α2
+θ1∨θ2(θ1,θ2,e1,e2) =ξ,∀ξFθ1∨θ2−measurable .
+We are also given a collection of maps Γ1,1, Γ1,2, onR+×Ω×Rd×A0×E, Γ2,1on
+R+×Ω×Rd×A1,1×Eand Γ2,2onR+×Ω×Rd×A1,2×Esuch that
+(t,ω,x,a,e )/mapsto→Γ1,1
+t(ω,x,a,e),Γ1,2
+t(ω,x,a,e)
+areP(F)⊗B(Rd)⊗B(A0)⊗B(E)−measurable
+(t,ω,x,a,e )/mapsto→Γ2,1
+t(ω,x,a,e) isP(F)⊗B(Rd)⊗B(A1,1)⊗B(E)−measurable
+(t,ω,x,a,e )/mapsto→Γ2,2
+t(ω,x,a,e) isP(F)⊗B(Rd)⊗B(A1,2)⊗B(E)−measurable
+18The controlled state process is then given by the mapping
+(x,α)∈Rd×AG/mapsto−→Xx,α∈ OG,
+where forα= (α0,α1,1,α1,2,α2),Xx,αis the process equal to
+Xx,α
+t=¯X0
+t1t<ˆτ1+¯X1,1
+t(τ1,ζ1)1τ1≤t<τ2+¯X1,2
+t(τ2,ζ2)1τ2≤t<τ1+¯X2
+t(τ1,τ2,ζ1,ζ2)1t≥ˆτ2,
+with (¯X0,¯X1,1,¯X1,2,¯X2)∈ OF×O1
+F(R+,E)×O1
+F(R+,E)×O2
+F(R2
++,E2) given by
+¯X0=X0,x,α0
+¯X1,1(θ1,e1) =X1,1,Γ1,1
+θ1(¯X0
+θ1,α0
+θ1,e1),α1,1(θ1,e1)
+¯X1,2(θ2,e2) =X1,2,Γ1,2
+θ2(¯X0
+θ2,α0
+θ2,e2),α1,2(θ2,e2)
+¯X2(θ1,θ2,e1,e2) =/braceleftBigg
+X2,Γ2,2
+θ2(¯X1,1
+θ2,α1,1
+θ2,e2),α2(θ1,θ2,e1,e2) ifθ1≤θ2
+X2,Γ2,1
+θ1(¯X1,2
+θ1,α1,2
+θ1,e1),α2(θ1,θ2,e1,e2) ifθ2<θ1.
+The interpretation is the following: X0is the controlled state process before any default,
+X1,1(resp.X1,2) is the controlled state process between τ1andτ2(resp. between τ2andτ1)
+if the default of index 1 (resp. index 2) occurs first, and X2is the controlled state process
+after both defaults. Moreover, Γ1,1(resp. Γ1,2) represents the jump of X0atτ1(resp.τ2) if
+the default of index 1 (resp. index 2) occurs first, and Γ2,2(resp. Γ2,1) represents the jump
+ofX1,1(resp.X1,2) atτ2(resp.τ1) when the default of index 2 (resp. index 1) occurs in
+second after index 1 (resp. index 2).
+For a fixed finite horizon T <∞, we are given a nonnegative map GTon Ω×Rdsuch
+that (ω,x)/mapsto→GT(ω,x) isGT⊗B(Rd)-measurable, thus in the form
+GT(x) =G0
+T(x)1T<ˆτ1+G1,1
+T(x,τ1,ζ1)1τ1≤T<τ2+G1,2
+T(x,τ2,ζ2)1τ2≤T<τ1
++G2
+T(x,τ1,τ2,ζ1,ζ2)1ˆτ2≤T,
+whereG0
+TisFT⊗B(Rd)-measurable, G1,1
+T,G1,2
+TareFT⊗B(Rd)⊗B(R+)⊗B(E)-measurable,
+andG2
+TisFT⊗B(Rd)⊗B(R2
++)⊗B(E2)-measurable. Therunninggain function is given by a
+nonnegative map fon Ω×Rd×Asuch that (t,ω,x,a)/mapsto→ft(ω,x,a) isO(G)⊗B(Rd)⊗B(A)-
+measurable, and which may be decomposed as
+ft(x,a) =f0
+t(x,a0)1t<ˆτ1+f1,1
+t(x,a1,1,τ1,ζ1)1τ1≤t<τ2+f1,2
+t(x,a1,2,τ2,ζ2)1τ2≤t<τ1
++f2
+t(x,a2,τ1,τ2,ζ1,ζ2)1ˆτ2≤T,
+fora= (a0,a1,1,a1,2,a2)∈A=A0×A1,1×A1,2×A2, wheref0isO(F)⊗B(Rd)⊗B(A0)-
+measurable, and f1,1isO(F)⊗B(Rd)⊗B(A1,1)⊗B(R+)⊗B(E)-measurable, f1,2isO(F)⊗
+B(Rd)⊗B(A1,2)⊗B(R+)⊗B(E)-measurableand f2isO(F)⊗B(Rd)⊗B(A2)⊗B(R2
++)⊗B(E2)-
+measurable.
+The value function for the stochastic control problem is the n defined by
+V0(x) = sup
+α∈AGE/bracketleftBig/integraldisplayT
+0ft(Xx,α
+t,αt)dt+GT(Xx,α
+T)/bracketrightBig
+, x∈Rd.
+19The main result of this section provides a decomposition of t he value function in the
+reference filtration, which is analog to the decomposition i n Theorem 4.1. To alleviate the
+notations, weomit thedependenceof thestate processin the controls andin theparameters
+θ,e, when there is no ambiguity.
+Theorem 5.1 The value function V0is obtained from the backward induction formula
+V2(x,θ1,θ2,e1,e2) = esssup
+α2∈A2
+FE/bracketleftBig/integraldisplayT
+θ1∨θ2f2
+t(X2,x
+t,α2
+t,θ1,θ2,e1,e2)γt(θ1,θ2,e1,e2)dt
++G2
+T(X2,x
+T,θ1,θ2,e1,e2)γT(θ1,θ2,e1,e2)/vextendsingle/vextendsingle/vextendsingleFθ1∨θ2/bracketrightBig
+V1,1(x,θ1,e1) = esssup
+α1,1∈A1,1
+FE/bracketleftBig/integraldisplayT
+θ1f1,1
+t(X1,1,x
+t,α1,1
+t,θ1,e1)γ1,1
+t(θ1,e1)dt
++G1,1
+T(X1,1,x
+T,θ1,e1)γ1,1
+T(θ1,e1)
++/integraldisplayT
+θ1/integraldisplay
+EV2/parenleftbig
+Γ2,2
+θ2(X1,1,x
+θ2,α1,1
+θ2,e2),θ1,θ2,e1,e2/parenrightbig
+η(de2)dθ2/vextendsingle/vextendsingle/vextendsingleFθ1/bracketrightBig
+V1,2(x,θ2,e2) = esssup
+α1,2∈A1,2
+FE/bracketleftBig/integraldisplayT
+θ2f1,2
+t(X1,2,x
+t,α1,2
+t,θ2,e2)γ1,2
+t(θ2,e2)dt
++G1,2
+T(X1,2,x
+T,θ2,e2)γ1,2
+T(θ2,e2)
++/integraldisplayT
+θ1/integraldisplay
+EV2/parenleftbig
+Γ2,1
+θ1(X1,2,x
+θ1,α1,2
+θ1,e1),θ1,θ2,e1,e2/parenrightbig
+η(de1)dθ1/vextendsingle/vextendsingle/vextendsingleFθ2/bracketrightBig
+V0(x) = sup
+α0∈A0
+FE/bracketleftBig/integraldisplayT
+0f0
+t(X0,x
+t,α0
+t)γ0
+tdt+G0
+T(X0,x
+T)γ0
+T
++/integraldisplayT
+0/integraldisplay
+EV1,1/parenleftbig
+Γ1,1
+θ1(X0,x
+θ1,α0
+θ1,e1),θ1,e1/parenrightbig
+η(de1)dθ1
++/integraldisplayT
+0/integraldisplay
+EV1,2/parenleftbig
+Γ1,2
+θ2(X0,x
+θ2,α0
+θ2,e2),θ2,e2/parenrightbig
+η(de2)dθ2/bracketrightBig
+,
+for all(θ1,θ2)∈[0,T]2,(e1,e2)∈E2.
+Remark 5.1 As mentioned in Remark 4.1, the valuefunctions V2,V1,1andV1,2correspond
+to standard stochastic control problem in the F-filtration. This is also the case for V0in the
+decomposition formulaofTheorem5.1. Indeed, denoteby V1themaponΩ ×[0,T]×Rd×A0:
+V1(x,θ,a0) =/integraldisplay
+EV1,1(Γ1,1
+θ(x,a0,e),θ,e)+V1,2(Γ1,2
+θ(x,a0,e),θ,e)η(de).
+Then,V0is computed from the stochastic control problem in the F-filtration with the
+terminal gain function G0
+Tweighted by the FT-measurable random variable γ0
+T, and with
+the running gain functions f0γ0andV1:
+V0(x) = sup
+α0∈AFE/bracketleftBig
+G0
+T(X0,x
+T)γ0
+T+/integraldisplayT
+0f0
+t(X0,x
+t,α0
+t)γ0
+t+V1(X0,x
+t,t,α0
+t)dt/bracketrightBig
+.
+206 Applications in mathematical finance
+6.1 Indifference pricing of defaultable claims
+We consider a stock subject to a single counterparty default at a random time τ, which
+induces a jump of random relative size ζvalued inE⊂(−1,∞). The price process of the
+stock is described by
+St=S0
+t1t<τ+S1
+t(τ,ζ)1t≥τ,
+whereS0is governed by
+dS0
+t=S0
+t/parenleftbig
+b0
+tdt+σ0
+tdWt/parenrightbig
+,
+and the indexed process S1(θ,e), (θ,e)∈R+×Eis given by
+dS1
+t(θ,e) =S1
+t(θ,e)/parenleftbig
+b1
+t(θ,e)dt+σ1
+t(θ,e)dWt/parenrightbig
+, t≥θ,
+S1
+θ(θ,e) =S0
+θ.(1+e).
+HereWis a (P,F)-Brownian motion, b0,σ0>0 areF-adapted processes, b1,σ1>0∈
+O1
+F(R+,E). The market information is represented by the progressive enlarged filtration
+G=F∨D, withD= (Dt)t≥0,Dt=∩ε>0{σ(ζ1τ≤s,s≤t+ε)∨σ(1τ≤s,s≤t+ε)}. An
+investor can trade in a riskless bond with zero interest rate , and in the defaultable stock.
+Her trading strategy is a G-predictable process α= (α0,α1)∈ PF×P1
+F(R+,E) representing
+the amount traded in the stock. We allow constraints on tradi ng strategy by considering
+closed setsA0andA1in which the controls α0andα1take values. Notice also that A0and
+A1may differ. The controlled wealth process of the investor is th en given by
+Xt=X0
+t1t<τ+X1
+t(τ,ζ)1t≥τ, (6.1)
+whereX0is the wealth process before the default, and governed by
+dX0
+t=α0
+tdS0
+t
+S0
+t=α0
+t(b0
+tdt+σ0
+tdWt),
+andX1(θ,e) is the wealth indexed process after-default, governed by
+dX1
+t(θ,e) =α1
+t(θ,e)dS1
+t(θ,e)
+S1
+t(θ,e)=α1
+t(θ,e)/parenleftbig
+b1
+t(θ,e)dt+σ1
+t(θ,e)dWt/parenrightbig
+, t≥θ
+X1
+θ(θ,e) =X0
+θ+α0
+θe.
+Let us now consider a defaultable contingent claim with payo ff at maturity Tgiven by
+HT=H0
+T1T<τ+H1
+T(τ,ζ)1τ≤T,
+whereH0
+Tis a bounded FT-measurable random variable, and H1
+T(,) is a bounded FT⊗
+B(R+)⊗B(E)-measurable map. We usethepopularindifferencepricingcri terion forvaluing
+this defaultable claim. We are then given an exponential uti lity function UonR, i.e.
+U(x) =−exp(−px), x∈R,
+21for somep >0, and we consider the optimal investment problem for an agen t delivering
+the defaultable claim at maturity T:
+VH
+0(x) = sup
+α∈AGE/bracketleftbig
+U(Xx,α
+T−HT)/bracketrightbig
+. (6.2)
+HereXx,αis the wealth process in (6.1) controlled by the trading stra tegyα, and starting
+fromxat time 0. We denote by V0the value function for the optimal investment problem
+without the defaultable claim, i.e. when HT= 0 in (6.2), and the indifference price for HT
+is the amount of initial capital such that the investor is ind ifferent between holding or not
+the defaultable claim. It is then defined as the unique number πsuch that
+VH
+0(x+π) =V0(x).
+A similar problem (without unpredictable mark ζ) was recently considered in [15] and
+[1] by using a global G-filtration approach under (H)hypothesis, see also [20]. The paper
+[14] studied an optimal investment problem with power utili ty functions under a single
+counterparty default by using a density approach for decomp osing the problem in the
+F-filtration. We follow this methodology and solve the stocha stic control problem (6.2)
+by applying the F-decomposition method. From Theorem 4.1, the value functio nVH
+0is
+obtained in two steps via the resolution of the after-defaul t problem
+VH
+1(x,θ,e) = esssup
+α1∈A1
+FE/bracketleftBig
+U/parenleftbig
+X1,x
+T(θ,e)−H1
+T(θ,e)/parenrightbig
+γT(θ,e)/vextendsingle/vextendsingle/vextendsingleFθ/bracketrightBig
+, (6.3)
+and then via the resolution of the before-default problem
+VH
+0(x) = sup
+α0∈A0
+FE/bracketleftBig
+U(X0,x
+T−H0
+T)γ0
+T+/integraldisplayT
+0/integraldisplay
+EVH
+1(X0,x
+θ+α0
+θe,θ,e)η(de)dθ/bracketrightBig
+.(6.4)
+•Solution to the after-default problem.
+For fixed (θ,e)∈[0,T]×E, problem (6.3) is a classical utility maximization problem with
+random endowment in the complete market model after default described by the indexed
+price process S1(θ,e). Indeed, notice that we can remove the positive term γT(θ,e) in (6.3)
+by defining the “modified claim” ˜H1
+T(θ,e) =H1
+T(θ,e)+1
+plnγT(θ,e) so that
+VH
+1(x,θ,e) = esssup
+α1∈A1
+FE/bracketleftBig
+U/parenleftbig
+X1,x
+T(θ,e)−˜H1
+T(θ,e)/parenrightbig/vextendsingle/vextendsingle/vextendsingleFθ/bracketrightBig
+. (6.5)
+This problem was addressed by several methods in the literat ure, and we know from dy-
+namic programming and BSDE methods (see [22] or [8])) that
+VH
+1(x,θ,e) =U/parenleftbig
+x−Y1,H
+θ(θ,e)/parenrightbig
+whereY1,H(θ,e) is the unique bounded solution to the BSDE
+Y1,H
+t(θ,e) =H1
+T(θ,e)+1
+plnγT(θ,e)+/integraldisplayT
+tf1(r,Z1,H
+r,θ,e)dr−/integraldisplayT
+tZ1,H
+rdWr
+22and the generator f1is theP(F)⊗B(R)⊗B(R+)⊗B(E)-measurable map defined by
+f1(t,z,θ,e) =−b1
+t(θ,e)
+σ1
+t(θ,e)z−1
+2p/parenleftBigb1
+t(θ,e)
+σ1
+t(θ,e)/parenrightBig2
++p
+2inf
+a∈A1/vextendsingle/vextendsingle/vextendsingle/parenleftBig
+z+1
+pb1
+t(θ,e)
+σ1
+t(θ,e)/parenrightBig
+−aσ1
+t(θ,e)/vextendsingle/vextendsingle/vextendsingle2
+.
+•Global solution
+The global solution is finally obtained from the resolution o f the before-default problem,
+which is then reduced to
+VH
+0(x) = sup
+α0∈A0
+FE/bracketleftBig
+U(X0,x
+T−H0
+T)γ0
+T+/integraldisplayT
+0/integraldisplay
+EU(X0,x
+θ+α0
+θe−Y1,H
+θ(e))η(de)dθ/bracketrightBig
+.
+From the additive dependence of the wealth process X0,xin function of x, and the ex-
+ponential form of the utility function U, we know that the value function VH
+0is in the
+form
+VH
+0(x) =U(x−Y0,H
+0),
+for some quantity Y0,H
+0independent of x, and which may be characterized by dynamic
+programming methods in the F-filtration. This can be achieved either via PDE methods in
+a Markovian setting, or via BSDE methods in the general case. The BSDE associated to
+Y0,His
+Y0,H
+t=H0
+T+1
+plnγ0
+T+/integraldisplayT
+tf0,H(r,Y0,H
+r,Z0,H
+r)dr−/integraldisplayT
+tZ0,H
+rdWr,(6.6)
+where the generator f0,His theO(F)⊗B(R)⊗B(R)-measurable map defined by
+f0,H(t,y,z) =−b0
+t
+σ0
+tz−1
+2p/parenleftBigb0
+t
+σ0
+t/parenrightBig2
+(6.7)
++p
+2inf
+a∈A0/vextendsingle/vextendsingle/vextendsingle/parenleftBig
+z+1
+pb0
+t
+σ0
+t/parenrightBig
+−aσ0
+t+2
+pU(y)/integraldisplay
+EU(ae−Y1,H
+t(t,e)η(de)/vextendsingle/vextendsingle/vextendsingle2
+.
+The solution to the optimal investment problem without defa ultable claim is obtained
+similarly as for the case with claim, by considering H= 0. We thus have V0(x) =R(x−Y0
+0),
+where the BSDE associated to Y0is given by
+Y0
+t=1
+plnγ0
+T+/integraldisplayT
+tf0(r,Y0
+r,Z0
+r)dr−/integraldisplayT
+tZ0
+rdWr,
+with a generator f0as in (6.7) for H= 0, i.e.Y1,Hreplaced by Y1solution to the BSDE
+Y1
+t(θ,e) =1
+plnγT(θ,e)+/integraldisplayT
+tf1(r,Z1
+r,θ,e)dr−/integraldisplayT
+tZ1
+rdWr.
+Finally, the indifference price is given by
+π=Y0,H
+0−Y0
+0.
+23Remark 6.1 Notice that the quadratic generator f0,Hin (6.7) of the BSDE (6.6) is not
+standard dueto theadditional term arisingfrom theintegra l gain involving Y1,H. However,
+one can prove existence and uniqueness of this BSDE and obtai n a verification theorem
+relating the solution of this BSDE to the original value func tion by choosing a suitable set
+of admissible controls AG=A0
+F× A1
+F. The details are provided in the companion paper
+[13]. Actually, in this related paper, we consider a multi-d imensional extension of the above
+model with assets subject to successive counterparty defau lt times, and we apply the F-
+decomposition method for solving the indifference pricing of defaultable claims, including
+credit derivatives such as k-th default swap.
+6.2 Optimal investment under bilateral counterparty risk
+We consider a portfolio with two names, each one subject to an external counterparty
+default, but also to the default of the other one due to a conta gion effect. We denote by
+S1andS2the value process of these two names, by τ1andτ2their default times, not
+necessarily ordered, and by ˆ τ1= min(τ1,τ2), ˆτ2= max(τ1,τ2). Once the name idefaults at
+random time τi, meaning that the value of Sidrops to zero, it also incurs a jump (drop or
+gain) on the other value process Sj,i,j∈ {1,2},i/ne}ationslash=j.
+The reference filtration Fis the filtration generated by a two-dimensional Brownian
+motionW= (W1,W2), driving the evolution of the names in absence of defaults, and the
+global market information is represented by G=F∨D1∨D2, withDi= (Di
+t)t≥0,Di
+t=
+∩ε>0σ(1τi≤s,s≤t+ε),i= 1,2.
+TheG-adapted value processes Siof namesi= 1,2, are given by
+Si
+t=Si,0
+t1t<ˆτ1+Si,j
+t(τj)1τj≤t<τi, t≥0, i,j= 1,2, i/ne}ationslash=j,
+whereS0= (S1,0,S2,0) is the vector price process of the two names in absence of any
+default, governed by
+dS0
+t= diag(S0
+t)/parenleftbig
+b0
+tdt+σ0
+tdWt/parenrightbig
+,
+b0= (b1,0,b2,0) isF-adapted,σ0is the 2×2-diagonal F-adapted matrix with diagonal diffu-
+sion coefficients σ1,0>0,σ2,0>0, and the indexed process Si,j(θj),θj∈R+, representing
+the price process of name iafter the default of name jat timeθj, is given by
+dSi,j
+t(θj) =dSi,j
+t(θj)/parenleftbig
+bi,j
+t(θj)dt+σi,j
+t(θj)dWi
+t/parenrightbig
+, t≥θj,
+Si,j
+θj(θj) =Si,0
+θj.(1+ei,j),
+whereei,jrepresents the proportional jump induced by the default of n amejon namei,
+and assumed constant for simplicity and valued in ( −1,∞) . The coefficients bi,0,σi,0>0
+areF-adapted processes, and bi,j,σi,j>0 are inO1
+F(R+).
+Thetradingstrategyoftheinvestorisa G-predictablemeasurableprocess αrepresenting
+the fraction of wealth invested in thetwo names. It is then de composed in four components:
+the first component α0is a pair of F-predictable processes representing the fraction investe d
+in the two names before any default, the second component α1,1is an indexed F-predictable
+process representing the fraction invested in the name2 whe n thename 1 defaults, the third
+24component α1,2is an indexed F-predictable process representing the fraction invested i n
+the name 1 when the name 2 defaults, and the fourth component i s zero when both names
+default. The wealth process of the investor is then given by
+Xt=X0
+t1t<ˆτ1+X1,1
+t(τ1)1τ1≤t<τ2+X1,2
+t(τ2)1τ2≤t<τ1+X2
+t(τ1,τ2)1t≥ˆτ2,
+whereX0is the wealth process before any default, governed by
+dX0
+t=X0
+t(α0
+t)′diag(S0
+t)−1dS0
+t
+=X0
+t/parenleftbig
+α0
+t.b0
+tdt+(α0
+t)′σ0
+tdWt/parenrightbig
+,
+X1,1(θ1) is the wealth indexed process after default of name 1, gover ned by
+dX1,1
+t(θ1) =X1,1
+t(θ1)α1,1
+t(θ1)dS2,1
+t(θ1)
+S2,1
+t(θ1), t≥θ1
+X1,1
+θ1(θ1) =X0
+θ1.(1+α0
+θ1.(−1,e2,1)),
+X1,2(θ2) is the wealth indexed process after default of name 2, gover ned by
+dX1,2
+t(θ2) =X1,2
+t(θ2)α1,2
+t(θ2)dS1,2
+t(θ2)
+S1,2
+t(θ2), t≥θ2
+X1,2
+θ2(θ2) =X0
+θ2./parenleftbig
+1+α0
+θ2.(e1,2,−1)/parenrightbig
+,
+andX2(θ1,θ2) is thewealth indexed processafter bothdefaults, henceco nstant after θ1∨θ2,
+and then given by
+X2
+t(θ1,θ2) =/braceleftBigg
+X1,1
+θ2(θ1)./parenleftbig
+1−α1,1
+θ2(θ1)/parenrightbig
+, θ1≤θ2≤t
+X1,2
+θ1(θ2)./parenleftbig
+1−α1,2
+θ1(θ2)/parenrightbig
+, θ2<θ1≤t
+In order to ensure that the wealth process is strictly positi ve, we assume that α0is valued
+in a closed subset A0⊂ {a∈R2: 1+a.(−1,e2,1)>0,and 1+a.(e1,2,−1)>0}, andα1,1,
+α1,2are valued respectively in closed subsets A1,1,A1,2⊂(−∞,1).
+We are next given a utility function UonR+, over a finite horizon T, and we consider
+the optimal investment problem
+V0(x) = sup
+α∈AGE/bracketleftbig
+U(Xx,α
+T)/bracketrightbig
+. (6.8)
+We use the F-decomposition method of Section 5 for the resolution of (6. 8). From The-
+orem 5.1, the value function V0is obtained via the following backward induction formula:
+V2(x,θ1,θ2) =U(x)E/bracketleftbig
+γT(θ1,θ2)/vextendsingle/vextendsingleFθ1∨θ2/bracketrightbig
+:=U(x)¯γ(θ1,θ2)
+V1,1(x,θ1) = esssup
+α1,1∈A1,1
+FE/bracketleftBig
+U(X1,1,x
+T)γ1,1
+T(θ1)+/integraldisplayT
+θ1U/parenleftbig
+X1,1
+θ2./parenleftbig
+1−α1,1
+θ2/parenrightbig/parenrightbig
+¯γ(θ1,θ2)dθ2/vextendsingle/vextendsingle/vextendsingleFθ1/bracketrightBig
+V1,2(x,θ2) = esssup
+α1,2∈A1,2
+FE/bracketleftBig
+U(X1,2,x
+T)γ1,2
+T(θ2)+/integraldisplayT
+θ2U/parenleftbig
+X1,2
+θ1./parenleftbig
+1−α1,2
+θ1/parenrightbig/parenrightbig
+¯γ(θ1,θ2)dθ1/vextendsingle/vextendsingle/vextendsingleFθ1/bracketrightBig
+V0(x) = sup
+α0∈A0
+FE/bracketleftBig
+U(X0,x
+T)γ0
+T
++/integraldisplayT
+0V1,1/parenleftbig
+X0
+θ./parenleftbig
+1+α0
+θ.(−1,e2,1)/parenrightbig
+,θ)+V1,2/parenleftbig
+X0
+θ./parenleftbig
+1+α0
+θ.(e1,2,−1)/parenrightbig
+,θ/parenrightbig
+dθ/bracketrightBig
+.
+25In the sequel, we consider power utility functions
+U(x) =1
+pxp, x≥0, p<1, p/ne}ationslash= 0,
+and we use dynamic programming and BSDE methods in the F-filtration to solve the above
+stochastic control problems. The value functions V1,1andV1,2are then in the form
+V1,1(x,θ1) =U(x)Y1,1
+θ1(θ1), V1,2(x,θ2) =U(x)Y1,2
+θ2(θ2),
+whereY1,1(θ1) andY1,2(θ2) are solutions to the BSDEs:
+Y1,1
+t(θ1) =γ1,1
+T(θ1)+/integraldisplayT
+tf1,1(r,Y1,1
+r(θ1),Z1,1
+r(θ1),θ1)dr−/integraldisplayT
+tZ1,1
+r(θ1)dW2
+r,
+Y1,2
+t(θ2) =γ1,2
+T(θ2)+/integraldisplayT
+tf1,2(r,Y1,2
+r(θ2),Z1,2
+r(θ2),θ2)dr,−/integraldisplayT
+tZ1,2
+r(θ2)dW1
+r
+with generators
+f1,1(t,y,z,θ 1) =psup
+a∈A1,1/bracketleftBig/parenleftbig
+b2,1
+t(θ1)y+σ2,1
+t(θ1)z/parenrightbig
+a−1−p
+2y|σ2,1
+t(θ1)a|2
++ ¯γ(θ1,t)(1−a)p
+p/bracketrightBig
+f1,2(t,y,z,θ 2) =psup
+a∈A1,2/bracketleftBig/parenleftbig
+b1,2
+t(θ2)y+σ1,2
+t(θ2)z/parenrightbig
+a−1−p
+2y|σ1,2
+t(θ2)a|2
++ ¯γ(t,θ2)(1−a)p
+p/bracketrightBig
+.
+Finally, we have
+V0(x) =U(x)Y0,
+whereY0is the solution to the BSDE
+Y0
+t=γ0
+T+/integraldisplayT
+tf0(r,Y0
+r,Z0
+r)dr−/integraldisplayT
+tZ0
+r.dWr,
+with a generator
+f0(t,y,z) =psup
+a∈A0/bracketleftBig/parenleftbig
+yb0
+t+σ0
+tz).a−1−p
+2y|σ0
+ta|2
++Y1,1
+t(t)(1+a.(−1,e2,1))p
+p+Y1,2
+t(t)(1+a.(e1,2,−1))p
+p/bracketrightBig
+.
+Thedetails andrigorous mathematical treatment of theabov e derivation arestudied in[13],
+where we prove the existence and uniqueness of the solutions to these BSDEs, and that
+they are indeed related to the original value functions of ou r optimal investment problem.
+26References
+[1] Ankirchner S., Blanchet-Scalliet C. and A. Eyraud-Lois el (2009): “Credit risk premia
+and quadratic BSDEs with a single jump”, preprint arXiv:090 7.1221
+[2] Bielecki T., Jeanblanc M. and M. Rutkowski (2004): Model ing and valuation of credit
+risk. InStochastic methods in finance , vol. 1856 of Lecture Notes in Math., pages 27-126.
+Springer.
+[3] Bielecki T. and M. Rutkowski (2002): Credit risk: modell ing, valuation and hedging,
+Springer Finance.
+[4] Duffie D. and K. Singleton (2003): Credit risk: Pricing, me asurement and management.
+Princeton University Press.
+[5] El Karoui N., Hamad` ene and A. Matoussi (2008): “BSDEs an d applications”, Indiffer-
+ence pricing, theory and applications , R. Carmona ed., Princeton University Press.
+[6] El Karoui N., Jeanblanc M. and Y. Jiao (2009): “What happe ns after a default: the
+conditional density approach”, to appear in Stoc. Proc. Appli .
+[7] Fleming W. andM. Soner(2006): Controlled Markov Proces ses and Viscosity Solutions,
+2nd edition, Springer Verlag.
+[8] HuY., Imkeller P. andM. M¨ uller(2005): “Utility maximi zation inincompletemarkets”,
+Annals of Applied Probability , 15, 1691-1712.
+[9] JacodJ.(1987): Grossissementinitial, hypoth` eseH’e tth´ eor` eme deGirsanov, S´ eminaire
+de calcul stochastique , 1982-1983, Lect. Notes in Maths, vol. 1118.
+[10] Jeanblanc M. and Y. Le Cam (2009): “Progressive enlarge ment of filtrations with
+initial times”, to appear in Stochastic Proc. and their Appli.
+[11] Jeulin T. (1980): Semimartingales et grossissements d ’une filtration, Lect. Notes in
+Maths, vol. 833, Springer.
+[12] Jeulin T. and M. Yor (1985): Grossissements de filtratio n: exemples et applications,
+Lect. Notes in Maths, vol 1118, Springer.
+[13] Jiao Y., Kharroubi I. and H. Pham (2009): “Pricing and op timal investment under
+multiple defaults risk”, work in progress.
+[14] Jiao Y. and H. Pham (2009): “Optimal investment under co unterparty risk: a default-
+density approach”, to appear in Finance and Stochastics .
+[15] Lim, T. and M.C. Quenez (2008): “Utility maximization i n incomplete markets with
+default”, preprint available from http://hal.archives-ouvertes.fr/hal-00342531/en/ .
+[16] Mansuy R. and M. Yor (2006): Random times and enlargemen ts of filtrations in Brow-
+nian setting, Lect. Notes in Mathematics, Springer.
+27[17] Musiela M. and T. Zariphopoulou (2008): “Portfolio cho ice under dynamic investment
+performance criteria”, Quantitative Finance , 9, 161-170.
+[18] Oksendal B. and A. Sulem (2007): Applied Stochastic Con trol of Jump-diffusion pro-
+cesses, Springer Verlag.
+[19] PhamH. (2008): Continuous-timeStochastic Control an dOptimization with Financial
+Applications, Springer.
+[20] Peng S. and X. Xu (2009): “BSDEs with random default time and their applications
+to default risk”, paper available at: http://arxiv.org/ab s/0910.2091
+[21] Protter P. (2005): Stochastic integration and different ial equations, 2nd edition,
+Springer.
+[22] Rouge R. and N. El Karoui (2000): “Pricing via utility ma ximization and entropy”,
+Mathematical Finance , 10, 259-276.
+[23] Sch¨ onbucher P. (2003): Credit derivatives pricing mo dels. Wiley Finance.
+[24] Soner M. and N. Touzi (2002): “Dynamic programming for s tochastic target problems
+and geometric flow”, J. Eur. Math. Soc. ,4, 201-236.
+[25] Wagner D. (1980): “Survey of measurable selection theo rems: an update”, Lect. Notes
+in Math., 794, Springer Verlag.
+[26] Yong J. and X.Y. Zhou (1999): Stochastic Controls: Hami ltonian Systems and HJB
+Equations, Springer Verlag.
+28
\ No newline at end of file
diff --git a/1001.0207.txt b/1001.0207.txt
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+arXiv:1001.0207v2  [math.NT]  7 Jan 2010ON A GENERALIZATION OF THE FROBENIUS NUMBER
+ALEXANDER BROWN, ELEANOR DANNENBERG, JENNIFER FOX, JOSHUA HANNA,
+KATHERINE KECK, ALEXANDER MOORE, ZACHARY ROBBINS, BRANDON SAMPLES,
+AND JAMES STANKEWICZ
+Abstract. We consider a generalization of the Frobenius Problem where the
+object of interest is the greatest integer which has exactly jrepresentations by
+a collection of positive relatively prime integers. We prov e an analogue of a
+theorem of Brauer and Shockley and show how it can be used for c omputation.
+Thelinear diophantine problem of Frobenius has long been a celebrated problem
+in number theory. Most simply put, the problem is to find the Frobenius Number
+ofkpositive relatively prime integers ( a1,...,a k), i.e., the greatest integer Mfor
+which there is no way to express Mas the non-negative integral linear combination
+of the given ai.
+Ageneralization,whichhasdrawninterestbothfromclassicalstud yoftheFrobe-
+nius Problem ([Al05, Problem A.2.6]) and from the perspective of partit ion func-
+tions and integer points in polytopes (as in Beck and Robins [BR04]), is to ask
+for the greatest integer Mwhich can be expressed in exactly jdifferent ways. We
+make this precise with the following definitions:
+Arepresentation ofMby ak-tuple (a1,...,a k) of non-negative, relatively prime
+integers is a solution ( x1,...,x k)∈Zk
+≥0to the equation M=/summationtextk
+i=1aixi.
+We define the j-Frobenius Number of ak-tuple (a1,...,a k) of relatively prime
+positive integers to be the greatest integer Mwith exactly jrepresentations of M
+by (a1,...,a k) if such a positive integer exists and zero otherwise. We refer to th is
+quantity as gj(a1,...,a k).
+Finally we define fj(a1,...,a k) exactly as we defined gj(a1,...,a k), except that
+we consider only positive representations (x1,...,x k)∈Zk
+>0.
+Note that the 0-Frobenius of ( a1,...,a k) is the classical Frobenius Number. The
+purpose of this paper is to show the following generalization of a resu lt of Brauer
+and Shockley [BS62] on the classical Frobenius Number.
+Theorem 1. Ifd= gcd(a2,...,a k)andj≥0, then either
+gj(a1,a2,...,a k) =d·gj(a1,a2
+d,...,ak
+d)+(d−1)a1
+orgj(a1,a2,...,a k) =gj(a1,a2
+d,...,ak
+d) = 0.
+The research done in this note was completed as part of the underg raduate
+number theory VIGRE research seminar directed by Professor Din o Lorenzini and
+assisted by graduate students Brandon Samples and James Stank ewicz at the Uni-
+versity of Georgia in Fall 2008. Support for the seminar was provide d by the
+Mathematics Department’s NSF VIGRE grant.
+12 ALEXANDERBROWN, ELEANOR DANNENBERG, JENNIFERFOX, JOSHU A HANNA, KATHERINE KECK,ALEXANDER MOORE,ZACHARY ROBBINS, BRANDON SAMPLES,AND JAMESSTANKEWICZ
+Lemma 1. Iffj(a1,...,a k)is nonzero, there exist integers x2,...,x k>0such
+that
+fj(a1,...,a k) =k/summationdisplay
+i=2aixi.
+Proof.Letfj:=fj(a1,...,a k). By the definition of fj, we can write fj=k/summationdisplay
+i=1aixi,ℓ
+withxi,ℓ>0 for 1≤ℓ≤j. Since
+fj+a1=k/summationdisplay
+i=1aixi,ℓ+a1=a1(x1,ℓ+1)+k/summationdisplay
+i=2aixi,ℓ,
+we obtain at least jpositive representations of fj+a1. Asfjis the largest number
+with exactly jpositive representations, there must be at least j+1 distinct ways
+to represent fj+a1. Specifically, we have fj+a1=k/summationdisplay
+i=1aix′
+i,ℓwithx′
+i,ℓ>0 for
+all 1≤ℓ≤j+1. Subtract a1from both sides of these j+1 equations to obtain
+fj= (x′
+1,ℓ−1)a1+k/summationdisplay
+i=2aix′
+i,ℓ. Evidently, there exists some ℓ0∈[1,j+1] for which
+x′
+1,ℓ0−1 = 0 because fjcannot have j+ 1 positive representations. Therefore,
+fj(a1,...,a k) =k/summationdisplay
+i=2aix′
+i,ℓ0. /square
+Theorem 2. Ifgcd(a2,...,a k) =d, then
+fj(a1,a2,...,a k) =d·fj(a1,a2
+d,...,ak
+d).
+Proof.Letai=da′
+ifori= 2,...,kandN=fj(a1,...,a k).
+Assuming N >0, we know by Lemma 1 that
+N=k/summationdisplay
+i=2aixi=dk/summationdisplay
+i=2a′
+ixi
+withxi>0. LetN′=/summationtextk
+i=2a′
+ixi. We want to show that N′=fj(a1,a′
+2,...,a′
+k)
+and will do this in three steps.
+Step 1: First, we knowthat N′does not have j+1ormore positiverepresentations
+bya1,a′
+2,...,a′
+k. IfN′could be so represented, then for 1 ≤l≤j+1 we would
+have
+N′=a1y1,ℓ+k/summationdisplay
+i=2a′
+iyi,ℓ.
+Multiplying this equation by dimmediately produces too many representations of
+Nand thus a contradiction.
+Step 2: Next, we know that
+fj(a1,...,a k) =N=a1x1,ℓ+k/summationdisplay
+i=2aixi,ℓON A GENERALIZATION OF THE FROBENIUS NUMBER 3
+for 1≤l≤jandxi>0, so
+N
+d=a1x1,ℓ
+d+k/summationdisplay
+i=2aixi,ℓ
+d.
+Sinced|Nandd|aifori≥2, we must have d|a1x1,ℓfor 1≤ℓ≤j. In addition,
+gcd(a1,d) = 1 so we must have d|x1,ℓfor 1≤ℓ≤j. So
+N′=a1x1,ℓ
+d+k/summationdisplay
+i=2a′
+ixi,ℓ,
+henceN′hasatleast jdistinct positiverepresentations. Butwehavealreadyshown
+thatN′cannot have j+1 or more positive representations, thus N′has exactly j
+positive representations.
+Step 3: Finally we will show that N′is the largest number with exactly jpositive
+representations by a1,a′
+2,...,a′
+k. Consider any n > N′. Sincedn > dN′=N, we
+know that dncan be represented as a linear combination of a1,...,a kin exactly X
+ways with X/negationslash=j.
+Thus, for 1 ≤l≤XandX/negationslash=jwe have
+dn=a1x1,ℓ+k/summationdisplay
+i=2aixi,ℓ
+and as in Step 2,
+n=a1(x1,ℓ
+d)+k/summationdisplay
+i=2a′
+ixi,ℓ.
+IfX > jthen we certainly do not have exactly jrepresentations, so assume X < j.
+Assume now that we can write n=a1y1+/summationtextk
+i=2a′
+iyiwhereyi/negationslash=xi,ℓfor any such
+ℓ. By multiplying by dwe get a new representation for dn, which is a contradiction
+becausednis represented in exactly X/negationslash=jways.
+Therefore N′is the greatest number with exactly jpositive representations and
+so
+N′=fj(a1,a′
+2,...,a′
+k).
+Thus
+fj(a1,a2,...,a k) =d·fj(a1,a2
+d,...,ak
+d).
+/square
+Having established our results about fj(a1,...,a k), we show that we can trans-
+late these results to results about the j-Frobenius Numbers.
+Lemma 2. Eitherfj(a1,...,a k) =gj(a1,...,a k) = 0or,
+fj(a1,...,a k) =gj(a1,...,a k)+k/summationdisplay
+i=1ai.
+Proof.For ease, write fjforfj(a1,...,a k),gjforgj(a1,...,a k), andK=/summationtextk
+i=1ai.
+Any representation ( y1,...,y k) ofMgives a representation ( y1+ 1,...,y k+ 1)4 ALEXANDERBROWN, ELEANOR DANNENBERG, JENNIFERFOX, JOSHU A HANNA, KATHERINE KECK,ALEXANDER MOORE,ZACHARY ROBBINS, BRANDON SAMPLES,AND JAMESSTANKEWICZ
+ofM+K. Moreover, adding or subtracting Kpreserves the distinctness of rep-
+resentations because it adjusts every coefficient yiby 1. Therefore if Mhasj
+representations, M+Khas at least jpositive representations. Likewise, every
+positive representation of M+Kgives a representation of M. Thusfj= 0 if and
+only ifgj= 0. Assume now that fjandgjare both nonzero.
+Suppose that fj< gj+K. By definition, we can find exactly jrepresentations
+(y1,...,y k) forgjandgjhas exactly jrepresentations if and only if gj+Khas ex-
+actlyjpositiverepresentations ( x1,...,x k). However, by assumption gj+K > f j
+andgj+Khas exactly jpositive representations. This contradicts the definition
+offj, hencefj≥gj+K.
+Suppose that fj> gj+K. By definition, we can find exactly jpositive representa-
+tions(x1,...,x k) forfj. Thesameargumentasaboveshowsthat fj−Khasexactly
+jrepresentations in contradiction to the definition of gj. Thusfj≤gj+K./square
+Proof of Theorem 1: Combine Theorem 2 with Lemma 2.
+Corollary 1. Leta1,a2be coprime positive integers and let mbe a positive integer.
+Suppose that gj=gj(a1,a2,ma1a2)/negationslash= 0. Then
+•gj= (j+1)a1a2−a1−a2forj < m+1
+•gm+1= 0and
+•gm+2= (m+2)a1a2−a1−a2.
+Proof.Theorem 1 tells us that if gj(1,1,m)/negationslash= 0 then
+gj(a1,a2,ma1a2) =a2(gj(a1,1,ma1))+(a2−1)a1
+=a2(a1gj(1,1,m)+(a1−1)1)+(a2−1)a1
+=a1a2(gj(1,1,m)+2)−a1−a2.
+Following Beck and Robins in [BR04], we can use the values of the restric ted
+partitionfunction p1,1,m(k)todetermine gj(1,1,m). Furthermorewecandetermine
+the relevant values with the Taylor series1
+(1−t)2(1−tm)=/summationtext∞
+k=0p1,1,m(k)tk. Now
+recall that for k < m,p1,1,m(k) =p1,1(k) =k+1 butp1,1,m(m) =m+2 and for all
+k > m,p1,1,m(k)> m+2. Note that no number is represented m+1 times. Thus
+gm+1(1,1,m) = 0,gj(1,1,m) =j−1 forj < mandgm+2(1,1,m) =m. /square
+Remark. It is a consequence of the asymptotic in Nathanson [Na00] that fo r a
+given tuple, there may be many jfor which gj= 0, so the ordering g0< g1< ...
+may not hold. In the processof discoveringthese equalities, we not ed the somewhat
+stranger occurrence of tuples where 0 < gj+1< gj.
+Take for instance, the 3-tuple (3, 5, 8). The order g0< g1< ...holds until
+g14= 52 and g15= 51. As should also be the case, the 3-tuple increased by a factor
+ofd= 2 creates the new “dependent” 3-tuple (3, 10, 16), which fails to hold order
+in the same position with g14= 107 and g15= 105. A few independent examples
+are as follows:
+g17(2,5,7) = 43 and g18(2,5,7) = 42,
+g38(2,5,17) = 103 and g39(2,5,17) = 102,
+g35(4,7,19) = 181 and g36(4,7,19) = 180, andON A GENERALIZATION OF THE FROBENIUS NUMBER 5
+g38(9,11,20) = 376 and g39(9,11,20) = 369.
+We do not as of yet know a lower bound on jfor the above to occur. Indeed,
+in every case we have computed, if g0,g1>0 theng1> g0, but to date neither a
+proof or a counterexample has presented itself.
+References
+[Al05] J. L. R. Alfons´ ın, The Diophantine Frobenius Problem, Oxford University Press, 2005.
+[BR04] M. Beck and S. Robins, A formula related to the Frobenius Problem in two dimensions ,
+Number Theory (New York 2003), 17-23, Springer, New York, 20 04.
+[BS62] A. Brauer and J.E. Shockley, On a problem of Frobenius ,J. Reine Angew. Math. 211
+(1962) 215-220.
+[Na00] M. Nathanson, Partitions with parts in a finite set, Proc. Amer. Math. Soc. ,128
+(2000) 1269-1273.
+2000Mathematics Subject Classification : Primary 11D45, Secondary 45A05.
+Keywords: Counting solutions of Diophantine Equations, Linear Integral Equa -
+tions
+E-mail address :stankewicz@gmail.com
+Department of Mathematics, University of Georgia, Athens, G A 30602
\ No newline at end of file
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+arXiv:1001.0208v2  [cs.CC]  3 Feb 2010Symposium on TheoreticalAspects of Computer Science 2010( Nancy, France), pp. 275-286
+www.stacs-conf.org
+INTRINSIC UNIVERSALITY IN SELF-ASSEMBLY
+DAVID DOTY1AND JACK H. LUTZ2AND MATTHEW J. PATITZ2AND SCOTT M. SUMMERS2
+AND DAMIEN WOODS3
+1Department of Computer Science, University of Western Onta rio
+London, Ontario, Canada, N6A5B7.
+E-mail address , David Doty: ddoty@csd.uwo.ca
+2Department of Computer Science, Iowa State University
+Ames, IA 50011 USA.
+E-mail address , Jack H. Lutz: lutz@cs.iastate.edu
+E-mail address , Matthew J. Patitz: patitz@cs.iastate.edu
+E-mail address , Scott M. Summers: summers@cs.iastate.edu
+3California Institute of Technology, Pasadena, CA 91125, US A.
+E-mail address :woods@caltech.edu
+Abstract. We showthattheTile Assembly Model exhibits astrongnotion ofuniversality
+where the goal is to give a single tile assembly system that si mulates the behavior of any
+other tile assembly system. We give a tile assembly system th at is capable of simulating
+a very wide class of tile systems, including itself. Specific ally, we give a tile set that
+simulates the assembly of any tile assembly system in a class of systems that we call locally
+consistent : each tile binds with exactly the strength needed to stay att ached, and that
+there are no glue mismatches between tiles in any produced as sembly.
+Our construction is reminiscent of the studies of intrinsic universality of cellular au-
+tomata by Ollinger and others, in the sense that our simulati on of a tile system Tby
+a tile system Urepresents each tile in an assembly produced by Tby ac×cblock of
+tiles inU, wherecis a constant depending on Tbut not on the size of the assembly T
+produces (which may in fact be infinite). Also, our construct ion improves on earlier sim-
+ulations of tile assembly systems by other tile assembly sys tems (in particular, those of
+Soloveichik and Winfree, and of Demaine et al.) in that we sim ulate the actual process of
+self-assembly, not just the end result, as in Soloveichik an d Winfree’s construction, and we
+do not discriminate against infinite structures. Both previ ous results simulate only tem-
+perature 1 systems, whereas our construction simulates til e assembly systems operating at
+temperature 2.
+1998 ACM Subject Classification: Theory.
+Key words and phrases: Biomolecular computation, intrinsic universality, self- assembly.
+This research was supported in part by National Science Foun dation Grants 0652569, 0728806 and
+0832824, by the Spanish Ministry of Education and Science (M EC) and the European Regional Devel-
+opment Fund (ERDF) under project TIN2005-08832-C03-02, by Junta de Andaluc´ ıa grant TIC-581, and by
+Natural Sciences and Engineering Research Council of Canad a (NSERC) Discovery Grant R2824A01 and
+the Canada Research Chair Award in Biocomputing.
+c/circlecopyrtD.Doty,J.H.Lutz,M.J.Patitz,S.M. Summers,andD.Woods
+CC/circlecopyrtCreativeCommons Attribution-NoDerivsLicense276 D. DOTY, J.H. LUTZ, M.J. PATITZ, S.M. SUMMERS, AND D. WOOD S
+1. Introduction
+The development of DNA tile self-assembly has moved nanotec hnology closer to the
+goal of engineering useful systems that assemble themselve s from molecular components.
+Since Seeman’s pioneering work in the 1980s [21], many labor atory experiments have shown
+that DNA tiles can be designed to spontaneously assemble wit h one another into desired
+structures [20]. As physical and mathematical error-suppr ession techniques improve [3,8,
+13,22,24], this molecular programming of matter will becom e practical at ever larger scales.
+The Tile Assembly Model, developed by Winfree [19,26], is a d iscrete mathematical
+model of DNA tile self-assembly that enables us to explore th e potentialities and limitations
+ofthiskindofmolecular programming. Itisessentially an“ effectivization” of classical Wang
+tiling [25] in which the fundamental components are un-rota table, but translatable square
+“tile types” whose sides are labeled with glue “colors” and “ strengths.” Two tiles that are
+placed next to each other interactif the glue colors on their abutting sides match, and they
+bindif the strength on their abutting sides matches with total st rength at least a certain
+ambient “temperature.” Extensive refinements of the abstra ct Tile Assembly Model were
+given by Rothemund and Winfree in [18,19]. (Consult the tech nical appendix for full details
+of the abstract Tile Assembly Model.) The model deliberatel y oversimplifies the physical
+realities of self-assembly, but Winfree proved that it is Tu ring universal [26], implying that
+self-assembly can be algorithmically directed.
+In this paper we investigate whether the Tile Assembly Model is capable of a much
+stronger notion of universality where the goal is to give a si ngle tile assembly system that
+simulates the behavior of any other tile assembly system. We give a tile assembly system
+that is capable of simulating a very wide class of tile system s, including itself. Our notion
+of simulation is inspired by, but somewhat stronger than, in trinsic universality in cellular
+automata [2,7,14–16]. In our construction a simulated tile assembly system is encoded in
+a seed assembly of the simulating system. This encoding is do ne in a very simple (logspace
+computable) way. The seed assembly then grows to form an asse mbly that is a re-scaled
+(larger) version of the simulated assembly, where each tile in the latter is represented by a
+supertile(square of tiles) in thesimulator. Not only this, but each of the possible (nondeter-
+ministically chosen)assemblysequencesof thesimulated t ilesystemismodeledbyapossible
+assembly sequence in the simulating system (also nondeterm inistically chosen). The latter
+property of our system is important and highlights one way in which this work distinguishes
+itself from other notions of intrinsic universality found i n the cellular automata literature:
+not only do we want to simulate the final assembly but we also wa nt the simulator to have
+the ability to dynamically simulate each of the valid growth processes that could lead to
+that final assembly.
+A second distinguishing property of our universal tile set i s that it simulates nonde-
+terministic choice in a “fair” way. An inherent feature of th e Tile Assembly Model is the
+fact there are often multiple (say k) tiles that can go into any one position in an assembly
+sequence, and one of these kis nondeterministically chosen. One way to simulate this fe a-
+ture is to nondeterministically choose which of ksupertiles should grow in the analogous
+(simulated) position. However, due to the size blowup in sup ertiles caused by encoding an
+arbitrary-sized simulated tile set into a fixed-sized unive rsal tile set, it seems that we need
+to simulate one nondeterministic choice by using a sequence of nondeterministic choices
+within the supertile. Interpreting the nondeterministic c hoice to be made according to uni-
+form random selection, if the selection by the simulating ti le set is implemented in a na¨ ıveINTRINSIC UNIVERSALITY IN SELF-ASSEMBLY 277
+way, this can lead to unfair selection: when selecting 1 supe rtile out of k, some supertiles
+are selected with extremely low probability. To get around t his problem, our system uses
+a random number selector that chooses a random tile with prob ability Θ(1 /k) and so we
+claim that we are simulating nondeterminism in a “fair” way.
+Thirdly, the Tile Assembly Model has certain geometric cons traints that are not seen
+in cellular automata, and this adds some difficulty to our cons truction. Existing techniques
+for constructing intrinsically universal cellular automa ta are not directly applicable to tile
+assembly. For example, when a tile is placed at a position, th at position can not be reused
+for further “computation” and this presents substantial di fficulties when trying to fit the
+various components of our construction into a supertile. Ea ch supertile encodes the entire
+simulated tile set and has the functionality to propagate th is information to other (yet to
+be formed) supertiles. Not only this, each supertile must de cide which tile placement to
+simulate, whilst making (fair) nondeterministic choices i f necessary. Finally, each supertile
+should correctly propagate (output) sides that are consist ent with the chosen supertile.
+We give a number of figures to illustrate how these goals were m et within the geometric
+constraints of the model.
+Our main result presented in this paper is, in some sense, a co ntinuation of some previ-
+ous results in self-assembly. For instance, Soloveichik an d Winfree [23] exhibit a beautiful
+connection between the Kolmogorov complexity of a finite sha peXand the minimum num-
+ber of tiles types needed to assemble X. It turns out that their construction can be made
+to be “universal” in the following sense: there exists a tile setT, such that for every “tem-
+perature 1” tile assembly system that produces a finite shape whose underlying binding
+graph is a spanning tree, Tsimulates the given temperature 1 tile system with a corre-
+sponding blow-up in the scale. Note that this method restric ts the simulated tile system to
+be temperature 1, i.e., a non-cooperative tile assembly sys tem, which are conjectured [6] to
+produce “simple” shapes and patterns in the sense of Presbur ger arithmetic [17].
+A similar result, recently discovered by Demaine, Demaine, Fekete, Ishaque, Rafalin,
+Schweller, andSouvaine[4], establishedtheexistenceofa general-purpose“staged-assembly”
+system that is capable of simulating any temperature1 tile a ssembly system that produces a
+“fully connected” finite shape. Note that, in this construct ion, the scaling factor is propor-
+tional to O(log|T|), where Tis the simulated tile set. This construction has the desirab le
+property that the set of tile types belonging to the simulato r is general purpose (i.e., the
+size of the simulator tile set is independent of the to-be-si mulated tile set) and all of the
+information needed to carry out the simulation is, in some se nse, encoded in a sequence
+of laboratory steps. An open question in [4] is whether or not their construction can be
+augmented to handle temperature 2 tile assembly systems.
+Our construction is general enough to be able to simulate pow erful and interesting
+tile sets, yet sufficiently simple so that it actually belongs to the class of tile assembly
+systems that it can simulate, a class we term locally consistent . Systems in this class
+have the properties that each tile binds with exactly streng th 2, and there are no glue
+mismatches in any producibleassembly. This captures a wide class of tile assembly systems,
+including counters, square-builders and other shape-buil ding tile assembly systems, and the
+tile assembly systems described in [1,12,19,23]. Modulo re -scaling, our universal tile set
+can be said to display the characteristics of the entire coll ection of tile sets in its class. Our
+construction is a direct simulation in that the technique does not involve the simulation
+of intermediate models (such as circuits or Turing machines ), which have been used in
+intrinsically universal cellular automata constructions [16].278 D. DOTY, J.H. LUTZ, M.J. PATITZ, S.M. SUMMERS, AND D. WOOD S
+One of the nice properties of intrinsic universality [16] is that it provides a clear def-
+inition that facilitates proofs that a given tile set is not u niversal. We leave as an open
+problem the intrinsic universality status of the Tile Assem bly Model in its full generality.
+Lafitte and Weiss [9–11] have also studied universality in th e related model of Wang
+tiling [25]. Some of their definitions, particularly in [10] , are similar to our definitions of
+simulation and universality, and also to those of Ollinger [ 16]. However, Wang tiling is not
+a model of self-assembly, as it is concerned with the ability of finite tile sets to tile the whole
+plane (with no mismatches), without regard to the processby which these tiles are placed.
+What is important is simply the existence of some valid tilin g. In the TAM, which takes the
+order in which tiles are placed, one by one, into account, it m ust be shown that not only is
+there a sequence by which tiles could be individually and sta bly added to form the output
+assembly, but that everypossible such sequence leads to the desired output. Further more,
+in the TAM a tile addition can be valid even if it causes mismat ches as long as it is stable.
+MostattemptstoadapttheconstructionsofWangtilingstud ies(suchasthosein[9–11])
+to self-assembly result in a tile assembly system in which ma ny junk assemblies are formed
+due to incorrect nondeterministic choices being made that a rrest any further growth and/or
+result in assemblies which are inconsistent with the desire d output assembly. We therefore
+require novel techniques to ensure that no nondeterminism i s introduced, other than that
+already present in the tile system being simulated, and that the only produced assemblies
+are those that represent the intended result or valid partia l progress toward it.
+2. Intrinsic Universality in Self-Assembly
+In this section, we define our notion of intrinsic universali ty of tile assembly systems.
+It is inspired by, but distinct from, similar notions for cel lular automata [16]. Where
+appropriate, we identify where some part of our definition di ffers from the “corresponding”
+parts in [16], typically due to a fundamental difference betwe en the abstract Tile Assembly
+Model and cellular automata models.
+Intuitively, a tile set Uis universal for a class Cof tile assembly systems if Ucan
+“simulate” any tile assembly system in C, where we use an appropriate seed assembly to
+give a tile assembly system U.Uis intrinsically universal if the simulation of TbyUcan be
+done according to a simple “block substitution scheme” wher e equal-size square blocks of
+tiles in assemblies produced by Urepresent tiles in assemblies produced by T. Furthermore,
+since we wish to simulate the entire process of self-assembl y, and not only the final result, it
+is critical that thesimulation besuch that the“local trans ition rules”involving intermediate
+producible (and nonterminal) assemblies of Tbe faithfully represented in the simulation.
+In the subsequent definitions, given two partial functions f,g, we write f(x) =g(x) iff
+andgarebothdefinedandequalon x, oriffandgarebothundefinedon x. Letc,c′∈N, let
+[c:c′] denote the set {c,c+1,...,c′−1}, and let [ c] denote the set [0 : c] ={0,1,...,c−1},
+so that [c]2forms ac×csquare with the origin as the lower-left corner.
+The natural analog of a configuration of a cellular automaton is an assembly of a tile
+assembly system. However, unlike cellular automata in whic h every cell has a well-defined
+state, in tile assembly, there is a fundamental difference bet ween a point being empty space
+and being occupied by a tile. Therefore we keep the conventio n of representing an assembly
+as a partial function α:Z2/axisshort/axisshort/arrowaxisrightT(for some tile set T), rather than treating empty space
+as just another type of tile.INTRINSIC UNIVERSALITY IN SELF-ASSEMBLY 279
+LetT= (T,σT,τ) andS= (S,σS,τ) be tile assembly systems. For simplicity, assume
+thatσT(0,0) is defined, and σTis undefinedon Z2−{(0,0)}(i.e.,Tis singly-seeded with the
+seed tile placed at the origin). We will use this assumption o f a single seed throughout the
+paper, but it is not strictly necessary and is only used for si mplicity of discussion. Define a
+representation function to be a partial function of the form r: ([c]2/axisshort/axisshort/arrowaxisrightS)/axisshort/axisshort/arrowaxisrightT. That is, r
+takes a pattern p: [c]2/axisshort/axisshort/arrowaxisrightSof tile types from Spainted onto a c×csquare (with locations
+at which pis undefined representing empty space), and (if ris defined for input p) gives a
+single tile type from T. Intuitively, rtells us how to interpret c×cblocks within assemblies
+ofSas single tiles of T. We write REPRfor the set of all representation functions.
+WesayS(intrinsically) simulates Twith resolution loss cifthereexistsarepresentation
+function r: ([c]2/axisshort/axisshort/arrowaxisrightS)/axisshort/axisshort/arrowaxisrightTsuch that the following conditions hold.
+(1) dom σS⊆[c]2andr(σS) =σT(0,0), i.e., the seed assembly of Srepresents the seed
+ofT.
+(2) For every producible assembly αT∈ A[T] ofT, there is a producible assembly
+αS∈ A[S] ofSsuch that, for every x,y∈Z,
+r((αS↾([cx:c(x+1)]×[cy:c(y+1)]))+( −cx,−cy)) =αT(x,y).
+That is, the c×cblock at (relative) position ( x,y) (relative to the other c×cblocks;
+theabsolute position is ( cx,cy)) of assembly αSrepresentsthe tile typeat (absolute)
+position ( x,y) of assembly αT. In this case, write r∗(αS) =αT; i.e.,rinduces a
+function r∗:A[S]→ A[T].
+(3) For all αT,α′
+T∈ A[T], it holds that αT→Tα′
+Tif and only if there exist αS,α′
+S∈
+A[S] such that r∗(αS) =αT,r∗(α′
+S) =α′
+T(in the sense of condition (2)), and
+αS→Sα′
+S. That is, every valid assembly sequence of Tcan be “mimicked” by S,
+but no other assembly sequences can be so mimicked, so that th e meaning of the
+relation→is preserved by r∗.
+LetCbe a class of singly-seeded tile assembly systems, and let Ube a tile set (with tile
+assembly systems having tile set Unot necessarily elements of C). Note that every element
+ofC,REPR, andFIN(U) is a finite object, hence can be represented in a suitable for mat for
+computation in some formal system such as Turing machines. W e sayUis(intrinsically)
+universal forCif there are computable functions R:C−→REPRandA:C−→FIN(U)
+such that, foreach T= (T,σT,τ)∈C, thereis a constant c∈Nsuchthat, letting r=R(T),
+σ=A(T), andUT= (U,σ,τ),UTsimulates Twith resolution loss cand representation
+function r. That is, R(T) outputs a representation function that interprets assemb lies of
+UTas assemblies of T, andA(T) outputs the seed assembly used to program tiles from U
+to represent the seed tile of T.
+3. An Intrinsically Universal Tile Set
+In this section, we exhibit an intrinsically universal tile set for any “nice” tile assembly
+system. Before proceeding, we must first define the notion of a “nice” tile assembly system.
+LetT= (T,σ,2) be a tile assembly system, and /vector αbe an assembly sequence in Twhose
+result is denoted as α. We say that Tislocally consistent if the following conditions hold.
+(1) For all /vector m∈domα−domσ,/summationtext
+/vector u∈IN/vector α(/vector m)strα(/vector m)(/vector u) = 2,where IN/vector α(/vector m) is the set of
+sides on which the tile that /vector αplaces at location /vector minitially binds. That is, every
+tile initially binds to the assembly with exactly bond stren gth equal to 2 (either a
+single strength 2 bond or two strength 1 bonds).280 D. DOTY, J.H. LUTZ, M.J. PATITZ, S.M. SUMMERS, AND D. WOOD S
+(2) For all producible assemblies α∈ A[T],/vector u∈U2, and/vector m∈domα, ifα(/vector m+/vector u) is
+defined, then the following condition holds:
+strα(/vector m)(/vector u)>0⇒labelα(/vector m)(/vector u) = label α(/vector m+/vector u)(−/vector u) and str α(/vector m)(/vector u) = str α(/vector m+/vector u)(−/vector u).
+While condition (1) of the above definition is reminiscent of the first condition of local
+determinism [23], the second condition says that there are n o (positive strength) label mis-
+matches between abutting tiles. However, we must emphasize that a locally consistent tile
+assembly system need not bedirected, and moreover, even a lo cally deterministic tile assem-
+bly system need not be locally consistent because of the lack of any kind of “determinism
+restriction” in the latter definition. Our main result is the following.
+Theorem 3.1 (Main theorem) .LetCbe the set of all locally consistent tile assembly sys-
+tems. There exists a finite tile set Uthat is intrinsically universal for C.
+In the remainder of this section, we prove Theorem 3.1, that i s, we show that for every
+locally consistent tile assembly system T= (T,σ,2), there exists a seed assembly σT, such
+that the tile assembly system UT= (U,σT,2) simulates Twith a resolution loss c∈N
+that depends only on the glue complexity of T. Instead of giving an explicit (and tedious)
+definition of the tile types in U, we implicitly define Uby describing how UTsimulates T.
+3.1. High-Level Overview
+Intuitively, Usimulates Tby growing “supertiles” that correspond to tile types in T. In
+other words, every supertile is a c×cblock of tiles that is mappedto a tile type t∈T. To do
+this, each supertilethat assembles in UTcontains the full specification of Tas alookup table
+(a long row of tiles that encodes all of the information in the set of tile types T), analogous
+to the genome of an organism being fully replicated in each ce ll of that organism, no matter
+how specialized the function of the cell. This lookup table i s carefully propagated through
+each supertile in UTvia a series of “rotation” and “copy” operations – both of whi ch are
+well-known self-assembly primitives.
+In the table, we represent each (glue,direction) pair as a bi nary string, and represent
+the tile set as a table mapping 1-2 input glue(s) to 0-3 output glue(s). Since each tile type
+ofTmay not have well-defined input sides, when two supertiles re presenting tiles of Tmust
+potentially cooperate to place a new supertile within a bloc k adjacent to both of them, it
+is imperative that each grows into the block in such a way as to remain unobtrusive to the
+other supertile. This is done with a “probe” that grows towar d the center of the block, as
+shown in Figure 1. At the moment the probes meet in the middle, they “find out” in what
+direction the other input supertile lies, and at that point d ecide in which direction to grow
+the rest of the forming supertile. so as to avoid the tiles tha t were already placed as part of
+the probes. We do not know how to deal with three probes at once , which is the reason both
+parts of the definition of locally consistent, which imply th at only two input probes will
+ever be present at one time. The next step is to bring the value s of two input glues together
+before doing a lookup on the table, because they are both need ed to simulate cooperation.
+The table must be read and copied at the same time, otherwise t he planarity of the tiles
+would hide the table as it is read and it could not be propagate d to the output supertiles.
+Many choices made in the construction, such as the relative p ositioning of glues/table, or
+thecounter-clockwise orderof assembly, are choices that s implywereconvenient and seemed
+to work, but are not necessarily required.INTRINSIC UNIVERSALITY IN SELF-ASSEMBLY 281
+3.2. Construction of the Lookup Table
+In order to simulate the behavior of TwithUT, we must first encode the definition of
+Tusing tiles from U. We will do this by constructing a “glue lookup table,” denot ed as
+TT, and is essentially the self-assembly version of a kind of ha sh table. Informally, TTis a
+(very) long string (of tiles from U) consisting of two copies of the definition of the tile set
+Tseparated by a small group of spacer symbols. The left copy of the lookup table is the
+reverseof the right copy. The lookup table maps all possible sets of i nput sides for each
+tile type t∈Tto the corresponding sets of output sides.
+3.2.1.Addresses. The lookup table TTconsists of a contiguous sequence of “addresses,”
+which are formed from the definition of T. Namely, for each tile type t∈T, we create a
+unique binary key for each combination of sides of twhose glue strengths sum to exactly
+2. Each of these combinations represents a set of sides which couldpotentially serve as the
+input sides for a tile of type tin a producible assembly in T.
+We say that a padis an ordered triple ( g,d,s) where gis a glue label in T,d∈
+{N,S,E,W }is an edge direction, and s∈ {0,1,2}is an allowable glue strength. Note that
+a set of four pads – one for each direction d– fully specifies a tile type. We use Pad( t,d) to
+denote the pad on side dof the tile type t∈T
+Let Bin(p) be the binary encoding of a pad p= (g,d,s), consisting of the concatenation
+of the following component binary strings:
+(1)g(glue specification ): LetGbethe set of glue types from all edges with positive glue
+strengthsin T∪{gnull}(a.k.a., thenullglue). Fixsomeordering gnull≤g0≤g1≤ ···
+of the set G. The binary representation of giis the binary value of ipadded with
+0’s to the left (as necessary) to ensure that the string is exa ctly⌈log(|G|+1)⌉bits.
+(2)d(direction ): Ifd=N(E,S, orW), append 00 (01, 10, or 11, respectively).
+(3)s(strength): Ifs= 1 (2) append 0 (1).
+Note that ⌈log(|G|+1)⌉+2+1 is the length of the binary string encoding an arbitrary
+padp, and is a constant that depends only on T.
+Anaddressis a binary string that represents a set of pads which, themse lves, can
+potentially serve as the input sides of some tile type t∈T. It can be composed of one of
+the two following binary strings:
+(1) A prefix of zeros, 0⌈log(|G|+1)⌉+3, followed by Bin( p) forp= (g,d,2), or
+(2) the concatenation of Bin( p1) and Bin( p2) forp1= (g1,d1,1) andp2= (g2,d2,1).
+The ordering of Bin( p1) and Bin( p2) in an address must be consistent with the
+following orderings: EN,SE,WS,NW,NS,EW .
+Note that it is possible for more than one tile type t∈Tto share a set of input pads
+and therefore an address.
+3.2.2.Encoding of T.We will now construct the string wT, which will represent the defini-
+tion ofT. Intuitively, wTwill be composed of a series of “entries.” Each entry is assoc iated
+to exactly one address of a tile type t∈Tand specifies the pads for the output sides of t.
+In this way, once the input sides for a supertile have formed, the corresponding pads can
+be used to form an address specifying (a set of) appropriate o utput pads. Note that since
+more than one tile type may share an address in a nondetermini stic tile assembly system,
+more than one tile type may share a single entry.282 D. DOTY, J.H. LUTZ, M.J. PATITZ, S.M. SUMMERS, AND D. WOOD S
+We define an entryto be a string beginning with ‘#’ followed by zero or more “sub -
+entries”, each corresponding to a different tile type, separa ted by semicolons. Let Abe the
+set of all binary strings representing every address create d for each t∈T. The string wT
+will consist of 1 + max Aentries for addresses 0 to max A. Theithentry, denoted as ei,
+corresponds to the ithaddress, which may or may not be in A(if it is not, then eiis empty).
+We say that a sub-entry consists of a string specifying the pads for the output sides
+of a tile type t∈T. Leteibe the entry containing a given sub-entry (note that iis the
+address of ei), andTi⊆Tbe the set of tile types addressable by i(i.e., the set of tile types
+for which iis a valid address). The entry eiwill be comprised of exactly |Ti|sub-entries.
+For 0≤k < j, thekthsub-entry in ei, wheretk∈Tiis thekthelement of Ti(relative to
+some fixed ordering), is the string OUT( N),OUT(E),OUT(S),OUT(W) (the commas in
+the previous string are literal) with OUT( d) = Bin(Pad( tk,d))Rif the glue for Pad( tk,d)
+is notgnullanddis not a component of the address i, otherwise OUT( d) =λ. Intuitively,
+a sub-entry is a comma-separated list of the (reversed) bina ry representations of the pads
+for an addressed tile type, but including only pads whose glu es are not gnulland whose
+directions are not a part of the address (and therefore input sides). We will now use the
+stringwTto construct the lookup table TT.
+3.2.3.Full specification of TT.We now give the full specification for the lookup table TT.
+First, define the following strings: w0= ‘>’,w1= ‘<%%>’,w2= ‘<’. Now let TT
+be as follows: TT= sb(w0◦wT◦w1◦(wT)R◦w2),where, for strings xandy,x◦y
+is the concatenation of xandy, and sb : Σ∗→Σ∗is defined to “splice blanks” into its
+input: between every pair of adjacent symbols in the string x, a single ‘ /rightanglesw/rightanglese’ (blank) symbol
+is inserted to create sb( x). This splicing of blanks is required to be able to read from t he
+table without “locking it from view”, when reading the table for operations that require
+growing a column of tiles in towards the table (as opposed to a way from it), a blank column
+is used, and for growing a column away from the table, a symbol column is used so that
+the symbol can be propagated to the top of the column for later copying.
+3.2.4.The Lookup Procedure. In our construction, when a supertile t∗that is simulating a
+tile type t∈Tforms, we must overcome the following problem: once we combi ne the input
+pads (given as the output pads of the supertiles to which t∗attaches), how do we use TT
+to lookup the output pads for t∗? In what follows, we briefly describe how we achieve this.
+In other words, we show how an address, a string of random bits , and a copy of TTare
+used to compute the pad values for the non-input sides of a sup ertile. A detailed figure and
+example of this procedure can be found in the technical appen dix.
+For ease of discussion and without loss of generality, we ass ume that the row of tiles
+encoding TT(assembled West to East) and the column of tiles encoding an a ddress and a
+randomstringof bits(assembled NorthtoSouthat theWest en dofTT)arefullyassembled,
+forming an ‘L’ shape with no tiles in the area between them. Fo r other orientations of the
+table and address the logical behavior is identical, simply rotated.
+Intuitively, the assembly of the lookup procedure assemble s column wise in a zig-zag
+fashion from left to the right. In the “first phase,” a counter initialized to 0 is incremented
+in each column where the value of the tile in the representati on ofTTis a ‘;’, thus counting
+up at each entry contained in TT. Once that number matches the value of the given
+address (which, along with the random bits is copied through this procedure), the entry eINTRINSIC UNIVERSALITY IN SELF-ASSEMBLY 283
+corresponding to that address has been reached and a new coun ter begins which counts the
+number of sub-entries nin that entry. Note that for directed tile systems, n≤1. Once the
+end of that entry is encountered, yet another counter, initi alized to 0, begins and increments
+on each remaining entry until the end of the first copy of wTis reached (the number nis
+propagated to the right). This counts the number of entries, denoted as m, between eand
+the end of the lookup table. The “second phase” is used to perf orm, in some sense, an
+operation equivalent to calculating p=bmodn, wherebis the binary value of the string
+of random bits required for the lookup procedure (this is how we simulate nondeterministic
+assemblies). This selects the index of the sub-entry in ewhich will be used, completing the
+random selection of one of the possibly many tile types conta ined in entry e.
+In the current version of our construction, we merely use a ra ndom number selection
+procedurereminiscent ofthemoreinvolved (but moreunifor m)randomselection procedures
+discussed in [5]. Although it is possible to incorporate the se more advanced techniques
+into our construction (and thus achieve a higher degree of un iformity in the simulation of
+randomized tile systems), we choose not to do so for the sake o f simplicity.
+Next, a reverse counter, a.k.a., a subtractor, counts down a t each entry from mto 0,
+and by the way we constructed TT, this final counter obtains the value 0 at the entry e(in
+the reverse of wT). Now, another subtractor counts from pto 0 to locate the correct sub-
+entry that was selected randomly. Finally, each pad in the su b-entry is rotated “up and to
+the right,” and the group of pads is propagated through the re mainder of the lookup table,
+thus ending with the values of the non-input pads represente d in the rightmost column.
+3.3. Supertile design
+A supertile sis a subassembly in UTconsisting of a c×cblock of tiles from T, wherec
+depends on the glue complexity of T. Eachscan be mapped to a unique tile type t∈T. In
+our construction there are two logical supertile designs. T he first, denoted type-0, simulates
+tile additions in Tin which there are 2 input sides, each with glue strength = 1. T he
+second, denoted type-1, simulates the addition of tiles via a single strength 2 bond .
+While there are several differences in the designs of type-0andtype-1supertiles, one
+commonality is how their edges are defined. Namely each input or output edge of any
+supertileisdefinedbythesamesequenceofvariablevalues. Sincetheedgesforeachdirection
+are rotations of each other, we will discuss only the layout o f the south side of a supertile.
+From left to right, the tiles along the south edge of a superti le will represent a string formed
+by the concatenation (in order) of the strings: TT, Bin(Pad( t,S)), 0c′, Bin(Pad( t,S)), and
+TT. Note that c′is a constant that depends on the glue complexity of T.
+3.3.1.Type-0 Supertiles (i.e., simulating tiles that attach via t wo single-strength bonds).
+When a tile binds to an assembly in Twith two input sides whose glues are each single
+strength, there are/parenleftbig4
+2/parenrightbig
+= 6 possible combinations of directions for those input side s: north
+and east (NE), north and south (NS), north and west (NW), east and south (ES), east
+and west (EW), and south and west (SW). These combinations ca n be divided into two
+categories, those in which the sides are opposite each other (NS and EW), and those in
+which the sides are adjacent to each other (NE, NW, ES, and SW) .
+Opposite Input Sides: Supertiles which represent tile additions with two opposit e
+input sides, NS and EW, are logically identical to rotations of each other, so here we will
+only describe the details of a supertile with NS input sides. Figure 1 shows a detailed image284 D. DOTY, J.H. LUTZ, M.J. PATITZ, S.M. SUMMERS, AND D. WOOD S
+1 11 1 10 0 00 0 00 0 01 1 1
+ge gs gsgn gnge gwge gw
+T TT T
+TT
+gs gs TT T
+Tgn
+gegegn
+0 0TTT
+15 5
+679101112 13
+8
+22
+314 4
+Look□up
+output□gluesMirror□of□tiles
+on□the□east□side
+Figure 1. NS supertile
+(a) (b) (c) (d)
+(e) (f) (g) (h)
+Figure 2. Intuitive depiction of (a portion of) the self-assembly of a type-0
+supertile. Note that the lookup procedure is performed in (d) and ( e).INTRINSIC UNIVERSALITY IN SELF-ASSEMBLY 285
+depicting the formation of an NS supertile, with arrows givi ng the direction of growth
+for each portion and numbers specifying the order of growth. For ease of discussion and
+without loss of generality, we assume that the rows of tiles w hich form the input sides of
+a supertile have fully formed before any other part of the sup ertile assembles. The first
+portions to assemble are the center blocks to the interior of each input side, labeled 1.
+This subassembly forms a square in which a series of nondeter ministic selections of tile
+types is used to generate a random sequence of bits. These bit s are propagated to the left
+and right sides of the block, to ensure that each side uses the same random bits for the
+randomized selection after the sides have been “sealed off” f rom each other by “probes”
+described next. Once that block has completed, a log-width b inary subtractor, which is
+half the width of the block, assembles. The subtractors from the north and south count
+down from a specified value (that depends on Tand is encoded into the seed supertile) to
+0, and shrink in width until they terminate at positions adja cent to the center square of the
+block. These subtractors are “probes” that grow to the cente r where the direction of the
+input sides (the type) is detected. It is at this point that the central (black in th e figure)
+tile can attach. It is this tile which determines the typeof the supertile (NS in this case)
+because it is unique to the combination of directions from wh ich the inputs came. At this
+point, symmetry is broken and two paths of tiles assemble fro m the center back towards the
+north side. They in turn initiate the growth of subassemblie s which propagate the value of
+the north input pad down towards the South of the supertile. O nce that growth nears the
+southern side, the two input pads are rotated and brought tog ether, with this combination
+of input pads forming an addressin the lookup table. In the manner described previously,
+this address along with the random bits generated within blo ck 1 (which are also passed
+through block 5) is used to form the subassembly of block 6 who se southern row contains a
+representation of TTand results in the correct output pads being represented in t he final
+column of that block. Note that Figure 1 only shows the detail s of the east side of the block
+since the West side is an identical but rotated version. Fina lly, subassemblies 7 through
+13 form which rotate and pass the necessary information to th e locations where it must be
+correctly deposited to form the output sides of the supertil e. Every side of a supertile that
+is not an input side receives an output pad, even if it is for th e null glue (in which case it
+does not initiate the growth of the input side of a possible ad jacent supertile).
+References
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+ber selection in self-assembly , Proceedings of The Eighth International Conference on Unc onventional
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+6. David Doty, Matthew J. Patitz, and Scott M. Summers, Limitations of self-assembly at temperature 1 ,
+Proceedings of The Fifteenth International Meeting on DNA C omputing and Molecular Programming
+(Fayetteville, Arkansas, USA, June 8-11, 2009), 2009, to ap pear.
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+l’Informatique du Parall´ elisme, Ecole Normale Sup´ erieu re de Lyon, January 1998.
+8. Kenichi Fujibayashi, David Yu Zhang, Erik Winfree, and Sa toshi Murata, Error suppression mechanisms
+for dna tile self-assembly and their simulation , Natural Computing: an international journal 8(2009),
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+Lecture Notes in Computer Science, vol. 4393, Springer, 200 7, pp. 367–380.
+10. ,Simulations between tilings , Tech. report, University of Athens, 2008.
+11. ,An almost totally universal tile set , TAMC (Jianer Chen and S. Barry Cooper, eds.), Lecture
+Notes in Computer Science, vol. 5532, Springer, 2009, pp. 27 1–280.
+12. JamesI.Lathrop, JackH.Lutz,MatthewJ.Patitz, andSco ttM.Summers, Computability and complexity
+in self-assembly , Proceedings of The Fourth Conference on Computability in E urope (Athens, Greece,
+June 15-20, 2008), 2008.
+13. Urmi Majumder, Thomas H LaBean, and John H Reif, Activatable tiles for compact error-resilient
+directional assembly , 13th International Meeting on DNA Computing (DNA 13), Memp his, Tennessee,
+June 4-8, 2007., 2007.
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+Philadelphia, 2007.
+15. Nicolas Ollinger, The intrinsic universality problem of one-dimensional cel lular automata , 20th Annual
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+vol. 2607, Springer, 2003, pp. 632–641.
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+N. Murphy, eds.), EPTCS, vol. 1, 2009, arXiv:0906.3213v1 [c s.CC], pp. 199–204.
+17. Moj˙ zesz Presburger, U¨ber die vollst¨ andigkeit eines gewissen systems der arithm etik ganzer zahlen ,
+welchem die Addition als einzige Operation hervortritt. Co mpte Rendus du I. Congrks des Mathe-
+maticiens des pays Slavs, Warsaw, 1930, pp. 92–101.
+18. Paul W. K. Rothemund, Theory and experiments in algorithmic self-assembly , Ph.D. thesis, University
+of Southern California, December 2001.
+19. Paul W. K. Rothemund and Erik Winfree, The program-size complexity of self-assembled squares (ex -
+tended abstract) , STOC ’00: Proceedings of the thirty-second annual ACM Symp osium on Theory of
+Computing (New York, NY, USA), ACM, 2000, pp. 459–468.
+20. Paul W.K. Rothemund, Nick Papadakis, and Erik Winfree, Algorithmic self-assembly of DNA Sierpinski
+triangles , PLoS Biology 2(2004), no. 12, 2041–2053.
+21. Nadrian C. Seeman, Nucleic-acid junctions and lattices , Journal of Theoretical Biology 99(1982), 237–
+247.
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+The eleventh International Meeting on DNA Computing, 2005.
+23. ,Complexity of self-assembled shapes , SIAM Journal on Computing 36(2007), no. 6, 1544–1569.
+24. Thomas LaBean Urmi Majumder, Sudheer Sahu and John H. Rei f,Design and simulation of self-
+repairing DNA lattices , DNA Computing: DNA12, Lecture Notes in Computer Science, v ol. 4287,
+Springer-Verlag, 2006.
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+This work is licensed under the Creative Commons Attributio n-NoDerivsLicense. To view a
+copyofthislicense,visit http://creativecommons.org/licenses/by-nd/3.0/ .
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+arXiv:1001.0209v1  [math.AP]  1 Jan 2010EXPONENTIAL ENERGY DECAY FOR
+DAMPED KLEIN-GORDON EQUATION WITH
+NONLINEARITIES OF ARBITRARY GROWTH
+L. ALOUI, S. IBRAHIM, AND K. NAKANISHI
+Abstract. We derive a uniform exponential decay of the total energy for the
+nonlinear Klein-Gordon equation with a damping around spatial infinity inRNor
+in the exterior of a star-shaped obstacle. Such a result was first p roved by Zuazua
+[40, 41] for defocusing nonlinearity with moderate growth, and late r extended to
+theenergysubcriticalcasebyDehman-Lebeau-Zuazua[7], usinglin earapproxima-
+tion and unique continuation arguments. We propose a different app roach based
+solely on Morawetz-type a prioriestimates, which applies to defocusing nonlinear-
+ity of arbitrary growth, including the energy critical case, the sup ercritical case
+and exponential nonlinearities in any dimensions. One advantage of o ur proof,
+even in the case of moderate growth, is that the decay rate is indep endent of the
+nonlinearity. We can also treat the focusing case for those solution s with energy
+less than the one of the ground state, once we get control of the nonlinear part
+in Morawetz-type estimates. In particular this can be achieved whe n we have the
+scattering for the undamped equation.
+Contents
+1. Introduction 2
+1.1. Defocusing case 3
+1.2. Focusing case 5
+2. Proof in the defocusing case: Theorem 1.1 7
+2.1. Truncation of nonlinearity 7
+2.2. Uniform decay for subcritical nonlinearity 10
+3. Proof in the focusing case: Theorem 1.2 13
+3.1. Variational properties 13
+3.2. Global Strichartz bound from the scattering 14
+3.3. Morawetz estimate from the Strichartz bound 15
+3.4. Concluding the proof of Theorem 1.2. 17
+References 18
+Date: May 28, 2018.
+2000Mathematics Subject Classification. 35L70, 35Q55, 35B60, 35B33, 37K07.
+Key words and phrases. Nonlinear Klein-Gordon equation, energy decay, stabilization.
+L. A. is partially supported by LAMSIN.
+S. I. is partially supported by NSERC# 371637-2009grant and a sta rt up fund from University
+of Victoria.
+K.N. is partially supported by Grant-in-Aid for Scientific Research 21 740095.
+12 L. ALOUI, S. IBRAHIM, AND K. NAKANISHI
+1.Introduction
+We study total energy decay for the damped nonlinear Klein-Gordo n equation
+(NLKG) for u(t,x) : [0,∞)×Ω→R,
+utt+a(x)ut−∆u+u+f′(u) = 0 (t,x)∈[0,∞)×Ω,
+u= 0 (t,x)∈(0,∞)×∂Ω,(1.1)
+where Ω is a C1(exterior) domain in RN(N≥1), the damper a: Ω→[0,∞) is
+effective uniformly around the spatial infinity
+a(x)≥0, a∈L∞(Ω),liminf
+|x|→∞a(x)>0,(1.2)
+and the nonlinear energy f:R→RisC2satisfying1
+f(u) =f(|u|), f(0) =f′(0) =f′′(0) = 0. (1.3)
+Thisequationmayberegardedasasimplemodelfornonlineardissipat ivehyperbolic
+equations with purely dispersive spatial regions. The stabilization pr oblem consists
+in deriving sufficient conditions on the dissipation a(x) guaranteeing uniform decay
+of the total energy, defined by
+E(u;t) :=/integraldisplay
+Ω|ut|2+|∇u|2+|u|2+2f(u)dx. (1.4)
+When the nonlinear energy is not positive definite in the whole energy s pace, we
+have to distinguish it from the free energy
+EF(u;t) :=/integraldisplay
+RN|˙u|2+|∇u|2+|u|2dx. (1.5)
+Multiplying (1.1) by ˙ uand integrating over the slab ( t1,t2)×Ω, one formally gets
+the energy identity
+E(u;t2)−E(u;t1) =−2/integraldisplayt2
+t1/integraldisplay
+Ωa(x)|˙u(t,x)|2dxdt. (1.6)
+Heuristically, one achieves the stabilization when certain portion of t he energy keeps
+flowing into thekinetic partonthe regionofeffective dissipation. The n one typically
+obtains exponential decay in the form
+E(u;t)≤Ce−γtE(u;0), (1.7)
+with some constants C,γ >0.
+The two major factors in this problem are the geometric structure of the domain,
+the boundary condition and the dissipation on one hand, and the non linear inter-
+action on the other hand. The geometric aspects have been exten sively studied
+especially in the linear setting, with deep insights into wave propagatio n properties,
+see e.g., [19, 29, 3].
+However the nonlinear interaction becomes the bigger obstruction for the finer
+investigation of the wave propagation. The standard strategy is t o start with some
+1These assumptions are quite natural for the nonlinear Klein-Gordo n energy. f(0) = 0 is
+necessary to have finite energy, f′(0) = 0 follows from the evenness, while f′′(0) = 0 means that
+the mass term is exactly u. It can be replaced with mu(m >0), but in the massless case m= 0
+we do not have the exponential decay. See e.g. [27] for the algebra ic decay in the massless case.EXPONENTIAL ENERGY DECAY FOR DAMPED NLKG 3
+fairly weak decay properties which are accessible even in the nonlinea r setting, and
+then through some limiting procedures and approximation argument s, to reduce the
+problem to the linear case, where much finer informations are availab le. In this type
+of arguments, the linear approximation poses essential limitations o n the generality
+of the nonlinear interaction. We refer to [40, 41, 5, 7, 6, 1] for the stabilization
+results in the current setting (1.1).
+1.1.Defocusing case. Our observation is that one can skip the latter part of the
+argument, i.e. the linear approximation, at least in the simplest geome tric setting,
+thereby obtaining the exponential decay for fairly general nonline arity with the
+repulsive sign. Our nonlinearity condition is a sort of self-coercivene ss, given in
+terms of
+g(u) :=uf′(u)−2f(u). (1.8)
+The first main result of this paper is
+Theorem 1.1 (defocusing case) .LetΩ⊂RN(N∈N) be either the whole space or
+the exterior of a star-shaped obstacle with a C1boundary. Assume that a: Ω→R
+satisfies for some constants M,R,a 0>0
+0≤a≤M,inf
+|x|>Ra(x)≥a0,(1.9)
+f:R→RisC2satisfying f(0) =f′(0) =f′′(0) = 0,f(u) =f(|u|), and
+g(u)≥0,0≤f(u)≤C0(|u|2+g(u)), (1.10)
+for some constant C0>0. Then there exists γ >0determined only by N,M,
+R,a0andC0, such that for every initial data (u(0,x),ut(0,x))with finite energy
+E(u;0)<∞, there exists a global weak solution uof(1.1)satisfying
+E(u;t)≤2e−γtE(u;0) (1.11)
+for allt≥0. If(1.10)is weakened to
+g(u)≥0,0≤f(u)≤C0(|u|2+|u|q+g(u)) (1.12)
+with some 2≤q≤2⋆:= 2N/(N−2)(2≤q <∞ifN≤2), then(1.11)still holds,
+butγ >0depends (non-increasingly) on E(u;0)andqtoo.
+Note that the rate γis uniform in the whole energy space under the condition
+(1.10), while under (1.12) it is uniform on each ball of the energy spac e. In any
+dimension N≥1, examples of the nonlinearity include all defocusing powers and
+exponentials
+f(u) =λexp(µ|u|ν)|u|2+α,(λ,µ,ν,α ≥0), (1.13)
+and their convex combinations. The global wellposedness in the ener gy space is so
+far available under the growth condition ( H1subcritical or critical case)
+|f′(u1)−f′(u2)| ≤C[|u1|+|u2|]p−1|u1−u2|, p+1≤2⋆=2N
+N−2,(1.14)
+with suitable modifications when 2⋆<3 (N >6), and exponential growth condition
+forN= 2 (see, e.g. [11, 32, 12]). Once we have global wellposedness, the above4 L. ALOUI, S. IBRAHIM, AND K. NAKANISHI
+uniform decay of course applies to the unique solution, but our theorem applies even
+if we do not know unique existence (i.e. in the H1supercritical case) .
+Let us compare our result with the preceding ones for the stabilizat ion problem
+of (1.1). Zuazua in [40, 41] first proved the exponential decay by u sing the unique
+continuation result by Ruiz [31], in bounded and unbounded domains fo r nonlinear
+termf′mapping H1intoL2, i.e. (1.14) with p≤N/(N−2), and with a(x) effective
+on the whole boundary and around the infinity. Then Dehman in [5] rela xed the
+nonlinearity condition by making use of the Strichartz estimates, an d propagation
+of microlocal defect measures, introduced by G´ erard [9] to des cribe possible lack
+of compactness in H1bounded sequence of linear solutions. The latter result was
+further extended to the full range of H1subcritical case, i.e. (1.14) with p+1<2⋆,
+by Dehman-Lebeau-Zuazua [7]. They used a linearization argument e nabling them
+to propagate the possible singularities (lack of compactness) along bi-characteristics
+to the region where the damper is effective. However this linear appr oximation
+might fail in the H1critical case p+ 1 = 2⋆, see for example G´ erard [10]. The
+H1critical case was finally solved by Dehman-G´ erard [6] using the nonlin ear profile
+decomposition of Bahouri-G´ erard [2] instead.
+We point out that these proofs do not explicitly provide any bound on the decay
+rateγ. Recently T´ ebout [38] gave another proof based on a Carleman e stimate
+due to Ruiz [31]. His proof is more quantitative, but also requires the condition
+p≤N/(N−2) to bound the nonlinear term in the Carleman estimate.
+Our proof of Theorem 1.1 gives explicit estimates on γ, and moreover it is very
+elementary, especially compared with the above mentioned argumen ts. It consists
+mainly of three ingredients: (1) a space-time a priori estimate of th e Morawetz type,
+(2) a weighted Sobolev-type inequality adapted to the Morawetz-t ype estimate, and
+(3) an energy equipartition estimate adjusted to the weight of the Morawetz-type
+estimate.
+The Morawetz space-time integral estimate was originally derived in [2 1] to have
+some weak decay of local energy for the undamped nonlinear Klein-G ordon equation
+¨u−∆u+u+f′(u) = 0, (1.15)
+inR3, and then later applied to the nonlinear scattering theory of the un damped
+equation by Strauss-Morawetz [35]. In this paper we use a modified v ersion of the
+estimate derived by Nakanishi [23] for the undamped equation, in th e form
+/integraldisplay/integraldisplay
+|x|<t/2|x˙u+t∇u|2
+|(t,x)|3+g(u)
+|t|dxdt≤CE(u;0), x∈Ω∈RN, N≥1,(1.16)
+which is available in all spatial dimensions. We emphasize that it is essent ial to use
+the quadratic term (via the weighted Sobolev inequality), instead of the nonlinear
+termg(u), in order to get decay estimates independent both of the nonlinea rity and
+of the energy size. This idea was first used by Nakanishi in [23] for th e nonlinear
+scattering theory.
+The energy equipartition property was also used in the preceding wo rks for the
+stabilization, but the mass energy term |u|2has prevented them to use it in a
+completely nonlinear way. Introduction of the Morawetz-type weig ht as well as the
+weighted Sobolev inequality enables us to control that problematic t erm directly.EXPONENTIAL ENERGY DECAY FOR DAMPED NLKG 5
+However, our reliance on the Morawetz-type estimate brings cert ain limitations of
+our approach. Since it represents the global dispersive property of the solutions, it
+seems difficult to use in bounded domains, or more precisely in presenc e of trapped
+rays. This is a big disadvantage compared with the preceding works, which equally
+apply to bounded domains (provided that one has enough linear estim ates).
+Another, more technical restriction in our theorem is the condition (1.10) posed
+ong. The preceding works require similar conditions to get the global sta bilization
+(namely a decay rate independent of the energy size), but they ca n also work with
+weaker conditions such as uf′(u)≥0 for local stabilization (decay rate dependent
+of the energy size).
+An example of fwhich satisfies (1.12) but not (1.10) is given by
+f(u) =|u|2log(1+|u|2), (1.17)
+i.e. a logarithmic perturbation of a linear term, whereas similar pertur bations of
+superlinear powers satisfy (1.10), for example
+f′(u) =u5log(1+|u|2), (1.18)
+which was considered by Tao in [37] as an H1supercritical case in R3, getting global
+solutions for smooth radial initial data (cf. [30] for the non radial c ase).
+1.2.Focusing case. In Theorem 1.1 for the defocusing case, our assumption (1.10)
+onfis used only to discard nonlinear terms in the Morawetz-type estimat es. That
+term can be controlled, if we have the nonlinear scattering for the u ndamped NLKG
+(1.26), and a global perturbation argument for the Strichartz no rms. The first
+argument provides us with global bounds on the Strichartz space- time norms of the
+undamped equation, while the second one allows us to transfer the g lobal Strichartz
+bounds to the damped equation. The two arguments are indeed sat isfied, even for a
+focusing nonlinearity, if we restrict to the solutions with energy less than the ground
+state (a solution of the static equation with possibly a modified mass) and inside
+“the potential well”. We should point out, however, that the previo us works in
+the defocusing case [41, 5, 7, 6] could also extend to the same sett ing, as far as
+the nonlinearity satisfies their growth conditions. See [8] for a res ult for the wave
+equation on a bounded domain.
+For simplicity of presentation, we consider only the whole space as th e domain,
+and typical nonlinearities of one of the following examples, though ou r argument
+applies to more general nonlinearities.
+(1)N≥1 andfisH1subcritical: We assume
+f(u) =λ1|u|p1+···+λk|u|pk, (1.19)
+for some λ1,...,λ k>0 and 2 ⋆< p1<···< pk<2⋆, where 2 ⋆= 2+4/Nis
+theL2critical power and 2⋆= 2N/(N−2) is the H1critical one.
+(2)H1critical case. We assume N≥3 and for some λ,
+f(u) =|u|2⋆. (1.20)
+(3) 2Dexponential case. We assume that N= 2 and,
+f(u) =e4π|u|2−1−4π|u|2−(4π|u|2)2/2. (1.21)6 L. ALOUI, S. IBRAHIM, AND K. NAKANISHI
+Note that the scattering for the lower critical power 2 ⋆is still open for large data
+even in the defocusing case, and the subtracted |u|4term in (1.21) corresponds to
+that power.
+Global solutions to (1.26) do not always exist in these cases. On the o ne hand,
+when the data is small in the energy space, it leads to a global solution which
+scatters. On the other hand, when the data is sufficiently large, th en the solution
+can blow up in finite time, for example if the energy is negative. The thr eshold
+between these two dynamics can be constructed as follows.
+LetJbe the static energy functional given by
+J(ϕ) :=/integraldisplay
+Rd[|∇ϕ|2+|ϕ|2]dx−2F(ϕ), F(ϕ) :=/integraldisplay
+Rdf(ϕ)dx, (1.22)
+and denote its derivative with respect to amplification by
+K(u) := 2/integraldisplay
+RN|∇u|2+|u|2−uf′(u)dx. (1.23)
+Consider the constrained minimization problem
+m= inf{J(ϕ)|0/ne}ationslash=ϕ∈H1(Rd), K(ϕ) = 0}, (1.24)
+and define
+K+={(u0,u1)∈H1(RN)×L2(RN)|E(u0,u1)< m, K(u0)≥0},
+K−={(u0,u1)∈H1(RN)×L2(RN)|E(u0,u1)< m, K(u0)<0}.(1.25)
+Then for the undamped NLKG
+utt−∆u+u+f′(u) = 0, (1.26)
+the solutions with energy below mare divided as follows:
+•If (u(0),ut(0))∈ K+, then the solution uof (1.26) is global.
+•If (u(0),ut(0))∈ K−, then the solution uof (1.26) blows up in finite time.
+In the subcritical case, this follows essentially from Payne-Satting er [28] and the
+local wellposedness in the energy space.
+Recently, Ibrahim-Masmoudi-Nakanishi [16] extended it to the crit ical and 2D
+exponential cases. Moreover, they proved that in all the cases ( 1)-(3), the solutions
+of (1.26) in K+scatter as t→ ±∞with global Strichartz bounds, implementing
+Kenig-Merle’s argument [17, 18] to the equation without scaling invar iance. In
+addition, they gave the following characterization of m(see [16, Proposition 1.2]),
+which is again well known in the subcritical case: There exist Q(x)andc∈[0,1]
+such that
+m=Jc(Q) =/integraldisplay
+|∇Q|2+c|Q|2dx−2F(Q), (1.27)
+−∆Q+cQ=f′(Q),/ba∇dbl∇Q/ba∇dbl2
+L2+c/ba∇dblQ/ba∇dbl2
+L2<∞, (1.28)
+andJc(Q)is the minimum among the solutions of (1.28), i.e.Qis the ground state
+for(1.28).c= 1in the subcritical case (1.19)and the exponential case (1.21), and
+c= 0in the critical case (1.20).
+Hence transferring the global Strichartz bounds to the damped N LKG, we obtain
+the following stabilization in the set K+.EXPONENTIAL ENERGY DECAY FOR DAMPED NLKG 7
+Theorem 1.2. LetN≥1,Ω =RNandf(u)given by either (1), (2) or (3). Assume
+thata:RN→Rsatisfies (1.9)with some M,R,a 0>0. Let0< E0< m. Then
+there exists γ >0determined by E0,f,M,R,a0andNsuch that for any initial
+data satisfying E(u;0)≤E0andK(u(0))≥0, we have a unique global solution u
+of(1.1)with exponential decay
+2
+N+2EF(u;t)≤E(u;t)≤2e−γtE(u;0), (1.29)
+for allt >0. Recall that EFis the free energy given in (1.5).
+In the above Theorem, the dependence of γuponE0is inevitable, since Qis a
+solution of the damped NLKG without any decay.
+In the sequel, A/lessorsimilarBorB/greaterorsimilarAmeansA≤CBfor some positive constant C.
+2.Proof in the defocusing case: Theorem 1.1
+In this section, we prove the uniform exponential decay in the defo cusing case,
+i.e. Theorem 1.1. The proof relies solely on the multiplier argument, with out any
+estimate for the linearized equation. However in order to justify so me energy-type
+computations, we first approximate the nonlinearity so that we hav e the global well-
+posedness, derive the uniform decay for the solutions of the appr oximate equations
+with the same initial data, and then take the weak limit to the original e quation.
+Note that it is in general hard to get any information for a given weak solution in
+the absence of the wellposedness.
+2.1.Truncation of nonlinearity. First we reduce to the H1subcritical case by
+approximation of the nonlinearity. Note that this procedure is not e ntirely trivial.
+Ontheonehand, we should notsimply cutoff thenonlinearity, since gvanishes when
+f′is linear. In other words, approximate nonlinearity must be at least s uperlinear.
+On the other hand, we need global wellposedness for the approxima te equations to
+justify our energy-type computations. The global wellposedness is available in the
+energy subcritical and critical cases, for which growth condition s hould be imposed
+onf′′, rather than on forf′. Consequently, contrary to the classical methods as
+in [33, 19, 34], we need two step approximations of the nonlinearity, o ne from below
+and another from above.
+LetV(u) =f(u)/|u|2be the nonlinear potential energy. Then we have f(u) =
+V(u)|u|2andg(u) =V′(u)|u|2u, so our conditions on fare rewritten as
+V(0) = 0, uV′(u)≥0, V(u)≤C0(1+uV′(u)), (2.1)
+and the weaker condition (1.12) is given by
+V(u)≤C0(1+uV′(u)+|u|q−2). (2.2)
+We construct a sequence of approximate nonlinear potentials Vkas follows. First
+we fixθ∈(0,1), depending only on the dimension N, such that 2+ θ <2⋆. If there
+exists some k∈Nsuch that for all |z| ≥k
+zV′(z)≤kV′(k)|z/k|θ, (2.3)8 L. ALOUI, S. IBRAHIM, AND K. NAKANISHI
+thenfis subcritical, i.e. |f′(z)|/lessorsimilar|zV′(z)z|+|V(z)z|/lessorsimilar|z|1+θ. In this case we
+putfk=f. Otherwise, there exists a sequence of k∈Nalong which V′(k)k1−θis
+increasing. For those kwe define the first approximation Vk:R→Rby
+Vk(z) =V(z) (|z| ≤k), zV′
+k(z) = min(zV′(z),kV′(k)|z/k|θ) (|z| ≥k),(2.4)
+andfk:R→Rby
+fk(z) =Vk(z)|z|2. (2.5)
+Then clearly we have Vk≥0,V′
+k≥0, andf′
+kis locally Lipschitz. Our choice of
+kimplies that zV′
+k(z),Vk(z) andfk(z) are all increasing in kat eachz. TheV′
+dominance is inherited by Vkas follows, where the constants may depend also on θ:
+forz > kwe have
+Vk(z)≤V(k)+min(/integraldisplayz
+kV′(y)dy,/integraldisplayz
+kV′(k)|y/k|θ−1dy)
+/lessorsimilarmin(V(z),V(k)+V′(k)|z|θk1−θ)/lessorsimilar1+zV′
+k(z),(2.6)
+and similarly for z <−k, while it is trivial for |z| ≤k. (1.12) is inherited in the
+same way.
+Althoughthegrowthfor fkandf′
+kisH1subcritical, f′′
+kmightnotbe H1subcritical
+whenf′′isoscillatingforlarge u. Hencetohaveglobalwellposednessforapproximate
+equations, we need further approximation Vkl:R→Rdefined for l > kby
+Vkl(z) =Vk(z) (|z| ≤l)zV′
+kl(z) =lV′
+k(l)|z/l|θ(|z| ≥l), (2.7)
+andfkl:R→Rbyfkl(z) =Vkl(z)|z|2. Thenf′
+klis locally Lipschitz and
+|f′
+kl(z1)−f′
+kl(z2)|/lessorsimilar(|z1|+|z2|)θ|z1−z2|. (2.8)
+Now that the approximate nonlinearity is H1subcritical (in fact we can make it as
+close to be linear as we want), we have the global wellposedness for
+∂2
+tu+a(x)∂tu−∆u+u+f′
+kl(u) = 0. (2.9)
+Now letuklbe the global solution for each k < lwith the same initial data, and
+assume that we have
+Ekl(ukl;t)≤2e−γtEkl(ukl;0), (2.10)
+for some γ >0, allt >0, and all k < l, whereEkldenotes the energy for the
+approximate equation
+Ekl(v;t) =/integraldisplay
+Ω|˙v|2+|∇v|2+|v|2+fkl(v)dx. (2.11)
+Thenuklis uniformly bounded in H1×L2, and so the classical weak compactness ar-
+gument (cf. [34, 36]) gives us a subsequence, which we still denote byukl, converging
+asl→ ∞to some function uk(t,x) in the sense
+ukl(t)→uk(t) inC([0,∞);w-H1
+0(Ω)),
+˙ukl→˙ukin w-L2
+loc((0,∞);L2(Ω)),(2.12)
+where w- Xdenotes the space endowed with the weak topology, and also ukl(t,x)→
+uk(t,x) pointwise for almost every x∈Ω at allt >0. Note that in this limit l→ ∞,EXPONENTIAL ENERGY DECAY FOR DAMPED NLKG 9
+the nonlinear energy fkland its limit fkareH1subcritical, and so are dominated by
+theH1norm through the Sobolev embedding. Hence the limit ukis a global weak
+solution to
+∂2
+tu+a(x)∂tu−∆u+u+f′
+k(u) = 0, (2.13)
+and moreover, by Fatou’s lemma, the uniform decay for ukl, and the Lebesgue
+dominated convergence theorem respectively, we have
+Ek(uk;t)≤liminf
+l→∞Ekl(ukl;t)≤liminf
+l→∞2e−γtEkl(u;0) = 2e−γtEk(uk;0).(2.14)
+Next for a subsequence of k→ ∞we have a similar convergence uk→uas above,
+and by Fatou, the uniform decay of ukand the monotone convergence theorem
+respectively we get
+E(u;t)≤liminf
+k→∞Ek(uk;t)≤liminf
+k→∞2e−γtEk(u;0) = 2e−γtE(u;0),(2.15)
+anduis a global weak solution of the original damped NLKG. Note that we ca nnot
+use the dominated convergence theorem in this step, since the non linear energy in
+the limit is no longer controlled by the Sobolev embedding.
+It remains to prove weak convergence of the nonlinear term in the e quation.
+Multiplying the equation of uklwithukland integrating it in space-time, we get
+[/an}b∇acketle{t˙ukl|ukl/an}b∇acket∇i}htL2x]T
+0+/integraldisplayT
+0/ba∇dbl∇ukl/ba∇dbl2
+L2x−/ba∇dbl˙ukl/ba∇dbl2
+L2xdt+/integraldisplayT
+0/an}b∇acketle{tfkl(ukl)|ukl/an}b∇acket∇i}htL2xdt= 0,(2.16)
+which implies through the energy identity that
+/integraldisplayT
+0/an}b∇acketle{tfkl(ukl)|ukl/an}b∇acket∇i}htL2xdt/lessorsimilarE(u;0). (2.17)
+Then by the dominated convergence theorem for the limit l→ ∞, and by the
+monotone convergence theorem for the limit k→ ∞, we get
+/integraldisplayT
+0/an}b∇acketle{tfk(uk)|uk/an}b∇acket∇i}htL2xdt+/integraldisplayT
+0/an}b∇acketle{tf(u)|u/an}b∇acket∇i}htL2xdt/lessorsimilarE(u;0). (2.18)
+For anyL≫1, letfL(u) = min( f(u),f(L)) andfL
+k(u) = min( fk(u),fk(L)). Then
+for any bounded domain K⊂Ω we have
+/ba∇dblfk(uk)−f(u)/ba∇dblL1([0,T]×K)≤ /ba∇dblfL
+k(uk)−fL(u)/ba∇dblL1
+t,x([0<t<T]×K)
++/ba∇dblukfk(uk)/L/ba∇dblL1
+t,x(0<t<T,|uk|>L)
++/ba∇dbluf(u)/L/ba∇dblL1
+t,x(0<t<T,|u|>L).(2.19)
+On the right, the first term tends to 0 as k→ ∞by the dominated convergence
+theorem, while the other two terms are bounded by E(u;0)/Lthanks to (2.18).
+Hencefk(uk)→f(u) inL1
+loc((0,∞)×Ω), and so uis a global weak solution of the
+original equation satisfying the uniform exponential decay.10 L. ALOUI, S. IBRAHIM, AND K. NAKANISHI
+2.2.Uniform decay for subcritical nonlinearity. Now it suffices to prove the
+uniform exponential decay in the H1subcritical (and defocusing) case, with the
+decay rate independent of the nonlinear term and the initial data. S ince we have
+global wellposedness in this case, energy-type computations used below are easily
+justified by standard approximation argument. In particular we ha ve the energy
+identity (1.6) for any finite energy solution u. Denote the energy decrement on the
+interval [0 ,T] by
+A(u;T) :=/integraldisplayT
+0/integraldisplay
+Ωa(x)|˙u|2dxdt. (2.20)
+We are going to show that for some quantitative T >0 andδ >0, and for all
+solutions uwith finite energy we have
+A(u;T)≥δE(u;T). (2.21)
+Then the energy identity implies that
+E(u;t+T)≤E(u;t)−δE(u;t+T), (2.22)
+for anyt >0, and hence by repeated use of it we get the exponential decay (1 .11)
+with the rate
+γ=T−1log(1+δ). (2.23)
+Hence we are mostly concerned with the case where A(u;T) is small compared
+withE(u;0). The main ingredients of our argument are the energy equipart ition
+property, and the following Morawetz-type estimate, which is a triv ial modification
+of that [23, 24] for the undamped equation. Let
+λ=|(t,x)|=/radicalbig
+t2+|x|2, q=(N−1)
+2λ+t2−|x|2
+λ3,
+m(u) =−tut+x·∇u
+λ+uq.(2.24)
+Multiplying equation (1.1) with m(u) and integrating it in the light cone, we get
+the Morawetz-type estimate with a space-time weight:
+/integraldisplayT
+S/integraldisplay
+x∈Ω
+|x|<t|x˙u+t∇u|2
+λ3+g(u)q+a˙um(u)dxdt/lessorsimilarE(u;0) (2.25)
+for any 1 < S < T andany solution u. See [23, 24] for the details of the computation
+in the whole space RN. Without loss of generality, we may now choose
+1≤R < S= 3R < T, (2.26)
+whereRis the radius for effective damping given in (1.9).
+The damping term in (2.25) is dominated by the energy decrement
+/integraldisplayT
+S/integraldisplay
+Ω|a˙um(u)|dxdt≤ /ba∇dbla/ba∇dbl1/2
+L∞x/radicalbig
+A(u;T)/ba∇dblm(u)/ba∇dblL2
+t,x(S<t<T)
+/lessorsimilarM1/2/radicalbig
+A(u;T)T1/2/radicalbig
+E(u;0)
+/lessorsimilarE(u;0)+MTA(u;T).(2.27)EXPONENTIAL ENERGY DECAY FOR DAMPED NLKG 11
+The kinetic part in (2.25) is discarded by
+/integraldisplayT
+S/integraldisplay
+Ω|x˙u|2
+λ3dxdt≤/integraldisplayT
+S/integraldisplay
+x∈Ω
+|x|<RR2
+t3|˙u|2dxdt+/integraldisplayT
+S/integraldisplay
+x∈Ω
+|x|>Ra(x)|˙u|2
+a0(R+t)dxdt
+/lessorsimilarE(u;0)+A(u;T)
+a0R.(2.28)
+Using that g≥0 as well, we thus get
+/integraldisplayT
+S/integraldisplay
+x∈Ω
+|x|<t|∇u|2
+t+(t2−|x|2)g(u)
+t3dxdt/lessorsimilarE(u;0)[1+µ(u;T)]. (2.29)
+where we put
+µ(u;T) := [MT+(a0R)−1]A(u;T)/E(u;0). (2.30)
+Next, we fix a smooth cut-off function χ(x)∈C∞
+0(RN) such that χ(x) = 1 for
+|x|< Randχ(x) = 0 for |x|>2R, and multiply (1.1) with χun. Integrating it in
+Ω, we get
+∂t/an}b∇acketle{t˙u|χu/an}b∇acket∇i}htL2x=/integraldisplay
+Ωχ/bracketleftbig
+|˙u|2−|∇u|2−|u|2−uf′(u)/bracketrightbig
+dx
+−/integraldisplay
+Ω[a˙uχu+u∇u·∇χ]dx.(2.31)
+Then by partial integration in t, and using the support property of χ,
+/integraldisplayT
+S/integraldisplay
+Ωχ|˙u|2
+tdxdt
+≤/integraldisplayT
+S∂t/an}b∇acketle{t˙u|χu/an}b∇acket∇i}htL2x
+tdt+/integraldisplayT
+S/integraldisplay
+x∈Ω
+|x|<2R|∇u|2+|u|2+|uf′(u)|+|a˙u|2
+tdxdt
+/lessorsimilarE(u;0)+MA(u;T)+/integraldisplayT
+S/integraldisplay
+x∈Ω
+|x|<2R|∇u|2+|u|2+|uf′(u)|
+tdxdt.(2.32)
+For the last term we use the assumption on fto replace it by
+|uf′(u)| ≤g(u)+2f(u)≤(1+C0)g(u)+C0|u|2. (2.33)
+Then the gterm as well as the gradient term is treated by the Morawetz-type
+estimate. For the remaining |u|2part, we apply a weighted Sobolev inequality to
+the quadratic part of the Morawetz-type estimate, which implies
+/integraldisplayT
+S/integraldisplay
+x∈Ω
+|x|<t|u|p
+tdxdt/lessorsimilarE(u;0)p/2−1/bracketleftBigg
+E(u;0)+/integraldisplayT
+S/integraldisplay
+x∈Ω
+|x|<t|x˙u+t∇u|2
+λ3dxdt/bracketrightBigg
+,(2.34)
+for all 2+4 /N≤p≤2N/(N−2). The proof was given in [26] for the undamped
+NLKG, which works verbatim for the damped NLKG. Then applying H¨ o lder and12 L. ALOUI, S. IBRAHIM, AND K. NAKANISHI
+the support property of χ, we get
+/integraldisplayT
+S/integraldisplay
+Ωχ|u|2
+tdxdt≤/bracketleftBigg/integraldisplayT
+S/integraldisplay
+x∈Ω
+|x|<t|u|p
+tdxdt/bracketrightBigg2/p/bracketleftBigg/integraldisplay
+x∈Ω
+|x|<2R/integraldisplayT
+Sdt
+tdx/bracketrightBigg1−2/p
+/lessorsimilarE(u;0)[1+µ(u;T)]2/p(RNlogT)1−2/p.(2.35)
+On the other hand for |x|> R, ˙uis estimated by
+/integraldisplayT
+S/integraldisplay
+x∈Ω
+|x|>R|˙u|2
+tdxdt≤/integraldisplayT
+S/integraldisplay
+Ωa(x)|˙u|2
+a0tdxdt≤A(u;T)
+a0S. (2.36)
+Putting together (2.32), (2.29), (2.35) and (2.36), we obtain
+/integraldisplayT
+S/ba∇dbl˙u/ba∇dbl2
+L2x
+tdt/lessorsimilar(1+C0)E(u;0)[1+µ(u;T)]
++E(u;0)[1+µ(u;T)]2/p(RNlogT)1−2/p.(2.37)
+This is an upper bound. To get a lower bound, we go back to (2.31) and replaceχ
+by 1. Then we obtain
+∂t[/an}b∇acketle{t˙u|u/an}b∇acket∇i}htL2x+/an}b∇acketle{tau|u/an}b∇acket∇i}htL2x/2] =/ba∇dbl˙u/ba∇dbl2
+L2x−K(u), (2.38)
+where as before we denote
+K(u) =/integraldisplay
+Ω|∇u|2+|u|2+uf′(u)dx=E(u;t)−/ba∇dbl˙u(t)/ba∇dbl2
+L2x+/integraldisplay
+Ωg(u)dx
+≥E(u;t)−/ba∇dbl˙u(t)/ba∇dbl2
+L2x.(2.39)
+Hence we get
+2/ba∇dbl˙u(t)/ba∇dbl2
+L2x≥E(u;t)+∂t/an}b∇acketle{t˙u|u/an}b∇acket∇i}htL2x+/an}b∇acketle{ta˙u|u/an}b∇acket∇i}htL2x, (2.40)
+and integrating it by dt/t,
+/integraldisplayT
+Sdt
+t2/ba∇dbl˙u(t)/ba∇dbl2
+L2x≥/integraldisplayT
+Sdt
+t/braceleftBig
+E(u;t)+∂t/an}b∇acketle{t˙u|u/an}b∇acket∇i}htL2x+/an}b∇acketle{ta˙u|u/an}b∇acket∇i}htL2x/bracerightBig
+≥E(u;T)logT
+S+[/an}b∇acketle{t˙u|u/t/an}b∇acket∇i}htL2x]T
+S+/integraldisplayT
+S/an}b∇acketle{t˙u|u/an}b∇acket∇i}htL2xdt
+t2
+−MA(u;T)−/integraldisplayT
+S/ba∇dblu/ba∇dbl2
+L2xdt
+t2
+≥E(u;T)logT
+3R−E(u;0)/R−MA(u;T).(2.41)
+Combining it with (2.37), we obtain
+E(u;T)logT
+3R/lessorsimilar(1+C0)E(u;0)[1+µ(u;T)]
++E(u;0)[1+µ(u;T)]2/p(RNlogT)1−2/p.(2.42)
+Hence if
+[1+MT+(a0R)−1]A(u;T)≤E(u;0)/2, (2.43)EXPONENTIAL ENERGY DECAY FOR DAMPED NLKG 13
+then we have µ(u;T)≤1/2,E(u;T)≥E(u;0)/2 and
+logT/lessorsimilar1+C0+logR+(RNlogT)1−2/p, (2.44)
+which implies
+logT/lessorsimilar1+C0+logR+RN(p/2−1)/lessorsimilar1+C0+RN(p/2−1).(2.45)
+In other words, there exists an absolute constant C∗>0 such that we have (2.21)
+and hence (1.11) with
+logT=C∗(1+C0+R2), δ= [1+MT+(a0R)−1]−1/2,(2.46)
+where we chose p= 2+4/N.
+Finally we consider the weaker assumption (1.12). Then (2.33) is chan ged to
+|uf′(u)| ≤(1+C0)g(u)+C0(|u|2+|u|q). (2.47)
+where without loss of generality, we may assume that q≥2+4/N. Then to bound
+(2.32) we need also (2.34) with p=q, and consequently (2.37) is modified to
+/integraldisplayT
+S/ba∇dbl˙u/ba∇dbl2
+L2x
+tdt/lessorsimilar(1+C0)E(u;0)[1+µ(u;T)]+C0E(u;0)q/2[1+µ(u;T)]
++E(u;0)[1+µ(u;T)]2/p(RNlogT)1−2/p,(2.48)
+and so the decay rate in this case is given by
+logT=C∗(1+C0+R2+C0E(u;0)q/2−1), δ= [1+MT+(a0R)−1]−1/2.(2.49)
+/square
+3.Proof in the focusing case: Theorem 1.2
+In this section, we prove the exponential decay in the focusing cas e, namely Theo-
+rem 1.2. In order to proceed in the same way as in Section 2.2, we need the following
+three ingredients:
+(1) Upper bound on EF(u;t) in terms of E(u;t).
+(2) Lower bound on K(u(t)) in terms of E(u;t).
+(3) Upper bound on the space-time integral of |g(u)|/λin terms of the energy
+and Strichartz norms of the undamped equation.
+3.1.Variational properties. The first two are well-known and coming from the
+variational characterization of Q. In fact, since |uf′(u)| ≥(2+4/N)|f(u)|, we have
+(N+2)E(u;t)≥NK(u(t))+2EF(u;t)+/integraldisplay
+N|˙u|2dx >2EF(u;t),(3.1)
+and for any E0∈(0,J(Q)) there exists ν >0 determined by N,fandE0such that
+for anyϕ∈H1(RN) satisfying J(ϕ)≤E0andK(ϕ)>0 we have
+K(ϕ)≥ν/ba∇dblϕ/ba∇dbl2
+H1. (3.2)
+For a proof of this lower bound, see [16, Lemma 2.12]. In the exponen tial case
+f=fexp(N= 2), we have another important property [16, Lemma 2.11]
+/ba∇dbl∇u(t)/ba∇dbl2
+L2x+/ba∇dbl˙u(t)/ba∇dbl2
+L2x≤E(u;t)≤E0< m≤1, (3.3)14 L. ALOUI, S. IBRAHIM, AND K. NAKANISHI
+which ensures that the solution stays in the subcritical regime for t he Trudinger-
+Moser inequality
+/ba∇dbl∇ϕ/ba∇dblL2(R2)<1 =⇒/integraldisplay
+R2(e4π|ϕ(x)|2−1)dx/lessorsimilar/ba∇dblϕ/ba∇dbl2
+L2(R2)
+1−/ba∇dbl∇ϕ/ba∇dbl2
+L2(R2).(3.4)
+3.2.Global Strichartz bound from the scattering. The integral bound on
+|g(u)|/λrequires transfer of global Strichartz norms from the undamped equation.
+Specifically, we use the following Strichartz norms:
+/ba∇dblu/ba∇dblStT:=/ba∇dblu/ba∇dblL2⋆
+t(0,T;B1/2
+2⋆,2(RN))+/ba∇dblu/ba∇dblL2+4/(N−1)
+t(0,T;B1/2
+2+4/(N−1),2(RN)),(3.5)
+ifN≥3,
+/ba∇dblu/ba∇dblStT:=/ba∇dblu/ba∇dblL2⋆
+t(0,T;B1/2
+2⋆,2(R2))+/ba∇dblu/ba∇dblL4
+t(0,T;B1/4
+∞,2(R2)), (3.6)
+ifN= 2, and
+/ba∇dblu/ba∇dblStT:=/ba∇dblu/ba∇dblL6
+t(0,T;B1/2
+6,2(R)), (3.7)
+ifN= 1, where Bs
+p,qdenotes the inhomogeneous Besov space.
+The main result of [16] asserts that under the assumptions of Theo rem 1.2, there
+exists a unique global solution v(t,x) :R1+N→Rfor the undamped NLKG (1.26),
+which satisfies
+sup
+t>0EF(v;t)≤A <∞,/ba∇dblv/ba∇dblSt∞≤B <∞,(3.8)
+for some AandBdetermined by N,fandE0. Moreover, one has the following
+global perturbation result [16, Lemma 4.5]: There exists ε >0, determined by N,f
+andE0, such that for any T >1and any u(t,x) : [0,T]×RN→Rsatisfying
+EF(u−v;0)1/2+/ba∇dbl¨u−∆u+u+f′(u)/ba∇dblL1
+t(0,T;L2x)≤ε, (3.9)
+we have
+/ba∇dblu−v/ba∇dblStT≤C <∞, (3.10)
+whereCis also determined by N,fandE0.
+We apply the above perturbation to the solution uof the damped NLKG (1.1)
+with the same initial data EF(u−v;0) = 0. Since
+/ba∇dbla˙u/ba∇dblL1
+t(0,T;L2x)≤M1/2T1/2A(u;T)1/2, (3.11)
+we get
+/ba∇dblu/ba∇dblStT≤C+B <∞, (3.12)
+provided that
+A(u;T)≤(MT)−1ε. (3.13)
+Otherwise we get the exponential decay (2.23) with δ= (MT)−1ε/E0.EXPONENTIAL ENERGY DECAY FOR DAMPED NLKG 15
+3.3.Morawetz estimate from the Strichartz bound. By using the Strichartz
+bound (3.12), we can bound the nonlinear part in the Morawetz estim ate
+/integraldisplayT
+1/integraldisplay
+RN|g(u)|
+λdxdt, (3.14)
+as follows. Denote N:= [(N+1)E0]1/2+C+Bso that we have
+EF(u;t)1/2+/ba∇dblu/ba∇dblStT≤ N. (3.15)
+First we consider the case N≥3. For the H1critical power, we get by the H¨ older,
+Sobolev and interpolation inequalities
+/integraldisplayT
+0/integraldisplay
+RN|u|2⋆
+|x|dxdt/lessorsimilar/ba∇dbl|x|−1/ba∇dblLN,∞
+x/ba∇dblu/ba∇dbl2⋆
+L2⋆
+t(0,T;L2⋆N/(N−1),2
+x )
+/lessorsimilar/ba∇dblu/ba∇dbl2⋆
+L2⋆
+t(0,T;B1/2⋆
+2⋆,2)/lessorsimilar/ba∇dblu/ba∇dbl4
+(N−1)(N−2)
+L∞
+t(0,T;H1x)/ba∇dblu/ba∇dbl2(N+1)
+N−1
+StT/lessorsimilarN2⋆,(3.16)
+whereLp,q
+xdenotes the Lorentz space on RN. For the L2critical power, we have
+/integraldisplayT
+0/integraldisplay
+RN|u|2⋆
+|x|dxdt/lessorsimilar/ba∇dbl|x|−1/ba∇dblLN,∞
+x/ba∇dblu/ba∇dbl2⋆
+L2⋆
+t(0,T;L(2N+4)/(N−1),2
+x )
+/lessorsimilar/ba∇dblu/ba∇dbl2⋆
+L2⋆
+t(0,T;BN/(2N+4)
+2⋆,2)/lessorsimilar/ba∇dblu/ba∇dbl2⋆
+StT/lessorsimilarN2⋆.(3.17)
+Next we consider the case N= 2 with fp,p≥4 = 2⋆. Then we have
+/integraldisplayT
+0/integraldisplay
+R2|u|p
+|x|dxdt/lessorsimilar/ba∇dbl|x|−1/ba∇dblL2,∞
+x/ba∇dblu/ba∇dblp
+Lp
+t(0,T;L2p,2
+x)
+/lessorsimilar/ba∇dblu/ba∇dblp
+Lp
+t(0,T;B1/p
+p,2)/lessorsimilar/ba∇dblu/ba∇dblp−4
+L∞
+t(0,T;B0
+∞,2)/ba∇dblu/ba∇dbl4
+L4
+t(0,T;B1/4
+4,2)
+/lessorsimilar/ba∇dblu/ba∇dblp−4
+L∞
+t(0,T;H1)/ba∇dblu/ba∇dbl4
+StT/lessorsimilarNp.(3.18)
+For the case N= 1 with fpandp≥6 = 2⋆, we have
+/integraldisplayT
+1/integraldisplay
+R|u|p
+λdxdt≤ /ba∇dblu/ba∇dblp−6
+L∞(0,T;L∞x)/ba∇dblu/ba∇dbl6
+L6
+t(0,T;L6x)/lessorsimilarN6. (3.19)
+Finally we consider the exponential case f=fexp(N= 2), which requires a bit
+more work. We start with H¨ older as in the previous cases
+/integraldisplayT
+0/integraldisplay
+R2|g(u)|
+|x|dxdt/lessorsimilar/ba∇dbl|x|−1/ba∇dblL2,∞
+x/ba∇dblg(u)/ba∇dblL1
+t(0,T;L2,1
+x), (3.20)
+and to the L2,1
+xnorm we apply the following exponential estimate. Its L2
+xversion
+was proved in [15].
+Lemma 3.1. Letg=uf′(u)−2f(u)andf(u) =fexp(u)as given in (1.21). Then
+for anyA∈(0,1), there exist κ∈(0,1/8), and a continuous increasing function
+CA: [0,∞)→[0,∞), such that for any ϕ∈H1(R2)satisfying
+/ba∇dbl∇ϕ/ba∇dblL2≤A, (3.21)
+we have,
+/ba∇dblg(ϕ)/ba∇dblL2,1≤CA(/ba∇dblϕ/ba∇dblH1)/ba∇dblϕ/ba∇dbl4
+C1/4−κ/ba∇dblϕ/ba∇dbl4
+Bκ/2
+2/κ,2+/ba∇dblϕ/ba∇dbl6
+L6,2. (3.22)16 L. ALOUI, S. IBRAHIM, AND K. NAKANISHI
+Note that (3.21) is satisfied by u(t) thanks to the variational property (3.3),
+and the coefficient CA(/ba∇dblu(t)/ba∇dblH1) is uniformly bounded. Thus we can continue the
+estimate (3.20) using the above lemma,
+/integraldisplayT
+0/integraldisplay|g(u)|
+|x|dxdt/lessorsimilar/ba∇dblu/ba∇dbl4
+L2/κ
+t(0,T;Bκ/2
+2/κ,2)/ba∇dblu/ba∇dbl4
+L4/(1−2κ)
+t(0,T;C1/4−κ)+/ba∇dblu/ba∇dbl6
+L6
+t(0,T;B1/6
+6,2)
+/lessorsimilar/ba∇dblu/ba∇dbl4
+L∞
+t(0,T;H1)/ba∇dblu/ba∇dbl8κ
+L4
+t(0,T;B1/4
+4,2)/ba∇dblu/ba∇dbl4−8κ
+L4
+t(0,T;B1/4
+∞,2)
++/ba∇dblu/ba∇dbl2
+L∞
+t(0,T;H1)/ba∇dblu/ba∇dbl4
+L4
+t(0,T;B1/4
+4,2)
+/lessorsimilarN8+N6.(3.23)
+It remains to prove the lemma. It is based on a sharp logarithmic inequ ality
+proved in [13, Theorem 1.3].
+Lemma 3.2. Let0< α <1. For any λ >1
+2παand any 0< µ≤1, there exists a
+constant Cλ>0such that, for any function u∈H1(R2)∩Cα(R2)
+/ba∇dblu/ba∇dbl2
+L∞≤λ/ba∇dblu/ba∇dbl2
+Hµlog/parenleftbigg
+Cλ+8αµ−α/ba∇dblu/ba∇dblCα
+/ba∇dblu/ba∇dblHµ/parenrightbigg
+, (3.24)
+where/ba∇dblu/ba∇dbl2
+Hµ:=/ba∇dbl∇u/ba∇dbl2
+L2+µ2/ba∇dblu/ba∇dbl2
+L2.
+The above logarithmic inequalities constitute a refinement (with resp ect to the
+best constant) of that appeared in Br´ ezis-Gallouet [4].
+Proof of Lemma 3.1. Without loss of generality, we may assume that A∈(1/2,1).
+Letϕ∈H1satisfying (3.21). We define µ >0 by
+2µ2(1+/ba∇dblu/ba∇dbl2
+H1x) = 1−A2, (3.25)
+so that we have
+/ba∇dblϕ/ba∇dbl2
+Hµ≤A1:= (1+A2)/2<1. (3.26)
+Now we can choose κ∈(0,1/8) andλ >2/π, depending only on A1, such that
+2πλ(1/4−κ)>1 = 2πλ(1/4+κ/2)A2
+1. (3.27)
+For instance, we can choose κ= (1−A2
+1)/8. Note that
+|g(u)|/lessorsimilar|u|4(e4π|u|2−1). (3.28)
+In particular, if /ba∇dblϕ/ba∇dblL∞≤A1, then|g(ϕ)|/lessorsimilar|ϕ|6and thanks to Sobolev embedding
+/ba∇dblg(ϕ)/ba∇dblL2,1/lessorsimilar/ba∇dblϕ/ba∇dbl6
+L12,6/lessorsimilar/ba∇dblϕ/ba∇dbl6
+L12,2/lessorsimilar/ba∇dblϕ/ba∇dbl6
+B1/6
+6,2. (3.29)
+If/ba∇dblϕ/ba∇dblL∞> A1, then we have
+/ba∇dblg(ϕ)/ba∇dblL2,1≤ /ba∇dble4π|ϕ|2−1/ba∇dbl1/2−κ
+L1/ba∇dble4π|ϕ|2−1/ba∇dbl1/2+κ
+L∞/ba∇dblϕ/ba∇dbl4
+L4/κ,4/(1/2+κ)
+≤ /ba∇dble4π|ϕ|2−1/ba∇dbl1/2−κ
+L1/ba∇dble4π|ϕ|2−1/ba∇dbl1/2+κ
+L∞/ba∇dblϕ/ba∇dbl4
+L4/κ,2.(3.30)EXPONENTIAL ENERGY DECAY FOR DAMPED NLKG 17
+The first factor on the right hand side in the above inequality is bound ed by the
+sharp Trudinger-Moser inequality. Since /ba∇dblϕ/ba∇dblC1/4−κ> A >1/2 andCλ≥1, then the
+second factor is bounded by using the logarithmic inequality (3.24)
+/ba∇dble4π|ϕ|2−1/ba∇dbl1/2+κ
+L∞≤e2π(1+2κ)/bardblϕ/bardbl2
+L∞≤/bracketleftbigg
+Cλ+(8/µ)1/4/ba∇dblϕ/ba∇dblC1/4−κ
+/ba∇dblϕ/ba∇dblHµ/bracketrightbigg2π(1+2κ)λ/bardblϕ/bardbl2
+Hµ
+≤[Cλ+(8/µ)1/4/ba∇dblϕ/ba∇dblC1/4−κ/A1]2π(1+2κ)λA2
+1
+≤(8/µ)[4Cλ/ba∇dblϕ/ba∇dblC1/4−κ]4.
+(3.31)
+Thus, using Sobolev embedding we obtain
+/ba∇dblg(ϕ)/ba∇dblL2,1/lessorsimilar/ba∇dblϕ/ba∇dbl6
+B1/6
+6,2+CA(/ba∇dblϕ/ba∇dblH1)/ba∇dblϕ/ba∇dbl4
+C1/4−κ/ba∇dblϕ/ba∇dbl4
+Bκ/2
+2/κ,2. (3.32)
+/square
+3.4.Concluding the proof of Theorem 1.2. Thus we have so far either
+A(u;T)≥(MT)−1ε, (3.33)
+or
+/integraldisplayT
+1/integraldisplay
+RN|g(u)|
+λdxdt≤C(N)E(u;0)<∞, (3.34)
+whereC(N)>0 is a certain uniform bound in terms of E0,fandN. In the latter
+case, we obtain from (2.25) in the same way as in Section 2.2, using the free energy
+bound (3.1) as well,
+/integraldisplayT
+S/integraldisplay
+|x|<t|∇u|2+|x˙u+t∇u|2
+λ+|g(u)|
+λdxdt
+/lessorsimilarE(u;0)[1+µ(u;T)+C(N)],(3.35)
+for 1≤R < S= 3R < T, whereµis as defined in (2.30). Then proceeding in the
+same way as before, we obtain
+/integraldisplayT
+S/ba∇dbl˙u/ba∇dbl2
+L2x
+tdt/lessorsimilar(1+C0)E(u;0)[1+µ(u;T)+C(N)]
++E(u;0)[1+µ(u;T)+C(N)]2/p(RNlogT)1−2/p,(3.36)
+instead of (2.37). Further tracking the same way, we replace (2.39 ) by the following
+bound due to (3.2)
+K(u)≥ν/ba∇dblu(t)/ba∇dbl2
+H1≥ν[E(u;t)−/ba∇dblu(t)/ba∇dbl2
+L2x], (3.37)
+hence we have
+(1+ν)/ba∇dbl˙u(t)/ba∇dbl2
+L2x≥νE(u;t)+∂t/an}b∇acketle{t˙u|u/an}b∇acket∇i}htL2x+/an}b∇acketle{ta˙u|u/an}b∇acket∇i}htL2x. (3.38)
+Integrate it by dt/t, we obtain the following instead of (2.41):
+/integraldisplayT
+Sdt
+t(1+ν)/ba∇dbl˙u(t)/ba∇dbl2
+L2x≥νE(u;T)logT
+3R−E(u;0)/R−MA(u;T).(3.39)18 L. ALOUI, S. IBRAHIM, AND K. NAKANISHI
+Combining it with (3.36), we obtain
+νE(u;T)logT
+3R/lessorsimilar(1+C0)E(u;0)[1+µ(u;T)+C(N)]
++E(u;0)[1+µ(u;T)+C(N)]2/p(RNlogT)1−2/p.(3.40)
+Hence if
+[1+MT+(a0R)−1]A(u;T)≤E(u;0)/2, (3.41)
+then we have µ≤1/2,E(u;T)≥E(u;0)/2 and
+νlogT/lessorsimilarlogR+(1+C0)[1+C(N)]+[1+ C(N)]2/p(RNlogT)1−2/p,(3.42)
+which implies that
+logT/lessorsimilarν−1(1+C(N))(1+C0+R2), (3.43)
+where we have chosen p= 2⋆= 2+4/N. Thus in conclusion there exists an absolute
+constant C∗>0 such that we have (1.11) with
+logT=C∗ν−1(1+C(N))(1+C0+R2),
+δ= min([1+ MT+(a0R)−1]−1/2,(MT)−1ε/E0),(3.44)
+where the last factor takes care of the case (3.33). /square
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+Department of Mathematics, University of Bizerte, Bizerte , Zarzouna 7021, Tunisia.
+E-mail address :Lassaad.Aloui@fsg.rnu.tn
+Department of Mathematics and Statistics,, University of V ictoria, PO Box 3060
+STN CSC, Victoria, BC, V8P 5C3, Canada
+E-mail address :ibrahim@math.uvic.ca
+URL:http://www.math.uvic.ca/ ∼ibrahim/
+Department of Mathematics, Kyoto University
+E-mail address :n-kenji@math.kyoto-u.ac.jp
\ No newline at end of file
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@@ -0,0 +1,1783 @@
+arXiv:1001.0210v2  [cs.IT]  4 Apr 2011MAHDAVIFAR AND VARDY: ACHIEVINGTHESECRECYCAPACITYOF WIRETAPCHANNELSUSING P OLARCODES 1
+Achieving theSecrecy Capacity
+ofWiretap ChannelsUsingPolarCodes
+HessamMahdavifar, StudentMember,IEEE , andAlexanderVardy, Fellow, IEEE
+Abstract—Suppose that Alice wishes to send messages to Bob
+through a communication channel C1, but her transmissions also
+reach an eavesdropper Eve through another channel C2. This is
+thewiretapchannelmodelintroducedbyWynerin1975.Thego al
+is to design a coding scheme that makes it possible for Alice t o
+communicate both reliablyandsecurely. Reliability is measured
+in terms of Bob’s probability of error in recovering the mess age,
+while securityis measured in terms of the mutual informatio n be-
+tween the message and Eve’s observations. Wyner showed that
+the situation is characterized by a single constant Cs, called the
+secrecy capacity , which has the following meaning: for all ε>0,
+there exist coding schemes of rate R/greaterorequalslantCs−εthat asymptotically
+achieve both the reliability and the security objectives. H owever,
+his proof of this result is based upon a nonconstructive rand om-
+coding argument. To date, despite a considerable research e ffort,
+theonlycasewhereweknowhowto construct codingschemesthat
+achieve secrecy capacity is when Eve’s channel C2is an erasure
+channel,or acombinatorial variation thereof.
+Polar codes were recently invented by Arıkan; they approach
+thecapacityofsymmetricbinary-inputdiscretememoryles s chan-
+nelswithlow encodinganddecodingcomplexity.In thispape r,we
+use polar codes to construct a coding scheme that achieves th e se-
+crecycapacityforawiderangeofwiretapchannels.Ourcons truc-
+tion works for any instantiation of the wiretap channel mode l, as
+longas both C1andC2aresymmetric andbinary-input,and C2is
+degradedwithrespectto C1.Moreover,weshowhowtomodifyour
+construction in order to provide strong security , in the sense de-
+finedbyMaurer,whilestilloperatingataratethatapproach esthe
+secrecy capacity. In this case, we cannot guarantee that the relia-
+bilityconditionwill also be satisfied unlessthe main chann elC1is
+noiseless,althoughwebelieveitcanbealwayssatisfiedinp ractice.
+IndexTerms —channelpolarization,information-theoreticsecu-
+rity,polarcodes,secrecycapacity,strongsecurity,wire tapchannel
+I. INTRODUCTION
+THE notion of wiretap channels was introduced by Aaron
+Wyner [41] in 1975. In this setting, Alice wishes to send
+messages to Bob through a communication channel C1, called
+themainchannel ,buthertransmissionsalsoreachanadversary
+Eve through another channel C2, called the wiretap channel .
+This is illustrated in Figure1, wherein Udenotes a k-bit mes-
+sage that Alice wishes to communicate to Bob. We think of U
+as a random variable that takes values in {0, 1}k; unlike most
+papersonwiretapchannels,we donotassumeanything regard-
+ingtheaprioridistributionof U.Whilemakinguseofauxiliary
+randombits, theencodermaps Uintoasequence Xofnchan-
+The material in this paper was presented in part at the IEEE In ternational
+Symposium on Information Theory, Austin, Texas, June2010.
+HessamMahdavifariswiththeDepartmentofElectricalandC omputerEngi-
+neering, University ofCalifornia San Diego, LaJolla, CA 92 093–0407, U.S.A.
+(e-mail:hessam@ucsd.edu ).
+Alexander VardyiswiththeDepartmentofElectrical andCom puter Engine-
+ering, the Department of Computer Science and Engineering, and the Depart-
+mentofMathematics, University ofCalifornia SanDiego,La Jolla,CA92093–
+0407, U.S.A.(e-mail: avardy@ucsd.edu ).Encoder
+random bitsDecoderAlice Bob
+ChannelWiretapMain
+Channel
+EvePSfragreplacements U X Y
+Z/hatwideUC1
+C2
+Figure1. Blockdiagram of a generic wiretap-channel system
+nelsymbols.Thissequenceistransmittedacrossthemainch an-
+nelandthewiretapchannelresultinginthecorrespondingc han-
+neloutputs YandZ.Finally,thedecodermaps Y(deterministi-
+cally)intoanestimate /hatwideUoftheoriginalmessage.
+Thegoalis to designa codingscheme— namely,an encod-
+ingalgorithmanda decodingalgorithm— that makesit possi-
+ble to communicate both reliablyandsecurely, as the message
+length ktendstoinfinity.Reliabilityismeasuredintermsofthe
+probabilityoferror inrecoveringthemessage.Specifically,the
+objectiveistosatisfy thefollowing
+ReliabilityCondition: lim
+k→∞Pr/braceleftbig/hatwideU/\e}atio\slash=U/bracerightbig=0(1)
+where the probability is over all the relevant coin tosses in the
+system: in the generationof U, in the encoder,and in the main
+channel. Security is usually measured in terms of the normal-
+ized mutual information between the message Uand Eve’s
+observations Z.Specifically,oneisinterestedinencodingalgo-
+rithmsthatsatisfy thefollowing
+SecurityCondition: lim
+k→∞I(U;Z)
+k=0 (2)
+Notethat I(U;Z)isequaltothedifferencebetweentheapriori
+entropy H(U)andtheconditionalentropy H(U|Z).Thus,intu-
+itively,(2)meansthatobserving Zdoesnotprovidemuchinfor-
+mationabout Ubeyondwhat is availablea priori,as compared
+tothemessagelength k.Maurerarguedin[27,28]thatthecon-
+ventional notion of security (2) is much too weak. Indeed,it is
+easytoconstructexampleswhere k1−εoutofthe kmessagebits
+aredisclosedtoEve,whilestillsatisfying(2).Thisiscle arlyun-
+acceptable.ThusMaurerintroducedin [27]analternative
+StrongSecurityCondition: lim
+k→∞I(U;Z) = 0(3)
+Noticethatbothsecurityconditions(2)and(3)areinforma tion-
+theoreticratherthancomputational:theadversaryisassu medto
+be computationally unbounded, and security does not depend
+oncomputationalhardnessassumptionsofanykind.2 MAHDAVIFAR AND VARDY: ACHIEVINGTHESECRECYCAPACITYOFWIRETAPCHANNELSUSING P OLARCODES
+A. PriorWork
+In 1975,Wyner [41]considereda special case of the system in
+Figure1 where both C1andC2are discrete memoryless chan-
+nels(DMCs)and,moreover, C2isdegradedwithrespectto C1.
+He proved that such a system is characterized by a single con-
+stantCs, called the secrecy capacity , which has the following
+meaning.Forall ε>0,thereexistcodingschemesofinforma-
+tionrate R/greaterorequalslantCs−εthatsatisfy(1)and(2);conversely,itisnot
+possibletosatisfyboth(1)and(2)atratesgreaterthan Cs.Since
+1975, Wyner’s results have been extended to a variety of con-
+texts, most notably Gaussian channels [22], general broadc ast
+channelswithconfidentialmessages[11],andchannelsthat im-
+pose a combinatorial (rather than probabilistic) constrai nt on
+the adversary[7,32].In fact, the literature on wiretap cha nnels
+encompasses,bynow,hundredsofpapers.
+However, the vast majority of this work relies on noncon-
+structive random-coding arguments to establish the main re -
+sults. Such results show that there exist codes that achieve se-
+crecy capacity, but are of little use if one’s goal is to desig n
+specificpolynomial-timeencoding/decodingalgorithms.T othe
+best of our knowledge, constructive solutions to the wiretap-
+channel problem are available only in two special cases. The
+first special case is when the main channel is noiseless and th e
+wiretap channelis the binaryerasure channel(BEC). A codin g
+scheme forthis case, using LDPC codesforthe BEC, was pre-
+sented in [36,38] and proved to achieve secrecy capacity. Th e
+other special case is when the adversary is constrained comb i-
+natorially: Eve can select to observe some tout of the ntrans-
+mittedsymbols,while theremaining n−tsymbolsareerased.
+This situation, studied by Ozarow and Wyner in [32], may be
+regarded as a combinatorial variation of an erasure channel .
+Provablyoptimalcodingschemesforthiscasecanbeconstru c-
+ted fromMDS codes [40],or using extractors[7]. We observe,
+however, that even for the simple situation where C1is noise-
+lessand C2isa binarysymmetricchannel,it isnotknownhow
+toexplicitlyconstructcodesthat achievesecrecycapacit y.
+We pointoutthatageneralmethodofcodingforthewiretap
+channel, often referred to as coset-coding orsyndrome-coding ,
+iswellknown.ThismethodgoesbacktotheworkofWyner[32,
+41], although it was significantly extended and generalized in
+[8,9] and other papers. Assume, for simplicity, that the inp ut
+alphabet of both C1andC2is binary. In this case, the coset-
+codingmethodutilizestwobinarylinearcodes:an“outer”c ode
+C∗andan“inner”code C,suchthat C⊂C∗andthedifference
+dim(C∗)−dim(C)between their dimensions is k. This con-
+dition implies that C∗can be partitioned into 2kcosets of C.
+A message u∈{0, 1}kis conveyed by Alice via the choice of
+oneofthese 2kcosets,say a+C.WhatistransmittedbyAlice
+is a vector Xthat is selected uniformlyat randomfrom a+C.
+Loosely speaking, the outer code C∗serves to correct the er-
+rorsonthemainchannel,andthusensuresreliability,whil ethe
+inner code C, over which Xis randomized, ensures security.
+The trouble is that it is not known how to explicitly construct
+asequenceofoutercodes C∗andinnercodes Cthatsatisfycon-
+ditions (1) and (2) at a rate that approaches the secrecy capa c-
+ity as n→∞. Theremarkableworkof CohenandZ´ emor[8,9]
+showsthata randomchoice oftheinnercode Csufficestoachi-evestrongsecurity,inaverygeneralsetting.Notably,the proof
+ofthisresultin[8,9]does notassumethatthemessagesareuni-
+formlyrandoma priori.Still, to the best of our knowledge,t he
+onlycaseswhereexplicitconstructionsof CandC∗areknown
+arethosedescribedintheforegoingparagraph(cf.[36,38] ).
+B. OurContributions
+In this paper,we present a coding scheme that achievesthe se -
+crecy capacity of wiretap channels whenever C1andC2are
+binary-inputsymmetricDMCsand C2isdegradedwithrespect
+toC1. This is the situation originally studied by Wyner; it in-
+cludestheimportantspecialcasewhere C1andC2arearbitrary
+binarysymmetricchannels.Weareabletosatisfythereliab ility
+and security conditions (1) and (2) with explicit polynomia l-
+time encoding/decodingalgorithms. In fact, the number of o p-
+erationsrequiredforencodinganddecodingisonly O(nlogn).
+Our construction is based upon key results in the literature on
+polarcodes ,recentlyinventedbyArıkan[3].
+Itisprovedin[3]thatpolarcodesachievethecapacityofar -
+bitrary binary-input symmetric DMCs, with low encoding and
+decodingcomplexity.Theproofofthisresultisbasedonaph e-
+nomenoncalled channelpolarization .Let
+G=/bracketleftbigg
+1 0
+1 1/bracketrightbigg
+(4)
+andlet G⊗mdenotethe m-th Kroneckerpowerof G. Let Wbe
+a symmetric binary-input DMC, and let V= (V1,V2, . . . , Vn)
+be a block of n=2mbits chosen uniformly at random from
+{0, 1}n. Suppose Vis encodedas X=VPnG⊗m, where Pnis
+then×nbit-reversal permutation matrix. Finally, Xis trans-
+mittedthrough nindependentcopiesof W, asshownbelow:
+...
+...PSfragreplacements
+U
+X
+Y
+Z
+/hatwideU
+C1
+C2
+V1
+V2
+VnY1
+Y2
+YnX1
+X2
+XnWW
+WPnG⊗m (5)
+Arıkan[3]consideredthe nchannels“seen”byeachofthe nin-
+dividualbits V1,V2, . . . , Vnastheyundergothe transformation
+in (5). Let uscall them the bit-channels — for a precise defini-
+tion of the notion of a bit-channel, see [3] and SectionIII. I t is
+shown in [3] that as mgrows, the bit-channelsstart polarizing:
+they approach either a noiseless channel or a pure-noise cha n-
+nel.Wewillsaythattheformerbit-channelsare goodwhilethe
+latter are bad(again, see SectionIII for a rigorous definition).
+One of the key results of [3] is that the fraction of bit-chann els
+thataregoodapproachesthecapacityof Wasn→∞.
+Giventhechannelpolarizationphenomenon,thegeneralide a
+of our construction is quite simple. We will transmit random
+bitsoverthosebit-channelsthataregoodforbothEveandBo b,
+information bits over those bit-channels that are good for B ob
+but bad for Eve, and zeros over those bit-channelsthat are ba d
+for both Bob and Eve. In the rest of this paper, we make this
+ideapreciseandprovethatit works.Toappear in the IEEE T RANSACTIONS ON INFORMATION THEORY, 2011 3
+InSectionII,webrieflyrecaprelevantresultsfromthelite ra-
+tureonwiretapchannels,inordertoobtainasimpleexpress ion
+forthesecrecycapacity Csinthecasewhere C1andC2aresym-
+metric DMCs and C2is degraded with respect to C1. In Sec-
+tionIII, we provide the necessary background on polar codes
+and establish a certain property of channel polarization th at is
+crucialforourconstruction(Lemma4).Theconstructionit self,
+namelytheproposedcodingscheme,ispresentedinSectionI V.
+In SectionV, we prove that the proposed coding scheme satis-
+fies the reliability and security conditions (1) and (2). We a lso
+show in SectionV that the rate k/nof our coding scheme ap-
+proachesthesecrecycapacity Csasn→∞.
+InSectionVI,weconsiderthestrongernotionofsecurity(3 ).
+ItwasshownbyMaurer-Wolf[28]thatanycodingschemethat
+satisfiesthe“weak”securitycondition(2)canbeconverted into
+a coding scheme that satisfies the stronger condition (3). Th is
+is accomplished using an ingenious information reconcilia tion
+and privacy amplification protocol [6]. Although, in princi ple,
+the rate overhead necessary for privacy amplification can be
+madearbitrarilysmall,thisisunlikelytobethecaseinpra ctice.
+In SectionVI, we show how to modify the coding scheme of
+SectionIV in order to guarantee strong security directly, with-
+outtheneedforprivacyamplification.Thisisachievedbysu it-
+ably modifyingour definition of “bad” bit-channels, in a man -
+nerthatdiffersfromthegenerallyacceptednotions[3].As are-
+sult, under the modified definition, a vanishing fraction of b it-
+channelscouldbegoodforEvebutbadforBob,evenwhenthe
+wiretap channel is degraded with respect to the main channel .
+In this situation, the rate of the coding scheme of SectionVI
+still approachesthe secrecycapacity Cs, but we cannotguaran-
+tee that the reliabilitycondition(1) issatisfied, unlesst he main
+channel is noiseless. Nevertheless, we believe that, in pra ctice,
+acceptably low probabilities of error could be achieved on t he
+mainchannel(usinga moreelaboratedecodingalgorithm).
+WeconcludethepaperinSectionVIIwithabriefdiscussion
+offurtherresults.Inparticular,weexplaininSectionVII thatour
+constructiongeneralizesstraightforwardlytothecasewh erethe
+channels C1andC2arenotsymmetric,althoughinthiscasepo-
+larcodesbecomelessexplicitandonlythe“symmetricsecre cy
+capacity”canbeachieved.Afewopenproblemsthatstemfrom
+ourresultshereinarealso discussedinSectionVII.
+C. RelatedWork
+Followingthepublicationofapreliminaryversionofthisp aper
+in[24]and[25],severalrelatedpapershaveappeared[1,16 ,20].
+Most notably, the work of Hof and Shamai [16] on polar cod-
+ing for wiretap channels is independent and contemporaneou s
+toours.Whilesomeofthemainresultsin[16]andinthispape r
+are similar, there are important differences that we would l ike
+to emphasize. One key difference is that Hof and Shamai [16]
+analyzetheirpolar-codingschemeindetail,includingrec ursive
+channel combining and splitting, whereas we treat polar cod es
+essentiallyasablackbox.Webelievethismakesourproofbo th
+shorterandclearer.
+Therearealso significantdifferencesbetweentheresultse s-
+tablishedin[1,16,20]andinthispaper.Inparticular,iti sshown
+in[1,16]thatpolarcodingachievestheentire rate-equivocationregion(see[23]foradefinition),whereasweareinterestedonly
+in the extreme point of this region that corresponds to secre cy
+capacity.Ontheotherhand,inseveralotherrespects,ourr esults
+arestrongerthanthoseof[1,16,20].First,theproofin[1, 16]is
+contingentontheassumptionthatthemessage Uisaprioriuni-
+form over {0, 1}k, whereas we do not place any constraints on
+theaprioridistributionof U.Assumingthatmessagesareapri-
+oriuniformiscommonininformationtheory,butsuchassump -
+tionsarecompletelyunacceptableincryptography[5,14]. Even
+moreimportantly,weshowhowpolarcodingshouldbeusedto
+providestrong security , whereas the work of [1,16,20] pro-
+videsweaksecurityonly.Again,incryptographicapplicat ions,
+conventionalweaksecurityis usuallyunacceptable.
+II. SECRECYCAPACITY
+Inthissection,wefirstestablishsomerelevantterminolog y.We
+thenbrieflyrecapthe resultsof [11,21]to providea simple e x-
+pressionforthesecrecycapacity Csinthecasewhere C1andC2
+aresymmetricDMCsand C2is degradedwith respectto C1.
+We will limitourconsiderationtofinite-inputandfinite-ou t-
+put discrete memoryless channels throughout. Such a channel
+is a triple /a\}bracketle{tX,Y,W/a\}bracketri}ht, where X,Yare finite sets and Wis
+an|X|×|Y|matrixwith W[x,y]beingtheprobabilityofre-
+ceiving y∈Ygiven that x∈Xwas sent. We will follow the
+conventionof[3,4,18]andwrite W(y|x)insteadof W[x,y].
+A matrix Mis stronglysymmetricif the rows of Mare per-
+mutations of each other and the columns of Mare permuta-
+tionsofeachother.Achannel /a\}bracketle{tX,Y,W/a\}bracketri}htisstronglysymmet-
+ricifWisastronglysymmetricmatrix.Following[3,4,13,18],
+wewillsaythat /a\}bracketle{tX,Y,W/a\}bracketri}htissymmetric (oftencalled output-
+symmetric ) if the columnsof Wcan be partitionedinto subsets
+such that each subset forms a strongly symmetric matrix. The
+capacityofasymmetricchannel /a\}bracketle{tX,Y,W/a\}bracketri}htisgivenby
+C(W)def=H(X)−H(X|Y) = log2|X|−H(X|Y)(6)
+wherethe randomvariable Xat the inputto the channelis uni-
+form over X, and Yis the corresponding random variable at
+the channel output (for a proof of this fact, see [13, p.94]). An
+important example of a symmetric channel is the binary sym-
+metricchannel BSC(p) =/angbracketleftbig{0, 1},{0, 1},W/angbracketrightbig
+with
+W=/bracketleftbigg1−p p
+p1−p/bracketrightbigg
+Givenachannel C1=/a\}bracketle{tX,Y,W1/a\}bracketri}ht,wesaythatanotherchan-
+nelC2=/a\}bracketle{tX,Z,W2/a\}bracketri}htisdegraded with respect to C1if there
+exists a third channel C3=/a\}bracketle{tY,Z,W3/a\}bracketri}htsuch that C2is the
+cascadeof C1andC3. Specifically,it isrequiredthat
+W2(z|x) =∑
+y∈YW1(y|x)W3(z|y) (7)
+forall x∈Xandz∈Z.Notethatwhenever p2/greaterorequalslantp1,thechan-
+nelC2=BSC(p2)isdegradedwithrespectto C1=BSC(p1).
+Thesecrecycapacity Csofthewiretap-channelsysteminFig-
+ure1 is defined as follows. First, assume that the message Uis
+uniformlyrandomover {0, 1}k. ThenCsis the supremumover
+allrates R=k/n(inbitsperchanneluse)suchthatthereexist
+codingschemesof rate Rsatisfying conditions(1) and (2). For
+thegeneralcasewhere C1andC2arearbitraryDMCs,comput-4 MAHDAVIFAR AND VARDY: ACHIEVINGTHESECRECYCAPACITYOFWIRETAPCHANNELSUSING P OLARCODES
+ingthesecrecycapacityisadifficultproblem.Let Xdenotethe
+single-letter input to C1andC2, letYandZdenote the corre-
+sponding single-letter outputs. The best known expression for
+thesecrecycapacity Cs,givenbyCsisz´ ar andK¨ ornerin[11],is
+Cs=max
+U/parenleftBig
+I(U;Y)−I(U;Z)/parenrightBig
+where the maximumis taken overall randomvariables Usuch
+thatU→X→(Y,Z)is a Markov chain. The problem is that
+this maximization is often difficult to evaluate, and there i s no
+simplerexpressionforthesecrecycapacityevenwhen C1andC2
+are both strongly symmetric, unless additional constraint s are
+satisfied. See[23,39]formoredetailsonthis.
+However,when C1=/a\}bracketle{tX,Y,W∗/a\}bracketri}htandC2=/a\}bracketle{tX,Z,W/a\}bracketri}htare
+symmetric andC2is degradedwith respect to C1, a simple ex-
+pression for Cswas given by Leung-Yan-Cheong in [21]. It is
+shownin [21,Theorem4]that inthiscase
+Cs=C(W∗)−C(W) = H(X|Z)−H(X|Y)(8)
+where Xisuniformover X.Inparticular,ifthemainchannelis
+BSC(p1)whilethewiretapchannelis BSC(p2),with p2/greaterorequalslantp1,
+then the secrecy capacity is given by h2(p2)−h2(p1), where
+h2(·)isthebinaryentropyfunction.
+III. POLARCODES
+This section provides a concise overview of the groundbreak -
+ingworkofArıkan[3]andothers[4,18,19]onpolarcodesand
+channelpolarization.Weestablishonlythoseresultsthat arees-
+sential forthecodingschemespresentedinthispaper.
+Asin[15,19],weconsiderexclusively binary-inputsymmet-
+ric memoryless (BSM) discrete channels. Such a channel is
+a symmetric DMC, as defined in the previoussection, with in-
+put alphabet X={0, 1}. With a slight abuse of notation, we
+willoftenfollow[3,4,19]andsimplywrite WtodenoteaBSM
+channel/angbracketleftbig{0, 1},Y,W/angbracketrightbig
+.TheBhattacharyyaparameter ofWis
+Z(W)def=∑
+y∈Y/radicalBig
+W(y|0)W(y|1)
+It can be shown that Z(W)always takes values in [0, 1]. Intu-
+itively, channels with Z(W)/lessorequalslantεare almost noiseless, while
+channels with Z(W)/greaterorequalslant1−εare almost pure-noise channels.
+Thisintuitionismadeprecisein[3,Proposition1].
+Arıkan [3] introduces a number of channels that are associ-
+ated with the transformation in (5). First, there is the chan nel/angbracketleftbig{0, 1}n,Yn,Wn/angbracketrightbig
+givenby
+Wn(y|x)def=n
+∏
+i=1W(yi|xi) (9)
+where x=(x1,x2, . . . , xn)andy=(y1,y2, . . . , yn).Thisisthe
+channelthatresultsfrom nindependentusesofthechannel W.
+Next,forall n=2m,letusdefinethe Arıkantransformmatrix
+Gndef=PnG⊗m,where Gisthematrixin(4)and Pnisthebit-re-
+versalpermutationmatrixdefinedin[3,SectionVII-B].Arı kan
+thenintroducesthe“combined”channel/angbracketleftbig{0, 1}n,Yn,/tildewideW/angbracketrightbig
+with
+transitionprobabilitiesgivenby
+/tildewideW(y|v)def=Wn/parenleftbig
+y/vextendsingle/vextendsinglevGn/parenrightbig=Wn/parenleftbig
+y/vextendsingle/vextendsinglevPnG⊗m/parenrightbig
+(10)
+Thisisthechannelseenbytherandomvector (V1,V2, . . . , Vn)
+asitundergoesthetransformationin(5).Arıkan[3]alsode finesthechannel/angbracketleftbig{0, 1},Yn×{0, 1}i−1,Wi/angbracketrightbig
+thatisseenbythe i-th
+bitVi, for i=1, 2, . . . , n, as follows.Let vi= (v1,v2, . . . , vi)
+denote a binary vector of length i, with the convention that v0
+istheemptystringandthat {0, 1}0={v0}. Then
+Wi/parenleftbig
+y,vi−1|vi)def=1
+2n−1∑
+v∈{0,1}n−i/tildewideW/parenleftBig
+y/vextendsingle/vextendsingle(vi−1,vi,v)/parenrightBig
+(11)
+where(·,·)denotesvectorconcatenation.Itiseasytoshow(cf.
+Lemma14) that Wi/parenleftbig
+y,vi−1|vi)is indeedtheprobabilityofthe
+eventthat (Y1,Y2, . . . , Yn)=yand(V1,V2, . . . , Vi−1)=vi−1
+giventheevent Vi=vi,provided V= (V1,V2, . . . , Vn)isapri-
+oriuniformover {0, 1}n.Consequently,ifoneconsidersa“hy-
+potheticaldecoder”thatattemptstoestimatethe i-thbit Vihav-
+ing observed yandvi−1, then Wiis the effective channel seen
+by such decoder (again, provided Vis a priori uniform). We
+will referto Wiasthe i-th bit-channel ,for i=1, 2, . . . , n.
+Observe that the optimal decision rule for the hypothetical
+decoderoftheforegoingparagraphis trivial:decide /hatwidevi=0if
+Wi/parenleftbig
+y,vi−1|0)/greaterorequalslantWi/parenleftbig
+y,vi−1|1) (12)
+and/hatwidevi=1otherwise. One can invokethis decision rule iterati-
+velyforall i=1, 2, . . . , n,whilesubstitutingthefirst i−1de-
+cisions(/hatwidev1,/hatwidev2, . . . ,/hatwidevi−1)in place of the hypotheticalobserva-
+tions vi−1.Uptoasmallmodificationdescribedlater,thisisthe
+successive cancellationdecoder inventedbyArıkan[3].
+Following[4,18],letuspartitionthe nbit-channelsinto good
+channels andbadchannels asfollows.Let [n]def={1, 2, . . . , n}
+and let β<1/2be a fixed positive constant. Then the index sets
+ofthegoodandbadchannelsaregivenby
+Gn(W,β)def=/braceleftBig
+i∈[n]:Z(Wi)<2−nβ/n/bracerightBig
+(13)
+Bn(W,β)def=/braceleftBig
+i∈[n]:Z(Wi)/greaterorequalslant2−nβ/n/bracerightBig
+(14)
+One of the key results of [3,4] is that the fraction of the good
+channelsapproachesthe channel capacity C(W), given by (6),
+asn→∞.We state thisresultpreciselyasfollows.
+Theorem1. For any BSM channel Wand any constant β<1/2
+wehave
+limn→∞|Gn(W,β)|
+n=C(W)
+Theorem1 readily leads to a constructionof capacity-achie -
+vingpolar codes . The general idea is to transmit the informa-
+tionbitsoverthegoodbit-channelswhilefixingtheinputto the
+badbit-channelstoaprioriknownvalues,sayzeros1.Formally,
+givenavector voflength nandasetA ⊆[n],letvAdenotethe
+projectionof vonthecoordinatesin A.Eachsubset Aof[n]of
+size|A|=kspecifiesa polarcode Cn(A)ofrate k/n.Wede-
+fineCn(A)via its encoder map E:{0, 1}k→ { 0, 1}n. Given
+amessage u∈{0, 1}k,theencoderproceedsintwosteps.First,
+theencoderconstructsthevector v∈{0, 1}n,bysetting vA=u
+andvAc=0, whereAcis the complementof Ain[n]and0is
+1In thework of[3,4],which deals with symmetric aswellas non -symmetric
+DMCs, it is important to allow an arbitrary choice of these “f rozen” values.
+However, it is also shown in [3] that in the case of symmetric c hannels, any
+choice is as good as any other. Since we are concerned exclusi vely with sym-
+metric channels in this paper, wewill usezeros for notation al convenience.Toappear in the IEEE T RANSACTIONS ON INFORMATION THEORY, 2011 5
+the all-zerovector.Next, it outputs E(u) =vGnas in (5). The
+decoder we will use for Cn(A)is the successive cancellation
+decoderofArıkan[3].Thisdecoderworksasalreadydescrib ed
+in (12), with one straightforward modification: for i∈Ac, the
+decisionruleissimply /hatwidevi=0.
+The key property of the encoder-decoderpair of the forego-
+ingparagraphissummarizedinthefollowingtheorem.Thist he-
+oremis(thesecondpartof)Proposition2ofArıkan[3].
+Theorem2. LetWbe a BSM channel and let Abe an arbit-
+rarysubsetof [n]ofsize|A|=k.Supposethatamessage Uis
+chosen uniformlyatrandom from{0, 1}k, encoded as a code-
+word of Cn(A), and transmitted over W. Then the probability
+that the channel output is not decoded to Uunder successive
+cancellationdecodingsatisfies
+Pr/braceleftbig/hatwideU/\e}atio\slash=U/bracerightbig/lessorequalslant∑
+i∈AZ(Wi) (15)
+Inthispaper,weneedaresultthatissomewhatstrongerthan
+Theorem2, since we do notassume that messages are chosen
+uniformlyatrandomfrom {0, 1}k.Fortunately,sucharesultcan
+bereadilyestablishedusingthemachinerythatwasalready de-
+velopedbyArıkan[3]forsymmetricchannels.Indeed,forsy m-
+metricchannels,itiswellknownthattheerrorprobability isin-
+dependentof the transmitted codeword;hence,the input dis tri-
+butionshouldnotmatter. Thefollowingpropositionmakest his
+observationprecise.We includea proof,forcompleteness.
+Proposition3. LetWbe a BSM channeland let Abe anarbit-
+rarysubsetof [n]ofsize|A|=k.Supposethatamessage Uis
+chosen accordingto anarbitrarydistribution from{0, 1}k, en-
+codedasacodewordof Cn(A),andtransmittedover W. Then
+the probability Pethat the channel output is not decoded to U
+undersuccessivecancellationdecodingsatisfies
+Pe/lessorequalslant∑
+i∈AZ(Wi) (16)
+Proof.FollowingArıkan[3,SectionV],weconsiderthesam-
+plespace Ωn={0, 1}n×Ynwith theprobabilitymeasure
+Pr/braceleftbig(v,y)/bracerightbigdef=2−n/tildewideW(y|v) (17)
+forall v∈{0, 1}nandy∈Yn.Onthisprobabilityspace,Arı-
+kan [3] definesthe event Eof blockerroras the set of all pairs
+(v,y)inΩnsuchthatthechanneloutput yisnotdecodedto vA
+under successive cancellation decoding. Let us further defi ne,
+forall w∈{0, 1}n, theevent
+Vwdef=/braceleftBig
+(v,y)∈Ωn:v=w/bracerightBig
+(18)
+Itisshownin [3,Proposition2]thatundertheprobabilitym ea-
+surein(17)we have
+Pr/braceleftbig
+E/bracerightbig/lessorequalslant∑
+i∈AZ(Wi) (19)
+It is furthermoreshown in [3, SectionVI-B] that, providedt ies
+inthedecisionrule(12)arebrokenatrandom,theevents Eand
+Vwareindependentforall w. Inotherwords,
+Pr/braceleftbig
+E/vextendsingle/vextendsingleVw/bracerightbig=Pr/braceleftbig
+E/bracerightbig
+forall w∈ {0, 1}n(20)
+Nowconsiderthesituationwheretheaprioridistributiono nthe
+messagesin {0, 1}kis a delta-function.Thatis, a specific mes-sage u∈{0, 1}kis always chosen with probability 1, and the
+input to the transformationin (5) is the vector wwith wA=u
+andwAc=0. Let Pe(u)denotetheprobabilitythat the succes-
+sive cancellation decoder does not decode the correspondin g
+channeloutput ytowA.Thenit followsfrom(10) that
+Pe(u) =∑
+y∈F(wA)/tildewideW(y|w) (21)
+whereF(wA)isthesetofchanneloutputsthatarenotdecoded
+towAbythe successivecancellationdecoder.Observethat
+E∩Vw=/braceleftBig
+(v,y)∈Ωn:v=wand y∈F(wA)/bracerightBig
+Therefore,theprobabilityoftheevent E∩Vw,undertheprob-
+abilitymeasurein(17),canbeexpressedas
+Pr/braceleftbig
+E∩Vw/bracerightbig=∑
+y∈E∩Vw2−n/tildewideW(y|v) = 2−n∑
+y∈F(wA)/tildewideW(y|w)(22)
+Itcanbereadilyseenfrom(18)that Pr/braceleftbig
+Vw/bracerightbig=2−nforall w.
+Hence,it followsfrom(20)that
+Pr/braceleftbig
+E∩Vw/bracerightbig=Pr/braceleftbig
+E/vextendsingle/vextendsingleVw/bracerightbig
+Pr/braceleftbig
+Vw/bracerightbig=2−nPr/braceleftbig
+E/bracerightbig
+(23)
+Combining(21),(22),(23),we concludethat Pe(u) =Pr/braceleftbig
+E/bracerightbig
+.
+Sincethisholdsforall u∈{0, 1}k,we have
+Pe=∑
+u∈{0,1}nPe(u)Pr/braceleftbig
+U=u/bracerightbig=Pr/braceleftbig
+E/bracerightbig
+foranyprobabilitydistribution Pr/braceleftbig
+U=u/bracerightbig
+on{0, 1}k.Hence,
+thepropositionnowfollowsfrom(19).
+In order to establish our main result in SectionV, we also
+needtoconsideraslightlydifferentencodinganddecoding sce-
+nario. In this variation,the encoder E′is no longer determinis-
+tic. Rather, it has access to random bits and selects VAcat ran-
+dom, according to some fixed (but otherwise arbitrary) proba -
+bilitydistributionon {0, 1}n−k.Inallotherrespects, E′isiden-
+ticalto theencoder EforCn(A)describedabove.Thespecific
+realization vAcofVAcisrevealedtothedecoderbyagenie.The
+decoder uses successive cancellation, with the suitable mo difi-
+cation: for i∈Ac, the value of /hatwideviis not set to zero, but rather
+tothecorrespondingcoordinateintherealization vAc(revealed
+by the genie). Let P′
+edenote the probability of block error in
+thisscenario. It is shown in [3,SectionVI] that Theorem2 st ill
+appliesinthiscase,namely
+P′
+e/lessorequalslant∑
+i∈AZ(Wi) (24)
+The coding scheme presented in the next section relies cru-
+ciallyononemoreresultfromtheliteratureonpolarcodes. The
+followinglemmawasprovedbyKoradain[18,Lemma4.7].
+Lemma4. LetWandW∗beBSM channelssuchthat Wisde-
+graded with respect to W∗. For n=2m, let W1,W2, . . . , Wn
+andW∗
+1,W∗
+2, . . . , W∗
+ndenotethe ncorrespondingbit-channels.
+Then Wiisdegradedwithrespectto W∗
+iforall i=1, 2, . . . , n,
+andtherefore C(Wi)/lessorequalslantC(W∗
+i)andZ(Wi)/greaterorequalslantZ(W∗
+i).
+It followsimmediatelyfromLemma4and(13)that if Wis de-
+gradedwithrespectto W∗,thenthesetofgoodbit-channelsfor
+Wisasubsetofthesetofgoodbit-channelsfor W∗. Morepre-
+cisely,wehave Gn(W,β)⊆ G n(W∗,β)forall constants β.6 MAHDAVIFAR AND VARDY: ACHIEVINGTHESECRECYCAPACITYOFWIRETAPCHANNELSUSING P OLARCODES
+IV. THECODINGSCHEME
+We consider a special case of the wiretap-channel system of
+Figure1,whereinboththemainchannel C1=/a\}bracketle{t{0, 1},Y,W∗/a\}bracketri}ht
+and Eve’swiretap channel C2=/a\}bracketle{t{0, 1},Z,W/a\}bracketri}htare symmetric
+DMCs, and C2is degraded with respect to C1. The proposed
+codingschemeis illustratedinformallybelow:
+..
+bit−channels bad for
+both Bob and EveBob but bad for Eveboth Bob and Eve
+bit−channels good forbit−channels good for
+.PSfragreplacements
+U
+X
+Y
+Z
+/hatwideU
+C1
+C2random
+bits
+zerosinformation
+bits
+row-permutedversionof G⊗mX1
+X2
+Xn(25)
+The generalidea is to transmitinformation onlyoverthose bit-
+channelsthatarebadforEve,whilefloodingthosebitchanne ls
+thatare goodforEvewith randombits. Formally,we fixa pos-
+itiveconstant β<1/2anddefinethreesubsetsof [n]asfollows:
+Rdef=Gn(W,β) (26)
+Adef=Gn(W∗,β)\Gn(W,β) (27)
+Bdef=Bn(W∗,β) (28)
+Noticethatthesets R,A,Baredisjointand R∪A∪B = [n].
+Thisisso since Gn(W∗,β)andBn(W∗,β)arecomplementsof
+each other by definition,and Gn(W,β)⊆ G n(W∗,β)by Lem-
+ma4. Let |R|=rand|A|=k. We are now ready to describe
+theproposedencodinganddecodingalgorithms.
+EncodingAlgorithm: Formally,theencoderisa function
+E:{0, 1}k×{0, 1}r→ { 0, 1}n.Itacceptsasinputames-
+sage u∈{0, 1}kandavector e∈{0, 1}r.Wemakenoas-
+sumptions about uat this point, but we assume that eis
+selectedbyAliceuniformlyat randomfrom {0, 1}r.The
+encoder first constructsthe vector v∈{0, 1}n, by setting
+vR=e,vA=u, and vB=0. The encoderthen outputs
+E(u,e):=vGn=vPnG⊗masin(5).
+DecodingAlgorithm: Formally,thedecoderisafunction
+D:Yn→ {0, 1}k.Itacceptsasinputavector y∈Ynat
+the output of the main channel C1=/a\}bracketle{t{0, 1},Y,W∗/a\}bracketri}ht. It
+theninvokessuccessivecancellationdecodingforthepo-
+lar code Cn(A∪R), used over W∗, to producethe vec-
+tor/hatwidev∈{0, 1}n.Thedecoderoutputs D(y):=/hatwidevA.
+Wedefertheproofthatthiscodingschemesatisfiestherelia bil-
+ity and security conditions(1), (2) to the next section. The rest
+ofthissectionisdevotedtotworemarksaboutourconstruct ion.
+Remark. We point out that our encoding algorithm can be re-
+gardedasaspecialcaseofthecoset-codingmethoddescribe din
+SectionI-A. Recall that the coset-codingscheme is basedup on
+an outer code C∗that provides error-correction on the main
+channelandaninnercode C⊂C∗thatensuressecurityforthe
+wiretap channel. In our encoding algorithm, the outer code i s
+C∗=Cn(A∪R)andtheinnercodeis C=Cn(R)./squareRemark. Giventhechannelpolarizationphenomenon,it’sintu-
+itivelyclearfrom(25)whytheproposedcodingschemeshoul d
+work. The information bits U= (U1,U2, . . . , Uk)reach Bob
+via good (almost noiseless) bit-channels. Thus Bob should b e
+abletoreconstructthemwithveryhighprobability.Ontheo ther
+hand,thesesamebitspassthroughbad(almostpure-noise)b it-
+channelsontheirwaytoEve.ThusEveshouldnotbeabletode-
+duce much information about Ufrom her observations Z, and
+H(U|Z)shouldbecloseto H(U).
+However,thissimpleintuitionismisleading,becauseit do es
+notshowhowtherandombitsin(25)helpkeepEveignorant.It
+mayappearthatthisrandomnessisnotreallyneeded.Forexa m-
+ple,whatwouldhappenifthevector ethatservesasthesecond
+input to our encoder function E(·,·)is not chosen at random
+from{0, 1}rbut rather set to an a priori fixed value? Since the
+channelsaresymmetric,anyfixedvalueisasgoodasanyother ,
+so we may as well assume e=0. This does not seem to affect
+the argumentin the foregoingparagraphand, accordingto th is
+argument, H(U|Z)wouldstill becloseto H(U).
+In fact, this is not true. The reason is that channels seen by
+individual input bits as they undergo the transformation in (5)
+depend on the distribution of other input bits . Specifically, if
+e=0oreisfixed,theresultingencoderwill notbesecure.This
+isanimportantpointthatwewouldliketoestablishrigorou sly.
+Todoso,we first needthefollowingsimplelemma.
+Lemma5. LetSbeanarbitrarysubsetof [n]ofsize k,andsup-
+pose that the polar code Cn(S)is used to communicate over
+a BSM channel /a\}bracketle{t{0, 1},Z,W/a\}bracketri}ht. Further, assume that the mes-
+sage Uat the inputto the encoderfor Cn(S)is uniformlyran-
+dom over {0, 1}k, and let Zdenote the random vector at the
+channeloutput.Then I(U;Z)/greaterorequalslantkC(W).
+Proof.In fact, the lemma is true not only for Cn(S)but for
+anybinarylinearcodeofdimension k.Theonlypropertyofthe
+transformmatrix Gnthatwe needisthatit isnonsingular.
+LetVbetherandomvectorobtainedbysetting VS=Uand
+VSc=0.Thenthecodewordtransmittedoverthechannelis
+X=VGn=UM (29)
+where Mis ak×nrow submatrix of Gn. Since Gnis nonsin-
+gular, rank(M) =kandthereexistsasubset Tof[n]ofsize k
+such that the corresponding kcolumns of Mare linearly inde-
+pendent.Thisimpliesthatthereisaone-to-onecorrespond ence
+between UandXT.Hence, I(U;Z) =I(XT;Z).Furthermore,
+since the random vector Uis uniform over {0, 1}k, so is the
+vector XT.Equivalently,itscomponents {Xi:i∈ T }arei.i.d.
+Ber(1/2)randomvariables,andwecanfurtherconcludethat
+I(XT;Z)/greaterorequalslantI(XT;ZT) =∑
+i∈TI(Xi;Zi) = kC(W)
+wherethelasttwoequalitiesfollowfromthefactthatthech an-
+nel/a\}bracketle{t{0, 1},Z,W/a\}bracketri}htismemorylessandsymmetric.
+Nowsupposethattheinputtoourencoderfunction E(·,·)is
+amessagechosenuniformlyatrandomfrom {0, 1}kalongwith
+e=0.ThisisaspecialcaseofthesituationconsideredinLem-
+ma5, with the set Sgiven by (27). Hence I(U;Z)/greaterorequalslantkC(W),
+andsecuritycondition(2)cannotbesatisfied:asignificant frac-
+tionofmessagebits,at least C(W),ispotentiallyexposed. /squareToappear in the IEEE T RANSACTIONS ON INFORMATION THEORY, 2011 7
+Wenotethattheforegoingremarkillustratesa generalresult .
+Itisknownthat(2)cannotbesatisfiedunlesstheencodermak es
+useofatleast I(X;Z)randombits,where I(X;Z)isthemutual
+informationbetweentheinputandoutputofEve’schannel[2 6].
+V. WEAKSECURITY
+Inthissection,weprovethatthecodingschemeoftheprevio us
+sectionsatisfiesthereliabilityandsecurityconditions( 1)and(2)
+whileitsrate k/napproachesthesecrecycapacity.
+The reliability of this coding scheme follows immediately
+from Proposition3. Let /hatwideVdenote the randomvector at the out-
+put of the successive cancellation decoder for Cn(A∪R)in-
+vokedbyourdecodingalgorithm,andnotethatthecorrespon d-
+ing probability of block error is upper bounded by (16). Sinc e
+/hatwideU=/hatwideVAandA∪R=Gn(W∗,β)bydesign,we seethat
+Pr/braceleftbig/hatwideU/\e}atio\slash=U/bracerightbig/lessorequalslant∑
+i∈A∪RZ(W∗
+i)/lessorequalslant2−nβ(30)
+Since n/greaterorequalslantk,thisclearlyimpliesthat lim k→∞Pr/braceleftbig/hatwideU/\e}atio\slash=U/bracerightbig=0
+asrequiredin (1),andthereliabilityconditionissatisfie d.
+We now turn to the proof of security. For the remainder of
+this section, the sets R,A,Bare given by (26)–(28), Ude-
+notes Alice’s message, Vdenotes the intermediate vector con-
+structedbyourencodingalgorithm(with VA=U,VB=0,and
+VRuniform over {0, 1}r), and Zdenotes Eve’s observations.
+Also|A|=k,|R|=r,and h2(·)isthebinaryentropyfunction.
+Lemma6.
+H/parenleftbig
+VR|Z,VA/parenrightbig/lessorequalslanth2/parenleftbig
+2−nβ/parenrightbig+r2−nβ
+Proof.Suppose that in addition to her observations Z, a ge-
+nie revealsto Evethe realization vAofVAandasks her to pro-
+duceanestimateof VR.Evealsoknowsthat VB=0.Thusshe
+knowsallthebitsof VRc.SinceR=Gn(W,β),thisisprecise-
+ly the scenario considered in (24). Consequently, Eve can us e
+successive cancellation decoding to deterministically co mpute
+anestimate /hatwideVR=f(Z,VA)suchthat
+λdef=Pr/braceleftbig/hatwideVR/\e}atio\slash=VR/bracerightbig/lessorequalslant∑
+i∈RZ(Wi)/lessorequalslant2−nβ(31)
+We now invoke Fano’s inequality [10, p.38] to bound the con-
+ditionalentropy H(VR|Z,VA)in termsof λasfollows
+H/parenleftbig
+VR|Z,VA/parenrightbig/lessorequalslanth2(λ) + rλ (32)
+wherewehavealsousedthefactthat VRtakesvaluesintheset
+{0, 1}rofsize 2r.Thelemmanowfollowsfrom(31),(32),and
+thefact that h2(·)isincreasingonthe interval [0,1/2].
+Let us define ǫndef=C(W)−|R| /n. SinceR=Gn(W,β),
+Theorem1impliesthat lim n→∞ǫn=0.
+Lemma7.
+I(U;Z)/lessorequalslantnǫn+h2/parenleftbig
+2−nβ/parenrightbig+(n−k)2−nβ(33)
+Proof.The lemma is proved via a long sequence of simple
+equalitiesandinequalities,asfollows:
+I(U;Z) =I(VA;Z) = I(VA∪ B ;Z) (34)
+=I(V;Z)−I(VR;Z|VA∪B)(35)=I(V;Z)−I(VR;Z|VA) (36)
+=I(V;Z)−H(VR|VA)+H(VR|Z,VA)(37)
+=I(V;Z)−H(VR)+H(VR|Z,VA)(38)
+=I(V;Z)−r+H(VR|Z,VA) (39)
+/lessorequalslantnC(W)−r+H(VR|Z,VA) (40)
+=nǫn+H(VR|Z,VA) (41)
+/lessorequalslantnǫn+h2/parenleftbig
+2−nβ/parenrightbig+r2−nβ(42)
+/lessorequalslantnǫn+h2/parenleftbig
+2−nβ/parenrightbig+(n−k)2−nβ(43)
+Theequalitiesin(34)holdsince VA=UandVB=0.Nowob-
+serve that A∪BandRare complementsof each other in [n].
+Hence,anydistributionof Vcanbe thoughtofas a joint distri-
+bution of VRandVA∪ B. Given this observation, (35) follows
+fromthechainruleformutualinformation.Theequalityin( 36)
+is trivial from VB=0, while (37) is the definition of conditio-
+nalmutualinformation.Theequalities(38)and(39)holdsi nce
+VRandVAare independent, and VRis a priori uniform over
+{0, 1}r.Inequality(40)isimmediatefromthefactthat C(W)is
+thecapacityof W,while(41)followsfromthedefinitionof ǫn.
+Finally,(42)followsfromLemma6and(43) istrivial.
+Theorem8. Theencodingalgorithmoftheprevioussectionsat-
+isfiestheweaksecuritycondition (2),namely
+lim
+k→∞I(U;Z)
+k=0
+Proof.ThisfollowsimmediatelyfromLemma7.Divideboth
+sidesof(33) by nto get
+I(U;Z)
+n/lessorequalslantǫn+h/parenleftbig
+2−nβ/parenrightbig
+n+n−k
+n2−nβ(44)
+Itisclearthatthelasttwotermsin(44)tendtozeroas n→∞,
+andlim n→∞ǫn=0byTheorem1.Alongwiththeobviousfact
+thatk=Ω(n),thiscompletestheproofofthetheorem.
+Recallthatforthewiretap-channelsystemsconsideredint his
+paper, the secrecy capacity is Cs=C(W∗)−C(W)(cf. Sec-
+tionII). Let Rn=k/ndenote the rate of our coding scheme.
+Theorem9.
+lim
+n→∞Rn=C(W∗)−C(W)
+Proof.Observethat
+Rn=|A|
+n=|Gn(W∗,β)|
+n−|Gn(W,β)|
+n(45)
+wherewehaveusedthedefinitionof Ain(27)andthefactthat
+Gn(W,β)isasubsetof Gn(W∗,β)forall β<1/2.Thetheorem
+nowfollowsfromTheorem1.
+Theorem9 does not directly imply that our coding scheme
+achieves secrecy capacity, because the rate of communicati on
+fromAlicetoBob,measuredininformationbitsperchannelu se,
+could be much less than k/nwhen H(U)<k. But this is true
+foranyencoderthatconverts kinputbitsto ncodedoutputbits.
+Iftheencoderaccommodatesanarbitrarydistributiononit sin-
+putU,itcanachievecapacity onlywhen H(U) =k/parenleftbig
+1−o(1)/parenrightbig
+.
+If this necessary conditionis satisfied, then our coding sch eme
+doesachievesecrecycapacitybyTheorem9.8 MAHDAVIFAR AND VARDY: ACHIEVINGTHESECRECYCAPACITYOFWIRETAPCHANNELSUSING P OLARCODES
+VI. STRONGSECURITY
+This section shows how polar coding could be used to provide
+strong security whenever the main channel C1and the wiretap
+channel C2are symmetric binary-input DMCs, and C2is de-
+gradedwithrespectto C1.Specifically,wedescribeapolarcod-
+ingschemethatsatisfiesthestrongsecuritycondition(3), while
+operatingat a rate k/nthatapproachesthesecrecycapacity.
+First, we will introducea subtle but important change in the
+coding scheme of SectionIV (see SectionVI-B). In order to
+show that this change suffices to guarantee strong security, we
+need to replace the proof in the previoussection by a more in-
+tricate argument (see SectionVI-D). This argument relies c ru-
+cially on the fact that a certain composite channel induced b y
+ourconstructionissymmetric(SectionVI-C).Aidedbyarec ent
+result ofHassani andUrbanke[15],we thenprovethat the rat e
+oftheproposedcodingschemeapproachesthesecrecycapaci ty
+(SectionVI-E). Unfortunately, we can show that the reliabi lity
+condition(1)issatisfiedonlyforthecasewherethemaincha n-
+nel is noiseless. Nevertheless, we believe that, in practic e, low
+probabilitiesofblockerrorcanbeachievedalsowhenthema in
+channel is not noiseless, using a modification of Arıkan’s su c-
+cessivecancellationdecoder(SectionVI-F).
+A. Analysisofthe Weak-SecurityCodingScheme
+Henceforth,letusrefertothecodingschemeintroducedinS ec-
+tionIV and analyzed in the previoussection as the weak-secu-
+rity coding scheme . A natural question is whether this coding
+scheme does, in fact, provide strong security. Although we d o
+nothaveadefinitiveanswertothisquestion,weconjecturet hat
+it doesnot.Asbefore,let
+ǫndef=C(W)−|R|
+n=C(W)−|Gn(W,β)|
+n(46)
+with the sets RandGn(W,β)defined as in (26) and (13). The
+followingresultprovidessomeevidenceforthisconjectur e.
+Proposition10. Wheneverthewiretapchannelisabinary-input
+symmetric DMC and the main channel is noiseless, the weak-
+securitycodingschemeachievesstrongsecurityif andonly if
+limn→∞nǫn=0 (47)
+Proof.Thefactthat(47)issufficientforstrongsecurityisob-
+vious from Lemma7, since the last two terms in (33) tend to
+zero exponentiallyfast. In fact, it is clear that (47) is suf ficient
+forstrongsecurity,whetherthemainchannelisnoiselesso rnot.
+Wenowshowthatthisconditionisalsonecessary,atleastin the
+casewherethemainchannelisnoiseless.Inthiscase,these tB
+in (28) is empty, and the vector Vat the input to the transfor-
+mation(5)consistsof VA=UandVR,with VRbeinguniform
+over{0, 1}r. Hence, if the message Uis a priori uniform over
+{0, 1}k,then Visuniformover {0, 1}n. Let X=VGnasin(5).
+Since the Arıkantransformmatrix Gnis nonsingular, Xis also
+uniformover {0, 1}n. Consequently,we have
+I(V;Z) = I(X;Z) =n
+∑
+i=1I(Xi,Zi) = nC(W)(48)
+wherethesecondequalityfollowsbynotingthat X1,X2, . . . , Xn
+arei.i.d. Ber(1/2)randomvariablesand Wismemoryless,whilethe last equality follows from the fact that Wis symmetric.
+This implies that the inequality (40) in the proof of Lemma7
+becomesanequalityin thiscase. Therefore
+I(U;Z) = nǫn+H(VR|Z,VA)/greaterorequalslantnǫn(49)
+Itisnowclearthat lim n→∞nǫn=0isnecessaryforthemutual
+information I(U;Z)tovanishasymptotically.
+Givena BSM channel W, is it true that lim n→∞nǫn=0for
+thischannel?Unfortunately,theanswertothisquestionis nega-
+tive.Itisknown[33,35]thatforanydiscretememorylessch an-
+nelWandanycode of length nand rate Rthat achieveserror-
+probability PeonW, we have
+C(W)−R/greaterorequalslantconst(Pe,W)√n−O/parenleftbigglogn
+n/parenrightbigg
+where the constant (which is given explicitly in [33]) depen ds
+onWandPe, but not on n. This implies that nǫn=Ω(√n),
+and the weak-security coding scheme does notprovide strong
+security. Consequently, in order to provide strong securit y, the
+polarcodingschemeofSectionIV hasto bemodified.
+B. Strong-SecurityCodingScheme
+Intuitively,themainreasonthatthecodingschemeofSecti onIV
+fails to providestrong security is this: the bit-channelst hat are
+deemed bad for Eve are not bad enough. Indeed, according to
+the definition of Bn(W,β)in (14), a bit-channel Wiis consid-
+ered bad for Eve whenever Z(Wi)/greaterorequalslant2−nβ/n. For example, if
+n=210andβ=0.499,a bit-channelmaybe declaredbadfor
+Eveevenwhenitscapacityisgreaterthan 1−10−9whileEve’s
+probabilityoferroronthischannelislessthan 2·10−10.Itisob-
+viousthatsucha bit-channeldoesnotpreventEvefromdeduc -
+ingtheinformationatits inputwithhighprobability.
+Theproblemisthatthegenerallyaccepteddefinitionsofgoo d
+andbadbit-channels—forexample,(13)and(14)—aremoti-
+vatedbyBob’spointofview .First,acriterionfor“goodness”is
+established,motivatedbytheprobabilityoferroronBob’s side,
+thenthebit-channelsthatdonotsatisfythiscriterionare deemed
+bad.Inordertoachievestrongsecurity,wewillre-defineth ings
+fromEve’s point of view . First, we introduce a strong criterion
+for“badness,”andthenmakesurethatrandombitsaresentov er
+allthebit-channelsthatdonotsatisfythiscriterion.Spe cifically,
+given a BSM channel Wand a positive δ<1, we define the
+indexset of δ-poorbit-channels asfollows:
+Pn(W,δ)def=/braceleftBig
+i∈[n]:C(Wi)/lessorequalslantδ/bracerightBig
+(50)
+Further,weleavethedefinitionofthegoodbit-channelsin( 13)
+unchanged,butre-definethesets R,A, andBasfollows:
+Rdef=[n]\P n(W,δn) (51)
+Adef=Pn(W,δn)∩G n(W∗,β) (52)
+Bdef=Pn(W,δn)\Gn(W∗,β) (53)
+where, for the time being, δnis an arbitrary function from the
+positiveintegerstotheinterval (0, 1).Wewillspecifythisfunc-
+tionpreciselylaterinthissection(cf.Theorem17andProp osi-Toappear in the IEEE T RANSACTIONS ON INFORMATION THEORY, 2011 9
+bad for Bobgood for Bob
+poor for Eve
+for EvepoornotPSfragreplacements
+U
+X
+Y
+Z
+/hatwideU
+C1
+C2
+δn- δn-
+Pn(W,δn)XY
+BA
+Bn(W∗,β)Gn(W∗,β)
+zerosrandom
+randombitsbits
+bitsinformation
+Figure2. Informal sketch of the strong-security coding scheme
+tion20). In the meantime, notice that the sets R,A, andB, as
+definedin (51)–(53),arestill disjointand R∪A∪B = [n].
+Thestrong-securitycodingscheme of(51)–(53)isillustrat-
+edschematicallyinFigure2,whereintheboldsquarerepres ents
+the set of nbit-channels. This set is partitioned two ways: into
+bit-channelsthat aregoodforBobandbadforBob(asbefore) ,
+and into bit-channelsthat are δn-poorand not δn-poorfor Eve.
+ThesetsXandY,withX ∪Y=R,will bediscussedlaterin
+thissection(see(94),(95)andSectionVI-F).
+With the new definitions of the sets R,A,Bin (51)–(53),
+the encoding and decoding algorithms for the strong-securi ty
+coding scheme are exactly those given in SectionIV for the
+weak-securitycodingscheme(althoughwewillmodifythede -
+codingalgorithmsomewhatinSectionVI-F).
+C. TheInducedChannelIsSymmetric
+We prove in the next subsection that the strong-security cod -
+ing scheme indeed provides strong security. In order to do so ,
+we first introduce and study a certain composite channel in-
+duced by our construction. Informally, this channel descri bes
+the transformation in (5) in the case where some rof the bits
+V1,V2, . . . , Vnare set independentlyand uniformlyat random,
+while the remaining n−rbits serve as the input to the chan-
+nel.ThissituationisdepictedinFigure3.Formally,the induced
+channelQn(W,R)is specified in terms of an arbitrary BSM
+channel Wwithoutputalphabet Z,andasubset Rof[n]ofsize
+|R|=r. Theinputalphabetof Qn(W,R)is{0, 1}n−randits
+output alphabet is Zn. To describe the transition probabilities
+ofQn(W,R), let us introduce the following notation. Hence-
+forth,givena vector x∈{0, 1}n−randa vector e∈{0, 1}r, let
+(x;e)denotethe vector v∈{0, 1}nwith vR=eandvRc=x.
+Withthis,referringtothedefinitionof Wnin(9),the 2n−r×|Z|n
+transition-probabilitymatrix QofQn(W,R)isgivenby
+Q(z|x)def=1
+2r∑
+e∈{0,1}rWn/parenleftbig
+z/vextendsingle/vextendsingle(x;e)Gn/parenrightbig
+(54)
+forall x∈{0, 1}n−randall z∈Zn. It canbe readilyseen that
+ifV=(V1,V2, . . . , Vn−r)is the random vector at the input tothechannelinFigure3and Z=(Z1,Z2, . . . , Zn)istherandom
+vectoratthe channeloutput,thenindeed
+Q(z|x) = Pr/braceleftbig
+Z=z/vextendsingle/vextendsingleV=x/bracerightbig
+(55)
+Our main goal in this subsection is to prove that the induced
+channelQn(W,R)inFigure3issymmetric.
+In order to do so, we start with a general result that uses el-
+ementarytoolsfromgrouptheorytoprovideasufficientcond i-
+tionforagivenchanneltobesymmetric.Recallthata groupac-
+tionofanabeliangroup Aonaset Yisafunctionfrom A×Y
+toY, denoted (a,y)/ma√sto→a.y, withthe followingproperties:
+P1.0.y=yforall y∈Y,where 0istheidentityof A;
+P2.(a+b).y=a.(b.y)forall a,b∈Aandall y∈Y,
+where+denotesthegroupoperation.
+The orbit of y∈Yis the set of all pointsof Yto which ycan
+bemovedbytheelementsof A. Explicitly,the orbit of yis
+O(y)def=/braceleftbig
+a.y:a∈A/bracerightbig
+(56)
+Itiswellknownthatorbitsofpointsin Yformapartitionof Y
+intoequivalenceclasses(undertheequivalencerelation y1∼y2
+iffthereexistsan a∈Awith y2=a.y1).
+Theorem11. Let/angbracketleftbig
+X,Y,W/angbracketrightbig
+bea DMC,andsupposethat X
+is anabeliangroupunderthe binaryoperation +. Further,sup-
+posethatthereexistsa groupaction .ofXonYsuchthat
+W(y|a+x) = W(a.y|x) (57)
+forall a,x∈Xandall y∈Y. Thenthe DMC/angbracketleftbig
+X,Y,W/angbracketrightbig
+is
+necessarilya symmetricchannel.
+Proof.Wepartitiontheset Yintoorbitsformedbythegroup
+action .ofX. LetObe an orbit, and let Mbe the|X|×|O|
+column submatrix of Wconsisting of those columns that are
+indexedbythe elementsof O. It wouldsuffice toprovethat M
+isa stronglysymmetricmatrix,regardlessofthechoiceof O.
+Consider two arbitrary rows of Mindexed, say, by the ele-
+ments x1∈Xandx2∈X. Set a=x2+(−x1),where−x1is
+the inverse of x1in the group X. Then we have x2=a+x1.
+Therefore,(57) impliesthat
+W(y|x2) = W(a.y|x1)forall y∈Y(58)
+Itiseasytoseefrom(56)andpropertyP2thatthemap y/ma√sto→a.y
+is bijective on O. Together with (58), this shows that the rows
+ofM,indexedby x1andx2, arepermutationsofeachother.
+Nowconsidertwoarbitrarycolumnsof Mindexedbytheel-
+ements y1∈Oandy2∈O. Then,by the definitionofan orbit,
+thereexistsan a∈Xsuchthat y2=a.y1,and(57)impliesthat
+W(y2|x) = W(y1|a+x)forall x∈X(59)
+Itisclearthatthemap x/ma√sto→a+xisbijectiveon X.Thus(59)
+showsthatthecolumnsof Marepermutationsofeachother.
+Notice that the inputalphabet {0, 1}n−rofQn(W,R)is an
+abelian group, with the group operation +being the compo-
+nentwise modulo 2addition of vectorsin Fn−r
+2. Consequently,
+in order to prove that the channel Qn(W,R)is symmetric, it
+would suffice to construct a group action of {0, 1}n−ronZn10 MAHDAVIFAR AND VARDY: ACHIEVINGTHESECRECYCAPACITYOFWIRETAPCHANNELSUSING P OLARCODES
+..
+......
+.PSfragreplacements
+U
+X
+Y
+Z
+/hatwideU
+C1
+C2
+δn-
+δn-
+Pn(W,δn)
+X
+Y
+B
+A
+Bn(W∗,β)
+Gn(W∗,β)
+zeros
+random
+bits
+bits
+bits
+information
+V1
+V2
+Vn−rZ1
+Z2
+ZnX1
+X2
+XnWWW
+Gn=PnG⊗mrandom bits
+
+R
+Figure3. Blockdiagram of the induced channel Qn(W,R)
+thatsatisfies (57).To doso, we will usethe factthat Witself is
+symmetric. As noted by Arıkan [3], a binary-input channel W
+with outputalphabet Zis symmetricif andonlyif thereexists
+a permutation π1onZsuchthat
+π1=π−1
+1(π1isaninvolution) (60)
+W/parenleftbig
+z|0/parenrightbig=W/parenleftbig
+π1(z)|1/parenrightbig
+forall z∈Z(61)
+Letπ0betheidentitypermutationon Z.FollowingArıkan[3],
+letusdefineagroupactionoftheadditivegroupof F2={0, 1}
+ontheset Zasfollows: x.z=πx(z)forall x∈F2andz∈Z.
+Itistrivialtoverifythat x.zhastherequiredgroup-actionprop-
+ertiesP1andP2,andthat forall a,x∈F2andz∈Z, wehave
+W(z|a+x) = W(a.z|x) (62)
+Asin[3],wecanextendthisfunctioncomponentwisetoagrou p
+actionoftheadditivegroupof Fn
+2ontheset Znasfollows:
+x.zdef= (x1.z1,x2.z2, . . . , xn.zn) (63)
+ThefollowinglemmawasprovedbyArıkanin[3,Propositions
+12and13].Weprovideasimpleproofherein,forcompletenes s.
+Lemma12. LetGn=PnG⊗mbetheArıkantransformmatrix.
+Thenforall a,x∈Fn
+2andall z∈Zn,we have
+Wn/parenleftbig
+z|(a+x)Gn/parenrightbig=Wn/parenleftbig
+aGn.z|xGn/parenrightbig
+(64)
+Proof.Infact,thelemmaistrueforanarbitrary n×nbinary
+matrix(inotherwords,anylineartransformationfrom Fn
+2toit-
+self).First, let usshowthat
+Wn(z|b+c) = Wn(b.z|c) (65)
+for all b,c∈Fn
+2and all z∈Zn. This followsdirectly from the
+definitionof Wn. Indeed,expandingbothsidesof(65) as
+n
+∏
+i=1W(zi|bi+ci) =n
+∏
+i=1W(bi.zi|ci)
+weconcludethat(65)isimpliedby(62).Letusnowset b=aGn
+andc=xGn.Thenthelemmafollowsfrom(65)alongwiththe
+fact that multiplicationby a matrix is a linear operation,t hat is
+(a+x)Gn=aGn+xGn=b+c.WenowdepartfromArıkan[3],andintroduceagroupaction
+◦oftheadditivegroupof Fn−r
+2onZn,definedasfollows.Re-
+call that(x,e)denotesthe vector v∈{0, 1}nwith vR=eand
+vRc=x.With this,we defineforall x∈Fn−r
+2andall z∈Zn
+x◦zdef= (x;0)Gn.z (66)
+wherethegroupaction .ontheright-handsideistheonedefin-
+edin(63).Again,itiseasytoverifythat(66)satisfiesP1an dP2.
+Ourmainresult inthissubsectionisthefollowingproposit ion.
+Proposition13. Theinducedchannel Qn(W,R)issymmetric.
+Proof.In light of Theorem11, it would suffice to prove that
+thetransition-probabilitymatrix Qdefinedin (54) satisfies
+Q(z|a+x) = Q(a◦z|x)
+forall a,x∈Fn−r
+2andz∈Zn. Expandingthevector (a+x;e)
+as(a;0)+( x;e),andsubstitutingin(54),we obtain
+Q(z|a+x) =1
+2r∑
+e∈{0,1}rWn/parenleftbig
+z/vextendsingle/vextendsingle/parenleftbig(a;0)+( x;e)/parenrightbig
+Gn/parenrightbig
+(67)
+=1
+2r∑
+e∈{0,1}rWn/parenleftbig(a;0)Gn.z/vextendsingle/vextendsingle(x;e)Gn/parenrightbig
+(68)
+=1
+2r∑
+e∈{0,1}rWn/parenleftbig
+a◦z/vextendsingle/vextendsingle(x;e)Gn/parenrightbig
+(69)
+where(68)followsfromLemma12and(69)followsfrom(66).
+But(69) isprecisely Q(a◦z|x)by(54),andwe aredone.
+D. ProofofStrongSecurity
+Letusbeginwithasimplelemmathatrelatesthecapacityoft he
+bit-channelsin (11) to the transformationin (5). Although this
+lemmaiswell-known,weincludea proofforcompleteness.
+Lemma14. LetWbe an arbitrary BSM channel. Suppose the
+vector V= (V1,V2, . . . , Vn)at the input to the transformation
+in(5)isuniformover {0, 1}n,andlet Z=(Z1,Z2, . . . , Zn)be
+the random vector at the output of the transformation.Furth er,
+letW1,W2, . . . , Wnbe the correspondingbit-channels,defined
+in(11).Thenforall i∈[n], thecapacityof Wiisgivenby
+C(Wi) = I(Vi;Z,V1,V2, . . . , Vi−1)
+Proof.LetVidenote the random vector (V1,V2, . . . , Vi)for
+alli∈[n], as before. It is shown in [3] that if Wis symmetric,
+then so is Wifor all i∈[n]. Hence, the capacity of Wiis the
+mutualinformationbetweenitsinputandoutputwhentheinp ut
+isuniformover {0, 1}.Thusitwouldsufficetoshowthat
+Wi/parenleftbig
+z,v|x) = Pr/braceleftbig
+Z=z,Vi−1=v/vextendsingle/vextendsingleVi=x/bracerightbig
+(70)
+forall z∈Zn,v∈{0, 1}i−1,and x∈{0, 1}.Since Viisuniform
+wecanre-writetheright-handsideof(70)asfollows:
+Pr/braceleftbig
+Z=z,Vi−1=v,Vi=x/bracerightbig
+Pr/braceleftbig
+Vi=x/bracerightbig =2 Pr/braceleftbig
+Z=z,Vi=(v,x)/bracerightbig
+Observe that the event/braceleftbig
+Vi=(v,x)/bracerightbig
+is the union of 2n−idis-
+joint events/braceleftbig
+V=(v,x,v)/bracerightbig
+, asvranges over {0, 1}n−i(orvToappear in the IEEE T RANSACTIONS ON INFORMATION THEORY, 2011 11
+is the emptystring, if i=n). Since Vis uniformover {0, 1}n,
+theprobabilityofeachsucheventis 2−n. Consequently
+2 Pr/braceleftbig
+Z=z,Vi=(v,x)/bracerightbig=
+=2∑
+v∈{0,1}n−iPr/braceleftbig
+Z=z,V=(v,x,v)/bracerightbig
+(71)
+=1
+2n−1∑
+v∈{0,1}n−iPr/braceleftbig
+Z=z/vextendsingle/vextendsingleV=(v,x,v)/bracerightbig
+(72)
+Since ZandVare, respectively,the outputand the inputto the
+transformationin(5),forall w∈ {0, 1}nwe have
+Pr/braceleftbig
+Z=z/vextendsingle/vextendsingleV=w/bracerightbig=Wn/parenleftbig
+z/vextendsingle/vextendsinglewPnG⊗m/parenrightbig=/tildewideW(z|w)
+bythedefinitionofthe“combined”channel /tildewideWin(10).Together
+with (72) and the definition of Wiin (11), this shows that the
+right-handsideof(70)isindeedequalto Wi/parenleftbig
+z,v|x).
+ThenextlemmacombinesProposition13withLemma14to
+upper-boundthe capacityoftheinducedchannel Qn(W,R).
+Lemma15. LetWbeanarbitraryBSM channel. For n=2m,
+letW1,W2, . . . , Wndenotethecorrespondingbit-channels.Then
+forallR ⊂[n],thecapacityoftheinducedchannel Qn(W,R),
+definedin (54), isupper-boundedasfollows:
+C/parenleftbigQn(W,R)/parenrightbig/lessorequalslant∑
+i∈RcC(Wi) (73)
+Proof.Consideragainthetransformationin(5),withtheinput
+vector V=(V1,V2, . . . , Vn)beinguniformover {0, 1}n,andlet
+Z= (Z1,Z2, . . . , Zn)denote the output of the transformation,
+asinLemma14.Then
+C/parenleftbigQn(W,R)/parenrightbig=I(VRc;Z) (74)
+This is so because Qn(W,R)is symmetric by Proposition13,
+and the capacity of a symmetric DMC is given by the mutual
+information between its input and output, under uniform input
+distribution [13,Theorem4.5.2].Next,write Rc= [n]\Ras
+Rc=/braceleftbig
+i1,i2, . . . , in−r/bracerightbig
+where r=|R|,andassumew.l.o.g.that i1<i2<···<in−r.
+Thelemma canbe nowprovedvia a sequenceofsimple equal-
+itiesandinequalities,asfollows:
+I(VRc;Z) = I/parenleftbig
+Vi1,Vi1, . . . , Vin−r;Z/parenrightbig
+(75)
+=n−r
+∑
+j=1I/parenleftbig
+Vij;Z/vextendsingle/vextendsingleVi1,Vi2, . . . , Vij−1/parenrightbig
+(76)
+=n−r
+∑
+j=1I/parenleftbig
+Vij;Z,Vi1,Vi2, . . . , Vij−1/parenrightbig
+(77)
+/lessorequalslantn−r
+∑
+j=1I/parenleftbig
+Vij;Z,V1,V2, . . . , Vij−1/parenrightbig
+(78)
+The equality (76) is the chain rule for mutual information.T he
+equality(77)followsfromthefactthat I(X;Z|Y)=I(X;Z,Y)
+forallrandomvariables X,Y,Zsuchthat XandYareindepen-
+dent.Toestablish(78),weadjointothesetofrandomvariab les
+{Vi1,Vi2, . . . , Vij−1}itscomplementintheset {V1,V2, . . . , Vij}.Clearly,thiscannotdecreasethemutualinformation.Last ly,we
+observe that the summation in (78) is equal to the summation
+ontheright-handside of(73) byLemma14.
+WithLemma15inhand,wearefinallyreadytoestablishour
+mainresultinthissection.
+Proposition16. LetUbe the message at the input to the enco-
+der for the strong-security coding scheme. Then, regardles s of
+thea prioridistributionof U,we have
+I(U;Z)/lessorequalslantδn|Pn(W,δn)| (79)
+where Zistheoutputofthewiretapchannel,and Pn(W,δn)is
+theindexset of δn-poorbit-channels,asdefinedin (50).
+Proof.Recall that our encoder first constructs the vector V
+with VA=U,VB=0,and VRuniformover {0, 1}r.Hence
+I(U;Z) = I(VA;Z) = I(VA∪ B ;Z)
+Since the sets A,B,Rpartition[n], the vector VA∪Bcan be
+regardedastheinputtotheinducedchannel Qn(W,R).More-
+over,since Rc=Pn(W,δn)by(51),Lemma15impliesthat
+I(VA∪B ;Z)/lessorequalslantC/parenleftbigQn(W,R)/parenrightbig/lessorequalslant∑
+i∈P n(W,δn)C(Wi)
+Thepropositionnowfollowsbyobservingthat C(Wi)/lessorequalslantδnfor
+alli∈P n(W,δn),bythedefinitionof Pn(W,δn)in(50).
+Note that we are still free to specify the function δnin (51)
+and (79). This means that the security of our coding scheme
+istunable. Let us henceforth refer to δnas thesecurity func-
+tion; thisfunctionis a designparameter in ourscheme.Propo-
+sition16impliesthatchoosingdifferentsettingsforthes ecurity
+functionguaranteesdifferentlevelsofsecurity.
+Theorem17. Foranysecurityfunctionsuchthat δn=o(1/n),
+thestrong-securitycodingschemeguaranteesstrongsecur ity.
+Proof.FollowsfromProposition16,alongwiththedefinition
+ofstrongsecurityin(3) andthefact that |Pn(W,δn)|/lessorequalslantn.
+In fact, we shall see in the next subsection (cf. Theorem21)
+thatwecanachievethesecrecycapacity,whilesettingthes ecu-
+rity function to be as small as δn=2−nβfor any positive con-
+stant β<1/2.Inthiscase,ourcodingschemeguaranteesthatthe
+mutualinformationbetweenthemessage UandEve’sobserva-
+tions Zscalesroughlyas
+I(U;Z) = o/parenleftBig
+2−√n/nε/parenrightBig
+(80)
+forany ε>0.Notethatthisholdsregardlessofthea prioridis-
+tributionofthemessage.
+E. RateoftheStrong-SecurityCodingScheme
+LetRn=k/n=|A|/ndenote the rate of the strong-security
+codingscheme,where Ais theset definedin(52).Notethat
+A=Pn(W,δn)\B n(W∗,β)
+sincethesets Gn(W∗,β)andBn(W∗,β)ofgoodandbadchan-
+nelsin(13),(14)arecomplementsofeachother.Itfollowst hat
+Rn/greaterorequalslant|Pn(W,δn)|
+n−|Bn(W∗,β)|
+n(81)12 MAHDAVIFAR AND VARDY: ACHIEVINGTHESECRECYCAPACITYOFWIRETAPCHANNELSUSING P OLARCODES
+Theasymptoticbehaviorofthefraction |Bn(W∗,β)|/nisgiven
+by Theorem1. Therefore, in order to prove that the rate of the
+strong-security coding scheme approaches secrecy capacit y, it
+remainsto analyzetheasymptoticbehaviorof |Pn(W,δn)|/n.
+In this regard, a recent result of Hassani and Urbanke [15]
+willbeuseful.Todescribethisresult,weneedtointroduce some
+notation.Givena BSM channel Wandapositive γ<1,let
+P′
+n(W,γ)def=/braceleftBig
+i∈[n]:Z(Wi)/greaterorequalslant1−γ/bracerightBig
+(82)
+Thisissimilarto thedefinitionoftheset Pn(W,δ)in(50),ex-
+ceptthat(82)usestheBhattacharyyaparameters Z(Wi)instead
+ofthechannelcapacities C(Wi).Givenapositiveinteger mand
+a real number ξin the open interval (0, 1),leta=a(m,ξ)de-
+notetheuniquepositiveintegersuchthat
+m
+∑
+i=a/parenleftbiggm
+i/parenrightbigg
+/lessorequalslantξ2m<m
+∑
+i=a−1/parenleftbiggm
+i/parenrightbigg
+(83)
+anddefine α(m,ξ)def=a(m,ξ)/m.Further,we saythat a func-
+tion f(m)from the positive integers to the interval (0, 1)isin
+theintersectionof o/parenleftbig
+1/√m/parenrightbig
+andω(1/m)if
+limm→∞/parenleftbig√
+m f(m)/parenrightbig=0and limm→∞/parenleftbig
+m f(m)/parenrightbig=∞
+Thefollowingis(thesecondpartof)Theorem3ofHassaniand
+Urbanke [15]. Although this result is more general than what
+we need,westate it belowexactlyasin[15,Theorem3].
+Theorem18. LetWbeaBSMchannel,andlet ξ<1beaposi-
+tiveconstant.Fix anarbitraryfunction f(m)intheintersection
+ofo/parenleftbig
+1/√m/parenrightbig
+andω(1/m),andforall n=2mdefine
+γndef=2−nα(m,ξ)(1+f(m))(84)
+where α(m,ξ)isthefunctiondefinedin (83).Thentheasymp-
+toticbehaviorofthefraction |P′n(W,γn)|/nisgivenby
+limn→∞|P′n(W,γn)|
+n=ξ/parenleftbig
+1−C(W)/parenrightbig
+(85)
+InCorollary19andProposition20,wespecializethegenera l
+resultofTheorem18to ourneeds.
+Corollary19. LetWbe an arbitrary BSM channel. Then for
+anypositiveconstant β<1/2we have
+limn→∞/vextendsingle/vextendsingleP′
+n/parenleftbig
+W, 2−nβ/parenrightbig/vextendsingle/vextendsingle
+n=1−C(W) (86)
+Proof.Applyingthe Stirling formulato both sides of (83), it
+canbeshown(cf.[15])that
+a(m,ξ) =m
+2+Q−1(ξ)
+2√
+m+o/parenleftbig√
+m/parenrightbig
+where Q(x)is the probability that a standard normal random
+variable will obtain a value largerthan x. Consequently,for all
+positive ξ<1,thereexistsa positiveconstant cξsuchthat
+a(m,ξ)/greaterorequalslantm
+2−cξ√
+m
+for all sufficiently large m. This implies that for any positive
+constant β<1/2, any function f(m)from the positive integerstotheinterval (0, 1),andforallsufficientlylarge m,thefollow-
+inginequalityholds
+α(m,ξ)/parenleftbig
+1+f(m)/parenrightbig>a(m,ξ)
+m/greaterorequalslant1
+2−cξ√m/greaterorequalslantβ
+This, in turn, implies that for all sufficiently large n=2m, we
+have γn<2−nβwhere γnisthefunctiondefinedin(84).There-
+fore,the set P′n(W,γn)is a subset of the set P′n(W, 2−nβ)for
+all sufficientlylarge n.
+Let us assume for a moment that the limit on the left-hand
+side of (86) exists, call it L. If so, we can concludefrom Theo-
+rem18,alongwiththefactthat P′
+n(W,γn)⊂ P′
+n(W, 2−nβ)for
+all sufficientlylarge n, that
+lim
+n→∞/vextendsingle/vextendsingleP′n/parenleftbig
+W, 2−nβ/parenrightbig/vextendsingle/vextendsingle
+n/greaterorequalslantlim
+n→∞|P′
+n(W,γn)|
+n=ξ/parenleftbig
+1−C(W)/parenrightbig
+Sincethisholdsforallpositive ξ<1,thelowestpossiblevalue
+ofLis1−C(W). Now observe that the set P′
+n(W, 2−nβ)and
+the setGn(W,β)of good channels, defined in (13), do not in-
+tersect.Therefore,forall n=2mwehave
+/vextendsingle/vextendsingleP′
+n/parenleftbig
+W, 2−nβ/parenrightbig/vextendsingle/vextendsingle
+n/lessorequalslant1−|Gn(W,β)|
+n
+TogetherwithTheorem1,thisimpliesthatthelimit Lindeedex-
+ists, andisequalto 1−C(W).
+Proposition20. LetWbe an arbitrary BSM channel, and let
+β<1/2beapositiveconstant.Further,let Pn(W,δn)bethein-
+dexsetof δn-poorbit-channels,asdefinedin (50),andsuppose
+thatthereexistpositiveconstants c1andc2suchthat
+c12−nβ/lessorequalslantδn/lessorequalslant1−c2 (87)
+forall sufficientlylarge n. Thenthe asymptoticbehaviorofthe
+fraction|Pn(W,δn)|/nisgivenby
+limn→∞|Pn(W,δn)|
+n=1−C(W) (88)
+Proof.LetW1,W2, . . . , Wndenote the bit-channels, as be-
+fore.ItwasshownbyArıkanin[3,Proposition1]that
+C(Wi)/lessorequalslant/radicalbig
+1−Z(Wi)2(89)
+for all i∈[n]. Fix a positive constant αsuch that β<α<1/2.
+Since Z(Wi)/greaterorequalslant1−2−nαforall i∈P′n(W, 2−nα)bydefinition,
+Arıkan’sbound(89) impliesthat
+C(Wi)/lessorequalslant2−(nα−1)/2(90)
+fori∈P′
+n(W, 2−nα).Forallsufficientlylarge n,theright-hand
+sideof(90)islessthantheleft-handsideof(87),andthere fore
+P′
+n(W, 2−nα)⊆ P n(W,δn) (91)
+Alsonotethatthecondition δn/lessorequalslant1−c2ontheright-handside
+of(87) impliesthatforall sufficientlylarge n, wehave
+Pn(W,δn)∩G n(W,β) =∅ (92)
+The proposition now follows by combining (91) and (92) with
+Corollary19andTheorem1, respectively.Toappear in the IEEE T RANSACTIONS ON INFORMATION THEORY, 2011 13
+We are now ready to prove that for a wide range of security
+functions,our strong-securitycoding scheme operates at a rate
+thatapproachesthe secrecycapacity.
+Theorem21. Foranysecurityfunction δnthatsatisfies (87),the
+rateRnofthecorrespondingstrong-securitycodingschemeap-
+proachesthesecrecycapacity,namely
+limn→∞Rn=C(W∗)−C(W) (93)
+Proof.A lower boundon the rate Rnis given in (81). Along
+withProposition20andTheorem1,thisimmediatelyshowsth at
+lim n→∞Rn/greaterorequalslantC(W∗)−C(W). In order to establish equality
+in(93),letuspartitiontheset R=[n]\P n(W,δn)in(51)into
+twosubsets,asinFigure2.Thesesubsetsaredefinedasfollo ws:
+Xdef=R∩ B n(W∗,β) (94)
+Ydef=R∩ G n(W∗,β) (95)
+Notethattheset Xin(94)andtheset Bin(53)formapartition
+ofBn(W∗,β),asillustratedinFigure2.Itfollowsthat
+Rn=|Pn(W,δn)|
+n−|Bn(W∗,β)|
+n+|X|
+n(96)
+We will show in Proposition22 of the next subsection that the
+fraction|X|/nvanishesas n→∞.Withthis,thetheoremfol-
+lowsfrom(96),Proposition20, andTheorem1.
+F. Reliabilityof theStrong-SecurityCodingScheme
+If the main channel W∗is noiseless, Bob can trivially recover
+themessagewithprobability 1asfollows.Bobreceivesthevec-
+torY=VGn.SincetheArıkantransformmatrix Gn=PnG⊗m
+is its own inverseover F2, Bob can compute V=YGnand set
+/hatwideU=VA.Clearly/hatwideU=U,andcondition(1)istriviallysatisfied.
+Whathappensifthemainchannel W∗isnotnoiseless?Sup-
+posethatBobattemptsto usethesuccessivecancellationde co-
+der, as in SectionV. Then, according to Theorem2 and Propo-
+sition3,Bob’sprobabilityoferrorisupper-boundedbythe sum
+of the Bhattacharyya parameters Z(W∗
+i)of those bit-channels
+thatarenotfixed(tozero).Theindexsetofthebit-channels that
+arenotfixedinourstrong-securitycodingschemeisgivenby
+A∪R=Gn(W∗,β)∪X
+whereXisthesetdefinedin(94).ThesumoftheBhattacharya
+parameters Z(W∗
+i)overtheset Gn(W∗,β)isboundedby 2−nβ
+bydefinition,andtherefore
+Pr/braceleftbig/hatwideU/\e}atio\slash=U/bracerightbig/lessorequalslant2−nβ+∑
+i∈XZ(W∗
+i) (97)
+Unfortunately, we are not aware of any useful bounds on the
+sum ∑i∈XZ(W∗
+i).Thuswedonothaveaproofthatthestrong-
+securitycodingschemesatisfies thereliabilitycondition (1).
+Observe, however, that the security and reliability requir e-
+mentsare fundamentallydifferent.Whetherwe’reinterest ed in
+the theory or in the practice of wiretap channels, security a l-
+ways requires a proof. It cannot be established througha com -
+putationalprocedure,suchassimulation.Ontheotherhand ,re-liability can be (and often is) established through computa tion
+in practice. For example, it is very common to rely on simu-
+lations to verify the reliability performance of error-cor recting
+codes. We believe that, in practice, our strong-security co ding
+schemecanbedecodedtoachievelow probabilitiesoferror.
+First, the following propositionshows that if Wis degraded
+with respect to W∗, thenthe set Xthat givesus troublein suc-
+cessivecancellationdecodingissmall.
+Proposition22. LetXbe the index set of bit-channelsthat are
+notδn-poorforEveyet badforBob,asdefinedin (94).Then
+lim
+n→∞|X|
+n=0
+for any security function δnthat satisfies (87), provided Eve’s
+channel Wisdegradedwithrespectto Bob’schannel W∗.
+Proof.Weclaimthatthesets X,Gn(W∗,β),andPn(W∗,δn)
+arepairwisedisjoint,andtherefore
+|X|
+n+|Gn(W∗,β)|
+n+|Pn(W∗,δn)|
+n/lessorequalslant1(98)
+SinceX ⊆ B n(W∗,β)bydefinition,and Bn(W∗,β)isthecom-
+plementof Gn(W∗,β),itisclearthat X ∩G n(W∗,β) =∅.Itis
+also clear that Pn(W∗,δn)∩G n(W∗,β) =∅, as in (92). Fur-
+thermore,since X ⊆RandR=[n]\Pn(W,δn)bydefinition,
+wehaveX ∩P n(W,δn) =∅.Consequently,inordertoprove
+ourclaimin (98), it wouldsufficetoshowthat
+Pn(W∗,δn)⊆ P n(W,δn)
+ButthisfollowsimmediatelyfromLemma4alongwiththedef-
+initionoftheset of δn-poorchannelsin(50).Nowobservethat
+asn→∞,thefraction |Gn(W∗,β)|/nconvergestothecapac-
+ityC(W∗)byTheorem1,whereasthefraction |Pn(W∗,δn)|/n
+convergesto 1−C(W∗)byProposition20.Togetherwith(98),
+thiscompletestheproofoftheproposition.
+Dependinguponhowsmalltheset Xturnsouttobeinprac-
+tice,variousdecodingsolutionsarepotentiallyapplicab le.
+First, it is quite possible that X=∅in manysituations, es-
+pecially if the main channel is much better than the wiretap
+channel. In this case, successive cancellation decoding ca n be
+used“asis,” andBob’sprobabilityoferrorisat most 2−nβ.
+The following example illustrates this situation. In order to
+obtainthenumericalvaluesgiveninthisexample,wehaveus ed
+themethodsof[37]toevaluatethepolarbit-channels.
+Example. Suppose that both the main channel and the wire-
+tapchannelarebinarysymmetricchannels,say C1=BSC(p1)
+andC2=BSC(p2).Letusfurtherassumethat p1=10−3and
+thattheerror-raterequiredattheoutputofthemain-chann elde-
+coder is 10−9. This is often the case in optical fiber communi-
+cations[31].We will use the polartransformation(5) oflen gth
+n=220. Indeed, codes of this length are already in use today
+in proprietary 100GbE fiber-optic systems. We also adopt the
+following stringent security criterion: we require that th e mu-
+tual information I(U;Z)between messages at the input to our
+encoderandobservationsattheoutputofthewiretapchanne lis
+less than 10−30. Using the methods developed in this section,
+we can simultaneously guarantee reliability of 10−9and secu-14 MAHDAVIFAR AND VARDY: ACHIEVINGTHESECRECYCAPACITYOFWIRETAPCHANNELSUSING P OLARCODES
+rityof 10−30atcommunicationratescloseto thesecrecycapa-
+city.Thefollowingtable:
+p2Rate R% ofCs
+0.45 0.933 95.1%
+0.40 0.882 91.9%
+0.35 0.817 88.5%
+0.30 0.738 84.8%
+0.25 0.647 80.9%
+0.20 0.543 76.4%
+0.15 0.425 71.1%
+0.10 0.293 64.0%(99)
+summarizes these rates as a function of the bit error-rate p2of
+the wiretapchannel.The thirdcolumnin the table givesthe r a-
+tioR/Cs, expressed as a percentage. Notably, in all of these
+cases, we have X=∅. In fact, the set Xremains empty un-
+til the bit error-rate on the wiretap channel decreases down to
+p2=0.066, in which case |X|=1. But the secrecy capacity
+forp2/lessorequalslant0.066islessthan 0.3395,whichisprobablytoosmall
+tobeofpracticalinterest(infiber-opticcommunications) ./square
+Now suppose that the set Xis nonempty. Observe that this
+setisfixedandknownaprioritoalltheparties(Alice,Bob,a nd
+Eve). In successive cancellation decoding, Bob makes his de -
+cisions/hatwidev1,/hatwidev2, . . . ,/hatwidevnsequentially using the decision rule (12).
+Bobwillknowapriorithatthisdecisionruleisunreliablew hen-
+ever an index i∈Xis reached. Therefore, Bob could follow
+bothalternatives /hatwidevi=0and/hatwidevi=1for all i∈X. Doing so in-
+creasesthedecodingcomplexitybya factorof 2|X|.But if|X|
+is a small constant (say, Xcontains only a couple of bit-chan-
+nels),thisisnotunreasonable.
+WhatcanBobdoiftheset Xislarger?Itiswellknownthat
+in successivecancellationdecoding,a singleincorrectde cision
+affectsall thefollowingdecisions,makingthemunreliabl e.We
+propose to take advantage of this phenomenon in order to re-
+ducethedecodingcomplexity.Let i1bethesmallestindexin X.
+Supposethat uponbranchingwith /hatwidevi1=0and/hatwidevi1=1, thede-
+coder begins to compute the channel noise (e.g. the Hamming
+distancetothe receivedvectorona BSC channel)accumulate d
+along each of the two decision paths being followed. Due to
+the “errorpropagation”inducedby the incorrectdecision a ti1,
+we expect this estimated noise to accumulate rapidly along t he
+incorrect path. On the other hand, along the correct path, th e
+channelnoiseshouldaccumulateslowly,governedbythesta tis-
+ticsof W∗thatareknownapriori.Thismeansthatthedecoder
+can detect, with high probability,which of the two paths bei ng
+followedis incorrect.Once thedecoderfindsthat the pathwi th
+thehigheraccumulatednoiseissufficientlyunlikely,acco rding
+tothe channelstatistics, thispathcanbe safelydiscarded .
+Ofcourse,itispossiblethatthesecondsmallestindex i2∈X
+is reached beforeone of the two pathsopenedat i1∈Xcan be
+discarded.Inthiscase,thedecoderwouldneedtobeginfoll ow-
+ing four paths. Once the third index i3∈Xis reached, the de-
+codermightbefollowing 1,2,3,or4paths.Andsoon.Inprac-
+tice, one could design the decoder to follow at most Mpaths,
+where Misapre-determinedlimitdictatedbythedecodercom-
+plexityconsiderations.Thesituationisquitesimilartod ecision-
+feedbackequalizationonISI channelsusinga bankof Mzero-forcing DFEs. That scenario was analyzed in [42], where it is
+shownthaterror-propagationcausedbyincorrectdecision scan
+beused todiscarderroneousdecision-feedbackpaths.It is also
+shown in [42] that, in practice, small values of Moften suffice
+toachieveverygoodperformance.
+We have limited our consideration herein to successive can-
+cellation decoding, or variants thereof. It is also possibl e that
+othermethodsofdecodingpolarcodes(suchasbeliefpropag a-
+tion [17] or recursive-list decoding [12]) may be relativel y ro-
+bust to not fixing a small set Xof bad channels. Analysis of
+suchdecodersisaresearchproblemofindependentinterest .
+VII. D ISCUSSION AND OPENPROBLEMS
+Webrieflymentioncertainstraightforwardextensionsofou rre-
+sults.Sofar,wehaveconsideredexclusivelybinary-input sym-
+metricwiretapchannels.However,itiswellknownthatthep ol-
+arization phenomenonextends to other types of channels. Ar ı-
+kanshowsin[3]that,givenanon-symmetricbinary-inputch an-
+nelW,polarcodesachieveitssymmetriccapacity I(W)inthe
+averagesense .Thisimpliesthatourcodingschemeachievesthe
+symmetriccapacitydifference I(W∗)−I(W), also inthe av-
+eragesense.Specifically,supposewemodifyourencodingal go-
+rithm in SectionIV as follows. Insteadof constructingthe v ec-
+torv∈{0, 1}nbysetting vR=e,vA=u, and vB=0,we set
+vR=e,vA=u, and vB=s, where sis a fixed binary vector
+knowna priori to all the parties. Then there exists some choice
+ofssuch that Theorem8 and Theorem9 hold. Based upon the
+resultsofArıkanin[3],exactlythesameproofasbefore(Le m-
+ma6andLemma7)applies.
+Ourresultsalsoextendtodiscretememorylesschannelswit h
+non-binary input. It was recently proved in [34] that channe ls
+withaninputalphabetof primesize qarepolarizedbythesame
+transformation (5), and the corresponding versions of Theo -
+rem1 and Theorem2 hold. The probability of error under suc-
+cessive cancellation decoding still scales as O(2−nβ)for all
+prime q. This means that our proof of Lemmas 6 and 7 goes
+through essentially “as is” (as long as ris replaced by rlog2q
+throughout).Ifthesize qoftheinputalphabetisnotprime,po-
+larizationrequireseitherarandomizedpermutationonthe input
+ormultilevelcoding(see[34]formoredetails).Itcanbesh own
+thatourresultsinSectionV extendtothiscaseaswell.
+Itisnotclearwhetherthestrong-securityresultsofthepr evi-
+oussection can be similarly extendedto non-symmetricand/ or
+tonon-binary-inputwiretapchannels.We believetheycan, and
+poseaproofofthisasan openproblem.
+Anotheropenproblemofgreatinterestishowtocodeforthe
+situationwherethewiretapchannel Winnotdegradedwithre-
+specttothemainchannel W∗.Notethatchanneldegradationis
+sufficient but, to the best of our knowledge, not necessary fo r
+ourcodingschemetowork.Whatseemstobenecessaryisthat
+the set of bit-channels that are “good” for Eve but “bad” for
+Bob is either empty (as in SectionV) or at least very small (as
+in SectionVI). Unfortunately, in the general case, there is no
+reasonwhythenumberofsuchbit-channelscouldnotbelarge .
+Finally, we point out that all the constructions in this pape r
+are only as explicit as the polar codesthemselves. An exact a l-
+gorithmforcomputingthesets Gn(W,β)andPn(W,δ)in(13)Toappear in the IEEE T RANSACTIONS ON INFORMATION THEORY, 2011 15
+and(50)wasgivenbyArıkanin[3].However,thisalgorithmr e-
+quirestime and memorythat growexponentiallywith the code
+length n. Since then, severalheuristic algorithmsforthis prob-
+lem have been proposed [2,29,30]. However, these algorithm s
+do not provideuseful guaranteeson the quality of their outp ut.
+Suchguaranteesareclearlyessentialtoestablishthesecu rityof
+ourcodingscheme.Fortunately,theproblemhasbeenresolv ed
+in [37].The algorithm of [37]runs in linear time, and makes i t
+possibletocomputeupperandlowerboundsonthecapacityof
+polarbit-channelswithanarbitrarydegreeofprecision.F orex-
+ample,toestablish the resultsreportedin (99),we haverun the
+algorithmof[37]withaprecisionof 300bits(whichisnecessa-
+rytoprovidemeaningfulguaranteesofsecuritydownto 10−30).
+ACKNOWLEDGMENT
+We thank Mihir Bellare, Hamed Hassani, Satish Korada, Ryu-
+taroh Matsumoto, Ido Tal, Emre Telatar, and R¨ udiger Urbank e
+forveryhelpfuldiscussions.Weareespeciallygratefulto Satish
+KoradaforpointingustoLemma4.7ofhisPh.D.thesis,andto
+IdoTal forhishelpwiththe examplein SectionVI-F.
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\ No newline at end of file
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+OPTIMAL CONTROL WITH MODERATION INCENTIVES
+DEBRA LEWIS
+Abstract. A purely state-dependent cost function can be modied by introducing a control-dependent term rewarding
+submaximal control utilization. A moderation incentive is identically zero on the boundary of the admissible control
+region and non-negative on the interior; it is bounded above by the inmum of the state-dependent cost function, so
+that the instantaneous total cost is always non-negative. The conservation law determined by the Maximum Principle,
+in combination with the condition that the moderation incentive equal zero on the boundary of the admissible control
+region, plays a crucial role in the analysis; in some situations, the initial and nal values of the auxiliary variable are
+uniquely determined by the condition that the conserved quantity equal zero along a solution of the arbitrary duration
+synthesis problem. Use of an alternate system of evolution equations, parametrized by the auxiliary variable, for one-
+degree of freedom controlled acceleration systems, can signicantly simplify numerical searches for solutions of the arbitrary
+duration synthesis problem. A one-parameter family of `elliptical' moderation incentives is introduced; the behavior of the
+well-known quadratic control cost and its incentive analog is compared to that of the elliptical incentives in two simple
+controlled acceleration examples. The elliptical incentives yield smooth solutions with controls remaining in the interior of
+the admissible region, while the quadratic incentive allows piecewise smooth solutions with controls moving on and o the
+boundary of the admissible region; in these examples, the arbitrary duration synthesis problem for the traditional quadratic
+control cost has no solution|the total cost is a monotonically decreasing function of the duration.
+1. Introduction. Optimal control problems typically involve constraints on both the state variables
+and control. For example, (bio)mechanical systems cannot generate or withstand arbitrarily large forces
+or accelerations. For some cost functions, trajectories approaching the boundary of the admissible region
+are so extravagant that the boundary can safely be left out of the mathematical model. However, when a
+task is to be executed as quickly as possible, the bounds on the possible play a crucial role in determining
+the optimal process|cost considerations would drive the solution outside the admissible region if these
+bounds were not explicit imposed.
+If the relevant constraints are explicitly incorporated in the state space and admissible control region,
+the cost function for a time minimization problem is constant. Given the degeneracy of the constant cost
+function, the optimal control values are sought on the boundary of the admissible controls set. In some
+situations of interest, geometric optimization and integration methods can be used (see, e.g., [10]) to
+work directly on the boundary. If geometric methods are not available or desirable, penalty functions
+can be used to construct algorithms on an ambient vector space that respect the boundary due to the
+prohibitive (possibly innite) expense of crossing it (see, e.g., [4, 2], and references therein). In many
+situations, particularly in biological models, a close approach to the boundary of the admissible region
+is undesirable|stresses on joints, muscles, and bones are severe near the breaking or tearing points of
+these structures|but sometimes justied. An animal may be willing to push itself to its physical limits
+to escape a high-risk situation; many machines are engineered to execute certain tasks rapidly, even if
+this involves high energy consumption and rapid wear of mechanical parts. In such cases, selection of an
+appropriate penalty function is essential, as too severe a penalty will yield overly conservative solutions.
+An important class of optimal control problems can be interpreted as modied time minimization
+problems, in which certain states are more costly than others. When modeling a conscious agent, the unit
+cost function of a traditional time minimization problem can be interpreted as representing a uniform
+stress and/or risk throughout the task, while a modied time minimization cost function models a com-
+bination of instantaneous stresses and risks that explicitly depend on the current state. This formulation
+may be more natural than an admissible/inadmissible dichotomy, particularly for biological systems. For
+example, consider the Kane-Scher model of the classic `falling cat' problem, in which a cat is suspended
+Mathematics Department, University of California, Santa Cruz, Santa Cruz, CA 95064. lewis@ucsc.edu . Supported
+by NSF DMS-0405610
+1arXiv:1001.0211v1  [math.OC]  1 Jan 20102 D. Lewis
+upside down and then released; a typical cat can right itself without net angular momentum from heights
+of approximately one meter. (See, e.g., [12, 5, 14, 10].) Kane and Scher [5] proposed a two rigid body
+model of a cat; to eliminate the mechanically ecient but biologically unacceptable solution in which
+the front and back halves counter-rotate, resulting in a 360twist in the `cat', Kane and Scher imposed
+a no-twist condition in their model. However, actual cats can and do signicantly twist their bodies;
+replacing the no-twist condition with a deformation-dependent term in the cost function that discourages
+excessively large relative motions allows more realistic motions. We will refer to optimal control problems
+with cost functions depending only on the state variables as modied time minimization problems.
+Given a modied time minimization problem, we are interested in modifying the cost function to take
+the control eort into account. We assume here that a cost function modeling the do-or-die, `whatever
+it takes' approach is known, and construct a new cost function by subtracting a control-dependent term.
+Our approach is to regard this term not as a penalty or cost, but a deduction rewarding submaximal
+control eorts. Hence we specify that this term equal zero on the boundary of the admissible control set
+and be bounded above by the minimum of the original cost function, so that the total instantaneous cost
+function remains non-negative. We can construct parametrized families of such functions and adjust the
+urgency of the task by adjusting the parameter. The incentive function may allow controls to move on
+and o the boundary of the admissible control region, or may approach zero suciently rapidly as the
+control approaches the boundary that controls starting in the interior of the admissible control region
+will remain there throughout the maneuver.
+The notion of a moderation incentive can guide the modication of familiar cost functions. As we shall
+see in the examples, simply modifying the cost function by a constant can make the dierence between the
+existence and absence of solutions of the arbitrary duration synthesis problem. The cost functions we use
+here to illustrate this property are quadratic in the control. Quadratic control cost (QCC) minimization,
+with cost functions of the form C(x;u) =Qx(u) for some smooth family of quadratic forms Qxdetermined
+by an inner product or Riemannian metric, has played an important role in geometric optimal control
+theory. (See, e.g., [1], [3], [13], and references therein.) If the admissible control region is unbounded,
+QCC functions yield relatively simple evolution equations: if the state space is a subset of a Riemannian
+manifoldM, the space of admissible controls has full rank, _ x=u, andC(x;u) =1
+2juj2
+x, then the traces
+inMof the optimal trajectories are geodesics. If the control uis constrained to lie in a distribution of
+less than full rank, the corresponding QQC problem leads to sub-Riemannian geometry. (See, e.g. [15]
+or [1].) Thus many existing results from geometric mechanics and (sub-)Riemannian geometry can be
+utilized in the analysis of simple QCC control problems. QCC optimization sometimes follows a `the
+slower, the better' strategy: in some important QCC problems the total cost is a decreasing function of
+the maneuver duration; hence there is no optimal solution of the arbitrary duration QCC problem. If
+there is a range of durations [ Tmin;1) for which unique specied duration QQC solutions exist, then the
+QCC trajectory of duration Tminmay be of interest as the fastest of the slow. However, it is unclear in
+what sense these trajectories are optimal. Modifying the quadratic control cost by a constant, so as to
+satisfy the condition that the moderation incentive equal zero on the boundary of the admissible control
+region, can yield arbitrary duration synthesis problems for which unique solutions doexist in situations
+where the QCC function (nonzero on the boundary of the sphere) lacks such solutions.
+We introduce a one-parameter family of `elliptical' moderation incentives eC: [0;1]![0;],2
+(0;1], by
+eC(s) :=p
+1s2: (1.1)
+(The graph of p
+1s2, 0s1, is a segment of an ellipse of eccentricity .)eC0is the trivial
+incentive associated to the unmoderated modied time minimization problem. For 2(0;1], the control
+values determined by eCalways lie in the interior of the unit ball; if the unmoderated cost function isOptimal control with moderation incentives 3
+smooth, the state variables and control will also be smooth. (In contrast, some of the solutions for the
+quadratic incentive and QCC we nd in the examples are only piecewise smooth, moving on and o the
+unit sphere.) However, the penalty imposed by eCis not prohibitive; as we shall see in the examples, we
+can come arbitrarily close to the control region boundary by adjusting . The elliptical incentives have
+some simple properties that make them particularly convenient to work with in certain kinds of analytic
+and numerical calculations. Finally, in the examples treated here, there are some qualitative resemblances
+between the trajectories determined by the quadratic incentive eCqand the elliptical incentives for values
+ofnear 1.
+We make several simplifying assumptions in the present work. We assume that the admissible control
+region is the Euclidean unit ball and our incentives are nonincreasing functions of the magnitude of the
+control. We restrict our attention to problems in which the state variables consist of a position vector in
+Rnand its rst k1 derivatives; the k-th derivative is fully controlled and the unmoderated cost function
+depends only on the position. These assumptions are not central to the formulation of the moderated
+problem, but they lead to particularly simple expressions in some key constructions. More general control
+systems, with more complex admissible control regions (including ones determined in part by the state
+variables) and more general controls, will be considered in future work.
+We consider two examples that illustrate some of the key features of the moderated control problems
+and suggest directions of future research. The rst example is a very simple one-dimensional controlled
+acceleration problem: a particle at rest at one position is to be moved a unit distance by controlling the
+acceleration; the initial and terminal velocities are zero. This classic starter problem is treated in [17, 6],
+and other texts. The well-known time minimizing solution is the `bang-bang' solution, with acceleration
+equal to 1 for the rst half of the maneuver and 1 for the second half; the solution of the arbitrary
+duration problem with quadratic incentive has linear acceleration; the solutions for the elliptical incentives
+have smooth accelerations approaching the bang-bang solution as the moderation parameter approaches
+zero, and approaching the QCC solution as the parameter approaches one.
+The second example is a generalization of the rst, with a position penalty added to the control
+cost. The position penalty is monotonically decreasing and equals zero at the destination. This example
+can be interpreted as a very simple model of `spooking' (
ight reaction), in which the position penalty
+models aversion to a localized stimulus and the destination is the position at which the animal rst
+feels entirely safe or comfortable. The re
ectional symmetry seen in the rst example is broken: all of
+the cost functions studied here, with the exception of the trivial moderation incentive, yield asymmetric
+solutions, with relatively strong initial accelerations and relatively weak decelerations. The quadratic
+incentive yields only piecewise smooth solutions, while the elliptical incentive solutions are smooth for all
+nonzero values of the moderation parameter. For small-to-middling values of the moderation parameter ,
+the solutions for the elliptical incentives show little response to the intensity of the position penalty|the
+solutions remain close to the corresponding solutions for the corresponding problem without a position
+penalty even when the position penalty is high. Roughly speaking, if little or no incentive to take it easy
+is added to a time-pressured task, the optimal strategy is to get it all over with (almost) as quickly as
+possible; there's little room for modication of the strategy if additional discomfort or risk is introduced.
+On the other hand, if there's a signicant reward for moderate eort, the strategy in the absence of a
+position penalty will be to take it slowly, and the introduction of some variable risk or discomfort can yield
+dramatic speed-ups in overall execution times, as well as signicant variations in control magnitudes.The
+classic quadratic control cost (QCC) function, which is nonzero on the boundary of the admissible control
+region, determines a total cost that is a monotonically decreasing function of the maneuver duration; as
+the specied duration is increased, the solutions perform an increasing number of oscillations about the
+destination before coming to rest.4 D. Lewis
+2. Thek-th order moderated synthesis problem. We apply Pontryagin's Maximum Principle
+to a special class of optimal control problems that illustrate some of the features of moderated control
+problems, but are relatively easily analysed. We focus on the k-th order evolution equation x(k)=u,
+x: [0;tf]!Rn, with specied initial and nal values of the x(j),j= 1;:::;k1, and unmoderated
+cost function depending only on the position x, not the derivatives of x. We further simplify the analysis
+by assuming that the admissible control region is the unit ball in Rnand that the cost depends on the
+control only through its norm.
+Consider a control problem with state variable z2Rm, controlu2URk, evolution equation
+_z=V(z;u), boundary conditions z(0) =z0andz(tf) =zf, and cost function C:RmU!R.
+Assume that both VandCare continuous, with continuous derivatives with respect to the state variable
+z; letAdenote the set of triplets ( z;u;tf), such that tf>0, (z;u) : [0;tf]!RmUsatises the
+evolution equation and boundary equations, zis continuous with piecewise continuous derivative _ z, and
+uis piecewise continuous. Pontryagin's Maximum Principle states that if ( z;u;t )2Aminimizes the total
+cost over A, i.e.
+Ztf
+0C(z(t);u(t))dt= min
+(~z;~u;~tf)2AZ~tf
+0C(~z(t);~u(t))dt;
+then there is a continuous curve  : [0;tf]!Rmand constant 0 such that
+_z=@H
+@ (z; ;u ); _ =@H
+@z(z; ;u ); andH(z; ;u ) = max
+2UH(z; ; ) (2.1)
+forH(z; ;u ) :=h ;V(z;u)iC(z;u). The Hamiltonian His constant along a curve satisfying (2.1);
+if the curve minimizes the curve on A, then constant is zero. (See [17] for the precise statement and proof
+of the Maximum Principle.)
+Pontryagin's conditions are necessary, but not sucient, for optimality; their appeal lies in their
+constructive nature: known results and techniques for boundary value problems and Hamiltonian systems
+can be used in constructing the triplets ( z;u;tf) satisfying Pontryagin's conditions. This construction is
+referred to as the synthesis problem in [17]. We will restrict our attention to the synthesis problem, setting
+aside the rigorous analysis of actual optimality of the trajectories we obtain. It suces to consider the
+cases= 1 and= 0, since we can rescale  by6= 0.H0is clearly independent of the cost function C;
+Hamilton's equations for H0equal those for a constant cost function, used in determined the minimum-
+time admissible curves. Hence we will focus on H1, simply noting that when searching for the optimal
+solution, it is necessary to consider the possibility that the total cost is minimized by the minimum
+duration trajectory.
+Here we consider control problems of the form x(k)=u, wherex: [0;tf]!Rnandu: [0;tf]!
+Bn=fu2Rn:juj1g, for some (possibly specied) tf>0, with specied boundary values x(j)(0) and
+x(j)(tf),j= 0;:::;k1. We restrict our attention to cost functions that are the dierence of a term
+depending only on the position xand a term depending only on the magnitude of the control. We assume
+that the position-dependent term is bounded below (for simplicity, we take the bound to be 1), and the
+control-dependent term has range contained within [0,1]; thus the instantaneous cost is non-negative.
+GivenC2C1(Rn;[1;1)) andeC2C0([0;1];[0;1]), we seek xwith piecewise continuous k-th derivative
+and piecewise continuous uminimizing the total cost
+Ztf
+0
+C(x(t))eC(ju(t)j)
+dt
+over all such state/control variable pairs.Optimal control with moderation incentives 5
+To apply the Pontryagin Maximum Principle to our control problem, we rst convert the k-th order
+evolution equation x(k)=uinto a rst order system of ODEs by introducing the auxilliary state variables
+dj:=x(j),j= 1;:::;k1, and setting z= (x;d1;:::;dk1)2(Rn)kRnk. The resulting rst order
+evolution equation is
+_z=V(z;u) := (d1;:::;dk1;u): (2.2)
+If we let = (1;:::;k1;)2(Rn)kRnk, thenH1is equivalent to the Hamiltonian H: (Rn)2k+1!
+Rgiven by
+H(x;d1;:::;dk1;1;:::;k1;;u) :=k1X
+j=1hj;dji+h;ui+eC(juj)C(x): (2.3)
+The control uis chosen at each time tso as to maximize the Hamiltonian. Since (2.3) depends on u
+andonly through the term h;ui+eC(juj), the optimal value of usatisesjju=juj, and
+H(x;d1;:::;dk1;1;:::;k1;;u) = max
+2BnH(x;d1;:::;dk1;1;:::;k1;;) (2.4)
+if and only if
+jujjj+eC(juj) =() := max
+01jj+eC(): (2.5)
+IfeC(0)>eC(s) for alls2(0;1], then= 0 implies u= 0; ifeCachieves its maximum at any point other
+than the origin, uis not uniquely determined when = 0. (In most of the cases considered here, uis
+uniquely determined at = 0 but this is not true for the time minimization problem.)
+Hamilton's equations
+_x=@H
+@1=d1 _1=@H
+@x=rC(x)
+_dj=@H
+@j+1=dj+1 _j=@H
+@dj1=j1; j = 2;:::;k1
+_dk1=@H
+@=u _=@H
+@dk1=k1
+for the Hamiltonian (2.4) are equivalent to x(j)=djand(j)= (1)jkj,j= 1;:::;k1,x(k)=u,
+and(k)= (1)k1rC(x). Inserting these expressions into the Hamiltonian (2.3) yields
+()C(x) +k1X
+j=1(1)jD
+x(kj);(j)E
+(2.6)
+Pontryagin's Maximum Principle implies that (2.6) is constant along a curve ( x;;u ) satisfying Hamilton's
+equations and maximizing the Hamiltonian.
+Definition 2.1. x2Ck1([0;tf];Rm), with piecewise continuous k-th derivative, satisfying the given
+boundary conditions on x(j)(0)andx(j)(tf),j= 0;:::;k1, is a solution of the k-th order synthesis
+problem if there exists : [0;tf]!Rmsatisfying
+(k)= (1)k1rC(x) andjx(k)jjj+eC(jx(k)j) =() (2.7)
+for0ttf. Ifx(k)is discontinuous at t, thenx(k)(t)agrees with the left hand limit, i.e. x(k)(t) =
+limt!t
+x(k)(t).6 D. Lewis
+If, in addition, (2.6) equals zero along the curve (x;),xis a solution of the k-th order synthesis
+problem of arbitrary duration.
+We follow the convention of [17] in specifying that x(k)is continuous to the left at a discontinuity;
+one could as well choose the right hand limit.
+We introduce a class of functions eCfor which the optimal value of the control uis explicitly given
+as the gradient of a function of when6= 0, determining a system of k-th order system of ODEs
+forxand. The functions equal zero on the sphere Sn1bounding the admissible control region; the
+instantaneous control cost equals the position-dependent term at peak control values.
+Definition 2.2. IfeC2C0([0;1];[0;1])is dierentiable on (0;1),eC(1) = 0 , and there is a unique
+non-decreasing function 2C0(R+;(0;1]), dierentiable on 1(0;1), satisfying
+(s)s+eC((s)) =e(s) := max
+01s+eC(); (2.8)
+we say thateCis a moderation incentive, with moderation potential :Rn![0;1)given by() :=
+e(jj).
+Lemma 1. IfeCis a moderation incentive, then eis continuously dierentiable and strictly increasing
+onR+, withe0=.
+Proof . If(s)<1, then(s) is a critical point of 7!seC(), and hence s=eC0((s)). In
+addition,eis dierentiable at s, with derivative
+e0(s) =(s) + (seC0((s))0(s) =(s):
+If1(1)6=;, then1(1) = [s;1) for somes, sinceis non-decreasing. Since e(s) =seC(1)
+forss,eis clearly dierentiable and satises e0(s) = 1 =(s) fors > s. Continuity of and the
+Mean Value Theorem imply that eis dierentiable at s, withe0(s) = 1 =(s).
+We focus our attention on a one-parameter family of moderation incentives, which includes the trivial
+incentiveeC00, and a quadratic polynomial moderation incentive that diers from a kinetic energy
+term by a constant in the case of controlled velocity.
+Proposition 1. The functions eC(s) =p
+1s2,01, are moderation incentives, with
+moderation potentials
+() =p
+2+jj2: (2.9)
+2C1(Rm;R+)if > 0;0is continuously dierentiable everywhere except at the origin, where the
+equation determining the optimal control value is completely degenerate.
+The quadratic polynomial eCq(s) =1
+2
+1s2

+is a moderation incentive function, with moderation
+potential
+q() =1
+2
+1 +jj2

+jj1
+jj j j>1(2.10)
+q2C1(Rm;R+).
+Proof . If>0, dierentiating
+s+eC() =s+p
+12
+with respect to yields the criticality condition s==p
+12, with unique solution
+=sp
+2+s2: (2.11)Optimal control with moderation incentives 7
+The inequality
+s2
+p
+2+s2+eC 
+sp
+2+s2!
+=p
+2+s2>maxf;sg= maxn
+eC(0);s+eC(1)o
+for > 0 ands > 0 implies that (2.11) is the optimal value of , and hence is given by (2.9).
+s+eC0(s) =sachieves its maximum s=p
+02+s2at the boundary point = 1.
+We now consider the quadratic polynomial eCq:
+s+eCq() =s+12
+2= 1(s)2
+2
+achieves its maximum on [0 ;1] at= minfs;1g.
+The evolution equations for xandin the synthesis problem associated to a moderation incentive
+are a pair of k-th order skewed gradient equations.
+Proposition 2. LetC2C1(Rn;[1;1))andeCbe a moderation incentive , with moderation potential
+.x: [0;tf]!Rnsatisfying the given boundary conditions is a solution of the synthesis problem for the
+cost function C(x)eC(juj)if and only if there exists : [0;tf]!Rnsatisfying
+x(k)=r() and(k)= (1)k1rC(x): (2.12)
+If, in addition, the conserved quantity (2.6) equals zero, (x;)is a solution of the arbitrary duration
+synthesis problem.
+If(t) = 0 at some time tandr(0)is undened, the rst equation in (2.12) is replaced with
+x(k)= limt!t
+r((t)).
+Proof . Assume that xis a solution of the synthesis problem, with auxilliary function . If6= 0,
+uniqueness of the maximizer and Lemma 1 imply that
+x(k)=(jj)
+jj=e(jj)
+jj=r():
+On the other hand, if exists such that ( x;) satises (2.12), then x(k)=r(), then the same argument
+shows that (2.7) is satised.
+SinceeisC1onR+, and hence isC1onRnnf0g, the left handed limit limt!t
+r((t)) is well-
+dened when (t) = 0, and equals limt!t
+x(k)(t).
+Ifris invertible, then the rst equation in (2.12) can be solved for . For example, the elliptic
+moderation incentives have a potential with invertible gradient:
+r() =
+(); with (r)1(u) =u
+1juj2:
+In this case, (2.12) is equivalent to a 2 k-order ODE in x.
+Corollary 1. LetC2C1(Rn;[1;1))andeCbe a moderation incentive with moderation potential .
+Ifris dened everywhere and invertible, then x: [0;tf]!Rnsatisfying the given boundary conditions
+is a solution of the synthesis problem for the cost function C(x)eC(juj)if and only if
+dk
+dtk(r)1
+x(k)
+= (1)k1rC(x) (2.13)8 D. Lewis
+for0ttf. If, in addition,
+C(x) =
+(r)1
+x(k)
++k1X
+j=1(1)j
+x(kj);dj
+dtj(r)1
+x(k)
+; (2.14)
+xis a solution of the arbitrary duration synthesis problem.
+3. One dimensional controlled acceleration systems. In the arbitrary duration problem the
+nal timetfis generally not known a priori; if a closed form expression for the solutions of the synthesis
+problem cannot be found, an iterative numerical procedure may be needed be nd the appropriate tf. In
+some situations it may be both possible and desirable to reformulate the problem to avoid this diculty.
+As an example, we present an approach suitable for a class of one dimensional controlled acceleration
+problems.
+If_is known a priori to be nonzero, we can reparametrize the evolution equations and use the
+conservation of (2.6) to replace the pair of autonomous second order of equations (2.12) with a pair of
+rst order nonautonomous ODEs, with independent variable , and a subordinate rst order equation
+relatingandt. The induced boundary conditions for this problem may be more convenient than those
+of the original synthesis problem. As we shall show below, the pair of rst order ODEs for xand the
+auxiliary variable qcan be formulated without a priori knowledge of the behavior of ; if a solution ( x;q)
+is found that satises the relevant equalities and inequalities, it determines a solution of the synthesis
+problem.
+Proposition 3. If
+(i) there are functions r: [0;f]!Randq: [0;f]!R+satisfying the evolution equations
+qr0+Cr= constant; andq0=2C0(r); (3.1)
+and the boundary conditions r(0) =x0,r(f) =xf,p
+q(0)r0(0) =v0sgn(f0), andp
+q(f)r0(f) =vfsgn(f0)
+(ii) the solution of the IVP
+_=sgn(f0)p
+q() and(0) =0 (3.2)
+passes through fat some positive time tf,
+thenx=r: [0;tf]!Rnis a solution of the one dimensional controlled acceleration synthesis problem
+with boundary data x(0) =x0,x(tf) =xf,_x(0) =v0, and _x(tf) =vf. If the constant in (3.1) is zero,
+thenxis a solution of the arbitrary duration synthesis problem.
+Proof . Dierentiation of _= sgn (f0)pqyields
+= sgn (f0)_q
+2pq=_q
+2_=q0
+2=C0(r):
+Dierentiation of the rst equation in (3.2) yields
+0 = (qr0+Cr)0=qr00+q0r0+ (Cr)00=qr00(Cr)00:
+Hence
+d2
+dt2(r) =r00(_)2+r0=qr00(Cr)0=0:
+Thus (x;) satisfy the evolution equations (2.12). The boundary conditions in (i)guarantee that r
+satises the boundary conditions of the synthesis problem.Optimal control with moderation incentives 9
+0.51.01.52.02.5/Minus1.0/Minus0.50.51.0
+quadratictimeminimizationt¨x
+incentive
+0.51.01.52.02.50.20.40.60.81.0x
+ttimeminimizationquadraticincentive
+Fig. 3.1 .Solutions of the arbitrary duration synthesis problem for the time minimization problem, with trivial moder-
+ation incentive and duration tf= 2, and for the quadratic moderation incentive eCq, withtf=p
+6. Left: acceleration x(t);
+right: position x(t).
+If the constant in the rst equation in (3.2) is zero, then
+=qr0+Cr=_2r0+Cr=__r+Cr;
+and hence (2.6) is zero.
+In the arbitrary duration case, the velocity initial condition is satised if and only if
+(0) =C(x0) v0= 0
+q(0) =v2
+0andx0(0)v0(0f)>0v06= 0; (3.3)
+entirely analogous conditions hold for the terminal velocity. For example, if the initial and terminal
+velocities are both zero and eis one-to-one, then j0j=e1(C(x0)) andjfj=e1(C(xf)).
+Remark: If it is known a priori that a solution xof the synthesis problem must satisfy _ x6= 0 for allt, we
+can reduce the fourth order system (2.13) to a third order one by solving (2.14) ford
+dtxp
+1x2, obtaining
+d
+dtxp
+1x2_x=p
+1x2C(x);
+and hence
+x(3)=1x2
+_x
+1C(x)
+p
+1x2
+:
+3.1. Warm-up example: constant cost controlled acceleration. As a simple illustrative ex-
+ample, we consider the one dimensional controlled acceleration problem  x=u, with boundary conditions
+x(0) = 0; x (tf) = 1; and _x(0) = _x(tf) = 0 (3.4)
+for some nal time tf>0, and constant cost function C1. The trivial incentive version of this problem
+appears as the introductory example in [17], and appears in several other control texts. The `bang-bang'
+solution
+x(t) =t+1
+2(j1tj(1t)1) =8
+<
+:t2
+20t1
+t2
+2+ 2t1 1<t2; (3.5)10 D. Lewis
+with peak control eort throughout the maneuver, is optimal; see, e.g. [6]. Given a moderation incentive
+eC: [0;1]![0;1], we seek a solution of the synthesis problem with boundary conditions (3.4) and
+unmoderated cost C1. Our treatment is a straightforward application of Proposition 3.
+Proposition 4. IfeCis a moderation incentive, then for any 0>0
+q= 2Z0
+0(e(0)e(s))ds; r () =1
+qZ0
+(e(0)e(jsj))ds; andtf=20pq(3.6)
+determine a solution
+x(t) :=r
+0
+12t
+tf
+; 0ttf;
+of the synthesis problem with boundary conditions (3.4) and cost function 1eC. Ife(0) = 1 , thenxis
+also a solution of the arbitrary duration synthesis problem.
+Proof . The constant qand function rgiven by (3.6) clearly satisfy the evolution equations q0= 0 =
+2C0(x) and
+qr0+Cr=e(jj)e(0) + 1() = 1e(0)
+and boundary conditions r(0) = 0,r(0) = 1, and r0(0) = 0. The auxiliary function (t) :=
+0(12t=tf) satises(0) =0,(tf) =0and _=pq= sgn (f0)pq. Hence Proposition 3
+implies that xis a solution of the synthesis problem.
+We now apply Proposition 4 to the trivial, quadratic, and elliptical incentives. For the time mini-
+mization problem, with trivial incentive eCtm0, we haveetm(s) =s, and hence
+rtm() =f+1
+2
+f2jj

+; andtf=2fp
+rtm(f)= 2:
+Thus we obtain the well-known `bang-bang' solution (3.5) for any positive f. The total cost is simply
+the duration, 2. Note that the condition e(f) = 1 is not necessary|the criticality (with respect to
+duration) condition leading to e(f) = 1 need not be satised at the end point of the range of possible
+durations.
+The quadratic incentive, eCq(s) =1
+2
+1s2

+has the piecewise smooth scalar incentive potential
+eq(s) =(1
+2
+1 +s2

+0<s1
+s s> 1:
+It follows that if 0 < 01, then (3.6) takes the form
+rq() =3
+0
+6
+2
+0
+1 +
+02
+andtf=r
+6
+0;
+and hence
+xq(t;tf) =t
+tf2
+32t
+tf
+; t fp
+6: (3.7)
+The arbitrary duration synthesis problem solution is given by 0= 1; note that tf=p
+6 is the minimum
+duration for the smooth eCqsolutions. (If 0>1, the corresponding solutions of the synthesis problem
+are piecewise smooth; we do not construct these solutions here.)Optimal control with moderation incentives 11
+0.51.01.52.0/Minus0.03/Minus0.02/Minus0.010.010.020.03sxµ−x0
+µ=13µ=23µ=.99µ=.01
+0.51.01.52.0/Minus0.03/Minus0.02/Minus0.010.010.020.03sµ=.99µ=23µ=13µ=.01xµ−xq
+0.51.01.52.00.20.40.60.81.0
+s˙xµ
+µ=.99µ=23µ=13µ=.01
+0.51.01.52.0/Minus1.0/Minus0.50.51.0sµ=.99µ=23µ=13µ=.01¨xµ
+Fig. 3.2 .Elliptical moderation incentive (EMI) solutions, plotted with respect to the rescaled time s:=2t
+tf():
+= 102,1
+3,2
+3,1102. Upper left: dierences between EMI solutions x(stf())and the time minimizing solution
+x0(s); upper right: dierences between EMI solutions x(stf())and the rescaled time quadratic incentive solution xq(s);
+lower left: velocities _x(stf()); lower right: accelerations: x(stf()).
+The total cost for the trajectory (3.7) is
+cost(tf) =Ztf
+0(1eCq(x(t;tf)))dt=1
+2Ztf
+0
+1 + x(t;tf)2

+dt=tf
+2+6
+t3
+f
+iftfp
+6 (and hence 01). Note that if we replace the moderated cost function 1 eCq(u) =1
+2
+1 +u2

+with the kinetic energy-style cost functionu2
+2, the evolution equation is unchanged, but the total cost
+6
+t3
+fis now a strictly decreasing function of the maneuver duration tf; hence none of the solutions of this
+synthesis problem are solutions of the arbitrary duration problem. The convention that the moderation
+incentive equal zero on the boundary of the admissible control region determines the constant that `selects'
+the fastest smooth solution of the evolution equation as the solution of the arbitrary duration problem.
+We now turn to the elliptic moderation incentives eC(s) =p
+1s2, 0< 1, withe(s) =p
+2+s2. (Note thateC0=eCtm.) If we let
+g() :=1
+2
+() +2ln (() +)

+denote an anti-derivative of e, then
+r() =(f)(+f) +g(f)g()12 D. Lewis
+=1
+2
+(f)(+f) + ((f)())2ln() +
+(f)f
+:
+The expression for rsimplies somewhat for the solution to the arbitrary duration problem, for which
+(f) = 1 and hence
+r() =1
+2 
++p
+12+ (1())+2ln 
+1p
+12
+() +!!
+: (3.8)
+In this case,
+r(f) =p
+12+2ln
+1 +p
+12: (3.9)
+We plot some information about the solutions xfor some sample values of in Figure 3.2. To
+facilitate comparison, we plot xand its derivatives using the scaled time s:=2t
+tf(). Figure 3.3 shows
+the total cost and duration for the solutions of the arbitrary time synthesis problem, plotted as functions
+of the moderation parameter .
+As the moderation parameter goes to zero, the EMI control problem approaches the time minimization
+one:
+lim
+!0(eC;;x) = (eCtm;tm;xtm):
+In the limit !1, the solutions xof the arbitrary duration EMI problem approach those of equal
+duration of the quadratic incentive synthesis problem. Specically,
+lim
+!1x(t)
+xq(t;tf())= 1 for 0 <ttf();
+whilef=p
+12and (3.9) imply that
+lim
+!1tf()2p
+12= lim
+!14f3
+r(f)= 6:
+(Recall thatp
+6 is the duration of the solution of the arbitrary duration quadratic incentive problem.)
+As suggested by these limits and Figure 3.2, the family of EMI solutions can regarded as linking those of
+the trivial and quadratic incentive problems.
+Remark: For the EMI synthesis problem with 0 < 1, the pair of second order ODEs (2.12) is
+equivalent to the fourth order ODE
+d2
+dt2xp
+1x2=rC(x); (3.10)
+see Corollary 1. In the present example, with constant C, the (nonzero) moderation parameter plays
+no role in (3.10), and hence has no in
uence on the solution of the specied duration problem. The
+parameter does, however, eect the solution of the arbitrary duration problem. The analog of the
+conditione(j0j) =C(x0) for an EMI controlled acceleration problem with zero initial velocity is the
+initial acceleration condition
+p
+1x(0)2=C(x0); i.e.jx(0)j=p
+C(x0)22:Optimal control with moderation incentives 13
+0.20.40.60.81.01234
+µtotalcostduration
+Fig. 3.3 .EMI solutions: duration, tf(), and total cost, cost ().
+The solution to the specied duration problem with cost function C, 0< 1, and duration T > 2
+isxt1
+f(T). Thus the expressions (3.8) and (3.9) are sucient to determine the solutions for both the
+specied and arbitrary duration problems; as in the time minimization case, there is some redundancy
+in the (x;) formulation.
+3.2. A one-dimensional controlled acceleration example: spooking. We consider a gener-
+alization of the controlled acceleration example from Section 3.1, adding a position penalty to the cost
+function; specically, C:R![1;1), withC(1) = 1 (the target is a `no-cost' position) and C(x)>1
+forx2[0;1). This can be regarded as a very simple model of spooking|the reaction of, e.g., a grazing
+herbivore to unexpected abrupt noise or motion. When startled, the animal will initially rush away from
+the disturbance; when it has reached its `comfort zone', it will either turn to examine the threat or resume
+grazing. In our simple one-dimensional model, the function C(x) acts as a `fear factor', modeling the
+undesirability of remaining near the perceived threat and providing an incentive to move rapidly to the
+comfort point x= 1.
+We consider the position-dependent cost function
+C(x) = 1 +c
+2(1x)2:
+The boundary conditions x(0) = _x(0) = _x(tf) = 0 and x(tf) = 1 imply that a solution ( x;) of the
+arbitrary duration problem satises
+(0) =C(0) = 1 +c
+2and(f) = 1: (3.11)
+In addition, must change sign at least once, since the boundary conditions for ximply that sgn  x= sgn
+must change at least once. We assume that  x(0) is positive, and hence 0=e1
+1 +c
+2

+.
+We rst analyse the arbitrary duration synthesis problem for the moderation incentives eC. Lacking
+closed form solutions for either the second order system
+x=p
+2+2and =c(1x): (3.12)
+derived from (2.12) for C(x) = 1 +c
+2(1x)2andeCor the fourth order ODE
+d2
+dt2xp
+1x2=c(1x) (3.13)14 D. Lewis
+123456/Minus1.0/Minus0.50.51.0µ=1µ=.75µ=.25µ=.5c=15
+123456/Minus1.0/Minus0.50.51.0
+µ=.5µ=.25µ=.75µ=1accelerationpositionc=1
+123456/Minus1.0/Minus0.50.51.0
+µ=.25µ=.75µ=.5µ=1c=5
+Fig. 3.4 .Position and accleration plots for the solutions of the EMI synthesis problem for c=1
+5, 1, and 5, and
+=1
+4,1
+2,3
+4, and 1.
+derived from (2.13), we turn to the reparametrized form of the system (3.1), which can be implemented
+using standard numerical BVP routines. (Our numerical approximations were computed using the built-in
+Mathematica function NDSolve .) The boundary conditions on take the form
+0=0(c;) :=r
+1 +c
+22
+2 andjfj=p
+12: (3.14)
+A solution of the reparametrized problem determines a solution of the synthesis problem such that the
+auxillary function has nonzero rst derivative; changes sign at most once (and hence exactly once)
+in this situation. Hence we take f=p
+12. The reparametrized problem takes the form of seeking
+a solution ( r;q) : [0;f]!RR+of the BVP
+qr0+ 1 +c
+2(1r)2=p
+2+2 andq0= 2c(1r);
+with boundary conditions r(0) = 0 andr(f) = 1 for0=0(c;) andf=p
+12. This BVP can
+be solved numerically (using, e.g., a shooting method). Once a solution has been found, the elapsed time
+can be found as a function of by numerically computing the integral
+t() =Z
+0dsp
+q(s):
+The desired solution xis given by x=rt1. Figure 3.4 shows some sample plots, with c=1
+5, 1, and
+5, and=1
+4,1
+2,3
+4, and 1.
+Remark: We shall see that, as in the previous example, the total cost function for the QCC problem is
+a monotonically decreasing function of the maneuver duration, with countably many in
ection points,
+corresponding to trajectories that oscillate about the target before coming to rest. We do not rule out
+the possibility that such oscillations may also lead to a decrease in cost for the elliptical moderation
+incentives; our numerical searches are directed only towards the identication of trajectories that do not
+overshoot the target.
+Note that Figure 3.4 shows little response to the position penalty strength cfor small values of |the
+solutions remain close to the corresponding solutions for c= 0. In Figure 3.5 we plot  x(t;c)x(t; 0) forOptimal control with moderation incentives 15
+=1
+8,1
+4, and1
+2, andc=1
+5and 1. On the other hand, the solutions for near 1 are strongly in
uenced
+by the position penalty; the initial acceleration increases dramatically with cand the acceleration curve
+changes from concave to convex as cincreases. A suciently frightening event will provoke an initially
+strong response even when the moderation incentive is high, but the later stages of the recovery from
+that initial response are largely determined by the moderation incentive. This roughly agrees with
+actual spooking behavior: when startled, an inexperienced animal will often respond by rst bolting,
+then abruptly halting and whirling about to examine the apparent threat while still in a state of high
+excitement; a more experienced one can still be spooked, but quickly regains its composure in the absence
+of real danger and gradually decelerates, reducing the signicant skeletomuscular stresses of a hard stop
+and risk of self-in
icted injury during rapid motion.
+0.51.01.52.0
+/Minus0.04/Minus0.03/Minus0.02/Minus0.010.000.010.02µ=.5µ=.25µ=.125
+0.51.01.52.0
+/Minus0.15/Minus0.10/Minus0.050.05µ=.5µ=.25µ=.125
+Fig. 3.5 . x(t;c)x(t; 0)for=1
+8,1
+4, and1
+2. left:c=1
+5, right:c= 1.
+The quadratic polynomial moderation incentive eCqyields a piecewise smooth solution of the arbitrary
+time synthesis problem that is simpler in some regards, but less convenient in others, than the solutions for
+the elliptical incentives. As we shall see, the controls for the eCqsolutions satisfying (2.12) are continuous,
+but not everywhere dierentiable if the parameter cin the position-dependent component of the cost
+function is nonzero; the solution behaves like the time-minimization solution, with control identically
+equal to one, for the rst part of the manuever, then satises a linear fourth order ODE for the remainder
+of the manuever.
+Equation (2.10) implies that the restriction of e1
+qto [1;1) is the identity map; hence the initial
+conditionq(0) =C(0) = 1 +c
+2implies thatj0j= 1 +c
+2andjx(0)j= 1. A similar argument shows
+that the terminal condition q(f) =C(1) = 1 implies that jfj=jx(tf)j= 1. Since _x(0) = _x(tf) implies
+that x, and hence changes sign, must pass through zero; hence there exists t>0 such that (t)1
+for 0ttand(t)<1 on some interval ( t;t+). We nd solutions to the arbitrary duration
+problem satisfying 1 >>1 on (t;tf), with(tf) =1. (As we shall show later, there are solutions
+to the specied duration synthesis problem that overshoot and oscillate about the target.)
+The second order system (2.12) equals that for eC0whenjj>1:x(t) satises x1 on the interval
+[0;t]; the initial conditions x(0) = _x(0) = 0 imply that x(t) =t2
+2on [0;t]. On (t;tf],
+x=0
+q() =; and hence x(4)==c(1x):
+The linear fourth order PDE y0000+ 4y= 0 has the general solution
+y(s) =coshs
+sinhsT
+Mcoss
+sins
+(3.15)16 D. Lewis
+0.51.01.52.0
+/Minus1.0/Minus0.50.00.51.0c=15
+¨x˙xx
+t
+0.51.01.52.0
+/Minus1.0/Minus0.50.00.51.0
+0.51.01.52.0
+/Minus1.0/Minus0.50.00.51.0c=1x˙x
+¨xt
+0.51.01.52.0
+/Minus1.0/Minus0.50.00.51.0c=5x˙x
+¨xt
+Fig. 3.6 .Position, velocity, and acceleration plots for the solutions of the synthesis problem with moderation incentive
+eCq,c=1
+5, 1, 5.
+for an arbitrary matrix M2R22; (3.15) satises the initial conditions y(0) =y0,y0(0) =v0, and
+y00(0) =a0if and only if
+M=0
+B@y0v0+m
+2
+v0m
+2a0
+21
+CA (3.16)
+for somem2R.
+We seekt,tf, andMsuch that
+x(t) =t2
+20tt
+1y
+c1=4=p
+2 (tt)

+t<ttf(3.17)
+is twice dierentiable and satises the boundary conditions x(tf) = 1, _x(tf) = 0, and x(tf) =1. Hence
+we require
+y0= 1t2
+
+2; v 0=t; anda0=
+c1=4p
+22
+x(t) =2pc:
+If we setsf:=c1=4=p
+2 (tft), then the terminal conditions are
+y(sf) =y0(sf) = 0 and y00(sf) =
+c1=4p
+22
+x(tf) =2pc: (3.18)
+Solving (3.18) for y0,v0, andm, givena0=2pc, yields
+pc0
+@y0
+v0
+m1
+A=0
+@y0(sf)
+v0(sf)
+m(sf)1
+A
+:=1
+coshsfsinsf+ cossfsinhsf0
+@(coshsf+ cossf)(sinhsfsinsf)
+(sinhsfsinsf)2
+(coshsf+ cossf)21
+A:
+The matching condition
+v0(sf)2
+4pc=_x(t)2
+2=t2
+
+2=x(t) = 1y0(sf)pc;Optimal control with moderation incentives 17
+2468100.10.20.30.40.50.6
+µt∗
+02468102.152.202.252.302.352.402.45
+µtf
+Fig. 3.7 .Transition time tand total duration tffor the quadratic moderation incentive eCq.
+equivalently
+pc=y0(sf) +v0(sf)2
+4;
+implicitly determines a function sf:R+!(0;~sf), where tan ~ sf+ tanh ~sf= 0. The functions
+t(c) :=v0(sf(c))p
+2c1=4andtf(c) :=p
+2sf(c)
+c1=4+t(c) =p
+2
+c1=4
+sf(c)v0(sf(c))
+2
+give the desired transition time and duration. The curves t(c) andtf(c) are shown in Figure 3.7; plots
+of the position, velocity, and acceleration for some representative values of care shown in Figure 3.6.
+We now brie
y consider the alternative cost function bCqcc(x;u) =u2
+2+c
+2(1x)2. This cost function,
+in which the termu2
+2is naturally interpreted as a positive control cost added to the position-dependent
+cost, diers from the eCqmoderated problem analysed above only by the constant1
+2. Thus solutions
+of the synthesis problem for one cost function are solutions for the other, as well. A solution of the
+synthesis problem for the moderation incentive eCqis a solution for the arbitrary duration synthesis
+problem forbCqccif and only if the Hamiltonian determined by bCqccis equal to zero along the solution;
+equivalently: if and only if  x(0)2=cand x(tf) = 0. If we restrict our attention to c1, then this
+condition on the initial acceleration is compatible with the general control constraint jxj1. We seek
+M2R22andsf2R+such thaty(s) given by (3.15) satises the boundary conditions y(0) = 1,
+y0(0) =y(sf) =y0(sf) =y00(sf) = 0, andy00(0) =2, and hence
+x(t) = 1y
+c1=4=p
+2t
+satises the boundary conditions of the arbitrary duration synthesis problem for bCqcc, wheretf=
+c1=4p
+2sf, as before. After substituting y0= 1,v0= 0, anda0=2 into (3.16), we nd that the termi-
+nal conditions y(sf) =y0(sf) =y00(sf) = 0 are satised if and only if m= 2 cothsfand sinsf= 0. Thus
+the solutions of the arbitrary duration synthesis problem for bCqcchave duration tf(c;k) :=c1=4p
+2k,
+k2N. Ifk > 1, the solution of duration tf(c;k) overshoots the destination, makingk1
+2oscillations
+about the target before stopping.
+We now show that none of these solutions of the arbitrary duration problem for bCqccactually minimize
+the total cost;
+bCtot(tf) :=Ztf
+0bCqcc(x;x)dt=cm
+2; with derivative eC0
+tot(tf) =2cx(tf)2:18 D. Lewis
+123456/Minus0.20.20.40.60.81.0
+tposition
+accelerationc=1.75.5.25
+Fig. 3.8 .Solutions of the arbitrary duration synthesis problem for bCqcc, forc=:25,:5,:75,1andk= 1.
+0.20.40.60.81.00.10.20.30.4k=1k=2˙x
+x
+1.011.021.031.041.05
+/Minus0.020/Minus0.015/Minus0.010/Minus0.005˙xxk=2k=3
+Fig. 3.9 .Parametric plots (x;_x)of QCC solutions for c= 1,tf=kp
+2. Left:k= 1and2; right: close-up of k= 2
+and3
+Thus the cost is nonincreasing and the solutions of duration tf(k) are all in
ection points, satisfying
+bCtot(tf(c;k)) = 2ccothk. Hence increases in the total maneuver time yield exponentially small reduc-
+tions in cost.
+For relatively small values of c, Figures 3.4 and 3.8 show a qualitative resemblance between the
+moderated solutions for eC1and the solutions for bCqcc. To facilitate the comparison of these solutions, we
+reparametrize the eC1solutions using the rescaled time s:=c1=4
+p
+2t. The rescaling of the fourth evolution
+equation (3.13) takes the form
+d2
+ds2x00
+p
+1c
+4(x00)2= 4(1x):
+Hence the moderated solution approaches that of the QCC problem as !1 andc!0, but the two
+families are not equal for nonzero c. See Figure 3.10 for a comparison of the controls for the two families
+for some representative values of c.
+4. Conclusions and future work. Our choices of state space, vector elds, admissible control re-
+gions, and unmoderated cost functions were intended to be the simplest possible. We intend to generalize
+each of these components of the synthesis problem.
+Optimal control on nonlinear manifolds has received signicant attention in recent years, particularly
+situations in which the controls can be modeled as elements of a distribution within the tangent bundle
+of the state manifold, corresponding to (partially) controlled velocities. See, e.g. [16, 18, 19, 3], and refer-
+ences therein. Analogous constructions for (partially) controlled higher order derivatives (e.g. controlledOptimal control with moderation incentives 19
+0.51.01.52.02.53.0
+/Minus0.30/Minus0.25/Minus0.20/Minus0.15/Minus0.10/Minus0.050.000.05
+c=12c=14c=18←−←sx/prime/prime1−x/prime/primeqcc
+Fig. 3.10 .Dierences in acceleration for rescaled time trajectories: x00
+1(s;c)x00
+qcc(s;c)forc=1
+8,1
+4, and1
+2.
+acceleration) can be implemented using jet bundles, but can be unwieldy in practical implementations.
+We are particularly interested in Lie groups, since these manifolds possess additional structure that fa-
+cilitates the identication of the controls with elements of a single vector space, the Lie algebra. Results
+for conservative systems on Lie groups, homogeneous manifolds, and associated bundles should be easily
+extended to optimal control problems. Geometric integration schemes for the numerical integration of
+Hamiltonian systems can be used to approximate the solutions of synthesis problems on such manifolds;
+see [11, 7, 20, 9, 10] and references therein.
+The conservation law (2.6) can play a crucial role in the exact or approximate solution of the synthesis
+problem. In [8] we develop several results, including a reduction of the evolution equation for the auxiliary
+variables to the sphere, that exploit the conservation law. We show that a simple vertical take-o
+interception model with controlled velocities can be reduced to quadratures using this approach. We
+intend to generalize these results and apply them to more complex systems in future work.
+The assumption that the admissible control set is the unit ball can be relaxed to fu2V:f(u)cg
+for some vector space V, dierentiable function f:V![fmin;1) withrfeverywhere nonzero on
+@U, and constant c2R. The analog of eCqwould beeC(u) =cf(u); the analogs of eCwould be
+p
+cf(u). Clearly the property that the control uwould be a rescaling of the auxiliary variable 
+would not hold; hence the determination of the value of umaximizing the Hamiltonian would, in general,
+be more complicated than in the case considered here. More generally, the incentive could be a function
+of both control and state variables, retaining the property that the function rewards avoidance of the
+boundary of the admissible control set.
+We intend to investigate the skewed gradient equations (2.12) in greater detail, seeking both analytic
+properties and ecient numerical schemes. We hope to model various biomechanical systems using
+optimal control formulations of the type described here, investigating the utility of moderation incentives
+in the interpretation of animal motion and behavior. The construction of families of optimal solutions
+parametrized by moderation or urgency may shed light on aspects of motion planning that are not easily
+understood using a single cost function. For example, most of us utilize a range of strategies when lifting
+and carrying objects: a newborn, a laptop, and a phone book merit dierent levels of caution. Signicant
+expenditure of energy is required to capture prey or evade a predator, but exhaustion and lameness leave
+both predator and prey vulnerable to future attacks and starvation|long-term survival depends on the
+adjustment of resource consumption to the demands of each encounter. Even very simple mathematical
+models can add to our understanding of the adaptability of natural control systems.20 D. Lewis
+REFERENCES
+[1] J. Baillieul and J.C. Willems. Mathematical Control Theory . Springer, 1999.
+[2] Enrico Bertolazzi, Francesco Biral, and Mauro Da Lio. Real-time motion planning for multibody systems: real life
+application examples. Multibody Syst. Dyn. , 17:119{139, 2007.
+[3] Anthony Bloch. Nonholonomic Mechanics and Control . Springer, 2003.
+[4] J. F. Bonnans and Th. Guilbaud. Using logarithmic penalties in the shooting algorithm for optimal control problems.
+Optimal Control Appl. Methods , 24:257{278, 2003.
+[5] T.R. Kane and M.P. Scher. A dynamical explanation of the falling cat phenomenon. Int. J. Solids Structures ,
+5:663{670, 1969.
+[6] Donald E. Kirk. Optimal Control Theory: An Introduction . Dover Publications, Inc., 2004.
+[7] Benjamin Leimkuhler and Sebastian Reich. Simulating Hamiltonian Dynamics . Cambridge University Press, 2005.
+[8] Debra Lewis. Utilization of the conservation law in optimal control with moderation incentives. 2007.
+[9] Debra Lewis and Peter Olver. Geometric integration algorithms on homogeneous manifolds. Foundations of Compu-
+tational Mathematics , 2:363{392, 2002.
+[10] Debra Lewis and Peter Olver. Geometric integration and optimal control. 2007.
+[11] Debra Lewis and J.C. Simo. Conserving algorithms for the dynamics of Hamiltonian systems on Lie groups. J. Nonlin.
+Sci., 4:253{299, 1994.
+[12] E.-J. Marey. M echanique animale. La Nature , 1119:569{570, 1894.
+[13] Richard Montgomery. Optimal control of deformable bodies and its relation to gauge theory. Math. Sci. Res. Inst.
+Publ. , 22:403{438, 1991.
+[14] Richard Montgomery. Gauge theory of the falling cat. Fields Institute Communications , 1:193{218, 1993.
+[15] Richard Montgomery. Survey of singular geodesics. Progr. Math. , 1445:325{339, 1996.
+[16] H. Nijmeijer and A.J. van der Schaft. Nonlinear Dynamical Control Systems . Springer-Verlag, 1990.
+[17] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mishchenko. The Mathematical Theory of Optimal
+Processes . Interscience Publishers, 1962.
+[18] Eduardo D. Sontag. Integrability of certain distributions associated with actions on manifolds and applications to
+control problems. In Nonlinear controlability and optimal control , pages 81{131. Marcel Dekker, Inc., 1990.
+[19] Eduardo D. Sontag. Mathematical Control Theory: Deterministic Finite Dimensional Systems . Springer, 1998.
+[20] SYNODE publication database. http://www.math.ntnu.no/num/synode/bibliography.php , 2003.
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+arXiv:1001.0212v2  [math.GT]  12 Feb 2010QUASI-ISOMETRIC CLASSIFICATION OF NON-GEOMETRIC
+3-MANIFOLD GROUPS
+JASON A. BEHRSTOCK AND WALTER D. NEUMANN
+Abstract. We describethe quasi-isometricclassificationoffundamen tal groups
+of irreducible non-geometric 3-manifolds which do not have “too many” arith-
+metic hyperbolic geometric components, thus completing th e quasi-isometric
+classification of 3–manifold groups in all but a few exceptio nal cases.
+1.Introduction
+In [1] we discussed the quasi-isometry classification of fundamenta l groups of
+3–manifolds (which coincides with the bilipschitz classification of the un iversal cov-
+ers of three-manifolds). This classification reduces easily to the ca se of irreducible
+manifolds. Moreover, no generality is lost by considering only orienta ble mani-
+folds. So from now on we only consider compact connected orientab le irreducible
+3–manifolds of zero Euler characteristic (i.e., with boundary consist ing only of tori)
+since these are the orientable manifolds which, by Perelman’s Geomet rization The-
+orem [9, 10, 11], decompose along tori and Klein bottles into geometr ic pieces (this
+decomposition removes the boundary tori and a family of embedded tori, so the
+pieces of the decomposition are without boundary). The minimal suc h decomposi-
+tion is what is called the geometric decomposition .
+We described (loc. cit.) the classification for geometric 3–manifolds a s well as
+for non-geometric 3–manifolds with no hyperbolic pieces in their geom etric decom-
+position (i.e., graph-manifolds). For geometric manifolds this was a su mmary of
+work of others; our contribution was in the non-geometric case. I n this paper we
+extend to allow hyperbolic pieces. However, our results are still not quite complete:
+at present we exclude manifolds with “too many” arithmetic hyperbo lic pieces1,
+and some of our results are only proved assuming the “cusp coverin g conjecture” in
+dimension 3 (see below and Section 5).
+In the bulk of this paper we restrict to non-geometric manifolds, all of whose
+geometric components are hyperbolic and at least one of which is non -arithmetic
+(we will call these NAH-manifolds for short). In the final section 8 we extend to
+the case where Seifert fibered pieces are also allowed.
+The classificationfor graph-manifoldsin [1] wasin terms offinite labelle d graphs;
+the labelling consisted of a color black or white on each vertex and the classifying
+objects were such two-colored graphs which are minimal under a re lation called
+bisimilarity . ForNAH-manifolds the classificationis againin termsof finite labelled
+2000Mathematics Subject Classification. Primary 20F65; Secondary 57N10, 20F36.
+Key words and phrases. three-manifold, quasi-isometry, commensurability.
+Research supported under NSF grant no. DMS-0604524.
+1The essential remaining case to address is when allpieces are arithmetic hyperbolic. These
+behave rather differently from the other cases—more like ari thmetic hyperbolic manifolds.
+12 JASON A. BEHRSTOCK AND WALTER D. NEUMANN
+graphs, and the the classifying objects are again given by labelled gr aphs that are
+minimal in a similar sense. The labelling is more complex: each vertex is labe lled
+by the isomorphism type of a hyperbolic orbifold2and each edge is labelled by
+a linear isomorphism between certain 2-dimensional Q–vectorspaces. We will call
+these graphs NAH-graphs . The ones that classify are the ones that are minimal
+andbalanced. All these concepts will be defined in Section 3.
+Finally, when both Seifert fibered and hyperbolic pieces occur the cla ssifying
+graphs are a hybrid of the two-coloredgraphs and the NAH-graph s, so we call them
+H-graphs . In this case we need that every component of the manifold obtaine d by
+removing all Seifert fibered pieces is NAH. We will call irreducible non-g eometric
+manifolds of this type good. For example, 3–manifolds which contain a hyperbolic
+piece but no arithmetic hyperbolic piece provide a large family of good m anifolds.
+The following three theorems summarize our main results. The secon d two the-
+orems complement the first, by making the classification effective, a nd then relat-
+ing the quasi-isometric and commensurability classicationfor some NA H-manifolds;
+bothoftheselattertworesultscurrentlyneedCCC 3: the CuspCoveringConjecture
+(Conjecture 5.1) in dimension 3.
+Classification Theorem. Each good 3–manifold has an associated minimal H-
+graph and two such manifolds have quasi-isometric fundamen tal groups (in fact,
+bilipschitz equivalent universal covers) if and only if the ir minimal H-graphs are
+isomorphic.
+Realization Theorem. The minimal H-graph associated to a good 3–manifold
+is balanced. Assuming CCC 3, the converse is true, namely if a minimal H-graph
+is balanced, then it is the minimal H-graph for some quasi-is ometry class of good
+3–manifold groups.
+CommensurablityTheorem. Assuming CCC 3, if twoNAH-manifolds have quasi-
+isometric fundamental groups and their common minimal NAH- graph is a tree with
+manifold labels then they (and in particular, their fundame ntal groups) are com-
+mensurable.
+See Remark 5.4 for a discussion of our use of CCC 3. Although CCC would
+follow from RFCH (residual finiteness conjecture for word hyperb olic groups), it is
+not clear how much confidence one should have in RFCH. We feel that CCC is more
+plausible, and of independent interest. In any case, we expect the conclusionsof the
+Realization and Commensurability Theorems to be true regardless. O n the other
+hand,wedonotknowtowhatextentourrestrictiveconditionsont heminimalNAH-
+graph in the Commensurability Theorem are needed; whether “NAH- manifolds are
+quasi-isometric if and only if they are commensurable” holds in complet e generality
+remains a very interesting open question.
+A slightly surprising byproduct of this investigation are the minimal orbifolds of
+section 2, which play a role similar to commensurator quotients but ex ist also for
+cusped arithmetic hyperbolic orbifolds. Although their existence is e asy to prove,
+they were new to us. Minimal orbifolds are precisely the orbifolds tha t can appear
+as vertex labels of minimal NAH-graphs.
+2“Hyperbolic orbifold” always means an orientable complete hyperbolic 3–orbifold of finite
+volume.QUASI-ISOMETRIC CLASSIFICATION OF NON-GEOMETRIC 3-MANIF OLD GROUPS 3
+2.Minimal orbifolds
+We consider only orientable manifolds and orbifolds. Let Nbe a hyperbolic
+3–orbifold (not necessarily non-arithmetic) with at least one cusp. Then each cusp
+ofNhas a smallest cover by a toral cusp (one with toral cross section) and this
+cover has cyclic covering transformation group FCof order 1, 2, 3, 4, or 6. We call
+this order the “orbifold degree of the cusp.”
+Proposition 2.1. Among the orbifolds N′covered by Nhaving the same number
+of cusps as Nwith each cusp of N′having the same orbifold degree as the cusp of
+Nthat covers it, there is a unique one, N0, that is covered by all the others. We
+call thisN0aminimal orbifold .
+More generally, for each cusp of Nspecify a “target” in {1,2,3,4,6}that is
+a multiple of the orbifold degree and ask that the correspond ing cusp of N′have
+orbifold degree dividing this target. The same conclusion t hen holds.
+Proof.Wefirst“neuter” Nbyremovingdisjointopenhoroballneighborhoodsofthe
+cusps to obtain a compact orbifold with boundary which we call N0. If we prove
+the proposition for N0, with boundary components interpreted as cusps, then it
+holds for N.
+Let/tildewideN0be the universal cover of N0. Any boundary component CofN0is
+isomorphic to the quotient of a euclidean plane /tildewideCby the orbifold fundamental
+groupπorb
+1(C), which is an extension of a lattice Z2by the cyclic group FCof
+order 1, 2, 3, 4, or 6. We choose an oriented foliation of this plane by parallel
+straight lines and consider all images by covering transformations o f this foliation
+on boundary planes of /tildewideN0. Each boundary plane of /tildewideN0that covers Cwill then
+have|FC|oriented foliations, related by the action of the cyclic group FC. If the
+target degree nCfor the given cusp is a proper multiple of |FC|, we also add the
+foliations obtained by rotating by multiples of 2 π/nC.
+We give the boundary planes of /tildewideN0different labels according to which boundary
+component of N0they cover, and we construct foliations on them as above. So
+each boundary plane of /tildewideN0carries a finite number (1, 2, 3, 4, or 6) of foliations and
+the collection of all these oriented foliations and the labels on bounda ry planes are
+invariant under the covering transformations of the covering /tildewideN0→N0. LetGNbe
+the group of all orientation preserving isometries of /tildewideN0that preserve the labels of
+the planes and preserve the collection of oriented foliations. Note t hatGNdoes not
+depend on choices: The only relevant choices are the size of the hor oballs removed
+when neutering and the direction of the foliation we first chose on a b oundary
+plane of /tildewideN0. If we change the size of the neutering and rotate the direction of
+the foliation then the size of the neutering and direction of the image foliations at
+all boundary planes with the same label change the same amount, so the relevant
+data are still preserved by GN. It is clear that GNis discrete (this is true for any
+group of isometries of H3which maps a set of at least three disjoint horoballs to
+itself).N0
+0:=/tildewideN0/GNis thus an oriented orbifold; it is clearly covered by N0, has
+the same number of boundary components as N0, and the boundary component
+covered by a boundary component CofN0has orbifold degree dividing the chosen
+target degree nC. Moreover if N0
+1is any cover of N0
+0with the same property, then
+/tildewideN0
+1=/tildewideN0and the labellings and foliations on /tildewideN0
+1and/tildewideN0can be chosen the same,
+soN0
+0is the minimal orbifold also for N0
+1. The proposition thus follows. /square4 JASON A. BEHRSTOCK AND WALTER D. NEUMANN
+Note that a minimal orbifold may have a non-trivial isometry group (in contrast
+with commensurator quotients of non-arithmetic hyperbolic manifo lds).
+3.NAH-graphs
+The graphs we need will be finite, connected, undirected graphs. W e take the
+viewpoint that an edge of an undirected graph consists of a pair of o ppositely
+directed edges. The reversal of a directed edge ewill be denoted ¯ eand the initial
+and terminal vertices of an edge will be denoted ιeandτe(=ι¯e).
+We will label the vertices of our graph by hyperbolic orbifolds so we fir st intro-
+duce some terminology for these. A horosphere section Cof a cusp of a hyperbolic
+orbifoldNwill be called the cusp orbifold . Although the position of Cas a horo-
+sphere section of the cusp involves choice, as a flat 2–dimensional o rbifold,Cis
+canonically determined up to similarity by the cusp. We thus have one c usp orb-
+ifold for each cusp of N.
+Since a cusp orbifold Cis flat, its tangent space TCis independent of which
+point on Cwe choose (up to the action of the finite cyclic group FC).TCnaturally
+contains the maximal lattice Z2⊂πorb
+1(C), so it makes sense to talk of a linear
+isomorphism between two of these tangent spaces as being rational, i.e., given by
+a rational matrix with respect to oriented bases of the underlying in tegral lattices.
+Moreover, such a rational linear isomorphism will have a well-defined d eterminant.
+Definition 3.1 (NAH-graph) .AnNAH-graph is a finite connected graph with the
+following data labelling its vertices and edges:
+(1) Each vertex vis labelled by a hyperbolic orbifold Nvplus a map e/mapsto→Ce
+from the set of directed edges eexiting that vertex to the set of cusp
+orbifolds CeofNv. This map is injective, except that an edge which begins
+and ends at the same vertex may have Ce=C¯e.
+(2) The cusp orbifolds CeandC¯ehave the same orbifold degree.
+(3) Eachdirected edge eislabelled bya rationallinearisomorphism ℓe:TCe→
+TC¯e, withℓ¯e=ℓ−1
+e. Moreover ℓereverses orientation (so det( ℓe)<0).
+(4) We only need ℓeup to right multiplication by elements of Fe:=FCe; in
+other words, the relevant datum is really the coset ℓeFerather than ℓeitself.
+This necessitates:
+(5)ℓeconjugates the cyclic group Feto the cyclic group F¯e, i.e.,ℓeFe=F¯eℓe
+(this holds automatically if the orbifold degree is ≤2; otherwise it is equiv-
+alent to saying that ℓeis a similarity for the euclidean structures).
+(6) At least one vertex label Nvis a non-arithmetic hyperbolic orbifold.
+Definition 3.2. We callanNAH-graph balanced ifthe product ofthe determinants
+of the linear maps on edges around any closed directed path is equal to±1.
+We call the NAH-graph integralif eachℓeis an integral linear isomorphism (i.e.,
+an isomorphism of the underlying Z–lattices). Integral clearly implies balanced.
+Definition 3.3. An integralNAH-graph containspreciselythe information to spec-
+ify how to glue neutered versions N0
+vof the orbifolds Nvtogether alongtheir bound-
+ary components to obtain a NAH-orbifold M, called the associated orbifold . Con-
+versely, any NAH-orbifold Mhas anassociated integral NAH-graph Γ(M), which
+encodes its decomposition into geometric pieces.
+We next want to define morphisms of NAH-graphs.QUASI-ISOMETRIC CLASSIFICATION OF NON-GEOMETRIC 3-MANIF OLD GROUPS 5
+Definition 3.4 (Morphism) .Amorphism of NAH-graphs, Γ →Γ′, consists of the
+following data:
+(1) an abstract graph homomorphism φ: Γ→Γ′, and
+(2) for each vertex vof Γ, a covering map πv:Nv→Nφ(v)of the orbifolds
+labelling vandφ(v) which respects cusps (so for each departing edge eat
+vone hasπv(Ce) =Cφ(e)), subject to the condition:
+(3) for each directed edge eof Γ the following diagram commutes
+TCe
+/d15/d15ℓe/d47/d47TC¯e
+/d15/d15
+TCφ(e)ℓφ(e)/d47/d47TCφ(¯e)
+Here the vertical arrowsare the induced maps of tangent spaces and commutativity
+of the diagram is up to the indeterminacy of item (4) of Definition 3.1.
+Morphisms compose in the obvious way, so a morphism is an isomorphism if and
+only ifφis a graph isomorphism and each πvhas degree 1.
+Definition 3.5. An NAH-graph is minimal if every morphism to another NAH-
+graph is an isomorphism.
+The relationship of existence of a morphism between NAH-graphs ge nerates an
+equivalence relation which we call bisimilarity .
+Theorem 3.6. Every bisimilarity class of NAH-graphs contains a unique (u p to
+isomorphism) minimal member, so every NAH-graph in the bisi milarity class has
+a morphism to it.
+NAH-graphs play an analogous role to the two-colored graphs that we used in
+[1] to classify graph-manifold groups up to quasi-isometry. For tho se graphs the
+morphisms were called “weak coverings”—they were color preservin g open graph
+homomorphisms (i.e., open as maps of 1–complexes). The term “bisimila rity” was
+coined in that context, since the concept is known to computer scie nce under this
+name. Theorem 3.6 has a proof analogous to the proof we gave in [1] for two-
+colored graphs. We give a different proof, which we postpone to the next section,
+since it follows naturally from the discussion there (an analogous pro of works in
+the two-colored graph case).
+Theorem 3.7. IfΓ→Γ′is a morphism of NAH-graphs and Γis balanced, then
+so isΓ′. In particular, a bisimilarity class contains a balanced NA H-graph if and
+only if its minimal NAH-graph is balanced.
+Proof.We will denote the negative determinant of the linear map labelling an ed ge
+eof an NAH-graph by δe, soδe=δ−1
+¯e>0.
+Assume Γ is balanced. We will show Γ′is balanced. We only need to show
+that the product of the δe’s along any simple directed cycle in Γ′is 1. Let C=
+(e′
+0,e′
+1,...,e′
+n−1), be such a cycle, so τe′
+i=ιe′
+i+1for each i(indices modulo n).
+Denote
+(1) D:=n−1/productdisplay
+i=0δe′
+i,
+so we need to prove that D= 1.6 JASON A. BEHRSTOCK AND WALTER D. NEUMANN
+From now on we will restrict attention to one connected component Γ0of the
+full inverse image of Cunder the map Γ →Γ′.
+Letebe an edge of Γ 0which maps to an edge e′ofC(e′may be an e′
+ior an
+¯e′
+i). Its start and end correspond to cusp orbifolds which cover the cusp orbifolds
+corresponding to start and end of e′with covering degrees that we shall call deand
+d¯erespectively. We claim that
+(2) deδe′=δed¯e.
+Indeed, if the cusp orbifolds are all tori, this equation (multiplied by −1) just repre-
+sents products of determinants for the two ways of going around the commutative
+diagram in part (3) of Definition 3.4. In general, deandd¯eare the determinants
+multiplied by the order of the cyclic group Fe, but this is the same factor for each
+ofdeandd¯eso the equation remains correct.
+Let/tildewideC → Cbe the infinite cyclic cover of C, and/tildewideΓ0→Γ0the pulled back infinite
+cyclic cover of Γ 0. Define deorδefor an edge of either of these covers as the value
+for the image edge.
+If (e1,e2,...,e m) is a directed path in /tildewideΓ0then the product of the equations (2)
+over all edges of this path gives/producttextm
+i=1deiδπei=/producttextm
+i=1δeid¯ei, so
+m/productdisplay
+i=1δπei
+δei=m/productdisplay
+i=1d¯ei
+dei.
+Since/tildewideΓ0and/tildewideCare both balanced, it follows that/producttextm
+i=1d¯ei
+deionly depends on the
+start and end vertex of the path. Thus, if we fix a base vertex v0of/tildewideΓ0and define
+(3) c(v) :=m/productdisplay
+i=1δπei
+δei=m/productdisplay
+i=1d¯ei
+deifor any path from v0tov,
+we get a well defined invariant of the vertices of /tildewideΓ0. For an edge ethis invariant
+satisfies
+c(ι¯e)
+c(ιe)=c(τe)
+c(ιe)=d¯e
+de,
+whence
+ζ(e) :=de
+c(ιe)
+is an invariant of the undirected edge: ζ(e) =ζ(¯e).
+Denote the map /tildewideΓ0→/tildewideCbyπ. Number the vertices sequentially along /tildewideCby
+integersi∈Z, and for each vertex vof/tildewideΓ0leti(v)∈Zbe the index of π(v). Let
+h:/tildewideΓ0→/tildewideΓ0be the covering transformation: h(v) is the vertex of /tildewideΓ0with the same
+image in Γ 0asvbut with i(h(v)) =i(v)+n.
+Note that the equations (1) and (3) and the fact that Γ 0is balanced implies that
+c(h(v)) =Dc(v) for any vertex vof/tildewideΓ0, so
+(4) ζ(h(e)) =D−1ζ(e).
+From now on consider the edges of /tildewideΓ0directed only in the direction of increasing
+i(v). Denote by Z(j) the sum of ζ(e) over all edges with i(τe) =j.
+The sum of the de’s overoutgoing edges at a vertex vof/tildewideΓ0equals the sum of d¯e’s
+overincomingedges at v, since each sum equalsthe degreeofthe coveringmap from
+the orbifold labelling vto the one labelling vertex π(v) of/tildewideC. Sinceζ(e) =de/c(v)QUASI-ISOMETRIC CLASSIFICATION OF NON-GEOMETRIC 3-MANIF OLD GROUPS 7
+for an outgoing edge and ζ(¯e) =d¯e/c(v) for an incoming one, the sum of ζ(e) over
+outgoing edges at vequals the sum of ζ(e) for incoming ones. Thus Z(j) is the
+sum ofζ(e) over all edges with i(ιe) =j. These are the edges with i(τe) =j+1,
+soZ(j) =Z(j+1). Thus Z(j) is independent of j. Clearly Z(j)>0. Equation (4)
+impliesZ(j+n) =D−1Z(j), soD= 1, as was to be proved. (We are grateful to
+Don Zagier for help with this proof.) /square
+We close this section with an observation promised in the introduction .
+Proposition 3.8. An orbifold can be a vertex label in a minimal NAH-graph if and
+only if it is a minimal orbifold.
+Proof.We first prove “only if.” In condition (3) of the definition of a morphism ,
+ℓφ(e)is determined by ℓesince the vertical arrows are isomorphisms (but ℓemay
+not be determined by ℓφ(e), since the indeterminacy Fφ(e)ofℓφ(e)may be greater
+than the indeterminacy Feofℓe). Thus for any NAH-graph there is an outgoing
+morphism to a new NAH-graph for which the underlying graph homomo rphism
+is an isomorphism and every orbifold vertex label is simply replaced in th e new
+graph by the corresponding minimal orbifold; the new edge labels are then as just
+described.
+Fortheconverse,if Nisaminimalorbifold, anyNAH-graphwhichisstar-shaped,
+withNlabelling the middle vertex and with one-cusp non-arithmetic commens ura-
+tor quotients labelling the outer ones, is minimal. One needs a one-cus p commen-
+surator quotient for each cusp degree in {1,2,3,4,6}for this construction; these
+are not hard to find. /square
+4.Minimal NAH-graphs classify for quasi-isometry
+LetM=M3be a NAH-manifold. For simplicity of exposition we first discuss
+the casethat Mhasnoarithmetic pieces, and then discuss the modificationsneeded
+when arithmetic pieces also occur.
+Mis then pasted together from pieces Mi, each of which is a neutered non-
+arithmetic hyperbolic manifold. We can choose neuterings in a consist ent way,
+by choosing once and for all a neutering for the commensurator qu otient in each
+commensurability class of non-arithmetic manifolds, and choosing ea chMito cover
+one of these “standard neutered commensurator quotients.”
+The pasting identifies pairs of flat boundary tori with each other by affine maps
+which may not be isometries. To obtain a smooth metric on the result w e glue a
+toral annulus T2×Ibetween the two boundaries with a metric that interpolates
+between the flat metrics at the two ends in a standard way (if g0andg1are the flat
+product metrics induced on T2×Iby the flat metric on its left and right ends we
+use (1−ρ(t))g0+ρ(t)g1whereρ: [0,1]→[0,1] is some fixed smooth bijection with
+derivatives 0 at each end). This specifies the metric on Mup to rigid translations
+of the gluing maps, so we get a compact family of different metrics on M.
+The universal cover /tildewiderMis glued from infinitely many copies of the /tildewiderMi’s with
+“slabs”R2×Iinterpolating between them. Each slab admits a full R2of isometric
+translations. The pieces /tildewiderMiwill be called “pieces” and each boundary component
+of a piece will be called a “flat” and will be oriented as part of the bound ary of the
+piece it belongs to.
+LetM′be another such manifold and /tildewiderM′its universal cover, metrized as above.
+Kapovich and Leeb [5] show that any quasi-isometry f:/tildewiderM→/tildewiderM′is a bounded8 JASON A. BEHRSTOCK AND WALTER D. NEUMANN
+distance from a quasi-isometry that maps slabs to slabs and geomet ric pieces to
+geometric pieces. Then a theorem of Schwartz [12] says that fis a uniformly
+bounded distance from an isometry on each piece /tildewiderMi. Our uniform choice of neu-
+terings assures that we can change fby a bounded amount to be an isometry on
+each piece and an isometry followed by a shear map on each slab (a “sh ear map”
+R2×I→R2×Iwill mean one of the form ( x,t)/mapsto→(x+ρ(t)v,t) withv∈R2and
+ρ:I→Ias described earlier). We then say fisstraightened .
+Consider now the group3
+I(/tildewiderM) :={f:/tildewiderM→/tildewiderM|fis a straightened quasi-isometry }.
+I(/tildewiderM) acts on the set of pieces of /tildewiderM; the pieces /tildewiderMithat cover a given piece Mi
+ofMare all in one orbit of this action, but the orbit may be larger. The sub group
+I/tildewiderMi(/tildewiderM) ofI(/tildewiderM) that stabilizes a fixed piece /tildewiderMiof/tildewiderMacts discretely on /tildewiderMi, so
+/tildewiderMi/I/tildewiderMi(/tildewiderM) is an orbifold (which clearly is covered by Mi)
+The subgroup IS(/tildewiderM)⊂ I(/tildewiderM) that stabilizes a slab S⊂/tildewiderMacts onSby isome-
+tries composed with shear maps. It acts discretely on each bounda ry component
+ofSbut certainly not on S. But an element that is a finite order rotation on one
+boundary component must be a similar rotation on the other bounda ry component
+(and is in fact finite order on the slab; the group IS(/tildewiderM) is abstractly an extension
+ofZ2×Z2by a finite cyclic group FSof order 1, 2, 3, 4, or 6; the two Z2’s are the
+translation groups for the two boundaries of S).
+We form an NAH-graph Ω( /tildewiderM) as follows. The vertices of Ω( /tildewiderM) correspond
+toI(/tildewiderM)–orbits of pieces and the edges correspond to orbits of slabs—th e edge
+determined by a slab connects the vertices determined by the abut ting pieces. We
+label each vertex by the corresponding orbifold /tildewiderMi/I/tildewiderMi(/tildewiderM) and each edge by the
+derivative of the affine map between the flats that bound a corresp onding slab S
+(this map is determined up to the cyclic group FS).
+Proposition 4.1. The NAH-graph Ω(/tildewiderM)constructed above is determined up to
+isomorphism by /tildewiderM. The manifold /tildewiderMis determined up to bilipschitz diffeomorphism
+byΩ(/tildewiderM).
+Proof.The first sentence of the proposition is true by construction.
+We describe how to reconstruct /tildewiderMfrom Ω(/tildewiderM). Ω(/tildewiderM) gives specifications for
+inductively gluing together pieces that are the universal covers of the orbifolds cor-
+responding to its vertices, and slabs between these pieces, accor ding to a tree: we
+start with a piece /tildewideNcorresponding to a vertex vof Γ and glue slabs and adjacent
+pieces on all boundary components as specified by the outgoing edg es atvin Ω(/tildewiderM),
+and repeat this process for each adjacent piece, and continue ind uctively. (There
+is an underlying tree for this construction which is the universal cov er of the graph
+obtained by replacing each edge of Γ by a countable infinity of edges.) The con-
+struction involves choices, since each gluing map is only determined up to a group
+of isometries of the form R2⋊FS, whereSis the slab. We need to show that the
+resulting manifold is well defined up to bilipschitz diffeomorphism.
+3Note that I(/tildewiderM)→ QI(/tildewiderM)is an isomorphism,where QIdenotes the group ofquasi-isometries
+(defined by identifying maps that differ by a bounded distance ).QUASI-ISOMETRIC CLASSIFICATION OF NON-GEOMETRIC 3-MANIF OLD GROUPS 9
+The constructed manifold Xand the original /tildewiderMcan both be constructed in the
+same way from Ω( /tildewiderM), but they potentially differ in the choices just mentioned. We
+can construct a bilipschitz diffeomorphism f:X→/tildewiderMinductively, starting with
+an isometry from one piece of Xto one piece of /tildewiderMand extending repeatedly over
+adjacent pieces. At any point in the induction, when extending to an adjacent piece
+across an adjacent slab S, we use an isometry of the adjacent piece XiofXto the
+adjacent piece Miof/tildewiderMthat takes the boundary component Xi∩SofXito the
+boundary component Mi∩SofMi. Restricted to this boundary component E, this
+isometry is well defined up to the action of a lattice Z2, so there is a choice that can
+be extended across the slab Swith an amount of shear bounded by the diameter
+of the torus E/Z2. Since only finitely many isometry classes of such tori occur in
+the construction, we can inductively construct the desired diffeom orphism using a
+uniformly bounded amount of shear on slabs. This diffeomorphism the refore has a
+uniformly bounded bilipschitz constant, as desired. /square
+We now describe how the abovearguments must be modified if Mhas arithmetic
+pieces. Let M1be an arithmetic hyperbolic piece which is adjacent to a non-
+arithmetic hyperbolic piece M2. We have already discussed how M2is neutered;
+we take an arbitrary neutering of M1(and any other arithmetic pieces) which we
+will adjust later. As before, the notation Mirefers to the neutered pieces and we
+glueMfrom these pieces mediating with toral annuli T2×Ibetween them.
+We aim to show that we can adjust the neutering of the arithmetic pie ces so that
+any quasi-isometry of /tildewiderMcan be straightened as in the beginning of this section to
+be an isometry on pieces and a shear map on slabs.
+Consider /tildewiderM1as a subset of H3obtained by removing interiors of infinitely many
+disjoint horoballs. Schwartz [12] shows that any quasi-isometry of /tildewiderM1is a bounded
+distance from an isometry of /tildewiderM1to a manifold obtained by changing the sizes of
+the removed horoballs by a uniformly bounded amount.
+In the universal cover /tildewiderMchoose lifts /tildewiderM1and/tildewiderM2glued to the two sides ∂1Sand
+∂2Sof a slab S∼=R2×I. Consider a quasi-isometry fof/tildewiderMwhich maps Sto a
+bounded Hausdorff distance from itself. We can assume that fis an isometry on
+/tildewiderM2. By the previous remarks, the map frestricted to /tildewiderM1is a bounded distance
+from an isometry of /tildewiderM1which moves its boundary component ∂1Sto a parallel
+horosphere (if we consider /tildewiderM1as a subset of H3); by inverting fif necessary, we
+can assume the diameter of the horosphere has not decreased. T hus/tildewiderM1can be
+positioned in H3={(z,y)∈C×R:y >0}so that∂1Sis the horosphere y= 1 and
+f(∂1S) is the horosphere y=λfor some λ≤1. Using a smooth isotopy of fwhich
+is supported in an ǫ–neighborhood of the region between f(∂1S) and∂1S, and
+which moves f(∂1S) to∂1S, we can adjust fto map∂1Sto itself. This moves each
+point off(∂1S) to its closest point on ∂1Sby a euclidean similarity, scaling distance
+uniformly by a factor of λ. The resulting adjusted fis still a quasi-isometry, so
+restricted to Swe then have a quasi-isometry which scales metric on ∂1Sbyλand
+is an isometry on ∂2S. This is only possible if λ= 1, sof, once straightened on
+/tildewiderM1, maps∂1Sto itself.
+Now consider the subgroupof the groupof quasi-isometriesof /tildewiderMwhich takes /tildewiderM1
+to itself, and just consider its restriction to /tildewiderM1, which we can think of as embedded
+inH3. By straightening, we have a group of isometries of H3which preserves a10 JASON A. BEHRSTOCK AND WALTER D. NEUMANN
+family of disjoint horoballs (the ones that are bounded by images of ∂1S). Any
+subgroup of Isom( H3) which preserves an infinite family of disjoint horoballs is
+discrete. Thus, from the point of view of the construction above, M1behaves like a
+non-arithmetic piece. By repeating the argument, this behavior pr opagates to any
+adjacentarithmeticpieces, hence, solongasat leastonepiece is no n-arithmetic, the
+constructionofthe graphΩ( /tildewiderM) goesthroughasbefore and the proofofProposition
+extends.
+Proposition 4.2. Ω(/tildewiderM)is balanced, and is the minimal NAH-graph in the bisim-
+ilarity class of the NAH-graph Γ(M)associated with M(Definition 3.3).
+Proof.By construction, there is a morphism Γ( M)→Ω(/tildewiderM). In particular, Ω( /tildewiderM)
+is in the bisimilarity class of Γ( M). It is balanced by Theorem 3.7, since Γ( M) is
+integral. It remains to show that it is the minimal NAH-graph in its class .
+We have not yet proved Theorem 3.6, which says that there is a uniqu e minimal
+graph in each bisimilarity class. The proof that Ω( /tildewiderM) is minimal will follows from
+that proof, so we do that first.
+Proof of Theorem 3.6. The construction of the proof of Proposition 4.1 works for
+any NAH-graph Γ, gluing together infinitely may copies of the univers al covers /tildewideNi
+of the orbifolds that label the vertices, with slabs between them, a ccording to an
+infinite tree (the universal cover of the graph obtained by replacin g each edge of
+Γ by countable-infinitely many). We get a simply connected riemannian manifold
+X(Γ) which, as long as Γ is finite, is well defined up to bilipschitz diffeomorp hism
+by the same argument as before.
+The group of straightened self-diffeomorphisms I(X(Γ)), when restricted to a
+piece/tildewideNi, includes the coveringtransformationsfor the covering /tildewideNi→Ni. It follows
+that the construction of a NAH-graph from X(Γ), as given in the first part of this
+section, yields an NAH-graph Ω( X(Γ)) together with a morphism Γ →Ω(X(Γ)).
+If Γ→Γ′is a morphism of NAH-graphs, then X(Γ) and X(Γ′) are bilipschitz
+diffeomorphic, since the instructions for assembling them are equiva lent. Hence
+Ω(X(Γ)) = Ω( X(Γ′)); call this graph m(Γ). Since m(Γ) =m(Γ′) and the existence
+of a morphism generates the relation of bisimilarity of NAH-graphs, m(Γ) is the
+same for every graph in the bisimilarity class. It is also the target of a morphism
+from every Γ in this class. Thus it must be the unique minimal element in t he class,
+so Theorem 3.6 is proved. /square
+IfMis an NAH-orbifold and Γ = Γ( M) its associated integral NAH-graph, then
+X(Γ) reconstructs /tildewiderM, so Ω(/tildewiderM) = Ω(X(Γ)) =m(Γ), and is hence minimal by the
+previous proof, so Proposition 4.2 now follows. /square
+Proof of Classification Theorem for NAH-manifolds. Since/tildewiderMis quasi-isometric to
+π1(M), a quasi-isometrybetween fundamental groupsof MandM′induces a quasi-
+isometry /tildewiderM→/tildewiderM′. We have already explained how this can then be straightened
+and thus give an isomorphism of the corresponding minimal NAH-grap hs. /square
+For the Classification Theorem to be a complete classification of quas i-isometry
+types of fundamental groups of good manifolds we will need to know that every
+minimal balanced NAH-graph is realized by an NAH-manifold. We formula te this
+as a conjecture (the Realization Theorem says this conjecture fo llows from the the
+cusp covering conjecture CCC 3).QUASI-ISOMETRIC CLASSIFICATION OF NON-GEOMETRIC 3-MANIF OLD GROUPS 11
+Conjecture 4.3. Every minimal balanced NAH-graph is the minimal NAH-graph
+for some NAH-manifold. Equivalently, every bisimilarity c lass of NAH-graphs
+which contains a balanced NAH-graph contains integral NAH- graphs (Definition
+3.2).
+5.Covers and RFCH
+For ourrealizationtheorem, and alsofor ourcommensurabilitytheo rem, we need
+to produce covers of 3–manifolds with prescribed boundary behav ior. The covers
+we need can be summarized by the following purely topological conjec ture, which
+we find is of independent interest.
+Cusp Covering Conjecture 5.1 (CCCn).LetMbe a hyperbolic n-manifold.
+Then for each cusp CofMthere exists a sublattice ΛCofπ1(C)such that, for any
+choice of a sublattice Λ′
+C⊂ΛCfor each C, there exists a finite cover M′ofM
+whose cusps covering each cusp CofMare the covers determined by Λ′
+C.
+As we rely on this conjecture for n= 3, we shall now show that it follows from
+the well-known residual finiteness conjecture for hyperbolic grou ps (RFCH).
+Theorem 5.2. The residual finiteness conjecture for hyperbolic groups (R FCH)
+implies CCC nfor alln.
+Proof.The Dehn surgery theorem for relatively hyperbolic groups of Osin [ 8] and
+Groves and Manning [2] guarantees the existence of sublattices Λ Cofπ1(C) for
+each cusp so that, given any subgroups Λ′
+C⊂ΛCfor each C, the result of adding
+relations to π1(M) which kill each Λ′
+Cgives a group Ginto which the groups
+π1(C)/Λ′
+Cinject and which is relatively hyperbolic relative to these subgroups. If
+the Λ′
+Care sub-lattices, then Gis relatively hyperbolic relative to finite (hence
+hyperbolic) subgroups, and is hence itself hyperbolic. By RFCH we ma y assume G
+is residually finite, so there is a homomorphism of Gto a finite group Hsuch that
+each of the finite subgroups π1(C)/Λ′
+CofGinjects. The kernel Kof the composite
+homomorphism π1(M)→G→Hthus intersects each π1(C) in the subgroup Λ′
+C.
+The covering of Mdetermined by Ktherefore has the desired property. /square
+Let Γ be a NAH-graph. For an edge eof Γ letTebe the tangent space of the
+cusp orbifold corresponding to the start of e. We can identify TewithT¯eusing the
+linear map ℓe, so we will generally not distinguish TeandT¯e. ThenTecontains
+twoZ–lattices, the underlying lattices for the orbifolds at the two ends o fe, and
+we will denote their intersection by Λ e. Thus, a torus Te/Λ is a common cover of
+the cusp orbifolds at the two ends of eif and only if Λ ⊂Λe.
+Assuming CCC 3, we can now choose a sublattice Λ′
+eof Λefor each edge eof Γ
+such that for each vertex vof Γ the corresponding orbifold Nvhas a cover Mvwith
+the following property:
+Property 5.3. For each cusp of Nvcorresponding to an edge edeparting v, all
+cusps of Mvwhich cover it are of type Te/Λ′
+e.
+This property is precisely the consequence of the CCC 3that we use in our proofs
+of the Realization and Commensurability Theorems.
+Remark 5.4. It clearly is desirable to avoid using CCC 3, but the current state
+of the art in constructing covers of a hyperbolic manifold Mwhich restrict to12 JASON A. BEHRSTOCK AND WALTER D. NEUMANN
+prescribed covers on the cusps does not go far enough. For exam ple, Hempel
+observed in [4] that one can do this for characteristic covers of th e cusps of degrees
+avoiding a finite set of primes. A characteristic cover is one given by a sublattice of
+the form qπ1(C) for some q >0. This is much too restrictive (a cover of the cusps
+corresponding to ends of an edge ein an NAH-graph can be characteristic for both
+cusps only if map ℓeis a rational multiple of a Z–isomorphism). By filling a cusp
+by Dehn surgery and then applying results of E. Hamilton [3] to try t o prescribe
+covers of the resulting geodesic one can get a little closer, but still f ar from what is
+needed.
+6.Realizing graphs
+Proof of Realizability Theorem for NAH-manifolds. LetΓbeabalancedNAH-graph.
+We want to show there is some NAH-manifold which realizes a graph in its bisimi-
+larity class. As pointed out in the previous section, this is equivalent t o finding an
+integral NAH-graph in the bisimilarity class.
+Since we assume CCC 3, after we choose a sublattice Λ′
+eof Λefor each edge eof
+Γ, we may assume Property 5.3 holds.
+Letdvbe the degree of the cover Mv→Nv. For an edge edeparting vletde
+be the degree of the corresponding cover of cusp orbifolds of MvandNv, i.e., the
+index of the lattice Λ′
+ein the fundamental group of the boundary component Ce
+ofNvcorresponding to e(this is slightly different from the usage in the proof of
+Theorem 3.7). Since the cusps of the Mvcorresponding to the two ends of eare
+equal, we have
+(5) deδe=d¯e.
+(Recall that δedenotes the determinant of the linear map ℓe.)
+Since Γ is balanced, we can assign a positiverational number m(v) to each vertex
+with the property that for any edge eone hasm(τe) =δem(ιe). Thus, by (5),
+(6)m(τe)
+d¯e=m(ιe)
+de.
+Choose a positive integer bsuch that n(v) :=bm(v)
+dvis integral for every vertex of
+Γ (sobis some multiple of the lcm of the denominators of the numbersm(v)
+dv). Let
+M′
+vbe the disjoint union of n(v) copies of Mv, soM′
+vis abm(v)–fold cover of Nv.
+Letπv:M′
+v→Nvbe the covering map.
+For an edge eof Γ from v=ιetow=τe, the number of boundary components
+ofM′
+vcovering the boundary component CeofNvcorresponding to eisbm(v)/de.
+By (6) this equals bm(w)/d¯e, which is the number of boundary components of M′
+w
+covering the boundary component C¯eofNw. Thusπ−1
+vCeandπ−1
+wC¯ehave the
+same number of components, and each component is Te/Λ′, so we can glue M′
+vto
+M′
+walong these boundary components using any one-one matching bet ween them.
+Doing this for every edge gives a manifold Mwhose NAH-graph has a morphism
+to Γ; ifMis disconnected, replace it by a component (but one can always do th e
+construction so that Mis connected). This proves the theorem. /square
+7.Commensurability
+Proof of Commensurability Theorem. LetM1andM2betwoNAH-manifoldswhose
+NAH-graphs are bisimilar. Assume that their common minimal NAH-gra ph is aQUASI-ISOMETRIC CLASSIFICATION OF NON-GEOMETRIC 3-MANIF OLD GROUPS 13
+tree and all the vertex labels are manifolds. We want to show that M1andM2are
+commensurable.
+Let Γ be the common minimal NAH-graph for M1andM2. Since we assume
+CCC3, we may assume Property 5.3 holds. Accordingly, for each vertex o f Γ we
+may take a common cover N′
+vof all the pieces of M1that cover Nvand then choose
+the lattices Λ′
+e, as in the previous section, to be subgroups of the cusp groups of
+these manifolds N′
+v. In this way we arrange that the pieces Mvof the manifold
+Mconstructed in that section are covers of N′
+v, and hence of the pieces of M1.
+Elementary arithmetic as in the proof of the Realization Theorem sho ws that there
+is ab0so that if the number bof that proof is a multiple of b0then we can choose
+the glueing in that proof to make Ma covering space of M1. Call the resulting
+manifold M′
+1. If we initially choose N′
+vto also cover the type vpieces of M2then
+we can also construct a covering space M′
+2ofM2out of copies of pieces Mv. The
+decompositions of M′
+1andM′
+2then give NAH-graphs whose vertices are labelled
+byMv’s and whose edges are labelled by Z–isomorphisms. It suffices to show that
+M′
+1andM′
+2are commensurable.
+We can encode the information needed to construct M′
+1in a simplified version
+Γ0(M′
+1) of its NAH-graph. The underlying graph is still just the graph desc ribing
+the decomposition of M′
+1into pieces, but the labelling is simplified as follows. Each
+vertexwof Γ0(M′
+1) corresponds to a copy of some Mv, which is a normal covering
+of the orbifold Nv. We say vertex wis oftypev. The coveringtransformation group
+for the covering Mv→Nvinduces a permutation group Pvon the edges of Γ 0(M′
+1)
+exitingw. We record the type and permutation action for each vertex of Γ 0(M′
+1).
+It is easy to see that the graph Γ 0(M′
+1) with these data records enough information
+to reconstruct M′
+1up to diffeomorphism from its pieces. A covering of such a graph
+induces a covering of the same degree for the manifolds they encod e.
+A graph with fixed permutation actions at vertices as above is called symmetry
+restricted in [7]. The graphs Γ 0(M′
+1) and Γ 0(M′
+2) have the same universal covering
+as symmetry restricted graphs.
+So we would like to know that when two such graphs have a common univ ersal
+covering, then they have a common finite covering. A generalization , proved in
+[7] of Leighton’s theorem [6] (which deals with graphs without the ext ra structure)
+shows that this is true if the underlying graph is a tree, so we are don e. /square
+The results of [7] allow one to carry out the above proof under slight ly weaker
+assumptions on the minimal graph than being a tree, but given the ot her strong
+assumption (CCC 3) used in the proof, it does not seem worth going into details.
+8.Adding Seifert fibered pieces
+We will modify Definition 3.1to allowthe inclusion ofSeifert fibered space pieces.
+The graphs we use are called H–graphs, and we define them below.
+We will need to consider Seifert fibered orbifolds among the pieces, s o we first
+describe a coarse classification into types. As always, the manifolds and orbifolds
+we consider are oriented. We will distinguish two types: the oriented Seifert fibered
+orbifold Nis type “o” or “n” according as the Seifert fibers can be consistent ly
+oriented or not. Nis type “o” if and only if the base orbifold Sof the Seifert
+fibration is orientable. This can fail in two ways: the topological surf ace underlying
+Smay be non-orientable, or Smay be non-orientable because it has mirrors. The
+latter arises when parts of Nlook locally like a Seifert fibered solid torus D2×S114 JASON A. BEHRSTOCK AND WALTER D. NEUMANN
+factored by the involution ( z1,z2)/mapsto→(z1,z2) (using coordinates in C2with|z1| ≤1
+and|z2|= 1). Thefiberswith z1∈Rin thislocaldescriptionareintervals(orbifolds
+of the form S1/(Z/2)), and the set of base points of such interval fibers of Nform
+mirror curves in Swhich are intervals and/or circles embedded in the topological
+boundary of S. If any of these mirror curves are intervals, so they merge with
+part of the true boundary of S(image of boundary of N), then the corresponding
+boundary component of Nis a pillow orbifold (topologically a 2–sphere, with four
+2–orbifold points).
+Iftwotype“o”Seifertfiberedpiecesinthedecompositionof Mareadjacentalong
+a torus, and we have oriented their Seifert fibers, then there is a s ense in which
+these orientations are compatible or not. We say the orientation is compatible or
+positiveif, when viewing the torus from one side, the intersection number in t he
+torus of a fiber from the near side with a fiber from the far side is pos itive. Note
+that this is well defined, since if we view the torus from the other side , both its
+orientation and the order of the two curves being intersected hav e changed, so the
+intersection number is unchanged.
+We define H–graphs below, with the geometric meanings of the new ing redients
+in square brackets. But there is a caveat to these descriptions. J ust an an NAH–
+graph can be associated to the geometric decomposition of an NAH– manifold, an
+H–graph can be associated with the decomposition of a good manifold or orbifold
+M(see the Introduction for the definition of “good”). However, th e NAH–graph
+associatedwith a geometric decompositionis a special kind ofNAH–gr aph (it is “in-
+tegral”in the terminologyofSection 3), and anH–graphcomingfrom the geometric
+decomposition of a good manifold is similarly special. The geometric expla nations
+therefore only match precisely for special cases of H–graphs.
+Definition 8.1. AnH–graph Γ is a finite connected graph with decorations on its
+vertices and edges as follows:
+(1) Vertices are partitioned into two types: hyperbolic vertices andSeifert ver-
+tices. The full subgraphsof Γ determined by the hyperbolic vertices, re spec-
+tively the Seifert vertices, are called the hyperbolic subgraph , respectively
+theSeifert subgraph .
+(2) The hyperbolic subgraph is labelled as in Definition 3.1, so that each of its
+components is an NAH–graph. In particular, each hyperbolic verte xvis
+labelled by a hyperbolic orbifold Nvand there is a map e/mapsto→Cefrom the
+set of directed edges eexiting that vertex to the set of cusp orbifolds Ceof
+Nv. This map is defined on the set of alledges exiting e, not just the edges
+in the hyperbolic subgraph. As before, it is injective, except that a n edge
+which begins and ends at the same vertex may have Ce=C¯e.
+(3) Each Seifert vertex is labelled by one of two colors, black or white [for an
+H–graph coming from a geometric decomposition this encodes wheth er the
+Seifertfiberedpieceinthegeometricdecompositionof Mcontainsboundary
+components of the ambient 3–manifold or not, as in [1]]. It is also labelle d
+by a Seifert fibration type “o” or “n”, as described above.
+(4) For an edge estarting at a Seifert vertex the group Fe=Feis{1}or{±1}.
+IfFeis{±1}then the Seifert vertex is type “n”.
+(5) For each edge efrom a hyperbolic vertex to a Seifert vertex the group
+Feassociated to the cusp section Ceis either trivial or {±1}. The edge is
+labelled byanon-zerorationalvectorin TCecalledthe slopese, determinedQUASI-ISOMETRIC CLASSIFICATION OF NON-GEOMETRIC 3-MANIF OLD GROUPS 15
+up to the action of Fe. [seencodes the direction and length of the fibers of
+the adjacent Seifert fibered piece.]
+(6) Each edge econnecting a type “o” Seifert vertex with a type “o” Seifert
+vertex or a hyperbolic vertex has a sign label ǫe=±1 withǫe=ǫe[this
+describes compatibility of orientations of Seifert fibers of adjacen t pieces
+or—if the edge connects a Seifert and a hyperbolic vertex—of Seife rt fiber
+and slope].
+(7) The data described in items (5) and (6) are subject to the equiv alence
+relation generated by the following moves:
+(a) For any type “o” Seifert vertex the signs at all edges adjacen t to it
+may be multiplied by −1 [reversal of orientation of the Seifert fibers].
+(b) The slope seof item (5) can be multiplied by −1 while simultaneously
+multiplying ǫeby−1.
+(c) For any Seifert vertex, the slopes at all adjacent hyperbolic v ertices
+may be multiplied by a fixed non-zero rational number.
+Note that the data encoded by the sign weights modulo the equivalen ce relation
+of item (7a) are equivalent to an element of H1(Γ\Γn,Γh;Z/2), where Γ his the
+hyperbolic subgraph and Γ nthe full subgraph on Seifert vertices of type “n”.
+We define a morphism of H–graphs ,π: Γ→Γ′, to be an open graph homo-
+morphism which restricts to a NAH–graph morphism of the hyperbolic subgraphs
+(Definition 3.4), preserves the black/white coloring on the Seifert v ertices, and on
+the edges between hyperbolic and Seifert vertices preserves the slope (in the sense
+that the slope at the image edge is the image under the tangent map o n cusp orb-
+ifolds of the slope at the source edge). Moreover, it must map type “n” vertices to
+type “n” vertices, and when an “o” vertex vis mapped to an “n” vertex w, then
+the preimage of each edge at wmust either include edges of different signs, or an
+edge terminating in a type “n” Seifert vertex.
+As in Section 3, the existence of morphisms between H–graphs gene rates an
+equivalence relation which we call bisimilarity .
+Proof of Classification Theorem. To prove the Classification Theorem we will show
+that each bisimilarity class of H–graphs has a minimal element, and if a H –graph
+comes from a non-geometric manifold Mthen the minimal H–graph determines
+and is determined by /tildewiderMup to quasi-isometry.
+We first explain why the qi-type of the universal cover /tildewiderM(or equivalently of
+π1(M)) determines a minimal H–graph Ω( /tildewiderM).
+As in Section 4, we can straighten any quasi-isometry /tildewiderM→/tildewiderM′and assume
+it takes geometric pieces to geometric pieces and slabs to slabs. We m ay also
+assume it is an isometry on hyperbolic pieces. A Seifert piece in /tildewiderMis bi-Lipschitz
+homeomorphic to a fattened tree times R, so the fibration by Rfibers is coarsely
+preserved, and we can straighten it so it is actually preserved. Mor eover, if an
+adjacent piece is hyperbolic, then the straightened quasi-isometr y is an isometry
+on the corresponding flat, so the affine structure on fibers of the Seifert piece is
+coarsely preserved, and we can straighten so that it is actually pre served. However,
+where a Seifert piece is adjacent to a Seifert piece the R×Rproduct structure on
+the correspondingflat (given bySeifert fiberson the two sides) is c oarselypreserved,
+but the affine structureson the Rfibers need onlybe preservedup toquasi-isometry.16 JASON A. BEHRSTOCK AND WALTER D. NEUMANN
+Considering straightened quasi-isometries in the above sense, we d enote again
+I(/tildewiderM) :={f:/tildewiderM→/tildewiderM|fis a straightened quasi-isometry }.
+As in Section 4, the underlying graph for our minimal H–graph Ω( /tildewiderM) has a
+vertex for each orbit of the action of I(/tildewiderM) on the set of pieces of /tildewiderMand edge for
+eachorbit of the action on the set ofslabs. The labelling of the hyper bolic subgraph
+is as before; in particular, any vertex corresponding to an orbit of hyperbolic pieces
+is labelled by the hyperbolic orbifold obtained by quotienting a represe ntative piece
+in the orbit by its isotropy subgroup in I(/tildewiderM). A Seifert vertex of the H–graph is
+of type “n” if some element of I(/tildewiderM) takes a corresponding Seifert fibered piece to
+itself reversing orientations of fibers, and is otherwise of type “o” . For each type
+“o” vertex we choose an orientation of the fibers of one piece in the corresponding
+orbit and then extend equivariantly to the other pieces in the orbit.
+For a Seifert vertex adjacent to at least one hyperbolic vertex th e fibers of the
+corresponding pieces in /tildewiderMcarry an affine structure which is defined up to affine
+scaling. We choose a specific scale for each such vertex, so we can s peak of length
+along fibers, and then the slopeof item (5) of Definition 8.1 is given by a tangent
+vector of unit length, viewed in the adjacent cusp.
+The sign weights of item (6) of Definition 8.1 are then defined, and item (7) of
+that definition reflects the choices of orientation and scale which we re made.
+By construction, the isomorphism type of Ω( /tildewiderM) is determined by /tildewiderMand thus
+two manifolds with quasi-isometric fundamental group have the sam e associated
+graph. It remains to show that the isomorphism type of Ω( /tildewiderM) determines the
+bilipschitz homeomorphism type of /tildewiderM.
+Construct a labelled graph /tildewideΩ from Ω = Ω( /tildewiderM) by first replacing each edge of Ω
+by infinitely many edges, keeping the weights on edges, but adding sig n weights to
+edges at type “n” vertices with infinitely many of each sign, and then taking the
+universal cover of the resulting weighted graph. Finally, “o” and “n ” labels are now
+irrelevant and can be removed.
+To associate a manifold X=X(/tildewideΩ) to this labelled graph, we must glue together
+appropriate pieces according to the tree /tildewideΩ, with appropriate choices for the gluing
+between slabs. The pieces for hyperbolic vertices will be universal c overs of the hy-
+perbolic orbifolds which label them, while for a Seifert vertex we take the universal
+coverof some fixed Seifert fibered manifold with base of hyperbolic t ype and having
+a boundary component for each incident edge in Ω and—if the vertex is a black
+vertex—an additional boundary component to contribute to boun dary ofX. (The
+universal cover Yof this Seifert piece is then a fattened tree times R, and, as in [1],
+it is in fact only important that there is a bound Bsuch that for each boundary
+component of Ythere are boundary components of all “types” within distance B
+of the give boundary component.)
+The choices in gluing depend on the types of the abutting pieces: Bet ween hy-
+perbolic pieces, the gluing map is, as before, determined up to a grou p of isometries
+of the form R2⋊FS, whereSis the slab. Between Seifert fibered pieces the gluing
+will be an affine map such that the fibers from the two pieces then inte rsect in the
+intervening flat with sign given by the sign label of the edge. Finally, be tween a
+Seifert fibered and a hyperbolic piece the gluing will be an affine map mat ching
+unit tangent vector along fibers with the slope vector for the hype rbolic piece. ForQUASI-ISOMETRIC CLASSIFICATION OF NON-GEOMETRIC 3-MANIF OLD GROUPS 17
+each edge of Ω we make a fixed choice of how to do the gluing subject t o the above
+constraints and do it this way for every corresponding edge of /tildewideΩ.
+To complete the proof, it remains to show that, independent of the se choices,
+there exists a bilipschitz homeomorphism from /tildewiderMtoX.
+As in theproofofProposition4.1, the desiredbilipschitzhomeomorph ismisbuilt
+inductively, starting with a homeomorphism from one piece of /tildewiderMto a piece of X
+and then extending via adjacent slabs to adjacent pieces. There a re four cases: (1),
+when both adjacent pieces are hyperbolic, this is exactly the case o f Proposition 4.1;
+(2), when both pieces are Seifert fibered; (3), extending from a h yperbolic piece to
+an adjacent Seifert fibered piece; and (4), extending from a Seife rt fibered piece to
+an adjacent hyperbolic piece. For Case (2) we use [1, Theorem 1.3] ( as in the proof
+of [1, Theorem 3.2]) to extend over the adjacent Seifert fibered pie ce, respecting the
+“types”ofboundarycomponents(i.e., belongingtoboundaryof Mornot, andifnot,
+then the “type”is givenby the edge ofΩ that the boundary compon entcorresponds
+to). Case (3) is essentially the same argument, and Case (4) is immed iate.
+Thus we obtain the desired bilipschitz homeomorphism, completing the proof.
+/square
+Proof of Realization Theorem. The construction of Ω( /tildewiderM), given above, has built in
+a morphism from the H-graph associated to M. Since the graph associated to M
+is balanced, it follows from Proposition 4.2 that Ω( /tildewiderM) is balanced.
+The balanced condition is only a constraint on NAH-graph component s of an
+H-graph. Thus, from the case of NAH-graphs which we established in Section 6,
+we may conclude that, assuming CCC3, every balanced minimal H-graph is the
+minimal H-graph of some quasi-isometry class of good 3–manifold gro up. /square
+References
+[1] J. Behrstock and W.D. Neumann, Quasi-isometric classifi cation of graph manifolds groups.
+Duke Math. J. 141(2008), 217–240.
+[2] D. Groves and J.F. Manning, Dehn filling in relatively hyp erbolic groups. Israel J. Math. 168
+(2008), 317–429.
+[3] Emily Hamilton, Abelian subgroup separability of Haken 3-manifolds and closed hyperbolic
+n-orbifolds. Proc. London Math. Soc. 83(2001), 626–646.
+[4] J. Hempel, Residual finiteness for 3-manifolds, in Combinatorial group theory and topology
+(Alta, Utah, 1984) , Ann. of Math. Stud., 111(Princeton Univ. Press, Princeton, NJ, 1987)
+379–396
+[5] M. Kapovich and B. Leeb. Quasi-isometries preserve the g eometric decomposition of Haken
+manifolds. Invent. Math. 128(1997), 393–416.
+[6] Frank Thomson Leighton, Finite Common Coverings of Grap hs, J. Comb. Theory, Series B
+33(1982), 231–238.
+[7] Walter D Neumann, On Leighton’s graph covering theorem. Preprint.
+[8] Denis V. Osin, Peripheral fillings of relatively hyperbo lic groups, Invent. Math. 167(2007),
+no. 2, 295–326.
+[9] G. Perelman. The entropy formula for the Ricci flow and its geometric applications. Preprint,
+arXiv:math.DG/0211159 , 2002.
+[10] G.Perelman. Ricciflow with surgeryon three-manifolds .Preprint, arXiv:math.DG/0303109 ,
+2003.
+[11] G. Perelman. Finite extinction time for the solutions t o the Ricci flow on certain three-
+manifolds. Preprint, arXiv:math.DG/0307245 , 2003.
+[12] R. Schwartz. The quasi-isometry classification of rank one lattices. IHES Sci. Publ. Math. ,
+82(1996) 133–168.18 JASON A. BEHRSTOCK AND WALTER D. NEUMANN
+Department of Mathematics, Lehman College, CUNY
+E-mail address :jason.behrstock@lehman.cuny.edu
+Department of Mathematics, Barnard College, Columbia Unive rsity
+E-mail address :neumann@math.columbia.edu
\ No newline at end of file
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+1 TESTING RELATIVISTIC GRAVITY AND  
+DETECTING  GRAVITATIONAL WAVES IN SPACE * 
+WEI-TOU NI 
+Center for Gravitation and Cosmology, Purple Mounta in Observatory,  
+Chinese Academy of Sciences, No. 2, Beijing W. Rd.,  Nanjing, 210008 China  
+National Astronomical Observatories, Chinese Academ y of Sciences, 
+Beijing, 100012 China 
+  e-mail: wtni@pmo.ac.cn   
+For testing gravity and detecting gravitational wav es in space, deep-space laser ranging 
+using drag-free spacecraft is a common method. Deep  space provides a large arena and a 
+long integration time. Laser technology provides me asurement sensitivity, while drag-
+free technology ensures that gravitational phenomen on to be measured with least 
+spurious noises. In this talk, we give an overview of motivations and methods of various 
+space missions/proposals testing relativistic gravi ty and detecting gravitational waves, 
+and refer to various references. 
+1.   Introduction - Mapping the Gravitational Field  in the Solar System 
+The solar-system gravitational field is determined by three factors: the dynamic 
+distribution of matter in the solar system; the dyn amic distribution of matter 
+outside the solar system (galactic, cosmological, e tc.) and gravitational waves 
+propagating through the solar system. Different rel ativistic theories of gravity 
+make different predictions of the solar-system grav itational field. Hence, precise 
+measurements of the solar-system gravitational fiel d test these relativistic 
+theories, in addition to enabling gravitational wav e observations, determination 
+of the matter distribution in the solar-system and determination of the observable 
+(testable) influence of our galaxy and cosmos. The tests and observations from 
+various missions with various configurations could include: (i) a precise 
+determination of the relativistic parameters with 3 -5 orders of magnitude 
+improvement over previous measurements; (ii) a 1-2 order of magnitude 
+improvement in the measurement of /uni0120; (iii) a precise determination of any 
+anomalous, constant acceleration A a directed towards the Sun; (iv) a 
+measurement of solar angular momentum via the Lense -Thirring effect; (v) the 
+                                                           
+*This work is supported by the National Natural Scie nce Foundation of China (Grant Nos. 
+10778710 and 10875171).  2
+detection of solar g-mode oscillations via their ch anging gravity field, thus, 
+providing a new eye to see inside the Sun; (vi) pre cise determination of the 
+planetary orbit elements and masses; (vii) better d etermination of the orbits and 
+masses of major asteroids; (viii) exploring the dyn amic distribution of matter 
+outside the solar system (galactic, cosmological, e tc.) and testing large scale 
+gravitational theories; (ix) detection and observat ion of gravitational waves from 
+massive black holes and galactic compact binary sta rs in the low and middle 
+frequency range 100 nHz to 10 Hz; and (x) exploring  background gravitational-
+waves [1].  
+2.   Methods of Measurement 
+For testing gravity and detecting gravitational wav es in space, deep-space laser 
+ranging using drag-free spacecraft is a common meth od. Deep space provides a 
+large arena and a long integration time. Laser tech nology provides measurement 
+sensitivity, while drag-free technology ensures tha t gravitational phenomenon to 
+be measured with least spurious noises. Up to now, microwave technology is 
+used for measuring the solar-system and testing sol ar-system gravity. However, 
+this will be replaced by more precise laser technol ogy. The precision has already 
+been demonstrated by lunar laser ranging [2]. For l ast forty years, we have seen 
+great advances in the dynamical testing of relativi stic gravity.  This is largely due 
+to interplanetary radio ranging and lunar laser ran ging [3]. Interplanetary radio 
+ranging and tracking provided more stimuli and prog resses at first. However 
+with improved accuracy of 1 mm from 20-30 cm and lo ng-accumulation of 
+observation data, lunar laser ranging reaches simil ar accuracy in determining 
+relativistic parameters as compared to interplaneta ry radio ranging. 
+There are various interplanetary optical missions p roposed. As an example 
+of interplanetary missions, we mention ASTROD; the baseline scheme of 
+ASTROD is to have two spacecraft in separate solar orbits and one spacecraft 
+near the Earth-Sun L1/L2 point carrying a payload o f a proof mass, two 
+telescopes, two 1-2 W lasers with spares, a clock a nd a drag-free system ranging 
+coherently among one another using lasers. [1] 
+For technology development, we refer the readers to  LISA proceedings and 
+ASTROD proceedings.  
+3.   Demonstration of Interplanetary Laser Ranging 
+Interplanetary laser ranging has been demonstrated by MESSENGER 
+(MErcury Surface, Space ENvironment, GEochemistry, and Ranging) [4-6]. The 
+MESSENGER spacecraft, launched on 3 August 2004, is  carrying the Mercury  3
+Laser Altimeter (MLA) as part of its instrument sui te on its 6.6-year voyage to 
+Mercury. Between 24 May, 2005 and 31 May, 2005 in a n experiment performed 
+at about 24 million km before an Earth flyby, the M LA on board MESSENGER 
+spacecraft performed a raster scan of Earth by firi ng its Q-switched Nd:YAG 
+laser at an 8 Hz rate. Pulses were successfully rec eived by the 1.2 m telescope 
+aimed at the MESSENGER spacecraft in the NASA Godda rd Geophysical and 
+Astronomical Observatory at Gaddard Space Flight Ce nter (GSFC) when the 
+MLA raster scan passed over the Earth station. Simu ltaneously, a ground based 
+Q-switched Nd:YAG laser at GSFC’s 1.2 m telescope w as aimed at the 
+MESSENGER spacecraft. Pulses were successfully exch anged between the two 
+terminals. From this two-way laser link, the range as a function of time at the 
+spacecraft over 2.39 × 10 10  m (~ 0.16 AU) was determined to ± 0.2 m (± 670 ps) : 
+a fractional accuracy of better than 1 × 10 -11 .  
+A similar experiment was conducted by the same team  to the Mars Orbiter 
+Laser Altimeter (MOLA) on board the Mars Global Sur veyor (MGS) spacecraft 
+in orbit about Mars [6]. At that time, the MOLA las er was no longer operable 
+after a successful topographic mapping mission at M ars. The experiment was 
+one way (uplink) and the MOLA detector saw hundreds  of pulses from 8.4 W Q-
+switched Nd:YAG laser at GSFC. 
+4.   Missions for Testing Relativistic Gravity 
+In this section, we briefly review various missions  for testing relativistic 
+gravity and illustrate by looking into various ongo ing / proposed experiments 
+related to the determination of the PPN space curva ture parameter γ.  Some 
+motivations for determining γ precisely to 10 -5 – 10 -9 are given in [7, 8].  
+First, we mention two recently completed experiment s --- Cassini 
+experiment and GP-B experiment. Cassini experiment is the most precise 
+experiment measuring the PPN space curvature parame ter γ up to now. In 2003, 
+Bertotti, Iess and Tortora [9] reported a measureme nt of the frequency shift of 
+radio photons due to relativistic Shapiro time-dela y effect from the Cassini 
+spacecraft as they passed near the Sun during the J une 2002 solar conjunction. 
+From this measurement, they determined γ to be 1.000021 ± 0.000023. GP-B 
+experiment used quartz gyro at low-temperature to m easure the Lense-Thirring 
+precession and the geodetic precession in a polar o rbit of the earth. The geodetic 
+precession gives a measure of γ. The precision from their current analysis is 
+around 0.003 [10]. 
+Bepi-Colombo [11] is planned for a launch in 2013 t o Mercury. A 
+simulation predicts that the determination of γ can reach 2×10 -6 [12].  4
+GAIA (Global Astrometric Interferometer for Astroph ysics) [13] is an 
+astrometric mission concept aiming at the broadest possible astrophysical 
+exploitation of optical interferometry using a mode st baseline length (~3m). 
+GAIA is planned to be launched in 2013. At the pres ent study, GAIA aims at 
+limit magnitude 21, with survey completeness to vis ual magnitude 19-20, and 
+proposes to measure the angular positions of 35 mil lion objects (to visual 
+magnitude V=15) to 10 µas accuracy and those of 1.3 billion objects (to V= 20) 
+to 0.2 mas accuracy. The observing accuracy of V=10  objects is aimed at 4 µas. 
+To increase the weight of measuring the relativisti c light deflection parameter γ, 
+GAIA is planned to do measurements at elongations g reater than 35° (as 
+compared to essentially 47° for Hipparcos) from the  Sun. With all these, a 
+simulation shows that GAIA could measure γ to 1×10 -5 – 2 ×10 -7 accuracy [14]. 
+ASTROD I is a first step towards ASTROD. Its scheme  is to have one 
+spacecraft in a Venus-gravity-assisted solar orbit,  ranging optically with ground 
+stations with less ambitious, but still significant  scientific goals in testing 
+relativistic gravity. In the ranging experiments, t he retardations (Shapiro time 
+delays) of the electromagnetic waves are measured t o give γ. In the astrometric 
+experiments, the deflections of the electromagnetic  waves are measured to give γ. 
+These two kinds of experiments complement each othe r in determining γ. The 
+Cosmic Vision ASTROD I (Single Spacecraft  Astrodynamical Space Test of 
+Relativity using Optical Devices) mission concept [ 15] is to use a drag-free 
+spacecraft orbiting around the Sun using 2-way (bot h uplink and downlink) laser 
+pulse ranging between Earth and spacecraft to measu re γ and other relativistic 
+parameters precisely. The γ parameter can be separated from the study of the 
+Shapiro delay variation. The uncertainty on the Sha piro delay measurement 
+depends on the uncertainties introduced by the atmo sphere, timing systems of the 
+ground and space segments, and the drag-free noise.   A simulation shows that an 
+uncertainty of 3 × 10 -8 on the determination of γ is achievable.   
+LATOR (Laser Astrometric Test Of Relativity) [16] p roposed to use laser 
+interferometry between two micro-spacecraft in sola r orbits, and a 100 m 
+baseline multi-channel stellar optical interferomet er placed on the ISS 
+(International Space Station) to do spacecraft astr ometry for a precise 
+measurement of γ. 
+For ASTROD (Astrodynamical Space Test of Relativity ) [1], 3 spacecraft, 
+advanced drag-free systems, and mature laser interf erometric ranging will be 
+used and the resolution is subwavelength.  The accu racy of measuring γ and 
+other parameters will depend on the stability of th e lasers and/or clocks. An  5
+uncertainty of 1 × 10 -9 on the determination of γ is achievable in the time frame 
+of 2025-2030.  
+Space mission proposals to the outer solar system f or testing relativistic 
+gravity and cosmology theories include Super-ASTROD  [17], SAGAS [18] and 
+Odyssey [19]. We refer the readers to the refereces  mentioned. 
+5.   Missions for Detecting Gravitational Waves 
+A complete classification of gravitational waves ac cording to their frequencies 
+is: (i) Ultra high frequency band (above 1 THz); (i i) Very high frequency band 
+(100 kHz – 1 THz); (iii) High frequency band (10 Hz  – 100 kHz);  (iv) Middle 
+frequency band ( 0.1 Hz – 10 Hz); (v) Low frequency  band (100 nHz – 0.1 Hz); 
+(vi) Very low frequency band (300 pHz – 100 nHz); ( vii) Ultra low frequency 
+band (10 fHz – 300 pHz); (viii) Hubble (extremely l ow) frequency band (1 aHz 
+– 10 fHz); (ix) Infra-Hubble frequency band (below 1 aHz). The aims of 
+gravitational-wave space missions are for detection  of gravitational waves in the 
+low frequency band (LISA, ASTROD, ASTROD-GW, and Su per-ASTROD) 
+and middle frequency band (DECIGO and Big Bang Obse rver). The space 
+detectors are complimentary to the ground detectors  which aim at detection of 
+gravitational waves in the high frequency band. 
+LISA has nearly equilateral triangular spacecraft f ormation of armlength 5 
+× 10 6 km in orbit 20° behind earth.  
+ASTROD has a large variation in its triangular form ation with armlength 
+varying up to about 2 AU.  
+ASTROD-GW (ASTROD [Astrodynamical Space Test of Rel ativity using 
+Optical Devices] optimized for Gravitation Wave det ection) is an optimization 
+of ASTROD to focus on the goal of detection of grav itational waves. The 
+detection sensitivity is shifted 52 times toward la rger wavelength compared to 
+that of LISA. The scientific aim is focused for gra vitational wave detection at 
+low frequency. The mission orbits of the 3 spacecra ft forming a nearly 
+equilateral triangular array are chosen to be near the Sun-Earth Lagrange points 
+L3, L4 and L5. The 3 spacecraft range interferometr ically with one another with 
+arm length about 260 million kilometers. After miss ion-orbit optimization, the 
+changes of arm length are less than 0.0003 AU or, f ractionally, less than ±10 -4 in 
+ten years, and the Doppler velocities for the three  spacecraft are less than ±4m/s. 
+Both fit the LISA requirement and a number of techn ologies developed by LISA 
+could be applied to ASTROD-GW. For the purpose of p rimordial GW detection, 
+a 6-S/C formation for ASTROD-GW will be used for co rrelated detection of 
+stochastic GWs. 
+The science goals for these missions are detection of Gravitational Waves 
+(GWs) from (i) Supermassive Black Holes; (ii) Inter mediate-Mass Black Holes;  6
+(iii) Extreme-Mass-Ratio Black Hole Inspirals; (iv)  Galactic Compact Binaries; 
+and (v) Primordial Gravitational Wave Sources, Stri ngs, Boson Stars etc.  
+For direct detection of primordial (inflationary, r elic) GWs in space, one 
+may go to frequencies lower or higher than the LISA  [20] bandwidth, where 
+there are potentially less foreground astrophysical  sources to mask detection. 
+DECIGO [21] and Big Bang Observer [22-23] look for GWs in the higher 
+frequency range while ASTROD [1], ASTROD-GW [24] an d Super-ASTROD  
+[17] look for GWs in the lower frequency range. In the following section, we 
+address to the issue of detectability of primordial  gravitational waves. 
+6.   The Detectability of Primordial Gravitational Waves 
+Inflationary cosmology is successful in explaining a number of outstanding 
+cosmological issues including the flatness, the hor izon and the relic issues. More 
+spectacular is the experimental confirmation of the  structure as arose from the 
+inflationary quantum fluctuations. However, the phy sics in the inflationary era is 
+unclear. Polarization observations of Cosmic Microw ave Background (CMB) 
+missions may detect the tensor mode effects of infl ationary gravitational waves 
+(GWs) and give an energy scale of inflation. To pro be the inflationary physics, 
+direct observation of gravitational waves generated  in the inflationary era is 
+needed. The following figure shows the sen that the  direct observation of these 
+GWs with sensitivity down to Ωgw  ~ 10 -20-10 -23  is possible using present 
+projected technology development if foreground coul d be separated (Figure 1). 
+7.   Outlook 
+Space missions using optical devices will be import ant in testing relativistic 
+gravity, measuring solar-system parameters and dete cting gravitational waves. 
+Laser Astrodynamics in the solar system envisages u ltra-precision tests of 
+relativistic gravity, provision of a new eye to see  into the solar interior, precise 
+measurement of /uni0120, monitoring the solar-system mass loss, and detect ion of low-
+frequency gravitational waves to probe the early Un iverse and study strong-field 
+black hole physics together with astrophysics of bi naries. One spacecraft and 
+multi-spacecraft mission concepts are in line for m ission opportunities. In view 
+of their importance both in fundamentals and in tec hnology developments, 
+mission concepts of this kind will be a focus in th e near future. 
+ 
+ 
+ 
+ 
+ 
+  7
+ 
+ 
+ 
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+ 
+ 
+ 
+ 
+ 
+ 
+ 
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+ 
+ 
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+ 
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+ 
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+ 
+ 
+ 
+Figure 1. The stochastic backgrounds with bounds an d GW detector sensitivities. (Adapted from 
+Figure 3 and Figure 4 of Ref’s 17 and 25 with sensi tivity curve of DECIGO/BBO-grand added; the 
+extragalactic foreground and the extrapolated foreg round curves are from Ref. 27; see Ref’s 17, and 
+25-28 for explanations). 
+ 
+References 
+1. W.-T. Ni, Int. J. Mod. Phys. D  17, 921 (2008); and references therein.  
+2. T. W. Murphy, et al. , Publ. Astron. Soc. Pac.  120 , 20 (2008); and references 
+therein. 
+3.  W.-T. Ni, Int. J. Mod. Phys. D  14 , 901 (2005); and references therein. 
+4. D.E. Smith, M.T. Zuber, X. Sun, G.A. Neumann, J. F. Cavanaugh, J.F. 
+McGarry and T.W. Zagwodzki, Science  311 , 53 (2006). 
+5. X. Sun, G. A. Neumann, J. F. McGarry, T. W. Zagw odzki, J. F. Cavanaugh, J. 
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\ No newline at end of file
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+arXiv:1001.0214v4  [cond-mat.stat-mech]  27 Jul 2010The entropy in finite N-unit nonextensive systems: The normal
+average and q-average
+Hideo Hasegawa∗
+Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan
+(Dated: November 17, 2018)
+Abstract
+We have discussed the Tsallis entropy in finite N-unit nonextensive systems, by using the mul-
+tivariate q-Gaussian probability distribution functions (PDFs) deri ved by the maximum entropy
+methods with the normal average and the q-average ( q: the entropic index). The Tsallis entropy
+obtained by the q-average has an exponential Ndependence: S(N)
+q/N≃e(1−q)N S(1)
+1for large
+N(≫1
+(1−q)>0). In contrast, the Tsallis entropy obtained by the normal a verage is given by
+S(N)
+q/N≃[1/(q−1)N] for large N(≫1
+(q−1)>0).Ndependences of the Tsallis entropy obtained
+by theq- and normal averages are generally quite different, although the both results are in fairly
+good agreement for |q−1| ≪1.0. The validity of the factorization approximation to PDFs w hich
+has been commonly adopted in the literature, has been examin ed. We have calculated correlations
+defined by Cm=∝an}bracketle{t(δxiδxj)m∝an}bracketri}ht−∝an}bracketle{t(δxi)m∝an}bracketri}ht∝an}bracketle{t(δxj)m∝an}bracketri}htfori∝ne}ationslash=jwhereδxi=xi−∝an}bracketle{txi∝an}bracketri}ht, and the bracket
+∝an}bracketle{t·∝an}bracketri}htstands for the normal and q-averages. The first-order correlation ( m= 1) expresses the intrinsic
+correlation and higher-order correlations with m≥2 include nonextensivity-induced correlation,
+whose physical origin is elucidated in the superstatistics .
+PACS numbers: 89.70.Cf, 05.70.-a, 05.10.Gg
+∗hideohasegawa@goo.jp
+1I. INTRODUCTION
+In the last decade, much attention has been paid to the nonextens ive statistics since Tsal-
+lis proposed the so-called Tsallis entropy [1]-[6]. The Tsallis entropy fo rN-unit nonextensive
+systems is defined by
+S(N)
+q=kB
+1−q/parenleftbigg/integraldisplay/bracketleftbig
+p(N)
+q(x)/bracketrightbigqdx−1/parenrightbigg
+, (1)
+whereqis the entropic index, kBthe Boltzmann constant ( kB= 1 hereafter), p(N)
+q(x)
+denotes the N-variate probability distribution function (PDF), x={xi}(i= 1 toN), and
+dx=/producttextN
+i=1dxi. The Tsallis entropy is a one-parameter generalization of the Boltzm ann-
+Gibbs (BG) entropy, to which the Tsallis entropy reduces in the limit of q→1.0,
+S(N)
+BG=S(N)
+1=−/integraldisplay
+p(N)
+q(x)lnp(N)
+q(x)dx. (2)
+The BG entropy is additive because for two independent subsystem sAandBwhere the
+PDF is assumed to be given by
+p(2)
+q(x1,x2) =p(1)
+q(x1)p(1)
+q(x2) for x1∈A, x2∈B, (3)
+we obtain
+S(2)
+1(A+B) =S(1)
+1(A)+S(1)
+1(B). (4)
+In contrast, the Tsallis entropy is pseudoadditive (non-additive) [1–3, 6] because Eqs. (1)
+and (3) lead to
+S(2)
+q,FA(A+B) =S(1)
+q(A)+S(1)
+q(B)+(1−q)S(1)
+q(A)S(1)
+q(B), (5)
+where the subscript of FA is attached for a later purpose. Similarly, when the PDF for
+N-unit independent subsystems is assumed to be given in a factorized form,
+p(N)
+q(x) =N/productdisplay
+i=1p(1)
+q(xi), (6)
+we obtain the pseudoadditive Tsallis entropy, S(N)
+q,FA, expressed by
+ln/bracketleftBig
+1+(1−q)S(N)
+q,FA/bracketrightBig
+=N/summationdisplay
+i=1ln/bracketleftbig
+1+(1−q)S(1)
+q(i)/bracketrightbig
+. (7)
+2We should note, however, that Eqs. (5) and (7) are not correct in the strict sense because
+the PDF derived by the maximum-entropy method (MEM) cannot be e xpressed by Eq. (3)
+or (6), as will be shown shortly [Eqs. (27)-(30)]. Indeed, for ident ical, independent systems,
+Eq. (1) with the use of exact multivariate PDFs yields
+S(2)
+q= 2S(1)
+q+(1−q)/bracketleftbig
+S(1)
+q/bracketrightbig2+∆S(2)
+q, (8)
+S(N)
+q=S(N)
+q,FA+∆S(N)
+q, (9)
+where ∆S(N)
+q(=S(N)
+q−S(N)
+q,FA) expresses a correction term. Equations (8) and (9) imply
+thatS(N)
+qdoes not satisfy the pseudoadditivity because the correction ter m of ∆S(N)
+qis not
+negligibly small, as will be shown in this study.
+It is worthwhile to explain the nonextensivity which is a different concept from the non-
+additivity [7]. An entropy S(N)forN-unit system is said to be extensive if
+0<lim
+N→∞S(N)
+N<∞. (10)
+For independent (or short-ranged interaction) systems, the BG entropy is extensive: S(N)
+BG=
+NS(1)
+BG, while the Tsallis entropy is nonextensive. In contrast, the BG entr opy is nonex-
+tensive for systems with long-ranged interactions. The Tsallis entr opy may be extensive
+for correlated systems such as the probabilistic system [8], some fe rmionic system [9], and
+Gaussian PDFs with correlation [10].
+The PDF is commonly evaluated by the MEM for the Tsallis entropy with im posing some
+constraints. At the moment, there are four possible MEMs: (a) or iginal method [1], (b) un-
+normalized method [4], (c) normalized method [2], and (d) the optimal L agrange multiplier
+(OLM) method [5]. A comparison among the four MEMs is made in Ref. [6]. In relation
+to the issue on the MEMs, two kinds of definitions have been consider ed for an expectation
+valueofphysical quantities: oneisthenormalaverage intheMEM(a ) andtheother isthe q-
+average using the escort probability in the MEMs (c) and (d). Variou s arguments have been
+given that we should employ the q-average [6, 11–13]. Recently, however, it has been pointed
+outthatforasmall change ofthePDF,thermodynamical average s obtainedbythe q-average
+are unstable whereas those obtained by the normal average are s table [14–16]. In contrast,
+Ref. [17] has claimed that for the escort PDF, the Tsallis entropy an d thermodynamical
+averages are robust. This issue on the stability (robustness) of t hermodynamical averages
+as well as the Tsallis entropy is currently controversial [18].
+3Inourpreviouspapers[19,20], wediscussed theTsallisentropyand thegeneralizedFisher
+information in nonextensive systems with and without spatial corre lation, calculating the
+multivariate q-Gaussian PDFs. The purpose of the present paper is twofold: (1) to make
+a comparison between the Tsallis entropies evaluated by the normal average [1] and the q-
+average [2, 5], and (2) to examine the validity of the FA and pseudoad ditivity of the Tsallis
+entropy. One of the advantages of a use of the q-Gaussian PDF is that it is free from an
+open problem on ambiguity in defining the physical temperature in con formity with the
+zeroth law of thermodynamics [21]-[25]. By using the derived PDFs, we may calculate the
+correlation induced by nonextensivity. Some related issues of the d egree of freedom Non
+the PDF have been discussed in Refs. [26–28]. The nonextensivity- induced correlation has
+been calculated for classical ideal gas [29–31] and harmonic oscillat or [30]. Our calculations
+will provide some insight to the current controversy on the normal versusq-averages and
+show the importance of effects which are not taken into account in t he FA.
+The superstatistics is one of alternative approaches to the nonex tensive statistics besides
+the MEM [32–34] (for a recent review, see [35]). In the superstatis tics, it is assumed that
+locallythe equilibrium state of a given system is described by the Boltzmann- Gibbs statistics
+and its global properties may be expressed by a superposition over the fluctuating intensive
+parameter ( i.e.,the inverse temperature) [32]-[35]. The superstatistics has been adopted
+in many kinds of subjects such as hydrodynamic turbulence, cosmic ray and solar flares
+[35]. The physical origin of the nonextensivity-induced correlation m ay be elucidated in the
+superstatistics.
+The paper is organized as follows. In Sec. II, we calculate the Tsallis e ntropy, by using
+multivariate PDFs for correlated nonextensive systems derived by the OLM-MEM with the
+q-average [5][19, 20]. In Sec. III, we derive the multivariate PDF by th e original MEM [1]
+withthenormalaverageinordertocalculatetheTsallisentropy. In Sec. IV,wepresent some
+numerical calculations of the Tsallis entropy evaluated by the norma l andqaverages. In Sec.
+V we calculate correlation induced by nonextensivity, whose physica l origin is elucidated in
+the superstatistics [32, 33]. Sec. VI is devoted to our conclusion.
+4II. OLM-MEM WITH THE q-AVERAGE
+A.q-Gaussian PDF
+We consider N-unit nonextensive systems whose PDF, p(N)
+q(x), is derived with the use of
+theOLM-MEM[5] fortheTsallis entropy givenby Eq. (1)[1, 2]. Weimpos e fourconstraints
+given by (for details, see Appendix B of Ref. [20])
+1 =/integraldisplay
+p(N)
+q(x)dx, (11)
+µ=1
+NN/summationdisplay
+i=1[xi]q, (12)
+σ2=1
+NN/summationdisplay
+i=1[(xi−µ)2]q, (13)
+sσ2=1
+N(N−1)N/summationdisplay
+i=1N/summationdisplay
+j=1(/negationslash=i)[(xi−µ)(xj−µ)]q. (14)
+Hereµ,σ2andsexpress the mean, variance, and degree of intrinsic correlation, r espectively,
+and the bracket [ ·]qdenotes the q-average over the escort PDF, P(N)
+q(x),
+[Q]q=/integraldisplay
+P(N)
+q(x)Q(x)dx, (15)
+with
+P(N)
+q(x) =/parenleftBig
+p(N)
+q(x)/parenrightBigq
+c(N)
+q, (16)
+c(N)
+q=/integraldisplay/parenleftbig
+p(N)
+q(x)/parenrightbigqdx, (17)
+whereQ(x) stands for an arbitrary function of x.
+The OLM-MEM with the constraints given by Eqs. (11)-(14) leads to the PDF given by
+[20, 36]
+p(N)
+q(x) =1
+Z(N)
+qexpq/bracketleftBigg
+−/parenleftBigg
+1
+2ν(N)
+qσ2/parenrightBigg
+Φ(x)/bracketrightBigg
+, (18)
+5where
+Φ(x) =N/summationdisplay
+i=1N/summationdisplay
+j=1[a0δij+a1(1−δij)](xi−µ)(xj−µ), (19)
+a0=[1+(N−2)s]
+(1−s)[1+(N−1)s], (20)
+a1=−s
+(1−s)[1+(N−1)s], (21)
+Z(N)
+q=
+
+r(N)
+s/bracketleftBig
+2πν(N)
+qσ2
+q−1/bracketrightBigN/2Γ(1
+q−1−N
+2)
+Γ(1
+q−1)forq >1,
+r(N)
+s(2πσ2)N/2forq= 1,
+r(N)
+s/bracketleftBig
+2πν(N)
+qσ2
+1−q/bracketrightBigN/2Γ(1
+1−q+1)
+Γ(1
+1−q+N
+2+1)forq <1,(22)
+r(N)
+s={(1−s)N−1[1+(N−1)s]}1/2, (23)
+ν(N)
+q=N
+2(1−q)+1. (24)
+HereB(x,y) and Γ( z) denote the beta and gamma functions, respectively, and expq(x)
+expresses the q-exponential function defined by
+expq(x) = [1+(1 −q)x]1/(1−q)
++, (25)
+with [x]+= max(x,0), which becomes expq(x)→exforq→1.0. The entropic index qmay
+take a value,
+0< q <1+2
+N≡qU, (26)
+becausep(N)
+q(x) given by Eq. (18) has the probability properties with ν(N)
+q>0 forq < qU
+and because the Tsallis entropy is stable for q >0 [11]. Evaluations of Z(N)
+qand [Q]qwith
+the use of the exact approach [40, 41] are discussed in Appendix A.1 .
+In the absence of the intrinsic correlation ( s= 0), Eq. (18) reduces to [19]
+p(N)
+q(x) =1
+Z(N)
+qexpq/bracketleftBigg
+−/parenleftBigg
+1
+2ν(N)
+qσ2/parenrightBiggN/summationdisplay
+i=1(xi−µ)2/bracketrightBigg
+. (27)
+On the other hand, the PDF in the FA is given by
+p(N)
+q,FA(x) =N/productdisplay
+i=1p(1)
+q(xi), (28)
+=1
+/parenleftBig
+Z(1)
+q/parenrightBigNN/productdisplay
+i=1expq/bracketleftBigg
+−/parenleftBigg
+1
+2ν(1)
+qσ2/parenrightBigg
+(xi−µ)2/bracketrightBigg
+. (29)
+6In Eqs. (27) and (29), Z(N)
+qis given by Eq. (22) with r(N)
+s= 1.0. From a comparison
+between Eqs. (27) and (29), it is evident that
+p(N)
+q(x)∝ne}ationslash=p(N)
+q,FA(x), (30)
+except for q= 1.0 orN= 1.
+B. Tsallis entropy
+Substituting the PDF given by Eqs. (18)-(24) to Eq. (1), we obtain the Tsallis entropy
+given by [19, 20]
+S(N)
+q=1−c(N)
+q
+q−1, (31)
+with
+c(N)
+q=ν(N)
+q/bracketleftbig
+Z(N)
+q/bracketrightbig1−q. (32)
+Thes-dependence of the Tsallis entropy was previously discussed (see F ig. 1 of Ref. [20]).
+With increasing s, the Tsallis entropy is decreased as given by
+S(N)
+q(s) =S(N)
+q(0)−N(N−1)c(N)
+q
+4s2for|s| ≪2//radicalbig
+N(N−1).(33)
+We pay hereafter our attention to identical, independent systems withs= 0. The Tsallis
+entropy for q= 1.0 (BG entropy) is given by
+S(N)
+1=NS(1)
+1, (34)
+with
+S(1)
+1=/parenleftbigg1
+2/parenrightbigg
+[ln(2πσ2)+1] = 1 .418 for σ2= 1.0. (35)
+Employing Eq.(29), we obtain the Tsallis entropy in the FA given by
+S(N)
+q,FA=1−(c(1)
+q)N
+q−1, (36)
+=N/summationdisplay
+k=1N!
+(N−k)!k!(1−q)k−1(S(1)
+q)k, (37)
+=NS(1)
+q+N(N−1)(1−q)
+2(S(1)
+q)2+··. (38)
+7In particular for N= 2, Eq. (38) becomes
+S(2)
+q,FA=1−(c(1)
+q)2
+q−1= 2S(1)
+q+(1−q)(S(1)
+q)2. (39)
+By using the formula:
+ln/parenleftbiggΓ(z+a)
+Γ(z)za/parenrightbigg
+≃ −a(1−a)
+2zfor|z| → ∞, (40)
+we obtain the entropy for |q−1| ≪1.0 given by
+S(N)
+q≃S(N)
+q,FA≃NS(1)
+1−(q−1)/bracketleftbiggN2
+2(S(1)
+1)2−N
+4/bracketrightbigg
++··. (41)
+The (q−1) term in Eq. (41) includes O(N2) contributions showing the nonadditivity.
+For large N, we obtain the Tsallis entropy expressed by
+S(N)
+q≃/parenleftbiggN
+2/parenrightbigg
+e(1−q)N S(1)
+1, (42)
+S(N)
+q,FA≃[ν(1)
+q]N[Z(1)
+q](1−q)N
+(1−q)forN≫1
+1−q>0, (43)
+employing the relation: lnΓ( z)≃(z−1/2)lnz−z+(1/2)ln2π+··for|z| → ∞.S(N)
+qgiven
+by Eq. (42) leads to a very large figure. In the case of q= 0.5 andσ2= 1.0, for example,
+S(N)
+q/Nis 5.99×102, 3.09×1030and 4.11×10307forN= 10, 100 and 1000, respectively.
+It yields an astronomical figure for Avogadro’s number of N= 6.022×1023.
+III. ORIGINAL MEM WITH THE NORMAL AVERAGE
+A.q-Gaussian PDF
+In preceding Sec. II, we have made calculations by using the OLM-ME M with the q-
+average [5]. In this section, we employ the original MEM with the norma l average [1],
+imposing the constraints,
+1 =/integraldisplay
+˜p(N)
+q(x)dx, (44)
+µ=1
+NN/summationdisplay
+i=1∝an}bracketle{txi∝an}bracketri}htq, (45)
+σ2=1
+NN/summationdisplay
+i=1∝an}bracketle{t(xi−µ)2∝an}bracketri}htq, (46)
+sσ2=1
+N(N−1)N/summationdisplay
+i=1N/summationdisplay
+j=1(/negationslash=i)∝an}bracketle{t(xi−µ)(xj−µ)∝an}bracketri}htq, (47)
+8where the bracket ∝an}bracketle{t·∝an}bracketri}htqdenotes the normal average (relevant quantities being expresse d with
+tilde hereafter),
+∝an}bracketle{tQ∝an}bracketri}htq=/integraldisplay
+˜p(N)
+q(x)Q(x)dx. (48)
+The original MEM yields [1]
+˜p(N)
+q(x) =1
+˜Z(N)
+qExpq/bracketleftBigg
+−/parenleftBigg
+1
+2˜ν(N)
+qσ2/parenrightBigg
+Φ(x)/bracketrightBigg
+, (49)
+with
+˜Z(N)
+q=
+
+r(N)
+s/bracketleftBig
+2πq˜ν(N)
+qσ2
+(q−1)/bracketrightBigN/2Γ(1
+q−1+1)
+Γ(1
+q−1+N
+2+1)forq >1,
+r(N)
+s(2πσ2)N/2forq= 1,
+r(N)
+s/bracketleftBig
+2πq˜ν(N)
+qσ2
+(1−q)/bracketrightBigN/2Γ(1
+1−q−N
+2)
+Γ(1
+1−q)forq <1,(50)
+˜ν(N)
+q=N
+2/parenleftbigg
+1−1
+q/parenrightbigg
++1 =ν(N)
+1/q. (51)
+Here Φ(x),r(N)
+sandν(N)
+qare given by Eqs. (19), (23) and (24), respectively, and Expq(x) is
+theq-exponential function defined by [15]
+Expq(x) =/bracketleftbigg
+1+/parenleftbigg
+1−1
+q/parenrightbigg
+x/bracketrightbigg1/(q−1)
++, (52)
+which is different from expq(x) given by Eq. (25). The two q-exponential functions, expq(x)
+and Expq(x), have the relation [15]:
+expq(x) = Exp2−q((2−q)x),Expq(x) = exp2−q(x/q). (53)
+The condition of ˜ ν(N)
+q>0 implies that a conceivable qvalue is
+q >1−2
+N+2≡qL=1
+qU, (54)
+whereqUis given by Eq. (26). The PDF given by Eqs. (49)-(51) is flat-tailed fo rqL< q <1
+andcompactsupportfor q >1, whichisincontrastwiththePDFgivenbyEq. (27)obtained
+by the OLM-MEM. Evaluations of ˜Z(N)
+qand∝an}bracketle{tQ∝an}bracketri}htqwith the use of the exact approach [40, 41]
+are discussed in Appendix A.2.
+9B. Tsallis entropy
+By using the PDF given by Eqs. (49)-(51), we obtain the Tsallis entro py given by
+˜S(N)
+q=1−˜c(N)
+q
+q−1, (55)
+with
+˜c(N)
+q=/integraldisplay/bracketleftbig
+˜p(N)
+q(x)/bracketrightbigqdx, (56)
+=/bracketleftBig
+˜Z(N)
+q/bracketrightBig1−q
+˜ν(N)
+q. (57)
+The Tsallis entropy in the FA is given by
+˜S(N)
+q,FA=1−[˜c(1)
+q]N
+q−1. (58)
+For|q−1| ≪1.0, the entropy is given by
+˜S(N)
+q≃˜S(N)
+q,FA≃NS(1)
+1−(q−1)/bracketleftbiggN2
+2(S(1)
+1)2+N
+4/bracketrightbigg
++··, (59)
+which is the same as Eq. (41) except for the O(N) term in the bracket.
+For large N, the Tsallis entropy is expressed by
+˜S(N)
+q≃1
+(q−1)/bracketleftBigg
+1−2q e−(q−1)NS(1)
+1
+N(q−1)/bracketrightBigg
+, (60)
+˜S(N)
+q,FA≃1−[˜ν(1)
+q]−N[˜Zq(1)]−(q−1)N
+q−1forN≫1
+q−1>0, (61)
+which lead to ˜S(N)
+q≃˜S(N)
+q,FAand ∆˜S(N)
+q≃0 for large ( q−1)N. We note that the N- and
+q-dependences of ˜S(N)
+qand˜S(N)
+q,FAin the normal average are quite different from those of S(N)
+q
+[Eq. (42)] and S(N)
+q,FA[Eq. (43)] in the q-average.
+IV. NUMERICAL CALCULATIONS
+We show some model calculations in which we hereafter set µ= 0.0 andσ2= 1.0.
+Equations (26) and (54) show that the results obtained by the q- and normal averages are
+valid for 0 < q <1 + 2/N≡qUandq >1−2/(N+ 2)≡qL, respectively (see the inset
+of Fig. 1). Although for large N|q−1|, theNdependence of the Tsallis entropy of the
+10q-average [Eq. (42)] is quite different from that of ˜S(N)
+qof the normal average [Eq. (60)],
+both entropies are in fairly good agreement for qL< q < q U. Figure 1 shows the Tsallis
+entropy for N= 10 as a function of qcalculated by the q-average (solid curve), normal
+average (chian curve) and by the FA in the normal average (dashe d curve). The differences
+between them are small except for q/lessorsimilarqLwhere˜S(N)
+qdivergently increases.
+Figures 2(a) and 2(b) show three-dimensional plots of S(N)
+q/Nand ∆S(N)
+q/S(N)
+q, respec-
+tively, calculated by the q-average as functions of qandN, where ∆ S(N)
+q=S(N)
+q−S(N)
+q,FA.
+We note in Fig. 2(a) that S(N)
+qis exponentially increased with increasing (1 −q) and/or
+N. Figure 2(b) shows that ∆ S(N)
+q/S(N)
+q→1.0 for large (1 −q)N, where the Tsallis entropy
+is given by S(N)
+q∼=∆S(N)
+q. This implies that the FA is not a good approximation and the
+pseudoadditivity is violated.
+Figures 3(a) and 3(b) show three-dimensional plots of ˜S(N)
+q/Nand ∆˜S(N)
+q/˜S(N)
+q, respec-
+tively, calculatedbythenormalaverageasfunctionsof qandN, where∆ ˜S(N)
+q=˜S(N)
+q−˜S(N)
+q,FA.
+It is shown that ˜S(N)
+qis decreased with increasing ( q−1) and/or N, where ∆ ˜S(N)
+q/˜S(N)
+qis
+considerably decreased. The maximum value of ∆ ˜S(N)
+q/˜S(N)
+qis about 0 .002, which means
+that the pseudoadditivity of ˜S(N)
+qis approximately satisfied with an accuracy better than
+0.998, and then the FA is a good approximation.
+V. DISCUSSION
+A. Correlations
+1.q-average
+We will calculate the mth-order correlation for i∝ne}ationslash=jdefined by
+Cm≡[(δxiδxj)m]q−[(δxi)m]q[(δxi)m]q, (62)
+which is evaluated by the q-average ( δxi=xi−µ). With the use of the PDF given by
+Eqs. (18)-(24), we obtain the first- and second-order correlat ions given by (for details, see
+Appendix A.1)
+C1= [δxiδxj]q=σ2s, (63)
+C2=C2s+C2n, (64)
+11with
+C2s=2[(N+2)−Nq]σ4s2
+[(N+4)−(N+2)q], (65)
+C2n=2(q−1)σ4
+[(N+4)−(N+2)q]for 0< q <1+2
+N+2, (66)
+where we adopt [ δxi]q= 0. We note that C1andC2sarise from intrinsic correlation s, and
+thatC2nexpresses correlation induced by nonextensivity which vanishes fo rq= 1.0 and
+which approaches −2σ4/NasN→ ∞. In particular for N= 2,C2sandC2nare given by
+C2s=2(2−q)σ4s2
+(3−2q), (67)
+C2n=(q−1)σ4
+(3−2q)for 0< q <3
+2, (68)
+whereC2s≥0 for 0< q <1.5 whileC2n≤0 for 0< q≤1 andC2n>0 for 1.0< q <1.5.
+On the contrary, the PDF in the FA given by Eq. (29) yields
+CFA
+1=CFA
+2s=CFA
+2n= 0. (69)
+By using Eq. (27), we obtain the mth-order correlation for arbitrary mwiths= 0,
+Cm=
+
+A2
+m/bracketleftBig
+2ν(N)
+qσ2
+(q−1)/bracketrightBigm/bracketleftBigg
+Γ(q
+q−1−N
+2−m)
+Γ(q
+q−1−N
+2)−/parenleftbigg
+Γ(q
+q−1−N
+2−m
+2)
+Γ(q
+q−1−N
+2)/parenrightbigg2/bracketrightBigg
+for 1.0< q < q m,
+A2
+m/bracketleftBig
+2ν(N)
+qσ2
+(1−q)/bracketrightBigm/bracketleftBigg
+Γ(q
+1−q+N
+2+1)
+Γ(q
+1−q+N
+2+m+1)−/parenleftbigg
+Γ(q
+1−q+N
+2+1)
+Γ(q
+1−q+N
+2+m
+2+1)/parenrightbigg2/bracketrightBigg
+forq <1.0,
+(70)
+where
+Am=
+
+Γ(1
+2+m
+2)
+Γ(1
+2)for evenm,
+0 for odd m,(71)
+qm= 1+2
+N+2(m−1). (72)
+The nonextensivity yields higher-order correlation of Cmwithm≥2 forq∝ne}ationslash= 1.0 in indepen-
+dent nonextensive systems where s=C1= 0.0.
+Figure 4 shows C2as functions of qandsforN= 2. At the origin of ( q,s) = (0.0,0.0),
+we obtain C2=−0.33. The parabolic-like line starting from ( q,s) = (1.0,0.0) expresses the
+12intersection of C2with the zero-level plane. With increasing qand/ors,C2is monotonously
+increased. As qapproaches 1.5, C2is divergently increased.
+Figure 5(a), 5(b) and 5(c) show Cmwithm= 2, 4 and 6, respectively, as functions of q
+andN. Magnitudes of Cmare significantly increased with increasing m.C6shows peculiar
+qandNdependences.
+2. Normal average
+We adopt the normal average to evaluate the mth-order correlation for j∝ne}ationslash=jdefined by
+˜Cm≡ ∝an}bracketle{t(δxiδxj)m∝an}bracketri}htq−∝an}bracketle{t(δxi)m∝an}bracketri}htq∝an}bracketle{t(δxi)m∝an}bracketri}htq. (73)
+With the use of the PDF given by Eqs. (49)-(51), the first- and sec ond-order correlations
+are given by (for details, see Appendix A.2)
+˜C1=∝an}bracketle{tδxiδxj∝an}bracketri}htq=σ2s, (74)
+˜C2=˜C2s+˜C2n, (75)
+with
+˜C2s=2[(N+2)q−N]σ4s2
+(N+4)q−(N+2), (76)
+˜C2n=2(1−q)σ4
+(N+4)q−(N+2)forq >1−2
+N+4, (77)
+where∝an}bracketle{tδxi∝an}bracketri}htq= 0. In the case of N= 2, Eqs. (76) and (77) become
+˜C2s=2(2q−1)σ4s2
+3q−2, (78)
+˜C2n=(1−q)σ4
+3q−2forq >2
+3. (79)
+By using Eq. (49), we obtain correlation of ˜Cmfor arbitrary mwiths= 0,
+˜Cm=
+
+A2
+m/bracketleftBig
+2q˜ν(N)
+qσ2
+(q−1)/bracketrightBigm/bracketleftBigg
+Γ(1
+q−1+N
+2+1)
+Γ(1
+q−1+N
+2+m+1)−/parenleftbigg
+Γ(1
+q−1+N
+2+1)
+Γ(1
+q−1+N
+2+m
+2+1)/parenrightbigg2/bracketrightBigg
+forq >1.0,
+A2
+m/bracketleftBig
+2q˜ν(N)
+qσ2
+(1−q)/bracketrightBigm/bracketleftBigg
+Γ(1
+1−q−N
+2−m)
+Γ(1
+1−q−N
+2)−/parenleftbigg
+Γ(1
+1−q−N
+2−m
+2)
+Γ(1
+1−q−N
+2)/parenrightbigg2/bracketrightBigg
+forq <1.0,
+(80)
+13whereAmis given by Eq. (71). When comparing Eqs. (74), (76), (77) and (80 ) with
+Eqs. (63), (65), (66) and (70), respectively, we note that ˜CmandCmhave the reciprocal
+symmetry: q↔1/q. Thenqdependences of ˜Cmare given by those of Cmin Figs. 4 and 5
+if we read q→1/q.
+B. PDF in the superstatistics
+The physical origin of the nonextensivity-induced correlation is eas ily understood in the
+superstatistics [32–34]. It is straightforward to apply a concept o f the superstatistics to
+composite systems [10]. We consider the N-unit Langevin model subjected to additive noise
+given by [20]
+dxi
+dt=−λxi+√
+2D ξi(t)+Ifori= 1 toN, (81)
+whereλdenotes the relaxation rate, ξi(t) the white Gaussian noise with the intensity D,
+andIan external input. The PDF of π(N)(x) for the system is given by
+π(N)(x) =N/productdisplay
+i=1π(1)(xi), (82)
+where the univariate PDF of π(1)(xi) obeys the Fokker-Planck equation,
+∂π(1)(xi,t)
+∂t=∂
+∂xi[(λxi−I)π(1)(xi,t)]+D∂2
+∂x2
+iπ(1)(xi,t). (83)
+The stationary PDF of π(1)(xi) is given by
+π(1)(xi) =1√
+2πσ2exp/parenleftbigg
+−(xi−µ)2
+2σ2/parenrightbigg
+, (84)
+with
+µ=I/λ, σ2=D/λ. (85)
+After the concept in the superstatistics [32–35], we assume that a model parameter of ˜β
+(≡λ/D) fluctuates, and that its distribution is expressed by the χ2-distribution with rank
+n[32, 33],
+f(˜β) =1
+Γ(n/2)/parenleftbiggn
+2β0/parenrightbiggn/2
+˜βn/2−1e−n˜β/2β0, (86)
+14where Γ( x) is the gamma function. Average and variance of ˜βare given by ∝an}bracketle{t˜β∝an}bracketri}ht˜β=β0and
+(∝an}bracketle{t˜β2∝an}bracketri}ht˜β−β2
+0)/β2
+0= 2/n, respectively. Taking the average of π(N)(x) overf(˜β), we obtain the
+stationary PDF given by [19]
+p(N)
+q(x) =/integraldisplay∞
+0π(N)(x)f(˜β)d˜β, (87)
+∝/bracketleftBigg
+1+β0
+nN/summationdisplay
+i=1(xi−µ)2/bracketrightBigg−(N+n)/2
+, (88)
+which is rewritten as
+p(N)
+q(x) =1
+Z(N)
+qexpq/bracketleftBigg
+−/parenleftBigg
+β0
+2ν(N)
+q/parenrightBiggN/summationdisplay
+i=1(xi−µ)2/bracketrightBigg
+, (89)
+with
+Z(N)
+q=/bracketleftBigg
+2ν(N)
+q
+(q−1)β0/bracketrightBiggN/2N/productdisplay
+i=1B/parenleftbigg1
+2,1
+q−1−i
+2/parenrightbigg
+, (90)
+q= 1+2
+(N+n), (91)
+ν(N)
+qbeing given by Eq. (24). In the limit of n→ ∞(q→1.0) where f(˜β)→δ(˜β−β0), the
+PDF reduces to the multivariate Gaussian distribution given by
+p(N)
+q(x) =1
+(Z(1)
+1)NN/productdisplay
+i=1exp/bracketleftbigg
+−β0
+2(xi−µ)2/bracketrightbigg
+, (92)
+which agrees with Eqs. (82) and (84) for β0=λ/D= 1/σ2.
+We note that the PDF given by Eq. (89) is equivalent to that given by E q. (27) derived
+by the MEM when we read β0= 1/σ2, although the former is valid for 1 ≤q <1+2/(N+1)
+while the latter for 0 < q <1+2/N. The nonextensivity-induced correlation arises from the
+common fluctuating field of ˜β, because p(N)
+q(x)∝ne}ationslash=/producttext
+ip(1)
+q(xi) forq∝ne}ationslash= 1.0 despite π(N)(x) =
+/producttext
+iπ(1)(xi).
+Alternatively we may rewrite the PDF given by Eq. (88) as to conform to the PDF in
+the normal average for s= 0 [Eq. (49)],
+p(N)
+˜q(x)∝Exp˜q/bracketleftBigg
+−/parenleftBigg
+β0
+2˜ν(N−2)
+˜q/parenrightBiggN/summationdisplay
+i=1(xi−µ)2/bracketrightBigg
+, (93)
+with
+˜q= 1−2
+N+n, (94)
+where ˜ν(N)
+qis given by Eq. (51). The PDF given by Eqs. (93) and (94) is valid for
+1−2/(N+n)<˜q≤1.
+15TABLE I: A comparison between the Tsallis entropies obtaine d by the q- and normal averages
+[S(1)
+1= (1/2){ln(2πσ2)+1},qU= 1+2/NandqL= 1−2/(N+2) = 1/qU].
+method range Sq(for|q−1| ≪1.0) Sq(forN≫1)
+q-average 0< q < q UNS(1)
+1−(q−1)/parenleftBig
+N2
+2(S(1)
+1)2−N
+4/parenrightBig
+N
+2e(1−q)NS(1)
+1
+normal average q > qLNS(1)
+1−(q−1)/parenleftBig
+N2
+2(S(1)
+1)2+N
+4/parenrightBig
+1
+(q−1)/bracketleftBig
+1−2q
+N(q−1)e−(q−1)NS(1)
+1/bracketrightBig
+VI. CONCLUDING REMARKS
+Table 1 shows a comparison between the q- and normal averages, which have been ob-
+tained with the use of exact N-variateq-Gaussian PDFs. Our calculation has shown the
+followings:
+(i) the MEMs with the q- and normal averages are valid for 0 < q < q U= 1 + 2/Nand
+q > qL= 1−2/(N+2), respectively (see the inset of Fig. 1),
+(ii)theTsallisentropyof S(N)
+qobtainedbythe q-average[5]hasanexponential Ndependence
+for large Nwhere the FA is not a good approximation and the pseudoadditivity do es not
+hold because ∆ S(N)
+q(=S(N)
+q−S(N)
+q,FA) is considerably large,
+(iii) theNdependence of ˜S(N)
+qfor large Nobtained by the normal average is quite different
+from that of S(N)
+qobtained by the q-average, and the FA in the normal average becomes a
+good approximation because ∆ ˜S(N)
+q(=˜S(N)
+q−˜S(N)
+q,FA) is very small,
+(iv) theq- and normal averages yield nearly the same results for qL< q < q Uwhere the two
+averages are valid,
+(v) the nonextensivity-induced correlation is realized in higher-ord er correlations ( Cmand
+˜Cm) withm≥2 while the first-order correlation expresses the intrinsic correlat ion, and
+(vi) the nonextensivity-induced correlation is elucidated as arising f romcommon fluctuating
+field introduced in the superstatistics [32–34].
+Items (i)-(iv) clarify the difference and similarity between the q- and normal averages. Items
+(ii)and(iii)maysuggest thatthenormalaverageismorefavorablet hantheq-average, which
+is consistent with Refs. [14–16][18]. We should be careful in employing t he FA with the
+q-average which is shown not to be a good approximating method exce pt for|q−1| ≪1.0
+16although it has been widely adopted in nonextensive classical statist ics [37]. The situation is
+the same also in the FA for nonextensive quantum systems as recen tly pointed out in Refs.
+[38, 39].
+Acknowledgments
+Thiswork ispartlysupportedby aGrant-in-AidforScientific Resear ch fromtheJapanese
+Ministry of Education, Culture, Sports, Science and Technology.
+Appendix: A. Evaluations of averages by the exact approach
+1. The q-average
+We first discuss an evaluation of the q-average given by
+Z(N)
+q(α)≡/integraldisplay
+[1−(1−q)αΦ(x)]1
+1−qdx, (A1)
+Q(N)
+q(α) = [Q(x)]q≡1
+ν(N)
+qZ(N)
+q/integraldisplay
+Q(x) [1−(1−q)αΦ(x)]q
+1−qdx,(A2)
+by using the exact expressions for the gamma function [38, 40, 41]:
+y−s=1
+Γ(s)/integraldisplay∞
+0us−1e−yudu fors >0, (A3)
+ys=i
+2πΓ(s+1)/integraldisplay
+C(−t−s−1)e−ytdtfors >0. (A4)
+Hereα= 1/(2ν(N)
+qσ2), Φ(x) is given by Eq. (19), Q(x) denotes an arbitrary function of x,
+Cthe Hankel path in the complex plane, and Eq. (32) is employed. We ob tain [38, 40, 41]
+Z(N)
+q(α) =
+
+1
+Γ(1
+q−1)/integraltext∞
+0u1
+q−1−1e−uZ(N)
+1[(q−1)αu]du forq >1.0,
+i
+2πΓ/parenleftBig
+1
+1−q+1/parenrightBig/integraltext
+C(−t)−1
+1−q−1e−tZ(N)
+1[−(1−q)αt]dtforq <1.0,
+(A5)
+17Q(N)
+q(α) =
+
+1
+ν(N)
+qZ(N)
+qΓ(q
+q−1)/integraltext∞
+0uq
+q−1−1e−uZ(N)
+1[(q−1)αu]Q(N)
+1[(q−1)αu]duforq >1.0,
+i
+2πν(N)
+qZ(N)
+qΓ/parenleftBig
+q
+1−q+1/parenrightBig/integraltext
+C(−t)−q
+1−q−1e−tZ(N)
+1[−(1−q)αt]
+×Q(N)
+1[−(1−q)αt]dt forq <1.0,
+(A6)
+where
+Z(N)
+1(α) =/integraldisplay
+e−αΦ(x)dx, (A7)
+Q(N)
+1(α) =1
+Z(N)
+1(α)/integraldisplay
+Q(x)e−αΦ(x)dx. (A8)
+Thus we may evaluate the qaverage of Q(x) from its average over the Gaussian PDF.
+For example, simple calculations lead to
+∝an}bracketle{t(xi−µ)2∝an}bracketri}ht1=1
+2α, (A9)
+∝an}bracketle{t(xi−µ)(xj−µ)∝an}bracketri}ht1=s
+2αfori∝ne}ationslash=j (A10)
+∝an}bracketle{t(xi−µ)2(xj−µ)2∝an}bracketri}ht1=1+2s2
+4α2fori∝ne}ationslash=j, (A11)
+∝an}bracketle{t(xi−µ)m∝an}bracketri}ht1=Am
+αm/2, (A12)
+∝an}bracketle{t(xi−µ)m(xj−µ)m∝an}bracketri}ht1=A2
+m
+αmfori∝ne}ationslash=j,s= 0, (A13)
+with
+Am=
+
+Γ(1
+2+m
+2)
+Γ(1
+2)for evenm,
+0 for odd m.(A14)
+Employing Eqs. (A5), (A6) and (A9)-(A13), we obtain
+[(xi−µ)2]q=σ2, (A15)
+[(xi−µ)(xj−µ)]q=σ2s fori∝ne}ationslash=j, (A16)
+[(xi−µ)2(xj−µ)2]q=(N+2−Nq)(1+2s2)σ4
+(N+4)−(N+2)qfori∝ne}ationslash=j, (A17)
+18[(xi−µ)m]q=
+
+Am/bracketleftBig
+2ν(N)
+qσ2
+q−1/bracketrightBigm/2Γ(q
+q−1−N
+2−m
+2)
+Γ(q
+q−1−N
+2)forq >1,
+Am/bracketleftBig
+2ν(N)
+qσ2
+1−q/bracketrightBigm/2Γ(q
+1−q+N
+2+1)
+Γ(q
+1−q+N
+2+m
+2+1)forq <1,(A18)
+and fors= 0 and i∝ne}ationslash=j,
+[(xi−µ)m(xj−µ)m]q=
+
+A2
+m/bracketleftBig
+2ν(N)
+qσ2
+q−1/bracketrightBigmΓ(q
+q−1−N
+2−m)
+Γ(q
+q−1−N
+2)forq >1,
+A2
+m/bracketleftBig
+2ν(N)
+qσ2
+1−q/bracketrightBigmΓ(q
+1−q+N
+2+1)
+Γ(q
+1−q+N
+2+m+1)forq <1,(A19)
+which yield Eqs. (63), (65), (66) and (70).
+2. The normal average
+Next we discuss an evaluation of the normal average given by
+˜Z(N)
+q(α)≡/integraldisplay
+[1−(q−1)αΦ(x)]1
+q−1dx, (A20)
+˜Q(N)
+q(α) =∝an}bracketle{tQ(x)∝an}bracketri}htq≡1
+˜Z(N)
+q/integraldisplay
+Q(x) [1−(q−1)αΦ(x)]1
+q−1dx,(A21)
+whereα= 1/(2q˜ν(N)
+qσ2). By using Eqs. (A3) and (A4), we obtain
+˜Z(N)
+q(α) =
+
+i
+2πΓ/parenleftBig
+1
+q−1+1/parenrightBig/integraltext
+C(−t)−1
+q−1−1e−tZ(N)
+1[−(q−1)αt]dtforq >1.0,
+1
+Γ[1/(1−q)]/integraltext∞
+0u1
+1−q−1e−uZ(N)
+1[(1−q)αu]du forq <1.0,
+(A22)
+˜Q(N)
+q(α)
+=
+
+iΓ[1/(q−1)+1]
+2π˜Z(N)
+q/integraltext
+C(−t)−1
+q−1−1e−tZ(N)
+1[−(q−1)αt]Q1[−(q−1)αt]dtforq >1.0,
+1
+˜Z(N)
+qΓ[1/(1−q)]/integraltext∞
+0u1
+1−q−1e−uZ(N)
+1[(1−q)αu]Q1[(1−q)αt]duforq <1.0,
+(A23)
+whereZ(N)
+1(α) andQ(N)
+1(α) are given by Eqs. (A7) and (A8), respectively.
+19Employing Eqs. (A9)-(A13), (A22) and (A23), we obtain
+∝an}bracketle{t(xi−µ)2∝an}bracketri}htq=σ2, (A24)
+∝an}bracketle{t(xi−µ)(xj−µ)∝an}bracketri}htq=σ2s fori∝ne}ationslash=j, (A25)
+∝an}bracketle{t(xi−µ)2(xj−µ)2∝an}bracketri}htq=[(N+2)q−N](1+2s2)σ4
+(N+4)q−(N+2)fori∝ne}ationslash=j,(A26)
+∝an}bracketle{t(xi−µ)m∝an}bracketri}htq=
+
+Am/bracketleftBig
+2q˜ν(N)
+qσ2
+q−1/bracketrightBigm/2Γ(1
+q−1+N
+2+1)
+Γ(1
+q−1+N
+2+m
+2+1)forq >1,
+Am/bracketleftBig
+2q˜ν(N)
+qσ2
+1−q/bracketrightBigm/2Γ(1
+1−q−N
+2−m
+2)
+Γ(1
+1−q−N
+2)forq <1,(A27)
+and fors= 0 and i∝ne}ationslash=j,
+∝an}bracketle{t(xi−µ)m(xj−µ)m∝an}bracketri}htq=
+
+A2
+m/bracketleftBig
+2q˜ν(N)
+qσ2
+q−1/bracketrightBigmΓ(1
+q−1+N
+2+1)
+Γ(1
+q−1+N
+2+m+1)forq >1,
+A2
+m/bracketleftBig
+2q˜ν(N)
+qσ2
+1−q/bracketrightBigmΓ(1
+1−q−N
+2−m)
+Γ(1
+1−q−N
+2)forq <1,(A28)
+which lead to Eqs. (74), (76), (77) and (80).
+20FIG. 1: (Color online) The q-dependence of S(N)
+q/Ncalculated by the normal average (N-av: solid
+curve) and q-average (q-av.: chain curve) and in the FA by the normal aver age (dashed curve)
+withN= 10 for which qL= 0.833 and qU= 1.20. The inset shows the validity range of the two
+averages in the q-Nspace: the q-average and normal average are valid for 0 < q < q UandqL< q,
+respectively, and for qL< q < q Uthe both averages are valid: the dashed line expresses N= 10.
+FIG. 2: (Color online) (a) S(N)
+q/Nand (b) ∆ S(N)
+q/S(N)
+qof the Tsallis entropy calculated by the q
+average as functions of qandN(s= 0): note that the ordinate of (a) is in the logarithmic scale .
+[1] C. Tsallis: J. Stat. Phys. 52, 479 (1988).
+[2] C. Tsallis, R. S. Mendes, and A. R. Plastino: Physica A 261, 534 (1998).
+[3] C. Tsallis, in Nonextensive Statistical Mechanics and Its Application , edited by S. Abe and Y.
+Okamoto (Springer-Verlag, Berlin, 2001), p 3.
+[4] E. M. F. Curado and C. Tsallis, J. Phys. A 24(1991) L69; 24, 3187 (1991); 25, 1019 (1992).
+[5] S. Martinez, F. Nicolas, F. Pennini, and A. Plastino, Phy sica A286, 489 (2000).
+[6] C. Tsallis: Physica D 193, 3 (2004).
+[7] C. Tsallis and U. Tirnakli, arXiv:0911.1263.
+[8] C. Tsallis, M. Gell-Mann and Y. Sato, Proc. Natl. Acad. Sc . USA102, 15377 (2005).
+[9] F. Caruso and C. Tsallis, Phys. Rev. E 78, 021102 (2008).
+[10] G. Wilk and Z.Wlodarczyk, Physica A 376, 279 (2007).
+[11] S. Abe, Phys. Rev. E 66, 046134 (2002).
+[12] S. Abe and G. B. Bagci, Phys. Rev. E 71, 016139 (2005).
+FIG. 3: (Color online) (a) ˜S(N)
+q/Nand (b) ∆ ˜S(N)
+q/˜S(N)
+qof the Tsallis entropy calculated by the
+normal average as functions of qandN(s= 0).
+21FIG. 4: (Color online) (a) C2as functions of qandsforN= 2 obtained by the q-average, the
+flat plane expressing the zero level.
+FIG. 5: (Color online) (a) C2, (b)C4and (c)C6as functions of qandNobtained by the q-average
+(s= 0.0).
+[13] S. Abe, Astrophysics and Space Science 305, 241 (2006).
+[14] S. Abe, Europhys. Lett. 84, 60006 (2008).
+[15] S. Abe, Phys. Rev. E 79, 041116 (2009).
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+[17] R. Hanel, S. Thurner and C. Tsallis, Europhys. Lett. 85, 20005 (2009).
+[18] J. F. Lutsko, J. P. Boon and P. Grosfils, Europhys. Lett. 86, 40005 (2009).
+[19] H. Hasegawa, Phys. Rev. E 77, 031133 (2008).
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+[21] S. Abe, Physica A 300, 417 (2001)
+[22] S. Abe, Phys. Rev. E 63, 061105 (2001).
+[23] S. Abe, S. Martinez, F. Pennini, and A. Plastino, Phys. L ett. A281, 126 (2001).
+[24] H. Hasegawa, Physica A 351, 273 (2005).
+[25] S. Abe, Physica A 368, 430 (2006).
+[26] Q. A. Wang, M. Pezeril, L. Nivanen, and A. L. M´ ehaut´ e, C haos, Solitons and Fractals 13,
+131 (2002).
+[27] Q. A. Wang, Physics Lettrs A 300, 169 (2002).
+[28] D. Jiulin, Chinnese Phys. B 19(2010) 070501.
+[29] S. Abe, Physica A 269, 403 (1999).
+[30] L. Liyan and D. Jiulin, Physica A 387, 5417 (2008).
+[31] Zhi-Hui Feng and Li-Yan Liu, Physica A 389, 237 (2010).
+[32] G. Wilk and Z. Wlodarczyk, Phys. Rev. Lett. 84, 2770 (2000).
+[33] C. Beck, Phys. Rev. Lett. 87, 180601 (2001).
+[34] C. Beck and E. G. D. Cohen, Physica A 322, 267 (2003).
+22[35] C. Beck, in Anomoulous Transport: Foundations and Applications , edited by Radons et al.
+(Wiley, New York, 2007).
+[36] Although the expression for Z(N)
+qgiven by Eq. (22) are different from that given by Eqs.
+(36)-(38) in Ref. [20], the both are equivalent.
+[37] Lists of many applications of the nonextensive statist ics are available at URL:
+(http://tsallis.cat.cbpf.br/biblio.htm)
+[38] H. Hasegawa, Phys. Rev. E 80, 011126 (2009).
+[39] H. Hasegawa, Physica A 389, 2358 (2010).
+[40] D. Prato, Phys. Lett. A 203, 165 (1995).
+[41] A. K. Rajagopal, R. S. Mendes and E. K. Lenzi, Phys. Rev. L ett.80, 3907 (1998).
+23This figure "fig1.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0214v4This figure "fig2.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0214v4This figure "fig3.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0214v4This figure "fig4.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0214v4This figure "fig5.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0214v4
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+arXiv:1001.0215v1  [astro-ph.HE]  1 Jan 2010Canthe excess intheFeXXVILy γlinefrom the GalacticCenter provide evidencefor17 keV ste rile
+neutrinos?
+D. A. Prokhorov∗
+Korea Astronomy and Space Science Institute, Hwaam-dong, Y useong-gu, Daejeon, 305-348, Republic of Korea
+Joseph Silk
+Astrophysics, Department of Physics, University of Oxford , Keble Road Ox1 3RH, Oxford, United Kingdom and
+Institut d'Astrophysique de Paris, 98bis Blvd Arago, Paris 75014, France
+The standard model of particle physics assumes that neutrin os are massless, although adding non-zeros is
+requiredbytheexperimentally establishedphenomenon ofn eutrino oscillationsrequires neutrinos tohave non-
+zeromass. Sterileneutrinos(orright-handedneutrinos)a reagoodwarmdarkmattercandidate. Wefindthatthe
+excessoftheintensityinthe8.7keVline(attheenergyofth eFeXXVILyγline)inthespectrumoftheGalactic
+center observed by the Suzaku X-ray mission cannot be explai ned by standard ionization and recombination
+processes. We suggest that the origin of this excess is via de cays of sterile neutrinos with a mass of 17.4 keV
+andestimatethevalueofthemixingangle. Theestimatedval ueofthemixinganglesin2(2θ)=(4.1±2.2)×10−12
+liesinthe allowedregion of the mixingangle of darkmatter s terileneutrino witha mass of 17-18 keV.
+I. INTRODUCTION
+Several astrophysical observations, such as the indicatio ns
+ofcentralcoresinlow-massgalaxies,thelownumberofsate l-
+lites observed around the Milky Way, and the near constant
+cores of the least luminous satellites, have revived intere st
+in sterile neutrinos as a warm dark matter candidate. Elec-
+troweak singlet right-handed (sterile) neutrinos with mas ses
+inthefewkeVrangenaturallyariseinmanyextensionsofthe
+Standard Model and could be produced in the early universe
+through the Dodelson-Widrow mechanism involving (non-
+resonant)oscillationswithactiveneutrinospecies[1]. T heex-
+perimentallyestablishedphenomenonofneutrinooscillat ions
+requires neutrinos to have non-zero mass, and sterile neutr i-
+nos(orright-handedneutrinos)areanaturalwarmdarkmatt er
+candidate.
+Direct constraints on masses and mixing angles are ob-
+tained both from the Lyman alpha forest power spectrumand
+X-ray observations of the radiative decay channel, the latt er
+providing a photon with energy E =msc2/2, where msis the
+sterile neutrino mass. Most recently, X-ray observations o f
+the local dwarf Wilman 1 have shown marginal evidence for
+a 5 keV sterile neutrino [2]. The inferred mixing angle lies
+ina narrowrangeforwhichneutrinooscillationscanproduc e
+all of the dark matter and for which sterile neutrino emissio n
+fromcoolingneutronstarscanexplainpulsarkicks.
+In fact if sterile neutrinosprovidea significantfraction( al-
+though not necessarily all, see [3],) of the dark matter, the
+GalacticCenterprovidesanevenmoreattractiveenvironme nt
+to search for radiative decay signals. If the sterile neutri no
+massisindeedaround17keV,weexpectaX-raylinenear8 .5
+keV. ThediffuseX-rayemissionfromthe GalacticCenter us-
+ing the X-ray imaging spectrometer on Suzaku was analyzed
+by[4],whodetectthe8.7keVlinecorrespondingtheFeXXVI
+Lyγ. We have reanalysed the data on hydrogen-likeiron line
+∗phdmitry@gmail.comstrengths and find that there is an unexpected excess in the
+8.7 keV line from the Galactic center in the Suzakudata. We
+demonstratethat thisexcesscannotbe explainedby meansof
+standardionizationandrecombinationprocesses. Wepropo se
+an explanation of the excess in the 8.7 keV line in terms of
+17.4keVneutrinodecays.
+II. THE EXCESSIN THE8.7 KEVLINEFROMTHE
+GALACTIC CENTERANDITSORIGIN
+X-rayemissionfromtheGalacticcenterhasbeenobserved
+for almost 30 years. A component of the di ffuse emission
+is thermal and is produced by a high temperature plasma in
+the inner 20 parsecs of the Galactic Center ( ∼8 keV). The
+most pronounced features in the emission lines are Fe I K α
+at 6.4 keV, and the K-shell lines 6.7 and 6.9 keV from the
+helium-like (FeXXV K α) and hydrogen-like (FeXXVI Ly α)
+ions of iron, respectively. An analysis of the ratio of the 6. 7
+keV to 6.9 keV lines is an interesting test of plasma com-
+ponents (see, e.g. [5], [6]). For the first time, the FeXXVI
+Lyγat8.7keVwasdetectedbytheSuzakuX-raymission[4].
+The observed line intensities by Suzaku are listed in Table 2
+of [4]. For convenience, we list below the most important
+lines(forouranalysis)takenfromthepaperby[4]. Themea-
+sured intensities of the lines of the hydrogen-like iron ion s
+are:ILyα=1.66+0.09
+−0.11×10−4ph/(cm2s),ILyβ=2.29+1.35
+−1.31×10−5
+ph/(cm2s),andILyγ=1.77+0.62
+−0.56×10−5ph/(cm2s). Theerrors
+areat 90%confidencelevel[4].
+The measured ratio of the FeXXVI Ly βto FeXXVI Lyα
+iron lines equals≈0.138±0.059 and is in agreement with
+the theoretical value of ≈0.14 in the gas temperature range
+between5and15keV(see linelist [7]; fora review,see [8]).
+We note that the measuredintensity of the FeXXVI Ly γ=
+1.77+0.62
+−0.56×10−5ph/(cm2s) iron line has an significant ex-
+cess above the value derived from the theoretical model [7]
+andthemeasuredintensity ILyαofthe FeXXVILyαironline.
+The ratio of the the FeXXVI Ly γto FeXXVI Lyαiron lines
+equaled 0.038 in the gas temperature range between 5 and2
+15 keV is from the theoretical model (see, [7]). Therefore
+the expectedvalue of the intensity in the FeXXVI Ly γline is
+0.038×ILyα=6.3+0.4
+−0.4×10−6ph/(cm2s) and is much smaller
+than the measuredintensityby Suzaku. Therefore,the exces s
+oftheintensityinthe8.7keVlineequals ≈(1.1±0.6)×10−5
+ph/(cm2s).
+Todemonstratethatthisexcesscannotbeexplainedbyion-
+izationandrecombinationprocesses,wecalculatetherati oof
+the fully stripped (FeXXVII) ionic fraction to the hydrogen -
+like iron ionic fraction using the ratio of the intensities o f the
+FeXXVILyγtoLyαlinesasa functionoftemperature T.
+Theratio rγαofthe FeXXVILyγto FeXXVI Lyαironline
+intensitiesis givenby
+rγα=Eγ(T)NFeXXVI+Rγ(T)NFeXXVII
+Eα(T)NFeXXVI+Rα(T)NFeXXVII(1)
+whereEγandEαare the impact excitation rate coe fficients,
+andRγandRαare the rate coefficients for the contribution
+from recombination to the Ly γand Lyαspectral lines, re-
+spectively. Excitation and recombinationrate coe fficients are
+takenfrom[8].
+FromEq. (1)theratioofthefullystripped(FeXXVII)ionic
+fractiontothe hydrogen-likeironionicfractionis
+NFeXXVII
+NFeXXVI=Eγ(T)−rγαEα(T)
+rγαRα(T)−Rγ(T)(2)
+Note that we do not use the assumption of collisional ioniza-
+tionequilibrium.
+For the best fit value of the intensity ratio of the Ly γto
+Lyαiron lines (1.77×10−5/(1.66×10−4)≈0.107) foundby
+[4], we find that the ratio of the fully stripped(FeXXVII) ion
+fractiontothehydrogen-likeironionfractionliesinther ange
+(−40,−20) when the electron temperature is in the range (5
+keV, 15 keV). Since the ratio of the fullystripped (FeXXVII)
+to hydrogen-like(FeXXVI) ion fractionsshould not be nega-
+tive, we concludethat the excess cannotbe explainedby ion-
+ization and recombination processes. A more physically ac-
+ceptable explanation of the excess is that of sterile neutri no
+decays, under the assumption that sterile neutrinosconsti tute
+a significant fraction of dark matter. In the next section, we
+calculate the mixing angle using our inferred value of the in -
+tensityexcessin the8.7keV line.
+III. RADIATIVEDECAYSOFSTERILENEUTRINOS
+Thesterileneutrinopossessesaradiativedecaychannelde -
+caying to an active neutrino and a photon with energy E=
+msc2/2. Since the excessin the 8.7 keV line correspondsto a
+sterile neutrino mass of 17 .4 keV, in this section we estimate
+themixinganglesin2(2θ)anddecayrateforsuchasterileneu-
+trino.
+Using the excess in the 8.7 keV line intensity equal to
+Iexcess≈(1.1±0.6)×10−5ph/(cm2s) and the definition of
+the energy flux Fs=Iexcess×E, we find that the value of the
+energy flux excess in the 8.7 keV line equals (9 .6±5.2)×
+10−5keV/(cm2s).Decays of sterile neutrinos in the dark matter halo of the
+Milky Way are a promising way to explain the excess in the
+8.7keVline. Theamountdarkmatterwithinthefieldofview
+of the Suzaku observation(see Fig. 1 of [4]) is only a minute
+fractionof thetotal massof thedarkmatterhaloof theMilky
+Way. We consider the model of the Milky Way dark halo
+presented in [9] and used to derive constraints on the steril e
+neutrino parameters by [10]. The Milky Way halo density is
+describedbythe Navarro-Frenk-White(NFW)profile
+ρNFW(r)=Mvir
+4πα1
+r(rs+r)2, (3)
+where the dark matter halo parameters of preferred models
+obtained in [9] correspond to Mvir=1012M⊙,rs=21.5 kpc
+and numerical constant α≃1.64 (see [10]). A mass within
+the field of view of Mfov≃2.5×106M⊙is derived from this
+model.
+Thedecayfluxintoasolidangle Ω(withinthefieldofview)
+isgivenby(see,e. g. [11])
+Fs=/integraldisplay
+ΩρNFW(/vectorr)
+4π(/vectorr0−/vectorr)2Γc2
+2d3/vectorr, (4)
+where|/vectorr0|=8.5kpcisthedistancetotheGalacticcenter. The
+decayrateΓisgivenby(e.g. [11]andreferencestherein)
+Γ=1.38×10−22sin2(2θ)/parenleftbiggms
+1keV/parenrightbigg5
+s−1.(5)
+Toestimatethevalueofthemixingangle,weusetheinferred
+valueofms=17.4keVandthevalueoftheenergyfluxexcess.
+Then,fromEqs. (4),(5)thevalueofthemixingangleisgiven
+by
+sin2(2θ)=(4.1±2.2)×10−12. (6)
+Using Eq. (5) we derive the following constraint on the
+decayrate:
+Γ=(9.0±4.8)×10−28s−1. (7)
+Note that the derived values of the mixing angle of
+sin2(2θ)=(4.1±2.2)×10−12anddecayrateofΓ=(9.0±4.8)×
+10−28s−1lieintheallowedregionforadarkmattersterileneu-
+trino with a mass of 17-18keV (see [12]). Such a neutrino is
+capable of accounting for a substantial fraction, if not all , of
+thedarkmatter.
+Anadditionalcontributionofcomparablestrengthcomesin
+apoint-likesourceifacuspofdarkmatterdevelopsaroundt he
+centar black hole. The dark matter cusp mass is constrained
+observationally by stellar orbit measurements and is taken to
+be3×106M⊙withinthecentral0.001pc. Thisiscomparable
+to the mass of the central black hole. Such a cusp is natu-
+rallyproducedviaadiabaticcontractionofdarkmatteraro und
+the black hole within its sphere of influence, about 3 pc for
+theMilkyWayblackhole[13],[14]. Typicallyaboutasmuch
+massiscapturedinthespikeasisinthecentralblackholean d
+with a density profile of ρ∝r−9/4or even steeper. Dynami-
+cal historiesof mergingcomplicatethispicture. In partic ular,3
+binary black hole mergers, had they occurred, would destroy
+sucha spike(see[15],[16],[17]).
+However an alternative and more conservative scenario is
+cuspregeneration. Thiscannotoccurdirectlyinthedarkma t-
+ter because its relaxation time is extremely long. However,
+thisisnotthecaseforthecentralstars,andonceacuspform s
+in the stars, scattering of dark matter particleso ffof stars can
+redistribute the dark matter in phase space on a time scale of
+orderthe relativelyshortstar-star relaxationtime [18], result-
+inginaρ∝r−3/2densitycuspinthecentralparsecofthedark
+matter[19].
+Sterile neutrinos would form such a dark matter cusp, al-
+thoughthe spike density is limited by phase space considera -
+tions for non-degenerateneutrinos(see, [20]). Using the v al-
+ues of the velocity dispersion for the Milky Way of 100 km /s
+[21]andasterileneutrinomassof17.4keV,wefindalimiton
+the densityρcr<1.4×109M⊙/(pc3). This would guaranteea
+point-likesourceif thereis indeedacusp.
+IV. CONCLUSIONS
+We havefoundthatthereisanexcessintheintensityinthe
+8.7 keV line observed from the Galactic center over the the-
+oretical value expected for the intensity in the FeXXVI Ly γ
+line. Thisintensityexcessequals(1 .1±0.6)×10−5ph/(cm2s),
+wheretheerrorsareatthe90%confidencelevel. Weshowthat
+this excess cannot be explained by means of standard ioniza-
+tion and recombination processes, since the ratio of the ful ly
+stripped (FeXXVII) to FeXXVI ion fractions becomes (un-
+physically) negative in the temperature range of (5 keV, 15
+keV), when the observed ratio of FeXXVI Ly γto FeXXVI
+Lyαironlinesisapplied.
+We suggest that a physically acceptable explanation of the
+intensity excess in the 8.7 keV line is due to decays of ster-
+ile neutrinos that that are in a halo of dark matter around
+the Milky Way. The dark matter density profile we adopt
+is obtained by numerical simulations [9]. The dark matter
+mass from the Milky Way dark matter halo within the field
+of view of the Suzaku observation is small compared withthe total mass of the dark halo ≃1012M⊙, however it suffices
+≃2.5×106M⊙toproduceasignificantflux(1 .1±0.6)×10−5
+ph/(cm2s)viadecaysofsterile neutrinos.
+Nearbylowsurfacebrightnessdwarfgalaxieshaveacentral
+dark matter density as high as ∝angbracketleftρ∝angbracketright∼5M⊙/pc3(200GeV/cm3)
+and are consistent with a common mass scale of ∼107M⊙
+[22] and even a universal dark matter profile with a limiting
+centraldarkmattersurfacedensityoflog(r 0ρ0)=2.15±0.2,in
+unitsoflog(M ⊙/pc2).[23]. Thecorrespondingdensityiswell
+withinthephasespaceconstraintsfor17keVsterileneutri nos.
+Oneexplanationofthecommonmassscaleisthatdarkmat-
+ter haloes do not exist at lower masses. Warm dark matter
+particleshavelargeenoughfreestreaminglengthstoerase all
+densityfluctuationsthat might have formedlower mass halos
+if the mass of the warm dark matter particle is approximately
+5 keV for a minimum halo mass of 107M⊙. In the present
+case, the massof suchsterile neutrinosis 2 ×8.7=17.4keV.
+Theminimumhalomassscaleisapproximately105M⊙. Note
+thattheactualdwarfmassescouldextendtolowervaluestha n
+evaluatedatthe300pcscale by[22].
+We estimated the mixing angle of sin2(2θ)=(4.1±2.2)×
+10−12from the excess in the intensity and the decay rate is
+found to be decay rate: Γ =(9.0±4.8)×10−28s−1.These
+valueslieintheallowedregionofthemixingangleanddecay
+rate parameter space for a dark matter sterile neutrino with a
+mass of 17-18keV (see [12]). Sterile neutrinosin this regio n
+of mass, decay rate and mixing angle parameter space can
+contribute significantly to dark matter. Such neutrinoswou ld
+be virtuallyindistinguishablefromcolddarkmatter excep tin
+thelowestmassdwarfgalaxieswherecoreswouldbeformed.
+High resolution x-ray observations should reveal a point-l ike
+line source of comparable strength to the Suzaku line excess
+ifa cuspispresentaroundthecentralblackhole.
+ACKNOWLEDGMENTS
+We are grateful to Kwang-il Seon and Vladimir Dogiel for
+valuablediscussions.
+[1] S. Dodelson and L. M. Widrow, L. M., Phys. Rev. Let. 72, 17
+(1994)
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\ No newline at end of file
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+arXiv:1001.0216v1  [astro-ph.EP]  1 Jan 2010accepted for ApJ Letters, 2009 December 21
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+INSTRUMENT PERFORMANCE IN KEPLER ’S FIRST MONTHS
+Douglas A. Caldwell1
+SETI Institute/NASA Ames Research Center, MS 244-30, Moffet t Field, CA 94035
+Jeffery J. Kolodziejczak
+NASA Marshall Space Flight Center, VP60, Huntsville, AL 358 12
+Jeffrey E. Van Cleve, Jon M. Jenkins, Paul R. Gazis
+SETI Institute/NASA Ames Research Center, MS 244-30, Moffet t Field, CA 94035
+Vic S. Argabright, Eric E. Bachtell
+Ball Aerospace & Technologies Corp.,1600 Commerce Street, Boulder, CO 80301
+Edward W. Dunham
+Lowell Observatory, 1400 West Mars Hill Road, Flagstaff, AZ 8 6001
+John C. Geary
+Smithsonian Astrophysical Observatory, 60 Garden St., Cam bridge, MA 02138
+Ronald L. Gilliland
+Space Telescope Science Institute, 3700 San Martin Drive, B altimore, MD 21218
+Hema Chandrasekaran, Jie Li, Peter Tenenbaum, Hayley Wu
+SETI Institute/NASA Ames Research Center, MS 244-30, Moffet t Field, CA 94035
+William J. Borucki, Stephen T. Bryson, Jessie L. Dotson, Mic hael R. Haas, David G. Koch
+NASA Ames Research Center, MS 244-30, Moffett Field, CA 94035
+accepted for ApJ Letters, 2009 December 21
+ABSTRACT
+TheKepler Mission relies on precisedifferential photometryto detect the 80partspe r million (ppm)
+signal from an Earth–Sun equivalent transit. Such precision requir es superb instrument stability on
+time scales up to ∼2 days and systematic error removal to better than 20 ppm. To th is end, the
+spacecraft and photometer underwent 67 days of commissioning, which included several data sets
+taken to characterize the photometer performance. Because Keplerhas no shutter, we took a series
+of dark images prior to the dust cover ejection, from which we meas ured the bias levels, dark current,
+and read noise. These basic detector properties are essentially un changed from ground-based tests,
+indicating that the photometer is working as expected. Several ima ge artifacts have proven more
+complex than when observed during ground testing, as a result of t heir interactions with starlight and
+the greater thermal stability in flight, which causes the temperatu re-dependent artifact variations to
+be on the timescales of transits. Because of Kepler’s unprecedented sensitivity and stability, we have
+also seen several unexpected systematics that affect photomet ric precision. We are using the first 43
+days ofscience data to characterizethese effects and to develop detection and mitigation methods that
+will be implemented in the calibration pipeline. Based on early testing, we expect to attain Kepler’s
+planned photometric precision over 80%–90% of the field of view.
+Subject headings: instrumentation: photometers — planetary systems — space vehic les: instruments
+— techniques: photometric
+1.INTRODUCTION
+Keplerwas launched on 2009 March 6, beginning a
+3 1/2 year mission to detect transiting exoplanets and
+determine the frequency of Earth-sizeplanets in the hab-
+1douglas.daldwell@nasa.govitable zones of solar-like stars. The objectives and early
+results of the Kepler Mission are reviewed by Borucki,
+et al. (2010) and the mission design and overall per-
+formance are reviewed by Koch et al. (2010). Before
+beginning science operations, Keplerunderwent a com-
+missioning period to ensure it was operating correctly2 Caldwell et al.
+after the rigors of launch, to verify that ground-based
+characterizations were still valid, and to perform charac-
+terizations that could only be done in space (Haas et al.
+2010). In this Letter we describe the instrument char-
+acteristics relevant for understanding Kepler’s raw pixel
+data. In addition to standard CCD detector properties
+(§3.1), we discuss the characterizations and data prod-
+ucts resultingfrom Kepler’s unique design and operation
+(§3.2). In §4, we describe several image artifacts that
+are present in Keplerdata and discuss their impact on
+photometric precision. Detailed descriptions of the pho-
+tometer beyond the scope of this Letter can be found in
+Argabright et al. (2008) and in the “Kepler Instrument
+Handbook” (Van Cleve & Caldwell 2009).
+2.OBSERVATION MODES
+Keplerscience data are available at either short ca-
+dence (∼1 minute) for 512 targets, or long cadence ( ∼30
+minutes) for 170,000 targets. All science data are col-
+lected with an integration time of 6.02 s. In science col-
+lection mode, the full single integration CCD frames are
+coadded together, then at the end of the short and long
+cadence period pre-specified pixels for each target are se-
+lected from the coadd, processed, and stored on board.
+Due to data storage and transmission limitations, only
+about 6% of the 96 million pixels are stored for eventual
+transmission to the ground.
+The focal plane consists of 84 separate science readout
+channels (identified as module#.output#) and four fine
+guidance sensor (FGS) channels all of which are read out
+synchronously. Each channel has several regions avail-
+abletocollectcalibration,or“collateral”data(Figure1).
+There are two sets of columns of virtual pixels: (1) 12
+columns of bias-only pixels resulting from 12 leading pix-
+els in the serial register (“leading black”), and (2) a 20
+column serial over-scan region (“trailing black”). There
+are also two sets of rows of collateral pixels: (1) the
+first 20 rows, which are covered by an aluminum mask
+(“masked smear”), and (2) a 26 row parallel over-scan
+region (“virtual smear”). During science data collection,
+a coadded sum of specified columns of the trailing black
+androwsofboththemaskedandvirtualsmeararestored
+at each cadence for each channel.
+SinceKeplerhas no shutter, we cannot take standard
+dark frames. Instead, the CCDs can be reverse-clocked
+so that none of the signal from the stars and sky reaches
+the CCD output, allowing us to measure the bias level
+throughout the image.
+Finally, a full frame image (FFI) mode is available, in
+whichallthepixelsinthefocalplanearestored. FFIsare
+invaluable for examining detector properties, verifying
+pointing, and verifying the target aperture definitions;
+however, at 380 megabytes each, only a limited num-
+ber can be processed and stored. Reverse-clocked data
+and FFIs are taken periodically throughout the mission
+(Haas et al. 2010).
+3.DETECTOR PROPERTIES
+EachofKepler’s detectorchannels hasdistinct proper-
+ties important for understanding and analyzing the pixel
+data, roughly divided into standard CCD detector char-
+acteristics and those unique to Kepler’s design. Where
+possible instrument characterization was performed on
+the ground (Argabright et al. 2008), though some had tobe done during commissioning. A significant portion of
+commissioning was spent with the dust-cover on, in or-
+dertomeasurecharacteristicsthatwouldlaterbemasked
+by star light. The latter half of commissioning included
+measurementsofthe photometer’spoint spreadfunction,
+pixel response, and focal plane geometry parameters, as
+discussed in Bryson et al. (2010). Commissioning obser-
+vations resulted in a series of focal plane models that are
+usedthroughoutthepipeline processingandareavailable
+at the data archive2.
+3.1.Standard Detector Properties
+Because of the large number of photons collected from
+our typical targets, low read noise is not critical and
+the focal plane median value of 95 e−read−1, or ap-
+proximately 1 digital number (DN) per read, is suffi-
+cient. The focal plane is maintained at −85◦C, reducing
+dark current effectively to zero. The CCDs are operated
+to guarantee that the full-well of 1.1 million electrons
+is not clipped by the 14-bit analog-to-digital converter
+(ADC) range, resulting in a median gain of 112 elec-
+trons DN−1. The high quantum efficiency (QE) back-
+illuminated CCDs combine with the broad bandpass so
+that stars with Keplermagnitude /lessorsimilar11.3 saturate de-
+pending on the field of view (FOV) location. The ob-
+served nonlinearity over the full range of input up to and
+beyond saturation is on the order of ±3% after account-
+ing for charge bleeding.
+SinceKepler’s goal is not absolute photometry, an ac-
+curate global flat field image is not required, but we do
+usealocalflat, orpixelresponsenon-uniformity(PRNU)
+map for calibration. The PRNU image maps each pixel’s
+relative brightness variation from the local mean, ex-
+pressed in percent (Van Cleve & Caldwell 2009). The
+median standard deviation of the pixel values in the
+PRNU image across the focal plane is 0 .96%. A “bad
+pixel” map, constructed by thresholding the PRNU map
+to find>5σoutliers, shows that ≤0.5% of the FOV is
+affected by pixel or column defects.
+Table 1 summarizes the standard CCD detector prop-
+erties as measured during ground testing and updated
+during photometer commissioning.
+3.2.Kepler-Specific Instrument Properties
+Kepler’s design and operation result in several non-
+standard properties that influence the content of the raw
+pixels: readout smear, charge injection, and digital data
+requantization.
+Because there is no shutter, stars shine on the CCDs
+during readout, resulting in trails along columns that
+contain stars. Each pixel in a given column of the im-
+age —including the masked and virtual smear rows—
+receives the same smear signal (Figure 1). The column-
+by-column smear level in each image is measured in the
+smear regions. Smear signals are typically small, since
+each pixel only “sees” a star for the readout time (0.52
+s) divided by the total number of rows (1070); there-
+fore smear is not a significant contributor to photometric
+noise.
+The science CCDs are operated with an electrical
+charge injection feature that injects charge at the top of
+2http://archive.stsci.edu/kepler/INSTRUMENT PERFORMANCE IN KEPLER’S FIRST MONTHS 3
+each CCD for four consecutive rows at a signal level ap-
+proximately 40% of full well. The signal appears entirely
+in the virtual smear. Charge injection serves the dual
+purpose of filling radiation induced traps in the CCDs
+and providing a stable signal for monitoring the readout
+electronics, or “local detector electronics” (LDE), under-
+shoot artifact ( §4.1).
+In order to store and downlink pixel data for 170,000
+targets, the CCD output must be compressed from 23
+bits pixel−1to 4–5 bits pixel−1. Because of the Poisson
+noise intrinsic in the data, we can afford to requantize
+(after the initial analog-to-digitalconversion) so that the
+effective noise due to quantization is a constant percent-
+age of the intrinsic noise for all signal levels. For higher
+signals with more shot noise, more ADC output values
+are mapped on to a single requantized value. With ∆ Q
+the step size for a signal with intrinsic variance σ2
+measured,
+the total variance, σ2
+total, is the quadrature sum of the
+observational noise and the quantization noise:
+σ2
+total=σ2
+measured+∆2
+Q/12. (1)
+(Note that the variance of a uniform random variable of
+unit width is 1/12.) Keplerdata are requantized such
+that the quantization noise is at most 1/4 of the intrin-
+sic noise, ∆ Q/√
+12≤σmeasured/4, resulting in a total
+noise increase of 3%. Requantization reduces the num-
+ber of bits pixel−1from 23 to 16 and greatly improves
+compressibility for subsequent lossless steps in the com-
+pression process (Haas et al. 2010).
+4.INSTRUMENT ARTIFACTS
+Ground testing uncovered several instrumental arti-
+facts, each of which was investigated to understand the
+cause, impact, and cost to fix or mitigate. Based on
+reviews by the KeplerTeam and outside experts, the
+project dispositioned each of these artifacts. There were
+several for which the KeplerTeam and review boards
+decided that the potential impact to the mission did not
+warrant the risk of fixing them. These artifacts were
+extensively characterized on the ground and then again
+during commissioning. For those with the largest im-
+pact, the data processing pipeline either already cor-
+rects for them, or corrections are under development
+(Jenkins et al. 2010b). Table 2 summarizes the FOV po-
+tentially affected by each artifact. Values for artifact
+levels given below are based on 43 days of science data
+collection.
+4.1.LDE Undershoot
+Testing of the LDE signal chain on star-like images
+revealed a large signal-dependent trailing undershoot in
+the video. The primary cause was traced to the use of a
+bipolar-input AD8021 operational amplifier in the corre-
+lated double-sample circuit. Corrective action to replace
+this amplifier with the junction field effect transistor-
+input AD8065 in all video channels resulted in greatly
+reducing this artifact, from an initial 2% amplitude and
+12 pixels duration to a median amplitude of 0.34% and
+3 pixels duration, as measured in flight data (Figure 2).
+The undershoot distortion can be modeled as an invert-
+ible, linear, shift-invariant digital filter, meaning we can
+correct the pixel values provided we have enough pixels
+upstream(lowercolumnnumbers)ofthepixelofinterest.To this end, a column of pixels is prepended to each tar-
+get aperture, and two targetswereadded to eachchannel
+to monitor the undershoot response to both an impulse
+and a step change. An undershoot correction is included
+in the pixel calibration pipeline (Jenkins et al. 2010b).
+4.2.FGS Clocking Cross Talk
+Cross talk from the FGS clocks to the science CCD
+video signals injects a complex pattern into the bias im-
+age of every science channel with an amplitude up to
+20 DN read−1(Argabright et al. 2008). Because the
+FGS and science CCDs share the same master clock, the
+pattern is spatially fixed; however, the amplitude of the
+cross talk is dependent on the temperature of the LDE.
+The cross talk has three distinct components based on
+the state of the FGS CCDs as the science pixel is read
+out (see Figure 1): FGS CCD frame transfer, parallel
+transfer, and serial transfer (which shows no cross talk).
+Approximately 20% of targets have at least one of the
+parallel or frame-transfer cross talk pixels in their aper-
+ture. Without mitigation, the cross talk introduces a
+small time-varying bias into a target’s flux time series as
+the LDE temperature changes with orbital position.
+In orderto measure the crosstalk thermal dependence,
+aseriesofdarkFFIs weretakenatdifferent temperatures
+during commissioning. Both the levels and their thermal
+dependenceareconsistentwithgroundcharacterizations.
+The signal in each cross talk pixel type can be modeled
+as an offset from the bias level that is linear in time and
+LDE board temperature (Van Cleve & Caldwell 2009).
+This model removes the cross talk effect to the level of
+the read noise for all but a few pixel types on the most
+affected channels. The temporal component decreased
+significantly during commissioning with a damping time
+ontheorderofaweek,indicativeofatransienteffectsuch
+as out-gassing. With the dust cover on, the focal plane
+was warmer during the dark data collection than during
+science operations, so the model is used to extrapolate
+the measured cross talk levels to those we see during
+operations, allowing us to generate a bias image, or “two
+dimensional (2D) black” for calibration that includes the
+clocking cross talk.
+4.3.2D Black Artifacts
+Ground tests revealed several artifacts associated with
+the 2D structure of the dark images. Their extent on
+the focal plane was observed to be small, as was the
+likelihood that they would worsen or spread. The dark
+images obtained in flight are completely consistent with
+ground test results. Nevertheless, the analysis pipeline
+does not yet include a means of identifying changes in
+these features, so time series that are released with the
+current processing version may contain artifact features
+at signal levels discussed below.
+4.3.1.High-frequency Oscillations
+A temperature-sensitive amplifier oscillation at
+>1 GHz was detected in some CCD video channels
+during the artifact investigation. We suspect that the
+origin of this oscillation is from the AD8021 operational
+amplifiers used extensively in the video signal chain,
+which may show subtle layout-dependent instability
+when used at low gains. The oscillation’s frequency4 Caldwell et al.
+range, rate of change, and pattern among the channels
+matched closely those characteristics in the dark images,
+strongly suggesting that the artifact is a moir´ e pattern
+generated by sampling the high-frequency oscillation at
+the 3MHz serial pixel clocking rate. Since the character-
+istic source frequency drifts with time and temperature
+of the electronic components by as much as 500kHz /◦C,
+the signal from a given pixel in a series of dark images
+has a time varying signature. This signature may be
+highly correlated with neighboring pixels and yet poorly
+correlated with slightly more distant pixels. When the
+oscillation frequency is a harmonic of the serial clocking
+frequency, a DC shift occurs producing a horizontal
+band offset from the mean bias-level in the image. As
+the frequency drifts with temperature, the point on the
+image where this DC shift occurs moves up or down
+from sample-to-sample, producing a rolling band.
+Forty-six of the 84 readout channels have never ex-
+hibited this behavior and an additional 9 channels have
+thus far not exhibited this behavior at a detectable level
+in flight. Typically, in the remaining 29 channels, 20% of
+the FOV exhibits the moir´ e pattern with peak-to-peak
+amplitudes >0.1 DN read−1pixel−1, while another 18%
+exhibits between 0 .02 and 0.1 DN read−1pixel−1. Two
+output channels, 9.2and 17.2, typicallyexhibit >0.1DN
+read−1pixel−1moir´ e pattern peak-to-peak amplitudes
+over their entire fields-of-view. The resulting total FOV
+fraction typically affected by moir´ e signal at or above a
+level of 0.1 DN read−1pixel−1(0.02 DN read−1pixel−1)
+is 9% (15%). For comparison, 0.1 DN read−1pixel−1is
+the change in signalper pixel in a typical 12thmagnitude
+star aperture for an Earth-size planet transit. We have
+adopted the conservative threshold of 0.02 DN read−1
+pixel−1, or roughly 1 /4 of the per pixel signal from an
+Earth-sizetransit, in orderto put an upper bound on the
+potential impact from image artifacts. While the moir´ e
+amplitude per pixel in these channels is significant, how
+the artifact affects our ability to detect small planets de-
+pends on its frequency, sum within a target aperture,
+and variations over time-scales of interest to transit de-
+tection. Based on the first 33.5 days of data from science
+operations, Jenkins et al. (2010a) find the instrument is
+meeting the 6-hour precision requirement across the fo-
+cal plane for the quietest 30% of stars. The two worst
+moir´ e channels, 9.2 and 17.2, exhibit a ∼20% increase in
+6-hour noise over the focal plane average at 12thmagni-
+tude, as measured by the standard deviation of 6-hour
+binned flux time series. Such an increase is small com-
+pared with the factor of 1.5 spread in the distribution
+of dwarf star precision at 12thmagnitude (Jenkins et al.
+2010a).
+4.3.2.Scene-dependent Artifacts
+As a consequence of the sensitivity of the oscillating
+LDE component to temperature, the thermal transient
+introduced during readout by the signal from a bright
+starcausesadditionallocalizedchangesinbias-level(Fig-
+ure 1). These scene dependent artifacts persist over
+a range of hundreds of pixels in the columns following
+bright stars. In the 29 output channels exhibiting themoir´ e pattern, approximately 20% of the FOV may be
+affected by these artifacts above the 0.02 DN read−1
+pixel−1level.
+A second scene dependent effect is observed on all
+channels. A bright star produces a strong undershoot
+signal as discussed above, but the undershoot signal ex-
+tends for hundreds of pixels at a very low level rather
+than the 20 pixels currently used for the inverse filter.
+This extended undershoot signal varies in proportion to
+variations in the source star’s light curve. We estimate
+that 36% of CCD rows contain stars bright enough to in-
+troduce extended undershoot signals which suggests that
+20%ofpixelsareatrisktobe affectedbythisartifact. Of
+these, we conservatively estimate that 50% are actually
+affected above the 0.02 DN read−1pixel−1level. This
+means 10% of the total FOV is subject to this effect.
+4.3.3.Start-of-line Ringing
+A transient signal initiated at the onset of serial clock-
+ing of each row is well-modeled as a seriesof ∼5 superim-
+posed, slightly under-damped oscillations which extend
+for roughly 150 pixels. This start-of-line ringing is evi-
+dent on all output channels, but shows much less ther-
+mal sensitivity than the oscillations discussed previously.
+The initial amplitude of the oscillations is >1 DN read−1
+pixel−1; however, the pattern is static, so the current
+processing algorithms adequately remove the artifact to
+accuracies <0.02 DN read−1pixel−1. We do not expect
+this artifact to impact photometric precision.
+5.SUMMARY
+Wehaveusedcommissioningandthe firstmonthofsci-
+ence operations to characterize Kepler’s instrument per-
+formance. The basic properties of the photometer are
+unchanged from ground testing. Image artifacts are con-
+sistent with ground observations, though star light cre-
+ates scene dependence of the moir´ e pattern signal, com-
+plicating mitigation plans. Corrections in the current
+analysis pipeline for static FGS clocking cross talk, LDE
+undershoot, and start-of-lineringing arefound to remove
+these artifacts to a level sufficient to meet Kepler’s preci-
+sion requirements. We are currently testing mitigations
+for the thermally varying FGS cross talk, moir´ e pat-
+tern, and extended undershoot, which will subsequently
+be implemented in the analysis pipeline. Results indi-
+cate that we can reduce the levels of these artifacts to
+<0.02 DN read−1pixel−1over 80%–90% of the focal
+plane, with the remainder flagged for use in subsequent
+analysis steps (Jenkins et al. 2010b). The photometer is
+currently providing measurements of unprecedented pre-
+cision and time coverage, giving us great confidence that
+theKepler Mission will meet its science goals and make
+many unexpected discoveries.
+We gratefully acknowledge the years of work by the
+many hundred members of the KeplerTeam who con-
+ceived, designed, built, and now operate this wonderful
+mission. Funding for this Discovery mission is provided
+by NASA’s Science Mission Directorate.
+Facilities: Kepler .
+REFERENCES
+Argabright, V. S., et al. 2008, Proc. SPIE, 7010, 70102L Boru cki, W. J., et al. 2010, Science, submittedINSTRUMENT PERFORMANCE IN KEPLER’S FIRST MONTHS 5
+Bryson, S. T., et al. 2010, ApJ, this issue
+Haas, M. R., et al. 2010, ApJ, this issue
+Jenkins, J. M., et al. 2010a, ApJ, this issue
+—. 2010b, ApJ, this issueKoch, D. G., et al. 2010, ApJ, this issue
+Van Cleve, J., & Caldwell, D. A. 2009, Kepler Instrument
+Handbook, KSCI 19033-001 (Moffett Field, CA: NASA Ames
+Research Center), http://archive.stsci.edu/kepler/6 Caldwell et al.
+Column numberRow number
+200 400 600 800 10001002003004005006007008009001000
+Column numberRow number
+96098010001020104010601080110011208408608809009209409609801000
+Fig. 1.— Raw FFI of channel 17.2 (left) and a zoomed portion of the imag e (right). The left panel shows the collateral data regions:
+leading black columns (left edge), trailing black columns ( right edge), masked smear rows (bottom), and virtual smear r ows (top). The
+four bright charge injection rows can be seen in the virtual s mear. The smear signal is visible as bright columns. The FGS f rame-transfer
+clocking cross talk signals are the five faint horizontal ban ds near rows 1, 220, 430, 640, and 860. The right panel shows a c lose-up near
+two bright stars. The smear signal, frame-transfer cross ta lk at row 860, and trailing black beginning at Column 1112 are visible. The
+FGS parallel-transfer cross talk signals are the ∼16 pixels wide segments spaced seven rows apart beginning ne ar the lower left and offset
+toward the upper right. Scene-dependent moir´ e pattern is v isible in the columns following the two bright stars in rows 8 65 and 930. This
+commissioning FFI (KPLR2009108050033) used 270 coadds of 2 .59 s integrations each, for a total exposure time of 700 s. Th e display uses
+a linear stretch from the 7thto 80thpercentile.
+Column numberRow number
+1000 1010 1020 1030 1040 1050520530540550560570580
+1000 1010 1020 1030 1040 1050−0.5−0.4−0.3−0.2−0.100.10.20.3
+Column numberFlux relative to peak height (%)
+Fig. 2.— Portion of a raw FFI of channel 11.1 near a saturating star tha t spills charge ±30 rows (left) and the mean of three rows
+(521-523) cutting through a saturated column (right), scal ed to be relative to the saturation peak height and expressed in percent. The
+dotted line with ‘x’ symbols is from the raw FFI, and the solid line is from the calibrated FFI, demonstrating the efficacy of the undershoot
+correction (Jenkins et al. 2010b). The undershoot level is 0 .4% for the first pixel and 0 .1% for the second. This FFI (KPLR2009231194831)
+used 270 co-adds of 6.02 s integrations, for a total exposure time of 1625 s. The display uses a linear stretch from the 5thto 95thpercentile.
+An optical ghost from the field flattener lens is clearly visib le around this bright star.INSTRUMENT PERFORMANCE IN KEPLER’S FIRST MONTHS 7
+TABLE 1
+Detector Properties Summary
+Parameter Where Measured Minimum Maximum Median
+Read noise (e−read−1) Flight 81 307 [149]a95
+Dark current (e−pix−1s−1) Flight -8.1[-3.2]a7.5 0.25
+QE at 600 nm Ground 0.81 0.92 0.87
+Gain (e−DN−1) Ground 94 120 112
+Saturation (Kepler Mag)bFlight 11.6 10.3 11.3
+PRNUc(%) Ground 0.82 1.20 0.96
+LDE Undershoot (%) Flight 0.08 1.92 0.34
+Note. — Minimum, maximum, and median are taken across all 84 chann els of the focal plane.
+aFlight measurements of read noise and dark current are conta minated by image artifacts for several of the noisiest chann els (§4). Values for
+non-artifact channels are given in square brackets, where d ifferent.
+bSaturation range is calculated using the measured QE variat ions, a vignetting model, and the observed central pixel flux fraction range of
+0.28–0.64.
+cPRNU is a measure of local pixel response variation and does n ot include large-scale optical effects such as vignetting ( §3.1).
+TABLE 2
+Field of View at Risk from Artifacts.
+Potential Extent FOV at Risk
+Artifact on FOVa(%) >0.02 DN/reada(%)
+LDE undershoot 100 0
+FGS cross talk 20 0
+Moir´ e pattern 45 15b
+Scene dependent moir´ e 45 7b
+Extended undershoot 100 10b
+a“Potential extent” indicates the fraction of the FOV where t he artifact could potentially be seen. “FOV at risk” indicat es the fraction where
+we could see time varying signals at or above 0.02 DN read−1after existing and planned mitigations in the analysis pipe line. See the text for
+discussion.
+bThe FOV values do not add since artifacts overlap. Scene depe ndent moir´ e pattern drift adds ∼4% unique FOV and extended undershoot adds
+∼8%.
\ No newline at end of file
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@@ -0,0 +1,700 @@
+arXiv:1001.0217v2  [math.MG]  31 Aug 2010Local minimality of the volume-product at the
+simplex
+Jaegil Kim∗and Shlomo Reisner†
+Abstract
+It is proved that the simplex is a strict local minimum for the volume prod-
+uct,P(K) = min z∈int(K)|K||Kz|, in the Banach-Mazur space of n-dimensional
+(classes of)convexbodies. Linearlocal stability inthene ighborhoodofthesim-
+plex is proved as well. The proof consists of an extension to t he non-symmetric
+setting of methods that were recently introduced by Nazarov , Petrov, Ryabo-
+gin and Zvavitch, as well as proving results of independent i nterest, concerning
+stability of square order of volumes of polars of non-symmet ric convex bodies.
+1 Introduction and Preliminaries
+Abodyis a compact set which is the closure of its interior and, in particular, a convex
+bodyinRnis a compact convex set with nonempty interior. If Kis a convex body
+inRnandzis an interior point of K, then the polar body KzofKwith center of
+polarityzis defined by
+Kz={y∈Rn:/a\}bracketle{ty,x−z/a\}bracketri}ht/lessorequalslant1 for allx∈K}
+where/a\}bracketle{t·,·/a\}bracketri}htisthecanonicalscalarproduct in Rn. Inparticular, ifthecenter ofpolarity
+is taken to be the origin, we denote by K◦the polar body of Kand we clearly have
+Kz= (K−z)◦.
+IfAis a measurable set in Rnandkis the minimal dimension of a flat containing
+A, wedenoteby |A|thek-dimensionalvolume (Lebesguemeasure) of A. Thereshould
+2000 Mathematics Subject Classification 52A40.
+Key words and phrases: convex bodies, convex polytopes, polar b odies, Santal´ o point, volume
+product.
+∗Supported in part by U.S. National Science Foundation grant DMS-0 652684.
+†Supported in part by the France-Israel Research Network Prog ram in Mathematics contract
+#3-4301
+1be no confusion of the last notation with the notation for the Euclide an norm of a
+vectorx∈Rnwhich is |x|=/radicalbig
+/a\}bracketle{tx,x/a\}bracketri}ht. A well known result of Santal´ o [S] states that
+in every convex body KinRnthere exists a unique point s(K), called the Santal´ o
+pointofK, such that/vextendsingle/vextendsingleKs(K)/vextendsingle/vextendsingle= min
+z∈int(K)|Kz|.
+Thevolume product ofKis defined by
+P(K) = inf{|K||Kz|:z∈int(K)}.
+A well known conjecture, called sometimes Mahler’s conjecture ([Ma 1, Ma2]), states
+that, for every convex body KinRn,
+P(K)/greaterorequalslantP(S) =(n+1)n+1
+(n!)2(1)
+whereSis ann-dimensional simplex. It is also conjectured that equality in (1)
+is attained only if Kis a simplex. The inequality (1) for n= 2 was proved by
+Mahler [Ma1] with the case of equality proved by Meyer [Me91]. Other cases, like e.g.
+bodies of revolution, were treated in [MR98]. Several special case s in the centrally
+symmetric setting can be found in [SR, R86, GMR, Me86, R87]. Not man y special
+cases in which (1) is true seem to be known, one such is proved in [MR0 6]: all n
+dimensional polytopes with at most n + 3 vertices (or facets). For m ore information
+on Mahler’s conjecture, see an expository article [Tao] by Tao.
+The (non-exact) reverse Santal´ o inequality of Bourgain and Milman [BM] is
+P(K)/greaterorequalslantcnP(Bn
+2)
+wherecis a positive constant and Bn
+2is the Euclidean ball (or any ellipsoid) in
+Rn. Kuperberg [Ku] reproved this result with an improved constant. T his should be
+compared with the Blaschke-Santal´ o inequality
+P(K)/lessorequalslantP(Bn
+2)
+with equality only for ellipsoids ([S], [P], see [MP] or also [MR06] for a simple p roof
+of both the inequality and the case of equality)
+The volume product is affinely invariant, that is, P(A(K)) =P(K) for every
+affine isomorphism A:Rn→Rn. Thus, in order to deal with local behavior of the
+volume product we need the following affine-invariant (the Banach-Mazur ) distance
+between convex bodies:
+dBM(K,L) = inf/braceleftBig
+c:A(L)⊂B(K)⊂cA(L),for affine isomorphisms A,BonRn/bracerightBig
+2If bothKandLare symmetric convex bodies, this is just the classical Banach-Maz ur
+distance.
+In a recent paper [NPRZ], the following result, connected to the sym metric form
+of Mahler’s conjecture, is proved:
+Theorem [NPRZ]LetKbe an origin-symmetric convex body in Rn. Then
+P(K)/greaterorequalslantP(Bn
+∞),
+provided that dBM(K,Bn
+∞)/lessorequalslant1+δ, andδ=δ(n)>0is small enough (where Bn
+∞is
+theℓn
+∞unit ball). Moreover, the equality holds only if dBM(K,Bn
+∞) = 1, i.e., ifKis
+a parallelopiped.
+In this paper we prove the analogous result for the n-dimensional simplex.
+Theorem 1. There exists δ(n)>0such that the following holds: Let Sbe a simplex
+inRnandKa convex body in RnwithdBM(K,S) = 1+δfor0< δ < δ(n). Then
+P(K)/greaterorequalslantP(S)+Cδ,
+whereC=C(n)is a positive constant.
+Therearesomeprofounddifferencesbetweenthesymmetricandt henon-symmetric
+cases. The most important one is, perhaps, the changed location o f the Santal´ o point
+when the body changes even slightly. Section 2 of this paper deals wit h this change
+(it is shown that it obeys linear stability) and its implication on the volume of the
+polar body (square-order stability). We believe that the results of Section 2 have
+importance for their own sake.
+Section 3 presents the necessary changes to the methods of [NPR Z], Among these
+we mention, in particular, the proof of Lemma 4 here, which is the ana logue of
+Section 5 of [NPRZ]. This Lemma required a new proof that will work also in the
+non-symmetric setting. We provide here a coordinate-free proof that has a potential
+of being useful in other settings as well.
+¿From now on we fix a regular simplex ∆ nto be the convex hull of n+1 vertices
+v0,...,v nwherev0,...,v nare points on Sn−1satisfying
+/a\}bracketle{tvi,vj/a\}bracketri}ht=/braceleftbigg1,ifi=j
+−1
+n,otherwise.
+Note that ∆◦
+n=−n∆n. Most of the constants throughout the proofs depend on the
+dimension n. They do not depend on the body K. We use the same lette r (usually C,
+cetc.) to denote different constants in different paragraphs or eve n in different lines.
+Acknowledgment . It is with great pleasure that we thank Karoly Boroczky, Math-
+ieu Meyer, Dmitry Ryabogin and Artem Zvavitch for very helpful adv ice during the
+preparation of this paper.
+32 Continuity of the Santal´ o map
+The following volume formula is known (using polar coordinates). For e very interior
+pointzofK,
+|Kz|=1
+n/integraldisplay
+Sn−1(hK(θ)−/a\}bracketle{tz,θ/a\}bracketri}ht)−ndσ(θ).
+whereσis the spherical Lebesgue measure and hKis the support function of K. By
+the minimum property of |Kz|at the Santal´ o point s(K), it turns out that z=s(K)
+is auniquepoint satisfying the condition (see [Sc])
+/integraldisplay
+Sn−1(hK(θ)−/a\}bracketle{tz,θ/a\}bracketri}ht)−n−1θdσ(θ) = 0. (2)
+This is equivalent to the fact that the centroid of Kzis the origin.
+Denote by Knthe space of convex bodies in Rnendowed with the Hausdorff
+metricdH. The space ( Kn,dH) is isometrically embedded in the space C(Sn−1) of
+continuous functionsonthesphere Sn−1bytheisometry K/ma√sto→hK, thatis, dH(K,L) =
+/bardblhK−hL/bardbl∞for every K,LinKn.
+Proposition 1. The Santal´ o map s: (Kn,dH)→Rnis continuous. Furthermore,
+for every convex body K0, there exist positive constants C=C(K0)andδ=δ(K0)
+such that
+dH(K,K0)/lessorequalslantδ⇒ |s(K)−s(K0)|/lessorequalslantCdH(K,K0).
+Proof.The continuity of s(K) is proved using a standard argument (the dominated
+convergence theorem and the uniqueness of s(K) in (2)).
+For the second part, fix a convex body K0and letKbe any convex body which
+is close to K0in the Hausdorff metric. Since s(K0) is in the interior of K0, there is a
+r0>0 such that the ball B(s(K0),r0) with center s(K0) and radius r0is contained
+inK0. Then, since
+Ks(K0)
+0= (K0−s(K0))◦⊂B(0,r0)◦=B(0,1/r0),
+we have, for every θ∈Sn−1,
+hK0(θ)−/a\}bracketle{ts(K0),θ/a\}bracketri}ht=/bardblθ/bardblKs(K0)
+0/greaterorequalslantr0.
+Define three functions a,x,yonSn−1by
+a(θ) =hK0(θ)−/a\}bracketle{ts(K0),θ/a\}bracketri}ht,
+x(θ) =/a\}bracketle{ts(K),θ/a\}bracketri}ht−/a\}bracketle{ts(K0),θ/a\}bracketri}htand
+y(θ) =hK(θ)−hK0(θ).
+4Note that, for every θ∈Sn−1,
+|a(θ)|/greaterorequalslantr0,|x(θ)|/lessorequalslant|s(K)−s(K0)|,|y(θ)|/lessorequalslantdH(K,K0)
+andhK(θ)−/a\}bracketle{ts(K),θ/a\}bracketri}ht=a(θ)−x(θ)+y(θ). By the Talyor formula, we can write as
+(a−x+y)−n−1=a−n−1/bracketleftBig
+1−x−y
+a/bracketrightBig−n−1
+=a−n−1/bracketleftBig
+1+(n+1)x−y
+a+f/parenleftbiggx−y
+a/parenrightbigg/bracketrightBig
+wheref(t) := (1−t)−n−1−1−(n+1)tisO(t2) for small t. Then (2) implies
+0 =/integraldisplay
+Sn−1(hK(θ)−/a\}bracketle{ts(K),θ/a\}bracketri}ht)−n−1θdσ(θ) =/integraldisplay
+Sn−1(a−x+y)−n−1θdσ(θ)
+=/integraldisplay
+Sn−1/parenleftBig
+a−n−1+(n+1)(x−y)
+an+2+f/parenleftbigx−y
+a/parenrightbig
+an+1/parenrightBig
+θdσ(θ)
+=: (1)+( n+1)/bracketleftBig
+(2)−(3)/bracketrightBig
++(4)
+where (1) ,(2),(3) and (4) are:
+(1)
+/integraldisplay
+Sn−1a(θ)−n−1θdσ(θ) =/integraldisplay
+Sn−1(hK0(θ)−/a\}bracketle{ts(K0),θ/a\}bracketri}ht)−n−1θdσ(θ) = 0.
+(2)
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Sn−1x(θ)
+a(θ)n+2θdσ(θ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Sn−1/a\}bracketle{ts(K)−s(K0),θ/a\}bracketri}ht
+[hK0(θ)−/a\}bracketle{ts(K0),θ/a\}bracketri}ht]n+2θdσ(θ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+/greaterorequalslant/angbracketleftbigg
+ζ,/integraldisplay
+Sn−1/a\}bracketle{ts(K)−s(K0),θ/a\}bracketri}ht
+[hK0(θ)−/a\}bracketle{ts(K0),θ/a\}bracketri}ht]n+2θdσ(θ)/angbracketrightbigg
+=|s(K)−s(K0)|/integraldisplay
+Sn−1|/a\}bracketle{tζ,θ/a\}bracketri}ht|2dσ(θ)
+[hK0(θ)−/a\}bracketle{ts(K0),θ/a\}bracketri}ht]n+2
+/greaterorequalslant|s(K)−s(K0)|/integraldisplay
+Sn−1|/a\}bracketle{tζ,θ/a\}bracketri}ht|2dσ(θ)
+|diam(K0)|n+2
+=C1|s(K)−s(K0)|,
+whereζ=s(K)−s(K0)
+|s(K)−s(K0)|∈Sn−1andC1=|Bn
+2||diam(K0)|−n−2.
+(3)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Sn−1y(θ)
+a(θ)n+2θdσ(θ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant/integraldisplay
+Sn−1|y(θ)|
+|a(θ)|n+2|θ|dσ(θ)/lessorequalslant1
+rn+2
+0dH(K,K0).
+5(4)
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Sn−1f/parenleftbigx−y
+a/parenrightbig
+an+1θdσ(θ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant/integraldisplay
+Sn−1c/vextendsingle/vextendsingle/vextendsinglex(θ)−y(θ)
+a(θ)/vextendsingle/vextendsingle/vextendsingle2
+|a(θ)|n+1|θ|dσ(θ)
+/lessorequalslant2c/integraldisplay
+Sn−1|x(θ)|2+|y(θ)|2
+|a(θ)|n+3dσ(θ)
+/lessorequalslant2c/integraldisplay
+Sn−1|s(K)−s(K0)|2+dH(K,K0)2
+rn+3
+0dσ(θ)
+=C2/parenleftBig
+|s(K)−s(K0)|2+dH(K,K0)2/parenrightBig
+,
+whereCis an absolute constant such that |f(t)|/lessorequalslantC|t|2fortnear 0 and
+C2= 2cn|Bn
+2|r−n−3
+0.
+Finally we have
+C1|s(K)−s(K0)|/lessorequalslant|(2)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(3)−1
+n+1/parenleftBig
+(1)+(4)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+/lessorequalslant1
+rn+2
+0dH(K,K0)+C2
+n+1/parenleftBig
+|s(K)−s(K0)|2+dH(K,K0)2/parenrightBig
+.
+By continuity of s(K) (and, infact, local uniformcontinuity at K0),|s(K)−s(K0)| →
+0 whenever dH(K,K0)→0. Thus the two quadratic terms in the inequality above
+can be ignored whenever δis small enough. Therefore
+|s(K)−s(K0)|/lessorequalslantCdH(K,K0)
+whereCis a constant greater than |diam(K0)|n+2|Bn
+2|−1r−n−2
+0.
+Proposition 2. LetKbe a convex body in Rn. Ifz∈int(K)is close enough to s(K)
+then
+|Kz| ≤ |Ks(K)|/parenleftbigg
+1+c
+r2
+0|z−s(K)|2/parenrightbigg
+,
+wherer0>0is such that B(s(K),r0)⊂Kandcis a constant independent of K.
+Proof.Assume that 0 is the Santal´ o point of K. ThenKs(K)=K◦,
+|K◦|=/integraldisplay
+K◦dyand|Kz|=/integraldisplay
+K◦dy
+(1−/a\}bracketle{tz,y/a\}bracketri}ht)n+1
+(cf. e.g. Lemma 3 of [MW]). Note that K◦⊂B(0,r−1
+0), Hence |/a\}bracketle{tz,y/a\}bracketri}ht| ≤|z|
+r0for
+y∈K◦. We represent (1 −t)−(n+1)as 1+(n+1)t+g(t). We have
+/integraldisplay
+K◦/a\}bracketle{tz,y/a\}bracketri}htdy= 0
+6because 0 is the Santal´ o point of K. Thus
+|Kz|=|K◦|+/integraldisplay
+K◦g(/a\}bracketle{tz,y/a\}bracketri}ht)dy
+and we get
+|Kz| ≤ |K◦|/parenleftBigg
+1+∞/summationdisplay
+j=2(n+1)(n+2)...(n+j)
+j!/parenleftbigg|z|
+r0/parenrightbiggj/parenrightBigg
+.
+We finally have
+|Kz| ≤ |K◦|/parenleftBigg
+1+c/parenleftbigg|z|
+r0/parenrightbigg2/parenrightBigg
+if|z| ≤r0
+2(say).
+Remark 1. Under the assumptions of Proposition 2, if |z−s(K)|< r0andK
+contains a ball B(z,2r0)then it contains B(s(K),r0)this will be used later in the
+application of Proposition 2.
+3 Construction of auxiliary Polytopes
+In this section we prove an analogue of [NPRZ] for the n-dimensional simplex. Thus
+most of the ideas and tools that are used in the proofs in this section are basically
+adaptations of those from [NPRZ]. Lemma 4, which replaces Section 4 of [NPRZ],
+had to be worked out anew and to be put on a less coordinate depend ent basis.
+LetFbe ak-dimensional face of ∆ nfor 0/lessorequalslantk < nand denote by cFits centroid.
+Consider theaffinesubspace HF=cF+F⊥whereF⊥={y∈Rn:/a\}bracketle{tx,y/a\}bracketri}ht= 0∀x∈F}.
+Take at >0 such that tHFis tangent to K. In case that (1 −δ)∆n⊂K⊂∆n, it
+should be 1 −δ/lessorequalslantt/lessorequalslant1. LetxFbe such a tangent point, that is, xF∈tHF∩∂K
+and putyF=tcF. Denote the dual face of FbyF∗={y∈∆◦
+n:/a\}bracketle{tx,y/a\}bracketri}ht= 1∀x∈F}.
+By the same way as above, we have points x∗
+Fandy∗
+Fby replacing F,Kand ∆ nby
+F∗,K◦and ∆◦
+n, respectively. These four points xF,yF,x∗
+Fandy∗
+Fhave the following
+properties.
+Lemma 1. LetFbe a face of ∆n. Suppose that (1−δ)∆n⊂K⊂∆n. Then
+1./a\}bracketle{txF,x∗
+F/a\}bracketri}ht= 1 =/a\}bracketle{tyF,y∗
+F/a\}bracketri}ht
+2./a\}bracketle{txF−yF,cF/a\}bracketri}ht= 0 =/a\}bracketle{tx∗
+F−y∗
+F,cF/a\}bracketri}ht
+3.|xF−yF|<2δand|x∗
+F−y∗
+F|<2nδ.
+7Proof.1. Lett,s >0 be such that xF∈tHF∩∂Kandx∗
+F∈sHF∗∩∂K◦. Then we
+can check easily that
+HF∗:=cF∗+(F∗)⊥
+={z∈Rn:/a\}bracketle{tz,h/a\}bracketri}ht= 1∀h∈HF}.
+Consider a hyperplane Gcontaining tHFwhich is tangent to KatxFand letαbe
+the dual point of G, that is, /a\}bracketle{tα,z/a\}bracketri}ht= 1 for every z∈G. So/a\}bracketle{tα,th/a\}bracketri}ht= 1 for all h∈HF
+which implies tα∈HF∗. Sinceα∈∂K◦by construction of G, we getα∈1
+tHF∗∩∂K◦
+which implies s= 1/t. After all, /a\}bracketle{txF,x∗
+F/a\}bracketri}ht= 1 since xF∈tHFandx∗
+F∈1
+tHF∗, and
+/a\}bracketle{tyF,y∗
+F/a\}bracketri}ht=/angbracketleftbig
+tcF,1
+tcF∗/angbracketrightbig
+= 1.
+2. Note that /a\}bracketle{txF−yF,cF/a\}bracketri}ht=/a\}bracketle{txF,cF/a\}bracketri}ht−/a\}bracketle{tyF,cF/a\}bracketri}ht=t−t= 0. Similarly, we have
+/a\}bracketle{tx∗
+F−y∗
+F,cF∗/a\}bracketri}ht= 0. Thus /a\}bracketle{tx∗
+F−y∗
+F,cF/a\}bracketri}ht= 0 since cF∗=1
+|cF|2cF.
+3. Write F= conv(v0,...,v k). ThenF⊥is in the linear span of vk+1,...,v nand
+hencetHF=tcF+F⊥should be in the linear span of vk+1,...,v nandcF. Thus
+tHF∩∆n= (tcF+F⊥)∩conv(v0,v1,...,v n)
+= (tcF+F⊥)∩conv(cF,vk+1,...,v n)
+⊂tcF+(1−t)conv(vk+1,...,v n).
+Therefore
+|xF−yF|/lessorequalslantdiam(tHF∩∆n)/lessorequalslant(1−t)diam/parenleftbig
+conv(vk+1,...,v n)/parenrightbig
+/lessorequalslantdiam(∆ n)δ.
+Similarly, we get |x∗
+F−y∗
+F|/lessorequalslantdiam(∆◦
+n)δ.
+LetFbe the set of all faces of ∆ n. A family FofnfacesF0,...,F n−1inFis
+called aflagoverFif eachFkis ak-dimensional face in FandF0⊂F1⊂ ··· ⊂Fn−1.
+For each face F∈ F, we constructed four points xF,x∗
+F,yFandy∗
+Fin the previous
+paragraph. These points induce the following four polytopes (in gen eral, not convex):
+P=/uniondisplay
+Fconv/parenleftbig
+0,xF0,...,x Fn−1/parenrightbig
+, P′=/uniondisplay
+Fconv/parenleftbig
+0,x∗
+F0,...,x∗
+Fn−1/parenrightbig
+,
+Q=/uniondisplay
+Fconv/parenleftbig
+0,yF0,...,y Fn−1/parenrightbig
+, Q′=/uniondisplay
+Fconv/parenleftbig
+0,y∗
+F0,...,y∗
+Fn−1/parenrightbig
+whereF:={F0,...,F n−1}runs over all flags of F. Under the assumption (1 −
+δ)∆n⊂K⊂∆n, they clearly satisfy P⊂K,P′⊂K◦, (1−δ)∆n⊂Q⊂∆nand
+∆◦
+n⊂Q′⊂1
+1−δ∆◦
+n.
+Lemma 2. |Q|·|Q′|/greaterorequalslant|∆n|·|∆◦
+n|
+8The proof is essentially the same as the proof of Lemma 7 of [NPRZ].
+Lemma 3. Suppose that (1−δ)∆n⊂K⊂∆n. Then there exist constants C1and
+C2such that/vextendsingle/vextendsingle|P|−|Q|/vextendsingle/vextendsingle/lessorequalslantC1δ2and/vextendsingle/vextendsingle|P′|−|Q′|/vextendsingle/vextendsingle/lessorequalslantC2δ2
+Proof.We can check that Lemma 4 of [NPRZ] is also true for the simplex ∆ n. This
+fact, together with Lemma 1 here and Lemma 5 of [NPRZ] (taking X0={cF},
+X1={xF},X2={yF}and similarly for the starred points), completes the proof of
+the lemma.
+Suppose that all the centroids of facets of ∆ nbelong to K. Then, for every facet
+Fof ∆n,
+xF=yF=cFandx∗
+F=y∗
+F=cF∗.
+This is helpful in the proof of the following lemma.
+Lemma 4. There exists c′>0such that if δ= min{d >0 : (1−d)∆n⊂K⊂∆n}is
+small enough and if all the centroids of facets of ∆nbelong to K, then|K|/greaterorequalslant|P|+c′δ
+or|K◦|/greaterorequalslant|P′|+c′δ.
+Proof.Webeginbyproving thatthereexists a constant c1>0 suchthat (1+ c1δ)P′/\e}atio\slash⊃
+P◦. Sinceδis the minimal number that satisfies (1 −δ)∆n⊂K, we can find a vertex
+vjof∆n, sayv0, such that(1 −δ)v0∈∂K. Taking F0={v0}inLemma 1, weconclude
+the existence of
+x0=tv0+h∈∂K,withh∈v⊥
+0,|h|< Cδ,1−δ≤t≤1,
+and
+x∗
+0=sv0∈∂K◦,with 1≤s≤1
+1−δ,
+such that /a\}bracketle{tx0,x∗
+0/a\}bracketri}ht=ts= 1.
+Letz∗∈∂K◦be such that /a\}bracketle{tz∗,(1−δ)v0/a\}bracketri}ht= 1. Then H={x;/a\}bracketle{tz∗,x/a\}bracketri}ht= 1}
+is a support hyperplane of Kat (1−δ)v0. Thus/a\}bracketle{tx0,z∗/a\}bracketri}ht ≤1. Since His also a
+support hyperplane of (1 −δ)∆nat (1−δ)v0, it follows that x0lies below the one-
+sided cone Cwith vertex (1 −δ)v0, which is the complimentary half of the cone with
+the same vertex, spanned by (1 −δ)∆n. Take a typical facet Gof the cone C. Say
+G⊂ {x;/a\}bracketle{tx,−nv1/a\}bracketri}ht= 1−δ}. The highest point (with respect to the direction v0) of
+G∩∆nis the intersection of Gwith the line segment [ v0,v1]. A simple calculation
+shows that this is the point βv0+ (1−β)v1withβ= 1−δ
+n+1. The height of this
+point is found by computing its projection on the altitude [ v0,−v0
+n] of ∆ n. This is
+βv0+(1−β)(−v0
+n) = (1−δ
+n)v0. we conclude that
+t≤1−δ
+nands≥1
+1−δ
+n.
+9Thus
+x∗
+0=sv0with1
+1−δ
+n≤s≤1
+1−δ.
+We look now at the vector h∈v⊥
+0that was found above (with x0=tv0+h).
+There exists one of the vectors −nvj−v0,j= 1,...,n, which are vertices in v⊥
+0of
+a regular simplex with center 0, such that /a\}bracketle{t−nvj−v0,h/a\}bracketri}ht ≤0. We may assume that
+thisjis 1 and denote v=γ(−nv1−v0)∈v⊥
+0for some γ >0 whose size will be
+determined later. Note that |v|=γ√
+n2−1 and that /a\}bracketle{tv,h/a\}bracketri}ht ≤0.
+Define ˜x=x∗
+0+v. We claim that if γis chosen correctly then ˜ x∈P◦and, for
+somec1>0, ˜x/\e}atio\slash∈(1+c1)P′. To verify that ˜ x∈P◦we have to check that /a\}bracketle{t˜x,xF/a\}bracketri}ht ≤1
+for all the vertices xFofP. These vertices are of the form xF=1
+k/summationtextk
+i=1vji+g,
+1≤k≤n,|g|< Cδandg= 0 ifk=n.
+1) LetxF0=x0=tv0+h. Then/a\}bracketle{t˜x,xF0/a\}bracketri}ht=/a\}bracketle{tx∗
+0,x0/a\}bracketri}ht+/a\}bracketle{tv,x0/a\}bracketri}ht= 1+/a\}bracketle{tv,h/a\}bracketri}ht ≤1.
+2) LetxF=1
+k/summationtextk
+i=1vji+g, 1≤k≤n,|g|< Cδ. Assume first that the index
+0 is not among the ji-s. SayxF=1
+k/summationtextk
+j=1vj+g(it is true that v1plays a
+somewhat different role than the other indices j≥1, but the result of the
+coming evaluation comes up to be the same). Then
+/a\}bracketle{t˜x,xF/a\}bracketri}ht=1
+kk/summationdisplay
+j=1s/a\}bracketle{tv0,vj/a\}bracketri}ht+/a\}bracketle{tv,1
+kk/summationdisplay
+j=1vj/a\}bracketri}ht+/a\}bracketle{tsv0,g/a\}bracketri}ht+/a\}bracketle{tv,g/a\}bracketri}ht.
+We have |1
+k/summationtextk
+j=1vj|=/radicalBig
+n−k+1
+nkthus
+/a\}bracketle{t˜x,xF/a\}bracketri}ht ≤s/parenleftbigg
+−1
+n+Cδ/parenrightbigg
++|v|/parenleftBigg/radicalbigg
+n−k+1
+nk+Cδ/parenrightBigg
+<1
+for small δ, ifγ <c2
+nfor an appropriate constant c2.
+3) LetxFbe as in 2) above, now with 0 among the ji-s, sayxF=1
+k/summationtextk−1
+j=0vj+g
+(same remark about v1). The calculation now gives
+/a\}bracketle{t˜x,xF/a\}bracketri}ht ≤s
+k−s(k−1)
+nk+|v||1
+kk−1/summationdisplay
+j=0vj|+(s+|v|)|g| ≤
+s/parenleftbigg1
+k−k−1
+kn+Cδ/parenrightbigg
++|v|/parenleftBigg/radicalbigg
+n−k+1
+nk+Cδ/parenrightBigg
+<1
+ifδis small and γ≤c2
+n(note that in this case k≥2).
+10We fix now the constant γthat was introduced above to be preciselyc2
+nwith the
+constant c2obtained above. Then ˜ x∈P◦(provided that δis small enough). As ˜ x
+is a positive linear combination of x∗
+0=sv0and−nv1, the line segment connecting
+the origin to ˜ xmust cross the edge [ sv0,−nv1] ofP′. Thus we look for M >0 and
+0< θ <1 such that the equality
+sv0+v=M(θsv0+(1−θ)(−nv1))
+will hold. Substituting v=γ(−nv1−v0) we get M= 1+γ(1−1
+s). As we had the
+evaluation1
+s=t≤1−δ
+n, we get
+M≥1+γ
+nδ= 1+c2
+n2δ.
+That is, if c1<c2
+n2then ˜x/\e}atio\slash∈(1+c1δ)P′and we get (1+ c1δ)P′/\e}atio\slash⊃P◦.
+Assume that K⊂(1+c1
+2δ)conv(P). Then (1 −c1
+2δ)P◦⊂1
+1+c1
+2δP◦⊂K◦. Let
+˜˜x= (1−c1
+2δ)˜x∈(1−c1
+2δ)P◦⊂K◦.
+By the preceding paragraph we have ˜˜x/\e}atio\slash∈(1+c1δ)(1−c1
+2δ)P′. As (1+ c1δ)(1−c1
+2δ)>
+1+c1
+4δifδ <1
+2c1, we conclude that for δsmall enough, either K/\e}atio\slash⊂(1+c1
+2δ)conv(P);
+in which case, by Lemma 2 of [NPRZ], |K| ≥ |P|+c3δ; or there exists ˜˜x∈K◦, such
+that the line segment [0 ,˜˜x] intersects the edge [ x∗
+0,−nv1] ofP′, but˜˜x/\e}atio\slash∈(1+c1
+4δ)P′.
+That is, K◦/\e}atio\slash⊂(1 +c4δ)P′. Lemma 2 of [NPRZ] completes now the proof. We
+remark, that, since P′is, in general, not convex the assumption of the uniform lower
+bound on the ( n−1)-dimensional volume of its facets should be verified using the
+δ-approximation.
+Proposition 3. LetKbe a convex body in Rnwhich is close to ∆nin the sense
+thatδ= min{d >0 : (1−d)∆n⊂K⊂∆n}is small enough. Suppose that all the
+centroids of facets of ∆nbelong to K. Then we have
+|K||K◦|/greaterorequalslant|∆n||∆◦
+n|+Cδ.
+Proof.We assume that |K|/greaterorequalslant|P|+cδby Lemma 4. Moreover, Lemma 3 implies
+|K||K◦|/greaterorequalslant(|P|+cδ)|P′|
+/greaterorequalslant(|Q|−c1δ2+cδ)(|Q′|−c2δ2)
+=|Q||Q′|+|Q′|(cδ−c1δ2)−c2|Q|δ2−c2δ2(cδ−c1δ2)
+/greaterorequalslant|Q||Q′|+|∆◦
+n|(cδ−c1δ2)−c2|∆n|δ2−c2δ2(cδ−c1δ2)
+Sinceδis small enough, the above inequality implies that |K||K◦|/greaterorequalslant|Q||Q′|+Cδ
+for a constant 0 < C <|∆◦
+n|c. Finally, Lemma 2 completes the proof.
+114 Proof of Theorem 1
+For the proof of main theorem, let us start with the following lemma.
+Lemma 5. LetLbe a convex body in Rncontaining the origin. Then, for every
+convex body KwithdBM(K,L)<1+δ, there are a constant C=C(L)and an affine
+isomorphism A:Rn→Rnsuch that
+(1−Cδ)L⊂A(K)⊂L.
+In particular, if L= ∆nandδ >0is small enough, then such CandAcan be chosen
+to satisfy that every centroid of facets of Lbelongs to A(K).
+Proof.By definition, there are affine isomorphisms A,B:Rn→Rnsuch that
+(1−δ)A(L)⊂B(K)⊂A(L).
+ClearlyA(L) should contain the origin 0. Put a=A−1(0). Then it is in Land we
+can write A(x) =T(x−a), (x∈Rn) for some linear transformation TonRn. Note
+that, for every point x,
+A−1/parenleftBig
+(1−δ)A(x)/parenrightBig
+=A−1/parenleftBig
+(1−δ)T(x−a)/parenrightBig
+=A−1T/parenleftBig
+(1−δ)x+aδ−a/parenrightBig
+=A−1A/parenleftBig
+(1−δ)x+aδ/parenrightBig
+= (1−δ)x+aδ,
+which implies A−1/parenleftBig
+(1−δ)A(L)/parenrightBig
+= (1−δ)L+aδ. Take a constant c >1 such that
+−L⊂cL. We have
+/parenleftBig
+1−(1+c)δ/parenrightBig
+L−aδ⊂/parenleftBig
+1−(1+c)δ/parenrightBig
+L−δL
+⊂/parenleftBig
+1−(1+c)δ/parenrightBig
+L+cδL= (1−δ)L
+The above two facts imply
+/parenleftBig
+1−(1+c)δ/parenrightBig
+L⊂A−1/parenleftBig
+(1−δ)A(L)/parenrightBig
+⊂A−1B(K)⊂L.
+For the case L= ∆n, note that −∆n⊂n∆n. Thus we have
+(1−(n+1)δ)∆n⊂A−1B(K)⊂∆n.
+LetSbe a simplex of minimal volume containing K1:=A−1B(K). It is proved
+in [Kl] that all the centroids of facets of Sbelong to K1. On the other hand, if
+12x∈S\(1+κ)/parenleftBig
+(1−(n+1)δ)∆n/parenrightBig
+, then by Lemma 2 of [NPRZ],
+|S|/greaterorequalslant/vextendsingle/vextendsingle/vextendsingle/parenleftBig
+1−(n+1)δ/parenrightBig
+∆n/vextendsingle/vextendsingle/vextendsingle+κ/parenleftbig
+1−(n+1)δ/parenrightbign1
+n/parenleftBig
+n2
+n+1|∆n|/parenrightBig
+n
+=|∆n|/parenleftBig
+1−(n+1)δ/parenrightBign/parenleftbigg
+1+κ
+n+1/parenrightbigg
+.
+Ifκ > κ0= (n+1)[(1−(n+1)δ)−n−1], the existence of such ximplies|S|>|∆n|
+which is a contradiction. Hence, for κ > κ0,
+(1−κ)S⊂(1−κ)/bracketleftbig
+(1+κ)/parenleftbig
+1−(n+1)δ/parenrightbig
+∆n/bracketrightbig
+⊂/parenleftbig
+1−(n+1)δ/parenrightbig
+∆n
+⊂K1⊂S.
+Note also that there exists a unique affine isomorphism A1satisfying S=A1(∆n).
+Applying the argument of the first part again, we have
+/parenleftbig
+1−(n+1)κ/parenrightbig
+∆n⊂A−1
+1(K1)⊂∆n.
+whereκ >(n+1)/parenleftBig
+(1−(n+1)δ)−n−1/parenrightBig
+≈n(n+1)2δifδ >0 is small enough.
+Proof of Theorem 1. LetKbe a convex body with dBM(K,S) = 1 + δfor suf-
+ficiently small δ >0. By Lemma 5, (replacing Sby ∆n), there is a constant
+C=C(n)>0 such that
+(1−Cδ)∆n⊂A(K)⊂∆n.
+and all the centroids of facets of ∆ nbelong to A(K). Since the volume product is
+invariant under affine isomorphisms of Rn, we may assume that all the centroids of
+facets of ∆ nbelong to K. We may also “include” the constant Cintoδand assume
+that
+δ:= min{d >0 : (1−d)∆n⊂K⊂∆n}.
+This implies that dH(K,∆n)/lessorequalslantδ. From Proposition 1 we now conclude that |s(K)|/lessorequalslant
+c1δfor some c1>0. By Proposition 2, and the remark following it, we get the
+inequality
+|K◦|/lessorequalslant/vextendsingle/vextendsingleKs(K)/vextendsingle/vextendsingle(1+c2|s(K)|2)
+/lessorequalslant/vextendsingle/vextendsingleKs(K)/vextendsingle/vextendsingle(1+c1c2δ2)
+for some constant c2>0 (a-priory, c2would depend on the radius of a ball centered
+ats(K) and contained in K. The remark following Proposition 2 allows us to use
+13instead a ball centered at 0. The relation (1 −δ)∆n⊂Kthen allows us to use a ball
+contained in, say,1
+2∆ninstead. Thus c2may be considered as independent of K).
+Hence/vextendsingle/vextendsingleKs(K)/vextendsingle/vextendsingle≥ |K◦|−c1c2/vextendsingle/vextendsingleKs(K)/vextendsingle/vextendsingleδ2≥ |K◦|(1−c1c2δ2),
+and the volume product of Ksatisfies
+P(K) =|K|/vextendsingle/vextendsingleKs(K)/vextendsingle/vextendsingle/greaterorequalslant|K||K◦|(1−c1c2δ2).
+Proposition 3 implies that |K||K◦|/greaterorequalslant|∆n||∆◦
+n|+cδ. Finally, we have
+P(K)/greaterorequalslant/parenleftBig
+|∆n||∆◦
+n|+cδ/parenrightBig
+(1−c1c2δ2)
+/greaterorequalslantP(S)+Cδ.
+for sufficiently small δ >0 and a constant C >0.
+References
+[BM]J. Bourgain, V. D. Milman, New volume ratio properties for convex sym-
+metric bodies in Rn. Invent. Math. 88 (1987), 319-340.
+[GMR]Y. Gordon, M. Meyer and S. Reisner ,Zonoids with minimal volume–
+product - a new proof , Proc. Amer. Math. Soc. 104 (1988), 273-276.
+[Kl]V. Klee ,Facet-centroids and volume minimization , Studia Sci. Math. Hun-
+gar. 21 (1986), 143-147.
+[Ku]G. Kuperberg ,From the Mahler Conjecture to Gauss Linking Integrals ,
+Geometric And Functional Analysis 18 (2008), 870-892.
+[Ma1]K. Mahler ,Ein Minimalproblem f¨ ur konvexe Polygone , Mathematica (Zut-
+phen) B7 (1939), 118-127.
+[Ma2]K. Mahler ,Ein¨Ubertr¨ agungsprinzip f¨ ur konvexe K¨ orper ,ˇCasopisPˇ est. Mat.
+Fys. 68 (1939), 93-102.
+[Me86]M. Meyer ,Une caract´ erisation volumique de certains espaces norm´ e s,Israel
+J. Math. 55 (1986), 317-326.
+[Me91]M. Meyer ,Convex bodies with minimal volume product in R2, Monatsh.
+Math. 112 (1991), 297-301.
+[MP]M. Meyer and A. Pajor ,On Santal´ o inequality , Geometricaspects offunc-
+tional analysis (1987-88), Lecture Notes in Math. 1376, Springer Ver. (1989),
+261-263.
+14[MR98]M. Meyer and S. Reisner ,Inequalities involving integrals of polar-
+conjugate concave functions , Monatsh. Math. 125 (1998), 219-227.
+[MR06]M. Meyer and S. Reisner ,Shadow systems and volumes of polar convex
+bodies, Mathematika 53 (2006), 129-148.
+[MW]M. Meyer and E. Werner ,The Santal´ o-regions of a convex body , Trans.
+A.M.S. 350 (1998), 4569-4591.
+[NPRZ] F. Nazarov, F. Petrov, D. Ryabogin, A. Zvavitch ,A remark on the
+Mahler conjecture: local minimality of the unit cube , to appear in Duke Math.
+J. .
+[P]C. M. Petty ,Affine isoperimetric problems , Ann. New York Acad. Sci. 440
+(1985), 113-127.
+[R86]S. Reisner ,Zonoids with minimal volume–product , Math. Zeitschrift 192
+(1986), 339-346.
+[R87]S. Reisner ,Minimal volume product in Banach spaces with a 1-unconditio nal
+basis, J. London Math. Soc. 36 (1987), 126-136.
+[S]L. A. Santal ´o,Un invariante afin para los cuerpos convexos del espacio de
+ndimensiones , Portugal. Math. 8 (1949), 155-161.
+[Sc]R. Schneider ,Convex bodies: The Brunn-Minkowski theory , Encyclop.
+Math. Appl. 44, Cambridge University Press (1993).
+[SR]J. Saint Raymond ,Sur le volume des corps convexes sym´ etriques . S´ eminaire
+d’Initiation ` a l’Analyse, 1980-81, Universit´ e Paris VI, 1981.
+[Tao]T. Tao,Structure and Randomness: pages from year one of a mathemati cal
+blog, Amer. Math. Soc. (2008), 216-219.
+J. Kim: Department of Mathematics, Kent State University, K ent, OH 44242, USA.
+Email: jkim@math.kent.edu
+S. Reisner: Department of Mathematics, University of Haifa , Haifa 31905, Israel.
+Email: reisner@math.haifa.ac.il
+15
\ No newline at end of file
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+ 1The free states-related Fermi pocket of cuprate superconductors  
+Tian De Cao* 
+Department of Physics, Nanjing University of Information Science & Technology, Nanjing 210044, China 
+ 
+Abstract 
+To stress the effect of the pairing position deviating from the Fermi level, we must investigate the 
+pairs in the wave vector space, and then we us e the dynamic equation to study some correlation 
+functions. This article shows that the Fermi pocket is related to the effect of free electron states on 
+the ARPES experiment. This also leads us to understand why the Fermi arc appears in Bi2212 while 
+the Fermi pocket appears in Bi2201 with the va lence bandwidth and the work function known for 
+them.  
+PACS  numbers : 79.60.-i; 74.72.-h; 74.20.Mn; 74.20.Rp 
+Keywords:  pseudogap; superconductivity; Fermi arc; Fermi pocket 
+ 
+Ⅰ. Introduction 
+The Fermi pocket is suggested by the d-density wa ve [1,2] or other order-based mechanisms, and 
+it is observed in the underdoped Bi 2201[3]. However, there are some  deviation between the theory 
+prediction and the experiment result. Moreover, the Fermi arc is discovered in the underdoped 
+Bi2212 [4,5]. Why does the Fermi pocket appear in  the underdoped Bi2201 not  in the Bi2212?  In 
+the previous paper [6], it is s hown that the elect ronic structures of cuprat e superconductors favor the 
+pairs-based pseudogap appearing in the antinodal region, and the relation between 
+superconductivity and pseudogap can be understood we ll, despite some physicists suggested the 
+order-based pseudogap [7,8]. On the basis of same ideas, this articl e shows that the Fermi arc and 
+pocket can be understood.  
+ 
+Ⅱ. Review of ARPES principle 
+ 
+The quantum measured by the ARPES experiment  is the hopping probability from a localized 
+                                                        
+*Corresponding author. 
+*E-mail address: tdcao@nuist.edu.cn (T. D. Cao). 
+*Tel: 011+86-13851628895   2orbital to a free state, while both the localized orbital and the free state all belong to the same 
+Hamiltonian0Has shown in Eq.(2). The total Hamiltonian is '0H H H+= , 'H describes the 
+interaction between electrons and photons. Because the photons can in crease the probability of the 
+electrons occupying free-particle states, when we in tend to discuss the electron states which are not 
+too lower than the Fermi energy level, we should also consider the free electron states, and we 
+derived the formula of photoemission intensity [9] 
+) ,(kin
+kEkIr
+= kMC2
+int') ( νφh Enkin
+ka
+F−+  ) ,,( νφσ h E kAkin
+k a −+ ⋅r
+                       ( 1 )  
+where kinE is the kinetic energy of the photoelectron, νhis the photon energy, and φ is the work 
+function. It is n ecessary to note that ),,(ωσkAar
+is the one-particle spectral function for the electron 
+systems both free states and localized el ectron orbitals with the Hamiltonian 0H as discussed in 
+the previous literature [10](the so-called “state outside a solid” in the pr evious paper seems vague, 
+because the free state is also  the state inside a solid.), and the energy conservation law 
+kin
+kE+φνh−k
+BEr
+− =0 is used, where k
+BEr
+ are the energies of electrons (k
+BEr
+=0 at FE) inside the 
+solid.  
+To understand the spectral functionaA, we take the simple Hamiltonian 
+=0Hσσ
+σωk k
+kkccrr
+r+∑
+,+ .). (
+,ch cbMk
+kkk+∑+
+σ
+σσr
+rrr +σσ
+σξk k
+kkbbrr
+r+∑
+,+bbH−                        ( 2 )  
+wherekω= μφ−+kin
+kE , krξ=με−kr , σqbrdestroy an electron in qr state of spin σinside the solid, 
+σkcrdestroy an electron in the free states, andbbH−represent the interactions between electrons inside 
+the solid. Eq.(2) requires⊥k=0, while we extend it to⊥kk<<, and this is met in some experiments. 
+The b-band (it does not strictly co rrespond to the Cud or Op states) in Eq.(2) expresses the energy 
+band near the Fermi energy in the CuO 2 planes of the p-type cupr ate superconductors. Although the 
+properties of the CuO 2 planes may be described by a multi-band model, the electronic feature near 
+the Fermi surface could be described affectively by the a-model.   
+Ⅲ. Calculation 
+To find the obvious effect of the overlap matrix element kMr(another element, intM, is not 
+discussed in this article) in the pseudogap phase, we take the model  3=0Hσσ
+σωk k
+kkcc+∑
+,+ .). (
+,ch cbMk
+kkk+∑+
+σ
+σσ +σσ
+σξk k
+kkbb+∑
+,+↓+
+↓↑+
+↑∑ lll
+llbbbb U                 ( 3 )  
+where we have denoted wave vector kr
+ as k, kkr
+≡. To compare with experiments in detail, this 
+model should be improved. However, we are only in terested in the qualitative relation of some 
+quanta. If we introdu ce the charge operator σ
+σσ ρk
+kqkb b q∑+
++=
+,21)(ˆ  and the spin operator 
+σ
+σσσk
+kqkbb qS∑+
++=
+,21)(ˆ , Eq. (3) is rewritten in the form 
+=0Hσσ
+σωk k
+kkcc+∑
+,+ .). (
+,ch cbMk
+kkk+∑+
+σ
+σσ +σσ
+σξk k
+kkbb+∑
+,+ )(ˆ)(ˆ q qU
+q−∑ρρ )(ˆ)(ˆ qSqSUz z
+q− −∑    (4) 
+To discuss the effect of kM, we define the functions 
+)' ,,(ττσ− kGb = > <−+)'()(ττσ στ k k b bT  
+)' ,,(ττσ−+kF = > <+ +)'()(ττσ στ k k b bT  
+)' ,,(ττσ− kF = > < )'()(ττσ στ k kb bT  
+The model (4) is similar to the Hubbard model, but  it includes the free states which have energy 
+levels higher above the chemical potential. Th e Hubbard model has not superconductivity on the 
+basis of the evidence of off- diagonal long range order (ODLRO) [11], while other calculations 
+found superconductivity. Moreover, we suggest that th e superconductivity is due to the pairs on the 
+Fermi surface and the ODLRO evidence is not reliable.  
+The doubly occupied states should be neglected for →U∞. However, we are interested in the 
+states near the Fermi surface, thus Uis not too large for the doped cuprate superconductors, we 
+establish their dynamic equati ons and find the tw o-particle Green’s functions such as 
+> <+
++ )'( )(ˆ τσσ τ k qkbbqST  and > <+
++ )'( )(ˆ τ ρσσ τ k qkb bqT  in these equations. Moreover, we proceed to 
+establish the dynamic equations of > <+
++ )'( )(ˆ τσσ τ k qkbbqST  and > <+
++ )'( )(ˆ τ ρσσ τ k qkb bqT  and use the 
+cut-off approximation. This is not the Hertree-Fock  approximation because we consider the effect of 
+correlations, and we obtain 
+k niξω~[+− +
+k nk
+iM
+ωω−2
++
+qk n qiqkP
++−∑ξωσ),,(),(]n b ikGωσ  
+=1−+
+k niV
+ξωρ
++−>< )0(ˆ)0(+
+qk nk qk
+q i++
++−−∑ξωξξ),()0, (nikF qkUF ωσ τσ+=+                    ( 5 )   4k niξω~[−− +
+qk n qiqkP
+++∑ξωσ),,(),(]nikFωσ+= ),()0 ,, (n b
+q qk nk qkikG qkFUiωσ τσξωξξ=−−−−+
+++∑      ( 6 )  
+To arrive at the Eq. (5) and (6),  we introduced theses functions 
+kξ~=kξ+bUn21 
+),,()( nikRωσ± =
+qk n qiqkP
++±∑ξωσ),,( 
+),,()( nikωσ±Δ =
+qk nk qk
+q i++
+±−−∑ξωξξ)0, ( =+τσqkUF  
+),,()( nikωσ+
+±Δ =
+qk nk qk
+q i++
+±−−∑ξωξξ)0, ( =++τσqk UF  
+),,(σqkP = >−< )(ˆ)(ˆ 2qSqS U + >−< )(ˆ)(ˆ2q q U ρρ >−<− )(ˆ)(ˆ 22qSq Uρσ  
+wherebnis the b-electron number at each site, >−< )(ˆ)(ˆ qSqS >− −≡< )0,(ˆ),(ˆ τττ qSqST and so on. It 
+is shown that the function G, Fand +Fdepend on the spin index σ due to the spin-charge 
+correlation function ><Sˆˆρ , while the spin dependence does not affect our main result for 
+non-magnetic materials, and it will be  neglected below. Thus we find 
+k niξω~{−
+k nk
+iM
+ωω−−2
+),()( nikRω −− ),( ),()( )( n n ik ik ω ω++
+−Δ Δ− /k niξω~[+ ),()( nikRω +− ),(]}n bikGω 
+=1+
+k nb
+iUn
+ξω−2/                                                                ( 7 )  
+k niξω~{[+ ),()( nikRω+− ]k niξω~[−
+k nk
+iM
+ωω−−2
+),()( nikRω −− ] ),( ),()( )( n n ik ik ω ω++
+−Δ Δ− ),(}nikFω+ 
+= ),()( nikω+
+−Δ− [1+
+k nb
+iUn
+ξω−2/]                                                     ( 8 )  
+A detail analysis shows that a non-zero pairing temperature pairT>0K can be found with Eq. (8), 
+and the highest pairing temperature will appear when the gap function )()(++
+−ΔΔ have the d-wave or 
+the similar non-s wave symmetry. A detail di scussion can be found in the previous paper6 if we note 
+that the pairing temperature is dominated by kM=0. Therefore, to discuss the electronic structure 
+near the node, we assume )()(++
+−ΔΔ =0 in the node (y x k k±= ) even if for pairTT< , this electronic  5structure can be extended to the small value of )()(++
+−ΔΔ in the node, and we arrive at 
+),(n bikGω=[1+
+k nb
+iUn
+ξω−2/]/k niξω~[−
+k nk
+iM
+ωω−−2
+)],()( nikRω −−                         ( 9 )  
+The form of bGin Eq.(9) shows that the excitation ener gies near (or on ) the Fermi surface are 
+determined by 
+k kEξ~−
+k kk
+EM
+ω−−2
+),()( kEkR−− = 0                                               ( 1 0 )  
+The Eq.(10) can be solved by successive iteration. Let k kEξ~)0(− ) /()0( 2
+k k kE M ω− − =0, we get 
+)0(
+)(,±kE = ] )~(~[21 2 2
+k k k k k M+−±+ ξωξω , thus )(
+)(,n
+kE±=kξ~+
+kn
+kk
+EM
+ω−−
+±)1(
+)(,2
++ ) ,()1(
+)(,)(−
+±−n
+kEkR . We find 
+)(
+)(,1n
+kE+)(
+)(,2n
+kE−−  =0 if )0(
+)(,1+kE)0(
+)(,2−−kE =0, thus we will pay close attention to the zero order 
+approximation)0(
+)(,±kE . To clearly see the related result, one can first take kM=0 and find 
+)0(
+)(,1+kE)0(
+)(,2−−kE =
+2 1~
+k kξω− =
+2 1 kkin
+kEεφ−+ . Because the width of 
+2kεis pt2 (the bandwidth of the 
+electron energy band in CuO 2 planes of cuprate superconductors), thus it could be )0(
+)(,1+kE)0(
+)(,2−−kE =0 
+for some wave vectors whenpt2≤φ . Therefore, )0(
+)(,+kE =0 for k=1k, 2k, 3k…if )0(
+)(,−kE =0 for 
+k='
+1k, '
+2k, '
+3k…around the node, and this case will lead to the Fermi pocket. However, 
+)0(
+)(,1+kE)0(
+)(,2−−kE >0 for pt2>>φ , and this case will lead to the Fermi arc.  
+ 
+Ⅳ. Explanation of experiment results 
+The Fermi arc is an open-ended gapless sectio n in the large Fermi surface, while the pocket 
+shows the small closed Fermi surface feat ures. We arrive at these results: 
+The Fermi arc can be understood. As discussed above, )0(
+)(,1+kE)0(
+)(,2−−kE >0 for bt2>φ , this 
+means )(
+)(,n
+kE+>0 for any wave vector if )(
+)(,n
+kE−=0 for k='
+1k, '
+2k, '
+3k…around the node, thus only one 
+Fermi segment could be found in  each nodal region. It shoul d be noted that the energy kE=0 means 
+Fkk=. For the p-type cuprate superconductors, we can explain σqbr as the operator which destroys 
+an hole in the CuO 2 planes, thus a hole Fermi arc could be observed in the hole doped cuprates (for  6the appropriate structure of krε). The valence bandwidth of Bi2212 is estimated to be 1.5 eV , the 
+work function of Bi2212 is estimated to be 4.3eV10, and the Fermi arc feature of Bi2212 has been 
+observed [12] (their arguments al so support our results). These cons istencies in the Fermi arc, the 
+bandwidth and the work functi on support our results above. 
+The Fermi pocket in the nodal region could be expl ained, too. As discussed above, it could be 
+)0(
+)(,+kE =0 and)0(
+)(,−kE =0 , this results in )(
+)(,n
+kE+=0 for k=1k, 2k, 3k… when)(
+)(,n
+kE−=0 for k='
+1k, '
+2k, 
+'
+3k…in the nodal region, and this gives two Fermi segments at each nodal region. That the two 
+Fermi segments form a close pocket is because'
+N Nk k= for few wave vectors if the pairs are far from 
+the nodal region. The bandwidth of  Bi2201 is estimated to be 6.5 eV[13], the work function of 
+Bi2212 is estimated to be 4.1eV [14], and the Fermi pocket of Bi2201 is observed in ARPES, these consistencies in ARPES measuremen ts also support our results. 
+The Eq.(10) may have three solutions, and the co existence of Fermi arc and Fermi pocket may be 
+understood. However, this case is so comple x that we will not discuss it in detail.  
+ 
+Ⅴ. Conclusion 
+In the discussion above, our results are based on  a basic suggestion that the pseudogap appears 
+in the antinodal region. This idea is also consis tent with other experime nts as discussed in our 
+previous articles [6]. Although the Hubbard-like Hamiltonian has been examined many times in both pseudogap and superc onductivity, our work is based on a different idea for the moderate 
+on-site interaction. 
+In summary, the Fermi arc, the Fermi pocket, and the pseudogap can be explained consistently 
+with the electronic structures of superconductors. Moreover, this  article supports further that 
+whether the pairs lead to superconductivity or pseudogap is determined by the pairing position. 
+  
+     7 
+REFERENCES: 
+[1] C. Nayak, Phys Rev B 62 (2000)4880-4889.   
+[2] S. Chakravarty, R. B. Laughlin, D. K. Morr, et al, Phys Rev B 63 (2001) 094503-094513. 
+[3] J. Q. Meng, G. D. Liu, W. T. Zhang, et al, Nature 462 (2009)335-338. 
+[4] K. Tanaka, W. S. Lee, D. H. Lu, et al, Science 314 (2006)1910-1913. 
+[5] U. Chatterjee, M. Shi, A. Ka minski, et al, Phy. Rev. Lett. 96 (2006)107006-107010. 
+[6] T. D. Cao, to be published in Chin. Phys. B (or see arXive: 1004.1792). 
+[7] J. R. Schrieffer, X. G. Wen,  S. C. Zhang, Phys. Rev. Lett. 60 (1988) 944-947. 
+[8] J. Schmalian, D. Pines, B. Stojkovic, Phys. Rev. Lett. 80 (1988) 3839-3842.  
+[9] T. D. Cao, Physica B: Condense Matter 405 (2010)1394-1396. 
+[10] D. L. Feng, C. Kim, Phys. Rev. B 65 (2002) 220501-220505®. 
+[11] G. Su, Phys. Rev. Lett. 86 (2001)3690. 
+[12] H. B. Yang, J. D. Rameau, P. D. Johnson, et al, Nature 456 (2008)77-80. 
+[13] C. O. Rodriguez, Ruben We ht, Mariana Weissmann, N. E. Christensen, M. S. Methfessel, 
+Physica C 258 (1996) 360-364. 
+[14] J. D. Lee, A. Fujimori, Phys. Rev.B 66 (2002)144509-144516. 
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+arXiv:1001.0219v1  [hep-th]  1 Jan 2010December 2009 DFUB 12/09
+Version 1
+The gauging of BV algebras
+by
+Roberto Zucchini
+Dipartimento di Fisica, Universit` a degli Studi di Bologna
+V. Irnerio 46, I-40126 Bologna, Italy
+I.N.F.N., sezione di Bologna, Italy
+E–mail: zucchinir@bo.infn.it
+Abstract
+A BV algebra is a formal framework within which the BV quantization al-
+gorithm is implemented. In addition to the gauge symmetry, encoded in the
+BV master equation, the master action often exhibits further glob al symmetries,
+which may be in turn gauged. We show how to carry this out in a BV algeb raic
+set up. Depending on the nature of the global symmetry, the gaug ing involves
+coupling to a pure ghost system with a varying amount of ghostly sup ersymme-
+try. Coupling to an N= 0 ghost system yields an ordinary gauge theory whose
+observables are appropriately classified by the invariant BV cohomo logy. Cou-
+pling to an N= 1 ghost system leads to a topological gauge field theory whose
+observables are classified by the equivariant BV cohomology. Couplin g to higher
+Nghost systems yields topological gauge field theories with higher top ological
+symmetry. Inthelattercase, however, problemsofacompletely n ewkindemerge,
+which call for a revision of the standard BV algebraic framework.
+1Contents
+1Introduction 3
+2BV algebras 9
+3Quantum BV master actions and observables 13
+4N=0 BV gauging and N=0 ghost system 17
+5N=1 BV gauging and N=1 ghost system 25
+6Higher N BV gaugings and ghost systems 38
+7Applications and examples 45
+8Conclusions 59
+ADifferential Lie modules and their invariant cohomology 60
+BLie operations and their equivariant cohomology 67
+CSuperfield formulation of the N=1 ghost system 75
+DThe modular class of a Lie algebroid 76
+21Introduction
+The Batalin–Vilkovisky (BV) approach [1,2] is the most general and p ower-
+ful quantization algorithm presently available. It is suitable for the q uantization
+of ordinary gauge theories, such as Yang–Mills theory, as well as mo re compli-
+cated gauge theories with open and/or reducible gauge symmetries . Its main
+feature consists in the introduction of ghost fields from the outse t automatically
+incorporating in this way BRST symmetry.
+The general structure of the BV formalism is as follows [3,4]. Given a c lassical
+field theory with gauge symmetries, one introduces an antifield with o pposite
+statistics foreach field, including ghost fields, thereforedoubling t hetotalnumber
+of fields. The resulting field/antifield space Fis equipped with an odd Poisson
+bracket{·,·}, called antibracket, and acquires an odd phase space structure, in
+which fields and antifields arecanonically conjugate. At tree level in t he quantum
+theory, the original classical action is extended to a new action S0defined on the
+whole content of Fand exhibiting an off shell odd symmetry corresponding to
+the gauge symmetry of the original field theory. The gauge fixing is c arried out
+by restricting the action S0to a suitable Lagrangian submanifold LinF. Gauge
+independence, that is independence from the choice of L, is ensured if S0satisfies
+the classical BV master equation
+{S0,S0}= 0. (1.1)
+At loop level, quantum corrections modify the action S0and turn it into a quan-
+tum action S/planckover2pi1. Gauge independence is then ensured provided S/planckover2pi1satisfies the
+quantum BV master equation
+/planckover2pi1∆S/planckover2pi1+1
+2{S/planckover2pi1,S/planckover2pi1}= 0, (1.2)
+where ∆ is a suitably regularized odd functional Laplacian in F. Violations of
+this correspond to gauge anomalies.
+3The observables of the field theory constructed in this way are cha racterized
+by having gauge independent correlators. The gauge independenc e of a correlator
+/an}bracketle{tψ/planckover2pi1/an}bracketri}htis ensured if ψ/planckover2pi1satisfies the equation
+δ/planckover2pi1ψ/planckover2pi1:=/planckover2pi1∆ψ/planckover2pi1+{S/planckover2pi1,ψ/planckover2pi1}= 0 (1.3)
+The solutions ψ/planckover2pi1of (1.3) are called quantum BV observables. The quantum BV
+operatorδ/planckover2pi1is nilpotent. Therefore, there is a cohomology associated with it,
+the quantum BV cohomology. Since correlators of BV exact observ ables vanish,
+effectively distinct BV observables are in one–to–onecorresponde nce with the BV
+cohomology classes.
+After this very brief review of BV theory, let us come to the topic of the
+paper. The algebraic structure consisting of the graded algebra o f functionals on
+the field/antifield space F, theantibracket {·,·}andthe oddLaplacian∆ iscalled
+aBValgebra. ItprovidestheformalframeworkwithinwhichtheBVq uantization
+algorithm is implemented. This has motivated a number of mathematica l studies
+of BV algebras [5–7].
+The classical field theory originally considered, even if it is a gauge the ory,
+may still have global symmetries. In certain cases, one may wish to g auge these
+latter. In a BV framework, the gauging of a global symmetry consis ts in the
+coupling of the ungauged “matter” field theory and a suitable pure “ ghost” field
+theorycorrespondingtothesymmetry. (Ordinaryghostandgau gefieldsnormally
+combine in ghost superfields.) Two procedures of concretely workin g this out are
+possible in principle.
+i) One couples the matter and the ghost field theories at the classica l level, by
+adding suitable interaction terms, obtaining a gauged classical field t heory. Then,
+one quantizes this latter using the BV algorithm, by constructing th e appropriate
+BV algebra and quantum BV master action.
+ii) One separately quantizes the matter and the ghost field theories , by con-
+4structing the appropriate BV algebra and quantum BV master actio n of each of
+them. Then, one embeds the matter and ghost BV algebra structu res so obtained
+in a minimal gauged BV algebra structure and constructs a gauged q uantum BV
+master action by adding the matter and ghost actions and suitable in teraction
+terms in a way consistent with the quantum BV master equation.
+We call these two approaches classical gauging andBV algebra gauging , re-
+spectively. Superficially, it may look like that classical gauging is more n atural:
+after all, BV theory was devised precisely to quantize classical gaug e theories. In
+fact, in certain cases, BV gauging is more advantageous.
+In a prototypical example, one efficient way of generating a sigma mo del on
+a non trivial manifold Xis the gauging of a sigma model on a simpler manifold
+Ycarrying the action of a Lie group Gsuch thatX≃Y/G[8,9]. The target
+space of the gauged model turns out to be precisely X. In a BV formulation of
+the ungauged sigma model, Gacts as a group of global symmetries. The gauging
+of these is performed by coupling the ungauged model to a suitable g host sigma
+model, yielding in a natural way a BV formulation of the gauged model [1 0–12].
+The Alexandrov–Kontsevich–Schwartz–Zaboronsky (AKSZ) for malism of ref.
+[13] is a method of constructing solutions of the classical BV master equation
+directly, without starting from a classical action with a set of symme tries, as is
+originally done in the BV framework. When building models with gauged glo bal
+symmetries in a AKSZ framework, BV algebra gauging is definitely more natural
+and transparent than classical gauging.
+In this paper, we study in great detail the BV algebra gauging of a ma tter
+field theory with global symmetries. For a certain global symmetry, the ghost
+field theory to be coupled to the matter theory may have a varying a mount
+of “ghostly supersymmetry”. Coupling, if feasible, to an N= 0 ghost system
+yields an ordinary gauge field theory. Coupling to an N= 1 ghost system leads
+to a topological gauge field theory. Coupling to higher Nghost systems yields
+5topological gauge field theories with higher topological supersymme try. In the
+latter case, however, problems of a completely new kind show up, wh ich may
+require a major revision of the standard BV algebraic framework.
+Though BV algebra gauging is ultimately carried out within the framewo rk
+of BV theory, ordinary BV cohomology is not adequate for the class ification of
+observables of the field theories constructed in this way. If gis the global sym-
+metry Lie algebra, g–invariant BV cohomology in the N= 0 case, g–equivariant
+BV cohomology in the N= 1 case and presumably some higher g–equivariant
+BV cohomologies for larger Nare required.
+We shall carry out our analysis of BV gauging in a finite dimensional set ting
+as in [5–7]. This has its advantages and disadvantages. It allows one t o focus on
+the essential features of gauging, especially those of an algebraic and geometric
+nature, on one hand, but it is of course no substitute for full–fledg ed field theory,
+which is essentially infinite dimensional, on the other. Nevertheless, w ith the due
+caution, one can presumably extend our considerations to realistic BV field the-
+ories. Further, it is known that certain BV field theories have finite d imensional
+reductions which capture some of their relevant structural feat ures [14–16].
+The plan of this paper is as follows. In sect. 2, we review the basics of
+BV algebra theory and set the notation used in the subsequent sec tions. In
+sect. 3, we recall the definition and the main properties of the BV ma ster action
+and observables. In sects. 4, 5, we illustrate how to carry out the N= 0
+andN= 1 gauging of BV algebras and identify the relevant versions of BV
+cohomology. In sect. 6, we tackle the problem of higher NBV algebra gauging
+highlighting the conceptual problems arising in this case. In sect. 7, we illustrate
+a number of examples and applications of the theory developed in the preceding
+sections, showing in particular its relevance for the finite dimensiona l reduction
+of the gauged Poisson sigma model of refs. [10,11]. In sect. 8, we provide some
+concluding remarks. Finally, in the appendices, we conveniently collec t details
+6on a few technical issues involved in our analysis.
+Acknowledgments
+We thank F. Bastianelli and P. Sundell for useful discussions.
+Remarks on conventions and notation
+In this paper, we use the following notations. All spaces and algebra s are over
+the field /CA.
+a) LetEbe /CI–graded vector space. Ekis the subspace of Eof degreek∈ /CI. If
+x∈ Ekfor somek,xis said homogeneous and ∂x=kis the degree of x. Ifp∈ /CI,
+E[p] is the /CI–graded vector space such that E[p]k=Ek−p. Similar conventions
+hold fora /CI–graded algebra A. Inthis case, we denote by Avthe /CI–gradedvector
+space underlying A.
+b) LetE,Fbe /CI–graded vector spaces. E⊗Fis the /CI–graded tensor product
+ofE,F. Its grading is given by ( E⊗F)k=⊕l+m=kEl⊗Fmfork∈ /CI. The tensor
+products A ⊗ Bof two /CI–graded algebras A,Bis defined in the same fashion
+with the graded multiplication
+x⊗uy⊗v= (−1)∂y∂uxy⊗uv, (1.4)
+for homogeneous x,y∈ A,u,v∈ B.
+c) LetE,Fbe /CI–graded vector spaces. Hom( E,F) is the /CI–graded vector
+space of vector space homomorphisms of EintoF. Its grading is defined so that
+X∈Homk(E) if, for all l∈ /CI,XEl⊂ Fk+l. The space End( E) = Hom( E,E),
+the set Iso( E,F) and the group Aut( E) = Iso(E,E) are defined accordingly. Sim-
+ilar notions hold for two /CI–graded algebras A,Bwith the proviso that algebra
+homomorphisms are concerned.
+d) LetEbe a /CI–graded vector space. For homogeneous X,Y∈End(E), the
+graded commutator of X,Yis given by
+[X,Y] =XY−(−1)∂X∂YYX. (1.5)
+7All commutators will always be assumed to be graded, unless otherw ise stated.
+e) LetAbe a /CI–graded algebra. Der( A) is the graded vector subspace of
+End(Av) of graded derivations of E. Ifk∈ /CIandD∈Derk(A), then
+D(xy) =Dxy+(−1)k∂xxDy, (1.6)
+for homogeneous x,y∈ A.
+f) Ifxis a formal graded variable, then ∂Lx= (∂/∂x)Land∂Rx= (∂/∂x)R,
+the subfixes L,Rindicating left, right graded differentiation. If φis a function of
+x, then∂Lxφ= (−1)(∂φ+1)∂x∂Rxφ.
+g) A differential space is a pair ( E,δ), where Eis a /CI–graded vector space,
+δ∈End1(E) andδ2= 0. The associated cohomology H∗(E,δ) is a space. A
+differential algebra is a pair ( A,δ), where Ais a /CI–graded algebra, δ∈Der1(E)
+andδ2= 0. The associated cohomology H∗(A,δ) is then an algebra.
+82BV algebras
+Batalin–Vilkovisky (BV) algebras are the formal structure underly ing the
+BV quantization algorithm in quantum field theory [1,2]. The BV algebr a of a
+field theory consists of a graded algebra of functions of fields and a ntifields, an
+odd Poisson bracket defining the canonical structure of the field t heory at the
+classical level and an odd Laplacian required for implementing the field theory’s
+quantization. BV algebras however can be treated in a completely fo rmal setting
+without invoking any concrete field theoretic realization [3,4].
+ABV algebra is a triple ( A,∆,{·,·}) consisting of the following elements.
+1) A /CI–graded commutative associative unital algebra A.
+2) ABV Laplacian , i. e. an element ∆ ∈End1(A) that is nilpotent,
+∆2= 0. (2.1)
+3) ABV antibracket , i.e. an /CA–bilinear map {·,·}:A×A → A such that
+∆(φψ) = ∆φψ+(−1)∂φφ∆ψ+(−1)∂φ{φ,ψ}, (2.2)
+{φ,ψυ}={φ,ψ}υ+(−1)(∂φ+1)∂ψψ{φ,υ}, (2.3)
+for all homogeneous φ,ψ,υ∈ A.
+We notice that, by (2.2), {·,·}is determined by ∆. So, the notion of BV
+algebra could be defined in terms of A, ∆ only.
+Several properties can be derived from the BV algebra axioms.
+a) One has∂{φ,ψ}=∂φ+∂ψ+1 and
+{φ,ψ}+(−1)(∂φ+1)(∂ψ+1){ψ,φ}= 0, (2.4)
+(−1)(∂φ+1)(∂υ+1){φ,{ψ,υ}}+(−1)(∂ψ+1)(∂φ+1){ψ,{υ,φ}} (2.5)
++(−1)(∂υ+1)(∂ψ+1){υ,{φ,ψ}}= 0,
+9for all homogeneous φ,ψ,υ∈ A. These relations follow from (2.2). Awith the
+multiplicative structure given by the bracket {·,·}is a /CI–graded commutative
+algebraAG, calledGerstenhaber (odd Poisson) algebra . The gradings of A,AG
+are such that AGv=Av[1].
+b) On account of (2.2), ∆ /ne}ationslash∈Der1(A). The Gerstenhaber bracket {·,·}measures
+the failure of ∆ being so. However, one has
+∆{φ,ψ}={∆φ,ψ}+(−1)∂φ+1{φ,∆ψ}, (2.6)
+for homogeneous φ,ψ∈ A. This relation follows from combining (2.1), (2.2).
+Thus, ∆ ∈Der1(AG).
+c) A derivation D∈Derk(A) such that
+[D,∆] = 0 (2.7)
+is called a BV derivation . IfD∈Derk(A), one hasD/ne}ationslash∈Derk(AG) in general.
+However, if Dis a BV derivation, then D∈Derk(AG) as well. This follows
+straightforwardly from combining (2.2), (2.7).
+d) Forα∈ Ak−1, let us set
+adαφ={α,φ}, φ∈ A. (2.8)
+Then, simultaneously ad α∈Derk(A), adα∈Derk(AG). These properties follow
+directly from (2.3), (2.5), respectively. A derivation D∈Derk(A) is called BV
+inner, if it is of the form D= adαfor someα∈ Ak−1such that
+∆α= 0. (2.9)
+Else, it is BV outer . By (2.6), (2.9), a BV inner Dfulfils (2.7) and, so, is BV.
+A BV algebra ( A,∆,{·,·}) is aBV subalgebra of a BV algebra ( A′,∆′,{·,·}′)
+ifAis a subalgebra of A′such that ∆′A ⊂ A,{A,A}′⊂ Aand ∆ = ∆′|A,
+{·,·}={·|A,·|A}′.
+10There is a natural notion of homomorphism of BV algebras. Let ( A,∆,{·,·}),
+(A′,∆′,{·,·}′) be BV algebras. A map T:A → A′is aBV algebra homomor-
+phism, if
+1)T∈Hom0(A,A′).
+2)Tintertwines the BV Laplacians ∆, ∆′,
+T∆ = ∆′T. (2.10)
+3)Tintertwines the brackets {·,·},{·,·}′,
+T{φ,ψ}={Tφ,Tψ}′, (2.11)
+forφ,ψ∈ A.
+As a matter of fact, (2.11) is not an independent condition, as it follo ws from
+(2.2), (2.10). One can also define a BV algebra monomorphism, epimorphism,
+isomorphism, endomorphism andautomorphism in obvious fashion.
+A few properties can bededuced fromthe BValgebra homomorphism axioms.
+a) By (2.11), T∈Hom0(AG,AG′) as well. Indeed, Tis a homomorphism of
+the odd Poisson structures of AG,AG′.
+b) kerTis a subalgebra of Asuch that ∆ker T⊂kerT,{kerT,kerT} ⊂kerT
+and, so, with the BV algebra structure induced by A, a BV subalgebra of A.
+Likewise, im Tis a subalgebra of A′such that ∆′imT⊂imT,{imT,imT}′⊂
+imTand, so, with the BV algebra structure induced by A′, a BV subalgebra of
+A′. This follows immediately from (2.10), (2.11).
+Homomorphisms describe the natural relationships of BV algebras.
+a) Let (A,∆,{·,·})bea BVsubalgebra of theBValgebra ( A′,∆′,{·,·}′). Then,
+the natural injection I:A → A′is a BV algebra monomorphism.
+b) The automorphisms of a BV algebra ( A,∆,{·,·}) represent the symmetries
+of this latter. Let α∈ A−1satisfy (2.9). Define a map Tα:A → Aby
+Tα= exp(adα), (2.12)
+11the right hand side being defined by the usual exponential series. I t is assumed
+that that either the series terminates after a finite number of ter ms by algebraic
+reasons or it converges in some natural topology of End( Av). Then,Tαis a BV
+algebra automorphism. The automorphisms of this type are called BV inner ,
+since adαis a BV inner derivation of A. Correspondingly, all other BV algebra
+automorphisms are called BV outer .
+The set of BV algebras can be organized as a category having BV alge bras
+homomorphisms as morphisms. One can define natural operations in this cate-
+gory. In particular, there is a notion of tensor product of BV algebras that will
+be extensively used in the following. Let ( A′,∆′,{·,·}′), (A′′,∆′′,{·,·}′′) be BV
+algebras. Construct a triple ( A,∆,{·,·}) as follows.
+1)A=A′⊗A′′, a tensor product of graded algebras.
+2) ∆∈End1(A) is defined by the relation
+∆(φ′⊗φ′′) = ∆′φ′⊗φ′′+(−1)∂φ′φ′⊗∆′′φ′′, (2.13)
+for homogeneous φ′∈ A′,φ′′∈ A′′.
+3){·,·}:A×A → A is defined by the relation
+{φ′⊗φ′′,ψ′⊗ψ′′} (2.14)
+= (−1)(∂ψ′+1)∂φ′′{φ′,ψ′}′⊗φ′′ψ′′+(−1)(∂φ′′+1)∂ψ′φ′ψ′⊗{φ′′,ψ′′}′′,
+for homogeneous φ′,ψ′∈ A′,φ′′,ψ′′∈ A′′. Then, ( A,∆,{·,·}) is a BV algebra.
+The verification of the basic relations (2.1)–(2.3) is straightforwar d. (A,∆,{·,·})
+is called the tensor product of the BV algebras ( A′,∆′,{·,·}′), (A′′,∆′′,{·,·}′′).
+The mapsI′:A′→ A,I′′:A′′→ Adefined by I′φ′=φ′⊗1′′,φ′∈ A′,
+I′′φ′′= 1′⊗φ′′,φ′′∈ A′′are BV algebra monomorphisms. Indeed, I′,I′′satisfy
+(2.10), (2.11) on account of (2.13), (2.14). In this way, A′,A′′can be considered
+as BV subalgebras of A.
+Examples of BV algebras will be illustrated in the following sections.
+123Quantum BV master actions and observables
+Let (F,∆,{·,·}) be the BV algebra relevant for a BV quantization prob-
+lem. In general, quantization can be viewed as the addition to a classic al quan-
+tity of a quantum correction expressed perturbatively as a forma l power series
+in the Planck constant /planckover2pi1. For this reason, BV quantization requires working
+with the graded algebra F((/planckover2pi1)) of formal power series φ/planckover2pi1=/summationtext
+k≥0/planckover2pi1kφ(k)with
+φ(k)∈ F, where/planckover2pi1is treated as a degree 0 formal parameter. The BV Laplacian
+∆ and antibracket {·,·}extend by formal linearity to F((/planckover2pi1)). (F((/planckover2pi1)),∆,{·,·})
+is then also a BV algebra. The natural injection of FintoF((/planckover2pi1)), defined by
+φ→/summationtext
+k≥0/planckover2pi1kδk,0φ, is a BV algebra monomorphism and, so, Fcan be viewed
+as a BV subalgebra of F((/planckover2pi1)). The quantum BV master action S/planckover2pi1and observ-
+ablesψ/planckover2pi1are the solutions of eqs. (1.2) and (1.3) in F((/planckover2pi1)). The corresponding
+classical approximations Sandψare obtained by truncating S/planckover2pi1andψ/planckover2pi1to their
+components in F.
+Keeping explicit the /planckover2pi1dependence of the relevant quantities in the following
+analysis would lead to unnecessary notational complication. For this reason, we
+shall treat the problems of quantization and classical approximatio n thereof more
+formally in the framework of a given BV algebra ( A,∆,{·,·}). It is tacitly under-
+stood that, in any physical realization, Amust be correspondingly interpreted as
+eitherF((/planckover2pi1)) orFfor the relevant BV algebra F.
+In the constructions of the following sections, other endomorphis msf∈
+End(A) will be considered beside ∆. It is tacitly understood that, in any phy sical
+realization, the fare independent from /planckover2pi1.
+Let (A,∆,{·,·})be a BV algebra. An element S∈ A0is called a quantum BV
+master action of the BV algebra if Ssatisfies the quantum BV master equation
+∆S+1
+2{S,S}= 0, (3.1)
+13An element ψ∈ Ais aquantum BV observable , if it satisfies the equation
+δψ= 0, (3.2)
+whereδis thequantum BV operator
+δ= ∆+adS. (3.3)
+From the definition, using the master equation (3.1), it can be easily v erified that
+δ∈End1(Av) and thatδis nilpotent,
+[δ,δ] = 2δ2= 0. (3.4)
+Hence, (A,δ) is a differential space. The associated cohomology is the quantum
+BV cohomology spaceHBV∗(A). We note that δis not a derivation, as ∆ is not.
+So, even though Ais an algebra, ( A,δ) is only a differential space. Correspond-
+ingly,HBV∗(A) is only a cohomology space.
+The classical counterpart of the above is as follows. Let ( A,∆,{·,·}) be a BV
+algebra. An element S∈ A0is called a classical BV master action of the BV
+algebra ifSsatisfies the classical BV master equation
+{S,S}= 0. (3.5)
+An element ψ∈ Ais aclassical BV observable , if it satisfies the equation
+δcψ= 0, (3.6)
+whereδcis theclassical BV operator
+δc= adS. (3.7)
+From the definition, using the master equation (3.1), it can be easily v erified that
+δc∈End1(Av) and thatδcis nilpotent,
+[δc,δc] = 2δc2= 0. (3.8)
+14So, (A,δc) is a differential algebra. The associated cohomology is the classical
+BV cohomology algebraHcBV∗(A). Recall that, in the quantum case, ( A,δ) and
+HBV∗(A) are merely spaces.
+Let (A,∆,{·,·}), (A′,∆′,{·,·}′) be BV algebras and let T:A → A′be a BV
+algebra homomorphism (cf. sect. 2). If Sbe a quantum BV master action of the
+BV algebra ( A,∆,{·,·}), then
+S′=TS (3.9)
+is a quantum BV master action of the BV algebra ( A′,∆′,{·,·}′). This follows
+easily from (3.1), (2.10), (2.11). Similarly, if ψis a quantum BV observable of
+the BV algebra ( A,∆,{·,·}) and master action S, then
+ψ′=Tψ (3.10)
+is a quantum BV observable of the BV algebra ( A′,∆′,{·,·}′) and master action
+S′. This follows from (3.2), (3.3), (2.10), (2.11). In fact, one has
+Tδ=δ′T. (3.11)
+Tis therefore a chain map of the differential spaces ( A,δ), (A′,δ′) and, so,
+it induces a homomorphism of the corresponding cohomology spaces HBV∗(A),
+HBV∗(A′). Analogous statements hold also in the classical case.
+Let (A′,∆′,{·,·}′), (A′′,∆′′,{·,·}′′) be BV algebras and let ( A,∆,{·,·}) be
+their tensor product (cf. sect. 2). If S′,S′′are quantum BV master actions of
+(A′,∆′,{·,·}′), (A′′,∆′′,{·,·}′′), respectively, then
+S=S′⊗1′′+1′⊗S′′(3.12)
+is a quantum BV master action of ( A,∆,{·,·}). This property follows straight-
+forwardly from applying (2.13), (2.14). Analogously, if ψ′,ψ′′are quantum BV
+observables of the BV algebras ( A′,∆′,{·,·}′), (A′′,∆′′,{·,·}′′) and actions S′,S′′,
+15respectively, then
+ψ=ψ′⊗1′′+1′⊗ψ′′(3.13)
+is a quantum BV observable of ( A,∆,{·,·}) andS. The verification of this
+property is also straightforward. Again, similar statements hold in t he classical
+case.
+In quantum field theory, the above construction is simply the adjoin ing of
+two field theories with no mutual interaction. From a physical point o f view is
+therefore rather trivial. In interesting models, one requires addin g to the non
+interacting action Sof eq. (3.12) interaction terms in a consistent way, that
+is without spoiling the quantum BV master equation (3.1). BV gauging o f a
+given field theory, discussed in the next sections, is an important ex ample of this
+procedure.
+Examples of BV master actions will be given in the following sections.
+164N=0 BV gauging and N=0 ghost system
+Now, we are ready for starting the study of N= 0 BV gauging and the N= 0
+ghost system. This will set the paradigm for N= 1 and higher Ngaugings.
+N= 0g–actions
+Let (A,∆,{·,·}) be a BV algebra. Let gbe a Lie algebra. An N= 0g–action
+on the BV algebra is a linear map l:g→Der0(A) such that
+[lx,ly] =l[x,y], (4.1a)
+[lx,∆] = 0, (4.1b)
+forx,y∈g. By (4.1b), for x∈g,lx∈Der0(A) is a BV derivation (cf. sect. 2).
+LetSbe a quantum BV master action of the BV algebra ( A,∆,{·,·}).Sis
+saidinvariant under the g–action if
+lxS= 0, (4.2)
+for allx∈g. This condition is compatible with (4.1a), (4.1b) and the quantum
+BV master equation (3.1). When Sis invariant, one has
+[δ,lx] = 0, (4.3)
+whereδis the quantum BV operator (cf. sect. (3), eq. (3.3)). By (3.4), ( 4.1a),
+(4.3), (A,g,l,δ) is an algebraic structure known as a differential g–module [19]
+(see appendix A for a review of differential Lie modules). By (4.3), it is possible
+to define a g–invariant quantum BV cohomology, that is the cohomology of the
+differential space ( Ainv,δ), where Ainv=∩x∈gkerlx⊂ A. The same statements
+hold also for the classical BV operator and its cohomology.
+Theg–action is called BV Hamiltonian , if there is a linear map λ:g→ A−1,
+calledBV moment map , such that
+lx= adλx, (4.4)
+17withx∈g, and that
+{λx,λy}=λ[x,y], (4.5a)
+∆λx= 0, (4.5b)
+withx,y∈g. (4.4) together with (4.5a), (4.5b) are indeed sufficient for (4.1a),
+(4.1b) to hold. By (4.4), (4.5b), lx∈Der0(A) is a BV inner derivation (cf. sect.
+2). Below, we consider only BV Hamiltonian g–actions.
+IfSis a quantum BV master action of the BV algebra ( A,∆,{·,·}) invariant
+under the g–action, then
+{λx,S}= 0, (4.6)
+for allx∈g. By (4.5b), (4.6), λxis a cocycle of both the classical and the
+quantum BV cohomology (cf. sect. 3).
+Gauging of a global N= 0g–symmetry
+Consider a matterBV algebra ( AM,∆M,{·,·}M) carrying a BV Hamiltonian
+N= 0g–actionlMwith BV moment map λMand a matter quantum BV master
+actionSMinvariant under the g–action. By (4.6), we may say that SMenjoys
+aglobalN= 0g–symmetry and thatλMis the corresponding symmetry charge .
+We want to find a meaningful way of gauging these symmetries.
+The gauging proceeds in three steps.
+1.We construct an N= 0ghostBV algebra ( Ag|0,∆g|0,{·,·}g|0) with a BV
+Hamiltonian N= 0g–actionlg|0with BV moment map λg|0and anN= 0 ghost
+quantum BV master action Sg|0invariant under the g–action. The construction
+is canonical, in that it depends solely on g.
+2.We construct an N= 0gauged matter BV algebra ( Ag|0M,∆g|0M,{·,·}g|0M)
+and equip it with an appropriate BV Hamiltonian N= 0g–actionlg|0Mwith BV
+moment map λg|0M.
+3.We construct an N= 0 gauged matter quantum BV master action Sg|0M
+18of the gauged matter BV algebra invariant under the g–action.
+Step 1.Given the Lie algebra g, define
+Ag|0= Fun(g∨[−2]⊕g[1]). (4.7)
+Denote by bi,ci,i= 1,...,dimg, the coordinates of g∨[−2],g[1], respectively,
+corresponding to a chosen basis {ti}ofg. Then, Ag|0can be viewed as the /CI
+graded commutative associative unital algebra of polynomials in the bi,ci. Define
+further the 2nd order differential operator
+∆g|0=∂Lbi∂Lci (4.8)
+and the bilinear brackets
+{φ,ψ}g|0=∂Rbiφ∂Lciψ−∂Rciφ∂Lbiψ, φ,ψ ∈ Ag|0. (4.9)
+Then, it is simple to check that relations (2.1)–(2.3) are verified. It f ollows that
+(Ag|0,∆g|0,{·,·}g|0) is a BV algebra, the N= 0 ghost BV algebra.
+Letfijkbe the structure constants of gwith respect to the basis {ti}. Set
+λg|0i=fj
+kibjck. (4.10)
+Sinceλg|0i∈ Ag|0−1, it defines via (4.4) a linear map lg|0:g→Der0(Ag|0). If the
+Lie algebra gisunimodular , that is
+fj
+ji= 0, (4.11)
+lg|0is a BV Hamiltonian N= 0g–action on the ghost BV algebra having λg|0
+as BV moment map. Indeed, using (4.8), (4.9), (4.10), one finds tha t (4.5a) is
+verified and that ∆ g|0λg|0i=fjji. So, (4.5b) is also verified, if (4.11) holds.
+The action of lg|0onbi,ciis given by
+lg|0ibj=fk
+ijbk, (4.12a)
+lg|0icj=−fj
+ikck, (4.12b)
+19as follows readily from (4.4), (4.9), (4.10).
+TheN= 0 ghost algebra Ag|0contains an element Sg|0∈ Ag|00given by
+Sg|0=−1
+2fi
+jkbicjck. (4.13)
+Sg|0satisfies the classical BV master equation (3.5) and, if gis unimodular, also
+the quantum BV master equation (3.1). Indeed, using (4.8), (4.9), (4.13), one
+finds that (3.5) is verified and that ∆ g|0Sg|0=−fiijcj. So, (3.1) is also verified, if
+(4.11) holds. Sg|0is theN= 0 ghost quantum BV master action. Sg|0is invariant
+under the g–actionlg|0. (4.9), (4.10), (4.13) indeed imply (4.6).
+The action of the quantum BV operator δg|0onbi,ciis given by
+δg|0bi=fk
+jibkcj, (4.14a)
+δg|0ci=−1
+2fi
+jkcjck, (4.14b)
+as follows from the definition (3.3) and from (4.8), (4.9), (4.13). Rela tions (4.14)
+are also the expressions of the action of the classical BV operator δg|0cdefined
+according to (3.7). Recall however that the actions of δg|0andδg|0con higher
+degree polynomials in bi,ciare different because ∆ g|0acts non trivially on them
+in general.
+The fulfillment of the unimodularity condition (4.11) is required by lg|0being
+a Hamiltonian g–action and Sg|0a quantum master action; it is thus crucial in
+the above BV construction. In full–fledged quantum field theory, ( 4.11) would be
+a quantum anomaly cancellation condition.
+Step 2. TheN= 0 gauged matter BV algebra ( Ag|0M,∆g|0M,{·,·}g|0M)
+is the tensor product of the N= 0 ghost BV algebra ( Ag|0,∆g|0,{·,·}g|0) and
+the matter BV algebra ( AM,∆M,{·,·}M) (cf. sect. 2). Via (4.4), the element
+λg|0Mi∈ Ag|0M−1given by the expression
+λg|0Mi=λg|0i⊗1M+1g|0⊗λMi (4.15)
+20defines a linear map lg|0M:g→Der0(Ag|0M). If, again, (4.11) is satisfied, lg|0M
+is a BV Hamiltonian N= 0g–action on the gauged matter BV algebra having
+λg|0Mas BV moment map. One just notices that λg|0Msatisfies (4.5a), (4.5b) if
+simultaneously λg|0,λMdo, by (2.13), (2.14). The g–actionlg|0Mextends trivially
+theg–actionslg|0,lM, in the sense that
+lg|0Mx=lg|0x⊗1M+1g|0⊗lMx, (4.16)
+forx∈g.
+Step 3. TheN= 0 gauged matter algebra Ag|0Mcontains a distinguished
+elementSg|0M∈ Ag|0M0given by
+Sg|0M=Sg|0⊗1M+1g|0⊗SM+ci⊗λMi. (4.17)
+The first two terms correspond to the trivial non interacting ghos t–matter action
+(3.12). The third term is a genuine ghost–matter interaction term. By explicit
+calculation, one can verify that, assuming again that (4.11) holds, Sg|0Msatisfies
+the quantum BV master equation (3.1). One notice, using systemat ically (2.13),
+(2.14), that Sg|0Msatisfies (3.1), if simultaneously Sg|0,SMsatisfy (3.1), SM
+satisfies (4.6) and λMsatisfies (4.5a), (4.5b). Sg|0Mis theN= 0 gauged matter
+quantum BV master action. Proceeding in a similar fashion, we find tha tSg|0M
+satisfies also (4.6), so that Sg|0Mis invariant under the g–actionlg|0M.
+The coupling of ghosts and matter in the quantum master action Sg|0Mmod-
+ifies the action of their respective quantum BV operators: δg|0Mextends non
+triviallyδg|0,δM, that isδg|0M/ne}ationslash=δg|0⊗1M+1g|0⊗δM. One has instead
+δg|0M(bi⊗1M) =δg|0bi⊗1M+1g|0⊗λMi, (4.18a)
+δg|0M(ci⊗1M) =δg|0ci⊗1M, (4.18b)
+δg|0M(1g|0⊗φ) = 1g|0⊗δMφ+ci⊗lMiφ, (4.18c)
+21whereδg|0bi,δg|0ciare given by (4.14a), (4.14b), respectively.
+Analysis of BV cohomology .
+On physical grounds, not all the observables of the original matte r system
+remain such upon gauging the global symmetry. Only those which are invariant
+under the symmetry do. They represent classes of the matter inv ariant quantum
+BV cohomology HBVinv∗(AM). So,it is the invariant BV cohomology that is
+relevant rather than the ordinary one .
+The map Υ 0:AM→ Ag|0Mdefined by
+Υ0φ= 1g|0⊗φ, φ ∈ AM, (4.19)
+yields a natural embedding of AMintoAg|0M. It is immediate to check that Υ 0
+is a monomorphism of BV algebras (cf. sect. 2). Further, we have
+lg|0MxΥ0= Υ0lMx, (4.20)
+forx∈g, and
+δg|0MΥ0= Υ0δM+ci⊗1M·Υ0lMi. (4.21)
+By (4.20), Υ 0mapsthematter invariant subalgebra AMinvinto thegaugedmatter
+invariant subalgebra Ag|0Minv. By (4.21), Υ 0|AMinvis a chain map of the matter
+and gauged matter invariant differential spaces ( AMinv,δM), (Ag|0Minv,δg|0M). So,
+Υ0|AMinvinduces a homomorphism of the matter and gauged matter invariant
+quantum BV cohomology spaces HBVinv∗(AM),HBVinv∗(Ag|0M). The homomor-
+phism is not a monomorphism in general and, so, HBVinv∗(AM) is not naturally
+embedded in HBVinv∗(Ag|0M). This renders the study of the observables in the
+gauged matter theory a bit problematic. The way out is the following.
+From (4.7), we notice that the N= 0 ghost algebra Ag|0contains as a subal-
+gebra the Chevalley–Eilenberg algebra
+CE(g) = Fun(g[1]) (4.22)
+22[17,18].CE(g) is generated by the ci. By (4.12b), CE(g) is stable under
+theg–actionlg|0. By (4.14b), CE(g) is also stable under the BV operator δg|0.
+Thus, (CE(g),g,lg|0,δg|0) is a differential g–module (see appendix A). Inspect-
+ing (4.14b), we realize that the BV cohomology HBV∗(CE(g)) is the Chevalley–
+Eilenberg cohomology HCE∗(g) ofg. Similarly, from (4.12b), (4.14b), we see
+that the invariant BV cohomology HBVinv∗(CE(g)) is the invariant Chevalley–
+Eilenberg cohomology HCEinv∗(g) ofg.HCE∗(g) is not known in general, but it is
+under the weak assumption that gis reductive, i. e. the direct sum of a semisim-
+ple and an Abelian Lie algebra, in which case HCE∗(g)≃CE(g)inv, the invariant
+subalgebra of CE(g). We recall that reductive Lie algebras are unimodular. So,
+this result fits usefully in the theory developed above. HCEinv∗(g)≃CE(g)inval-
+ways. We note that the classical and quantum BV operators are eq ual onCE(g),
+since, by (4.8), ∆ g|0vanishes on CE(g) and, so, the classical and the quantum
+BV cohomologies coincide.
+TheN= 0 gauged matter algebra Ag|0Mcontains as a subalgebra
+A+
+g|0M=CE(g)⊗AM. (4.23)
+By (4.16), (4.12b), A+g|0Mis stable under the g–actionlg|0M. Similarly, by
+(4.18b), (4.18c), (4.14b), A+g|0Mis stable under the BV operator δg|0M. Thus,
+(A+g|0M,g,lg|0M,δg|0M) is a differential g–module. By (4.18b), (4.18c), (4.14b),
+the quantum BV cohomology HBV∗(A+g|0M) is the Chevalley–Eilenberg coho-
+mologyHCE∗(g,AM) ofgwith coefficients in the differential space ( AM,δM).
+Similarly, by (4.16), (4.12b), (4.18b), (4.18c), (4.14b), the invarian t quantum BV
+cohomology HBVinv∗(A+g|0M) is the invariant Chevalley–Eilenberg cohomology
+HCEinv∗(g,AM) ofgwith coefficients in the differential g–module ( AM,g,lM,δM).
+Unlike for the pure ghost system, the quantum and classical BV ope rators are
+generally different in the matter sector and, so, it is necessary to d istinguish the
+classical and the quantum BV cohomologies. Anyway, analogous sta tements hold
+23in the classical case, with the proviso that ( AM,δMc) is a differential algebra in
+this case.
+Let us now come back to the problem of the cohomological analysis of ob-
+servables in the gauged matter theory. We notice that the range o f the BV
+algebra homomorphism Υ 0:AM→ Ag|0Mis contained in A+g|0M. By (4.20),
+(4.21), Υ 0|AMinvis a chain map of the invariant differential spaces ( AMinv,δM),
+(A+g|0Minv,δg|0M). Thus, Υ 0|AMinvinduces a homomorphism of the invariant BV
+cohomology spaces HBVinv∗(AM),HBVinv∗(A+g|0M), which can be shown to be
+a monomorphism. So, HBVinv∗(AM) is naturally embedded in HBVinv∗(A+g|0M).
+In this way, the study of the observables in the gauged matter theory is na turally
+framed in that of the invariant BV cohomology HBVinv∗(A+g|0M)ofA+g|0M. In
+fact, more can be shown [19]. If the Lie algebra gis reductive, then
+HBVinv∗(A+
+g|0M)≃CE(g)inv⊗HBVinv∗(AM). (4.24)
+A self–contained proof of (4.24) is given in appendix A. (4.24) indicate s thatthe
+gauged matter algebra Ag|0Mcontains objects which may reasonably considered to
+be observables, but which do not arise from the original matt er algebra AM. They
+are thepure gauge theoretic observables .
+245N=1 BV gauging and N=1 ghost system
+N= 1 gauging follows in outline the same steps as N= 0 gauging, though
+theN= 1 ghost system has a larger amount of ghost supersymmetry tha n its
+N= 0 counterpart. However, there are some significant differences , the most
+conspicuous of which is that the unimodularity of the symmetry Lie alg ebragis
+no longer required for the consistency of the construction.
+N= 1g–actions
+Let (A,∆,{·,·}) be a BV algebra. Let gbe a Lie algebra. An N= 1g–action
+on the BV algebra is a pair of linear maps i:g→Der−1(A),l:g→Der0(A)
+satisfying the commutation relations
+[ix,iy] = 0, (5.1a)
+[lx,iy] =i[x,y], (5.1b)
+[lx,ly] =l[x,y], (5.1c)
+[ix,∆] = 0, (5.1d)
+[lx,∆] = 0, (5.1e)
+withx,y∈g. Note that an N= 1 action is automatically also an N= 0 action
+(cf. sect. 4). By (5.1d), (5.1e), for x∈g,ix∈Der−1(A),lx∈Der0(A) are both
+BV derivations (cf. sect. 2).
+LetSbe a quantum BV master action of the BV algebra ( A,∆,{·,·}).Sis
+saidinvariant under the g–action if
+lx= adixS, (5.2a)
+lxS= 0, (5.2b)
+forallx∈g. Theseconditionsarecompatiblewith(5.1a)–(5.1e)andthequantu m
+25BV master equation (3.1). Note that this notion of invariance is more restrictive
+than the corresponding one of the N= 0 case: it is not simply a condition on S,
+but also on l. WhenSis invariant, one has
+[δ,ix] =lx, (5.3a)
+[δ,lx] = 0, (5.3b)
+whereδis the quantum BV operator (cf. sect. (3), eq. (3.3)). By (3.4),
+(5.1a)–(5.1c), (5.3a), (5.3b), ( A,g,i,l,δ) is an algebraic structure known as a g–
+operation [19] (see appendix B for a review of Lie operations). By (5 .3a), (5.3b),
+it is possible to define a g–basic quantum BV cohomology, that is the cohomology
+of the differential space ( Abas,δ), where Abas=∩x∈g(kerix∩kerlx)⊂ A. The
+same statements hold also for the classical BV operator and its coh omology.
+The action is called BV Hamiltonian , if there exists a pair of linear maps
+ι:g→ A−2,λ:g→ A−1, called below BV premoment andmoment map ,
+respectively, such that
+ix= adιx, (5.4a)
+lx= adλx, (5.4b)
+withx∈gand that
+{ιx,ιy}= 0, (5.5a)
+{λx,ιy}=ι[x,y], (5.5b)
+{λx,λy}=λ[x,y], (5.5c)
+∆ιx= 0, (5.5d)
+∆λx= 0, (5.5e)
+withx,y∈g. (5.4a), (5.4b) together with (5.5a)–(5.5e) are indeed sufficient fo r
+26(5.1a)–(5.1e) to hold. By (5.4a), (5.4b), (5.5d), (5.5e), ix∈Der−1(A),lx∈
+Der0(A) are both BV inner derivations (cf. sect. 2). Note that the under lying
+N= 0g–action is also Hamiltonian (cf. sect. 4). Below, we consider only BV
+Hamiltonian N= 1 actions.
+Aquantum BVmaster action SoftheBValgebra( A,∆,{·,·})isHamiltonian
+invariant under the g–action, ifSsatisfies
+{ιx,S}=λx, (5.6a)
+{λx,S}= 0, (5.6b)
+for allx∈g. Hamiltonian invariance is stricter than simple invariance. If S
+were simply invariant, (5.6a) would hold only up to a central element of the
+Gerstenhaber algebra AG, as (5.2a), (5.4a), (5.4b) imply only that λx− {ιx,S}
+is central. In the N= 0 case, there is no similar distinction between simple and
+Hamiltonian invariance. (5.5d), (5.6a) combined imply that λxis a coboundary of
+both the classical and the quantum BV cohomology (cf. sect. 3). R ecall that in
+theN= 0 case,λxis a cocycle in general. (5.6b) is not an independent relation;
+it follows from (5.5d), (5.5e), (5.6a) and the master equation (3.1).
+Gauging of a global N= 1g–symmetry
+Consider a matterBV algebra ( AM,∆M,{·,·}M) carrying a BV Hamiltonian
+N= 1g–actioniM,lMwithBV(pre)momentmaps ιM,λMandamatterquantum
+BV master action SMinvariant under the g–action. By (5.6a), (5.6b), we may
+say thatSMenjoys aglobalN= 1g–symmetry and thatλMis the corresponding
+symmetry charge , extending, perhaps with some abuse, the terminology of the
+N= 0 case. However, here, the symmetry is derived in the sense that (5.6b) is
+actually a consequence of the more basic relation (5.6a), unlike for N= 0. We
+want to gauge this symmetry in a way that reflects this richer struc ture.
+As the inN= 0 case, the gauging proceeds in three steps.
+271.We construct an N= 1ghostBV algebra ( Ag|1,∆g|1,{·,·}g|1) with a BV
+Hamiltonian N= 1g–actionig|1,lg|1with BV (pre)moment map ιg|1,λg|1and an
+N= 1 ghost quantum BV master action Sg|1invariant under the g–action. The
+construction is canonical, depending on gonly.
+2.We construct an N= 1gauged matter BV algebra ( Ag|1M,∆g|1M,{·,·}g|1M)
+and equip it with an appropriate BV Hamiltonian N= 1g–actionig|1M,lg|1M
+with BV (pre)moment maps ιg|1M,λg|1M.
+3.We construct an N= 1 gauged matter quantum BV master action Sg|1M
+of the gauged matter BV algebra invariant under the g–action.
+Step 1. For a Lie algebra g, define
+Ag|1= Fun(g∨[−2]⊕g[1]⊕g∨[−3]⊕g[2]) (5.7)
+Denote bybi,ci,Bi,Ci,i= 1,...,dimg, the coordinates of g∨[−2],g[1],g∨[−3],
+g[2], respectively, corresponding to a chosen basis {ti}ofg. Then, Ag|1can be
+viewed as the /CIgraded commutative associative unital algebra of polynomials in
+thebi,ci,Bi,Ci. Define next the 2nd order differential operator
+∆g|1=∂Lbi∂Lci−∂LBi∂LCi (5.8)
+and the bilinear bracket
+{φ,ψ}g|1=∂Rbiφ∂Lciψ−∂Rciφ∂Lbiψ (5.9)
++∂RBiφ∂LCiψ−∂RCiφ∂LBiψ, φ,ψ ∈ Ag|1.
+Then, it is simple to check that relations (2.1)–(2.3) are verified. It f ollows that
+(Ag|1,∆g|1,{·,·}g|1) is a BV algebra, the N= 1 ghost BV algebra.
+Letfijkbe the structure constants of gwith respect to the basis {ti}. Set
+ιg|1i=bi (5.10a)
+λg|1i=fj
+kibjck+fj
+kiBjCk. (5.10b)
+28Sinceιg|1i∈ Ag|1−2,λg|1i∈ Ag|1−1, they define via (5.4a), (5.4b) linear maps ig|1:
+g→Der−1(Ag|1),lg|1:g→Der0(Ag|1). The pair ig|1,lg|1is a BV Hamiltonian
+N= 1g–action on the ghost BV algebra having ιg|1,λg|1as BV (pre)moment
+maps. Indeed, by (5.8), (5.9), (5.10a), (5.10b), relations (5.5a)– (5.5e) areverified.
+The Lie algebra gno longer needs to be unimodular (cf. eq. (4.11)), as in the
+N= 0 case, due to the cancellation of the offending terms fjjiof thebcandBC
+sectors.
+The action of ig|1,lg|1onbi,ci,Bi,Ciis given by
+ig|1ibj= 0, (5.11a)
+ig|1icj=δij, (5.11b)
+ig|1iBj= 0, (5.11c)
+ig|1iCj= 0, (5.11d)
+lg|1ibj=fk
+ijbk, (5.11e)
+lg|1icj=−fj
+ikck, (5.11f)
+lg|1iBj=fk
+ijBk, (5.11g)
+lg|1iCj=−fj
+ikCk, (5.11h)
+as follows readily from (5.4a), (5.4b), (5.9), (5.10a), (5.10b).
+TheN= 1 ghost algebra Ag|1contains an element Sg|1∈ Ag|10given by
+Sg|1=−1
+2fi
+jkbicjck+biCi+fi
+jkBicjCk. (5.12)
+Sg|1satisfies both the classical and the quantum BV master equation (3 .5). This
+follows from the definition (5.12) using (5.8), (5.9). Again, gneeds not to be
+unimodular, as in the N= 0 case, due to the cancellation of the anomalous terms
+29fiijcjoriginating in the bcandBCsectors.Sg|1is theN= 1 ghost quantum
+BV master action. Sg|1is invariant under the g–actionig|1,lg|1. (5.9), (5.10a),
+(5.10b), (5.12) indeed imply (5.6a), (5.6b).
+The action of the quantum BV operator δg|1onbi,ci,Bi,Cireads
+δg|1bi=fk
+jibkcj+fk
+jiBkCj, (5.13a)
+δg|1ci=Ci−1
+2fi
+jkcjck. (5.13b)
+δg|1Bi=−bi−fk
+jiBkcj, (5.13c)
+δg|1Ci=−fi
+jkcjCk, (5.13d)
+as follows by direct application of the definition (3.3). Relations (5.13) are also
+the expressions of the action of the classical BV operator δg|1cdefined according
+to (3.7). Analogously to the N= 0 case, the action of δg|1andδg|1con higher
+polynomials in bi,Bi,ci,Ciis different because ∆ g|1acts on them non trivially
+in general.
+TheN= 1 ghost system has an elegant superfield formulation that is illus-
+trated in appendix C.
+Step 2.TheN= 1 gauged matter BV algebra ( Ag|1M,∆g|1M,{·,·}g|1M) is
+the tensor product of the N= 1 ghost BV algebra ( Ag|1,∆g|1,{·,·}g|1) and the
+matter BV algebra ( AM,∆M,{·,·}M) (cf. sect. 2), analogously to the N= 0
+case. The elements ιg|1Mi∈ Ag|1M−2,λg|1Mi∈ Ag|1M−1given by
+ιg|1Mi=ιg|1i⊗1M, (5.14a)
+λg|1Mi=λg|1i⊗1M+1g|1⊗λMi (5.14b)
+define via (5.4a), (5.4b) linear maps ig|1M:g→Der−1(Ag|1M),lg|1M:g→
+Der0(Ag|1M). The pair ig|1M,lg|1Mis a BV Hamiltonian N= 1g–action on the
+gauged matter BV algebra having ιg|1M,λg|1Mas BV (pre)moment maps. One
+30just notices that ιg|1M,λg|1Msatisfy (5.5a)–(5.5e) if ιg|1,λg|1,λMdo, by (2.13),
+(2.14). The g–actionig|1Mlg|1Mextends the g–actionsig|1,lg|1,iM,lMas
+ig|1Mi=ig|1i⊗1M, (5.15a)
+lg|1Mi=lg|1i⊗1M+1g|1⊗lMi (5.15b)
+forx∈g. Unlike the N= 0 case, this extension is non trivial: in the right hand
+side of (5.15a), a term 1 g|1⊗iMiis absent.
+Step 3. TheN= 1 gauged matter algebra Ag|1Mcontains a distinguished
+elementSg|1M∈ Ag|1M0given by
+Sg|1M=Sg|1⊗1M+1g|1⊗SM+ci⊗λMi−Ci⊗ιMi.(5.16)
+As in theN= 0 case, the first two terms correspond to the trivial non inter-
+acting ghost–matter action (3.12) while the third and fourth terms are genuine
+ghost–matter interaction terms. By explicit calculation, one can ve rify thatSg|1M
+satisfies the quantum BV master equation (3.1), noticing that, by s ystematic use
+of (2.13), (2.14), that Sg|1Msatisfies (3.1), if simultaneously Sg|1,SMsatisfy (3.1),
+SMsatisfies (5.6a), (5.6b), and ιM,λMsatisfy (5.5a)–(5.5e). Sg|1Mis theN= 1
+gauged matter quantum BV master action. As Sg|1Msatisfies also (5.6a), (5.6b),
+Sg|1Mis invariant under the g–actionlg|1M.
+As in theN= 0 case, the coupling of ghosts and matter in the quantum
+masteraction Sg|1Mmodifiestheactionoftheirrespective quantumBVoperators:
+δg|1M/ne}ationslash=δg|1⊗1M+ 1g|1⊗δMand, so,δg|1Mextends non trivially δg|1,δM. One
+has instead
+δg|1M(bi⊗1M) =δg|1bi⊗1M+1g|1⊗λMi, (5.17a)
+δg|1M(ci⊗1M) =δg|1ci⊗1M, (5.17b)
+δg|1M(Bi⊗1M) =δg|1Bi⊗1M+1g|1⊗ιMi, (5.17c)
+31δg|1M(Ci⊗1M) =δg|1Ci⊗1M, (5.17d)
+δg|1M(1g|1⊗φ) = 1g|1⊗δMφ+ci⊗lMiφ−Ci⊗iMiφ. (5.17e)
+whereδg|0bi,δg|0ci,δg|0Bi,δg|0Ciare given by the expressions (5.13a)–(5.13d),
+respectively.
+Analysis of BV cohomology
+Before beginning the study of BV cohomology, the following remarks are in
+order. The construction illustrated above is modeled on topological gauge field
+theory. In the Mathai–Quillen formulation of topological field theory [20], the
+computationofatopologicalcorrelatorisreducedtothatofanint egraloftheform
+/integraltext
+Z(s)ω,whereωisaclosedformofthefieldspace MandZ(s)isthesubmanifoldof
+Mofsolutionsofacertainfieldequation s= 0(aphenomenoncalledlocalization).
+Now, it turns out that/integraltext
+Z(s)ω=/integraltext
+Mω∧e(E), wheree(E) is a closed form of M
+representing the Euler class of an oriented Riemannian vector bund leEoverM,
+of whichsis a section. It is known that e(E) =s∗t(E), wheret(E) is a closed
+formofErepresenting theThomclass of E(adistinguished element ofthevertical
+rapid decrease cohomology of E).t(E) in turn yields the closed form π∗t(E) of
+the natural principal G–bundleπ:P×V→E, wherePandVare the oriented
+orthogonal frame principal bundle and the typical fiber of E, respectively, and
+G≃SO(V) is the structure group of P.Pis endowed with a canonical g–
+operation, where gis the Lie algebra of G[19]. The operation allows one to
+define basic forms of P×V.π∗t(E) is closed and basic. In this way, the problem
+of the computation of the original topological correlator can be fo rmulated in
+terms of the basic cohomology and the closely related equivariant co homology of
+the principal bundle P×V. See [21,22] for up to date reviews of this subject
+matter.
+As observed at the beginning of this section, N= 1g–actions on BV algebras
+32are instances of g–operations. So, it is reasonable to suppose that, in a BV
+algebraic formulation, a topological field theory should be realized as matter BV
+algebra with an N= 1g–action and an invariant BV master action. For the
+reasons explained above, among all the observables of the matter system, those
+which are basic under the g–action have a central role. In certain topological
+field theories, the relevant basic observables turn out to be non loc al. The way
+to restore locality is precisely the gauging of the N= 1g–symmetry.
+Let us thus assume that the observables of the original matter th eory which
+arerelevant upongauging arethebasic ones. They represent clas ses of thematter
+basic quantum BV cohomology HBVbas∗(AM). So,it is the basic BV cohomology
+that is relevant rather than the ordinary one .
+We now proceed similarly as we did in the N= 0 case (cf. sect. 4). The map
+Υ1:AM→ Ag|1Mdefined by
+Υ1φ= 1g|1⊗φ, φ ∈ AM, (5.18)
+yields a natural embedding of AMintoAg|1M, which is a monomorphism of BV
+algebras (cf. sect. 2). Further, we have
+ig|1MxΥ1= 0, (5.19a)
+lg|1MxΥ1= Υ1lMx, (5.19b)
+forx∈g, and
+δg|1MΥ1= Υ1δM+ci⊗1M·Υ1lMi−Ci⊗1M·Υ1iMi.(5.20)
+By (5.19a), (5.19b), Υ 1maps the matter basic subalgebra AMbasinto the gauged
+matter basic subalgebra Ag|1Mbas. By (5.20), Υ 1|AMbasis a chain map of the
+matter and gauged matter basic differential spaces ( AMbas,δM), (Ag|1Mbas,δg|1M).
+So, Υ 1|AMbasinduces a homomorphism of the matter and gauged matter basic
+33quantum BV cohomologies HBVbas∗(AM),HBVbas∗(Ag|1M). As in the N= 0 case,
+the homomorphism is not a monomorphism in general and, so, HBVbas∗(AM) is
+not naturally embedded in HBVbas∗(Ag|1M). As in the N= 0 case again, this
+renders the study of the observables in the gauged matter theor y problematic.
+The way out is similar in spirit.
+From (5.7), we observe that the N= 1 ghost algebra Ag|1contains as a
+subalgebra the Weil algebra
+W(g) = Fun(g[1]⊕g[2]) (5.21)
+[23–25].W(g) is generated by the ci,Ci. By (5.11b), (5.11d), (5.11f), (5.11h),
+W(g) is stable under the g–actionig|1,lg|1. By (5.13b), (5.13d), W(g) is also
+stable under the BV operator δg|1. Thus, (W(g),g,ig|1,lg|1,δg|1) is ag–operation
+(see appendix B). Upon inspecting (5.13b), (5.13d), we recognize t hat the BV
+cohomology HBV∗(W(g)) is the Weil algebra cohomology HW∗(g) ofg. Similarly,
+from (5.11b), (5.11d), (5.11f), (5.11h), (5.13b), (5.13d), we see that the basic BV
+cohomology HBVbas∗(CE(g)) coincides with the basic Weil algebra cohomology
+HWbas∗(g) ofg. It is known that HW∗(g)≃ /CAδ∗,0, i.e. the Weil cohomology is
+trivial.H∗Wbas(g) is instead non trivial and concentrated in even degree, namely
+HWbas∗(g)≃W(g)bas= Fun(g[2])inv, the basic subalgebra of W(g). As in the
+N= 0 case, there is no distinction between classical and quantum BV op erators,
+since, by (5.8), ∆ g|1vanishes on W(g) and, so, there is also no distinction between
+classical and quantum BV cohomologies.
+TheN= 1 gauged matter algebra Ag|1Mcontains as a subalgebra
+A+
+g|1M=W(g)⊗AM. (5.22)
+By (5.15a), (5.15b), (5.11b), (5.11d), (5.11f), (5.11h), A+g|1Mis stable under the
+g–actionig|1M,lg|1M. Similarly, by (5.17b), (5.17d), (5.17e), A+g|1Mis stable un-
+der the BV operator δg|1M. Thus, ( A+g|1M,g,ig|1M,lg|1M,δg|1M) is ag–operation.
+34From (5.17b), (5.17d), (5.17e), we realize that the quantum BV coh omology
+HBV∗(A+g|1M) is the Weil cohomology HW∗(g,AM) ofgwith coefficients in the
+differential space ( AM,δM). Similarly, from (5.15a), (5.15b), (5.11b), (5.11d),
+(5.11f), (5.11h), (5.17b), (5.17d), (5.17e), we find that the quan tum basic BV co-
+homologyHBVbas∗(A+g|1M)isthebasicWeil cohomology HWbas∗(g,AM)ofgwith
+coefficients in the g–operation ( AM,g,iM,lM,δM). Unlike for the pure ghost sys-
+tem, the quantum and classical BV operators aregenerally differen t in the matter
+sector and, so, it is necessary to distinguish the classical and the q uantum BV
+cohomologies, as in the N= 0 case. Analogous statements hold in the classical
+case, (AM,δMc) being a differential algebra in this case.
+Now, we can solve the problem of the cohomological analysis of obser vables
+in the gauged matter theory. We notice that the range of the BV alg ebra homo-
+morphism Υ 1:AM→ Ag|1Mis contained in A+g|1M. From (5.19a), (5.19b),
+(5.20), Υ 1|AMbasis a chain map of the basic differential spaces ( AMbas,δM),
+(A+g|1Mbas,δg|1M). Thus, Υ 1|AMbasinduces a homomorphism of the basic BV
+cohomology spaces HBVbas∗(AM),HBVbas∗(A+g|1M). It can be shown that this is
+in fact as an isomorphism [19,22],
+HBVbas∗(A+
+g|1M)≃HBVbas∗(AM) (5.23)
+under the mild assumption that the g–operation AMadmits a connection . A
+self–contained proof of (5.23) is given in appendix B. (5.23) is to be co mpared
+with itsN= 0 counterpart, eq. (4.24), from which it differs qualitatively in two
+ways. First, (5.23) holds with no restriction on the Lie algebra g, whilst (4.24)
+holds provided gis reductive. Second, by (5.23), HBVbas∗(AM) is actually nat-
+urally isomorphic to HBVbas∗(A+g|1M), whilst, by (4.24), HBVinv∗(AM) is simply
+naturally embedded in HBVinv∗(A+g|0M). The reason for this can be ultimately
+traced back to the triviality of the Weil cohomology. In this way, the study of the
+observables in the gauged matter theory is fully reduced to t hat of the basic BV
+35cohomology HBVbas∗(A+g|1M)ofA+g|1M.
+HBVbas∗(A+g|1M)≃HWbas∗(g,AM) is known as the g–equivariant cohomology
+Hequiv∗(AM)ofAM[19]. Equivariantcohomologyisdefinedusuallyfordifferential
+algebras AM. In our case, AMis a differential algebra in the classical but not in
+the quantum case (cf. sect. 3). However, Hequiv∗(AM) can still be defined.
+As is well–known, there are three models of equivariant cohomology: the
+original models of Weil and Cartan of refs. [23–25] and the so-called BRST model
+of ref. [26]. The three models are in fact equivalent. The most direct and efficient
+way to show this was found in ref. [27], where the author proves tha t the Cartan
+and Weil model can be obtained from the BRST model via reduction of and
+application of a suitable inner automorphism to the algebra A+g|1M, respectively.
+The formal structure of the underlying algebra A+g|1M,g–actionig|1M,lg|1Mand
+differential δg|1Mreproduces very closely that of the corresponding objects of th e
+BRST model of equivariant cohomology. Thus, mimicking the treatme nt of [27],
+one may try to generate the counterparts of the Cartan and Weil model in the
+present BV algebraic framework by reduction and action of a suitable BV inner
+automorphism (cf. sect. 2), respectively.
+The Cartan model relies on the algebra C+g|1M= Fun(g[2])⊗ AMinstead
+ofA+g|1M.C+g|1Mis the subalgebra of the elements of A+g|1Mcontaining no
+occurrences of the ci. The Cartan model can be obtained from BRST model by
+observing that A+g|1Mbas=C+g|1Minv. So, the Cartan model is in a sense an
+“effective” reduction of the BRST model in which the cihave been eliminated
+from the outset.
+The Weil model can be derived from the BRST model as follows. Define
+αg|1M=ci⊗ιMi. (5.24)
+Clearly,αg|1M∈ Ag|1M−1. Further, by (5.5d), (5.8), we have
+∆g|1Mαg|1M= 0. (5.25)
+36Therefore, as shown in sect. 2,
+Tg|1M= exp(−adαg|1M) (5.26)
+is a BV algebra inner automorphism. It is indeed the BV algebra analog o f the
+automorphism defined and exploited in ref. [27] to show the equivale nce of the
+BRST and Weil models. By an elementary calculation, we find
+ι′
+g|1Mi:=Tg|1Mιg|1Mi=ιg|1i⊗1M+1g|1⊗ιMi, (5.27a)
+λ′
+g|1Mi:=Tg|1Mλg|1Mi=λg|1i⊗1M+1g|1⊗λMi, (5.27b)
+whereιg|1Mi,λg|1Miare given by (5.10a), (5.10b). In this way, the g–actioni′g|1M,
+l′g|1Mresulting from the application of Tg|1Mis a trivial extension of the actions
+ig|1,lg|1andiM,lM. Another simple calculation shows that
+S′
+g|1M=Tg|1MSg|1M=Sg|1⊗1M+1g|1⊗SM, (5.28)
+whereSg|1Mis given by (5.16). So, the BV master action S′g|1Mresulting from
+the application of Tg|1Mis the trivial non interacting one. Correspondingly, the
+quantum BV operator δ′g|1Myielded by Tg|1Mis a trivial extension of the BV
+operatorsδg|1,δM. The formal structure of the underlying algebra A+g|1M,g–
+actioni′g|1M,l′g|1Mand differential δ′g|1Mobtained in this way reproduces closely
+that of the corresponding objects of the Weil model of equivarian t cohomology.
+In this way, the interaction of the matter and gauge sector can be absorbed by
+means of an inner BV automorphism. This would seem to trivialize the ga uged
+matter model. However, recall that in the quantum field theoretic r ealizations of
+the construction, the automorphism may introduce non locality.
+376Higher N BV gaugings and ghost systems
+In sect. 5, we found out that N= 1 BV gauging is at the basis of topological
+gaugefieldtheory. Thetopologicalmodelsconcerned herehave N= 1topological
+supersymmetry. There are also topological models having N= 2 topological
+supersymmetry, which were first systematically studied by Dijkgra af and Moore
+inref.[28], wheretheywerecalledbalanced. Theproblemthenarises ofdescribing
+their gauging in a BV framework as done in the N= 1 case. However, when
+attempting this, problems of a new kind show up, as we now explain.
+The basic elements of N= 1 BV gauging treated in sect. 5 are a BV algebra
+(A,∆,{·,·}) equipped a quantum BV operator δand aN= 1g–actioni,l
+organizedinanalgebraicstructure, calleda g–operationintheterminologyof[19].
+This structure underlies the Mathai–Quillen formulation of N= 1 topological
+field theory [20]. From now on, we shall refer to it as an N= 1g–operation.
+In ref. [28], the authors showed that the Mathai–Quillen formulation can be
+generalized to N= 2 topological field theory. Their construction hinges on
+an algebraic framework generalizing that of N= 1g–operation and thus called
+N= 2g–operation henceforth.
+If we tried to implement N= 2 BV gauging following [28] and mimicking the
+N= 1 case, the basic elements would be a BV algebra ( A,∆,{·,·}) equipped
+with a doublet of quantum BV operator δA,A= 1,2 satisfying
+[δA,δB] = 0. (6.1)
+In addition, we would have an N= 2g–action, which is a set of linear maps
+j:g→Der−2(A),iA:g→Der−1(A),A= 1,2,l:g→Der0(A) satisfying the
+following commutation relations
+[jx,jy] = 0, (6.2a)
+38[jx,iAy] = 0, (6.2b)
+[iAx,iBy] =ǫABj[x,y], (6.2c)
+[lx,jy] =j[x,y], (6.2d)
+[lx,iAy] =iA[x,y], (6.2e)
+[lx,ly] =l[x,y], (6.2f)
+withx,y∈g, whereǫABis the two dimensional antisymmetric symbol. Finally,
+the derivations jx,iAx,lxwould be related as
+[δA,jx] =iAx, (6.3a)
+[δA,iBx] =−ǫABlx, (6.3b)
+[δA,lx] = 0, (6.3c)
+withx∈g. Relations(6.2a)–(6.2f)and(6.3a)–(6.3c)definean N= 2g–operation.
+(Compare with relations (5.1a)–(5.1c) and (5.3a), (5.3b) defining an N= 1g–
+operation). N= 2g–operations were systematically studied in ref. [29]. One
+of their main properties is the existence of an internal sl(2, /CA)⊕ /CAalgebra of
+automorphisms, an ” R–symmetry” in physical parlance.
+In a BV framework, the existence of a doublet of quantum BV opera torsδA
+is intriguing. It apparently implies the corresponding existence of a d oublet of
+quantum BV master actions SA. However, a relation of the form
+δA= ∆+adSA (6.4)
+is incompatible with the internal sl(2, /CA)–symmetry. This indicates that the or-
+dinary approach based on BV algebras is inadequate for the constr uction we are
+attempting. If we wish to remedy this changing as little as possible our BV
+39framework, a doublet of degree 1 BV Laplacians ∆ Arather a single one ∆ is
+required in addition to the bracket {·,·}. So, instead of a customary BV algebra
+(A,∆,{·,·}), we should have some structure of the form ( A,∆A,{·,·}).
+∆Aand{·,·}should fulfill certain conditions generalizing those defining a BV
+algebra in natural fashion. Presumably, they are the following. Firs t, the bracket
+{·,·}satisfy the graded Leibniz relation (2.3) and the Gerstenhaber rela tions
+(2.4), (2.5). Second, the BV Laplacians ∆ Aare nilpotent and anticommute
+[∆A,∆B] = 0 (6.5)
+(compare with (2.1)). Third, the ∆ Aare degree 1 derivations of AG,
+∆A{φ,ψ}={∆Aφ,ψ}+(−1)∂φ+1{φ,∆Aψ}, (6.6)
+withφ,ψ∈ A(compare with (2.6)). There is no extension of relation (2.2).
+In the resulting extended BV algebraic framework, (6.4) is improved as
+δA= ∆A+adSA. (6.7)
+In order (6.1) to be satisfied, it is sufficient that
+∆ASB+∆BSA+{SA,SB}= 0. (6.8)
+This is theresulting generalizationof themaster equation (3.1). Its fieldtheoretic
+origin, if any, is not clear at all.
+Let us assume that the N= 2g–action is BV Hamiltonian in the following
+sense. There exist linear BV moment maps η:g→ A−3,ιA:g→ A−2,A= 1,2,
+λ:g→ A−1such that
+jx= adηx, (6.9a)
+iAx= adιAx, (6.9b)
+lx= adλx, (6.9c)
+40and satisfying the relations
+{ηx,ηy}= 0, (6.10a)
+{ηx,ιAy}= 0, (6.10b)
+{ιAx,ιBy}=ǫABη[x,y], (6.10c)
+{λx,ηy}=η[x,y], (6.10d)
+{λx,ιAy}=ιA[x,y], (6.10e)
+{λx,λy}=λ[x,y], (6.10f)
+∆Aηx= 0, (6.10g)
+∆AιBx= 0, (6.10h)
+∆Aλx= 0, (6.10i)
+withx,y∈g. Assuming that (6.9a)–(6.9c) hold, (6.10a)–(6.10f) ensure that
+(6.2a)–(6.2f) are fulfilled. On account of (6.7), (6.10g)–(6.10i) ens ure that (6.3a)–
+(6.3c) are also fulfilled if the master action doublet SAsatisfies
+{SA,ηx}=ιAx, (6.11a)
+{SA,ιBx}=−ǫABλx, (6.11b)
+{SA,λx}= 0, (6.11c)
+withx∈g. We may take this as the definition of invariance of the action doublet
+SAunder theN= 2 BV Hamiltonian g–action.
+We shall not attempt to fully generalize the constructions of sects . 4, 5 to
+obtainN= 2 BV gauging. The construction is algebraically complicated, on
+one hand, and its eventual relevance in field theoretic applications s till doubtful,
+41on the other. Moreover, the definitions of the relevant structur es do not seem
+to be unique. We shall limit ourselves to a broad outline of the N= 2 gauging
+procedure and the structure of the N= 2 ghost system and its coupling to a
+matter system.
+Consider a matterextended BV algebra ( AM,∆MA,{·,·}M) carrying a BV
+Hamiltonian N= 2g–actionjM,iMA,lMwith BV moment maps ηM,ιMA,
+λMand a matter quantum BV master action doublet SMAinvariant under the
+g–action. We want to gauge the g–symmetry.
+The gauging proceeds in three steps, as usual.
+1.We construct an N= 2ghostextended BV algebra ( Ag|2,∆g|2A,{·,·}g|2)
+with a BVHamiltonian N= 2g–actionjg|2,ig|2A,lg|2with BVmoment maps ηg|2,
+ιg|2A,λg|2and anN= 2 ghost quantum BV master action doublet Sg|2Ainvariant
+under the g–action. The construction is canonical, depending on gonly.
+2.Weconstructan N= 2gauged matter extendedBValgebra( Ag|2M,∆g|2MA,
+{·,·}g|2M)andequipitwithanappropriateBVHamiltonian N= 2g–actionjg|2M,
+ig|2MA,lg|2Mwith BV moment maps ηg|2M,ιg|2MA,λg|2M.
+3.We construct an N= 2 gauged matter action doublet Sg|2MAof the gauged
+matter BV algebra invariant under the g–action.
+TheN= 2 ghost system was introduced originally in ref. [28] and studied
+in detail in ref. [29]. It consists of a degree 1 g–valued doublet cAi, a degree
+2g–valued singlet ci, a degree 2 g–valued triplet CABisymmetric in A,Band
+a degree 3 g–valued doublet CAi. In the extended BV framework, these are
+conjugated to a degree −2g∨–valued doublet bAi, a degree −3g∨–valued singlet
+bi, a degree −3g∨–valued triplet BABisymmetric in A,Band a degree −4g∨–
+valued doublet BAi, respectively. They span a graded algebra Ag|2. Apparently,
+the only consistent choice of the BV Laplacians ∆ g|2AinAg|2is the trivial one
+∆g|2A= 0. (6.12)
+42Ag|2has instead a natural non trivial bracket
+{φ,ψ}g|2=∂RbAiφ∂LcA
+iψ−∂RcA
+iφ∂LbAiψ (6.13)
++∂RBABiφ∂LCAB
+iψ−∂RCAB
+iφ∂LBABiψ
++∂Rbiφ∂Lciψ−∂Rciφ∂Lbiψ
++∂RBAiφ∂LCA
+iψ−∂RCA
+iφ∂LBAiψ, φ,ψ ∈ Ag|2.
+The construction of the moment maps ηg|2,ιg|2A,λg|2of the appropriate Hamil-
+tonianN= 2g–action on Ag|2and of the correct N= 2 ghost master action
+doubletSg|2Asatisfying the invariance conditions (6.11) and the master equation
+(6.8) is an open problem. A superfield formulation of the N= 2 ghost system is
+possible in principle, as in the N= 1 case.
+TheN= 2 gauged matter extended BV algebra ( Ag|2M,∆g|2MA,{·,·}g|2M)
+is the tensor product of the N= 2 ghost BV algebra ( Ag|2,∆g|2A,{·,·}g|2) and
+the matter BV algebra ( AM,∆MA,{·,·}M). The tensor product of extended BV
+algebras is defined by a straightforward generalization of the defin ition of tensor
+product of ordinary BV algebras given in sect. 2. We expect that, in a BRST
+model, the appropriate Hamiltonian N= 2g–action of the gauged matter BV
+algebra to be some non trivial extension of those of its ghost and ma tter factors,
+as in theN= 1 case. The precise definition of the corresponding moment maps
+ηg|2M,ιg|2MA,λg|2Mis a further open problem.
+If we tried to generalize (5.16) in the extended BV framework illustra ted
+above, the gauged matter action doublet would be something like
+Sg|2MA=Sg|2A⊗1M+1g|2⊗SMA+cAi⊗λMi (6.14)
+−ǫBC(CABi−ǫABci)⊗ιMCi+CAi⊗ηMi,
+the last three terms being interaction terms. The fulfilment of the in variance con-
+43ditions (6.11) and the master equation (6.8) cannot be ascertained as long as the
+explicit form of the ghost BV action Sg|2Ais not known.
+It is reasonable to expect that the appropriate classification of th e observables
+of a theory described by an extended BV algebra ( A,∆A,{·,·}) and a quantum
+BV master action doublet SAis encoded in the cohomology of the bidifferential
+space (A,δA). However, this cohomology cannot have the customary form of a/CI–bigraded cohomology. Ahas no /CI–bigrading such that there are two inde-
+pendent linear combinations of the δAeach of which raises one of the underlying/CI–gradings by one unit and leaves invariant the other one. Rather, t he observ-
+ables are classified by the cohomology of any non vanishing linear comb ination of
+theδA, the internal sl(2, /CA)⊕ /CAguaranteeing the independence of the cohomology
+from the choice of the combination.
+If the extended BV algebra ( A,∆A,{·,·}) is equipped with N= 2g–actionj,
+iA,lunder which the BV action doublet SAis invariant in the sense that (6.3a)–
+(6.3c) are satisfied, one may define an N= 2g–basic quantum BV cohomology.
+This is the cohomology, as defined in the previous paragraph, of the bidifferential
+space (Abas,δA), where Abas=∩x∈g(kerjx∩ ∩AkeriAx∩kerlx)⊂ A. When
+carrying out the gauging of a matter extended BV algebra with an inv ariant
+matter action doublet as outlined above, a corresponding notion of N= 2g–
+equivariant cohomology should appear.
+The above analysis presumably generalizes to higher values of N. To the
+best of our knowledge, virtually nothing is known about N≥3g–operations
+and ghost systems. However, we expect the inadequacy of the cu stomary BV
+algebraic framework to emerge again.
+447Applications and examples
+Inthissection, weshall present afewapplicationsoftheformalismd eveloped in
+the preceding sections. Our examples are drown from Lie algebroid a nd Poisson
+geometry, which cover a broad spectrum of cases. We concentra te on the well
+understood N= 0 andN= 1 gauging.
+The BV algebra of a Lie algebroid and its gauging
+ALie algebroid is a vector bundle Eover a manifold Mequipped with a
+bundle map ρE:E→TM, called the anchor, and an /CA–linear bracket [ ·,·]E:
+Γ(E)×Γ(E)→Γ(E) with the following properties.
+1) [·,·] is a Lie bracket so that Γ( E) is a Lie algebra:
+[X,Y]E+[Y,X]E= 0, (7.1)
+[X,[Y,Z]E]E+[Y,[Z,X]E]E+[Z,[X,Y]E]E= 0, (7.2)
+forX,Y,Z∈Γ(E).
+2)ρdefines a Lie algebra homomorphism of Γ( E) into Γ(TM):
+ρ([X,Y]E) = [ρ(X),ρ(Y)]TM, (7.3)
+forX,Y∈Γ(E), where [ ·,·]TMis the usual Lie bracket of vector fields of M.
+3) The generalized Leibniz rule holds:
+[X,fY]E=f[X,Y]E+(ρ(X)f)Y, (7.4)
+forf∈C∞(M) andX,Y∈Γ(E).
+The prototype Lie algebroid over Mis the tangent bundle TM: the anchor is
+the identity id TMand the bracket is the usual Lie bracket [ ·,·]TM. Lie algebroids
+generalize Lie algebras: a Lie algebra can be viewed as a Lie algebroid ov er the
+singleton manifold M= pt.
+45Let{er}be a local frame of E. Then, one has
+ρE(er) =ρra∂a, (7.5)
+[er,es]E=ct
+rset. (7.6)
+Here,a,b,c,... are base coordinate indices while r,s,t,... are fiber coordinate
+indices.ρra,ctrsare called the anchor and structure functions of E, respectively.
+From (7.1)–(7.4), they satisfy
+cr
+st+cr
+ts= 0, (7.7a)
+cr
+svcv
+tu+cr
+tvcv
+us+cr
+uvcv
+st+ρsa∂acr
+tu+ρta∂acr
+us+ρua∂acr
+st= 0,(7.7b)
+ρrb∂bρsa−ρsb∂bρra−ct
+rsρta= 0. (7.7c)
+(7.7a)–(7.7c) are the structure relations of E.
+The Lie algebroid Eis characterized by a natural cohomology. We shall define
+this conveniently in the language of graded geometry [30]. Consider t he parity
+shifted bundle E[1] and the /CIgraded algebra Fun( E[1]) of functions on E[1].
+There exists a degree 1 derivation dEof Fun(E[1]) defined by
+dE=ρra(x)ξr∂La−1
+2cr
+st(x)ξsξt∂Lr, (7.8)
+wherexa,ξrare the base and fiber coordinates of a generic trivialization of E[1]
+with degree 0 ,1, respectively, and ∂a=∂/∂xa,∂r=∂/∂ξr. Using the relations
+(7.7), one checks easily that dEis nilpotent and is therefore a differential
+dE2= 0. (7.9)
+The cohomology of the differential space (Fun( E[1]),dE) is the Lie algebroid
+cohomology of E,HLA∗(E). WhenE=TM,dEreduces to the ordinary de
+Rham differential and HLA∗(E) reduces to the familiar de Rham cohomology.
+46With any section X∈Γ(E), there are associated two derivations of Fun( E[1])
+of degree −1, 0 defined by
+iEX=Xr(x)∂Lr (7.10a)
+lEX=ρraXr(x)∂La+(ρra∂aXs+cs
+rtXt)(x)ξr∂Ls. (7.10b)
+They generalize the interior and Lie derivatives of de Rham theory an d reduce to
+those when E=TM.
+It is simple to check that the above derivations satisfy
+[dE,dE] = 0, (7.11a)
+[dE,iEX] =lEX, (7.11b)
+[dE,lEX] = 0, (7.11c)
+[iEX,iEY] = 0, (7.11d)
+[lEX,iEY] =iE[X,Y]E, (7.11e)
+[lEX,lEY] =lE[X,Y]E, (7.11f)
+withX,Y∈Γ(E), generalizing the well–known Cartan relations.
+Letgbe a Lie algebra and let ϕ:g→Γ(E) be a fiducial Lie algebra homo-
+morphism. Then, for x∈g, the degree −1, 0 derivations of Fun( E[1])
+iEx=iEϕ(x), (7.12a)
+lEx=lEϕ(x) (7.12b)
+are defined. By (7.11a), (7.11c), (7.11f), (Fun( E[1]),g,lE,dE) is a differential
+g–module (cf. app. A). The associated invariant cohomology is the inv ariant
+Lie algebroid cohomology HLAinv∗(E) ofE[31]. Analogously, by (7.11a)–(7.11f),
+(Fun(E[1]),g,iE,lE,dE) is ag–operation (cf. app. B). The basic cohomology as-
+47sociated with it is the basic Lie algebroid cohomology HLAbas∗(E) ofE[32].
+The /CI–graded algebra Fun( T∗[−1]E[1]) of functions on the parity shifted co-
+tangent bundle T∗[−1]E[1] ofE[1] can be given a structure of BV algebra. This
+BValgebraextends, inanappropriatesensetobespecified, thealg ebraFun(E[1])
+considered above.
+T∗[−1]E[1] has the canonical degree −1 symplectic structure
+ωE=dyadxa+dηrdξr, (7.13)
+wherexa,ξr,ya,ηrare the base and fiber coordinates of a generic trivialization
+ofT∗[−1]E[1] with degree 0 ,1,−1,−2, respectively. Let us assume now that the
+orientation line bundle QE=∧nT∗M⊗∧qE, wheren= dimMandq= rankE
+is trivial. There then exists a nowhere vanishing section γ∈Γ(QE), which can
+be used to construct a volume form on T∗[−1]E[1],
+µEγ=γ2(x)dx1···dxndξ1···dξrdy1···dyndη1···dηr.(7.14)
+These geometrical objects allow us to endow Fun( T∗[−1]E[1]) with the structure
+of BV algebra. The construction is standard and is illustrated in the lit erature
+(see e. g. ref. [7]). The BV Laplacian ∆ Eγis given by
+∆Eγ=γ−1(x)∂Laγ(x)∂La−∂Lr∂Lr(7.15)
+=/parenleftbig
+∂La+∂alnγ(x)/parenrightbig
+∂La−∂Lr∂Lr,
+where∂a=∂/∂xa,∂r=∂/∂ξr,∂a=∂/∂ya,∂r=∂/∂ηr. The BV antibracket
+has the standard form
+{φ,ψ}E=∂Raφ∂Laψ−∂Raφ∂Laψ+∂Rrφ∂Lrψ−∂Rrφ∂Lrψ, (7.16)
+withφ,ψ∈Fun(T∗[−1]E[1]). Itiseasytocheck thatthetriple(Fun( T∗[−1]E[1]),
+∆Eγ,{·,·}E) satisfies (2.1)–(2.3) and is therefore a BV algebra as announced.
+48Thebundle projection πE:T∗[−1]E[1]→E[1]induces adegree 0gradedalge-
+bra monomorphism πE∗: Fun(E[1])→Fun(T∗[−1]E[1]). In this way, Fun( E[1])
+can be viewed as a subalgebra of Fun( T∗[−1]E[1]). From (7.15), (7.16), one
+has ∆Eγ|Fun(E[1])= 0 and {·|Fun(E[1]),·|Fun(E[1])}E= 0. It follows that Fun( E[1]),
+equipped with the trivial BV algebra structure, is a BV subalgebra of the BV
+algebra Fun( T∗[−1]E[1]) (cf. sect. 2). Indeed, the BV antibracket structure of
+Fun(T∗[−1]E[1]) is closely related to the bracket structure of the “big bracket”
+formulation of Lie algebroid theory [33–35].
+With applications of the theory of the preceding sections in mind, we w ant to
+equip the BV algebra Fun( T∗[−1]E[1]) with a quantum BV master action with
+global symmetries. This is achieved by the following construction.
+Fun(T∗[−1]E[1]) contains the degree 0 element
+SE=ρra(x)yaξr+1
+2cr
+st(x)ξsξtηr (7.17)
+and, forX∈Γ(E), the degree −2,−1 elements
+ιEX=−Xr(x)ηr, (7.18a)
+λEX=−ρraXr(x)ya−(ρra∂aXs+cs
+rtXt)(x)ξrηs. (7.18b)
+By a straightforward calculation, one finds the brackets
+{SE,SE}E= 0, (7.19a)
+{ιEX,SE}E=λEX, (7.19b)
+{λEX,SE}E= 0, (7.19c)
+{ιEX,ιEY}E= 0, (7.19d)
+{λEX,ιEY}E=ιE[X,Y]E, (7.19e)
+49{λEX,λEY}E=λE[X,Y]E. (7.19f)
+One also shows that the relations
+∆EγSE= 0, (7.20a)
+∆EγιEX= 0, (7.20b)
+∆EγλEX= 0. (7.20c)
+hold, provided γsatisfies the condition
+∂aρra+ρra∂alnγ−cs
+sr= 0. (7.21)
+In general, the chosen γ∈Γ(QE) does not fulfil (7.21). Note, however, that γ
+is determined only up to a rescaling by a factor of the form efwithf∈Fun(M).
+Hence, if, instead of (7.21), γsatisfies the weaker condition
+∂aρra+ρra∂alnγ−cs
+sr+ρra∂af= 0 (7.22)
+for some function f∈Fun(M), then, after redefining γinto efγ, one can make γ
+fulfil (7.21). It can be shown that this is the case precisely when the Lie algebroid
+Eisunimodular , i. e. its modular class θE, a distinguished element of the degree
+1 cohomology HLA1(E), vanishes [36]. Indeed, the first three terms in left hand
+side of (7.22) constitute the local expression of a generic represe ntative ofθE
+and (7.22) is the statement that this representative is exact. See appendix D
+for a review of the definition and the main properties of the modular c lass. The
+relevance ofunimodularity in BVtheory hasbeen recently emphasise d inref. [16].
+When aγ∈Γ(QE) satisfying (7.21) exists, it may not be unique. We are still
+free to redefine γinto efγfor any function f∈Fun(M) such that
+ρra∂af= 0. (7.23)
+50Note that eq. (7.23) reads compactly as dEf= 0. So, its solutions span the
+degree 0 cohomology HLA0(E).
+Henceforth, we assume that a nowhere vanishing γ∈Γ(QE) satisfying (7.21)
+exists and has been chosen. Naturalness requires that all the rele vant BV struc-
+tures do not depend on this choice, a property that must be caref ully checked.
+TheBValgebraFun( T∗[−1]E[1])isnowequippedwiththedegree1derivation
+dE= adESE, (7.24)
+and, forX∈Γ(E), the degree −1, 0 derivations
+iEX= adEιEX, (7.25a)
+lEX= adEλEX, (7.25b)
+where ad Eis defined according to (2.8). By (7.19a)–(7.19f), dE,iEX,lEXsatisfy
+the Cartan relations (7.11a)–(7.11f). Further, by (7.20a)–(7.20 c),dE,iEX,lEX
+are BV inner derivations (cf. sect. 2).
+Inspecting (7.8), (7.10a), (7.10b), we observe that dE=dE|Fun(E[1]),iEX=
+iEX|Fun(E[1]),lEX=lEX|Fun(E[1]), withX∈Γ(E). Therefore, the derivations dE,
+iE,lEextenddE,iE,lEfrom Fun(E[1]) to Fun( T∗[−1]E[1]).
+By (7.19a), (7.20a), SEsatisfies the quantum BV master equation (3.1) and is
+therefore a quantum BV master action of the BV algebra Fun( T∗[−1]E[1]). The
+quantum BV operator is δEγ= ∆Eγ+ adESE(cf. eq. (3.3)). δEγdepends ex-
+plicitly onγ. The quantum BV cohomology HBV∗(Fun(T∗[−1]E[1])), conversely,
+does not up to isomorphism, since, for f∈Fun(M) satisfying (7.23), one has
+δEefγ= e−fδEγef. The classical BV operator is δEγc= adESE=dE(cf. eq.
+(3.7)). It is manifestly independent from γ. Hence, the classical BV cohomology
+HcBV∗(Fun(T∗[−1]E[1])) also is.
+SinceδEγ|Fun(E[1])=δEγc|Fun(E[1])=dE, the algebra inclusion πE∗: Fun(E[1])
+→Fun(T∗[−1]E[1]) induces a homomorphism of the Lie algebroid cohomology
+51HLA∗(E) into the quantum BV cohomology HBV∗(Fun(T∗[−1]E[1])) as well as
+the classical BV cohomology HcBV∗(Fun(T∗[−1]E[1])). Thus, each Lie algebroid
+cohomology class gives rise to a well–defined BV observable.
+Letgbe a Lie algebra and let ϕ:g→Γ(E) be a fiducial Lie algebra homo-
+morphism. For any x∈g, let us define
+ιEx=ιEϕ(x), (7.26a)
+λEx=λEϕ(x), (7.26b)
+and then define derivations iEx,lExon Fun(T∗[−1]E[1]) via (7.25a), (7.25b).
+Suppose we keep only the Lie derivations lExand forget about the interior
+derivations iEx. From (7.25b), (7.19f), (7.20c), it follows immediately that (4.4),
+(4.5) are satisfied. Hence, the BV algebra Fun( T∗[−1]E[1]) carries an N= 0 BV
+Hamiltonian g–actionlEhavingλEas BV moment map. Further, by (7.19c),
+the master action SEsatisfies (4.6) and is thus invariant under the g–action.
+Therefore, we can perform the N= 0 gauging of the BV algebra following the
+scheme described in sect. 4.
+The relevant invariant quantum BV cohomology HBVinv∗(Fun(T∗[−1]E[1]))
+(cf. sect. 4) is independent from the choice of γ, likeHBV∗(Fun(T∗[−1]E[1])). In
+fact, forf∈Fun(M) satisfying (7.23), one has δEefγ= e−fδEγefand, forx∈g,
+lEx= e−flExef, aslExf= 0. Obviously, the invariant classical BV cohomology
+HcBVinv∗(Fun(T∗[−1]E[1])) is independent from the choice of γ.
+SinceδEγ|Fun(E[1])=δEγc|Fun(E[1])=dEand, forx∈g,lEx=lEx|Fun(E[1]), the
+algebra inclusion πE∗: Fun(E[1])→Fun(T∗[−1]E[1]) induces a homomorphism
+of the invariant Lie algebroid cohomology HLAinv∗(E) into the invariant quan-
+tum BV cohomology HBVinv∗(Fun(T∗[−1]E[1])) as well as the invariant classical
+BV cohomology HcBVinv∗(Fun(T∗[−1]E[1])). Thus, each invariant Lie algebroid
+cohomology class gives rise to an invariant BV observable.
+52Suppose we keep both the interior derivations iExand Lie derivations lEx.
+From (7.25a), (7.25b), (7.19d)–(7.19f), (7.20b), (7.20c), it follow s immediately
+that (5.4), (5.5) are satisfied. Hence, the BV algebra Fun( T∗[−1]E[1]) carries
+anN= 1 BV Hamiltonian g–actioniE,lEhavingιE,λEas BV (pre)moment
+maps. Further by (7.19b), (7.19c), the master action SEsatisfies (5.6) and is thus
+(Hamiltonian) invariant under the g–action. Therefore, the N= 1 gauging of the
+BV algebra can be carried out along the lines illustrated in sect. 5,
+The relevant basic quantum BV cohomology HBVbas∗(Fun(T∗[−1]E[1])) (cf.
+sect. 5) is independent from the choice of γ, likeHBV∗(Fun(T∗[−1]E[1])), anal-
+ogously to the N= 0 case. In fact, for f∈Fun(M) satisfying (7.23), one
+hasδEefγ= e−fδEγefand, forx∈g,iEx= e−fiExef,lEx= e−flExef, asiExf
+=lExf= 0. The basic classical BV cohomology HcBVbas∗(Fun(T∗[−1]E[1])) is of
+course independent from the choice of γ.
+AsδEγ|Fun(E[1])=δEγc|Fun(E[1])=dEand, forx∈g,iEx=iEx|Fun(E[1]),lEx=
+lEx|Fun(E[1]), the algebra inclusion πE∗: Fun(E[1])→Fun(T∗[−1]E[1]) induces a
+homomorphism of the basic Lie algebroid cohomology HLAbas∗(E) into the basic
+quantum BVcohomology HBVbas∗(Fun(T∗[−1]E[1])) aswell asthebasic classical
+BV cohomology HcBVbas∗(Fun(T∗[−1]E[1])), analogously to the N= 0 case. So,
+each basic Lie algebroid cohomology class gives rise to a basic BV obser vable.
+The general construction expounded above exhibits a richgeomet ry, but, from
+the point of view of BV gauging, is kind of trivial: one can gauge any Lie a lgebra
+gunder the mild assumption that a Lie algebra homomorphism ϕ:g→Γ(E)
+is available. In physical problems, there virtually always are restrict ions on the
+symmetries that one can gauge. The Poisson Lie algebroid is a special case of the
+above general construction, in which such restrictions emerge na turally.
+The Poisson Lie algebroid BV algebra and its gauging
+Suppose that Mis a Poisson manifold and that P∈Γ(∧2TM) is its Poisson
+53bivector [37]. Then, Psatisfies the Poisson condition
+Pad∂dPbc+Pbd∂dPca+Pcd∂dPab= 0. (7.27)
+As is well–known, the Poisson structure of Mendows the cotangent bundle
+T∗MofMwith the structure of Lie algebroid. For simplicity, we shall mark
+all objects referring to this algebroid with a suffix P. The anchor and structure
+functions of T∗Mare given by ρab=Pabandcaab=∂aPbc. The Lie algebroid
+cohomology of T∗Mis the Poisson–Lichnerowicz cohomology of P.
+Proceeding as explained in detail above, we construct the associat ed BV alge-
+bra (Fun(T∗[−1]T∗[1]M),∆Pγ,{·,·}P), the quantum BV master action SPand,
+forα∈Γ(T∗M), the (pre)moments ιPα,λPαof the interior and Lie derivations
+iPα,lPαof Fun(T∗[−1]T∗[1]M).
+The master action SPof, defined according to (7.17), reads as
+SP=Pab(x)ybξa+1
+2∂aPbc(x)ξbξcηa. (7.28)
+Similarly, for α∈Γ(T∗M), the (pre)moments ιPα,λPαdefined according to
+(7.18a), (7.18b), are given by
+ιPα=−αa(x)ηa, (7.29a)
+λPα=−Pabαa(x)yb−(Pac∂cαb+∂bPacαc)(x)ξaηb.(7.29b)
+Theorientationlinebundle QP= (∧nT∗M)⊗2isalwaystrivial. Letting γ∈Γ(QP)
+be a nowhere vanishing section, the unimodularity condition (7.21) re ads
+−2γ−1/2∂b(γ1/2Pba) = 0., (7.30)
+(7.30) determines γonly up to a rescaling by a factor ef, wheref∈Fun(M)
+is a Casimir function of P(that isPab∂bf= 0, cf. eq. (7.23)). The action SP
+coincides with the reduced action used in the semiclassical computat ion of the
+54correlators of quantum observables for the Poisson sigma model o n the sphere in
+ref. [16].
+Now, we shall assess whether it is possible to perform a non trivial ga uging of
+the Poisson BV algebra just constructed on the lines of ref. [11]. To this end, we
+make the following assumptions.
+1. A compact connected Lie group Gwith Lie algebra gis given.
+2.Mcarries a smooth effective left G–action.
+3. TheG–action is Hamiltonian.
+As is well–known, the fundamental vector fields of the G–action organize as a
+sectionu∈Γ(TM⊗g∨).uisG–equivariant, that is
+uib∂buja−ujb∂buia=fk
+ijuka, (7.31)
+wherefkijare the structure constants of g. As theG–action is Hamiltonian,
+there exists a moment map µ∈Γ(g∨) of it.µisG–equivariant, that is
+uib∂bµj=fk
+ijµk, (7.32)
+and has the property that
+uia=−Pab∂bµi. (7.33)
+Being the fundamental vector fields Hamiltonian, they leave the Pois son 2–vector
+invariant,lMuiPab= 0.
+Now, define a section ϕ∈Γ(T∗M⊗g∨) by
+ϕia=∂aµi (7.34)
+A simple calculation based on (7.32), (7.33) shows that
+[ϕi,ϕj]Pa=Pbc(ϕib∂cϕja−ϕjb∂cϕia)+∂aPbcϕibϕjc=fk
+ijϕka.(7.35)
+55Therefore,ϕ:g→Γ(T∗M) is a Lie algebra homomorphism. The (pre)moments
+ιPi,λPi, defined according to (7.26a), (7.26b), are obtained by substitut ingϕifor
+αin (7.29a),(7.29b),
+ιPi=−∂aµi(x)ηa, (7.36a)
+λPi=Pab∂bµi(x)ya−∂b(Pac∂cµi)(x)ξaηb. (7.36b)
+As explained in the first part of this section, we can construct in this way an
+N= 0 and an N= 1 gauging of the BV algebra Fun( T∗[−1]T∗[1]M) with in-
+variant master action SP. In theN= 0 case, the invariant Poisson–Lichenrowicz
+cohomology of Pis contained, in the sense precisely defined above, in the in-
+variant BV cohomology of Fun( T∗[−1]T∗[1]M). Similarly, in the N= 1 case,
+the basic Poisson–Lichenrowicz cohomology of Pis contained in the basic BV
+cohomology of Fun( T∗[−1]T∗[1]M).
+Relation to the Poisson–Weil sigma model
+ThePoisson–Weil sigmamodelisagaugedversionofthePoissonsigma model.
+It has been studied in an AKSZ framework in refs. [10,11] and furth er generalized
+in ref. [12]. The target space of the model is a Poisson manifold Mwith a
+Hamiltonian effective left G—action as described above. The fields of the model
+are de Rham superfields, that is sections of suitable bundles on the p arity shifted
+tangent bundle T[1]Σ of the 2–dimensional world sheet Σ. In the simplest version
+of the model, the field content is as follows
+1.b∈Γ(T[1]Σ,g∨[0]).
+2.c∈Γ(T[1]Σ,g[1]).
+3.B∈Γ(T[1]Σ,g∨[−1]).
+4.C∈Γ(T[1]Σ,g[2])
+565.x∈Map(T[1]Σ,M).
+6.y∈Γ(T[1]Σ,x∗T∗[1]M).
+The classical BV master action of the Poisson–Weil sigma model is
+SPW=/integraldisplay
+T[1]Σ̺/bracketleftBig
+bi/parenleftBig
+dci−1
+2fi
+jkcjck+Ci/parenrightBig
+−Bi/parenleftbig
+dCi−fi
+jkcjCk/parenrightbig
+(7.37)
++ya/parenleftbig
+dxa+uia(x)ci/parenrightbig
+−µi(x)Ci−1
+2Pab(x)yayb/bracketrightBig
+,
+where̺is the invariant supermeasure on T[1]Σ. It is not known whether SPW
+satisfies also the appropriate quantum BV master equation, thoug h it is known
+this to be the case for the pure Poisson sigma model [38]1.
+The Poisson–Weil sigma model has a finite dimensional reduction defin ed as
+follows. Denote by 1andωthe unit and a volume form of Σ, viewed respectively
+as a degree 0 element and a nowhere vanishing degree 2 element of Fu n(T[1]Σ).
+We assume further that ωis normalized as
+/integraldisplay
+T[1]Σ̺ω= 1. (7.38)
+Take the superfields of the model to be of the form
+bi=biω, (7.39a)
+ci=ci1, (7.39b)
+Bi=Biω, (7.39c)
+Ci=Ci1, (7.39d)
+xa=xa1−ηaω, (7.39e)
+ya=ξa1+yaω, (7.39f)
+1In [10,11], µiandPabhave opposite sign.
+57where (bi,ci), (Bi,Ci), (xa,ya), (ξa,ηa) are BV conjugate pairs of variables of de-
+grees (−2,1), (−3,2), (0,−1), (1,−2), respectively. Substituting (7.39a)–(7.39f)
+into (7.37), we get a finite dimensional reduction SPW0ofSPW. This reads as
+SPW0=Sg|1+SP+ciλPi−CiιPi, (7.40)
+whereSg|1,SP,ιPi,λPiare given by (5.12), (7.28), (7.36a), (7.36b), respectively.
+Upon comparing with (5.16), we immediately realize that SPW0is nothing but
+theN= 1 gauged matter BV master action of the finite dimensional Poisson Lie
+algebroid model described above
+SPW0=Sg|1P. (7.41)
+(In eq. (7.40), the tensor product symbol ⊗is omitted.) Presumably, the finite
+dimensional model can be used to compute correlators of the Poiss on–Weil model
+along the lines described in ref. [16].
+588Conclusions
+In this paper, we have explored certain less known features of BV a lgebras,
+which have not been the object of a systematic study so far. We ha ve pointed out
+thataBVmaster actionmaypossess globalsymmetries notdirectly relatedtothe
+gaugesymmetries which underlie the BV symmetry and which may be int eresting
+togaugeforavarietyofreasons. Wehaveseenthatthegaugingc anbecarriedout
+inapurelyBVframework. Theglobalsymmetry ofthemaster action organizesas
+a Lie algebra action with a varying amount of supersymmetry, which d etermines
+directly the amount of ghost supersymmetry and the procedure o f gauging. We
+have found that N= 0 andN= 1 gauging correspond to ordinary gauging and
+to topological gauging, respectively. For higher N, the situation is not clear yet.
+The ordinary formal structure of BV algebras seems to be inadequ ate to treat
+these cases and, though sensible algebraic constructions can be c arried out, their
+eventual field theoretic origin or underpinning is not clear. This may b e the
+object of future investigation.
+We feel that the BV algebraic framework is more versatile than it has so far
+been realized. It would be certainly worth the effort to explore the f ull range of
+its applications.
+59ADifferential Lie modules and their invariant cohomology
+In this appendix, we recall the basic properties of differential Lie mo dules and
+their cohomology. We further provide a self-contained proof of th e important
+cohomology isomorphism (A.6). See ref. [19] for background mater ial.
+Adifferential Lie module is a quadruplet ( E,g,l,δ), where Eis a /CI–graded
+vector space, gis a Lie algebra and l:g→End0(E) is a linear map and δ∈
+End1(E) satisfying the graded commutation relations
+[li,lj] =fk
+ijlk, (A.1a)
+[δ,li] = 0, (A.1b)
+[δ,δ] = 0, (A.1c)
+with respect to a chosen basis {ti}ofg. We note that neither Eis supposed to
+be an algebra nor li,δare supposed to be graded derivations. If F ⊂ Eis a
+subspace, Finv=F ∩(∩ikerli) is called the invariant component of F.
+The pairs ( E,δ), (Einv,δ) are both differential spaces. Their associated co-
+homologies H∗(E) =H∗(E,δ),Hinv∗(E) =H∗(Einv,δ) are the ordinary and the
+invariant cohomology of the differential Lie module ( E,g,l,δ).
+With the Lie algebra g, there is associated a canonical differential Lie module
+(CE(g),g,lg,δg), called the Chevalley–Eilenberg Lie module .CE(g) is
+CE(g) = Fun(g[1]), (A.2)
+the algebra of polynomials of the coordinates ciofg[1] with respect to the basis
+{ti}.lg,δgare defined by the relations
+lgicj=−fj
+ikck, (A.3a)
+δgci=−1
+2fi
+jkcjck. (A.3b)
+60For a given differential Lie module ( E,g,l,δ), let us set
+E′=CE(g)⊗E. (A.4)
+We can define endomorphisms of E′by
+l′
+i=lgi⊗1+1g⊗li, (A.5a)
+δ′=δg⊗1+1g⊗δ+ci⊗li. (A.5b)
+Then, (E′,g,l′,δ′) is a differential Lie module.
+Ifgis a reductive Lie algebra, then
+Hinv∗(E′)≃CE(g)inv⊗Hinv∗(E). (A.6)
+Recall that gis reductive if gis the direct sum of an Abelian and a semisimple
+Lie algebra. The rest of this appendix is devoted to the sketch of th e proof of the
+above result.
+To begin with, we note that
+E′
+inv= (CE(g)⊗E)inv. (A.7)
+This suggests defining the following subspaces of E′inv
+Cn= (CE(g)n⊗E)inv, (A.8)
+Dn=/circleplusdisplay
+0≤m≤nCm, (A.9)
+wheren≥0. Then, Einv≃ D0⊂ D1⊂ D2⊂...⊂ Dh=E′inv,h= dimg, is a
+filtration of the vector space E′inv. One can show the following two properties.
+Let{zkx}be a basis of CE(g)invsuch thatzkx∈CEk(g) for allx. (Such a
+basis exists as lgiCEk(g)⊂CEk(g) for allk.)
+61i) Letµ∈ Dnbe such that
+δ′µ= 0. (A.10)
+Then, there are ν∈ Dn−1,αkx∈ Einvwith 0≤k≤nsuch that
+µ=δ′ν+/summationdisplay
+0≤k≤nzkx⊗αkx, (A.11)
+δαkx= 0. (A.12)
+ii) Letν∈ Dn−1,αkx∈ Einvwith 0≤k≤nbe such that
+δ′ν+/summationdisplay
+0≤k≤nzkx⊗αkx= 0. (A.13)
+Then, there are βkx∈ Einvwith 0≤k≤n, such that
+αkx=δβkx. (A.14)
+Byi,ii, there is a homomorphism q:H∗(E′inv,δ′)→CE(g)inv⊗H∗(Einv,δ)
+defined by the expression
+q([µ]) =/summationdisplay
+0≤k≤nzkx⊗[αkx], (A.15)
+whereµis expressed as in (A.11). By (A.3b), as δgzkx=1
+2cilgizkx= 0, one has
+δ′/summationdisplay
+0≤k≤nzkx⊗γkx=/summationdisplay
+0≤k≤n(−1)kzkx⊗δγkx, (A.16)
+forγkx∈ Einv. It follows that qis an isomorphism. Thus, if one shows i,ii, (A.6)
+is shown as well.
+Proof ofi.ii.gacts onCE(g) via (A.3a) and so, CE(g) is a representation of
+g. From Lie algebra theory, since gis reductive, this representation is semisimple.
+Thus, forany g–stablesubspace U⊂CE(g)andforany g–stablesubspace V⊂U
+there is a g–stable subspace W⊂Usuch thatU≃V⊕W. In particular, if U⊂
+62CE(g) is ag–stable subspace, then U=Uinv⊕lgU, whereUinv=U∩(∩ikerlgi)
+andlgU= spanilgiU=U∩(spaniimlgi).
+ConsiderZn(CE(g)) = kerδg∩CEn(g).Zn(CE(g)) isg–stable and, there-
+fore,Zn(CE(g)) =Zn(CE(g))inv⊕lgZn(CE(g)). Sinceδg=1
+2cilgi, by (A.3b),
+CEn(g)inv⊂Zn(CE(g))inv⊂CEn(g)invand, hence, Zn(CE(g))inv=CEn(g)inv.
+Now, letBn(CE(g)) = imδg∩CEn(g). Sincelgi=igiδg+δgigi, whereigiis the de-
+gree−1 derivation of CE(g) defined by igicj=δji, andδg=1
+2lgici, by (A.3b), as
+fjji= 0 for a reductive Lie algebra g,lgZn(CE(g))⊂Bn(CE(g))⊂lgZn(CE(g))
+and, thus,lgZn(CE(g)) =Bn(CE(g)). In conclusion,
+Zn(CE(g)) =CEn(g)inv⊕Bn(CE(g)). (A.17)
+Further, as Zn(CE(g)),CEn(g) areg–stable and Zn(CE(g))⊂CEn(g),
+CEn(g) =Zn(CE(g))⊕/tildewidestCEn(g), (A.18)
+for some g–stable subspace /tildewidestCEn(g)⊂CEn(g). As a consequence, δg:/tildewidestCEn(g)→
+Bn+1(CE(g)) is an isomorphism.
+From the above discussion, it follows that, for each n≥0, there is a basis
+{znx,rnu,sns}ofCEn(g) such that {znx},{rnu},{sns}are bases of CEn(g)inv,
+Bn(CE(g)),/tildewidestCEn(g), respectively, with the property that
+lgiznx= 0, l girnu=−Aniv
+urnv, l gisns=−Bnit
+ssnt, (A.19)
+δgznx= 0, δ grnu= 0, δ gsns=−Qnu
+srn+1u, (A.20)
+whereAni,BniandQnare square matrices with Qninvertible. The relation
+[lgi,δg] = 0 implies further the matrix relation
+An+1iQn−QnBni= 0. (A.21)
+Next, combining (A.5a), (A.5b) and the relation δg=1
+2cilgi,δ′can be cast as
+δ′=−δg⊗1+1g⊗δ+ci⊗1·l′
+i (A.22)
+63It follows that, when restricting to E′inv,
+δ′=δ′
+1+δ′
+2, (A.23)
+whereδ′1,δ′2are given by
+δ′
+1=−δg⊗1, (A.24a)
+δ′
+2= 1g⊗δ. (A.24b)
+Next, we have the following result. Let n≥0. Ifµn∈ Cnis such that
+δ′
+1µn= 0, (A.25)
+thenµnis of the special form
+µn=δ′
+1νn−1+znx⊗αx, (A.26)
+for certainνn−1∈ Cn−1,αx∈ Einv. To see this, we write µnas
+µn=znx⊗αx+rnu⊗βu+sns⊗γs, (A.27)
+whereαx,βu,γs∈ E. By (A.5a), (A.19), the condition l′iµn= 0 implies that
+liαx= 0, liβu=Aniu
+vβv, liγs=Bnis
+tγt. (A.28)
+By the first relation (A.28), αx∈ Einv. By the 3rd relation (A.20), one has
+rnu=−Qn−1−1suδgsn−1s. Hence, on account of (A.24a), one has
+rnu⊗βu=δ′
+1νn−1, (A.29)
+whereνn−1is given by
+νn−1=Qn−1−1s
+usn−1s⊗βu. (A.30)
+Using the 2nd relation (A.28), the 3rd relation (A.19) and (A.21), one finds that
+l′iνn−1= 0. Hence, νn−1∈ Cn−1. By (A.24a), (A.20) and the invertibility of the
+matrixQnthe condition δ′1µn= 0 implies that the γsall vanish,
+64γs= 0. (A.31)
+From (A.27), (A.29), (A.31), we get (A.26).
+The proof of iproceeds by induction on n. Letµ∈ D0satisfy (A.10). Then,
+µ= 1g⊗αfor someα∈ Einv. Further, by (A.5b), one has δα= 0. Therefore,
+(A.11), (A.12) hold with ν= 0. So,iholds forn= 0. Suppose now iholds for
+n−1 withn≥1. Letµ∈ Dnsatisfy (A.10). Write µ=µn+ ˜µ, whereµn∈ Cn,
+˜µ∈ Dn−1. Sinceδ′1Cm⊂ Cm+1δ′2Cm⊂ Dmform≥0, condition (A.10) implies
+thatδ′1µn= 0. So,µnsatisfies (A.25) and, so, by (A.26), there are νn−1∈ Cn−1
+andαnx∈ Einvsuch thatµn=δ′1νn−1+znx⊗αnx=δ′νn−1+znx⊗αnx−δ′2νn−1.
+Settingµ∗= ˜µ−δ′2νn−1∈ Dn−1, we have then
+µ=δ′νn−1+znx⊗αnx+µ∗. (A.32)
+Next, since δ′µ= 0 andδ′(znx⊗αnx) = (−1)nznx⊗δαnxby (A.23), (A.24) and
+the 1st relation (A.20), ( −1)nznx⊗δαnx+δ′µ∗= 0. Asδ′µ∗has no components
+of the form znx⊗γx, one hasδαnx= 0. Thus, δ′µ∗= 0. So,µ∗satisfies (A.10),
+and, so, by the inductive hypothesis, (A.11), (A.12) hold, yielding
+µ∗=δ′ν∗+/summationdisplay
+0≤k≤n−1zkx⊗αkx, (A.33)
+withν∗∈ Dn−2,αkx∈ Einvsuch thatδαkx= 0 for 0 ≤k≤n−1. Substituting
+(A.33) into (A.32) and setting ν=νn−1+ν∗∈ Dn−1, we find that µis of the
+form (A.11) with (A.12) satisfied. By induction on n,iis shown.
+The proof of iialso proceeds by induction on n. Letν∈ D0,α0,α1x∈ Einv
+satisfy (A.13). (Note that {z0x}={1}.) Then,ν=−1g⊗β, for someβ∈ Einv.
+Further, by (A.5b), one has α0=δβandα1x= 0. Hence, (A.14) holds. So,
+iiholds forn= 1. Suppose iiholds forn−2 withn≥2. Letν∈ Dn−1,
+αkx∈ Einvwith 0≤k≤nsatisfy (A.13). Write ν=νn−1+˜ν, whereνn−1∈ Cn−1,
+˜ν∈ Dn−2. Sinceδ′1Cm⊂ Cm+1δ′2Cm⊂ Dmform≥0, condition (A.13) implies
+65thatδ′1νn−1+znx⊗αnx= 0. Asδ′1νn−1has no components of the form znx⊗γx,
+one hasαnx= 0. Hence, δ′1νn−1= 0. So,νn−1satisfies (A.25) and, so, by (A.26),
+there areνn−2∈ Cn−2andβx∈ Einvsuch thatνn−1=δ′1νn−2+zn−1x⊗βx. Then,
+by (A.23), (A.24), δ′νn−1=δ′2νn−1= (−1)n−1zn−1x⊗δβx−δ′δ′2νn−2. Setting
+ν∗= ˜ν−δ′2νn−2∈ Dn−2, we have then
+δ′ν=δ′ν∗+(−1)n−1zn−1x⊗δβx. (A.34)
+Substituting (A.34) in (A.13) and recalling that αnx= 0, we find
+δ′ν∗+/summationdisplay
+0≤k≤n−1zkx⊗α∗
+kx= 0, (A.35)
+whereα∗kx=αkx+(−1)n−1δk,n−1δβx∈ Einv. Thus,ν∗,α∗kxsatisfy (A.13) and,
+so, by the inductive hypothesis, α∗kx=δβ∗kxfor certain β∗kx∈ Einv. Thus,
+(A.14) holds. By induction on n,iiis shown. QED
+66BLie operations and their equivariant cohomology
+In this appendix, we recall the basic properties of Lie operations an d their coho-
+mology. We further provide a self-contained proof of the importan t cohomology
+isomorphism (B.16). See refs. [19,22], for background material.
+ALie operation is a quintuplet ( E,g,i,l,δ), where Eis a /CI–graded vector
+space,gis a Lie algebra and i:g→End−1(E),l:g→End0(E) are linear maps
+andδ∈End1(E) satisfying the commutation relations
+[ii,ij] = 0, (B.1a)
+[li,ij] =fk
+ijik, (B.1b)
+[li,lj] =fk
+ijlk, (B.1c)
+[δ,ii] =li, (B.1d)
+[δ,li] = 0, (B.1e)
+[δ,δ] = 0, (B.1f)
+with respect to a chosen basis {ti}ofg. We note that neither Eis supposed to
+be an algebra nor ii,li,δare supposed to be graded derivations. If F ⊂ Eis a
+subspace, Fhor=F ∩(∩ikerii),Finv=F ∩(∩ikerli) andFbas=F ∩(∩i(kerii∩
+kerli)) =Fhor∩Finvare called the horizontal, invariant andbasiccomponent of
+F, respectively.
+The pairs ( E,δ), (Ebas,δ) are both differential spaces. Their associated coho-
+mologiesH∗(E) =H∗(E,δ),Hbas∗(E) =H∗(Ebas,δ) are the ordinary and thebasic
+cohomology of the Lie operation ( E,g,i,l,δ).
+Lie operations can be equipped with connections. A connection of the Lie op-
+eration (E,g,i,l,δ) is a linear map θ:g∨→End1(E) satisfying the commutation
+relations
+67[ij,θi] =δi
+j, (B.2a)
+[lj,θi] =−fi
+jkθk. (B.2b)
+Thecurvature of the connection θis the linear map Θ : g∨→End2(E) defined by
+Θi= [δ,θi]+1
+2fi
+jkθjθk. (B.3)
+Θ satisfies the commutation relations
+[ij,Θi] = 0, (B.4a)
+[lj,Θi] =−fi
+jkΘk. (B.4b)
+The Bianchi identities
+[δ,θi] = Θi−1
+2fi
+jkθjθk, (B.5a)
+[δ,Θi] =−fi
+jkθjΘk(B.5b)
+hold.
+With the Lie algebra g, there is associated a canonical Lie operation ( W(g),g,
+ig,lg,δg), called the Weil operation .W(g) is
+W(g) = Fun(g[1]⊕g[2]), (B.6)
+the algebra of polynomials of the coordinates ci,Ciofg[1],g[2] with respect to
+the basis {ti}.ig,lg,δgare defined by the relations
+igicj=δij, (B.7a)
+igiCj= 0, (B.7b)
+68lgicj=−fj
+ikck, (B.7c)
+lgiCj=−fj
+ikCk, (B.7d)
+δgci=Ci−1
+2fi
+jkcjck. (B.7e)
+δgCi=−fi
+jkcjCk. (B.7f)
+Note that (multiplication by) cidefines a connection of the Weil operation having
+Cias its curvature.
+For a given Lie operation ( E,g,i,l,δ), let us set
+E′=E′′=W(g)⊗E. (B.8)
+We can define endomorphisms of E′by
+i′
+i=igi⊗1, (B.9a)
+l′
+i=lgi⊗1+1g⊗li, (B.9b)
+δ′=δg⊗1+1g⊗δ+ci⊗li−Ci⊗ii. (B.9c)
+Then, (E′,g,i′,l′,δ′) is a Lie operation. By definition, the equivariant cohomology
+of the operation ( E,g,i,l,δ) (in the BRST model) is
+Hequiv∗(E) :=Hbas∗(E′). (B.10)
+When the operation ( E,g,i,l,δ) has a connection θwith curvature Θ, we can
+define endomorphisms of E′′by
+i′′
+i= 1g⊗ii, (B.11a)
+l′′
+i=lgi⊗1+1g⊗li, (B.11b)
+δ′′=δg⊗1+1g⊗δ+lgi⊗θi−igi⊗Θi. (B.11c)
+69Then, (E′′,g,i′′,l′′,δ′′) is also a Lie operation. A crucial technical result is that
+the basic cohomologies of EandE′′are isomorphic
+Hbas∗(E)≃Hbas∗(E′′). (B.12)
+We shall give a sketch of its proof momentarily.
+The Lie operations ( E′,g,i′,l′,δ′), (E′′,g,i′′,l′′,δ′′), which we have just con-
+structed are isomorphic, since i′,l′,δ′andi′′,l′′,δ′′are related as
+i′′
+i=I−1i′
+iI, (B.13a)
+l′′
+i=I−1l′
+iI, (B.13b)
+δ′′=I−1δ′I, (B.13c)
+whereI∈Iso0(E′′,E′) is given by
+I= exp(ci⊗ii)exp(igi⊗θi). (B.14)
+(The exponential are well defined as the exponential series termin ate after a finite
+number of terms.) It follows that
+Hbas∗(E′)≃Hbas∗(E′′). (B.15)
+From (B.10), (B.12), (B.15), we conclude that
+Hequiv∗(E)≃Hbas∗(E). (B.16)
+Thus,if the Lie operation (E,g,i,l,δ)admits a connection, the basic and equiv-
+ariant cohomologies of Eare equivalent . The above fundamental result hinges on
+the isomorphism (B.12), whose proof we shall now sketch.
+To begin with, we note that, by (B.11a), (B.11b),
+E′′
+bas= (W(g)⊗Ehor)inv. (B.17)
+70This suggests defining the following subspaces of E′′bas
+Cn= (W(g)n⊗Ehor)inv, (B.18)
+Dn=/circleplusdisplay
+0≤m≤nCm, (B.19)
+wheren≥0. Then, Ebas≃ D0⊂ D1⊂ D2⊂...E′′basis a filtration of the vector
+spaceE′′bas. One can now show the following two properties.
+i) Letµ∈ Dnbe such that
+δ′′µ= 0. (B.20)
+Then, there are ν∈ Dn−1,α∈ Ebassuch that
+µ=δ′′ν+1g⊗α, (B.21)
+δα= 0. (B.22)
+ii) Letν∈ Dn−1,α∈ Ebasbe such that
+δ′′ν+1g⊗α= 0. (B.23)
+Then, there is β∈ Ebassuch that
+α=δβ. (B.24)
+Byi,ii, there is a homomorphism q:H∗(E′′bas,δ′′)→H∗(Ebas,δ) defined by
+q([µ]) = [α], (B.25)
+whereµis expressed as in (B.21). Since
+δ′′(1g⊗γ) = 1g⊗δγ, (B.26)
+forγ∈ E, by (B.11c), qis an isomorphism. Thus, if one shows i,ii, (B.12) is
+shown as well.
+71Proof ofi,ii.By (B.11c), δ′′can be split as
+δ′′=δ′′
+1+δ′′
+2, (B.27)
+whereδ′′1,δ′′2are given by
+δ′′
+1=δg⊗1, (B.28a)
+δ′′
+2= 1g⊗δ+lgi⊗θi−igi⊗Θi. (B.28b)
+The following property holds. Let n≥0. Ifµn∈ Cnis such that
+δ′′
+1µn= 0, (B.29)
+thenµnis of the special form
+µn=δ′′
+1νn−1+δn,01g⊗α, (B.30)
+for certain νn−1∈ Cn−1,α∈ Ebas. To see this, we notice preliminarily that ci,
+˜Ci:=Ci−1
+2fijkcjckare generators of W(g) such that lgicj=−fjikck,lgi˜Cj=
+−fjik˜Ckandδgci=˜Ci,δg˜Ci= 0. Now, being µn∈ Cn, we have
+µn=/summationdisplay
+p≥0,q≥0,p+2q=nci1···cip˜Cj1···˜Cjq⊗αp,q
+i1...ip;j1...jq,(B.31)
+whereαp,qi1...ip;j1...jq∈ E. By (B.11a), (B.11b), the conditions i′′iµn= 0,l′′iµn= 0
+imply that
+ikαp,q
+i1...ip;j1...jq= 0, (B.32a)
+lkαp,q
+i1...ip;j1...jq−/summationdisplay
+rfl
+kirαp,q
+i1...l...ip;j1...jq−/summationdisplay
+sfl
+kjsαp,q
+i1...ip;j1...l...jq= 0.(B.32b)
+By (B.28a), the condition δ′′1µn= 0 implies further that
+αp,q
+i1...{ip;j1...jq}= 0, p≥1, (B.33)
+where{...}stands for complete symmetrization of the enclosed indices. Let us
+72define
+νn−1=/summationdisplay
+p≥0,q≥1,p+2q=n(−1)pq
+p+qci1···cipcip+1˜Cj1···˜Cjq−1⊗αp,q
+i1...ip;ip+1j1...jq−1.(B.34)
+Then, by (B.11a), (B.11b), (B.32a), (B.32b), we have i′′iνn−1= 0,l′′iνn−1= 0 so
+thatνn−1∈ Cn−1. Further, by (B.28a), (B.33),
+δ′′
+1νn−1=µn−δn,01g⊗α, (B.35)
+whereα=α0,0. By (B.32a), (B.32b), α∈ Ebas. (B.30) follows.
+The proof of iproceeds by induction on n. Letµ∈ D0satisfy (B.20). Then,
+µ= 1g⊗αfor someα∈ Ebas. Using (B.11c), one has δα= 0. Hence, (B.21),
+(B.22) hold with ν= 0. So,iholds forn= 0. Suppose now iholds forn−1
+withn≥1. Letµ∈ Dnsatisfy (B.20). Write µ=µn+ ˜µ, whereµn∈ Cn,
+˜µ∈ Dn−1. Sinceδ′′1Cm⊂ Cm+1δ′′2Cm⊂ Dmform≥0, condition (B.20)
+implies that δ′′1µn= 0. So, µnsatisfies (B.29) and, so, by (B.30), there is
+νn−1∈ Cn−1such thatµn=δ′′1νn−1=δ′′νn−1−δ′′2νn−1. Thus,µ=δ′′νn−1+µ∗
+withµ∗= ˜µ−δ′′2νn−1∈ Dn−1. Asδ′′µ= 0,δ′′µ∗= 0 as well. So, µ∗satisfies
+(B.20) and, so, by the inductive hypothesis, µ∗=δ′′ν∗+1g⊗α, withν∗∈ Dn−2,
+α∈ Ebassuch thatδα= 0. Hence, µ=δ′′ν+1g⊗α, whereν=νn−1+ν∗∈ Dn−1.
+By induction on n,iis shown.
+The proof of iialso proceeds by induction on n. Letν∈ D0,α∈ Ebassatisfy
+(B.23). Then, ν=−1g⊗βfor someβ∈ Ebas. Using (B.11c), one has α=δβ.
+Hence, (B.24) holds. So, iiholds forn= 1. Suppose iiholds forn−2 withn≥2.
+Letν∈ Dn−1,α∈ Ebassatisfy (B.23). Write ν=νn−1+ ˜ν, whereνn−1∈ Cn−1,
+˜ν∈ Dn−2. Sinceδ′′1Cm⊂ Cm+1δ′′2Cm⊂ Dmform≥0, condition (B.23) implies
+that thatδ′′1νn−1= 0. So,νn−1satisfies (B.29) and, so, by (B.30), there is
+νn−2∈ Cn−2such thatνn−1=δ′′1νn−2=δ′′νn−2−δ′′2νn−2. Thus,ν=δ′′νn−2+ν∗
+withν∗= ˜ν−δ′′2νn−2∈ Dn−2. By (B.23), δ′′ν∗+1g⊗α= 0. So,ν∗satisfies (B.23)
+73and, so, by the inductive hypothesis, α=δβfor someβ∈ Ebas. By induction on
+n,iiis shown. QED
+RemarkAlthough to prove the isomorphism (B.16) we had to make explicit
+use at several points of a fixed connection θof the operation ( E,g,i,l,δ), the
+isomorphism can be shown to be independent from the choice of θand is thus
+canonical.
+74CSuperfield formulation of the N=1 ghost system
+In this appendix, we show that the N= 1 ghost system described in sect. 5
+has an elegant superfield formulation.
+N= 1ghost superfields areelements ofthealgebra Ag|1((θ)), whereθaformal
+odd variable such that ∂θ=−1. Define the superfields/BUi=Bi+θbi, (C.1a)
i=ci−θCi. (C.1b)
+We have /BUi∈ Ag|1((θ))−3, 
i∈ Ag|1((θ))1. In terms of these, the g–action (5.11)
+reads succinctly as
+ig|1i
+/BUj= 0, (C.2a)
+ig|1i
+
j=δij(C.2b)
+lg|1i
+/BUj=fk
+ij
+/BUk, (C.2c)
+lg|1i
+
j=−fj
+ik
+
k. (C.2d)
+The superfield expression of ghost master action (5.12) is
+Sg|1=−/integraldisplay
+dθ/bracketleftBig/BUi∂θ
+
i+1
+2fi
+jk
+/BUi
+
j
k/bracketrightBig
+. (C.3)
+Similarly the quantum BV variations (5.13) read concisely as
+δg|1
+/BUi=−∂θ
+/BUi−fk
+ji
+/BUk
+
j, (C.4a)
+δg|1
+
i=−∂θ
+
i−1
+2fi
+jk
+
j
k. (C.4b)
+The superfield formalism was originally worked out in ref. [39].
+75DThe modular class of a Lie algebroid
+The modular class of a Lie algebroid was first introduced in [36]. Let Ebe a
+Lie algebroid over the manifold Mwith anchor ρEand Lie bracket [ ·,·]E. Then,
+the real line bundle over M
+QE=∧nT∗M⊗∧qE, (D.1)
+wheren= dimMandq= rankE, is defined. QEis called the orientation bundle
+ofE.
+Forγ∈Γ(QE),X∈Γ(E), we set
+DXγ= (lMρE(X)⊗1∧qE+1∧nT∗M⊗lEX)γ, (D.2)
+wherelMρ(X)is the ordinary Lie derivative along the vector field ρE(X) andlEX
+is defined by
+lEX(Y1∧...∧Yd) =d/summationdisplay
+r=1Y1∧...∧[X,Yr]E∧...∧Yd, Yr∈Γ(E).(D.3)
+Clearly,DXγ∈Γ(QE). It can be verified that the map X→DXdefines a
+representation of the Lie algebroid EinQE,
+DfXγ=fDXγ, (D.4a)
+DX(fγ) =fDXγ+(ρE(X)f)γ, (D.4b)
+[DX,DY]γ−D[X,Y]Eγ= 0, (D.4c)
+forf∈Fun(M) andX,Y∈Γ(E),γ∈Γ(QE).
+Suppose that the line bundle QEis trivial. Then, there is a nowhere vanishing
+sectionγ∈Γ(QE). Further, there is a section θγ∈Γ(E∗) such that
+DXγ=θγ(X)γ, (D.5)
+76forX∈Γ(E). It can be shown that θγsatisfies
+dEθγ= 0. (D.6)
+Further, if one rescales γintoγ′= efγwithf∈Fun(M), then one has
+θγ′=θγ+dEf. (D.7)
+Thus,θγis a representative of a well defined cohomology class θE∈H1(E),
+themodular class ofE, independent from the choice of γ. IfQEis not trivial,
+the modular class of Ecan still be defined as follows. One notes that QE⊗2
+is trivial and one repeats a similar construction with γreplaced by a nowhere
+vanishing section ν∈Γ(QE⊗2) andDXreplaced by DX⊗1QE+1QE⊗DX. Then,
+θE=1
+2[θν]H1(E).
+In general, Eis saidunimodular ifθE= 0.
+A straightforward calculation shows that, when QEis trivial,θγis given in
+any local trivialisation of Eby
+θγr=∂aρra+ρra∂alnγ−cs
+sr, (D.8)
+whereρra,ctrsare the anchor and structure functions of E, respectively. Upon
+rescalingγintoγ′= efγwithf∈Fun(M), one has
+θγ′r=θγr+ρra∂af. (D.9)
+Therefore, it is possible to chose γin such a way that θγ= 0 precisely when Eis
+unimodular.
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+arXiv:1001.0220v1  [hep-th]  1 Jan 2010Reconstruction of k-essence model
+Jiro Matsumoto and Shin’ichi Nojiri
+Department of Physics, Nagoya University, Nagoya 464-8602 , Japan
+We explicitly construct the k-essence models which reprodu ce the arbitrary FRW cosmology,
+that is, the arbitrary time-development of the scale factor or the Hubble rate. The k-essence model
+includes scalar quintessence model, tachyon dark energy mo del, ghost condensation model as special
+cases. Explicit formulas of the reconstruction are given.
+First we consider the case that the action only contains the k inetic term. In this case, we find
+that the model reproducing the development of the universe i n the exact ΛCDM model cannot be
+constructed although there is a model which reproduces the d evelopment infinitely closing to ΛCDM
+model. We find, however, the solution is not stable.
+Another is more general case including potential etc., wher e we find that there appear infinite
+number of arbitrary functions of the scalar fields, which are redundant to the time development of
+the scale factor. By adjusting one of the redundant function s, however, we can obtain the model
+where any solution we need could be stable.
+PACS numbers: 95.36.+x, 98.80.Cq, 04.50.Kd, 11.10.Kk, 11. 25.-w
+I. INTRODUCTION
+Several observations tell that the expansion of the present universe is accelerating [1–3]. In order to explain the
+acceleration, many kinds of models have been proposed. So-c alled k-essence model [4–6] is one of these models. The
+k-essence model is derived from k-inflation model [7, 8]. It i s possible to regard the tachyon dark energy model [9–12],
+ghost condensation model [17, 18], and scalar field quintess ence model [13–16] as variations of the k-essence model.
+In this paper we consider the reconstruction of k-essence mo del. We explicitly construct the k-essence models which
+reproduce the arbitrary FRW cosmology, that is, the arbitra ry time-development of the scale factor or the Hubble
+rate. For general reconstruction, see [21, 22]. We consider two cases: One is the case that the action only contains
+the kinetic term and another is more general case including p otential etc. In the former case, we find that the exact
+ΛCDM model cannot be constructed although there is a model infi nitely closing to ΛCDM model. In the model,
+however, the solution corresponding to ΛCDM model is unfortunately not stable. In the latter case, we find that
+there appear infinite number of redundant functions of the sc alar fields, which are not directly related with the time
+development of the scale factor. By adjusting one of them, ho wever, we can obtain the model where the solution we
+need could be stable. At present, the roles of the other funct ions are not clear but we may obtain a model satisfying
+other constraints from cosmology by adjusting these functi ons.
+II. PURE KINETIC MODEL
+We may start with the following action:
+S=/integraldisplay
+d4x√−g/parenleftbiggR
+2κ2−K(q(φ)∂µφ∂µφ)/parenrightbigg
+, (1)
+whereKandqare adequate functions. In the FRW background
+ds2=−dt2+a(t)2/summationdisplay
+i=1,2,3(dxi)2, (2)
+we may assume the scalar field φonly depends on the time coordinate t. Since the redefinition of φcan be absorbed
+into the redefinition of q(φ), we further identify the scalar field φwith the time coordinate t. Then we obtain the
+following FRW equation:
+3
+κ2H2=K(−q(t))+2K′(−q(t))q(t), (3)
+and the equation given by the variation of φ, we obtain
+0 = 2K′′(−q(t))q(t)q′(t)−(6Hq(t)+q′(t))K′(−q(t)). (4)2
+HereHis the Hubble rate defined by H= ˙a(t)/a(t). Eq. (4) can be integrated as
+a(t)6=1
+q(t)/parenleftbiggK0
+K′(−q(t))/parenrightbigg2
+. (5)
+By differentiating (3) with respect to tand using (5), we find
+6
+κ2˙H=−6K0a(t)−3/radicalbig
+q(t), (6)
+which gives
+q(t) =a(t)6(H′(t))2
+κ4K2
+0. (7)
+Eq. (7) gives the explicit form of q(t)and can be solved with respect to tby using an adequate function fas
+t=f(q). (8)
+Then (5) gives an explicit form of K′(−q)as
+K′(−q) =K0√qa(f(q))−3. (9)
+Then for the arbitrary time-development of aorHgiven by a=a(t), Eqs. (7) and (9) give the forms of the functions
+Kandqrealizing the time-development. We should note, however, ˙Hcannot change its sign as we can see from (6).
+Then the transition between non-phantom phase ( ˙H <0) and phantom phase ( ˙H >0) does not occur.
+As an example, we consider the model corresponding to the Ein stein gravity where the universe is filled with only
+one kind of perfect fluid whose equation of state (EoS) parame terwis constant. In the universe, the scale factor a
+and the Hubble rate Hbehaves as
+a(t) =a0t2
+3(1+w), H=2
+3(1+w)
+t, (10)
+with a constant a0. Then Eq. (7) gives
+q(t) =q0t4w
+1+w, q0≡a6
+0
+κ4K2
+0/parenleftbigg2
+3(1+w)/parenrightbigg2
+, (11)
+and Eq. (9) has the following form:
+K′(−q) =K0q−1
+2w
+0a−3
+0q1
+2w−1
+2. (12)
+We can integrate (12) as
+K(−q) =K1−2w
+w+1K0q−1
+2w
+0a−3
+0q1+w
+2w, (13)
+whereK1is a constant of the integration. By using (11), we find
+K(−q) =K1−4w
+3(1+w)2κ2t2,2K′(−q)q=4
+3(1+w)κ2t2. (14)
+Then we can check that Eq. (3) is satisfied if K1= 0and we find the Einstein gravity where the universe filled with
+only one kind of perfect fluid can be reproduced by the model (1 ) if we choose q(t)andK(−q)by (11) and (13) with
+K1= 0.
+We now extend the model (1) to include the matters with consta nt EoS parameters wi. Then since the energy
+density of the matters is given by/summationtext
+iρ0ia−3(1+wi)with constants ρ0i, the FRW equation (3) is modified as
+3
+κ2H2=K(−q(t))+2K′(−q(t))q(t)+/summationdisplay
+iρ0ia−3(1+wi). (15)3
+Eq. (4) given by the variation of φis not changed and we obtain (5) again. On the other hand, Eq. ( 6) is changed to
+6
+κ2˙H=−6K0a(t)−3/radicalbig
+q(t)−/summationdisplay
+i3(1+wi)ρ0ia−3(1+wi), (16)
+which gives
+q(t) =a(t)6(H′(t)+κ2
+2/summationtext
+i(1+wi)ρ0ia−3(1+wi))2
+κ4K2
+0. (17)
+Eq. (17) gives the explicit form of q(t)as in (7) and can be solved with respect to tby using an adequate function f
+as in (8), again. Then Eq.(5) gives an explicit form of K′(−q)by (9).
+If we substitute the evolution of scale factor a(t),
+a(t) =Asinh2
+3[αt], (18)
+which corresponds to the ΛCDM model in the Einstein gravity, to Eqs. (17) and (12), then these equations give
+constant qandK. This result seems to be very trivial since Kis nothing but the cosmological constant. Then we
+now consider the evolution of scale factor which differs a lit tle bit from that of ΛCDM model in the Einstein gravity.
+For this purpose, we slightly modify (18) as
+a(t) =Aexp/bracketleftBig
+ln(sinh2
+3[αt])+ǫf(t)/bracketrightBig
+, (19)
+whereA= (ρ0m
+Λ)1
+3andα=κ
+2√
+3Λare positive constants which reproduce the universe evolut ion inΛCDM model, f
+is an arbitrary function of t, andǫis a small constant. Then Eq. (17) gives
+q(t)∼ǫ2
+κ4K2
+0/parenleftbigg9
+4κ4ρ2
+0mf2−3κ2ρ2
+0m
+Λsinh2[αt]ff′′+ρ2
+0m
+Λ2sinh4[αt](f′′)2/parenrightbigg
+, (20)
+where we neglected the higher order terms of ǫ. As an examples, we may consider the following function as f,
+f(t) = ln/parenleftBig
+sinh2
+3[αt]/parenrightBig
+s.t.a(t) =A(sinh2
+3[αt])1+ǫ. (21)
+By substituting (21) to (20), we find
+t∼1
+αsinh−1/bracketleftbigg
+exp/bracketleftbigg
+−1
+2+K0
+ρ0mǫ√q/bracketrightbigg/bracketrightbigg
+. (22)
+Then by further substituting (22) into Eq. (5), we obtain
+K(−q)∼K0Λ
+ρ0m/integraldisplayq
+dq′q′−1
+2/parenleftbigg
+exp/bracketleftbigg
+−1
+2+K0
+ρ0mǫ/radicalbig
+q′/bracketrightbigg/parenrightbigg−2(1+ǫ)
+=ǫΛe1+ǫ
+1+ǫ/parenleftbigg
+C−exp/bracketleftbigg
+−2K0(1+ǫ)√q
+ρ0mǫ/bracketrightbigg/parenrightbigg
+. (23)
+HereCis a constant of the integration. Then we find the existence of the functions qandK, which give a(t)very
+close to that of ΛCDM model.
+We now investigate the stability of the solution. So we now st art with the two FRW equations and the field equation
+obtained from Eq. (1),
+3
+κ2H2=K(−q˙φ2)+2K′(−q˙φ2)q˙φ2, (24)
+1
+2κ2(3H2+2˙H) =1
+2K(−q˙φ2), (25)
+d
+dt(2a3q1
+2˙φK′(−q˙φ2)) = 0. (26)
+From Eqs. (24) and (25), we obtain
+K′(−Q)Q=−˙H
+κ2. (27)4
+HereQ≡q˙φ2. Inserting this equation to Eq. (26), we find
+d
+dt/parenleftBig
+2a3Q−1
+2˙H
+κ2/parenrightBig
+= 0, (28)
+which can be integrated and gives
+1
+κ2a3Q−1
+2˙H≡K0. (29)
+HereK0is a constant of the integration. We now write the form of a(t)asa(t) =a0eh(t)and consider the perturbation
+of the function h(t):h(t) =h0(t)+δh. Then from Eq. (29), we find the variation of Qis given by
+δQ=a6
+0
+κ4K2
+0e6h0/parenleftBig
+6δh¨h2
+0+2¨h0δ¨h/parenrightBig
+. (30)
+By combining Eq.(30) and Eq. (27), we obtain
+K′′(−Q0)Q2
+0=−¨h0
+κ2+...
+h0
+κ2/parenleftBig
+6˙h0+2...
+h0
+¨h0/parenrightBig. (31)
+HereQ0is defined by substituting h(t) =h0(t)into (29): Q0≡a6
+0e6h0(t)¨h0(t)2/K2
+0κ4. Substituting Eqs. (27), (30),
+and (31) into the equation,6
+κ2˙h0δ˙h=/parenleftBig
+K′(−Q0)−2K′′(−Q0)Q0/parenrightBig
+δQ, which is given by the variation of Eq. (24), we
+obtain a rather simple differential equation for δh(t),
+δ¨h−3˙h0¨h0+...
+h0
+¨h0δ˙h+3¨h0δh= 0. (32)
+The above equation only contains h0and its derivatives. Then if we have h0, even if we do not know the explicit
+forms of qandK, we can find the stability of the solution. In case that soluti onh0is given by (19), by choosing
+a0=A,
+h0= ln(sinh2
+3[αt])+ǫf(t), (33)
+Eq. (32) has the following form:
+δ¨h−2α2
+sinh2[αt]δh+O(ǫ) = 0. (34)
+Since2α2
+sinh2[αt]>0, the solution (33) is unstable. Then unfortunately the solu tion which behaves close to the ΛCDM
+model is not stable.
+By defining x≡δhandy=δ˙h, Eq.(32) can be rewritten as
+d
+dt/parenleftbigg
+x
+y/parenrightbigg
+=/parenleftBigg
+0 1
+−3¨h03˙h0¨h0+...
+h0
+¨h0/parenrightBigg/parenleftbigg
+x
+y/parenrightbigg
+. (35)
+In order that the solution could be stable, the real part of al l the eigenvalues of the 2 ×2 matrix in (35) should be
+negative:
+ℜλ±<0, λ±≡1
+2
+3˙h0¨h0+...
+h0
+¨h0±/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftBigg
+3˙h0¨h0+...
+h0
+¨h0/parenrightBigg2
+−12¨h0
+, (36)
+which requires ¨h0>0. This tells that the solution corresponding to non-phantom era, like quintessence era or matter
+dominance era, is not stable.5
+We now investigate the stability when we include the matters as in (15). Then the equations corresponding to (24)
+- (26) have the following form:
+3
+κ2H2=K(−Q)+2K′(−Q)Q+/summationdisplay
+iρ0ia−3(1+wi), (37)
+1
+2κ2(3H22˙H) =1
+2K(−Q)−1
+2/summationdisplay
+iwiρ0ia−3(1+wi), (38)
+d
+dt(2a3Q1
+2K′(−Q)) = 0. (39)
+Then by considering the perturbation h→h0+δh, we find
+0 =δ¨h−bδ˙h−cδh, (40)
+b≡31
+κ2(3˙h0¨h0+...
+h0)−3
+2/summationtext
+iwi(1+wi)ρ0ia−3(1+wi)
+0e−3(1+wi)h0˙h0
+3
+κ2¨h0+3
+2/summationtext
+i(1+wi)ρ0ia−3(1+wi)
+0e−3(1+wi)h0, (41)
+c≡ −κ2
+218
+κ4¨h2
+0−3/summationtext
+i(1+wi)ρ0ia−3(1+wi)
+0e−3(1+wi)h01
+κ2/parenleftBig
+6wi¨h0+(1+wi)...
+h0
+˙h0/parenrightBig
+3
+κ2¨h0+3
+2/summationtext
+i(1+wi)ρ0ia−3(1+wi)
+0e−3(1+wi)h0
+−κ2
+29
+2/summationtext
+i,jwi(1+wi)(1+wj)ρ0iρ0ja−3(2+wi+wj)
+0 e−3(2+wi+wj)h0
+3
+κ2¨h0+3
+2/summationtext
+i(1+wi)ρ0ia−3(1+wi)
+0e−3(1+wi)h0. (42)
+The above expression is rather complicated but in principle , we can check the stability by using the above equations.
+As long as we check numerically, the solution (33) which beha ves as almost ΛCDM solution in the Einstein gravity
+seems to be unstable.
+III. MORE GENERAL MODELS
+Since in the model (1), the solution which reproduces almost ΛCDM is unstable, we consider a more general model,
+whose action is given by
+S=/integraldisplay
+d4x√−g/parenleftbiggR
+2κ2−K(φ,X)+Lmatter/parenrightbigg
+, X≡∂µφ∂µφ. (43)
+Then the FRW equations are given by
+3
+κ2H2= 2X∂K(φ,X)
+∂X−K(φ,X)+ρmatter,−1
+κ2/parenleftBig
+2˙H+3H2/parenrightBig
+=K(φ,X)+pmatter(t). (44)
+As in (15), we consider the matters which have constant EoS pa rameters wi. We now choose φ=t, again, then we
+can rewrite the equations in (44) in the following form
+K(t,−1) =−1
+κ2/parenleftBig
+2˙H+3H2/parenrightBig
+−/summationdisplay
+iwiρ0ia−3(1+wi),∂K(φ,X)
+∂X/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+X=−1=1
+κ2˙H+1
+2/summationdisplay
+i(1+wi)ρ0ia−3(1+wi).(45)
+By using the appropriate function g(φ), if we choose
+K(φ,X) =−1
+κ2/parenleftbig
+2g′′(φ)+3g′(φ)2/parenrightbig
+−/summationdisplay
+iwiρ0ia−3(1+wi)
+0e−3(1+wi)g(φ)
++(X+1)/braceleftBigg
+1
+κ2g′′(φ)+1
+2/summationdisplay
+i(1+wi)ρ0ia−3(1+wi)
+0e−3(1+wi)g(φ)/bracerightBigg
++∞/summationdisplay
+n=2(X+1)nK(n)(φ),(46)
+we find the following solution for the FRW equations (44),
+H=g′(t)/parenleftBig
+a=a0eg(t)/parenrightBig
+. (47)6
+HereK(n)(φ)withn= 2,3,···can be arbitrary functions. The case that K(n)(φ)’s withn= 2,3,···vanish was
+studied in [19, 20] and the instability was investigated.
+We now investigate the stability of the equations without ma tter. From Eqs. (44), we can derive the following
+equation which does not contain the variable g′′,
+31−y2
+1+XX=−˙H
+H2+κ2
+H2∞/summationdisplay
+n=2((n−1)X−n−1)X(X+1)n−2K(n)(φ), (48)
+wherey=g′
+H. Using Eq. (48), we can rewrite dy/dN= (1/H)dy/dt (Nis called e-foldings and the scale factor is
+given in terms of Nasa∝eN) in the form which does not contain g:
+dy
+dN= 3X1−y2
+1+X/parenleftBigg˙φ
+X+y/parenrightBigg
+−κ2
+H2∞/summationdisplay
+n=2/bracketleftBig
+(˙φ+yX)((n−1)X−n−1)+˙φn(X+1)/bracketrightBig
+(X+1)n−2K(n)(φ).(49)
+We now consider the perturbation from a solution φ=tby putting φ=t+δφin (49). Then we obtain
+dδ˙φ
+dN=/bracketleftbigg
+−3−g′′
+g′2−d
+dN/braceleftbiggκ2
+6g′2(8K(2)−2
+κ2g′′)/bracerightbigg/bracketrightbigg
+δ˙φ. (50)
+If the quantity inside [ ]is negative, the fluctuation δ˙φbecomes exponentially smaller with time and therefore the
+solution becomes stable. Note that the stability is determi ned only in terms of K(2)and does not depend on other
+K(n)(n∝ne}ationslash= 2). Then if we choose K(2)properly, the solution corresponding to arbitrary develop ment of the universe
+becomes stable.
+We now investigate the stability when we include the matter. Then the equation corresponding to Eq.(48) has the
+following form:
+31−y2
+1+XX=−˙H
+H2+κ2
+H2∞/summationdisplay
+n=2/parenleftbigg
+(n−1)X−n−1/parenrightbigg
+X(X+1)n−2K(n)(φ)
++κ2
+H2X−1
+2(X+1)ρmatter−κ2
+2H2pmatter−κ2
+H2X
+X+1/summationdisplay
+iρ0ia−3(1+wi)
+0e−3(1+wi)g(φ). (51)
+Then we find
+dy
+dN= 3X1−y2
+1+X/parenleftBigg˙φ
+X+y/parenrightBigg
+−κ2
+H2∞/summationdisplay
+n=2/bracketleftBig
+(˙φ+yX)((n−1)X−n−1)+˙φn(X+1)/bracketrightBig
+(X+1)n−2K(n)(φ)
++κ2
+2H2X/parenleftbigg
+−X−1
+X+1/parenleftBig
+˙φ+yX/parenrightBig
+−˙φ/parenrightbigg
+ρmatter+κ2
+2H2ypmatter
++κ2
+2H2/summationdisplay
+i/parenleftbigg/parenleftBig
+˙φ+yX/parenrightBig2
+X+1−˙φ(1+wi)/parenrightbigg
+ρ0ia−3(1+wi)
+0e−3(1+wi)g(φ). (52)
+When we include the matter, the situation becomes a little bi t complicated since not only Hbut the scale factor a
+appears in the equation. Then we need equation to describe th e time development of a. By defining δλfor convenience
+as
+δλ≡3/summationdisplay
+i(1+wi)ρ0ia(t)−3(1+wi)κ2
+6g′2/parenleftbiggδa
+a−g′δφ/parenrightbigg
+. (53)
+we obtain the following equations,
+d
+dN/parenleftbigg
+δ˙φ
+δλ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+φ=t,H=g′(t)=/parenleftbigg
+A B
+C D/parenrightbigg/parenleftbigg
+δ˙φ
+δλ/parenrightbigg
+. (54)7
+Here
+A≡ −3+g′′
+g′2+κ2
+2g′2/summationdisplay
+i(1+wi)ρ0ia(t)−3(1+wi)−d
+dNln/braceleftBigg
+8K(2)−2
+κ2g′′−/summationdisplay
+i(1+wi)ρ0ia(t)−3(1+wi)/bracerightBigg
+,
+B≡3−24K(2)
+8K(2)−2
+κ2g′′−/summationtext
+i(1+wi)ρ0ia(t)−3(1+wi),
+C≡3/summationdisplay
+i(1+wi)ρ0ia(t)−3(1+wi)/parenleftbiggκ2
+6g′2/parenrightbigg2/braceleftBigg
+8K(2)−6g′2
+κ2−2
+κ2g′′−/summationdisplay
+i(1+wi)ρ0ia(t)−3(1+wi)/bracerightBigg
+,
+D≡d
+dNln/braceleftBigg
+κ2
+2g′2/summationdisplay
+i(1+wi)ρ0ia(t)−3(1+wi)/bracerightBigg
+−κ2
+2g′2/summationdisplay
+i(1+wi)ρ0ia(t)−3(1+wi). (55)
+Generally, 2 ×2 matrix must have negative trace and positive determinant i n order that two eigenvalues could be
+negative since the two eigenvalues are given by1
+2{trM±/radicalbig
+(trM)2−4(detM)}. So we just need to calculate the
+determinant and the trace of the matrix for investigating th e stability of the fixed point φ=t,H=g′(t). Since the
+expressions in (55) are very complicated, we consider the ca se that the solution is given by the behavior mimicing
+ΛCDM solution (18) in the Einstein gravity and the matter cont ents are given in the present universe. Then we find
+a(t)∼Asinh2
+3[αt], (56)
+g′(t)∼2
+3αcoth[αt], (57)
+κ2
+α2/summationdisplay
+i(1+wi)ρ0ia(t)−3(1+wi)∼2
+31
+sinh2[αt], (58)
+κ2
+α2/summationdisplay
+iwi(1+wi)ρ0ia(t)−3(1+wi)∼4
+9×1.86×10−41
+sinh8
+3[αt], (59)
+whereA≡(ρm0/ρΛ)1
+3andα≡κ√3ρΛ/2. Note that in (56), (57), and (58), we neglect the contributi on from
+radiation, and in (59), there only appears the contribution from radiation. Therefore the expressions in (56) - (59)
+could be valid at least when t≥109years. By using (56) - (59), we find the following expressions of the determinant
+and trace of the matrix in Eq.(54):
+tr/parenleftbigg
+A B
+C D/parenrightbigg
+∼ −6+3
+21
+cosh2[αt]−8κ2
+α3K(2)′−4
+3cosh[αt]
+sinh3[αt]
+2
+3coth[αt]/parenleftBig
+8κ2
+α2K(2)+2
+31
+sinh2[αt]/parenrightBig, (60)
+det/parenleftbigg
+A B
+C D/parenrightbigg
+∼/braceleftbigg
+8κ2
+α2K(2)+2
+31
+sinh2[αt]/bracerightbigg−1/bracketleftbigg/parenleftbigg
+−27sinh[αt]
+cosh3[αt]+36tanh[ αt]/parenrightbiggκ2
+α3K(2)′
++/parenleftBigg
+72−361
+cosh2[αt]−181
+cosh4[αt]−12×1.86×10−41
+sinh8
+3[αt]/parenrightBigg
+κ2
+α2K(2)
+−6×1.86×10−4 1
+cosh2[αt]sinh8
+3[αt]−3
+21
+cosh4[αt]sinh2[αt]
++31
+sinh2[αt]cosh2[αt]+4×1.86×10−41
+sinh8
+3[αt]/bracketrightBigg
+. (61)
+Note that 1<cosh[αt]≤2and0<sinh[αt]≤1.7in evolution of the universe, For example, if we consider the case
+thatK(2)is constant, then the trace of the matrix is always negative a nd the determinant is positive when K(2)≥0
+since72κ2
+α2K(2)and3/(sinh2[αt]cosh2[αt])are dominant terms in Eq.(61). Therefore even if K(2)is constant, the
+fixed point solution mimicing ΛCDM solution in the Einstein gravity becomes stable as long a sK(2)≥0.
+IV. TACHYON MODEL AS A GENERALIZED K-ESSENCE MODEL
+We now consider the tachyon model [9–12] as a variation of the generalized k-essence model.8
+The action of the tachyon model is given by
+SBI=τ3/integraldisplay
+d4x√−gV(T)/radicalBig
+1+l2sf(T)∂µT∂µT . (62)
+Herelsis the string scale and τ3is defined by using the string coupling constant gsasτ3≡1/(8π3gsl4
+s). Since the
+scalar field Tis dimensionless, we define a new field φbyφ≡κTand redefine the functions V(T)andf(T)as
+V(T) =−v(φ)/τ3andf(T) = ˜ω(φ)κ2/l2
+s. Then the action (62) can be rewritten as
+SBI=−/integraldisplay
+d4x√−gv(φ)/radicalBig
+1+ ˜ω(φ)∂µφ∂µφ. (63)
+We now consider the FRW background (2). Since the redefinitio n of the scalar field φcan be absorbed into the
+redefinition of ˜ω(φ), later we may identify φwith the cosmological time t. Then by expanding the square root in the
+action (62) by the power of 1+∂µφ∂µφ∼0, we find,
+/radicalbig
+1+ ˜ω(φ)X=/radicalbig
+1−˜ω(φ)+ ˜ω(φ)(X+1)
+=/radicalbig
+1−˜ω(φ)+˜ω(φ)
+2/radicalbig
+1−˜ω(φ)(X+1)−˜ω2(φ)
+8(1−˜ω(φ))3
+2(X+1)2+O/parenleftBig
+(X+1)3/parenrightBig
+, (64)
+whereX≡∂µφ∂µφ. Then by comparing SBIin (63) with Eq.(43) and Eq.(46), we can identify
+v(φ) =/braceleftBigg
+−1
+κ2/parenleftbig
+2g′′(φ)+3g′(φ)2/parenrightbig
+−/summationdisplay
+iwiρ0ia−3(1+wi)
+0e−3(1+wi)g(φ)/bracerightBigg1
+2
+×/braceleftBigg
+−3
+κ2g′(φ)2+/summationdisplay
+iρ0ia−3(1+wi)
+0e−3(1+wi)g(φ)/bracerightBigg1
+2
+, (65)
+˜ω(φ) =2
+κ2g′′(φ)+/summationtext
+i(1+wi)ρ0ia−3(1+wi)
+0e−3(1+wi)g(φ)
+−3
+κ2g′(φ)2+/summationtext
+iρ0ia−3(1+wi)
+0e−3(1+wi)g(φ), (66)
+K(2)(φ) =−1
+8˜ω2(φ)
+(1−˜ω(φ))3
+2v(φ)
+=1
+8/braceleftBig
+2
+κ2g′′(φ)+/summationtext
+i(1+wi)ρ0ia−3(1+wi)
+0e−3(1+wi)g(φ)/bracerightBig2
+1
+κ2(2g′′(φ)+3g′(φ)2)+/summationtext
+iwiρ0ia−3(1+wi)
+0e−3(1+wi)g(φ). (67)
+If we consider the case that the action SBIreproduces the ΛCDM behavior in the Einstein gravity, by using Eqs.(56)-
+(59), we find
+K(2)(t)∼α2
+6κ21
+−4sinh2[αt]+4cosh2[αt]sinh2[αt]+1.86×10−4sinh4
+3[αt]>0, (68)
+K(2)′(t)∼ −α3
+κ22cosh[αt]
+9sinh3[αt]/braceleftbigg
+−4+4cosh2[αt]+1.86×10−41
+sinh2
+3[αt]/bracerightbigg2
+×/parenleftBigg
+−12+12cosh2[αt]+1.86×10−41
+sinh2
+3[αt]/parenrightBigg
+<0. (69)
+Then we can find the stability of the fixed point corresponding to theΛCDM model (18) by substituting the equations
+in (68) into Eq.(60) and (61). Since the expression is, howev er, very complicated, it is difficult to check the signs of the
+determinant and the trace analytically. Numerical calcula tion, however, tells that both of the determinant and the
+trace of the matrix are negative and therefore the fixed point solution corresponding to the ΛCDM model is unstable.
+So this model cannot reproduce the stable evolution of exact ΛCDM model.9
+V. SUMMARY
+In this paper, we consider the reconstruction of k-essence m odel which includes scalar quintessence model, tachyon
+dark energy model, ghost condensation model as special case s. Explicit formulas of the reconstruction were given.
+First we considered the case that the action only contains th e kinetic term. In this case, we find that the model
+reproducing the development of the universe in the exact ΛCDM model cannot be constructed although there is a
+model which reproduce the development infinitely closing to that ofΛCDM model. We find, however, the solution is
+not stable.
+Another is more general case including potential etc., wher e we find that there appear infinite number of functions
+of the scalar fields, which are denoted by K(n)(φ)(n= 2,3,4,···) and redundant to the time development of the scale
+factor. Although K(2)(φ)is also redundant to the time development of the scale factor , we found that by adjusting
+K(2)(φ), we can obtain the model where any solution we need could be st able. At present, the roles of the other
+functions K(n)(φ)(n= 3,4,···) are not clear but we expect that we may obtain a model satisfy ing other constraints
+from cosmology by adjusting these functions.
+Acknowledgments
+We are grateful to S. D. Odintsov for very helpful discussion s. The work by S.N. is supported by Global COE
+Program of Nagoya University provided by the Japan Society f or the Promotion of Science (G07).
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+[3] S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J.517, 565 (1999); A. G. Riess et al.
+[Supernova Search Team Collaboration], Astron. J. 116, 1009 (1998); P. Astier et al. [The SNLS Collaboration], Astron.
+Astrophys. 447, 31 (2006); A. G. Riess et al., Astrophys. J. 659, 98 (2007).
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+arXiv:1001.0221v3  [hep-ph]  1 Dec 2010Triple-Top Signal of New Physics at the LHC
+Vernon Barger1, Wai-Yee Keung2, Brian Yencho1
+1Department of Physics,
+University of Wisconsin,
+Madison, WI 53706
+2Department of Physics,
+University of Illinois, Chicago, IL 60607
+Abstract
+We present leading-order (LO) cross sections for the produc tion of three top quarks ( tt¯t,t¯t¯t) at the
+Large Hadron Collider (LHC). We find a total LO cross section f or triple-top production in the
+Standard Model of σ≈2 fb at√s= 14 TeV and we give examples of two new physics models
+which have a significant enhancement to this channel. In the M inimal Supersymmetric Standard
+Model (MSSM), there are regions of parameter space where the decays of gluino pairs into final
+states including three tops has a cross section σ≈41 fb. In a leptophobic Z′model featuring
+right-handed couplings of the u-quark to the top, we find σ≈28 fb. With efficient identification
+and reconstruction of the top quarks, the triple-top signal could potentially provide evidence for
+new physics at the LHC.
+1I. INTRODUCTION
+The top quark, with a measured mass of mt= 171.3±1.1±1.2 GeV [1], is the most
+massive of the elementary particles. It is this distinction that makes the top quark so
+interesting to phenomenology as it is typically the most closely related to proposals of new
+physics beyond the Standard Model (SM).
+As a leading potential mode of new physics discovery, top-pair prod uction has been
+studied extensively forboththeFermilabTevatronandforthe Lar geHadronCollider (LHC)
+(see [2] and [3] and references therein for a thorough review of LH C top physics). Single top
+production is also of interest, particularly for the Vtbmixing element of the SM, and was
+recently observed at the Tevatron [4]. In this paper, we discuss th e observation of triple-top
+events at the LHC and how this channel can be significantly affected by new physics.
+In Section II, we evaluate the leading-order (LO) SM production cr oss sections at the
+LHC for up to four tops while highlighting the triple-top signal. We then describe two new
+physics models in Section III and discuss the results in Section IV.
+II. STANDARD MODEL TOP PRODUCTION
+In the Standard Model, triple-top production, like single-top produ ction, comes from
+threedistinctprocesses atLO: pp→3t+W±,3t+b,3t+jets, where tgenericallydenotestop
+and anti-top quarks. The typically dominant diagram from each of th ese processes is given
+in Figures 1(a)-(c) and a summary of the total cross sections for each, calculated from all
+contributing diagrams, is given in Table I at three LHC center-of-ma ss energies,√s= 7,10,
+and14TeV.Wecalculateallcrosssectionstoanaccuracyof1%usin gMadGraph/MadEvent
+version 4.4.24 [5, 6], allowing the renormalization and factorization sca les to vary with the
+process and using the default values on all cuts except ηmaxof the jets, which we set to 4.0,
+assuming forward-jet tagging at ATLAS and CMS [7, 8]. We choose mt= 171.4 GeV and a
+Higgs boson mass of mh= 130 GeV.
+Unlike top-pair and even four-top production1, which are dominated at the LHC by the
+strong coupling gg→t¯tprocess, the production of an odd number of tops requires a Wtb
+1The productionoffour-heavy-quarkfinal statesat hadroncollid ers, including t¯tt¯tat supercolliderenergies,
+was first calculated in [9].
+2/A1/CV
+/CQ/AM/D8
+/D8/D8
+/CF/A1
+/CF/CF/CI/D9
+/CQ/CS
+/AM/D8/D8
+/D8(a) (b)/A1
+/CS/CF
+/CV/D9
+/CS/AM/CQ
+/D8/D8
+/AM/D8
+(c)
+FIG. 1: Leading diagrams contributing to SM triple-top prod uction at the LHC with√s= 14 TeV
+corresponding to (a) 3 t+W±(b) 3t+jets (c) 3 t+b. The total number of uniquediagrams (not
+summing over quark flavors or distinguishing between order o f initial state partons) contributing
+to each process are (a) 236 (b) 144 and (c) 72. All of these are i ncluded in our calculations.
+vertex in every diagram and often involves a b-quark in the initial state of the hard process,
+both of which result in a significant suppression compared to the str ong processes. Table I
+compares the total LO cross sections for 1,2,3, and 4 top product ion for the three center-of-
+mass energies. Figure 2 plots this information while summing over all po ssible channels and,
+inthecaseofsingle- andtriple-tops, thecontributions fromboth t/¯tandtt¯t/t¯t¯t, respectively.
+At the design LHC energy of√s= 14 TeV, the triple-top cross section, with σ= 1.9 fb, is
+five orders of magnitude less than that of top-pair production, th e dominant mode of top
+creation at the LHC. This relatively low SM production is precisely what makes three tops
+an interesting channel for investigating new physics, as there may be possibilities for notable
+enhancements above the SM.
+3Inclusive Cross Sections at the LHC (fb)
+Process√s= 7 TeV 10 TeV 14 TeV
+pp→1t(Total) 66×103138×103258×103
+pp→t+... 42×10385×103154×103
+pp→tW−5.4×10314×10332×103
+pp→tj 35×10368×103117×103
+pp→t¯b1.8×1033.1×1034.8×103
+pp→¯t+... 24×10353×103104×103
+pp→¯tW+5.4×10314×10332×103
+pp→¯tj 18×10337×10369×103
+pp→¯tb 0.99×1031.8×1033.0×103
+pp→t¯t 98×103255×103581×103
+pp→3t(Total) 0.11 0.52 1.9
+pp→tt¯t+... 0.072 0.31 1.1
+pp→tt¯tW−0.027 0.16 0.69
+pp→tt¯tj 0.024 0.091 0.27
+pp→tt¯t¯b0.021 0.054 0.11
+pp→t¯t¯t+... 0.042 0.21 0.83
+pp→t¯t¯tW+0.028 0.16 0.68
+pp→t¯t¯tj 0.008 0.033 0.10
+pp→t¯t¯tb0.0065 0.019 0.045
+pp→tt¯t¯t 0.53 2.7 11
+TABLE I: LO inclusive cross sections for multi-top producti on in the SM for three different center-
+of-mass energies at the LHC.
+III. NEW PHYSICS MODELS
+One of the most studied and believed by many to be the most likely cand idate of physics
+beyond the Standard Model is supersymmetry. It successfully st abilizes the quantum cor-
+rections to the Higgs mass, provides a suitable dark matter candida te, is consistent with
+47 10 14
+Ecm (TeV)10-2100102104106σ (fb)2 tops
+1 top
+4 tops
+3 tops
+FIG. 2: Inclusive LO cross sections for multi-top productio n in the Standard Model with mH=
+130 GeV for three LHC center-of-mass energies. Single- and t riple-top curves represent the sum
+over all contributing final states, which are given in Table I .
+gauge coupling unification, and is characterized by a space-time sym metry that is a natural
+extension of the Poincar´ e group. The generic softly-broken sup ersymmetric model with the
+least number of new particles is the Minimal Supersymmetric Standar d Model (MSSM). As
+the MSSM makes no assumptions on the precise method of supersym metry breaking and
+instead includes all allowed soft SUSY-breaking terms, many of the p arameters defining the
+masses, mixings, and couplings of the new particles are left undeter mined, introducing over
+100 arbitrary parameters.
+The phenomenology of the MSSM is dependent on the choice of these parameters. Spec-
+ifying the method of SUSY breaking, or choosing parameters with re lationships motivated
+by a particular preference, makes the problem of quantifying MSSM expectations more
+tractable. The “Snowmass Points and Slopes” (SPS) are a collection of benchmarks derived
+from different SUSY-breaking methods that all provide viable, and y et markedly different,
+phenomenological behavior [10].
+The parameter point of interest to the current study is SPS 2. This is in the so called
+“focus point” region of SUSY parameter space [11] and is defined in the mSUGRA model
+5by the point:
+m0= 1450 GeV , m1/2= 300 GeV , A0= 0,tanβ= 10, µ >0 (1)
+wherem0is the universal scalar mass, m1/2is the universal gaugino mass, A0is the common
+trilinearcoupling, tan βistheratiooftheHiggsVEVs, and µistheHiggsinomassparameter,
+all of which are evaluated at MGUT∼1016GeV and run down with the renormalization
+group equations to determine all the low-scale parameters of the t heory. A feature of this
+benchmark point is that the gluino mass, m˜g= 782 GeV, is less that that of the squarks.
+Gluino decays to quark-squark pairs (˜ g→q¯˜qL,R,¯q˜qL,R) are kinematically inaccessible,
+opening up three-body decays via virtual squarks:
+˜g→χ0
+iq¯q
+˜g→χ±
+iq¯q′ (2)
+These decays can result in both single- and double-top final states with considerable branch-
+ing fractions of ≈10%−20% when summing over all neutralinos and charginos, an enhance-
+ment that has been studied for similar regions of parameter space [1 2–14]. These branching
+fractions are given in Table II.
+Because gluinos are strongly interacting, the production of gluino p airs at the LHC will
+be very large, typically several hundred fb. With one gluino decaying to a single top via
+˜g→t¯bχ−
+ior¯tbχ+
+iand the other to a top pair through ˜ g→χ0
+it¯t, the rate of triple-top events
+can therefore be quite large in the MSSM in this region of parameter s pace. The diagram
+for the leading contribution is given in Figure 3.
+The major limitations to the rate of triple-top production in the SM, a s discussed in
+Section II, are the Wtbvertex and the initial state b-quark needed to create the single top
+(though the latter is not involved for pp→tt¯t¯b). Models that can produce this single top
+with greater ease should therefore have a larger triple-top signal as well. An example of one
+such model is the addition of a U(1)′symmetry in which a leptophobic Z′couples the top
+directly to the u-quark. The additional interaction terms of the Lagrangian are giv en by
+L⊃(gXZ′
+µ¯uγµPRt+h.c.)+ǫUgXZ′
+µ¯uiγµPRui (3)
+6Gluino Branching Fractions
+˜g→1t+... 0.21
+˜g→t¯bχ−
+10.080
+˜g→¯tbχ+
+10.080
+˜g→t¯bχ−
+20.024
+˜g→¯tbχ+
+20.024
+˜g→2t+... 0.11
+˜g→t¯tχ0
+10.099
+˜g→t¯tχ0
+20.012
+˜g→t¯tχ0
+30
+˜g→t¯tχ0
+40
+TABLE II: Branching fractions for gluino decay modes to tops in the focus point region of the
+MSSM (SPS 2). Here the gluino has a mass of m˜g= 782 GeV and a total decay width of
+Γ˜g= 2.6 MeV.
+whereǫ <1 and the diagonal term is summed over generations. This model was discussed
+in [15] as a candidate for describing the observed 2 σdeviation from the SM prediction of
+forward-backward asymmetry in the top-pair signal at the CDF de tector of the Tevatron
+collider [16]. The small diagonal couplings characterized by the param eterǫUexist only
+to escape bounds on like-sign top quark events from the decay of t woZ′s by forcing the
+dominant decay Z′→u¯u. This study found the best match to the asymmetry and to the
+invariant mass distribution of the top pair with MZ′= 160 GeV and αX= 0.024, with
+any small ǫU/negationslash= 0 giving comparable results. We take these values and choose ǫU= 0.1.
+The dominant process for triple-top production in this model (for s mallǫU) is the t-channel
+exchange of the Z′, shown in Figure 4. This diagram illustrates the unique topology of the se
+events in this Z′model, with the three tops produced at LO.
+IV. RESULTS AND CONCLUSIONS
+The LO triple-top production cross sections for the two new physic s models discussed
+are calculated with MadGraph according to the prescriptions in Sect ion II. The Z′model is
+7/A1
+/DI /CV
+/DI /CV/DI /CV/DI/D8/CX
+/DI/D8/CX/CV/CVꜸ
+/BC/CX
+/D8
+/AM/D8/D8
+/AM/CQ
+Ꜹ
+/A0/CXFIG. 3: Leading diagram contributing to triple-top product ion in the focus point region of the
+MSSM through the decay of two gluinos. There are a total of 8 uniquediagrams (not summing
+over quark flavors or distinguishing between the order of the initial state partons) contributing to
+gluino pair production. We include all of these in our calcul ations./A1
+/CI
+/BC/CV
+/D9/AM/D8
+/D8
+/D8
+FIG. 4: Leading diagram contributing to triple-top product ion in a leptophobic Z′in which the top
+quark can couple directly to a u-quark. There are a total of 16 uniquediagrams (not distinguishing
+between the order of the initial state partons) involved in t his process. We include all of these in
+our calculations.
+implemented with the “usermod” format and the cross section for t he three-top final state
+is found directly. For the MSSM we use the standard MadGraph MSSM implementation
+and the parameter card for SPS 2 from the MadGraph webpage2. We calculate the cross
+section for the on-shell gluino pair and then determine the triple-to p cross section using the
+branching fractions in Table II. The results for three different LHC running energies are
+given in Table III and the sum of all processes for each model is plott ed with the SM results
+in Figure 5(a).
+At each energy studied, the new physics models lie approximately an o rder of magnitude
+2It should be noted that while the current value of the top mass, mt= 171.4 GeV, was used for the SM
+andZ′model calculations, the MSSM at SPS 2 uses mt= 175 GeV.
+8Inclusive Cross Sections at the LHC (fb)
+pp→3t(Total)√s= 7 TeV 10 TeV14 TeV
+Z′model 4.0 1228
+pp→tt¯t 3.8 1126
+pp→t¯t¯t 0.20 0.752.2
+MSSM 0.97 7.941
+pp→tt¯t+¯bχ−
+iχ0
+j0.49 4.021
+pp→t¯t¯t+bχ+
+iχ0
+j0.49 4.021
+TABLE III: LO inclusive cross sections for triple-top produ ction at the LHC for three different
+center-of-mass energies. In MSSM, the final state neutralin os and charginos have been summed
+over.
+above the SM prediction. Each model will also involve different topolog ies and kinematics
+which may allow them to be distinguished with this signal, given enough da ta. The SM
+triple-top events will often be associated with an extra W-boson or extra tagged b-quark,
+while the MSSM events would feature large /negationslashETdue to the neutralinos escaping the detector.
+TheZ′model would best be characterized by the absence of these additio nal states and may
+also be distinguished by the presense of a broad pseudo-rapidity dis tribution of one top due
+to the t-channel exchange of the Z′-boson3
+However, despite the large increase in cross section at all energies and distinct topologies,
+the actual detection of the triple-top signal for the models discus sed here would likely not
+be possible for the initial run of the LHC at 7 TeV center-of-mass en ergy and with 1 fb−1
+of integrated luminosity. Indeed, the given BSM models may be discov ered through other
+signals before triple-top events can even be identified. For example , in theZ′model, there
+are large cross sections for same-sign top production at 7 TeV ( ∼50 pb at LO) that would
+likely lead to discovery, and the MSSM could possibly be inferred at the se lower energies
+from generic missing energy searches. If supersymmetry has elud ed detection during this
+initial run, however, the triple-top signal could prove to be an impor tant part of detection
+3When the tops are ordered by pT, we find that the low- pTt(as opposed to the ¯t) in thett¯tevents occurs
+with|η|>2 in≈60% of the events in the Z′model, compared to ≈30% in the SM and ≈16% in the
+focus point region of the MSSM. This was based on an analysis of 10,00 0 unweighted events.
+97 10 14
+Ecm (TeV)10-210-1100101102σ (fb)
+SMMSSM (FP)Z´
+7 10 14
+Ecm (TeV)10-210-1100101102σ (fb)Z´
+3 tops
+4 topsMSSM (FP)
+SM
+(a) (b)
+FIG. 5: (a) Inclusive LO cross sections for triple-top produ ction at three LHC center-of-mass
+energies in the Standard Model, the Z′model, and the focus point region of the MSSM through
+the decay of two gluinos. Each curve represents the sum over a ll contributing final states, which
+are given in Tables I and III. In (b), these same curves are plo tted (solid lines) along with the
+corresponding four-top cross sections of each model (dotte d lines), which are similarly summed
+over all contributing final states.
+at 14 TeV. Indeed, the branching fractions of light gluinos to multi-t op final states can be
+made much larger than those presented here if the 1st and 2nd gen eration squarks are much
+more massive than those of the 3rd, providing a significant boost to the triple-top cross
+section. The discovery potential of the MSSM using multi-top signals in such a scenario is
+discussed in detail in [14].
+In addition to providing evidence for physics beyond the SM, knowled ge of the triple-
+top cross sections can also aid in the understanding of the underlyin g models. In the Z′
+model, while the two- and four-top production cross sections are e ssentially independent of
+the parameter ǫU, every diagram contributing to the production of three tops has a linear
+dependence on this parameter and therefore the total cross se ction goes as ǫ2
+U. Knowledge
+of the triple-top signal in addition to the two- and four-top signals c ould therefore allow ǫU
+to be determined. In the focus-point region of the MSSM, the relat ive sizes of the various
+multi-top signals are determined by the relative branching fractions of the gluinos to one-
+and two-top final states. In certain scenarios, the ratios of the se branching fractions can
+provide insight into the 3rd generation squark mass parameters an d the composition of the
+lightest neutralino, whiletheoverall ratesandtotaleffective mass distributions canconstrain
+10the gluino mass itself [14].
+An important question to then consider is whether a BSM triple-top s ignal could be faked
+by SM four-top production with one top going unidentified. In Figure 5(b), we compare
+the triple-top cross sections predicted by each model with the cor responding four-top cross
+sections calculated at LO. Both the Z′and MSSM triple-top cross sections lie well above
+the SM four-top cross sections at all energies. This reduces the c hance that a SM four-top
+signal could fake the larger BSM triple-top one. Each model also lies a bove its own four-top
+cross sections, which are themselves enhanced from the SM predic tion. It should be noted,
+however, that these conclusions are not generic features of the models; slightly different
+values of ǫUand the gluino branching fractions can flip the relative importance of the three-
+and four-top channels. It should be understood, therefore, th at the triple-top signal is most
+valuable in its relation to top signals of other multiplicities.
+Means of distinguishing between two-, three-, and four-top final states are then critical
+to properly utilizing these signals. Direct reconstruction of every t op quark in a multi-top
+event, after including detector cuts, background, and showerin g, can be extremely difficult
+due to large combinatorics [14]. Other methods have been developed for tagging high- pT
+top quarks at the LHC by investigating the jet substructure of th e decay products [17–21].
+These studies have focused on top-pair signals but perhaps could fi nd applicability to three-
+and four-top events as well. While there are certainly practical cha llenges involved, the
+results presented here indicate that new physics may be inferred f rom the triple-top signal
+and therefore warrant further, more detailed simulations.
+V. ACKNOWLEDGEMENTS
+The authors would like to thank the referee for valuable comments t hat have contributed
+to quality of this manuscript and also Y. Gao, I. Lewis, and M. McCask ey for helpful
+discussions. This work was supported in part by the U.S. Departmen t of Energy under
+grants Nos. DE-FG02-95ER40896 and DE-FG02-84ER40173 and in part by the Wisconsin
+Alumni Research Foundation.
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+13
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+arXiv:1001.0222v1  [hep-th]  1 Jan 2010Casimir Energy of the Universe and New
+Regularization of Higher Dimensional Quantum Field
+Theories
+Shoichi Ichinose
+Laboratory of Physics, School of Food and Nutritional Scien ces, University of Shizuoka, Yada
+52-1, Shizuoka 422-8526, Japan
+E-mail:ichinose@u-shizuoka-ken.ac.jp
+Abstract. Casimir energy is calculated for the 5D electromagnetism an d 5D scalar theory
+in thewarpedgeometry. It is compared with the flat case. A new regularizat ion, called
+sphere lattice regularization , is taken. In the integration over the 5D space, we introduce two
+boundary curves (IR-surface and UV-surface) based on the minimal area principle . It is a
+directrealization of the geometrical approach to the renormalization group . The regularized
+configuration is closed-string like . We do nottake the KK-expansion approach. Instead,
+the position/momentum propagator is exploited, combined w ith theheat-kernel method . All
+expressions are closed-form (not KK-expanded form). The generalized P/M propagators are
+introduced. We numerically evaluate Λ(4D UV-cutoff), ω(5D bulk curvature, warp parameter)
+andT(extra space IR parameter) dependence of the Casimir energy . We present two new ideas
+in order to define the 5D QFT: 1) the summation (integral) regi on over the 5D space is restricted
+by two minimal surfaces (IR-surface, UV-surface) ; or 2) we i ntroduce a weight function and
+require the dominant contribution, in the summation, is giv en by the minimal surface . Based
+on these, 5D Casimir energy is finitelyobtained after the proper renormalization procedure. The
+warp parameter ωsuffers from the renormalization effect . The IR parameter Tdoes not. We
+examine the meaning of the weight function and finally reach a new definition of the Casimir
+energy where the 4D momenta( or coordinates) are quantized with the extra coordinate as the
+Euclidean time (inverse temperature). We examine the cosmo logical constant problem and
+present an answer at the end. Dirac’s large number naturally appears.
+1. Introduction
+In the dawn of the quantum theory, the divergence problem of the specific heat of the radiation
+cavity was the biggest one (the problem of the blackbody radi ation). It is historically so famous
+that the difficulty was solved by Planck’s idea that the energy is quantized. In other words,
+the phase space of the photon field dynamics is not continuous but has the ”cell” or ”lattice”
+structure with the unit area (∆ x·∆p) of the size 2 π¯h(Planck constant). The radiation energy
+is composed of two parts, ECasandEβ:
+E4dEM=ECas+Eβ, ECas=/summationdisplay
+mx,my,n∈Z˜ωmxmyn, Eβ= 2/summationdisplay
+mx,my,n∈Z˜ωmxmyn
+eβ˜ωmxmyn−1,
+˜ω2
+mxmyn= (mxπ
+L)2+(myπ
+L)2+(nπ
+l)2, l≪L , (1)where the parameter βis the inverse temperature, lis the separation length between two
+perfectly-conducting plates, and Lis the IR regularization parameter of the plate-size. The
+second part Eβis, essentially, Planck’s radiation formula. The first one ECasis the vaccuum
+energy of the radiation field, that is, the Casimir energy. It is a very delicate quantity. The
+quantity is formally divergent , hence it must be defined with careful regularization .ECas/(2L)2
+does depend only on the boudary parameterl. The quantity is a quantum effect and , at the
+same time, depends on the global(macro) parameter l.
+ECas
+(2L)2=π2
+(2l)3B4
+4!=−π2
+7201
+(2l)3, B4(the fourth Bernoulli number) = −1
+30,(2)
+In Fig.1, Planck’s radiation spectrum distribution is show n. Introducing the axis of the
+1
+1.2
+1.4
+1.6
+1.8
+2be246810
+k00.050.1PL
+1
+1.2
+1.4
+1.6
+1.8
+2be
+Figure 1. Graph of Planck’s radiation
+formula. P(β,k) =1
+(c¯h)31
+π2k3/(eβk−1) (1≤
+β≤2,0.01≤k≤10).0.2
+0.4
+0.6
+0.85000100001500020000
+-1.475-1.45-1.425-1.4
+0.2
+0.4
+0.6
+0.8
+Figure 2. Behavior of ln |1
+2F−(˜k,z)|=
+ln|˜k G−
+k(z,z)/(ωz)3|.ω= 104,T= 1,Λ =
+2×104. 1.0001/ω≤z≤0.9999/T. ΛT/ω≤
+˜k≤Λ. Note ln |(1/2)×(1/2)| ≈ −1.39.
+inversetemperature( β), besidesthephotonenergyorfrequency( k), itisshownstereographically.
+Although we will examine the 5D version of the zero-point par t (the Casimir energy), the
+calculated quantities in this paper are much more related to this Planck’s formula. We see,
+near theβ-axis, a sharply-rising surface, which is the Rayleigh-Jea ns region (the energy density
+is proportional to the squareof the photon frequency). The damping region in high kis the
+Wien’s region.1The ridge (the line of peaks at each β) forms the hyperbolic curve (Wien’s
+displacement law). When we will, in this paper, deal with the energy distribution over the 4D
+momentum and the extra-coordinate, we will see the similar b ehavior (although top and bottom
+appear in the opposite way).
+In the quest for the unified theory, the higher dimensional (H D) approach is a fascinating one
+fromthegeometrical point. Historically theinitial succe ssfuloneis theKaluza-Klein model[1, 2],
+which unifies the photon, graviton and dilaton from the 5D spa ce-time approach. The HD
+theories , however, generally have the serious defect as the quantum field theory(QFT) : un-
+renormalizability. The HD quantum field theories, at presen t, are not defined within the QFT.
+One can take the standpoint that the more fundamental formul ation, such as the string theory
+1We recall that the old problem of the divergent specific heat was solved by the Wien’s formula. This fact
+strongly supports the present idea of introducing the weight function (see Sec.6).and D-brane theory, can solve the problem. In the present pap er, we have the new standpoint
+that the HD theories should be defined by themselves within th e QFT. In order to escape the
+dimension requirement D=10 or 26 from the quantum consisten cy (anomaly cancellation)[3], we
+treat the gravitational (metric) field only as the backgroun d one. This does notmean the space-
+time is not quantized. See later discussions (Sec.7). We pre sent a way to define 5D quantum
+field theory through the analysis of the Casimir energy of 5D e lectromagnetism.
+In 1983, the Casimir energy in the Kaluza-Klein theory was ca lculated by Appelquist and
+Chodos[4]. They took the cut-off (Λ) regularization and foun d the quintic (Λ5) divergence and
+the finite term. The divergent term shows the unrenormalizability of the 5D theory, but the
+finite term looks meaningful2and, in fact, is widely regarded as the right vacuum energy wh ich
+showscontraction of the extra axis.
+In thedevelopment of the string and D-brane theories, a new a pproach to the renormalization
+group was found. It is called holographic renormalization [6, 7, 8, 9, 10, 11]. We regard the
+renormalization flow as a curve in the bulk (HD space). The flow goes along the extra axis. The
+curve is derived as a dynamical equation such as Hamilton-Ja cobi equation. It originated from
+the AdS/CFT correspondence[12, 13, 14]. Spiritually the pr esent basic idea overlaps with this
+approach. The characteristic points of this paper are: a) We donotrely on the 5D supergravity;
+b) We do notquantize the gravitational(metric) field; c) The divergenc e problem is solved by
+reducing the degree of freedom of the system, where we requir e, not higher symmetries, but
+some restriction based on the minimal area principle ; d) No local counterterms are necessary. @
+In the previous paper[15], we investigated the 5D electroma gnetism in the flatgeometry. The
+results show the renormalization of the compactification size l.
+EW
+Cas/Λl=−α
+l4(1−4cln(lΛ)) =−α
+l′4, β=∂
+∂(lnΛ)lnl′
+l=c , (3)
+whereαandcare some numbers. They are, at present, not fixed, but are nume rically obtained
+depending on the weight function W. The aim of this paper is to examine how the above results
+change for the 5D warped geometry case. One additional massi ve parameter, that is, the warp
+(bulk curvature) parameter ωappears. This introduction of the ”thickness” 1 /ωcomes from
+the expectation that it softens the UV-singularity, which i s the same situation as in the string
+theory.
+2. Kaluza-Klein expansion approach
+In order to analyze the 5D EM-theory, we start with 5D massivevector theory.
+S5dV=/integraldisplay
+d4xdz√
+−G(−1
+4FMNFMN−1
+2m2AMAM), FMN=∂MAN−∂NAM,
+ds2=1
+ω2z2(ηµνdxµdxν+dz2) =GMNdXMdXN, G≡detGAB.(4)
+The 5D vector mass, m, is regarded as a IR-regularization parameter. In the limit ,m= 0,
+the above one has the 5D local-gauge symmetry. Casimir energ y is given by some integral
+where the (modified) Bessel functions, with the index ν=/radicalBig
+1+m2
+ω2, appear. Hence the
+5D EM limit is given by ν= 1 (m= 0). We consider, however, the imaginary mass case
+m=iω(m2=−ω2, ν= 0) mainly for the simplicity. We can simplify the model furt hermore.
+Instead of analyzing the m2=−ω2of the massive vector (4), we take the 5D massive scalar
+theory on AdS 5withm2=−4ω2, ν=/radicalbig
+4+m2/ω2= 0.
+L=√
+−G(−1
+2∇AΦ∇AΦ−1
+2m2Φ2), ds2=GABdXAdXB,∇A∇AΦ−m2Φ+J= 0,(5)
+2The gauge independence was confirmed in Ref.[5].where Φ(X) = Φ(xa,z) is the 5D scalar field. The integral region is given by
+−1
+T≤z≤ −1
+ωor1
+ω≤z≤1
+T(−l≤y≤l ,|z|=1
+ωeω|y|),1
+T≡1
+ωeωl,(6)
+where we take into account Z2symmetry: z↔ −z.ωis the bulk curvature (AdS 5parameter)
+andT−1is the size of the extra space (Infrared parameter). In this s ection, we present the
+standard expressions, that is, those obtained by the Kaluza -Klein expansion. The Casimir
+energyECasis given by
+e−T−4ECas=/integraldisplay
+DΦexp{i/integraldisplay
+d5XL}
+=/integraldisplay
+DΦp(z)exp/bracketleftBigg
+i/integraldisplayd4p
+(2π)42/integraldisplay1/T
+1/ωdz/braceleftbigg1
+2Φp(z)s(z)(s(z)−1ˆLz−p2)Φp(z)/bracerightbigg/bracketrightBigg
+=/integraldisplay/productdisplay
+ndcn(p)exp/bracketleftBigg/integraldisplayd4pE
+(2π)4/summationdisplay
+n{−1
+2cn(p)2(p2
+E+M2
+n)}/bracketrightBigg
+= exp/summationdisplay
+n,p{−1
+2ln(p2
+E+M2
+n)},(7)
+whereΦ p(z) is thepartially(4D world only)-Fourier-transformed one of Φ(X). Φp(z) is expressed
+in the expansion form using the eigen functions, ψn(z), of this AdS 5system.
+Φp(z) =/summationdisplay
+ncn(p)ψn(z),{s(z)−1ˆLz+Mn2}ψn(z) = 0,ˆLz≡d
+dz1
+(ωz)3d
+dz−m2
+(ωz)5,
+ψn(z) =−ψn(−z) forP=−;ψn(z) =ψn(−z) forP= +.(8)
+wheres(z) =1
+(ωz)3. The expression (7) is the familiar one of the Casimir energy .
+3. Heat-Kernel Approach and Position/Momentum Propagator
+Eq.(7) is the expression of ECasby the KK-expansion. In this section, the same quantity is
+re-expressed in a closedform using the heat-kernel method and the P/M propagator. Fi rst we
+can express it, using the heat equation solution, as follows .
+e−T−4ECas= (const) ×exp/bracketleftBigg
+T−4/integraldisplayd4p
+(2π)42/integraldisplay∞
+01
+2dt
+tTrHp(z,z′;t)/bracketrightBigg
+,
+TrHp(z,z′;t) =/integraldisplay1/T
+1/ωs(z)Hp(z,z;t)dz ,{∂
+∂t−(s−1ˆLz−p2)}Hp(z,z′;t) = 0.(9)
+The heat kernel Hp(z,z′;t) is formally solved, using the Dirac’s bra and ket vectors ( z|,|z), as
+Hp(z,z′;t) = (z|e−(−s−1ˆLz+p2)t|z′). (10)
+We here introduce the position/momentum propagators G∓
+pas follows.
+G∓
+p(z,z′)≡/integraldisplay∞
+0dt Hp(z,z′;t) =/summationdisplay
+n∈Z1
+M2n+p21
+2{ψn(z)ψn(z′)∓ψn(z)ψn(−z′)}.(11)
+They satisfy the following differential equations of propagators .
+(ˆLz−p2s(z))G∓
+p(z,z′) =/summationdisplay
+n∈Z{ψn(z)ψn(z′)∓ψn(z)ψn(−z′)}
+2=/braceleftBigg
+ǫ(z)ǫ(z′)ˆδ(|z|−|z′|) P=−1
+ˆδ(|z|−|z′|) P=1(12)Therefore the Casimir energy ECasis given by
+−E−
+Cas(ω,T) =/integraldisplayd4pE
+(2π)42/integraldisplay∞
+0dt
+t2/integraldisplay1/T
+1/ωdz s(z)HpE(z,z;t)
+=/integraldisplayd4pE
+(2π)42/integraldisplay∞
+0dt
+t2/integraldisplay1/T
+1/ωdz s(z)
+
+/summationdisplay
+n∈Ze−(M2
+n+p2
+E)tψn(z)2
+
+, (13)
+wheres(z) = 1/(ωz)3. The momentum symbol pEindicates Euclideanization. This expression
+leads to the same treatment as the previous section. Note tha t the above expression shows the
+negative definiteness ofE−
+Cas. Finally we obtain the following useful expression of the Ca simir
+energy forP=∓.
+−E∓
+Cas(ω,T) =/integraldisplayd4pE
+(2π)4/integraldisplay1/T
+1/ωdz s(z)/integraldisplay∞
+p2
+E{G∓
+k(z,z)}dk2. (14)
+The P/M propagators G∓
+pin (11) and (14) can be expressed in a closedform. Taking the
+Dirichlet condition at all fixed points, the expression for the fundame ntal region (1 /ω≤z≤
+z′≤1/T) is given by
+G∓
+p(z,z′) =∓ω3
+2z2z′2{I0(˜p
+ω)K0(˜pz)∓K0(˜p
+ω)I0(˜pz)}{I0(˜p
+T)K0(˜pz′)∓K0(˜p
+T)I0(˜pz′)}
+I0(˜p
+T)K0(˜p
+ω)−K0(˜p
+T)I0(˜p
+ω),(15)
+where ˜p≡/radicalbig
+p2,p2≥0. We can express Casimir energy in terms of the following fun ctions
+F∓(˜p,z).
+−EΛ,∓
+Cas(ω,T) =/integraldisplayd4pE
+(2π)4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+˜p≤Λ/integraldisplay1/T
+1/ωdz F∓(˜p,z),
+F∓(˜p,z)≡s(z)/integraldisplayΛ2
+p2
+E{G∓
+k(z,z)}dk2=2
+(ωz)3/integraldisplayΛ
+˜p˜k G∓
+k(z,z)d˜k≡/integraldisplayΛ
+˜pF∓(˜k,z)d˜k , (16)
+whereF∓(˜k,z) are the integrands of F∓(˜p,z) and ˜p=/radicalBig
+p2
+E. Here we introduce the UV cut-off
+parameter Λ for the 4D momentum space. In Fig.2, we show the be havior of F−(˜k,z). The
+table-shape graph says the ”Rayleigh-Jeans” dominance. Th at is, for the wide-range region
+(˜p,z) satisfying both ˜ p(z−1
+ω)≫1 and ˜p(1
+T−z)≫1,
+F−(˜p,z)≈1
+2,F+(˜p,z)≈1
+2,(˜p,z)∈ {(˜p,z)|˜p(z−1
+ω)≫1 and ˜p(1
+T−z)≫1}.(17)
+4. UV and IR Regularization Parameters and Evaluation of Casimir Energy
+The integral region of the above equation (16) is displayed i n Fig.3. In the figure, we introduce
+the regularization cut-offs for the 4D-momentum integral, µ≤˜p≤Λ. As for the extra-
+coordinate integral, it is the finite interval, 1 /ω≤z≤1/T= eωl/ω, hence we need not
+introduce further regularization parameters. For simplic ity, we take the following IR cutoff
+of 4D momentum : µ= Λ·T
+ω= Λe−ωl. Hence the new regularization parameter is Λ only.
+Let us evaluate the (Λ ,T)-regularized value of (16).
+−EΛ,∓
+Cas(ω,T) =2π2
+(2π)4/integraldisplayΛ
+µd˜p/integraldisplay1/T
+1/ωdz˜p3F∓(˜p,z), F∓(˜p,z) =2
+(ωz)3/integraldisplayΛ
+˜p˜k G∓
+k(z,z)d˜k .(18)zp = 
+1/T
+Figure 3. Space of (z,˜ p) for the integration.
+The hyperbolic curve will be used in Sec.5.0.2
+0.4
+0.6
+0.8200040006000800010000
+-2·1014-1·10140
+0.2
+0.4
+0.6
+0.8
+Figure 4. Behaviour of ( −1/2)˜p3F−(˜p,z)
+(18).T= 1,ω= 104,Λ = 104. 1.0001/ω≤
+z <0.9999/T, ΛT/ω≤˜p≤Λ.
+The integral region of (˜ p,z) is therectangle shown in Fig.3. Note that eq.(18) is the rigorous
+expression of the (Λ ,T)-regularized Casimir energy. We show the behavior of ( −1/2)˜p3F−(˜p,z)
+taking the values ω= 104,T= 1 in Fig.4(Λ = 104).3Behavior along ˜ p-axis does not so much
+depend on z. A valley runs parallel to the z-axis with the bottom line at the fixed ratio of
+˜p/Λ∼0.75. The depth of the valley is proportional to Λ4. BecauseECasis the (˜p,z) ’flat-plane’
+integral of ˜ p3F(˜p,z) , thevolumeinside the valley is the quantity ECas. Hence it is easy to
+seeECasis proportional to Λ5. This is the same situation as the flat case. Importantly, (18 )
+shows the scalingbehavior for large values of Λ and 1 /T. From a closenumerical analysis of
+(˜p,z)-integral (18), we have confirmed : Eq.(4A) EΛ,−
+Cas(ω,T) =2π2
+(2π)4×/bracketleftBig
+−0.0250Λ5
+T/bracketrightBig
+. It does not
+depend onωand has no lnΛ
+T-term. (Note: 0 .025 = 1/40.) Compared with the flat case, we see
+the factorT−1plays the role of IR parameter of the extra space. We note that the behavior of
+Fig.4is similartotheRayleigh-Jeans’s region (small mome ntumregion) of thePlanck’s radiation
+formula (Fig.1) in the sense that ˜ p3F(˜p,z)∝˜p3for ˜p≪Λ.
+Finally we notice, from the Fig.4, the approximate form of F(˜p,z) for the large Λ and 1 /T
+is given by : Eq.(4B) F∓(˜p,z)≈f
+2Λ(1−˜p
+Λ), f= 1. It does notdepend onz,ωandT.fis the
+degree of freedom. The above result is consistent with (17).
+5. UV and IR Regularization Surfaces, Principle of Minimal Area and
+Renormalization Flow
+The advantage of the new approach is that the KK-expansion is replaced by the integral of
+the extra dimensional coordinate zand all expressions are written in the closed(not expanded)
+form. The Λ5-divergence, (4A), shows the notorious problem of the highe r dimensional theories,
+as in the flat case. In spite of all efforts of the past literature , we have not succeeded in defining
+the higher-dimensional theories. (The divergence causes p roblems. The famous example is the
+divergent cosmological constant in the gravity-involving theories. [4] ) Here we notice that the
+divergence problem can be solved if we find a way to legitimately restrict the integral region in
+(˜p,z)-space.
+3The requirement for the three parameters ω,T,Λ is Λ≫ω≫T. See ref.[18] for the discussion about the
+hierarchy Λ ,ω,T.One proposal of this was presented by Randall and Schwartz[1 6]. They introduced the
+position-dependent cut-off ,µ <˜p <Λ/ωu , u ∈[1/ω,1/T] , for the 4D-momentum integral
+in the ”brane” located at z=u. See Fig.3. The total integral region is the lower part of the
+hyperbolic curve ˜p= Λ/ωz. They succeeded in obtaining the finiteβ-function of the 5D warped
+vector model. We have confirmed that the value ECasof (18), when the Randall-Schwartz
+integral region (Fig.3) is taken, is proportional to Λ5. The close numerical analysis says
+E−RS
+Cas(ω,T) =2π2
+(2π)4/integraldisplayΛ
+µdq/integraldisplayΛ/ωq
+1/ωdz q3F−(q,z) =2π2
+(2π)4/integraldisplay1/T
+1/ωdu/integraldisplayΛ/ωu
+µd˜p˜p3F−(˜p,u)
+=2π2
+(2π)4Λ5
+ω/braceleftbigg
+−1.58×10−2−1.69×10−4lnΛ
+ω/bracerightbigg
+,(19)
+which is independent ofT. This shows the divergence situation does notimprove compared
+with the non-restricted case of (4A). Tof (4A) is replaced by the warp parameter ω.This is
+contrasting with the flat case where ERS
+Cas∝ −Λ4. The UV-behavior, however, does improve if
+we can choose the parameter Λ in the way: Λ ∝ω. This fact shows the parameter ω”smoothes”
+the UV-singularity to some extent.
+Although they claim the holography is behind the procedure, the legitimateness of the
+restriction looks less obvious. We have proposed an alterna te approach and given a legitimate
+explanation within the 5D QFT[17, 19, 15, 20]. Here we closel y examine the new regularization .
+On the ”3-brane” at z= 1/ω, we introduce the IR-cutoff µ= Λ·T
+ωand the UV-cutoff Λ
+zUV
+IR’
+’
+1/T
+Figure 5. Space of (˜p,z) for the
+integration (present proposal).1/µ’1/Λ’
+1/T
+1/ωzr ( z)IR
+r ( z)UV
+r IR
+r ( z)UV(z)
+Figure 6. Regularization Surface BIRandBUVin the
+5D coordinate space ( xµ,z).
+(µ≪Λ) in the way : Eq.(5A) µ≪Λ (T≪ω). See Fig.5. This is legitimate in the sense
+that we generally do this procedure in the 4D renormalizable thoeries. (Here we are considering
+those 5D theories that are renormalizable in ”3-branes”. Examples are 5D free theories (presentmodel), 5D electromagnetism[15], 5D Φ4-theory, 5D Yang-Mills theory, e.t.c..) In the same
+reason, on the ”3-brane” at z= 1/T, we may have another set of IR and UV-cutoffs, µ′and Λ′.
+We consider the case: Eq.(5B) µ′≤Λ′,Λ′≪Λ, µ∼µ′. This case will lead us to introduce
+therenormalization flow . (See the later discussion.) We claim here, as for the regula rization
+treatment of the ”3-brane” located at other points z(1/ω < z ≤1/T), the regularization
+parameters are determined by the minimal area principle . To explain it, we move to the 5D
+coordinate space ( xµ,z). See Fig.6. The ˜ p-expression can be replaced by√xµxµ-expression by
+thereciprocal relation : Eq.(5C)/radicalBig
+xµ(z)xµ(z)≡r(z)↔1
+˜p(z). The UV and IR cutoffs change
+their values along z-axis and their trajectories make surfaces in the 5D bulk space ( xµ,z). We
+requirethe two surfaces do not cross for the purpose of the renormalization group interpretatio n
+(discussed later). We call them UV and IR regularization (or boundary) surfaces( BUV,BIR).
+Thecross sections of the regularization surfaces at zare thespheres S3with theradii rUV(z) and
+rIR(z). Here we consider the Euclidean space for simplicity. TheU V-surface is stereographically
+shown in Fig.7 and reminds us of the closed string propagation. Note that the boundary surface
+BUV(and B IR) is the 4 dimensional manifold.
+z1/T
+changing along zUV,ε
+UV, 1/TUV, z
+Figure 7. UV regularization surface ( BUV) in 5D
+coordinate space.0.2
+0.4
+0.6
+0.85000100001500020000
+-4·1014 -3·1014-2·1014-1·10140
+0.2
+0.4
+0.6
+0.8
+Figure 8. Behavior of
+(−N1/2)˜p3W1(˜p,z)F−(˜p,z)(elliptic
+suppression). Λ = 20000 , ω=
+5000, T= 1 . 1.0001/ω≤z≤
+0.9999/T, µ= ΛT/ω≤˜p≤Λ.
+The 5D volume region bounded by BUVandBIRis the integral region of the Casimir energy
+ECas. The forms of rUV(z) andrIR(z) can be determined by the minimal area principle .
+3+4
+zr′r−r′′r
+r′2+1= 0, r′≡dr
+dz, r′′≡d2r
+dz2,1/ω≤z≤1/T . (20)
+We have confirmed, by numerically solving the above differenti al eqation (Runge-Kutta), those
+curves that show the flow of renormalization really occur. Th e results imply the boundary
+conditions determine the property of the renormalization flow.
+The present regularization scheme gives the renormalization group interpretation to the
+change of physical quantities along the extra axis. See Fig. 6.4In the ”3-brane” located
+4This part is contrasting with AdS/CFT approach where the ren ormalization flow comes from the Einstein
+equation of 5D supergravity.atz, the UV-cutoff is rUV(z) and the regularization surface is the sphere S3with the radius
+rUV(z). The IR-cutoff is rIR(z) and the regularization surface is the another sphere S3with
+the radiusrIR(z). We can regard the regularization integral region as the sphere lattice of the
+following properties: a) A unit lattice (cell) is the sphere S3with radius rUV(z) and its inside; b)
+Total lattice is the sphere S3with radius rIR(z) and its inside; c) The integration region of this
+regularization is made of many cells and the total numberof c ells is const. ×/parenleftBigrIR(z)
+rUV(z)/parenrightBig4. Thetotal
+number of cells changes from (Λ
+µ)4atz= 1/ωto (Λ′
+µ′)4atz= 1/T. Along the z-axis, the number
+increases or decreases as : Eq.(5D)/parenleftBigrIR(z)
+rUV(z)/parenrightBig4≡N(z). For the ”scale” change z→z+∆z,N
+changes as : Eq.(5E) ∆(ln N) = 4∂
+∂z{ln(rIR(z)
+rUV(z))}·∆z. When the system has some coupling g(z),
+the renormalization group ˜β(g)-function (along the extra axis) is expressed as
+˜β=∆(lng)
+∆(lnN)=1
+∆(lnN)∆g
+g=1
+41
+∂
+∂zln(rIR(z)
+rUV(z))1
+g∂g
+∂z, (21)
+whereg(z) is a renormalized coupling at z.5
+We have explained, in this section, that the minimal area principle determines the flow of
+the regularization surfaces.
+6. Weight Function and Casimir Energy Evaluation
+In the expression (14), the Casimir energy is written by the i ntegral in the (˜ p,z)-space over the
+range: 1/ω≤z≤1/T,0≤˜p≤ ∞. In Sec.5, we have seen the integral region should be properly
+restricted because the cut-off region in the 4D world changes along the extra-axis obeying the
+bulk (warped) geometry ( minimal area principle ). We can expect the singular behavior (UV
+divergences) reduces by the integral-region restriction, but the concrete evaluation along the
+proposed prescription is practically not easy. In this sect ion, we consider an alternate approach
+which respects the minimal area principle and evaluate the Casimir energy.
+We introduce, instead of restricting the integral region, a weight function W(˜p,z) in the
+(˜p,z)-space for the purpose of suppressing UV and IR divergences of the Casimir Energy.
+−E∓W
+Cas(ω,T)≡/integraldisplayd4pE
+(2π)4/integraldisplay1/T
+1/ωdz W(˜p,z)F∓(˜p,z),˜p=/radicalBig
+p2
+4+p2
+1+p2
+2+p2
+3,
+Examples of W(˜p,z) :W(˜p,z) =
+
+(N1)−1e−(1/2)˜p2/ω2−(1/2)z2T2≡W1(˜p,z), N1= 1.711/8π2elliptic suppr.
+(N2)−1e−˜pzT/ω≡W2(˜p,z), N2= 2ω3
+T3/8π2hyperbolic suppr.1
+(N8)−1e−1/2(˜p2/ω2+1/z2T2)≡W8(˜p,z), N8= 0.4177/8π2reciprocal suppr.1(22)
+whereF∓(˜p,z) are defined in (16). In the above, we list some examples expec ted for the weight
+functionW(˜p,z).W2is regarded to correspond to the regularization taken by Ran dall-Schwartz.
+How to specify the form of Wis the subject of the next section. We show the shapeof the ene rgy
+integrand ( −1/2)˜p3W1(˜p,z)F−(˜p,z) in Fig.8. We notice the valley-bottom line ˜ p≈0.75Λ, which
+appeared in the un-weighted case (Fig.4), is replaced by a ne w line: ˜p2+z2×ω2T2≈const. It
+is located away from the original Λ-effected line (˜ p∼0.75Λ).
+5Here we consider an interacting theory, such as 5D Yang-Mill s theory and 5D Φ4theory, where the coupling
+g(z) is the renormalized one in the ’3-brane’ at z.We can check the divergence (scaling) behavior of E∓W
+Casbynumerically evaluating the (˜ p,z)-
+integral (22) for the rectangle region of Fig.3.
+−EW
+Cas=
+
+ω4
+TΛ×1.2/braceleftBig
+1+0.11 lnΛ
+ω−0.10 lnΛ
+T/bracerightBig
+forW1
+T2
+ω2Λ4×0.062/braceleftBig
+1+0.03 lnΛ
+ω−0.08 lnΛ
+T/bracerightBig
+forW2
+ω4
+TΛ×1.6/braceleftBig
+1+0.09 lnΛ
+ω−0.10 lnΛ
+T/bracerightBig
+forW8(23)
+The suppression behavior of W2improves, compared with (19) by Randall-Schwartz. The
+quintic divergence of (19) reduces to the quartic divergenc e in the present approach of W2. The
+hyperbolic suppressions, however, are still insufficient fo r the renormalizability. After dividing
+by the normalization factor, Λ T−1, the cubic divergence remains. The desired cases are others .
+The Casimir energy for each case consists of three terms. The first terms give finitevalues after
+dividing by the overall normalization factor Λ T−1. The last two terms are proportional to logΛ
+and show the anomalous scaling . Their contributions are order of 10−1to the firstleading terms.
+The second ones (lnΛ
+ω) contribute positively while the third ones (lnΛ
+T) negatively. They give,
+after normalizing the factor Λ /T,onlythelog-divergence .
+EW
+Cas/ΛT−1=−αω4/parenleftbig1−4cln(Λ/ω)−4c′ln(Λ/T)/parenrightbig, (24)
+whereα,candc′can be read from (23) depending on the choice of W. This means the 5D
+Casimir energy is finitelyobtained by the ordinary renormalization of the warp factor ω. (See
+the final section.) In the above result of the warped case, the IR parameter lin the flat result
+(3) is replaced by the inverse of the warp factor ω.
+So far as the legitimate reason of the introduction of W(˜p,y) is not clear, we should regard
+this procedure as a regularization to define the higher dimensional theories. We give a clear
+definition of W(˜p,y) and a legitimate explanation in the next section. It should be done, in
+principle, in a consistent way with the bulk geometry and the gauge principle.
+7. Meaning of Weight Function and Quantum Fluctuation of Coordinates and
+Momenta
+In the previous work[15], we have presented the following id ea to define the weight function
+W(˜p,z). In the evaluation (22):
+−EW
+Cas(ω,T) =/integraldisplayd4pE
+(2π)4/integraldisplay1/T
+1/ωdz W(˜p,z)F∓(˜p,z) =2π2
+(2π)4/integraldisplay
+d˜p/integraldisplay1/T
+1/ωdz˜p3W(˜p,z)F∓(˜p,z),(25)
+the (˜p,z)-integral is over the rectangle region shown in Fig.5 (with Λ→ ∞andµ→0).F∓(˜p,z)
+is explicitly given in (16). Following Feynman[21], we can r eplace the integral by the summation
+over all possible pathes ˜ p(z).
+−EW
+Cas(ω,T) =/integraldisplay
+D˜p(z)/integraldisplay1/T
+1/ωdz S[˜p(z),z], S[˜p(z),z] =2π2
+(2π)4˜p(z)3W(˜p(z),z)F∓(˜p(z),z).(26)
+There exists the dominant path ˜pW(z) which is determined by the minimal principle : δS= 0.
+Dominant Path ˜ pW(z) :d˜p
+dz=−∂ln(WF)
+∂z
+3
+˜p+∂ln(WF)
+∂˜p. (27)
+Hence it is fixed by W(˜p,z). An example is the valley-bottom line in Fig.8. On the other hand,
+there exists another independent path: the minimal surface curverg(z).
+Minimal Surface Curve rg(z) : 3+4
+zr′r−r′′r
+r′2+1= 0,1
+ω≤z≤1
+T,(28)which is obtained by the minimal area principle :δA= 0 where
+ds2= (δab+xaxb
+(rr′)2)dxadxb
+ω2z2≡gab(x)dxadxb,A=/integraldisplay/radicalbig
+detgabd4x=/integraldisplay1/T
+1/ω1
+ω4z4/radicalBig
+r′2+1r3dz.(29)
+Hencerg(z) is fixed by the induced geometry gab(x). Here we put the requirement [15]: Eq.(7A)
+˜pW(z) = ˜pg(z), where ˜pg≡1/rg. This means the following things. We requirethe dominant
+path coincides with the minimal surface line ˜ pg(z) = 1/rg(z) which is defined independently
+ofW(˜p,z). In other words, W(˜p,z) is defined here by the induced geometry gab(x). In this
+way, we can connect the integral-measure over the 5D-space w ith the (bulk) geometry. We have
+confirmed the (approximate) coincidence by the numerical me thod.
+In order to most naturally accomplish the above requirement , we can go to a new step .
+Namely, we proposetoreplacethe 5D space integral with the weight W, (25), by the following
+path-integral . Wenewly define the Casimir energy in the higher-dimensional theory as foll ows.
+−ECas(ω,T,Λ)≡/integraldisplay1/µ
+1/Λdρ/integraldisplay
+r(1/ω)
+=r(1/T)
+=ρ/productdisplay
+a,zDxa(z)F(1
+r,z) exp/bracketleftBigg
+−1
+2α′/integraldisplay1/T
+1/ω1
+ω4z4/radicalBig
+r′2+1r3dz/bracketrightBigg
+,(30)
+whereµ= ΛT/ωand the limit Λ T−1→ ∞is taken. The string (surface) tension parameter
+1/2α′is introduced. (Note: Dimension of α′is [Length]4. ) The square-bracket ([ ···])-parts
+of (30) are −1
+2α′Area =−1
+2α′/integraltext√detgabd4xwheregabis the induced metric on the 4D surface.
+F(˜p,z) is defined in (22) or (16) and shows the field-quantization of the bulk scalar (EM) fields.
+In the above expression, we have followed the path-integral formulation of the density matrix
+(See Feynman’s text[21]). The validity of the above definiti on is based on the following points:
+a) When the weight part (exp [ ···]-part) is 1, the proposed quantity ECasis equal to EW
+Cas,
+(25), withW= 1 ; b) The leading path is given by rg(z) = 1/pg(z), (28); c) The proposed
+definition, (30), clearly shows the 4D space-coordinates xaor the 4D momentum-coordinates pa
+arequantized (quantum-statistically, not field-theoretically) with th e Euclidean time zand the
+”areaHamiltonian” A=/integraltext√detgabd4x. Note that F(˜p,z) orF(1/r,z) appears, in (30), as the
+energy density operator in the quantum statistical system of {pa(z)}or{xa(z)}.
+In the view of the previous paragraph, the treatment of Sec.6 is aneffective action approach
+using the (trial) weight function W(˜p,z). Note that the integral over ( pµ,z)-space, appearing
+in (16), is the summation over all degrees of freedom of the 5D space(-time) points using the
+”naive” measure d4pdz. An important point is that we have the possibility to take an other
+measure for the summation in the case of the higher dimension al QFT. We have adopted, in
+Sec.6, the new measure W(pµ,z)d4pdzin such a way that the Casimir energy does not show
+physical divergences . We expect the directevaluation of (30), numerically or analytically, leads
+to the similar result.
+8. Discussion and Conclusion
+The log-divergence in (24) is the familiar one in the ordinar y QFT. It can be renormalized in
+the following way.
+EW
+Cas
+ΛT−1=−αω4/parenleftbigg
+1−4cln(Λ
+ω)−4c′ln(Λ
+T)/parenrightbigg
+=−α(ωr)4,ωr=ω4/radicalBigg
+1−4cln(Λ
+ω)−4c′ln(Λ
+T),(31)
+whereωris therenormalized warp factor and ωis thebareone. No local counterterms are
+necessary. Note that this renormalization relation is exact(not a perturbative result). In the
+familiar case of the 4D renormalizable theories, the coeffici entscandc′depend on the coupling,but, in the present case, they are pure numbers . It reflects the interaction between (EM) fields
+and the boundaries . Whencandc′are sufficiently small we find the renormalization group
+function for the warp factor ωas
+|c| ≪1,|c′| ≪1, ωr=ω(1−cln(Λ/ω)−c′ln(Λ/T)), β≡∂
+∂(lnΛ)lnωr
+ω=−c−c′.(32)
+We should notice that, in the flat geometry case, the IR parame ter (extra-space size) lis
+renormalized. In the present warped case, however, the corr esponding parameter Tisnot
+renormalized , but the warp parameter ωis renormalized . Depending on the sign of c+c′, the 5D
+bulk curvature ωflowsas follows. When c+c′>0, the bulk curvature ωdecreases (increases) as
+the the measurement energy scale Λ increases (decreases). W henc+c′<0, the flow goes in the
+opposite way. When c+c′= 0,ωdoes not flow ( β= 0) and is given by ωr=ω(1+cln(ω/T)).
+Thefinalresult(31) is the newtypeCasimir energy, −ω4.ωappearsasaboundaryparameter
+likeT. The familiar one is −T4in the present context. In ref.[22], another type T2ω2was
+predicted using a ”quasi” Warped model (bulk-boundary theo ry).
+Through the Casimir energy calculation, in the higher dimen sion, we find a way to quantize
+the higher dimensional theories within the QFT framework. T hequantization with respect to the
+fields(except the gravitational fields GAB(X)) is done in the standard way. After this step, the
+expression has the summation over the 5D space(-time) coordinates or momenta/integraltextdz/producttext
+adpa.
+We have proposed that this summation should be replaced by th epath-integral/integraltext/producttext
+a,zDpa(z)
+with the areaaction (Hamiltonian) A=/integraltext√detgabd4xwheregabis theinducedmetric on the
+4D surface. This procedure says the 4D momenta pa(or coordinates xa) arequantum statistical
+operators and the extra-coordinate zis the inverse temperature (Euclidean time). We recall the
+similar situation occurs in the standard string approach. T he space-time coordinates obey some
+uncertainty principle[23].
+Recently the dark energy (as well as the dark matter) in the un iverse is a hot subject. It
+is well-known that the dominant candidate is the cosmologic al term. We also know the proto-
+type higher-dimensional theory, that is, the 5D KK theory, h as predicted so far the divergent
+cosmological constant[4]. This unpleasant situation has b een annoying us for a long time. If we
+applythepresentresult, thesituation drastically improv es. Thecosmological constant λappears
+as:Eq.(8A) Rµν−1
+2gµνR−λgµν=Tmatter
+µν,S=/integraltextd4x√−g{1
+GN(R+λ)}+/integraltextd4x√−g{Lmatter},g=
+detgµν, whereGNis the Newton’s gravitational constant, Ris the Riemann scalar curvature.
+We consider here the 3+1 dim Lorentzian space-time ( µ,ν= 0,1,2,3). The constant λ
+observationally takes the value : Eq.(8B)1
+GNλobs∼1
+GNRcos2∼m4
+ν∼(10−3eV)4,λobs∼
+1
+R2cos∼4×10−66(eV)2, whereRcos∼5×1032eV−1is the cosmological size (Hubble length),
+mνis the neutrino mass.6On the other hand, we have theoretically so far : Eq.(8C)
+1
+GNλth∼1
+GN2=Mpl4∼(1028eV)4. This is because the mass scale usually comes from the
+quantum gravity. (See ref.[26] for the derivation using the Coleman-Weinberg mechanism.) We
+have the famous huge discrepancy factor : Eq.(8D)λth
+λobs∼N2
+DL,NDL≡MplRcos∼6×1060,
+whereNDLis the Dirac’s large number[27]. If we use the present result (31), we can obtain
+a natural choice of T,ωand Λ as follows. By identifying T−4ECas=−α1ΛT−1ω4/T4with/integraltextd4x√−g(1/GN)λob=R2
+cos(1/GN), weobtain thefollowingrelation: Eq.(8E) N2
+DL=R2
+cos1
+GN=
+−α1ω4Λ
+T5,α1: some coefficient. The warped (AdS 5) model predicts the cosmological constant
+negative, hence we have interest only in its absolute value. We take th e following choice for Λ
+andω: Eq.(8F) Λ = Mpl∼1019GeV,ω∼1
+4√
+GNRcos2=/radicalBig
+Mpl
+Rcos∼mν∼10−3eV. The choice
+6The relation mν∼/radicalbig
+Mpl/Rcos=/radicalbig
+1/Rcos√GN, which appears in some extra dimension model[24, 25], is
+used. The neutrino mass is, at least empirically, located at thegeometrical average of two extreme ends of the
+mass scales in the universe.for Λ is accepted in that the largest known energy scale is the Planck energy. The choice for ω
+comes from the experimental bound for the Newton’s gravitat ional force.
+As shown above, we have the standpoint that the cosmological constant is mainly made from
+the Casimir energy. We do not yet succeed in obtaining the val ueα1negatively, but succeed in
+obtaining the finiteness of the cosmological constant and it s gross absolute value. The smallness
+of the value is naturally explained by the renormalization fl ow as follows. Because we already
+know the warp parameter ωflows(32), theλobs∼1/R2
+cosexpression (8F), λobs∝ω4, says
+that the smallness of the cosmological constant comes from the renor malization flow for the non
+asymptotic-free case ( c+c′<0 in (32)).
+The IR parameter T, the normalization factor Λ /Tin (24) and the IR cutoff µ= ΛT
+ωare
+given by : Eq.(8G) T=R−1
+cos(NDL)1/5∼10−20eV,Λ
+T= (NDL)4/5∼1050,µ=MplN−3/10
+DL∼
+1GeV∼mN, wheremNis the nucleon mass. The Fig.6 strongly suggests that the deg ree of
+freedom of the universe (space-time) is given by : Eq.(8H)Λ4
+µ4=ω4
+T4=N6/5
+DL∼1074∼(Mpl
+mN)4.
+References
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+arXiv:1001.0223v1  [math.AG]  1 Jan 2010COMBINATORIAL CUBIC SURFACES
+AND RECONSTRUCTION THEOREMS
+Yu. I. Manin
+Max–Planck–Institut f¨ ur Mathematik, Bonn, Germany,
+and Northwestern University, Evanston, USA
+Abstract. This note contains a solution to the following problem: reco n-
+struct thedefinition field and theequationof a projectivecu bic surface, using
+only combinatorial information about the set of its rationa l points. This in-
+formation is encoded in two relations: collinearity and cop lanarity of certain
+subsets of points. We solve this problem, assuming mild “gen eral position”
+properties.
+This study is motivated by an attempt to address the Mordell– Weil prob-
+lem for cubic surfaces using essentially model theoretic me thods. However,
+the language of model theory is not used explicitly.
+Contents
+0. Introduction and overview
+1. Quasigroups and cubic curves
+2. Reconstruction of the ground field and a cubic surface from combina-
+torics of tangent sections
+3. Combinatorial and geometric cubic surfaces
+4. Cubic curves and combinatorial cubic curves over large fie lds
+APPENDIX. Mordell–Weil and height: numerical evidence
+References
+0. Introduction and overview
+0.1. Cubic hypersurfaces. LetKbe a field, finite or infinite. In the
+finite case we assume cardinality of Kto be sufficiently large, the exact lower
+boundary depending on various particular combinatorial co nstruction.
+LetP=PN
+Kbe a projective space over K, with a projective coordinate
+system ( z1:z2:···:zN+1). Acubic hypersurface V⊂Pdefined over K
+is, by definition, the closed subscheme defined by an equation c= 0 where
+c∈K[z1:z2,···:zN+1] is a non–zero cubic form. There is a bijection
+12
+between the set of such subschemes and the set P9(K) of coefficients of c
+moduloK∗.
+We will say that Visgenerically reduced if after extending Kto an alge-
+braic closure K,cdoes not acquire a multiple factor.
+In this paper, I will be interested in the following problem:
+0.1.1. Problem. Assuming Vgenerically reduced, reconstruct Kand the
+subscheme V⊂P=PN
+Kstarting with the set of its K–pointsV(K)endowed
+with some additional combinatorial structures of geometri c origin.
+The basic combinatorial data that I will be using are subsets of smooth
+points of V(K) lying upon various sections of Vby projective subspaces of P
+defined over K. Thus, for the main case treated here, that of cubic surfaces
+(N= 3), I will deal combinatorially with the structure, consis ting of
+a) The subset of smooth (reduced, non–singular) points S:=Vsm(K).
+b) A triple symmetric relation “collinearity”: L ⊂S3:=S×S×S.
+c) A set Pof subsets of Scalled “plane sections”.
+In the first approximation, one can imagine L(resp.P) as simply subsets
+of collinear triples (resp. K–points of K–plane sections) of V. However,
+various limiting and degenerate cases must be treated with c are as well.
+For example, as a working definition of Lwe will adopt the following
+convention: ( p,q,r)∈S3belongs to Lif eitherp+q+ris the full intersection
+cycle ofVwith aK–linel⊂PN(with correct multiplicities), or else if there
+exists aK–linel⊂Vsuch that p,q,r∈l.
+0.2. Geometric constraints. If an instance of the set–theoretic com-
+binatorial structure such as ( S,L,P) above, comes from a cubic surface V
+defined over a field K, we will call such a structure geometric one .
+Geometric structures satisfy additional combinatorial co nstraints.
+The reconstruction problem in this context consists of two parts:
+(i) Find a list of constraints ensuring that each (S,L,P)satisfying these
+constraints is geometric.
+(ii) Devise a combinatorial procedure that reconstructs KandV⊂P3
+realizing (S,L,P)as a geometric one.
+Besides, ideally we want the reconstruction procedure to be functorial:
+certain maps of combinatorial structures, in particular, t heir isomorphisms,3
+must induce/be induced by morphisms of ground fields and K–linear maps
+ofP.
+Inthesubsection0.4, Iwilldescribeaclassicalarchetype ofreconstruction,
+– combinatorial characterization of projective planes. I w ill also explain the
+main motivation for trying to extend this technique to cubic surfaces: the
+multidimensional weak Mordell-Weil problem.
+0.3. Reconstruction of Kfrom curves and configurations of
+curves. One cannot hope to reconstruct the ground field K, ifVis zero–
+dimensional or one–dimensional. Only starting with cubic s urfaces (N=3),
+this prospect becomes realistic.
+In fact, if N= 1, we certainly cannot reconstruct Kfrom any combinato-
+rial information about one K–rational cycle of degree 3 on P1
+K.
+IfN= 2, then for a smoothcubic curve V, the setV(K) endowed withthe
+collinearity relation is the same as V(K) considered as a principal homoge-
+neous space over the “Mordell–Weil” abelian group, unambig uously obtained
+from (V(K),L) as soon as we arbitrarily choose the identity (or zero) poin t:
+cf. a recollection of classical facts in sec. 1 below. Genera lly, this group does
+not carry enough information to get hold of K, ifKis finitely generated over
+Q.
+However, the situation becomes more promising, if we assume Vgeomet-
+rically irreducible and having just one singular point which is defined over
+K. More specifically, assume that this point is either an ordin ary double
+point with two different branches/tangents defined over Keach, or a cusp
+with triple tangent line, which is then automatically define d overK.
+In the first case, we will say that Vis a curve of multiplicative type , in the
+second, of additive type.
+Then we can reconstruct, respectively, the multiplicative or the additive
+group of K, up to an isomorphism. In fact, these two groups are canonica lly
+identified with Vsm(K) as soon as one smooth K–point is chosen, in the same
+way as the Mordell–Weil group is geometrically constructed from a smooth
+cubic curve with collinearity relation.
+Finallyfor N= 3, nowallowing Vtobesmooth, andunder mildgenericity
+restrictions, we can combine these two procedures and recon struct both K
+and a considerable part of the whole geometric picture.
+The idea, which is the main new contribution of this note, is t his. Choose
+two points ( pm,pa) inVsm(K), not lying on a line in V, whose tangent4
+sections ( Cm,Ca) are, respectively, of multiplicative and additive type. ( To
+find such points, one might need to replace Kby its finite extension first).
+Now, one can intersect the tangent planes to pmandpaby elements of a
+K–rational pencil of planes, consisting of all planes contai ningpmandpa.
+This produces a birational identification of CmandCa.
+Thecombinatorialinformation, used inthisconstruction, canbeextracted
+from the data LandP. The resulting combinatorial object, carrying full
+information about both K∗andK+, can be then processed into K, if a set
+of additional combinatorial constraints is satisfied.
+Using four tangent plane section in place of two, one can then unambigu-
+ously reconstruct the whole subscheme V.
+For further information, cf. the main text.
+0.4. Combinatorial projective planes and weak Mordell–Wei l
+problem. My main motivation for this study was an analog of Mordell–
+Weil problem for cubic surfaces: cf. [M3], [KaM], [Vi].
+Roughly speaking, the classical Mordell–Weil Theorem for e lliptic curves
+can be stated as follows. Consider a smooth plane cubic curve C, i. e. a
+plane model of an elliptic curve, over a field Kfinitely generated over its
+prime subfield. Then the whole set C(K) can be generated in the following
+way: start with a finitesubsetU⊂C(K) and iteratively enlarge it, adding
+to already obtained points each point p◦q∈C(K) that is collinear with two
+pointsp,q∈C(K) that were already constructed. If p=q, then the third
+collinear point, by definition, is obtained by drawing the ta ngent line to C
+atp.
+In the case of a cubic surface V, say, not containing K–lines, there are
+two versions of this geometric process (“drawing secants an d tangents”). We
+may allow to consecutively add only points collinear to p,q∈V(K) when
+p/\e}atio\slash=q. Alternatively, we may also allow to add all K–points of the plane
+section of Vtangent to Vatp=q.
+I will call the respective two versions of finite generation c onjecture strong,
+resp. weak, Mordell–Weil problem for cubic surfaces.
+Computer experiments suggest that weak finite generation mi ght hold at
+least for some cubic surfaces defined over Q: see [Vi] for the latest data. The
+same experiments indicate however, that the “descent” proc edure, by which
+Mordell–Weil is proved for cubic curves, will not work in two –dimensional
+case: a stable percentage of Q–points of height ≤Hremains not expressible
+in the form p◦qwithp,qof smaller height.5
+In view of this, I suggested in [M3], [KaM] to use a totally diff erent ap-
+proach to finite generation, based on the analogy with classi cal theory of
+abstract, or combinatorial, projective planes.
+The respective finite generation statement can be stated as f ollows.
+For any field Kof finite type over its prime subfield, the whole set P2(K)
+can be obtained by starting with a finite subset U⊂ P2(K)and consecu-
+tively adding to it lines through pairs of distinct points, a lready obtained, and
+intersection points of pairs of constructed lines.
+Thestrategyofproofcanbepresented asa sequence ofthefol lowingsteps.
+STEP 1. Define a combinatorial projective plane S,Las an abstract set
+Swhose elements are called (combinatorial) points , endowed with a set of
+subsets of points Lcalled(combinatorial) lines , such that each two distinct
+points are contained in a single line, and each two distinct l ines intersect at
+a single point.
+STEP 2. Find combinatorial conditions upon ( S,L), that are satisfied
+forK–points of each geometric projective plane P2(K), and that exactly
+characterize geometric planes, so that starting with ( S,L) satisfying these
+conditions, one can reconstruct from ( S,L) a fieldKand an isomorphism of
+(S,L) with (P2(K), projective K −lines) unambiguously.
+In fact, this reconstruction must be also functorial with re spect to embed-
+dings of projective planes S⊂S′and the respective combinatorial lines.
+These conditions are furnished by the beautiful Pappus Theorem/Axiom
+(at least, if cardinality of Sis infinite or finite but large enough).
+STEP 3. Given a geometric projective plane ( P2(K), projective K −
+lines), start with four points in general position U0⊂ P2(K) and generate
+the minimal subset SofP2(K) stable with respect to drawing lines through
+two points and taking intersection point of two lines.
+This subset, with induced collinearity structure, is a comb inatorial projec-
+tive plane. It satisfies the Pappus Axiom, because it was sati sfied forP2(K).
+Itisnotdifficulttodeducethenthat Sisisomorphicto P2(K0),withK0⊂K
+the prime subfield, and the embeddings K0→KandS→ P2(K) are com-
+patible with geometry.
+STEP 4. Finally, one can iterate this procedure as follows. If Kis finitely
+generated, thereexistsafinitesequenceofsubfields K0⊂K1⊂...⊂Kn=K
+suchthateach Kiisgeneratedover Ki−1byoneelement, say θi. Ifwealready6
+know a finite generating set of points Ui−1⊂ P2(Ki−1), define Ui⊂ P2(Ki)
+asUi−1∪{(θi: 1 : 0)}. One easily sees that Uigenerates P2(Ki).
+0.5. Results of this paper. As was explained in 0.3, results of this
+paper give partial versions for cubic surfaces of Steps 1 and 2 in the finite
+generation proof, sketched above. I can now reconstruct the ground field K
+and the total subscheme V⊂ P3
+K, under appropriate genericity assumptions,
+from the combinatorics of V(K) geometric origin.
+However, these results still fall short of a finite generatio n statement.
+Thereadermustbeawarethatthisapproachisessentiallymo del–theoretic,
+and it was inspired by yhe successes of [HrZ] and [Z].
+My playground is much more restricted, and I do not use explic itly the
+(meta)languageofmodeltheory, workingintheframeworkof Bourbakistruc-
+tures.
+Moreprecisely, constructions, explainedinsec. 2and3, ar eorientedtothe
+reconstruction offields of finitetypeand cubic surfaces ove r them. According
+to [HrZ] and [Z], if one works over an algebraically closed gr ound field, one
+can reconstruct combinatorially (that is, in a model theore tic way) much of
+the classical algebraic geometry.
+In sec. 4, I introduce the notion of a large field , tailor–made for cubic (hy-
+per)surfaces, and show that large fields can be reconstructe d even from (sets
+of rational points of) smooth plane cubic curves, endowed wi th collinearity
+relation and an additional structure consisting of pencils of collinear points
+onsuch a curve. Any field Khaving no non–trivialextensions ofdegree 2 and
+3 is large, hence large fields lie between finitely generated a nd algebraically
+close ones.
+1. Quasigroups and cubic curves
+1.1. Definition. LetSbe a set and L ⊂S×S×Sbe a subset of triples
+with the following properties:
+(i)Lis invariant with respect to permutations of factors S.
+(ii) Each pair p,q∈Suniquely determines r∈Ssuch that (p,q,r)∈ L.
+Then(S,L)is called a symmetric quasigroup.
+This structure in fact defines a binary composition law
+◦:S×S→S:p◦q=r⇐⇒(p,q,r)∈ L. (1.1)7
+Properties of Lstated in the Definition 1.1 can be equivalently rewritten in
+terms of ◦: for allp,q∈S
+p◦q=q◦p, p◦(p◦q) =q. (1.2)
+The structure ( S,◦), satisfying (1.2), will also be called a symmetric quasi-
+group. The importance of Lfor us is that, together with its versions, it
+naturally comes from geometry.
+In terms of ( S,◦), we can define the following groups. For each p∈S, the
+maptp:q/mapsto→p◦qis an involutive permutation of S:t2
+p= idS.
+Denote by Γ = Γ( S,L) the group generated by all tp,p∈S. Let Γ0⊂Γ
+be its subgroup, consisting of products of an even number of i nvolutions tp.
+1.2. Theorem–Definition. A symmetric quasigroup (S,◦)is called
+abelian, if it satisfies any (and thus all) of the following eq uivalent conditions:
+(i) There exists a structure of abelian group on S,(p,q)/mapsto→pq, and an
+elementu∈Ssuch that for all p,q∈Swe have p◦q=up−1q−1.
+(ii) The group Γ0is abelian.
+(iii) For all p,q,r∈S,(tptqtr)2= 1.
+(iv) For any element u∈S, the composition law pq:=u◦(p◦q)turnsS
+into an abelian group.
+(v) The same as (iv) for some fixed element u∈S.
+Under these conditions, Sis a principal homogeneous space over Γ0.
+For a proof, cf. [M1], Ch. I, sec. 1,2, especially Theorem 2.1 .
+1.3. Example: plane cubic curves. LetKbe a field, C⊂P2
+Kan
+absolutely irreducible cubic curve defined over K. Denote by S=Csm(K)⊂
+C(K) the set of non–singular K–points of C. Define the collinearity relation
+Lby the following condition:
+(p,q,r)∈ Liff p+q+r is the intersection cycle of C with a K −line.
+(1.3)
+Then (S,L) is an abelian symmetric quasigroup. This is a classical res ult.
+More precisely, we have the following alternatives. Cmight be non–
+singular over an algebraic closure of K. ThenCis the plane model of an
+abstract elliptic curve defined over K, the group Γ0can be identified with8
+K–points of its Picard group. We call the latter also the Morde ll–Weil group
+ofCoverK.
+Singular curves will be more interesting for us, because the y carry more
+information about the ground field K. Each geometrically irreducible sin-
+gular cubic curve has exactly one singular geometric point, sayp, and it is
+rational over K. More precisely, we will distinguish three cases.
+(I)Cis ofmultiplicative type . This means that pis a double point two
+tangents to which at pare rational over K.
+(II)Cis ofadditive type . This means that pis a cusp: a point with triple
+tangent.
+(III)Cis oftwisted type . This means that pis a double point ptwo
+tangents to which at pare rational and conjugate over a quadratic extension
+ofK.
+The structure of quasigroups related to singular cubic curv es is clarified
+by the following elementary and well known statement.
+1.3.1. Lemma. (i) IfCis of multiplicative type, Γ0is isomorphic to
+K∗.
+(ii) IfCis of additive type, Γ0is isomorphic to K+.
+(iii) IfCis of twisted type, Γ0is isomorphic to the group of K–points of
+a form of GmorGathat splits over the respective quadratic extension of K.
+The first case occurs when charK/\e}atio\slash= 2, the second one when charK= 2.
+Proof. (Sketch.) In all cases, the group law pq:=u◦(p◦q), for an
+arbitrary fixed u∈Sdetermines the structure of an algebraic group over
+Kupon the curve C0which can be defined as the normalization of Cwith
+preimage(s) of pdeleted. An one–dimensional geometrically connected alge -
+braic group becomes isomorphic to GmorGaover any field of definition of
+its points “at infinity”.
+In the next section, we will recall more precise information about the
+respective isomorphisms in the non–twisted cases.
+2. Reconstruction of the ground field and a cubic surface
+from combinatorics of tangent sections
+2.1. The key construction. LetKbe a field of cardinality ≥4.Then
+the setH:=P1(K) consists of ≥5 points.9
+Consider a family of five pairwise distinct points in Hfor which we choose
+the following suggestive notation:
+0a,∞a,0m,1m,∞m∈P1(K). (2.1)
+In view of its origin, the set H\{∞a}has a special structure of abelian
+groupA(written additively, with zero 0 a). In fact, the choice of any affine
+coordinate xaonP1
+Kwith zero at 0 aand pole at ∞adefines this structure:
+it sendsp∈H\{∞a}to the value of xaatp, and addition is addition in K+.
+The structure does not depend on xa, butxadetermines the isomorphism of
+GawithK+, and this isomorphism doesdepend on xa: the set of all xa’s is
+the principal homogeneous space over K∗.
+Similarly, the set H\{0m,∞m}has a special structure of abelian group
+M, with identity 1 m. A choice of affine coordinate xmonP1, with divisor
+supported by(0 m,∞m) and taking value 1 ∈Kat 1m, defines this structure.
+Again, it does not depend on xm, butxmdetermines its isomorphism with
+K∗, and this isomorphism does depend on xm. There are, however, only two
+choices: xmandx−1
+m. They differ by renaming 0 m↔ ∞m.
+Having said this, consider now an abstract set Hwith a subfamily of five
+elements denoted as in (2.1). Moreover, assume in addition t hat we are given
+composition laws + on H\{∞a}and·onH\{0m,∞m}turning these sets
+into two abelian groups, A(written additively, with zero 0 a) andM(written
+multiplicatively, with identity 1 m). Define the inversion map i:M→M
+using this multiplication law: i(p) =p−1.
+Wewillencodethisextendedversionof(2.1),withaddition aldatarecorded
+in the notation M,A, as a bijection
+µ:M∪{0m,∞m} →A∪{∞a} (2.2)
+It is convenient to extend the multiplication and inversion , resp. addition
+and sign reversal, to commutative partial composition laws on two sets (2.2)
+by the usual rules: for p∈M,q∈A, we set
+p·0m:= 0m, p·∞m:=∞m, i(0m) :=∞m, i(∞m) := 0m,(2.3)
+q±∞a:=∞a. (2.4)
+The following two lemmas are our main tool in this section.10
+2.2. Lemma. If (2.2) comes from a projective line as above, then the
+map
+ν:M∪{0m,∞m} →A∪{∞a},
+ν(p) :=µ{µ−1[µ(p)−µ(0m)]·i◦µ−1[µ(p)−µ(∞m)]}(2.5)
+is a well defined bijection.
+Moreover,
+ν(0m) = 0a:= 0, ν(∞m) =∞a:=∞. (2.6)
+Finally, identifying M∪{0m,∞m}andA∪{∞a}with the help of νand
+combining addition and multiplication, now (partially) de fined on H, we get
+uponH\{∞}a structure of the commutative field, with zero 0and identity
+1 :=ν(1m). This field is isomorphic to the initial field K.
+Proof.In the situation (2.1), if Ais identified with K+using an affine
+coordinate xa, andMis identified with K∗using another affine coordinate
+xmas above, these coordinates are connected by the evident fra ctional linear
+transformation, bijective on P1(K):
+xa=c·(xm−xm(0m))·(xm−xm(∞m))−1, c∈K∗.
+The definition (2.5) is just a fancy way to render this relatio n, taking into
+account that now we have to add and to multiply in two different locations,
+passing back and forth via µandµ−1. Instead of multiplying by c, we
+normalize multiplication so that ν(1∞) becomes identity.
+This observation makes all the statements evident.
+The same arguments read in reverse direction establish the f ollowing re-
+sult:
+2.3. Lemma on Reconstruction. Conversely, let MandAbe two
+abstract abelian groups, extended by “improper elements” t o the sets with
+partial composition laws M∪ {0m,∞m}andA∪ {∞a}, as in (2.3), (2.4).
+Assume that we are given a bijection µas in (2.2), mapping 1,0m, and∞m
+toA. Assume moreover that:
+(i) The respective mapping νdefined by (2.5) is a well defined bijection.
+(ii) The set Aendowed with its own addition, and multiplication trans-
+ported by νfromM, is a commutative field K.
+Then we get a natural identification H=P1(K). This construction is
+inverse to the one described in sec. 2.1.11
+2.4. Combinatorial projective lines and functoriality. Let us call
+an instance of the data (2.2)–(2.4),satisfying the constra ints of Lemma 2.2, a
+combinatorial projective line (this name will be better justified in the remain-
+der of this section). Let us call triples ( K,P1(K),j) wherejis a subfamily
+of five points in P1(K) as in (2.1), geometric projective lines.
+The constructions we sketched above are obviously functori al with respect
+to various natural maps such as:
+a)On the geometric side: Morphisms of fields, naturally extended to pro-
+jective lines with marked points. Fractional linear transf ormations of P1(K),
+naturally acting upon jand identical on K.
+b)On the combinatorial side: Embeddings of groups M→M′,A→A′,
+compatible with ( µ,µ′) and on improper points. Automorphisms of ( M,A),
+supplied with compatibly changed µand improper points.
+These statements can be made precise and stated as equivalen ce of cate-
+gories. We omit details here.
+Now we turn to the description of a bare–bones geometric situ ation, that
+can be obtained (in many ways) from a cubic surface, directly producing
+combinatorial projective lines.
+2.5.(Cm,Ca)–configurations . Consider a family of subschemes in P3
+K,
+that we will call a configuration:
+Conf:= (pm,pa;Cm,Ca;Pm,Pa) (2 .7)
+It consists of the following data:
+(i) Two distinct K–pointspm,pa∈P3(K).
+(ii) Two distinct K–planesPm,Pa⊂P3such that pm∈Pm,pm/∈Paand
+pa∈Pa,pa/∈Pm.
+(iii) Two geometrically irreducible cubic K–curvesCm⊂Pm,Ca⊂Pa.
+We impose on these data the following constraints:
+(A)pm∈Cm(K) is a double point, and Cmif of multiplicative type, in
+the sense of 1.3.
+(B)pa∈Ca(K) is a cusp, and Cais of additive type.
+(C) Letl:=Pm∩Pa. Denote by 0 m,∞m∈lthe intersection points with
+lof two tangents to Cmatxm(in the chosen order). Denote by 0 a∈lthe12
+intersection point with lof the tangent to Caatxa. These three points are
+pairwise distinct.
+LetM:=Cm,sm(K),A:=Ca,sm(K) be the respective sets of smooth
+points, with their group structure, induced by collinearit y relation and a
+choice of 1 m, resp. 0 a, as in sec. 1.
+Define the bijection α:/tildewideCm(K)→l(K), where /tildewideCmis the normalization
+ofCm, by mapping each smooth point q∈C(K) to the intersection point
+withlof the line, passing through pmandq. The two tangent lines at pm
+define the images of two points of ˜Cmlying over pm.
+Similarly,definethebijection β:Ca(K)→l(K),bymappingeachsmooth
+pointq∈C(K) to the intersection point with lof the line, passing through
+paandq. The point on lwhere the triple tangent at cusp intersects it, is
+denoted ∞a.
+Finally, put
+µ:=β−1◦α:M∪{0m,∞m} →A∪{∞a} (2.8)
+Thusl(K) acquires both structures: of a combinatorial line and of a g eo-
+metric line.
+2.6.(Cm,Ca)–configurations from cubic surfaces. LetVbe a
+smooth cubic surface defined over K. At each non–singular point p∈V(K),
+there exists a well defined tangent plane to Vdefined over K. The intersec-
+tion of this plane with V, forpoutside of a proper Zariski closed subset, is
+a geometrically irreducible curve C, having pas its single singular point.
+Again, generically it is of twisted multiplicative type, if charK/\e}atio\slash= 2, and
+of twisted additive type, when plies on a curve in V.
+Therefore, under these genericity conditions , replacing Kby its finite ex-
+tension if need be, and renaming this new field K, we can find two tangent
+plane sections of Vthat form a ( Cm,Ca)–configuration in the ambient pro-
+jective space.
+2.6.1. Example. Consider the diagonal cubic surface/summationtext4
+i=1aiz3
+i= 0
+over a field Kof characteristic /\e}atio\slash= 3. Then the discriminant of the quadratic
+equation defining directions of two tangents of the tangent s ection at ( z1:
+z2:z3:z4), up to a factor in K∗2, is
+D:=4/productdisplay
+i=1aizi.13
+Hence the set of points of (twisted) additive type consists o f four elliptic
+curves
+Ei:zi=/summationdisplay
+j/negationslash=iajz3
+j= 0, i= 1,...,4.
+The remaining points (outside 27 lines) are of (twisted) mul tiplicative type.
+Those for which D∈K∗2are of purely multiplicative type.
+2.7. Reconstruction of the configuration itself. Returning to the
+map (2.8), we see that Kcan be reconstructed from the ( Cm,Ca) configura-
+tion, using onlythe collinearity relation on the set
+/tildewideCm(K)∪Ca(K)∪l(K). (2.9)
+Moreover, we get the canonical structure of a projective lin e overKonl,
+together with the family of five K–points on it.
+To reconstruct the whole configuration, as a K–scheme up to an isomor-
+phism, from the same data, it remains to give in addition two 0 –cycles on l:
+its intersection with CmandCarespectively. Again, passing to a finite ex-
+tension of K, if need be, we may and will assume that all intersection poin ts
+inCm∩l,Ca∩lare defined over K. This again means that these cycles
+belong to the respective collinearity relation on Cm(K)∪Ca(K)∪l(K).
+To show that knowing these cycles, we can reconstruct CmandCain their
+respective projective planes, let us look at the equations o f these curves.
+InPm, choose projective coordinates ( z1:z2:z3) overKin such a way
+thatlis given by the equation z3= 0,pmis (0 : 0 : 1), equations of two
+tangents at pmarez1= 0,z2= 0, and the points 0 m,∞mare respectively
+(0 : 1 : 0) and (1 : 0 : 0) Then the equation of Cmmust be of the form
+z1z2z3+c(z1,z2) = 0
+wherecis a cubic form. To give the intersection Cm∩lis the same as to give
+the linear factors of c. Sinceziare defined up to multiplication by constants
+fromK∗, this defines ( Cm,Pm) up to isomorphism.
+Similar arguments work for Ca; its equation in coordinates ( z′
+1:z′
+2:z′
+3)
+onPasuch that lis defined by z′
+3= 0, will now be
+z′2
+1z3+c′(z′
+1,z′
+2) = 0.14
+We may normalize z′
+2by the condition that 0 a= (1 : 0 : 0), and then
+reconstruct linear factors of c′from the respective intersection cycle Ca∩l.
+2.8. Reconstruction of Vfrom a tangent tetrahedral configura-
+tion.Let now Vbe a cubic surface over K. Assume that V(K) contains
+four points pi,i= 1,...,4, such that tangent plains Piat them are pairwise
+distinct. Moreover, assume that tangent sections Ciare either of multiplica-
+tive, or of additive type, and each of these two types is repre sented by some
+Ci. One can certainly find such pidefined over a finite extension of K.
+We will call such a family of subschemes ( pi,Ci,Pi)a tetrahedral configu-
+ration, even when we do not assumed a priori that it comes from a V. If it
+comes from a V, we will say that it is a tangent tetrahedral configuration.
+Without restricting generality, we may choose in the ambien tP3
+Ka co-
+ordinate system ( z1:···:z4) in such a way that zi= 0 is an equation of
+Pi.
+If the configuration is tangent to V, letF(z1,...,z 4) = 0 be the equation
+ofV. HereFis a cubic form with coefficients in Kdetermined by Vup to
+a scalar factor. For each i∈ {1,...,4}, writeFin the form
+F=3/summationdisplay
+a=0za
+if(i)
+3−a(zj|j/\e}atio\slash=i), (2.10)
+wheref(i)
+bis a form of degree bin remaining variables.
+Clearly, f(i)
+3= 0 is an equation of Ciin the plane Pi. Hence Kand
+this equation can be reconstructed, up to a common factor, fr om a part of
+the tetrahedral configuration consisting of Pi, another plane Pjwith tangent
+section of different type, and the induced relation of collin earity on them.
+Considerthegraph G=G(V;p1,...,p 4)withfourverticeslabeled(1 ,...,4),
+in which iandj/\e}atio\slash=iare connected by an edge, if there is a cubic monomial
+in (zk|k/\e}atio\slash=i,j), that enters with nonzero coefficients in both f(i)
+3andf(j)
+3.
+We want this graph to be connected. This will hold, for exampl e, if inF
+all four coefficients at z3
+ido not vanish. It is clear from this remark that
+connectedness of Gis an open condition holding on a Zariski dense subset of
+all tangent configurations.
+2.8.1. Proposition. If the tetrahedral configuration is tangent to V, with
+connected graph G, then this Vis unique.15
+Proof.Letg(i)be a cubic form in zk,k/\e}atio\slash=i,such that zi= 0,g(i)= 0
+are equations of Ci. We may change g(i)multiplying them by non–vanishing
+constants ci∈K. If our configuration is tangent to V, given by (2.10), we
+may find ciin such a way that cig(i)=f(i)
+3. The obtained family of forms
+{cig(i)}iscompatible in the following sense: if a cubic monomial in only
+two variables has non–zero coefficients in two g(i)’s, then these coefficients
+coincide. In fact, they are equal to the coefficient of the resp ective monomial
+inF.
+Conversely, if such a compatible system exists, and moreove r, the graph
+Gis connected, then ( ci) is unique up to a common factor. From such cig(i)
+one can reconstruct a cubic form of four variables, which wil l be necessarily
+proportional to F: coefficient at any cubic monomial min (z1,...,z 4) in it
+will be equal to the coefficient of this monomial in any of cig(i), for which zi
+does not divide m.
+2.9. Summary. This section was dedicated to several key constructions
+that show how and under what conditions a cubic surface Vconsidered as a
+scheme, together with a ground field K, can be reconstructed from its set of
+K–points, endowed with some combinatorial data.
+The main part of the data was the collinearity relation on Vsm(K), and
+this relation, when it came from geometry, satisfied some str ong conditions
+stated in Lemma 2.2.
+However, this Lemma and the data used in 2.8 made appeal also t o in-
+formation about points on the lines of intersections of tang ent planes: cf.
+specifically constructions of maps αandβbefore formula (2.8).
+We want to get rid of this extra datum and work onlywith points of
+Vsm(K).
+This must be compensated by taking in account, besides the co llinearity
+relation, an additional coplanarity relation onV(K), essentially given by the
+sets ofK–points of (many) non–tangent plane sections.
+The next section is dedicated principally to a description o f the relevant
+abstract combinatorial framework. The geometric situatio ns are used mainly
+to motivate or illustrate combinatorial definitions and axi oms.
+3. Combinatorial and geometric cubic surfaces
+3.1. Definition. A combinatorial cubic surface is an abstract set S
+endowed with two structures:16
+(i) A symmetric ternary relation “collinearity”: L ⊂S3. We will say that
+triples(p,q,r)∈ Lare collinear.
+(ii) A set Pof subsets C⊂Scalled plane sections.
+These relations must satisfy the axioms made explicit below in the sub-
+sections 3.2 and 3.3. Until all the axioms are stated and impo sed, we may
+call a structure ( S,L,P) acubic pre–surface.
+3.2. Collinearity Axioms. (i) For any (p,q)∈S2, there exists an r∈S
+such that (p,q,r)∈L.
+Call the triple (p,q,r)strictly collinear, if ris unique with this property,
+andp,q,rare pairwise distinct.
+(ii) The subset Ls⊂ Lof strictly collinear triples is a symmetric ternary
+relation.
+(iii) Assume that p/\e}atio\slash=qand that there are two distinct r1,r2∈Swith
+(p,q,r1)∈ Land(p,q,r2)∈ L. Denote by l=l(p,q)the set of all such r’s.
+Thenl3⊂ L, that is any triple (r1,r2,r3)of points in lis collinear.
+Such sets lare called lines in S.
+3.2.1. Example: combinatorial cubic surfaces of geometric ori-
+gin.LetKbe a field, and Va cubic surface in P3overK. Denote by
+S=Vsm(K) the set of nonsingular K–points of V.
+We endow Swith the following relations:
+(a) (p,q,r)∈ Liff either p+q+ris the complete intersection cycle of V
+with a line in P3defined over K(K–line), or else if p,q,rlie on aK–line
+P1
+K, entirely contained in V.
+(b) LetP⊂P3be aK–plane. Assume that it either contains at least
+two distinct points of S, or is tangent to a K–pointp, or else contains the
+tangent line to one of the branches of the tangent section of m ultiplicative
+type. Then C:=P(K)∩Sisan element of P. All elements of Pare obtained
+in this way.
+3.3. Plane sections. We now return to the general combinatorial situ-
+ation. Let ( S,L,P) be a cubic pre–surface.
+For anyp∈S, put
+Cp=Cp(S) :={q|(p,p,q)∈ L}∪{p}. (3.1)
+3.3.1. Tangent Plane Sections Axiom. For each p∈S, we have
+Cp∈ P.Such plane sections are called tangent ones.17
+The next geometric property of plane sections of geometric c ubics can now
+be rephrased combinatorially as follows.
+3.3.2. Composition Axiom. (i) LetC∈ Pbe a non–tangent plane
+section containing no lines in S. Then the collinearity relation Linduces on
+suchCa structure of Abelian symmetric quasigroup (cf. Theorem–D efinition
+1.2).
+(ii) LetCp=Cp(S)be a tangent plane section containing no lines. Then
+Linduces on C0
+p:=Cp\{p}a structure of Abelian symmetric quasigroup.
+Choosing a zero/identity point in C, resp.Cp\{p}, we get in this way a
+structure of abelian group on each of these sets.
+3.3.3. Pencils of Plane Sections Axiom. Letλ:= (p,q,r)∈ L.
+Assume that at least two of the points p,q,rare distinct. Denote by Πλ⊂ P
+the set
+Πλ:={C∈ P|p,q,r∈P}. (3.2)
+and call such Πλ’s pencils of plane sections.
+Then we have:
+(i) If(p,q,r)do not lie on a line in S, then
+S\{p,q,r}=/coproductdisplay
+C∈Πλ(C\{p,q,r}) (3 .3)
+(disjoint union).
+(ii) If(p,q,r)lie on a line l, then
+S\l=/coproductdisplay
+C∈Πλ(C\l) (3 .4)
+(disjoint union).
+3.4. Combinatorial plane sections Cpof multiplicative/additive
+types.First of all we must postulate ( p,p,p)∈ L, since in the geometric
+case (p,p,p)/∈ Lcan happen only in a twisted case.
+There are two different approaches to the tentative distinct ion between
+multiplicative and additive types. In one, we may try to prefi gure the future
+realization of CmandCaas essentially the multiplicative (resp. additive)
+groups of a field Kto be constructed.18
+Then, restricting ourselves for simplicity by fields of char acteristic zero,
+we see that Cp\{p}which is ofadditive typeafter a choice of 0 amust become
+a vector space over Q(be uniquely divisible), whereas the respective group
+of multiplicative type is never uniquely divisible.
+However, these restrictions are too weak.
+Instead, we will define pairsof combinatorial tangent plane sections mod-
+eled on ( Cm,Ca)–configurations of sec. 2. After this is done, we will be able
+to “objectively”, independently of another member of the pa ir, distinguish
+between CmandCausing e.g. the divisibility criterion.
+3.5. Combinatorial (Cm,Ca)–configurations. We can now give a
+combinatorialversionofthose( Cm,Ca)–configurations, that inthegeometric
+case consist of two tangent plane sections of a cubic surface , one of additive,
+another of multiplicative type.
+The main point is to see, how to use combinatorial plane secti ons in place
+of “external” lines l=Pm∩Pa. This is possible, because the set of points of
+thisline willnow be replaced by bijective to it set of plane s ections, belonging
+to a pencil, defined in terms of ( Cm,Ca), and geometrically consisting just
+of all sections containing pmandpa.
+Let (S,L,P) be a combinatorial pre–surface, satisfying Axioms 3.2, 3. 3.1,
+3.3.2, 3.3.3.
+Start with two distinct points of S, not lying on a line in S, and respective
+tangent sections of S
+(pm,pa;Cpm,Cpa) (3 .5)
+Letr∈Sbe the unique third point such that ( pm,pa,r)∈ L,λ:=
+{pm,pa,r}.PutC0
+pm:=Cpm\{pm},C0
+pa:=Cpa\{pa}.
+Denote by Π λthe respective pencil of plane sections. Consider the follo w-
+ing binary relation:
+R⊂Cpa×Cpm: (p,q)∈R⇐⇒ ∃P∈Πλ, p,q∈P. (3.6)
+3.5.1. Definition. (pm,pa;Cpm,Cpa)is called a (Cm,Ca)–configuration,
+if the following conditions are satisfied.
+(i)Ris a graph of some function
+λ:Cpa→Cpm (3.7)19
+This function must be a bijection outside of two distinct poi nts0m,∞m∈C0
+pa
+which are mapped to pm. Besides, we must have λ(pa)∈C0
+pm.
+Assuming (i), put
+A:=C0
+pa, M:=C0
+pm
+Introduce on these sets the structures of abelian groups usi ng the the Com-
+position Axiom 3.3.2 and some choices of zero and identity
+0a∈C0
+pa,1m∈C0
+pm.
+Define the map
+µ:M∪{0m,∞m} →A∪{∞a} (3.8)
+which is λ−1onMand identical on 0m,∞m. Then
+(ii) Conditions of Lemma 2.2 must be satisfied for this µ.
+(iii)Cpm∩Cpaconsists of three pairwise distinct points.
+Thus, if ( pm,pa;Cpm,Cpa) is a (Cm,Ca)–configuration, then we can com-
+binatorially reconstruct the ground field and the isomorphi c geometric con-
+figuration.
+However, passing to tetrahedral configurations, we have to i mpose ad-
+ditional combinatorial compatibility conditions, that in the geometric case
+were automatic.
+They are of two types:
+(a) If two planes Pi,Pjof the tetrahedron carry plane sections of the
+same type (both additive or both multiplicative), we must wr ite combina-
+torially maps, establishing their isomorphism, and postul ate this fact in the
+combinatorial setup.
+This can be done similarly to the case of ( Cm,Ca)–configurations.
+(b) If a schematic tangent plane section Cican be reconstructed from two
+different pairs of tetrahedral plane sections Ci,Cjand (Ci,Ck), the results
+must be naturally isomorphic.
+It is clear, how to do it in principle, and the respective cons traints must
+be stated explicitly.
+I abstain from elaborating all details here for the followin g reason.20
+If, asamainapplicationofthistechnique, onetriestoimit atetheapproach
+to weak Mordell–Weil problem following the scheme of 0.4, th en the neces-
+sary combinatorial constraints will probably hold automat ically for finitely
+generated combinatorial subsurfaces of an initial geometr ic surface.
+The real problem is: how to recognize that a given (say, finite ly gener-
+ated) combinatorial subsurface of a geometric surface is ac tually the whole
+geometric surface.
+I do not know any answer to this problem.
+It is well known, however, that such proper combinatorial su bsurfaces do
+exist.
+For example, when K=RandV(R) is not connected, one of the compo-
+nents can be a combinatorial cubic surface in its own right. M ore generally,
+some unions of classes of the universal equivalence relatio n ([M1]) are closed
+with respect to collinearity and coplanarity relations: th is can be extracted
+from the results of [SwD1].
+4. Cubic curves and combinatorial cubic curves over large fie lds
+4.1. Large fields and smooth cubic curves. Consider a smooth cubic
+curveC⊂ P2
+Kdefined over K. PutS:=C(K) and endow Swith the
+collinearity relation L ⊂S3defined by (1.3). Let L0:=L/S3, the set of
+orbits of Lwith respect to the permutations. We may and will represent t he
+image in L0of (p,q,r)∈ Las the 0–cycle p+q+r.
+Now assume that Khas no non–trivial extensions of degree 2 and 3. Then
+all intersection points of any K–line with Clie inC(K).Therefore, we have
+the canonical bijection
+ξ:{K−lines in P2(K)} → L0:l/mapsto→intersection cycle l ∩C.(4.1)
+In this approach, K–points of P2(K)}have to be characterized in terms
+of pencils of all lines passing through a given point. Theref ore, it is more
+natural to work with the dual projective plane from the start .
+Let/hatwideP2be the projective plane dual to the plane in which Clies. Combi-
+natorially, K–points/hatwidel(resp. lines /hatwidep) of/hatwideP2areK–linesl(resp. points p) of
+P2
+K, with inverted incidence relation: /hatwidel∈/hatwidepiffp∈l. Thus, (4.1) turns into a
+bijection
+/hatwideξ:{K−points in /hatwideP2(K)} → L0:/hatwidel/mapsto→intersection cycle l ∩C.(4.2)21
+Thus,/hatwideξsends lines in /hatwideP2to certain subsets in L0that we also may call
+pencils.
+This geometric situation motivates the following definitio n.
+Let (S,L) be an abelian symmetric quasigroup. Put L0=L/S3.
+Assume that L0is endowed with a set of its subsets P0, calledpencils,
+which turns it into a combinatorial projective plane, with p encils as lines.
+This means that besides the trivial incidence conditions, P appus Axiom is
+also valid. Hence we can reconstruct from ( L0,P0) a field Ksuch that
+L0=P2(K),P0= the set of K–lines in P2(K).
+The following Definition, inspired by geometry, encodes the interaction
+between the structures ( S,L) and (L0,P0).
+4.2. Definition. The structure (S,L,P0)is called a combinatorial cubic
+curve over a large field, if the following conditions are sati sfied.
+(i) For each fixed p∈S, the set of cycles p+q+r∈ L0,q,r∈S, is a
+pencilΠp.
+(ii) If a pencil Πis not of the type Πp, then any two distinct elements in
+Πdo not intersect (as unordered triples of S–points).
+(iii) For each pencil Πq(resp. each pencil not of type Πq) and any p∈S,
+(resp. any p/\e}atio\slash=q) there exists a unique cycle in L0contained in Πand
+containing p.
+Obviously, each geometric smooth cubic curve over a large fie ld defines
+the respective combinatorial object.
+4.3. Question. Are there such combinatorial curves not coming from a
+geometric one? In particular, are fields Kcoming from such combinatorial
+objects necessarily “large” in the sense of algebraic defini tion above (closed
+under taking square and cubic roots)?
+Similar constructions can be done and question asked for cub ic surfaces.
+Notice that over a large field, any point on a smooth surface, n ot lying on a
+line, is of either multiplicative, or additive type.
+APPENDIX. Mordell–Weil and height: numerical evidence
+LetVbe a geometrically irreducible cubic curve or a cubic surfac e over
+a fieldK, with the standard geometric collinearity relation (1.3) f or curves,
+(3.2.1a) for surfaces, and the binary composition law (1.1) for curves. For
+surfaces, we will state the following fancy definition.22
+A.1. Definition. LetS⊂V(K), andX1,...,X N,...free commuting
+but nonassociative variables,
+w= (...(Xi1◦Xi2)◦(Xi3◦...(···◦Xik)...)
+a finite word in this variables,
+ev:{X1,...,X N,...} →S
+an evaluation map.
+a) A point p∈V(K) is called the strong value of wat(S,ev) if during
+the inductive calculation of
+p=ev(W) := (...(ev(Xi1)◦ev(Xi2))◦(ev(Xi3))◦...(···◦ev(Xik)...)
+we never land in a situation where the result of composition i s not uniquely
+defined, that is x◦xwith singular xfor a curve, or x◦ywherey=xor the
+line through x,ylies inVfor a surface.
+b) A point p∈S(K) is called a weak value of wat(S,ev) if during the
+inductive calculation of
+p:=evweak(W) = (...(ev(Xi1)◦ev(Xi2))◦(ev(Xi3))◦...(···◦ev(Xik)...)
+whenever we land in a situation where ◦is not defined, we are allowed to
+choose as a value of y◦z(resp.y◦y) any point of the line yz(resp. any
+point of intersection of a tangent line to VatywithV.)
+Thus, weak evaluation produces a whole set of answers.
+A.2. Definition. (i) A subset S⊂V(K)strongly generates V(K), if
+V(K)coincides with the set of all strong S–values of all words was above.
+(i) A subset S⊂V(K)weakly generates V(K), ifV(K)coincides with
+the set of all weak S–values of all words w.
+Now we can state two versions of Mordell–Weil problem for cub ic surfaces.
+Strong Mordell–Weil problem for V:Is there a finite Sthat strongly
+generates V(K)?
+Weak Mordell–Weil problem for V:Is there a finite Sthat weakly
+generates V(K)?23
+For curves, one often calls the weak Mordell–Weil theorem th e statement
+thatC(K)/2C(K)isfinite (referring to the groupstructure p+q=e◦(p◦q)).
+A.3. Proving strong Mordell–Weil for smooth cubic curves ov er
+number fields. The classical strategy of proof includes two ingredients.
+(a) Introduce an arithmetic height function h:P2(K)→R. E.g. for
+K=Q, p= (x0:x1:x2)∈ P2(Q), xi∈Z,g.c.d.(xi) = 1
+put
+h(p) := max i|xi|.
+(b)Prove the descent property :∃H0such that if h(p)> H0, p∈C(K),then
+p=q◦rfor some q,r∈C(K) withh(q),h(r)< h(p).
+The same strategy works for general finitely generated fields . For larger
+fields, the strong Mordell–Weil generally fails, but the wea k one might sur-
+vive.
+LetKbe algebraically closed, or R, or a finite extension of Qp. LetCbe
+a smooth cubic curve, Va smooth cubic surface over K.
+Then:
+–V(K) is weakly finitely generated, but not strongly.
+–C(K), if non–empty, is not finitely generated.
+A.4. Point count on cubic curves. It is well known that as H→ ∞,
+card{p∈C(K)|h(p)≤H}=const·(logH)r/2(1+o(1)),
+r:= rkC(K) = rkPic C. (A.1)
+A.5. Point count on cubic surfaces. Here we have only a conjecture
+and some partial approximations to it:
+Conjecture: asH→ ∞,
+card{p∈V0(K)|h(p)≤H}=const·H(logH)r−1(1+o(1)),
+V0:=V\{all lines}, r:= rkPic V (A.2)24
+A proof of (A.1) can be obtained by a slight strengthening of t he tech-
+nique used in the finite generation proof. Namely, one shows t hat logh(p) is
+“almost a quadratic form” on C(K). In fact, it differs from a positive defined
+quadratic form by O(1), so that (A.1) follows from the count of lattice point
+in an ellipsoid.
+The descent property used for Mordell–Weil ensures that thi s quadratic
+form is positive definite.
+How could oneattack thisconjecture? For the circlemethod, there are too
+few variables. Moreover, connections with Mordell–Weil fo r cubic surfaces
+are totally missing.
+Nevertheless, the inequality
+card{p∈V0(K)|h(p)≤H}> const·H(logH)r−1
+is proved in [SlSw–D] for cubic surfaces over Qwith two rational skew lines.
+There are also results for singular surfaces: cf. [Br], [BrD 1], [BrD2].
+A.6. Some numerical data. HereIwillsurvey somenumerical evidence
+computed by Bogdan Vioreanu, cf. [Vi].
+In the following tables, the following notation is used.
+Input/table head: code [a1,a2,a3,a4] of the surface
+V:4/summationdisplay
+i=1aix3
+i= 0.
+Outputs:
+(i)GEN: Conjectural list of weak generators
+p:= (x1:x2:x3:x4)∈V(Q)
+.
+(ii)Nr: The length of the list Listgoodof all points x, ordered by the in-
+creasing height h(p) :=/summationtext
+i|xi|, such that any point of the height ≤maximal
+height in Listgood, is weakly generated by GEN.
+(iii)Hbad: the height of the first point that was NOT shown to be gener-
+ated byGEN.25
+(iv)L: the maximal length of a non–associative word with generato rs in
+(GEN,◦) one of whose weak values produced an entry in Listgood.
+Example: ForV= [1,2,3,4], we have:
+GEN={p0:= (1 :−1 :−1 : 1)}
+Nr= 8521: the first 8521 points in the list of points of increasin g height
+are weakly generated by the single point p0.
+L= 13: the maximal length of a non–associative word in ( p0,◦) repre-
+senting some point of the Listgoodwas 13.
+Hbad= 24677: the first point that was not found to be generated by p0
+was of height 24677.
+SELECTED DATA
+[1,2,3,5],rkPic = 1
+—————————
+GEN Nr L H bad
+—————————————————————–
+(0:1:1:-1) 15222 12 23243
+(1:1:-1:0)
+(2:-2:1:1)
+[1,1,5,25],rkPic = 2
+—————————
+(1:-1:0:0) 32419 9 30072
+(1:4:-2:-1)
+[1,1,7,7],rkPic = 2
+—————————
+(0:0:1:-1) 16063 7 2578
+(1:-1:0:0)
+(1:-1:-1:1)
+(1:-1:1:-1)
+A.7. Discussion of other numerical data. Bogdan Vioreanu studied
+all in all 16 diagonal cubic surfaces V; he compiled lists of all points up to
+height 105, for some of them up to height 3 ·105.26
+The conjectural asymptotics (A.2) seems to be confirmed.
+There is a good conjectural expression for the constant in (A.2) (for ap-
+propriately normalized height, not the naive one we used). I t goes back to
+works of E. Peyre. Its theoretical structure very much remin ds the Birch–
+Swinnerton–Dyer constant for elliptic curves. For theory a nd numerical evi-
+dence, see [PeT1], [PeT2], [Sw–D2], [Ch-L].
+The (weak) finite generation looks confirmed for most of the co nsidered
+surfaces, butsomestubbornlyresist, mostnotably[ 17,18,19,20],[4,5,6,7],
+[9,10,11,12].
+If one is willing to believe in weak finite generation (as I am) , the reason
+for failure might be the following observable fact:
+When one manages to represent a “bad” point pof large height as a
+non–associative word in the generators ( GEN,◦), the height of intermediate
+results (represented by subwords) tends to be much higher th anh(p), and
+hence outside of the compiled list of points.
+Finally,therelativedensityofpoints pforwhich“one–stepdescent” works,
+that is,
+p=q◦r, h(q),h(r)< h(p),
+seems to tend to a certain value 0 < d(V)<1.
+Question: Can one guess a theoretical expression for d(V)?
+Notice that on each smooth cubic curve C, the “one–step descent” works
+for all points of sufficiently large height, so that d(C) = 1.
+References
+[Br] T. D. Browning. The Manin conjecture in dimension 2.
+arXiv:0704.1217
+[BrD1]T. D. Browning, U. Derenthal. Manin’s conjecture for a quartic del
+Pezzo surface with A4singularity. Ann. Inst. Fourier (Grenoble) 59 (2009),
+no. 3, 1231–1265.
+[BrD2] T. D. Browning, U. Derenthal. Manin’s conjecture for a cubic
+surface with D5singularity. Int. Math. Res. Not. IMRN 2009, no. 14,
+2620–2647.
+[Ch-L] A. Chambert-Loir. Lectures on height zeta functions: At the con-
+fluence of algebraic geometry, algebraic number theory, and analysis.27
+arXiv:0812.0947
+[HrZ] E. Hrushovski, B. Zilber. Zariski geometries. Journ. AMS, 9:1
+(1996), 1–56.
+[K1] D. S. Kanevski. Structure of groups, related to cubic surfaces , Mat.
+Sb. 103:2, (1977), 292–308 (in Russian); English. transl. i n Mat. USSR
+Sbornik, Vol. 32:2 (1977), 252–264.
+[K2] D. S. Kanevsky, On cubic planes and groups connected with cubic
+surfaces. J. Algebra 80:2 (1983), 559–565.
+[KaM] D. S. Kanevsky, Yu. I. Manin. Composition of points and Mordell–
+Weil problem for cubic surfaces. In: Rational Points on Algebraic Vari-
+eties (ed. by E. Peyre, Yu. Tschinkel), Progress in Mathemat ics, vol. 199,
+Birkh¨ auser, Basel, 2001, 199–219. Preprint math.AG/0011 198
+[M1] Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. North
+Holland, 1974 and 1986.
+[M2]Yu.I.Manin. On some groups related to cubic surfaces . In: Algebraic
+Geometry. Tata Press, Bombay, 1968, 255–263.
+[M3]Yu.I.Manin. Mordell–Weil problem for cubic surfaces . In: Advances
+in the Mathematical Sciences—CRM’s 25 Years (L. Vinet, ed.) CRM Proc.
+and Lecture Notes, vol. 11, Amer. Math. Soc., Providence, RI , 1997, pp.
+313–318.
+[PeT1] E. Peyre, Yu. Tschinkel. Tamagawa numbers of diagonal cubic
+surfaces, numerical evidence. In: Rational Points on Algebraic Varieties,
+Progr. Math., 199. Birkh¨ auser, Basel, 2001, 275–305. arXi v:9809054
+[PeT2] E. Peyre, Yu. Tschinkel. Tamagawa numbers of diagonal cubic
+surfaces of higher rank. arXiv:0009092
+[Pr] S. J. Pride. Involutary presentations, with applications to Coxeter
+groups, NEC-Groups, and groups of Kanevsky . J. of Algebra 120 (1989),
+200–223.
+[SlSw–D] J.Slater, H. P. F. Swinnerton–Dyer. Counting points on cubic
+surfaces I. In: Ast´ erisque 251 (1998), 1–12.
+[Sw–D1] H. P. F. Swinnerton–Dyer. Universal equivalence for cubic sur-
+faces over finite and local fields. Symp. Math., Bologna 24 (1981), 111–143.
+[Sw–D2] H. P. F. Swinnerton–Dyer. Counting points on cubic surfaces II.
+In: Geometric Methods in Algebra and Number Theory. Progr. M ath., 235,
+Birkh¨ auser, Boston, 2005, pp. 303–309.28
+[Sw–D3] H. P. F. Swinnerton–Dyer. Universal equivalence for cubic sur-
+faces over finite and local fields. Symp. Math., Bologna 24 (1981), 111–143.
+[Vi] B. G. Vioreanu. Mordell–Weil problem for cubic surfaces, numerical
+evidence. arXiv:0802.0742
+[Z] B. I. Zilber. Algebraic geometry via model theory. Contemp. Math.,
+vol. 131, Part 3 (1992), 523–537.
\ No newline at end of file
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+arXiv:1001.0224v1  [math.RT]  1 Jan 2010Publ. RIMS, Kyoto Univ.
+4x(200x), 000–000
+Algebraic analysis of minimal representations
+Dedicated to Mikio Sato whose pioneering work in
+algebraic analysis has been an inspiration for me.
+By
+Toshiyuki Kobayashi∗
+Abstract
+Small representations of a group bring us to large symmetrie s in a representation
+space. Analysis on minimal representations utilises large symmetries in their geomet-
+ric models, and serves as a driving force in creating new inte resting problems that
+interact with other branches of mathematics.
+This article discusses the following three topics that aris e from minimal repre-
+sentations of the indefinite orthogonal group:
+1. construction of conservative quantities for ultra-hype rbolic equations,
+2. quantative discrete branching laws,
+3. deformation of the Fourier transform
+with emphasis on the prominent roles of Sato’s idea on algebr aic analysis.
+§1. Introduction
+The aim of this article is to highlight the prominent roles of Sato’s idea on
+hyperfunctions and D-modules in the new developments on analysis of minimal
+representations [4, 35, 36, 40].
+Communicated by JJ, December 31, 2009; Revised *.
+2010 Mathematics Subject Classification(s): Primary 22E46 , Secondary 22E45, 53A30,
+46F15, 58J15
+Key Words : minimal representations, hyperfunction, branching law, reductive group,
+generalized Fourier transform, holomorphic semigroup, co nservative quantity, D-module.
+Partiallysupported byGrant-in-AidforScientific Researc h (B)(18340037), Japan Society
+for the Promotion of Science
+∗Graduate School of Mathematical Sciences, The University o f Tokyo, 3-8-1 Komaba,
+Meguro, Tokyo, 153-8914 Japan.
+c/circlecopyrt200x Research Institute for Mathematical Sciences, Kyoto U niversity. All rights reserved.2 Toshiyuki Kobayashi
+Minimal representations are the simplest, infinite dimensional ‘unipot ent
+representations’. They are building blocks of unitary representat ions. Segal–
+Shale–Weil representation is a classic example for the split simple grou p of type
+C. There has been an active study on minimal representations of red uctive
+groups, mostly by algebraic approaches since 1990s both in the real and in the
+p-adic fields [5, 6, 10, 12, 22, 23, 37, 42, 43, 46, 50, 51, 52].
+On the other hand, I believe that geometric analysis on minimal represen-
+tations is also a promising area, and have been advocating its study b ased on
+the following change of viewpoints:
+smallrepresentations of a group
+=largesymmetries in a representation space . (1.1)
+The terminology ‘minimal representations’ is defined inside represen tation the-
+ory (i.e. the annihilator in the universal enveloping algebra is the Jose ph ideal,
+see e.g. [10]), and the corresponding ‘largest symmetries’ are expe cted to serve
+as a driving force in creating new interesting areas of mathematics e ven outside
+representation theory.
+The ‘largest symmetries’ in representation spaces of minimal repre sen-
+tations may be also observed in branching laws. Indeed, as we shall s ee in
+Theorem 3.4, it may well happen that broken symmetries of minimal repre-
+sentations reduce to analysis on certain semisimple symmetric space s (see also
+[30,39,41]). Thisobservationindicatesthatanalysisonminimalrepr esentation
+involves higher symmetries than those for (traditional) analysis on s ymmetric
+spaces.
+We focus on the minimal representation of a simple group of type D. T his
+is just a single irreducible representation, however, it turns out th at geometric
+analysis on its various models is surprisingly rich. Indeed, papers dev oted to
+this single representation in very recent years already exceed 500 pages, giving
+rise to the interactions with the following topics:
+•conformal geometry for general pseudo-Riemannian manifolds [30 , 38],
+•Dolbeault cohomologies on open complex manifolds [37].
+•conservative quantities for ultra-hyperbolic equations [40],
+•breaking symmetries and discrete branching laws [39],
+•Schr¨ odinger model and the unitary inversion operator [34, 36],
+•deformation of Fourier transforms [4],Algebraic analysis of minimal representations 3
+•holomorphic semigroup [33, 35],
+•new special function theory for fourth order differential operat ors [17, 18].
+In this article, we choose three topics among them, and try to expla in
+their flavours in Sections 2, 3 and 4, respectively with emphasis on th e role of
+Sato’s idea on algebraic analysis, both in philosophy and in techniques. For
+the reader’s convenience, we list some representationtheoretic p roperties of our
+minimal representation in the Appendix.
+§2. Conservative quantities for D-modules.
+The energy of a wave is a conservative quantity for the wave equat ion,
+namely,itisinvariantundertime-translations. Inthissection, wedis cusshigher
+symmetriescomingfromconformaltransformations. Byusingthe ideaofSato’s
+hyperfunctions [20, 47], we construct conservative quantities fo r specific ultra-
+hyperbolic equations (see Theorem 2.6).
+§2.1. Yamabe operator and conformal geometry
+A diffeomorphism hof a Riemannian manifold ( X,g) is said to be confor-
+malif there exists a positive-valued function Ω( h,·) onXsuch that
+h∗ghx= Ω(h,x)2gxforx∈X.
+It is isometry if Ω( h,·)≡1. We write
+Isom(X,g)⊂Conf(X,g)
+for the groups consisting of isometries and conformal diffeomorph isms, respec-
+tively. The same notation will be applied to a more general setting whe re
+gis a non-degenerate symmetric tensor, namely, to an indefinite-Rie mannian
+manifold.
+TheinvariancefortheLaplacian∆ Xcharacterizesisometriesamongdiffeo-
+morphismsof X. In other words, anon-isometric transformationon ( X,g) does
+not preserve ∆ X. However, the Laplacian ∆ Xis still subject to the following
+covariance under conformal transformations:
+̟n+2
+2(h)◦/tildewide∆X=/tildewide∆X◦̟n−2
+2(h) for any h∈Conf(X,g),(2.1)
+wherenis the dimension of X, ScalXis the scalar curvature, and
+/tildewide∆X:=∆X−n−2
+4(n−1)ScalX(the Yamabe operator),
+̟λ(h)f(x) :=Ω(h−1,x)λf(h−1x) for f∈C∞(X).4 Toshiyuki Kobayashi
+The formula(2.1)implies that the operator∆ X(or/tildewide∆X) is not conformally
+invariant, but the D-module generated by /tildewide∆Xis conformally invariant ! As far
+as the solutions are concerned, the invariance of the D-module is sufficient.
+Namely, by putting
+Sol(/tildewide∆X) :={f∈C∞(X) : ∆Xf= Scal Xf}, (2.2)
+we get:
+Fact 2.1. The conformal group Conf(X,g)preservesSol(/tildewide∆X)via̟n−2
+2.
+See [38, Theorem 2.5] for the proof.
+Remark2.2.
+1) Other eigenspaces Sol(/tildewide∆X−λ) are not conformally invariant if λ/\e}atio\slash= 0.
+2) It may be better formulated if we use the ring of twisted different ial oper-
+ators acting on sections of the line bundle Ln−2
+2.
+3) The differential equation /tildewide∆Xf= 0, namely, ∆ Xf= Scal Xf, is ellip-
+tic, hyperbolic, or ultra-hyperbolic, respectively, if ( X,g) is Riemannian,
+Lorentzian, or of general signature, respectively.
+Then a general problem is:
+Problem 2.3 (see [30, Problem C]) .
+1)Does there exist an invariant inner product on an appropriat e subspace of
+Sol(/tildewide∆X)?
+2)If yes, construct it explicitly.
+Such an inner product may be seen as a conservative quantity for t he
+solution to the equation /tildewide∆Xf= 0.Problem 2.3 does not find a final answer in
+the general setting. We shall give a partial answer in the flat case ( see Theorem
+2.6 below).
+§2.2. Conservative quantities
+LetRp,qbe the Euclidean space Rp+qendowed with the flat indefinite-
+Riemannian structure
+ds2=dx2
+1+···+dx2
+p−dx2
+p+1−···−dx2
+p+q.Algebraic analysis of minimal representations 5
+Then, the corresponding Laplace–Beltrami operator takes the f orm:
+/squarep,q:=∂2
+∂x2
+1+···+∂2
+∂x2p−∂2
+∂x2
+p+1−···−∂2
+∂x2
+p+q.
+Obviously,thescalarcurvatureon Rp,qvanishesidentically. Hence, theYamabe
+operator on Rp,qcoincides with /squarep,q. The space of solutions to /squarep,qf= 0,
+denoted bySol(/squarep,q), is obviously invariant under the motion group
+Isom(Rp,q)≃O(p,q)⋉Rp+q.
+It was proved in [40, Theorem 4.7] that Sol(/squarep,q) has even larger symmetries
+ifp+qis even, namely, by the indefinite orthogonal group
+G:=O(p+1,q+1) ={g∈GL(p+q+2,R) :tgIp+1,q+1g=Ip+1,q+1}
+acting on Rp+qas M¨ obius transforms. (To be more precise, Gpreserves the
+spaceSol0(/squarep,q) of smooth solutions with certain decay conditions at infinity
+together with their derivatives.)
+Remark2.4.
+1) The parity condition on p+qis crucial. In fact, a theorem of Howe and
+Vogan [51] asserts that there does not exist an infinite dimensional repre-
+sentation of Gwhose Gelfand–Kirillov dimension is p+q−1 ifp+qis odd
+andp,q >3.
+2)Sol0(/squarep,q) is defined as the twisted pull-back of smooth functions on the
+conformal compactification of Rp+q. See [40] for details.
+Problem 2.3 in this specific setting is stated as:
+Problem 2.5. Find aG-invariant inner product on Sol0(/squarep,q)if exists.
+§2.3. Unitarizability versus unitarization.
+Ifp,q >0 andp+qis even and greater than two, then we can tell a priori
+that the representation on Sol0(/squarep,q) is unitarizable and irreducible (e.g. [5,
+38])byalgebraictechniques. Namely, weknowtheexistenceandthe uniqueness
+of aG-invariant inner product on Sol0(/squarep,q) in this case.
+What we seek for in Problem 2.5 is not merely an abstract unitarizabil-
+itybut theunitarization of the representation space for a concrete geometric6 Toshiyuki Kobayashi
+model, namely, the construction of the invariant inner product. Th en, there
+are two approaches to the unitarization — one is easier, and the oth er is more
+challenging as discussed below.
+An easier approach to Problem 2.5 is to write the inner product by usin g
+the integral representation of solutions. Such an integral formu la was given in
+[40, Theorem 4.7] by using an explicit formula of the Green kernel [30, 40]. The
+disadvantage of this approach is that the formula of the inner prod uct involves
+a preimage of the integral representation, which is not canonically g iven.
+A second approach is to use an expansion of solutions into a countab le
+sum of better understood solutions, and then to give a Parseval–P lancherel
+type theorem. We shall discuss this approach in Section 3.
+A more intrinsic approach is to find a formula of the inner product dire ctly
+without using the integral representationof solutions. A hint to th is is the well-
+knownformulaofenergyto the waveequation, which isgivenby the in tegration
+of the Cauchy data on the hyperplane ( t= constant) in the space-time, see
+(2.6). (We note that, however, the energy is not conformally invar iant but
+invariant only under time-translations.)
+In order to explain the second approach, let us set up some notatio n. We
+recall that any non-characteristic hyperplane in Rp,qis written as
+α≡αv,c:={x∈Rp+q: (x,v)Rp,q=c} (2.3)
+forsome c∈Randv∈Rp,qsuchthat ( v,v)Rp,q=±1. Fixsuch v, andexpressa
+function fonRp+qas Sato’s hyperfunction ([47]) in the direction of v, namely,
+f(x) = lim
+ε↓0(f+(x+√
+−1εv)−f−(x−√
+−1εv)). (2.4)
+Here,f±(x+tv) is a holomorphic function of one variable tnear the real axis
+in±Imt >0.
+We set
+∂f±
+∂ν(x) :=∂
+∂t|t=0f(x+tv) (normal derivative),
+and introduce a function Qαfon the hyperplane α≡αv,cby
+Qαf:=1√−1(f+∂f+
+∂ν−f−∂f−
+∂ν).
+Finally, we define
+(f,f) :=/integraldisplay
+αQαf. (2.5)Algebraic analysis of minimal representations 7
+The right-hand side of (2.5) does not always converge, but it makes sense
+iffsatisfies suitable decay conditions, say, f∈Sol0(/squarep,q). Then, we can give
+an answer to Problem 2.5 as follows:
+Theorem 2.6 (see [40, Theorem 6.2], also [30]) .
+1)Forf∈Sol0(/squarep,q),(2.5)is independent of the choice of the pair (f+,f−)
+in the expression (2.4)and of the hyperplane α.
+2) (f,f)≥0for anyf∈Sol0(/squarep,q). The equality holds if and only if f= 0.
+3)The polarization of the norm (2.5)yields aG-invariant inner product on
+Sol0(/squarep,q).
+We denote bySol0(/squarep,q) for the Hilbert space obtained as the completion
+ofSol0(/squarep,q). Then, we get a unitary representation of G=O(p+1,q+1), to
+bedenoted by ̟≡̟p+1,q+1, onSol0(/squarep,q). Itturnsout thatthis isirreducible
+and a minimal representation of G. See Section 5 for representation theoretic
+properties of ̟.
+Remark2.7. The assertion 1) in Theorem 2.6 is a part of the invariance
+ofthe innerproduct( ,)becauseanynon-characteristichyperplaneisconjugate
+to either x1= 0 orxp+q= 0 by the motion group Isom( Rp,q)≃O(p,q)⋉Rp+q.
+We note that Gcontains Isom( Rp,q) as a proper subgroup.
+The proof of Theorem 2.6 was given in [40] by using some representat ion
+theoretic results of the representation ̟. It might be interesting to find a proof
+that does not depend on group theory but only on geometry such a s Stokes’
+theorem. We then pin down this as an open problem:
+Problem 2.8. Give a purely geometric proof to Theorem 2.6.
+§2.4. Energy generator
+Our conformally invariant inner product (2.5) is very close to the ene rgy
+of the wave, where one integrates Cauchy data on the zero time hy perplane.
+We end this section by making more explicit its connection.
+Forp= 1, let us introduce time and space coordinates ( t;x) instead of
+the previous coordinates ( x1,···,xp;xp+1,···,xp+q). Then, the energy of the
+wavefis given by
+E(f) =1
+2/integraldisplay
+Rq(|ft|2+|∇f|2)dx. (2.6)8 Toshiyuki Kobayashi
+Then, in terms of the inner product (2.5), E(f) is written as
+(f,|H|f) = (f+,Hf+)−(f−,Hf−)
+whereH=i∂tis the energy generator (infinitesimal time translations). Since
+the energy generator His invariant under time-translations (i.e. invariant by a
+one-dimensional subgroup of G) and the inner product ( ,) is invariant under
+the wholegroup G,E(f)isalsoinvariantundertime-translations. Thisexplains
+the classical fact that the energy (2.6) is a conservative quantity in the narrow
+sense that it is independent of which constant-time hyperplane we in tegrate
+over.
+§3. Quantative branching laws
+In Section 1, we have given a concrete formula of the conformally inv ariant
+inner product on the minimal representation ( conservative quantities ). It is
+given by the integral over hyperplanes. Yet another formula of th e same inner
+product will be given as a countable sum of well-understood quantitie s.
+This is a Parseval-type theorem (see Theorem 3.4), which is built on a
+‘good expansion theorem’ of solutions. Such an expansion theorem is obtained
+as a special case of the general theory of discretely decomposab le restrictions of
+unitary representations (see Theorem 3.2). We will see that algebr aic analysis
+providesa powerful method to branching problems in representat iontheory (cf.
+Problem 3.1 below).
+§3.1. Breaking symmetries and discrete decomposability
+Suppose π:G→GL(H) is a unitary representation of a Lie group G.
+Given a subgroup G′ofG, and consider the broken symmetry, namely, the re-
+striction π|G′. In general, the restriction π|G′decomposes into a direct integral
+of irreducible representations of G′. Our concern here is with:
+Problem 3.1 (see [24, 25]) .For which triple (G,G′,π)does the restric-
+tionπ|G′decompose discretely with finite multiplicities?
+It often happens that the irreducible decomposition of the restric tionπ|G′
+(branching law ) contains continuous spectrum if G′is non-compact. Even
+worse, each irreducible representation of G′may occur in the branching law
+with infinite multiplicities. Thus, Problem 3.1 seeks for a very nice class o f
+branching laws.Algebraic analysis of minimal representations 9
+Now, let us fix some notation for a real reductive group G. LetKbe
+a maximal compact subgroup of G,Ta maximal torus of K, andt,kthe Lie
+algebrasof T,K, respectively. We choose the set ∆+(k,t) of positive roots, and
+denote the dominant Weyl chamber by t+(⊂√−1t∗). We also fix a K-invariant
+inner product on k, and regard√−1t∗as a subset of√−1k∗.
+Suppose that K′is a closed subgroup of K. The group Kacts on the
+homogeneous space K/K′from the left, and then on the cotangent bundle
+T∗(K/K′) in a Hamiltonian fashion. We write
+µ:T∗(K/K′)→√
+−1k∗
+for the momentum map, and define the following closed cone by
+CK(K′) := Image µ∩t+.
+For a subgroup G′ofG, we shall consider CK(K′) by setting K′:=K∩G′.
+Next, suppose that πisa (reducible) representationofacompact Liegroup
+K. The asymptotic K-support of π, to be denoted by AS K(π), was introduced
+by Kashiwara and Vergne [21] as the asymptotic cone of the K-types of π.
+From definition AS K(π) ={0}if dimπ <∞. For a representation πofG, we
+can define AS K(π) for the restriction π|K.
+We are ready to state an answer to Problem 3.1:
+Theorem 3.2 (see [26]) .Suppose that πis a unitary representation of
+Gof finite length, and that G′is a closed subgroup of G. We set K′=K∩G′.
+If
+CK(K′)∩ASK(π) ={0}, (3.1)
+then the restriction π|G′decomposes discretely into a direct sum of irreducible
+unitary representations of G′with finite multiplicities.
+An upper estimate of the singularity spectrum of the hyperfunctio n char-
+acter ofπplays a crucial role in the proof of Theorem 3.2. In particular, the
+assumption (3.1) assures
+Restriction and Trace ( hyperfunction character ) commute. (3.2)
+Here, we remark that the character of an infinite dimensional repr esentation π,
+Traceπ(g) (g∈G)
+does not make sense as an ordinary function because Trace π(e) = dimπ=∞.
+Harish-Chandra proved that Trace πis well-defined as a distribution on Gifπ10 Toshiyuki Kobayashi
+is an irreducible unitary representation of a real reductive group G, and proved
+further that Trace πbelongs to L1
+loc(G). On the other hand, the restriction
+Traceπ|Kis not locally integrable on Kany more (see Atiyah [1]). What (3.2)
+means is that
+Trace(π|K′) = Trace( π)|K′
+as an identity of hyperfunctions (or distributions) on K′. See [26, Theorem 2.8]
+for the proof. We also refer to the lecture notes [32] for heuristic ideas of the
+proof.
+Recently, Hansen, Hilgert, and Keliny [13] has given an alternative pr oof
+of Theorem 3.2 by replacing Sato’s hyperfunctions with Schwartz’s d istribu-
+tions. See also [27, 28] for the necessary condition of discrete dec omposability
+of branching laws, where the associated variety of an infinite dimens ional rep-
+resentation π(an analogue of the characteristic variety of a D-module) plays
+an important role. The references [29, 31] discuss some application s of discrete
+branching laws.
+Loosely, Theorem 3.2 says that if CK(K′) and AS K(π) are not ‘large’ then
+the restriction π|G′is discretely decomposable. We note that CK(K′) ={0}if
+K′=K, and consequently, the assumption (3.1) is automatically satisfied. In
+this case, Theorem 3.2 is nothing but Harish-Chandra’s admissibility th eorem
+([14]). For any minimal representation πof a reductive group G, we get from
+[51] that AS K(π) is one-dimensional, i.e. AS K(π) =RvorR+vwherevis the
+highest root. Thus we can expect that there is a rich family of subgr oupsG′
+ofGfor which the restriction of the minimal representation of Gdecomposes
+discretely.
+§3.2. Space forms of indefinite Riemannian manifolds
+Before applying Theorem 3.2 to actual branching problems, we revie w
+quickly known results about the geometry and global analysison spa ce forms of
+indefinite-Riemannian manifolds (referred also to as pseudo-hyper bolic spaces,
+generalized hyperboloids, etc.).
+We set
+Xp,q
++:={(x,y)∈Rp+1⊕Rq:||x||2−||y||2= 1} ≃O(p+1,q)/O(p,q),
+Xp,q
+−:={(x,y)∈Rp⊕Rq+1:||x||2−||y||2=−1}≃O(p,q+1)/O(p,q).
+We note that Xp,0
++≃SpandX0,q
+−≃Sq. By switching the factor, we have
+Xp,q
++≃Xq,p
+−.
+We induce an indefinite Riemannian structure on Xp,q
++andXp,q
+−from the
+ambient space Rp+1,qandRp,q+1, respectively. Then, Xp,q
++andXp,q
+−haveAlgebraic analysis of minimal representations 11
+constant sectional curvatures. Here is a summary of indefinite-R iemannian
+manifolds Xp,q
++andXp,q
+−:
+sectional curvature κsignature of metric tensor
+Xp,q
++ κ≡+1 ( p,q)
+Xp,q
+− κ≡−1 ( p,q)
+LetL2(Xp−1,q
++) be the Hilbert space of square integrable functions on
+Xp−1,q
++with respect to the induced volume element. For λ∈C, we set
+Vp,q
+λ:={f∈L2(Xp−1,q
++) :/tildewide∆Xp−1,q
++f= (1
+4−λ2)f},
+where the Yamabe operator /tildewide∆Xp,q
++takes the following form:
+/tildewide∆Xp,q
++= ∆Xp,q
++−1
+4(p+q)(p+q−2). (3.3)
+Clearly, the isometry group Isom( Xp−1,q
++)≃O(p,q) preserves Vp,q
+λfor any
+λ∈C. The representations on Vp,q
+λare called discrete series representations
+forXp−1,q
++ifVp,q
+λ/\e}atio\slash={0}, which were studied by Gelfand, Graev, Vilenkin,
+Shintani, Molchanov, Faraut, and Strichartz among others. We su mmarise:
+Proposition 3.3.
+1) (p= 1)Vp,q
+λ={0}for anyλ∈C.
+2) (p/\e}atio\slash= 1)Vp,q
+λ/\e}atio\slash={0}⇔λ∈p+q
+2+2Zandλ/\e}atio\slash= 0.
+Furthermore, O(p,q)acts irreducibly on each Vp,q
+λ, when it is non-zero.
+The resulting representation in Proposition 3.3 2) will be denoted by πp,q
+λ.
+SinceVp,q
+λ=Vp,q
+−λ, we may and do assume Re λ≥0 without loss of generality.
+By the coherent continuation of πp,q
+λforλ >0 such that λ∈p+q
+2+2Z, we can
+define irreducible unitary representations πp,q
+0(p+q:even) and πp,q
+−1
+2(p+q:odd)
+ofO(p,q). These representations do not lie in L2(Xp−1,q
++) but enjoy analogous
+algebraic properties to πp,q
+λ(λ >0) (see [24,§6] or [39,§5.4] the vanishing
+results on cohomologies in details).
+§3.3. Quantative branching laws12 Toshiyuki Kobayashi
+We return to the setting of Section 2.2. The flat indefinite-Riemannia n
+manifold Rp,qmay be seen as the direct product of two flat spaces:
+(Rp,dx2
+1+···+dx2
+p)×(Rq,−dx2
+p+1−···−dx2
+p+q).
+Likewise, the direct product of two space forms:
+Y:=Xp′,q′
++×Xp′′,q′′
+−
+is locally conformal to Rp,qfor anyp′,q′,p′′,q′′such that
+p′+p′′=p, q′+q′′=q.
+Thislocalconformalmapisgivenasfollows: For u= ((ξ0,ξ′,η′),(ξ′′,η′′,η0))∈
+R1+p′+q′⊕Rp′′+q′′+1,we set
+Φ(u) :=2
+ξ0+η0(ξ′,η′,ξ′′,η′′).
+Then, the restriction of Φ to Yis conformal (see [38, Lemma 3.3], for example).
+More precisely, the map
+Φ :Xp′,q′
++×Xp′′,q′′
+−→Rp,q(3.4)
+is well-defined and conformalin the open dense set Y′ofY, defined by ξ0+η0/\e}atio\slash=
+0. Correspondingly, if we set
+(/tildewideΦ∗f)(u) := (2
+ξ0+η0)p+q−2
+2f(Φ(u)) (3.5)
+then/tildewideΦ∗fsolves/tildewide∆Y/tildewideΦ∗f= 0 onY′if/squarep,qf= 0 (see [38, Proposition 2.6]).
+Here,/tildewide∆Yis the Yamabe operator on Y, which amounts to
+/tildewide∆Y=/tildewide∆Xp′,q′
++−/tildewide∆Xp′′,q′′
+−
+=∆Xp′,q′
++−∆Xp′′,q′′
+−−1
+4(p′+q′−p′′−q′′)(p+q−2).
+Hence, we can realize the minimal representation ̟ofO(p+1,q+1) on the
+solution space /tildewide∆YF= 0 as well through /tildewideΦ∗.
+We note that the map (3.4) is two to one at generic points. In ordert o give
+a global action of the group O(p+1,q+1) on the solution space to /tildewide∆YF= 0,
+we need to define F=/tildewideΦ∗fby (3.5) for ξ0+η0>0, and by the parity condition
+forξ0+η0<0 so that F(−u) = (−1)p−q
+2F(u) holds (see [40, (4.4.2a)]).Algebraic analysis of minimal representations 13
+In light that the isometry group of Y=Xp′,q′
++×Xp′′,q′′
+−is the reductive
+groupO(p′+1,q′)×O(p′′,q′′+1), it is natural to consider the branching law
+of the minimal representation ̟with respect to the following symmetric pair
+O(p+1,q+1)↓O(p′+1,q′)×O(p′′,q′′+1)
+by using the geometric model Y.
+In this setting, the criterion(3.1) ofTheorem3.2holds if andonly if p′′= 0
+orq′= 0 (see [39, Theorem 4.2]). Then, it follows from Theorem 3.2 that ̟
+decompose discretely. For the description of the irreducible decom position, we
+define the space of spherical harmonics of degree lby
+Hl(Rm) :={ϕ∈C∞(Sm−1) : ∆Sm−1ϕ=−l(l+m−2)ϕ}
+=/braceleftBigg
+ϕ∈C∞(Sm−1) :/tildewide∆Sm−1ϕ=/parenleftBigg
+1
+4−/parenleftbigg
+l+m−2
+2/parenrightbigg2/parenrightBigg
+ϕ/bracerightBigg
+.(3.6)
+The orthogonal group O(m) acts irreducibly on Hl(Rm) for any l∈N.
+Here is the branching law together with quantative information on th e
+invariant inner product:
+Theorem 3.4 (see [39, Theorem B]) .Suppose p+q(>2)is even,q=
+q′+q′′, andp,q >0. Then the twisted pull-back /tildewideΦ∗of the conformal map
+Φ :Y→Rp,qinduces the following quantative branching law:
+1)(branching law; O(p+1,q+1)↓O(p+1,q′)×O(q′′+1)).
+̟p+1,q+1|O(p+1,q′)×O(q′′+1)≃∞/summationdisplay
+l=0⊕
+πp+1,q′
+l+q′′
+2−1
+2⊗Hl(Rq′′+1) (3.7)
+Here, the right-hand side of (3.7)is a multiplicity-free Hilbert direct sum
+of irreducible representations of O(p+1,q′)×O(q′′).
+2)(Parseval-type theorem). For f∈ Sol0(/squarep,q), we expand /tildewideΦ∗finto the
+series/summationtext
+lFlaccording to the discrete decomposition (3.7). Then we have:
+||f||2
+Rp,q=∞/summationdisplay
+l=0(l+q′′
+2−1
+2)||Fl||2
+L2(Y) (3.8)
+Here||||Rp,qis the norm defined in Theorem 2.6.
+In view of (3.6), the self-adjoint operator1
+4−/tildewide∆Sq′′is non-negative, and
+therefore we can define a pseudo-differential operator/parenleftBig
+1
+4−/tildewide∆Sq′′/parenrightBig1
+4onY=
+Xp,q′
++×Sq′′as well as on Sq′′.14 Toshiyuki Kobayashi
+Hence, wegetanotherexpressionontheinvariantinnerproducto fthemin-
+imal representation in the geometric model Yby means of a pseudo-differential
+operator:
+Corollary 3.1. Suppose p+q(>2)is even,q=q′+q′′, andp,q′′>0.
+Let/parenleftBig
+1
+4−/tildewide∆Sq′′/parenrightBig1
+4be the pseudo-differential operator on Y=Xp,q′
++×Sq′′. We
+setF=/tildewideΦ∗fforf∈Sol0(/squarep,q). Then
+/ba∇dblf/ba∇dbl2
+Rp,q=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1
+4−/tildewide∆Sq′′/parenrightbigg1
+4
+F/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2
+L2(Y). (3.9)
+Remark3.5.
+1) Forq′′= 0 or 1, l+q′′
+2−1
+2≤0 ifl= 0. In this case Vp+1,q′
+l+q′′
+2−1
+2={0}.
+Nevertheless, we can justify the summand in (3.8) by using the argu ment
+of the analytic continuation.
+2) In the case p′′=q′= 0, the branching law (3.7) is nothing but the K-type
+formula of the minimal representation ̟, and (3.8) was proved earlier by
+Kostant [42] for p=q= 3, and by Binegar and Zierau [5] for general p,q
+such that p+qis even and greater than 2.
+3) In the case q′′= 0, we have Y≃Xp,q
++×S0, namely, Yconsists of two
+copies of Xp,q
++. Then Theorem 3.4 asserts that the minimal representation
+splits into two components, namely,
+̟p+1,q+1|O(p+1,q)≃πp+1,q
+−1
+2⊕πp+1,q
+1
+2
+becauseHl(R1) = 0 for l≥2.
+4) In the case p′′= 0 and p′=q′= 1, we are dealing with the branching law
+for the pair
+O(2,q+1)↓O(2,1)×O(q).
+The branching law (3.7) in this special case yields a setting of the defo r-
+mation of the Fourier transform (see Section 4).
+§4. Deformation of Fourier transforms
+Minimal representations give us also a hint to define a generalization o f
+the Fourier transform. In this section, we introduce a holomorphic semigroup
+Ik,a(z) consisting of Hilbert–Schmidt operators with three parameters:Algebraic analysis of minimal representations 15
+a: interpolating minimal representations of simple groups of type CandD,
+k: Dunkl deformation parameter (multiplicities on the root system),
+z: complex number,
+such that the operator-valued boundary value
+lim
+Rez↓0Ik,a(z)
+of Hilbert–Schmidt operators yields a one-parameter group of unit ary opera-
+tors. The underlying idea may be seen as a descendant of Sato’s hyp erfunc-
+tion theory [47] and also that of the Gelfand–Gindikin program [11, 44 , 48]
+for unitary representations of real reductive groups. We shall s ee in Dia-
+gram 4.2 that the Euclidean Fourier transform, the Hankel-type tr ansform,
+and the Dunkl transform, etc. arise naturally as the special value s ofIk,a(πi
+2) =
+limε↓0Ik,a(πi
+2+ε).
+§4.1.L2-model of minimal representations
+We return to the setting of Section 2.2. If a tempered distribution f∈
+S′(Rp+q) satisfies the differential equation /squarep,qf= 0, then it is easy to see that
+its Fourier transform Ffis supported on the characteristic variety
+Ξ :={ξ∈Rp+q:ξ2
+1+···+ξ2
+p−ξ2
+p+1−···−ξ2
+p+q= 0}.(4.1)
+Much more than this, the following theorem holds (see [39, Theorem 6 .2]):
+Theorem 4.1. Forp+q >2even and p,q >0, the Euclidean Fourier
+transformF≡F Rp+qinduces the bijection:
+F:Sol0(/squarep,q)∼→L2(Ξ).
+It is an isometry up to the scalar multiplication by 2p+q+2
+2πp+q+1
+2.
+Here,Sol0(/squarep,q)is the Hilbert spacewith respecttothe conservative quan-
+tity(,) definedin Theorem2.6, and L2(Ξ) denotesthe Hilbert spaceconsisting
+of square integrable functions with respect to the canonical meas ure on Ξ. The
+non-trivial part of Theorem 4.1 is to show that Image F∩L2(Ξ)/\e}atio\slash={0}. See
+[39, Theorem 6.2] for the proof.
+It follows from Theorem 4.1 that we can realize the minimal represent ation
+of the indefinite orthogonal group O(p+ 1,q+1) on the Hilbert space L2(Ξ)
+(L2-model) from the one on Sol0(/squarep,q) (conformal model ).16 Toshiyuki Kobayashi
+At this moment, we remark a distinguishing feature of minimal repres enta-
+tions (see Appendix in Section 5). Unlike well-understood family of irre ducible
+unitary representations of real reductive groups such as unitar y principal series
+representations or discrete series representations, minimal rep resentations are
+too ‘small’ that there is no existing geometric model for which both gr oup ac-
+tions and the Hilbert structure are given in a simple manner (cf. [6, 50 ]). We
+pin down the advantages of the aforementioned two models:
+Group action Hilbert structure
+Conformal model Sol(/squarep,q)simple 1/ci∇cleco√y∇t
+L2-modelL2(Ξ) 2/ci∇cleco√y∇t simple
+Finding the missing parts 1/ci∇cleco√y∇tand2/ci∇cleco√y∇tis interesting, particularly because it
+interacts with other branches of mathematics. Representation t heoretic con-
+sideration plays a guiding role in formalizing problems there. In fact, w e have
+seen in Theorem 2.6 that 1/ci∇cleco√y∇tbrought us to the construction of conservative
+quantities for ultra-hyperbolic equations, whereas 2/ci∇cleco√y∇tleads us to the notion of
+a Fourier transform on the isotropic cone Ξ [3, 35, 36], as discussed below.
+From now, we consider the missing part 2/ci∇cleco√y∇t. In order to find the global
+formula of group actions on the L2-model, let us clarify what is trivial and
+what will be the crucialoperator. We observethat there is a maxima l parabolic
+subgroup PofG=O(p+1,q+1) that contains the conformal transformation
+group
+Conf(Rp,q)≃(R>0×O(p,q))⋉Rp+q
+as a subgroup of index two. Then we have the Bruhat decomposition
+G=P∐PwP,
+wherew=/parenleftBigg
+Ip+10
+0−Iq+1/parenrightBigg
+. In fact, the Euclidean Fourier transform FRNap-
+pear as the unitary inversion operator of the Segal–Shale–Weil representation
+of the metaplectic group Mp(N,R), which is also a minimal representation.
+See [37, Chapter 1] for the comparison of FΞandFRNfrom this point of view.
+In theL2-model of the minimal representation πofGonL2(Ξ), the P-
+action is simple, namely, it is given just by translations and multiplication s
+[40]. Hence, it is enough to find the single unitary operator ( unitary inversion
+operator)π(w) in order to fill the missing part 2/ci∇cleco√y∇t. We set
+FΞ:=cπ(w), (4.2)Algebraic analysis of minimal representations 17
+wherecis the phase factor. Algebraically, FΞintertwines the multiplication of
+coordinate functions ξj(1≤j≤p+q) with the Bargmann–Todorov operators
+Rj(1≤j≤p+q) which are mutually commuting differential operators of
+second order on Ξ (see [2], [36, Chapter 1]).
+This algebraic feature is similar to the classical fact that the Euclidea n
+Fourier transform FRNintertwines the multiplication operators ξjand the dif-
+ferential operators√−1∂j(1≤j≤N).
+The goal of this section is to explain these operators FΞandFRNfrom
+a broader point of view, by constructing continuous family of opera tors that
+includeFΞandFRNas their special values.
+For this, we limit ourselves to the case p= 1. Then, the light cone Ξ (see
+(4.1)) splits into the forward light cone Ξ +and the backward light cone Ξ −
+according as x1>0 andx1<0. The unitary inversion operator FΞpreserves
+the direct sum
+L2(Ξ) =L2(Ξ+)⊕L2(Ξ−). (4.3)
+For later purpose, we set q=N. Then the projection to the second factor,
+R1⊕RN→RN, induces the following isomorphism between the Hilbert spaces:
+L2(Ξ+)≃L2(RN,/ba∇dblx/ba∇dbl−1dx). (4.4)
+Via(4.4), theunitaryinversionoperator FΞonL2(Ξ+)maybeseenasaunitary
+operator on L2(RN,/ba∇dblx/ba∇dbl−1dx). The explicit formula of FΞin the coordinates
+ofRNwas given in [35]. In this framework, we can construct a holomorphic
+family of bounded operators so that the unitary operator FΞis obtained as the
+limit of holomorphic objects. Deformation in the Dunkl setting [3, 4] is also
+built on this formulation. We will discuss those operators in this gener ality in
+Section 4.4.
+An alternative approach was taken in [36] based on the Barnes–Mellin
+integral to find the kernel function of FΞfor general pandq.
+§4.2. Hermite semigroup and Fourier transform
+We begin with recalling a general fact on the classical Hermite operat or
+onRN(e.g. [9, 19]):
+∆−/ba∇dblx/ba∇dbl2=N/summationdisplay
+j=1∂2
+∂x2
+j−N/summationdisplay
+j=1x2
+j. (4.5)18 Toshiyuki Kobayashi
+Then, ∆−/ba∇dblx/ba∇dbl2extends to a self-adjoint operator on L2(RN). We normalize
+the Euclidean Fourier transform FRNonL2(RN) as
+(FRNf)(ξ) =1
+(2π)N
+2/integraldisplay
+RNf(x)e−i/angbracketleftx,ξ/angbracketrightdx.
+Then,FRNis written as a special value of the one-parameter group of unitary
+operators
+χ(t) := exp/parenleftbiggit
+2(∆−/ba∇dblx/ba∇dbl2)/parenrightbigg
+,
+namely, we have
+FRN=e1
+4πiNexp/parenleftbiggπi
+4(∆−/ba∇dblx/ba∇dbl2)/parenrightbigg
+. (4.6)
+Further, the one-parameter group χ(t) of unitary operators extends to a holo-
+morphic semigroup I(z) defined by
+I(z) = expz
+2(∆−/ba∇dblx/ba∇dbl2) for Re z >0. (4.7)
+The semigroup I(z) is called the Hermite semigroup , and it is expressed as
+an integral transform against the Mehler kernel [9, 19], a Gaussian type kernel.
+Next, we consider another differential operator on RN,
+/ba∇dblx/ba∇dbl∆−/ba∇dblx/ba∇dbl.
+It turns out that this operator has a self-adjoint extension on th e Hilbert space
+L2(RN,/ba∇dblx/ba∇dbl−1dx) (see [35, Section 1.1]). Moreover, an analogous formula to
+(4.6) holds: via the identification (4.4), the ‘Fourier transform’ FΞon the
+forward light cone Ξ +can be expressed as
+FΞ=cexp/parenleftbiggπi
+2(/ba∇dblx/ba∇dbl∆−/ba∇dblx/ba∇dbl)/parenrightbigg
+, (4.8)
+wherec=e1
+2πi(N−1)is the phase factor. Then, the expression (4.8) allows
+us to seeFΞas the limit of the following holomorphic semigroup ( Laguerre
+semigroup )
+I(z) = expz(/ba∇dblx/ba∇dbl∆−/ba∇dblx/ba∇dbl),for Rez >0, (4.9)
+asz→πi
+2+ 0. The kernel function of I(z) is given in terms of the Bessel
+function [33].
+Interpolating∆−/ba∇dblx/ba∇dbl2and/ba∇dblx/ba∇dbl∆−/ba∇dblx/ba∇dbl,namely, theinfinitesimalgenerators
+of the Hermite semigroup (4.7) and the Laguerre semigroup (4.9), w e consider
+the differential operator
+∆0,a:=/ba∇dblx/ba∇dbl2−a∆−/ba∇dblx/ba∇dbla.Algebraic analysis of minimal representations 19
+It might not be so obvious that the symmetric operator ∆ 0,ahas a self-adjoint
+extension on the Hilbert space L2(RN,/ba∇dblx/ba∇dbla−2dx). In fact, it is the case. The
+proof uses representationtheory (see Proposition 4.6), and the same idea works
+in a more general setting of Dunkl’s differential-difference operator s. Thus, we
+shall introduce a holomorphic semigroup Ik,a(z) with infinitesimal generator
+∆k,a(see (4.10) below for the definition) for Re z >0 with parameters kanda
+in Section 4.3.
+In Diagram 4.2, we have summarised some of the deformation proper ties
+by indicating the limit behaviour of the holomorphic semigroup Ik,a(z). The
+specializationIk,a(πi
+2) leads us to a ( k,a)-generalized Fourier transform Fk,a
+(up to a phase factor), which reduces to the Fourier transform ( a= 2 and
+k≡0), the Dunkl transform Dk(a= 2 and k≡0), and the Hankel-type
+transform ( a= 1 and k≡0).
+(k,a)-generalized Fourier transform Fk,a
+a→2 a→1/arrowtpz→πi
+2
+holomorphic semigroup Ik,a(z)
+a→2
+←−−−−−−−−−−−−−−−−−−−−→
+a→1
+Ik,2(z) Ik,1(z)
+z→πi
+2
+←−−−−−−−−−−→
+k→0 k→0
+←−−−−−−−−−−→
+z→πi
+2
+Dunkl transform
+DkHermite semigroup
+I(z)Laguerre semigroup Fk,1
+k→0−−−−−→
+←−−−−−z→πi
+2z→πi
+2−−−−−→
+←−−−−−k→0
+Fourier transform Hankel transform.....⇐‘unitary inversion operator’ ⇒
+.....
+the Weil representation of
+Mp(N,R)the minimal representation of
+O(2,N+1)
+Diagram 4.2. Special values of holomorphic semigroup Ik,a(z)
+§4.3. Holomorphic semigroup Ik,a(z)with two parameters kanda20 Toshiyuki Kobayashi
+This subsection introduces a holomorphic semigroup, denoted by Ik,a(z),
+of which the infinitesimal generator is a self-adjoint differential-diffe rence op-
+erator.
+LetCbe the Coxeter group associated with a reduced root system Rin
+RN. For aC-invariant function k≡(kα) (multiplicity function ) onR, we set
+/a\}b∇acketle{tk/a\}b∇acket∇i}ht:=1
+2/summationdisplay
+α∈Rkα,
+and write ∆ kfor the Dunkl Laplacian on RN(see [16]). This is a differential-
+difference operator, which reduces to the Euclidean Laplacian ∆ whe nk≡0.
+We take a >0 to be yet another deformation parameter, and define
+∆k,a:=/ba∇dblx/ba∇dbl2−a∆k−/ba∇dblx/ba∇dbla. (4.10)
+We define a density on RNby
+ϑk,a(x) :=/ba∇dblx/ba∇dbla−2/productdisplay
+α∈R|/a\}b∇acketle{tα,x/a\}b∇acket∇i}ht|kα.
+The volume of the unit ball with respect to the measure ϑk,a(x)dxis explicitly
+known in terms of the gamma function owing to the work by Selberg, M ac-
+donald, Heckman, and Opdam among others but we do not go into det ails (see
+Etingov [8]).
+In the case a= 2 and k≡0,ϑ0,2(x)≡1 and ∆ 0,2is the Hermite operator
+(4.5) on RN.
+Here are basic properties of our differential-difference operator ∆ k,a:
+Theorem 4.2 (see [4, Theorem A]) .Assumea >0anda+2/a\}b∇acketle{tk/a\}b∇acket∇i}ht+N−
+2>0.
+1) ∆k,aextends toa self-adjoint operator on the Hilbert space L2(RN,ϑk,a(x)dx).
+2)There is no continuous spectrum of ∆k,a.
+3)All the discrete spectrum of ∆k,ais negative.
+We introduce the following operators on L2(RN,ϑk,a(x)dx) by
+Ik,a(z) := exp/parenleftBigz
+a∆k,a/parenrightBig
+,for Rez≥0. (4.11)
+Correspondingly to the properties of the infinitesimal generator1
+a∆k,ain The-
+orem 4.2, we get:Algebraic analysis of minimal representations 21
+Theorem 4.3 (see [4, Theorem B]) .Retain the assumption of Theorem
+4.2.
+1){Ik,a(z) : Rez >0}forms a holomorphic semigroup in the complex right-
+half plane{z∈C: Rez >0}in the sense that Ik,a(z)is a Hilbert–Schmidt
+operator on L2(RN,ϑk,a(x)dx)satisfying
+Ik,a(z1)◦Ik,a(z2) =Ik,a(z1+z2),(Rez1,Rez2>0),
+and that the scalar product (Ik,a(z)f,g)is a holomorphic function of zfor
+Rez >0, for any f,g∈L2(RN,ϑk,a(x)dx).
+2)Ik,a(z)is a one-parameter group of unitary operators on the imagina ry axis
+Rez= 0.
+We shall callIk,a(z) as the ( k,a)-generalized Laguerre semigroup Ik,a(z).
+We note thatI0,2(z) is the Hermite semigroup (4.7) (see [9, 19]), and I0,1(z)
+is the Laguerre semigroup (4.9) (see [33]).
+§4.4.(k,a)-generalized Fourier transforms Fk,a
+Theorem 4.3 2) asserts that the ‘boundary value’ of the holomorph ic semi-
+groupIk,a(z) produces a one-parameter family of unitary operators.
+The case z= 0 gives the identity operator, namely, Ik,a(0) = id. The
+particularly interesting case is when z=πi
+2. We set
+c:= exp(iπN+2/a\}b∇acketle{tk/a\}b∇acket∇i}ht+a−2
+2a) (phase factor) ,
+and define the ( k,a)-generalized Fourier transform by
+Fk,a:=cIk,a/parenleftBigπi
+2/parenrightBig
+=cexp/parenleftBigπi
+2a(/ba∇dblx/ba∇dbl2−a∆k−/ba∇dblx/ba∇dbla)/parenrightBig
+.
+Then, this operator Fk,afor general aandksatisfies the following significant
+properties:
+Theorem 4.4 ([4, Theorem D]) .Retain the assumption of Theorem 4.2.
+1)Fk,ais a unitary operator on L2(RN,ϑk,a(x)dx).
+2)Fk,a◦Hk,a=−Hk,a◦Fk,a. Here, Hk,ais the differential operator of first
+order defined in (4.12).22 Toshiyuki Kobayashi
+3)Fk,a◦/ba∇dblx/ba∇dbla=−/ba∇dblx/ba∇dbl2−a∆k◦Fk,a,
+Fk,a◦(/ba∇dblx/ba∇dbl2−a∆k) =−/ba∇dblx/ba∇dbla◦Fk,a.
+4)Fk,ais of finite order if and only if a∈Q. Its order is 2mifa=m
+nsuch
+that(m,n) = 1. In particular,Fk,1is of order 2, andFk,2is of order 4.
+As indicated in Diagram 4.2, Fk,areduces to the Euclidean Fourier trans-
+formFonRNifk≡0 anda= 2; to the Dunkl transform Dkintroduced by
+C. Dunkl himself if k >0 anda= 2. The unitary operator F0,1arises as the
+unitary inversion operator FΞonL2(Ξ+) of the minimal representation of the
+conformal group (see Section 4.1).
+Our study also contributes to the theory of special functions, in p articular
+orthogonal polynomials; indeed we derive several new identities, fo r example,
+the (k,a)-deformation of the classical Bochner–Hecke identity where the Gaus-
+sian function and harmonic polynomials in the classical setting ( k≡0 and
+a= 2) are replaced respectively with exp( −1
+a/ba∇dblx/ba∇dbla) and polynomials annihi-
+lated by the Dunkl Laplacian. The ( k,a)-generalized Fourier transform Fk,a
+also satisfies a Heisenberg-type inequality. This generalizes the clas sical case
+(k≡0 anda= 2) and R¨ osler’s Heisenberg inequality [45] ( k >0 anda= 2).
+We refer to [4] for full details.
+§4.5. Hidden symmetries in the Hilbert space L2(RN,ϑk,a(x)dx)
+The key idea of the proof for Theorem 4.1, 4.2, and 4.3 is to use more
+operators rather than the single operator ∆ k,a, and then to appeal representa-
+tion theory of sl2, in particular, the theory of discretely decomposable unitary
+representations.
+Lemma 4.5. Letkbe a multiplicity-function on a root system, and
+a∈C×. Then, the following differential-difference operators on RN\{0}
+E+
+k,a:=i
+a/ba∇dblx/ba∇dbla,
+E−
+k,a:=i
+a/ba∇dblx/ba∇dbl2−a∆k,
+Hk,a:=2
+aN/summationdisplay
+i=1xi∂i+N+2/a\}b∇acketle{tk/a\}b∇acket∇i}ht+a−2
+a(4.12)
+form an sl2-triple, namely, we have:
+[Hk,a,E+
+k,a] = 2E+
+k,a,[Hk,a,E−
+k,a] =−2E−
+k,a,[E+
+k,a,E−
+k,a] =Hk,a.Algebraic analysis of minimal representations 23
+Special cases of Lemma 4.5 was previously known: the case k≡0 and
+a= 2 is the classical harmonic sl2-triple (e.g. Howe [19]), the case k >0 and
+a= 2 by Heckman [16], and k≡0 anda= 1 by Kobayashi and Mano [33].
+The operator ∆ k,a(see (4.10)) takes the form,
+∆k,a=ai(E+
+k,a−E−
+k,a),
+which may be seen as an element of sl(2,C).
+Lemma 4.5 fits well into the framework of discretely decomposable re pre-
+sentations of reductive groups [25, 26, 27] as we discussed in Sect ion 3.1:
+Proposition 4.6 (see [4, Theorem 3.31]) .Ifa >0anda+2/a\}b∇acketle{tk/a\}b∇acket∇i}ht+N−
+2>0, then the representation of sl(2,R)lifts to a unitary representation of
+the simply-connected group on L2(RN,ϑk,a(x)dx). The resulting unitary rep-
+resentation is discretely decomposable, and commutes the o bvious action of the
+Coxeter group C.
+This unitary representation plays the central role in the key to the proof of
+Theorems4.2, 4.3and 4.4. An explicit formulaofthe irreducible decomp osition
+ofL2(RN,ϑk,a(x)) is found in [4, Theorem 3.28]. In the special cases k≡0 and
+a= 1 or 2, this formula may be regarded as the branching law of the minim al
+representations of O(2,N+ 1)/tildewideorMp(N,R), respectively (see Diagram 4.5
+below). Correspondingly, all the spectrum of ∆ k,ais also obtained explicitly.
+In the case a= 2, thesl2-actionalsoinduces the representationof SL(2,C)
+on the algebra generated by Dunkl’s operators, multiplication opera tors, and
+the Coxeter group. The restriction of this action to SL(2,Z) coincides with a
+special case of the SL(2,Z)-action discovered by Cherednik [7] on the (degen-
+erate) rational DAHA (double affine Hecke algebra).
+Theorem 4.6 asserts that the Hilbert space L2(RN,ϑk,a(x)dx) has a sym-
+metry of the direct product group C×/tildewiderSL(2,R) for allkanda. This symmetry
+becomes larger for special values of kandaas below:
+O(2,N+1)/tildewidea→1
+−−−−−→C×/tildewiderSL(2,R)k→0−−−−→O(N)×/tildewiderSL(2,R)
+(k,a: general)−−−−−→
+Mp(N,R)a→2
+Diagram 4.5. Hidden symmetries in L2(RN,ϑk,a(x)dx)
+Fora= 2, this symmetry is given by the Segal–Shale–Weil representation24 Toshiyuki Kobayashi
+of the metaplectic group Mp(N,R). Fora= 1, it is given by the irreducible
+unitary representation of the double covering O(2,N+ 1)/tildewideof the conformal
+group on L2(RN,/ba∇dblx/ba∇dbl−1dx). Here, as we saw in Theorem 4.1, we do not need
+to take a double cover when Nis odd. Both of them are minimal represen-
+tations and, in particular, they attain the minimum of their Gelfand–K irillov
+dimensions among the unitary dual. In this sense, our continuous pa rameter
+a >0 interpolates the L2-models of two minimal representations of different
+reductive groups by keeping smaller symmetries O(N)×/tildewiderSL(2,R). In view of
+Lemma 4.5, the ( k,a)-generalized Fourier transform Fk,a(k≡0,a= 1,2) arise
+as the unitary operators (up to phase factors) corresponding t o the following
+element,
+expπ
+2/parenleftBigg
+0 1
+−10/parenrightBigg
+∈/tildewiderSL(2,R).
+§5. Appendix — representation theoretic properties of ̟
+For the reader’s convenience, we list some representation theore tic prop-
+erties of the irreducible unitary representation ̟of the indefinite orthogonal
+groupG=O(p+1,q+1), on which geometric analysis is the motif throughout
+this article.
+In what follows, we assume
+p,q≥1 andp+qis an even integer greater than two.
+We write K≃O(p+ 1)×O(q+ 1) for a maximal compact group of G,g≃
+o(p+1,q+1)fortheLiealgebraof G,gC≃o(p+q+2,C)foritscomplexification,
+andgC=kC+pCfor the complexified Cartan decomposition.
+Here are some properties of ̟from representation theoretic viewpoints.
+1)̟is an irreducible unitary representation of G.
+2) (minimal representation) The representation ̟is a minimal representa-
+tion in the sense that the annihilator of the underlying ( gC,K)-module ̟K
+in the universal enveloping algebra U(gC) is the Joseph ideal if p+q≥6
+([5, 42]). See [10] for algebraic aspects of minimal representations of reduc-
+tive groups and the definition of the Joseph ideal.
+3) (restriction to the identity component) The group Ghas four connected
+components, and we write G0=SO0(p+ 1,q+ 1) for the identity com-
+ponent. Then, ̟stays irreducible when restricted to G0if and only if
+p,q >1.Algebraic analysis of minimal representations 25
+4) (highest weight module case) If p= 1 orq= 1, then the restriction ̟|G0
+is a direct sum of two irreducible representations, one is a highest we ight
+representation ̟+and the other is a lowest weight representation ̟−. As
+we have seen in (4.3), this decomposition ̟|G0=̟+⊕̟−corresponds to
+the direct sum
+L2(Ξ) =L2(Ξ+)⊕L2(Ξ−)
+in theL2-model. Both ̟+and̟−are minimal representations of the
+connected group G0.
+To fix the notation, we suppose p= 1. Then, Gis the conformal group
+O(2,q+1) of the Minkowski space R1,q, namely, the Euclidean space R1+q
+equipped with the flat Lorentz metric of signature (1 ,q). In this case
+our representation ̟has a long history of study, also in physics (see e.g.
+Todorov [49]). The minimal representation ̟+may be interpreted as the
+symmetry of the solution space to the mass-zero spin-zero wave e quation.
+The representation ̟+arises also on the Hilbert space of bound states of
+the Hydrogen atom.
+5) (spherical case) The underlying ( gC,K)-module ̟Khas the following K-
+type formula:
+̟K≃/circleplusdisplay
+a,b∈N,
+a+p
+2=b+q
+2Ha(Rp+1)⊠Hb(Rq+1). (5.1)
+In particular, the representation ̟is spherical (i.e. contains a non-zero
+K-fixed vector) if and only if p=q.
+6) (infinitesimal character) Let Z(gC) be the center of U(gC). Then, the in-
+finitesimal character of ̟Kis given by
+(1,p+q
+2−1,p+q
+2−2,···,1,0).
+Here, we normalize the Harish-Chandra isomorphism for the simple Lie
+algebragCof typeDn(n=p+q
+2+1),
+HomC-algebra(Z(gC),C)≃Cn/W(Dn),
+in a way that the infinitesimal character of the trivial one dimensiona l
+representation is
+(p+q
+2,p+q
+2−1,p+q
+2−2,···,1,0).26 Toshiyuki Kobayashi
+7) (theta correspondence) The representation ̟is obtained also as the theta
+correspondence of the trivial one-dimensional representation o fSL(2,R)
+with respect to the following reductive dual pair
+O(p+1,q+1)·SL(2,R)⊂Sp(p+q+2,R).
+See [52].
+8) (Gelfand–Kirillov dimension) The Gelfand–Kirillov dimension of the rep -
+resentation ̟, to be denoted by DIM( ̟), attains its minimum among all
+unitary representations of G, that is,
+DIM(̟) =p+q−1.
+The associated variety of the underlying ( gC,K)-module ̟Kis given by
+AV(̟) =OC
+min∩pC,
+see [39, Lemma 4.4]. Here, OC
+minis the minimal nilpotent coadjoint orbit
+ing∗
+Cidentified with the Lie algebra gC.
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+[52] C.-B. Zhu and J.-S. Huang, On certain small representations of indefinite orthogonal
+groups, Represent. Theory 1(1997), 190–206.
\ No newline at end of file
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+arXiv:1001.0225v1  [physics.atom-ph]  1 Jan 2010Restricted Thermalization for Two Interacting Atoms in a Mu ltimode Harmonic
+Waveguide
+V. A. Yurovsky
+School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Is rael
+M. Olshanii
+Department of Physics, University of Massachusetts Boston , Boston MA 02125, USA
+(Dated: October 27, 2018)
+In this article, we study the thermalizability of a system co nsisting of two atoms in a circular,
+transversely harmonic waveguide in the multimode regime. W hile showing some signatures of the
+quantum-chaotic behavior, the system fails to reach a therm al equilibrium in a relaxation from
+an initial state, even when the interaction between the atom s is infinitely strong. We relate this
+phenomenon to the previously addressed unattainability of a complete quantum chaos in the ˇSeba
+billiard [P. ˇSeba, Phys. Rev. Lett., 64, 1855 (1990)], and we conjecture the absence of a complete
+thermalization tobeageneric propertyofintegrable quant umsystems perturbedbyanon-integrable
+but well localized perturbation.
+PACS numbers: 67.85.-d, 37.10.Gh, 05.45.Mt, 05.70.Ln
+Introduction .– The ultracold quantum gases have been
+long proven to be an ideal testbench for studying the
+fundamental properties of quantum systems. One of the
+most interesting and frequently addressed topics is the
+behavior of quantum systems in the vicinity of an inte-
+grable point. Experimental results already include the
+suppression of relaxation of the momentum distribution
+[1] and the modified decay of coherence [2]. Note that
+the fully developed quantum chaos with ultracold atoms
+has been investigated in experiments [3].
+Traditionally, the underlying integrable system is rep-
+resented by the Lieb-Liniger gas [4] that is well suited
+to describe the dynamics of Van-der-Waals interacting
+bosons in a monomode atomic guide (see [5] for a re-
+view). The nontrivial integrals of motion there are inti-
+mately related to both the one-dimensional character of
+the atomic motion and to the effectively zero value of the
+two-body interaction range. The causes for lifting inte-
+grability include the virtual excitation of the transverse
+modes during the collision [6] and the coupling between
+two parallel weakly interacting Lieb-Liniger gases [2].
+One would expect that in the multimode regime , when
+the motion becomes substantially multidimensional, no
+remnants of the former integrals of motion will survive.
+However, in this article we show that for the case of only
+two interacting atoms a strong separation between the
+transverse and longitudinal degrees of freedom remains,
+even in the case of the infinitely strong interaction be-
+tween the atoms. We attribute this effect to the short-
+range nature of the interaction.
+For our system we study both the degree of the eigen-
+state thermalization and the actual thermalization in an
+expansion from a class of realistic initial states. The
+eigenstate thermalization [7–9] – the suppression of the
+eigenstate-by-eigenstate variance of quantum expecta-
+tion values of simple observables – provides an ultimateupper bound for a possible deviation of the relaxed value
+of an observable from its thermodynamical expectation,
+for any initial state in principle. However, recently a new
+direction of research has emerged: quantum quench in
+many-body interacting systems [1, 2, 9, 10]. In this class
+of problems, the initial state is inevitably decomposed
+into a large superposition ofthe eigenstates ofthe Hamil-
+tonian governing the dynamics, and the discrepancy be-
+tween the result of the relaxation and thermal values is
+expected to be diminished. In our article we show that
+in the case of two short-range-interactingatoms in a har-
+monic waveguide, a complete thermalization can never
+be reached under either scenario.
+We relate the absence of thermalization in our system
+to the previously addressed unattainability of a complete
+quantumchaosinthe ˇSebabilliard–flattwo-dimensional
+rectangular billiard with a zero-range scatterer in the
+middle [11]. Similarly to our system, the ˇSeba billiard
+does show some signatures of the quantum-chaotic be-
+havior, although both the level statistics [12–14] and the
+momentum distributions in individual eigenstates [15]
+show substantial deviations from the quantum chaos pre-
+dictions. Both the ˇSeba billiard (first suggested and
+solved in Ref. [11]) and our system allow for an exact
+analytic solution, in spite of the absence of a complete
+set of integrals of motion.
+The system .– Consider two short-range-interacting
+atoms in a circular, transversally harmonic waveguide.
+In this case the center-of-mass and the relative motion
+canbe separated. Notethat the harmonicapproximation
+for the transverse confinement has been proven to model
+reallife atomwaveguideswith highaccuracy. Predictions
+of the analogous model involving an infinite waveguide
+[16,17]hasbeenconfirmedbyadirectexperiment-theory
+comparison of the energies of the quasi-one-dimensional
+dimers [18] and numerical calculations involving anhar-2
+monic potentials (see review [5] for references).
+The unperturbed Hamiltonian of the relative motion
+is given by the sum of the longitudinal and transverse
+kinetic energies and the transverse trapping energy ˆU=
+µω2
+⊥ρ2/2,
+ˆH0=−¯h2
+2µ∂2
+∂z2−¯h2
+2µ∆ρ+ˆU−¯hω⊥,(1)
+whereω⊥isthetransversefrequency, ∆ ρisthetransverse
+two-dimensional Laplacian, and µis the reduced mass.
+In our model, the transverse and longitudinal degrees of
+freedomarecoupledbyapotentialofaFermi-Huangtype
+[5]:
+ˆV=2π¯h2as
+µδ3(r)(∂/∂r)(r·), (2)
+were,asis the three-dimensional s-wave scattering
+length. This approximation is valid at low collision en-
+ergies (e.g., below 300 mK for Li, see a recent review
+[19], while the ultracold atom energies lie well below
+1µK). The model (2) stands in a excellent agreement
+with the experimentally measured equation of state of
+the waveguide-trapped Bose gas [20] and with numerical
+calculations with finite-range potentials [17, 21].
+The ring geometry of the waveguide imposes period-
+Lboundary conditions along z-direction. In what fol-
+lows, we will restrict the Hilbert space to the states that
+have zeroz-component of the angular momentum and
+that are even under the z↔ −zreflection; the inter-
+action has no effect on the rest of the Hilbert space.
+The unperturbed spectrum is given by (see [22]) Enl=
+2¯hω⊥n+ ¯h2(2πl/L)2/(2µ), wheren≥0 andl≥0 are
+the transverseand longitudial quantum numbers, respec-
+tively.
+The interacting eigenfunctions with the eigenenergy
+Eαcan be expressed as (see [22])
+∝angbracketleftρ,z|α∝angbracketright=Cα∞/summationdisplay
+n=0cos/parenleftbig
+2√ǫα−λnζ/parenrightbig
+√ǫα−λnsin√ǫα−λne−ξ
+2L(0)
+n(ξ),(3)
+where the rescaled energy ǫαis given by ǫα≡
+λEα/(2¯hω⊥),λ≡(L/a⊥)2is the aspect ratio, ζ≡
+z/L−1/2,ξ≡(ρ/a⊥)2,L(0)
+n(ξ) are the Legendre poly-
+nomials,a⊥= (¯h/µω⊥)1/2is the size of the transverse
+ground state, and Cαis the normalization constant. The
+eigenenergies are solutions of the following transcenden-
+tal equation (see [22]):
+√
+λ∞/summationdisplay
+n=0cot√ǫα−λn+i√ǫα−λn−ζ/parenleftbigg1
+2,−ǫα
+λ/parenrightbigg
+=a⊥
+as,(4)
+whereζ(ν,x) is the Hurwitz ζ-function (see [5]). Similar
+solutions were obtained for two atoms with a zero-range
+interaction in a cylindrically-symmetric harmonic poten-
+tial [23] (that system was analysed numerically in Ref.[24]). Higher partial wave scatterers were analyzed in
+the Ref. [25].
+At rational values of λ/π2, the unperturbed energy
+spectrum shows degeneracies that are not fully lifted in
+the full [deduced from (4)] spectrum. To minimize the
+effect of the degeneracies, we, following Ref. [11], fix the
+length of the guide to ( L/a⊥)2≡λ= 2φgrπ7≈9774,
+whereφgr= (1+√
+5)/2 is the golden ratio.
+Standard quantum-chaotic tests .– The results of the
+study of the level spacing distribution in our system are
+fully consistent with the analogous results for the ˇSeba
+billiard [12, 13]. For the energy range E>∼100¯hω⊥
+and the aspect ratio ( L/a⊥)2= 2φgrπ7, the distribu-
+tion quickly converges to the ˇSeba distribution [12] at
+as>∼10−1a⊥. This distribution does show a gap at
+small level spacings but fails to reproduce the Gaussian
+tail predicted by the Gaussian Orthogonal Ensemble. At
+smallasthe level spacing distribution tends to the Pois-
+son one. We havealsoverified that in the unitary regime,
+as≫a⊥, the statistics of the point-by-point variations
+of the eigenstate wavefunction in the spatial representa-
+tion is closeto the Gaussianone, accordingto the general
+prediction [26] and a particular observation in the case
+of theˇSeba billiard [11].
+Eigenstate thermalization .– According to the eigen-
+state thermalization hypothesis [7–9], the ability of an
+isolated quantum system (with few or many degrees of
+freedom regardless) to thermalize follows from the sup-
+pression of the eigenstate-by-eigenstate fluctuations of
+the quantum expectation values of relevant observables.
+Scattered dots at the Fig. 1 show the quantum expec-
+tation values of the transverse trapping energy, ∝angbracketleftα|ˆU|α∝angbracketright
+as a function of the eigenstate energy. The variation
+of∝angbracketleftα|ˆU|α∝angbracketrightremains (in contrast to the quantum-chaotic
+billiards [27]) of the order of the mean, even for the en-
+ergies much larger than any conceivable energy scale of
+the system. Here and below, we work in the regime of
+the infinitely strong interactions, as= 106a⊥.
+For a given eigenstate, quantum expectation values
+of the occupation probabilities for the transverse (Fig.
+2a) and longitudinal (Fig. 2b) modes show no conver-
+genceto their microcanonicalexpectation values, P(n)∝/integraltext
+dlδ(Enl−E) andP(l)∝/integraltext
+dnδ(Enl−E) respectively.
+Both distributions are heavily dominated by only a few
+peaks(cf. scarsin themomentum spacefor ˇSebabilliards
+[15]).
+Relaxation from an initial state .– Now we are going
+to address directly the ability of our system to ther-
+malize from an initial state |ψ(t= 0)∝angbracketright. In this case,
+theinfinite time average of the quantum expectation
+value of an observable ˆAwill be given by Arelax.≡
+limtmax→∞(1/tmax)/integraltexttmax
+0∝angbracketleftψ(t)|ˆA|ψ(t)∝angbracketright=/summationtext
+α|∝angbracketleftα|ψ(t=
+0)|2∝angbracketleftα|ˆA|α∝angbracketright, where|α∝angbracketrightare the eigenstates of the system
+(see [8]). For a quantum-chaotic system, it is expected
+that the above infinite time average coincides with the3
+ 0 0.2 0.4 0.6 0.8 1 1.2 1.4
+ 100 200 300 400 500 600Trapping energy, Urelax./Utherm.
+Energy, E// h ωperp*l
+nl
+nl
+n
+l
+n
+FIG. 1. The time average of the transverse trapping energy
+Urelax.≡limtmax→∞(1/tmax)/integraltexttmax
+0/angbracketleftψ(t)|ˆU|ψ(t)/angbracketrightafter relax-
+ation from an initial state as a function of the energy of the
+initial state E[see Hamiltonian (1)]. Four families of the ini-
+tial states are considered. Long-dashed (green) line: stat e (5)
+withn0= 0,δ= 0.99, and the energy being controlled via
+scanningl0; short-dashed (blue) line: the same as the previ-
+ous one, except for δ=.1; dotted (purple) line: the state (5)
+withδ= 0.1 andl0= 0 and the energy control via n0; solid
+(red) line: state (6) with κ2= 2κ1,δ= 0.99,l0= 0 and the
+energy control via κ1. Legends schematically show the distri-
+bution of the initial states over the quantum numbers; arrow s
+indicate the direction in which the parameters controlling the
+energy are scanned. The asterisk (*) corresponds to the set
+of parameters used to produce the Fig. 2. Grey dots show
+the quantum expectation value of the trapping energy Ufor
+every hundreds eigenstate of the full Hamiltonian (1,2). Al l
+the values of the trapping energy are shown with respect to
+its microcanonical expectation value Utherm.≈E/3. For all
+the points, ( L/a⊥)2= 2φgrπ7≈9774 andas= 106a⊥.
+thermal prediction.
+Consider the following initial state:
+∝angbracketleftz|ψ(t=0)∝angbracketright ∝cosπζ
+δθ/parenleftbiggδ
+2−|ζ|/parenrightbigg
+cos(2πl0ζ)|n0∝angbracketright.(5)
+Longitudinally, the state is represented by the ground
+state of a length Lδhard-wall box split initially by an
+ideal beamsplitter with momenta ±2πl0/L; the box is
+centered at the maximal interatomic distance. The state
+is distributed among approximately π/δaxial modes lo-
+calized about l=±l0. The initial transverse state is
+limited to a single mode n0. Note that for n0= 0 and
+δ≈1 the state is conceptually similar to the initial state
+used in the equilibration experiments [1].
+After relaxation, both transverse (Fig. 2a) and longi-
+tudinal (Fig. 2b) distributions remain very far from the
+equilibrium predictions. Fig. 1 shows the equilibrium
+value of the transverse trapping energy U(see (1)) for
+three sequences of the initial states of the type (5). The
+first two, with low transverse energy, show an approx-
+imately -40% deviation of the relaxed value of Ufroml
+n
+ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
+ 0 50 100 150 200 250 300 350Occupation probability, P(l)
+Longitudinal mode, lbeamsplitterinitial
+relaxed
+eigenstate
+microcanon. x 10(b)l
+n
+ 0 0.2 0.4 0.6 0.8 1 1.2
+ 0 20 40 60 80 100Occupation probability, P(n)
+Transverse mode, ninitial
+relaxed
+eigenstate
+microcanon. x 100(a)
+FIG. 2. Distribution of the mean occupation numbers for
+the transverse (a) and longitudinal (b) modes of the unper-
+turbed Hamiltonian. The long-dashed (green) and solid (red )
+lines show the initial and time-averaged distributions for an
+initial state (5) with n0= 0,δ= 0.99, andl0= 292.8. En-
+ergy of the initial state is E≡(2/λ)¯hω⊥ǫα= 173.2¯hω⊥. The
+short-dashed (blue) line corresponds to the 17096’s eigens tate
+(closest in energy to Efrom the above, with a relative dis-
+crepancy of 10−5) of the interacting Hamiltonian. The dotted
+(purple) line shows the microcanonical prediction. The res t
+of the parameters is the same as at the Fig. 1.
+the thermal prediction, thus being seemingly correlated
+with its initial, negative value. Note that the correla-
+tion between the final and initial values of observables
+in quantum systems is addressed in Ref. [28]. Also, note
+that the initial states of the second sequence havea much
+greater spread over the longitudinal modes than the first
+one; consistently, the energy-to-energy variation of the
+relaxedvaluesof Uis lessthan forthe firstsequence. The
+third sequence, with low longitudial energy, shows devi-
+ations from the thermal prediction that range between
++50% and +5% but never reach zero.4
+Another type of the initial state
+∝angbracketleftρ, z|ψ(t=0)∝angbracketright ∝cosπζ
+δθ/parenleftbiggδ
+2−|ζ|/parenrightbigg
+cos(2πl0ζ)
+×/parenleftbig
+e−κ1ξ−e−κ2ξ/parenrightbig
+(6)
+allowsforanadditionalspreadoverthetransversemodes.
+The transverse wavefunction vanishes at the waveguide
+axis. The mode occupationhas a minimum at n= 0, and
+the occupation of the ground transverse mode tends to
+zero forκ1<κ2≪1. The results for a sequence similar
+to the third sequence described above are shown at Fig.
+1 as well. In spite of the significant initial spread over
+both transverse and longitudinal modes, the deviations
+from the equilibrium are close to the ones for the third
+sequence, while the energy-to-energy variations for the
+former are indeed less than for the latter.
+Note that we have also successfully tested the conver-
+gence of the instantaneous quantum expectation values
+of the observables to their infinite-time averages.
+Summary and interpretation of results .– In this arti-
+cle, we solve analytically the problem of two short-range-
+interacting atoms in a circular, transversely harmonic
+multimode waveguide. We assess the ability of the sys-
+tem to thermalize from an initially excited state; a broad
+class of initial states has been analyzed. We find a sub-
+stantial suppression of thermalization, even for the in-
+finitely strong interactions. We associate this effect with
+the previously demonstrated unattainability of complete
+quantum chaos in the ˇSeba-type billiards [11–15].
+Weconjecturethattheeffectofsuppressionofthermal-
+ization is generic for the integrable systems perturbed,
+no matter how strong, by a well localized perturbation.
+The following reasoning applies. In a quantum chaotic
+system, a giveneigenstate |α∝angbracketrightconsistsof a largesuperpo-
+sition of the eigenstates |/vector n∝angbracketrightof the underlying integrable
+system,whicharedrawnindiscriminatelyfromthemicro-
+canonical shell. Number of principal components in such
+asuperpositionisasensitivemeasureoftheapproachtoa
+complete chaos and the subsequent thermalizability (see,
+for example [29]). The zero-point-localized perturbation,
+being a particular case of a separable perturbation, gen-
+erates the eigenstates of a form ∝angbracketleft/vector n|α∝angbracketright ∝1/(Eα−E/vector n)
+(see [22]). There is always one and only one perturbed
+energyEαin between any two unperturbed energies E/vector n
+(see [13]). In addition, for the caseofthe infinitely strong
+perturbation (where the approachto chaos is expected to
+be the closest) Eαtends to the middle position between
+the twoE/vector n’s. In this case, the only energy scale that
+remains is the distance between the unperturbed levels,
+and, as a result, the number of principal components a
+given eigenstate |α∝angbracketrightconsists of is always of the order of
+unity. This property extends to the case of finite-range
+interactions, whenever the interaction range is much less
+then the de Broglie wavelength of the colliding atoms.
+Our study is the first attempt to address thermaliz-
+ability of a quantum system with separable interactions.We are grateful to Felix Werner for the enlightening
+discussions on the subject. This work was supported by
+a grant from the Office of Naval Research ( N00014-09-1-
+0502).
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+79, 4365 (1997).arXiv:1001.0225v1  [physics.atom-ph]  1 Jan 2010Supplementary material to: Restricted Thermalization for Two Interacting Atoms in
+a Multimode Harmonic Waveguide
+V. A. Yurovsky
+School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Is rael
+M. Olshanii
+Department of Physics, University of Massachusetts Boston , Boston MA 02125, USA
+(Dated: October 27, 2018)
+This supplementary material contains a detailed derivatio n of the eigenenergies and eigenstates
+for a system consisting of two atoms in acircular, transvers ely harmonic waveguide in the multimode
+regime.
+PACS numbers: 67.85.-d, 37.10.Gh, 05.45.Mt, 05.70.Ln
+The waveguide system [1], as well as the ˇSeba billiard
+[2], belongs to the class of problems where an integrable
+system of Hamiltonian ˆH0is perturbed by a separable
+rank I interaction V|L∝angbracketright∝angbracketleftR|(seee.g.[3]). Eigenstates |α∝angbracketright
+of the interacting system are solutions of the Schr¨ odinger
+equation
+[ˆH0+V|L∝angbracketright∝angbracketleftR|]|α∝angbracketright=Eα|α∝angbracketright, (1)
+whereEαare the corresponding eigenenergies. Let us
+expand the eigenstate |α∝angbracketrightover the eigenstates |/vector n∝angbracketright(of en-
+ergyE/vector n) of the non-interacting hamiltonian ˆH0. The
+Schr¨ odinger equation leads to the following equations for
+the expansion coefficients
+(Eα−E/vector n)∝angbracketleft/vector n|α∝angbracketright=∝angbracketleft/vector n|L∝angbracketright∝angbracketleftR|α∝angbracketright ∝ ∝angbracketleft/vector n|L∝angbracketright,(2)
+where the omitted factor in the second equality is inde-
+pendent of /vector n.
+As a result, any eigenfunction |α∝angbracketrightfunctionally co-
+incides with the Green function of the non-interacting
+Hamiltonian taken at the energy of the eigenstate Eα,
+|α∝angbracketright ∝/summationdisplay
+/vector n|/vector n∝angbracketright∝angbracketleft/vector n|L∝angbracketright
+Eα−E/vector n. (3)
+The omitted factor is determined by the normalization
+conditions.
+Substitution of this expression to the Schr¨ odinger
+equation (1) leads to the following eigenenergy equation:
+/summationdisplay
+/vector n∝angbracketleftR|/vector n∝angbracketright∝angbracketleft/vector n|L∝angbracketright
+Eα−E/vector n=1
+V. (4)
+Similar expressions were obtained in Refs. [2, 4] for the
+case of the ˇSeba billiard and its generalizations.
+In the case of the relative motion of two short-range-
+interacting atoms in a circular, transversely harmonic
+waveguide, the derivation uses ideas of the analogous
+model involving an infinite waveguide [5–7]. The unper-
+turbed Hamiltonian hereis givenbyEq. (1) in [1] and theFermi-Huang interaction (see Eq. (2) in [1]) can be writ-
+tenin theseparableform[3]with theinteractionstrength
+and formfactors given by
+V=2π¯h2as
+µ,|L∝angbracketright=δ3(r),∝angbracketleftR|=δ3(r)∂
+∂r(r·).(5)
+In what follows, we will restrict the Hilbert space to
+the states of zero z-component of the angular mo-
+mentum and even under the z↔ −zreflection; the
+interaction has no effect on the rest of the Hilbert
+space. The non-interacting eigenstates are products
+of the transverse and longitudial wavefunctions. The
+transversetwo-dimensionalzero-angular-momentumhar-
+monic wavefunctions,
+∝angbracketleftρ|n∝angbracketright=1√πa⊥L(0)
+n(ξ)exp(−ξ/2), (6)
+labeled by the quantum number n≥0, are expressed
+in terms of the Legendre polynomials, L(0)
+n(ξ), where
+ξ= (ρ/a⊥)2anda⊥= (¯h/µω⊥)1/2is the transverse os-
+cillator range. The longitudial wavefunctions, labeled by
+l≥0, are the symmetric plane waves satisfying periodic
+conditions with the period L:
+∝angbracketleftz|k∝angbracketright= (2/L)1/2cos2πkζ, ∝angbracketleftz|0∝angbracketright=L−1/2,(7)
+whereζ≡z/L−1/2. The unperturbed spectrum is
+therefore given by
+Enl= 2¯hω⊥n+¯h2(2πl/L)2/(2µ) (8)
+Substituting the above non-interactingeigenstates and
+eigenenergies to Eq. (3) and using Eq. (1.445.8) in [8]
+for summation over l, one obtains Eq. (3) in [1] for the
+interacting eigenstates. The normalization factor Cαis
+determined by the condition
+2πL/integraldisplay
+0dz∞/integraldisplay
+0ρ2dρ∝angbracketleftα′|ρ,z∝angbracketright∝angbracketleftρ,z|α∝angbracketright=δαα′.(9)2
+Using orthogonality of the Legendre polynomials it can
+be expressed as
+Cα=2
+a⊥√
+πL/bracketleftbigg∞/summationdisplay
+n=0/parenleftbiggcot√ǫα−λn
+(ǫα−λn)3/2
++1
+(ǫα−λn)sin2√ǫα−λn/parenrightbigg/bracketrightbigg−1/2
+, (10)
+where the rescaled energy ǫαis given by ǫα≡
+λEα/(2¯hω⊥) andλ≡(L/a⊥)2is the aspect ratio.
+Forthe Fermi-Huanginteractionthe eigenenergyequa-
+tion (4) attains the form
+∂
+∂r
+r/summationdisplay
+n,l∝angbracketleftρ|n∝angbracketright∝angbracketleftz|l∝angbracketright∝angbracketleftn|0∝angbracketright∝angbracketleft0|0∝angbracketright
+E−Enl
+
+r=0=1
+V(11)
+In the limit r→0, the eigenstate is spherically symmet-
+ric. This allows us to deal with the z-axis only (see [6]).
+Substitution of the noninteracting wavefunctions (6) and
+(7) and energies (8) with the subsequent summation over
+lleads to
+√
+λ∂
+∂z/bracketleftBigg
+z∞/summationdisplay
+n=0cos/parenleftbig
+2√ǫα−λn(z/L−1/2)/parenrightbig
+√ǫα−λnsin√ǫα−λn/bracketrightBigg
+z=0=as
+a⊥.
+(12)
+Thissum containsboth the regularpartand the irregular
+one, the latter being proportional to z−1. The regular
+part can be extracted using the identity
+lim
+z→0/bracketleftBigg∞/summationdisplay
+n=0(λn−ǫ)−1/2exp(−2√
+λn−ǫz/L)−L
+λz/bracketrightBigg=−1√
+λζ/parenleftbigg1
+2,−ǫ
+λ/parenrightbigg
+(13)
+(see [6]) where ζ(ν,α) is the Hurwitz zeta function (see,
+e. g., [7, 8]. Finally we arriveat the transcendental equa-
+tion for the eigenenergies (4) in [1]. The summands in
+the sums Eqs. (3) and (4) in [1] and in Eq. (10) decay
+exponentially with n, leading to the fast converging se-
+ries. Note that the imaginary parts of the two terms
+in the left hand side of Eq. (4) in [1] cancel each other
+automatically.
+[1] V. A. Yurovsky and M. Olshanii, Restricted Thermaliza-
+tion for Two Interacting Atoms in a Multimode Harmonic
+Waveguide.
+[2] P.ˇSeba, Phys. Rev. Lett. 64, 1855 (1990).
+[3] S. Albeverio and P. Kurasov., Singular perturbations of
+differential operators : solvable Schr¨ oinger type operator s,
+no.271inLondonMathematical Societylecturenoteseries
+(University Press, Cambridge, 2000).
+[4] S. Albeverio and P. ˇSeba, J. Stat. Phys. 64, 369 (1991).
+[5] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).
+[6] M. G. Moore, T. Bergeman, and M. Olshanii, J. Phys. IV
+(France) 116, 69 (2004).
+[7] V. A. Yurovsky, M. Olshanii, and D. S. Weiss, in Adv.
+At. Mol. Opt. Phys. (Elsvier Academic Press, New York,
+2008), vol. 55, pp. 61–138.
+[8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Se-
+ries, and Products (Academic Press, New York, 1994).
\ No newline at end of file
diff --git a/1001.0226.txt b/1001.0226.txt
new file mode 100644
index 0000000000000000000000000000000000000000..0eea0f5a5557db4b746e4eab85d500a4a06a43ee
--- /dev/null
+++ b/1001.0226.txt
@@ -0,0 +1,545 @@
+arXiv:1001.0226v2  [physics.gen-ph]  23 Jul 2010Derivation of Gell-Mann-Nishijima formula from
+the electromagnetic field modes of a hadron
+Huai-yang Cui
+Department of Physics, Beihang University
+Beijing, 100191, China, E-mail: hycui@buaa.edu.cn
+July 26, 2010
+Abstract
+When an electron probes another elementary particle Q, the w ave
+function of the electron can be separated into two independe nt parts,
+the first part represents the electronic motion, the second p art represents
+the electromagnetic field mode around the particle Q. In anal ogy with
+optical modes TEM nlmfor a laser resonator, when the electromagnetic
+fieldaroundtheparticle Qforms intoamode, thequantumnumb ersofthe
+mode satisfy the Gell-Mann-Nishijima formula, these quant um numbers
+are recognized as the charge number, baryon number and stran geness
+number. The modes are used as a visual model to understand the abstract
+baryon number and strangeness number of hadrons.
+PACS numbers: 11.30.Fs, 12.20.Ds,31.15.ae
+1 Introduction
+The Gell-Mann-Nishijima formularelatesthe baryonnumber B, the strangeness
+numberS, the isospin I3of hadrons to the charge number Q. It was originally
+given by Tadao Nakano and Kazuhiko Nishijima in 1953[1, 2], Murray Gell-
+Mann proposed the formula independently in 1956[3]. The modern ver sion of
+the formula relates all flavorquantum numbers with the baryon num ber and the
+electriccharge. Wecanunderstandthe chargenumber QthroughtheCoulomb’s
+interaction, but failed to understand the baryon number Band the strangeness
+numberSwith reality, so B,SandI3are abstract. Since BandSappear in
+the formula with the charge number Q, they must relate to the electromagnetic
+interaction in some way, this clue guide us to study the possible electr omagnetic
+field modes around a hadron, in analogy with optical modes TEMnlmfor a
+laser resonator. As a result, it is found that the Gell-Mann-Nishijima formula
+is the critical condition of the electromagnetic field modes around a h adron. It
+is important to realize that the electromagnetic field modes around a hadron
+provide us a visual model to understand the abstract baryon num berBand
+strangeness number Sin reality.
+12 Electromagnetic field around a particle
+Consider an electron probing an elementary particle Q, the wave function of the
+electron is given by[4]
+(pµ−eAµ)ψ=−i¯h∂µψ (1)
+ψ(x) =exp[i
+¯h/integraldisplay
+l(pµ−eAµ)dxµ] (2)
+WhereAµis the 4-vector potential around the particle Q. Frequently, pµin
+Eq.(1) is understood as momentum operator of the electron, but h ere we regard
+pµas momentum itself as shown in Eq.(2) for calculating the wave functio n.
+Because of path independence, lis an arbitrary path from the initial point x0to
+the observed point x, we can choose two path landl+Lto calculate the wave
+function, as shown in Figure 1, Lis a closed loop linking the observed point x
+around the particle Q, then
+ψ(x)l=ψ(x)l+L (3)
+exp[i
+¯h/integraldisplay
+l(pµ−eAµ)dxµ] =exp[i
+¯h/integraldisplay
+l+L(pµ−eAµ)dxµ] (4)
+We obtain a quantization equation:
+1
+¯h/integraldisplay
+L(pµ−eAµ)dxµ= 2πn n= 0,±1,±2,... (5)
+x3 x1 x2 
+r x     l      ∞ 
+L 
+O  Q 
+Figure1: Thecalculationofthewavefunctionaroundaparticlebypa thintegral.
+The second part of the loop integral,1
+¯h/integraltext
+L(−eAµ)dxµ, is independent from
+the first part1
+¯h/integraltext
+L(pµ)dxµ, no mater how much the electron energy εis (ε=
+1MeVorε= 500MeV), it always hold the quantization equation, so we have
+1
+¯h/integraldisplay
+L(−eAµ)dxµ= 2πn′n′= 0,±1,±2,... (6)
+2This part represents the properties of the particle Q, and defining
+ξ(x) =exp[i
+¯h/integraldisplay
+l(−eAµ)dxµ] (7)
+We callξ(x) as the potential-wave-function of the particle Q, Eq.(6) is the
+quantization equation of the potential-wave-function.
+3 Magnetism of particle
+If the particle Qhas magnetic field around it without electric field, E= 0, then
+A4(x) =i
+cV(x) =i
+c/integraldisplay∞
+xE·dl= 0 (8)
+The quantized Eq.(6) reduces to
+−e
+¯h/integraldisplay
+LA·dl= 2πn n= 0,±1,±2,... (9)
+Becauseofmagnetic field B=∇×A, we find that the magnetic flux Φ mthrough
+the closed loop Lis quantized by
+Φm=/integraldisplay
+SB·ds=/integraldisplay
+S(∇×A)·ds=/contintegraldisplay
+LA·dl=−2πn¯h
+e(10)
+It is important to note that the loop shape Lor surface area Sare arbitrary
+because of path independence for the potential-wave-function, the magnetic
+flux must concentrate into the particle interior as a tiny solenoid, th e particle
+position is a singularity in mathematics. Experiments have made numer ous
+measurements on the magnetic moments of baryons, but lacking at tentions to
+magnetic fluxes of baryons.
+4 Gell-Mann-Nishijima formula for hadrons
+In order to describe the 4D electromagnetic field around the partic leQ, we
+choose a big closed loop Las shown in Figure 2, saying A-B-C-D-A, where A
+contains the particle at the origin and C at the infinity, from Eq.(6) we have
+−e
+¯h/contintegraldisplay
+LAr·dr+−e
+¯h/contintegraldisplay
+LAθ·rdθ+−e
+¯h/contintegraldisplay
+LAφ·rsinθdφ+−e
+¯h/contintegraldisplay
+LA4·d(ict) = 2πn(11)
+n= 0,±1,±2,...
+The big closed loop Lmakes that the angle θvaries from 0 to 2 πand the
+angleφvaries from 0 to 2 π, the radical distance rvaries from 0 to R=∞then
+returns to 0, time t varies from 0 to T=∞then returns to 0. Note that Lis
+not the path of the probing electron, is a calculation path for wave f unction.
+We can imagine a conformal transformation for the electromagnet ic field
+around the nucleus, as shown in Figure 3, from 4D Ain the circular region to
+4DAin the conformal box. (conformal theory in mathematics is a classic al
+branch). We use this conformal box to emphasize that we must fair ly treatAr,
+3x3 
+ x1 
+ x2 
+ 
+r L 
+θ 
+φ A B  (at infinity)  C 
+D 
+Figure 2: The big closed loop ABCDA around a particle.
+Aθ,AφandA4. Regarding this 4D conformal box, the boundary conditions for
+the electromagnetic field are given by
+−e
+¯h/integraldisplayR=∞
+0Ar·dr= 2πQmQm= 0,±1,±2,... (12)
+−e
+¯h/integraldisplay2π
+0Aθ·rdθ= 2πBmBm= 0,±1,±2,... (13)
+−e
+¯h/integraldisplay2π
+0Aφ·rsinθdφ= 2πSmSm= 0,±1,±2,... (14)
+−e
+¯h/integraldisplayT=∞
+0A4·d(ict) = 2πQ Q= 0,±1,±2,... (15)
+These quantization equations are obvious when thinking about that the four
+components of the vector potential are independent from each o ther.
+The path from the corner ( r= 0,θ= 0,φ= 0,t= 0) in the 4D conformal
+box to the corner ( R,π,π,T ) corresponds to the half loop of the big closed
+loopLas shown in Figure 2, saying A-B-C. Considering the half loop A-B-C in
+Figure 2, then Eq.(11) becomes
+−e
+¯h/integraldisplayR
+0Ar·dr+−e
+2¯h/integraldisplay2π
+0Aθ·rdθ+−e
+2¯h/integraldisplay2π
+0Aφ·rsinθdφ+−e
+¯h/integraldisplayT
+0A4·d(ict) =2πn
+2(16)
+n= 0,±1,±2,...
+Substituting Eq.(12-15) into Eq.(16), we obtain
+Qm+Bm+Sm
+2+Q=n
+2n= 0,±1,±2,... (17)
+We immediately recognize that this equation is the Gell-Mann-Nishijima r e-
+lation for hadrons: According to hadron proprieties, we have to de fineQas
+4↑ Ar  
+Aθ ↑Ar  
+Aθ ↑ Ar  
+Aθ ↑ Ar  
+→ Aθ 
+Conformal transformation Conformal box  
+Figure 3: A conformal transformation from the circular region to t he conformal
+box.
+the charge number, to define −Bmas the baryon number B, to define −Bm
+as strangeness number S, to definen
+2as the third component of isospin I3,
+typically,Ar= 0,Qm= 0, Eq.(17) becomes the Gell-Mann-Nishijima formula
+Q−B+S
+2=I3I3= 0,±1
+2,±1,... (18)
+To note that the charge number Qis conserved, therefore we deduce that the
+baryonnumber Bisconservedandthestrangenessnumber Sisconservedtoofor
+all hadron reactions. It is important to realizethat the electromag neticfield of a
+hadron can be regarded as the higher order mode in terms of its pot ential-wave-
+function, marked by three quantum numbers: chargenumber Q, baryonnumber
+Band strangeness number S. This result has three significations: (1) the for-
+malism we developed about the potential-wave-function is success a nd useful for
+elementary particle classification. (2) the potential-wave-functio n provides us a
+visual model for abstract baryon number and strangeness numb er, in analogy
+with the optical mode TEMqlmin a laser resonator, it provides us an impor-
+tant work platform for extending our knowledge. (3) the Gell-Mann -Nishijima
+formula is the critical condition of the electromagnetic field modes ar ound a
+hadron.
+5 Charge quantization
+Consider the fourth component quantization Eq.(15) and the path indepen-
+dence, we have
+−e
+¯h/integraldisplayT
+0A4(r)·d(ict) =−eA4(r)
+¯h/integraldisplayT
+0d(ict) =−eA4(r)icT
+¯h(19)
+A4(r) =i
+cV(r) =i
+c/integraldisplay∞
+rE·dl=ikeq
+cr(20)
+5Where we have used the Coulomb’s law E=keq/r2, we have
+−e
+¯h/integraldisplayT
+0A4(r)·d(ict) =keeq
+r·T·1
+¯h= 2πQ Q= 0,±1,±2,...(21)
+From the Heisenberg’s uncertainty principle, we immediately recogniz e that the
+potential energy is the uncertain energy △E=keeq/rfor the observed point
+x, and the time limit Tfor the observed point xis the uncertain time △t=T,
+both satisfies the charge quantization and the uncertainty princip le, because
+from Eq(21):
+q=Qe Q= 0,±1,±2,... (22)
+△E·△t= (keee
+r)·T= 2π¯h (23)
+The charge qof the particle Qhas been quantized by Eq.(22). Here for the
+first time, the charge quantization can be derived from the quantu m mechanics.
+In fact, the uncertainty principle Eq.(23) uses the energy keee/rto determine
+time limitTwithout regarding the elementary particle charge number Q, it is
+in minimum case for the Heisenberg’s uncertainty principle, or:
+△E·△t=|keeQe
+r|·T≥2π¯h Q=±1,±2,... (24)
+Where we discard Q= 0.
+6 Confinement of quarks
+Quarks are abstract particles; here we focus on how to form the q uarks by using
+our theory. The charge number Q, baryon number Band strangeness number
+Sare three freedoms for classifying elementary particles, we can bu ild a frame
+in whichQ,BandSare the three independent axes as shown in Figure 4. But
+mathematicians have rights to rotate the frame ( Q,B,S) to a new orientation
+and let it becomes a new frame ( u,d,s) as shownin Figure 4, the transformation
+equations between the old coordinates ( Q,B,S) and new coordinates ( u,d,s)
+are given by
+Q
+B
+C
+=
+R11R12R13
+R21R22R23
+R31R32R33
+·
+u
+d
+s
+ (25)
+It is an easy thing to get the transformation matrix Rafter reading the basic
+data of baryons and three quarks, finally it is
+
+Q
+B
+C
+=
+2/3−1/3−1/3
+1/3 1/3 1/3
+0 0 −1
+·
+u
+d
+s
+ (26)
+Where the axis urepresents the up-quark number, the axis drepresents the
+down-quark number; the axis srepresents the strange-quark number. In our
+viewpoint, the three quark types are another mathematical repr esentation for
+the higher order modes of the electromagnetic field around a partic le.
+6B    d u 
+ 
+Q s   S  
+O Particle status 
+Figure 4: visual model for forming quarks.
+The motivation of this section is to make readers to understand with real-
+ity: (1) The electromagnetic filed modes around a hadron have thre e quantum
+numbers:Q,BandS, therefore the quark type number through the mathe-
+matical transformation in Eq.(26) are three. (2) The three quark types have
+been confined in Eq.(26), we have no way to let the three quark type s being free
+particles mathematically or physically, experiments have no chance t o discover
+free quark in reality, like virtual phonons in metals.
+7 Six components of electromagnetic field
+Generally speaking, the electromagneticfield around the particle Qhas six com-
+ponent: (Br,Bθ,Bφ,Er,Eθ,Eφ), we can not naively think that Eθ=Eφ= 0,
+specially the lift time for some hadrons has only 10−10seconds or even shorter.
+The electrical potential Vin the fourth component quantization Eq.(15) is writ-
+ten as
+−e
+¯h/integraldisplayT
+0A4·d(ict) =eT
+¯h/integraldisplay∞
+rE·dl (27)
+It is also path independent about the electric field, so we have
+eT
+¯h/contintegraldisplay
+LE·dl= 2πn n= 0,±1,±2,... (28)
+WhereLis the big close loop linking to the observedpoint x, as shown in Figure
+2, saying A-B-C-D-A. We are now in a position to realize: (1) the elect ric field
+around the particle has been quantized by Eq.(28) mathematically. ( 2) if the
+quantum number n= 0, thenEθ=Eφ= 0, then Eq.(28) is the Ampere’s law
+for the electrostatic field. (3) if n/negationslash= 0, then, Eθ/negationslash= 0 orEφ/negationslash= 0, the electric
+fieldEis recognized as a higher order mode ( ndenotes the mode rank) with
+7a mathematical singularity at the particle. (3) regarding the poten tial-wave-
+function, the boundary conditions for the electric field are
+eT
+¯h/integraldisplay∞
+rEr·dr= 2πQ Q= 0,±1,±2,... (29)
+eT
+¯h/integraldisplay2π
+0Eθ·rdθ= 2πBeBe= 0,±1,±2,... (30)
+eT
+¯h/integraldisplay2π
+0Eφ·rsinθdφ= 2πSeSe= 0,±1,±2,... (31)
+Considering the half loop A-B-C in Figure 2, then Eq.(15) becomes
+−e
+¯h/integraldisplayR
+0Ar·dr+−e
+2¯h/integraldisplay2π
+0Aθ·rdθ+−e
+2¯h/integraldisplay2π
+0Aφ·rsinθdφ
++eT
+¯h/integraldisplay∞
+REr·dr+eT
+2¯h/integraldisplay2π
+0Eθ·rdθ+eT
+2¯h/integraldisplay2π
+0Eθ·rdθ=2πn
+2(32)
+n= 0,±1,±2,...
+Substituting Eq.(12-15) and Eq.(29-31) into Eq.(32), we obtain
+Qm+Bm+Sm
+2+Q+Be+Se
+2=n
+2n= 0,±1,±2,... (33)
+We immediately recognize that this equation is the Gell-Mann-Nishijima r ela-
+tion for hadrons.
+Therefore, the electromagnetic field mode around a particle has six quantum
+numbers: ( Q,Be,Se,Qm,Bm,Sm), thus the independent quark type number
+relating to the six quantum numbers are actually six. Recalling up to no w we
+have known that the quarks have six types ( u,d,s,c,b,t ) and the leptons have
+six type too, and the electromagnetic field has six components, we r ealize that 6
+is a magic number for physics: all physical quantities are sharing the 6 freedoms
+originating from the 6 components of electromagnetic field in explicit o r implicit
+ways[4].
+Because of the charge conservation, we deduce that these six qu antum num-
+bers are conserved, in agreement with the experiments of hadron s in conser-
+vation law number: six. Of cause, the details of the relationships con cerning
+quarks seem to be complex and abstract.
+8 Gell-Mann-Nishijima formula in hydrogen atom
+Hydrogenatom consists of a single electron bound to its centralnu cleus, a single
+proton, by the attractive Coulomb force that acts between them . In the theory
+of relativistic quantum mechanics, the 4-vector momentum of an ele ctronpis
+given by
+(pµ+qAµ)ψ=−i¯h∂µψ µ= 1,2,3,4 (34)
+WhereAis the 4-vector potential of the electromagnetic field, ψis the wave-
+function of the electron. We have adopted momentum prather than momentum
+8operator ˆpin Eq.(34)[4]. In our viewpoint, the wave-function ψis the potential-
+wave-function of the atom, because
+ψ= exp[i
+¯h/integraldisplay
+l(pµ+qAµ)·dxµ] (35)
+Wherelisanarbitrarypathfromthe observedpointxtotheinfinity. Inana logy
+with the electric field modes discussed in the preceding sections, for any closed
+loopLcontaining the nucleus in Figure 1, the wave function ψfor the electron
+in a hydrogen atom is quantized by
+1
+¯h/contintegraldisplay
+L(pµ+qAµ)·dxµ= 2πn n= 0,±1,±2,... (36)
+The nucleus of the hydrogen atom provides the 4-vector potentia l for the elec-
+tron:
+Ar=Aθ=Aφ= 0;A4=ie
+cr(37)
+Where we take ke= 1 in the Coulomb’s law: Gaussian units. Because the
+electron keeps the angular momentum conservation, the angular m omentum
+magnitude J=r/radicalBig
+p2
+θ+p2
+φis a constant and its z-axis component magnitude
+Jz=rsinθpφis a constant too. The energy of the electron Ealso is a constant
+for its stationary state: ψ=ψ(r)e−iEt/¯h, thus
+exp(−iEt/¯h) = exp[i
+¯h/integraldisplay
+l(p4+qA4)·dx4] (38)
+p4=−E−e2/r
+icx4=ict (39)
+Frompµpµ=−m2
+ec2, wheremeis the mass of the electron, we obtain
+pr=/radicalBig
+−m2ec2−p2
+θ−p2
+φ−p2
+4=/radicalbigg
+−m2ec2−J2
+r2+1
+c2(E+e2
+r)2(40)
+pθ=/radicalBig
+(p2
+θ+p2
+φ)−p2
+φ=1
+r/radicalbigg
+J2−J2z
+sin2θ(41)
+Therefore, quantization Eq.(36) for the loop in Figure 2 becomes
+1
+k/contintegraldisplay
+Lpr·dr+1
+k/contintegraldisplay
+Lpθ·rdθ+1
+k/contintegraldisplay
+Lpφ·rsinθdφ= 2πn (42)
+n= 0,±1,±2,...
+Where the loop integral about the time tcomponent vanishes for the constant
+energyEin Eq.(38), while the loop becomes three quantization equations
+1
+¯h/integraldisplay∞
+0pr·dr=πf f= 0,±1,±2,... (43)
+1
+¯h/integraldisplay2π
+0pθ·rdθ= 2πg g= 0,±1,±2,... (44)
+1
+¯h/integraldisplay2π
+0pφ·rsinθdφ= 2πm m= 0,±1,±2,... (45)
+9They also can be written as
+1
+¯h/integraldisplay∞
+0/radicalbigg
+−m2ec2−J2
+r2+1
+c2(E+e2
+r)2·dr=πf f= 0,±1,±2,...(46)
+1
+¯h/integraldisplay2π
+0/radicalbigg
+J2−J2z
+sin2θ·dθ= 2πg g= 0,±1,±2,...(47)
+1
+¯h/integraldisplay2π
+0Jzdφ= 2πm m= 0,±1,±2,...(48)
+These definite integrals have been evaluated on the ranges: these integrands
+take real values in the previous paper[4], as the result, they are
+πEα/radicalbig
+m2ec4−E2−π/radicalbig
+(g+|m|)2−α2=πf f= 0,±1,±2,...(49)
+2π
+¯h(J−|Jz|) = 2πg g= 0,±1,±2,...(50)
+2π
+¯hJz= 2πm m= 0,±1,±2,...(51)
+Where the α=e2/(¯hc) is known as the fine structure constant. It is very easy
+to get the energy levels of the electron from the Eq.(49):
+E=mec2/bracketleftBigg
+1+α2
+(/radicalbig
+(g+|m|)2−α2+f)2/bracketrightBigg−1
+2
+(52)
+This result, Eq.(52), is completely the same as the calculation of Dirac wave
+equationforhydrogenatom,itisjustthefinestructureofhydro genatomenergy.
+Considering the half loop A-B-C in Figure 2 about the nucleus, then Eq .(42)
+becomes
+1
+k/integraldisplay∞
+0pr·dr+1
+2k/integraldisplay2π
+0pθ·rdθ+1
+2k/integraldisplay2π
+0pφ·rsinθdφ=2πn
+2(53)
+n= 0,±1,±2,...
+Substituting Eq.(43-45) into Eq.(53), we find that the Gell-Mann-Nis hijima for-
+mula also exists in hydrogen atom:
+f
+2+g+m
+2=n
+2n= 0,±1,±2,.. (54)
+In the stationary state spectrum of hydrogen atom, frequently f= 0. This Gell-
+Mann-Nishijima formula is the critical condition of the wave function m odes
+around a hydrogen atom.
+Among Eq.(43-45), it is unfair to ψ(r),ψ(θ) andψ(φ), because we let ψ(r)
+form a standing wave (half integral multiple) from r= 0 tor=∞whileψ(θ)
+andψ(φ) are simple wave (integral multiple). As we have mentioned, we must
+fairly treat the three components in their conformal box (Figure 3 ). Therefore,
+Fairly, if we let they all are integral multiple, then their quantum numb ers
+satisfy the standard Gell-Mann-Nishijima formula.
+109 Visual Model of hadrons
+Consider a baryon of charge number Q, Baryon number Band strangeness
+numberS, its potential-wave-function can be separated into three parts a s
+ξ(r) =ξ1(r)ξ2(θ)ξ3(φ) (55)
+Wherervaries in the range(0 ,∞) withQ+1nodes,θvaries in the range(0 ,2π)
+with 2B+1 nodes,φvaries in the range (0 ,2π) with 2S+1 nodes, as shown in
+Figure 5, where only depicting each fragments for them. The pictur e in Figure
+5 is the visual model for hadrons.
+x3 
+ x1 
+ x2 
+ 
+ξ2(θ) ξ3(φ) ξ1(r) 
+Figure 5: The visual model for hadrons is represented by three po tential-wave-
+functions:ξ1(r),ξ2(θ),ξ3(φ) and their nodes, where only depicting their frag-
+ments.
+According to Eq.(26), the visual model for quarks can established from
+
+u
+d
+s
+=
+2/3−1/3−1/3
+1/3 1/3 1/3
+0 0 −1
+−1
+·
+Q
+B
+S
+ (56)
+10 Conclusions
+When an electron probes another elementary particle Q, the wave f unction of
+the electron can be separated into two independent parts, the fir st part rep-
+resents the electronic motion, the second part represents the e lectromagnetic
+field mode around the particle Q. In analogy with optical modes TEMnlmfor a
+laser resonator, when the electromagnetic field around the partic le Q forms into
+a mode, the quantum numbers of the mode satisfy the Gell-Mann-Nis hijima
+formula, these quantum numbers are recognized as the charge nu mber, baryon
+number and strangeness number. As a result, it is found that the G ell-Mann-
+Nishijima formula is the critical condition of the electromagnetic field m odes
+around a hadron. It is important to realize that the electromagnet ic field modes
+around a hadron provide us a visual model to understand the abst ract baryon
+numberBand strangeness number Sin reality.
+11References
+[1] Nakano, Tadao; Nishijima, Kazuhiko. ”Charge Independence fo r V-
+particles”. Progress of Theoretical Physics 10 (5): 581-582. (1 953).
+[2] Nishijima, Kazuhiko. ”Charge Independence Theory of V Particle s”.
+Progress of Theoretical Physics 13 (3): 285-304. (1955).
+[3] Gell-Mann, Murray. ”The Interpretation of the New Particles as Displaced
+Charged Multiplets”. Il Nuovo Cimento 4: 848. (1956).
+[4] Cui, H.Y.”Discoveryofquantumhiddenvariable”,arXiv:0711.124 7.(2007).
+12
\ No newline at end of file
diff --git a/1001.0227.txt b/1001.0227.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9b67291dbf514eea8674c8f2a2c2b8d8c0629f12
--- /dev/null
+++ b/1001.0227.txt
@@ -0,0 +1,839 @@
+arXiv:1001.0227v3  [hep-ph]  17 May 2010Analysis of the Λb→Λℓ+ℓ−decay in QCD
+T. M. Alieva∗†, K. Azizib‡, M. Savcıa§
+aPhysics Department, Middle East Technical University, 065 31 Ankara, Turkey
+bPhysics Division, Faculty of Arts and Sciences, Do˘ gu¸ s Uni versity
+Acıbadem-Kadık¨ oy, 34722 Istanbul
+Abstract
+Taking into account the Λ baryon distribution amplitudes an d the most general
+form of the interpolating current of the Λ b, the semileptonic Λ b→Λℓ+ℓ−transition
+is investigated in the framework of the light cone QCD sum rul es. Sum rules for
+all twelve form factors responsible for the Λ b→Λℓ+ℓ−decay are constructed. The
+obtained results for the form factors are used to compute the branching fraction. A
+comparison of the obtained results with the existing predic tions of the heavy quark
+effective theory ispresented. Theresultsofthebranchingra tioshowsthedetectability
+of this channel at the LHCb in the near future is quite high.
+PACS number(s): 11.55.Hx, 13.30.-a, 14.20.Mr
+∗e-mail: taliev@metu.edu.tr
+†permanent address:Institute of Physics,Baku,Azerbaijan
+‡e-mail: kazizi@dogus.edu.tr
+§e-mail: savci@metu.edu.tr1 Introduction
+Experimentally, the detection and isolation of the heavy baryons is s imple comparing to the
+light systems since having the heavy quark makes their beam narrow . In the recent years,
+considerable experimental progress has been made in the identifica tion and spectroscopy
+of the heavy baryons containing a heavy bottom or charm quark [1– 8]. These evidences
+can be considered as a good signal to search also the decay channe ls of the heavy baryons
+like, Λ b→Λℓ+ℓ−at LHCb. This rare channel, induced by the flavor changing neutral
+currents (FCNC) of b→stransition, serves testing ground for the standard model at loop
+level and is very sensitive to the new physics effects [9], such as supe rsymmetric particles
+[10], light dark matter [11] and also fourth generation of the quarks and extra dimensions,
+etc. Moreover, this channel can be inspected as a useful tool in e xact determination of the
+Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, VtbandVts, CP and T violations,
+polarization asymmetries.
+Theoretically, there are some works devoted to the analysis of the heavy baryon decays,
+where practically in all of them the predictions of the heavy quark eff ective theory (HQET)
+for form factors have been used. Transition form factors of the Λb→Λcand Λ c→Λ decays
+have been studied in three points QCD sum rules in [12], the Λ b→plνtransition form
+factors have also been calculated via three point QCD sum rules in the context of the heavy
+quark effective theory (HQET) in [13] and in the framework of the SU (3) symmetry and
+HQET in [14]. In the present work, using the most general form of th e interpolating current
+for the Λ band also the distribution amplitudes of Λ baryon, all form factors re lated to the
+electroweak penguin and weak box diagrams describing the Λ b→Λℓ+ℓ−are calculated in
+the frame work of the light cone QCD sum rules in full theory. The obt ained results for
+the form factors are used to estimate the decay rate and branch ing ratio. Regard that this
+transition has been investigated in [15] and [16] also in the context of the HQET but the
+same frame work using the distribution amplitudes of the Λ and Λ b, respectively. Moreover,
+form factors, branching ratio and di–lepton forward–backward a symmetries are studied in
+[17–19] also within the context of the HQET. In [20–22], Σ b,cand Λ b,cto nucleon transitions
+are also evaluated using the nucleon wave functions in light cone QCD s um rules approach.
+The plan of the paper is as follows: in section II, the light cone QCD sum rules for the
+form factors are obtained using the Λ DA’s. The HQET relations amon g all form factors
+are also discussed in this section. Section III is dedicated to the num erical analysis of the
+sum rules for the form factors as well as numerical results of the d ecay rate and branching
+ratio.
+2 Theoretical Framework
+The Λ b→Λℓ+ℓ−channel proceeds via FCNC b→stransition at quark level. The effective
+Hamiltonian describing the electroweak penguin and weak box diagram s related to this
+transition can be written as :
+Heff=GFαemVtbV∗
+ts
+2√
+2π/braceleftbigg
+Ceff
+9¯sγµ(1−γ5)b¯lγµl+C10¯sγµ(1−γ5)b¯lγµγ5l
+1−2mbC71
+q2¯siσµνqν(1+γ5)b¯lγµl/bracerightbigg
+. (1)
+To find the amplitude, we need to sandwich this effective Hamiltonian be tween the initial
+and final baryon states, i.e., /an}bracketle{tΛb(p+q)|Heff|Λ(p)/an}bracketri}ht. From Eq. (1) we see that in calculation
+of the Λ b→Λℓ+ℓ−decay amplitude, the matrix elements, /an}bracketle{tΛb(p+q)|¯bγµ(1−γ5)s|Λ(p)/an}bracketri}htand
+/an}bracketle{tΛb(p+q)|¯biσµνqν(1+γ5)s|Λ(p)/an}bracketri}htare appeared. These matrix elements can be parametrized
+in terms of the twelve form factors, fi,gi,fT
+iandgT
+iin the following manner:
+/an}bracketle{tΛ(p)|¯sγµ(1−γ5)b|Λb(p+q)/an}bracketri}ht= ¯uΛ(p)/bracketleftBig
+γµf1(Q2)+iσµνqνf2(Q2)+qµf3(Q2)
+−γµγ5g1(Q2)−iσµνγ5qνg2(Q2)−qµγ5g3(Q2)/bracketrightBig
+uΛb(p+q), (2)
+and
+/an}bracketle{tΛ(p)|¯siσµνqν(1+γ5)b|Λb(p+q)/an}bracketri}ht= ¯uΛ(p)/bracketleftBig
+γµfT
+1(Q2)+iσµνqνfT
+2(Q2)+qµfT
+3(Q2)
++γµγ5gT
+1(Q2)+iσµνγ5qνgT
+2(Q2)+qµγ5gT
+3(Q2)/bracketrightBig
+uΛb(p+q), (3)
+whereQ2=−q2. For calculation of these form factors we use the QCD sum rules app roach.
+To obtain the sum rules for the form factors in this approach, the f ollowing correlation
+functions, the main objects in this approach, are considered:
+ΠI
+µ(p,q) =i/integraldisplay
+d4xe−iqx/an}bracketle{t0|T{JΛb(0),¯b(x)γµ(1−γ5)s(x))} |Λ(p)/an}bracketri}ht,
+ΠII
+µ(p,q) =i/integraldisplay
+d4xe−iqx/an}bracketle{t0|T{JΛb(0),¯b(x)iσµνqν(1+γ5)s(x)} |Λ(p)/an}bracketri}ht, (4)
+where,prepresents the Λ’s momentum and qis the transferred momentum and the JΛb
+is interpolating current of Λ b. The most general form of the interpolating current of Λ b
+baryon can be written as:
+JΛb(x) =1√
+6ǫabc/braceleftbigg
+2/bracketleftBig
+(qaT
+1(x)Cqb
+2(x))γ5bc(x)+β(qaT
+1(x)Cγ5qb
+2(x))bc(x)/bracketrightBig
++ (qaT
+1(x)Cbb(x))γ5qc
+2(x)+β(qaT
+1(x)Cγ5bb(x))qc
+2(x)
++ (baT(x)Cqb
+2(x))γ5qc
+1(x)+β(baT(x)Cγ5qb
+2(x))qc
+1(x)/bracerightbigg
+, (5)
+whereq1andq2aretheuanddquarks, respectively, a, bandcarecolorindexes and Cisthe
+charge conjugation operator. The βis an arbitrary parameter with β=−1 corresponding
+to the Ioffe current.
+In order to obtain the sum rules for the transition form factors, w e will calculate the
+aforementioned correlation functions in two different ways, namely , physical (phenomeno-
+logical) and theoretical (QCD) sides and equate these two represe ntations isolating the
+2ground state through the dispersion relation. Finally, to suppress the contribution of the
+higher states and continuum, we will apply the Borel transformatio n and continuum sub-
+traction to both sides of the correlation function and impose the qu ark hadron duality
+assumption.
+The first stepistocalculatethephysical side ofthecorrelationfun ctions. Saturatingthe
+correlation functions with complete set of the intermediate states with the same quantum
+numbers as the initial state, for the physical part of the correlat ion function we obtain,
+ΠI
+µ(p,q) =/summationdisplay
+s/an}bracketle{t0|JΛb(0)|Λb(p+q,s)/an}bracketri}ht/an}bracketle{tΛb(p+q,s)|¯bγµ(1−γ5)s|Λ(p)/an}bracketri}ht
+m2
+Λb−(p+q)2+···,(6)
+ΠII
+µ(p,q) =/summationdisplay
+s/an}bracketle{t0|JΛb(0)|Λb(p+q,s)/an}bracketri}ht/an}bracketle{tΛb(p+q,s)|¯biσµνqν(1+γ5)s|Λ(p)/an}bracketri}ht
+m2
+Λb−(p+q)2+···,(7)
+where the dots ···represent the contribution of the higher states and continuum. T he
+vacuum to the baryon matrix element of the interpolating current, /an}bracketle{t0|JΛb(0)|Λb(p+q,s)/an}bracketri}ht
+is written in terms of the residue, λΛbas:
+/an}bracketle{t0|JΛb(0)|Λb(p+q,s)/an}bracketri}ht=λΛQ¯uΛQ(p+q,s). (8)
+Putting Eqs. (2), (3) and (8) in Eqs. (6) and (7) and performing su mmation over spins of
+the Λbbaryon using
+/summationdisplay
+suΛb(p+q,s)uΛb(p+q,s) =/ne}ationslashp+/ne}ationslashq+mΛb, (9)
+we get the following expressions for the correlation functions
+ΠI
+µ(p,q) =λΛb/ne}ationslashp+/ne}ationslashq+mΛb
+m2
+Λb−(p+q)2/braceleftBig
+γµf1−iσµνqνf2+qµf3
+−γµγ5g1−iσµνqνγ5g2+qµγ5g3/bracerightBig
+uΛ(p), (10)
+ΠII
+µ(p,q) =λΛb/ne}ationslashp+/ne}ationslashq+mΛb
+m2
+Λb−(p+q)2/braceleftBig
+γµfT
+1−iσµνqνfT
+2+qµfT
+3
++γµγ5gT
+1+iσµνγ5qνgT
+2−qµγ5gT
+3/bracerightBig
+uΛ(p). (11)
+Using the equation of motion and Eqs. (10) and (11), we get the follo wing final expressions
+for the phenomenological sides of the correlation functions:
+ΠI
+µ(p,q) =λΛb
+m2
+Λb−(p+q)2/braceleftBig
+2f1(Q2)pµ+2f2(Q2)pµ/ne}ationslashq+/bracketleftBig
+f2(Q2)+f3(Q2)/bracketrightBig
+qµ/ne}ationslashq
+−2g1(Q2)pµγ5+2g2(Q2)pµ/ne}ationslashqγ5+/bracketleftBig
+g2(Q2)+g3(Q2)/bracketrightBig
+qµ/ne}ationslashqγ5
++ other structures/bracerightBig
+uΛ(p), (12)
+3ΠII
+µ(p,q) =λΛb
+m2
+Λb−(p+q)2/braceleftBig
+2fT
+1(Q2)pµ+2fT
+2(Q2)pµ/ne}ationslashq+/bracketleftBig
+fT
+2(Q2)+fT
+3(Q2)/bracketrightBig
+qµ/ne}ationslashq
++ 2gT
+1(Q2)pµγ5−2gT
+2(Q2)pµ/ne}ationslashqγ5−/bracketleftBig
+gT
+2(Q2)+gT
+3(Q2)/bracketrightBig
+qµ/ne}ationslashqγ5
++ other structures/bracerightBig
+uΛ(p). (13)
+To compute the form factors or their combinations, f1,f2,f2+f3,g1,g2andg2+g3,
+we will choose the independent structures pµ,pµ/q,qµ/q,pµγ5,pµ/qγ5, andqµ/qγ5from Eq.
+(12), respectively. The same structures are selected to calculat e the form factors or their
+combinations labeled by T in the second correlation function, Eq. (13 ).
+The next step is to calculate the correlation functions from QCD side in deep Euclidean
+region where ( p+q)2≪0. For this aim, we expend the time ordering products of the
+interpolating current of the Λ band transition currents in the correlation functions (see
+Eq. (4)) near the light cone, x2≃0 via operator product expansion, where the short
+and long distance effects are separated. The former is calculated u sing QCD perturbation
+theory, whereas the latter are parameterized in terms of the Λ DA ’s. Mathematically, this
+is equivalent to contract out all quark pairs in the time ordering prod uct of theJΛband
+transition currents via the Wick’s theorem. As a result of this proce dure, we obtain the
+following representations of the correlation functions in QCD side:
+ΠI
+µ=−i√
+6ǫabc/integraldisplay
+d4xeiqx/braceleftBig/bracketleftBig
+2(C)ηφ(γ5)ρβ+(C)ηβ(γ5)ρφ+(C)βφ(γ5)ηρ/bracketrightBig
++β/bracketleftBig
+2(Cγ5)ηφ(I)ρβ+(Cγ5)ηβ(I)ρφ+(Cγ5)βφ(I)ηρ/bracketrightBig/bracerightBig/bracketleftBig
+γµ(1−γ5)/bracketrightBig
+σθ
+×Sb(−x)βσ/an}bracketle{t0|ua
+η(0)sb
+θ(x)dc
+φ(0)|Λ(p)/an}bracketri}ht, (14)
+ΠII
+µ=−i√
+6ǫabc/integraldisplay
+d4xeiqx/braceleftBig/bracketleftBig
+2(C)ηφ(γ5)ρβ+(C)ηβ(γ5)ρφ+(C)βφ(γ5)ηρ/bracketrightBig
++β/bracketleftBig
+2(Cγ5)ηφ(I)ρβ+(Cγ5)ηβ(I)ρφ+(Cγ5)βφ(I)ηρ/bracketrightBig/bracerightBig/bracketleftBig
+iqνσµν(1+γ5)/bracketrightBig
+σθ
+×Sb(−x)βσ/an}bracketle{t0|ua
+η(0)sb
+θ(x)dc
+φ(0)|Λ(p)/an}bracketri}ht, (15)
+The heavy quark propagator, Sb(x) is calculated in [23]:
+Sb(x) =Sfree
+b(x)−igs/integraldisplayd4k
+(2π)4e−ikx/integraldisplay1
+0dv/bracketleftbigg/ne}ationslashk+mQ
+(m2
+Q−k2)2Gµν(vx)σµν
++1
+m2
+Q−k2vxµGµνγν/bracketrightbigg
+. (16)
+where,
+Sfree
+b=m2
+b
+4π2K1(mb√
+−x2)√
+−x2−im2
+b/ne}ationslashx
+4π2x2K2(mb√
+−x2), (17)
+4andKiare the Bessel functions. The terms proportional to the gluon fie ld strength are
+contributed mainly to the four and five particle distribution function s [23–27] and expected
+to be very small in our case, hence when doing calculations, these te rms are ignored. The
+matrix element ǫabc/an}bracketle{t0|ua
+η(0)db
+θ(0)sc
+φ(x)|Λ(p)/an}bracketri}htappearing in Eqs. (14,15) represents the Λ’s
+wave functions, which are calculated in [27] and we list them out for th e completeness of
+this paper in the Appendix. Using the Λ wave functions and the expre ssion of the heavy
+quark propagator, and after performing the Fourier transform ation, the final expressions
+of the correlation functions for both vertexes are found in terms of the Λ DA’s in QCD or
+theoretical side.
+In order to obtain the sum rules for the form factors, f1,f2,f3,g1,g2,g3,fT
+1,fT
+2,fT
+3,
+gT
+1,gT
+2andgT
+3, we equate the coefficients of the corresponding structures fro m both sides of
+the correlation functions through the dispersion relations and app ly Borel transformation
+with respect to ( p+q)2to suppress the contribution of the higher states and continuum.
+The expressions for the sum rules of the form factors are very len gthy, so we will give only
+extrapolation formulas to explore their dependency on the transf erred momentum squared
+q2.
+The explicit expressions of the sum rules for the form factors depic t that to calculate
+the values of the form factors, we need also the expression of the residue,λΛb. This residue
+is calculated in [28].
+Few words about the relations among the form factors in HQET are in order. In HQET,
+the number of independent form factors is reduced to two, F1andF2, so the transition
+matrix element can be parameterized in terms of these two form fac tors as [28,29]:
+/an}bracketle{tΛ(p)|¯sΓb|Λb(p+q)/an}bracketri}ht= ¯uΛ(p)[F1(Q2)+/ne}ationslashvF2(Q2)]ΓuΛb(p+q), (18)
+Here, Γ refers to any Dirac matrices and /ne}ationslashv= (/ne}ationslashp+/ne}ationslashq)/mΛb. Comparing this matrix element
+with definitions of the form factors in Eqs. (2) and (3), the following relations among the
+form factors are obtained (see also [30,31]):
+f1=g1=fT
+2=gT
+2=F1+mΛ
+mΛbF2,
+f2=g2=f3=g3=F2
+mΛb,
+fT
+1=gT
+1=F2
+mΛbq2,
+fT
+3=−F2
+mΛb(mΛb−mΛ),
+gT
+3=F2
+mΛb(mΛb+mΛ). (19)
+3 Numerical Analysis
+This section is devoted to the numerical analysis of the form factor s, their extrapolation
+in terms of the momentum transferred square and calculation of th e total decay rate and
+branching ratio for rare Λ b→Λℓ+ℓ−transition in QCD.
+5Someinputparametersusedinthenumericalcalculationsare: /an}bracketle{t¯uu/an}bracketri}ht(1GeV) =/an}bracketle{t¯dd/an}bracketri}ht(1GeV) =
+−(0.243±0.01)3GeV3,m2
+0(1GeV) = (0.8±0.2)GeV2[32],mΛ= (1115.683±0.006)MeV,
+mΛb= (5620.2±1.6)MeVandmb= (4.7±0.1)GeV. Sum rules for the form factors de-
+pict that the Λ DA’s are the main input parameters (see the Appendix ). They contain 4
+independent parameters which are given as [27]:
+fΛ= (6.0±0.3)×10−3GeV2, λ 1= (1.0±0.3)×10−2GeV2,
+|λ2|= (0.83±0.05)×10−2GeV2, |λ3|= (0.83±0.05)×10−2GeV2.(20)
+It is well known that, the Wilson coefficient Ceff
+9receives long distance contributions
+fromJ/ψfamily, in addition to short distance contributions. In the present w ork, we do
+not take into account the long distance effects. From the explicit ex pressions for the form
+factors, it is clear that they depend on three auxiliary parameters , continuum threshold
+s0, Borel mass parameter M2
+Band the parameter βentering the most general form of the
+interpolating current of the Λ b. The form factors should be independent of these auxiliary
+parameters. Therefore, we look for working regions for these pa rameters, where the form
+factors are practically independent of them. To determine the wor king region for the Borel
+mass parameter the procedure is as follows: the lower limit is obtained requiring that the
+higher states and continuum contributions constitute a small perc ent of the total dispersion
+integral. The upper limit of M2
+Bis chosen demanding that the series of the light cone
+expansion with increasing twist should be convergent. As a result, t he common working
+region ofM2
+Bis found to be 15 GeV2≤M2
+B≤30GeV2. As an example, we present the
+dependence of the form factor f1on Borel mass parameter, M2
+Bat two fixed values of q2
+in Fig. 1. From this figure it follows that the form factor f1exhibits a good stability with
+respect to the variations of M2
+B. The continuum threshold s0is correlated to the first exited
+state with quantum numbers of the interpolating current of the Λ band is not completely
+arbitrary. Numerical analysis leads to the interval, ( mΛb+0.3)2≤s0≤(mΛb+0.5)2, where
+the form factors weakly depend on the continuum threshold. In or der to attain the working
+region for the parameter, β, we look for the variation of the form factors with respect to
+cosθ, whereβ=tanθ. After performing numerical calculations, we obtained that in the
+interval−0.6≤cosθ≤0.3 all form factors weakly depend on β. As an example, we
+show the dependence of the form factor, f1on cosθat two fixed values of the q2and at
+M2
+B= 22GeV2in Fig. 2 . From this figure indeed we see that in the aforementioned re gion
+of cosθ, the form factor f1weakly depends on β.
+The analysis of the sum rules, as has already been explained above, is based on, so
+called, the standard procedure, i.e., the continuum threshold s0is independent of M2
+Band
+q2. However, in [33], instead of the standard procedure, namely, inde pendence of the s0
+fromM2
+Bandq2, it is assumed that the continuum threshold depends on M2
+Bandq2and
+this leads to large realistic errors. Following [33], in the present work t he systematic error
+is taken to be around 15%.
+In calculating the branching ratio of the Λ b→Λℓ+ℓ−decay, the dependence of the form
+factorsfi(q2),gi(q2),fT
+i(q2),andgT
+i(q2)onq2inthephysicalregion4 m2
+ℓ≤q2≤(mΛb−mΛ)2
+are needed. But unfortunately, sum rules predictions for the for m factors are not reliable
+in the whole physical region. Therefore, in order to obtain the q2dependence of the form
+factors from sum rules we consider a range of q2where the correlation function can reliably
+be calculated. For this aim we choose a region which is approximately 1 GeVbelow the
+6QCD sum rules
+a bm2
+fit
+f1−0.0460.36839.10
+f20.0046−0.01726.37
+f30.006−0.02122.99
+g1−0.2200.53848.70
+g20.005−0.01826.93
+g30.035−0.05024.26
+fT
+2−0.1310.42645.70
+fT
+3−0.0460.10228.31
+gT
+2−0.3690.66459.37
+gT
+3−0.026−0.07523.73
+Table 1: Parameters appearing in the fit function of the form facto rs,f1,f2,f3,g1,g2,g3,
+fT
+2,fT
+3,gT
+2andgT
+3in full theory for Λ b→Λℓ+ℓ−. In this Table only central values of the
+parameters are presented.
+perturbative cut, i.e., up to q2≃12GeV2. To be able to extend the results for the form
+factors to the whole physical region, we look for a parameterizatio n of the form factors in
+such a way that, in the region 4 m2
+ℓ≤q2≤12GeV2this parameterization coincides with
+the sum rules predictions.
+The next step is to present the q2dependency of the form factors. Our numerical
+calculations show that the best parameterization for the depende nce of the form factors f1,
+f2,f3,g1,g2,g3,fT
+2,fT
+3,gT
+2andgT
+3onq2is as follows:
+fi(q2)[gi(q2)] =a/parenleftbigg
+1−q2
+m2
+fit/parenrightbigg+b/parenleftbigg
+1−q2
+m2
+fit/parenrightbigg2, (21)
+where the fit parameters a, bandm2
+fitin full theory are given in Table 1. On the other
+hand, we find that the best fit for the form factors fT
+1andgT
+1is of the following form,
+fT
+1(q2)[gT
+1(q2)] =c/parenleftbigg
+1−q2
+m′2
+fit/parenrightbigg−c/parenleftbigg
+1−q2
+m′′2
+fit/parenrightbigg2. (22)
+Theresults fortheparameters c,m′2
+fitandm′′2
+fitarepresented inTable2. Inextractionofthe
+values of the fit parameters presented in both Tables 1 and 2, the v alues of the continuum
+threshold,s0= 35GeV2, Borel mass parameter, M2
+B= 22GeV2and cosθ= 0.2 have been
+used.
+7QCD sum rules
+cm′2
+fitm′′2
+fit
+fT
+1−1.19123.8159.96
+gT
+1−0.65324.1548.52
+Table 2: Parameters appearing in the fit function of the form facto rsfT
+1andgT
+1in full
+theory for Λ b→Λℓ+ℓ−.
+The values of form factors at q2= 0 are also presented in Table 3. In this table we also
+present the numerical results obtained from HQET, using the value s for the form factors
+F1(0) = 0.462 andF2=−0.077 predicted in [17], and relations in Eq. (19) at HQET
+limit. The errors in the values of the form factors at q2= 0 are due to the uncertainties
+coming from M2
+B,s0, the parameter β, errors in the input parameters, as well as from the
+systematic errors. From this Table we see that, the predictions of the HQET on the form
+factors are changed more than 40%for the formfactors f1(0),g1(0),fT
+2(0), andgT
+1(0), while
+the results of both approaches are very close to each other for t he remaining form factors.
+Present work HQET ([13])
+f1(0)0.322±0.112 0.446
+f2(0)−0.011±0.004 −0.013
+f3(0)−0.015±0.005 −0.013
+g1(0)0.318±0.110 0.446
+g2(0)−0.013±0.004 −0.013
+g3(0)−0.014±0.005 −0.013
+fT
+1(0) 0±0.0 0.0
+fT
+2(0)0.295±0.105 0.446
+fT
+3(0)0.056±0.018 0.061
+gT
+1(0) 0±0.0 0.0
+gT
+2(0)0.294±0.105 0.446
+gT
+3(0)−0.101±0.035 −0.092
+Table 3: The values of the form factors at q2= 0 for Λ b→Λℓ+ℓ−.
+The final task is to calculate the total decay rate of the Λ b→Λℓ+ℓ−transition in the
+8whole physical region, 4 m2
+ℓ≤q2≤(mΛb−mΛ)2. The differential decay rate is obtained as:
+dΓ
+ds=G2α2
+emmΛb
+8192π5|VtbV∗
+ts|2v√
+λ/bracketleftbigg
+Θ(s)+1
+3∆(s)/bracketrightbigg
+, (23)
+wheres=q2/m2
+Λb,r=m2
+Λ/m2
+Λb,λ=λ(1,r,s) = 1+r2+s2−2r−2s−2rs,GF= 1.17×10−5
+GeV−2is the Fermi coupling constant and v=/radicalBig
+1−4m2
+ℓ
+q2is the lepton velocity. For the
+element of the CKM matrix |VtbV∗
+ts|= 0.041 has been used [34]. The functions Θ( s) and
+∆(s) are given as:
+Θ(s) = 32m2
+ℓm4
+Λbs(1+r−s)/parenleftbig
+|D3|2+|E3|2/parenrightbig
++ 64m2
+ℓm3
+Λb(1−r−s)Re[D∗
+1E3+D3E∗
+1]
++ 64m2
+Λb√r(6m2
+ℓ−m2
+Λbs)Re[D∗
+1E1]
++ 64m2
+ℓm3
+Λb√r/parenleftBig
+2mΛbsRe[D∗
+3E3]+(1−r+s)Re[D∗
+1D3+E∗
+1E3]/parenrightBig
++ 32m2
+Λb(2m2
+ℓ+m2
+Λbs)/braceleftBig
+(1−r+s)mΛb√rRe[A∗
+1A2+B∗
+1B2]
+−mΛb(1−r−s)Re[A∗
+1B2+A∗
+2B1]−2√r/parenleftBig
+Re[A∗
+1B1]+m2
+ΛbsRe[A∗
+2B2]/parenrightBig/bracerightBig
++ 8m2
+Λb/braceleftBig
+4m2
+ℓ(1+r−s)+m2
+Λb/bracketleftBig
+(1−r)2−s2/bracketrightBig/bracerightBig/parenleftbig
+|A1|2+|B1|2/parenrightbig
++ 8m4
+Λb/braceleftBig
+4m2
+ℓ/bracketleftBig
+λ+(1+r−s)s/bracketrightBig
++m2
+Λbs/bracketleftBig
+(1−r)2−s2/bracketrightBig/bracerightBig/parenleftbig
+|A2|2+|B2|2/parenrightbig
+−8m2
+Λb/braceleftBig
+4m2
+ℓ(1+r−s)−m2
+Λb/bracketleftBig
+(1−r)2−s2/bracketrightBig/bracerightBig/parenleftbig
+|D1|2+|E1|2/parenrightbig
++ 8m5
+Λbsv2/braceleftBig
+−8mΛbs√rRe[D∗
+2E2]+4(1−r+s)√rRe[D∗
+1D2+E∗
+1E2]
+−4(1−r−s)Re[D∗
+1E2+D∗
+2E1]+mΛb/bracketleftBig
+(1−r)2−s2/bracketrightBig/parenleftbig
+|D2|2+|E2|2/parenrightbig/bracerightBig
+,(24)
+∆(s) =−8m4
+Λbv2λ/parenleftbig
+|A1|2+|B1|2+|D1|2+|E1|2/parenrightbig
++ 8m6
+Λbsv2λ/parenleftBig
+|A2|2+|B2|2+|D2|2+|E2|2/parenrightBig
+, (25)
+where
+A1=1
+q2/parenleftbig
+fT
+1+gT
+1/parenrightbig
+(−2mbC7)+(f1−g1)Ceff
+9
+A2=A1(1→2),
+A3=A1(1→3),
+B1=A1/parenleftbig
+g1→ −g1;gT
+1→ −gT
+1/parenrightbig
+,
+B2=B1(1→2),
+B3=B1(1→3),
+D1= (f1−g1)C10,
+D2=D1(1→2), (26)
+9D3=D1(1→3),
+E1=D1(g1→ −g1),
+E2=E1(1→2),
+E3=E1(1→3). (27)
+Integrating the differential decay rate on sin the whole physical region 4 m2
+ℓ/m2
+Λb≤s≤
+(1−√r)2and using the life time of the Λ bbaryon,τΛb= 1.383×10−12s[34], we obtain
+the results for the branching ratio which are presented in Table 4.
+Present work HQET([19])
+Br(Λb→Λe+e−)(4.6±1.6)×10−6(2.23÷3.34)×10−6
+Br(Λb→Λµ+µ−)(4.0±1.2)×10−6(2.08÷3.19)×10−6
+Br(Λb→Λτ+τ−)(0.8±0.3)×10−6(0.179÷0.276)×10−6
+Table 4: Values of the Branching ratio for Λ b→Λℓ+ℓ−in full theory and HQET for
+different leptons .
+In this Table we also present the values of the branching ratio obtain ed in HQET [19].
+Comparing the results of both approaches, we see that our predic tions on the branching
+ratios for the Λ b→Λe+e−, Λb→Λµ+µ−channels are larger approximately by a factor
+of two than the ones predicted by the HQET, while for the Λ b→Λτ+τ−channel our
+prediction is four times larger than the result of the HQET. Since 1010÷1011pairs are
+expected to be produced per year at LHCb [35], the results presen ted in Table–4 show that
+detectability of Λ b→Λℓ+ℓ−(ℓ=e,µ,τ) decays in this machine is quite high.
+In conclusion, we calculate all twelve form factors responsible for t he Λb→Λℓ+ℓ−decay
+within light cone sum rules. It is obtained the maximum difference betwe en our results and
+HQET predictions on the form factors is about 40%. Using the param etrization for the
+form factors, the branching ratio of the Λ b→Λℓ+ℓ−decay is estimated, and the result we
+obtain allows us to conclude that the delectability of this decay at LHC b is quite high.
+10References
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+12Appendix
+InthisAppendix, thegeneraldecompositionofthematrixelement, ǫabc/an}bracketle{t0|ua
+η(0)sb
+θ(x)dc
+φ(0)|Λ(p)/an}bracketri}ht
+entering Eqs. (14,15) as well as the Λ DA’s are given [27]:
+4/an}bracketle{t0|ǫabcua
+α(a1x)sb
+β(a2x)dc
+γ(a3x)|Λ(p)/an}bracketri}ht
+=S1mΛCαβ(γ5Λ)γ+S2m2
+ΛCαβ(/xγ5Λ)γ
++P1mΛ(γ5C)αβΛγ+P2m2
+Λ(γ5C)αβ(/xΛ)γ+(V1+x2m2
+Λ
+4VM
+1)(/pC)αβ(γ5Λ)γ
++V2mΛ(/pC)αβ(/xγ5Λ)γ+V3mΛ(γµC)αβ(γµγ5Λ)γ+V4m2
+Λ(/xC)αβ(γ5Λ)γ
++V5m2
+Λ(γµC)αβ(iσµνxνγ5Λ)γ+V6m3
+Λ(/xC)αβ(/xγ5Λ)γ+(A1
++x2m2
+Λ
+4AM
+1)(/pγ5C)αβΛγ+A2mΛ(/pγ5C)αβ(/xΛ)γ+A3mΛ(γµγ5C)αβ(γµΛ)γ
++A4m2
+Λ(/xγ5C)αβΛγ+A5m2
+Λ(γµγ5C)αβ(iσµνxνΛ)γ+A6m3
+Λ(/xγ5C)αβ(/xΛ)γ
++ (T1+x2m2
+Λ
+4TM
+1)(pνiσµνC)αβ(γµγ5Λ)γ+T2mΛ(xµpνiσµνC)αβ(γ5Λ)γ
++T3mΛ(σµνC)αβ(σµνγ5Λ)γ+T4mΛ(pνσµνC)αβ(σµρxργ5Λ)γ
++T5m2
+Λ(xνiσµνC)αβ(γµγ5Λ)γ+T6m2
+Λ(xµpνiσµνC)αβ(/xγ5Λ)γ
++T7m2
+Λ(σµνC)αβ(σµν/xγ5Λ)γ+T8m3
+Λ(xνσµνC)αβ(σµρxργ5Λ)γ. (A.1)
+The calligraphic functions in the above expression have not definite t wists but they can be
+written in terms of the Lambda distribution amplitudes (DA’s) with defi nite and increasing
+twists via the scalar product pxand the parameters ai,i= 1,2,3. The explicit expressions
+for scalar, pseudo-scalar, vector, axial vector and tensor DA’s for Lambda are given in
+Tables 5, 6, 7, 8 and 9, respectively.
+S1=S1
+2pxS2=S1−S2
+Table 5: Relations between the calligraphic functions and Lambda sca lar DA’s.
+P1=P1
+2pxP2=P1−P2
+Table 6: Relations between the calligraphic functions and Lambda pse udo-scalar DA’s.
+Every distribution amplitude F(aipx)=Si,Pi,Vi,Ai,Tican be represented as:
+F(aipx) =/integraldisplay
+dx1dx2dx3δ(x1+x2+x3−1)eipxΣixiaiF(xi). (A.2)
+where,xiwithi= 1,2 and 3 are longitudinal momentum fractions carried by the partici-
+pating quarks.
+13V1=V1
+2pxV2=V1−V2−V3
+2V3=V3
+4pxV4=−2V1+V3+V4+2V5
+4pxV5=V4−V3
+4(px)2V6=−V1+V2+V3+V4+V5−V6
+Table 7: Relations between the calligraphic functions and Lambda vec tor DA’s.
+A1=A1
+2pxA2=−A1+A2−A3
+2A3=A3
+4pxA4=−2A1−A3−A4+2A5
+4pxA5=A3−A4
+4(px)2A6=A1−A2+A3+A4−A5+A6
+Table 8: Relations between the calligraphic functions and Lambda axia l vector DA’s.
+The explicit expressions for the Λ DA’s up to twist 6 are given as: twist -3 DA’s:
+V1(xi) = 0, A 1(xi) =−120x1x2x3φ0
+3,
+T1(xi) = 0. (A.3)
+twist-4 DA’s:
+S1(xi) = 6x3(1−x3)(ξ0
+4+ξ′0
+4), P 1(xi) = 6(1−x3)(ξ0
+4−ξ′0
+4),
+V2(xi) = 0, A 2(xi) =−24x1x2φ0
+4,
+V3(xi) = 12(x1−x2)x3ψ0
+4, A 3(xi) =−12x3(1−x3)ψ0
+4,
+T2(xi) = 0, T 3(xi) = 6(x2−x1)x3(−ξ0
+4+ξ′0
+4),
+T7(xi) =−6(x1−x2)x3(ξ0
+4+ξ′0
+4). (A.4)
+Ttwist-5 DA’s:
+S2(xi) =3
+2(x1+x2)(ξ0
+5+ξ′0
+5), P 2(xi) =3
+2(x1+x2)(ξ0
+5−ξ′0
+5),
+V4(xi) = 3(x2−x1)ψ0
+5, A 4(xi) =−3(1−x3)ψ0
+5,
+V5(xi) = 0, A 5(xi) =−6x3φ0
+5,
+T4(xi) =−3
+2(x1−x2)(ξ0
+5+ξ′0
+5), T 5(xi) = 0,
+T8(xi) =−3
+2(x1−x2)(ξ0
+5−ξ′0
+5). (A.5)
+and twist-6 DA’s:
+V6(xi) = 0, A 6(xi) =−2φ0
+6,
+T6(xi) = 0. (A.6)
+14T1=T1
+2pxT2=T1+T2−2T3
+2T3=T7
+2pxT4=T1−T2−2T7
+2pxT5=−T1+T5+2T8
+4(px)2T6= 2T2−2T3−2T4+2T5+2T7+2T8
+4pxT7=T7−T8
+4(px)2T8=−T1+T2+T5−T6+2T7+2T8
+Table 9: Relations between the calligraphic functions and Lambda ten sor DA’s.
+The following functions are encountered to the above amplitudes an d they can be defined
+in terms of the 4 independent parameters, namely fΛ,λ1,λ2andλ3:
+φ0
+3=φ0
+6=−fΛ, φ0
+4=φ0
+5=−1
+2(fΛ+λ1),
+ψ0
+4=ψ0
+5=1
+2(fΛ−λ1), ξ0
+4=ξ0
+5=λ2+λ3,
+ξ′0
+4=ξ′0
+5=λ3−λ2. (A.7)
+15Figure 1: The dependence of form factor, f1on Borel mass parameter at two fixed values
+of theq2, and ats0= 35GeV2andβ= 5./D5
+/BE/BP /BI /BZ/CT/CE
+/BE
+/D5
+/BE/BP /BC /BZ/CT/CE
+/BE/C5
+/BE/BU
+/B4 /BZ/CT/CE
+/BE/B5/CU
+/BD/BF/BC/BA/BC /BE/BH/BA/BC /BE/BC/BA/BC /BD/BH/BA/BC
+/BD/BA/BC/BC/BA/BK/BC/BA/BI/BC/BA/BG/BC/BA/BE/BC/BA/BC
+16Figure 2: The dependence of form factor, f1on cosθparameter at two fixed values of the
+q2, and ats0= 35GeV2andM2
+B= 22GeV2./D5
+/BE/BP /BI /BZ/CT/CE
+/BE
+/D5
+/BE/BP /BC /BZ/CT/CE
+/BE
/D3/D7 /AI/CU
+/BD/BD/BA/BC/BC /BC/BA/BJ/BH /BC/BA/BH/BC /BC/BA/BE/BH /BC/BA/BC/BC /B9/BC/BA/BE/BH /B9/BC/BA/BH/BC /B9/BC/BA/BJ/BH /B9/BD/BA/BC/BC
+/BD/BA/BC/BC/BA/BK/BC/BA/BI/BC/BA/BG/BC/BA/BE/BC/BA/BC
+17
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+arXiv:1001.0228v2  [math.KT]  3 Feb 2010SYMMETRIC MONOIDAL STRUCTURE ON
+NON-COMMUTATIVE MOTIVES
+by
+Denis-Charles Cisinski & Gon¸ calo Tabuada
+Abstract . —In this article we further the study of non-commutative moti ves, ini-
+tiated in [ 11,42]. Our main result is the construction of a symmetric monoida l
+structure on the localizing motivator Motloc
+dgof dg categories. As an application, we
+obtain: (1) a computation of the spectra of morphisms in Motloc
+dgin terms of non-
+connective algebraic K-theory; (2)a fully-faithfulembedding of Kontsevich’s ca tegory
+KMMkof non-commutative mixed motives into the base category Motloc
+dg(e) of the
+localizing motivator; (3) a simple construction of the Cher n character maps from
+non-connective algebraic K-theory to negative and periodic cyclic homology; (4) a
+precise connection between To¨ en’s secondary K-theory and the Grothendieck ring of
+KMMk; (5) a description of the Euler characteristic in KMM kin terms of Hochschild
+homology.
+Contents
+Introduction........................................ ................ 2
+1. Preliminaries........................................ ............. 7
+2. Background on dg categories.......................... ........... 8
+3. Homotopically finitely presented dg categories.............. .....14
+4. Saturated dg categories............................. ............. 17
+5. Simplicial presheaves................................. ............ 20
+6. Monoidal stabilization................................. .......... 25
+7. Symmetric monoidal structure......................... .......... 26
+8. Applications........................................ ............. 35
+A. Grothendieck derivators............................. ............ 45
+2000Mathematics Subject Classification . —19D35, 19D55, 18D10, 18D20, 19E08.
+Key words and phrases . —Non-commutative motives, Non-commutative algebraic geom etry,
+Non-connective algebraic K-theory, Secondary K-theory, Hochschild homology, Negative cyclic ho-
+mology, Periodic cyclic homology.
+The first named author was partially supported by the ANR (gra nt No. ANR-07-BLAN-042). The
+second named author was partially supported by the Est´ ımul o ` a Investiga¸ c˜ ao Award 2008 - Calouste
+Gulbenkian Foundation.2 D.-C. CISINSKI & G. TABUADA
+References......................................... ................. 54
+Introduction
+Dg categories. — Adifferential graded (=dg) category , over a commutative base
+ringk, is a category enriched over complexes of k-modules (morphisms sets are such
+complexes) in such a way that composition fulfills the Leibniz rule: d(f◦g) = (df)◦
+g+(−1)deg(f)f◦(dg). Dg categoriesenhance and solvemany ofthe technical problems
+inherent to triangulated categories; see Keller’s ICM adress [ 30]. Innon-commutative
+algebraic geometry in the sense of Bondal, Drinfeld, Kapranov,Kontsevich, To¨ en, Va n
+den Bergh, ...[3,4,16,17,33,34,35,48], they areconsideredasdg-enhancements
+of derived categories of (quasi-)coherent sheaves on a hypothe tic non-commutative
+space.
+Localizing invariants. — All the classical (functorial) invariants, such as
+Hochschild homology, cyclic homology and its variants (periodic, nega tive,...),
+algebraic K-theory, and even topological Hochschild homology and topological cyclic
+homology (see [ 45]), extend naturally from k-algebras to dg categories. In order
+to study allthese classical invariants simultaneously, the second named autho r
+introduced in [ 42] the notion of localizing invariant . This notion, that we now
+recall, makes use of the language of Grothendieck derivators [ 28], a formalism which
+allows us to state and prove precise universal properties; consult Appendix A. Let
+L:HO(dgcat)→Dbe a morphism of derivators, from the derivator associated to
+the Morita model structure of dg categories (see §2.3), to a triangulated derivator
+(in practice, Dwill be the derivator associated to a stable model category M, and
+Lwill come from a functor dgcat→Mwhich sends derived Morita equivalences to
+weak equivalences in M). We say that Lis alocalizing invariant (see Definition 7.3)
+if it preserves filtered homotopy colimits as well as the terminal obje ct, and sends
+Drinfeld exact sequences of dg categories
+A−→B−→C /ma√sto→ L(A)−→L(B)−→L(C)−→L(A)[1]
+to distinguished triangles in the base category D(e) ofD. Thanks to the work
+of Keller [ 31,32], Schlichting [ 41], Thomason-Trobaugh [ 47], and Blumberg-
+Mandell [ 2], all the mentioned invariants satisfy localization(1), and so give rise to
+localizing invariants. In [ 42], the second named author proved that there exists a
+localizing invariant
+Uloc
+dg:HO(dgcat)−→Motloc
+dg,
+with values in a strong triangulated derivator (see §A.3), such that given any strong
+triangulated derivator D, we have an induced equivalence of categories
+(Uloc
+dg)∗:Hom!(Motloc
+dg,D)∼−→Homloc(HO(dgcat),D).
+(1)In the case of algebraic K-theory we consider its non-connective version.SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 3
+The left-hand side denotes the category of homotopy colimit prese rving morphisms
+of derivators, and the right-hand side the category of localizing inv ariants.
+Because of this universality property, which is a reminiscence of mot ives,Uloc
+dg
+is called the universal localizing invariant , Motloc
+dgthelocalizing motivator , and the
+base category Motloc
+dg(e) of the localizing motivator the category of non-commutative
+motives over k.
+Symmetric monoidalstructure. — Thepurposeofthisarticleistodevelopanew
+ingredient in the theory of non-commutativemotives: symmetric monoidal structures .
+The tensor product extends naturally from k-algebras to dg categories, giving rise to
+a symmetric monoidal structure on HO(dgcat); see Theorem 2.23. Therefore, it is
+natural to consider localizing invariants which are symmetric monoida l. Examples
+include Hochschild homology and the mixed and periodic complex constr uctions; see
+Examples 7.9-7.11. The main result of this article is the following.
+Theorem 0.1 . —(see Theorem 7.5) The localizing motivator Motloc
+dgcarries a
+canonical symmetric monoidal structure −⊗L−, making the universal localizing in-
+variantUloc
+dgsymmetric monoidal. Moreover, this tensor product preserv es homotopy
+colimits in each variable and is characterized by the follow ing universal property:
+given any strong triangulated derivator D, endowed with a monoidal structure which
+preserves homotopy colimits, we have an induced equivalenc e of categories
+(Uloc
+dg)∗:Hom⊗
+!(Motloc
+dg,D)∼−→Hom⊗
+loc(HO(dgcat),D),
+where the left-hand side stands for the category of symmetri c monoidal homotopy
+colimit preserving morphisms of derivators, while the righ t-hand side stands for the
+category symmetric monoidal morphisms of derivators which are also localizing invari-
+ants; see§A.5. Furthermore, Motloc
+dgadmits an explicit symmetric monoidal Quillen
+model.
+The proof of Theorem 0.1is based on an alternative description of Motloc
+dg, with two
+complementaryaspects: a constructive one, and anothergivenby universal properties .
+The constructive aspect, i.e.the construction of an explicit symmetric monoidal
+Quillen model for Motloc
+dg, is described in the main body of the text. The key start-
+ing point is the fact that homotopically finitely presented dg categor ies are stable
+under derived tensor product; see Theorem 3.4. This allows us to obtain a small
+symmetric monoidal category which “generates” the entire Morita homotopy cate-
+gory of dg categories. Starting from this small monoidal category , we then construct
+a specific symmetric monoidal Quillen model for each one of the deriva tors used in
+the construction of Motloc
+dg; see§7.1.
+The characterizationof Motloc
+dgby its universalproperty (as stated in Theorem 0.1),
+relies on general results and constructions in the theory of Groth endieck derivators,
+and is described in the appendix. We develop some general results co ncerning the
+behavior of monoidal structures under classical operations: Kan extension (see The-
+oremA.3), left Bousfield localization (see Proposition A.9) and stabilization (see4 D.-C. CISINSKI & G. TABUADA
+Theorem A.15). Using these general results, we then characterize by a precise uni-
+versal property each one of the Quillen models used in the construc tion of the new
+symmetric monoidal Quillen model for Motloc
+dg; see§7.1.
+Let us now describe some applications of Theorem 0.1.
+Non-connective K-theory. — As mentioned above, non-connective algebraic K-
+theoryI K(−) is an example of a localizing invariant. In [ 11] the authors proved that
+this invariant becomes co-representable in Motloc
+dg(e) by the unit object Uloc
+dg(k) (where
+kcorresponds to k, seen as a dg category with one object). In other words, given an y
+dg categoryA, we have a natural isomorphism in the stable homotopy category of
+spectra:
+(0.0.1) RHom(Uloc
+dg(k),Uloc
+dg(A))≃I K(A).
+A fundamental problem of the theory of non-commutative motives is the computation
+of the (spectra of) morphisms in the category of non-commutativ e motives between
+any two objects. Using the monoidal structure of Theorem 0.1we extend the above
+natural isomorphism ( 0.0.1), and thus obtain a partial solution to this problem.
+Theorem 0.2 . —(see Theorem 8.2) LetBbe a saturated dg category in the sense of
+Kontsevich, i.e.its complexes of morphisms are perfect and Bis perfect as a bimodule
+over itself; see Definition 4.1. Then, for every small dg category A, we have a natural
+isomorphism in the stable homotopy category of spectra
+RHom(Uloc
+dg(B),Uloc
+dg(A))≃I K(rep(B,A)),
+whererep(−,−)denotes the internal Hom-functor in the Morita homotopy cat egory
+of dg categories; see §2.4.
+Given a quasi-compact and separated k-scheme X, there is a natural dg category
+perf(X) which enhances the category of perfect complexes ( i.e.of compact objects)
+in the (unbounded) derived category Dqcoh(X) of quasi-coherent sheaves on X; see
+Example 4.5. Moreover, when Xis smooth and proper, the dg category perf(X) is a
+saturated dg category. In this geometrical situation, we have th e following computa-
+tion.
+Proposition 0.3 . —(see Proposition 8.3) Given smooth and proper k-schemes X
+andY, we have a natural isomorphism in the stable homotopy catego ry of spectra
+RHom(Uloc
+dg(perf(X)),Uloc
+dg(perf(Y)))≃I K(X×Y),
+whereI K(X×Y)denotes the non-connective algebraic K-theory spectrum of X×Y.
+Kontsevich’s category of non-commutative mixed motives. — In hisnon-
+commutative algebraic geometry program [ 33,34,35], Kontsevich introduced the
+category KMM kofnon-commutative mixed motives ; see§8.2. Roughly, KMM kis
+obtained by taking a formal Karoubian triangulated envelope of the category of satu-
+rated dg categories (with algebraic K-theory of bimodules as morphism sets). Using
+Theorem 0.2, we prove the following result.SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 5
+Proposition 0.4 . —(see Proposition 8.5) There is a natural fully-faithful embedding
+(enriched over spectra) of Kontsevich’s category KMMkof non-commutative mixed
+motives into the base category Motloc
+dg(e)of the localizing motivator. The essential im-
+age is the thick triangulated subcategory spanned by motive s of saturated dg categories.
+Note that, in contrast with Kontsevich’s ad hocdefinition, the category Motloc
+dg(e)
+of non-commutative motives is defined purely in terms of precise univ ersal properties.
+Chern characters. — Let
+E:HO(dgcat)−→D
+be a symmetric monoidal localizing invariant. Thanks to Theorem 0.1there is a
+(unique) symmetric monoidal homotopy colimit preserving morphism o f derivators
+Egmwhich makes the diagram
+HO(dgcat)E/d47/d47
+Uloc
+dg
+/d15/d15D
+Motloc
+dgEgm/d59/d59/d119/d119/d119/d119/d119/d119/d119/d119/d119/d119
+commute (up to unique 2-isomorphism). We call Egmthegeometric realization ofE.
+IfE(k)≃1denotes the unit of D, we can also associate to Eitsabsolute realization
+Eabs:=RHomD(1,Egm(−)) (see Definition 8.6)
+Proposition 0.5 . —(see Proposition 8.7) The geometric realization of E induces a
+canonical Chern character
+I K(−)⇒RHom(1,E(−))≃Eabs(Uloc
+dg(−)).
+Here,I K(−)and Eabs(Uloc
+dg(−))are two morphisms of derivators defined on HO(dgcat).
+LetAbe a small dg category. When Eis given by the mixed complex construction
+(see Example 7.10), the absolute realization of Uloc
+dg(A) identifies with the negative
+cyclic homology complex HC−(A) ofA. Therefore by Proposition 0.5, we obtain a
+canonical Chern character
+I K(−)⇒HC−(−)
+from non-connective K-theory to negative cyclic homology; see Example 8.10. When
+Eis given by the composition of the mixed complex construction with the periodiza-
+tion procedure (see Example 7.11), the absolute realization of Uloc
+dg(A) identifies with
+the periodic cyclic homology complex HP(A) ofA. Therefore by Proposition 0.5, we
+obtain a canonical Chern character
+I K(−)⇒HP(−)
+from non-connective K-theory to periodic cyclic homology; see Example 8.11.6 D.-C. CISINSKI & G. TABUADA
+To¨ en’s secondary K-theory. — In hissheaf categorification program [ 50,52],
+To¨ en introduced a “categorified” version of algebraic K-theory, named secondary K-
+theory. Given a commutative ring k, thesecondary K-theory ring K(2)
+0(k)ofkis
+roughly the quotient of the free abelian group on derived Morita isom orphism classes
+of saturated dg categories, by the relations [ B] = [A] + [C] coming from Drinfeld
+exact sequencesA→B→C . The multiplication is induced from the derived tensor
+product; see§8.4. As To¨ en pointed out in [ 50], one of the motivations for the study of
+this secondary K-theoryisits expected connection with anhypothetical Grothend ieck
+ring of motives in the non-commutative setting. Thanks to the mono idal structure of
+Theorem 0.1we are now able to make this connection precise.
+Definition 0.6 . — (seeDefinition 8.16)Givenacommutativering k,theGrothendieck
+ringK0(k)of non-commutative motives over kis the Grothendieck ring of Kontsevich
+mixed motives ( i.e.of the thick triangulated subcategory of Motloc
+dg(e) generated by
+the objectsUloc
+dg(A), whereAruns over the family of saturated dg categories).
+Kontsevich’s saturated dg categories can be characterized conc eptually as the du-
+alizable objects in the Morita homotopy category; see Theorem 4.8. Therefore, since
+the universal localizing invariant Uloc
+dgis symmetric monoidal we obtain a ring homo-
+morphism
+Φ(k) :K(2)
+0(k)−→K 0(k).
+Moreover, the Grothendieck ring of Definition 0.6is non-trivial (see Remark 8.17),
+functorial in k(see Remark 8.18), and the ring homomorphism Φ( k) is functorial in
+kand surjective “up to cofinality” (see Remark 8.19). Furthermore, any realization
+ofK(2)
+0(k) (i.e.ring homomorphism K(2)
+0(k)→R), which is induced from a sym-
+metric monoidal localizing invariant, factors through Φ( k). An interesting example is
+provided by To¨ en’s rank map
+rk0:K(2)
+0(k)−→K0(k) (see Remark 8.19).
+Euler characteristic. — Recall that, in any symmetric monoidal category, we have
+the notion of Euler characteristic χ(X) of a dualizable object X(see Definition 8.20).
+In the symmetric monoidal category of non-commutative motives w e have the follow-
+ing computation.
+Proposition 0.7 . —(see Proposition 8.24) LetAbe a saturated dg category. Then,
+χ(Uloc
+dg(A))is the element of the Grothendieck group K0(k)which is associated to the
+(perfect) Hochschild homology complex HH (A)ofA.
+Whenkis the field of complex numbers, the Grothendieck ring K0(C) is naturally
+isomorphic to Zand the Hochschild homology of a smooth and proper k-scheme X
+agrees with the Hodge cohomology H∗(X,Ω∗
+X) ofX. Therefore, when we work over
+C, and ifperf(X) denotes the (saturated) dg category of perfect complexes ov erX,
+the Euler characteristic of Uloc
+dg(perf(X)) is the classical Euler characteristic of X.SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 7
+Acknowledgments: The authors are very grateful to Bertrand To¨ en for sharing
+his insights on saturated dg categories and to Maxim Kontsevich for comments on a
+previous version which motivated the creation of subsection 8.3. The second named
+author would like also to thank Paul Balmer and Christian Haesemeyer for some
+conversations. This article was initiated at the Institut Henri Poinc ar´ e in Paris. The
+authors would like to thank this institution for its hospitality and stimu lating working
+environment.
+1. Preliminaries
+1.1. Notations. — Throughout the article we will work over a fixed commutative
+and unital base ring k.
+We will denote by C(k) be the category of (unbounded) complexes of k-modules;
+see [24,§2.3]. We will use co-homological notation, i.e.the differential increases the
+degree. The category C(k) is a symmetric monoidal model category (see [ 24, Defini-
+tion4.2.6]), where one uses the projective model structure for which weak equivalences
+are quasi-isomorphisms and fibrations are surjections; see [ 24, Proposition4.2.13].
+The category of sets will be denoted by Set, the category of simplicial sets by sSet,
+and the category of pointed simplicial sets by sSet•; see [20,§I]. The categories sSet
+andsSet•are symmetric monoidal model categories; see [ 20, Proposition4.2.8]. The
+weak equivalences are the maps whose geometric realization is a weak equivalence of
+topological spaces, the fibrations are the Kan-fibrations, and th e cofibrations are the
+inclusion maps.
+Wewilldenoteby SpNthecategoryofspectraandby SpΣthecategoryofsymmetric
+spectra (of pointed simplicial sets); see [ 27].
+Finally, the adjunctions will be displayed vertically with the left (resp. right)
+adjoint on the left- (resp. right-) hand side.
+1.2. Triangulated categories. — Throughout the article we will use the language
+of triangulated categories. The reader unfamiliar with this language is invited to
+consult Neeman’s book [ 39] or Verdier’s original monograph [ 53]. Recall from [ 39,
+Definition 4.2.7] that given a triangulated category Tadmitting arbitrary small co-
+products, an object GinTis called compact if for each family {Yi}i∈Iof objects in
+T, the canonical morphism
+/circleplusdisplay
+i∈IHomT(G,Yi)∼−→HomT(G,/circleplusdisplay
+i∈IYi)
+is invertible. We will denote by Tcthe category of compact objects in T.
+1.3. Quillen model categories. — Throughout the article we will use freely the
+language of Quillen model categories. The reader unfamiliar with this la nguage is
+invited to consult Goerss & Jardine [ 20], Hirschhorn [ 23], Hovey [ 24], or Quillen’s8 D.-C. CISINSKI & G. TABUADA
+original monograph [ 40]. Given a model category M, we will denote by Ho(M) its
+homotopy category and by
+Map(−,−) :Ho(M)op×Ho(M)−→Ho(sSet)
+its homotopy function complex; see [ 23, Definition 17.4.1].
+1.4. Grothendieck derivators. — Throughout the article we will use the lan-
+guage of Grothendieck derivators. Derivators allow us to state an d prove precise
+universal properties and to dispense with many of the technical pr oblems one faces
+in using Quillen model categories. Since this language may be less familiar to the
+reader, we revise it in the appendix. Given a model category M, we will denote by
+HO(M) its associatedderivator; see §A.2. Any triangulated derivator Dis canonically
+enriched over spectra; see §A.9. We will denote by RHomD(X,Y) the spectrum of
+maps from XtoYinD. When there is no ambiguity, we will write RHom(X,Y)
+instead of RHomD(X,Y).
+2. Background on dg categories
+In this section we collect and recall the notions and results on the (h omotopy)
+theory of dg categories which will be used throughout the article; s ee [43,44,48].
+At the end of the section we prove a new result: the derivator asso ciated to the
+Morita model structure carries a symmetric monoidal structure; see Theorem 2.23.
+This result is the starting point of the article, since it allow us to define symmetric
+monoidal localizing invariants; see Definition 7.6.
+Definition 2.1 . — Asmall dg category AisaC(k)-category;see[ 5, Definition6.2.1].
+Recall that this consists of the following data:
+- a set of objects obj( A) (usually denoted by Aitself);
+- for each ordered pair of objects ( x,y) inA, a complex of k-modulesA(x,y);
+- for each ordered triple of objects ( x,y,z) inA, a composition morphism in C(k)
+A(y,z)⊗A(x,y)−→A(x,z),
+satisfying the usual associativity condition;
+- for each object xinA, a morphism k→ A(x,x), satisfying the usual unit
+condition with respect to the above composition.
+Definition 2.2 . — Adg functor F:A → B is aC(k)-functor; see [ 5, Defini-
+tion6.2.3]. Recall that this consists of the following data:
+- a map of sets F: obj(A)−→obj(B);
+- for each ordered pair of objects ( x,y) inA, a morphism inC(k)
+F(x,y) :A(x,y)−→B(Fx,Fy),
+satisfying the usual unit and associativity conditions.SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 9
+Notation 2.3 . — We denote by dgcatthe category of small dg categories.
+2.1. Dg cells. —
+(i) Letkbe the dg category with one object ∗and such that k(∗,∗) :=k(in degree
+zero). Note that given a small dg category B, there is a bijection between the
+set of dg functors from ktoBand the set of objects of B.
+(ii) Forn∈Z, letSnbe the complex k[n] (withkconcentrated in degree n) andDn
+the mapping cone on the identity of Sn−1. We denote byS(n) the dg category
+with two objects 1 and 2 such that S(n)(1,1) =k,S(n)(2,2) =k,S(n)(2,1) =
+0,S(n)(1,2) =Snandcompositiongivenbymultiplication. Wedenoteby D(n)
+the dg categorywith two objects 3 and 4 such that D(n)(3,3) =k,D(n)(4,4) =
+k ,D(n)(4,3) = 0,D(n)(3,4) =Dnand composition given by multiplication.
+(iii) For n∈Z, letι(n) :S(n−1)→D(n) be the dg functor that sends 1 to 3, 2 to
+4 andSn−1toDnby the identity on kin degree n−1:
+S(n−1)ι(n)/d47/d47D(n)
+1k
+/d5/d5
+Sn−1
+/d15/d15/d31 /d47/d473k
+/d5/d5
+Dn
+/d15/d15incl/d47/d47
+2
+k/d69/d69/d31 /d47/d474
+k/d69/d69whereSn−1incl/d47/d47Dn
+0/d47/d47
+/d15/d150
+/d15/d15
+0/d47/d47
+/d15/d15k
+id/d15/d15
+kid/d47/d47
+/d15/d15k
+/d15/d15(degree n−1)
+0/d47/d470
+Notation 2.4 . — Let Ibe the set consisting of the dg functors {ι(n)}n∈Zand the
+dg functor∅→k(where the empty dg category ∅is the initial object in dgcat).
+Definition 2.5 . — A dg category Ais called a dg cell(resp. afinite dg cell ) if the
+unique dg functor ∅→Acan be expressed as a transfinite (resp. finite) composition
+of pushouts of dg functors in I; see [23, Definition 10.5.8(2)].
+2.2. Dg modules. — LetAbe a small (fixed) dg category.
+Definition 2.6 . — - The category H0(A) has the same objects as Aand mor-
+phisms given by H0(A)(x,y) := H0(A(x,y)), where H0(−) denotes the 0-th
+co-homology group functor.
+- Theopposite dg categoryAopofAhas the same objects as Aand complexes of
+morphisms given by Aop(x,y) :=A(y,x).
+- Aright dgA-moduleM(orsimplyaA-module)isadgfunctor M:Aop→Cdg(k)
+with values in the dg category Cdg(k) of complexes of k-modules.10 D.-C. CISINSKI & G. TABUADA
+We denote byC(A) the category of right dg A-modules. Its objects are the right
+dgA-modules and its morphisms are the natural transformations of dg functors; see
+[30,§2.3]. The differential graded structure of Cdg(k) makesC(A) naturally into a
+dg category; see [ 30,§3.1] for details. We denote by Cdg(A) this dg category of
+right dgA-modules. Recall from [ 30, Theorem 3.2] that C(A) carries a standard
+projective model structure. A morphism M→M′inC(A) is aweak equivalence ,
+resp. afibration, if for any object xinA, the induced morphism M(x)→M′(x) is
+a weak equivalence, resp. a fibration, in the projective model stru cture onC(k). In
+particular, every object is fibrant. Moreover, the dg category Cdg(A) endowed with
+this model structure is a C(k)-model category in the sense of [ 24, Definition 4.2.18].
+Notation 2.7 . — - We denote by Ddg(A) the full dg subcategory of fibrant and
+cofibrant objects in Cdg(A).
+- We denote byD(A) thederived category of A,i.e.the localization of C(A) with
+respect to the class of weak equivalences. Note that D(A) is a triangulated
+category with arbitrary small coproducts. Recall from §1.2that we denote by
+Dc(A) the category of compact objects in D(A).
+SinceCdg(A) is aC(k)-model category, [ 48, Proposition 3.5] implies that we have
+a natural equivalence of (triangulated) categories
+(2.2.1) H0(Ddg(A))≃D(A).
+We denote by
+h:A−→C dg(A)x/ma√sto→A(−,x),
+the classical Yoneda dg functor ; see [5,§6.3]. Since theA-modulesA(−,x),xinA,
+are fibrant and cofibrantin the projectivemodel structure, the Yoneda functor factors
+through the inclusion Ddg(A)⊂Cdg(A). Thanks to the above equivalence ( 2.2.1), we
+obtain an induced fully-faithful functor
+h:H0(A)֒→D(A)x/ma√sto→A(−,x).
+Finally, let F:A→Bbe a dg functor. As shown in [ 48,§3] it induces a restric-
+tion/extension of scalars Quillen adjunction (on the left-hand side)
+C(B)
+F∗
+/d15/d15D(B)
+F∗
+/d15/d15
+C(A)F!/d79/d79
+D(A),LF!/d79/d79
+which can be naturally derived (on the right-hand side).SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 11
+2.3. Morita model structure. —
+Definition 2.8 . — A dg functor F:A→Bis a called a derived Morita equivalence
+iftherestrictionofscalarsfunctor F∗:D(B)∼→D(A) isanequivalenceoftriangulated
+categories. Equivalently, Fis a derived Morita equivalence if the extension of scalars
+functorLF!:D(A)∼→D(B) is an equivalence of triangulated categories.
+Theorem 2.9 . —The category dgcatcarries a cofibrantly generated Quillen model
+structure (called the Morita model structure ), whose weak equivalences are the de-
+rived Morita equivalences. Moreover, its set of generating cofibrations is the set Iof
+Notation 2.4.
+Proof. — See [ 43, Th´ eor` eme 5.3].√
+Notation 2.10 . — We denote by Hmothe homotopy category hence obtained.
+Recall from [ 43, Remarque 5.4] that the fibrant objects of the Morita model stru c-
+ture (called the Morita fibrant dg categories ) are the dg categories Afor which the
+image of the induced functor
+h:H0(A)֒→D(A)x/ma√sto→A(−,x)
+is stable under (co-)suspensions, cones and direct factors.
+2.4. Monoidal structure. —
+Definition 2.11 . — LetA1andA2be small dg categories. The tensor product
+A1⊗A2ofA1withA2is defined as follows: the set of objects of A1⊗A2is obj(A1)×
+obj(A2) and given objects ( x,x′) and (y,y′) inA1⊗A2, we set
+(A1⊗A2)((x,x′),(y,y′)) =A1(x,y)⊗A2(x′,y′).
+Remark 2.12 . — The tensor product of dg categories gives rise to a symmetric
+monoidal structure on dgcat, with unit object the dg category k(see§2.1). Moreover,
+this model structure is easily seen to be closed. However, the mode l structure of
+Theorem 2.9endowed with this symmetric monoidal structure is nota symmetric
+monoidal model category, as the tensor product of two cofibran t objects in dgcatis
+not cofibrant in general; see [ 48,§4]. Nevertheless, the tensor product can be derived
+into a bifunctor
+−⊗L−:Hmo×Hmo−→Hmo (A1,A2)/ma√sto→Q(A1)⊗A2=:A1⊗LA2,
+whereQ(A1) is a Morita cofibrant resolution of A1.
+Now, letBandAbe small dg categories. For any object xinB, we have a dg
+functorA→Bop⊗A. It sends an object yinAto (x,y) and for each ordered pair of
+objects ( y,z) inA, the morphism in C(k)
+A(y,z)−→(Bop⊗A)((x,y),(x,z)) =Bop(x,x)⊗A(y,z)12 D.-C. CISINSKI & G. TABUADA
+is given by the tensor product of the unit morphism k→Bop(x,x) with the identity
+morphism onA(y,z). Take a Morita cofibrant resolution Q(Bop) ofBopwith the same
+set of objects; see [ 48, Proposition 2.3(2)]. Since BopandQ(Bop) have the same set
+of objects, one sees that for any object xinB, we have a dg functor
+(2.4.1) ix:A−→Q(Bop)⊗A=:Bop⊗LA.
+Definition 2.13 . — LetAbe a small dg category. A A-module Mis called perfect
+if it is compact as an object in the derived category D(A). We denote by Cdg(A)pe
+the dg category of perfect A-modules, and put
+perf(A) =Cdg(A)pe∩Ddg(A).
+We thus have an equivalence of triangulated categories
+H0(perf(A))≃Dc(A).
+LetBandAbesmalldgcategories. A( Bop⊗LA)-module Xissaidtobe locally perfect
+overBif, for any object xinB, theA-module i∗
+x(X) (see (2.4.1) and§2.2) is perfect.
+We denote byCdg(Bop⊗LA)lpethe dg category of locally perfect ( Bop⊗LA)-modules
+overB, and put
+rep(B,A) =Cdg(Bop⊗LA)lpe∩Ddg(Bop⊗LA).
+Note that H0(rep(B,A)) is canonically equivalent to the full triangulated subcategory
+ofD(Bop⊗LA) spanned by the locally perfect ( Bop⊗LA)-modules overB.
+Remark 2.14 . — For any dg category A, the Yoneda embedding h:A→perf(A)
+is a derived Morita equivalence, and perf(A) is Morita fibrant. This construction thus
+provides us a canonical fibrant replacement functor for the Morit a model structure.
+Theorem 2.15 (To¨ en). —Given small dg categories BandA, we have a natural
+bijection
+HomHmo(B,A)≃IsoH0(rep(B,A))
+(whereIsoCstands for the set of isomorphism classes of objects in C). Moreover,
+given small dg categories A1,A2, andA3, the composition in Hmocorresponds to the
+derived tensor product of bimodules:
+IsoH0(rep(A1,A2))×IsoH0(rep(A2,A3))−→IsoH0(rep(A1,A3))
+([X],[Y]) /ma√sto−→ [X⊗L
+A2Y].
+Proof. — See [ 48, Corollary 4.8] and [ 43, Remarque 5.11].√
+Theorem 2.16 (To¨ en). —The derived tensor product −⊗L−onHmoadmits the
+bifunctor
+rep(−,−) :Hmoop×Hmo−→Hmo
+as an internal Hom-functor.
+Proof. — See [ 48, Theorem 6.1] and [ 43, Remarque 5.11].√SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 13
+Corollary 2.17 . —Given small dg categories A1,A2andA3, we have a natural
+isomorphism in Ho(sSet)
+(2.4.2) Map(A1⊗LA2,A3)≃Map(A1,rep(A2,A3)).
+Proof. — See [ 48, Corollary 6.4] and [ 43, Remarque 5.11].√
+Remark 2.18 . — Observe that, when B=k, the dg category Bop⊗LAidentifies
+withA, which gives a natural isomorphism in Hmo:
+(2.4.3) rep(k,A)≃perf(A).
+We finish this section by showing how to endow the derivator associat ed to the
+Morita model structure on dg categories with a symmetric monoidal structure; see
+Theorem 2.23. Recall from§2.4that the Morita model structure is nota monoidal
+model structure and so we cannot apply the general Proposition A.2. We sidestep
+this difficulty by introducing the notion of k-flat dg category .
+Definition 2.19 . — A complex of k-modules is called k-flatif for any acyclic com-
+plexN, the complex M⊗Nis acyclic.
+A dg category is called k-flatif for each ordered pair of objects ( x,y) inA, the
+complexA(x,y) isk-flat.
+Notation 2.20 . — We denote by dgcatflatthe category of k-flat dg categories.
+Remark 2.21 . — Clearly dgcatflatis a full subcategory of dgcat. Since complexes of
+k-flat modules are stable under tensor product, the category dgcatflatis a symmetric
+monoidal full subcategory of dgcat(see§2.4). Moreover, the tensor product
+−⊗−:dgcatflat×dgcatflat−→dgcatflat
+preserves derived Morita equivalences (in both variables).
+Notation 2.22 . — We denote by HO(dgcat) the derivator associated to the Morita
+model structure on dg categories; see Theorem 2.9.
+Theorem 2.23 . —The derivator HO(dgcat)carries a symmetric monoidal struc-
+ture−⊗L−; see§A.5. Moreover, the induced symmetric monoidal structure on
+HO(dgcat)(e) =Hmocoincides with the one described in Remark 2.12.
+Proof. — Let HO(dgcatflat) =dgcatflat[Mor−1] be the prederivator associated to the
+classMorof derived Morita equivalences in dgcatflat; see§A.1. Since dgcatflatis
+a symmetric monoidal category, and the tensor product preserv es the classMor,
+i.e.Mor⊗Mor⊂Mor, the prederivator HO(dgcatflat) carries a natural symmetric
+monoidal structure; see §A.5. Now, choose a functorial Morita cofibrant resolution
+functor
+Q:dgcat−→dgcat Q⇒Id.14 D.-C. CISINSKI & G. TABUADA
+Thanks to [ 48, Proposition 2.3(3)] every cofibrant dg category is k-flat, and so we
+obtain functors
+i:dgcatflat֒→dgcat Q:dgcat−→dgcatflat
+and natural transformations
+i◦Q⇒IdQ◦i⇒Id. (2.4.4)
+Since the functors iandQpreservederived Moritaequivalencesand the abovenatural
+transformations ( 2.4.4) are objectwise derived Morita equivalences, we obtain by 2-
+functoriality (see §A.1) morphisms of derivators
+i:HO(dgcatflat)−→HO(dgcat)Q:HO(dgcat)−→HO(dgcatflat)
+which are quasi-inverse to each other. Using this equivalence of der ivators, we trans-
+port the monoidal structure from HO(dgcatflat) toHO(dgcat). The fact that the
+induced monoidal structure on HO(dgcat)(e) =Hmocoincides with the one described
+in Remark 2.12is now clear.√
+Proposition 2.24 . —The tensor product on HO(dgcat)of Theorem 2.23preserves
+homotopy colimits in each variable.
+Proof. — This follows immediately from Corollary 2.17.√
+3. Homotopically finitely presented dg categories
+In this section we give a new characterization of homotopically finitely presented
+dg categories; see Theorem 3.3(3). Using this new characterization we show that
+homotopicallyfinitely presenteddgcategoriesarestableunderder ivedtensorproduct;
+see Theorem 3.4. This stability result will play a key role in the construction of a
+symmetric monoidal model structure on the category of non-com mutative motives;
+see Section 7.
+Proposition 3.1 . —For any small dg category Aandn∈Z, we have natural iso-
+morphisms in Hmo
+S(n)op⊗A≃S (n)op⊗LA≃rep(S(n),A).
+Proof. — SinceS(n)≃S(n)opis cofibrant, we obtain the following isomorphism in
+Hmo
+S(n)op⊗LA≃S(n)op⊗A.
+By construction, the dg category S(n) is also locally perfect ,i.e.its complexes of
+morphisms are perfect complexes of k-modules; see [ 51, Definition 2.4(1)]. We obtain
+then by [ 51, Lemma2.8] the following inclusion
+S(n)op⊗A≃perf(S(n)op⊗LA)⊂rep(S(n),A).SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 15
+Letusnowprovethe converseinclusion. Observethatwehavetwo naturaldgfunctors
+i1:A −→ S (n)op⊗A
+x/ma√sto−→ (1,x)i2:A −→ S (n)op⊗A
+x/ma√sto−→ (2,x),
+and the category of ( S(n)op⊗A)-modules identifies with the category of morphisms
+of degree ninC(A),i.e.to the category of triples ( M,M′,f), where MandM′areA-
+modules, and f:M−→M′[n] is a morphismof A-modules. Under this identification,
+we obtain the following extension of scalars functors (see §2.2):
+(i1)!:C(A)−→C(S(n)op⊗A)M/ma√sto−→(M,M[−n],1M)
+(i2)!:C(A)−→C(S(n)op⊗A)M/ma√sto−→(0,M,0).
+Now, let Xbe an object in rep(S(n),A). Note that Xcorresponds to a cofibration
+f:M/d47/d47 /d47/d47M′[n]
+inC(A) between cofibrant and perfect A-modules (see Definition 2.13). Consider the
+following short exact sequence of morphisms of A-modules
+(3.0.5)
+(M[−n])[n] =M/d47/d47f/d47/d47M′[n] /d47/d47coker(f) = (coker(f)[−n])[n]
+M1M/d79/d79
+M/d79/d79f/d79/d79
+/d47/d470/d79/d79
+wherecoker(f) denotes the cokernel of fin the categoryC(A). Since MandM′[n]
+are cofibrant and perfect A-modules, and fis a cofibration, the A-module coker(f)
+is also cofibrant and perfect. Perfect modules are stable under su spension, and so
+coker(f)[−n] is a perfect (and cofibrant) A-module. We have natural isomorphisms:
+(i1)!(M)≃(M,M[−n],1M) and ( i2)!(coker(f)[−n])≃(0,coker(f)[−n],0).
+Since the extension of scalars functors L(i1)!andL(i2)!preserve perfect modules,
+thesetwoobjectsareperfect. Finally, since perf(S(n)op⊗A)isstableunder extensions
+inDdg(S(n)op⊗A), and in the above short exact sequence ( 3.0.5) the left and right
+vertical morphisms belong to perf(S(n)op⊗A), we conclude that the object Xalso
+belongs to perf(S(n)op⊗A).√
+Definition 3.2 ([51, Definition 2.1(3)] ). — LetMbe a Quillen model category.
+An object XinMis called homotopically finitely presented if for any filtered system
+of objects{Yj}j∈JinM, the induced map
+hocolim
+j∈JMap(X,Yj)−→Map(X,hocolim
+j∈JYj)
+is an isomorphism in Ho(sSet).
+Theorem 3.3 . —LetBbe a small dg category. In the Morita model structure (see
+Theorem 2.9) the following conditions are equivalent:16 D.-C. CISINSKI & G. TABUADA
+(1)The dg categoryBis homotopically finitely presented;
+(2)The dg categoryBis a retract in Hmoof a finite dg cell (see Definition 2.5);
+(3)For any filtered system {Aj}j∈Jof dg categories, the induced morphism
+(3.0.6) hocolim
+j∈Jrep(B,Aj)∼−→rep(B,hocolim
+j∈JAj)
+is an isomorphism in Hmo.
+Proof. — (1)⇔(2) : This equivalence follows from [ 42, Proposition 5.2] and [ 42,
+Example 5.1].
+(3)⇒(1) : Let{Aj}j∈Jbe a filtered system of dg categories. By hypothesis, we
+have an induced isomorphism in Hmo
+(3.0.7) hocolim
+j∈Jrep(B,Aj)∼−→rep(B,hocolim
+j∈JAj).
+Thanks to Corollary 2.17we have, for any dg category A, natural isomorphisms in
+Ho(sSet):
+Map(B,A)≃Map(k⊗LB,A)≃Map(k,rep(B,A)).
+The dg category kis a finite dg cell, and so using equivalence (1) ⇔(2) we conclude
+thatkis homotopically finitely presented. Therefore, by applying the func tor
+Map(k,−) :Hmo−→Ho(sSet)
+to the above isomorphism ( 3.0.7), we obtain an induced isomorphism in Ho(sSet)
+hocolim
+j∈JMap(B,Aj)∼−→Map(B,hocolim
+j∈JAj).
+This shows thatBis homotopically finitely presented.
+(2)⇒(3) : The class of objects in Hmowhich satisfy condition (3) is clearly stable
+under retracts. Moreover, given a small dg category A, the functor
+rep(−,A) :Hmoop−→Hmo
+sends homotopy colimits to homotopy limits. Since by construction ho motopy pull-
+backs in Hmocommute with filtered homotopy colimits, we conclude that the class o f
+objects in Hmowhich satisfy condition (3) is also stable under homotopy pushouts.
+Therefore, it is enough to verify condition (3) for the domains and c odomains of the
+elements of the set I; see Notation 2.4. IfB=∅this is clear. The case where B=k
+was proved in [ 51, Lemma2.10]. If B=D(n),n∈Z, we have a natural derived
+Morita equivalence
+k∐L
+∅k≃k∐k∼−→D(n).
+Therefore, it remains to prove the case where B=S(n),n∈Z. Since by Proposi-
+tion2.24the derived tensor product preserves homotopy colimits, this cas e follows
+automatically from Proposition 3.1.√SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 17
+Theorem 3.4 . —LetB1andB2be homotopically finitely presented dg categories.
+ThenB1⊗LB2is a homotopically finitely presented dg category.
+Proof. — Let{Aj}j∈Jbe a filtered system of dg categories. The proof is a conse-
+quence of the following weak equivalences:
+Map(B1⊗LB2,hocolim
+j∈JAj)≃Map(B1,rep(B2,hocolim
+j∈JAj)) (3.0.8)
+≃Map(B1,hocolim
+j∈Jrep(B2,Aj)) (3.0.9)
+≃hocolim
+j∈JMap(B1,rep(B2,Aj)) (3.0.10)
+≃hocolim
+j∈JMap(B1⊗LB2,Aj). (3.0.11)
+Equivalences ( 3.0.8) and (3.0.11) follow from Corollary 2.17, Equivalence ( 3.0.9) fol-
+lows from Theorem 3.3, and Equivalence ( 3.0.10) follows from Definition 3.2.√
+4. Saturated dg categories
+InthissectionweintroduceKontsevich’snotionofsaturateddgca tegory. Following
+To¨ en’swork,wecharacterizethesedgcategoriesasthedualizab leobjectsintheMorita
+homotopy category; see Theorem 4.8. This conceptual characterization will play an
+important role in our applications; see Section 8.
+Definition 4.1 (Kontsevich) . — ([33,35,36])
+- A small dg category Ais called smoothif the right dg (Aop⊗LA)-module
+A(−,−) :A⊗LAop−→Cdg(k) (x,y)/ma√sto→A(y,x) (4.0.12)
+is perfect; see Definition 2.13.
+- A small dg category Ais called properif for each ordered pair of objects ( x,y)
+inA, the complex of k-modulesA(x,y) is perfect.
+- A small dg category Ais called saturated if it is smooth and proper.
+Remark 4.2 . — Given a dg functor F:B→A, we have a natural ( Bop⊗LA)-
+module
+B⊗LAop−→Cdg(k) (x,y)/ma√sto→A(y,Fx),
+which belongs to rep(B,A) (see Definition 2.13). The above (Aop⊗LA)-module
+A(−,−) (4.0.12) is obtained by taking B=AandFthe identity dg functor.
+Notation 4.3 . — Note that the class of smooth (resp. proper) dg categories is
+invariantunder derivedMoritaequivalences(see Definition 2.8). We denoteby Hmosat
+the full subcategory of Hmo(see Notation 2.10) whose objects are the saturated dg
+categories.18 D.-C. CISINSKI & G. TABUADA
+Example 4.4 . — Considerthe dgcategories S(n),n≥0, from§2.1(ii). Byconstruc-
+tion these dg categories are proper. Thanks to Proposition 3.1, we have a natural
+isomorphismS(n)op⊗LS(n)≃rep(S(n),S(n)) inHmo. Since the (S(n)op⊗LS(n))-
+moduleS(n)(−,−) belongsto rep(S(n),S(n)) (see Remark 4.2) andwe havea natural
+derived Moritaequivalence S(n)op⊗LS(n)→perf(S(n)op⊗LS(n)) (see Remark 2.14),
+we conclude that the dg categories S(n) are also smooth. Therefore, they are satu-
+rated.
+Example 4.5 . — (i) Let Xbeaquasi-compactandseparated k-scheme. Consider
+the categoryC(QCoh(X)) of (unbounded) complexes of quasi-coherent sheaves
+onX. Thanks to [ 26],C(QCoh(X)) is a model category with monomorphisms
+as cofibrations and quasi-isomorphisms as weak equivalences. More over, when
+endowed with its natural C(k)-enrichment,C(QCoh(X)) becomes aC(k)-model
+categoryin the sense of [ 24, Definition 4.2.18]. Let Lqcoh(X) be the dg category
+of fibrant objects in C(QCoh(X)). Note that H0(Lqcoh(X)) is naturally equiva-
+lent to the (unbounded) derived category Dqcoh(X) of quasi-coherentsheaveson
+X. Finally, let perf(X) be the full dg subcategoryof Lqcoh(X) whose objects are
+the perfect complexes. Note that H0(perf(X)) is naturally equivalent to the cat-
+egory of compact objects in Dqcoh(X). Thanks to To¨ en (see [ 51, Lemma 3.27]),
+whenXis a smooth and proper k-scheme, perf(X) is a saturated dg category.
+(ii) LetAbe ak-algebra, which is projective of finite rank as a k-module, and of
+finite global cohomological dimension. Then, the dg category of per fect com-
+plexes of A-modules is a saturated dg category.
+(iii) For examples coming from deformation quantization , we invite the reader to
+consult [33].
+Definition 4.6 . — LetCbe a symmetric monoidal category with monoidal product
+⊗and unit object 1. An object XinCis called dualizable (orrigid) if there exists
+an object X∨inC, and maps ev : X⊗X∨→1andδ:1→X∨⊗Xsuch that the
+composites
+(4.0.13) X≃X⊗1id⊗δ−→X⊗X∨⊗Xev⊗id−→1⊗X≃X
+and
+(4.0.14) X∨≃1⊗X∨δ⊗id−→X∨⊗X⊗X∨id⊗ev−→X∨⊗1≃X∨
+are identities. The object X∨is called the dualofX, the map ev is called the
+evaluation map , and the map δis called the co-evaluation map .
+Remark 4.7 . — (i) Given (( X∨)1,ev1,δ1) and (( X∨)2,ev2,δ2) as in Defini-
+tion4.6, there is a unique isomorphism ( X∨)1∼→(X∨)2making the natural
+diagrams commute. Therefore, the dual of X, together with the evaluation and
+the co-evaluation maps, is well-defined up to unique isomorphism.SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 19
+(ii) LetXbe a dualizable object in C. Thanks to Equations ( 4.0.13) and (4.0.14),
+the evaluation and co-evaluation maps give rise to an adjunction:
+(4.0.15) C
+X∨⊗−
+/d15/d15
+C−⊗X/d79/d79
+In fact, an object XofCis dualizable if and only if there exists an object X′,
+together with a functorial isomorphism
+HomC(Y⊗X,Z)≃HomC(Y,X′⊗Z)
+for any objects YandZinC(and of course such an X′is a dual of X). In
+other words, Xis dualizable with dual X∨if and only if, for any object ZofC,
+X∨⊗Zis the internal Hom object from XtoZ.
+(iii) Let XandYbe dualizable objects in C. ThenX⊗Yis also a dualizable object
+with dual Y∨⊗X∨.
+(iv) Let F:C→C′be a symmetric monoidal functor. Then, if Xis a dualizable
+object inC,F(X) is a dualizable object in C′with dual F(X∨).
+We are now ready to state the following folklore result:
+Theorem 4.8 . —The dualizable objects in the Morita homotopy category Hmo(see
+§2.4) are the saturated dg categories. Moreover, the dual of a sat urated dg category A
+is its opposite dg category Aop.
+Proof. — Let us start by introducing the category DGCAT. Its objects are the small
+dg categories. Given small dg categories BandA, the set of morphisms from BtoA
+is the set of isomorphism classes in D(Bop⊗LA). Given small dg categories A1,A2,
+andA3, the composition corresponds to the derived tensor product of b imodules:
+IsoD(Aop
+1⊗LA2)×IsoD(Aop
+2⊗LA3)−→IsoD(Aop
+1⊗LA3)
+([X],[Y]) /ma√sto→[X⊗L
+A2Y].
+As in the case of Hmo, the derived tensor product of dg categories gives rise to a
+symmetric monoidal structure on DGCAT, with unit object the dg category k. The
+key property of DGCATis that all its objects are dualizable: given a small dg category
+A, take for dual its opposite dg category Aop, for evaluation map the isomorphism
+class inD((A⊗LAop)op⊗Lk)≃D(Aop⊗A) of the (Aop⊗LA)-module
+A(−,−) :A⊗LAop−→Cdg(k) (x,y)/ma√sto→A(y,x), (4.0.16)
+and for co-evaluation map the isomorphism class in D(kop⊗L(Aop⊗LA))≃D(Aop⊗
+A) of the same (Aop⊗LA)-module ( 4.0.16). With this choices, both composites
+(4.0.13) and (4.0.14) are identities. Note also that, by construction of DGCAT, and
+by Theorem 2.15, we have a natural symmetric monoidal functor
+Hmo−→DGCATA/ma√sto→A, (4.0.17)20 D.-C. CISINSKI & G. TABUADA
+which is faithful but not full.
+LetAbe adualizableobject in Hmo, with dualA∨, evaluationmap ev : A⊗LA∨→
+kand co-evaluation map δ:k→ A∨⊗LA. Since the above functor ( 4.0.17) is
+symmetric monoidal, Remark 4.7(iv) implies that Ais dualizable in DGCAT. By
+unicity of duals (see Remark 4.7(i)),A∨is the opposite dg category Aop, and ev and
+δare the isomorphism class in D(Aop⊗LA) of the above (Aop⊗LA)-moduleA(−,−)
+(see4.0.16). The morphism ev belongs to Hmo, and soA(−,−) takesvalues in perfect
+complexes of k-modules. We conclude then that Ais proper. Similarly, since δis a
+morphism in Hmo,A(−,−) belongs to perf(Aop⊗LA). In this case we conclude that
+Ais smooth. In sum, we have shown that Ais saturated and that A∨=Aop.
+Now, letAbe a saturated dg category. Since Ais properA(−,−) takes values
+in perfect complexes of k-modules. Similarly, since Ais smoothA(−,−) belongs
+toperf(Aop⊗LA). We conclude that the evaluation and co-evaluation maps of Ain
+DGCATbelongtothe subcategory Hmo. This impliesthat Aisdualizablein Hmo.√
+We finish this section with a comparison between saturated and homo topically
+finitely presented dg categories (see §3).
+Lemma 4.9 . —LetBbe a saturated dg category. Then, for every dg category A,
+we have a functorial isomorphism in Hmo
+Bop⊗LA≃rep(B,A).
+Proof. — SinceB∨=Bop, the general theory of dualizable objects implies the claim;
+see Remark 4.7(iii).√
+Proposition 4.10 . —Every saturated dg category is homotopically finitely pre-
+sented.
+Proof. — LetBbe a saturated dg category. Since the derived tensor product of
+dg categories preserves homotopy colimits in each variable (see Pro position 2.24),
+Lemma4.9implies that the functor rep(B,−) commutes with homotopy colimits.
+Using Theorem 3.3(3) the proof is achieved.√
+5. Simplicial presheaves
+In this section we present a general theory of symmetric monoidal structures on
+simplicial presheaves. These general results will be used in the cons truction of an
+explicit symmetric monoidal structure on the category of non-com mutative motives;
+see Section 7.
+Starting from a small symmetric monoidal category C, we recall that the tensor
+product ofCextends to simplicial presheaves on C; see§5.4. Then, we prove that
+this symmetric monoidal structure is compatible with the projective model structure;
+see Theorem 5.2. Finally, we study its behavior under left Bousfield localizations.SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 21
+We describe in particular a “minimal” compatibility condition between the tensor
+product and a localizing set; see Theorem 5.7.
+5.1. Notations. — Throughout this section Cwill denote a (fixed) small category.
+(i) We denote by /hatwideCthecategory of presheaves of sets on C,i.e.the category of
+contravariant functors from CtoSet.
+(ii) Given an object αinC, we still denote by αthe presheaf represented by α
+α:Cop−→Set, β/ma√sto→HomC(β,α)
+(i.e.we consider the Yoneda embedding as a full inclusion).
+(iii) Let ∆ be the category of simplices, i.e.the full subcategory of the category of
+ordered sets spanned by the sets ∆[ n] ={0,...,n}forn≥0. We set
+sSet=/hatwide∆.
+(iv) Wedenoteby s/hatwideC≃/hatwider∆×Cthecategoryofsimplicialobjectsin /hatwideC,i.e.thecategory
+of contravariant functors from ∆ to /hatwideC.
+(v) Finally, we consider /hatwideCas a full subcategory of s/hatwideC. A presheaf of sets on Cis
+identified with a simplicially constant object of s/hatwideC.
+5.2. Simplicial structure. — Recall that we have a bifunctor
+−⊗−:/hatwideC×Set−→/hatwideC
+defined by
+X⊗K=/coproductdisplay
+k∈KX.
+Thisdefinesanactionofthecategory Seton/hatwideC. Thisconstructionextendstosimplicial
+objects
+s/hatwideC×sSet−→s/hatwideC(F,K)/ma√sto→F⊗K,
+where for n≥0:
+(F⊗K)n=Fn⊗Kn.
+This makes s/hatwideCinto a simplicial category; see for instance [ 20,§II Definition 2.1].
+5.3. Quillen Model structure. — The generating cofibrations of the classical
+cofibrantly generated Quillen model structure on sSetare the boundary inclusions
+in:∂∆[n]−→∆[n]n≥0
+and the generating trivial cofibrations are the horn inclusions
+jk
+n: Λ[k,n]−→∆[n] 0≤k≤n, n≥1.
+We have the projective model structure ons/hatwideC: the weak equivalences are the termwise
+simplicial weak equivalences, and the fibrations are the termwise Kan fibrations; see
+for instance [ 6, page 314] or [ 23, Theorem 11.6.1]. The projective model structure22 D.-C. CISINSKI & G. TABUADA
+is proper and cellular/combinatorial. In particular, it is cofibrantly ge nerated, with
+generating cofibrations
+1α⊗in:α⊗∂∆[n]−→α⊗∆[n], α∈C, n≥0,
+and generating trivial cofibrations
+1α⊗jk
+n:α⊗Λ[k,n]−→α⊗∆[n], α∈C,0≤k≤n, n≥1.
+In particular, observe that representable presheaves are cofib rant ins/hatwideC.
+5.4. Day’s convolution product. — Throughout this subsection, and until the
+end of Section 5, we will assume that our (fixed) small category Ccarries a symmetric
+monoidal structure, with tensor product ⊗and unit object 1. Under this assumption,
+the general theory of left Kan extensions in categories of preshe aves implies formally
+that/hatwideCis endowed with a unique closed symmetric monoidal structure which m akes
+the Yoneda embedding a symmetric monoidal functor. We will also den ote by⊗the
+corresponding tensor product on /hatwideC; the reader who enjoys explicit formulas is invited
+to consult [ 13,§3].
+This monoidal structure extends to the category s/hatwideCin an obvious way: given two
+simplicial presheaves FandGonC, we define F⊗Gby the formula
+(F⊗G)n=Fn⊗Gn, n≥0.
+The functor
+sSet−→s/hatwideC, K/ma√sto−→1⊗K
+is naturally endowed with a structure of symmetric monoidal functo r (where sSetis
+considered as a symmetric monoidal category with the cartesian pr oduct as tensor
+product).
+Definition 5.1 ([24, Definition 4.2.1] ). — Given maps f:X→Yandg:A→B
+in a symmetric monoidal category (with tensor product ⊗), thepushout product map
+f/squaregoffandgis given by:
+f/squareg:X⊗B/coproductdisplay
+X⊗AY⊗A−→Y⊗B.
+Theorem 5.2 . —The category s/hatwideC, endowed with the projective model category struc-
+ture is a symmetric monoidal model category (see [24, Definition 4.2.6] ).
+Proof. — As the model category of simplicial sets is a symmetric monoidal mo del
+category (with the cartesian product as tensor product), the r esult follows from the
+explicit description of the generating cofibrations and generating t rivial cofibrations
+ofs/hatwideC: given two objects αandα′inCand two maps i:K→Landi′:K′→L′in
+sSet, we have
+(1α⊗i)/square(1α′⊗i′)≃1α⊗α′⊗(i/squarei′).SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 23
+Since for any object αinC, the functor K/ma√sto→α⊗Kis a left Quillen functor from
+sSettos/hatwideC, the proof is finished.√
+Lets/hatwideC•be thecategoryofpointed simplicialpresheaveson C. The forgetfulfunctor
+U:s/hatwideC•→s/hatwideChas a left adjoint
+(5.4.1) s/hatwideC−→s/hatwideC•, F/ma√sto−→F+,
+whereF+denotes the pointed simplicial presheaf F∐⋆, with⋆the terminal object of
+s/hatwideC. The category s/hatwideC•is then canonically a cofibrantly generated model category, in
+such a way that the functor ( 5.4.1) is a left Quillen functor; see [ 24, Proposition 1.1.8
+and Lemma 2.1.21].
+Furthermore, there is a unique symmetric monoidal structure on s/hatwideC•making the
+functor ( 5.4.1) symmetric monoidal. The unit object is 1+, and the tensor product
+⊗•is defined as follows: for two pointed simplicial presheaves FandG, their tensor
+product is given by the following pushout in the category s/hatwideCof unpointed simplicial
+presheaves:
+(F⊗⋆)∐(⋆⊗G) /d47/d47
+/d15/d15F⊗G
+/d15/d15
+⋆ /d47/d47F⊗•G
+In particular, for two simplicial presheaves FandG, we have
+(5.4.2) ( F⊗G)+≃F+⊗•G+.
+Proposition 5.3 . —With the above definition, the model category s/hatwideC•is a symmet-
+ric monoidal model category.
+Proof. — The generating cofibrations (resp. generating trivial cofibrat ions) ofs/hatwideC•
+arethe maps ofshape A+→B+forA→Ba generatingcofibration(resp. generating
+trivial cofibration) of s/hatwideC. As the functor ( 5.4.1) is a left Quillen functor, the result
+follows immediately from Formula ( 5.4.2) and from Theorem 5.2.√
+Remark 5.4 . — In practice, we shall refer to the pointed tensor product ⊗•as the
+canonicaltensorproduct on s/hatwideC•associatedto the monoidalstructure on C. Whenever
+there is no ambiguity, we denote the pointed tensor product simply b y⊗.
+5.5. Left Bousfield localization. —
+Definition 5.5 . — ([23, Definition 3.1.4]) Let Mbe a model category and Sa set
+of morphisms inM. An object XinMis called S-localif it is fibrant and for every
+elements:A→Bof the set S, the induced map of homotopy function complexes
+s∗:Map(B,X)−→Map(A,X)24 D.-C. CISINSKI & G. TABUADA
+is a weak equivalence. A map g:X→YinMis called a S-local equivalence if for
+everyS-local object W, the induced map of homotopy function complexes
+g∗:Map(Y,W)−→Map(X,W)
+is a weak equivalence
+Recall that, ifMis cellular (or combinatorial) and left proper, the left Bousfield
+localization ofMis the model category LSMwhose underlying category is M, whose
+cofibrations are those of M, and whose weak equivalences are the S-local weak equiv-
+alences; see [ 23, Definition 9.3.1(1)]. The fibrant objects of LSMare then the objects
+ofMwhich are both fibrant and S-local, and the fibrations between fibrant objects
+inLSMare the fibrations of M.
+Proposition 5.6 . —LetMbe a left proper, cellular (or combinatorial), symmetric
+monoidal model category (with tensor product ⊗), andSa set of morphisms in M.
+Assume that the following conditions hold:
+(i)Madmits generating sets of cofibrations and of trivial cofibra tions consisting of
+maps between cofibrant objects;
+(ii)every element of Sis a map between cofibrant objects;
+(iii)given a cofibrant object X, the functor X⊗(−)sends the element of StoS-local
+weak equivalences.
+(iv)the unit object of the monoidal structure on Mis cofibrant.
+Then the left Bousfield localization LSMofMwith respect to the set Sis a symmetric
+monoidal model category.
+Proof. — The left Bousfield localization LSMofMwith respect to the set Sis
+cofibrantly generated (see [ 23, Theorem 4.1.1(3)]) and thanks to condition (iv) the
+unit object is cofibrant. Therefore, by [ 24, Lemma4.2.7] it is enough to verify the
+pushout product axiom on the sets of generating (trivial) cofibrat ions. The class of
+cofibrations in LSMand inMis the same, and so half of the pushout product axiom
+is automatically verified. Now, let g:A/d47/d47 /d47/d47Bbe a generating cofibration in
+LSMandf:X/d47/d47∼/d47/d47Ya generating trivial cofibration in LSM. By condition (i),
+we may assume, that the objects X,Y,AandBare cofibrant. Moreover, condition
+(iii) implies that tensoring by a cofibrant object preserves S-local weak equivalences.
+Consider the following commutative diagram:
+A⊗X
+/d15/d15
+g⊗idX
+/d15/d15idA⊗f/d47/d47A⊗Y
+/d15/d15
+g⊗idY
+/d15/d15i(A⊗Y)
+/d117/d117/d108/d108/d108/d108/d108/d108/d108
+B⊗X∐
+A⊗XA⊗Y
+/d42/d42f/squareg
+/d42/d42/d85/d85/d85/d85
+B⊗Xi(B⊗X)/d52/d52/d105/d105/d105/d105/d105/d105
+idB⊗f/d47/d47B⊗Y .SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 25
+Using the two-out-of-three property for S-local weak equivalences, we conclude that
+f/squaregis anS-local equivalence.√
+Theorem 5.7 . —LetSbe a set of morphisms between cofibrant objects in s/hatwideC. As-
+sume that the following condition holds:
+(C)given an object αinCand a map G→HinS, the morphism α⊗G→α⊗H
+is anS-local equivalence.
+Then the left Bousfield localization LSs/hatwideCofs/hatwideCwith respect to the set S, is a symmetric
+monoidal model category.
+Proof. — We shall apply Proposition 5.6. It is sufficient to prove condition (iii) of
+loc. cit. In other words, we need to prove that for any object Fofs/hatwideCand any map
+G→HinS, the map
+F⊗LG−→F⊗LH
+inHo(s/hatwideC) is sent to an isomorphism in Ho(LSs/hatwideC). Thanks to Condition (C)this is
+the case when Fis representable. As the functors ( −)⊗LFcommute with homotopy
+colimits, the general case follows from the fact that the functor Ho(s/hatwideC)→Ho(LSs/hatwideC)
+commutes with homotopy colimits, and that any simplicial presheaf is a homotopy
+colimit ofrepresentablepresheaves(see forinstance [ 15, Proposition2.9]or[ 8, Propo-
+sition 3.4.34]).√
+Corollary 5.8 . —Assume that Sis a set of maps in Cwhich is closed under tensor
+product inC(up to isomorphism). Then, by considering Sas a set of maps in s/hatwideCvia
+the Yoneda embedding, the left Bousfield localization LSs/hatwideCis a symmetric monoidal
+model category.
+Proof. — Condition (C)of the Theorem 5.7is trivially satisfied.√
+6. Monoidal stabilization
+In this section we relate the general theory of spectra with the ge neral theory of
+symmetric spectra; see Theorem 6.1. This will be used in the construction of an
+explicit symmetric monoidal structure on the category of non-com mutative motives;
+see Section 7.
+LetMbe a cellular (or combinatorial) pointed simplicial left proper model cat e-
+gory. There is then a natural action of the category of pointed sim plicial sets
+M×sSet•−→M,(X,K)/ma√sto−→X⊗K
+(in the literature, X⊗Kis usually denoted by X∧K). We denote by SpN(M) the
+stable model category of S1-spectra onM, whereS1= ∆[1]/∂∆[1] is the simplicial
+circle, seen as endofunctor X/ma√sto→X⊗S1ofM; see [25,§1].
+Assume thatMis asymmetric monoidalmodel categorywith cofibrantunit object
+1. We write SpΣ(M) for the stable symmetric monoidal model category of symmetric26 D.-C. CISINSKI & G. TABUADA
+S1⊗1-spectra onM; see [25,§7]. In this situation we have a symmetric monoidal
+left Quillen functor
+Σ∞:M−→SpΣ(M).
+Theorem 6.1 . —Under the above assumptions, the model categories SpN(M)and
+SpΣ(M)are canonically Quillen equivalent.
+Proof. — By applying [ 25, Theorem 10.3 and Corollary 10.4], it is sufficient to prove
+thatS1⊗1is symmetric in Ho(M),i.e.that the permutation (1 ,2,3) acts trivially on
+(S1⊗1)⊗3. Using [ 9, Corollaire 6.8], we see that ( S1⊗1)⊗3≃S3⊗1inHo(M), and
+that it is sufficient to check this condition in the case where Mis the model category
+of pointed simplicial sets. Finally, in this particular case, the result is w ell known; see
+for instance [ 24, Lemma 6.6.2].√
+7. Symmetric monoidal structure
+In this section we motivate, state and prove our main result: the loc alizing moti-
+vator carries a canonical symmetric monoidal structure; see The orem7.5.
+Definition 7.1 . — A sequence of triangulated categories
+RI−→SP−→T
+is calledexactif the composition is zero, the functor Iis fully-faithful and the induced
+functor from the Verdier quotient S/RtoTiscofinal,i.e.it is fully-faithful and every
+object inTis a direct summand of an object of S/R; see [39,§2] for details. A
+sequence of dg categories
+AF−→BG−→C
+is called exactif the induced sequence of triangulated categories (see §2.2)
+D(A)LF!−→D(B)LG!−→D(C)
+is exact.
+Recall from [ 42,§10] the construction of the universal localizing invariant .
+Theorem 7.2 . —([42, Theorem 10.5] ) There exists a morphism of derivators
+Uloc
+dg:HO(dgcat)−→Motloc
+dg,
+with values in a strong triangulated derivator (see §A.3), which has the following
+properties:
+flt)Uloc
+dgpreserves filtered homotopy colimits;
+p)Uloc
+dgpreserves the terminal object;SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 27
+loc)Uloc
+dgsatisfies localization ,i.e.sends exact sequence of dg categories
+A−→B−→C
+to distinguished triangles in Motloc
+dg(e)
+Uloc
+dg(A)−→Uloc
+dg(B)−→Uloc
+dg(C)−→Uloc
+dg(A)[1].
+Moreover,Uloc
+dgis universal with respect to these properties, i.e.given any strong
+triangulated derivator D, we have an induced equivalence of categories
+(Uloc
+dg)∗:Hom!(Motloc
+dg,D)∼−→Homloc(HO(dgcat),D),
+where the left-hand side stands for the category of homotopy colimit preserving mor-
+phisms of derivators, while the right-hand side stands for t he full subcategory of
+Hom(HO(dgcat),D)spanned by the morphisms of derivators which verify the thre e
+conditions above.
+Definition 7.3 . — The objects of the category Homloc(HO(dgcat),D) are called lo-
+calizing invariants andUloc
+dgis called the universal localizing invariant . Because of its
+universal property, which is a reminiscence of motives, Motloc
+dgis called the localizing
+motivator . Its base category Motloc
+dg(e) is a triangulated category and is morally what
+we would like to consider as the category of non-commutative motives .
+Example 7.4 . — Examples of localizing invariants include Hochschild homology
+and cyclic homology (see [ 42, Theorem 10.7]), non-connective algebraic K-theory
+(see [42, Theorem 10.9]), and even topological Hochschild homology and topo logical
+cyclic homology (see [ 2, Theorem 6.1] and [ 45,§8]).
+In this section we introduce a new ingredient in Theorem 7.2: symmetric monoidal
+structures. As shown in Theorem 2.23the derivator HO(dgcat) carries a symmetric
+monoidal structure. It is therefore natural to consider localizing invariants which are
+symmetric monoidal; see Examples 7.9-7.11. Our main result is the following.
+Theorem 7.5 . —The localizing motivator Motloc
+dgcarries a canonical symmetric mo-
+noidal structure−⊗L−, making the universal localizing invariant Uloc
+dgsymmetric
+monoidal (see§A.5). Moreover, this tensor product preserves homotopy colimi ts in
+each variable, and is characterized by the following univer sal property: given any
+strong triangulated derivator D(see§A.3), endowed with a symmetric monoidal struc-
+ture which preserves homotopy colimits in each variable, we have an induced equiva-
+lence of categories
+(Uloc
+dg)∗:Hom⊗
+!(Motloc
+dg,D)∼−→Hom⊗
+loc(HO(dgcat),D),
+where the left-hand side stands for the category of symmetri c monoidal homotopy
+colimit preserving morphisms of derivators, while the righ t-hand side stands for the
+category of symmetric monoidal morphisms of derivators whi ch belong to the category28 D.-C. CISINSKI & G. TABUADA
+Homloc(HO(dgcat),D). Furthermore, Motloc
+dgadmits an explicit symmetric monoidal
+Quillen model.
+Definition 7.6 . — The objects of the category Hom⊗
+loc(HO(dgcat),D) are called
+symmetric monoidal localizing invariants .
+Corollary 7.7 . —Any dualizable object of Motloc
+dg(e)is compact; see Definition 4.6
+and§1.2. In particular, given any saturated dg category A(see Definition 4.1), the
+objectUloc
+dg(A)is compact.
+Proof. — Let Mbe a dualizable object of Motloc
+dg(e). We need to prove that the
+functorHom(M,−)commuteswitharbitrarysums. Since Misdualizable,thisfunctor
+is isomorphic to Hom(Uloc
+dg(k),M∨⊗L−). The unit object Uloc
+dg(k) is known to be
+compact (see [ 11, Theorem 7.16]), and so the first assertion is proven. The second
+assertion follows from the fact that Uloc
+dgis symmetric monoidal, and that Uloc
+dg(A)
+is dualizable for any saturated dg category A(see Remark 4.7(iv) and Theorem
+4.8).√
+Remark 7.8 . — Although we do not know if the triangulated category Motloc
+dg(e) is
+compactly generated, Corollary 7.7implies that the localizing triangulated subcate-
+gory of Motloc
+dg(e) generated by dualizable objects is compactly generated.
+Before proving Theorem 7.5, let us give some examples of symmetric monoidal
+localizing invariants.
+Example 7.9 (Hochschild homology) . — LetAbeasmall k-flatdgcategory;see
+Definition 2.19. We can associate to Aa simplicial object in C(k),i.e.a contravariant
+functor from ∆ to C(k) : itsn-th term is given by
+/circleplusdisplay
+(x0,...,xn)A(xn,x0)⊗A(xn−1,xn)⊗···⊗A (x0,x1),
+where (x0,...,x n) is an ordered sequence of objects in A. The face maps are given
+by
+di(fn,...,f i,fi−1,...,f 0) =/braceleftbigg(fn,...,f i◦fi−1,...,f 0) ifi >0
+(−1)(n+σ)(f0◦fn,...,f 1) ifi= 0
+whereσ= (degf0)(degf1+...+degfn−1), and the degenerancies maps are given by
+sj(fn,...,f j,fj−1,...,f 0) = (fn,...,f j,idxj,fj−1,...,f 0).
+Associated to this simplicial object we have a chain complex in C(k) (by the Dold-
+Kan equivalence), and so a bicomplex of k-modules. The Hochschild complex HH (A)
+ofAis the sum-total complex associated to this bicomplex. If Ais an arbitrary dg
+category, its Hochschild complex is obtained by first taking a k-flat (e.g.cofibrant)
+resolution ofA; see Definition 2.19. This construction furnishes us a functor
+HH:dgcat−→C(k),SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 29
+which by [ 42, Theorem 10.7] gives rise to a localizing invariant
+(7.0.1) HH:HO(dgcat)−→HO(C(k)).
+Given small dg categories AandB, we have a functorial quasi-isomorphism
+sh:HH(A)⊗HH(B)−→HH(A⊗B)
+given by the shuffle product map; see [ 38,§4.2.3]. The localizing invariant ( 7.0.1),
+endowed with the shuffle product map, becomes then a symmetric mo noidal localizing
+invariant.
+Example 7.10 (Mixed complexes) . — Following Kassel [ 29,§1] we denote by Λ
+the dg algebra k[ǫ]/ǫ2, whereǫis of degree−1 andd(ǫ) = 0. Under this notation, a
+mixed complex is a right dg Λ-module (see Definition 2.6).
+LetAbe a small dg category. The Hochschild complex HH(A) ofA(see Exam-
+ple7.9), endowed with the cyclic operator
+tn(fn−1,...,f 0) = (−1)n+σ(f0,fn−1,fn−2,...,f 1),
+gives rise to a mixed complex C(A); see [31,§1.3]. The assignment A/ma√sto→C(A) yields
+a localizing invariant
+(7.0.2) C:HO(dgcat)−→HO(C(Λ))
+with values in the derivator associated to right dg Λ-modules; see [ 42, Theorem 10.7].
+Recall from [ 29,§1] that the category C(Λ) carries a natural symmetric monoidal
+structurewhoseunitobjectis k: giventwomixedcomplexesthereisacanonicalmixed
+complex structure on the tensor product of the underlying comple xes. Moreover, this
+symmetric monoidal structure is compatible with the projective mod el structure (see
+§2.2). Thanks to [ 29, Theorem 2.4] the localizing invariant ( 7.0.2) becomes then a
+symmetric monoidal localizing invariant.
+This example will be used in the construction of a canonical Chern cha racter
+map from non-connective algebraic K-theory to negative cyclic homology; see Ex-
+ample8.10.
+Example 7.11 (Periodic complexes) . — In this example we assume that our
+base ring kis a field. Let k[u] be the cocommutative Hopf algebra of polynomials
+in one variable uof degree 2; see [ 29,§1]. Consider the symmetric monoidal model
+category k[u]-Comod of k[u]-comodules; see [ 24, Theorem 2.5.17]. The monoidal
+structure is given by the cotensor product −/squarek[u]−of comodules, with unit k[u].
+Given a mixed complex M(see Example 7.10) we denote by P(M) thek[u]-
+comodule, whose underlying complex is M⊗L
+Λk, obtained by iteration of the map
+(M⊗L
+Λk)[−2]S−→M⊗L
+Λk,30 D.-C. CISINSKI & G. TABUADA
+see [29, Proposition 1.4]. Using [ 29, Theorem 1.7] and [ 19, Proposition 9.2], we
+conclude that we have a symmetric monoidal morphism of derivators
+P:HO(C(Λ))−→HO(k[u]-Comod) .
+By composing Pwith the localizing invariant ( 7.0.2) we obtain then a symmetric
+monoidal localizing invariant
+(P◦C) :HO(dgcat)−→HO(k[u]-Comod) .
+ThisexamplewillbeusedintheconstructionofacanonicalCherncha ractermapfrom
+non-connective algebraic K-theory to periodic cyclic homology; see Example 8.11.
+7.1. Proof of Theorem 7.5. —We will use freely the theory of derivators which
+is recalled and developed in the appendix. Recall from [ 42,§10] thatUloc
+dgis obtained
+by the following composition:
+HO(dgcat)Rh−→LΣHotdgcatfΦ−→LΣ,PHotdgcatfstab−→St(LΣ,PHotdgcatf)γ−→Motloc
+dg.
+The morphism stabcorresponds to a stabilization procedure (see §A.8) and the mor-
+phismγto a left Bousfield localization procedure (see §A.7). Since these procedures
+commute (see Proposition A.12), we can also obtain Uloc
+dgby the following composition
+(7.1.1) HO(dgcat)Rh−→LΣHotdgcatfΦ−→LΣ,PHotdgcatfγ−→Motuloc
+dgstab−→Motloc
+dg,
+where Motuloc
+dgis theunstable analogue of the localizing motivator.
+The proof of Theorem 7.5will consist on the concatenation of Propositions 7.12-
+7.16followed by Remark 7.18. In each one of these propositions we construct an ex-
+plicit symmetric monoidal Quillen model for the corresponding (interm ediate) deriva-
+tor of the composition ( 7.1.1).
+We start by fixing on dgcata fibrant resolution functor R, a cofibrant resolution
+functorQ, a left framing Γ ∗(i.e.a well-behaved cosimplicial resolution functor; see
+[24, Definition 5.2.7 and Theorem 5.2.8]), as well as a small full subcategor ydgcatf
+ofdgcat, satisfying the following properties:
+(a) any finite dg cell (see Definition 2.5) is indgcatf;
+(b) any object in dgcatfis homotopically finitely presentated (see Definition 3.2);
+(c) given any object Aindgcatf,Q(R(A)) andQ(A) belong to dgcatf;
+(d) for any cofibrant object Aofdgcatf, if Γ∗(A) denotes the given cosimplicial
+frame ofA, then Γ n(A) belongs to dgcatffor alln≥0.
+We let Σ be the set ofderivedMoritaequivalencesin dgcatf. The derivator LΣHotdgcatf
+is simply the derivator HO(LΣs/hatwiderdgcatf) associated to the left Bousfield localization of
+the projective model structure on s/hatwiderdgcatf(see§5.3), with respect to the set Σ. Note
+that, up to Quillen equivalence, this construction does not depend o n the choice of
+the category dgcatfbut only on the Dwyer-Kan localization of dgcatfby Σ (see [ 18]).
+The above stability properties imply that the Dwyer-Kan localization o fdgcatfby ΣSYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 31
+is simply (equivalent to) the full simplicial subcategory of the Dwyer- Kan localization
+of the model category dgcatspanned by the homotopically finitely presented dg cate-
+gories. In order to obtain a symmetric monoidal structure, we hav e then the freedom
+to add the following properties to dgcatf:
+(e) any dg category in dgcatfisk-flat (see Definition 2.19);
+(f) given any dg categories AandBindgcatf,A⊗Bbelongs to dgcatf(this makes
+sense because of Theorem 3.4).
+In the sequel, we assume that a small full subcategory dgcatfofdgcatsatisfying all
+the above properties (a)–(f) has been chosen; for instance, on e might consider the
+smallest one relatively to R,Qand Γ∗. The morphism
+Rh:HO(dgcat)−→LΣHotdgcatf=HO(LΣs/hatwidedgcatf)
+is induced by the functor
+h:dgcat−→s/hatwidedgcatf,
+which associates to any dg category Athe simplicial presheaf on dgcatf:
+Rh(A) :B/ma√sto−→Map(B,A) =Hom(Γ∗(Q(B)),R(A)).
+Proposition 7.12 . —The tensor product of dg categories in dgcatfextends
+uniquelly to a closed symmetric monoidal structure on the ca tegory of simplicial
+presheaves on dgcatf, making LΣs/hatwidedgcatfinto a symmetric monoidal model category.
+As a consequence, the derivator LΣHotdgcatfcarries a symmetric monoidal structure,
+making the morphism Rhsymmetric monoidal. Moreover, given any derivator D,
+the category of filtered homotopy colimit preserving symmet ric monoidal morphisms
+fromHO(dgcat)toDis canonically equivalent to the category of homotopy colim it
+preserving symmetric monoidal morphisms from LΣHotdgcatftoD.
+Proof. — As derived Morita equivalences are stable under derived tensor p roduct,
+it follows immediately from Corollary 5.8thatLΣs/hatwidedgcatfis a symmetric monoidal
+model category.
+Let us now show that the morphism Rhis symmetric monoidal. Recall from [ 42,
+§5] that the morphism Rhpreserves filtered homotopy colimits and that we have a
+commutative diagram
+(7.1.2) dgcatf[Σ−1]
+/d15/d15i/d47/d47HO(dgcat)
+Rh/d120/d120/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112
+LΣHotdgcatf,
+wheredgcatfstands for the prederivator represented by dgcatf, anddgcatf[Σ−1] for
+its formal localization by Σ. By construction, the left vertical morp hism in the
+above diagram ( 7.1.2) is symmetric monoidal and the symmetric monoidal struc-
+ture onLΣHotdgcatfpreserves homotopy colimits in each variable. Moreover, thanks32 D.-C. CISINSKI & G. TABUADA
+to Lemma 2.24, the symmetric monoidal structure on HO(dgcat) preserves filtered
+homotopy colimits.
+Now, recall the universal property of HO(dgcat): it is the free completion of
+the prederivator dgcatf[Σ−1] by filtered homotopy colimits. In other words, given
+any derivator D, the category of filtered homotopy colimit preserving morphisms
+fromHO(dgcat) toDis canonically equivalent to the category of morphisms from
+dgcatf[Σ−1] toD; see [42,§5]. Replacing Dby the derivator of filtered homotopy col-
+imit preserving morphisms from HO(dgcat) toD, we deduce that the category of mor-
+phisms from HO(dgcat)×HO(dgcat) toDwhich preservefiltered homotopy colimits in
+eachvariableisequivalenttothecategoryofmorphismsfrom dgcatf[Σ−1]×dgcatf[Σ−1]
+toD. By induction, we prove similarly that, for any n≥0, the category of morphisms
+fromHO(dgcat)ntoDwhich preserve filtered homotopy colimits in each variable is
+equivalent to the category of morphisms from dgcatf[Σ−1]ntoD. As the morphism i
+in the above diagram ( 7.1.2) is symmetric monoidal, this implies that the morphism
+Rhis symmetric monoidal as well. Similarly, we see that, given any derivato rD, the
+categoryof symmetric monoidal morphisms from dgcatf[Σ−1] toDis equivalent to the
+categoryoffiltered homotopy colimit preservingsymmetric monoida l morphismsfrom
+HO(dgcat) toD. The last assertion of this proposition thus follows from Theorem A.3
+and Proposition A.9.√
+Leth:dgcatf−→s/hatwidedgcatfbetheYonedaembedding. Wedenoteby P:∅−→h(∅)
+the canonical map. Then, the derivator LΣ,PHotdgcatfis simply the left Bousfield
+localization of LΣHotdgcatfbyP. Thus, it can be described as
+LΣ,PHotdgcatf=HO(LΣ,Ps/hatwidedgcatf),
+whereLΣ,Ps/hatwidedgcatfis the left Bousfield localization of the model category LΣs/hatwidedgcatf
+by the map P.
+Proposition 7.13 . —The model category LΣ,Ps/hatwidedgcatfis symmetric monoidal, and
+the localization functor
+LΣs/hatwidedgcatf−→LΣ,Ps/hatwidedgcatf
+is a symmetric monoidal left Quillen functor. In particular , the derivator
+LΣ,PHotdgcatfis symmetric monoidal, and the localization morphism
+Φ :LΣHotdgcatf−→LΣ,PHotdgcatf
+is symmetric monoidal.
+Proof. — For any dg category A, we haveA⊗∅≃∅. We deduce easily from this
+formula that condition (C)of Theorem 5.7(withM=LΣs/hatwidedgcatf) is satisfied, and
+so the proof is finished.√SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 33
+Lets/hatwidedgcatf,•be the model category of pointed simplicial presheaves on s/hatwidedgcatf.
+By virtue of Proposition 5.3, this is a symmetric monoidal model category, and the
+functor
+s/hatwidedgcatf−→s/hatwidedgcatf,•, F/ma√sto−→F+
+is a symmetric monoidal left Quillen functor. We define the pointed mod el category
+LΣ,Ps/hatwidedgcatf,•as the left Bousfield localization of s/hatwidedgcatf,•with respect to the set of
+maps Σ +∪{P+}.
+Proposition 7.14 . —The model category LΣ,Ps/hatwidedgcatf,•is symmetric monoidal,
+and the symmetric monoidal left Quillen functor
+LΣ,Ps/hatwidedgcatf−→LΣ,Ps/hatwidedgcatf,•
+is a Quillen equivalence. In particular, we have a canonical equivalence of symmetric
+monoidal derivators
+LΣ,PHotdgcatf≃HO(LΣ,Ps/hatwidedgcatf,•).
+Proof. — The first assertion is a direct application of Theorem 5.7, while the second
+one follows from [ 42, Remark 8.2].√
+Note that the initial and terminal dg categoriesare Moritaequivalen t. This implies
+that the dg category 0 is sent to the point (up to weak equivalence) inLΣ,Ps/hatwidedgcatf,•.
+LetEbe the set of morphisms of LΣ,Ps/hatwidedgcatf,•of shape
+cone[Rh(A)−→Rh(B)]−→Rh(C),
+associated to each exact sequence of dg categories
+A−→B−→C ,
+withBindgcatf(whereconemeans homotopy cofiber). We define Motuloc
+dgas the left
+Bousfield localization of LΣ,Ps/hatwidedgcatf,•byE. The derivator Motuloc
+dgis defined as
+Motuloc
+dg=HO(Motuloc
+dg).
+Proposition 7.15 . —The model category Motuloc
+dgis symmetric monoidal, in such
+a way that the left Quillen functor
+LΣ,Ps/hatwidedgcatf,•−→Motuloc
+dg
+is symmetric monoidal. Under the identification of Proposit ion7.13, the induced
+morphism of derivators
+γ:LΣ,PHotdgcatf−→Motuloc
+dg
+is symmetric monoidal.
+Proof. — As tensoring by a k-flat dg category preserves exact sequences of dg cate-
+gories (see [ 16, Proposition 1.6.3]), and as Rhis symmetric monoidal (see Proposition
+7.12), the proof follows from Theorem 5.7.√34 D.-C. CISINSKI & G. TABUADA
+Finally, since by construction the model category Motuloc
+dgis symmetric monoidal
+and simplicially enriched, we can consider its stabilization Motloc
+dg,i.e.the stable
+model category of symmetric spectra in Motuloc
+dg(see§6):
+Motloc
+dg=SpΣ(Motuloc
+dg).
+The derivator Motloc
+dgis defined as
+Motloc
+dg=HO(Motloc
+dg).
+Proposition 7.16 . —The model category Motloc
+dgis symmetric monoidal, and the
+left Quillen functor
+Σ∞:Motuloc
+dg−→Motloc
+dg
+is symmetric monoidal. The induced morphism of derivators
+stab=LΣ∞: Motuloc
+dg−→Motloc
+dg
+is symmetric monoidal.
+Proof. — This is true by construction (see Proposition A.2).√
+Remark 7.17 . — In the construction ofMotloc
+dggiven in [ 42], the definition of Motloc
+dg
+wasHO(SpN(Motuloc
+dg)),i.e.it used non-symmetric spectra. However, thanks to The-
+orem6.1, both definitions agree up to a canonical equivalence of derivators .
+Remark 7.18 . — The concatenation of Propositions 7.12-7.16show us that the lo-
+calizing motivator Motloc
+dgcarries a symmetric monoidal structure −⊗L−, making
+the universal localizing invariant Uloc
+dgsymmetric monoidal. By Proposition A.2the
+associated symmetric monoidal structure preserves homotopy c olimits. Therefore, in
+order to conclude the proofof Theorem 7.5, it remains to show the universal property.
+LetDbe a strong triangulated derivator endowed with a symmetric monoid al struc-
+ture which preserves homotopy colimits in each variable. Thanks to T heorem7.2, we
+have an induced equivalence of categories
+(Uloc
+dg)∗:Hom!(Motloc
+dg,D)∼−→Homloc(HO(dgcat),D).
+This implies that the induced functor
+(7.1.3) (Uloc
+dg)∗:Hom⊗
+!(Motloc
+dg,D)−→Hom⊗
+loc(HO(dgcat),D).
+is faithful. More precisely, by Proposition 7.12, the category of filtered homotopy
+colimit preserving symmetric monoidal morphisms from HO(dgcat) toDis equivalent
+to the category of homotopy colimit preserving symmetric monoidal morphisms from
+LΣs/hatwidedgcatftoD. Using the universal properties of Bousfield localization and stabi-
+lization in the setting of derivators (see Theorem A.4and Corollary A.13), we can
+apply Proposition A.9, and Theorem A.15to conclude, by construction of Motloc
+dg,
+that (7.1.3) is an equivalence of categories. This ends the proof of Theorem 7.5.SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 35
+8. Applications
+In this section we describe several applications of Theorem 7.5.
+8.1. Non-connective K-theory. — Recall from Example 7.4that non-connective
+algebraic K-theory is an example of a localizing invariant of dg categories. In [ 11] the
+authors proved that this localizing invariant becomes co-represen tablein the category
+Motloc
+dg(e) of non-commutative motives (see Definition 7.3).
+Theorem 8.1 . —([11, Theorem 7.16] ) For every small dg category A, we have a
+natural isomorphism in the stable homotopy category of spec tra
+RHom(Uloc
+dg(k),Uloc
+dg(A))≃I K(A).
+Here,kdenotes the dg category with one object ∗such that k(∗,∗) =kin degree zero
+(see§2.1(i)), and I K(A)the non-connective algebraic K-theory spectrum of A. In
+particular, we obtain isomorphisms of abelian groups
+Hom(Uloc
+dg(k)[n],Uloc
+dg(A))≃I Kn(A)n∈Z.
+A fundamental problem of the theory of non-commutatives motive s is the compu-
+tation of the (spectra of) morphisms between two object in the loc alizing motivator.
+Using Theorem 7.5we give a partial solution to this fundamental problem.
+Theorem 8.2 . —LetBbe a saturated dg category; see Definition 4.1. For every
+small dg category A, we have a natural isomorphism in the stable homotopy catego ry
+of spectra
+RHom(Uloc
+dg(B),Uloc
+dg(A))≃I K(rep(B,A)).
+Hererep(−,−)denotes the internal Hom-functor in Hmo; see Theorem 2.16. In
+particular, we obtain isomorphisms of abelian groups
+Hom(Uloc
+dg(B)[n],Uloc
+dg(A))≃I Kn(rep(B,A))n∈Z.
+Proof. — The proof is a consequence of the following weak equivalences:
+RHom(Uloc
+dg(B),Uloc
+dg(A))≃RHom(Uloc
+dg(k)⊗LUloc
+dg(B),Uloc
+dg(A)) (8.1.1)
+≃RHom(Uloc
+dg(k),Uloc
+dg(B)∨⊗LUloc
+dg(A)) (8.1.2)
+≃RHom(Uloc
+dg(k),Uloc
+dg(Bop⊗LA)) (8.1.3)
+≃RHom(Uloc
+dg(k),Uloc
+dg(rep(B,A))) (8.1.4)
+≃I K(rep(B,A)). (8.1.5)
+Equivalence ( 8.1.1) follows from the fact that Uloc
+dg(k) is the unit object in Motloc
+dg(e);
+see Remark 2.12and Theorems 2.23and7.5. SinceBis a saturated dg category,
+Theorem 4.8implies thatBis a dualizable object in Hmo. Therefore, Equivalence36 D.-C. CISINSKI & G. TABUADA
+(8.1.2) follows from the fact that Uloc
+dg(B) is a dualizable object in Motloc
+dg(e) (see Re-
+mark4.7(iv) andTheorem 7.5)andfromthe adjunction( 4.0.15)ofRemark 4.7(ii) (see
+§A.9for its spectral enrichment). Equivalence ( 8.1.3) follows from Remark 4.7(iv),
+from Theorem 4.8, and from the fact that the universal localizing invariant is sym-
+metric monoidal. Equivalence ( 8.1.4) follows from Lemma 4.9. Finally, Equivalence
+(8.1.5) follows from Theorem 8.1.√
+Proposition 8.3 . —LetXandYbe smooth and proper k-schemes. Then, we have
+a natural isomorphism in the stable homotopy category of spe ctra
+RHom(Uloc
+dg(perf(X)),Uloc
+dg(perf(Y)))≃I K(X×Y).
+Here,I K(X×Y)denotes the non-connective algebraic K-theory spectrum of X×Y(see
+[41,§8]), andperf(−)the dg category constructed in Example 4.5(i). In particular,
+we obtain isomorphisms of abelian groups
+Hom(Uloc
+dg(perf(X))[n],Uloc
+dg(perf(Y)))≃I Kn(X×Y)n∈Z.
+Proof. — Since XandYare smooth and proper k-schemes, [ 51, Lemma 3.27] implies
+thatperf(X) andperf(Y) are saturated dg categories. Therefore, by Theorem 8.2we
+have a natural isomorphism in the stable homotopy category of spe ctra
+RHom(Uloc
+dg(perf(X)),Uloc
+dg(perf(Y)))≃I K(rep(perf(X),perf(Y))).
+Moreover, by [ 48, Theorem 8.9] we have a natural isomorphism
+perf(X×Y)≃rep(perf(X),perf(Y))
+inHmo. Finally, thanks to [ 41,§8 Theorem 5] we have a natural isomorphism
+I K(perf(X×Y))≃I K(X×Y)
+and so the proof is finished.√
+Remark 8.4 . — Let Zbe a noetherian regular scheme. Thanks to [ 1, Exp. I,
+Cor. 5.9 and Exp. II, Cor. 2.2.2.1] we have a derived Morita equivalenc e
+perf(Z)∼−→Db
+dg(Coh(Z)),
+where the left-hand side is the saturated dg category of Example 4.5(i) and the
+right-hand side is the bounded derived (dg) category of coherent sheaves on Z.
+SinceCoh(Z) is a noetherian abelian category (see [ 41,§10.1]), we conclude by [ 41,
+§10.1 Theorem 7] that
+I Kn(Z) =I Kn(perf(Z)) = 0 n <0.
+In particular, if in Proposition 8.3the base ring kis regular and noetherian, the
+negative stable homotopy groups of the spectrum I K(X×Y) vanish.SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 37
+8.2. Kontsevich’s non-commutative mixed motives. — Kontsevich intro-
+duced in [ 33,34,35] the category of non-commutative mixed motives. His
+construction decomposes in three steps:
+(1) First, consider the following category KPM k, enriched over symmetric spectra:
+the objects are the dualizable dg categories (see Definition 4.1); given saturated
+dgcategoriesAandB, the symmetric spectrum ofmorphismsfrom AtoBisthe
+non-connective K-theory spectrum I K(Aop⊗LB); the composition corresponds
+to the derived tensor product of bimodules(2).
+(2) Then, take the formal triangulated envelope tri(KPM k) of KPM k. Objects in
+this new category are formal finite extensions of formal shifts of objects in
+KPMk.
+(3) Finally, add formal direct summands for projectors in tri(KPM k). The resulting
+category KMM kis what Kontsevich named the category of non-commutative
+mixed motives(3).
+A precise wayto perform these constructions consists on seeing K MMkas the spectral
+category of perfect KPM k-modules (KMM kis the Morita completion of KPM k; see
+[46,§5.2] for a precise exposition of these constructions).
+Thanks to Theorem 8.2, we are now able to construct a fully-faithful embedding of
+KMMkinto ourcategoryMotloc
+dg(e) ofnon-commutativemotives, i.e.the basecategory
+of the localizing motivator. Note that, in contrast with Kontsevich’s ad hocdefini-
+tion, our category of non-commutative motives is defined purely in t erms of precise
+universal properties.
+LetMotloc
+dgbethe modelcategoryunderlyingthederivatorMotloc
+dg. As, byconstruc-
+tion,Motloc
+dgis a left Bousfield localization of a category of presheaves of symmet ric
+spectra over some small category, this model category is canonic ally enriched over
+symmetric spectra. We can thus consider the category Motloc
+dg(e) as a category en-
+richedoversymmetricspectra(byconsideringfibrantandcofibra ntobjectsinMotloc
+dg).
+Proposition 8.5 . —There is a natural fully-faithful embedding (enriched over sym-
+metric spectra) of Kontsevich’s category of non-commutati ve motives KMMkinto the
+category Motloc
+dg(e). The essential image is the thick triangulated subcategory spanned
+by motives of saturated dg categories.
+Proof. — Given saturated dg categories AandB, Lemma 4.9implies that we have a
+natural isomorphism in Hmo
+Aop⊗LB≃rep(A,B).
+(2)This category is the non-commutative (and derived) analogu e of Grothendieck’s category of pure
+motives: PM stands for Pure Motives, while K stands for both K ontsevich and K-theory.
+(3)MM stands for Mixed Motives, while K stands for both Kontsevi ch andK-theory.38 D.-C. CISINSKI & G. TABUADA
+Therefore, using Theorem 8.2we obtain a natural fully-faithful spectral functor
+KPMk−→Motloc
+dg(e)A/ma√sto→Uloc
+dg(A).
+By construction of KMM k, [46, Proposition 5.3.1] implies that this functor extends
+(uniquely) to a spectral functor
+KMMk−→Motloc
+dg(e).
+In order to show that this functor is (homotopically) fully-faithful, it is sufficient to
+prove that its restriction to a generating family of KMM kis fully faithful, which holds
+by construction.√
+8.3. Chern characters. — In [11] the authors used the co-representability Theo-
+rem8.1to classify all natural transformations out of non-connective K-theory. More
+precisely,theyprovedin[ 11, Theorem8.1]thatgivenalocalizinginvariant L, withval-
+ues in the derivators of spectra, the data of a natural transfor mationI K(−)⇒L(−)
+is equivalent to the datum of a single class in the stable homotopy grou pπ0L(k).
+From this result they obtained higher Chern characters (resp. hig her trace maps),
+from non-connective K-theoryto (topological) cyclic homology(resp. to (topological)
+Hochschild homology); see [ 11, Theorem 8.4].
+However, negative cyclic homology HC−and periodic cyclic homology HPdo not
+preserve filtered homotopy colimits since they are defined using infin ite products; see
+[38,§5.1]. Therefore, they are not examples of localizing invariants and so the theory
+developed in [ 11] is not directly applicable in these cases. Nevertheless, we shall
+explain below why and how negative cyclic homology and periodic cyclic ho mology
+fit naturally in our framework; see Examples 8.10and8.11.
+LetDbe a strong triangulated derivator endowed with a symmetric monoid al
+structure (with unit 1) which preserves homotopy colimits in each variable (see §A.5),
+and
+E:HO(dgcat)−→D
+asymmetricmonoidallocalizinginvariant(seeDefinition 7.6). ThankstoTheorem 7.5
+there is a (unique) symmetric monoidal homotopy colimit preserving m orphism of
+derivators Egmwhich makes the diagram
+HO(dgcat)E/d47/d47
+Uloc
+dg
+/d15/d15D
+Motloc
+dgEgm/d59/d59/d119/d119/d119/d119/d119/d119/d119/d119/d119/d119
+commute (up to unique 2-isomorphism).SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 39
+Definition 8.6 . — Themorphism Egmiscalledthe geometric realization ofE. Since
+by hypothesis Dis triangulated, we have a natural morphism of derivators
+RHom(1,−) :D−→HO(SpN) (see§A.9).
+The composed morphism
+Eabs:=RHom(1,Egm(−)) : Motloc
+dg−→HO(SpN)
+is called the absolute realization ofE.
+Given a symmetric monoidal localizing invariant E, we have two objects associated
+to a non-commutative motive M∈Motloc
+dg(e): its geometric realization Egm(M) and
+its absolute realization Eabs(M). Although the morphism Egmalways preserves ho-
+motopy colimits, this is not always the case for the morphism Eabs; a sufficient (and
+almost necessary) condition for Eabsto preserve homotopy colimits is that the unit 1
+ofDis a compact object.
+Proposition 8.7 . —The geometric realization of E induces a canonical Chern cha r-
+acter
+I K(−)⇒RHom(1,E(−))≃Eabs(Uloc
+dg(−)).
+Here,I K(−)and Eabs(Uloc
+dg(−))are two morphisms of derivators defined on HO(dgcat).
+Proof. — The geometric realization of Eis symmetric monoidal and so it sends the
+unit objectUloc
+dg(k) to1∈D. Therefore, given a small dg category A, we obtain an
+induced map
+I K(A)≃RHomMotloc
+dg(Uloc
+dg(k),Uloc
+dg(A))−→RHomD(1,Egm(Uloc
+dg(A))) =Eabs(Uloc
+dg(A)),
+where the left-hand side equivalence follows from Theorem 8.1. Since this induced
+map is functorial in A, the proof is finished.√
+Let us now give some examples which illustrate Proposition 8.7.
+Example 8.8 (Non-connective K-theory) . — The tautological version of the
+situation above is: for E=Uloc
+dg,Egmis by definition the identity of Motloc
+dg(see
+Theorem 7.5), whileEabs=I Kis non-connective K-theory (see Theorem 8.1). The
+corresponding Chern character is the identity, and this is in this pre cise sense that
+non-connective K-theory is initial among absolute homology theories.
+Example 8.9 (Hochschild homology) . — Takefor Ethe symmetricmonoidallo-
+calizing invariant
+HH:HO(dgcat)−→HO(C(k))
+ofExample 7.9. Inthiscase,thereisnodifference(uptotheDold-Kancorrespon dance
+relating complexes of k-modules and spectra) between the geometric and the absolute
+realization: if we consider HO(C(k)) as enriched over itself, then the morphism
+RHom(k,−) :HO(C(k))−→HO(C(k))40 D.-C. CISINSKI & G. TABUADA
+is (isomorphic to) the identity. Therefore by Proposition 8.7, we obtain a canonical
+Chern character
+I K(−)⇒HH(−).
+Example 8.10 (Negative cyclic homology) . — Take for Ethe symmetric
+monoidal localizing invariant
+C:HO(dgcat)−→HO(C(Λ))
+of Example 7.10. Given a small dg category A, we have an equivalence
+Cabs(Uloc
+dg(A)) =RHom(k,C(A))≃HC−(A),
+whereHC−(A) denotes the negative cyclic homology complex of A; see [32,§2.2].
+Therefore by Proposition 8.7, we obtain a canonical Chern character
+I K(−)⇒HC−(−).
+Example 8.11 (Periodic cyclic homology) . — Take for Ethe symmetric
+monoidal localizing invariant
+(P◦C) :HO(dgcat)−→HO(k[u]-Comod)
+of Example 7.11. Assuming that the ground ring kis a field, for any small dg category
+A, we have a natural identification
+(P◦C)abs(Uloc
+dg(A)) =RHom(k[u],(P◦C)(A))≃HP(A),
+whereHP(A) denotes the periodic cyclic homology complex of A. This can be seen
+as follows. Given a mixed complex M(see Example 7.10), a map k[u]→P(M) in
+k[u]-Comod corresponds to a collection of maps k→(M⊗L
+Λk)[2n],n≥0, inC(k)
+which are compatible with the operator S. In other words, these data correspond to
+a map inC(k) fromkto the tower
+···S−→(M⊗L
+Λk)[−2n]S−→(M⊗L
+Λk)[−2n+2]S−→···S−→(M⊗L
+Λk)[−2]S−→M⊗L
+Λk.
+In other words, we have:
+RHom(k[u],P(M))≃holim
+n(M⊗L
+Λk)[−2n].
+The Milnor short exact sequence [ 24, Proposition 7.3.2] applied to this homotopy
+limit corresponds to the short exact sequence
+0−→lim←−n1Hi+2n−1(M⊗L
+Λk)−→Hom(k[u],P(M)[−i])−→lim←−nHi+2n(M⊗L
+Λk)−→0.
+Now, letAbe a small dg category. The above arguments, with M=C(A), allow us
+to deduce the formula
+RHom(k[u],(P◦C)(A))≃holim
+n(C(A)⊗L
+Λk)[−2n]≃HP(A).
+Therefore, by Proposition 8.7, we obtain a canonical Chern character
+I K(−)⇒HP(−).SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 41
+8.4. To¨ en’s secondary K-theory. — To¨ enintroduced in [ 49,50] a“categorified”
+version of algebraic K-theory named secondary K-theory; see [52] for a survey article.
+Definition 8.12 ([49,§5.4]). — Given a commutative ring k, letZ[Hmosat,k] be the
+freeabeliangroupontheisomorphismclassesofobjectsin Hmosat,k(seeNotation 4.3).
+Thesecondary K-theory group K(2)
+0(k)ofkis the quotient of Z[Hmosat,k] by the
+relations [B] = [A]+[C] associated to exact sequences (see Definition 7.1)
+A−→B−→C
+of saturated dg categories.
+Remark 8.13 . — (i) ThankstoTheorem 4.8the category Hmosat,kcoincideswith
+the category of dualizable objects in Hmok. Therefore, by Remark 4.7(iii), the
+derived tensor product in Hmokrestricts to a bifunctor
+−⊗L−:Hmosat,k×Hmosat,k−→Hmosat,k.
+By [16, Proposition 1.6.3] the derived tensor product preserves exact s equences
+(in both variables), and so we obtain a commutative ring structure o nK(2)
+0(k).
+(ii) Given a ring homomorphism k→k′, we have a derived base change functor
+−⊗L
+kk′:Hmok−→Hmok′A/ma√sto→A⊗L
+kk′.
+This functor preserves exact sequences and is symmetric monoida l. Therefore,
+by Theorem 4.8and Remark 4.7(iv), we obtain a ring homomorphism
+K(2)
+0(k)−→K(2)
+0(k′).
+In conclusion, secondary K-theory is a functor K(2)
+0(−) from the category of com-
+mutative rings to itself.
+One of the motivations for the study of this secondary K-theory was its expected
+connectionwithanhypotheticalGrothendieckringofmotivesinthe non-commutative
+setting; see [ 50, page 1]. Thanks to Theorem 7.5, we are now able to make this
+connection precise; see Remarks 8.17-8.19. We shall use the following well known
+property of dualizable objects in a triangulated category.
+Proposition 8.14 . —LetCbe a closed symmetric monoidal triangulated category.
+Then, the category C∨of dualizable objects in C(see Definition 4.6) is a symmetric
+monoidal thick triangulated subcategory of C.
+Proof. — The fact that dualizable objects are stable under tensor produ ct is clear;
+see Remark 4.7(iii). For an object XinC, setX∨=Hom(X,1). Given two objects
+XandYinC, we have a canonical map
+uX,Y:X∨⊗Y−→Hom(X,Y)
+which corresponds by adjunction to the map X∨⊗X⊗Y→Yobtained by tensoring
+Ywith the evaluation map X⊗X∨→1. The object Xis dualizable if and only if42 D.-C. CISINSKI & G. TABUADA
+the map uX,Yis invertible for any object Y. Since for any fixed object Ythe map
+uX,Yis a natural transformation of triangulated functors, we conclud e thatC∨is a
+thick triangulated subcategory of C.√
+Notation 8.15 . — Thanks to Theorem 7.5the localizing motivator carries a
+symmetric monoidal structure, and so its base category Motloc
+dg,k(e) is a symmet-
+ric monoidal triangulated category. Therefore by Proposition 8.14, the category
+Motloc
+dg,k(e)∨of dualizable objects is a symmetric monoidal thick triangulated subc at-
+egory of Motloc
+dg,k(e).
+Let KMM kbe Kontsevich’s category of non-commutative mixed motives. Than ks
+to Proposition 8.5, we can identify it with the thick triangulated subcategory of
+Motloc
+dg,k(e)∨generated by objects of shape Uloc
+dg(A), whereAruns over the family
+of saturated dg categories over k. Therefore, KMM kis naturally a rigid symmetric
+monoidal triangulated category.
+Definition 8.16 . — Let kbe a commutative ring. The Grothendieck ring K0(k)of
+non-commutative motives over kis the Grothendieck ring K0(KMM k).
+Remark 8.17 (Non-triviality) . — Recall from Example 7.9the construction of
+the symmetric monoidal localizing invariant
+HH:HO(dgcat)−→HO(C(k)).
+By restricting its geometric realization (see Definition 8.6) to the base category, we
+obtain a symmetric monoidal triangulated functor
+HHgm(e) : Motloc
+dg,k(e)−→HO(C(k))(e) =D(k).
+Recall that the dualizable objects in D(k) are precisely the perfect complexes of k-
+modules. Therefore, by Remark 4.7(iv),HHgm(e) sends dualizable objects to perfect
+complexes and so it induces a ring homomorphism
+(8.4.1) rK0:K0(k) =K0(KMM k)−→K0(Dc(k)) =K0(k).
+Finally, since K0(k) is non-trivial we conclude that K0(k) is also non-trivial.
+Remark 8.18 (Functoriality) . — Given a ring homomorphism k→k′, we have a
+base change functor
+−⊗kk′:dgcatk−→dgcatk′A/ma√sto→A⊗ kk′.
+This functor gives rise to a morphism of derivators
+−⊗L
+kk′:HO(dgcatk)−→HO(dgcatk′),
+which is symmetric monoidal, preserves homotopy colimits (and the po int), and sat-
+isfies localization; see Theorem 7.2. Therefore, the composition
+HO(dgcatk)−⊗L
+kk′
+−→HO(dgcatk′)Uloc
+dg−→Motloc
+dg,k′SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 43
+is a symmetric monoidal localizing invariant; see 7.6. Using Theorem 7.5, we obtain
+a (unique) symmetric monoidal morphism, which we still denoted by −⊗L
+kk′, making
+the diagram
+HO(dgcatk)
+Uloc
+dg/d15/d15−⊗L
+kk′
+/d47/d47HO(dgcatk′)
+Uloc
+dg/d15/d15
+Motloc
+dg,k−⊗L
+kk′/d47/d47Motloc
+dg,k′(8.4.2)
+commute (up to 2-isomorphism). By restricting ourselves to the ba se categories, we
+have a symmetric monoidal triangulated functor
+−⊗L
+kk′: Motloc
+dg,k(e)−→Motloc
+dg,k′(e).
+As the (derived) changeofscalarsfunctor preservessaturate ddg categories,we obtain
+then an induced ring homomorphism
+K0(k)−→K 0(k′).
+In conclusion, the Grothendieck ring of non-commutative motives is a functorK0(−)
+from the category of commutative rings to itself.
+Remark 8.19 (Connection) . — Thanks to Theorem 7.5, the functor
+Uloc
+dg:Hmok−→Motloc
+dg,k(e)
+is symmetric monoidal. By construction it sends exact sequences to distinguished
+triangles, and so it induces a ring homomorphism
+(8.4.3) Φ( k) :K(2)
+0(k)−→K 0(k).
+Note that this ring homomorphism is not necessarily surjective beca use of step (3)
+in the construction of KMM k. However, the image of Φ( k) can be described as
+the Grothendieck group of the triangulated category tri(KPM k): by cofinality the
+Grothendieck ring K0(tri(KPM k)) is a subring of K0(k) and by d´ evissage Φ( k) sur-
+jects on K0(tri(KPM k)). Moreover, the above (up to 2-isomorphism) commutative
+square (8.4.2) shows us that the ring homomorphism ( 8.4.3) gives rise to a natural
+transformation of functors
+K(2)
+0(−)⇒K0(−), k/ma√sto→Φ(k).
+Now, let Rbe a commutative ring and l:K(2)
+0(k)→Rarealization of K(2)
+0(k),
+i.e.a ring homomorphism. Note that if there exists a symmetric monoidal localizing
+invariant
+HO(dgcatk)−→D,
+whose induced ring homomorphism (see Proposition 8.14)
+K(2)
+0(k)−→K0(D(e)∨)44 D.-C. CISINSKI & G. TABUADA
+identifies with l, thenlfactors through Φ( k). An interesting example is proved by
+To¨ en’s rank map (see [ 49,§5.4])
+rk0:K(2)
+0−→K0(k).
+Thanks to [ 49,§5.4 Lemma 3] this rank map is induced from the symmetric monoidal
+localizing invariant
+HH:HO(dgcatk)−→HO(C(k))
+of Example 7.9. Therefore, it corresponds to the following composition
+K(2)
+0(k)Φ(k)−→K 0(k)rK0−→K0(k),
+whererK0is the ring homomorphism ( 8.4.1) of Remark 8.17.
+8.5. Euler characteristic. —
+Definition 8.20 . — LetCbe a symmetric monoidal category with monoidal prod-
+uct⊗and unit object 1. Given a dualizable object XinC(see Definition 4.6) its
+Euler characteristic χ(X) is the following composition
+χ(X) :1δ−→X∨⊗Xτ−→X⊗X∨ev−→1,
+whereτdenotes the symmetry isomorphism.
+Remark 8.21 . — LetCisawellbehavedsymmetricmonoidaltriangulatedcategory;
+e.g.C=D(e) for some symmetric monoidal triangulated derivator D. Then, thanks
+to [37, Theorem 1.9], the Euler characteristic gives rise to a ring homomorp hism
+χ:K0(C∨)−→HomC(1,1).
+Proposition 8.22 . —LetAbe a saturated dg category. Then its Euler characteris-
+ticχ(A)inHmois the isomorphism class of Dc(k)which is associated to the (perfect)
+Hochschild homology complex HH (A)ofA(see Example 7.9).
+Proof. — From Theorem 4.8(and its proof) we see that the dual of Ais its opposite
+dg categoryAop, and that the following composition in Hmo
+χ(A) :k[A(−,−)]−→ Aop⊗LAτ−→A⊗LAop[A(−,−)]−→k
+corresponds to the complex
+A(−,−)L/circlemultiplydisplay
+Aop⊗LAA(−,−).
+By [38, Proposition1.1.13]this complex of k-modules computes Hochschild homology
+ofA(with coefficients in itself), which achieves the proof.√SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 45
+Proposition 8.23 . —LetF:C →C′be a symmetric monoidal functor between
+symmetric monoidal categories with unit objects 1and1′. Then, given a dualizable
+objectXinC, the Euler characteristic χ(F(X))ofF(X)agrees with F(χ(X))on
+1′≃F(1).
+Proof. — It is a straightforward consequence of the definitions. The det ails are left
+as an exercise for the reader.√
+Proposition 8.24 . —LetAbe a saturated dg category. Then, χ(Uloc
+dg(A))is the el-
+ement of the Grothendieck group K0(k)which is associated to the (perfect) Hochschild
+homology complex HH (A)ofA.
+Proof. — Thanks to Theorem 7.5, the universal localizing invariant Uloc
+dgis symmetric
+monoidal. Using Theorem 8.1, we see that the map
+Uloc
+dg(e) : IsoDc(k)≃HomHmo(k,k)−→Hom(Uloc
+dg(k),Uloc
+dg(k))≃K0(k)
+sends an element in Iso Dc(k) to the corresponding class in the Grothendieck group
+K0(Dc(k))≃K0(k). Hence, Theorem 4.8and Proposition 8.23achieve the proof.√
+Example 8.25 . — Recall from Example 4.5that, given a smooth and proper k-
+schemeX, we have a saturated dg category perf(X) which enhances the category of
+compact objects in Dqcoh(X). Thanks to Keller [ 31,32] the Hochschild homology of
+perf(X) (see Example 7.9) agrees with the Hochschild homology of Xin the sense of
+Weibel [54]. Therefore, by Proposition 8.24the Euler characteristic of Uloc
+dg(perf(X))
+is the element of the Grothendieck group K0(k) which is associated to the (perfect)
+Hochschild homology complex HH(X) ofX.
+Whenkis the field of complex numbers, the Grothendieck ring K0(C) is naturally
+isomorphic to Zand the Hochschild homology of Xagrees with the Hodge cohomol-
+ogyH∗(X,Ω∗
+X) ofX. Therefore, when we work over C, the Euler characteristic of
+Uloc
+dg(perf(X)) is the classical Euler characteristic of X.
+A
+Grothendieck derivators
+TheoriginalreferenceforthetheoryofderivatorsisGrothendie ck’smanuscript[ 28]
+and Heller’s monograph [ 21]. See also [ 7,9,12,42].
+A.1. Prederivators. — Aprederivator Dconsists of a strict contravariant 2-
+functor from the 2-category of small categories to the 2-categ ory of categories
+D:Catop−→CAT.
+Prederivatorsorganizethemselvesnaturallyina2-category: the 1-morphisms(usually
+called morphisms) are the pseudo natural transformations and th e 2-morphisms are46 D.-C. CISINSKI & G. TABUADA
+the modifications; see [ 12,§5] for details. Given prederivators DandD′, we denote
+byHom(D,D′) the category of morphisms.
+Given a category M, we denote byMthe prederivator defined for every small
+category Xby
+M(X) :=Fun(Xop,M),
+whereFun(Xop,M) is the category of presheaves on Xwith values inM. IfWis a
+class of morphisms in M, we denote byM[W−1] the prederivator defined for every
+small category Xby
+M[W−1](X) :=Fun(Xop,M)[W−1].
+HereFun(Xop,M)[W−1] is the localization of Fun(Xop,M) with respect to the class
+of morphism which belong termwise to W. Note that the assignment ( M,W)/ma√sto→
+M[W−1] is 2-functorial ,i.e.given a natural transformation
+(M,W)F/d42/d42
+G/d52/d52⇓(N,V)
+between functors, such that F(W)⊂(V) andG(W)⊂(V), we obtain an induced
+2-morphism
+M[W−1]F/d42/d42
+G/d52/d52⇓N[V−1]
+of prederivators.
+A.2. Derivators. — Aderivator is a prederivator which is subject to certain con-
+ditions, the main ones being that for any functor u:X→Ybetween small categories,
+theinverse image functor
+u∗=D(u) :D(Y)−→D(X)
+has a left adjoint, called the homological direct image functor ,
+u!:D(X)−→D(Y),
+as well as right adjoint, called the cohomological direct image functor
+u∗:D(X)−→D(Y).
+See [7] for details. Similarly to the case of prederivators, derivators org anize them-
+selves in a 2-category. Given derivators DandD′, we denote by Homflt(D,D′) the
+category of morphisms of derivators which preserve filtered homo topy colimits, and
+byHom!(D,D′) the category of morphisms of derivators which commute with all ho -
+motopy colimits; see [ 7,9].SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 47
+The essential example of a derivator to keep in mind is the derivator D=HO(M)
+associated to a (complete and cocomplete) Quillen model category M(see [7, Theo-
+rem 6.11]), which is defined for every small category Xby
+HO(M)(X) :=Ho/parenleftbig
+Fun(Xop,M)/parenrightbig
+.
+In this case, any colimit (resp. limit) preserving left (resp. right) Qu illen functor
+induces a morphism of derivators which preserves homotopy colimits (resp. limits);
+see [7, Proposition 6.12].
+Finally, we denote by ethe 1-point category with one object and one (identity)
+morphism. Heuristically, the category D(e) is the basic “derived” category under
+consideration in the derivator D. For instance, if D=HO(M) thenD(e) =Ho(M) is
+the usual homotopy category of M.
+A.3. Properties. —
+(i) A derivator Dis called strongif for every finite free category Xand every
+small category Y, the natural functor D(X×Y)→Fun(Xop,D(Y)) is full and
+essentially surjective.
+(ii) A derivator Dis called regularif sequential homotopy colimits commute with
+finite products and homotopy pullbacks.
+(iii) A derivator Dis called pointedif for any closed immersion i:Z→XinCat
+the cohomological direct image functor i∗:D(Z)→D(X) has a right adjoint,
+and if, dually, for any open immersion j:U→Xthe homological direct image
+functorj!:D(U)→D(X) has a left adjoint; see [ 12, Definition 1.13].
+(iv) A derivator Dis called triangulated orstableif it is pointed and if every global
+commutative square is cartesian exactly when it is cocartesian; see [12, Defini-
+tion 1.15].
+A strong derivator is the same thing as a small homotopy theory in th e sense of
+Heller [22]. Thanks to [ 10, Proposition 2.15], if Mis a Quillen model category
+its associated derivator HO(M) is strong. Moreover, if sequential homotopy colimits
+commute with finite products and homotopy pullbacks in M, the associated derivator
+HO(M) is regular. Notice that if Mis pointed, then the derivator HO(M) is pointed.
+Finally, a pointed Quillen model category Mis stable if and only if its associated
+derivator HO(M) is triangulated.
+A.4. Kan extensions. — Given a small category A, we denote by HotA=HO(s/hatwideA)
+the derivator associated to the projective model category stru cture on the category
+of simplicial presehaves. We then have a Yoneda embedding
+(A.4.1) h:A−→HotA.
+LetDbe a derivator. A 2-functorial version of the Yoneda lemma gives a c anonical
+equivalence of categories
+Hom(A,D)≃D(Aop).48 D.-C. CISINSKI & G. TABUADA
+Theorem A.1 . —The morphism of prederivators (A.4.1)is the universal morphism
+fromAto a derivator. In other words, given any derivator D, the induced functor
+h∗:Hom!(HotA,D)∼−→Hom(A,D)
+is an equivalence of categories.
+Proof. — See [ 9, Corollaire 3.26].√
+A.5. Monoidal structures. — Thanks to [ 9, Proposition5.2] the 2-category of
+prederivators form a closed symmetric monoidal 2-category with r espect to the carte-
+sian product. Given two prederivators DandD′, we denote by Hom(D,D′) the
+corresponding internal Hom; see [ 9,§5.1].
+Given a prederivator D, by asymmetric monoidal structure on Dwe mean a struc-
+ture of symmetric pseudo monoid on D. In other words, for every small category X,
+D(X) is a symmetric monoidal category, and for every functor u:X→Ybetween
+small categories, the inverse image functor
+u∗:D(u) :D(Y)−→D(X)
+is symmetric monoidal; see [ 9,§5.4]. Asymmetric monoidal prederivator is a pred-
+erivator endowed with a symmetric monoidal structure. Given symm etric monoidal
+prederivators DandD′, we denote by Hom⊗(D,D′) the category of symmetric
+monoidal morphisms; see [ 9,§5.11] for details. A symmetric monoidal derivator is
+a symmetric monoidal prederivator Dwhich is also a derivator, and such that the
+tensor product preserves homotopy colimits in each variable ,i.e.sucht that, for any
+objectX∈D(e) the induced morphism
+X⊗−:D−→D
+preserves homotopy colimits.
+Given symmetric monoidal derivators DandD′, we denote by Hom⊗
+!(D,D′) the
+category of symmetric monoidal morphisms which preserve homoto py colimits.
+A basic example of a symmetric monoidal prederivator is given as follow s: letM
+be a symmetric monoidal category (with monoidal product −⊗−) andWa class
+of morphisms in M. If the monoidal product preserves the class W,i.e.if we have
+an inclusionW⊗W ⊆W , then the prederivator M[W−1] of§A.1is naturally a
+symmetric monoidal prederivator. Moreover, if F: (M,W)→(N,V) is a symmetric
+monoidal functor such that F(W)⊂V, then the induced morphism
+M[W−1]−→N[V−1]
+is symmetric monoidal.
+As for examples of symmetric monoidal derivators, most of them ar e obtained from
+symmetric monoidal model categories [ 24, Definition 4.2.6].SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 49
+Proposition A.2 . —LetMbe a symmetric monoidal model category. Then its as-
+sociated derivator HO(M)carries a symmetric monoidal structure. Moreover, any
+symmetric monoidal left Quillen functor between symmetric monoidal model cate-
+gories induces a symmetric monoidal morphism between the as sociated derivators.
+Proof. — See [ 9, Proposition 6.1].√
+A.6. Derived Day convolution product. — LetAbe a small symmetric
+monoidal category. Then Ais a symmetric monoidal prederivator. Moreover, as s/hatwideA
+is then a symmetric monoidal model category (see Theorem 5.2), the derivator HotA
+is then a symmetric monoidal derivator, in such a way that the Yoned a embedding
+(A.4.1) is a symmetric monoidal morphism of prederivators. Given a symmet ric
+monoidal derivator D, we thus have an induced functor
+(A.6.1) h∗:Hom⊗
+!(HotA,D)−→Hom⊗(A,D).
+Theorem A.3 . —The functor (A.6.1)is an equivalence of categories.
+Proof. — This is simply a variation on Theorem A.1. Given two derivators D′and
+D′′, the derivator Hom!(D′,D′′) is defined by
+Hom!(D′,D′′)(A) =Hom!(D′,D′′
+A),
+whereD′′
+Ais in turn the derivator of “presheaves on Awith values in A”,i.e.the
+derivator defined by
+D′′
+A(X) =D′′(A×X).
+For a third derivator D, the data of a morphism of prederivators
+D×D′−→D′′
+which preserves homotopy colimits in each variable is equivalent to the data of an
+object in Hom!(D,Hom!(D′,D′′)); see [9, Lemme 5.18]. Moreover, when D′is of the
+formHotB, withBa small category, it follows immediately from Theorem A.1that
+we have equivalences of derivators
+Hom!(HotB,D′′)≃Hom(B,D′′)≃D′′
+Bop.
+Hence, if Ais another small category, we obtain canonical equivalences of cat egories:
+Hom!(HotA,Hom!(HotB,D′′))≃Hom(A,Hom!(HotB,D′′))
+≃Hom(A,Hom(B,D′′))
+≃Hom(A×B,D′′) =D′′(Aop×Bop).
+ForA=BandD′′=HotA, we note that the tensor product on HotAcorresponds, un-
+der these equivalences of categories, to the tensor product ⊗:A×A−→Acomposed
+with the Yoneda embeding ( A.4.1).
+More generally, we obtain (by induction on n≥0) that for any n-tuple of small
+categories ( A1,...,A n), the category of morphisms HotA1×···×HotAn−→D′′50 D.-C. CISINSKI & G. TABUADA
+which preserve homotopy colimits in each variable is canonically equivale nt to the
+category of morphisms A1×···×An−→D′′. This fact implies that the symmetric
+monoidal structure on Aextends uniquely to a symmetric monoidal structure on the
+derivator HotA. Moreover,the categoryof symmetric monoidal morphisms from HotA
+toD′′, which preserve homotopy colimits, is canonically equivalent to the ca tegory of
+symmetric monoidal morphisms from AtoD′′.√
+A.7. Left Bousfield localization. — LetDbe a derivator and Sa class of mor-
+phisms in the base category D(e). We say that the derivator Dadmits a left Bousfield
+localization with respect to the class S, if there exists a morphism of derivators
+γ:D−→LSD,
+which commutes with homotopy colimits, sends the elements of Sto isomorphisms
+inLSD(e), and satisfies the following universal property: given any derivat orD′, the
+morphism γinduces an equivalence of categories
+γ∗:Hom!(LSD,D′)∼−→Hom!,S(D,D′),
+whereHom!,S(D,D′) denotes the category of morphisms of derivators which commute
+with homotopy colimits and send the elements of Sto isomorphisms in D′(e).
+Theorem A.4 . —LetMbe a left proper cellular model category and Sa set of
+maps in the homotopy category Ho(M)ofM. Consider the left Bousfield localization
+LSMofMwith respect to the set S,i.e.to perform the localization we choose in
+Ma representative for each element of S. Then, the induced morphism of derivators
+HO(M)→HO(LSM)is a left Bousfield localization of the derivator HO(M)with
+respect to the set S. Moreover, we have a natural adjunction of derivators:
+HO(M)
+/d15/d15
+HO(LSM)./d79/d79
+Proof. — See [ 42, Theorem 4.4].√
+Remark A.5 . — If the domains and codomains of the elements of the set Sare
+homotopically finitely presented (see Definition 3.2), the morphism HO(LSM)→
+HO(M) (right adjoint to the localizing functor) preserves filtered homot opy colimits.
+Therefore, under these hypothesis, if HO(M) is regular so it is HO(LSM).
+By [42, Lemma 4.3], the Bousfield localization LSDof atriangulated derivator D
+remains triangulated as long as Sis stable under the loop space functor. For more
+generalS, to remain in the world of triangulated derivators, one has to localize with
+respect to the set Ω( S) generated by Sand loops, as follows.SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 51
+Proposition A.6 . —LetDbe a triangulated derivator and Sa class of morphisms
+inD(e). Let us denote by Ω(S)the smallest class of morphisms in D(e)which con-
+tainsSand is stable under the loop space functor Ω :D(e)→D(e). Then for any
+triangulated derivator T, we have an equality of categories
+Hom!,Ω(S)(D,T) =Hom!,S(D,T).
+As a consequence, whenever LΩ(S)Dexists, this is the triangulated left Bousfied local-
+ization of Dwith respect to S.
+Proof. — For Fan element of Hom!(D,T), the functor F(e) :D(e)→T(e) commutes
+with homotopy colimits, hence it commutes in particular with the suspe nsion functor.
+Since both DandTare triangulated, suspension and loop space functors are inverse
+to each other. Hence F(e) also commutes with Ω. It is then obvious that F(e) sends
+Sto isomorphisms if and only if it does so with Ω( S).√
+Theorem A.7 (Dugger) . —LetMbe a combinatorial model category. Then, there
+exists a small category Aand a small set of maps SinHotA(e) =Ho(s/hatwideA), such that
+HO(M)is equivalent to LSHotA.
+Proof. — This follows from [ 14, Proposition 3.3] and from Theorem A.4applied to
+the projective model structure on the category of simplicial pres heaves of a small
+category.√
+Remark A.8 . — It follows immediately from Theorem A.7that the statement of
+Theorem A.4holds also for left proper combinatoriel model categories. In part icular,
+any derivator which is equivalent to a derivator associated to a comb inatorial model
+category admits a left Bousfield localization with respect to any small set of maps.
+Proposition A.9 . —LetDbe a symmetric monoidal derivator, and Sa class of
+maps in D(e). Assume that Sis closed under tensor product in D, and that the left
+Bousfield localization of DbySexists. Then, LSDis symmetric monoidal, and the
+localization morphism γ:D−→LSDis symmetric monoidal. Moreover, given any
+symmetric monoidal derivator D′, the induced functor
+γ∗:Hom⊗
+!(LSD,D′)−→Hom⊗
+!(D,D′)
+is fully-faithful, and its essential image consists of the s ymmetric monoidal homotopy
+colimit preserving morphisms which send Sto isomorphisms.
+Proof. — This is an immediate consequence of the universal property of LSD. The
+details are left as an exercise for the reader.√52 D.-C. CISINSKI & G. TABUADA
+A.8. Stabilization. — LetDbe a derivator. Then there is a universal pointed
+derivator D→D•: given any pointed derivator D′, the induced functor
+Hom!(D•,D′)∼−→Hom!(D,D′)
+is an equivalence of categories; see [ 9, Corollaire 4.19]. Using the explicit construction
+ofD•(see [9,§4.5]) it is easy to see that when D•is strong (resp. regular) so is D. If
+D=HO(M) for some model category M, thenD•is equivalent to HO(M•), where
+M•denotes the model category of pointed objects in M; see [24, Proposition 1.1.8].
+LetDbe a pointed derivator. A stabilization ofDis a homotopy colimit preserving
+morphism stab:D→St(D), withSt(D) a triangulated strong derivator, which is
+universal for these properties: given any triangulated strong de rivatorT, the induced
+functor
+stab∗:Hom!(St(D),T)∼−→Hom!(D,T).
+is an equivalence of categories.
+Theorem A.10 (Heller [22]). —Any pointed regular strong derivator admits a
+stabilization.
+Given a pointed simplicial model category M, the second named author compared
+in [42,§8] the derivator associated to the model category SpN(M) ofS1-spectra on
+Mwith the stabilization of the derivator associated to the model cate goryM.
+Proposition A.11 . —LetMbe a pointed, simplicial, left proper, cellular model
+category. Assume that sequential homotopy colimits commut e with finite products and
+homotopy pullbacks. Then, the induced morphism of triangul ated strong derivators
+St(HO(M))∼−→HO(SpN(M))
+is an equivalence.
+Proof. — See [ 42, Theorem 8.7].√
+LetDbe a pointed strong derivator (see §A.3) andSbe a class of morphisms in
+D(e). Assume that Dadmits a left Bousfield localization LSDwith respect to S; see
+§A.7. Assume also that the stablization St(D) ofDexists.
+We then have two homotopy colimit preserving morphisms:
+LSDγ←−−Dstab−−→St(D).
+By examining the relevant universal properties, we obtain the follow ing result.
+Proposition A.12 . —Under the above assumptions, the derivator LΩ(stab(S))St(D)
+exists if and only if the derivator St(LSD)exists. Moreover, if this is the case, then
+LΩ(stab(S))St(D)≃St(LSD)underD.SYMMETRIC MONOIDAL STRUCTURE ON NON-COMMUTATIVE MOTIVES 53
+Corollary A.13 . —LetMbe a pointed left proper combinatorial model category.
+Then the stabilization St(HO(M))ofHO(M)exists and is equivalent to the derivator
+associated to a stable combinatorial model category.
+Proof. — Thanks to Theorem A.7we may assume that HO(M)≃LSHotA,•. Using
+Proposition A.12and Theorem A.4, we see it is sufficient to treat the case where Sis
+the empty set. Proposition A.11allow us then to conclude the proof.√
+Remark A.14 . — A careful analysis of the proof of the Corollary A.13will lead
+to a proof of Proposition A.11for any simplicial combinatorial model category M.
+Note that since any combinatorial model categoryis equivalent to a simplicial one, we
+conclude the existence of stabilizations for any derivatorassociat ed to a combinatorial
+model category; however, we will not need this level of generality.
+Theorem A.15 . —LetAbe a small symmetric monoidal category. Then there is a
+unique symmetric monoidal structure on the triangulated de rivatorSt(HotA,•)whose
+tensor product preserves homotopy colimits in each variabl es, such that the composed
+morphism
+A−→HotA−→St(HotA,•)
+is symmetric monoidal. Moreover, given any strong triangul ated derivator D, the
+induced functor
+Hom⊗
+!(St(HotA,•),D)∼−→Hom⊗(A,D)
+is an equivalence of categories.
+Proof. — It follows immediately from Theorem A.1and from the universal property
+ofSt(HotA,•) that, for any small category Aand any strong triangulated derivator D,
+we have canonical equivalences of categories:
+Hom!(St(HotA,•),D)≃Hom(A,D)≃D(Aop).
+Starting from this point on, the proof of Theorem A.3holds here mutatis mutandis .√
+Remark A.16 . — The derivator St(HotA,•) is equivalent to the derivator associ-
+ated to the model category of symmetric spectra in the category of pointed simpli-
+cial presheaves on A. This equivalence defines a symmetric monoidal structure on
+St(HotA,•), making the morphism
+A−→HotA−→St(HotA,•)
+symmetric monoidal; see Proposition 5.3, Theorem 6.1, and Proposition A.11. This
+monoidal structure coincides with the one of Theorem A.15, thanks to the uniqueness
+of the latter.54 D.-C. CISINSKI & G. TABUADA
+A.9. Spectral enrichment. — Recall from [ 11, Appendix A.3] that any triangu-
+lated derivator Dis canonically enriched over spectra, i.e.we have a morphism of
+derivators
+RHom(−,−) :Dop×D−→HO(SpN).
+Moreover, this enrichment over spectra is compatible with adjunct ions: given an
+adjunction
+D′
+Ψ
+/d15/d15
+DΦ/d79/d79
+we have a canonical isomorphism in the stable homotopy category of spectra:
+RHomD′(ΦX,Y)≃RHomD(X,ΨY)X∈D, Y∈D′.
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+Url :http://www-math.univ-paris13.fr/~cisinski/
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+Portugal •E-mail : tabuada@fct.unl.pt
\ No newline at end of file
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+arXiv:1001.0229v2  [astro-ph.SR]  22 Jan 2010Accepted by ApJ Letters
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+SOLAR-LIKE OSCILLATIONS IN LOW-LUMINOSITY RED GIANTS: FIR ST RESULTS FROM KEPLER
+T. R. Bedding,1D. Huber,1D. Stello,1Y. P. Elsworth,2S. Hekker,2T. Kallinger,3,4S. Mathur,5B. Mosser,6
+H. L. Preston,7,8J. Ballot,9C. Barban,6A. M. Broomhall,2D. L. Buzasi,7W. J. Chaplin,2R. A. Garc ´ıa,10
+M. Gruberbauer,11S. J. Hale,2J. De Ridder,12S. Frandsen,13W. J. Borucki,14T. Brown,15
+J. Christensen-Dalsgaard,13R. L. Gilliland,16J. M. Jenkins,17H. Kjeldsen,13D. Koch,14K. Belkacem,18
+L. Bildsten,19H. Bruntt,6,13T. L. Campante,13,20S. Deheuvels,6A. Derekas,1,23M.-A. Dupret,18M.-J. Goupil,6
+A. Hatzes,21G. Houdek,4M. J. Ireland,1C. Jiang,22C. Karoff,2L. L. Kiss,1,23Y. Lebreton,6A. Miglio,18
+J. Montalb ´an,18A. Noels,18I. W. Roxburgh,24V. Sangaralingam,2I. R. Stevens,2M. D. Suran,25
+N. J. Tarrant,2and A. Weiss26
+Accepted by ApJ Letters
+ABSTRACT
+We have measured solar-like oscillations in red giants using time-series photometry from the first 34
+days of science operations of the Kepler Mission . The light curves, obtained with 30-minute sampling,
+revealclearoscillationsinalargesampleofGandKgiants,extendingin luminosityfromtheredclump
+down to the bottom of the giant branch. We confirm a strong corre lationbetween the large separation
+of the oscillations (∆ ν) and the frequency of maximum power ( νmax). We focus on a sample of 50
+low-luminosity stars ( νmax>100µHz,L/lessorsimilar30L⊙) having high signal-to-noise ratios and showing the
+unambiguous signature of solar-like oscillations. These are H-shell-b urning stars, whose oscillations
+should be valuable for testing models of stellar evolution and for cons training the star-formation rate
+in the local disk. We use a new technique to compare stars on a single´ echelle diagram by scaling their
+frequencies and find well-defined ridges corresponding to radial an d non-radial oscillations, including
+clear evidence for modes with angular degree l= 3. Measuring the small separation between l= 0
+andl= 2 allows us to plot the so-called C-D diagram of δν02versus ∆ ν. The small separation δν01of
+l= 1 from the midpoint of adjacent l= 0 modes is negative, contrary to the Sun and solar-type stars.
+The ridge for l= 1 is notably broadened, which we attribute to mixed modes, confirm ing theoretical
+predictions for low-luminosity giants. Overall, the results demonstr ate the tremendous potential of
+Keplerdata for asteroseismology of red giants.
+Subject headings: stars: oscillations
+1Sydney Institute for Astronomy (SIfA), School of
+Physics, University of Sydney, NSW 2006, Australia;
+bedding@physics.usyd.edu.au
+2School of Physics and Astronomy, University of Birming-
+ham, Birmingham B15 2TT, UK
+3Department of Physics and Astronomy, University of British
+Columbia, Vancouver, Canada
+4Institute of Astronomy, University of Vienna, 1180 Vienna,
+Austria
+5Indian Institute of Astrophysics, Koramangala, Bangalore
+560034, India
+6LESIA, CNRS, Universit´ e Pierre et Marie Curie, Universit´ e
+Denis, Diderot, Observatoire de Paris, 92195 Meudon cedex,
+France
+7Eureka Scientific, 2452 Delmer Street Suite 100, Oakland,
+CA 94602-3017, USA
+8Department of Mathematical Sciences, University of South
+Africa, Box 392 UNISA 0003, South Africa
+9Laboratoire d’Astrophysique de Toulouse-Tarbes Universi t´ e
+de Toulouse, CNRS 14 av E. Belin 31400 Toulouse, France
+10Laboratoire AIM, CEA/DSM-CNRS, Universit´ e Paris 7
+Diderot, IRFU/SAp, Centre de Saclay, 91191, Gif-sur-Yvett e,
+France
+11Department of Astronomy and Physics, Saint Mary’s
+University, Halifax, NS B3H 3C3, Canada
+12Instituut voor Sterrenkunde, K.U.Leuven, Belgium
+13Danish AsteroSeismology Centre (DASC), Department of
+Physics and Astronomy, Aarhus University, DK-8000 Aarhus C ,
+Denmark
+14NASA Ames Research Center, MS 244-30, Moffett Field,
+CA 94035, USA
+15Las Cumbres Observatory Global Telescope, Goleta, CA
+93117, USA
+16Space Telescope Science Institute, 3700 San Martin Drive,
+Baltimore, Maryland 21218, USA17SETI Institute/NASA Ames Research Center, MS 244-30,
+Moffett Field, CA 94035, USA
+18Institut d’Astrophysique et de G´ eophysique de l’Universi t´ e
+de Li` ege, All´ ee du 6 Aoˆ ut 17 - B 4000 Li` ege, Belgium
+19Kavli Institute for Theoretical Physics and Department
+of Physics, University of California, Santa Barbara, CA 931 06,
+USA
+20Centro de Astrof´ ısica da Universidade do Porto, Rua das
+Estrelas, 4150-762 Porto, Portugal
+21Thueringer Landessternwarte Tautenburg, Sternwarte 5,
+D-07778, Tautenburg, Germany
+22Department of Astronomy, Beijing Normal University,
+China
+23Konkoly Observatory of the Hungarian Academy of Sci-
+ences, H-1525 Budapest, P.O. Box 67, Hungary
+24Queen Mary University of London, Mile End Road, London
+E1 4NS, UK
+25Astronomical Institute of the Romanian Academy, Str.
+Cutitul de Argint, 5, RO 40557,Bucharest, Romania
+26Max-Planck-Institut f¨ ur Astrophysik, Karl-Schwarzschi ld-
+Str. 1, 85748 Garching, Germany2 T. R. Bedding et al.
+1.INTRODUCTION
+Studying solar-like oscillations is a powerful way to
+probe the interiors of stars (e.g., Brown & Gilliland
+1994;Christensen-Dalsgaard2004). Oscillationsinmain-
+sequence stars and subgiants have been measured us-
+ing ground-based spectroscopy (for a recent review see
+Aerts et al. 2008) and more recently by space missions
+such as CoRoT(e.g., Michel et al. 2008). Red giants
+oscillate at lower frequencies and so demand long and
+preferably uninterrupted time series to resolve their os-
+cillations.
+The first indications of solar-like oscillations in G
+and K giants were based on ground-based observa-
+tions in radial velocity (see Merline 1999 and ref-
+erences therein; Frandsen et al. 2002; De Ridder et al.
+2006) and photometry (Stello et al. 2007), and on
+space-based photometry from the Hubble Space Tele-
+scope(HST; Edmonds & Gilliland 1996; Gilliland 2008;
+Stello & Gilliland 2009), WIRE(Buzasi et al. 2000;
+Retter et al. 2003; Stello et al. 2008), Microvariability
+and Oscillations of Stars (MOST; Barban et al. 2007;
+Kallinger et al. 2008a,b), and Solar Mass Ejection Im-
+ager(SMEI; Tarrant et al. 2007). A major break-
+through came from observations over about 150 days
+with the CoRoTspace telescope, which produced clear
+detections in numerous stars of both radial and non-
+radial oscillations in the frequency range 10–100 µHz
+(De Ridder et al. 2009; Hekker et al. 2009; Carrier et al.
+2010). This increased the number of known pulsating G
+and K giants to nearly 800, and enabled the first sys-
+tematic study of oscillations in a large population of
+red clump stars (Hekker et al. 2009; Miglio et al. 2009;
+Kallinger et al. 2010). Note that the red clump com-
+prises core He-burning stars and is the metal-rich coun-
+terpart to the horizontal branch.
+Detecting oscillations in lower-luminosity red giants,
+which are H-shell-burning stars at the base of the giant
+branch, is very desirable for testing models of stellar evo-
+lutionandalsoforconstrainingthe star-formationratein
+the local disk (Miglio et al. 2009). However, such detec-
+tions are even more challenging than for clump stars be-
+causetheamplitudesaresubstantiallylower. Inaddition,
+the oscillation frequencies are higher and are comparable
+to the orbital frequency of a satellite in low-Earth orbit.
+These factors made it difficult for CoRoTto achievesuch
+detections.
+In this Letter, we present data from the aster-
+oseismology program of the NASA Kepler Mission
+(Gilliland et al. 2010) that reveal clear oscillations in a
+large sample of red giants, extending in luminosity from
+the red clump down to the bottom of the giant branch.
+2.GLOBAL OSCILLATION PARAMETERS
+The observations were obtained over the first 34 days
+of science operations of the KeplerMission (Q1 data).
+We analyzed light curves having 29.4-minute sampling
+(long-cadence mode) for about 1500 stars listed as red
+giants in the Kepler Input Catalog (KIC; Latham et al.
+2005). About 20% of the Q1 light curves showed large
+variations ( >1%) on long timescales, mostly due to M-
+giant pulsations, and are not analyzed here.27For the
+27One of these was subsequently found to be an oscillating red
+giant in an eclipsing binary system (Hekker et al. 2010a).remaining sample, we used the light curves to measure
+the global oscillation parameters, most notably the fre-
+quency of maximum power ( νmax) and the mean large
+frequency separation(∆ ν). We were able to find a power
+excessin ∼1000stars(see alsoFigure 5 in Gilliland et al.
+2010) and measure the large separation for ∼700 stars.
+In this Letter we focus on the low-luminosity red giants
+(corresponding to νmax>100µHz) for which, as dis-
+cussed in the introduction, CoRoTresults have not yet
+been reported. As discussed by Miglio et al. (2009), such
+stars generally have luminosities below about 30 L⊙.
+Figure 1 shows the measured values of ∆ νandνmax
+for these stars. The symbols show results obtained
+using the pipeline described by Huber et al. (2009),
+who also described the method used to fit and sub-
+tract the background. Similar results were found using
+pipelines developed by four other groups (Hekker et al.
+2010b; Mosser & Appourchaux 2009; Kallinger et al.
+2010; Mathur et al. 2010), and the dotted lines in Fig-
+ure 1 show the region inside which ∼80% of those mea-
+surements lay. A detailed comparison of the results from
+the different pipelines is ongoing.
+The tight correlationin Figure 1 between ∆ νandνmax
+was discussed by Hekker et al. (2009) and Stello et al.
+(2009), andisseenheretoextendtoredgiantswithmuch
+higherνmax. This and other correlations between global
+oscillation parameters, and their connection to funda-
+mental stellar parameters, will be discussed in future pa-
+pers.
+3.RESULTS
+3.1.Frequency analysis
+We now consider a subset of 50 of the low-luminosity
+red giants for more detailed analysis, chosen as having
+the best signal-to-noise ratios. These stars are indicated
+in Figure 1 with filled symbols. Figure 2 (left panel)
+shows power spectra for six of them, spanning the full
+range of νmaxbeing considered. Each power spectrum
+shows a regular series of peaks, which is the clear signa-
+ture of solar-like oscillations.
+Solar-like oscillations are p modes with high order
+and low degree, and the observed frequencies in main-
+sequence stars are reasonably well-described by the fol-
+lowing relation:
+νn,l≈∆ν(n+1
+2l+ǫ)−l(l+1)D0.(1)
+Here,n(the radial order) and l(the angular degree)
+are integers. The form of Equation 1 is motivated by
+theoreticalcalculations: forrelativelyunevolvedstars,an
+asymptotic expansion (Tassoul 1980; Gough 1986) shows
+that ∆νis approximately the inverse of the sound travel
+time across the star, while D0is sensitive to the sound
+speed gradient near the core and ǫis sensitive to the
+surface layers.
+We conventionallydefine three small frequency separa-
+tions:δν02is the spacing between adjacent modes with
+l= 0 and l= 2,δν13is the spacing between adjacent
+modes with l= 1 and l= 3, and δν01is the amount by
+whichl= 1 modes are offset from the midpoint between
+thel= 0 modes on either side. If Equation 1 holds then
+it follows that δν02= 6D0,δν13= 10D0andδν01= 2D0.
+We shall use Equation 1 as a guide to the analysis of the
+observedfrequencies, althoughits physicalinterpretationOscillations in low-luminosity red giants with Kepler 3
+may be open to question in the case of red giants.
+To determine oscillation frequencies for each of the
+50 stars in our subset, we needed to locate the highest
+peaks in the power spectra. We did this using conven-
+tional iterative sine-wave fitting, sometimes called pre-
+whitening or CLEAN. This involves finding the highest
+peak in the power spectrum, subtracting the correspond-
+ing sinusoidal variation from the time series and then re-
+calculating the power spectrum in an iterative process.
+We retained only those peaks that lay in the frequency
+rangeνmax±5∆νand had heights greaterthan 4.5 times
+the fitted background level (in amplitude).
+Damping of solar-like oscillations causes the observed
+power from each mode to be spread into multiple peaks
+under a Lorentzian envelope. If the length of the ob-
+servations is significantly greater than the typical mode
+lifetime, one therefore expects several peaks to be ex-
+tracted for each mode. In fact, the modes are only barely
+resolved(see Section 3.3) and so we expect iterative sine-
+wave fitting to give good results.
+3.2.Semi-scaled ´Echelle Diagram
+A powerful method to study the mode frequencies of
+solar-like oscillations is to plot them modulo the large
+separation, in a so-called ´ echelle diagram (Grec et al.
+1983). We have displayed the frequencies of all 50 stars
+in a single ´ echelle diagram by using the scaling tech-
+nique described by Bedding & Kjeldsen (2010), in an ef-
+fort to carry out “ensemble asteroseismology”. The re-
+sult is shown in Figure 3 a. To make this diagram, we
+first divided the frequencies of each star by ∆ ν(that
+is, scaling them to have a large separation of unity). We
+used the´ echelle diagramto identify the radialmodes and
+then fine-tuned the scaling factor (∆ ν) by up to a few
+percent to align them on a single line in the ´ echelle dia-
+gram, shown as a solid line in Figure 3 a. As discussed by
+Bedding & Kjeldsen (2010, their Method 2), this equates
+to assigning all stars the same value of ǫ. The fact that
+all the stars can be aligned indicates this is a valid ap-
+proximation. This alignment is also clearly seen in the
+right-hand panel of Figure 2, which shows power spectra
+plotted against scaled frequencies (i.e., divided by ∆ ν).
+The ´ echelle diagram in Figure 3 adiffers from those
+shown by Bedding & Kjeldsen (2010) in one important
+respect: only the horizontal axis has been scaled. In the
+vertical direction, we have plotted the original frequen-
+cies without scaling, which spreads out the points and
+allows us to look for variations with νmax. Such a plot
+might be called a semi-scaled ´ echelle diagram.
+3.3.Collapsed ´Echelle Diagram and Mode Lifetimes
+We can collapse the ´ echelle diagram in the vertical di-
+rection in order to define the ridges more clearly. Fig-
+ure 3bshows this using a simple histogram of the peaks
+in Figure 3 a, and the ridges are clearly visible. Rather
+than counting the number of peaks above a threshold,
+another approach is to sum the total power. To do this,
+we restricted each background-correctedpower spectrum
+to the same frequency rangeas before ( νmax±5∆ν), then
+dividedthe frequencyscalebythelargeseparation(using
+the same fine-tuned value as above) and folded the spec-
+trum modulo unit spacing. These folded spectra were
+summed forall 50starsand then smoothed slightly(witha boxcar of width 0.01∆ ν) to give the result shown in
+Figure 3c.
+The clarity of the ridges in Figure 3 allows us to
+make some statements about mode lifetimes. As men-
+tioned in Section 3.1, damping causes each mode to
+be spread into multiple peaks under a Lorentzian en-
+velope with a full-width at half maximum (FWHM) of
+1/(πτ), where τis the mode lifetime. There has been
+considerable discussion of the lifetimes of red-giant os-
+cillations, particularly regarding whether the modes are
+sufficiently long-lived to allow their frequencies to be
+measured with sufficient accuracy to be useful for aster-
+oseismology (Houdek & Gough 2002; Stello et al. 2006;
+Barban et al. 2007; Dupret et al. 2009). The CoRoTre-
+sultshaveclearlydismissedthisworryforredclumpstars
+(De Ridder et al. 2009; Carrier et al. 2010) and our Ke-
+plerdata do the same for the low-luminosity stars. This
+is excellent news for asteroseismology.
+Figure 3 was constructed by aligning the radial modes
+on the vertical solid line. However, the resulting width
+of thel= 0 ridge is not zero, which arises from at least
+three factors: (1) the resolution limit set by the duration
+of the observations (34d), which leads to peaks in the
+power spectrum having FWHM 0.3 µHz; (2) variations
+in ∆νwith frequency in individual stars over the range
+being considered (curvature in the ´ echelle diagram); and
+(3) broadening of power due to finite mode lifetimes (see
+above). The observed narrowness of the l= 0 ridge sets
+upper limits on the second and third of these effects.
+Here, as discussed above, we are particularly interested
+in the mode lifetimes.
+The FWHM of the l= 0 peak in Figure 3 c, found
+by subtracting the background and fitting a Lorentzian,
+is 0.033∆ ν. This sets an upper limit on the intrinsic
+linewidths of these modes. However, so far our measure-
+ment is in terms of ∆ ν, which varies over the sample.
+The median ∆ νis 11.5µHz, in which case the measured
+width equates to a lower limit on the mode lifetimes of
+about 10days. To investigate further, we divided the
+sample into four bins according to ∆ νand formed the
+collapsed ´ echelle diagram for each subset, as before. We
+found the widths of the l= 0 ridgesin all four casesto be
+about the same in absolute terms ( ∼0.4µHz), which is
+only slightly greater than the FWHM set by the resolu-
+tion of the data (see above). We conclude that the mode
+lifetimes in these red giants is at least 10days, possibly
+much greater.
+3.4.Small Separations and the C-D Diagram
+Thel= 2 ridge is closely parallel to the l= 0 ridge,
+implying that δν02is a nearly constant fraction of ∆ ν
+for the stars in our sample, with δν02≈0.125∆ν(Fig-
+ure 3c). This frequency scaling indicates that, to a good
+approximation, the stars are homologous.
+We were able to measure the mean small separation
+δν02for 38 stars in our sample. Figure 4 ashows the re-
+sults in a so-called C-D diagram (Christensen-Dalsgaard
+1988), which plots δν02versus ∆ ν. The dashed line in
+Figure 4ashows a linear fit, which has parameters
+δν02= (0.122±0.006)∆ν+(0.05±0.08)µHz.(2)
+The spread of points about this relation presum-
+ably reflects a spread in stellar masses, and is better4 T. R. Bedding et al.
+seen by plotting the ratio of the two separations, as
+shown in Figure 4 b(see Roxburgh & Vorontsov 2003;
+Ot´ ı Floranes et al. 2005; Mazumdar 2005).
+The dotted line in Figure 3 marks the midpoint be-
+tweenl= 0 modes. Interestingly, the l= 1 ridge is
+centered slightly to the right of this line. This indicates
+that the average small separation δν01in these red gi-
+ants is negative, whereas it is positive in the Sun and
+the handful of other main-sequence stars for which it has
+been measured: αCen A (Bedding et al. 2004), αCen B
+(Kjeldsen et al. 2005), γPav (Mosser et al. 2008), and
+τCet (Teixeira et al. 2009). A negative value for δν01
+also indicates that Equation 1 is not satisfied in detail
+for these red giants, a point already noted for the G8
+giant HR 7349 by Carrier et al. (2010), based on CoRoT
+observations.
+3.5.Mixed Modes with l= 1
+Thel= 1 ridge in Figure 3 is notably broader than
+the others. We do not attribute this to shorter mode life-
+times, but rather to the presence of mixed modes whose
+frequencies are shifted by avoided crossings. Mixed
+modes occur in evolved starsand havethe characteristics
+of p modes in the envelope of the star and of g modes in
+the interior (e.g., Aizenman et al. 1977). A typical ex-
+ample is the star KIC 5356201 (see Figure 2 and filled
+blue symbols in Figure 3 a). Thel= 0 and 2 modes are
+quite close to their respective ridges, whereas there is
+much more scatter about l= 1. A more extreme exam-
+ple is the star KIC 4350501 (see Figure 2 and filled red
+symbols in Figure 3 a), which shows a particularly high
+peak at 87 µHz, well below νmax. The proximity of this
+peak to the l= 1 ridge indicates that it is likely to be a
+mixed mode. A similar situation is seen in the F5 sub-
+giant Procyon, for which ground-based velocity observa-
+tions show a narrow peak well below νmaxthat lies close
+to thel= 1 ridge in the ´ echelle diagram (Bedding et al.
+2010). Thelargepeakheightcanbeexplainedbythefact
+that mixed modes are expected to have longer lifetimes
+(smaller linewidths) than pure p modes because they
+have larger mode inertias (e.g., Christensen-Dalsgaard
+2004).
+Another good example is KIC 5006817, shown in Fig-
+ure 5, where we clearly see multiple l= 1 peaks in each
+order (see also KIC 9904059 = KOI 145 in Figure 1 of
+Gilliland et al. 2010). Each of these clusters of peaks
+is reminiscent of a single mode that is heavily damped
+(see Section 3.3). However, as mentioned above, mixed
+modes are expected to have longer lifetimes than pure p
+modes and hence to produce narrower – not broader –
+peaks. Indeed, this phenomenon of multiple l= 1 peaks
+is probably what led previous authors to report short
+mode lifetimes in red giants (see Section 3.3). Guided by
+the theoretical work of Dupret et al. (2009), we instead
+interpret the observed properties of the l= 1 ridge in
+terms of mixed modes.
+Dupret et al. (2009) discussed theoretical amplitudes
+of solar-like oscillations for a range of models on the
+red giant branch. Their models show mostly regular
+frequency patterns for evolved giants ( νmax<80µHz),
+but predict more complex spectra for low-luminosity gi-
+ants. This is mainly explained by the less efficient ra-
+diative damping in the cores of these stars, leading to
+a stronger interaction of p-mode and g-mode cavities,and by the less efficient trapping due to the smaller
+density contrast between the core and the envelope.
+As a consequence, more non-radial modes could be ob-
+served than only those trapped in the envelope. This
+is particularly expected to affect l= 1 modes because
+the evanescent region just above the H-burning shell
+is smaller for these modes (see also Dziembowski et al.
+2001; Christensen-Dalsgaard 2004). Our observation
+of a broadened l= 1 ridge with a small cluster of
+modes per order supports the theoretical predictions by
+Dupret et al. (2009). Observations of frequency spectra
+in a sample of higher luminosity red giants would be a
+desirable follow-up study.
+3.6.Detection of l= 3Modes
+In addition to ridges corresponding to l= 0, 1 and 2,
+we see clear evidence in Figure 3 for modes with l= 3.
+Thedetection ofthese modeswith photometryisdifficult
+because of the very low amplitudes that result from ge-
+ometric cancellation. The only previous report of solar-
+like oscillations with l= 3 from photometry (except for
+the Sun) is a probable detection using CoRoTin the G0
+main-sequence star HD 49385 (Deheuvels et al. 2010).
+The ability to detect l= 3 modes from Keplerobser-
+vations of red giants is a significant bonus.
+In Figures 3 aandbthere are 12 peaks that fall along
+the ridge that we identify with l= 3, and these arise
+from 10 stars in the sample of 50. These are presumably
+peaksthat, duetothe stochasticnatureoftheexcitation,
+happen to rise above our detection threshold in these
+observations. A more objective approach is to sum the
+powerfrom all stars, as shown in Figure 3 c, and we again
+see clear evidence for the l= 3 ridge.
+How does the position of the l= 3 ridge compare with
+expectations? As mentioned in Section 3.1, it is conven-
+tional to measure l= 3 relative to l= 1, via the small
+separation δν13. However,giventhatthepositionof l= 1
+for these red giants is anomalous, it seems sensible to in-
+stead measure l= 3 relative to l= 0. We therefore
+suggest using a new small separation, δν03, defined (by
+analogyto δν01) as the amount by which the l= 3 modes
+are offset from the midpoint between the l= 0 modes on
+either side. If follows that δν03=δν01+δν13. From Fig-
+ure 3cwe see that δν03≈0.28∆ν. If the asymptotic rela-
+tion held (Equation1), we would expect δν03= 12D0. In
+that case the ratio between δν03andδν02would be 2.0,
+whereas we observe a ratio of 2.2. These results should
+provide valuable tests for theoretical models.
+4.CONCLUSIONS
+These results, based on the analysis of only 34 days
+of data, show the tremendous potential of Keplerlong-
+cadence data for the study of solar-like oscillations in
+red giants. As more data become available, the longer
+time series will allow detailed studies of stars covering
+the whole range of evolutionary states along the red gi-
+ant branch and provide stringent observational tests for
+theories of stellar structure and evolution in this part of
+the H-R diagram.
+We gratefully acknowledge the entire Keplerteam,
+whose outstanding efforts have made these results pos-
+sible. Funding for this Discovery mission is providedOscillations in low-luminosity red giants with Kepler 5
+Fig. 1.— Large frequency separation versus frequency of maximum pow er for 78 low-luminosity red giants. Filled symbols indicat e the
+50 stars that were selected for more detailed analysis. The d otted lines enclose the region containing ∼80% of measurements by the five
+pipelines (see text).
+by NASA’s Science Mission. Grants were received
+from the European Research Council (ERC/FP7/2007–
+2013 n◦227224, PROSPERITY), and K.U.Leuven(GOA/2008/04).
+Facilities: Kepler
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+Fig. 2.— Left:Power spectra of six representative low-luminosity red gia nts.Right:The same power spectra plotted against scaled
+frequency (see Section 3.2). The dotted lines are equally sp aced, having unit separation and being aligned with the l= 0 modes. Stars are
+labelled with identification numbers from the Kepler Input Catalog (KIC; Latham et al. 2005).
+Fig. 3.— (a) Semi-scaled ´ echelle diagram for the sample of 50 low-lumi nosity red giants (see Section 3.2). Symbols indicate extra cted
+frequencies, with symbol sizes proportional to amplitude. Filled blue and red symbols are frequencies of the stars KIC 5 356201 and
+KIC 4350501, respectively (see Figure 2 and Section 3.5). Th e value of ∆ νfor each star was fine-tuned in order to align the l= 0 modes on
+the left-most vertical dotted line. The other dotted line li es exactly 0.5 away, marking the midpoint between l= 0 modes. ( b) Histogram
+of the points shown in panel a. (c) Folded and scaled power spectra, collapsed over all stars i n the sample and smoothed slightly (see
+Section 3.2). The arrows show the definitions of the small sep arations δν01,δν02,δν03, andδν13, with the sign convention that left-pointing
+arrows correspond to positive separations (see Section 3.1 ).Oscillations in low-luminosity red giants with Kepler 7
+Fig. 4.— (a) The so-called C-D diagram (Christensen-Dalsgaard 1988) o fδν02versus ∆ νfor the 38 stars in our sample for which both
+quantities were reliably determined. The dashed line shows a linear fit. ( b) The ratio δν02/∆νversus ∆ νfor the same stars, to better show
+the scatter about the trend.
+Fig. 5.— Power spectrum and ´ echelle diagram for the star KIC 5006817 , illustrating multiple l= 1 peaks per order. Note that the highest
+peak in the power spectrum extends beyond the plot limits (to 8400ppm2).This figure "fig02.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0229v2This figure "fig03.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0229v2This figure "fig04_final.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0229v2This figure "fig05.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0229v2This figure "fig01_v7.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0229v2
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+arXiv:1001.0230v1  [math.AC]  1 Jan 2010CUBIC RINGS AND THEIR IDEALS
+YURIY A. DROZD AND RUSLAN V. SKURATOVSKII
+Abstract. We give an explicit description of cubic rings over a
+discrete valuation ring, as well as a description of all ideals of such
+rings.
+Introduction
+Ideals of local rings have been studied by a lot of authors from quite
+different viewpoints. One of the questions that arise with this respe ct
+is on the number of parameters par(C) defining the ideals of such a
+ringCup to isomorphism, especially when it is reduced and of Krull
+dimension 1. Certainly, it makes sense if the residue field kis infinite.
+In [8] it was shown that par( C) = 0, i.e. Chas a finite number of
+ideals (up to isomorphism), if and only if CisCohen–Macaulay finite ,
+i.e. has a finite number of indecomposable non-isomorphic Cohen–
+Macaulay modules (in the 1-dimensional reduced case they coincide
+with torsion free modules). Then Schappert [12] proved that a plan e
+curve singularity has at most 1-parameter families of ideals if and only
+if it dominates one of the strictly unimodal plane curve singularities
+in the sense of [14], or, the same, unimodal andbimodalplane curve
+singularities in the sense of [1]. In [7] this result was generalized to all
+curve singularities. Note that this time par( C) = 1 doesnot imply that
+CisCohen–Macaulay tame , i.e. has at most 1-dimensional families of
+indecomposable Cohen–Macaulay modules. Tameness means that C
+dominates a singularity of type Tpq[5]. The case par( C)>1 had
+not been studied before the second author described the one bra nch
+singularities of type Wsuch that par( C)≤2 [13].
+In this paper we study the cubic rings . We describe all such rings,
+their ideals and, in particular, establish the value par( C) for any cubic
+ringC. As a consequence, we show that a cubic ring is Gorenstein if
+and only if it is a plane curve singularity (i.e. its embedding dimension
+equals 2).
+1.Generalities
+Wedenoteby Dadiscretevaluationringwiththeringoffractions K,
+the maximal ideal m=tDand the residue field k=D/tD. Acubic
+ringoverDis, by definition, a D-subalgebra Cin a 3-dimensional
+2000Mathematics Subject Classification. Primary 13C05, Secondary 16G60.
+12 YURIY A. DROZD AND RUSLAN V. SKURATOVSKII
+semisimple K-algebra L, which is a free D-module of rank 3. We
+also denote Athe integral closure of DinLand always suppose that
+Ais finitely generated as C-module. Equivalent condition (see, for
+instance, [3]): the m-adic completion ˆCof the ring Chas no nilpotent
+elements. It isalways thecaseifthealgebra Lisseparable , forinstance,
+if charK= 0. We also set Am=tmA+DandJm=tAm−1=
+radAm(m >0).
+In what follows, an idealmeans a fractional C-idealinK, i.e. a
+finitely generated C-submodule M⊆Ksuch that KM=K. Then
+Mis a freeD-module of rank 3. We are going to describe all ideals of
+cubic rings up to isomorphism. It is known (see, for instance, [9]) tha t
+there is a one-to-one correspondence between C-ideals and ˆC-ideals,
+mapping Mto itsm-adic completion. This correspondence reflects
+isomorphisms , i.e. maps non-isomorphic ideals to non-isomorphic. So,
+in what follows we may (and will) suppose that Discomplete with
+respect to the m-adic topology.
+Recall also that the embedding dimension edimCof a local noether-
+ian ringCwith the maximal ideal Jand the residue filed kis defined
+as dim kJ/J2. IfCis of Krull dimension 1 and edim C= 2,Cis called
+aplane curve singularity . In the geometric case, when Ccontains a
+subfield of representatives of k, it actually means that there is a plane
+curveCsuch that Cis the completion of the local ring of a singular
+pointx∈C.
+From the general theory of ramification in finite extensions we see
+that the following cases can happen:
+One branch, ramified case: L is a field, the maximal ideal of
+AequalsτA,A/τA≃kandtA=τ3A.
+One branch, non-ramified case: L isafield, themaximalideal
+ofAequalstAandA/tA=k[¯θ] is a cubic extension of the
+fieldk, where ¯θis a root of an irreducible cubic polynomial
+f(x)∈k[x].
+Two branches, ramified case: L =K1×K, where K1is a
+quadratic extension of K,A=D1×D, the maximal ideal of
+D1isτD1,D1/τD1≃kandtD1=τ2D1.
+Two branches, non-ramified case: L =K1×K, whereK1is
+a quadratic extension of K,A=D1×D, the maximal ideal of
+D1istD1andD1/τD1=k[¯θ] is a quadratic extension of the
+fieldk, where¯θis a root of an irreducible quadratic polynomial
+f(x)∈k[x].
+Three branches case: L =K3,A=D3.
+We recall [10, 2] that, for any cubic ring C, every ideal of Cis
+isomorphic either to an over-ring ofC, i.e. a cubic ring Bsuch that
+C⊆B⊂L, or to the dual ideal B∗= Hom D(B,D) of such an
+over-ring. Hence, to describe all ideals of C, we only need to describeCUBIC RINGS AND THEIR IDEALS 3
+over-rings of C. Obviously, any cubic ring in Lcontains some Am.
+Therefore, to describe all cubic rings (so their ideals as well), we hav e
+to describe the over-rings of Am. IfBis an over-ring of C, they also
+say that Bdominates C.
+Since the unique (up to isomorphism) A-ideal isAitself, we proceed
+by induction: supposing that all over-rings of Amare known, we find
+all over-rings of Am+1. IfCis an over-ring of Am+1, thenB=CAmis
+an over-ring of Am,tB⊂CandC/tBis ak-subalgebra in B/tB. If
+B⊇Am−1, thentB⊇Jm, hence,C⊇Jm+D=Am. Therefore, the
+following procedure gives all over-rings of Am+1which are not over-
+rings ofAm:
+Procedure.
+•For every over-ring BofAm, which is not an over-ring of Am−1,
+calculate ¯B=B/tB. Set¯A= (Am+tB)/tB⊆¯B.
+•Find all proper subalgebras S⊂¯Bsuch that ¯AS=¯B. Equiv-
+alently, the natural map S→B/BJmmust be surjective.
+•For each such Stake its preimage in B.
+2.Calculations
+2.1.One branch, ramified case. We set
+C2r(α) =D+trαD+t2rA,wherev(α) = 1,
+C2r+1(α) =D+trαD+t2r+1A,wherev(α) = 2,
+wherevis the discrete valuation related to the ring A, i.e.v(α) =k
+means that α∈τkD\τk+1D. Note that C0(α) =A. Obviously, αcan
+be uniquely chosen as τ+aτ2forC2rand asτ2+atτforC2r+1, where
+a∈Dis defined modulo tr.
+Theorem 2.1. Every over-ring of Amcoincides with tkCr(α)+Dfor
+somek,rsuch that r+k≤mand some α. The rings Cr(α)are just
+all plane curve singularities in this case.
+Proof.Form= 1 itis easyandknown [8,11]. So, we usethe Procedure
+form >1, setting B=tkCr(α) +D, wherek+r=m. Then the
+basis of¯Bconsists of the classes of the elements {1,thα,tmτs}, where
+h=k+ [r/2],s∈ {1,2}ands≡r(mod 2). Since thα /∈Jm, the
+subalgebra Snecessarily contains the class of thα+ctmτsfor some
+c∈D. Ifk= 0, then m=randv(tmτs) = 2v(thα). Therefore, ¯B
+has no proper subalgebra containing the class of thα+ctmτs. Ifk >0,
+the preimage of SisD+ (thα+ctmτs)D+tm+1A. It coincides with
+tk−1Cr+2(α′)+Dwhereα′=α+ctm−hτs.
+Now one easily checks that edim Cr(α) = 2, while edim C= 3 for all
+other rings. /square4 YURIY A. DROZD AND RUSLAN V. SKURATOVSKII
+2.2.One branch, non-ramified case. We setCr(α) =D+trαD+
+t2rA0, whereα∈A×\D. Again C0(α) =A0. Note that αcan be
+uniquely chosen as θ+aθ2, whereθis a fixed preimage of ¯θinD1and
+a∈Dis defined modulo tr.
+Theorem 2.2. Every over-ring of Amcoincides with tkCr(α)+Dfor
+somek,randαwith2r+k≤m. The rings Cr(α)are just all plane
+curve singularities in this case.
+Proof.Form= 1 it is obvious. So, using the Procedure for m >1, we
+setB=tkCr(α)+Dwith 2r+k=m. Then a basis of ¯Bconsists of
+the classes of elements {1,tr+kα,tmα2}for some α2∈A×\(D+αD).
+Sincetr+kα /∈Jm,Smust contain the class of an element tr+kα′=
+tr+kα+ctmα2for some c∈D. As above, it is impossible if k= 0. If
+k >0, thenthepreimageof SisD+tr+kα′+tm+1A=tk−1Cr+1(α′)+D.
+Now one easily checks that edim Cr(α) = 2, while edim C= 3 for all
+other rings. /square
+2.3.Two branches, ramified case. We denote by vthe valuation
+defined by the ring D1, byethe idempotent in Asuch that eA=D1
+and set
+Cl,q(α) =D+tl(e+tqα)D+trA,wherer= 2l+q,
+Cr(α) =D+trαD+t2r+1A.
+In both cases α∈D1andv(α) = 1, where vis the valuation defined by
+the ringD1. Obviously, αcan be uniquely chosen as aτ, wherea∈D
+is defined modulo r. Note that C0,q(α) =D+e1D+tqAare just all
+decomposable rings in this case and C0,0(α) =A.
+Theorem 2.3. Every over-ring of Amcoincides with either tkCl,r(α)+
+DortkCr(α)+D, wherek+r≤m. The rings Cl,q(α)andCr(α)are
+just all plane curve singularities in this case.
+Proof.The case m= 1 is obvious. So, using the Procedure, we suppose
+thatm >1 andk+r=m. IfB=tkCl,q(α)+D, a basis of ¯Bconsists
+of the classes of {1,tk+l(e+tqα),tmτ}. Sincetk+l(e+tqα)/∈Jm, the
+subalgebra Smust contain the classe of tk+l(e+tqα′) for some α′∈D1
+withv(α′) = 1. Again the case k= 0 is impossible. If k >0, the
+preimage of Scoincides with tk−1Cl+1,q+D. IfB=tkCr(α)+D, the
+calculations are quite similar.
+Now one easily checks that edim Cl,q(α) = edim Cr(α) = 2, while
+edimC= 3 for all other rings. /square
+2.4.Two branches, non-ramified case. We set
+Cl,q(α) =D+tl(e1+tqα)D+trA,wherer= 2l+q
+andα∈D1\(e1D+tD).CUBIC RINGS AND THEIR IDEALS 5
+Thenαcan be chosen as aθ, whereθis a fixed preimage of ¯θinD1and
+a∈Dis uniquely defined modulo tl. AgainC0,q(α) =D+e1D+tqA
+are just all decomposable rings in this case. Especially, C0,0(α) =A.
+Theorem 2.4. Every over-ring of Amcoincides with one of the rings
+tkCl,q(α)+D, wherek+r≤m. The rings Cl,q(α)are just all plane
+curve singularities in this case.
+We omit the proof in this case, since it practically repeats the cal-
+culations in the other cases.
+2.5.Three branches case. We set
+Cl,q(α) =D+tlαD+trA,
+whereα=e+tqae′,e/\e}atio\slash=e′are primitive idempotent in A,r= 2l+q,
+a∈D×anda/\e}atio\slash≡1(modt) ifq= 0. Obviously, ais unique modulo
+tl. AgainC0,q(α) =D+eD+tqAare just all decomposable rings in
+this case and C0,0=A. Note also that if C=D+tlαD+trA, where
+α=e+ae′as above with a≡1(modt), then, for a≡1(modtl),
+C=tlC0,q(1−e−e′) +D, and for a≡1(modtq) with 0 < q < l,
+C=Cl,q(α′) for some α′.
+Theorem 2.5. Every over-ring of Amcoincides with tkCl,q(α)+Dfor
+someαand some l,qwithk+r≤m. The rings Cl,q(α)are just all
+plane curve singularities in this case.
+We also omit the proof in this case, since it practically repeats the
+calculations in the other cases.
+2.6.Table of plane curve cubic singularities. Wepresent in Table
+1 below all plane curve cubic singularities. In this table sis the num-
+ber of branches,∗marks the unramified cases (related to the residue
+field extensions, hence impossible if kis algebraically closed); x,yare
+generators of the maximal ideal, v(a) denotes the multivaluation of an
+elementa∈A, i.e. the vector of valuations of its components with re-
+spect to the decomposition of Ainto the product of discrete valuation
+rings. The column “type” shows the correspondence with the Arno ld’s
+classification [1, §15]. If char k= 0 and Ais ramified, it actually
+shows the place of the rings in this classification. If char k= 0 and C
+is non-ramified, it shows the place of the ring in this classification afte r
+the natural extension of the field k. The validation of this column is
+given in [7, Section 2.3]. Note that, following [7], we denote by El,qthe
+singularities Jl,qin the sense of [1]. Such notations seem more uniform.
+Note also that the singularities of types E1andE2are actually not
+cubic, but quadratic, and coincide with those of types A1andA2of
+[1]. Finally, the last column, “par” shows the number of parameters p
+from the residue field kwhich define a unique ring of this type. We
+will consider this value in the last section. It does not coincide with
+themodality in the sense of [1]; the latter equals p−1.6 YURIY A. DROZD AND RUSLAN V. SKURATOVSKII
+Table 1.
+sname v(x)v(y)typepar
+1C2r(α)(3) (3r+1) E6rr
+C2r+1(α)(3) (3r+2)E6r+2r
+1∗Cr(α)(1) (r)E∗
+r,0r
+2Cr(α)(2,1)(2r+1,∞)E6r+1r
+Cl,q(α)(2,1)(2l,∞)El,2q+1l
+2∗Cl,q(α)(1,1)(l,∞)E∗
+l,2ql
+3Cl,q(α)(1,1,1)(l,l+q,∞)El,2ql
+Remark. Thetamecubic plane curve singularities T3,q(q≥6) [4, 5]
+coincide with those of types E2,q−6.
+3.Ideals
+As we have mentioned above, every ideal of a cubic ring Cis isomor-
+phic either to an over-ring B⊇Cor to its dual B∗= Hom D(B,D). If
+CisGorenstein (for instance, if it is a plane cubic singularity), then
+C∗≃C, thusB∗≃HomC(B,C). Therefore, to calculate B∗, one has
+to choose a Gorenstein subring C⊆Band to calculate
+HomC(B,C)≃ {λ∈L|λB⊆C}={λ∈C|λB⊆C}
+(the latter equality holds since 1 ∈B). This remark easily leads to the
+following result.
+Theorem 3.1. The duals to the cubic rings are as follows:
+One branch ramified case: IfB=D+tkCr(α), thenB∗≃
+D+t[r/2]αD+tk+rA.
+One branch non-ramified case: IfB=D+tkCr(α), then
+B∗≃D+trαD+tk+2rA.
+Two branches ramified case: (1)IfB=D+tkCl,q(α), then
+B∗≃D+tl(e+tqα)D+tk+2l+qA.
+(2)IfB=D+tkCr(α), thenB∗≃D+trαD+tk+2r+1A.CUBIC RINGS AND THEIR IDEALS 7
+Two branches non-ramified case: IfB=D+tkCl,q(α), then
+B∗≃D+tl(e+tqα)D+tk+2l+qA.
+Three branches case: IfB=D+tkCl,q(α), thenB∗≃D+
+tlαD+tk+2l+qA.
+Proof.The proof is immediate if we choose for a Gorenstein subring
+C⊆Btheplanecurvesingularity C=Ck+r(α)orCk+l,q(α)depending
+on the shape of B. For instance, in two branches ramified case, if
+B=D+tkCl,q(α) andC=D+Cl+k,q(α), then
+B∗≃ {λ∈C|λB⊆C}=tkD+tk+l(e+tqα)D+t2k+2l+qA
+≃D+tl(e+tqα)D+tk+2l+qA. /square
+Corollary 3.2. If a cubic ring is Gorenstein, it is a plane curve sin-
+gularity.
+Note that it is no more the case for the extensions of bigger degree s.
+For instance, the rings Ppqfrom [5], which are quartic, are Gorenstein
+(they are complete intersections) but of embedding dimension 3.
+4.Geometric case. Number of parameters
+In this section we suppose that our rings are of geometric nature ,
+i.e.D=k[[t]], where kis algebraically closed. Then one can consider
+thenumber of parameters par(C) defining C-ideals (see [4, Section
+2.2] or [6, Section 3], where it is denoted by par(1; C,A)). Actually, it
+coincides with the minimal possible number pfor which there is a finite
+set offamilies of ideals Ik(1≤k≤m) of dimensions at most psuch
+that every C-ideal is isomorphic to one belonging to some family Ik.
+Equivalently, it is the maximal possible psuch that is a p-dimensional
+family of ideals Iwhere every isomorphism class of ideals only occurs
+finitely many times. In [7] a criterion was established in order that
+par(C)≤1. For cubic rings it means that Cdominates a singularity
+of typeEm(18≤m≤20) orE3,i. The following results give the exact
+value of par( C) for all cubic rings of geometric nature. (Note that no
+unramified case can occur for such rings.)
+Theorem 4.1. IfCis a cubic ring of geometric nature, par(C)≤n
+if and only if Cdominates one of the singularities of type E12n+i(6≤
+i≤8)orE2n+1,q(q≥0).
+Proof.Certainly, we have to prove that
+(1) every ring of one of the listed types have at most n-parameter
+families of ideals;
+(2) ifCdominates no ring of the listed types, it has ( n+ 1)-
+parameter families of ideals.
+Since the calculations in all cases are similar, we only consider the
+one branch ramified case. Note first that the rings C2r(α) as well as8 YURIY A. DROZD AND RUSLAN V. SKURATOVSKII
+C2r+1(α) form a r-parametric family. Indeed, we can choose in the
+first case α=τ+aτ2, and in the second one α=τ2+aτ4, where
+a∈Dis defined modulo tr, and such a presentation is unique. The
+same is true also for tkC2r(α)+DandtkC2r+1(α)+Dfor anyk. Since
+C2r(α)⊇A2rfor allα, we get par( A2r)≥r.
+LetCdominate neither a ring of type E12n+6, i.e.C4n+2(α), nor
+a ring of type E12n+8, i.e.C4n+3(α). Then it contains no element of
+valuation smaller than 6 n+6, soC⊆A2n+2. Hence, par( C)≥n+1.
+On the other hand, consider the ring C2r+q(α), where q∈ {0,1}.
+Its over-rings are of the kind D+tkC2m+q(β), where k+m≤rand
+k+2m≤2r. Moreover, let α=τq+1+aτ2q+2andβ=τq+1+bτ2q+2.
+Thenbis defined modulo tmandb≡a(modtr−m−k). Therefore, the
+over-rings with the fixed m,kform ap-parameter family, where p=
+min(m,r−m−k). Hence, 2 p≤randp≤[r/2]. If we set r= 2n+1,
+we get that par( C4n+2(α))≤nand par(C4n+3(α))≤nfor all possible
+α. It accomplishes the proof. /square
+Obvious considerations give the number of parameters for special
+rings.
+Corollary 4.2.
+par(Cr(α)) = [r/2],
+par(Cl,q(α)) = [l/2],
+par(Am) = [m/2].
+References
+[1] V.I.Arnold, A.N.Varchenko,S.M.Gusein-Zade.Singularitieso fDifferentiable
+Maps. Vol. 1. Nauka, Moscow, 1982.
+[2] Y.A.Drozd. Ideals of commutative rings. Matem. Sbornik, 101 ( 1976), 334–
+348.
+[3] Y.A.Drozd. On the existence of maximal orders, Matem. Zamet ki, 37 (1985),
+313–315.
+[4] Y.A.Drozd. Cohen-Macaulay modules over Cohen-Macaulay alge bras. CMS
+Conference Proceedings, 19 (1996), 25–53.
+[5] Y.A.Drozd, G.-M.Greuel. Cohen-Macaulay module type, Compos itio Math.,
+89 (1993) 315-338.
+[6] Y.A.Drozd, G.-M.Greuel. Semi-continuity for Cohen-Macaulay m odules.
+Math. Ann., 306 (1996), 371–389.
+[7] Y.A.Drozd, G.-M.Greuel. On Schappert characterization of un imodal
+plane curve singularities. Singularities: The Brieskorn Anniversary V olume.
+Birkh¨ auser, 1998, 3-26.
+[8] Y.A.Drozd, A.V.Roiter. Commutative rings with a finite number o f integral
+indecomposable representations. Izvestia Acad. Nauk SSSR. Ser . matem., 31
+(1967), 783-798.
+[9] D.K.Faddeev. Introduction to multiplicative theory of modules o f integral
+representations. Trudy Mat. Inst. Steklova, 80 (1965), 145–1 82.
+[10] D.K.Faddeev. On the theory of cubic Z-rings. Trudy Matem. Inst. Steklova
+80, (1965), 183–187.CUBIC RINGS AND THEIR IDEALS 9
+[11] H.Jacobinski. Sur les ordres commutatifs avec un nombre fini d e r´ eseaux
+ind´ ecomposables. Acta Math., 118 (1967), 1-31.
+[12] A.Schappert. A characterizationof strictly unimodal plane cu rve singularities.
+Lecture Notes in Math. 1273 (1987), 168–177.
+[13] R.Skuratovskii. Ideals of singularities of type W. Ukrainian J. Math.
+[14] C.T.C.Wall. Classification of unimodal isolated singularities of comple te inter-
+sections. Proc. Symp. Pure Math. 40(2) (1983), 625–640.
+Institute of Mathematics, National Academy of Sciences of U kraine,
+Tereschenkivska 3, 01601 Kyiv, Ukraine
+E-mail address :drozd@imath.kiev.ua
+URL:www.imath.kiev.ua/ ∼drozd
+Kyiv National Taras Shevchenko University, Department of M e-
+chanics and Mathematics, Volodymyrska 60, 01601 Kyiv, Ukra ine
+E-mail address :ruslcomp@mail.ru
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+arXiv:1001.0232v4  [math.SP]  6 Jan 2010The Spectral Mapping Theorem
+Narinder S Claire
+Abstract
+We give a direct non-abstract proof of the Spectral Mapping T heorem for the Helffer-Sj¨ ostrand
+functional calculus for linear operators on Banach spaces w ith real spectra and consequently give
+a new non-abstract direct proof for the Spectral Mapping The orem for self-adjoint operators on
+Hilbert spaces. Our exposition is closer in spirit to the pro of by explicit construction of the ex-
+istence of the Functional Calculus given by Davies. We apply an extension theorem of Seeley to
+derive a functional calculus for semi-bounded operators.
+AMS Subject Classification : 47A60
+Keywords : Functional Calculus, Spectral Mapping Theorem, Spectrum,
+1 Introduction
+The Helffer-Sj¨ ostrand formula was established in [6] in the following p roposition
+Proposition 1.1 ([6] Proposition 7.2) .LetHbe a self-adjoint operator (not necessarily bounded) on a
+Hilbert space H. Supposefis inC∞
+0(R)and˜finC∞
+0(C)is an extension of fsuch that∂˜f
+∂¯z= 0onR.
+Then we have
+f(H) =i
+2π/integraldisplay/integraldisplay
+C∂˜f
+∂z(z−H)−1d¯z∧dz=−1
+π/integraldisplay/integraldisplay
+C∂˜f
+∂z(z−H)−1dxdy (1.1)
+whereL(C)is the Lebesgue measure on C.
+The existence of the functional calculus was assumed by the autho rs. Davies[2] showed that the formula
+(equation 1.1) yielded a new approach to constructing the functional calculus for linear operators on
+Banach spaces under the following hypothesis
+Hypothesis 1.2. His a closed densely defined operator on a Banach space Bwith spectrum σ(H)⊆R.
+The resolvent operators (z−H)−1are defined and bounded for all z /∈Rand
+/bardbl(z−H)−1/bardbl ≤c|Imz|−1/parenleftbigg/angbracketleftz/angbracketright
+|Imz|/parenrightbiggα
+(1.2)
+for someα≥0and allz /∈R, where/an}bracketle{tz/an}bracketri}ht:= (1+|z|2)1
+2.
+1His functional calculus, for operators on Banach spaces, was defi ned for an algebra of slow decreasing
+smoothfunctions. Davies[2]pointedoutthatafunctionalcalculus baseduponalmostanalyticextensions
+was also constructed by Dyn’kin[5]. However, the two approaches w ere quite different and that Davies’
+approach was more appropriate for differential operators.
+The Spectral Mapping theorem for the Helffer-Sj¨ ostrand funct ional calculus was also independently
+proved by B´ atkai and Faˇ sanga [1]. They applied methods from ab stract functional analysis and their
+primary tool was an existing abstract Spectral Mapping Theorem f rom the theory of Banach algebras :
+Theorem 1.3 ([1] Theorem 4.1) .LetB1be a commutative, semisimple, regular Banach algebra, B2be
+a Banach algebra with unit, Θ :B1→ B2be a continuous algebra homomorphism and a∈ B1. Then
+σB1(Θ(a)) =ˆa(Sp(θ))whereSp(Θ) :=∩b∈KerΘKerˆb
+andˆdenotes the Gelfand transform.
+Our exposition, part of the Ph.D thesis referred to in the introduct ion of [1], takes a very non-abstract
+and direct approach to the proof. In particular an existing spectr al mapping is not assumed. Our sole
+ingredients, supplementing the tools provided in Davies[2], are the ve ry elementary observations:
+•It is possible to join two non-zero points in Csmoothly without passing through the origin.
+•([4] Problem 8.1.11) If His a closed operator and λlies in the topolgical boundary of the spectrum
+ofHthen for every ǫ>0 there is a vector vwith length 1 such that /bardblHv−λv/bardbl<ǫ
+•Stokes Formula has similarities to the Cauchy Integral Forumla.
+A compelling argument for a direct proof that does not rely on spect ral mapping results from the theory
+of Banach algebras follows from the claim by Davies[2] that all the calc ulations in the construction
+of his functional calculus can all be carried out at a Banach algebra le vel rather than at an operator
+level, provided one has a resolvent family in the algebra satifying the o bvious analogue of hypothesis 1.2.
+In the last part of our exposition we derive a functional calculus for operators with spectra bounded on
+one side. Our main tool here is an extension operator of Seeley :
+E:C∞[0,∞)−→C∞(R)
+1.1 Functional Calculus
+We summarize some of the main aspects of the Helffer-Sj¨ ostrand f unctional calculus presented in
+Davies[3] and some properties of the algebra. Let ψa,ǫbe a smooth function such that
+ψa,ǫ(x) :=
+
+1 ifx≥a
+0 ifx≤a−ǫ
+2Then given an interval [ a,b] we define the approximate characteristic function Ψ[a,b],ǫ
+Ψ[a,b],ǫ(x) =ψa,ǫ(x)−ψb+ǫ,ǫ(x)
+which has support [ a−ǫ,b+ǫ] and is equal to 1 in [ a,b] and is smooth.
+Definition 1.4. Forβ∈RletSβto be the set of all complex-valued smooth functions defined o nR
+where for every nthere is a positive constant cnsuch that where
+|dnf
+dxn| ≤cn/an}bracketle{tx/an}bracketri}htβ−n
+We then define the Algebra A:=/uniontext
+β<0Sβ
+Lemma 1.5 (Davies [2, 3]) .Ais an algebra under point-wise multiplication. For any finAthe
+expression
+/bardblf/bardbln:=n/summationdisplay
+r=0∞/integraldisplay
+−∞|drf
+dxr|/an}bracketle{tx/an}bracketri}htr−1dx (1.3)
+defines a norm on Afor eachn. Moreover C∞
+0(R)is dense in Awith this norm.
+Lemma 1.6. The function /an}bracketle{tx/an}bracketri}htβis inAfor eachβ <0
+Proof.The statement follows from the observations that if β <0 andm≥nthen
+xn/an}bracketle{tx/an}bracketri}htβ−m≤ /an}bracketle{tx/an}bracketri}htβ
+and
+d/parenleftbig
+xn/an}bracketle{tx/an}bracketri}htβ−m/parenrightbig
+dx=nxn−1/an}bracketle{tx/an}bracketri}htβ−m+2(β−m)xn+1/an}bracketle{tx/an}bracketri}htβ−m−2
+Lemma 1.7. Lets∈R. Iffis inAthen the function
+gs(x) :=
+
+f(x)−f(s)
+x−sx/ne}ationslash=s
+f′(s)x=s
+is also in A
+Proof.When|x−s|is large then
+1
+|x−s|≤cs/an}bracketle{tx/an}bracketri}ht−1
+for somecs>0. Moreover
+g(r)
+s(x) =r/summationdisplay
+m=0crf(m)(x) (x−s)m−r−1+cf(s)(x−s)−r−1
+and
+lim
+x→sg(m)
+s(s) =f(m+1)
+s(s)
+3Lemma 1.8. Iff∈Sβforβ <0andg∈S0thenfg∈ A
+Proof.
+|(fg)(r)(x)| ≤crr/summationdisplay
+m=0|g(r−m)(x)||f(m)(x)| ≤cr,φ/an}bracketle{tx/an}bracketri}htβ−r
+The following concept of almost analytic extensions is due to H¨ orman der[7, p63].
+Definition 1.9. Letτ(x,y)be a smooth function such that
+τ(x,y) :=
+
+1if|y| ≤ /an}bracketle{tx/an}bracketri}ht
+0if|y| ≥2/an}bracketle{tx/an}bracketri}ht
+Then given f∈ Awe define an almost analytic extension ˜fas
+˜f(x,y) :=/parenleftBiggn/summationdisplay
+r=0drf(x)
+dxr(iy)r
+r!/parenrightBigg
+τ(x,y) (1.4)
+Moreover we define
+∂˜f
+∂z:=1
+2/parenleftBigg
+∂˜f
+∂x+i∂˜f
+∂y/parenrightBigg
+(1.5)
+The following lemma establishes the construction of the new function al calculus
+Lemma 1.10 (Davies[2]) .Letf∈ Athen define
+f(H) :=−1
+π/integraldisplay/integraldisplay
+C∂˜f
+∂z(z−H)−1dxdy (1.6)
+where˜fis an almost-analytic version of fas defined in definition 1.9. Then
+i. Ifn>αthen subject to hypothesis 1.2 the integral (1.6)is norm convergent for all finAand
+/bardblf(H)/bardbl ≤c/bardblf/bardbln+1
+ii. The operator f(H)is independent of nand the cut-off function τ, subject to n>α
+iii. Iffis a smooth function of compact support disjoint from the spe ctrum ofHthenf(H) = 0
+iv. Iffandgare inAthen(fg)(H) =f(H)g(H)
+v. Ifz/ne}ationslash∈Randgz(x) := (z−x)−1for allx∈Rthengz∈ Aandgz(H) = (z−H)−1
+41.2 Preliminaries
+Definition 1.11. Givenz,ωinCwe define the curve Γin the complex plane
+Γ(z,ω,α) := ((1−α)|z|+α|ω|)ei(1−α)Arg(z)+iαArg(ω)
+whereα∈[0,1]andz,w∈C
+The important property of Γ is that it is able to connect two non-zer o points in the complex plane
+without intersection with the origin.
+Theorem 1.12. Letλ∈C. Iffis a smooth complex valued function in the interval [a,b]where
+f(a)/ne}ationslash=λandf(b)/ne}ationslash=λthen there is a smooth function hinC∞([a.b])such that
+{x∈[a,b] :h(x) =λ}is empty
+Moreoverf−gand all derivatives of f−gvanish ataandb.
+Proof.Let
+g(x) := Γ/parenleftbigg
+f(a)−λ,f(b)−λ,x−a
+b−a/parenrightbigg
++λ (1.7)
+Sincefis continuous we know there is an 0 <ǫ<b−a
+2such that
+{x∈[a,b]/(a+ǫ,b−ǫ) :f(x) =λ}=∅
+Then we can define
+h:=/parenleftbig
+1−Ψ[a+ǫ,b−ǫ],ǫ/parenrightbig
+f+Ψ[a+ǫ,b−ǫ],ǫg
+Lemma 1.13. Givenf∈ A, letλbe a non-zero point in Cand letAλ:={x:f(x) =λ}
+IfAλ∩σ(H)is empty then there is a function h∈ Asuch thath(x)/ne}ationslash=λfor allx∈Rand
+h(H) =f(H)
+Proof.IfAλis empty then we put h=f.
+IfAλis not empty then Aλis a compact subset of ρ(H). Moreover Aλcan be covered by a finite set
+of closed disjoint intervals [ ai,bi] which are also subsets of ρ(H). By applying theorem 1.12 to each
+interval we can find a function hinAsuch
+h(x) =f(x)for allx∈σ(H)
+andh(x)/ne}ationslash=λfor allx∈R. Moreover since ( f−h) has compact support in ρ(H) then it follows from
+lemma 1.10(iii) that h(H) =f(H).
+52 Bounded Operators
+We letBbe a bounded operator satisfying hypothesis (1.2). Moreover let
+u:= supσ(B) andl:= infσ(B)
+Lemma 2.1. For anyf∈ Aandǫ>0
+fΨ[l′,u′],ǫ(B) =f(B)
+wherel′≤landu′≥u
+Proof.Supposefhas compact support then f−fΨ[l′,u′],ǫhas compact support disjoint from the spec-
+trum ofBhence by lemma 1.10(iii) the statement of the lemma is true for functio ns inC∞
+0(R). The
+statement for all f∈ Afollows from density of C∞
+0(R) inA.
+Lemma 2.2. Letf∈ A. Ifǫ>0and
+Dǫ:={z:|z−u+l
+2|<u−l
+2+ǫ}and∂Dǫ:={z:|z−u+l
+2|=u−l
+2+ǫ}
+then
+f(B) =1
+2πi/integraldisplay
+∂Dǫ˜f(z)(z−B)−1dz−1
+π/integraldisplay
+Dǫ∂˜f
+∂z(z−B)−1dxdy
+Proof.By lemma 2.1 we can assume that fhas compact support in [ l−ǫ,u+ǫ]
+IfR>u−l
+2+ǫandARis the annulus {z:u−l
+2+ǫ<|z−u+l
+2|<R}then
+/integraldisplay
+|z−u+l
+2|<R∂˜f
+∂z(z−B)−1dxdy=/integraldisplay
+AR∂˜f
+∂z(z−B)−1dxdy+/integraldisplay
+Dǫ∂˜f
+∂z(z−B)−1dxdy
+Applying Stokes’ theorem
+/integraldisplay
+AR∂˜f
+∂z(z−B)−1dxdy=1
+2i/integraldisplay
+|z−u+l
+2|=R˜f(z−B)−1dz−1
+2i/integraldisplay
+∂Dǫ˜f(z−B)−1dz
+and letting Rbe large enough for ˜fto vanish on {z:|z−u+l
+2|=R}completes the proof.
+Lemma 2.3. Letǫ>0. Ifl′<landu′>uthen
+Ψ[l′,u′],ǫ(B) = 1
+Proof.Let 0<δ<1 and define Ω as the open rectangle
+{z∈C:|Rez−u′+l′
+2|<u′−l′
+2,|Imz|<δ}
+Using a similar argument to that given in the proof of lemma 2.2 we see th at
+Ψ[l′,u′],ǫ(B) =1
+2πi/integraldisplay
+∂Ω/tildewideΨ[l′,u′],ǫ(z,z)(z−B)−1dz−1
+π/integraldisplay
+Ω∂/tildewideΨ[l′,u′],ǫ
+∂z(z−B)−1dxdy
+6Whenl′≤x≤u′then Ψ[l′,u′],ǫ(x) = 1. Moreover when l′≤x≤u′then Ψ(n)
+[l′,u′],ǫ(x) = 0 for all n>0.
+Recalling definition (1.4) we can see that
+/tildewideΨ[l′,u′],ǫ(z,z) = 1 for allz∈Ω0
+hence
+Ψ[l′,u′],ǫ(B) =1
+2πi/integraldisplay
+∂Ω(z−B)−1dz
+and we conclude with an application of Cauchy’s integral formula.
+u+l
+2
+R l
 l 0 u u u+l
+2+RiR
+ii iR
+x
+	
+
+x
+
+
+Figure 1: Integral domain for lemma 2.3
+3 Enlargement of A
+We extend the algebra Aof slow decaying functions in a trivial but necessary way.
+7Definition 3.1. Let
+ˆA:={(z,f) :z∈C,f∈ A}
+where for each x∈Rwe define
+(z,f)(x) :=z+f(x)
+Moreover we define point-wise addition and multiplication:
+(ω,f)◦(z,g) := (ωz,ωg+zf+fg)
+(ω,f)+(z,g) := (ω+z,f+g)
+It is clear that (1,0)the multiplicative identity and (0,0)the additive identity are in ˆAand the algebra
+is closed under these operations.
+For anyz∈Cwe will denote (z,0)∈ˆAsimply byz.
+Givenφ= (z,f)∈ˆA, let
+πA,φ:=fandπC,φ:=z
+and let
+/bardblφ/bardbln:=|πC,φ|+/bardblπA,φ/bardbln
+Definition 3.2. We have the extended functional calculus. For φ∈ˆAlet
+φ(H) :=πA,φ(H)+πC,φI
+along with the implied norm
+/bardblφ(H)/bardbl:=|πC,φ|+/bardblπA,φ(H)/bardbl
+≤ |πC,φ|+/bardblπA,φ/bardbln+1
+=/bardblφ/bardbln+1
+Definition 3.3. Forφ∈ˆAlet:
+µ(φ) :=1
+πC,φ+πA,φ−1
+πC,φ
+Lemma 3.4. Ifφ∈ˆAand−πC,φis not inRan(πA,φ)thenµ(φ)is inAand
+φ−1=/parenleftbigg1
+πC,φ,µ(φ)/parenrightbigg
+Proof.By re-writing
+µ(φ) =1
+πC,φ+πA,φ−1
+πC,φ=−πA,φ
+πC,φ/parenleftbig
+πC,φ+πA,φ/parenrightbig
+then it is routine exercise in differentiation to show that
+−1
+πC,φ/parenleftbig
+πC,φ+πA,φ/parenrightbig
+is inS0. Then since πA,φis inA, lemma 1.8 implies the statement.
+Corollary 3.5. Givenφ∈ˆAandλ∈Csuch thatφ(x)/ne}ationslash=λfor allx∈Rthen
+(φ−λ)−1∈ˆA
+84 Spectral Mapping Theorem
+Lemma 4.1. Ifφis inˆAthen
+σ(φ(H))⊆Ran(φ)
+Proof.Givenλ∈Cwhich is not in Ran(φ) we have by by corollary 3.5
+(φ−λ)−1∈ˆA
+hence (φ(H)−λ)−1exists and is bounded and therefore λ/ne}ationslash∈σ(φ(H)).
+Lemma 4.2. Ifφis inˆAthen
+σ(φ(H))⊆φ(σ(H))∪{πC,φ}
+Proof.Letλ∈Cbe such that λ/ne}ationslash=πC,φand let
+Aλ={x:φ(x) =λ}
+IfAλ∩σ(H) =∅then by lemma 1.13 we have that there is function hinAsuch that
+h(x) =πA,φ(x)for allx∈σ(H)
+and
+h(x)/ne}ationslash=λ−πC,φfor allx∈R
+moreover
+h(H) =πA,φ(H)
+Ifθ:=/parenleftbig
+πC,φ,h/parenrightbig
+∈ˆAit follows from lemma 1.10 we have
+φ(H) =θ(H)
+Sinceλ /∈Ran(θ), the statement of the lemma follows from lemma 4.1.
+Lemma 4.3. Letφ∈ˆA. IfHis bounded and
+{x:φ(x) =πC,φ}∩σ(H)is empty
+thenπC,φ/∈σ(φ(H))
+Proof.Letu:= supσ(H) andl:= infσ(H).
+Let 0<ǫ≪1 such that πA,φis not zero on [ l−ǫ,l] and on [u,u+ǫ].
+Then letu′:=u+ǫandl′:=l−ǫ.
+The set
+{x∈[l′,u′] :πA,φ(x) = 0}
+can be covered by a finite number of disjoint intervals [ ai,bi] which are all disjoint from σ(H) and are
+all in [l′,u′]. Applying lemma 1.12 to each [ ai,bi] we can find a function f∈ Asuch that
+{x∈[l′,u′] :f(x) = 0}=∅
+9andf=πA,φfor all x in R/[l′,u′].
+Letgbe any function in Asuch thatg(x) =1
+f(x)for allx∈[l′,u′]
+By lemma 1.10(iii) we have
+πA,φ(H)g(H) =f(H)g(H)
+and by lemma 2.1 we have
+f(H)g(H) = (fgΨ[l′,u′],ǫ)(H) = Ψ[l′,u′],ǫ(H)
+hence by lemma 2.3 we have
+πA,φ(H)g(H) = 1
+and consequently/parenleftbig
+πC,φ−φ(H)/parenrightbig
+g(H) = 1
+Theorem 4.4. IfφinˆAthenσ(φ(H))⊆φ(σ(H))
+Proof.IfHis unbounded then φ(σ(H)) =φ(σ(H))∪{πC,φ}and the theorem follows from lemma 4.2.
+IfHis bounded and there is an x∈σ(H) such that φ(x) =πC,φthenφ(σ(H)) =φ(σ(H))∪{πC,φ}
+and again the theorem follows from 4.2. If His bounded and
+φ(x)/ne}ationslash=πC,φ
+for allx∈σ(H) thenφ(σ(H)) =φ(σ(H)) by lemmas 4.2 and 4.3.
+Lemma 4.5. Givens∈Rand a function f∈ A, letks(x) :=/parenleftBig
+1,−s+i
+x+i/parenrightBig
+∈ˆAand let the function gs
+be defined as in lemma 1.7 then
+(f(H)−f(s))(H+i)−1=gs(H)ks(H)
+Proof.This statement follows directly from the functional calculus and the observation
+(−f(s),f(x))/parenleftBig
+0,(x+i)−1/parenrightBig
+=/parenleftbigg
+0,f(x)−f(s)
+x−s/parenrightbigg/parenleftbigg
+1,−s+i
+x+i/parenrightbigg
+Theorem 4.6. Letfbe a function in Athen
+f(σ(H))⊆σ(f(H))
+Proof.We observe the identity
+H−x= (H+i)−(x+i) =/parenleftBig
+1−(x+i)(H+i)−1/parenrightBig
+(H+i) =kx(H)(H+i) (4.8)
+for somex∈R.
+Lets∈R. Supposethereisasequenceofunitlengthvectors {vm} ⊂Dom(H)suchthat lim
+m→∞(H−s)vm=
+0. Using identity (4.8 ) we have lim
+m→∞gs(H)ks(H)(H+i)vm= 0. By applying lemma 4.5 we can con-
+clude that lim
+m→∞(f(H)−f(s))vm= 0.
+The accumulation points of f(σ(H)) are inσ(f(H)) since the latter is closed.
+10Corollary 4.7. Letφbe a function ˆAthen
+φ(σ(H))⊆σ(φ(H))
+5 Self-Adjoint Operators
+We now assume that His self-adjoint and Bis a Hilbert space.
+The following theorem of Davies extends the Helffer-Sj¨ ostrand fu nctional calculus to C0(R) for self-
+adjoint operators.
+Theorem 5.1 (Davies[2] Theorem 9) .The functional calculus may be extended to a map from f∈
+C0(R)tof(H)∈ L(B)with the follwoing properties:
+i.f→f(H)is an algebra homomorphism.
+ii.f(H) =f(H)∗
+iii./bardblf(H)/bardbl ≤ /bardblf/bardbl∞
+iv. Ifz/ne}ationslash∈Randgz(x) := (z−x)−1for allx∈Rthengz(H) = (z−H)−1
+Moreover the functional calculus is unique subject to these conditions.
+Lemma 5.2. Iff∈C0(R)then
+f(σ(H))⊆σ(f(H))
+Proof.This is a consequence of the density of AinC0(R). By the Stone-Weierstrass theorem the linear
+subspace
+{n/summationdisplay
+i=1λi
+x−ωi:λ∈Cωi/∈R}
+is dense in C0(R). Iffǫ∈ Ais close tofand ifv∈ Bis of length 1 then
+/bardblf(H)v−f(s)v/bardbl ≤ /bardblf(H)−fǫ(H)/bardbl+/bardblfǫ(H)v−fǫ(s)v/bardbl+/bardblfǫ−f/bardbl∞
+The statement then follows from lemma 5.1(iii)
+Lemma 5.3. Iff∈C0(R)then
+σ(f(H))⊆f(σ(H))
+Proof.Let
+f:=∞/summationdisplay
+i=1λi
+x−ωiandfn:=n/summationdisplay
+i=1λi
+x−ωi
+Supposeλ∈Cis not in the closure of f(σ(H)). Then there is δ>0 such that
+inf
+s∈σ(H)|f(s)−λ|=δ
+11Also for all large enough nwe have /bardblfn−f/bardbl∞<δ
+2. Then from
+|f(s)−fn(s)+fn(s)−λ|>δ
+we can deduce that
+|fn(s)−λ|>δ−/bardblfn−f/bardbl∞
+hence
+inf
+s∈σ(H)|fn(s)−λ|>δ
+2
+andλ /∈σ(fn(H)).
+From the identity
+/bardbl(f(H)−λ)(fn(H)−λ)−1−1/bardbl=/bardbl(f(H)−fn(H))(fn(H)−λ)−1/bardbl
+we can deduce that λ /∈σ(f(H)).
+6 Functional Calculus for Semi-Bounded Operators
+We modify our main hypothesis(1.2) by assumingthat the spectrum o fHis bounded belowand without
+loss of generality σ(H)⊆[0,∞).
+We introduce a new ring of functions A+
+Definition 6.1. Sβ
++is the set of smooth functions on R+∪{0}with the same decaying property as Sβ
+that is for every nthere is positive constant cnsuch that
+|dnf
+dxn| ≤cn/an}bracketle{tx/an}bracketri}htβ−n
+ThenA+is defined appropriately and similarly we define the Banach sp aceA+
+nwith norm
+/bardblf/bardblA+
+n:=n/summationdisplay
+r=0∞/integraldisplay
+0|drf
+dxr|/an}bracketle{tx/an}bracketri}htr−1dx (6.9)
+We present a theorem due to Seeley [8] which gives a linear extension operator for smooth functions
+from the half space to the whole space.
+Theorem 6.2 (Seeley’s Extension Theorem) .There is a linear extension operator
+E:C∞[0,∞)−→C∞(R)
+such that for all x>0
+(Ef)(x) =f(x)
+The extension operator is continuous for many topologies including u niform convergence of each deriva-
+tive. The proof of the theorem relies on the following lemma.
+12Lemma 6.3 ([8]).There are sequences {ak},{bk}such that
+i.bk<0
+ii.∞/summationtext
+k=0|ak||bk|n<∞for all non-negative integers n
+iii.∞/summationtext
+k=0ak(bk)n= 1for all non-negative integers n
+iv.bk→ −∞
+The proof to Seeley’s extension theorem is by construction and it is in formative to give explicitly the
+extension. First we need a to define two linear operators.
+Definition 6.4. Givenf∈ A+,φ∈ Aand realawe define two operators on A+,
+(Taf)(x) =f(ax)
+(Sφf)(x) =φ(x)f(x)
+Proof of Seeley’s Extension Theorem. Letφ∈C∞
+c(R) such that
+φ(x) =
+
+1x∈[0,1]
+0x≥2
+0x≤ −1
+Then define Esuch that
+(Ef)(x) :=
+
+∞/summationtext
+k=0ak(TbkSφf)(x)x<0
+f(x) x≥0
+Lemma 6.5. Ifa>1then/bardblTa/bardblA+
+n→A+
+n≤an
+Proof.Follows from
+/bardblTaf/bardblA+
+n=n/summationdisplay
+r=1∞/integraldisplay
+0|drf(ax)
+dxr|/an}bracketle{tx/an}bracketri}htr−1dx≤n/summationdisplay
+r=1ar∞/integraldisplay
+0|drf(x)
+dxr|/an}bracketle{tx/an}bracketri}htr−1dx
+Lemma 6.6. Ifφ∈ AthenSφis a bounded operator with respect to each norm /bardbl/bardblA+
+n
+Proof.A simple application of Leibnitz gives
+dr(φ(x)f(x))
+dxr=r/summationdisplay
+m=0crdr−m(φ(x))
+dxr−mdm(f(x))
+dxm
+13then
+|dr(φ(x)f(x))
+dxr| ≤crr/summationdisplay
+m=0dr−m,φ/an}bracketle{tx/an}bracketri}htβ−(r−m)dm(f(x))
+dxm
+≤cr,φr/summationdisplay
+m=0/an}bracketle{tx/an}bracketri}htm−rdm(f(x))
+dxm
+we integrate to give
+∞/integraldisplay
+0|dr(φ(x)f(x))
+dxr|/an}bracketle{tx/an}bracketri}htr−1dx≤cr,φr/summationdisplay
+m=0∞/integraldisplay
+0|dm(f(x))
+dxm|/an}bracketle{tx/an}bracketri}htm−1dx
+=cr,φ/bardblf/bardblA+
+r
+and hence we have our estimate
+/bardblSφf/bardbln=n/summationdisplay
+r=0∞/integraldisplay
+0|d(φ(x)f(x))r
+drx|/an}bracketle{tx/an}bracketri}htr−1dx
+≤cn,φn/summationdisplay
+r=0/bardblf/bardblA+
+r
+≤cn,φ/bardblf/bardblA+
+n
+Theorem 6.7. Seeley’s Extension Operator is a bounded operator on each of the normed vector spaces
+A+
+n
+Proof.
+/bardblEf/bardblAn=n/summationdisplay
+r=0∞/integraldisplay
+−∞|dr(Ef)
+dxr|/an}bracketle{tx/an}bracketri}htr−1dx
+=n/summationdisplay
+r=0∞/integraldisplay
+0|drf(x)
+dxr|/an}bracketle{tx/an}bracketri}htr−1dx+n/summationdisplay
+r=00/integraldisplay
+−∞|∞/summationdisplay
+0akdr(φ(bkx)f(bkx))
+dxr|/an}bracketle{tx/an}bracketri}htr−1dx
+=/bardblf/bardblA+
+n+/bardbl∞/summationdisplay
+k=0akT−bkSφf/bardblA+
+n
+≤ /bardblf/bardblA+
+n+∞/summationdisplay
+k=0|ak|/bardblSφ/bardbl/bardbl|T−bk/bardbl/bardblf/bardblA+
+n
+≤ /bardblf/bardblA+
+n+/parenleftBigg∞/summationdisplay
+k=0|ak||bk|n/parenrightBigg
+cn,φ/bardblf/bardblA+
+n
+and hence the extension operator is continuous.
+Iffandgare elements of Asuch thatf|[0,∞]=g|[0,∞]and the spectrum of His [0,∞) then it is not
+necessary that supp(f−g)∩σ(H) is empty, since supp(f−g)∩σ(H) ={0}is possible.
+14Lemma 6.8. Iffis a smooth function on Rof compact support such that
+supp(f) = [−a,0]
+andHis an operator satisfying our modified hypothesis with σ(H)⊆[0,∞]then
+f(H) = 0
+Proof.Letǫ∈(0,1) and define
+fǫ(x) :=f(x+ǫ)
+so thatsupp(fǫ) = [−(a+ǫ),−ǫ].
+By lemma 1.10(iii) fǫ(H) = 0. For all nthere are constants cn≥0 such that
+/bardbldnf
+dxn−dnfǫ
+dxn/bardbl∞≤cnǫ
+then
+/bardblf(H)/bardbl=/bardblf(H)−fǫ(H)/bardbl
+≤n/summationdisplay
+r=00/integraldisplay
+−(a+1)|drf(x)
+dxr−drfǫ(x)
+dxr|/an}bracketle{tx/an}bracketri}htr−1dx
+≤n/summationdisplay
+r=0ǫcr0/integraldisplay
+−(a+1)/an}bracketle{tx/an}bracketri}htr−1dx
+=ǫkn,f
+hence our result.
+Corollary 6.9. Iffandgare inAsuch thatf|[0,∞]=g|[0,∞]andσ(H)⊆[0,∞]thenf(H)−g(H) = 0
+Theorem 6.10. IfHsatisfies our modified hypothesis with spectrum σ(H)⊆[0,∞)then there is a
+functional calculus γH:A+→ L(B)and moreover for all f∈ A+∩A
+γH(f) =−1
+π/integraldisplay/integraldisplay
+C∂˜f
+∂z(z−H)−1dxdy
+Proof.Letf+∈ A+, then by Seeley’s Extension Theorem there exists an extension f∈ A. We define
+γH(f+) :=f(H). This definition is independent of the particular extension by corolla ry 6.9. The
+functional analytic properties are inherited from the extension.
+Theorem 6.11 (Refinement of Theorem 10 of [2]) .Letn≥1be an integer and t>0.
+If we denote the operator γH/parenleftbig
+e−snt/parenrightbig
+bye−Hntthen
+e−Hn(t1+t2)=e−Hnt1e−Hnt2
+for alln≥1and0<t≤1
+15Acknowledgements
+This research was funded by an EPSRC Ph.D grant 95-98 at Kings Colle ge, London. I am very grateful
+to E. Brian Davies for giving me this problem, his encouragement since and for continuing to be a
+mentor in Mathematics long after having finished supervising my Ph.D. I am indebted to Anita for all
+her support.
+References
+[1] A. B´ atkai and E. Faˇ sanga The Spectral Mapping Theorem for Davies’ functional calcul us.
+Rev. Roumaine Math. Pures Appl. 48 (2003), no. 4, 365–372.
+[2] E.B. Davies, The Functional Calculus ,
+J. London Math. Soc Vol 52, Number 1 166-176
+[3] E.B. Davies, Spectral Theory and Differential Operators ,
+Cambridge Studies in Adv. Math. 42, Cambridge Univ. Press, 1995.
+[4] E.B. Davies, Linear Operators and their Spectra ,
+Cambridge Studies in Adv. Math. 106, Cambridge Univ. Press, 2007.
+[5] E.M.Dyn’kin, An operator calculus based upon the Cauchy-Green formula, a nd the quasi-analyticity
+of the classes D(h)
+Sem. Math. V A Steklov Math. Inst. Leningrad 19 (1972) 128-131.
+[6] B. Helffer and J. Sjstrand, quation de Schrdinger avec champ magntique et quation de Har per,
+Schrdinger Operators
+(Snderborg, 1988) eds. H. Holden and A. Jensen, Lecture Notes in Phys., vol. 345, Springer-Verlag,
+Berlin, 1989, pp. 118-197
+[7] L. H¨ ormander, The Analysis of Linear Partial Differential Operators Vol 1,
+Springer, New York 1993
+[8] S.T. Seeley Extensions of C∞functions defined on a half space ,
+Proc. Amer. Math. Soc 15 1964
+Narinder Claire
+Global Equities & Commodity Derivatives Quantitative Research
+BNP Paribas London
+10 Harewood Avenue
+London
+NW1 6AA
+e-mail: narinder.claire@uk.bnpparibas.com
+16
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+arXiv:1001.0233v2  [math.OA]  22 Apr 2013A homomorphism theorem and a Trotter product
+formula for quantum stochastic flows with
+unbounded coefficients
+Biswarup Das,1Debashish Goswami2and Kalyan B. Sinha3
+Abstract
+We give a new method for proving the homomorphic property of
+a quantum stochastic flow satisfying a quantum stochastic differen tial
+equation with unbounded coefficients, under some further hypoth eses.
+As an application, we prove a Trotter product formula for quantum
+stochastic flows and obtain quantum stochastic dilations of a class o f
+quantum dynamical semigroups generalizing results of [5] .
+1 Introduction
+The theory of quantum stochastic calculus and of the quantum stochas-
+tic differential equations have been developed over the last t hree decades
+([12], [16]). These methods have many possible application s in the study
+of non-equilibrium and dissipative physical systems, for e xample, modeling
+of damped quantum harmonic oscillator [16]. They are also of independent
+mathematical interest, particularly those with unbounded coefficients.
+However, a difficulty in dealing with quantum stochastic calc ulus with
+unbounded coefficients is the absence of a convenient method f or proving
+the homomorphic property of a quantum stochastic flow. Suppo se that
+A ⊆B(h) is a von-Neumann or C*-algebra and A0⊆ Ais a dense (in
+the appropriate topology) ∗-subalgebra, such that there exists a family of
+completely positive, contractive cocycles jt:A → A′′⊗B(Γ(L2(R+,k))) (t≥
+0),satisfying an equation of the form
+jt(x) =x+/summationdisplay
+µ,ν/integraldisplayt
+0js(θµ
+ν(x))Λν
+µ(ds),
+wherekis a Hilbert space, Γ( L2(R+,k)) is the symmetric Fock space over
+L2(R+,k) and (θµ
+ν)µ,νare linear maps on A0. In general, theredoes not exist
+a convenient method for showing that such a map jtis a∗-homomorphism.
+Using the quantum Itˆ o formula, one can easily write down alg ebraic
+relations which are necessary for a quantum stochastic flow ( jt)t≥0to be
+1Indian Statistical Institute, Kolkata; partially support ed by UKIERI.
+2Indian Statistical Institute, Kolkata; partially support ed bySwarnajayanti Fellowship
+and Grant from DST, Govt. of India.
+3J.N. Centre for Advanced Scientific Research and Indian Inst itute of Science, Ban-
+galore .
+1∗-homomorphic in general. But it is not known whether such alg ebraic con-
+ditions are also sufficient. In this paper, we prove that such a lgebraic con-
+ditions will be sufficient, if we furthermore assume some anal ytic conditions
+on the flow as well as the underlying (C* or von-Neumann) algeb ra. The
+crucial aspect of this proof is the implementation of an indu ctive procedure
+on the Itˆ o formula which is more natural in the set-up of quan tum stochastic
+flows with unbounded coefficients than the usual iterative pro cedure.
+We give a number of applications of the method, including a ne w Trotter
+product formula, generalizing the formulas obtained in [7] .
+2 Notation and terminologies.
+2.1 Some generalities on Hilbert spaces and completely pos-
+itive maps
+We shall refer the reader to [12], [3], [16] and references th erein for the basics
+of the formalism of quantum stochastic calculus, which we br iefly review
+here. All the Hilbert spaces appearing in this article will b e separable. We
+adopt the convention that inner-product is linear in the rig ht and conjugate
+linear in the left. Let CandINbe the set of complex and natural numbers
+respectively. For n∈IN,Mn(C) will denote the set of n×nmatrices
+with complex entries. For A∈Mn(C), we will often write A≡((aij)),
+where{aij,i,j= 1,...,n}are the entries of A. For a Hilbert space H,
+letIHbe the identity operator on H; writeIn:=ICn. We denote the
+symmetric Fock space over Hby Γ(H) and the exponential vector over
+f∈ Hbye(f). The vector e(0) is called the vacuum vector. We adopt
+the convention that for u∈ H,|u/an}bracketri}ht ∈ Hand/an}bracketle{tu| ∈ H∗. Foru,v∈ H, the
+rank one operator |u/an}bracketri}ht/an}bracketle{tv| ∈B(H) is given by |u/an}bracketri}ht/an}bracketle{tv|(|w/an}bracketri}ht) :=/an}bracketle{tv,w/an}bracketri}ht|u/an}bracketri}ht, for all
+|w/an}bracketri}ht ∈ H.Lin(V,W) will denote the space of linear maps from a subspace
+D(L) (called the domain of L) of a vector space Vto another vector space
+W. The tensor product of Hilbert spaces or of operators will us ually be
+denoted by ⊗, whereas ⊗algwill be used for the algebraic tensor product.
+We shall also use the projective tensor product ⊗γof Banach spaces, which
+will be explained later.
+LetH1,H2be two Hilbert spaces and A∈Lin(H1,H1⊗H2).Suppose
+thatD:=D(A). For each f∈ H2, we define a linear operator /an}bracketle{tf,A/an}bracketri}htwith
+domainDand taking values in H1such that
+/an}bracketle{tu,/an}bracketle{tf,A/an}bracketri}htv/an}bracketri}ht=/an}bracketle{tu⊗f,Av/an}bracketri}ht (1)
+forv∈ D, u∈ H1. We denote by /an}bracketle{tA,f/an}bracketri}htthe adjoint of /an}bracketle{tf,A/an}bracketri}ht, whenever it
+exists.
+Forφ∈Lin(D0⊗algV0,H1⊗H2),whereD0andV0are subspaces of H1
+andH2respectively, f∈V0andg∈ H2, define the partial trace of φwith
+2respect to the rank one operator |f ><g|, denoted by /an}bracketle{tg,φf/an}bracketri}ht:D0−→ H 1
+by:
+/an}bracketle{tu,/an}bracketle{tg,φf/an}bracketri}htv/an}bracketri}ht=/an}bracketle{tu⊗g),φ(v⊗f)/an}bracketri}ht,(v∈D0,u∈ H1).
+We also denote by φf∈Lin(D0,H1⊗H2) given byφf(v) =φ(v⊗f).
+LethandHbe Hilbert spaces, A ⊆B(h) be aC∗or von-Neumann
+algebra, and ψt:A →B(h⊗Γ(L2(R+,H))),t≥0 be a family of completely
+positive, contractive maps. For c,d∈ H, we define a map ψc,d
+t:A →B(h)
+as follows:
+ψc,d
+t(x) :=/an}bracketle{te(c1[0,t)),ψt(x)e(d1[0,t))/an}bracketri}ht(x∈ A). (2)
+We conclude this subsection with the following proposition , which will
+be needed to define a quantum stochastic cocycle.
+Proposition 2.1. LetH1,H2andhbe Hilbert spaces and A ⊆B(h)be
+a C*-algebra. Suppose that the maps ψi:A → A′′⊗B(Hi),i= 1,2are
+completely positive, contractive and satisfy /an}bracketle{tξi,ψi(x)ηi/an}bracketri}ht ∈ Afor allx∈ A
+andξi,ηi∈ Hi, i= 1,2. Then there exists a completely positive, contractive
+map, denoted by ψ1•ψ2fromAtoA′′⊗B(H1⊗H2)satisfying
+(i)
+/an}bracketle{tξ,(ψ1•ψ2)(x)η/an}bracketri}ht ∈ A(x∈ A,ξ,η∈ H1⊗H2).
+(ii)
+/an}bracketle{tξ1⊗ξ2,(ψ1•ψ2)(·)(η1⊗η2)/an}bracketri}ht=/an}bracketle{tξ1,ψ1(/an}bracketle{tξ2,ψ2(·)η2/an}bracketri}ht)η1/an}bracketri}ht,
+forξi,ηi∈ Hi(i= 1,2).
+Proof.Letn≥1,ω:=/summationtextn
+i=1ui⊗ξi⊗ηi∈h⊗H1⊗H2, whereui∈h,ξi∈ H1
+andηi∈ H2for alli= 1,2,...,n. Note that the map IEη:A′′⊗B(H2)→
+Mn(A′′) (∼=A′′⊗Mn) given by IEη(X) = ((/an}bracketle{tηi,Xηj/an}bracketri}ht))n
+i,j=1is completely
+positive. Thus, for X≥0, we have
+0≤IEη(X)≤ /bardblX/bardblIEη(1) =/bardblX/bardbl1A′′⊗((/an}bracketle{tηi,ηj/an}bracketri}ht)). (3)
+Moreover, if X(≥0) is such that /an}bracketle{tηi,Xηj/an}bracketri}ht ∈ Afor alli,j, thenIEη(X)∈
+Mn(A) andY:= (ψ1⊗idMn)(IEη(X)) is well-defined. As ψ1is completely
+positive and contractive, we get the following from (3):
+0≤(ψ1⊗idMn)((eα⊗1)IEη(X)(eα⊗1))≤ /bardblX/bardbl1A′′⊗B(H1)⊗((/an}bracketle{tηi,ηj/an}bracketri}ht))
+where{eα}αis any approximate unit for A. Taking limit with respect to α,
+we get
+0≤Y≤ /bardblX/bardbl1A′′⊗B(H1)⊗((/an}bracketle{tηi,ηj/an}bracketri}ht)). (4)
+3Clearly,Y∈Mn(A′′⊗B(H1)) has the matrix form Y= ((Yij)) where
+Yij=ψ1(/an}bracketle{tηi,Xηj/an}bracketri}ht), and by (4) for ui∈h(i= 1,2,...,n),
+0≤n/summationdisplay
+i,j=1/an}bracketle{tui⊗ξi,ψ1(/an}bracketle{tηi,Xηj/an}bracketri}ht)(uj⊗ξj)/an}bracketri}ht ≤ /bardblX/bardbln/summationdisplay
+i,j=1/an}bracketle{tui,uj/an}bracketri}ht/an}bracketle{tξi,ξj/an}bracketri}ht/an}bracketle{tηi,ηj/an}bracketri}ht.
+Asn,uiandηiare arbitrary, we conclude that there is a unique bounded
+positive operator TX(say) inA′′⊗B(H1)⊗B(H2) such that /bardblTX/bardbl ≤ /bardblX/bardbl
+and/an}bracketle{tω,TXω/an}bracketri}ht=/summationtextn
+i,j=1/an}bracketle{tui⊗ξi,ψ1(/an}bracketle{tηi,Xηj/an}bracketri}ht)(uj⊗ξj)/an}bracketri}htforω=/summationtextn
+i=1ui⊗
+ξi⊗ηi. TakingX=ψ2(x) wherex∈ Ais positive, we denote by ( ψ1•
+ψ2)(x) the operator Tψ2(x), and then it is easy to extend the map x/ma√sto→
+(ψ1•ψ2)(x) linearly to all x∈ A. Properties (i) and (ii) in the statement of
+the proposition now follow easily, thereby completing the p roof.✷
+Remark 2.2. In caseAis a von-Neumann algebra and ψiis normal for
+i= 1,2, we have
+ψ1•ψ2= (ψ1⊗idB(H2))◦ψ2.
+In the more general case treated in [8], ψ1•ψ2coincides with (ψ1⊗M
+idB(H2))◦ψ2, where⊗Mdenotes the matrix space tensor product, introduced
+in [8].
+2.2 Quantum stochastic analysis in symmetric Fock space
+For this subsection and rest of the paper, we fix a separable Hi lbert space
+k0, to be called the noise or multiplicity space. Suppose that {ei}i∈Iis
+an orthonormal basis for k0, where I={1,2,...,dim(k0)}ifk0is finite
+dimensional and I=INotherwise. Let Edenote the subspace spanned
+by the exponential vectors in Γ( L2(R+,k0)). Set Γ := Γ( L2(R+,k0)), Γs
+r:=
+Γ(L2([s,r),k0))(s<r), Γt:= Γ(L2([0,t),k0))(t≥0)andΓt:= Γ(L2([t,+∞),k0)).
+We recall the four fundamental martingales of quantum stoch astic cal-
+culus (see [12]):
+Time:
+{Λ0
+0(t)}t≥0∈Lin(E,Γ) : Λ0
+0(t)e(f) :=te(f) (t≥0,e(f)∈ E);
+Annihilation:
+{Λi
+0(t)}t≥0∈Lin(E,Γ) : Λi
+0(t)e(f) :=/integraldisplayt
+0dsfi(s)e(f) (t≥0,i∈ I,e(f)∈ E),
+wherefi(s) :=/an}bracketle{tei,f(s)/an}bracketri}ht.
+Creation:
+{Λ0
+i(t)}t≥0∈Lin(E,Γ) : Λ0
+i(t) := Λi
+0(t)∗(t≥0,i∈ I)
+4i.e. fore(f),e(g)∈ E,
+/an}bracketle{te(f),Λ0
+i(t)e(g)/an}bracketri}ht:=/integraldisplayt
+0dsfi(s)/an}bracketle{te(f),e(g)/an}bracketri}ht,
+wherefi(s) :=fi(s).
+Number:
+{Λi
+j(t)}t≥0∈Lin(E,Γ) : Λi
+j(t)e(f) :=/integraldisplayt
+0dsfi(s)Λ0
+j(s)e(f) (t≥0,i,j∈ I,e(f)∈ E).
+Thefundamentalmartingales {Λα
+β}α,β∈I∪{0}satisfy thequantum-Itˆ o for-
+mula (see page 127 of [16]):
+Λα
+β(dt)Λα′
+β′(dt) =ˆδα
+β′Λα′
+β(dt)
+forα,α′,β,β′= 0,1,2,3..., where
+ˆδα
+β:= 0if α= 0or β= 0
+:=δα,βotherwise,(5)
+δα,βbeing the Kr¨ onecker delta symbol.
+Remark 2.3. It follows by a direct computation that
+/an}bracketle{te(f),Λµ
+ν(t)e(g)/an}bracketri}ht=/an}bracketle{tΛν
+µ(t)e(f),e(g)/an}bracketri}ht,
+or in other words, we have (Λµ
+ν(t))∗= Λν
+µ(t),t≥0.
+Let(θt)t≥0denotesthesemigroupofunilateralshiftson L2(R+,k0), given
+by:
+(θtf)(s) =f(s−t)1[t,+∞)(s) (t≥0,s∈R+,f∈L2(R+,k0)).
+Define (Γ(θt))t≥0∈B(Γ) as follows:
+Γ(θt)e(f) :=e(θtf) (t≥0,s∈R+,f∈L2(R+,k0)), (6)
+and extending linearly. It can be shown (see p. 169, Section 7 .1 of [16]) that
+(Γ(θt))t≥0becomes aC0-semigroup of isometries in Γ.
+Let{Ii}i∈Jbe a collection of disjoint sub-intervals of R+. It can be
+shown (page 31, Corollary 2.4.4 of [16]), that
+Γ(L2(/uniondisplay
+iIi,k0))∼=/circlemultiplydisplay
+iΓ(L2(Ii,k0)),
+where in the right hand side, the tensor product is with respe ct to the
+stabilizing sequence {ΩIi}i∈J, ΩIibeing the vacuum vector in Γ( L2(Ii,k0)),
+5i∈ J. Thus forr,s∈R+,r<s, we can view Γr
+sas a subspace of Γ via the
+identification :
+Γ(L2([r,s),k0))∼=Ω[0,r)⊗Γ(L2([r,s),k0))⊗Ω[s,+∞),
+where the Hilbert space on the right hand side is a subspace of Γr⊗Γr
+s⊗Γs.
+Lethbe a separable Hilbert space and A ⊆B(h) be a C* or von-
+Neumann algebra. For X∈ A′′⊗B(Γs), it can be seen that ( Ih⊗Γ(θt))(X⊗
+IΓs)(Ih⊗Γ(θ∗
+t)) is of the form P12(|Ω[0,t)><Ω[0,t)|⊗X1⊗IΓt+s)P∗
+12, where
+X1∈ A′′⊗B(Γt
+t+s) andP12: Γt⊗h⊗Γt−→h⊗Γt⊗Γtis the unitary flip
+between first and second tensor components. Set Ξ t(X) :=X1.
+Proposition 2.4. (I) Suppose that A ⊂B(h)is a C*-algebra. If X∈
+A′′⊗B(Γs)such that /an}bracketle{tξ,Xη/an}bracketri}ht ∈ Afor allξ,η∈Γs, then/an}bracketle{tξ′,Ξt(X)η′/an}bracketri}ht ∈
+A, for allξ′,η′∈Γt
+t+s.
+(II) Suppose that jt:A → A′′⊗B(Γ);t≥0is a family of completely
+positive maps such that
+jt=/hatwidejt⊗IΓt, jc,d
+t(A)⊆ A(t≥0,c,d∈k0),
+where/hatwidejtis a completely positive map from AtoA′′⊗B(Γt).
+Then the map Js+t:=/hatwidejs•(Ξs◦/hatwidejt)is a well-defined map from Ato
+A′′⊗B(Γs⊗Γs
+t+s).
+Proof.Letξ′,η′∈Γt
+s+t. By virtue of the discussions on Fock space and shift
+operators preceding Proposition 2.4, it follows that
+/angbracketleftbig
+ξ′,Ξt(X)η′/angbracketrightbig
+=/angbracketleftbig
+Γ(θ∗
+t)(Ω[0,t)⊗ξ′⊗Ω[s+t,+∞)),(X⊗IΓs)Γ(θ∗
+t)(Ω[0,t)⊗η′⊗Ω[s+t,+∞))/angbracketrightbig
+.
+The right hand side of the above equation belongs to A, which proves (I).
+(II) follows by combining (I) and Proposition 2.1. ✷
+Definition 2.5. A family of contractive and completely positive maps (jt)t≥0
+fromAtoA′′⊗B(Γ)is called a quantum stochastic cocycle on Aif the family
+(jt)t≥0satisfies:
+(i)Adaptedness:
+jt=/hatwidejt⊗IΓt, jc,d
+t(x)∈ A(x∈ A,c,d∈k0),
+where/hatwidejtis a completely positive map from AtoA′′⊗B(Γt).
+(ii)Cocycle property:
+/hatwidejt+s(X) ={/hatwidejs•(Ξs◦/hatwidejt)}(X) (X∈ A).
+6Remark 2.6. Note that by virtue of Proposition 2.4, the right hand side of
+(ii) is well-defined. It also follows from (ii) of Definition 2. 5 that{jc,d
+t}t≥0
+is a semigroup of maps on A(see [1]).
+If the C*-algebra Ais not closed in the ultraweak topology of B(h), then
+wewill refer tothenormtopology on Aas its naturaltopology. Ontheother
+hand, ifAis a von-Neumann algebra, the ultraweak topology will be cho sen
+as the natural topology. We call a semigroup ( Tt)t≥0of bounded maps on A,
+aC0-semigroup if the map t→Tt(x) is continuous in the natural topology
+ofAfor every fixed x∈ A.
+We now introduce a special class of quantum stochastic cocyc les which
+are of interest to us in this paper.
+Definition 2.7. A quantum stochastic cocycle (jt)t≥0on a C*-algebra A
+is called a (C*-algebraic) quantum stochastic flow on Awith the domain
+algebraA0, structure maps {θµ
+ν}µ,ν∈Lin(A,A)and noise space k0, where
+A0is a∗-subalgebra of A,µ,ν∈ {0}∪I, if the following hold :
+(i) The map t→jt(x)is weakly measurable for every x∈ A, in the sense
+that
+t/ma√sto→ /an}bracketle{tu⊗ξ,jt(x)(v⊗η)/an}bracketri}ht(u,v∈h,ξ,η∈Γ)
+is a Borel-measurable map from R+toC.
+(ii) The domain algebra A0is norm-dense in A,A0⊆D(θµ
+ν),µ,ν≥0, and
+the family {jt(x)}t≥0satisfies a weak quantum stochastic differential
+equation of the following form: ∀u,v∈h, f,g∈L2(R+,k0)and
+x∈ A0,
+/an}bracketle{tjt(x)ue(f),ve(g)/an}bracketri}ht
+=/an}bracketle{txue(f),ve(g)/an}bracketri}ht+/summationdisplay
+µ,ν/integraldisplayt
+0ds/an}bracketle{tjs(θµ
+ν(x))ue(f),ve(g)/an}bracketri}htgµ(s)fν(s),
+(7)
+or symbolically,
+djt(x) =/summationdisplay
+µ,νjs(θµ
+ν(x))Λν
+µ(ds);
+j0(x) =x⊗IΓ,(8)
+where we set f0(s) =f0(s) =g0(s) =g0(s) = 1.
+7IfAis a von-Neumann algebra, we define a (von-Neumann algebraic ) quan-
+tum stochastic flow on it in a similar way, replacing the norm- density of A0
+by the ultra-weak density and also assuming the normality of jtfor eacht.
+A quantum stochastic flow (jt)t≥0is called a ∗-homomorphic flow if fur-
+thermorejtis a∗-homomorphism for each t≥0.
+Often it is convenient to write the family of structure maps a s a matrix
+Θ of maps given by:
+Θ :=/parenleftbiggLδ†
+δ σ/parenrightbigg
+, (9)
+whereσ:=/summationtext
+i,jθi
+j(x)⊗ |ej>< ei|, δ(x) :=/summationtext
+iθi
+0(x)⊗ |ei/an}bracketri}ht,δ†(x) :=/summationtext
+iθ0
+i(x)⊗ /an}bracketle{tei|andL(x) :=θ0
+0(x),forx∈ A0.We call Θ the structure
+matrix associated with the flow ( jt)t≥0.
+Remark 2.8. Suppose that a family of contractive and completely positiv e
+flow(jt)t≥0satisfies (i) of Definition 2.5. If furthermore (jt)t≥0satisfies the
+equation (7)of Definition 2.7 with bounded structure maps, then it follow s
+that(jt)t≥0satisfies (ii) of Definition 2.5 (see page 171, Lemma 7.1.3 in
+[16]).
+Lemma 2.9. If(jt)t≥0is a∗-homomorphic flow, then the structure maps
+{θµ
+ν}µ,νsatisfy :
+θµ
+ν(xy) =θµ
+ν(x)y+xθµ
+ν(y)+/summationdisplay
+i∈Iθi
+ν(x)θµ
+i(y), θµ
+ν(x)∗=θν
+µ(x∗),(10)
+forx∈ A0.
+Proof.The proof is an adaptation of the arguments given in Proposit ion
+28.1 in page 234 of [12]. Since t/ma√sto→jt(x) is weakly measurable for all x∈ A
+and (jt)t≥0is a quantum stochastic cocyle, we have strong measurabilit y of
+the mapt/ma√sto→jt(x)ve(g) for all fixed x∈ A,v∈handg∈L2(R+,k0). Hence
+we can re-phrase equation (7) as
+/an}bracketle{tue(f),jt(x)(ve(g))/an}bracketri}ht=/an}bracketle{tu,xv/an}bracketri}hte/an}bracketle{tf,g/an}bracketri}ht+/summationdisplay
+µ,ν/integraldisplayt
+0ds/an}bracketle{tue(f),js(θµ
+ν(x))ve(g)/an}bracketri}htfµ(s)gν(s).
+(11)
+Using the quantum Itˆ o formula as discussed in Subsection 2. 2 we have:
+0 =/an}bracketle{tue(f),(jt(xy)−jt(x)jt(y))ve(g)/an}bracketri}ht
+=/summationdisplay
+µ,ν/integraldisplayt
+0ds/angbracketleftBigg
+ue(f),js/parenleftBigg
+θµ
+ν(xy)−θµ
+ν(x)y−xθµ
+ν(y)−/summationdisplay
+i∈Iθi
+ν(x)θµ
+i(y)/parenrightBigg
+ve(g)/angbracketrightBigg
+fµ(s)gν(s)
+from which it follows that θµ
+ν(xy) =θµ
+ν(x)y+xθµ
+ν(y) +/summationtext
+i∈Iθi
+ν(x)θµ
+i(y).
+Similarly, it follows from jt(x∗) =jt(x)∗thatθi
+j(x∗) = (θj
+i(x))∗for alli,j.
+✷
+8Remark 2.10. The relations (10)are equivalent to the following identities:
+π(x) :=σ(x)+x⊗Ik0(x∈ A0)
+is a∗-homomorphism from A0toA′′⊗B(k0).
+δ(xy) =δ(x)y+π(x)δ(y) (x,y∈ A0);
+L(x∗) = (L(x))∗(x∈ A0);
+δ(x)∗δ(x) =L(x∗x)−L(x∗)x−x∗L(x) (x∈ A0).
+Remark 2.11. It follows from Lemma 7.1.3 of page 171 in [16] that if the
+structure maps in Definition 2.7 are norm bounded, then relat ions(10)are
+also sufficient for (jt)t≥0to be a∗-homomorphic flow.
+Lemma 2.12. Let(jt)t≥0be a C* (respectively von-Neumann) algebraic
+quantum stochastic flow on Awith the noise space k0, domain algebra A0and
+the structure matrix Θ =/parenleftbiggLδ†
+δ σ/parenrightbigg
+. Moreover in the von-Neumann algebraic
+case, we assume that (j0,0
+t)t≥0isC0. Then(jc,d
+t(·))t≥0is aC0-semigroup on
+Awith respect to the natural topology. Furthermore, the rest riction of the
+generator of (jc,d
+t(·))t≥0toA0isL+/an}bracketle{tc,δ/an}bracketri}ht+δ†
+d+/an}bracketle{tc,σd/an}bracketri}ht+/an}bracketle{tc,d/an}bracketri}htidA,where
+/an}bracketle{tc,δ/an}bracketri}ht(x) :=/an}bracketle{tc,δ(x)/an}bracketri}ht,δ†
+d(x) :=/an}bracketle{td,δ(x∗)/an}bracketri}ht∗and/an}bracketle{tc,σd/an}bracketri}ht(x) :=/an}bracketle{tc,σ(x)d/an}bracketri}ht, for
+x∈ A0.
+Proof.It follows from Remark 2.6 that ( jc,d
+t)t≥0is a semigroup of maps on
+A. To prove the C0property, we proceed as follows. We shall most often
+writectforc1[0,t),wherec∈k0,and obtain for x∈ A0
+|/angbracketleftBig
+u,{jc,d
+t(x)−x}v/angbracketrightBig
+|=|/an}bracketle{tue(ct),jt(x)ve(dt)/an}bracketri}ht−/an}bracketle{tu,xv/an}bracketri}ht|
+≤ |/an}bracketle{tu(e(ct)−e(0)),jt(x)ve(0)/an}bracketri}ht|+|/an}bracketle{tue(0),jt(x)ve(0)/an}bracketri}ht −/an}bracketle{tu,xv/an}bracketri}ht|
++|/an}bracketle{tue(ct),jt(x)v(e(dt)−e(0))/an}bracketri}ht|
+≤ /bardblu/bardbl/bardblv/bardbl/bardblx/bardbl/braceleftBig/radicalbig
+et/bardblc/bardbl2−1+et
+2/bardblc/bardbl2/radicalbig
+et/bardbld/bardbl2−1/bracerightBig
++|/angbracketleftBig
+u,(j0,0
+t(x)−x)v/angbracketrightBig
+|.
+(12)
+If (jt)t≥0is a C*-algebraic flow, it follows from equation (7) that for
+x∈ A0,
+/bardblj0,0
+t(x)−x/bardbl ≤t/bardblθ0
+0(x)/bardbl.
+The norm density of A0inA, contractivity of the flow ( jt)t≥0and the above
+estimate together imply that ( j0,0
+t)t≥0isC0in the norm topology of A.
+Combining this with the estimate (12), it follows that /an}bracketle{tu,jc,d
+t(x)v/an}bracketri}ht → /an}bracketle{tu,xv/an}bracketri}ht
+ast→0+ uniformly for u,v∈hsuch that /bardblu/bardbl ≤1 and/bardblv/bardbl ≤1. This
+implies that ( jc,d
+t)t≥0isC0in the norm topology.
+9On the other hand, if ( jt)t≥0is a von-Neumann algebraic flow, then by
+hypothesis, the semigroup ( j0,0
+t)t≥0isC0in the ultra-weak topology of A.
+Thus estimate (12) implies that jc,d
+t(x)→xast→0+ in the weak operator
+topology, hence also in the ultra-weak topology as /bardbljc,d
+t(x)/bardblis uniformly
+bounded in t. This implies that t/ma√sto→jc,d
+t(x) is ultra-weakly continuous for
+everyx∈ A.
+Now forx∈ A0andu,v∈h,combining (7) and (9), we have
+/angbracketleftbig
+ue(c1[0,t)),jt(x)ve(d1[0,t))/angbracketrightbig
+=/an}bracketle{tu,xv/an}bracketri}htet/an}bracketle{tc,d/an}bracketri}ht
++/angbracketleftbigg
+ue(c1[0,t)),{/integraldisplayt
+0dτ jτ(L(x)+/an}bracketle{tc,δ/an}bracketri}ht(x)+δ†
+d(x)+/an}bracketle{tc,σd/an}bracketri}ht(x))}ve(d1[0,t))/angbracketrightbigg
+.
+(13)
+Thus
+limt→0+1
+t/an}bracketle{tu,(jc,d
+t(x)−x)v/an}bracketri}ht=/an}bracketle{tu,xv/an}bracketri}htlim
+t→0+et/an}bracketle{tc,d/an}bracketri}ht−1
+t
++limt→0+/an}bracketle{tue(c1[0,t)),{1
+t/integraldisplayt
+0dτ jτ(L(x)+/an}bracketle{tc,δ/an}bracketri}ht(x)+δ†
+d(x)+/an}bracketle{tc,σd/an}bracketri}ht(x))}ve(d1[0,t))/an}bracketri}ht
+=/an}bracketle{tu,{L(x)+/an}bracketle{tc,δ/an}bracketri}ht(x) +δ†
+d(x)+/an}bracketle{tc,σd/an}bracketri}ht(x)+/an}bracketle{tc,d/an}bracketri}htx}v/an}bracketri}ht,
+(14)
+from which the conclusion follows. ✷
+Definition 2.13. AC0-semigroup of contractive (also normal in case of
+von-Neumann algebra) maps (Tt)t≥0onAis called a quantum dynamical
+semigroup if each Ttis completely positive. In case Ais unital, then a
+quantum dynamical semigroup (Tt)t≥0is called conservative if Tt(1) = 1
+also holds for all t.
+Remark 2.14. For a quantum stochastic flow satisfying the hypotheses of
+Lemma 2.12 the semigroup (j0,0
+t)t≥0is a quantum dynamical semigroup. It
+is called the vacuum expectation semigroup of the flow jt.
+Remark 2.15. It is well known that if a classical stochastic process has
+chaotic decomposition property ([10],[12]), then one can a ssociate a quantum
+stochastic ∗-homomorphic flow with the process (see [10],[12]). The class of
+stochastic processes having this property is rich and inclu des processes driven
+by Brownian motion and Poisson process.
+2.3 Tensor product of Banach spaces
+Here we collect a few facts about the projective tensor produ ct of Banach
+spaces which is an important technical tool, needed to prove our main re-
+sult. Recall that (see [18]) for two Banach spaces E1andE2, the projective
+10tensor product E1⊗γE2is the completion of the algebraic tensor product
+E1⊗algE2in the cross-norm /bardbl · /bardblγgiven by /bardblX/bardblγ= inf/summationtext
+i/bardblxi/bardbl/bardblyi/bardblfor
+X∈E1⊗algE2, where the infimum is taken over all possible expressions of
+Xof the form X=/summationtextp
+i=1xi⊗yi, xi∈E1, yi∈E2andp≥0.
+Lemma 2.16. Suppose that Tj∈B(Ej,Fj)whereEj,Fjforj= 1,2are
+Banach spaces. Then T1⊗algT2extends to a bounded operator
+T1⊗γT2:E1⊗γE2−→F1⊗γF2
+with the bound
+/bardblT1⊗γT2/bardbl ≤ /bardblT1/bardbl/bardblT2/bardbl.
+Proof.The proof of Lemma 2.16 is an easy consequence of the estimate
+/bardbl(T1⊗algT2)(k/summationdisplay
+i=1xi⊗yi)/bardblγ≤k/summationdisplay
+i=1/bardblT1(xi)⊗T2(yi)/bardblγ
+≤k/summationdisplay
+i=1/bardblT1(xi)/bardblF1/bardblT2(yi)/bardblF2≤ /bardblT1/bardbl/bardblT2/bardblk/summationdisplay
+i=1/bardblxi/bardblE1/bardblyi/bardblE2.(15)
+✷
+Lemma 2.17. Suppose that (Tt)t≥0and(St)t≥0are twoC0-semigroups of
+bounded operators on Banach spaces E1andE2, with generators L1andL2
+respectively. Then (Tt⊗γSt)t≥0becomes aC0-semigroup of operators on
+E1⊗γE2whose generator is the closed extension of the operator L1⊗alg
+IE2+IE1⊗algL2(defined on D(L1)⊗algD(L2)), the closure being taken
+with respect to /bardbl·/bardblγ.
+Proof.We have (( Tt⊗algSt)◦(Ts⊗algSs))(X) = (Tt+s⊗algSt+s)(X) for
+X∈E1⊗algE2. Both sides being continuous in /bardbl · /bardblγ, the above identity
+extends by Lemma 2.16 to E1⊗γE2. Thus we have the semigroup property:
+(Tt⊗γSt)◦(Ts⊗γSs) = (Tt+s⊗γSt+s).
+By similar arguments ( Tt⊗γIE2)◦(IE1⊗γSt) =Tt⊗γStand thus the
+strong continuity of Tt⊗γIE2andIE1⊗γSt, as a function of t yields the
+strong continuity of Tt⊗γSt.Hence (Tt⊗γSt)t≥0is aC0-semigroup on
+E1⊗γE2. Moreover Tt⊗γStkeepsD(L1)⊗algD(L2) invariant. Thus
+D(L1)⊗algD(L2) is a core for the generator of Tt⊗γSt(see [2]). We will
+denote the generator by L1⊗γIE2+IE1⊗γL2. Clearly, this is the closure
+of the operator L1⊗algIE2+IE1⊗algL2.✷
+Corollary 2.18. In the notation of Lemma 2.17, suppose that D1⊂E1
+andD2⊂E2are cores for the generators L1andL2respectively. Then
+D1⊗algD2is a core for L1⊗γIE2+IE1⊗γL2.
+11Proof.Recall that D(L1)⊗algD(L2) is a core for the generator L1⊗γIE2+
+IE1⊗γL2. LetX:=/summationtextk
+i=1xi⊗yi∈D(L1)⊗algD(L2), wherexi∈D(L1)
+andyi∈D(L2),i= 1,...,k. Get (x(i)
+n)n∈ D1and (y(i)
+n)n∈ D2,i= 1,...,k
+such thatx(i)
+n→xi,L1(x(i)
+n)→L1(xi),y(i)
+n→yi,L2(y(i)
+n)→L2(yi). Let
+Xn:=/summationtextk
+i=1x(i)
+n⊗y(i)
+n. Then it follows that Xn→Xand (L1⊗γIE2+
+IE1⊗γL2)(Xn)→(L1⊗γIE2+IE1⊗γL2)(X) in/bardbl · /bardblγ. This proves the
+result.✷
+We conclude this section with the following result about ope rators on
+Banach spaces, which will play a crucial role in the proof of h omomorphism
+property in Section 3.
+Lemma 2.19. LetEbe aBanach space, and let AandTbelong toLin(E,E)
+with dense domains D(A)andD(T)respectively. Suppose that there is a
+total setD⊂D(A)∩D(T)with the properties :
+(i)A(D)is total inE,(ii)/bardblT(x)/bardbl</bardblA(x)/bardblfor allx∈D.
+Then(A+T)(D)is also total in E.
+Proof.SetF:=span{(A+T)(D)}and suppose that F/ne}ationslash=E. Then noting
+that ifA(D)⊆(A+T)(D), one hasF⊃spanA(D) =E(by hypothesis),
+we conclude that there exists a y0(/ne}ationslash= 0) inA(D), such that y0/∈F. Let
+y0=A(x0) for somex0∈D.Then by Hahn-Banach theorem, there exists
+Λ∈E∗,the topological dual of E, such that /bardblΛ/bardbl= 1,|Λ(y0)|=/bardbly0/bardblas well
+as Λ((A+T)(D)) = 0.Then/bardbly0/bardbl=|Λ(A(x0))|and|Λ(A(x0))|=|Λ(T(x0))|.
+But|Λ(T(x0))| ≤ /bardblT(x0)/bardbl</bardblA(x0)/bardbl=/bardbly0/bardblwhich leads to a contradiction.
+ThereforeF=E.✷
+3 A sufficient condition for a quantum stochastic
+flow to be ∗-homomorphic
+3.1 Assumptions and statement of the main result
+Let (jt)t≥0be a quantum stochastic flow with the noise space k0, domain
+algebraA0and structure matrix Θ =/parenleftbiggLδ†
+δ σ/parenrightbigg
+. Our aim is to give a set
+of natural and sufficient conditions which will imply that the flow is∗-
+homomorphic. In the following, note that the first assumptio nA(1)is
+nothing but those given in Remark 2.10 which are known to be ne cessary
+by Lemma 2.9.
+We now state the assumptions:
+A(1)The mapπ:A0→ A′′⊗B(k0) given by:
+π(x) :=σ(x)+x⊗Ik0(x∈ A0)
+12is a∗-homomorphism from A0toA′′⊗B(k0).
+L(x∗) = (L(x))∗(x∈ A0);
+δ(xy) =δ(x)y+π(x)δ(y) (x,y∈ A0);
+δ(x)∗δ(x) =L(x∗x)−L(x∗)x−x∗L(x) (x∈ A0).
+A(2)There exists a semifinite, faithful and lower-semicontinuo us traceτon
+A, such that A0⊂D(τ). Settingh:=L2(τ), the G.N.S. Hilbert space
+ofAwith respect to τand viewing Aas a C*-subalgebra of B(h),A0
+is furthermore assumed to be dense in h.
+A(3)(i) For each t≥0, the semigroup of maps ( Tt)t≥0extends to a C0
+contractive semigroup ( T(2)
+t)t≥0onh. Its generator is denoted by
+L2.
+Furthermore, A0⊆D(L)∩D(L2) and is a core for L2.
+(ii) Forx∈ A0,L(x∗x)∈ A ∩L1(τ) andτ(L(x∗x))≤0 (a kind of
+weak dissipativity).
+(iii) The semigroup ( T(2)
+t)t≥0is analytic.
+A(4)
+sup0≤s≤t|/an}bracketle{tuf⊗mjs(x)vg⊗n/an}bracketri}ht| ≤C(m,n,u,v,t,f,g )/bardblx/bardbl1(u,v∈h,x∈ A∩L1(τ)),
+wherem,n∈IN,C(m,n,u,v,t,f,g ) =O(eβt) forβ >0 andf,g∈ W
+for some total subset WofL2(R+,k0).
+We now state the main theorem of this paper:
+Theorem 3.1. Let(jt)t≥0be a quantum stochastic flow with noise space
+k0and structure matrix Θ =/parenleftbiggLδ†
+δ σ/parenrightbigg
+. Furthermore, suppose that the flow
+satisfies the assumptions A(1)–A(4). Thenjtis a∗-homomorphism for
+eacht≥0.
+3.2 Remarks on the assumptions
+We begin with conditions in terms of the semigroups ( jc,d
+t)t≥0associated
+with a quantum stochastic cocycle ( jt)t≥0, which will imply the condition
+given in assumption A(4).
+Lemma 3.2. LetWbe a total subset of k0such thatzW ⊆ W for allz∈C.
+Suppose that a quantum stochastic cocycle (jt)t≥0satisfies :
+13A(4)′
+/bardbljc,d
+t(x)/bardbl1≤exp(tM)/bardblx/bardbl1 (16)
+forx∈ A ∩L1(τ), c,d∈ Wand where M≥0depends only on /bardblc/bardbl,/bardbld/bardbl.
+Then the estimate in A(4)holds.
+Proof.For a partition 0 = s0< s1< s2< ... < s n=tand for functions
+of the form f=/summationtext
+j1[sj−1,sj)cj, g=/summationtext
+j1[sj−1,sj)dj,(cj,dj∈ Wfor j=
+1,2,...), the cocycle property of jt(·) and (16) together imply:
+/bardbl/an}bracketle{te(f),jt(x)e(g)/an}bracketri}ht/bardbl1≤exp(tM)/bardblx/bardbl1,
+whereM=maxj(Mj),and eachMjdepends only on /bardblcj/bardbland/bardbldj/bardbl.Let
+Λ(z) :=/an}bracketle{tue(¯zf),jt(x)ve(g)/an}bracketri}ht. Wehave |/an}bracketle{tue(¯zf),jt(x)ve(g)/an}bracketri}ht| ≤exp(tM)/bardblx/bardbl1
+for|z|= 1.Clearly Λ is entire in z since z/ma√sto→e(zf) is strongly entire and by
+considering a unit disc centered at zero and applying Cauchy ’s estimate to
+this function, we obtain:
+(m!)1
+2|/angbracketleftbig
+uf⊗m,jt(x)ve(g)/angbracketrightbig
+| ≤ /bardblu∗v/bardbl∞m!exp(tM)/bardblx/bardbl1,(17)
+foru,v∈ A ∩L2(τ) andx∈ A ∩L1(τ).A similar calculation with the
+functionβ(z) :=/angbracketleftbig
+uf⊗m,jt(x)ve(zg)/angbracketrightbig
+yields:
+|/angbracketleftbig
+uf⊗m,jt(x)vg⊗n/angbracketrightbig
+| ≤ /bardblu∗v/bardbl∞(m!n!)1
+2exp(tM))/bardblx/bardbl1,(18)
+which proves that the cocycle ( jt)t≥0satisfiesA(4), if we take
+C(m,n,u,v,t,f,g ) :=/bardblu∗v/bardbl∞(m!n!)1
+2exp(tM).✷
+For certain special types of von-Neumann algebras, the esti mate inA(4)
+is automatic, as the next result shows. Note that a type-I von -Neumann
+algebraAwith atomic center and acting on a separable Hilbert space, i s a
+direct sum of the form A∼=/circleplustext
+m∈TAm⊗l∞(Γm) whereTis some index set
+such that for each m∈T,Amis isomorohic to B(H) (whereHis a separable
+Hilbert space of finite or infinite dimension) and Γ mis a discrete, at most
+countable set. There is a natural faithful and semifinite tra ceτonA, which
+is a direct sum (over T) of the tensor product of the canonical semifinite
+trace on B(H) and the trace on l∞(Γm) coming from the counting measure
+on Γm. It is clear that /bardblx/bardbl∞≤ /bardblx/bardbl1for allx∈ A/intersectiontextL1(τ).
+Lemma 3.3. LetAbe a type-I von-Neumann algebra with atomic center,
+acting on a seperable Hilbert space. Suppose that τis the natural trace on A
+as discussed above. Then any quantum stochastic flow (jt)t≥0onAsatisfying
+the assumtions A(1)–A(3)must also satisfy the estimate in A(4).
+Proof.Using/bardblx/bardbl∞≤ /bardblx/bardbl1and the contractivity of jt, we have for x∈L1(τ)
+14sup
+0≤s≤t|/angbracketleftbig
+uf⊗m,jt(x)vg⊗n/angbracketrightbig
+|
+≤ /bardblx/bardbl∞/bardblf⊗m/bardbl/bardblg⊗n/bardbl/bardblu/bardbl2/bardblv/bardbl2
+≤ /bardblx/bardbl1/bardblf⊗m/bardbl/bardblg⊗n/bardbl/bardblu/bardbl2/bardblv/bardbl2,(19)
+which implies that the flow ( jt)t≥0satisfiesA(4)withC(m,n,u,v,t,f,g ) :=
+/bardblf⊗m/bardbl/bardblg⊗n/bardbl/bardblu/bardbl2/bardblv/bardbl2.✷
+Let usnowdiscussafewspecial cases toshowthat theassumpt ionsmade
+above are general enough to cover many interesting and natur al examples
+arising in classical probability.
+We state without proof the following proposition (see pp.63 , Lemma
+3.2.28 of [16]).
+Proposition 3.4. LetAbe a unital C*-algebra with a faithful, semifinite
+and lower-semicontinuous trace τand(jt)t≥0be a quantum stochastic flow
+with the noise space k0and structure matrix Θ =/parenleftbiggLδ†
+δ σ/parenrightbigg
+. Suppose fur-
+thermore that with respect to the trace τ:
+Tt(1) = 1, τ(Tt(x)y) =τ(xTt(y)) (t≥0;x,y∈D(τ)),
+where(Tt)t≥0is the vacuum expectation semigroup associated with the flow
+(jt)t≥0. Also suppose that the assumptions A(1)andA(2)hold. Then A(3)
+follows.
+The following result shows the abundance of classical as wel l as noncom-
+mutative quantum stochastic flows for which the estimate A(4)′holds.
+Lemma 3.5. LetAbe a C*-algebra equipped with a faithful, semifinite,
+lower semicontinuous trace τand assume that there is a strongly continuous,
+∗-automorphic action αgof a Lie group GonAwhich isτ-preserving, that is
+τ(αg(a)) =τ(a). Extendαgto a unitary operator UgonL2(τ)and we extend
+αtoA′′as a normal ∗-automorphism given by αg(x) =UgxU∗
+g.Let(gt)t≥0be
+aG-valued L´ evy process, defined on some probability space (Ω,F,P).Define
+jt:A′′→L∞(Ω,A′′)⊆B(L2(τ)⊗L2(Ω)),byjt(x)(ω) :=αgt(ω)(x)and let
+Tt(x) =IE(αgt(ω)(x)).
+(i) Then for f,g∈L2(Ω), x∈ A∩L1(τ),we have
+τ(|/an}bracketle{tf,jt(x)g/an}bracketri}ht|)≤ /bardblf/bardbl2/bardblg/bardbl2/bardblx/bardbl1. (20)
+(ii)(Tt)t≥0is a normal quantum dynamical semigroup on A′′and if its
+restriction on AleavesAinvariant, it is a quantum dynamical semi-
+group on A. Furthermore if gtand(gt)−1have the same distribution
+for eacht, then the semigroup is τ-symmetric.
+15Proof.To prove (i), it suffices to show the inequality for positive x∈ A.For
+suchx,we have
+τ(|/an}bracketle{tf,jt(x)g/an}bracketri}ht|)
+≤/integraldisplay
+ΩdPτ/parenleftbig/vextendsingle/vextendsinglef(ω)g(ω)jt(x)(ω)/vextendsingle/vextendsingle/parenrightbig (21)
+from which the result follows.
+To prove (ii) we proceed as follows:
+from the defining property of L´ evy processes, the semigroup property
+ofTtfollows; while the normality of Ttis a consequence of the fact that jt
+is implemented by a normal automorphism of A′′.Moreover, if gtandg−1
+t
+have the same distribution, we have for a,b∈L1(τ)/intersectiontextA
+τ(Tt(a)b) =τ[IE{αgt(ω)(a)b}]
+=τ[IE{αgt(ω)(aαgt(ω)−1(b))}] =IE[τ{αgt(ω)(aαgt(ω)−1(b))}]
+=IE[τ{(aαgt(ω)−1(b))}] =τ[aIE{αgt(ω)(b)}] =τ(aTt(b)).(22)
+✷
+Note that if the L´ evy process in Lemma 3.5 has the chaotic dec omposi-
+tion property, for example if it is a Brownian motion or a Pois son process,
+then we can realize the flow jtas a quantum stochastic flow in a suitable
+Fock space. To see a concrete example, assume furthermore th atGbe a
+second countable, compact Lie group of dimension k, acting smoothly on
+A. Denote by A∞the dense ∗-subalgebra of Agenerated by elements x
+such thatg→αg(x) is norm-smooth, where g→αgis the group action.
+Let{χℓ}k
+ℓ=1be a basis for the Lie algebra of Gand letgtbe theG-valued
+standard Brownian motion. Then gtandg−1
+thave the same distribution,
+so the vacuum expectation semigroup of jtis symmetric. Moreover taking
+A0=A∞, the assumptions A(1),A(2),A(3)are easily seen to hold. By
+Lemma 3.5 A(4)′and hence A(4)holds too. The flow jtcan be viewed as
+a possibly noncommutative Brownian motion on A.
+Remark 3.6. Consider a typical diffusion process in Rwhose generator is
+of the form:
+L=1
+2d
+dxa2(x)d
+dx+b(x)d
+dx.
+The coefficients aandbare assumed to be smooth and ais assumed to be
+non-vanishing everywhere. By a change of variable x/ma√sto→φ(x), whereφ(x) =
+/integraltextx
+0dse/integraltexts
+0dt2b(t)
+a2(t)+C(where C is a constant), the generator Lcan be made
+symmetric with respect to the trace τ′given byτ′(f) =/integraltext
+f(x)φ′(x)dx.Thus
+the assumption of symmetry can accommodate the semigroup co rresponding
+to an arbitrary one-dimensional diffusion with smooth coeffic ients.
+16Remark 3.7. On the other hand, for some of the most common Markov
+processes arising in classical probability for which the co rresponding quantum
+stochastic flow (see Remark 2.15) satisfies assumptions A(1)–A(3), the
+vacuum expectation semigroup (Tt)t≥0associated with the flow cannot be
+made symmetric even by a change of measure on the underlying f unction
+algebra. For example, consider the flow associated to the sta ndard Poisson
+process on Z+,realized on the commutative von-Neumann algebra l∞(Z+),
+equipped with the trace given by the counting measure. Here t he generator
+L2is the bounded operator l−I,where l denotes the unilateral shift operator
+onl2(Z+).It is straightforward to see that for φ∈l∞(Z+), φ≥0, τ((l−
+I)(φ))≤0.However, there does not exist any faithful positive trace fo r which
+Lis symmetric, as seen from the following argument:
+IfLis symmetric with respect to some measure µ,say given by a sequence
+{pi=µ({i})}of non-negative numbers, then the symmetry condition will
+imply (since 1∈D(L)) that/summationtext
+i≥0(φ(i+1)−φ(i))pi= 0for allφ∈l∞(Z+).
+Hence we have pi= 0for all i.
+3.3 Proof of the main theorem
+Throughout this subsection let us fix a quantum stochastic flo w (jt)t≥0with
+the noise space k0and structure matrix Θ =/parenleftbiggLδ†
+δ σ/parenrightbigg
+, satisfying the as-
+sumptions A(1)–A(4).
+We need a few preparatory lemmas to prove Theorem 3.1. We begi n
+with the following observations:
+Remark 3.8. LetEbe a Banach space, F:R→Ebe a strongly measurable
+map and let µbe a measure on R. Suppose that the integrals/integraltext
+Rdµ(t)F(t)
+and/integraltext
+Rdµ(t)T(F(t))exist, where Tis a closed densely defined operator in
+E.Then/integraltext
+Rdµ(t)F(t)∈D(T)and
+T(/integraldisplay
+Rdµ(t)F(t)) =/integraldisplay
+Rdµ(t)T(F(t)).
+Remark 3.9. We observe that because of analyticity in assumption A(3),
+the real part of the operator (−L2)exists as an unbounded, densely defined
+and non-negative operator (see pages 322 and 336 of [6]).
+Throughout this subsection, we will be working in the projec tive tensor
+producth⊗γh. We fix the following notations:
+ˆL:=L2⊗γI+I⊗γL2;
+L:= (−2Re(L2))1
+2;
+17L⊗γL:= (L⊗γI)◦(I⊗γL) = (I⊗γL)◦(L⊗γI);
+F:=A0⊗algA0;
+Y:={(λ−ˆL)−1(x⊗y)|x,y∈ A0}(λ>0).
+The next two lemmas set the stage for the application of Lemma 2.19 to
+our problem, leading to the proof of Theorem 3.1.
+Lemma 3.10. Forx∈ A0withx/ne}ationslash= 0andλ >0, we have the following
+strict inequality:/integraldisplay∞
+0e−λtdt/bardblL(Tt(x))/bardbl2</bardblx/bardbl2.
+Proof.For x in A0,
+d
+dt/bardblTt(x)/bardbl2=/an}bracketle{tL2(Tt(x)),Tt(x)/an}bracketri}ht+/an}bracketle{tTt(x),L2(Tt(x))/an}bracketri}ht=−/bardblL(Tt(x))/bardbl2.(23)
+Forλ>0, we have
+/integraldisplay∞
+0e−λt/bardblL(Tt(x))/bardbl2dt=−/integraldisplay∞
+0e−λtd
+dt/bardblTt(x)/bardbl2dt
+=/braceleftbigg
+/bardblx/bardbl2−λ/integraldisplay∞
+0e−λt/bardblTt(x)/bardbl2dt/bracerightbigg
+,(24)
+which implies that
+/integraldisplay∞
+0e−λt/bardblL(Tt(x))/bardbl2dt≤ /bardblx/bardbl2(x∈ A0).
+If/integraltext∞
+0dt e−λt/bardblTt(x)/bardbl2= 0 for some λ >0, we have /bardblTt(x)/bardbl= 0 for all
+t. By the continuity of /bardblTt(x)/bardblas a function of t, this implies that x= 0,
+which leads to a contradiction. ✷
+Lemma 3.11. /bardbl(L⊗γL)(X)/bardblγ≤ /bardbl(λ−ˆL)(X)/bardblγfor allXinD(ˆL)and
+λ>0. Furthermore we have strict inequality if Xis inY.
+Proof.LetX∈ Fsuch thatX=/summationtextk
+i=1xi⊗yi. It is obvious that
+(I⊗γL)(F)⊂D((L⊗γI)). So using Lemma 3.10 we have:
+/integraldisplay∞
+0dt e−λt/bardbl{(L⊗γL)(Tt⊗γTt)}(X)/bardblγ
+=/integraldisplay∞
+0dt e−λt/bardblk/summationdisplay
+i=1L(Tt(xi))⊗L(Tt(yi))/bardblγ
+≤k/summationdisplay
+i=1(/integraldisplay∞
+0dt e−λt/bardblL(Tt(xi))/bardbl2)1
+2(/integraldisplay∞
+0dt e−λt/bardblL(Tt(yi))/bardbl2)1
+2<k/summationdisplay
+i=1/bardblxi/bardbl/bardblyi/bardbl.
+(25)
+18Equation (25) and Remark 3.8 together yield:
+/bardbl(L⊗γL)((λ−ˆL)−1(X))/bardblγ≤ /bardblX/bardblγ.
+Thus(L⊗γL)◦(λ−ˆL)−1isacontractiveoperatorin h⊗γh. Asaconsequence,
+L⊗γLextends to D(ˆL) by (i) of A(3)and Corollary 2.18. This gives us
+the required inequality.
+LetX=x⊗y,wherex,y∈ A0. The above equations give:
+/bardbl(L⊗γL)((λ−ˆL)−1(x⊗y))/bardblγ</bardblx/bardbl/bardbly/bardbl=/bardblx⊗y/bardblγ
+or/bardbl(L⊗γL)(Y)/bardblγ</bardbl(λ−ˆL)(Y)/bardblγforY∈ Y,(26)
+which proves the second claim. ✷
+Lemma 3.12. Forx∈ A0we have:
+(i)
+θi
+0(x)∈h(=L2(τ)) (i≥1).
+(ii) The series/summationtext
+i≥1θi
+0(x)⊗|ei/an}bracketri}htis convergent in the norm of h⊗k0. Thus
+we have:
+δ(x) =/summationdisplay
+i≥1θi
+0(x)⊗|ei/an}bracketri}ht ∈h⊗k0.
+(iii) There exists a densely defined positive operator Csuch thatD(L2)is
+a core forCand
+Cu=Lu(u∈D(L2));
+Re/an}bracketle{tu,(−L2)v/an}bracketri}ht=/an}bracketle{tCu,Cv/an}bracketri}ht(u,v∈D(L2)).
+For eachi≥1,θi
+0extends to a linear map from D(C)toh.
+(iv) There exists a linear map B:D(ˆL)(⊆h⊗γh)→h⊗γh, satisfying
+(a)
+/bardblB(X)/bardblγ≤ /bardbl(λ−ˆL)(X)/bardblγ(X∈D(ˆL)),
+the inequality being strict if X∈ Y.
+(b)
+B(x⊗y) =/summationdisplay
+i≥1θi
+0(x)⊗θi
+0(y) (x,y∈ A0).
+Proof.Forx∈ A0, we have
+τ(θi
+0(x)∗θi
+0(x))≤τ(/summationdisplay
+i≥1θi
+0(x)∗θi
+0(x)) =τ(δ(x)∗δ(x)).
+19The last identity in A(1)implies that:
+τ(δ(x)∗δ(x)) =τ(L(x∗x))−τ(L(x∗)x)−τ(x∗L(x)) (x∈ A0).
+Combining this with (ii) of A(3)we have:
+τ(δ(x)∗δ(x))≤ −τ(L(x∗)x)−τ(x∗L(x))<∞(x∈ A0),
+which proves (i) and (ii).
+To prove (iii) note that the last identity in A(2)implies the following:
+/bardblθi
+0(x)/bardbl2
+2≤∞/summationdisplay
+j=1/bardblθj
+0(x)/bardbl2
+2≤τ(δ(x)∗δ(x))
+≤τ((−L)(x∗)x)+τ(x∗(−L)(x))≤ /bardblL(x)/bardbl2
+2≤ /bardbl(L+ǫ)(x)/bardbl2
+2,
+(27)
+for all x in A0, ǫ>0 andi≥1, asLis a non-negative operator. By Remark
+3.9, we have
+Re/an}bracketle{tu,(−L2)v/an}bracketri}ht=/an}bracketle{tu,L2v/an}bracketri}ht=/an}bracketle{tLu,Lv/an}bracketri}ht(u,v∈D(L2)).
+Thus by Theorem 1.27 of page 318 in [6], the densely defined, no n-negative
+sesquilinear form q(u,v) :=/an}bracketle{tu,L2v/an}bracketri}ht,u,v∈D(L2) is closable. Let Tbe the
+positive operator associated with the form q, as obtained in Theorem 2.23
+of page 331 in [6]. Set C:=T1
+2. Note that Tu=L2ufor allu∈D(L2) and
+D(L2) is a core for C. Thus from equation (27), we have
+/bardblθi
+0(x)/bardbl2
+2≤∞/summationdisplay
+i=1/bardblθi
+0(x)/bardbl2
+2≤τ(δ(x)∗δ(x))
+≤τ((−L)(x∗)x)+τ(x∗(−L)(x))≤ /bardblC(x)/bardbl2
+2≤ /bardbl(C+ǫ)(x)/bardbl2
+2,
+(28)
+forx∈ A0andi≥1, from which (iii) follows.
+Finally we prove (iv):
+SetC⊗γC:= (I⊗γC)◦(C⊗γI) = (C⊗γI)◦(I⊗γC). Define a map
+Bbelonging to Lin(D(C)⊗algD(C),h⊗γh) by:
+B(x⊗y) =/summationdisplay
+i≥1θi
+0(x)⊗θi
+0(y) (x,y∈D(C),i≥1),
+and extending linearly. This operator is well-defined becau se forǫ>0,
+/summationdisplay
+i≥1/bardblθi
+0(x)/bardbl/bardblθi
+0(y)/bardbl ≤ {(/summationdisplay
+i≥1/bardblθi
+0(x)/bardbl2)(/summationdisplay
+i≥1/bardblθi
+0(y)/bardbl2)}1
+2
+≤ /bardbl(C+ǫ)x/bardbl/bardbl(C+ǫ)y/bardbl<∞.(29)
+20It follows that
+/bardblB{(C+ǫ)−1⊗γ(C+ǫ)−1}(x⊗y)/bardblγ≤ /bardblx⊗y/bardblγ, (30)
+which implies that B◦{(C+ǫ)−1⊗γ(C+ǫ)−1}extends to a contraction
+onh⊗γh. Thus we have
+/bardblB(X)/bardblγ≤ /bardbl{(C+ǫ)⊗γ(C+ǫ)}(X)/bardblγ (31)
+for allX∈D(C)⊗algD(C) andǫ>0. Lettingǫ→0 we get
+/bardblB(X)/bardblγ≤ /bardbl(C⊗γC)(X)/bardblγ (32)
+for all X in D(C)⊗algD(C).Note that the conclusion of Lemma 3.11 holds
+withLreplaced by C. AsC⊗γCextends to D(ˆL), we can also extend B
+toD(ˆL).So we have:
+/bardblB(X)/bardbl ≤ /bardbl(C⊗γC)(X)/bardblγ≤ /bardbl(λ−ˆL)(X)/bardblγfor allX∈D(ˆL). (33)
+SinceY ⊆D(ˆL), we have:
+/bardblB(Y)/bardblγ≤ /bardbl(C⊗γC)(Y)/bardblγ</bardbl(λ−ˆL)(Y)/bardblγ(Y∈ Y).(34)
+This proves (iv). ✷
+We are now in a position to prove Theorem 3.1.
+Proof of Theorem 3.1:
+Throughout the proof we adopt Einstein’s summation convent ion for
+convenience.
+Forf,ginW,the flow equation (7) leads to :
+/an}bracketle{tjt(x)ue(f),ve(g)/an}bracketri}ht=/an}bracketle{txue(f),ve(g)/an}bracketri}ht+/integraldisplayt
+0ds/an}bracketle{tjs(θµ
+ν(x))ue(f),ve(g)/an}bracketri}htgµ(s)fν(s).
+(35)
+Using the quantum Itˆ o formula we get:
+/an}bracketle{tjt(x)ue(f),jt(y)ve(g)/an}bracketri}ht
+=/an}bracketle{txue(f),yve(g)/an}bracketri}ht+/integraldisplayt
+0ds[/an}bracketle{tjs(θµ
+ν(x))ue(f),js(y)ve(g)/an}bracketri}htgµ(s)fν(s)
++/an}bracketle{tjs(x)ue(f),js(θµ
+ν(y))ve(g)/an}bracketri}htfµ(s)gν(s)
++/angbracketleftbig
+js(θi
+µ(x))ue(f),js(θi
+ν(y))ve(g)/angbracketrightbig
+fµ(s)gν(s)].(36)
+For fixedu,vinA∩h, f,ginW,we define for each t≥0,
+φt:A0×A0→Cby
+φt(x,y) :=/an}bracketle{tjt(x)ue(f),jt(y)ve(g)/an}bracketri}ht −/an}bracketle{tjt(y∗x)ue(f),ve(g)/an}bracketri}ht(x,y∈ A0).
+(37)
+21Using (7), (36) and (10) we have:
+φt(x,y) =/integraldisplayt
+0ds[φs(θ0
+0(x),y)+φs(x,θ0
+0(y))+φs(θi
+0(x),θi
+0(y))
++gi(s)φs(θi
+0(x),y)+gi(s)φs(x,θ0
+i(y))+fi(s)φs(θ0
+i(x),y)
++fi(s)φs(x,θi
+0(y))+gi(s)fj(s)φs(θi
+j(x),y)+gi(s)fj(s)φs(x,θj
+i(y))
++gi(s)φs(θm
+0(x),θm
+i(y))+fi(s)φs(θm
+i(x),θm
+0(y))+fj(s)gi(s)φs(θm
+j(x),θm
+i(y))].
+(38)
+Next we follow the ideas indicated in the pages 178-181 in [16 ] and define
+for m,n inIN∪0,
+φm,n
+t(x,y) :=1
+(m!n!)1
+2[/angbracketleftbig
+jt(x)uf⊗m,jt(y)vg⊗n/angbracketrightbig
+−/angbracketleftbig
+jt(y∗x)uf⊗m,vg⊗n/angbracketrightbig
+]
+=1
+m!n!∂m
+∂ρm∂n
+∂ηn{/an}bracketle{tjt(x)ue(ρf),jt(y)ve(ηg)/an}bracketri}ht −/an}bracketle{tjt(y∗x)ue(ρf),ve(ηg)/an}bracketri}ht}|ρ,η=0.
+(39)
+Differentiating (38) with respect to ρandηand setting ρ=η= 0,we get a
+recursive integral relation amongst φm,n
+t(x,y) as follows:
+φm,n
+t(x,y) =/integraldisplayt
+0ds[φm,n
+s(θ0
+0(x),y)+φm,n
+s(x,θ0
+0(y))+φm,n
+s(θi
+0(x),θi
+0(y))
++gi(s)φm,n−1
+s(θi
+0(x),y)+gi(s)φm,n−1
+s(x,θ0
+i(y))
++fi(s)φm−1,n
+s(θ0
+i(x),y)+fi(s)φm−1,n
+s(x,θi
+0(y))
++gi(s)fj(s)φm−1,n−1
+s(θi
+j(x),y)+gi(s)fj(s)φm−1,n−1
+s(x,θj
+i(y))
++gi(s)φm,n−1
+s(θk
+0(x),θk
+i(y))+fi(s)φm−1,n
+s(θk
+i(x),θk
+0(y))
++fj(s)gi(s)φm−1,n−1
+s(θk
+j(x),θk
+i(y))],
+(40)
+whereφ−1,n
+t(x,y) :=φm,−1
+t(x,y) := 0 for all m,nandx,y∈ A0. We set in
+(40)m=n= 0 to get
+φ0,0
+t(x,y) =/integraldisplayt
+0ds{φ0,0
+s(θ0
+0(x),y)+φ0,0
+s(x,θ0
+0(y))+φ0,0
+s(θi
+0(x),θi
+0(y))}.
+Thus we consider an equation of the form
+φm,n
+t(x,y) =/integraldisplayt
+0ds[φm,n
+s(θ0
+0(x),y)+φm,n
+s(x,θ0
+0(y))+φm,n
+s(θi
+0(x),θi
+0(y))],
+(41)
+forx,y∈ A0,m,n≥0. Our aim is to show that the hypothesis of Theorem
+3.1 and equation (41) imply that φm,n
+t(x,y) = 0. Having achieved this, we
+can embark on our induction hypothesis as
+φk,l
+t(x,y) = 0 fork+l≤m+n−1.
+22Under the induction hypothesis, equation (40) reduces to an equation of the
+form:
+φm,n
+t(x,y) =/integraldisplayt
+0ds[φm,n
+s(θ0
+0(x),y)+φm,n
+s(x,θ0
+0(y))+φm,n
+s(θi
+0(x),θi
+0(y))] (42)
+forx,y∈ A0,which is an equation similar to (41), leading to φm,n
+t(x,y) = 0
+as earlier and this will complete the induction process. Thu s it only remains
+to show that the assumptions of this theorem lead to a trivial solution to
+an equation of the type (41). Omitting the indices m,n, define a mapψt
+belonging to Lin(A0⊗algA0,C) by:
+ψt(x⊗y) :=φm,n
+t(x,y) (x,y∈ A0,m,n≥0)
+and extending linearly. Thus equation (41) leads to:
+ψt(X) =/integraldisplayt
+0ds[ψs((θ0
+0⊗1+1⊗θ0
+0+(θi
+0⊗algθi
+0))(X))] (X∈ F).(43)
+The complete positivity of the map jtimplies that
+/an}bracketle{tjt(x)ξ,jt(x)ξ/an}bracketri}ht ≤ /an}bracketle{tjt(x∗x)ξ,ξ/an}bracketri}ht (44)
+forξ∈h⊗Γ and hence by A(4), we get that
+|/angbracketleftbig
+jt(x)uf⊗m,jt(y)vg⊗n/angbracketrightbig
+|≤(|/angbracketleftbig
+jt(x∗x)uf⊗m,uf⊗m/angbracketrightbig/angbracketleftbig
+jt(y∗y)vg⊗n,vg⊗n/angbracketrightbig
+|)1
+2
+≤(C(m,m,u,u,t,f,f )C(n,n,v,v,t,g,g ))1
+2/bardblx/bardbl2/bardbly/bardbl2
+=O(eβt)/bardblx/bardbl2/bardbly/bardbl2.
+(45)
+The assumption A(4)and (45) together yield:
+|ψt(X)| ≤O(eβt)/bardblX/bardblγ,forX∈ F, (46)
+which proves (by virtue of denseness of Finh⊗γh) thatψtextends as a
+bounded map from h⊗γhtoC.If we letG=ˆL+B,then forX∈ F,the
+equation (43) becomes:
+ψt(X) =/integraldisplayt
+0ψs(G(X))ds.
+Equation (46) leads to/integraltext∞
+0dte−λt|ψt(X)|<∞forλ≥β, which implies that
+/integraldisplay∞
+0dte−λtψt(X) =/integraldisplay∞
+0dte−λt/integraldisplayt
+0dsψs(G(X)).
+An integration by parts leads to
+/integraldisplay∞
+0dte−λtψt((G−λ)(X)) = 0,forX∈ F. (47)
+23Now recall that Fis a core for ˆL, and moreover by Lemma 3.12, Bis
+relatively bounded with respect to ˆL. ThusFis a core for Gas well. So for
+Y∈span{Y},let{Xn∈ F}nbe a sequence such that G(Xn) goes toG(Y).
+We have /integraldisplay∞
+0dte−λtψt((G−λ)(Y)) = 0 (Y∈ Y), (48)
+by an application of the dominated convergence theorem. In t he notation
+of Lemma 2.19, let A:= (ˆL −λ),D:=YandT:=B. The inequality
+(34) implies that we are in the set-up of Lemma 2.19. Thus the d enseness
+of (G−λ)(span{Y}) follows by applying Lemma 2.19. Therefore equation
+(48) and (46) lead to
+/integraldisplay∞
+0dte−λtψt(X) = 0 (X∈h⊗γh,λ>β).
+This implies that ψt(X) = 0 for X∈h⊗γh. In particular we have
+φm,n
+t(x,y) = 0 forx,y∈ A0, t≥0 andm,n≥0. Hence the result fol-
+lows.
+Corollary 3.13. Suppose that the trace τon the algebra Ais finite. Let
+(jt)t≥0be a quantum stochastic flow with the noise space k0and structure
+matrixΘ =/parenleftbiggLδ†
+δ σ/parenrightbigg
+, satisfying A(1)–A(2), (i)–(ii) of A(3)and either
+A(4)orA(4)′. Moreover, assume the following condition instead of (iii)
+inA(3):
+A0⊆D(L2)∩D(L∗
+2).
+Then(jt)t≥0is a∗-homomorphic flow.
+Proof.Define a symmetric form q(x,y) =−/an}bracketle{tL2(x),y/an}bracketri}ht − /an}bracketle{tx,L2(y)/an}bracketri}htfor all
+x,y∈ A0,with domain D(q) :=A0.This form is non-negative by A(1)
+and (ii) of A(3). Sinceq(x,y) =/an}bracketle{tx,(−L2−L∗
+2)y/an}bracketri}ht ∀x,y∈D(q),we may
+proceed along the lines of the standard proof for the Friedri ch extension (see
+[15], vol-II, page-177), which gives a positive self-adjoi nt operator Zwith
+D(q)⊆D(Z) such that q(x,y) =/an}bracketle{tx,Z(y)/an}bracketri}ht.SetC=Z1
+2.We have
+d
+dt/bardblTt(x)/bardbl2=/an}bracketle{tL2(Tt(x)),Tt(x)/an}bracketri}ht+/an}bracketle{tTt(x),L2(Tt(x))/an}bracketri}ht=−/bardblC(Tt(x))/bardbl2.(49)
+Now we may proceed as in Lemma 3.11 and Theorem 3.1 to conclude the
+result.✷
+The following corollary is a straightforward application o f Lemma 3.2
+and Theorem 3.1.
+Corollary 3.14. Let(jt)t≥0be a quantum stochastic flow on A, with the
+noise space k0and structure matrix Θ =/parenleftbiggLδ†
+δ σ/parenrightbigg
+, satisfying A(1)–A(3).
+24Suppose that the associated semigroups (jc,d
+t)t≥0,c,d∈k0satisfy the esti-
+mateA(4)′of Lemma 3.2 with a suitable total subset W. Then(jt)t≥0is a
+∗-homomorphic flow.
+Combining Lemma 3.3 with Theorem 3.1, we get the following:
+Corollary 3.15. LetAbe a type-I von-Neumann algebra with atomic center
+with the semifinite trace τas in the Lemma 3.3. Suppose that (jt)t≥0is a
+quantum stochastic flow on A, with the noise space k0and structure matrix
+Θ =/parenleftbiggLδ†
+δ σ/parenrightbigg
+, satisfying A(1)–A(3). Then (jt)t≥0is a∗-homomorphic
+flow.
+4 Applications
+In this section we give some applications of Theorem 3.1, inc luding a strong
+version of the Trotter product formula for quantum stochast ic flows with
+unbounded coefficients and an extension of a previously known dilation re-
+sult, as obtained in [5], for a class of quantum dynamical sem igroup on UHF
+algebras ([9]). Using the strong version of the Trotter prod uct formula, we
+also give a new construction of Brownian motion on compact Li e groups and
+a construction of random walk on discrete groups.
+4.1 A strong Trotter product formula for quantum stochas-
+tic flows with unbounded coefficients
+Throughout this section, we fix a C* or von-Neumann algebra Aequipped
+with a faithful, semifinite and lower-semicontinuous trace τsatisfying the
+assumption A(2), with a dense ∗-subalgebra A0and also two C* or von-
+Neumann ∗-homomorphic quantum stochastic flows ( j(1)
+t)t≥0and (j(2)
+t)t≥0,
+with noise spaces k1andk2and structure matrices θ(1):=/parenleftbiggL(1)δ†(1)
+δ(1)σ(1)/parenrightbigg
+andθ(2):=/parenleftbiggL(2)δ†(2)
+δ(2)σ(2)/parenrightbigg
+respectively, with the common domain algebra
+A0.
+FollowingthediscussionsinSection3of[7], wedefinetheTr otterproduct
+formula of two quantum stochastic flows.
+Definition of Trotter product of quantum stochastic flows
+LetΓl:= Γ(L2(R+,kl)),l= 1,2. SetΓ := Γ 1⊗Γ2,sothatΓ := Γ( L2(R+,k1⊕
+k2)). Let Ξ t,t≥0 be the map defined in Section 2.2, with k0:=k1⊕k2.
+Forx∈ A, let
+ηt(x) :=/parenleftBig
+/hatwidej(1)
+t•/hatwidej(2)
+t/parenrightBig
+(x), (50)
+25where•is the product defined in Proposition 2.1 and /hatwidej(l)
+t,l= 1,2,t≥0 is
+the map, as given in Definition 2.5. Take a dyadic partition of the whole
+real line Rand fors < t, consider the part of the partition in the interval
+[s,t] as described in the picture below:
+|[2ns]·2−n−−/bracketleftBig
+s−−/vextendsingle/vextendsingle([2ns]+1)·2−n−−−−−−−−−/vextendsingle/vextendsingle
+[2nt]·2−n−−t/bracketrightBig
+,
+where [t] := greatest integer ≤tfor realtandn∈INis sufficiently large.
+Definition 4.1. Set
+φ(n)
+[s,t]:=Φs,([2ns]+1)2−n•Φ([2ns]+1)2−n,2−n•Φ([2ns]+2)2−n,2−n•...
+•Φ([2nt]−1)2−n,2−n•Φ[2nt]2−n,t−[2nt]2−n,(51)
+whereΦs,t:= Ξs◦ηt. Setφ(n)
+t:=φ(n)
+[0,t].The mapφ(n)
+twill be called the n-fold
+Trotter product of the flows (j(1)
+t)t≥0and(j(2)
+t)t≥0.
+Note that the map φ(n)
+[s,t]is a∗-homomorphism for each n.
+Theorem 4.2. Suppose that (j(l)
+t)t≥0,l= 1,2are C*-algebraic flows satis-
+fyingA(1)–A(3)and the conditions of Lemma 3.2 with the corresponding
+total sets W1,W2respectively. Furthermore assume the following:
+(a)L(1)
+2+L(2)
+2is a pre-generator of a C0contractive and analytic semi-
+group inhsuch that A0is a core for the generator.
+(b) For each cj,djbelonging to Wj(j= 1,2),
+2/summationdisplay
+j=1/parenleftBig
+L(j)+/angbracketleftBig
+cj,δ(j)/angbracketrightBig
++δ†(j)
+dj+/angbracketleftbig
+cj,σdj/angbracketrightbig
++/an}bracketle{tcj,dj/an}bracketri}ht/parenrightBig
+is a pre-generator of a C0-semigroup in A.
+Thenφ(n)
+t(x)as in the Definition 4.1 converges in the strong operator
+topology of B(h⊗Γ)to a quantum stochastic flow (jt)t≥0with the noise
+spacek1⊕k2and structure matrix
+Θ :=
+L(1)+L(2)δ†(1)δ†(2)
+δ(1)σ(1)0
+δ(2)0σ(2)
+.
+26Proof.Following the line of arguments given in Section 3 of [7], it f ollows
+that the Trotter product φ(n)
+t(x) forx∈ A, converges in the weak operator
+topology of B(h⊗Γ) to a quantum stochastic flow ( jt)t≥0with the noise
+spacek1⊕k2and structure matrix Θ.
+Let ∆ 0:= [s,[2ns]+1
+2n), ∆j:= [[2ns]+j
+2n,[2ns]+j+1
+2n) for 1≤j≤[2nt]−
+[2ns]−1 and ∆′:= [[2nt]
+2n,t). Letχ0:= 1∆0χj:= 1∆jandχ′:= 1∆′.
+Suppose that Ml≡Ml(/bardblcl/bardbl,/bardbldl/bardbl),l= 1,2 is the constant in the estimate
+A(4)′forj(l)
+t, i.e./bardblj(l)cl,dl
+t(x)/bardbl1≤exp(tMl)/bardblx/bardbl1forcl,dl∈ Wl. Then for
+c:=c1⊕c2,d:=d1⊕d2andx∈ Awe have:
+/angbracketleftBig
+e(c1[s,t]),φ(n)
+[s,t](x)e(d1[s,t])/angbracketrightBig
+=/angbracketleftBigg
+e(cχ0)[2nt]−[2ns]−1/circlemultiplydisplay
+j=1e(cχj)⊗e(cχ′), φ(n)
+[s,t](x)e(dχ0)[2nt]−[2ns]−1/circlemultiplydisplay
+j=1e(dχj)⊗e(dχ′)/angbracketrightBigg
+=/parenleftBig
+ηc,d
+([2ns]+1)2−n−s◦(ηc,d
+2−n)[2nt]−[2ns]−1◦ηc,d
+t−[2nt]2−n/parenrightBig
+(x).
+(52)
+TakeWto be the total subset of k1⊕k2consisting of vectors of the form
+c1⊕c2(ci∈ Wi) with either c1orc2is zero. A direct computation implies
+thatηc,d
+w(x) =/parenleftBig
+j(1)c1,d1w◦j(2)c2,d2w/parenrightBig
+(x).Thus forx∈ A∩L1(τ) andc,d∈ W,
+/bardblηc,d
+w(x)/bardbl1≤ew(M1+M2)/bardblx/bardbl1, (53)
+whereM1,M2depend on /bardblc/bardbl,/bardbld/bardbl. Thus
+/bardbl/angbracketleftBig
+e(c1[s,t]),φ(n)
+[s,t](x)e(d1[s,t])/angbracketrightBig
+/bardbl1≤e(t−s)(M1+M2)/bardblx/bardbl1.
+From this, it follows that the limiting quantum stochastic fl ow (jt)t≥0satis-
+fiesthehypothesisofLemma3.2. As( j(1)
+t)t≥0and(j(2)
+t)t≥0are∗-homomorphic,
+they satisfy the assumption A(1), from which it follows by an easy compu-
+tation that ( jt)t≥0satisfies A(1)too. The assumption A(3)follows from
+the condition (a) of the present theorem and A(2)is the standing assump-
+tion throughout this subsection. Thus by Corollary 3.14 of T heorem 3.1,
+(jt)t≥0is a∗-homomorphic flow. Hence φ(n)
+t(x) converges to jt(x) for each
+x∈ Aandt≥0 in the strong operator topology of B(h⊗Γ).✷
+In the von-Neumann algebraic case, we have the following the orem for a
+special class of quantum stochastic flows:
+Theorem 4.3. Let the quantum stochastic flow (j(l)
+t)t≥0,l= 1,2be a von-
+Neumann algebraic flow, satisfying A(1)–A(3)and the estimate in Lemma
+3.2. Furthermore assume the following:
+27(a)σ(j)= 0forj= 1,2.
+(b) The closure of the operator L(1)
+2+L(2)
+2generates a C0contractive and
+analytic semigroup in hsuch that A0is a core for the generator.
+Thenφ(n)
+t(x)as in Definition 4.1, converges in the strong operator topolo gy
+ofB(h⊗Γ)to a quantum stochastic flow (jt)t≥0with the noise space k1⊕k2
+and structure matrix
+Θ :=
+L(1)+L(2)δ†(1)δ†(2)
+δ(1)0 0
+δ(2)0 0
+.
+Proof.Setθi,(1)
+0(x) :=/angbracketleftbig
+ei,δ(1)/angbracketrightbig
+(x), θi,(2)
+0(x) :=/angbracketleftbig
+li,δ(2)/angbracketrightbig
+(x) fori≥1, where
+{ei}iand{li}iare orthonormal bases for k1andk2respectively. For x∈ A0
+and every positive integer n, we have
+/bardblδ(j)(x)/bardbl2
+2=/summationdisplay
+i/bardblθi,(j)
+0(x)/bardbl2
+2≤2/bardblL(j)
+2(x)/bardbl2/bardblx/bardbl2
+≤21√
+2n/bardblL(j)
+2(x)/bardbl21√
+2n/bardblx/bardbl2
+≤/braceleftbigg1√
+2(1
+n/bardbl(L(j)
+2(x))/bardbl2+n/bardblx/bardbl2)/bracerightbigg2
+forj= 1,2 .(54)
+Thus the operators θi,(j)
+0are relatively bounded with respect to L(j)
+2with
+the bound less than 1. Similar calculations hold for θ0,(j)
+i(x)(≡θi,(j)
+0(x∗)∗),
+j= 1,2. SinceL(j)
+2arethe pre-generators of contractive analytic semigroups
+inh, we see that the operators
+θi,(j)
+0+θ0,(j)
+k+L(j)
+2,
+forj= 1,2,i,k≥1, are pre-generators of C0-semigroups (see [6] Theorem
+2.4 and Corollary 2.5, p 497-498). This implies that for c,d∈ Gwhere
+G:={(ei,0),(0,lj) :i,j≥1},j(l)c,d
+t(x)∈ A ∩h, fort≥0 andx∈ A0.
+Moreover we have: for x∈ A0,c,d∈ G,
+/bardbl/angbracketleftBig
+c,δ(1)⊕δ(2)/angbracketrightBig
+(x)/bardbl2≤/braceleftbigg
+(1√
+2n/bardbl(L(1)
+2+L(2)
+2)(x)/bardbl2)+n√
+2/bardblx/bardbl2/bracerightbigg
+/bardblc/bardbl,
+/bardbl(δ(1)⊕δ(2))†
+d(x)/bardbl2≤/braceleftbigg
+(1√
+2n/bardbl(L(1)
+2+L(2)
+2)(x)/bardbl2)+n√
+2/bardblx/bardbl2/bracerightbigg
+/bardbld/bardbl.
+Thus by hypothesis (b), for c,d∈ G, the operator
+/angbracketleftBig
+c,δ(1)⊕δ(2)/angbracketrightBig
++(δ(1)⊕δ(2))†
+d+L(1)
+2+L(2)
+2+/an}bracketle{tc,d/an}bracketri}ht
+28generates a C0-semigroup in h.
+Letu,v∈L∞(τ)∩L2(τ) andx∈ A0⊂ A. We have:
+/an}bracketle{tu,(j(1)c,d
+t
+n◦j(2)c,d
+t
+n)n(x)v/an}bracketri}ht=/an}bracketle{tv∗u,(j(1)c,d
+t
+n◦j(2)c,d
+t
+n)n(x)/an}bracketri}ht(x∈ A0).
+Note that if we let n→ ∞, the term on the right hand side of the above
+expression converges, by our previous discussions. From th is, it follows that
+φ(n)
+t(x) converges in the weak operator topology of B(h⊗Γ), forx∈ A0
+and hence for x∈ Aby the density of A0inA. The strong convergence
+follows by an application of Corollary 3.14 of Theorem 3.1 as in the proof
+of Theorem 4.2. ✷
+Remark 4.4. The conclusions of Theorem 4.2 and Theorem 4.3 also hold
+under the weaker assumption A(4)replacing the estimate of Lemma 3.2.
+Remark 4.5. By repeating the above arguments, the conclusions of Theorem
+4.2 and Theorem 4.3 hold for finitely many quantum stochastic fl ows, each
+satisfying the corresponding hypotheses.
+4.2 Construction of classical and non-commutative stochas -
+tic processes:
+We shall now illustrate how to construct various multidimen sional processes
+as random Trotter product limits of the corresponding “marg inals”. Our
+examples will include Brownian motion on compact Lie groups and random
+walk on discrete groups. We begin with some general facts abo ut stochastic
+processes on locally compact groups.
+LetGbe a second countable locally compact group. It is well-know n
+that the topology of such a group is metrizable by a metric whi ch makes it
+a Polish (complete and separable) space. A choice of such a me tric is given
+by
+ρ(g,g′) :=∞/summationdisplay
+n=1{|φn(g)−φn(g′)|
+2n(1+|φn(g)−φn(g′)|)}, (55)
+where{φi}∞
+i=1is a countable family of functions from C0(G) which separates
+points ofG.
+Lemma 4.6. Let(Xn)nbe aG-valued random variable on some probability
+space(Ω,F,P)and suppose that for all ψinL2(G)and for all φinC0(G),
+/integraldisplay
+Gdg/integraldisplay
+Ωdω|ψ(g)|2|φ(g.Xn)−φ(g.Xm)|2−→0
+asn,m−→ ∞,wheredgis the left-invariant Haar measure on G.Then there
+exists a random variable X: Ω−→Gsuch thatXn−→Xin probability.
+29Proof.Wechooseandfixsome ψinL2(G)with/bardblψ/bardbl2= 1,andletdIP(ω,g) :=
+dP(ω)⊗|ψ(g)|2dg.Since/integraltext
+Gdg|ψ(g)|2/integraltext
+ΩdP(ω)|φi(g.Xn)−φi(g.Xm)|2→0,
+for everyi,it follows by setting Yn(g,ω) :=g.Xn(ω),and using the domi-
+nated convergence theorem that for every ǫ>0,
+IP(ρ(Yn,Ym)≥ǫ)≤1
+ǫ2IEIP(ρ(Yn,Ym))→0,
+asm,n−→ ∞, whereIEIPdenotes the expectation with respect to the
+probability measure IP. Thus there is a G-valued random variable Y defined
+on Ω,such that
+YnIP−→Y.
+So
+IP(ρ(Yn,Y)≥ǫ) =/integraldisplay
+Gdg|ψ(g)|2P(ρ(g.Xn,Y)≥ǫ)≡/integraldisplay
+Gdg|ψ(g)|2fn(g)−→0,
+wherefn(g) :=P(ρ(g.Xn,Y)≥ǫ).By Egoroff’s theorem, there exists a
+measurable set say ∆ of positive Haar-measure such that for a ll g in ∆,
+we havefn(g)−→0 and the proof of the theorem is complete by taking
+X(ω) =g−1Y(g,ω),for any fixed g in ∆. ✷
+We now give a few concrete examples.
+(A) Classical and non-commutative Brownian motion: Assume, as
+in the Subsection 3.2 (Lemma 3.5 and the discussion followin g it), that
+Gbea second countable and compact Lie group of dimension k, ac ting
+smoothly on a C*-algebra Aandτis a lower-semicontinuous, faithful
+and finite trace on A. LetA∞,Ugand{χℓ}k
+ℓ=1be as in the Subsection
+3.2 and let Gℓdenote the one-parameter subgroup exp( tχl), t∈R.
+Definej(ℓ)
+t:A → A′′⊗B(L2(W(l)))(∼=A′′⊗B(Γ(L2(R+)))),by
+j(ℓ)
+t(x)(ω) :=αexp(W(ℓ)
+t(ω)χl)(x),whereW(ℓ)
+tis the standard Brown-
+ian motion on R.It follows from the discussion of Subsection 3.2, re-
+placingGbyGℓ,that eachj(ℓ)
+tsatisfies the assumptions A(1)–A(3).
+Thus, to verify the hypotheses of Theorem 4.2 for the flows ( j(l)
+t)t≥0,
+l= 1,2,...,k, we need to check only the conditions (a) and (b) of
+that theorem. To this end, let us first verify that L=/summationtextk
+ℓ=1L(ℓ)
+2is the
+pre-generator of a C0-semigroup. For this we proceed as follows:
+Letδℓbethenormgeneratoroftheautomorphismgroup( αexp(tχℓ))t∈R.
+Thenδℓextends to an unbounded, densely defined skew-adjoint oper-
+ator inL2(τ), which generates the unitary group ( Uexp(tχℓ))t∈Rwhere
+αgis implemented by the group of unitaries Ugas discussed in Subsec-
+tion 3.2. By an abuse of notation, we again denote this extens ion by
+30δℓ.Note that L(ℓ)
+2=1
+2δ2
+ℓonA∞for allℓ. Thus/summationtextk
+ℓ=1L(ℓ)
+2=1
+2/summationtextk
+ℓ=1δ2
+ℓ
+is a densely defined, negative and symmetric operator. By Nel son’s
+analytic vector theorem for the representations of the Lie a lgebra [11,
+Theorem 3, p. 591] and [15, Theorem X.39],/summationtextk
+ℓ=1L(ℓ)
+2is essentially
+self-adjoint in L2(τ) and hence its closure generates a C0contraction
+and analytic semigroup. This verifies condition (a) of Theor em 4.2.
+The condition (b) follows by noting that σ(ℓ)= 0 in this case and each
+of the maps δ(l)≡δℓis relatively compact with respect to L. So The-
+orem 4.2 can be applied and the limiting flow gives a ∗-homomorphic
+quantum stochastic flow on Awith generator/summationtextk
+l=1L(l). Specializing
+this to the case when A=C(G), also applying Lemma 4.6, we get
+the convergence in probability of the following sequence of random
+variables:
+X(n)
+t:=k/productdisplay
+i=1[2nt]/productdisplay
+l=0exp((W(i)
+[2nl]+1
+2n−W(i)
+l
+2n)χi), (56)
+where the limiting random variable is clearly a Brownian mot ion on
+G,giving a result similar to that in [13].
+In case ofG=T2andAthe irrational-rotational C*-algebra Aθ
+(see page 254 of [16]), the quantum Brownian motion describe d in
+page-275 of [16] can be constructed using the method describ ed here.
+(B) Random walk on discrete group: Let G be a discrete and finitely
+generated group, generated by a symmetric set of torsion fre e genera-
+tors, say {g1,g2,...,g2k}. Let e be the identity element of G, g1gk+1=
+eandg/ma√sto→αgfor eachgbe the automorphism obtained by the action
+ofGon itself. Take A=C0(G) andτto be the trace with respect
+to the counting measure. Consider 2k mutually independent P oisson-
+processes (N(i)
+t)t≥0, i= 1,...,2k,onIN∪{0},with intensity param-
+eter (λi)2k
+i=1respectively. Let Z(i)
+t:=N(i)
+t−N(k+i)
+t, i= 1,2,...,k.
+Definej(l)
+t:A →L∞(G)⊗B(L2(N(l)
+t,N(k+l)
+t)),(l= 1,2,...,k), by
+j(l)
+t(φ)(ω) =αZ(l)
+t(ω)(φ).Since the generator L(l)of the vacuum ex-
+pectation semigroup associated with j(l)
+tis bounded, so are the other
+structure maps. This implies that all the hypotheses of Theo rem 4.2
+are satisfied. So by Lemma 4.6, we have convergence in probabi lity of
+the following sequence of random variables
+X(n)
+t:=[2nt]/productdisplay
+l=0k/productdisplay
+i=1G(i)
+l+1
+2n(G(i)
+l
+2n)−1,
+31whereG(l)
+t(ω) :=gZ(l)
+t(ω)
+l.The limit is a random variable Xtwhich is
+a time homogeneous continuous time simple random walk.
+4.3 An Application to a class of stochastic processes on a
+UHF algebra
+We begin this subsection with the following definition:
+Definition 4.7. LetAbe a C* or von-Neumann algebra and (Tt)t≥0be a
+quantum dynamical semigroup on A. A∗-homomorphic C*-algebraic ( or
+von-Neumann algebraic) quantum stochastic flow (jt)t≥0onAwith domain
+algebraA0and noise space k0is called a quantum stochastic dilation of
+(Tt)t≥0if we have:
+(i)A0⊆D(L), whereLis the generator of (Tt)t≥0.
+(ii)
+j0,0
+t(x) =Tt(x) (x∈ A,t≥0).
+In this subsection, we apply the results obtained in Section 4.1 to con-
+struct quantum stochastic dilation (in the sense of Definiti on 4.7) of a class
+of quantum dynamical semigroups on a uniformly hyperfinite C *-algebra.
+LetAbe such an algebra, generated by the infinite tensor product o f finite
+dimensional matrix algebras MN(C),i.e. the C*-completion of the algebraic
+tensor product ⊗j∈ZdMN(C) whereNanddare two fixed positive inte-
+gers. The unique normalized trace tronAis given by tr(x) =1
+NnTr(x) for
+x∈MNn(C).For a simple tensor a∈ A, leta(j)be thejthcomponent of a.
+We define support of a, denoted by supp(a) as:
+supp(a) :={j∈Zd|a(j)/ne}ationslash= 1}.
+LetAlocbe the∗-algebra generated by finitely supported simple tensors in
+A.ClearlyAlocis dense in A.Fork∈Zd,the translation τkonAis an
+automorphism determined by τk(x(j)) =x(j+k).
+Note thatMN(C) is generated by a pair of non-commutative represen-
+tatives of the finite discrete group ZN={0,1,2,...,N−1}such that
+UN=VN= 1∈MN(C) andUV=ωVUwhereωis theNthroot of
+unity. Using this fact we get a unitary representation of G=/producttext
+j∈ZdGfor
+G=ZN×ZNin the Hilbert space L2(tr) as follows:
+G ∋g→Ug=/productdisplay
+j∈ZdU(j)αjV(j)βj∈ A,
+forg=/producttext
+j∈Zd(αj,βj), αj,βj∈ZN.We set|g|:={j∈Zd: (αj,βj)/ne}ationslash=
+(0,0)}. For a given completely positive map ψonA, formally we define the
+32Linbladian L:=/summationtext
+k∈ZdLkwhere forx∈ Aloc,Lk(x) :=τkL0(τ−kx) with
+L0(x) :=−1
+2{ψ(1)x+xψ(1)}+ψ(x). Consider the Linbladian Lfor the
+completely positive map ψ(x) :=/summationtextp
+l=1r(l)∗xr(l), x∈ A, wherer(l)∈ A,l=
+1,2,...,p. Forg(j)= (αj,βj)∈G, j∈Zd,we setWj,g(j):=U(j)αjV(j)βj∈
+Alocand define a set C1(A) as:
+C1(A):={x∈ A:/summationdisplay
+j,g/bardblWj,g(j)xW∗
+j,g(j)−x/bardbl<∞}.
+Matsui ([9]) proved that the Linbladian Lthus constructed is well-defined
+onC1(A), closable and a pre-generator of a quantum dynamical semig roup
+(denoted by ( Tψ
+t)t≥0) onA. Furthermore C1(A) is left invariant by Tψ
+tfor
+eacht≥0.
+In[5]theauthorsconsideredtheproblemofconstructingqu antumstochas-
+tic dilations of quantum dynamical semigroups arising as ab ove. However
+they could construct quantum stochastic dilations for only a special class
+of semigroups, namely those for which the associated comple tely positive
+mapψis of the form: ψ(x) =r∗xrforr:=/summationtext
+g∈/producttext
+j∈ZdZNcgWgwith
+Wg:=/producttext
+j∈Zd(UaVb)αj, whereg:=/producttext
+j∈Zdαjsuch that/summationtext
+g|cg||g|2<∞,
+a,b∈ZNbeing fixed. Our aim is to generalize this result. First of all
+we will construct dilations by considering ψof the form given earlier i.e.
+ψ(x) :=/summationtextp
+m=1r(m)∗xr(m), eachr(m)satisfying either (I)or(II)below:
+(I)r(m)is a normal element of A;
+(II)r(m)∈ Alocwith [r(m),r(m)∗]≤0.
+For eacht≥0 setTt:=Tψ
+tandT(m)
+t:=Tψm
+t, whereψm(x) :=r(m)∗xr(m),
+x∈ A. We also denote by LψandL(m)the norm generators of ( Tt)t≥0and
+(T(m)
+t)t≥0respectively.
+Before proving the main theorem of this subsection, we prove a few
+preparatory lemmas. As before, we will view Aas a C*-subalgebra of
+B(L2(tr)).
+Lemma 4.8. Suppose that (St)t≥0is a quantum dynamical semigroup on a
+C*-algebra B, equipped with a finite trace τ. LetLbe the norm generator
+of(St)t≥0and suppose that A0⊆D(L)is a dense ∗-subalgebra of B. Let
+τ(L(y))≤0for ally≥0, y∈ A0. Then(St)t≥0extends to L1(τ)as a
+contractive C0-semigroup.
+Proof.In what follows, L1
+Rwill denote the real Banach space obtained by
+taking theL1-closure of the real vector space of self adjoint elements of B,
+whereasL1or equivalently L1(tr) will denote the usual non-commutative
+complexL1-space of B. Letf(t) =τ(St(y)), y≥0 andy∈ A0. Then
+33f′(t) =τ(L(St(y)))≤0, which implies that f(t) is monotone decreasing.
+Hence we have f(t)≤f(0) i.e.
+/bardblSt(y)/bardbl1≤ /bardbly/bardbl1(y∈ A0,y≥0). (57)
+Lety∈ Bbe positive so that y=x∗xfor somex∈ B. SinceA0is dense
+inB, there is a sequence ( xn)n∈ A0such thatxn→xin/bardbl · /bardbl∞,hence
+x∗
+nxn→x∗xin/bardbl·/bardbl∞.Asτis finite we have /bardbl·/bardbl1≤ /bardbl·/bardbl ∞. Thus we have
+x∗
+nxn→x∗xin/bardbl·/bardbl1, which implies that the inequality (57) extends to all
+the positive elements of B. Decomposing a general self-adjoint xinBas
+x=x+−x−such that |x|=x++x−withx+≥0,x−≥0, we have
+/bardblSt(x)/bardbl1=/bardblSt(x+)−St(x−)/bardbl1≤ /bardblSt(x+)/bardbl1+/bardblSt(x−)/bardbl1≤ /bardblx+/bardbl1+/bardblx−/bardbl1=/bardblx/bardbl1.
+(58)
+This allows us to extend Stas a contractive map on the real Banach space
+L1
+R.We denote this extension by Ssa
+tand consider its complexification S′
+t:
+L1→L1given byS′
+t(x) =Ssa
+t(Re(x))+iSsa
+t(Im(x)), wherex∈L1∩Band
+Re(x), Im(x) are the real and imaginary parts of xrespectively. Clearly,
+S′
+t|B=St.As/bardblRe(x)/bardbl1≤ /bardblx/bardbl1and/bardblIm(x)/bardbl1≤ /bardblx/bardbl1,S′
+tis a bounded (not
+necessarily contractive) map on L1.It follows that the dual map S′,∗
+t:L∞→
+L∞is a weak- ∗continuous map (i.e. ultraweakly continuous in this case).
+Moreover, observe that for a fixed positive x∈ Band an arbitrary positive
+y∈ B ∩L1=B, we have:
+τ(S′,∗
+t(x)y) =τ(xS′
+t(y)) =τ(xSt(y))≥0 (asSt(y)≥0∀y≥0),(59)
+henceS′,∗
+t(x)≥0,i.e.S′,∗
+tis a positive map. Thus we have:
+/bardblS′,∗
+t/bardbl=/bardblS′,∗
+t(1)/bardbl∞=/bardblSsa,∗
+t(1)/bardbl∞=sup/bardblρ/bardbl1≤1,ρ∈L1
+R|τ(Ssa,∗
+t(1)ρ)|
+=sup/bardblρ/bardbl1≤1,ρ∈L1
+R|τ(Ssa
+t(ρ))| ≤1.(60)
+Thus for each t≥0,S′,∗
+tis a contractive map on L∞and hence its
+predual map S′
+tis a contractive map on L1. The semigroup property as well
+as theC0property of ( S′
+t)t≥0onL1follows from the similar properties of
+St=S′
+t|Bwith respect to /bardbl·/bardbl∞and the fact that /bardbl·/bardbl1≤ /bardbl·/bardbl ∞.✷
+Lemma 4.9. Each of the semigroups (Tt)t≥0and(T(m)
+t)t≥0(m= 1,...,p)
+extends toC0-semigroup on L2(tr).
+Proof.For simplicity we will dropthe index mand writerforr(m)andLfor
+L(m). Letrk:=τk(r) andA0:=C1(A).Suppose that y∈ A0andy≥0.A
+simple computation using assumption (I)or(II)yieldstr(L(y))≤0.Thus
+by Lemma 4.8, ( T(m)
+t)t≥0extends toL1as aC0contractive semigroup.
+Letx∈ A0. Then using contractivity and complete positivity of T(m)
+t
+for eachm, we have:
+/bardblT(m)
+t(x)/bardbl2
+2=tr(T(m)
+t(x∗)T(m)
+t(x))≤tr(T(m)
+t(x∗x)) =/bardblT(m)
+t(x∗x)/bardbl1≤tr(x∗x) =/bardblx/bardbl2
+2,
+(61)
+34hence each T(m)
+textends to L2(tr) as a contractive map. The semigroup
+propertyandthestrongcontinuity inthe L2-normnowfollowsbythedensity
+ofA0inL2(tr) and the inequality /bardbl·/bardbl2≤ /bardbl·/bardbl ∞.
+Similar arguments will work for the semigroup ( Tt)t≥0.✷
+Lemma 4.10. LetL2denotes the L2-generator of any of the semigroups
+(Tt)t≥0or(T(m)
+t)t≥0,m= 1,2,...,p. Then we have:
+C1(A)⊆D(L2)∩D(L∗
+2).
+Proof.For simplicity, we drop the superscript mand writerforr(m). Let
+rk:=τk(r). We see that formally L∗
+2(x) =1
+2/summationtext
+k∈Zd{rk[x,r∗
+k] + [rk,x]r∗
+k+
+x[rk,r∗
+k]+[rk,r∗
+k]x}. Forx∈ C1(A) we have:
+/bardblL∗
+2(x)/bardbl ≤/bardblr/bardbl
+2/summationdisplay
+k∈Zd{/bardblδ†
+k(x)/bardbl+/bardblδk(x)/bardbl}+/bardblx/bardbl∞/bardbl/summationdisplay
+k∈Zd[rk,r∗
+k]/bardbl,(62)
+whereδk(x) := [x,rk],x∈ A,k= 1,2,...,∞. But/summationtext
+k∈Zd{/bardblδ†
+k(x)/bardbl+
+/bardblδk(x)/bardbl}<∞(see [5] and [9]) and thus it suffices to show the conver-
+gence of the third series in (62). Note that if rsatisfies the assumption
+(I), then [rk,r∗
+k] = 0 for all k= 1,2,...,∞which proves the required
+convergence. Suppose now that rsatisfies the assumption (II). Asr=/summationtext
+g∈Gcg/producttext
+j∈ZdU(j)αjV(j)βj,
+/summationdisplay
+k∈Zdτk{[r,r∗]}=/summationdisplay
+k∈Zd/summationdisplay
+g,h∈Gcgchτk{[/productdisplay
+j∈ZdU(j)αjV(j)βj,/productdisplay
+j∈ZdV(j),−β′
+jU(j),−α′
+j]}
+=/summationdisplay
+k∈Zd/summationdisplay
+g,h∈Gcgch[/productdisplay
+j∈ZdU(j+k)αjV(j+k)βj,/productdisplay
+j∈ZdV(j+k),−β′
+jU(j+k),−α′
+j]
+=/summationdisplay
+k∈Zd/summationdisplay
+g,h∈Gcgch[/productdisplay
+j∈ZdU(j)αj−kV(j)βj−k,/productdisplay
+j∈ZdV(j),−β′
+j−kU(j),−α′
+j−k].
+(63)
+Asr∈ Aloc, we haveαj−k=βj−k=α′
+j−k=β′
+j−k= 0∈ZNfor|k| ≥M
+for sufficiently large integer M. Hence [rk,r∗
+k] = 0 for such k.Thus the
+series is actually finite and hence /bardblL∗
+2(x)/bardbl<∞,i.e.A0⊆D(L2)∩D(L∗
+2)
+as required. ✷
+Lemma 4.11. For eachm= 1,2,...,p, there exists a C*-algebraic quantum
+stochastic flow (j(m)
+t)t≥0onAwith domain algebra A0:=C1(A), which
+gives a quantum stochastic dilation of the semigroup (T(m)
+t)t≥0. Furthermore
+(j(m)
+t)t≥0satisfies assumptions A(1),A(2), (i) and (ii) of A(3)and the
+hypotheses of Lemma 3.2 and Corollary 3.13, with A0=C1(A).
+35Proof.The existence of the dilation ( j(m)
+t)t≥0is the main result in [5]. Since
+theflow(j(m)
+t)t≥0isa∗-homomorphicflowforeach m= 1,2,...,p, byLemma
+2.9 the flow satisfies assumption A(1). OnA,tris a faithful and finite trace
+andA0is a dense ∗-subalgebra of A, which implies A(2). The statement
+in the first part of (i) of A(3)follows from Lemma 4.9. From [9] it follows
+thatA0is a core for L(m), asA0is left invariant by the semigroup ( T(m)
+t)t≥0
+for eachm= 1,2,...,p. Furthermore, since A0is dense in L2(tr) and the
+L2-extension of ( T(m)
+t)t≥0leaves it invariant for each m, it follows that A0
+is also a core for L2. Thus the second part in (i) of A(3)follows. The
+statement in (ii) of A(3)follows by a simple computation. From Lemma
+4.10 it follows that the hypotheses of Corollary 3.13 hold fo r (j(m)
+t)t≥0. We
+prove that it also satisfies the hypothesis of Lemma 3.2 as fol lows:
+Letδ(m)
+j(x) := [x,τj(r(m))], x∈ A0.Then
+/bardblδ(m)
+j(x)/bardbl1≤2/bardblτj(r(m))/bardbl∞/bardblx/bardbl1. (64)
+Thusδ(m)
+jextends to a bounded operator on L1for eachm. Similar result
+holds forδ†,(m)
+j. Since (j(m)
+t)t≥0is a C*-algebraic quantum stochastic flow,
+it follows from Lemma 2.12 that the semigroup ( j(m)ei,ej
+t)t≥0isC0and its
+generator restricted to A0is given by L(m)+δ(m)
+i+δ†(m)
+j, where{ej}jis an
+orthonormal basis for the associated noise space k0(see [5]). By Lemma 4.8,
+(T(m)
+t)t≥0extends to a contractive C0-semigroup on L1(tr). Let us denote
+the generator of the extended semigroup by L(m)
+1. Clearly, as A0is dense
+inL1(tr) and is left invariant by the semigroup ( T(m)
+t)t≥0, it is a core for
+L(m)
+1. As the maps δ(m)
+iandδ†(m)
+jare bounded maps on L1(tr), it follows
+that the map L(m)
+1+δ(m)
+i+δ†(m)
+jgenerates a C0-semigroup on L1(tr) and
+moreover, A0is a core for this map as well. This proves that the semigroup
+(j(m)ei,ej
+t)t≥0extends to a C0-semigroup on L1(tr). Similar conclusion will
+hold if we replace δi,δ†
+jbyzδi,zδ†
+j’s for any fixed z∈C, hence thehypothesis
+of Lemma 3.2 is satisfied with the total set Wbeing{zei;z∈C, i≥1}.✷
+Now we prove the main theorem of this subsection.
+Theorem 4.12. Suppose that (j(m)
+t)t≥0is the quantum stochastic flow with
+structure maps L(m),δ(m)
+i,δ†(m)
+i,i= 1,2,...,∞, as obtained in Lemma 4.11
+form= 1,2,...,p. Then the quantum stochastic differential equation:
+djt(x) =/summationdisplay
+j∈Zdp/summationdisplay
+m=1jt(δ†(m)
+j(x))da(m)
+j(t)+/summationdisplay
+j∈Zdp/summationdisplay
+m=1jt(δ(m)
+j(x))da†(m)
+j(t)+p/summationdisplay
+m=1jt(L(m))dt,
+j0(x) =x⊗IΓ,
+(65)
+36wherex∈ A0andΓis a symmetric Fock space, admits a ∗-homomorphic
+solution (jt)t≥0which is a C*-algebraic quantum stochastic flow and gives a
+quantum stochastic dilation of the quantum dynamical semig roup(Tψ
+t)t≥0.
+Proof.We will apply Theorem 4.2 to the collection of quantum stocha stic
+flows: (j(m)
+t)t≥0,m= 1,2,...,p(see Remark 4.5). By Lemma 4.11 the flows
+satisfy assumptions A(1),A(2), (i) and (ii) of A(3)and the hypothesis of
+Corollary 3.13. By virtue of Remark 4.4, it suffices to verify c onditions (b)
+and the core property of A0as in (a) of Theorem 4.2.
+Observe that the map/summationtextp
+m=1L(m)is the pre-generator of the quantum
+dynamical semigroup ( Tψ
+t)t≥0(by [9]). Moreover, the maps δ(m)
+j,δ†(m)
+j,m=
+1,2,...,p,j = 1,2,...,∞areboundedmapson A. Thusthemap/summationtextp
+m=1(L(m)+
+δ(m)
+i+δ†(m)
+j) generates a C0-semigroup on A. Thuscondition (b) of Theorem
+4.2 holds.
+From [9] it follows that the dense ∗-subalgebra A0:=C1(A) is a core for
+the map/summationtextp
+m=1L(m).
+Hence by Theorem 4.2 we get a quantum stochastic flow ( jt)t≥0which
+satisfies the required quantum stochastic differential equat ion.✷
+Corollary 4.13. Letr(m)=/summationtext
+g∈ZNcgWgfor eachm,whereWg=/producttext
+j∈Zd(UaVb)αj
+forg=/producttext
+j∈Zdαj(as in [5]). Then the hypotheses of Theorem 4.12 are sat-
+isfied and the same conclusion follows, which generalizes th e dilation result
+obtained in [5].
+Acknowledgment: The second author is partially supported by a project
+on ‘Non-commutative Geometry and Quantum Groups’ funded by Indian
+National ScienceAcademy and bySwarnajayanti Fellowship a nd Grant from
+DST, Govt. of India. The third author is supported by the Bhat nagar
+Fellowship of CSIR. All the authors also acknowledge the sup port of UK-
+India Education and Research Initiative (UKIERI).
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+38
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new file mode 100644
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@@ -0,0 +1,817 @@
+arXiv:1001.0234v1  [astro-ph.CO]  4 Jan 2010Mon. Not. R. Astron. Soc. 000, 1–11 (2002) Printed 1 November 2018 (MN L ATEX style file v2.2)
+Ionized gas outflow in the isolated S0 galaxy NGC 4460⋆
+Alexei Moiseev †, Igor Karachentsev and Serafim Kaisin
+Special Astrophysical Observatory, Russian Academy of Sci ences, Nizhnii Arkhyz, Karachaevo-Cherkesskaya Republic , 369167 Russia
+Accepted December 29; Received 2009 December 8; in original form 2009 October 1
+ABSTRACT
+Weuseintegral-fieldandlong-slitspectroscopytostudythebright extendednebulosity
+discovered in the isolated lenticular galaxy NGC 4460 during a recent H αsurvey of
+nearbygalaxies.AnanalysisofarchivalSDSS, GALEX,andHSTimagesindicatesthat
+current star formation is entirely concentrated in the central kilo parsec of the galaxy
+disc. The observed ionized gas parameters (morphology, kinematic s and ionization
+state) can be explained by a gas outflow above the plane of the galax y caused by a
+star formation in the circumnuclear region. Galactic wind parameter s in NGC 4460:
+outflow velocity, total kinetic energy – are several times smaller co mparing with the
+known galactic wind in NGC253, which is explained substantially lower tot al star
+formation rate. We discuss the cause of the star formation proce sses in NGC 4460
+and in two other known isolated S0 and E galaxies of the Local volume: NGC 404
+and NGC 855. We provide evidence suggesting that feeding of isolate d galaxies by
+intergalactic gas on a cosmological time scale is a steady process with out significant
+variations.
+Key words: galaxies: elliptical and lenticular – galaxies: ISM – galaxies: kinematics
+and dynamics – galaxies: starburst – galaxies: individual: NGC 4460
+1 INTRODUCTION
+Isolated elliptical (E) and lenticular (S0) galaxies are ra ther
+rare objects, whose properties differ appreciably from E
+and S0 galaxies in clusters and groups. Among the 513
+most isolated galaxies of the Local Universe with radial
+velocities VLG<3500 only 19 systems (i.e., about 4 per
+cent) belong to types E and S0 (Karachentsev et al. 2009).
+Isolated E and S0 galaxies are characterized by low lu-
+minosity ( MB≃ −17.m6), bluer-than-average colors, and
+appreciable presence of gas and dust. The nearby lentic-
+ular galaxy NGC 404 with MB=−16.m6, which was
+found to host an extended HI shell (Del Rio et al 2004) and
+ultraviolet-bright ring of young stars (Thilker 2009), is a
+typical example. Among ∼450 Local-volume (LV) galaxies
+with heliocentric distances D <10 Mpc (Karachentsev et al.
+2004) there are only threeE and S0 galaxies with ab-
+solute magnitudes [ −16.0> M B>−18.0] and nega-
+tive ‘tidal indices’ TI <0, which allowed these galaxies —
+NGC 404, NGC 855, and NGC 4460 — to be classified as
+isolated objects. These objects seem to be local analogues
+of recently identified by Kannappan, Guie & Baker (2009)
+⋆Based on observations collected with the 6m telescope of the
+Special Astrophysical Observatory of the Russian Academy o f
+Sciences which is operated under the financial support of Sci ence
+Department of Russia (registration number 01-43)
+†moisav@gmail.com‘blue-sequence E/S0’ galaxies which might form a transitio n
+population between late-type spirals/iregulars and early -
+type red-sequence galaxies.
+An extensive H αsurvey of LV galaxies has been carried
+out in recent years with the 6-m telescope of the Special As-
+trophysical Observatory of the Russian Academy of Sciences
+(SAORAS)inordertodeterminerates ofstar formation ina
+representative distance-limited sample of galaxies. An an al-
+ysis of the population of Canes Venatici I scattered cloud of
+nearby galaxies revealed (Kaisin & Karachentsev 2008) that
+the circumnuclear region in NGC 4460 hosts the bright H α
+line emission (see Fig. 1), which must be indicative of ongo-
+ing star formation. Such a phenomenon appears to be non-
+typical in an isolated early-type galaxy. This bright emiss ion
+may be a result of a recent merger of NGC 4460 with a gas-
+rich companion, or of an impact of a massive intergalactic
+HI cloud onto the central region of NGC 4460. In this case,
+the Hαnebulosity, which extends along the major axis of
+the galaxy, may be a product of a recent interaction. An
+analysis of the kinematics and ionization state of the gas
+in NGC 4460 helps us understand the nature of this mys-
+terious phenomenon. With this aim in view, we analyzed
+the available archival images of the galaxy and performed
+spectroscopic observations of NGC 4460 with the SAO RAS
+6-m telescope. Following Tonry et al. (2001), we adopt the
+distance to the galaxy to be 9.59 Mpc that corresponds to
+a scale of 46 .5pcarcsec−1.
+c/circlecop†rt2002 RAS2Moiseev et al.
+Figure 1. Images of NGC 4460 in logarithmic brightness scale. The uppe r row shows the SDSS r-band image (left) and the H α+[Nii]
+image (right) taken with the 6-m telescope (Kaisin & Karache ntsev 2008). The locations of the spectrograph slits are sho wn. Hereafter
+the black square marks the MPFS field of view. The lower row con tains the images of the center of the galaxy with H αline contours
+superimposed based on GALEX FUV data (left), HSTWFPC2/F606W optical map (center) and map of the ( g−r) color index according
+to SDSS (right). The cross marks the galaxy nucleus that is co incided with the kinematic center derived from MPFS data. Th e arrow
+shows the position of second circumnuclear knot (see text).
+2 PHOTOMETRIC STRUCTURE OF THE
+GALAXY
+2.1 Circumnuclear region
+A comparison of the H α+[Nii] image of the galaxy with
+broadband SDSS images (Fig. 1) convincingly demonstrates
+that most of the emission-line radiation emerges from a
+compact region in the disc inside the radius r <20 arc-
+sec. That is where all HII regions are located, whereas dif-
+fuse Hα+[Nii] emission extends along the minor axis of the
+galaxy on both sides of its nucleus. An analysis of opti-
+cal and UV images shows that ongoing star formation inNGC 4460 is concentrated entirely inside the central kilo-
+parsec. First, it follows from the distribution of FAR-UV
+radiation ( λ1344−1786˚A) according to GALEX archival
+data, that is exactly coincident with the circumnuclear HII
+regions (Fig. 1). Second, the young stellar population foun d
+in this region stands out by its bluer color index in SDSSim-
+ages (see Fig. 1).Star-formingregions makeanalmost close d
+ring in the disc of the galaxy. The r-band image reveals a
+compact nucleus with ( g−r) = 0.65, whereas a condensa-
+tion located 1 .5−2 arcsec North of the nucleus (marked
+with arrow on Fig. 1) becomes brighter in blue filters with
+the color ( g−r) = 0.45. On the HSToptical image this
+c/circlecop†rt2002 RAS, MNRAS 000, 1–11Ionized gas outflow in NGC 4460 3
+circumnuclear knot corresponds to compact stellar cluster
+with a linear size FWHM = 17×10 pc whose luminosity
+is only twice lower than that of the compact cluster in the
+‘main’ nucleus. We note that the dynamical center of the gas
+velocity field (marked with cross on Figs. 1 and 3) coincides
+with the main nucleus of the galaxy (see below Sect. 5). The
+circumnuclear star-forming regions have evidently differe nt
+evolutionary status, because only NE of the nucleus does the
+peak of H αemission coincide with the ‘blue’ region on the
+(g−r) map. At the same time, the two blue regions near the
+minor axis (including the second nucleus) are unassociated
+with Hα+[Nii] peaks. These regions must have already lost
+most of their gas or lack OB stars needed for its ionization.
+The SDSS images exhibit individual dust lanes in the
+inner part of the region. On the HSTimage these lanes can
+be easily seen toborder HII regions, and they are most prob-
+ably be associated with dense gas compressed by shocks pro-
+duced by star-forming processes (SN explosions and winds
+from massive stars).
+2.2 Orientation of the disc and the brightness
+distribution
+Using isophotal analysis of SDSS images we found the posi-
+tion angle and ellipticity of the outer randi-band isophotes
+to bePA= 39.5±0.4◦,ǫ= 0.71±0.02. According to empir-
+ical relation (Hubble 1926), this implies a galaxy inclinat ion
+ofi= 77±1◦.Weusedtheinferredorientation parametersto
+construct the azimuthally averaged surface-brightness pr o-
+files ofubvriSDSS images. In all filters the brightness distri-
+bution is almost exactly exponential at r >30−40 arcsec,
+with the brightness excess at r <20 arcsec being more con-
+spicuous in blue filters – it is represented by star-forming r e-
+gions. Figure 2 shows decompositions of the r-band surface
+brightness profile into components. After the subtraction o f
+the outer disc ( µ0= 19.4m,h= 29.5 arcsec), brightness ex-
+cess still remains at r <30−40 arcsec. However, it is not a
+bulge, because after the subtraction of the two-dimensiona l
+disc model the brightness contours of the residual image
+have the same ellipticity as those of the outer disc. The
+brightness profile at the center can be described fairly well
+by a second exponential function with almost the same cen-
+tral brightness ( µ0= 19.2m), but with a three times shorter
+scale length h= 11.5 arcsec. At the center one can see a
+low-contrast compact nucleus described above.
+It thus follows that only some of the morphological
+features (smooth outer isophotes, lack of spiral arms) sug-
+gest that NGC 4460 may be a lenticular galaxy, whereas
+the bulge/disc ratio is indicative of a later morphological
+type, because the spherical component is barely visible in
+the brightness distribution. Inner exponential discs are o f-
+ten referred to as ‘pseudo-bulges’ (Kormendy & Kennicutt
+2004). However, the corresponding feature in NGC4460 it is
+indeed a bona fide disc whose thickness, according to pho-
+tometric data, coincides with that of the outer disc.
+Multitiered (antitruncation according the classification
+by Erwin, Beckman & Pohlen 2005) discs have been found
+increasingly often in spiral and lenticular galaxies, and t hey
+are attributed either to slow internal secular evolution th at
+can only develop in the presence of a bar and/or a spiral (see
+Kormendy & Kennicutt 2004, for review), or to a recent mi-
+nor merging event. Both cases imply the loss of angular mo-mentum by a part of gas moved toward the center, where a
+burst of star formation currently occurs, and the newly born
+stars form the inner disc with a short scale length. Recent
+formation of the inner disc is also evidenced by the fact that
+its contribution to the combined brightness profile (Fig. 2)
+is almost everywhere smaller than that of the main disc of
+the galaxy.
+2.3 Bar or ring?
+In Sect 2.1 we mentioned that the star-forming regions make
+an almost closed ring in the central kpc. At first sight, the
+observed distribution of the HII regions has another inter-
+pretation, where the two brightest condensations, located
+almost equidistant from the nucleus ∼10 arcsec to north-
+east and south-west, belong to the ends of a stellar bar. In
+this case, the two blue regions appeared on the color-index
+map (Fig. 1) at r= 2−4′′(included the off-nuclear com-
+pact cluster mentioned above) can be considered as part of
+nuclear compact ring formed on the resonances of this bar.
+However, this assumption contradicts to other morpho-
+logical features. First is a significant asymmetry in the pro p-
+erties of two brightest HII regions. They have different size ,
+Hα-luminosity and color index, i.e. different history of their
+formation. That is strange in the case of lobes on the ends
+of a bar which should be formed simultaneously. Secondly,
+the dust in the circumnuclear region has a chaotic distribu-
+tion (Sect 2.1) instead sharp curved dust lanes expected in a
+bar. Finally, the isophotal analysis of SDSS red images does
+not support any significant changes of ellipticity and PAof
+isophotes at the distances corresponded to the possible bar .
+The effect of non-circular gas motions (radial inflow)
+could be also insufficient in the observed gas velocity field,
+because the major axis of possible bar coincides with the
+galaxy line of nodes. Therefore, the regular velocity pat-
+tern in the circumnuclear region of ionized gas velocity
+field (Sect. 5) can not be use as pro or contra arguments
+about bar. We think that only data on stellar kinemat-
+ics can evident the existence (or absent) a stellar bar in
+NGC 4460. However, our analysis of galactic photometry
+presented above suggests the star-forming pseudo-ring 2 kp c
+in diameter nested in the inner stellar disc as more reliable
+description of the observed morphology.
+3 OBSERVATIONS AND DATA REDUCTION
+Long-slit and integral-field (3D) spectral observations we re
+made at the prime focus of the SAO RAS) 6-m telescope.
+The 2048 ×2048 EEV 42-40 CCD was used as a detector in
+both cases.
+The central region of NGC 4460 was observed with
+the MultiPupil Fiber Spectrograph (MPFS). The MPFS
+(Afanasiev, Dodonov & Moiseev 2001) takes simultaneous
+spectra from 256 spatial elements (constructed in the
+shape of square lenses) that form on the sky an ar-
+ray of 16 ×16 elements (‘spaxels’) with the angular size
+1arcsecspaxel−1. A bundle of 17 fibers placed at the dis-
+tance about 3.5 arcmin from the lens array provides the
+night-sky background spectra simultaneously with objects
+exposition. The spectral range included bright emission li nes
+of ionized gas: H α, [Nii]λλ6548,6583 and [S ii]λλ6717,6731.
+c/circlecop†rt2002 RAS, MNRAS 000, 1–114Moiseev et al.
+Table 1. Log of spectral observations
+Date Device position Texpsp. range sp. resol. seeing
+(s) ( ˚A) ( ˚A) ( arcsec)
+27/28 Mar 2007 MPFS nucleus 1800 5800-7300 3.5 1.7
+15/16 May 2007 SCORPIO/LS PA= 128◦, offset 5 arcsec to SW 1200 6050-7100 2.5 1.7
+15/16 May 2007 SCORPIO/LS PA= 128◦, offset 9 arcsec to NE 900 6050-7100 2.5 1.8
+Figure 2. Decomposition of the r-band surface brightness profile
+(dots) into two discs (the dashed line). The solid line marks the
+sum of both disc models.
+Fig 1 shows the locations of MPFS lens array on the
+galaxy image, the log of observations is presented in Ta-
+ble 1. The spectra were reduced using the IDL-based soft-
+ware, the data reduction sequence is briefly described in
+Moiseev, Vald´ es & Chavushyan (2004). The data reduction
+results in a data cube in which a spectrum corresponds to
+each image ‘spaxel’ in two-dimensional field 16 ×16 arcsec2.
+The data cube was flux-calibrated using the standard star
+observations taken on the same night, just after the galaxy.
+The maps and velocity fields of the H α, [Nii] and [Sii] emis-
+sion lines were constructed using a single-Gaussian fitting .
+The long-slit observations have been made with the
+multi-mode focal reducer SCORPIO (Afanasiev & Moiseev
+2005). We obtained two cross-sections in direction near
+perpendicular to the galactic major axis, along extended
+Hαfilaments and though bright HII-regions in the galaxy
+disc (hereafter slits #1 and #2, see Fig. 1). The slit width
+was 1 arcsec, the scale along slit was 0 .35arcsecpx−1, the
+log of observations is given in Table 1. The slit spec-
+tra were reduced and calibrated using the IDL-based soft-
+ware, developed in SAO RAS. The parameters of the emis-
+sion lines (FWHM, flux, velocity) were calculated from
+a single-Gaussian fitting. A multicomponent structure of
+the emission line profiles was not detected as well as in
+MPFS spectra. The long-slit spectra are deeper, because
+SCORPIO quantum efficiency is significantly higher than
+MPFS. This fact allows us to detect the weak emission lines
+[OI]λλ6300,6360 and He I λ6678 in the brighter HII regions
+in the galactic disc.4 PARAMETERS OF IONIZED GAS
+Fig. 3 shows several maps based on MPFS data. The H α
+brightness distribution agrees well with the narrow-band fi l-
+ter image discussed above. The continuum image exhibits
+a central condensation extending along the N-S direction.
+Given the lower spatial resolution of the MPFS compared
+to SDSS data, the condensation must correspond to the
+double nucleus described in Sect. 2.1. The MPFS field of
+view contains three bright HII regions. The maps of the
+[Nii]/Hαand [Sii]/Hαflux ratios indicate a slight en-
+hancement of forbidden-line emission around star-forming
+regions1, whereas the corresponding ratios are minimal at
+the centers of HI regions (see Fig. 3). In the space be-
+tween star-forming regions we most likely observe hot gas
+ejected from these regions and ionized, among other factors ,
+by shocks. The presence of shock fronts around HII regions
+is further evidenced by dust lines seen at these locations
+(Sect. 2.1). In these places the [S ii]/Hαratio reaches 0.45,
+which is close to the usually adopted border between shock
+ionization and photoionization (see below).
+Figure 4 shows variations of the emission-line param-
+eters (flux, radial velocity, FWHM corrected for instru-
+mental broadening) along the slits. The maximum of the
+emission lines brightness is therefore shifted relative to the
+peak in brightness of stellar continuum, because bright HII
+regions in the circumnuclear ring do not lie on the major
+axis of the galaxy. Beyond the disc, surface brightness in
+the nebulosity decreases almost exponentially and emissio n
+lines can be detected at even greater distance from the disc
+than on the H α+[Nii] image. The [S ii]/Hα, intensity ratio
+increases with increasing distance from HII regions, where as
+the corresponding increase of the [N ii]/Hαratio is less ap-
+preciable (see Fig. 4).
+To identify the source of gas ionization, we constructed
+the diagnostic diagram showing the ratios of the fluxes of
+lines with different excitation mechanisms and similar wave -
+lengths so that the result would not depend on dust extinc-
+tion. Unfortunately, we can construct only one such dia-
+gram in the wavelength interval considered: [N ii]/Hαversus
+[Sii]/Hα(Fig 5). The dashed lines separate regions with dif-
+ferent excitation mechanisms indicated as ‘photoionizati on’
+and ‘shock’. We assume, in accordance with Stasi´ nska et al.
+(2006), that in the case of ionization by young stars
+the line rations follow to relations: log[S ii]/Hα <−0.4,
+log[Nii]/Hα <−0.4. Higher relative intensity of forbid-
+den lines means that shocks, or even a nonthermal source
+(AGN), contribute appreciably to ionization. When inter-
+1Hereafter by the [N ii]/Hαand [Sii]/Hαratios we mean the
+[Nii]λ6583/Hαand [Sii](λ6717+λ6731)/H αline flux ratios, re-
+spectively.
+c/circlecop†rt2002 RAS, MNRAS 000, 1–11Ionized gas outflow in NGC 4460 5
+pretingthe positions of datapoints on the diagram one must
+bear in mind that: (1) boundaries of the photoionization do-
+main are somewhat controversial. Namely, the boundaries
+are determined not only by particular models, but also by
+the statistics of observations of different galaxies sample s;
+(2) in real objects the combined effect of several ioniza-
+tion sources is present. Thus, according to Stasi´ nska et al .
+(2006),the −0.2<log[Nii]/Hα <−0.4domaincorresponds
+to a composite source of ionization in LINER-type galaxies:
+photoionization bystars plus theeffect of shocks, and shock s
+dominate only at log[N ii]/Hα >−0.4. Note that many re-
+searchers use even harder criterion (e.g., log[N ii]/Hα >0
+in Veilleux, Cecil & Bland-Hawthorn 2005).
+Figure 5 shows that the gas in the circumnuclear region
+(i.e. central part observed with the MPFS, and the r <5
+arcsec regions in slit #1 and r <10 arcsec regions in slit
+#2) is ionized by young stars. In the outer filaments shock
+ionization begins to dominate with increasing distance fro m
+the disc plane, according to the [S ii]/Hαcriterion. At the
+sametime, the[N ii]/Hαcriterionindicates thatall ourmea-
+surements, except several data points with r≈30 arcsec in
+slit #2, lie in the photoionization domain. However, in this
+case we consider the latter criterion to be less important, b e-
+cause the observed line ratio agrees well with the results of
+model computations Allen et al. (2008) for shock+precursor
+ionization with number density n= 1cm−3and solar metal-
+licity. Figure 5 shows the grid of parameters for this model.
+It is evident from this figure that with increasing rthe ob-
+served data points lie along the lines of the increasing shoc k
+velocity from 200 to 300kms−1for small values of magnetic
+parameter 0 −1µGcm2/3. For estimation of gas mettalicity
+we used the data of slit #2 that passes through the center
+of the brightest HII region, where the [N ii]/Hαratio is min-
+imal and equal to 0 .16±0.03. If we assume that the shock
+contribution is minimal at this location then the approxi-
+mation of Stasi´ nska et al. (2006) implies a gas metallicity
+of 0.74±0.04Z⊙, i.e. the solar-metallicity approximation is
+quite applicable to NGC 4460.
+It thus follows that gas ionization in the extended H α
+nebulosity can be explained by the combined effect of shocks
+and photoionization by young stars with the latter dominat-
+ing in the inner regions. The galaxy lacks an active nucleus
+and therefore the likely cause of shocks is star formation.
+Gas is blown above the plane of the galaxy by the com-
+bined action of supernovae and stellar winds from young
+massive stars. This phenomenon is known as a ‘galactic
+wind’ (see a review by Veilleux et al. 2005). For compari-
+son, we present in Fig. 5 the line ratios for three well stud-
+ied galaxies where the existence of galactic wind has long
+been proven: M 82, NGC 253, and NGC 1569. For M 82
+we give two average values obtained by Westmoquette et al.
+(2009), who used DensePak and GMOS instruments. We
+adopt the galactic-wind data for NGC 253 from Fig. 5 in
+Matsubayashi et al. (2009), and for NGC 1569 we give sev-
+eral estimates for external emission regions adopted from
+Fig. 7 in Heckman et al (1995), for slit PA= 70◦andr=
+±20,±50,±70 arcsec from the center. Onthe diagnostic dia-
+gram the published data for these well-known galactic winds
+scattered in relative wide range ( ±0.5 dex), which also in-
+cludes the data points for NCC 4460 . Shock contribution to
+gas ionization in the filaments of NCC 4460 is comparable to
+galactic winds for all three galaxies hitherto studied. Mor e-
+Figure 3. Results of MPFS observations. The top panel shows
+the continuum image taken near λ6400˚A(left) and the H αmap
+(right). The bottom panel shows the H αvelocity field (left) and
+the map of the [S ii](λ6717+λ6731)/H αflux ratio with H αcon-
+tours superimposed (right). The coordinate origin coincid es with
+the dynamic center (the cross).
+over, the [S ii]/Hαratio in NGC 4460 is comparable with the
+values observed in NGC 253 and NGC 1569. That is not for
+[Nii]/Hα, because the intensity of the nitrogen doublet also
+depends from from metallicity. For instance, [N ii]/Hαratio
+inNGC253 is most likely duetothenitrogen overenrichment
+of the interstellar medium (Matsubayashi et al. 2009).
+We use the [S ii] line doublet ratio to estimate the elec-
+trondensity neinaccordancewiththerelation ofOsterbrock
+(1989) for Te= 10000K. Figure 6 shows the radial varia-
+tions of of the line ratio and corresponded values of ne. Un-
+fortunately, Osterbrock’s relation can be applied with con fi-
+dence only at ne>50−100cm−3, whereas at low densities
+the slope of the ([S ii]λ6717/λ6731 vs ne) relation flattens
+and therefore leads to large uncertainties. Although the ob -
+served line ratio lies mainly inside this ‘inconvenient’ in ter-
+val, it is safe to conclude that the density of emitting gas is
+low than 100cm−3in the disc of the galaxy. Outer filaments
+exhibit a large scatter of density values ranging from zero
+to 200−300cm−3. It might be supposed that in the outer
+regions the spectrograph slit crosses individual clumps wi th
+highly nonuniform density distribution.
+5 THE GAS KINEMATICS
+The Hα-line velocity field based on MPFS measurements
+(Fig. 3) can be described fairly well in terms of the model of
+circular rotation of the thin disc. The line-of-sight veloc ities
+measured by the [N ii] and [S ii] lines show no appreciable
+deviations from H αline-of-sight velocities. The center of
+c/circlecop†rt2002 RAS, MNRAS 000, 1–116Moiseev et al.
+Figure 4. Measurements of parameters of the H α, [Nii]λ6583 and [S ii]λ6717 + 6731 emission lines along the slits #1 (left) and #2
+(right). From top to bottom: surface brightness, line-of-s ight velocity, FWHM (corrected for the widths of the instrum ental profile), and
+the line-to-H αflux ratio. The dashed and solid lines show the systemic veloc ity (according to MPFS data) and the cross section of the
+model velocity field, respectively.
+rotation defined as the symmetry center of the velocity field
+coincides with the center of the continuum image.
+We constructed the model of rotation using the method
+described by Moiseev et al. (2004) with the position angle
+and inclination of the disc fixed in accordance with photo-
+metric data (Sect. 2.2). The PAmeasured from the velocity
+field agrees with the photometric position angle within the
+errors. The systemic velocity is Vsys= 508±kms−1. The
+rotation curve shows solid-rotation increase up to vmax=
+67kms−1atr= 20 arcsec that is a border of MPFS field-
+of view. This value is close to the maximum rotation ve-
+locity according to HI measurements (Sage & Welch 2006)
+Observed velocities deviate from the corresponding modelvalues by up to 20kms−1along the minor axis of the galaxy,
+and this behavior is most probably indicative of gas ejectio n
+above the plane of the galaxy. It could also be due to the
+uncertainties in the model: the thin-disc approximation ma y
+perform poorly at i= 77◦, internal absorption should be
+taken into account, etc.
+The ionized gas velocity field in the compact star clus-
+ter region noted on optical images at ∼2 arcsec North of
+the nucleus (see Sect. 2.1) does not show any kinematic pe-
+culiarities, like non-circular motions or shift of the rota tion
+center. Therefore, our data do not support that this clus-
+ter is a nucleus of a dwarf satellite – a merging remnant.
+c/circlecop†rt2002 RAS, MNRAS 000, 1–11Ionized gas outflow in NGC 4460 7
+Figure 5. The diagram of the [N ii]/Hαvs [Sii]/Hαflux ratios. The dashed line separates domains with different ionization mechanisms.
+The blue lines show the grid of shock+precursor ionization m odels according to Allen et al. (2008) for n= 1cm−3and solar elemental
+abundances. The thin and bold blue lines mark the contours of constant magnetic parameter 0 .001,0.5,1,5µGcm2/3(from bottom to
+top) and contours of constant shock velocity (which is label ed in kms−1), respectively. The circles show the measurements made alo ng
+slits #1 (left) and #2 (right). Different colors correspond t o different distances along the slits. The diamond signs indi cate the results
+of MPFS measurements for the circumnuclear region. The aste risks in the left-hand figure show the measurements for the re gions of
+galactic wind in M 82 (Westmoquette et al. 2009), NGC 253 (Mat subayashi et al. 2009), and NGC 1569 (Heckman et al 1995).
+New high-resolution stellar kinematics data are needed for
+understanding the origin of this off-set cluster.
+To estimate the peculiar velocities of the outer fila-
+ments, we extrapolate our model out to greater distances
+assuming that the rotation curve reaches a plateau. Figure 4
+compares the model extrapolation and observed velocities
+in slit cross sections. Slit #1 shows a good agreement be-
+tween the model and observations at r <15 arcsec in the
+NW half of the nebulosity, whereas its more extended SE
+half exhibits appreciable deviations already beyond 7 −10
+arcsec from the center with the discrepancies amounting to
+30−35kms−1. Slit #2 passes farther from the center, along
+the edge of the field of view of the MPFS, and therefore here
+deviations from the model extrapolation are more conspicu-
+ous in the disc region. On the whole, we may conclude that
+in the outer filaments systematic deviations of the observed
+line-of-sight velocities from the corresponding model val ues
+do not exceed, on the average, 30kms−1. This implies an
+outflow velocity of vout≈130kms−1, if we assume that the
+observed peculiar gas motions are directed perpendicularl y
+to the plane of the disc and take projection effect into ac-
+count. This value is comparable to the observed halfwidth
+of emission lines, which amounts to 150 −170kms−1at a
+distance of r= 20−30 arcsec (0.9-1.4 kpc), whereas the
+FWHM is half in the disc. The fact that the amplitudes of
+regular velocities are approximately equal tothose of chao tic
+velocities are indicative of strong turbulence in the filame nt
+gas.6 DISCUSSION
+6.1 Galactic wind
+We showed above that the peculiarities of the distribution
+of the parameters of ionized gas (kinematics, density, and
+state of ionization) observed in NGC 4460 can be most eas-
+ily explained in terms of the ‘galactic wind’ hypothesis, i. e.,
+assuming gas outflow above the plane of the galaxy pro-
+duced by a central burst of star formation. The alterna-
+tive assumption suggesting that we observe remnants of a
+tidally disrupted gas-rich companion is ruled out, because it
+would imply that line-of-sight velocities in filaments shou ld
+deviate significantly from the systemic velocity. Besides t he
+above indicators, the shape of the emission nebulosity also
+suggests the presence of galactic wind. The nebulosity re-
+sembles a butterfly or, rather, two bubbles expanding above
+theplane ofthe galaxy. Note thatthe diameter ofthe bubble
+has become greater than the halfwidth of the gaseous disc, it
+has open vertex, because the latter is no longer compressed
+by the ambient gas. Moreover, the emission nebulosity re-
+sembles the well-known superwind galaxy NGC 1482 (see
+review by Veilleux et al. 2005, for this and other examples).
+The nonuniform distribution of the density in the emission
+nebulosity is not surprising, because the optical emission
+lines are thought to originate from a boundary between hot
+bubble gas and cool ambient ISM, fragmented under the ac-
+tion of Kelvin-Helmholtz and Rayleigh-Taylor instabiliti es.
+Regions of modern star formation are located at the
+c/circlecop†rt2002 RAS, MNRAS 000, 1–118Moiseev et al.
+base of the wind and we showed that ongoing star formation
+is concentrated almost entirely inside a ring less than 2 kpc
+in diameter. Gas is ionized not only by the radiation of OB
+stars, but also by shocks produced by the combined effect
+of supernovae and winds of massive stars, which transfer
+kinetic energy to the interstellar medium. The observed lin e-
+of-sight velocities in filaments are small and close to the
+systemic velocity, suggesting that these features consist of
+gas that was ejected from central regions and has preserved
+its angular momentum.The gas outflow occurs mostly in the
+sky plane and therefore the line-of-sight projection of the
+outflow velocity is small.
+We adopt the characteristic length of the brightest fil-
+amentsl= 30−35 arcsec (1.3-1.6 kpc) to derive the dy-
+namic age of the emission feature τdyn=l/vout≈10−
+12Myr. The total H αluminosity of the galaxy is LHα=
+1.7×1040ergs−1based on the observed H α+[Nii] flux from
+Kaisin & Karachentsev (2008) and the average line ratio of
+log[Nii]/H =−0.7. We assume that the wind radiation in-
+cludes the flux from the regions located outside the ellipse
+with the semi-major axis 25 arcsec and axial ratio 0.4, which
+includes all HII regions in the disc. The wind luminosity is
+LHα(wind) = 2.7×1039ergs−1, it makes up for 16 per cent
+of the total LHα, and the SE and NW parts of the bipolar
+structure have almost equal luminosities (9 and 7 per cent,
+respectively). We now adopt, in accordance with Fig. 6, a
+mean electron density of ne= 50cm−3to infer, in the same
+way as Matsubayashi et al. (2009), the total mass of ionized
+gas ejected from the disc2,Mwind= 1.7×105M⊙, and its
+kinetic energy E= 5.8×1052. We describe the emitting
+volume of the SE part of the outflow wind by a truncated
+cone with the bases of diameters 30 and 40 arcsec to derive
+a characteristic filling factor for this volume f≈3×10−5.
+This value of fis indicative of the highly nonuniform nature
+of the emitting medium – H αemitting gas is located mostly
+in the walls of the bubble, which fragment under the action
+of characteristic instabilities (Veilleux et al. 2005).
+We now compare our inferred parameters with the
+wind parameters in two well-known nearby galaxies:
+M82 (Shopbell & Bland-Hawthorn 1998) and NGC 253
+(Matsubayashi et al. 2009). The mass of the ejected gas in
+NGC 253 is almost the same as in our case, although the ki-
+netic energy of the gas is twice lower, whereas the formation
+time scale of the feature is almost six times shorter. In M82
+MwindandEare greater by almost a factor of 30 and 360,
+respectively, with closer τdyn= 3.4 Myr. The wind outflow
+velocities in these galaxies are three to four times higher
+than in NGC 4460. Given our remark about the uncertainty
+of measured vout, it can be concluded that, on the whole, the
+wind parameters in NGC 4460 are close to those observed in
+NGC 253, except for the fact that the emission nebulosity in
+NGC 4460 took much longer time to form. This is not sur-
+prising, because the total SFR formally inferred from LHα
+is almost 10 times higher in NGC 253. This fact most likely
+2In this analysis we used information about single phase of th e
+wind, because H αemission traces only the cooler interaction zone
+between hot wind fluid and the ambient medium. Therefore the
+values of MwindandEare underestimated. However, we present
+its for comparison with the similar values taken from simila r anal-
+yses in other galaxies.Figure 6. Variation of the [S ii] line doublet ratio for slit #1
+(top) and slit #2 (bottom). The horizontal dashed lines indi cate
+the values of electron density corresponded which are label ed near
+the lines.
+determines the relatively lower outflow velocity. In partic u-
+lar, we do not know whether the kinetic energy of the wind
+is sufficient for the gas to escape from the galaxy. A sim-
+ple estimate of the escape velocity obtained in terms of the
+isothermal sphere model with a cutoff radius of 2 arcmin
+yieldsvesc≥2.7vmax= 180kms−1for regions located at
+galactocentric distances r <8 arcsec, which is appreciably
+greater than vout. Therefore, most probably, after cooling
+down, the gas of the wind will fall back onto the galactic
+plane.
+6.2 Cause of the burst of star formation
+Thus the central kiloparsec in NGC4460 is engulfed
+in on-going star formation ( SFR= 0.3M⊙/yr, see
+Kaisin & Karachentsev 2008), in contrast to the rather old
+and‘quiet’ discofthegalaxy.Whatcouldhavetriggered sta r
+formation in this lenticular galaxy? Tidal interaction wit h
+a companion is unlikely since the galaxy appears isolated.
+Neither thereis anyevidence for a merger of a gas-rich dwarf
+companion – the velocity field lacks kinematically decouple d
+components to be expected in such a case. No tidal features
+can be seen on optical images or published low-resolution HI
+maps (Sage & Welch 2006). The inner disc that we found in
+the brightness distribution proves that star formation has
+been going on for quite a long time in the center, however
+it is not indicative of interaction.
+We already pointed out above that isolated E and S0
+galaxies are rather rare objects, their integrated parame-
+ters differ appreciably from normal E and S0 galaxies in
+c/circlecop†rt2002 RAS, MNRAS 000, 1–11Ionized gas outflow in NGC 4460 9
+groups and clusters. Table 2 lists the basic parameters of al l
+the three representatives of this category of galaxies foun d
+within 10 Mpc from the Sun. Row (2) gives the tidal index
+according to Karachentsev et al. (2004); distances in row (3 )
+are from Tonry et al. (2001) for except NGC 404 for which
+meanvaluefrom Tonry et al.(2001)andKarachentsev et al.
+(2002) is accepted; Rows (1), (4) and (5) give the ob-
+served parameters of the galaxies adopted from the NED
+database; Row (6) presents the HI mass from HyperLEDA
+database adopted for the accepted distances. Row (7) gives
+the estimated global star-formation rate in galaxies infer red
+from their H αfluxes from Kaisin & Karachentsev (2008) for
+NGC4460 andfrom Karachentsev & Kaisin (2010) for other
+two galaxies. The last rows give two dimensionless and dis-
+tance independent parameters (see Karachentsev & Kaisin
+(2007) and Kaisin & Karachentsev (2008) for details): P∗
+K=
+log([SFR]×T0/LK) andF∗= log(MHI/[SFR]×T0). The
+former characterizes the proportion of its luminosity the
+galaxy would produce during the Hubble time T0= 13.7
+Gyr at the current rate of star formation and the mass-to-
+(K-band)-luminosity ratio M/LK= 1M⊙/L⊙. The param-
+eterF∗shows how much Hubble time the galaxy will need
+in future to spend the present supply of gas if star formation
+proceeds at the currently observed rate.
+Distribution of 420 Local volume galaxies in the ‘past-
+future’ diagram (Karachentsev & Kaisin 2010) is presented
+in Fig. 7. The discussed isolated early-type galaxies of mod -
+erate luminosity are distinguished on the diagram by red
+color. Here, galaxies of different morphological types occu py
+essentially different regions in the diagnostic diagram. Th e
+median values of P∗
+KandF∗for the total LV sample are
+−0.40 and−0.25, respectively. These quantities mean that
+a typical LV galaxy requires 2 −3 times higher SFR in
+the past to reproduce its observed luminosity, having also
+enough amount of gas to continue star formation with the
+observed rate during the next 8 Myrs.
+The NGC 404, NGC 855, and NGC 4460 galaxies differ
+appreciably in luminosity, but have almost similar hydroge n
+masses∼1×108M⊙. Specific star-formation rate per unit
+luminosity differs little amongthese galaxies. The paramet er
+P∗
+Kis only slightly smaller than zero for all three galaxies,
+and this indicates that the galaxies are capable of reproduc -
+ing most of their stellar mass during cosmological time T0
+provided that star formation is at the current level. At the
+same time, the available gas reserves in these galaxies can
+maintain the current rate of star formation only for a short
+time scale ranging from 1/40 (NGC 4460) to 1/6 (NGC 404)
+of the age of the Universe.
+The location of isolated E and S0 galaxies on the diag-
+nostic diagram {P∗
+K,F∗}does not allow us to view their
+observed evolutionary stage as a burst of star formation
+triggered by a merger of a companion or a single hydro-
+gen cloud. Should this be the case, one would expect in
+the Local Volume, in addition to the three isolated emission
+E and S0 galaxies mentioned above, an order of magnitude
+greater number of similar objects at the quiescent interbur st
+stage. However, the currently available H α-flux data for 420
+galaxies of the Local Volume Karachentsev & Kaisin (2010)
+do not support the existence of such a population. Besides
+NGC 4460, NGC 404, and NGC 855, the Local Volume con-
+tains only two other isolated galaxies: NGC 2787 (S0-a) and
+NGC 4600 (S0) where very weak H αemission is presented.Table 2. Integrated parameters of isolated E/S0 galaxies within
+10 Mpc
+Line Parameter NGC 4460 NGC 855 NGC 404
+(1) Type SB(s)0+ E SA(s)0-
+(2) TI -0.7 -0.8 -1.0
+(3) Distance (Mpc) 9.59 9.73 3.24
+(4) M B(mag) -17.89 -17.07 -16.61
+(5) M K(mag) -20.87 -20.15 -19.00
+(6) logM(HI) (sun) 7.92 8.13 7.97
+(7) log[SFR](M ⊙/yr) -0.59 -1.07 -1.37
+(8)P∗
+K-0.11 -0.30 -0.14
+(9)F∗-1.63 -0.94 -0.80
+These galaxies gave blue absolute magnitudes −18.5 and
+−15.7, respectively.
+The predominance of objects with current central star
+formation among isolated E and S0 galaxies and their capa-
+bility of reproducing the available stellar mass over the co s-
+mological time scale with the observed SFR combined with
+the scarcity of available gas reserves ( ∼1×108M⊙) leads
+us to suggest that ongoing star formation in this galaxies
+is fed by an external source, i.e. intergalactic gas clouds o r
+filaments. The process of external gas accretion should be
+have aregular steady character on a cosmological time scale ,
+without significant bursts.
+Recently Kannappan et al. (2009) described a popula-
+tion of ‘blue-sequence E/S0s’ resided on the locus of late-
+type galaxies in color vs. stellar mass parameter space. The y
+argue that these galaxies actively (re)growing their stell ar
+discs and therefore may form a bridge between late-type spi-
+rals/iregulars and early-type red-sequence galaxies. Man y
+of low-to-intermediate mass blue-sequence E/S0s reside in
+low density environments, often have a blue center and sub-
+stantial fraction of cold gas. However their gas consumptio n
+time usually smaller than 3 Gyr (Wei et al. 2009). All these
+properties are similar with the Local Volume isolated E/S0
+galaxies described above, therefore NGC 4460, NGC 404,
+and NGC 855 seem to be belonging to the population of
+blue-sequenceE/S0s. Kannappan et al. (2009) suggest these
+galactic population connects with the disc rebuilding afte r
+major/minor merging or companions interactions. In the
+present paper we stress that accretion of external gas is als o
+important in a particular case of the evolution of isolated
+early-type galaxies.
+7 SUMMARY
+We used available UV and optical images to explore the
+structure of the isolated lenticular galaxy NGC 4460. Also
+we obtained long-slit and3D spectroscopic datatoprobe the
+origin of the bright H αnebulosity surrounding the central
+region of the galaxy. We consider the following points to be
+the most important.
+(i) All ongoing star formation with the rate about
+0.3M⊙/yrresides in a pseudo-ring with radius about one
+kiloparsec. Together with antitruncated shape of the surfa ce
+brightness profile it suggests a recent formation of the inne r
+exponential disc, whose radial scale three times smaller th an
+global old stellar disc.
+c/circlecop†rt2002 RAS, MNRAS 000, 1–1110Moiseev et al.
+Figure 7. Evolutional plane ‘past-future’ for 420 Local Volume
+galaxies (Karachentsev & Kaisin 2010). The galaxies observ ed
+and detected in H αare shown by filled symbols, and galaxies with
+onlyupper limitoftheir H αflux areindicated byopen circles.The
+isolated E/S0 galaxies of moderate luminosity are distingu ished
+by red squares and marked with their NGC numbers.
+(ii) Whereas gas in the circumnuclear disc is photoion-
+ized by radiation of young stars, the external regions of
+Hαnebulosity are ionized by shocks. This fact, in combi-
+nation with a bi-conical shape of the ionized gas nebulosity ,
+with the radial distributions of the emission lines velocit ies
+and width can be explained as a galactic-scale outflow (su-
+perwind) induced by a central star formation. We estimated
+outflow velocity as ≥130kms−1. Galactic wind parameters,
+such as outflow velocity, formation time and total kinetic
+energy are several times smaller comparing with the known
+galactic wind in NGC253, which is explained by substan-
+tially different star formation rates.
+(iii) Alsowe discussedtheevolutional status ofNGC4460
+together with other well-isolated E/S0 galaxies in the
+local volume: NGC 404, and NGC 855. We considered
+the position of the galaxies on the ‘past-future’ diagram
+(Karachentsev & Kaisin 2010) and suggested that continu-
+ous accretions of an intergalactic gas is most probable sour ce
+of their current star formation. In contrast to merging sce-
+nario, the external gas accretion seems to be a sluggish pro-
+cess without violent events.
+The research is partly based on data obtained from the
+Multimission Archive at the Space Telescope Science Insti-
+tute (MAST). STScI is operated by the Association of Uni-
+versities for Research in Astronomy, Inc., under NASA con-
+tract NAS5-26555. Support for MAST for non- HSTdata is
+provided by the NASA Office of Space Science via grant
+NAG5-7584 and by other grants and contracts. Funding
+for the SDSS and SDSS-II has been provided by the Al-
+fred P. Sloan Foundation, the Participating Institutions,
+the National Science Foundation, the US Department ofEnergy, the National Aeronautics and Space Administra-
+tion, the Japanese Monbukagakusho, the Max Planck So-
+ciety and the Higher Education Funding Council for Eng-
+land. The SDSS web site is http://www.sdss.org/. This
+research has made use of the NASA/IPAC Extragalactic
+Database (NED) which is operated by the Jet Propulsion
+Laboratory, California Institute of Technology, under con -
+tract with the National Aeronautics and Space Administra-
+tion. We acknowledge the usage of the HyperLeda database
+(http://leda.univ-lyon1.fr). This work was supported by t he
+Russian Foundation for Basic Research (projects no. 09-02-
+00870, no. 07-02-00005, and no. 08-02-00627). AM is also
+grateful to ‘Dynasty’ Fund. The authors thank the anony-
+mous referee for his constructive advice that helped us to
+improve the paper.
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+arXiv:1001.0236v3  [cs.CG]  3 Feb 2010Symposium on TheoreticalAspects of Computer Science 2010( Nancy, France), pp. 239-250
+www.stacs-conf.org
+THE TRAVELING SALESMAN PROBLEM
+UNDER SQUARED EUCLIDEAN DISTANCES
+MARK DE BERG1AND FRED VAN NIJNATTEN1AND REN ´E SITTERS2AND
+GERHARD J. WOEGINGER1AND ALEXANDER WOLFF3
+1Department of Mathematics and Computer Science, TU Eindhov en, the Netherlands.
+E-mail address :mdberg@win.tue.nl
+E-mail address :f.s.b.v.nijnatten@tue.nl
+E-mail address :gwoegi@win.tue.nl
+2Faculty of Economics and Business Administration, VU Amste rdam, the Netherlands.
+E-mail address :rsitters@feweb.vu.nl
+3Lehrstuhl f¨ ur Informatik I, Universit¨ at W¨ urzburg, Germ any.
+URL:http://www1.informatik.uni-wuerzburg.de/en/staff/wo lffalexander
+Abstract. LetPbe a set of points in Rd, and let α/greaterorequalslant1 be a real number. We define the
+distance between two points p,q∈Pas|pq|α, where|pq|denotes the standard Euclidean
+distance between pandq. We denote the traveling salesman problem under this distan ce
+function by Tsp(d,α). We design a 5-approximation algorithm for Tsp(2,2) and generalize
+this result to obtain an approximation factor of 3α−1+√
+6α/3 ford= 2 and all α/greaterorequalslant2.
+We also study the variant Rev- Tspof the problem where the traveling salesman is
+allowed to revisit points. We present a polynomial-time app roximation scheme for Rev-
+Tsp(2,α) withα/greaterorequalslant2, and we show that Rev- Tsp(d,α) isapx-hard if d/greaterorequalslant3 andα >1.
+Theapx-hardness proof carries over to Tsp(d,α) for the same parameter ranges.
+1. Introduction
+Motivated by a power-assignment problem in wireless networ ks (see below for a short
+discussion of this application) Funke et al. [12] studied th e following special case Tsp(d,α)
+of the Traveling Salesman Problem ( Tsp) which is specified by an integer d/greaterorequalslant2 and a real
+numberα>0. The cities are npoints ind-dimensional space Rd, and the distance between
+two points pandqis|pq|α, where|pq|denotes the standard Euclidean distance between p
+andq.
+•The objective in problem Tsp(d,α) is to find a shortest tour (under distances |·|α)
+that visits every city exactlyonce.
+1998 ACM Subject Classification: I.1.2 Algorithms, F.2.2 Nonnumerical Algorithms and Probl ems.
+Key words and phrases: Geometric traveling salesman problem, power-assignment i n wireless networks,
+distance-power gradient, NP-hard, APX-hard.
+c/circlecopyrtM. deBerg,F. vanNijnatten,R. Sitters,G.J.Woeginger,and A. Wolff
+CC/circlecopyrtCreativeCommons Attribution-NoDerivsLicense240 M. DE BERG, F. VAN NIJNATTEN, R. SITTERS, G. J. WOEGINGER, AND A. WOLFF
+•In the closely related problem Rev- Tsp(d,α), the objective is to find a shortest tour
+that visits every city at leastonce; thus the salesman is allowed to revisit cities.
+Note that Tsp(2,1) is the classical two-dimensional Euclidean Tspand that Tsp(d,∞) is
+the so-called bottleneck TspinRd, where the goal is to find a tour whose longest edge has
+minimum length. We are, however, mainly interested in the ca se whereαis some small
+constant, and we will not touch the case α=∞.
+Similarities and differences to the classical Euclidean TSP. The classical Euclidean Tspis
+np-hard even in two dimensions, but it is relatively easy to app roximate. In particular, it
+admits a polynomial-time approximation scheme: Given a par ameterε>0 and a set of n
+points ind-dimensional Euclidean space, one can find in 2(d/ε)O(d)+(d/ε)O(d)nlogntime a
+tour whose length is at most 1+ εtimes the optimal length [23].
+A crucial property of the Euclidean Tspis that the underlying Euclidean distances
+satisfy the triangle inequality. The triangle inequality i mplies that no reasonable salesman
+would ever revisit the same city: Instead of returning to a ci ty, it is always cheaper to skip
+the city and to travel directly to the successor city. All pos itive approximation results for
+the Euclidean Tsprely heavily on the triangle inequality. In strong contrast to this, for
+exponentsα >1 the distance function |·|αdoes not satisfy the triangle inequality. Thus
+the combinatorial structure of the problem changes signific antly—for example, revisits may
+suddenly become helpful—and the existing approximation al gorithms for Euclidean Tsp
+cannot be applied.
+Another nice property of the classical Euclidean problem Tsp(2,1) is that, sloppily
+speaking, instances with many cities have long optimal tour s. Consider for instance a set P
+ofnpoints in the unit square. Then there exists a tour whose Eucl idean length is bounded
+byO(√n) [15]. This bound is essentially tight since there are point sets for which every
+tour has Euclidean length Ω(√n). Interestingly, these results do not carry over to Tsp(2,2)
+withsquaredEuclidean distances. Problem #124 in the book by Bollob´ as [ 8] shows that
+there always exists a tour for Psuch that the sum of the squared Euclidean distances is
+bounded by 4, and that this bound of 4 is best possible. Since, as a rule of thumb, large
+objective values are easier to approximate than small objec tive values, this already indicates
+a substantial difference in the approximability behaviors of Tsp(2,1) and Tsp(2,2).
+Previous work and our results. Funke et al. [12] note that the distance function |·|αsatisfies
+the so-called τ-relaxed triangle inequality with parameter τ= 2α−1(see Section 2 for a
+definition). The classical TSP under the τ-relaxed triangle inequality has been extensively
+studied [2, 3, 6, 7], and all the corresponding machinery fro m the literature can be applied
+directly to Tsp(d,α). For instance, Andreae [6] derives a ( τ2+τ)-approximation for the
+classical Tspunder theτ-relaxed triangle inequality (∆ τ-Tsp, for short). This result trans-
+lates into a (4α−1+ 2α−1)-approximation for Tsp(·,α). Forτ >3, it is better to apply
+Bender and Chekuri’s 4 τ-approximation [2] for ∆ τ-Tsp, which yields a 2α+1-approximation
+forTsp(·,α). Funke et al. derive a (2 ·3α−1)-approximation algorithm for Tsp(·,α), which
+for the range 2 < α <log3/23≈2.71 is better than applying the known results [6, 2].
+The best result for α <2 is obtained by B¨ ockenhauer et al. [7] whose Christofides-b ased
+(3τ2/2)-approximation for ∆ τ-Tspyields a (3·22α−3)-approximation for Tsp(·,α).
+We will demonstrate in Section 2 that essentially everyvariant of the original T3-
+algorithm byAndreaeandBandelt [3] already gives a(2 ·3α−1)-approximation for Tsp(d,α).
+The bottom-line of all this, and the actual starting point of our paper, is that the machineryTHE TRAVELING SALESMAN PROBLEM UNDER SQUARED EUCLIDEAN DIS TANCES 241
+around the τ-relaxed triangle inequality only yields a bound of roughly 2·3α−1. This
+raises the following questions: How much can geometry help u s in getting even better
+approximation ratios? Can we beat the 6-approximation for Tsp(2,2) of Funke et al.? We
+answer these questions affirmatively: We develop a new varian t of the T3-algorithm which
+we call the geometric T3-algorithm . An intricate analysis in Section 3 shows that this yields
+a 5-approximation for Tsp(2,2). We then extend our analysis to Tsp(2,α) withα >2,
+and thus obtain a (3α−1+√
+6α/3)-approximation; see Section 4. This new bound is always
+better than the bound 2 ·3α−1of Funke et al. and of our analysis of the T3-algorithm.
+Finally, in Section 5, we turn our attention to the following two questions: (a) How
+does the approximability of Tspbehave when we make αlarger than one? (b) Does al-
+lowing revisits change the complexity or the approximabili ty of the problem? As we know,
+classical Euclidean Tsp(that is, Tsp(d,1)) isnp-hard [19] and has a polynomial-time ap-
+proximation scheme (PTAS) in any fixed number dof dimensions [4]. On the other hand,
+Rev-Tsp(d,α) has—to the best of our knowledge—not been studied before. C oncerning
+question (b), complexity behaves as expected: Rev- Tsp(d,α) is NP-hard for any d/greaterorequalslant2 and
+anyα>0, and our (straightforward) hardness argument also works f orTsp(d,α). In terms
+of approximability, we show that whereas the two-dimension al problem Rev- Tsp(2,α) still
+has a PTAS for all values α/greaterorequalslant2, the problem becomes apx-hard for all α >1 in three
+dimensions. We were surprised that the apx-hardness proof, too, carried over to Tsp(3,α)
+for allα>1. This inapproximability result stands in strong contrast to the behavior of the
+classical Euclidean Tsp(the caseα= 1).
+The connection to wireless networks. Consider a wireless network whose nodes are equipped
+with omni-directional antennas. The nodes are modeled as po ints in the plane, and every
+node can communicate with all other nodes that are within its transmission radius. The
+power (that is, the energy) needed to achieve a transmission radius ofris roughly propor-
+tional torαfor some real parameter αcalled the distance-power gradient . Depending on
+environmental conditions, αtypically is in the range 2 to 6 [13, Chapter 1]. The goal is to
+assign powers to the nodes such that the resulting network ha s certain desirable properties,
+while the overall power consumption is minimized. A widely s tudied variant has the objec-
+tive to make the resulting network strongly connected [1, 11 , 16]. Other variants (finding
+broadcast trees; having small hop diameter; etc) have been s tudied as well. Funke et al. [12]
+suggest that it is useful to have a tour through the network, w hich can be used to pass a
+virtual token around. The resulting power-assignment problem is Tsp(2,α).
+Another setting related to Tsp(2,α) is the following. Instead of omni-directional anten-
+nas, some wireless networks use directional antennas. This achieves the same transmission
+radius under a smaller energy consumption [17, 22]. To model directional antennas, Cara-
+giannis et al. [9] assume that a node can communicate with oth er nodes in a circular sector
+of a given angle (where the sector’s radius is still determin ed by the power of the node’s
+signal). For directional antennas one not only has to assign a power level to each node, but
+also has to decide on the direction in which each node transmi ts. If the opening angle tends
+to zero and the points are in general position, a strongly con nected network becomes a tour.
+Hence, our results on Tsp(2,α) may shed some light on the difficulty of power assignment
+for directional antennas with small opening angles.242 M. DE BERG, F. VAN NIJNATTEN, R. SITTERS, G. J. WOEGINGER, AND A. WOLFF
+2. Approximating Tsp(·,α)
+In this section we lay the basis for our main contribution, a 5 -approximation for
+Tsp(2,2) in Section 3. We review known algorithms for a related vers ion ofTsp, which
+can be applied to our setting. As it turns out, these algorith ms already yield the same
+worst-case bounds as the algorithm that Funke et al. [12] gav e recently.
+We recall some definitions. Let Sbe a set, let dist( ·,·) :S×S→R/greaterorequalslant0be a distance
+function on S, and letτ/greaterorequalslant1. We say that dist( ·,·) fulfills the τ-relaxed triangle inequality
+if any three elements p,q,r∈Ssatisfy dist( p,r)/lessorequalslantτ·(dist(p,q) +dist(q,r)). Recall that
+we denote by ∆ τ-TsptheTspproblem on complete graphs whose weight function (when
+viewed as a distance function on the vertices) fulfills the τ-relaxed triangle inequality. The
+following lemma, which has beenobserved by Funke et al. [12] , allows us toapply algorithms
+for ∆τ-Tspto our problem. The proof relies on H¨ older’s inequality.
+Lemma 2.1 ([12]).Letα>0be a fixed constant. The distance function |·|α:Rd×Rd→
+R/greaterorequalslant0,(p,q)/ma√sto→|pq|αfulfills theτ-relaxed triangle inequality for τ= 2α−1.
+Andreae and Bandelt [3] gave an approximation algorithm for ∆τ-Tsp. Their T3-
+algorithm is an adaptation of the well-known double-spanni ng-tree heuristic for Tsp. This
+heuristic finds a minimum spanning tree (MST) in the given gra phG, doubles all edges,
+finds an Euler tour in the resulting multigraph, and finally co nstructs a Hamiltonian cycle
+from the Euler tour by skipping all nodes that have already be en visited. The weight of
+the MST is a lower bound for the length of a Tsp-tour since removing any edge from a
+tour yields a spanning tree whose weight is at least the weigh t of the MST. Note that
+this statement holds for arbitrary weight functions. If the triangle inequality holds, the
+heuristic yields a 2-approximation since then skipping ove r visited nodes never increases
+the length of the tour, which initially equals twice the weig ht of the MST. For the weight
+function|·|α, however, the heuristic can perform arbitrarily badly—con sider a sequence of
+nequally-spaced points on a line.
+The T3-algorithm of Andreae and Bandelt also creates a Hamiltonia n tour by short-
+cutting the MST, but their algorithm never skips more than tw o consecutive nodes. It is
+never necessary to skip more than two consecutive nodes beca use the cube T3of a treeTis
+always Hamiltonian by a result of Sekanina [24]. Recall that the cube of a graph Gcontains
+an edgeuvif there is a path from utovinGthat uses at most three edges. The proof of
+Sekanina is constructive; Andreae and Bandelt use it to cons truct a tour in MST3.
+The recursive procedure of Sekanina [24] to obtain a Hamilto nian cycle in T3intuitively
+works as illustrated in Fig. 1; for the pseudo-code, see Algo rithm 1. The algorithm is
+applied to a tree Tand an edge e=u1u2ofT. Removing the edge esplits the tree into
+two components T1andT2. In each component Ti(i= 1,2), the algorithm selects an
+arbitrary edge ei=uiwiincident to uiand recursively computes a Hamiltonian cycle of Ti
+that includes the edge ei. The algorithm returns a Hamiltonian cycle of Tthat includes e.
+The cycle consists of the cycles in T1andT2without the edges e1ande2, respectively. The
+two resulting paths are stitched together with the help of eand the new edge w1w2.
+Note that different choices of the edge eiin line 5 give rise to different versions of the
+algorithm. ThestandardT3-algorithm takes anarbitrary suchedge, whileAndreae’s re fined
+version [2] makes a specific choice, which gives a better resu lt. (In the next section we will
+chooseeibased on the local geometry of the MST, which will lead to an im proved result
+for our problem.) Andreae’s tour in MST3has weight at most ( τ2+τ) times the weight of
+the MST, which is worst-case optimal [3]. Combining his resu lt with Lemma 2.1 yields thatTHE TRAVELING SALESMAN PROBLEM UNDER SQUARED EUCLIDEAN DIS TANCES 243
+u1u2w1 w2
+T1 T2 ee1 e2
+Figure 1: Recursively find-
+ing a Hamiltonian
+cycle in the cube
+of the treeT.Algorithm 1 :CycleInCube (T,e=u1u2)
+fori←1to2do 1
+Ti←component of T−ethat contains ui 2
+if|Ti|= 1thenPi←∅;wi←ui 3
+else4
+pick an edge ei=uiwiincident to uiinTi 5
+if|Ti|= 2thenΠi←ei 6
+elseΠi←CycleInCube (Ti,ei)−ei 7
+returnΠ1+e+Π2+w1w2 8
+the refined T3-algorithm is a (4α−1+2α−1)-approximation for Tsp(·,α). We now improve
+on this with the help of a simple argument. We will frequently use the following definition.
+LetTbe a tree and let v0,...,vkbe a simple path in T. Then we call v0vkak-shortcut
+ofT. We say that a shortcut vwusesan edgeeifelies on the path connecting vandw
+inT. It is not hard to see that the weight of a k-shortcut can be bounded as follows.
+Lemma 2.2. Letα/greaterorequalslant1and letebe ak-shortcut using edges e1,...,ek. Then|e|α/lessorequalslant
+kα−1/summationtextk
+i=1|ei|α.
+Given a tree T, the tour constructed by the T3-algorithm consists of edges of Tand 2-
+and 3-shortcuts that use edges of T. Note that in this tour each edge of Tis used exactly
+twice. Thus, for α/greaterorequalslant2, the original T3-algorithm does actually better than the bound we
+obtained above for the refined T3-algorithm.
+Corollary 2.3. Every version of the T3-algorithm is a (2·3α−1)-approximation for Tsp(·,α).
+Note that our improved analysis of the T3-algorithm yields the same result as the
+algorithm of Funke et al. [12].
+Bender and Chekuri [6] designed a 4 τ-approximation for ∆ τ-Tspusing a different lower
+bound: the optimal Tsptour is a biconnected subgraph of the original graph. The wei ght
+of the optimal Tsptour is at least that of the minimum-weight biconnected subg raph.
+The latter is np-hard to compute [10], but can be approximated within a facto r of 2 [21].
+Moreover, the square of a biconnected subgraph is always Ham iltonian. Thus using only
+edges of the biconnected subgraph and two-shortcuts yields a 4 τ-approximation for ∆ τ-
+Tsp. Combining the result of Bender and Chekuri with Lemma 2.1 im mediately yields the
+following result, which is better than Corollary 2.3 for α>log3/23≈2.71.
+Corollary 2.4. The algorithm of Bender and Chekuri is a 2α+1-approximation for Tsp(·,α).
+3. A 5-Approximation for T SP(2,2)
+In the previous section we have used graph-theoretic argume nts to determine the per-
+formance of the T3-algorithm. By Corollary 2.3, the T3-algorithm yields a 6-approximation
+forα= 2, independently of the dimension of the underlying Euclid ean space. We now
+define what we call the geometric T3-algorithm and show that it yields a 5-approximation
+forTsp(2,2). The geometric T3-algorithm simply chooses in line 5 of Algorithm 1 the edge
+eithat makes the smallest angle with the edge e.244 M. DE BERG, F. VAN NIJNATTEN, R. SITTERS, G. J. WOEGINGER, AND A. WOLFF
+ba
+c
+b1b2s(a,b,c)
+ψba ψbc
+∆x∆y
+(a)aandclie on the same side of the
+line through bba
+cb1b2s(a,b,c)
+ψba
+ψbc
+∆x∆y
+(b)aandclie on different sides of the line
+through b
+Figure 2: Two cases for computing the length of the 3-shortcu ts(a,b,c).
+The idea behind taking advantage of geometry is as follows. I n Corollary 2.3 we have
+exploited the fact that each edge is used in two ( /lessorequalslant3)-shortcuts. The weight of a 3-shortcut
+is maximum if the corresponding points lie on a line. For the c ase of the Euclidean MST it
+is well-known that edges make an angle of at least π/3 if they share an endpoint. The same
+proof also works for the MST w.r.t. |·|α. This guarantees that in line 5 of Algorithm 1, we
+can pick an edge eithat makes a relatively small angle with e—if the degree of uiis larger
+than 2. Otherwise, it is easy to see that eiis used by a ( /lessorequalslant2)- and a ( /lessorequalslant3)-shortcut, which
+is favorable to being used by two 3-shortcuts, see Lemma 2.2.
+Although the intuition behind our geometric T3-algorithm is clear, its analysis turns
+out to be non-trivial. We start with the following lemma that can be proved with some
+elementary trigonometry. Given two line segments sandtincident to the same point, we
+denote the smaller angle between sandtby∠stand defineψst=π−∠st.
+Lemma 3.1. Given a tree T, the 3-shortcut s(a,b,c)that uses the edges a,b,cofTin this
+order has weight
+|s(a,b,c)|2=|a|2+|b|2+|c|2+2|a||b|cosψba+2|b||c|cosψbc+2|a||c|cos(ψba+δ·ψbc),
+whereδ= +1ifaandclie on the same side of the line through b, andδ=−1ifaandclie
+on opposite sides. Moreover, |s(a,b,c)|2/lessorequalslant2|a|2+|b|2+2|c|2+2|a||b|cosψba+2|b||c|cosψbc.
+Lemma 3.1 (illustrated in Fig. 2) expresses the weight of a 3- shortcut in terms of the
+lengths of the edges and the angles between them. Now we show t hat if an edge ais used in
+two 3-shortcuts, two of these angles are related. Note that t he T3-algorithm generates the
+two 3-shortcuts that use ain two consecutive recursive calls, see Fig. 3. The T3-algorithm
+is first applied to edge band then recursively to edge a. In the recursive call, the shortcut
+s(e,a,d) is generated where dis an edge incident to both aandb. Then the algorithm
+returns from the recursion and generates the 3-shortcut s(a,b,c). Thusais the middle edge
+in one 3-shortcut and the first or last edge in the other 3-shor tcut. We rely on the following.
+Lemma 3.2. If the geometric T3-algorithm generates the two 3-shortcuts s(a,b,c)and
+s(e,a,d)in two recursive calls and dis incident to both aandb, thenψba/greaterorequalslant(π−ψad)/2.
+Now we are ready to prove the main result of this section.
+Theorem 3.3. The geometric T3-algorithm yields a 5-approximation for Tsp(2,2).
+Proof.We express the length of each shortcut sof the T3-tour in terms of the lengths of the
+MST edges that suses. Changing the perspective, for each MST edge a, we use contrib( a)
+to denote the sum of all terms that contain the factor |a|. The edge ais used in at mostTHE TRAVELING SALESMAN PROBLEM UNDER SQUARED EUCLIDEAN DIS TANCES 245
+two shortcuts. Bounding their lengths yields an upper bound on contrib(a). The sum of all
+contributions relates the length of the T3-tour to that of the MST (w.r.t. |·|α), which in
+turn is a lower bound for the length of an optimal Tsptour.
+Due to Lemma 2.2, contrib( a)/lessorequalslant5|a|2ifais used in a ( /lessorequalslant2)-shortcut on one side and a
+(/lessorequalslant3)-shortcut on the other side. So we focus on the case that ais used in two 3-shortcuts,
+see Fig. 3. We rewrite the composite terms in the bound for s(a,b,c) in Lemma 3.1 using
+Young’s inequality with ε, which, given x,y∈Randε>0, states that xy/lessorequalslantx2/(2ε)+y2ε/2.
+Letvbe the vertex that is incident to edges aandb. If there are multiple 3-shortcuts
+that use edges that are incident to vthen the T3-algorithm generates these in consecutive
+recursive calls. We renumber the edges incident to vsuch that the algorithm is first applied
+tovv1, then recursively to vv2etc. Then there is some i/greaterorequalslant1 such that b=vviand
+a=vvi+1because the algorithm is first applied to band then recursively to a. We define
+ψi=ψvvi,vvi+1(=ψba). We rewrite the term 2 |a||b|cosψbain the bound for |s(a,b,c)|2in
+Lemma 3.1 as follows.
+2|a||b|cosψba= 2|vvi||vvi+1|cosψi/lessorequalslantf(|vvi+1|,|vvi|,ψi), (3.1)
+where
+f(|vvi+1|,|vvi|,ψi) =
+
+0 if ψi/greaterorequalslantπ
+2,
+|vvi|2+|vvi+1|2cos2ψiifψi<π
+2and/parenleftbig
+i= 1 or/parenleftbig
+i>1 andψi−1/greaterorequalslantπ
+2/parenrightbig/parenrightbig
+,/parenleftbig
+|vvi|2+|vvi+1|2/parenrightbig
+cosψiifψi<π
+2andi>1 andψi−1<π
+2.
+The second case of inequality (3.1) follows from Young’s ine quality with ε= 1/cosψiand
+the third case from Young’s inequality with ε= 1. Replacing 2|b||c|cosψbcin the bound for
+|s(a,b,c)|2in Lemma 3.1 is analogous. Together, the two replacements yi eld the bound
+|s(a,b,c)|2/lessorequalslant2|a|2+|b|2+2|c|2+f(|a|,|b|,ψba)+f(|c|,|b|,ψbc). (3.2)
+We use (3.2) to bound the weights of all 3-shortcuts. The weig ht of the final tour is the
+sum of the weights of all shortcuts. In this sum we can take the two occurrences of an
+edgea=vvi+1together and analyze the contribution of ato the tour. Note that the result
+of (3.2) is still at most 3( |a|2+|b|2+|c|2). So if an edge ais used in a ( /lessorequalslant3)-shortcut on one
+side and a ( /lessorequalslant2)-shortcut on the other side, then we still have that contri b(a)/lessorequalslant5|a|2. It
+remains to consider the case that ais used in two 3-shortcuts. Let s(a,b,c) ands(e,a,d) be
+these 3-shortcuts. The algorithm is first applied to edge band generates shortcut s(a,b,c),
+whereais the first or the third edge of the shortcut. Then the algorit hm is recursively
+applied to edge aand generates shortcut s(e,a,d), whereais the middle edge. Fig. 3 shows
+how the vertices are numbered in this case.
+bac
+de s(a,b,c)s(e,a,d
+)ψi=ψba ψbcψae
+v
+vivi+1
+vi+1ψi+1=ψad
+Figure 3: Two 3-shortcuts that use edge a.a
+v v i−1vi
+vi+1
+vj/negationslashi
+/negationslashi+1/negationslashi−1
+Figure 4: Illustration of case III.246 M. DE BERG, F. VAN NIJNATTEN, R. SITTERS, G. J. WOEGINGER, AND A. WOLFF
+Letσabe a function that takes a sum of terms and returns the sum of al l terms that
+contain|a|. We derive the following expression for contrib( a).
+contrib(a) =σa(weight(s(a,b,c)))+σa(weight(s(e,a,d)))
+/lessorequalslantσa/parenleftbig
+2|a|2+|b|2+2|c|2+f(|a|,|b|,ψba)+f(|c|,|b|,ψbc)/parenrightbig
++σa/parenleftbig
+2|e|2+|a|2+2|d|2+f(|e|,|a|,ψae)+f(|d|,|a|,ψad)/parenrightbig
+/lessorequalslant4|a|2+σa(f(|vvi+1|,|vvi|,ψi))+σa(f(|vvi+2|,|vvi+1|,ψi+1)) (3.3)
+By definition of fwe have to consider three cases in (3.3) for contrib( a).
+Case I:ψi/greaterorequalslantπ/2 orψi+1/greaterorequalslantπ/2.
+We assume w.l.o.g. that ψi/greaterorequalslantπ/2. Then we know that f(|vvi+1|,|vvi|,ψi)) = 0 and in
+the worst case σa(f(|vvi+2|,|vvi+1|,ψi+1))/lessorequalslant|a|2. Thus we have that contrib( a)/lessorequalslant5|a|2.
+Case II:ψi<π/2 andψi+1<π/2 and (i= 1 or (i>1 andψi−1/greaterorequalslantπ/2)).
+By definition of fwe have:
+σa(f(|vvi+1|,|vvi|,ψi)) = σa/parenleftbig
+|vvi|2+|vvi+1|2cos2ψi/parenrightbig
+=|a|2cos2ψi
+σa(f(|vvi+2|,|vvi+1|,ψi+1)) =σa/parenleftbig
+(|vvi+1|2+|vvi+2|2)cosψi+1/parenrightbig
+=|a|2cosψi+1
+Lemma 3.2 states that ψi/greaterorequalslant(π−ψi+1)/2. We also know that ψi/lessorequalslantπby definition. Thus
+we have
+contrib(a)/lessorequalslant/parenleftBig
+4+cos2π−ψi+1
+2+cosψi+1/parenrightBig
+|a|2/lessorequalslant5|a|2.
+Case III:ψi<π/2 andψi+1<π/2 andi>1 andψi−1<π/2.
+It can be shown that this leads to a contradiction, see Fig. 4 ( on page 245).
+In cases I and II, the contribution of any edge |a|to the tour is at most 5 |a|2. The
+theorem follows by summing up the contributions of all edges .
+When using the MST as a lower bound in the analysis, there is no t much room for
+improvement. There are instances of Tsp(2,2) where the T3-algorithm yields a tour whose
+weight is 44
+11times that of the MST; see also [18, Theorem 4.19].
+4. Approximating Tsp(2,α)withα/greaterorequalslant2
+In this section we generalize the main result of the previous section toα/greaterorequalslant2. Our new
+bound is always better than the bound 2 ·3α−1of Funke et al. [12], see also Corollary 2.3.
+Forα <3.41 our bound is better than the bound 2α+1that follows from the algorithm of
+Bender and Chekuri [6], see Corollary 2.4.
+Theorem 4.1. The geometric T3-algorithm yields a (3α−1+√
+6α/3)-approximation for
+Tsp(2,α)ifα/greaterorequalslant2.
+Proof.If an edgeais used in a ( /lessorequalslant2)-shortcut on one side and a ( /lessorequalslant3)-shortcut on the other
+side then the total contribution of ato the tour is at most (2α−1+3α−1)|a|αby Lemma 2.2.
+So we will focus our analysis again on the case that ais used in two 3-shortcuts. For α= 2
+we can express the weight of a 3-shortcut by Lemma 3.1 and rewr ite the composite termsTHE TRAVELING SALESMAN PROBLEM UNDER SQUARED EUCLIDEAN DIS TANCES 247
+as in inequality (3.1). For α>2 we apply H¨ older’s inequality.
+|s(a,b,c)|α=/parenleftbig
+|s(a,b,c)|2/parenrightbigα/2
+/lessorequalslant/parenleftbig
+2|a|2+|b|2+2|c|2+f(|a|,|b|,ψba)+f(|c|,|b|,ψbc)/parenrightbigα/2
+=/parenleftbig
+βa|a|2+βb|b|2+βc|c|2/parenrightbigα/2(4.1)
+/lessorequalslant3α/2−1/parenleftBig
+βα/2
+a|a|α+βα/2
+b|b|α+βα/2
+c|c|α/parenrightBig
+(4.2)
+We introduced the constants of type βto shorten the expression. Note that the last in-
+equality holds only if α>2.
+In order to bound the contribution of an edge athat is used in two 3-shortcuts we
+follow the proof of Theorem 3.3. Since the assumptions of cas e III in that proof led to a
+contradiction, it suffices to consider cases I and II.
+Case I:ψi/greaterorequalslantπ/2 orψi+1/greaterorequalslantπ/2.
+contrib(a)/lessorequalslant3α/2−1/parenleftBig
+(2+cosψi)α/2+(2+cosψi+1)α/2/parenrightBig
+|a|α
+/lessorequalslant3α/2−1/parenleftBig
+2α/2+3α/2/parenrightBig
+|a|α=/parenleftBig
+3α−1+√
+6α/3/parenrightBig
+|a|α
+Case II:ψi<π/2 andψi+1<π/2 and (i= 1 or (i>1 andψi−1/greaterorequalslantπ/2)).
+contrib(a)/lessorequalslant3α/2−1/parenleftBig
+(2+cosψi+1)α/2+/parenleftbig
+2+sin2ψi+1/2/parenrightbigα/2/parenrightBig
+/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright
+gα(ψi+1)|a|α
+Now we use the fact that the function h: [0,2π]→R,x/ma√sto→(2 + cosx)k+ (2 + sin2x/2)k
+attains its maximum value at x= 0. Thusgαalso attains its maximum in the range [0 ,π/2)
+inx= 0. This yields
+contrib(a)/lessorequalslant3α/2−1·gα(0)·|a|α/lessorequalslant/parenleftbig
+3α−1+√
+6α/3/parenrightbig
+|a|α.
+In both cases we showed that contrib( a)/lessorequalslant(3α−1+√
+6α/3)|a|α. The theorem follows for
+α >2 by summing up the contributions of all edges. The case α= 2 corresponds to
+Theorem 3.3.
+5. The Approximability of T SPand Rev-T SP
+In this section we discuss complexity and approximability o fTspand its variant Rev-
+Tsp, where the salesman is allowed to revisit the cities. Recall that for any fixed dimension
+d/greaterorequalslant2,Tsp(d,1) isnp-hard [19] and admits a PTAS [4].
+Theorem 5.1. Tsp(d,α)and Rev- Tsp(d,α)arenp-hard for any d/greaterorequalslant2andα>0.
+Proof.Itai et al. [14] showed that, given npoints in the unit grid, it is np-hard to decide
+whether there is a Tsptour of Euclidean length n. Thus for both of our problems it is
+np-hard to distinguish between opt=nandopt/greaterorequalslantn−1+√
+2α.248 M. DE BERG, F. VAN NIJNATTEN, R. SITTERS, G. J. WOEGINGER, AND A. WOLFF
+Theorem 5.2. Tsp(d,α)and Rev- Tsp(d,α)areapx-hard for any d/greaterorequalslant3and anyα>1.
+Proof.We only discuss the case d= 3 andα= 2—all other cases can be settled by slightly
+modified arguments— Tsp. and we only consider Rev- Tsp; a similar reduction can be used
+forTsp. We reduce from {1,2}-Tsp, theTspon the complete graph where the weight
+of every edge is either 1 or 2; this problem is apx-hard [20]. An instance of {1,2}-Tsp
+consists of the complete graph Kn= (Vn,En) with vertex set Vn={v1,...,vn}, edge
+setEn={e1,...,em}wherem=n(n−1)/2, and edge lengths that are specified by a
+weight function w:En→{1,2}. GivenKnandw, we construct a corresponding instance
+Pn,w⊂R3of Rev-Tsp(3,2).
+We start our construction by introducing several auxiliary line segments. For each
+vertexvi∈Vnwe define its spineto be the vertical line segment going from point ( ni,ni,n)
+to point (ni,ni,nm ). For each edge ek=vivj∈Enwithi<j, we define two corresponding
+line segments that are parallel to the xy-plane and that are called bones. The first bone
+connects point ( ni,ni,nk ) on the spine of vito the point ( nj,ni,nk ). The other bone
+connects point ( nj,nj,nk ) on the spine of vjto the point ( nj,ni−δk,nk), whereδk= 1 if
+w(ek) = 1 andδk=√
+2 ifw(ek) = 2. Note that these two bones do not quite touch; they
+are separated by a gap of length δk.
+In order to get the instance Pn,wof Rev-Tsp(3,2), we subdivide every single (spine
+or bone) line segment introduced above by a dense, evenly dis tributed set of points—we
+call these points citiesfrom now on—so that every unit-length piece receives n5cities.
+The distance between adjacent cities is 1 /n5, and so the cost for going from one city to
+an adjacent city is 1 /n10. All these cities together form instance Pn,w, and this completes
+our construction. Since we have introduced line segments wi th a total length of at most
+n·n(m−1)+m·2n(n−1)<2n4, the overall number of cities is at most 2 n9.
+For 1/lessorequalslanti/lessorequalslantnwe call the cities on the spine of viand on all bones incident to this spine
+thecity cluster ofvi. Traversing all cities within such a city cluster is very che ap; even if
+we visit every city twice, this costs at most 2 ·2n9/n10= 4/nfor all cities in all city clusters
+together. In a traveling salesman tour, the only expensive s teps occur when the salesman
+jumps from one city cluster to another city cluster. By the ab ove definition of δk, when
+jumping from bone to bone across the gap corresponding to edg eekthe incurred cost is
+exactlyw(ek). Note that jumping from city cluster to city cluster in any o ther way would
+be much more expensive and would thus not reduce the total cos t of the tour.
+Finally, let us show that our reduction is approximation pre serving. Fix an εwith
+0< ε <1. Consider an instance Knandwof{1,2}-Tsp, and assume without loss of
+generality that n >4/ε. Consider an optimal tour π0for this instance. If π0usesℓ/greaterorequalslant0
+edges of length 2 and n−ℓedges of length 1, then it has cost n+ℓ. Given a PTAS for
+Rev-Tsp, we show how to compute in polynomial time a tour of cost at mos t (1+ε)(n+ℓ)
+forKnandw.
+First note that the tour π0can be transformed into a tour π1throughPn,wthat makes ℓ
+jumps of cost 2 and n−ℓjumps of cost 1. That tour π1costs at most n+ℓ+4/n. Using
+our hypothetical PTAS for Rev- Tsp, we can compute for any ε′>0 in polynomial time
+a tourπ2throughPn,wof cost at most (1 + ε′)copt, wherecoptis the cost of an optimal
+Rev-Tsptour. Theexistence of π1yieldscopt/lessorequalslantn+ℓ+4/n. Thetourπ2can betransformed
+into a tour π3throughKn: Just map the jumps of π2to the corresponding edges of Kn.
+Since this mapping cannot increase the cost, tour π3costs at most (1 + ε′)(n+ℓ+ 4/n).THE TRAVELING SALESMAN PROBLEM UNDER SQUARED EUCLIDEAN DIS TANCES 249
+Choosingε′=ε/2 and using 4 /n<ε< 1, we can bound the cost of π3from above by/parenleftBig
+1+ε
+2/parenrightBig
+(n+ℓ)+/parenleftBig
+1+ε
+2/parenrightBig
+ε=/parenleftBig
+1+ε
+2/parenrightBig
+(n+ℓ)+ε
+2(2+ε) = (1+ε)(n+ℓ)
+as desired. Like π2, the tourπ3may visit vertices more than once. This can be fixed by
+greedily introducing shortcuts. The shortcuts do not incre ase the cost of the tour since the
+weight function w(trivially) fulfills the triangle inequality.
+Theorem 5.3. There exists a PTAS for Rev- Tsp(2,α)for anyα/greaterorequalslant2.
+Proof.Given a set Pof points in the plane, consider the Gabriel graph GPthat has a vertex
+for each point in P. There is an edge between points pandq, if the open disk with diameter
+pqis empty, in other words, if for all points r∈P\{p,q}, the angle ∠prqis at mostπ/2.
+The weight of the edge is |pq|α. Note that|pr|α+|rq|α/lessorequalslant|pq|αif∠prqis at leastπ/2.
+Therefore, there is an optimal Tsptour with revisits through Pthat only uses the edges
+ofGP: Indeed, if a tour uses an edge pqfor which there is a point rwith∠prq/greaterorequalslantπ/2,
+then replacing pqbyprandrqwould shorten the tour. Such a replacement is feasible since
+revisiting city ris allowed. The Gabriel graph is planar. Hence we end up with a n instance
+of theTspon weighted planar graphs, for which a PTAS is known [5].
+Recall that a quasi-PTAS is an approximation scheme with running time npolylogn,
+wherenis the size of the input. The following result follows immedi ately from the facts
+that (a) the metric |·|αhas boundeddoublingdimension and (b) Tspon metrics of bounded
+doubling dimension admits a quasi-PTAS [25].
+Theorem 5.4. There exists a quasi-PTAS for Rev- Tsp(d,α)for anyα∈(0,1]andd/greaterorequalslant1.
+6. Conclusions
+In order to construct considerably better approximation al gorithms for Tsp(d,α), we
+expect that substantially different methods of analysis have to be found. A result of Van
+Nijnatten [18, Theorem 4.19] indicates that there is not muc h room left for improvement
+as long as we compare to the MST.
+The approximability of Rev- Tsp(2,α) for 1< α <2 is an interesting open question.
+We believe that a (quasi)-PTAS may be obtained using the fram ework of the PTAS for
+weighted planar graph Tspby Arora et al. [5]. A simple reduction shows that deriving a
+PTAS for our problem is at least as hard as deriving a PTAS for w eighted planar graph
+Tsp. Assume we have a PTAS for Rev- Tsp(2,α) for someα>1. Given a weighted planar
+graph and a planar embedding, we replace each edge by a dense s et of points such that
+traversing a subedge basically costs zero. By making one sub edge of each edge elonger, we
+can make the cost of that subedge (and thus of e) in Rev- Tspproportional to the weight
+ofe. Then, the costs of the optimal solutions of the two problems will be the same up to an
+arbitrarily small constant factor of 1+ ε. Such a reduction is polynomially bounded if all
+weights are polynomially bounded, which can be achieved by a standard rounding scheme.
+A PTAS for Rev- Tsp(2,α) for anyα>1 would be an interesting generalization of the
+existing PTAS’s for weighted planar graphs. Ideally, one wo uld have a PTAS with running
+time independent of αsince it would contain both Euclidean Tspand weighted planar
+graphTspas special cases.250 M. DE BERG, F. VAN NIJNATTEN, R. SITTERS, G. J. WOEGINGER, AND A. WOLFF
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+This work is licensed under the Creative Commons Attributio n-NoDerivsLicense. To view a
+copyofthislicense,visit http://creativecommons.org/licenses/by-nd/3.0/ .
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+arXiv:1001.0237v2  [math.CO]  18 Jan 2010TROPICAL TYPES AND
+ASSOCIATED CELLULAR RESOLUTIONS
+ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+Abstract. An arrangement of finitely many tropical hyperplanes in the t ropical torus Td−1
+leads toanotionof‘type’datafor pointsin Td−1, withtheunderlyingunlabeledarrangement
+givingrise to‘coarse type’. Itis shown thatthedecomposit ion ofTd−1inducedbytypesgives
+rise to minimal cocellular resolutions of certain associat ed monomial ideals. Via the Cayley
+trick from geometric combinatorics this also yields cellul ar resolutions supported on mixed
+subdivisions of dilated simplices, extending previously k nown constructions. Moreover, the
+methods developed lead to an algebraic algorithm for comput ing the facial structure of
+arbitrary tropical complexes from point data.
+1.Introduction
+The study of convexity over the tropical semiring has been an area of active research in
+recent years. Fundamental propertiesof tropical convexit y, inparticular froma combinatorial
+perspective, were established by Develin and Sturmfels in [ 7]. There the notion of a tropical
+polytope was defined as the tropical convex hull of a finite set of points in the tropical torus
+Td−1. Fixing the set of generating points yields a decomposition of the tropical polytope
+called the tropical complex , and in [7] it was shown that the collection of such complexes are
+in bijection with the regular subdivisions of a product of si mplices. The origin of tropical
+convexity can be traced back to the study of ‘max-plus linear algebra’; see [18, 5] and the
+references therein for a proper account.
+A tropical complex can be realized as the subcomplex of bound ed cells of the polyhedral
+complex arising from an arrangement of tropical hyperplane s, and it is this perspective that
+we adopt in this paper. We study combinatorial properties of arrangements of tropical
+hyperplanes, and in particular their relation to algebraic properties of associated monomial
+ideals. A tropical hyperplane in Td−1is defined as the locus of ‘tropical vanishing’ of a linear
+form, and can be regarded as a fan polar to a ( d−1)-dimensional simplex. In this way each
+tropical hyperplane divides the ambient space Td−1intodsectors. Given an arrangement A
+of tropical hyperplanes and a point p∈Td−1, one can record the position of pwith respect
+to each sector of each hyperplane. This tropical analog of th e covector data of an oriented
+matroid is called the typedata, and the combinatorial approach to tropical convexity taken
+in [7] is based on this concept. Here we consider a coarsening of the type data (which we call
+coarse type ) arising from an arrangement Aof tropical hyperplanes, amounting to neglecting
+the labels on the individual hyperplanes.
+The connection between tropical polytopes/complexes and r esolutions of monomial ideals
+was first exploited by Block and Yu in [4]. There the authors as sociate a monomial ideal to
+Research of Dochtermann supported by an Alexander von Humbo ldt postdoctoral fellowship.
+Research of Joswig supported by Deutsche Forschungsgemein schaft, DFG Research Unit “Polyhedral Sur-
+faces”.
+Research of Sanyal supported by the Konrad-Zuse-Zentrum f¨ ur Informationstechnik (ZIB) and a Miller
+Research Fellowship .
+12 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+a tropical polytope with generators in general position, an d use algebraic properties of its
+minimal resolution to determine the facial structure of the bounded subcomplex. Further
+progress along these lines was made in [8]. The primary tool e mployed in this context is that
+of acellular resolution of a monomial ideal I, where the i-th syzygies of Iare encoded by the
+i-dimensional faces of a polyhedral complex (see Section 3). Theideals from [4] are squarefree
+monomials ideals generated by the cotypedata, i.e. the complements of the tropical covectors,
+arising from the associated arrangement of hyperplanes, an d hence can be seen as a tropical
+analog of the (oriented) matroid ideals studied by Novik, Postnikov and Sturmfels in [22].
+In this paper we study the polyhedral complex CA(and its bounded subcomplex BA)
+induced by the type data of an arrangement Aofnhyperplanes in Td−1. Both complexes are
+naturally labeled by fine and coarse type and cotype data, and we show how the resulting
+labeled complexes supportminimal(co)cellular resolutio ns of associated monomial ideals. We
+pay special attention to labels given by coarse type. For ins tance, we show that CAsupports
+a minimal cocellular resolution of the ideal It(A)generated by monomials corresponding to
+the set of all coarse types. The proof involves a considerati on of the topology of certain
+subsets of CAas well as the combinatorial properties of the coarse type la belings. When
+the arrangement Aissufficiently generic we show that the resulting ideal is always given by
+/a\}bracketle{tx1,...,x d/a\}bracketri}htn, then-th power of the maximal homogeneous ideal; in general, It(A)is some
+Artinian subideal. Our results in this area are all independ ent of the characteristic of the
+coefficient field.
+Via the connection to products of simplices and the Cayley tr ick we interpret these results
+in the context of mixed subdivisions of dilated simplices. I n particular we obtain a minimal
+cellularresolution of It(A)supported on a subcomplex of the dilated simplex n∆d−1. One
+other direct consequence is that anyregular fine mixed subdivision of n∆d−1supports a
+minimal resolution of /a\}bracketle{tx1,...,x d/a\}bracketri}htn. This extends a result of Sinefakopoulos from [29] where
+a particular subdivision is considered (although much less explicitly), and also complements
+a construction of Engstr¨ om and the first author from [9] wher e such complexes are applied
+to resolutions of hypergraph edge ideals. The duality betwe en tropical complexes and mixed
+subdivisions of dilated simplices was established in [7], a nd we show how this extends to the
+algebraic level in terms of Alexander duality of our resolut ions of the coarse type and cotype
+ideals.
+Finally, we show how these algebraic results lead to observa tions regarding the combina-
+torics oftropical polytopes/complexes andmixed subdivis ionsof dilated simplices. Weobtain
+a formula for the f-vector of the bounded subcomplex of an arbitrary tropical h yperplane ar-
+rangement in terms of the Betti numbers of the associated coa rse type ideal. The uniqueness
+of minimal resolutions also implies that for any sufficiently generic arrangement A, the multi-
+set of coarse types is independent of the arrangement. Furth ermore, we present an algorithm
+for determining the incidence face structure of a tropical c omplex from the coordinates of the
+generic set of vertices, utilizing the fact that such a compl ex supports a minimal resolution
+of the square-free monomial cotype ideal. This approach was first introduced by Block and
+Yu in [4] for the case of sufficiently generic arrangements, an d we extend the algorithm to
+the general case.
+The rest of the paper is organized as follows. In Section 2 we r eview the basic notions of
+tropical convexity including tropical hyperplanes and typ e data, and discuss the polytopal
+complex that arises from an arrangement of tropical hyperpl anes. We introduce the notion
+ofcoarse type and establish some of the basic properties that will be used l ater in the paper.
+In Section 3 we introduce the monomial ideals that arise from an arrangement of hyperplanes
+and show that the polytopal complexes labeled by fine and coar se (co)type support cocellularTROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 3
+(and cellular) resolutions. In Section 4 we recall the const ruction of the Cayley trick and use
+this to interpret our results in terms of mixed subdivisions of dilated simplices. In Section 5
+we discuss some examples of our construction, in particular the case of the (generic) staircase
+triangulation (recovering a result of [29]) as well as a family of non-gener ic arrangements
+corresponding to tropical hypersimplices . In Section 6 we show our results give rise to certain
+consequences for the combinatorics (e.g., the f-vector) of the bounded subcomplexes of trop-
+ical hyperplane arrangements, and also describe an algorit hm for determining the entire face
+poset from the coordinates of the arrangement. This strengt hens a result from [4]. Finally,
+we end in Section 7 with some concluding remarks and open ques tions.
+We are grateful to Kirsten Schmitz for very careful proof rea ding.
+2.Tropical convexity and coarse types
+In order to fix our notation we begin in this section with a brie f review of the foundations of
+tropical convexity as layed out by Develin and Sturmfels in [ 7]. We then define ‘coarse types’
+and establish some combinatorial and topological results r egarding the type decomposition
+of the tropical torus induced by a finite set of points. While s ome of these observations
+may be worthwhile in their own right, their main interest for us will be their applications to
+subsequent constructions of (co)cellular resolutions.
+2.1.Tropical convexity and tropical hyperplane arrangements. Tropical convexity
+is concerned with linear algebra over the tropical semi-ring (R,⊕,⊙), where
+x⊕y:= min(x,y) andx⊙y:=x+y.
+We will sometimes replace the operation min with max, and alt hough the two resulting
+semi-rings are isomorphic via
+−max(x,y) = min( −x,−y),
+it will be useful for us to consider both structures on the set Rsimultaneously. To avoid
+confusion we will therefore use the terms min-tropical semi-ring andmax-tropical semi-ring ,
+respectively. Componentwise tropical addition and tropic al scalar multiplication turn Rd
+into a semi-module. The tropical torus Td−1is the quotient of Euclidean space Rdby the
+linear subspace R /BD, where /BD∈Rdis the all-ones vector. By interpreting this quotient in the
+category of topological spaces, Td−1inherits anatural topology which is homeomorphicto the
+usual topology on Rd−1. A setS⊂Td−1istropically convex if it contains ( λ⊙x)⊕(µ⊙y) for
+allx,y∈Sandλ,µ∈R. For an arbitrary set S⊂Td−1thetropical convex hull tconv(S) is
+defined as the smallest tropically convex set containing S. If the setSis finite, then tconv( S)
+is called a tropical polytope . In this paper, tropical convexity and related notions will be
+studied with respect to both min and max, and hence we will als o talk about max-tropically
+convexsets and the like.
+Thetropical hyperplane withapex−a∈Td−1is the set
+H(−a) :=/braceleftig
+p∈Td−1: (a1⊙p1)⊕(a2⊙p2)⊕···⊕(ad⊙pd) is attained at least twice/bracerightig
+.
+That is, a tropical hyperplane is the tropical vanishing loc us of a polynomial homogeneous
+of degree 1 with real coefficients. If we want to explicitly dis tinguish between the min-
+and max-versions we write Hmin(−a) andHmax(−a), respectively. Any two min-tropical
+(resp. max-tropical) hyperplanes are related by an ordinar y translation, and hence a tropical
+hyperplane is completely determined by its apex. Furthermo re, the min-tropical hyperplane
+with apex 0 is a mirror image of the max-tropical hyperplane w ith apex 0 under the map4 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+x/mapsto→ −x. Thecomplement ofany tropical hyperplanein Td−1consists ofprecisely dconnected
+components, its open sectors . Each open sector is convex, in both the tropical and ordinar y
+sense. The k-th (closed) sector of the max-tropical hyperplane with apex ais the set
+Smax
+k(a) :=/braceleftig
+p∈Td−1:ak−pk≤ai−pifor alli∈[d]/bracerightig
+.
+Similarly we have
+Smin
+k(a) :=/braceleftig
+p∈Td−1:ak−pk≥ai−pifor alli∈[d]/bracerightig
+for the min-version. Notice that x∈Smax
+k(y) if and only if y∈Smin
+k(x). Each closed sector is
+thetopological closureofanopensector. Againthecloseds ectorsaretropically andordinarily
+convex. A sequence of points V= (v1,v2,...,vn) inTd−1gives rise to the arrangement
+A(V) :=/parenleftbig
+Hmax(v1),Hmax(v2),...,Hmax(vn)/parenrightbig
+ofnlabeled max-tropical hyperplanes. The position of points i nTd−1relative to each hyper-
+plane in the arrangement furnishes combinatorial data and l eads to the following definition.
+Definition 2.1 (Fine type) .LetA=A(V) be the arrangement of max-tropical hyperplanes
+given byV= (v1,v2,...,vn) inTd−1. Thefine type (or sometimes simply type) of a point
+p∈Td−1with respect to Ais the table TA(p)∈ {0,1}n×dwith
+TA(p)ik= 1 if and only if p∈Smax
+k(vi)
+fori∈[n] andk∈[d]. Thefine cotype TA(p)∈ {0,1}n×dis defined as the complementary
+matrix, that is,
+TA(p)ik= 1 if and only if TA(p)ik= 0.
+Let us now fix a sequence of points Vand the corresponding arrangement A=A(V) in
+Td−1. We write T(p) instead of TA(p) when no confusion arises. For a fixed type T=T(p),
+the set
+C◦
+T:=/braceleftig
+q∈Td−1:T(q) =T/bracerightig
+of points in Td−1with that type is a relatively open subset of Td−1=Rd−1; called the
+relatively open cell of typeT. The setC◦
+Tas well as its topological closure CTare tropically
+and ordinarily convex. The set of all relatively open cells C◦=C◦(V) partitions the tropical
+torusTd−1. Moreover, CT, theclosed cell of type T, is the intersection of finitely many closed
+sectors. Since each closed sector is the intersection of fini tely many polyhedral cones, this
+implies that CT, provided it is bounded, is a polytope in both the classical a nd the tropical
+sense; in [14] these were called polytropes . The collection of all closed cells yields a polyhedral
+subdivision C=C(V) ofTd−1. The collection of types with the componentwise order is
+anti-isomorphic to the face lattice of C(V), that is,TA(D)≤TA(C) whenever C⊆Dare
+closed cells of C(V). The cells which are bounded form the bounded subcomplex B=B(V).
+Following Ardila and Develin [1], a fine type should be though t of as the tropical equivalent
+of a covector in the setting of tropical oriented matroids, w ith the cells of maximal dimension
+playing the role of the topes. The following result highligh ts the relation of min-tropical
+polytopes and max-tropical hyperplane arrangements.
+Theorem 2.2 ([7, Thm. 15 and Prop. 16]) .Themin-tropical polytope tconv(V)is the union
+of cells in the bounded subcomplex B(V)of the cell decomposition of Td−1induced by the
+max-tropical hyperplane arrangement A(V).TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 5
+The polytopal complex B(V) induced by types is a subdivision of the tropical polytope
+tconv(P) called the tropical complex generated by V. The points Vare in tropically general
+position if the combinatorial type of C(V) (or equivalently B(V)) is invariant under small
+perturbations. In Section 4, we will elaborate on the import ant connection between trop-
+ical hyperplane arrangements and mixed subdivisions of dil ated simplices. In light of this
+connection, the points Vare in general position if the associated mixed subdivision isfine.
+At this point we introduce our running example, borrowed fro m [4, Ex. 10].
+Example 2.3. The points v1= (0,3,6),v2= (0,5,2),v3= (0,0,1), andv4= (1,5,0)
+give rise to the max-tropical hyperplane arrangement shown in Figure 1. It decomposes the
+tropical torus T2into 15 two-dimensional cells, three of which are bounded. N ote that there
+is precisely one bounded cell of dimension one (incident wit h the 0-cell v1) which is maximal
+with respect to inclusion. This shows that the polytopal com plexB(V) need not be pure.
+Moreover, one can check that the dual regular subdivision of ∆3×∆2is a triangulation, and
+hence these four points are sufficiently generic.
+2.2.Coarse types. As we have seen, the fine type records the position of a point re lative
+to a labeled tropical hyperplane arrangement. Neglecting t he labels on the hyperplanes leads
+to the following coarsening of the type information.
+Definition 2.4 (Coarse type) .LetA=A(V) be an arrangement of nmax-tropical hy-
+perplanes in Td−1. Thecoarse type of a point p∈Td−1with respect to Ais given by
+tA(p) = (t1,t2,...,td)∈Ndwith
+tk=n/summationdisplay
+i=1TA(p)ik
+fork∈[d].
+The coarse type entry tkrecords for how many hyperplanes in Athe pointplies in the
+k-th closed sector. Again we will write t(p) when no confusion can arise. By definition t
+is constant when restricted to a relatively open cell C◦
+T. The induced map C◦→Ndis also
+denoted by t.
+Objects in tropical geometry can be lifted to objects in clas sical algebraic geometry by
+considering fields of formal Puiseux series and their valuat ions. Following this path leads
+to another interpretation of the coarse types: For vi∈Td−1the max-tropical hyperplane
+Hmax(vi) is thetropicalization tropmax(hi) of a hyperplane given as the vanishing of a ho-
+mogeneous linear form hi∈C{{z}}[x1,x2,...,x d] defined over the field of Puiseux series.
+Let
+(1) h=h1·h2···hn
+be the product of these nlinear forms, so that his a homogeneous polynomial of degree nin
+dindeterminates. Notice that there is no canonical choice fo r the polynomials hiand hence
+not forh. In the computation below we choose hito bez−vi1x1+z−vi2x2+···+z−vidxd.
+Remark 2.5. Note that the classical Puiseux series have rational expone nts, and therefore
+the valuation map takes rational values only. For this reaso n the tropical hypersurface of
+a tropical polynomial is often defined as the topological clo sure of the vanishing locus of
+a tropical polynomial; e.g., see Einsiedler, Kapranov, and Lind [11]. By extending C{{z}}
+to a field of generalized Puiseux series one can directly deal with real exponents, as the
+corresponding valuation map is onto the reals; for a constru ction see Markwig [19]. In6 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+-1 0 1 2 3 4 5 6-2-101234567
+v1
+v2
+v3
+v4400 310 220 130 040301 211 121031202103
+112022004 013
+Figure 1. Coarsely labeled type decomposition of T2induced by four max-
+tropical lines. Bounded cells are shaded.
+the sequel we will make use of such a field of generalized Puise ux series, and we denote it
+byC{{z}}gen.
+Proposition 2.6. The tropical hypersurface defined by tropmax(h)is the union of the max-
+tropical hyperplanes in A.
+The tropical hypersurface defined by tropmax(h) is the orthogonal projection of the co-
+dimension-2-skeleton of an unbounded ordinary convex poly hedronPAinRdwhose facets
+correspond to monomials of the polynomial h; see [24, Thm. 3.3].
+Proposition 2.7. Letp∈Td−1\ Abe a generic point. Then its coarse type tA(p)is the
+exponent of the monomial in hwhich defines the unique facet of PAabovep.
+Proof.Sincepis generic, it is contained in a unique sector Smax
+τi(vi) with respect to the
+hyperplane with apex vi. Hence,psatisfies the strict inequalities
+(2) vi,τi−pτi< vi,j−pj
+for alli∈[n] andj∈[d]\τi. Equivalently,
+n/summationdisplay
+i=1pτi−vi,τi
+is the evaluation of the tropical polynomial tropmax(h) and the corresponding monomial in h
+atwhichtheuniquemaximumisattainedis xτ1xτ2···xτnwithcoefficient z−v1,τ1−v2,τ2−···−vn,τn.
+/square
+Example 2.8 (continued) .For the points in Example 2.3 we set
+h1=x1+z−3x2+z−6x3,
+h2=x1+z−5x2+z−2x3,
+h3=x1+x2+z−1x3,and
+h4=z−1x1+z−5x2+x3.TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 7
+Thenh=h1·h2·h3·h4equals
+h=z−1x14+(z−6+z−5+z−4+z−1)x13x2+(z−7+z−3+z−2+1)x13x3+
+(z−10+z−9+z−8+z−6+z−5+z−4)x12x22+(z−12+z−11+3z−7+2z−6+
+2z−5+2z−3+1)x12x2x3+(z−9+z−8+z−6+z−4+z−2+z−1)x12x32+(z−13+
+z−10+z−9+z−8)x1x23+(z−16+z−12+2z−11+2z−10+z−9+z−8+z−7+
+z−6+z−5+z−3)x1x22x3+(2z−13+z−12+z−11+z−9+z−8+z−7+2z−6+
+z−5+z−4+z−2)x1x2x32+(z−10+z−8+z−7+z−3)x1x33+z−13x24+(z−16+
+z−14+z−10+z−8)x23x3+(z−17+z−13+2z−11+z−9+z−5)x22x32+(z−14+
+z−12+z−8+z−6)x2x33+z−9x34.
+Its max-tropicalization is
+tropmax(h) = (−1⊙x1⊙4)⊕(−1⊙x1⊙3⊙x2)⊕(x1⊙3⊙x3)⊕(−4⊙x1⊙2⊙
+x2⊙2)⊕(x1⊙2⊙x2⊙x3)⊕(−1⊙x1⊙2x3⊙2)⊕(−8⊙x1⊙x2⊙3)⊕(−3⊙x1⊙
+x2⊙2⊙x3)⊕(−2⊙x1⊙x2⊙x3⊙2)⊕(−3⊙x1⊙x3⊙3)⊕(−13⊙x2⊙4)⊕(−8⊙
+x2⊙3⊙x3)⊕(−5⊙x2⊙2⊙x3⊙2)⊕(−6⊙x2⊙x3⊙3)⊕(−9⊙x3⊙4).
+The polynomial hand its max-tropicalization have 15 terms each, one for each maximal cell
+of the arrangement shown in Figure 1. For instance, the point p= (0,1,0) is generic with fine
+typeT(p) = (12,3,4). Evaluating tropmax(h) atpgives−1, and this maximum is attained
+for the unique tropical monomial x1⊙2⊙x2⊙x3= 2x1+x2+x3. We have t(p) = (2,1,1)
+for the coarse type.
+Recall that a composition ofnintodparts is a sequence ( t1,t2,...,td)∈Ndof nonnegative
+integers suchthat t1+t2+···+td=n. Suchcompositionsbijectively correspondtomonomials
+of total degree nindvariables, of which there are exactly/parenleftbign+d−1
+n/parenrightbig
+=/parenleftbign+d−1
+d−1/parenrightbig
+.
+Theorem 2.9. For an arrangement A=A(V), the map tfrom the set of cells in CAof
+maximal dimension d−1to the set of compositions of nintodparts is injective. Moreover,
+if the points Vare sufficiently generic then the map tis bijective.
+Proof.The injectivity of tfollows immediately from Proposition 2.7.
+Now suppose that the points in Vare sufficiently generic. All coordinates are finite, and
+hence all linear monomials are present in the linear form hiwith non-zero coefficients. Since
+his the product of h1,h2,...,h nall possible monomials of degree nin thedindeterminates
+x1,x2,...,x dactually occurin h. Thereforealltropicalmonomialsoccurinthetropicaliza tion
+tropmax(h). As we assumed that the coefficients in Vare chosen sufficiently generic, the
+coefficients in the tropical polynomial tropmax(h) are generic. Equivalently, each monomial
+ofhdefines a facet of PA, and hence the claim. /square
+Proposition4.2providesaninterpretationoftheaboveres ultintermsofmixedsubdivisions
+of dilated simplices.
+Corollary 2.10. For an arrangement A=A(V), the number of cells in CAof maximal
+dimensiond−1does not exceed/parenleftbign+d−1
+n/parenrightbig
+. If the points Vare sufficiently generic then this
+bound is attained.
+Note that the injectivity of tdoes not extend to the lower-dimensional cells, even in the
+sufficiently generic case. For instance, in the arrangement c onsidered in Examples 2.3 and 2.8
+thepoints (0 ,2,3) and (0,3,5) lie inthe relative interiors of two distinct 1-cells; yet they share
+the same coarse type (1 ,1,3). However coarse types are locallydistinct in the following sense.
+Proposition 2.11. LetCandDbe two distinct cells in CA. IfCis contained in the closure
+ofDthenCandDhave distinct coarse types.8 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+Proof.By [7, Cor. 13] we have TA(D)≤TA(C) andTA(D)/\e}atio\slash=TA(C). This, in particular
+implies that Cis contained in more closed sectors with respect to Aand hence CandD
+cannot have the same coarse type. /square
+Note also that the boundedness of a cell in CAcan be read off its coarse type.
+Proposition 2.12 ([7, Cor. 12]) .LetA=A(V)be a tropical hyperplane arrangement and
+letC∈ CAbe a cell in the induced decomposition. Then Cis bounded if and only if t(C)i>0
+for alli= 1,...,n.
+Proof.We noted previously that for two points p,q∈Td−1we havep∈Smax
+k(q) if and only
+ifq∈Smin
+k(p). Now, the point pis contained in an unbounded cell of CAif there is a ksuch
+thatSmin
+k(p)∩V=∅. This is the case if and only if tA(p)k= 0, that is, there is no hyperplane
+Hifor whichpis contained in the k-th sector. /square
+Remark 2.13. There is another perspective on Proposition 2.7 that we wish to share. The
+standard valuation ν: (C{{z}}gen)∗→Ron the field of (generalized) Puiseux series maps
+an element to its order. For ω∈Rd, theω-weightoff=/summationtext
+acaxa∈C{{z}}gen[x1,...,x d]
+isω(f) = max {ν(ca) +/a\}bracketle{tω,a/a\}bracketri}ht:ca/\e}atio\slash= 0}and the initial form of the polynomial fis given
+by inω(f) :=/summationtext{xa:ν(ca) +/a\}bracketle{tω,a/a\}bracketri}ht=ω(f)}. Finally, for an ideal I⊂C{{z}}gen[x1,...,x d]
+theinitial ideal inω(I) is generated by all in ω(f) forf∈I. An alternative characterization
+of the tropical variety was given by Speyer and Sturmfels [30 , Thm. 2.1] as the collection of
+weightsω∈Rdsuch that in ω(I) does not contain a monomial. For a tropical hyperplane
+arrangement A=A(V) with defining polynomial h=hA∈C{{z}}gen[x1,...,x d] the coarse
+typet(p) of a generic point p∈Td−1\Asatisfies
+/a\}bracketle{txt(p)/a\}bracketri}ht= inp(/a\}bracketle{th/a\}bracketri}ht).
+Remark 2.14. The evaluation of the tropical polynomial tropmax(h) at some point pcan
+be modeled as a min-cost-flow problem as follows: Define a dire cted graph with nodes α,
+β1,β2,...,β n,γ1,γ2,...,γd, andδ. For eachi∈[n] and eachk∈[d] there are the following
+arcs: from αtoβiwith cost 0 and capacity 1, from βitoγkwith costvikand capacity 1,
+fromγktoδwith costpkand unbounded capacity. Now tropmax(h)(p) equals the minimal
+cost of shipping nunits of flow from the unique source αto the unique sink δ. This problem
+has a strongly polynomial time solution in the parameters nandd; see [28, Chapter 12].
+This is remarkable in view of the fact that the tropical polyn omial tropmax(h) may have
+exponentially many monomials.
+2.3.Topology of types. We next investigate topological properties of certain subs ets of
+Td−1induced by a tropical hyperplane arrangement. The motivati on for such a study comes
+from our applications to (co)cellular resolutions as descr ibed in Section 3. However, since the
+methods used to establish the desired properties are based o n notions from (coarse) tropical
+convexity, we include them here. In particular, we show that certain subsets of Td−1are
+contractible by proving that they are, in fact, tropically c onvex. Let us emphasize that none
+of the results that follow require the hyperplanes to be in ge neral position.
+We begin with the following observation, which was establis hed in [7, Thm. 2]. We include
+a short proof for the sake of completeness.
+Proposition 2.15. A tropically convex set is contractible.
+Proof.Thedistance of two points p,q∈Td−1is defined as
+dist(p,q) := max
+1≤i<j≤d|pi−pj+qj−qi|.TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 9
+We have the equations
+/parenleftbig
+dist(p,q)⊙p/parenrightbig
+⊕q=qandp⊕/parenleftbig
+dist(p,q)⊙q/parenrightbig
+=p.
+Now letpbe a point in some tropically convex set S. The map
+η:S×[0,1]→S: (q,t)/mapsto→/parenleftbig
+((1−t)·dist(p,q))⊙p/parenrightbig
+⊕/parenleftbig
+(t·dist(p,q))⊙q/parenrightbig
+is continuous, and it contracts the set Sto the point p. /square
+Recall that for two types T,T′∈ {0,1}n×dwe writeT≤T′for the componentwise induced
+partial order. We let min( T,T′) and max( T,T′) denote the tables with entries given by the
+componentwise minimum and maximum, respectively.
+Proposition 2.16. LetA=A(V)be amax-tropical hyperplane arrangement in Td−1and
+p,q∈Td−1. Then
+min(TA(p),TA(q))≤TA(r)≤max(TA(p),TA(q))
+for every point r∈tconvmax{p,q}on the max-tropical line segment between pandq.
+Proof.Letr= (λ⊙p)⊕(µ⊙q) = max {λ /BD+p,µ /BD+q}forλ,µ∈Rand letk∈[d] be
+arbitrary but fixed. We treat each inequality separately but in each case we assume without
+loss of generality that rk=λ+pk≥µ+qk.
+For the first inequality suppose that both pandqare in thek-th sector of some hyperplane
+H(v), so thatpk−pi≥vk−viandqk−qi≥vk−vifor alli∈[d]. Now, for j∈[d] we
+distinguish two cases. If rj=λ+pj≥µ+qj, thenrj−rk=pj−pk, so thatris in thek-th
+sector ofH(v). Ifrj=µ+qj≥λ+pj, thenrk−rj≥µ+qk−rj=µ+qk−(µ+qj) =qk−qj.
+Hencerk−rj≥vk−vjfor allj∈[d], and we conclude that ris in thek-th sector of H(v).
+For the second inequality supposethat ris contained in the k-th sector of some hyperplane
+H(v), so thatrk−rj≥vk−vjfor allj∈[d]. Sincerk=λ+pk≥µ+qkwe have that
+λ+pk≥vk−vj+rjfor allj∈[d]. Also,rj≥λ+pjand henceλ+pk≥vk−vj+λ+pj. We
+concludepk−pj≥vk−vjfor allj∈[d]. Hencepis in thek-th sector of H(v), as desired. /square
+From the definition the coarse type we obtain the following st atement regarding coarse
+types.
+Corollary 2.17. LetA=A(V)be amax-tropical hyperplane arrangement in Td−1and
+p,q∈Td−1. Then
+tA(r)≤max(tA(p),tA(q))
+forr∈tconvmax{p,q}.
+From Proposition 2.16 and Corollary 2.17 we obtain the follo wing result regarding the
+topology of regions of bounded fine and coarse (co)type. Thes e will be the main tools for
+establishing results regarding (co)cellular resolutions in Section 3.
+Corollary 2.18. LetA=A(V)be an arrangement of nmax-tropical hyperplanes in Td−1,
+and letB∈ {0,1}n×dandb∈Nd. With labels determined by fine (respectively, coarse) type
+the following subsets of Td−1aremax-tropically convex and hence contractible:
+(CA,T)≤B:=/braceleftbig
+p∈Td−1:TA(p)≤B/bracerightbig
+=/uniontext{C∈ CA:TA(C)≤B}
+(CA,t)≤b:=/braceleftbig
+p∈Td−1:tA(p)≤b/bracerightbig
+=/uniontext{C∈ CA:tA(C)≤b}.10 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+Similarly, with labels determined by fine (respectively, co arse)cotype the following subsets of
+Td−1aremin-tropically convex and hence contractible:
+(CA,T)≤B:=/braceleftbig
+p∈Td−1:TA(p)≤B/bracerightbig
+=/uniontext/braceleftbig
+C∈ CA:TA(C)≤B/bracerightbig
+(CA,t)≤b:=/braceleftbig
+p∈Td−1:tA(p)≤b/bracerightbig
+=/uniontext/braceleftbig
+C∈ CA:tA(C)≤b/bracerightbig
+.
+As a consequence, the two subsets of Td−1obtained by replacing the complex CAin the
+above pair of formulas with the bounded complex BAaremin-tropically convex and hence
+contractible.
+Proof.The max-tropical convexity of ( CA,T)≤Bfollows from Proposition 2.16, and Corol-
+lary 2.17 establishes the same property for the coarse varia nt (CA,t)≤b.
+Ifris a point in the min-tropical line segment between pandqa reasoning similar to
+the proof of Proposition 2.16 shows that TA(r)≥min(TA(p),TA(q)). Passing to comple-
+ments yields TA(r)≤max(TA(p),TA(q)) and thus tA(r)≤max(tA(p),tA(q)) for the coarse
+types. We conclude that the sets ( CA,T)≤Band (CA,t)≤bare min-tropically convex. For
+the last claim, we note that the tropical complex BAitself is min-tropically convex; in view
+of Proposition 2.12 it is a down-set of CAunder the cotype labeling. Hence both ( BA,T)≤B
+and (BA,t)≤bare intersections of min-tropically convex sets. /square
+Note that the sets of bounded (coarse) type need not be closed or bounded. In Figure 2
+we illustrate an example of a coarse down-set from our runnin g example.
+-1 0 1 2 3 4 5 6-2-101234567
+v1
+v2
+v3
+v4400 310 220 130 040301 211 121031202103
+112022004 013
+Figure 2. Coarse down-set ( CA)≤(2,2,2)for the arrangement Afrom Example 2.3.
+3.Resolutions
+In this section we show how the polyhedral complexes CAandBAarising from a tropical
+hyperplane arrangement A=A(V) support resolutions for associated monomial ideals. We
+begin with a few definitions.TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 11
+Definition 3.1. LetA=A(V) be an arrangement of ntropical hyperplanes in Td−1. The
+fine type andfine cotype ideal associated to Aare the squarefree monomial ideals
+IT(A)=/a\}bracketle{txT(p):p∈Td−1/a\}bracketri}ht ⊂ /CZ[x11,x12,...,x nd]
+IT(A)=/a\}bracketle{txT(p):p∈Td−1/a\}bracketri}ht ⊂ /CZ[x11,x12,...,x nd]
+wherexT(p)=/producttext{xij:T(p)ij= 1}. Analogously, the coarse type andcoarse cotype ideal
+associated to Aare given by
+It(A)=/a\}bracketle{txt(p):p∈Td−1/a\}bracketri}ht ⊂ /CZ[x1,x2,...,x d]
+It(A)=/a\}bracketle{txt(p):p∈Td−1/a\}bracketri}ht ⊂ /CZ[x1,x2,...,x d].
+wherext(p)=xt1
+1xt2
+2···xtd
+dwitht(p) = (t1,t2,...,td).
+An analogous construction of fine cotype ideals for classica l hyperplane arrangements first
+appears in Novik et al. [22] in the form of oriented matroid ideals , where monomial generators
+are given by (complements of) covector data. As remarked ear lier, the fine type of a tropical
+hyperplane arrangement can be thought of as the covector dat a of a tropical oriented matroid
+and, from this perspective, a tropical analog of [22] is give n in [4]. In Sections 3 and 6 we
+will establish further connections between our work and the results and constructions of [4].
+A common generalization in the form of fan arrangements will be given in [27]. In [22] the
+authors also consider matroid ideals given by specializing the underlying oriented matroid
+ideal, in effect not distinguishing between different sides of a (classical) hyperplane. We point
+out in the case of our coarse type ideals, the covector data co ming from the different sectors
+induced by a (tropical) hyperplane (the tropical analog of t he ‘side’) is maintained, whereas
+thelabelson the hyperplanes are no longer distinguished.
+3.1.Cellular and cocellular resolutions. The relation between the decompositions of
+Td−1and the various ideals of Definition 3.1 is given via (co)cell ular resolutions. Whereas
+cellular resolutions are by now a standard tool in (combinat orial) commutative algebra,
+cocellular resolutions seem to be less popular. As they arise n aturally in connection with
+cotypes, we take the opportunity to discuss cocellular reso lutions alongside cellular resolu-
+tions in some detail. Our presentation is based on the book [2 1] of Miller and Sturmfels.
+For a fixed field /CZwe letS= /CZ[x1,...,x m] be the polynomial ring equipped with the
+Zm-grading given by deg xa=a∈Zm. A free Zm-graded resolution F•of aZm-graded
+moduleMis an algebraic complex of Zm-gradedS-modules
+F•:···φk+1−→Fkφk−→ ···φ2−→F1φ1−→F0→0
+whereFi∼=/circleplustext
+a∈ZmS(−a)βi,aare freeZm-gradedS-modules, the maps φiare homogeneous,
+and such that the complex is exact except for coker φ1∼=M. The resolution is called minimal
+exactly when βi,a= dim/CZTorS
+i(S/I, /CZ)aand the numbers βi,aare called the fine graded Betti
+numbers.
+An efficient way of encoding Zm-graded resolutions of monomial ideals is given by cellular
+andcocellular resolutions , introduced in [2] and [20], respectively. Let Pbe an oriented
+polyhedral complex and let ( aH)H∈P∈Zmbe a labeling of the cells of Psuch that
+aH= max{aG: forG⊂Ha face}.
+The labeled complex ( P,a) gives rise to an algebraic complex of free Zm-gradedS-modules
+in the following way: Let ( C•,∂•) be the cellular chain complex for Pand for two cells12 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+G,H∈ Pwith dimH= dimG+1 denote by ε(H,G)∈ {0,±1}the coefficient of Gin the
+cellular boundary of H. Now define free modules
+Fi:=/circleplusdisplay
+H∈P,dimH=i+1S(−aH).
+The differentials φi:Fi→Fi−1are given on generators by
+φi(eH) :=/summationdisplay
+dimG=dimH−1ε(H,G)xaH−aGeG.
+It can be verified that this defines an algebraic complex FP
+•. Forb∈Zmdenote by P≤bthe
+subcomplex given by all cells H∈ PwithaH≤b, that is ( aH)i≤bifor alli∈[m].
+Lemma 3.2 ([21, Prop. 4.5]) .LetFP
+•be the algebraic complex obtained from the labeled
+polyhedral complex (P,a). If for every b∈Znthe subcomplex P≤bis acyclic over /CZ, then
+FP
+•resolves the quotient of Sby the ideal /a\}bracketle{txav:v∈Pvertex/a\}bracketri}ht. Furthermore, the resolution
+is minimal if aH/\e}atio\slash=aGfor any two faces G⊂HwithdimH= dimG+1.
+The complex FP
+•is called a cellular resolution if it meets the criterion above, and we say
+that the polyhedral complex Psupports the resolution. If the labeling is such that
+aH= max{aG: forG⊃Ha face}
+thenPis said to be colabeled and gives rise to an algebraic complex utilizing the cellula r
+cochaincomplex of P. For this, let F•
+Pdenote the algebraic complex with free S-modules
+Fi:=Fias defined above and differentials φi:Fi−1→Fiwith
+φi(eH) :=/summationdisplay
+dimG=dimH+1δ(H,G)xaH−aGeG
+whereδ(H,G) records the corresponding coefficient in the coboundary map forP. If the
+algebraic complex is acyclic, the resulting resolution is c alledcocellular . Forb∈Zmthe
+collection P≤bof relatively open cells HwithaH≤bisnota subcomplex. However, as a
+topological space it is the union of the relatively open star s of cellsGfor which aG≤bis
+minimal andthecochain complex of thenerveisisomorphicto thedegree bcomponent of F•
+P.
+This yields an analogous criterion regarding the exactness ofF•
+P. The proof of Lemma 3.2
+given in [21] essentially proves the following criterion.
+Lemma 3.3. IfP≤bis acyclic over /CZfor every b∈ZnthenF•
+PresolvesS/IwhereI=
+/a\}bracketle{txaH:H∈ Pmaximal cell /a\}bracketri}ht. The resolution is minimal if aH/\e}atio\slash=aGfor any two faces G⊂H
+withdimH= dimG+1.
+3.2.Resolutions from the arrangement. As usual let A=A(V) be a max-tropical
+hyperplanearrangement in Td−1and letCAbetheinducedpolyhedraldecomposition of Td−1.
+As we have seen, every cell in CAis naturally assigned a matrix and avector determined by its
+fine and coarse type, respectively. The next result states th at these assignments are actually
+colabelings in the sense of Section 3.1.
+Proposition 3.4. LetA=A(V)be an arrangement of tropical hyperplanes in Td−1. For
+every cellC∈ CAof codimension ≥1we have
+TA(C) = max {TA(D) :C⊂D},and
+tA(C) = max {tA(D) :C⊂D}.
+Thus, both the fine type and the coarse type yield a colabeling f or the complex CA.TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 13
+Proof.As we remarked before C⊆DimpliesTA(D)≤TA(C) and thus we have to show
+thatTA(C) is not strictly larger. For k∈[d] consider the set Qk⊂Rdgiven by all points
+x∈Rdsuch thatxk≤xifor alli/\e}atio\slash=k. The setQkis an ordinary polyhedron of dimension d
+and it can be checked that for p∈Td−1we have that p∈Smax
+k(vi) impliesp+Qk⊆Sk(vi).
+Now, since codim( C)≥1 we get that C+Qkmeets the star of Cthereby showing that if
+TA(C)ik= 1 for some i, then there is a cell D⊃CwithTA(D)ik= 1.
+The same argument proves the purported equality for the coar se type. /square
+As an immediate consequence we obtain sets of generators for the fine and the coarse type
+ideals.
+Corollary 3.5. For a tropical hyperplane arrangement Aboth the fine and coarse type ideals
+are generated by monomials corresponding to the respective types on the inclusion-maximal
+cells ofCA.
+We now have all the ingredients to establish our first main res ult regarding the relation
+between fine/coarse type ideals and the polyhedral decompos itionCA.
+Theorem 3.6. LetA=A(V)be a tropical hyperplane arrangement, and let CAbe the
+decomposition of the tropical torus Td−1induced by A. Then with labels given by fine type
+(respectively, coarse type) the labeled complex CAsupports a minimal cocellular resolution of
+the fine type ideal IT(A)(respectively, the coarse type ideal It(A)).
+Proof.By Proposition 3.4 the polyhedral complex CAis colabeled by both fine and coarse
+type. It follows from Lemma 3.3 and Corollary 2.18 that this y ields a cellular resolution of
+the respective type ideal. The minimality is a consequence o f Proposition 2.11. /square
+A key point is that all of the above is valid for point configura tionsVwhich are not
+necessarily in general position. In the case of hyperplanes in general position, the coarse type
+ideal is well known and, in particular, is independent of the choice of hyperplanes.
+Corollary 3.7. LetA=A(V)be asufficiently generic arrangement of ntropical hyperplanes
+inTd−1. ThenCAsupports a minimal cocellular resolution of
+It(A)=/a\}bracketle{tx1,...,x d/a\}bracketri}htn
+then-th power of the homogeneous maximal ideal.
+Since the minimal resolution of an ideal is unique up to isomo rphism, we have that the
+resolution arising from Corollary 3.7 is always isomorphic (as a chain complex), to the well-
+known Eliahou-Kervaire resolution [21, §2.3]. However, Corollary 3.7 shows that there is a
+multitude of colabeled complexes coming from tropical hype rplane arrangements that give
+rise to a cellular description of the minimal resolution of /a\}bracketle{tx1,...,x n/a\}bracketri}htd.
+We next turn to the other class of ideals introduced in the beg inningof this section, namely
+the fine and coarse cotype ideals. As was the case with the type ideals, we first cons ider the
+labeling of the relevant complexes.
+Proposition 3.8. LetA=A(V)be an arrangement of tropical hyperplanes. For every cell
+D∈ CAof dimension ≥1we have
+TA(D) = max {TA(C) :C⊂D},and
+tA(D) = max {tA(C) :C⊂D}.
+Thus, both the fine and coarse types yield labelings for the com plexCA.14 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+Proof.The assertion follows by showing that
+TA(D)≥min{TA(C) :C⊂D}.
+Wemimictheargumentintheproofof Proposition3.4. Itfoll ows fromthefull-dimensionality
+ofQkthatC+QkintersectsDfor every cell C⊂Dand thusT(D)ik≥T(C)ikfor allCin
+the boundary of D. /square
+The previous proposition implies that the cotype ideals are generated by the inclusion-
+minimal faces (namely, the 0-dimensional cells) of CA. As a consequence (together with
+the last part of Corollary 2.18) we see that resolutions of th ese ideals are supported on the
+collection of bounded faces of CA, and we obtain our next main result. This generalizes [4,
+Theorem 1].
+Theorem 3.9. LetA=A(V)be an arrangement of tropical hyperplanes and let BAbe the
+subcomplex of bounded cells of CA. ThenBA, with labels given by fine cotype (respectively,
+coarse cotype) supports a minimal resolution of the fine coty pe idealIT(A)(respectively, coarse
+cotype ideal It(A)).
+Remark 3.10. For fixednanddwe have a ring map ψ: /CZ[x11,...,x nd]→ /CZ[x1,...,x d]
+determined by xij/mapsto→xj. This map takes fine (co)type ideals to coarse (co)type ideal s. In
+either case, both the fine and the coarse ideals are resolved b y the same polyhedral complex,
+and one might hope that the resolution of the fine ideal descen ds to the coarse ideal via
+ψ. Such an approach is employed in [22] in the passage from the o riented matroid to the
+matroid ideal. There it is shown that any oriented matroid id ealJis Cohen-Macaulay, with
+(x11−x21,x21−x22,...,x n1−xn2) forminga linear system of parameters (and hence a regular
+sequence) in S/J. Unfortunately, this approach does notwork in our context. It turns out
+that the fine cotype ideals IT(A)are not in general Cohen-Macaulay. Indeed, regarding IT(A)
+as a Stanley-Reisner ideal we see that S/IT(A)has dimension dn−2, whereas an application
+of the Auslander-Buchsbaum formula (and knowledge of the mi nimal resolution) shows that
+its depth is dn−d. The coarsening sequence {x11−x21,...,x 11−xn1,...,x 1d−xnd}is also
+of lengthdn−d, but one can show that it is not a regular sequence (not even a s ystem of
+parameters).
+4.The mixed subdivision picture
+As mentioned in Section 2.2, an arrangement A=A(V) ofnhyperplanes in Td−1gives
+rise to a regular subdivision of the product of two simplices ∆n−1×∆d−1. To see this,
+we let ∆ n−1⊂Rndenote the convex hull of the standard basis vectors {ei: 1≤i≤n}.
+IfV= (v1,...,vn) is the ordered sequence of the apices of the arrangement A, we lift each
+vertex (ei,ej) of ∆n−1×∆d−1to the height ( vi)j, thej-th coordinate of the i-th apex. Taking
+the lower convex hull of the resulting configuration of point s induces a (by definition regular)
+subdivision of ∆ n−1×∆d−1. A main result from [7] is that the combinatorial types of the
+tropical complexes generated by a set of nvertices in Td−1, seen as polytopal complexes, are
+in bijection with the regular polyhedral subdivisions of ∆ n−1×∆d−1.
+At this point, the reader may find it useful to contemplate the schematic diagram in
+Figure 3 below. It sketches how various relevant combinator ial concepts are related. Again
+Vdenotes our ordered sequence of npoints in Td−1, which can be thought of as the apices
+of a max hyperplane arrangement A(where the hyperplane with apex vicorresponds to
+the tropical vanishing of the form −vi⊙x). We have BAgiven by the min-tropical convex
+hull tconvmin(V), which is the bounded part of the polyhedral complex CAdetermined byTROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 15
+the coarse types of the arrangement A. The correspondence described above (here denoted
+‘DS’) associates to BAa certain regular polyhedral subdivision ∆ Aof ∆n−1×∆d−1; this
+subdivision is dual to the complex CA. The Cayley trick (described in more detail below,
+here denoted ‘Cay’) then associates to that regular subdivi sion a regular mixedsubdivision
+ΣAof the dilated simplex n∆d−1. Moreover, we have the ‘switch’ isomorphism obtained
+by exchanging coordinates of the product; applying the Cayl ey trick then gives a regular
+mixed subdivision Σ∗
+Aofd∆n−1. Finally, observe that n∆d−1is the Newton polytope of the
+polynomial hdefined in (1). The mixed subdivision Σ Ais theprivileged regular subdivision
+ofn∆d−1defined by the coefficients of tropmax(h). The corresponding arrow in the diagram
+is marked ‘New’ for ‘Newton’.
+V
+NewBA= tconvmin(V)
+boundedDS
+C=Cmax(V)dual∆Aof ∆n−1×∆d−1Cay
+∼= switchΣAofn∆d−1
+∆Aof ∆d−1×∆n−1CayΣ∗
+Aofd∆n−1
+Figure 3. Diagram explaining how the various combinatorial concepts are related.
+4.1.The Cayley trick and mixed subdivisions. TheCayley trick gives a bijection be-
+tween the polyhedral subdivisions of ∆ n−1×∆d−1and the mixedsubdivisions of n∆d−1.
+More generally, the Cayley trick relates mixed subdivision s of the Minkowski sum of several
+polytopesP1,P2,...,P kwith the polyhedral subdivisions of a certain polytope know n as the
+Cayley embedding of those polytopes. We refer to [26] for a good account of the C ayley trick
+in the context of products of simplices, but we wish to discus s some of the basics here. This
+way it will also become apparent in which way products of simp lices arise. The collection of
+all polyhedral subdivisions of a polytope Pform a poset under refinement, and the minimal
+elements are the triangulations of P.
+Recall that if P1,P2,...,P nare polytopes in Rd, then the Minkowski sum is defined to be
+the polytope
+P1+P2+···+Pn:=/braceleftbig
+x1+x2+···+xn:xi∈Pi/bracerightbig
+.
+IfP=P1+P2+···+Pnis a Minkowski sum of polytopes, a mixed cellB⊆Pis defined to
+be a Minkowski sum B1+···+Bn, where each Biis a polytope with vertices among those of
+Pi. Amixed subdivision ofPis then a subdivision of Pconsisting of mixed cells. Once again,
+the mixed subdivisions of Pform a poset under refinement, and in this case the minimal
+elements are called finemixed subdivisions.
+Now, if as above P1,P2,...,P nare polytopes in Rd, we again let {e1,...,en}be the
+standard basis of Rnand letιi:Rd→Rd×Rndenote the inclusion x/mapsto→(x,ei). TheCayley
+embedding ofP1,P2,...,P nis defined to be
+Cayley(P1,P2,...,P n) := conv/parenleftbig/uniondisplay
+ιi(Pi)/parenrightbig
+,16 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+the convex hull of all these inclusions. The intersection of Cayley(P1,P2,...,P n) with the
+affine subspace Rd×{/summationtext1
+nei}equals the (scaled) Minkowski sum (1
+n/summationtextPi)×{/summationtext1
+nei}. This
+way each polyhedral subdivision of the Cayley embedding Cay ley(P1,P2,...,P n) induces a
+polyhedralsubdivisionoftheMinkowskisum/summationtextPi. TheCayley tricksaysthatthiscorrespon-
+dence gives a poset isomorphism between the polyhedral subd ivisions of Cayley( P1,...,P n)
+and the mixed subdivisions of/summationtextPi, with regular subdivisions corresponding to coherent
+mixed subdivisions.
+Herewehave P1=P2=···=Pn= ∆d−1, sothatCayley(∆ d−1,...,∆d−1) = ∆n−1×∆d−1,
+and hence the Cayley trick provides a bijection between the s ubdivisions of the product
+∆n−1×∆d−1andmixedsubdivisionsofthedilated simplex n∆d−1. Inthiscase, amixedcell τ
+in a mixed subdivision of n∆d−1is given byτ=/summationtextn
+j=1∆Ijwhere each ∆ Ij= conv{ei:i∈Ij}
+is a face of ∆ d=1withIj⊆[d]. Hence we can represent the cell as simply τ=I1+I2+···+In.
+We have seen that (regular) subdivisions of ∆ n−1×∆d−1are closely related to the com-
+binatorics of tropical complexes, and the aim of this sectio n is to interpret our results from
+the previous sections in terms of mixed subdivisions of dila ted simplices. For this it will be
+convenient to have a notion of coarse type of a cell defined pur ely in terms of the mixed
+subdivision. Of course these are obtained by applying the Ca yley trick to the coarse types
+introduced above.
+Definition 4.1. Supposeτ=I1+I2+···+Inis ani-dimensional mixed cell in a mixed
+subdivision of n∆d−1, with each Ij⊆[d]. Thecoarse type t(τ)∈Ndis a vector whose i-th
+coordinate is given by # {Ij:i∈Ij}, the number of occurrences of iin the decomposition
+ofτ. Thedual coarse type d(τ)∈Nnis a vector whose i-th coordinate is given by # Ii, the
+number of elements of Ii.
+The following can be seen as an interpretation of Theorem 2.9 in the context of mixed
+subdivisions. Here we provide a complete proof here which do es appeal to tropical geometry
+and hence applies to the more general situation of not necess arily regular subdivisions.
+Proposition 4.2. In any fine mixed subdivision Σofn∆d−1, the set of 0-dimensional cells
+are precisely the lattice points n∆d−1∩Zd, and the collection of coarse types of these cells are
+in bijection with the set of compositions of nintodparts.
+Proof.As above, we denote by ∆ d−1= conv{ei:i∈[d]} ⊂Rdthe standard simplex. By
+definition, each cell τof the subdivision Σ is of the form τ=/summationtextn
+j=1∆IjwhereIj⊆[d] and
+∆Ij= conv{ei:i∈Ij}is a face of ∆ d−1. Since we assumed the mixed subdivision Σ to be
+fine this yields dim τ=/summationtext
+jdim∆Ij=/summationtext
+j(|Ij|−1).
+Any0-dimensional cell ofΣisoftheform {ei1}+···+{ein}, where1 ≤ij≤d. Anysuchcell
+is a lattice point in n∆d−1∩Zd, and the collection of coarse types of such cells correspond to
+the set of compositions of nintodparts. In order to show that all such lattice points arise as
+0-dimensional cells, let t∈n∆d−1∩Zdand letτ=/summationtextn
+j=1∆Ijbe the inclusion minimal mixed
+cell containing t. By [23, Prop. 14.12] every lattice point of τis of the form/summationtextn
+j=1{eij}with
+ij∈Ijfor allj. However, as the mixed subdivision is fine, τis combinatorially isomorphic
+to the product/producttextn
+j=1∆Ijand hence every sum of vertices is a vertex. Therefore, τis a vertex
+of the mixed subdivision Σ. /square
+4.2.Resolutions supported by mixed subdivisions. In this section we discuss our re-
+sults regarding cellular resolutions in the context of mixe d subdivisions of dilated simplices.
+Although these results are more or less translations of the a bove via the Cayley trick, we findTROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 17
+004
+013
+022
+031
+040103
+112
+121
+130202
+211
+220301
+310 4001
+2 3
+4
+Figure 4. Mixed subdivision of 4∆ 2corresponding to Example 2.3.
+it useful to make this transition explicit. It seems that the mixed subdivision picture allows
+for more natural statements whereas the tropical convexity picture allows for more natural
+proofs. We refer the reader back to Definition 4.1 for the defin ition of coarse type of a mixed
+cell.
+Corollary 4.3. LetΣbe any regular mixed subdivision of n∆d−1. Consider Σto be a labeled
+polytopal complex with each face σlabeled by the least common multiple of the vertices that
+it contains. Then for any field /CZ, the complex ΣAsupports a minimal cellular resolution of
+the coarse type ideal It(A)=/a\}bracketle{txt(p):p∈Td−1/a\}bracketri}htin /CZ[x1,...,x d].
+Proof.The fact that Σ Ais a labeled complex follows from Proposition 3.4. As a poset ΣA
+is isomorphic to the corresponding regular subdivision of ∆ n−1×∆d−1via the Cayley trick.
+By [7, Lemma 22] this regular subdivision is dual to the cell d ecomposition CAofTd−1. In
+this way the labeling of Σ Aturns into the colabeling of BAby coarse types. Theorem 3.6
+now establishes the claim. /square
+It is now straightforward to derive the mixed subdivision re sult corresponding to Corol-
+lary 3.7.
+Corollary 4.4. LetΣbe anyfinemixed subdivision of n∆d−1. Then ΣA, as a labeled
+polyhedral complex, supports a minimal cellular resolutio n of/a\}bracketle{tx1,...,x d/a\}bracketri}htn.
+Proof.This follows from Corollary 4.3 and Proposition 4.2. /square
+We note that mixed subdivisions of dilated simplices have be en used to obtain cellular
+resolutions in previous work. In [9] the authors study appli cations to resolutions of edge
+ideals of graphs and hypergraphs. In [29], Sinefakopoulos s hows that the mixed subdivision
+ofn∆d−1corresponding to the staircase triangulation of ∆ n−1×∆d−1supports a cellular
+resolution of /a\}bracketle{tx1,...,x d/a\}bracketri}htn, although his construction is much less explicit. We discus s this
+example further in Section 5.
+Remark 4.5. It seems to be a challenging task to characterize which monom ial ideals
+I⊂ /CZ[x1,...,x d] arise asIt(A)for some arrangement Aof tropical hyperplanes in Td−1.
+Some necessary conditions are obvious, e.g., Ishould be homogeneous of some degree n, and
+thatxd
+ishould be contained in Ifor alli. In particular, this means that the coarse type ideal
+is necessarily Artinian.18 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+4.3.Alexander duality of ideals and resolutions. We have seen how the bounded sub-
+complexes of tropical hyperplane arrangements are related to mixed subdivisions of dilated
+simplices in terms of a geometric duality. This duality exte nds to the algebraic level of
+our resolutions in the context of Alexander duality of resolutions . For this we will need the
+following notion of the Alexander dual of a (not necessarily square-free) monomial ideal.
+Definition 4.6. SupposeIis a monomial ideal in the polynomial ring /CZ[x1,...,x d] and let
+a∈Nd. TheAlexander dual of Iwith respect to ais given by the intersection
+I[a]=/intersectiondisplay/braceleftig
+ma\b:xbis a minimal generator of I/bracerightig
+,
+wherea\bdenotes the vector whose i-th coordinate is ai+1−biifbi≥1, and is 0 if bi= 0.
+Here we borrow the notation ma:=/a\}bracketle{txai
+i:ai≥1/a\}bracketri}ht.
+Note that if Iis a square-free monomial ideal (and hence the Stanley-Reis ner ring of some
+simplicial complex) and a= /BDis taken to be the all-ones vector, then this notion recovers the
+morefamiliarnotionofAlexanderduality ofsimplicial com plexes. Themainresultconcerning
+duality of resolutions, relevant for us, is the following ([ 21, Theorem 5.37]).
+Theorem 4.7. SupposeIis a monomial ideal in degrees preceding some a∈Ndand suppose
+FP
+•is a minimal cellular resolution of S/(I+ma+ /BD)such that all face labels on Pprecede
+a+ /BD. LetQdenote the labeled complex with the same underlying complex Pbut with labels
+tF=a+ /BD−tF. ThenFQ≤a
+•is a minimal cocellular resolution of I[a].
+Applying this theorem we obtain the following dual resoluti on of the coarse cotype ideal.
+Forσa face of a mixed subdivision Σ, the coarse cotype ofσis defined to be n /BD−t(σ), where
+t(σ) is the coarse type of σdefined in Definition 4.1.
+Proposition 4.8. Given any arrangement Aofntropical hyperplanes in Td−1, letΣAde-
+note the associated mixed subdivision of n∆d−1with labels given by coarse cotype. Then ΣA
+supports a minimal cocellular resolution of the coarse coty pe idealIt(A)in /CZ[x1,...,x d]. Con-
+sequently the associated tropical complex BA, with labels given by coarse cotype, supports a
+minimal cellularresolution of It(A).
+Proof.We apply Theorem 4.7 with a:= (n−1) /BD, and withIdefined to bethe ideal generated
+by all monomials fofIt(A)withf≤a(in other words, throw out all generators of the form
+xn
+ifor 1≤i≤d, all of which show up in It(A)regardless of the arrangement A). We then
+have from Corollary 4.3 that Σ Asupports a minimal cellular resolution of It(A)=I+ma+1.
+The conditions of Theorem 4.7 are met and we conclude that ( ΣA)≤asupports a minimal
+cocellular resolution of I[a].
+We next determine I[a], the Alexander dual of Iwith respect to a= (n−1) /BD. In [21] it is
+shown that if b≤athenxblies outside Iif and only if xa−blies insideI[a]. Hence to find a
+set of generators for I[a]it suffices to determine the maximal monomials which lie outsi deI.
+But these monomials correspond to the minimal cotypes that a rise in the complex CA, and
+these are given by the collection of monomials t(x), forxa 0-dimensional cell in CA. Hence
+we conclude that I[a]=It(A).
+For the second part of the claim, we note that a face σ∈ΣAwith label bsatisfiesb≤aif
+and only if n /BD−t(σ)≤(n−1) /BDfor the coarse type label t(σ) in ΣA. But this occurs exactly
+whent(σ)≥ /BD, which happens if and only if the face σis not contained in the boundary of
+ΣA. Hence the cocellular resolution of I[a]that we obtain, supported on ( ΣA)≤a, is given
+by the relative cocellular complex of (Σ A,∂ΣA). By duality, the relative cochain complexTROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 19
+C∗(ΣA,∂ΣA) is isomorphic to the chain complex C∗(BA) of the tropical complex BA(the
+bounded subcomplex of the decomposition of Td−1induced by A). Hence BA, with labels
+given by coarse cotype, supports a minimal cellularresolution of the ideal I[a]=It(A)./square
+5.Examples
+In this section we discuss some examples of our construction s and results. We begin with
+a family of generic arrangements which correspond to well-k nown objects in the theory of
+subdivisions (staircase triangulations) and tropical con vexity (cyclic polytopes). In the con-
+text of the associated coarse type ideal we recover a constru ction of a cellular resolution first
+described by Sinefakopoulos in [29]. We then discuss a famil y of non-generic examples which
+arise from tropical hypersimplices; in this we are able to ex plicitly describe the generators of
+the coarse type ideal.
+5.1.The staircase triangulation. One particularly well-behaved fine mixed subdivision
+ofn∆d−1comes from applying the Cayley trick (discussed in Section 4 .1 above) to the so-
+calledstaircase triangulation of ∆ n−1×∆d−1. The origins of the staircase triangulation date
+back at least to work of Eilenberg and Zilber in [10], where th e authors introduce algebraic
+operations which they call ‘shuffles’ to study the homology gr oups of products of spaces. We
+wish to recall the construction here in language suitable fo r our purposes.
+As above, we use {e1,...,ed}to denote the vertices of the simplex ∆ d−1and consider fine
+mixed cells of the following kind. Given a sequence of intege rs (b1,b2,...,bn+1) satisfying
+1 =b1≤b2≤ ··· ≤bn≤bn+1=dwe letBi:={ebi,ebi+1,...,ebi+1}for 1≤i≤n, and use
+(b1,b2,...,bn+1) to denote the corresponding (fine) mixed cell B1+B2+···+Bnofn∆d−1.
+Note that each such sequence can be thought of as a ‘staircase ’ of length nand height d,
+where at the i-th step one should climb a step of height bi+1−bi.
+We claim that the collection of fine mixed cells correspondin g to (b1,b2,...,bn+1) for
+1 =b1≤b2≤ ··· ≤bn≤bn+1=dforms a fine mixed subdivision of the complex n∆d−1. To
+see this, we once again employ the Cayley trick and consider t he (proposed) triangulation of
+the product ∆ n−1×∆d−1. As discussed above, the vertices of ∆ n−1×∆d−1are pairs (ei,ej),
+where 1≤i≤nand 1≤j≤d. A simplex in any triangulation of ∆ n−1×∆d−1corresponds
+to a tree in the complete bipartite graph Kn,d, where(vi,wj) is an edge of the graph whenever
+(ei,ej) is a vertex of the simplex. The corresponding mixed cell B1+···+Bnofn∆d−1has
+as thei-th summand the face {ej:wj∈N(vi)}, given by all vertices that are adjacent to vi.
+Note that the coarse type of the mixed cell is given by the degr ee sequence of the dvertices
+in the second vertex set partition.
+Working with ∆ n−1×∆d−1has the advantage that one can readily verify if a collection
+of simplices does, in fact, give a triangulation of the space , in terms of the corresponding
+subgraphs. We omit the details here, but point out that the st aircase triangulation also
+arises from the so-called ‘cyclic arrangement’ of ntropical hyperplanes in Td−1as discussed
+in [4]. We call the associated fine mixed subdivision of n∆d−1thestaircase mixed subdivision .
+As with any fine mixed subdivision, the 0-dimensional cells o f the staircase mixed subdivision
+ofn∆n−1are labeled by the monomial generators of /a\}bracketle{tx1,...,x d/a\}bracketri}htn, and from Corollary 4.4 we
+know that this labeled complex supports a minimal cellular r esolution of /a\}bracketle{tx1,...,x d/a\}bracketri}htn. In
+fact, the case of n=d= 3 is hinted at in [21, Example 2.20].
+Remark 5.1. In [29], Sinefakopoulos constructs a labeled polyhedral co mplex which he
+callsPn(x1,...,x d), and shows that it supports the minimal resolution of /a\}bracketle{tx1,...,x d/a\}bracketri}htn. The
+complexPn(x1,...,x d) is constructed inductively, with each Pn(xk+1,...,x d) a subcomplex20 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+ofPn(xk,...,x d) for allk<d. In [29] it is shown that Pn(x1,...,x d) can in fact berealized as
+a subdivisionof thedilated simplex n∆d−1, and herewe claim that this complex is isomorphic
+(as a labeled complex) to the staircase mixed subdivision.
+Mimicking the construction of Pn(x1,...,x d) from [29], we proceed by induction on n. If
+n= 1, bothP1(x1,...,x d) and the staircase mixed subdivision correspond to the stan dard
+(d−1)-dimensional simplex with vertex labels {x1,...,x d}, denoted ∆ d−1(x1,...,x d). We
+assume that for n≥1, the complex Pn(x1,...,x d) is isomorphic as a labeled complex to the
+staircase subdivision of n∆d−1, with this isomorphism restricting to an isomorphism on eac h
+Pn(xk,...,x d). In [29] the inductive step of the construction is obtained as
+Pn+1(x1,...,x d) :=C1∪···∪Cd,
+where each Ckis defined as
+Ck:= ∆k−1(x1,...,x k)×Pn(xk,...xd).
+For anyk, we claim that the complex Ckcorrespondsto thecomplex composed of the set of
+fine mixed cells ( b1,b2,...,bn+1) withb2=k. This would establish the desired isomorphism
+since the set of fine mixed cells in the staircase mixed subdiv ision correspond to the all such
+sequences with 1 ≤b2≤d. To prove the claim, we note that for any labeled complex Pthe
+product ∆ k−1(x1,...,x k)×Pis isomorphic to ∆ k−1(x1,...,x k) +P, assuming that these
+complexes lie in affinely independent subspaces. Furthermor e, the vertices of the product
+complex are labeled by sums of vertices from each summand. No w, by induction we have
+thateachPn(xk,...,x d)isisomorphictothecorrespondingsubcomplexofthestair casemixed
+subdivision of n∆d−1. In particular, the set of mixed cells correspond to the set o f sequences
+(b1,b2,...,bn+1) withk=b1≤b2··· ≤bn≤bn+1=d. The facets of Ckare obtained by
+taking the Minkowski sum of the simplex ∆ k−1(x1,...,x k) with these mixed cells, and hence
+Ckis isomorphic to thecomplex composed of all finemixed cells ( b1,b2,...,bn+1) withb2=k.
+Hence our constructions recover the result from [29] (with a more explicit description of
+the underlying polyhedral complex). Once again, we wish to e mphasize that from Corollary
+4.4 we know that in fact any(regular) fine mixed subdivision of n∆d−1supports a minimal
+cellular resolution of the homogeneous ideal /a\}bracketle{tx1,...,x d/a\}bracketri}htn.
+5.2.Tropical hypersimplices. The hypersimplex ∆( k,n) is an ordinary convex polytope
+which, e.g., naturally turns up in the study of tropical Gras smannians of tropical k-planes in
+Tn−1. Its tropical counterpart, the tropical hypersimplex ∆trop(k,n) is defined as the tropical
+convex hull of all 0 /1-vectors of length nwith exactly kzeros. Notice that we have the strict
+inclusions
+∆trop(1,n)/supersetnoteql∆trop(2,n)/supersetnoteql···/supersetnoteql∆trop(n−1,n)
+of subsets of Tn−1. We want to determine the coarse type ideal corresponding to the configu-
+ration of/parenleftbign
+k/parenrightbig
+points in Tn−1given by the tropical vertices of ∆trop(k,n). Thengenerators of
+∆trop(1,n) are in general position. Hence the coarse type ideal is the h omogeneous maximal
+ideal/a\}bracketle{tx1,x2,...,x n/a\}bracketri}htin this case. The second tropical hypersimplex ∆trop(2,n) is contained
+in the min-tropical hyperplane with the origin as its apex. I n particular, ∆trop(2,n), seen as
+a polytopal complex in Rn−1=Tn−1, is of dimension n−2. This implies that all maximal
+cells in the type decomposition of Tn−1induced by the tropical vertices of ∆trop(k,n) are
+unbounded if 2 ≤k < n. Equivalently, the coarse type ideal only has minimal gener ators
+with the property that at least one variable is missing (that is, the exponent of this variable
+is zero). The symmetric group Sym( n) acts on the set of tropical vertices of ∆trop(k,n), and
+hence it also acts on the maximal cells of the type decomposit ion.TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 21
+Proposition 5.2. Let2≤k<n. Then up to the Sym(n)action the coarse types of the max-
+imal cells in the type decomposition of Tn−1induced by the/parenleftbign
+k/parenrightbig
+tropical vertices of ∆trop(k,n)
+are given by
+(/parenleftbiggn−α
+k/parenrightbigg
++/parenleftbiggn−1
+k−1/parenrightbigg
+,/parenleftbiggn−2
+k−1/parenrightbigg
+,...,/parenleftbiggn−α
+k−1/parenrightbigg
+,0,...,0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+n−α)
+where1≤α≤n−k+1.
+This result has been obtained by Katja Kulas, and a complete p roof will appear in [15].
+Hereweonlysketchtheargument. Let Adenotethearrangementofmax-tropicalhyperplanes
+induced by the tropical vertices of ∆trop(k,n). The maximal dimension of a bounded cell
+or, equivalently, the tropical rank of the matrix whose colu mns are the tropical vertices of
+∆trop(k,n), equalsn−k[6, Proposition 7.2]. All cells in the arrangement A, bounded or
+not, are convex polyhedra in R/parenleftbign
+k/parenrightbig
+−1which are pointed, that is, they do not contain any
+affine line. This implies that each cell must contain a bounded cell as a face. Let Cbe
+some maximal cell in the arrangement A. From now on we assume that k >1 whenceC
+is necessarily unbounded. Let α−1 be the maximal dimension of a bounded cell in the
+boundary of C; and we call αtheclassofC. From the bound on the tropical rank it follows
+that 1≤α≤n−k+ 1. All these values for αactually occur. It turns out that Sym( n)
+acts transitively on the set of maximal cells of class α. The coarse type of one representative
+of each class is given in Proposition 5.2. Due to Corollary 3. 5 this yields generators for the
+corresponding coarse type ideal.
+6.Face counting and incidence structure
+of the tropical complex
+In Section 3 we saw how the polyhedral complexes CAandBAassociated to an arrangement
+Agave rise to resolutions of the coarse type ideal It(A)and the fine cotype ideal IT(A),
+respectively. The minimality of our resolution also leads to some important implications
+regardingthe combinatorics of CAandBAthemselves. Inthis section we discussface numbers
+of tropical complexes, as well as an algorithm for determini ng the facial structure of BAgiven
+the arrangement A=A(V). The latter generalizes a result of [4], where a similar alg orithm
+for the case of sufficiently generic arrangements was provide d.
+6.1.Counting faces. As a first application we point out that the f-vector of CAcan be
+determined fromthe Z-graded (‘coarse’) Betti numbersof It(A). We noted inProposition 2.12
+that from the coarse type it is possible to distinguish bound ed from unbounded cells in CA.
+Thus, we can also recover the numerical behavior of the bound ed complex BA.
+Corollary 6.1. LetA=A(V)be a tropical hyperplane arrangement in Td−1and letIt(A)
+be its coarse type ideal. Then the number of cells in CAof dimension kis
+fk(CA) =βd−1−k(It(A)) =/summationdisplay
+b∈Zdβd−1−k,b(It(A)).
+The number of bounded k-cells inCAis given by as the sum of Betti numbers βd−1−k,b(It(A))
+for which b>0.
+Note that for the k-cells we need to consider the ( d−1−k)-th Betti numbers. This is due
+to the fact that CAsupports a cocellular resolution.22 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+Furthermore, we can use the uniqueness of minimal resolutions to derive further results
+regarding face numbers of arrangements. For this, suppose Ais a sufficiently generic arrange-
+ment ofntropical hyperplanes in Td−1, and let CAdenote the induced polyhedral subdivision
+ofTd−1determined by type. In Corollary 3.7 we saw that CA, with labels given by coarse
+type, supports a minimal cocellular resolution of the ideal /a\}bracketle{tx1,...,x d/a\}bracketri}htn. Any two such reso-
+lutions are isomorphic as chain complexes, and in particula r the finely graded Betti numbers
+βi,σdo not depend on the resolution. By construction, βi,σis precisely the number of cells
+inCAwith monomial label σ. But the monomial labels are given by the coarse types, and
+hence we obtain the following.
+Corollary 6.2. LetAbe a sufficiently generic arrangement of nhyperplanes in Td−1. For
+every0≤i≤d−1the collection of coarse types t(CT)fordimCT=i, counted with
+multiplicities, is independent of the arrangement.
+Putting the above result in perspective with the second stat ement of Corollary 6.1, this
+proves the following result without appealing to the equide composability of the product of
+simplices.
+Corollary 6.3. The number of cells of CAfor a tropical hyperplane arrangement Ain gen-
+eral position is independent of the choice of hyperplanes. M ore precisely, the number of
+k-dimensional cells induced by the arrangement of ntropical hyperplanes in Td−1equals
+fk(CA) =k/summationdisplay
+ℓ=0/parenleftbiggn+d−2−ℓ
+n−1/parenrightbigg/parenleftbiggd−1−ℓ
+d−1−k/parenrightbigg
+.
+Proof.By Corollaries 6.1 and 3.7 wehave fi(CA) =βd−1−i(/a\}bracketle{tx1,...,x n/a\}bracketri}htn). TheBetti numbers
+of the power of the homogeneous maximal ideal are well-known . For example, they can be
+determined as follows.
+The ideal mn=/a\}bracketle{tx1,...,x d/a\}bracketri}htnisstrongly stable , that is,xi
+xjxb∈mnfor every xbmonomial
+divisible by xjandi<j. In particular, mnis Borel-fixed, and hence the Betti numbers are
+given by [21, Thm. 2.18], and we have
+βi(mn) =/summationdisplay
+a∈Nd,|a|=n/parenleftbiggmax(xa)−1
+i/parenrightbigg
+where max( xa) = max{i:ai>0}. Now if max( xa) =ℓ, then
+a= (a1,a2,...,aℓ+1,0,...,0)
+witha1,...,a ℓ≥0 and/summationtext
+iai=n−1. Hence, the number of generators xawith max( xa) =ℓ
+is the number of monomials in ℓvariables of total degree n−1. This yields
+βi(mn) =d/summationdisplay
+ℓ=1/parenleftbiggn−2+ℓ
+n−1/parenrightbigg/parenleftbiggℓ−1
+i/parenrightbigg
+Now substitute iin the formula with d−1−k. /square
+In the context of mixed subdivisions of dilated simplices, t he previous two corollaries yield
+the following.
+Corollary 6.4. In any regular fine mixed subdivision Σofn∆d−1the collection of coarse
+types, counted with multiplicities, is independent of the s ubdivision. In particular, fi(Σ), theTROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 23
+number ofi-dimensional faces of Σis independent of the subdivision, and is given by
+fi(Σ) =d/summationdisplay
+ℓ=1/parenleftbiggn−2+ℓ
+n−1/parenrightbigg/parenleftbiggℓ−1
+i/parenrightbigg
+.
+Remark 6.5. For the case of vertices, Corollary 6.4 was already establis hed in 4.2, where
+the it was shown that the 0-dimensional cells in any fine mixed subdivision correspond to the
+lattice points of n∆d−1. The case of facets can also be proven directly using the foll owing
+combination ofmixedvolumecalculations andtheswitchdua lityoftropicalcomplexes. Given
+a fine mixed subdivision Σ of n∆d−1we consider the corresponding fine mixed subdivision Σ∗
+ofd∆n−1induced by applying the Cayley trick and switching the facto rs in ∆ n−1×∆d−1. For
+nonnegative real numbers λ1,...,λ d, we have that the linear functional given by the volume
+Vol(λ1∆n−1+···+λd∆n−1) can be expressed as a homogeneous polynomial of degree n−1
+in the variables λ1,...,λ d. The coefficient of the monomial λi1
+1λi2
+2···λid
+dis called a mixed
+volume, and for any fine mixed subdivision of d∆n−1it is equal to the sum of the volumes
+of the mixed cells of the corresponding dualcoarse type. In our case all fine mixed cells of a
+certain dual coarse type have the same volume, since the rele vant data is given by the number
+of vertices in each factor, regardless of the labels on the ve rtices. This implies that in any fine
+mixed subdivision the number of fine mixed cells with a certai n dual coarse type is always
+the same. But the collection of dual coarse types for this fine mixed subdivision coincides
+with collection of coarsetypes for the given fine mixed subdivision of n∆d−1, proving our
+claim.
+In fact (for k= 0), each coarse type shows up at most once and the collection of coarse
+types can be seen to coincide with the monomial generators of the ideal
+x1x2···xd/a\}bracketle{tx1,x2,...,x d/a\}bracketri}htn−1.
+This follows from the fact that in this case the volume polyno mial of the dual mixed subdivi-
+sion is given by ( λ1+···+λd)n−1and the volume of a mixed cell of a particular dual coarse
+type is exactly the coefficient of the corresponding monomial . We do not know of a similar
+mixed volume interpretation of the other face numbers.
+6.2.Incidence structure of the bounded complex from the fine cotype ideal. In [4]
+Block and Yu develop an algorithm to compute the bounded comp lex of a generic tropical
+hyperplane arrangement A=A(V) that draws from computational commutative algebra.
+There it is shown that the bounded complex supports a minimal cellular resolution of a
+monomial ideal which we have called the fine cotype ideal IT(A)associated to A. It turns
+out that this ideal is an initial ideal of the toric ideal In,dfor the vertices of ∆ n−1×∆d−1.
+The results in [4] rely on the genericity of the arrangement i n two ways: i) the fact that the
+bounded complex supports a resolution of the fine cotype idea l is proved by appealing to the
+polyhedral detour to tropical convexity presented in [7], and ii) the generici ty is needed to
+guarantee that the initial ideal is indeed monomial.
+In this section we extend their algorithm to the case of non-g eneric hyperplane arrange-
+ments. We bypass the two mentioned dependences on genericit y as follows. In Section 3 we
+have already shown that the bounded complex resolves the fine cotype ideal for an arbitrary
+arrangement, using first principles in tropical convexity. As for ii), we next show that the
+fine cotype ideal is the maximal monomial ideal contained in t he initial ideal associated to
+the weights V. In terms of polyhedral geometry, this corresponds to the pa ssage from the
+polyhedral subdivision Σ Vof ∆n−1×∆d−1induced by Vto thecrosscut complex .24 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+The results of the previous sections cannot directly be turn ed into a practical algorithm
+as the fine cotype ideal can only be determined aftercomputing the bounded complex or, at
+least, the vertices of CA. In the case of a generic arrangement of tropical hyperplane s this
+problem is resolved in [4] as follows. The ideal
+In,d=/a\}bracketle{txikxjl−xilxjk:i,j∈[n],k,l∈[d]/a\}bracketri}ht
+of 2×2-minors of a general n×d-matrix in /CZ[x11,...,x nd] is the toric ideal associated to the
+vertices of the ordinary lattice polytope ∆ n−1×∆d−1. The initial ideal in V(In,d) with respect
+to the weights Vis a squarefree monomial ideal and, by a celebrated result of Sturmfels [31,
+Thm. 8.3], equal to the Stanley-Reisner ideal IΣof the triangulation Σ of ∆ n−1×∆d−1
+induced by V; see Section 4. The face poset of the bounded complex BAis anti-isomorphic
+to the subposet of interior cells of Σ. In short, there is a bijection between i-cells ofCAand
+(n+d−2−i)-cells of Σ which lie in the interior of ∆ n−1×∆d−1. The bijection takes a cell
+C◦
+Tof fine type Tto the cell of Σ with vertices ( ei,ej) forTij= 1. Finally, the Alexander
+dual ofIΣis the monomial ideal generated by the fine cotypes of the vert ices ofCA.
+If the placement Vof the hyperplanes is not generic, then the induced subdivis ion Σ is
+not a triangulation. However, the above bijection is unaffect ed and, in particular, the fine
+type of a vertex of CAcan be read off the corresponding facet of Σ. We define the crosscut
+complex CrossCut(Σ) ⊆2[n]×[d]to be the unique simplicial complex with the same vertices-
+in-facets incidences as the polyhedral complex Σ. The cross cut complex is a standard notion
+in combinatorial topology and can be defined in more generali ty (see Bj¨ orner [3, pp. 1850ff]).
+The following observation is immediate from the definitions .
+Proposition 6.6. ForVan ordered sequence of npoints in Td−1letA=A(V)be the
+corresponding tropical arrangement and let Σbe the regular subdivision of ∆n−1×∆d−1
+induced byV. Then the fine cotype ideal IT(A)is Alexander dual to the Stanley-Reisner ideal
+ofCrossCut(Σ) .
+The crosscut complex encodes the information of which colle ctions of vertices lie in a
+common face. Hence, the crosscut complex is a purely combina torial object and does not see
+the affine structure of the underlying polyhedral complex.
+Algebraically, the initial ideal in V(In,d) is a coherent A-graded ideal in the sense of [31,
+Ch. 10] and encodes the corresponding polyhedral subdivisi on Σ induced by V(see [31,
+Thm. 10.10]). The ideal in V(I) of a toric ideal Iis generated by monomials and binomials.
+Intuitively, the binomials encode the affine structure withi n cells of Σ, that is, the affine
+dependencies, whilethemonomial generatorsencodetheSta nley-Reisnerdataforthecrosscut
+complex. For an arbitrary ideal J, denote by M(J) the largest monomial ideal contained in
+J.
+Lemma 6.7. LetIAbe the toric ideal for A∈Nd×nand forω∈RnletJ= inω(IA)andΣ
+the regular subdivision of Ainduced by ω. Then the radical of M(J)is the Stanley-Reisner
+ideal of the crosscut complex of Σω, that is
+IRad(M(J))=ICrossCut(Σ).
+Proof.By Theorem 10.10 of [31], the ideal Jis the intersection of ideals Jσindexed by the
+facesσ∈Σ andJσis torus isomorphic to Iσ=IA+/a\}bracketle{txi:i/\e}atio\slash∈σ/a\}bracketri}ht. Hence, we have
+M(J) =/intersectiondisplay
+σ∈ΣM(Iσ).TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 25
+Under the projection map /CZ[x1,...,x n]→ /CZ[xi:i∈σ] that takes xi/mapsto→0 fori/\e}atio\slash∈σ,Iσis
+isomorphic to the toric ideal corresponding to the columns o fAindexed by σ⊆[n]. Hence,
+a monomial xais contained in Iσif and only if supp( xa)/\e}atio\slash⊆σ. Therefore M(Iσ) =/a\}bracketle{txi:i/\e}atio\slash∈σ/a\}bracketri}ht
+and forτ⊆[n] we have that xτ∈Rad(M(J)) if and only if τis not contained in any cell
+of Σ. /square
+A special case of this lemma appears in [25, Lem. 4.5.4]. Sinc e the toric ideal In,dis
+unimodular, the ideal M(inV(In,d)) is automatically squarefree and we obtain the following
+corollary.
+Corollary 6.8. LetA=A(V)be a tropical hyperplane arrangement and let J= inV(In,d)
+the initial ideal of In,dfor the weights V. Then the Alexander dual of the squarefree monomial
+idealM(J)is the fine cotype ideal of A.
+In the case that Ais generic, the ideal M(J) coincides with the initial ideal in V(In,d) and
+hence recovers the main result of [4]. Moreover, it entails t he modification of the algorithm in
+[4] by replacing the ideal in V(In,d). We describe our Algorithm A below. For an overview of
+other algorithms to produce the same output via techniques f rom polyhedral combinatorics
+and algorithmic geometry see [13].
+input: matrixV∈Rn×d
+output: face poset of BA(V)
+calculate the initial ideal J= inV(In,d) 1
+calculateM(J) =/a\}bracketle{txa:xa∈J/a\}bracketri}ht 2
+calculate the Alexander dual M(J)∗3
+find a minimal fine graded resolution to determine the face pos et ofBA(V) 4
+Algorithm A : Computing the face poset of the tropical complex
+An algorithm for calculating M(·) for a toric ideal is given in [25, Algo. 4.4.2] in terms
+of elimination theory. Computing the Alexander dual of a fini te simplicial complex can be
+accomplished via an algorithm of Lawler [17]. In [16] La Scal a and Stillman give an overview
+of methods to compute minimal resolutions. To illustrate th e results in this section, we show
+an example computed with the computer algebra system Macaulay2 [12].
+Example 6.9. Consider the points v1= (0,1,1),v2= (0,0,1), andv3= (0,1,0) inT2. The
+induced type decomposition is shown in Figure 5. We have
+I3,3=/a\}bracketle{tx11x22−x12x21, x11x33−x13x31, x12x31−x11x32,
+x12x33−x13x32, x13x21−x11x23, x13x22−x12x23,
+x21x33−x23x31, x22x31−x21x32, x22x33−x23x32/a\}bracketri}ht,
+where the initial forms of the generators with respect to the weight matrix V=/parenleftigv1v2v3/parenrightig
+∈R3×3
+are underlined; these generate the toric ideal J. Its maximal monomial subideal equals
+M(J) =/a\}bracketle{tx12x21, x12x23, x13x31, x23x31, x13x32, x21x32, x23x32/a\}bracketri}ht,
+and this is squarefree. Its Alexander dual is
+M(J)∗=/a\}bracketle{tx13x21x23, x12x13x23x32, x12x31x32, x21x23x31x32/a\}bracketri}ht.26 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+-1 0 1 2-1012
+v1 v2
+v3
+(123,−,−) (13,2,−) (−,123,−)(12,−,3) (1,2,3)(−,12,3)(−,−,123) ( −,2,13)
+Figure 5. Non-generic arrangement in T2with labeling by fine type.
+These four generators of M(J)∗encode the fine cotypes of the points v3, (0,0,0),v2, andv1,
+respectively. Setting S= /CZ[x11,x12,...,x 33] we obtain the minimal free resolution
+0→Sφ3−→S4φ2−→S4φ1−→I→0,
+where the non-trivial differentials φiare given by the matrices
+φ1=/parenleftbig
+x13x21x23x12x31x32x21x23x31x32x12x13x23x32/parenrightbig
+φ2=
+0−x31x32−x12x320
+−x21x230 0 −x13x23
+x12x13 0 0
+0 0 x21x31
+
+φ3=
+−x13
+x12
+−x31
+x21
+.
+These matrices are to be multiplied to column vectors from th e left. The non-zero finely
+graded Betti numbers are
+β0,(12,13,23,32)=β0,(12,31,32)=β0,(21,23,31,32)=β0,(13,21,23)= 1
+β1,(12,13,23,31,32)=β1,(12,21,23,31,32)=β1,(13,21,23,31,32)=β1,(12,13,21,23,32)= 1
+β2,(12,13,21,23,31,32)= 1,
+where, for example, (12 ,13,23,32) is the squarefree monomial x12x13x23x32corresponding
+to the point (0 ,0,0). Note that we are resolving the ideal M(J)∗rather than the quotient
+S/M(J)∗(which would yield a shift of +1 in the first coordinate of each Betti number). The
+non-zero coarsely graded Betti numbers are then
+β0,3=β0,4= 2, β1,5= 4, β2,6= 1.TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 27
+7.Further remarks and open questions
+Having constructed cellular resolutions of ideals arising from regular mixed subdivisions of
+dilatedsimplices, anaturalquestiontoaskisiftheassump tionofregularityisreallynecessary.
+The relevant properties of our subdivisions were establish ed by considering them as induced
+by arrangements of tropical hyperplanes, and hence these su bdivisions were always regular.
+However, theconstruction of alabeled complex from an arbit rary mixedsubdivisionof n∆d−1
+still makes sense, and it is an open question (as far as we know ) whether these also support
+cellular resolutions. As a special case, in light of Proposi tion 4.2 we can ask whether anyfine
+mixed subdivision of n∆d−1supports a minimal cellular resolution of /a\}bracketle{tx1,...,x d/a\}bracketri}htn.
+A connection to tropical geometry is provided by the tropical oriented matroids of Ardila
+and Develin from [1]. There the authors introduce an axiomat ic approach to the study of
+(fine) types, with a list of properties which they show are sat isfied by the collection of fine
+types arising from an arrangement of tropical hyperplanes. It is conjectured that all abstract
+oriented matroids are realized by arbitrary subdivisions, and if this were the case we might
+thinkof theideals describedin theprevious paragraphas ‘t ropical oriented matroid ideals’. A
+further task would be to relate the algebraic properties of t hese ideals with the combinatorial
+properties of the underlying matroid, in the spirit of [22].
+As mentioned above, another unresolved question is to chara cterize which monomial ideals
+arise as coarse type ideals for some tropical hyperplane arr angement. We have seen that
+certain necessary properties are easy to deduce but it seems difficult to provide a complete
+classification. Do these ideals fit into some other well-know n class?
+References
+[1]F. Ardila and M. Develin ,Tropical hyperplane arrangements and oriented matroids , Mathematische
+Zeitschrift, (to appear).
+[2]D. Bayer and B. Sturmfels ,Cellular resolutions of monomial modules , J. Reine Angew. Math., 502
+(1998), pp. 123–140.
+[3]A. Bj¨orner,Topological methods , in Handbook of combinatorics, Vol. 1, 2, Elsevier, Amsterd am, 1995,
+pp. 1819–1872.
+[4]F. Block and J. Yu ,Tropical convexity via cellular resolutions , J.Algebraic Combin., 24(2006), pp.103–
+114.
+[5]G. Cohen, S. Gaubert, J.-P. Quadrat, and I. Singer ,Max-plus convex sets and functions , inIdempo-
+tent mathematics and mathematical physics, vol. 377 of Cont emp. Math., Amer. Math. Soc., Providence,
+RI, 2005, pp. 105–129.
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+[12]D. R. Grayson and M. E. Stillman ,Macaulay2, a software system for research in algebraic geom etry.
+Available at http://www.math.uiuc.edu/Macaulay2/.
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+and S. N. Sergeev, eds., vol. 495 of Contemporary Mathematic s, Amer. Math. Soc., 2009.
+[14]M. Joswig and K. Kulas ,Tropical and ordinary convexity combined , Adv. Geometry, (to appear).
+preprint arXiv:0801.4835 .
+[15]K. Kulas ,Combinatorics of Tropical Polytopes , PhD thesis, TU Darmstadt, to appear.28 ANTON DOCHTERMANN, MICHAEL JOSWIG, AND RAMAN SANYAL
+[16]R. La Scala and M. Stillman ,Strategies for computing minimal free resolutions , J. Symbolic Comput.,
+26 (1998), pp. 409–431.
+[17]E. L. Lawler ,Covering problems: Duality relations and a new method of sol ution, SIAM J. Appl. Math.,
+14 (1966), pp. 1115–1132.
+[18]G. L. Litvinov, V. P. Maslov, and G. B. Shpiz ,Idempotent functional analysis. An algebraic approach ,
+Mat. Zametki, 69 (2001), pp. 758–797.
+[19]T. Markwig ,A field of generalised Puiseux series for tropical geometry , 2007. preprint
+arXiv.org:0709.378 .
+[20]E. Miller ,Alexander duality for monomial ideals and their resolution s, Rejecta Math., 1 (2009), pp. 18–
+57.
+[21]E. Miller and B. Sturmfels ,Combinatorial commutative algebra , vol. 227 of Graduate Texts in
+Mathematics, Springer-Verlag, New York, 2005.
+[22]I. Novik, A. Postnikov, and B. Sturmfels ,Syzygies of oriented matroids , Duke Math. J., 111 (2002),
+pp. 287–317.
+[23]A. Postnikov ,Permutohedra, Associahedra, and Beyond , Int Math Res Notices, 2009 (2009), pp. 1026–
+1106.
+[24]J. Richter-Gebert, B. Sturmfels, and T. Theobald ,First steps in tropical geometry , in Idempotent
+mathematics and mathematical physics, vol. 377 of Contemp. Math., Amer. Math. Soc., Providence, RI,
+2005, pp. 289–317.
+[25]M. Saito, B. Sturmfels, and N. Takayama ,Gr¨ obner deformations of hypergeometric differential
+equations , vol. 6 of Algorithms and Computation in Mathematics, Sprin ger-Verlag, Berlin, 2000.
+[26]F. Santos ,The Cayley trick and triangulations of products of simplice s, in Integer points in polyhedra—
+geometry, number theory, algebra, optimization, vol. 374 o f Contemp. Math., Amer. Math. Soc., Provi-
+dence, RI, 2005, pp. 151–177.
+[27]R. Sanyal ,Geometry and combinatorics of fan arrangements , in preparation.
+[28]A. Schrijver ,Combinatorial optimization. Polyhedra and efficiency. Vol. A, vol. 24 of Algorithms and
+Combinatorics, Springer-Verlag, Berlin, 2003. Paths, flow s, matchings, Chapters 1–38.
+[29]A. Sinefakopoulos ,On Borel fixed ideals generated in one degree , J. Algebra, 319 (2008), pp. 2739–2760.
+[30]D. Speyer and B. Sturmfels ,The tropical Grassmannian , Adv. Geom., 4 (2004), pp. 389–411.
+[31]B. Sturmfels ,Gr¨ obner Bases and Convex Polytopes , vol. 8 of University Lecture Series, AMS, Provi-
+dence, RI, 1996.
+Anton Dochtermann, Department of Mathematics, 6188 Kemeny Hall, Dartmouth College,
+Hanover, NH 03755, USA
+E-mail address :anton.dochtermann@gmail.com
+Michael Joswig, Fachbereich Mathematik, TU Darmstadt, Dol ivostr. 15, 64293 Darmstadt,
+Germany
+E-mail address :joswig@mathematik.tu-darmstadt.de
+Raman Sanyal, Department of Mathematics, University of Cal ifornia, Berkeley, California
+94720, USA
+E-mail address :sanyal@math.berkeley.edu
\ No newline at end of file
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+ 1A Raman-type optical frequency comb 
+adiabatically generated in  an enhancement cavity 
+  
+ 
+*M. Katsuragawa1, 2, R. Tanaka1, H. Yokota1, and T. Matsuzawa1 
+ 
+1Department of Applied Physics and Chemistry,  University of El ectro-Communications, 
+1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan 
+ 
+2 PRESTO, Japan Science and Technology Agency,  
+4-1-8 Honcho, Kawaguchi, Saitama, Japan 
+ 
+Abstract 
+We demonstrate efficient generation of a Raman-type optical frequency comb by 
+employing adiabatic Raman excitation in an  enhancement cavity. A broad frequency 
+comb spanning over 130 THz is realized wi th an excitation power reduction exceeding 
+three orders of magnitude compared  with a single-p ass configuration.  
+PACS numbers:  42.65.Dr, 42.25.Kb, 42.60.Da, 42.79.Nv  
+ 
+A method that drives a nonlin ear optical process in a cavity has been employed as a key 
+technique for overcoming the weak field power  that prevents the realization of an 
+efficient nonlinear optical process1. This method was first demonstrated 40 years ago. It 
+realized the second harmonic generation of continuous wave (cw) He-Ne laser 
+radiation2. In the 1980’s, the technique of freque ncy locking to a cavity was introduced, 
+and thus single-frequency second harmonic generation was achieved3. A recent 
+significant study would be the generation of  a high-harmonic comb in the vacuum 
+ultraviolet region that was realized by conf ining a femtosecond mode-locked laser in a 
+high finesse cavity4, 5. Here, we demonstrate adiabatic generation of frequency comb by 
+placing a Raman medium in an enhancemen t cavity and confining nanosecond pulses in 
+it. 
+Figure 1a is a conceptual schematic of this study. A pair of single frequency 
+nanosecond pulses whose difference frequency is near resonant to a Raman transition is 
+produced from a single laser resonator. We co nfine these nanosecond pulses stably in an 
+enhancement cavity filled with a Raman medium, and then adiabatically drive 
+parametric stimulated Raman scattering in the enhancement cavity.  2Fig. 1: Scheme for adiabati c Raman sideband gene ration in an enhancement cavity. a, 
+Conceptual illustration. b, Detailed configuration for Raman sideband generation in an 
+enhancement cavity. The following must satisfy triple resonance conditions, the Ti:sapphire 
+(Ti:S) laser cavity, the enhancement cavity, and optimum two-photon detuning for the 
+adiabatic excitation of Raman coherence. c, Typical example depicting optimal two-photon 
+detuning without the enhancement cavity. The black curve is a resonance profile of the 
+rotational Raman transition (J = 2 <- 0) in parahydrogen. The circles and  triangles indicate 
+the generated intensities of ±3 and ±4 sidebands (normalized), shown as a function of 
+two-photon detuning.  
+The right panel in Fig. 1b shows the scheme  of the adiabatic parametric stimulated 
+Raman scattering. The states of |g> and |u> form a two-level system in which a Raman 
+transition is allowed. We apply a pair of single-frequency pulses, Ω0 and Ω-1, which are 
+near resonant to these two levels, and drive th is Raman system from th e initial state, |g>. 
+When the two-photon detuning δ, is set so that the Raman transition and the difference 
+frequency of the excitation laser pulses , including their spectral broadenings, δνm and 
+ 3δνL, respectively, do not overlap, i.e., δ > 1/2 ( δνm + δνL), the excitation process 
+becomes adiabatic6, 7. When we control δ at the minimum values that satisfy this 
+adiabatic condition, i.e., | δopt| ~ 1/2 ( δνm + δνL), a realistic excitation power  can satisfy 
+Ωgu/δopt >> 1  to produce the theoretic al-limit coherence, | ρgu| = 0.5, where Ωgu and ρgu 
+are the two-photon Rabi frequency and the of f-diagonal term of the density matrix 
+expressing the two-level system of |g> and |u>6, 7.      
+The resulting molecular oscillation with very high coherence,  in turn, deeply 
+modulates the two excitation pulses, and gene rates a high order series of parametric 
+stimulated Raman scatterings (w e call these ‘Raman sidebands’)6-9. A noteworthy 
+feature of this Raman sideband generation is that efficient generation can occur within a 
+unit phase-slip length, since th e produced coherence is extremely high. This results in 
+coaxial generation of the sidebands over a wi de spectral range, w ithout any restriction 
+being imposed by the phase matching condition6-9. 
+Figure 1c shows a case employing a pure rotational transition of J = 2 <- 0  in 
+parahydrogen (density: 4.8 x 1019 cm-3, Raman transition frequency: 10.62380 THz, 
+spectral width: 110 MHz at HWHM)10. The highest coheren ce is produced at a 
+two-photon detuning of + 120 MHz from the exact Raman resonance, corresponding 
+approximately to 1/2 ( δνm + δνL), and thereby the coaxial high-order sidebands ( Ω±3: 
+±3 orders, Ω±4: ±4 orders) are efficiently generated. 
+In this study, we carry out this adiaba tic Raman sideband generation inside an 
+enhancement cavity. The concept is already outlined in Fig. 1a. A more quantitative 
+configuration is shown in Fig. 1b. Both  of the excitation nanosecond pulses, Ω0 and Ω-1, 
+which provide the optimal two-photon det uning for the adiabatic Raman excitation, 
+must simultaneously match each longitudinal mode of the excitation laser (free spectral range (FSR): 1.24 GHz) and the enhancement cavity (FSR: 1.98 GHz); namely the triple resonant condition, so as to be stably confined in the enhancement cavity.  4Fig. 2: Schematic illustrating the experi mental setup for Rama n sideband generation 
+in an enhancement cavity.  LD, laser diode, EOM, electrooptic modulator, FG, function 
+generator, DBM, double balanced mixer, PD, photo diode, PET, piezoelectric transducer, 
+CL, coupling lens, LN-cryostat, liquid nitrogen cryosta t. 
+Figure 2 is a conceptual i llustration of the experime ntal setup. The complete 
+system consists of three parts: a system  for generating cw laser radiation at two 
+frequencies stabilized to the enhancement cav ity (highlighted in blue), a system for 
+generating nanosecond pulses at two frequencie s as carriers (highlighted in yellow), and 
+an enhancement cavity filled with gaseous parahydrogen. 
+The simultaneous confinement of the nanos econd pulses at two frequencies in the 
+enhancement cavity was realized as follows. LD1( Ω-1：808 ±3 nm) and LD2( Ω0：785 ±3 
+nm) are external-cavity-controlled lase r diodes. Each oscillation frequency was 
+simultaneously stabilized to the enhan cement cavity by employing two independent 
+feedback loops based on the Pound-Drever-Hall (PDH) method11. These cw laser 
+radiations were also extracted from the other outlet of the cubic beam splitter and introduced into the in jection-locked nanosec ond pulsed Ti:S laser
+12, 13 as seeds 
+(typically 1 mW). By referencing these two se ed frequencies, we finely adjusted the 
+Ti:S cavity length and stabilized it in a doubl y resonant condition with a mismatch of 
+less than a few MHz. The mism atch between the resonant conditions of the two seed 
+frequencies was compensated by tuning the cav ity length of the Ti:S laser itself. The 
+compensation amounted to 1/36 FSR for ev ery 1 FSR cavity-length change. In the 
+doubly resonant condition, the pulsed outputs we re simultaneously injection-locked to 
+ 5the two seeds. Thus the nanosecond pulses we re produced with carrier frequencies, Ω-1 
+and Ω0, both stabilized to the enhancement cav ity. Finally, the nanosecond pulses were 
+transformed so as to match the transverse mode of the enhancement cavity, and coupled 
+into it. 
+We employed a Fabry-Perot configuration for the enhancement cavity. It consisted 
+of a pair of concave (r = 200 mm) and flat mirrors, each with a reflectivity of 98.75% 
+for a 730 - 900 nm spectral range. The cav ity length, FSR, and finesse were 75.00 mm, 
+2.00 GHz, and ~ 250, respectively. The waist di ameter of the fundamental transverse 
+mode of this cavity, TEM 00, was 0.31 mm. This corresponds  to a Rayleigh length of 94 
+mm, provided a Gaussian beam in free space.  The enhancement cavity was installed in a 
+sample cell and placed in a liquid nitrog en cryostat to realize Raman sideband 
+generation at the optimal temperature (50 - 100 K). The sample cell was filled with pure 
+parahydrogen (> 99.9%) 14, and its density was controll ed in the (2.9 ~ 0.48) × 1020 cm-3 
+range. 
+To match the optimum condition needed for the adiabatic Raman process with the 
+requirements for the enhancement cavity sideband generation (Fig. 1b), we 
+systematically investigated the spectroscopi c properties of the target rotational Raman 
+transition in advance, namely the density de pendence of the resonant frequency and the 
+decoherence time. Based on this fundamental  data, we designed and manufactured the 
+enhancement cavity length to the target value with a precision of better than 10 μm. The 
+design included the effects of the refrac tive index of parahydr ogen and the thermal 
+shrinking of the cavity length from room temp erature to liquid nitrogen temperature. 
+Finally, around the target densities, we adju sted the parahydrogen density slightly (a 
+few %) to achieve fine control to  the optimal adiabatic conditions.  
+Raman sideband generation was carried out  in the enhancement cavity around the 
+triple resonant condition shown in Fig. 1b. The seed laser frequencies were monitored 
+with a wavemeter (Burleigh, WA1500, resolution: 10-7). The spectrum and temporal 
+profiles of the generated Raman sidebands were measured with an optical multichannel analyzer (OMA) and biplanar phototubes (Hamamatsu Photonics K.K., R1328U, 
+build-up time: 60 ps), respectively. 
+ 
+   6Fig. 3: Confinement of two-frequency exci tation nanosecond pulses in an enhancement 
+cavity.  The reflected waveforms are shown for the two excitation pulses, Ω0 (black), Ω-1 
+(red). The inset depicts the reflected and intr acavity waveforms calculated for the incident 
+pulse (duration: 25 ns at FWHM). 
+Figure 3 shows the reflected waveforms of  the excitation lase r pulses introduced 
+into the enhancement cavity (black: Ω0, 382.4216 THz, 783.9319 nm; red: Ω-1, 
+371.7991 THz, 806.3292 nm). Both incident pulses had durations of 25 ns at full width 
+at half maximum (FWHM). The pul se energies were set to be so weak that they did not 
+generate any Raman sidebands. It is clearly  seen that the reflected waveforms are 
+identical for the two-frequency pulses. They have two peaks and a dip that approaches 
+zero between the two peaks. The shapes of their waveforms were stable shot-by-shot. 
+The inset shows the waveforms calculated  using the conditions employed for the 
+experiment. The black dots, and red and green solid curves correspond to incident, reflected, and intracavity waveforms, respectively
+15. The observed reflected waveform 
+agrees well with the expected one. Based on the system shown in Fig. 2, the two-frequency excitation pulses were simu ltaneously and stably confined in the 
+enhancement cavity.
+ 
+As depicted in the inset, here the intracavity radiation forms a waveform 
+appropriate for the adiabatic excitation that  smoothly builds up and decreases with the 
+cavity lifetime (pulse duration, 43 ns at FWHM). Furthermore, due to this confinement, the peak power of the intracavity waveform is  97 times greater than that of the incident 
+excitation pulse. 
+ 7Fig. 4: Raman sidebands adiabatically generated in enhancement cavity. The density of 
+parahydrogen and the excitation power introduced into the enhancement cavity were 4.8  x 
+1019 cm-3 and 0.27 kW, respectively . For the sideband spectrum above, the two-photon 
+detuning was set at + 80 MHz . The dotted curve represents the frequency dependence of the 
+finesse of the enhancement cavity. Typical beam profiles of the Raman sidebands are shown 
+below, where the two-photon detuning was set at + 180 MHz . 
+In the density control range, (2.9 ~ 0.48) x 1020 cm-3, we were able to select six 
+different conditions that satisfied the tr iple resonances shown in Fig. 1b. At the 
+respective conditions, we raised the excita tion power, and performed Raman sideband 
+generation with the enhancement cavity. The sidebands were observed both forward and 
+backward and they were similar. The dependence on the two-photon detuning was nearly the same as that found in a single -pass configuration. On  the other hand, the 
+dependence on the parahydrogen density was ra ther different. As expected, more 
+efficient sidebands were observed for lowe r densities at which the optimum two-photon 
+detuning could be small, while  in a single-pass c onfiguration, it was nearly independent 
+of density. 
+Figure 4 is a typical Raman sideband spectrum observed with the 
+enhancement-cavity configuration. The medium density was set at the lowest value that 
+ 8satisfied the triple resonance condition (4.8 ｘ1019 cm-3). Here, the two-photon detuning 
+was set at + 80 MHz ( Ω0 - Ω-1: 10.62372 THz). This value wa s almost in accordance 
+with the optimum two-photon detuning of 120 MHz found in a single-pass 
+configuration (Fig. 1c). 
+Fourteen sidebands are clearly seen that spread over a broad spectral range of 656 
+– 941 nm ( Ω+7: 656.32 nm, 456.78 THz – Ω-6: 940.73 nm, 318.69 THz). The sidebands 
+have an equidistant frequency spacing that is near resonant to the molecular Raman 
+transition. This is like a Ra man-type optical frequency comb. Although a very broad 
+spectrum was produced, the p eak power of each excita tion laser was only 0.27 kW 
+(energy: 6.8 μJ, duration: 25 ns). This is three orders of magnit ude less than that of a 
+typical single-pass configuration.16, 17 
+The dotted curve in Fig. 4 represents th e finesse of the employed enhancement 
+cavity. The nearly flat intensity distributi on of the generated si debands well reflects 
+behavior similar to the spectral dependence of the finesse. This is a manifestation of the 
+typical aspect that the confinement in th e enhancement cavity dominates the sideband 
+generation process. 
+The lower panel in Fig. 4 shows typical  beam profiles of the generated Raman 
+sidebands (two-photon detuning: 180 MHz). These were measured with a charge 
+coupled device-based beam profiler. We s ee that all the sideband components have 
+round, high quality beam profiles reflectin g the fundamental tr ansverse mode (TEM 00) 
+of the enhancement cavity.  9Fig. 5: Raman sideband generation, Enhancement cavity vs. Single pass.  The left panel 
+shows the sideband spectra for a, enhancement cavity (excitation power, 0.19 kW), b, single 
+pass (0.54 MW), and c, single pass (41 kW). The right panel shows modulation indexes, estimated from the generated sideband spectra as a function of the excitation power, for enhancement cavity (blue circles) and singl e-pass (brown circles) configurations. 
+Finally, in Fig. 5, we show the sideba nd generation dependence on excitation 
+power, which focuses on a comparison with a single-pass configuratio n. Panel 5a is a 
+sideband spectrum with the enhancement cavity  configuration, obtaine d at an excitation 
+power of 0.19 kW (4.7 μJ, 25 ns). We also find a similar spectrum in Fig. 5b, but this 
+was obtained with a single-pass configura tion. The excitation power was 0.54 MW (3.5 
+mJ, 6.5 ns), which is 2800 times more intense th an that in 5a. If we reduce the excitation 
+power by one order of magnitude to 41 kW (260 μJ, 6.5 ns) as shown in Fig. 5c, only 
+±1 orders of sidebands were se en. The higher orders were not  generated at all. We find 
+that the sidebands in 5a with the enhancement cavity were realized at an excitation power 200 times further lower than those in 5b. 
+The generation process of the Raman sidebands is analogous to optical phase 
+modulation using an electrooptic  modulator. That is, the in clination of the intensity 
+distribution of the Raman sidebands, expres sed in a log scale, can be in inverse 
+proportion to the product of the phase modulation index, 
+β, and the cavity finesses, F, 
+βF 18. Figure 5d plots the estimated inclinations  as a function of the excitation power. 
+This is examined by focusing on the high fr equency (anti-Stokes) side. An example is 
+ 10shown by the dotted line in Fig. 5a. It is clea rly seen that due to the enhancement cavity 
+generation, the required excita tion power is certainly redu ced by three orders (~ 1 / 
+3000). 
+In the enhancement cavity configuration employed in this study, the excitation 
+laser power ( ∝β ) can be enhanced 96.7 times. We can  consider the effective finesse ( F) 
+for the nanosecond excitation puls es to be similar. Using th ese values, when we briefly 
+estimate the Raman sideband generation in the enhancement cavity, we can expect a 
+reduction factor of 9,350 ( βF ~ 96.7 x 96.7) for the required excitation power. The result 
+in Fig. 5d is reasonably consiste nt with this brief estimation.  
+In summary, we have demonstrated extremely efficient Raman sideband 
+generation (Raman type optical frequency comb) based on adiabatic Raman excitation 
+in an enhancement cavity. We have s hown that optimization for both of the 
+adiabatic-excitation condition in the enhan cement cavity and the medium temperature 
+and density, realized broad Raman sidebands with a nearly flat intensity distribution 
+over 130 THz (656 – 941 nm). We have also  shown that this enhancement cavity 
+technique provides an excitation power reduc tion exceeding three orders of magnitude 
+compared with a single-pass configuration. 
+The present study can be extended to quasi-cw and/or cw regimes19, 20. This would 
+be very attractive as a novel method for realizing an optical  frequency comb originating 
+from single-frequency lasers21-23, and also for generating arbitrary optical waves24 with 
+an ultrahigh frequency repetition rate. We thank K. Hakuta for valuable comments. We 
+also thank T. Suzuki and A. Mio for helpful discussions and support with the 
+experiments.    11References  
+1. Y. R. Shen, The Principle of Nonlinear Optics (Wiley-Interscience, New Jersey, 
+2003).; R.W. Boyd, Nonlinear Optics (Academic, San Diego,2003). 
+2. A. Ashkin, G. D. Boyd, and J. M. Dziedzic, IEEE J. Quantum Electron. , QE-2 , 109 
+(1966). 
+3. M. Brieger, et al. , Opt. Commun. 38, 423 (1981). 
+4. C. Gohle, et al. , Nature  436, 234 (2005). 
+5. R. J. Jones, et al. , Phys. Rev. Lett.  94, 193201 (2005). 
+6. S. E. Harris and A. V . Sokolov, Phys. Rev. A  55, R4019 (1997). 
+7. L. K. Fam, et al.  Phys. Rev. A  60, 1562 (1999). 
+8. A. V. Sokolov, et al.  Phys. Rev. Lett.  85, 562 (2000). 
+9. J. Q. Liang, et al. , Phys. Rev. Lett.  85, 2474 (2000). 
+10. This spectral width include s those of the lasers (~  13 MHz) used in this 
+measurement, implying to be 1/2 ( δνm + δνL). 
+11. R. W. P. Drever, et al., Appl. Phys. B: Lasers and Optics , 31, 97 (1983). 
+12. M. Katsuragawa, and T. Onose, Opt. Lett. 30, 2421 (2005). 
+13. T. Onose, and M. Katsuragawa, Opt. Exp.  15, 1600 (2007). 
+14. M. Suzuki, et al. , J. Low Temp. Phys.  111, 463 (1998). 
+15. R. Tanaka, et al. , Opt. Exp.  16, 18667 (2008). 
+16. M. Katsuragawa, et al. , Opt. Exp.  13, 5628 (2005).; M. Katsuragawa, et al. , 
+CLEO/QELS 2006, QELS Technical Digest, QFE1, Long Beach, California, USA, 
+May. 21-26 (2006). 
+17. T. Suzuki, M. Hirai, and M. Katsuragawa, Phys. Rev. Lett.  101, 243602 (2008). 
+18. M. Kourogi, K. Nakagawa, and M. Ohtsu, IEEE J. Quantum Electron.  29, 2693 
+(1993). 
+19. J. K. Brasseur, K. S. Repasky, and J. L. Carlsten, Opt. Lett. 23, 367 (1998). 
+20. D. D. Yavuz, Phys. Rev.  A 76, 011805(R) (2007). 
+21. S. Uetake, et al. , Phys. Rev. A  61, R 011803 (2000). 
+22. F. Benabid, et al.  Nature  434, 499 (2005). 
+23. P. Del’Haye, et al.  Nature  450, 1214 (2007). 
+24. Z. Jiang, et al. , Nature Photon.  1, 463 (2007). 
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+arXiv:1001.0239v4  [math.RT]  21 Sep 2011COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS
+IVAN LOSEV
+Abstract. In this paper we study the structure of completions of symplectic r eflection
+algebras. Our results provide a reduction to smaller algebras. We ap ply this reduction to
+the study of two-sided ideals and Harish-Chandra bimodules.
+1.Introduction
+1.1.Setting. The goalof thispaper isto describe thestructure ofcompletions o fsymplectic
+reflection algebras (SRA) and to apply this description to the study of their two-sided ideals
+and Harish-Chandra bimodules and to some other problems, as well.
+SRA’s were introduced by Etingof and Ginzburg, [EG]. Let us recall th eir definition.
+Our base field is the field Cof complex numbers. Let Vbe a vector space equipped with
+a symplectic (=non-degenerate skew-symmetric) form ωand Γ be a finite group of linear
+symplectomorphisms of V. Recall that an element γ∈Γ is said to be a symplectic reflection
+if rk(γ−id) = 2 (this is the smallest possible rank for a nontrivial element γ∈Sp(V)). For
+any symplectic reflection s∈Γ letωsbe the skew-symmetric form on Vdefined by
+ωs(x,y) =ω(x,y) ifx,y∈im(s−id),
+ωs(x,y) = 0 ifxory∈ker(s−id).(1.1)
+LetSdenote the set of symplectic reflections in Γ. Note that Γ acts on Sby conjugation,
+letS1,...,Srbe the orbits. Pick independent variables t,c1,...,crand define c(s) to beci
+fors∈Si. Below for convenience we will sometimes write c0instead of t. Letcdenote the
+vector space with basis c0,...,cr. Consider the symmetric algebra S(c) =C[c∗] of the vector
+spacec. Then define the S(c)-algebra H(:=H(V,Γ)) as the quotient of the smash-product
+S(c)⊗CT(V)#Γ
+by the relations
+(1.2) [ x,y] =c0ω(x,y)+/summationdisplay
+s∈Sc(s)ωs(x,y)s, x,y∈V.
+HereT(V) stands for the tensor algebra of VandT(V)#Γ is the smash-product ofT(V) and
+CΓ, i.e., is the algebra that coincides with T(V)⊗CΓ as a vector space with the product
+(a1⊗g1)(a2⊗g2) =a1(g1.a2)⊗g1g2, whereg1.a2stands for the image of a2under the action
+ofg1.
+Theorem 1.3 in [EG] can be interpreted in the following way.
+Proposition 1.1.1. H is a flatS(c)-algebra.
+Supported by the NSF grant DMS-0900907.
+MSC 2010: 16G99, 16S99.
+Address: Northeastern University, Dept. of Math., 360 Huntingt on Avenue, Boston MA02115, USA.
+E-mail: i.loseu@neu.edu.
+12 IVAN LOSEV
+Specializing t:=t,ci:=ci, wherei= 1,...,r,andt,ci∈C,we get the C-algebraHt,c.
+It is clear thatH0,0=SV#Γ(=C[V∗]#Γ). The algebra Ht,chas a natural filtration, where
+CΓ has degree 0, while Vhas degree 1. Proposition 1.1.1 means that the associated graded
+algebra grHt,ccoincides withH0,0. Also note that for any nonzero athe algebrasHt,cand
+Hat,acare naturally isomorphic. So we get two (essentially) different cases :t= 1 andt= 0.
+One of the most crucial differences is that H0,cis finite over its center Zc. Moreover, grZc
+is canonically identified with the invariant subalgebra ( SV)Γ⊂SV, see [EG], Theorem 3.3.
+On the other hand, the center of H1,cis always trivial, see [BrGo], Proposition 7.2.
+Let us describe briefly the content of the paper. It is divided into fo ur sections. Section
+2 describes the structure of completions of H. The most straightforward version of a com-
+pletion we are interested in together with the statement of the cor responding result is given
+in Subsection 1.2. Our starting point here was a result of Bezrukavn ikov and Etingof, [BE],
+Theorem 3.2.
+In Section 3 we obtain some sort of “reduction theorems” for ideals and Harish-Chandra
+bimodules of SRA’s. Our results regarding ideals are described in Subs ection 1.3. This part
+is mostly inspired by the author’s work on W-algebras, [Lo1],[Lo2].
+In Section 4 we give some miscellaneous applications of results of the fi rst two sections.
+These applications are described in Subsection 1.4.
+Finally, in Section 5 we consider perhaps the most important special c lass of SRA, the
+so called rational Cherednik algebras. We relate our work to that of Bezrukavnikov and
+Etingof, [BE], and strengthen some of our results. Also we show tha t the definition of
+a Harish-Chandra bimodule for a rational Cherednik algebra given in [ BEG2] agrees with
+ours. Finally, we give a complete classification of two-sided ideals in the rational Cherednik
+algebras of type A.
+In the first subsections of Sections 2,3,5 their contents are descr ibed in more detail.
+1.2.Completions. We are going to study completions of the algebra H. Let us explain
+what kind of completions we are interested in.
+Letπdenote the quotient map V∗→V∗/Γ. Pick a point b∈V∗. LetIbdenote the ideal
+ofC[V∗] generated by the maximal ideal of π(b) inC[V∗]Γ. ThenIb#Γ is a two-sided ideal
+inH0,0=C[V∗]#Γ. Let mbdenote the preimage of Ib#Γ under the canonical epimorphism
+H։H0,0. This is a two-sided ideal in H. Consider the completion
+(1.3) H∧b:= lim←−nH/mn
+b.
+Letusremarkthat H∧bisaC[[c∗]]-algebrabecause mb∩S(c)coincides withtheaugmentation
+idealcS(c).
+We want to understand the structure of the algebra H∧bin terms of a similar but simpler
+algebra. Define the algebra Has the quotient of S(c)⊗CT(V)#Γbby the relations
+(1.4) [ x,y] =c0ω(x,y)+/summationdisplay
+s∈S∩Γbc(s)ωs(x,y)s, x,y∈V.
+We can define the algebra H∧0analogously to H∧b. In more detail, we can take the
+maximal ideal I0⊂C[V∗] corresponding tothepoint 0 ∈V∗. Then weconsider thepreimage
+m0ofI0#ΓbinHand form the completion H∧0:= lim←−nH/mn
+0.
+Now we are ready to state one of the main results of this paper comp aring the completions
+H∧bandH∧0.COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 3
+Theorem 1.2.1. There is an isomorphism
+Θb:H∧b→Mat|Γ/Γb|(H∧0)
+of topological C[[c∗]]-algebras. Here Mat|Γ/Γb|stands for the algebra of square matrices of size
+|Γ/Γb|.
+In fact, we will prove a much more precise (but also much more techn ical) statement,
+Theorem 2.5.3, which will “globalize” Θband will give an explicit formula for Θb“modulo
+c”.
+One special case of Theorem 1.2.1 was known before. Namely, Bezru kavnikov and Etingof
+proved a similar statement in [BE] for the special class of SRA called rational Cherednik
+algebras. This special class is described as follows. Suppose that we have a de composition
+V=h⊕h∗into the sum of two pairwise dual Γ-stable subspaces. The algebra Ht,cin this
+case is known as a rational Cherednik algebra. This case is much simple r than the general
+one: one can even get an explicit formula for the isomorphism Θb(in fact, for some special
+pointsb). It seems to be unlikely that one can find an explicit formula in the gen eral case.
+1.3.General results on two-sided ideals. We will apply Theorem 1.2.1 (more precisely,
+its enhanced version, Theorem 2.5.3) to the study of ideals and Haris h-Chandra bimodules
+of the algebras H,H1,c,Zc. We remark that both our results and our proofs are inspired by
+those for W-algebras, see [Lo1],[Lo2]. The definition of a Harish-Chan dra bimodule is a bit
+technical so we postpone it (together with the statement of the c orresponding result) until
+Subsection 3.4. The main result for Harish-Chandra bimodules is Theo rem 3.4.6.
+Now let us describe our results on ideals. Consider a two-sided ideal J ⊂H 1,c. Then
+H1,c/Jhas a natural filtration. It is easy to see that the actions of ( SV)Γon gr(H1,c/J)
+by left and by right multiplications coincide. Let V( H1,c/J)⊂V∗/Γ denote the support of
+the (SV)Γ-module gr(H1,c/J). One can show (Subsection 3.4) that V( H1,c/J) is a Poisson
+subvariety in V∗/Γ. Similarly, for two two-sided ideals J1⊂ J2⊂ H1,cwe can define
+V(J1/J2)⊂V∗/Γ and it is also a Poisson subvariety.
+The Poisson variety V∗/Γ hasfinitely many symplectic leaves. The leaves can bedescribed
+as follows, see, for example, [BrGo], Subsection 7.4. Consider the se t of all conjugacy classes
+of subgroups in Γ that are stabilizers of elements of V(equivalently, of V∗). Then there
+is a bijection between this set and the set of symplectic leaves in V∗/Γ. Namely, given
+a conjugacy class, pick a subgroup Γ ⊂Γ in it. Then the corresponding symplectic leaf
+coincides with the π/parenleftbig
+(V∗
+0)fr/parenrightbig
+, whereπ:V∗→V∗/Γ denotes the quotient morphism, V0:=
+VΓ,(V∗
+0)fr:={v∈V∗|Γv= Γ}.
+Fix a symplectic leaf L⊂V∗/Γ and let Γ⊂Γ be a subgroup in the conjugacy class of
+subgroups corresponding to L. We consider the set IdL(H1,c) consisting of all two-sided
+idealsJ⊂H 1,csuch that V(H1,c/J) =L. We will relate IdL(H1,c) to a certain set of ideals
+of finite codimension in a “smaller” SRA to be specified in a moment.
+There is a unique Γ -stable decomposition V=V0⊕V+. LetH+be the quotient of
+S(c)⊗TV+#Γmodulo the relations (1.3). In particular, we have the decomposition H=
+At(V0)⊗C[t]H+. HereAt(V0) is the homogeneous Weyl algebra of the symplectic vector
+spaceV0, in other words, At(V0) :=H(V0,{1}).
+Note that the group /tildewideΞ :=NΓ(Γ) acts on H+by automorphisms, for V+⊂Vis/tildewideΞ-stable.
+The action of Γ⊂/tildewideΞ coincides with that by inner automorphisms. So we can consider the set
+Id0(H+
+1,c) of two-sided ideals of H+
+1,cof finite codimension, andits subset IdΞ
+0(H+
+1,c) consisting
+of all Ξ := /tildewideΞ/Γ-stable ideals (in fact, /tildewideΞ acts on Id(H+
+1,c) via its quotient Ξ).4 IVAN LOSEV
+We have the following result that is completely parallel to Theorem 1.2.2 from [Lo2].
+Theorem 1.3.1. There are maps IdL(H1,c)→IdΞ
+0(H+
+1,c),J /ma√sto→ J †, andId0(H+
+1,c)→
+IdL(H1,c),I/ma√sto→I‡,with the following properties:
+(1)The image ofJ/ma√sto→J †coincides with IdΞ
+0(H+
+1,c).
+(2)J⊂(J†)‡andI⊃(I‡)†for allJ∈IdL(H1,c),I∈Id0(H+
+1,c).
+(3)We have V((J†)‡/J)⊂∂L:=L\L.
+(4)Consider the restriction of the map I/ma√sto→I‡to the set of all primitive (=maximal)
+ideals in Id0(H+
+1,c). The image of this restriction is the set of all primitive ide als in
+IdL(H1,c)and each fiber is a single Ξ-orbit.
+Recall that a two-sided ideal Iin an associative unital algebra Ais calledprimitive if it is
+the annihilator of a simple module.
+In[G2]Ginzburgprovedthatanyprimitiveidealsof H1,ciscontainedinsomeset IdL(H1,c).
+This is an analog of the Joseph irreducibility theorem, [Jo], from the r epresentation theory
+of universal enveloping algebras. We will rederive Ginzburg’s result in the present paper. So
+Theorem 1.3.1 allows one to reduce the study (in particular, the class ification) of primitive
+ideals in symplectic reflection algebras to the study of the annihilator s of irreducible finite
+dimensional representations.
+We also have an analog of Theorem 1.3.1 for t= 0. Recall that Zcdenotes the center
+ofH0,c. It was shown by Etingof and Ginzburg in [EG] that Zchas a natural Poisson
+bracket{·,·}whose definition will be recalled in Subsection 2.8. Recall that Zcis a filtered
+algebra, let F iZcdenote the corresponding filtration. Then {FiZc,FjZc}⊂Fi+j−2Zcand
+the induced Poisson bracket on Z0= (SV)Γcoincides with the restriction of the bracket
+fromSV.
+Again fix a symplectic leaf L⊂V∗/Γ. Consider the set IdL(Zc) consisting of all Poisson
+idealsJ ⊂Zcwhose associated variety (= the variety of zeros of gr J) coincides withL.
+Recall that an ideal Jin a Poisson algebra Ais called Poisson if {A,J}⊂J.
+As above, we can form the subgroups Γ ,/tildewideΞ⊂Γ, and the algebras H0,c,Zc,Z+
+c. Note that
+(1.5) H0,c=SV0⊗H+
+0,c,
+(1.6) Zc=SV0⊗Z+
+c,
+(1.6) is an equality of Poisson algebras, the Poisson structure on SV0is induced from the
+symplectic form on V0. LetIdΞ
+0(Z+
+c) denote the set of all Ξ-stable Poisson ideals of finite
+codimension inZ+
+c.
+The following theorem is, in a sense, a quasiclassical analogue of Theo rem 1.3.1.
+Theorem 1.3.2. There are maps IdL(Zc)→IdΞ
+0(Z+
+c),J/ma√sto→J †,Id0(Z+
+c)→IdL(Zc),I/ma√sto→
+I‡,with the following properties:
+(1)The mapJ/ma√sto→J †is surjective.
+(2)J⊂(J†)‡andI⊃(I‡)†for allJ∈IdL(Zc),I∈Id0(Z+
+c).
+(3)We have V((J†)‡/J)⊂∂L.
+(4)Consider the restriction of the map I/ma√sto→I‡to the set of all prime (=maximal) ideals
+inId0(Z+
+c). The image of this restriction is the set of all prime ideals i nIdL(Zc)
+and each fiber is a single Ξ-orbit.
+Let us make a remark regarding a geometric interpretation of part (4). Maximal ideals
+inZ+
+cthat are Poisson are in one-to-one correspondence with zero-dim ensional symplecticCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 5
+leaves of the variety C+
+c:= Spec(Z+
+c). On the other hand, the set of prime ideals in IdL(Zc)
+can be described as follows. Consider the variety Spec( R/planckover2pi1(Zc)). Here and below R/planckover2pi1(A)
+denotes the Rees algebra of a filtered algebra A. In more detail, if F iAis an increasing
+filtration on A, then, by definition, R/planckover2pi1(A) :=/circleplustext
+i/planckover2pi1iFiA⊂A[/planckover2pi1−1,/planckover2pi1].
+The inclusion C[/planckover2pi1]֒→R/planckover2pi1(Zc) gives rise to the dominant morphism Spec( R/planckover2pi1(Zc))→C.
+The preimage of C×is naturally identified with Spec( Zc)×C×, while the preimage of 0
+is Spec(Z0) =V∗/Γ. The set of prime ideals in IdL(Zc) is identified with the set of all
+symplectic leaves Ysuch that the closure of Y×C×in Spec(R/planckover2pi1(Zc)) intersects V∗/Γ exactly
+inL. As Martino checked in [M], for any symplectic leaf in Spec( Zc) its ideal lies in IdL(Zc)
+for someL(our techniques allows to give an alternative proof of this result).
+1.4.Applications. Our first result concerns the structure of H0,cas an algebra over its
+centerZc. More precisely, since H0,cis finite overZc, we can consider H0,cas the algebra of
+global sections of an appropriate coherent sheaf of algebras ove r Spec(Zc). Let us describe
+the fibers of the restriction of this sheaf to a symplectic leaf.
+Theorem 1.4.1. LetˆLbe a symplectic leaf of Spec(Zc)andq⊂Zcthe prime ideal defining
+the closure of ˆL. LetLbe a unique symplectic leaf of V∗/Γsuch that q∈IdL(Zc). Pick
+a pointy∈ˆL, letnybe its maximal ideal in Zc. Finally, let nbe a maximal ideal of
+Z+
+ccontaining q†(by Theorem 1.3.2 all such ideals are Ξ-conjugate). Then H0,c/H0,cnyis
+isomorphic to Mat|Γ/Γ|(H+
+0,c/H+
+0,cn).
+While this paper was in preparation, a proof in the case of rational Ch erednik algebras
+appeared, [B]. The proof was based on the Bezrukavnikov-Etingof theorem on isomorphisms
+of completions. Also note that Theorem 1.4.1 is similar to the Kac-Weisf eiler conjecture on
+modular Lie algebras proved by Premet, [P], and to the De Concini-Ka c-Procesi conjecture
+on quantized universal enveloping algebras at roots of unity prove d by Kremnizer, [K] (his
+proof is rather sketchy and imposes some restrictions on the cent ral character).
+Together Theorems 1.3.2,1.4.1 allow one to reduce the study of repre sentations of the
+algebrasH0,c/H0,cnyto the case when yis a (zero dimensional) symplectic leaf.
+Oursecondapplicationisthefollowingtheoremwhichwascommunicate dtous(essentially
+with a the proof) by P. Etingof.
+Theorem 1.4.2. For general cthe algebraH1,cis simple.
+The meaning of a “general c” will be made precise in Subsection 4.2.
+1.5.Conventions and notation. In this subsection we gather some conventions and no-
+tation we use. Each of Sections 2,3 contains its own list of convention s and notation, Sub-
+sections 2.2, 3.2.
+Sheaves. LetXbe an algebraic variety and Sbe a sheaf (of abelian groups) on X. For
+an open subset U⊂Xwe denote byS(U) the group of sections of SonU. For a morphism
+Y→Xwe denote byS|Ythe sheaf-theoretic pull-back of StoY.
+In this paper we mostly consider quasi-coherent and pro-coheren t sheaves. By a pro-
+coherent sheaf we mean a sheaf Sof vector spaces (algebras, modules) on Xadmitting
+a decreasing filtration F iS(by subspaces, two-sided ideals, submodules) such that Sis
+complete with respect to this filtration, i.e., S= lim←−iS/FiS, and FiS/Fi+1Sis a coherent
+OX-module.
+The following table contains the list of some standard notation used in the paper.6 IVAN LOSEV
+/hatwide⊗ completed tensor product of complete topological vector spaces /modules.
+(V) the two-sided ideal in an associative algebra generated by a subse tV.
+Der(A) the Lie algebra of derivations of an algebra A.
+Gx the stabilizer of xin a groupG.
+grA the associated graded vector space of a filtered vector space A.
+R/planckover2pi1(A) :=/circleplustext
+i∈Z/planckover2pi1iFiA,the Rees vector space of a filtered vector space A.
+RΓ the group algebra of a group Γ with coefficients in a ring R.
+V(M) the associated variety of M.
+Acknowledgements. This paper would never have appeared without P. Etingof’s help
+that is gratefully acknowledged. Apart from inspiration and many fr uitful ideas, Etingof
+contributed several results to the present paper, including Theo rem 1.4.2. Also I would
+like to thank G. Bellamy, R. Bezrukavnikov, A. Braverman, I. Gordo n and A. Premet for
+stimulating discussions andfor useful remarks ontheprevious ver sions ofthis paper. Finally,
+I would like to thank the referee for numerous remarks that allowed me to improve the
+exposition.
+2.Completion theorem
+2.1.Structure of this section. In this section we will study completions of the algebra
+H. The main result of this section is Theorem 2.5.3, which is an enhanced ( in particular,
+“globalized”) version of Theorem 1.2.1.
+InSubsection 2.2 we explain conventions andsome notationused in th epresent section. In
+Subsection 2.3 we gather various results on the centralizer constr uction introduced in [BE].
+This construction provides a convenient language for our consider ations.
+The next two subsections, 2.4,2.5 are devoted to our principal resu lt, Theorem 2.5.3. The
+theorem deals with certain sheaves on symplectic leaves. The sheav es of interest are “local-
+ized completions” of the algebras H,Mat|Γ/Γb|(H). These sheaves are defined in Subsection
+2.4. Theorem 2.5.3 claims that thereis anisomorphism withsome special properties between
+(certain twists of) these sheaves.
+The proof of Theorem 2.5.3 is rather indirect. As it happens, it is more convenient to work
+with another version of sheafified completions. This version is introd uced in Subsection 2.6.
+For instance, on the Hside we have a pro-coherent sheaf H∧on a symplectic leaf, whose
+fiber atbisH∧b. The sheaves under consideration come equipped with flat connect ions.
+In Subsection 2.7 we state an isomorphism theorem for the sheaves with flat connections of
+interest, Theorem 2.7.3.
+In Subsection 2.8 we mostly recall some basic and standard facts ab out the spherical
+subalgebras and the centers of the SRA’s. Subsections 2.9,2.10 are quite technical. Their
+goal is to make sure that in our setting we have nice properties of co mpletions that are
+standard in the commutative situation.
+The proof of Theorem 2.7.3 occupies two subsections, 2.11,2.12. In t he former we study
+derivations of algebras of sections of H∧. The latter subsection completes the proof.
+Theorem 2.5.3 is deduced from Theorem 2.7.3 in the last Subsection 2.13 of the section.
+The idea, roughly speaking, is that to get Theorem 2.5.3 from Theore m 2.7.3 we just need
+to pass to flat sections of the sheaves considered in the latter.
+2.2.Notation and conventions. Let us introduce some notation to be used throughout
+the section.COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 7
+Varieties and morphisms. SetX:=V∗/Γ. This is a Poisson variety with finitely many
+symplectic leaves. Let π:V∗→Xdenote the quotient morphism. Let Lbe a symplectic
+leaf ofX. Pick a point b∈V∗withπ(b)∈L. Set Γ:= Γb,V0:=VΓ. ThenL=π(V∗
+0). Set
+L:=V∗
+0∩π−1(L),X:=V∗/Γ, then we can consider Las a subvariety in X. We remark
+thatLis open inV∗
+0.
+Letπdenote the quotient morphism V∗→Xandπ′:X→Xbe the natural morphism
+so thatπ=π′◦π. LetX0be the open subset of Xconsisting of all points where π′is ´ etale.
+In other words,X0=π({v∈V|Γv⊂Γ}). Further, setX0=π′(X0).
+It is clear thatLis a closed subvariety in X0. LetX∧
+Ldenote the formal neighborhood of
+LinX0. Similarly,X∧
+Ldenotes the formal neighborhood of LinX0.
+Groups. Weset/tildewideΞ :=NΓ(Γ),Ξ :=/tildewideΞ/Γ. WeremarkthatΞactsnaturallyon X. Moreover,
+the restriction of π′toLis a coveringL→Lwith fundamental group Ξ.
+Algebras.
+Recall the algebras H,H,At,H+defined in the Introduction.
+Letc(i)(resp.,c(i)),i= 0,...,r+ 1,denote the subspace in cspanned by c0,...,ci−1
+(resp., by ci,...,cr). We write H(l),l= 0,...,r+ 1,for the quotient H/(c(l)) (soH(0)=
+H,H(r+1)=SV#Γ). We remark that H(l)is aS(c(l))-algebra. Let ρl,l= 0,...,r,denote
+the natural projection H(l)→H(l+1), set ˆρ=ρr◦...◦ρ0:H→H(r+1). The analogous
+projections for H(l)are denoted by ρl,ˆρ.
+Centers. LetAbe an associative unital algebra. We denote the center of Abyz(A).
+Now suppose Atis an associative C[t]-algebra. By zt(At) we denote the inverse image of
+z(At/(t)) inAtunder the natural projection. In other words, zt(At) consists of all elements
+a∈Atsuch that [a,b]∈(t) for allb∈At.
+2.3.The centralizer construction. In this subsection we will recall the centralizer con-
+struction of Bezrukavnikov and Etingof, [BE].
+LetG⊃Hbe finite groups and Abe an algebra containing CH. Set
+FunH(G,A) :={f:G→A|f(hg) =hf(g),∀g∈G,h∈H}.
+Clearly, Fun H(G,A) is a free right A-module of rank|G/H|. By definition, the centralizer
+algebraZ(G,H,A) is End A(FunH(G,A)). In particular, Z(G,H,A) is isomorphic to the
+matrix algebra Mat |G/H|(A). An isomorphism depends on a choice of representatives in
+the cosets from H\G, and there is no canonical isomorphism. We have a monomorphism
+CG֒→Z(G,H,A) given by ( g.f)(g1) =f(g1g),f∈FunH(G,A),g,g1∈G. Also we have an
+embedding of AHintoZ(G,H,A):AHacts on Fun H(G,A) by (a.f)(g1) =af(g1). Under
+a choice of identification Z(G,H,A)∼=Mat|G/H|(A) as explained above, AHis identified
+with the subalgebra consisting of all matrices of the form diag( a,a,...,a ), wherea∈AH.
+In particular, the center ZofAis contained in AH. So we get the embedding of Zinto
+Z(G,H,A) that identifies Zwith the center of Z(G,H,A).
+Now letMbe anA-bimodule. We want to construct a Z(G,H,A)-bimodule Z(G,H,M)
+fromM. Set
+Z(G,H,M) := Hom H×Hop(G×G,M) = FunH(G,A)⊗AM⊗AHomA(FunH(G,A),A).
+The notation Hopin the subscript indicates that the second copy of Hacts on the second
+copy ofGfrom the right. If we choose representatives in the right cosets H\Gand iden-
+tifyZ(G,H,A) with Mat |G/H|(A), theZ(G,H,A)-bimodule Z(G,H,M) gets identified with
+Mat|G/H|(M).8 IVAN LOSEV
+Let us explain how to recover AfromZ(G,H,A) andMfromZ(G,H,M). Pick a left
+cosetx∈H\G. Definee(x)∈End(Fun H(G,A)) in the following way:
+(2.1) [ e(x).f](g) =/braceleftBigg
+f(g),ifg∈x,
+0,else.
+Clearly,e(x)∈Z(G,H,A) ande(x)2=e(x). Moreover,
+(2.2)/summationdisplay
+x∈H\Ge(x) = 1, ge(x)g−1=e(xg).
+Further, we see that
+(2.3) ae(x) =e(x)a,∀a∈AH.
+We have a natural isomorphism A∼=Z(H,H,A) =e(H)Z(G,H,A)e(H). On the other
+hand,Z(G,H,A)e(H) is naturally identified with Fun H(G,A) as aZ(G,H,A)-A-bimodule.
+Namely, to ϕ∈Z(G,H,A)e(H) we assign the map g/ma√sto→e(H)gϕ∈FunH(G,A). Therefore
+the bimodule Z(G,H,A)e(H) gives rise to a Morita equivalence between AandZ(G,H,A).
+Thus the assignment N/ma√sto→e(H)Ne(H) is a quasi-inverse equivalence to M/ma√sto→Z(G,H,M).
+Lemma 2.3.1. LetBbe a unital associative algebra and ι:Z(G,H,CH)֒→Bbe an algebra
+monomorphism(mapping 1to1). ThenBis naturallyidentifiedwith Z(G,H,ι(e(H))Bι(e(H))).
+Proof.Sete:=ι(e(H)) for brevity. Let us identify Bewith Fun H(G,eBe). Toϕ∈Bewe
+assign the map g/ma√sto→egϕ. An inverse map is given by f/ma√sto→1
+|H|/summationtext
+g∈Gg−1f(g). The claim
+that these maps are mutually inverse follows from (2.2).
+This defines an action of Bon FunH(G,eBe) by righteBe-module endomorphisms and
+hence gives rise to a homomorphism B→Z(G,H,eBe ). To check that this homomor-
+phism is an isomorphism we need to show that the bimodule Beproduces a Morita equiv-
+alence between BandeBe, equivalently, that 1 ∈BeB. This stems from the inclusion
+1∈Z(G,H,CH)e(H)Z(G,H,CH). /square
+Let us discuss the compatibility of the centralizer construction with certain group actions.
+Suppose that /tildewideHis a subgroup of GcontainingHand thatAis equipped with an action of
+/tildewideHby automorphisms, whose restriction to Hcoincides with the adjoint action of H⊂A.
+Further, suppose that His normal in /tildewideH. Then/tildewideHacts on Fun H(G,A) by
+(2.4) ( /tildewideh.f)(g) =/tildewideh.f(/tildewideh−1g/tildewideh),
+where on the right hand side /tildewideh.means the action of /tildewidehonA. So we have also an /tildewideH-action
+onZ(G,H,A) = EndA(FunH(G,A)) by
+(2.5) ( /tildewideh.a)f=/tildewideh.(a(/tildewideh−1.f)).
+Note that the restriction of the last /tildewideH-action toHcoincides with the adjoint H-action on
+Z(G,H,A). Also note that the element e(H) is/tildewideH-invariant and the induced /tildewideH-action on
+A=e(H)Z(G,H,A)e(H) coincides with the initial one.
+Also we can consider the action of /tildewideHon FunH(G,A) given by /tildewideh.f(g) :=/tildewideh.f(/tildewideh−1g) and
+the induced action on Z(G,H,A). Note that Hacts trivially. So we get the action of /tildewideH/H
+onZ(G,H,A) by automorphisms.
+Suppose now that Mis anA-bimodule. We say that Mis an/tildewideH-equivariant A-bimodule,
+if there is an action of /tildewideHonMsuch that:COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 9
+•The restriction of this action to Hcoincides with the adjoint action of HonM.
+•The natural map A⊗M⊗A→Mis/tildewideH-equivariant.
+Then we can define natural /tildewideH- and/tildewideH/H-actions on Z(G,H,M) similarly to the above.
+Now letDbe aCH-linear derivation of A. Define the operator Don FunH(G,A) by
+(D.f)(g) =D.(f(g)) and the derivation Dof End(Fun H(G,A)) byD.ϕ= [D,ϕ]. Clearly,
+D.(fa) = (D.f)a+f(D.a) fora∈A,f∈FunH(G,A). Therefore DpreservesZ(G,H,A)⊂
+End(Fun H(G,A)). Note that the derivation Dof EndA(FunH(G,A)) isCG-linear. We
+remark that under an identification Z(G,H,A)∼=Mat|G/H|(A) the derivation Dof this
+algebra corresponds to the entry-wise derivation, whose compon ents coincide with D∈
+Der(A).
+Finally, let us provide an alternative description of Z(G,H,A) in a special case.
+Supposethat A:=A0#Hforsomealgebra A0actedonby H. EquipthespaceFun( G,A0) =
+C[G]⊗A0with the algebra structure (with respect to the pointwise multiplicat ion of func-
+tions) and with the diagonal H-action, where the action on C[G] is induced by the left
+G-action. The invariant subalgebra ( C[G]⊗A0)His equipped with a G-action induced from
+the action of Gon itself by right translations.
+Lemma 2.3.2. There is a natural isomorphism (C[G]⊗A0)H#G∼− →Z(G,H,A 0#H).
+Proof.Let us define maps ϑ:G,(C[G]⊗A0)H→Z(G,H,A 0#H) as follows:
+[ϑ(g)f](g′) :=f(g′g),
+[ϑ(F)f](g′) :=F(g′)f(g′),
+g,g′∈G,f∈FunH(G,A0#H),F∈(C[G]⊗A0)H= FunH(G,A0).(2.6)
+It is easy to see that ϑis well-defined and extends to a homomorphism ( C[G]⊗A0)H#G→
+Z(G,H,A 0#H).
+Let us show that this homomorphism is anisomorphism. Choose repre sentativesg1,...,gk
+of the leftH-cosets inG. This choice identifies Z(G,H,A 0#H) with Mat |G/H|(A0#H). It is
+easytoseethat ϑalsogivesrisetoanidentificationof( C[G]⊗A0)H#GwithMat |G/H|(A0#H).
+/square
+Remark 2.3.3. All constructions of this subsection deal with algebras and their bim odules.
+However, they work for sheaves of algebras and sheaves of bimod ules without any noticeable
+modifications.
+2.4.Sheafified versions of algebras, I. The goal of this section is to introduce certain
+sheavesH∧L
+(l)|LandH∧L
+(l)|Lof topological algebras on Land onL, respectively.
+First of all, let us define the sheaf of algebras H|XonX=V∗/Γ. The procedure we
+are going to use is very similar to the microlocalization procedure, see , for example, [G1],
+Section 1, or [G2], the proof of Theorem 2.1. It is enough to specify t he algebras of sections
+on principal open subsets.
+Namely, pickaprincipalopensubset U⊂Xcorrespondingto f∈C[X]. SetS:={fk}k/greaterorequalslant0.
+Consider the algebra H/(c)k. We have a natural epimorphism ˆ ρ:H/(c)k→H(r+1). Since
+ada:H/(c)k→H/(c)kis locally nilpotent for any a∈ˆρ−1(S), we see that ˆ ρ−1(S) is
+an Ore subset of H/(c)k. Consider the localization H/(c)k(U) ofH/(c)kwith respect to
+ˆρ−1(S). The algebras H/(c)k(U) glue to a sheaf H/(c)k|Xin the Zariski topology. Then set
+H|X:= lim←−kH/(c)k|X.10 IVAN LOSEV
+There is a natural C×-action on H:t.γ=γ,t.v=tv,t.c=t2c,γ∈Γ,v∈V,c∈c∗,t∈C×.
+This action naturally extends to an action on H|X. Also we note that the filtration ( c)kon
+H|Xis complete and the quotients of this filtration are coherent OX-modules.
+Similarly to the analogous properties of the microlocalization, see [G1], Section 1, we have
+the following lemma.
+Lemma 2.4.1. (1)H|X(X)is thec-adic completion H∧cofHin the(c)-adic topology.
+(2)H(U)is a flat (left or right) H-module for any principal open subset U⊂X.
+(3)The functor (H|X)⊗H•from the category of finitely generated left H-modules to the
+category of left H|X-modules is exact.
+Consider the ideal Jof (SV)Γconsisting of all functions vanishing on L. This is a Poisson
+ideal in (SV)Γ. Setp(r+1):=H(r+1)J,p:= ˆρ−1(p(r+1)). Clearly, pis a two-sided ideal in H.
+Localizing poverX, wegetasheaf p|Xoftwo-sidedidealsin H|X. Formally, p|X:=H|X⊗Hp.
+Now consider the restriction H|X0of the sheaf H|XtoX0. We have the sheaf of ideals
+p|X0⊂H|X0. We remark that p|X0=H|X0⊗Hp=H|X0pand similarly p|X0=pH|X0.
+Introduce the completion H∧L:= lim←−kH|X0/(p|X0)k= lim←−k(H/pk)|X0. This is a sheaf of
+algebrasonX∧
+Lcompleteinthe p∧L-adictopology,where p∧L:= lim←−k(p/pk)|X0. SincepisC×-
+stable, thegroup C×stillactson H∧L. Finally, let H∧L|L,p∧L|Ldenotethe(completed)sheaf-
+theoretic inverse images of the topological sheaves H∧L,p∧Lunder the embedding L֒→X∧
+L.
+Alternatively, one can consider the pull-backs H|L,p|LofH|X,p|XtoL. ThenH∧L|L=
+lim←−kH|L/(p|L)k.
+Again,H∧L|Lis aC×-equivariant sheaf of algebras on Lcomplete in the p∧L|L-adic topol-
+ogy. Wealso remarkthatthequotients forthe p∧L|L-adicfiltrationarecoherent OL-modules.
+We have the following standard property of completions. More subt le properties based on
+the Artin-Rees lemma will be established later in Subsection 2.10.
+Lemma 2.4.2. The functor•∧L|L:M/ma√sto→(lim←−kM/pkM)|Lfrom the category of finitely
+generated left H|X0-modules to the category of H∧L|L-modules is right exact.
+Similarly, we can define the sheaves H|X0,p|X0onX0,H∧L,p∧LonX∧
+LandH∧L|LonL.
+The latter is a C×-equivariant sheaf of algebras. Also it has an action of /tildewideΞ induced from the
+action on H, as described in the introduction.
+In fact, thanks to the decomposition,
+(2.7) H=At⊗C[t]H+,
+the structure of H∧L|Lis pretty simple. Namely, we can sheafify the Weyl algebra AtoverL
+similarly to the above. As a sheaf of algebras At|LisOL[[t]] equipped with the Moyal-Weyl
+∗-product. Let us recall that the latter is defined as follows: f∗g:=µ(exp(t
+2ω|V0)f⊗g) for
+sectionsf,gofOL. Hereµis the multiplication map O⊗2
+L→OL. The form ω|V0∈/logicalandtext2V∗
+0
+can be considered as the contraction map O⊗2
+L→O⊗2
+L. The decomposition (2.7) implies
+(2.8) H∧L|L=At|L/hatwide⊗C[[t]](H+)∧0
+Here and below /hatwide⊗stands for the completed tensor product of topological algebras /sheaves.
+We equip At|Lwith the t-adic topology and ( H+)∧0with the m+
+0-adic topology.
+Below we write C( •) instead of Z(Γ,Γ,•).
+Consider the sheaf C( H∧L|L) =At|L/hatwide⊗C[[t]]C(H+∧0). As explained in Subsection 2.3, Ξ
+acts on this sheaf by automorphisms. The action clearly preserves the tensor factors and theCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 11
+corresponding action on At|Lis induced from the Ξ-action on L. Abusing the notation, we
+write C(H∧L|L)Ξinstead of [π′
+∗(C(H∧L|L))]Ξ. This is a sheaf on L.
+Remark 2.4.3. Wecandefinethesheaves H(l)|X,H∧L
+(l),H∧L
+(l)|Letc. similarlyto H|X,H∧L,H∧L|L.
+However, Lemmas 2.4.1,2.4.2 imply that H(l)|X=H|X/(c(l)),HL
+(l)=H∧L/(c(l)).
+Our goal is to establish a relationship between the sheaves H∧L|Land C(H∧L|L)Ξ. We will
+see below that these sheaves become isomorphic sheaves of topolo gical algebras if we twist
+one of them by a 1-cocycle. The precise statement will be given below in Subsection 2.5. As
+we will explain there, this result is an enhanced version of Theorem 1.2 .1.
+2.5.Isomorphism of completions theorem, I. First of all, we will describe an isomor-
+phism
+θ0:H∧L
+(r+1)|L→C(H∧L
+(r+1)|L)Ξ.
+Recall that the algebras H(r+1),H(r+1)are nothing else but SV#Γ andSV#Γand C(•)
+stands forZ(Γ,Γ,•). The construction of θ0will be given after some preliminary considera-
+tions.
+The proof of the following lemma is straightforward.
+Lemma 2.5.1. There is a unique homomorphism θ0:H(r+1)→C(H(r+1))such that
+[θ0(γ)f](γ′) =f(γ′γ),
+[θ0(v)f](γ′) =γ′(v)f(γ′),
+γ,γ′∈Γ,v∈V,f∈FunΓ(Γ,H(r+1)),(2.9)
+The sheafOX0is the center of H(r+1)|X0and so also of C( H(r+1)|X0). Thanks to Lemma
+2.5.1, we have a natural map
+(2.10) π′∗(H(r+1)|X0) =OX0⊗(SV)ΓH(r+1)→C(H(r+1)|X0).
+Lemma 2.5.2. The homomorphism (2.10) is an isomorphism.
+Proof.It is enough to check that (2.10) is an isomorphism fiberwise. Pick b∈ X0and
+b′∈π−1(b) such that Γ b′⊂Γ.
+Let us describe the fiber of H(r+1)atb. This fiber is naturally identified with Ab#Γ,
+whereAb:=C[V∗]/C[V∗]nb,nbstanding for the maximal ideal of binC[V∗]Γ. Thanks to
+the slice theorem, Abis naturally identified with/summationtext
+v∈π−1(b)A0
+v, whereA0
+v:=C[V∗]/C[V∗]n0
+v,
+withn0
+vbeing the maximal ideal of vinC[V∗]Γv. SetA0:=A0
+b′. Let us identify Abwith
+FunΓb′(Γ,A0). Namely, to a∈A0
+vwe assign a map f: Γ→A0that mapsgto 0 ifgv/\e}atio\slash=b′and
+tog.aifgv=b′. It is easy to see that the assignment a/ma√sto→fais a Γ-equivariant isomorphism
+Ab∼− →FunΓb′(Γ,A0). SoAb#Γ = Fun Γb′(Γ,A0)#Γ. Further, we have a natural identification
+FunΓb′(Γ,A0)∼=FunΓ(Γ,FunΓb′(Γ,A0))
+induced by restricting a function Γ →A0to the left Γ -cosets.
+Similarly, we see that the fiber of C( H(r+1)) is identified with C(Fun Γb′(Γ,A0)#Γ). Set
+A0:= Fun Γb′(Γ,A0) so that the fibers of interest are identified with Fun Γ(Γ,A0)#Γ and
+C(A0#Γ). It is not difficult to see that under our identifications the map betw een the fibers
+induced by (2.10) coincides with ϑfrom Lemma 2.3.2. /square12 IVAN LOSEV
+So we get an isomorphism θ0:π′∗(H∧L
+(r+1))→C(H∧L
+(r+1)) of coherent sheaves of OX∧
+L-
+algebras induced by (2.10). But, since π′is etale onLand it induces a covering L→L
+with fundamental group Ξ, we see that π′:X∧
+L→X∧
+Lis just the quotient morphism for the
+(free) action of Ξ. So by restricting θ0to Ξ-invariants, we get an isomorphism θ0:H∧L
+(r+1)→
+C(H∧L
+(r+1))Ξ. Finally, restricting everything to L, we get the required isomorphism
+(2.11) θ0:H∧L
+(r+1)|L→C(H∧L
+(r+1)|L)Ξ.
+In fact, one can show that the sheaves H∧L|L,C(H∧L|L)Ξcannot be isomorphic for the
+open symplectic leaf L⊂X. To get an isomorphism we need to “twist” one of the sheaves
+with a cocycle.
+Namely, let us choose an open C×-stable affine covering L=/uniontext
+iWi. For alli,jchoose
+Ξ×Γ-invariant elements Xij∈czt(C(H∧L|L(Wij)) having degree 2 with respect to C×. Here
+Wij:=Wi∩Wj. We may (and will) assume that Xij=−Xji. Suppose that the elements
+exp(1
+tXij) form a 1-cocycle, i.e.
+(2.12) exp(1
+tXij)exp(1
+tXjk)exp(1
+tXki) = 1.
+A problem with (2.12) is that the elements exp(1
+tXij) do not make sense, because, at least
+if we are working algebraically, the series defining exp(1
+tXij) diverges even if we extend the
+sheaf of algebras under consideration. However, the expression s like
+ln(exp(1
+tXij)exp(1
+tXjk)exp(1
+tXki))
+still make sense (thanks to the Campbell-Hausdorff formula), the r esulting expression lies in
+1
+tczt(C(H∧L|L)(Wijk) (whereWijk:=Wi∩Wj∩Wk). And so when we write (2.12), we mean
+that
+ln/parenleftbigg
+exp(1
+tXij)exp(1
+tXjk)exp(1
+tXki)/parenrightbigg
+= 0.
+Given sections Xijas above we can form the twist C( H∧L|L)twof C(H∧L|L) by the cocycle
+exp(1
+tXij). Namely, C( H∧L|L)tw(Wi) := C(H∧L|L)(Wi) but the transition function from
+C(H∧L|L)(Wj) to C(H∧L|L)(Wi) onWijisu/ma√sto→exp(1
+tadXij)u(= exp(1
+tXij)uexp(−1
+tXij)).
+We note that exp(1
+tadXij)uconverges although exp(1
+tXij) does not. We also remark that
+C(H∧L|L)tw/(c) still identifies naturally with C( H∧L
+(r+1)|L). This is because1
+tadXijis zero
+modulo ( c).
+Theorem 2.5.3. Fora suitable 1-cocycle Xijas above there is a C×-equivariantisomorphism
+θ:H∧L|L→C(H∧L|tw
+L)Ξof sheaves of C[[c∗]]Γ-algebras onLmaking the following diagram
+commutative and such that θ(p∧L|L)coincides with the twist of C(p∧L|L)Ξ.
+H∧L
+(r+1)|LH∧L|L
+C(H∧L
+(r+1)|tw
+L)ΞC(H∧L|L)Ξ
+❄ ❄
+✲✲
+θ0θ
+ˆρ C(ˆρ)
+Let us see why this theorem implies Theorem 1.2.1. For b∈Lthe ideal mbofHgives rise
+to the sheaf of ideals m∧L
+b|L⊂H∧L|L. The completion of H∧L|Latb(i.e., the completionCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 13
+with respect to ( mb)∧L|L-adic topology) is nothing else but H∧b. Now let a point blie in
+π′−1(b)∩L. The completion of C( H∧L|L)Ξatbis the same as the completion of C( H∧L|L) at
+b. The decomposition (2.8) implies that this completion is naturally isomor phic to C( H∧0).
+Theorem 1.2.1 follows.
+The proof of Theorem 2.5.3 will be given in Subsection 2.13. It is rather indirect. We
+derive Theorem 2.5.3 from Theorem 2.7.3 below. The latter is another e nhanced version of
+Theorem 1.2.1.
+2.6.Sheafified versions of algebras, II. SetH:=OL⊗CH. This is a quasi-coherent
+sheaf ofOL-algebras. Define a global counterpart mof the ideals mbas follows. Let Idenote
+the sheaf of ideals in OL⊗CC[V∗] vanishing in all points ( b,γb),b∈L,γ∈Γ. Letmdenote
+the inverse image of I#Γ inH. Finally, we can define the completion H∧:= lim←−nH/mn. We
+can view H∧as a pro-coherent sheaf on Lwhose fiber at b∈LisH∧b. This claim follows
+from the exact sequence mb⊗OLH→H→H→0 and the right exactness of the completion
+functor, compare with Lemma 2.4.2. If U⊂Lis an open affine subvariety then the space
+of sections H∧(U) is the completion of C[U]⊗CHin them(U)-adic topology. Define the
+sheavesH(l),m(l),H∧
+(l)in a similar way.
+The Ξ-action onLgives rise to an action of Ξ on Hbyξ.r⊗h= (ξ.r)⊗h,ξ∈Ξ,r∈
+OL,h∈H. This action extends to an action of Ξ on H∧.
+Also we need a C×-action on H. Namely, recall the C×-action on H. Next, we have a
+C×-action onV∗:t.α=t−1α,α∈V∗,that gives rise to the C×-action onOL. We consider
+the diagonal C×-action on H. We note that the ideal I⊂OL⊗CC[V∗] is stable with respect
+to this action, so m(l)is also stable.
+Now we introduce a sheafified version of H∧0. Namely, set H:=OL⊗CH. We have the
+S(c)-linear action of /tildewideΞ onHgiven by /tildewideξ.γ=/tildewideξγ/tildewideξ−1,/tildewideξ.v=/tildewideξv. So we get an action of /tildewideΞ onH,
+/tildewideξ.(r⊗h) = (/tildewideξ.r)⊗(/tildewideξ.h). Also analogously to the above, we get a C×-action on H.
+Define the completion H∧ofH: consider the ideal Iof the zero section L→L×V∗,
+construct the ideal m⊂HfromIby analogy with m⊂H, and set H∧:= lim←−mH/mm. We
+have a natural isomorphism H∧=OL/hatwide⊗H∧0. The actions of /tildewideΞ,C×onHextend to H∧. As
+explained in Subsection 2.3, we get an action of Ξ on C( H∧).
+The sheaves of algebras we introduced have flat connections. We h ave a linear map
+α/ma√sto→Lα,basefromV∗
+0to the space of derivations of OL, mappingαto the Lie derivative −∂α
+corresponding to α. We can extend Lα,baseto the derivation LαofHbyLα=Lα,base⊗1.
+Then we can extend Lα,baseto the completion H∧uniquely. Similarly, we get derivations Lα
+ofH,H∧,C(H∧).
+Remark 2.6.1. The algebras H(l),H∧
+(l)have the flat connections α/ma√sto→Lα,(l)defined anal-
+ogously to Lα. We have H∧
+(l)=H∧/(c(l)), compare with Remark 2.4.3. A similar remark
+applies to H(l), etc.
+2.7.Isomorphism of completions theorem, II. We are going to establish an isomor-
+phism between H∧and a certain twist of C( H∧). Again, we start with H∧
+(r+1),C(H∧
+(r+1)). We
+will see that these two sheaves are isomorphic. Their isomorphism is a baby version of the
+isomorphism constructed in Theorem 3.2 from [BE].
+Letβdenote the tautological section of V∗
+0⊗OLgiven byb/ma√sto→b. Note that βisC×-
+invariant.14 IVAN LOSEV
+Proposition 2.7.1. A mapΘ0defined by
+[Θ0(γ)f](γ′) =f(γ′γ),
+[Θ0(v)f](γ′) = (γ′(v)+/a\}b∇acketle{tγ′(v),β/a\}b∇acket∇i}ht)f(γ′),
+γ,γ′∈Γ,v∈V,f∈FunΓ(Γ,H∧
+(r+1)),(2.13)
+extends to a unique homomorphism H(r+1)→C(H(r+1))of sheaves ofOL-algebras. This
+homomorphism is Ξ×C×- equivariant. Moreover, its natural extension to the compl etions
+H∧
+(r+1)→C(H∧
+(r+1))is an isomorphism.
+Proof.Theclaimabouttheextensiontoahomomorphismfollowsfromtherela tionΘ 0(γ)Θ0(v) =
+Θ0(γ.v)Θ0(γ). The equivariance is checked directly. To prove that the extensio n to the com-
+pletions is an isomorphism it is enough to show that it is an isomorphism fib erwise. This
+can be proved analogously to Lemma 2.5.2. /square
+In the sequel we will need a relation between Lα,(r+1)and Θ 0◦Lα,(r+1)◦Θ−1
+0. Forα∈V0
+let ˇα=ω(α,·). This is an element of V∗
+0. By the definition of the Poisson bracket on OL
+we have{ˇα,f}=−Lα,baseffor a section fofOL. SetˆLα,(r+1):=Lα,(r+1)+{ˇα,·}. Extend
+ˆLα,(r+1)toH(r+1),H∧
+(r+1),C(H∧
+(r+1)) in a natural way.
+Lemma 2.7.2. We have Θ0◦Lα,(r+1)◦Θ−1
+0=ˆLα,(r+1)as linear operators on C(H∧
+(r+1)).
+Proof.NotethatbothΘ 0◦Lα,(r+1)◦Θ−1
+0andˆLα,(r+1)arederivationsofC( H∧
+(r+1))thatcoincide
+onOL. So it is enough to verify that the two derivations coincide on some to pological
+generators of the sheaf C( H∧
+(r+1)) ofOL-algebras, for example, on Θ 0(γ) and Θ 0(v). This
+follows directly from (2.13). /square
+Now we are ready to state the second version of our isomorphism of completions result,
+Theorem 2.7.3. This theorem says that the flat (=equipped with flat c onnections) sheaves
+of algebras H∧,C(H∧) are Ξ×Γ×C×-equivariantly isomorphic up to a twist and, moreover,
+modulo ( c) the isomorphism coincides with Θ 0. Pick an open covering UiofLconsisting of
+C××Ξ-stable affine subsets.
+Theorem 2.7.3. There are isomorphisms Θi:H∧(Ui)→C(H∧)(Ui)and elements Xij∈
+czt(C(H∧(Uij))), whereUij:=Ui∩Uj, with the following properties:
+(1)Modulocthe isomorphism Θicoincides with Θ0for alli.
+(2)XijisΞ×Γ-invariant and has degree 2with respect to C×.
+(3) Θi◦(Θj)−1= exp(1
+tadXij).
+(4) Θi◦Lα◦(Θi)−1=ˆLα, whereˆLα:=Lα+1
+tad ˇα.
+(5)Xij=−Xji.
+(6) exp(1
+tXij)exp(1
+tXjk)exp(1
+tXki) = 1(see the discussion before Theorem 2.5.3).
+(7)ˆLαXij= 0.
+2.8.Spherical subalgebras and the centers. Lete=1
+|Γ|/summationtext
+γ∈Γγ∈CΓ be the trivial
+idempotent. By definition, the spherical subalgebras U,U(l)inH(l)areeHe,eH(l)e. Clearly,
+U(l)=U/(c(l)). Similarly, we can define the spherical subalgebra U(l)⊂H(l)(using the
+trivial idempotent efor Γ).COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 15
+LetZ(l),l= 1,...,r+1,(resp.,Z(l)) denote the center of H(l)(resp.,H(l)). In particular,
+Z(r+1)= (SV)Γ⊂SV#Γ =H(r+1). SetZ(=Z(0)) :=zt(H). Define the algebra Z(0)
+similarly.
+Also we can introduce the sheafified versions U(l):=eH(l)e=OL⊗U(l),Z(l):=OL⊗
+Z(l),U(l),Z(l)of the algebras under consideration.
+Proposition 2.8.1. (1)Forl>0the mapz/ma√sto→ezinduces isomorphism Z(l)→U(l),Z(l)
+→U(l)(the so called Satake isomorphisms). Similarly, the map z/ma√sto→ezinduces
+isomorphisms Z(l)→U(l),Z(l)→U(l).
+(2)Forl= 0,1,...,rone hasρl(Z(l)) =Z(l+1),ρl(Z(l)) =Z(l+1),ρl(Z(l)) =Z(l+1),ρl(Z(l)) =
+Z(l+1).
+Proof.Assertion (1) was essentially verified by Etingof and Ginzburg in [EG], T heorem 3.1.
+Now assertion (1) easily implies assertion (2). /square
+Corollary 2.8.2. LetIbe an ideal in Z(l),l>0. ThenH(l)I∩Z(l)=I. An analogous claim
+holds for Z(l)⊂H(l)etc.
+Proof.Clearly,I⊂H(l)I∩Z(l). Thanks to theSatake isomorphism, it isenough to show that
+eI⊃e(H(l)I∩Z(l)). Bute(H(l)I∩Z(l)) =e(H(l)I∩Z(l))e⊂eH(l)Ie=eH(l)eeI=eI./square
+Proposition 2.8.3. (1)The algebra H(l)is finite over Z(l)forl>0.
+(2)The center of Hcoincides with S(c).
+The similar claims hold for the algebras H,H(l).
+Proof.The first assertion is essentially due to Etingof and Ginzburg, [EG], Th eorem 3.3.
+The second one follows from results of Brown and Gordon, [BrGo], Pr oposition 3.2. /square
+Wefinishthesubsection byrecalling thePoisson structures on Z(l),U(l)thatareessentially
+due to Etingof and Ginzburg. We start with Z(1). Choose an arbitrary embedding ι:Z(1)→
+Hsuch thatρ0◦ι= id. Then it is easy to see that ρ0([ι(a),ι(b)]) = 0 for any a,b∈Z(1)
+and that1
+t[ι(a),ι(b)]∈zt(H). So an element {a,b}:=ρ0(1
+t[ι(a),ι(b)]) is well-defined. It is
+straightforward to check that {·,·}does not depend on the choice of ιand that{·,·}is a
+S(c(1))-bilinear Poisson bracket on Z(1). In particular, we have the induced brackets on Z(l).
+In the sequel we will need an embedding Z(1)→Hwith some special properties.
+Lemma 2.8.4. We have a graded (= C×-equivariant) S(c(1))-linear section ι:H(1)→Hof
+ρ0.
+Proof.TheS(c)-algebra His a free graded (=graded flat) S(c)-module. So H∼=H(1)[t] as a
+gradedS(c)-module. Inparticular,wehaveagraded S(c(1))-linearembedding H(1)֒→H./square
+This lemma allows us to define (non-canonical) maps Z(l)×H(l)→H(l)induced from
+(a,b)/ma√sto→ρ0(1
+t[ι(a),ι(b)]),a∈Z(1),b∈H(1).
+Lemma 2.8.4 shows that the induced bracket on Z(r+1)= (SV)Γcoincides with the one
+coming from the symplectic form on V. Indeed, the bracket on ( SV)Γis obtained using a
+sectionSVΓ→H/(c(1)) =At#Γ ofAt#Γ→SV#Γ, where Atstands for the Weyl algebra
+ofV. With this interpretation of the bracket our claim is easy.
+We can equip U(1)with a Poisson bracket using a section U(1)→U. It is easy to see that
+the isomorphism in Proposition 2.8.1 preserves the Poisson brackets .16 IVAN LOSEV
+Similarly, we can equip Z(l),U(l),l>0,with Poisson structures. Also the sheaves Z(l),U(l),
+Z(l),U(l)become sheaves of Poisson algebras. We also can extend the brack et{·,·}:Z(l)⊗
+H(l)→H(l)toZ(l)⊗H(l)→H(l)byOL-bilinearity.
+2.9.Compatibility of filtrations. Recall that two decreasing filtrations F mV,F′
+mVof a
+vectorspacearecalled compatible ifforanymthereisnwithFnV⊂F′
+mVandF′
+nV⊂FmV.
+In this subsection we will, roughly speaking, prove that all filtrations onH,Z(resp., on
+H,Z) related to the m-adic filtration on H(resp., the p-adic filtration on H) are compatible.
+Proposition 2.9.1. The following descending filtrations are compatible:
+(1) FmH(l):=H(l)(Z(l)∩m(l))m,F′
+mH(l):=mm
+(l)(for anyl= 0,1,...,r+1). We remark
+thatFmH(l)is actually a two-sided ideal in H.
+(2) FmZ(l):= (Z(l)∩m(l))m,F′
+mZ(l):=Z(l)∩mm
+(l),l/greaterorequalslant1.
+The similar claims hold for H(l),Z(l)and the ideals related to mb.
+In the proof we will need a technical lemma.
+Lemma 2.9.2. LetIbe a two-sided ideal in Zcontaining tH. ThenHIm= (HI)mfor all
+m. The similar claim holds for Z⊂H.
+Proof.This follows from [ H,I]⊂[H,Z]⊂tH⊂I. /square
+Proof of Proposition 2.9.1. We remark that the filtrations mnand (H(m∩Z))nonHare
+compatible. Indeed, it is enough to show that mn⊂H(m∩Z) for somen. Since the r.h.s.
+contains c, it is enough to prove the analogous inclusion modulo ( c). There it is enough to
+prove the inclusion fiberwise. This reduces to checking that In
+b⊂SV(Ib)Γfor sufficiently
+largen. But this is clear.
+It is enough to prove (1) for l= 0. Thanks to the previous paragraph, (1) follows from
+Lemma 2.9.2.
+(2) follows from (1) and Corollary 2.8.2. /square
+We can sheafify Z(l)onXanalogously to Subsection 2.4. The proof of the following
+proposition is completely analogous to that of Proposition 2.9.1.
+Proposition 2.9.3. The following descending filtrations are compatible:
+(1) FmH(l)|X0:=H(l)(Z(l)∩p(l))m|X0,F′
+mH(l)|X0:=pm
+(l)|X0(for anyl= 0,1,...,r+1).
+(2) FmZ(l)|X0:= (Z(l)∩p(l))m|X0,F′
+mZ(l)|X0:=Z(l)∩pm
+(l)|X0,l/greaterorequalslant1.
+Define the completion Z∧
+(l)ofZ(l)with respect to any of the equivalent filtration of Propo-
+sition 2.9.1. Alternatively, Z∧
+(l)is the closure of Z(l)inH(l). Also we can define the completion
+Z∧bofZ.
+Similarly, we can define the sheaf Z∧L
+(l)|L⊂H∧L
+(l)|L. By the construction and the properties
+of completions, [Ei], Chapter 7, we have the following lemma.
+Lemma 2.9.4. Forl>0we have the following statements:
+(1)The sheaf Z∧
+(l)is flat over Z(l). AlsoZ∧b
+(l)is flat over Z(l).
+(2)Z∧
+(l)/(cl)∼=Z∧
+(l+1)andZ∧b
+(l)/(cl)∼=Z∧b
+(l+1).
+(3)Z∧L
+(l)|Lis flat over Z(l).
+(4)Z∧L
+(l)|L/(cl)∼=Z∧L
+(l+1)|L.COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 17
+Also we remark that the analogs of Propositions 2.9.1,2.9.3 hold also for H,Z,H,Z. There
+thecorrespondingstatementsareeveneasier, thankstothede compositions m=OL⊗m0,p=
+At⊗C[t]m+
+0.
+Corollary 2.9.5. (1)ρl(Z∧
+(l)) =Z∧
+(l+1)andρl(Z∧b
+(l)) =Z∧b
+(l+1).
+(2)Forl>0we haveZ∧
+(l)=z(H∧
+(l)),Z∧b
+(l)=z(H∧b
+(l)).
+Proof.Assertion (1) follows from assertions (1) and (2) of Lemma 2.9.4.
+Let us proceed to assertion 2. We have the inclusion Z∧
+(l)⊂z(H∧
+(l)). It is an inclusion of
+procoherentOL-sheaves. Therefore it is enough to check the equality fiberwise. B y assertion
+1,ρl(Z∧b
+(l)) =Z∧b
+(l−1). It follows easily from Proposition 2.8.1 that the algebra Z(l)is flat over
+S(c(l)). By assertion (1) of Lemma 2.9.4, the algebra Z∧b
+(l)is flat over C[[c∗
+(l)]].
+Suppose that we know that Z∧b
+(l+1)=z(H∧b
+(l+1)). Then we have z(H∧b
+(l)) =Z∧b
+(l)+clz(H∧b
+(l)).
+Sincez(H∧b
+(l)) is closed in H∧b
+(l), the last equality easily implies Z∧b
+(l)=z(H∧b
+(l)).
+This reduces the proof to the case l=r+ 1. Recall the isomorphism Θb
+0:H∧b
+(r+1)→
+C(H∧0
+(r+1)). Under this isomorphism Z∧b
+(r+1)goes to (SV∧0)Γ. Now our claim is clear. /square
+An analogous argument, of course, applies to Z(l)|X⊂H(l)|X,Z(l)|L⊂H(l)|L,Z∧L
+(l)|L⊂
+H∧L
+(l)|L. We will need the following corollary of the analog of assertion (2).
+Corollary 2.9.6. Z |X0(resp.,Z|L) is dense in zt(H∧L)(resp.,zt(H∧L|L)).
+2.10.Blow-ups and flatness. We are going to prove that H∧
+(l)is flat over H(l), whileH∧L|L
+is flat over H|L(=the sheaf-theoretic pull-back of H|X0toL).
+A standard technique to prove such statements is to consider the blow-up algebras. For
+an associative algebra Aand a two-sided ideal J⊂Aone can form the blow-up algebra
+BlJ(A) =/circleplustext∞
+i=0Ji. This algebra is Z/greaterorequalslant0-graded. To ensure nice properties of the completion
+A∧:= lim←−iA/Jiwe need the blow-up algebra Bl J(A) to be Noetherian.
+More generally, given a sequence Ji,i= 1,2,...,of two-sided ideals in AwithJiJj⊂
+Ji+jone can consider the completion A∧:= lim←−iA/Jiand the blow-up algebra Bl (Ji)(A) =/circleplustext∞
+i=0Ji, whereJ0=A.
+The following lemma is proved completely analogously to the correspon ding statements
+for commutative algebras, see, for example, [Ei], Chapter 7.
+Lemma 2.10.1. LetA,Jibe as above. Suppose Bl(Ji)(A)is Noetherian. Then
+(1)The algebra/circleplustext∞
+i=0Ji/Ji+1is Noetherian.
+(2)The algebra A∧is Noetherian.
+(3)The algebra A∧is a flat (left or right) A-module.
+(4)The completion functor M/ma√sto→M∧:= lim←−iM/JiMfrom the category of finitely gen-
+erated leftA-modules to the category of left A∧-modules is exact. Moreover, M∧is
+canonically isomorphic to A∧⊗AM.
+Of course, Lemma 2.10.1 can be generalized to sheaves in a straightf orward way.
+In [Lo2], Lemma 2.4.2, we have proved the following statement.
+Lemma 2.10.2. LetAbe aC[t]-algebraand Ibe a two-sidedidealof Acontaining t. Suppose
+thatAis complete and separated with respect to the t-adic topology. Further, suppose that
+the algebra A/(t)is commutative and Noetherian. Finally, suppose that [J,J]⊂tJ. Then
+the algebra BlJ(A)is Noetherian.18 IVAN LOSEV
+Although we can prove that Bl m(H) is Noetherian, it is easier to do this for a related sheaf,
+which is also enough for our purposes.
+Setm′
+i:=H(Z∩m)2i. By Proposition 2.9.1, the m-adic topology on Hcoincides with the
+one defined by the sequence m′
+i. Remark that m′
+im′
+j⊂m′
+i+jbecause [ H,Z∩m]⊂tH⊂Z∩m.
+LetH∧tdenote the t-adic completion of H. Abusing the notation, we write m′
+ifor the
+completion of m′
+i⊂HinH∧t.
+Proposition 2.10.3. The sheaf of algebras/circleplustext
+i/greaterorequalslant0m′
+iis Noetherian.
+Proof.The proof is analogous to that of Lemma A2 in [Lo3]. For reader’s conv enience we
+will provide the proof here.
+Fix an open affine subset U⊂L. We need to prove that/circleplustext
+i/greaterorequalslant0m′
+i(U) is a Noetherian
+algebra. But this algebra equals H(U)BlI(A), whereA:=Z(U),I:= (Z(U)∩m(U))2. The
+algebraH(U) is finite over A⊂BlI(A). So/circleplustext
+i/greaterorequalslant0m′
+i(U) is finite over Bl I(A) and it is enough
+to show that Bl I(A) is Noetherian.
+According to Lemma 2.10.2, to do this we need to check that the algeb raA/tAis com-
+mutative and finitely generated and that [ I,I]⊂tI.
+The fact that the algebra A/tAis commutative has been already proved when we dis-
+cussed the Poisson bracket on Z(1)(orZ(1)). To show that the algebra A/tAis finitely
+generated consider the epimorphism A/tA։Z(1)(U). Its kernel is naturally identified with
+tH(U)/tZ(U). In particular, its square is zero so we can view the kernel as a Z(1)(U)-module.
+This module is finitely generated. The algebra Z(1)(U) is finitely generated as well. So we
+see thatA/tAis finitely generated.
+Letuscheckthat[ I,I]⊂tI. Thisboilsdowntocheckingthat {[Z(1)(U)∩m(1)(U)]2,[Z(1)(U)∩
+m(1)(U)]2}⊂[Z(1)(U)∩m(1)(U)]2. But this is clear. /square
+We have the following corollary of Lemma 2.10.1 and Proposition 2.10.3.
+Corollary 2.10.4. (1)The sheaf H∧
+(l)and the algebra H∧bare Noetherian.
+(2)H∧
+(l)(resp.,H∧b
+(l)) is a flat (left or right) H(l)-module (resp., H(l)-module).
+A similar construction can be applied to p|X0⊂H|X0. Namely, set pi:=H(Z∩p)2i.
+By Proposition 2.9.3, the p|X0-adic topology on H|X0coincides with the one defined by the
+sequence pi|X0.
+Proposition 2.10.5. The sheaf of algebras/circleplustext
+i/greaterorequalslant0pi|X0is Noetherian.
+Proof.The proof basically repeats that of Proposition 2.10.3. The most ess ential difference
+is that the algebra Z(1)(U) is not finitely generated. However, this algebra is still Noetherian.
+Indeed,Z(1)(U) is complete in the ( c(1))-adic topology and its quotient by ( c(1)),Z(r+1)(U),
+is finitely generated. /square
+Again, this proposition implies the following corollary.
+Corollary 2.10.6. (1)H∧L
+(l)|Lis a flat sheaf of (left or right) H(l)-modules.
+(2)H∧L
+(l)|L/(cl)∼=H∧L
+(l+1)|L.
+We will need some further corollaries in Subsection 3.6.
+Finally, we remark that all results in this subsection hold also for H∧,H∧betc. They are
+even simpler, compare with the remark at the end of the previous su bsection.COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 19
+2.11.Derivations. Throughout this subsection Uisanopen C××Ξ-stable affine subvariety
+ofL. We setR:=C[U]. In this subsection we are going to study derivations of the algebra
+H∧(U). We will show that all R-linear derivations of this algebra are “almost inner” in
+the sense of (A) of Proposition 2.11.1. Also we will establish the existe nce of an “Euler”
+derivation ofH∧(U). This derivation is crucial in the “local” construction of an isomorph ism
+Θ from Theorem 2.7.3.
+First of all, one can consider the space Der R[[c∗]](H∧(U)) ofR[[c∗]]-linear derivations. Fur-
+ther, we say that a R-linear derivation dofH∧isweakly Euler ifd|c= 2id. Clearly, weakly
+Euler derivations form an affine space, whose associated vector sp ace is Der R[[c∗]](H∧(U)).
+This affine space is non-empty. Indeed, let E(l)denote the derivation of H(l)induced by
+theC×-action. Set Ef,(l):=E(l)−/summationtextdimV0
+i=1αiLαi,(l), whereα1,...,α dimV0is a basis of V0, and
+αiare the dual basis vectors in V∗
+0. ThenEf,(l)isOL-linear,Ef,(l)v=v,v∈V,Ef,(l)ci= 2ci.
+ExtendEf,(l)toH∧
+(l). Clearly,Ef,(l)is a weakly Euler derivation of H∧
+(l). Similarly, we can
+define the derivation Ef,(l)ofH(l),H∧
+(l),C(H∧
+(l)).
+Now let us define Euler derivations. We say that a derivation dofH∧(U) isEulerif it
+is weakly Euler, C××Γ×Ξ-equivariant, and the induced derivation of H∧
+(r+1)(U) coincides
+with Θ−1
+0◦Ef,(r+1)◦Θ0.
+Proposition 2.11.1. The following statements hold:
+(A)All elements of DerR[[c∗]](H∧(U))are of the form1
+c0ada, wherea∈Z∧(U). Recall
+that, by definition, c0=t.
+(B)There is an Euler derivation of H∧(U).
+Proof.First, we reduce (A),(B) to the similar claims for the algebras H∧
+(l)(U),l= 1,...,r+1.
+Recall that Z∧
+(l)(U) is a Poisson algebra for any l>0 and coincides with the center of H∧
+(l)(U)
+(the latter follows from Corollary 2.9.5). We say that an R-linear derivation of H∧
+(l)(U) is:
+•Poisson, if it is R[[c∗
+(l)]]-linear and annihilates the Poisson bracket on Z∧
+(l)(U) (we say
+that a derivation Dof an algebra Aannihilates a bracket {·,·}ifD{a,b}−{Da,b}−
+{a,Db}= 0 for alla,b∈A).
+•weakly Euler, if it acts on c(l)by 2·id and also multiplies the Poisson bracket {·,·}
+onZ∧
+(l)(U) by−2.
+The notion of an Euler derivation of H∧
+(l)is introduced exactly as above.
+AnyR[[c∗]]-linear(resp., weaklyEuler, Euler)derivationof H∧(U)inducesaPoisson(resp.,
+weakly Euler, Euler) derivation of H∧
+(l)(U).
+Consider the following statements, where l= 1,...,r+1,
+(Al) Any Poisson derivation d0ofH∧
+(l)(U) has the form d0(c) ={a1,c}+ [a2,c],a1∈
+Z∧
+(l)(U),a2∈H∧
+(l)(U). Moreover, any such derivation lifts to H∧
+(l−1)(U). Here{·,·}is
+defined as the continuous extension of {·,·}:Z(l)⊗H(l)→H(l)see the concluding
+remarks in Subsection 2.8.
+(Bl) There is an Euler derivation of H∧
+(l)(U).
+(Cl) Any derivation d0ofH∧
+(l)(U) vanishing on Z∧
+(l)(U) is inner.
+(A1)&(C1)⇒(A): Letd∈DerR[[c∗]](H∧(U)) and letd0be the corresponding derivation
+ofH∧
+(1)(U). Choose a1as in (A 1) so that the restriction of d0toZ∧
+(1)(U) coincides with
+{a1,·}. Lifta1to an element a′∈H∧(U). Replacing dwithd:=d−1
+c0ad(a′) we get that
+d0vanishes on Z∧
+(1)(U). So by (C 1),d0is inner, i.e., there is a2such thatd0= ad(a2).20 IVAN LOSEV
+Lifta2to an element a′′ofH∧(U). Replacing dwithd−ad(a′′) we getd0= 0. In other
+words, imd⊂c0H∧(U). SinceH∧(U) is a flat C[[c0]]-module (Corollary 2.10.4), we see that
+d=c0d1for somed1∈DerR[[c∗]](H∧(U)), which completes the proof of the lifting property.
+We can apply the same argument to d1and replace c0d1withc2
+0d2and so on. Since H∧(U)
+is complete in the c0-adic topology, we see that dhas the required form.
+The proofs of the implications (C l)⇒(Cl−1) and (A l)&(Cl)⇒(Al−1) forl >1 are very
+similar and so we omit them (one needs to consider {a′,·}instead of1
+c0ad(a′)).
+Let us now prove (A r+1) and (C r+1). Both stem from the following result.
+Lemma 2.11.2. The following statements hold:
+(1)The restriction of derivations from H∧
+(r+1)(U)toZ∧
+(r+1)(U)gives rise to an isomor-
+phismHH1
+R(H∧
+(r+1)(U))→HH1
+R(Z∧
+(r+1)(U)) = Der R(Z∧
+(r+1)(U)), whereHHdenote the
+Hochschild cohomology.
+(2)Any Poisson derivation of Z∧
+(r+1)(U)is inner.
+Proof of Lemma 2.11.2. RecalltheisomorphismΘU
+0:H∧
+(r+1)(U)→C(H∧
+(r+1)(U))fromPropo-
+sition 2.7.1. This isomorphism restricts to an isomorphism Z∧
+(r+1)(U)→Z∧
+(r+1)(U). As we
+explained in Subsection 2.3, there is a natural map
+(2.14) Der R(H∧
+(r+1)(U))→DerR(C(H∧
+(r+1)(U))).
+ThealgebraC( H∧
+(r+1)(U))isisomorphictoamatrixalgebraover H∧
+(r+1)(U). Itfollowsthatthe
+map (2.14) yields an isomorphism of the 1st Hochshild cohomology grou ps. Since this map
+intertwinestherestrictionmapsDer R(H∧
+(r+1)(U))→DerR(Z∧
+(r+1)(U)),DerR(C(H∧
+(r+1)(U)))→
+DerR(Z∧
+(r+1)(U)), weseethatitisenoughtoprovetheanalogsof(1),(2)for H∧
+(r+1)(U),Z∧
+(r+1)(U).
+Let us prove (1). Recall the fiberwise Euler derivation Ef,(r+1)ofH∧
+(r+1)(U). Note that
+any element of Der R(H∧
+(r+1)(U)) is uniquely determined by its restrictions to CΓandV.
+These subspaces of H∧
+(r+1)(U) have degrees 0,1 with respect to Ef,(r+1). It follows that any
+elementd∈DerR(H∧
+(r+1)(U)) can be uniquely written as the (automatically converging)
+sum/summationtext∞
+i=−1di, where [Ef,(r+1),di] =i·di. Note that each diis a derivation of H(r+1)(U) =
+R⊗SV#Γ. It is easy to see that Der R(R⊗SV#Γ) =R⊗Der(SV#Γ), DerR(R⊗(SV)Γ) =
+R⊗Der((SV)Γ).
+Analogously to the proof of Theorem 9.1 in [Et], one can show that the natural map
+(Der(SV))Γ→Der(SV#Γ) gives rise to an isomorphism (Der( SV))Γ→HH1(SV#Γ).
+Since the morphism V→V/Γis ´ etale in codimension 1, we see that the restriction map
+(Der(SV))Γ→Der((SV)Γ) is an isomorphism. (1) follows.
+The proof of (2) is similar. Namely, we reduce the proof to checking t hat any Poisson
+R-linear derivation of Z(r+1)(U) =R⊗(SV)Γis Hamiltonian. We have PDer( Z(r+1)(U)) =
+R⊗PDer((SV)Γ), wherePDerdenotesthespaceof R-linearPoissonderivations. Theisomor-
+phism(Der( SV))Γ→Der((SV)Γ)restrictstoanisomorphism[PDer( SV)]Γ→PDer((SV)Γ).
+However, any Poisson derivation of SVis Hamiltonian. /square
+(B1)⇒(B): LetE1be an Euler derivation of H∧
+(1)(U). ThenEf,(1)−E1is a Poisson
+derivation of H∧
+(1)(U). By (A 1), being a Poisson derivation of H∧
+(1)(U),Ef,(1)−E1can be
+liftedtoanelement d∈DerR[[c∗]]Γ(H∧(U)). Replace dwithitsC××Γ×Ξ-invariantcomponent
+(thisispossible because theactionof C×is pro-algebraic). Then Ef+disanEuler derivation
+ofH∧(U).COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 21
+The proof of (B l)⇒(Bl−1) forl>1 is completely analogous.
+The claim (B r+1) is vacuous, for Θ−1
+0◦Ef,(r+1)◦Θ0is an Euler derivation of H∧
+(r+1)(U)./square
+The following proposition explains why the existence of an Euler deriva tion is important.
+Proposition 2.11.3. LetEbe an Euler derivation of H∧(U). SetHi(U) :={a∈H∧(U) :
+Ea=ia},i∈Z,Hi
+(r+1)(U) :={a∈H∧
+(r+1)(U) : Θ−1
+0◦Ef,(r+1)◦Θ0(a) =ia}. Then the
+following claims hold:
+(1)Hi(U) ={0}fori<0.
+(2) ˆρ:H∧(U)→H∧
+(r+1)(U)gives isomorphisms Hi(U)→Hi
+(r+1)(U),i= 0,1.
+(3)The map/producttext∞
+i=0Hi(U)→H∧(U),/producttext∞
+i=0v(i)/ma√sto→/summationtext∞
+i=0v(i),is well-defined (meaning that
+the r.h.s. converges) and is a bijection.
+Proof.Letusdefinetheactionof EonH∧
+(r+1)(U)[[c∗]]asfollows: EactsonH∧
+(r+1)(U)byΘ−1
+0◦
+Ef,(r+1)◦Θ0and by 2 on c. We remark that the analogs of claims (1),(3) hold for H∧
+(r+1)(U).
+Recallthat H∧(U)isC[[c∗]]-flatandwehavethenaturalisomorphism H∧(U)/(c)∼=H∧
+(r+1)(U)
+intertwining the action of E. Now (1) for H∧
+(r+1)(U) implies that there is a E-equivariant
+C[[c∗]]-linear isomorphism H∧(U)∼=H∧
+(r+1)(U)[[c∗]].
+All three claims follow easily from here. /square
+In the next subsection we will need the following result.
+Lemma 2.11.4. LetU⊂Lbe an open affine subset, and X∈zt(C(H∧(U))). Suppose that
+the image of the derivation1
+tadXlies inczt(C(H∧(U))). ThenX∈czt(C(H∧(U)))+C[U].
+Proof.We will prove by the induction on i= 0,...,r+1 that
+X∈(c0,...,ci−1)zt(C(H∧(U)))+(ci,...,cr)C(H∧(U)).
+LetXidenote the image of Xin C(H∧(U))/(ci,...,cr). Since C[U] coincides with the
+Poisson center of C( H∧
+(r+1)(U)), we see that X0∈C[U] and so we have the base of induction.
+Now suppose that we have proved our claim for i. Let us prove it for i+ 1. The
+natural homomorphism zt(C(H∧(U)))→zt(C(H∧(U))/(ci+1,...,cr)) is surjective (com-
+pare with Corollary 2.9.5). This reduces the claim for i+ 1 to checking that Xi+1∈
+(c0,...,ci)zt(C(H∧(U))/(ci+1,...,cr)). LetX′
+idenote some lifting of Xito the algebra
+zt(C(H∧(U))/(ci+1,...,cr)). ThenX′
+i−Xi+1=ciYfor someY∈C(H∧(U))/(ci+1,...,cr).
+Of course, ciY∈zt(C(H∧(U))/(ci+1,...,cr)). Since C( H∧(U))/(ci+1,...,cr) is flat over
+C[c0,...,ci], we see that Y∈zt(C(H∧(U))/(ci+1,...,cr)), and we are done. /square
+2.12.Proof of Theorem 2.7.3. In this subsection we will prove Theorem 2.7.3. The first
+stepistoshowthatΘexistslocally. Asbefore, let UbeaC××Ξ-stableopenaffinesubvariety
+ofLandR:=C[U].
+Proposition 2.12.1. There is an R[[c∗]]-linearC××Γ×Ξ-equivariant isomorphism ΘU:
+H∧(U)→C(H∧(U))liftingΘU
+0:H∧
+(r+1)(U)→C(H∧
+(r+1)(U)).
+Proof.Recall the idempotent e(Γ)∈C(H∧
+(r+1)(U)). In the notation of Proposition 2.11.3,
+we have C( CΓ)⊂H0
+(r+1)(U). So we can use ˆ ρto get an embedding C( CΓ)֒→H0(U). It
+is clear that ˆ ρ:H0(U)→H0
+(r+1)(U) is an isomorphism of algebras. So we may consider
+C(CΓ) as a subalgebra in H0(U)⊂H∧(U). Thanks to Lemma 2.3.1, the algebra H∧(U)
+is naturally identified with C( e(Γ)H∧(U)e(Γ)). So we need to establish an isomorphism22 IVAN LOSEV
+e(Γ)H∧(U)e(Γ)→H∧(U). Consider the subspace e(Γ)H1(U)e(Γ)⊂e(Γ)H∧(U)e(Γ). It is
+identified with R⊗(V⊗CΓ)by means of ˆ ρ. Letι:V ֒→e(Γ)H1(U)e(Γ) bethecorresponding
+embedding.
+Let us check that
+(2.15) [ ι(u),ι(v)] =c0ω(u,v)+/summationdisplay
+s∈S∩Γc(s)ωs(u,v)s.
+Considerv∈V ֒→H(U). We can write the decomposition v=/summationtext∞
+i=0v(i)withv(i)∈Hi(U).
+The formulas (2.13) imply that e(Γ)Θ0(v) = Θ0(v)e(Γ) =v+/a\}b∇acketle{tβ,v/a\}b∇acket∇i}htfor allv∈V. Therefore
+(2.16) e(Γ)ve(Γ)≡e(Γ)v≡ve(Γ)≡/a\}b∇acketle{tβ,v/a\}b∇acket∇i}ht+ι(v) mod/productdisplay
+i/greaterorequalslant2Hi(U).
+Recall that [ u,v] =c0ω(u,v)+/summationtext
+s∈Sc(s)ωs(u,v)s. It follows that
+(2.17) e(Γ)[u,v]e(Γ) =c0ω(u,v)+/summationdisplay
+s∈S∩Γc(s)ωs(u,v)s.
+On the other hand, from (2.16) we deduce that
+(2.18) [ ι(u),ι(v)]≡e(Γ)[u,v]e(Γ)≡c0ω(u,v)+/summationdisplay
+s∈S∩Γcsωs(u,v)smod/productdisplay
+i/greaterorequalslant3Hi(U).
+Since [ι(u),ι(v)]∈H2(U), (2.15) is proved.
+(2.15) implies that there is a unique R[[c∗]]Γ-linear isomorphism ΘU:e(Γ)H∧(U)e(Γ)→
+H∧(U) coinciding with ιonV. The natural extension of this isomorphism
+ΘU:H∧(U) = C(e(Γ)H∧(U)e(Γ))→C(H∧(U))
+is the isomorphism we need in the proposition. This isomorphism is C××Γ×Ξ-equivariant
+because the Euler derivation EisC××Γ×Ξ-equivariant, by definition. /square
+To finish theproof of Theorem 2.7.3 (gluethe isomorphisms ΘUtogether in anappropriate
+way) we need two technical lemmas.
+Lemma 2.12.2. There are
+(i)ElementsXij∈czt(C(H∧(Uij)))satisfying (2),(3),(5) of Theorem 2.7.3,
+(ii)and aΓ×Ξ-equivariant map V∗
+0→czt(C(H∧))(Ui),α/ma√sto→Yi
+α,of degree 2 with respect
+toC×
+such that for all α,β∈V∗
+0the following hold
+ΘUi◦Lα◦(ΘUi)−1=ˆLα+1
+tad(Yi
+α). (2.19)
+cijk:=tln/parenleftbigg
+exp(1
+tXij)exp(1
+tXjk)exp(1
+tXki)/parenrightbigg
+∈cC[Uijk][[c∗]], (2.20)
+cij(α) := ˇα+Yi
+α−exp(1
+tXij)(ˇα+Yj
+α)exp(−1
+tXij)−exp(1
+tXij)[Lαexp(−1
+tXij)] (2.21)
+∈cC[Uij][[c∗]],
+ci(α,β) :=Lα(ˇβ+Yi
+β)−Lβ(ˇα+Yi
+α)+1
+t[ˇα+Yi
+α,ˇβ+Yi
+β]−ω(α,β)∈cC[Ui][[c∗]]. (2.22)COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 23
+Proof.The automorphism ΘUi◦(ΘUj)−1of C(H∧(Uij)) is the identity modulo ( c). Therefore
+d:= ln(ΘUi◦(ΘUj)−1) converges and is a C××Γ×Ξ-equivariant C[Uij][[c∗]]-linear derivation
+of C(H∧(Uij)). Clearly, dis zero modulo ( c), and ΘUi◦(ΘUj)−1= exp(d). By Proposition
+2.11.1,d=1
+tad(Xij) for some Xij∈zt(C(H∧))(Uij). By Lemma 2.11.4, since dis trivial
+modulo ( c), we haveXij∈C[Uij][[c∗]]+czt(C(H∧))(Uij). SinceC[Uij][[c∗]]⊂z(C(H∧))(Uij),
+we may assume that actually Xij∈czt(C(H∧)(Uij)). Further, after replacing Xijwith its
+appropriate isotypic component for the action of C××Γ×Ξ, we may assume that Xij
+satisfies (2) of Theorem 2.7.3. Also we may assume that Xij=−Xji. Since
+(2.23) exp(1
+tadXij) = ΘUi◦(ΘUj)−1,
+we see that exp(1
+tadXki)exp(1
+tadXjk)exp(1
+tadXij) = id. Therefore the left hand side
+of (2.20) lies in the center of C( H∧(Uijk)). Proposition 2.8.3 implies that the center of
+C(H∧(Uij)) coincides with C[Uij][[c∗]] hence (2.20).
+Similarly, we see that there is a map α/ma√sto→Yi
+αsatisfying (ii) and (2.19). To prove (2.22) we
+notice that [ Lα,Lβ] = 0 and repeat the argument in the previous paragraph. (2.21) fo llows
+in a similar way from (2.23) and (2.19) together. /square
+It is pretty easy to see that cij(α) =−cji(α) andcijk=−cjik=−cikj. So the triple
+c= (ci(•,•),cij(•),cijk) is a 2-cochain in the ˇCech-De Rham complex of L.
+Lemma 2.12.3. The 2-cochain cis a coboundary.
+Proof.Below we write ˆYi
+αfor ˇα+Yi
+α.
+The varietyLis the complement to the union of certain subspaces of even codimen sion in
+V∗
+0. It follows that H2
+DR(L) = 0. Therefore it is enough to check that cis a 2-cocycle. This
+reduces to checking the following four equalities:
+Lαci(β,γ)+Lβci(γ,α)+Lγci(α,β) = 0, (2.24)
+ci(α,β)−cj(α,β) =Lαcij(β)−Lβcij(α), (2.25)
+cij(α)+cjk(α)+cki(α) =Lαcijk, (2.26)
+cijk−cijl+cikl−cjkl= 0. (2.27)
+(2.24) reduces to the Jacobi identity for Yi
+α,Yi
+β,Yi
+γ.
+Below we write Aijinstead of exp(1
+tXij). Again, although Aijis not well-defined (because
+it diverges) all expressions involving Aijbelow will be well-defined. More precisely, we will
+need expressions of the form AijxAji,AjiD(Aij) for some derivation D. The former just
+equals exp(1
+tadXij)x, while the latter is expressed as some convergent series in Xij,DXij
+and their brackets, with DXijappearing only ones, thanks to the Campbell-Haussdorff
+formula.
+Let us prove (2.25). Rewrite the l.h.s. as
+ci(α,β)−Aijcj(α,β)Aji=
+=LαˆYi
+β−LβˆYi
+α+1
+t[ˆYi
+α,ˆYi
+β]−AijLαˆYj
+βAji+AijLβˆYj
+αAji−1
+t[AijˆYj
+αAji,AijˆYj
+βAji].
+From (2.21) it follows that
+[ˆYi
+α,ˆYi
+β] = [AijˆYj
+αAji+AijLαAji,AijˆYj
+βAji+AijLβAji].24 IVAN LOSEV
+So the l.h.s. of (2.25) equals
+LαˆYi
+β−LβˆYi
+α−AijLαˆYj
+βAji+AijLβˆYj
+αAji+
++1
+t[AijˆYj
+αAji,AijLβAji]+1
+t[AijLαAji,AijˆYj
+βAji]+1
+t[AijLαAji,AijLβAji].
+It is pretty straightforward to see that the last expression coinc ides with the r.h.s. of (2.25)
+(we remark that AijLαAji=−(LαAij)AjibecauseAijAji= 1).
+Let us check (2.26). We can rewrite the l.h.s. as
+cij(α)+Aijcjk(α)Aji+AijAjkcki(α)AkjAji=
+−AijLαAji−AijAjk(LαAkj)Aji−AijAjkAki(LαAik)AkjAji.
+On the other hand, the r.h.s. of (2.26) equals
+[Lα(AijAjkAki)]AikAkjAji=
+= (LαAij)Aji+Aij(LαAjk)AkjAji+AijAjk(LαAki)AikAkjAji.
+Again, it is easy to see that these two expressions are the same.
+Finally, let us prove (2.27).
+ln(AijAjkAki)+ln(AikAklAli)−ln(AijAjlAli)−ln(AjkAklAlj) =
+= ln(AijAjkAklAli)−ln(AjkAklAlj)−ln(AjlAliAij) =
+= ln(AijAjkAklAli)−ln(AjkAklAliAij) = 0.
+/square
+Proof of Theorem 2.7.3. LetXij,α/ma√sto→Yi
+αbe as in Lemma 2.12.2. Thanks to Lemma 2.12.3,
+we may assume that cijk,cij(α),ci(α,β) vanish. We are going to show that there are Γ ×Ξ-
+invariant elements Xi∈czt(C(H∧))(Ui) of degree 2 with respect to C×-action such that the
+following condition (*) holds
+(*) Θi:= exp(1
+tadXi)ΘUi,Xij:=tln(exp(1
+tXi)exp(1
+tXij)exp(−1
+tXj)) satisfy condi-
+tions (1)-(7) of Theorem 2.7.3, while Yi
+α:= exp(1
+tadXi)Yi
+α+exp(1
+tXi)ˆLαexp(−1
+tXi)
+is zero.
+The proof is in several steps.
+Step 1.We can rewrite (2.21),(2.22) as
+cij(α) =Yi
+α−exp(1
+tadXij)Yj
+α+exp(1
+tXij)ˆLαexp(−1
+tXij), (2.28)
+ci(α,β) =ˆLαYβ−ˆLβYi
+α+1
+t[Yi
+α,Yi
+β]. (2.29)
+We remark that Θi,Xij,Yi
+αobtained from ΘUi,Xij,Yi
+αby using the formulas above still
+satisfy conditions (2),(3),(5) of Theorem 2.7.3, equality (2.19), and the vanishing conditions
+cijk= 0,cij(α) = 0,ci(α,β) = 0. This is checked by computations similar in spirit to those
+of Lemma 2.12.3. Clearly, the equality cijk= 0 is just condition (6) of Theorem 2.7.3.
+Step 2.We consider α/ma√sto→ˆLαas a connection ˆLon the sheaf C( H∧), i.e., as a sequence of
+mapsˆLi: C(H∧)⊗Ωi
+L→C(H∧)⊗Ωi+1
+L. The connection ˆLis flat.
+The flat sheaf C( H∧) is just the product of several copies of the sheaf/producttext∞
+i=0SiΩ1
+Lwith the
+“diagonal” connection: ˆLαis the difference of the “fiberwise” and the “base” derivations inCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 25
+the direction of α. We claim that all higher cohomology of ˆLon/producttext∞
+i=0SiΩ1
+Lvanish, while
+the 0th cohomology is identified with OL, an isomorphism OL→H0(/producttext∞
+i=0SiΩ1
+L,ˆL) maps a
+sectionfofOLto its “Taylor expansion”
+(2.30) T(f) :=/summationdisplay
+i1
+i!∂if
+∂αiαi,
+wherei= (i1,...,in),i! :=/producttextn
+j=1ij!,αi=/producttextn
+j=1(αj)ij, etc.
+Tocompute thecohomologywe extend themap T:C[U]→/producttext∞
+i=0SiΩ1
+L(U) toa continuous
+automorphism of/producttext∞
+i=0SiΩ1
+L(U) by requiring that it is the identity on the subspace V0⊂
+S1Ω1
+L(U) of constant 1-forms. Then T◦ˆLα◦T−1is (up to a sign) the fiberwise derivation
+in the direction of α. Now the claim is clear.
+Step3.Letussupposethat Yi
+α∈ckzt(C(H∧))(Ui),k/greaterorequalslant1. Then1
+t[Yi
+α,Yi
+β]∈ck+1zt(C(H∧))(Ui).
+From (2.29) it follows that modulo ck+1zt(C(H∧))(Ui) we have ˆLαYi
+β−ˆLβYi
+α= 0. This means
+that modulo ck+1zt(C(H∧))(Ui) the mapα/ma√sto→Yi
+αis a 1-cocycle in the complex considered
+on Step 2. Thus there is Xi
+k∈ckzt(C(H∧))(Ui)) such that Yi
+α+ˆLαXi
+k∈ck+1zt(C(H∧))(Ui).
+In addition, we can assume that Xi
+ksatisfies the required equivariance conditions. Let us
+replaceYi
+αwith the expression analogous to Yi
+αwithXi
+kinstead ofXi. So we achieve
+Yi
+α∈ck+1zt(C(H∧))(Ui).
+SetXi=tln(...exp(1
+tXi
+2)exp(1
+tXi
+1)),theexpressionconvergesbecausewehavechosen Xi
+k
+inckzt(C(H∧))(Ui)). With this Xithe element Yi
+αis 0. Also it is clear from the construction
+that (2.19) becomes condition (4) of Theorem 2.7.3.
+Step 4.It remains to show that (7) of Theorem 2.7.3 holds, i.e., ˆLαXij= 0. The equalities
+(2.19) imply that bij
+α:=ˆLαXij∈cC[Uij][[c∗]]. The equality cij(α) = 0 can be rewritten as
+bij
+α= 0. /square
+2.13.Proof of Theorem 2.5.3. Define the sheaf of algebrasFl( H∧) onLas follows. For an
+open subset U⊂Llet Fl(H∧)(U) denote the space of flat (with respect to the connection L•)
+sections of H∧(U). This is a C××Ξ-equivariant sheaf of C-algebras (but not of OL-algebras)
+onL.
+Similarly, define the sheaf Fl( H∧) using the connection α/ma√sto→ˆLα.
+We remark that H∧L|L=At|L/hatwide⊗C[[t]]H+∧0,Fl(H∧) = Fl(OL/hatwide⊗At|L)/hatwide⊗C[[t]]H+∧0. Following
+the argument of Step 2 of the proof of Theorem 2.7.3, we get an isom orphism At|L=
+OL[[t]]∼− →Fl(OL/hatwide⊗A∧t
+t). This gives rise to an isomorphism
+(2.31) T⊗id :H∧L=A∧t
+t/hatwide⊗C[[t]]H+∧0→Fl(OL/hatwide⊗A∧t
+t)/hatwide⊗C[[t]]H+∧0= Fl(H∧),
+which will also be denoted by T.
+Recall the elements Xij∈Fl(C(H∧(Uij))) from Theorem 2.7.3. To simplify the notation
+we will write Xijinstead ofXij. Use these elements to twist the sheaf C( H∧L)|Las explained
+in the discussion preceding Theorem 2.5.3. Theorem 2.7.3 implies that th e mapsT−1◦ΘU
+induceanisomorphism η: Fl(H∧)→C(H∧L)|tw
+L. Thisisomorphismis C××Γ×Ξ-equivariant.
+We have a natural embedding H֒→H(L). Its image consists of flat and Ξ-invariant
+sections. So Hembeds into Fl( H∧)(L)Ξ. Composing this homomorphism with η, we obtain
+a homomorphism
+(2.32) θ:H→/bracketleftbig
+C(H∧L)|tw
+L(L)/bracketrightbigΞ.26 IVAN LOSEV
+We are going to prove that this homomorphism extends to an isomorp hismθ:H∧L|L→/bracketleftbig
+C(H∧L)|tw
+L/bracketrightbigΞthat equals θ0modulo ( c).
+First of all, we remark that θ0:H(r+1)→C(H∧L
+(r+1)|L) coincides with T−1◦Θ0and so
+also coincides with the map induced by (2.32). Localizing the map (2.32) overL, we get a
+homomorphism
+(2.33) θ:H|L→/bracketleftbig
+C(H∧L)|tw
+L/bracketrightbigΞ
+of sheaves of algebras that coincides with θ0:H(r+1)|L→/bracketleftBig
+C(H∧L
+(r+1))|tw
+L/bracketrightBigΞ
+modulo ( c).
+We claim that (2.33) is continuous with respect to the p|L-adic topology on H|L. To show
+this, it is enough to verify that (2.32) is continuous (with respect to thep-adic topology on
+Hand the topology on Fl(C( H∧)tw)Ξinduced from the C( m)-adic topology on C( H∧)). This
+will follow if we check that θ(p)⊂C(m)(L). Since both ideals contain c, it is enough to show
+thatθ0(p(r+1))⊂C(m(r+1))(L). But this is straightforward from the definition of θ0.
+The sheaf H∧L|Lis the completion of H|Lwith respect to the ideal p|L. So we see that
+(2.33) extends to a homomorphism θ:H∧L|L→/bracketleftbig
+C(H∧L)|tw
+L/bracketrightbigΞthat coincides with θ0:
+H∧L
+(r+1)|L→/bracketleftBig
+C(H∧L
+(r+1))|tw
+L/bracketrightBigΞ
+modulo ( c). But we have seen that θ0is an isomorphism. Since
+(C(H∧L)|tw
+L)ΞisC[[c∗]]-flat and H∧L|Lis complete in the ( c)-adic topology, we see that θis
+an isomorphism. The equality θ(p∧L|L) = [C(p∧L)|tw
+L]Ξfollows from the construction.
+3.Harish-Chandra bimodules
+3.1.Content of the section. The goal of this section is to study Harish-Chandra bimod-
+ules over SRA’s.
+In Subsection 3.3 we introduce notions of noncommutative Poisson algebras and their
+Poisson bimodules. These arealgebrasandbimodules equipped withan additionalstructure:
+a Poisson bracket. A similar notion already appeared, for instance, in [BeKa]. After giving
+all necessary definitions we study some simple properties of Poisson bimodules.
+In Subsection 3.4 we define Harish-Chandra (shortly, HC) H-bimodules as graded finitely
+generated Poisson H-bimodules. Then we define HC H1,c-bimodules. Also in this subsection
+we state the main result in our study of HC bimodules, Theorem 3.4.5, a nd its version for the
+H-algebras: Theorem 3.4.6. These theorems claim that there are fun ctors between certain
+categories of Harish-Chandra bimodules similar to the functors in [Lo 2], Theorem 1.3.1.
+The proof of Theorem 3.4.5 occupies the next four subsections who se content will be
+described in Subsection 3.4. In the Subsection 3.9 we derive Theorem 3.4.6 as well as
+Theorems 1.3.1,1.3.2 from Theorem 3.4.5. Finally, in Subsection 3.10 we will provide more
+simple minded versions of our functors. The constructions of this s ubsection will be used in
+Section 5.
+3.2.Notation and conventions. Algebras. Set˜H:=H[/planckover2pi1]/(t−/planckover2pi12). This is a flat S(˜c)-
+algebra, where ˜cis a vector space with basis /planckover2pi1,c1,...,cr. Similarly define ˜H,˜H+,A/planckover2pi1:=
+At[/planckover2pi1]/(t−/planckover2pi12). Recall that C( H) stands for Z(Γ,Γ,H). We have the two-sided ideal ˜pin˜H
+generated by pand/planckover2pi1. Similarly, we have the ideals ˜p⊂˜H,˜p+⊂˜H+.COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 27
+Thenwecanformthesheafified(andcompleted)versions ˜H∧L|L:=H∧L|L[/planckover2pi1]/(t−/planckover2pi12),˜H∧L|L∼=
+A/planckover2pi1|L/hatwide⊗C[[/planckover2pi1]]˜H+∧0of˜H,˜H. We have an isomorphism
+˜θ:˜H∧L|L→/parenleftBig
+C(A/planckover2pi1|L/hatwide⊗C[[/planckover2pi1]]˜H+∧0)tw/parenrightBigΞ
+induced by θfrom Theorem 2.5.3. Modulo ( ˜c) the isomorphism ˜θcoincides with
+θ0:H∧L
+(r+1)|L→C(H∧L
+(r+1)|L)Ξ,
+see Subsection 2.4.
+The reason why we need to consider the extensions ˜Hetc. is the following. Recall that
+the algebraH1,cis filtered. So we can form the Rees algebra R/planckover2pi1(H1,c). There is a unique
+homomorphism ˜H→R/planckover2pi1(H1,c) given byγ/ma√sto→γ,/planckover2pi1/ma√sto→/planckover2pi1,v/ma√sto→/planckover2pi1v,ci/ma√sto→ci/planckover2pi12,i= 1,...,r. It is
+clear that this homomorphism is surjective. So R/planckover2pi1(H1,c) is represented as a quotient of ˜H.
+D-modules. For a smooth algebraic variety XletDXdenote the sheaf of linear differential
+operators on X. For aDX-moduleMlet Fl(M) denote the space of flat sections of M.
+Recall that a section is called flat if it is annihilated by all vector fields.
+LetMbe aDX-module and Xbe equipped with an action of a group G. Recall thatM
+is said to be weakly G-equivariant ifMis equipped with an action of Gmaking the action
+mapDX⊗M→M equivariant.
+Annihilators. For anA-bimoduleMlet LAnn(M),RAnn(M) denote the left and the
+right annihilators of MinA.
+3.3.Poisson algebras and bimodules. In this subsection we will introduce the notions
+of (not necessarily commutative) Poisson algebras and their Poisso n bimodules.
+Definition 3.3.1. LetAbe an associative unital C[t]-algebra. We say that AisPoissonif
+it is equipped with a C[t]-bilinear map zt(A)⊗A→A,z⊗a/ma√sto→{z,a}such that zt(A) is
+closed with respect to {·,·}and
+{z,z}= 0, (3.1)
+{ta,b}= [a,b], (3.2)
+{z,ab}={z,a}b+a{z,b}, (3.3)
+{z1z2,a}={z1,a}z2+z1{z2,a}, (3.4)
+{{z1,z2},a}={z1,{z2,a}}−{z2,{z1,a}}, (3.5)
+∀z,z1,z2∈zt(A),a,b∈A.
+Definition 3.3.2. LetAbe a Poisson C[t]-algebra andMbe anA-bimodule such that the
+left and right actions of C[t] onMcoincide. We say that Mis a PoissonA-bimodule if it
+is equipped with a C[t]-bilinear map zt(A)⊗M→M satisfying the following equalities:
+{ta,m}= [a,m], (3.6)
+{z,am}={z,a}m+a{z,m},{z,ma}={z,m}a+m{z,a}, (3.7)
+{z1z2,m}={z1,m}z2+z1{z2,m}, (3.8)
+{{z1,z2},m}={z1,{z2,m}}−{z2,{z1,m}}, (3.9)
+∀z,z1,z2∈zt(A),a∈A,m∈M.28 IVAN LOSEV
+For instance, suppose Ais commutative with zero action of t. Then we get the usual
+definition of a Poisson algebra. Further, if Mis anA-module (that can be considered as an
+A-bimodule, where the left and right actions coincide) and tacts onMby 0, then we get a
+more standard notion of a Poisson A-module.
+As the other extreme, suppose AisC[t]-flat. Then the bracket on Ais uniquely recovered
+from the multiplication: {z,a}=1
+t[z,a] (compare with Subsection 2.8). Similarly, if Mis
+C[t]-flat, then{z,m}=1
+t[z,m].
+Examples of Poisson algebras (in our sense) include H,H,C(H) and also Weyl algebras.
+By a Poisson ideal in a Poisson algebra Awe mean a two-sided ideal Ithat is a Poisson
+sub-bimodule inA. We remark that the quotient A/Iis, of course, a Poisson A-bimodule
+but has no natural structure of a Poisson algebra.
+Lemma 3.3.3. LetAbe a Poisson C[t]-algebra andMbe a PoissonA-bimodule.
+(1)IfNis another Poisson A-bimodule, thenM⊗ANhas a natural structure of a
+PoissonA-bimodule.
+(2)IfI⊂Ais a Poisson two-sided ideal, then IM⊂M is a Poisson sub-bimodule.
+(3)Left and right annihilators of MinAare Poisson ideals.
+(4)The left and right actions of zt(A)/tAonM/tMcoincide.
+Proof.(1): Define the bracket on M⊗ANby{z,m⊗n}={z,m}⊗n+m⊗{z,n}. It
+is easy to see that the bracket is well-defined and turns M⊗ANinto a Poisson bimodule.
+The proofs of (2)-(4) follow directly from the definitions. /square
+Now suppose that C×acts on a Poisson algebra Aby aC-algebra automorphisms such
+thatthas degree 2, while the bracket has degree −2. We say that a Poisson A-bimodule
+MisC×-equivariant if it is equipped with a C×-action such that the multiplication map
+A⊗M⊗A→M isC×-equivariant, and the bracket map zt(A)⊗M→M has degree
+−2. We say that a C×-equivariant Poisson bimodule is graded if the C×-action comes from
+a grading.
+LetAbe a Poisson algebra. Set ˜A:=A[/planckover2pi1]/(t−/planckover2pi12). Then we still say that ˜Ais a
+Poisson algebra (the bracket is extended to zt(A)⊗˜A→˜Aby the right C[/planckover2pi1]-linearity).
+By a Poisson ˜A-bimodule we mean a bimodule M, where the left and right actions of C[/planckover2pi1]
+coincide, equipped with a map zt(A)⊗M→M that isC[t]-linear in the first argument,
+C[/planckover2pi1]-linear in the second one and satisfies (3.6)-(3.9).
+Now suppose that Ais a flat (Poisson) C[t]-algebra and e∈Ais an idempotent such that
+we have a “Satake isomorphism”, i.e., the map z(A/tA)e·− →z(e(A/tA)e) is an isomorphism.
+It follows that zt(eAe) =ezt(A)e.
+We are going to relate Poisson eAe- andA-bimodules.
+First of all, let us note that eAehas a natural Poisson bracket. Namely, we set
+(3.10) {eze,eae}=e{z,eae}e,z∈zt(A),a∈A.
+The bracket on eAeis well-defined. To check this let z,z′∈zt(A) be such that eze=ez′e.
+We need to show that e{z,eae}e=e{z′,eae}e. Thanks to theSatake isomorphism above, we
+see thatz−z′∈tA. Our claim follows easily from {ta,b}= [a,b]. Also it is straightforward
+to check that (3.10) satisfies (3.1)-(3.5).
+Now letMbe a PoissonA-bimodule. We claim that eMecan be equipped with a natural
+Poisson bracket. Namely, we set
+(3.11) {eze,eme}=e{z,eme}eCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 29
+foranyz∈zt(A),m∈M. Letuscheck thatthisbracketiswell-defined. Itisstraightforwa rd
+to verify that the bracket satisfies (3.6)-(3.7). So we get a funct orM/ma√sto→eMefrom the
+category of Poisson A-bimodules to the category of Poisson eAe-bimodules. This functor is
+exact becauseMis completely reducible as a SpanC(1,e)-bimodule.
+The functorM/ma√sto→eMehas a right inverse. Let Nbe a Poisson eAe-bimodule. Set
+/tildewideN:=Ae⊗eAeN⊗eAeeA. Let us equip this bimodule with a Poisson bracket. For z∈zt(A)
+we set{z,ae⊗n⊗eb}={z,ae}e⊗n⊗eb+ae⊗{eze,n}⊗eb+ae⊗n⊗e{z,eb}. The
+verification that this bracket is well-defined is analogous to the prev ious paragraph. The
+bracket on /tildewideNsatisfies (3.6)-(3.9). Also it is easy to see that e/tildewideNe=eA⊗A/tildewideN⊗AAeis
+canonically identified with N. On the other hand, /tildewideeMeis naturally embedded into Mwith
+imageAeMeA.
+3.4.Harish-Chandra bimodules. In this subsection we will introduce the notion of a
+Harish-Chandra bimodule over ˜H. Using this notion we will define Harish-Chandra H1,c-
+H1,c′bimodules.
+Recall that the algebra ˜His a graded Poisson algebra in the sense of the previous subsec-
+tion.
+Definition 3.4.1. By a Harish-Chandra (shortly, HC) ˜H-bimodule we mean a graded Pois-
+son˜H-bimodule Mthat is finitely generated as an ˜H-bimodule.
+The category of HC bimodules is denoted by HC( ˜H). By definition, a morphism between
+two objects in HC( ˜H) is a grading preserving homomorphism of graded Poisson bimodules.
+Remark 3.4.2. Since the grading on ˜His positive, each graded subspace Miis finite
+dimensional, and is zero for isufficiently small.
+In the sequel we will also need the following simple lemma.
+Lemma 3.4.3. M ∈HC(˜H)is finitely generated both as a left and as a right ˜H-module.
+Proof.By definition, Mis finitely generated as an ˜H-bimodule. But ˜His finite as a left (or
+right)Z-module. So there is a finite dimensional graded subspace M⊂MwithM=˜HMZ.
+Thanks to (3.6), [ Z,M]⊂tM. SoM=˜HM+tM. It follows that M=˜HM+tnMfor
+anyn. Since˜HMis a graded subspace in M, we deduce that any graded component of M
+lies in˜HM. So˜HM=M. Similarly, M=M˜H. /square
+Definition 3.4.4. LetMbe anH1,c-H1,c′-bimodule. We say that Mis HC if there is a
+filtration F iMcompatible with the filtrations F jH1,c,FjH1,c′and such that R/planckover2pi1(M) con-
+sidered as an ˜H-bimodule (recall that R/planckover2pi1(H1,c),R/planckover2pi1(H1,c′) are quotients of ˜H) is HC, where
+/planckover2pi1imhas degree i, and the bracket is uniquely recovered from the bimodule structur e, see
+remarks after Definition 3.3.2.
+The category of HC H1,c-H1,c′-bimodules will be denoted by cHC(H)c′.
+The category HC( ˜H) is monoidal, the tensor product functor is just tensoring over ˜H.
+Similarly, we have the bifunctor
+cHC(H)c′×c′HC(H)c′′→cHC(H)c′′
+given by tensoring over H1,c′. To see that the product is HC one needs to notice that for
+M1∈cHC(H)c′,M2∈c′HC(H)c′′the natural morphism R/planckover2pi1(M1)⊗˜HR/planckover2pi1(M2)/(/planckover2pi1−1)→
+M1⊗H1,c′M2is an isomorphism, compare with [Lo2], Proposition 3.4.1.30 IVAN LOSEV
+The category HC( ˜H) has a direct sum decomposition that will be of importance later.
+Namely, let M∈HC(˜H). Consider the linear operators Ci:={ci,·}onM. Since{ci,x}= 0
+for allx∈˜H, we see that Ciis an endomorphism of the Poisson bimodule M. Moreover,
+the degree of ciis 2, while{·,·}decreases degrees by 2. So Ciis a graded endomorphism
+ofM. SinceMis generated by a finite number of the graded components, we see t hatM
+decomposes into the finite direct sum M=/circleplustext
+λMλ, whereλ∈c∗
+(1)and
+Mλ:={m∈M|(Ci−/a\}b∇acketle{tλ,ci/a\}b∇acket∇i}ht)nm= 0 for some n}.
+This decomposition produces the category decomposition
+(3.12) HC( ˜H) =/circleplusdisplay
+λHC(˜H)λ.
+Let ΛHdenote the subspace in c∗
+(1)generated by all λ∈c∗
+(1)such that HC( ˜H)λis non-zero.
+Let us proceed to defining the associated varieties. Now let M∈HC(˜H). Thanks to
+assertion 4 of Lemma 3.3.3, we can consider M/˜cMas aZ(r+1)= (SV)Γ-module. Since
+H(r+1)is finite over Z(r+1), we get that M/˜cMis finitely generated as a Z(r+1)-module. So
+we can define the associated variety V( M) as the support of M/˜cMin Spec(Z(r+1)) =V∗/Γ
+(the corresponding ideal in Z(r+1)is the radical of the annihilator of M/˜cM). Assertion (3)
+of Lemma 3.3.3 implies that the annihilator of M/˜cMin˜His a Poisson ideal. From this it
+is easy to deduce that the annihilator of M/˜cMinZ(r+1)is a Poisson ideal. In particular,
+V(M) is a Poisson subvariety in V∗/Γ.
+ForM1,M2∈HC(˜H) we have M1⊗˜HM2/˜c(M1⊗˜HM2) = (M1/˜cM1)⊗SV#Γ(M2/˜cM2).
+In particular, M1⊗˜HM2/˜cM1⊗˜HM2is a quotient of ( M1/˜cM1)⊗SVΓ(M1/˜cM1). It follows
+that
+(3.13) V( M1⊗˜HM2)⊂V(M1)∩V(M2).
+Similarly, we can give the definitions of HC bimodules for the algebras ˜H,H1,c,˜H+,H+
+1,c.
+Also we consider the category HCΞ(˜H) consisting of all /tildewideΞ-equivariant bimodules M, i.e.,
+those equipped with an action of /tildewideΞ enjoying the following two properties:
+•the structure maps ˜H⊗M⊗˜H→M,Z⊗M→Mare/tildewideΞ-equivariant,
+•the restriction of the /tildewideΞ-action to Γ⊂/tildewideΞ coincides with the adjoint action of Γ .
+For instance, ˜H∈HCΞ(˜H).
+Let us state our main results concerning Harish-Chandra bimodules .
+For a Poisson subvariety Y⊂V∗/Γ define the full-subcategory HC Y(˜H)⊂HC(˜H) con-
+sisting of all M∈HC(˜H) such that V( M)⊂Y. Define the full subcategory cHCY(H)c′in
+cHC(H)c′similarly. By (3.13), the categories HC Y(˜H),cHCY(H)c′are closed with respect
+to the tensor product functor.
+Recall that we have fixed a symplectic leaf L⊂V∗/Γ. Consider the category HCL(˜H). It
+has the Serre subcategory HC ∂L(˜H), where∂L:=L\L. Form the quotient category
+HCL(˜H) = HCL(˜H)/HC∂L(˜H).
+The category HC L(˜H) inherits the tensor product functor from HCL(˜H). Also the decom-
+position (3.12) descends to a decomposition HC L(˜H) =/circleplustext
+λHCL(˜H)λ.
+Similarly, we can define the category cHCL(H)c′. Also we can consider categories of the
+formHCΞ
+Y+(˜H+),cHCΞ
+Y+(H+)c′,whereY+isaPoissonsubvarietyin V∗
++/Γ. WeareparticularlyCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 31
+interested in the case when Y+={0}. The category HCΞ
+0(˜H+) consists of all M∈HCΞ(˜H+)
+that are finitely generated over S(˜c), while cHCΞ
+0(H+)c′consists of all finite dimensional
+/tildewideΞ-equivariantH+
+1,c-H+
+1,c′-bimodules.
+As in (3.12), we have the decomposition HCΞ(˜H+) =/circleplustext
+λHCΞ(˜H+)λ. Set HCΞ
+0(˜H+)ΛH:=/circleplustext
+λ∈ΛHHCΞ
+0(˜H+)λ.
+Theorem 3.4.5. There are functors
+•†: HCL(˜H)→HCΞ
+0(˜H+)ΛH,
+•†: HCΞ
+0(˜H+)ΛH→HCL(˜H)
+with the following properties:
+(1)•†is exact and descends to HCL(˜H).
+(2)•†is right adjoint to •†and(•†)†∼=idinHCL(˜H).
+(3)The functor•†intertwines the tensor product functors.
+(4) LAnn( M†) = LAnn( M)†andRAnn(M†) = RAnn( M)†for anyM∈HCL(˜H).
+(5)•†is an equivalence from HCL(˜H)to some full subcategory of HCΞ
+0(˜H+)ΛHclosed
+under taking subquotients.
+Theorem 3.4.6. Suppose that c,c′∈c∗
+(1)are such that c−c′∈ΛH. There are functors
+•†:cHCL(H)c′→cHCΞ
+0(H+)c′,
+•†:cHCΞ
+0(H+)c′→cHCL(H)c′.
+with the following properties:
+(1)•†is exact and descends to cHCL(H)c′.
+(2)•†is right adjoint to •†and(•†)†∼=idincHCL(H)c′.
+(3)The functor•†intertwines the tensor product functors.
+(4) LAnn(M†) = LAnn(M)†andRAnn(M†) = RAnn(M)†for anyM∈cHCL(H)c′.
+(5)•†is an equivalence from cHCL(H)c′to some full subcategory of cHCΞ
+0(H+)c′closed
+under taking subquotients.
+The proof of Theorem 3.4.5 occupies basically the next four subsect ions. Subsection 3.5
+is preparatory, there we discuss some facts about D-modules we need in our construction.
+In (very long) Subsection 3.6 we construct the functor •†and study some of its properties.
+In Subsection 3.7 we construct the functor •†and study its properties and its relation to •†.
+The constructions are very similar to the constructions of the ana logous functors in [Lo2],
+but they aremoreinvolved technically. The reasonisthat in[Lo2] we c onsidered completions
+at a point, while here we need to deal with completions along subvariet ies. In Subsection 3.8
+we prove Theorem 3.4.5. A key ingredient in the proof is Theorem 3.8.2, which is a complete
+analog of Theorem 4.1.1 from [Lo2].
+3.5.D-modules and related objects. Wewillneedsomefactsaboutweakly C×-equivariant
+D-modules on certain C×-varieties. Namely, let Ube a vector space and U0be aC×-stable
+open subset of U.
+Proposition 3.5.1. Suppose that codimUU\U0>1. LetMbe a weakly C×-equivariant
+DU0-module that is coherent as a OU0-module. Then there is an isomorphism M∼=OU0⊗
+Fl(M)ofDU0-modules.32 IVAN LOSEV
+Proof.We remark that U0is simply connected because codim UU\U0>1.
+By [HHT], Corollary 5.3.10, it is enough to check that any weakly C×-equivariantDU0-
+module that is coherent as a OU0-module has regular singularities. Set U:=P(U⊕C).
+EquipUwith aC×-action induced from the following action on U⊕C:t.(u,x) = (tu,x).
+So we have C×-equivariant inclusions U0⊂U⊂U.
+Being a coherent OU0-module, aDU0-moduleMis a vector bundle equipped with a flat
+(C×-invariant) connection, say, ∇. We need to check that ∇has regular singularities. The
+only divisor in U\U0isP(U) =U\U. Pick a point x∈P(U) and let [x] by the line in U
+corresponding to x. By [HHT], Theorem 5.3.7, we only need to check that the restriction
+∇|[x]of∇to [x] has regular singularities. The restriction M|[x]×ofMto [x]×= [x]\{0}
+equivariantly trivializes. So the connection ∇|[x]×is given by∇|[x]×:=d+A(y)dy, where
+ydenotes the coordinate on [ x]∼=CandA(y) is aC[y,y−1]-valued matrix. Since ∇|[x]is
+C×-invariant, we see that A(y) =y−1A, whereAis a constant matrix. Therefore ∇|[x]has
+regular singularities. /square
+We will mostly use a corollary of this proposition.
+Corollary 3.5.2. LetU0be as in Proposition 3.5.1 and Mbe a weakly C×-equivariantDU0-
+module. Suppose that there is a C×-stable decreasing DU0-module filtration FiMsuch that
+M/FiMis a coherentOU0-module andMis complete with respect to the filtration. Then
+M∼=OU0/hatwide⊗CFl(M).
+In the sequel we will need some results about sheaves of TDO. By a s heaf of twisted
+differential operators (TDO) on a smooth algebraic variety Xwe mean a sheaf of algebras D′
+equipped with a monomorphism OX֒→D′that is locally isomorphic to the monomorphism
+OX֒→DX. “Locally isomorphic” means that there is a covering XiofXby open subsets
+such that there is an isomorphism ιi:D′|Xi→DXiintertwining the embeddings OXi֒→
+DXi,OXi֒→D′|Xi.
+It is known (and is easy to show) that the set of isomorphism classes of the sheavesD′
+of TDO (an isomorphism is supposed to intertwine the embeddings of OX) is naturally
+identified with H1(X,Ω1
+cl), where Ω1
+cldenotes the sheaf of closed 1-forms on X.
+Namely, pick an open cover X=/uniontextXiand consider a 1-cocycle ˜ α= (˜αij) in Ω1
+cl. Define
+the sheafD˜α
+Xas follows:D˜α
+X|Xi=DXiand the transition function from XjtoXimaps a
+vector filed ξtoξ+/a\}b∇acketle{tξ,˜αij/a\}b∇acket∇i}ht. Up to an isomorphism D˜α
+Xdepends only on the class αof ˜αin
+H1(X,Ω1
+cl) and we writeDα
+Xinstead ofD˜α
+X.
+A result about TDO we need is the following proposition.
+Proposition 3.5.3. LetUbe a vector space and U0be its open subset such that codimUU\
+U0>1. Suppose that a sheaf Dα
+U0of TDO has aOU0-coherent module M. Thenα= 0.
+Proof.The proof is in two steps.
+Step 1.LetM1,M2beDα1
+U0,Dα2
+U0-modules. ThenM1⊗M2is aDα1+α2
+U0-module. Indeed,
+pick an affine covering X=/uniontextXiand fix identifications ιl
+i:Dαl
+Xi→DXi. Define the action of
+a vector field ξonM1⊗M2|Xibyιi(ξ) =ι1
+i(ξ)⊗1+1⊗ι2
+i(ξ). Pick a cocycle ˜ αl
+ijrepresenting
+αl. Ifιl
+i(ξ)−ιl
+j(ξ) =/a\}b∇acketle{t/tildewideαl
+ij,ξ/a\}b∇acket∇i}htforl= 1,2, thenιi(ξ)−ιj(ξ) =/a\}b∇acketle{t/tildewideα1
+ij+/tildewideα2
+ij,ξ/a\}b∇acket∇i}ht.
+Step 2.Now letMbe as in the statement of the proposition and let mdenote the rank
+ofM. ThenM⊗mis aDmα
+U0-module. We remark that/logicalandtextmM(the sheaf of anti-symmetric
+sections ofM⊗m) is aDmα
+U0-submodule inM⊗m. But/logicalandtextmMis a line bundle on U0. However
+Pic(U0) ={1}because codim UU\U0/greaterorequalslant2. SoOU0is aDmα
+U0-module. From this it is easy
+to deduce that mα= 0. Since H1(X,Ω1
+cl) is a complex vector space, we see that α= 0./squareCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 33
+Till the end of the subsection suppose Uis a symplectic vector space and let U0be an
+arbitrary open subset (without restrictions on codim UU\U0).
+LetMbe a Poisson At|U0-bimodule (as usual, t=/planckover2pi12). Suppose that Madmits a
+decreasing filtration F iMbyAt|U0-submodules such that
+•Mis complete with respect to this filtration,
+•tFiM⊂Fi+1M,
+•and FiM/Fi+1Mis a coherentOU0-module for all i.
+For instance, ifMis finitely generated as a left At|U0-module, then we can take the t-adic
+filtration. The claim that this filtration is complete is proved analogous ly to [Lo2], Lemma
+2.4.4.
+We want to equip Mwith a structure of a DU0-bimodule. Denote the product map
+At|U0⊗M⊗At|U0bya⊗m⊗b/ma√sto→a∗m∗b. Foru,v∈Uand a section mofMwe set
+u·m=1
+2(u∗m+m∗u),∂vm={ˇv,m}, where ˇvis defined by/a\}b∇acketle{tˇv,u/a\}b∇acket∇i}ht=ω(v,u),v,u∈U. Then
+the operators u·,v·,∂u,∂vsatisfy the commutation relations [ u·,v·] = 0 = [∂u,∂v],[∂u,v·] =
+/a\}b∇acketle{tu,v/a\}b∇acket∇i}htid, soMbecomes a sheaf of DU(U)-modules. Now let U′be a principal open subset
+ofUcontained in U0defined by a polynomial f. We have f·m≡f∗mmodtand sof
+is invertible onM(U0). It follows that Mis aDU0-module. We remark that F iMis a
+DU0-submodule ofMand the quotientM/FiMis a coherentOU0-module.
+This discussion has an interesting corollary.
+Proposition 3.5.4. Suppose that U0is stable with respect to the usual action of C×on
+UandcodimUU\U0/greaterorequalslant2. LetMbe aC×-equivariant Poisson At|U0-bimodule equipped
+with a filtration FiMas above. Then the natural map At|U0/hatwide⊗C[[t]]M(U0)adU→Mis an
+isomorphism; here M(U0)adUstands for the kernel of {U,·}inM(U0).
+Proof.ViewMas aDU0-module and apply Corollary 3.5.2. /square
+3.6.Functor•†: HC(˜H)→HCΞ(˜H+)ΛH.This functor is obtained as a composition of
+several functors between some intermediate categories. Let us list these functors. The
+definitions of the categories involved will be given later.
+F1: HC(˜H)→HC(˜H∧L|L)ΛH,
+F2: HC(˜H∧L|L)ΛH→HC/parenleftBig
+(C(˜H∧L|L)tw)Ξ/parenrightBigΛH,
+F3: HC/parenleftBig
+(C(˜H∧L|L)tw)Ξ/parenrightBigΛH→HCΞ/parenleftBig
+C(˜H∧L|L)tw/parenrightBigΛH,
+F4: HCΞ/parenleftBig
+C(˜H∧L|L)tw/parenrightBigΛH→HCΞ/parenleftBig
+C(˜H∧L|L)/parenrightBigΛH,
+F5: HCΞ/parenleftBig
+C(˜H∧L|L)/parenrightBigΛH→HCΞ/parenleftBig
+C(˜H+∧0)/parenrightBigΛH,
+F6: HCΞ/parenleftBig
+C(˜H+∧0)/parenrightBigΛH→HCΞ(˜H+∧0)ΛH,
+F7: HCΞ(˜H+∧0)ΛH→HCΞ(˜H+)ΛH.
+All these functors but F1turn out to be category equivalences.
+The functorF1: the (localized) completion.
+PickM∈HC(˜H). Define the (micro)localization M|X0similarly to H|X0in Subsection
+2.4. According to the discussion preceding Corollary 2.9.6, Z|X0=zt(H|X0). Therefore the34 IVAN LOSEV
+localization M|X0ofMhas a natural structure of a Poisson ˜H|X0-bimodule. We can form
+the completion M∧L:= lim←−kM|X0/M|X0(˜p|X0)k. Then, by definition, F1(M) :=M∧L|Lis
+the sheaf-theoretic restriction of M∧LtoL.
+The following lemma describes the properties of F1(M) we need.
+Lemma 3.6.1. LetM∈HC(˜H).
+(1)F1(M)has a natural structure of a C×-equivariant Poisson ˜H∧L|L-bimodule. This
+module is finitely generated both as a left and as a right ˜H∧L|L-module.
+(2)The natural maps ˜H∧L|L⊗˜HM,M⊗˜H˜H∧L|L→M∧L|Lare isomorphisms.
+(3)The completion functor M/ma√sto→F1(M)is exact and intertwines the tensor products (of
+Poisson bimodules).
+(4)As aZ∧L
+(r+1)|L-moduleF1(M)/˜cF1(M)coincides with the usual (commutative) comple-
+tion of the Z(r+1)-moduleM/˜cM. In particular,F1(M) = 0if and only if V(M)∩L=
+∅.
+Proof.We have [ ˜p,M]|X0⊂/planckover2pi1M|X0. From this it follows that the filtrations M˜pn|X0and
+˜pnM|X0coincide. So M∧Lhas a natural structure of a ˜H∧L-bimodule (a unique structure
+extended from H|X0⊗M|X0⊗H|X0→M|X0by continuity). Also, thanks to Corollary
+2.9.6, we can extend the bracket from Z|X0⊗M|X0→M|X0tozt(H∧L)⊗M∧L→M∧Lby
+continuity. This bracket then transfers to the pull-backs to L.
+(2) and the exactness part in (3) are standard corollaries of Prop osition 2.10.5, see Lemma
+2.10.1. The compatibility with tensor products in (3) follows from (2).
+(4) follows from the construction and Proposition 2.9.3. /square
+Now let us define the category HC( ˜H∧L|L). By definition it consists of C×-equivariant
+Poisson˜H∧L|L-bimodules that are finitely generated as left ˜H∧L|L-modules. Since ˜H∧L|Lis
+a sheaf of Noetherian algebras complete in the p∧L|L-adic topology, we can use the claim
+similar to Lemma 2.4.2 from [Lo2] to show that HC( ˜H∧L|L) is an abelian category and
+that any object in HC( ˜H∧L|L) is complete and separated in the p∧L|L-adic topology. The
+categoryHC( ˜H∧L|L)isequippedwithatensorproductfunctor: thetensorproducto fPoisson
+bimodules. So the functor F1:M/ma√sto→M∧L|L: HC(˜H)→HC(˜H∧L|L) is monoidal.
+We want to introduce an analog of the decomposition (3.12) for HC( ˜H∧L|L). LetM∈
+HC(˜H∧L|L) and setMn:=M/M(˜p∧L|L)n. We have the endomorphisms Ci={ci,·}of
+a Poisson ˜H∧L|L-bimodule. The quotients Mn/Mn+1are coherentOL-modules and Pois-
+son˜H∧L|L-bimodules. The annihilator of such a module in H∧L|Land hence inOLis a
+Poisson ideal. But OLhas no nontrivial Poisson ideals for Lis a symplectic variety. It
+follows that any quotient Mn/Mn+1has finite length. Therefore we have the decomposi-
+tionMn/Mn+1:=/circleplustext
+λ(Mn/Mn+1)λinto the sum of the generalized eigensheaves for the
+operatorsCi. We remark that the natural homomorphism ( ˜p∧L|L)n/(˜p∧L|L)n+1⊗M/M1→
+Mn/Mn+1is surjective. Hence ( ˜p∧L|L)n/(˜p∧L|L)n+1⊗(M/M1)λ→(Mn/Mn+1)λis surjec-
+tive. SetMλ:= lim←−n(Mn)λ. Then we have the direct sum decomposition M=/circleplustext
+λMλand
+the corresponding category decomposition HC( ˜H∧L|L) =/circleplustext
+λHC(˜H∧L|L)λ. The subcategory
+HC(˜H∧L|L)ΛH⊂HC(˜H∧L|L) is now defined in an obvious way. Clearly, the image of F1lies
+in HC(˜H∧L|L)ΛH.
+Remark 3.6.2. In fact, one can show that HC( ˜H∧L|L)ΛHcoincides with the whole category
+HC(˜H∧L|L). This follows from the construction above and Proposition 3.7.2 belo w.COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 35
+The functorF2: the push-forward with respect to ˜θ.
+Recalltheisomorphism ˜θ:˜H∧L|L→(C(˜H∧L|L)tw)Ξ. DefinethecategoryHC((C( ˜H∧L|L)tw)Ξ)
+analogously to HC( ˜H∧L|L). We setF2:=˜θ∗. The functorF2restricts to an equivalence
+HC(˜H∧L|L)ΛH→HC((C(˜H∧L|L)tw)Ξ)ΛH.
+The functorF3: lifting toL.
+LetM∈HC((C(˜H∧L|L)tw)Ξ). Set
+(3.14) M|L:= C(˜H∧L|L)tw⊗(C(˜H∧L|L)tw)ΞM.
+By definition,M|Lis a C(˜H∧L|L)tw- (C(˜H∧L|L)tw)Ξ-bimodule. It is equipped with a bracket
+withzt((C(H∧L|L)tw)Ξ) defined using the Leibnitz rule.
+Lemma 3.6.3. (1)There is a unique bracket map zt(C(H∧L|L)tw)⊗M| L→M| Lex-
+tending the bracket with zt((C(H∧L|L)tw)Ξ)and satisfying (3.7)-(3.9).
+(2)There is a unique right multiplication map C(˜H∧L|L)tw⊗M| L→M| Lsatisfying
+(3.6) so thatM|Lbecomes a Poisson C(˜H∧L|L)tw-bimodule.
+(3)The natural map M→(M|L)Ξis an isomorphism.
+Proof.First of all, we remark that the right product map is recovered uniqu ely from the left
+product and the bracket map. So (1) implies (2).
+(1) and (3) are local so it is enough to prove the similar claims for the r estrictions of the
+sheaves of interest to a Ξ-stable affine open subset Ui⊂L. But there
+C(˜H∧L|L)tw|Ui∼=C(˜H∧L|Ui).
+We also remark that the natural homomorphism
+At|Ui⊗(At|Ui)ΞC(˜H∧L|Ui)Ξ→C(˜H∧L|Ui)
+is an isomorphism. This again follows from the fact that π′:X∧
+L→X∧
+Lis the quotient map
+for the free action of Ξ. (3) follows. Also to prove (1) it is enough to show that the bracket
+can be uniquely extended from ( At|Ui)Ξ⊗M|Ui→M|UitoAt|Ui⊗M|Ui→M|Ui.
+Further, consider the filtration M|Ui/parenleftbig
+C(p∧L|Ui)Ξ/parenrightbignofM|Ui. The quotient
+Mn:=M|Ui/M|Ui/parenleftbig
+C(p∧L|Ui)Ξ/parenrightbign
+is an (At|Ui)Ξ/(t)n-bimodule that is finitely generated as a left (or right) module. If we
+prove assertion (1) for each Mm, then (1) forMwill follow. So it is enough to consider the
+case whenMis finitely generated as a left module over ( At|Ui)Ξ.
+Pick a point u∈ L. Choose a Ξ-stable affine open neighborhood UofuinUi. To
+simplify the notation we will write Minstead ofM(U) andAinstead of At(U). Shrinking
+Uif necessary (so that still u∈U) we may assume that C[U] is a free C[U]Ξ-module.
+Pick a free basis f1,...,fk∈C[U] overC[U]Ξ. We can shrink Uagain to find elements
+ai
+j∈C[U]Ξ,i= 1,...,k,j = 0,1,...,k−1 such that Pi(fi) :=fk
+i+ai
+k−1fk−1+...+ai
+0
+vanishes in C[U] butP′
+i(fi) is invertible in C[U]. We remark that f1,...,fkis still a basis of
+the (left or right) AΞ-moduleA. Set/tildewiderM:=A⊗AΞMso thatM=/tildewiderMΞ.
+Now recall an identification A∼=C[U][[t]], see Subsection 2.4. Let ˆPi(fi),ˆP′
+i(fi) be the
+polynomials, where the commutative products are replaced with ∗(the order of the factors
+is the same). So in Awe have ˆPi(fi) =tgifor somegi∈AandˆP′
+i(fi) is still invertible. For
+m∈/tildewiderMsetQ(fi,m) =fk−1
+i{ai
+k−1,m}+...+fi{ai
+1,m}+{ai
+0,m}.36 IVAN LOSEV
+Suppose for a moment that there is lifting of {·,·}:AΞ⊗/tildewiderM→/tildewiderMtoA⊗/tildewiderM→/tildewiderMsuch
+that
+{a,bm}={a,b}m+b{a,m},∀a,b∈A, (3.15)
+{ab,m}=a{b,m}+b{a,m}−t{b,{a,m}}, (3.16)
+∀a,b∈A,m∈/tildewiderM.
+Thenwehave t{gi,m}=ˆP′
+i(fi){fi,m}+Q(fi,m)+tF(fi,(aj
+i)k−1
+j=0,m)whereF(fi,(aj
+i)k−1
+j=0,m)
+is an expression involving fi,aj
+i,m, various brackets and products. Then we clearly have
+(3.17){fi,m}=ˆP′
+i(fi)−1[t({gi,m}−F(fi,(aj
+i)k−1
+j=0,m))−Q(fi,m)].
+Conversely, given (3.17) we can extend {·,·}:AΞ⊗/tildewiderM→/tildewiderMto a unique mapA⊗/tildewiderM→/tildewiderM
+using the equalities
+(3.18) {fia,m}=fi{a,m}+a{fi,m}−t{a,{fi,m}}
+and theAΞ-linearity (recall that f1,...,fkis a basis of the right AΞ-moduleA).
+We can use (3.17),(3.18) to define a map {·,·}:A⊗M→M recursively. This is possible
+becauseMis complete and separated in the t-adic topology. For f∈Awe get{f,m}in
+the form
+(3.19) {f,m}=/summationdisplay
+iFi{bi,m},
+wherebi∈AΞand the sequence Ficonverges to 0 in the t-adic topology.
+It remains to prove that the map {·,·}:A⊗/tildewiderM →/tildewiderMdefined in this way satisfies
+(3.7)-(3.9).
+Pick a point b∈U. It follows from the definition of the map m/ma√sto→{f,m},f∈A,that this
+map extends toA∧b⊗/tildewiderM∧b→/tildewiderM∧b, whereA∧b,/tildewiderM∧bdenote the completions of A,/tildewiderMat
+b. These completions are defined as the completions above. Namely, le tnb⊂A=C[U][[t]]
+denote the maximal ideal of b. Then we can form the completions A∧b:= lim←−kA/nk
+b,/tildewiderM∧b:=
+lim←−k/tildewiderM/nk
+b/tildewiderM. Using the machinery of Subsection 2.10, we see that the functor /tildewiderM/ma√sto→/tildewiderM∧b
+is exact and /tildewiderM∧
+b=A∧b⊗A/tildewiderM.
+Similarly, we can consider the completion ( AΞ)∧π′(b)ofAΞatπ′(b). We remark that the
+inclusionAΞ֒→Ainduces an isomorphism ( AΞ)∧π′(b)∼=A∧b. Now/tildewiderM∧b=A∧b⊗A/tildewiderM=
+A∧b⊗AA⊗AΞM= (AΞ)∧π′(b)⊗AΞM=M∧π′(b).
+AsinLemma3.6.1,thereisauniquecontinuousbracketmap( AΞ)∧π′(b)⊗M∧π′(b)→M∧π′(b)
+extending the map AΞ⊗M→M . We have a natural embedding A֒→A∧b∼=(AΞ)∧π′(b),
+which gives rise to a bracket map A⊗/tildewiderM∧b→/tildewiderM∧b. By construction, this map satisfies
+(3.7)-(3.9). Hence this map satisfies (3.17),(3.18). So it coincides wit h the map induced
+(=extended by continuity) from A⊗/tildewiderM→/tildewiderM.
+So it remains to show that /tildewiderM→/tildewiderM∧bis injective. LetNbe the kernel. Then N∧b={0}.
+We want to show that N={0}. Since the completion functor is exact and the t-adic
+filtration onNis separated, we may replace NwithN/tNand assume that tacts trivially
+onN. By a discussion preceding Proposition 3.5.4, Ncan be equipped with a structure of
+aDL-module. ThenNis also a coherent OL-module and, in particular, a vector bundle. So
+N∧b={0}impliesN={0}. /squareCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 37
+Introduce the category HCΞ/parenleftBig
+(C(˜H∧L|L)tw/parenrightBig
+similarly to HC/parenleftBig
+(C(˜H∧L|L)tw)Ξ/parenrightBig
+with the
+only difference that the Poisson bimodules in the new category are su pposed to be ad-
+ditionally Ξ-equivariant. So F3:M/ma√sto→M| Lis an equivalence HC/parenleftBig
+(C(˜H∧L|L)tw)Ξ/parenrightBig
+→
+HCΞ/parenleftBig
+(C(˜H∧L|L)tw/parenrightBig
+, the quasi-inverse functor is given by taking Ξ-invariants. The claim s
+about the equivalences comes from the fact that Ξ acts freely on L.
+These equivalences restrict to equivalences between the categor ies HC/parenleftBig
+(C(˜H∧L|L)tw)Ξ/parenrightBigΛH
+and HCΞ/parenleftBig
+(C(˜H∧L|L)tw/parenrightBigΛH.
+The functorF4: untwisting.
+Recall the elements Xij∈czt(C(H∧L|L)(Uij)) from Theorem 2.5.3. We have isomorphisms
+C(˜H∧L|tw
+L)|Ui∼=C(˜H∧L|Ui) andthetransitionfunctions from UjtoUifor thesheaf C( ˜H∧L|tw
+L)
+are exp(1
+tadXij) = exp({Xij,·}). Of course, the endomorphism {Xij,·}ofM|Uijmakes
+sense but its exponential does not because, in general, it diverges (with the exception of the
+case when all Ciare nilpotent).
+However, we have the following result:
+Lemma 3.6.4. There are elements bl
+i∈At(Ui)C××Ξ,l= 1,...,r,such that the operator
+m/ma√sto→ {Xij,m}−/summationtextr
+l=1(bl
+i−bl
+j)Ci(m)on(M/C(p∧L)M)|Uijis nilpotent for any M ∈
+HCΞ/parenleftBig
+(C(˜H∧L|tw
+L)/parenrightBig
+.
+Proof.We can represent XijastX0
+ij+/summationtextr
+l=1ciXl
+ij, whereX0
+ij,...,Xr
+ij∈zt(C(H∧L|L)(Uij)).
+We remark that the elements Xl
+ij,l= 1,...,rare not unique but their classes modulo
+tH+c(1)zt(C(H∧L|L)(Uij)) are. This follows from the fact that z(C(H∧L
+(1)|L)(Uij)) is flat
+overC[[c∗
+(1)]]. To verify the last statement one uses the descending induction on kto show
+thatz(C(H∧L
+(k)|L)(Uij)) is flat over C[[c∗
+(k)]] (compare with the proofs of Lemma 2.11.4 and
+Corollary 2.9.5).
+Consider the image xl
+ijof the cochain Xl
+ijinOLand letaldenote its class in H1(L,OL).
+Thanks to the uniqueness property for Xl
+ijfrom the previous paragraph and the fact that
+the collection exp(1
+tXij) is a cocycle, we deduce that the elements xl
+ijform a cocycle for each
+l. We also remark that for an object Min HCΞ((C(˜H∧L|L)tw) annihilated by C( ˜p∧L|L)twthe
+(left or right) multiplication by Xl
+ijis the same as the multiplication by xl
+ij.
+We are going to check that there are bl
+isuch that
+(3.20)r/summationdisplay
+l=1(xl
+ij−bl
+i+bl
+j)λl= 0,
+for any eigenvalue λ= (λ1,...,λr) of (C1,...,Cr) on an objectMas in the previous
+paragraph (such an object is automatically a coherent OL-module).
+The existence of the cochains bl
+isatisfying (3.20) is equivalent to aλ:=/summationtextr
+l=1λlal= 0. We
+have a natural map d:H1(L,OL)→H1(L,Ω1
+cl) induced by the De Rham differential d.
+This map is injective because H1(L,C) = 0 (all open subsets intersect).
+Letλbe an eigenvalue of ( C1,...,Cr) inMandMλbe the corresponding eigensheaf.
+Pickv∈V0and let{v,·}idenote the Poisson bracket with vonMλ|Ui. So{v,·}i−{v,·}j=
+{/summationtextr
+l=1cl∂vxl
+ij,·}=/summationtextr
+l=1λl(∂vxl
+ij). Recall the construction of the correspondence between38 IVAN LOSEV
+Poisson bimodules and D-modules in Subsection 3.5. From this construction it follows that
+Mλis aDdaλ
+L-module. ButMλis a coherentOL-module. Proposition 3.5.3 implies that
+daλ= 0. Hence aλ= 0.
+Let us check that the elements bl
+iwe have foundsatisfy the condition of the lemma. Let M
+be such as in the lemma statement. We may replace MwithM/C(p∧L)|tw
+LMand assume
+thatMis annihilated by C( p∧L)|tw
+L. The statement of the lemma will follow if we check
+thatMhas finite length. The action of zt(C(H∧L)|tw
+L) onMreduces to the action of OL
+so thatMbecomes a coherent OL-module. Consider M|Uias a Poisson At|Ui-bimodule.
+This module is annihilated by tAt|Uiand so, according to Proposition 3.5.4, is a coherent
+DUi-module. It follows that the Poisson C( ˜H∧L)|tw
+L-bimoduleMhas finite length. /square
+Fixbl
+ias in Lemma 3.6.4. Set Bi:=/summationtextr
+l=1clbl
+i.
+Now we are ready to construct an “untwisting” functor
+F4: HCΞ(C(˜H∧L|L)tw)ΛH→HCΞ(C(˜H∧L|L))ΛH.
+Set
+(3.21) ˆXij=tln(exp(1
+tBj)exp(1
+tXij)exp(−1
+tBi)).
+We remark that ˆXij∈czt(C(H∧L|L))(Uij) and moreover,
+(3.22) ˆXij≡−Bi+Xij+Bjmodc2zt(C(H∧L|L))(Uij).
+(3.22) holds because all brackets appearing in the Lie series (3.21) lie inc2zt(C(H∧L|L))(Uij).
+We deduce from (3.22) and the definition of the elements Bithat exp({ˆXij,·}) is a well-
+defined map onM|Uij. LetF4(M) be the sheaf obtained from Mby regluing the sheaves
+M|Uiusing the transition functions exp( {ˆXij,·}). Introduce a new structure of a Poisson
+C(˜H∧L|Ui)-bimodule onM|Uiby sending a section xof C(˜H∧L|Ui) to exp({Bi,·})x. Then
+these Poisson bimodule structures glue together to a Poisson C( ˜H∧L|L)-bimodule structure
+on F4(M). Since all transition functions are C××Γ×Ξ-equivariant, we see that F 4(M)∈
+HCΞ(C(˜H∧L|L))ΛH.
+Conversely, letN∈HCΞ(C(˜H∧L|L)ΛH. Then exp(−{ˆXij,·}) is well-defined on N|Uij. So
+reversing the untwisting procedure explained before we get a func tor
+HCΞ(C(˜H∧L|L))ΛH→HCΞ(C(˜H∧L|tw
+L)ΛH.
+It is clear that this functor is (quasi)inverse to F4.
+The functorF5: taking flat sections. LetM∈HCΞ(C(˜H∧L|L)). LetF5(M) be the
+subspace ofM(L) consisting of all sections annihilated by {V0,·}. We remark that being a
+finitely generated Poisson At|L-bimodule,Msatisfies the assumptions of Proposition 3.5.4
+(the filtrationMC(p∧L|L)nhas the required properties). So M=At|L/hatwide⊗C[[t]]F5(M).
+Define the category HCΞ(C(˜H+∧0)) by analogy with the category HCΞ((C(˜H∧L|L)tw). In
+particular, all Poisson bimodules in HCΞ(C(˜H+∧0)) are complete in the C( p+∧0)-adic topol-
+ogy.
+So we see thatF5induces an equivalence HCΞ(C(˜H∧L|L))→HCΞ(C(˜H+∧0)), a quasi-
+inverse equivalence is given by ( A/planckover2pi1|L)/hatwide⊗C[[/planckover2pi1]]•. Clearly,F5,F−1
+5restrict to equivalences of
+HCΞ(C(˜H∧L|L))ΛHand HCΞ(C(˜H+∧0))ΛH.
+The functorF6(•) =e(Γ)•e(Γ).COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 39
+Recall the element e(Γ)∈CΓ⊂C(˜H+∧0) defined in Subsection 2.3 and the natural
+isomorphism ˜H+∧0∼=e(Γ)C(˜H+∧0)e(Γ).
+So we can defineF6: HCΞ(C(˜H+∧0))→HCΞ(˜H+∧0) by sendingMtoe(Γ)Me(Γ), see the
+discussion at the end of Subsection 3.3. The functor F6is an equivalence. Indeed, the right
+inverse functor constructed in Subsection 2.3 is nothing else but N/ma√sto→C(N). It is inverse
+toF6by the remarks there.
+The functorF7: taking finite vectors.
+LetM∈HCΞ(˜H+∧0). We say that m∈Mis locally finite with respect to C×(shortly,
+C×-l.f.) if it lies in a finite dimensional C×-stable subspace of M. The space of all C×-l.f.
+elements inMwill be denoted by MC×−l.f.. It is clear thatMC×−l.f.is aC××/tildewideΞ-equivariant
+Poisson ˜H+-bimodule. We claim that MC×−l.f.is dense inMand is finitely generated as
+a˜H-bimodule. So we get the functor F7: HCΞ(˜H+∧0)→HCΞ(˜H+),M/ma√sto→M C×−l.f.. This
+functor is an equivalence, a quasiinverse functor sends N∈HCΞ(˜H+) to its completion in
+the˜p+-adic topology.
+All claims are quite standard and follow easily from the fact that ˜H+is positively graded,
+compare with the proof of Proposition 3.3.1 in [Lo2].
+It is clear thatF7gives an equivalence HCΞ(˜H+∧0)ΛH→HCΞ(˜H+)ΛH.
+The functor•†: HC(H)→HCΞ(H+)ΛH.
+Fori>jsetFi,j:=Fi◦Fi−1◦...Fj. Let•†:=F7,1. Let us list some properties of •†.
+Proposition 3.6.5. (1)The functor•†isS(˜c)-bilinear, exact and monoidal. Also it
+commutes with the operators Ci.
+(2)ForM∈HC(H)we haveM†= 0if and only ifL∩V(M) =∅.
+(3)IfMis annihilated by ˜p(from the left or from the right, these two conditions are
+equivalent), then M†is annihilated by ˜p+.
+(4) V(M†)can be recovered from V(M)in the following way. Let V(M) =/uniontextl
+i=1Libe the
+decomposition into irreducible components and let Γibe a subgroup in Γcorresponding
+toLi. LetΓ1
+i,...,Γji
+ibe all subgroups in Γconjugate to ΓiinsideΓ. Finally, letLj
+i
+be the symplectic leaf in V+/Γcorresponding to Γj
+i. ThenV(M†) =/uniontext
+i,jLj
+i.
+When we say that •†isS(˜c)-bilinear we mean the following. For M∈HC(˜H) letM[k]
+denotethesamemodulewiththegradingshiftedby k:M[k]n=Mn+k. Similarly, onedefines
+thegrading shiftsfor objects inHCΞ(˜H+). It isclear fromthedefinition that M[k]†=M†[k].
+Now we have morphisms M[−2]→Mgiven by left and right multiplications by ci(and also
+the morphism M[−1]→Mgiven by the multiplication by /planckover2pi1). When we say that the functor
+•†isS(˜c)-bilinear, we mean that it is additive and it sends the morphism [ m/ma√sto→cim] to
+[n/ma√sto→cin] (and similarly for the right multiplications and for the multiplication by /planckover2pi1).
+Proof.All functorsFiareS(˜c)-bilinear and commute with the Ci’s. The functorF1is exact
+and the other functors Fiare equivalences. Further, all functors Fiare monoidal (the tensor
+product functors on the intermediate categories are tensor pro ducts of (sheaves of) Poisson
+bimodules). Hence (1).
+(3)followsfromtheobservationthatallfunctors Fipreserve theconditionthat abimodule
+is annihilated by an appropriate ideal.
+Let us prove (4). Since •†is exact, we may replace MwithM/˜cMand assume that
+˜cannihilates M. The associated formal scheme of F1(M) is nothing else but V( M)∧L:=40 IVAN LOSEV
+V(M)∩X∧
+L. This follows from assertion (4) of Lemma 3.6.1. The associated form al scheme
+ofF4,1(M) is the preimage of V( M)∧LinX∧
+L=L×(V∗
++)∧
+0/Γ. It equalsL×V(F5,1(M)).
+The functorF6does not change the associated schemes. Finally, V( F6,1(M)) is just the
+intersection of V( M†)⊂V∗
++/Γwith (V∗
++)∧
+0/Γ.
+The claim of (4) easily follows from the description in the previous para graph.
+(2) stems from assertion (4). /square
+Remark 3.6.6. Wecanextend•†toafunctorbetweentheind-completions /hatwiderHC(˜H),/hatwiderHCΞ(˜H+).
+The category /hatwiderHC(˜H) consists of all graded Poisson ˜H-bimodules that are sums of their HC
+bimodules and the category /hatwiderHCΞ(˜H+) admits a similar description.
+We will need the following remark in the proof of Theorem 1.3.2.
+Remark 3.6.7. Pickl= 1,...,r+1 and let /planckover2pi1lbe a new independent variable. Set ˜H(l)=
+H(l)[/planckover2pi1l]/(cl−/planckover2pi12
+l),˜Z(l)=z(˜H(l)) =Zl[/planckover2pi1l]/(cl−/planckover2pi12
+l) (the latter equality stems, for example,
+from the Satake isomorphism, Proposition 2.8.1 and from the fact th atH(l)is a flatS(c(l))-
+module). Similarly, we can introduce the algebras ˜H+
+(l),˜Z+
+(l).
+Let/hatwiderHC(˜Z(l)) denote the category of locally finitely generated graded Poisson ˜Z(l)-modules
+(=bimodules). Define /hatwiderHCΞ(˜Z+
+(l)) in a similar way. We can define the functor •†:/hatwiderHC(˜Z(l))→
+/hatwiderHCΞ(˜Z+
+(l)) completely analogously to •†:/hatwiderHC(˜H)→/hatwiderHCΞ(˜H+) (the functorF6should be
+replaced with the identity functor). All properties of •†including Proposition 3.6.5 are
+transferred to the present situation without any noticeable modifi cations.
+3.7.Functor•†: HCΞ(˜H+)ΛH→HC(˜H).In this subsection we will construct a functor •†
+from/hatwiderHCΞ(˜H+)ΛHto/hatwiderHC(˜H). This functor equals G1◦F−1
+7,2, whereG1is defined as follows.
+PickM∈HC(˜H∧L|L). ThenitsspaceM(L)ofglobalsectionsisa C×-equivariantPoisson
+˜H-bimodule. LetM(L)l.f.denotethesum ofallHCsub-bimodules of M(L). Note, however,
+thatM(L)l.f.isnotaprioriHCitself becausewedo notknowatthemoment that M(L)l.f.is
+finitely generated. Set G1(M) =M(L)l.f.. This is a functor from HCΞ(˜H+) to the category
+/hatwiderHC(˜H)fromRemark3.6.6. Belowwewillseethat N†∈HCL(˜H)provided N∈HCΞ
+0(˜H+)ΛH.
+It is easy to see that •†naturally extends to a functor /hatwiderHCΞ(˜H+)→/hatwiderHC(˜H).
+Let us list some properties of the functor •†.
+Proposition 3.7.1. (1)The functor•†is left exact, S(˜c)-bilinear and commutes with
+the operators Ci.
+(2)Thefunctor•†:/hatwiderHCΞ(˜H+)ΛH→/hatwiderHC(˜H)is rightadjointto•†:/hatwiderHC(˜H)→/hatwiderHCΞ(˜H+)ΛH.
+Proof.To prove (1) we remark that G1is easily seen to be left exact, while Fi,i= 2,...,7 are
+equivalences (of abelian categories). Also all functors under cons ideration are S(˜c)-bilinear
+and commute with the Ci’s.
+Let us prove (2). It is enough to show that for M∈HC(˜H) andM∈HC(˜H∧L|L)ΛHthe
+Hom-spaces Hom( M,M(L)l.f.) and Hom(F1(M),M) are naturally isomorphic.
+Let us remark that Hom( M,M(L)l.f) = Hom( M,M(L)). Any homomorphism ϕ∈
+Hom(M,M(L)) can be localized to a unique homomorphism ϕ|L:M|L→M(hereM|L
+stands for the sheaf theoretic pull-back of M|XtoL). It is clear that ϕ|Lis continuous
+in the˜p|L-adic topology. The completion of M|Lwith respect to the ˜p|L-adic topologyCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 41
+coincides withF1(M). Soϕ|Lextends to a unique continuous homomorphism F1(M)→M.
+Summarizing, we get a natural map Hom( M,M(L)l.f)→Hom(M∧L|L,M).
+Conversely, pick ψ∈Hom(F1(M),M). Then we get the corresponding homomorphism
+ψ(L) :F1(M)(L)→M(L) between the spaces of global sections. Compose ψ(L) with a
+natural homomorphism M→F1(M)(L). So we get the natural map Hom( F1(M),M)→
+Hom(M,M(L)l.f.). It is clear that the maps we have constructed are inverse to eac h other.
+/square
+Now we are going to study the restriction of •†to a certain subcategory of HCΞ(˜H+)ΛH.
+Namely, for λ∈ΛHlet HCΞ
+˜p+(˜H+)λdenote the full subcategory of HCΞ(˜H+)ΛHconsisting
+of all bimodules satisfying the following two conditions
+(i) they are annihilated by ˜p+from the left (and then automatically also from the right).
+(ii) The operator Ci={ci,·}is the scalar λifor everyi= 1,...,r.
+Also let HC ˜p(˜H)λbe the full subcategory of HC( ˜H) consisting of all modules annihilated by
+˜pfrom the left (and from the right) and satisfying (ii). Tautologically, HC˜p(˜H)λ⊂HCL(˜H).
+Set HC ˜p,∂L(˜H)λ:= HC ˜p(˜H)λ∩HC∂L(˜H),HC˜p,L(˜H)λ:= HC ˜p(˜H)λ/HC˜p,∂L(˜H)λ. According
+to Proposition 3.6.5, •†induces a functor HC ˜p,L(˜H)λ→HC˜p+(˜H+)λ. The induced functor
+will also be denoted by •†. We remark that ˜p+has finite codimension in ˜H+and so all
+modules in HC˜p+(˜H+)λare finite dimensional.
+Proposition 3.7.2. The image of HC˜p+(˜H+)λunder•†lies inHC˜p(˜H)λ. The induced
+functor•†: HC˜p+(˜H+)λ→HC˜p,L(˜H)λis quasiinverse to •†: HC˜p,L(˜H)λ→HC˜p(˜H+)λ.
+Proof.The proof is in several steps.
+Step 1. LetN∈HC˜p+(˜H+)λ. The left and right multiplication actions of the sheaf
+C(A/planckover2pi1|L/hatwide⊗C[[/planckover2pi1]]˜H+∧0) onF−1
+7,5(N) factor through C( OL#Γ). The latter sheaf is naturally iden-
+tified withOL⊗C(CΓ) because Γ acts trivially onOL. Furthermore, as a Ξ ×C×-equivariant
+C(OL#Γ)-bimodule,F−1
+7,5(N) is justOL⊗C(N).
+The multiplication actions of H∧L|LonF−1
+7,4(N) still factors through OL⊗C(CΓ). This
+follows directly from the construction of the functor F4. Moreover, we claim that F−1
+7,4(N)
+is still isomorphic to OL⊗C(N) as a Ξ×C×-equivariantOL⊗C(CΓ)-bimodule. This again
+follows from the construction of F4and (ii): indeed, (3.20) implies that F−1
+4is just a regluing
+by means of a coboundary with the necessary invariance propertie s.
+So as a (OL⊗C(CΓ))Ξ-moduleF−1
+7,3(N) is just (OL⊗C(N))Ξ. NowF−1
+2is just the
+pull-back with respect to the isomorphism
+θ0:ι∗(SV#Γ|X)∼− →(OL⊗C(CΓ))Ξ
+induced from (2.11); here ιis the inclusionL֒→X. In particular, we see that the space of
+global sections of F−1
+7,2(N) is just (SV0⊗C(N))Ξ. The algebra SV0is finite over C[L] and so
+(SV0⊗C(N))Ξis a finitely generated C[L]-module. So we have checked that
+•N†is naturally identified with ( SV0⊗C(N))Ξ,
+•the left and right multiplication actions of ˜HonN†factor through ι∗(SV#Γ), where
+ι:L֒→Xis the inclusion,42 IVAN LOSEV
+•the multiplication action of ι∗(SV#Γ) on (SV0⊗C(N))Ξis obtained from the natural
+homomorphism ι∗(SV#Γ)֒→(SV0⊗C(CΓ))Ξ.
+•N†is a finitely generated ι∗(SV#Γ)-module.
+In particular, we have checked that •†maps HC˜p+(˜H+)λto HC ˜p(˜H)λ.
+Step 2. Now let M∈HC˜p(˜H)λ. The action of ˜HonMfactors through ι∗(SV#Γ).
+The natural map M→ι∗(M) is therefore an isomorphism. Then F1(M) is just the pull-
+backι∗(M). SinceF7,2is a category equivalence, we deduce from the previous step that
+G1(F1(M)) =ι∗(ι∗(M)) is a finitely generated C[L]-module. We are going to prove that
+both the kernel and the cokernel of the natural map M→ι∗(ι∗(M)) are supported on
+∂L. This homomorphism is nothing else but the natural homomorphism M→κ∗(κ∗(M)),
+whereκstands for the inclusion L֒→L. Its (sheaf-theoretic) restriction to Lis just the
+isomorphism κ∗(M)→κ∗(M).
+So we see that (•†)†is isomorphic to the identity functor of HC ˜p,L(˜H)λ.
+Step 3. Now it remains to prove that ( •†)†is isomorphic to the identity functor of
+HC˜p+(˜H+)λ. Again, this reduces to checking that the natural homomorphism F1(G1(M))→
+Misanisomorphismforany M∈HC(˜H∧L|L)λannihilatedby ˜p∧L|L. Butonthelevelof OL-
+modules the functor F1◦G1is justκ∗◦κ∗. Sinceκ∗(M) is a finitely generated C[L]-module
+(seestep1), weseethatthenaturalhomomorphism M→κ∗(κ∗(M))isanisomorphism. /square
+Proposition3.6.5implies that •†restricts toa functor HCL(˜H)→HCΞ
+0(˜H+)ΛH. The latter
+induces a functor •†: HCL(˜H)→HCΞ
+0(˜H+)ΛH.
+Proposition 3.7.3. (1)N†∈HCL(˜H)forN∈HCΞ
+0(˜H+)ΛH.
+(2)For any M∈HCL(˜H)the kernel and the cokernel of the natural homomorphism
+M→(M†)†lie inHC∂L(˜H). In other words, •†: HCΞ
+0(˜H+)ΛH→HCL(˜H)is the left
+inverse to•†: HCL(˜H)→HCΞ
+0(˜H+)ΛH.
+(3)The functor•†: HCL(˜H)→HCΞ
+0(˜H)ΛHis a fully faithful embedding.
+Proof.(1). We are going to prove the following statement using the descen ding induction
+onl.
+(∗l) Suppose Nis annihilated by c(l). ThenN†∈HCL(˜H).
+The base is l=r. In this case Nis annihilated by cand hence is finite dimensional. So it
+has a finite filtration, whose quotients are objects from HCΞ
+p+(˜H+)λfor various λ∈ΛH. To
+prove (∗r) use Proposition 3.7.2 and the left exactness of •†.
+Now suppose that ( ∗l+1) is proved and let Nbe annihilated by c(l). Consider the homo-
+morphism N[−2]→Ngiven by the left multiplication by cl(recall that c0:=t). Clearly,
+•†commutes with the grading shifts and is a S(c)-bilinear functor, Proposition 3.7.1. Let
+N0stand for the cl-torsion in N. Applying (∗l+1) and a filtration argument, we see that
+N0†∈HCL(˜H). Since•†is left exact, this reduces the proof to the case when NisC[cl]-flat.
+So we get a monomorphism N†[−2]cl·−→N†. Its cokernel is contained in ( N/clN)†. The
+latter lies in HCL(˜H) by (∗l+1). It remains to show that N†is a finitely generated H-
+bimodule (or a left module). This is done completely analogously to the p roof of Lemma
+3.3.3 in [Lo2].
+(2). Let M′,M′′denote the kernel and the cokernel of the natural homomorphis mM→
+(M†)†. First of all, let us check that M′∈HC∂L(˜H). By Proposition 3.6.5, the latterCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 43
+is equivalent to M′
+†= 0. But since •†is exact, we have the following exact sequence:
+0→M′
+†→M†→[(M†)†]†. FromProposition3.7.1itfollowsthat •†: HCΞ
+0(˜H+)→HCL(˜H)
+is right adjoint to •†: HCL(˜H)→HCΞ
+0(˜H+). In particular, the natural homomorphism
+M†→[(M†)†]†is injective. So M′
+†= 0.
+It remains to show that M′′
+†= 0. First of all, suppose that Mis annihilated by ˜p. Then
+the claim follows from Proposition 3.7.2. Next, consider the case when Mis annihilated by
+c. In this case Madmits a finite filtration whose quotients lie in HC˜p+(˜H+)λ. Since both
+functors•†,[(•†)†]†are left exact, we see that M′′
+†= 0, compare with the proof of [Lo2],
+Proposition 3.3.4.
+Consider the general case. By the previous paragraph, the natu ral morphism M†/cM†∼=
+[M/cM]†→[([M/cM]†)†]†is an isomorphism. Now a simple diagram chase shows that the
+projection of c[(M†)†]†toM′′
+†is surjective. Therefore cM′′
+†=M′′
+†and soM′′
+†= 0.
+(3) follows from (2) and some standard abstract nonsense. /square
+Remark 3.7.4. We use the notation introduced in Remark 3.6.7.
+Similarly to the above we can define a functor •†: HCΞ(˜Z+
+(l))→/hatwiderHC(˜Z(l)). Propositions
+3.7.1,3.7.2 and Corollary 3.7.3 generalize to the present situation in a str aightforward way.
+In particular, we have a functor •†: HCΞ
+0(˜Z+
+(l))→HCL(˜Z(l)).
+3.8.Proof of Theorem 3.4.5. LetM∈HC(˜H). LetN⊂M†be an HC sub-bimodule.
+LetN†Mdenote the preimage of N†⊂(M†)†under the natural homomorphism M→(M†)†.
+The following lemma describes the properties of N†M.
+Lemma 3.8.1. (1)M′:=N†Mis the largest (with respect to inclusion) HC sub-bimodule
+inMwithM′
+†⊂N.
+(2)SupposeM1⊂N†M. ThenN†M/M1= (N/M1†)†M/M1.
+Proof.(1): Letuscheckthat( N†M)†⊂Nor, inotherwords, thatthecomposition( N†M)†֒→
+M†։M†/Nis zero. By the adjointness property, the composition ( N†M)†֒→M†։M†/N
+gives rise to a morphism N†M→[M†/N]†and it is enough to check that the latter is zero.
+By the construction, the last morphism factors as
+N†M→M→[M†]†→[M†]†/N†→[M†/N]†.
+The composition of the first three morphisms in the sequence above is zero by the definition
+ofN†M. So (N†M)†⊂N.
+Now let M′⊂Mbe such that M′
+†⊂N. We need to show that the image of M′in
+(M†)†lies inN†. This follows easily from the adjointness property similarly to the pre vious
+paragraph.
+(2): This follows from the exactness of •†and assertion (1). /square
+The main result of this subsection and a crucial step in the proof of T heorem 3.4.5 is the
+following result. This theorem (together with a proof) is a complete a nalog of Theorem 4.1.1
+from [Lo2].
+Theorem 3.8.2. LetM∈HC(˜H)andN⊂M†be such that M†/N∈HCΞ
+0(˜H+). Then
+N= (N†M)†.
+Proof.Thanks to the exactness of •†and assertion (2) of Lemma 3.8.1, we may assume that
+N†M= 0. SoMembeds into ( M†/N)†. The latter is an object in HCL(˜H) by Proposition44 IVAN LOSEV
+3.7.3. So Mlies in HCL(˜H). Since M†= [(M†)†]†, below in the proof we may (and will)
+assume that M= (M†)†. SoN†= 0. We want to show that N= 0.
+We will prove the following claim using the descending induction on l.
+(∗l)N= 0 provided Mis annihilated by the left multiplication by c(l).
+The base is the case l=r. In this case Mis annihilated by c.
+We remark that since Nlies in HCΞ
+0(˜H+) and is annihilated by c, there is a nonzero HC
+sub-bimodule N1inNannihilated by ˜p+. But then N†
+1is nonzero by Proposition 3.7.2 and
+embeds into N†. Contradiction.
+Now suppose we have proved ( ∗l+1). LetMbe annihilated by c(l). First of all, consider
+the case when Mis annihilated, in addition, by the left multiplication by cn
+lfor somen. Let
+M1be the annihilator of clinM. Since•†is exact and cl-linear, we see that M1†is the
+annihilator of clinM†, compare with Corollary 2.4.3 in [Lo2] (it does not matter whether
+we consider left or right actions). Also clacts locally nilpotently on N. So ifN/\e}atio\slash= 0, then
+N∩M1†/\e}atio\slash= 0. It follows that ( N∩M1†)†/\e}atio\slash= 0 by (∗l+1) and hence N†/\e}atio\slash= 0. So (∗l) is proved
+provided Mis annihilated by cn
+lfor somen.
+Consider the general case. By the above, we may assume that Nis a flat left C[cl]-module.
+For sufficiently large nthe module cn
+lMhas no torsion. So we may assume that MisC[cl]-
+flat. Therefore N†
+1= 0 for any HC sub-bimodule N1⊂M†withclN1⊂N. So we may
+assume, in addition, that N⊂M†iscl-saturated, i.e., clm∈Nimpliesm∈N, equivalently
+M†/NisC[cl]-flat.
+SetNn:=N+cn
+lM†. We remark that ( N†Mn)†=Nn. Indeed, apply assertion (2) of
+Lemma 3.8.1 to cn
+lM⊂N†
+nand use the equality [( Nn/cn
+lM†)†M/cn
+lM]†=Nn/cn
+lM†that
+follows from the above.
+SetTk:=N†
+k/N†
+k+1. We have the natural maps Nk[−2]→Nk+1given by the left multi-
+plication by cl. Applying•†, we get a morphism
+(3.23) N†
+k[−2]→N†
+k+1
+againgiven by the multiplication by cl. We deduce that N†
+k+1contains clN†
+k. In other words,
+clacts trivially on Tk. Also the map (3.23) gives rise to a morphism
+(3.24) Tk[−2]→Tk+1.
+We can form the graded H(l+1)-moduleT:=/circleplustext∞
+k=0Tk. EquipTwith the structure of C[cl]-
+module by setting clmform∈Tkto be the image of munder the homomorphism (3.24).
+SoTbecomes an H(l+1)[cl]-module.
+Suppose that we know that Tis finitely generated as an H(l+1)[cl]-module. In particular,
+there isN∈Nsuch thatTN+n=cn
+lTNfor anyn/greaterorequalslant0 or, equivalently, N†
+N+n=cn
+lN†
+N+
+N†
+N+n+1. From this using the induction on kwe can check that
+(3.25) N†
+N+n=cn
+lN†
+N+N†
+N+n+k.
+SetM′:=/intersectiontext
+nN†
+n. ThenM′is an HC sub-bimodule in M, whose image in M/cn
+lMcoincides
+with the image of N†
+m,m≫0. The latter is nonzero since ( N†
+m)†=Nm. Therefore M′is
+nonzero. Applying assertion (1) of Lemma 3.8.1, we see that M′
+†⊂Nmfor anym. But the
+intersection of Nmis justN, soM′
+†⊂N. Applying Lemma 3.8.1 again, we get M′⊂N†,
+contradiction with N†= 0.COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 45
+It remains to verify that Tis finitely generated as an H(l+1)[cl]-module. Set C:=
+(M†/N1)†. This is an object in HCL(˜H) annihilated by c(l+1)and, in particular, a graded
+H(l+1)-bimodulethatisfinitely generatedasaleft H(l+1)-module. Theclaimthat Tisfinitely
+generated stems from the following lemma. The proof of the lemma will complete that of
+the theorem. /square
+Lemma 3.8.3. There is an embedding T ֒→C[cl]ofH(l+1)[cl]-modules.
+Proof.We will construct embeddings ιi:Ti→Csuch thatιi+1(clm) =ιi(m). SinceM†/N
+is a flatC[cl]-module, we have the following exact sequence
+(3.26) 0 →M†/N1cn
+l·−→M†/Nn+1→M†/Nn→0.
+Applying•†to (3.26), we get the exact sequence
+(3.27) 0 →(M†/N1)†cn
+l·−→(M†/Nn+1)†→(M†/Nn)†.
+We have a natural embedding Tn֒→M/N†
+n+1֒→(M†/Nn+1)†, whose image in ( M†/Nn)†
+vanishes. This gives rise to an embedding ιn:Tn֒→C. The claim on the compatibility with
+the multiplication by clstems from the following commutative diagram.
+0
+C
+(M†/Nn+1)†
+(M†/Nn)†0
+C
+(M†/Nn+2)†
+(M†/Nn+1)†Tn Tn+1❄
+❄
+❄❄
+❄
+❄✲
+✲
+✲
+✲
+ ✠❅
+❅❅ ❘❍❍❍ ❨✟✟✟ ✯id
+cl·
+cl·
+cl·
+/square
+Let us complete the proof of Theorem 3.4.5.
+Proof of Theorem 3.4.5. (1) and (3) follow from Proposition 3.6.5. (2) follows from Propo-
+sitions 3.7.1, 3.7.3. The claim that the image of •†is closed under taking quotients follows
+from Theorem 3.8.2. Now (5) follows easily from (2) and the exactnes s of•†.
+The proof of (4) is completely analogous to the proof of assertion ( 4) of Theorem 1.3.1 in
+[Lo2]. /square
+Remark 3.8.4. WepreservethenotationofRemarks3.6.7, 3.7.4. Again, allconstru ctionsof
+this subsection apply to the functors •†: HC(˜Z(l))→HCΞ(˜Z+
+(l)),•†: HCΞ(˜Z+
+(l))→/hatwiderHC(˜Z(l)).
+The straightforward analogs of Theorems 3.8.2, 3.4.5 hold for these functors.
+3.9.Proofs of Theorems 1.3.1,1.3.2,3.4.6. We start by constructing functors required
+in Theorem 3.4.6.
+LetM∈cHC(H)c′. Pick a filtration F iMas in Definition 3.4.4 and set M:=R/planckover2pi1(M).
+Then setM†:=M†/(/planckover2pi1−1)M†. It is easy to see that M†∈cHCΞ(H+)c′. As in Subsec-
+tion 3.4 of [Lo2], we see that M†does not depend on the choice of the filtration F iM(or46 IVAN LOSEV
+more precisely, the objects MF
+†,MF′
+†constructed from different filtrations F ,F′are isomor-
+phic via a distinguished isomorphism) so the assignment M/ma√sto→M †does define a functor
+cHC(H)c′→cHCΞ(H+)c′. We remark that •†is a tensor functor, this is proved completely
+analogously to assertion (1) of Proposition 3.4.1 in [Lo2].
+Similarly, forN∈cHCΞ
+0(H+)c′one can defineN†∈cHCL(H)c′.
+Now Theorem 3.4.6 can be deduced from Theorem 3.4.5 in a straight-fo rward way.
+To prove Theorems 1.3.1,1.3.2 we will need their analogs for the algebra s˜H,˜H+and
+˜Z(l),˜Z+
+(l). For a Poisson two-sided ideal I⊂˜H+setI‡:= (/intersectiontext
+γ∈Ξγ.I)†˜H. In other words,
+I‡is the kernel of the map ˜H→(˜H+//intersectiontext
+γ∈Ξγ.I)†. LetId(˜H) denote the set of graded
+Poisson two-sided ideals of ˜H. LetIdL(˜H) stand for the subset of Id(˜H) consisting of all
+J∈Id(˜H) with V( ˜H/J) =L. Define the sets Id(˜H+),Id0(˜H+) in a similar way. Finally, let
+IdΞ
+0(˜H+) denote the set of Ξ-invariants in Id0(˜H+). It is clear that J†∈IdΞ
+0(˜H+) provided
+J∈IdL(˜H) and that I‡∈IdL(˜H) whenever I∈Id0(˜H+).
+Theorem 3.9.1. The maps introduced above enjoy the following properties.
+(1)J∩S(˜c)⊂J†∩S(˜c)andI∩S(˜c)⊂I‡∩S(˜c).
+(2)J⊂(J†)‡,I⊃(I‡)†for anyJ∈Id(˜H),I∈IdΞ(˜H+).
+(3){J†:J⊂IdL(H)}=IdΞ
+0(˜H+).
+(4)We have V((J†)‡/J)⊂∂L.
+(5)Considerthe restriction of the map I/ma√sto→I‡to the setof all maximalideals in IdΞ
+0(˜H+).
+The image of this restriction is the set of all prime ideals J∈IdL(˜H)such that
+S(˜c)∩Jis a maximal homogeneous ideal in S(˜c).1Further, each fiber of the restriction
+is a single Ξ-orbit.
+Recall that a two-sided ideal Iin an associative algebra Ais called prime if aAb⊂I
+impliesa∈Iorb∈I.
+Proof of Theorem 1.3.1. Assertion (1) follows easily from the fact that the functors •†and
+•†areS(˜c)-bilinear. Assertion (2) follows from Lemma 3.8.1. Assertion (3) follo ws from
+assertion (3) of Theorem 3.4.5 applied to I/(I‡)†⊂(˜H/I‡)†. In fact, Theorem 3.8.2 implies
+that (I‡)†=Iprovided I∈IdΞ
+0(˜H). Assertion (4) follows from assertion (2) of Theorem
+3.4.5.
+Let us prove assertion (5). Our argument closely follows the lines of the proof of the
+similar statements for W-algebras, see [Lo1], the proof of assertio n (viii) of Theorem 1.3.1,
+and [Lo2], the proof of Conjecture 1.2.1. The proof is in several ste ps.
+LetId(˜H)cc,Id(˜H+)ccdenote the subsets in Id(˜H),Id(˜H+) consisting of all ideals whose
+intersections with S(˜c) are maximal homogeneous.
+Step 1.We claim that for any prime Poisson ideal J∈Id(˜H)ccthe associated variety
+V(˜H/J) is irreducible. Our claim can be easily deduced from Ginzburg’s results , [G2], but
+we will present a proof based on our techniques. Let L1be a symplectic leaf of maximal
+dimension in V( ˜H/J). Then we have an inclusion J⊂J1:= (J†)‡, where•†,•‡are taken
+with respect toL1. Since the Gelfand-Kirillov dimensions of ˜H/Jand˜H/J1coincide (and
+1Of course, S(˜c) has the augmentation ideal ( ˜c) that contains any other homogeneous ideal. When we
+speak about maximal homogeneous ideals we mean homogeneous idea ls that are strictly contained in ( ˜c) and
+are maximal among such.COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 47
+coincide with dim L1+ 1), Corollar 3.6. from [BoKr] implies J=J1. This argument also
+implies the surjectivity statement in (5).
+Step 2.Now we claim that an element J∈IdL(˜H)ccis a prime ideal if and only if Jis
+maximal in IdL(˜H)ccwith respect to inclusion. Indeed, a non-maximal element cannot be
+prime thanks to Corollar 3.6. from [BoKr]. Now let Jbe a maximal element in IdL(˜H)cc.
+Consider the radical√
+J. For anyz∈Zthe mapb/ma√sto→{z,b}is a derivation of ˜H. If an ideal
+is stable with respect to some derivation d, then its radical is also d-stable, see [D], Lemma
+3.3.3. So we see that√
+Jis a Poisson graded ideal and hence√
+J∈IdL(˜H)cc. It follows that
+J=√
+J. Therefore we have the prime decomposition J=/intersectiontextn
+i=1Ji. AllJiare again Poisson
+and graded, this is proved as above using [D], Lemma 3.3.3. So V( ˜H/Ji) is the closure of a
+symplectic leaf, thanks to Step 1. It follows that some Jilies inIdL(˜H)ccand hence J=Ji.
+Step 3.To complete the proof it remains to show that for a given maximal (=p rime) ideal
+J∈IdL(˜H)ccthe setAof the maximal ideals I∈Id0(˜H+)ccwithI‡=Jis a single Ξ-orbit
+(it is clear from the definition that this set is Ξ-stable). Pick I∈A. Then, as we have
+noticed in the proof of (3), J†= (I‡)†. The r.h.s. coincides with/intersectiontext
+γ∈Ξγ.I. It follows that/intersectiontext
+γ∈Ξγ.I1=/intersectiontext
+γ∈Ξγ.I2for anyI1,I2∈A. SoI1,I2are conjugate under the Ξ-action. /square
+Proof of Theorem 1.3.1. EmbedId(H1,c),Id(H+
+1,c)intoId(˜H),Id(˜H+),respectively, bysend-
+ing, say,J ∈Id(H1,c) toR/planckover2pi1(J). Assertion (1) of Theorem 3.9.1 implies that the maps
+in Theorem 3.9.1 restrict to maps between IdL(H1,c),Id0(H+
+1,c). The proofs of assertions
+(1),(2),(3) of Theorem 1.3.1 now follow from the corresponding asse rtions of Theorem 3.9.1.
+It is easy to see that J ∈Id(H1,c) is prime provided R/planckover2pi1(J) is. Assertion (5) of Theorem
+3.9.1 implies the analog of assertion (4) of Theorem 1.3.1, where “primit ive” is replaced with
+“prime”. Now it remains to use the fact that any prime two-sided idea l inH1,cis primitive,
+see Theorem A1 in [Lo3]. /square
+Proof of Theorem 1.3.2. Choose the minimal number lsuch thatcl/\e}atio\slash= 0. We may assume
+thatcl= 1. Then R/planckover2pi1l(Zc) is the quotient of ˜Z(l)by the ideal generated by ci−/planckover2pi12
+lci,i=
+l,...,r.
+Thanks to Remarks 3.6.7, 3.7.4, 3.8.4 we can follow the proof of Theore m 3.9.1 to see that
+there are maps•†:IdL(Zc)→IdΞ
+0(Z+
+c),•‡:Id0(Z+
+c)→IdL(Zc) enjoying the following
+properties.
+(1) The image of J/ma√sto→J †coincides with IdΞ
+0(Z+
+c).
+(2) We have V((J†)‡/J)⊂∂L.
+(3) Consider the restriction of the map I /ma√sto→ I‡to the set of all maximal ideals in
+Id0(Z+
+c). The image of this restriction is the set of all prime ideals J ∈IdL(Zc).
+Further, each fiber of the restriction is a single Ξ-orbit.
+Now the proof of Theorem 1.3.2 basically repeats that of Theorem 1.3 .1.
+/square
+Remark 3.9.2. As in the proof of Theorem 3.9.1, V( J) is irreducible for any prime ideal
+J∈Id(Zc). Alternatively, this follows from Martino’s results, [M].
+3.10.Definitions of functors using Θb.For applications in Section 5 we will need more
+simple minded versions of the functors •†,•†. Namely, above we used the isomorphism
+θ:˜H∧L|L∼− →C(H∧L|tw
+L)Ξto pass between the H- andH-sides. Now we are going to use the
+isomorphism ˜Θb:˜H∧b∼− →C(˜H∧0), whereb∈L. The last isomorphism can be obtained from48 IVAN LOSEV
+˜θby passing to completions, compare with the discussion in the end of S ubsection 2.5. The
+main disadvantage of using ˜Θbis that we do not see the action of Ξ. An advantage, however,
+is that the construction using ˜Θbis easier to deal with.
+Weare goingto construct functors •†,b: HC(˜H)→HC(˜H+) and•†,b: HC(˜H)→/hatwiderHC(˜H+).
+The functor•†,bis given by•†,b=F7◦F6◦F′
+5◦˜Θb
+∗(•∧b). Here•∧bis the comple-
+tion functor at bmapping from HC( ˜H) to HC( ˜H∧b), where the latter is defined anal-
+ogously to HC( ˜H∧L|L) (theC×-equivariance condition is replaced with the equivariance
+with respect to the derivation Einduced from the Euler derivation on ˜H). The functor
+F′
+5: HC(C( ˜H)∧0)→HC(C(˜H+)∧0) is constructed analogously to F5(see Subsection 3.6)
+and is a category equivalence.
+The functor•†,bis defined as [( ˜Θb
+∗)−1◦F′−1
+5◦F−1
+6◦F−1
+7(•)]l.f., where the subscript “ l.f.”
+has the same meaning as in the beginning of Subsection 3.7.
+The following lemma describes some properties of the functors •†,b,•†,b.
+Lemma 3.10.1. (1)Thebifunctors Hom(•†,b,⋄)andHom(•,⋄†,b)from/hatwiderHC(˜H)op×/hatwiderHC(˜H+)
+to the category of vector spaces are isomorphic.
+(2)There is a functor isomorphism between •†,band the composition of •†with the for-
+getful functor HCΞ(˜H+)→HC(˜H+).
+(3)There is an embedding of •†(or, more precisely, of the composition of •†with the
+corresponding forgetful functor) into •†,b.
+Proof.The first assertion is proved in the same way as assertion (2) of Pro position 3.7.1.
+Let us prove the second assertion. For M∈HC(˜H) the object Θb
+∗(M∧b) is naturally
+identified with the completion of F4,1(M) at a point b∈Llying overb. So we have a natural
+mapF5,1(M)→F′
+5◦Θb
+∗(M∧b). The proof that F5is an equivalence given in Subsection 3.6
+implies that the map under consideration is an isomorphism. Assertion (2) follows.
+To prove the third assertion we notice that ( ˜Θb
+∗)−1◦F′−1
+5◦F−1
+6◦F−1
+7(N) is naturally
+identified with the completion of F−1
+7,2(N) atb. The only global section of F−1
+7,2(N) that
+vanishes at the formal neighborhood of bis zero. This implies assertion (3). /square
+Remark 3.10.2. Proposition3.7.3 implies that N†is a finitely generated bimodule provided
+N∈HCΞ
+0(H+). In fact, N†,bis finitely generated for any finite dimensional Nas well. The
+proof is completely analogous to that given in Propositions 3.7.2,3.7.3.
+For an idealI⊂˜H+we can define the ideal I‡,b⊂˜Hsimilarly toI‡, see the beginning
+of Subsection 3.8. The following statement follows from assertions ( 2),(3) of Lemma 3.10.1
+and the constructions of I‡,b,I‡.
+Corollary 3.10.3. I‡,b=I‡for any idealI⊂˜H+.
+Corollary 3.10.4. IfI⊂H+
+1,cis a primitive ideal, then so is I‡⊂H1,c.
+Proof.It is enough to show that I†is prime, see the main theorem from [Lo3]. The proof
+basically repeats that of assertion (iv) of Theorem 1.2.2 in [Lo1]. We will provide the proof
+for reader’s convenience.
+First of all, we note that I‡is prime provided R/planckover2pi1(I‡) =R/planckover2pi1(I)‡is. The ideal in R/planckover2pi1(H1,c)∧b
+corresponding to R/planckover2pi1(I) under the bijections explained above is prime. So we need to check
+thatJ/planckover2pi1:=R/planckover2pi1(H1,c)∩J′
+/planckover2pi1is prime for any prime ideal J′
+/planckover2pi1⊂R/planckover2pi1(H1,c)∧b. Assume the contrary:
+letJ1
+/planckover2pi1,J2
+/planckover2pi1be two-sided ideals with J1
+/planckover2pi1J2
+/planckover2pi1⊂J/planckover2pi1andJ/planckover2pi1/subsetnoteqlJ1
+/planckover2pi1,J2
+/planckover2pi1. Then (J1
+/planckover2pi1)∧b(J2
+/planckover2pi1)∧b⊂J′
+/planckover2pi1COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 49
+hence, say, (J1
+/planckover2pi1)∧b⊂J′
+/planckover2pi1. ThereforeJ1
+/planckover2pi1⊂(J1)∧b
+/planckover2pi1∩R/planckover2pi1(H1,c)⊂J/planckover2pi1, a contradiction. So J/planckover2pi1is
+a prime ideal. /square
+4.Applications to general SRA
+4.1.Proof of Theorem 1.4.1. We begin the proof with the following algebro-geometric
+lemma that should be standard, compare with [BrGo], the proof of T heorem 4.2. The proof
+was essentially suggested to me by I. Gordon.
+Lemma 4.1.1. LetX1,X2be varieties, Y1,Y2irreducible divisors of X1,X2contained in all
+irreducible components of X1,X2, respectively, and Aibe anOXi-coherent sheaf of associative
+algebras on Xi,i= 1,2. Suppose that:
+(1)A2is a trivial sheaf of algebras on X2\Y2.
+(2)There is an isomorphism ϑ: (X1)∧
+Y1→(X2)∧
+Y2of the completions such that the sheaves
+of algebras ϑ∗(ι∗
+2(A2)),ι∗
+1(A1)are isomorphic. Here ιiis the canonical embedding
+(Xi)∧
+Yi֒→Xi.
+Then there is a neighborhood U1ofY1inX1such that the fiber of A1at any point u∈U1\Y1
+is isomorphic as an algebra to a fiber of A2.
+Proof.ShrinkingX1,X2if necessary we may (and will) assume that A1is a free sheaf on
+X1\Y1of rank, say, r. Also we can assume that X1is affine. Let f1denote a function in
+C[X1] that vanishes on Y1.
+Consider the space Mof all algebra structures on Cr, i.e., maps Cr⊗Cr→Cr. LetZ
+denote the locally closed subvariety in Mconsisting of all algebra structures isomorphic to
+a fiber ofA2(at any point of X2\Y2). LetI,˜I⊂C[M] denote the ideals of the closure Z
+and of the boundary ∂Z. Letzbe a point in Z. It corresponds to a choice of a basis in some
+fiber ofA2.
+Fix a section sofA1. It defines a morphism s:X1\Y1→M. We claim that s∗(I) is zero
+ands∗(˜I) is not. The algebra C[X1]f1maps toA:= (C[X1]∧
+Y1)f1. (2) can be interpreted as
+follows: there is an element g∈GLr(A) such that s∗(f) =f(gz) (the equality in A).
+First of all, let us show that f(gz) = 0 forf∈I. We have f(gz) = 0 (in C) for all
+f∈GLr(C) and hence f(gz) = 0 (in C[X]f1) for allgwith coefficients in C[X]f1(not
+necessarily invertible). Any g∈GLr(A) can be written as fN
+1g1, whereg1has entries in
+C[X1]∧
+Y1. Now any g1∈GLr(C[X1]∧
+Y1) is the limit of a sequence of elements with entries in
+C[X]. It follows that f(gz) = 0 for all g∈GLr(A). Sos∗(f) = 0 and hence s∗(I) ={0}.
+The proof that s∗(˜I)/\e}atio\slash={0}is similar. /square
+Proof of Theorem 1.4.1. We preserve the notation used in the proof of Theorem 1.3.2. Let
+Lbe such that q∈IdL(Zc).
+Set˜X1= Spec(R/planckover2pi1l(Zc)),˜X2= Spec(R/planckover2pi1l(Zc)Ξ) =V∗
+0×Spec(R/planckover2pi1l(Z+
+c)Ξ). Further, let
+˜A1,˜A2be the sheaves on ˜X1,˜X2corresponding to R/planckover2pi1l(H0,c),C(R/planckover2pi1l(H0,c))Ξ. The latter sheaf
+is just the product of OV∗
+0and the sheaf on Spec( R/planckover2pi1l(Z+
+c)Ξ) corresponding to C( R/planckover2pi1l(H+
+0,c))Ξ.
+Theorem 2.5.3 implies that there are open subsets ˜X0
+1,˜X0
+2of˜X1,˜X2such that there is an
+isomorphism ϑ: (˜X0
+1)∧
+L→(˜X0
+2)∧
+Lsatisfyingϑ∗(ι∗
+2(A2))∼=ι∗
+1(A1). Hereι1,ι2have similar
+meanings to Lemma 4.1.1 and ( ˜X0
+i)∧
+Lstand for the formal neighborhood of L∩˜X0
+iin˜X0
+i
+(we assume that ˜X0
+i∩L⊂L).
+Now let us introduce X1,X2,Y1,Y2,A1,A2for which we are going to apply Lemma 4.1.1.
+LetX1be the intersection of ˜X0
+1with the set of zeros of R/planckover2pi1l(q). SetX2:= (V∗
+0×X′
+2)/Ξ,50 IVAN LOSEV
+whereX′
+2is the set of zeros of R/planckover2pi1l(q†). ForYiwe take the divisor of zeros of /planckover2pi1linXi. These
+divisors are isomorphic to open subsets in Lhence are irreducible. Finally, let Ai,i= 1,2,
+be the restrictions of ˜AitoXi.
+The claim that ϑrestricts to an isomorphism ( X1)∧
+Y1→(X2)∧
+Y2follows directly from the
+construction of R/planckover2pi1l(q)†.
+So we see that for a general point yofLwe have
+H0,c/H0,cny∼=C(H+
+0,c/H+
+0,cn)
+(the r.h.s. is a typical fiber of A2). Thanks to [BrGo], Theorem 4.2, the algebras in the l.h.s.
+are pairwise isomorphic for all y∈ˆL. This proves Theorem 1.4.1 for any y∈ˆL. /square
+4.2.Simplicity ofH1,cfor general c.The following theorem and its proof (modulo The-
+orem 1.2.1) were communicated to me by P. Etingof. This was the first motivation to state
+Theorem 1.2.1.
+LetQdenote the field of algebraic numbers.
+Theorem 4.2.1. There is a finitely generated subgroup Λ⊂Qrsuch that the algebra H1,c
+is simple whenever/summationtextr
+i=1λici/\e}atio\slash∈Zfor all(λi)r
+i=1∈Λ\{0}.
+Proof. Step 1. First of all, let us study the set of all parameters csuch thatH1,chas a finite
+dimensional representation. We claim that there is a lattice Λ0⊂Qrsuch that the algebra
+H1,chas no finite dimensional representations whenever/summationtextr
+i=1λici/\e}atio\slash∈Zfor all (λi)r
+i=1∈Λ0.
+Let us reduce the proof to the case when Γ is generated by symplec tic reflections and is
+symplectically irreducible. The reduction is standard but we provide it for reader’s conve-
+nience.
+First, we claim that H1,c(V,Γ) has a finite dimensional representation if and only if
+H1,c(V,Γ′) does, where Γ′is the subgroup of Γ generated by all symplectic reflections. In-
+deed, it is easy to see from (1.2) that H1,c(V,Γ) =H1,c(V,Γ′)#Γ′Γ. Here the meaning of the
+right hand side is as follows. Let Abe an associative algebra containing CΓ′and acted on by
+Γ in such a way that the restriction of the Γ-action to Γ′coincides with the adjoint action of
+Γ′⊂A. Then as a vector space A#Γ′Γ is justA⊗CΓ′CΓ and the product is defined in the
+same way as forA#Γ. Now if Vis a non-zero leftA-module, then CΓ⊗CΓ′Vis a non-zero
+A#Γ′Γ-module. Since Ais included intoA#Γ′Γ, this shows the equivalence stated in the
+beginning of the paragraph.
+Let us now explain the reduction to the case when the Γ-module Vis symplectically
+irreducible. The Γ-module Vis uniquely decomposed into the direct sum of symplecti-
+cally irreducible Γ-submodules V=/circleplustextl
+i=1Vi(Viis said to be symplectically irreducible
+if it cannot be decomposed into the direct sum of two proper symplectic submodules).
+Moreover, since Γ is generated by symplectic reflections, it decomp oses into the direct
+product/producttextl
+i=1Γi, where Γ i⊂Sp(Vi) is generated by symplectic reflections. In particular,
+S=/coproducttextl
+i=1(S∩Γi). As a consequence of this discussion, we have a tensor product d ecom-
+positionH1,c(V,Γ) =/circlemultiplytextl
+i=1H1,ci(Vi,Γi), whereciis the restriction of ctoS∩Γi. Clearly,
+H1,chas a finite dimensional module if and only if every H1,ci(Vi,Γi) does. So it is enough
+to assume that Vis symplectically irreducible.
+LetMbe a finite dimensional H1,c(V,Γ)-module. Then the trace of [ x,y] onMis 0 for
+anyx,y∈V⊂H1,c(V,Γ). Therefore (1.2) implies that
+(4.1) (dim M)ω(x,y)+r/summationdisplay
+i=1ni(M)ciωi(x,y) = 0,COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 51
+whereni(M) denotes the trace of s∈SiinM, andωi:=/summationtext
+s∈Siωs. We remark that ωiis a
+Γ-invariant skew-symmetric form. Therefore ωiis proportional to ω, sayωi=miω. So we
+can rewrite (4.1) as
+(4.2) dim M+r/summationdisplay
+i=1ni(M)mici= 0.
+Consider the subgroup Λ0⊂Qrgenerated by ( mini(M′))r
+i=1, whereM′runs over all
+irreducible Γ-modules. (4.2) implies that Λ0satisfies the condition in the beginning of the
+step.
+Step 2.Now letLbe a symplectic leaf of V∗/Γ and Γ⊂Γ be such as above. Let cL
+denote the quotient of c(1)by the space of all ciwithSi∩Γ=∅. The space ( cL)∗is a
+subspace of c∗
+(1)spanned by some basis vectors. So the form ( cL)∗(Q) makes sense and it
+is a subspace in c∗
+(1)(Q) =Qr. Consider the algebra H+
+1,cconstructed from L. According
+to step 1, there is a finitely generated subgroup ΛL⊂(cL)∗(Q) such thatH1,chas no finite
+dimensional representations provided/summationtextr
+i=1ciλi/\e}atio\slash∈Zfor every (λi)r
+i=1∈ΛL. Let Λ be the
+subgroup in Qrgenerated by ΛLfor all symplectic leaves L⊂V∗/Γ. Let us prove that this
+Λ satisfies the conditions of the theorem.
+LetJ⊂H 1,cbe a two-sided ideal. Pick a symplectic leaf Lsuch thatLis an irreducible
+component of V(H1,c/J). ThenJ†⊂H1,cis of finite codimension. It follows that H1,chas a
+finite dimensional module. So some linear combination like in the stateme nt of the theorem
+should be integral. /square
+5.Rational Cherednik algebras
+5.1.Content. In this section we consider the case of rational Cherednik algebras in more
+detail. Recall that a rational Cherednik algebra is a special case of a n SRA corresponding
+to a special pair ( V,Γ). Namely, let hbe a complex vector space and Wbe a finite subgroup
+in GL(h). Then we set V:=h⊕h∗and take the image of Wunder a natural embedding
+GL(h)֒→Sp(V) for Γ. The corresponding SRA is given by the following relations, see [EG].
+[x,x′] = [y,y′] = 0, x,x′∈h∗,y,y′∈h.
+[y,x] =t/a\}b∇acketle{ty,x/a\}b∇acket∇i}ht−/summationdisplay
+s∈Sc(s)/a\}b∇acketle{tx,α∨
+s/a\}b∇acket∇i}ht/a\}b∇acketle{ty,αs/a\}b∇acket∇i}hts.(5.1)
+Hereαs∈h∗,α∨
+s∈hare elements that vanish on the fixed point hyperplanes hs,(h∗)sand
+satisfy/a\}b∇acketle{tαs,α∨
+s/a\}b∇acket∇i}ht= 2. We remark that the independent variables c(s) here are not exactly the
+same as in (1.2) – to get that setting we need to multiply c(s) from (5.1) by−1/2. Since
+it is common to present the rational Cherednik algebras in the form ( 5.1), we will use this
+version of c(s)’s from now on.
+Many questions considered in the present paper in the case of Cher ednik algebras were
+studied previously. One goal of this section is to compare our result s and constructions with
+the existing ones. Another goal is to strengthen some of our resu lts in this special case.
+There is a version of the isomorphism of completions theorem for rat ional Cherednik
+algebras due to Bezrukavnikov and Etingof, [BE]. In the first subse ction we will compare
+their result with ours. In Subsection 5.3 we will use the explicit form of the completions
+isomorphism obtained in [BE] to relate the associated varieties of the idealsI,I‡.52 IVAN LOSEV
+In Subsection 5.4 we will show that our notion of Harish-Chandra bimo dules agrees with
+one from [BEG2]. Subsection 5.5 contains an alternative definitions of functors•†,b,•†,bthat
+is based on the coincidence of the definitions. In [BE] the isomorphism of completions was
+used to construct certain induction andrestriction functors. In Subsection 5.6 we will apply
+the description of Subsection 5.5 to establish a relationship between those functors and the
+•†,•‡-maps between the sets of ideals. We also apply that description in Su bsection 5.7 to
+study the Harish-Chandra part of the space of maps between two modules in the category
+O.
+Finally,inSubsection5.8wedescribethetwo-sidedidealsintherationa lCherednikalgebra
+of typeA, i.e., corresponding to the symmetric group Sn.
+Below we will write Wfor Γ, andWfor Γ.
+5.2.Bezrukavnikov-Etingof’s isomorphism of completions. As before take the sym-
+plectic leafL⊂V∗/Wcorresponding to a subgroup W⊂W. Pick a point b∈hwith
+Wb=W. In[BE]EtingofandBezrukavnikovdiscoveredanisomorphism ϑb:H∧b→C(H∧0).
+The homomorphism ϑbis the only continuous homomorphism satisfying
+[ϑb(u)f](w) =f(wu),
+[ϑb(xα)f](w) = (xwα+/a\}b∇acketle{tb,wα/a\}b∇acket∇i}ht)f(w),
+[ϑb(ya)f](w) =ywaf(w)+r/summationdisplay
+i=1/summationdisplay
+s∈Si\W2ci
+1−λsαs(wa)
+xαs+/a\}b∇acketle{tb,αs/a\}b∇acket∇i}ht(f(sw)−f(w)).
+u,w∈W,α∈h∗,a∈h,f∈FunW(W,H∧0).(5.2)
+Let us explain the notation used in the formula. By xα,xαwe mean the images of α∈h∗
+under the embeddings h∗֒→H,H. Byλswe denote a unique non-unit eigenvalue of son
+h∗.
+The homomorphism ϑbis an isomorphism. Indeed, modulo cthe homomorphism ϑbco-
+incides with the isomorphism induced by θ0constructed in Subsection 2.5. Since H∧bis
+complete in the c-adic topology, ϑbis surjective, and since C( H∧0) is flat over C[[c∗]],ϑbis
+injective.
+We remark that we have a ( C×)2-action on Hgiven by (t1,t2).x=t2
+1x,(t1,t2).y=
+t2
+2y,(t1,t2).w=w,(t1,t2).ci=t2
+1t2
+2ciforx∈h∗,y∈h,w∈W. There is an analogous
+action on H. The isomorphism ϑbis equivariant with respect to the second copy of C×.
+This is checked directly from the formulas above. Also tracking the c onstruction of the iso-
+morphismθ:H∧L|L→C(H∧L|tw
+L)Ξ, we see that it can be made ( C×)2-equivariant. So the
+isomorphism θb:H∧b→C(H∧0) induced by θis also equivariant with respect to the second
+copy ofC×.
+We do not know whether it is possible to find θwithθb=ϑb. However, the following
+claim holds.
+Lemma 5.2.1. There is an element f∈c⊗C[h/W]∧0such thatθb= exp(1
+tad(f))ϑb.
+Proof.Consider the automorphism ρ:=θb◦(ϑb)−1of the algebra C( H∧0). This automor-
+phism is the identity modulo cand isC××W-equivariant. So ρ= exp(d), wheredis a
+c-linearC××W-equivariant derivation of C( H∧0). As in the proof of Proposition 2.11.1, we
+see thatd=1
+tad(f)for somef∈czt(C(H∧0)). We may assume that1
+tfisC××W-invariant.
+But the latter just means that f∈c⊗C[h/W]∧0. /squareCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 53
+In particular, we see that θb(x) =ϑb(x) forx∈h∗(in fact, this follows directly from the
+C×-equivariance condition).
+5.3.Associated varieties and •‡.In this subsection we will use the results of the previous
+one to relate the associated varieties of the ideals I⊂˜H+andI‡⊂˜H.
+In the sequel we will need a remarkable Eulerelement in ˜H. It is given by the following
+formula:
+(5.3) h=n/summationdisplay
+i=1xiyi+dimh
+2−/summationdisplay
+s∈S2c(s)
+1−λss.
+Herey1,...,ynis a basis in h, andx1,...,xnis a dual basis in h∗. A crucial property of h
+is that [h,xi] =xi,[h,yi] =−yifor alli. We can define the Euler element h+∈˜H+in a
+similar way.
+Proposition 5.3.1. LetIbe a Poisson ideal in ˜H+. Suppose that V(˜H+/I)coincides with
+the closure of the symplectic leaf L0⊂V∗
++/Wcorresponding to a subgroup W0⊂W. Then
+V(˜H/I‡)coincides with the closure of the leaf L0⊂V∗/Wcorresponding to W0(⊂W).
+Proof.Seth+:=h∩V+. First, we are going to show that the associated variety of I∩C[h+]
+coincides with WhW0
++.
+To this end let us show that the associated graded ideal gr IofIwith respect to the usual
+grading coincides with the associated graded gryIfor the grading induced by the C×-action
+considered in the previous subsection (we will call it the y-grading because the degree of
+y∈h+is 2, while the degree of x∈h∗
++is 0).
+TheC×-action on ˜H+given byt.x=tx,t.y=t−1y,t.w=w,t./planckover2pi1=/planckover2pi1,t.ci=ci,x∈
+h∗
++,y∈h+is inner, the corresponding grading is the inner grading given by eigen values of
+1
+tad(h+). Being Poisson (and so1
+tad(h+)-stable), the ideal Iis graded with respect to the
+inner grading. The standard grading and the y-grading differ by the inner grading. This
+implies the claim in the previous paragraph.
+NowI∩C[h+]Wis the 0th component in gry(˜Z+∩I). To establish the statement in the
+beginning oftheproofitisenoughtocheck thattheassociated var ietyofthe C[h+]W-module
+C[h+]/C[h+]∩Icoincides with πW(hW0
++). The latter is nothing else but the projection of
+V(˜H+/I) =πW(VW0
++)toh+/Wunderthenaturalmorphism V+/W։h+/W. Soitcoincides
+with the variety of zeros of I∩C[h+]W. This implies the claim in the beginning of the proof.
+Now let us show that the variety of zeros of I‡∩C[h]Wis contained in πW(hW0). LetAt
+stand for the Weyl algebra of VWand letIbe the ideal of πW(hW0)⊂C[h]W. By the above,
+(At⊗C[t]I)∩C[h]W⊃Infor somen. By the construction of I‡usingθbgiven in Subsection
+3.10, we see that I‡⊃(θb)−1/bracketleftbig
+(At⊗C[t]I)∩C[h]W/bracketrightbig
+. Now we can use Lemma 5.2.1 and the
+explicit form of ϑbto see that I‡⊃In∩C[h]W. This implies the claim in the beginning of
+the paragraph.
+So we see that V( ˜H/I‡)⊂πW(VW0). Let us prove the equality. The dimension of
+V(˜H+/(I‡)†) equals dimV( ˜H/I‡)−dimVWby assertion 4 of Proposition 3.6.5. On the
+other hand, dim VW0−dimVW= dimVW0
++= dimV( ˜H+/I). It remains to notice that
+(I‡)†⊂Iand so dimV( ˜H+/(I‡)†)/greaterorequalslantdimV(˜H+/I). /square54 IVAN LOSEV
+5.4.Equivalence of definitions of Harish-Chandra bimodules. Let us recall the no-
+tionofaHarish-Chandra H1,c-H1,c′-bimodulefrom[BEG2]. Considerthesubalgebras( Sh)W,
+(Sh∗)W⊂H1,c,H1,c′. In [BEG2] anH1,c-H1,c′-bimoduleMis called Harish-Chandra if the
+operators ad( a),ad(b) are locally nilpotent on Mfor anya∈(Sh∗)Wandb∈(Sh∗)W. The
+goal of this subsection is to show that this definition is equivalent to D efinition 3.4.4 (we
+remark that no analog of the Berest-Etingof-Ginzburg definition is known for general SRA).
+The easier implication is that a HC bimodule in the sense of Definition 3.4.4 is also HC
+in the sense of [BEG2]. This is based on the following proposition.
+Proposition 5.4.1. LetM∈HC(˜H). Then the operators ad(a),ad(b)are locally nilpotent
+for anya∈(Sh∗)W,b∈(Sh)W.
+Proof.Recall the Euler element h∈˜Hdefined by (5.3). Hence1
+t[h,·] is a grading preserving
+linear map M→M. So we have the decomposition of the graded component Miinto the
+direct sum/circleplustext
+αMi,αof generalized eigenspaces of1
+tad(h). Forβ∈CsetM(β):=/circleplustext
+iMi,β+i.
+ThenxM(β)⊂M(β)for anyx∈h∗, whileyM(β)=M(β−2)fory∈h. SinceMis a finitely
+generated graded H-module, we see that there are finitely many elements βi∈Cwith the
+property that M(β)= 0 whenever βi−β/\e}atio\slash∈Z/greaterorequalslant0.
+On the other hand, for a∈(Sh∗)Wwe have ad( a)M(β)⊂M(β)and hence1
+tad(a)M(β)⊂
+M(β+2). It follows that the operator1
+tad(a) is locally nilpotent. The proof for b∈(Sh)Wis
+similar. /square
+The following claim follows directly from Proposition 5.4.1 and the definitio n ofcHC(H)c′.
+Corollary 5.4.2. AnyM∈cHC(H)c′is a Harish-Chandra bimodule in the sense of [BEG2].
+Proposition 5.4.3. Let aH1,c-H1,c′-bimoduleMbe HC in the sense of [BEG2]. Then there
+exists a filtration FiMsatisfying the conditions of Definition 3.4.4.
+Proof. Step 1. Before constructing a filtration on Mlet us introduce some notation.
+Abusing the notation, by hwe denote the images of hinH1,c,H1,c′.
+Pick free homogeneous generators a1,...,anof (Sh∗)Wandb1,...,bnof (Sh)W. Let
+d1,...,dnbe their degrees. Further, let c1,...,cmbe bi-homogeneous generators of the
+(Sh∗)W⊗(Sh)W-module C[h⊕h∗]W. Let their degrees (of polynomials on h⊕h∗) be
+d′
+1,...,d′
+m. Liftc1,...,cmtoW-invariant ad( h)-eigenvectors in H1,c,H1,c′. Abusing the
+notation, we denote both liftings by c1,...,cm. We remark that if we have another collec-
+tion of liftings ˜ c1,...,˜cm, thenci−˜ci∈Fd′
+i−2H•. Finally, let us choose bi-homogeneous
+generators e1,...,ekofC[h⊕h∗]#WoverC[h⊕h∗]Wand lift them to ad( h)-eigenvectors
+e1,...,ekofH1,c,H1,c′. Letd′′
+1,...,d′′
+kbe the degrees of e1,...,ekviewed as polynomials.
+On the next steps we construct a filtration F iMonMsuch that grMis a finitely
+generated left SV#Γ-module and
+[z,v]∈Fi+j−2M, (5.4)
+vh⊂Fi+jM, (5.5)
+hv⊂Fi+jM. (5.6)
+for anyv∈FjM,h∈FiH1,•and anyzthat is an ordered monomial in ck,bj,aiof total
+degreei(with the degrees of ai,bj,ckas above). This filtration has the properties required
+in Definition 3.4.4.
+Step 2.We will define a filtration on Musing the inductive procedure below.COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 55
+LetVjbe a finite dimensional subspace in M, andAi,i∈I,be linear operators on M.
+SetVj+1:=/summationtext
+i∈IAiVj. Suppose that Vjis equipped with an increasing filtration F mVj. To
+eachAiwe assign an integral degree ni. Then we can define a filtration on Vj+1by defining
+FnVj+1to be the linear span of elements Aiv0, wherev0∈FmVjandm+ni/lessorequalslantn.
+Step 2.1. LetV0be a subspace that generates Mas a leftH1,c-module. SinceMis
+a finitely generated left H1,c-module we can choose V0to be finite dimensional. For the
+operatorsAiwe take all monomials in ad( ai),i= 1,...,n. For the degree of ad( ai) we take
+di−2, and the degree of a monomial is defined in an obvious way. Since ad( ai) are pairwise
+commuting locally nilpotent operators on M, we see that V1is finite dimensional.
+Step 2.2. To defineV2we useV1and the operators that are monomials in ad( bi), whose
+degrees are defined analogously to Step 3.1. The space V2is finite dimensional for the same
+reason asV1.
+Step 2.3. Now consider the operators 1 ,ad(ci),i= 1,...,m. To ad(ci) we assign degree
+d′
+i−2, and 1 is supposed to be of degree 0. We use the filtered subspace V2⊂Mand the
+operators 1 ,ad(ci) to defineV3, which is obviously finite dimensional.
+Step 2.4. We defineV4usingV3and the operators of the right multiplication by 1 ,ei. The
+operator corresponding to eihas degree d′′
+i.
+Step 2.5. Consider the basis of the form wxi1
+1...xinnyj1
+1...yjnninH1,c, wherex1,...,xn
+is a basis in h∗,y1,...,ynis a basis in h. DefineV5fromV4using the operators of left
+multiplication by the basis elements. The degree of an operator comin g from the basis
+element above is/summationtextn
+l=1(il+jl). Of course, V5=M.
+Let Fi
+•be the filtration on Vi,i= 1,2,3,4,5 defined on the corresponding step above. By
+our definition, Fj
+iVj⊂Fj+1
+iVjfor alli,j.
+On the next step we are going to prove that the filtration F •:= F5
+•onMsatisfies all
+conditions indicated in the end of Step 1.
+Step 3.Consider a collection C= (i1,w,i2,α,β,j1,j2), where i1,i2,j1,j2aren-tuples of
+non-negative integers, w∈W,αis either an element of {1,2,...,m}or the symbol∅, andβ
+is either an element of {1,2,...,k}, or∅. Tov0∈V0andCas above we assign the element
+v(C,v0)∈Mby
+v(C,v0) :=wxi1yi2[ad(c)αad(b)j1ad(a)j1v0]eβ,
+where, for instance ad( a)j2means/producttextn
+i=1ad(ai)j2
+i, and ad(c)αis 1 whenα=∅, and ad(cα)
+otherwise.
+Further, define the “degree”
+d(C) :=n/summationdisplay
+r=1(i1
+r+i2
+r+(j1
+r+j2
+r)(dr−2))+d′
+α−2+d′′
+β,
+whered′
+∅:= 2,d′′
+∅:= 0 so that v(C,v0)∈Fd(C)M(but it may happen that v(C,v0)∈
+Fd(C)−1M). Finally, define the “length”
+l(C) =n/summationdisplay
+r=1(i1
+r+i2
+r+j1
+r+j2
+r)+lα+lβ+lw,
+wherelα= 0,lβ= 0,lw= 0 whenα=∅,β=∅,w= 1 andlα= 1,lβ= 1,lw= 1 otherwise.
+By our choice of V0, the elements v(C,v0) spanMas a vector space. Moreover, by our
+definition of the filtration on M, the images of v(C,v0) in Fd(C)M/Fd(C)−1Mspan grM.
+From here it is easy to see that gr Mis a finitely generated left C[h⊕h∗]#W-module. It56 IVAN LOSEV
+remains to check (5.4)-(5.6). It is enough to do this for those v(C,v0) that do not lie in
+Fd(C)−1M.
+First of all, let us notice that (5.6) follows directly from the construc tion of the filtration.
+We are going to prove (5.4),(5.5) by induction on l(C).
+We start with l(C) = 0 (so that v(C,v0) =v0). Assume that (5.4),(5.5) are proved for all
+z,hof degree less than Nand let us prove the inclusions for degree N(the base is N= 0,
+here there is nothing to prove). Each h∈H1,•can be uniquely written as the sum of ordered
+monomials in cα,bj,ai,eβ(withcα,eβoccurring at most once). For h∈FiH1,•the degrees
+of monomials do not exceed i. Also ifhis an eigenvector for ad( h) and its image in gr H1,•
+is in the center, then we see that all monomials of degree exactly iwill be incα,bj,ai(or we
+can also write the variables in the opposite order: ai,bj,cα), and all other monomials will
+have degree /lessorequalslanti−2.
+To prove [z,v]∈FN−2Mwe may assume that zis an ordered monomial in ai,bj,cα
+(in this order). If zis a single variable, then we are done by the definition of the fil-
+tration. If not, write zas a product z=z1z2, wherez1is a variable. Then rewrite
+[z,v0] =z1[z2,v0] + [z1,v0]z2=z1[z2,v0] +z2[z1,v0]−[z2,[z1,v0]]. Then the first two sum-
+mands lie in F N−2Mthanks to the inductive assumption and (5.6). To show that the thir d
+summand lies in F N−2Mwe apply the same procedure for z2=z′
+2z′′
+2and [z1,v0] instead
+ofv0. We get [z2,[z1,v0]] =z′
+2[z′′
+2,[z1,v0]] +z′′
+2[z′
+2,[z1,v0]] + [z′′
+2,[z′
+2,[z1,v0]]]. So we need to
+show that ad( z′
+2)ad(z1)v0,ad(z′′
+2)ad(z1)v0,ad(z′′
+2)ad(z′
+2)ad(z1)v0lie in the appropriate filtra-
+tion components. For the first expression this follows directly from the definition of the
+filtration. We remark that the variables z1,z′
+2still precede all variables in z′′
+2in the order
+prescribed in the beginning of the paragraph. So we apply the same p rocedure as before to
+z′′
+2and so on. Finally, we arrive at the sum of expressions of the form z0ad(zk)...ad(z1)v0,
+wherezk,...,z1are ordered correctly. Tracking the construction and using the d efinition of
+the filtration, we see that all these expressions are in F N−2M.
+The inclusion (5.5) for v∈V0is proved in a similar way.
+Now suppose that (5.4),(5.5) hold for all z,hand allv(C,v0) withl(C)<l. Let us prove
+(5.5)forl(C) =l, theproofof(5.4)issimilar. If βinCisnot∅, thenv(C,v0)h=v(C′,v0)eβh
+withl(C′)< l(C) and we are done by the inductive assumption. Suppose now that β=
+∅. Thenv(C,v0) = [z,v(C′,v0)] for appropriate zandl(C′)< l(C). Thenv(C,v0)h=
+z(v(C′,v0)h)−v(C′,v0)zh. By the inductive assumption, v(C′,v0)h,v(C′,v0)zhlie in the
+appropriate filtration component. Now the right hand side itself lies in the appropriate
+filtration component by (5.6).
+/square
+5.5.Alternative definitions of •†,b,•†,b.LetMbeaHarish-Chandra H1,c-H1,c′-bimodule.
+Since the adjoint action of ( Sh∗)W=C[h/W] onMis locally nilpotent, the tensor prod-
+uctM∧b:=C[h/W]∧b⊗C[h/W]Mhas a natural structure of a bimodule over H∧b
+1,c:=
+C[h/W]∧b⊗C[h/W]H1,candH∧b
+1,c′. Soθb
+∗(M′) is a bimodule over C( H∧0
+1,c)-C(H∧b
+1,c′). Consider
+the subspaceM♥,b⊂e(W)θb
+∗(M∧b)e(W) consisting of all elements vthat commute with
+hW,(h∗)Wand such that ad( a)Nv= ad(b)Nv= 0 for alla∈S(h∗
++)W,b∈S(h+)W,N≫0.
+Proposition 5.5.1. There is a functorial isomorphism M†,b∼− →M ♥,b.
+Proof.First of all, let us define a derivation DofMthat is compatible with ad( h), is
+diagonalizable, and has integral eigenvalues.COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 57
+For this fix a lifting ι:C/Z→Cof the natural projection C։C/Z. The action of ad( h)
+(where recall we write hfor the images of h∈HinH1,c,H1,c′) onMis locally finite, this
+follows from Definition 3.4.4, because1
+tad(h) is a degree preserving map for any Harish-
+Chandra ˜H-bimodule. Define Dby making it act by λ−ι(λ) on the generalized eigenspace
+of ad(h) with eigenvalue λ. LetM(j) denote the jth eigenspace of DinM.
+Now choose a filtration F iMonMas in Definition 3.4.4. Consider the filtration FD
+iM
+defined by FD
+iM:=/circleplustext
+jFi−jM(j). The filtration FD
+•is compatible with the doubled y-
+filtration onH1,c,H1,c′. Moreover, the associated graded grDMis still a finitely generated
+S(h⊕h∗)#W-module. Also let us remark that the Rees bimodules RD
+/planckover2pi1(M),R/planckover2pi1(M) are
+naturally identified (the gradings differ by twisting with D). We writeM/planckover2pi1for both these
+bimodules. Unlessspecifiedotherwise, weconsider thegradingon M/planckover2pi1comingfrom RD
+/planckover2pi1(M/planckover2pi1).
+The bimoduleM∧bhas they-filtration inherited from M. The corresponding Rees bi-
+moduleR/planckover2pi1(M∧b) coincides with the subspace of C×-finite vectors inM∧b
+/planckover2pi1, where the latter
+completion is defined as in Subsection 3.10.
+LetN′
+/planckover2pi1denote the centralizer of ( h⊕h∗)Wine(W)θb
+∗(M∧b
+/planckover2pi1)e(W). Consider two subspaces
+N1
+/planckover2pi1,N2
+/planckover2pi1⊂N′
+/planckover2pi1defined as follows. The subspace N1
+/planckover2pi1is defined as the subspace of all vectors
+thatarelocallyfinitefortheEulerderivationof N′
+/planckover2pi1. ThebimoduleM†,bisjustthequotientof
+N′
+/planckover2pi1by/planckover2pi1−1. Further, thesubspace N2
+/planckover2pi1consists ofall vectors thatarelocallyfiniteforthe C×-
+action and locally nilpotent for the operators ad( a),ad(b) for alla∈S(h∗
++)W,b∈S(h+)W. It
+is easy to see that M♥,bis the quotient ofN2
+/planckover2pi1by/planckover2pi1−1. So it remains to show that N1
+/planckover2pi1=N2
+/planckover2pi1.
+SinceN1
+/planckover2pi1=R/planckover2pi1(M†), and the action of ad( h) onM†,bis locally finite, we see that C×
+acts locally finitely on N1
+/planckover2pi1. Since, in addition, the actions of ad( a),ad(b) onN1
+/planckover2pi1are locally
+nilpotent, it follows that N1
+/planckover2pi1⊂N2
+/planckover2pi1.
+Let us check that N2
+/planckover2pi1⊂N1
+/planckover2pi1. Again, it is enough to show that1
+tad(h) acts locally finitely
+onN2
+/planckover2pi1. It is enough to show that1
+tad(h) acts locally finitely on the C×-eigenspaces inN2
+/planckover2pi1.
+But each such eigenspace embeds into M♥,bunder the /planckover2pi1= 1 quotient map. So it is enough
+to show that ad( h) is locally finite on M♥,bor on any its finitely generated sub-bimodule.
+But any such sub-bimodule is Harish-Chandra in the sense of Definitio n 3.4.4 by Proposition
+5.4.3. And hence ad( h) acts locally finitely. /square
+Weremarkthatalthoughweusedalifting C/Z֒→Ctoconstructafunctorialisomorphism
+M†,b∼− →M ♥,b, this isomorphism is independent of the choice of ι. Also we remark that the
+functor•♥,bdoes not change (up to an isomorphism) if we identify H∧b
+1,•with C(H∧0
+1,•) using
+the isomorphism ϑbinstead ofθb. This follows from Lemma 5.2.1.
+We can construct a functor •♥,b:cHC(H+)c′→c/hatwiderHC(H)c′in a similar way. Namely, let
+N∈cHC(H+)c′. SetN′= (θb
+∗)−1/parenleftbig
+C(W,W,D(hW∧0)⊗N∧0)/parenrightbig
+. ForN♥,btake the subspace
+of all elements, where ad( a),ad(b),a∈S(h)W,b∈S(h∗)Wact locally nilpotently. As in the
+proof of Proposition 3.7.1, we see that •♥,bis right adjoint to •♥,b. Therefore•♥,b=•†,b.
+5.6.Induction and restriction functors and the annihilators. The main application
+of the isomorphism of completions theorem in [BE] is the constructio n of functors between
+the categoriesOof appropriate Cherednik algebras.
+Namely, consider the algebra H1,c. By definition, the corresponding category Ois the
+full subcategory of H1,c-modules consisting of all modules Msatisfying the following two
+conditions:
+•Mis finitely generated over C[h] (or, equivalently, over C[h/W]).
+•h⊂H1,cacts onMby locally nilpotent endomorphisms.58 IVAN LOSEV
+LetO+,Odenote the similar categories for H+
+1,c,H1,c. Pick a point b∈hwithWb=W.
+Following [BE], let us construct certain functors Res b:O→O+,Indb:O+→O.
+Recall the (partial) completion H∧b
+1,c:=C[h/W]∧b⊗C[h/W]H1,c. Consider the category
+O∧bof allH∧b
+1,c-modules that are finitely generated over C[h/W]∧b. We have the completion
+functorO→O∧b,M/ma√sto→C[h/W]∧b⊗C[h/W]M. There is an equivalence between O∧bandO+
+established in [BE].
+Namely, recall the Bezrukavnikov-Etingof isomorphism ϑb:H∧b
+1,c→C(H∧0
+1,c), whereH∧0
+1,c:=
+C[h/W]∧0⊗C[h/W]H1,c. ThemodulecategoriesofC( H∧0
+1,c)andofH∧0
+1,careequivalent, anequiv-
+alenceρ: C(H∧0
+1,c)-Mod→H∧0
+1,c-Mod is given by M/ma√sto→e(W)M. So we have an equivalence
+ρ◦ϑb
+∗:O∧b→O∧0. Now let us recall an equivalence O∧0→O+∧0from [BE] sending a
+moduleM∈O∧0to the joint kernel of the operators from hW. Next, we have an equivalence
+O+∧0→O+sending a module M∈O+∧0to its subspace of all vectors, where h+acts
+locally nilpotently. Composing the equivalences we have just constru cted we get a required
+equivalenceO∧b→O+. Now the functor Res bis obtained by taking the composition of the
+completion functor O→O∧bwith the equivalence O∧b→O+.
+Let us recall the construction of the functor Ind b:O+→O. Consider the equivalence
+O+→O∧bthat is inverse to the equivalence O∧b→O+. Then, as was checked in [BE], for
+any module M∈O∧bits subspace Ml.n.consisting of all vectors, where h⊂H1,cacts locally
+nilpotently, is a finitely generated H1,c-module. So we have a functor O∧b→O,M/ma√sto→Ml.n..
+Composing this functor with the equivalence O+→O∧bwe get the functor Ind b.
+In a subsequent paper we will need the following result on annihilators .
+Proposition 5.6.1. ForM∈O,N∈O+we have the inclusions
+AnnH1,c(M)†⊂AnnH+
+1,c(Resb(M)),AnnH+
+1,c(N)†⊂AnnH1,c(Indb(N)).
+Proof.The first inclusion follows from the isomorphism •†,b∼=•♥,band the construction of
+•♥,b(withϑbinstead ofθb, see the remarks after the proof Proposition 5.5.1). The second
+inclusion follows similarly from the isomorphism •†,b=•♥,b. /square
+5.7.Bifunctor L(•,•).LetM,Nbe modules over the algebras H1,c′,H1,c, respectively.
+The space Hom C(M,N) is anH1,c-H1,c′-bimodule. Consider the subspace L(M,N)⊂
+HomC(M,N)ofallvectorsthatarelocallynilpotentwithrespecttotheoperat orsad(a),ad(b)
+for alla∈(Sh∗)W,b∈(Sh)W. An analogous construction is very important in the represen-
+tation theory of semisimple Lie algebras.
+From the definition it follows that the bimodule L(M,N) is the direct limit of its Harish-
+Chandra sub-bimodules. The following proposition, which is the main re sult of this subsec-
+tion, implies that L(M,N) is Harish-Chandra itself.
+Proposition 5.7.1. For allM,Nin the categoryOthe bimodule L(M,N)is finitely gen-
+erated.
+The proof is based on the following two claims.
+Lemma 5.7.2. LetM,Nbe simples inOwith different supports. Then L(M,N) = 0.
+We remark that the support of a simple in O(considered as a C[h/W]-module) is irre-
+ducible, see [G2], Theorem 6.8.
+Proposition 5.7.3. LetMbe a module in the category OforH1,c′, andNbe a finite
+dimensional module in the category OforH+
+1,c, wherebis chosen in the open stratum inCOMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 59
+the support of Mso thatResb(M)is finite dimensional. Then there is a (bi)functorial
+isomorphism
+L(M,Indb(N))∼=HomC(Resb(M),N)†,b.
+Proof of Lemma 5.7.2. SupposeL(M,N)/\e}atio\slash= 0.
+LetIbe the annihilator of Min (Sh∗)W. Pickϕ∈L(M,N). Then the image of ϕconsists
+of vectors that are locally nilpotent for all a∈I. It follows that/summationtext
+ϕimϕis a submodule
+inNthat is annihilated by Inforn≫0. SinceNis simple, this means that Nitself is
+annihilated by In. So we see that Supp( N)⊂Supp(M).
+Now letJbe the annihilator of Nin (Sh∗)W. Then for any ϕ∈L(M,N) the inclusion
+JnM⊂kerϕholdsforn≫0. Pickanonzerofinitelygeneratedsub-bimodule X⊂L(M,N).
+ThenXis a Harish-Chandra bimodule, and so is finitely generated as a left H1,c-module.
+Letϕ1,...,ϕkbe generators. Then/intersectiontextk
+i=1kerϕi⊂kerϕfor anyϕ∈X. So/intersectiontextk
+i=1kerϕiis
+anH1,c′-submodule in M. SinceMis simple, we have/intersectiontextk
+i=1kerϕi={0}. But now Jacts
+locally nilpotently on M, which implies Supp( M)⊂Supp(N). /square
+Remark 5.7.4. One can strengthen the claim of Lemma 5.7.2 (with the same proof) as
+follows. Let Mbe a simple module and let N0be the maximal submodule of Nsupported
+on the support of M. Then the natural homomorphism L(M,N0)→L(M,N) is an isomor-
+phism. Analogously, let Nbe simple, and M0be the maximal quotient of Msupported on
+the support of N. ThenL(M,N)∼=L(M0,N).
+Proof of Proposition 5.7.3. Recall the category O∧bconsidered in Subsection 5.6, the com-
+pletion functor•∧b:O→O∧band the functor•l.n.:O∧b→Oof taking locally nilpotent
+vectors.
+Step 1.LetM∈Oc′,M′∈O∧bc. Weclaimthatthebimodules L(M,M′
+l.n.)andL(M∧b,M′)
+are naturally identified.
+Pickϕ∈L(M,M′
+l.n.). We can view ϕas an element of L(M,M′). Since the action of
+ad(a),a∈(Sh∗)WonL(M,M′) is locally nilpotent, we see that ϕis continuous in the b-adic
+topology. So ϕuniquely extends by continuity to M∧b. The extension ψϕobviously lies in
+L(M∧b,M′).
+Conversely, take ψ∈L(M∧b,M′) and consider the composition of ψwith the natural
+mapM→M∧b. The composition ϕψlies inL(M,M′). Since ad( b),b∈(Sh)W, acts locally
+nilpotently on L(M,M′), we see that the image of ϕψlies inM′
+l.n.. So we can view ϕψas an
+element ofL(M,M′
+l.n.). The maps ϕ/ma√sto→ϕψ,ψ/ma√sto→ψϕas mutually inverse.
+Step 2.Now forM′take the image of Nunder the equivalence O+∼− →O∧b. We claim
+that the bimodules L(M∧b,M′) and Hom(Res b(M),N)†,bare naturally identified. This will
+complete the proof.
+First of all, let us embed Hom(Res b(M),N)†,binto Hom C(M∧b,M′). For this recall, that,
+by the discussion after Proposition 5.5.1, Hom(Res b(M),N)†,bis the same as
+(θb
+∗)−1[D(hW)∧0⊗C(Hom(Res b(M),N))]l.n.
+On the other hand,
+θb
+∗(M∧b) =C[[hW]]⊗FunW(W,Resb(M)),
+θb
+∗(M′) =C[[hW]]⊗FunW(W,N).
+So Hom(M∧b,M′) = Hom( C[[hW]],C[[hW]])⊗C(Hom(Res b(M),N)) and the natural action
+ofD(hW)∧0onC[[hW]] defines an embedding Hom(Res b(M),N)†,b֒→Hom(M∧b,M′). We60 IVAN LOSEV
+need to check that the Harish-Chandra (=locally nilpotent) part of Hom(M∧b,M′) lies in
+Hom(Res b(M),N)†,b. For this we notice that ad( a) acts locally nilpotently on L(M∧b,M′)
+for anya∈C[h/W]∧b. In particular, the adjoint action of C[hW]⊂θb
+∗(C[h/W]∧b) on
+θb
+∗(L(M∧b,M′)) is locally nilpotent. But it is well-known that the locally nilpotent part o f
+Hom(C[[hW]],C[[hW]]) is nothing else but D(hW)∧0. This completes the proof. /square
+Proof of Proposition 5.7.1. Let us remark that the bifunctor L(•,•) is left exact. So it is
+enough to prove that L(M,N) is finitely generated for simple M,N. Thanks to Lemma
+5.7.2, here we only need to consider the case when M,Nhave the same support. Pick
+bin the open stratum of the support. Since Ind bis a right adjoint for Res b, we have a
+natural homomorphism N→Indb◦Resb(N). SinceNis simple, this homomorphism is an
+inclusion, and so L(M,N)֒→L(M,Indb◦Resb(N)). According to Proposition 5.7.3, the
+latter bimodule is L(Resb(M),Resb(N))†,b. It is finitely generated by Remark 3.10.2. Being
+a sub-bimodule in a finitely generated bimodule, L(M,N) is finitely generated as well. /square
+5.8.Ideals in therational Cherednik algebra of type A.Inthissectionwewilldescribe
+the structure of the set of two-sided ideals in the rational Chered nik algebraH1,cof type
+A. “Type A” means that Γ = Snis a symmetric group in nletters,V=h⊕h∗, where
+h∼=Cn−1is the reflection Γ-module. In this case cis a single number. It is well known, see,
+for example [BEG1], that H1,chas a finite dimensional module if and only if c=r
+n, where
+n∈Z>1andris an integer mutually prime with n. In the latter case there is a unique (up
+to an isomorphism) indecomposable finite dimensional module.
+The following theorem describes the structure of the set Id(H1,c).
+Theorem 5.8.1. (1)The algebraH1,cis not simple if and only if c=q
+m, whereq,mare
+mutually prime integers, and 1<m/lessorequalslantn. Below we suppose that chas this form.
+(2)Sets= [n/m]. The set of proper two-sided ideals of H1,cconsists of selements, say
+Ji,i= 1,...,s, andJ1/supersetnoteqlJ2/supersetnoteql.../supersetnoteqlJs.
+(3) V(Jj) =π(V∗Γj), whereΓj=S×j
+m֒→Sn.
+Part (1) of this theorem as well as the fact that the associated va rieties of two-sided
+ideals inH1,chave the form described in (3) seem to be known before, although w e could
+not find a precise reference (for example, these facts are easily p roved using the claim on
+finite dimensional representations and techniques from [BE]). The s tatement of part (2) was
+communicated to me by Etingof.
+Proof. Step 1. LetJbe a proper two-sided ideal of H1,c. Choose a symplectic leaf Lin
+V∗/Γ that is open in V( J). Let Γ,/tildewideΞ,H1,cbe such as in Subsection 1.3. The subgroup
+Γhas the form/producttextj
+i=1Sni, wheren=n1+...+nj. ThenJ†is a/tildewideΞ-stable ideal of finite
+codimension inH+
+1,c. The algebraH+
+1,cis the tensor product of the algebras H1,c(Cni−1,Sni)
+(we suppose that for ni= 1 the latter algebra is C). Each of these algebras has a finite
+dimensional representation. From the discussion preceding Theor em 5.8.1 we see that ni
+is either some m >1 or 1, and c=q
+mwith GCD( q,m) = 1. In particular, Γ =S×j
+m,
+and the group Ξ is naturally identified with Sj×Sn−mj. There is a section Ξ ֒→/tildewideΞ of the
+projection /tildewideΞ→Ξ, the corresponding subgroup Sj⊂/tildewideΞ permutes the tensor factors of H1,c.
+SetA:=H1,c(Cm−1,Sm).
+Step 2.Forj= 1,...,s(= [n/m]) letLjbe the open symplectic leaf in π(V∗Γj), where Γj
+is as in part (3).COMPLETIONS OF SYMPLECTIC REFLECTION ALGEBRAS 61
+We claim that there is a unique maximal element JjinIdLj(H1,c) and that V(Jj/J)⊂
+∂Lj(=∪i>jLi) for anyJ∈IdLj(H1,c).
+Let us prove this claim. If J∈IdLj(H1,c), thenJ†is an ideal of finite codimension in the
+corresponding slice algebra H+
+1,c=A⊗j. But the latter algebra has a unique maximal ideal
+of finite codimension. Indeed, if Iis such an ideal, then the kernels of all homomorphisms
+A→ A⊗j/Icoincide with a unique ideal of finite codimension in A(the uniqueness of
+such an ideal follows from the fact that Ahas a unique indecomposable finite dimensional
+module). Now the claim follows from the observation that the tensor product of the matrix
+algebras is again a matrix algebra and so is simple.
+So the ideal (J†)‡∈IdLj(H1,c) does not depend on Jand containsJ. By Theorem 1.3.1,
+(J†)‡/Jis supported on ∂Lj. It remains to put Jj:= (J†)‡.
+Step 3.Fix somej. On this step we are going to describe the set of proper Sj-stable ideals
+inA⊗j. We can apply the results of Step 2 to Ato see thatAhas only one proper two-sided
+ideal, say n. This ideal has finite codimension. We claim that any Sj-stable ideal inA⊗jis
+of the formJp(j) =/summationtext
+i1<...<i pni1...nip,p= 1,...,j,whereni:=A⊗...⊗n⊗...⊗A, with
+non thei-th place.
+First, we claim that any prime=(primitive) ideal in A⊗jhas the formJI:=/summationtext
+i∈Inifor
+someI⊂{1,...,j}. First of all, being the tensor product of a matrix algebra and of sev eral
+copies ofA, the algebraA⊗j/JIis prime. Now let Jbe a prime ideal in A⊗jandL′be the
+open symplectic leaf in V( A⊗j/J)⊂V+/Γ= (C2m−2/Sm)×j. The algebraA⊗jis an SRA so
+we can use an argument of Step 1. We see that L′is the product of 0’s or C2m−2/Sm’s and
+moreoverJ=I†, whereIis a unique ideal of finite codimension in the slice algebra (which
+is again the tensor product of several copies of A). In particular, we have a unique prime
+ideal with a given associated variety and this prime ideal must be some JI. This completes
+the proof of the claim in the beginning of the paragraph.
+NowletJbeanarbitrary Sj-stableidealinH1,c. LetJI,I∈F,beallminimalprimeideals
+ofJ, whereFis some (finite) index set. Then FisSj-stable. We claim that/intersectiontext
+σ∈SjJσ.I=
+Jp(j), wherep=j−#I+1.
+The proof is by the induction on j. Set
+J1:=/intersectiondisplay
+σ∈Sj,j/\egatio\slash∈σ.IJσ.I,J2:=/intersectiondisplay
+σ∈Sj,j∈σ.IJσ.I.
+Then, using the inductive assumption, we get J1=Jp−1(j−1)⊗AandJ2=Jp(j−1)⊗
+A+nj. The first equality is trivial, let us explain why the last one holds. It is cle ar that
+J2⊃nj. Using the inductive assumption, we see that J2/nj=Jp(j−1)⊗(A/n). The
+equality forJ2follows.
+SinceJp−1(j−1)⊗A∩nj=Jp−1(j−1)⊗n, we see thatJ1∩J2=Jp(j−1)⊗A+
+([Jp−1(j−1)⊗A]∩nj) =Jp(j).
+It follows from the previous paragraph that the radical of any Sj-stable ideal inH1,c
+coincides with some Jp(j) (we remark that J1(j)⊃J2(j)⊃...⊃Jj(j)). Note however,
+thatJp(j)2=Jp(j), because n2=n. So we have proved that any proper Sj-stable ideal in
+H1,ccoincides with some Jp(j). Moreover, we remark that Jp(j)Jq(j) =Jmax(p,q)(j).
+Step 4.Consider the (smallest) symplectic leaf L1and the algebraH1,ccorresponding to
+this leaf. By assertion (4) of Proposition 3.6.5, J†/Jj(s) = 0 provided V( J) =Lj. Also
+note thatJ⊂J′,J†=J′
+†impliesJ=J′. Indeed, V(J′/J)⊂∂L1and soJ′/J= 0.62 IVAN LOSEV
+Let us check that Ji1Ji2=Jmax(i1,i2). Consider the ideals Ji1†,Ji2†. The idealsJi1†,Ji2†
+are proper Ss-stable ideals inA⊗sand so are of the form Ji1(s),Ji2(s). Also we remark
+that (Ji1Ji2)†=Ji1†Ji2†. It follows that ( Ji1Ji2)†=Jmax(i1,i2)†and so, by the previous
+paragraph,Ji1Ji2=Jmax(i1,i2). In particular, it follows that J1/supersetnoteqlJ2/supersetnoteql.../supersetnoteqlJs.
+It remains to check that any ideal JofH1,ccoincides with some Ji. Let V(J) =Lj.
+ThenJ⊂Jjby Step 2. However, J†=Jj(s) =Jj†and henceJ=Jj. /square
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+arXiv:1001.0240v1  [math.RA]  1 Jan 2010Fundamental representations and algebraic properties of
+biquaternions or complexified quaternions
+Stephen J. Sangwine∗
+School of Computer Science and Electronic Engineering,
+University of Essex, Wivenhoe Park,
+Colchester, CO4 3SQ, United Kingdom.
+Email:S.Sangwine@IEEE.org
+Todd A. Ell†
+5620 Oak View Court, Savage, MN 55378-4695, USA.
+Email:T.Ell@IEEE.org
+Nicolas Le Bihan
+GIPSA-Lab
+D´ epartement Images et Signal
+961 Rue de la Houille Blanche, Domaine Universitaire
+BP 46, 38402 Saint Martin d’H` eres cedex, France.
+Email:nicolas.le-bihan@gipsa-lab.inpg.fr
+October 22, 2018
+Abstract
+The fundamental properties of biquaternions (complexified quaternions) are presented including
+several different representations, some of them new, and defi nitions of fundamental operations such
+as the scalar and vector parts, conjugates, semi-norms, pol ar forms, and inner and outer products.
+The notation is consistent throughout, even between repres entations, providing a clear account of
+the many ways in which the component parts of a biquaternion m ay be manipulated algebraically.
+1 Introduction
+It is typical of quaternion formulae that, though they be difficult to find, once found they are
+immediately verifiable.
+J. L. Synge (1972) [ 43, p34]
+The quaternions are relatively well-known but the quaternions with c omplex components (complexified
+quaternions, or biquaternions1) are less so. This paper aims to set out the fundamental definitions
+of biquaternions and some elementary results, which, although elem entary, are often not trivial. The
+emphasis in this paper is on the biquaternions as an applied algebra – th at is, a tool for the manipulation
+∗This paper was started in 2005 at the Laboratoire des Images e t des Signaux (now part of the GIPSA-Lab), Grenoble,
+France with financial support from the Royal Academy of Engin eering of the United Kingdom and the Centre National
+de la Recherche Scientifique (CNRS). Thanks are due to Dr Seba stian Miron, who gave a short series of seminars on
+biquaternions in Grenoble in 2005, and thus started us on the road to this paper. We also acknowledge the contribution
+made by Daniel Alfsmann who, working with one of us, and using his knowledge of hypercomplex algebras in general,
+greatly clarified the topic of divisors of zero [ 35,36].
+†Dr T. A. Ell is a Visiting Fellow at the University of Essex, fu nded by grant numbers GR/S58621 and EP/E010334/1
+from the United Kingdom Engineering and Physical Sciences R esearch Council from September 2003 – September 2009.
+1Biquaternions was a word coined by Hamilton himself [ 21] and [17,§669, p664]. (The word was used 18 years later by
+Clifford [ 5] for a different concept, which is unfortunate.)
+1of algebraic expressions and formulae to allow deep insights into scien tific and engineering problems. It is
+not a study of the abstract properties of the biquaternion algebr a, nor its relations with other algebras.
+Throughout the paper ‘quaternion’ means a quaternion with real c omponents (a quaternion over the
+reals,R), and ‘biquaternion’ means a quaternion with complex components ( a quaternion over the field
+of complex numbers, C). We denote the set of quaternions by H, and the set of biquaternions by B.H
+is, of course, a subset of B.
+Some of the material in the paper is based on the book by Ward [ 45, Chapter 3] which is one of
+the few sources of detail on the biquaternions. We have not followe d Ward’s notation in this paper,
+preferring instead a scheme based on bold or plain symbols without ha ts and underlines.
+We have also drawn upon the paper by Sangwine and Alfsmann [ 35,36] which sets out comprehen-
+sive results on the divisors of zero, and their subsets the idempote nts and nilpotents. Sangwine and
+Alfsmann’s paper uses the same notations as this paper (having ben efited from access to this paper in
+draft).
+The quaternions themselves (with real elements) are well-covered in various books, for example [ 1,2,
+27,28,29]. Hamilton’s works on quaternions were published in book form in [ 17,18,14], and many are
+also now available freely on the Internet in various digital repositorie s. The paper by Coxeter [ 6] is also
+a useful source.
+We begin by setting out various ways in which quaternions and biquate rnions may be represented. We
+start with representations for quaternions, for reference, an d because these representations are generalized
+for biquaternions. In what follows, we use notation as consistently as we can. In particular, the complex
+operator usually denoted by i(or in electrical engineering j) is represented in this paper as Iin every
+case. This is to keep the complex operator distinct from the first of the three quaternion operators i,
+since it is independent. The independence of iandIis perhaps the most fundamental axiomatic aspect
+of the biquaternions that must be understood. Bold symbols denot e vectors and bivectors2, whereas
+normal weight symbols denote scalar or complex quantities or quate rnions.
+Throughout the paper3we use the term norm or the more specialized term semi-norm , both denoted
+/bardblq/bardbl, to mean the sum of the squares of the components of a quaternio n or biquaternion, and modulus ,
+denoted |q|, to mean the square root of the norm and thus the Euclidean magnit ude. This is not
+universally accepted terminology, many sources using norm where we use modulus . However, our usage
+is consistent with several authors who have written on quaternion s, including Synge [ 43] and Ward [ 45],
+but it does require care when using statements made about norms in other sources.
+Many of the concepts given in this paper are implemented in numerical form in a matlab R/circleco√yrttoolbox
+[37] which two of the authors first developed in 2005 and are built upon in a toolbox for handling linear
+quaternion systems, first developed in 2007 [ 10]. The toolbox [ 37] was essential to the development of
+this paper and the results presented within it, since otherwise, err ors in algebra would go unnoticed. In
+many cases we have established results first by using the toolbox, a nd then derived the algebraic proofs
+or statements which appear here.
+2 Quaternions
+Classically, quaternions are represented in the form of hypercomp lex numbers with three imaginary
+components. In Cartesian form this is:
+q=w+xi+yj+zk (1)
+wherei,jandkare mutually perpendicular unit bivectors4obeying the famous multiplication rules:
+i2=j2=k2=ijk=−1, discovered by Hamilton in 1843 [ 19], andw,x,y,z, are real. Quaternions
+are generalized to biquaternions by permitting w,x,yandzto be complex, as discussed in §3, but in
+this paper we reserve the symbols w,x,y,zfor the real case. A quaternion with w= 0 is known as a
+pure quaternion (Hamilton’s terminology, but still used and widely underst ood).
+2Bivectors represent directed areas, and are explained in Ta ble3and§3.6.
+3Except in equation 11, where we use the notation of the cited reference.
+4Classic texts often refer to the operators i,jandkasvectors, a misconception that has caused considerable confusion
+over many years, but is understandable, since it could not be cleared up without the concept of geometric algebra and
+bivectors. We discuss this in §3.6.
+2The conjugate of a quaternion is given by negating the three imagina ry components: q=w−xi−
+yj−zk. It is easily shown that qp=pqfor general quaternions pandq. Indeed the formula may
+be generalized to more than two quaternions (the generalized form ula was first noted by Hamilton [ 20,
+§20, p238], and was also included in [ 28, p60,§31]):pqrst =tsrqp. The quaternion conjugate may be
+expressed in terms of multiplications and additions [ 11, Theorem 11] using any system of three mutually
+orthogonal unit pure quaternions (here i,j,k):
+q=−1
+2(q+iqi+jqj+kqk)
+Similar formulae, based on involutions [ 9,11], exist for extracting the four Cartesian components of a
+quaternion5[41]:
+w=1
+4(q−iqi−jqj−kqk)
+x=1
+4i(q−iqi+jqj+kqk)
+y=1
+4j(q+iqi−jqj+kqk)
+z=1
+4k(q+iqi+jqj−kqk)(2)
+The norm of a quaternion is given by the sum of the squares of its com ponents: /bardblq/bardbl=w2+x2+y2+
+z2,/bardblq/bardbl ∈R. It can also be obtained by multiplying the quaternion by its conjugat e, in either order since
+a quaternion and its conjugate commute: /bardblq/bardbl=qq=qq. The modulus of a quaternion is the square
+root of its norm: |q|=/radicalbig
+/bardblq/bardbl.
+Every non-zero quaternion has a multiplicative inverse6given by its conjugate divided by its norm:
+q−1=q//bardblq/bardbl.
+The quaternion algebra His a normed division algebra, meaning that for any two quaternions pand
+q,/bardblpq/bardbl=/bardblp/bardbl /bardblq/bardbl, and the norm of every non-zero quaternion is non-zero (and pos itive) and therefore
+the multiplicative inverse exists for any non-zero quaternion.
+Of course, as is well known, multiplication of quaternions is not commu tative, so that in general for any
+two quaternions pandq,pq/ne}ationslash=qp. This can have subtle ramifications, for example: ( pq)2=pqpq/ne}ationslash=p2q2.
+Alternative representations for quaternions are given in Table 1, expressed in terms of the Cartesian
+form given above in equation 1, and in selected cases, in terms of other representations given he re.µis
+a unit pure quaternion and is known as the axis of the quaternion. It expresses the direction in 3-space
+of the vector part7,V(q). Hamilton himself showed that any unit pure quaternion is a square r oot of
+−1 [22, pp 203, 209][ 17,§167, p179] and a proof is also given in [ 9, Lemma 1]. This is why we call the
+forma+µbthe ‘complex’ form, since it is isomorphic to a complex number a+Ib. This means that
+the modulus and argument of the quaternion are identical to those ofa+Ib(one could think of them
+as having similar Argand diagrams, where in the case of quaternions, the Argand diagram represents
+a plane section of 4-space defined by the ‘axis’ µand the scalar quaternion axis – along which wis
+measured).
+The polar form of a quaternion is analogous to the polar form of a com plex number, with one
+exception. The argument, θ, is confined to the interval [0 ,π) because the modulus of the vector part is
+always taken to be positive (there is no convenient way to define an o rientation in 3-space which would
+permit the sign of the vector part to be determined). If we negate the argument of the exponential in the
+polar form, therefore, the negation is conventionally applied to the axis,µand not to the argument θ.
+Whenθis computed numerically, the result is always in [0 ,π) because we have to use the (non-negative)
+modulus of the vector part to compute it (using an atan2 function, typically).
+The Cayley-Dickson polar form [ 38,39] has a complex modulus and a complex argument (both in
+fact are degenerate quaternions of the form w+ix, isomorphic to complex numbers).
+5These formulae, and that for the quaternion conjugate, gene ralize to biquaternions, even with mutually orthogonal
+unit pure biquaternions in place of i,jandk, although this latter point has not been thoroughly checked .
+6This does not apply to biquaternions.
+7We use the term ‘vector part’ throughout this paper to mean th at part of the quaternion consisting of the three
+components containing i,jandk. It does not necessarily correspond to the concept of a 3-spa ce vector, since it could be a
+vector, bivector or a combination of both, using the languag e of geometric algebra described later. The term ‘vector par t’
+is well-established in the literature and, lacking a good al ternative which would be readily understood, we retain it.
+3Designation Representation Details ( w,x,y,z,a,b,r,θ ∈R,q∈H,
+µ∈H,S(µ) = 0,µ2=−1)
+Cartesian q=w+xi+yj+zk
+Scalar + vector q= S(q) +V(q) S(q) =w
+V(q) =xi+yj+zk
+‘Complex’ form q=a+µb a=w
+b=/radicalbig
+x2+y2+z2=|V(q)|
+µ=V(q)/|V(q)|
+Cayley-Dickson q= (w+xi) + (y+zi)j This multiplies out to the Cartesian
+representation.
+Polar form q=rexp (µθ) =r(cosθ+µsinθ)r=|q|
+rcosθ=a
+rsinθ=b
+Cayley-Dickson q=Aexp(Bj) A,B ∈Hwithy=z= 0
+polar form (isomorphic to C)
+See [38,39] for formulae defining AandB.
+Table 1: Representations for quaternions.
+3 Biquaternions
+To generalize the quaternions to biquaternions we simply permit the f our elements to be complex rather
+than real, thus giving us the Cartesian representation:
+q=W+Xi+Yj+Zk (3)
+wherei,jandkare exactly as in §2andW,X,Y,Z, are complex. This generalization was first studied
+by Hamilton himself [ 21] and [ 17,§669, p664], and was also discussed by Cayley [ 44].
+In equation 3each of the four elements is of the form W=ℜ(W) +Iℑ(W), where I2=−1 is the
+usual complex operator, distinct from i. Axiomatically, Icommutes with the three quaternion operators
+i,jandk, that is iI=Ii,jI=IjandkI=Ik. Since reals commute with the three quaternion
+operators, so do all complex numbers. It is also worth noting that a ny quaternion with only one non-
+zero component in the vector part, for example w+ixis not a complex number, but a degenerate
+quaternion. Such numbers are isomorphic to the complex numbers, but it is best not to confuse them
+with complex numbers when working with biquaternions.
+Some familiar rules of algebra apply to biquaternions. Since the real a nd imaginary parts are quite
+separate from the concept of the four quaternion components, real and imaginary parts may be equated
+just as when working with complex equations. However, some of the elementary properties of quaternions
+become non-elementary when the quaternions are complexified. Fo r example, generalizations of the norm
+and modulus of a quaternion are complex in general for biquaternion s, and so is the argument in one of
+the polar forms. We discuss each of these non-trivial properties in a later section.
+The existence of complex generalizations of the norm, modulus and in ner product, yields a problem
+of terminology, since conventionally these quantities are real, and s atisfy properties that their complex
+generalizations cannot (the triangle inequality /bardblp+q/bardbl ≤ /bardblp/bardbl+/bardblq/bardbl, for example, requires ordering, but
+complex numbers lack ordering, hence the triangle inequality cannot be applicable to a ‘norm’ with a
+complex value). Rather than invent new terms, we use the existing a ccepted terms (with the exception
+ofnorm , where we substitute semi-norm , for reasons discussed in §3.3), but caution the reader that
+because these quantities are complex, they cannot be assumed to satisfy all the usual properties of their
+conventional real equivalents. In taking this approach, we are fo llowing Synge [ 43, p 9], and to some
+extent Ward [ 45], both of whom use the term norm to refer to a complex generalization. Synge also
+refers to a scalar product with a complex value, and Ward uses the t erminner product for a complex
+generalization of the concept in his section on the Minkowski metric [ 45,§3.3, p 115].
+Although a biquaternion commutes with its quaternion conjugate, a nd complex numbers commute
+(including with their complex conjugates), somewhat surprisingly, a biquaternion does not necessarily
+commute with its complex conjugate (Proposition 1in§3.1).
+4Alternative representations for biquaternions are shown in Table 2, expressed in terms of the Cartesian
+form given above in equation 3, and in selected cases, in terms of other representations given he re. In
+Designation Representation Details: wa,xa,ya,za∈R,a∈ {r,i},
+W,X,Y,Z,A,B,R, Θ∈C,
+qr,qi,Q,Ψ∈H;q∈B,
+ξ∈B,ξ2=−1 [34,33]
+Cartesian q=W+Xi+Yj+Zk
+Scalar + vector q= S(q) +V(q)S(q) =W
+V(q) =Xi+Yj+Zk
+‘Complex’ form I q=A+ξBA=W= S(q)
+ξB=V(q)
+B=/radicalbig
+X2+Y2+Z2=|V(q)|
+ξ= (Xi+Yj+Zk)/B
+‘Complex’ form II q=qr+Iqi=ℜ(q) +Iℑ(q)qr=wr+xri+yrj+zrk
+qi=wi+xii+yij+zik
+Expanded formq=wr+xri+yrj+zrk+
+I(wi+xii+yij+zik)wr=ℜ(W),wi=ℑ(W)
+xr=ℜ(X),xi=ℑ(X)
+yr=ℜ(Y),yi=ℑ(Y)
+zr=ℜ(Z),zi=ℑ(Z)
+Hamilton polar formq=Rexp (ξΘ)
+=R(cos Θ + ξsin Θ)R=|q|
+A=Rcos Θ
+B=Rsin Θ
+Complex polar formq=Qexp (IΨ)
+=Q(cos Ψ + Isin Ψ)Ψ = tan−1(q−1
+rqi)
+Q=q/exp (IΨ) =qexp (−IΨ)
+qr=Qcos Ψ
+qi=Qsin Ψ
+Table 2: Representations for biquaternions.
+the scalar/vector form, the scalar part is complex, and the vecto r part is a pure biquaternion.
+Complex form I corresponds to the ‘complex’ form in Table 1. The differences are that AandBare
+now complex, whereas aandbwere real, and the imaginary unit ξis a pure biquaternion root of −1 of
+the form ξ=bµ+dIν, whereb2−d2= 1 andµandνare mutually perpendicular unit pure quaternions
+(themselves also roots of −1) [33,34]. Note that Bcan vanish – as discussed in §3.5.2 and in detail in
+[35,36,§2] – although its components ( X,YandZ), and therefore q, do not vanish. ξcan be thought
+of as a complex axis, in the sense that it has real and imaginary parts which each define directions in
+3-space. The geometric interpretation of biquaternions is discuss ed further in §3.6.
+Complex form II has quaternions in the real and imaginary parts, an d is perhaps the most obvious
+representation for a biquaternion other than the Cartesian form . It is related to the complex polar form
+described below.
+In the expanded form, the biquaternion is represented as a comple x number with quaternion real and
+imaginary parts expressed in Cartesian form.
+The polar form of a quaternion depends on Euler’s formula exp( Iθ) = cosθ+Isinθwhich generalizes
+by replacing the complex operator Iwith any root of −1. In the case of quaternions, the set of unit pure
+quaternions provides an infinite number of roots of −1 and the general polar form rexp(µθ), as given
+in Table 1is therefore a straightforward extension of Euler’s formula. In th e case of biquaternions there
+are two possibilities for the root of −1: the complex root I, or any one of the biquaternion roots of −1
+defined in §3.4. Thus we have two possible fundamental polar forms.
+The first (‘Hamilton’) polar form generalizes the polar form in the real case.R, the ‘modulus’ of the
+biquaternion, is complex; ξis a biquaternion root of −1; and Θ, the ‘angle’ in the exponential, is also
+5complex8. The interpretation of complex angles is discussed further in §3.2.
+In the second (‘complex’) polar form, the standard complex operat orIserves as the root of −1 in the
+exponential, but in this form, the ‘argument’ of the exponential, Ψ, is a quaternion, and the exponential
+is scaled by a quaternion ‘modulus’ Q. We thus have a polar form with a ‘modulus’ and ‘argument’
+inH. Note that it is not possible to find Qby the obvious direct route of Q=/radicalbig
+q2r+q2
+ibecause
+qr=Qcos Ψ and squaring this gives Qcos ΨQcos Ψ and not Q2cos2Ψ. However, it is the case that
+cos2Ψ + sin2Ψ = 1, as would be expected. Further, note that care is needed in co mputing the inverse
+tangent in order to find Ψ: it is important that the real part is divided on the left. This is because
+tan Ψ = ( Qcos Ψ)−1(Qsin Ψ) = cos−1ΨQ−1Qsin Ψ = cos−1Ψ sin Ψ
+whereas with a division on the right we have:
+tan Ψ/ne}ationslash= (Qsin Ψ)(Qcos Ψ)−1=Qsin Ψ cos−1ΨQ−1
+andQand its inverse are at opposite ends of the product and do not canc el. (Note that cos Ψ and
+sin Ψ commute, so their quotient is the same whether divided on the lef t or right.) The exponential is a
+biquaternion because of the presence of Iand it has a complex modulus. The modulus of this complex
+modulus is 1. We discuss both polar forms further in §3.1in the context of conjugation. Finally, note
+that in the complex polar form, Qand the exponential do not commute. It is therefore possible to de fine
+a variant by placing Qon the right of the exponential. The variant is related to the form in T able 2by
+the conjugate rule.
+De Leo and Rodrigues [ 7,8] discussed polar forms of biquaternions but apparently did not see that
+there were two simple polar forms as here, with different imaginary un its. Instead they described a single
+polar form containing the product of two exponentials. We can do th is with either of our polar forms,
+representing the ‘modulus’ in each case in its own polar form. Thus th e ‘Hamilton’ polar form can be
+written:
+q=Rexp (ξΘ) =rexp (Iφ) exp (ξΘ) (4)
+wherer,φ∈Rare the modulus and argument of R, the complex ‘modulus’ of q. Note that, because I
+andξcommute, the two exponentials commute, and it is possible to write th is polar form as:
+q=rexp(Iφ+ξΘ)
+where the argument of the exponential is now a biquaternion with Iφas scalar part and ξΘ as vector
+part. (In general, with quaternions as well as biquaternions, epeq/ne}ationslash=epqbecause of non-commutativity.)
+Similarly, the ‘complex’ polar form can also be written in this way, expan ding the quaternion ‘mod-
+ulus’,Q, into the standard polar form of a quaternion as given in Table 1:
+q=Qexp (IΨ) =rexp (µθ) exp (IΨ) (5)
+whereµ∈Handr,θ∈R. Notice that the single real modulus ris the same in each case, but the various
+‘angles’ are different in value and type ( φ,θ∈R,Θ∈C,Ψ∈H). The real modulus, r, is the absolute
+value of the square root of the semi-norm as discussed in §3.3. In this case, the arguments of the two
+exponentials (specifically µand Ψ) do not commute, hence the two exponentials cannot be comb ined by
+addingµθtoIΨ, and neither can the order of the exponentials be changed.
+We can usefully combine the ‘complex’ representation of a quaternio n from Table 1with ‘complex’
+form II from Table 2to give the following representation, which was used in [ 34,33] to derive the
+biquaternion roots of −1:
+q= (α+µβ) +I(γ+νδ) (6)
+in which µandνare real pure unit quaternions, and α,β,γandδare real. The four real coefficients
+in this representation may be related to the coefficients in the other representations given above. For
+8The cosine and sine of a complex angle are simply defined in ter ms of Euler’s formula as cos z=1
+2/parenleftbigeIz+e−Iz/parenrightbigand
+sinz=−1
+2I/parenleftbig
+eIz−e−Iz/parenrightbig
+; or, ifwe write z=x+Iy: cosz= cosxcoshy−Isinxsinhyand sinz= sinxcoshy+Icosxsinhy
+[4, Seecomplexes ].
+6example: α+Iγ=A=W. The correspondence between ξ, andµandνis not so simple. Equating the
+vector part of equation 6with the vector part of ‘complex’ form I, we get:
+µβ+Iνδ=ξB=ξ/radicalbig
+X2+Y2+Z2
+and we can see that dividing µβ+Iνδby its (complex) modulus B, will yield ξprovided that Bdoes
+not vanish, as discussed in §3.5.2 .
+We may relate each of the terms in equation 6to a concept from geometric algebra [ 24,40], as shown
+in Table 3with equivalent definitions based on various representations from T able 2.
+Geometric algebra concept Term in equation 6 Terms from Table 2
+Scalar – undirected quantity α ℜ(W) =ℜ(A) = S(qr) =wr
+Bivector – directed area µβ ℜ(ξB) =ℜ(V(q)) =V(qr)
+Vector – directed magnitude Iνδ Iℑ(ξB) =Iℑ(V(q)) =IV(qi)
+Pseudoscalar – undirected volume Iγ Iℑ(W) =Iℑ(A) =IS(qi) =Iwi
+Table 3: Correspondence between geometric algebra concepts an d biquaternion components.
+These equivalences are given by Ward [ 45,§3.2, p 112] and they are discussed in more detail in §3.6.
+Note carefully that geometric vectors are represented by imaginary pure quaternions, and that real pu re
+quaternions are bivectors . This is because the product of two perpendicular vectors must yie ld a bivector.
+The product of two bivectors gives a bivector (and a scalar, unless the bivectors are perpendicular).
+The representation in equation 6was used in [ 34,33] to derive the solutions of the equation q2=−1
+(that is the biquaternion roots of −1) when qis a biquaternion, and it was shown that the solutions
+required α=γ= 0,µ⊥ν, andβ2−δ2= 1. Clearly any of the other representations given in this
+section would not permit this result to be expressed so clearly, nor t o be easily derived. We return to
+this result in Theorem 2.
+The biquaternion algebra Bis not a division algebra because non-zero biquaternions exist that la ck
+a multiplicative inverse. The set of such biquaternions is known as the divisors of zero [35,36] and is
+defined in §3.5.
+3.1 Conjugates
+The conjugate of a biquaternion may be defined exactly as for a qua ternion by negating the vector part.
+Thus we have
+q=W−Xi−Yj−Zk= S(q)−V(q) =qr+Iqi (7)
+We call this the ‘Hamiltonian’ or quaternion conjugate , in agreement with Synge [ 43, Equation 3.4, p 8].
+The conjugate rule qp=pqand its generalization to more than two quaternions applies equally to
+biquaternions. Similarly, a biquaternion commutes with its quaternion conjugate, and the product of
+the two is the semi-norm, as discussed in §3.3.
+However, there is another possible conjugate – that obtained by t aking the complex conjugate of the
+complex components of the quaternion. We denote this complex con jugate by a superscript star, which
+is a common convention with complex numbers. Thus the complex conjugate is given by:
+q⋆=W⋆+X⋆i+Y⋆j+Z⋆k= S(q)⋆+V(q)⋆=qr−Iqi (8)
+Our definition in equation 8agrees with that of Synge [ 43, p8, equation 3.4] and appears to be the obvious
+way to define the complex conjugate, but it differs from that of War d [45] who defines a complex conjugate
+which is a quaternion conjugate with complex conjugate component s. There is no way to define the
+complex conjugate in terms of additions and multiplications as can be d one with the quaternion conjugate
+— if a means existed, it would also apply to complex numbers because a d egenerate biquaternion of the
+formα+Iγwithα,γ∈R, is isomorphic to a complex number. Instead the complex conjugate must
+be seen as a fundamental operation. Note very carefully that a biq uaternion does not commute with its
+complex conjugate, a surprising result that we examine in Propositio n1below.
+7Of course, we can apply both conjugates [ 45] (we call this a total conjugate, but the term biconjugate
+has also been suggested in [ 12], and we consider it apposite). It is not difficult to show the following
+results:
+q⋆=q⋆
+(pq)⋆=p⋆q⋆
+In terms of the various representations above, the total or bico njugate is:
+q⋆=wr−xri−yrj−zrk−I(wi−xii−yij−zik) =qr−Iqi= S(q)⋆−V(q)⋆(9)
+However, the ramifications of non-commutativity are deep, as sho wn by the following Proposition.
+Proposition 1. A biquaternion does not, in general, commute with its comple x conjugate: reversing the
+order of the product yields a complex conjugate result.
+Proof. Represent an arbitary biquaternion in complex form I, as given in Tab le2:q=qr+Iqi. Then
+the two products of qwith its complex conjugate q⋆are:
+qq⋆= (qr+Iqi) (qr−Iqi) =q2
+r+q2
+i+I(qiqr−qrqi)
+q⋆q= (qr−Iqi) (qr+Iqi) =q2
+r+q2
+i−I(qiqr−qrqi)
+The results are a complex conjugate pair, and therefore differ.
+Note that important exceptions to Proposition 1are:
+•Quaternions, that is biquaternions with qi= 0 (trivial).
+•Imaginary biquaternions, with qr= 0. In this case the product is q2
+i, regardless of order.
+•Biquaternions with real and imaginary parts that commute (co-plan ar, with a common axis). In
+this case the imaginary part of the product vanishes and the result isq2
+r+q2
+i, regardless of order.
+The Hamilton and complex conjugates correspond to negation of th e argument of the exponential in
+the two polar forms in Table 2. In the ‘Hamilton’ polar form, negating the argument of the expone ntial
+negates the sine term, which corresponds to the vector part of q, and thus yields the classical Hamilton
+conjugate. In the ‘complex’ polar form, negating the argument of the exponential again negates the sine
+term, but in this case the sine term corresponds to the imaginary pa rt of the quaternion, and thus it is
+the complex conjugate that is obtained. A third possibility would be a p olar form that would correspond
+(in the sense just outlined) to the total conjugate. However, th is would require the sine term to represent
+all the components in equation 9exceptwr, which would be represented in the cosine term. This appears
+improbable.
+The three types of conjugate allow us to construct formulae for e xtracting the geometric components
+of a biquaternion, as defined in Table 3. The formulae in Table 4are obtained by substitution from four
+formulae for the scalar/vector and real/imaginary parts of a quat ernion based on sums and differences
+of the Hamilton or complex conjugates. Unlike the formulae in equatio n2however, which are based
+on involutions, and therefore require multiplication and addition only, these formulae contain complex
+conjugates, which are primitive operations that cannot be expres sed in terms of multiplications and
+additions. Similarly, and fairly trivially, by combining the formulae in equa tion 2with formulae based
+on sums and differences of complex conjugates, one can obtain for mulae for extracting any of the eight
+real components of a biquaternion9. For example, that for the real part of the scalar part is wr=
+1
+8(q+q⋆−iqi−iq⋆i−jqj−jq⋆j−kqk−kq⋆k).
+9The utility of such formulae is not in numeric computation, w here explicit access to the components is a much better
+approach, but in algebraic manipulation, where the formula e may allow an algebraic solution to an otherwise seemingly
+difficult derivation.
+8Scalar ℜ(S(q))1
+4(q+q+q⋆+q⋆)
+Bivector ℜ(V(q))1
+4(q−q+q⋆−q⋆)
+Vector ℑ(V(q))I1
+4(q−q−q⋆+q⋆)
+Pseudoscalar ℑ(S(q))I1
+4(q+q−q⋆−q⋆)
+Table 4: Formulae for the geometric components of a biquaternion. (See also Table 3.)
+3.2 Inner Product
+The definition of the inner product in the classic quaternion case was given by Porteous [ 32, Prop. 10.8,
+p 177] and we take this as the basis for defining the inner product of two biquaternions, since it is
+consistent with the quaternion case, and also yields the semi-norm ( see the next section) in the case of the
+inner product of a biquaternion with itself. The same formula is given b y Synge [ 43, Equation 3.8, p 9] in
+the context of a discussion of basic properties of biquaternions. W ard, in his discussion of biquaternions
+and the Minkowski metric [ 45,§3.3, p114], utilises the same definition, but he also discusses other
+definitions of the inner product (see [ 45,§3.2, p109]). The definitions of Porteous and Synge are:
+/an}bracketle{tp,q/an}bracketri}ht=1
+2(pq+qp) =1
+2(pq+qp) (10)
+where the overbar represents a (Hamilton or quaternion) conjug ate. The result of this expression will
+have zero vector part (since the vector parts of the two terms in side the parentheses cancel). In general,
+the scalar part of the result, and therefore the inner product, w ill be complex. The inner product may
+also be defined in terms of a simple elementwise product of the two qua ternions, as can be seen by
+expanding out equation 10. If we let p=Wp+Xpi+Ypj+Zpkandq=Wq+Xqi+Yqj+Zqk, then
+/an}bracketle{tp,q/an}bracketri}ht=WpWq+XpXq+YpYq+ZpZq
+which is, of course, complex. (The result will be equal to the scalar p art of the quaternion with zero
+vector part resulting from equation 10.)
+As already discussed in §3, the use of the term inner product here does not imply that all the usual
+properties of an inner product will be satisfied. Ward [ 45,§3.3, p 115] states the following properties
+that are satisfied by the inner product defined in equation 10, and these are easily verified:
+/an}bracketle{tp,q/an}bracketri}ht=/an}bracketle{tq,p/an}bracketri}ht, p,q ∈B
+/an}bracketle{tp,q+r/an}bracketri}ht=/an}bracketle{tp,q/an}bracketri}ht+/an}bracketle{tp,r/an}bracketri}ht, r ∈B
+α/an}bracketle{tp,q/an}bracketri}ht=/an}bracketle{tαp,q/an}bracketri}ht=/an}bracketle{tp,αq/an}bracketri}ht, α ∈C
+Conventionally, but not always [ 31, See:Scalar product ], the inner product is positive definite , that is
+greater than or equal to zero, or non-negative, which cannot ap ply to a complex-valued inner product.
+Classically, the inner product of two vectors is given by |v1||v2|cosθwhereθis the angle between
+the two vectors. If extended to quaternions, it is not difficult to se e that the angle is that between
+the two quaternions in 4-space, in the common plane defined by the t wo quaternions. When we extend
+this concept to biquaternions, the angle between them, and their m oduli, must be complex, in general,
+so the geometric interpretation of the inner product is slightly more difficult. In the quaternion case,
+orthogonality arises from the angle between the two quaternions ( having a vanishing cosine), but in the
+biquaternion case there is the additional possibility that one or both moduli are zero, resulting in a
+vanishing inner product. Expressing the inner product of two biqua ternions in terms of inner products
+of the real and imaginary parts makes it possible to understand wha t the inner product represents
+geometrically. Representing pas follows: p=pr+Ipiand similarly for q, the inner product may be
+expanded as:
+/an}bracketle{tp,q/an}bracketri}ht=/an}bracketle{tpr,qr/an}bracketri}ht−/an}bracketle{tpi,qi/an}bracketri}ht+I(/an}bracketle{tpr,qi/an}bracketri}ht+/an}bracketle{tpi,qr/an}bracketri}ht)
+Certain special cases are apparent after careful inspection:
+9• /an}bracketle{tp,q/an}bracketri}ht= 0 indicates that the two biquaternions are orthogonal or ‘perpen dicular’, and it requires
+the real and imaginary parts of the inner product to vanish separa tely. The detailed conditions
+required for this are many, since the four inner product componen ts (two real, two imaginary) may
+have positive or negative real values and can cancel in several way s. We list here four different
+ways in which the scalar product can vanish:
+The strongest constraint is when all four real and imaginary parts of the two biquaternions ar e
+mutually orthogonal. A simple example is p= 1+Iiandq=j+Ik, but it is easy to construct
+a more general example starting from a random quaternion. Let p1be a randomly chosen
+(pure or full) quaternion. Then let p2=p1i,p3=p1jandp4=p1k. The four quaternions
+thus constructed have the same four numeric components permu ted with sign changes in such
+a way that any two will have a vanishing inner product10. Two biquaternions p=p1+Ip2
+andq=p3+Ip4(or any other permutation) will have /an}bracketle{tp,q/an}bracketri}ht= 0. This pair of biquaternions
+will be found to be divisors of zero (see §3.5), but scaling the real and imaginary parts with
+different scale factors will result in biquaternions that are not diviso rs of zero, but are still
+orthogonal (orthogonality not being dependent on norm). Choos ing the initial random value
+to be pure does not result in the pair of biquaternions being pure, be cause of permutation of
+the components.
+Weaker constraint I
+1. The real parts of the two biquaternions are orthogonal, and th e imaginary parts of the two
+biquaternions are orthogonal, hence the real part of the scalar p roduct vanishes because
+it is the difference of two vanishing scalar products.
+2. The real part of each biquaternion is not orthogonal to the imag inary part of the other.
+However, the scalar products of the real part of one biquaternio n with the imaginary part
+of the other have the same values, but opposite signs . This means the imaginary part of
+the scalar product of the two biquaternions vanishes by cancellation .
+A pair of biquaternions satisfying this constraint can be construct ed from an arbitrary bi-
+quaternion p, by choosing q=pi(and two others also orthogonal to the first may be con-
+structed by choosing q=pjorq=pk)11.
+Weaker constraint II
+1. The real parts of the two biquaternions are not orthogonal, an d neither are the imaginary
+parts, but the scalar product of the two real parts has the same valueand sign as the
+scalar product of the two imaginary parts. This means the real par t of the scalar product
+of the two biquaternions vanishes by cancellation .
+2. The real part of each biquaternion is orthogonal to the imaginar y part of the other, hence
+the imaginary part of the scalar product of the two biquaternions v anishes because it is
+the sum of two vanishing scalar products.
+A pair of biquaternions satisfying this constraint can be construct ed from an arbitrary bi-
+quaternion p, by choosing q=pIi(and ditto, mutatis mutandis. ).
+The weakest constraint is when none of the real and imaginary parts of the two biquaternion s
+are orthogonal, but the real and imaginary parts of the scalar pro duct of the two biquaternions
+vanish by cancellation. A pair of biquaternions satisfying this constr aint can be constructed
+from an arbitrary biquaternion p, by choosing q=p(i+Ij) or similar. Note that this results
+inqbeing a divisor of zero (see §3.5) because ( i+Ij) is a divisor of zero.
+• /an}bracketle{tp,q/an}bracketri}htis real, that is with zero imaginary part. There is a trivial case: the tw o biquaternions
+may be imaginary with zero real parts. Otherwise, the imaginary par t of the inner product can
+vanish in two ways, as in the weaker cases above of /an}bracketle{tp,q/an}bracketri}ht= 0: either the imaginary part vanishes
+by cancellation, or it vanishes because the real part of each biquat ernion is orthogonal to the
+imaginary part of the other. It is easy to construct a biquaternion to satisfy the second condition
+using the technique outlined above for the strongest constraint: construct two pairs of orthogonal
+quaternions, and then construct two biquaternions using the com ponents of these pairs.
+10Three other permutations can be obtained by multiplication on the right, or by division on either side by i,jandk.
+11As in the stronger case, it is also possible to multiply by ion the left, etc.
+10• /an}bracketle{tp,q/an}bracketri}htis imaginary, that is with zero real part. The conditions required for this are very similar to
+the previous case, with the appropriate changes. A trivial case is w here one biquaternion is real
+and the other is imaginary.
+Clearly, from the above analysis, orthogonality of biquaternions is not as simple as orthogonality of real
+quaternions where a geometrical interpretation in terms of a plane and a real angle is straightforward.
+In the biquaternion case, interpreting the inner product /an}bracketle{tp,q/an}bracketri}htas|p||q|cos Θ, where the two moduli and
+the angle are complex, is not at all obvious. This remains a topic for fu rther work.
+3.3 Semi-norm
+It is possible to define more than one norm for the biquaternions (se e for example [ 45,§§3.2 and 3.3]).
+In this section we consider the generalization of the quaternion nor m/bardblq/bardbl=w2+x2+y2+z2,/bardblq/bardbl ∈Rto
+the biquaternion case by allowing the four Cartesian components to become complex. Conventionally a
+norm is real and positive definite, that is non-negative, but in the case of biquaternions, this convention
+must be relaxed because the norm has a complex value. Additionally, t he existence of divisors of zero
+(see§3.5) means that a non-zero biquaternion can have a vanishing norm. Fo r this reason, we use the
+termsemi-norm . A semi-norm is a generalization of the concept of a norm, with no req uirement that
+the norm be zero only at the origin (of a vector space) [ 3, See:semi-norm ]. As with our use of the term
+inner product in§3.2for a complex-valued generalization of the inner product, we are he re extending
+the term semi-norm to a complex-valued quantity analogous to a norm, with the additiona l property
+of vanishing in the case of divisors of zero. We also use the term modulus for the square root of the
+semi-norm – it is also of course complex, in general.
+The semi-norm can be real and negative in special cases, as well as p urely imaginary. The semi-norm
+can be defined in terms of the inner product of a biquaternion with its elf, that is /bardblq/bardbl=/an}bracketle{tq,q/an}bracketri}htor directly
+in terms of the four complex components: /bardblq/bardbl=|q|2=W2+X2+Y2+Z2. Although the semi-norm
+is complex-valued the result given by Coxeter for quaternions [ 6,§2] still holds: /bardblq/bardbl=|q|2=qq=qq.
+This is, of course, a special case of the formula for the inner produ ct, in this case of qwith itself (equation
+10).
+Lemma 1. [35,36, Lemma 1] Letq=qr+Iqibe a non-zero biquaternion with real part qr∈Hand
+imaginary part qi∈H. The real part of /bardblq/bardblis equal to the difference between the norms of qrandqi,
+and the imaginary part of /bardblq/bardblis equal to twice the inner product of qrandqi.
+Proof. We express the semi-norm of qas the product of qwith its quaternion conjugate: /bardblq/bardbl=qq.
+Writing this explicitly:
+/bardblq/bardbl= (qr+Iqi)(qr+Iqi) = (qr+Iqi)(qr+Iqi)
+and multiplying out we get:
+/bardblq/bardbl=qrqr−qiqi+I(qrqi+qiqr)
+The real part of /bardblq/bardblcan be recognised as /bardblqr/bardbl−/bardblqi/bardblfrom Coxeter’s result [ 6,§2]. The imaginary part
+of/bardblq/bardblis twice the inner product of qrandqias given by Porteous [ 32, Prop. 10.8, p 177] and equation
+10.
+There are some special cases of the result given in Lemma 1:
+•biquaternions with perpendicular real and imaginary parts have a re al semi-norm (because the
+imaginary part vanishes), but the semi-norm may be negative if the n orm of the imaginary part
+exceeds the norm of the real part;
+•biquaternions with real and imaginary parts with equal norms have a n imaginary semi-norm (be-
+cause the real part vanishes).
+•imaginary biquaternions (with qr= 0) have a negative real semi-norm, and therefore an imaginary
+modulus.
+11•biquaternions with perpendicular real and imaginary parts with equa l norms have a vanishing
+semi-norm. This important special case is covered in more detail in §3.5.
+The semi-norm is invariant under quaternion conjugation, that is /bardblq/bardbl=/bardblq/bardbl, but not under complex
+conjugation: /bardblp/bardbl=/bardblp⋆/bardbl⋆. This can be seen easily from Lemma 1, since quaternion conjugation does not
+affect the norms of the real and imaginary parts, nor their inner pr oduct. However, although complex
+conjugation does not alter the norms of the real and imaginary par ts, it does negate their inner product
+and therefore negates the imaginary part of the semi-norm.
+The semi-norm as defined here (and by Ward [ 45,§3.3, p 115]) obeys the ‘rule of the norms’12so
+that for any two biquaternions, the semi-norm of their product eq uals the product of their semi-norms:
+/bardblpq/bardbl=/bardblp/bardbl/bardblq/bardbl. This is true even in the case where one or both of pandqis a divisor of zero, with
+vanishing semi-norm, as discussed in §3.5(but the result will be zero, computed as the semi-norm of
+the product, or the product of the semi-norms). Hence Bis anormed algebra using the definition of the
+semi-norm used in this section. The non-zero biquaternions that ha ve zero semi-norm (the divisors of
+zero) lack a multiplicative inverse.
+Anorm is usually required to satisfy the following three properties [ 3, See:norm ] (caution13):
+/bardbl−x/bardbl=/bardblx/bardbl
+/bardblλx/bardbl=|λ|/bardblq/bardbl
+/bardblx+y/bardbl ≤ /bardblx/bardbl+/bardbly/bardbl(Triangle inequality)(11)
+The first of these properties is easily seen to be satisfied for all q∈Bbecause negating a complex number
+does not change its square, and therefore does not alter the sum of the squares of the four components
+of a biquaternion, nor the square root of the sum of the squares. The second property requires rather
+more careful analysis, but when the notation is correctly interpre ted, it can be seen to be satisfied by the
+square root of the semi-norm of a biquaternion. The quantity λis usually taken to be a scalar, that is in
+the biquaternion context, a complex number. If we write the expre ssion explicitly in biquaternion terms,
+using the Cartesian form, we have: |λq|=/radicalbig
+(λW)2+ (λX)2+ (λY)2+ (λZ)2=√
+λ2|q|. Taking the
+square root of the square of a scalar quantity conventionally yields the positive square root, or absolute
+value. In the case of a complex λ, taking the square root of the square yields a result in the right
+half-plane, rather than the absolute value. Thus to decide whethe r the biquaternion semi-norm satisfies
+the property as conventionally stated requires some deeper unde rstanding of the reason for the property.
+The triangle inequality, as has already been noted in §3is not applicable to the biquaternion semi-norm,
+because ordering is not defined for complex numbers.
+Finally we note that it is possible to define a real norm and modulus for t he biquaternions. G¨ urlebeck
+and Spr¨ ossig [ 13, Lemma 1.30] state that there is a unique real norm satisfying /bardblpq/bardbl=/bardblp/bardbl /bardblq/bardbland
+/bardblµσ0/bardbl=|µ|,µ∈Rwhereσ0is a basis element (Pauli matrix). This norm is simply the absolute value
+of the square root of the semi-norm, in other words, rin equations 4and5, or|Q|in Table 2.
+3.4 Roots of −1
+This section draws on and builds on results first presented by Sangw ine in [ 33,34]. These results have
+been generalized to Clifford algebras by Hitzer and Ab/suppress lamovicz [ 25,26] where the concept is aptly referred
+to asgeometric roots of−1.
+Biquaternion roots of -1 appear as ξin Table 2, in the ‘complex’ and ‘Hamilton polar’ forms.
+Theorem 1. Any non-zero quaternion or biquaternion with a non-vanishi ng modulus, divided by its
+modulus has a unit real norm.
+Proof. Letqbe an arbitrary quaternion or biquaternion and let p=q/|q|. For any quaternion or
+biquaternion xwe have /bardblx/bardbl=xxand|x|=/radicalbig
+/bardblx/bardbl. Therefore p=q/√qqand we can write the norm of
+12The ‘rule of the norms’ holds also for the modulus (the square root of the norm), which is what the term ‘norm’ refers
+to in many sources in the literature.
+13The properties given in equation 11are stated using the notation used in the cited reference whi ch does not conform
+to the usage elsewhere in this paper. In particular, /bardblx/bardblin equation 11means what we call the modulus or square root of
+the norm in this paper.
+12pas:
+/bardblp/bardbl=pp=q√qq/parenleftbiggq√qq/parenrightbigg
+Since√qqis complex, it is unaffected by the quaternion conjugate and we can s implify this to:
+=q√qqq√qq=qq
+qq= 1
+Theorem 2. Any pure quaternion or biquaternion with a non-vanishing mo dulus, divided by its modulus,
+is a root of −1.
+Proof. Letqbe an arbitrary quaternion or biquaternion with |q| /ne}ationslash= 0. Then ( q/|q|)2=q2/qq=q/q. If
+we restrict qto have zero scalar part, the conjugate reduces to negation and we have ( q/|q|)2=q/−q=
+−1.
+Theorem 2is simply a restatement of the well-known fact that unit pure quater nions are roots of
+−1 [17,§167, p179], but in the case of biquaternions, the ramifications of th e result are not so obvious.
+Theorem 2is also easily demonstrated using the Cartesian form:
+/parenleftbiggXi+Yj+Zk√
+X2+Y2+Z2/parenrightbigg2
+=−X2+XY(ij+ji) +XZ(ik+ki)−Y2+YZ(jk+kj)−Z2
+X2+Y2+Z2
+=−/parenleftbig
+X2+Y2+Z2/parenrightbig
+X2+Y2+Z2=−1 (12)
+It was shown in [ 34, Theorem 2.1]14that any pure biquaternion ξsatisfying the constraints ℜ(ξ)⊥
+ℑ(ξ) and/bardblℜ(ξ)/bardbl−/bardblℑ (ξ)/bardbl= 1 is a root of −1 and that no other biquaternions are roots of −1. It follows
+therefore that dividing an arbitrary pure biquaternion by its (comp lex) modulus produces a result that
+satisfies the constraints stated. Since these constraints are by no means obvious from equation 12,
+it is interesting to verify them directly. To verify the constraints st ated, we take an arbitrary pure
+biquaternion p, divide it by its (complex) modulus and show that the result satisfies t he constraints
+stated. Dividing pby its modulus gives a root of −1:ξ=p/|p|. We have to show that the real and
+imaginary parts of ξare orthogonal, that is /an}bracketle{tℜ(ξ),ℑ(ξ)/an}bracketri}ht= 0, and also that the difference between the
+norms of the real and imaginary parts of ξis 1. We can express the real and imaginary parts of ξas
+follows, by taking the sum and difference of ξwith its complex conjugate:
+ℜ(ξ) =1
+2(ξ+ξ⋆),ℑ(ξ) =1
+2I(ξ−ξ⋆)
+The inner product of two quaternions is given by equation 10, but since we are dealing with pure
+quaternions, the conjugates reduce to negation, and the inner p roduct simplifies to:
+/an}bracketle{tu,v/an}bracketri}ht=−1
+2(uv+vu) (13)
+Substituting the results above for the real and imaginary parts of ξ, we obtain the following:
+/an}bracketle{tℜ(ξ),ℑ(ξ)/an}bracketri}ht=−1
+2/bracketleftbigg1
+2(ξ+ξ⋆)1
+2I(ξ−ξ⋆) +1
+2I(ξ−ξ⋆)1
+2(ξ+ξ⋆)/bracketrightbigg
+=−1
+8I[(ξ+ξ⋆) (ξ−ξ⋆) + (ξ−ξ⋆) (ξ+ξ⋆)]
+=−1
+8I/bracketleftBig
+ξ2+ξ⋆ξ−ξ⋆2−ξξ⋆+ξ2−ξ⋆ξ−ξ⋆2+ξξ⋆/bracketrightBig
+=−1
+4I/bracketleftBig
+ξ2−ξ⋆2/bracketrightBig
+14Also in [ 33, Theorem 1].
+13Sinceξis a root of −1 by Theorem 2, its complex conjugate must also be a root of minus one15, thus
+ξ2=ξ⋆2=−1 and therefore /an}bracketle{tℜ(ξ),ℑ(ξ)/an}bracketri}ht= 0 as expected.
+To show that the real and imaginary parts of ξhave norms differing by 1, we can use a similar
+approach. Starting with the fact that the norm is the inner produc t of a quaternion with itself, as stated
+in§3.3, the formula in equation 13simplifies to /bardblu/bardbl=−u2whenuis pure and v=u. Thus the difference
+between the norms of the real and imaginary parts of ξis given by:
+/bardblℜ(ξ)/bardbl−/bardblℑ (ξ)/bardbl=−ℜ(ξ)2+ℑ(ξ)2=−/bracketleftbigg1
+2(ξ+ξ⋆)/bracketrightbigg2
++/bracketleftbigg1
+2I(ξ−ξ⋆)/bracketrightbigg2
+=−1
+4/bracketleftBig
+ξ2+ξ⋆2+ξξ⋆+ξ⋆ξ/bracketrightBig
+−1
+4/bracketleftBig
+ξ2+ξ⋆2−ξξ⋆−ξ⋆ξ/bracketrightBig
+=−1
+2/bracketleftBig
+ξ2+ξ⋆2/bracketrightBig
+which is 1 because ξand its conjugate are roots of −1.
+3.5 Divisors of zero
+It is possible for the semi-norm, /bardblq/bardbl, of a biquaternion to be zero, even though q/ne}ationslash= 0. In modern
+terminology, the biquaternions with vanishing semi-norm are better known as divisors of zero . The ‘rule
+of the norms’ given in §3.3shows that multiplying an arbitrary biquaternion by a divisor of zero ( with
+vanishing semi-norm) yields a result which is also a divisor of zero (with v anishing semi-norm).
+The conditions for the semi-norm to vanish were discovered by Hamilt on [17, Lecture VII, §672,
+p669]16— if we represent a non-zero biquaternion in the form: q=qr+Iqithen its norm is zero
+iff/bardblqr/bardbl=/bardblqi/bardbland/an}bracketle{tqr,qi/an}bracketri}ht= 0, that is, the real and imaginary parts must have equal norms an d be
+perpendicular in 4-space. This result follows easily from Lemma 1in§3.3. The conditions stated are
+equivalent to the condition W2+X2+Y2+Z2= 0, which can be expanded into a pair of simultaneous
+equations each equivalent to one of the two conditions above, by eq uating the real and imaginary parts
+to zero separately [ 35,36, Proposition 1]. Notice that the conditions required for a vanishing s emi-
+norm may be satisfied in the case of a pure biquaternion (the real an d imaginary components are then
+perpendicular in 3-space). It follows from these conditions that an y non-zero quaternion or any non-zero
+imaginary biquaternion must have a non-zero norm, since the const raint of real and imaginary parts
+having equal norm is violated.
+Sangwine and Alfsmann have shown in [ 35,36] that every biquaternion divisor of zero is one of:
+•an idempotent as defined in the next section in Theorem 3;
+•a complex multiple of an idempotent (dividing the divisor of zero by twice its scalar part yields an
+idempotent);
+•a nilpotent as defined in §3.5.2.
+3.5.1 Idempotents
+An idempotent is a value that squares to give itself. An example of a biq uaternion idempotent is given
+in [12] (quoting a paper of Lanczos) but no general case is given. We pre sent here a derivation of the set
+of idempotents in B, drawing on results first presented in [ 35,36] by Sangwine and Alfsmann.
+Theorem 3. [35,36, Theorem 2] Any biquaternion of the form q=1
+2±1
+2ξI, whereξ∈Bis a root of
+−1, is an idempotent. There are no other idempotents in B.
+Proof. The proof is by construction from an arbitrary biquaternion repre sented in the form q=A+ξB
+(A,B∈C). Squaring and equating to qgives:
+q2=A2−B2+ 2ABξ=A+ξB=q
+15This is easily shown using the ‘complex’ form ξ=ξr+Iξi, since the imaginary part must be zero in both ξ2andξ⋆2
+if they are each equal to −1.
+16In [17, Lecture VII, §673, p671], Hamilton rather quaintly refers to an evanescent tensor — a tensor being what we
+here call a semi-norm.
+14Equating coefficients of 1 and ξ(that is, equating the real and imaginary parts with respect to ξ) gives:
+A=A2−B2
+B= 2AB
+The second of these equations requires that A=1
+2. Making this substitution into the first equation
+gives:1
+2=1
+4−B2, from which B2=−1
+4. SinceBis complex, the only solutions are B=±I/2. Thus
+q=1
+2±ξI
+2.
+The roots of −1 inBinclude two trivial cases, as noted in [ 34, Theorem 2.1]. These are ±Iwhich
+yields the trivial idempotents q= 0 and q= 1; and ±µwhereµis a real pure quaternion root of −1.
+Since every biquaternion idempotent, q, is a solution of q2= 0, it must also be a solution of q(q−1) =
+0. Therefore every biquaternion idempotent is also a divisor of zero and must satisfy the conditions stated
+in§3.5. This is demonstrated in detail in [ 35,36,§4, Proposition 2].
+All non-pure biquaternion divisors of zero have been shown to be of the form p=αqwhereαis a
+non-zero complex number, and qis a biquaternion idempotent [ 35,36,§4, Theorem 3]. Further, dividing
+a non-pure biquaternion divisor of zero by twice its scalar part yields an idempotent [ 35,36,§4, Corollary
+1].
+3.5.2 Nilpotents
+A nilpotent is a quantity whose square vanishes. Hamilton discovered nilpotent biquaternions and was
+aware that all such biquaternions were divisors of zero [ 17, Lecture VII, §674, pp671-3].
+We first present the case of a pure biquaternion with vanishing semi- norm and show a result due to
+Hamilton [ 17, Lecture VII, §672, p669] that the square of such a biquaternion vanishes.
+Theorem 4. Letp=pr+Ipibe a non-zero pure biquaternion with vanishing semi-norm. T henp2= 0.
+Proof. From the fact that phas a vanishing semi-norm, /bardblpr/bardbl=/bardblpi/bardbl, therefore we may divide by their
+common norm, and obtain p//bardblpr/bardbl=µ+Iνwhereµandνare perpendicular unit pure quaternions.
+Then:/parenleftbiggp
+/bardblpr/bardbl/parenrightbigg2
+= (µ+Iν)2=µ2−ν2+I(µν+νµ)
+which vanishes because the squares of the two unit pure quaternio ns are−1 and the imaginary part
+vanishes because the products of two perpendicular unit pure qua ternions changes sign when the order
+of the product is reversed.
+Sangwine and Alfsmann, in [ 35,36,§5, Lemma 1] have shown that all nilpotent biquaternions are
+pure, and in [ 35,36,§5, Proposition 3] that all nilpotent biquaternions are divisors of zer o. It follows
+that all nilpotent biquaternions have real and imaginary parts with e qual norm and therefore that any
+nilpotent biquaternion can be normalised to the form µ+Iνas in Theorem 4. We see therefore that all
+nilpotent biquaternions are constructed from a pair of mutually ort hogonal unit pure quaternions.
+A nilpotent biquaternion can be written in the form ξBwhereξis a root of −1 andBis complex
+(the square root of the semi-norm: B=√
+X2+Y2+Z2). Since a nilpotent biquaternion is a divisor of
+zero, its semi-norm is zero. This means that Bis zero, even though ξBis not. A consequence is that it
+is not possible to compute the ‘axis’ of a nilpotent biquaternion by divid ing byB.
+3.6 The biquaternions as a geometric algebra
+In this section we discuss equivalence of the components of a biquat ernion to the elements of a geometric
+algebra [ 24,40,23,30,42] orClifford algebra . Jaap Suter’s tutorial paper [ 42] is particularly recom-
+mended. A full discussion of the biquaternions as a geometric algebr a is a large and non-trivial topic,
+and will be left to a later paper or papers.
+Hamilton himself, in his Lectures on Quaternions [17, Lectures I, II, III] was interested in an algebra
+of points, lines and planes, and clearly was motivated by a desire to ha ve an algebra applicable to 3-
+dimensional geometry. The modern notion of a geometric algebra us es different language, and some of
+15the concepts are more sophisticated, but essentially the aim is the s ame: to represent and manipulate
+geometric objects algebraically.
+A geometric algebra contains four types of element, as already intr oduced in §3and Tables 3and4:
+scalars, vectors, bivectors and pseudoscalars. Scalars are qua ntities without geometric form (or points),
+vectors represent directed magnitudes (or lines), bivectors rep resent directed areas, and pseudoscalars
+represent signed volumes. The product of two perpendicular vect ors is a bivector, representing the area
+swept out by one vector when moved along the other [ 42,§2.1]. This is why vectors are identified
+with pure imaginary biquaternions. Given two perpendicular pure qua ternions, µandν, such that
+/an}bracketle{tµ,ν/an}bracketri}ht= 0, the two vectors IµandIνhave a product −µνwhich is perpendicular to both µandν.
+The negative sign is a consequence of squaring I. Changing the order of the two vectors changes the
+sign of the resulting bivector (it is oppositely directed), so the minus sign is of little consequence. It is
+usual in geometric algebra to have to choose the sense by convent ion. Here the rules of biquaternion
+multiplication define the convention for us. The product of three or thogonal vectors defines a volume,
+represented by a pseudoscalar. For example, ( Ii)(Ij)(Ik) =I, as given by Ward [ 45, p 113].
+There has been much confusion caused by the use of the term vector when in fact the objects under
+consideration were bivectors . In physics the concepts of axial andpolar vectors have been used, causing
+further confusion, because the two terms suggest different typ es of vector, rather than fundamentally
+different types of quantity. Even the present authors, in their ea rlier works, used the term vector to refer
+to quantities which are now clearly seen to be bivectors. Jaap Suter in hisGeometric Algebra Primer
+states clearly:
+A quaternion is a scalar plus a bivector.
+and he is right [ 42, p 50]. Unfortunately, the scalar/vector part terminology commo nly applied to quater-
+nions has this wrong, and we are stuck with it.
+In geometric algebra a composite quantity consisting of two or more of the four types of quantity
+(scalar, vector, bivector, pseudoscalar) is called a multivector . A biquaternion may contain all four
+types of geometric element, and except in degenerate cases (suc h as a pure imaginary biquaternion),
+corresponds to the concept of a multivector.
+Once we start to multiply multivectors, the geometric interpretatio n becomes more difficult, although
+the rules of multiplication are well-understood and easily described: t he full biquaternion product can
+be expressed, exactly as in the quaternion case, as:
+pq= S(p) S(q) + S(p)V(q) + S(q)V(p) +V(p)V(q)
+and the last term can be written as V(p)·V(q) +V(p)×V(q) or/an}bracketle{tV(p),V(q)/an}bracketri}ht+V(p)×V(q), that is
+the sum of a dot or scalar or inner product, and a cross product. A deeper analysis of the biquaternions
+as a geometric algebra requires further work.
+Although multivectors in geometric algebra are regarded as compos ites of scalar, vector, bivector and
+pseudoscalar, a biquaternion is not constructed in this way. Inste ad it has four complex components,
+but as we have seen in §3, it is possible to interpret these in multiple ways, for example as a comp lex
+number with quaternion components. It is also possible to consider t he biquaternions as 8-dimensional
+quantities with basis (1 ,iI,jI,kI,i,j,k,I) where we place the vector basis first. The multiplication
+table is then easily derived from the rules of quaternion multiplication ( Table 5).
+From the rules of quaternion multiplication, it is fairly easy to draw up a table showing how the
+geometric components of a biquaternion multiply together. Table 6shows the result. The four centre
+entries apply in the general case, but in the specific cases where th e bivector/vector are perpendicular,
+the scalar or pseudoscalar part of the product will be zero. Multiplic ation by the pseudoscalar is known
+as thedual operation: it maps a bivector into a vector, and a scalar into a pseud oscalar or vice versa .
+3.6.1 The wedge or outer product
+In geometric algebra, the algebraic or geometric product of two ve ctors is the sum of the scalar product
+and the wedge ( ∧) or outer product (also known as the cross product in three dimen sions):
+pq=p·q+p∧q
+161iIjIkIi j k I
+1 1iIjIkIi j k I
+iIiI 1−k j −IkI−jI−i
+jIjIk 1−i−kI−IiI−j
+kIkI−j i 1jI−iI−I−k
+ii−IkI−jI−1k−jiI
+jj−kI−IiI−k−1ijI
+kkjI−iI−Ij−i−1kI
+II−i−j−kiIjIkI−1
+Table 5: Biquaternion basis multiplication table.
+S B V P
+S – scalar S B V P
+B – bivector B S + B P + B V
+V – vector V P + B S + B B
+P – pseudoscalar P V B S
+Table 6: Multiplication table for components of a biquaternion multivec tor
+It is possible to define a wedge product for quaternions and biquate rnions using a formula widely used
+in geometric algebra [ 42, Equation 3.10]:
+p∧q=1
+2(pq−qp)
+This gives the cross product of the vector parts (because the co mponents of the products involving
+the scalar parts, and the dot product of the vector parts, comm ute and therefore cancel). If applied
+to a pair of vectors (i.e. pure biquaternions with zero real part), t he result is a bivector (i.e. a pure
+biquaternion with zero imaginary part). Applied to a pair of bivectors (pure quaternions) the result is
+another bivector. Applied to a vector and a bivector, the result is a vector.
+4 Further work
+There is much further interpretation needed of the biquaternions as a geometric algebra. At present
+formulae are lacking for vector/bivector products to yield pseudo scalars and for generalisations of the
+wedge product.
+The interpretation of complex angles in the polar form and in the cont ext of the inner product requires
+further work.
+References
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+1986.
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+tional exercises and material by H.B. Griffiths.
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+the idempotents and nilpotents. Advances in Applied Clifford Algebras , 10pp, December 2009.
+Publisher: Birkh¨ auser Basel, in press.
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+http://qtfm.sourceforge.net/ , 2005. Software library, licensed under the GNU General
+Public License.
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+complex argument inspired by the Cayley-Dickson form. Advances in Applied Clifford Algebras ,
+10pp, 2008. Publisher: Birkh¨ auser Basel, in press.
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+20
\ No newline at end of file
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@@ -0,0 +1,442 @@
+arXiv:1001.0241v1  [gr-qc]  1 Jan 2010On the Origin of Gauge Symmetries and Fundamental Constants
+S.G. Rubin
+National Research Nuclear University ”MEPhI” ,
+Kashirskoe sh. 31, Moscow, 115409 Russia
+E-mail: sergeirubin@list.ru
+A statistical mechanism is proposed for symmetrization of a n extra space. The conditions and
+rate of attainment of a symmetric configuration and, as a cons equence, the appearance of gauge
+invariance in low-energy physics is discussed. It is shown t hat, under some conditions, this situation
+occurs only after completion of the inflationary stage. The d ependencies of the constants /planckover2pi1and G
+on the geometry of the extra space and the initial parameters of the Lagrangian of the gravitational
+field with higher derivatives are analyzed.
+PACS numbers: 04.50.-h; 04.05.Cd; 11.30.Ly
+I. INTRODUCTION
+In spite of the advances made in fundamental physics, the exi stence of a large number of theories and approaches
+suggests that there are problems in describing observation al and experimental data. The idea of compact extra
+dimensions allows one to effectively explain a considerable number of phenomena and indicates the direction of
+further development in the theory despite the absence of dir ect experimental confirmation of its existence.
+This paper investigates the problems associated with the em ergence of gauge symmetries and fundamental constants
+at the early stage of the evolution of the Universe using the a pproach proposed in our previous works в [1, 2].
+This approach is based on the assumption that extra dimensio ns exist forming a compact space with the properties
+determining the observed low-energy physics. The solution to the first problem within the Kaluza - Klein approach
+is well known: gauge symmetries arise as a consequence of the corresponding symmetries of the extra space (see,
+for example, [3]). Therefore, the problem is reduced to just ification of the choice of symmetric spaces among a set
+of extra spaces with an arbitrary geometry. More precisely, we assume that, at some instant of time on the Planck
+scale, the four-dimensional Riemannian space M4[4–6] arises as a result of quantum fluctuations. Simultaneo usly,
+there arises a compact extra d- dimensional space Md. The set of possible geometries of the extra space Mdis at
+least a continuum. However, there exists a geometry with a hi gh degree of symmetry, because the existence of gauge
+symmetries is beyond question. The probability of this even t is negligible, and, hence, there should exist a mechanism
+of selection that separates appropriate geometries. One of the possible variants based on statistical considerations will
+be analyzed below.
+The second problem discussed in our work is related to the fun damentality of the parameters /planckover2pi1,G[7] 7] and consists
+in the following. The geometric approach to the theory impli es the presence of only one scale of the dimension of
+lengthℓ. The parameters of the initial Lagrangian constructed only from the metric tensor are proportional to ℓn,n,
+where n is an integer number. The question arises as to the ins tant of time at which the Planck and gravitational
+constants appear as independent fundamental parameters an d as to their relation to the initial parameters of the
+theory. The answer to this question for the gravitational co nstant is well known and, as a rule, is represented in the
+form
+M2
+P=mD−2
+DVd. (1)
+Here,MPis the Planck mass, mDis the parameter with the dimension of mass or, more exactly, inverse length, and
+Vd<∞is the volume of the extra space Md. The similar problem of the fundamentality of the Planck con stant
+/planckover2pi1is less well understood, even though interesting works have appeared in this direction. In particular, Volovik [8]
+discussed the place of the Planck constant in modern theory, analyzed different variants of including this constant
+in field equations, and considered the consequences of the hy pothetical spatial dependence of the Planck constant.
+However, specific implementation of the aforementioned hyp othetical variants was not given.
+In this paper, we propose the mechanism of emergence of the Pl anck constant together with the gravitational
+constant without detailed discussion of the possibilities this entails. As a result, we can reveal the relations of the
+constants /planckover2pi1andGdetermined from low-energy experiments to the parameters o f the initial Lagrangian describing
+multidimensional gravity with higher derivatives.
+As a continuation of the ideology developed in [2], we assume that spaces with an appropriate geometry arise in a
+spacetime foam with some, even though small, probability. W e are interested in spaces with a geometry of the direct
+product
+MD=MD1⊗Md1;D1≥4;d1≥2 (2)2
+and the relationship of the volumes V(D1)>> V(d1). The transitions with a change in the geometry are convenien tly
+described in terms of the path integral technique [4, 5]. For this purpose, the superspace MD= (MD;gij)is defined
+as a set of metrics gijin the space MDto within diffeomorphisms. On a spacelike section Σwe introduce the metric
+hij(for details, see [6]) and define the space of all Riemannian (D−1)metrics in the form
+Riem(Σ) ={hij(x)|x∈Σ}
+The transition amplitude from the section Σinto the section Σfis the integral over all geometries allowable by the
+boundary conditions:
+Af,in=/an}b∇acketle{thf,Σf|hin,Σin/an}b∇acket∇i}ht=/integraldisplayhf
+hinDgexp[iS(g)]. (3)
+The absence of the Planck constant in the exponential functi on is a result of choosing the appropriate units of
+measurement. Nonetheless, it will be shown below that the Pl anck constant naturally emerges after the definition
+of the dimensional units simultaneously with gravitationa l constant G. However, the statement on the unification of
+gravity and quantum theory would be premature. The essence o f quantum mechanics is based on the rule of summation
+of the transition amplitudes (3), which is postulated origi nally.
+II. WHY IS THE EXTRA SPACE SYMMETRIC?
+The transition amplitude (3), generating a four-dimension al space, as a rule, has been calculated under the
+assumption of a special form of the metric hfon a hypersurface Σfwith an interval of the type [6, 9]
+ds2=σ2/bracketleftbig
+N(t)2dt2−a(t)2dΩ2
+3/bracketrightbig
+, (4)
+σ2=1
+12π2M2
+Pl.
+In this case, it is implied that the space with a metric that we akly differs from the metric written above asymptotically
+tends to (4) in the course of cosmological expansion. If the t ransition amplitude (3) describes the birth of a space of
+type (2), where D1= 4, no preferred geometry exists for the subspace Md1. In this section, we propose the mechanism
+of symmetrization of compact spaces.
+Let us postulate that the evolution of any closed system is ac companied by an increase in its entropy. It should be
+noted that the entropy of a subsystem can decrease. In partic ular, the Hawking evaporation of a black hole decreases
+its entropy but increases the entropy of the Universe as a who le. Below, we will consider a space with symmetry (2)
+under the assumption that the volume of the subspace is consi derably larger than the volume of the compact subspace
+Md1.
+A. Entropy of the Compact Space
+First and foremost, we will demonstrate that the entropy of a compact space reaches a minimum on a class of
+maximally symmetric spaces. The initial definition of the en tropy in our work coincides with the known definition
+of Boltzmann, who related the entropy to the number Ωof states of a system: s=kBlnΩ. The discussion of other
+possibilities can be found, for example, in [10, 11]. The not ion of the number of states is correctly defined at the
+quantum level, when the set of energy levels and the degree of their degeneracy are known. However, the quantization
+of the space Mdmeans the quantization of a gravitational field, which by its elf remains an unsolved problem. In
+this respect, we will restrict ourselves to the calculation of the number of states from the classical viewpoint. This is
+sufficient because we are interested in relative quantities ( see also [12]).
+The entropy of an arbitrary compact space Md, in which fields of matter are absent is a functional s[G]of the metric
+tensorG. The number of states is determined by an observer outside th e system. It is intuitively clear that the higher
+the symmetry of an object (in our case, the compact space), th e smaller the statistical weight of the object. Let us
+prove this statement. In order to calculate the statistical weightΩ, the space Mdis embedded in the space RN, which
+can be done if the space Mdis sufficiently smooth and Nis sufficiently large [13]. It is assumed that the external
+observer is located in the space RN. Each point of the space Mdis described by the internal coordinate y∈Mdand
+the external coordinate x∈RN. We choose a specific point P∈Mdand fix its coordinate xP∈RN. Then, we choose
+the set of basis vectors ek,k= 1,2,...,N in the space RN. We require that all dtangent linearly independent vectors3
+at the point P,e(P)
+a,a= 1,2,...,d, which form the coordinate basis in the space Md, should be included in this set.
+With the use of this basis, we calculate the components of the metric tensor G(P)
+ab(xP)of the space Md. We fix the
+tangent space T(P), spanned by the vectors e(P)
+a.
+Then we choose the second point Q∈Md, with the coordinates xQ∈RNand the tangent space T(Q). By moving
+along the curve lPQ⊂Md, the point Qis displaced to the point with the coordinates xPso that the tangent space
+T(Q)coincides with the tangent space T(P).
+The new components of the metric tensor G(Q)
+ab(xP), of the space Mdare calculated at the same point xP∈RN. If
+the metric tensors do not coincide with each other G(Q)
+ab(xP)/ne}ationslash=G(P)
+ab(xP), the observer in the space RNwill fix a new
+”microstate” and increase the statistical weight of the spa ceMdby unity. When the equality
+G(Q)
+ab(xP) =G(P)
+ab(xP), (5)
+is valid, the number of microstates remains unchanged. It sh ould be noted that condition (5) corresponds to the
+condition for the existence of the Killing vector along the c urvePQ. Therefore, the presence of Killing vectors
+decreases the statistical weight of the compact space. Maxi mally symmetric spaces have a minimum entropy.
+B. Decay of Excitations of the Compact Space
+The compact subspace Md1can be considered as a subsystem of the space MD, (see relationships (2)). Now, we
+show that, if the volumes of the subspaces satisfy the inequa lityVD1>> Vd1, there is an entropy flow from the
+subspace Md1to the subspace MD1the entropy of the subspace Md1tends to a minimum, and its geometry tends to
+a maximally symmetric geometry. Furthermore, the entropy o f the entire system, i.e., the space MDincreases.
+As a rule, the entropy of a closed system changes with conserv ation of the system energy. Since the determination
+of the gravitational energy depends on the space topology, w e will restrict our consideration to transitions without
+changes in it. In [14, 15], it was demonstrated that, in the fr amework of gravity nonlinear in the Ricci scalar, at least
+local minima of the energy density exist. In this case, there is a set of energy levels and each geometry of the extra
+space can be represented in the form of an expansion in eigenf unctions of the d’Alembert operator of the background
+metric. In terms of the Kaluza - Klein theory, the eigenvalue s of this operator contribute to the mass of excitations,
+which are interpreted as particles in the space M4. For a compact space with the geometry of a circle of radius r,
+the mass of the lightest particle is m1= 1/rand it is experimentally limited from below by a value of seve ral TeV.
+It is evident that its decay into light particles propagatin g in our space should be accompanied by an increase in
+the entropy. In the process, the geometry of the extra space r elaxes to a state that has a minimum entropy and is
+characterized by the absence of excitations.
+As an illustration, we consider a space of type (2), where MD1=M4⊗Md2,Md2=S1,Md1=S1with the metric
+gMN, which differs from the diagonal metric ηMN=diag(1,−1,−1,−1,−r1,−r2), only slightly, so that gMN=
+ηMN+hMN(x,y1,y2). The size of one of the extra spaces is considerably larger th an the size of the other extra space:
+r2=βr1, β >> 1. The dynamic equation for the field hMNcan be written in the form [3]
+ηAB∂A∂BhMN(x,y1,y2) = 0.
+By substituting the field hin the form of the expansion in eigenfunctions of the d’Alemb ert operators Yn(y1),Yn(y2)
+оof both subspaces (in our case, circles)
+hMN(x,y1,y2) =/summationdisplay
+n1,n2h(n1,n2)
+MN(x)Yn1(y1),Yn2(y2),
+we obtain the equation for the components
+/parenleftbigg
+/squarex+n2
+1
+r2
+1+n2
+2
+r2
+2/parenrightbigg
+h(n1,n2)
+MN(x) = 0. (6)
+The microstates of the subspaces Md1andMd2are characterized by the integer numbers n1иn2. The first excited
+state of the subspace Md1has a mass m1= 1/r1. The decay of this state into two excited states of the subspa ceMd2
+with masses m2=n2/r2andm′
+2=n′
+2/r2,n2,n′
+2= 0,1,2,...,[β]and zero momenta occurs with energy conservation,
+m1=m2+m′
+2 (7)
+and an increase in the entropy of the entire system. This incr ease is associated with the fact that the number of
+different microstates of the space Md2, that satisfy condition (7) is of the order of Ω2=β/2∼r2/r1. In this case, the4
+geometry of the subspace Md1tends to a maximally symmetric geometry, because the number of excitations in the
+subspace tends to zero and the entropy tends to a minimum.
+It is obvious that the number of microstates increases with a n increase in the volume of the subspace Md2. The
+inclusion of states with different four-momenta in the Minko wski space M4substantially enhances the effect.
+Let us increase the dimension of the compact space with a larg e volume and consider the manifold of type(2), where
+MD1=M4⊗Md2,Md2=S2,Md1=S1. It is known that the excitation mass in S2ism(l) =/radicalbig
+l(l+1)/r2, and
+the degeneracy factor is 2l+1. The decay of excitation of the space S1into two excited states of the space S2also
+proceeds with energy conservation: m1=m(l)+m(l′). It is easy to see that, with number of allowable energy state s
+with zero momentum is also proportional to the volume of the l arge space: Ω2≃β2=r2
+2/r2
+1.
+Therefore, if quantum fluctuations generate a space of the ty peMa⊗Mb, the entropy flow is directed
+toward the subspace with a larger size. The geometry of the su bspace with a smaller size tends to a
+maximally symmetric geometry compatible with its topology . The existence of gauge symmetries in the
+main space appears to be a purely statistical effect.
+C. Rate of the Transition to the Symmetric State
+The entropy of the entire system consisting of two subspaces (2), increases as a result of the transformation of
+particles (excitations) of the compact subspace Md1to particles propagating in the subspace MD1. The compact extra
+space that, at the initial instant of time, has an arbitrary g eometry acquires a maximally symmetric shape during the
+entropy transfer to the main space. In this case, the entropy of the entire system (the extra and main spaces) increases
+and the entropy of the subsystem (the compact extra space) te nds to a minimum. The situation resembles the third
+law of thermodynamics, according to which the entropy of a bo dy tends to zero with a decrease in the thermostat
+temperature.
+The subspace MD1in (2), should not necessarily have a direct-product struct ure. The main requirement for this
+subspace is the presence of a large number of energy levels th at contribute to the statistical weight of the system at
+a fixed temperature. The Minkowski space M4itself exhibits this property in full measure.
+Let us estimate the rate of ”symmetrization” of the extra spa ce. Weak deviations of the geometry from an equilibrium
+configuration can be interpreted as excited states with the m assm1,(see, for example, [16]). Since this is the only
+scale, it should be expected that the probability of decay wi ll satisfy the relationship Γ∼m1∼1/Ld, whereLd
+- is the characteristic size of the extra space. Setting Ld≤10−17cm, we find that the lifetime of the excited state
+ist1∼Ld≤10−27s. Therefore, the extra space transformed into the most symm etric state long before the onset
+of the primordial nucleosynthesis but, possibly, after com pletion of the inflationary stage. However, the states that
+correspond to the first excited level of the Kaluza - Klein tow er with a lifetime of the order of 105s or more is also
+considered [17]. Under these conditions, the theory acquir es the gauge invariance well after the nucleosynthesis stag e,
+which creates new possibilities and problems. In order for a Kaluza - Klein particle to be stable, it is necessary to
+make additional assumptions that complicate the structure of the extra space [18].
+III. RELATION OF THE FUNDAMENTAL CONSTANTS TO THE PROPERTIE S OF THE EXTRA
+SPACE
+Now, we discuss the problem associated with the emergence of physical parameters in the modern Universe. In
+the sequential implementation of the Kaluza - Klein idea, bo son fields represent individual metric components of
+the extra space. The only scale possible in this situation is the length scale I, I; however, the use of even this unit
+is complicated because quantum fluctuations in the spacetim e foam continuously renormalize any parameters in an
+unpredictable manner. The situation becomes better as soon as a classical spacetime region arises due to the same
+quantum fluctuations. This is also accompanied by the appear ance of independent stationary quantities that can be
+chosen as units of measurement, for example, the volume of th e compact space Vd1and the vacuum energy density Um.
+Only in this case, dimensional physical constants, includi ng fundamental constants, can be fixed. A possible variant
+of this scenario will be considered below.
+The action is chosen in the form (see, for example, [19–21]),
+S=N0/integraldisplay
+dDy√
+−GF(R;an);F(R;an) =/summationdisplay
+nanRn, a1= 1. (8)
+The parameters an,N0take on specific values in the birth of this spatial region [2] after the choice of the length unit.
+The appropriate choice of the parameters ancan provide the boundedness of the effective action from belo w [1, 15].5
+The dimension of the function F(R)is conveniently chosen so that it coincides with the dimensi on ofR; i.e., it is I−2.
+Then, the dimension of N0isI2−D.
+The metric of the space MDis written in the form [15, 22]
+ds2=GABdXAdXB=gab(x)dxadxb−e2β(x)γij(y)dyidyj(9)
+= Ndt2−gµν(x)dxµdxν−e2β(x)γij(y)dyidyj.
+Heregabis the metric of the subspace M4=R⊗M3with the signature (+−−−),e2β(x)is the radius of curvature of
+the compact subspace Md, andγij(y)is its positive-definite metric. For a specified foliation of the space by spacelike
+surfaces, we can always choose the normal Gaussian coordina tes, which is used in the last equality in expression(9).
+We assume that the coordinates x,yhave dimension I, the time is measured in seconds, and the lapse function Nis
+an unknown parameter with the dimension of (I/s)2. The metrics gabandγijare dimensionless.
+According to [1, 15], the topology and metric on the spacelik e section Σfof amplitude (3), are defined by imposing
+the following conditions.
+i) The topology of the space MDhas the form of the direct product
+MD=M4⊗Md, (10)
+wheredis the dimension of the compact extra space.
+ii) The curvature of the subspace Mdsatisfies the condition
+R4(gab)<< Rd(γij). (11)
+iii) As was shown above, the extra space with an arbitrary geo metry evolves to the space with a maximum number
+of Killing vectors that is possible for the topology under co nsideration. In this respect, in the set of subspaces Mdwe
+choose the maximally symmetric spaces with a constant curva tureRd, which is related to the curvature parameter k
+in a typical manner
+Rd(γij) = e−2β(x)kd(d−1)I−2. (12)
+Owing to the special form of the chosen metric (9), the follow ing relationships hold true [1, 22]
+R=R4+φ+fder,
+φ=kd(d−1)e−2β(x)I−2
+fder= 2dgµν∇µ∇νβ+d(d+1)gµν∂µβ∂νβ, (13)
+where the field φis defined by explicitly introducing its dimension I−2. The covariant derivative ∇acts in the space
+M4. The volume Vdof the internal space of unit curvature depends on its geomet ry, because it is expressed through
+the intrinsic metric
+Vd=/integraldisplay
+ddy√γ. (14)
+In what follows, we will use the slow-change approximation p roposed in [1]
+|φ| ≫ |R4|,|fder| (15)
+which is valid already at the onset of the inflationary stage. Then, by expanding the expression F(R;an) =F(φ+
+R4+fder;an)в into a Taylor series and integrating over the coordinates o f extra dimensions, we obtain the expression
+S≃ VdN0/integraldisplay/radicalbig
+4gd4xedβ[F′(φ;an)R4+F(φ;an)+F′(φ;an)fder], (16)
+which is typical of the scalarЏtensor theory of gravity in th e Jordan frame. By using the conformal transformation
+gµν/ma√sto→/tildewidegµν=|f(φ)|gµν, f (φ) =edβF′(φ;an), (17)6
+we change over to the Einstein frame
+S=VdN0/integraldisplay
+d4x/radicalbig
+/tildewideg(signF′)L, (18)
+L=RI+1
+2K(φ)(∂φ)2−U(φ), (19)
+K(φ) =1
+2φ2/bracketleftBigg
+6φ2/parenleftbiggF(φ;an)′′
+F(φ;an)′/parenrightbigg2
+−2dφF(φ;an)′′
+F(φ;an)′+1
+2d(d+2)/bracketrightBigg
+, (20)
+U(φ) =−(signF(φ;an)′)/bracketleftbigg|φ|·I2
+d(d−1)/bracketrightbiggd/2F(φ;an)
+F′(φ;an)2(21)
+whereF(φ;an)′=dF/dφ .
+By assuming the existence of a minimum of potential (21) at φ=φmand using the slow-change approximation,
+action (18), in the vicinity of this minimum can be represent ed in the form
+S≃N0VdcI/integraldisplay
+dtd3xI/bracketleftbigg
+RI+1
+2K(φm)(∂Iφ)2−U(φm)−1
+2U′′(φm)(φ−φm)2/bracketrightbigg
+, (22)
+where
+(∂Iφ)2= (∂φ/cI∂t)2−(∂φ/∂x I)2
+. Here, we took into account that√g=cIin the four-dimensional Minkowski space. The index Idesignates the used
+unit of length, and cIis the speed of light in the chosen units.
+The units of measurement are related by the expression
+I =α·cm. (23)
+гдеαis as yet unknown parameter. In relationship (22), we change over to the standard units of length
+S=N0Vdc/integraldisplay
+dtd3x/bracketleftbigg
+R4+1
+2K(φm)(∂φ)2−α2U(φm)−α21
+2U′′(φm)(φ−φm)2/bracketrightbigg
+. (24)
+Here,xI=αx,cI=αc,RI=R4/α2and(∂Iφ)2= (∂φ)2/α2.
+Since expression (8), is the low-energy limit of action (24) , it should adequately describe purely gravitational
+phenomena. Moreover, expression (24) makes it possible to e xplain the origin of the inflaton potential and cosmological
+constant. The effective action (24) contains the initial par ameters of the theory without using fundamental parameters ,
+such as the Planck constant /planckover2pi1and the gravitational constant G. However, from the practical and historical viewpoints,
+it is more convenient to explicitly introduce these paramet ers. In terms of the developed approach, this can be done
+by imposing the constraints
+N0Vdc=c4
+16πG/planckover2pi1, (25)
+α2U(φm) =16πG
+c4Λ. (26)
+HereΛis the observable vacuum energy (dark energy) density. It is important that the quantities, such as the volume
+of the extra space Vdand the minimum value of the potential U(φm), acquire specific values only at φ=const. If the
+fieldφis identified with the inflaton, a nontrivial situation arise s. The inflationary stage is completed before the scalar
+field (inflaton) reaches a potential minimum. Consequently, at the inflationary stage, when the Universe formed and
+the field φvaried with time, the quantities Gand/planckover2pi1should also depend on time. In this case, the theory of gravit y is
+an effective theory valid at low energies (see also [23]).
+Let us introduce the definition
+mΦ≡α/radicalBigg
+U′′(φm)
+K(φm)(27)7
+and, under the assumption that K(φm)>0, change the variables
+/radicalbigg
+c4
+16πGK(φm)(φ−φm) = Φ.
+As a result, we arrive at the conventional form of the action f or the scalar field Φinteracting with gravity; that is,
+S=1
+/planckover2pi1/integraldisplay
+dtd3x/parenleftbiggc4
+16πGR4+1
+2(∂Φ)2−Λ−1
+2m2
+ΦΦ2/parenrightbigg
+. (28)
+Now, it becomes clear that the parameter mΦhas the meaning of the scalar field mass.
+Relationships (25) and (26) allow us to solve the problem of t he separate emergence of the gravitational and Planck
+constants, which appear to be functions of the parameters of the theory N0,{an}. The set of parameters {an}is also
+implicitly included in the expressions for the quantities Vd,U(φm). In this paper, we do not consider the origin of
+the lapse function and, hence, the speed of light. By elimina ting the insignificant parameter αfrom relationships(25),
+(26) and (27), we obtain
+U(φm)K(φm)
+U′′(φm)=16πG
+c4Λ
+m2
+Φ(29)
+N0Vd(φm) =c3
+16πG/planckover2pi1. (30)
+Here, the right-hand sides contain measurable quantities, whereas the left-hand sides involve quantities dependent o n
+the initial parameters of the theory. At energies on the Plan ck scale, when the field φdid not reach the potential
+minimum, the use of the modern values of the fundamental cons tantsGand/planckover2pi1requires care. Moreover, by assuming
+that the vacuum energies Λare different in different regions of the Universe [2, 24, 25], the fundamental constants G
+и/planckover2pi1are also different according to expressions (29), (30).The d iscussion of this problem and references can be found
+in [8].
+At the inflationary stage when φ=var, the quantities Gand/planckover2pi1also vary. The dependencies of these quantities on
+the field can be determined from (29) and (30) through the chan geφm→φ, which holds true for weak deviations of
+the field from the equilibrium position. As a result, we have
+G=G(φ)≃c4m2
+Φ
+16πΛU(φ)K(φ)
+U′′(φ)(31)
+/planckover2pi1=/planckover2pi1(φ)≃c3
+16πG(φ)Vd(φ)N0.
+The time dependence of the inflaton field φfor different inflationary models is well understood [26], he nce, the dynamics
+of the fundamental parameters can be described by choosing a specific inflationary model.
+The factor of conversion between the units of length α= I/cmcan also be found by another method. Indeed, the
+characteristic size of the extra space is determined to be V1/d
+din units of I. Moreover, by designating the size of the
+hypothetical extra space as Ld, expressed in terms of cm, we obtain α=Ld/V1/d
+d.Values of Ld≤10−17do not
+contradict experimental data. With due regard for expressi on (27), we find the constraint on the initial parameters
+mΦV1/d
+d/radicalBigg
+K(φm)
+U′′(φm)=Ld<10−17. (32)
+Relationships (29), (30) and (32) allow us to determine the p arameters of the theory N0,an(see expressions (8))
+from the known constants c,/planckover2pi1,Gand the energy density Λ, the mass of the scalar field mΦ, and the volume of the
+extra space Vd. The implementation of this program is a matter for the futur e. Actually, the fundamental constants are
+measured with a high accuracy, whereas the vacuum energy den sity is determined with a considerably lower accuracy.
+This is especially true in regard to the mass of the hypotheti cal scalar field, even though this field is identified with
+the inflaton. The greatest uncertainty is associated with th e volume of the extra space, which is yet to be revealed.
+IV. DISCUSSION
+In this paper, we have considered the problems of the origin o f the gauge symmetries and fundamental parameters
+/planckover2pi1andG. It has been shown that the modern values of the fundamental p arameters and the gauge invariance begin8
+to be established in the inflationary period, i.e., at energy densities substantially lower than the Planck densities.
+The revealed dependence of the constants /planckover2pi1,Gon the geometry of the extra space and the initial parameters of the
+Lagrangian of the gravitational field with higher derivativ es indicates that these parameters are not fundamental.
+According to string theory [27, 28], and within the framewor k of the cascade birth of the Universe[2], the continuous
+formation of spatial domains generates domains characteri zed by different values of the effective parameters of the
+theory [24, 25] and, in particular, by different values of the cosmological constant Λ. According to relationship (29),
+this means that the gravitational and Plank constants are al so subjected to spatial fluctuations.
+The choice of the specific symmetry is a necessary step in the c onstruction of the theory. The use of gauge symmetries
+makes it possible to effectively describe a wide class of phen omena. It is worth noting that the existence of symmetries
+themselves, as a rule, is postulated and that their origin is not refined. However, it is well known that, within the
+multidimensional approach, the gauge symmetries are consi dered as a consequence of the symmetries of the extra
+space. If we accept the hypothesis that the extra space exist s, the problem is reduced to the elucidation of the factors
+responsible for the existence of the Killing vectors in this space. As has been shown in the present paper, the extra
+space evolves to the maximally symmetric geometry, which is allowable by the corresponding topology due to the
+transition to the state with a minimum entropy. Meantime, th e entropy of the entire system, including the extra and
+main spaces, increases. The conditions and rate of the symme trization of the extra space and emergence of the gauge
+invariance in low-energy physics have been discussed. It ha s been demonstrated that, under certain conditions, this
+situation occurs only after the completion of the inflationa ry stage.
+V. ACKNOWLEDGMENTS
+The author is grateful to A. Berkov and D. Singleton for their interest expressed in this work and S.V. Bolokhov,
+K.A. Bronnikov, V.D. Ivashchuk, M.I. Kalinin, and V.N. MelЎ nikov for helpful discussions. This study was supported
+by the Russian Foundation for Basic Research (project no. 09 -02-00677-a).9
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\ No newline at end of file
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+arXiv:1001.0242v1  [math.AG]  1 Jan 2010Genus-Zero Mirror Principle For Two Marked
+Points
+Luke Cherveny
+December 27, 2009
+Abstract
+We study a generalization of Lian-Liu-Yau’s notion of Euler data [LLY97]
+in genus zero and show that certain sequences of multiplicat ive equivariant
+characteristic classes on Kontsevich’s stable map moduli w ith markings induce
+data satisfying the generalization. In the case of one or two markings, this data
+is explicitly identified in terms of hypergeometric type cla sses, constituting a
+complete extension of Lian-Liu-Yau’s mirror principle in g enus zero to the case
+of two marked points and establishing a program for the gener al case. We give
+several applications involving the Euler class of obstruct ion bundles induced by
+a concavex bundle on Pn.
+1 Introduction
+GivenXbeasmoothprojectivevarietyover C,letMg,m(X,β)beKontsevich’smoduli
+space of stable maps to X. Its points are triples ( f;C;z1,...,zm), representing a
+holomorphic map f:C→Xfrom a genus gcurve with smooth markings z1,...,zm∈
+Cand such that f∗[C] =β∈H2(X,Z), modulo the obvious equivalence. For an
+excellent introduction to stable maps see [FP96]. The Gromov-Witten theory ofX
+with primary insertions concerns integrals of the form
+KX
+g,d(γ1,...,γm) =/integraldisplay
+[Mg,m(X,β)]virm/productdisplay
+i=1ev∗
+iγi, (1)
+where integration is over the virtual fundamental class [ Mg,m(X,β)]virof Li-Tian
+[LT98] (also Behrend-Fantechi [BF97]), eviis the map
+evi:Mg,m(X,β)→X
+given by evaluation of fat thei-th marked point, and γi∈H∗(X,Z). AsMg,m(X,β)
+is a Deligne-Mumford stack, the Gromov-Witten invariants are in gen eralQ-valued.
+12 L. Cherveny
+WhenXis Calabi-Yau, the duality between type IIA and IIB string theories
+postulatesthatacertaingenerating series fortheGromov-Witte ninvariants of X(the
+A-model partitionfunction) iscomputable in termsof periodintegra lsonthecomplex
+moduli of a mirror space (the B-model), for instance yielding the mirr or conjectures
+of [CdlOGP91] and [BCOV93] for the quintic. For a thorough history a nd overview
+of mirror symmetry, see either [CK99] [HKK+03]. We will restrict our attention to
+the mirror principle program of Bong Lian, Kefeng Liu, and S.-T. Yau w hich seeks to
+provemirrorconjecturesinasimpleandunifiedframeworkbyexhibit ing certainseries
+of equivariant multiplicative characteristic classes on Kontsevich’s m oduli of stable
+maps in terms of hypergeometric type classes [LLY97] [LLY99a] [LLY 99b] [LLY01].
+The original paper [LLY97] develops mirror principle in genus zero for Xa com-
+pleteintersection andmoregenerallya localCalabi-Yauspace arising froma concavex
+bundleVoverPn(direct sum of a positive and negative bundles). Such a bundle
+V=V+⊕V−naturally induces a sequence of obstruction bundles VdonM0,m(Pn,d)
+whose fiber at ( f;C;z1,...zm) is given by H0(C,f∗V+)⊕H1(C,f∗V−). The calcula-
+tion of Gromov-Witten invariants (1) for Xreduces to enumeration of twisted Euler
+classes of these obstruction bundles:
+KX
+d(Hk1,...,Hkm) =/integraldisplay
+M0,m(Pn,d)Euler(Vd)m/productdisplay
+i=1ev∗
+iHki(2)
+HereHis the hyperplane class on Pnand we have identified β=d[H]. The power
+of this formulation is that a torus Tacting on Pninduces aT-action onM0,m(Pn,d),
+which permits the localization methods of Atiyah-Bott [AB84] to be ap plied to an
+equivariant version of (2). We will review the essentials of equivarian t cohomology
+and localization in Section 2, as well as introduce Lian-Liu-Yau’s notion of Euler data
+and give a more precise formulation of their program. The idea is that one should
+study sequences of equivariant classes on a large projective spac e (the so-called linear
+model) that satisfy certain quadratic relations on fixed points unde r a torus action.
+These relations reflect the geometry of fixed components of a com plicated moduli
+space (the nonlinear model) which collapses onto the linear model. One instance of
+Euler data is induced by the concavex bundle Vand corresponds to the A-model;
+another instance, corresponding to the B-model, is given in terms of linear classes on
+the linear model. Their equality after certain mirror transformation s is established
+in [LLY97] in the case m= 0. When m>0, Lian-Liu-Yau give a general formulation
+of Euler data but do not explicitly identify the proper linear classes fo r comparison
+[LLY01].
+In section 3 we introduce the notion of ( u,v)-Euler data, which can be interpreted
+as a deformation of Lian-Liu-Yau’s Euler data (seen to be the (0 ,0) case). We show
+that a concavex bundle on Pnindeed induces ( u,v)-Euler data, and prove a key
+uniqueness lemma for the specific case of (0 ,1)-Euler data. Section 4 gives a natural
+extension of (0 ,0)-Euler data to (0 ,1)-Euler data and discusses certain invertible
+maps on (u,v)-Euler data known as mirror transformations. The goal is to prov e3
+Theorem 4.5, which gives an extension to (0 ,1)-Euler data with controlled growth
+in a certain equivariant weight critical to our uniqueness lemma. In Se ction 5 we
+combine the results of previous sections to prove Theorems 5.2 and 5.3, the one-point
+and two-point mirror theorems establishing the equality of Euler dat a induced by
+Vwith explicite linear data. When V=O(n+ 1) on Pnand working with the
+equivariant Euler class this agrees with the results of [Zin07]. Section 6 discusses the
+explicit recovery of Gromov-Witteninvariants in general, while Sectio n 7 gives several
+concrete examples.
+We note that integrality conjectures have been advanced connec ting the numbers
+produced in (1) with the enumerative geometry of X. In dimension 3, the Aspinwall-
+Morrison formula [AM93] conjectures integer invariants in the case g= 0,m= 0
+whenXis a Calabi-Yau after multiple coverings are properly taken into accou nt.
+This was generalized by Gopakumar and Vafa to a full integratlity con jecture in all
+genera for CY 3-folds. New integrality conjectures for Calabi-Yau spaces in genus
+1 for dimensions 4 and 5 and an extension of the Aspinwall-Morrison fo rmula ton
+dimensions have also recently been advanced [KP08] [PZ08]. To be sp ecific, in genus
+0 it is conjectured for Calabi-Yau n-folds that the invariants ηdefined via
+/summationdisplay
+β>0Kd(Hk1,...,Hkm)qd=/summationdisplay
+β>0ηd(Hk1,...,Hkm)∞/summationdisplay
+d=1d−3+mqdβ(3)
+are integers. The author has verified the integrality of these invar iants inm= 1 or
+m= 2 for a number of different Vin low degree by computer.
+Acknowledgments The author wishes to thank his advisor, Prof. Kefeng Liu, for
+his advice and support, as well as Prof. Jun Li for his interest in the project and
+assistance.
+2 Preliminaries
+2.1 Equivariant Cohomology and Localization
+We briefly summarize equivariant cohomology for T-spaces and the powerful tech-
+nique of localization. For a thorough discussion, the standard refe rence is [AB84].
+Let T be a compact Lie group. There exists a contractible space ET, unique
+up to homotopy equivalence, on which Tacts freely, and with Tclassified by the
+principle bundle ET→BT. IfXis a topological space with T-action, we define
+XT=X×TET, which is a bundle over the classifying space BTwith fiberX.
+Definition Theequivariant cohomology ofXis defined to be
+H∗
+T(X) =H∗(XT),4 L. Cherveny
+whereH∗(XT) is the ordinary cohomology of XT.
+We will always take Tto be an algebraic torus ( C∗)n, in which case one may check
+thatBT∼=(P∞)nandET=π∗
+1S⊗ ···⊗π∗
+nS, whereπiis projection of BTto the
+i-th copy of P∞whileS∼=O(−1) is the tautological line bundle on P∞. WhenXis
+a point we recover the ordinary cohomology of the classifying space BT:
+H∗
+T(pt) =H∗(BT) =C[λ1,...,λn],
+whereλi=c1(O(λi)) are the weights of the torus action.
+Many of the standard concepts in ordinary cohomology translate d irectly to the
+equivariant setting, for instance the notions of pullback for T-maps and pushforward
+by a proper T-map. If we denote inclusion of the fiber by iX:X ֒→XT, then
+in particular the equivariant pullback induces a map i∗
+X:H∗
+T(X)→H∗(X) called
+thenonequivariant limit . Furthermore, pullback by the contraction map X→pt
+realizesH∗
+T(X) as aH∗(BT)-module. The notions of equivariant vector bundle and
+equivariant characteristic classes are also readily defined.
+2.2 Atiyah-Bott Localization
+IfXbe a smooth manifold acted on by a torus T, theT-fixed components of Xare
+known to be a union of smooth submanifolds {Zj}(e.g. [Ive72]). We will denote
+inclusion of a fixed component into XbyiZj:Zj֒→Xand let the (equivariant)
+normal bundle of Zj⊆Xbe denoted by Nj. A key observation is that the equivariant
+Euler class Euler T(Nj)∈H∗
+T(Zj), which we may pushforward by the Gysin map
+iZj!:H∗
+T(Zj)→H∗
+T(X), is invertible in H∗
+T(X)⊗RTwhereRT∼=C(λ1,...,λn) is the
+field of fractions of H∗(BT).
+A fundamental result in the subject is that there is an isomorphism b etween the
+equivariant cohomology of Xproperly localized and that of its fixed components
+H∗
+T(X)⊗RT∼=/circleplusdisplay
+jH∗
+T(Zj)⊗RT,
+given explicitly by the map
+α→/summationdisplay
+ji∗
+Zjα
+Euler(NZj).
+In integrated form,
+/integraldisplay
+XTα=/summationdisplay
+Zj/integraldisplay
+(Zj)Ti∗
+Zjα
+Euler(NZj)
+for anyα∈H∗
+T(X)⊗RT.5
+More generally, let f:X→Ybe a proper equivariant map of T-spaces. IfF⊆Y
+is aT-fixed component of Y,E⊂f−1(F) fixed components of f−1(F), andf0:=f|E,
+thenfunctorial localization [LLY97] says for ω∈H∗
+T(X),
+f0∗/parenleftbiggi∗
+Eω
+eT(E/X)/parenrightbigg
+=i∗
+F(f∗ω)
+eT(F/Y).
+Localization techniques hold in the case where the spaces support s upport virtual
+fundamental classes. For instance, see the discussion in Ch. 27 of [HKK+03].
+2.3 Lian-Liu-Yau Euler Data
+We now give a brief overview of Lian, Liu, and Yau’s mirror principle, in pa rticular
+focusing on their notion of Euler data. Ultimately one wishes to estab lish the equiv-
+alence of two sequences of equivariant classes on projective spac es satisfying certain
+gluing relations on the fixed points. Such sequences are known as Eu ler data and are
+the primary item of interest in mirror principle. We refer the reader t o the original
+paper for further discussion [LLY97].
+LetMd=M0,0(Pn×P1,(d,1)). This space compactifies the space of all unmarked
+rational mapsof degree dtoPnandis oftencalled the nonlinear model (themapto Pn
+is the image of the the map given by each point in Md).Mdhas aG=C∗×T-action,
+whereT= (C∗)n+1, induced by the G-action
+(t,t0,...,tn)·([w0,w1],[z0,...,zn]) = ([t−1w0,w1],[t−1
+0z0,...,t−1
+nzn]) (4)
+onP1×Pn. Furthermore, there is a natural (equivariant) forgetful morp hismπto
+M0,0(Pn,d):
+π:Md→M0,0(Pn,d).
+Also define Nd=P(H0(P1,O(d))n+1). Since an unmarked degree drational map
+toPnmay be given up to scalar multiple as an ( n+1)-tuple of degree dhomogeneous
+polynomials in two variables with no common factor, Ndconstitutes a different com-
+pactification of unmarked rational curves to Pnof degreed, which we call the linear
+model. It too has a G-action, given by
+(t,t0,...,tn)·[α0(w0,w1),...,αn(w0,w1)] = [t−1
+0α0(t−1w0,w1),...,t−1
+nαn(t−1w0,w1)]
+where theαiare degreedhomogeneous polynomials in two variables.
+TheG-fixed components of Ndare isolated fixed points, which we denote by {pi,r}
+where 0≤i≤nand 0≤r≤d, representing the point [0 ,...,wr
+1wd−r
+0,...,0] where the
+nonzero term is in the i-th slot. We similarly label the T-fixed points of Pnby{qi}
+where 0≤i≤n. One may compute the equivariant cohomology ring of Ndto be6 L. Cherveny
+H∗
+G(Nd) =C[κ,λ0,...,λn,/planckover2pi1]/producttextn
+l=0/producttextd
+m=1(κ−λl−m/planckover2pi1)
+whereκis the equivariant hyperplane class, the λlare weights of the T-action, and
+/planckover2pi1is the weight of the final C∗-action [CK99]. Let RTH∗
+G(Nd) denote the localization
+ofH∗
+G(Nd) where polynomials in the λiare inverted.
+Definition A sequence of equivariant classes Q={Qd}withQd∈ RTH∗
+G(Nd) is
+called Ω-Euler data if there is an invertible class Q0= Ω∈ RTH∗
+T(Pn) soQsatisfy
+the relations
+i∗
+qi(Ω)·i∗
+pi,r(Qd) =i∗
+pi,r(Qr)·i∗
+pi,0(Qd−r)
+for each 0 ≤i≤n,d>0,and 0<r<d.
+We will casually refer to the data asEuler data when Ωis understood o runimportant.
+Because pullback to pi,rof a classω∈H∗
+G(Nd) is given by evaluation in the
+polynomial ring of the equivariant hyperplane class κat (λi+r/planckover2pi1), it is customary to
+writei∗
+pi,r(ω) asω(λi+r/planckover2pi1). This way, a sequence of equivariant classes Qis Ω-Euler
+data if
+Ω(λi)·Qd(λi+r/planckover2pi1) =Qr(λi+r/planckover2pi1)·Qd−r(λi). (5)
+We will also need the notion of linked Euler data:
+Definition Two Ω-Euler data PandQare said to be linkedifPd(λi) =Qd(λi) at
+/planckover2pi1= (λi−λj)/dfor alli,j/ne}ationslash=i, andd.
+Geometrically, Euler data are linked if they agree on multiple coverings of coor-
+dinate lines in Nd. The following uniqueness lemma for linked Euler data is proved
+in [LLY97]:
+Lemma 2.1 IfQandPare linked Ω-Euler data such that deg/planckover2pi1(Q(λi)−P(λi))≤
+(n+1)d−2, thenQ=P.
+A key result from [LLY97] is the existence of an equivariant collapsing map
+ϕ:M0,0(Pn×P1,(d,1))→Nd. (6)
+We maytake thepullback of anymultiplicative equivariant characteris tic classbof
+the bundles VdonM0,0(Pn,d) viaπand then pushforward via ϕto obtain a sequence
+ˆQ={ˆQd}withˆQd∈H∗
+G(Nd):
+ˆQd=ϕ∗(π∗(b(Vd)))∈H∗
+G(Nd),
+Unless stated otherwise, bwill always be the equivariant Euler class in our applica-
+tions, although the total Chern class is also interesting [LLY97]. As a special case of
+the more general Lemma 3.1, ˆQis Euler data. To be more precise, if V=V+⊕V−
+is the given concavex bundle, then ˆQturns out to bee(V+)
+e(V−)-Euler data.7
+3(u,v)-Euler Data
+For anym≥0,u+v=m, andki≥0, let
+/vectork= (k1,...,km)
+/vectork′= (k1,...,kv,0,...,0)
+/vectork′′= (0,...,0,kv+1,...,km)
+wherem,u,v, and thekiare nonnegative integers. Also let |/vectork|=/summationtextki.
+Definition A (u+v+ 1)-sequence Q={Qd,/vectork}withQd,/vectork∈ RTH∗
+G(Nd) is called
+(u,v,β,Ω)-Euler data if the following hold:
+1.Q0,/vectork=β|/vectork|Ω for some β∈H∗
+T(Pn) and all/vectork.
+2.Qsatisfies the Euler condition
+Ω(λi)·Qd,/vectork(λi+r/planckover2pi1) =Qr,/vectork′′(λi+r/planckover2pi1)·Qd−r,/vectork′(λi). (7)
+We will call the sequence at some fixed /vectorktheheight/vectorkdata and denote it by Q/vectork. As
+before, weoftensuppress βandΩwhenunimportant, casuallyreferringto( u,v)-Euler
+data.
+Example Lian-Liu-Yau’s Euler data coincides with (0 ,0,β,Ω)-Euler data. More
+generally, the height /vector0 part of any ( u,v)-Euler data obeys their gluing relation (5).
+However, height /vectork/ne}ationslash=/vector0 data is not quite Ω-Euler data nor β|/vectork|Ω-Euler data but rather
+a deformation in some sense, and we will say that QextendsQ/vector0. This constitutes a
+different generalization of Euler data from the one given in [LLY01]. Th e motivating
+idea is that we wish to work within a framework that allows one to tie mar ked data
+coming from a concavex bundle to unmarked data rather than stud y the problem for
+each number of markings independently.
+3.1(u,v)-Euler data and Concavex Bundles
+We will now show that a concavex bundle naturally induces ( u,v)-Euler data when
+defined through restriction to certain subspaces of an appropria te nonlinear moduli
+space.
+LetMu,v
+d⊂M0,u+v(Pn×P1,(d,1)) be the subspace of the Deligne-Mumford com-
+pactification of maps P1→Pn×P1of bi-degree ( d,1) withu+vmarked points such
+that the first umarked points are sent to [0 ,1] under projection to the P1factor and
+the finalvmarked points are sent to [1 ,0] under that projection. We call Mu,v
+dthe8 L. Cherveny
+nonlinear model and again give it a G-action induced by multiplication on the image
+as in (4). Note that M0,0
+dis the nonlinear moduli Mdused by Lian-Liu-Yau. The
+natural projection to M0,u+v(Pn,d) will again be denoted by π, suppressing the u,v
+dependence:
+π:Mu,v
+d−→M0,u+v(Pn,d)
+As in the unmarked case, there exists an equivariant collapsing map t oNd, still
+to be denoted ϕ:Mu,v
+d→Nd, defined as composition of the Lian-Liu-Yau collapsing
+map (6) from M0,0(Pn×P1,(d,1)) toNdwith the forgetful morphism(s) from Mu,v
+d
+toM0,0(Pn×P1,(d,1)) forgetting the markings.
+Fixu+v=m. For/vectork= (k1,...,km), let
+χV
+d,/vectork=π∗/parenleftBigg
+b(Vd)m/productdisplay
+j=1ev∗
+1(pkj)/parenrightBigg
+∈H∗
+G(Mu,v
+d),
+wherepis the equivariant hyperplane class on Pnandbis any multiplicative equiv-
+ariant characteristic class. Also let ˆQ={ˆQd,/vectork}:
+ˆQd,/vectork=ϕ∗(χV
+d,/vectork)∈H∗
+G(Nd).
+Them= 0 case is precisely the Euler data ˆQstudied in [LLY97]. Our immediate
+goal is to prove the following:
+Lemma 3.1 ˆQis(u,v,p,Ω)-Euler data, where Ω =e(V+)
+e(V−)andpis the equivariant
+hyperplane class on Pn.
+We first give some setup andprove a crucial gluing lemma (Lemma 3.2). Although
+theobstructionbundles VdonM0,m(Pn,d)inducedbyaconcavexbundle V=V+⊕V−
+onPnexist on different spaces, there are nontrivial relations on multiplica tive equiv-
+ariant characteristic classes connecting them. For any 0 <r<d, defineMi,j(r,d−r)
+by the pullback diagram
+Mi,j(r,d−r)p1✲M0,i+1(Pn,r)
+M0,j+1(Pn,d−r)p2
+❄
+evj+1✲Pnevi+1
+❄
+wherep1andp2areprojectionsand evkevaluationatthe k-thmarkedpoint. Wethink
+ofMi,j(r,d−r) as consisting of pairs of an ( i+1)-pointed stable map of degree rand9
+a (j+1)-pointed stable map of degree d−r, ((f1,C1,w1,...,wi+1),(f2,C2,z1,...,zj+1)),
+such thatf1(wi+1) =f2(zj+1) =q. We also have the diagram
+Mi,j(r,d−r)ψ✲M0,i+j(Pn,d)
+Pnev
+❄
+where the image of evisq, and the image of ψis the stabilization of the map formed
+by connecting C1atwi+1withC2atzj+1of degreedgiven byf|Cl=fl.
+In the case that r= 0 andi= 0 or 1,M0,i+1(Pn,r) is empty. To remedy this
+we replace M0,i+1(Pn,0) with a copy of Pn, and replace evi+1withidPn. This way,
+Mi,j(0,d)∼=M0,j+1(Pn,d). We letψforget the ( j+ 1)-th point when i= 0 and be
+the identity when i= 1. This way ψresults in an ( i+j)-point curve in all situations.
+The caser=dis similarly modified when j= 0 or 1.
+Lemma 3.2 (Gluing Lemma) Ifbis a multiplicativeequivariantcharacteristic class
+then the following relation on Mi,j(r,d−r)holds for 0<r<d:
+ev∗Ω·ψ∗b(Vd) =p∗
+1ev∗
+i+1b(Vr)·p∗
+2ev∗
+j+1b(Vd−r)
+whereΩ =b(V+)
+b(V−).
+ProofSuppose((f1,C1,w1,...,wi+1),(f2,C2,z1,...,zj+1))mapto(f,C,w 1,...,wi,z1,...,zj)
+underψ. This gives rise to the exact sequence
+0→f∗V→f∗
+1V⊕f∗
+2V→V|q→0
+In the case that Vis convex, passing to cohomology gives the exact sequence
+0→H0(C,f∗V)→H0(f∗
+1V)⊕H0(f∗
+2V)→V|q→0,
+from which the lemma follows easily. Likewise, if Vis concave we have the exact
+sequence
+0→V|q→H1(C,f∗V)→H1(f∗
+1V)⊕H1(f∗
+2V)→0,
+and again the corresponding gluing lemma follows. The general conca vex case is then
+an easy combination of the two cases.
+We now prove Lemma 3.1:10 L. Cherveny
+ProofFirst observe that ˆQd,/vectorkisautomatically polynomial in /planckover2pi1, asit is anequivariant
+class onNd. We show that ˆQsatisfies (7) by pulling the calculation back to Md:
+ˆQd,/vectork(λi+r/planckover2pi1) =/integraldisplay
+(Nd)Gφpi,rˆQd,/vectork=/integraldisplay
+(Mu,v
+d)Gϕ∗(φpi,r)χV
+d,/vectork, (8)
+whereφpi,r=/producttext
+(j,s)/ne}ationslash=(i,r)(κ−(λj+s/planckover2pi1)) is the equivariant Poincar´ e dual to the fixed
+pointpi,r. We wish to calculate (8) via localization and so must understand the
+G-fixed components. Each fixed component MΓ⊆Mdis labeled by a decorated
+graph Γ in the usual way [CK99] [HKK+03], although we will not directly need this
+description. Furthermore, we must have ϕ(MΓ) =pi,r.
+We first consider the case 0 < r < d . To obtain a better description of these
+G-fixed components and in particular their normal bundles inside of Mu,v
+d, we build
+a variation ψof the map ψgiven above as follows: for each point of Mu,v(r,d−r)
+represented by the pair (( f1,C1,w1,...,wu+1),(f2,C2,z1,...,zv+1)) withf1(wu+1) =
+q=f2(zv+1), letC=C0∪C1∪C2whereC0∼=P1, with [1,0]∈C0glued to
+wu+1∈C1, and [0,1]∈C0glued tozv+1∈C2. Then define f:C→P1×Pnby
+f|C0(z) = (z,q)
+f|C1(z) = ([1,0],f1(z))
+f|C2(z) = ([0,1],f2(z))
+This induces a map Mu,v(r,d−r)→M0,u+v(Pn×P1,(d,1)). Moreover, the image is
+contained in Mu,v
+dand so defines a map ¯ψ
+¯ψ:Mu,v(r,d−r)֒→Mu,v
+d.
+Fix 0≤i≤nfor the moment and let {Fj
+k}denote the set of T-fixed components
+ofM0,j(Pn,k) such that the j-th marked point is sent to qi∈Pn. In the present case,
+it is clear that F1
+r×F2
+d−r⊆M(r,d−r), and we identify this subset with its image
+under¯ψ, writingF1
+r×F2
+d−r⊆Mu,v
+d. It is not hard to see that all the fixed components
+MΓofMu,v
+dwithϕ(MΓ) =pi,rarise in this way. Hence by functorial localization, (8)
+becomes
+ˆQd,/vectork(λi+r/planckover2pi1) =/summationdisplay
+{Fu+1
+r×Fv+1
+d−r}/integraldisplay
+(Fu+1
+r×Fv+1
+d−r)Gi∗
+Γ(ϕ∗(φpi,r)χV
+d,/vectork)
+EulerG(NFu+1
+r×Fv+1
+d−r).
+Wewant toidentify theclassEuler G(N(Fu+1
+r×Fv+1
+d−r))intheequivariant K-theory.
+By considering the various inclusions at hand it is straightforward to find
+N(Fu+1
+r×Fv+1
+d−r) =N(Fu+1
+r)+N(Fv+1
+d−r)−2TqiPn+TqiPn+N(¯ψ)−N(J),11
+whereJis the inclusion map J:Mu,v
+d֒→M0,u+v(Pn×P,(d,1)). Thus, omitting
+pullbacks for simplicity,
+EulerG(N(Fu+1
+r×Fv+1
+d−r)) = Euler G(N(¯ψ))EulerT(N(Fu+1
+r))EulerT(N(Fv+1
+d−r))
+EulerG(N(J))EulerT(TqiPn).
+These pieces may be identified as
+EulerT(TqiPn) =/productdisplay
+j/ne}ationslash=i(λi−λj)
+EulerG(N(J))) = (−1)v/planckover2pi1u+v
+EulerG(N(¯ψ)) =−/planckover2pi1−2
+(−/planckover2pi1−cG
+1(Ld−r,v+1))(/planckover2pi1−cG
+1(Lr,u+1))
+whereLr,jis the line bundle on M0,j(Pn,d) given by the cotangent line to the j-th
+marked point on each curve. Furthermore, by Lemma 3.2, on Fu+1
+r×Fv+1
+d−r
+Ω(λi)b(Vd) =b(Vr)b(Vd−r)
+Putting all this together, we have:
+ˆQd,/vectork(λi+r/planckover2pi1) =/integraldisplay
+(Mu,v
+d)Gϕ∗(φi,r)χV
+d,/vectork
+= (−1)v/planckover2pi1u+v−2(Ω(λi))−1i∗
+pi,r(φi,r)/productdisplay
+j/ne}ationslash=i(λi−λj)×
+/summationdisplay
+{Fu+1
+r}/integraldisplay
+(Fu+1
+r)Tπ∗
+u+1(b(Vr)/producttextu
+j=1ev∗
+jpkj+v)
+EulerT(N(Fu+1
+r))(/planckover2pi1−cG
+1(Lr,u+1))×
+/summationdisplay
+{Fv+1
+d−r}/integraldisplay
+(Fv+1
+d−r)Tπ∗
+v+1(b(Vd−r)/producttextv
+j=1ev∗
+jpkj)
+EulerT(N(Fv+1
+d−r))(/planckover2pi1+cG
+1(Ld−r,v+1)).(9)
+A similar analysis for r= 0 yields
+ˆQd,/vectork(λi) = (−1)vi∗
+pi,0(φi,0)/planckover2pi1v−1u/productdisplay
+j=1λkv+j
+i/summationdisplay
+{Fv+1
+d}/integraldisplay
+(Fv+1
+d)Tπ∗
+v+1(b(Vd)/producttextv
+j=1ev∗
+jpkj)
+EulerG(N(Fv+1
+d))(/planckover2pi1+cG
+1(Ld,v+1))
+(10)
+so that12 L. Cherveny
+ˆQd−r,/vectork′(λi) = (−1)vi∗
+pi,0(φi,0)/planckover2pi1v−1/summationdisplay
+{Fv+1
+d−r}/integraldisplay
+(Fv+1
+d−r)Tπ∗
+v+1(b(Vd−r)/producttextv
+j=1ev∗
+jpkj)
+EulerG(N(Fv+1
+d−r))(/planckover2pi1+cG
+1(Ld−r,v+1)).
+(11)
+Likewise, for r=dwe find
+ˆQr,/vectork′′(λi+r/planckover2pi1) =i∗
+pi,r(φi,r)/planckover2pi1u−1/summationdisplay
+{Fu+1
+r}/integraldisplay
+(Fu+1
+r)Tπ∗
+u+1(b(Vr)/producttextu
+j=1ev∗
+jpkj+v)
+EulerG(N(Fu+1r))(/planckover2pi1−cG
+1(Lr,u+1)),
+(12)
+Combining (9), (11), and (12), it is then straightforward to check that that ˆQsatisfies
+(7) and thus is ( u,v,p,Ω)-Euler data. (Note that in (9), i∗
+pi,r(φi,r) is onNdbut in (11)
+i∗
+pi,0(φi,0) is onNrand in (12) i∗
+pi,r(φi,r) onNd−r.)
+3.2 Uniqueness for (0,1)-Euler data
+In this section we will restrict ourselves to studying a uniqueness lem ma for (0,1)-
+Euler data. The discussion has a natural analog for (1 ,0)-Euler data, which we will
+not state, that will only be used briefly in the discussion on two markin gs.
+For notational simplicity, we will refer to the height using krather than /vectork= (k1).
+Lemma 3.3 (Uniqueness Lemma) IfQ={Qd,k}andP={Pd,k}are(0,1,β,Ω)-
+Euler data such that Q0=P0and deg /planckover2pi1(Qd,k(λi)−Pd,k(λi))≤(n+1)d−1for alli,
+k, andd, thenQ=P.
+ProofWe will prove Qd,k=Pd,kfor eachkby induction on d. Notice that Q0,k=
+βkΩ =P0,kby assumption. Suppose that for any fixed k >0,Qd′,k=Pd′,kfor
+0< d′< d. Since the intersection pairing is nondegenerate, it will be sufficient t o
+show that
+/integraldisplay
+(Nd)Gκs(Qd,k−Pd,k) = 0 for all s≥0,
+whereκis the equivariant hyperplane class on Nd. We compute this integral by
+localization:13
+/integraldisplay
+(Nd)Gκs(Qd,k−Pd,k) =/summationdisplay
+0≤i≤n/summationdisplay
+0≤r≤d(λi+r/planckover2pi1)si∗
+pi,r(Qd,k−Pd,k)
+EulerG(Npi,r)
+=n/summationdisplay
+i=0/bracketleftbiggλs
+i(Qd,k(λi)−Pd,k(λi))
+EulerG(Npi,0)+(λi+d/planckover2pi1)s(Qd,k(λi+d/planckover2pi1)−Pd,k(λi+d/planckover2pi1))
+EulerG(Npi,d)/bracketrightbigg
+=n/summationdisplay
+i=0λs
+i(Qd,k(λi)−Pd,k(λi))
+EulerG(Npi,0)
+The second line follows from the first because Qsatisfies the Euler condition,
+allowingQd,k(λi+r/planckover2pi1) to be expressed in terms of Q1,k,...,Qd−1,kandQr,0whenever
+0<r<d, and likewise for Pd,k(λi+r/planckover2pi1). By the inductive hypothesis these agree, and
+theonlytermsremaininginthedifference arethe r= 0andr=dcases. Furthermore,
+using the assumption Q0=P0whenr=d,
+Qd,k(λi+d/planckover2pi1) = Ω(λi)−1Qd,0(λi+d/planckover2pi1)Q0,k(λi)
+= Ω(λi)−1Pd,0(λi+d/planckover2pi1)P0,k(λi)
+=Pd,k(λi+d/planckover2pi1),
+so this difference vanishes as well giving the final line. Computing the n ormal bundle
+in the final line, we find that
+/integraldisplay
+(Nd)Gκs(Qd,k−Pd,k) =n/summationdisplay
+i=0λs
+iAi(/planckover2pi1)
+/planckover2pi1d(13)
+where
+Ai(/planckover2pi1) =(−1)d
+d!/producttext
+j/ne}ationslash=i(λi−λj)Qd,k(λi)−Pd,k(λi)/producttext
+j/ne}ationslash=i/producttextd
+m=1(λi−(λj+m/planckover2pi1)).
+We claim that Aiis in fact polynomial in /planckover2pi1. It is easy to see that (0 ,1,β,Ω)-Euler
+data automatically satisfies the self-linking condition thatQd,k(λi) =β(λj)kQd,0(λi)
+at/planckover2pi1= (λi−λj)/dfor alli,j/ne}ationslash=i,k, andd. It follows that at /planckover2pi1= (λi−λj)/d
+Qd,k(λi)−Pd,k(λi) =β(λj)k(Qd,0(λi)−Pd,0(λi)) = 0,
+which cancels out some zeros of the denominator. The other zeros at/planckover2pi1= (λi−λj)/s
+where 0< s < d cancel as well again by repeated use of the self-linking condition
+(geometrically if Qd,kandQs,kare both multiples of Qd,0on covers of coordinate lines14 L. Cherveny
+then they are multiples of each other). We have shown that Aiis indeed a polynomial
+in/planckover2pi1. However, the left side of (13) is naturally a polynomial in /planckover2pi1. The degree bound
+implies that
+deg/planckover2pi1Ai≤(n+1)d−1−nd=d−1
+Since/planckover2pi1ddividesAiwe conclude that Ai, and hence the left hand side of (13), always
+vanish. The lemma then follows.
+4 Hypergeometric Data
+Throughoutthissectionandthenext, wewilltake β∈H∗
+T(Pn)tobep,theequivariant
+hyperplane class. The goal is first to extend (0 ,0)-Euler data to (0 ,1)-Euler data in
+a natural way, and then to transform (0 ,1)-Euler data into “linked” data so certain
+hypergeometric series associated to the data obey simple relations . These notions
+will be made precise shortly. Ultimately we prove Theorem 4.5 showing t hat this can
+be done in a way that controls the growth of /planckover2pi1.
+4.1 An Extension Lemma
+LetS0denote the set of all sequences B={Bd}such thatBd∈H∗
+T(Pn)(/planckover2pi1) for all
+d, where/planckover2pi1is now a formal parameter. The hypergeometric function associat ed to
+B∈ S0is a formal function taking values in H∗
+T(Pn)⊗C(/planckover2pi1) given by
+HG[B](t) =e−pt//planckover2pi1/bracketleftBigg
+B0+∞/summationdisplay
+d=1edt Bd/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1)/bracketrightBigg
+.
+LetId:Pn֒→Ndbe the equivariant embedding given by
+[a0,...,an]/mapsto→[a0wd
+1,...,anwd
+1].
+Also let Sbe the set of all sequences Q={Qd}of equivariant classes with Qd∈
+RTH∗
+G(Nd). We then have a natural map I:S → S 0, defined via ( I(Q))d=I∗
+d(Qd)
+for everyd. With this notation, the hypergeometric series associated to the h eight/vectork
+sequence in ( u,v,p,Ω)-Euler data is given by
+HG[I(Q/vectork)](t) =e−pt//planckover2pi1/bracketleftBigg
+Ωpk+∞/summationdisplay
+d=1edtI∗
+d(Qd,/vectork)
+/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1)/bracketrightBigg
+.(14)
+For the remainder of this section we will deal only with (0 ,1)-Euler data, and
+again adopt the notation Qkin place ofQ/vectorkfor the data at a particular height.15
+Lemma 4.1 (Extension Lemma) There is a map ρ:S → Ssuch thatρgives a
+natural extension of (0,0,−,Ω)-Euler data Q0∈ Sto(0,1,p,Ω)-Euler data Qin such
+a way that
+Qk=ρk(Q0)
+for allk>0and with the property that
+HG[I(Qk)](t) =/braceleftbigg
+−/planckover2pi1∂
+∂t/bracerightbiggk
+HG[I(Q0)](t). (15)
+ProofGivenQ0={Q0,k}, defineρk(Q0) ={Qd,k}via
+Qd,k(λi+r/planckover2pi1) = (λi−(d−r)/planckover2pi1)kQd,0(λi+r/planckover2pi1).
+That this extension satisfies 7 is easily checked, and as the restrict ion at each fixed
+point ofNdis polynomial in /planckover2pi1so is the class Qd,kitself. Furthermore, i∗
+qiI∗
+d(Qd,k) =
+Qd,k(λi) = (λi−d/planckover2pi1)kQd,0(λi) for every fixed point qi∈Pn, from which we conclude
+I∗
+d(Qd,k) = (p−d/planckover2pi1)kI∗
+d(Qd,0).
+Equation (15) is now verified by straightforward power series manip ulation.
+4.2 Mirror Transformations
+Lemma4.1implieswemayalwaysextendsomeinitial(0 ,0,Ω)-Eulerdatato(0 ,1,p,Ω)-
+Euler datainasimple way. However, hypergeometric dataassociate d totheextension
+will behave badly in that it grows in powers of /planckover2pi1as the height kincreases. It turns
+out the correct way to control the hypergeometric data is trans form Euler data into
+new “linked” data. More generally, one may wish to manipulate ( u,v)-Euler data.
+We now give some natural extensions to definitions found in [LLY97]:
+Definition Two (u,v,β,Ω)-Euler data PandQare called linkedifQ0andP0are
+linked (0,0,Ω)-Eulerdata, meaning that Qd,0(λi)andPd,0(λi) agreeat /planckover2pi1= (λi−λj)/d
+for alli,j, andd.
+LetAΩdenote the set of all ( u,v,p,Ω)-Euler data.
+Definition Amirror transformation on (u,v,β,Ω)-Euler data is an invertible map
+µ:AΩ→ AΩso thatµ(Q) is linked to Qfor allQ∈ AΩ. We will sometimes deal
+with mirror transformations that alter the Euler data at one partic ular height /vectorkwhile
+fixing all other heights; in this case, we call µamirror transformation of height /vectork.
+Clearly the composition of mirror transformations of different heigh ts will be a mirror
+transformation.16 L. Cherveny
+Lemmas4.2,4.3,and4.4belowdescribespecificmirrortransformatio nson(u,v,p,Ω)-
+Euler data that will be needed to establish Theorem 4.5 or in the examp les. The
+existence of a mirror transformation described at each height by a relation on hy-
+pergeometric data is established by first showing that there is a well- defined image
+sequence in S0satisfying the relation, then that this sequence lifts well to a seque nce
+inSso the lift satisfies the Euler condition and is polynomial in /planckover2pi1, and of course is
+linked to the original data.
+If one ignores the higher height data, Lemmas 4.2 and 4.3 exactly red uce to
+Lemma 2.15 of [LLY97]. Their proofs are essentially the same argumen t with small
+modifications; we produce those arguments for Lemma 4.2; Lemma 4 .3 is simpler and
+left as an exercise.
+Lemma 4.2 There exists a mirror transformation µ:AΩ→ AΩso that for any
+g∈etRT[[et]],
+HG[I(µ(Q)/vectork)](t) =HG[I(Q/vectork)](t+g)
+for allQ∈ AΩandallheights/vectork.
+Lemma 4.3 There exists a mirror transformation ν:AΩ→ AΩofarbitrary height
+/vectorksuch that for any f∈etRT[[et]],
+HG[I(ν(Q)/vectork)](t) =ef//planckover2pi1HG[I(Q/vectork)](t).
+ProofTo prove Lemma 4.2, let I(Q/vectork) =B/vectork∈ S0so that
+HG[B/vectork](t+g) =e−pt//planckover2pi1e−pg//planckover2pi1∞/summationdisplay
+d=0edtedgBd,/vectork/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1).
+Expandedg=/summationtext
+s≥0gd,sestande−pg//planckover2pi1=/summationtext
+s≥0g′
+sest, wheregd,s∈ RTandg′
+s∈
+RT[p//planckover2pi1]. It is straightforward to find that
+˜Bd,/vectork=B′
+d,/vectork+d−1/summationdisplay
+r=0g′
+d−rB′
+r,/vectorkn/productdisplay
+l=0d/productdisplay
+m=r+1(p−λl−m/planckover2pi1),
+where
+B′
+d,/vectork=Bd,/vectork+d−1/summationdisplay
+r=0gr,d−rBr,/vectorkn/productdisplay
+l=0d/productdisplay
+m=r+1(p−λl−m/planckover2pi1),
+defines a unique ˜B/vectork={˜Bd,/vectork}such that
+HG[˜B/vectork](t) =HG[B/vectork](t+g).17
+For each/vectork, we want to lift ˜B/vectorkto a sequence ˜Q/vectork={˜Qd,/vectork}with˜Qd,/vectork∈ RTH∗
+G(Nd)
+in such a way that ˜Q/vectorkis the height /vectorkdata in (u,v,p,Ω)-Euler data ˜Qlinked toQ.
+We define the Lagrange map Lby
+(L(B/vectork))d(λi+r/planckover2pi1) = Ω(λi)−1Br,0(λi)Bd−r,/vectork(λi) (16)
+for eachk≥0. Here overline denotes the conjugate map on Pnsending/planckover2pi1to−/planckover2pi1(and
+more generally on Nd) studied in [LLY97]. The relevant property of the conjugate
+map here is that Qd,0(λi+d/planckover2pi1) =Qd,0(λi), from which one sees that the image of the
+Lagrange map satisfies the Euler condition 7. One may check that I ◦L= idS0, and
+thatL◦ I= idSwhen restricted to Euler data. Define ˜Qkto beL(˜Bk). It is clear
+from the definition of ˜Bd,/vectorkthat˜Qd,0(λi) =˜Bd,0(λi) agrees with Qd,0(λi) =Bd,0(λi) at
+/planckover2pi1= (λi−λj)/d,so˜QislinkedtoQ. Wenowneedonlytoshowthat ˜Qd,k∈ RTH∗
+G(Nd).
+Broadly, if Qis (u,v,p,Ω)-Euler data, then multiplying both sides of (16) by the
+identity
+edτ e(λi+r/planckover2pi1)(t−τ)//planckover2pi1
+/producttextn
+j=0/producttextd
+m=0,(j,m)/ne}ationslash=(i,r)(λi+r/planckover2pi1−λj−m/planckover2pi1)=1/producttext
+j/ne}ationslash=i(λi−λj)×ert
+/producttextn
+j=0/producttextr
+m=1(λi−λj+m/planckover2pi1)
+×eλi(t−τ)//planckover2pi1 e(d−r)τ
+/producttextn
+j=0/producttextr
+m=1(λi−λj−m/planckover2pi1)
+and summing over 0 ≤i≤n, 0≤r≤d, and then finally over d= 0 to∞yields (via
+localization) the identity
+/summationdisplay
+d≥0edτ/integraldisplay
+(Nd)Geκ(t−τ)//planckover2pi1Qd,/vectork=/integraldisplay
+(Pn)T(Ω−1HG[B0](t)HG[B/vectork](τ)) (17)
+withB0=I(Q0) andB/vectork=I(Q/vectork). Equation (17) is generally useful in establishing
+that lifts under the Lagrange map are polynomial in /planckover2pi1.
+In the present case, 17 gives
+/summationdisplay
+d≥0edτ/integraldisplay
+(Nd)Geκ(t−τ)//planckover2pi1˜Qd,/vectork
+=/integraldisplay
+(Pn)TΩ−1HG[I(µ(Q)0)](t)HG[I(µ(Q)/vectork)](τ)
+=/integraldisplay
+(Pn)TΩ−1HG[I(Q0)](t+g(et))HG[I(Q/vectork)](τ+g(eτ))
+=/summationdisplay
+d≥0edτ/integraldisplay
+(Nd)Geκ(t+¯g(et)−τ−g(eτ))//planckover2pi1Qd,/vectork18 L. Cherveny
+Decompose g=g++g−, where ¯g±=±g±, and setq=eτandζ= (t−τ)//planckover2pi1, so
+thatg+(qeζ/planckover2pi1)−g+(q)∈/planckover2pi1RT[[q,ζ]]. Moreover, g−(q),g−(qeζ/planckover2pi1)∈/planckover2pi1RT[[q,ζ]] naturally.
+SinceQd,/vectork∈ RTH∗
+G(Nd), the final summation above lies in RT[[q,ζ]], and so too
+much the initial summation. It follows that
+/integraldisplay
+(Nd)Gκs˜Qd,/vectork∈ RT
+for alls≥0. A priori ˜Qd,/vectork=aNκN+...+a0, withai∈ RG, and pairing this
+expression with various powers of κand integrating over ( Nd)Ggivesai∈ RT. Thus
+˜Qd,/vectork∈ RTH∗
+G(Nd) is (u,v)-Euler data and we may define the mirror transformation
+µviaµ(Q) =˜Q.
+The proof of Lemma 4.3 is carried out by a similar argument.
+Thesetwolemmasdescribemirrortransformationsthatgeneralize thosein[LLY97]
+used to manipulate the first two terms in the /planckover2pi1−1-expansion of hypergeometric data
+so a uniqueness result applies. We will need a new class of mirror trans formations
+that allow one to further manipulate the degree 0 term of the /planckover2pi1−1-expansion of higher
+derivatives of hypergeometric data in such a way as to eliminate all th e purely equiv-
+ariant terms that appear after applying Lemma 4.1. The following lemm a provides
+that new class of transformations:
+Lemma 4.4 There exists a mirror transformation η:AΩ→ AΩof arbitrary height
+/vectorkso that for any f∈etRT[[et]]and/vectork′,
+HG[I(η(Q)/vectork)](t) =HG[I(Q/vectork)](t)+f·HG[I(Q/vectork′)](t)
+for allQ∈ AΩ.
+ProofWriteI(Q/vectork) =B/vectork={Bd,/vectork}and likewise I(Q/vectork′) =B/vectork′={Bd,/vectork′}. We need to
+show that there is a sequence ˜B/vectork={˜Bd,/vectork} ∈ S0so that
+HG[˜B/vectork](t) =HG[B/vectork](t)+f·HG[I(Q/vectork′)](t)
+and that this sequence lifts well under the Lagrange map to linked ( u,v)-Euler data.
+It is straightforward to calculate that if we write f(t) =/summationtext
+s≥1fs(t)est, the desired
+sequence ˜Bk∈ S0is given by
+˜Bd,/vectork=Bd,/vectork+d/summationdisplay
+s=1fs(t)Bd−s,/vectork′n/productdisplay
+l=0d/productdisplay
+m=d−s+1(p−λl−m/planckover2pi1).
+The Lagrange lift L(˜B/vectork), given as before by
+(L(˜B/vectork))d(λi+r/planckover2pi1) = Ω(λi)−1Br,0(λi)˜Bd−r,/vectork(λi)19
+will satisfy the Euler condition (7) by design, and will be linked to the or iginal data as
+˜Bd,/vectork(λi)andBd,/vectork(λi)agreeat /planckover2pi1= (λi−λj)/dforalli,j, andd. Defineη(Q)/vectork=L(B/vectork).
+We need to check that that ν(Q)/vectorkis in fact in RTH∗
+G(Nd). Applying (17), we have
+/summationdisplay
+d≥0edτ/integraldisplay
+(Nd)Geκ(t−τ)//planckover2pi1(η(Q)d,/vectork)
+=/integraldisplay
+(Pn)TΩ−1HG[I(ν(Q)0)](t)HG[I(η(Q)/vectork)](τ)
+=/integraldisplay
+(Pn)TΩ−1HG[I(Q0)](t)/parenleftbig
+HG[I(Q/vectork)](τ)+f·HG[I(Q/vectork′)](τ)/parenrightbig
+=/summationdisplay
+d≥0edτ/integraldisplay
+(Nd)Geκ(t−τ)//planckover2pi1/bracketleftBig
+Qd,/vectork+f·Qd,/vectork′/bracketrightBig
+BothQd,/vectorkandQd,/vectork′are inRTH∗
+G(Nd) asQis Euler data. An argument similar to
+that of Lemma 4.2 shows that ν(Q)d,/vectorkmust also be in RTH∗
+G(Nd).
+We now have all the ingredients necessary to prove:
+Theorem 4.5 Suppose that P0is(0,0,Ω)-Euler data whose hypergeometric data has
+series expansion in /planckover2pi1−1of the form
+HG[I(P0)](t) = Ω/bracketleftBigg
+1+∞/summationdisplay
+q=1q/summationdisplay
+r=0(−1)qyr
+0,q(t)pq−r/planckover2pi1−q/bracketrightBigg
+,
+whereyr
+0,q(t)are degree-rsymmetric polynomials in {λi}. ThenP0may be extended
+to(0,1,p,Ω)-Euler data Pin such a way that the hypergeometric data at each height
+khas series expansion of the form
+HG[I(Pk)](t) = Ωpk/bracketleftBigg
+1+∞/summationdisplay
+q=1q+k/summationdisplay
+r=0(−1)qyr
+k,q(t)pq−r/planckover2pi1−q/bracketrightBigg
+,
+whereyr
+k,q(t)are degree-rsymmetric polynomials in {λi}and they0
+i,q(t)are recursively
+defined by
+y0
+i,q(t) =y0
+i−1,q+1′(t)
+y0
+i−1,1′(t)
+for1≤i≤q.20 L. Cherveny
+ProofWe construct the (0 ,1)-Euler data Pby induction on height. The existence of
+height 0 data satisfying the theorem is presupposed. Suppose tha t for all 0 ≤k′≤k,
+Phas been constructed with hypergeometric data
+HG[I(Pk)](t) = Ωpk/bracketleftBigg
+1+∞/summationdisplay
+q=1q+k/summationdisplay
+r=0(−1)qyr
+k,q(t)pq−r/planckover2pi1−q/bracketrightBigg
+,
+and we wish to construct the height k+1 data. The operator ρ:S → Sfrom Lemma
+4.1 onPkgives height k+1 dataρ(Pk) such that
+HG[I(ρ(Pk))](t) =−/planckover2pi1∂
+∂t(HG[I(Pk)](t))
+= Ωpk/bracketleftBiggk+1/summationdisplay
+r=0yr
+k,1′(t)p1−r+∞/summationdisplay
+q=2q+k/summationdisplay
+r=0(−1)q+1yr
+k,q′(t)pq−r/planckover2pi1−q+1/bracketrightBigg
+.
+Since Ω is assumed to be invertible, we see from the definition for hype rgeometric
+data thaty0
+k,q(t)−tq
+q!∈etRT[[et]] and in particular y0
+k,1(t)−t∈etRT[[et]], which
+impliesy0
+k,1′(t)−1∈etRT[[et]]. Thusf=/planckover2pi1ln(y0
+k,1′(t))∈etRT[[et]] and we may use
+Lemma 4.3 to obtain a mirror transformation νsuch that
+HG[I(ν(ρ(Pk)))](t) =e−f//planckover2pi1HG[I(ρ(Pk))](t)
+= Ω/bracketleftBigg
+pk+1+k+1/summationdisplay
+r=1yr
+k,1′(t)
+y0
+k,1′(t)pk+1−r/bracketrightBigg
++Ωpk∞/summationdisplay
+q=2q+k/summationdisplay
+r=0(−1)q+1yr
+k,q′(t)
+y0
+k,1′(t)pq−r/planckover2pi1−q+1.
+The previous observation also implies that−yr
+k,1′(t)
+y0
+k,1′(t)∈etRT[[et]] for allr= 1...k+1.
+We may then use Lemma 4.4 (repeatedly) to obtain another mirror tr ansformation η
+such that
+HG[I(η(ν(ρ(Pk))))](t) =HG[I(ν(ρ(Pk)))](t)+k+1/summationdisplay
+r=1−yr
+k,1′(t)
+y0
+k,1′(t)HG[I(Pk+1−r)](t)
+= Ωpk+1+Ωpk∞/summationdisplay
+q=2q+k/summationdisplay
+r=0(−1)q+1yr
+k,q′(t)
+0
+k,1′(t)pq−r/planckover2pi1−q+1
++k+1/summationdisplay
+r=1Ωpk+1−r−yr
+k,1′(t)
+y0
+k,1′(t)∞/summationdisplay
+q=1q+k+1−r/summationdisplay
+s=0(−1)qys
+k+1−r,q(t)pq−s/planckover2pi1−q
+= Ωpk+1/bracketleftBigg
+1+∞/summationdisplay
+q=1q+k+1/summationdisplay
+r=0(−1)qyr
+k+1,q(t)pq−r/planckover2pi1−q/bracketrightBigg21
+for some functions yr
+k+1,q(t). It is clear that yr
+k+1,q(t) is symmetric of degree rin{λ},
+and thaty0
+k+1,q(t) is given by
+y0
+k+1,q(t) =y0
+i−1,q+1′(t)
+y0
+i−1,1′(t).
+Thus we define Pk+1to beη(ν(ρ(Pk))) and the theorem is proved.
+5 Mirror Theorems
+In this section we combine the results of previous sections to estab lish mirror theo-
+rems, which express the Euler data coming from a concavex bundle in terms of linear
+classes onNdin the case of one and two marked points. We first show that the
+(0,1)-Euler data induced by a concavex bundle Vand the extension constructed in
+Theorem 4.5 are the same. The case of two markings is solved by gluing together the
+results of the one point case.
+5.1 One-point Mirror Theorem
+The mirror theorems in the case of one marked point will follow from th e following
+lemma, which explicitly identifies I∗
+d(ˆQ/vectork).
+Lemma 5.1 Letφd=/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1)∈H∗
+T(Pn)denote the denominator
+of the coefficient of edtin the hypergeometric series HG[I(ˆQ/vectork)](t). Then:
+1. Ifπj,evjare as before then
+I∗
+d(ˆQd,/vectork) = (−1)vφdp|/vectork′′|/planckover2pi1v−1evv+1∗/parenleftBigg
+π∗
+v+1(EulerT(Vd)/producttextv
+j=1ev∗
+jpkj)
+/planckover2pi1+cT
+1(Ld,v+1)/parenrightBigg
+2.ˆQsatisfies deg/planckover2pi1I∗
+d(ˆQd,/vectork)≤(n+1)d+v−2for every/vectork.
+ProofTo prove the first part of the lemma, we show that both sides of the equation
+have equal localizations at qi∈Pnfor all 0≤i≤n. SinceId(qi) =pi,0∈Nd, we have
+i∗
+qi(I∗
+d(ˆQd,/vectork)) =i∗
+pi,0(ˆQd,/vectork) =ˆQd,/vectork(λi).
+From the proof of Lemma 3.1, we know
+ˆQd,/vectork(λi) = (−1)vi∗
+pi,0(φi,0)/planckover2pi1v−1u/productdisplay
+j=1λkj+v
+i/summationdisplay
+{Fv+1
+d}/integraldisplay
+(Fv+1
+d)Tπ∗
+v+1(EulerT(Vd)/producttextv
+j=1ev∗
+jpkj)
+EulerT(N(Fv+1
+d))(/planckover2pi1+cT
+1(Ld,v+1))22 L. Cherveny
+It follows from the definition of φdandφqi=/producttext
+j/ne}ationslash=i(p−λj) that
+i∗
+pi,0(φpi,0) =i∗
+qi(φqi)i∗
+qi(φd),
+which, when combined with the above expression for ˆQd,k(λi), gives
+ˆQd,k(λi) = (−1)vi∗
+qi(φqi)i∗
+qi(φd)/planckover2pi1v−1u/productdisplay
+j=1λkj+v
+i/summationdisplay
+{Fv+1
+d}/integraldisplay
+(Fv+1
+d)Tπ∗
+v+1(EulerT(Vd)/producttextv
+j=1ev∗
+jpkj)
+EulerT(N(Fv+1
+d))(/planckover2pi1+cT
+1(Ld,v+1)),
+where the summation is over fixed components Fv+1
+d⊂M0,v+1(Pn,d) such that the
+final marked point is mapped to qi∈Pn.
+On the other hand, if we let
+αd=π∗
+v+1(EulerT(Vd)/producttextv
+j=1pkj)
+/planckover2pi1+cT
+1(Ld,v+1),
+then
+i∗
+qi((−1)v/planckover2pi1v−1p|/vectork′′|φdevv+1∗(αd)) = (−1)v/planckover2pi1v−1/integraldisplay
+(Pn)Tφqip|/vectork′|φdevv+1!(αd)
+= (−1)v/planckover2pi1v−1/integraldisplay
+(M0,v+1(Pn,d))Tev∗
+v+1(p|/vectork′|φqiφd)αd
+= (−1)v/planckover2pi1v−1i∗
+qi(φqi)i∗
+qi(φd)u/productdisplay
+j=1λkj+v
+i/integraldisplay
+(M0,v+1(Pn,d))Tαd.
+The last line follows since ev∗
+v+1φqivanishes unless the image of that last point is
+qi. Hence, in calculating this expression via localization we only need cons ider fixed
+components of M0,v+1(Pn,d) such that the last marked point is mapped to qi. Such
+components are precisely the ones considered in the proof of Lemm a 3.1 to obtain the
+expression for ˆQd,/vectork(λi). Thus both sides have equal localizations at each fixed point;
+the lemma follows.
+To show the degree bound in the second statement of the lemma, we know that
+ˆQd,/vectork, and hence I∗
+d(ˆQd,/vectork), are both automatically polynomials in /planckover2pi1sinceˆQis (u,v)-
+Euler data. It is clear that deg/planckover2pi1φd= (n+1)d, so part two of the lemma then follows
+immediately from the first part.
+As an immediate application, we have the following theorem:
+Theorem 5.2 (Mirror Theorem I) The(0,1,p,Ω)-Eulerdata ˆQ={Qd,k}induced
+by concavex bundle V→Pnfrom Lemma 3.1 and the extension ˆPof its height 0data
+ˆP0(=ˆQ0)given by Theorem 4.5 are equal.23
+ProofWe confirm that the two conditions for uniqueness in Lemma 3.3 are me t:ˆQ
+andˆPhave the same height 0 data by assumption, which satisfies the hypo theses of
+Theorem 4.5 by the second part of Lemma 5.1. For k >0 the extension of Theorem
+4.5 satisfies
+HG[I(ˆPk)](t)≡Ωpkmod/planckover2pi1−1.
+The second statement in Lemma 5.1 also implies that in the case of (0 ,1)-Euler data,
+HG[I(ˆQk)](t)≡Ωpkmod/planckover2pi1−1.
+The degree bound in Lemma 3.3 is equivalent to requiring that the hype rgeometric
+series agree modulo /planckover2pi1−1, so the theorem follows.
+Remark The entire development of (0 ,1)-Euler data can also be carried out for
+(1,0)-Euler data with the modification that the map Id:Pn֒→Ndbe replaced with
+I′
+d:Pn֒→Nd, given by
+I′
+d([a0,...,an]) = [a0wd
+0,...,anwd
+0].
+Any statement concerning (0 ,1)-Euler data may then be read as a statement about
+(1,0)-Euler data by using I′
+d. In particular, an analogous one-point mirror theorem
+concerning the (1 ,0)-Euler data from a concavex bundle holds.
+5.2 Two-point Mirror Theorem
+Explicitly identifying the (1 ,1)-Euler data induced by Vin terms of linear classes is
+acomplished by gluing together (1 ,0) and (0,1)-Euler data.
+Theorem 5.3 (Mirror Theorem II) The(1,1)-Euler data ˆQinduced by a con-
+cavex bundle Vis expressible in terms of linear classes by gluing together its(0,1)-
+Euler data, identified by Theorem 5.2 in terms of linear class es, and its analog for
+(1,0)-Euler data as prescribed by the Euler condition (7). As a con sequence, the
+associated hypergeometric data at height /vectork= (k1,k2)is given by
+HG[I(ˆQd,/vectork)](t) =pk2HG[I(ˆQd,k1)](t)
+ProofAs an equivariant class is fully determined by its values at fixed compon ents,
+the Euler condition (7) identifies any (1 ,1)-Euler data in terms of its (1 ,0)-Euler data
+and (0,1)-Euler data. A good solution to the one-point problem as given by T heorem
+5.2 then identifies ˆQin terms of linear classes on Nd. The second statement follows
+from the fact that I∗
+d(Qk) =pkI∗
+d(Q0) for any (1 ,0)-Euler data Q.24 L. Cherveny
+6 Recovery Lemmas
+Recall that the Gromov-Witten invariants Kd,/vectorkassociated with the concavex bundle
+VonPnare defined as
+Kd,/vectork=/integraldisplay
+M0,m(Pn,d)Euler(Vd)m/productdisplay
+j=1ev∗Hkj(18)
+for/vectork= (k1,...,km) whenever
+dimM0,m(Pn,d) = dimVd+|/vectork|,
+in which case we say that the integrand is of critical dimension .
+The next lemma is useful for extracting the invariants Kd,/vectorkfrom hypergeometric
+data.
+Lemma 6.1 IfˆQis(0,v)-Euler data induced by a concavex bundle Vand/vectork=
+(k1,...,kv)is such that the integrand in (18) is of critical dimension, t hen
+lim
+λi→0/integraldisplay
+(Pn)Tpse−pt//planckover2pi1I∗
+d(ˆQd,/vectork)
+/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1)=
+
+(−1)v/planckover2pi1v−3(2−v−dt)Kd,/vectorks= 0
+(−1)v/planckover2pi1v−2dKd,/vectorks= 1
+0 s>1
+
+
+(19)
+ProofDenote the left-hand side of (19) by As. By Lemma 5.1,
+lim
+λi→0I∗
+dˆQd,/vectork= (−1)v/planckover2pi1v−1d/productdisplay
+m=1(H−m/planckover2pi1)n+1evv+1∗/parenleftBigg
+π∗
+v+1(Euler(Vd)/producttextv
+j=1ev∗
+jHkj)
+/planckover2pi1+c1(Ld,v+1)/parenrightBigg
+.
+Inserting this expression into As, we have
+As= (−1)v/planckover2pi1v−1/integraldisplay
+PnHse−Ht//planckover2pi1evv+1∗/parenleftBigg
+π∗
+v+1(Euler(Vd)/producttextv
+j=1ev∗
+jHkj)
+/planckover2pi1+c1(Ld,v+1)/parenrightBigg
+= (−1)v/planckover2pi1v−1/integraldisplay
+M0,v+1(Pn,d)ev∗
+v+1Hse−ev∗
+v+1Ht//planckover2pi1·π∗
+v+1(Euler(Vd)/producttextv
+j=1ev∗
+jHkj)
+/planckover2pi1+c1(Ld,v+1)
+= (−1)v/planckover2pi1v−1/integraldisplay
+M0,v(Pn,d)Euler(Vd)v/productdisplay
+j=1ev∗
+jHkjπv+1∗/bracketleftBigg
+ev∗
+v+1Hse−ev∗
+v+1Ht//planckover2pi1
+/planckover2pi1+c1(Ld,v+1)/bracketrightBigg
+.25
+Since Euler( Vd)/producttextv
+j=1ev∗
+jHkjhas top degree, this integral is just Kd,/vectorktimes a
+scalar factor given by integrating the expression in brackets over a generic fiber of
+πv+1(which is isomorphic to P1). Hence,
+As= (−1)v/planckover2pi1v−1Kd,/vectork/integraldisplay
+P1ev∗
+v+1Hs
+/planckover2pi1/parenleftbigg−c1(Ld,2)
+/planckover2pi1−ev∗
+v+1Ht
+/planckover2pi1/parenrightbigg
+(20)
+The classc1(Ld,v+1) restricted to a fiber is just the cotangent bundle of P1with
+vmarked points, while the evaluation map on the fiber is just fwhich has degree d.
+By the Gauss-Bonnet theorem, the former class gives a contribut ion of
+/integraldisplay
+P1i∗c1(Ld,v+1) =−2+v,
+while the latter contributes
+/integraldisplay
+P1i∗ev∗
+v+1H=d.
+Inserting these into (20) and considering the various possibilities fo rsestablishes
+the lemma.
+We will need a general recovery lemma for (1 ,v)-Euler data as well. Fix /vectork=
+(k1,...,kv,kv+1) such that the integrand in
+Kd,/vectork=/integraldisplay
+M0,v+1(Pn,d)Euler(Vd)v+1/productdisplay
+j=1ev∗
+jHkj
+is of critical dimension. Let 0 ≤k∗
+v+1≤kv+1and/vectork∗= (k1,...,k∗
+v+1). Define
+Ki
+d,/vectork=/integraldisplay
+M0,v+1(Pn,d)Euler(Vd)v/productdisplay
+j=1ev∗
+jHkj·ev∗
+v+1Hi·ψkv+1−i
+v+1 (21)
+where as before ψiis the first Chern class of the cotangent line bundle. In particular,
+K0
+d,/vectork=Kd,/vectork.
+Lemma 6.2 IfˆQis(1,v)-Euler data induced by a concavex bundle VonPnand
+kv+1,k∗
+v+1,/vectork, and/vectork∗are defined as above and Ki
+d,/vectorkas in (21), then
+lim
+λi→0/integraldisplay
+(Pn)Tpse−pt//planckover2pi1I∗
+d(ˆQd,/vectork∗)
+/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1)= (−1)v+a/planckover2pi1v−2−aa/summationdisplay
+j=0Ka+j
+d,/vectorktj
+j!,
+wherea=kv+1−k∗
+v+1−swhen0≤s≤kv+1−k∗
+v+1and is0otherwise.26 L. Cherveny
+ProofWe again calculate the integral by pulling back to M0,v+1(Pn,d) and using the
+expression for I∗
+d(ˆQd,/vectork∗) given in Lemma (5.1):
+lim
+λi→0/integraldisplay
+(Pn)Tpse−pt//planckover2pi1I∗
+d(ˆQd,/vectork∗)
+/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1)
+= (−1)v/planckover2pi1v−1/integraldisplay
+M0,v+1(Pn,d)ev∗
+v+1Hs+k∗
+v+1e−ev∗
+v+1Ht//planckover2pi1·π∗
+v+1(Euler(Vd)/producttextv
+j=1ev∗
+jHkj)
+/planckover2pi1+c1(Ld,v+1)
+= (−1)v/planckover2pi1v−2−kv+1+k∗
+v+1+skv+1−(k∗
+v+1+s)/summationdisplay
+j=0(−1)kv+1−(k∗
+v+1+s)Kkv+1−(k∗
+v+1+j+s)
+d,/vectorktj
+j!,
+where we have expanded everything by power series and taken the terms of critical
+dimension. The lemma then follows.
+7 Applications
+7.1 One-Point Candelas Formula
+We will prove a one-point generalization for hypersurfaces of the f amous Candelas
+formula, which expresses mirror symmetry for the quintic in the unm arked case.
+LetV=O(n+1)→Pnwithn≥4. The (n+1)p-Euler data
+ˆP0=(n+1)d/productdisplay
+m=0((n+1)κ−m/planckover2pi1) (22)
+has hypergeometric data given in the non-equivariant limit in power se ries expansion
+as
+lim
+λ→0HG[I(ˆP0)](t) = (n+1)p∞/summationdisplay
+q=0(−1)qfq(t)pq/planckover2pi1−q.
+Defineyi,q(t) recursively by
+y0,q(t) =fq(t)
+f0(t);y0
+i,q(t) =y0
+i−1,q+1′(t)
+y0
+i−1,1′(t)i≥1 (23)
+For dimensional reasons, the one-point Gromov-Witten invariants of interest are
+Kd(Hn−3) =/integraldisplay
+M0,1(Pn,d)Euler(Vd)ev∗
+1Hn−3.27
+Lemma 7.1 (One-point Candelas Formula for Hypersurfaces) Letthe Gromov-
+Witten potential Φfor the degree n+1hypersurface in Pnbe defined
+Φ(t) =(n+1)
+2t2+/summationdisplay
+Kd(Hn−3)edt.
+Then
+Φ(T) = (n+1)[y0,1(t)yn−3,1(t)−yn−3,2(t)]
+where theyi,q(t)are defined by (23) and the mirror change of variables is
+T(t) =y0,1(t).
+ProofLetˆQ0be the (0,0)-Euler data induced by V=O(n+ 1) and ˆP0be as in
+(22). As a consequence of Lemma 3 .3 of [LLY97], mirror transformations of ˆQ0and
+ˆP0agree, which we will denote µ(ˆQ0) andν(ˆP0), in such a way that
+HG[I(µ(ˆQ0))(t) =HG[I(ˆQ0)(T(t)) =1
+f0HG[I(ˆP0)(t) =HG[I(ν(ˆP0))(t)
+in the nonequivariant limit. By Theorem 5.2, the extension of ν(ˆP0) to (0,1)-Euler
+dataν(ˆP) given by Theorem 4.5 and the transformed (0 ,1)-Euler data µ(ˆQ) agree
+(hereµnow refers to the extension given in Lemma 4.2 of the mirror transfo rmation
+of [LLY97] to all heights). In particular, again working in the nonequ ivariant limit,
+HG[I(µ(ˆQn−3))(t) =HG[I(ˆQn−3)(T)
+= Ωpn−3/bracketleftbigg
+1−1
+n+1Φ′(T)p
+/planckover2pi1+1
+n+1(TΦ′(T)−Φ(T))p2
+/planckover2pi12/bracketrightbigg
+agrees with
+HG[I(ν(ˆPn−3)) = Ωpn−3/bracketleftbigg
+1−yn−3,1p
+/planckover2pi1+yn−3,2p2
+/planckover2pi12/bracketrightbigg
+.
+The above expression for HG[I(ˆQn−3)(T) is a direct consequence of Lemma 6.1. The
+theorem then follows easily by equating terms in these expansions.
+7.2 One-Point Closed Formula for rk(V−)≥2
+LetV=V+⊕V−be a split concavex bundle on Pnsuch/summationtextli+/summationtextki=n+1, and
+the rank of the concave part is at least two:
+V=/circleplusdisplay
+li>0O(li)/circleplusdisplay
+ki>0O(−ki),28 L. Cherveny
+with at least two kifactors. When
+k=n−2−rk(V+)+rk(V−)>0 (24)
+the integrand in
+Kd(Hk) =/integraldisplay
+M0,1(Pn,d)Euler(Vd)ev∗
+1Hk(25)
+has the correct dimension.
+Lemma 7.2 The one-point Gromov-Witten invariants Kd(Hk)in 25 withkas in 24
+and rk(V−) = 2are given by
+Kd(Hk) = (−1)d(/summationtextki)/producttextli/producttext
+li(lid)!/producttext
+ki(kid−1)!
+(d!)n+1
+and are0if rk(V−)>2.
+As a special case, when V=O(−1)⊕ O(−1)→P1, we recover the multiple
+covering formula for one marked point mentioned in the introduction :
+Kd,1=1
+d2
+Although this contribution follows easily from the original Aspinwall-Mo rrison for-
+mula and the Divisor Equation [HKK+03], the methods here give another simple
+proof. Other specific cases agree with calculations appearing elsew here, for instance
+V=O(−1)⊕O(−2)→P2considered in Section 3.2 of [KP08] (where the authors’
+ultimate interest was genus 1 invariants).
+We now prove Lemma 7.2:
+ProofAs before, let Ω =e(V+)
+e(V−). IfˆQ0is the (0,0,p,Ω)-Euler data induced by Vand
+ˆP0is the (0,0,p,Ω)-Euler data given by
+ˆPd=/productdisplay
+lilid/productdisplay
+m=0(liκ−m/planckover2pi1)/productdisplay
+kikid−1/productdisplay
+m=1(−kiκ+m/planckover2pi1)
+then by the results of [LLY97], ˆQ0andˆP0are linked and, because rk(V−)≥2,
+immediately satisfy
+HG[I(Q0)](t)≡HG[I(P0)](t) mod/planckover2pi1−2,
+from which one concludes that ˆQ0=ˆP0(Lemma 2.1).
+Now let ˆQbe the (1,0,p,Ω)-Euler data induced by V(Lemma 3.1). Notice that
+by Lemma 5.1,29
+I∗
+d(ˆQk) =pkI∗
+d(ˆQ0).
+By inserting this into Lemma 6.2 in the case of (1 ,0)-Euler data, we find
+/planckover2pi1−2Kd(Hn) = lim
+λi→0/integraldisplay
+(Pn)Te−pt//planckover2pi1I∗
+d(ˆQd,k)/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1)
+= lim
+λi→0/integraldisplay
+(Pn)Te−t//planckover2pi1pkI∗
+d(ˆQd,0)/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1)
+= lim
+λ→0/integraldisplay
+(Pn)Te−pt//planckover2pi1pk/producttext
+li/producttextlid
+m=0(lip−m/planckover2pi1)/producttext
+ki/producttextkid−1
+m=1(−kip+m/planckover2pi1)
+/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1)
+= (−1)d(/summationtextki)/producttextli/producttext
+li(lid)!/producttext
+ki(kid−1)!
+/planckover2pi12(d!)n+1
+where the last line follows by picking off the term of proper degree whe n rk(V−) = 2.
+If rk(V−)>2 then the contribution is 0.
+7.3 Generalized Multiple Covering Formula
+LetV=O(−1)n+1onPn, which induces on M0,2(Pn,d) a sequence of obstruction
+bundlesVdofdimension( n+1)d−n−1. SinceM0,2(Pn,d)hasdimension( n+1)d+n−1,
+the only two-point invariants for dimensional reasons are
+Kd(Hn,Hn) =/integraldisplay
+M0,2(Pn,d)Euler(Vd)ev∗
+1Hnev∗
+2Hn.
+Lemma 7.3
+Kd(Hn,Hn) = (−1)(n+1)(d−1)1
+d
+This agrees with the calculation made in [LLW07].
+ProofThe(0,0,(−p)−n−1)-Eulerdata ˆP0=/producttextd−1
+m=1(−p+m/planckover2pi1)n+1isimmediately equal
+to the (0,0,(−p)−n−1)-Euler data ˆQ0induced by Vas they are linked and their
+hypergeometric series agree mod /planckover2pi1−2[LLY97]. The simple extension of ˆP0to (0,1)-
+Euler data of Lemma 4.1 given by I∗
+dˆPd,k= (p−d/planckover2pi1)kI∗
+dˆPd,0already has the property
+that
+HG[I(ˆPn)](t)≡(−1)n+1
+pmod/planckover2pi1−1.
+On the other hand, the (0 ,1)-Euler data ˆQinduced by Vsatisfies30 L. Cherveny
+HG[I(ˆQn)](t)≡(−1)n+1
+pmod/planckover2pi1−1
+aswellbyLemma5.1. Thesetwo(0 ,1)-EulerdatathenagreebyLemma3.3. Theorem
+5.3 then yields that the (1 ,1)-Euler data ˆQinduced by VsatisfiesHG[I(ˆQn,n)](t) =
+pnHG[I(ˆPn)](t) =pnHG[I(ˆPn)](t). One may then calculate
+−/planckover2pi1−1Kd(Hn,Hn) = lim
+λi→0/integraldisplay
+(Pn)Te−pt//planckover2pi1I∗
+d(ˆQd,n,n)/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1)
+= lim
+λ→0/integraldisplay
+(Pn)Te−pt//planckover2pi1pn(p−d/planckover2pi1)n/producttextd−1
+m=1(−p+m/planckover2pi1)n+1
+/producttextn
+l=0/producttextd
+m=1(p−λl−m/planckover2pi1)
+= (−1)(n+1)(d−1)+11
+d/planckover2pi1
+per Lemma 6.2, from which the lemma follows.
+7.4 Example of General Case
+We now give an example typical of the general case computable by th e methods in
+this paper. Consider thebundle V=O(3)⊕O(−3)→P5, which defines a sequence of
+bundlesVdonM0,1(P5,d) (orM0,2(P5,d)) of rank 6 d. SinceM0,1(P5,d) has dimension
+6d+3 andM0,1(P5,d) has dimension 6 d+4, we should calculate
+Kd(τi(H3−i)) =/integraldisplay
+M0,1(P5,d)Euler(Vd)ev∗
+1H3−iψi
+1.
+for 0≤i≤3 and
+Kd(H2,τi(H2−i)) =/integraldisplay
+M0,2(P5,d)Euler(Vd)ev∗
+1H2ev∗
+2H2−iψi
+2.
+for 0≤i≤2. The other invariants computable by the methods in this paper follo w
+quickly from the divisor property or symmetry and are not included.
+LetˆQ={ˆQd,k}be the (0,1,p,−1)-Euler data induced by Vas in Lemma 3.1.
+Takeg(t) =/summationtext
+d≥1(−1)d(3d)!2
+d(d!)6edt∈etRT[[et]]; by Lemma 4.2 there exists a mirror
+transformation µso that for every height k,
+HG[I(µ(ˆQ)k)](t) =HG[I(ˆQk)](t+g(t)). (26)
+In particular, by using Lemma 5.1, at height k= 0 (corresponding to the (0 ,0)-Euler
+data induced by V) the associated hypergeometric data has the expansion31
+HG[I(µ(ˆQ)0)](t) =HG[I(ˆQ0)](t+g(t))
+=−1/bracketleftBig
+1−p
+/planckover2pi1(t+g(t))/bracketrightBig
++O(/planckover2pi1−2).
+Now consider the (0 ,0)-Euler data ˆP0defined by
+ˆPd,0=3d/productdisplay
+m=0(3κ−m/planckover2pi1)3d−1/productdisplay
+m=1(−3κ+m/planckover2pi1),
+which is linked to ˆQ0(and hence µ(Q)0) by [LLY97]. The hypergeometric data for
+ˆP0has series expansion of the form
+HG[I(ˆP0)](t) = Ω·∞/summationdisplay
+q=0q/summationdisplay
+r=0(−1)qyr
+0,q(t)pq−r/planckover2pi1−q
+=−1/bracketleftbig
+1−/planckover2pi1−1(p·y0
+0,1(t)+y1
+0,1(t))/bracketrightbig
++O(/planckover2pi1−2)
+Letf(t) =y1
+0,1(t)∈etRT[[et]]. ByLemma4.3,thereexistsamirrortransformation
+νso that
+HG[I(ν(ˆP0))](t) =ef//planckover2pi1HG[I(ˆP0)](t)
+=−1/bracketleftBig
+1−p
+/planckover2pi1·y0
+0,1(t)/bracketrightBig
++O(/planckover2pi1−2).
+One may explicitly check that y0
+0,1=g(t)+t, so that
+HG[I(µ(ˆQ)0)](t+g(t))≡HG[I(ν(ˆP0))](t) mod/planckover2pi1−2,
+which, by Lemma 2.1, implies that µ(ˆQ)0=ν(ˆP0) as (0,0)-Euler data.
+By Theorem 5.2, the extension of ˆP0of Theorem 4.5 and the transformed (0 ,1)-
+Euler data µ(ˆQ) induced by Vagree. For instance, this implies
+HG[I(ˆQ3)](t+g(t)) =−/planckover2pi1
+y0
+3,3∂
+∂t/bracketleftbigg−/planckover2pi1
+y0
+2,2∂
+∂t/bracketleftbigg−/planckover2pi1
+y0
+1,1∂
+∂tHG[I(ˆP0)](t)/bracketrightbigg/bracketrightbigg
+.
+from which one can calculate the one-point Gromov-Witten invariant sKd(H3).
+More generally, Theorem 5.3 along with the recovery lemmas 6.1 and 6.2 may
+be used to recover one and two-point Gromov-Witten with descend ents, which are
+calculated by computer and given in the accompanying figures.32 L. Cherveny
+d Kd(H3) ηd(H3)
+1 144 144
+2 −15228 -15264
+3 3387832 3387816
+4 −1033328799 -1033324992
+59395106912144
+25375804276480
+6152957189840958 -152957190686220
+73299075934458784120
+4967328080295077224
+8−125586661840964581023
+4-31396665459982813056
+9137738185029530693381824
+915304242781058965554888
+10−193162880799109330140903228
+25-7726515231964467156704640
+Figure 1: One-point Gromov-Witten invariants
+dKd(τ1(H2)) Kd(τ2(H)) Kd(τ3(1))
+1 27 207 -414
+2136485
+8−339471
+1638799
+16
+3 −471281351696073
+18137491061
+108
+4100423232037
+64−150374595087
+256−268540355185
+512
+5−74841919774848
+1251483860161171187
+1000041652053617157379
+200000
+650249578265872917
+200−168846749581461909
+4000−21002329435853044853
+240000
+7−968660782674268810158
+857530592323981030547843299
+240100013092926237088181035302337
+336140000
+833559516570446549225317133
+627200−678590997243490327156420149
+175616000−447547582585761567954290766657
+24586240000
+9−873077405632389938867605433
+330754338715330778600822100647627393
+400075200089279689020110822840228699723304167
+10081895040000
+102354511971663254407450821413
+175−5193643584353587057315746796417
+23520000−791809569634596848121327607679163149
+177811200000
+Figure 2: One-point Gromov-Witten invariants with descendents
+d Kd(H2,H2) ηd(H2,H2)
+1 261 117
+2−141669
+2−70965
+3 28141053 28140966
+4−52415855109
+4−13103928360
+533295202406036
+56659040481155
+6−7152693165982701
+2−3576346597038222
+713967902006221361284
+71995414572317337289
+8−9158575282812536703813
+8−1144821910345015106088
+9670934720575777355387210 670934720575777346006859
+10−1999257192955989705434315922
+5−399851438591201270607089595
+Figure 3: Two-point Gromov-Witten invariants33
+dKd(H2,τ1(H)) Kd(H2,τ2(1))
+1 -117 -27
+2−31995
+4239355
+8
+3 6726101 −10610038
+4−62339468859
+16292304178171
+64
+555140229054008
+25−1097087319721233
+500
+6−25282225051035849
+20113016673920662199
+100
+7180503575429845806631
+245−10471104834998308828011
+17150
+8−976993199472105955290897
+224042829575696798982686013239
+125440
+982446713274250766320045042
+315−14815808389405172763875125873
+75600
+10−5561428230594376375028839092
+354499962304088589402305334576101
+39200
+Figure 4: Two-point Gromov-Witten invariants with descendents
+References
+[AB84] M. Atiyah and R. Bott, The moment map and equivariant cohomology ,
+Topology 23(1984), 1–28.
+[AM93] P. Aspinwall and D. Morrison, Topological field theory and rational
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+[BCOV93] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Holomorphic
+anomolies in topological field theories , Nucl. Phys. B405(1993), 279–
+304.
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+128(1997), 45–88.
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+Yau manifolds as an exactly soluble superconformal theory , Nucl. Phys.
+B359(1991), 21–74.
+[CK99] D. Cox and S. Katz, Mirror symmetry and algebraic geometry , Mathe-
+matical Surveys and Monographs, vol. 68, Amer. Math. Soc., 1999 .
+[FP96] W. Fulton and R. Pandharipande, Notes on stable maps and quantum
+cohomology , preprint, arXiv:alg-geom/9608011, 1996.
+[HKK+03] K. Hori, S. Katz, A. Klemm, R Pandharipande, R. Thomas, C. Vaf a,
+R. Vakil, and E. Zaslow, Mirror symmetry , Clay Mathematics Mono-
+graphs, vol. 1, Amer. Math. Soc., 2003.
+[Ive72] B. Iversen, A fixed point formula for action of tori on algebraic varietie s,
+Inv. Math. 16(1972), 229–236.34 L. Cherveny
+[KP08] A. Klemm and R. Pandharipande, Enumerative geometry of Calabi-Yau
+4-folds, Comm. Math. Phys. 281(2008), no. 3, 621–653.
+[LLW07] Yuan-Pin Lee, Hui-Wen Lin, and Chin-Lung Wang, Flops, motives, and
+invariance of quantum rings , to appear in Annals Math.
+[LLY97] B. Lian, K. Liu, and S.T. Yau, Mirror principle I , Asian J. Math. 1
+(1997), no. 4, 729–763.
+[LLY99a] ,Mirror principle II , Asian J. Math. 3(1999), no. 1, 109–146.
+[LLY99b] ,Mirror principle III , Asian J. Math. 3(1999), no. 4, 771–800.
+[LLY01] ,Mirror principle IV , Surveys Diff. Geo. 7(2001), 475–496.
+[LT98] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants
+of algebraic varieties , J. AMS 11(1998), 119–174.
+[PZ08] R. Pandharipande and A. Zinger, Enumerative geometry of Calabi-Yau
+5-folds, preprint, arXiv:0802.1640v1, 2008.
+[Zin07] A. Zinger, Genus-zero two-point hyperplane integrals in the Gromov-
+Witten theory , preprint, arXiv:0705.2725v2, 2007.
\ No newline at end of file
diff --git a/1001.0243.txt b/1001.0243.txt
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+arXiv:1001.0243v1  [cond-mat.mes-hall]  1 Jan 2010Magnetic Response in Mesoscopic Rings and Moebius Strips: A
+Theoretical Study
+Santanu K. Maiti†,‡,1
+†Theoretical Condensed Matter Physics Division, Saha Insti tute of Nuclear Physics,
+1/AF, Bidhannagar, Kolkata-700 064, India
+‡Department of Physics, Narasinha Dutt College, 129 Belilio us Road, Howrah-711 101, India
+Abstract
+We investigate magnetic response in mesoscopic rings and moebius st rips penetrated by magnetic flux φ.
+Based on a simple tight-binding framework all the calculations are per formed numerically which describe
+persistent current and low-field magnetic susceptibility as function s of magnetic flux φ, total number of
+electrons Ne, system size Nand disorder strength W. Our exact analysis may provide some important
+signatures to study magnetic response in nano-scale loop geometr ies.
+PACS No. : 73.22.-f; 73.23.-b; 73.23.Ra; 75.20.-g
+Keywords : Mesoscopic ring; Moebius strip; Persistent current; Low-field ma gnetic susceptibility;
+Disorder.
+1Corresponding Author : Santanu K. Maiti
+Electronic mail: santanu.maiti@saha.ac.in
+11Introduction
+More advanced nanoscience and technology has
+made it possible to fabricate devices whose dimen-
+sions are comparable to mean free path of an elec-
+tron and electron transport in such systems gives
+several novel and interesting new phenomena. In
+the mesoscopic regime, phase coherence of the elec-
+tronic states are of fundamental importance, and
+the phenomenon of persistent current is a spectac-
+ular consequence of quantum phase coherence in
+this regime. Experimental investigations on these
+systems have provided several surprising quantum
+behaviors in contrast to those anticipated from the
+classical theory of metals. At much low tempera-
+tures, two aspects of the new quantum regime ap-
+pear that are of particular interest.
+(i) The phase coherence length Lφ, the length scale
+over which an electron maintains its phase memory,
+increases significantly with the decrease of temper-
+ature and is comparable to the system size L.
+(ii) The energy levels of these small finite size sys-
+tems are discrete.
+These two are the essential criteria for the existence
+of persistent current in these systems upon the ap-
+plication of an external magnetic flux φ. Many the-
+oretical works [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
+13, 14, 15, 16, 17, 18] on persistent current in one-
+dimensional (1D) mesoscopic rings are available in
+literature, yet they cannot clearly explain several
+results those have been obtained experimentally.
+The discovery of persistent current in mesoscopic
+ringshasaddressednewinterestingquestionsonthe
+thermodynamics of these systems. Although such
+an effect was predicted several years ago, the un-
+expectedly large amplitudes of measured currents
+lead to many important questions. It has been pro-
+posed that the electron-electron (e-e) interactions
+contribute significantly to the average currents. Al-
+though, an explanation based on the perturbative
+calculation both in interaction and disorder, seems
+to give a quantitative estimate closer to the ex-
+perimental results but still it is less than the mea-sured currents by an order of magnitude, and the
+interaction parameter used in the theory is not well
+understood physically. Though the enhancement
+of current amplitude can be understood by some
+theoretical arguments [16, 17, 18] but the sign of
+the currents cannot be predicted precisely. Levy
+et al.[19] have observed diamagnetic response for
+the measured currents at low fields in the experi-
+ment on 107isolated mesoscopic Cu rings. While,
+Chandrasekhar et al.[20]havemeasured φ0periodic
+currents in Ag rings with paramagnetic response
+near zero fields. In a theoretical work Cheung et
+al.[4] have predicted that the direction of current
+is random depending on the total number of elec-
+trons in the system and the specific realization of
+the random potentials. On the other hand, Yu and
+Fowler [12] have shown both diamagnetic and para-
+magnetic responses in mesoscopic Hubbard rings.
+In an experiment, Jariwala et al.[21] have obtained
+diamagnetic persistent currents with both h/eand
+h/2eflux periodicities in an array of 30-diffusive
+gold rings. Similar diamagnetic sign of the cur-
+rents in the vicinity of zero magnetic field were also
+found in an experiment [22] on 105disconnected
+Ag rings. Thus we can emphasize that the predic-
+tion of the sign of low-field currents is a major chal-
+lenge in this area. It has been studied that only for
+single-channel rings, the sign of the low-field cur-
+rents can be mentioned exactly [23, 24]. While,
+in all other cases i.e., for multi-channel rings and
+cylinders, sign of the low-field currents cannot be
+predicted exactly. Hence we see that the sign of
+low-field currents is not very clear and the theoret-
+ical and experimental results still do not agree very
+well. Therefore, we can say that the phenomenon
+of persistent current in metallic systems gives an
+exciting curiosity and is an open subject.
+The paper is organized as follows. Following the
+brief introduction (Section 1), in Section 2, we de-
+scribe the behavior of persistent current in strictly
+one-channel mesoscopic rings. The effect of intra-
+chain interaction on persistent current in moebius
+strips is discussed in Section 3. Section 4 illus-
+2trates the behavior of low-field magnetic suscep-
+tibility both for one-channel mesoscopic rings and
+moebius strips, and finally, we summarizeourstudy
+in Section 5.
+2Persistent current in one-
+channel mesoscopic rings
+Hereweexplorethebehaviorofpersistentcurrentin
+strictly one-channel mesoscopic rings. Let us start
+by referring to Fig. 1 where a mesoscopic ring is
+penetrated by a magnetic flux φ. To describe the
+Φ
+Figure 1: Schematic view of a 1D mesoscopic ring
+penetrated by a magnetic flux φ. The filled black
+circles correspond to the position of atomic sites. A
+persistent current Iis established in the ring.
+system we use a tight-binding framework. Within a
+non-interacting electron picture, the tight-binding
+Hamiltonian for a N-site ring, enclosing a magnetic
+fluxφ(inunitsoftheelementaryfluxquantum φ0=
+ch/e), can be expressed in Wannier basis as,
+H=/summationdisplay
+iǫic†
+ici+/summationdisplay
+<ij>v/parenleftBig
+eiθc†
+icj+e−iθc†
+jci/parenrightBig
+(1)
+where,ǫi’s are the on-site energies, vis the hop-
+pingstrengthbetweennearest-neighboratomicsites
+andθ= 2πφ/Nis the phase factor due to the flux
+threaded by the ring. As the magnetic field associ-
+ated with flux φdoes not penetrate anywhereof the
+circumference of the ring, we neglect Zeeman term
+in the above Hamiltonian (Eq. 1). Throughout our
+calculations we set the hopping strength v=−1
+and use the units where c=e=h= 1.In order to study current-flux ( I-φ) characteris-
+tics first it is necessary to know the variation of
+energy levels with φ. For an ordered ring, we cal-
+culate the energy eigenvalues analytically. Mathe-
+matically, the energy eigenvalue for n-th eigenstate
+can be written in the form,
+En(φ) = 2vcos/bracketleftbigg2π
+N/parenleftbigg
+n+φ
+φ0/parenrightbigg/bracketrightbigg
+(2)
+where,nisanintegerboundedintherange −N/2≤
+n < N/ 2. But as long as impurities are intro-
+duced in the ring i.e., the ring becomes a disordered
+one, energy eigenvalues cannot be obtained analyt-
+ically. In that case we do exact numerical diagonal-
+ization of the tight-binding Hamiltonian to achieve
+the eigenvalues. As illustrative examples, in Fig. 2
+/Minus1/Minus0.5 0.5 1Φ
+/Minus2/Minus112E/LParen1Φ/RParen1
+Figure 2: Energy-flux characteristics for ordered
+(solid line) and disordered (dotted line) mesoscopic
+rings with N= 10. For the disordered case, we
+chooseW= 1.
+we plot the energy-flux ( E-φ) characteristics for a
+mesoscopic ring considering N= 10. The solid and
+dotted curves correspond to the perfect and disor-
+dered rings, respectively. For the disordered ring,
+we setW= 1. In absence of any impurity ( W= 0),
+energy levels intersect with each other at integer
+or half-integer multiples of φ0/2, while gaps open
+at these points in presence of impurity. In both
+the two cases energy levels vary periodically as a
+function of φshowing φ0(= 1, in our chosen unit
+3c=e=h= 1) flux-quantum periodicity.
+The current Incarried by n-th energy level hav-
+ing energy Enis obtained by taking the 1-st order
+derivative of Enwith flux φ. Mathematically it can
+expressed as,
+In=−∂En
+∂φ(3)
+At absolute zero temperature ( T= 0K), total cur-
+rent of a system, described with fixed number of
+electrons Ne, will be obtained by taking the sum of
+/Minus1.0 1.0Φ
+/Minus0.060.06I/LParen1Φ/RParen1
+/LParen1b/RParen1/Minus1.0 1.0Φ
+/Minus0.060.06I/LParen1Φ/RParen1
+/LParen1a/RParen1
+Figure3: Persistentcurrentasafunctionof φforor-
+dered(solid line) anddisordered(dotted line) meso-
+scopic rings with N= 150, where (a) Ne= 50and
+(b)Ne= 55. For the disorderedcase, we fix W= 1.
+individual contributions from the lowest Neenergy
+levels. Thus, we can write the total current in the
+form,
+I(φ) =/summationdisplay
+nIn(φ) (4)
+As representative examples, in Fig. 3 we show the
+variation of persistent current for a typical meso-
+scopic ring considering N= 150, where (a) and (b)
+represent the results for even ( Ne= 50) and odd(Ne= 55) number of electrons, respectively. The
+solid and dotted curves correspond to the perfect
+(W= 0) and disordered ( W= 1) rings, respec-
+tively. For the ordered ring, we set ǫi= 0 for all i,
+while for the disordered case site energies ( ǫi) are
+chosen randomly from a “Box” distribution func-
+tion of width W= 1. From the results it is ob-
+served that, in impurity free rings current exhibits
+saw-tooth like variation as a function of φprovid-
+ing sharp transitions, associated with the crossing
+ofenergylevels,at φ=±nφ0or±nφ0/2,depending
+on whether the ring contains even or odd number
+of electrons. On the other hand, this saw-tooth like
+behavior completely disappears as long as impuri-
+ties are introduced in the ring. This is due to the
+fact that, in presence of impurities all the degenera-
+cies move out, and accordingly, energy levels vary
+continuously with φwhich provide continuous vari-
+ation of the current. Addition to this feature, it is
+also important to note that in disordered ring cur-
+rent amplitude gets reduced significantly compared
+to the ordered one. Such a behavior can be well un-
+derstood from the theory of Anderson localization,
+where we get more localization with the increase of
+disorderstrength W[25]. Bothforperfect anddirty
+rings, persistent current varies periodically with φ
+showing only φ0flux-quantum periodicity.
+3Persistent current in moe-
+bius strips
+This section describes the behavior of persistent
+current in moebius strips. Persistent current in
+a moebius strip is highly sensitive on the intra-
+chain interaction strength and here we will focus
+our study in this particular aspect. The schematic
+view of a moebius strip, penetrated by a magnetic
+fluxφ, is shown in Fig. 4. The vertical line connect-
+ing the atomic sites in upper and lower strands is
+called as ‘rung’. In presence of magnetic flux φ, the
+tight-binding Hamiltonian for a moebius strip with
+Nrungs (i.e., 2 Natomic sites since in each strand
+4there are Natomic sites) can be written within the
+non-interacting electron picture as,
+H=2N/summationdisplay
+i=1ǫic†
+ici+v2N/summationdisplay
+i=1/parenleftBig
+eiθc†
+ici+1+e−iθc†
+i+1ci/parenrightBig
++v⊥2N/summationdisplay
+i=1c†
+ici+N (5)
+where,ǫirepresentstheon-siteenergyofanelectron
+in siteiandθ= 2πφ/Nis the phase factor associ-
+ated with flux φ.vgives the nearest-neighbor hop-
+ping integral in each strand and v⊥corresponds to
+the hopping strength between the nearest-neighbor
+sites in the two strands, the so-called intra-chain
+interaction strength. It ( v⊥) significantly controls
+the periodicity of persistent current in such a spe-
+Figure 4: Schematic representation of a moebius
+strip penetrated by a magnetic flux φ.
+cial type of geometric model which we will describe
+in forthcoming sub-section.
+Now we start our discussion with the energy-flux
+characteristics of a moebius strip. For illustrative
+examples, in Fig. 5 we plot the energy levels as a
+function of φfor a moebius strip considering N= 5
+in the typical case where v⊥= 0. The solid and
+dotted lines correspond to the perfect and disor-
+dered moebius strips, respectively. The presence
+(disordered strip) or absence (ordered strip) of W
+provides exactly the similar behavior i.e., gap or
+gap less spectrum as we observe in the case of one-
+channel mesoscopic rings. Most importantly we no-
+tice that the energy levels have extremas at inte-
+ger multiples of φ=±nφ0/4 or±nφ0/2. Additionto this, it is also observed that the energy levels
+vary periodically with flux φproviding φ0/2 flux-
+/Minus1/Minus0.5 0.5 1Φ
+/Minus2/Minus112E/LParen1Φ/RParen1
+Figure 5: Energy-flux characteristics for ordered
+(solid curve)and disordered(dotted curve)moebius
+strips (N= 5) withv⊥= 0.
+quantum periodicity instead of simple φ0period-
+icity, as observed in traditional one-channel meso-
+scopic rings or multi-channel cylinders. This phe-
+nomenon of half flux-quantum periodicity can be
+explained as follow. In the typical case where
+/Minus1/Minus0.5 0.5 1Φ
+/Minus3/Minus113E/LParen1Φ/RParen1
+Figure 6: Energy-flux characteristics for ordered
+(solid line) and disordered (dotted line) moebius
+strips (N= 5) withv⊥=−1.
+v⊥= 0, electrons cannot able to hop along the ver-
+tical direction. Therefore, in a moebius strip if an
+electron starts to move from any point, it will come
+back at its initial position after traversing twice the
+path length, and accordingly, it encloses 2 φflux.
+5This provides φ0/2 flux-quantum periodicity and it
+isthe specialityofsuchatwistedgeometry. Now, as
+longaselectronsareallowedtohopbetweenthetwo
+strands i.e., v⊥/negationslash= 0, energy levels get back φ0flux-
+quantum periodicity. This is quite obvious since for
+the case when v⊥/negationslash= 0 one can treat the moebius
+strip as a conventional cylinder. For illustrations,
+/Minus1.0 1.0Φ
+/Minus0.20.2I/LParen1Φ/RParen1
+/LParen1b/RParen1/Minus1.0 1.0Φ
+/Minus0.20.2I/LParen1Φ/RParen1
+/LParen1a/RParen1
+Figure 7: I-φcharacteristics for ordered (solid
+curve)anddisordered(dotted curve)moebius strips
+(N= 50) withv⊥= 0. (a)Ne= 40and (b)
+Ne= 35. For the disordered case, W= 1.
+see the results plotted in Fig. 6, where the solid and
+dotted lines correspond to the identical meaning as
+in Fig. 5.
+With the above energy-flux characteristics now
+we concentrate our study on the current-flux spec-
+tra. Let us begin with the results plotted in Fig. 7,
+where the currents are evaluated for some moebius
+strips with v⊥= 0. The system size Nis fixed at
+50, where (a) and (b) correspond to Ne= 40 (even)
+and 35 (odd), respectively. The solid and dotted
+curves represent the currents for ordered and disor-
+dered moebius strips, respectively, where the disor-derstrength Wissetto1. Likesinglechannelmeso-
+scopicring, persistent currentrevealssaw-toothlike
+behavior as a function of φboth for odd and even
+number of electrons. Similarly, in presence of disor-
+der current varies continuously, as studied earlier,
+/Minus1.0 1.0Φ
+/Minus0.30.3I/LParen1Φ/RParen1
+/LParen1b/RParen1/Minus1.0 1.0Φ
+/Minus0.20.2I/LParen1Φ/RParen1
+/LParen1a/RParen1
+Figure 8: I-φcharacteristics for ordered (solid
+curve)anddisordered(dotted curve)moebiusstrips
+(N= 50) withv⊥=−1. (a)Ne= 40and (b)
+Ne= 35. For the disordered case, Wis fixed at 1.
+and gets much reduced value than the perfect case.
+These features are clearly understood from our pre-
+vious discussion. The main point is that, for this
+particular case where v⊥= 0, current varies peri-
+odically with φshowing φ0/2 periodicity, instead
+of conventional φ0periodicity. This unconventional
+period halving behavior of persistent current may
+be clearly visible from our energy-flux spectrum,
+for instance see Fig. 5.
+The situation becomes quite different when elec-
+trons can able to hop along the transverse direction
+i.e., for v⊥/negationslash= 0. In this case a moebius can be
+regarded as an ordinary mesoscopic cylinder, and
+therefore, we getthe conventionalfeaturesofpersis-
+6tent current. For illustrative purposes, in Fig. 8 we
+show the current-flux characteristics for some typi-
+cal moebius strips ( N= 50) considering v⊥=−1,
+where (a) and (b) correspond to the results for
+Ne= 40 and 35, respectively. The solid and dotted
+linesrepresentthe similarmeaningasin Fig.7. The
+saw-tooth like behavior is still preserved in perfect
+case. But, some additional kinks may appear in I-
+φspectrum at different values of φdepending on
+the choices of Ne. For instance, the current shows
+a kink across φ= 0 when Ne= 40. These kinks
+are associated with the crossing of energy levels for
+the cylindricalsystems. While, fordisorderedcases,
+current varies continuously with much reduced am-
+plitude, as expected, and no kink will appear.
+4Low-field magnetic suscep-
+tibility
+Next, we focus our attention on the determination
+of magnetic susceptibility χ(φ) in the limit φ→0.
+Evaluating the sign of χ(φ) one can able to predict
+whetherthe currentis paramagneticordiamagnetic
+in nature. It was a long standing problem over the
+past few many years, and, here we try to emphasize
+about it both for one-channel rings and moebius
+strips described with fixed number of electrons Ne.
+The generalexpressionof magnetic susceptibility at
+any flux φis expressed in the form,
+χ(φ) =N3
+16π2/parenleftbigg∂I(φ)
+∂φ/parenrightbigg
+(6)
+where,Ncorrespondstothe totalnumberofatomic
+sites in the system. Here we will determine χ(φ)
+only in the limit φ→0 i.e., we are interested only
+on the low-field magnetic susceptibility.
+As representative examples, in Fig. 9 we display
+the variation of low-field magnetic susceptibility as
+a function of Nefor strictly one-channel mesoscopic
+rings, where (a) and (b) correspond to the perfect
+(W= 0) and disordered ( W= 1) rings, respec-
+tively. The system size Nis fixed at 200. Our re-sults predict that in absence of any impurity χ(φ) is
+always negative irrespective of the total number of
+electrons Nei.e., whether Neis odd or even. It re-
+vealsthatforperfectringslow-fieldcurrentprovides
+only the diamagnetic response. This diamagnetic
+behavior of low-field currents can be clearly under-
+stoodbynotingthe slopeof I-φcharacteristicsplot-
+50100150200Ne
+/Minus2.5/Multiply1032.5/Multiply1037.5/Multiply103Χ/LParen1Φ/RParen1
+/LParen1b/RParen12575125175Ne/Minus6/Multiply103/Minus4/Multiply1030
+/Minus2/Multiply103Χ/LParen1Φ/RParen1
+/LParen1a/RParen1
+Figure 9: Low-field magnetic susceptibility χ(φ)as
+a function of Nefor (a) ordered rings and (b) dis-
+ordered ( W= 1) rings considering N= 200. The
+solid and dotted curves in (b) correspond to the
+results for even and odd Ne, respectively.
+ted by the solid lines in Figs. 3(a) and (b). The
+response drastically changes as long as impurities
+are introduced in the system (9(b)). For odd and
+evenNe, it shows completely opposite behavior.
+The low-field current gives paramagnetic response
+for the rings with even Ne(solid line), while for
+the rings with odd Neit shows diamagnetic na-
+ture (dotted line). These diamagnetic and para-
+magnetic natures of persistent current for the dis-
+ordered case can be easily understood by observing
+the slopes of I-φcurves presented by the dotted
+7lines in Figs. 3(a) and (b). All these features are
+equally valid irrespective of disorder strength and
+disordered configurations. Thus, in brief we can
+say that for one-channel rings described with fixed
+number of electrons, sign of low-field currents can
+be precisely determined.
+At the end, we focus on the behavior of low-field
+magneticresponseinmoebiusgeometries. Forthese
+systems, sign of low-field currents strongly depends
+on the strength of intra-chain interaction ( v⊥). For
+v⊥= 0, a moebius strip becomes a single channel
+ring, and therefore, sign of low-field currents can be
+mentioned exactly according to the above prescrip-
+tion. On the other hand for the case where v⊥is
+finite, sign of low-field currents cannot be predicted
+accurately as it strongly depends on the number of
+electrons and specific realization of disordered con-
+figuration in the strips.
+5Concluding remarks
+To summarize, we have studied the behavior of
+persistent current and low-field magnetic suscepti-
+bility in strictly one-channel mesoscopic rings and
+moebius strips penetrated by magnetic flux φ. A
+tight-binding framework is given to describe the
+system, where all calculations are done numerically
+to study the magnetic response. In perfect rings
+current shows saw-tooth like variation as a func-
+tion ofφ, while it varies continuously in disordered
+rings. Both for perfect and disordered rings cur-
+rent varies periodically with φexhibiting φ0flux-
+quantum periodicity. In moebius strips a uncon-
+ventional φ0/2 periodic nature of persistent current
+is observed for the particular case when vbot= 0.
+While, current gets back its φ0periodicity when
+electrons are able to hop along the transverse direc-
+tion i.e., v⊥/negationslash= 0. In the study of low-field magnetic
+response we have seen that for one-channel meso-
+scopicringsdescribed withconstantnumberofelec-
+trons, sign of low-field currents can be mentioned
+precisely. While, for moebius strips with v⊥/negationslash= 0,sign of low-field currents cannot be predicted ex-
+actly as it depends on the number of electrons as
+well as disordered configurations. Our exact analy-
+sis can provide some physical insight to study mag-
+netic response in mesoscopic loop geometries.
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+and R. A. Webb, Phys. Rev. Lett. 86, 1594
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+and D. Mailly, Phys. Rev. Lett. 89, 206803
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+9
\ No newline at end of file
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@@ -0,0 +1,842 @@
+ 1
+All-Optical Arithmetic and Combinatorial Logic Circ uits with High-Q 
+Bacteriorhodopsin Coated Microcavities 
+ 
+Sukhdev Roy *, Mohit Prasad  
+Department of Physics and Computer Science 
+ Dayalbagh Educational Institute (Deemed University ) 
+ Dayalbagh, Agra 282 110 INDIA 
+Ph.: +91-562-2801545, Fax: +91-562-2801226 
+*E -mail: sukhdevroy@dei.ac.in   
+ 
+Juraj Topolancik 1 and Frank Vollmer 
+Biofunctional Photonics Group 
+The Rowland Institute, Harvard University 
+100 Edwin H. Land Blvd., Cambridge, MA 02142U.S.A. 
+Ph.: +1-617 497 4681, Fax: +1-617-497-4627 
+E-mail: vollmer@rowland.harvard.edu  
+ 
+1Present Address: Department of Electrical and Compu ter Engineering 
+Northeastern University 
+360 Huntington Avenue, Boston, MA 02115 
+Ph.: +1-617-373-4159, Fax: +1-617-373-8970 
+E-mail: jtopolan@ece.neu.edu   
+ 
+Abstract 
+ 
+We present designs of all-optical computing circuit s, namely, half-full adder/subtractor, 
+de-multiplexer, multiplexer, and an arithmetic unit , based on bacteriorhodopsin (BR) protein 
+coated microcavity switch in a tree architecture. T he basic all-optical switch consists of an input 
+infrared (IR) laser beam at 1310 nm in a single mod e fiber (SMF-28) switched by a control 
+pulsed laser beam at 532 nm, which triggers the cha nge in the resonance condition on a silica 
+beed coated with BR between two tapered fibers. We show that fast switching ~50 µs can be 
+achieved by injecting a blue laser beam at 410 nm t hat helps in truncating the BR photocycle at 
+the M intermediate state. Realization of all-optica l switch with BR coated microcavity switch has 
+been done experimentally. Based on this basic switc h configuration, designs of all-optical higher 
+computing circuits have been presented. The design requires 2 n-1 switches to realize n bit 
+computation. The proposed designs require less numb er of switches than terahertz optical 
+asymmetric demultiplexer based interferometer desig ns. The combined advantages of high Q- 2
+factor, tunability, compactness and low power contr ol signals, with the flexibility of cascading 
+switches to form circuits, makes the designs promis ing for practical applications. The design 
+combines the exceptional sensitivities of BR and mi crocavities for realizing low power circuits 
+and networks. The designs are general and can be im plemented (i) in both fiber-optic and 
+integrated optic formats, (ii) with any other coate d photosensitive material, or (iii) an externally 
+controlled microresonator switch. 
+ 
+Index Terms —All-optical switching, microcavities, bacteriorhod opsin, computing, multiplexer 
+  3
+I. INTRODUCTION 
+The anticipated requirement for ultrahigh speed ult rahigh bandwidth information 
+processing has provided tremendous impetus for real ization of all-optical devices and circuits. 
+The advent of nano and bio technologies in the desi gn, synthesis and characterization of novel 
+materials and structures that include nanophotonics , biophotonics, plasmonics, organic and 
+silicon photonics, photonic crystals and metamateri als, high-Q microresonators, slow and fast 
+light and quantum information processing has opened  exciting new possibilities for generation, 
+manipulation and detection of light along with inte gration of multiple components and devices to 
+achieve all-optical information processing [1-3].  
+Recent years have witnessed a renewed interest in o ptical computing due to two 
+important advancements, namely, the development of (i) efficient conservative and reversible 
+logic, algorithms and architectures that exploit th e strengths of optics, and (ii) novel materials 
+that exhibit high nonlinearities and structures tha t confine light to low dimensions that facilitates 
+low power nonlinear optics [4-8]. These advancement s offer tremendous scope to overcome the 
+impediments to realize optical computing such as ca scading components and devices to fabricate 
+circuits and networks in the form of 2D/3D arrays t hat are small enough to have low switching 
+energies and high speeds [9, 10]. 
+The flexibility of electronic processing stems from  its ability to perform non-linear 
+operations such as thresholding. In optical process ing, non-linear optical mechanisms play 
+important role in ultrafast optical switches and al l-optical logic gates. The integration of photonic 
+components is expected to increase significantly du e to emerging novel photonic structures such 
+as microresonators, photonic crystals and plasmonic s [11]. 
+Microcavities have emerged as extremely sensitive a nd versatile device configurations 
+for a variety of operations due to their high Q-fac tor, low switching threshold and ultra 
+compactness [12, 13]. Optical microcavities confine  light to small volumes by resonant  4
+circulation. For a coupled, input power P in , the circulating intensity within the resonator is  given 
+by I = P in  (λ/2 πn) (Q/V) where n is the group index. For a cavity o f Q ~10 8 and a mode volume 
+of 500 µm3 (both obtainable in spheres roughly 40 µm in diameter), the circulating intensity 
+exceeds 1 GW/cm 2 with less than 1 mW of coupled input power [12]. 
+A very wide range of microresonator shapes have bee n explored over the years for 
+various applications. The most widely used are rota tionally symmetric structures such as Fabry-
+Perot cavities, spheres, cylinders, disks, torroids , and photonic crystals which have been shown 
+to support very high-Q whispering gallery (WG) mode s, whose modal field intensity distribution 
+is concentrated near the dielectric-air interface [ 12-18].  
+Devices based on microcavities are already indispen sable for a wide range of applications 
+and studies. By tailoring the microcavity shape, si ze or material composition, the microcavity 
+can be tuned to support a spectrum of optical modes  with required polarization, frequency and 
+emission patterns [12-15]. Some key optical microre sonator material systems are Si, SOI, 
+SiO xNy, GaAs, InP, GaN and LiNbO 3 [14]. Various all-optical logic operations have be en shown 
+using Si, GaAs and InGaAs microcavities [19-20]. Th e nonlinear optical mechanism 
+implemented in the gates is the change in the refra ctive index from free charge carriers generated 
+by two photon absorption (TPA). Although the change  in refractive index in these microcavities 
+is high, the absorption generates heat inside the m icrocavity due to pump power. The pump and 
+probe power is in mWs and the Q-factor ~ 10 4 [19-20]. Recently, a spatially non-blocking optica l 
+router with a footprint of 0.07 mm 2 has been demonstrated [21]. The device is dynamica lly 
+switched using thermo-optically tuned silicon micro ring resonators with a wavelength shift to 
+power ratio of 0.5nm/mW. The design can route four optical inputs to four outputs with 
+individual bandwidths of upto 38.5 GHz. These confi gurations can route a single-wavelength 
+laser and provides a maximum extinction ratio large r than 20dB [21].   5
+Silica microcavities have an inherent advantage of high Q-factor, relatively simple 
+fabrication, possible on chip integration and contr ol of the coupling efficiency through taper by 
+the change of the fiber thickness [12-14, 17]. Fibe r optic tapers have been proposed as a means 
+to couple quantum states to or from a resonator ont o a fiber [12]. Also the recent demonstration 
+of a fiber-taper-coupled ultrahigh-Q microtoroid on  a chip enables integration of wafer based 
+functions with ultralow-loss fibre-coupled quantum devices. The bulk optical loss from silica is 
+also exceptionally low and record Q factors of 8 x 10 9 (and finesse of 2.3 * 10 6) have been 
+reported [12]. Ultra high-Q microtoroidal silica re sonators represent a distinct class of optical 
+microresonators with Q’s in excess of 10 8 [17]. Due to the high Q-factor and the small 
+dimensions, switching at low power is feasible. Mor eover, coating the microcavity with a 
+photosensitive material can further lead to switchi ng at ultra-low powers.  
+Recently, all-optical switching in the near infrare d with bacteriorhodopsin (BR) coated 
+silica microcavities has been reported, with a Q-fa ctor on BR adsorption ~5 x 10 5 [22]. The state 
+of the BR in optical microcavities is controlled by  a low power (< 200 µW) continuous green 
+pump laser coupled to the microsphere cavity using a tapered fiber [22-24]. The photochromic 
+protein bacteriorhodopsin (bR) found in the purple membrane fragments of halobacterium 
+halobium,  has emerged as an outstanding photonic material fo r practical applications due to its 
+unique multifunctional photoresponse and properties  [22-28]. By absorbing green–yellow light, 
+the wild-type BR molecule undergoes several structu ral transformations in a complex photocycle 
+that generates a number of intermediate states [29- 32]. The main photocycle of BR is as shown 
+in Fig.1 After excitation with green–yellow light a t 570 nm, the molecules in the initial B-state 
+get transformed into J-state within about 0.5 ps. The species in the J-state thermally transforms 
+in 3 ps into the intermediate K-state, which in turn transforms in about 2 µs into the L-state. 
+From the L-state, BR thermally relaxes to the MI-state within 8 µs and undergoes irreversible  6
+transition to the MII-state. The molecules then relax through the N and O intermediates to the 
+initial B-state within about 10 ms. An important feature of all the intermediate states is their 
+ability to be photo-chemically switched back to the  initial B-state by shining light at a 
+wavelength that corresponds to the absorption peak of the intermediate in question [29-32]. The 
+wavelength in nm of the absorption peak of each spe cies is shown as a subscript in Fig.1 
+Photoinduced molecular transitions in BR can be use d to reversibly configure an ultrasensitive 
+micron scale photonic component in which the optica l response is resonantly enhanced.  
+Cascading to integrate a large number of all-optica l components such as switches, logic 
+gates etc., is a complex problem and a major obstac le in the development of a complete all-
+optical computing system [1, 8]. Branching of signa ls all-optically among various logic devices 
+is a critical task. Theoretical schemes suggesting alternative solutions for parallel generation of 
+logic gates have been recently reported in literatu re [33-35]. An effective method for cascading 
+involves the tree architecture [36]. It is a multip lying system of a single straight path into several  
+distributed branches and sub-branch paths. Shen and  Wu have demonstrated that re-
+configurability can be introduced into designs of a ll-optical logic circuits by electo-optic 
+switches [37]. Recently, Kodi and Louri have demons trated that optical based system 
+architecture shows better performance than electric al interconnects for uniform and non-uniform 
+patterns without the application of reconfiguration  techniques [38]. Even with the application of 
+reconfiguration techniques, the dynamically reconfi gurable optoelectronic provides much better 
+performance for all communication patterns. Caulfie ld et al. have also proposed an electro-
+optical logic system with silicon-on-insulator (SOI ) resonant structures [39]. Thus, low power 
+BR based microcavity switch in a suitable architect ure can be useful to achieve all-optical 
+computing. 
+In this paper, we show fast switching (~50 µs) in BR coated microcavity switch by 
+injecting an blue laser beam (405 nm) in synchroniz ation with green laser beam (532 nm) and  7
+further combine the advantages of BR coated microca vity switches in a tree architecture to 
+realize reconfigurable all-optical logic and arithm etic operations. The key object of this paper is 
+to propose general designs for optically controlled  microrcavity based computing circuits, i.e., 
+all-optical arithmetic and combinatorial circuits t hat can be configured to realize different logic 
+and arithmetic operations in parallel. The whole lo gic unit consists of only BR coated 
+microcavity switches and the horizontal and vertica l extension of the designs can also be easily 
+realized.  
+II. EXPERIMENTAL SETUP OF BR COATED MICROCAVITY SWI TCH 
+The basic configuration of the all-optical switch i n the design has been considered to be 
+the BR based microcavity switch [22]. The silica mi cro-cavity in contact between two tapered 
+fibers serves as a four-port tunable resonant coupl er. The diameter of the silica microcavity is 
+300 µm. The switch shown in Fig.2 represents the si mplified schematic representation of the 
+resonant coupler. Here, Port 1 acts as port for inc oming as well as for control signal. Port 2 acts 
+as lower output port and port 3 acts as an output p ort. The various states have been listed in 
+Table I. In the BR coated microcavity switch, the s witching of incoming signal operating at 
+wavelength 1310 nm between the output ports (2 and 3) is photo-induced with a fiber-coupled 
+green pump laser (at 532 nm) which controls the con formational state of the adsorbed BR. The 
+molecularly functionalized microcavity thus redirec ts the flow of near-infrared light beam 
+between two optical fibers. With the pump OFF, the probing light from input port 1 is detuned 
+from resonance and is directly transmitted into the  output port 2. The pump evanescently excites 
+whispering gallery modes (WGM) propagating around t he microsphere’s equator, inducing 
+photoisomerization along their path. A low green cw  laser (< 200 µW at 532 nm) is sufficient for 
+this purpose as its effective absorption is resonan tly enhanced. Isomerization reduces the retinal 
+polarizability, tuning the peak/trough of the reson ance to match the wavelength of the infrared 
+probe which is then rerouted into the output port 3  [22-24].   8
+The schematic diagram of the experimental setup for  all-optical switching with BR 
+coated microcavity is shown in the Fig. 3. Resonant  modes were excited with a distributed 
+feedback laser, operating around 1310 nm, connected  to port 1 via. single mode fiber (SMF-28, 
+Dow-Corning, Midland MI) through fiber coupler. Gre en pump beam at wavelength 532 nm and 
+a blue pump beam at wavelength 405 nm were also inj ected into port 1 of the switch. A 300 µm 
+silica microsphere was formed by melting the tip of  a single mode fiber in a butane nitrous oxide 
+flame. Three layers of BR mutant D96N (Munich Innov ative Biomaterials, Munich, Germany) 
+were adsorbed onto the microsphere surface using al ternate electrostatic deposition of cationic 
+poly(dimethyldaiallyl)ammoniumchloride (PDAC) and a nionic BR membranes. In each cycle a 
+single oriented PDAC/BR monolayer was self-assemble d onto the microcavity surface. The BR 
+adsorption process slightly degraded the microcavit y Q to ~5 x 10 5, which was caused by the 
+introduction of the scattering impurities during ea ch drying process. Two parallel, single mode 
+fibers held at 250 µm apart in a standard 1 cm acid resistant polystyre ne cuvette were tapered by 
+hydrofluoric acid erosion. Once etched, the fibers were immersed in 0.01 M phosphate buffered 
+saline with pH = 7.4. The BR coated microsphere was  then spring loaded between the two 
+tapered fibers. To determine the resonant wavelengt h, the modulation current scanned 
+periodically scanned at 100 Hz with a sawtooth shap ed function. Photodiodes PD1, PD2 
+connected to fiber ports 2 and 3 were used to monit or the transmitted intensity of the probe and a 
+spectrum containing 1000 points was recorded every ~200 ms with a LABVIEW program, as 
+shown in the schematic diagram in Fig.3. The transm itted spectra exhibited an extinction of –9.4 
+dB in port 2 and a 9.8 dB increase in transmission in port 3. In order to achieve fast switching, a 
+blue light beam is injected into the BR coated micr ocavity switch in synchronization with a 
+green pump laser beam. The blue light beam operatin g at wavelength 405 nm helps in truncating 
+the photocycle of the BR molecules at the M interme diate state, which is near to its peak  9
+absorption wavelength of 410 nm. Fig.4 shows the va riation of the signal probe beam operating 
+at 1310 nm in BR coated microcavity with the modula ting signals green and blue pump beams 
+operating at wavelengths 532 nm and 405 nm respecti vely. Switching takes place in µs as 
+compared to the previous experimental results where  switching occurs in ms. 
+III. DESIGN OF ALL-OPTICAL COMPUTING CIRCUITS 
+1. HALF-FULL ADDER/SUBTRACTOR 
+A half-adder/subtractor is the basic building block  of computing circuits. Fig.5 shows the 
+architecture of a half-adder/subtractor based on BR  coated microcavity switches S1, S2, and S3. 
+Here X and Y act as optical control signals operati ng at wavelength 532 nm with a power of 200 
+µW. A laser source (LS) is coupled at port (1) of sw itch S1. Considering the detected output 
+signal power ~ 200 µW, with an extinction of 9.8 dB [22], LS of 0.25 mW  is enough to act as an 
+incoming signal for the half-adder/subtractor. The cw light from the LS acts as an incoming 
+signal to the switch S1. When the LS is switched OF F there is no light detected in any of the 
+output ports of the architecture. When the LS is sw itched ON and both the optical control signals 
+are zero i.e. X = 0 and Y = 0, no light is incident  on the BR microcavity switches. BR is in its 
+ground state hence, the incoming signal passes from  the port (1) of switch S1 to port (1) of 
+switch S3. As Y = 0, therefore light emerges from t he port (2) of the switch S3 i.e. output port 
+OP1. The logic generated at output port OP1= X ′Y′, as shown in Table II.  
+Now, when X = 0 and Y = 1, light is incident on BR microcavity switches S2 and S3 
+respectively. It triggers the conformational change s in the BR. Thus, when an incoming signal is 
+incident on port (1) of switch S1 it emerges from p ort (2) of switch S1, and reaches incoming 
+port (1) of switch S3. Light at 532 nm is incident on switch S3; it redirects the incoming signal 
+from port (1) to the port (3) of switch S3 via whis pering gallery modes. The signal emerges from 
+the output port OP2. The logic generated at output port OP2= X′Y, as shown in Table II.  10 
+Considering the values taken by optical control sig nals as X = 1, Y = 0, when cw light is 
+incident on the incoming port of switch S1, presenc e of the light (at 532 nm) at switch S1 
+triggers the switch to route the incoming signal fr om the port (1) to the port (3) of switch S1. The 
+incoming signal reaches port (1) of the switch S2. As Y = 0, no light is incident on the switch 
+and hence the incoming signal is directed to the ou tput port OP3, via port (2) of switch S2. Thus 
+light emerges from the output port OP3. The logic g enerated at output port OP3= XY ′, as shown 
+in Table II. 
+Similarly, when optical control signals X = 1, and Y = 1 i.e. light (at 532 nm) is incident 
+on all the switches. BR in all the microcavity swit ches will undergo conformational change and 
+hence incoming signal is directed from port (1) to the port (3) of the switch S1. From there it 
+reaches the incoming port (1) of switch S2. As Y = 1, the incoming signal is again rerouted to the 
+port (3) of the switch S2 and hence to the output p ort OP4. The logic generated at output port 
+OP4= XY, as shown in Table II.   
+Logic operations generated at output ports OP1, OP2 , OP3 and OP4 are X ′Y′, X ′Y, XY ′ 
+and XY respectively as shown in Table II. Different  logic operations can be derived by 
+combining output of these ports. For e.g. if we com bine signals of output ports OP1 with OP2 it 
+results into logic operation X ′, combining OP2 with OP3 results into logic operati on of XOR. 
+Sixteen different logic operations can be derived u sing this procedure. The logic operations are 
+X, Y, X ′, Y ′ XOR ′, XOR, X+Y ′, X ′+Y, X ′+Y ′, X+Y, X ′Y, XY, X ′Y′, XY ′, T and F.  
+Working on this same principle, this tree architect ure in combination with beam splitters 
+and combiners can result in an all-optical half add er and subtractor as shown in Fig.5. Table II 
+clearly depicts the various logic operations genera ted at output ports OP1, OP2, OP3, and OP4 
+respectively. Considering Table II carefully, if we  combine light signals from output ports OP2 
+and OP3 through a beam combiner, it results into an  XOR operation which is “sum” bit of an 
+half adder. “Carry” bit is given by output port OP4 . In the case of half subtractor, the  11 
+“difference” bit is obtained by combining port OP2 and OP3, whereas “borrow” bit is given by 
+the output port OP2. Truth table of half adder and subtractor is shown in Table III. 
+To implement tree architecture for triple-input-bin ary logic we have to incorporate 
+another four BR based microcavity optical switches S4, S5, S6 and S7 as shown in Fig 6. There 
+are eight different input combinations for implemen ting this logic. Depending on the state of 
+input variables f (X, Y, Z) the output is obtained from output port OP1 to output port OP8. The 
+truth table of full adder and subtractor for three input binary variables is given in Table IV.  In 
+the case of the full addition we have two outputs o ne is “sum” and another is “carry”. Here 
+“sum” takes the expression ∑f (1, 2, 4, 7) and “carry” takes the expression ∑f (3, 5, 6, 7). In the 
+case of full subtraction we have two outputs, one i s difference and the other is “borrow”. Here 
+difference can be expressed as ∑f (1, 2, 4, 7) and “borrow” can be written as ∑f (1, 2, 3, 7). 
+Tree architecture, which has three optical control signals X, Y and Z and eight outputs 
+shown in Fig. 6, can successfully be used to implem ent an all-optical full-adder and full 
+subtractor for triple input binary logic. If we com bine the output of ports OP2, OP3, OP5 and 
+OP8, we can obtain the “sum”, whereas combination o f output ports OP4, OP6, OP7 and OP8 
+gives the carry. Similarly, the combination of outp ut ports OP2, OP3, OP5 and OP8 gives the 
+“difference” whereas the combination of output port s OP2, OP3, OP4 and OP8 also gives the 
+“borrow” in case of full subtraction. Thus, this ar chitecture works as an all-optical full adder and 
+subtractor.  
+2.  DESIGN OF DE-MULTIPLEXER AND MULTIPLEXER  
+A de-multiplexer is a circuit which is used to de-m ultiplex an incoming signal from one 
+input channel to one of the many output channels vi a select lines. The select lines decide the 
+routing of the incoming signal to the desired outpu t channel. Fig.7 shows the architecture of a de-
+multiplexer based on BR coated microcavity switches  S1, S2, and S3. Here X and Y act as select 
+lines operating at wavelength 532 nm with a power o f 200 µW. A laser source (LS) is coupled at  12 
+port (1) of switch S1. Considering the detected out put signal power ~ 200 µW, with an extinction 
+of 9.8 dB [22], LS of 0.25 mW is enough to act as a n incoming signal for the de-multiplexer. The 
+cw light from the LS acts as an incoming signal to the switch S1. When the LS is switched OFF 
+there is no light detected in any of the output por ts of the architecture. When the LS is switched 
+ON and both the select lines are zero i.e. X = 0 an d Y = 0, no light is incident on the BR 
+microcavity switches. BR is in its ground state hen ce, the incoming signal passes from the port 
+(1) of switch S1 to port (1) of switch S3. As Y = 0 , therefore light emerges from the port (2) of 
+the switch S3 i.e. output port OP1 as shown in Tabl e V.  
+Now, when X = 0 and Y = 1, light is incident on BR microcavity switches S2 and S3 
+respectively. It triggers the conformational change s in the BR. Thus, when an incoming signal is 
+incident on port (1) of switch S1 it emerges from p ort (2) of switch S1, and reaches incoming 
+port (1) of switch S3. Light at 532 nm is incident on switch S3; it redirects the incoming signal 
+from port (1) to the port (3) of switch S3 via whis pering gallery modes. The signal emerges from 
+the output port OP2. 
+Considering the values taken by select lines as X =  1, Y = 0, when cw light is incident on 
+the incoming port of switch S1, presence of the lig ht (at 532 nm) at switch S1 triggers the switch 
+to route the incoming signal from the port (1) to t he port (3) of switch S1. The incoming signal 
+reaches port (1) of the switch S2. As Y = 0, no lig ht is incident on the switch and hence the 
+incoming signal is directed to the output port OP3,  via port (2) of switch S2. Thus light emerges 
+from the output port OP3. 
+Similarly, when X = 1, and Y = 1 i.e. light (at 532  nm) is incident on all the switches. BR 
+in all the microcavity switches will undergo confor mational change and hence incoming signal is 
+directed from port (1) to the port (3) of the switc h S1. From there it reaches the incoming port (1) 
+of switch S2. As Y = 1, the incoming signal is agai n rerouted to the port (3) of the switch S2 and 
+hence to the output port OP4. Table V clearly depic ts the various states of the routing signals of  13 
+1:4 de-multiplexer according to the select lines. T o realize 1:4 de-multiplexer we need three 
+switches. The scheme can be easily extended for n l ines with 2n-1 switches. The whole circuit 
+operates with very low power.   
+On the other hand multiplexer is a combinatorial ci rcuit which is used to multiplex 
+incoming signals from a number of input channels to  a single output channel depending on the 
+value of the select lines. Fig.8 shows a schematic of the architecture of a low power all-optical 
+multiplexer. Consider IP1, IP2, IP3 and IP4 as inco ming ports and X and Y as select lines. The 
+incoming signals are multiplexed and sent to the ou tput port according to the select lines. The 
+output port acts as the port for the outgoing signa l in the multiplexer case. The architecture can 
+be expanded to multiplex n number of lines, for whi ch we need 2 n-1 switches. Table VI 
+represents the various states of the multiplexer. I t works in a similar manner as the de-
+multiplexer. 
+Let us consider the case when all the input ports h ave incoming signals i.e. IP1 = IP2 = 
+IP3 = IP4 = LS, and X = 0, Y = 0. LS operate at wav elength 1311 nm. As Y = 0, incoming signal 
+from incoming port IP1 moves to the port (2) of the  switch S3, from there it is directed to the 
+incoming port (1) of the switch S1. Here, X = 0 the refore, incoming signal from IP1 emerges 
+from the port (2) of switch S1. Incoming signal is detected at the output port OP. Incoming 
+signal from IP2 doesn’t reach incoming port (1) of switch S1 as Y = 0. Incoming signal from IP3 
+moves from port (1) to port (2) of switch S2, but w hen it reaches switch S1, it is not directed to 
+the output port as X= 0. Similarly, input signal fr om IP4 is not directed to the port (2) of the 
+switch S2 as Y = 0 and it is not able to reach the output port OP of the switch S1.  
+Now, let us consider X = 0 and Y = 1 state. In this  case, the incoming signal moves from 
+port IP2 to the incoming port of the switch S1, as Y = 1, switch S3 directs the incoming signal 
+from IP2 to the incoming port (1) of switch S1. Her e X = 0, thus incoming signal flows from  14 
+incoming port (1) to the port (2) of switch S1 and the signal is detected at output port OP. 
+Incoming signal from IP1, IP3, and IP4 does not rea ch the output port of the switch S1. 
+When X = 1 and Y = 0, the incoming signal moves fro m port IP3 to the port (2) of the 
+switch S2, from where it is directed to the output port OP of the switch S1 via whispering gallery 
+modes. Other incoming signal from IP1, IP2, and IP4  does not reach output port of the switch 
+S1. Similarly, when X = 1 and Y = 1, the incoming s ignal from IP4 is directed to the port (2) of 
+the switch S2, from where it is directed to the out put port OP of switch S1. Incoming signals 
+from other ports are not directed to the output por t OP of the switch S1. The select lines direct 
+the flow of the desired incoming signal to the outp ut channel and thus incoming signals can be 
+easily multiplexed.  
+3. ARITHMETIC UNIT  
+Arithmetic unit (AU) is used to perform various ari thmetic operations combined in one 
+circuit. The number of functions performed by the A U depends on the number of select inputs. 
+The operational principle of an all-optical arithme tic unit is explained by exploiting BR 
+microcavity based all-optical switches, full-adders  and multiplexers. The output of a BR based 
+microcavity switch can be used as a control switch for the other. The optical circuit shown in the 
+Fig. 9 can perform eight different arithmetic funct ions with two four bit numbers X (X 3X2X1X0) 
+and Y (Y 3Y2Y1Y0). It gives the output Z (Z 3Z2Z1Z0) depending on the value of three select inputs 
+(S 1, S 0 and C in ). This scheme can be extended upto n bit numbers b y using BR based microcavity 
+switches, MUX and full-adders. Let us consider an e xample where X=1011 (X 3X2X1X0) and Y = 
+1001 (Y 3Y2Y1Y0). Eight different operations with four subsidiary functions are described and 
+shown in Table VII in detail. 
+For AU to perform four bit operations as shown in t he Fig.9 consist of BR based 
+microcavity switches S1, S2,…, S13, multiplexers MU X-1,…, MUX-4, and full adders FA-1,…, 
+FA-4 respectively. Select inputs S 1, S 0, C in  operate at wavelength 532 nm. Input variable X als o  15 
+operates at wavelength 532 nm. Input variable Y ope rates with wavelength 1311 nm. 
+Wavelength converters are used to realize the desig n of AU. Select inputs determine the 
+arithmetic operation performed by the AU. 
+Case: 1  a) When S 1 = 0, S 0 = 0 and C in  = 0, as the select lines are 0 hence input line IP 0 of all the 
+multiplexers (MUXs) will be active. IP0 of all the MUX is connected to the value of Y. It directs 
+the value of Y 0 to the output port OP0. In the same way values of Y 1, Y 2 and Y 3 will be sent to 
+the output ports OP1, OP2, OP3 respectively via. in put ports. 
+The output of each MUX OP0, OP1, OP2, and OP3 is co nnected to one input (i.e. Y) of the full 
+adder, FA-1, FA-2, FA-3 and FA-4 respectively. All the full adders receive the other input i.e. X 
+directly. In this case X 0 is directly connected to one input (i.e. X) of the  FA-1, carry in (C in  = 0). 
+Here the select input C in  = 0 and the other two inputs X 0 = 1, Y 0 = 1. So Z 0 takes the value zero 
+i.e. Z 0 = 0 and carry out i.e. C out  = 1. Now the carry out of the FA-1 is fed into the  input of the 
+FA-2. In this way C in  of the full-adders FA-2, FA-3, FA-4, receive the v alue C out  of the FA’s FA-
+1, FA-2 and FA-3 respectively. So, in FA-2, C in =1 and the other two inputs (X and Y) receive 
+the value 1 and 0 (as X 1 = 1, Y 1 = 0). In this case output Z 1 = 0 and C out  = 1. Similarly in FA-3, 
+Cin  = 1, and the other two inputs (X and Y) receive th e values 0 and 0 (as X 2 = 0 and Y 2 = 0). 
+Hence Z 2 = 1 and C out  = 0. Finally in FA-4, C in  = 0, and the other two inputs (X and Y) receive 
+the values 1 and 1 (as X 3 = 1 and Y 3 = 1), so Z 3 = 1 and C out  = 1. Here the output Z 4 takes the 1 
+as C out  = 1. The final output (Z) is 10100 (Z 4Z3Z2Z1Z0) which verifies the binary addition (X+Y) 
+of two four bit numbers for select inputs S 1 = 0, S 0 = 0 and C in  = 0. 
+b) When S 1 = 0, S 0 = 0 and C in  = 0, and one of the input i.e. X = 0000 than the o utput is merely 
+Y. 
+Case: 2  a) When S 1 = 0, S 0 = 0 and C in  = 1. Here the select inputs are S 1 = 0, S 0 = 0, so all the 
+operations are similar to that of case 1 a), with a  difference that the carry in C in  of the FA is 1 as 
+the select input C in  = 1. The other two inputs receive the values 1 (X 0 = 1, Y 0 = 1). So, the output  16 
+of FA-1 is 1 Z 0 = 1 and C out  = 1. Other FA’s perform in a similar fashion and t he final output is 
+10101 (Z 4Z3Z2Z1Z0). So it performs the operation of addition with ca rry. 
+b) When S 1 = 0, S 0 = 0 and C in  = 1, and one of the input i.e. X = 0000 than the f unction generated 
+by the arithmetic unit is Y+1. 
+Case: 3  a) When S 1 = 0, S 0 = 1 and C in  = 0. As the select inputs S 1 = 0, S 0 = 1, the input data line 
+IP1 will be selected for all the MUXs. Light form L S at wavelength 1311 nm is incident on 
+switch S2. As the select input Y 0 = 1 the light emerges from the port P4. In this cas e output port 
+P4 receives the light whereas P3 does not receive a ny light. So, P3 is in the zero state. i.e. P3 = 0.  
+It is the complement of Y 0. Similarly, Y 1, Y 2, Y 3 acts as control input of the switches S5, S8 and 
+S11. The ports P7, P11 and P15 receive the values t hat are the complement values of Y 0, Y 1, Y 2 
+and Y 3 respectively. So the complement values of Y 0, Y 1, Y 2 and Y 3 are fed into one input Y 
+(i.e.Y) of full adder FA-1, FA-2, FA-3 and FA-4 res pectively. Here X 0, X 1, X 2 and X 3 act as one 
+input i.e. X of the FA’s FA-1, FA-2, FA-3 and FA-4.  Now in FA-1, carry in C in  = 0 and the input 
+X receives the value 1 (as X 0 = 1), and the input Y receives the value 0, which is the complement 
+of Y 0 (as Y 0 = 1). According to the operation principle of the full adder the output sum of FA-1 
+is one (i.e. Z 0 = 1) and the carry out is zero (i.e. C out  = 0). Again in FA-2, C in  = 0 (as C out of the 
+FA-1 is 0). The input X receives the value 1 (as X 1 = 1) and the input Y receives the value 1, 
+which is the complement of Y 1 (as Y 1 = 0). So, Z 1 = 0 and C out  = 1. Similarly in FA-3, C in  = 0 
+and the input X receives the value 0 (as X 2 = 0), and the input Y receives the value 1 which i s the 
+complement of Y 2 (as Y 2 = 0). Hence IP2 = 0, and C out =1. Finally in FA-4, C in  = 1 i.e. Z 4 = 1. So 
+the final output Z is 10001 (Z 4Z3Z2Z1Z0) which also verifies the third operation (X + Y ′) as given 
+in table for select inputs S 1 = 0, S 0 = 1 and C in  = 0. 
+b) When S 1 = 0, S 0 = 1 and C in  = 0, and one of the input i.e. X = 0000, the funct ion generated at 
+the output is Y ′.  17 
+Case: 4  a) When S 1 = 0, S 0 = 1 and C in  = 1. All the operations are similar to that of cas e 3 a), 
+with a difference that the carry in of the full add er FA-1 is 1 as the select input C in  = 1. 
+Therefore, the input X receives the value 1 (as X 0 = 1) and the input Y receive the value 0, which 
+is the complement of Y 0 (as Y 0 = 1). So output of Z 0 of FA-1 is zero i.e. Z 0 = 0, and the carry out 
+is one i.e. C out  = 1. Other output Z as 10100. In this case the opt ical circuit uses 2’s complement 
+method of subtraction. As the final carry (C out ) from FA-4 is one (i.e. Z 4 = 1) that means the 
+result will be positive and hence the final carry i s to be discarded. Hence, the final result is 
+positive as its value is 0010 (Z 3Z2Z1Z0), which also verifies the 2’s complement subtracti on 
+operation (X + Y ′ + 1) for select inputs (S 1 = 0, S 0 = 1 and C in  = 1) as given in table. Now if we 
+consider X = 1001 (X 3X2X1X0) and Y = 1011 (Y 3Y2Y1Y0), then the optical circuit generates the 
+output Z as 01110 (Z 4Z3Z2Z1Z0). As the final carry is 0 (i.e. Z 4 = 0) that means the result will be 
+negative and hence the final carry Z 4 is to be discarded. Also taking the 2’s complement  of 1110 
+(Z 3Z2Z1Z0). So, the final result is negative as its value is  1110 (2’s complement of 1110). 
+b) When S 1 = 0, S 0 = 1 and C in  = 0, and one of the input i.e. X = 0000, the funct ion generated at 
+the output is Y ′ + 1. 
+Case: 5  When S 1 = 1, S 0 = 0 and C in  = 0. As the select inputs S 1 = 1, S 0 = 0, the input data line 
+IP2 will be selected for all the MUXs. Light from L S is incident on switch S1. According to the 
+switching principle, Port P1 will receive the value , which will be the complement of C in  (i.e. 
+Cin ′). Again X 0, X 1, X 2 and X 3 act as control inputs for switches S3, S6, S9 and S 12 respectively. 
+Port P1 (i.e. C in ′) is connected to switches S4, S7, S10 and S13. Acc ording to the switching 
+principle, ports P6, P10, P14 and P18 receive the v alues which will be equal to XC in ′ and 
+connected to IP2 line of all the MUXs receiving the  values equal to X 0, X 1, X 2 and X 3 
+respectively (because XC in ′ = X as C in  = 0), and also generate the values as X 0, X 1, X 2 and X 3 
+respectively. Now in FA-1 carry in i.e. C in  = 0 (as select inputs C in  = 0), and the other two inputs 
+(X and Y) receive the same value (as X 0 = 1 as independent of Y 0). So, the FA-1 generates the  18 
+output as zero i.e. Z 0 = 0, and the carry out C out  as one i.e. C out  = 1. Again in FA-2 C in  =1 and the 
+other two inputs receive the same value 1 (as X 1 = 1). So in this case Z 1 takes the value 1 i.e. Z 1 = 
+1 and the carry out receives the value 1 i.e. C out  = 1. Similarly in FA-3 and FA-4 values of X and 
+Y propagate in the same manner. So the final output  Z is 10110 (Z 4Z3Z2Z1Z0), which verifies the 
+double operation (2X) of a four bit number (X) for select inputs (S 1 = 1, S 0 = 0 and C in  = 0) as 
+given in Table VII. 
+Case: 6  When S 1 = 1, S 0 = 0 and C in  = 1. Here also the select inputs S 1 = 1, S 0 = 0, the input data 
+line IP2 will be selected for all the MUXs. As the select inputs C in  = 1, the input data line IP2 of 
+all the MUXs receive the value 0. (XC in  = 0 as select input C in = 1). Hence one input i.e. Y of all 
+the full adders receives the value 0. So, in FA-1, C in =1 (as the select input C in  =1), and the two 
+other inputs (X and Y) receive the values 1 and 0 ( as X 0 = 1 and X 0Cin ′ = 0. So the output of the 
+FA-1 is 0 i.e. Z 0 = 0 and the carry out C out  is one. Again in FA-2, C in  = 1, and the other two 
+inputs (X and Y) receive the values 1 and 0 (as X 1 = 1 and X 1Cin ′ = 0). So in this case Z 1 takes 
+the value 0, Z 1 = 0 and the carry out receives the value 1 i.e. C out  = 1. Similarly in FA-3, Cin = 1, 
+and the other two inputs (X and Y) receives the val ues 0 and 0 (as X 2 = 0 and X 2Cin ′ = 0). Hence 
+Z2 = 1 and C out = 0. Finally in FA-4, Cin = 0, and the other two in puts (X and Y) receive the 
+values 1 and 0 i.e. Z 4 = 0. So, the final output Z is 01100 (Z 4Z3Z2Z1Z0), which also verifies the 
+increment operation (X + 1) of a four bit number (X ) for select inputs (S 1 = 1, S 0 = 0 and C in  =1) 
+as given in Table VII. 
+Case: 7  When S 1 = 1, S 0 = 1 and C in  = 0. As the select inputs S 1 = 1, S 0 = 1, the input data line 
+IP3 will be selected for all the MUXs. Here the IP3  line of all the MUXs is connected to the 
+control signal which is always in one (high) state.  So all the MUXs generate the output as 1. 
+Thus one input (Y) of all the full adders receives the value 1. Now in FA-1, C in  = 0 (as the select 
+input C in  = 0), and input X receives the value 1 (X 0 = 1) and input Y receives the value 1 (as 
+MUX-1 generates output 1). According to the operati onal principle of the full adder, it generates  19 
+the output as zero i.e. Z 0 = 0 and the carry out as one i.e. C out  = 1. Again in FA-2, C in  = 1 and 
+input X receives the value 1 (as MUX-2 gives output  1). In this case it gives output as zero, i.e. 
+Z1 = 1 and the carry out i.e. C out  = 1. Similarly in FA-3, C in  = 1, and input X receives the value 0 
+(as X 2 = 0) and input Y receives the value 1 (as MUX-3 ge nerates output 1). Hence the output Z 2 
+= 0 and C out  = 1. Finally in FA-4 C in  = 1, and input X receives the value 1 (as X 3 = 1) and input 
+Y receives the value 1 (as MUX-4 gives output 1). S o Z 3 = 1 and C out  = 1, i.e. Z 4 = 1. The circuit 
+generates the output Z as 11010 (Z 4Z3Z2Z1Z0). In this case circuit uses 2’s complement method 
+for subtraction for decrement (X - 1) of a four bit  number for select inputs (S 1 = 1, S 0 = 1, C in  = 
+0) as given in the Table VII.  
+Case: 8  When S 1 = 1, S 0 = 1 and C in  = 1. Here also the select inputs are 1: so all the  operations 
+are similar to that of case 7 with a difference tha t the carry in C in  of the FA-1, C in  = 1 (as the 
+select input C in  = 1), and input X receives the value 1 (as X 0 = 1) and input Y receives the value 
+1 (as MUX-1 generates output 1), it generates the o utput one, i.e. X 0 = 1 and the carry out is one 
+i.e. C out  = 1. Again in FA-2, C in  = 1 and the other two inputs (X and Y) receive the  values 1 and 1 
+(as X 1 = 1 and MUX-2 gives output = 1). In this case it g ives the output 1, i.e. Z 0 = 1 and carry 
+out i.e. C out  = 1. Similarly in FA-3 and FA-4 the value of X and  Y propagate in the same manner. 
+The circuit gives the output Z as 11011 (Z 4Z3Z2Z1Z0). As Z (Z 4Z3Z2Z1Z0) is a four bit number 
+and the final carry is to be discarded. So the fina l result is 1101 (Z 3Z2Z1Z0) which verifies the 
+transfer operation of a four bit number for select inputs (S 1 = 1, S 0 = 1 and C in  = 1) as given in 
+Table VII. In this way we can perform twelve (eight  main and four subsidiary) operations.  20 
+IV. RESULTS AND DISCUSSION  
+All-optical switching in BR coated microsphere has been demonstrated experimentally. 
+In the BR coated microcavity switch, the switching of incoming signal operating at wavelength 
+1310 nm between the output ports (2 and 3) is photo -induced with a fiber-coupled green pump 
+laser (at 532 nm) which controls the conformational  state of the adsorbed BR. In order to achieve 
+fast switching, blue light beam is injected into th e BR coated microcavity switch in 
+synchronization with a green pump laser beam (at 53 2 nm). The blue light beam operating at 
+wavelength 405 nm helps in truncating the photocycl e of the BR molecules at the M intermediate 
+state, which is near to its peak absorption wavelen gth of 410 nm.  
+This basic all-optical BR coated microcavity switch  has been used to design higher 
+computing circuits. All the designs presented in th is paper are all-optical in nature. The proposed 
+designs are simple and can easily implement multipl e functions. The designs have good 
+scalability. The proposed design of AU does not req uire erbium-doped fibre amplifiers for signal 
+regeneration and uses less number of wavelength con verters. The switching of near IR 
+wavelength (1310 nm) probe with the controlled pump  operating at 532 nm may find potential 
+applications in the telecommunication as well. The microresonator cavity can be tailored to 
+effectively switch at a desired wavelength. Moreove r, with the powerful capabilities of nano-
+biotechnological techniques, the desired response o f BR molecules can also be tailored to meet 
+device specifications. Cascading can be easily done  as BR based microcavity switches dissipate 
+very less power. Moreover, Q-factor of these microc avity switches is very high which makes 
+these designs highly sensitive. Coating the microca vities with a photosensitive material is of 
+critical value, as the material should exhibit high  sensitivity, absorption, fast dynamics, photo 
+and thermal stability and potential to tailor its p roperties. BR protein is a natural photochromic 
+material that exhibits this unique combination of p roperties for practical realization. Although 
+proposed designs can be realized with dual-ring swi tches and other interferometric  21 
+configurations, single BR based microcavity switch has advantages in terms of simple geometry, 
+ease of fabrication, high thermal and photo stabili ty, high fan out, cost-effectiveness and low 
+power operation. 
+V. CONCLUSION  
+In this paper, we have shown experimentally all-opt ical switching in BR coated 
+microcavity switch. Blue light beam (at 405 nm) is injected into the BR coated microcavity 
+switch in synchronization with a green pump laser b eam (at 532 nm) for fast switching. The blue 
+light beam operating at wavelength 405 nm helps in truncating the photocycle of the BR 
+molecules at the M intermediate state, which is nea r to its peak absorption wavelength of 410 
+nm. We have also presented a theoretical scheme whi ch combines the advantages of both BR 
+based microcavity switch and tree architecture for designing low power all-optical, half 
+adder/subtractor, de-multiplexer, multiplexer and a rithmetic unit. These schemes can easily be 
+extended and implemented for any higher number of i nput digits, by proper interconnection of 
+BR based microcavity switches using vertical and ho rizontal extension of the tree and by suitable 
+branch selection. Due to the high-Q factor and smal l dimensions, switching at low power is 
+possible with these BR coated microcavities. They r epresent an alternative to the waveguide 
+based techniques providing exceptional sensitivity and straightforward optical integration on 
+micron scales. The proposed designs can yield large  computing circuits and networks within 
+mW-W power budget. The architectures with BR microc avity switches have an advantage that 
+they require less number of switches and have low p ower consumption as compared to other tree 
+based architectures for realization of these higher  computing circuits. To realize n bit 
+computation the proposed design requires only 2 n-1 switches. 
+The change in the refractive index in a BR based mi crocavity switch depends on the 
+wavelength of the control beam making the switches tunable. The combined advantages of high 
+Q-factor, tunability, compactness and low power con trol signals of BR coated microcavities  22 
+along with the flexibility of cascading switches in  tree architectures to form circuits makes the 
+designs promising for practical applications. The p roposed designs proposes a new paradigm for 
+all-optical computing based on hybrid nano-bio-phot onic photonic integrated devices employing 
+biomolecules to perform photonic functions.   23 
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+ 
+TABLE I. TRUTH TABLE OF FIG. 2 
+Incoming Signal Control Signal Port (3) Port (2) 
+0 0 0 0 
+0 1 0 0 
+1 0 0 1 
+1 1 1 0 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+TABLE II. STATE OF DIFFERENT OUTPUT PORTS FOR DIFFE RENT VALUES OF X 
+AND Y IN A TREE ARCHITECTURE. 
+Input Output at different Ports 
+X Y OP1 
+(X ′Y′) OP2 
+(X ′Y) OP3 
+(XY ′) OP4 
+(XY) 
+0 0 1 0 0 0 
+0 1 0 1 0 0 
+1 0 0 0 1 0 
+1 1 0 0 0 1 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  27 
+ 
+ 
+ 
+TABLE III. TRUTH TABLE OF HALF-ADDER AND SUBTRACTOR . 
+ 
+ 
+ 
+ 
+        
+ 
+ 
+ 
+TABLE IV. TRUTH TABLE OF FULL ADDER AND SUBTRACTOR.  
+ 
+Input Output 
+X Y Z Full-Adder Full-Subtractor 
+   Sum Carry Difference  Borrow 
+0 0 0 0 0 0 0 
+0 0 1 1 0 1 1 
+0 1 0 1 0 1 1 
+0 1 1 0 1 0 1 
+1 0 0 1 0 1 0 
+1 0 1 0 1 0 0 
+1 1 0 0 1 0 0 
+1 1 1 1 1 1 1 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ Input Output of Half adder Output of Half Subtracto r 
+X Y Sum Carry Difference Borrow 
+0 0 0 0 0 0 
+0 1 1 0 1 1 
+1 0 1 0 1 0 
+1 1 0 1 0 0  28 
+ 
+ 
+ 
+TABLE V. TRUTH TABLE OF 1: 4 DE-MULTIPLEXER 
+ 
+Incoming 
+Signal Select 
+Inputs/Control 
+Signals Data Outputs 
+LS X Y OP1  OP2  OP3  OP4  
+0 0 0 0 X X X 
+1 0 0 1 X X X 
+0 0 1 X 0 X X 
+1 0 1 X 1 X X 
+0 1 0 X X 0 X 
+1 1 0 X X 1 X 
+0 1 1 X X X 0 
+1 1 1 X X X 1 
+ 
+ 
+ 
+ 
+TABLE VI. TRUTH TABLE FOR 4 : 1 MULTIPLEXER 
+  
+Data Inputs Select 
+Inputs/Control 
+Signals Outgoing 
+Signal 
+IP1 IP2 IP3 IP4 X Y O/S 
+0 X X X 0 0 0 
+1 X X X 0 0 1 
+X 0 X X 0 1 0 
+X 1 X X 0 1 1 
+X X 0 X 1 0 0 
+X X 1 X 1 0 1 
+X X X 0 1 1 0 
+X X X 1 1 1 1 
+ 
+ 
+  29 
+ 
+ 
+ 
+TABLE VII. ARITHMETIC FUNCTIONS REALIZED USING ALL- OPTICAL AU 
+ 
+Functions Generated Select Inputs Inputs : X, Y 
+Output : Z (Z 4 Z3 Z2 Z1 Z0) S1 S0 C in  
+Binary-addition 0 0 0 Z = X + Y, 
+Z = Y; if X = 0 
+Addition with Carry 0 0 1 Z = X + Y + 1, 
+Z = Y + 1; if X = 0 
+Subtract with borrow 0 1 0 Z = X + Y ′, 
+Z = Y′; if X = 0 
+Subtract (2’s complement) 0 1 1 Z = X + Y ′ + 1, 
+Z = Y′ + 1; if X = 0 
+Double of X 1 0 0 Z = 2 X 
+Increment of X 1 0 1 Z = X + 1 
+Decrement of X 1 1 0 Z = X – 1 
+Transfer of X 1 1 1 Z = X 
+   30 
+FIGURE CAPTIONS 
+1.  Schematic of the photochemical cycle of BR molecule . Subscripts indicate absorption 
+peaks in nm. Solid and dashed arrows represent ther mal and photo-induced transitions 
+respectively. 
+2.  Schematic of BR coated microcavity switch. 
+3.  Experimental setup of all-optical switching in BR c oated microcavity switch. 
+4.  Variation of the signal probe beam (operating at 13 10 nm) with the modulating green and 
+blue laser pump beams operating at wavelengths 532 nm and 405 nm respectively. 
+5.  Design of an all-optical half-adder/half-subtractor  circuit (B.S = Beam Splitter, B.C = 
+Beam Combiner, LS = Laser Source) 
+6.  Design of an integrated all-optical full-adder and subtractor circuit (B.S = Beam Splitter, 
+*B.C = Beam Combiner, OP = Output Port, LS = Laser Source).  
+7.  Design of de-multiplexer with BR based microcavity switch (LS = laser source). 
+8.  Design of multiplexer with BR based microcavity swi tch (B.S = Beam Splitter, LS = 
+Laser Source). 
+9.  Design of all-optical arithmetic unit (MUX = Multip lexer, FA = Full-Adder).  31 
+ 
+ 
+Fig.1.  32 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+  
+                                                                Fig.2 
+Port (3) 
+Port (1) Port (2) BR coated 
+ Microsphere 
+Control Signal  33 
+ 
+ 
+ 
+ 
+ 
+   
+  PD1   
+Oscilloscope 
+  
+Pump Laser (532 nm)  Probe Laser (1310 nm)  
+ Tunable DFB   
+   PD2  
+Fiber  
+Coupler  SMF -28  
+Cell BR coated Microsphere 
+ Pump Laser (405 nm)  
+Fig. 3  34 
+ 
+Fig. 4 
+  
+ 
+  35 
+ 
+ 
+ 
+ Output Port (OP4) 
+(Carry) 
+Output Port (OP3 ) 
+ (Sum/Diff) 
+Output Port (OP1) 
+Signal X Signal Y LS  (S1)  (S2) 
+ (S3) B.S 
+. 
+B.S 
+. 
+B.S.   
+Fig. 5 200 µW 
+200 µW 
+400 µW 200 µW Incoming  
+Signal B.C 
+. 
+Optical Control Signals 
+Output Port (OP2) 
+Borrow 
+ 36 
+ Sum/  
+Difference Signal X Signal Y (S1) 
+(S3) (S2)  
+B.S.  
+Signal Z B.C *B.C 
+Borrow Carry 
+*B.C = Beam Combiner, OP = Output Port  S4  
+S5  
+S6 
+S7  OP1  OP2 OP3 OP5 OP6 OP7 OP8  
+Fig. 6 200 µW 
+200 µW 
+200 µW 
+800 µW 200 µW 
+200 µW 200 µW 
+400 µW 200 µW 
+LS  
+Incoming  
+Signal OP4 
+Optical Control Signals 
+ 37 
+ Output Port (OP1) Output Port (OP2) Output Port (OP3 ) Output Port (OP4)  
+ Signal X Signal Y (S1) 
+ (S3)  (S2) 
+Beam Splitter LS   
+Fig. 7 200 µW 
+200 µW 
+400 µW 200 µW 
+Select Lines Incoming  
+Signal 
+ 38 
+ Input Port (IP4) 
+ Signal X Signal Y (S1) 
+ (S3)  (S2) 
+Beam Splitter  
+Fig . 8  200 µW 
+200 µW 
+400 µW 200 µW 
+Select Lines Output 
+Port (OP) 
+LS  
+Incoming Signal 
+(at each Input Port) Input Port (IP1) Input Port (IP2) Input Port (IP3 ) 
+   
+ 
+ 39 
+ 
+Fig. 9 
+ 
\ No newline at end of file
diff --git a/1001.0245.txt b/1001.0245.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6f510f82345b4b4c955f5d9dd8f2faf55a9b76c7
--- /dev/null
+++ b/1001.0245.txt
@@ -0,0 +1,447 @@
+arXiv:1001.0245v1  [math.PR]  1 Jan 2010Bayesian nonparametric analysis for a species sampling mod el
+with finitely many types∗
+Annalisa Cerquetti†
+Department of Geoeconomical, Linguistic, Statistical and Historical Studies ,
+Sapienza University of Rome, Italy
+Abstract
+We derive explicit Bayesian nonparametric analysis for a species samp ling model with
+finitely many types of Gibbs form of type α=−1 recently introduced in Gnedin (2009).
+Our results complement existing analysis under Gibbs priors of type α∈[0,1) proposed
+in Lijoi et al. (2008). Calculations rely on a groups sequential const ruction of Gibbs
+partitions introduced in Cerquetti (2008).
+1 Introduction
+In thespecies sampling problem a random sample is drawn from an hypothetically infin ite
+population of individuals to make inference on the unknown t otal number of different species.
+A Bayesian nonparametric approach to this problem has been r ecently proposed in Lijoi et
+al. (2007, 2008) to derive posterior predictive inference o n species richness for an additional
+sample under the assumption that the vector of multipliciti es of different species observed is
+a random sample from an exchangeable Gibbs partition of type α∈[0,1).
+Exchangeable Gibbs partitions of type α∈[−∞,1) (Gnedin and Pitman, 2006) are models
+for random partitions of the positive integers which extend the Ewens-Pitman two-parameter
+family and are characterized by a probability function (EPP F) with the following structure
+p(n1,...,n k) =Vn,kk/productdisplay
+j=1(1−α)nj−1↑, (1)
+with weights ( Vn,k) being solution to the backward recursion Vn,k= (n−αk)Vn+1,k+Vn+1,k+1
+withV1,1= 1. Gnedin and Pitman even provide a constructive result for this class (cfr. Th.
+12) showing that each element arises as a unique probability mixture of extreme partitions
+which are
+∗AMS (2000) subject classification . Primary: 60G58. Secondary: 60G09.
+†Corresponding author: Annalisa Cerquetti, Dept. Studi Geo economici, Facolt` a di Economia, Sapienza Uni-
+versit` adiRoma, ViadelCastroLaurenziano, 9, 00161, Rome , Italy. E-mail: annalisa.cerquetti@uniroma1.it
+1a)PD(α,ξ|α|)partitions with ξ= 1,...,∞forα∈[−∞,0),
+b)PD(0,θ)partitions with θ∈[0,∞) forα= 0,
+c)PK(ρα|t)partitions with t∈[0,∞) forα∈(0,1).
+HerePD(·,·) stands for the two-parameter Poisson-Dirichlet distribu tion (Pitman and Yor,
+1997) and PK(ρα|t) for the conditional Poisson-Kingman distribution derive d from the stable
+subordinator, (cfr. Pitman, 2003).
+Lijoi et al. (2008) establish general distributional resul ts and properties for EPPFs belonging
+to subclass c) to make conditional predictions according to a Bayesian nonparametric pro-
+cedure. Focusing on class c) their analysis relies on the hyp othesis that the total unknown
+number of species is so large to be assumed infinite. An assump tion that may be unrealistic in
+concrete applications. Here we focus on a class of random par titions with finite but random
+number of different species, recently introduced in Gnedin (2 009), which belongs to subclass
+a) and contribute, in view of possible future Bayesian imple mentations, deriving posterior
+predictive results analogous to that in Lijoi et al. (2008).
+First recall that for α <0,θ=|α|ξandξ= 1,2,3,... PD(α,ξ|α|) model has EPPF (see
+e.g. Pitman, 1996, 2006)
+p(n1,...,n k) =(ξ|α|−|α|)k−1↓|α|
+(ξ|α|+1)n−1↑1k/productdisplay
+j=1(1+|α|)nj−1↑ (2)
+or equivalently
+p(n1,...,n k) =(ξ)k↓
+(|α|ξ)n↑1k/productdisplay
+j=1(|α|)nj↑,
+and arises by the following sequential procedure. Given the partition of [ n] inKn=kblocks
+with occupancy counts n= (n1,...,n k), the partition of [ n+1] is obtained through one-step
+prediction rules
+pj(n) =nj+|α|
+n+|α|ξand p0(n) =|α|(ξ−k)
+ξ|α|+n
+wherepj(n) forj= 1,...,kstands for the probability of randomly placing n+ 1 in an old
+blockj, whilep0(n) stands for the probability of n+1 to form a new block k+1. For α=−1
+the EPPF in (2) reduces to
+p(n1,...,n k) =(ξ−1)k−1↑−1
+(ξ+1)n−1k/productdisplay
+j=1nj↑
+or alternatively
+p(n1,...,n k) =(ξ)k↓
+(ξ)n↑1k/productdisplay
+j=1nj↑
+2with one-step prediction rules
+pj(n) =nj+1
+n+ξand p0(n) =(ξ−k)
+n+ξ.
+The corresponding limit frequencies ( ˜Pξ,1,...,˜Pξ,ξ) have a stick-breaking representation
+˜Pξ,j=Wjj−1/productdisplay
+i=1(1−Wi),with independent Wi∼Beta(2,ξ−i)
+fori= 1...,ξand Beta (2,0) a Dirac mass at 1.
+2 Gnedin’s model with finitely many types
+As from Gnedin and Pitman result, PD(α,ξ|α|) models are extreme points of a convex set of
+Gibbs partitions of type α <0, whose elements are in one to one correspondence with a set o f
+mixing distributions over the set of the positive integers. Gnedin (2009) studies the particular
+model arising by mixing a PD(α,|α|ξ) model for α=−1 overξwith
+P(K=ξ) =γ(1−γ)ξ−1
+ξ!
+forξ= 1,2,...andγ∈(0,1) and shows it has weights
+Vn,k=(k−1)!
+(n−1)!(1−γ)k−1(γ)n−k
+(1+γ)n−1=(k−1)!
+(n−1)!γ(1−γ)k−1
+(γ+n−k)k, (3)
+hence EPPF
+p(n1,...,n k) =(k−1)!
+(n−1)!(1−γ)k−1(γ)n−k
+(1+γ)n−1k/productdisplay
+j−1nj! (4)
+obtained by sequential construction with one-step predict ion rules
+pj(n) =(n−k+γ)(nj+1)
+n(n+γ)forj= 1,...,k andp0(n) =k(k−γ)
+n(n+γ).
+Gnedin also derives analogous of the Ewens sampling formula and further results on the
+vector of frequencies and of exchangeable sequences induce d by sampling from this model.
+For what follows it is worth to notice that Gnedin’s one-step prediction rules may be
+equivalently expressed as m-steps prediction rules as in Cerquetti (2008, Prop. 3), whi ch here
+we formulate as a group sequential random allocation of ball s labelled 1 ,2...in a series of
+boxes. First from (3) we obtain
+Vn+m,k+k∗=(k+k∗−1)!
+(n+m−1)!(1−γ)k+k∗−1(γ)n+m−k+k∗
+(1+γ)n+m−1,
+3then by basic properties of rising factorials specializati on of the general formulas for Gibbs
+partitions easily follow. Start with box B1,1with a single ball. Given the placement of the first
+group of nballs in a ( n1,...,n k) configuration in kboxes, the newgroup of mballs labelled
+{n+1,..., n+m }is:
+a) allocated in the oldkboxes in configuration ( m1,...,m k), formj≥0,/summationtextk
+j=1mj=m,
+with probability
+pm(n) =1
+(n)m(γ+n−k)m
+(γ+n)mk/productdisplay
+j=1(nj+1)mj (5)
+b) allocated in k∗newboxes in configuration ( s1,...,sk∗), for/summationtextk∗
+j=1sj=m, 1≤k∗≤m,
+sj≥1, with probability
+ps(n) =(k)k∗
+(n)m(k−γ)k∗(γ+n−k)m−k∗
+(γ+n)mk∗/productdisplay
+j=1sj! (6)
+c)s < mballs are allocatedat k∗newboxes in configuration ( s1,...,sk∗) and the remaining
+m−sballs in the oldboxes in configuration ( m1,...,m k) for/summationtextm
+j=1mj=m−s, 1≤s≤m,/summationtextk∗
+j=1sj=s,mj≥0,sj≥1 with probability
+ps,m(n) =(k)k∗
+(n)m(k−γ)k∗(γ+n−k)m−k∗
+(γ+n)mk/productdisplay
+j=1(nj+1)mjk∗/productdisplay
+j=1sj! (7)
+Thesem-steps prediction rules allows to readily obtain Bayesian p osterior predictive dis-
+tributional results for the random partition induced by an a dditional m-sample from Gnedin’s
+model. By exploiting the definition of central and non-centr al Lah numbers as particular case
+of central and non-central generalized Stirling numbers of the first kind, the results of next
+section are obtained specializing results in Lijoi et al. (2 008) by means of expressions de-
+rived in Cerquetti (2008). See the Appendix for the relation ship between generalized Stirling
+numbers and generalized factorial coefficients both central and non-central. For an explicit
+example of application of this kind of results see e.g. Secti on 4. in Lijoi et al. (2008).
+3 Posterior predictive analysis of Gnedin’s model
+First notice that generalized Stirling numbers of the first k indS−1,−α
+n,kforα=−1 admit an
+explicit expression known as Lah numbers
+S−1,1
+n,k=/parenleftbiggn−1
+k−1/parenrightbiggn!
+k!
+which are connection coefficients defined by
+(x)n↑=n/summationdisplay
+k=0n!
+k!/parenleftbiggn−1
+n−k/parenrightbigg
+(x)k↑−1
+4hence, by an application of (10) in Gnedin and Pitman (2006), (see Gnedin, 2009, eq. (2)) the
+distribution of the number Knof occupied boxes for Gnedin’s model is readily calculated a s
+Pr(Kn=k) =Vn,kS−1,1
+n,k=/parenleftbiggn
+k/parenrightbigg(1−γ)k−1(γ)n−k
+(1+γ)n−1.
+Then recall the definition of non-central Lah numbers with pa rameter of non-centrality r
+(see e.g. Charalambides, 2005) which correspond to non cent ral generalized Stirling numbers
+forα=−1
+S−1,1,r
+n,k=n!
+k!/parenleftbiggn−r−1
+n−k/parenrightbigg
+.
+Now a Bayesian nonparametric posterior predictive analysi s of the class (4) is readily de-
+rived. Notice that for the sake of generality here we mantain the treatment in terms of random
+allocation of balls in boxes. The obvious traslation in term s of random partition of individuals
+among species follows easily.
+Given the sufficiency of the number Knof boxes induced by the basic sample, the joint
+distribution of the number Sof balls allocated in new k∗boxes in a specific configuration
+(s1,...,sk∗) given (n1,...,n k) is obtained marginalizing (7) with respect to ( m1,...,m k) and
+by an application of the multinomial theorem for rising fact orials (see the Appendix) as in
+Lijoi et al. (2008) (cfr. eq. (27) in Cerquetti, 2008),
+Pr(s1,...,sk∗|Kn=k) =(k)k∗
+(γ+n)m(k−γ)k∗(γ+n−k)m−k∗
+(n)m/parenleftbiggm
+s/parenrightbigg
+(n+k)m−s↑k∗/productdisplay
+j=1sj!.(8)
+The joint distribution of the number of new boxes Kmand of the total number of balls
+falling in new boxes SgivenKn(cfr. eq. (28) in Cerquetti, 2008) is given by
+Pr(Km=k∗,S=s|Kn=k) =(k)k∗
+(n)m(k−γ)k∗(γ+n−k)m−k∗
+(γ+n)m/parenleftbiggm
+s/parenrightbigg
+(n+k)m−s↑/parenleftbiggs
+k∗/parenrightbigg(s−1)!
+(k∗−1)!
+(9)
+and arises by (8) by summing over the space of all partitions o fselements in k∗blocks and
+exploiting the definition of Lah numbers.
+Marginalizing (9) with respect to Km, a probability distribution for the total number S
+of balls in new boxes is obtained in terms of Lah numbers accor ding to eq. (29) in Cerquetti
+(2008) and eq. (11) in Lijoi et al. (2008)
+Pr(S=s|n1,...,n k) =/parenleftbiggm
+s/parenrightbigg(n+k)m−s↑
+(n)m(γ+n)ms/summationdisplay
+k∗=0/parenleftbiggs
+k∗/parenrightbigg(s−1)!
+(k∗−1)!(k)k∗(k−γ)k∗(γ+n−k)m−k∗.
+(10)
+The probability distribution of the number Kmof new blocks induced by the additional
+sample given the basic sample follows marginalizing (9) wit h respect to Sand exploiting the
+5definition of non-central Lah numbers with parameter of non c entrality r=−(n+k). An
+application of eq. (4) in Lijoi et al. (2007) yields
+Pr(Km=k∗|Kn=k) =(k−γ)k∗(γ+n−k)m−k∗
+(n+γ)m/parenleftbiggm
+k∗/parenrightbigg(n+k+k∗)m−k∗(k)k∗
+(n)m.(11)
+The expected value of the number of new boxes in the m-sample conditioned to the basic
+sample, which provides the Bayes estimator for Kmunder quadratic loss function, results
+E(Km|Kn=k) =(k)n+m
+(n+γ)mm/summationdisplay
+k∗=0/parenleftbiggm
+k∗/parenrightbiggk∗
+(k+k∗)n(k−γ)k∗(γ+n−k)m−k∗
+(n)m.(12)
+Notice that for m→ ∞, (11) agrees with the posterior result for the total number o f boxes
+obtained in Gnedin (2009). In fact, by standard asymptotic Γ (n+a)/Γ(n+b)∼na−band
+recalling the definition of rising and falling factorials in terms of Gamma function
+(a)b↑=Γ(a+b)
+Γ(a)and (a)b↓=Γ(a+1)
+Γ(a−b+1),
+equation (11) reduces to
+Pr(K=k∗|Kn=k) =(n−1)!
+(k−1)!(γ+n−k)k↑(k−γ)k∗↑Γ(k+k∗)
+Γ(k∗+1)Γ(n+k+k∗). (13)
+The posterior distribution obtained in Gnedin (2009), in te rms ofκ=k+k∗, is
+Pr(Ξ =κ|Kn=k) =(n−1)!
+(k−1)!(κ+n−1)!k−1/productdisplay
+i=1(κ−i)k/productdisplay
+j=1(γ+n−j)κ−1/productdisplay
+l=k(l−γ)
+for 1≤k≤n,κ≥kand may be re-written as
+=(n−1)!
+(k−1)!(κ+n−1)!(κ−1)k−1↓(k−γ)κ−k↑(γ+n−k)k↑.
+By substitution κ=k+k∗the result easily follows
+Pr(K=k∗|Kn=k) =(n−1)!
+(k−1)!(γ+n−k)k↑Γ(k∗+k)
+Γ(k+k∗+n)(k−γ)k∗
+k∗!.
+The corresponding expected value results
+E(K|Kn=k) =(n−1)!
+(k−1!)(n+γ−1)k↓/summationdisplay
+k∗=0∞1
+(k∗−1)!(k−γ)k∗
+(k+k∗)n
+which expressed in terms of κ=k+k∗yields
+E(Ξ|Kn=k) =(n−1)!
+(k−1)!(n+γ−1)k↓
+(k−γ−1)!∞/summationdisplay
+κ=k(κ−γ−1)!
+(κ−k−1)!1
+(κ)n.
+6The distribution of the number Sof balls in the new m-sample which belong to new boxes,
+given the number of boxes in the basic sample Knand the number of new boxes Km, follows
+by an application of Eq. 12 in Lijoi et al. (2008)
+Pr(S=s|Km=k∗,Kn=k) =/parenleftbiggs−1
+k∗−1/parenrightbigg/parenleftbiggn+k+m−s−1
+m−s/parenrightbigg
+//parenleftbiggn+k+m−1
+m−k∗/parenrightbigg
+(14)
+By Proposition 2. in Lijoi et al. (2008) the mean number of bal ls in the subsequent m
+sample in given by
+E(S|Kn=k) =mVn+1,k+1
+Vn,k=mk
+n(k−γ)
+(n+γ)(15)
+and by Proposition 4 in Lijoi et al. (2008) (cfr. also Corolla ry 10, in Cerquetti, 2008) the
+probability that the mnew balls don’t occupy a subset of ( k−r) old boxes arises from (7) by
+summing over the ways to choose sballs from the mof the new group, by summing over the
+ways to partition sballs in a subset of k∗boxes, and over the ways to allocate m−sballs in
+at mostrold boxes and is equal to
+m/summationdisplay
+k∗=1(k)∗
+k
+(n)m(k−γ)∗
+k(γ+n−k)m−k∗
+(γ+n)m/parenleftbiggm
+k∗/parenrightbigg1
+(r+/summationtext
+jnj+m)k∗−m. (16)
+The conditional Gibbs structure characterizing Gnedin’s m odel as from Proposition 3. in
+Lijoi et al. (2008), which may be obtained by the operation of deletion of the first kclasses
+(Pitman, 2003) as clarified in Cerquetti (2008, Prop. 12) wil l be as follows
+p(s1,...,sk∗|m,n) =Γ(k+k∗)Γ(k+k∗−γ)Γ(γ+n+m−k−k∗)/summationtextm
+k∗=1/parenleftbigs
+k∗/parenrightbig
+Γ(k∗)kΓ(s)Γ(k+k∗−γ)Γ(γ+n+m−k−k∗)k∗/productdisplay
+j=1sj!.
+Finally a Bayesian nonparametric estimator for the probabi lity of ball n+m+1th to fall
+in a new box given Kn=kis readily derived by eq. (6) in Lijoi et al. (2007)
+ˆDn,k
+m=m/summationdisplay
+k∗=0(k)k∗+1
+(n)m+1(k−γ)k∗+1(γ+n−k)m−k∗
+(γ+n)m+1/parenleftbiggm
+k∗/parenrightbiggm+n+k∗−1!
+(n−1)!.
+4 Appendix
+Forn= 0,1,2,...,and arbitrary real xandh, let (x)n↑hdenote the nth factorial power of x
+with increment h(also called generalized risingfactorial)
+(x)n↑h:=x(x+h)···(x+(n−1)h) =n−1/productdisplay
+i=0(x+ih) =hn(x/h)n↑, (17)
+where (x)n↑stands for ( x)n↑1, (x)h↑0=xhand (x)0↑h= 1, and for which the following
+multiplicative law holds
+(x)n+r↑h= (x)n↑h(x+nh)r↑h. (18)
+7From e.g. Normand (2004, cfr. eq. 2.41 and 2.45) a binomial fo rmula also holds, namely
+(x+y)n↑h=n/summationdisplay
+k=0/parenleftbiggn
+k/parenrightbigg
+(x)k↑h(y)n−k↑h, (19)
+as well as a generalized version of the multinomial theorem, i.e.
+(p/summationdisplay
+j=1zj)n↑h=/summationdisplay
+nj≥0,/summationtextnj=nn!
+n1!···np!p/productdisplay
+j=1(zj)nj↑h. (20)
+We recall the notion of generalized Stirling numbers , (for a comprehensive treatment see
+Hsu and Shiue, 1998; see also Pitman, 2006). For arbitrary di stinct reals ηandβ, these are
+the connection coefficients Sη,β
+n,kdefined by
+(x)n↓η=n/summationdisplay
+k=0Sη,β
+n,k(x)k↓β
+where (x)n↓hare generalized fallingfactorials and ( x)n↓−h= (x)n↑h. Hence for η=−1,
+β=−α, andα∈(−∞,1),S−1,−α
+n,kis defined by
+(x)n↑1=n/summationdisplay
+k=0S−1,−α
+n,k(x)k↑α, (21)
+or specializing partial Bell polynomials as follows
+Bn,k((1−α)•−1↑) =/summationdisplay
+{A1,...,Ak}∈Pk
+[n]k/productdisplay
+i=1(1−α)ni−1↑=n!
+k!/summationdisplay
+(n1,...,nk)k/productdisplay
+i=1(1−α)ni−1↑
+ni!=S−1,−α
+n,k.
+(22)
+Referring to formulas in Lijoi et al. (2007, 2008) it is convi enent to recall that their
+treatment is in terms of generalized factorial coefficients , which are the connection coefficients
+Cα
+n,kdefined by ( αy)n↑1=/summationtextn
+k=0Cα
+n,k(y)k↑1,(cfr. Charalambides, 2005). From (17) and (21),
+ifx=yαthen
+(yα)n↑1=n/summationdisplay
+k=0S−1,−α
+n,k(yα)k↑α=n/summationdisplay
+k=0S−1,−α
+n,kαk(y)k↑1,
+henceS−1,−α
+n,k=α−kCα
+n,k.
+It is also worth to clarify the relationship between non central generalized Stirling num-
+bers of the first kind as defined in Hsu and Shiue (1998) and non central generalized factorial
+coefficients as in Charalambides (2005).
+First recall that non central generalized Stirling numbers of the first kind are connection
+coefficients defined by
+(x)n↑=n/summationdisplay
+k=0S−1,−α,γ
+n,k(x−γ)k↑α
+8or by the following convolution relation
+S−1,−α,γ
+n,k=n/summationdisplay
+s=0/parenleftbiggn
+k/parenrightbigg
+S−1,−α
+s,kS−1,−α,γ
+n−s,0. (23)
+Since as a convention we assume S−1,−α
+s,k= 0 fors < kand it is known that S−1,−α,γ
+n−s,0=
+(γ)n−s↑then
+S−1,−α,γ
+n,k=n/summationdisplay
+s=k/parenleftbiggn
+s/parenrightbigg
+S−1,−α
+s,k(−γ)n−s↑1, (24)
+Now, for parameter of non centrality −γ, andx=ya
+(yα−γ)n↑=n/summationdisplay
+k=0S−1,−α,−γ
+n,k(yα)k↑α=n/summationdisplay
+k=0/bracketleftBiggn/summationdisplay
+s=k/parenleftbiggn
+s/parenrightbigg
+S−1,−α
+s,k(−γ)n−s↑/bracketrightBigg
+(yα)k↑α.
+Then exploiting the relation between central generalized S tirling numbers and central
+generalized factorial coefficients
+(yα−γ)n↑=n/summationdisplay
+k=0/bracketleftBigg
+α−kn/summationdisplay
+s=k/parenleftbiggn
+s/parenrightbigg
+Cα
+s,k(−γ)n−s↑/bracketrightBigg
+(yα)k↑α
+and the definition of non central factorial coefficients as in C haralambides (2005) follows
+(yα−γ)n↑=n/summationdisplay
+k=0α−kCα,γ
+s,k(ya)k↑α=n/summationdisplay
+k=0Cα,γ
+s,k(y)k↑.
+References
+Cerquetti, A. (2008) Generalized Chinese restaurant construction of exc hangeable Gibbs
+partitions and related results. arXiv:0805.3853v1 [math. PR]
+Charalambides, C. A. (2005)Combinatorial Methods in Discrete Distributions . Wiley,
+Hoboken NJ.
+Gnedin, A. (2009) A species sampling model with finitely many types. arX iv:0910.1988v1
+[math.PR]
+Gnedin, A. and Pitman, J. (2006) Exchangeable Gibbs partitions and Stirling triangl es.
+Journal of Mathematical Sciences , 138, 3, 5674–5685.
+Hsu, L. C, & Shiue, P. J. (1998) A unified approach to generalized Stirling numbers. Adv.
+Appl. Math. , 20, 366-384.
+Lijoi, A., Mena, R. and Pr ¨unster, I. (2007) Bayesian nonparametric estimation of the
+probability of discovering new species Biometrika , 94, 769–786.
+9Lijoi, A., Pr ¨unster, I. and Walker, S.G. (2008) Bayesian nonparametric estimator
+derived from conditional Gibbs structures. Annals of Applied Probability , 18, 1519–
+1547.
+Normand, J.M. (2004) Calculation of some determinants using the s-shifted factorial. J.
+Phys. A: Math. Gen. 37, 5737-5762.
+Pitman, J. (1996) Some developments of the Blackwell-MacQueen urn sch eme. In T.S.
+Ferguson, Shapley L.S., and MacQueen J.B., editors, Statistics, Probability and Game
+Theory, volume 30 of IMS Lecture Notes-Monograph Series , pages 245–267. Institute of
+Mathematical Statistics, Hayward, CA.
+Pitman, J. (2003) Poisson-Kingman partitions. InD.R. Goldstein, edi tor,Science and Statis-
+tics: A Festschrift for Terry Speed , volume 40 of Lecture Notes-Monograph Series, pages
+1–34. Institute of Mathematical Statistics, Hayward, Cali fornia.
+Pitman, J. (2006)Combinatorial Stochastic Processes . Ecole d’Et´ e de Probabilit´ e de Saint-
+Flour XXXII - 2002. Lecture Notes in Mathematics N. 1875, Spr inger.
+Pitman, J. and Yor, M. (1997) The two-parameter Poisson-Dirichlet distribution derived
+from a stable subordinator. Ann. Probab. , 25:855–900.
+10
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+arXiv:1001.0247v1  [hep-ph]  1 Jan 2010Dihadron Azimuthal Correlation from Collins Effect in
+Unpolarized Hadron Collisions
+Zhong-Bo Kang1,∗and Feng Yuan1,2,†
+1RIKEN BNL Research Center, Brookhaven National Laboratory , Upton, NY 11973, USA
+2Nuclear Science Division, Lawrence Berkeley
+National Laboratory, Berkeley, CA 94720, USA
+(Dated: October 24, 2018)
+Abstract
+We study the dihadron azimuthal correlation produced nearl y back-to-back in unpolarized
+hadron collisions, arising from the product of two Collins f ragmentation functions. Using the
+latest Collins fragmentation functions extracted from the global analysis of available experimental
+data, we make predictions for the azimuthal correlation of t wo-pion production in ppcollisions at
+RHIC energies. We find that the correlation is sizable in the m id-rapidity region for moderate jet
+transverse momentum.
+PACS numbers: 12.38.Bx, 12.39.St, 13.85.Qk, 13.88+e
+∗Electronic address: zkang@bnl.gov
+†Electronic address: fyuan@lbl.gov
+1I. INTRODUCTION
+The transverse momentum dependent (TMD) distributions and fra gmentation functions
+have received much attention recently [1]. They are believed to be r esponsible for several
+azimuthal asymmetries observed in experiments, such as the single transverse spin asym-
+metry (SSA) in semi-inclusive hadron production in deep inelastic scat tering (SIDIS) [2, 3]
+and in hadronic collisions [4], as well as the large cos(2 φ) anomalous azimuthal asymmetry
+in back-to-back dihadron production in e+e−annihilation [5].
+Among these TMD parton distributions and fragmentation function s, the Sivers quark
+distribution [6] and the Collins fragmentation function [7] are mostly d iscussed in the last
+few years. The Sivers quark distribution represents a distribution of unpolarized quarks
+in a transversely polarized nucleon, through a correlation between the quark’s transverse
+momentum and the nucleon polarization vector. On the other hand, the Collins fragmenta-
+tion function describes a transversely polarized quark jet fragme nting into an unpolarized
+hadron, whose transverse momentum relative to the jet axis corr elates with the transverse
+polarization vector of the fragmenting quark.
+Though both of them belong to the so-called “naive-time-reversal- odd” (T-odd) func-
+tions, they have very different universality properties. For the Siv ers functions, it has been
+shown that they differ by a sign for the SIDIS and Drell-Yan (DY) pro cesses [8], and those
+in the hadronic collisions have even more nontrivial relation to that in S IDIS and DY pro-
+cesses [9–13]. On the other hand, the Collins fragmentation functio n is universal between
+different processes, in the SIDIS, e+e−and hadronic collisions [14–16]. The effect of the
+Collins fragmentation function has been recently explored by one of us in the azimuthal
+asymmetric distribution of hadrons inside a jet in p↑pcollision [17]. It is demonstrated that
+the asymmetry is sizable at RHIC, therefore, the experimental st udy of this process could
+provide an important information on the universality of the Collins fra gmentation function.
+Another difference between these two functions is that the Collins f ragmentation function
+is chiral-odd whereas the Sivers function is chiral-even. Because of its chiral-odd nature, the
+Collins effect can only be observed when it is coupled to another chiral- odd distribution
+or fragmentation function. In SIDIS, the chiral-odd quark trans versity [18] can couple to
+the Collins fragmentation function and leads to nonzero azimuthal S SA [7]. This SSA has
+been studied by the HERMES [2] and COMPASS [3] collaborations, and v ery interesting
+2results on the Collins fragmentation function have been found. In e+e−annihilation pro-
+cess, two Collins fragmentation functions couple to each other in th e back-to-back dihadron
+production, results into to a cos(2 φ) azimuthal asymmetry [19]. This anomalous cos(2 φ)
+asymmetry has been measured by the BELLE Collaboration [5], and w as found consistent
+with the HERMES and COMPASS measurements on the Collins fragment ation functions.
+Recently a global analysis of these experimental data has been per formed and the Collins
+fragmentation functions have been extracted [20].
+Inthispaper weinvestigate thepossibility ofexploring theCollins frag mentationfunction
+in unpolarized ppcollision by studying the azimuthal correlation in back-to-back dihad ron
+production, following the same wisdom of dihadron production in e+e−annihilation. We
+show that the asymmetry is proportional to the product of two Co llins fragmentation func-
+tion, same as that in e+e−annihilation. Using the latest Collins fragmentation function
+extracted from the global analysis of available data on SIDIS and e+e−experiments, we es-
+timate the asymmetry for dihadron production at RHIC energy. We find that the azimuthal
+asymmetry is sizable at mid-rapidity region for moderate jet transv erse momentum. We
+argue that this process shall provide addtional important informa tion on the Collins frag-
+mentation function and its universality properties.
+The rest of the paper is organized as follows. In Sec. II, we derive t he theoretical results
+for the dihadron azimuthal correlation produced nearly back-to- back in unpolarized hadron
+collision. In Sec. III, we present our numerical predictions for the azimuthal correlation in
+unpolarized ppcollisions for RHIC kinematics. Finally, we summarize our findings and th e
+corresponding conclusions in Sec. IV.
+II. DIHADRON AZIMUTHAL CORRELATION IN UNPOLARIZED HADRON
+COLLISION
+We study the azimuthal correlation of two hadrons h1andh2produced nearly back-to-
+back in a hadronic collision,
+A(P1)+B(P2)→h1(Ph1)+h2(Ph2)+X, (1)
+where both of the incident hadrons AandBare unpolarized. The momenta of the initial
+hadronsaredenotedby P1andP2,andthoseofthefinalhadronsby Ph1andPh2,respectively.
+3a
+bc
+dk1
+k2Ph1
+Ph2P1
+P2q1
+q2H H∗∆(q1)
+∆(q2)Φ(k1)
+Φ(k2)
+FIG. 1: The leading order contribution to the cross section o fA(P1)+B(P2)→h1(Ph1)+h2(Ph2)+
+Xwith the 2 →2 partonic process a(k1)+b(k2)→c(q1)+d(q2).
+The leading order contribution to the scattering cross section com es from partonic 2 →2
+sub-processes, a(k1)+b(k2)→c(q1)+d(q2), as shown in Fig. 1. The parton momenta are
+expanded as follows,
+k1=x1P1+k1T, (2a)
+k2=x2P2+k2T, (2b)
+Ph1=z1q1+p1T, (2c)
+Ph2=z2q2+p2T, (2d)
+wherex1andx2arethelongitudinalmomentumfractions, and k1Tandk2Tarethetransverse
+momentum of the parton relative to the corresponding incident had ron.q1andq2are the
+momenta of the nearly back-to-back jets J1andJ2, which has a polar angle θ1andθ2relative
+to the incoming hadron P1, respectively. The momenta of the incoming hadrons and the
+final state two jets form the so-called reaction plane (approximat ely). Besides carrying a
+longitudinal momentum fraction z1(z2) of the jet J1(J2), the hadron h1(h2) also has
+a transverse momentum p1T(p2T) relative to the jet J1(J2) direction, which defines an
+azimuthal angle with the reaction plane: φ1(φ2), as shown in Fig. 2. Due to the Collins
+effect, there will be a correlation between these two azimuthal ang lesφ1andφ2, which
+is proportional to the product of the two Collins fragmentation fun ctions as we will show
+below.
+4P1 P2J1
+J2φ1
+φ2θ1
+θ2xy
+zh1
+h2
+FIG. 2: Illustration of the kinematics for dihadron product ionAB→h1h2+X.
+From Fig. 1, the cross section of dihadron production can be writte n as [13],
+dσ=1
+2S/integraldisplay
+dx1d2k1Tdx2d2k2Tdz1d2p1Tdz2d2p2Td3q1
+(2π)32E1d3q2
+(2π)32E2(2π)4δ4(k1+k2−q1−q2)
+×Tr[Φ(x1,k1T)Φ(x2,k2T)∆(z1,p1T)∆(z2,p2T)H(k1,k2,q1,q2)H∗(k1,k2,q1,q2)],(3)
+whereS= (P1+P2)2, and Φ(x,kT) and ∆( z,pT) are the distribution and fragmentation
+correlation functions, and H(k1,k2,q1,q2) are the hard part amplitude.
+For the unpolarized hadron, the correlation functions Φ( x,kT) can be simply decomposed
+as [21, 22],
+Φ(P,x,k T) =1
+2/bracketleftbigg
+f(z,k2
+T)P /+h⊥
+1(z,k2
+T)σµνkTµPν
+M/bracketrightbigg
+, (4)
+wheref(x,k2
+T) is the unpolarized TMD parton distribution function, and h⊥
+1(z,k2
+T) is the
+Boer-Mulders function [22]. Similarly, we can parameterize the gluon d istributions from the
+incominghadrons. TheeffectofBoer-Muldersfunctioninunpolarize dhadroniccollisionshas
+been extensively studied previously [23–25], which will beneglected ino ur current study. We
+will concentrate on the effect coming from the fragmentation corr elation function ∆( z,pT),
+which can be expanded as [26]
+∆(Ph,z,pT) =1
+2/bracketleftbigg
+D(z,p2
+T)P /h+H⊥
+1(z,p2
+T)σµνPhµpTν
+zMh/bracketrightbigg
+, (5)
+whereD(z,p2
+T) is the unpolarized TMD fragmentation function, and H⊥
+1is the Collins
+function. From its definition, we can see that the Collins function des cribes a transversely
+polarized quark jet fragmenting into an unpolarized hadron [7]. It is t his Collins function
+that generates a non-vanishing azimuthal correlation between th e final state two hadrons.
+Since Collins function is a chiral-odd TMD function, the azimuthal corr elation will depend
+on the product of two Collins functions.
+5The phase space integral in Eq. (3) can be simplified by using
+d3q1
+(2π)32E1d3q2
+(2π)32E2(2π)4δ4(k1+k2−q1−q2)
+=dy1dy2dP2
+⊥/parenleftbigg1
+8πS/parenrightbigg
+δ/bracketleftbigg
+x1−P⊥√
+S(ey1+ey2)/bracketrightbigg
+δ/bracketleftbigg
+x2−P⊥√
+S/parenleftbig
+e−y1+e−y2/parenrightbig/bracketrightbigg
+,(6)
+wherey1andy2are rapidities for the jet J1andJ2,P⊥is the jet transverse momentum.
+Sincep1T,p2T≪P⊥, the rapidity of the hadron is approximately equal to that of the pa rent
+jet. As stated above, we neglect the intrinsic transverse moment um effects for the incoming
+partons. Finally we obtain the cross section for this process as
+dσ
+dPS=πα2
+s
+S2/summationdisplay
+abcdfa(x1)
+x1fb(x2)
+x2/bracketleftbig
+Dc→h1(z1,p2
+1T)Dd→h2(z2,p2
+2T)HU
+ab→cd/parenleftbig
+ˆs,ˆt,ˆu/parenrightbig
++/parenleftbigg
+p1T·p2T−x1x2
+P2
+⊥(P1·p1TP2·p2T+P1·p2TP2·p1T)/parenrightbigg
+×H⊥
+1(z1,p2
+1T)
+z1MhH⊥
+1(z2,p2
+2T)
+z2MhHCollins
+ab→cd/parenleftbig
+ˆs,ˆt,ˆu/parenrightbig/bracketrightbig
+, (7)
+wheredPS ≡dy1dy2dP2
+⊥dz1dz2d2p1Td2p2Tis the phase space for this process, fa(x1) and
+fb(x2) are the standard unpolarized parton distribution functions, and ˆs,ˆt, and ˆuare the
+usual partonic Mandelstam variables. The parton momentum fract ionsx1andx2are fixed
+by the delta functions in Eq. (6),
+x1=P⊥√
+S(ey1+ey2), (8a)
+x2=P⊥√
+S/parenleftbig
+e−y1+e−y2/parenrightbig
+. (8b)
+6The normal partonic cross sections HU
+ab→cdare well-known [27],
+HU
+gg→q¯q=1
+2Nc/bracketleftbiggˆt
+ˆu+ˆu
+ˆt/bracketrightbigg
+−Nc
+N2
+c−1/bracketleftbiggˆt2+ ˆu2
+ˆs2/bracketrightbigg
+, (9)
+HU
+q¯q→q′¯q′=N2
+c−1
+2N2
+c/bracketleftbiggˆt2+ ˆu2
+ˆs2/bracketrightbigg
+, (10)
+HU
+q¯q→q¯q=N2
+c−1
+2N2c/bracketleftbiggˆt2+ ˆu2
+ˆs2+ˆs2+ ˆu2
+ˆt2/bracketrightbigg
+−N2
+c−1
+N3c/bracketleftbiggˆu2
+ˆsˆt/bracketrightbigg
+, (11)
+HU
+qq′→qq′=N2
+c−1
+2N2
+c/bracketleftbiggˆs2+ ˆu2
+ˆt2/bracketrightbigg
+, (12)
+HU
+qq→qq=N2
+c−1
+2N2c/bracketleftbiggˆs2+ ˆu2
+ˆt2+ˆs2+ˆt2
+ˆu2/bracketrightbigg
+−N2
+c−1
+N3c/bracketleftbiggˆs2
+ˆtˆu/bracketrightbigg
+, (13)
+HU
+q¯q→gg=4(N2
+c−1)
+N3c/bracketleftbiggˆt
+ˆu+ˆu
+ˆt/bracketrightbigg
+−N2
+c−1
+Nc/bracketleftbiggˆt2+ ˆu2
+ˆs2/bracketrightbigg
+, (14)
+HU
+gq→gq=−N2
+c−1
+2N2c/bracketleftbiggˆs
+ˆu+ˆu
+ˆs/bracketrightbigg
++/bracketleftbiggˆs2+ ˆu2
+ˆt2/bracketrightbigg
+, (15)
+HU
+gg→gg=4N2
+c
+N2c−1/bracketleftbigg
+3−ˆtˆu
+ˆs2−ˆsˆu
+ˆt2−ˆsˆt
+ˆu2/bracketrightbigg
+. (16)
+The new hard parts HCollins
+ab→cdthat are responsible for the azimuthal correlation are given by,
+HCollins
+gg→q¯q=1
+Nc−Nc
+N2c−1/bracketleftbigg2ˆtˆu
+ˆs2/bracketrightbigg
+, (17)
+HCollins
+q¯q→q′¯q′=N2
+c−1
+N2
+c/bracketleftbiggˆtˆu
+ˆs2/bracketrightbigg
+, (18)
+HCollins
+q¯q→q¯q=N2
+c−1
+N2
+c/bracketleftbiggˆtˆu
+ˆs2/bracketrightbigg
+−N2
+c−1
+N3
+c/bracketleftbiggˆu
+ˆs/bracketrightbigg
+, (19)
+HCollins
+qq′→qq′= 0, (20)
+HCollins
+qq→qq=N2
+c−1
+N3
+c, (21)
+and the hard parts for the channels with gluon in the final state, q¯q→gg,qg→qgand
+gg→ggvanish since there is no gluon Collins function.
+III. PHENOMENOLOGY STUDY
+In this section we first properly define the azimuthal asymmetry to be measured in the
+experiments. We then use the latest Collins fragmentation function to estimate this asym-
+metry for RHIC kinematics.
+7From Eq. (7), the explicit form of the azimuthal correlation depend s on the following
+function,
+h(p1T,p2T,φ1,φ2)≡p1T·p2T−x1x2
+P2
+⊥(P1·p1TP2·p2T+P1·p2TP2·p1T).(22)
+Function h(p1T,p2T,φ1,φ2) is reduced to its simplest form in the partonic center of mass
+frame,
+h(p1T,p2T,φ1,φ2) =−p1Tp2Tcos(φ1+φ2). (23)
+To take advantage of this simplicity, one could boost from the Lab fr ame to the partonic
+CM frame experimentally. Or equivalently, one can select the events withy1+y2≈0, where
+the Lab frame coincides with the partonic CM frame. In the rest of o ur paper, we will
+takey1+y2= 0 in our calculations and present the numerical estimate for the az imuthal
+asymmetry.
+Integrating over the intrinsic transverse momentum p1Tandp2T, we have
+dσ
+dy1dy2dP2
+⊥dz1dz2=πα2
+s
+S2/summationdisplay
+abcdfa(x1)
+x1fb(x2)
+x2Dc→h1(z1)Dd→h2(z2)HU
+ab→cd/parenleftbig
+ˆs,ˆt,ˆu/parenrightbig
+(24)
+While integrating over the moduli of the intrinsic momenta p1Tandp2T, and over the
+azimuthal angle φ1, one obtain
+dσ
+dy1dy2dP2
+⊥dz1dz2d(φ1+φ2)=α2
+s
+2S2/summationdisplay
+abcdfa(x1)
+x1fb(x2)
+x2/bracketleftbig
+Dc→h1(z1)Dd→h2(z2)HU
+ab→cd/parenleftbig
+ˆs,ˆt,ˆu/parenrightbig
+−cos(φ1+φ2)δˆq(1/2)
+c→h1(z1)δˆq(1/2)
+d→h2(z2)HCollins
+ab→cd/parenleftbig
+ˆs,ˆt,ˆu/parenrightbig/bracketrightBig
+,(25)
+whereδˆq(1/2)(z) is the so-called half-moment of the Collins function given by,
+δˆq(1/2)(z) =/integraldisplay
+d2pTpTH⊥
+1(z,p2
+T)
+zMh. (26)
+Then following the normal analysis in e+e−→h1h2+X, we define
+Ah1h2(y1,y2,P⊥,z1,z2,φ1+φ2)
+≡dσ
+dy1dy2dP2
+⊥dz1dz2d(φ1+φ2)/slashbigg1
+2πdσ
+dy1dy2dP2
+⊥dz1dz2
+= 1−cos(φ1+φ2)/summationtext
+abcdfa(x1)fb(x2)δˆq(1/2)
+c→h1(z1)δˆq(1/2)
+d→h2(z2)HCollins
+ab→cd/summationtext
+abcdfa(x1)fb(x2)Dc→h1(z1)Dd→h2(z2)HU
+ab→cd.(27)
+8Again to eliminate the false asymmetries [5], we introduce the ratio of t he unlike-sign to
+like-sign pion production, AUandAL, given by
+R≡AU
+AL=1−cos(φ1+φ2)PU
+1−cos(φ1+φ2)PL≈1−cos(φ1+φ2)(PU−PL)
+≡1−cos(φ1+φ2)A12(y1,y2,P⊥,z1,z2) (28)
+with
+PU=/summationtext
+abcdfa(x1)fb(x2)/bracketleftBig
+δˆq(1/2)
+c→π+(z1)δˆq(1/2)
+d→π−(z2)+δˆq(1/2)
+c→π−(z1)δˆq(1/2)
+d→π+(z2)/bracketrightBig
+HCollins
+ab→cd/summationtext
+abcdfa(x1)fb(x2)[Dc→π+(z1)Dd→π−(z2)+Dc→π−(z1)Dd→π+(z2)]HU
+ab→cd,(29)
+PL=/summationtext
+abcdfa(x1)fb(x2)/bracketleftBig
+δˆq(1/2)
+c→π+(z1)δˆq(1/2)
+d→π+(z2)+δˆq(1/2)
+c→π−(z1)δˆq(1/2)
+d→π−(z2)/bracketrightBig
+HCollins
+ab→cd/summationtext
+abcdfa(x1)fb(x2)[Dc→π+(z1)Dd→π+(z2)+Dc→π−(z1)Dd→π−(z2)]HU
+ab→cd,(30)
+A12(y1,y2,P⊥,z1,z2) =PU−PL. (31)
+This way the true asymmetry due to Collins effect is encoded in the so- called double
+ratio asymmetry parameter A12(y1,y2,P⊥,z1,z2). To evaluate A12for the dihadron produc-
+tion in unpolarized ppcollision at√
+S= 200GeV at RHIC, we use the Collins fragmentation
+functions [20] extracted froma combined fit to the experimental d ata fromHERMES, COM-
+PASS and BELLE collaborations. We use CTEQ5L parton distributions [28] and Kretzer
+unpolarized fragmentation function obtained in [29].
+00.010.020.030.040.05
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9P⊥=4 GeV
+z2A12
+00.010.020.030.040.05
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9P⊥=6 GeV
+z2A12
+FIG. 3: Azimuthal asymmetry ratios A12(y1,y2,P⊥,z1,z2) (defined in Eq. (28)) of unlike-sign to
+like-sign pion production in unpolarized proton-proton sc attering at RHIC energy√
+S= 200GeV,
+as functions of z2withy1=y2= 0,P⊥= 4 GeV (left) or P⊥= 6 GeV (right). The curves are:
+solid (0.7< z1<0.9), dashed (0 .5< z1<0.7) and dotted (0 .3< z1<0.5).
+9In Fig. 3, we plot A12(y1,y2,P⊥,z1,z2) for the dihadron production in mid-rapidity y1=
+y2= 0 region at RHIC energy√
+S= 200 GeV. In the figure on the left, we plot A12as a
+function of z2at the jet transverse momentum P⊥= 4 GeV with z1integrated from three
+different ranges 0 .3< z1<0.5 (dotted), 0 .5< z1<0.7 (dashed), and 0 .7< z1<0.9 (solid),
+respectively. On the right, we present the same plot but with P⊥= 6 GeV. We find that the
+asymmetry A12is largest when both z1andz2become large, same as what has beenobserved
+ine+e−experiments [5]. On the other hand, A12decreases when increasing P⊥. This is also
+consistent with what BELLE observed if one realizes that ˆ s/4P2
+⊥= sin2θin parton CM
+frame. Though it has similar features as that in e+e−collision, the asymmetry in hadronic
+collision is smaller. This is due to the fact that there is copious gg→ggandqg→qg
+contribution to the azimuthal angle independent cross section, wh ile they do not contribute
+to the azimuthal dependent part since there is no gluon Collins funct ion. However, the
+asymmetry is still around several percent and shall be measurable at RHIC.
+These results canbeextended togeneral kinematics, forexample , intwo different rapidity
+regions:|y1| /negationslash=|y2|. In this case, although the azimuthal angular dependence is not ex actly
+as cos(φ1+φ2) in Eq. (23), the Collins fragmentation functions will nevertheless le ad to a
+nonzero mean value of /angbracketleftcos(φ1+φ2)/angbracketright. This can be seen from the differential cross section
+expression in Eq. (7). The experimental observation of this nonze ro effects can be used
+as signal of the Collins effects, since the normal fragmentation fun ctionsD(z,pT) will not
+contribute to a nonzero /angbracketleftcos(φ1+φ2)/angbracketright. We hope that the future RHIC experiments can
+carry out these measurements, and provide more information on t he Collins fragmentation
+functions, which will help us to pin down the mechanism for the single sp in asymmetry in
+hadronic collisions as we discussed in the Introduction.
+IV. SUMMARY
+In this paper, we have studied the dihadron azimuthal correlation p roduced nearly back-
+to-back in unpolarized hadron collision, arising from the product of t wo Collins fragmen-
+tation functions. Using the latest Collins fragmentation function ex tracted from the global
+analysis of available experimental data, we make predictions for the azimuthal correlation
+of two-pion production in unpolarized ppcollisions at RHIC energies. We find that the
+feature of the asymmetry is similar to those observed in e+e−annihilation. The asymmetry
+10parameter A12is sizable in the mid-rapidity region for moderate jet transverse mom entum,
+and could be measurable in the experiments. The experimental stud y of this process could
+provide the important information on the size of Collins fragmentatio n function in hadronic
+collision, at the same time, it could also be used to test the universality properties of the
+Collins fragmentation function in different processes.
+Acknowledgments
+We thank M. Grosse Perdekamp, J. Qiu and R. Seidl for helpful discu ssions. This work
+was supported in part by the U. S. Department of Energy under Gr ant No. DE-FG02-
+05CH11231. We are grateful to RIKEN, Brookhaven National Lab oratory, and the U.S.
+Department of Energy (Contract No. DE-AC02-98CH10886) for providing the facilities es-
+sential for the completion of this work.
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+13
\ No newline at end of file
diff --git a/1001.0248.txt b/1001.0248.txt
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@@ -0,0 +1,545 @@
+arXiv:1001.0248v2  [math.NT]  17 Apr 2010On Apery’s Constant and Catalan’s Constant
+Akhila Raman
+University of California at Berkeley, CA-94720. Email: akh ila.raman@berkeley.edu. Ph:
+510-540-5544
+Abstract
+In this paper, Riemann’s Zeta function with odd positive integer argu ment
+is represented as an infinite summation of integer powers of πwith rational
+coefficients. Specific values for Apery’s Constant and Catalan’s Con stant are
+then derived.
+Keywords:
+1. Introduction
+It is well known that Riemann’s Zeta function with even positive intege r
+argument is given by
+ζ(2n) =∞/summationdisplay
+k=11
+k2n=B2n
+2(2π)2n(−1)n+1
+!(2n)(1)
+whereB2nis a Bernoulli number. No such simple expression is known
+for odd positive integer argument. In this section, Riemann’s Zeta f unction
+with odd positive integer argument is represented as an infinite summ ation
+of integer powers of πwith rational coefficients.
+Preprint submitted to Number Theory April 20, 2010It is well known from the theory of divergent series[1] that in the int erval
+|θ|< π,
+sin(θ)−sin(2θ)+sin(3θ)−sin(4θ)+...= (1/2)tan(θ/2) (2)
+The Right Hand Side of this equation can be expanded using Maclaurin’s
+series[2] as follows:
+(1/2)tan(θ/2) = (1/2)∞/summationdisplay
+n=1cn(θ/2)2n−1(3)
+where the coefficients cnare given in terms of Bernoulli numbers B2nas
+cn=B2n(22n−1)22n(−1)n+1
+!(2n).
+Step 1
+Integrating Eq.2 over [0 ,θ] in Step 1 [Integrating over [0 ,φ] and substituting
+φ=θ],
+−[cos(θ)
+1−cos(2θ)
+2+cos(3θ)
+3−cos(4θ)
+4+...]+[1
+1−1
+2+1
+3−1
+4+...] = (1/2)∞/summationdisplay
+n=1cn(θ)2n
+(2n)22n−1
+(4)
+Puttingθ=π
+2in Eq.4, we get,
+A1= [1
+1−1
+2+1
+3−1
+4+...] =∞/summationdisplay
+n=1En(1)(π
+2)2n(5)
+whereEn(1) =Dn(1) andDn(k) =cn
+22n−1(2n+k−1)PkandDn(k) =Dn(k−1)
+(2n+k−1).
+2[cos(θ)
+1−cos(2θ)
+2+cos(3θ)
+3−cos(4θ)
+4+...] =A1−1
+2∞/summationdisplay
+n=1Dn(1)(θ)2n(6)
+Step 2
+Integrating Eq.6 over [0 ,θ] in Step 2,
+[sin(θ)
+12−sin(2θ)
+22+sin(3θ)
+32−sin(4θ)
+42+...] =A1θ−1
+2∞/summationdisplay
+n=1Dn(2)(θ)2n+1(7)
+Puttingθ=π
+2in Eq.7, we get,
+A2= [1
+12−1
+32+1
+52+...] =A1π
+2−1
+2∞/summationdisplay
+n=1Dn(2)(π
+2)2n+1(8)
+A2=∞/summationdisplay
+n=1En(2)(π
+2)2n+1(9)
+whereEn(2) =En(1)−Dn(2)
+2
+Step 3
+Integrating Eq.7 over [0 ,θ] in Step 3,
+−[cos(θ)
+13−cos(2θ)
+23+cos(3θ)
+33−cos(4θ)
+43+...]+[1
+13−1
+23+1
+33−1
+43+...] =A1θ2
+!2−1
+2∞/summationdisplay
+n=1Dn(3)(θ)2n+2
+(10)
+Puttingθ=π
+2, we get,
+A3= [1
+13−1
+23+1
+33−1
+43...] =1
+1−1
+23[A1
+!2(π
+2)2−1
+2∞/summationdisplay
+n=1Dn(3)(π
+2)2n+2] (11)
+3A3=∞/summationdisplay
+n=1En(3)(π
+2)2n+2(12)
+whereEn(3) =−Dn(3)
+2+En(1)
+!2
+1−1
+23
+we have
+[cos(θ)
+13−cos(2θ)
+23+cos(3θ)
+33−cos(4θ)
+43+...] =A3−A1θ2
+!2+1
+2∞/summationdisplay
+n=1Dn(3)(θ)2n+2
+(13)
+Step 4
+Integrating Eq.13 over [0 ,θ] in Step 4,
+[sin(θ)
+14−sin(2θ)
+24+sin(3θ)
+34−sin(4θ)
+44+...] =A3θ−A1θ3
+!3+1
+2∞/summationdisplay
+n=1Dn(4)(θ)2n+3
+(14)
+Puttingθ=π
+2, we get,
+A4= [1
+14−1
+34+1
+54+...] =A3π
+2−A1(π
+2)3
+!3+1
+2∞/summationdisplay
+n=1Dn(4)(π
+2)2n+3(15)
+A4=∞/summationdisplay
+n=1En(4)(π
+2)2n+3(16)
+4whereEn(4) =Dn(4)
+2+En(3)
+!1−En(1)
+!3
+Step 5
+Integrating Eq.14 over [0 ,θ] in Step 5,
+−[cos(θ)
+15−cos(2θ)
+25+cos(3θ)
+35−cos(4θ)
+45+...]+[1
+15−1
+25+1
+35−1
+45+...] =
+A3θ2
+!2−A1θ4
+!4+1
+2∞/summationdisplay
+n=1Dn(5)(θ)2n+4
+(17)
+Puttingθ=π
+2, we get,
+A5= [1
+15−1
+25+1
+35−1
+45...] =1
+1−1
+25[A3
+!2(π
+2)2−A1
+!4(π
+2)4+1
+2∞/summationdisplay
+n=1Dn(5)(π
+2)2n+4]
+(18)
+A5=∞/summationdisplay
+n=1En(5)(π
+2)2n+4(19)
+whereEn(5) =Dn(5)
+2+En(3)
+!2−En(1)
+!4
+1−1
+25
+In general, we can derive the following results Eq.20 to Eq.23 using the
+principle of mathematical induction as shown in Section 2. In Section 3 , it is
+shownthattheseriesexpansionof A2kandA2k+1converges. For k= 1,2,3,...
+5A2k= [1
+12k−1
+32k+1
+52k+...] =∞/summationdisplay
+n=1En(2k)(π
+2)2n+2k−1(20)
+A2k+1= [1
+12k+1−1
+22k+1+1
+32k+1−1
+42k+1...] =∞/summationdisplay
+n=1En(2k+1)(π
+2)2n+2k(21)
+En(2k) =(−1)kDn(2k)
+2+(−1)k+1k−1/summationdisplay
+r=0(−1)rEn(2r+1)
+!(2(k−r)−1)(22)
+En(2k+1) =(−1)kDn(2k+1)
+2+(−1)k+1/summationtextk−1
+r=0(−1)rEn(2r+1)
+!(2(k−r))
+1−1
+22k+1(23)
+Thus we see that A2kandA2k+1can be expressed as an infinite summa-
+tion of integer powers of πwith rational coefficients En(2k) andEn(2k+1).
+And, we have
+B2k+1= [1
+12k+1+1
+22k+1+1
+32k+1+1
+42k+1...] =A2k+1
+1−2−2k(24)
+Using the above results, we can deduce specific values of Apery’s con-
+stantζ(3)and Catalan’s constant K as follows:
+K=∞/summationdisplay
+n=0(−1)n
+(2n+1)2=A2=∞/summationdisplay
+n=1En(2)(π
+2)2n+1(25)
+ζ(3) =∞/summationdisplay
+n=11
+n3=B3=/summationtext∞
+n=1En(3)(π
+2)2n+2
+1−2−2(26)
+2. Section 2
+Let us assume that A2k,A2k+1,En(2k),En(2k+ 1) are given by Eq. 20
+- Eq.23. for some k and that Eq.14 and Eq.17 can be generalized as fol-
+6lows(Inductive hypothesis). Let S1(k) = [sin(θ)
+12k−sin(2θ)
+22k+sin(3θ)
+32k−sin(4θ)
+42k+...]
+and letS2(k) = [cos(θ)
+12k+1−cos(2θ)
+22k+1+cos(3θ)
+32k+1−cos(4θ)
+42k+1+...]
+S1(k) = (−1)k[1
+2∞/summationdisplay
+n=1Dn(2k)θ2n+2k−1]+k−1/summationdisplay
+r=0(−1)k−r−1A2r+1θ2k−2r−1
+!(2k−2r−1)(27)
+S2(k) = (−1)k+1[1
+2∞/summationdisplay
+n=1Dn(2k+1)θ2n+2k]+k/summationdisplay
+r=0(−1)k−rA2r+1θ2k−2r
+!(2k−2r)(28)
+Wewillprovethataboveequationsholdtruefork=k+1(Inductive r esult).
+Integratingabove equation28oncefrom[0 ,θ], wegetS3(k) = [sin(θ)
+12k+2−sin(2θ)
+22k+2+
+sin(3θ)
+32k+2−sin(4θ)
+42k+2+...]
+S3(k) = (−1)k+1[1
+2∞/summationdisplay
+n=1Dn(2k+1)θ2n+2k+1
+2n+2k+1]+k/summationdisplay
+r=0(−1)k−rA2r+1θ2k−2r+1
+(2k−2r+1)[!(2k−2r)]
+(29)
+SinceDn(2k+1)
+2n+2k+1=Dn(2k+2) and (2 k−2r+1)[!(2k−2r)] =!(2k−2r+1),
+above equation S3(k) =S1(k+1), thus proving the inductive hypothesis for
+k=k+1.
+Similarly, Integrating S3(k) once from [0 ,θ], we get
+F4(k) =−[cos(θ)
+12k+3−cos(2θ)
+22k+3+cos(3θ)
+32k+3−cos(4θ)
+42k+3+...]+[1
+12k+3−1
+22k+3+1
+32k+3−1
+42k+3+...]
+(30)
+F4(k) = (−1)k+1[1
+2∞/summationdisplay
+n=1Dn(2k+2)θ2n+2k+2
+2n+2k+2]+k/summationdisplay
+r=0(−1)k−rA2r+1θ2k−2r+2
+(2k−2r+2)[!(2k−2r+1)]
+(31)
+7SinceDn(2k+2)
+2n+2k+2=Dn(2k+3)and(2 k−2r+2)[!(2k−2r+1)] =!(2 k−2r+2),
+and [1
+12k+3−1
+22k+3+1
+32k+3−1
+42k+3+...] =A2k+3we can define S4(k) =
+A2k+3−F4(k) as follows.
+S4(k) =A2k+3−F4(k) = [cos(θ)
+12k+3−cos(2θ)
+22k+3+cos(3θ)
+32k+3−cos(4θ)
+42k+3+...] (32)
+S4(k) =A2k+3+(−1)k+2[1
+2∞/summationdisplay
+n=1Dn(2k+3)θ2n+2k+2]−k/summationdisplay
+r=0(−1)k−rA2r+1θ2k−2r+2
+[!(2k−2r+2)]
+(33)
+Puttingk=k+1 in Eq. 28, we have
+S2(k+1) = (−1)k+2[1
+2∞/summationdisplay
+n=1Dn(2k+3)θ2n+2k+2]−k/summationdisplay
+r=0(−1)k−rA2r+1θ2k−2r+2
+!(2k−2r+2)+A2k+3
+(34)
+We can see that S4(k) =S2(k+1), thus proving the inductive hypothesis
+for k=k+1.
+Substituting θ=π/2 in above equations 27 and 28, we get
+A2k= [1
+12k−1
+32k+1
+52k...] = (−1)k[1
+2∞/summationdisplay
+n=1Dn(2k)(π
+2)2n+2k−1]+k−1/summationdisplay
+r=0(−1)k−r−1A2r+1(π
+2)2k−2r−1
+!(2k−2r−1)
+(35)
+A2k+1= 22k+1[(−1)k+1[1
+2∞/summationdisplay
+n=1Dn(2k+1)(π
+2)2n+2k]+k/summationdisplay
+r=0(−1)k−rA2r+1(π
+2)2k−2r
+!(2k−2r)]
+(36)
+8whereA2k+1= [1
+12k+1−1
+22k+1+1
+32k+1−1
+42k+1+...]. These show that A2k
+andA2k+1can be expressed as linear combination of powers of π/2. Let us
+assume the following inductive hypothesis for some k:
+A2k+1=∞/summationdisplay
+n=1En(2k+1)(π
+2)2n+2k(37)
+En(2k+1) =(−1)kDn(2k+1)
+2+(−1)k+1/summationtextk−1
+r=0(−1)rEn(2r+1)
+!(2(k−r))
+1−1
+22k+1(38)
+We will prove that this hypothesis holds for k=k+1. Putting k=k+1 in
+Eq.36,
+A2k+3= 22k+3[(−1)k[1
+2∞/summationdisplay
+n=1Dn(2k+3)(π
+2)2n+2k+2]+k/summationdisplay
+r=0(−1)k+1−rA2r+1(π
+2)2k−2r+2
+!(2k−2r+2)+A2k+3]
+(39)
+This can be written as follows:
+A2k+3=1
+1−1
+22k+3[(−1)k+1[1
+2∞/summationdisplay
+n=1Dn(2k+3)(π
+2)2n+2k+2]−k/summationdisplay
+r=0(−1)k+1−rA2r+1(π
+2)2k−2r+2
+!(2k−2r+2)]
+(40)
+Using Eq. 37 to replace A2r+1in above equation, interchanging order of
+summation and taking out common factor (π
+2)2n+2k+2, we get
+A2k+3=∞/summationdisplay
+n=1En(2k+3)(π
+2)2n+2k+2(41)
+En(2k+3) =(−1)k+1Dn(2k+3)
+2+(−1)k/summationtextk
+r=0(−1)rEn(2r+1)
+!(2(k−r+1))
+1−1
+22k+3(42)
+9Thus we have proved the inductive result for k=k+1. Eq. 37 and Eq. 38
+imply above results.
+Similarly, A2kinEq.35 canbeextended to A2k+2by replacing k withk+1:
+A2k+2= (−1)k+1[1
+2∞/summationdisplay
+n=1Dn(2k+2)(π
+2)2n+2k+1]+k/summationdisplay
+r=0(−1)k−rA2r+1(π
+2)2k−2r+1
+!(2k−2r+1)
+(43)
+Letusassume thefollowing inductive hypothesis for A2kandEn(2k)some
+k:
+A2k= [1
+12k−1
+32k+1
+52k+...] =∞/summationdisplay
+n=1En(2k)(π
+2)2n+2k−1(44)
+En(2k) =(−1)kDn(2k)
+2+(−1)k+1k−1/summationdisplay
+r=0(−1)rEn(2r+1)
+!(2(k−r)−1)(45)
+Using Eq. 37 to replace A2r+1in above equation 43, interchanging order
+of summation and taking out common factor (π
+2)2n+2k+1, we get
+A2k+2=∞/summationdisplay
+n=1En(2k+2)(π
+2)2n+2k+1(46)
+En(2k+2) =(−1)k+1Dn(2k+2)
+2+(−1)kk/summationdisplay
+r=0(−1)rEn(2r+1)
+!(2(k−r)+1)(47)
+Thus we have proved the inductive result for k=k+1. Eq. 44 and Eq. 45
+imply above results.
+103. Section 3
+In thissection, it will beshown that theseries expansion of A2kandA2k+1
+in Eq. 20 and Eq. 21 converges.
+We know that Dn(k) =cn
+22n−1(2n+k−1)Pk, which can be rewritten as follows:
+Dn(k) =Dn(1)
+(2n+k−1)Pk−1(48)
+We will write En(2k) andEn(2k+ 1) in Eq. 22 and Eq.23 in terms of
+Dn(1) as follows:
+En(1) =Dn(1) (49)
+En(2) =En(1)−Dn(2)
+2=Dn(1)Fn(2) (50)
+whereFn(2) = 1−1
+2(2n+1)and lim n→∞Fn(2) =K(2) where K(2) is a con-
+stant.
+En(3) =−Dn(3)
+2+En(1)
+!2
+1−1
+23=Dn(1)Fn(3) (51)
+whereFn(3) =1
+!2−1
+2((2n+2)P2)
+1−1
+23and lim n→∞Fn(3) =K(3) where K(3) is a con-
+stant.
+En(4) =Dn(4)
+2+En(3)
+!1−En(1)
+!3=Dn(1)Fn(4) (52)
+11whereFn(4) =−1
+!3+1
+2((2n+3)P3)+1
+!2−1
+2((2n+2)P2)
+1−1
+23and lim n→∞Fn(4) =K(4)
+whereK(4) is a constant.
+En(5) =Dn(5)
+2+En(3)
+!2−En(1)
+!4
+1−1
+25=Dn(1)Fn(5) (53)
+whereFn(5) =−1
+!4+1
+!2−1
+2((2n+2)P2)
+(!2)(1−1
+23)+1
+2((2n+4)P4)
+1−1
+25andlim n→∞Fn(5) =K(5)where
+K(5) is a constant.
+Let us assume the Inductive Hypothesis that En(2k) =Dn(1)Fn(2k)
+andEn(2k+1) =Dn(1)Fn(2k+1) and that lim n→∞Fn(2k) =K(2k) and
+limn→∞Fn(2k+1) =K(2k+1) where K(2k) andK(2k+1) are constants.
+Substituting this in Eq.22 and Eq.23, we can write Eq.42 and Eq.47 as
+En(2k+ 2) =Dn(1)Fn(2k+ 2) and En(2k+ 3) =Dn(1)Fn(2k+ 3) where
+limn→∞Fn(2k+2) =K(2k+2) and lim n→∞Fn(2k+3) =K(2k+3) where
+K(2k+2) and K(2k+3) are constants, thus proving the Inductive Result.
+Hence we can write
+lim
+n→∞En+1(2k)
+En(2k)=K0lim
+n→∞Dn+1(1)
+Dn(1)=K0
+4lim
+n→∞cn+1
+cn2n
+2n+2(54)
+whereK0= lim n→∞Fn+1(2k)
+Fn(2k)= 1. Given that lim n→∞|cn+1
+cn|<1 in the
+series expansion of tan( θ) in Eq. 3, we get the result
+lim
+n→∞|En+1(2k)
+En(2k)|<1 (55)
+12Similarly it can be shown that
+lim
+n→∞|En+1(2k+1)
+En(2k+1)|<1 (56)
+Hence the the series expansion of A2kandA2k+1in Eq. 20 and Eq. 21
+converges.
+4. Conclusion
+It has been shown that Riemann’s Zeta function with odd positive inte ger
+argument can be represented as an infinite summation of integer po wers ofπ
+with rational coefficients. Specific values for Apery’s Constant and Catalan’s
+Constant have been derived.
+5. References
+[1] Hardy, G. H. Divergent Series. New York: Oxford University Pre ss,
+1949.
+[2] Abramowitz, M. and Stegun, I. A. (Eds.). ”Circular Functions.” 4 .3
+in Handbook of Mathematical Functions with Formulas, Graphs, and Math-
+ematical Tables, 9th printing. New York: Dover, p. 75, 1972.
+13
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diff --git a/1001.0249.txt b/1001.0249.txt
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+arXiv:1001.0249v2  [math.HO]  11 Jan 2010On Taylor seriesexpansionof(1 +z)Afor|z|>1
+AkhilaRaman
+University ofCalifornia at Berkeley, CA-94720. Email: akh ila.raman@berkeley.edu. Ph: 510-540-5544
+Abstract
+ItiswellknownthattheTaylorseriesexpansionof(1 +z)Adoesnotconvergefor |z|>1where
+A is a realnumberwhichisnot equalto zeroora positiveinteg er. A limited seriesexpansionof
+thisexpressionisobtainedin thispaperfor |z|>1asaproductofconvergentseries.
+Keywords:
+1. Introduction
+Itiswell knownthattheTaylorseriesexpansionof(1 +z)Aisgivenby
+(1+z)A=∞/summationdisplay
+n=0/parenleftBiggA
+n/parenrightBigg
+zn(1)
+for|z|<1 where/parenleftBigA
+n/parenrightBig
+is the binomial choose function. It is well known that the ser ies ex-
+pansion does not converge for |z|>1 where A is a real number which is not equal to zero or a
+positiveinteger.
+We could obtain a limited series expansion for |z|>1 by writing the above expression as
+follows
+(1+z)A=(1+z
+2+z
+2)A=(1+z
+2)A(1+z
+2
+(1+z
+2))A=(1+z
+2)A(1+z
+z+2)A(2)
+The second term in the above equation has a convergent series representation, given that
+|z
+z+2|<1. If|z
+2|>1,we canwrite
+(1+z
+2)A=(1+z
+4)A(1+z
+z+4)A(3)
+Repeatingthis procedureiteratively,if m0is the minimumvaluefor which |z
+2m0|<1, we can
+write
+(1+z)A=(1+z
+2m0)Am0/productdisplay
+r=1(1+z
+z+2r)A(4)
+Eachofthetermsintheaboveproductoftermshasaconvergen tseriesrepresentation. Given
+that we can write the convergent series expansion for each of the terms above as (1 +z
+2m0)A=
+Preprint submitted to Number Theory November 15, 2018/summationtext∞
+n=0/parenleftBigA
+n/parenrightBig
+(z
+2m0)nand(1+z
+(z+2r))A=/summationtext∞
+m=0/parenleftBigA
+m/parenrightBig
+(z
+(z+2r))m,where/parenleftBigA
+n/parenrightBig
+representstheChoosefunction[1],
+we havethe seriesexpansionfor (1+z)Aexpressedasa productofconvergentseries, which
+convergesfor|z|>1 asfollows:
+(1+z)A=[∞/summationdisplay
+n=0/parenleftBiggA
+n/parenrightBigg
+(z
+2m0)n][m0/productdisplay
+r=1∞/summationdisplay
+m=0/parenleftBiggA
+m/parenrightBigg
+[z
+z+2r]m] (5)
+2. Section2
+Let us take the case of m0=1 for 1<|z|<2. Expanding the term1
+(z+2r)min the above
+equation5asTaylorseriesaroundapoint z=0, wehavefor m>0
+1
+(z+2r)m=∞/summationdisplay
+j=0b(j,r,m)zj(6)
+whereb(0,r,m)=1
+(2r)mandb(j,r,m)isgivenasfollowsfor j=1,2,3...
+b(j,r,m)=/parenleftBiggm+j−1
+j/parenrightBigg(−1)j
+(2r)m+j; (7)
+Form=0,1
+(z+2r)m=1. Nowwecanwritethetheseriesexpansionof(1 +z)Awhichconverges
+for1<|z|<2,asaproductoftermsexpandedin Taylorseriesasfollows:
+(1+z)A=[∞/summationdisplay
+n=0/parenleftBiggA
+n/parenrightBigg
+(z
+2)n][∞/summationdisplay
+m=0/parenleftBiggA
+m/parenrightBigg
+zm∞/summationdisplay
+j=0b(j,1,m)zj] (8)
+Forthecaseof m0>1 for|z|>2,wecanwrite asfollows:
+(1+z)A=[∞/summationdisplay
+n=0/parenleftBiggA
+n/parenrightBigg
+(z
+2m0)n][∞/summationdisplay
+m=0/parenleftBiggA
+m/parenrightBigg
+zm∞/summationdisplay
+j=0b(j,m0,m)zj][m0−1/productdisplay
+r=1∞/summationdisplay
+m=0/parenleftBiggA
+m/parenrightBigg
+zm[1
+z+2r]m] (9)
+Thelast termintheaboveequation(1
+z+2r)mcanbe expressedasfollows:
+(1
+z+2r)m=(z+2r)−m=2−r∗m(1+z
+2r)−m(10)
+The term(1+z
+2r)−mcan be recursively expanded using Eq.9 by substituting z→z
+2rand
+A→−mandm0→m0−rtoobtaintheseriesexpansionof(1 +z)Awhichconvergesfor |z|>1
+asa productoftermsexpandedinTaylorseries.
+3. Section3
+Letusconsiderthefollowingbinomialexpression
+2(x+y)A(11)
+whereAisarealnumberwhichisnotequaltozeroorapositiveintege randz=x
+yand|z|>1.
+Writing(x+y)A=(1+x
+y)AyA=(1+z)AyA, wecanwritetheseriesexpansionofthisexpression
+usingresultsobtainedin equations5 and9 asfollows:
+(x+y)A=yA∞/summationdisplay
+n=0/parenleftBiggA
+n/parenrightBigg
+(z
+2m0)nm0/productdisplay
+r=1∞/summationdisplay
+m=0/parenleftBiggA
+m/parenrightBigg
+[z
+z+2r]m(12)
+(x+y)A=yA[∞/summationdisplay
+n=0/parenleftBiggA
+n/parenrightBigg
+(z
+2m0)n][∞/summationdisplay
+m=0/parenleftBiggA
+m/parenrightBigg
+zm∞/summationdisplay
+j=0b(j,m0,m)zj][m0−1/productdisplay
+r=1∞/summationdisplay
+m=0/parenleftBiggA
+m/parenrightBigg
+zm[1
+z+2r]m] (13)
+4. Conclusion
+It has been shown that the Taylor series expansionof (1 +z)Acan be expandedas a product
+ofconvergentseries,for |z|>1whereAisa realnumberwhichisnotequalto zeroora positiv e
+integer.
+5. Acknowledgements
+The author would like to thank Michael Schlosser(Universit at Wien) for his constructive
+commentsonthepaper.
+6. References
+[1] Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Math ematical Functions with
+Formulas,Graphs,andMathematicalTables, 9thprinting. N ewYork: Dover,1972.
+3
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+Morlet wavelets in quantum mechanicsJohn AshmeadAbstractWavelets offer significant advantages for the analysis of problems in quantum mechanics. Because wavelets are localized in both time and frequency they avoid certain subtle but potentially fatal conceptual errors that can result from the use of plane wave or δ function decomposition. Morlet wavelets are particularly well-suited for this work: as Gaussians, they have a simple analytic form and they work well with Feynman path integrals. To take full advantage of Morlet wavelets we need an explicit form for the inverse Morlet transform and a manifestly covariant form for the four-dimensional Morlet wavelet. We supply both here.In theory, theory and practice are the same. In practice they are not. – A. EinsteinIntroductionWavelet transforms represent a natural development of Fourier transforms and may be used for similar purposes. Where the Fourier transform lets us decompose a wave function into its component plane waves, a wavelet transform lets us decompose a wave function into its component wavelets. If we think of the plane waves as corresponding to pure tones, we may think of the wavelets as corresponding to the notes produced by physical instruments: of finite duration and spanning a finite range of tones.Wavelets have two advantages over plane waves:1.They are localized in time and frequency. This can make them a better fit to the wave forms found in nature, which are always localized in both time and frequency. As a result, wavelet series will often converge faster than corresponding Fourier series.2.There are many different wavelets to choose from: we can tailor our wavelets to our problem.These advantages have resulted in their application to a wide variety of practical problems in acoustics, astronomy, medical imaging, computer graphics, meteorology, and so on1.Wavelets also have significant – if less numerous – application on the theory side. Examples: canonical quantization of the electromagnetic field using a discrete wavelet basis (Havukainen 2006), analysis of localization properties of photons using windowed wavelets (Kim 1996), regularization of Euclidean field theories (Altaisky 2003), and use of  wavelets to provide “Lorentz covariant, singularity-free, finite energy, zero action, localized solutions to the wave equation” (Visser 2003).Wavelets offer significant benefits for the study of foundational questions in quantum mechanics as well. We will focus here specifically on Morlet wavelets. These are Gaussians, so are both easy to work with and a natural fit to path integrals, which typically consist of long series of Gaussian integrations. Use of Morlet wavelets can let us:1.Avoid any need to invoke the notorious “collapse of the wave function” in the analysis of the Stern-Gerlach experiment,2.Avoid the use of artificial convergence factors or Wick rotation in computing path integrals,3.Compute path integrals in a time symmetric way.But to prepare Morlet wavelets for their new responsibilities we need to:1.Supply an explicit form for the “admissibility constant” discussed below. This is needed to define the inverse Morlet transform.2.Provide a manifestly covariant extension of Morlet wavelets to four dimensions.Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

1 of 171 A search on the preprint archive arXiv.org turned up over one thousand preprints with wavelet in title or abstract.Morlet's original reference is (Morlet, Arens et al. 1982). Wavelets are discussed in: (Chui 1992; Meyer 1992; Kaiser 1994; van den Berg 1999; Addison 2002; Bratteli and Jørgensen 2002; Antoine, Murenzi et al. 2004). Path integrals are discussed in: (Feynman and Hibbs 1965; Schulman 1981; Swanson 1992; Khandekar, Lawande et al. 1993; Marchewka and Schuss 2000; Kleinert 2004; Zinn-Justin 2005).Measurements, path integrals, and timeStern-Gerlach experimentIn the original Stern-Gerlach experiment (Gerlach and Stern 1922; Gerlach and Stern 1922; Gerlach and Stern 1922) a beam of silver atoms is sent through an inhomogeneous magnetic field. The beam is split into two: those atoms with spin up getting a kick in one direction; those with spin down in the opposite. This was striking both because it demonstrated the existence of spin and because a classical system would have shown a continuous range of values for the spin, not just up and down. This split is regarded as a classic demonstration of the measurement problem, explained – in the Copenhagen interpretation (von Neumann 1955) – by the “collapse of the wave function” into up and down components, and gotten much attention since.Typically the initial wave function is modeled as a plane wave times a spin vector. But recently Gondran and Gondran (Gondran and Gondran 2005) have shown that if you model the initial wave function of the silver atom as a Gaussian, and turn the crank on the Schrödinger equation, you see the spin up and spin down components separate without any need to invoke a “collapse”. It works a bit like a diffraction experiment: there is coherent interference at two spots, incoherent at the rest. One may think of this as an internal diffraction effect.Gondran and Gondran intended their work at least partly in support of the Bohm interpretation, however the math is independent of the interpretation. Standard quantum mechanics – if done in sufficient detail, i.e. with Gaussian test functions rather than plane waves – can explain the Stern-Gerlach effect without further assistance.The use of a single Gaussian test function is not of itself general. But with the use of the Morlet wavelet transform we can write an arbitrary square-integrable wave function as a sum over Gaussian test functions, making the Gondran and Gondran result completely general.To be sure, we could attempt to restore the honor of the plane wave by arguing that we could build up a Gaussian test function as a sum over such. But then why not “eliminate the middleman” and start with Gaussian test functions?I am aware of several related examinations of the Stern-Gerlach effect2. Cruz-Barrios and Gómez-Camacho (Cruz-Barrios and Gómez-Camacho 1998; Cruz-Barrios and Gómez-Camacho 2001) argued that if the atoms could be modeled using “coherent internal states” (CIS), we would see the effect. And Venugopalan et al (Venugopalan, Kumar et al. 1995; Venugopalan 1997; Venugopalan 1999) argued we would see the effect as an effect of decoherence. The Gondran and Gondran result is simpler in that it posits no additional structure (CIS) or additional interaction (decoherence); standard quantum mechanics of its own gives the effect.Convergence of path integralsMorlet wavelets can assist in establishing convergence of Feynman path integrals without recourse to convergence factors as used in (Feynman and Hibbs 1965; Schulman 1981) or Wick rotation as in (Zinn-Justin 2005); convergence of the slice-by-slice integrals in the path integral is a side-effect of the initial wave function being a (sum of) Gaussians, for which convergence is automatic. Step by step in the free case:1.We start with the free Schrödinger equation: iddτψτx()=−12m∇2ψτx()2.The path integral expression for the kernel is given by: Kτx;′x()=limN→∞m2πiε3Ndx1…dxN−1expiεm2xj+1−xjε⎛⎝⎜⎞⎠⎟2j=0N−1∑⎛⎝⎜⎞⎠⎟∫3.A typical integral is: Kjxj+1;xj−1()=m2πiε3dxjexpiεm2xj+1−xjε⎛⎝⎜⎞⎠⎟2+m2xj−xj−1ε⎛⎝⎜⎞⎠⎟2⎛⎝⎜⎞⎠⎟⎛⎝⎜⎜⎞⎠⎟⎟∫Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

2 of 172 See also Ashmead, J. (2003, Tue, 7 Jan 2003 08:30:07 GMT). "Uncollapsing the wave function." from http://lanl.arxiv.org/pdf/quant-ph/0301016.Where ε is the width of a time slice:ε≡τN4.The typical integral does not converge. We can force convergence by adding a small imaginary part iσ to the mass:m→m+iσ. We could also add the small imaginary part to the time step ε or Wick rotate into imaginary time: t→it.5.Now, focus attention on the first step: ψ1x1()=dx0K1x1;x0()ψ0x0()∫6.Assume the initial wave function is a Gaussian: ψ1x1()=m2πiε3dx0expiεm2x1−x0ε⎛⎝⎜⎞⎠⎟2⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟1πσ64exp−x0−x0()22σ2⎛⎝⎜⎜⎞⎠⎟⎟∫This integral is convergent of itself. The result is a (slightly wider) Gaussian. We can do an infinite series of these, with the initial wave function showing an increasing amount of middle-aged spread but with all of the integrals converging.As an arbitrary wave function may be written, via Morlet wavelets, as a sum over Gaussian test functions, we have convergence in the general case3, without the introduction of artificial convergence factors. This by no means eliminates all of the technical problems with path integrals; for instance, there is still the curious question of the mid-point rule, as discussed in (Schulman 1981). But one step at a time.Symmetric analysis of time in path integralsOne immediate benefit of not needing convergence factors or Wick rotation is that we can treat time in a more symmetric way. One case where we might want to do this is in setting up a path integral analysis of the Stückelberg-Schrödinger equation:idψux()du=Hψux()Here u is a formal parameter – a scalar of some kind, perhaps the particle’s proper time -- and H is a Lorentz invariant Hamiltonian. There are examples in Feynman (Feynman 1950; Feynman 1951) and more recently in work by Land, Horwitz, and Seidewitz (Land and Horwitz 1996; Horwitz 1998; Seidewitz 2005). This approach has been sufficiently interesting that there is an ongoing conference devoted to this & related questions: (Gill, Horwitz et al. 2010).In the free case H might be given by:H=−12mi∂∂xµ⎛⎝⎜⎞⎠⎟i∂∂xµ⎛⎝⎜⎞⎠⎟=12m∂2∂t2−∂2∂x2−∂2∂y2−∂2∂z2⎛⎝⎜⎞⎠⎟Note that because of the Lorentz invariance the time and space parts enter into H with opposite sign, so in the path integral will have a problem converging in a Lorentz covariant way:1.Path integral form for the kernel: Kτx;′x()=limN→∞m2πε4Ndt1dx1…dtN−1dxN−1exp−iεm2tj+1−tjε⎛⎝⎜⎞⎠⎟2−xj+1−xjε⎛⎝⎜⎞⎠⎟2⎛⎝⎜⎞⎠⎟j=0N−1∑⎛⎝⎜⎜⎞⎠⎟⎟∫Note the pre-factor from the time part is slightly different from that for space:Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

3 of 173 I have elided some technical difficulties here; discussed at (much) greater length in a work in progress Ashmead, J. (2009). Quantum Time. Philadelphia. m2πiε→im2πε2.A typical slice: Kjxj+1;xj−1()=m2πε4dxjexp−iεm2tj+1−tjε⎛⎝⎜⎞⎠⎟2−xj+1−xjε⎛⎝⎜⎞⎠⎟2+tj−tj−1ε⎛⎝⎜⎞⎠⎟2−xj−xj−1ε⎛⎝⎜⎞⎠⎟2⎛⎝⎜⎞⎠⎟⎛⎝⎜⎜⎞⎠⎟⎟∫3.But now the addition of a small imaginary part to mass or time fails; any change that causes the time integrals to converge will cause the space integrals to diverge and vice versa. If we use different signs for time and space, then we break covariance.4.Wick rotation fails for the same reason. No matter which sign we choose for the Wick rotation, either the past or the future side will blow up.5.Again, look at the first step: ψ1t1,x1()=dt0dx0K1t1,x1;t0,x0()ψ0t0,x0()∫6.Assume our initial wave function is given by a Gaussian test function: ψ1t1,x1()=m2πε4dt0dx0exp−iεm2t1−t0ε⎛⎝⎜⎞⎠⎟2−x1−x0ε⎛⎝⎜⎞⎠⎟2⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟1πσ84exp−t022σ2−x0−x0()22σ2⎛⎝⎜⎜⎞⎠⎟⎟∫7.Again, the integrals converge step by step. As any square-integrable wave function may be written as a sum over such (see below) we have convergence. Of course to do this, we need covariant Morlet wavelets (see further below).In many cases, an asymmetric treatment of time is harmless. But if we are analyzing time itself, then we do not want to wire the assumption that it is asymmetric into the maths. To do so would result in circular reasoning (see Price for an amusing discussion: (Price 1991)). The use of small imaginary factors/Euclidean time/Wick rotation will not work for analyses that are of time itself, as such approaches implicitly prejudge the conclusion.Morlet wavelets in one dimensionϕt()=e−it−1ee−12t2-3-2-1123-0.4-0.20.20.4imaginaryrealt
+Mother Morlet wavelet, with f = 1To generate a set of wavelets we start with a mother wavelet ϕt(). We get the general wavelet ϕsdt() by scaling the mother wavelet by a scale factor s and displacing her by a displacement d:ϕsdt()≡1sϕt−ds⎛⎝⎜⎞⎠⎟.As with life, so with wavelets: correct choice of your mother is essential for success. For Morlet wavelets the mother wavelet is given by:Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

4 of 17ϕt()≡e−ift−e−f22⎛⎝⎜⎞⎠⎟e−12t2.The second term is needed to satisfy the admissibility condition, discussed below. t is often the time and f may be thought of as a reference frequency.As f goes to infinity, the second term becomes less and less relevant – in many practical applications it is dropped (see for instance (Johnson 2009)). At the other extreme, when f is zero, the mother wavelet is zero. We keep f a variable to help in calculating the value of the admissibility constant Cf.The general Morlet wavelet is created from the mother Morlet wavelet by scaling by s and displacing by d:ϕsdt()≡1se−ift−ds⎛⎝⎜⎞⎠⎟−e−f22⎛⎝⎜⎞⎠⎟e−12t−ds⎛⎝⎜⎞⎠⎟2.Both scale and displacement run from -∞ to ∞. A negative scale just gives the complex conjugate of the Morlet wavelet with positive scale:ϕ−s,dt()=ϕs,d*t().The mother Morlet wavelet herself is given in this notation as the Morlet wavelet with scale factor one, displacement zero:ϕt()=ϕ1,0t().Any square integrable function ψ may be expressed as a sum over Morlet wavelets. In principle this excludes δ functions and plane waves. We will see below that they are handled correctly however. The Morlet wavelet transform of a wave function ψ (Morlet wavelet transform of ψ) is given by: ψsd=dtϕsd*t()ψt()−∞∞∫.We get the original ψ back (inverse Morlet wavelet transform of ψ) by integrating over the displacement and scale: ψt()=1Cfdsdds2ddϕsdt()ψsd−∞∞∫−∞∞∫.The admissibility constant is given by an integral over the square of the Fourier transform of the mother wavelet:Cf≡2πdwwˆϕw()2−∞∞∫.In the general case we could use a different set of wavelets for the forward and the inverse transforms; it is one of the attractions of Morlet wavelets that we do not need to do this.The wavelet decomposition fails if Cf is not finite. For Cf to be finite, we see we need the zero frequency component of the Fourier transform of the Morlet wavelet mother to be zero:ˆϕ0()=0⇒dtϕt()−∞∞∫=0.The Fourier transform of the Morlet mother is real:ˆϕw()=expfw()−1()exp−12f2−12w2⎛⎝⎜⎞⎠⎟.The Fourier transform of the general Morlet wavelet is not:ˆϕsdw()=seidwexpsfw()−1()exp−12f2−12s2w2⎛⎝⎜⎞⎠⎟but may be written in terms of the Fourier transform of the mother:Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

5 of 17ˆϕsdw()=seidwˆϕsw().NormalizationMorlet wavelets are not wave functions, but do not object to being treated as such. Their normalization is independent of their scale and displacement:dtϕsd*t()ϕsdt()∫=e−f2−2e−3f24+1⎛⎝⎜⎞⎠⎟π.We can therefore write normalized Morlet wavelets as:ϕsdnormalized()t()=1π4e−f2−2e−3f24+1ϕsdt().Resolution of unityWe can establish the completeness of the wavelet transform by very general methods, see (Kaiser 1994). But if we are only concerned with Morlet wavelets, we can take advantage of their specific character to give a less general but more immediate proof4.If we substitute the integral for  ψsd in the integral for the inverse Morlet wavelet transform we get:ψt()=1Cfdsddd′ts2ϕsdt()ϕsd*′t()ψ′t()−∞∞∫.This will be true if we have5:δt−′t()=1Cfdsdds2ϕsdt()ϕsd*′t()−∞∞∫.This looks like a familiar decomposition in terms of a set of states weighted by 1s2. If we can show this directly, we will have shown we have a resolution of unity. To do this, we define the integral I:It,′t()≡1Cfdsdds2ϕsdt()ϕsd*′t()−∞∞∫.We wish to show that this integral gives the δ function. We write the Morlet wavelets in terms of their Fourier transforms to get:It,′t()=1Cfdsdds2dw2πexp−iwt()ˆϕsdw()∫d′w2πexpi′w′t()ˆϕsd*′w()∫−∞∞∫.Then we write the Fourier transforms of the wavelets in terms of the Fourier transform of the mother wavelet:1Cfdsddsdw2πexp−iwt−d()()ˆϕsw()∫d′w2πexpi′w′t−d()()ˆϕ*s′w()∫−∞∞∫.We recognize the integral over d as a δ function in w and ′w:dd2πexpiw−′w()d()∫=δw−′w().We use this hitherto disguised δ function to do the integral over ′w:Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

6 of 174 We are using the word “proof” in a relaxed rather than a rigorous sense.5 We are assuming we can freely interchange orders of integration.It,′t()=1Cfdsdwse−iwteiw′tˆϕsw()ˆϕ*sw()∫∫.We change the variable of integration to′s=sw, then rename ′s to s, break up the integral into two parts, with scale s positive and s negative, and flip the sense of s in the negative s plane:It,′t()=1Cfdssdwe−iwteiw′tˆϕs()ˆϕ*s()−∞∞∫0∞∫+1Cfdssdwe−iwteiw′tˆϕ−s()ˆϕ*−s()−∞∞∫0∞∫.We identify the w integration as another δ function, one which can come outside of the integrals:It,′t()=δt−′t()2πCfdssˆϕs()ˆϕ*s()+ˆϕ−s()ˆϕ*−s()()0∞∫.We recognize the remaining integral as Cf2πso we have:It,′t()=δt−′t().To show completeness we do not need the actual value of Cf, only that it is finite.Calculation of admissibility constant
+-10-5510123456Cf=2πe−f2dwwefw−1()2e−w2−∞∞∫
+fAdmissibility constant as a function of fNow, we compute the actual value of the admissibility constant. By substituting the explicit form of the Fourier transform of the mother Morlet wavelet in the formula for the admissibility constant we get:Cf=2πe−f2dwwefw−1()2e−w2−∞∞∫.For convenience, we define a new integral I:If()≡dwwewf−1()2e−w2−∞∞∫.For f equal to zero, I is zero by inspection. This is expected given that the original mother wavelet is zero when f is zero. As w goes to zero, the integrand goes as f2w so I is well-behaved in the small w limit. As w goes to ∞ the integral is obviously convergent; the exponential with argument quadratic in w ensures this. Therefore we can write I as:If()=d′fdI′f()d′f0f∫.The advantage of taking the derivative with respect to f is that it gets rid of the troubling factor of w in the denominator. The derivative of I with respect to f is:Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

7 of 17dIf()df=2dwe2wf−ewf()−e−2wf−e−wf()()e−w20∞∫.Or:dIf()df=4dwsinh2wf()−sinhwf()()()e−w20∞∫.Giving:dIf()df=2πef2erff()−ef24erff2⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟.We integrate this with respect to f to get a pair of hypergeometric functions:If()=f222F21,1;32,2;f2⎛⎝⎜⎞⎠⎟−2F21,1;32,2;f24⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟.Therefore we have for Cf:Cf=2πe−f2f222F21,1;32,2;f2⎛⎝⎜⎞⎠⎟−2F21,1;32,2;f24⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟.The F's are generalized hypergeometric functions. For f set to one we have:C1=2πe−122F21,1;32,2;1⎛⎝⎜⎞⎠⎟−2F21,1;32,2;14⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟≈4.1636.This can be checked by doing the original integral numerically.For small f, Cf goes as:limf→0Cf→2πf2.For large f, Cf goes as:limf→∞Cf→exp−f2().At this point we have an explicit form for the inverse Morlet transform, so have reached our objective. We now apply the Morlet wavelet transform to some interesting cases.Gaussian test functionsGaussian test functions (squeezed states) are the most important case:ψt()=1πσ24exp−iEt−τ()−t−τ()22σ2⎛⎝⎜⎞⎠⎟.The Fourier transform of a Gaussian test function is:ˆψw()=1πσ24expiwτ−E−w()22σ2⎛⎝⎜⎞⎠⎟.AnalysisThe Morlet wavelet transform of a Gaussian test function is: ψsd=d′t1seif′t−ds−exp−f22⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟e−12′t−ds⎛⎝⎜⎞⎠⎟21πσ24exp−iE′t−τ()−′t−τ()22σ2⎛⎝⎜⎞⎠⎟−∞∞∫.The integral is elementary:Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

8 of 17 ψsd=π1/4σs2+σ2s2e−ifss2+σ2d−τ()−12Es−f()2σ2s2+σ2−e−f22e−12E2s2σ2s2+σ2⎛⎝⎜⎞⎠⎟e−iEσ2s2+σ2d−τ()−12d−τ()2s2+σ2.As expected, the leading term is greatest when the displacement d = τ and when the scale s=fE. When f is zero, the wavelet transform is zero.Inverse Morlet wavelet transformAs H. Dumpty might have put it, the tricky part isn’t cutting the original wave function into parts, it’s putting the parts back together again6. We expect the original ψ will be given by: ψt()=1Cfdsdds2ϕsdt()ψsd−∞∞∫.Without loss of generality, we simplify by assuming τ is zero:ψt()=1Cfdsdds2π1/4σs2+σ22e−ift−ds⎛⎝⎜⎞⎠⎟−e−f22⎛⎝⎜⎞⎠⎟e−12t−ds⎛⎝⎜⎞⎠⎟2×e−ifss2+σ2d−12Es−f()2σ2s2+σ2−e−f22e−12E2s2σ2s2+σ2⎛⎝⎜⎞⎠⎟e−iEσ2s2+σ2d−12d2s2+σ2⎛⎝⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟−∞∞∫.The integral over d is straightforward, as all the terms are Gaussians in d:ψt()=1Cfdss2π3/4σ2s2+σ2exp−2E2s2σ2+2f22s2+σ2()+2iEσ2t+t222s2+σ2()⎛⎝⎜⎞⎠⎟−2exp−3f2s2+2f2σ2−2Efsσ2+2E2s2σ2+2ifst+2iEσ2t+t222s2+σ2()⎛⎝⎜⎞⎠⎟+exp−2(f−Es)2σ2+2i2fs+Eσ2()t+t222s2+σ2()⎛⎝⎜⎞⎠⎟⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟−∞∞∫.The limit as s goes to zero is:2e−f2−iEt−t22σ2f2π3/41σ⎛⎝⎜⎞⎠⎟9/2σ2+E2σ4−2iEσ2t−t2()⎛⎝⎜⎞⎠⎟s+Os2⎡⎣⎤⎦.The limit as s goes to±∞ is:±2e−f2−E02σ221−2ef24+ef2⎛⎝⎜⎞⎠⎟π3/4σs2+O1s⎡⎣⎢⎤⎦⎥3.Our integral is therefore neither singular at the origin nor divergent at infinity. Of course, we expect this since we are guaranteed by the decomposition theorem that this integral will give the original Gaussian. To show explicitly we get the original Gaussian we take the Fourier transform of both sides, with respect to t. The simplification is dramatic and most of the factors come outside of the integral over s. On the right we get:1Cf2πexp−f2()σ2π4exp−E−w()2σ22⎛⎝⎜⎞⎠⎟dsse−s2w2−1+efsw()2−∞∞∫.Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

9 of 176 For H. Dumpty’s role in the analysis of the Stern-Gerlach experiment see Englert, B.-G., J. Schwinger, et al. (1988). "Is Spin Coherence Like Humpty-Dumpty? I. Simplified Treatment." Found. Phys. 18: 1045--1056, Schwinger, J., M. O. Scully, et al. (1988). "Spin coherence and Humpty-Dumpty. II." Z. Phys. D 10: 135, Scully, M. O., B.-G. Englert, et al. (1989). "Spin coherence and Humpty-Dumpty. III. The effects of observation." Phys. Rev. A 40: 1775--1784.We change variables ′s=sw and note the integral is essentially the admissibility constant. Factors cancel giving:σ2π4exp−E−w()2σ22⎛⎝⎜⎞⎠⎟1Cf2πexp−f2()d′s′s2e−′s2−1+ef′s()2−∞∞∫⎛⎝⎜⎞⎠⎟=σ2π4exp−E−w()2σ22⎛⎝⎜⎞⎠⎟=ˆψw().Other ApplicationsWe will compute the Morlet wavelet transforms of δ functions, plane waves, and – to achieve maximum self-referentiality – a Morlet wavelet itself.δ functionsSince the δ function is not a square-integrable function, we are not guaranteed the wavelet transform will work. We therefore write the δ function as a limit of Gaussian test functions:δx()=limσ→0+12πσ2exp−x22σ2⎛⎝⎜⎞⎠⎟.This lets us use the result for a Gaussian test function: δsdτ()=limσ→0+1s2+σ2se−ifss2+σ2d−τ()−12f2σ2s2+σ2−e−f22⎛⎝⎜⎞⎠⎟e−12d−τ()2s2+σ2.Taking the limit as σ goes to zero: δsdτ()=1se−ifd−τs⎛⎝⎜⎞⎠⎟−e−f22⎛⎝⎜⎞⎠⎟e−12d−τ()2s2.This is itself a Morlet wavelet: δsdτ()=ϕsd*τ().We get the same result by computing the Morlet wavelet transform directly: δsdτ()=d′tϕsd*′t()δ′t−τ()−∞∞∫=1seifτ−ds−exp−f22⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟e−12τ−ds⎛⎝⎜⎞⎠⎟2.Since the demonstration of the resolution of unity only applies to square-integrable functions, we verify the inverse transform. We want to show:δt−τ()=1Cfdsdds2ϕsdt()ϕsd*()τ()−∞∞∫.However this is just what we showed when we computed the admissibility constant, so we are done.Plane wavesThe Morlet wave transform of a plane wave:φt()≡12πexp−iEt() is given by: φsdE()=dt1seift−ds−exp−f22⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟e−12t−ds⎛⎝⎜⎞⎠⎟212πexp−iEt()∫.We do the integral, discovering we get the Fourier transform of a Morlet wavelet: φsdE()=ˆϕsd*()E().For the inverse transform to be valid we require:Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

10 of 17φt()=1Cfdsdds2ϕsdt()ˆϕsd*()E()∫.To show this, we take the Fourier transform of each side. On the left side we get:δw−E().On the right side we have (writing the Fourier transforms of the Morlet wavelets in terms of the Fourier transforms of their mothers):1Cfdsdds2sexpidw()ˆϕsw()sexp−idE()ˆϕ*sE()∫.The integral over d is a δ function, which we pull out of the integral, leaving the now familiar admissibility constant behind:2πCfdssˆϕsw()ˆϕ*sE()∫δw−E()→2πCfdssˆϕsw()ˆϕ*sw()∫δw−E()=δw−E().Morlet wavelet transform of a Morlet wavelet:We look at the Morlet wavelet transform of a Morlet wavelet. We use a normalized Morlet wavelet, with σ replacing s and E replacing f:ψσEt()≡NσE1πσ24e−iEt−τ()−e−σ2E22⎛⎝⎜⎞⎠⎟e−12t−τσ⎛⎝⎜⎞⎠⎟2NσE≡π4e−σ2E2−2e−3σ2E24+1.The Morlet wavelet transform is given by: ψsdσE()=d′tϕsd*t()−∞∞∫ψσEt().To apply the results for Gaussian test functions break out the incoming Morlet wavelet into its two Gaussian test functions; read off the results: ψsdσE()=2πσs2+σ2se−σ2E2−2e−3σ2E24+1e−ifss2+σ2d−τ()−12Es−f()2σ2s2+σ2−e−f22e−12E2s2σ2s2+σ2⎛⎝⎜⎜⎞⎠⎟⎟e−iEσ2s2+σ2d−τ()−e−ifss2+σ2d−τ()−12f2σ2s2+σ2−σ2E22−e−f22−σ2E22⎛⎝⎜⎞⎠⎟⎛⎝⎜⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟⎟e−12d−τ()2s2+σ2.Covariant Morlet waveletsStrategyWe would like to generalize Morlet wavelets to four dimensions (one time, three space) in a way that is manifestly covariant. We will do this by taking the direct product of Morlet wavelets in time and the three space dimensions. The natural generalization of the Gaussian part of the one-dimensional Morlet wavelet is:exp−x22⎛⎝⎜⎞⎠⎟→exp−xµxµ2⎛⎝⎜⎞⎠⎟=expt22−x2+y2+z22⎛⎝⎜⎞⎠⎟.This clearly diverges in t. We have to fix this without losing manifest covariance.We will assume we start in a specific frame M. We will define the four-dimensional Morlet wavelet as the product of  four one-dimensional Morlet wavelets, then write our results in a way that is Lorentz-invariant.Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

11 of 17ConstructionWe take the four-dimensional mother Morlet wavelet as the direct product of four one-dimensional mother Morlet wavelets, one for each coordinate:ϕt()→ϕt()ϕx()ϕy()ϕx().We write the mother Morlet wavelet out explicitly:ϕt,x,y,z()=exp−if0t()−exp−f022⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟expif1x()−exp−f122⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟expif2y()−exp−f222⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟expif3z()−exp−f322⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟×exp−t2+x2+y2+z22⎛⎝⎜⎞⎠⎟.By scaling and displacing each component separately we get:ϕsdt,x,y,z()=1s0s1s2s3exp−if0t−d0s0⎛⎝⎜⎞⎠⎟−exp−f022⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟expif1x−d1s1⎛⎝⎜⎞⎠⎟−exp−f122⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟×expif2y−d2s2⎛⎝⎜⎞⎠⎟−exp−f222⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟expif3z−d3s3⎛⎝⎜⎞⎠⎟−exp−f322⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟×exp−12t−d0s0⎛⎝⎜⎞⎠⎟2+x−d1s1⎛⎝⎜⎞⎠⎟2+y−d2s2⎛⎝⎜⎞⎠⎟2+z−d3s3⎛⎝⎜⎞⎠⎟2⎛⎝⎜⎞⎠⎟⎛⎝⎜⎜⎞⎠⎟⎟.The Fourier transform of the mother Morlet wavelet is:ˆϕE,px,py,pz()=expf0E()−1()expf1px()−1()expf2py()−1()expf3pz()−1()×exp−f02+f12+f22+f322−E2+px2+py2+pz22⎛⎝⎜⎞⎠⎟.The Fourier transform of the general Morlet wavelet is:ˆϕsdE,px,py,pz()=s0s1s2s3expid0E−d1px−d2py−d3pz()()×exps0f0E()−1()exps1f1px()−1()exps2f2py()−1()exps3f3pz()−1()×exp−f02+f12+f22+f322−s02E2+s12px2+s22py2+s32pz22⎛⎝⎜⎞⎠⎟.Now we have to promote various non-covariant bits to covariant bits.The scale factors enter into the inverse Morlet integral in a slightly awkward way:ds0s02ds1s12ds2s22ds3s32∫.The simplest approach to this is to treat the four scale factors as so many scalars.The obvious choices for the displacement d and the reference frequency f are to treat them as four vectors. For the displacement a single four vector will suffice:d=d0,d1,d2,d3().We will need one four vector for each reference frequency:F0()≡f0,0,0,0()F1()≡0,f1,0,0()F2()≡0,0,f2,0()F3()≡0,0,0,f3().For convenience, we define the sum over all four F’ s as:Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

12 of 17F≡Fn()n=03∑=f0,f1,f2,f3().This is also a four vector. Note that the raw frequencies f0, f1, f2, f3 are themselves scalars since they are defined with respect to the specific frame M.To represent the sums as Lorentz invariants we define a set of second rank tensors (with their inverses):Σµn()ν≡s0−n0000−s1−n0000−s2−n0000−s3−n⎛⎝⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟,1Σn()⎛⎝⎜⎞⎠⎟µν=s0n0000−s1n0000−s2n0000−s3n⎛⎝⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟.The first three – all we need – are:Σµ0()ν≡10000−10000−10000−1⎛⎝⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟,Σµ1()ν≡1s00000−1s10000−1s20000−1s3⎛
+⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎞
+⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟,Σµ2()ν≡1s020000−1s120000−1s220000−1s32⎛
+⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎞
+⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟.By having the signature for each of these tensors be 1,−1,−1,−1() we are ensuring that our Gaussian integrals will converge.Written with these definitions the mother Morlet wavelet is:ϕxµ()=exp−iFn()µxµ()−exp−Fn()µFµ2⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟n=03∏⎛⎝⎜⎜⎞⎠⎟⎟exp−xµΣµ0()ν2xν⎛⎝⎜⎞⎠⎟and the general Morlet wavelet is:ϕΣdxµ()=detΣ1()()exp−iFn()µΣµ0()ϖΣϖ1()νxν−dν()()−exp−Fn()µΣµ0()νFνn()2⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟n=03∏⎛⎝⎜⎜⎞⎠⎟⎟exp−xµ−dµ()Σµ2()ν2xν−dν()⎛⎝⎜⎞⎠⎟.While we have worked this out in frame M, as it is written in terms of covariant quantities it is valid in all frames. We have therefore guaranteed Lorentz covariance of the Morlet wavelets.Note that the choice of frame defines a set of Morlet wavelets; with each frame there is a distinct set of Morlet wavelets. If we have multiple frames we wish to work with we will need to tag each Morlet wavelet with the frame it comes from. Usually there is an obvious choice of frame, i.e. the center-of-mass frame.With these definitions, the Fourier transform of the mother Morlet wavelet is:ˆϕp()=expFn()1Σ0()p⎛⎝⎜⎞⎠⎟−1⎛⎝⎜⎞⎠⎟n=03∏⎛⎝⎜⎞⎠⎟exp−F12Σ0()F−12p12Σ0()p⎛⎝⎜⎞⎠⎟.The Fourier transform of the general Morlet wavelet is:ˆϕΣdp()=1detΣ1()()expipd()expFn()1Σ1()p⎛⎝⎜⎞⎠⎟−1⎛⎝⎜⎞⎠⎟n=03∏⎛⎝⎜⎞⎠⎟exp−F12Σ0()F−12p12Σ2()p⎛⎝⎜⎞⎠⎟.Resolution of unityAny square integrable function ψt,x,y,z()may be expressed as a sum over these Morlet wavelets. The covariant Morlet wavelet transform is given by:Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

13 of 17 ψΣd=d4xϕΣd*x()ψx()−∞∞∫.And the inverse is given by: ψx()=1Cf0Cf1Cf2Cf3dsnsn2n=03∏⎛⎝⎜⎞⎠⎟d4dϕΣdx()ψΣd−∞∞∫−∞∞∫.The resolution of unity and the values of the constants of admissibility follow directly from the results for one dimension.The solutions for Gaussian test functions, δ functions, and plane waves are merely the direct products of the corresponding one-dimensional wave functions.We have therefore reached our goal: to generalize the Morlet wavelet transform to four dimensions in a way which is manifestly covariant.Alternative approachesAlternative (and more sophisticated) lines of attack are possible. For instance in (Antoine, Murenzi et al. 2004) or in (Perel and Sidorenko 2007) two dimensional wavelets are generated from the mother wavelet by using displacements, rotations (in the x-y plane), and a single scale factor: ϕRdsx,y()=1sϕmom()R⋅r−d()s⎛⎝⎜⎞⎠⎟where R is a rotation matrix (in two dimensions).By analogy, we could generalize one-dimensional wavelets to four dimensions by using displacements d, Lorentz transformations Λ, and a single dilation s:ϕΛdsxµ()=1s2ϕmom()1sΛµνxν−dν()⎛⎝⎜⎞⎠⎟.But establishing convergence, verifying the resolution of unity, and computing the admissibility constant for these wavelets would be a new project. Our immediate requirement is merely to establish that there is at least one set of covariant Morlet wavelets.SummaryThe naive use of plane wave/δ function decomposition can create artificial difficulties in the analysis of foundational questions of quantum mechanics7. The use of Morlet wavelet decomposition avoids these difficulties. With the explicit calculation of the admissibility constant and the demonstration of covariant Morlet wavelets, we have eliminated two of the barriers to full use of this powerful technology for the analysis of foundational questions in quantum mechanics.Conventions for the Fourier transformFor the Fourier transform from time to frequency we are using:ˆfw()≡12πdtexpiwt()ft()−∞∞∫with inverse Fourier transform:ft()=12πdwexp−iwt()ˆfw()−∞∞∫.In the Fourier transform in four dimensions time and space enter with opposite signs:Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

14 of 177 That the use of unlocalized wave functions creates artificial difficulties suggests that the localization of the wave function in time and frequency/space and momentum should be regarded as a fundamental attribute. ˆfw,k()≡14π2dtdxexpiwt−ik⋅x()ft,x()−∞∞∫with inverse Fourier transform: ft,x()=14π2dwdkexp−iwt+ik⋅x()ˆfw,k()−∞∞∫.Morlet wavelets and quantum mechanics
 
 John Ashmead
+January 1, 2010

15 of 17BibliographyAddison, P. S. (2002). The illustrated wavelet transform handbook : introductory theory and applications in science, engineering, medicine and finance. Bristol, Institute of Physics Publishing.Altaisky, M. V . (2003). Wavelet based regularization for Eucldean field theory. Proc. Int. Conf.LLN - GROUP 24: Physical and Mathematical Aspects of Symmetries, Bristol, IOP Publishing.Antoine, J. P., R. Murenzi, et al. (2004). Two-dimensional wavelets and their relatives. Cambridge, UK ; New York, Cambridge University Press.Ashmead, J. (2003, Tue, 7 Jan 2003 08:30:07 GMT). "Uncollapsing the wave function." from http://lanl.arxiv.org/pdf/quant-ph/0301016.Ashmead, J. (2009). Quantum Time. Philadelphia.Bratteli, O. and P. E. T. Jørgensen (2002). Wavelets through a looking glass : the world of the spectrum. Boston, Birkhäuser.Chui, C. K. (1992). An introduction to wavelets. Boston, Academic Press.Cruz-Barrios, S. and J. Gómez-Camacho (1998). "Semiclassical description of scattering with internal degrees of freedom." Nucl. Phys. A 636: 70--84.Cruz-Barrios, S. and J. Gómez-Camacho (2001). "Semiclassical description of Stern-Gerlach experiments." Phys. Rev. A 63: 012101.Englert, B.-G., J. Schwinger, et al. (1988). "Is Spin Coherence Like Humpty-Dumpty? I. Simplified Treatment." Found. Phys. 18: 1045--1056.Feynman, R. P. (1950). "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction." Physical Review 80(3): 440-457.Feynman, R. P. (1951). "An Operator Calculus Having Applications to Quantum Electrodynamics." Physical Review 84(1): 108-128.Feynman, R. P. and A. R. Hibbs (1965). Quantum Mechanics and Path Integrals, McGraw-Hill.Gerlach, W. and O. Stern (1922). "Das magnetische Moment des Silberatoms." Zeitschrift für Physik 9: 353--55.Gerlach, W. and O. Stern (1922). "Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld." Zeitschrift für Physik 9: 349--352.Gerlach, W. and O. Stern (1922). "Der experimentelle Nachweis des magnetischen Moments des Silberatoms." Zeitschrift für Physik 8: 110--111.Gill, T. L., L. P. Horwitz, et al. (2010). International Assocation for Relativistic Dynamics. The 7th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields. Hualien, Taiwan.Gondran, M. and A. Gondran. (2005, Wed, 30 Nov 2005 20:58:41 GMT). "A complete analysis of the Stern-Gerlach experiment using Pauli spinors." from http://lanl.arxiv.org/abs/quant-ph/0511276.Havukainen, M. (2006). "Wavelets as basis functions in canonical quantization."Horwitz, L. P. (1998). Second Quantization of the Stueckelberg Relativistic Quantum Theory and Associated Gauge Fields. First International Conference of Parametrized Relativistic Quantum Theory, PRQT '98, Houston, US.Johnson, R. W. (2009, Sun, 6 Dec 2009 18:52:01 GMT). "Symmetrization and enhancement of the continuous Morlet transform."   Retrieved Fri, 1 Jan 2010, from http://arxiv.org/abs/0912.1126.Kaiser, G. (1994). A friendly guide to wavelets. Boston, Birkhäuser.Khandekar, D. C., S. V . Lawande, et al. (1993). Path-integral Methods and their Applications. Singapore, World Scientific Publishing Co. Pte. Ltd.Kim, Y . S. (1996). Wavelets and Information-preserving Transformations. 3rd International Conference on Quantum Communications and Measurements, Fiji-Hakone Land, Japan.Kleinert, H. (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets. Singapore, World Scientific.Land, M. C. and L. P. Horwitz (1996). Off-Shell Quantum Electrodynamics, Tel Aviv University.Marchewka, A. and Z. Schuss (2000). "A Path Integral Approach to Current." Phys. Rev. A 61: 052107.Meyer, Y . (1992). Wavelets and operators. Cambridge, Cambridge University Press.Morlet, J., G. Arens, et al. (1982). "Wave propagation and sampling theory." Geophysics 47: 203-236.Perel, M. V . and M. S. Sidorenko (2007). "New physical wavelet 'Gaussian Wave Packet'." J. Phys. A: Math. Theor. 40: 3441-3461.Price, H. (1991). "The asymmetry of radiation: reinterpreting the Wheeler-Feynman argument." Found. Phys. 21: 959--75.Schulman, L. S. (1981). Techniques and Applications of Path Integrals. New York, John Wiley and Sons, Inc.Schwinger, J., M. O. Scully, et al. (1988). "Spin coherence and Humpty-Dumpty. II." Z. Phys. D 10: 135.Scully, M. O., B.-G. Englert, et al. (1989). "Spin coherence and Humpty-Dumpty. III. The effects of observation." Phys. Rev. A 40: 1775--1784.Morlet wavelets and quantum mechanics
 
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+January 1, 2010

16 of 17Seidewitz, E. (2005, Sun, 21 Aug 2005 20:01:30 GMT). "A Spacetime path formalism for relativistic quantum physics." from http://lanl.arxiv.org/pdf/quant-ph/0507115.Swanson, M. (1992). Path Integrals and Quantum Processes, Academic Press, Inc.van den Berg, J. C., Ed. (1999). Wavelets in Physics. Cambridge, Cambridge University Press.Venugopalan, A. (1997). "Decoherence and Schrödinger-cat states in a Stern-Gerlach-type experiment." Phys. Rev. A 56: 4307-4310.Venugopalan, A. (1999). "Pointer states via Decoherence in a Quantum Measurement." Physical Review A, submitted to.Venugopalan, A., D. Kumar, et al. (1995, Fri, 27 Jan 1995 22:21:09 +0530 (GMT+5:30)). "Analysis of the Stern-Gerlach Measurement." from http://lanl.arxiv.org/pdf/quant-ph/9501022.Visser, M. (2003). "Physical wavelets: Lorentz covariant, singularity-free, finite energy, zero action, localized solutions to the wave equation." Phys. Lett. A 315: 219-224.von Neumann, J. (1955). Mathematical Foundations of Quantum Mechanics. Princeton, Princeton University Press.Zinn-Justin (2005). Path Integrals in Quantum Mechanics. Oxford, Oxford University Press.Morlet wavelets and quantum mechanics
 
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+January 1, 2010

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+arXiv:1001.0252v1  [math.NA]  1 Jan 2010Blended General Linear Methods based on
+Boundary Value Methods in the
+Generalized BDF family
+Luigi Brugnano∗Cecilia Magherini†
+Final revised form March 28, 2009.
+Dedicated to John Butcher on the occasion of his 75th birthda y
+Abstract
+Among the methods for solving ODE-IVPs, the class of General
+Linear Methods (GLMs) is able to encompass most of them, ranging
+from Linear Multistep Formulae (LMF) to RK formulae. Moreover, it
+is possible to obtain methods able to overcome typical drawbacks of
+the previous classes of methods. For example, order barriers for stable
+LMF and the problem of order reduction for RK methods. Neverthe -
+less, these goals are usually achieved at the price of a higher compu-
+tational cost. Consequently, many efforts have been made in orde r to
+derive GLMs with particular features, to be exploited for their efficie nt
+implementation.
+In recent years, the derivation of GLMs from particular Bound-
+ary Value Methods (BVMs), namely the family of Generalized BDF
+(GBDF), has been proposed for the numerical solution of stiff ODE-
+IVPs [11]. In particular, in [8], this approach has been recently devel-
+oped, resulting in a new family of L-stable GLMs of arbitrarily high
+order, whose theory is here completed and fully worked-out. More over,
+for each one of such methods, it is possible to define a correspondin g
+Blended GLM which is equivalent to it from the point of view of the
+stability and order properties. These blendedmethods, in turn, allow
+the definition of efficient nonlinear splittings for solving the generate d
+discrete problems.
+A few numerical tests, confirming the excellent potential of such
+blendedmethods, are also reported.
+∗Dipartimento di Matematica “U.Dini”, Viale Morgagni 67/A, 50134 Firenze (Italy),
+e-mail:luigi.brugnano@unifi.it
+†Dipartimento di Matematica Applicata “U.Dini”, Via Buonar roti 1/C , 56127 Pisa
+(Italy), e-mail:cecilia.magherini@dma.unipi.it
+1Keywords: Numerical methods for ordinary differential equations,
+General Linear Methods, Boundary Value Methods (BVMs), Gener -
+alized Backward Differentiation Formulae (GBDF), Blended Implicit
+Methods, blended iteration.
+MSC:65L05, 65L20, 65H10, 65Y20.
+1 Introduction
+General Linear Methods, in the form introduced by Burrage an d Butcher
+[13], arenumericalmethodsforsolvingODE-IVPswhich,ino rdertoadvance
+the integration one step, require some information from the previous step
+(external stages ), along with some internal stages to be computed at the
+current step. Such methods are able to describe, as extreme c ases, both RK
+methods and LMF (the latter, when used as Initial Value Metho ds (IVMs))
+[14]. A GLM with rexternal stages and sinternal stages, when applied with
+stepsizehfor solving the IVP
+y′=f(t,y), y(t0) =y0∈Rm, (1)
+is usually described as
+Y[n]=h(A⊗Im)F(Y[n])+(U⊗Im)y[n−1], (2)
+y[n]=h(B⊗Im)F(Y[n])+(V⊗Im)y[n−1], n= 1,2,...,
+where:
+•the matrices A∈Rs×s,U∈Rs×r,B∈Rr×s,V∈Rr×rcharacterize
+the method;
+•Y[n],F(Y[n])∈Rsmare the vector of the internal stages and the cor-
+responding values of the function f, respectively;
+•y[n−1],y[n]∈Rrmare the vectors of the external stages.
+Many papers have been devoted, across the years, to the const ruction and
+the analysis of GLMs (see, e.g., the review article [16]; see also [15, 17, 18,
+19, 20, 21, 22, 23]). Clearly, the efficient implementation of GLMs depends
+on the properties of the matrix A. In particular, we shall here consider
+implicit GLMs for which the external and the internal stages coincide, i.e.,
+A=B, U=V∈Rr×r,andY[n]=y[n]∈Rrm.
+2Moreover, matrix Ais nonsingular and the order of accuracy of each entry of
+y[n]is equal to p(to be specified later), which is, therefore, the global orde r
+of the method. We mention that sometimes GLMs with approxima tions
+having the same order have been also called “peer methods” (s ee, e.g., [29]).
+Theapproach thatweshallconsider istheoneintroducedin[ 11, Chapter11,
+Section6]andisbasedonBoundaryValueMethods(BVMs). Inm oredetail,
+the family of BVMs which we shall consider is that of Generali zed Backward
+Differentiation Formulae (GBDF) [10, 11]. For the efficient sol ution of the
+generated discrete problems, we shall consider the blendedimplementation
+of such methods. This implementation, introduced in [2] for block implicit
+methods (see also [3, 4, 6, 9, 12]), naturally induces an effici ent splitting
+procedure for the solution of the generated discrete proble ms. The linear
+analysis of convergence for this splitting will be done acco rding to [7].
+With these premises, the structure of the paper is the follow ing: in Sec-
+tion 2 Boundary Value Methods (BVMs) are briefly sketched, al ong with
+their block form; in Section 3 the family of Generalized Back ward Differ-
+entiation Formulae (GBDF), in the BVMs class, is considered , in order to
+obtainL-stable GLMs of arbitrarily high order; in Section 4 the corr espond-
+ingBlendedGLMs are described; Section 5 is devoted to the implementati on
+details of the methods; Section 6 contains some numerical te sts; finally, Sec-
+tion 7 contains a few concluding remarks.
+2 BoundaryValueMethods (BVMs)and their block
+form
+Boundary Value Methods (BVMs) are a relatively new class of n umerical
+methods for ODEs based on an unconventional use of LMF. Even t hough
+recent developments of such methods have been also obtained (see, e.g.,
+[26, 27]), nevertheless, the main reference for such method s remains the
+book [11]. The basic idea, on which BVMs rely, can be easily ex plained
+by just applying a k-step LMF, which can be described, by using the usual
+notation, as
+k/summationdisplay
+i=0αiyn+i=hk/summationdisplay
+i=0βifn+i, (3)
+to the standard test equation
+y′=λy, Re (λ)<0. (4)
+3If we assume we know the first ν∈ {1,...,k}values of the discrete solution,
+as well as the last k−ν, say
+y0,...,yν−1,andyN−k+ν+1,...,yN, (5)
+then, under suitable conditions for the method, if
+|z1(q)| ≤ |z2(q)| ≤ ··· ≤ | zk(q)|
+are the zeros of the characteristic polynomial of the method ,
+ρ(z)−qσ(z),whereq=hλ,andρ(z) =k/summationdisplay
+i=0αizi, σ(z) =k/summationdisplay
+i=0βizi,(6)
+then the discrete solution is approximately given by
+yn≈c[zν(q)]n,for 0≪n≪N, (7)
+with the constant cindependent of N[11]. When ν=kwe have the
+usual way in which LMF are used; i.e., we approximate the cont inuous IVP
+by means of a discrete IVP. We speak, in such a case, of an Initial Value
+Method (IVM) . On the other hand, when ν < kwe are approximating the
+continuous IVP by means of a discrete BVP . The latter defines a Boundary
+Value Method (BVM) usedwith( ν,k−ν)-boundary conditions . Thisfreedom
+in the choice of the number νof the initial conditions in (5), allows us to
+overcome the usual Dahlquist’s barriers for stable LMF (see [11], for further
+details).
+Nevertheless, the IVP (1) only provides the initial conditi ony0, whereas
+the remaining set of k−1additional conditions in (5) are not known. How-
+ever, they can be retrieved implicitly, by introducing a sui table set of ν−1
+additional initial methods ,
+k/summationdisplay
+i=0α(j)
+iyi=hk/summationdisplay
+i=0β(j)
+ifi, j= 1,...,ν−1, (8)
+andk−νadditional final methods ,
+k/summationdisplay
+i=0α(j)
+k−iyN−i=hk/summationdisplay
+i=0β(j)
+k−ifN−i, j=ν+1,...,k, (9)
+which are independent of the main method (3). Moreover, in such a way,
+BVMs can be also implemented in block form ( block BVMs [11]), which
+4is computationally more appealing: as a matter of fact, bloc k BVMs have
+been successfully implemented in the computational codes G AM [24] and
+GAMD [32]. In more detail, the block method, with blocksize r, is used in
+a sequential fashion by solving, at the nth step, a discrete problem in the
+form
+A⊗Imy−hB ⊗Imf=0,
+where
+A=
+α(1)
+0α(1)
+1... ... α(1)
+k.........
+α(ν−1)
+0α(ν−1)
+1... ... α(ν−1)
+k
+α0α1... ... α k
+.........
+α0α1... ... α k
+α(ν+1)
+0α(ν+1)
+1... ... α(ν+1)
+k.........
+α(k)
+0α(k)
+1... ... α(k)
+k
+∈Rr×r+1,
+(10)
+matrixBis similarly defined by replacing the α-s with the β-s,
+y= (y(n−1)r, y(n−1)r+1, ..., y nr)T,f= (f(n−1)r, f(n−1)r+1, ..., f nr)T,
+andy(n−1)r,f(n−1)rare known from the previous step (when n= 1,y0is the
+initial condition for problem (1)). Block BVMs allow an easi er implemen-
+tation of variable stepsize, since only the stepsize hof the current “block”
+can be varied. Several families of block BVMs have then been d efined (see
+[11], for full details).
+3 Generalized Backward Differentiation Formulae
+(GBDF) and GLMs
+Let us consider the particular class of k-step LMF having the polynomial
+σ(z) in its simplest form, i.e.,
+σ(z) =zj,
+wherej∈ {0,1,...,k}, with the coefficients of the corresponding polyno-
+mialρ(z) (see (6)) uniquely defined by imposing that the method has th e
+5maximum possible order p=k. When j=k, one obtains the well-known
+family of BDF which, however, provides 0-stable methods onl y up tok= 6
+andA-stable methods up to k= 2. Nevertheless, when
+j=ν≡ ⌈(k+2)/2⌉,
+and the formula is used as a BVM with ( ν,k−ν)-boundary conditions,
+i.e., by fixing the values (5) of the discrete solution, then i t turns out that
+the resulting methods are stable for all values of k, and have been called
+Generalized BDF (GBDF) [10, 11, 1]. By the way, when k= 1,2, one
+obtains the usual first two BDF. In view of the implementation of GBDF as
+GLMs, one may assume they know the initial values of the discr ete solution.
+On the other hand, the final k−νvalues can be retrieved implicitly by using
+a suitable set of additional final methods (9), consisting of LMF having the
+following characteristic polynomials:
+σ(j)(z) =zj, ρ(j)(z) =k/summationdisplay
+i=0α(j)
+izi, j=ν+1,...,k,
+with the coefficients {α(j)
+i}uniquely defined by imposing the order p=k
+conditions. If we assume, for the sake of simplicity, that a c onstant stepsize
+his used, then by introducing the vectors
+ynew=
+yn+1
+...
+yn+r
+,fnew=
+fn+1
+...
+fn+r
+,yold=
+yn−r+1
+...
+yn
+,
+and ther×rmatrices A1andA2such that
+[A1|A2] =
+α0... ... ... ... α k
+......
+α0...... ... ... α k
+0 α(ν+1)
+0...... ... ... α(ν+1)
+k......
+α(k)
+0...... ... ... α(k)
+k
+,(11)
+it turns out that the discrete problem, at a given point tn=nh, can be cast
+in matrix form as
+(A2⊗Im)ynew=hfnew−(A1⊗Im)yold. (12)
+6This corresponds to a GLM of the form (2) with coinciding inte rnal and
+external stages, and A=A−1
+2,U=−A−1
+2A1(indeed, matrix A2turns out
+to be nonsingular). Such methods were called Block BVMs with Memory
+(B2VM2s)in [11]. In the GLM notation, the abscissae defining this GLM
+areci=i,i= 1,2,...,r. However, in order to guarantee the L-stability of
+the corresponding method, it is sometimes required to chang e them into the
+following set of abscissae:
+ci=i, i= 1,...,ℓ−1,
+cℓ+j=ℓ−1+j/summationdisplay
+m=0ξm, j= 0,...,r−ℓ, (13)
+with positive {ξm}s.t.cr=ℓ.
+A first possible choice, as also suggested in [11], is that of s etting
+ξm=ζm+1, m= 0,...,r−ℓ, (14)
+whereζis the positive root of the polynomial
+p(z) =r−ℓ/summationdisplay
+m=0zm+1−1.
+However, in general such a value of ζis an irrational number, which im-
+plies that the coefficients of the corresponding GLM are irrat ional numbers
+as well. In order to have methods whose coefficients are always rational
+numbers, a slightly different choice for the abscissae can be m ade, i.e.:
+ξm=2r−ℓ−m
+2r−ℓ+1−1, m= 0,...,r−ℓ. (15)
+Clearly, when ℓ=rwe have a uniformmesh, whereas for ℓ < rthelast
+r−ℓ+1 stepsizes are geometrically decreasing, in the case of ch oice (14), or
+approximately halved, in the case of choice (15). In such a ca se, the points
+correspondingto the abscissae {c0,...,cℓ−1,cr}are equally spaced and will
+betheonesneededforthesubsequentintegrationstep. This impliesthat r−ℓ
+zero columns must be appropriately inserted in matrix A1in (11), i.e., those
+referring to the approximations at the abscissae {cℓ,...,cr−1}(according to
+[11], such points are called auxiliary points ). Consequently, such a GLM
+based on GBDF is uniquely determined by the triple of integer s (k,r,ℓ).
+Clearly, the order of the corresponding method is p=k. In addition, for
+all practical values of k, by appropriately choosing the values of ( r,ℓ), it is
+7possible to guarantee the L-stability of such methods. However, it turns out
+that, for each value of k, the couples ( r,ℓ) are not unique. Consequently, an
+additional criterion will be considered, in the next sectio n, which is aimed
+to speed-up the iterative solution of the corresponding dis crete problems,
+via theblended implementation of the methods.
+4 Blended General Linear Methods
+In the previous section, we devised a procedurefor obtainin g a whole class of
+implicit GLMs based on GBDF, containing L-stable methods of arbitrarily
+high order. We now consider the problem of efficiently solving the generated
+discrete problems which, at a given time step, assume the for m (see (2)),
+y−h(A⊗Im)f=η, (16)
+where, according to (11) and (12),
+y=ynew,f=fnew, A=A−1
+2,η=−(A−1
+2A1⊗Im)yold.
+A discrete problem in the form (16) can be efficiently solved, u nder suitable
+hypotheses, via the blended implementation of the method. We recall that
+the blended implementation of block implicit methods has be en previously
+considered in the framework of one-step methods [2, 3, 4, 6, 7 , 9, 12] and
+has been implemented in the computational codes BiM[4, 31] and BiMD
+[9, 31]. In order to describe the blended implementation of the GLM (16), it
+is convenient to consider its application for solving the us ual test equation
+(4). In such a case, in fact, the discrete problem reduces to a linear system
+of dimension r:
+(I−qA)y=η, q=hλ, (17)
+where, hereafter, Idenotes the identity matrix of dimension r. Such a linear
+system is clearly equivalent to
+γ(A−1−qI)y=γA−1η≡η1, (18)
+withγ >0 a free parameter. By introducing the weight function
+θ(q) = (1−γq)−1I, (19)
+we can then obtain an equivalent discrete problem by combini ng, with
+weightsθ(q) andI−θ(q), respectively, the linear systems (17)-(18):
+M(q)y≡/parenleftbig
+θ(q)(I−qA)+γ(I−θ(q))(A−1−qI)/parenrightbig
+y
+=θ(q)η+(I−θ(q))η1≡η(q). (20)
+8Equation (20) defines the Blended General Linear Method (Blended GLM)
+corresponding to the original GLM (17). Then, in view of the f act that
+M(q)≈/braceleftigg
+I,forq≈0,
+−γqI,for|q| ≫1,
+the following blended iteration for solving the discrete problem (20) is nat-
+urally induced:
+N(q)y(i+1)≡(I−γqI)y(i+1)= (N(q)−M(q))y(i)+η(q), i= 0,1,....
+(21)
+Remark 1 By considering that in (21) N(q)−1=θ(q)(see (19)), it is quite
+straightforward to realize that, in the case of problem (1), the corresponding
+blended iteration formally becomes
+δ(i)=N−1/parenleftig
+θ/parenleftig
+(I−γA−1)⊗Im(y(i)−η)−h(A−γI)⊗Imf(i)/parenrightig
++γ/parenleftig
+A−1⊗Im(y(i)−η)−hI⊗Imf(i)/parenrightig/parenrightig
+, (22)
+y(i+1)=y(i)−δ(i), i= 0,1,...,
+where
+N≡θ−1=I⊗(Im−hγJ), (23)
+withJthe Jacobian of fevaluated at the last known point. Consequently,
+only the factorization of one matrix of the size of the continuous problem is
+required.
+Coming back to the iteration (21), we observe that (see [3, 7] ) the it-
+eration matrix corresponding to the blended iteration (21) turns out to be
+given by
+Z(q)≡I−N(q)−1M(q) =q
+(1−γq)2A−1(A−γI)2.(24)
+According to the linear analysis described in [7], we shall n ow study the
+spectralpropertiesof Z(q), inordertoobtainalinearanalysisofconvergence
+for theiteration (21). For this purpose, hereafter let ρ(q) denote the spectral
+radius of Z(q). Clearly, the iteration will be convergent if and only if ρ(q)<
+1; moreover, the set
+Γ ={q∈C:ρ(q)<1}
+is theregion of convergence of the iteration. For the sake of completeness
+(see [7] for details), we recall that the iteration is:
+9•0-convergent , ifρ(0) = 0;
+•A-convergent , ifC−⊆Γ;
+•L-convergent withindexν∞, if it isA-convergent,
+Z∞≡lim
+q→∞Z(q) =O,
+andν∞is the index of nilpotency of Z∞.
+Clearly, the iteration (21) is 0 -convergent , sinceZ(0) =O. Moreover, one
+easily verifies (from (24)) that
+Z(q)→O,asq→ ∞,
+so thatA-convergence and L-convergence (with index 1) are equivalent. In
+addition to this,
+ρ(q)≈/braceleftigg
+≈˜ρq, forq≈0,
+≈˜ρ∞q−1,for|q| ≫1,
+where ˜ρand ˜ρ∞arethenonstiff amplification factor andthestiff convergence
+factorof the iteration, respectively. Finally, for a 0-convergen t iteration,
+A-convergence is equivalent to requiring that the maximum amplification
+factor,
+ρ∗= max
+x>0ρ(ix),
+(withidenoting, as usual, theimaginary unit)is notgreater than1 . Accord-
+ing to [7], ˜ ρ, ˜ρ∞, andρ∗are evaluation parameters for the blended iteration
+(21) (the smaller they are, the better the properties of the i teration). Conse-
+quently, if ρ∗≤1, the iteration is L-convergent and, therefore, appropriate
+forL-stable methods [7], like the GLMs defined in Section 3.
+Because of the form (24) of the iteration matrix, it turns out that (see
+also [7]):
+˜ρ=ρ/parenleftbig
+A−1(A−γI)2/parenrightbig
+,˜ρ∞=γ−2˜ρ, ρ∗= (2γ)−1˜ρ,
+withρ/parenleftbig
+A−1(A−γI)2/parenrightbig
+denoting the spectral radius of that matrix. Con-
+sequently, the value of the free positive parameter γis chosen in order to
+minimize ρ∗. In Table 1, the relevant figures are reported for selected GL Ms
+based on GBDF, each characterized by the corresponding trip le (k,r,ℓ),
+when the choice (14) for the auxiliary points is considered. Similarly, in
+10Table 2, the same results, obtained with the choice (15) of th e auxiliary
+points, are listed. In both tables, we list r−ℓ(i.e., the number of the auxil-
+iary points) in place of ℓ. The specified values of the blocksize r, have been
+here chosen as small as possible, in order to minimize the com putational
+cost per step.
+As it can be seen, the blended iteration corresponding to eac h method
+turns out to be L-convergent. Moreover, the value of the parameter γ(with
+the only exception of that corresponding to k= 4, for both the choices of
+the auxiliary steps) turns out to coincide with the value
+γ∗= min
+µ∈σ(A)|µ|,
+suggested in [3] for a class of Blended Implicit Methods (see also [7]).
+As an example, we list the matrices AandUwhich define the third
+order GLM corresponding to the triple (3 ,2,2), requiring no auxiliary steps
+(hereafter, let c∈Rrdenote the vector of the abscissae),
+c= (1,2)T, A=1
+23/parenleftbigg22−4
+36 6/parenrightbigg
+, U=1
+23/parenleftbigg−5 28
+−4 27/parenrightbigg
+,(25)
+andthosedefiningthefourthorderGLM correspondingtothet riple(4,4,3),
+with the choice (14),
+c=/parenleftigg
+1,2,3+√
+5
+2,3/parenrightiggT
+,
+A=
+
+0.85795933248329 −0.22588594615620 0 .09292560797716 −0.02384741384935
+1.14013555838905 0 .75295315385401 −0.30975202659054 0 .07949137949784
+1.16454145194449 0 .91413597782316 0 .27338586108581 −0.07015876367984
+1.16304178987375 0 .90057211551936 0 .50490826434742 0 .09507592794253
+,
+U=
+0.07149661104027 −0.44184164162566 0 1 .37034503058538
+0.09501129653242 −0.52719452791448 0 1 .43218323138206
+0.09704512099537 −0.53021970356702 0 1 .43317458257165
+0.09692014915615 −0.53024220062923 0 1 .43332205147308
+.
+and with the choice (15),
+11c=/parenleftbigg
+1,2,8
+3,3/parenrightbiggT
+,
+A=1
+6336684
+5429268 −1381941 570807 −174960
+7249176 4606470 −1902690 583200
+7421568 5690784 2388672 −732160
+7415388 5637357 3628233 198936
+,
+U=1
+6336684
+452439 −2798388 0 8682633
+604098 −3345408 0 9077994
+618464 −3365888 0 9084108
+617949 −3366036 0 9084771
+,
+which slightly differ from each other.
+Finally, in Figures 1 and 2, we plot the boundary of the stabil ity regions
+of the methods listed in Table 1, whereas in Figures 3 and 4, we plot the
+boundary of the stability regions of the methods listed in Ta ble 2.
+5 Implementation details
+In theactual implementation of the above Blended GLMs, thre emain points
+need to be clarified:
+•the choice of a suitable starting procedure;
+•an efficient local error estimate;
+•the variable stepsize implementation of the methods.
+All of them are briefly sketched here.
+Concerning the first issue, namely the definition of a startin g procedure
+to obtain the first vector of approximations, from the initia l condition y0of
+problem (1), a natural candidate is the block GBDF of order kand blocksize
+r=k. In more detail, at the very beginning, one solves the discre te problem
+A⊗Imy−hB ⊗Imf=0, (26)
+where
+y= (y0, y1, ..., y k)T,f= (f0, f1, ..., f k)T,
+12krr−ℓγ ˜ρ˜ρ∞ρ∗
+3200.7223 0.2272 0.4355 0.1573
+4410.6195 0.3802 0.9908 0.3069
+6510.6063 0.5734 1.5600 0.4729
+8610.5769 0.6380 1.9170 0.5530
+10710.5502 0.6626 2.1887 0.6021
+12920.5271 0.7345 2.6438 0.6968
+141020.5127 0.7366 2.8022 0.7183
+161120.4999 0.7345 2.9393 0.7347
+Table1: Parameters oftheblendediteration associated wit hBlendedGLMs,
+based on GBDF, characterized by the triple ( k,r,ℓ) and the choice (14).
+krr−ℓγ ˜ρ˜ρ∞ρ∗
+3200.7223 0.2272 0.4355 0.1573
+4410.6249 0.3827 0.9801 0.3062
+6510.6082 0.5740 1.5520 0.4719
+8610.5778 0.6381 1.9113 0.5522
+10710.5507 0.6625 2.1845 0.6015
+12920.5274 0.7345 2.6407 0.6964
+141020.5130 0.7366 2.7998 0.7180
+161120.5000 0.7345 2.9374 0.7344
+Table2: Parameters oftheblendediteration associated wit hBlendedGLMs,
+based on GBDF, characterized by the triple ( k,r,ℓ) and the choice (15).
+and, by denoting with {α(j)
+i}theα-s coefficients of the additional initial and
+final methods (8) and (9), the matrices AandBare defined as (see (10)):
+A=
+α(1)
+0α(1)
+1... α(1)
+k.........
+α(ν−1)
+0α(ν−1)
+1... α(ν−1)
+k
+α0α1... α k
+α(ν+1)
+0α(ν+1)
+1... α(ν+1)
+k.........
+α(k)
+0α(k)
+1... α(k)
+k
+,B= (0|Ik)∈Rk×k+1.
+For the efficient solution of equation (26), a correspondingb lendedmethods,
+withtheassociated blendediteration, canbeconveniently andeasilydefined.
+13The estimate of the local error is obtained by deferred correction , as
+described in [11, Chapter 10] (see also [4, 5]), by first “plug ging” the local
+discrete solution (see (12)) ( yold,ynew)Tin the discrete problem defined
+by a higher order method. In such a way, one obtains an estimat eτof
+the local truncation error. The same is done by considering a n equivalent
+formulation of the same method (compare with (16) and (17)-( 18)), thus
+obtaining a new approximation τ1=γA−1⊗Imτ. After that, the estimate
+eof the local error is obtained by performing one blended iter ation, namely
+by formally solving the block diagonal linear system (see (2 3))
+Ne=θτ+(I−θ)τ1.
+In more detail, when using the GLM defined by the triple ( k,r,ℓ), the cor-
+responding method used for estimating τis the GBDF defined by the triple
+(k+1,r,ℓ). Consequently, the cost for estimating eis the same as that for
+carrying out one blended iteration. This can be done for all m ethods listed
+in Tables 1 and 2, with the only exception being the triple (3, 2,2), for which
+the value of rshould be increased. For this reason, such a method has not
+been considered for carrying out the numerical tests in Sect ion 6.
+Finally, for the efficient variation of the stepsize, we have c onsidered the
+Nordsiek implementation of the methods, firstly introduced in [28] and later
+modified according to, e.g., [25, 30]. For the sake of complet eness, we also
+mention that the initial guess for the blended iteration (22 )-(23) is obtained
+by extrapolation, through the interpolating polynomial on (see (12)) yold.
+6 Numerical Tests
+In this section, we report a few numerical tests comparing a M atlab fixed-
+order implementation of theBlendedGLMs presented here, wi th someof the
+most reliable codes currently available, on selected stiff t est problems. Both
+the problems and the codes have been taken from the current re lease (re-
+lease 2.4) of Test Set for IVP Solvers [32]. In particular, we have considered
+the following solvers:
+•BiMD,whichusesablendediteration forsolvingthegenerat eddiscrete
+problems;
+•GAMD, which uses a nonlinear splitting for solving the discr ete prob-
+lems, generated by block BVMs in the Generalized Adams Metho ds
+(GAMs) family;
+14•DASSL, which implements standard BDF methods.
+The problems considered are:
+•Pollution (of dimension 20);
+•Elastic Beam (of dimension 80);
+•Emep (of dimension 66);
+•Ring Modulator (of dimension 15).
+The comparisons have been summarized in corresponding work-precision
+diagrams [32], where the computational cost is plotted versus accura cy. A
+standardized cost has been computed as the number of floating -point oper-
+ations for the factorizations and the system solvings requi red by each code,
+while accuracy has been measured in terms of mixed-error significant correct
+digits[32]. For all codes, the used tolerances are essentially tho se specified
+in the Test Set. Figures 5–8 summarize the results obtained, where the la-
+bel “ORDER k” is used for the fixed-order implementation of the Blended
+GBDF of order k(see Table2): as onecansee, for each problem, theselected
+fixed-order implementations of the Blended GBDF presented h ere, appear
+to be competitive with the above mentioned solvers.
+7 Conclusions
+In this paper, a straightforward approach for deriving L-stable General Lin-
+ear Methods (GLMs) of arbitrarily high order has been presen ted. Such an
+approach relies on the class of Boundary Value Methods (BVMs ) for ODEs.
+In the same framework correspondingstarting procedures ar e easily derived,
+as well as appropriate error estimates.
+The generated discrete problems can be efficiently solved by m eans of
+the blended implementation of the methods, thus defining cor responding
+Blended GLMs, equivalent to theoriginal methods, from thep oint of view of
+the stability and accuracy properties. The corresponding b lended iterations
+are allL-convergent, thus appropriate for the underlying L-stable methods.
+A number of numerical tests, on problems taken from the Test S et for
+IVP Solvers, prove that the obtained methods are competitiv e with some of
+the best codes currently available.
+The availability of methods having arbitrarily high-order makes them
+good candidates for an efficient variable-order implementat ion.
+15References
+[1] L.Aceto, D.Trigiante. On the A-stable methods in the GBDF class.
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+cal implicit methods for ODEs. Jour. Comput. Appl. Mathematics 116
+(2000) 41–62.
+[3] L.Brugnano, C.Magherini. Blended Implementation of Bl ock Implicit
+Methods for ODEs. Appl. Numer. Math. 42(2002) 29–45.
+[4] L.Brugnano, C.Magherini. The BiM code for the numerical solution of
+ODEs.Jour. Comput. Appl. Mathematics 164–165 (2004) 145–158.
+[5] L.Brugnano, C.Magherini. Economical Error Estimates f or Block Im-
+plicit Methods for ODEs via Deferred Correction. Appl. Numer. Math.
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+and DAE problems, and their extension for second order probl ems.
+Jour. Comput. Appl. Mathematics 205(2007) 777–790
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+Initial and Boundary Value Methods , GordonandBreach SciencePubl.,
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+plicitmultistage integration methodsforordinarydifferen tialequations.
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+with Runge-Kutta stability properties. Numer. Alg. 36(2004) 53–72.
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+approach to the construction of general linear methods of hi gh order.
+Jour. Comput. Appl. Math. 81(1997) 181–196.
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+methods. BIT43(2003) 695–721.
+[24] F.Iavernaro, F.Mazzia. Solving Ordinary Differential E quations by
+Generalized Adams Methods: properties and implementation tech-
+niques.Appl. Num. Math. 28(1998) 107–126.
+[25] J.D.Lambert. Numerical Methods for Ordinary Differential Equations ,
+John Wiley & Sons, Chichester, 1993.
+17[26] F.Mazzia, A.Sestini, D.Trigiante. B-spline multiste p methods and
+their continuous extensions. SIAM Jour. Numer. Anal. 44(2006) 1954–
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+uniformmesh. Jour. Numer. Anal., Industrial and Appl. Math. 1(2006)
+91–112.
+[28] A.Nordsiek. On Numerical Integration of Ordinary Differ ential Equa-
+tions.Math. Comp. 16(1962) 22–49.
+[29] H.Podhaisky, R.Weiner, B.A.Schmitt. Linearly-impli cit two-step
+methods and their implementation in Nordsieck form. Appl. Numer.
+Math.56(2006) 374–387.
+[30] K.Radhakrishnan, A.Hindmarsh. Description and Use of LSODE, the
+Livermore Solver for Ordinary Differential Equations. LLNLReport
+UCRL-ID-113855, 1993.
+[31] Codes BiMandBiMDhomepage:
+http://www.math.unifi.it/~brugnano/BiM
+[32] Test Set for IVP Solvers: http://pitagora.dm.uniba.it/~testset
+180 1 2 3 4 5 6 7 8−5−4−3−2−1012345
+Re(q)Im(q)k=3
+k=4k=6k=8k=10
+Figure 1: Boundary loci of the GLMs obtained with the choice ( 14),k=
+3,4,6,8,10.
+0 1 2 3 4 5 6 7 8−5−4−3−2−1012345
+k=12k=14k=16
+Re(q)Im(q)
+Figure 2: Boundary loci of the GLMs obtained with the choice ( 14),k=
+12,14,16.
+190 1 2 3 4 5 6 7 8−5−4−3−2−1012345
+k=10k=8k=6k=4k=3
+Re(q)Im(q)
+Figure 3: Boundary loci of the GLMs obtained with the choice ( 15),k=
+3,4,6,8,10.
+0 1 2 3 4 5 6 7 8−5−4−3−2−1012345
+k=12k=14
+k=16
+Re(q)Im(q)
+Figure 4: Boundary loci of the GLMs obtained with the choice ( 15),k=
+12,14,16.
+204 6 8 10 12 14 16105106107
+mixed−error significant correct digits# flops
+ORDER 4
+ORDER 6
+ORDER 8
+BiMD
+GAMD
+DASSL
+Figure 5: Numerical results for the Pollution Problem.
+2 2.5 3 3.5 4 4.5 5 5.5 6 6.5107108109
+mixed−error significant correct digits# flops
+ORDER 4
+ORDER 6
+ORDER 8
+BiMD
+GAMD
+DASSL
+Figure 6: Numerical results for the Elastic Beam Problem.
+211 2 3 4 5 6 7 8 9 10108109
+mixed−error significant correct digits# flops
+ORDER 4
+ORDER 6
+BiMD
+GAMD
+DASSL
+Figure 7: Numerical results for the Emep Problem.
+0 2 4 6 8 10 12108109
+mixed−error significant correct digits# flops
+ORDER 6
+ORDER 8
+ORDER 10
+BiMD
+GAMD
+DASSL
+Figure 8: Numerical results for the Ring Modulator Problem.
+22
\ No newline at end of file
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+arXiv:1001.0253v4  [cs.DM]  24 Aug 2011Symposium on TheoreticalAspects of Computer Science 2010( Nancy), pp. 441–452
+www.stacs-conf.org
+REVISITING THE RICE THEOREM OF CELLULAR AUTOMATA
+PIERRE GUILLON1AND GA ´ETAN RICHARD2
+1DIM - CMM, UMI CNRS 2807, Universidad de Chile, Av. Blanco Enc alada 2120, 8370459
+Santiago, Chile
+E-mail address :pguillon@dim.uchile.cl
+2Greyc, Universit´ e de Caen & CNRS, boulevard du Mar´ echal Ju in, 14000 Caen, France
+E-mail address :grichard@info.unicaen.fr
+Abstract. A cellular automaton is a parallel synchronous computing mo del, which con-
+sists in a juxtaposition of finite automata whose state evolv es according to that of their
+neighbors. It induces a dynamical system on the set of configu rations, i.e.the infinite
+sequences of cell states. The limit set of the cellular autom aton is the set of configurations
+which can be reached arbitrarily late in the evolution. In th is paper, we prove that all
+properties of limit sets of cellular automata with binary-s tate cells are undecidable, except
+surjectivity. This is a refinement of the classical “Rice The orem” that Kari proved on
+cellular automata with arbitrary state sets.
+Introduction
+Among all results on undecidability, Rice’s Theorem [Ric53 ] is probably one of the most
+important. It can be seen as stating the following: for any pr operty on the functions com-
+putedbyTuringmachines, theset of correspondingmachines iseither trivial orundecidable.
+Following Church-Turing thesis, it is often thought that th is result should remain true for
+other computational systems. It has, for instance, been ext ended with various restrictions
+to general dynamical systems [DB04], tilings [LW08] or, in a weaker form [CD04].
+In this paper, we shall focus on a specific model known as cellu lar automata, introduced
+by Von Neumann [vN66]. Cellular automata are made of infinite ly many cells endowed with
+a finite state and interacting locally and synchronously wit h each other. As this system
+does not have any way to give output, study of dynamics often u ses the limit set, that
+consists of configurations which can appear arbitrarily lat e [Hur87, ˇCul´ ık IIPY89]. In this
+domain, Jarkko Kari has already proved an equivalent of Rice theorem [Kar94b] for limit
+set. A similar, “perpendicular”, result is also known for th e trace, which consists on the
+evolution of only one fixed cell [CG07].
+On the other hand, it is known that the property of being surje ctive (i.e.having a full
+limit set) is decidable and not trivial for one-dimensional cellular automata [AP72]. Such
+1998 ACM Subject Classification: F.1.1 Models of Computation.
+Key words and phrases: cellular automata, undecidability.
+Thanks to the Projet Blanc ANR Sycomore and Program ECOS-Sud .
+c/circlecopyrtP. GuillonandG. Richard
+CC/circlecopyrtCreativeCommonsAttribution-NoDerivsLicense2 P. GUILLON AND G. RICHARD
+a statement is not contradictory with Kari’s theorem since i t is not, properly speaking, a
+property of the limit set: a surjective CA can have the same li mit set than a non-surjective
+one if the alphabets are distinct. Nevertheless, when fixing the alphabet, surjectivity be-
+comes a property of the limit set. This leads to the question w hether there exist other
+decidable properties on limit sets of cellular automata wit h fixed alphabet [Kar05, DFM00].
+In this paper, we shall answer negatively to this question by extending the result of
+Kari when fixing the number of states, showing that all proper ties on limit sets other
+than surjectivity are either trivial or undecidable. Note t hat surjectivity is undecidable for
+higher dimensional cellular automata [Kar94a]. Our proofs use borders (example of similar
+constructions can be found in [DFV03, CFG07, Pou08]). The id ea here is to restrict our
+study to nonsurjective cellular automata, since surjectiv ity is decidable. The (computable)
+forbidden words of the image can be used as border words.
+The paper is organised as follows: first we give all the necess ary definitions in Section 1
+and some first properties of limit sets in Section 2. After tha t, we detail the core encoding
+of our construction in Section 3. With all this, we state the m ain Rice theorem in Section 4
+before giving some concluding remarks in Section 5.
+1. Definitions
+We denote the set with two elements as /BE={0,1}. For any alphabet B, we denote
+asB
+/CIthe set of configurations (all bi-infinite sequences over B). The length of some word
+u∈B∗will be noted |u|. Auniform word or configuration is one where a single letter
+appears, with repetitions. For any configuration x∈B
+/CIor any word x∈B∗,l,k∈ /CI,x/llbracketl,k/llbracket
+denotes the finite pattern xlxl+1...xk−1. This notation is extended to the case where lor
+kis infinite.
+Acylinder is the subset of configurations [ u]i={x∈B
+/CI|x/llbracketi,i+|u|/llbracket=u}sharing the
+common pattern u∈B∗at position i∈ /CI. Similarly, if E⊂Bkfor some k∈ /C6\{0}, then
+[E]iwill stand for the set of configurations x∈B
+/CIsuch that x/llbracketi,i+k/llbracket∈E.
+The set of configurations B
+/CIis a compact metric space when endowing it with the
+metric induced by the Cartesian product of the discrete topo logy onB. In this setting,
+open sets correspond to unions of cylinders.
+Ifb∈B, thenwenote∞b∞theconfigurationconsistinginauniformbi-infinitesequen ce
+ofb. IfE⊂Bk, withk∈ /C6\{0}, then we will represent the set of bi-infinite sequences of
+words of Eas∞E∞=/braceleftbig
+x∈B
+/CI/vextendsingle/vextendsingle∃i < k,∀j∈ /CI,x/llbracketi+jk,i+(j+1)k/llbracket∈E/bracerightbig
+.
+The following definition will be very helpful for future cons tructions: it allows to build
+borders so that some particular nonoverlapping zones of con figurations can be recognized.
+Definition 1.1. LetBbe an alphabet and n∈ /C6\{0}. Astrongly freezing word u∈Bn
+is a word such that for all i∈/llbracket1,n/llbracket,uBi∩Biu=∅. Equivalently, [ u]0∩[u]i=∅.
+A setE⊂Bnisstrongly freezing if for all i∈/llbracket1,n/llbracket,EBi∩BiE=∅.
+One first remark is that any word ucan be extended to some strongly freezing word:
+simply take ubk, whereb/\e}atio\slash=u0andk∈ /C6\{0}such that bkdoes not appear in u.
+Theshiftσ:B
+/CI→B
+/CIis defined for all x∈B
+/CIandi∈ /CIbyσ(x)i=xi+1.
+AsubshiftΣ is a closed subset of B
+/CIwhich is strongly invariant by shift, i.e.σ(Σ) = Σ.
+Equivalently, a subshift can be defined as the set of configura tions avoiding a particular set
+L⊂B∗offinitepatterns,called forbidden language :/braceleftbig
+x∈B
+/CI/vextendsingle/vextendsingle∀i∈ /CI,∀u∈L,x/llbracketi,i+|u|/llbracket/\e}atio\slash=u/bracerightbig
+.REVISITING THE RICE THEOREM OF CELLULAR AUTOMATA 3
+If the forbidden language Lcan be taken finite, then we say that Σ is of finite type ; if
+it is empty it is the full shift. A subshift of finite type has orderk∈ /C6\{0}if it admits a
+forbidden language L⊂Bkcontaining only words of length k— or equivalently, of length
+at mostk.
+For any subshift Σ and −∞ ≤l≤m≤+∞, denote L/llbracketl,m/rrbracket(Σ) =/braceleftbig
+x/llbracketl,m/rrbracket/vextendsingle/vextendsinglex∈Σ/bracerightbig
+.
+Note that, when l−mis finite, it only depends on this difference, justifying the de finition
+Lk(Σ) =L/llbracket0,k/llbracket(Σ) fork∈ /C6. We note L(Σ) =/uniontext
+k∈ /C6Lk(Σ) =/braceleftbig
+x/llbracketl,m/rrbracket/vextendsingle/vextendsinglex∈Σ,l,m∈ /CI/bracerightbig
+the
+language of the subshift Σ.
+Definition 1.2. A (one-dimensional) cellular automaton is a triplet ( B,r,f) whereBis a
+finitealphabet (or state set), r∈ /C6is the neighborhood radiusandf:B2r+1→Bis the
+local transition function .
+A cellular automaton acts onelements of B
+/CI(calledconfigurations ) bysynchronousand
+uniform application of the local transition function, indu cing the global transition function
+F:B
+/CI→B
+/CI, formallydefinedforall x∈B
+/CIandi∈ /CIbyF(x)i=f(xi−r,xi−r+1,...,x i+r).
+We will assimilate the cellular automaton with its global fu nction.
+It is easy to see that any cellular automaton commutes with th e shift. In a more general
+way, Curtis, Hedlund and Lyndon proved that cellular automa ta correspond exactly to
+continuous self-maps of B
+/CIwhich commute whith the shift [Hed69].
+Note that a local rule f:B2r+1→Bcan be extended in f:B∗→B∗byf(u) =
+f(u/llbracket0,2r+1/llbracket)...f(u/llbracket|u|−2r−1,|u|/llbracket).
+Apartial cellular automaton is the restriction of the global function of some cellular
+automaton to some subshift of finite type Σ. Note that it can be defined from an alphabet
+B, a radius r∈ /C6and a local rule f:L2r+1(Σ)→B.
+For a cellular automaton ( B,r,f), a state b∈Bis said to be quiescent iff(b2r+1) =b.
+It is said to be spreading iff(u) =bwhenever the letter bappears in the word u.
+Note that if Fis a cellular automaton on alphabet B, thenF(B
+/CI) is a subshift. In
+particular, either Fis surjective, or F(B
+/CI) admits (at least) a forbidden pattern. It is easy
+to see that if j∈ /C6\{0}, thenFjis also a cellular automaton, and the subshift Fj(B
+/CI) is
+included in Fj−1(B
+/CI).
+The evolution being parallel and synchronous, we can see tha t the image of any uniform
+configuration remains uniform. The set of uniform configurat ion is then a finite subsystem,
+with an ultimate period p≤ |B|. In particular, Fpadmits some quiescent state.
+Definition 1.3. Thelimit set of a cellular automaton Fis the set
+ΩF=/intersectiondisplay
+j∈ /C6Fj(B
+/CI)
+of the configurations that can be reached arbitrarily late.
+From the remark above, the limit set of the cellular automato nFalways contains
+(at least) one uniform configuration. It is closed, and stron gly invariant by F. More
+precisely, the restriction of Fover Ω Fis its maximal surjective subsystem. In particular, F
+is surjective if and only if Ω F=B
+/CI.
+Moreover, it can be seen from the definition that Ω F= ΩFk: the configurations that
+can be reached arbitrarily late are the same.4 P. GUILLON AND G. RICHARD
+2. Preliminary results
+In this section, we shall recall some classical results that will be needed in the proof.
+The “Firing Squad” is a problem on algorithmics over cellula r automata, introduced in
+1964 by Moore and Myhill in [Moo64]: the goal is to synchroniz e cells of arbitrarily wide
+zones sothat they all take thesamegiven state at thesame tim e. Itled to different solutions
+(see [Maz96]); when dealing with infinitely many cells, we ob tain that it is possible to make
+themget thisstatearbitrarilylateintime, andKari’stheo remwasthefirstextrinsicpurpose
+to this construction; we will reuse it as it was claimed.
+Proposition 2.1 ([Kar94b]) .There exist some cellular automaton Son some alphabet B
+and some states κ,γ∈B, withκspreading, such that:
+(1)For any J∈ /C6, there is some configuration z∈B
+/CIsuch that FJ(z) =∞γ∞and
+∀i∈ /CI,j < J,Fj(z)i/\e}atio\slash=γ;
+(2) ΩS∩[γ]0⊂ {κ,γ}
+/CI.
+To prove the undecidability of some property, we need to redu ce to it some other
+property which is already known to be undecidable. It is clas sical to reduce the nilpotency
+problem, which was proved undecidable in [Kar92]. This proo f reduced some tiling problem
+to the nilpotency, but actually, the CA involved all admitte d some spreading state. Hence
+the following stronger result can be derived directly.
+Theorem 2.2. The problem
+Instance: a cellular automaton Nwith some spreading state θ.
+Question: isNnilpotent?
+is undecidable.
+The restriction to cellular automata with spreading state i s very convenient to allow
+constructions of products of cellular automata, thanks to t he following result (see for in-
+stance [CG07] for a simple proof).
+Proposition 2.3. A cellular automaton Non some alphabet Awith some spreading state
+θ∈Ais nilpotent if and only if ∀x∈A
+/CI,∃i∈ /CI,j∈ /C6,Nj(x)i=θ.
+3. Binary simulation
+The main construction in Kari’s proof is based on a simultane ous simulation of several
+cellular automata thanks to some complex alphabet. In order to keep a fixed alphabet, we
+now need to encode additionnal information into binary confi gurations. This can be done
+thanks to the fact that one of the cellular automata is assume d to be non-surjective. The
+non-reachable portions of configurations can be used for the complex encodings.
+Lemma 3.1. LetCbe an alphabet, Σa subshift on alphabet /BEdistinct from /BE
+/CI. Then we
+can build some strongly freezing language E⊂ /BEk\ Lk(Σ), withk∈ /C6\ {0}, and some
+bijection ξ:Lk(Σ)×C→E.
+Proof.The basic idea is to use the space outside Σ to compress the wor d ofL(Σ) and make
+space for the additional information v∈C. However, to construct it freezing, we shall
+compress only the second half on the word. Let ube a forbidden pattern for Σ. Should we
+extend it as stated before, we can suppose that it is strongly freezing. Should we renameREVISITING THE RICE THEOREM OF CELLULAR AUTOMATA 5
+the letters, we can suppose that C⊂ /BEl, withl∈ /C6\ {0}. As a subset of ( /BE|u|\ {u})m,
+Lm|u|(Σ) has cardinal less than (2|u|−1)m, and admits thus a bijection ˜ξfromLm|u|(Σ)
+onto some subset of /BEnwhenever 2n≥(2|u|−1)m. Let us take m≥2|u|+l
+|u|−log(2|u|−1)and
+n=m|u|−2|u|−l. We now take k= 2m|u|and define ξ: (z,v)/mapsto→uz/llbracket0,m|u|/llbracket˜ξ(z/llbracketm|u|,k/llbracket)vu
+(see Fig. 1) and E=ξ(Lk(Σ)×C).
+z v
+ξ
+u z/llbracket0,m|u|/llbracket ˜ξ(z/llbracketm|u|,k/llbracket)vu
+Figure 1: Encoding into strongly freezing alphabet
+Now let w∈E /BEi∩ /BEiEwith 1≤i < k. Note that w/llbracketi,i+|u|/llbracket=u. But we also have
+u=w/llbracket0,|u|/llbracket=w/llbracketk−|u|,k/llbracketanduis strongly freezing, so |u| ≤i < k−2|u|. Moreover,
+w/llbracket|u|,(m+1)|u|/llbracketis inLm|u|(Σ) and therefore does not contain the pattern u. Hencei > m|u|.
+Similarly, w/llbracketi+|u|,i+(m+1)|u|/llbracketcannot contain the pattern u, sok− |u|/∈/llbracketi+|u|,i+m|u|/llbracket,
+i.e.i > k−2|u|, which gives a contradiction.
+The language Ewill then be used as a particular alphabet, over which we can b uild
+configurations in E
+/CI. This full shift can be more or less seen – up to a short initial shift
+– as the system (∞E∞,σk), but must not be confused with the subshift (∞E∞,σ) over /BE.
+The key point in that construction is that the inclusion of th e information of another shift
+can be done by a constant-space simulation: Σ and C
+/CIare, in an independent way, factors
+of respectively (∞E∞,σ) and of E
+/CI– or, thanks to freezingness, of (∞E∞,σk). This could
+not be done in the absence of any forbidden word u.
+Given somepartial cellular automaton Gon somesubshiftof finitetypeΣ, somecellular
+automata NandSon alphabets A∋θandB∋γ,κ. Considering C=A×B, we
+can build E,k,ξas in Lemma 3.1. As a local rule of radius 1, and with a little ab use
+of notation corresponding to commuting the products, ξcan be extended to injections
+ξ:Lik(Σ)×Ai×Bi→Ei. LetδG,δNandδSbe the local rules of G,NandS, and
+assume, without loss of generality, that SandNhave same radius rSand that Ghas radius
+rG< rSk. Define some cellular automaton ∆ G,N,Sof radius r= (rS+ 1)k−1 and local
+ruleδ: /BE2r+1→ /BEdefined as:6 P. GUILLON AND G. RICHARD
+δ(y) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleδG(y/llbracketr−rG,r+rG/rrbracket) ify∈ L(2r+1)(Σ) (1)
+zi if/braceleftbigg
+y∈ /BEiErSξ(z,v,γ)ErS/BEk−1−i
+0≤i < k,z∈ Lk(Σ),v∈A(2)
+ξ(z,δN(v),δS(w))iif
+
+y∈ /BEiξ(z,v,w) /BEk−1−i
+0≤i < k,z∈ L(2rS+1)k(Σ)
+v∈ArS×A\{θ}×ArS
+w∈BrS×B\{γ,κ}×BrS(3)
+0 otherwise (4)
+This rule is well-defined since the freezingness of Eimposes the unicity of iin the
+cases (2) and (3). Intuitively, the constructed automaton b ehaves as Gon Σ (1) and uses
+the freezing alphabet to simulate both automata NandSwhile keeping “compressed”
+an element of Σ (3). This element is uncompressed when automa tonSreaches state γ
+(2). When Nreaches state θ,Sreaches state κor when the encoding is invalid, the local
+transition goes to 0 (4).
+Throughtheendof thesection, thecellular automaton Swill beaFiringSquadsolution
+as built in Proposition 2.1. This will allow to make any config uration of Σ appear arbitrarily
+late during the evolution, since before the synchronizatio n ofS, the configuration of Σ will
+not be altered.
+We willalso assumethat θis aspreadingstate forthecellular automaton N. Intuitively,
+we wonder if Nis nilpotent, and show that we can get an answer if we assume th at some
+property over ∆ G,N,Sis decidable.
+Finally, we assume that the domain Σ of Gis the subshiftof finite type avoiding a single
+forbidden pattern u(of length less than k) such that u0/\e}atio\slash= 0/\e}atio\slash=u|u|−1. This last property
+allows that 0∗L(Σ)⊂ L(Σ) and L(Σ)0∗⊂ L(Σ).
+The following lemma shows that the non-encoding patterns wi ll give words in Σ after
+one evolution step.
+Lemma 3.2. Letxbe such that ∆G,N,S(x)∈[u]0. Then there is some i∈/rrbracket−k,0/rrbracketsuch that
+x∈[E2rS+1]i−rSk.
+Proof.
+•If case (4) of the rule is applied to xin cell 0, then ∆ G,N,S(x)0= 0/\e}atio\slash=u0, hence
+∆G,N,S(x)/∈[u]0.
+•If case (1) is applied to xin all cells of /llbracket0,k/llbracket, then ∆ G,N,S(x)/llbracket0,k/llbracket∈[Lk(Σ)]0, hence
+∆G,N,S(x)/∈[u]0.
+•Ifxapplies case (1) in cell 0, but there exists some i∈/rrbracket0,k/llbracket(say minimal) which
+applies some other rule. This means that the neighborhood x/llbracketi−1−r,i−1+r/rrbracketis in
+L2r+1(Σ) whilst the neighborhood x/llbracketi−r,i+r/rrbracketis not,i.e.x∈[u]i+r−|u|. As a result,
+all cells of /llbracketi,k/llbracket(and many more) will see a non-homogeneous neighborhood and
+apply (4). x/llbracket0,k/llbracket∈ Li(Σ)0k−i⊂ Lk(Σ), hence ∆ G,N,S(x)/∈[u]0.
+•Ifxapplies either (2) or (3) in cell 0, then we get the result.
+The following lemma completes the previous one: not only can notuappear from
+scratch, but no encoding pattern can appear from a non-encod ing pattern.REVISITING THE RICE THEOREM OF CELLULAR AUTOMATA 7
+Lemma 3.3. Letxbe such that ∆G,N,S(x)∈[E]0. Thenx∈[E2rS+1]−rSk.
+Proof.
+•If case (3) of the rule is applied in some cell j∈/llbracket0,k/llbracket, then ∆ G,N,S(x)∈[E]j−ifor
+somej∈/llbracket0,k/llbracket.Ebeing freezing, we get ∆ G,N,S(x)∈[E]0.
+•Since all words of Econtain some occurrence of u, Lemma 3.2 gives some i∈
+/rrbracket−k,k−|u|/rrbracketsuch that x/llbracketi−rSk,i+(rS+1)k/llbracket=ξ(z,v,w), for some z∈ L(2rS+1)k(Σ),v∈
+A2rS+1,w∈B2rS+1. Assume that i≥0 – the symmetric case is similar. If the
+previous point does not occur, then case (2) is applied to cel ls of/llbracketi,i+k/llbracket, and
+either case (2) or case (3) to cells of /llbracketi−k,i/llbracket. Since 0kLk(Σ)⊂ L2k(Σ), both
+cases imply that ∆ G,N,S(x)/llbracketi−k,i+k/llbracket∈ L2k(Σ). This contradicts the assumption that
+∆G,N,S(x)/llbracket0,k/llbracket∈E.
+To study more in details the limit set of the constructed auto maton, let us first consider
+the set
+Λ =/uniondisplay
+0≤i<k
+−∞≤l<m≤+∞/braceleftBig
+x∈ /BE
+/CI/vextendsingle/vextendsingle/vextendsinglex/llbracketi+lk,i+mk/llbracket∈Em−land∀j /∈/llbracketi+lk,i+mk/llbracket,xj= 0/bracerightBig
+of configurations or pieces of configurations of∞E∞surrounded by 0. Note that this set
+does not depend on G– only on the subshift Σ. These partially encoding configurat ions
+correspond exactly to those of the limit set of our cellular a utomaton which are not in Σ,
+as proved below.
+Lemma 3.4. Letx∈Ω∆G,N,Sandi,j∈ /CIsuch that i < jandx/llbracketi,i+k/llbracket=ξ(zi,(ai,γ)),x/llbracketj,j+k/llbracket=
+ξ(zj,(aj,bj))∈E. Thenbj∈ {γ,κ}.
+Proof.Assume on the contrary that bj/∈ {γ,κ}. Let (xt)t∈ /CIbe a biorbit of x=x0,i.e.
+a bisequence of configurations such that ∀t∈ /CI,∆G,N,S(xt) =xt+1. By an easy recur-
+rence and Lemma 3.3, we can see that for any t∈ /C6,x−t
+/llbracketi−rStk,i+(rSt+1)k/rrbracketcan be written
+ξ(z−t
+i−rSt,(a−t
+i−rSt,b−t
+i−rSt))...ξ(z−t
+i+rSt,(ai+rSt,bi+rSt))∈E(2rSt+1)kandδt
+S(b−t
+i−rSt...b−t
+i+rSt) =
+bi; similarly, x−t
+/llbracketj−rSt,j+rSt/rrbracketcanbewritten ξ(z−t
+j−rSt,(a−t
+j−rSt,b−t
+j−rSt))...ξ(z−t
+j+rSt,(aj+rSt,bj+rSt))∈
+(A×B)(2rSkt+1)k, andδt
+S(b−t
+j−rSt...b−t
+j+rSt) =bj.Ebeing strongly freezing, we can see that
+j−i=klfor some l∈ /C6, and for any t >l−1
+2rS,x−t
+/llbracketi−rSkt,j+rSkt/rrbracketis inEl−1+2rtand the image
+δt
+S(b−t
+i−rSt...b−t
+j+rSt) contains biandbj. In other words, the cylinder [ biBj−i−1bj]iintersects
+any of the St(B
+/CI), and by compactness intersects Ω S, which contradicts Proposition 2.1.
+Lemma 3.5. Ω∆G,N,S⊂Σ∪Λ.
+Proof.From Lemma 3.2, the image of the subshiftwhich avoids all pat terns ofEis included
+inΣ, whichitself isinvariant. Byshift-invariance, it ist hus sufficient toprovethat Ω ∆G,N,S∩
+[E]0⊂Λ. One can remark that the patterns of/braceleftbig
+v∈E /BEk/BE∗/vextendsingle/vextendsinglev/llbracketk,2k/llbracket/∈Eandv/llbracketk,2k/llbracket/\e}atio\slash= 0k/bracerightbig
+areforbiddenintheimage∆ G,N,S( /BE
+/CI). Indeed,ifyouapplycase(3)oftheruleinthecentral
+cell and (1) in another cell, then between these two cells the re will be at least a range of
+kcells seeing a non-homogeneous neighborhood and applying c ase (4). Now if you apply
+case (3) in the central cell and (2) in another cell, this mean s that you had a configuration
+which involved simultaneously a state of A×(B\ {γ,κ}and a state of A× {gamma},
+which contradicts Lemma 3.4. By induction on n≥1, we can prove that the patterns of/braceleftbig
+v∈E /BEnk/BE∗/vextendsingle/vextendsinglev/llbracketk,2k/llbracket/∈Eandv/llbracketk,2k+n−1/llbracket/\e}atio\slash= 0k+n−1/bracerightbig
+are forbidden in ∆n
+G,N,S( /BE
+/CI), since at8 P. GUILLON AND G. RICHARD
+least the rSextremal encoding patterns of Edisappear at each step, whereas the non-zero
+patterns of Σ can spread only by rG< rSkcells every step. In the limit, we obtain that all
+configurations of Ω ∆G,N,Scontaining a pattern of Eare in Λ.
+In the case where Nis nilpotent, we can see that the second part of the limit set i s
+empty, and therefore we obtain the limit set of the original c ellular automaton G.
+Lemma 3.6. IfNis nilpotent, then Ω∆G,N,S= ΩG.
+Proof.From Lemma 3.5 and the fact that (∆ G,N,S)|Σ=G|Σ, it is sufficient to prove the
+emptyness of Ω ∆G,N,S∩Λ. Letx∈Ω∆G,N,S∩[E]0andJ∈ /C6. There exists yJ∈ /BE
+/CIsuch
+that ∆J
+G,N,S(yJ) =x. Applying inductively Lemma 3.3, we obtain that yJ
+/llbracket−JrSk,(JrS+1)k/llbracket∈
+E2JrS+1. By compactness, there is some configuration ysuch that for any i∈ /CIand any
+j∈ /C6,Fj(y)/llbracketik,(i+1)k/llbracket∈E. Clearly, in the successive evolution step from y, case (3) of
+the local rule is always applied, which implies that there is a configuration in A
+/CIin the
+evolution of which θnever appears, hence contradicting the nilpotency of N.
+WhenNis not nilpotent, we can see that there is a way to let the Firin g Squad inject
+any configurations of Σ at any time, hiding the action of G: the limit set does not depend
+onG.
+Lemma 3.7. IfNis not nilpotent, then Σ⊂Ω∆G,N,S.
+Proof.Letx∈Σ. From Proposition 2.3, there is some configuration y∈A
+/CIsuch that for
+anyi∈ /CIand any j∈ /C6,Nj(x)i/\e}atio\slash=θ. LetJ∈ /C6\ {0}. From Proposition 2.1, there
+is some configuration z∈B
+/CIsuch that SJ−1(z) =∞γ∞and for any j < J−1 and any
+i∈ /CI,Sj(z)i/∈ {γ,κ}(sinceκis spreading). Consider now the configuration ˜ xdefined
+by∀i∈ /CI,˜x/llbracketik,(i+1)k/llbracket=ξ(x/llbracketik,(i+1)k/llbracket,yi,zi). By a quick induction on j < J, we can see
+that for any cell i∈ /CI, only case (3) of the local rule is used, and ∆j
+G,N,S(˜x)/llbracketik,(i+1)k/llbracket=
+ξ(x/llbracketik,(i+1)k/llbracket,Nj(y)i,Sj(z)i). At time J, sinceSJ−1(y) =∞γ∞, the second part of the rule
+is applied and ∆J
+G,N,S(˜x)/llbracketik,(i+1)k/llbracket=x/llbracketik,(i+1)k/llbracket. As a result, x∈/intersectiontext
+J∈ /C6\{0}∆J
+G,N,S( /BE
+/CI).
+4. Rice Theorem
+The construction of the previous section allows us to separa te the cases whether Nis
+nilpotent in the same time as we separate properties of the li mit set.
+Lemma 4.1. For any nontrivial property Pover the limit sets of nonsurjective cellular
+automata on /BE, there exist two cellular automata G0,G1on alphabet /BEsharing the same
+quiescent state q∈ /BE, and such that ΩG0∈ P,ΩG1/∈ P.
+Proof.Take any nonsurjective cellular automaton Mon /BEwhich has both 0 and 1 quies-
+cent (such as a minimum cellular automaton). If its limit set satisfiesP, then take some
+nonsurjective cellular automaton Gon /BEwhose limit set does not satisfy P. ThenG2has
+the same limit set and some quiescent state q∈ /BE. If the limit set of Mdoes not satisfy the
+property P, then we can do the same with some nonsurjective cellular aut omatonGwhose
+limit set does satisfy P.REVISITING THE RICE THEOREM OF CELLULAR AUTOMATA 9
+Lemma 4.2. LetG0,G1be two nonsurjective cellular automata on alphabet /BEsharing the
+same quiescent state q∈ /BE, andNa cellular automaton with a spreading state θ. Then
+we can build two cellular automata Fi,i∈ {0,1}such that ΩFi= ΩGi,i∈ {0,1}, ifNis
+nilpotent; ΩF0= ΩF1otherwise.
+Proof.Should we invert 0 and 1 in the construction, we can assume tha tq= 0. Let uibe
+a forbidden pattern of the (non-full) shift Gi( /BE
+/CI), and consider the word u= 1u0u11. The
+restrictions ˜GiofGion the subshift Σ forbidding {u}are partial cellular automata with
+the same limit sets than the respective Gi. Define Fi= ∆˜Gi,N,SwithSbeing as obtained
+in Proposition 2.1. Note that, except in the first case of the l ocale rule, the definitions of
+these two cellular automata are equivalent since they are ba sed on the same subshift. If N
+is not nilpotent, then by Lemmas 3.5 and 3.7, Ω ∆˜Gi,N,S= Σ∪Ω(∆˜Gi,N,S)|Λ. It can be noted
+that the restrictions of ∆ ˜G0,N,Sand ∆ ˜G1,N,Son Λ are equal. Hence Ω ∆˜G0,N,S= Ω∆˜G1,N,S.
+Now ifNis nilpotent, Lemma 3.6 gives the statement.
+Here is now the main result.
+Theorem 4.3. LetPbe a property satisfied by the limit set of at least one nonsurj ective
+cellular automaton on /BE, but not all. Then the problem
+Instance: a cellular automaton Fon /BE.
+Question: ΩF∈ P?
+is undecidable.
+Proof.Assume such a property Pis decidable. Let Gi,i∈ {0,1}be as in Lemma 4.1. Let
+us show a procedure to decide whether a given cellular automa tonNon alphabet Awith
+spreading state θis nilpotent or not, which will contradict Theorem 2.2. We bu ild the two
+cellular automata Fias in Lemma 4.2 and we algorithmically check whether their li mit sets
+satisfy property P. If ΩF0∈ Pand ΩF1/∈ P, thenNis nilpotent (otherwise the two limit
+sets would be equal). Otherwise, we know that one Ω Fiis not equal to Ω Gi, soNis not
+nilpotent.
+Fromthedecidability ofthesurjectivityproblem, establi shedin[AP72], wecanrephrase
+the previous theorem as follows: surjectivity is the only no ntrivial property of the limit sets
+of cellular automata on alphabet /BEto be decidable. Of course, this can be translated to
+any other fixed alphabet (of at least two letters).
+5. Perspectives
+Thisresultisaverycompleteone, sinceitstatesthatnothi ngcanbesaidalgorithmically
+withrespecttohowthelong-timeconfigurationslooklike. I nspiteofthis, concreteexamples
+of propertiesconcernedarenot sonumerous, except nilpote ncyor apparitionof agiven state
+or pattern.
+This is due to the fact that it does not include any dynamical i dea. Various results
+have been obtained in this direction, about some properties of the restriction of the cellular
+automaton to the limit set [dLM09], the properties of the seq uences of states taken by a
+particular cell [CG07], or the regularity of the languages o btained this way [dL06].
+Among the properties that are not known to be concerned by our result, an important
+open problem consists in asking whether stability, which co rresponds to the fact that the
+limit set is reached whithin a finite number of states (and the undecidability of which is not10 P. GUILLON AND G. RICHARD
+very hard to establish anyway), is a property of the limit set s or not. This issue is linked
+to the understanding of the different types of limit sets we can get with cellular automata,
+treated in particular in [Maa95].
+Note that space-time diagrams of cellular automata, which r epresent the superposition
+of successive configurations in its application, are two-di mensional subshifts of finite type,
+i.e.drawings defined by some local constraints. Hence our result directly implies some
+kind of Rice theorem on subshift projections (multidimensi onal subshifts can be defined
+similarly).
+Corollary 5.1. LetPbe a property satisfied by the limit set of at least one non-sur jective
+cellular automaton on /BE, but not all, and π: /BE
+/CI2→ /BE
+/CIdefined by π((xij)i,j∈ /CI) = (x0j)j∈ /CI.
+Then the problem
+Instance: a subshift of finite type Σ⊂ /BE
+/CI2.
+Question: π(Σ)∈ P?
+is undecidable.
+The previous collary can obviously be generalized to any pro jection defined similarly
+from some k-dimensional tiling to some q-dimensional tiling, where 0 < q < k. This does
+not include the property of being the full shift, but the unde cidability of this property was
+already a consequence of the “perpendicular” Rice theorem i n [CG07].
+One may also wonder what happens for higher-dimensionnal ce llular automata. In this
+case, ourconstruction seemstoextendwell. Moreover, surj ectivity isalsoundecidablewhich
+makes any non-trivial property undecidable.
+Moreover, one can ask the same question on another character istic set of cellular au-
+tomata: the ultimate set, containing all the adhering value s of orbits, studied for instance
+in [GR08]. One can notice that, when shifting enough a cellul ar automaton, the limit set
+is unchanged but the ultimate set becomes equal to the limit s et. Hence, any nontrivial
+property of limit sets of cellular automata, except being a f ull shift, is an undecidable prop-
+erty of the ultimate set. We can wonder if it is the case for oth er nontrivial properties of
+ultimate sets.
+More generally, the Firing Squad can be seen as a very powerfu l tool to touch the limit
+set. Our binary simulation can help hide its evolution withi n any alphabet. This could
+allow other complex constructions desolidarizing the simu lation of a cellular automaton
+and the structure of its limit set. For instance, could we bui ld an intrinsically universal
+cellular automaton ( i.e.that can simulate any other cellular automaton) whose limit set is
+any given subshift of finite type?
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diff --git a/1001.0254.txt b/1001.0254.txt
new file mode 100644
index 0000000000000000000000000000000000000000..17238e92c0f0040c7296e037f83a2d3f85e850d0
--- /dev/null
+++ b/1001.0254.txt
@@ -0,0 +1,604 @@
+Spin flexoelectricity in multiferroics  
+Alexander P. Pyatakov1,2*, Anatoly K. Zvezdin1 
+ 
+1) A. M. Prokhorov  General Physics Institute, 38, Vavilova st., Moscow, 119991, Russia 
+2) Physics Department, M.V. Lomonosov MSU, Leninskie gori, Moscow, 119992, Russia 
+ 
+PACS: 75.80.+q, 75.50.Ee  
+* pyatakov@phys.msu.ru 
+ 
+Abstract 
+ Various phenomena related to inhomoge neous magnetoelectric interaction are 
+considered. The interrelation between spatial modulation of order parameter and electric 
+polarization, known as flexoelectric effect in liquid crystals, in the case of magnetic media 
+appears in a form of electric polarization induced by spin m odulation and vice versa. This 
+flexomagnetoelectric interaction is  also related to the effect of ferroelectric domain structure on  
+antiferromagnetic structure, and to  the magnetoelectric properties of micromagnetic structures. 
+The influence of inhomogeneous magnetoel ectric interaction on dynamic properties of 
+multiferroics, particularly magnon spectra is also considered.  
+ 
+1. Introduction 
+ 
+In the last few years there was a remark able progress in physics of multiferroics, 
+particularly in the mechanisms of magnetoelectric coupling [1-11]. In classical review [12] that 
+summarizes the achievement of the initial period  of multiferroic era the emphasis was put on the 
+magnetoelectric interaction biquadratic on the order parameters ∑
+′′ ′ −=
+ssl
+sk
+sji ijkl
+ssMEMMPP F γ21, 
+where P is polarization and Ms is magnetization of sublattices  (s is the number of magnetic 
+sublattice). This term does not require any sp ecial conditions except for existence of magnetic 
+and electric orders. Nowadays ot her types of interactions intr oduced in [12], namely the ones 
+linear in order parameters are brought to the forefr ont. In particular the relation between electric 
+polarization and existence of spatially modul ated structures in the magnetic media was 
+established [13-17], and the eff ects of electric field control of  magnetic structure odd with 
+respect to the electric field we re revealed [18-21]. In all th ese cases the nonvanishing spatial 
+derivative of magnetic order parameter j iM∇  creates necessary prerequisites for existence of 
+inhomogeneous magnetoelectric interaction .  
+ 
+2. Inhomogeneous magnetoelectric (fl exomagnetoelectric) interaction 
+ 
+The contribution to energy of the crystals that is linear in jiη∇ (η is order parameter 
+vector) was introduced in condense matter physics long ago [22]. In cont ext of magnetoelectric 
+effect it appeared as inhomogeneous magnetoel ectric interaction  in theoretical papers [23,24] 
+that was treated as the possible mechanism of l ong range spatially modulated  spin structures. It 
+was also introduced as a mechanism of the elec tric polarization genera ted at magnetic domain 
+boundaries [25].  
+Experimentally the existence of spatially m odulated spin structur e (spin cycloid) has 
+been proved in multiferroic bismuth ferrite BiFeO 3 [26,27]. Later it was shown that the 
+mechanism of the spin modulation in BiFeO 3 is of magnetoelectric nature  [14]. The origin of the 
+spin cycloid is inhomogeneous magnetoelectric in teraction whose contribution to the free energy 
+in crystal with R3c symmetry can be writ ten in the Lifshitz invariant-like form: 
+( )( ) ( ) MEz z zF PL L L Lγ=⋅ ⋅ ⋅ ∇ ⋅ − ⋅ ∇ ,   (1a) where L stands for the antiferromagnetic vector, P z is the spontaneous polarization directed 
+along the c-axis. At temperatures below the temperature of magnetic ordering T<<T N  one can 
+use the unit vector n as order parameter.  
+In the case of isotropic or cubic sy mmetry the inhomogeneous magnetoelectric 
+interaction can be expressed in an elegant way* [15]:  
+() () [ ] ( )n n n nP curl div ×+ ⋅⋅⋅=γFlexoF .  (1b) 
+for the bismuth ferrite γP~ 0.6 erg/cm2 (estimated from the wave vector of spin cycloid ~106cm-1   
+and its energy ~ 6 105 erg/cm3 [28]). 
+It is interesting to note that magnetoelectric interaction in the form (1b) is very similar 
+to the expression used for the flexoelectri c effect in nematic liquid crystal, where n stands for the 
+director [15]. The electric fiel d induces inhomogenous distribution of director in liquid crystals, 
+and vice versa, the modul ation of the director n induces electric polarization. This profound 
+analogy gives grounds to name inhomoge neous magnetoelectric interaction the 
+flexomagnetoelectric  one, so we will use this  terms as synonyms below.  
+The most vivid example that illustrates the relation between electric polarization P, wave 
+vector q of the spin modulated struct ure and spin rotation vector Ω is introduced in [7]: the 
+vectors form the vector triple [ ]Ωq×~P  (fig. 1).  
+It follows from this simple image that in magnetic ferroelectrics the spatially modulated 
+structures is of cycloidal type ( q⊥Ω), not helical one ( q||Ω). It follows also from this rule that 
+only Neel-type domain walls can exhi bit ferroelectric polarization.  
+ 
+ 
+Fig. 1 a) The spin cycloid and the right-hand triple of the orthogonal vectors: the spin rotation axis Ω, the wave 
+number q, and the polar vector P. In the inset: spin cycloid in vertical plane L(x) is associated in bismuth ferrite with 
+the wave of magnetization  M(x) in horizontal plane xy. 
+ 
+ 
+3. Phase transitions in bismuth ferrite BiFeO 3 
+The aforesaid spatial modulated (incommens urate) structure in multiferroic BiFeO 3 is the 
+spin cycloid with a period 62 nm. The incomm ensurate-homogeneous (IC -H) phase transition 
+can be considered as spontaneous nucleation of  domain walls in the H-phase. The domain wall 
+surface energy is: 
+s u DW P AK F πγ− =4 ,       ( 2 )  
+                                                 
+* The expression (1) is given to full derivatives of order parameter. where P s is spontaneous polarization, A and Кu are constants determine the exchange stiffness 
+and uniaxial magnetic anisotropy:  
+() () () ( )2 2 2
+,,2sin ϕθ θ ∇ + ∇ = ∇ =∑
+=A n A F
+zyxii exch     (3) 
+θ2cosu an K F −= ,         ( 4 )  
+θ is polar angle with respect to the z-axis directed along с-axis of crystal (fig .1). The second term 
+in domain wall energy (2) is θπ
+dFflexo∫
+0. Thus the condition of the phase transition F DW=0 takes 
+the form:  
+4
+u
+sAKPγπ=  .      (5) 
+This expression can be derived also from the comparison of the thermodynamical 
+potential for homogeneous and spin modulated phases, in which magnetic order parameter 
+distribution is described by elliptic functions [27,28]. 
+In literature another approach to the phase tr ansition is used that is based on a harmonic 
+approximation [13], that implies the component of n to depend on coordinates in the following 
+way: n i(x)~sin(qx+ φi). This technique allows to make th e estimates with ~10% accuracy [28]. 
+The condition (5) can be also used for estimation of the critical field HС of magnetic 
+field induced phase transition. In this case the uniaxial anisotropy in (5) at external magnetic 
+field H||c should be substituted by effective anisotropy Keff(H)=K u-χ⊥H2/2, where χ⊥ is 
+perpendicular magnetic susceptibilit y. In general case of arbitrar y magnetic field orientation the 
+value  HС depends on the orientation with respect  to the cycloid plane (see Appendix). The 
+effective magnetic anisotropy can  be also modulated by othe r factors: rare earth doping, 
+magnetostriction caused by epitaxial strain in thin films etc. 
+The occurrence of spin cycloid in BiFeO 3 explains the absence of weak ferromagnetism 
+allowed by the symmetry [5]†. The existence of incommensurate spin structure in bismuth ferrite 
+results in oscillating weak ferromagnetic mome nt that is proportional to x-component of 
+antiferromagnetic vector L (fig.1 inset) so its averaged va lue is zero. The weak ferromagnetic 
+moment appears only when the spin cycloid is suppressed that occurs in high magnetic fields 
+[13, 29,30], in compounds doped by rare earth ions [31], and in thin films [32]. The later raise 
+hopes for applications of multife rroics in spintronics [33-39]. 
+ 
+4. Antiferromagnetic ordering in bismuth ferri te films with ferroelec tric stripe domain 
+structure  
+ 
+ It is shown in numerous studies of bismuth ferrite thin films (thi ckness <500nm) that the 
+incommensurate structure is suppr essed [32] (see also reviews [40,41] and reference therein). 
+That does not mean, however, that inhomogeneous ma gnetoelectric interaction is not relevant for 
+the case of thin films. Even in the homogeneou s antiferromagnetic state there is ferroelectric 
+stripe domain structure [42- 44] that induces modulation of antiferromagnetic vector L via the 
+flexomagnetoelectric interaction, as shown in [45]. This interacti on (1) manifests itself in a jump 
+of spatial derivative ∇θ at a boundary between the ferro electric domains with electric 
+polarization oriented upwa rd «+» and downward «-»: 
+()SP P A γ γ θ 2=−+= ∇+
+−    (6) 
+                                                 
+† The origin of weak ferromagnetism (whether it is ferroelectrically induced or Dzyaloshinskii Moria-like) have 
+remained disputable issue fo r years. See C. Fennie PRL 100, 167203, the comment on it in PRL 102, 249701 and 
+the response to the comment in PRL 102, 249702. The graphical image of antiferromagnetic vector modulation θ(x) is shown  in figure 2. 
+Thus, in presence of ferroelectric domain struct ure the ground state of antiferromagnetic system 
+is not uniform anymore. If antiferromagnetic and ferroelectric dom ain structures coexist, then the 
+above-mentioned effect will manifest as a pinn ing of antiferromagnetic domain structure at 
+ferroelectric domain walls. Such a coupling of  ferroelectric and antiferromagnetic domains, 
+particularly in manganites,  was reported in [46]. 
+      
+   
+a )         b )   
+Fig. 2 The modulation of the antiferromagnetic vector orientation induced by ferroelectric structure in multiferroics 
+with flexomagnetoelectric interaction: a) schematic view (solid arrows ex emplifies antiferromagnetic vector, open 
+arrows are the electric polarization vectors) b) the dependence of the polar angle on coordinate θ(x) (dotted line) 
+superimposed on the ferroelectri c structure (solid line) [45]. 
+ 
+ 5. The influence of flexomagnetoelectric inte raction on magnon spectra in multiferroics
+ 
+ 
+The existence of flexomagnetoelectric interact ion in the homogeneous  state, though in a 
+latent form, manifest itself not only in cycloid and the aforementi oned static structures in films 
+but also in dynamical properties of multiferroics, particularly  in magnon (electromagnon [47]) 
+spectra. The influence of the flexomagnetoelect ric interaction on magnon spectra in materials 
+with incommensurate structures was considered  in [48-51]. In thin films spin modulated 
+structure is usually absent and th e magnetoelectric interaction (1) seem s to be irrelevant here (see 
+[52], for example).  However, it was shown in [53; 54] th at the flexomagnetoelectric interaction 
+(1) influences considerably on magnon spectra even in the homoge neous state: th e dispersion 
+curve has minimum at finite wave vectors for wa ves propagating in the direction normal to the 
+(P,L) plane in the case of multiferroic [54] or pe rpendicular to the extern al electric field in the 
+case of centrosymmetric ferromagnetic media [53].  
+Thus in thin films of bism uth ferrite inhomogeneous magneto electric interaction causes 
+the coupling between modes of spin wave propagating along weak ferromagnetic moment k||M0. 
+This coupling appears as a minimu m of dispersion curve w(q) at the wave vector of the spin 
+cycloid q 0 (fig. 3). For BiFeO 3 q0 is ~106 cm-1. Another interesting feat ure of magnons in BiFeO 3 
+consists in nonreciprocity of magnon propagation along and opposite to antiferromagnetic vector 
+that is parallel with toroidal moment [54]. Thes e peculiarities can appear  in Raman and Brillouin 
+light scattering that can be used to obtain th e parameters of magnetoelectric interactions.  
+Of special interest are practical aspects, as  magnons in multiferroics can be excited and 
+controlled by electric field. Th e promising design of ultrasmall logic devices based on 
+electromagnon excitations was proposed [55,56] . The conventional semiconductor technology at 
+the present level of miniaturizat ion encounters the problem of larg e electric fields approaching 
+the threshold one. The alternative approach base d on the spin transfer effects requires large 
+current densities (~106 
+A/сm2 
+or even more). The implementati on of logic devices based on long 
+living magnons in magnetic dielectrics mi ght be a solution of this problem.    qy/q0
+ 
+Fig. 3 The minimum in the dispersion curve for magnon in multiferroic with flexomagnetoelectric interaction in 
+homogeneous magnetic state. q 0 is the wave vector of spin cy cloid in volume crystal. [54] 
+ 
+ 6. Surface flexomagnetoelectric effect 
+ 
+Special conditions realized in th in films of bismuth ferrite le ad to suppression of spatially 
+modulated structures that are obs erved in volume crystals. Howeve r the inverse situation is not 
+only possible but is even more natural:  the exis tence of center of symmetry in the symmetry 
+group of crystal forbids flexomagnetoelectric in teraction while at th e surface the inversion 
+symmetry is broken and the symmet ry restriction is lifted.  
+The possibility of spontaneous spin modulat ed structure occurren ce at the surface of 
+centrosymmetric magnetic crystal and in thin films was predicted in [57], the condition of phase transition between homogeneous and incommensura te phase was formulated (for details see 
+Appendix B) .  
+This effect was discovered in Mn monolay ers [58] and in tw o monolayers of Fe 
+epitaxially grown on (110) tungsten substrate [5 9]. Using polarized electron scanning tunneling 
+microscope they observed magnetic modulation with  a period ~0.5 nm [58] . Measurements with 
+probes of different magnetic moment orientation enabled to determine that the structure in 
+monolayer corresponds to the spin cycloid [58], while in two monolayers of Fe it was the domain 
+wall of Neel type with certain chirality [59].  
+ 
+ 7. Flexomagnetoelectric effect as the origin  of improper ferroelectri city in multiferroics 
+ 
+As was mentioned above, the central symmetry breaking leads to the formation of 
+spatially modulated structures. However the inve rse effect is possible:  spatially modulated 
+structure lowers the symmetry of the crysta l and ferroelectric polarization appears.  
+This mechanism is supposed to cause electric polarization in orthorhombic manganites, 
+RMnO 3 (R=Tb, Dy, Gd) [9,16-19], in vanadate Ni 3V2O8 [60], and in hexaferrite Ba 2Mg 2Fe12O22 
+[17]. In multiferroic MnWO 4 the ferroelectric domains formed  by spin spiral with opposite 
+chiralities was observed [61].  The magnetoelectri c effects such as magnetic control of electric 
+polarization [16,17]  as well as transformation of spatially modul ated structure under influence 
+of electric field [18,19] are al so attributed to the flexomagnetoelectric interaction.  
+Indeed, it follows from the figure 1 that the re versal of electric polar ization should lead to 
+the reorientation of the rotation vector Ω with respect to the wave vector q, i.e. to the reversal of 
+the chirality. This effect was demonstrated in [18, 19].       In the case of ferroelectrics such as BiFeO 3 and BaMnF 4, in which large spontaneous 
+electric polarization is present the additio nal polarization induced by spin modulation‡ can be 
+detected only under special circumstances, namely  at the IC-H phase transitions expressed by 
+equation (4).  
+Flexomagnetoelectric polarization can be found from the contribution to the 
+thermodynamic potential: 
+dxd
+EFPFlexo θγκ=∂∂−=Δ , (6) 
+where κ  is electric susceptibility: P= κE, E is electric field. 
+The polarization averaged over the period of cycloid is: 
+γκλπθθ λπ2)(12
+0= Δ =Δ ∫dddxxP P , (7) 
+where λ is the period of the cycloid.  
+In figure 4 the magnetoelectric anomalies ar e shown that appear as jumps of electric 
+polarization in certain cr itical field. These anomalies can be explained as additional polarization 
+that is related to the spin cycloid.  
+ 
+ 
+ 
+a) b) 
+Fig. 4 The magnetoelectric anomalies at phase transitions: а) incommensurate phase-hom ogenous state in BiFeO 3 
+[13] б) homogenous state – incomm ensurate state in BaMnF 4 [62] (in the inset there is the phase diagram in the 
+coordinates ( Hy, Hz), H is homogenous state, IC is incommensurate phase). 
+ 
+In figure 4 a) the magnetoelectric curve for the BiFeO 3 is shown. The jump of 
+polarization is observed in the field ~200kOe when cycloid is suppressed.  
+Interesting situation occurs  in magnetoelectric BaMnF 4 . The jump of electric polarization 
+was observed in the magnetic field oriented in bc -plane when the angle of magnetic field  with 
+respect to the b-axis is in the vicinity of 45 ˚ (fig.4b). This effect occurs in H C ~ 5kOe and still 
+lacking theoretical interpretation. We believe th is effect is a flexomagnetoelectric one.   
+Indeed the symmetry of BaMnF 4 (class 2, space group A2 1am) allows inhomogeneous 
+magnetoelectric interaction: 
+()xPH
+xPHH Fx x z y Flexo∂∂
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛Φ + −=∂∂+ −=θ γγθγ γ 2sin22
+11
+01 11 01   (8)  
+                                                 
+‡ In general case the calculation of electric polarization in modulated magnetics is a self-consistent problem. 
+However in the case of proper ferroelectric T C >>T N the electric polarization can be considered as a sum of 
+spontaneous polarization P S and perturbation ΔP caused by spin modulation.  where Φ is the angle of the magnetic field with respect to the b-axis of the crystal.  
+It differs from the conventional form (1) in  the way that the magnetoelectric coefficient γ depend 
+on the value and orientation of the magnetic field. One can easily find from (8) that 
+magnetoelectric coefficient is maximum at the angle Φ=45 ˚, so the critical magnetic field at this 
+orientation is minimum and is equal to 4.5kOe (fig. 4b). 
+ 
+8. Thin films of iron garnets and flexomagn etoelectric effect at room temperatures  
+ 
+If one compares the number of papers devoted  to bismuth ferrite with the reports on other 
+room temperature magnetoelectric the striking disproportion of research activity will become 
+evident. For example the potential of iron garnets films seems to be underestimated as               
+the linear magnetoelectric effect measured by electric field indu ced Faraday rotation in this 
+material is an order larger th an in classical magnetoelectric Cr 2O3 [63]. Iron garnet films are 
+probably the most convenient obj ect to study micromagnetism, so the relation between the 
+micromagnetic structure and flexomagnetoelectric in teraction can be illustrated in this material. 
+The influence of electric field on micromagneti c structure was predic ted theoretically in 
+the series of works [7, 15, 25, 64, 65]. In th is context the magnetic domain walls [25,65], 
+magnetic vortices [7] and vertical Bloch line [64]  were considered and it has been shown that 
+various electric charge distribut ions are associated with them . In the recent paper [65] the 
+nucleation of the Neel-type domai n wall in magnetic media at critical electric field has been 
+predicted. This phenomenon, by no means, is inte resting as prototype el ectric write/magnetic 
+read memory.  
+ 
+ 
+a)  
+ 
+b) 
+ 
+c) 
+Fig. 5. Electric field control of the domain wall in epitaxi al iron garnet films: a) th e magnetooptical image of iron 
+garnet film in the transmitted light: dark lines are the magnetic domain boundaries, 1 is the tip electrode, 2 is a 
+domain wall [20].  
+b) The stripe domain head motion induced by electric field [21] c) the time dependence of the  displacement of the 
+domain boundary at different potential at the electrode [21]. 
+ 
+Although the electric field i nduced nucleation of Neel domain wall was not confirmed 
+experimentally so far, the motion of domain walls in magnetic material under influence of 
+electric field has been observed in the 10 μm thickness epitaxial iron ga rnet films, grown on the 
+GdGG substrate [20; 21]. In figure 5 the effect of electric field fr om the tip electrode is shown. 
+The tip electrode potential positive with respect  to the substrate causes the attraction of the 
+domain wall to the tip (fig. 5 a), and the negati ve one results in repulsi on. When the voltage is 
+switched off the domain wall like a stretched stri ng goes to the initial equilibrium state. Not 
+always the transformation of micromagnetic struct ure had a reverse charac ter: if the modified 
+state were more stable than the initial one th e domain walls would remain in the new position. 
+The basic features of the effect are:  
+(i) The change of the sign of the e ffect at electric polarity reversal;  
+(ii) The independence of the effect from th e magnetic polarity of the domain over which 
+the tip was located (T-even effect); 
+(iii) the crucial role of anisotropy of the f ilms (effect was observed in films with (210) 
+and (110) substrate orientations a nd was not observed in (111) films). 
+These properties evidences for its flexomagnetoelectric nature.  
+Indeed, the dependence on the electric polarity and Т-evenness follow directly from the 
+equation (1), while the dependenc e on the substrate orientation is related to the difference 
+between the micromagnetic configuration in hi ghly symmetrical (111) films and ones in the 
+films with low symmetry (110), (210) substrat es. In the (111) films the direction of the 
+anisotropy axis is along the normal to the fi lm, the boundaries between domains are of Bloch 
+type ( 0=Mdiv ; [ ]0= × M Mcurl , provided that | M|=const) and therefore the effect is not 
+observed. At the same time in the (210) and (110)  films the anisotropy ax is is inclined to the 
+normal that causes the Neel component in domain walls so the flexomagnetoelectric interaction 
+is nonvanishing [21]. 
+The possibility of the domain wall motion in elec tric field is also discussed in [65], and it 
+is shown that the domain wall velocity should be proportional to the electr ic field gradient. The 
+experimental studies of domain wall dynamics in pulse electric field (fig. 5 b, c) show that the 
+velocity increases with increasing tip electrode potential. The ultimate displacement of the wall 
+demonstrates similar dependence on the voltage. 
+Comparison of the results of measurements in pulse electric field wi th their analogues in 
+pulse magnetic field gives us the quantitative m easure of the effect: the voltage 500 V (i.e. the 
+electric field 1 MV/cm) creates the effect similar to the one of  50 Oe magnetic field [21]. From 
+this data we find the threshold fi eld of single domain wall nucleation πγKA Et4=  
+~200 MV/cm. This value is too large to  be obtained at normal circumstances. 
+More practicable in this c ontext looks the novel concept of  magnetic storage based on 
+domain wall shift [66]. At the size of tec hnological junction ~100 nm and the domain wall 
+velocity 100 m /s the switching time of the element might be ~1 ns.  
+ 
+There are other interesting manifestation of flexomagnetoelectric interaction in Nuclear 
+Magnetic Resonance [67], Electron Spin Res onance [68] and probably in Second Harmonic 
+Generation [69,70] but they are be yond the scope of this paper.  
+ 
+Authors are grateful to A.M. Kadomtseva, Yu.F. Popov, G.P. Vorob’ev, A.A. Mukhin, Z.V. 
+Gareeva, A.S. Logginov, A.V. Nikolaev, H. Schmid, D.I. Khomskii and M. Bibes for 
+collaboration and valuable discussions. The support of RFBR № 08-02-01068- а and Progetto 
+Lagrange-Fondazione CRT is acknowledged.  
+ 
+ Appendix A: Magnetic field induced phase transition  
+Let us find the dependence of critical magnetic field on the orientation with respect to the 
+cycloid plane in harmonical a pproximation. That means anisotropy does not influence on the 
+shape of the cycloid and the dependence of  polar angle is linear in coordinate: 
+qx=θ       (A.1) 
+The wave vector of the spin cycloid 0q that correspond to the mi nimum of energy can be 
+found from the following condition:  
+( )( )02
+=∂⋅− ∂=∂+ ∂
+qqP Aq
+qF Fz flexo exch γ,   (A.2) 
+therefore:  
+APqz
+20⋅=γ.       (A.3) 
+In external magnetic field H=(0, H y, H z) the contribution to the free energy in cycloidal 
+state will be:  
+2sin2cos2
+22
+2
+02
+0y z
+u z cycloidH HK qP Aq F⊥ ⊥− − − ⋅− =χθχθ γ ,        (A.4)  
+using (A.3) we obtain for the value of energy averaged over the period  λ of the cycloid:  
+2 4 22 2
+2
+0y z u
+cycloidH H KAq F⊥ ⊥− − − −=χ χ
+λ.  (A.5) 
+For homogeneous state at which antiferromagnetic vector is perpendicular to the 
+magnetic field and the c-axis ( θ=90, ϕ=0) the free energy is: 
+2 22 2
+homy z
+y sH HHm F⊥ ⊥− − −=χ χ,   (A.6)  
+where m s the weak ferromagnetic moment.  
+The critical field of  phase transition H C  can be found from the equation (A.5) and (A.6):  
+( )
+ψ χψ χψ ψ
+22
+02 2 2
+cos2 cos sin 2 sin2
+⊥⊥ + + + −=u s s
+CK Aq m mH  (A.7)  
+where ψ is the angle of magnetic field with respect to the c-axis:  HС=(0, H С sinψ, HС  cosψ). 
+For BiFeO 3 А=3·10-7 erg/cm, q 0= 106cm-1 
+(corresponds to λ=62 nm),5107.4−
+⊥ ⋅ =χ , 
+ms~5G  [45] and 35
+cmerg103⋅=uK  (that 
+include magnetic anisotropy 
+35 0
+cmerg106⋅=uK , and the contribution of 
+weak ferromagnetism  
+352
+cmerg1032⋅−≈ −=
+⊥χs
+DMmK  [68]).  The 
+calculated dependence is shown in figure 6. It 
+should be stressed that these results are obtained neglecting the deformation of the 
+cycloid shape in the external field, i.e. the 
+cycloid remained harmonical described by equation A.1 and the wave vector q
+0 (A.3).  
+ 
+ 03 0 6 0 9 0050100150200Hc, kOe
+ψ, deg.
+Fig. 6 The orientational dependence of critical field 
+of phase transition from spin modulated to 
+homogeneous field. The ψ is the angle of magnetic 
+field with respect to the c-axis.   
+Appendix B: Spatially modulated structures in thin films  
+ 
+ In this appendix we analyze the possibility 
+of the formation in thin films the spatially modulated structure induced not by volume 
+interaction but by the presen ce of a Lifshitz type 
+invariant in the surface energy. Let us consider a film medium with the orie ntational order parameter 
+n (|n|=1). In the case of a ferromagnet this is a unit 
+vector directed along a lo cal magnetization, while 
+in the case of antiferromagnetic media it is vector of 
+antiferromagnetism. The geometry of the film 
+deposited onto an isotropic substrate and a Cartesian coordinate system are presented in Fig.7.   
+ 
+Let us present the energy density of the film as  
+() () ( )()( )2
+12 2 2sinx snK KIx AF + − β+ϕ∇θ +θ∇ = ,  (B.1) 
+where the first summand is the exchange energy, the second summand is a Lifshitz-like invariant  
+ynnynnznnznnIx
+yy
+xz
+xx
+z∂∂+∂∂−∂∂−∂∂= , (B.2) 
+and the third summand is the energy density of the anisotropy involving the general case of the 
+bulk energy K1 and the near surface energy Ks(x). In magnetic films Ks is the Neel magnetic 
+surface anisotropy. The coefficients β(x) and KS(x) are nonzero in only narro w intervals near the 
+upper and lower film surface. When integrated over  x they give the “real” surface coefficients βS 
+and KS. The polar θ and azimuthal φ angles of the director are determined in the usual manner in 
+the coordinated system plotted in Fig. 7, the angle θ  being counted from the z-axis and the angle 
+φ  from the x-axis. In angular variable s the second summand assumes the form  
+()⎥⎦⎤
+⎢⎣⎡ϕ∇⋅θ −ϕ∇⋅ϕθ−ϕ θ∇ β= βy z z x Ix2sin sin22sincos )( . (B.3) 
+ We shall treat the anisotropy energy as a disturbance, the energy density F0 and the 
+energy E0  integrated over the volume of the sample as being equal to  
+() () ( )()Ix A F β ϕθ θ + ∇ + ∇ =2 2 2 0sin  (B.4) 
+dxdydzF E∫=0 0.   (B.5) 
+The Euler-Lagrange equation for the functional (B.5) have the form  
+() ( ) () ( )0 cos sin2 sin2cos cos sin2 cos 22 2= ∇⋅ − ∇⋅ −+ ∇ + ∇− ∇ − ϕ θ θ ϕ ϕ θ β ϕθ θ ϕ β θy z z S A   (B.6)  
+() ( ) () 0 cos22sinsin sin sin cos sin sin 22 2 2=⎟
+⎠⎞⎜
+⎝⎛∇⋅ − ∇⋅ − + ∇+ ∇+ ∇ ∇ − ϕ ϕθθ ϕ β θ β ϕ θ θ β ϕθz z y z A . (B.7) 
+The Euler-Lagrange equation (B.6) and (B.7) obviously have trivial solution θ=π/2, ϕ=0 (K>0) 
+and θ=π/2, ϕ=π/2 (K<0). In the case of spatially homogeneous structure, the total film energy 
+with allowance for anis otropy is equal to  
+() DK K ES 1hom+ −=  .    (B.8) 
+We shall now consider the inhom ogeneous states. One can readil y see that equation (B.6) has 
+solution θ=π/2. Substitution of θ=π/2 into (B.7) brings equation (B.7) to the form  
+02= ∇ϕ  .    (B.9) 
+Without loss of generality, the solution of  equation (B.9) can be represented as ϕ=ϕ(y). So, 
+equation (B.6),(B.7) have the following particular solution  
+ 0 
+-D 21 x 
+y 
+z 
+Fig.7 Geometry of the problem: (1) is a film of 
+thickness D, (2) is isotropic substrate.  θ=π/2; ϕ=qy.    (B.10) 
+ 
+This solution describes the spatially modulated (c ycloidal) structure and q is its vector. To find 
+the constant q, we shall substitu te (B.10) into (B.4) to obtain 
+ 
+ ()zys zy
+Dyz LLqLDLAq dydz dxx q LDLAq E β β − = − = ∫∫ ∫
+−20
+2 0 , (B.11) 
+ 
+where ()dxx
+Ds∫
+−=0
+β β  and L yLz is the area over which the integration was taken. 
+Minimization of the energy (B.11) yields  
+ADqs
+2β=     ( B . 1 2 )  
+from which the structure period λ is equal to  
+sAD
+q βπ πλ4 2= =     (B.13) 
+Including the energy of anisotropy by  thermodynamic perturbation theory  
+      anF EE + =0, 
+where ()2
+1 x s an nK K F + = and ... means averaging over the film volume if the angle θ and ϕ are 
+determined from equation in zero approximation (i n this case from equations (B.6), (B.7)), we 
+arrive at the total film energy (per unit area):  
+ ⎟
+⎠⎞⎜
+⎝⎛+ − −= + − = =∫
+2 2 8112
+2 DK K
+AD LLdVF
+qDAqELLs s
+zyan
+s
+yzββ σ . (B.14) 
+Comparing the energy (B.14) of the spatially m odulated (cycloidal) stru cture with the energy 
+(B.8) of the homogenous structur e, one can establish the phase existence regions for this 
+structure. The interfaces are defined by the inequalities  
+() DK KDK K
+ADss s
+112
+2 2 8+ −≤⎟
+⎠⎞⎜
+⎝⎛+ − −β,  (B.15) 
+ 
+It is convenient to represent the inequalities (B.15)  in the form  
+   ⎟
+⎠⎞⎜
+⎝⎛+ ≥12 24 KDKAsλ π     (B.16) 
+Where all the parameters involved in the inequality can be found experimentally.  
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+arXiv:1001.0255v2  [math.QA]  12 Oct 2010POTENTIALS OF HOMOTOPY CYCLIC A∞-ALGEBRAS
+CHEOL-HYUN CHO AND SANGWOOK LEE
+Abstract. For a cyclic A∞-algebra, a potential recording the structure con-
+stants can be defined. We define an analogous potential for a ho motopy cyclic
+A∞-algebra and prove its properties. On the other hand, we find a nother
+different potential for a homotopy cyclic A∞-algebra, which is related to the
+algebraic analogue of generalized holonomy map of Abbaspou r, Tradler and
+Zeinalian.
+1.Introduction
+We first recall the definition of cyclic inner products due to Kontsev ich [Ko],
+which may be understood as constant invariant symplectic structu res in the non-
+commutative geometry.
+Definition 1.1. AnA∞-algebra(A,{m∗}) is said to have a cyclic inner product
+if there exists a skew symmetric non-degenerate, bilinear map
+<,>:A⊗A→k
+such that for all integer k≥1,
+<mk(x1,···,xk),xk+1>= (−1)K(/vector x)<mk(x2,···,xk+1),x1>. (1.1)
+Here, (−1)K(/vector x)denotes the sign given by Koszul sign convention. Namely,
+(−1)K(/vector x)= (−1)|x1|′(|x2|′+···+|xk+1|′), (1.2)
+where|x|′is the shifted degree of x.
+This notion for the A∞-algebras and A∞-categories is crucial in homological
+mirror symmetry, for example, as in the work of Kontsevich-Soibelm an[KS] or
+Costello[Cos]. In particular, Costello has proved in [Cos] that the ca tegory of open
+topologicalconformal field theory is homotopy equivalent to the ca tegoryof Calabi-
+Yau categories, where the Calabi-Yau category is a categorical ge neralization of a
+cyclicA∞-algebra.
+Thefirstapplicationofthisgadgetistodefineapotentialforacyclic A∞-algebra,
+which in physics, is called an action of a string field theory: Let ( A,mA
+∗) be a cyclic
+A∞-algebra. Let eibe generators of Aas a vector space, which is assumed to
+be finite dimensional. Define x=/summationtext
+ieixiwherexiare formal parameters with
+deg(xi) =−deg(ei).
+Definition 1.2. Define
+ΦA(x) =∞/summationdisplay
+k=11
+k+1<mA
+k(x,x,···,x),x> (1.3)
+This work was supported by the Korea Research Foundation Gra nt funded by the Korean
+Government (MOEHRD, Basic Research Promotion Fund) (KRF-2 008-C00031).
+12 CHEOL-HYUN CHO AND SANGWOOK LEE
+This may be considered as a systematic way of gathering structure constants of
+a cyclicA∞-algebra. In the case of toric manifolds, this potential when restr icted
+to the Maurer-Cartan elements, becomes the Landau-Ginzburg s uperpotential of
+the mirror B-model (see [CO],[FOOO1]).
+The notion of cyclicity is not a homotopy invariant notion. For example , anA∞-
+algebra which is homotopy equivalent to a cyclic A∞-algebra may not be cyclic.
+Instead, it has a strong homotopy inner product, which was define d by the first
+author in [C]: for this, a cyclic inner product on Amay be understood as a special
+kind ofA∞-bimodule map A→A∗(Lemma 3.1, [C]). An A∞-bimodule quasi-
+isomorphism A→A∗is called as an infinity inner product by Tradler (see [T],[TZ]
+for example).
+Definition 1.3 ([C], Definition 3.6) .LetAbe anA∞-algebra. We call an A∞-
+bimodule map φ:A→A∗astrong homotopy inner product if there exists a
+cyclicA∞-algebraBwithψ:B→B∗and anA∞-quasi-isomorphism f:A→B
+such that the following diagram of A∞-bimodules over Acommutes
+A
+g=/tildewidef/d47/d47
+φ
+/d15/d15B
+ψcyc
+/d15/d15
+A∗B∗
+g∗/d111/d111(1.4)
+Here byg:A→B, we denote the induced A∞-bimodule map /tildewidef=gwhereBis
+considered as an A∞-bimodule over A.
+In this paper, we give a definition of the potential for strong homot opy inner
+products and prove its properties. It turns out that the definitio n of the potential
+3.1 is very similar to that of (1.2), but the prove that they are indeed related is
+non-trivial and involves quite combinatorial arguments. Beyond th e fact that it is
+quitenaturaltoworkwith homotopynotionswhendealingwith homot opyalgebras,
+sometimes it is necessary to work with homotopy notions directly. Fo r example,
+in the work of Kontsevich and Soibelman [KS], they find a relation betwe en cyclic
+cohomology of an A∞-algebraAand cyclic symmetry. Given a cyclic cohomology
+class, one obtains first a homotopy inner product on A, and then cyclic inner
+product in the minimal model. We refer readers to [CL] for the explicit formulas
+of this correspondence in terms of negative cyclic cohomology HC•
+−(A) and strong
+homotopy inner products.
+Now, let us assume that the A∞-algebra is unital(see definition 4.1), and assume
+that theA∞-bimodule maps are also unital. Then, from the strong homotopy
+inner products{<,>p,q}we can define another potential as follows. (Here, <,>p,q
+is obtained from the ( p,q)-component of the bimodule map φ. See (3.1))
+Definition 1.4. Define
+ΨA(x) =/summationdisplay
+p,q≥01
+p+q+1<x,···,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+p,x,x,···,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+q|I >p,q (1.5)
+We prove that this potential is in fact invariant under the gauge equivalence
+for Maurer-Cartan elements. We also find its relation to the work of Abbaspour,
+Tradler and Zeinalian [ATZ], where this map corresponds to the algebr aic ana-
+logue of the generalized holonomy map from the negative cyclic cohom ology to the
+function ring of Maurer-Cartan elements.POTENTIALS OF HOMOTOPY CYCLIC A∞-ALGEBRAS 3
+(a) (b) (c)xxxx
+xxxxx
+x xI x xx
+xx x x
+x
+Figure 1. (a) Potential Ψ, (b) Cyclic Potential Φ, (c) Homotopy
+cyclic potential Φ
+The following Figure 1 explains the differences of the expressions use d in these
+potentials (without the coefficients). In the figure, the circle repr esent the strong
+homotopy inner product (following that of Tradler [T]) whose horizon tal arrowsare
+for the inputs from modules. The filled circle represent the A∞-operationm.
+This paper may be considered as a continuation of the paper [C] to wh ich we
+refer readers for the notations and further introductions, esp ecially about the signs.
+Throughout the paper we assume that H•(A) is finite dimensional. We thank H.
+Kajiura for sending us the unpublished manuscript (with Y. Terashim a) on the
+decomposition theorem of A∞-algebras.
+2.Strong homotopy inner products
+We begin by proposing a modified definition of strong homotopy inner p roducts,
+and discuss their equivalences and pull-backs.
+We first make an observationthat there exists certain subtlety in t he direction of
+arrowsin the diagram 1.4 in the definition of strong homotopy inner pr oducts. One
+could try to make the definition with the arrow A←−Binstead ofA−→B, but
+the resultingdiagramwouldbecomeweakerastheremayexists eleme ntsofAwhich
+is not covered by the image of the map A←−Bin general. The subtlety actually
+disappears if we have non-degeneracy in the chain level. The correc t definition
+(which corresponds to exactly the non-commutative invariant sym plectic two form)
+is rather in between these two definitions: to make the correct defi nition, we first
+recall the following characterization theorem of strong homotopy inner products
+from [C].
+Theorem 2.1 ([C], Theorem 5.1) .AnA∞-algebraAhas a strong homotopy inner
+product if and only if there exists an A∞-bimodule map φ:A→A∗, satisfying the
+following three conditions.
+(1)(Skew symmetry) For any ai,v,bj,w∈A,
+φk,l(/vector a,v,/vectorb)(w) =−(−1)Kφl,k(/vectorb,w,/vector a)(v),
+with|K|= (/summationtextk
+i=1|ai|′+|v|′)(/summationtextl
+j=1|bj|′+|w|′)
+(2)(Closedness) For any choice of a family (a1,···,al+1)and any choice of
+indices1≤i<j <k≤l+1, we have
+(−1)Kiφ(..,ai,..)(aj)+(−1)Kjφ(..,aj,..)(ak)+(−1)Kkφ(..,ak,..)(ai) = 0,4 CHEOL-HYUN CHO AND SANGWOOK LEE
+where the arguments inside φare uniquely given by the cyclic order of the
+family(a1,···,al+1), and the signs K∗are given by the Koszul convention:
+K∗= (|a1|′+···+|a∗|′)(|a∗+1|′+···+|ak|′). (2.1)
+(3)(Homological non-degeneracy) For any non-zero [a]∈H•(A)witha∈A,
+there exists a [b]∈H•(A)withb∈A, such that φ0,0(a)(b)/ne}ationslash= 0.
+For non-degeneracy on the chain level, φitself gives the strong homotopy inner
+product, otherwise the inner product obtained φ′:A→A∗is only equivalent to φ.
+The second condition is called closed condition since it is equivalent to the closed
+conditionoftherelatednon-commutativesymplectic2-form,andt hisplaysacrucial
+role in proving the properties of the potential defined in this paper.
+We also remark that in the proof of the Theorem 2.1, φsatisfying the three
+conditions, does not always become exactly a strong homotopy inne r product (in
+the sense of definition 1.3), but only equivalent to a strong homotop y inner product
+(the equivalence is defined below).
+Hence, we propose to define the strong homotopy inner products by the The-
+orem 2.1 because such a definition is equivalent to that of the non-co mmutative
+symplectic form as explained in [C].
+Definition 2.1. LetAbe anA∞-algebra. We call an A∞-bimodule map φ:
+A→A∗astrong homotopy inner product if it is skew-symmetric, closed and
+homologically non-degenerate as in Theorem 2.1.
+AndAis called homotopy cyclic A∞-algebra , if there exists a strong homo-
+topy inner product of A.
+Then, the main result of [C] can be phrased as the following theorem.
+Theorem 2.2. Letφ:A→A∗be anA∞-bimodule map.
+(1)Ifφis a strong homotopy inner product in the sense of definition 2 .1, then
+there exists an A∞-algebraBwith a cyclic inner product ψ:B→B∗and
+anA∞-quasi-isomorphism ι:B→Asatisfying the following commutative
+diagram of A∞-bimodule homomorphisms
+A
+φ
+/d15/d15B˜ι/d111/d111
+cycψ
+/d15/d15
+A∗˜ι∗/d47/d47B∗(2.2)
+(2)If there exists a cyclic A∞-algebraBwithψ:B→B∗and anA∞-quasi-
+isomorphism f:A→Bsuch that the following diagram of A∞-bimodules
+overAcommutes
+A
+g=/tildewidef/d47/d47
+φ
+/d15/d15B
+ψcyc
+/d15/d15
+A∗B∗
+g∗/d111/d111(2.3)
+then,φis a strong homotopy inner product in the sense of definition 2 .1.
+Ifφ0,0is non-degenerate in the chain level, the old and the new defin itions of a
+strong homotopy inner product are equivalent.POTENTIALS OF HOMOTOPY CYCLIC A∞-ALGEBRAS 5
+Remark 2.2. Hence the new definition of the strong homotopy inner product is
+a little stronger than the diagram using A←−B, a little weaker than the diagram
+usingA−→Band equivalent to the non-commutative symplectic two form.
+Proof.In the non-degenerate case, from the proof of Theorem 2.1, one can findB
+with anA∞-isomorphism f:A→Bwith the commuting diagram (1.4). Hence
+one can find exact inverse of fto make the commuting diagram (2.2).
+Also, the statement (2) can be checked without much difficulty from the com-
+muting diagram, so we only consider the statement (1). We explain th at the proof
+ofthe theorem 2.1given in [C] is enoughto provethe existence ofthe diagram(2.2):
+We recall from [C] that the first step of the construction of cyclic A∞-algebraB
+whenAis only homologically non-degenerate was considering the minimal mode l
+ι:H•(A)→Aand consider the pull back ι∗φ.
+A
+φ
+/d15/d15H•(A)/tildewideι /d111/d111
+ι∗φ
+/d15/d15/tildewidef/d47/d47H•(A)
+cyc
+/d15/d15
+A∗/tildewideι∗/d47/d47(H•(A))∗(H•(A))∗/tildewidef∗/d111/d111(2.4)
+Thenι∗φis non-degenerate and skew symmetric and closed, and one proves the
+theorem for ι∗φto findf:H•(A)→H•(A) with the above commutative diagram.
+As the quasi-isomorphism fonH•(A) is in fact an isomorphism, hence there exists
+explicit inverse f−1and we obtain the diagram (2.2). /square
+We can also prove the following corollary.
+Corollary 2.3. Letφ:A→A∗be a strong homotopy inner product. Suppose we
+have anA∞-quasimorphism f:A→H•(A)with the commuting diagram
+A
+g=/tildewidef/d47/d47
+φ
+/d15/d15H•(A)
+ψcyc
+/d15/d15
+A∗ H•(A)∗
+/tildewideg∗/d111/d111(2.5)
+then, there exists an A∞-quasimorphism h:H∗(A)→Awith the commuting dia-
+gram (with the same ψas the above)
+A
+φ
+/d15/d15H•(A)˜h/d111/d111
+ψcyc
+/d15/d15
+A∗˜h∗/d47/d47(H•(A))∗(2.6)
+Proof.Bythedecompositiontheoremof A∞-algebras,the map fhasarightinverse
+A∞-quasi-homomorphism, say h:H∗(A)→Asuch thatf◦h=id. To see this,
+consider an A∞-isomorphism η
+η:A→Adc:=AH⊕Alc
+to the direct sum of the minimal A∞-algebraAHand the linear contractible Alc.
+Letπ:Adc→AHbe the projection and i:AH→Adcbe the inclusion where
+the both are A∞-quasimorphisms with π◦i=id. Asfis anA∞-quasimorphism,
+f◦η−1◦i:AH→H•(A) is anA∞-isomorphism, hence has an A∞-inverse say ξ.6 CHEOL-HYUN CHO AND SANGWOOK LEE
+Then, we define the right A∞inverseh=η−1◦i◦ξ. The property f◦h=idcan be
+checked immediately. The second diagram then follows from the first commuting
+diagram. /square
+Now, we define equivalences between strong homotopy inner produ cts.
+Definition 2.3. Letφ:A→A∗andψ:B→B∗be strong homotopy inner
+products. They arecalled equivalent ifthere existsa cyclicsymmetric A∞-algebra
+Hwith a commutative diagram:
+A
+φ
+/d15/d15H
+cyc
+/d15/d15qis/d111/d111qis/d47/d47B
+ψ
+/d15/d15
+A∗ /d47/d47H∗B∗ /d111/d111
+One can actually choose Hto be a minimal(or canonical) model.
+Givenastronghomotopyinnerproduct φ:B→B∗, andanA∞-quasi-isomorphism
+f:A→B, we may define a pullback f∗φ:A→A∗as a composition : f∗φ=
+/tildewidef∗◦/hatwideφ◦/hatwide/tildewidef
+A
+f∗φ
+/d15/d15/tildewidef/d47/d47B
+φ
+/d15/d15
+A∗B∗
+/tildewidef∗/d111/d111
+Proposition 2.4. f∗φdefines a strong homotopy inner product on Awhich is
+equivalent to φ.
+Proof.Sinceφ:B→B∗is skew-symmetric and closed, so is f∗φby lemma 5.6
+of [C]. It is not hard to check that f∗φis also homologically non-degenerate as f
+is a quasi-isomorphism. Hence, f∗φ, by the proposition 2.2 (1), is a strong homo-
+topy inner product. Hence there exist an A∞-algebraCwhich is cyclic symmetric
+(ψ:C→C∗), andA∞-quasi-homomorphism h:C→Awith the following com-
+mutative diagrams.
+C/tildewideh/d47/d47
+ψ
+/d15/d15A
+f∗φ
+/d15/d15/tildewidef/d47/d47B
+φ
+/d15/d15
+C∗A∗
+/tildewideh∗/d111/d111 (B)∗
+/tildewidef∗/d111/d111. (2.7)
+From the diagram, it is easy to see that φandf∗φis equivalent in the sense of
+definition 2.3. /square
+3.Potentials
+In this section we define a potential of a homotopy cyclic A∞-algebra and prove
+its properties. Let ( A,mA
+∗) be given a strong homotopy inner product φ:A→A∗.
+Recall that an A∞-bimodule map φis given by a family of maps
+φp,q:A⊗p⊗A⊗A⊗q→A∗,POTENTIALS OF HOMOTOPY CYCLIC A∞-ALGEBRAS 7
+where the underlined Ais to emphasize that it is an A-bimodule for reader’s con-
+venience. We denote by
+<x1,···,xp,v,y1,···,yq|w>p,q:=φ(x1,···,xp,v,y1,···,yq)(w).(3.1)
+As in the cyclic case, let eibe generators of Aas a vector space, which is assumed
+to be finite dimensional. (One may use pull-back defined in the previous section
+using the inclusion ι:H∗(A)→Ain the case that H∗(A) is finite dimensional).
+Definex=/summationtext
+ieixiwherexiare formal parameters with deg(xi) =−deg(ei).
+Now we give a definition a potential for the strong homotopy inner pr oducts.
+Definition 3.1. Thepotential ofanA∞-algebra(A,mA
+∗) with a strong homotopy
+inner product φ:A→A∗is defined as
+ΦA(x) =∞/summationdisplay
+N=1ΦA
+N(x)
+:=∞/summationdisplay
+N=1∞/summationdisplay
+p+q+k=N1
+N+1<x,x,···,x,mA
+k(x,x,···,x),x,···,x|x>p,q
+(3.2)
+The definition itself is somewhat similar to that of cyclic case(1.2). But in (1.2),
+the fraction 1 /kwas to cancel out repetitive contribution to the potential due to
+cyclic symmetry (1.1), whereas in the strong homotopy case, such cyclic symmetry
+of the rotation of arguments do not exist. Namely, in general
+<e1,···,mi(ej,···,ej+i−1),···,ek|ek+1>/ne}ationslash=<e2,···,mi(ej+1,···,ej+i),···,ek+1|e1>.
+We later show that the combination of A∞-bimodule equation, skew-symmetry and
+closed condition will compensate the absence of the strict cyclic sym metry.
+We explain how the potential behaves under pull-backs, and this will s how the
+relation between the potentials of equivalent strong homotopy inne r products. For
+A∞-quasi-isomorphism h:B→Athe pull-back of a potential is defined as follows:
+We assume Bis finite dimensional as a vector space, and denote by {f∗}its basis,
+and introduce corresponding formal variables y∗as before. Suppose
+hk(fj1,···,fjk) =hi
+j1,···,jkei, hi
+j1,···,jk∈k.
+Then, we set
+xi/mapsto→hi
+j11yj11+hi
+j21,j22yj21yj22+···+hi
+jl1,···,jlkyjl1···yjlk+···.(3.3)
+Then, one define the pull-back h∗ΦAby using the above change of coordinate
+formula.
+Theorem 3.1. Letφ:A→A∗be a strong homotopy inner products. Let Bbe a
+cyclicA∞-algebra with a quasi-isomorphism h:B→Aproviding the commutative
+diagram (2.2). Then, we have
+ΦB=h∗ΦA
+Proof.The overall scheme of the proof, which is first to differentiate and t hen to
+compare, follows that of [C] (idea due to Kajiura [Kaj] in the unfiltere d case). The
+main difficulty, and the essential part of the proof is the first step w here we take
+(formal) partial derivatives on each side. The following lemma shows t hat after
+partial differentiation, the fraction on each summand disappears.8 CHEOL-HYUN CHO AND SANGWOOK LEE
+Lemma 3.2.
+∂
+∂xiΦA
+N(x) =∂
+∂xi∞/summationdisplay
+p+q+k=N1
+N+1<x,x,···,x,mA
+k(x,x,···,x),x,···,x|x>p,q
+=∞/summationdisplay
+k=1<x,x,···,x,mA
+k(x,x,···,x),x,···,x|ei>p,q.
+We assume the lemma for a moment and show the proofofthe theore m using the
+lemma. Let{fi}be basis ofH∗(A), and let{yi}be corresponding formal variables
+for{fi}, namely y:=/summationtext
+iyifi.
+We leth(y) :=/summationtext
+k≥1hk(y⊗k). Then
+∂
+∂yiΦH∗(A)=/summationdisplay
+k≥1<mH∗(A)
+k(y,···,y),fi>
+by cyclic symmetry, and
+∂
+∂yih∗ΦA=∂
+∂yi/summationdisplay
+k≥1
+p+q+1=N1
+N+1<h(y)⊗p,mA
+k(h(y),···,h(y)),h(y)⊗q|h(y)>
+=/summationdisplay
+N≥1
+p+q+1=N<h(y)⊗p,mA
+k(h(y),···,h(y)),h(y)⊗q|∂
+∂yih(y)>
+by above lemma. From the diagram 2.2, we have ψ=/tildewideh∗◦/hatwideφ◦/hatwide/tildewideh, where all maps
+areH∗(A)-bimodule homomorphisms, consider following:
+/summationdisplay
+p,q≥0
+k≥1ψ(y⊗p,mH∗(A)
+k(− →y),y⊗q)(fi) (3.4)
+=/summationdisplay
+p,q≥0
+k≥1(/tildewideh∗◦/hatwideφ◦/tildewideh)(y⊗p,mH∗(A)
+k(− →y),y⊗q)(fi)
+=/summationdisplay
+p,q≥0
+k≥1/summationdisplay
+p1+p2+p3=p
+q1+q2+q3=q/tildewideh∗(y⊗p3,φ(/hatwideh(y⊗p2),h(y⊗p1,mH∗(A)
+k(− →y),y⊗q1),/hatwideh(y⊗q2)),y⊗q3)(fi)
+=/summationdisplay
+p,q≥0
+k≥1/summationdisplay
+p1+p2+p3=p
+q1+q2+q3=qφ(/hatwideh(y⊗p2),h(y⊗p1,mH∗(A)
+k(− →y),y⊗q1),/hatwideh(y⊗q2))(h(y⊗q3,fi,y⊗p3))
+=/summationdisplay
+p,q≥0
+k≥1/summationdisplay
+p1+p2+p3=p
+q1+q2+q3=q</hatwideh(y⊗p2),h(y⊗p1,mH∗(A)
+k(− →y),y⊗q1),/hatwideh(y⊗q2)|h(y⊗q3,fi,y⊗p3)>
+=/summationdisplay
+N≥1
+p+q+1=N<h(y)⊗p,mA
+k(h(y),···,h(y)),h(y)⊗q|∂
+∂yih(y)>
+=∂
+∂yih∗ΦA
+Here, we denote by mk(− →y) the expression mk(y,···,y) for simplicity. The last
+identity holds because the sum is over all p1+p2+p3=pandq1+q2+q3=qPOTENTIALS OF HOMOTOPY CYCLIC A∞-ALGEBRAS 9
+wherepandqrun over all nonnegative integers, and there is A∞-bimodule relation
+/hatwidestmA◦/hatwideh=/hatwideh◦/hatwidermH∗(A).
+The summands of (3.4) are all zero except for ( p,q) = (0,0) becauseψis a cyclic
+symmetric inner product. Hence,
+∂
+∂yih∗ΦA=/summationdisplay
+k≥1ψ(mH∗(A)
+k(− →y))(fi) =/summationdisplay
+k≥1<mH∗(A)
+k(y,···,y),fi>=∂
+∂yiΦH∗(A).
+This proves the theorem. /square
+Proof.We prove the lemma 3.2. Before we proceed, we give some remarks on the
+signs. The sign convention used in this paper and in [C] is the Koszul co nvention
+after the degree one shift. For simplicity, we omit the Koszul sign fa ctor and the
+expressions will appear with + if it agrees with the Koszul sign rule, −if it is the
+negative of the Koszul sign. We illustrate this for two examples, fro m which the
+general convention can be easily understood. The first example is t heA∞-equation
+with two inputs. We write
+m1m2(x1,x2)+m2(m1(x1),x2)+m2(x1,m1(x2)) = 0 (3.5)
+whereas the actual equation is
+m1m2(x1,x2)+m2(m1(x1),x2)+(−1)|x1|′m2(x1,m1(x2)) = 0.
+The equation (3.5) will be also written as
+m1m2(x1,x2) =−m2(m1(x1),x2)−m2(x1,m1(x2)).
+The second example is the equation for < m2(x1,x2)|x3>. Note that φbeing
+A∞-bimodule map φ:A→A∗with the induced A∞-bimodule structure on A∗(see
+expression (3.3) [C] for the precise definition) implies the following act ual equation.
+<m2(x1,x2)|x3>+<m1(x1),x2|x3>+(−1)|x1|′<x1,m1(x2)|x3>
++(−1)|x1|′+|x2|′<x1,x2|m1(x3)>+(−1)|x1|′<x1|m2(x2,x3)>= 0.
+In this paper, the above equation will be written simply as
+<m2(x1,x2)|x3>+<m1(x1),x2|x3>+<x1,m1(x2)|x3>
++<x1,x2|m1(x3)>+<x1|m2(x2,x3)>= 0.10 CHEOL-HYUN CHO AND SANGWOOK LEE
+Now, we begin the proof of the lemma. From now on, we replace mA
+kbymkif
+there is no ambiguity. By taking a derivative, the expression become s:
+∂
+∂xi∞/summationdisplay
+p+q+k=N<x,x,···,x,mA
+k(x,x,···,x),x,···,x|x>p,q(3.6)
+=/summationdisplay
+p+q+k=N
+r+s=k−1<x,···,x,mk(r/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownrightx,···,x,ei,s/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownrightx,···,x),x,···,x|x>p,q(3.7)
++/summationdisplay
+p+q+k=N
+r+s=p−1<x,···,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+r,ei,x,···,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+s,mk(x,···,x),x,···,x|x>p,q(3.8)
++/summationdisplay
+p+q+k=N
+r+s=q−1<x,···,x,mk(x,···,x),x,···,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+r,ei,x,···,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+s|x>p,q(3.9)
++/summationdisplay
+p+q+k=N<x,···,x,mk(x,···,x),x,···,x|ei>p,q. (3.10)
+Now, the lemma can be proved by the following lemma. /square
+Lemma 3.3. The sum of the terms in (3.7),(3.8) and (3.9) equals to Ntimes of
+the expression (3.10)
+Proof.To prove the lemma, we recall the notion of A∞-bimodule equation. The
+equation for A∞-bimodule homomorphism A→A∗is
+φ◦/hatwiderbA=bA∗◦/hatwideφ (3.11)
+withbA=mAwhenAis considered to be an A∞-bimodule, and bA∗is defined by
+canonical construction of the dual of the A∞-bimoduleA. Here/hatwideφis the coalgebra
+homomorphism induced from φ(We refer readers to [C],[T] or [GJ] for details). Let
+us restrict the equation (3.11) to the case ( x,···,x,ei,x,···,x)∈A⊗n⊗A⊗A⊗m
+wheren+m+1 =N. Then it becomes
+/summationdisplay
+p+j1=n
+j2+q=m<x,···,x,mj1+j2+1(j1/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownrightx,···,x,ei,j2/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownrightx,···,x),x,···,x|x>p,q(3.12)
++/summationdisplay
+k1+k2+j=n
+p=k1+k2+1<x,···,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+k1,mj(x,···,x),x,···,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+k2,ei,x,···,x|x>dum
+p,m(3.13)
++/summationdisplay
+l1+l2+h=m
+q=l1+l2+1<x,···,x,ei,x,···,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+l1,mh(x,···,x),x,···,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+l2|x>dum
+n,q(3.14)
+=/summationdisplay
+p+k1=m
+k2+q=n<x,···,x,mk1+k2+1(k1/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownrightx,···,x,x,k2/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownrightx···x),x,···,x|ei>p,q.(3.15)
+Remark 3.2. It is important to note that the expression in the summand (3.15)
+is obtained in k:=k1+k2+ 1 different ways according to the position of the
+(underlined) bimodule element x. Namely, different choices of a bimodule element
+still give rise to equivalent expressions.POTENTIALS OF HOMOTOPY CYCLIC A∞-ALGEBRAS 11
+Remark 3.3. Herethe terms (3.12)and (3.15) in the above A∞-bimodule equation
+do appear in the process of derivation (3.6) but the terms (3.13) an d (3.14) do not
+appear in (3.6). Hence we marked them as <,>dumfor reader’s convenience to
+indicate that they are dummy parts. We will show how all the dummy pa rts are
+canceled out or used in the subsequent process.
+We say an expression such as in (3.12), ···, (3.15) to be of ( n,m)-type as it is
+obtained from the input A⊗n⊗A⊗A⊗m. And for convenience, we will denote
+the summands as in (3.12), ···, (3.15) to be/summationtext
+(n,m)−typeinstead of writing down
+specific conditions.
+Note that the expression (3.12) equals (3.7) and (3.15) provides ktimes the
+expression (3.10) from the remark 3.3. Hence, we may use the abov eA∞-bimodule
+equation to turn (3.7) into k-times (3.10) together with dummy terms. Hence, to
+prove the Lemma 3.3, we need to find N−ktimes the expression (3.10) from what
+are left out in (3.6) together with the new dummy terms.
+Now, we explain the dummy expressions we add to the equation. We se t
+mj(/vectorx) :=mj(x,···,x) just for simplicity. The following are the dummy expres-
+sions to be added to the expression (3.6).
+(i)/summationtext
+p+j+k1+k2+1=N<x⊗p,mj(/vectorx),x⊗k1,ei,x⊗k2|x>dum,
+(ii)/summationtext
+p+j+k1+k2+k3+2=N<x⊗p,ei,x⊗k1,mj(/vectorx),x⊗k2,x,x⊗k3|x>dum,
+(iii)/summationtext
+p+j+k1+k2+k3+2=N<x⊗p,mj(/vectorx),x⊗k1,ei,x⊗k2,x,x⊗k3|x>dum,
+(iv)/summationtext
+p+j+k1+k2+k3+2=N<x⊗p,ei,x⊗k1,x,x⊗k2,mj(/vectorx),x⊗k3|x>dum,
+(v)/summationtext
+p+k1+k2+l1+l2+2=N<x⊗p,mj(x⊗l1,ei,x⊗l2),x⊗k1,x,x⊗k2|x>dum,
+(vi)/summationtext
+p+j+k1+k2+1=N<x⊗p,mj(/vectorx),x⊗k1,x,x⊗k2|ei>dum,
+(vii)/summationtext
+p+j+k1+k2+1=N<x⊗p,ei,x⊗k1,mj(/vectorx),x⊗k2|x>dum,
+(viii)/summationtext
+p+j+k1+k2+k3+2=N<x⊗p,x,x⊗k1,ei,x⊗k2,mj(/vectorx),x⊗k3|x>dum,
+(ix)/summationtext
+p+j+k1+k2+k3+2=N<x⊗p,x,x⊗k1,mj(/vectorx),x⊗k2,ei,x⊗k3|x>dum,
+(x)/summationtext
+p+j+k1+k2+k3+2=N<x⊗p,mj(/vectorx),x⊗k1,x,x⊗k2,ei,x⊗k3|x>dum,
+(xi)/summationtext
+p+k1+k2+l1+l2+2=N<x⊗p,x,x⊗k1,mj(x⊗l1,ei,x⊗l2),x⊗k2|x>dum,
+(xii)/summationtext
+p+j+k1+k2+1=N<x⊗p,x,x⊗k1,mj(/vectorx),x⊗k2|ei>dum,
+The following is easy to check.
+Lemma 3.4. By applying skew symmetry condition, we have
+(i)+(xii) = 0,(ii)+(ix) = 0,(iii)+(viii) = 0,
+(iv)+(x) = 0,(v)+(xi) = 0,(vi)+(vii) = 0
+Hence, the overall sum also vanishes:
+(i)+(ii)+···+(xi)+(xii) = 0.
+Hence we add all the dummy terms above to the expression (3.6) with out chang-
+ing the value. Notice that the expression ( i) and (vii) already appeared in (3.13)
+and (3.14) and is used to turn (3.7) into ktimes (3.10).12 CHEOL-HYUN CHO AND SANGWOOK LEE
+Inadditionweneedthefollowing A∞-bimoduleequation(3.11)whichisobtained
+consideringthe case that the input eiisnotthe bimodule element of the expression:
+Givenm,n∈Nwithn+m=N−1, we have
+/summationdisplay
+(n,m)−type/parenleftbig
+<x,···,x,ei,x,···,x,mj(/vectorx),x,···,x,x,x,···,x|x>dum(3.16)
++<x,···,x,mj(/vectorx),x,···,x,ei,x,···,x,x,x,···,x|x>dum(3.17)
++<x,···,x,ei,x,···,x,x,x,···,x,mj(/vectorx),x,···,x|x>dum(3.18)
++<x,···,x,mj(x,···,x,ei,x,···,x),x,···,x,x,x,···,x|x>dum(3.19)
++<x,···,x,x,x,···,x,ei,x,···,x,mj(/vectorx),x,···,x|x>dum(3.20)
++<x,···,x,x,x,···,x,mj(/vectorx),x,···,x,ei,x,···,x|x>dum(3.21)
++<x,···,x,mj(/vectorx),x,···,x,x,x,···,x,ei,x,···,x|x>dum(3.22)
++<x,···,x,x,x,···,x,mj(x,···,x,ei,x,···,x),x,···,x|x>dum/parenrightbig
+(3.23)
+=−/summationdisplay
+(n,m)−type/parenleftbig
+<x,···,x,ei,x,···,x,···,x|mj(x,···,x,···,x)>(3.24)
++<x,···,x,ei,x,···,x,mj(x,···,x,x,x,···,x),x,···,x|x>(3.25)
++<x,···,x,mj(x,···,x,x,x,···,x),x,···,x,ei,x,···,x|x>/parenrightbig
+(3.26)
++<x,···,x,x,···,x,ei,x,···,x|mj(x,···,x,x,x,···,x)>(3.27)
++<x,···,x,x,x,···,x|mj(x,···,x,x,x,···,x,ei,x,···,x)>(3.28)
++<x,···,x,x,x,···,x|mj(x,···,x,ei,x,···,x,x,x,···,x)>(3.29)
++<x,···,x,mj(x,···,x,ei,x,···,x,x,x,···,x),x,···,x|x>(3.30)
++<x,···,x,mj(x,···,x,x,x,···,x,ei,x,···,x),x,···,x|x>/parenrightbig
+(3.31)
+In the case that n=m, we consider only the above equation, but if n/ne}ationslash=m, then
+we also consider the similar equation for ( m,n)-type also. Observe the following
+facts.
+•For later purposes, we have separated certain dummy expression s in the
+LHS.
+•Some of dummy expressions already appeared before.
+(3.16)∼(ii),(3.17)∼(iii),(3.18)∼(iv),(3.19)∼(v),
+(3.20)∼(viii),(3.21)∼(ix),(3.22)∼(x),(3.23)∼(xi).
+•As (i),(vii) has been used in (3.13),(3.14), the dummy expressions (among
+the list above the Lemma 3.4 which has not used elsewhere (yet) are ( vi)
+and (xii).
+•Note the equivalence in (3.30) and (3.31) if the number of xahead ofeiin
+the following expression are the same:
+mj(x,···,x,ei,x,···,x,x,x,···,x)=mj(x,···,x,x,x,···,x,ei,x,···,x)
+Now, we show that in the right hand side of the equation, all the term s cancel
+out from skew symmetry. In the case that n=m, we have the cancellations (from
+skew symmetry)
+(3.24)+(3.26) = 0 ,(3.27)+(3.25) = 0 ,(3.28)+(3.30) = 0 ,(3.29)+(3.31) = 0 .POTENTIALS OF HOMOTOPY CYCLIC A∞-ALGEBRAS 13
+Hence we may consider the case that n/ne}ationslash=m. In this case, there are similar can-
+cellations between ( n,m)-type and ( m,n)-type terms. Namely, if we use subscript
+(n,m),(m,n)to denote (n,m)-type and ( m,n)-type, we have
+(3.24)(n,m)+(3.26) (m,n)= 0,(3.24)(m,n)+(3.26) (n,m)= 0,
+(3.27)(n,m)+(3.25) (m,n)= 0,(3.27)(m,n)+(3.25) (n,m)= 0,
+(3.28)(m,n)+(3.30) (n,m)= 0,(3.28)(n,m)+(3.30) (m,n)= 0,
+(3.29)(m,n)+(3.31) (n,m)= 0,(3.29)(n,m)+(3.31) (m,n)= 0.
+Hence, the right hand side always vanishes. ) Consequently, if we co llect all the
+remainingterms, thereare(3.8)and(3.9)and( vi)and(xii). Now, weshowthatthe
+addition of these expressions produce N−ktimes (3.9), which proves the theorem.
+Let us list the remaining terms first.
+(3.9)/summationtext
+p+k+j1+j2+1=N<x⊗p,mk(/vectorx),x⊗j1,ei,x⊗j2|x>,(3.32)
+(3.8)/summationtext
+p+k+j1+j2+1=N<x⊗p,ei,x⊗j1,mk(/vectorx),x⊗j2|x>,(3.33)
+(vi)/summationtext
+p+k+j1+j2+1=N<x⊗p,mk(/vectorx),x⊗j1,x,x⊗j2|ei>,(3.34)
+(xii)/summationtext
+p+k+j1+j2+1=N<x⊗p,x,x⊗j1,mk(/vectorx),x⊗j2|ei>.(3.35)
+Now we use closed condition with these terms to obtain (3.10).
+(1) By applying the closed condition in the theorem 2.1 to (3.9) and ( xii), we
+obtain (here ( ai,aj,ak) corresponds to ( ei,mk(/vector x),x))
+<x,···,x,x,x,···,x/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright
+s,mk(/vectorx),x⊗r|ei>
++<x,···,x,mk(/vectorx),x,···,x,ei,x,···,x|x>
++<x⊗r,ei,x⊗s|mk(/vectorx)>= 0
+In fact, we obtain sdifferent such equations depending on the position of
+xin the first line. Hence, the sum of expressions (3.9) and ( xii) produces
+stimes that of (3.10) as the last term equals the minus of (3.10):
+<x⊗r,ei,x⊗s|mk(/vectorx)>=−<x⊗s,mk(/vectorx),x⊗r|ei
+(2) Similarly by applying the closed condition to (3.8) and ( vi),
+<x⊗s,mk(/vectorx),x,···,x,x,x,···,x/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright
+r|ei>
++<x,···,x,ei,x,···,x,mk(/vectorx),x,···,x|x>
++<x⊗r,ei,x⊗s|mk(/vectorx)>
+= 0.
+we obtainrdifferent such equations depending on the position of xin the
+first line.
+Hence we obtain r+s=N−ktimes the expression of (3.10), which proves the
+lemma 3.3. /square14 CHEOL-HYUN CHO AND SANGWOOK LEE
+4.Potential Ψand the generalized holonomy map
+In this section, we consider another potential Ψ defined in the defin ition 1.4
+for a unital homotopy cyclic A∞-algebra. We discuss its gauge invariance and its
+relationship with the algebraic analogue of generalized holonomy map in [ATZ].
+Let us first recall the definition of a unit for A∞-algebra.
+Definition 4.1. An element I∈C0=C−1[1] is called a unit if
+/braceleftBigg
+mk+1(x1,···,I,···,xk) = 0 fork≥2 ork= 0
+m2(I,x) = (−1)degxm2(x,I) =x.(4.1)
+We assume that the strong homotopy inner product φ:A→A∗is an unital
+A∞-bimodule map, or φk,l(/vector a,v,/vectorb)(w) vanishes if one of ai’s orbi’s is a constant
+multiple of I.
+We also recall the Maurer-Cartan elements and its gauge equivalenc es.
+Definition 4.2. LetAbe anA∞-algebra. An element b∈A1satisfyingm(eb) =/summationtext
+kmk(a,...,a) = 0is called the Maurer-Cartanelements and wedenote by MC(A)
+the set of all Maurer-Cartan elements. Let MC:=MC/∼be the moduli space of
+Maurer-Cartan elements, whose gauge equivalence is defined as fo llows(definition
+2.3 of [Fu]): bisgauge equivalent to/tildewidebif there are one-parameter families b(t)∈
+A1[t],c(t)∈A0[t] such that
+•b(0) =b,b(1) =/tildewideb,and
+•d
+dtb(t) =/summationdisplay
+k≥1mk(b(t),...,b(t),c(t),b(t),...,b(t)).
+We remark that b(t) is also a Maurer-Cartan element for any t(Lemma 4.3.7 of
+[FOOO]). Now, weprovethe gaugeinvarianceofthepotential ΨforM aurer-Cartan
+elements.
+Proposition 4.1. The potential Ψ(x) =/summationtext
+p,q≥01
+p+q+1<x⊗p⊗x⊗x⊗q|I >when
+restricted to the Maurer-Cartan elements MCis invariant under gauge equiva-
+lences. i.e. if x(t)is a one-parameter family in the Maurer-Cartan solution spa ce,
+thend
+dtΨ(x(t)) = 0.
+Proof.We prove this proposition with the help of following lemmas.
+Lemma 4.2. Ψ(x)equals the following expression. Ψ(x) =/summationtext
+k≥0<x⊗x⊗k|I >.
+Proof.By the closedness condition of φ, for anypandqwe have
+<x⊗p⊗x⊗x⊗q|I >+<x⊗p+q⊗I|x>
++<x⊗q⊗I⊗x⊗x⊗p−1|x>= 0.
+By definition of unital A∞-bimodule homomorphisms, we have
+<x⊗q⊗I⊗x⊗x⊗p−1|x>= 0,
+and the above equation gives
+<x⊗p⊗x⊗x⊗q|I >=−<x⊗p+q⊗I|x>=<x⊗x⊗p+q|I >,
+where the last equality follows from the skew-symmetry of φ. This proves the
+lemma. /squarePOTENTIALS OF HOMOTOPY CYCLIC A∞-ALGEBRAS 15
+Lemma 4.3./summationtext
+σ∈Z/nZ<aσ(1),aσ(2),...,aσ(n−1)|aσ(n)>= 0.
+Proof.Fixa1,···,anand denote [ i,j] :=< ...,ai,...|aj> .Then what we need to
+prove is
+[1,n]+[2,1]+···+[n,n−1] = 0.
+The closedness condition of strong homotopy inner products gives
+[i,j]+[j,k] = [i,k].
+Hence, it follows that
+[1,n]+[n,n−1]+···+[2,1] = [1,n]+[n,1] = 0.
+/square
+Now we prove the above proposition. First, assume
+d
+dtx(t) =/summationdisplay
+i+j=k≥0mk+1(x(t)⊗i⊗c(t)⊗x(t)⊗j).
+We denote xbyx(t) andcbyc(t), for it causes no problem in this proof.
+Applying lemma 4.2, the fraction disappears and we get
+d
+dtΨ(x) =/summationdisplay
+l≥0</summationdisplay
+i+j=k≥0mk+1(x⊗i⊗c⊗x⊗j)⊗x⊗l|I > (4.2)
++/summationdisplay
+l,m≥0<x⊗x⊗l⊗/summationdisplay
+i+j=k≥0mk+1(x⊗i⊗c⊗x⊗j)⊗x⊗m|I >.(4.3)
+To prove that it is zero, we use the A∞-bimodule equation. Namely, we compute
+(φ◦/hatwidem−m∗◦/hatwideφ)(/summationdisplay
+l≥0c⊗x⊗l+/summationdisplay
+l,m≥0x⊗x⊗l⊗c⊗x⊗m)(I),
+which is a priori zero.
+(φ◦/hatwidem)(/summationdisplay
+i≥0c⊗x⊗i)(I) =/summationdisplay
+l≥0</summationdisplay
+k≥0mk+1(c⊗x⊗k)⊗x⊗l|I > (4.4)
++/summationdisplay
+l,m≥0<c⊗x⊗l⊗(/summationdisplay
+k≥1mk(x⊗k))⊗x⊗m|I >(4.5)
+and (4.5) is zero by Maurer-Cartan equation.
+(φ◦/hatwidem)(/summationdisplay
+i,j≥0x⊗x⊗i⊗c⊗x⊗j)(I)
+=/summationdisplay
+l,m≥0</summationdisplay
+k≥1mk(x⊗k)⊗x⊗l⊗c⊗x⊗m|I > (4.6)
++/summationdisplay
+l≥0</summationdisplay
+i≥1,j≥0mk(x⊗i⊗c⊗x⊗j)⊗x⊗l|I > (4.7)
++/summationdisplay
+l,m≥0<x⊗x⊗l⊗/summationdisplay
+i+j=k≥0mk+1(x⊗i⊗c⊗x⊗j)⊗x⊗m|I >(4.8)
++/summationdisplay
+l,m,n≥0<x⊗x⊗l⊗c⊗x⊗m⊗/summationdisplay
+k≥1mk(x⊗k)⊗x⊗n|I >. (4.9)16 CHEOL-HYUN CHO AND SANGWOOK LEE
+Remark again, that (4.6) and (4.9) vanish by Maurer-Cartan equat ion. Observe
+also that
+•(4.4)+(4.7)=(4.2),
+•(4.8)=(4.3).
+It remains to show that
+(m∗◦/hatwideφ)(/summationdisplay
+l≥0c⊗x⊗l+/summationdisplay
+l,m≥0x⊗x⊗l⊗c⊗x⊗m)(I) = 0.
+SinceIis the unit, we may easily verify that
+(m∗◦φ)(/summationdisplay
+l≥0c⊗x⊗l)(I) =/summationdisplay
+l≥0<c⊗x⊗l|x>, (4.10)
+(m∗◦φ)(/summationdisplay
+l≥0x⊗x⊗l⊗c)(I) =/summationdisplay
+l≥0<x⊗x⊗l|c>, (4.11)
+(m∗◦φ)(/summationdisplay
+l≥0,m≥1x⊗x⊗l⊗c⊗x⊗m)(I) =/summationdisplay
+l,m≥0<x⊗x⊗l⊗c⊗x⊗m|x>.
+In (4.10) and (4.11), for l= 0, we have
+<c|x>+<x|c>= 0
+by skew-symmetry. For remaining parts, we collect terms appropr iately and use
+closedness condition to show that they all vanish. More precisely, f ork≥1, we
+claim that
+<c⊗x⊗k|x>+<x⊗x⊗k|c>+/summationdisplay
+l+m=k−1<x⊗x⊗l⊗c⊗x⊗m|x>= 0
+But this follows from the previous lemma 4.3, by setting a1=c,a2=···=
+ak+2=x. /square
+Now, we discuss the potential Ψ and the algebraic generalized holono my map.
+We refer readers to [ATZ] or [CL] for the relevant definitions of this construction.
+First, recall from Proposition6.1 of [CL] that given a negative cyclic c ohomology
+classαofanA∞-algebraA, oneobtainsa bimodule map /tildewideα:A→A∗. This provides
+a strong homotopy inner product, if αis in addition homologically non-degenerate.
+The definition 1.4 thus provides the potential Ψαusingα. Combined with the
+above proposition, we prove
+Theorem 4.4. The potential Ψprovides a map Ψ :HC•
+−(A)→O(MC)defined
+byα/mapsto→Ψα|MC. Furthermore, this agrees with the algebraic analogue of ge neralized
+holonomy map of Abbaspour, Tradler and Zeinalian [ATZ].
+Proof.We only need to prove the relation with that of [ATZ] and we recall the
+construction of a map ρ:HC•
+−(A)→O(MC). Here we always work with reduced
+versions of negative cyclic or Hochschild (co)homologies.
+Given a Maurer-Cartan element aof a unital A∞-algebraA, consider the ex-
+pression (Definition 8 of [ATZ])
+P(a) :=/summationdisplay
+i≥0I⊗a⊗i= (I⊗I)+(I⊗a)+(I⊗a⊗a)+···.
+One can check that P(a) is a Hochschild homologycycle from the unital property of
+Iand the Maurer-Cartan equation. Note that Connes-Tsygan ope ratorBofP(a)POTENTIALS OF HOMOTOPY CYCLIC A∞-ALGEBRAS 17
+vanishes on the reduced complex, due to the unit I. Hence,P(a) can be considered
+as a negative cyclic homology cycle.
+Hence, given a negative cyclic cohomology cycle α∈HC•
+−(A), one can use the
+pairing<,>:HC•
+−(A)⊗HC−
+•(A)→kto define the map ρas
+ρ([α])([a]) :=/an}bracketle{tα,/summationdisplay
+i≥0I⊗a⊗i/an}bracketri}ht (4.12)
+Now, we compare the above expression with that of Lemma 4.2. We re call the
+following proposition from [CL].
+Proposition 4.5 (Proposition 6.1 [CL]) .Letα∈C•
+red(A,A∗)be a negative cyclic
+cocycle. We define
+/tildewiderα0(/vector a,v,/vectorb)(w) :=α0(/vector a,v,/vectorb)(w)−α0(/vectorb,w,/vector a)(v).
+Then/tildewiderα0is anA∞-bimodule map from AtoA∗, satisfying the skew-symmetry and
+closedness condition.
+Hereα0is the part of αwhich is dual to the inclusion of Hochschild homology
+cycles. Also, from the unital property, we have
+/tildewiderα0(a,a,···,a)(I) =α0(a,···,a)(I)−α0(a,···,a,I)(a) =α0(a,···,a)(I)
+Hence,
+/an}bracketle{tα,I⊗a⊗i/an}bracketri}ht=/an}bracketle{tα0,I⊗a⊗i/an}bracketri}ht=α0(a,···,a)(I) =/tildewiderα0(a,a,···,a)(I) =<a,a,···,a|I >
+where the equality in the middle follows from the identification
+Hom(A⊗(A[1]/k·1)⊗n,k)∼=Hom((A[1]/k·1)⊗n,A∗).
+Hence, each term of the function ρof [ATZ] equals the potential Ψ in the paper
+given in the Lemma 4.2. This proves the theorem. /square
+References
+[ATZ] H. Abbaspour, T. Tradler, M. Zeinalian, Algebraic string bracket as a Poisson bracket
+K-theory, 38 (2007), no.1, 59-82
+[C] C.-H. Cho Strong homotopy inner products of an A∞-algebra, IMRN. ID 41. (2008),
+[CL] C.-H. Cho, S.-W. Lee Notes on Kontsevich-Soibelman’s theorem on cyclic A∞-algebras,
+Imrn. (2010).
+[CO] C.-H. Cho, Y.-G. Oh Floer cohomology and disc instantons of Lagrangian torus fib ers
+in toric Fano manifolds Asian Journ. Math. 10 (2006), 773-814
+[Cos] K. Costello Topological conformal field theories and Calabi-Yau catego ries.Adv. Math.
+210 (2007), no. 1, 165-214.
+[Fu] K. Fukaya, Counting pseudo-holomorphic discs in Calabi-Yau 3 fold
+[FOOO] K.Fukaya, Y.-G.Oh, H.Ohta and K.Ono, Lagrangian intersection Floer theory-anomaly
+and obstruction, Part I, II, AMS/IP Studies in Advanced Mathematics, 46.1. Am erican
+Mathematical Society, Providence, RI; International Pres s, Somerville, MA, 2009.
+[FOOO1] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on toric manifolds,
+[GJ] E. Getzler and J. Jones A∞-algebras and the cyclic bar complex Illinois Journal Math.
+34 no. 2 (1990) 256-283
+[Kaj] H. Kajiura Noncommutative homotopy algebras associated with open str ings,Reviews in
+Mathematical Physics. 1 (2007), 1-99.
+[Ko] M. Kontsevich Formal (non)commutative symplectic geometry The Gelfand Mathemat-
+ical seminars 1990-1992, Birkhauser Boston (1993) 173-187
+[KS] M. Kontsevich and Y. Soibelman Notes on A-infinity algebras, A-infinity categories and
+non-commutative geometry. I Preprint, arXiv:math/0606241
+[T] T. Tradler Infinity inner products J.Homotopy Relat. Struct. 3 (2008), no.1, 245-271.18 CHEOL-HYUN CHO AND SANGWOOK LEE
+[TZ] T. Tradler, M. Zeinalian Infinity structure of Poincare duality spaces Algebr. Geom.
+Topol. 7 (2007), 233-260
+E-mail address :chocheol@snu.ac.kr
+Department of Mathematical Sciences, Research Institute o f Mathematics, Seoul
+National University, Gwanak-gu, Seoul, 151-747 South Korea
+E-mail address :leemky7@snu.ac.kr
+Department of MathematicalSciences, Seoul National Univers ity, Gwanak-gu, Seoul,
+151-747 South Korea
\ No newline at end of file
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+arXiv:1001.0256v1  [astro-ph.EP]  1 Jan 2010Received 2009 November 14; accepted 2009 December 16
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+INITIAL CHARACTERISTICS OF KEPLER LONG CADENCE DATA
+FOR DETECTING TRANSITING PLANETS
+Jon M. Jenkins1, Douglas A. Caldwell1, Hema Chandrasekaran1, Joseph D. Twicken1, Stephen T. Bryson2,
+Elisa V. Quintana1, Bruce D. Clarke1, Jie Li1, Christopher Allen3, Peter Tenenbaum1, Hayley Wu1, Todd C.
+Klaus1, Jeffrey Van Cleve1, Jessie A. Dotson2, Michael R. Haas2, Ronald L. Gilliland4, David G. Koch2, and
+William J. Borucki2
+1SETI Institute/NASA Ames Research Center, M/S 244-30, Moffett Field, CA 94035, USA
+2NASA Ames Research Center, M/S 244-30, Moffett Field, CA 94035, USA
+3Orbital Sciences Corporation/NASA Ames Research Center, M/S 244-30, M offett Field, CA 94035, USA and
+4Space Telescope Science Institute, Baltimore, MD 21218, USA
+Received 2009 November 14; accepted 2009 December 16
+ABSTRACT
+TheKepler Mission seeks to detect Earth-size planets transiting solar-like s tars in its ∼115 deg2
+field of view over the course of its 3.5 year primary mission by monitoring the brightness of each
+of∼156,000 Long Cadence stellar targets with a time resolution of 29.4 minutes. We discuss the
+photometricprecisionachievedontimescalesrelevanttot ransitdetectionfordataobtainedinthe33.5-
+day long Quarter 1 (Q1) observations that ended 2009 June 15. The lower envelope of the photometric
+precision obtained at various timescales is consistent wit h expected random noise sources, indicating
+thatKeplerhas the capability to fulfill its mission. The Keplerlight curves exhibit high precision
+over a large dynamic range, which will surely permit their us e for a large variety of investigations in
+addition to finding and characterizing planets. We discuss t he temporal characteristics of both the
+raw flux time series and the systematic error-corrected flux t ime series produced by the KeplerScience
+Pipeline, and give examples illustrating Kepler’s large dynamic range and the variety of light curves
+obtained from the Q1 observations.
+Subject headings: techniques: photometric — methods: data analysis
+1.INTRODUCTION
+TheKepler Mission is designed to be capable of find-
+ing∼50 Earth-like planets transiting solar-like stars
+in the solar neighborhood if such planets are common
+(Borucki et al. 2010). Specifically, Kepleris required
+to obtain a signal-to-noise ratio (S/N) of 4 σfor an
+84 ppm deep, 6.5-hr transit of a G2V star with a Ke-
+plermagnitude ( Kp) of 12 (Koch et al. 2010). This
+implies that the rms noise on 6.5-hr intervals must be
+below 20 ppm, including contributions from stellar vari-
+ability. Since 2009 May 12, Keplerhas been observing
+∼156,000 stars at 29.4 minute intervals as Long Cadence
+(LC) targets for the primary purpose of detecting tran-
+siting planets. Of these targets, 512 are also sampled at
+1 minute intervals to support asteroseismic characteriza-
+tion of planetary host stars (Gilliland et al. 2010a,b), as
+well as precise timing of transits.
+In Sectoin 2 we examine the photometric precision ob-
+tained to date on the first 33.5 days of data obtained
+during Quarter 1 (Q1) which began May 13 and ended
+June 151. We discuss temporal characteristics of the raw
+and systematic error-corrected flux time series in Section
+3, andpresentsamplelightcurvesinSection4. Summary
+remarks appear in Section 5.
+2.PHOTOMETRIC PRECISION
+Jon.Jenkins@nasa.gov
+1Kepler’s science operations are broken into quarterly seg-
+ments, each of which is ∼93 days long and is subdivided into three
+monthly segments. Data are downlinked at these monthly intervals
+(Haas et al. 2010).To address the question of the photometric precision
+obtained from the KeplerLC data, we examined system-
+atic error-corrected flux time series furnished by the Pre-
+search Data Conditioning (PDC) component of the Sci-
+ence Operations Center (SOC) pipeline2, hereafter called
+PDC time series. At the end of each quarter, the space-
+craft rolls 90◦about the boresight in order to reorient
+the solar arrays toward the Sun, and the target stars ro-
+tate onto new detectors in the Focal Plane Array (FPA;
+Haas et al. 2010). Each of these quarterly segments is
+processed by first calibrating the raw pixels and then
+the Photometric Analysis (PA) component subtracts the
+sky background and extracts simple aperture photomet-
+ric brightness estimates for each star. Systematic errors
+and instrumental effects are identified and removed from
+the raw flux time series by PDC.
+The PDC flux time series for all 156,097 stars were pre-
+pared for analysis by subtracting the median flux value,
+normalizing by the same, and then removing a fitted 4th
+order polynomial across all 1639 LC samples. The collec-
+tion of standard deviations of the normalized, detrended
+flux time series represents the photometric precision on
+29.4 minute intervals as a function of stellar magnitude,
+and this is plotted in panel A of Figure 1.
+The standard propagation of errors is applied to each
+data processing step so that flux uncertainties are avail-
+ablealongwiththefluxmeasurementsthemselves. These
+uncertainties include shot noise from photon counting
+statistics from the direct stellar flux, sky background
+2Details of the SOC science pipeline are discussed in
+Jenkins et al. (2010).2 Jenkins et al.
+and shutterless readout smear, terms such as read noise
+and quantization noise, and the effects of processing the
+data, for example, pixel-level calibrations and sky back-
+ground removal. The uncertainties depend on the de-
+tails of the size of the photometric aperture, the Pixel
+Response Function, and other considerations. It is im-
+portant to note that these uncertainties do notinclude
+any contributions from intrinsic stellar variability.
+Plotting the individual uncertainty estimates
+would clutter the figure, so we fit a simple an-
+alytic expression ”by eye” to the lower enve-
+lope of these uncertainties at the LC timescale:
+ˆσlower=/radicalbig
+c+7×106max(1,Kp/14)4/cwhere
+c= 3.46×10[0.4∗(12−Kp)+8]is the number of de-
+tectede−per LC sample, and Kpis the Kepler Input
+Catalog (KIC) magnitude for each star. The component
+cis a Poisson term while the ( Kp/14)4accounts heuris-
+tically for the increased importance of sky background
+and other noise sources for the dimmer stars with
+Kp≥14. We also fit ”by eye” an analytic expression
+to the upper envelope of this distribution as a function
+ofKp: ˆσupper=√
+c+7×107/c. In this case a simple
+constant background term was sufficient. The lower and
+upper uncertainty envelopes appear in Figure 1 as solid
+and dashed green curves, respectively.
+Three populations of stars can be seen in panel A. We
+take stars with KIC entries of log g <4 to be giants
+or sub-giants3. For the most part, the main sequence
+stars float just above and follow the profile of ˆ σloweras
+a dense ”finger” of points extending over the full mag-
+nitude range of the stars. The fractional width of this
+”finger” is approximately 1.5 near Kp= 12 and flares
+out to a factor of 2 at Kp= 16. This is not unex-
+pected as the dim stars are more sensitive to background
+sky noise, point spread function variations, and detec-
+tor properties, which vary significantly from detector to
+detector. A high ”cloud” of stars, composed of both gi-
+ants and dwarfs, hovers from 3,000 to 10,000 ppm across
+nearly the full magnitude range. These are variable stars
+of all flavors, and possibly some new ones observed for
+the first time. There is a sharp change in the density
+of points for Kp≥14 (Batalha et al. 2010). This in-
+cludes the fact that no stars classified as giants with
+Kp >14 were selected. Giant stars cluster on a hori-
+zontal band centered between ∼400 and∼700 ppm with
+Kp <14. The fact that giant stars exhibit significantly
+more photometric variability on timescales of minutes
+to hours compared to dwarf stars has been previously
+observed by Gilliland (2008) in Hubble Space Telescope
+observations of bulge stars, and more recently, by CoRot
+(Aigrain et al. 2009). Giant stars universally appear to
+exhibit solar-like oscillations, and their increased size rel-
+ative to solar-type stars causes these oscillations to occu r
+at much lower frequencies and larger amplitudes than
+observed for solar-type stars (Gilliland et al. 2010b). A
+total of 350 points lie below the ˆ σlower. These stars suf-
+fer from excess flux in their apertures likely from nearby
+brighter stars or bleeding charge from saturated stars in
+the same column. When these points are plotted at their
+3The KIC is 90% reliable for the purposes of separating giants
+from dwarf stars for Kp <13 and the corresponding uncertainties
+in loggare at the 0.5 dex level (Batalha et al. 2010).instrumental magnitudes (i.e., their measured counts are
+converted to magnitudes via the inverse of the relation
+given above), rather than by catalog magnitude, Kp,
+they are all above ˆ σlower.
+As the central transit of an Earth analog lasts ∼13 hr,
+weexaminethephotometricprecisionrelevantfordetect-
+ing Earth-like transits lasting 6 hr or more. Each PDC
+flux time series considered was first high pass filtered to
+remove variations on timescales of 2 days and longer and
+then convolved with a 6-hr wide ”boxcar” averaging fil-
+ter. The high pass filtering was accomplished by fitting
+a 2nd order polynomial to a sliding analysis window 2
+days wide about each point and removing the resulting
+polynomial at the central point for each window loca-
+tion. We calculated the standard deviations of the time
+series resulting from this process as estimates of the 6 hr
+precision of these data and present these in panel B of
+Figure 1. The LC uncertainty envelopes have been nor-
+malizedby√
+12toaccountforsquarerootstatistics. The
+giant stars (red points) occupy a band between 200 and
+500 ppm. The main sequence stars continue to lie in a
+dense ”finger” just above ˆ σlower/√
+12. The bottom enve-
+lope of the main sequence population tracks the expected
+measurement uncertainties quite well. A total of ∼2000
+points lie below the ˆ σlower/√
+12, but since there are fewer
+independent measurements at the 6 hr level, this is ex-
+pected. 90% of the 2,665 stars with 11 .5≤Kp≤12.5
+in the dense main sequence ”finger” are below 30 ppm,
+65% are below 25 ppm, 50% are below 23 ppm, and 31%
+are below 21 ppm (which corresponds to 20 ppm at 6.5
+hours).
+Kepleris achieving extremely high precision over a re-
+markably large dynamic range considering that the mis-
+sion design was driven largely by the 20 ppm, 6.5-hr pre-
+cision requirement for Kp= 12 G2V stars. Clearly, stars
+do exhibit stellar variability, but there are stars that fal l
+near the ˆ σlower/√
+12 down to at least Kp= 9 and there
+is aKp= 8 star exhibiting noise on 6 hr intervals of 4
+ppm! We believe that further improvements in photo-
+metric precision will be made as we gain experience with
+the spacecraft and the photometric data.
+3.TEMPORAL CHARACTERISTICS OF LC LIGHT CURVES
+Although the photometric precision is a key consid-
+eration in assessing Kepler’s ability to detect transiting
+terrestrial planets, it is not the only one. The precision
+metrics discussed in Section 2 only speak to the mean
+noise power per 29.4 minuts or per 6 hr interval. Here
+we discuss temporal characteristics of the raw and PDC
+light curves of importance for transit detection and for
+non-exoplanet investigations.
+There are systematic effects that occur on timescales
+much longer than those relevant for detecting transits.
+These include Differential Velocity Aberration (DVA),
+whichcausesstarstomoveasmuchas0.6pixelsovera93
+dayperiodduetothelargesizeofthefieldofview(FOV).
+Keplerhas also experienced secular pointing drifts of ∼
+0.05 pixels over the Q1 observations, and there have been
+thermal systematic effects occurring on timescales of sev-
+eral days and longer seasonal variations (Jenkins et al.
+2010) as a consequence of the Sun’s sky position rotating
+about the telescope by about 1◦day−1. A bulk change
+in focus (including tip-tilt terms as well as a ”piston”Long Cadence KeplerData 3
+term) occurred in part due to desorption of water from
+the telescope structure and seasonal thermal effects. The
+gradual change in focus has caused the measured flux of
+stars on the outer periphery of the FOV to drop by about
+0.1% over Q1 while stars toward the interior (on the op-
+posite side of best focus) have experienced corresponding
+increases in flux.
+Systematic errors on these timescales are apparent es-
+pecially in the raw photometric data, but are largely
+suppressed in the PDC time series. Astrophysical sig-
+natures occurring on timescales similar to the long term
+seasonal variations in Kepler’s systematic errors will in-
+evitably be attenuated by PDC. We do not intention-
+ally remove astrophysical signatures on any time scale.
+However, the conditioning of the data simply does not
+allow us to distinguish between astrophysical signatures
+and systematic errors if the former are correlated with
+ancillary engineering data and science data diagnostics
+used in the systematic error removal process. Neverthe-
+less, the PDC time series display a remarkable wealth
+of astrophysics at virtually all timescales probed by the
+observations.
+3.1.Pointing Drift and Focus Changes
+As reported in Jenkins et al. (2010) the photometer
+is quite sensitive even to the slightest change in the ther-
+mal environment. Figure 2 shows changes in a) the focus
+distance local to module 2.1, b) mean row, and c) mean
+columnpositionsoftheensembleofstarslocatedonmod-
+ule 2.1 near the outside corner of the FPA over a 15 day
+interval during Q1. These time series have been filtered
+to remove trends on timescales longer than 2 days, which
+are not relevant for transit detection. The focus distance
+of this module appears to be changing by about 1 µm ev-
+ery time a heater on one reaction wheel assembly turns
+on to keep the wheels above the setpoint temperature.
+These are the downward spikes in the focus distance,
+which represents the change in the relative distances be-
+tween the stars over the observations. The mean row and
+columnpositionsofthesestarsalsoexperiencechangesin
+response to this heater cycle. Additionally, they experi-
+enced excursions every 1.7 days lasting about 8 hr due to
+an eclipsing binary on the Fine Guidance Sensor (FGS)
+target list that was modulating the spacecraft pointing
+by∼1 mpix during eclipses (Haas et al. 2010).
+Similar features appear in the raw flux time series of
+many LC targets. Some respond to the pointing excur-
+sions, whileothersdonot. Mostrespondtofocuschanges
+at some level. Fortunately, PDC was designed to remove
+systematic effects of this kind and does a good job of re-
+moving such signatures for relatively quiet stars and for
+stars that exhibit coherent oscillations with slowly vary-
+ing frequency components. The harmonic components of
+the latter type star are removed prior to removing sig-
+natures correlated with information such as presented in
+Figure 2, and are then restored at the end of the cor-
+rection process. There are incoherently variable targets
+for which it is difficult to separate the intrinsic stellar
+variability from systematic effects. PDC detects when
+this is the case and provides a much simpler detrending
+scheme for such targets. Figure 3 exhibits one example
+of raw and PDC flux time series for one star located on
+module 2.1 which is one of the modules most sensitive to
+the short-term thermal effects.We are making excellent progress with respect to un-
+derstanding the behavior of the photometer. However,
+it is certain that much future work will consist of mon-
+itoring the systematics available through ancillary engi-
+neering data and science data diagnostics so that we can
+best remove the effects from the flux time series while
+retaining as much intrinsic stellar variability as possibl e.
+3.2.Cosmic Rays
+Pre-launch expectations for the cosmic rays Kepler
+would encounter on orbit were informed by a radiation
+model developed by Ball Aerospace Technologies Cor-
+poration (Neal Nickles, private communication 2003).
+Based on this model, we adopted a conservative hit rate
+of 5 cm−2sec−1, and the mode of the distribution of to-
+tal charge deposited per hit was expected to be 2500 e−.
+These facts translate into an expectation that every pixel
+will receive a direct cosmic-ray hit rate of ∼3 day−1and
+that each such event would contribute charge to ∼4 ad-
+jacent pixels. Analysis of dark frames acquired prior to
+releasing Kepler’s dust cover indicate a cosmic ray depo-
+sition rate of 10 cm−2s−1(Van Cleve & Caldwell 2009),
+consistent with our pre-launch expectations. Thus, cos-
+mic rays do not pose a significant noise problem for tran-
+sit detection.
+3.3.Image Artifacts
+Image artifacts introduced by some circuits in the ana-
+log electronics for Kepler’s Focal Plane Array were iden-
+tified during pre-flight testing (Caldwell et al. 2010).
+Although a small fraction of the FOV is impacted at any
+one time, the effects are either amenable to correction in
+data processing or can be identified and excluded from
+consideration at little risk to the primary mission goals.
+The risk to the hardware and cost of attempting a fix
+to the electronics were too high to proceed with a direct
+intervention with the flight hardware. The KeplerSci-
+ence Office and SOC are engaged in developing, testing
+and implementing software mitigations for these image
+artifacts (Jenkins et al. 2010). Typical amplitudes of
+these artifacts are <1 ADU pixel−1read−1, so that in
+most cases their presence in the flux time series of stars
+can be detected only by sophisticated analyses. Once
+the mitigations are in place we expect to recover the re-
+quiredphotometricperformanceforover90%oftheFOV
+(Caldwell et al. 2010).
+3.4.CCD Defects
+A few otherwise good targets land on column or pixel
+defects. Secular drift in pointing together with DVA car-
+ries stars by up to ∼0.7 pixels over each quarter, and
+as an affected star moves across a column defect, short-
+term pointing errors can lead to abrupt changes in the
+measured flux value so that the flux appears to ”toggle”
+between two states for a period of time. Currently we
+make no attempt to identify and correct this behavior.
+Panels A and B of Figure 4 show the light curve for KID
+5286794 illustrating this behavior. Very few targets are
+affected by this phenomenon, given the paucity of CCD
+defects, as documented in Caldwell et al. (2010).
+4.EXAMPLES OF LIGHT CURVES
+In this section we present a potpourri of light curves
+obtained during Q1 observations that illustrate Kepler’s4 Jenkins et al.
+stunning photometric precision and the remarkable di-
+versity of stellar photometric behavior across a wide
+range of amplitudes and timescales. Panels (B) through
+(H) of Figure 4 display large amplitude variable stars’
+light curves. Low amplitude variable stars are exhibited
+in Figure 5. The light curve featured in panels (C) and
+D) is one example of the fact that Keplerachieves a pre-
+cision sufficient to find transit-like features on the order
+of 200 ppm ”by eye”.
+5.CONCLUSIONS
+Kepler’s grand voyage of discovery has begun and in
+the short while since Science Operations commenced we
+are already obtaining the photometric precision so in-
+dispensable to the task of discovering Earth-like plan-
+ets transiting solar-like stars. There are a number of
+instrumental systematics driven by changes in the tem-
+perature of the telescope and spacecraft that are both
+extrinsic to the photometer, such as the seasonal varia-
+tionsduetotheheliocentricorbitandquarterlyrolls, and
+intrinsic to hardware on the spacecraft, such as the re-
+action wheel heater cycles. We are adjusting operationalparameters in order to eliminate or reduce the ampli-
+tude of intrinsic systematics, or to move the timescale
+for these instrumental signatures outside of those rele-
+vant to transit detection, namely from 2 to 16 hours.
+These systematic errors represent nuisances that can be
+filtered out by data processing at the expense of some
+non-exoplanet astrophysics. The light curves obtained
+to date by Keplerare a delight to behold and promise
+unforeseen discoveries and a wealth of information about
+star’sphotometricbehaviorataprecisionandoveracon-
+tiguous and extended period of time not heretofore pos-
+sible. We look forward to establishing a tight constraint
+on the frequency of earth-like planets, a quantum leap
+in our understanding of the formation of planetary sys-
+tems, major strides from asteroseismic results and many
+serendipitous discoveries wholly unrelated to exoplanet
+science.
+Funding for this Discovery Mission is provided by
+NASA’s Science Mission Directorate. We thank the
+thousands of people whose efforts made Keplerpossible.
+REFERENCES
+Aigrain, S., et al. 2009, A&A 506, 425
+Batalha, N. B., et al. 2010, ApJ, this issue
+Borucki, W. J., et al. 2010, Science, submitted
+Caldwell, D. A., et al. 2010, ApJ, this issue
+Gilliland, R. L. 2008, ApJ, 136, 566
+Gilliland, R. L., et al. 2010a, ApJ, this issue
+Gilliland, R. L., et al. 2010b, PASP, in press
+Jenkins, J. M., et al. 2010, ApJ, this issueHaas, M., et al. 2010, ApJ, this issue
+Koch, D. G., et al. 2010, ApJ, this issue
+Van Cleve, J., & D. A. Caldwell, 2009, Kepler Instrument
+Handbook, KSCI 19033-001, (Moffett Field, CA: NASA Ames
+Research Center)Long Cadence KeplerData 5
+Fig. 1.— Scatter plot of the 0.5 hr-to-0.5 hr precision (A) and the 6 hr-to-6 hr precision (B) obtained in the Q1 LC data set. The dashed
+and solid green curves are the upper and lower envelopes of the measurement un certainties, respectively, propagated through the data
+processing steps used to construct the flux time series. Dwarf stars (log g≥4) appear as black points, while giants (log g <4) appear as
+red points. There is a strong separation between the dwarfs and the gian ts in their photometric behavior at both timescales. Kepleris
+delivering near intrinsic measurement-limited noise performance across t he full dynamic range of its target stars.6 Jenkins et al.
+4984 4986 4988 4990 4992 4994 4996−5−4−3−2−101234
+Time, JD−2450000Drift, mpix/Change in Focus, µ m
+ba
+c
+Fig. 2.— Time series of changes in a) focus distance, b) mean row position and c) mean column position for the stars located on module
+2.1 near the outside corner of the FPA over a 10 day interval during Q1 observations.Long Cadence KeplerData 7
+0.990.99511.0051.011.015f/〈 f 〉A
+4980 4982 4984 4986 4988 4990−5000500100015002000
+Time JD − 2450000∆ f/〈 f 〉, ppmB
+Fig. 3.— Raw and systematic error-corrected flux time series for one star on module 2 .1 observed during Q1. The curves in panel A
+are the relative intensity of the raw flux (upper) and the systematic er ror-corrected flux time series (lower) for this star, which exhibits
+variability on timescales upwards of several days. Panel B displays the results of high pass filtering each of these time series to remove long
+term trends. The heater cycle pulses so prevalent in the raw light curve are su ppressed in the systematic error-corrected light curve, and
+the long term variations are preserved since they are not correlated with instrumental systematic variables.8 Jenkins et al.
+4970 4975 4980 4985−0.1−0.0500.050.10.15A
+4975 4980 4985−0.200.20.40.6B
+4960 4970 4980 4990 500000.20.40.60.81C
+4960 4970 4980 4990 5000−0.015−0.01−0.00500.0050.010.015D
+4960 4970 4980 4990 5000−0.01−0.00500.0050.01E∆ f / 〈 f 〉
+4975 4980 4985−0.01−0.00500.0050.01F
+4960 4970 4980 4990 5000−0.5−0.4−0.3−0.2−0.100.1
+Time, JD−2450000G
+4960 4970 4980 4990 5000−0.01−0.00500.0050.01H
+Time, JD−2450000
+Fig. 4.— Relative flux variations for a variety of large amplitude varia ble stars and other flux time series exhibiting the broad dynamic
+range of Kepler’s sensitivity and stellar variability. The left column displays the ent ire Q1 light curve, while the right column shows close
+ups, except where otherwise noted. (A) Light curve for a star whose apertur e is located on a column defect and ”toggles” as its image
+drifts. (B) Close up on an RR-Lyrae star with an evolving envelope. (C a nd D) A flare star with flares that almost double the flux for
+this star on top of quasi-periodic variations of a few percent. (E and F) A star exhibiting high frequency oscillations on top of longer term
+quasi-periodic variations. (G and H) a highly eccentric eclipsing bina ry.Long Cadence KeplerData 9
+4960 4970 4980 4990 5000−4000−2000020004000A
+4970 4972 4974 4976 4978 4980−3000−2000−100001000200030004000B
+4960 4970 4980 4990 5000−400−300−200−1000100C
+4977 4977.2 4977.4 4977.6 4977.8 4978−400−300−200−1000100D
+4960 4970 4980 4990 5000−60−40−200204060∆ f / 〈 f 〉, ppm
+E
+4970 4972 4974 4976 4978 4980−60−40−200204060F
+4960 4970 4980 4990 5000−150−100−50050100150G
+Time, JD−24500004970 4972 4974 4976 4978 4980−150−100−50050100150
+Time, JD−2450000H
+Fig. 5.— Relative flux variations for a variety of small amplitude variab le stars and other flux time series exhibiting the broad dynamic
+range of Kepler’s sensitivity and stellar variability. The left column displays the ent ire Q1 light curve, while the right column shows
+close ups and zooms. (A and B) A star with high frequency oscillations on to p of mmag variations. (C and D) Light curve for a star
+exhibiting 300 ppm transit-like features that can be seen easily by eye. (E and F) A star with variations well below 50 ppm, and coherent,
+small amplitude oscillations. (G and H) A quiet, main sequence star with a random fluctuations with an rms below 50 ppm on half-hour
+timescales.
\ No newline at end of file
diff --git a/1001.0257.txt b/1001.0257.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3a90f53a13a359caa03dbbe600264f64d6cc3cb9
--- /dev/null
+++ b/1001.0257.txt
@@ -0,0 +1,320 @@
+arXiv:1001.0257v1  [math.FA]  1 Jan 2010On the maximal monotonicity of the sum
+of a maximal monotone linear relation
+and the subdifferential operator
+of a sublinear function
+Heinz H. Bauschke∗, Xianfu Wang†, and Liangjin Yao‡
+January 1, 2010
+Abstract
+ThemostimportantopenprobleminMonotoneOperatorTheory concernsthemaximal
+monotonicity of the sum of two maximal monotone operators pr ovided that Rockafel-
+lar’s constraint qualification holds.
+Inthisnote, weprovideanewmaximalmonotonicityresultfo rthesumofamaximal
+monotone relation and the subdifferential operator of a prope r, lower semicontinuous,
+sublinear function. The proof relies on Rockafellar’s form ula for the Fenchel conjugate
+of the sum as well as some results on the Fitzpatrick function .
+2000 Mathematics Subject Classification:
+Primary 47A06, 47H05; Secondary 47B65, 49N15, 52A41, 90C25
+Keywords: Constraint qualification, convex function, convex set, Fenchel c onjugate, Fitz-
+patrick function, linear relation, maximal monotone operator, mult ifunction, monotone op-
+erator, set-valued operator, subdifferential operator, subline ar function, Rockafellar’s sum
+theorem.
+∗Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, Br itish Columbia V1V 1V7, Canada.
+E-mail:heinz.bauschke@ubc.ca .
+†Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, Br itish Columbia V1V 1V7, Canada.
+E-mail:shawn.wang@ubc.ca .
+‡Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, Br itish Columbia V1V 1V7, Canada.
+E-mail:ljinyao@interchange.ubc.ca .
+11 Introduction
+Throughout this paper, we assume that Xis a real Banach space with norm /bardbl·/bardbl, thatX∗
+is the continuous dual of X, and that XandX∗are paired by /an}bracketle{t·,·/an}bracketri}ht. LetA:X⇒X∗be
+aset-valued operator (also known as multifunction) from XtoX∗, i.e., for every x∈X,
+Ax⊆X∗, and let gra A=/braceleftbig
+(x,x∗)∈X×X∗|x∗∈Ax/bracerightbig
+be thegraphofA. Recall that A
+ismonotone if
+(1)/parenleftbig
+∀(x,x∗)∈graA/parenrightbig/parenleftbig
+∀(y,y∗)∈graA/parenrightbig
+/an}bracketle{tx−y,x∗−y∗/an}bracketri}ht ≥0,
+andmaximal monotone ifAis monotone and Ahas no proper monotone extension (in
+the sense of graph inclusion). We say Ais alinear relation if graAis a linear subspace.
+Monotone operators have proven to be a key class of objects in mo dern Optimization and
+Analysis; see, e.g., the books [8, 9, 10, 13, 18, 19, 17, 26] and the r eferences therein. (We also
+adopt standard notation used in these books: dom A=/braceleftbig
+x∈X|Ax/ne}ationslash=∅/bracerightbig
+is thedomain
+ofA. Given a subset CofX, intCis theinteriorofC, andCis theclosureofC. We set
+C⊥:={x∗∈X∗|(∀c∈C)/an}bracketle{tx∗,c/an}bracketri}ht= 0}andS⊥:={x∗∗∈X∗∗|(∀s∈S)/an}bracketle{tx∗∗,s/an}bracketri}ht= 0}for a
+setS⊆X∗. Theindicator function ofC, written as ιC, is defined at x∈Xby
+ιC(x) :=/braceleftBigg
+0,ifx∈C;
+∞,otherwise .(2)
+Givenf:X→]−∞,+∞], we set dom f=f−1(R) andf∗:X∗→[−∞,+∞] :x∗/mapsto→
+supx∈X(/an}bracketle{tx,x∗/an}bracketri}ht −f(x)) is the Fenchel conjugate off. Iffis convex and dom f/ne}ationslash=∅, then
+∂f:X⇒X∗:x/mapsto→/braceleftbig
+x∗∈X∗|(∀y∈X)/an}bracketle{ty−x,x∗/an}bracketri}ht+f(x)≤f(y)/bracerightbig
+is thesubdifferential
+operator off. Recall that fissublinear iff(0) = 0,f(x+y)≤f(x)+f(y), andf(λx) =
+λf(x) for all x,y∈domfandλ >0. Finally, the closed unit ball inXis denoted by
+BX:=/braceleftbig
+x∈X| /bardblx/bardbl ≤1/bracerightbig
+.) Throughout, we shall identify Xwith its canonical image in
+the bidual space X∗∗. Furthermore, X×X∗and (X×X∗)∗=X∗×X∗∗are likewise paired
+via/an}bracketle{t(x,x∗),(y∗,y∗∗)/an}bracketri}ht=/an}bracketle{tx,y∗/an}bracketri}ht+/an}bracketle{tx∗,y∗∗/an}bracketri}ht, where ( x,x∗)∈X×X∗and (y∗,y∗∗)∈X∗×X∗∗.
+LetAandBbemaximalmonotoneoperatorsfrom XtoX∗. Clearly, the sum operator A+
+B:X⇒X∗:x/mapsto→Ax+Bx=/braceleftbig
+a∗+b∗|a∗∈Axandb∗∈Bx/bracerightbig
+is monotone. Rockafellar’s
+[16, Theorem 1] guarantees maximal monotonicity of A+Bunder the classical constraint
+qualification domA∩intdomB/ne}ationslash=∅whenXis reflexive. The most famous open problem
+concerns the behaviour in nonreflexive Banach spaces. See Simons ’ monograph [19] for a
+comprehensive account of the recent developments.
+Now we focus on the special case when Ais alinear relation andBis the subdifferential
+operator of a sublinear function f. We show that the sum theorem is true in this setting.
+Recently, linear relations have increasingly been studied in detail; see , e.g., [1, 2, 3, 4, 5, 6,
+7, 14, 21, 23, 24, 25] and Cross’ book [11] for general backgrou nd on linear relations.
+2The remainder of this paper is organized as follows. In Section 2, we c ollect auxiliary
+results for future reference and for the reader’s convenience. The main result (Theorem 3.1)
+is proved in Section 3.
+2 Auxiliary Results
+Fact 2.1 (Rockafellar) (See [15, Theorem 3], [19, Corollary 10.3 and Theorem 18.1], or
+[26, Theorem 2.8.7(iii)].)
+Letf,g:X→]−∞,+∞]be proper convex functions. Assume that there exists a point
+x0∈domf∩domgsuch that gis continuous at x0. Then for every z∗∈X∗, there exists
+y∗∈X∗such that
+(3) ( f+g)∗(z∗) =f∗(y∗)+g∗(z∗−y∗).
+Furthermore, ∂(f+g) =∂f+∂g.
+Fact 2.2 (Fitzpatrick) (See [12, Corollary 3.9].) LetA:X⇒X∗be maximal monotone,
+and set
+(4)FA:X×X∗→]−∞,+∞] : (x,x∗)/mapsto→sup
+(a,a∗)∈graA/parenleftbig
+/an}bracketle{tx,a∗/an}bracketri}ht+/an}bracketle{ta,x∗/an}bracketri}ht−/an}bracketle{ta,a∗/an}bracketri}ht/parenrightbig
+,
+which is the Fitzpatrick function associated with A. Then for every (x,x∗)∈X×X∗, the
+inequality /an}bracketle{tx,x∗/an}bracketri}ht ≤FA(x,x∗)is true, and equality holds if and only if (x,x∗)∈graA.
+Fact 2.3 (Simons) (See [19, Theorem 24.1(c)].) LetA,B:X⇒X∗be maximal monotone
+operators. Assume/uniontext
+λ>0λ[PX(domFA)−PX(domFB)]is a closed subspace, where PX:
+(x,x∗)∈X×X∗→x. If
+(5) ( x,x∗)is monotonically related to gra(A+B)⇒x∈domA∩domB,
+thenA+Bis maximal monotone.
+Fact 2.4 (Simons) (See[19, Lemma 19.7andSection 22].) LetA:X⇒X∗be a monotone
+linear relation such that graA/ne}ationslash=∅. Then the function
+(6) g:X×X∗→]−∞,+∞] : (x,x∗)/mapsto→ /an}bracketle{tx,x∗/an}bracketri}ht+ιgraA(x,x∗)
+is proper and convex.
+3Fact 2.5 (Simons) (See [20, Lemma 2.2].) Letf:X→]−∞,+∞]be proper, lower
+semicontinuous, and convex. Let x∈Xandλ∈Rbe such that inff < λ < f (x)≤+∞,
+and set
+K:= sup
+a∈X,a/ne}ationslash=xλ−f(a)
+/bardblx−a/bardbl.
+ThenK∈]0,+∞[and for every ε∈]0,1[, there exists (y,y∗)∈gra∂fsuch that
+/an}bracketle{ty−x,y∗/an}bracketri}ht ≤ −(1−ε)K/bardbly−x/bardbl<0. (7)
+Fact 2.6 (See [26, Therorem 2.4.14].) Letf:X→]−∞,+∞]be a sublinear function.
+Then the following hold.
+(i)∂f(x) ={x∗∈∂f(0)| /an}bracketle{tx∗,x/an}bracketri}ht=f(x)},∀x∈domf.
+(ii)∂f(0)/ne}ationslash=∅⇔fis lower semicontinuous at 0.
+(iii)Iffis lower semicontinuous, then f= sup/an}bracketle{t·,∂f(0)/an}bracketri}ht.
+Fact 2.7 (See [13, Proposition 3.3 and Proposition 1.11].) Letf:X→]−∞,+∞]be a
+lower semicontinuous convex and intdomf/ne}ationslash=∅. Thenfis continuous on intdomfand
+∂f(x)/ne}ationslash=∅for every x∈intdomf.
+Lemma 2.8 Letf:X→]−∞,+∞]be a sublinear function. Then domf+ intdom f=
+intdomf.
+Proof. The result is trivial when intdom f=∅so we assume that x0∈intdomf. Then
+there exists δ >0 such that x0+δBX⊆domf. By sublinearity, ∀y∈domf, we have
+y+x0+δBX⊆domf. Hence
+y+x0∈intdomf.
+Then dom f+ intdom f⊆intdomf. Since 0 ∈domf, intdom f⊆domf+ intdom f.
+Hence dom f+intdom f= intdom f. /squaresolid
+Lemma 2.9 LetA:X⇒X∗be a maximal monotone linear relation, and let z∈X∩(A0)⊥.
+Thenz∈domA.
+Proof. Suppose to the contrary that z /∈domA. Then the Separation Theorem provides
+w∗∈X∗such that
+/an}bracketle{tz,w∗/an}bracketri}ht>0 and w∗∈domA⊥. (8)
+4Thus, (0,w∗) is monotonically related to gra A. SinceAis maximal monotone, we deduce
+thatw∗∈A0. By assumption, /an}bracketle{tz,w∗/an}bracketri}ht= 0, which contradicts (8). Hence, z∈domA./squaresolid
+The proof of the next result follows closely the proof of [19, Theore m 53.1].
+Lemma 2.10 LetA:X⇒X∗be a monotone linear relation, and let f:X→]−∞,+∞]
+be a proper lower semicontinuous convex function. Suppose t hatdomA∩intdom∂f/ne}ationslash=∅,
+(z,z∗)∈X×X∗is monotonically related to gra(A+∂f), and that z∈domA. Then
+z∈dom∂f.
+Proof. Letc0∈Xandy∗∈X∗be such that
+c0∈domA∩intdom∂fand (z,y∗)∈graA. (9)
+Takec∗
+0∈Ac0, and set
+M:= max/braceleftbig
+/bardbly∗/bardbl,/bardblc∗
+0/bardbl/bracerightbig
+, (10)
+D:= [c0,z], andh:=f+ιD. By (9), Fact 2.7 and Fact 2.1, ∂h=∂f+∂ιD. Set
+H:X→]−∞,+∞] :x/mapsto→h(x+z)−/an}bracketle{tz∗,x/an}bracketri}ht. It remains to show that
+0∈dom∂H. (11)
+If infH=H(0), then (11) holds. Now suppose that inf H < H(0). Letλ∈Rbe such that
+infH < λ < H (0), and set
+Kλ:= sup
+H(x)<λλ−H(x)
+/bardblx/bardbl. (12)
+We claim that
+Kλ≤M.
+By Fact 2.5, we have Kλ∈]0,∞[ and∀ε∈]0,1[, by gra ∂H= gra∂h−(z,z∗) there exists
+(x,x∗)∈gra∂hsuch that
+/an}bracketle{tx−z,x∗−z∗/an}bracketri}ht ≤ −(1−ε)Kλ/bardblx−z/bardbl<0. (13)
+Since∂h=∂f+∂ιD, there exists t∈[0,1] withx∗
+1∈∂f(x) andx∗
+2∈∂ιD(x) such that
+x=tc0+(1−t)zandx∗=x∗
+1+x∗
+2. Then/an}bracketle{tx−z,x∗
+2/an}bracketri}ht ≥0. Thus, by (13),
+/an}bracketle{tx−z,x∗
+1−z∗/an}bracketri}ht ≤ /an}bracketle{tx−z,x∗
+1+x∗
+2−z∗/an}bracketri}ht ≤ −(1−ε)Kλ/bardblx−z/bardbl<0. (14)
+Asx=tc0+(1−t)zandAis a linear relation, we have ( x,tc∗
+0+(1−t)y∗)∈graA. Since
+(z,z∗) is monotonically related to gra( A+∂f), by (10),
+/an}bracketle{tx−z,x∗
+1−z∗/an}bracketri}ht ≥ −/an}bracketle{tx−z,tc∗
+0+(1−t)y∗/an}bracketri}ht ≥ −M/bardblx−z/bardbl. (15)
+5Combining (15) and (14), we obtain
+−M/bardblx−z/bardbl ≤ −(1−ε)Kλ/bardblx−z/bardbl<0. (16)
+Hence, (1 −ε)Kλ≤M. Letting ε↓0, we deduce that Kλ≤M. Then, by (12) and letting
+λ↑H(0), we get
+H(y)+M/bardbly/bardbl ≥H(0),∀y∈X. (17)
+By [19, Example 7.1], 0 ∈dom∂H. Hence (11) holds and thus z∈dom∂f. /squaresolid
+3 Main Result
+Theorem 3.1 LetA:X⇒X∗be a maximal monotone linear relation, let f:X→
+]−∞,+∞]be a proper lower semicontinuous sublinear function, and su ppose that domA∩
+intdom∂f/ne}ationslash=∅. ThenA+∂fis maximal monotone.
+Proof. Let (z,z∗)∈X×X∗and suppose that
+(18) ( z,z∗) is monotonically related to gra( A+∂f).
+By Fact 2.2, dom A⊆PX(FA) and dom ∂f⊆PX(F∂f). Hence,
+/uniondisplay
+λ>0λ/parenleftbig
+PX(domFA)−PX(domF∂f)/parenrightbig
+=X. (19)
+Thus, by Fact 2.3, it suffices to show that
+(20) z∈domA∩dom∂f.
+We have
+/an}bracketle{tz,z∗/an}bracketri}ht−/an}bracketle{tz,x∗/an}bracketri}ht−/an}bracketle{tx,z∗/an}bracketri}ht+/an}bracketle{tx,x∗/an}bracketri}ht+/an}bracketle{tx−z,y∗/an}bracketri}ht
+=/an}bracketle{tz−x,z∗−x∗−y∗/an}bracketri}ht ≥0,∀(x,x∗)∈graA,(x,y∗)∈gra∂f. (21)
+By Fact 2.6(ii), ∂f(0)/ne}ationslash=∅. By (21),
+inf[/an}bracketle{tz,z∗/an}bracketri}ht−/an}bracketle{tz,A0/an}bracketri}ht−/an}bracketle{tz,∂f(0)/an}bracketri}ht]≥0.
+Thus,
+z∈X∩(A0)⊥. (22)
+6Then, by Fact 2.6(iii),
+/an}bracketle{tz,z∗/an}bracketri}ht ≥f(z).
+Thus,
+z∈domf. (23)
+By (22) and Lemma 2.9, we have
+z∈domA. (24)
+By Fact 2.6(i), y∗∈∂f(0) asy∗∈∂f(x). Then/an}bracketle{tx−z,y∗/an}bracketri}ht ≤f(x−z),∀y∗∈∂f(x). Thus,
+by (21), we have
+/an}bracketle{tz,z∗/an}bracketri}ht−/an}bracketle{tz,x∗/an}bracketri}ht−/an}bracketle{tx,z∗/an}bracketri}ht+/an}bracketle{tx,x∗/an}bracketri}ht+f(x−z)≥0,∀(x,x∗)∈graA,x∈dom∂f. (25)
+LetC:= intdom f. Then by Fact 2.7, we have
+/an}bracketle{tz,z∗/an}bracketri}ht−/an}bracketle{tz,x∗/an}bracketri}ht−/an}bracketle{tx,z∗/an}bracketri}ht+/an}bracketle{tx,x∗/an}bracketri}ht+f(x−z)≥0,∀(x,x∗)∈graA,x∈C. (26)
+Setj:= (f(·−z)+ιC)⊕ιX∗and
+(27) g:X×X∗→]−∞,+∞] : (x,x∗)/mapsto→ /an}bracketle{tx,x∗/an}bracketri}ht+ιgraA(x,x∗).
+By Fact 2.4, gis convex. Hence,
+(28) h:=g+j
+is convex as well. Let
+(29) c0∈domA∩C.
+By Lemma 2.8 and(23), z+c0∈intdomf. Then there exists δ >0 such that z+c0+δBX⊆
+domfandc0+δBX⊆domf. By (24), z+c0∈domAsince dom Ais a linear subspace.
+Thus there exists b∈1
+2δBXsuch that z+c0+b∈domA∩intdomf. Letv∗∈A(z+c0+b).
+Sincec0+b∈intdomf,
+(z+c0+b,v∗)∈graA∩/parenleftbig
+intC∩intdomf(·−z)×X∗/parenrightbig
+= domg∩intdomj/ne}ationslash=∅. (30)
+7By Fact 2.1 and Fact 2.7, there exists ( y∗,y∗∗)∈X∗×X∗∗such that
+h∗(z∗,z) =g∗(y∗,y∗∗)+j∗(z∗−y∗,z−y∗∗)
+=g∗(y∗,y∗∗)+ι{0}(z−y∗∗)+sup
+x∈C[/an}bracketle{tx,z∗−y∗/an}bracketri}ht−f(x−z)]
+≥g∗(y∗,y∗∗)+ι{0}(z−y∗∗)+ sup
+x∈z+C[/an}bracketle{tx,z∗−y∗/an}bracketri}ht−f(x−z)] (by Lemma 2.8 and (23))
+=g∗(y∗,y∗∗)+ι{0}(z−y∗∗)+/an}bracketle{tz,z∗−y∗/an}bracketri}ht+sup
+y∈C[/an}bracketle{ty,z∗−y∗/an}bracketri}ht−f(y)]
+=g∗(y∗,y∗∗)+ι{0}(z−y∗∗)+/an}bracketle{tz,z∗−y∗/an}bracketri}ht+ sup
+{y∈C,k>0}[/an}bracketle{tky,z∗−y∗/an}bracketri}ht−f(ky)]
+=g∗(y∗,y∗∗)+ι{0}(z−y∗∗)+/an}bracketle{tz,z∗−y∗/an}bracketri}ht+ sup
+{y∈C,k>0}k[/an}bracketle{ty,z∗−y∗/an}bracketri}ht−f(y)]
+≥g∗(y∗,y∗∗)+ι{0}(z−y∗∗)+/an}bracketle{tz,z∗−y∗/an}bracketri}ht.(31)
+By (26), we have, for every ( x,x∗)∈graA∩(C×X∗),/an}bracketle{t(x,x∗),(z∗,z)/an}bracketri}ht−h(x,x∗) =/an}bracketle{tx,z∗/an}bracketri}ht+
+/an}bracketle{tz,x∗/an}bracketri}ht−/an}bracketle{tx,x∗/an}bracketri}ht−f(x−z)≤ /an}bracketle{tz,z∗/an}bracketri}ht. Consequently,
+(32) h∗(z∗,z)≤ /an}bracketle{tz,z∗/an}bracketri}ht.
+Combining (31) with (32), we obtain
+(33) g∗(y∗,y∗∗)+/an}bracketle{tz,z∗−y∗/an}bracketri}ht+ι{0}(z−y∗∗)≤ /an}bracketle{tz,z∗/an}bracketri}ht.
+Therefore, y∗∗=z. Henceg∗(y∗,z) +/an}bracketle{tz,z∗−y∗/an}bracketri}ht ≤ /an}bracketle{tz,z∗/an}bracketri}ht. Sinceg∗(y∗,z) =FA(z,y∗), we
+deduce that FA(z,y∗)≤ /an}bracketle{tz,y∗/an}bracketri}ht. By Fact 2.2,
+(34) ( z,y∗)∈graA
+Hence
+z∈domA.
+Apply Lemma 2.10 to obtain z∈dom∂f. Thenz∈domA∩dom∂f. Hence A+Bis
+maximal monotone. /squaresolid
+Remark 3.2 Verona and Verona (see [22, Corollary 2.9(a)] or [19, Theorem 53.1]) showed
+the following: “Let f:X→]−∞,+∞] be proper, lower semicontinuous, and convex, let
+A:X⇒X∗be maximal monotone, and suppose that dom A=X. Then∂f+Ais maximal
+monotone.” Note that Theorem 3.1 cannot be deduced from this res ult because dom Aneed
+not have full domain.
+8Acknowledgment
+Heinz Bauschke was partially supported by the Natural Sciences an d Engineering Research
+Council of Canada and by the Canada Research Chair Program. Xian fu Wang was partially
+supported by the Natural Sciences and Engineering Research Cou ncil of Canada.
+References
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+tonicity, and monotonicity of the conjugate are all the same for continuous linear operators”,
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+[3] H.H. Bauschke, X. Wang, and L. Yao, “Autoconjugate repre senters for linear monotone oper-
+ators”,Mathematical Programming (Series B) , to appear;
+http://arxiv.org/abs/0802.1375v1 , February 2008.
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+[5] H.H. Bauschke, X. Wang, and L. Yao, “An answer to S. Simons ’ question on the maximal
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+[6] H.H. Bauschke, X. Wang, and L. Yao, “Examples of disconti nuous maximal monotone linear
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+linear relations”, submitted;
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+10
\ No newline at end of file
diff --git a/1001.0258.txt b/1001.0258.txt
new file mode 100644
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+++ b/1001.0258.txt
@@ -0,0 +1,522 @@
+Received 2009 November 14; accepted 2009 December 21
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+OVERVIEW OF THE KEPLER SCIENCE PROCESSING PIPELINE
+Jon M. Jenkins1, Douglas A. Caldwell1, Hema Chandrasekaran1, Joseph D. Twicken1, Stephen T. Bryson2,
+Elisa V. Quintana1, Bruce D. Clarke1, Jie Li1, Christopher Allen3, Peter Tenenbaum1, Hayley Wu1, Todd C.
+Klaus3, Christopher K. Middour3, Miles T. Cote3, Sean McCauliff3, Forrest R. Girouard3, Jay P. Gunter3,
+Bill Wohler3, Jeneen Sommers3, Jennifer R. Hall3, AKM K. Uddin3, Michael S. Wu5, Paresh A. Bhavsar2,
+Jeffrey Van Cleve1, David L. Pletcher2, Jessie A. Dotson2, Michael R. Haas2, Ronald L. Gilliland4, David G.
+Koch2, and William J. Borucki2
+1SETI Institute/NASA Ames Research Center, M/S 244-30, Moett Field, CA 94035, USA
+2NASA Ames Research Center, M/S 244-30, Moett Field, CA 94035, USA
+3Orbital Sciences Corporation/NASA Ames Research Center, M/S 244-30, Moett Field, CA 94035, USA
+4Space Telescope Science Institute, Baltimore, MD 21218, USA and
+5Bastion Technologies/NASA Ames Research Center, M/S 244-30, Moett Field, CA 94035, USA
+Received 2009 November 14; accepted 2009 December 21
+ABSTRACT
+The Kepler Mission Science Operations Center (SOC) performs several critical functions including
+managing the 156,000 target stars, associated target tables, science data compression tables and
+parameters, as well as processing the raw photometric data downlinked from the spacecraft each
+month. The raw data are rst calibrated at the pixel level to correct for bias, smear induced by a
+shutterless readout, and other detector and electronic eects. A background sky 
ux is estimated from
+4500 pixels on each of the 84 CCD readout channels, and simple aperture photometry is performed
+on an optimal aperture for each star. Ancillary engineering data and diagnostic information extracted
+from the science data are used to remove systematic errors in the 
ux time series that are correlated
+with these data prior to searching for signatures of transiting planets with a wavelet-based, adaptive
+matched lter. Stars with signatures exceeding 7 :1are subjected to a suite of statistical tests
+including an examination of each star's centroid motion to reject false positives caused by background
+eclipsing binaries. Physical parameters for each planetary candidate are tted to the transit signature,
+and signatures of additional transiting planets are sought in the residual light curve. The pipeline is
+operational, nding planetary signatures and providing robust eliminations of false positives.
+Subject headings: techniques: photometric | methods: data analysis
+1.INTRODUCTION
+The Kepler Mission seeks to detect Earth-like planets
+transiting solar-like stars by performing photometric ob-
+servations of 156,000 carefully selected target stars in
+Kepler's 115 deg2eld of view (FOV), as reviewed in
+Borucki et al. (2010) and Koch et al. (2010). These
+Long Cadence (LC) targets are sampled every 29.4 min-
+utes and include all the planetary targets for which we
+seek signatures of transiting planets. In addition, a to-
+tal of 512 Short Cadence (SC) targets are sampled at
+58.85 s intervals permitting further characterization of
+the planet-star systems for the brighter ( Kp < 12) stars
+via asteroseismology, and more precise transit timing.
+The Kepler Mission Science Operations Center (SOC)
+at NASA Ames Research Center performs nine major
+functions:
+1. Manage target aperture and denition tables spec-
+ifying which 5 :4106of the 95 106pixels in the
+CCD array are processed and stored on the Solid
+State Recorder for later downlink1.
+2. Manage the science data compression tables and
+parameters, including the length-limited Human
+coding table, and the requantization table.
+Jon.Jenkins@nasa.gov
+1Kepler's pointing stability requirement is 0".009, 3 , allowing
+us to preselect the pixels of interest for each star (Haas et al.
+2010).3. Report on the Kepler photometer's health and
+status semiweekly after each X-band contact and
+monthly after each Ka-band science data downlink.
+4. Monitor the pointing error and compute pointing
+tweaks when necessary to adjust the spacecraft
+pointing to ensure the validity of the uplinked sci-
+ence target tables,
+5. Process the science data each month to obtain
+calibrated pixels for all LC and SC targets, raw
+
ux time series, and systematic error-corrected 
ux
+time series,.
+6. Archive calibrated pixels, raw and corrected 
ux
+time series and centroid measurements to the Data
+Management Center (DMC).
+7. Search each 
ux time series for signatures of tran-
+siting planets,.
+8. Fit physical parameters and calculate error esti-
+mates for planetary signatures, and .
+9. Perform statistical tests to reject false positives and
+establish accurate statistical condence in each de-
+tection. .
+Kepler's observations are organized into three month
+intervals called quarters dened by the roll maneuversarXiv:1001.0258v1  [astro-ph.EP]  1 Jan 20102 Jenkins et al.
+the spacecraft executes about its boresight to keep the
+solar arrays pointed toward the Sun (Haas et al. 2010).
+Once each month, the accumulated science data are
+transmitted via the Deep Space Network2(DSN) to the
+Mission Operations Center3, which forwards them to the
+DMC4. The DMC packages them into FITS les and
+pushes them to the SOC. A selected set of ancillary en-
+gineering data are also delivered with the science data,
+containing any parameters likely to have a bearing on the
+quality of the science data, such as temperature measure-
+ments of the focal plane and readout electronics.
+The Science Pipeline is divided into several compo-
+nents in order to allow for ecient management and
+parallel processing of the data, as shown in Figure 1.
+Raw pixel data downlinked from the Kepler photometer
+are calibrated by the Calibration module (CAL) to pro-
+duce calibrated target and background pixels and their
+associated uncertainties. The calibrated pixels are pro-
+cessed by Photometric Analysis (PA) to t and remove
+sky background and extract simple aperture photometry
+from the background-corrected, calibrated target pixels.
+PA also measures the centroid locations of each star on
+each frame. The nal step to produce light curves hap-
+pens in Pre-search Data Conditioning (PDC) where sig-
+natures in the light curves correlated with systematic
+error sources such as pointing drift, focus changes, and
+thermal transients are removed. Output data products
+include raw and calibrated pixels, raw and systematic
+error-corrected 
ux time series, centroids and associated
+uncertainties for each target star, which are archived to
+the DMC and eventually made available to the public
+through the Multimission Archive at STScI.5
+In Transiting Planet Search (TPS) a wavelet-based,
+adaptive matched lter is applied to identify transit-like
+features with durations in the range of 1{16 hours. Light
+curves with transit-like features whose combined (folded)
+transit detection statistic exceeds 7.1 for some trial pe-
+riod and epoch are designated as Threshold Crossing
+Events (TCEs) and subjected to further scrutiny by Data
+Validation (DV). This threshold ensures that no more
+than one false positive will occur due to random 
uc-
+tuations over the course of the mission, assuming non-
+white, non-stationary Gaussian observation noise (Jenk-
+ins, Caldwell & Borucki 2002; Jenkins 2002). DV per-
+forms a suite of statistical tests to evaluate the condence
+in the detection, to reject false positives by background
+eclipsing binaries, and to extract physical parameters of
+each system (along with associated uncertainties and co-
+variance matrices) for each planet candidate. After the
+planetary signature has been tted, it is removed from
+the light curve and the residual is subjected to a search
+for additional transiting planets. This process repeats
+until no further TCEs are identied. The DV results
+and diagnostics are furnished to the Science Team to fa-
+cilitate disposition by the Follow-up Observing Program
+(FOP; Gautier et al. 2010).
+2.PIXEL LEVEL CALIBRATIONS
+2The DSN is operated by the Jet Propulsion Laboratory for
+NASA.
+3The MOC is located at LASP in Boulder, CO, USA.
+4The DMC is located at the Space Telescope Science Institute
+(STScI) in Baltimore, MD, USA.
+5http://stdatu.stsci.edu/kepler/The Pipeline module CAL corrects the raw Kepler
+photometric data at the pixel level prior to the extraction
+of photometry and astrometry. Several of the processing
+steps given in Figure 2 are familiar to ground-based pho-
+tometrists. However, a few are peculiar to Kepler due
+to the lack of a shutter and unique features in its analog
+electronics chains. Details of these instrument charac-
+teristics and how they were determined and updated in
+
ight are discussed in Caldwell et al. (2010) and are
+comprehensively documented in the Kepler Instrument
+Handbook ( ?).
+The sequence of processing steps in CAL that produce
+calibrated pixels and associated uncertainties is as fol-
+lows. (1) The two-dimensional black level (CCD bias
+voltage) structure (xed pattern noise) is removed, fol-
+lowed by tting and removing a dynamic estimate of the
+black level. (2) Gain and nonlinearity corrections are ap-
+plied. (3) The analog electronics chain exhibits memory,
+necessitating the application of a digital lter to remove
+this eect, called Local Detector Electronics (LDE) un-
+dershoot. (4) Cosmic ray events in the black and smear
+measurements are removed prior to subsequent correc-
+tions. (5) The smear signal caused by operating in the
+absence of a shutter and the dark current for each CCD
+readout channel are estimated from the masked and vir-
+tual smear collateral data measurements. (6) A 
at eld
+correction is applied.
+3.PHOTOMETRIC ANALYSIS
+Before photometry and astrometry can be extracted
+from the calibrated pixel time series, the Pipeline detects
+so-called "Argabrightening" events in the background
+pixel data. These mysterious transient increases in the
+background 
ux were identied early in Commissioning.
+The current hypothesis is that these transient events are
+due to small dust particles from Kepler achieving escape
+velocity after micrometeorite hits and re
ecting sunlight
+into the barrel of the telescope as they drift across the
+FOV. Argabrightenings that aect 10 or more CCD read-
+out channels occur 15 times per month, but the rate is
+dropping over time. The most egregious of these events
+cannot be perfectly corrected by the current background
+correction. We gap the data when the excess background
+
ux exceeds the 100 Median Absolute Deviation (MAD)
+level.
+PA then robustly ts a two-dimensional surface to
+4500 background pixels on each channel to estimate
+the sky background, which is evaluated at each target
+star pixel location and subtracted from the calibrated
+pixel values. Each target pixel time series is scanned for
+cosmic rays by rst detrending the time series with a
+moving median lter with a width of ve cadences (time
+steps) and examining the residuals for outliers compared
+to the MAD of the residuals for each pixel. Care is taken
+not to remove clusters of outliers that might be due to
+astrophysical signatures such as 
ares or transits that are
+intrinsic to the target star.
+The photocenters of the 200 brightest, unsaturated
+stars on each channel are tted using the pixel response
+functions (PRF; Bryson et al. 2010) and then used to
+dene the ensemble motion of the stars over the ob-
+servations. The aggregate star motion is used along
+with the PRFs reconstructed from Commissioning data
+to dene the optimal aperture as the collection of pix-Kepler Science Pipeline 3
+els that maximizes the mean signal-to-noise ratio of the
+
ux measurement for each star (Bryson et al. 2010).
+The background-corrected, cosmic ray-corrected pixels
+are then summed over the optimal aperture to dene a
+
ux estimate for each cadence frame.
+4.SYSTEMATIC ERROR CORRECTIONS
+PDC's task is to remove systematic errors from the
+raw 
ux time series. These include pointing errors, focus
+changes, and thermal eects on instrument properties.
+PDC co-trends each 
ux time series against ancillary en-
+gineering data such as temperatures of the focal plane
+and electronics, reconstructed pointing and focus varia-
+tions to remove signatures correlated with these proxy
+systematic error measurements. A Singular Value De-
+composition (SVD) is applied to the design matrix con-
+taining the ancillary data to identify the most signicant,
+independent components and to stabilize the matrix in-
+version inherent in the t to the data. Additionally, PDC
+identies residual isolated outliers, and lls intra-quarter
+gaps so that the data for each quarterly segment are con-
+tiguous when presented to TPS. Finally, PDC adjusts the
+light curves to account for the excess 
ux in the optimal
+apertures due to stareld crowding in order to make ap-
+parent transit depths uniform from quarter to quarter
+as the stars move from detector to detector with each
+roll maneuver. This is achieved by estimating the mean
+excess 
ux in each photometric aperture from sources
+other than the target star itself from knowledge of the
+PRF and background star positions and magnitudes and
+subtracting this value from each point in the time series
+(see Bryson et al. 2010).
+Signicant eort has been applied to PDC in order to
+achieve good results with 
ight data. There are a num-
+ber of phenomena that were signicantly dierent than
+expected, including focus variations, and the amount of
+pointing drift observed during the rst two quarters of
+operation. The systematic errors observed in 
ight ex-
+hibit a range of dierent time scales, from a few hours to
+several days to many days and weeks. Such phenomena
+include the intermittent modulation of the focus by 1
+m every 3.2 hr by a heater on one of the reaction wheel
+assemblies. One of the Fine Guidance Sensors' guide
+stars through the rst quarter (Q1) of observations was
+an eclipsing binary whose 30% eclipses induced a 1 mpix
+pointing excursion lasting 8 hr every 1.7 days (Haas et
+al. 2010). By far the strongest systematic eects in the
+data so far have occurred after each of two safe mode
+events (Haas et al. 2010) during which the photometer
+was shut o, the telescope cooled and the focus changed
+by2.2m perC. One of these occurred at the end of
+Q1 and the second 2 weeks into Q2. Thermal eects
+can be observed in the science data for 5 days after
+each safe mode recovery. The fact that most systematics
+such as these aect all the science data simultaneously,
+and that there is a rich amount of ancillary engineering
+data and science diagnostics available provides signicant
+leverage in dealing with these eects.
+Some systematic phenomena are specic to individual
+stars and cannot be corrected by co-trending against an-
+cillary data. The rst issue is the occasional, abrupt drop
+in pixel sensitivity that introduces a step discontinuity in
+an aected star's light curve (and associated centroids).
+This is often preceded immediately by a cosmic ray event,and is sometimes followed by an exponential recovery
+over a few hours, but usually not to the same 
ux level
+as before. The typical drop in sensitivity is 1%, which
+is unmistakable in the 
ux time series. Such step dis-
+continuities are identied separately from those due to
+operational activities, such as safe modes and pointing
+tweaks, and are mended by raising the light curve after
+the discontinuity for the remainder of the quarter. These
+events do not mimic transits since they do not recover to
+the same pre-event 
ux level, and few transits, if any, are
+aected by this correction. The second issue is that many
+stars exhibit coherent or slowly evolving oscillations that
+interfere with systematic error removal. The approach
+taken is to identify and remove strong coherent compo-
+nents in the frequency domain prior to co-trending, and
+then to restore these components to the residuals after
+co-trending.
+Figure 3 shows the results of running two 
ux time se-
+ries obtained during Quarter 2 through PDC on schedule
+for release early in 2010, demonstrating PDC's eective-
+ness. We expect that learning to deal with the various
+systematic errors will consume a great deal of eort over
+the lifetime of the mission as we push the detection limit
+to smaller and smaller planets.
+5.TRANSITING PLANET SEARCH
+TPS searches for transiting planets by "stitching" the
+quarterly segments of data together to remove gaps and
+edge eects and then applies the wavelet-based, adap-
+tive matched lter of Jenkins (2002). This approach is
+a time-adaptive approach that estimates the power spec-
+trum of the observational noise as a function of time.
+This approach was developed specically for solar-like
+stars with colored, broadband power spectra. Some mod-
+ications to the original approach have been developed
+to accommodate target stars that exhibit coherent struc-
+ture in the frequency domain. Similar to the approach
+adopted in PDC, we t and remove strong harmonics
+that are inconsistent with transit signatures prior to ap-
+plying the wavelet-based lter. This signicantly in-
+creases the sensitivity of the transit search for such stars
+and also provides photometric precision estimates (as by-
+products of the search) that are more realistic for such
+targets. If the transit-like signature of a given target star
+exceeds 7.1 then a TCE is recorded and sent to DV for
+additional scrutiny.
+6.DATA VALIDATION
+DV performs a suite of tests to establish or break con-
+dence in each TCE 
agged by TPS, as well as to t
+physical parameters to each transit-like signature. DV
+is currently under development and we anticipate its re-
+lease in early 2010 to support the next FOP observing
+season.
+The statistical condence in the TCE is examined by
+performing a bootstrap test (Jenkins 2002; Jenkins,
+Caldwell & Borucki 2002) to take into account non-
+Gaussian statistics of the individual light curves. A tran-
+siting planet model is tted to the transit signature as a
+joint noise characterization/parameter estimation prob-
+lem. That is, the observation noise is notassumed to
+be white and its characteristics are estimated using the
+wavelet-based approach employed in TPS, but as an es-
+timator, rather than as a detector. This process yields a4 Jenkins et al.
+set of physical parameters and an associated covariance
+matrix.
+To eliminate false positives due to eclipsing binaries,
+the planet model t is performed again only to the even
+transits, and then only to the odd transits, and the result-
+ing odd/even depths and epochs are compared in order
+to see if the results indicate the presence of secondary
+eclipses. After the multi-transiting planet search is com-
+plete, the periods are compared to detect eclipsing bina-
+ries with signicant eccentricity causing TPS to detect
+two transit pulse trains at essentially the same period,
+but at a phase other than 0.5.
+To guard against background eclipsing binaries, a cen-
+troid motion test is performed to determine whether the
+centroids shifted during the transit sequence. If so, the
+source right ascension and declination can be estimated
+by the measured in- versus out-of-transit centroid shift
+normalized by the fractional change in brightness of the
+system (i.e., the transit "depth"; Batalha et al. 2010b;
+Monet et al. 2010).
+Additional tests include checking whether the transit
+signature is consistent in the target pixels, whether the
+transit signature is correlated with any ancillary engi-
+neering data or any collateral data, and whether the dis-
+tribution of cosmic ray events during transit is signi-
+cantly dierent than that out-of-transit.
+7.FUTURE DEVELOPMENT
+Future development for the SOC includes implement-
+ing dierence image analysis photometry, completing
+DV, and developing and implementing mitigations for
+the instrument artifacts described in Caldwell et al.
+(2010). These artifacts aect a small portion of the Ke-
+pler FOV at any one time. The development schedule
+calls for delivery of these features to allow us to discover
+long period Earths by late 2010 and recover at least 90%
+of the FOV in 2011 in time to characterize the frequency
+of Earth-size transiting planets in the habitable zones of
+solar-like stars.
+The instrumental artifacts consist of two cate-
+gories of phenomena: (1) temperature-dependent
+two-dimensional bias image structure, and other
+temperature-dependent electronics eects, and (2) Moir e
+patterns caused by an unstable circuit with an oper-
+ational amplier oscillating at 1.5 GHz. Normally
+this latter feature appears as a high-frequency oscilla-
+tion on each readout row whose frequency changes with
+row number as the readout electronics heat up during
+readout. When this signal aliases to the sample rate of
+the CCD readout, a transient band appears in an af-
+fected channel and slowly rolls across the frame as the
+temperature changes. The Moir e pattern can interact
+with bright, saturated star signals and generate scene-
+dependent eects. The typical amplitude of these image
+artifacts is <1 ADU per pixel per read, comparable to
+or smaller than the typical readout noise. It is impor-
+tant to note that not all of these op amps are oscillating
+and that the perturbations to the images are very small.
+Our mitigation plan consists of two approaches, one for
+the temperature-dependent eects, and one for the Moir e
+pattern eects, and most of the eort takes place prior
+to pixel level calibrations.Before launch, we added pixels to the target table that
+allow us to sample and trend the artifacts simultane-
+ously with the science data. The Kepler Science Oce
+and SOC are developing and prototyping algorithms that
+use these image artifact pixels, together with other sci-
+ence data, to reconstruct the underlying temperature-
+dependent two-dimensional bias structure as a function
+of time over each quarter. The resulting dynamic model
+will allow the temperature-dependent bias signals to be
+removed directly from the data. Moreover, the ther-
+mal environment of the Kepler photometer is very sta-
+ble and changes slowly during nominal operations. Thus
+any residual thermal two-dimensional bias eects will be
+small after the corrections are in place and can be co-
+trended out of the data by PDC like other thermally
+driven, instrumental eects.
+Given that the Moir e pattern noise exhibits both high
+spatial frequencies and high temporal frequencies, the
+prospect of reconstructing a high delity model of the
+eects at the pixel level with an accuracy sucient to
+correct the aected data appears unlikely. We are de-
+veloping algorithms that identify when these Moir e pat-
+terns are present and mark the aected CCD regions as
+suspect on each aected LC. These suspect data 
ags
+will then be used to inform downstream modules so that
+the aected data can be appropriately weighted, and so
+that, for example, TPS can selectively ignore time inter-
+vals that are potentially contaminated with electronics-
+induced transients when searching for transit signatures.
+DV will produce a contamination report for each TCE
+indicating the fraction and severity of the Moir e pattern
+eect. The Pipeline will track and trend diagnostic met-
+rics re
ecting the prevalence and severity of the Moir e
+pattern as a diagnostic of this aspect of the photometer
+performance.
+In spite of the presence of these image artifacts, Ke-
+pleris already achieving photometric precision sucient
+to detect Earth-size planets transiting solar-like stars for
+the majority of the FOV at any given time (Jenkins et
+al. 2010). We are condent that these eorts will enable
+us to minimize the impact of these artifacts on exoplanet
+detection, and produce high-quality photometric and as-
+trometric time series for other scientic investigations by
+the greater astronomical community.
+8.CONCLUSIONS
+We have presented an overview of the Kepler SOC sci-
+ence pipeline processing. The output products include
+raw and calibrated pixel time series, raw and systematic
+error-corrected 
ux time series, centroid time series for
+each star, and associated uncertainties. These products
+will permit the detection and characterization of transit-
+ing planets in the Kepler FOV as well as enabling astro-
+physical investigations and serendipitous discoveries not
+contemplated in Kepler's driving design requirements.
+Funding for this Discovery Mission is provided by
+NASA's Science Mission Directorate. We thank thou-
+sands of people whose eorts made Kepler's grand voy-
+age of discovery possible.Kepler Science Pipeline 5
+REFERENCES
+Batalha, N. B., et al. 2010, ApJ, this issue
+Borucki, W. J., et al. 2010, Science, in press
+Bryson, S. T., et al. 2010, ApJ, this issue
+Caldwell, D. A., et al. 2010, ApJ, this issue
+Gautier, N. T., et al. 2010, ApJ, this issue
+Jenkins, J. M., D. A. Caldwell, & W. J. Borucki 2002, ApJ, 564,
+495
+Jenkins, J. M. 2002, ApJ, 575, 493
+Jenkins, J. M., et al. 2010, ApJ, this issueHaas, M., et al. 2010, ApJ, this issue
+Koch, D. G., et al. 2010, ApJ, this issue
+Monet, D, et al. 2010, ApJ, this issue
+Van Cleve, J., D. A. Caldwell, 2009, Kepler Instrument
+Handbook, KSCI 19033-001, (Moett Field, CA: NASA Ames
+Research Center)6 Jenkins et al.
+CAL
+Pixel Level
+CalibrationsRaw
+DataPA
+Photometric
+Analysis
+PDC
+Presearch
+Data
+ConditioningTPS
+Transiting
+Planet
+Search
+DV
+Data ValidationCalibrated
+Pixels
+Corrected
+Light
+Curves
+Threshold
+Crossing
+EventsDiagnosticsRaw Light
+Curves and
+Centroids
+Fig. 1.| Data 
ow diagram for the SOC Science Pipeline.Kepler Science Pipeline 7
+2-D Black
+Correction
+(static)Raw
+PixelsDynamic 1-D
+Black 
+Correction
+Gain & 
+Nonlinearity
+CorrectionLDE
+Undershoot
+CorrectionIdentify 
+Cosmic Rays 
+in Smear
+Smear + Dark 
+Current 
+CorrectionFlat Field 
+CorrectionCalibrated
+Pixels +
+Uncertainties
+Fig. 2.| Data 
ow diagram for the Calibration Pipeline module.8 Jenkins et al.
+11.021.041.061.081.11.121.14
+A
+5000 5010 5020 5030 5040 5050 5060 50700.991   1.011.021.031.041.051.06B
+Time, JD − 2450000Relative Intensity
+Pixel Sensitivity DropPointing
+Tweak
+Safe ModePointing
+Tweak
+Pixel Sensitivity Drop
+Fig. 3.| Raw and systematic error-corrected light curves for two dierent stars. The raw light curves exhibit step discontinuities due
+to pointing osets, thermal transients due to safe modes and pixel sensitivity changes, as well as pointing errors and focus changes, as
+indicated in the gure. Panel A shows the raw 
ux time series (top curve, oset) and the PDC 
ux time series (bottom curve) of a
+Kp= 15:6 dwarf star. Panel B shows the raw and PDC 
ux time series for a Kp= 14:9 dwarf star that displays less sensitivity to the
+thermal transients. Both stars' corrected 
ux time series are signicantly improved by the detrending aorded by PDC while retaining
+intrinsic stellar variability on timescales of several weeks.
\ No newline at end of file
diff --git a/1001.0259.txt b/1001.0259.txt
new file mode 100644
index 0000000000000000000000000000000000000000..b402c78e6c0d48bb57eea28a844e881494ea7cb5
--- /dev/null
+++ b/1001.0259.txt
@@ -0,0 +1,1554 @@
+arXiv:1001.0259v2  [astro-ph.CO]  13 Mar 2010Cosmic perturbations with running GandΛ
+Javier Grande, Joan Sol` a
+HEP Group, Dept. ECM and Institut de Ci` encies del Cosmos
+Univ. de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Ca talonia, Spain
+Julio C. Fabris
+Departamento de F´ ısica – CCE,
+Univ. Federal do Espir´ ıto Santo, CEP 29060-900, Vit´ oria, ES, Brazil
+Ilya L. Shapiro
+Departamento de F´ ısica - ICC
+Univ. Federal de Juiz de Fora, CEP 36036-330, MG, Brazil
+E-mail: jgrande@ecm.ub.es, sola@ecm.ub.es, fabris@pq.c npq.br, shapiro@fisica.ufjf.br
+Abstract. Cosmologies with running cosmological term ρΛand gravitational Newton’s
+couplingGmay naturally be expected if the evolution of the universe ca n ultimately
+be derived from the first principles of Quantum Field Theory o r String Theory. For
+example, if matter is conserved and the vacuum energy densit y varies quadratically
+with the expansion rate as ρΛ(H) =n0+n2H2, withn0∝negationslash= 0 (a possibility that has
+been advocated in the literature within the QFT framework), it can be shown that
+G must vary logarithmically (hence very slowly) with H. In this paper, we derive the
+general cosmological perturbationequations formodelswi thvariableGandρΛinwhich
+the fluctuations δGandδρΛare explicitly included. We demonstrate that, if matter is
+covariantly conserved, the late growth of matter density pe rturbations is independent
+of the wavenumber k. Furthermore, if ρΛis negligible at high redshifts and Gvaries
+slowly, we find that these cosmologies produce a matter power spectrum with the same
+shape as that of the ΛCDM model, thus predicting the same basi c features on structure
+formation. Despite this shape indistinguishability, the f ree parameters of the variable
+GandρΛmodels can still be effectively constrained from the observat ional bounds on
+the spectrum amplitude.
+PACS: 95.36.+x 04.62.+v 11.10.Hi1Introduction
+The current “standard model” (or “concordance model”) of ou r universe, being an homogeneous
+and isotropic FLRW cosmological model, consists of a remark ably small number of ingredients, to
+wit: matter, radiation and a cosmological constant (CC) ter m, Λ. The first two ingredients are
+dynamical and vary (decrease fast) with the cosmic time twhereas the third, Λ, remains strictly
+constant. After many years of theoretical insight (cf. e.g. the reviews [1,2] and references therein),
+the situation of the original FLRW cosmological models has n ot changed much, in the sense that
+we have not been able to make any fundamental advance in the co mprehension of the relationship
+between the matter energy density and the CC. Still, we have p erformed a major accomplishment
+at thephenomenological level bysimultaneously fittingthe modernindependentdatasets emerging
+from LSS galaxy surveys, supernova luminosities and the cos mic microwave background (CMB)
+anisotropies [3–7]. On the basis of this successful fit, in wh ich Λ enters as a free parameter, one
+claims that a non-vanishing and positive cosmological cons tant has been measured. Nevertheless,
+we still don’t know what is the true meaning of the fitted param eter Λ. Assuming that Newton’s
+gravitational coupling Gis strictly constant, the combined set of observational dat a determine the
+value
+ρΛ=Λ
+8πG≃(2.3×10−3eV)4, (1)
+which we interpret as the vacuum energy density. It correspo nds to Ω0
+Λ=ρΛ/ρ0
+c≃0.7 when the
+ρΛdensity is normalized with respect to the current critical d ensityρ0
+c= (3√
+h×10−3eV)4– for
+a reduced Hubble constant value of h≃0.70. Assuming (in the light of the same observational
+data) that the universe is spatially flat, this means that the current matter density normalized to
+the critical density is Ω0
+m=ρ0
+m/ρ0
+c≃0.3.
+All the efforts to deduce the value of the energy density (1) – a v ery small one for all particle
+physics standards, except if a very light neutrino mass is th e only particle involved [8] – have
+failed up to now. The main stumbling block to a solution is the fact that any approach based on
+the fundamental principles of quantum field theory (QFT) or s tring theory lead to some explicit
+or implicit form of severe fine-tuning among the parameters o f the theory. The reason for this is
+that these theoretical descriptions are plagued by large hi erarchies of energy scales associated to
+the existence of many possible vacua.
+This difficulty became clear already from the firstattempts to treat the dark energy component
+as a dynamical scalar field [9]. The idea was to let such a field s elect automatically the vacuum
+state in a dynamical way, especially one with zero value of th e energy density. More recently, this
+approach took the popular form of a “quintessence” field slow ly rolling down its potential and has
+adopted many different faces [10] – for a review, see [2]. But th e situation at present is even harder
+because the quintessence field – or, more generally, a dark en ergy (DE) field – should be able to
+naturally choose not zero but the very small number (1) as its ground state.
+Despite the fact that a working dynamical mechanism able to c hoose the correct vacuum state
+has to be found yet, another important motivation for the qui ntessence models is that they aim at
+explaining the puzzling coincidence between the present va lue ofρΛand the value of the matter
+density,ρ0
+m. In other words, why Ω0
+Λ/Ω0
+m=O(1)? Arguably, a dynamical DE should be a starting
+2point to understand this puzzle. Detailed analyses of these models exist in the literature, including
+their confrontation with the data [11–14].
+Ontheother hand, thepossibility that thecosmological ter m is arunningquantity, which could
+be sensitive to the quantum matter effects, seems a more appeal ing ansatz, as it could provide
+an interface between QFT and cosmology [8,15]. This fundame ntal possibility has been recently
+emphasized in [16]– for a review, see [17]. Actually, the ana lysis of the various observational data
+show (see Ref. [18,19]) that a wide class of dynamical CC mode ls models are indeed able to fit
+the combined observations to a level of accuracy comparable to the standard ΛCDM model. In
+some cases, the dynamical nature of Λ allows these models to p rovide a clue to the coincidence
+problem [20,21], and maybe eventually to the full cosmologi cal constant problem [22].
+In general, in this kind of dynamical CC scenarios we have Λ = Λ (ξ) whereξ=ξ(t) is
+a cosmological variable that evolves with time, and therefo re ultimately Λ = Λ( t) is also time
+varying. Originally, these models were purely phenomenolo gical, with no relation to QFT [23].
+Nowadays we know that not all models of this kind are allowed, and the fact that the observational
+data can discriminate which of them are good candidates and w hich are not so good gives some
+insight into the function ξ=ξ(t) [18]. A particularly interesting class of variable CC mode ls
+are those in which the gravitational coupling Gchanges very slowly (logarithmically) with the
+expansion of the universe [24,25]. This scenario is possibl e e.g. if matter is covariantly conserved.
+In practice, ξcould be the expansion rate H=H(t), the scale factor a=a(t), the matter
+energy density, etc. In this paper, we shall assume that ξ=H, similarly as it was done for models
+with a variable Λ interacting with matter [15,26,27]. With t his ansatz, we shall study the impact
+on the structure formation, showing that these models predi ct the same shape for the matter
+power spectrum as the ΛCDM model, a fact that would not apply e .g. for variable CC models in
+which the CC decays into matter. We find this feature remarkab le and we shall explore its possible
+consequences and also some possible phenomenological test s of this kind of models.
+The paper is organized as follows. In the next section, we cla ssify the possible scenarios with
+variableρΛandG. In section 3, we present the general set of coupled perturba tion equations
+involvingδρΛandδG, showing that the late growth of matter fluctuations becomes independent
+of the scale k. A class of models with variable cosmological term as a serie s function of the Hubble
+rate is introduced in section 4. In section 5, we concentrate on a particular model in this class,
+which is well motivated from the QFT point of view, and analyz e the constraints imposed by
+primordial nucleosynthesis and structure formation. In th e last section, we present and discuss
+our conclusions. Finally, an appendix is included at the end where we discuss gauge issues.
+2 Generic models with variable cosmological parameters
+We start from Einstein’s equations in the presence of the cos mological constant term,
+Rµν−1
+2gµνR= 8πG(Tµν+gµνρΛ), (2)
+whereTµνis the ordinary energy-momentum tensor associated to isotr opic matter and radiation,
+andρΛrepresents the vacuum energy density associated to the CC. L et us now contemplate the
+3possibility that G=G(t) and Λ = Λ( t) can beboth functions of the cosmic time within the context
+of the FLRW cosmology. It should be clear that the very precis e measurements of Gexisting in the
+literature refer only to distances within the solar system a nd astrophysical systems. In cosmology
+these scales are immersed into much larger scales (galaxies and clusters of galaxies) which are
+treated as point-like (and referred to as “fundamental obse rvers”, comoving with the cosmic fluid).
+Therefore, the variations of Gat the cosmological level could only be observed at much larg er
+distances where at the moment we have never had the possibili ty to make direct experiments. In
+practice, the potential variation of G=G(t) and Λ = Λ( t) should be expressed in terms of a
+possible cosmological redshift dependence of these functi ons,G=G(z) and Λ = Λ( z).
+Let us consider the various possible scenarios for variable cosmological parameters that appear
+when we solve Einstein’s equations (2) in the flat FLRW metric ,ds2=dt2−a2(t)dx2. To start
+with, one finds Friedmann’s equation with non-vanishing ρΛ, which provides Hubble’s expansion
+rateH= ˙a/a(˙a≡da/dt) as a function of the matter and vacuum energy densities:
+H2=8πG
+3(ρm+ρΛ). (3)
+On the other hand, the general Bianchi identity of the Einste in tensor in (2) leads to
+▽µ[G(Tµν+gµνρΛ)] = 0. (4)
+Using the FLRW metric explicitly, the last equation results into the following “mixed” local con-
+servation law:
+d
+dt[G(ρm+ρΛ)]+3GH(ρm+pm) = 0. (5)
+If ˙ρΛ∝negationslash= 0, thenρmis not generally conserved as there may be transfer of energy from matter-
+radiation into the variable ρΛor vice versa (including a possible contribution from a vari ableG,
+if˙G∝negationslash= 0). Thus this law mixes, in general, the matter-radiation e nergy density with the vacuum
+energyρΛ. However, the following particular scenarios are possible :
+•i)G=const. and ρΛ=const. This is the standard case of ΛCDM cosmology, implyin g the
+local covariant conservation law of matter-radiation:
+˙ρm+3H(ρm+pm) = 0; (6)
+•ii)G=const and ˙ ρΛ∝negationslash= 0, in which case Eq.(5) leads to the mixed conservation law
+˙ρΛ+ ˙ρm+3H(ρm+pm) = 0; (7)
+•iii)˙G∝negationslash= 0 andρΛ=const, implying ˙G(ρm+ρΛ)+G[˙ρm+3H(ρm+pm)] = 0;
+•iv)˙G∝negationslash= 0 and ˙ρΛ∝negationslash= 0. In this case, if we assume the standard local covariant co nservation
+of matter-radiation, i.e Eq.(6), it is easy to see that Eq.(5 ) boils down to
+(ρm+ρΛ)˙G+G˙ρΛ= 0. (8)
+4Notice that only in cases i) and iv) matter is covariantly sel f-conserved, meaning that matter
+evolves according to the solution of Eq.(6):
+ρm(a) =ρ0
+ma−αm=ρ0
+m(1+z)αm, αm= 3(1+ωm), (9)
+whereρ0
+mis the current matter density and ωm=pm/ρm= 0,1/3 are the equation of state (EOS)
+parameters for cold and relativistic matter, respectively . We have expressed the result (9) in terms
+of the scale factor a=a(t) and the cosmological redshift z= (1−a)/a.
+In cases ii) and iii), instead, matter is not conserved (if on e of the two parameters ρΛorG
+indeed is to be variable, respectively). Explicit cosmolog ical models with variable parameters as
+in case ii) have been investigated in detail in [15,19,27]. C osmic perturbations of this model have
+been considered in [28–32]. Case iii) has been studied at the background level in [33]. Finally, the
+background evolution of case iv) has been studied in different contexts in Ref. [24,25]1. In the
+next section, we shall address the calculation of cosmic per turbations in a general model where
+ρΛandGcan evolve with the expansion, and we will then specialize th e set of equations for the
+type iv) model in which matter is covariantly conserved. Onl y in section 4 we will further narrow
+down the obtained set of perturbation equations for a concre te model with running cosmological
+parameters [24].
+3 Perturbations with variable ΛandG
+For the analysis of the cosmic perturbations in the general r unningρΛandGmodel iv) of
+the previous section, we have to perturb all parts of Einstei n’s equations that may evolve with
+time; namely the metric, the energy-momentum tensor for bot h matter and vacuum, and finally
+we must perturb also the gravitational constant. Einstein’ s equations (2) can be conveniently cast
+as follows:
+Rµν= 8πG/parenleftbigg
+Tµν−1
+2gµνTλ
+λ/parenrightbigg
+. (10)
+As a background metric, we use the flat FLRW metric:
+ds2=gµνdxµdxν=dt2−a2(t)δijdxidxj. (11)
+In perturbing it, gµν→gµν+hµν, we adopt here the synchronous gauge2(h00=h0i= 0). The
+total energy momentum tensor of the cosmological fluid can be written as the sum of the matter
+and vacuum contributions:
+Tµ
+ν=Tµ
+ν(matter)+Tµ
+ν(Λ)=−pTδµ
+ν+(ρT+pT)UµUν, (12)
+where
+ρT=ρm+ρΛ
+pT=pm+pΛ=ωmρm−ρΛ, (13)
+1Some potential astrophysical implications of scenario iv) have been addressed first in [24] and recently in [34].
+2See the Appendix for an alternative calculation in the Newtonian gauge.
+5(notice that ωΛ≡pΛ/ρΛ=−1 even if the CC is running). As for the density and pressure, w e
+introduce the perturbations in the usual form, except that w e have to include also the perturbation
+for the vacuum energy density because, in the present contex t, Λ is an evolving variable. Thus,
+we haveρi→ρi+δρi, pi→pi+δpi, wherei=m,Λ. As warned above, if we allow for a variable
+gravitational coupling G, we must also consider perturbations for it:
+G→G+δG. (14)
+Obviously, the perturbations δGandδρΛare the two most distinguished dynamical components
+of the present cosmic perturbation analysis, as they are bot h absent in the ΛCDM model.
+Finally, we have to perturb the 4-velocity of the matter part icles,Uµ→Uµ+δUµ. For an
+observer moving with the fluid (i.e. a comoving observer) it r eads:Uµ= (1,δUi).
+We are now ready to find the perturbed equations of motion for t his running model. As
+the fundamental equations describing the perturbations fo r our model, we will take: i) the 00
+component of the Einstein equations (10); and ii) the genera lized Bianchi identity (4), which splits
+into energy conservation (notice that ˙G∝negationslash= 0 in the present framework)
+∇µ(GTµ
+0) =∂µ(GTµ
+0)+G/bracketleftbig
+Γµ
+σµTσ
+0−Γσ
+µ0Tµ
+σ/bracketrightbig
+= 0, (15)
+and momentum conservation:
+∇µ(GTµ
+i) =∂µ(GTµ
+i)+G/bracketleftbig
+Γµ
+σµTσ
+i−Γσ
+µiTµ
+σ/bracketrightbig
+= 0. (16)
+As for the 00 component of the Einstein equations, and taking into account that Ggets perturbed
+as in (14), we have:
+0th order: −3¨a
+a= 4πG(ρT+3pT),
+(17)
+1st order:˙ˆh+2Hˆh= 8π[G(δρT+3δpT)+δG(ρT+3pT)],
+wherewehavedefined ˆh=∂
+∂t/parenleftbigghkk
+a2/parenrightbigg
+andrepeatedLatinindicesareunderstoodtobesummedover
+the values 1, 2, 3. As energy-density components, we have mat ter and the running cosmological
+constant, with EOS parameters indicated in (13), and thus th e perturbed equation takes on the
+form:
+˙ˆh+2Hˆh= 8π/braceleftBig
+G/bracketleftbig
+(1+3ωm)δρm+δρΛ+3δpΛ/bracketrightbig
++δG/bracketleftbig
+(1+3ωm)ρm−2ρΛ/bracketrightbig/bracerightBig
+.(18)
+Let us now work out the equation of local covariant conservat ion of the energy, Eq.(15). After a
+straightforward calculation, the final equations read as fo llows. On the one hand,
+0th order: ˙G(ρm+ρΛ)+G(˙ρm+ ˙ρΛ)+3GHρm(1+ωm) = 0. (19)
+If we assume local covariant conservation of matter, ∇µTµ
+0 (matter)= 0, i.e. Eq.(6), we see that
+(19) indeed reduces to Eq.(8), as it should be. On the other ha nd,
+1st order: G/bracketleftBigg
+δ˙ρm+δ˙ρΛ+ρm(1+ωm)/parenleftBigg
+θ−ˆh
+2/parenrightBigg
++3H/bracketleftbig
+(1+ωm)δρm+(δρΛ+δpΛ)/bracketrightbig/bracketrightBigg
++
++˙G(δρm+δρΛ)+/bracketleftbig
+˙ρm+ ˙ρΛ+3H(1+ωm)ρm/bracketrightbig
+δG+(ρm+ρΛ)δ˙G= 0.(20)
+6In the previous equation, we have introduced the variable
+θ≡ ∇µδUµ=∂iδUi, (21)
+which represents the perturbation on the matter velocity gr adient, and we have used δU0= 0. If
+we invoke once more the local covariant conservation of matt er, both at 0th order – see Eq.(6) –
+and at 1st order,
+δ˙ρm+ρm(1+ωm)/parenleftBigg
+θ−ˆh
+2/parenrightBigg
++3H(1+ωm)δρm= 0, (22)
+equation (20) can be further reduced to the simpler form:
+G/bracketleftbig
+δ˙ρΛ+3H/parenleftbig
+δρΛ+δpΛ/parenrightbig/bracketrightbig
++˙G(δρm+δρΛ)+ ˙ρΛδG+(ρm+ρΛ)δ˙G= 0. (23)
+Finally, let us work out the equation of local covariant cons ervation of momentum, Eq.(16). In this
+instance, there are no 0th order terms (in the background, th e momentum conservation is trivial).
+The perturbed result reads
+a2∂t[Gρ(1+ω)δUi]+5a˙aGρ(1+ω)δUi+G∂i(δp)+p∂i(δG) = 0, (24)
+where it is understood that we should sum over all the energy c omponents, in this case, matter
+and Λ. By applying the operator ∂iand transforming to the Fourier space, we get:
+(1+ω)/bracketleftBig
+˙Gρθ+G(˙ρθ+ρ˙θ+5Hρθ)/bracketrightBig
+−k2
+a2[Gδp+ωρδG] = 0, (25)
+which, after summing over matter and CC, becomes:
+(1+ωm)/bracketleftBig
+˙Gρmθ+G(˙ρmθ+ρm˙θ+5Hρmθ)/bracketrightBig
+=k2
+a2[G(δpΛ+ωmδρm)+(ωmρm−ρΛ)δG].(26)
+The part corresponding to matter conservation/parenleftBig
+∇µTµ
+i(matter)= 0/parenrightBig
+is
+(1+ωm)/bracketleftBig
+˙ρmθ+ρm˙θ+5Hρmθ/bracketrightBig
+=k2
+a2ωmδρm, (27)
+so that Eq.(26) simplifies as follows:
+(1+ωm)˙Gρmθ=k2
+a2[GδpΛ+(ωmρm−ρΛ)δG]. (28)
+Summarizing, our final set of equations is given by (18),(22) ,(23),(27) and (28). It is particularly
+relevant for our purposes to rewrite these equations in the m atter dominated epoch ( ωm= 0) and
+assuming adiabatic perturbations for the CC ( δpΛ=−δρΛ). In a nutshell, we find:
+˙ˆh+2Hˆh= 8π/bracketleftbig
+ρm−2ρΛ/bracketrightbig
+δG+8πG/bracketleftbig
+δρm−2δρΛ/bracketrightbig
+(29)
+δ˙ρm+ρm/parenleftBigg
+θ−ˆh
+2/parenrightBigg
++3Hδρm= 0 (30)
+˙θ+2Hθ= 0 (31)
+δ˙G(ρm+ρΛ)+δG˙ρΛ+˙G(δρm+δρΛ)+Gδ˙ρΛ= 0 (32)
+k2[GδρΛ+ρΛδG]+a2ρm˙Gθ= 0. (33)
+7Since Eq.(31) will be important for the subsequent consider ations, let us note that it follows from
+Eq.(27) upon using the matter conservation equation (6).
+We are now ready to derive an important result that applies to all cosmological models of the
+type iv) in section 2, i.e. models with variable Λ and Gin which matter is covariantly conserved, to
+wit: forallthesemodels, theperturbationequationsthatw ehavejustderiveddo notdependinfact
+on the wavenumber k. To prove this remarkable result is very easy at this stage of the discussion,
+as it ensues immediately from equations(31) and (33). Indee d, if we change the variable from
+cosmic time tto the scale factor a(which is readily done by using ˙f≡df/dt=aHdf/da ≡aHf′),
+equation (31) can be cast as
+θ′+2
+aθ= 0. (34)
+Therefore,
+θ=θ0a−2. (35)
+It follows that the second term on the l.h.s.of Eq.(33) behaves as a2ρm˙Gθ=θ0ρm˙G. Notice
+thatθ0is the perturbation of the matter velocity at present, which is of course much smaller than
+any valueθithat this variable can take early on, more specifically at any time after the transfer
+function regime has finished (see below). Obviously, θi=θ0a−2
+i≪θ0(whereai≪a0= 1). Thus,
+taking into account that the matter perturbations δρmare growing rather than decaying, θwill
+be comparatively negligible. Let us also note that the θ-term in Eq.(33) is multiplied by the time
+derivative of G, which in all reasonable models (in particular, the one we ex plore in section 5)
+should be small. Finally, if we divide Eq.(33) by k2and take into account that in practice we
+are interested in deep sub-horizon scales , i.e. scales λthat satisfy λa≪H−1(or, equivalently,
+k≫Ha), it follows that the the θ-term in Eq.(33) is entirely negligible. Thus, in all practi cal
+respects, this is tantamount to set θ(a) = 0 (∀a). The outcome is that the full set of perturbation
+equations becomes independent of k, as announced3. To better assess the physical significance
+of this important result, we have repeated the calculation i n another gauge (the Newtonian or
+longitudinal gauge) and we have obtained the same result for scales well below the horizon (see
+the Appendix at the very end for a summarized presentation an d discussion).
+Having shown that the shape of the spectrum is not distorted a t late times in our model,
+we may ask ourselves if, in contrast, there are additional so urces of wavenumber dependence at
+early times. Recall that the transfer function T(k) parameterizes the important dynamical effects
+that the various k-modes undergo at the early epochs [35]. More specifically, i t accounts for
+the non-trivial evolution of the primordial perturbations through the epochs of horizon crossing
+and radiation/matter transition (the latter occurs at the s o-called “equality time”, teq). A most
+important feature in the ΛCDM case is that the scale dependen ce is fully encoded in T(k), and
+so the spectrum of the standard model evolves without any fur ther distortion during most of the
+matter epoch till the present time.
+We may ask ourselves if the evolution of our model with runnin g Λ andGat the early epochs
+follows also the same pattern as the ΛCDM, or if its dynamics c ould imprint some significant
+modification on the structure of T(k). In such a case, a distortion of the power spectrum with
+3This result is similar to the model of Ref. [21], where matter is also covariantly conserved.
+8respect to the ΛCDM would begenerated at early times and it wo uld freely propagate (i.e. without
+furthermodifications)till thepresent daysandact asakind of signatureof themodel. However, we
+deem that this possibility is not likely to be the case, for in the kind of models under consideration
+there is no production/decay of matter or radiation. Thus, t he value of the scale factor at equality,
+aeq=a(teq), must be identical to that of the standard ΛCDM model (cf. mo re details in section
+5.1). On the other hand, neither the value of ρΛnor of the perturbation in this variable are
+expected to play an important role in the past. Finally, for r easons that will become apparent
+later, it is important to restrict our discussion to models w herein the time variation of Gis very
+small, as this may also affect the previous consideration on th e transfer function (and on top of
+this there are important bounds from primordial nucleosynt hesis to be satisfied, see section 5.2).
+After insuring that these conditions are fulfilled by the adm itted class of variable Gmodels,
+we trust that the k-dependence of the transfer function for these models shoul d be the same as
+that for the ΛCDM model. Combining this fact with the above pr oven scale independence of the
+late time evolution of the coupled set of perturbations for δρΛandδG, we reasonably infer that
+the matter power spectrum of a model with variable ρΛand slowly variable G– and with self-
+conserved matter components – must generally have the very s ame spectral shape as that of the
+ΛCDM model. Fortunately, in spite of their shape invariance , such models can still be constrained
+and distinguished from the standard cosmological model by m eans of the spectrum amplitude (i.e.
+from the normalization of the matter fluctuation power spect rum). We shall further elaborate on
+these points in section 5, where a concrete model exhibiting these properties will be analytically
+and numerically analyzed.
+Let us now retake our analysis of cosmic perturbations by sho wing that the perturbations in
+Gare actually tightly linked to the perturbations in ρΛ. This feature is another consequence of
+the aforementioned scale invariance of the late time evolut ion of the perturbations. Indeed, from
+(33) and the neglect of the velocity perturbations (35), we o btain:
+δΛ≡δρΛ
+ρΛ=−δG
+G. (36)
+It follows that the two kind of perturbations are not ultimat ely independent. Furthermore, using
+Eq.(36), a straightforward calculation from (32) renders t he simple result:
+δm≡δρm
+ρm=−˙δG
+˙G=−(δG)′
+G′. (37)
+The last two equations show clearly that the perturbations i nGandρΛmay not be consistently
+set to zero, since they will be generated even if they are assu med to vanish at some initial instant
+of time.4From (30), and using the matter conservation (6), it is easy t o see that
+ˆh= 2˙δm. (38)
+We can now use this last expression and Eq.(29) to produce a se cond order differential equation
+4This is analogous to what happens in models with self-conser ved DE, where DE perturbations cannot be con-
+sistently neglected [36]
+9forδm. Using again differentiation with respect to the scale factor , we find
+δ′′
+m+A(a)δ′
+m=B(a)/parenleftbigg
+δm+δG
+G/parenrightbigg
+, (39)
+where we have defined:
+A=3
+a+H′(a)
+H(a)(40)
+B=3
+2a2˜Ωm(a) (41)
+˜Ωm(a) =ρm(a)
+ρc(a)=8πG(a)
+3H2(a)ρm(a). (42)
+Here,˜Ωm(a) is the “instantaneous” normalized matter density at the ti me where the scale factor
+isa=a(t). It is also convenient to define the corresponding instanta neous normalized CC density
+˜ΩΛ(a) =ρΛ(a)/ρc(a), and it is easy to check that the following sum rule
+˜Ωm(z)+˜ΩΛ(z) = 1 (43)
+is satisfied for all z. Clearly, the sum rule is a simple but convenient rephrasing of Eq.(3).
+If we would neglect the perturbations in G(δG∼0) and hence δρΛ∼0 too, owing to Eq.(36),
+it is immediate to see that Eq.(39) would reduce to the standa rd differential equation [35] for the
+growth factor for DE models with self-conserved matter unde r the assumption of negligible DE
+perturbations, i.e. explicitly Eq.(90) of Ref. [21]. In par ticular, if we take for Hthe standard
+form (3) with ρΛ=const., we arrive to the equation for the ΛCDM model. Howeve r, Eq.(39) with
+δG∝negationslash= 0 tells us precisely how to take into account the effect of non- vanishing perturbations in the
+gravitational constant G. We could now solve the coupled system formed by (39) and (37) , giving
+initial conditions for δ′
+m,δmandδG, or use those two equations to get a third order differential
+equationfor δm, inwhichcasewewouldneedtogivetheinitial condition for δ′′
+minstead. Thisseems
+to be the natural thing to do, because it involves boundary co nditions on δmand its derivatives
+only, and does not mix with the boundary conditions on other v ariables.
+In order to get the aforesaid third order differential equatio n forδm, we have to differentiate
+Eq.(39) and use this same equation to eliminate δG/G. Using also (37), we obtain
+/parenleftbiggδG
+G/parenrightbigg′
+=δG′
+G−G′
+GδG
+G=−G′
+Gδm−G′
+G/parenleftbiggδ′′
+m+Aδ′
+m
+B−δm/parenrightbigg
+=−G′
+GB/parenleftbig
+δ′′
+m+Aδ′
+m/parenrightbig
+,(44)
+and hence we finally arrive at the desired third order equatio n:
+δ′′′
+m+/parenleftbigg
+A−B′
+B+G′
+G/parenrightbigg
+δ′′
+m+/bracketleftbigg
+A′−B+A/parenleftbiggG′
+G−B′
+B/parenrightbigg/bracketrightbigg
+δ′
+m= 0. (45)
+Now we have to compute the coefficients of this equation. From t he definition of ˜Ωm, Eq. (42), we
+get the useful relation:
+˜Ω′
+m
+˜Ωm=G′
+G−2H′
+H−3
+a. (46)
+10On the other hand, differentiating Friedmann’s equation (3),
+2HH′=8π
+3[G′(ρm+ρΛ)+G(ρ′
+m+ρ′
+Λ) =8πG
+3ρ′
+m=−8πG
+aρm, (47)
+where in the last two steps we have taken into account the ener gy-conservation law (8) and (9) for
+αm= 3 (matter epoch). Combining this last equation with the defi nition of ˜Ωm, (42), we get:
+H′
+H=−3
+2a˜Ωm. (48)
+Finally, using (46) and (48) we can rewrite our third order di fferential equation (45) in terms of
+˜Ωmas:
+δ′′′
+m+1
+2/parenleftBig
+16−9˜Ωm/parenrightBigδ′′
+m
+a+3
+2/parenleftBig
+8−11˜Ωm+3˜Ω2
+m−a˜Ω′
+m/parenrightBigδ′
+m
+a2= 0. (49)
+Let us stress that this equation is valid for anymodel with variable G and ρΛin which matter
+is covariantly conserved, i.e. models of the type iv) in sect ion 2. However, a very particular case
+where it should give the correct results is for the ΛCDM and CD M models because matter is
+conserved and the equation (8) is trivially satisfied. For th e CDM model, for instance, ˜Ωm= 1
+and the equation reduces to:
+δ′′′
+m+7
+2δ′′
+m
+a= 0, (50)
+which has the general solution:
+δm=C1a+C2+C3a−3/2. (51)
+Barringtheconstant anddecayingmodes,whichwecanneglec t, therelevant solutionisthegrowing
+modeδm∝a, which is linear in the scale factor, as expected. Note that t he setting ˜Ωm= 1 implies
+that there is no CC term, and from Eq.(8) we infer that Gmust be strictly constant in such case.
+Finally, Eq.(36) entails that δGis then also constant, and this constant can be set to zero thr ough
+a redefinition of G. Let us also remark that, in this limit, one does not expect to recover the
+perturbative analysis of scenario ii) in section 2, i.e. the results obtained in [28]. The reason
+is that although in the last reference there are no perturbat ions onG, matter and vacuum are
+exchanging energy. Therefore the underlying physics is com pletely different and there is in general
+no simple connection between these two kind of models.
+We can use the general equation (49) to study the perturbatio ns in any model with variable G
+andρΛin which matter is conserved. In the next section, we conside r a well motivated non-trivial
+example.
+4 The class of vacuum power series models in H
+As we have mentioned in the introduction, there have been man y attempts to envisage a dynamical
+cosmological term ρΛ(t) [23]. However, in most cases the approach is purely phenome nological,
+with no reference to a potential connection with fundamenta l physics, via QFT or string theory.
+On the other hand, there are models wherein there is such a pos sible connection. Obviously, this
+11represents an additional motivation for their study. Consi der the class of models in which the CC
+evolves as a power series in the Hubble rate:
+ρΛ(H) =n0+n1H+n2H2+..., (52)
+whereni(i= 0,1,2,...) are dimensional coefficients (except n4, which is dimensionless). The
+background solution for this cosmological model up to i= 2 can be found in [18]. The higher
+order terms in (52), made out of powers of Hlarger than 2, are phenomenologically irrelevant at
+the present time and will not be discussed. Besides, let us st ress that from the fundamental point
+of view of QFT in curved space-time, the general covariance o f the effective action can only admit
+even powers of the expansion rate [15,19,37]. So the coefficie ntn1of the linear term in Hcannot
+be related to the properties of the effective action, but to som e phenomenological parameter of the
+theory (e.g. the viscosity of the DE fluid) which is not part of the fundamental principles. If we
+dispense with this kind of phenomenological coefficients, we obtain the subclass of models of the
+form
+ρΛ(H) =n0+n2H2, (53)
+which have been advocated as scenarios in which the CC evolut ion can be linked to the renormal-
+ization group (RG) running in QFT [15,16]. Notice that the co efficientn2has dimension of an
+effective mass squared, n2=M2
+eff. We shall further comment on it below.
+The form (53) is indeed crucially different from just consider ing that the vacuum energy is
+proportional to H2, in the sense that Eq.(53) is an “affine quadratic law” (i.e. n0∝negationslash= 0). While
+the pureH2law is not favored by the experimental data [18], the affine ver sion has been recently
+tested in a framework where Gis constant, specifically in the framework of the scenario ii ) in
+section 2, and it was found that it can provide a fit to the curre nt observational data of similar
+quality as the ΛCDM one – see [18] for details.
+Consider now the size of the n2coefficient in Eq.(53). In its absence, the the CC is strictly
+constant and n0just coincides with the current value ρ0
+Λ. However, in the presence of the H2
+correction, the boundary condition at H=H0becomesρ0
+Λ=n0+n2H2
+0. If the additional term
+is to play a significant role it should not be negligible as com pared ton0, and as the latter is the
+leading term it must still be of order ∼ρ0
+Λ. It follows that the effective mass Meffassociated to the
+coefficientn2cannot be small, but actually very large – specifically, of or der of a Grand Unified
+Theory scale (see below). This also explains why no other eve n power ofHcan play any significant
+role in the series expansion (52) at any stage of the cosmolog ical history below a typical GUT scale.
+The upshot is that, in practice, the evolution law (53) is the leading form throughout all relevant
+cosmic epochs (radiation dominated, matter dominated and l ate CC dominated epochs).
+Likewise, the possibility that the vacuum energy could be ev olving linearly with H– i.e. as if
+n0=n2= 0 in Eq.(52) – has also been addressed in the literature and c an be motivated through
+a possible connection of cosmology with the QCD scale of stro ng interactions [38–40]. However,
+as we have said, this option is not what one would expect from t he the general covariance of the
+effective action. Actually, the confrontation of the purely l inear model Hwith the data does not
+seem to support it [18,41], and therefore the linear term alo ne is unfavored. However, it could
+perhaps enter as a phenomenological term in a general power s eries vacuum model of the form
+12(52) – a possibility which is currently under study [42]. For some alternative recent models with
+variable cosmological parameters, see e.g. [43].
+Inthefollowing, wefocusonthequantumfieldvacuummodelof theform(53). Itisparticularly
+convenient to rewrite the coefficients of this equation as fol lows:n0=ρ0
+Λ−3νM2
+PH2
+0/(8π) and
+n2= 3νM2
+P/(8π). Therefore, the CC evolution law reads
+ρΛ(H) =ρ0
+Λ+3ν
+8πM2
+P(H2−H2
+0). (54)
+Hereνis a small coefficient ( |ν| ≪1) andMPis the Planck mass; it defines the current value of
+Newton’s constant: G0= 1/M2
+P. Clearly, the vacuum energy density (54) is normalized to th e
+present value, i.e.
+ρΛ(H0) =ρ0
+Λ≡3
+8πΩ0
+ΛH2
+0M2
+P, (55)
+where experimentally Ω0
+Λ≃0.7. The above parametrization satisfies the aforementioned c ondition
+thatn0is the leading term and is of order ρ0
+Λ, and at the same time the correction term is of order
+M2
+effH2, withMeff∼√νMPa large mass, even if νis as small as, say, |ν| ∼10−3or less.
+Itis now convenient to defineanew set ofcosmological energy densities normalized with respect
+to thecurrentcritical density ρ0
+c= 3H2
+0/(8πG0):
+Ωi(z)≡ρi(z)
+ρ0c=E2(z)
+g(z)˜Ωi(z) (i=m,Λ), (56)
+where we have introduced the ratios
+E(z)≡H(z)
+H0=/radicalbig
+g(z) [Ωm(z)+ΩΛ(z)]1/2, g(z)≡G(z)
+G0, (57)
+withG(z) Newton’s constant at redshift z. Notice that, in Eq. (56), we have explicitly related the
+newparameters Ω i(z) with the old ones ˜Ωi(z), thelatter beingdefinedin Eqs.(42),(43). Obviously,
+they all coincide at z= 0 with the normalized current densities Ω0
+i, i.e. Ω i(0) =˜Ωi(0) = Ω0
+i.
+Clearly, the coefficient νin Eq.(54) measures the amount of running of the CC. For any gi ven
+ν, we can compare the value of the dynamical CC term at a cosmic e poch characterized by the
+expansion rate H– or by the redshift z– with the current value (55). The relative correction can
+be conveniently expressed as follows:
+∆ΩΛ(z)≡ΩΛ(z)−Ω0
+Λ
+Ω0
+Λ=ν
+Ω0
+Λ/bracketleftbig
+E2(z)−1/bracketrightbig
+, (58)
+Of courseG(0) =G0. Moreover, since g= 1 forν= 0, it follows that for small ν,g(z) deviates
+little from 1, namely g(z) = 1+O(ν). Thus, expanding to order νin the matter epoch, it is easy
+to show from the previous equations that
+∆ΩΛ(z)≃νΩ0
+m
+Ω0
+Λ/bracketleftbig
+(1+z)3−1/bracketrightbig
+, (59)
+whereg(z)∼1 to this order. If we look back to relatively recent past epoc hs, e.g. exploring
+redshiftsz=O(1) relevant for Type Ia supernovae measurements, we see tha t ∆ΩΛ(z) can be
+of order of a few times ν. For example, for z= 1.5 andz= 2, we have ∆Ω Λ(1.5)≃6νand
+13∆ΩΛ(2)≃11νrespectively, assuming Ω0
+m= 0.3. The correction is thus guaranteed to be small,
+as desired, but is not necessarily negligible. We shall see, in the next sections, the potential
+implications on important observables, which will actuall y put tight bounds on the value of ν.
+Although the above parametrization of the CC running can be p urely phenomenological, let us
+recall that the dimensionless coefficient νcan be interpreted, in more fundamental terms, within
+the context of QFT in curved-space time; specifically it is pr oportional to the “ β-function” for the
+RG running of the CC term. The predicted value in this QFT fram ework is [15,19,24,25]
+ν=σ
+12πM2
+M2
+P, (60)
+whereMis an effective mass parameter, representing the average mass of the heavy particles of the
+Grand Unified Theory responsible for the CC running through q uantum effects (after taking into
+account the multiplicities of the various species of partic les). Obviously, M∼Meff. Sinceσ=±1
+(depending on whether bosons or fermions dominate in the loo p contributions), the coefficient ν
+can be positive or negative, but |ν|is naturally predicted to be smaller than one. For instance,
+if GUT fields with masses MinearMPdo contribute, then |ν|/lessorsimilar1/(12π)≃2.6×10−2, but
+we expect it to be even smaller in practice because the usual G UT scales are not that close to
+MP. By counting particle multiplicities in a typical GUT, a nat ural estimate lies in the range
+ν= 10−5−10−3(see [24,25] for details).
+In the next section, we shall concentrate on a specific runnin g QFT vacuum model of the type
+(54) where the vacuum energy and the gravitational constant vary simultaneously in accordance
+to the Bianchi identity (8). We shall also perform a detailed numerical analysis of our results.
+5 Application: running QFT vacuum model with variable G
+In this section, we consider a CC running vacuum model in whic hρΛ=ρΛ(H) depends on the
+Hubble rate through the affine quadratic law (54) and in which m atter is separately conserved –
+the time variation of the CC being compensated by that of the N ewton’s coupling, G=G(H).
+This setup corresponds to the scenario iv), as defined in sect ion 2 and was first considered at the
+background level with alternative motivations in Refs. [24 ,25]. The novelty here is to consider
+the detailed treatment of the cosmic perturbations in that s cenario. Namely, we apply to it the
+general treatment of cosmic perturbations with variable ρΛandGdeveloped in section 3. In this
+way, we can study the growth of matter density perturbations and obtain a LSS bound on the
+basic parameter ν, the one that controls the variation of GandρΛ. Actually, the tightly bounded
+region will appear in the ( ν,Ω0
+m) plane.
+Specifically, we will constrain the parameter ν(60) by requiring that the amount of growth of
+matter perturbations in our model does not deviate too much f rom the value deduced from the
+observations of the number density of local clusters. In fac t, let us recall that these observations
+allow to set the normalization of the matter power spectrum [ 44,45], and hence fix its amplitude.
+In view of the shape independence of the power spectrum for th e models under study, which we
+have discussed in section 3, the truly relevant parameter to constraint our model is the spectral
+14amplitude [35]. As we shall see, the constraints obtained in this way are in very good agreement
+with those arising from primordial nucleosynthesis. Throu ghout this section, we will be using the
+scale factor and the cosmological redshift z= (1−a)/ainterchangeably.
+The cosmological evolution of our model is determined by the Friedmann equation (3), the
+Bianchi identity (8) and the RG law for the CC (54). Using the n ormalized density parameters
+defined in Eq.(56), the basic cosmological equations can be f ormulated as
+E2(z) =g(z)[Ωm(z)+ΩΛ(z)], (61)
+(Ωm+ΩΛ)dg+gdΩΛ= 0, (62)
+ΩΛ(z) = Ω0
+Λ+ν/bracketleftbig
+E2(z)−1/bracketrightbig
+, (63)
+Ωm(z) = Ω0
+m(1+z)3(1+ωm), (64)
+where the last equation just reflects the covariant conserva tion of matter, i.e. it is a rephrasing of
+Eq.(9). Solving the remaining system for the function g=g(H), it is easy to arrive at
+g(H)≡G(H)
+G0=1
+1+νln/parenleftbig
+H2/H2
+0/parenrightbig. (65)
+We confirm that g(H) depends also on the parameter νand that, to first order, we have g=
+1+O(ν). Thus, in this model, νplays also the role of the β-function for the RG running of G.
+Compared to the quadratic running of the CC with the Hubble ra te, indicated in Eq.(63), the
+running of GwithHis indeed very slow, it is just logarithmic. The functions g(z) and Ω Λ(z)
+cannot be determined explicitly in an analytic form. Nevert heless, it is possible to derive an
+implicit equation for g(z) by combining (65) with (61) and (63). For the matter-domina ted epoch,
+the final result reads:
+1
+g(z)−1+νln/bracketleftbigg1
+g(z)−ν/bracketrightbigg
+=νln/bracketleftbig
+Ωm(z)+Ω0
+Λ−ν/bracketrightbig
+, (66)
+with Ω m(z) = Ω0
+m(1+z)3. The CC density follows from (63), and is given as a function o fg(z):
+ΩΛ(z) =Ω0
+Λ+ν[Ωm(z)g(z)−1]
+1−ν g(z). (67)
+As a simple check, we can see that Eq. (66) is satisfied at z= 0 for any value of ν, usingg(0) = 1
+and the sum rule Ω0
+m+Ω0
+Λ= 1 for flat space. Then from (67) we immediately obtain Ω Λ(0) = Ω0
+Λ,
+as expected.
+In Fig. 1, we show the evolution of different background quanti ties in this model in terms of
+the redshift z. For the plots, we have used Ω0
+m= 0.3 and both a positive ( ν= +4×10−3, red
+solid line) and a negative value ( ν=−4×10−3, blue dashed line) for the parameter ν. As we will
+see later in this section, according to LSS and nucleosynthe sis considerations, we expect νto be
+(at most) of order 10−3, so these are reasonable values. In Fig. 1a, we plot the evolu tion of the
+matter density fraction, ˜Ωm(z) (42), which rapidly approaches unity in the past. This mean s that
+˜ΩΛ(z) is negligible for high redshifts – recall the sum rule (43) – so that in this regime our model
+150 10 100 500z0.30.40.60.81
+~Ωmν < 0
+ν > 0
+20 100 500z1.0041.0042
+~ΩmDetaila)
+0 10 100 500z0.9511.05
+ν < 0
+ν > 0
+N
+ν < 0
+ν > 0H2
+H2
+Λ
+G
+G0b)
+Figure 1: Evolution of different background quantities for th e QFT cosmological model with
+running Λ and G, in terms of the redshift z. We take Ω0
+m= 0.3 and consider both a positive
+(ν= +4×10−3, red solid lines) and a negative value ( ν=−4×10−3, blue dashed lines) for the
+parameterν, which controls the variation of Gand Λ; a) the matter density fraction, ˜Ωm(z) (42),
+rapidly approaches unity in the past, meaning that for high r edshifts ˜ΩΛ(z) is negligible and our
+model closely resembles a CDM (for which ˜Ωm(z) isexactly1); b) evolution of G(z) (thin lines)
+andH2(z) (thick lines), normalized to the ΛCDM values, showing that the departure from the
+standard cosmological model is small. The evolution of both quantities is related by Eq. (71).
+closely resembles a CDM model (for which we would have ˜Ωm(z)exactly1.) This could have been
+anticipated from (63), which far in the past reads:
+ΩΛ(z)≃νE2(z). (68)
+Using this expression in (56), we obtain the corresponding a symptotic value of ˜ΩΛin the past:
+˜ΩΛ(z) =g(z)
+E2(z)ΩΛ(z)≃νg(z)∼ O(ν)≪1 (z≫1), (69)
+since indeed g(z)∼ O(1), as can be confirmed from Fig. 1b. Equation (69) tells us th at, for
+any reasonable value of ν, the contribution of the running CC in the past will be unimpo rtant.
+Nevertheless, as seen in the detail frame of Fig. 1a, ˜Ωm(z) in our model is indeed not exactly one,
+nor really constant. Note that for ν <0 the CC density decreases as we go into the past (the
+oppositeis truefor ν >0) until it eventually gets negative. Therefore, our model c an accommodate
+either originally positive or negative values for Λ, thanks to the running nature of this quantity.
+Notice that the previous results can be viewed as being a cons equence of the fact that, for flat
+space, the tilded normalized densities satisfy the sum rule (43).
+In Fig. 1b we show the evolution of g(z) andH2(z)/H2
+Λ(z) whereHΛis the (flat) ΛCDM
+Hubble function, simply given by:
+H2
+Λ=H2
+0/bracketleftbig
+Ωm(z)+Ω0
+Λ/bracketrightbig
+=H2
+0/bracketleftbig
+Ωm(1+z)3+Ω0
+Λ/bracketrightbig
+. (70)
+16Taking into account (61) and (67) we obtain, to order ν,
+H2(z)
+H2
+Λ=g(z)Ωm(z)+ΩΛ(z)
+Ωm(z)+Ω0
+Λ≃g(z)/parenleftbigg
+1+νΩm(z)−Ω0
+m
+Ωm(z)+Ω0
+Λ/parenrightbigg
+. (71)
+Therefore the evolutions of H2(z)/H2
+Λandg(z) are expected to be very similar, as indeed shown in
+Fig. 1b. Furthermore, both quantities stay close to 1, so the deviation from the standard ΛCDM
+evolution is reasonably small, although it maybe large enou gh so as to be detected in a future
+generation of precision cosmology experiments. For instan ce, for the values of νand Ω0
+mthat we
+are considering, the relative deviations of Ω Λ(z),g(z) andH2(z) atz= 2 with respect to the
+standard model values are (approximately) 4%, 1% and 0 .5%, respectively.
+5.1 Constraints from large-scale structure
+Let us now move to the detailed study of the growth of matter de nsity perturbations in our model,
+and further elaborate on some important issues raised in sec tion 3. The matter power spectrum
+for a sufficiently recent value of the scale factor ( a≫aeq) can be written as:
+P(k,a)≡ |δm(k,a)|2=AknT2(k)D2(k,a). (72)
+HereAis a normalization coefficient; nis the scalar spectral index (which gives us the shape of
+the primordial spectrum; e.g. n= 1 if we assume a Harrison-Zel’dovich spectrum); T(k) is the
+transfer function, which does not depend on the initial cond itions but only on the physics of the
+microscopic constituents; i.e. it encodes the modification s of the primordial spectrum that arise
+when taking into account the dynamical properties of the par ticles and their interactions. The
+T(k) function receives non-trivial contributions only from th e evolution of the different (comoving)
+wavelenghts λ∼1/kin the epochs of radiation, horizon crossing and radiation/ matter equality,
+i.e. it reflects the k-dependent features that occur at early times when amoves from a≪aeqto
+a≫aeq. In general, the form of T(k) depends on the cosmological model under consideration, an d
+determines to a large extent the final shape of the spectrum. H owever, for the late time evolution
+(a≫aeq), theremight bemoredistortions of thespectrumin models t hat departsignificantly from
+the standard ΛCDM scenario (for which there are no further k-dependence beyond that already
+encoded in the transfer function). These additional, model -dependent, effects are reflected in the
+k-dependence of the growth factor D(k,a) in Eq.(72). A dependence of this sort would appear e.g.
+in models of type ii) in section 2 where the CC decays into matt er, as it was previously studied
+in [28]. However, our claim (formulated in section 3) is that this is not the case for the variable
+CCandGmodels under consideration, i.e. models of type iv) in secti on 2 with self-conservation
+of matter, provided that Gis a slowly varying function of time. Obviously this is so for the model
+studied in the previous section, where Gvaries logarithmically with the expansion of the universe,
+see Eq.(65). For these models, and of course also for the stan dard ΛCDM model, the growth factor
+is independent of kand can be written as
+D(a) =δm(a)
+δCDM(a0), (73)
+whereδCDM(a0) is the present matter density contrast in a pure cold dark ma tter (CDM) scenario,
+taken as a fiducial model.
+175.1.1 More on the transfer function for variable ρΛandGmodels.
+A few additional comments on the transfer function are now in order. As previously commented,
+we expect the transfer function in our model to be the same as f or the ΛCDM. First of all, let us
+try to better justify this expectation, which implies that t he shape of the matter power spectrum
+in our model coincides with that of a ΛCDM with the same value o f Ω0
+m. Then we show that
+we can still constrain our model by means of the growth factor , obtaining bounds in very good
+agreement with those arising from primordial nucleosynthe sis.
+Thescale dependenceof thetransferfunctionfor CDMmodels is different for large scales (small
+k’s), whereT(k) = 1, and small scales, where it asymptotes to ln( k)/k2[35]. The turnaround
+occurs at the value k=keq, corresponding exactly to the scale that enters the Hubble h orizon
+(H−1) at the moment of matter-radiation equality, i.e. at the poi nta=aeqwhich satisfies
+Ωm(aeq) = Ωr(aeq), hence
+aeq=Ω0
+r
+Ω0m, (74)
+where Ω0
+r∼10−4is the present density of radiation. The modes that enter the horizon at
+the equality time, with comoving wavelength λeq, have a physical wavelength that follows upon
+multiplication with the scale factor (74), i.e. λeqaeq= 1/H(aeq), or, equivalently,
+keq=aeqH(aeq). (75)
+Obviously, since Hdecreases with time the perturbations with wavelenght shor ter thanλeq(i.e.
+those with with k>keq) will enter the Hubble horizon before the matter-radiation equality, i.e. in
+the radiation era ( t<teq). From this moment until t=teq, the growth of inhomogeneities in the
+cold DM component becomes suppressed because the expansion rate during the radiation epoch is
+faster than the characteristic collapsing rate of the CDM. A s a result, the modes in the radiation
+epochcan grow at most logarithmically withthescale factor . Onlyafter thecold component begins
+to dominate ( t>teq) the amplitude of the formerly inhibited modes starts growi ng linearly with
+the scale factor. Finally, as light can only cross regions sm aller than the horizon, the suppression in
+theradiation epoch does not affect the large-scale perturbat ions (k≪keq), which enter thehorizon
+in the matter epoch. Such different behavior of the perturbati ons according to their entrance in
+the horizon before or after the time of equality is the origin of the characteristic shape of the
+transfer function for CDM-like models in the various availa ble parameterizations [35].
+From the previous standard discussion, it is apparent that t he shape of the transfer function
+depends critically on the value of keq, which in turn depends on aeqandH(aeq). In the models we
+are studying, dark matter and radiation are separately cons erved, and therefore the value of aeq
+will not change with respect to the standard one, Eq. (74). So the remaining issue is to clarify if
+the change of H(aeq) in our model as compared to the corresponding ΛCDM value is s ignificant
+or not. The answer follows easily from Eq.(71). At the high re dshift where equality of matter and
+radiation occurs, z=O(103)≫1, the function that accompanies νon ther.h.s.of that equation
+is virtually equal to one, and we are left with
+H(aeq)≃/radicalBig
+g(aeq)(1+ν)HΛ(aeq). (76)
+18Here, as before, HΛis just the standard ΛCDM Hubble rate. For the maximal values ofνthat we
+will be considering ( |ν| ∼ O(10−3), see the subsequent sections), it is easy to see from (65) th at
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalBig
+g(aeq)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle≃ |ν|lnH(aeq)
+H0∼1%, (77)
+where the expansion rate at the time of equality is H(aeq)∼105H0. We see that the change of
+H(aeq) with respect to HΛ(aeq) is mainly caused by the variation of gfrom 1 att=teq. However,
+numerically, the effect is very small. These results allow us t o safely conclude that the value of keq
+expected in our model is essentially the same as that of the ΛC DM model, up to differences of 1%
+at most. Given that in the ΛCDM model the (comoving) wavenumb er at equality is
+keq=aeqHΛ(aeq) =/radicalBigg
+2
+Ω0
+RΩ0
+mH0∼10−2hMpc−1, (78)
+we see from this expression that a 1% change in/radicalbig
+g(aeq) – hence in H(aeq) – is equivalent to a 1%
+change in Ω0
+min the standard scenario (i.e. with g= 1). This variation is too small to be within
+reach of the present observations (see the next section), an d can be safely neglected.
+The final point is that, as argued throughout this section, ne itherρΛnor its perturbations or
+those ofGare important in the past. Therefore, the evolution of the pe rturbations both in the
+radiation and in the matter-dominated eras (and both in the c ase of sub-Hubble and super-Hubble
+perturbations) remains essentially unchanged.
+All in all, we expect that the transfer function in the model u nder consideration is very close
+to that of a ΛCDM with the same value of Ω0
+m. From the line of our argumentation that we have
+used, it is not difficulty to convince oneself that this result can be extended to any model with
+variable Λ and G, and self-conserved matter, as long as G(a) is not changing too fast. Adding
+this property to the fact – cf. section 3 – that the late growth of matter density perturbations in
+these models does not depend on the wavenumber, the final robu st conclusion is that the matter
+power spectrum for models within the scenario iv) of section 2 will present the same shape as
+in the standard ΛCDM case. Therefore, the spectrum shape wil l not be useful to constrain the
+additional free parameters of the model. This is in sharp con trast to models within scenario ii),
+in which there is an exchange between dark matter and vacuum e nergy. For these models there
+is an explicit dependence on kbeyond that of the transfer function and this produces a late time
+distortion of the power spectrum with respect to the ΛCDM. As indicated, this was exemplified
+in Ref. [28,29] for the case of an evolution law of the type (53 ).
+To summarize, there is a significant difference between the run ning QFT vacuum model (53)
+when studied either in scenario ii) or when considered in sce nario iv). While in Ref. [28,29] the
+shape of the spectrum was used to restrict the parameter νfor type ii) models, in the next section
+we show that for the alternative type iv) models, being them s hape-invariant with respect to the
+ΛCDM model, one can make use of the amplitude of the power spec trum in order to constrain the
+free parameters.
+195.1.2 The spectrum amplitude as a way to constrain running G and Λmodels.
+The normalization of the matter power spectrum on scales rel evant to large-scale structure can
+be performed through different methods [46], e.g. from the mic rowave-background anisotropies or
+by measuring the local variance of galaxy counts within cert ain volumes. One of the most robust
+ways to do it is through the number density of rich galaxy clus ters [44,45], which is very sensitive
+to the amplitude of the dark matter fluctuations that collaps ed to form them. The typical scales
+for these fluctuations are of order 10 Mpc, which are the small est ones still in the linear part of
+the spectrum. The cluster method determines the amplitude o f the power spectrum on just that
+length scale (corresponding to wavenumbers in the upper end of those explored with this method).
+The normalization is usually phrased in terms of σ8, the root mean square mass fluctuation in
+spheres with radius 8 h−1Mpc (i.e. ∼10 Mpc, for h≃0.7). By assuming a ΛCDM model, these
+studies are able to provide constraints in the σ8- Ω0
+mplane. An important feature is that the
+results are approximately independent of the spectrum shap e and galaxy bias [44]. For instance,
+in this last reference, it is found the relation:
+σ8=/parenleftbig
+0.495+0.034
+−0.037/parenrightbig
+(Ω0
+m)−0.60, (79)
+valid for spatially flat models with a wide range of shapes (in particular, valid for 0 .2≤Ω0
+m≤0.8).
+When combined with CMB analyses, cluster studies can determ ine bothσ8and Ω0
+m. In a recent
+work [45], local cluster counts are used in conjunction with WMAP5 data to find:
+Ω0
+m= 0.30+0.03
+−0.02(68%). (80)
+We will use this result to constrain our G-variable model. In order to do this, we first define the
+“amount of growth”, namely thesquareof thegrowth factor (7 3) at present, D2(a0), whichappears
+directly in the formula of the power spectrum, Eq.(72). The i dea is to compare the amount of
+growth in our model model with the amount of growth in the ΛCDM model with Ω0
+m= 0.30.
+From the standard expression for the growth factor in the ΛCD M model [35] one finds that a
+∼10% variation in Ω0
+mgiven by (80) represents a ∼5% change in the amount of growth. As the
+determination (80) entails only a 1 σconstraint, we will be conservative and allow up to a 10%
+deviation in the amount of growth of our model with respect to the ΛCDM model. We are thus
+asking our model to pass the following “F-test” [29,47]:
+F=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−D2(a0,Ω0
+m,ν)
+D2(a0,0.3,0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤0.1. (81)
+Notice that D2(a0,0.3,0) in the denominator is just the amount of growth in the ΛCDM f or the
+central value of (80), whereas in the numerator we have the am ount of growth for the model under
+consideration at a given non-vanishing value of the relevan t parameter ν, with Ω0
+mleft as a free
+parameter. As a result, the constraint (81) will generate co ntours in the ν- Ω0
+mplane which will
+define the allowed region in parameter space.
+The admissible values for Ω0
+mshould, in principle, be those compatible with the shape of t he
+spectrum. However, the shape has been measured by the 2dFGRS and SDSS surveys, existing
+20a significant difference between their results. On the one hand (assuming a Hubble parameter
+h= 0.72), the SDSS main galaxy analysis [4] favored the result
+Ω0
+m= 0.296±0.032. (82)
+Similar values around Ω0
+m≃0.3 were found by alternative analyses of the SDSS catalogue [4 8].
+On the other hand, the 2dFGRS collaboration [3] found a much l ower matter density
+Ω0
+m= 0.231±0.021, (83)
+obtained from measurements of clustering of blue-selected galaxies. The inclusion of Luminous
+Red Galaxies (LRGs) in the SDSS analysis seems to even increa se the discrepancy. For instance,
+the authors of [49] findΩ0
+m= 0.32±0.01, although a lower density is recovered when restricting t he
+analysis to large scales (Ω0
+m= 0.22±0.04 for 0.01<k<0.06hMpc−1.) It is widely believed that
+the differences are due to the scale-dependence of the galaxy b ias (which is apparently stronger
+for the red galaxies that dominate the SDSS catalogue) or eve n to some kind of systematic effect,
+but the problem has not been fully settled so far [50]. As an ex ample, the last result by the
+SDSS team [5], obtained from the analysis of a LRGs sample (in combination with WMAP5 data),
+yields Ω0
+m= 0.289±0.019, still much larger than (and incompatible with) the 2dfG RS result, (83),
+although if we make allowance for a 10% gaussian uncertainty inhin the 2dfGRS analysis can
+bring both results within 1 σ.
+At the end of the day, and taking into account the results from 2dFGRs and SDSS, we think
+that it would be premature to discard any value for Ω0
+min the range (0 .21,0.33) on the grounds
+of structure formation data – as long as it predicts the right amount of growth, e.g. by satisfying
+the F-test (81). Therefore, in our analysis, for illustrati ve purposes, we will consider values of Ω0
+m
+between 0.2 and 0.4.
+5.1.3 Numerical results.
+In this numerical section, and in view of the previous consid erations, we wish to determine the set
+of points in the ν- Ω0
+mplane for which the amount of growth, determined by the value ofD2(a0),
+in our running ρΛandGmodel deviates less than 10% from the central (Ω0
+m= 0.3) ΛCDM value
+in Eq.(80). To compute the growth factor in our model, we evol ve the solution δm(a) of the
+differential equation (49) from an initial value a=aiup to the present moment ( a0= 1), where
+ai≪1 is the scale factor at some early time, deep into the matter- dominated era but well after
+recombination (so that the transfer function regime has alr eady ended). We will take ai= 1/501
+(i.e.zi= 500), although we have checked that the results do not the de pend significantly on the
+specific value. As we have seen in Fig. 1a, early on at a=aiour model is very similar to the
+CDM model ( ˜Ωm≃1), for which the matter density perturbations grow linearl y with the scale
+factor, i.e. D(a) =a. Therefore, we will assume that this is also the case for our r unningGand
+ρΛmodel, and take D(ai) =ai,D′(ai) = 1,D′′(ai) = 0 as the initial conditions for the third-order
+differential equation (49).
+Theresults areshown in Fig. 2a. Theshaded areas represent t he pointsallowed by our analysis.
+Specifically, we compare the case with perturbations in ρΛandG(green band) with the case in
+21-0.01 0 0.01 0.02
+ν0.20.250.30.350.4
+Ωm0a)
+δG = 0δG = 0
+-0.01 0 0.01 0.02
+ν0.20.250.30.350.4
+Ωm0b)
+Scenario ii)
+Scenario iv)δG = 0
+Figure 2: Analysis of the parameter space of the runningQFT m odels. The shaded areas represent
+the points for which the amount of growth ( D2(a0)) of the model deviates less than 10% from the
+ΛCDM value. For the latter, we take Ω0
+m= 0.3, Eq.(80), on the basis of data on the matter
+power spectrum amplitude; a) Scenario iv) section 2 (runnin g Λ andG, self-conserved matter).
+The green (resp. yellow) band include (exclude) perturbati ons inρΛandG. With perturbations,
+the constraint on the parameter space becomes tighter; b) Co mparison between two scenarios of
+section 2 when perturbations in ρΛandGare neglected: scenario iv) (yellow band) and scenario
+ii) (orange band). The deviations from the ΛCDM case are larg er in scenario ii) owing to the
+production/decay of matter from the running ρΛat fixedG, and the constraint is correspondingly
+tighter. The blue and red points in the plane have coordinate s (ν=±4×10−3,Ω0
+m= 0.3) – used
+in Figs. 1 and 3. The dashed horizontal lines signal the 1 σlimits on Ω0
+mfrom Eq.(80).
+.
+which these perturbations are neglected (yellow band). In t he last situation, the growth factor
+is obtained by solving Eq. (39), although, as discussed in se ction 3, it is not possible to neglect
+them consistently. The most remarkable conclusion that eme rges from this numerical analysis is
+that by considering the effect of the perturbations results in narrower domains, meaning that the
+deviations with respect to the standard ΛCDM amount of growt h tend to be larger – which is
+why the restriction is accordingly tighter. Therefore, the outcome of the analysis with δG∝negationslash= 0 is
+that, for any Ω0
+min the range 0 .2<Ω0
+m<0.4, values of |ν|larger than 10−2are ruled out by
+the data on the number density of local clusters. In particul ar, this excludes the canonical value
+|ν|= 1/12π≃0.026 obtained from the simplest choice M=MPin Eq.(60). Notice that for
+Ω0
+min the narrower (1 σ) interval (80), the allowed values for |ν|are of order 10−3at most. For
+these values of ν, our model is on equal footing with the ΛCDM as far as structur e formation is
+concerned.
+In Fig. 2b we compare the results for two of the scenarios of se ction 2: scenario iv), represented
+by the yellow band (this is the scenario we have been analyzin g so far, with variable ρΛandGand
+self-conserved matter; and scenario ii), indicated by the o range band; for the latter, Gis constant
+22and the CC exchanges energy with matter. In both cases, we are neglecting the perturbations5in
+ρΛ, which in scenario iv) implies that δG= 0 as well, cf. Eq.(36). The effective equation for the
+matter density contrast in scenario ii) is the following:
+¨δm+/parenleftbigg
+2H+Ψ
+ρm/parenrightbigg
+˙δm−/bracketleftbigg
+4πGρm−2HΨ
+ρm−d
+dt/parenleftbiggΨ
+ρm/parenrightbigg/bracketrightbigg
+δm= 0, (84)
+Ψ≡˙ρm+3Hρm=−˙ρΛ. (85)
+This equation, whose primary derivation was performed with in the Newtonian formalism [51],
+can also can be derived from the general relativistic treatm ent of perturbations [21], as explained
+in [18]. It is convenient to express it in terms of the scale fa ctora:
+δ′′
+m+/parenleftbigg3
+a+H′
+H+Ψ
+ρmHa/parenrightbigg
+δ′
+m−/bracketleftbigg3
+2˜Ωm−2
+HΨ
+ρm−a
+H/parenleftbiggΨ
+ρm/parenrightbigg′/bracketrightbiggδm
+a2= 0. (86)
+For Ψ = 0, we recover equation (39) (with δG= 0), as expected. In Fig. 2b, we see that the
+differences in the amount of growth with respect to the ΛCDM cas e are larger for scenario ii); the
+natural interpretation is that this is caused by the product ion/decay of matter associated to the
+time evolution of ρΛ.
+The blue and red points in Figs. 2a,b correspond to the values analyzed in Fig. 1, i.e. Ω0
+m= 0.3
+andν=±4×10−3. We will use again these values to exemplify the evolution of the matter and
+Newton’s coupling perturbations. Fig. 3a shows the growth f actor as a function of the redshift,
+both when allowing for perturbations in GandρΛand when they are neglected. For ν <0, our
+model predicts more growth than the ΛCDM model. When δG= 0, this is just due to the fact that
+ρΛis decreasing when we go into the past, so its repulsive effect d iminishes. When considering the
+perturbations in Gand Λ, the deviation with respect to the standard case increa ses even more,
+as already commented in Fig. 2a. The opposite situation occu rs forν >0; here the suppression
+of growth is attributed to the enhanced repulsion of matter a ssociated to a positive ρΛ, which
+is increasing with z. Such suppression is larger when we include the Gand Λ perturbations. A
+detailed study of these effects for model ii) of section 2 was pe rformed in Ref. [29], in an effective
+approach with no perturbations in the CC.
+In Fig. 3b, the evolution of δG/G, as obtained from Eq.(37), is depicted. This equation only
+determines δG′, so we need to give an initial condition for δGata=ai. In order to do so, we note
+that the fact that ˜Ωm(z)≃1 in the past (reflected in Fig. 1a) ensures that the perturbat ions inG
+are not playing any important role, according to our discuss ion at the end of section 3. Therefore,
+the most natural condition seems to be δG(ai) = 0, and, besides, we expect δG/Gto remain
+negligible until very recent times ( z/lessorsimilar10), when ˜Ωmbegins to depart from 1. This is indeed what
+can be seen in Fig. 3b, giving additional support to our assum ption. Let us remind from (36) that
+δΛ=−δG/G. Thus, for ν <0 we haveδG >0 and hence δΛ<0, which explains the further
+enhancement in the growth of matter perturbations with resp ect to the case with δG=δρΛ= 0.
+The main conclusion of this section is that for νof order 10−3or smaller, our model is in
+perfect agreement with recent data on the normalization of t he power spectrum. For the time
+5Remember that within scenario ii), including the perturbat ions in Λ causes the growth factor to become scale-
+dependent [28], D=D(k,a), so the kind of analysis we are performing here would be no lo nger possible.
+230 10 1 20z0.20.40.60.8
+D(z)
+ν < 0
+ν > 0.
+ΛCDM.
+ν < 0
+ν > 00.1 0.2 0
+z0.70.750.8
+D(z)Detail
+δG = 0
+δG = 0a)
+0 10 100 500z-0.00500.005
+δG
+Gν < 0
+ν > 0 b)
+Figure 3: Evolution of the perturbations for the QFT model wi th running Λ and G, using the
+same values of Ω0
+mandνas in Fig. 1; a) The growth factor D(z), showing an enhancement
+(suppression) of the growth with respect to the ΛCDM case for negative (positive) ν. The effect is
+present even when we neglect the perturbations in GandρΛ, but it is larger if we consider them;
+b) Evolution of the perturbations in G, under the natural assumption that they are negligible
+atzi= 500. The perturbations remain unimportant until very rece nt times, when ˜Ωmbegins to
+depart significantly from 1 (cf. Fig. 1). Remembering that δΛ=−δG/G, we can explain the
+enhancement (suppression) of matter perturbations observ ed in a) on account of the opposite sign
+ofδΛ=δρΛ/ρΛ, see Eq.(39).
+being, there is still some controversy on the value of Ω0
+mresulting from the power spectrum shape
+measured by the two main galaxy surveys. For a given value of Ω0
+m, the power spectrum in our
+model presents, in very good approximation, the same shape a s in the ΛCDM model. However, a
+non-vanishing νcould become manifest through a difference in the amount of gro wth. Therefore, a
+future simultaneous precision cosmology measurement of th e shape and amplitude (normalization)
+ofthematter powerspectrumshouldhave thepotential todis criminatebetween amodelof variable
+ρΛandG, and the ΛCDM.
+In the following section, we show that ν∼ O(10−3) is also the maximum allowed value that
+we would expect from arguments related to primordial nucleo synthesis, fact that reinforces the
+validity of this bound.
+5.2 Constraints from primordial nucleosynthesis
+Big Bang nucleosynthesis (BBN) can also provide limits on th e possible variation of Newton’s
+couplingG. The BBN predictions for the light element abundances are se nsitive to a number of
+parameters, such as the baryon-to-photon ratio η=nB/nγ(in view of the fact that the nuclear
+reaction rates depend on the nucleon density) or the value of the Hubble function at the nucle-
+osynthesis time, HN≡H(z=zN) (given that the expansion rate competes against the reacti on
+rates). Since η∝ΩBcan be accurately determined through CMB measurements, one can use the
+observed abundances to constrain the expansion rate, and si nceH∝√
+Gthese constraints can be
+24directly translated6into bounds on gN≡g(zN), whereg(z) was defined in Eq.(57).
+The current constraints on gNavailable in the literature usually make use either of the de u-
+terium abundance ( D/H) or the4He mass fraction ( Yp). Deuterium has the advantage that it is
+not produced in significant quantities after nucleosynthes is; its primordial abundance can be deter-
+mined in a quite precise manner by studying the spectrum of th e light from distant quasars, which
+exhibits an absorption line corresponding to the deuterium present in (high-redshift) intervening
+neutral hydrogen systems. This turns deuterium into an exce llent probe of the universe at the
+time of BBN. However, its predicted abundance depends much m ore strongly on ηthan onHN.
+As a consequence, the constraints on gNbased on deuterium measurements are not very stringent.
+For instance, in [52] it is found gN= 1.01+0.20
+−0.16at the 68.3% confidence level or:
+gN= 1.01+0.42
+−0.30(95%). (87)
+The abundance of4He is much more sensitive to the expansion rate, but accurate observational
+valuesaredifficulttoobtain, sincetherearemany potential sources ofsystematic uncertainties [53].
+As a result, a wide range of values for Ypcan be found in the literature, see e.g. Table 12 in [54]. In
+this reference, the authors used Yp= 0.250±0.003 (in fact, they use Ypin combination with D/H,
+although the dominant effect is that of helium [54]) to obtain ( see section 8.2.5 of the latter):
+0.964<gN<1.086 (95%) , (88)
+whereas in [55] a more conservative range for the4He mass fraction is adopted ( Yp= 0.2495±
+0.0092), leading to:
+0.9<gN<1.13 (68%). (89)
+Applying (87)-(89) to our model, we can effectively constrain our parameter ν. In order to do so,
+we compute gNfrom (65):
+gN=1
+1+νln/parenleftbig
+H2
+N/H2
+0/parenrightbig≃1
+1+νln [Ω0r(1+zN)4], (90)
+where Ω0
+ris the radiation energy density fraction at present, we have neglected both the matter
+and CC contributions to the expansion rate at z=zN, and is evident that gNcan be neglected as
+well in the logarithm. Taking Ω0
+r∼10−4(which includes photons and three light neutrino species)
+together with zN∼109, and comparing the last expression to (87)-(89), we find:
+From (87) −→ −4.1×10−3/lessorsimilarν/lessorsimilar5.5×10−3
+From (88) −→ −1.1×10−3/lessorsimilarν/lessorsimilar5.1×10−4
+From (89) −→ −1.6×10−3/lessorsimilarν/lessorsimilar1.5×10−3
+Let us remind that (88) was derived from a value for Ypwith a (possibly unrealistic) very small
+error and that (89) is a 68% value (whereas the other two limit s are given at the 95% confidence
+level.) In any case, the conclusion arising from this nucleo synthesis analysis seems to be that the
+6Provided that the total energy density at z=zNis approximately the same as in the standard case. Eq. (69)
+shows that this condition is satisfied in our model.
+25parameterνcan be, at most, of order 10−3. This is in complete agreement with the constraint on ν
+obtained in section 5.1 from structure formation considera tions. Therefore, we have arrived to the
+same result by using two very different methods, which gives ad ditional credit to our conclusions.
+6 Conclusions
+Inthispaper,wehavederivedthegeneralsetofcosmologica l perturbationequationsforFLRW
+models with variable cosmological parameters ρΛandGin which matter is covariantly conserved.
+To our knowledge, this is the first time that a complete set of c oupled differential equations for δρΛ
+andδGispresentedintheliterature. Wehaveshownthatthelinear growthofmatterperturbations
+of this model, D(a)∝δm(a), is independent of the wave number k. Adding this property to the
+fact that we generally expect these models to present the sam e transfer function as the ΛCDM,
+the scaling dependence and hence the shape of the power spect rum will not change with the late
+time evolution and it will coincide with that of the ΛCDM mode l. This fact is remarkable as it is
+in contrast to the situation, more frequently studied in the literature, in which the time-evolving
+vacuum models exchange energy with matter at fixed Newton’s c ouplingG[18].
+We have exemplified the difference in the power spectrum of runn ing vacuum models, with and
+without matter conservation, by considering the class of co smological models characterized by the
+running CC law (53), which we call quantum field vacuum models because the evolution of the
+CC in them is of the form that one would naturally expect from Q FT, and more specifically from
+the renormalization group evolution of the cosmological pa rameters. Such quantum field vacuum
+models dependon a single parameter ν, which acts as the β-function for the runningof the vacuum
+energy density ρΛ=ρΛ(H) and the running gravitational coupling G=G(H). It is interesting to
+note that while the running of the vacuum energy ρΛ(H) is quadratical in the expansion rate, the
+running ofG(H) is logarithmic in H, Eq.(65), and therefore much milder. For ν >0, the running
+ofGis asymptotically free, hence Gdecreases (resp. increases) logarithmically with the reds hift
+(resp. with the expansion), whereas for ν <0 it decreases with the expansion. The background
+evolution of these models was studied in [24,25].
+In the present paper, we have concentrated on the study of the cosmic perturbation equations
+of the running ρΛ(H) andG(H) models. After confronting the predicted matter growth wit h the
+various cosmological data, we have found that the final regio n of the parameter space is a natu-
+ralness region in which νcould be, in absolute value, of order 10−3at most. We have also checked
+that this bound is compatible with the restriction on Gvariation that follows from primordial
+nucleosynthesis. Remarkably enough, the results emerging from the perturbations analysis are
+derived from the amplitude of the power spectrum rather than from its shape. Numerically, the
+obtained upper bound on νis perfectly compatible with the quantum field theoretical d efinition
+of this parameter, see Eq.(60), and it implies that the mass s cales that would enter the quantum
+runningof the cosmological parameters could be one order of magnitude below the Planck scale, at
+most. The result |ν|<O(10−3) is also compatible with previous analyses, using various i ndepen-
+dent procedures, of models with running ρΛin which the vacuum can decay into matter [18,28–30].
+This decay feature, however, is impossible for the models wi th running ρΛ(H)andG(H) that we
+26have studied here, and this is the basic reason why they can ex hibit the same power spectrum
+profile as the ΛCDM. We cannot exclude that this property coul d have been responsible for the
+fact that we have observationally missed this fundamental p ossibility up to now. As we have
+emphasized, we expect that these models should eventually b e testable in the next generation of
+precision cosmology observations from the analysis of the s pectrum amplitude, rather than of the
+spectral shape.
+Acknowledgments. JG and JS have been supported in part by MEC and FEDER under
+project FPA2007-66665, by the Spanish Consolider-Ingenio 2010 program CPAN CSD2007-00042
+and by DIUE/CUR Generalitat de Catalunya under project 2009 SGR502. JF thanks CNPq and
+FAPES, and ISh is grateful to CNPq, FAPEMIG, FAPES and ICTP fo r the partial support.
+7 Appendix: cosmic perturbations in the Newtonian gauge
+In this appendix, we sketch the derivation of the perturbati on equations in the Newtonian or
+longitudinal gauge [56]. Our aim is to explicitly check that the same physical consequences that
+we have derived in section 3 for the synchronous gauge are rec overed in another gauge. In this
+way, we confirm in our particular context the general expecta tion that calculations of cosmic
+perturbations at deep sub-horizon scales should not presen t significant gauge dependence.
+The most general perturbation of the spatially flat FLRW metr ic can be conveniently written
+as follows [56]
+ds2=a2(η)[(1+2ψ)dη2−ωidηdxi−((1−2φ)δij+χij)dxidxj], (91)
+whereηis the conformal time, defined through dη=dt/a. The above metric is expressed in the
+notation of[21] andconsists ofthe10degreesof freedomass ociated tothetwo scalar functions ψ,φ,
+thethree components of the vector function ωi(i= 1,2,3), and the fivecomponents of the traceless
+second-rank tensor χij. Clearly, the synchronous gauge that we used in section 3 is o btained by
+settingψ= 0,ωi= 0 and absorbing the function φinto the trace of χij, which then contributes
+six degrees of freedom. In this gauge, the metric part of the a nalysis of cosmic perturbations is
+tracked by the nonvanishing trace of the metric disturbance :h≡gµνhµν=gijhij=−hii/a2.
+The corresponding equations have been presented in detail i n section 3 after defining the “hat
+variable”: ˆh=−∂h/∂t=∂/parenleftbig
+hii/a2/parenrightbig
+/∂t.
+An alternative possibility is to use the (conformal) Newton ian gauge [56,57]. In this case, we
+setωi=χij= 0(∀i,j) in Eq.(91). A useful feature of this gauge is that the metric is diagonal
+and another is that, in the Newtonian limit, ψplays the role of the gravitational potential. It is
+now straightforward to repeat the calculation of the pertur bation equations in a similar way as in
+section 3. In the absence of anisotropic shear stress (which is always the case with non-relativistic
+particles, such as baryons and dark matter), the perturbed E instein’s equations for i∝negationslash=jgive
+ψ=φ[57], see also [58]. Indicating by a prime the derivatives wi th respect to the scale factor
+27(f′≡df/da), the final perturbation equations in that gauge read as foll ows:
+k2φ=−4πa2/braceleftbigg
+G/bracketleftbigg
+δρm+δρΛ+3Hρma2
+k2θ/bracketrightbigg
++(ρm+ρΛ)δG/bracerightbigg
+,
+δρ′
+m+ρm/parenleftbiggθ
+aH−3φ′/parenrightbigg
++3
+aδρm= 0,
+θ′+2
+aθ=k2
+Ha3φ, (92)
+δG′(ρm+ρΛ)+δGρ′
+Λ+G′(δρm+δρΛ)+Gδρ′
+Λ= 0,
+GδρΛ+ρΛδG+/parenleftbiggHa3
+k2/parenrightbigg
+ρmG′θ= 0.
+Notice that the role of ˆhin the synchronous gauge is taken up here by the variable φ=ψ, which
+turns out to satisfy an algebraic rather than a differential eq uation. Worth noticing is also the fact
+that the last two equations of (92) are identical to equation s (32) and (33) respectively, except that
+in the present case we used derivatives with respect to the sc ale factor rather than with respect to
+the cosmic time (as in this way length scales can be better com pared with the horizon).
+We are now ready to show that for deep sub-Hubble (sub-horizo n) perturbations, i.e. those
+satisfyingk/(aH)≫1, these equations lead to the same second and third-order di fferential equa-
+tions for the normalized matter overdensity δm=δρm/ρm, which we have previously found within
+the synchronous gauge in section 3. To start with, the first eq uation of (92) can be cast as:
+φ=−3
+2H2a2
+k2/braceleftbigg
+˜Ωmδm+˜ΩΛδΛ+δG
+G/bracerightbigg
+, (93)
+whereδΛ=δρΛ/ρΛ, and of course we use the same notation as in section 3 concern ing the
+parameters ˜Ωmand˜ΩΛ. In the previous equation, we have dropped the term 3 Hρma2θ/k2since it
+is negligible for sub-Hubble perturbations. In this same re gime of perturbations, Eq.(93) obviously
+implies that φ≪δm. Next, let us note that the second equation of (92) can be simp lified by using
+the conservation of matter, giving:
+δ′
+m+θ
+aH= 3φ′. (94)
+Differentiating this equation and using the third one in (92) t o substitute for θ′, we find:
+δ′′
+m−θ
+aH/parenleftbigg3
+a+H′
+H/parenrightbigg
+= 3φ′′−/parenleftbiggk
+Ha/parenrightbigg2φ
+a2. (95)
+Furthermore, if we eliminate θfrom this equation by means of (94) and also φusing (93), we get:
+δ′′
+m−3φ′′+(δ′
+m−3φ′)/parenleftbigg3
+a+H′
+H/parenrightbigg
+=3
+2a2/braceleftbigg
+˜Ωmδm+˜ΩΛδΛ+δG
+G/bracerightbigg
+. (96)
+Again we can neglect the terms that are small at sub-Hubble sc ales; for example, in the previous
+equation we can neglect φ′(φ′′) as compared to δ′
+m(δ′′
+m) on account of the previously discussed
+relationφ≪δm. In this way, we arrive at:
+δ′′
+m+/parenleftbigg3
+a+H′
+H/parenrightbigg
+δ′
+m=3
+2a2/braceleftbigg
+˜Ωmδm+˜ΩΛδΛ+δG
+G/bracerightbigg
+. (97)
+28The almost final step is to substitute δΛby means of the last equation of (92). Once more, we
+can argue (as we did in section 3) that apart from the damping f actorHa/k≪1 for the sub-
+Hubble modes, there are additional suppression effects, such as the presence of G′which is small
+as compared to G, and hence ( G′/G)θcan be considered almost second-order. Therefore, the last
+equation of (92) boils down in practice to GδρΛ+ρΛδG= 0, which simply renders δΛ=−δG/G.
+Using also the sum rule (43), the relation (97) finally takes o n the simpler form
+δ′′
+m+/parenleftbigg3
+a+H′
+H/parenrightbigg
+δ′
+m=3˜Ωm
+2a2/parenleftbigg
+δm+δG
+G/parenrightbigg
+. (98)
+This equation exactly coincides with Eq.(39), showing inde ed that our result was not gauge-
+dependent. Obtaining the third-order equation for δmis now straightforward. From the last two
+equations of (92) one can show that
+δm=−(δG)′
+G′. (99)
+From here we proceed as in section 3 for the synchronous gauge , i.e. we differentiate Eq.(98)
+with respect to the scale factor and we reach once more the thi rd order differential equation (49).
+(q.e.d.)
+Some discussion is now in order. Let us first note that the vari able cosmological constant
+scenario ii) of section 2, whose perturbation equations wer e originally analyzed in [28] in the
+synchronous gauge, has been reanalyzed in the Newtonian gau ge in [32] and the same conclusions
+have been attained. We should also mention a very recent pape r [59] where a nice discussion is
+made on comparing the synchronous and Newtonian gauges in th e context of a model with self-
+conserved DE and without DE perturbations. As it is well-kno wn, in that case the two gauges
+give the same evolution for the cosmological perturbations for scales well below the horizon – see
+also [57,60]. This feature was expected in our model too, eve n if it is a bit more complicated
+than the one considered in [59] (our DE is also self-conserve d, but we do have both DE and G
+perturbations), and in fact the physical discussion about t he gauge differences is very similar. In
+the same reference, it is also pointed out that the k-dependent effects are negligible for large values
+ofk, although they could start to be appreciable for k <0.01h(or, equivalently, for length scales
+larger than ℓ∼1/k≃100h−1Mpc.).
+Theissues about gauge dependenceare of courseimportant in general, becausescale-dependent
+effects may change from gauge to gauge. For instance, let us not e that there is no counterpart
+(within the synchronous gauge analysis considered in secti on 3) of the scale-dependent effects
+introduced by the first equation (92). At the same time, there may be scale dependent features
+which are tied to the particular cosmological model under co nsideration, and they need not show
+up in another model, even if we compare them within the same ga uge. Indeed, take once more the
+model discussed in Ref. [59], where the DE is self-conserved and there are no perturbations in this
+variable. Insuch situation, thecosmological perturbatio nsof matter in thesynchronousgauge turn
+out to be exactly scale-independent (without any approxima tion). In our case, in contrast, there is
+someresidual scale dependencein the synchronousgauge, al though it can beshown to benegligible
+– as we have extensively discussed in section 3. Gauge differen ces, however, can be significant in
+certain cases, specially for calculations involving very l arge scales, showing that in general it is
+29non-trivial to relate the perturbation variables to observ able quantities in different gauges [57,60].
+In that case, one must keep the appropriate k-dependence (as e.g. in the Newtonian gauge [59])
+or resort to a gauge-invariant formalism [61,62]. However, for the range of wavenumbers usually
+explored in the analysis of the matter power spectrum (viz. 0 .01hMpc−1<k<0.2hMpc−1[3]),
+the gauge differences, and in particular the k-dependence, should be small enough so as to be
+licitly neglected. In other words, for such scales, the sub- Hubble approximation that we have
+taken in this Appendix holds good and the calculations in the two gauges are found to be exactly
+coincident, as we expected.
+Finally, let us note that the reason why the physical discuss ion can be more transparent in
+the Newtonian gauge is because the time slicing in this gauge respects the isotropic expansion
+of the background. The synchronous gauge, instead, corresp onds to free falling observers at all
+points, entailing that its predictions are relevant only to length scales significantly smaller than
+the horizon, but at these scales it renders the same physics a s the Newtonian gauge [57,59,60].
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+arXiv:1001.0261v1  [astro-ph.HE]  1 Jan 2010The Inclination of the Soft X-ray Transient A0620–00
+and the Mass of its Black Hole
+Andrew G. Cantrell
+and
+Charles D. Bailyn
+Department of Astronomy, Yale University. PO box 208101, Ne w Haven, CT, 06520
+andrew.cantrell@yale.edu, charles.bailyn@yale.edu
+Jerome A. Orosz
+Department of Astronomy, San Diego State University, 5500 C ampanile Drive, San Diego,
+CA 92182-1221
+orosz@sciences.sdsu.edu
+Jeffrey E. McClintock
+Harvard-Smithsonian Center for Astrophysics, 60 Garden St reet, Cambridge, MA 02138
+jmcclintock@cfa.harvard.edu
+Ronald A. Remillard
+Kavli Institute for Astrophysics and Space Research, MIT, C ambridge, Massachusetts 02139
+rr@space.mit.edu
+Cynthia S. Froning
+Center for Astrophysics and Space Astronomy, University of Colorado, 593 UCB, Boulder,
+CO 80309-0593
+Cynthia.Froning@Colorado.EDU
+Joseph Neilsen
+Harvard University Department of Astronomy, 60 Garden Stre et, MS-10 Cambridge, MA
+02138, USA
+jneilsen@cfa.harvard.edu– 2 –
+Dawn M. Gelino
+NASA Exoplanet Science Institute, California Institute of Technology, MS 100-22, 770
+South Wilson Avenue, Pasadena, CA 91125
+dawn@ipac.caltech.edu
+Lijun Gou
+Harvard-Smithsonian Center for Astrophysics, 60 Garden st reet, Cambridge, MA 02138,
+USA
+lgou@cfa.harvard.edu
+ABSTRACT
+We analyzephotometry of theSoft X-rayTransient A0620-00spa nning nearly
+30 years, including previously published and previously unpublished da ta. Pre-
+vious attempts to determine the inclination of A0620 using subsets o f these data
+have yielded a wide range of measured values of i. Differences in the measured
+value of ihave been due to changes in the shape of the light curve and un-
+certainty regarding the contamination from the disk. We give a new t echnique
+for estimating the disk fraction and find that disk light is significant in a ll light
+curves, even in the infrared. We also find that all changes in the sha pe and
+normalization of the light curve originate in a variable disk component. After
+accounting for this disk component, we find that all the data, includ ing light
+curves of significantly different shapes, point to a consistent value ofi. Com-
+bining results from many separate data sets, we find i= 51.◦0±0.9, implying
+M= 6.6±0.25M⊙. Using our dynamical model and zero-disk stellar VIH magni-
+tudes, wefind d= 1.06±0.12kpc. Understanding thediskoriginofnon-ellipsoidal
+variability may assist with making reliable determinations of iin other systems,
+and the fluctuations in disk light may provide a new observational too l for un-
+derstanding the three-dimensional structure of the accretion d isk.
+Subject headings: X-rays: binaries — stars: individual (A0620-00) — accretion
+disks– 3 –
+1. Introduction
+The prototypical soft X-ray transient A0620–00 (hereafter A0 620) was discovered in its
+1975 outburst, when it reached nearly 50 Crab and became the brig htest X-ray nova ever
+observed (Elvis et al. 1975). In quiescence, the X-ray luminosity is 3 ×1030erg/s (Garcia et
+al. 2001), a tiny fraction of the stellar flux. Optical flux in quiescenc e is dominated by the
+secondary star, a K3-7 dwarf (Froning, Robinson & Bitner 2007 an d references therein). The
+mass function was first measured by McClintock & Remillard (1986, he nceforth MR86), who
+foundf(M) = 3.18±0.16M⊙. This number makes A0620 a strong black hole candidate and
+has been confirmed and refined by Marsh, Robinson & Wood (1994, h enceforth MRW94),
+Orosz et al. (1994), and Neilsen, Steeghs & Vrtilek (2008, hencefo rth NSV08). MRW94 and
+NSV08 also presented Doppler tomography of the accretion disk, r evealing an asymmetrical
+disk with a prominent bright spot at the stream-disk impact point. Th e quiescent disk has
+been found to vary photometrically on a range of timescales (e.g. MR W94, NSV08), and to
+be significant even in the infrared (Froning, Robinson & Bitner, 2007 , henceforth FRB07).
+The optical variability of the X-ray quiescent disk can be categorize d in three distinct states,
+which are distinguished by brightness, color, and aperiodic variability (Cantrell et al. 2008,
+henceforth CBMO08).
+A0620’s mass function makes it a strong black hole candidate, but a p recise determina-
+tion of the mass of the primary depends on knowing the inclination of A 0620’s orbit. Many
+attempts have been made to determine the inclination by fitting ellipso idal light curves, but
+the results have been inconsistent: Haswell et al. (1993) found i >62◦; Shahbaz et al.
+(1994) found i= 36.◦7; Gelino, Harrison & Orosz (2001, heceforth GHO01) found i= 40.◦8;
+Froning & Robinson (2001, henceforth FR01) found a wide range of inclinations in different
+epochs of data and concluded that 38◦< i <75◦. These measurements of iallow for masses
+ranging from 3.4 M⊙fori= 75◦to 13.2M⊙fori= 37◦(MRW 1994). In this paper, we
+give a comprehensive reanalysis of new and previously published light c urves and find that
+all available data are consistent with a single value of i.
+The disagreement between published values of ihas at least two origins, both of which
+have been discussed in the literature: secular changes in the shape of the light curve, and
+uncertainty in the contribution of the disk. We now describe each of these briefly.
+(1) A0620’s light curve has changed shape repeatedly over the yea rs (e.g. Leibowitz,
+Hemar, & Orio 1998, henceforth LHO98), and attempts to model t he different light curves
+have not producedconsistent results. Forexample, FR01modeled three different light curves
+using a model with an accretion disk and hot spot, and found inclinatio ns of 53, 70, and
+74 for the different curves. At least some of the changes in curve s hape are due to optical
+state changes identified by CBMO08. In what CBMO08 call the “pass ive state”, A0620– 4 –
+showed a consistent curve shape with no aperiodic variability from 19 99-2003. Other states
+are brighter, bluer, and show strong aperiodic variability on many tim escales.
+(2) The inclination derived from a given light curve depends sensitively on the disk
+fraction assumed in making the fit. For a single light curve, FR01 find in clinations ranging
+from from i= 44◦toi= 64◦by varying the disk fraction (defined as disk flux over total
+flux) from 0.2 to 0.5. Some authors have argued that the disk is neglig ible in the NIR and
+thus assumed a zero disk fraction in determining i(e.g. GHO01). However, FRB07 find
+a significant NIR disk fraction and instead determine iby using earlier light curves and
+assuming a constant disk fraction. Subsequent work has shown th at the disk fraction is
+highly variable: NSV08 find that the V-band disk fraction ranges fro m 0.48 to 0.76 over
+three nights of observations, and CBMO08 show that the infrared and optical show similar
+photometricvariability. Determining ithereforerequiresabetterunderstanding ofvariations
+in the disk fraction.
+In this paper, we address the issues of variable curve shape and un certain disk fractions
+and present a comprehensive reanalysis of light curves spanning alm ost 30 years. We find
+that once these two problems are addressed, the full data set ind icates a self-consistent value
+ofi. In Section 2, we describe the available data and select a subset whic h will give the
+most reliable determination of i. In Section 3, we present a method for determining the disk
+fraction in specific curves, and we derive the disk fraction for each of the good light curves
+identified in Section 2. In Section 4, we fit ellipsoidal models to these ligh t curves, fixing
+the disk fraction using the results of Section 3. Fitting each curve in dividually, we find that
+many curves give consistent determinations of i; combining these determinations of i, we
+findi= 50.◦98±0.87, implying M= 6.6±0.25M⊙. In Section 5, we use our firm dynamical
+model to make an improved estimate of the distance to A0620. In Se ction 6, we discuss our
+results, including the implications of this work for future determinat ions ofiusing ellipsoidal
+variations in other systems.
+2. Collection of Passive-State Light Curves
+CBMO08identifiedthreedistinctopticalstatesinX-rayquiescence andarguedthatonly
+data in what they call the passive state may be reliably used to deter mine the inclination.
+Whereas passive-state data in CBMO08 retain a consistent curve s hape across 4 years of
+observations, the active state data show a variable and poorly-de fined curve shape. In
+addition, active-state data have a large and variable disk fraction: In active state optical
+observations on 2006 December 14, NSV08 observed the V-band d isk fraction to change from
+59% to 75% in just 10 minutes (J. Neilsen, private communication). Su ch large fluctuations– 5 –
+in the disk fraction pose a serious challenge to a determination of ifrom ellipsoidal light
+curves; the problems with using active state data to determine iwill be discussed in detail
+in Section 6. We therefore carry out a reanalysis of existing data, u sing exclusively passive
+state data to determine i.
+In this section, we present a comprehensive collection of passive st ate light curves with
+well-definedcurveshapes. InSection2.1,wedescribethefullphot ometricdatasetconsidered
+in this paper and describe the calibration procedures used to combin e curves from different
+epochs. In Section 2.2, we restrict our attention to passive-stat e data, which we identify
+using a more general definition than that given in CBMO08. Within the p assive state data,
+we identify a limited set of curves which show a consistent light curve o ver at least a full
+orbit, and are therefore usable for determinations of i.
+2.1. Collection and Reduction of Data
+The data considered in this paper were obtained by many observers between 1981 and
+2008, and include most previously published light curves and some pre viously unpublished
+data; thesedataaresummarized inTable1. Ouranalysisincludes the fulldatasetspresented
+in MR86, LHO98, FR01, GHO01, and CBMO08. Details of the observat ions and reduction
+of these data sets may be found in the respective papers. Though some of these data sets
+have previously been published only as folded light curves binned in pha se, we were able to
+obtain unbinned versions of all data sets except GHO01. Unbinned d ata are necessary to
+reliably identify passive states and changes in curve shape, so we will postpone an analysis
+of the GHO01 data until Section 4.
+In order to compare data from different papers, we use common co mparison stars to
+perform a magnitude calibration between different data sets. MR86 includes both I-band
+data and data obtained using a Corning 9780 filter, which they refer to as “B+V” and which
+we call the W-band. The W-band filter has a FWHM covering 3740-575 0˚A comparable to
+the total range covered by the Johnson B and V filters. MR86 give r esults in the W- and
+I-bands as a ratio of A0620’s brightness to that of a single comparis on star, which they call
+star T. In the I-band, we convert these ratios to differential mag nitudes, ( IA−IT), where
+subscripts AandTdenote the magnitudes of A0620 and star T, respectively. We dete rmine
+ITusing SMARTS I-band photometry and add this to the differential ma gnitudes to get
+IA. We were unable to perform any color correction, as we have no I-b and comparison stars
+other than star T: McClintock & Remillard (1983) and MR86 include com parison stars in
+the W-band only. The MR86 I-band magnitude calibration could there fore be significantly
+off if the I-band filter used in MR86 differs from the SMARTS filter.– 6 –
+We normalize the W-band data so that its magnitude calibration is cons istent with the
+SMARTS V-band data; doing so requires a magnitude calibration and c olor correction. We
+do not attempt to adjust the shape of the curve to agree with V-b and curves: the W-band
+curve shapes still reflect the W-band filter, but we adjust their no rmalization so that the
+overall brightness of the data may be compared directly to V-band data. We begin by
+converting the ratios given in MR86 to differential W-band magnitude s, (WA−WT). We
+then compute a differential color correction (described in the next paragraph) to convert
+these values to differential V-band magnitudes, ( VA−VT). Adding the SMARTS value of VT
+then gives VA, so that the MR86 W-band data have the same normalization as the S MARTS
+V-band data. We believe that this magnitude calibration is more reliable than the I-band
+calibration, since we had multiple comparison stars and could therefo re determine a color
+correction.
+To determine the differential color correction relating ( WA−WT) to (VA−VT), we
+use three field stars with differential photometry in MR86, giving us ( W∗−WT) for each
+star. Using SMARTS V-band magnitudes, we find ( W∗−WT)−(V∗−VT) for each star and
+plot these values as a function of V-I color. We then fit a line and dete rmine the value of
+(W∗−WT)−(V∗−VT) corresponding to the V-I color of A0620. We find
+(WA−WT)−(VA−VT) = 0.0±0.1.
+The color correction is negligible for two reasons: (1) star T has almo st the same color as
+A0620 (V-I=1.48 for star T and 1.54 for A0620), so the differential magnitude depends only
+weakly on the filter, and (2) ( W∗−WT)−(V∗−VT) is a very weak function of V-I, with the
+three stars used in McClintock & Remillard (1983) giving statistically co nsistent values of
+(W∗−WT)−(V∗−VT). Inparticular, thoughthecolorofA0620isknowntobeslightlyvar iable
+(FR01), these small changes in color will not affect the magnitude ca libration. Though our
+color correction is based on only three stars, we believe it is trustwo rthy: point (1) would
+be true even if more stars were available, and point (2) is unlikely to be qualitatively wrong,
+given the overlap between the filters and the fact that for red obj ects like A0620, W-band
+fluxwill bedominated byemission intheV-portionofthe“B+V”filter. W ethereforebelieve
+the essential conclusion is robust and that the differential color co rrection is negligible.
+Similarly, we use common comparison stars to give a common magnitude calibration
+to the FR01 H-band data and the SMARTS H-band data. Specifically, we compute for
+each observation in FRB01 the differential magnitude between A062 0 and the mean of the
+comparison stars called ”star 1” and ”star 2” in Table 2 of FR01. We t hen add the mean
+calibrated magnitude of these two stars in the SMARTS images, to ob tain a common mag-
+nitude calibration. However, this magnitude calibration is unreliable, a s the comparison
+stars listed in FR01 do not give a self-consistent calibration relative t o the SMARTS data.– 7 –
+For example, their star 1 and star 2 differ by 1.43 mag, while they differ by 1.48 mag in
+the SMARTS images. With only three comparison stars to work with, it is impossible to
+identify the origin of this discrepancy, or estimate its impact on the c alibration of A0620.
+In addition to the previously published data discussed above, we inclu de two data sets
+obtained by JEM and RAR, that have not been presented elsewhere . The first of these was
+obtained using the McGraw-Hill 1.3m during 1987–1992(W- and I-ban d). These data are an
+extension ofthe photometrypublished inMR86 andwere gathered a ndreduced following the
+procedures described therein. We apply the same magnitude calibra tion and color correction
+used on data from MR86, so that all W- and I-band data may be comp ared directly to the
+V- and I-band data in other papers; as in the MR86 I-band data, we were unable to test
+for any color term in the I-band magnitude calibration, making the ma gnitude calibration
+particularly uncertain.
+The second of the previously unpublished data sets was obtained by JEM and RAR
+using the Fred Lawrence Whipple Observatory (FLWO) 1.2m during 19 93-1996 (W-, V- and
+I-band). The FLWO data were reduced using standard IRAF tasks . To give a consistent
+magnitude calibration across all I-band data, aperture photomet ry was performed using the
+same comparison stars and magnitude calibration as CBMO08.
+All data are folded on the ephemeris given in Johannsen, Psaltis, & Mc Clintock (2009),
+which is sufficiently accurate to determine T0to within 0.005 in phase throughout the 27
+years spanned by our data. We believe that our magnitude calibratio ns are robust, with the
+possible exceptions discussed above, namely JEM and RAR’s McGraw- Hill I-band data and
+the FR01 H-band data. We therefore have a consistent phasing an d magnitude calibration
+for a wide range of data sets, allowing these data to be combined and compared.
+2.2. Selection of Curves for Use in Determining i
+In this section, we identify light curves which may reliably be used to de terminei. We
+begin by expanding the definition of passive given by CBMO08, so that we may identify
+passive data in other data sets. We then identify all the passive nigh ts in the data described
+in Section 2.1. Even among passive-state light curves, we find that t here is significant
+variation in the shape of A0620’s light curve. We show that combining c urves of different
+shapes may produce spurious results, and argue that only curves with a verifiably consistent
+shape and normalization should be used. We therefore apply a secon d cut, eliminating light
+curves which lack compete phase coverage of a single shape and nor malization. After these
+two cuts, we are left with 12 passive-state, single-shape, single-n ormalization light curves,– 8 –
+Table 1. Photometric Data Setsa
+Data SetbDate Range FilterscndNumber of Nights
+MR86 1981 Oct 29-1985 Jan 17 WI 360 23
+MR1 1987 Jan 1-1991 Feb 7 WI 891 11
+LHO98 1991 Jan 08-1995 Nov 19 R 953 28
+MR2 1993 Jan 19-1996 Jan 15 WVI 349 10
+FR01 1995 Dec 15-1996 Dec 12 JHK 1479 11
+GHO01e1999 Feb 25-2000 Dec 11 JHK – 6
+CBMO08 1999 Sept 11-2007 April 1 VIH 4816 934
+aList of all data sets used in this paper; only a fraction of this data is s elected for use in determining i.
+See Section 2.2 for the selection criteria applied.
+bMR1=Data obtained by McClintock & Remillard at the McGraw Hill 1.3m, ex tending the data published
+in MR86; MR2=Data Obtained by McClintock & Remillard at FLWO, 1993-1 996. All other data sets are
+referenced by the paper in which they first appeared. See text fo r details.
+cSee Section 2.1 for a description of the W-band filter; all others are standard.
+dTotal number of exposures in all bands.
+eWe were unable to obtain an unbinned version of the GHO01 data set, so we do not know the total
+number of exposures in the set. The analysis of Section 2.2 cannot b e performed on binned data, so we
+postpone a discussion of the GHO01 data set until Section 4.– 9 –
+which are the most suitable for determining i.
+Our first cut, identifying passive-state data, is made based on brig htness, color, and
+aperiodic variability. For the SMARTS data, obtained nightly over a pe riod of many years,
+we use the same selection criterion as in CBMO08: we identify passive p eriods based on
+a 10-day running average of V and V-I: any 10 day period with a phas e-adjusted average
+V <16.57or(V−I)<1.52isidentified asnon-passive, whileallotherperiodsareconsidered
+passive; see CBMO08 for details. For data sets taken with a more tr aditional cadence than
+the SMARTS data, a 10-day running average is not practical for th e identification of passive
+data. For these data sets, we use the fact that non-passive dat a show substantial aperiodic
+variability, unlike their passive-state counterparts (CBMO08). Fo r all data other than the
+SMARTS data, we therefore identify as passive any night which show s negligible aperiodic
+variability and which, at some phase, lies near the lower envelope of da ta in a given band.
+Specifically, a night is considered passive if it meets two criteria: (1) A periodic variability,
+defined as the scatter of data in phase bins of width 0.03, has a stan dard deviation less than
+the photometric errors or 0.035 mag, whichever is larger; and (2) A t some phase, the data
+are within 0.1 mag of the lower envelope of all data in the same band and at the given phase.
+Table 2 lists all the nights on which we were able to determine whether A 0620 was
+passive, based on the two criteria above. With the exceptions of GH O01 and LHO98, all
+data sets listed in Table 1 appear in their entirety in Table 2, split into pa ssive and non-
+passive nights. We were able to obtain the GHO01 data only in binned fo rm, so we cannot
+reliably identify passive and active nights within their data set. The R- band data of LHO98
+consistently show aperiodic variability greater than 0.035 mag, makin g the entire data set
+non-passive by our definition. However, there is evidence for gene rally enhanced variability
+in the R-band (Haswell et al. 1993). We suspect that the enhanced aperiodic variability in
+the R-band is due to variation in the H- αline, rather than variability in the disk continuum
+flux. Since variations in the continuum drive the state changes obse rved in other bands,
+R-band variability does not necessarily correlate with these state t ransitions. It is therefore
+likely that A0620 was passive on some nights of LHO98 observations, but that the R-band
+cannot be used to identify the states apparent in other bands. We exclude R-band data from
+subsequent analysis because of the high level of aperiodic variability and the uncertainty in
+the identification of passive nights.
+The full collection of passive-state data includes many partial night s of data with in-
+complete phase coverage. It is tempting to combine these data to g et complete light curves,
+but we now argue that this can produce misleading results. In Figure 1, we show light
+curves obtained by JEM and RAR, on the nights of 1989 January 12, 14, and 15. Though
+the January 14-15 data are consistent, the curve shows a clear c hange of both shape and– 10 –
+Table 2. Identification of Passive and Active Data Setsa
+Date Range Passive? data set
+1981 Oct 29-Nov 03 N MR86
+1982 Dec 15-20 N MR86
+1983 Dec 6-16 N MR86
+1985 Jan 12-17 N MR86
+1987 Jan 1-4 N MR1
+1988 Jan 12-15 Y MR1
+1989 Jan 12-16 Y MR1
+1991 Feb 07 N MR1
+1992 Jan 01 N MR1
+1993 Jan 20-25 N MR2
+1994 Jan 5-8 Y MR2
+1995 Dec 16-18 N FR01
+1996 Jan 14-16 N MR2
+1996 Jan 27-29 Y FR01
+1996 Dec 7-12 Y FR01
+1999 Sept 12-22 N CBMO08
+1999 Sept 23-Oct 29 Y CBMO08
+1999 Oct 30-Nov 11 N CBMO08
+2000 Jan 26-2000 Feb 05 N CBMO08
+2000 Feb 13-April 09 Y CBMO08
+2000 Oct 1-2001 Feb 20 Y CBMO08
+2001 Feb 21-Mar 25 N CBMO08
+2001 Mar 26-Apr 10 Y CBMO08
+2001 Oct 13-2001 Nov 6 N CBMO08
+2001 Nov 10- 2002 Jan 25 Y CBMO08
+2002 Feb 08-2002 Mar 29 Y CBMO08
+2002 Mar 31-May 09 N CBMO08
+2003 Feb 04-Apr 04 Y CBMO08
+2003 Oct 03-Nov 07 Y CBMO08
+2003 Nov 08-Dec 10 N CBMO08
+2003 Dec 12-Dec 15 Y CBMO08
+2003 Dec 16-2007 Apr 1 N CBMO08
+aNightsonwhichA0620couldbereliablyidentified aspassiveornon-pas sive. Statescouldnotbe identified
+from the GHO01 data, as we obtained the data in binned form only. St ates also could not be identified from
+the LHO98 data because of the persistently high level of R-band va riability (see text).– 11 –
+normalization between January 12 and January 14. If all three nigh ts of data were binned
+together, we would get a spurious light curve with a shape dependen t on the orbital phases
+covered each night. In addition, if we had January 14 data only betw een phases 0.8 and
+0.35, we might combine the three nights and obtain a smooth curve co vering all phases,
+never realizing that the combined curve is drawn from different light c urves indicating dif-
+ferent physical conditions in the binary. Model light curves assume that all variation is due
+to our changing viewing angle of the system; spurious results will thu s be given by fitting
+curves which, like the January 12-15 curves, include variation intrin sic to the system.
+Considering the risk posed by combining data taken on different night s, we allow our-
+selves to combine data into a single curve only when there is clear evide nce that all the data
+being combined comes from a single-shape, single-normalization curv e. We therefore require
+that the data being combined meets one of two criteria: either (1) t here is a single night
+of observations which covers at least one full orbit, is self-consist ent, and is consistent with
+any data combined from consecutive nights; or (2) data taken on c onsecutive nights cover
+every orbital phase at least twice and all overlaps are consistent. For example, criterion
+1 allows us to combine the January 14 and 15 data shown in Figure 1; cr iterion 2 allows
+us to combine all passive SMARTS data, which shows a verifiably consis tent shape across
+several years of observations even though no one night covers a full orbit. Combining data
+only when permitted by one of the above criteria, we identify twelve lig ht curves from seven
+epochs which are usable for determining i. These light curves are listed in Table 3 and shown
+in Figure 2.
+The light curves listed in Table 3 include data in four wavebands from FR 01, CBMO08,
+and previously unpublished data obtained by JEM and RAR. Figure 2 sh ows substantial
+variability among the passive light curves, even within a single band: I- band curves range
+fromnearlysymmetric (I4), tohaving onevery deepminimum (I6), t ohaving uneven minima
+and maxima (I8). These curves will form the basis for our determina tion ofi, and the range
+of shapes fit by our models will serve as a consistency check on our r esults.
+3. The Variable Disk Fraction of A0620
+The inclination derived by fitting ellipsoidal light curves depends sensit ively on the
+disk fraction assumed in the fits (FR01), and authors have fit light c urves using various
+assumptions on the amount and variability of the disk light. FRB07 fou nd an H-band disk
+fraction of 18%, demonstrating that one cannot assume the disk t o be negligible, even in
+the infrared. NSV08 used the veiling of stellar lines to make a time-res olved determination
+of the V-band disk fraction, and found that the disk fraction can c hange substantially on– 12 –
+0.0 0.5 1.0 1.5 2. 016.75 16.65 16.55
+Orbital phaseI-ba nd ma gnit ude
+Fig. 1.— I-band photometry of A0620. Data were taken in 1989 on th e nights of January 12
+(black circles), January 14 (open circles), and January 15 (grey c ircles). The data show that
+the light curve may change on short timescales even in the passive st ate, which provides a
+caution about combining data obtained on different nights.– 13 –
+Table 3. Passive, Single-Shape, Single-normalization light curvesa
+Curve IDbData SetcDate Range n f 0.554die
+W1 MR1 1988 Jan 12 22 0 .34±0.03 48.79±2.73
+I1fMR1 1988 Jan 12 64 0 .24±0.03 53.36±1.66
+W2 MR1 1988 Jan 13 20 0 .34±0.03 44.42±4.60
+I2fMR1 1988 Jan 13 61 0 .20±0.03 48.14±1.20
+W3 MR1 1988 Jan 14 24 0 .40±0.03 52.56±4.31
+I3fMR1 1988 Jan 14 71 0 .26±0.03 45.92±1.32
+I4 MR2 1994 Jan 4- Jan 7 122 0 .19±0.03 56.19±3.27
+V6 CBMO08 1999 Sept 23-2003 Dec 15 769 0 .35±0.03 51.75±1.05
+I6 CBMO08 1999 Sept 23-2003 Dec 15 742 0 .25±0.03 50.13±1.35
+H6 CBMO08 2000 Sept 30-2003 Dec 15 858 0 .13±0.02 51.58±3.00
+H7fFR01 1996 Jan 27-29, 1996 Dec 7-12 907 0 .16±0.02 43.20±0.81
+I8 MR1 1989 Jan 14-Jan 15 115 0 .15±0.03 50.92±1.18
+aA comprehensive list of passive state light curves with a consistent s hape and normalization. See Section
+2.1 for a description of the full data set from which these curves we re taken and Section 2.2 for details of
+our selection criteria.
+bLetters indicate the wavebands in which the observations were mad e, while numbers identify overlapping
+observations. For example, W1 and I1 consist of W- and I-band obs ervations from the same night.
+cAs in Table 1.
+dPhase 0.554 disk fractions, determined assuming that the disk is the source of all photometric variability
+at this phase. Quoted errors include only the uncertainty inherited from the simultaneous spectroscopy and
+photometry used to determine the zero-diskmagnitude in each ban d. These errorsexclude uncertainty in the
+magnitude calibration, which are discussed in Sections 2.1 and 4. W- an d V-band factions are determined in
+Section 3.1, H-band fractions are determined in Section 3.2 and refin ed in Section 3.3, and I-band fractions
+are determined in Section 3.4.
+eInclination determined by using ELC to model the light curve. Each ligh t curve is modeled separately
+with a model including ellipsoidal variation and a spotted disk. In each c ase,f0.554is constrained to agree
+with the value given in this table, to within the uncertainty given in this t able. The fits and their residuals
+are shown in Figure 2. See Section 4 for details of the fits.
+fLight curves I1, I2, I3, and H7 come from data sets with particular ly uncertain magnitude calibrations.
+The values of f0.554andiare therefore particularly unreliable in these curves and should be t reated with
+caution. See Sections 2.1 and 4.2 for details– 14 –
+Fig. 2a.— Passive, single-shape, single-normalization light curves with full phase coverage.
+Details of each curve are listed in Table 3. To facilitate comparison, th e axes are identical
+for all curves in a given waveband (and for V- and W-band light curve s). Unbinned data are
+plotted for curves with fewer than 500 data points; V6, I6, H6, an d H7 are binned. Lines are
+best fit models to unbinned data, with the disk fractions constraine d by the values in Table
+3; see Section 4 for a description of the fits and Table 3 for the implied values of i. Residuals
+to the fits are shown beneath the respective plots. Figure 2a: cur ves W1, W2, W3, and V6;
+Figure 2b: curves I1, I2, I3, and I6; Figure 2c: curves I4, I8, H7 and H6.– 15 –
+Fig. 2b.—– 16 –
+Fig. 2c.—– 17 –
+timescales ranging from minutes to days. In addition, the broad ran ge of brightness observed
+in CBMO08 suggests that this variability extends to the infrared, an d occurs on timescales
+ranging from days to years. Since the disk is neither negligible nor con stant, a curve-by-
+curve determination of the disk fraction is necessary. In this sect ion, we use spectroscopic
+and photometric observations to determine the disk fraction in eac h light curve listed in
+Table 3.
+Throughout this section, we assume that the stellar flux at a given p hase is constant, so
+that any changein brightness ata given phase is interpreted as ach ange indisk flux. It isnot
+immediately obvious that this is a valid assumption: GHO01 and others h ave argued that
+star spots dominate changes in the shape of A0620’s light curve, imp lying that the stellar
+flux at a given phase is variable. However, we will show in two ways that the disk dominates
+nonellipsoidal variation, withstellar flux constant at a given phase: ( 1) inSection 3.3, we will
+show that nonellipsoidal variations have a color consistent with a disk origin and inconsistent
+with a stellar origin, and (2) in Section 4, we show that the disk fractio ns derived here give
+a consistent value of ifor a wide range of curves, which provides a consistency check on t he
+disk fractions themselves.
+In Section 3.1, we make determinations of disk fractions in W and V, ex trapolating
+from a spectroscopic determination of the disk fraction with simulta neous photometry. In
+Section 3.2, we similarly determine the H-band disk fractions. In Sect ion 3.3, we perform
+a consistency check, demonstrating that the color of nonellipsoida l variations is consistent
+with the disk color given by our V- and H-band disk fractions. Finally, in Section 3.4, we
+determine I-band disk fractions, using a different method based on the results of Section 3.3.
+We thus determine disk fractions for all curves in Table 3.
+3.1. V- and W-band Disk Fractions
+In this section, we make a determination of the disk fractions in the W - and V-band
+light curves listed in Table 1. Assuming constant stellar flux at a given p hase (see above), we
+use a spectroscopic determination of the disk fraction, together with simultaneous V-band
+photometry, to determine how bright A0620 would be at the given ph ase if all disk light
+disappeared. In other V- and W-band light curves, the disk fractio n at the same phase
+can then be determined by comparing the observed brightness to t he computed zero-disk
+brightness. The disk fractions at this phase are used to constrain our fits in Section 4.
+On the night of 2006 December 15, NSV08 observed a V-band disk fr action of 0 .516±
+0.009 in an 11.6-minute exposure ending at HJD 2454084.7187 (J. Neilse n, private communi-– 18 –
+cation). In a 6-minute exposure beginning 27 seconds after the en d of the NSV08 exposure,
+V-band SMARTS photometry found V=17 .83±0.03. The two exposures do not quite over-
+lap, but the disk was stable at the time: the two subsequent spectr a obtained by NSV08 give
+disk fractions of 0.503 and 0.526. The difference between these two values and 0 .516±0.009
+is of marginal statistical significance, reveals no systematic trend , and is small compared to
+the uncertainty in the photometry. We therefore adopt 0 .516±0.009 as the V-band disk
+fraction at the time of the SMARTS photometry, corresponding to phase 0.554. If all disk
+light disappeared, the system would be 48.4% as bright, and V-band m agnitude would in-
+crease by ∆ V= 0.787±0.02. The zero-disk V-band magnitude of the secondary at phase
+0.554 is thus (17 .83±0.03)+(0.787±0.02) = 18.62±0.04.
+Comparing the zero-disk magnitude of the secondary star at phas e 0.554 to observed
+passive-state magnitudes at this phase, we can now find the phase 0.554 disk fraction in each
+of the V- and W-band curves listed in Table 1. For example, in curve V6 , the passive state
+SMARTS data, V= 18.150±0.01 at phase 0.554. This is 0 .473±0.04 mag brighter than the
+zero-disk stellar magnitude, implying a disk fraction of 0 .35±0.03 at phase 0.554. Since we
+have performed a magnitude calibration and color correction to allow the direct comparison
+of V- and W-band data, we can similarly determine the phase 0.554 disk fraction for other
+V- and W-band curves. We denote the disk fraction at this phase by f0.554; the values of
+f0.554we derive are listed in Table 3. The phase 0.554 W- and V-band disk frac tions are
+consistently nonzero even in passive-state data: the smallest eve r observed is 0 .34±0.03.
+We note that our V-band disk fractions are somewhat higher than t hose found by
+Gonzales Hernandez et al. (2004) and by Marsh, Robinson & Wood (1 994). It is possible
+thattheseauthorshappenedtoobserveA0620onnightswhenth ediskwasparticularlyquiet,
+or that their observations coincide with orbital phases when the dis k fraction is relatively
+low. GonzalesHernandez et al. (2004)determined their disk fractio nfromspectra coinciding
+with data in curve V6. In Sections 3.3, 4 and 5, we will argue that the d isk fraction in curve
+V6 is phase-variable, ranging from 0.35 near phase 0.5 to 0.25 near ph ase 0.0. The latter
+value is within the errors of the disk fraction found near the V band b y Gonzales Hernandez
+(2004), who provide neither precise times nor phases for their exp osures. Thus, depending
+on the phase distribution of their spectra, their disk fraction near the V band could be
+consistent with the higher value listed for V6 in Table 3.
+3.2. H-band disk fractions
+We now determine H-band disk fractions. We use the spectroscopic disk fraction in
+FRB07, together with simultaneous SMARTS photometry. Unlike NSV 08, the spectroscopy– 19 –
+in FRB07 is not time-resolved. Rather, they give only an average H-b and disk fraction, from
+which we compute the average disk fraction in curve H6, the SMARTS passive data set. To
+allow direct comparison to the V- and W-band disk fractions, we conv ert this phase-averaged
+disk fraction to f0.554in H6, then extend the result to determine f0.554in H7 as well.
+FRB07 find that the average H-band disk fraction on the nights of 2 004 January 8-10
+was 0.18±0.02. Their data cover every phase at least once, with triple coverag e of phases
+0.92-0.47 and double coverage of phases 0.47-0.49, 0.59-0.61, and 0 .69-0.92. The three nights
+of the FRB07 observations include five H-band SMARTS observation s, all of which fall in
+the range of phases with triple coverage in the FRB07 data set. The five SMARTS data
+points are on average 9 ±3% brighter than data at the corresponding phases in H6. We
+assume that the average disk fraction during these five SMARTS ex posures was 0 .18±0.02,
+and that the disk accounts for all excess light relative to passive. U nder these assumptions,
+the stellar fraction in curve H6 is 9 ±3% greater than the stellar fraction in the FRB07 data.
+This implies that the average disk fraction in H6 is 0 .11±0.03.
+To allow direct comparison to V- and W-band disk fractions, we now co nvert the phase
+averaged H6 disk fraction to f0.554. To relate a phase-averaged disk fraction to the disk
+fraction at a specific phase, we need to determine the variation of t he disk as a function of
+phase. We do this by adopting a pure ellipsoidal model and attributing all deviations from
+the model to variation in the disk. This procedure potentially depend s on the inclination
+of the ellipsoidal model, so to test for inclination dependance we use m odels corresponding
+to two different values of i, spanning the range allowed by preliminary fits to the V- and
+W-band curves using the disk fractions of Section 3.1. Specifically, fi ts to all V- and W-band
+curves give 46◦< i <54◦, consistent withthe final determination of iwe will make inSection
+4. We find that within this range, the choice of idoes not make a statistically significant
+difference in the value of f0.554: adopting i= 46◦givesf0.554= 0.132±0.03, while i= 54◦
+givesf0.554= 0.124±0.03. We thus adopt f0.554= 0.13±0.03 for light curve H6.
+Having determined f0.554for H6, we now determine f0.554for H7, the other H-band light
+curve in Table 3. We use the same process as we did for the V- and W-b and light curves.
+Specifically, the mean magnitude of H6 at phase 0.554 is H= 14.92±0.02. Together with
+f0.554= 0.13±0.03, this implies a zero-disk magnitude of H= 15.07±0.04. For H7, we then
+determine f0.554by assuming that all additional light is nonstellar. The results are inclu ded
+in Table 3. The errors quoted include only the error inherited from th e determination of
+f0.554in H6; they exclude the uncertainty in the magnitude calibration of H7 with respect to
+H6. As discussed in Section 2.1, this relative calibration is particularly u ncertain, so the H7
+disk fraction is potentially subject to a large systematic error. The H-band disk fractions
+are smaller than the V- and W-band disk fractions, but they are still significantly nonzero– 20 –
+and therefore necessary to a robust determination of i.
+3.3. The Disk Origin of Nonellipsoidal Variations
+In this section, we will compare the color of nonellipsoidal variations in the SMARTS
+light curves, to the color of disk light implied by the disk fractions of §3.1 and§3.2. We will
+find that the nonellipsoidal variations have a color consistent with a d isk origin. In addition,
+the results of this section provide a consistency check on the resu lts of§3.1 and§3.2: if our
+determinations of V- and H-band disk fractions were off, the implied c olor would likely not
+agree with the observed deviations from an ellipsoidal light curve.
+Thediskfractionsderived inSections3.1and3.2givetheV-Hcolorofd isk light, relative
+to star light, at phase 0.554 in the SMARTS passive light curve. We will e xpress this color in
+terms of the ratio∆H
+∆V, where ∆ Vand ∆Hrepresent the difference between the star-plus-disk
+magnitude and the star-only magnitude. In Section 3.1, we found f0.554= 0.35±0.03 in the
+SMARTS passive V-band curve, implying ∆ V= 0.47±0.04; similarly, the SMARTS passive
+H-banddiskfractiongives∆ H=.15±.036. Together, thesetwovaluesgive∆H
+∆V= 0.32±0.08.
+The SMARTS passive light curves exhibit prominent and wavelength-d ependent devia-
+tions from pure ellipsoidal curves; we now express the color of thes e nonellipsoidal variations
+in terms of∆H
+∆V. In each of 50 phase bins, we compute the differences ∆ Vand ∆Hbetween
+the mean SMARTS passive V-band magnitude and the magnitude of a p ure ellipsoidal light
+curve. As in Section 3.2, we perform this calculation using pure ellipsoid al light curves for
+bothi= 46◦andi= 54◦and find that varying iwithin this range affects the results which
+follow by just a fraction of one standard deviation. In Figure 3, we p lot ∆Vagainst ∆ H
+The slope of the points in Figure 3 gives the color of nonellipsoidal varia tions in terms
+of∆H
+∆V. We determine this slope using a principle components analysis (PCA), because
+χ2minimization systematically underestimates slope when both variables have significant
+uncertainty. PCA may also be biased if ∆ Vand ∆Hhave different errors, so we use a
+Monte-Carlo model to estimate both the bias and the uncertainty in∆H
+∆V. The scatter in
+Figure 3 is consistent with photometric errors, which we therefore use in our Monte Carlo
+error determination. We find a negligible bias and a slope of∆H
+∆V= 0.31±0.065, giving the
+color of non-ellipsoidal periodic variations in the SMARTS passive light c urve.
+We have given two determinations of∆H
+∆V, corresponding to (1) the color of disk light
+identified in Sections 3.1 and 3.2 and (2) the color of periodic, non-ellips oidal variations.
+These two determinations agree to within a fraction of one standar d deviation, suggesting
+thatnonellipsoidal variationsoriginateinthedisk. Onecouldargueth atthis logiciscircular,– 21 –
+18.05 18.10 18.15 18.2014.84 14.88 14.92
+V, relative to pure ellipsoi dalH, relative to pure ellipsoidal
+Fig. 3.—Periodic, non-ellipsoidalvariabilityintheV-andH-bandpassive SMARTSdata. In
+each of 50 phase bins, we compute the difference between the obse rved SMARTS magnitude
+andthatofapureellipsoidal lightcurvewith i= 46◦andzeromeanmagnitude. Ifweassume
+i= 54◦instead, the individual points are displaced somewhat, but there is n o statistically
+significant effect on the slope of the data. The dashed line shows the slope implied by the
+points, as determined by a principle components analysis. The solid line is not determined
+using the points, but rather is the slope we would expect if the non-e llipsoidal variations
+are due to variation a disk component with the color implied by Sections 3.1 and 3.2. The
+agreement between the two lines suggests that the nonellipsodial v ariations originate in the
+disk; see text for a quantitative comparison. The normalization of b oth lines is arbitrary, so
+we have drawn them to agree at the left edge of the plot.– 22 –
+since method (1) assumed that the disk dominates all nonellipsoidal v ariations. However, if
+this assumption is wrong, it would not have a self-consistent effect o n the two determinations
+of∆H
+∆V. Specifically, method (1) depends strongly on the normalization of t he light curve and
+only weakly on changes in shape, while method (2) depends entirely on the curve shape: in
+terms of Figure 3, method (2) uses only the slope of the points, not their normalization. If
+stellar features produced a nonellipsoidal change in stellar flux, met hod (2) would merely
+give thecolorof these stellar featuresat thetimeof theSMARTSligh t curve. Method(1), by
+contrast, would give a result depending on the spectroscopic disk f ractions and the amount
+of excess stellar flux present in both the NSV08 data and the FRB07 data, relative to the
+SMARTS passive data; this amalgam of spectroscopic disk fraction a nd variable stellar flux
+would not have any simple physical interpretation, and would be unlike ly to agree with the
+color of features in the SMARTS light curve.
+The results of this section also provide a check on the spectroscop ic disk fractions from
+NSV08 and FRB07, on which all our disk fractions are based. Hynes, Robinson, & Bitner
+(2005) showed that there may be large systematic errors in deriva tions of disk fractions
+based on veiling. For example, they show that if the star and templat e are mismatched by
+100K, the derived disk fraction may be off by a factor of 2. However , these effects would
+be unlikely to produce a self consistent error in the resulting V- and H -band disk fractions.
+This is particularly true given that the NSV08 and FRB08 data sets we re non-simultaneous
+and active: in order to give the right color, any error in the disk frac tions would have to
+be consistent across a range of wavelengths and also be insensitive to variations in the disk
+between the two sets of observations.
+Finally, we use the slope in Figure 3, together with the value of ∆ Vfrom Section 3.1, to
+giveanewderivationof∆ H. Specifically, ∆ H= ∆V×∆H
+∆V= (0.47±0.04)(0.31±0.065) = 0.15±0.03.
+This value and error are identical to those derived using the disk fra ction of FRB07. We
+improve our error on the H-band disk fraction by combining the two d eterminations, giving
+f0.554= 0.13±0.02, listed in Table 3. In addition to indicating a disk origin of nonellipsoidal
+variations and providing a consistency check on our determinations of disk fractions, the
+results of this section have thus given us a new technique for deter mining disk fractions.
+3.4. I-band disk fractions
+In this section, we extend the method of Section 3.3 to determine th e I-band disk
+fraction of the SMARTS passive light curve, then extrapolate as in S ection 3.2 to determine
+disk fractions for all I-band light curves in Table 3. This is a key deter mination, as half of the
+lightcurves inTable3areI-banddata. FollowingSection3.3, wedivide t heSMARTSpassive– 23 –
+data into 50 phase bins and, in each bin, we define ∆ Vand ∆Ito be the difference between
+the SMARTS passive data and a pure ellipsoidal light curve with i= 46◦; as in Sections 3.2
+and 3.3, using i= 54◦instead affects the results by just a fraction of one standard dev iation.
+In Figure 4, we plot ∆ Ias a function of ∆ V, revealing a linear relationship between
+the two. As with Figure 3, we find that the scatter about the line in Fig ure 4 is consistent
+with photometric errors. Following Section 3.3, we use a principle comp onents analysis
+to determine the slope and a Monte-Carlo model to determine the er ror, giving∆I
+∆V=
+(0.63±0.055). Together with ∆ V= 0.47±0.04, determined in Section 3.1, we find that
+∆I= (0.47±0.04)×(0.63±0.055) = 0.31±0.04 at phase 0.554 in I6, the SMARTS passive
+light curve. This implies f0.554= 0.25±0.03 in curve I6.
+As in Section 3.2, we now use the SMARTS passive disk fraction to extr apolate disk
+fractions in other curves, attributing to the disk all photometric v ariability at a given phase.
+At phase 0.554, the SMARTS passive curve has I= 16.62±0.01 and ∆ I= (0.47±0.04).
+This implies a zero-disk I-band magnitude of 16 .93±0.04 at phase 0.554. Comparing this to
+the observed phase 0.554 magnitude of each I-band light curve in Ta ble 3, we compute f0.554
+for each of these curves. The results are listed in Table 3, along with the results of Sections
+3.1 and 3.2.
+The I6 and I8 disk fractions appear to be quite different, but this is a n artifact of the
+phase we have chosen for measuring the disk fraction. Both I6 and I8 have one very deep
+minimum due to the phase-variable disk, but the phasing of this minimum is opposite in
+the two cases: Phase 0.554 is near maximum disk brightness in I6 and n ear minimum disk
+brightness in I8. If we had instead determined disk fractions at pha se 0.0, the relative values
+of the disk fraction would be reversed.
+4. Determination of iby fitting ellipsoidal light curves
+We now determine iby using ELC (Orosz & Hauschildt 2000) to fit ellipsoidal models.
+ELC generates models of ellipsoidal light curves, with possible contrib utions from the disk,
+star spots, and X-ray heating of the secondary. For a given star and disk configuration,
+it computes the light curve which would be observed through a filter w ith a user-specified
+transmission curve. It has several optimizer routines to search t he wide parameter space of
+possible models, finding optimal fits to data based on χ2minimization. In this section, we
+use ELC to fit models to the data listed in Table 2 and thereby determin ei.
+We have found that all our light curves have a nonzero disk fraction , so we include
+a disk in all our fits. We constrain f0.554in each curve to agree with the value given in– 24 –
+18.05 18.10 18.1518.2016.5416.5816.62 16.66
+V, relative to pure  ellipsoidalI, relative to pure ellipsoidal
+Fig. 4.— Periodic, non-ellipsoidal variability in V- and I-band passive SMA RTS data. As
+in Figure 3, the points plotted come from assuming i= 46, but the slope is not significantly
+changed if we assume i= 54◦instead. The linear relationship indicates that the nonstellar
+light has a consistent color throughout the orbit. The dashed line sh ows the slope of the
+points, representing thecolorofdisklight usedinourdetermination s ofI-banddisk fractions.– 25 –
+Table 2. In addition, the results of Section 3.3 indicate that disk light d ominates the non-
+ellipsoidal component of variation; to allow for this phase-variable dis k component, we allow
+for a spotted disk in all models. We fit each curve individually, rather t han performing
+simultaneous fits on simultaneous curves. This provides a stronger consistency check on i:
+while it is good to find a single value of iconsistent with all curves, it is far more compelling
+to find that multiple curves independently indicate the same value of i. In Section 4.1,
+we describe the parameters involved in our models, and in Section 4.2 w e make a final
+determination of ibased on the results from individual curves.
+4.1. Input parameters used in determining i
+The ELC model as applied to A0620 has many parameters, several o f which can be
+set to reasonable values based on photometric and spectroscopic observations of the system.
+The most important parameters are related either to the basic geo metrical properties of the
+binary and the components therein or to the radiative properties o f the star and accretion
+disk.
+The parameters that set the scale of the binary and determine the shape of the star
+include the orbital period P, the orbital separation a, the ratio of the masses Q≡M/M2,
+whereMis the mass of the black hole and M2is the mass of the secondary star, the
+inclination i, and the Roche lobe filling factor f2. In the case of A0620, the orbital period
+Pis quite well known and was held fixed at P= 0.3230160 d (Johannsen et al. 2009). We
+assume the star fills its Roche lobe, and consequently we set f2= 1. We also assume the
+star rotates synchronously and that the orbit is circular. Finally, t he search of parameter
+space is more efficient if we use the mass of the secondary star M2(a free parameter) and
+the semiamplitude of its radial velocity curve K2(taken from NSV08) to set the scale of the
+binary.
+The parameters related to the radiative properties of the star inc lude its average tem-
+perature T2, its gravity darkening exponent β, and its bolometric albedo A. The value of
+T2was set to 4600 K (GHO01), and the gravity darkening exponent wa s set to 0.1 (Claret
+2000). There is some disagreement on the temperature of A0620 ( e.g. FRB07 find T=4000-
+44000), but this small ambiguity in T will not change out results signific antly, so we use only
+the GHO01 value. Since A0620 is exceedingly faint in X-rays in quiescen ce (Garcia et al.
+2001), the effects of X-ray heating were neglected. Since ELC use s specific intensities derived
+from model atmosphere computations, no parameterized limb dark ening law is needed. The
+specific intensities used for A0620 were derived from the NextGen grid (Hauschildt, et al.
+1999a, 1999b) with updates (Hauschildt private communication).– 26 –
+Since all curves in Table 2 have a nonzero disk fraction, all our fits inc lude a disk. The
+disk in ELC is circular and flared, and the parameters describing it inclu de the outer radius
+of the disk rout, the inner radius of the disk rin, which was fixed at 0.001 times the black
+hole’s Roche lobe radius, the opening angle of the outer disk rim βrim, the temperature of
+the inner edge Tdisk, and the power-law exponent on the disk temperature profile ξ, where
+T(r) =Tdisk(r/rin)ξ. Since optical emission is dominated by the outer disk, the precise va lue
+of the inner radius is unimportant to the optical emission discussed h ere.
+Our models all include a hotspot on the disk, since we have argued in Se ction 3.3 that
+disk light — not star light — is the dominant source of periodic, non-ellips oidal variability.
+In all cases, we find that one spot is sufficient to fit the data. In the ELC model, the spot
+is on the rim and extends down to a radius rcuton the disk face. The temperature on each
+pixel within the spot is some constant sspottimes the underlying temperature. The spot is
+centered at azimuth θspot, and has a radius wspot(in degrees). Thus one needs four additional
+parameters to model a spot on the disk.
+We note that at low inclinations, the spot is generally in view for most or all of the orbit:
+it is not visible at some phases, then hidden at others. Rather, the s pot sits on the surface
+of a flared disk and changes in projected intensity as our viewing ang le of the flared disk
+changes. This results in a smooth variation in brightness throughou t the orbit, in contrast
+to the sharp dip which would be present if the spot were ever eclipsed , by either the disk or
+the star. In addition, the spot color is consistent throughout the orbit since the variation in
+flux is due to a projection effect, not the appearance and disappea rance of extra-hot regions.
+These characteristics — smooth variation and constant spot color throughout the orbit —
+are apparent in Figures 3 and 4, where an eclipsed feature would app ear as a discontinuous
+jump in color and magnitude.
+To summarize, we have 11 free parameters: i,M2,K2,Tdisk,rout,βrim,ξ,rcut,sspot,θspot,
+wspot. We have an additional three extra constraints on the model. NSV0 8 measured a K-
+velocity of K2= 435.4±0.5 km s−1, and a projected rotational velocity of Vrotsini= 82±2
+km s−1. These two constraints are imposed by applying a penalty to χ2, equal to the weight
+of any one data point. Finally, in each light curve, we likewise constrain the disk fraction
+f0.554by applying a penalty to χ2for deviations from the value given in Table 2.
+The fits were optimized by using three of ELC’s available optimizers: a s cheme that uses
+a genetic algorithm, an algorithm using a Monte Carlo Markov Chain, an d a scheme using a
+downhill simplex method. These optimizers were used in combination, w ith the results from
+each contributing to the final χ2curve. We fit each curve individually, even in data sets
+which were obtained simultaneously. The inclinations given by our fits a re listed in Table 3;
+the best fits and residuals are shown in Figure 2.– 27 –
+4.2. Determination of i
+In this section, we give a determination of iwith the smallest uncertainty that can be
+achieved based on the twelve measurements of ilisted in Table 3. The values of iin Table
+3 are generally susceptible to four sources of error: (1) any misma tch between the SMARTS
+magnitude calibration and the magnitude calibration in other data set s, leading to an error
+inf0.554and thus in the derived value of i; (2) Systematic errors in our fitting procedure,
+e.g. imperfect modelling of the disk or star, or problems in our method of determining
+disk fractions; (3) the statistical errors quoted in Table 3; and (4 ) Systematic errors due to
+the fact that all disk fractions are derived from just two spectro scopic observations and are
+therefore not independent. We will consider each of these errors in turn.
+We begin by considering the error due to magnitude calibration relativ e to the SMARTS
+data set. Since we are concerned only with relativecalibration, the SMARTS data set is not
+susceptible to this error; data sets with calibration problems will the refore give determi-
+nations of iwhich differ systematically from the values given by SMARTS curves. T ak-
+ing a weighted mean of the values of igiven by the SMARTS passive light curves gives
+i= 51.◦13±0.85. Of the nine non-SMARTS curves in Table 3, six curves (W1, I1, W2 , W3,
+I4, and I8) give values of iconsistent with the SMARTS value: among these six curves, the
+biggest deviation from the SMARTS value is 1 .54σ, consistent with what one would expect
+from statistical uncertainty alone. Given that there is little room fo r any non-statistical
+source of disagreement among these curves, calibration errors, if any, must be small.
+The three curves which deviate significantly from the SMARTS value o fiall come from
+data sets identified in Section 2.1 as being particularly susceptible to m agnitude calibration
+errors: (1)H7comes fromFR01, inwhich differential magnitudes of variouscomparisonstars
+are inconsistent with the SMARTS values, and (2) I2 and I3 come fro m the McGraw-Hill
+I-band data set, in which the I-band data only had one comparison s tar and we therefore
+could not perform a color correction. Moreover, I1, I2 and I3 indic ate a self-consistent
+calibration error within this data set: I1, which agrees with the SMAR TS value of i, is the
+curve whose fit is least sensitive to the disk fraction and thus to the magnitude calibration.
+Specifically, if the magnitude calibration is off by 0.12 mag, all three cur ves would give
+inclinations within 1 .5σof the SMARTS value. A consistent calibration error thus puts
+these curve in better agreement with each other and with the SMAR TS curves. We conclude
+that magnitude calibrations and disk fractions in I1, I2, and I3 and H 7 are likely off, and we
+therefore exclude these curves from subsequent analysis of i.
+We now consider the second source of error mentioned above, nam ely the possibility
+that our fitting procedure gives spurious values of idue to either imperfect modeling by
+ELC or faulty assumptions in our method of determining disk fraction s. We have seen above– 28 –
+how spurious disk fractions lead to conspicuous errors in i, so the agreement between V6,
+I6, and H6 supports our determinations of the disk fractions. If, for example, we used the
+disk fractions given by either GHO01 or Gonzales Hernandez et al. (2 004), V6, I6 and
+H6 would give inconsistent results. Moreover, a self-consistent er ror in these disk fractions
+seems improbable: if the assumption of constant stellar flux at a give n phase were wrong,
+we would expect V6 and H6 to give inconsistent results, given that th e two spectra are from
+different epochs. Similarly, if the color of nonellipsoidal variations in th ese curves did not
+indicate the color of the disk, then the I-band fraction would be off a nd we would expect I6
+to give a different inclination than V6 and H6. Our assumptions are fur ther supported by
+the agreement between V6 and the W-band curves, the agreemen t between I6, I8, and I4,
+and the general agreement between W-band and I-band data.
+The agreement between different curves also suggests that any f ailures of the models
+used by ELC do not result in a significant error in i. Most obviously, if ELC’s simple disk
+model was flawed in a way which yielded a spurious ellipsoidal component , we would expect
+the curves with strong nonellipsoidal features — V6, I6, H6, and I8 — to give inclinations
+differentthancurveswithminordiskfeatures—W1, W2, W3, andI4. Inaddition, theagree-
+ment among the SMARTS curves indicates that the disk model produ ces no systematic error
+ini: the disk is more prominent in bluer bands, so if we were over- or unde r-compensating
+for the disk, we would expect a systematic trend in ibetween the different bands; no such
+trend is present. In addition, the agreement between curves in diff erent bands supports the
+robustness of the pure ellipsoidal light curves generated by ELC. F or example, if the code
+used a nonphysical model of limb darkening, it could produce a syste matic error in the am-
+plitude of light curve associated with a given i. Such an error, however, would probably not
+be consistent across several bands. The agreement between re sults from curves in different
+bands, with different nonellipsoidal components, therefore suppo rts the robustness of the
+models used in ELC.
+We now consider the statistical error, which we reduce by combining together the results
+from many curves. Reducing the statistical error in this manner is w orthwhile only if the
+systematics are small, but we have argued that they must be: while s imultaneous curves
+share some systematics, curves in the same band share others, a nd curves with a relatively
+large disk component share yet others. The agreement across th ese various groups indicates
+that the corresponding systematic systematic effects are all sma ll. The statistical errors are
+ingoodagreement withthescatterpresent inourvaluesof i, indicatingthattheformalerrors
+given by fitting are realistic. For example, the inclinations given by V6, I6, and I8 all have
+errors below 1 .5◦and give values of iranging from 50.13 to 51.75. The other five light curves
+with reliable magnitude calibrations have larger errors, but taking a w eighted mean of their
+results gives i= 50.◦93±1.5. We therefore have four statistically independent measurement s– 29 –
+ofi, each with errors near 1 .5◦, which span a range of just 1.62 degrees, smaller than would
+be expected given the statistical errors. This indicates that the f ormal errors given by the
+fits are realistic and, if anything, are overestimated. We therefor e compute a refined value
+ofiby taking a weighted mean of the results from W1, W2, W3, I4, V6, I6 , H6, and I8. The
+weighted mean of these eight measurements of iisi= 50.◦98±0.71, which we adopt as our
+final determination of iand the statistical error on i.
+Finally, we consider the systematic error due to the fact that all de terminations of
+ioriginate from just two spectroscopic measurements of the disk f raction. In particular,
+all W-, V- and I-band disk fractions are based on the same spectro scopic measurement in
+NSV08; seven of the eight curves used to determine iwill therefore suffer from the same
+systematic error if the result from NSV08 is off. To estimate this sys tematic error, we fit
+every V- and I-band curve with disk fractions offset by ±σ, then recompute the weighted
+mean inclination, once with systematically large disk fractions and onc e with systematically
+small disk fractions. We find that the weighted mean changes by ±0.5 when all V- and
+I-band disk fractions are systematically offset by 1 σ, and we adopt this as the systematic
+error due to the common origin of the disk fractions. Combining this in quadrature with the
+statistical error of 0.71, we find i= 50.◦98±0.87; this error includes both statistical error
+and the only systematic error which we believe is significant.
+As a final check on the value of i, we fit an ellipsoidal light curve with a spotted disk to
+the H-band light curve in GHO01. This curve was not included in Tables 1 and 2, because
+we were able to obtain the data only in binned form and therefore can not be sure that it is
+passive and consistent in shape throughout the observations. We therefore find these results
+less trustworthy than others, but view them as a useful check on our determination of i.
+Following the method of Section 3.2, we find a disk fraction of 0 .2±0.05. Assuming this
+disk fraction and fitting the curve with a model including a spotted dis k, we find i= 48◦±3,
+consistent with the value of i= 50.◦98±0.87 determined from the curves in Table 3. The
+lower value of idetermined by GHO01 is therefore completely explained by their decis ion to
+exclude the disk.
+In conclusion, we find the inclination of A0620 to be i= 50.◦98±0.87, implying a black
+hole mass of M= 6.61±0.25M⊙(NSV08) and a stellar mass of M2= 0.40±0.045M⊙. Of
+the twelve curves in Table 3, nine are consistent with this value. The o nly deviants — I2, I3,
+and H7 — are likely off due to problems with their magnitude calibration. T he curves with
+reliable magnitude calibrations have a spread consistent with that ex pected from statistical
+errors alone, indicating that most sources of systematic error ar e small; in particular, the
+assumption of constant stellar flux at a given phase, and the modelin g of the star and disk
+in ELC, appear to be robust against systematic errors in i. The data sets included in this– 30 –
+paper previously gave inclinations spanning nearly 40 degrees, but w e have brought them
+into agreement by excluding active state data, combining light curve s only if they show a
+consistent shape, and accounting for disk light on a curve-by-cur ve basis.
+5. The Distance to A0620-00
+In this section, we follow the method of Barret, McClintock & Grindlay (1996) to
+estimate the distance to A0620: we compute the absolute magnitud e implied by the spectral
+type and radius of the secondary, then use its apparent magnitud e and reddening to estimate
+the distance. We use Eggleton (1983) to compute R2, the effective radius of the Roche
+lobe. We then use Popper (1980) to compute the absolute magnitud e based on R2and the
+spectral type. Finally, we usethe reddening lawofFitzpatrick (199 9)tocompute dereddened
+apparent magnitudes and compute the distance. We note that Pop per (1980) assumed
+log(g/g⊙) = 0.1for aK5Vstar, while our dynamical model implies log( g/g⊙) =−0.05±0.05;
+we will argue later that A0620’s low surface gravity is of minimal import ance to the distance
+determination, and therefore use Popper (1980) and other work s without an adjustment for
+surface gravity. Our robust determinations of iand multi-band disk fractions eliminate two
+of the major uncertainties in the distance determination, namely th e radius of the secondary
+and the flux contribution of the disk (Barret, McClintock & Grindlay 1 996).
+Two major uncertainties remain in the computation of distance: bot h the reddening
+and the spectral type of A0620 have been disputed, and a range o f values for each has been
+quoted in the literature. We will show that these uncertainties have a relatively small effect
+on distance when constrained by the zero-disk color of A0620, but we begin by considering
+the full range of both values which have been claimed in the literature . The reddening of
+A0620 has been determined by many methods: use of the 2175 ˚A feature, Na D lines, dust
+maps, and comprehensive modeling of diffuse interstellar bands. The se various methods give
+values of E(B−V) ranging from 0.25 to 0.49. Hynes (2005, and references therein) argues
+that the most robust value is E(B−V) = 0.35±0.02 (Wu et al. 1983), and that the only
+measurement less than 0.3 is strongly model-dependant.
+The most reliable determinations of spectral type come fromNIR sp ectroscopy obtained
+by FRB07 and Harrison et al. (2007), who find spectral types of K5 and K5-7, respectively.
+These spectral types are in good agreement with the pioneering wo rk of Oke (1977), who
+also found a spectral type of K5-7. Spectral types of K4 (GHO01 ) and K2-3 (Gonzales
+Hernandez et al. 2004) have also been claimed, but there are reaso ns to be suspicious
+of both results. Specifically, GHO01 used only photometric data and neglected the disk
+contribution, while Gonzales Hernandez et al. (2004) determined th e spectral types and– 31 –
+disk fraction based exclusively on the FeI lines; FRB07 argued that t his may have biased
+their results. In addition, the results of Gonzales Hernandez et al. (2004) imply that the
+disk is negligible redward of 7500 ˚A , contrary to the significant I- and H-band disk fractions
+we have found. A spectral type of K5 is thus preferred by existing observations, but for the
+sake of completeness we will consider spectral types of K2, K4, K5 , and K7, spanning the
+full range of values which have been claimed in the literature.
+Though both the reddening and the effective temperature are unc ertain, these two
+uncertainties are not independent: given the zero-disk colors der ived in Section 3, a hotter
+star requires more reddening in order to match the observed color . Since the color of the
+secondaryvariesasafunctionofphase, wecompute E(B−V)atbothphase0.5(aminimum)
+andphase0.25(amaximum). InSection3, wecomputedzero-diskma gnitudesatphase0.554
+only. We therefore use ELC to model the variationinVIH stellar magn itudes asa function of
+phase, normalizing the model VIH stellar magnitudes to agree with ou r zero-disk magnitudes
+at phase 0.554. For a given spectral type and phase, we assume th at the star has the intrinsic
+colors given by Bessell & Brett (1988), assume the reddening law of Fitzpatrick (1999), and
+compute the value of E(B-V) implied by the apparent zero-disk V-H c olor. To check for
+consistency, we use the same procedure to compute the reddenin g implied by the observed
+V-I and I-H zero-disk colors. Where these determinations of E(B- V) are inconsistent, it
+indicates either a nonstandard reddening law or a mismatch between the assumed spectral
+type and A0620. Table 4 lists the values of E(B−V) derived fromvarious spectral types and
+phases, based on the apparent colors of A0620. We use subscript s to denote the color used
+in each determination of E(B−V). SinceE(B−V)V−Hhas the smallest statistical error,
+we use this value to compute the dereddened magnitude V0and distance, again assuming
+the reddening law of Fitzpatrick (1999). Our errors on E(B−V)V−H,V0, anddall include
+any disagreement between E(B−V)V−IandE(B−V)I−H.
+Table 4 shows that K5 is the only spectral type for which the V-I colo r and I-H color
+give consistent values of E(B−V). In addition, K4-5arethe only spectral types which imply
+values of E(B−V) consistent with any previous determination of E(B−V. The spectral
+type of K2 is strongly inconsistent with our data, implying inconsisten t and implausibly high
+values of E(B−V). For spectral types of K7 and K4, E(B−V)V−IandE(B−V)I−Hare
+inconsistent only at the 1 .3σlevel, which could plausibly be due to a nonstandard reddening
+law. However, assuming a spectral type of K7 implies an implausibly low v alue ofE(B−V).
+Though K4 cannot be strongly ruled out, K5 gives the best match to the observed colors for
+a standard reddening law, and is also the value preferred by the mos t robust spectroscopic
+studies (e.g. FRB07).
+Table 4 shows that the distance is only a weak function of the spectr al type assumed.– 32 –
+Table 4. Distance and Reddening of A0620
+MK Type Phase E(B−V)V−IaE(B−V)I−HaE(B−V)V−HbV0cMVdd(kpc)
+K2V 0.5 0 .45±0.04 0 .62±0.05 0 .52±0.085 17 .13±0.27 6.61±0.05 1.26±0.35
+K2V 0.25 0 .42±0.04 0 .59±0.05 0 .49±0.085 16 .98±0.27 6.61±0.05 1.17±0.33
+K4V 0.5 0 .34±0.04 0 .42±0.05 0 .38±0.04 17 .47±0.13 7.20±0.05 1.13±0.16
+K4V 0.25 0 .31±0.04 0 .39±0.05 0 .34±0.04 17 .32±0.13 7.20±0.05 1.05±0.14
+K5V 0.5 0 .30±0.04 0 .30±0.05 0 .30±0.02 17 .71±0.07 7.49±0.05 1.10±0.09
+K5V 0.25 0 .27±0.04 0 .27±0.05 0 .27±0.02 17 .55±0.07 7.49±0.05 1.02±0.08
+K7V 0.5 0 .16±0.04 0 .25±0.05 0 .19±0.04 18 .05±0.13 8.05±0.05 1.00±0.14
+K7V 0.25 0 .13±0.04 0 .21±0.05 0 .16±0.04 17 .89±0.13 8.05±0.05 0.92±0.13
+aE(B−V)V−Iis the extinction derived from the observed zero-disk V-I color at p hase 0.5, while E(B−
+V)I−His derived from the zero-disk I-H color at phase 0.5. In both cases, we assume for each spectral
+type that the star has the intrinsic colors given in Bessell & Brett (1 988) and we use the reddening law
+of Fitzpatrick(1999) to derive the value of E(B−V) implied by the observed zero-disk colors. The errors
+quoted are statistical errors due to uncertainty in our photomet ric calibrations. Inconsistencies between the
+two derivations of E(B−V) indicate either a nonstandard reddening law or a poor fit to the tru e spectral
+type.
+bE(B−V)V−His the extinction derived from the observed Zero-disk V-H color by t he same procedure as
+E(B−V)V−IandE(B−V)I−H. Since this value has the smallest statistical uncertainty and reflec ts color
+overthe full colorrange ofour observations, we use these values for the computation of V0. The quoted errors
+are the larger of (1) the statistical error due to photometric unc ertainty, and (2) half the range spanned by
+E(B−V)V−IandE(B−V)I−H. The latter error reflects uncertainty in E(B−V) due to inconsistencies
+between different bands; this is the dominant source of uncertaint y for all spectral types except K5.
+cDereddened, zero-disk V-band magnitude of A0620 at phase 0.5. C omputed using the observed zero-disk
+brightness at this phase and E(B−V)V−Has given in column 4. Errors on V0are propagated using the
+errors listed in column 4
+dAbsolute visual magnitude computed assuming the star has radius R2and using the relation of Pop-
+per(1980). Popper(1980) does not give a value of F′
+Vfor spectral type K4, so this value was computed by
+linear interpolation between K2 and K5– 33 –
+Assuming a hotter star makes the star more luminous, but requires greater reddening; these
+two changes have opposite effects on the derived distance. For sp ectral types K2-7, the
+distance determined from phase 0.5 ranges from 1.00-1.26kpc; with out the change in implied
+reddening, this range of spectral types would give nearly a factor of 2 uncertainty in the
+distance. Thus, our constraint on the color eliminates over 70% of t he uncertainty inherited
+from the uncertain spectral type. Similarly, the precise value of E(B−V) has a relatively
+small effect on the derived distance: given the observed zero-disk colors, larger values of
+E(B−V) imply hotter stars, counteracting the effect of the reddening on distance. Indeed,
+if we had computed spectral type as a function of E(B−V) for the full range of E(B−V)
+that has been claimed in the literature, we would actually have found a smaller range of
+distances than that listed in Table 4.
+Wenotethatthedeterminationofdistanceisalsoonlyweaklydepend entontheassumed
+value of surface gravity. As with changing the assumed spectral t ype, decreasing log( g/g⊙)
+makes the assumed star both less luminous and redder; this forces a lower value of E(B−V),
+counteracting the effect of the lower luminosity when distance is calc ulated. As an extreme
+example, consider computing the distance as in Table 4 assuming a K5 g iant star in place of
+a K5 dwarf: one finds that the distance differs by just 16%; at a con stant value of E(B−V),
+the difference in distance would be 65%. In A0620, log( g/g⊙) differs from a main sequence
+star by just 0.15, whereas assuming a K5III changes log( g/g⊙) by 2.8. Given that even this
+large change in the assumed log( g/g⊙) has a relatively small effect on the derived distance,
+we are confident that we have not introduced a significant error by assuming a normal main
+sequence log( g/g⊙) in using Popper (1980) and Bessell & Brett (1988).
+The derived distance is also a relatively weak function of phase: the d istance derived
+at the maximum is just 0.08 kpc closer than that derived at minimum, co mparable to the
+standard deviation on the measurement based on a spectral type of K5. Given that the
+precise orbital phase used to compute the distance is of relatively m inor importance, we
+estimate the distance simply by averaging the values given by phase 0 .5 and phase 0.25.
+We then incorporate this range into the error on the final value. Fo r our preferred spectral
+type of K5, this gives d= 1.06±0.12 kpc. Even if the spectral type is off, the effect on
+the distance is not likely to be large: spectral types of K2 and K7 are improbable given our
+data, and assuming a spectral type of K4 changes the derived dist ance by just 0 .25σ.
+6. Discussion
+The results of this paper have broad implications for determinations ofibased on mod-
+eling of ellipsoidal light curves. The state changes we have found in A0 620 are likely present– 34 –
+in many or all of the other seven black hole binaries with periods less th an half a day, such as
+Nova Mus 1991 and GS 2000+25 (Remillard & McClintock 2006). In addit ion, the presence
+of a significant and variable disk component is likely to be a feature of lig ht curves in many
+other systems. A broadband SED is not sufficient to detect the disk : though the infrared
+colors of A0620 are consistent with purely stellar light (GHO01), the disk is significant in
+these bands. Wetherefore believe thatfuture determinations of imust include identifications
+of states, attention to possible changes in the light curve, spectr oscopic determinations of
+disk fraction and models allowing for a phase-variable disk component .
+InSection3.3,weshowedthatthecolorofnonellipsoidalvariabilityint heSMARTSdata
+is consistent with the disk color, and we concluded that the disk domin ates nonellipsoidal
+variations. This conclusion is supported by the success of a phase- variable disk in explaining
+a wide rangeoflight curves witha consistent value of i(Section 4). Aphase-variable disk, far
+frommerely diluting alightcurve andthereby reducing itsamplitude, m ayhave fundamental
+effects on the shape of the light curve: in some cases it may even increase the light curve’s
+amplitude. To illustrate the range of these effects, Figure 5 shows t he best fit models of
+Figure 2, broken down into star and disk components. Included are all light curves except
+I1, I2, I3, and H7, which have questionable determinations of f0.554(see Sections 4.2 and
+2.1), and therefore may not yield accurate disk models.
+Figure 5 shows that even the curves which lack conspicuous nonellips oidal features are
+best fit with significant variation in the disk. There are two types of d isk light curves:
+ones like V6, I6, and I8 which are roughly sinusoidal, and ones like W2, W 3, and I4 which
+are double-humped; W1 and H6 are intermediate, with a flattened min imum but only one
+prominent maximum. The sinusoidal disk curves generally lead to more conspicuously nonel-
+lipsoidal light curves, not because they have a stronger effect on t he shape of the light curve,
+but because their single-waved form produces uneven maxima and m inima.
+The sinusoidal and double-waved disk curves are easily understood in the context of
+ELC’s disk model. All models include a single spot on the rim, extending pa rtway down
+the face of the disk; the relative importance of the face (sinusoida l) and rim (second peak)
+determine the type of light curve observed. The face component o f the spot is always visible,
+but its projected intensity changes sinusoidally with viewing angle, re aching a maximum
+when the spot is on the far side of the disk and therefore most direc tly facing the observer.
+By contrast, the rim component is eclipsed when the spot is on the fa r side of the disk, then
+produces a relatively peaked maximum when the spot is on the near sid e of the disk. The
+rim component is thus in antiphase with the face component, yielding a second maximum if
+the rim produces a significant fraction of the total spot emission. I n V6 and I6 the face is
+dominant, butinH6therimissignificant comparedtotheface. Thisre flectsthetemperature– 35 –
+Fig. 5a.— Components of best fit ELC models for all curves with reliable values of f0.554.
+Curves are the same as the models shown in Figure 2. In each panel, t he top line shows the
+total light curve, the middle line shows the ellipsoidal component, and the lower line shows
+the phase-variable disk contribution. All light curves are plotted in r elative flux units, scaled
+to give total flux 1 at phase 0.554. Figure 5a: W1, W2, W3 and V6; Figu re 5b: I4, I6, I8,
+and H6.– 36 –
+Fig. 5b.—– 37 –
+gradient in the disk: the face is relatively important in the optical, sinc e the spot extending
+down the face is hotter than the rim.
+We note that the double-waved disk features apparent in W2, W3, a nd I4 can easily
+appear to be part of the ellipsoidal modulation: to the eye, W3 and I4 look like purely
+ellipsoidal curves. Indeed, it is remarkable that ELC can pick out the double-waved disk
+component reliably enough to give a consistent value of iacross these curves. The double-
+waved disk feature also gives a sense of why these curves cannot b e well-modeled using a
+spotted star rather than a spotted disk: Though a single disk spot naturally produces a
+double-waved feature, a single starspot produces just a single-p eaked modulation. Thus,
+when a star spot is invoked, much of the double-waved disk feature is modeled as ellipsoidal
+variability, resulting in inconsistent determinations of iand large, phase-variable residuals
+in most curves; this is just what we found when we attempted to mod el these curves with a
+spotted star and a constant disk.
+Though we have given a roughphysical interpretation of the disk ligh t curves, we believe
+that moredetailed physical modelsare necessary in order to have r eal insight into thephysics
+ofthespotdiscussed here. Theactualdiskandspotarepresuma blynottheperfectgeometric
+figures assumed by ELC, and there are degeneracies which allow diffe rent configurations to
+produce identical disk light curves. Without more physical disk mode ls, we cannot know
+whether the spot we have found comes from the optically thin disk wh ich produces the spot
+seen in emission line studies (e.g. NSV08), or whether it is coming from a n optically thick
+component of the sort described by Hynes, Robinson, & Bitner (20 05). Nor can we say with
+certainty whether this spot is related to the stream-disk impact po int or is due to some other
+source of uneven disk heating. It would be fascinating to model the se light curves using full
+MHD disk simulations with radiative transfer. In particular, we note t hat the data in figure
+1 show that the configuration which produced I8 was apparently un stable, and changed on a
+timescale of days. A study of this time variability could be a valuable too l in understanding
+the nature of the spot.
+The disk features discussed here — like the constraint provided by p hase shifts (Cantrell
+& Bailyn, 2007) — reflect the third dimension of disk structure, which cannot be probed by
+Doppler tomography. While the properties of phase shifts depend p rimarily on the velocity
+field and opening angle of the disk, the modulations discussed here de pend primarily on the
+physical locationofbright spotsona three-dimensional disk. Modu lationsin disk brightness,
+phase shifts, and Doppler tomography therefore provide complem entary constraints on the
+structure of disks in accreting binary stars.
+We have demonstrated that a single determination of the disk fract ion with simulta-
+neous photometry may be sufficient to determine i, but future work would be improved by– 38 –
+many independent determinations of disk fraction: the ideal data s et for determining inclina-
+tions would be a passive light curve with simultaneous phase-resolved spectroscopy. In some
+objects, extended periods of activity make it unfeasible to obtain p assive data with simulta-
+neous spectroscopy: CBMO08 and ongoing SMARTS observations in dicate that A0620 has
+been active since 2003, and that Nova Mus 1991 has been persisten tly active for even longer.
+It would nonetheless be possible to obtain a disk-corrected light cur ve from a persistently ac-
+tive object by obtaining photometry with perfectly simultaneous sp ectroscopy corresponding
+to every photometric exposure. Then each photometric measure ment could be corrected for
+the disk contribution present in that exposure , yielding a disk-free light curve which should
+be well-fit by a pure ellipsoidal model.
+It would be tempting to simply bin active state data or use some other smoothing algo-
+rithm to produce a light curve for fitting, but such procedures hav e fundamental problems,
+beyond the challenges intrinsic to working with lower-quality data: (1 ) ELC and similar
+codes assume that the disk is not intrinsically variable, with all variatio n due to changing
+viewing angle. The active disk, however, varies rapidly in intrinsic brigh tness (NSV08), and
+smoothing over such fluctuations will not generally produce a curve which corresponds to an
+intrinsically constant disk. (2) Active state data generally have upp er and lower envelopes
+with distinct shapes. For example, in LHO98 Figures 2b and 2c, there is excess variability
+near phase 0.5, so the depth of this minimum changes by 0.1 magnitude depending on
+whether one considers the mean, the upper envelope, or the lower envelope. It is unclear
+which, if any, smoothing procedure would give a meaningful light curv e. (3) Active state
+data typically include many flaring events, which would produce spurio us features in any
+fitting or smoothing algorithm. For example, in Figure 2c of LHO98, th ere is a .2 magnitude
+flare apparent near phase 0.5, with smaller flares apparent near ph ases 0.2 and 0.3; the active
+light curves of CMBO08 contain many such flares. Though large flare s could be identified
+and removed, smaller flares are so numerous that they could not be identified individually
+and would have a major impact on the light curve shape.
+Small flares are present even in passive data. I8 shows a clear flare : Though its residual
+in Figure 2c is almost perfectly flat, there are three consecutive po ints near phase 0.25 which
+peak 0.04 mag above the fit. Curve I4 similarly has a flat residual with o ne point 0.05 mag
+above all the rest. These features are so isolated and peaked tha t it is easy to identify them
+as minor flares, separate from the smooth variation due to the cha nging viewing angle of
+the disk. Such small flares could not be identified in the SMARTS data, which have larger
+photometric errors and just one exposure per night. It is theref ore likely that many such
+flares exist within the SMARTS passive data. We believe this is the origin of the uneven
+residuals given by fits to the SMARTS curves V6 I6 and H6 in Figure 2: a veraging over
+many minor flares would lead to spurious features, which would appea r as uneven residuals– 39 –
+to the fit. If the flares are more prominent at some phases than ot hers, they would naturally
+produce the phase-modulated residuals we see. This scenario seem s likely, given that the
+only curves to show systematic residuals are those which average o ver a period of years.
+Our results put A0620 in good agreement with the results of Bailyn et al. (1998), who
+find that the masses ofblack holes intransient low-mass X-ray binar ies arestrongly clustered
+near 7M ⊙. In that paper, the mass of A0620’s black hole was only constrained to lie between
+4 and 25M ⊙. It is therefore remarkable that our determination of i, along with refined
+determinations of the mass function and mass ratio, put A0620 in su ch good agreement with
+the masses of the other objects discussed in Bailyn et al. (1998). A s discussed there, the
+clustering of masses at such a large value is unexpected: simple mode ls of stellar evolution
+would predict that lower-mass objects would be more common and ce rtainly do not predict
+clustering at any particular mass. Our result thus strengthens th e conclusion of Bailyn et al.
+(1998) and deepens the question of why black holes should prefere ntially form at a specific
+mass, especially one much larger than the 3M ⊙lower limit for the formation of stellar-mass
+black holes.
+7. Conclusions
+1. Previous disagreements on the inclination of A0620 have likely resu lted from three
+effects: (1) state changes were not recognized before CBMO08, and non-passive data were
+often included in curves to be fit for the determination of i, (2) the light curve may change
+shape from night-to night, and combining curves of different shape s can give misleading
+results, and (3) the disk contribution has not been adequately con strained in previous work.
+2. After correcting for thethree effects above, we find that the data consistently point to
+a single value of i. Based on fits to many light curves which had previously given conflict ing
+results, we find i= 50.◦98±0.87, implying M= 6.61±0.25M⊙andM2= 0.40±0.045M⊙.
+3. Based on our firm dynamical model and zero-disk magnitudes of A 0620, we estimate
+the distance to A0620 to be d= 1.06±0.12kpc.
+4. We find that all non-ellipsoidal variations are dominated by disk light : the disk
+dominates secular changes in brightness, the flickering present in n on-passive states, and
+periodic modulations in the shape of the light curve. The disk fraction is greater than 10%
+in all curves we study, including passive state, infrared light curves .
+5. The lessons learned from A0620 may help make robust determinat ions ofiin other
+objects. Specifically, we have shown the importance of states, dis k light, and night-to-night– 40 –
+changes in curve shape. These effects are probably not unique to A 0620 and our techniques
+for handling them may lead to more robust determinations of iin other objects.
+This work was supported by NSF Graduate Research Fellowship DGE- 0202738 to AGC
+and NSF/AST grants 0407063 and 0707627 to CDB.
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diff --git a/1001.0262.txt b/1001.0262.txt
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+arXiv:1001.0262v1  [hep-th]  1 Jan 2010The No-Boundary Measure in the Regime of Eternal Inflation
+James Hartle,1S.W. Hawking,2and Thomas Hertog3
+1Department of Physics, University of California, Santa Barb ara, 93106, USA
+2DAMTP, CMS, Wilberforce Road, CB3 0WA Cambridge, UK
+3APC, UMR 7164 (CNRS, Universit´ e Paris 7),
+10 rue A.Domon et L.Duquet, 75205 Paris, France
+and
+International Solvay Institutes, Boulevard du Triomphe,
+ULB – C.P. 231, 1050 Brussels, Belgium
+Abstract
+The no-boundary wave function (NBWF) specifies a measure for prediction in cosmology that
+selects inflationary histories and remains well behaved for spatially large or infinite universes. This
+paper explores the predictions of the NBWF for linear scalar fluctuations about homogeneous and
+isotropic backgrounds in models with a single scalar field mo ving in a quadratic potential. We treat
+boththespace-timegeometryoftheuniverseandtheobserve rsinhabitingitquantummechanically.
+We evaluate top-down probabilities for local observations that are conditioned on the NBWF and
+on part of our data as observers of the universe. For models wh ere the most probable histories do
+not have a regime of eternal inflation, the NBWF predicts homo geneity on large scales, a specific
+non-Gaussian spectrum of observable fluctuations, and a sma ll amount of inflation in our past. By
+contrast, for models where the dominant histories have a reg ime of eternal inflation, the NBWF
+predicts significant inhomogeneity on scales much larger th an the present horizon, a Gaussian
+spectrum of observable fluctuations, and a long period of infl ation in our past. The absence or
+presence of local non-Gaussianity therefore provides info rmation about the global structure of the
+universe, assuming the NBWF.
+1I. INTRODUCTION
+Inflation tends to make the universe so large that we can at bes t observe only a tiny
+part of it. Even for a closed universe it has been argued that w hen there is a regime
+of eternal inflation, inhomogeneities can lead to an infinite ly large reheating surface [1].
+Bubble nucleation in false vacuum models, for instance, lea ds to inhomogeneous universes
+that contain infinite open spatial slices of constant densit y inside the bubbles [2]. The issues
+that arise for prediction as a consequence of large or infinit e sized universes are loosely
+referred to as the measure problem1.
+This is one of a series of papers [4–6] devoted to the measure f or prediction provided
+by the no-boundary quantum state of the universe (NBWF) [7, 8] . If the universe is a
+quantum mechanical system it has a quantum state. This state predicts probabilities for
+alternative histories of theuniverseand everything in it, including thealternative histories of
+its spacetime geometry. It seems inevitable that any discus sion of prediction in fundamental
+cosmology should take these probabilities into account. In deed, it is plausible that, except
+for an assumption of the typicality of our data, no measure be yond that supplied by the
+NBWF is needed for any observational prediction.
+Previous papers [4–6] considered the NBWF’s predictions in h omogeneous, isotropic min-
+isuperspace models. We showed how the no-boundary measure f or observations is well de-
+fined even in the limit of very large universes provided that t he quantum nature of the
+observer making the observation is taken into account. In th is paper we extend this work to
+consider linear fluctuations in matter and geometry away fro m homogeneity and isotropy.
+This enables us to consider predictions for observable quan tities such as those connected
+with the cosmic microwave background (CMB). We continue to m odel the matter by a sin-
+gle scalar field moving in a quadratic potential. We also cont inue with our simple model of
+an observer as a physical system characterized by data Dand a probability pE(D) to exist
+in any one Hubble volume on spacelike surfaces specified by D.
+We will describe our assumptions and procedures for calcula tion in Section II. But one
+crucial distinction should be mentioned at the outset — the d ifference between top-down
+(TD ) and bottom-up (BU ) probabilities [9].
+By itself, the NBWF predicts the probabilities for the altern ative, four-dimensional, clas-
+sical histories that the universe may exhibit. We call these the BU probabilities for the
+classical ensemble. However, we do not observe entire histor ies. Instead our observations
+are restricted to a light cone located somewhere in the unive rse and extending over roughly
+a Hubble volume. Predictions for observations in cosmology a re necessarily conditioned on
+a description of the local observational situation in addit ion to the NBWF. For instance an
+observation of the CMB spectrum depends on whenandwherethe observation is made in
+the history of the universe. In general, probabilities for ourobservations are conditioned on
+some part of our data Dand predict other properties of the universe. Probabilitie s condi-
+tioned on some part of our data are called TD probabilities. T hey can differ significantly
+from the BU probabilities for the same alternatives as those for the number of efolds of
+inflation discussed in [5]. The probabilities for perturbat ions that are within our current
+horizon that will be calculated here provide another illust ration of this difference.
+1Although it is usually discussed in the context of inflation ( see e.g.[3] for recent work), the measure
+problem is not specific to inflationary cosmology. Similar is sues arise in any theory of cosmology that
+predicts spatially infinite universes.
+2AfterabriefstatementofourassumptionsandproceduresinS ectionIIthepaperproceeds
+to derive the TD probabilities for local observations relat ed to fluctuations as follows: In
+Sections III and IV we calculate the classical ensemble of fo ur-dimensional, Lorentzian,
+homogeneous and isotropic (homo/iso) histories with linea r scalar fluctuations predicted by
+the semiclassical approximation to the NBWF. The real part of the Euclidean action of
+the complex saddle-points corresponding to the different hi stories in the ensemble provides
+the BU NBWF probabilities of both the homo/iso backgrounds an d their perturbations
+viewed as global features of the universe. From these bottom -up probabilities one can
+obtain predictions for local observations such as the CMB te mperature anisotropies. This
+is done in Sections V where we calculate the top-down probabi lities for observing different
+perturbations inside our current horizon. We find a slightly non-Gaussian spectrum of
+perturbations on currently observable scales in models whe re the most probable histories do
+not have a regime of eternal inflation. By contrast, for model s where the dominant histories
+have a regime of eternal inflation, we find the NBWF predicts a Ga ussian spectrum of
+observable fluctuations. In Section VI we comment on backreac tion effects in the regime of
+eternal inflation, and argue that these are unlikely to chang e the above results. Finally in
+Section VII we present our conclusions.
+II. FROM THE NBWF TO PROBABILITIES FOR OUR OBSERVATIONS
+This section sets out our assumptions and procedures for cal culating the probabilities
+for our observations from a quantum state of the universe. Th ese are then illustrated in a
+simple model.
+A. Framework
+A quantum universe with a quantum state. We assume that the universe is a closed
+quantum mechanical system with a particular quantum state. That state is taken to be
+the NBWF. The state predicts probabilities for the individua l members of decoherent sets
+of alternative, coarse-grained, four-dimensional histor ies of the universe and its contents
+according to the principles of generalized quantum theory [ 10]. We call these bottom-up
+(BU) probabilities.
+Bottom-up probabilities for the classical ensemble. In particular, the state predicts the
+probabilities for the classical ensemble consisting of fou r-dimensional alternative histories
+with high probabilities for correlations in time governed b y classical equations of motion. In
+[4, 5, 11] we described how to calculate a semiclassical appr oximation for the probabilities
+of these histories from the semiclassical approximation to the NBWF assuming decoherence
+in an appropriate coarse graining. In this paper we restrict attention to probabilities of
+alternatives that can be defined in terms of these classical h istories. Simple examples are
+the probabilities for the number of inflationary efolds or fo r the size of fluctuations away
+from homogeneity and isotropy.
+Top-down probabilities for observations. Probabilities for our2observations are not prob-
+abilities for a four-dimensional history of the universe. R ather they are probabilities for
+2‘We’, ‘us’, ‘our’ etc refer loosely to the collection of huma ns engaged in scientific research on cosmology.
+We will not need a more precise definition.
+3local alternatives at a particular time and place in a classi cal history. They are constructed
+from the BU probabilities supplied by the NBWF by conditionin g on at least that part of
+our data that describes what we know of our location in spacet ime. We call probabilities
+conditioned on all or part of our data top-down (TD) probabil ities.
+Observers as quantum systems. As observers we are quantum systems within the uni-
+verse characterized at an appropriate coarse-grained leve l by the data Dthat we possess —
+including a physical description of ourselves. We arose fro m physical processes that occurred
+over the universe’s history. We are therefore not certain to exist in the universe, Indeed,
+there is only a very tiny probability pE(D) for an instance of the data Din any Hubble
+volume. However, in a very large universe the probability bec omes significant that the data
+Dare replicated elsewhere. All we know for sure about the unive rse is that it exhibits at
+least one instance of the data D— a situation we abbreviate by D≥1. TD probabilities can
+differ significantly from BU probabilities for the same alter natives when these facts about
+observers are taken into account as we now illustrate in a ver y simple model.
+B. Procedures Illustrated by a Simple Model
+Consider a toy model universe consisting of a number of boxes — ‘Hubble volumes’.
+We consider these at a single moment of time. There are Kphysical degrees of freedom
+z1,···zKeachconstrainedtobethesameinallHubblevolumes. Wedenot ethemcollectively
+byz≡(z1,···,zK). The quantum state supplies BU probabilities3p(z) for the values of
+thezi. The fields ziare crudely analogous to the fluctuations away from homogene ity and
+isotropy that we will consider later. The number of Hubble vol umesNhdepends on z,Nh(z),
+as it would for a fluctuation in geometry. Observers in the Hubb le volumes can measure z.
+The probability that there is an observer with data Din any Hubble volume is pE(D). With
+this simple model we will be able to illustrate many of our pro cedures and results without
+getting bogged down in the elegant but complex technology of cosmological perturbation
+theory.
+We distinguish between local and global predictions. Local predictions are for features of
+the universe inside a Hubble volume — features that we could in principle observe in ours.
+Examples are the CMB correlation functions. Global predict ions are for features of our
+universe that may extend outside our Hubble volume or beyond t he present time. Examples
+are predictions of the number of efolds of inflation in the pas t or inhomogeneities outside
+the present horizon.
+Probabilities for the results of our observations are for lo cal features of the universe
+conditioned on what we know about it — our data D. All we know for certain from our
+data is that the universe exhibits at least one instance of it ,D≥1. In the present model we
+can consider the probabilities for the value of zin a Hubble volume given D≥1. But we are
+not just interested in these probabilities in any Hubble volu me; we are interested in them
+inourHubble volume. We discuss how to handle such first person quest ions generally in
+Section IIE. But in cases where there is a symmetry between Hub ble volumes there is a
+shortcut to the answer.
+This simple model has such a symmetry — all the Hubble volumes a re the same. The
+probability for a value of zin our Hubble volume given D≥1is therefore the same as the
+3Here and throughout we do not distinguish notationally betw een probabilities and probability densities.
+4probability that the universe exhibits a value of zin any Hubble volume given D≥1. And
+sincezis constant over the universe that is the same as the probabil ityp(z|D≥1) that the
+universe has a value of zgivenD≥1.
+This can can be efficiently computed by starting from the relat ion
+p(z|D≥1) =p(z,D≥1)
+p(D≥1)=p(D≥1|z)p(z)
+p(D≥1). (2.1)
+The probability p(D≥1|z) that there is at least one instance of Din the universe given zis
+1 minus the probability that there are no instances in any Hubb le volume. This is
+p(D≥1|z) = 1−(1−pE(D))Nh(z). (2.2)
+Combining (2.1) and (2.2) gives
+p(z|D≥1) =[1−(1−pE(D))Nh(z)]p(z)/integraltext
+dz[1−(1−pE(D))Nh(z)]p(z). (2.3)
+This is the probability for our observations of zin this very simple model.
+C. Gaussianity and Non-Gaussianity
+Eq (2.3) for the probability of our observations of zsimplifies in two important limits.
+First, it simplifies when pE(D)Nh(z)≪1 for the whole range of z, that is, in the limit in
+which we are rare in the universe. Then we have
+p(z|D≥1)≈Nh(z)p(z)/integraltext
+dzNh(z)p(z). (2.4)
+The difficult to estimate4probability pE(D) has cancelled out. Nh(z) would also cancel
+were it independent of zleaving the probability for observation of a value zequal to the BU
+probability that the universe has that value.
+But ifNh(z) depends on zthe probabilities for observation will differ from the botto m-up
+probabilities. In particular suppose the bottom-up probab ilitiesp(z) are Gaussian, that is
+a product of terms of the form exp( −constz2
+i). Then the probabilities for observing zwill
+not be Gaussian. The TD probabilities for values of zthat make the universe larger are
+enhanced over their BU values because in a larger universe th ere are more places for our
+dataDto be.
+The second limit in which p(z|D≥1) is independent of pE(D) is whenpE(D)Nh(z)≫1
+for the whole range of z. This is the limit where our universe is so large that our data are
+common. Then,
+p(z|D≥1)≈p(z), (2.5)
+that is, TD probabilities equal BU probabilities. As a conseq uence, Gaussian BU probabil-
+ities imply a Gaussian distribution for the probabilities o f observing z.
+Thus, from local measurement of the zian observer confident of the validity of this
+simple model could infer something about the size of the univ erseNh. That does not violate
+4For a discussion of ways to bound pE(D) see [6].
+5causality. The data Dmay be assumed to be within our past light cone. But the quantu m
+state predicts non-local correlations between the propert ies of different Hubble volumes.
+These can be exploited to make predictions outside our Hubble volume from data inside it.
+This is not qualitatively different from assuming that the un iverse is homogeneous and then
+inferring the mean density outside our Hubble volume from obs ervations inside.
+The common limit (2.5) shows that predictions for observati ons can be defined even
+when there are an infinite number of Hubble volumes provided th at the BU probabilities
+are normalized. No ‘measure’ beyond that provided by the quan tum state is needed to deal
+with infinite volumes in these simple models.
+D. Detecting Gaussianity
+Suppose that the BU probabilities for the z’s are Gaussian. That is, suppose specifically
+that,
+p(z) =/productdisplay
+i(2πσ2)−1/2exp(−z2
+i/2σ2). (2.6)
+As (2.4) shows, Gaussian BU probabilities do not necessarily imply Gaussian TD probabili-
+ties for observation. But to understand a little more about t he tests for non-Gaussianity let
+us first consider the common limit (2.5) where the TD probabil ities are Gaussian.
+Acompletedescriptionofour universewill generally requi revariables other thanthosewe
+observe directly. In the absence of observation we may only h ave probabilities for these vari-
+ables and the resulting TD probabilities for observation ma y not have the simple Gaussian
+form. But, if a Gaussian distribution is predicted for all va lues of the unknown variables,
+Gaussian statistics for observation will still be predicte d. This elementary but important
+point can be illustrated with a modest extension of our simpl e model.
+Suppose that in addition to the z’s the widths of the Gaussian distributions in (2.6)
+depend on a variable φ0so thatσ=σ(φ0). Then the TD probabilities for observation will
+be given by
+p(z|D≥1) =/integraldisplay
+dφ0p(z|D≥1,φ0)p(D≥1|φ0)p(φ0) (2.7)
+wherep(φ0) is the BU probability for the unobserved φ0. Even ifp(z|D≥1,φ0) is a Gaussian
+distribution of form (2.6) with φ0dependent σ’s, the sum of them in (2.7) will not be.
+Consider functions of the zi’s whose expected value vanishes for Gaussian distribution s and
+thus test Gaussianity. An example is the correlation functio n
+Bkj≡1
+ℓ/summationdisplay
+izizi+kzi+j. (2.8)
+If the expected value of such functions vanish for each φ0it will also vanish for the sum.
+The point is that our universe is characterized by some value ofφ0even if we have not
+determined what it is. If Gaussianity is predicted for all va lues ofφ0we predict Gaussianity
+for our observations of the z’s despite our ignorance of φ0’s value.
+E. From 3rd Person to 1st Person
+The derivation of the probabilities for observation in our s imple model relied on a sym-
+metry — the equivalence of all Hubble volumes. That symmetry a llowed us to ignore all the
+6other instances of our data Dthat a large universe might exhibit and focus on our own. We
+will rely on an analogous underlying symmetry in our discuss ion of the fluctuations away
+from homogeneity and isotropy in Section V. But lest the reade r believe that a symmetry is
+essential to calculating probabilities for observations w e present in this section a derivation
+in the simple model that does not require a symmetry and expli citly takes into account the
+instances of Dbeyond our own.
+To begin let us calculate the probability p(z,n) that the universe has the value zandn
+Hubble volumes with the data D. This evidently is
+p(z,n) =/parenleftbiggNh(z)
+n/parenrightbigg
+(pE(D))n(1−pE(D))Nh(z)−np(z). (2.9)
+Since all the Hubble volumes are the same, the sum over locatio ns of theninstances has
+reduced to the binomial coefficient giving the number of ways o f pickingnHubble volumes
+with observers out of Nh(z) total Hubble volumes. The probability p(z,n) is an example of a
+third person probability — a probability for a feature the universe may exhibit indepe ndently
+of any relation to us. But we are interested in the first person probability of what value of z
+wewill observe. The theory, by itself, doesn’t predict such pr obabilities. We are one of the
+instances of Dbut the theory doesn’t say which one. Indeed, it has no notion of ‘we’.
+To connect first person probabilities for what we observe wit h third person probabilities
+of what the universe exhibits a further assumption is needed . This assumption — called a
+xerographic distribution [12] — specifies the probability t hat we are any one of the instances
+ofD. The simplest and least informative assumption is that we ar e equally likely to be any
+one of the instances of Dthat the universe exhibits. Put differently, it is the assump tion
+that we are typical of those instances. This assumption is ma de throughout this paper5.
+First personprobabilities for what we observeare necessar ily conditioned on theexistence
+of at least one instance of our data Din the universe — us! Thus we write p(1p)(z|D≥1)
+for the probability that we observe z. To calculate this, first calculate the joint probability
+p(1p)(z,D≥1) as follows: Suppose the universe exhibits ninstances of D. Use an index A
+running from 1 to nto distinguish these. Assuming typicality the xerographic d istribution
+isξA= 1/n. Multiply this by the probability (2.9) for ninstances and sum over A. Finally
+sum over the number of instances from n= 1 (at least one instance) to n=Nh(z). The
+factor ofnfrom the sum over Acancels with the xerographic distribution to give
+p(1p)(z,D≥1) =Nh(z)/summationdisplay
+n=1/parenleftbiggNh(z)
+n/parenrightbigg
+(pE(D))n(1−pE(D))Nh(z)−np(z),(2.10a)
+=[1−(1−pE(D))Nh(z)]p(z). (2.10b)
+The conditional probability p(1p)(z|D≥1) is this joint probability divided by the probability
+just forD≥1:
+p(1p)(z|D≥1) =[1−(1−pE(D))Nh(z)]p(z)/integraltext
+dz[1−(1−pE(D))Nh(z)]p(z). (2.11)
+This is the probability that we observe z, and it is exactly the same as (2.3) derived with
+the aid of the symmetry.
+5However sometimes assumptions of atypicality yield more predictive theories [12, 13].
+7III. QUANTUM AND CLASSICAL NBWF FLUCTUATIONS
+The NBWF Ψ is defined on the superspace of three-geometries and spatial matter field
+configurations. Here, we consider minisuperspace models defi ned by linearized perturba-
+tionsawayfromclosed, homogeneousandisotropicthree-ge ometriesandfieldconfigurations.
+Minisuperspace is spanned by the scale factor bof the homogeneous three-geometries, the
+homogeneous value of the scalar field χand the parameters defining the modes of perturba-
+tion. We denote the latter collectively by z= (z1,z2,...) and define these precisely in Section
+IV. Thus, Ψ = Ψ( b,χ,z)
+The NBWF is an integral of the exponential of minus the Euclide an actionIover com-
+plex four-geometries and field configurations that are regul ar on a four-disk with a three-
+sphere boundary on which the four-dimensional histories ta ke the real values ( b,χ,z) [7, 8].
+Schematically we can write
+Ψ(b,χ,z) =/integraldisplay
+Cδaδφδζexp(−I[a(τ),φ(τ),ζ(τ)]/¯h). (3.1)
+Here,a(τ) andφ(τ) are (complex) histories of scale factor and scalar field defi ning a homo-
+geneous, isotropic background. The quantities ζ(τ) = (ζ1(τ),ζ2(τ),···) denote histories of
+modes of fluctuation away from homogeneity and isotropy in bo th metric and matter field.
+I[a(τ),φ(τ),ζ(τ)] is the Euclidean action. The integral is over geometries a nd matter fields
+that are regular on a disk with only one boundary at which a(τ),φ(τ) andζ(τ) take the
+valuesb,χ, andz. The integration is carried out along a suitable complex con tourCwhich
+ensures the convergence of (3.1) and the reality of the resul t [14].
+We restrict to linear fluctuations when only up to quadratic t erms inζare retained in
+the action in (3.1):
+I=I(0)[a(τ),φ(τ)]+I(2)[a(τ),φ(τ),ζ(τ)]. (3.2)
+(There is no linear term for the models considered in this pap er.) ThenI(0)describes the
+homogeneous isotropic background and I(2)describes the linear and quadratic perturbations
+away from that background.
+Supposethatinsomeregionofsuperspacetheintegralin(3. 1)overa(τ)andφ(τ)defining
+thehomogeneousbackgroundcanbeapproximatedbythemetho dofsteepestdescents. Then
+the wave function Ψ will be a sum of terms of the form
+Ψ(b,χ,z)≈exp{[−I(0)
+R(b,χ)+iS(0)(b,χ)]/¯h}ψ(b,χ,z), (3.3)
+one such term for each history ( a(τ),φ(τ)) that extremizes the action I(0), matches ( b,χ)
+at the boundary of the disk, and is regular elsewhere. For eac h contribution I(0)
+R(b,χ) is the
+real part of the action I(0)[a(τ),φ(τ)] evaluated at the extremizing history and −S(0)(b,χ)
+is the imaginary part. The wave function ψis defined by the remaining integral over ζ
+ψ(b,χ,z)≡/integraldisplay
+Cδζexp(−I(2)[a(τ),φ(τ),ζ(τ)]/¯h). (3.4)
+As we showed6in [5], classical Lorentzian histories are predicted in regions of superspace
+whereS(0)(b,χ) varies rapidly when compared with I(0)(b,χ). Specifically, then Ψ predicts
+6And as we intend to show in more detail in [11].
+8an ensemble of suitably coarse-grained Lorentzian histori es (b(t),χ(t)) that with high prob-
+ability lie along the integral curves of S(0)(b,χ). Their relative probabilities are given by
+exp[−2IR(b,χ)], which is preserved along each history [5].
+When evaluated on one of these classical histories the wave f unction (3.4) becomes a
+function of time,
+ψ(z,t)≡ψ(b(t),χ(t),z). (3.5)
+As shown in a variety of ways [15] the Wheeler-DeWitt equation implies a Schr¨ odinger
+equation for ψ(z,t)
+i¯hdψ(z,t)/dt=H(t)ψ(z,t). (3.6)
+The time dependent Hamiltonian describes the evolution of th e state of the fluctuations
+in the background ( b(t),χ(t)). An inner product is induced from the generalized quantum
+mechanics on the full superspace [10]. Equation and product define the quantum mechanics
+of the fluctuation field zin the homogeneous, isotropic background.
+In this way, the fluctuation fields can be thought of as quantum fields on the possible
+backgroundclassical spacetimes. Thestateofthefieldsisd eterminedbytheNBWFthrough
+(3.4). There is no independent assumption of a “vacuum” stat e. However, the Euclidean
+integral defining the NBWF is analogous to the Euclidean integ ral defining the ground state.
+It is therefore reasonable to expect the NBWF to imply that fluc tuations are in something
+like a quantum field theory ground state early in the universe . This was shown explicitly
+in [16] and we will show it explicitly for our model in the next section7. Hence the NBWF
+provides a unified treatment of both classical homogeneous a nd isotropic backgrounds and
+the quantum fluctuations away from them.
+Theintegraldefiningthewavefunctionin(3.4)mayitselfbe approximatedbythemethod
+of steepest descents. Indeed, since the action is quadratic inζwe expect that it can be
+evaluated exactly when the measure is suitable. Either way, the result for a particular
+extremuma(τ),φ(τ) ofI(0)is
+ψ(b,χ,z) =A(2)(b,χ)exp{[−I(2)
+R(b,χ,z)+iS(2)(b,χ,z)]/¯h}. (3.7)
+The extremizing history ζ(τ) is regular on the manifold of integration and matches zat its
+one boundary. I(2)
+R(b,χ,z) and−S(2)(b,χ,z) are the real and imaginary parts of the action
+I(2)evaluated on this history and A(2)is a prefactor.
+This fully quantum mechanical theory of fluctuations around a classical background uni-
+verse will predict their classical behavior in regions of su perspace where S(2)(b(t),χ(t),z)
+varies rapidly in zcompared to I(2)(b(t),χ(t),z). The detailed conditions for this are
+called the ‘classicality conditions’ [5]. Specifically whe n they are satisfied the wave func-
+tion (3.7) predicts an ensemble of suitably coarse grained, classical, Lorentzian histories z(t)
+that with high probability lie along the integral curves of S(2)(b(t),χ(t),z). The probabil-
+ities of the classical fluctuations in a given homo/iso backg round are then proportional to
+exp[−2I(2)[b(t),χ(t),z(t)]. In general we can expect the regions of superspace where p ertur-
+bation modes behave classically to be different for different modes8and we will see this in
+detail in what follows.
+7This also opens the possibility that the NBWF can predict cor rections to popular assumptions about the
+vacuum of the fluctuation fields.
+8In inflationary cosmology it is sometimes said that the modes ‘become classical’ at a certain time as
+though there were a transition between quantum and classica l physics. This is incorrect. Classical physics
+9Withthesetechniqueswewillbeabletotreatboththequantu mmechanicsoffluctuations
+of the universe and their classical approximation.
+IV. BOTTOM-UP PROBABILITIES FOR PERTURBATIONS
+In this section we describe the calculation of the bottom-up probabilities for alternative
+four-dimensional classical histories of the universe that include linear fluctuations away from
+homogeneity and isotropy. These are the probabilities for c lassical behavior conditioned
+on the NBWF alone. They are the input to the calculation of top- down probabilities for
+observation described in the next section.
+A. Homogeneous Isotropic Histories
+We first review the bottom-up probabilities of the homogeneo us isotropic histories pre-
+dicted by the semiclassical NBWF (3.3). These were calculate d in [4, 5], in a simple model
+consisting of a single scalar field moving in a quadratic pote ntial. It was found that there is
+a one-parameter family of extremizing complex histories – f uzzy instantons – which obey the
+classicality conditions at the boundary where one evaluate s the wave function and therefore
+predict a Lorentzian history. The different histories can be labeled by the magnitude of
+the complex scalar field φ0≡ |φ(0)|at the ‘South Pole’ (SP) of the corresponding fuzzy
+instanton. It was found [4] that the classicality condition s requireφ0≥φc
+0≈1. The relative
+probabilities of the different histories are given by exp[ −2IR(φ0)], whereIR(φ0) is the real
+part of the Euclidean action of the fuzzy instanton.
+Astrikingfeatureoftheensembleofclassicalhistoriesin thismodelisthecloseconnection
+it reveals between classicality and inflation [5]. Specifica lly the histories have values of
+H≡(db/dt)/bandχ, whichalllie within a very narrow band around H=mχcharacteristic
+of Lorentzian slow roll inflationary solutions. It follows t hat a classical, homogeneous and
+isotropic universe must have an early inflationary state if the universe is in the no-bound ary
+state.
+For sufficiently large φ0there is an approximate analytic solution [17] for the fuzzy in-
+stanton,
+φ(τ)≈φ(0)+imτ
+3, a(τ)≈i
+2mφ(0)e−imφ(0)τ+m2τ2/6. (4.1)
+These solutions are the complex analogs of the standard ‘slo w roll’ inflationary solutions.
+They are valid in the region of the complex τ=x+itplane where tis not so large that the
+slow roll assumption breaks down, and where |a(τ)| ≫1 so that the spatial curvature is ex-
+ponentiallynegligible9. Bytuningthephaseof φ0attheSPsothatIm[ φ(0)] =−π/6Re[φ(0)]
+vertical lines given by τ=π/2mRe[φ(0)] +itare obtained along which both aandφare
+is not an alternative to quantum theory; it is an approximati on to it. The modes are always quantum
+mechanical but a classical approximation only holds in cert ain regimes. It would be better to say that
+the modes enter a region where a classical approximation hol ds with a suitable coarse graining. But we
+will use the less accurate terminology with this understand ing.
+9The constant multiplicative normalization of the scale fac tor is determined by matching these solutions
+to the ‘no-roll’ solutions φ(τ)≈φ(0), a(τ)≈sin[mφ(0)τ]/mφ(0) that are regular at the origin.
+10approximately real and describe Lorentzian inflating unive rses with the scalar field approx-
+imately equal to φ0at the start of inflation.
+The real part of the action of the fuzzy instantons in this app roximation is
+IR(φ0)≈ −π
+2(mφ0)2≈ −2π
+(3m2N(φ0))(4.2)
+whereN(φ0)≈3φ2
+0/2 is the number of inflationary efolds in the classical histor y labeledφ0.
+Hence the bottom-up probabilities conditioned only on the NBW F are largest for classical
+histories with a small amount of inflation.
+B. Semiclassical Wave Function for Quantum Fluctuations
+Following the analysis of [16, 18] we now calculate the wave f unction (3.4) for linear
+scalar fluctuations around the homogeneous isotropic histo ries predicted by the NBWF.
+We restrict attention to scalar perturbations, since these turn out to matter most for the
+top-down effects we are interested in here. We write the pertu rbed metric as
+ds2= (1+2ϕ)dτ2+2a(τ)B|idxidτ+a2(τ)[(1−2ψ)γij+2E|ij]dxidxj(4.3)
+whereγijis the metric of the unit radius three-sphere, xiare the coordinates on the three-
+sphere and a vertical bar denotes covariant differentiation with respect to γij. Expanding
+the perturbations in the standard, normalized scalar harmo nicsQn
+lm(xi) onS3gives the
+definitions
+ϕ=1√
+6/summationdisplay
+nlmgnlmQn
+lm, ψ=−1√
+6/summationdisplay
+nlm(anlm+bnlm)Qn
+lm, (4.4)
+B=1√
+6/summationdisplay
+nlmknlmQn
+lm
+(n2−1), E=1√
+6/summationdisplay
+nlm3bnlmQn
+lm
+(n2−1)(4.5)
+and the scalar field perturbation
+δφ(τ,x) =1√
+6/summationdisplay
+nlmfnlmQn
+lm. (4.6)
+From here onwards we denote the labels n, l, mcollectively by ( n). The expansion coeffi-
+cientsa(n),b(n),f(n),g(n),k(n)are functions of time only.
+From the above expansions we see there are five scalar degrees of freedom. However, the
+functionsg(n)andk(n)appear as Lagrange multipliers in the action. Variations of the action
+with respect to g(n)andk(n)result in the linear Hamiltonian and momentum constraints. I n
+quantum cosmology the NBWF satisfies the operator forms of the se constraints [19]. The
+wave function therefore depends only on the background vari ablesbandχand on a single
+linear combination of the (boundary values of the) perturba tion variables a(n),b(n),f(n)–
+the three functions that describe the perturbed three geome try. One can take this linear
+combination to be the following (Appendix A),
+ζ(n)=a(n)+b(n)−HE
+˙φf(n) (4.7)
+11whereHE≡˙a/aand the subscript Erefers to quantities constructed with Euclidean time.
+Hence one has ψ(b,χ,z), wherez≡(z(1),z(2),...) are the real values of ζ= (ζ(1),ζ(2),...)
+at the boundary. The variables zare invariant under linear gauge transformations and
+approximately conserved outside the horizon [20, 21].
+The wave function ψ(b,χ,z) can be found explicitly in the semiclassical approximatio n.
+To first order in perturbation theory (3.7) takes the form
+ψ(b,χ,z) =/productdisplay
+(n)ψ(n)(b,χ,z(n)). (4.8)
+The action I(2)
+(n)[b,χ,z(n)] of each mode is generally a positivequadratic function of z(n).
+Thus, in a regime where the perturbations are small and behav e classically the bottom-
+up probabilities from (3.7) will favor vanishing perturbat ions and homogeneous classical
+histories.
+An analytic approximation to the wave function (4.8) was obta ined in [18], by solving
+the complex perturbation equations in the slow roll backgro unds (4.1). In Appendix A we
+summarize this calculation and verify its accuracy by numer ically calculating the perturba-
+tions around several representative members of the ensembl e of exact complex extremizing
+geometries found in [4, 5]. We concentrate on perturbation m odes that leave the Hubble
+radius during inflation. As we will see, these are the modes tha t are amplified by the time-
+dependent background, become classical and, ultimately, l ead to the large-scale structures
+we observe today.
+The no-boundary condition of regularity at the SP requires f(n)anda(n)to vanish there.
+Ifτ→0 labels the SP then the field equations imply that to leading o rder inτone has
+ζ(n)=ζ(n)(0)τn, whereζ(n)(0)≡ |ζ(n)(0)|eiθ≡ζ(n)0eiθis a complex constant. Its phase
+θshould be fine-tuned such that ζ(n)is real at the boundary, and its amplitude ζ(n)0is
+determined by the value of the boundary perturbation z(n).
+At smallτthe modulus of the complex ‘wavelength’ a/nof a perturbation mode will
+be shorter than the horizon size since |aHE| →1 whenτ→0. In this regime we find
+the complex solution for ζ(n)oscillates and is independent of the nature of the potential .
+On the other hand we show in Appendix A that at larger τ, whenn≪ |aHE|, the general
+perturbationsolutionisacombinationofaconstantandade cayingmode. Henceoneexpects
+the wave function ψ(n)(b,χ,z(n)) depends only on the behavior of the potential for values of
+φnear the value taken by φ(τ) at the time the perturbation leaves the horizon. At horizon
+crossing the perturbation ζ(n)generally has an imaginary component. The requirement that
+ζ(n)be real at the boundary therefore means that the phase θofζ(n)(0) at the SP should be
+tuned such that the imaginary component of the subhorizon mo de function matches onto
+the decaying mode when the perturbation leaves the horizon. It turns out that this implies
+that a perturbation mode will become classical when its phys ical wavelength becomes much
+larger than the Hubble radius, as is evident from Fig 4 in Append ix A.
+As a consequence of the decay of the imaginary component of the perturbation the real
+part of the Euclidean action I(2)
+(n)(b,χ,z(n)) tends to a constant when the mode leaves the
+horizon. This determines the bottom-up probabilities of th e different classical perturbed
+histories predicted by the NBWF. Substituting the perturbat ion solutions (A5) in the action
+(A6) and normalizing one obtains, for all wavenumbers n<exp(3φ2
+0/2),
+p(z(n)|φ0)≈/radicalBigg
+ǫ∗n3
+2πH2
+∗exp/bracketleftbigg
+−ǫ∗
+2H2
+∗n3z2
+(n)/bracketrightbigg
+(4.9)
+12whereǫ≡˙χ2/H2is the usual slow-roll parameter. The subscript ∗on a quantity in (4.9)
+means it is evaluated at horizon crossing during inflation. E quation (4.9) specifies the
+bottom-up probabilities of linear, classical perturbatio ns around the homogeneous isotropic
+histories predicted by the NBWF. One sees the probabilities o fz(n)n3are Gaussian, with
+varianceH2
+∗/ǫ∗characteristic of inflationary perturbations.
+Although (4.9) was derived using the slow roll approximation for the fuzzy instantons, we
+have numerically verified (Appendix A) that this result is accu rate over most of the range
+ofφ0except near its lower bound φc
+0, and for all modes that become classical except those
+that leave the horizon towards the very end of inflation10.
+C. Perturbed Classical Histories
+The evolution of perturbations in a classical background un iverse (b(t),χ(t)) is in gen-
+eral given by a Schr¨ odinger equation (3.6). However in regio ns of superspace where
+S(2)(b(t),χ(t),z) varies rapidly in zcompared to I(2)(b(t),χ(t),z) the semiclassical wave
+function (4.8) predicts an ensemble of suitably coarse grai ned,classical, Lorentzian histo-
+riesz(t) that with high probability lie along the integral curves of S(2)(b(t),χ(t),z). Their
+relative probabilities are given by (4.9), which is preserv ed along each history [5].
+We have seen that in inflationary histories, perturbation mo des behave classically when
+their physical wavelength is larger than the Hubble radius. S ince the modes that left the
+horizon during inflation are responsible for the large-scal e structure we observe today, it is
+appropriate to evaluate the wave function of perturbations on a surface towards the end of
+inflation and to coarse-grain over modes that are inside the h orizon at that time11. The
+values of the perturbation modes at the boundary, together w ith their derivatives, provide
+Cauchy data for their future classical evolution12. The members of the classical ensemble of
+perturbation histories obtained in this way can be labeled b y (φ0,ζ0).
+Just after inflation ends the general solution for classical , long-wavelength ( n≪bH) per-
+turbations (see e.g.[21]) implies the scalar metric pertur bations remain essentially constant,
+with a small oscillatory component due to the oscillations o f the background scalar field.
+The matter perturbation starts oscillating again when the Hu bble radius becomes larger
+than the scalar field Compton wavelength ∼1/mb. This behavior can also be seen in Fig
+3 (form2=.05), where inflation ends around y∼ O(60). In realistic models the energy
+of the inflaton is then converted in ordinary matter and radia tion, reheating the universe.
+Hence the solutions for the scalar matter and metric perturba tions are not directly related
+to observations of the present universe. Fortunately, rehe ating occurs when the perturbation
+modes that are relevant for current observations are well ou tside the horizon, so that the
+variablez(n)is conserved. Hence the wave function ψ(z,t) provides initial conditions for the
+classical evolution of perturbation modes after they re-en ter the horizon at late times.
+To first approximation reheating takes place at a definite val ue ofχ. Hence one expects
+10These corrections to (4.9) can in principle be calculated sy stematically, opening up the way to study the
+small deviations from the Bunch-Davis vacuum implied by the NBWF as discussed in Section III.
+11The histories obtained by evolving forward Cauchy data take n at an earlier time, involving fewer modes,
+can be viewed as a coarse-graining of these.
+12The evolution of perturbations backwards in time, towards t he initial singularity or the bounce [5], is
+generally not classical everywhere and can be obtained usin g (3.6). This will be discussed elsewhere.
+13surfaces of constant scalar field during inflation to evolve t o surfaces of constant temperature
+after reheating. The surface of last scattering will be such a surface. Variations in the
+observed temperature of the CMB arise e.g. from variations i n the gravitational redshift
+of the surface of last scattering in different directions of o bservation, which are themselves
+determined by the perturbation z(t). Quantities of particular interest in cosmology are
+averages over a particular pattern of perturbations at the s urface of last scattering. The
+simplest examples are the multipole coefficients Clthat characterize the average of a product
+of two temperature fluctuations in two different directions. Expressed in terms of z(n)the
+Cl’s involve a sum over the wavenumber n. However for l≫1 the dominant contribution
+to this sum comes from perturbations with wavenumber ¯ n≈l/rL, whererLis the radial
+distance from us to the surface of last scattering in the Robe rtson-Walker geometry (see
+e.g. [21]). This means there is a direct relation between the Cl’s and the variance of the
+probability distributions (4.9). In particular CMB correl ations on a certain angular scale at
+the present time provide information about the inflaton pote ntial at the time of horizon exit
+of the relevant modes during inflation. For 10 ≤l≤50 the CMB anisotropies are dominated
+by the Sachs-Wolfe effect. In this range the Cl’s are to a good approximation given by [21]
+Cl≈ ∝an}bracketle{tz2
+(n)∝an}bracketri}htn3=8π2T2
+0H2
+∗
+9ǫ∗l(l+1)(4.10)
+whereT2
+0is the present mean value of the temperature of the CMB. In the model we
+have considered H2
+∗/ǫ∗≈m2χ4
+∗. Since galactic scales correspond to χ2
+∗∼ O(50) and since
+observations require the gravitational potential to be ∼10−5on these scales, the mass of
+the scalar field should be about ∼10−6in Planck units. Larger scales leave the horizon
+earlier during inflation. During inflation one has χ∗∼ln(be/b∗)∼ln(λph(n)H∗) wherebeis
+the scale factor at the end of inflation and λph=b/n. SinceHis approximately constant
+during inflation this leads to a slightly red spectrum. Where as this is a small effect on the
+range of currently observable scales, this has significant c onsequences on very large scales as
+we discuss below.
+V. TOP-DOWN PROBABILITIES FOR PERTURBATIONS
+In thissection wecalculate thetop-down probabilities p(z(n)|D≥1) for perturbationmodes
+z(n)that are relevant for observation of the CMB. The Gaussian bo ttom-up probabilities
+(4.9) are an input to this calculation. Our particular aim is to determine in what models
+and under what conditions the top-down corrections can lead to observable effects. We will
+find that this may be the case when the potential is such as to no t allow a regime of eternal
+inflation.
+We begin by reviewing the connection between bottom-up and t op-down probabilities.
+This is the same as the connection derived in Section II but wr itten out here with the full
+machinery necessary to describe perturbations.
+A. Top-Down from Bottom-Up
+From the bottom-up probabilities (4.9) we seek to construct the (top-down) probabilities
+p(z|D) for the present amplitudes of fluctuation observables z= (z1,z2,···) conditioned
+on a subset Dof our total data. Suppose that Dcan be divided into two parts: First,
+14a partDsconsisting of large scale observations that place the data Don one or more
+surfaces of homogeneity ti(Ds,φ0) in each classical spacetime. Observations of the present
+Hubble constant H0and local average energy density are an example. For simplic ity we
+restrict attention to a single surface that we denote by t. The generalization to more is
+straightforward.
+The second part, Dh, consists of local observations that are largely independe nt of the
+large scale features of the spacetimes. Thus D= (Ds,Dh). For each φ0divide the surface
+labeled byDsinto Hubble volumes and denote their total number by Nh(Ds,φ0,ζ0). Finally,
+denote bypE(D) the probability that the data Doccur in any one of the Hubble volumes on
+the surface tand assume that the probability of more than one occurrence i n any one volume
+is negligible. We can now follow the model in Section II to der ive the TD probabilities for
+our observations of fluctuations.
+All we know from our local observations is that there is at leas t one occurrence of Dh
+(abbreviated D≥1
+h) in one of the Hubble volumes (ours). The probability that the re is at
+least one instance of Dhin the classical spacetime labeled by ( φ0,ζ0) is [cf. (2.2)]
+p(D≥1
+h|Ds,φ0,ζ0) = 1−[1−pE(D)]Nh(t,φ0,ζ0). (5.1)
+Neitherφ0orζ0is directly observable. But in each classical spacetime we c an determine
+thevaluesof zonthesurfacesspecifiedby Ds:z=z(Ds,t,φ0,ζ0). Conversely, given zandφ0
+we can determine13the amplitude of the fluctuations at the South Pole ζs(z)≡ζ0(z,Ds,φ0)
+necessary to produce zon the surface t. Thus we can write for the (top-down) probabilities
+p(z|D≥1)
+p(z|D≥1) =/integraldisplay
+dφ0p(φ0,ζs(z))|D≥1). (5.2)
+This can be cast into a more useable form by using the joint pro bability [cf (2.1)]
+p(φ0,ζ0,D≥1
+h|Ds) =p(D≥1
+h|Ds,φ0,ζ0)p(φ0,ζ0|Ds) (5.3a)
+=p(D≥1
+h|Ds,φ0,ζ0)p(ζ0|Ds,φ0)p(φ0|Ds) (5.3b)
+Combining (5.2), (5.3), and(5.1)wefindthefollowingformu laforthetop-downprobabilities
+for fluctuations given at least one instance of the data D[cf (2.3)]
+p(z|D≥1)≈/integraltext
+dφ0p(ζs(z)|Ds,φ0){1−[1−pE(D)]Nh(Ds,φ0,ζs(z))}p(φ0|Ds)/integraltext
+dφ0dζ0p(ζ0|Ds,φ0){1−[1−pE(D)]Nh(Ds,φ0,ζ0)}p(φ0|Ds)(5.4)
+In (5.4) we expect the dependence of the probabilities p(ζ0|Ds,φ0) andp(φ0|Ds) onDsto
+be weak. They will be approximately proportional to p(ζ0|φ0) andp(φ0) respectively except
+when the spacetime specified by φ0does not contain a surface with data Ds. Then they are
+proportional to zero.
+The probabilities p(z|D≥1) are for the values of the fluctuations the universe may exhib it
+givenD≥1. (In the language of Section IIE they are third person probab ilities.) But we
+are interested in the (first person) probabilities for fluctu ations in a particular history and
+inside our Hubble volume, where our specific instance of Dis located. For each φ0, the
+underlying homogeneity is a symmetry that means that all pre dictions for observation will
+be the same in all Hubble volumes. The probability for any quan tities derived from the z’s
+in our Hubble volume is the same as that derived from p(z|D≥1) for any Hubble volume.
+13To compress the notation we will not always write out the depe ndence of ζs(z) onDsandφ0.
+15B. Non-Gaussianity from Volume Weighting
+We now evaluate the TD probabilities p(z(¯n)|D≥1) for different values of perturbation
+modesz(¯n)in the classical ensemble of homo/iso histories with linear ized perturbations. We
+are interested in particular in the modes that contribute to the CMB. As reviewed at the
+end of the last section these are modes with approximately th e same wavenumber that left
+the horizon the same number of efolds before the end of inflati on in all members of the
+ensemble. We denote the value of the relevant wavenumber in e ach history by ¯ n. This
+depends on the duration of inflation and therefore on φ0. In terms of the angular scale
+this is given by ¯ n≈l/rL, whererLis the radial distance to the surface of last scattering
+in the Robertson-Walker geometry (see e.g. [21]). To calcul ate the top-down probability
+p(z(¯n)|D≥1) for the CMB relevant modes requires summing (coarse-grain ing) (5.4) over all
+other modes.
+As before we assume part of our data locate us on a surface of con stant density in each
+member of the classical ensemble. The top-down probabiliti es (5.4) then involve the volume
+Nhof this surface. This is most easily calculated in the f(n)=b(n)= 0 gauge, where surfaces
+of constant density are constant time surfaces with volume [ cf.(4.3)]
+V=V0+δV=b3/integraldisplay
+d3x√γ(1−2ψ)3/2. (5.5)
+The leading correction to V0averages to zero over the surface, but the second order term
+leads to a change in volume. In terms of the gauge invariant va riablez(¯n)one hasδV/V0=/summationtext
+(n)z2
+(n)/8π2. Hence the number of present Hubble volumes in the different his tories of the
+ensemble is given by
+Nh(Ds,φ0,z) =N0
+h(Ds,φ0)/parenleftbigg
+1+/summationdisplay
+(n)z2
+(n)
+8π2/parenrightbigg
+exp/parenleftbigg9
+2φ2
+0/parenrightbigg
+, (5.6)
+whereN0
+h(Ds,φ0) varies slowly with φ0and depends on the present Hubble constant, the
+details of reheating etc. The range of nin the sum encompasses all modes that left the
+horizon during inflation and are therefore classical. Its up per limitnmtherefore depends
+onφ0and is approximately given by nm≈exp(3φ2
+0/2). Using the BU distribution (4.9) of
+znthe expected value of the sum in (5.6) can be bounded by the var iance of the longest
+wavelength perturbations in each history — with n=nm— yielding ∝an}bracketle{t/summationtext
+(n)z2
+(n)∝an}bracketri}ht ≤m2φ4
+0.
+The TD distribution is of the form (5.4). Using the analytic ap proximations (4.2) and
+(4.9) of the BU probabilities of homo/iso histories with lin earized perturbations one finds
+for the top-down probability of the CMB relevant modes14
+p(z(¯n)|D≥1)∝/integraldisplay
+dφ0
+/productdisplay
+(n)/negationslash=(¯n)dζ(n)0exp/parenleftBigg
+−z2
+(n)
+2σ2
+n/parenrightBigg
+/bracketleftbig
+1−(1−pE)Nh/parenrightbig
+]exp/parenleftBigg
+−z2
+(¯n)
+2σ2
+¯n/parenrightBigg
+exp/parenleftbigg4π
+3m2N/parenrightbigg
+.
+(5.7)
+14In (5.7) we have not taken in account the Jacobian that arises when one changes the integration measure
+fromdζ(n)0todzn, because this is polynomial in φ0(see Appendix and also [18]) and therefore hardly
+affects the TD probabilities.
+16Hereσ2
+n(φ0)≡H2
+∗/2ǫ∗n3and the product is taken over all wavenumbers nup tonm. The
+integralsover ζ(n)0canbeevaluatedanalyticallywithoutfurtherapproximati ons. Thisyields
+p(z(¯n)|D≥1)∝/integraldisplay
+dφ0
+/productdisplay
+(n)/negationslash=(¯n)/radicalbig
+2πσ2
+n−(1−pE)¯Nh/productdisplay
+(n)/negationslash=(¯n)/radicalbig
+2πσ2
+n/radicalbig
+1−(σ2
+n/4π2)N0
+he3Nlog(1−pE)
+
+×exp/parenleftBigg
+−z2
+(¯n)
+2σ2
+¯n/parenrightBigg
+exp/parenleftbigg4π
+3m2N/parenrightbigg
+(5.8)
+where¯Nh≡N0
+h(N)(1+z2
+(¯n)/8π2)exp(3N).
+In [4, 6] we have argued that for realistic values15ofpE, volume weighting applies in
+the ensemble of homogeneous isotropic histories even in mod els where the potential ad-
+mits inflationary solutions all the way up to the Planck scale , corresponding to values
+φpl
+0∼1/m. This is because one can easily find data Dfor whichpE≪1/Npl
+h, where
+log(Npl
+h)≈3N(φpl
+0) = 9/2m2≈1012. In this regime the top-down factor reduces to pENh
+and the probability pEcancels out, as discussed in Section II. A single perturbati on mode
+on currently observable scales hardly changes the volume Nh. Hence the factor (1 −pE)¯Nh
+in (5.8) is approximately given by 1 −¯NhpEfor realistic values of pE.
+The product in the second, non-Gaussian term in (5.8) furthe r simplifies in histories
+where
+pE</bracketleftbigg/parenleftBig/summationdisplay
+(n)σ2
+n/4π2/parenrightBig
+N0
+he9φ2
+0/2/bracketrightbigg−1
+. (5.9)
+Whenφ0<1/√mthis condition automatically holds when the data are rare in the back-
+groundhistorybecausethesumover σnissmallerthanone. Incontrast, ineternallyinflating
+histories this is a stronger condition than the requirement used above that the data be rare
+in the homo/iso background. Indeed, in histories with a regi me of eternal inflation and hence
+φ0>1/√mone finds (/summationtext
+(n)σ2
+n/8π2)≈m2φ4
+0≫1, due to long wavelength perturbations
+that leave the horizon when χ(t)>1/√m. This reflects the fact that in eternal inflation,
+perturbations can significantly change the volume of surfac es of constant scalar field and
+therefore the possible locations where our data can be. Howev er, based on the arguments
+in [6] it appears plausible that the condition (5.9) holds wi th realistic values of pE(D) even
+in eternally inflating histories. Hence the TD probabilities p(z(¯n)|D≥1) are approximately
+given by
+p(z(¯n)|D≥1)∝/integraldisplay
+dφ0
+/productdisplay
+(n)/negationslash=(¯n)/radicalbig
+2πσ2
+n
+
+1−1−pE¯Nh/radicalBig
+1+pE(/summationtext
+(n)/negationslash=(¯n)σ2
+n/4π2)N0
+he3N
+
+×exp/parenleftBigg
+−z2
+(¯n)
+2σ2
+¯n/parenrightBigg
+exp/parenleftbigg4π
+3m2N/parenrightbigg
+(5.10)
+15That is, assuming we are a typical instance of Dand conditioning on our actual observational situation.
+Top-down probabilities conditioned on fewer or altogether different data can be calculated as well and
+may be of interest. When pE>1/Nh, the first term in (5.8) provides the dominant contribution t o such
+TD probabilities which are therefore Gaussian
+17Expanding the square root and including the normalization f actor in (5.4) yields
+p(z(¯n)|D≥1) =/integraltext
+dφ0/parenleftBig/producttext/radicalbig
+2πσ2
+n/parenrightBig
+N0
+h/parenleftbigg
+1+/summationtextσ2
+n
+8π2+z2
+(¯n)
+8π2/parenrightbigg
+exp/bracketleftbigg
+−z2
+(¯n)
+2σ2
+¯n/bracketrightbigg
+exp/bracketleftbig
+3N+4π
+3m2N/bracketrightbig
+/integraltext
+dφ0/producttext
+(n)/radicalbig
+2πσ2
+nN0
+h/parenleftBig
+1+/summationtext
+(n)σ2
+n/8π2/parenrightBig
+exp/bracketleftbig
+3N+4π
+3m2N/bracketrightbig
+(5.11)
+where the product and sum in the numerator are taken over all c lassical modes except the
+mode labeled by (¯ n). The probability pEhas cancelled out. In models with a regime of
+eternal inflation, the volume weighting exp(3 N) implies that the dominant contribution to
+the integrals in (5.11) comes from histories with the larges t values16ofφ0and hence a long
+period of inflation. In histories of this kind/summationtext
+(n)σ2
+n/8π2≫1. Hence the normalizing factor
+in the denominator makes the non-Gaussian TD corrections in z(¯n)extremely small, yielding
+p(z(¯n)|D≥1)≈p(z(¯n)|D≥1,φpl
+o)≈1/radicalbig
+2πσ2
+nexp/parenleftbigg
+−ǫ∗
+H2
+∗¯n3z2
+(¯n)/parenrightbigg
+(5.12)
+Even in the context of quadratic potentials it is possible to construct models without a
+regime of eternal inflation for instance by restricting the p hysically allowed range of φ. In
+such models, where all histories have φ0<1/√m, the integral over the other perturbation
+modes has little effect and the non-Gaussian TD corrections i nz(¯n)remain relevant in con-
+trast to the result above. On the other hand, in this case the v olume weighting does not
+significantly change the BU distribution of histories with d ifferentφ0[6], so that the integral
+overφ0is dominated by histories with the smallest amount of inflati on compatible with the
+dataD. The TD distribution in a background of this kind is approxim ately given by
+p(z(¯n)|D≥1)≈1/radicalbig
+2πσ2
+¯n1+z2
+(¯n)/8π2
+1+σ2
+¯n/8π2exp/parenleftbigg
+−ǫ∗
+H2
+∗¯n3z2
+(¯n)/parenrightbigg
+. (5.13)
+Hence in models without a regime of eternal inflation the NBWF pr edicts we should observe
+a slightly non-Gaussian spectrum of perturbations even tho ugh their BU distribution is
+Gaussian.
+It is possible to calculate the BU probabilities for the fluct uations pertaining to the CMB
+byfocussingonlyontherelevantmodesandignoringallothe rsinarestrictedminisuperspace
+model. At the BU level all modes are independent in the linear approximation. However, we
+have seen here that this is not possible for the TD probabilit ies for CMB observations. The
+observations may only probe the wavelengths characteristi c of only a few modes, but the
+top-down weighting depends on all of them. In a minisuperspa ce approximation consisting
+of homo/iso histories with a singleperturbation mode z(¯n), we would have predicted non-
+Gaussianity even in models of eternal inflation. When we incl ude all modes in our analysis
+the answer is qualitatively different. Quantum mechanics th en instructs us to coarse-grain
+over perturbations we do not observe. In eternally inflating histories this reduces the non-
+Gaussianity as in eq (5.11).
+As discussed earlier, CMB temperature correlations on a give n angular scale provide
+an excellent probe of the TD distribution for z¯nespecially at large l, where cosmic vari-
+ance is limited and where the dominant contribution comes fr om modes with a particular
+16What these are depends on the model, i.e. where Vbecomes too steep for inflation to occur. Below we
+assume for simplicity this only happens at the Planck scale, corresponding to φpl
+0≈1/m.
+18wavenumber n. Hence the prediction of non-Gaussianity with a specific shap e in models
+without eternal inflation leads to the possibility of determ ining whether or not eternal in-
+flation took place. The fact that we can learn something about the global structure of the
+universe from local observations conditioned on local data Dcan be traced to the quantum
+statewhichpredictsnon-localcorrelations. Thepredicte dlevelofnon-Gaussianityinmodels
+without a regime of eternal inflation is small on currently ob servable scales. However even a
+small departure from a Gaussian spectrum may be detectable w ith future observations. We
+will therefore return in future work to a more detailed analy sis of top-down corrections in
+the CMB anisotropies.
+VI. BACKREACTION IN THE REGIME OF ETERNAL INFLATION
+The expected amplitude of long wavelength perturbations th at leave the horizon in the
+regime of eternal inflation is large. Indeed, it follows from (4.9) thatH2
+∗≥ǫ∗whenχ∗≥√m
+and hence ∝an}bracketle{tz2
+(n)∝an}bracketri}htn3>1. Since in models of eternal inflation histories with φ0>1/√m
+dominate the TD probabilities [4, 6], this means there is a si gnificant probability for our
+universe to be strongly inhomogeneous on the largest scales in models of this kind.
+This inhomogeneity has important implications for the poss ible locations of our data,
+becausethesetypicallyconfineustooneorseveralsurfaces ofconstantdensity. Acalculation
+in perturbation theory of the expected fractional change in the volume V(t) of a surface of
+constant scalar field, due to combined effect of all fluctuatio ns outside the horizon yields,
+from (5.5) and using (4.10),
+∝an}bracketle{tδV
+V0(t)∝an}bracketri}ht=1
+8π2/integraldisplaynm(t)
+d3n∝an}bracketle{tz2
+n∝an}bracketri}ht ≈1
+8π2H2(t)
+ǫ(t)(6.1)
+wherenm(t) =Hb(t) andV0(t) = 2π2b3(t) is the volume of a surface which is at time tin the
+unperturbed geometry. Hence, for instance, the expected vol ume of the reheating surface in
+perturbed histories with φ0>1/√mcan differ significantly from the reheating volume in the
+homogeneous isotropic background. This indicates perturb ation theory may be inadequate
+to calculate the precise shape of the reheating surface in et ernally inflating histories. In
+fact, it has been argued (see e.g. [1, 22, 23]) – albeit in part based on perturbation theory –
+that starting with a finite inflationary volume in the regime o f eternal inflation, backreaction
+effects give rise to a significant probability for developing constant scalar field surfaces of
+arbitrarily large or even infinite volume17.
+This implies that in models of eternal inflation it may not be c orrect to assume that
+our data is rare in every history of the ensemble18. Instead in a subset of histories the
+more general weighting (5.4), or even its common limit, may a pply in the calculation of TD
+probabilities rather than volume weighting.
+However this more general weighting is unlikely to change our results for the TD prob-
+abilitiesp(z(¯n)|D≥1) obtained in Section VB, as we now explain. Let us assume, as b efore,
+17Numerical simulations of perturbed classical universes in this regime using stochastic techniques [24–27]
+provide some support for this.
+18We note however that the connection between the volume of the reheating surface and that of the surface
+of constant present matter density is rather complicated, s ince large-scale perturbations are large. This
+is a caveat in the analysis of top-down probabilities in this model.
+19that the data are rare in all background histories, i.e. pE≪1/Npl
+h. The top-down weighting
+then implies that eternally inflating histories with large φ0provide the dominant contribu-
+tion to the TD distribution [4]. Volume weighting will apply in approximately homogeneous
+and isotropic histories of this kind, yielding the Gaussian contribution to the TD distri-
+bution given in (5.12). However, if backreaction leads to a si gnificant probability for the
+reheating surface to be infinite then the main contribution t o the TD distribution will come
+from significantly perturbed histories where our data are co mmon because Nhis large or
+infinite. But in such histories the TD weighting in (5.7) equa ls one. Hence predictions
+for observations are given by the bottom-up probabilities. One expects BU probabilities of
+observable fluctuations not to be affected by backreaction eff ects, since perturbation modes
+on currently observable scales leave the horizon well outsi de the regime of eternal inflation
+where these effects are negligible. Hence we expect the result (5.12) remains unchanged
+when backreaction is taken in account.
+Roughly speaking, one could say that in these models, by sele cting histories with a large
+numberofefolds, thetopdownweightingalsomakesitlikely fortheretobeaHubblevolume
+with any given local perturbation on surfaces of constant de nsity. Indeed in a sufficiently
+largeuniverseanythingwillhappensomewhere. Hencethepro babilitythatatypicalobserver
+sees a particular fluctuation is determined by the relative f requency with which different
+fluctuations occur. But this is precisely what is given by the BU probabilities. This is an
+example where the quantum state specifies a measure for local prediction in cosmology that
+is well behaved for spatially large or infinite universes.
+VII. CONCLUSION
+The approach of this paper to cosmology in the regime of etern al inflation is significantly
+different from many others [3]. We have started from the funda mental assumption that
+the universe, including all its contents, is a closed quantu m mechanical system. We have
+exploredtheconsequencesofthisforpredictionintheregi meofeternalinflationinsimplified
+models in the context of the low-energy approximate quantum theory of gravity.
+Like any other closed quantum system the universe has a quant um state. The NBWF is
+the model for this state used here. Bottom-up probabilities for the different, coarse-grained
+histories of the universe and its contentsfollow from this s tate and not from a further posited
+measure. Classical behavior of spacetime geometry is not as sumed. Rather the ensemble of
+possible classical histories of the universe is derived fro m its quantum state.
+Observers are not assumed to necessarily exist, nor to be uni que, nor to be essentially
+classical systems outside the reach of quantum mechanics. R ather they are quantum sub-
+systems of the universe described by certain data with a prob ability to exist in any Hubble
+volume and a probability to be exactly replicated elsewhere in the universe.
+Probabilities relevant for observations are top-down prob abilities that take in account
+the observing system as a quantum subsystem of the universe. The starting point for the
+calculation of top-down probabilities are the bottom up pro babilities for four-dimensional
+histories conditioned on just the NBWF — the universe sub spec ie aeternitatis.
+The NBWF predicts a particular ensemble of classical, inflati onary histories with a char-
+acteristic set of perturbations that emerge from quantum flu ctuations. The bottom-up
+probabilities favor histories with a small number of efolds [4]. The perturbations are Gaus-
+sian with variance V(χ)/ǫevaluated at horizon crossing, where ǫis the slow-roll parameter
+(eq. (4.9)). Therefore in histories with a regime where V(χ)> ǫ, significant probabilities
+20are predicted for large fluctuations that left the horizon wh ile this condition holds. This is
+called the regime eternal inflation. The NBWF thus predicts th at histories of this kind are
+inhomogeneous on the large scales that left the horizon during such a regime . In particular
+it predicts that any constant χsurface, such as the reheating surface, can differ significan tly
+from the same surface in the homo/iso background. This resul t resonates well with other
+discussions of eternal inflation as well as numerical simula tions using stochastic techniques
+[24–27].
+Top-down probabilities are constructed from bottom-up pro babilities by further condi-
+tioning on some part of our data that includes a description o f the observational situation
+within the universe. If one conditions on data Dthat localize the observer on one or several
+surfaces in each history then the general weighting (5.4) co nnects top-down probabilities
+to bottom-up ones. This weighting is not a choice, or a postul ate, or a proposal. Instead
+it arises necessarily from four considerations: 1) Our data Doccur within a given Hubble
+volume only with some quantum probability pE. 2) In a large universe our data may occur
+elsewhere with significant probability. 3) All we know about t he universe is that our history
+exhibits at least one instance of it. 4) An assumption that we a re equally likely to be any
+of the instances of Dthat our universe exhibits.
+Volume weighting arises as an approximation to (5.4) onlywhen our data are rare in all
+histories in the ensemble that are predicted with any signifi cant probability. For realistic
+values ofpE[6] we find this implies that top-down probabilities favor hi stories with a large
+number of efolds in models that have a parameter regime where V >ǫ, withǫthe slow-roll
+parameter [4, 5]. Unlike this approximation, the general wei ghting (5.4) is well behaved even
+when spatial volumes become infinite. In fact for very large v olumes, the quantum nature of
+the observational situation implies that the top-down prob abilities for observations converge
+to the bottom-up probabilities19. This is an important difference with other discussions of
+eternal inflation, usually not based on quantum cosmology. T here the infinite volume limit
+instead leads to ambiguities. In those cases, to predict the outcome of our observations
+unambiguously, a measure must be introduced that regulariz es the infinitely large spatial
+volumes that arise in the regime of eternal inflation. By cont rast, in quantum cosmology the
+wavefunctionprovidestheonlymeasureneededforunambigu ousprediction20. Furthermore,
+it does this as part of a unified framework that also explains t he origin of inflation and of
+classical spacetime itself.
+In this paper we have calculated the top-down probabilities for different fluctuations in
+models with a single scalar field χwith a quadratic potential m2χ2. We find that the
+NBWF predicts a significantly inhomogeneous universe on very large scales and a Gaussian
+spectrum of small perturbations on currently observable sc ales when there is a regime of
+eternal inflation, i.e. χ>1/√min the early universe. The inclusion of backreaction effects
+of perturbations may give rise to histories with a truly infin ite reheating surface, but we have
+no indications this leads to a breakdown of the calculationa l framework nor do we expect
+this to change this specific result.
+19This resonates with [28] where it is argued that the total num ber of locally distinguishable FRW universes
+generated by eternal inflation is finite. Here we have seen tha t the top-down probabilities for different
+values of local perturbations become indistinguishable in very large universes.
+20It would be of interest to compare top-down probabilities ca lculated from the NBWF with the predictions
+of other measures employed in the study of eternal inflation.
+21By contrast, in models where the scalar field takes values onl y in a restricted range21
+that does not include a regime where V > ǫ, we find the top-down probabilities predict
+large-scale homogeneity and a slightly non-Gaussian spect rum of observable fluctuations,
+for realistic values of pE. The predicted level of non-Gaussianity is small on observa ble
+scales but potentially detectable with future experiments . More generally we expect it to
+be true that the top-down weighting leads to some non-Gaussi anity only in models without
+eternal inflation, and therefore to the possibility to test w hether our universe exhibits a
+regime of eternal inflation.
+The differences between the TD and BU probabilities are strik ing. Bottom up probabili-
+ties favor past inflation but only in small amounts. In contra st, top-down probabilities favor
+a large number of efolds of past inflation. Bottom up probabil ities favor a homogeneous
+universe. Top-down probabilities predict a universe that i s significantly inhomogeneous on
+scales much larger than the present horizon in models with et ernal inflation.
+The top-down probabilities for prediction exemplified by (5 .4) depend only on data D
+within our past light cone. (In the present models this data i s approximated by data on
+a spacelike surface in our Hubble volume.) But they also depen d on the implications of
+the theory for the structure of the universe on scales much la rger than the present horizon.
+That is because top-down probabilities depend not only on wh at the data are on our past
+light cone, but also on where light cones with that data may be located in spacetime. This
+is determined in part by the quantum state, which predicts no n-local correlations and in
+particular specifies what the allowed classical spacetimes are.
+Turning this connection around we see that from local observ ations we may draw infer-
+ences about the structure of our universe outside the presen t horizon, assuming of course
+that the theoretical framework behind these predictions is secure. The TD predictions for
+the spectrum of primordial perturbations provide a strikin g example of this. These predict
+a specific form of non-Gaussianity, but only in histories whe re we are rare. Any observation
+of this non-Gaussianity would therefore provide valuable i nformation about the possible lo-
+cations of our data and place an upper bound on the size of our u niverse. If by contrast this
+non-Gaussianity turns out to be absent in the perturbation s pectrum this would be evidence
+for a much larger, eternally inflating and therefore possibl y infinite universe.
+Thus if the values of the top-down probabilities depend on th e large scale structure of
+the universe then the results of the observations they predi ct offer the opportunity to probe
+this structure. This striking connection between global st ructure and local observation is
+ultimately traceable to the NBWF which, like any quantum stat e, is defined globally not
+locally.
+The underlying homo/iso symmetry has of course greatly simp lified the calculation of
+TD probabilities in this paper. The symmetry means all Hubble volumes on the surface
+where the data Doccur are equivalent, which essentially allows one to ignor e all other
+instances of our data and focus on our own. In particular, the probabilities for different
+values of a perturbation mode z(n)in our own Hubble volume given D≥1are the same as the
+probabilities that the universe exhibits different values o f perturbations z(n)in any Hubble
+volume given D≥1. The symmetry therefore automatically ‘organizes’ the diff erent Hubble
+volumes, even when Doccurs on infinitely large surfaces.
+To apply the top-down approach to string theory which, it has been argued, at low
+21Or models where quantum corrections render the potential to o steep so that the regime of eternal inflation
+sets in only at the Planck scale.
+22energies predicts a potential landscape with finitely many v acua with different physics, one
+must generalize the calculations in this paper to models whe re the possible locations of D
+are not all connected by symmetry22. In models of this kind, one expects the NBWF to
+select inflating histories that roll down from flat patches in the landscape where the slow roll
+conditions hold. However it appears plausible that besides h istories where the background
+is homogeneous and isotropic, the ensemble also includes hi stories where our data occur on
+homogeneous surfaces in open FRW universes that are bubbles inside de Sitter space. This
+is because one can get histories of this kind from complex Col eman-De Luccia instantons
+that obey the no-boundary condition of regularity [2]. If on e neglects collisions between
+bubbles then all locations inside bubbles of the same type ar e equivalent, and only one
+‘representative’ location enters in the calculation of TD p robabilities. By contrast, the
+relative probability of finding our data in bubbles of differe nt types (or in histories without
+bubbles) is important. In the NBWF this is given by the ratio of the real part of the actions
+of the corresponding instantons, yielding a well-defined pr ediction. Thus, at the present
+moment, we see no obstacle of theory or practice to extending the results of this paper to
+more general and realistic models of the implications of a qu antum universe.
+Acknowledgments We thank Ben Freivogel, Jaume Garriga, Allan Guth, Ricardo Mo n-
+teiro and Alex Vilenkin for helpful discussions. We also thank the Theory Division of CERN
+for its hospitality during part of this work. The work of JH wa s supported in part by the
+National Science Foundation under grant PHY05-55669.
+Appendix A: Semiclassical Wave Function of Linear Perturbati ons
+In this Appendix we calculate the wave function of linearized perturbations around the
+homogeneous isotropic saddle point histories discussed in Section IVA.
+As discussed in Section IVB, the wave function of linear pertur bations depends on the
+background variables bandχ, and on a single linear combination of the perturbation vari -
+ablesa(n),b(n)andf(n)that describe the perturbed complex extremizing four geome try. We
+work with the following gauge-invariant linear combinatio n (see also [18, 29]),
+˜ζ(n)=a3[˙φ(a(n)+b(n))−HEf(n)], (A1)
+where˜ζtends to a real value ˜ zat the boundary. To calculate Ψ( b,χ,˜z) it is convenient
+to return to the original perturbation variables (4.3) and t o choose a particular gauge to
+find the solutions that extremize the action. One can then rew rite the result in terms of ˜ zn
+and therefore express the wave function in a gauge invariant way. (In Section IV we have
+written the wave function in terms of z= ˜z/a3˙φ, which is conserved outside the horizon and
+therefore closely related to physical (observable) quanti ties.)
+A general linear scalar gauge transformation allows one to s etE=B= 0 in (4.3), or
+b(n)=k(n)= 0 in terms of perturbation modes. This is the Newtonian gauge in which
+g(n)=−a(n), and the equations that govern the fluctuations read [18]
+¨a(n)+4HE˙a(n)−(3m2φ2−2/a2)a(n)=−3˙φ˙f(n)−3m2φf(n) (A2a)
+22Therangeofpossiblelocationsdependsonthesetofhistori esinvolvedandthereforeonthecoarse-graining.
+The latter is in turn determined by the question one asks.
+23102030405060y2./Multiply10104./Multiply10106./Multiply10108./Multiply1010Re/LParen1a/RParen1
+20406080100y1234Re/LParen1Φ/RParen1
+FIG. 1: The real part of the scale factor (left) and the scalar field (right) of the complex homoge-
+neous isotropic slow-roll solution labeled by φ0= 4, with m2=.05. This is shown here along the
+vertical part of a contour in the complex τ-plane that first runs from the origin to X≈π/2mφR(0)
+and then upward along the y-axis. The turning point Xand the phase of φ(0) have been fine-tuned
+so thataandφtend to real functions along the vertical part of the contour . This happens very
+rapidly, so that the solution behaves classically already a ty≥ O(1).
+¨f(n)+3HE˙f(n)−(m2+(n2−1)/a2)f(n)=−4˙φ˙a(n)−2m2φa(n) (A2b)
+˙a(n)+HEa(n)=−3˙φf(n) (A2c)
+We consider a coarse-graining in which we concentrate on per turbation modes that leave
+the Hubble radius during inflation. These are the modes that ge t amplified by the time-
+dependent background and, ultimately, lead to the large-sc ale structures we observe today.
+The no-boundary condition selects solutions of (A2) that are regular at the SP. This
+meansf(n)anda(n)must vanish as τ→0. From eqs (A2) and regularity of the background
+it follows that near the SP, the leading order term in τis given by
+f(n)=ζ(n)(0)τn−1, a (n)=−3m2φ(0)
+4(n+2)ζ(n)(0)τn+1(A3)
+whereζ(n)(0)≡ |ζ(n)(0)|eiθ≡ζ(n)0eiθis a complex constant. The phase θshould be fine-
+tuned such that ζ(n)is real at the endpoint υ. The amplitude ζ(n)0in turn is determined
+byz(n). At the SP ζ(n)0is thus a free parameter which can be used to label the differen t
+histories. The ensemble of perturbed histories can therefo re be labeled by ( φ0,ζ0).
+At early times, when the physical wavelength a/nof the perturbation mode is smaller
+than the Hubble radius H−1
+E, the metric perturbation a(n)does not significantly affect the
+evolution of the matter perturbation f(n). Specifically the terms on the right-hand side in
+(A2b) are negligible in slow-roll backgrounds (4.1) when n≫ |HEa|, so that for n≫1 the
+matter perturbation equation reduces to
+f′′
+(n)+2HEf′
+(n)−(n2−1)f(n)= 0. (A4)
+Here prime denotes the derivative with respect to conformal E uclidean time ηEandHE≡
+a′/a. In this regime the solutions that are regular at the SP take t he approximate analytic
+form
+f(n)=ζ(n)(0)
+aenηE, a (n)=−3φ′ζ(n)(0)
+naenηE, (A5)
+240.51.01.52.02.53.03.5y
+/Minus100/Minus50050100Re/LParen1f20/RParen1
+0.51.01.52.02.53.03.5y
+/Minus100000/Minus5000050000100000IR/LParen120/RParen1
+FIG. 2: Left panel : When the perturbation mode is inside the horizon the comple x scalar field
+fluctuation oscillates with amplitude ∝a−1, as illustrated here for the real part of the n= 20 mode
+in the background of Figure 1.
+Right panel : As a consequence of this, the Euclidean action of a perturba tion mode that is inside
+the horizon oscillates with an approximately constant ampl itude.
+where the constraint (A2c) was used to find the metric perturba tion. These solutions are
+valid in the complex η-plane in the regime n≫Ha. From (A1) it follows that in this regime,
+˜ζ(n)≈ −Ha3f(n).
+One can verify whether the analytic approximations (A5) are a ccurate by solving nu-
+merically for the perturbations simultaneously with the co mplex background. This can be
+done e.g. by integrating the field equations along a broken co ntourCB(X) in the complex
+τ-plane that runs along the real axis to a point X, and then up the imaginary y-axis. When
+φ0≥φc
+0one can adjust both the turning point Xand the phase angle γofφ(0) so that a
+andφtend to real functions b(y) andχ(y) along the vertical line given by τ=X+iyin
+the complex τ-plane [5]. These are the scale factor and scalar field of a cla ssical Lorentzian
+solution. An example of an exact complex background that tend s to a classical history is
+shown in Figure 1, for φ0= 4 andm2=.05.
+In Figure 2 (left panel) we plot the evolution of the n= 20 matter perturbation along
+the vertical part of the contour in this background. The rang e ofyshown here corresponds
+to the regime where the mode is inside the horizon. One sees it oscillates rapidly with
+decreasing amplitude ∝a−1, in good agreement with the analytic approximation (A5). The
+Euclidean action of a solution to the equations (A2) is just a b oundary term [18],
+I(n)=M˜z(n)˜z′
+(n)−N˜z2
+(n) (A6)
+where
+M≡(n2−4)
+2[(n2−4)a′2+3a2φ′2](A7)
+and
+N≡1
+4MUa3/bracketleftbigg
+Kn/parenleftbigg
+2a4−3a6m2φ2+3n2−1
+n2−4a4φ′2/parenrightbigg
++a12m4φ2+3a9φφ′a′/bracketrightbigg
+(A8)
+withU=Knaa′+a8m2φφ′andKn≡1
+3[(n2−4)a′2−(n2+5)a4φ′2−(n2−4)a6m2φ2].
+All quantities here are evaluated on the boundary surface whe re one calculates the wave
+2520406080100y
+/Minus20/Minus101020Re/LParen1f20/RParen1
+20406080100y
+/Minus2246Re/LParen1a20/RParen1
+FIG. 3: Numerical solution of the perturbation modes a(n)andf(n)forn= 20 in the exact complex
+φ0= 4backgroundshowninFigure1. Themodesarecomplexandosc illatewhentheabsolutevalue
+of their wavelength is smaller than the Hubble radius, or n >|aHE|. When the wavelength crosses
+the Hubble radius around y∼5 both the matter - and metric perturbation start slowly grow ing
+until the end of inflation around y∼60. The imaginary part of the gauge invariant combination
+ζ(n)decays away in this regime. After inflation ends the metric pe rturbation is essentially real and
+constant, with small oscillations due to the oscillating ba ckground scalar field. These primordial
+metric perturbations provide the seeds for structure forma tion in the corresponding Lorentzian
+cosmology.
+function. In the complex τ-plane this surface is given by a certain value τf=X+iyfwhere
+the variables take real values a(τf) =b,φ(τf) =χand˜ζ(τf) = ˜z.
+When the absolute value of the wavelength a/nof a complex perturbation mode becomes
+larger than the Hubble radius, both the scalar field and metric perturbations stop oscillating
+andstartslowlygrowing. Thistransitioncanbeclearlysee ninthenumericalsolutionsshown
+in Figure 3. It can also be understood analytically: Outside the horizon the gradient term is
+unimportant in the equations of motion (A2), which therefore admit growing and decaying
+solutions for a(n)andf(n). The growing solutions are given by
+fg
+(n)∼1
+φ, a (n)=1
+φfg
+(n)(A9)
+and the decaying modes are
+fd
+(n)∼1
+a3, a (n)=−m
+2Hfd
+(n). (A10)
+The general solution for ˜ζ(n)in this regime is a combination of a growing and decaying
+mode. The(complex)proportionalityconstantsmultiplyin geachtermcanbeapproximately
+determined in terms of ζ(n)(0) by matching the solution on subhorizon scales at horizon
+crossingn=a∗H∗. Here the subscript star means the quantity is to be evaluated at the time
+of horizon crossing of modes with wavenumber n. At horizon crossing ˜ζ(n)generally has an
+imaginary component, since the scalar field and metric pertu rbation are not simultaneously
+real. The requirement that ˜ζ(n)be real at the boundary essentially means that the phase
+θofζ(n)(0) at the SP should be tuned so that the imaginary component o f the subhorizon
+mode function matches onto the decaying mode when the pertur bation leaves the horizon.
+This also means perturbations behave classically when thei r wavelength exceeds the Hub-
+ble radius, since the information on their phase decays away . We illustrate this in Figure 4
+261015202530y
+/Minus0.0010/Minus0.00050.00050.00100.0015Im/LParen1z20/RParen1
+Re/LParen1z20/RParen1
+68101214161820y0.020.040.060.080.10/LBracketBar1/GradientIR/RBracketBar12
+/LBracketBar1/GradientS/RBracketBar12
+FIG. 4:Left panel : The phase of the perturbation mode at the SP should be tuned s o thatznis
+real at the boundary. This is illustrated here for the numeri cal perturbation solution z20along the
+vertical part of a contour in the complex τ-plane, in the complex φ0= 4 background.
+Right panel: The ratio of the gradients of the real part of the Euclidean ac tion to the imaginary
+part tends to zero when the wavelength of a perturbation mode becomes larger than the Hubble
+radius.
+whereθis fine-tuned so that the numerical solution ˜ζ(n), forn= 20, tends to a real function
+z(n)along the vertical part of the broken contour CB(X) in theφ0= 4 background and with
+m2=.05. One sees the ratio of the gradients of the real part of the E uclidean action to the
+imaginary part tends to zero.
+The real part of the action (A6) tends to a constant23, which is approximately given by
+its value when the mode leaves the horizon. Hence for the appro ximate analytic solutions
+(A5) we obtain
+I(n)
+R→n˜z2
+(n)(y∗)
+2b4
+∗H2
+∗(A11)
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+nally existing self-reproducing chaotic inflationary univ erse,Phys. Lett. B 175, 395 (1986).
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+ities in eternal inflation ,Phys. Rev. Lett. 97, 191302, (2006), arXiv:hep-th/0605263; A. D.
+23The asymptotic value of IRcan also be obtained from the approximate analytic form of th e superhorizon
+solutions (A9) and (A10). Indeed, while the growing term in ˜ζ(n)must be tuned to be real outside the
+horizon, the derivative z′
+(n)contains an imaginary component of order a2
+∗ζ(n)0aH2that arises from taking
+the derivative of the decaying mode. This is a subleading con tribution to the total action, but it gives
+rise to a real part IRof the correct magnitude.
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+29
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+arXiv:1001.0263v2  [math.MG]  18 Jul 2011Short paths for symmetric norms in the unitary group∗
+Jorge Antezana, Gabriel Larotonda and Alejandro Varela
+Abstract
+For a given symmetrically normed ideal Iφon an infinite dimensional Hilbert space
+H, we study the rectifiable distance in the classical Banach-Lie unitar y group
+Uφ={ua unitary operator in H, u−1∈ Iφ}.
+We prove that one-parameter subgroups of Uφare short paths, provided the spectrum
+of the exponent is bounded by π, and that any two elements of Uφcan be joined with a
+short path, thus obtaining a Hopf-Rinow theorem in this infinite dimen sional setting,
+for a wide and relevant class of (non necessarily smooth) metrics. T hen we prove that
+the one-parameter groups are the unique short paths joining give n endpoints, provided
+the symmetric norm considered is strictly convex. This preprint has been replaced
+by [5] where the general case of convex Lagrangians in the uni tary group
+is studied with similar techniques .1
+1 Introduction
+The group of unitary operators on Hilbert space carries, as a ny Banach-Lie group, a canon-
+ical connection without torsion ∇XY=1
+2[X,Y], whose geodesics are the one-parameter
+groupst/ma√sto→uetz(hereuis a unitary operator and zan anti-Hermitian operator). In the
+finite dimensional setting, the trace is available to introd uce a Riemannian metric on the
+unitary group in a standard fashion
+/an}b∇acketle{tx,y/an}b∇acket∇i}htg=Tr(u∗x(u∗y)∗) =Tr(xy∗),
+foru∗x,u∗yin the Lie algebra of the group, that is, for u∗x,u∗yanti-Hermitian matrices.
+It is well-known that the connection just introduced is in fa ct the Levi-Civita connection
+of the metric ginduced by the trace, and that geodesics are short provided t he spectrum
+ofzis bounded by π.
+Now consider the bi-invariant Finsler metric given by the un iform norm,
+/ba∇dblx/ba∇dblu=/ba∇dblu∗x/ba∇dbl=/ba∇dblx/ba∇dbl
+∗2010 MSC. Primary 47B10, 53C23; Secondary 51F25, 53C22
+1Keywords and phrases: geodesic segment, ideal of compact op erators, p-norm, short path, symmetric
+norm, unitary group.
+1for anyxtangent to a unitary operator u. Remarkably, if one keeps the connection but
+changes the metric, the geodesics of the connection are stil l short for the induced rectifiable
+distance (which, as in the Riemannian setting, is computed a s the infimum of the length
+of piecewise smooth curves joining given endpoints, and L(γ) =/integraltext1
+0/ba∇dbl˙γ/ba∇dbldt). The proof of
+this well-known fact deserves to be sketched here since is sh ort and elegant: if γis a path
+of unitaries such that γ(0) =u,γ(1) =v=uez, withzany anti-Hermitian logarithm of
+u∗vof norm bounded by π, letρbe a norming state for −z2≥0 andξbe a unit norm
+vector in the Hilbert space Hρof the Gelfand-Neimark-Segal representation induced by
+ρ, such that −z2·ξ=/ba∇dblz/ba∇dbl2ξ(chooseξto be the class of the identity operator). Then
+the map w/ma√sto→w·ξ, from the unitary group to the unit sphere of Hρ, with its standard
+Riemann-Hilbert metric, sends the one-parameter group δ(t) =uetzto a geodesic of the
+unit sphere, with equal lengths. Thus uetz·ξis shorter than γ·ξ, and
+L(γ) =/integraldisplay
+/ba∇dbl˙γ/ba∇dbl ≥/integraldisplay
+/ba∇dbl˙γ·ξ/ba∇dblHρ≥/integraldisplay
+/ba∇dbl(uetz·ξ)˙/ba∇dblHρ=/ba∇dblz/ba∇dbl=L(δ).
+This idea can betraced back to a paper by Porta and Recht [17], whereit was used to prove
+the minimality of certain paths in the Grassmann manifold of aC∗-algebra, see also [2] for
+a Hopf-Rinow theorem in the context of (infinite-dimensiona l) Grassmann manifolds.
+In recent years, there has been an increasing interest in the geometric properties of
+idealsIof compact operators on Hilbert space (see [6, 7, 9] and the re ferences therein),
+and in particular in the Lie and representation theory of the subgroups of the linear group
+GI, whichconsistof invertible operators gonHsuch that g−1∈ I. Ofparticular relevance
+is the group of unitary operators, since it acts isometrical ly on the ideals that arise from
+a symmetric norm, and gives rise to several homogeneous spac es such as the Grassmann
+manifold, which can be viewed as the coadjoint orbit of a fixed projection.
+In this paper we focus on the metric properties of the unitary groups arising from
+a symmetric norm on compact operators. We study short curves , and we prove what
+we think is a fundamental result in the interplay between the algebraic and geometric
+properties of these groups: one-parameter subgroups are sh ort geodesics for the rectifiable
+distance induced by the symmetric norm, thus extending the a bove mentioned result for
+the Finsler metric induced by the uniform norm. We also prove that these geodesics are
+unique among short paths joining its endpoints, provided th e symmetric norm involved is
+strictly convex. Since the paper is intended for those inter ested in metric geometry and
+operator algebras alike, we have tried to keep the prerequis ites to a minimum, and we also
+included the proof of some perhaps well-known facts in both a reas to make the paper more
+readable.
+Thispaperis organized as follows. In Section 2we fix ournota tion regardingsymmetric
+norms and the ideals of compact operators induced. In Sectio n 3 we recall the notions of
+rectifiable length and rectifiable distance associated to a s ymmetric norm in U(H). Then
+in Section 4 we introduce the notion of geodesic segment, whi ch is a one-parameter group
+inUφsuch that its initial speed has spectrum contained in [ −iπ,iπ]. We prove a certain
+triangle inequality valid for any symmetric norm (Proposit ion 4.3), the key tool to the
+proof of Theorem 4.7. This is one of the two main results of thi s paper, and states that
+2geodesics segments are short for any symmetric norm . In Section 5, we recall the notion of
+dual norm and prove some useful properties in connection wit h conditional expectations.
+We use these tools to prove Corollary 5.8, which is the other m ain result of this paper, and
+states that if the symmetric norm is strictly convex, then ge odesic segments are the unique
+short paths joining given endpoints in Uφ.
+2 Symmetric norms
+LetHbe a complex separable and infinite dimensional Hilbert spac e. In this paper B(H),
+K(H) andJ(H) stand for the sets of bounded linear operators, compact ope rators and
+finite rank operators in Hrespectively. The unitary group of B(H) is indicated by U(H),
+andP(H) is the set of self-adjoint projections in B(H). Ifx∈ B(H), then/ba∇dblx/ba∇dblstands for
+the usual uniform norm, and we will use | · |to indicate the modulus of an operator, i.e.
+|x|=√
+x∗x. We indicate with B(H)h(resp.B(H)ah) the real linear spaces of Hermitian
+(resp. anti-Hermitian) elements of B(H).
+If/ba∇dbl·/ba∇dblφ:B(H)→R∪{∞}is a norm, let
+Iφ={x∈ B(H) :/ba∇dblx/ba∇dblφ<∞}
+stand for the set of operators with finite norm. A norm /ba∇dbl·/ba∇dblφis called symmetric if
+/ba∇dblxyz/ba∇dblφ≤ /ba∇dblx/ba∇dbl/ba∇dbly/ba∇dblφ/ba∇dblz/ba∇dbl
+for anyx,y,z∈bh. The standard references on the subject are the books by Gohb erg and
+Krein [11] and Simon [18]. The set Iφis a bilateral ideal in B(H). Let
+I(0)
+φ=J(H)/bardbl·/bardblφ
+stand for the closure of the finite rank operators in the symme tric norm, which is again
+a bilateral ideal of compact operators. Clearly I(0)
+φ⊂ Iφ, while if I(0)
+φ=Iφwe say that
+/ba∇dbl · /ba∇dblφisregular. The ideals I(0)
+φ,Iφare theminimal andmaximal ideals related to the
+symmetric norm, according to the classical literature [18, Chapter 2]. It will be assumed
+(unless otherwise stated) that /ba∇dbl·/ba∇dblφisinequivalent to the uniform norm of B(H), soI(0)
+φ,Iφ
+are proper ideals of compact operators. Some elementary use ful properties include the
+following:
+•unitary invariance : for any u,v∈ U(H),/ba∇dbluxv/ba∇dblφ=/ba∇dblx/ba∇dblφ.
+•gauge invariance : ifx=u|x|is the polar decomposition of x, then/ba∇dblx/ba∇dblφ=/ba∇dbl|x|/ba∇dblφ.
+•invariance for the adjoint : for any x, it holds /ba∇dblx/ba∇dblφ=/ba∇dblx∗/ba∇dblφ.
+What we are interested in here is in the unitary groups of /ba∇dbl·/ba∇dblφ, that is
+Uφ={u∈ U(H) :u−1∈ Iφ}orU(0)
+φ={u∈ U(H) :u−1∈ I(0)
+φ}.
+3The topology in these groups is given by the metric
+d(u,v) =/ba∇dblu−v/ba∇dblφ=/ba∇dbl(u−1)−(v−1)/ba∇dblφ.
+Well-known examples of symmetric norms are the p-norms (p≥1) given by
+/ba∇dblx/ba∇dblp=Tr(|x|p)1
+p,
+whereTris the usual (infinite) trace of B(H), and also the Ky-Fan k-norms. In the case of
+thep-norms, the regular ideals induced are called p-Schatten ideals , and the unitary groups
+are examples of what Pierre de la Harpe calls classical Banach-Lie groups of operators on
+Hilbert space [13]. Note that for p=∞,Iφ=B(H) whileI(0)
+φ=K(H).
+3 The unitary group and its rectifiable distances
+The unitary group U(H) ofB(H) is a smooth manifold in the uniform norm of B(H), and it
+is in fact a real Banach-Lie subgroup of the full linear group . Its Banach-Lie algebra can be
+identified with B(H)ah, and ifγ: [0,1]→ U(H) is a smooth curve, then γ∗(t)˙γ(t)∈ B(H)ah
+for anyt∈[0,1]. Here smooth means C1and with non-vanishing derivative.
+It is well-known that if one endows the tangent spaces of U(H) with the Finsler metric
+that consists of measuring vectors with the uniform norm of B(H),
+/ba∇dbl˙γ/ba∇dblγ=/ba∇dblγ∗˙γ/ba∇dbl=/ba∇dbl˙γ/ba∇dbl,
+then the curves δ(t) =eitz, with Hermitian symbol z, are shorter than any other piecewise
+smooth curve joining its endpoints, provided /ba∇dblz/ba∇dbl ≤π(see the introduction for a proof).
+Thus the rectifiable distance related to the uniform norm is a lso known as exponential
+lengthin the unitary group (of a C∗-algebra), and in that setting it is a relevant invariant
+of the classification theory, see [16] and the references the rein.
+However, in B(H) (in fact, in any von Neumann algebra), any pair of elements u,v∈
+U(H) are conjugated by an Hermitian zwith/ba∇dblz/ba∇dbl ≤π(i.e. there exists z∈ B(H)hsuch
+that/ba∇dblz/ba∇dbl ≤πandueiz=v). This element is unique if /ba∇dblu−v/ba∇dbl<2, and in that case
+/ba∇dblz/ba∇dbl< π. Thus if one defines the rectifiable distance as the infimum of t he lengths of curves
+joining its endpoints,
+d∞(u,v) = inf/braceleftbigg/integraldisplay1
+0/ba∇dbl˙γ/ba∇dbl:γis piecewise smooth and joins utovinU(H)/bracerightbigg
+,
+thend∞(u,v) =/ba∇dblz/ba∇dbl.
+Remark 3.1. Recall that there exists a constant cφsuch that /ba∇dblx/ba∇dbl ≤cφ/ba∇dblx/ba∇dblφfor any
+symmetric norm /ba∇dbl·/ba∇dblφand any x∈ Iφ, see for instance [11, page 65 and the footnote
+at page 58]. The number cφis the value of /ba∇dbl·/ba∇dblφon any one-dimensional projection. So
+we might as well assume from now on that cφ= 1, i.e. /ba∇dblx/ba∇dbl ≤ /ba∇dblx/ba∇dblφfor anyx∈ Iφ. In
+particular, convergence in Iφimplies convergence in the uniform norm, and a continuous
+curve in Iφis continuous for the uniform norm.
+4We will consider piecewise smooth curves γ: [0,1]→ Uφ. Note that γ,˙γare continuous
+also in the uniform norm. Since ˙ γ∈ Iφ, it is legitimate to define
+/ba∇dbl˙γ/ba∇dblγ=/ba∇dbl˙γ/ba∇dblφ.
+Definition 3.2. The rectifiable length of γis
+Lφ(γ) =/integraldisplay1
+0/ba∇dbl˙γ/ba∇dblφ,
+and the rectifiable distance among u,v∈ Uφis
+dφ(u,v) = inf{Lφ(γ) :γ: [0,1]→ Uφis piecewise smooth and joins utovinUφ}.
+The function dφis in fact a distance, since /ba∇dblu−v/ba∇dblφ≤dφ(u,v) for any u,v∈ Uφ, see
+the related Proposition 4.8 below.
+One of the main features of this metric is that is invariant fo r the action of the unitary
+groupU(H), in fact it is a bi-invariant metric
+dφ(uv1w,uv2w) =dφ(v1,v2)
+foru,w∈ U(H) andv1,v2∈ Uφ.
+Remark 3.3. Observe that, for given u∈ Uφ,u−1 is a normal compact operator and
+thus admits a spectral decomposition relative to a family of disjoint projections {pk}k∈N,
+namelyu−1 =/summationtextαkpkfor some bounded sequence αk. In fact lim kαk= 0 since u−1 is
+compact. Now let p0= 1−/summationtextpkthus
+u=p0+/summationdisplay
+k(1+αk)pk.
+It is then clear that |1+αk|= 1 for any k∈N. Let{tk}k∈Nbe a sequence of real numbers
+such that |tk| ≤πandeitk= 1+αk. Inparticular, tk→k0. From theelementary inequality
+|eit−1|=/radicalbig
+2(1−cost)≥ |t|/radicalbigg
+1−t2
+12, (1)
+valid for any t∈[−π,π], and the fact that
+|t|/radicalbigg
+1−t2
+12≥1
+2|t|
+for sufficiently small t, we derive that |tk| ≤2|αk|for sufficiently large k. Thus the operator
+z∈ B(H) defined by z=/summationtext
+ktkpkis clearly compact and moreover z∈ Iφ, since|αk|are
+the singular values of u−1 (see [11, page 69]).
+5Remark 3.4. The previous remark shows that for any u∈ Uφthere exists an Hermitian
+z∈ Iφwith/ba∇dblz/ba∇dbl ≤πsuch that eiz=u. Moreover, if /ba∇dblu−1/ba∇dbl<2 (or equivalently /ba∇dblz/ba∇dbl< π),
+thiszis unique by the uniqueness of the principal branch of the ana lytic logarithm of uin
+B(H). Then any pair of unitaries u,v∈ Uφcan be conjugated with an Hermitian z∈ Iφ,
+ueiz=v, with/ba∇dblz/ba∇dbl ≤π, and one can consider the elementary smooth curve
+δ(t) =ueitz
+which joins them in Uφ. Its rectifiable length is /ba∇dblz/ba∇dblφ. It is known, for the p-norms of B(H),
+withp≥2, that these curves are shorter that any other piecewise smo oth curve joining u
+tov, and hence
+dp(u,v) =/ba∇dblz/ba∇dblp, p≥2.
+Moreover the curve is unique, among C2curves joining u,v, if/ba∇dblz/ba∇dbl< πandp <∞. The
+proof of these facts can be found in [4, Theorem 3.2]. See also [1] for related results on the
+unitary group of a finite von Neumann algebras and their homog eneous spaces.
+4 Minimality of geodesic segments
+Let us establish some facts concerning the curves t/ma√sto→ueitzfor Hermitian z∈ Iφjoiningu
+andv=ueizinUφ. We will call such paths geodesic segments . In this section we prove that
+the geodesic segments are short among piecewise smooth path s inUφ, when we measure
+them with the rectifiable distance. We begin with a technical lemma.
+Lemma 4.1. Letx,y∈ K(H)hand assume that eix=eiy,/ba∇dblx/ba∇dbl< π. Thenxcommutes
+withy, and|x| ≤ |y|. Moreover, if /ba∇dbly/ba∇dbl ≤π, thenx=y.
+Proof.Lett∈R. Theneityeixe−ity=eityeiye−ity=eiy. This is equivalent to
+exp(eityixe−ity) =eiy.
+Differentiating at t= 0 we obtain exp∗ix([y,x]) = 0. Since /ba∇dblix/ba∇dbl< π, the map exp∗ixis
+invertible (see for instance [4, Lemma 3.3]), and the first cl aim [y,x] = 0 follows. Now
+let{pk}k∈Z⊂ P(H) be a family of disjoint projections (i.e pipj= 0 ifi/ne}ationslash=j) such that
+x=/summationtextαkpk,y=/summationtextβkpk, withαk,βkreal numbers. Thus eix−iy= 1 implies that there
+exist integers {nj}j=1...m, and disjoint projections {qj}j=1...msuch that
+/summationdisplay
+k(αk−βk)pk= 2π/summationdisplay
+jnjqj.
+If/ba∇dbly/ba∇dbl ≤π, then multiplying by a fixed qjand taking norms gives
+2π|nj| ≤ /ba∇dblx−y/ba∇dbl<2π,
+which in turn implies nj= 0 for any j= 1...m, or equivalently x=y.
+In the general case, we have
+y−x= 2πi/summationdisplay
+njqj.
+6LetA ⊂ K(H) stand for the abelian C∗-algebra generated by the projections {pk}k∈N.
+Theny−x∈ A, thusqj∈ Afor anyjbecauseqjcan be obtained via a uniform limit of
+polynomials in x−y. In particular, qjcommutes with pkfor anyj,k.
+Fromypk=xpk+2π/summationtext
+jnjqjpk, squaring both sides and rearranging terms we obtain
+β2
+kpk=α2
+kpk+4π/summationdisplay
+jnj(πnj+αk)qjpk.
+Note that qjpkis positive and also that nj(πnj+αk)≥0 for any j,k, since|αk|< π. This
+implies
+β2
+kpk≥α2
+kpk,
+thus|βk|pk≥ |αk|pk, and summing over kyields|y| ≥ |x|.
+Remark 4.2. Note that if 0 ≤s≤t∈ Iφ, there exists a∈ B(H) such that /ba∇dbla/ba∇dbl ≤1 and
+s=at; in particular s∈ Iφand/ba∇dbls/ba∇dblφ≤ /ba∇dbla/ba∇dbl/ba∇dblt/ba∇dblφ≤ /ba∇dblt/ba∇dblφfor any symmetric norm /ba∇dbl·/ba∇dblφ.
+The following proposition is our fundamental tool to prove t hat geodesic segments are
+short. Following an idea introduced by Li, Qiu and Zhang in [2 0, 21] to study angular
+metrics in the finite dimensional Grassmannian, we prove a tr iangle inequality by using a
+remarkable result by Thompson [19], a result that until not s o lang ago had an incomplete
+proof since it depends on the validity of Horn’s conjecture o n the spectrum of the sum of
+two Hermitian matrices (see [10] and the references therein ).
+Proposition 4.3. Letx,y∈ K(H)h. Ifzis a Hermitian operator such that eixeiy=eiz,
+and/ba∇dblz/ba∇dbl< π, then there exist unitary operators ukandvk∈ B(H), fork∈N, such that
+eiz= lim
+k→∞eiukxu∗
+k+ivkyv∗
+k
+in the uniform topology. Moreover, if x,y∈ Iφfor some symmetric norm /ba∇dbl · /ba∇dblφ, then
+z∈ Iφand
+/ba∇dblz/ba∇dblφ≤ /ba∇dblx/ba∇dblφ+/ba∇dbly/ba∇dblφ.
+Proof.Ifaandbare Hermitian matrices in Mn(C) the problem was studied by Thompson
+[19]. In this case he proved that there exist unitary matrice sun,vn∈Mn(C) such that
+eiaeib=eiunau∗
+n+ivnbv∗
+n. (2)
+Let us approximate in the uniform norm the compact Hermitian operators xandywith
+finiterangeHermitianoperators xk,yk∈ J(H)fork∈N. Thatimpliesthatlim k→∞eixk=
+eix, limk→∞eiyk=eiyand
+lim
+k→∞eixkeiyk=eixeiy=eiz(3)
+uniformly. Explicitly, since xis compact, so is |x|, and let αj≥0 be a rearrangement
+in decreasing order of the eigenvalues of |x|counted with multiplicity. Let ejstand for
+7the corresponding unit norm eigenvector. Then {ej}j∈Nis an orthonormal set, in fact an
+orthonormal basis of ran|x|, that can be extended to an orthonormal basis {ej}∪{e′
+j}of
+H=ran|x|⊕ker(x).
+Then if|x|=/summationtext
+jαjej⊗ejis the singular value decomposition of |x|, whereej⊗ejstands
+for the one-dimensional projection /an}b∇acketle{tej,·/an}b∇acket∇i}htej, one obtains x=/summationtext
+jαjej⊗(uej). We let
+xk=k/summationdisplay
+j=1αjej⊗(uej),
+and with a similar argument we let
+yk=k/summationdisplay
+j=1βjfj⊗(vfj),
+wherey=v|y|,{fj}is an orthonormal basis of ran|y|, andβj≥0 are the singular values
+of|y|in decreasing order.
+Then, if we denote with R(xk) the finite dimensional range of xk, sinceR(xk) =
+Ker(xk)⊥, we can identify xkas an operator from R(xk) toR(xk). The same can be done
+withykandR(yk). If we define Sk=R(xk)+R(yk), thenxk(Sk)⊂ Skandyk(Sk)⊂ Sk,
+and therefore xk,yk∈ B(Sk)≃Mn(C) (where n=dim(Sk)).
+From the finite dimensional result (2), there exist uk,vkunitary linear transformations
+inSk(that means that uku∗
+k=pSkandvkv∗
+k=pSk, withpSkthe orthogonal projection on
+Sk) such that
+eixkeiyk=eiukxku∗
+k+ivkykv∗
+k. (4)
+We can extend uk,vk∈ B(Sk) to the unitaries ˜ uk=uk+pS⊥
+kand ˜vk=vk+pS⊥
+k∈ B(H).
+Then from the equality (4) valid in Skwe get the following in B(H)
+eixkeiyk=ei˜ukxk˜u∗
+k+i˜vkyk˜v∗
+k. (5)
+By hypothesis /ba∇dblz/ba∇dbl< π, then/ba∇dbleiz−1/ba∇dbl<2, and therefore /ba∇dbleixeiy−1/ba∇dbl<2. Then,
+takingk0large enough
+/ba∇dbleixkeiyk−1/ba∇dbl<2,fork≥k0.
+Letwkbe the unique Hermitian operator such that iwk= log/parenleftbig
+eixkeiyk/parenrightbig
+and/ba∇dblwk/ba∇dbl< π.
+Therefore using this and the expression (5)
+eiwk=eixkeiyk=ei˜ukxk˜u∗
+k+i˜vkyk˜v∗
+k. (6)
+Then using (3) and the fact that /ba∇dblz/ba∇dbl< πand/ba∇dblwk/ba∇dbl< πwe get that
+lim
+k→∞wk=z, (7)
+8in the uniform norm (in particular zis compact). Since (˜ ukx˜u∗
+k+ ˜vky˜v∗
+k)−(˜ukxk˜u∗
+k+
+˜vkyk˜v∗
+k)→0, then using (6) and (7) we get
+ei(˜ukx˜u∗
+k+˜vky˜v∗
+k)−eiz→0
+uniformly.
+Now, for each k∈N, from (6) and Lemma 4.1 we get
+|wk| ≤ |˜ukxk˜u∗
+k+ ˜vkyk˜v∗
+k|.
+Sincexk,ykarefiniterangeoperators, then wkis alsoafiniterangeoperator. From Remark
+4.2 above follows that
+/ba∇dblwk/ba∇dblφ≤ /ba∇dblxk/ba∇dblφ+/ba∇dblyk/ba∇dblφ. (8)
+Ifx,y∈ Iφ, sincexk=Pkxfor some orthogonal projection Pk∈ P(H), one obtains
+/ba∇dblxk/ba∇dblφ=/ba∇dblPkx/ba∇dblφ≤ /ba∇dblPk/ba∇dbl/ba∇dblx/ba∇dblφ=/ba∇dblx/ba∇dblφ.
+Similarly /ba∇dblyk/ba∇dblφ≤ /ba∇dbly/ba∇dblφ. Thus from (8) follows that
+/ba∇dblwk/ba∇dblφ≤ /ba∇dblx/ba∇dblφ+/ba∇dbly/ba∇dblφ,
+in particular
+sup
+k/ba∇dblwk/ba∇dblφ<∞.
+Sincewk→zuniformly, the noncommutative Fatou lemma [18, Theorem 2.7 (d)] gives us
+z∈ Iφand moreover
+/ba∇dblz/ba∇dblφ≤ /ba∇dblx/ba∇dblφ+/ba∇dbly/ba∇dblφ.
+Corollary 4.4. Letx,y,z∈ Iφbe Hermitian, eixeiy=eiz. Then if /ba∇dbly/ba∇dbl< πand/ba∇dblz/ba∇dbl ≤π,
+/ba∇dblz/ba∇dblφ≤ /ba∇dblx/ba∇dblφ+/ba∇dbly/ba∇dblφ.
+Proof.Consider the triangle spanned by 1 ,eix,eizinU(H). Note that there exists ǫ >0
+such that
+/ba∇dble−ixeisz−1/ba∇dbl<2
+if 1−ǫ < s≤1, due to the fact that /ba∇dbleiy−1/ba∇dbl<2. Thus ys∈ K(H)hdefined as iys=
+log(e−ixeisz) is well defined in the same interval, and depends smoothly on s; moreover it
+is easy to check that ys∈ Iφ, and then /ba∇dblys−y/ba∇dblφ→0 whens→1−. By the previous
+proposition, since /ba∇dblsz/ba∇dbl=s/ba∇dblz/ba∇dbl< πfors <1, we obtain
+/ba∇dblsz/ba∇dblφ≤ /ba∇dblx/ba∇dblφ+/ba∇dblys/ba∇dblφ
+forsclose to 1. Letting s→1−gives the claim.
+94.1 Geodesic segments are short
+Recall that we use the term segment for the curves ueitzinUφ, withz∈ IφHermitian. A
+polygonal path is a continuous curve that consists piecewise of segments.
+Proposition 4.5. Ifz∈ Iφis Hermitian, /ba∇dblz/ba∇dbl< π, andP: [0,1]→ Uφis a polygonal path
+joining1toeiz, thenLφ(P)≥ /ba∇dblz/ba∇dblφ.
+Proof.Iterating the argument, it suffices to prove the assertion for a path that consists
+of two segments, that is, assume eiz=eixeiyandPconsists of µ1(t) =eitxfollowed by
+µ2(t) =eixeity, with/ba∇dbly/ba∇dbl< π. NowLφ(P)≥ /ba∇dblz/ba∇dblφis the claim of Corollary 4.4.
+Now we approximate a smooth path with a polygonal path.
+Lemma 4.6. Letγ: [0,1]→ Uφbe piecewise smooth. Then for any ǫ >0there is a
+polygonal path Pǫ: [0,1]→ Uφsuch that for any t∈[0,1],
+/ba∇dblP∗
+ǫ(t)˙Pǫ(t)−γ∗(t)˙γ(t)/ba∇dblφ< ǫ.
+Proof.We may as well assume that γis smooth in [0 ,1]. Recall that γ,˙γare continuous
+in the uniform norm. Let ǫ >0, and choose a partition 0 = t0< t1<···< tn= 1 of the
+interval [0 ,1] such that, for any k= 0,1,···,n,
+/ba∇dblγ(t)−γ(s)/ba∇dblφ<2 and /ba∇dblγ∗(t)˙γ(t)−γ∗(s)˙γ(s)/ba∇dblφ<ǫ
+2
+ifs,t∈[tk,tk+1]. Then, by Remark 3.1 /ba∇dblγ(t)−γ(s)/ba∇dbl<2 ifs,t∈[tk,tk+1]. Let log denote
+the principal branch of the analytic logarithm in B(H). According to Remark 3.4, the
+unique anti-Hermitian logarithm of γ∗(tk)γ(tk+1) inB(H) such that /ba∇dblz/ba∇dbl< πis in fact an
+element of Iφ, due to the fact that
+/ba∇dblγ∗(tk)γ(tk+1)−1/ba∇dbl<2.
+Then put
+zk=log(γ∗(tk)γ(tk+1))
+tk+1−tk∈ Iφ.
+Now note that, for any fixed t∈[0,1], the map g:h/ma√sto→1
+hlog(γ∗(t)γ(t+h)), is well-defined
+and analytic, for sufficiently small h. Moreover, if h→0, then
+g(h)→dlog1γ∗(t)˙γ(t) =γ∗(t)˙γ(t).
+Then, taking a refinement of the partition if necessary, we ca n also assume that
+/ba∇dblzk−γ∗(tk)˙γ(tk)/ba∇dblφ<ǫ
+2
+for anyk= 0,1,2···,n. Consider the map Pǫ: [0,1]→ U(H) which is defined as
+Pǫ(t) =γ(tk)e(t−tk)zkfort∈[tk,tk+1].
+ThenPǫis certainly a polygonal path, and it is straightforward to s ee that verifies the
+claim of the lemma.
+10Theorem 4.7. Letu,v∈ Uφandv=ueiz, with/ba∇dblz/ba∇dbl ≤π,z∈ Iφand Hermitian. Then,
+the curve δ(t) =ueitzis shorter than any other piecewise smooth curve γinUφjoiningu
+tov, when we measure them with the norm /ba∇dbl·/ba∇dblφ. In particular, dφ(u,v) =/ba∇dblz/ba∇dblφ.
+Proof.It suffices to prove the theorem for u= 1. Assume first that /ba∇dblz/ba∇dbl< π. For given
+ǫ >0, letPǫbe a polygonal path in Uφas in the previous lemma, joining 1 to eiz, such
+that
+/ba∇dblγ∗˙γ−P∗
+ǫ˙Pǫ/ba∇dblφ< ǫ.
+Letδ(t) =eitz. Then by Proposition 4.5,
+/ba∇dblz/ba∇dblφ=Lφ(δ)≤Lφ(Pǫ) =/integraldisplay1
+0/ba∇dblP∗
+ǫ˙Pǫ/ba∇dblφ≤/integraldisplay1
+0/ba∇dblγ∗˙γ−P∗
+ǫ˙Pǫ/ba∇dblφ+Lφ(γ)< ǫ+Lφ(γ),
+and thus Lφ(γ)≥ /ba∇dblz/ba∇dblφ.
+If/ba∇dblz/ba∇dbl=π, consider for s∈(0,1) the path δswhich is the geodesic segment joining
+γ(s) toeisz. Then
+/integraldisplay1
+0/ba∇dbl˙γ/ba∇dblφ=/integraldisplays
+0/ba∇dbl˙γ/ba∇dblφ+/integraldisplay1
+s/ba∇dbl˙γ/ba∇dblφ=/integraldisplays
+0/ba∇dbl˙γ/ba∇dblφ+Lφ(δs)+/integraldisplay1
+s/ba∇dbl˙γ/ba∇dblφ−Lφ(δs).
+Note that for s <1, the path γ|[0,s]followed by δsis a piecewise smooth curve joining 1 to
+eisz. Since/ba∇dblsz/ba∇dbl=s/ba∇dblz/ba∇dbl< π, by what we have just proved,
+/integraldisplays
+0/ba∇dbl˙γ/ba∇dblφ+Lφ(δs)≥s/ba∇dblz/ba∇dblφ.
+Also note that, for sclose to 1, δs(t) =γ(s)eitzs, withizs= log(γ∗(s)eisz). Since /ba∇dblγ(s)−
+eiz/ba∇dblφ→0 assapproaches 1 from the left, then /ba∇dblzs/ba∇dblφ→0 ass→1−. Then
+Lφ(δs) =/integraldisplay1
+0/ba∇dbl˙δs/ba∇dblφ=/integraldisplay1
+0/ba∇dblzs/ba∇dblφ
+shows that Lφ(δs)→0 ass→1−. Now from
+Lφ(γ) =/integraldisplay1
+0/ba∇dbl˙γ/ba∇dblφ≥s/ba∇dblz/ba∇dblφ+/integraldisplay1
+s/ba∇dbl˙γ/ba∇dblφ−Lφ(δs)
+we obtain Lφ(γ)≥ /ba∇dblz/ba∇dblφlettings→1−.
+4.2 Relation with the linear metric
+Here we establish that the rectifiable distance is in fact uni formly equivalent to the linear
+metric in the unitary group Uφ.
+Proposition 4.8. Letu,v∈ Uφ. Then
+/radicalbigg
+1−π2
+12dφ(u,v)≤ /ba∇dblu−v/ba∇dblφ≤dφ(u,v).
+In particular, the function dφis a distance in Uφ, equivalent to the linear distance, and if
+the ideal Iφis complete, then the metric space (Uφ,dφ)is complete.
+11Proof.By the invariance of the metric, it suffices to check the inequa lities of the assertion
+forv= 1. From equation (1) in Remark 3.3, it clearly follows that i fu=eiz, with/ba∇dblz/ba∇dbl ≤π
+andz∈ IφHermitian, then the inequality on the left holds. The other i nequality is trivial.
+As for the completeness, it suffices to check that Uφis complete with the linear metric
+induced by the norm /ba∇dbl·/ba∇dblφ. This follows from the fact that if {un} ⊂ U φis a Cauchy
+sequence, then the sequence {un−1}is a Cauchy sequence in Iφ, and by assumption it
+converges to an element z∈ Iφ. Letu=z+ 1, we claim that u∈ U(H). Note that
+/ba∇dblun−u/ba∇dbl=/ba∇dbl(un−1)−z/ba∇dbl ≤ /ba∇dbl(un−1)−z/ba∇dblφ→0, hence
+/ba∇dbluu∗−1/ba∇dbl= lim
+n/ba∇dblunu∗
+n−1/ba∇dbl= 0,
+thusuu∗= 1. Likewise, u∗u= 1.
+5 Uniqueness of short geodesics
+In this section we prove uniqueness of short paths, assuming the symmetric norm is strictly
+convex.
+5.1 Dual norms and dual ideals
+Definition 5.1. LetTrstand for the infinite trace of B(H). Let/ba∇dbl·/ba∇dblφ′be the dual norm
+of/ba∇dbl·/ba∇dblφ, which is computed for x∈ K(H)as
+/ba∇dblx/ba∇dblφ′= sup{|Tr(xy)|:/ba∇dbly/ba∇dblφ≤1}.
+This formula defines a norm /ba∇dbl·/ba∇dblφ′:K(H)→R∪{∞}, all the properties of a norm are
+trivially verified with the exception that /ba∇dblx/ba∇dblφ′= 0impliesx= 0. But this follows from the
+singular values expansion
+x=u|x|=/summationdisplay
+jαjej⊗(uej).
+By picking y= (en⊗en)u∗(see Remark 3.1) one obtains αn= 0from/ba∇dblx/ba∇dblφ′= 0, thus
+x= 0.
+Remark 5.2. Directly from the definition it follows that
+|Tr(xy)| ≤ /ba∇dblx/ba∇dblφ/ba∇dbly/ba∇dblφ′.
+Let/ba∇dbl·/ba∇dblφ′′=/ba∇dbl·/ba∇dbl(φ′)′, andIφ′′the corresponding ideal. Then it also follows directly from
+the definitions that
+/ba∇dblx/ba∇dblφ′′≤ /ba∇dblx/ba∇dblφ.
+In particular, if x∈ Iφthenx∈ Iφ′′. Note that, by definition, Iφ′,Iφ′′⊂ K(H).
+Remark 5.3. Letx∈ K(H),x≥0. LetE:K(H)→C∗(x) stand for the unique
+conditional expectation compatible with the trace of B(H), which is obtained as follows:
+12let{ek}be a basis of ran(x), completed to a basis of H. Letpk=ek⊗ekbe the induced
+family of projections. Then for any y∈ K(H),
+Tr(pky) =/summationdisplay
+n/an}b∇acketle{tpkyen,en/an}b∇acket∇i}ht=/an}b∇acketle{tyek,ek/an}b∇acket∇i}ht
+thusTr(pky)→0 and
+E(y) =/summationdisplay
+kTr(pky)pk=/summationdisplay
+kpkypk,
+which is known as a pinching operator . It is well-known that pinchings in finite dimensions
+arecontractive forthesymmetricnorms, theresultextends easilytoourcontext, aproperty
+that turns out to be quite useful to estimate the norm of an ele ment.
+The following lemma collects a series of useful facts regard ing symmetric norms and
+their duals.
+Lemma 5.4. Let/ba∇dbl·/ba∇dblφby a symmetric norm in B(H), let/ba∇dbl·/ba∇dblφ′stand for its dual norm.
+Then
+1./ba∇dbl·/ba∇dblφ′is a symmetric norm in K(H).
+2. Letx∈ K(H)be positive, let Estand for the induced conditional expectation. Then
+for any symmetric norm /ba∇dbl·/ba∇dblφ, we have that E(y)∈ Iφwhenever y∈ Iφ, and
+moreover
+/ba∇dblE(y)/ba∇dblφ≤ /ba∇dbly/ba∇dblφ.
+3. Letx,y∈ B(H), then
+|Tr(xy)| ≤ /ba∇dblxy/ba∇dbl1≤ /ba∇dblx/ba∇dblφ/ba∇dbly/ba∇dblφ′.
+Ifx,y∈ B(H)h(orx,y∈ B(H)ah), thenTr(xy∗)∈R. Thus,
+−/ba∇dblx/ba∇dblφ/ba∇dbly/ba∇dblφ′≤Tr(xy∗)≤ /ba∇dblx/ba∇dblφ/ba∇dbly/ba∇dblφ′.
+4. Ifx≥0, then/ba∇dblx/ba∇dblφ′= sup{Tr(xy) :y≥0, yx=xy,/ba∇dbly/ba∇dblφ≤1}.
+5.Iφ=Iφ′′and/ba∇dbl·/ba∇dblφ′′=/ba∇dbl·/ba∇dblφ.
+6. Ifx∈ Iφ, then/ba∇dblx/ba∇dblφ=/ba∇dbl|x|/ba∇dblφ= sup{Tr(|x|y) :y≥0, y|x|=|x|y,/ba∇dbly/ba∇dblφ′≤1}.
+Proof. 1. Letx∈ K(H),u∈ U(H). Then, since y/ma√sto→yuis an isometric isomorphism
+for the norm /ba∇dbl·/ba∇dblφ, naming z=yuwe obtain
+/ba∇dblux/ba∇dblφ′= sup{|Tr(uxy)|:/ba∇dbly/ba∇dblφ≤1}= sup{|Tr(xz)|:/ba∇dblz/ba∇dblφ≤1}=/ba∇dblx/ba∇dblφ′,
+that is/ba∇dblux/ba∇dblφ′=/ba∇dblx/ba∇dblφ′, likewise, /ba∇dblxv/ba∇dblφ′=/ba∇dblx/ba∇dblφ′for anyv∈ U(H). Thus/ba∇dbl·/ba∇dblφ′is an
+unitarily invariant norm. Now assumethat /ba∇dblx/ba∇dbl<1, then [12] thereexists uk∈ U(H),
+k= 1...nsuch that x=1
+k/summationtextn
+k=1uk. Thus
+/ba∇dblxy/ba∇dblφ′≤1
+kn/summationdisplay
+k=1/ba∇dbluky/ba∇dblφ=/ba∇dbly/ba∇dblφ′,
+13hence/ba∇dbltz
+/bardblz/bardbly/ba∇dblφ′≤ /ba∇dbly/ba∇dblφ′for anyt∈(0,1), namely t/ba∇dblzy/ba∇dblφ′≤ /ba∇dblz/ba∇dbl/ba∇dbly/ba∇dblφ′. Letting t→1+
+gives/ba∇dblzy/ba∇dblφ′≤ /ba∇dblz/ba∇dbl/ba∇dbly/ba∇dblφ′, and with a similar argument one obtains the corresponding
+property for the right multiplication.
+2. Letxn=/summationtextn
+j=1αjej⊗ejbe the finite rank operators associated to the singular
+value decomposition of x,/ba∇dblxn−x/ba∇dbl →0 asn→ ∞. Letpj=ej⊗ej∈ P(H) and
+p=/summationtextn
+j=1pj∈ P(H). Note that Mp=pB(H)pwhich is a finite dimensional algebra.
+LetEn:Mp→ Mpstand for the pinching
+En(pyp) =n/summationdisplay
+j=1pjyp.
+Then, since any pinching is majorized by the identity (see fo r instance [8, page 97]),
+by considering the restriction of /ba∇dbl·/ba∇dblφtoMp, we obtain
+/ba∇dblEn(pyp)/ba∇dblφ≤ /ba∇dblpyp/ba∇dblφ≤ /ba∇dbly/ba∇dblφ.
+Note that, for any y∈ B(H),En(pyp) =En(y), thus if y∈ Iφ,
+sup
+n/ba∇dblEn(y)/ba∇dblφ≤ /ba∇dbly/ba∇dblφ.
+SinceEn(y)→E(y) strongly, by the noncommutative Fatou Lemma [18, Theorem
+2.7(d)], we obtain that E(y)∈ Iφand/ba∇dblE(y)/ba∇dblφ≤ /ba∇dbly/ba∇dblφ.
+3. Ifx,y∈ B(H), it is well-known that |Tr(xy)| ≤Tr|xy|=/ba∇dblxy/ba∇dbl1, and then
+/ba∇dblxy/ba∇dbl1= sup{|Tr(uxy)|:u∈ U(H)} ≤ /ba∇dblux/ba∇dblφ/ba∇dbly/ba∇dblφ′=/ba∇dblx/ba∇dblφ/ba∇dbly/ba∇dblφ′.
+4. Ifx≥0, considertheconditionalexpectation Easintheseconditem. Then Tr(xy) =
+Tr(E(xy)) =Tr(xE(y)) sinceEis abi-modulemapcompatible with thetrace. Since
+Eis contractive,
+{E(y) :/ba∇dbly/ba∇dblφ′≤1} ⊂ {z∈C∗(x) :/ba∇dblz/ba∇dblφ′≤1},
+hence
+/ba∇dblx/ba∇dblφ′= sup
+/bardbly/bardblφ≤1|Tr(xy)|= sup
+/bardbly/bardblφ≤1|Tr(xE(y))| ≤sup
+z∈C∗(x)
+/bardblz/bardblφ≤1|Tr(xz)|
+≤sup
+z∈C∗(x)
+/bardblz/bardblφ≤1/ba∇dblzx/ba∇dbl1= sup
+z∈C∗(x)
+/bardblz/bardblφ≤1Tr(x|z|),
+where we have used that xand|z|commute and thus x|z|=|xz| ≥0. Hence
+/ba∇dblx/ba∇dblφ′≤sup{Tr(xy) :y≥0, yx=xy,/ba∇dbly/ba∇dblφ≤1},
+and the other inequality is obvious.
+145. Letx∈ Iφ, then as we mentioned x∈ Iφ′′and/ba∇dblx/ba∇dblφ′′≤ /ba∇dblx/ba∇dblφfollow from the
+definition of dual norms. Now assume that x∈ Iφ′′, letxn,p,Mpbe as in the proof
+of the second item. Let Ip
+φ= (Mp,/ba∇dbl·/ba∇dblφ) which is a finite dimensional algebra. By
+the Hahn-Banach theorem, there exists ϕ∈(Ip
+φ)∗(the dual space) such that /ba∇dblϕ/ba∇dbl= 1
+andϕ(xn) =/ba∇dblxn/ba∇dblφ. Being a finite dimensional algebra, there exists z∈ Mpsuch
+thatϕ(·) =Tr(z·). It is easy to check (using the definition of dual norms) that
+/ba∇dblz/ba∇dblφ′=/ba∇dblϕ/ba∇dbl= 1.
+Thus
+/ba∇dblxn/ba∇dblφ=ϕ(xn) =Tr(xnz)≤ /ba∇dblxn/ba∇dblφ′′=/ba∇dblpxp/ba∇dblφ′′
+by definition of the second dual norm. Hence
+/ba∇dblxn/ba∇dblφ≤ /ba∇dblpxp/ba∇dblφ′′≤ /ba∇dblx/ba∇dblφ′′<∞.
+By the noncommutative Fatou lemma, since xn→xin the strong operator topology,
+x∈ Iφand
+/ba∇dblx/ba∇dblφ≤sup
+n/ba∇dblxn/ba∇dblϕ≤ /ba∇dblx/ba∇dblφ′′.
+6. This assertion follows directly from the two previous ite ms.
+LetI∗
+φstand for the dual space of the normed ideal Iφ.
+Corollary 5.5. Letx∗=x∈ Iφ. Then there exist a net {ξi} ⊂ Iφ′such that /ba∇dblξi/ba∇dblφ′≤1,
+ξ∗
+i=ξi,ξix=xξi, and a functional ηx∈ I∗
+φsuch that ηi=Tr(ξi·)→ηxin theω∗-topology,
+andηx(x) =/ba∇dblx/ba∇dblφ.
+Proof.For each y∈ Iφ′, letηy(z) =Tr(yz), thenηy∈ I∗
+φand/ba∇dblηy/ba∇dbl ≤ /ba∇dbly/ba∇dblφ′. Consider the
+family
+Fx={ηy:y∈ Iφ′, y≥0, y|x|=|x|y,/ba∇dbly/ba∇dblφ′≤1}.
+By the last item of the previous lemma, there exists a sequenc e{yj} ⊂ Fxsuch that
+/ba∇dblx/ba∇dblφ= lim
+jTr(|x|yj)
+Now ifx∗=x, write down x=|x|u=u|x|(polar decomposition) and put
+ξj=1
+2(uyj+yju)∈ Iφ′.
+Note that /ba∇dblξj/ba∇dblφ′≤1,ξ∗
+j=ξjsinceu∗=uand also that ξjcommutes with x. Moreover
+Tr(xξj) =1
+2[Tr(uyj|z|u)+Tr(yjuu|z|)] =Tr(|x|yj),
+thus/ba∇dblx/ba∇dblφ= limjTr(xξj). Since ηj∈ I∗
+φand/ba∇dblηj/ba∇dbl ≤1, by the Banach-Alaouglu-Bourbaki
+theorem, the family {ηj}has aω∗-convergent subnet ηito a functional ηx∈ I∗
+φ,
+ηx(z) = lim
+iηi(z) = lim
+iTr(ξiz)
+for anyz∈ Iφ. In particular ηx(x) =/ba∇dblx/ba∇dblφ.
+155.2 Uniqueness of short paths for strictly convex norms
+Definition 5.6. LetXbe a Banach space. A norm /ba∇dbl·/ba∇dblinXis
+1. strictly convex (or rotund) if /ba∇dblx+y/ba∇dbl=/ba∇dblx/ba∇dbl+/ba∇dbly/ba∇dblimpliesx=λy.
+2. smooth if it is Gateaux differentiable (at each x/ne}ationslash= 0∈X) as a map from XtoR.
+Equivalently, for any x∈Xthere exists a unique ϕx∈X∗such that /ba∇dblϕx/ba∇dbl= 1and
+ϕx(x) =/ba∇dblx/ba∇dbl.
+It is well-known that if the norm on the dual space X∗is rotund, the norm in Xis
+smooth, and that if the norm in the dual space X∗is smooth, then the norm in Xis
+rotund. The reciprocal of these affirmations are false, see fo r instance [15, 2.36] and the
+references therein.
+Theorem 5.7. Assume that the norm /ba∇dbl·/ba∇dblφis rotund. Let u,v,w∈ Uφsuch that
+dφ(u,v) =dφ(u,w)+dφ(w,v),
+and assume that d∞(u,v)< π(equivalently, /ba∇dblu−v/ba∇dbl<2). Then u,v,ware aligned in Uφ.
+That is, there exists z∗=z∈ Iφwith/ba∇dblz/ba∇dbl< πsuch that
+v=ueiz,whilew=ueit0z
+for some t0∈[0,1].
+Proof.We assume that u= 1. Write v=eizwith/ba∇dblz/ba∇dbl< π. Letw=eix=eize−iy, with
+/ba∇dbly/ba∇dbl ≤π,/ba∇dblx/ba∇dbl ≤πandx,yHermitian operators in Iφ. That is v=eiz=eixeiy. Then our
+assumption is
+/ba∇dblz/ba∇dblφ=/ba∇dblx/ba∇dblφ+/ba∇dbly/ba∇dblφ.
+Note that /ba∇dbly/ba∇dblφ≤ /ba∇dblz/ba∇dblφ, likewise, /ba∇dblx/ba∇dblφ≤ /ba∇dblz/ba∇dblφ. To prove the claim of the theorem, it
+suffices to show that y=λz, since in that case,
+w=eix=eize−iy=ei(1−λ)z,
+and we can pick t0= 1−λ. In fact, if y=λz, then since /ba∇dbly/ba∇dblφ≤ /ba∇dblz/ba∇dblφ, we obtain |λ| ≤1,
+thust0= 1−λ≥0, and on the other hand, since
+(1−λ)/ba∇dblz/ba∇dblφ=/ba∇dblx/ba∇dblφ≤ /ba∇dblz/ba∇dblφ
+we also have t0= 1−λ≤1.
+We will use a standard variation technique to show that y=λz. Consider the geodesic
+triangle in Uφspanned by 1 ,eixandeiz=eixeiy. Let
+β(s) =eixei(1−s)y=eize−isy.
+Since/ba∇dblz/ba∇dbl< π, ifs≥0 is small enough then
+/ba∇dblβ(s)−1/ba∇dbl ≤ /ba∇dbleiz−1/ba∇dbl+/ba∇dbleisy−1/ba∇dbl<2.
+16Thus for small s≥0, the element zs= (−i)log(β(s)) is an analytic function of s, is
+Hermitian in Iφand/ba∇dblzs/ba∇dbl< π. In particular, z0=z, and moreover from eizs=eize−isywe
+obtaind
+dss=0eizs=−eiziy.
+If we recall the formula for the differential of the exponentia l map in the unitary group
+dexpx(y) =/integraldisplay1
+0e(1−t)xyetxdt
+forx,y∈ B(H)ah(see for instance [4, Lemma 3.3]), we obtain
+−y=/integraldisplay1
+0e−tz˙z0etzdt. (9)
+Now by Theorem 4.7, /ba∇dblzs/ba∇dblφ=dφ(1,β(s)), and by the triangle inequality for the distance
+dφ, we have
+/ba∇dblx/ba∇dblφ+(1−s)/ba∇dbly/ba∇dblφ=/ba∇dblz/ba∇dblφ−s/ba∇dbly/ba∇dblφ≤ /ba∇dblzs/ba∇dblφ≤ /ba∇dblx/ba∇dblφ+(1−s)/ba∇dbly/ba∇dblφ.
+Thus, for any s≥0 sufficiently small,
+−/ba∇dblzs/ba∇dblφ+/ba∇dblz/ba∇dblφ
+s=/ba∇dbly/ba∇dblφ.
+Sincezs=z+ ˙z0s+o(s2), then/ba∇dblzs/ba∇dblφ≥ /ba∇dblz+ ˙z0s/ba∇dbl−o(s2) for small s≥0, and it follows
+that
+/ba∇dbly/ba∇dblφ≤/ba∇dblz/ba∇dblφ−/ba∇dblz+ ˙z0s/ba∇dbl
+s+o(s).
+Let{ξi} ⊂ Iφ′be a net as in Corollary 5.5 such that /ba∇dblz/ba∇dblφ= limiTr(zξi) =ηz(z). Then
+/ba∇dblz+ ˙z0s/ba∇dbl ≥τ(ξi(z+ ˙z0s)) =τ(ξiz)+sτ(˙z0ξi).
+Then
+/ba∇dbly/ba∇dblφ≤/ba∇dblz/ba∇dblφ−τ(ξiz)
+s−τ(ξi˙z0)+o(s).
+Since the ξiandzcommute, by equation (9) we have τ(ξiy) =−τ(ξi˙z0) for any i. Taking
+limit on iwe obtain
+/ba∇dbly/ba∇dblφ≤ηz(y)+o(s)≤ /ba∇dbly/ba∇dblφ+o(s)
+since/ba∇dblηz/ba∇dbl ≤1. Letting s→0+we obtain /ba∇dbly/ba∇dblφ=ηz(y). To finish, note that
+/ba∇dblz+y/ba∇dblφ≥ηz(z+y) =ηz(z)+ηz(y) =/ba∇dblz/ba∇dblφ+/ba∇dbly/ba∇dblφ,
+which shows that y=λz, since we are assuming that the norm /ba∇dbl·/ba∇dblφis rotund.
+Corollary 5.8. If the norm /ba∇dbl·/ba∇dblφis rotund, and zis a Hermitian element of Iφsuch that
+/ba∇dblz/ba∇dbl< π, thenδ(t) =ueitzis the unique piecewise C1curve in Uφjoiningutov=ueiz
+with length equal to dφ(u,v) =/ba∇dblz/ba∇dblφ.
+17Proof.Assume that γis any short, piecewise smooth curve joining 1 to eiz. Lett0∈(0,1)
+and letγ(t0) =eix=eize−iy, with/ba∇dbly/ba∇dbl ≤π,/ba∇dblx/ba∇dbl ≤π. By Proposition 4.5 and Theorem 4.7
+applied to each segment,
+/ba∇dblz/ba∇dblφ≤ /ba∇dblx/ba∇dblφ+/ba∇dbly/ba∇dblφ≤/integraldisplayt0
+0/ba∇dbl˙γ/ba∇dblφ+/integraldisplay1
+t0/ba∇dbl˙γ/ba∇dblφ=Lφ(γ) =/ba∇dblz/ba∇dblφ,
+hence/ba∇dblx/ba∇dblφ+/ba∇dbly/ba∇dblφ=/ba∇dblz/ba∇dblφ. Now the claim of the corollary follows form the previous
+theorem.
+Remark 5.9. This result generalizes [4, Theorem 3.2] for the p-norms ( p≥2), where it
+was proved using a particular variational calculus for the p-norms, see also [14].
+Remark 5.10. If/ba∇dblu−v/ba∇dbl= 2 (equivalently, if v=ueizwith/ba∇dblz/ba∇dbl=π), then the segment
+δ(t) =ueitzis not unique as a short path joining u,v. In fact, it suffices to consider
+the case v= 1. Since the condition /ba∇dblu−1/ba∇dbl= 2 is equivalent to −2∈σ(u−1), if
+x=u−1∈ Iφ⊂ K(H), this is equivalent to −1∈σ(x). Then, as in Remark 3.3,
+u= 1+x=p0+/summationdisplay
+k≥1eitkpk= 1+/summationdisplay
+k≥1(eitk−1)pk
+where{pk}k≥0are disjoint projections (finite dimensional for k≥1, and|tk| ≤π,tk→0).
+Letp1be the finite rank projection to the eigenspace of λ=−1, then we can assume that
+t1=π, and|tk|< πfork≥2. If the rank of p0is one, there are exactly two logarithms of
+uinIφ(with uniform norm equal to π), corresponding to
+z=∓iπp1+i/summationdisplay
+k≥2tkpk.
+However, if rank(p0) =n≥2, then therearean infinitenumberof decompositions p1=q+
+q′into proper disjoint projections, which allows to choose an infinite number of logarithms
+z∈ Iφ,
+z=iπq−iπq′+i/summationdisplay
+k≥1tkpk
+of norm equal to π.
+Remark 5.11. Clearly, these results of existence and uniqueness of short geodesics joining
+given endpoints (Theorem 4.7 and Corollary 5.8) hold verbat im if we replace UφbyU(0)
+φ,
+the group associated to the minimal ideal which is the closur e in the symmetric norm of
+finite rank operators. The same can be said for symmetric norm s in matrix algebras, that
+is, if the dimension of the Hilbert space His finite, a result which has interest on its own
+right.
+18References
+[1] E. Andruchow, E. Chiumiento, G. Larotonda, Homogeneous manifolds from noncom-
+mutative measure spaces . J. Math. Anal. Appl. (2010), in press.
+[2] E. Andruchow, G. Larotonda, Hopf-Rinow theorem in the Sato Grassmannian , J.
+Funct. Anal. 255 (2008), no.7, 1692-1712.
+[3] E. Andruchow, G. Larotonda, The rectifiable distance in the unitary Fredholm group .
+Studia Math. 196 (2010), 151-178.
+[4] E. Andruchow, G. Larotonda, L. Recht, Finsler geometry and actions of the Schatten
+unitary groups , Trans. Amer. Math. Soc. 62 (2010), 319-344.
+[5] J. Antezana, G. Larotonda, A. Varela, Optimal paths for symmetric actions in the
+unitary group , preprint arXiv:1107.2439v1 (2011).
+[6] D. Beltita, Iwasawa decompositions of some infinite-dimensional Lie grou ps. Trans-
+actions of the American Mathematical Society 361 (2009), no . 12, 6613-6644.
+[7] D. Beltita, T.S. Ratiu, Symplectic leaves in real Banach Lie-Poisson spaces . Geomet-
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+[8] R. Bhatia. Matrix analysis. Graduate Texts in Mathemati cs, 169. Springer-Verlag,
+New York, 1997.
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+ideals, Adv. in Math. 185 (1) (2004), 1–79.
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+Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209–249 (elec tronic).
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+erators. Translated from the Russian by A. Feinstein. Trans lations of Mathematical
+Monographs, Vol. 18 American Mathematical Society, Provid ence, R.I. 1969.
+[12] R.V. Kadison, G.K. Pedersen, Means and convex combinations of unitary operators .
+Math. Scand. 57 (1985), no. 2, 249–266.
+[13] P. de la Harpe. Classical Banach-Lie algebras and Banac h-Lie groups of operators in
+Hilbert space. Lecture Notes in Mathematics, Vol. 285. Spri nger-Verlag, Berlin-New
+York, 1972.
+[14] L.E. Mata-Lorenzo, L. Recht, Convexity properties of Tr[(a∗a)n]. Linear Algebra
+Appl. 315 (2000), no. 1-3, 25–38.
+[15] R. Phelps. Convex functions, monotone operators and di fferentiability. Second edi-
+tion. Lecture Notes in Mathematics, 1364. Springer-Verlag , Berlin, 1993.
+19[16] N.C. Phillips. A survey of exponential rank, C* -Algebr as: 1943-1993, A Fifty Year
+Celebration. Contemporary Math., Vol. 167, Amer. Math. Soc , Providence RI, 1994,
+353-399.
+[17] H. Porta, L. Recht, Minimality of geodesics in Grassmann manifolds , Proc. Amer.
+Math. Soc. 100 (1987), no. 3, 464-466.
+[18] B. Simon. Trace ideals and their applications. Second e dition. Mathematical Surveys
+and Monographs, 120. American Mathematical Society, Provi dence, RI, 2005.
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+Algebra 19 (1986), no. 2, 187-197.
+[20] C.-K. Li, L. Qiu, Y. Zhang, Unitarily invariant metrics on the Grassmann space .
+SIAM J. Matrix Anal. Appl. 27 (2005), no. 2, 507–531 (electro nic).
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+Appl. 421 (2007), no. 1, 163–170.
+Jorge Antezana:
+Universitat Aut´ onoma de Barcelona.
+Departamento de Matem´ atica,
+Facultad de Ciencias
+Edificio C Bellaterra (08193)
+Barcelona, Espa˜ na.
+e-mail: jaantezana@mat.uab.cat
+Gabriel Larotonda and Alejandro Varela:
+Instituto de Ciencias
+Universidad Nacional de General Sarmiento.
+J. M. Gutirrez 1150
+(B1613GSX) Los Polvorines,
+Buenos Aires, Argentina.
+e-mails: glaroton@ungs.edu.ar,
+avarela@ungs.edu.arJ. Antezana, G. Larotonda
+and A. Varela:
+Instituto Argentino de Matem´ atica
+“Alberto Calder´ on”, CONICET
+Saavedra 15, 3er piso
+(C1083ACA) Buenos Aires,
+Argentina.
+20
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+      Multiscaling for Systems with a Broad Continuum of Characteristic Lengths and Times:  Structural Transitions in Nanocomposites     S. Pankavicha,b and P. Ortolevaa   Center for Cell and Virus Theorya  Department of Chemistry  Indiana University  Bloomington, IN 47405   Department of Mathematicsb University of Texas at Arlington Arlington, TX 76019 Contact: ortoleva@indiana.edu  1 ABSTRACT The multiscale approach to N-body systems is generalized to address the broad continuum of long time and length scales associated with collective behaviors. A technique is developed based on the concept of an uncountable set of time variables and of order parameters (OPs) specifying major features of the system.  We adopt this perspective as a natural extension of the commonly used discrete set of timescales and OPs which is practical when only a few, widely-separated scales exist.  The existence of a gap in the spectrum of timescales for such a system (under quasi-equilibrium conditions) is used to introduce a continuous scaling and perform a multiscale analysis of the Liouville equation. A functional-differential Smoluchowski equation is derived for the stochastic dynamics of the continuum of Fourier component order parameters.  A continuum of spatially non-local Langevin equations for the OPs is also derived.  The theory is demonstrated via the analysis of structural transitions in a composite material, as occurs for viral capsids and molecular circuits.  Keywords: multiscale analysis, continuous scaling, Liouville equation, functional expansion, nanoparticles, nanomaterials, molecular circuits, self-assembly, structural transitions  2 1 INTRODUCTION There are many systems composed of multi-atom, nanoscale components that can assemble into a composite structure. Examples include: • nanoparticle arrays at solid or liquid interfaces • networks of protein filaments (e.g. the cytoskeleton or the spindle for chromosome separation) • composite three-dimensional materials made of macromolecules or nanoparticles  • clusters of cell surface receptors • a viral capsid assembled from proteins organized into protomers, pentamers, or hexamers. Due to the large number of atoms in each nanocomponent, the center-of-mass (CM) and orientation evolve much more slowly than the 10-14 second timescale of atomic vibration or collision. In addition to the atomistic high-frequency phenomena, the dynamics evolve on a wide range of long timescales associated with the collective modes of the multi-component system, including the coherent response of dimers, trimers, and larger clusters of components. These considerations suggest the existence of a gap between the short timescale (10-14 second) and the broad range of long timescales of the coherent motion of single or multiple nanometer size components, the longest timescale being associated with the coordinated motion of many components simultaneously. In this way, such composite systems can display a wide spectrum of timescales starting with the microscopic time, followed by a gap, and then a broad quasi-continuum of long timescales. For this reason, traditional two or three timescale approaches cannot be applied.   It has long been known that an N-body system can display phenomena spanning a wide range of scales in both space and time. For example, hydrodynamic modes can exist over a broad range of scales. Phase transitions in viral capsids are believed to nucleate with the displacement and rotation of a single protein or protomer and, subsequently, progress across the capsid as a structural transition front. Similarly there is a broad range of vibrational modes of nanocomposite materials corresponding to the vibrations of clusters of nanocomponents.  The traditional approach to multiscale systems is to introduce a timescale ratio ε (<<1) and an associated sequence of time variables tn = ε n t, n = 0, 1,··· that capture a set of processes, each with a discrete characteristic time and separated from nearby scales by a gap. However, this is not  3 appropriate when there are so many very closely packed timescales as to form a quasi-continuum extending over many orders of magnitude. In a sense, one expects an approach based on a more general set of time variables, tn indexed not by the set of non-negative integers, but instead an uncountable set (e.g., positive numbers).  One approach to multiscale systems that seems to avoid discrete scaling is the projection operator method7. This method accounts for slow variables not included in a reduced description of interest, e.g., hydrodynamic modes in a host fluid interacting with a nanoparticle. Unfortunately, the memory kernels introduced by this method cannot readily be evaluated, although insights can be obtained by computing them approximately or fitting them to experimental data. The evaluation of these memory kernels requires Molecular Dynamics (MD) simulations on the order of the timescale of the kernels.  If this is long, then the calculations are impractical.  Instead, if this is short then the approach presented in the following sections will directly yield the reduced equations of interest; thus removing the need to use projection operators.  Practical progress can be made within the projection operator framework when it is combined with mode-coupling theory.  However, such an analysis does not take advantage of the gap in the spectrum of timescales often displayed by N-body systems. For example, a disturbance created by launching a nanoparticle in a host medium of small molecules could excite only those modes of the host with wavelengths greater than or equal to the size of the nanoparticle or with timescales greater than or equal to the size of the nanoparticle divided by its maximum velocity.  The objective of the present study is to develop an analysis of N-body systems that captures the character of this gap/continuum scaling phenomena.  The multiple scale understanding of N-body systems dates back to Einstein, Langevin, Smoluchowski, Chandrasekhar and others5, 6, 8, 10. Deutch and Oppenheim7 presented an approach to Brownian motion based on projection operators and a perturbation scheme in the host particle/nanoparticle mass ratio. This approach set the stage for a series of analyses of Brownian particle dynamics based on the Liouville equation leading to the derivation of Fokker-Planck (FP) equations. Shea and Oppenheim9 derived FP and Langevin equations for a single structureless Brownian particle in a host medium using projection operators and perturbation techniques based on expansions in the mass ratio or macroscopic gradients in the host medium. The original approach of Ortoleva et al11-16 is based on a formal discrete multiple space-time scaling approach integrated with a statistical argument derived from the host particle/Brownian  4 particle size ratio that allows for a united asymptotic expansion to arrive at FP equations for single and multiple Brownian particles and intra-Brownian particle structural dynamics.  When the slow hydrodynamics of the host medium are accounted for, it was shown formally that the treatment leads to a set of coupled FP equations, one for each host mode. Shea and Oppenheim9 extended their work to the case of multiple nanoparticles, introducing a number of smallness parameters. In [11-16] a unified asymptotic expansion and a multiple time scale formulation was presented that captures transitional behaviors (e.g. when the timescale switches as the number of nanoparticles per volume increases so that the behavior becomes more like that of a solid porous matrix than a suspension of disconnected nanoparticles).  The notion of space-warping was introduced to treat nanoparticle deformation and rotation17. Additionally, we have placed the space-warping method in a self-consistent framework for analyzing single and multiple nanocomponents with internal atomistic structure.  This will be shown in the present study to facilitate the development of a continuum multiscaling theory12-16,18-19.   In all the above multiscale treatments, only a discrete set of time scales, ordered by powers of a small parameter , was considered. The existence of a broad continuum of scales has not been addressed even though such behavior is prevalent in many systems, so that the development of some generalization of discrete scaling is required.  In this study we address systems for which there is a set of short timescale processes, and also a broad continuous range of long timescales.  The discrete problem is briefly reviewed in Sect 2.  In Sect. 3, the continuum scaling problem for a set of interacting nanocomponents is formulated, and a perturbation analysis is performed.  In Sect. 4 we derive a spatially non-local Langevin equation that is equivalent to the functional Smoluchowski equation of Sect. 3 and discuss how it can be used to simulate systems with a broad spectrum of length and time scales.  Conclusions are drawn in Sect. V.  5 2 ORDER PARAMETERS, EFFECTIVE MASS, AND DISCRETE SCALING Consider a set of M identical nanocomponents, each containing many atoms.  Let the mass of each component scale as ε -1 times a typical atomic mass m for .  The components are immersed in a host fluid, the entire system constituting a total of N atoms. The objective of this section is to demonstrate the breakdown of classic multiscale techniques and the need for continuous scaling in such a system when M is large.  We do so in the course of deriving an equation for , the reduced probability density for , the set of CMs of the nanocomponents.    The reduced probability density  is defined via    , (2.1) where  is the N-atom probability density,  is an M-fold product of Dirac delta functions, and describes the state of the N-atom system and  is the N-atom state over which integration is taken;  is an expression for the  CMs in terms of the N-atom configuration . This latter, starred, notation will be used below to denote other quantities expressed in the integration variables.  Here,  satisfies the Liouville equation:         (2.2) where are the mass, momentum, position, and force on atom .   Using (2.1),(2.2), and integration by parts, we have    (2.3)    (2.4) where  is the set of total momenta of the M nanocomponents.  We determine conditions under which the conservation law (2.2) is closed in , i.e., involves only the reduced distribution  and not the full N-atom distribution .  6  Similar to methods developed earlier12-16,  is envisioned to depend on  directly and through the M nanocomponent center-of-mass (CM) positions , indirectly. For simplicity, we ignore the orientation of the nanocomponents in this exploratory study. With this ansatz on the dual dependence of  on  and the chain rule, the Liouville equation takes the multiscale form     (2.5)    (2.6)   . (2.7) Introducing -dependence in  does not imply a change in the number of degrees of freedom (i.e., 6N). Instead, the inclusion of these CM variables in the formulation is to acknowledge the multiple ways in which  depends on the all-atom state .  It has been assumed that each nanocomponent has total mass, where m is a typical atomic mass. The operators  and  follow from the chain rule and the assumed dependence of the N-atom probability density at time t.  A discrete multiscale development proceeds by introducing a short time variable  that tracks the 10-14 s atomistic fluctuations and a set of long-time variables , used to track the distinct ways in which  depends on time.  With this,  is cast as a power series in and determined via a perturbative solution of the Liouville equation.  The lowest-order solution  is assumed to be slowly varying (i.e., independent of ) and hence .  Thus,  is in the nullspace of , and in particular, is determined using entropy maximization12-16.  For the present problem, we obtain    (2.8)    (2.9)  7 where  is the Hamiltonian generating . This lowest-order solution is developed using an information theory approach to construct the isothermal quasi-equilibrium ensemble for fixed values of .  The development proceeds via an order-by-order analysis, eventually leading to a closed equation for , after inserting  to  in (2.3) and recognizing that as .   Unfortunately, this approach does not reveal the broad, essentially continuous, spectrum of timescales in the many-component system.  For example, clusters of nanocomponents of increasingly large size are expected to evolve on increasingly long timescales.  As M and N are very large for problems of interest, the development should reconstruct the -dependence of  for an extremely large set of n, while the classical, discrete timescale analysis ends at .  Thus we reformulate the problem in terms of a set of mode coordinates in favor of the component CM () description.  Consider the method of Jaqaman and Ortoleva17, as modified to construct order parameters for the multiscale analysis of N-atoms systems13,15,16,19. In this formulation, the position of atom i is written as a sum of coherent and residual parts.  The residual displacement accounts for fluctuations over-and-above coherent deformation. Hence the position of atom i is written in terms of a deformation from a reference position  using order parameters , and basis functions  via   . (2.10) Define a mass-weighted inner product (a,b) via     (2.11) If  for all i, then  , the total mass of the system. The are constructed to be orthogonal and normalized such that       , (2.12)  8  where  is a constant chosen to be an effective mass associated with the k-th order parameter. The label of the basis functions is chosen such that  decreases with .  For example, if  represents an inverse length characteristic of the spatial variation of , then in one period of , there is a volume of about  and hence a mass of similar magnitude.  Thus, we suggest that  behaves like  for large k since  is  for small  and about the mass of a single nanocomponent  for the maximum cutoff value . Therefore, .  With k having the character of a wavevector for the k-th mode, we suggest that  is qualitatively of the form  with A and B determined by imposing  when  and  when .   Upon minimizing the sum of the squares of residuals   as in Pankavich et al14-16, one finds    (2.13) If  is roughly one, then  acts like the CM of the entire collection of components. For a more localized basis function, i.e.,  is roughly , then  acts like the CM of a component.  We only choose the  that vary on a length scale much greater than the average nearest-neighbor atomic spacing and, in particular, greater than the size of a nanocomponent, so that their effective mass is high.  When the structural components are fairly rigid, their internal dynamics are fast. Assuming the host fluid dynamics are also fast relative to the translation of the M units, a gap exists that separates the timescale of the nanocomponent dynamics from that of the fluctuations at the atomic scale. As above, we introduce a parameter  for typical atomic mass m. Using arguments as in earlier studies11-16,18-19, the typical magnitude of the thermalized momentum of a nanocomponent is . For constant , is the CM of the entire collection of components. Thus, the evolve slowly when the associated basis function  varies on a  9 length scale greater than or equal to the size of a nanocomponent, i.e., when  for . If the number of components is , then  so that  is large and varies greatly over the range of k.  With this large range of effective mass, one expects there is an associated broad spectrum of timescales.  We postulate that for the phenomena of interest, two length scales exist on which units move: (1) the interatomic scale (i.e., as in close encounters between nanocomponents) that is characterized by , and (2) the long scale, , movement that is related to larger scale reorganization (i.e., ). With , the chain rule implies that the Liouville equation takes the form  where  is as in (2.6) and    . (2.14)  The order parameters and momenta  are labeled by the integer k.  However, if these order parameter disturbances are quasi-continuous, i.e., change smoothly in character with k, then we switch to labeling by a wavevector  with  and  determined by the volume of the system.  In developing the continuous scaling framework of the next section, we use the three-dimensional wavevector  to index order parameters and derive implications for the continuous scaling behavior of these large systems.  Since we consider only systems with large volume, and hence include arbitrarily small wavevector lengths, we take .  Letting  vary over a continuum of values implies the existence of order parameters , and conjugate momenta , that depend continuously on the wavevector, in addition to the N-atom state .   10 3 A FUNCTIONAL DIFFERENTIAL SMOLUCHOWSKI EQUATION The order parameter dynamics of the previous section naturally allow for the introduction of a continuous scaling and the derivation of associated Smoluchowski equations.  In the proposed generalization of the discrete scaling of Sect. 2, we introduce the function  which increases monotonically with .  In particular, we take  with  as in Sect. 2.  Since small  (i.e., long wavelength) disturbances have the greatest effective mass , these are the slowest excitations in the system. Recall from Set. 2 that the values of  are restricted by the typical nanocomponent mass so that  throughout our analysis. In the following, we introduce a continuous set of -dependent order parameters, use  to order a perturbation expansion, and arrive at a scaling theory for systems with a broad continuum of timescales.  Order parameters for continuous scaling are introduced by writing atomic positions as an integral transform analogue of (2.8) via a continuum of basis functions  which depend on the reference configuration :   . (3.1) In transitioning from the discrete case to the continuum, we replace the sum on the integer index by a -weighted integral, i.e.,  is the number of -index values in  for system volume .  As for the discrete case, the  represent the residual, “random” displacements from the coherent configuration generated by changes in the order parameters . For example, if , the order parameter  corresponds to the Fourier transform from  -space to -space.  Arguing by analogy with Sect. 2 and using (3.1), yields    (3.2)   . (3.3)  11  This constitutes a set of equations that determine the  in terms of the N-atom configuration.  These expressions are obtained via minimization of the total mass-weighted square residual over all order parameters , i.e., by solving .  One may derive an equation for computing the OPs from Newtonian dynamics by taking  of both sides of (3.2). Letting yields    (3.4) which determine the  in terms of the N-atom configuration. Integral equations for  and  emerge which simplify greatly for orthogonal basis functions.  For instance, if   with  from Sect. 2, then (3.2) becomes an explicit equation for the OPs:    (3.5) In this case, the expression for  simplifies as well. Introducing , one obtains  where   . (3.6) We take (3.5) and (3.6) as the definitions of the OPs and their conjugate momenta, though variations will occur if basis functions do not satisfy mass-weighted orthogonality.  The reduced probability density  is a functional of  and a function of .  In the discrete case, took the form   12      (3.7) where  was a product of delta functions, one for each order parameter.  Generalizing to the continuous case, we write the limit of this product of delta functions as   . (3.8) Hence,  is represented in the same form as in Sect. 2, but with  having functional dependence on  as defined by (3.8). Taking the time derivative of both sides of (3.7) and using the Liouville equation with the chain rule, one obtains the functional conservation law   . (3.9) We construct  as a functional expansion in  using a generalization of the power series in  for the discrete case, and then use (3.9) to derive a closed equation for  (i.e, one that does not depend on the full N-atom probability density , but only involves ).  In traditional multiscale theory, one assumes that an N-atom system can be characterized as evolving on a discrete set of timescales, each separated from nearby timescales with a finite gap. The width of these gaps scales as an inverse power of a small parameter (e.g., the ratio of the mass of a typical atom to that of a many-atom nanocomponent). But the discussion at the end of Sect. 2 suggests that for many systems there is a myriad of timescales, each of which is separated from those nearby by such a small gap that the scaling is essentially continuous, i.e., the underlying physics changes in an essentially continuous manner across these scales. Therefore, we now investigate a theory wherein a finite gap, as in Fig. 1, separates a discrete timescale (i.e., that of rapidly-fluctuating atomistic processes) from a quasi-continuum of scales associated with a spectrum of slow processes. The presence of a continuum of functional Liouville-type operators on the right side of (3.9), suggests there is a related continuum of operators that correspond to an uncountable set of time variables on the left side as follows.   13 
+ In analogy to the discrete case, we make the ansatz , so that  has an explicit functional dependence on the continuum of OPs, in addition to its explicit - dependence.  The Liouville operator  is then written as a sum of discrete and continuously scaled contributions:    (3.10)    (3.11) With these definitions and the chain rule, we obtain an explicit form for the  operator:    . (3.12) In the following, we treat the set of short scales as discrete and isolated. The fast timescale dynamics are driven by, accounting for atomic-scale processes whose timescale is that of atomic collision or vibration (i.e., 10-14 seconds). In contrast,  accounts for the long-length scale, slow dynamics of the order parameters, and thus involves functional derivatives with respect to the continuous family of order parameters.  In analogy with traditional discrete multiscale treatments of the Liouville equation11-16, we introduce a continuous spectrum of time variables to capture the effects of . As a consequence, the reformulation develops a continuum character, i.e., traditional sums over a discrete timescale index are replaced by a  integration and derivatives are replaced by functional derivatives. To achieve self-consistency, we first introduce times . The short time Fig. 1 The graph of a continuous scaling factor for slow modes that operate on a distinctly longer timescale than that of rapid atomic-scale fluctuations is shown. The gap in the time scale of modes that are active is created by the nature of the initial data.  14 variable  is the analogue of  in Sect. 2, while the continuum of times  is the analogue of the traditional discrete sequence .  It is thereby assumed that  depends on the discrete short time  and the uncountable continuum of long times .   In this formulation, the N-atom probability density is a function of  the set of N atomic positions and momenta, as well as, the short time variable , and a functional of a continuous set of order parameters  and long time variables .  Through the chain rule,  operates on the explicit -dependence, while the continuum term  operates on the indirect functional dependence, through , and drives the slow dynamics as implied by the smallness of the scaling function  over its range of values.  The scaling function  decreases to zero as , reflecting the fact that longer wavelength (smaller ) modes are slower varying.  Since  has direct dependence on the microscopic variables  and indirect dependence on them through the continuum of variables , the multiscale Liouville equation becomes a mixed differential - functional differential equation.  The weight  plays the role of the scaling factor  raised to various discrete powers.  For example, for  and  is an example of a scaling function.   The above formulation is not a violation of the number of degrees of freedom, as the 6N degrees of freedom constituting already provide a complete description of the state of a classical system. However, it was shown for discrete scaling systems12-16 that the dual dependence of  on  and OPs is simply a reflection of the multiple ways that  depends on the N-atom configuration.  Thus,  depends on   both directly and through the order parameters, indirectly. The chain rule and the functional dependence of  on the OPs and the continuum of time variables imply that (3.10) becomes  . (3.13)  15 With these arguments, the challenge is to demonstrate that (1) this is a self-consistent mathematical framework, (2) a perturbation analysis can be performed, and (3) a generalized functional Smoluchowski equation can be developed that captures the long-time, stochastic dynamics of systems like a large collection of nanoparticles. Since  is small (reflecting the long timescales involved), a Volterra-type functional expansion in  is introduced for :   . (3.14) The missing terms indicated on the LHS of (3.13) arise from the need to introduce additional types of time variables (e.g. ).  These higher-order time variables will arise because, in general, there is no simple relation between the times corresponding to differing wavevector values (e.g. ).  However, since our analysis need only be carried out to first order in the functional expansion of , we will not elucidate further.  Eqn (3.13) is then solved using (3.14) in an “order-by-order” fashion, i.e., in a manner analogous to Sect. 2 in which the equations were solved in powers of.  Using information theory to construct the lowest-order solution and the Gibbs-hypothesized equivalence of thermal and long-time averages to remove secular behavior in the solution at each order, we obtain expressions for  and the functions  thru .  Upon recomposing  and using these expressions in (3.9), we find    (3.15)    (3.16) We term this a functional Smoluchowski equation for continuous scaling systems.  Here, the  terms represent thermal-averaged forces, which can be computed using an order-parameter-constrained ensemble of atomistic configurations.  Details of these calculations are provided in the appendix.  16 4. THE NONLOCAL LANGEVIN EQUATION Since the Smoluchowski equation of the previous section cannot be solved analytically or numerically due to practical difficulties, the equivalent Langevin equations are the only practical way to simulate the stochastic processes of interest here. We now show that this Langevin equation is nonlocal and describes the stochastic dynamics of the continuum of order parameters.  We prove the Monte Carlo-equivalence by postulating the form of the Langevin equation and comparing the resulting Smoluchowski equation with that derived in Sect. 3.    Our postulated form of the Langevin equation is       (4.1) for .  Here,  is a diffusion kernel that we will relate to the diffusion coefficients  of (3.16),  is the thermal-averaged force for the -th order parameter, and  is a random force driving the stochastic dynamics of .  As described at the end of the previous section, the thermal-averaged forces are defined via an OP-constrained ensemble, and hence depend only upon the order parameters in the system. We now determine conditions under which (4.1) is Monte Carlo-equivalent to (3.15) with (3.16).  For simplicity, we have introduced units for the wavevector such that the  factors do not appear.   Let  be the probability density functional for the continuum of order parameters .  We assume the set of random forces  change much more rapidly over time than .  A time interval  is postulated to exist such that the system experiences a representative sample of variations in  during, and yet  changes only a small amount during  due to .  We adopt a Markov assumption such that can be advanced from t to solely knowing at time t and the transition probability for a change in  during. The evolution of  is thus determined by the statistics of all  time courses. Let  be the probability functional for a scenario of  during t to .  Then,  evolves via the Markov expression     (4.2)  17 Here,  is the solution of (4.1) at time , given that the state was  at t and  is defined by (3.8).  The functional integrals in (4.2) are over all order parameter states  (for all ) and scenarios of  for times between t and. A Volterra series expansion of  in the changes  (that vanish as) takes the form  . (4.3) Denote the averaging implied by the -weighted functional integrals in (4.2) as.  Under the postulate that  changes only a small amount over, one finds   . (4.4) Substituting this and (4.3) to second order into (4.2), and neglecting terms of order higher than  yields   (4.5) To better represent the -kernel, define the linear operator  by    (4.6) for any integrable function .  Integration of (4.1) yields    (4.7) To derive formulas for the  terms in (4.5), we assume the average random force vanishes for every , that is, .  If the process generating  is stationary, then  only depends on , and not on t and  independently, i.e, . Neglecting terms of order higher than, the statistical properties of  and (4.7) imply  18     (4.8)   . (4.9) For essentially all  in the time period , the upper limit of the s integral is much greater than the correlation time. Since  is negligible except for , one finds    (4.10) where   . (4.11) These values of  and  are inserted into (4.5) to yield the functional Smoluchowski equation:  (4.12) At long times, the closed, isothermal system reaches equilibrium.   Hence  where .  As this must be a time-independent solution of the functional Smoluchowski equation, we find .  Using (4.11) this relation constrains the statistics of  by     (4.13) for every ,.  Comparison of the functional Smoluchowski equation (3.15) with (4.12), derived from the postulated Langevin equation, implies a relation between the diffusion coefficients  and the kernel, namely      (4.14)  19 for every ,.  Combining (4.13) and (4.14), we can relate the statistics of  to .  Thus there is a well-defined relationship between the integrated, fluctuating force correlation functions and the diffusion coefficients , a classic fluctuation-dissipation relation.  The Langevin description can then be used to simulate many-component systems using -space finite element techniques and Monte Carlo sampling of the random force. An algorithm for co-evolving the OPs and quasi-equilibrium, atomic probability density can be constructed by advancing the OPs at each Langevin timestep and simultaneously reconstructing the thermal-averaged forces and diffusion coefficients, allowing for the next advancement of the OPs in the implicit Langevin dynamics.  This will avoid the need to impose simple phenomenological expressions for the thermal-averaged forces that may miss the interscale feedback between atomistic variables and OPs.   20 5. CONCLUSIONS The broad quasi-continuum of characteristic times displayed by a system of many interacting nanocomponents is but one example of physical systems for which one cannot use the paradigm of a few timescales separated by large gaps.  This latter classic approach has dominated multiscale analysis.  We have presented a novel approach wherein a continuum of characteristic lengths and times, and associated variables, enables the derivation of a functional generalization of the Smoluchowski equation.  This equation provides a framework for understanding the stochastic dynamics of order parameter field variables.   This multiscale approach, though similar to the projection operator method, contains subtle and important differences.  The projection operator technique starts with the identification of descriptive variables of interest and, like the multiscale procedure, derives equations for the reduced probability density for these variables. However, the former technique introduces memory kernels; if these kernels have long-time memory then for the large systems of interest they cannot practically be evaluated using computer molecular dynamics. Furthermore, the kernels involve a propagator which is generated by a modified Liouville operator that cannot be constructed without pre-determining the reduced probability. If the kernels only have short-time memory, then the projection operator technique indeed reduces to the multiscale formalism. Since the latter is more readily generalized to a variety of problems, and the associated computations lead more directly to the reduced equation without involving a heuristic assumption on the characteristics of the memory kernel, the present methodology is preferable for systems with short memory.  Key aspects of our formalism are the following:  • a set of order parameter field variables with an associated effective mass central to the scaling of their slow behavior • an uncountable set of time variables related to laboratory time via a wavevector-dependent scaling function • the ansatz that the N-atom probability density can be viewed as depending on the N-atom state  both directly and, via a continuum of wavevector-dependent order parameters, indirectly  21 • a functional expansion in the inverse effective mass with the consequent emergence of a Smoluchowski equation, and Monte Carlo-equivalent Langevin equation, for the dynamics of order parameter field variables. The theory has application to the self-assembly and dynamics of composite materials from nanocomponents, self-organization in cellular systems, flow in complex media, and, more generally, systems of many strongly interacting components evolving over a broad spectrum of timescales.   22 APPENDIX: MULTISCALE ANALYSIS AND CONTINUOUS TIME SCALING To further explore the analysis of Sect. 3, we begin with the ansatz on the probability density , so that  depends on implicitly via the continuum of OPs in addition to its explicit -dependence.  We rescale the wavevectors by letting  and then drop the prime for brevity.  We take the  integrals which appear below to be over the sphere  for cutoff where .  The Liouville operator  can then be written (via use of the chain rule) as a sum of discrete and continuously scaled contributions:    (A.1)    (A.2)   . (A.3) In this formulation, the N-atom probability density is a function of  the set of N atomic positions and momenta, and a functional of a continuous set of order parameters. Hence,  operates on the former while the  term operates on the latter, and the Liouville equation becomes a mixed differential/functional-differential equation.  As discussed in Sect. 3, we introduce a continuous spectrum of time variables  to capture the effects of  respectively. Hence,  is a function of  and the N-atom state , while it is a functional of the slow times .  The chain rule and the functional dependence of  imply that (3.1) becomes . (A.4) While the set of times  is satisfactory for the derivation of the Smoluchowski equation presented here, the fact that  cannot be related to , for related to  implies the need to introduce more complex time variables (e.g., ) as suggested by the omitted terms in (A.4).  23 Since  is small, a Volterra functional expansion in  is introduced for:   . (A.5) To lowest order in  the Liouville equation implies   . (A.6) For quasi-equilibrium states is independent of , but it does depend on the slow times .  Hence, (A.6) becomes   . (A.7) The form of  is developed in analogy to the nanocanonical ensemble approach12-16:     (A.8)  where  is the partition function    (A.9) and  is as yet unknown.  Here,  is defined by (3.8), and  represents the value of the order parameter field in terms of the integrated atomic variables. The functional derivative of  is related to the thermal-averaged force, i.e. .  The analysis proceeds by combining (A.4) and (A.5), taking the -th functional derivative of both sides of the result with respect to , and setting  to zero for .  To first order one obtains   . (A.10) Assuming the system begins in a quasi-equilibrium state given by , i.e.  vanishes at , we obtain  24   . (A.11)  Changing variables such that  yields    . (A.12) Using the definition of , we find the integral of the last term in (A.12) takes the form . To remove secular behavior, we utilize the fundamental postulate of equilibrium statistical mechanics:   ,  (A.13) i.e., the thermal and long-time averages of any quantity are equal.  The order parameters evolve slowly as compared to the 10-14 second atomistic timescale.  Hence, we expect that the momenta come to a slowly evolving equilibrium, i.e., their ensemble average as weighted by  is zero.  Using (A.13), it follows that the long time average of   is zero.  Upon dividing by , taking the limit as  tends to infinity, and using (A.13) in (A.12), we obtain   . (A.14)  Thus, vanishes and  is independent of the long times . Hence, (A.12) becomes    . (A.15) Next, one might expect to conduct a higher-order (i.e., ) analysis of the problem in order to determine an equation for .  However, even in the discrete case, the kinematic equation (3.9) can be used to reassemble the expansion of  and determine a closed equation for 14-16.  Using (A.8) and (A.15) in (3.9), we find  25    (A.16) and since , this simplifies to  . (A.17) Finally, expanding as we did for  in (A.5), it follows that  as  (i.e., when all slow processes operate on a significantly longer timescale than that of atomistic fluctuations), and thus we find    (A.18) where .  This is the functional Smoluchowski equation for the dynamics of systems with a broad continuum of time scales.  26 ACKNOWLEDGEMENTS This project was supported in part by the US Department of Energy, the National Institutes of Health (NIBIB), the METACyt Initiative, the Office of the Provost at the University of Texas at Arlington, and the College of Arts and Sciences at Indiana University through the Center for Cell and Virus Theory.  LITERATURE CITED 1. Bose, S. and P. Ortoleva (1979a). Reacting hard sphere dynamics: Liouville equation for condensed media. J. Chem. Phys. 70(6):3041-3056. 2. Bose, S. and P. Ortoleva (1979b). A hard sphere model of chemical reaction in condensed media. Physics Letters. A69(5):367-369. 3. Bose, S., S. Bose and P. Ortoleva (1980). Dynamic Padé approximants for chemical center waves. J. Chem. Phys. 72(8):4258-4263. 4. Bose, S., M. Medina-Noyola and P. Ortoleva (1981). Third body effects on reactions in liquids. J. Chem. Phys. 75(4):1762-1771. 5. Chandrasekhar, S. (1943). Dynamical Friction. I. General Considerations: the Coefficient of Dynamical Friction. Astrophysical J. Vol. 97, p.255. 6. Coffey, W.T., Yu.P. Kalmykov and J.T. Waldron (2004). The Langevin Equation With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering World Scientific Publishing Co. Pte. Ltd. 7. Deutch, J.M. and I. Oppenheim (1987). The Concept of Brownian Motion in Modern Statistical Mechanics. Faraday Discuss Chem Soc. 83, 1-20. 8. McQuarrie, D.A. (1976). Statistical Mechanics Harper and Row. 9. Shea, J-E., and I. Oppenheim (1998).  Fokker-Planck and non-linear hydrodynamic equations of several Brownian particles in a non-equilibrium bath. Physica A. 265-294. 10. Tokuyama, K. and I. Oppenheim (2003). Slow Dynamics in Complex Systems: 3rd International Symposium on Slow Dynamics in Complex Systems Sendai, Japan 3-8 Nov 2003 (Aip Conference Proceedings)  27 11. Ortoleva, P. (2005) Nanoparticle Dynamics: A Multiscale Analysis of the Liouville Equation. J. Phys. Chem., 109, 21258-21266. 12. Miao, Y., Ortoleva, P. (2006a) All-atom multiscaling and new ensembles for dynamical nanoparticles J. Chem. Phys. Vol. 125: 044901. 13. Miao, Y., Ortoleva, P. (2006b) All-atom theory of viral capsids with genome or other  embedded nanostructures J. Chem. Phys. 125: 214901. 14. Pankavich, S., Shreif, Z., Ortoleva, P. (2008). Multiscaling for Classical nanosystems: Derivation of Smoluchowski and Fokker-Planck Equations. Physica A. 387:4053-4069 15. Pankavich, S., Miao, Y., Ortoleva, J., Shreif, Z., Ortoleva, P. (2008). Stochastic dynamics of bionanosystems: Multiscale analysis and specialized ensembles. J. Chem. Phys. 128: 234908. 16. Pankavich, S., Shreif Z., Ortoleva P. (2009). Self-Assembly of Nanocomponents into Composite Structures: Derivation and Simulation of Langevin Equations. J. Chem. Phys. 130: 194115. 17. Jaqaman, K., Ortoleva, P. (2002). New space warping method for the simulation of large-scale macromolecular conformational changes. J. Comp. Chem. 23(4) 484-491. 18.  Shreif, Z., Adhangale, P., Cheluvaraja, S., Perera, R., Kuhn, R., and Ortoleva, P. (2008). Enveloped viruses understood via multiscale simulation: computer-aided vaccine design.  Scientific Modeling and Simulation 15: 363-380. 19. Miao, Y., Ortoleva, P. (2008). Molecular dynamics/order parameter extrapolation for bionanosystem simulations. J. Comp. Chem. 30(3) 423-437.  
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+++ b/1001.0265.txt
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+Physics Procedia 00 (2010) 1–17Physics Procedia, ,
+Diagnosis and Prediction of Tipping Points in Financial Markets:
+Crashes and Rebounds
+Wanfeng Yana, Ryan Woodarda, Didier Sornettea,b,
+aChair of Entrepreneurial Risks, Department of Management, Technology and Economics, ETH Zurich, CH-8001
+Zurich, Switzerland
+bSwiss Finance Institute, c /o University of Geneva, 40 blvd. Du Pont dArve, CH 1211 Geneva 4, Switzerland
+Abstract
+By combining (i) the economic theory of rational expectation bubbles, (ii) behavioral finance
+on imitation and herding of investors and traders and (iii) the mathematical and statistical physics
+of bifurcations and phase transitions, the log-periodic power law (LPPL) model has been devel-
+oped as a flexible tool to detect bubbles. The LPPL model considers the faster-than-exponential
+(power law with finite-time singularity) increase in asset prices decorated by accelerating oscil-
+lations as the main diagnostic of bubbles. It embodies a positive feedback loop of higher return
+anticipations competing with negative feedback spirals of crash expectations. The power of the
+LPPL model is illustrated by two recent real-life predictions performed recently by our group:
+the peak of the Oil price bubble in early July 2008 and the burst of a bubble on the Shanghai
+stock market in early August 2009. We then present the concept of “negative bubbles”, which
+are the mirror images of positive bubbles. We argue that similar positive feedbacks are at work
+to fuel these accelerated downward price spirals. We adapt the LPPL model to these negative
+bubbles and implement a pattern recognition method to predict the end times of the negative
+bubbles, which are characterized by rebounds (the mirror images of crashes associated with the
+standard positive bubbles). The out-of-sample tests quantified by error diagrams demonstrate the
+high significance of the prediction performance.
+Keywords: financial bubble, crash, negative bubble, rebound, prediction, log-periodic power
+law, positive feedback, error diagram
+Corresponding author. Address: KPL F 38.2, Kreuzplatz 5, ETH Zurich, CH-8032 Zurich, Switzerland. Phone: +41
+44 632 89 17, Fax: +41 44 632 19 14.
+Email addresses: wyan@ethz.ch (Wanfeng Yan), rwoodard@ethz.ch (Ryan Woodard), dsornette@ethz.ch
+(Didier Sornette)
+URL: http://www.er.ethz.ch/ (Didier Sornette)arXiv:1001.0265v2  [q-fin.GN]  6 Feb 2010/Physics Procedia 00 (2010) 1–17 2
+1. Introduction
+Bubbles and crashes in financial markets are of global significance because of their e ects
+on the lives and livelihoods of a majority of the world’s population. In spite of this, the science
+to correctly identify bubbles in advance of their associated crashes produces fewer successful
+results than that used to treat baldness, choose ‘quality’ videos to watch or find a date on the
+internet. Instead, pundits and experts alike line up after the fact to claim that a particular bubble
+was obvious in hindsight. We present here a report on recent progress of the on-going e ort of our
+group, the Financial Crisis Observatory at ETH Zurich ( www.er.ethz.ch/fco/ , to advance the
+science of understanding why, how and when bubbles form so that they can be identified before
+they spectacularly crash and spread misfortune to those who knowingly or unknowingly had bet
+on a long position.
+The basis for our approach contradicts the accepted wisdom of the E cient Market Hypoth-
+esis, which claims that large deviations from fundamental prices (i.e., bubbles and crashes) only
+exist when a new piece of information drops (exogenously) onto an unsuspecting market on a
+very short time scale. Instead, we claim that bubbles are the result of endogenous market dy-
+namics over a much longer time scale–weeks, months and years. Because of the long build-up
+of these e ects, bubbles can be identified by particular dynamical signatures predicted by our
+theoretical framework.
+Among the well-documented history of financial bubbles and crashes over the past 400 years,
+through countless significantly di erent countries and kings, empires and economies, regulations
+and reform, there has been one consistent ingredient in all booms and busts: humans. Only
+human behavior has survived all attempts at preventing repeats in the wake of disastrous crashes.
+Much of the dynamics of the long time scales mentioned above is due to humans acting like
+humans: those without the knowledge imitate one another in the absence of a clearly better
+alternative and take refuge in the comfort of the crowd (herding) while those with the knowledge
+refute the masses and claim these noise traders are wrong.
+We hypothesize that the signatures of this characteristic human behavior can be quantita-
+tively identified. The imitation and herding behavior creates positive feedback in the valuation
+of an asset, resulting in a greater-than-exponential (power law) growth of the price time series.
+The tension and competition between the learned experts and the noise traders creates decora-
+tions on this power law growth comprising oscillations that are periodic in the logarithm of time.
+Log-periodic oscillations appear to our clocks as peaks and valleys with progressively smaller
+amplitudes and greater frequencies that eventually reach a point of no return, where the unsus-
+tainable growth has the highest probability of ending in a violent crash or gentle deflation of the
+bubble.
+The log periodic power law (LPPL) model has thus been developed as a flexible tool to detect
+bubbles. This model combines (i) the economic theory of rational expectation bubbles, (ii) be-
+havioral finance on imitation and herding of investors and traders and (iii) the mathematical and
+statistical physics of bifurcations and phase transitions. The LPPL model considers the faster-
+than-exponential (power law with finite-time singularity) increase in asset prices decorated by
+accelerating oscillations as the main diagnostic of bubbles. It embodies a positive feedback loop
+of higher return anticipations competing with negative feedback spirals of crash expectations.
+The power of the LPPL model is illustrated by several recent real-life predictions performed
+by our group. We will present two examples in this paper: one is the peak of the Oil price
+bubble in early July 2008 and the other is the burst of a bubble on the Shanghai stock market
+in early August 2009. We then present the concept of “negative bubbles”, which are the mirror/Physics Procedia 00 (2010) 1–17 3
+images of positive bubbles. We argue that similar positive feedbacks are at work to fuel these
+accelerated downward price spirals. We adapt the LPPL model to these negative bubbles and
+implement a pattern recognition method to predict the end times of the negative bubbles, which
+are characterized by rebounds (the mirror images of crashes associated with the standard positive
+bubbles). The out-of-sample tests quantified by error diagrams demonstrate the high significance
+of the prediction performance.
+This paper is constructed as follows. We will give a general introduction to the LPPL model
+in section 2 . In section 3, two successful real-life predictions mentioned above are discussed.
+Then we describe the phenomenon of negative bubbles and their termination in the form of
+rebounds. We develop a pattern recognition method based on the LPPL model and test its pre-
+diction performance in section 4. Section 5 concludes.
+2. Log-periodic Power Law
+2.1. Long time scale fermentation of bubbles
+In sharp contrast to the E cient Market Hypothesis (EMH) that crashes result from novel
+negative information incorporated in prices at short time scales, we build on the radically di erent
+hypothesis summarized by Sornette [1], that the underlying causes of the crash should be found
+in the preceding year(s). We define a bubble as a market regime in which the price accelerates
+“super-exponentially”. The term “super-exponential” means that the growth rate of the price
+grows itself. A constant price growth rate (also called return) leads to an exponential growth,
+the normal average trajectory of most economic and financial time series. When the growth rate
+grows itself, the price accelerate hyperbolically. This growth of the growth rate is interpreted
+as being due to progressively increasing build-up of market cooperation between investors. As
+the bubble matures, its approaches a critical point, at which an instability can be triggered in the
+form of a crash, or more generally a change of regime. This critical point can also be called
+a phase transition, a bifurcation, a catastrophe or a tipping point. According to this “critical”
+point of view, the specific manner by which prices collapse is not the most important problem:
+a crash occurs because the market has entered an unstable phase and any small disturbance or
+process may reveal the existence of the instability. Think of a ruler held up vertically on your
+finger: this very unstable position will lead eventually to its collapse as a result of a small (or
+an absence of adequate) motion of your hand or due to any tiny whi of air. The collapse is
+fundamentally due to the unstable position; the instantaneous cause of the collapse is secondary.
+In the same vein, the growth of the sensitivity and the growing instability of the market close to
+such a critical point might explain why attempts to unravel the proximal origin of the crash have
+been so diverse. Essentially, anything would work once the system is ripe.
+What is the origin of the maturing instability? A follow-up hypothesis underlying this pro-
+posal is that, in some regimes, there are significant behavioral e ects underlying price formation
+leading to the concept of “bubble risks”. This idea is probably best exemplified in the context
+of financial bubble, where, fuelled by initially well-founded economic fundamentals, investors
+develop a self-fulfilling enthusiasm by an imitative process or crowd behavior that leads to the
+building of castles in the air, to paraphrase Malkiel [2].
+Our previous research suggests that the ideal economic view, that stock markets are both e -
+cient and unpredictable, may be not fully correct. We propose that, to understand stock markets,
+one needs to consider the impact of positive feedbacks via possible technical as well as behav-
+ioral mechanisms such as imitation and herding, leading to self-organized cooperation and the/Physics Procedia 00 (2010) 1–17 4
+development of possible endogenous instabilities. We thus propose to explore the consequences
+of the concept that most of the crashes have fundamentally an endogenous, or internal, origin and
+that exogenous, or external, shocks only serve as triggering factors. As a consequence, the origin
+of crashes is probably much more subtle than often thought, as it is constructed progressively by
+the market as a whole, as a self-organizing process. In this sense, the true cause of a crash could
+be termed a systemic instability.
+2.2. Imitation and Herding Among Humans as the Cause of Bubbles
+Humans are perhaps the most social mammals and they shape their environment to their
+personal and social needs. This statement is based on a growing body of research at the frontier
+between new disciplines called neuro-economics, evolutionary psychology, cognitive science
+and behavioral finance ([3], [4], [5]). This body of evidence emphasizes the very human nature
+of humans with its biases and limitations, opposed to the previously prevailing view of rational
+economic agents optimizing their decisions based on unlimited access to information and to
+computation resources.
+We hypothesize that financial bubbles are footprints of perhaps the most robust trait of hu-
+mans and the most visible imprint in our social a airs: imitation and herding (see Sornette [1]
+and references therein). Imitation has been documented in psychology and in neuro-sciences
+as one of the most evolved cognitive processes, requiring a developed cortex and sophisticated
+processing abilities. In short, we learn our basics and how to adapt mostly by imitation all
+through our life. It seems that imitation has evolved as an evolutionary advantageous trait, and
+may even have promoted the development of our anomalously large brain (compared with other
+mammals), according to the so-called “social brain hypothesis” advanced by R. Dunbar [6]. It
+is actually “rational” to imitate when lacking su cient time, energy and information to make a
+decision based only on private information and processing, that is, most of the time. Imitation,
+in obvious or subtle forms, is a pervasive activity of humans. In the modern business, economic
+and financial worlds, the tendency for humans to imitate leads in its strongest form to herding
+and to crowd e ects. Imitation is a prevalent form in marketing with the development of fashion
+and brands. Cooperative herding and imitation lead to positive feedbacks, that is, an action leads
+to consequences which themselves reinforce the action and so on, leading to virtuous or vicious
+circles.
+The methodology that we have developed consists in using a series of mathematical and com-
+putational formulations of these ideas, which capture the hypotheses that (1) bubbles can be the
+result of positive feedbacks and (2) the dynamical signature of bubbles derives from the interplay
+between fundamental value investment and more technical analysis. The former can be embod-
+ied in nonlinear extensions of the standard financial Black-Scholes model of log-price variations
+[7, 8, 9, 10]. The later requires more significant extensions to account for the competition be-
+tween (i) inertia separating analysis from decisions, (ii) positive momentum feedbacks and (iii)
+negative value investment feedbacks [8].
+2.3. Positive Feedback Among Traders Leads to Power Law Growth in Asset Price
+The idea of positive feedback has led us to propose that one of the hallmarks of a financial
+bubble is the faster-than-exponential growth of the price of the asset under consideration, as
+already mentioned. It is convenient to model this accelerated growth by a power law with a so-
+called finite-time singularity [11]. This feature is nicely illustrated by the price trajectory of the
+Hong Kong Hang Seng index from 1970 to 2000, as shown in Fig. 1. The Hong Kong financial/Physics Procedia 00 (2010) 1–17 5
+market is repeatedly rated as providing one of the most pro-economic, pro-entrepreneurship and
+free market-friendly environments in the world, and thus provides a textbook example of the
+behavior of weakly regulated liquid and striving financial markets. In Fig. 1, the logarithm of the
+price p(t) is plotted as a function of time (in linear scale), so that an upward trending straight line
+qualifies as exponential growth with a constant growth rate: the straight solid line corresponds
+indeed to an approximately constant growth rate of the Hang Seng index equal to 13.8% per year.
+The most striking feature of Fig. 1 is not this average behavior but instead the obvious fact
+that the real market is never following and abiding to a constant growth rate. One can observe
+a succession of price run-ups characterized by growth rates :::growing themselves: this is re-
+flected visually in Fig. 1 by transient regimes characterized by strong upward curvature of the
+price trajectory. Such an upward curvature in a linear-log plot is a first visual diagnostic of a
+faster than exponential growth (which of course needs to be confirmed by rigorous statistical
+testing). Such a price trajectory can be approximated by a characteristic transient finite-time
+singular power law of the form
+logp(t)=A+B(tct)m(1)
+where B<0, 0<m<1 and tcis the theoretical critical time corresponding to the end of the
+transient run-up (end of the bubble). Such transient faster-than-exponential growth of p(t)is
+our definition of a bubble. It has the major advantage of avoiding the conundrum of distinguish-
+ing between exponentially growing fundamental price and exponentially growing bubble price,
+which is a problem permeating most of the previous statistical tests developed to identify bubbles
+[12, 13]. The conditions B<0 and 0<m<1 ensure the super-exponential acceleration of the
+price, together with the condition that the price remains finite even at tc. Stronger singularities
+can appear for m<0.
+To see that faster-than-exponential growth is naturally related to positive feedback, let us
+consider the following simple presentation. Consider a population of animals of size xwhich
+grows with some constant rate k, i.e., dx=dt=k x. Then, growth is exponential in that x(t)=
+x(0)ekt. On the other hand, positive feedback in growth dynamics arises if the growth rate kitself
+depends on the population size xin that the growth rate k=k(x) increases with the population
+size. A particular simple example is the setting k(x)xm1, where m>1. Indeed, m1
+can be regarded as a measure of the degree of cooperation within the population: the higher m
+is the larger is the degree of cooperation. In the case of no cooperation, growth dynamics is
+exponential. Positive multiplicative feedback generates growth which is faster than exponential.
+Indeed, due to growth dynamics given by dx=dt=k xm, the size of the population growth exhibits
+a finite-time sigularity at tc. This singularity is attained according to x(t)=x(0)[1]1
+1m,
+0<< 1 where=t
+tc. Note that when m=1, no cooperation, growth is indeed exponential and
+when m>1, growth dynamics is in fact faster than exponential. It is remarkable that a critical
+time tcemerges apparently out of nowhere. Actually, tcis determined by the initial conditions
+and the structure of the growth equation. This emergence of tcwhich depends on the initial
+conditions justifies its name in mathematical textbooks as a “movable singularity.”
+Many systems exhibit similar transient super-exponential growth regimes, which are de-
+scribed mathematically by power law growth with an ultimate finite-time singular behavior. An
+incomplete list of examples includes: planet formation in solar systems by runaway accretion
+of planetesimals, rupture and material failures, nucleation of earthquakes modeled with the slip-
+and-velocity, models of micro-organisms interacting through chemotaxis aggregating to form
+fruiting bodies, the Euler rotating disk, and so on. Such mathematical equations can actually/Physics Procedia 00 (2010) 1–17 6
+Figure 1: Trajectory of the Hong-Kong Hang Seng index from 1970 to 2000. The vertical log-scale together with the
+linear time scale allows one to qualify an exponential growth with constant growth rate as a straight line. This is indeed
+the long-term behavior of this market, as shown by the best linear fit represented by the solid straight line, corresponding
+to an average constant growth rate of 13.8% per year. The 8 arrows point to 8 local maxima that were followed by a
+drop of the index of more than 15% in less than three weeks (a possible definition of a crash). The 8 small panels at
+the bottom show the upward curvature of the log-price trajectory preceding each of these local maxima, which diagnose
+unsustainable bubble regimes, each of which culminates at its peak before crashing./Physics Procedia 00 (2010) 1–17 7
+provide an accurate description of the transient dynamics not too close to the mathematical sin-
+gularity where new mechanisms come into play. The singularity at tcmainly signals a change
+of regime. In the present context, tcis the end of the bubble and the beginning of a new market
+phase, possibly a crash or a di erent regime.
+Such an approach may be thought at first sight to be inadequate or too naive to capture the
+intrinsic stochastic nature of financial prices, whose null hypothesis is the geometric random walk
+model ([2]). However, it is possible to generalize this simple deterministic model to incorporate
+nonlinear positive feedback on the stochastic Black-Scholes model, leading to the concept of
+stochastic finite-time singularities [7, 14, 15, 10, 16, 17]. Still much work needs to be done on
+this theoretical aspect.
+Coming back to Fig. 1, one can also notice that each burst of super-exponential price growth
+is followed by a crash, here defined for the eight arrowed cases as a correction of more than 15%
+in less than 3 weeks. These examples suggest that the non-sustainable super-exponential price
+growths announced a “tipping point” followed by a price disruption, i.e., a crash. The Hong-
+Kong Hang Seng index shows that the average exponential growth of the index is punctuated by
+a succession of bubbles and crashes, which seem to be the norm rather than the exception.
+2.4. Competition between di erent types of traders lead to log-periodic oscillations
+More sophisticated models than Eq. (1) have been proposed to take into account the inter-
+play between technical trading and herding (positive feedback) versus fundamental valuation
+investments (negative mean-reverting feedback). Accounting for the presence of inertia between
+information gathering and analysis on the one hand and investment implementation on the other
+hand [18], and taking additionally into account the coexistence of trend followers and value in-
+vesting [8], the resulting price dynamics develop second-order oscillatory terms and boom-bust
+cycles. Value investing does not necessarily cause prices to track value. Trend following may
+cause short-term trend in prices but, together with value investing and inertia, also causes longer-
+term oscillations.
+The simplest model generalizing (1) and including these ingredients is the so-called log-
+periodic power law (LPPL) model (see Sornette [1] and references therein). Formally, some of
+the corresponding formulas can be obtained by considering that the exponent mis a complex
+number with an imaginary part, where the imaginary part expresses the existence of a preferred
+scaling ratio 
describing how the continuous scale invariance of the power law (1) is partially
+broken into a discrete scale invariance [19]. The LPPL structure may also reflect the discrete hi-
+erarchical organization of networks of traders, from the individual to trading floors, to branches,
+to banks, to currency blocks. More generally, it may reveal the ubiquitous hierarchical organi-
+zation of social networks recently reported [20] to be associated with the social brain hypothesis
+[6].
+Examples of calibrations of financial bubbles with one implementation of the LPPL model
+are the 8 super-exponential regimes discussed above in Fig. 1: the 8 small insets at the bottom of
+the figure show the LPPL calibration on the Hang Seng index. Preliminary tests [1] suggest that
+the LPPL model provides a good starting point to detect bubbles and forecast their most probable
+end. Rational expectation models of bubbles a la Blanchard and Watson implementing the LPPL
+model [21, 22] have shown that the end of the bubble is not necessarily accompanied by a crash,
+but it is indeed the time where a crash is the most probable. But crashes can occur before (with
+smaller probability) or not at all. That is, a bubble can land smoothly, approximately one-third
+of the time, according to preliminary investigations [23]. Therefore, only probabilistic forecasts/Physics Procedia 00 (2010) 1–17 8
+can be developed. Probability forecasts are indeed valuable and commonly used in daily life,
+such as in weather forecast.
+3. Successful case studies
+3.1. The Oil Bubble of 2008
+In [24], we have presented the prediction performed ex-ante and its post-mortem analysis
+of the bubble that has developed on oil prices in USD and in other major currencies. The ex-
+ante diagnostic of the oil bubble was performed on the basis of the identification of unsustainable
+faster-than-exponential behavior during the course of 2007 until the mid-2008. We found support
+for the hypothesis that the oil price run-up in the first half of 2008 was amplified by speculative
+behavior of the type found during a bubble-like expansion. We also attempted to unravel the in-
+formation hidden in the oil supply-demand data reported by two leading agencies, the US Energy
+Information Administration (EIA) and the International Energy Agency (IEA). We suggested that
+the found increasing discrepancy between the EIA and IEA figures provides a measure of the es-
+timation errors. Rather than a clear transition to a supply restricted regime, we interpreted the
+discrepancy between the IEA and EIA as a signature of uncertainty. This is compatible with the
+idea that there is no better fuel than uncertainty to promote speculation. Our post-crash analysis
+confirmed that the oil peak in July 2008 occurred within the expected 80% confidence interval
+predicted ex-ante with data available in our pre-crash analysis.
+The main result of our post-crash analysis is shown in Fig. 2. We calculated the 80% con-
+fidence interval for the critical time tcof the end of the Oil bubble, which was found to be
+17 May 2008 to 14 July 2008 and is shown as the shaded box in the main and inset plots of
+Fig. 2. The actual peak oil price was observed to be 3 July and the steep descent in price began
+on 11 July. Both dates are within the confidence interval calculated with the LPPL model using
+data through the last week of May. This confirms that the simple LPPL model was a successful
+predictor of the 2008 oil price bubble.
+3.2. The Chinese Index Bubble of 2009
+In the midst of the global financial crisis of the past year, when the prices of many assets and
+indexes fell, the Shanghai Composite Index defied financial gravity and continued climbing as it
+had done since October 2008.
+On 10 July 2009 (using data through 9 July 2009), we submitted our prediction online to
+arXiv.org [25], in which we gave the 20% /80% (respectively 10% /90%) quantiles of the pro-
+jected crash dates to be 17-27 July 2009 (respectively 10 July - 10 August 2009). This corre-
+sponds to a 60% (respectively 80%) probability that the end of the bubble occurs and that the
+change of regime starts in the interval within those time windows. Redoing the analysis 5 days
+later with data through 14 July 2009, the predictions tightened up with a 80% probability for the
+change of regime to start between 19 July and 3 August 2009 (unpublished). Our forecasts were
+featured in the press at, among other places, http://www.technologyreview.com/blog/
+arxiv/23839/ .
+On 29 July 2009, Chinese stocks su ered their steepest drop since November 2008, with
+an intraday bottom of more than 8% and an open-to-close loss of more than 5%. The market
+rebounded with a peak on 4 August 2009 before plummeting the following weeks. The Shanghai
+Index slumped 22 percent in August, the biggest decline among 89 benchmark indices tracked
+world wide by Bloomberg, in stark contrast with being the best performing index during the first/Physics Procedia 00 (2010) 1–17 9
+2002 2003 2004 2005 2006 2007 2008 2009
+date050100150200250300price / USD
+Mar 2008 Sep 200850100150200250 Shaded region is
+80% confidence interval
+of projected peak date
+Original
+analysis date
+27 May 2008Actual
+peak date
+03 July 2008
+Figure 2: Time series of observed prices in USD of “NYMEX Light Sweet Crude, Contract 1” from the Energy Informa-
+tion Administration of the U.S. Government (see http://www.eia.doe.gov/emeu/international/Crude2.xls )
+and simple LPPL fits (see text for explanation). The oil price time series was scanned in multiple windows defined by
+(t1,t2), where t1ranged from 1 April 2003 to 2 January 2008 in steps of approximately 3 months (see text) and t2was
+fixed to 23 May 2008. Shaded box shows the 80% confidence interval of fit parameter tcfor fits with tcless than six
+months beyond t2. Also shown are dates of our original analysis in June 2008 and the actual observed peak oil price on
+3 July 2008. Reproduced from [24]./Physics Procedia 00 (2010) 1–17 10
+half of 2009. We thus successfully predicted time windows for this crash in advance with the
+same methods used to successfully predict the peak in mid-2006 of the US housing bubble [26]
+and the peak in July 2008 of the global oil bubble [24]. The more recent bubble in the Chinese
+indexes was detected and its end or change of regime was predicted independently by two groups
+with similar results, showing that the model has been well-documented and can be replicated by
+industrial practitioners.
+In [27], we presented a thorough post-crash analysis of this 2008-2009 Chinese bubble and,
+also, the previous 2005-2007 Chinese bubble in Ref. [28]. This publication also documents
+another original forecast of the 2005-2007 bubble (though there was not a publication on that,
+except a public announcement at a hedge-fund conference in Stockholm in October 2007). Also,
+it clearly lays out some of our many technical methods used in testing our analyses and forecasts
+of bubbles: the search and fitting method of the LPPL model itself, Lomb periodograms of
+residuals to further identify the log-periodic oscillation frequencies, (H, q)-derivatives [29, 30]
+and, most recently, unit root tests of the residuals to confirm the Ornstein-Uhlenbeck property of
+their stationarity [17]. Here, we reproduce the main figure documenting the advance prediction
+which included the peak of the bubble on 4 August 2009 in its 5-95% confidence limits. The
+curves are fitted by first order Laudau model:
+logp(t)=A+B(tct)m+C(tct)mcos(!ln(tct)) (2)
+4. Detection of Rebounds using Pattern Recognition Method
+Until now, we have focused our attention on bubbles, their peaks and the crashes that often
+follow. We have argued that bubbles develop due to positive feedbacks pushing the price upward
+towards an instability at which the bubble ends and the price may crash.
+But positive feedbacks do not need to be active just for price run-ups, i.e., for exuber-
+ant “bullish” market phases. Positive feedbacks may also be at work during “bearish” market
+regimes, when pessimism replaces optimism and investors sell in a downward spiral. Zhou and
+Sornette (2003) have developed a generalized Weierstrass LPPL formulation to model the 5 drops
+and their successive rebounds observed in the US S&P 500 index during the period from 2000 to
+2002 [31].
+Here, we go further and develop a methodology that combines a LPPL model of accelerating
+price drops, termed a “negative bubble,” and a pattern recognition method in order to diagnose
+the critical time tcof the rebound (the antithesis of the crash for a “positive bubble”). A negative
+bubble is modeled with the same tools as a positive bubble, that is, with the power law expression
+(1). But the essential di erence is that the coe cient Bis positive for a negative bubble (while
+it is negative for a normal positive bubble, as discussed above). The exponent mobeys the same
+condition 0 <m<1 as for the positive bubbles. The positivity of Btogether with the condition
+0<m<1 implies that the average log-price trajectory exhibits a downward curvature, express-
+ing a faster-than exponential downward acceleration. In other words, the log-price trajectory
+associated with a negative bubble is the upside-down mirror image of the log-price trajectory of
+a positive bubble. Additional log-periodic oscillations are added to the LPPL model, which also
+account for the competition between value investors and trend followers.
+We adapt the pattern recognition method of [32] to generate predictions of rebound times in
+financial markets. A similar method has been developed by Sornette and Zhou (2006) to combine/Physics Procedia 00 (2010) 1–17 11
+08/07/01 09/01/01 09/07/01 10/01/0 18.68.899.29.49.6t1fixed: 08/11/01
+t2min/max: 09/06/01−09/07/27
+peak date: 09/07/2820/80 q: 09/07/25−09/07/29
+5/95q:  09/07/25−09/08/05
+num of fits: 13(d)
+tlnp(t)
+  
+SZSC observations
+Figure 3: Daily trajectory of the logarithmic SSEC (a,b) and SZSC (c,d) index from Sep-01-2008 to Jul-31-2009 (dots)
+and fits to the LPPL formula (2). The dark and light shadow box indicate 20 /80% and 5 /95% quantile range of values
+of the crash dates for the fits, respectively. The two dashed lines correspond to the minimum date of t1and the fixed
+date of t2. (a) Examples of fitting to shrinking windows with varied t1and fixed t2=Jul-31-2009 for SSEC. The six
+fitting illustrations are corresponding to t1=Oct-15-2008, Nov-07-2008, Dec-05-2008, Jan-05-2008, Feb-06-2008, and
+Feb-20-2008. (b) Examples of fitting to expanding windows with fixed t1=Nov-01-2008 and varied t2for SSEC.
+The six fitting illustrations are associated with t2=Jun-01-2009, Jun-10-2009, Jun-19-2007, Jun-29-2007, Jul-13-2007,
+Jul-27-2007. (c) Examples of fitting to shrinking windows with varied t1and fixed t2=Jul-31-2009 for SZSC. The
+six fitting illustrations are corresponding to t1=Oct-15-2008, Nov-03-2008, Nov-26-2008, Dec-19-2008, Jan-14-2008,
+and Jan-23-2008. (d) Examples of fitting to expanding windows with fixed t1=Dec-01-2005 and varied t2for SZSC.
+The six fitting illustrations are associated with t2=Jun-01-2009, Jun-10-2009, Jun-19-2007, Jun-29-2007, Jul-13-2007,
+Jul-27-2007. Reproduced from Jiang et. al. [27]./Physics Procedia 00 (2010) 1–17 12
+the information provided by the LPPL model with a pattern recognition method for the prediction
+of the end of bubbles [33].
+We analyze the S&P 500 index prices, obtained from Yahoo! finance for ticker ‘ˆGSPC’
+(adjusted close price)1. The start time of our time series is 1950-01-05, which is very close to the
+first day of S&P 500 index (1950-01-03). The last day of our tested time series is 2009-06-03.
+We first divide our S&P 500 index time series into di erent sub-windows ( t1;t2) of length
+dtt2t1according to the following rules:
+1. The earliest start time of the windows is t10=1950-01-03. Other start times t1are calcu-
+lated using a step size of dt1=50 calendar days.
+2. The latest end time of the windows is t20=2009-06-03. Other end times t2are calculated
+with a negative step size dt2=50 calendar days.
+3. The minimum window size is dtmin=110 calendar days.
+4. The maximum window size is dtmax=1500 calendar days.
+These rules lead to 11,662 windows in the S&P 500 time series.
+Then we fit the log-price time series in each window with the LPPL model and get the corre-
+sponding parameters.
+A rebound is defined by:
+Rbd=fdjPd=minfPxg;8x2[d200;d+200]g (3)
+where Pdis the adjusted closing price on day d. According to this definition, a rebound is said to
+have occurred on day dif condition (3) holds.
+The prediction method uses a learning in-sample data set. Then, an out-of-sample dataset
+allows us to test the validity and success of the prediction method. We take all the fits before
+1975-01-01 as learning set. In the learning set, if the critical time of a fit is near some of the
+rebounds, then we select this fit into Class I. In contrast, if there is not any rebound near the
+critical time of a fit, this fit will be classified into Class II. We then construct the statistics of all
+the parameters of the fits from Class I and Class II separately. We then compare the properties
+of these two sets of statistics for the two classes. The statistically significant di erences between
+these properties allow us to define the features which are specific to each class.
+Using these features associated with each class, we analyze every trading day before 1975-
+01-01. We collect all the fits with a critical time near this trading day, and find out the features of
+these fits. If most of the features in these fits belong to Class I, we interpret this as evidence that
+this trading day has a high probability to be a rebound. Otherwise, this trading day is not likely
+to be a rebound.
+To quantify the probability that this trading day is a rebound, we develop the rebound alarm
+index as follows:
+RIt=(t;I
+t;I+t;IIift;I+t;II0
+0 ift;I+t;II=0(4)
+wheret;I;t;IIrepresent the numbers of features of Class I and Class II respectively. From this
+definition, we can see that RIt2[0;1]. If RItis high, then we can declare that this day has a high
+probability that the rebound will start.
+1http://finance.yahoo.com /q/hp?s=ˆGSPC/Physics Procedia 00 (2010) 1–17 13
+We use the features generated from the learning set, i.e. the fits before 1975-01-01, get the
+rebound alarm index before 1975-01-01 to check how the index performs. Fig. 4 shows the
+results. In this figure, the top panel shows the rebound alarm index for each trading day. The
+middle panel shows the times at which the real rebounds defined in (3) actually occurred. And
+the bottom panel is the log-price of the S&P 500 index. Fig. 4 shows that the rebound alarm
+index diagnosed the real rebounds quite well. A quantitation of what is meant by “quite well”
+will be given for the out-of-sample analysis performed below.
+1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 19740.00.20.40.60.81.0Rebound alarm indexS&P 500 rebound index. Use information before Jan. 1, 1975, back test rebounds before 1975
+1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 19740.00.20.40.60.81.0Target indexLocal minimums in range of [-200, 200] days. Thin vertical lines show rebound region.
+1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974
+Time2.53.03.54.04.55.0Logrithmic priceHistorical S&P 500 index
+Figure 4: Top panel: rebound alarm index of the back tests on each day of our dataset before 1975-01-01. The rebound
+alarm index is defined in [0,1]. The higher the rebound alarm index, the more likely the rebound will happen. Middle
+panel: the actual rebounds in this period, as defined by (3). Lower panel: log-price of the S&P 500 index.
+We now test how the rebound alarm index performs for ex-ante predictions. For this, we use
+the features learned from the learning set before 1975, then construct accordingly the rebound
+alarm index after 1975-01-01. The results are shown in Fig. 5. One can observe a satisfac-
+tory match between the periods when the rebound alarm index is large and the times when the
+rebounds actually occurred.
+In order to quantify systematically and rigorously the quality of the performed predictions,
+we construct the associated error diagram as follows. We set a threshold Thsuch that an alarm is
+declared when the rebound index passes above Th. When this occurs, the alarm is set to last for
+40 additional days. The total alarm set is thus the set of all days for which the rebound index is
+larger than Th, augmented by the 40 consecutive days following the threshold days. A rebound
+is declared as having been predicted when it falls inside the total alarm set. For a given threshold
+Th, we determine the fraction of days which fall in the alarm set (alarm period /total period). We
+also count the fraction of rebounds which have not been predicted. This defines the “failure to
+predict” equal to the fraction of missed rebounds. For a given threshold Th, this gives one point
+in the plane where “failure to predict” is plotted as a function of the (alarm period /total period).
+Changing Thfrom large to small values yields a line in the error diagram, which quantifies the/Physics Procedia 00 (2010) 1–17 14
+1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 20080.00.20.40.60.81.0Rebound alarm indexS&P 500 rebound index. Use information before Jan.1, 1975, predict rebounds until Jun. 3, 2009
+1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 20080.00.20.40.60.81.0Target indexLocal minimums in range of [-200, 200] days. Thin vertical lines show rebound region.
+1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008
+Time4.04.55.05.56.06.57.07.5Logarithmic priceHistorical S&P 500 index
+Figure 5: Same as figure 4 for the out-of-sample prediction period after 1975-01-01.
+quality of the predictor in the way it addresses the trade-o between never missing a rebound
+target and declaring the smallest number of alarms (not “crying wolf” too much).
+It is clear that, if an alarm is declared at all times, no rebound will be missed. This corre-
+sponds to the point (1 ;0). In contrast, declaring zero alarm will miss all targets. This corresponds
+to the point (0 ;1). In between, it is easy to see that the anti-diagonal line y=1xcorresponds
+to random predictions, such that the fraction of successfully predicted events is just the fraction
+of alarm time. In contrast, an ideal perfect predictor corresponds to the convergence to the ori-
+gin (0;0), for which there is no failure to predict any rebound, while using an asymptotically
+vanishing fraction of time in alarms. A prediction system better than random falls below the
+anti-diagonal line y=1x.
+Fig. 6 presents the error diagrams obtained for di erent types of features. Fig. 7 also plots
+the error diagrams obtained in the back tests. The fast drop of failure to predict when the (alarm
+period /total period) is increased from zero is the signature of a very significant predictive ability.
+5. Conclusion
+We have presented partial results of an on-going ambitious project, that aims to test sci-
+entifically, rigorously and systematically the hypotheses that certain anomalous stock market
+regimes can be diagnosed in real-time and their tipping point can be determined better than
+chance. Specifically, we have presented two case studies of the recent bubble on the Oil price
+that culminated in early July 2008 and on the Shanghai stock market that crashes in August 2009.
+We have then shown how the proposed log-periodic power law (LPPL) model can be extended
+to so-called “negative” bubbles, these bearish market regimes where the price spirals down in
+an accelerated way, before rebounding. We have presented statistical tests in- and out-of-sample
+that demonstrate the strong predictive power of the proposed methodology./Physics Procedia 00 (2010) 1–17 15
+0.0 0.2 0.4 0.6 0.8 1.0
+Alarm period / Total period0.00.20.40.60.81.0Failure to predictError Diagram for S&P 500 predictions.
+9 rebounds after 1975-01-01
+feature qualification10,300
+feature qualification10,400
+feature qualification10,500
+feature qualification20,500
+feature qualification20,1000
+Figure 6: Error diagrams for predictions performed after Jan. 1, 1975 with di erent types of feature qualifications.
+Feature qualification ;means that when the occurrence of a certain trait in Class I is more than and less than in
+Class II, then we call this trait a feature of Class I.
+0.0 0.2 0.4 0.6 0.8 1.0
+Alarm period / Total period0.00.20.40.60.81.0Failure to predictError Diagram for S&P 500 back tests.
+10 rebounds before 1975-01-01
+feature qualification10,300
+feature qualification10,400
+feature qualification10,500
+feature qualification20,500
+feature qualification20,1000
+Figure 7: Same as figure 6 for the back tests performed before Jan. 1, 1975./Physics Procedia 00 (2010) 1–17 16
+The research presented here is highly unconventional in financial economics and will swim
+against the convention that bubbles cannot be diagnosed in advance and crashes are somehow
+inherently impossible. But as Einstein once said: “Problems cannot be solved at the same level
+of awareness that created them.” We thus propose a kind of Pascal’s wager: is it really a big risk
+for the community to explore the possibility of changing the conventional wisdom and open new
+directions for the diagnostic of bubbles, ones that may eventually lead to important policy and
+regulatory implications?
+This research may have indeed global impacts. We confront directly the wide-spread belief
+that crises are inherently unpredictable. If one can convince that some crises can be diagnosed
+in advance and, what is even more important, if one can quantify the associated uncertainties,
+this may help economists and policy makers develop new approaches to deal with financial and
+economic crises.
+But piling up more and more case studies, we are convinced that the wealth of empirical
+results will foster many new theoretical insights for the understanding of financial bubbles. We
+envision that this will have a natural spill-over to the questions of how systemic risk and crisis
+depend on factors such as the increasing interdependencies in the global economic systems that
+arise from the development of new financial instruments. Central Banks, for example, could
+benefit from the models discussed here with a wide database of the national banks and of the
+firms and calculate the risks associated with these various economic agents. At a microeconomic
+level, banks could implement such models to supervise the activity of their branches to optimize
+risk management. The relatively small number of variables and parameters involved as well
+as the relative simplicity of the mathematics involved ensures the practical applicability of the
+model in any framework of network dependence and interactions. Banks could find it useful
+to monitor the exposition of correlation derivatives issued to immunize portfolios of liabilities.
+Financial institutions as well could appreciate the flexibility and real-time updating ability of our
+models to manage risk associated with large portfolios of assets.
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+[2] B. G. Malkiel, A random walk down wall street, Norton, New York.
+[3] A. Damasio, Descartes’ error, G.P. Putnam’s Sons, New York.
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+bolic bubbles, crashes and chaos, Quantitative Finance 2 (2002) 264–281.
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+492–506.
+URL http://arXiv.org/abs/physics/0301007
+[12] R. Gurkaynak, Econometric tests of asset price bubbles: Taking stock, Journal of Economic Surveys 22 (1) (2008)
+166–186.
+[13] T. Lux, D. Sornette, On rational bubbles and fat tails, Journal of Money, Credit and Banking 34 (3) (2002) 589–610.
+[14] H. Fogedby, Damped finite-time-singularity driven by noise, Physical Review E 68 (2003) 051105./Physics Procedia 00 (2010) 1–17 17
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+[19] D. Sornette, Discrete scale invariance and complex dimensions, Physics Reports 297 (5) (1998) 239–270.
+URL http://xxx.lanl.gov/abs/cond-mat/9707012
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+Soc. London 272 (2005) 439–444.
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\ No newline at end of file
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+arXiv:1001.0266v2  [cond-mat.str-el]  4 May 2010Realization of the Exactly Solvable Kitaev Honeycomb Latti ce Model in a Spin
+Rotation Invariant System
+Fa Wang1
+1Department of Physics, Massachusetts Institute of Technol ogy, Cambridge, MA 02139, USA
+The exactly solvable Kitaev honeycomb lattice model is real ized as the low energy effect Hamil-
+tonian of a spin-1/2 model with spin rotation and time-rever sal symmetry. The mapping to low
+energy effective Hamiltonian is exact, without truncation e rrors in traditional perturbation series
+expansions. This model consists of a honeycomb lattice of cl usters of four spin-1/2 moments, and
+contains short-range interactions up to six-spin(or eight -spin) terms. The spin in the Kitaev model
+is represented not as these spin-1/2 moments, but as pseudo- spin of the two-dimensional spin singlet
+sector of the four antiferromagnetically coupled spin-1/2 moments within each cluster. Spin corre-
+lations in the Kitaev model are mapped to dimer correlations or spin-chirality correlations in this
+model. This exact construction is quite general and can be us ed to make other interesting spin-1/2
+models from spin rotation invariant Hamiltonians. We discu ss two possible routes to generate the
+high order spin interactions from more natural couplings, w hich involves perturbative expansions
+thus breaks the exact mapping, although in a controlled mann er.
+PACS numbers: 75.10.Jm, 75.10.Kt
+Contents
+I. Introduction. 1
+II. Formulation of the Pseudo-spin-1/2 from
+Four-spin Cluster. 2
+III. Realization of the Kitaev Model. 3
+IV. Generate the High Order Physical Spin
+Interactions by Perturbative Expansion. 5
+A. Generate the High Order Terms by Coupling
+to Optical Phonon. 5
+B. Generate the High Order Terms by Magnetic
+Interactions between Clusters. 7
+V. Conclusions. 8
+Acknowledgments 8
+A. Coupling between Distortions of a
+Tetrahedron and the Pseudo-spins 8
+B. Derivation of the Terms Generated by
+Second Order Perturbation of Inter-cluster
+Magnetic Interactions 9
+References 10
+I. INTRODUCTION.
+Kitaev’s exactly solvable spin-1/2 honeycomb lattice
+model1(noted as the Kitaev model hereafter) has in-
+spired great interest since its debut, due to its exact
+solvability, fractionalized excitations, and the potentialto realize non-Abelian anyons. The model simply reads
+HKitaev=−/summationdisplay
+x−links<jk>Jxτx
+jτx
+k−/summationdisplay
+y−links<jk>Jyτy
+jτy
+k
+−/summationdisplay
+z−links<jk>Jzτz
+jτz
+k
+(1)
+whereτx,y,zare Pauli matrices, and x,y,z-links are de-
+fined in FIG. 1. It was shown by Kitaev1that this spin-
+1/2 model can be mapped to a model with one Majo-
+rana fermion per site coupled to Ising gauge fields on the
+links. And as the Ising gauge flux has no fluctuation, the
+model can be regarded as, under each gauge flux config-
+uration, a free Majorana fermion problem. The ground
+state is achieved in the sector of zero gauge flux through
+eachhexagon. The Majoranafermionsin this sectorhave
+Dirac-likegaplessdispersionresemblingthatofgraphene,
+as long as |Jx|,|Jy|, and|Jz|satisfy the triangular rela-
+tion, sum of any two of them is greater than the third
+one1. It was further proposed by Kitaev1that opening of
+fermion gap by magnetic field can give the Ising vortices
+non-Abelian anyonic statistics, because the Ising vortex
+will carry a zero-energy Majorana mode, although mag-
+netic field destroys the exact solvability.
+Great efforts have been invested to better understand
+the properties of the Kitaev model. For example, sev-
+eral groups have pointed out that the fractionalized Ma-
+jorana fermion excitations may be understood from the
+more familiar Jordan-Wigner transformation of 1D spin
+systems2,3. The analogy between the non-Abelian Ising
+vortices and vortices in p+ipsuperconductors has been
+raised in serveral works4–7. Exact diagonalization has
+been used to study the Kitaev model on small lattices8.
+And perturbative expansion methods have been devel-
+oped to study the gapped phases of the Kitaev-type
+models9.
+Many generalizations of the Kitaev model have been2
+yxzz zzz zzz z
+y yy
+xxxxxxyy y
+y
+xy
+x
+zz
+yyxxx
+yz
+x
+x
+FIG. 1: The honeycomb lattice for the Kitaev model. Filled
+and open circles indicate two sublattices. x,y,z label the links
+along three different directions used in (1).
+derived as well. There have been several proposals to
+open the fermion gap for the non-Abelian phase without
+spoiling exact solvability4,6. And many generalizations
+to other(even 3D) lattices have been developed in the
+last few years10–16. All these efforts have significantly
+enriched our knowledge of exactly solvable models and
+quantum phases of matter.
+However, in the original Kitaev model and its later
+generalizations in the form of spin models, spin rotation
+symmetry is explicitly broken. This makes them harder
+to realize in solid state systems. There are many pro-
+posals to realized the Kitaev model in more controllable
+situations, e.g. in cold atom optical lattices17,18, or in
+superconducting circuits19. But it is still desirable for
+theoretical curiosity and practical purposes to realize the
+Kitaev-type models in spin rotation invariant systems.
+In this paper we realize the Kitaev honeycomb lattice
+model as the low energy Hamiltonian for a spin rotation
+invariantsystem. The trickis notto use the physicalspin
+as the spin in the Kitaev model, instead the spin-1/2 in
+Kitaev model is from some emergent two-fold degener-
+ate low energy states in the elementary unit of physical
+system. This type of idea has been explored recently by
+Jackeli and Khaliullin20, in which the spin-1/2 in the Ki-
+taev model is the low energy Kramers doublet created by
+strong spin-orbit coupling of t2gorbitals. In the model
+presented below, the Hilbert space of spin-1/2 in the Ki-
+taev model is actually the two dimensional spin singlet
+sector of four antiferromagneticallycoupled spin-1/2 mo-
+ments, and the role of spin-1/2 operators(Paulimatrices)
+in the Kitaev model is replaced by certain combinations
+ofSj·Sk[or the spin-chirality Sj·(Sk×Sℓ)] between the
+four spins.
+One major drawback of the model to be presented is
+that it contains high order spin interactions(involves up
+to six or eight spins), thus is still unnatural. However it
+opens the possibility to realize exotic (exactly solvable)
+models from spin-1/2 Hamiltonian with spin rotation in-
+variant interactions. We will discuss two possible routes
+to reduce this artificialness through controlled perturba-
+tive expansions, by coupling to optical phonons or by
+magnetic couplings between the elementary units.
+The outline of this paper is as follows. In Section II
+we will lay out the pseudo-spin-1/2 construction. In Sec-4
+2 31z zx
+xy
+y2 3
+41
+FIG. 2: Left: the physical spin lattice for the model (8). The
+dash circles are honeycomb lattice sites, each of which is ac -
+tually a cluster of four physical spins. The dash straight li nes
+are honeycomb lattice bonds, with their type x,y,z labeled.
+The interaction between clusters connected by x,y,z bonds
+are theJx,y,z terms in (8) or (9) respectively. Note this is not
+the 3-12 lattice used in Ref.9,10. Right: enlarged picture of
+the clusters with the four physical spins labeled as 1 ,..., 4.
+Thick solid bonds within one cluster have large antiferroma g-
+netic Heisenberg coupling Jcluster .
+tion III the Kitaev model will be explicitly constructed
+using this formalism, and some properties of this con-
+struction will be discussed. In Section IV we will discuss
+two possible ways to generate the high order spin in-
+teractions involved in the construction of Section III by
+perturbative expansions. Conclusions and outlook will
+be summarized in Section V.
+II. FORMULATION OF THE PSEUDO-SPIN-1/2
+FROM FOUR-SPIN CLUSTER.
+In this Section we will construct the pseudo-spin-1/2
+from a cluster of four physical spins, and map the phys-
+ical spin operators to pseudo-spin operators. The map-
+ping constructed here will be used in later Sections to
+construct the effective Kitaev model. In this Section we
+will workentirely within the four-spin cluster, all unspec-
+ified physical spin subscripts take values 1 ,...,4.
+Consider a cluster of four spin-1/2 moments(called
+physical spins hereafter), labeled by S1,...,4, antiferro-
+magnetically coupled to each other (see the right bot-
+tom part of FIG. 2). The Hamiltonian within the clus-
+ter(up to a constant) is simply the Heisenberg antiferro-
+magnetic(AFM) interactions,
+Hcluster= (Jcluster/2)(S1+S2+S3+S4)2(2)
+The energy levels should be apparent from this form:
+one group of spin-2 quintets with energy 3 Jcluster, three
+groupsofspin-1tripletswithenergy Jcluster, andtwospin
+singlets with energy zero. We will consider large positive3
+Jclusterlimit. So only the singlet sector remains in low
+energy.
+The singlet sector is then treated as a pseudo-spin-1/2
+Hilbert space. From now on we denote the pseudo-spin-
+1/2 operators as T= (1/2)/vector τ, with/vector τthe Pauli matri-
+ces. It is convenient to choose the following basis of the
+pseudo-spin
+|τz=±1/angbracketright=1√
+6/parenleftBig
+| ↓↓↑↑/angbracketright+ω−τz| ↓↑↓↑/angbracketright+ωτz| ↓↑↑↓/angbracketright
++| ↑↑↓↓/angbracketright+ω−τz| ↑↓↑↓/angbracketright+ωτz| ↑↓↓↑/angbracketright/parenrightBig
+(3)
+whereω=e2πi/3is the complex cubic root of unity,
+| ↓↓↑↑/angbracketrightand other states on the right-hand-side(RHS) are
+basis states of the four-spin system, in terms of Szquan-
+tum numbers of physical spins 1 ,...,4 in sequential or-
+der. This pseudo-spin representation has been used by
+Harriset al.to study magnetic ordering in pyrochlore
+antiferromagnets21.
+We now consider the effect of Heisenberg-type inter-
+actionsSj·Skinside the physical singlet sector. Note
+that since any Sj·Skwithin the cluster commutes with
+the cluster Hamiltonian Hcluster(2), their action do not
+mix physical spin singlet states with states of other total
+physical spin. This property is also true for the spin-
+chirality operator used later. So the pseudo-spin Hamil-
+tonian constructed below will be exactlow energy Hamil-
+tonian, without truncation errors in typical perturbation
+series expansions.
+It is simpler to consider the permutation operators
+Pjk≡2Sj·Sk+ 1/2, which just exchange the states
+of the two physical spin-1/2 moments jandk(j/negationslash=k).
+As an example we consider the action of P34,
+P34|τz=−1/angbracketright=1√
+6/parenleftBig
+| ↓↓↑↑/angbracketright+ω| ↓↑↑↓/angbracketright+ω2| ↓↑↓↑/angbracketright
++| ↑↑↓↓/angbracketright+ω| ↑↓↓↑/angbracketright+ω2| ↑↓↑↓/angbracketright/parenrightBig
+=|τz= +1/angbracketright
+and similarly P34|τz=−1/angbracketright=|τz= +1/angbracketright. Therefore P34
+is justτxin the physical singlet sector. A complete list
+of all permutation operators is given in TABLE I. We
+can choose the following representation of τxandτy,
+τx=P12= 2S1·S2+1/2
+τy= (P13−P14)/√
+3 = (2/√
+3)S1·(S3−S4)(4)
+Many other representations are possible as well, because
+several physical spin interactions may correspond to the
+same pseudo-spin interaction in the physical singlet sec-
+tor, and we will take advantage of this later.
+Forτzwe can use τz=−iτxτy, whereiis the imagi-
+nary unit,
+τz=−i(2/√
+3)(2S1·S2+1/2)S1·(S3−S4) (5)physical spin pseudo-spin
+P12, andP34 τx
+P13, andP24 −(1/2)τx+ (√
+3/2)τy
+P14, andP23 −(1/2)τx−(√
+3/2)τy
+−χ234,χ341,−χ412, andχ123(√
+3/4)τz
+TABLE I: Correspondence between physical spin operators
+and pseudo-spin operators in the physical spin singlet sect or of
+the four antiferromagnetically coupled physical spins. Pjk=
+2Sj·Sk+1/2 are permutation operators, χjkℓ=Sj·(Sk×Sℓ)
+are spin-chirality operators. Note that several physical s pin
+operators may correspond to the same pseudo-spin operator.
+However there is another simpler representation of τz,
+by the spin-chirality operator χjkℓ=Sj·(Sk×Sℓ). Ex-
+plicit calculation shows that the effect of S2·(S3×S4) is
+−(√
+3/4)τzin the physical singlet sector. This can also
+be proved by using the commutation relation [ S2·S3,S2·
+S4] =iS2·(S3×S4). A complete list of all chirality
+operators is given in TABLE I. Therefore we can choose
+another representation of τz,
+τz=−χ234/(√
+3/4) =−(4/√
+3)S2·(S3×S4) (6)
+Theaboverepresentationsof τx,y,zareallinvariantunder
+global spin rotation of the physical spins.
+With the machinery of equations (4), (5), and (6), it
+will be straightforward to construct various pseudo-spin-
+1/2 Hamiltonians on various lattices, of the Kitaev vari-
+ety and beyond, as the exact low energy effective Hamil-
+tonianofcertainspin-1/2modelswith spin-rotationsym-
+metry. In these constructions a pseudo-spin lattice site
+actually represents a cluster of four spin-1/2 moments.
+III. REALIZATION OF THE KITAEV MODEL.
+In this Section we will use directly the results of the
+previous Section to write down a Hamiltonian whose low
+energy sector is described by the Kitaev model. The
+Hamiltonian will be constructed on the physical spin lat-
+tice illustrated in FIG. 2. In this Section we will use
+j,kto label four-spinclusters(pseudo-spin-1/2sites), the
+physical spins in cluster jare labeled as Sj1,...,Sj4.
+Apply the mappings developed in Section II, we have
+the desired Hamiltonian in short notation,
+H=/summationdisplay
+clusterHcluster−/summationdisplay
+x−links<jk>Jxτx
+jτx
+k
+−/summationdisplay
+y−links<jk>Jyτy
+jτy
+k−/summationdisplay
+z−links<jk>Jzτz
+jτz
+k(7)
+wherej,klabel the honeycomblattice sites thus the four-
+spin clusters, Hclusteris given by (2), τx,y,zshould be
+replaced by the corresponding physical spin operators in
+(4) and (5) or (6), or some other equivalent representa-
+tions of personal preference.4
+Plug in the expressions(4) and (6) into (7), the Hamil- tonian reads e xplicitly as
+H=/summationdisplay
+j(Jcluster/2)(Sj1+Sj2+Sj3+Sj4)2−/summationdisplay
+z−links<jk>Jz(16/9)[Sj2·(Sj3×Sj4)][Sk2·(Sk3×Sk4)]
+−/summationdisplay
+x−links<jk>Jx(2Sj1·Sj2+1/2)(2Sk1·Sk2+1/2)−/summationdisplay
+y−links<jk>Jy(4/3)[Sj1·(Sj3−Sj4)][Sk1·(Sk3−Sk4)]
+(8)
+While by the represenation (4) and (5), the Hamilto- nian becomes
+H=/summationdisplay
+j(Jcluster/2)(Sj1+Sj2+Sj3+Sj4)2
+−/summationdisplay
+x−links<jk>Jx(2Sj1·Sj2+1/2)(2Sk1·Sk2+1/2)−/summationdisplay
+y−links<jk>Jy(4/3)[Sj1·(Sj3−Sj4)][Sk1·(Sk3−Sk4)]
+−/summationdisplay
+z−links<jk>Jz(−4/3)(2Sj3·Sj4+1/2)[Sj1·(Sj3−Sj4)](2Sk3·Sk4+1/2)[Sk1·(Sk3−Sk4)]
+(9)
+This model, in terms of physical spins S, has full
+spin rotation symmetry and time-reversal symmetry. A
+pseudo-magnetic field term/summationtext
+j/vectorh·/vector τjterm can also be
+included under this mapping, however the resulting Ki-
+taev model with magnetic field is not exactly solvable.
+It is quite curious that such a formidably looking Hamil-
+tonian (8), with biquadratic and six-spin(or eight-spin)
+terms, has an exactly solvable low energy sector.
+We emphasize that because the first intra-cluster term/summationtext
+clusterHclustercommutes with the latter Kitaev terms
+independent ofthe representationused, theKitaevmodel
+is realized as the exactlow energy Hamiltonian of this
+model without truncationerrorsofperturbationtheories,
+namely no ( |Jx,y,z|/Jcluster)2or higher order terms will
+be generated under the projection to low energy clus-
+ter singlet space. This is unlike, for example, the t/U
+expansion of the half-filled Hubbard model22,23, where
+at lowest t2/Uorder the effective Hamiltonian is the
+Heisenberg model, but higher order terms ( t4/U3etc.)
+should in principle still be included in the low energy ef-
+fective Hamiltonian for any finite t/U. Similar compari-
+son can be made to the perturbative expansion studies of
+the Kitaev-type models by Vidal et al.9, where the low
+energy effective Hamiltonians were obtained in certian
+anisotropic (strong bond/triangle) limits. Although the
+spirit of this work, namely projection to low energy sec-
+tor, is the same as all previous perturbative approaches
+to effective Hamiltonians.Note that the original Kitaev model (1) has three-
+fold rotation symmetry around a honeycomb lattice site,
+combined with a three-fold rotation in pseudo-spin space
+(cyclic permutation of τx,τy,τz). This is not apparent
+in our model (8) in terms of physical spins, under the
+current representation of τx,y,z. We can remedy this by
+using a different set of pseudo-spin Pauli matrices τ′x,y,z
+in (7),
+τ′x=/radicalbig
+1/3τz+/radicalbig
+2/3τx,
+τ′y=/radicalbig
+1/3τz−/radicalbig
+1/6τx+/radicalbig
+1/2τy,
+τ′z=/radicalbig
+1/3τz−/radicalbig
+1/6τx−/radicalbig
+1/2τy
+With proper representation choice, they have a symmet-
+ric form in terms of physical spins,
+τ′x=−(4/3)S2·(S3×S4)+/radicalbig
+2/3(2S1·S2+1/2)
+τ′y=−(4/3)S3·(S4×S2)+/radicalbig
+2/3(2S1·S3+1/2)
+τ′z=−(4/3)S4·(S2×S3)+/radicalbig
+2/3(2S1·S4+1/2)
+(10)
+So the symmetry mentioned above can be realized by a
+three-foldrotationofthe honeycomblattice, with acyclic
+permutation of S2,S3andS4in each cluster. This is in
+fact the three-foldrotationsymmetry ofthe physicalspin
+lattice illustrated in FIG. 2. However this more symmet-
+ric representation will not be used in later part of this
+paper.5
+Another note to take is that it is not necessary to have
+such a highly symmetric cluster Hamiltonian (2). The
+mappings to pseudo-spin-1/2 should work as long as the
+ground states of the cluster Hamiltonian are the two-fold
+degenerate singlets. One generalization, which conforms
+the symmetry of the lattice in FIG. 2, is to have
+Hcluster= (Jcluster/2)(r·S1+S2+S3+S4)2(11)
+withJcluster>0 and 0 < r <3. However this is not
+convenient for later discussions and will not be used.
+We briefly describe some of the properties of (8). Its
+lowenergystatesareentirelyinthespacethateachofthe
+clusters is a physical spin singlet (called cluster singlet
+subspace hereafter). Therefore physical spin correlations
+are strictly confined within each cluster. The excitations
+carrying physical spin are gapped, and their dynamics
+are ‘trivial’ in the sense that they do not move from one
+cluster to another. But there are non-trivial low energy
+physicalspinsingletexcitations,describedbythepseudo-
+spins defined above. The correlationsof the pseudo-spins
+can be mapped to correlations of their corresponding
+physical spin observables (the inverse mappings are not
+unique, c.f. TABLE I). For example τx,ycorrelations
+become certain dimer-dimer correlations, τzcorrelation
+becomes chirality-chiralitycorrelation, or four-dimer cor-
+relation. It will be interesting to see the corresponding
+picture of the exotic excitations in the Kitaev model, e.g.
+the Majoranafermion and the Ising vortex. Howeverthis
+will be deferred to future studies.
+It is tempting to call this as an exactly solved spin liq-
+uid with spin gap ( ∼Jcluster), an extremely short-range
+resonating valence bond(RVB) state, from a model with
+spin rotation and time reversal symmetry. However it
+should be noted that the unit cell of this model contains
+an even number of spin-1/2 moments (so does the orig-
+inal Kitaev model) which does not satisfy the stringent
+definition of spin liquid requiring odd number of elec-
+trons per unit cell. Several parent Hamiltonians of spin
+liquids have already been constructed. See for example,
+Ref.24–27.
+IV. GENERATE THE HIGH ORDER PHYSICAL
+SPIN INTERACTIONS BY PERTURBATIVE
+EXPANSION.
+Onemajordrawbackofthepresentconstructionisthat
+it involves high order interactions of physical spins[see
+(8) and (9)], thus is ‘unnatural’. In this Section we will
+make compromises between exact solvability and natu-
+ralness. We consider two clusters jandkand try to
+generate the Jx,y,zinteractions in (7) from perturbation
+series expansion of more natural(lower order) physical
+spin interactions. Two different approaches for this pur-
+pose will be laid out in the following two Subsections. In
+Subsection IVA we will consider the two clusters as two
+tetrahedra, and couple the spin system to certain opti-
+cal phonons, further coupling between the phonon modes(a) (b) (c) (d)
+(b) (c) (d)QE
+2QE
+1
+(a)11 1 1
+1 1 12
+22
+22
+2 22
+33
+3 3 3
+3 34 44 4
+3
+44
+44
+1
+FIG. 3: Illustration of the tetragonal to orthorhombic
+QE
+1(top) and QE
+2(bottom) distortion modes. (a) Perspective
+view of the tetrahedron. 1 ,..., 4 label the spins. Arrows in-
+dicate the motion of each spin under the distortion mode. (b)
+Top view of (a). (c)(d) Side view of (a).
+of the two clusters can generate at lowest order the de-
+sired high order spin interactions. In Subsection IVB we
+will introduce certain magnetic, e.g. Heisenberg-type, in-
+teractions between physical spins of different clusters, at
+lowestorder(secondorder)of perturbation theory the de-
+sired high order spin interactions can be achieved. These
+approaches involve truncation errors in the perturbation
+series, thus the mapping to low energy effect Hamilto-
+nian will no longer be exact. However the error intro-
+duced may be controlled by small expansion parameters.
+In this Section we denote the physical spins on cluster
+j(k) asj1,...,j4 (k1,...,k4), and denote pseudo-spins
+on cluster j(k) as/vector τj(/vector τk).
+A. Generate the High Order Terms by Coupling to
+Optical Phonon.
+In this Subsection we regard each four-spin cluster
+as a tetrahedron, and consider possible optical phonon
+modes(distortions) and their couplings to the spin sys-
+tem. The basic idea is that the intra-cluster Heisen-
+berg coupling Jclustercan linearly depend on the dis-
+tance between physical spins. Therefore certain distor-
+tions of the tetrahedron couple to certain linear combi-
+nations of Sℓ·Sm. Integrating out phonon modes will
+then generate high order spin interactions. This idea has
+been extensively studied and applied to several magnetic
+materials28–34. More details can be found in a recent
+review by Tchernyshyov and Chern35. And we will fre-
+quently use their notations. In this Subsection we will
+use the representation (5) for τz.
+Consider first a single tetrahedron with four spins
+1,...,4. The general distortions of this tetrahedron can
+be classified by their symmetry (see for example Ref.35).
+Only two tetragonal to orthorhombic distortion modes,
+QE
+1andQE
+2(illustrated in FIG. 3), couple to the pseudo-
+spins defined in Section II. A complete analysis of all
+modes is given in Appendix A. The coupling is of the6
+form
+J′(QE
+1fE
+1+QE
+2fE
+2)
+whereJ′is the derivative of Heisenberg coupling Jcluster
+between two spins ℓandmwith respect to their distance
+rℓm,J′= dJcluster/drℓm;QE
+1,2are the generalized coor-
+dinates of these two modes; and the functions fE
+1,2are
+fE
+2= (1/2)(S2·S4+S1·S3−S1·S4−S2·S3),
+fE
+1=/radicalbig
+1/12(S1·S4+S2·S3+S2·S4+S1·S3
+−2S1·S2−2S3·S4).
+According to TABLE I we have fE
+1=−(√
+3/2)τxand
+fE
+2= (√
+3/2)τy. Then the coupling becomes
+(√
+3/2)J′(−QE
+1τx+QE
+2τy) (12)
+The spin-lattice(SL) Hamiltonian on a single cluster j
+is [equation (1.8) in Ref.35],
+Hcluster j,SL=Hcluster j+k
+2(QE
+1j)2+k
+2(QE
+2j)2
+−√
+3
+2J′(QE
+1jτx
+j−QE
+2jτy
+j),(13)
+wherek >0 is the elastic constant for these phonon
+modes,J′is the spin-lattice coupling constant, QE
+1jand
+QE
+2jare the generalized coordinates of the QE
+1andQE
+2
+distortion modes of cluster j,Hcluster jis (2). As al-
+ready noted in Ref.35, this model does not really break
+the pseudo-spin rotation symmetry of a single cluster.
+Now we put two clusters jandktogether, and in-
+clude a perturbation λHperturbation to the optical phonon
+Hamiltonian,
+Hjk,SL=Hcluster j,SL+Hcluster k,SL
++λHperturbation [QE
+1j,QE
+2j,QE
+1k,QE
+2k]whereλ(in factλ/k) is the expansion parameter.
+Consider the perturbation Hperturbation =QE
+1j·QE
+1k,
+which means a coupling between the QE
+1distortion
+modes of the two tetrahedra. Integrate out the optical
+phonons, at lowest non-trivial order, it produces a term
+(3J′2λ)/(4k2)τx
+j·τx
+k. This can be seen by minimizing
+separately the two cluster Hamiltonians with respect to
+QE
+1, which gives QE
+1= (√
+3J′)/(2k)τx, then plug this
+into the perturbation term. Thus we have produced the
+Jxterm in the Kitaev model with Jx=−(3J′2λ)/(4k2).
+Similarly the perturbation Hperturbation =QE
+2j·QE
+2k
+will generate (3 J′2λ)/(4k2)τy
+j·τy
+kat lowest non-trivial
+order. So we can make Jy=−(3J′2λ)/(4k2).
+Theτz
+j·τz
+kcoupling is more difficult to get. We
+treat it as −τx
+jτy
+j·τx
+kτy
+k. By the above reasoning, we
+need an anharmonic coupling Hperturbation =QE
+1jQE
+2j·
+QE
+1kQE
+2k. It will produce at lowest non-trivial or-
+der (9J′4λ)/(16k4)τx
+jτy
+j·τx
+kτy
+k. Thus we have Jz=
+(9J′4λ)/(16k4).
+Finally we have made up a spin-lattice model HSL,
+which involves only Sℓ·Sminteraction for physical spins,
+HSL=/summationdisplay
+clusterHcluster,SL+/summationdisplay
+x−links<jk>λxQE
+1j·QE
+1k
++/summationdisplay
+y−links<jk>λyQE
+2j·QE
+2k
++/summationdisplay
+z−links<jk>λzQE
+1jQE
+2j·QE
+1kQE
+2k
+whereQE
+1jis the generalized coordinate for the QE
+1mode
+on cluster j, andQE
+1k,QE
+2j,QE
+2kare similarly defined;
+λx,y=−(4Jx,yk2)/(3J′2) andλz= (16Jzk4)/(9J′4); the
+single cluster spin-lattice Hamiltonian Hcluster,SLis (13).
+Collect the results above we have the spin-lattice
+Hamiltonian HSLexplicitly written as,
+HSL=/summationdisplay
+cluster j/bracketleftBig
+(Jcluster/2)(Sj1+Sj2+Sj3+Sj4)2+k
+2(QE
+1j)2+k
+2(QE
+2j)2
++J′/parenleftBig
+QE
+1jSj1·Sj4+Sj2·Sj3+Sj2·Sj4+Sj1·Sj3−2Sj1·Sj2−2Sj3·Sj4√
+12
++QE
+2jSj2·Sj4+Sj1·Sj3−Sj1·Sj4−Sj2·Sj3
+2/parenrightBig/bracketrightBig
+−/summationdisplay
+x−links<jk>4Jxk2
+3J′2QE
+1j·QE
+1k−/summationdisplay
+y−links<jk>4Jyk2
+3J′2QE
+2j·QE
+2k+/summationdisplay
+z−links<jk>16Jzk4
+9J′4QE
+1jQE
+2j·QE
+1kQE
+2k(14)
+The single cluster spin-lattice Hamiltonian [first three
+lines in (14)] is quite natural. However we need someharmonic(on x-andy-linksofhoneycomblattice)andan-
+harmonic coupling (on z-links) between optical phonon7
+modes of neighboring tetrahedra. And these coupling
+constants λx,y,zneed to be tuned to produce Jx,y,zof
+the Kitaev model. This is still not easy to implement in
+solid state systems. At lowest non-trivialorder of pertur-
+bative expansion, we do get our model (9). Higher order
+terms in expansion destroy the exact solvability, but may
+be controlled by the small parameters λx,y,z/k.
+B. Generate the High Order Terms by Magnetic
+Interactions between Clusters.
+In this Subsection we consider more conventional per-
+turbations, magnetic interactions between the clusters,
+e.g. the Heisenberg coupling Sj·Skwithjandkbelong
+to different tetrahedra. This has the advantage over the
+previous phonon approach for not introducing additional
+degrees of freedom. But it also has a significant disad-
+vantage: the perturbation does not commute with the
+cluster Heisenberg Hamiltonian (2), so the cluster sin-
+glet subspace will be mixed with other total spin states.
+In this Subsection we will use the spin-chirality represen-
+tation (6) for τz.
+Again consider two clusters jandk. For simplicity
+of notations define a projection operator Pjk=PjPk,
+wherePj,kis projection into the singlet subspace of clus-
+terjandk, respectively, Pj,k=/summationtext
+s=±1|τz
+j,k=s/angbracketright/angbracketleftτz
+j,k=
+s|. For a given perturbation λHperturbation with small
+parameter λ(in factor λ/Jclusteris the expansion param-
+eter), lowest two orders of the perturbation series are
+λPjkHperturbation Pjk+λ2PjkHperturbation (1−Pjk)
+×[0−Hcluster j−Hcluster k]−1(1−Pjk)Hperturbation Pjk
+(15)
+With properchoiceof λandHperturbation we cangeneratethe desired Jx,y,zterms in (8) from the first and second
+order of perturbations.
+The calculation can be dramatically simplified by the
+following fact that any physical spin-1/2 operator Sx,y,z
+ℓ
+converts the cluster spin singlet states |τz=±1/angbracketrightinto
+spin-1 states of the cluster. This can be checked by
+explicit calculations and will not be proved here. For
+all the perturbations to be considered later, the above
+mentioned fact can be exploited to replace the factor
+[0−Hcluster j−Hcluster k]−1in the second order pertur-
+bation to a c-number ( −2Jcluster)−1.
+The detailed calculationsare given in Appendix B. We
+will only list the results here.
+The perturbation on x-links is given by
+λxHperturbation , x=λx[Sj1·Sk1+sgn(Jx)·(Sj2·Sk2)]
+−Jx(Sj1·Sj2+Sk1·Sk2).
+whereλx=/radicalbig
+12|Jx|·Jcluster, sgn(Jx) =±1 is the sign
+ofJx.
+The perturbation on y-links is
+λyHperturbation , y
+=λy[Sj1·Sk1+sgn(Jy)·(Sj3−Sj4)·(Sk3−Sk4)]
+−|Jy|(Sj3·Sj4+Sk3·Sk4)
+withλy=/radicalbig
+4|Jy|·Jcluster.
+The perturbation on z-links is
+λzHperturbation , z
+=λz[Sj2·(Sk3×Sk4)+sgn(Jz)·Sk2·(Sj3×Sj4)]
+−|Jz|(Sj3·Sj4+Sk3·Sk4).
+withλz= 4/radicalbig
+|Jz|·Jcluster.
+The entire Hamiltonian Hmagneticreads explicitly as,
+Hmagnetic=/summationdisplay
+cluster j(Jcluster/2)(Sj1+Sj2+Sj3+Sj4)2
++/summationdisplay
+x−links<jk>/braceleftbig/radicalbig
+12|Jx|·Jcluster/bracketleftbig
+Sj1·Sk1+sgn(Jx)·(Sj2·Sk2)/bracketrightbig
+−Jx(Sj1·Sj2+Sk1·Sk2)/bracerightbig
++/summationdisplay
+y−links<jk>/braceleftbig/radicalBig
+4|Jy|·Jcluster/bracketleftbig
+Sj1·(Sk3−Sk4)+sgn(Jy)Sk1·(Sj3−Sj4)/bracketrightbig
+−|Jy|(Sj3·Sj4+Sk3·Sk4)/bracerightbig
++/summationdisplay
+z−links<jk>/braceleftbig
+4/radicalbig
+|Jz|·Jcluster/bracketleftbig
+Sj2·(Sk3×Sk4)+sgn(Jz)Sk2·(Sj3×Sj4)/bracketrightbig
+−|Jz|(Sj3·Sj4+Sk3·Sk4)/bracerightbig
+.(16)
+In (16), we have been able to reduce the four spin in-
+teractions in (8) to inter-cluster Heisenberg interactions,
+and the six-spin interactions in (8) to inter-cluster spin-
+chirality interactions. The inter-cluster Heisenberg cou-
+plings in Hperturbation x,ymay be easier to arrange. Theinter-cluster spin-chirality coupling in Hperturbation zex-
+plicitly breaks time reversal symmetry and is probably
+harder to implement in solid state systems. However
+spin-chirality order may have important consequences
+in frustrated magnets36,37, and a realization of spin-8
+chirality interactions in cold atom optical lattices has
+been proposed38.
+Our model (8) is achieved at second order of the per-
+turbation series. Higher order terms become trunca-
+tion errors but may be controlled by small parameters
+λx,y,z/Jcluster∼/radicalbig
+|Jx,y,z|/Jcluster.
+V. CONCLUSIONS.
+WeconstructedtheexactlysolvableKitaevhoneycomb
+model1as the exact low energy effective Hamiltonian of
+a spin-1/2model [equations (8) or (9)] with spin-rotation
+and time reversal symmetry. The spin in Kitaev model is
+representedasthe pseudo-spin in the two-folddegenerate
+spin singlet subspace of a cluster of four antiferromag-
+netically coupled spin-1/2 moments. The physical spin
+model is a honeycomb lattice of such four-spin clusters,
+with certain inter-cluster interactions. The machinery
+for the exact mapping to pseudo-spin Hamiltonian was
+developed (see e.g. TABLE I), which is quite general
+and can be used to construct other interesting (exactly
+solvable) spin-1/2 models from spin rotation invariant
+systems.
+In this construction the pseudo-spin correlationsin the
+Kitaev model will be mapped to dimer or spin-chirality
+correlationsin the physical spin system. The correspond-
+ing picture of the fractionalized Majorana fermion exci-
+tations and Ising vortices still remain to be clarified.
+This exact construction contains high order physical
+spin interactions, which is undesirable for practical im-
+plementation. We described two possible approaches to
+reduce this problem: generating the high order spin in-
+teractions by perturbative expansion of the coupling to
+optical phonon, or the magnetic coupling between clus-
+ters. This perturbative construction will introduce trun-
+cation error of perturbation series, which may be con-
+trolled by small expansion parameters. Whether these
+constructions can be experimentally engineered is how-
+everbeyond the scope ofthis study. It is conceivablethat
+otherperturbativeexpansioncanalsogeneratethesehigh
+order spin interactions, but this possibility will be left for
+future works.
+Acknowledgments
+The author thanks Ashvin Vishwanath, Yong-Baek
+KimandArun Paramekantiforinspiringdiscussions, and
+TodadriSenthil forcriticalcomments. The authoris sup-
+ported by the MIT Pappalardo Fellowship in Physics.
+Appendix A: Coupling between Distortions of a
+Tetrahedron and the Pseudo-spins
+In this Appendix we reproduce from Ref.35the cou-
+plings of all tetrahedron distortion modes to the spinsystem. And convert them to pseudo-spin notation in
+the physical spin singlet sector.
+Consider a general small distortion of the tetrahedron,
+the spin Hamiltonian becomes
+Hcluster,SL= (Jcluster/2)(/summationdisplay
+ℓSℓ)2+J′/summationdisplay
+ℓ<mδrℓm(Sℓ·Sm)
+(A1)
+whereδrℓmis the change of bond length between spins
+ℓandm,J′is the derivative of Jclusterwith respect to
+bond length.
+There are six orthogonaldistortion modes of the tetra-
+hedron [TABLE 1.1 in Ref.35]. One of the modes Ais the
+trivial representation of the tetrahedral group Td; twoE
+modes form the two dimensional irreducible representa-
+tion ofTd; and three T2modes form the three dimen-
+sional irreducible representation. The Emodes are also
+illustrated in FIG. 3.
+The generic couplings in (A1) [second term] can be
+converted to couplings to these orthogonal modes,
+J′(QAfA+QE
+1fE
+1+QE
+2fE
+2+QT2
+1fT2
+1+QT2
+2fT2
+2+QT2
+3fT2
+3)
+whereQaregeneralizedcoordinatesofthe corresponding
+modes, functions fcan be read off from TABLE 1.2 of
+Ref.35. For the Amode,δrℓm=/radicalbig
+2/3QA, sofAis
+fA=/radicalbig
+2/3(S1·S2+S3·S4+S1·S3
++S2·S4+S1·S4+S2·S3).
+Thefunctions fE
+1,2fortheEmodeshavebeengivenbefore
+but are reproduced here,
+fE
+2= (1/2)(S2·S4+S1·S3−S1·S4−S2·S3),
+fE
+1=/radicalbig
+1/12(S1·S4+S2·S3+S2·S4+S1·S3
+−2S1·S2−2S3·S4).
+The functions fT2
+1,2,3for theT2modes are
+fT2
+1= (S2·S3−S1·S4),
+fT2
+2= (S1·S3−S2·S4),
+fT2
+3= (S1·S2−S3·S4)
+Now we can use TABLE I to convert the above cou-
+plings into pseudo-spin. It is easy to see that fAand
+fT2
+1,2,3areallzerowhenconvertedtopseudo-spins, namely
+projected to the physical spin singlet sector. But fE
+1=
+(P14+P23+P24+P13−2P12−2P34)/(4√
+3) =−(√
+3/2)τx
+andfE
+2= (P24+P13−P14−P23)/4 = (√
+3/2)τy. This
+has already been noted by Tchernyshyov et al.28, only
+theEmodes can lift the degeneracy of the physical spin
+singlet ground states of the tetrahedron. Therefore the
+general spin lattice coupling is the form of (12) given in
+the main text.9
+Appendix B: Derivation of the Terms Generated by
+Second Order Perturbation of Inter-cluster
+Magnetic Interactions
+In this Appendix we derive the second order pertur-
+bations of inter-cluster Heisenberg and spin-chirality in-
+teractions. The results can then be used to construct
+(16).
+Firstconsiderthe perturbation λHperturbation =λ[Sj1·
+Sk1+r(Sj2·Sk2)], where ris a real number to be tuned
+later. Due to the fact mentioned in Subsection IVB,
+the action of Hperturbation on any cluster singlet state
+will produce a state with total spin-1 for both cluster j
+andk. Thus the first order perturbation in (15) van-
+ishes. And the second order perturbation term can be
+greatly simplified: operator (1 − Pjk)[0−Hcluster j−
+Hcluster k]−1(1− Pjk) can be replaced by a c-number
+(−2Jcluster)−1. Therefore the perturbation up to second
+order is
+−λ2
+2JclusterPjk(Hperturbation )2Pjk
+This is true for other perturbations considered later in
+this Appendix. The cluster jand cluster kparts can be
+separated, this term then becomes ( a,b=x,y,z),
+−λ2
+2Jcluster/summationdisplay
+a,b/bracketleftbig
+PjSa
+j1Sb
+j1Pj·PkSa
+k1Sb
+k1Pk
++2rPjSa
+j1Sb
+j2Pj·PkSa
+k1Sb
+k2Pk
++r2PjSa
+j2Sb
+j2Pj·PkSa
+k2Sb
+k2Pk/bracketrightbig
+Then use the fact that PjSa
+jℓSb
+jmPj=δab(1/3)Pj(Sjℓ·
+Sjm)Pjby spin rotation symmetry, the perturbation be-
+comes
+−λ2
+6Jcluster/bracketleftBig9+9r2
+16+2rPjk(Sj1·Sj2)(Sk1·Sk2)Pjk/bracketrightBig
+=−λ2
+6Jcluster/bracketleftBig9+9r2
+16+(r/2)τx
+jτx
+k−r/2
+−rPjk(Sj1·Sj2+Sk1·Sk2)Pjk/bracketrightBig
+.
+Sowecanchoose −(rλ2)/(12Jcluster) =−Jx, andinclude
+the last intra-cluster Sj1·Sj2+Sk1·Sk2term in the first
+order perturbation.
+The perturbation on x-links is then (not unique),
+λxHperturbation , x=λx[Sj1·Sk1+sgn(Jx)·(Sj2·Sk2)]
+−Jx(Sj1·Sj2+Sk1·Sk2)
+withλx=/radicalbig
+12|Jx|·Jcluster, andr= sgn(Jx) is the sign
+ofJx. The non-trivial terms produced by up to second
+order perturbation will be the τx
+jτx
+kterm. Note that the
+last term in the above equation commutes with cluster
+Hamiltonians so it does not produce second or higher
+order perturbations.Similarly considering the following perturbation on y-
+links,λHperturbation =λ[Sj1·(Sk3−Sk4)+rSk1·(Sj3−
+Sj4)]. Following similar procedures we get the second
+order perturbation from this term
+−λ2
+6Jcluster/bracketleftBig9+9r2
+8
++2rPjk[Sj1·(Sj3−Sj4)][Sk1·(Sk3−Sk4)]Pjk
+−(3/2)Pjk(Sk3·Sk4+r2Sj3·Sj4)Pjk/bracketrightBig
+=−λ2
+6Jcluster/bracketleftBig9+9r2
+8+2r(3/4)τy
+jτy
+k
+−(3/2)Pjk(Sk3·Sk4+r2Sj3·Sj4)Pjk/bracketrightBig
+So we can choose −(rλ2)/(4Jcluster) =−Jy, and include
+the last intra-cluster Sk3·Sk4+r2Sj3·Sj4term in the
+first order perturbation.
+Therefore we can choose the following perturbation on
+y-links (not unique),
+λyHperturbation , y
+=λy[Sj1·Sk1+sgn(Jy)·(Sj3−Sj4)·(Sk3−Sk4)]
+−|Jy|(Sj3·Sj4+Sk3·Sk4)
+withλy=/radicalbig
+4|Jy|·Jcluster,r= sgn(Jy) is the sign of Jy.
+Theτz
+jτz
+kterm is again more difficult to get. We use
+the representation of τzby spin-chirality (6). And con-
+sider the following perturbation
+Hperturbation =Sj2·(Sj3×Sj4)+rSk2·(Sj3×Sj4)
+The first order term in (15) vanishes due to the same
+reason as before. There are four terms in the second
+order perturbation. The first one is
+λ2PjkSj2·(Sk3×Sk4)(1−Pjk)
+×[0−Hcluster j−Hcluster k]−1
+×(1−Pjk)Sj2·(Sk3×Sk4)Pjk
+For the cluster jpart we can use the same arguments
+as before, the Hcluster jcan be replaced by a c-number
+Jcluster. For the cluster kpart, consider the fact that
+Sk3×Sk4equals to the commutator −i[Sk4,Sk3·Sk4],
+the action of Sk3×Sk4on physical singlet states of kwill
+also only produce spin-1 state. So we can replace the
+Hcluster kin the denominator by a c-number Jclusteras
+well. Use spin rotation symmetry to separate the jand
+kparts, this term simplifies to
+−λ2
+6JclusterPjSj2·Sj2Pj·Pk(Sk3×Sk4)·(Sk3×Sk4)Pk.
+Use (S)2= 3/4 and
+(Sk3×Sk4)·(Sk3×Sk4)
+=/summationdisplay
+a,b(Sa
+k3Sb
+k4Sa
+k3Sb
+k4−Sa
+k3Sb
+k4Sb
+k3Sa
+k4)
+= (Sk3·Sk3)(Sk4·Sk4)−/summationdisplay
+a,bSa
+k3Sb
+k3[δab/2−Sa
+k4Sb
+k4]
+= 9/16+(Sk3·Sk4)(Sk3·Sk4)−(3/8)10
+this term becomes
+−λ2
+6Jcluster·(3/4)[3/16+(τx/2−1/4)2]
+=−(λ2)/(32Jcluster)·(2−τx
+k).
+Another second order perturbation term r2λ2PjkSk2·
+(Sj3×Sj4)(1− Pjk)[0−Hcluster j−Hcluster k]−1(1−
+Pjk)Sk2·(Sj3×Sj4)Pjkcan be computed in the similar
+way and gives the result −(r2λ2)/(32Jcluster)·(2−τx
+j).
+For one of the cross term
+rλ2PjkSj2·(Sk3×Sk4)(1−Pjk)
+×[0−Hcluster j−Hcluster k]−1
+×(1−Pjk)Sk2·(Sj3×Sj4)Pjk
+We can use the previous argument for both cluster jand
+k, so(1−PAB)[0−Hcluster j−Hcluster k]−1(1−Pjk)canbe
+replace by c-number ( −2Jcluster)−1. This term becomes
+−rλ2
+2JclusterPjk[Sj2·(Sk3×Sk4)][Sk2·(Sj3×Sj3)]Pjk.
+Spinrotationsymmetryagainhelpstoseparatetheterms
+for cluster jandk, and we get −(rλ2)/(32Jcluster)·τz
+jτz
+k.
+The other cross term rλ2PjkSk2·(Sj3×Sj4)(1−
+Pjk)[0−Hcluster j−Hcluster k]−1(1− Pjk)Sj2·(Sk3×
+Sk4)Pjkgives the same result.
+In summary the second orderperturbation from λ[Sj2·
+(Sj3×Sj4)+rSk2·(Sj3×Sj4)] is
+−rλ2
+16Jcluster·τz
+jτz
+k+λ2
+32Jcluster(τx
+k+r2τx
+j−2r2−2).Using this result we can choose the following pertur-
+bation on z-links,
+λzHperturbation , z
+=λz[Sj2·(Sk3×Sk4)+sgn(Jz)·Sk2·(Sj3×Sj4)]
+−|Jz|(Sj3·Sj4+Sk3·Sk4)
+withλz= 4/radicalbig
+|Jz|Jcluster,r= sgn(Jz) is the sign of Jz.
+The last term on the right-hand-sideis to cancel the non-
+trivial terms ( r2τx
+j+τx
+k)λ2
+z/(32Jcluster) from the second
+order perturbation of the first term. Up to second order
+perturbation this will produce −Jzτz
+jτz
+kinteractions.
+Finally we have been able to reduce the high order
+interactionsto at most three spin terms, the Hamiltonian
+Hmagneticis
+Hmagnetic=/summationdisplay
+jHcluster j+/summationdisplay
+x−links<jk>λxHperturbation x
++/summationdisplay
+y−links<jk>λyHperturbation y
++/summationdisplay
+z−links<jk>λzHperturbation z
+whereHcluster jare given by (2), λx,y,zHperturbation x,y,z
+are given above. Plug in relevant equations we get (16)
+in Subsection IVB.
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+ApJL Revised: 26 Jan 1  Kepler Mission Design, Realized Photometric Performance, and Early Science   David G. Koch1, William J. Borucki1, Gibor Basri2, Natalie M. Batalha3, Timothy M. Brown4, Douglas Caldwell5, Jørgen Christensen-Dalsgaard6, William D. Cochran7, Edna DeVore5, Edward W. Dunham8, Thomas N. Gautier III9, John C. Geary10, Ronald L. Gilliland11, Alan Gould12, Jon Jenkins5, Yoji Kondo13, David W. Latham10, Jack J. Lissauer1, Geoffrey Marcy2, David Monet14, Dimitar Sasselov10, Alan Boss15, Donald Brownlee16, John Caldwell17, Andrea K. Dupree10, Steve B. Howell18, Hans Kjeldsen6, Søren Meibom10, David Morrison1, Tobias Owen19, Harold Reitsema20, Jill Tarter5, Stephen T. Bryson1, Jessie L. Dotson1, Paul Gazis5, Michael R. Haas1, Jeffrey Kolodziejczak21, Jason F. Rowe1,23, Jeffrey E. Van Cleve5, Christopher Allen22, Hema Chandrasekaran5, Bruce D. Clarke5, Jie Li5, Elisa V. Quintana5, Peter Tenenbaum5, Joseph D. Twicken5, and Hayley Wu5  1NASA Ames Research Center, Moffett Field, CA 94035 2Dept. of Astronomy, University of California-Berkeley, Berkeley, CA 94720 3Dept. of Physics and Astronomy, San Jose State University, San Jose, CA 95192 4Las Cumbres Observatory Global Telescope, Goleta, CA 93117 5SETI Institute, NASA Ames Research Center, Moffett Field, CA 94035 6Aarhus University, DK-8000, Aarhus C, Denmark 7McDonald Observatory, University of Texas at Austin, Austin, TX 78712 8Lowell Observatory, Flagstaff, AZ 86001 9Jet Propulsion Laboratory/California Institute of Technology, Pasadena, CA 91109 10Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 11Space Telescope Science Institute, Baltimore, MD 21218 12Lawarence Hall of Science, Univ. California-Berkeley, Berkeley, CA 94720 13NASA Goddard Space Flight Center, Greenbelt, MD 20771 14United States Naval Observatory, Flagstaff, AZ 86002 15Carneige Institution of Washington, Washington, DC 20015 16Dept of Astronomy, University of Washington, Seattle, WA 98195 17 Dept of Physics and Astronomy, York University, Toronto, ON M3J 1P3, Canada 18National Optical Astronomy Observatory, Tucson, AZ 85726 19Institute for Astronomy, University of Hawaii, Honolulu, HI 96822 20Ball Aerospace and Technologies Corp., Boulder, CO, 80306 21 Space Science Office, NASA Marshall Space Flight Center, Huntsville, AL 35812 22Orbital Sciences Corp., NASA Ames Research Center, Moffett Field, CA 94035 23NASA Postdoctoral Program fellow,   Electronic address: D.Koch@NASA.govApJL Revised: 26 Jan 2 Abstract The Kepler Mission, launched on Mar 6, 2009 was designed with the explicit capability to detect Earth-size planets in the habitable zone of solar-like stars using the transit photometry method. Results from just forty-three days of data along with ground-based follow-up observations have identified five new transiting planets with measurements of their masses, radii, and orbital periods. Many aspects of stellar astrophysics also benefit from the unique, precise, extended and nearly continuous data set for a large number and variety of stars. Early results for classical variables and eclipsing stars show great promise. To fully understand the methodology, processes and eventually the results from the mission, we present the underlying rationale that ultimately led to the flight and ground system designs used to achieve the exquisite photometric performance. As an example of the initial photometric results, we present variability measurements that can be used to distinguish dwarf stars from red giants.  Subject headings: Instrumentation: photometers – planetary systems – space vehicles: instruments – stars: statistics – stars: variables – techniques: photometric Facilities: Kepler Mission      1. Introduction The foremost purpose of the Kepler Mission is to detect Earth-size planets in the habitable zone (Kasting et al.1993) of solar-like stars (F- to K-dwarfs), determine their frequency, and identify their characteristics. The method of choice is transit photometry (Pont et al. 2009), which provides the orbital period and size of the planet relative to its star. When combined with stellar parameters and radial velocity measurements, the mass, radius and density of the planet are obtained. Transit photometry requires high photometric precision with continuous time series data of a large number of stars over an extended period of time. Although designed for the explicit purpose of terrestrial planet detection, the nature of the Kepler data set is also of tremendous value for stellar astrophysics. Asteroseismology (Stello et al. 2009), gyrochronology (Barnes 2003), and astrometry, the studies of solar-like stars (Basri et al. 2010, Chaplin et al. 2010), eclipsing binaries (Gimenez et al. 2006), and classical variables (Gautschy & Saio 1996) all depend on precise, extended, continuous flux time series. Results from many of these fields in turn tie back into the interpretation of exoplanet science.  Kepler was launched on Mar 6, 2009, commissioned in sixty-seven days and began science operations on May 13. In this issue of Astrophysical Journal-Letters, we present results from the first two data sets: Q0 consisting of 9.7 days of data taken during commissioning and Q1 consisting of 33.5 days of data taken before the first quarterly roll of the spacecraft. The Q0 data are particularly unique and interesting, since all spectral types and luminosity classes of stars with V<13.6 were included.   2. Fundamental requirements ApJL Revised: 26 Jan 3 Borucki et al. (2008) have described the scientific goals and objectives of the Kepler Mission. Overviews of much of the hardware have been given elsewhere (Koch et al. 2004). This section describes how the scientific goals formed the basis for the design of the mission.   2.1 Mission duration The fundamental requirement for mission success is to reliably detect transits of Earth-size planets. To have confidence that the signatures are of planets, we require a sequence of at least three transits, all with a consistent period, brightness change and duration. To be able to detect three transits of a planet in the habitable zone of a solar-like star, led to the first requirement, a mission length of at least three years.  2.2 Number of stars to observe Transit photometry requires that the orbital plane of the exoplanet must be aligned along our line-of-sight to the exoplanet's parent star. Let θ be the angle between the line-of-sight and the orbital plane for a planet that just grazes the edge of its star. Then sin(θ)=R/a, where R is the sum of the radii of the star and the planet and a is the semi-major axis of the orbit, taken to be nearly circular. The solid angle integrated over the sky for all orbital pole positions where the planet can be seen transiting is 4πsin(θ). Dividing by the total area of the sky (4π) and substituting in for sin(θ) yields  ϕ=R/a       (1) as the geometric random probability for alignment of a transit, with no small angle approximation even down to where a=R. For an Earth-Sun analog, R/a≅0.5%. The second requirement is to observe at least one-hundred thousand solar-like stars with ~5000 having V<12. Assuming each solar-like star has just one R=1.0R⊕ planet in or near the habitable zone, one would expect twenty-five "earths", a statistically meaningful result (Borucki et al. 2010b).A null result would also be significant.   2.3 Photometric precision An Earth-Sun analog transit produces a signal of 84 parts per million (ppm), the ratio of their areas. A central transit of an Earth-Sun analog lasts for 13 hours. The third requirement is to reliably detect transits of 84 ppm in 6.5 hrs (half of a central transit duration for an Earth-Sun analog). There are three distinct types of noise that determine the detection threshold: 1) photon-counting shot noise, 2) stellar variability and 3) measurement noise. We define the combination of these sources as the combined differential photometric precision (CDPP).   CDPP=(shot noise2 + stellar variabiltiy2 + measurement noise2)1/2 .  (2) Equation 2 needs to be used with caution, since stellar variability and many of the measurement noise terms do not scale simply with time. CDPP is differential, since it is the difference in brightness from the near-term trend that is important - not long-term variations. For an 84 ppm signal to be detected at >4σ, the CDPP must be <20 ppm. In a well-designed experiment, little is gained by reducing any noise component to be significantly smaller than the total. We have no control over the stellar variability contribution other than selecting stars with low variability. Analysis of data from the Sun shows that variability during solar maximum on the time scale of a transit is typically 10 ppm (Jenkins, 2002). For the shot noise from a star to be 14 ppm in 6.5 hrs, the flux from ApJL Revised: 26 Jan 4 the star needs to result in 5x109 photoelectrons. Quadratic subtraction of these two components from 20 ppm, leaves ~10 ppm for the measurement noise. Measurement noise includes not just detector and electronic noise, but also such things as pointing jitter, image drift, thermal, optical, and integration-time stability, stray light, video and optical ghosting, and sky noise.   2.4 Stability - the key to success A technology demonstration was performed (Koch et al. 2000) to prove the feasibility of the Kepler Mission. Key lessons learned from these tests were that pointing stability had to be better than 0.003 arcsec/15 min and thermal stability of the CCD had to be better than 0.15K/day. This led to the fourth design requirement: photometric noise introduced by any source must either have a mean value of zero or not vary significantly on the time scale of a transit.  Several other aspects were also demonstrated: 1) the required precision can be achieved without a shutter, 2) optimum photometric apertures are necessary, 3) pixels with large full wells allow for longer times between readouts, 4) traps in the CCDs will remain filled if the CCDs are operated below about -85C and the integration times are kept <8 sec, 5) precision photometry works even with saturated pixels, if the upper parallel clock voltage is properly set, and 6) data at the individual pixel level should be preserved.  2.5 Pixel time series Preserving the individual pixel time series that compose each stellar image permits the ground analysis to: 1. Measure and decorrelate the effects of image motion (Jenkins et al. 2010a), 2. Remove cosmic rays at the pixel level (Jenkins et al. 2010a), 3. Remove instrumental features and systematic noise (Caldwell et al. 2010), 4. Measure the on-orbit pixel response function (PRF) (Bryson et al. 2010), 5. Fit and remove local background for each star (Jenkins et al. 2010a), 6. Measure any centroid shift during a transit at the sub-millipixel level (Batalha et al. 2010a), 7. Perform astrometry and obtain parallaxes and hence distances to the stars (Monet et al. 2010), and 8. Reprocess the data using improved understanding of the instrument and data.  3. Mission design choices 3.1 Observing strategy Three design considerations enter into the choice of how best to observe a large number of stars for transits: size of the field-of-view (FOV), aperture of the optics, and duty cycle. Concepts considered were large FOV fish-eye optics, multiple aperture optics, a single large aperture, and single versus multiple pointings during each cadence. After considering various combinations, the design that was settled upon was a one-meter class aperture with a FOV >100 square degrees and viewing of a single star field. Pointing to a single star field has the advantages of: 1) selecting the richest available star field, 2) minimizing the stellar classification necessary to select the desired stars, 3) optimizing the spacecraft design, 4) maximizing the duty cycle, 5) simplifying operations, data ApJL Revised: 26 Jan 5 processing and data accounting, and 6) continuous asteroseismic measurements over long periods of time. The combination of these factors results in a photometric database with unprecedented precision, duration, contiguity and number and variety of stars.  For an extended mission, there are many reasons to not change the star field. Beside the points above, additional reasons are: 7) extending the maximum orbital period for planet detection, 8) improving the photometric precision for the already detected planets, 9) improving the statistics for marginal candidates, 10) decreasing the minimum detectable size of planets, 11) providing a longer time base for transit and eclipsing binary timing to enable detection of unseen companions and 12) observing stellar activity cycles that approach the time scale of a solar cycle. Based solely on the current rate of propellant usage, the only expendable, the mission can operate for nearly ten years.  3.2 The photometer  The Delta II (7925-10L) launch vehicle provided by NASA’s Discovery program defined the combination of the size of the optics, the sunshade and solar avoidance angle, the spacecraft mass, and the orbit. In the final analysis, it was found that a one-meter class Schmidt telescope with a 16º diameter FOV and a 55º sun-avoidance angle could be launched into an Earth-trailing heliocentric orbit (ETHO). The shot noise of 14 ppm in 6.5 hr is met with the 0.95-m aperture for a V=12 solar-like star. The large FOV required a curved focal surface and the use of sapphire field-flattening lenses on each CCD module. The spectral bandpass is defined by the optics, the quantum efficiency of the CCDs and bandpass filters. To maximize the signal-to-noise ratio (SNR) for solar-like stars, the bandpass filters applied to the field-flattening lenses have a >5% response only from 423 to 897 nm. This is shown in figure 1. The blue cutoff was chosen to avoid the UV and the Ca II H&K lines. For the Sun, 60% of the irradiance variation is at <400nm, but photons <400nm only account for 12% of the total flux (Krivova et al. 2006). The red cutoff was selected to avoid fringing due to internal reflection of light in the CCDs. The spectral response is somewhat broader than a combination of V and R bands. Kepler magnitudes (Kp) are usually within 0.1 of an R magnitude for nearly all stars.   We shall refer to the single instrument on the Kepler Mission as the photometer. The heart of the photometer is the 95-megapixel focal plane composed of 42 science CCDs and four fine guidance sensor CCDs (figure 2). The science CCDs are thinned, back-illuminated and anti-reflection coated, four-phase devices from e2v Technologies Plc. The CCD controller, located directly behind the focal plane, provides the clocking signals and digitizes the analog signals from the 84 CCD outputs. Data from each CCD are co-added on board for 270 readouts to form a long cadence (LC) of ~30 min, the primary data for planet detection. At the end of each LC, only the pixels of interest (POI) for each star are extracted from the image, compressed and stored for later downlink. In parallel, a limited set of 512 POI are extracted from the image every nine readouts, to provide short cadence (SC) 1-min data. The SC data are used to provide improved timing of planetary transits and to perform asteroseismology (Gilliland et al. 2010a). In addition to the POIs for measuring stellar brightnesses, significant amounts of collateral data are also collected to enable calibration (Caldwell et al. 2010). Full-field images (FFI) are also recorded once per month, calibrated and archived. The design and testing of the focal plane ApJL Revised: 26 Jan 6 assembly is described by Argabright et al. (2008). A summary of the key characteristics is given in table 1.  3.3 The orbit An ETHO is significantly more benign and stable for precision photometry than an Earth orbit. Important features of an ETHO (nearly identical to the Spitzer ETHO, Werner, et al. 2004) versus a low-Earth orbit are: 1) the spacecraft is not passing in and out of the radiation belts or the South Atlantic Anomaly 2) the spacecraft is not passing in and out of the Earth’s shadow and heating from the Sun, 3) there is no atmospheric drag or gravity gradient to torque the spacecraft, and 4) there is no continuously varying earthshine getting into the telescope - all of these happening on the time scale of a transit. The largest disturbing torque in an ETHO is sunlight.   3.4 Star field selection The 55º sun avoidance angle limited the choice to a star field >55º from the ecliptic plane. Two portions of the galactic plane are accessible. Counting stars brighter than V=14 from the USNO star catalog showed the Cygnus region, looking along the Orion arm of our galaxy, to be the richest choice. After reviewing the confusion due to distant background giants in the galactic plane, the center of the FOV was placed at 13.5º above the galactic plane at an RA=19h22m40s and Dec=44°30' 00". For distances greater than a few kiloparsecs, stars in the field are above the galactic disk, thus minimizing the number of background giant stars. The distance to a V=12 solar-like star is ~270 pc, with the typical distance to terrestrial planets detected by Kepler being 200-600 pc. An important aspect of the northern over the southern sky is that the team resources needed for ground-based follow-up observing are in the North.   The final orientation of the focal plane was chosen to minimize the number of bright stars on the CCDs. Bright stars bloom and cause significant loss of useable pixels. Only a dozen stars brighter than V=6 are on silicon, with only one, θ Cyg, being brighter than V=5. The spacecraft needs to be rotated about the optical axis every one-quarter of an orbit around the Sun to keep the Sun on the solar panels and the radiator that cools the focal plane pointed to deep space. To keep the bright stars in the gaps between the CCDs and blooming in the same direction on the CCDs, the 42 CCDs were arranged with four-fold symmetry (except for the central two CCDs) and mounted to within +/-3 pixels of co-alignment.  Given the 16º diameter FOV and the large varying angle between the center of the star field and the velocity vector of the spacecraft as it orbits the Sun, the affect of velocity aberration on the locations of the stars on the CCDs is significant. This effect causes the diameter of the FOV to change on an annual basis by 6 arcsec, and is taken into account when computing the POI used for each star.  3.5 The Kepler Input Catalog To maximize the results, we preferentially observe solar-like stars. Most star catalogs provide sufficient information to approximate Teff, but the stellar size is generally not provided. We chose two approaches to distinguish dwarf from giant stars: The primary ApJL Revised: 26 Jan 7 approach was to perform multi-band photometric observations using a filter set similar to the Sloan Survey (g, r, i, z) with the addition of a filter for the Mg b line, that is especially sensitive to log(g), and then modeling of the observations to derive Teff and log(g). This resulted in the Kepler Input Catalog (KIC)1. The process of ranking and selecting stars based on multiple parameters using the KIC is described by Batalha et al. (2010b). The backup approach was to observe all the stars in the FOV brighter than Kp=15 early in the mission (the reason for the larger star handling capacity at the start of the mission) and, based on their measured variability, distinguish the dwarfs from giants. (This result is described below.)  4. Performance 4.1 Saturation and dynamic range The design of the photometer called for collecting 5x109 photoelectrons in 6.5 hrs for a V=12 star. In a single 6.02 sec integration this amounts to 1.4x106 e-. For the tightest PRFs, 60% of the energy can fall into one pixel (Bryson et al. 2010), so stars brighter than Kp~11.5 saturate, depending on FOV location. We have set the upper parallel clock voltage on the CCDs so that the overflow electrons are preserved and fill the adjacent pixels in the same column. No electron is left behind. The photometric aperture sizes are adjusted for saturated stars and the photometric precision is preserved.   To illustrate that photometric precision is attained in saturated stars, light curves for some of the brightest stars, which are saturated by as much as a factor of one hundred, are presented in figure 3. These data also illustrate that the measurement noise must be well under 10 ppm. Stars fainter than Kp=15 are also useful. Noise measurements of many of these are near shot-noise limited performance, indicating the instrument is not limiting the noise. The dynamic range is at least from Kp=7 to Kp=17 (Gilliland et al. 2010b).  4.2 The commissioning data set  As a final step during commissioning, 9.7 days of photometric data were taken. A special set of 52,496 stars was used. This set had oversize apertures to compensate for the +/-3 pixel prelaunch uncertainty in the focal plane geometry. The geometry was determined to <0.1 pixel as part of commissioning (Haas et al. 2010). This Q0 data set included all stars brighter than Kp=13.6 that were fairly uncrowded, irrespective of their classification with the exception of about 160 stars with Kp<8.4. Bright stars known from published data to be variable or not to be dwarf stars were excluded. The primary uses of these data were to: 1) obtain an early measure of CDPP to verify the performance of the photometer, and 2) identify quiet stars that may have been either excluded from the Kepler target list based on their KIC classification or were not classified with confidence. Two of the planet detections being reported were unclassified stars.   4.3 Variability of dwarf and giant stars From Hubble Space Telescope time series data it has been shown (Gilliland, 2008) that red giants are more variable than dwarfs and that the variations tend to be quasi-periodic                                                 1 http://archive.stsci.edu/kepler/kepler_fov/search.php  ApJL Revised: 26 Jan 8 with multiple simultaneous periods. Two control sets of 1000 stars each were formed from the Q0 data consisting of dwarfs and giants based on their KIC classification. They were selected to be bright, but not saturated (11.3<Kp<12.0), very uncrowded (very little or no light from a nearby star), and uniformly distributed over the focal plane. For the giants, Teff was required to be <5400K, since the radii given in the KIC is most secure for stars below this temperature. The data were not ensemble normalized or detrended beyond a fit to a cubic function. Impulsive outliers of greater than +5σ were suppressed. The modeled noise based on measured instrument performance (not including stellar variability) for stars in this narrow magnitude range is 26.5 ppm in 2 hr (14 ppm in 6.5 hrs) and almost entirely due to shot noise. The result of the analysis is shown in figure 4. The median value of CDPP for the dwarfs is less than twice the modeled noise level implying a median 2 hr stellar variability of <46 ppm, typical of the quiet Sun. An empirical fit of CoRoT data to their median noise from their equation 1 (Aigrain et al. 2009) for dwarfs at R=12 is 100 ppm in 2 hrs. Much, but certainly not all, of their higher noise level can be accounted for as due to a higher shot noise. The Kepler collecting area is 9.25 times greater than CoRoT's and the shot noise at a given magnitude is three times smaller.   The data in figure 4 show: 1) a clear distinction in variability between dwarfs and giants (>60% of the giants have variability >10 times the shot noise), 2) the observed variability of most dwarfs on the time scale of transits is small enough to permit detection of transits of Earth-size planets, and 3) the classification given in the KIC is quite reliable in distinguishing dwarfs from giants.  The data in figure 5 are the measured CDPP for all 52,496 of the stars in the Q0 data set. These data show an evident giant branch of higher variability distinct from the dwarfs. Also, the noise floor for Kp<13.6 is nearly coincident with the shot noise, indicating that the measurement noise is quite small. For fainter stars the ratio of the measurement noise to the smaller stellar flux becomes a significant part of the CDPP, but is still less than the shot noise.  5. Early results and prospects 5.1 Exoplanets Results from the first forty-three days of data include detection of five transiting exoplanets (Borucki et al. 2010a, Borucki et al. 2010b, Koch et al. 2010, Dunham et al. 2010, Latham et al. 2010, Jenkins et al. 2010b), detection of the occultation and phase variation in the light curve of the previously known transiting planet HAT-P-7b (Borucki, et al. 2009, Welsh et al. 2010), and detection of hot compact transiting objects (Rowe et al. 2010). Each of the planet discovery papers presents details on one or more aspect of the different analysis and evaluation processes used for all of the planet discoveries. The precision has been found to be so good that the vetting process of candidates (Batalha et al, 2010a) has been able to use the photometric data to weed out most false positives that heretofore could not be recognized with photometry alone. For example, analysis of the photometric data for centroid motion in and out of transit is at the one-tenth of a millipixel level, permitting the recognition of about 70% of the candidates as false ApJL Revised: 26 Jan 9 positives due to background eclipsing binaries. This led to an initial set of 21 candidates for the follow-up observing program (Gautier et al. 2010) with a low false-positive rate.  5.2 Oscillating, pulsating, and eclipsing stars A full discussion of the Kepler early results and extensive references placing these in the context of their fields may be found in Gilliland (2010a). Measurement of the p-mode oscillations in the star HAT-P-7 has provided refined stellar parameters for this system, reducing the radius uncertainty from 10% (Pal et al., 2008) to 1% (Christensen-Dalsgaard et al. 2010). Measurement of p-modes from three G-type stars (Chaplin et al. 2010) has revealed twenty oscillation modes providing radii, masses and ages for these stars as a sample of what is to come. With a year's worth of data the depth of the convection zone can be measured. Solar-like oscillations have been measured in low-luminosity red giants (Bedding et al. 2010). The observations should be valuable for testing models of these H-shell-burning stars and shed light on the star formation rate in the local disk. Especially interesting are stars that show both eclipses and oscillations (Gilliland et al. 2010a, Hekker et al. 2010) allowing two independent means of inferring stellar properties. Over the 3.5 year baseline mission, a few thousand stars are to be observed at the 1-min cadence rate for the purposes of measuring p-mode oscillations.  Analysis of stellar pulsations leads to an understanding of the stellar interior. The main-sequence δ Sct stars with periods of about two hours and γ Dor stars with periods of about one day are particularly useful. Hybrid stars pulsating with both ranges of periods provide complementary model constraints. Previously only four hybrids were known. A search of the Kepler data has found 150 δ Sct and γ Dor stars and two new hybrids with the prospect for more to come (Grigahcene et al. 2010).   RR Lyrae stars like Cepheid variables follow a period-luminosity relationship, but unlike Cepheids, RR Lyrae are lower mass and luminosity and much more common. A significant fraction of RRab exhibit the still unexplained Blazhko effect with a modulation in the light curve of tens to hundreds of days (Kolenberg et al. 2010). RR Lyrae itself is in the Kepler FOV and is being observed. A search for variability in a set of 2288 stars (the bulk of which are red giants by design) has identified many new variables: 27 RR Lyrae subtype ab, 28 β Cep and δ Sct, 28 slowly-pulsating Bs and γ Dor, 23 ellipsoidal variables and 101 eclipsing binaries (Blomme et al. 2010).  5.3 Anticipated results  Distances to the stars can be derived from trigonometric parallax, which along with their luminosities provides the sizes of the stars, essential for knowing the size of each planet. Using PRF fitting for astrometry requires good photometry. From Q1 data it can be seen that the Kepler measurements are in a new regime of high SNR for astrometry (Monet et al. 2010). Astrometric precision of individual 30-min cadences is already known to be at the millipixel level (4 milliarcsec). The ultimate precision of such high SNR data is yet to be realized. Parallaxes and proper motions require years of data to separate the two.   Knowing the mass of a star is essential for calculating the orbit of a planet from the period of the transits and for calculating the mass of the planet from radial velocity ApJL Revised: 26 Jan 10 observations. As solar-like stars age their rotation rates decrease (Skumanich 1972). For young clusters like the Hyades a mass-rotation-rate relationship has been derived (Barnes, 2003). Kepler observations of clusters NGC 6866 (0.5 Gyr), 6811 (1 Gyr), 6819 (2.5 Gyr), and 6791 (8-12 Gyr) will provide rotation rates of stars in much older clusters. From ground-based spectroscopic observations and asteroseismic analysis of the Kepler data (Stello et al. 2010) the masses will be derived and a mass-age-rotation rate (gyrochronology) calibration can be derived.   6. Summary The Kepler Mission design was based on a top-down scientific requirements flow to optimize the ability to detect Earth-size planets in the habitable zone of solar-like stars. The photometer and spacecraft were launched on March 6, 2009, commissioned in 67 days and have been gathering exquisite photometric data. Analysis of the initial data has shown: 1. The discovery of five transiting planets with measured radii, masses and orbits; 2. A wealth of results from oscillating, pulsating, and eclipsing stars; 3. High photometric precision is being achieved, even for extremely saturated stars; 4. The median variability for dwarf stars is similar to that of the Sun, with many examples of stars even quieter than the Sun; and 5. Dwarf and red-giant stars can readily be distinguished based on their inherent variability. The early data from Kepler have produced both new planet detections and new results in stellar astrophysics, which bodes well for future prospects.  Acknowledgements Kepler was selected as the tenth Discovery mission. Funding for this mission is provided by NASA's Science Mission Directorate. The Kepler science team cannot possibly name all of those who contributed to the design, development, and operation of the mission starting as far back as 1992, but we are nonetheless deeply indebted to and thankful to those hundreds of individuals. Managers at various times and institutions included: Charlie Sobeck, Marcie Smith, Dave Pletcher, Sally Cahill, Roger Hunter, Laura MacArthur-Hines, Dave Mayer, Mark Messersmith, Janice Voss, Larry Webster, John Troeltzsch, Jerome Stober, Alan Frohbieter, Monte Henderson, Len Andreozzi, Tim Kelly, Scott Tennant, Jim Fanson, Peg Frerking, Leslie Livesay, Chet Sasaki, Bill Possel and Dave Taylor; System engineers at various times and institutions included: Eric Bachtell, Dave Acton, Jeff Baltrush, Adam Harvey, Sean Lev-Tov, Dan Peters, Vic Argabright, Riley Duren, Brian Cook, Tracy Drain, Karen Dragon, Don Eagles, Rick Thompson, Steve Jara, Chris Middour, Rob Nevitt, Jeneen Sommers, Brett Strooza, Steve Walker, and Daryl Swade; and IPT leads at Ball Aerospace included: Bruce Bieber, Bill Bensler, Dan Berry, Susan Borutzki, Chris Burno, Brian Carter, Wayne Davis, Mike Dean, Bill Deininger, Mike Hale, Roger Lapthorne, Scot McArthur, Chris Miller, Rob Philbrick, Amy Puls, Charlie Schira, Dan Shafer, Beth Sholes, Chris Stewart, Dennis Teusch, Bryce Unruh, and Mike Weiss.    ApJL Revised: 26 Jan 11 References Aigrain, S., et al., 2009, A&A 506, 425 Argabright, V., et al., 2008, in Space Telescopes and Instrumentation 2008: Optical, Infrared, and Millimeter. Oschmann, J., et al. editors, SPIE Conf Series, 7010, 2L-1 Barnes, S. A., 2003, ApJ, 586, 464 Basri, G., et al., 2010, ApJL, this issue  Batalha, N., et al., 2010a, ApJL, this issue  Batalha, N., et al., 2010b, ApJL, this issue  Bedding, T., et al., 2010, ApJL, this issue Blomme, J., et al 2010, ApJL, this issue Borucki, W., et al., 2008, in Exoplanets: Detection, Formation and Dynamics IAU Symp, Eds. Y. Sun, S. Mello, & J. Zhou , Suzhou, 249, 17 Borucki, W., et al., 2009, Science 325,709 Borucki, W., et al., 2010a, accepted Science Borucki, W., et al., 2010b, ApJL, this issue,  Bryson, S., et al., 2010, ApJL this issue  Caldwell, J., et al., 2010, ApJL this issue  Chaplin, W., et al., 2010, ApJL, this issue Christensen-Dalsgaard, J., et al., 2010, ApJL, this issue Dunham, E., et al., 2010, ApJL this issue  Gautier, N. et al., 2010, ApJL, this issue Gautschy, A. and Saio, H., 1996, ARAA, 34, 551 Gilliland, R., 2008, ApJ 136, 566 Gilliland, R., et al., 2010a, PASP, in press Gilliland, R., et al., 2010b, ApJ, this issue  Gimenez, A., Guinan, E., Niarchos, P. and Rucinski, S., editors, 2006, Ap&SS, 304 Grigahcene, A., et al., 2010, ApJL, this issue Haas, M., et al., 2010, ApJL, this issue  Hekker, S., et al., 2010, this issue  Jenkins, J., 2002, ApJ 575, 493 Jenkins, J., et al, 2010a, ApJL, this issue  Jenkins, J., et al, 2010b, ApJ, submitted  Jenkins, J., et al, 2010c, ApJL, this issue  Kasting, J. F., Whitmire, D. P., Reynolds, R. T., 1993. Icarus, 101: 108 Koch, D., et al. 2000, in UV, Optical and IR Space Telescopes and Instruments, SPIE Conf. Series, J. B. and Jakobsen, P., editors, Breckinridge,4013, 508 Koch, D., et al., 2004, in Optical, Infrared, and Millimeter Space Telescopes, SPIE Conf Series, Mather, J. C. editor, 5487, 1491 Koch, D., et al., 2010, ApJL this issue,  Kolenberg, K., et al 2010, ApJL, this issue Krivova, N., et al., 2006, A&A, 452, 631 Latham, D., et al., 2010, ApJL this issue,  Monet, D., et al., 2010, ApJL, this issue  Pal, A., et al. 2008, ApJ, 680, 1450 Pont, F., Sasselov, D. & Holman, M., editors, 2009, Cambridge, Transiting Planets, IAU Sym 253, Cambridge Univ. Press Rowe, J, et al., 2010, ApJL, this issue  ApJL Revised: 26 Jan 12 Skumanich, A., 1972, ApJ, 171, 565 Stello, D., et al., 2009 ApJ, 700, 1589 Stello, D. et al., 2010, ApJL, this issue Werner, M. W., 2004, ApJSuppl, 154, 1      
+  Figure 1. Spectral response curves. An average value is shown for the 21 field-flattening lenses (FFL) and the bandpass (BP) filters, which are multi-layer coatings on the back of the sapphire FFLs. The CCD quantum efficiency (QE) is the average from all 42 CCDs. The combined response curve also includes a 4% loss for contamination  
+ApJL Revised: 26 Jan 13  Figure 2 Cross-sectional view of the photometer and inset of the focal plane. The overall height with the sunshade is 4.3 m. The spacecraft (not shown) is a hexagonal structure 0.8 m high that surrounds the base of the photometer.  
+ApJL Revised: 26 Jan 14 Figure 3 Flux time series for ten of the brighter G-dwarf stars. The CDPP is calculated by applying a moving-median filter that is 48 hours wide and then computing the RMS deviation for 6.5-hours. For reference, a grazing one-Earth-size transit (84 ppm) is shown to scale.   
+Figure 4 Distribution functions of variability for flux time series of stars classified as 
+ApJL Revised: 26 Jan 15 dwarfs and giants. Panels A & C show the ratio of RMS noise to model noise as a function of luminosity. Panels B & D show histograms for these ratios. The right hand bins include any noise ratios >10. Nineteen percent of the dwarfs in panel B are in the >10 bin. Of these 87% are recognized as intrinsic variable stars from their light curves and the remaining 13% have light curves characteristic of red giants, suggesting that the error in the classification of dwarfs in the KIC is of the order of 2.5%. Inspection of the light curves for the red giants does not show any stars that are likely to be dwarfs, based on the character of their variability.      
+Figure 5. Measured half-hour precision of the Q0 data set. A strong separation in photometric variability can be seen between the dwarfs (green points) and the red giants (red points). The two lines bound the upper and lower noise uncertainties propagated through the data processing pipeline (Jenkins, et al, 2010c), which is mostly shot noise for Kp<13.  
+ApJL Revised: 26 Jan 16 Table 1  Kepler Mission Characteristics Component Value Comment Optics Brashear & Tinsley Manufacturers Schmidt Corrector 0.99 m dia./ 0.95 m dia. stop Corning Fused silica Primary mirror F1 1.40 m dia., silver coated Corning ULE® Central obscuration 23.03% Due to focal plane and spider Field flattening lenses 2.5° square Sapphire Pixel Response Function 3.14-7.54 pixels, 95% encircled energy diameter Depends on FOV location Sunshade 55º sun avoidance  From center of FOV CCDs e2v Technologies Manufacturer Format 1024 rows x 2200 columns 2 outputs per CCD Pixel size 27 µm square Four phase Plate scale 3.98 arcsec/pixel  Full well 1.05 x106 electrons, typical Set by parallel clock voltage Dynamic range 7<Kp<17 Meets photometric precision Operating temp -85C 10mK stability Controller Ball Aerospace Design and manufacturer Channels 84 Multiplexed into 20 ADCs on 5 electronic board pairs CCD integration time 6.02 sec Selectable 2.5 to 8 sec CCD readout time 0.52 sec Fixed Long cadence period 1765.80 sec 270 integrations + reads Short cadence period 58.86 sec 9 integrations + reads Maximum LC targets 170,000 Average 32 pixels/target Maximum SC targets 512 Average 85 pixels/target Timing accuracy 50 msec For asteroseismology and transit timing System Ball Aerospace Design, integration, and test Spectral response 423 to 897 nm 5% points FOV 105 sq. degrees  <10% vignetted 115 sq. degrees of non-contiguous active silicon Pointing jitter 3 milliarcsec per 15 min 1σ per axis CDPP (total noise) 20 ppm in 6.5 hr  for >90% of FOV V=12 solar-like star including 10 ppm for stellar variability Data downlink period 31 day average <1 day observing gap Mission length 3.5 years Baseline. May be extended Orbital period 372 days by year 3.5 Earth-trailing heliocentric  
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+arXiv:1001.0269v3  [math.AP]  26 Apr 2011The elliptic Kirchhoff equation in RN
+perturbed by a local nonlinearity∗
+A. Azzollini†
+Abstract
+In this paper we present a very simple proof of the existence o f at least
+one non trivial solution for a Kirchhoff type equation on RN, forN/greaterorequalslant3. In
+particular, in the first part of the paper we are interested in studying the ex-
+istence of a positive solution to the elliptic Kirchhoff equ ation under the ef-
+fect of a nonlinearity satisfying the general Berestycki-L ions assumptions.
+In the second part we look for ground states using minimizing arguments
+on a suitable natural constraint.
+keywords: Kirchhoff equation, Pohozaev identity, natural constrain t, minimiz-
+ing sequence
+MSC: 35J20 35J60
+Introduction
+The multidimensional Kirchhoff equation is
+∂2u
+(∂t)2−(1+/integraldisplay
+Ω|∇u|2)∆u= 0 (1)
+whereΩ⊂RNandu: Ω→Rsatisfies some initial or boundary conditions.
+It arises from the following Kirchhoff’ nonlinear generali zation (see [5]) of the
+well known d’Alembert equation
+ρ∂2u
+(∂t)2−/parenleftBigg
+P0
+h+E
+2L/integraldisplayL
+0/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u
+∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+dx/parenrightBigg
+∂2u
+(∂x)2= 0. (2)
+Equation (2) describes a vibrating string, taking into acco unt the changes in
+length of the string during the vibration. Here, Lis the length of the string, h
+is the area of the cross section, Eis the Young modulus of the material, ρis the
+mass density and P0is the initial tension. In [6] the problem was proposed in
+the following form:
+
+
+∂2u
+(∂t)2−M(/integraltext
+Ω|∇u|2)∆u=finΩ×(0,T),
+u= 0 in∂Ω×(0,T)
+u(0) =u0, u′(0) =u1,
+∗The author is supported by M.I.U.R. - P .R.I.N. “Metodi varia zionali e topologici nello studio di
+fenomeni non lineari”
+†Dipartimento di Matematica ed Informatica, Universit` a de gli Studi della Basilicata, Via
+dell’Ateneo Lucano 10, I-85100 Potenza, Italy, e-mail: antonio.azzollini@unibas.it
+12 A. Azzollini
+whereM: [0,+∞[→Ris a continuous function such that M(s)/greaterorequalslantc >0for
+anys/greaterorequalslant0,andΩis a bounded set in RN,with smooth boundary.
+This hyperbolic problem has an elliptic version when we look for static solu-
+tions.
+In [14], it has been considered a class of problems among whic h the following
+elliptic Kirchhoff type equation was included
+/braceleftbigg
+−M(/integraltext
+Ω|∇u|2)∆u=finΩ,
+u= 0 in∂Ω
+(hereΩis an open subset of RN).
+Taking into account the original formulation of the equatio n given by Kirch-
+hoff, we assume the following
+Definition 0.1. If there exist two positive constants aandbsuch that M:R+→R
+can be written M(s) =a+bs,thenMis called Kirchhoff function.
+Recently, many authors have used variational methods to stu dy the Kirch-
+hoff equation perturbed by a local nonlinear term (see [7] fo r a short survey
+on the topic). By arguments based on the mountain pass theore m, in [1] the
+problem/braceleftbigg
+−M(/integraltext
+Ω|∇u|2)∆u=f(x,u)inΩ,
+u= 0 in∂Ω
+has been solved in a bounded domain of RNunder suitable growth conditions
+onf:RN×R→RandM: [0,+∞[→R.TakingN= 1,2or3, the problem
+has been treated also in [10] where Mis a Kirchhoff function, and the nonlin-
+earityf(x,t)has been supposed to behave linearly near 0 and like t3at infinity.
+The Yang index has been used to find a nontrivial solution. A si milar growth
+at infinity has been assumed in [15] where the authors have loo ked for sign
+changing solutions. They also have obtained a sign changing solution when
+the nonlinearity fsatisfies either the following growth condition
+|f(x,t)|/lessorequalslantC(|t|p−1+1),uniformly in x, forp <4
+or the following Ambrosetti-Rabinowitz condition
+νF(x,t)/lessorequalslanttf(x,t),for|t|large and ν >4.
+In [8], the equation has been studied assuming that the nonli nearity grows at
+infinity more than t3, without introducing the Ambrosetti-Rabinowitz hypoth-
+esis. Using a variational approach, a multiplicity result h as been showed in [4].
+Finally we recall the recent result obtained in [11], where t hree solutions have
+been found for the Kirchhoff type problem
+/braceleftbigg
+−M(/integraltext
+Ω|∇u|2)∆u=λf(x,u)+µg(x,t)inΩ,
+u= 0 in∂Ω
+whereλandµare two parameters.
+The joining point of all these papers is that they consider th e equation on a
+bounded domain, with Dirichlet conditions on the boundary.
+In this paper we study an autonomous Kirchhoff type equation on all the spaceKirchhoff equation in RN3
+RN,looking for the existence of positive solutions, namely we c onsider the
+problem/braceleftbigg
+−M(/integraltext
+Ω|∇u|2)∆u=g(u)inRN, N/greaterorequalslant3,
+u >0.(K)
+In this sense, the problem turns out to be a generalization of the well known
+Schr¨ odinger equation
+−∆u=g(u),inRN. (S)
+In the first part of the paper we are interested in studying the problem ( K)
+in presence of a Berestycki-Lions nonlinearity. In order to explain what this
+means, we provide the following definitions
+Definition 0.2. A function g:R→Ris called a Berestycki-Lions nonlinearity if it
+satisfies the following assumptions
+(g1)g∈C(R,R),g(0) = 0 ;
+(g2)−∞<liminf s→0+g(s)/s/lessorequalslantlimsups→0+g(s)/s=−m <0;
+(g3)−∞/lessorequalslantlimsups→+∞g(s)/s2∗−1/lessorequalslant0;
+(g4) there exists ζ >0such that G(ζ) :=/integraltextζ
+0g(s)ds >0.
+A function g:R→Ris called a zero mass Berestycki-Lions nonlinearity if it
+satisfies ( g1), (g3), (g4) and the following zero mass assumption
+(g2)’−∞<liminf s→0+g(s)/s,limsups→0+g(s)/s2∗−1/lessorequalslant0.
+Remark 0.3. Using the terminology inherited by [2], we refer to the const antmin
+(g2) calling it mass. This is the reason for which we say that a fun ctiongsatisfying
+(g2)’ instead of ( g2) is a zero mass nonlinearity.
+In the very celebrated paper [2], these types of nonlinearit ies appeared for
+the first time, and it was showed that the hypotheses ( g1),...,(g4) are almost
+optimal to get an existence result for the Schr¨ odinger equa tion.
+In the first section of this paper, we use a simple rescaling ar gument to
+establish a sufficient condition for the existence of a solut ion to (K). The first
+result we get is the following
+Theorem 0.4. LetM:R+→R+\{0}a continuous function such that
+liminf
+t→0tM(t2−N
+2) = 0 (3)
+and letg:R→Ra (possibly zero mass) Berestycki-Lions nonlinearity. The n the
+problem (K)has a solution in C2(RN).
+We point out that any Kirchhoff function satisfies our assump tions ifN= 3.
+On the other hand, when Mis a Kirchhoff type function we are able to refine
+our estimates and we get the following result
+Theorem 0.5. Letg:R→Ra (possibly zero mass) Berestycki-Lions nonlinearity,
+f:R+→R+a continuous function and M(s) =a+bf(s).Then for any N/greaterorequalslant3
+there exists a positive constant δ(depending on a) such that if b∈]0,δ],the problem
+(K)has a solution in C2(RN).If moreover
+liminf
+t→0t2
+2−Nf(t) = 0, (4)4 A. Azzollini
+then there exists a positive constant δ(depending on b) such that if a∈]0,δ],the
+problem (K)has a solution in C2(RN).
+We again remark that when Mis a Kirchhoff function, then f=id|R+and
+assumption (4) is automatically satisfied for N/greaterorequalslant5.
+In the second part of the paper we study the existence of the so called ground
+state solutions to the Kirchhoff equation
+−(a+b/integraldisplay
+RN|∇u|2)∆u=g(u). (5)
+We recall that a ground state is a solution which minimizes th e functional of
+the action among all the other solutions.
+The problem of finding such a type of solutions is a very classi cal problem:
+it has been introduced by Coleman, Glazer and Martin in [3], a nd reconsidered
+by Berestycki and Lions in [2] for a class of nonlinear equati ons including the
+Schr¨ odinger’s one. Here we will use a minimizing argument b ased on an idea
+developed in [13] (recent applications can be found in [9] an d in [12]). In that
+paper the author showed that the ground state for the Schr¨ od inger equation
+can be found as the minimizer of the functional of the action r estricted to a par-
+ticular natural constraint. This natural constraint actua lly is a manifold which
+is constituted by all non null functions satisfying the Poho zaev identity related
+to the Schr¨ odinger equation.
+Unfortunately, a similar manifold turns out to be a nice cons traint in order
+to look for ground state solutions of the Kirchhoff equation only ifN= 3 or
+N= 4 : whenN/greaterorequalslant5,we are no more able to establish if the restricted func-
+tional is bounded below. The final result we get is
+Theorem 0.6. IfN= 3 orN= 4 andgis a Berestycki-Lions nonlinearity, then
+equation (5)possesses a ground state solution.
+The paper is so organized:
+in Section 1 we show our rescaling argument to get a solution f or (K) in
+the general case described in Theorem 0.4 and in the particul ar situation of a
+Kirchhoff type function as in Theorem 0.5.
+In Section 2 we study the problem of the existence of a ground s tate solution
+for the Kirchhoff equation using a variational approach.
+1 Bound state solution
+In the sequel, we denote by va ground state solution of ( S) (respectively a
+bound state solution if gis a zero mass Berestycki-Lions nonlinearity).
+Proof of Theorem 0.4 Observe that, by hypothesis (3) and since
+lim
+t→+∞tM(t2−N
+2) = +∞,
+by continuity we have that there exists ¯t >0such that ¯t2M(¯t2−N/integraltext
+RN|∇v|2) =
+1.The function u:RN→Rdefined as follows:
+x∈RN→v(¯tx)∈R,Kirchhoff equation in RN5
+satisfies the equalities
+/braceleftbigg
+M(/integraltext
+RN|∇u|2) =1
+¯t2
+−∆u(x) =−¯t2∆v(¯tx) =¯t2g(v(¯tx)) =¯t2g(u(x)),
+and then it is a solution of ( K). /square
+Remark 1.1. We point out that, since the only moment in which we use hypoth esis(3)
+is to determine the rescaling parameter ¯t, we can relax our assumption just requiring
+that
+inf
+t/greaterorequalslant0tM/parenleftbigg
+t2−N
+2/integraldisplay
+RN|∇v|2/parenrightbigg
+/lessorequalslant1. (6)
+Proof of Theorem 0.5 As in the proof of Theorem 0.4 we look for a solution
+to the equation
+t2/parenleftbigg
+a+bf(t2−N/integraldisplay
+RN|∇v|2)/parenrightbigg
+= 1.
+Taking into account the previous remark, it is enough to prov e that
+inf
+t/greaterorequalslant0Ψ(t)/lessorequalslant1,
+whereΨ(t) :=t/parenleftBig
+a+bf(t2−N
+2/integraltext
+RN|∇v|2)/parenrightBig
+.Set
+¯h=f/parenleftbigg
+(2a)N−2
+2/integraldisplay
+RN|∇v|2/parenrightbigg
+andδ1=a
+¯h.It is easy to verify that, if b/lessorequalslantδ1,thenΨ(1/2a)/lessorequalslant1.
+Now suppose that (4) holds. We deduce that
+liminf
+t→+∞tf/parenleftbigg
+t2−N
+2/integraldisplay
+RN|∇v|2/parenrightbigg
+= 0.
+Let¯tsuch that ¯tf/parenleftBig
+¯t2−N
+2/integraltext
+RN|∇v|2/parenrightBig
+/lessorequalslant1
+2band choose a/lessorequalslantδ2=1
+2¯t.Again we
+have that Ψ(¯t)/lessorequalslant1. /square
+2 Ground state solution
+In this section we use a variational approach which requires some preliminar-
+ies. In next subsection we will use the same arguments as in [2 ] to modify
+the nonlinearity gin such a way we can study equation ( K) looking for critical
+points of a suitable functional.
+2.1 Functional framework
+Defines0:= min{s∈[ζ,+∞[|g(s) = 0}(s0= +∞ifg(s)/\e}atio\slash= 0 for anys/greaterorequalslantζ).
+We set˜g:R→Rthe function such that
+˜g(s) =
+
+g(s) on[0,s0];
+0 onR+\[0,s0];
+(g(−s)−ms)+−g(−s)onR−.(7)6 A. Azzollini
+By the strong maximum principle, if uis a nontrivial solution of ( K) with˜gin
+the place of g, then0< u < s 0and so it is a positive solution of ( K). There-
+fore we can suppose that gis defined as in (7), so that ( g1), (g2), (g4) and the
+following limit
+lim
+s→±∞g(s)
+s2∗−1= 0 (8)
+hold.
+We set
+g1(s) :=/braceleftbigg(g(s)+ms)+,ifs/greaterorequalslant0,
+0, ifs <0,
+g2(s) :=g1(s)−g(s),fors∈R.
+Since
+lim
+s→0g1(s)
+s= 0,
+lim
+s→±∞g1(s)
+s2∗−1= 0, (9)
+and
+g2(s)/greaterorequalslantms,∀s/greaterorequalslant0, (10)
+by some computations, we have that for any ε >0there exists Cε>0such that
+g1(s)/lessorequalslantCεs2∗−1+εg2(s),∀s/greaterorequalslant0. (11)
+If we set
+Gi(t) :=/integraldisplayt
+0gi(s)ds, i= 1,2,
+then, by (10) and (11), we have
+G2(s)/greaterorequalslantm
+2s2,∀s∈R (12)
+and for any ε >0there exists Cε>0such that
+G1(s)/lessorequalslantCε
+2∗s2∗+εG2(s),∀s∈R. (13)
+We define the functional
+I(u) :=1
+2˜M(/ba∇dblu/ba∇dbl2)−/integraldisplay
+R3G(u)
+where we are denoting by /ba∇dbl · /ba∇dbl the norm/parenleftbig/integraltext
+R3|∇·|2/parenrightbig1
+2of the space D1,2(RN),
+which is the closure of the compactly supported smooth funct ions with respect
+to the norm /ba∇dbl·/ba∇dbl.The previous functional is C1inH1(RN),beingH1(RN)the
+closure of the compactly supported smooth functions with re spect to the norm
+/ba∇dbl·/ba∇dbl2
+H1(RN)=/integraldisplay
+R3|∇·|2+/integraldisplay
+R3|·|2.Kirchhoff equation in RN7
+We will look for critical points of the functional Iinside
+H1
+r(RN) :={u∈H1(RN)|uis radial},
+which is a natural constraint for the functional Iby Palais’ principle of sym-
+metric criticality. By standard variational arguments, it is easy to prove that
+any critical point of Icorresponds to a weak solution of the equation. By the
+maximum principle we will get a positive solution.
+2.2 Existence of a ground state solution
+We look for a ground state solution to
+−(a+b/ba∇dblu/ba∇dbl2)∆u=g(u), u:RN→R, N= 3,4.
+A ground state of (5) is a nontrivial solution ¯u∈H1(RN)such that, if v∈
+H1(RN)is another nontrivial solution of (5), then
+I(¯u)/lessorequalslantI(v),
+whereI:H1(RN)→Ris the functional of the action related with (5), namely
+I(u) =1
+2/parenleftbigg
+a+b
+2/integraldisplay
+RN|∇u|2/parenrightbigg/integraldisplay
+RN|∇u|2−/integraldisplay
+RNG(u).
+Usually a standard technique to find a ground state consists i n looking for min-
+imizers of the functional of the action restricted to a natur al constraint which
+contains all the possible solutions. A candidate to play thi s role is the following
+Pohozaev set
+P={u∈H1(RN)\{0} |P(u) = 0}
+where for any u∈H1(RN)
+P(u) =aN−2
+2N/integraldisplay
+RN|∇u|2+bN−2
+2N/parenleftbigg/integraldisplay
+RN|∇u|2/parenrightbigg2
+−/integraldisplay
+RNG(u).
+Actually the equality P(u) = 0 is nothing but the Pohozaev identity related
+with equation.
+We will prove Theorem 0.6 following this scheme:
+step 1 : we show that Pis aC1manifold containing all the possible solutions of
+equation (5);
+step 2 : we prove that Pis a natural constraint, in the sense that every critical
+point ofIrestricted to Pis a critical point of I;
+step 3: we show that I|Pis bounded below and
+µ= inf
+u∈PI(u) = inf
+u∈P1
+N/parenleftBigg
+a/integraldisplay
+RN|∇u|2+(4−N)b
+4/parenleftbigg/integraldisplay
+RN|∇u|2/parenrightbigg2/parenrightBigg
+is achieved.8 A. Azzollini
+It is easy to see that Pis aC1functional. Moreover Pis nondegenerate in
+the following sense:
+∀u∈ P:P′(u)/\e}atio\slash= 0
+so thatPis aC1manifold of codimension one. Indeed, suppose by contradic-
+tion that u∈ P andP′(u) = 0,namelyuis a solution of the equation
+−/parenleftbigg
+aN−2
+N+2bN−2
+N/ba∇dblu/ba∇dbl2/parenrightbigg
+∆u=g(u). (14)
+As a consequence, usatisfies the Pohozaev identity referred to (14), that is
+a(N−2)2
+2N2/integraldisplay
+RN|∇u|2+b(N−2)2
+N2/parenleftbigg/integraldisplay
+RN|∇u|2/parenrightbigg2
+=/integraldisplay
+RNG(u). (15)
+SinceP(u) = 0,by (15) we get
+−2a/integraldisplay
+RN|∇u|2+b(N−4)/parenleftbigg/integraldisplay
+RN|∇u|2/parenrightbigg2
+= 0
+and we conclude that u= 0: absurd since u∈ P.SoPis aC1manifold.
+It obviously contains all the solutions to (5) since every so lution satisfies the
+Pohozaev identity P(u) = 0.
+Now we pass to prove that Pis a natural constraint for I.Suppose that
+u∈ P is a critical point of the functional I|P.Then, there exists λ∈Rsuch that
+I′(u) =λP′(u),
+that is
+−(a+b/ba∇dblu/ba∇dbl2)∆u−g(u) =−λ(aN−2
+N+2bN−2
+N/ba∇dblu/ba∇dbl2)∆u−λg(u).
+As a consequence, usatisfies the following Pohozaev identity
+P(u) =λa(N−2)2
+2N2/integraldisplay
+RN|∇u|2+λb(N−2)2
+N2/parenleftbigg/integraldisplay
+RN|∇u|2/parenrightbigg2
+−λ/integraldisplay
+RNG(u)
+which, since P(u) = 0 , can be written
+λ/parenleftBigg
+−2a/integraldisplay
+RN|∇u|2+b(N−4)/parenleftbigg/integraldisplay
+RN|∇u|2/parenrightbigg2/parenrightBigg
+= 0.
+Sinceu/\e}atio\slash= 0,we deduce that λ= 0,and we conclude.
+Now it remains to show that µis achieved.
+By the well known properties of the Schwarz symmetrization, we are al-
+lowed to work on the functional space H1
+r(RN)as showed by the following
+Lemma 2.1. For anyu∈ P there exists ˜u∈ P ∩H1
+r(RN)such that I(˜u)/lessorequalslantI(u)Kirchhoff equation in RN9
+Proof Letu∈ P and setu∗∈H1
+r(RN)its symmetrized. It is easy to see that
+there exists 0<˜θ/lessorequalslant1such that ˜u:=u∗(·/˜θ)∈ P ∩H1
+r(RN)and
+I(˜u) =1
+N/parenleftBigg
+a/integraldisplay
+RN|∇˜u|2+(4−N)b
+4/parenleftbigg/integraldisplay
+RN|∇˜u|2/parenrightbigg2/parenrightBigg
+=a˜θN−2
+N/integraldisplay
+RN|∇u∗|2+b(4−N)˜θ2(N−2)
+4N/parenleftbigg/integraldisplay
+RN|∇u∗|2/parenrightbigg2
+/lessorequalslanta
+N/integraldisplay
+RN|∇u∗|2+(4−N)b
+4N/parenleftbigg/integraldisplay
+RN|∇u∗|2/parenrightbigg2
+/lessorequalslantI(u).
+/square
+Before we proceed with the proof of the main result, another p reliminary result
+is required
+Lemma 2.2. µ:= inf{I(v)|v∈ P}>0.
+Proof Ifu∈ P, then, by (13), we have
+C/ba∇dblu/ba∇dbl2/lessorequalslantaN−2
+2/integraldisplay
+RN|∇u|2+bN−2
+2/parenleftbigg/integraldisplay
+RN|∇u|2/parenrightbigg2
++N(1−ε)/integraldisplay
+RNG2(u)
+/lessorequalslantNCε/integraldisplay
+RN|u|2∗/lessorequalslantC′/ba∇dblu/ba∇dbl2∗
+whereε <1,Cε,CandC′are suitable positive constants. We deduce that there
+exists a positive constant C′′such that /ba∇dblu/ba∇dbl/greaterorequalslantC′′for anyu∈ P.The conclusion
+then follows once one observes that I|P(u)/greaterorequalslant˜C/ba∇dblu/ba∇dbl2. /square
+Now let (un)nbe a minimizing sequence for I|PinH1
+r(RN),namely
+{un}n⊂ P ∩H1
+r(RN), I(un)→µ. (16)
+Obviously /ba∇dblun/ba∇dblis bounded. Moreover, since {un}n⊂ P,certainly, by (13),
+there exist 0< ε <1andCε>0such that
+aN−2
+2/integraldisplay
+RN|∇u|2+bN−2
+2/parenleftbigg/integraldisplay
+RN|∇u|2/parenrightbigg2
++N(1−ε)/integraldisplay
+RNG2(u)/lessorequalslantCεN/ba∇dblun/ba∇dbl2∗
+2∗,
+and then we deduce also the boundedness of the L2−norm of {un}nby the
+continuous Sobolev embedding D1,2(RN)֒→L2∗(RN)and (12).
+Letu∈H1
+r(RN)be the function such that, up to subsequences,
+un⇀ u, weakly in H1(RN). (17)
+We are going to prove that there exists ¯θ >0such that
+¯u∈ P andI(¯u) =µ
+where¯u:=u(·/¯θ).
+Actually, by compactness due to the radial symmetry, from th e weak conver-
+gence (17) we deduce
+lim
+n/integraldisplay
+RNG1(un) =/integraldisplay
+RNG1(u). (18)10 A. Azzollini
+Of course, u/\e}atio\slash= 0.Otherwise, by (18) and since un∈ P for anyn/greaterorequalslant1,we should
+have that
+0/lessorequalslantlimsup
+naN−2
+2/ba∇dblun/ba∇dbl2/lessorequalslantNlim
+n/integraldisplay
+RNG1(un) = 0,
+which, by (16), contradicts Lemma 2.2.
+By (18), the lower semicontinuity of the D1,2(RN)−norm and the Fatou lemma,
+we have
+aN−2
+2/integraldisplay
+RN|∇u|2+bN−2
+2/parenleftbigg/integraldisplay
+RN|∇u|2/parenrightbigg2
++N/integraldisplay
+RNG2(u)
+/lessorequalslantliminf
+n/parenleftBigg
+aN−2
+2/integraldisplay
+RN|∇un|2+bN−2
+2/parenleftbigg/integraldisplay
+RN|∇un|2/parenrightbigg2
++N/integraldisplay
+RNG2(un)/parenrightBigg
+= lim
+nN/integraldisplay
+RNG1(un) =N/integraldisplay
+RNG1(u).
+Let0<¯θ/lessorequalslant1such that ¯u=u(·/¯θ)∈ P.Using the lower semicontinuity of the
+D1,2(RN)−norm, we infer that
+I(¯u) =a¯θN−2
+N/integraldisplay
+RN|∇u|2+b(4−N)¯θ2(N−2)
+4N/parenleftbigg/integraldisplay
+RN|∇u|2/parenrightbigg2
+/lessorequalslantliminf
+na
+N/integraldisplay
+RN|∇un|2+b(4−N)
+4N/parenleftbigg/integraldisplay
+RN|∇un|2/parenrightbigg2
+= lim
+nI(un) =µ
+and then we conclude.
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+arXiv:1001.0270v1  [cond-mat.str-el]  4 Jan 2010Nematic, vector–multipole, andplateau–liquid statesint he classical O(3)pyrochlore
+antiferromagnet withbiquadratic interactions inapplied magnetic field
+Nic Shannon,1Karlo Penc,2and Yukitoshi Motome3
+1H. H. Wills Physics Laboratory, University of Bristol, Tynd all Av, BS8–1TL, UK.
+2Research Institute for Solid State Physics and Optics, H–15 25 Budapest, P.O.B. 49, Hungary
+3Department of Applied Physics, University of Tokyo, Bunkyo –ku, Tokyo 113–8656, Japan
+(Dated: November 6, 2018)
+The classical bilinear–biquadratic nearest–neighbor Hei senberg antiferromagnet on the pyrochlore lattice
+does not exhibit conventional N´ eel–type magnetic order at any temperature or magnetic field. Instead spin
+correlations decay algebraically over length scales r/lessorsimilarξc∼√
+T, behavior characteristic of a Coulomb phase
+arising from a strong local constraint. Despite this, its th ermodynamic properties remain largely unchanged
+if N´ eel order is restored by the addition of a degeneracy–li fting perturbation, e.g., further neighbor interac-
+tions. Here we show how these apparent contradictions can be resolved by a proper understanding of way in
+which long–range N´ eel order emerges out of well–formed loc al correlations, and identify nematic and vector–
+multipole orders hidden in the different Coulomb phases of t he model. So far as experiment is concerned, our
+results suggest that where long range interactions are unim portant, the magnetic properties of Cr spinels which
+exhibit half–magnetization plateaux may be largelyindepe ndent of the type of magnetic order present.
+PACS numbers: 75.10.-b, 75.10.Hk75.80.+q
+I. INTRODUCTION
+Frustrated magnets have long been studied as a paradigm
+for complex behavior in condensed matter and statistical
+physics1. The most widely studied systems are frustrated an-
+tiferromagnets (AF), where competing interactions suppre ss
+classical N´ eelorder. Insomehighlyfrustratedmagnets,s pins
+do not order at anytemperature, and the ground state retains
+only very short–ranged spin–spin correlations. The result ing
+state isgenerallytermeda “spinliquid”.
+Howeveritisalsopossibletoinvertthisparadigmandthink
+ofhighlyfrustratedmagnetsassystemswherelocal“order” is
+robust enough to survive, even where long range order has
+beenobliteratedbyfluctuations. ConventionalN´ eelorder can
+then easily be restored — albeit with a relatively low critic al
+temperature — by anyperturbation which forces long–range
+coherence on this preformed local order. Moreover, where
+quantities such as heat capacity and magnetic susceptibili ty
+are controlledbylocal fluctuations,the thermodynamicpro p-
+erties of the globally–disordered spin–liquid phase may be
+practically indistinguishable from those of the magnetica lly
+orderedphase.
+In this paper we explore this “bottom–up” formulation of
+frustration,showinghowthedifferentmultipolarandspin liq-
+uid states of a simple classical frustrated antiferromagne t in
+appliedmagneticfieldalreadycontaintheseedsoflongrang e
+N´ eel order. The model which we consider is the antiferro-
+magneticnearest–neighborHeisenbergmodelwithaddition al
+biquadraticinteractions b
+H=J1/summationdisplay
+/angbracketleftij/angbracketright1/bracketleftbig
+Si·Sj−b(Si·Sj)2/bracketrightbig
+−h·/summationdisplay
+iSi,(1)
+wherethesum /an}bracketle{tij/an}bracketri}ht1runsoverthenearestneighborbondsofa
+pyrochlorelattice(Fig.1). Alltheenergyscalesincludin gh≡
+|h|andtemperature Taremeasuredinunitsof J1hereafter.
+This model was introduced in Ref. [2] to explain the dra-FIG.1: (Coloronline)16-sitecubicunitcellofthepyrochl ore lattice
+—anetworkofcorner–sharingtetrahedra. Exchange interac tionsJ1
+are associated with the first neighbor bonds /an}bracketle{tij/an}bracketri}ht1, andJ3with the
+two(inequivalent) types of third–neighbor bond /an}bracketle{tij/an}bracketri}ht3.
+matic half–magnetization plateau observed in Cr spinels3–8.
+Inthiscasethe biquadraticinteraction boriginatesin a strong
+coupling to the lattice. However such terms can also be of
+electronic origin, and quite generally they can be taken to
+characterize the effects of quantum and/or thermal fluctua-
+tions in a frustrated magnet9–11. Thus we anticipate many of
+our results will also be relevant for the quantum model. Re-
+cent results for the S= 3/2XXZpyrochloreAF in applied
+magneticfieldsuggestthatthisisindeedthe case12,13.
+It is well known that the classical Heisenberg model with
+the nearest–neighborbilinear couplingsonly does not exhi bit
+N´ eel–type magnetic order on the pyrochlore lattice at any
+temperature16,17. As we shall see, these arguments are es-
+sentially unchanged by the introduction of magnetic field, o r
+by nearest–neighbor biquadratic interactions b. The system
+canhoweverbebroughttoorderbyintroducinganinteraction2
+FIG. 2: (Color online) (a) Magnetic phase diagram of the clas sical
+pyrochlore antiferromagnet with biquadratic interaction sb= 0.1,
+and additional third–neighbor interactions J3=−0.05, as deter-
+mined by classical Monte Carlo simulation14,15. The form of four–
+sublattice N´ eel order is illustrated, together with the ir reducible rep-
+resentation (irrep) of the tetrahedral symmetry group Tdto which it
+belongs. Fortheseparameters,themodelprovidesagooddes cription
+ofthehalf-magnetizationplateauseeninCdCr 2O43–8. (b)Equivalent
+magnetic phase diagram in the absence of any longer range int erac-
+tions. Canted N´ eel states are replaced by phases with multi polar or-
+der,whilethecollinear half-magnetizationplateaustate giveswayto
+a collinear spin liquid. Crosses denote the crossover at tem perature
+T∗from paramagnet to the plateau–liquid state, as determined by a
+peak in the heat capacity. In both (a) and (b) circles with sol id (red)
+lines denote first–order phase boundaries, while those with dashed
+(blue) lines denote second–order ones. Both Tandhare measured
+inunits of J1.
+whichlinksspinsin differenttetrahedra,forexample,
+HLRO=J3/summationdisplay
+/angbracketleftij/angbracketright3Si·Sj, (2)
+where/an}bracketle{tij/an}bracketri}ht3runs over the two (inequivalent) sets of third
+neighborbondshowninFig. 1.
+Forferromagnetic(FM) J3<0,thisspecificformof HLROleadstothefour–sublatticelong-rangeorder(LRO)descri bed
+in Ref. [2], and to the finite temperature transitions shown i n
+Fig. 2(a)14,15. Four sublattice order can also be stabilized by
+AF secondneighborinteraction J2. Moregenerally,however,
+thetypeoforderwhichresultsdependsonthedetailsofthei n-
+teractionHLRO18,19. Thesystemcanthereforebetunedatwill
+between different types of ordered state, simply by changin g
+HLRO. From this we concludethat, as a functionof magnetic
+fieldh, forHLRO= 0, there must be a line of second–order
+multicritical —orfirst–order multifurcative points—separat-
+ingahugeset ofdifferentorderedphases.
+The main purpose of this paper is to explorethe symmetry
+breaking which persistsin the limit of HLRO→0for finite
+biquadraticinteraction bandfinite temperature T. In orderto
+maketheproblemaccessibletolargescaleMonteCarlo(MC)
+simulation, we consider the classical S≡ |S| → ∞limit of
+Eq.(1),rescalingvariablessuchthat S≡1.
+Using a mixture of classical MC simulation, analytic low–
+Texpansion, and simple field theoretical arguments, we find
+aset ofphasesinthe h–Tplanewhichexhibitpower–lawde-
+cayofspincorrelationfunctions. Twoofthesephasesposse ss
+long–rangenematicorvector–multipoleorderand,mostint er-
+estingly, the magnetization plateau persists in the absenc e of
+conventionalmagneticorder. Weshowhowalloftheseresult s
+can be understood — and even anticipated — from a proper
+understanding of the geometry of the pyrochlore lattice, an d
+the way in which a single tetrahedron behaves in magnetic
+field. Ourfindingsaresummarizedbythe h–Tphasediagram
+shownFig.2(b).
+Sofarasexperimentisconcerned,ourmainconclusionwill
+be that the thermodynamic properties of the pyrochlore anti -
+ferromagnet in applied magnetic field are mostly determined
+bysymmetrybreakingatthelevelofsingletetrahedron. Loc al
+order is well–formed for HLRO= 0, and many properties of
+the system are therefore insensitive to the details of the LR O
+orderpresent. Thusthe verysimple phase diagramderivedin
+Ref. [2] and itsfinite temperaturegeneralizationin Refs. [ 14]
+and [15] [reproduced in Fig. 2(a)], are applicable for a wide
+varietyofdifferent HLRO.
+The paper is structured as follows: In Sec. II we briefly
+review the basic physics of the Heisenberg model on the py-
+rochlorelattice. Definitionsare givenof orderparameters for
+conventionalN´ eel (dipolar)order,and of rank–twotensor or-
+der parameters which can be used to signal multipolar order.
+Then, in Sec. III we use these tools to construct the
+h–Tphase diagram of the pyrochlore AF with additional bi-
+quadratic interactions [Fig. 2(b)]. Thermal fluctuations p re-
+serve the extensive degeneracies present in the ground stat e,
+and fail to select any conventional long–range dipolar or-
+der. Despite this, the thermodynamic properties of the sys-
+temandthetopologyofthephasediagramareessentiallyun-
+changed — the magnetization plateau survives and nematic
+and vector–multipolephases correspondingto the two diffe r-
+entcantedstatesareshowntoexistforfieldsbelowandabove
+themagnetizationplateau[Fig.2(a)].
+In Sec. IV we explore the way in which long–range N´ eel
+order is recovered as a FM third–neighbor interaction J3is3
+b2A1
+T2
+EE+T
+0.1 0 0.25 0.20.152
+0.05h/J1 8
+0.3 7
+ 6
+ 5
+ 4
+ 3
+ 2
+ 1
+ 0T
+421
+3
+421
+3
+421
+3Λ Λ
+Λ Λ ΛA E,1 E,2
+T  ,1 T  ,2 T  ,3Λ
+1
+2 2 221
+3
+421
+3
+421
+3
+4
+FIG. 3: (Color online) (Left panel) Gound–state phase diagr am of a single classical tetrahedron as a function of magneti c fieldhand di-
+mensionless coupling constant b, taken from Ref. [2]. Solid (red) lines denote first and dashe d (blue) lines second order transitions. Spin
+configurations and relevant irreducible representations ( irreps) are shown in each case. (Right panel) Symmetries of t heA1,E, andT2irreps
+of the tetrahedral group Tdused to classify different states. Solid (red) lines have ne gative weight; hollow (blue) lines have positive weight.
+Thin(black) lines have zeroweight. SeealsoEq. (7).
+“turned on”, focusing on the half–magnetization plateau fo r
+h≈4. For small |J3|, the system now exhibits two charac-
+teristic temperature scales — an upper temperature T∗≈b
+at which the gap protecting the magnetization plateau opens ,
+and a lower temperature TN≈ O(|J3|)at which the system
+exhibits long–range magnetic order. This is contrasted wit h
+the situation for h= 0, where the system also exhibits two
+characteristictemperaturescales,butthesecorrespondt osuc-
+cessive phase transitions: a nematic transitionat TQ∼band
+a N´ eel ordering at TN∼O(|J3|). We discuss the nature of
+these transitions for J3→0, identifying a line of first–order
+multifurcative pointsatJ3= 0. And, for J3= 0, we identify
+an unusual continuous transition from the coulombic platea u
+liquidtothevector-multipolephase. Atlowtemperaturest his
+transitionappearstohavemeanfield character.
+Finally, in Sec. V we conclude with a discussion of the
+broaderimplicationsoftheseresults.
+II. DEGENERACIESIN FINITEMAGNETICFIELD
+A. Geometrical arguments
+The pyrochlorelattice (Fig. 1) is the simplest exampleof a
+three–dimensional (3D) network of corner sharing complete
+graphs. Its elementary building block is the tetrahedron, i n
+whicheverysiteisconnectedtoeveryothersite,i.e. thete tra-
+hedronisacompletegraphoforderfour. Tetrahedrainthepy -
+rochlore lattice can be divided into A and B sublattices, wit h
+eachlatticesitesharedbetweenanA-andaB-sublatticetet ra-
+hedron. The centres of the two types of tetrahedra together
+form a (bipartite)diamondlattice.63The overallsymmetryof
+thelattice iscubic.As such, the pyrochlore lattice is a natural 3D analogue of
+the 2D kagome lattice, a corner sharing network of triangles
+(complete graphs of order three). In fact the [111] planes
+of the pyrochlore lattice are alternate kagome and triangu-
+lar lattices, composed of the triangular “bases” of tetrahe dra
+and their “points”, respectively. Much of the unusualphysi cs
+of the kagome lattice also extends to its higher dimensional
+cousin.
+Lattices composed of complete graphs have the special
+propertythatbilinearquantitiesonnearestneighborbond scan
+berecastasasumofsquares. Thusfor b= 0theHamiltonian
+Eq.(1)canbewritten
+H= 4/summationdisplay
+tetra/parenleftbigg
+M−h
+8/parenrightbigg2
+−h2
+16+const.,(3)
+wherethe sumrunsovertetrahedra,and
+M=1
+4(S1+S2+S3+S4) (4)
+is the magnetization (per site) of a given tetrahedron. For
+h= 0,asimpleclassicalcountingargumentshowsthattwoof
+theeightanglesneededtodeterminetheorientationofthef our
+spins in any given tetrahedronremain undetermined. Neares t
+neighborinteractionsdonotselectoneuniquegroundstate on
+the pyrochlore lattice but rather the entire manifold of states
+for which |M|= 0ineachtetrahedron. Thus at T= 0, the
+system is disordered. For fields h < h sat= 8this conclu-
+sion is unaltered by the presence of magnetic field. In this
+case the manifold of ground states is determined by the con-
+ditionM=h/8in each tetrahedron, and the magnetization
+is linear in hup to the saturation field hsat= 8. (We recall
+that magnetic field is measured in units of J1, so that in fact
+hsat= 8J1.)4
+In order to understand how nearest–neighbor biquadratic
+interactions bselect among this manifold of states, it is suf-
+ficient to solve the problem of a single tetrahedron embed-
+ded in the 3D lattice. This problem was considered in
+Ref. [2]. For b >0, biquadratic interactions select coplanar
+(and collinear) configurations from the larger ground–stat e
+manifold of Eq. (3). There are four dominant phases, illus-
+tratedinFig.3 :
+(i) a2:2coplanarcantedstate forlowfield
+(ii) a 3:1collinear( uuud)half–magnetizationplateaustate
+forintermediatefield
+(iii) a 3:1 coplanar canted state for fields approachingsatu -
+ration
+(iv) a saturated ( uuuu) state for large magnetic field
+h > hsat
+An exhaustive enumeration of possible states is given in
+Ref.[20].
+Up to this point, we have not been specific about howthe
+tetrahedron was embedded in the lattice. It could, triviall y,
+form part of a state with N´ eel order, e.g., the simple four–
+sublattice order favored by FM J3. However, there are in-
+finitely many other ways of joining 2:2 or 3:1 tetrahedra to-
+gether at the corners, and not all of them correspond to N´ eel
+orderedstates. Infactthegroundstatemanifoldretainsan ex-
+tensiveIsing–likedegeneracyfor all h < hsat, and asa resultthe system remains“disordered”. The nature of this degener -
+acy,anditsconsequences,areexploredinsomedetailbelow .
+B. Bondorder parameters
+Where N´ eel order is present, it can be detected in the re-
+ducedspin–spincorrelationfunction
+D(rij) =/an}bracketle{tSi·Sj/an}bracketri}ht−m2. (5)
+Herem2istheexpectationvalueofthesquaredmagnetization
+perspin,
+m2=/angbracketleftig1
+N/parenleftbig/summationdisplay
+iSi/parenrightbig2/angbracketrightig
+, (6)
+whichvanishesintheabsenceofmagneticfield. Nisthetotal
+numberofspins. Thesimplestformofordersupportedbythe
+pyrochlorelatticeisthefour–sublatticeN´ eelorderfavo redby
+FMJ3,asillustratedinFig. 2(a).
+Writtenintermsoftheminimalfour–siteunitcellofthepy-
+rochlore lattice, four–sublattice order has momentum q= 0,
+and different states can easily be classified using the A1,E,
+andT2irreducible representations (irreps) of the symmetry
+groupTdfora singletetrahedron:
+
+ΛA1
+ΛE,1
+ΛE,2
+ΛT2,1
+ΛT2,2
+ΛT2,3
+=
+1√
+61√
+61√
+61√
+61√
+61√
+61√
+3−1
+2√
+3−1
+2√
+3−1
+2√
+3−1
+2√
+31√
+3
+01
+2−1
+2−1
+21
+20
+0 0−1√
+21√
+20 0
+0−1√
+20 01√
+20
+−1√
+20 0 0 01√
+2
+
+S1·S2
+S1·S3
+S1·S4
+S2·S3
+S2·S4
+S3·S4
+, (7)
+where the spins Sibelong to a single tetrahedron2,20,21— cf.
+right panel of Fig. 3. We can use these irreps to define bond
+orderparameters
+λglobal
+ν=4
+N/parenleftig/summationdisplay
+tetraΛν/parenrightig2
+, (8)
+andassociatedgeneralizedsusceptibilities
+χglobal
+ν=N
+T/bracketleftbig
+/an}bracketle{t(λglobal
+ν)2/an}bracketri}ht−/an}bracketle{tλglobal
+ν/an}bracketri}ht2/bracketrightbig
+,(9)
+where the sum/summationtext
+tetraruns over all N/4independent A–
+sublattice tetrahedra, and Λνis the vector associated with
+theν={E,T2}irrepsof the tetrahedralsymmetry group Td,
+namely,ΛE= (ΛE,1,ΛE,2)andΛT2= (ΛT2,1,ΛT2,2,ΛT2,3).
+These order parameters allow us to distinguish the sharp
+first–order transition between orders in the T2andEirreps,and the transition fromthe A1toT2states at high field — cf.
+Fig. 2(a) — but not the more subtle second–order transition
+betweenthe T2symmetry uuudplateaustateandthe T2sym-
+metry3:1cantedstate. Thesearenonethelessdistinctphas es
+—thecollinearandcanted T2statesareconnectedbyazone–
+center(i.e., q=0)excitationwhichisgappedinthecollinear
+uuudstate, and becomes soft at the critical field marking the
+onset of the 3:1 canted state20. The condensation of this soft
+spinmodecorrespondstotheemergenceoforderinthetrans-
+verse spin components /an}bracketle{tSx/an}bracketri}ht,/an}bracketle{tSy/an}bracketri}ht— i.e., canting of spins
+awayfromthe zaxis(thedirectionofappliedmagneticfield).
+The bond order parameters defined in Eq. (7) couple di-
+rectlytothelattice,throughchangesinbondlength2. Theyare
+thereforeparticularlywellsuitedtodescribingsimpleN´ eelor-
+deredstates,wherethemagneticorderingisdrivenbythela t-
+tice effects. However the irreps on which they are based also5
+providea useful measure of the correlationwhich survivesi n
+theabsenceoflongrangeorder,acentralquestionforthisp a-
+per. Tothisend,we introducea measureoflocalcorrelation
+λlocal
+ν=4
+N/summationdisplay
+tetraΛ2
+ν, (10)
+anditsassociatedgeneralizedsusceptibility
+χlocal
+ν=N
+T/bracketleftbig
+/an}bracketle{t(λlocal
+ν)2/an}bracketri}ht−/an}bracketle{tλlocal
+ν/an}bracketri}ht2/bracketrightbig
+.(11)
+For a single tetrahedron, λlocal
+νandλglobal
+νare identical. On
+a lattice, λlocal
+νlacks crossterms between different tetrahedra
+present in λglobal
+ν, and is therefore a measure of correlation in
+the absence of long range order. We return to these points
+below.
+C. Rank–twotensor order parameters
+Not all of the phases supportedby the HamiltonianEq. (1)
+can be described using the bond order parameters Eq. (7). In
+Appendix A we formally classify the different types of sym-
+metry breakingwhich can arise in this modelat the level of a
+singlesite. Herewe restrictourselvestothe simplestposs ible
+generalizationfromN´ eeltomultipolarorder;boththe T2and
+Esymmetry canted states possess order of transverse (i.e., x
+andy)spincomponentswhichvanishesinthecollinear uuud
+state, and which can survive even in the absence of conven-
+tional(canted)N´ eelorder.
+To describe this, it is convenientto introducethe rank–two
+tensororderparameters
+Qα=1
+NN/summationdisplay
+i=1Qα
+i, (12)
+wherethelocalquadrupolemoments
+Q3z2−r2
+i=1√
+3/bracketleftbig
+2(Sz
+i)2−(Sx
+i)2−(Sy
+i)2/bracketrightbig
+,(13)
+Qx2−y2
+i= (Sx
+i)2−(Sy
+i)2, (14)
+Qxy
+i= 2Sx
+iSy
+i, (15)
+Qxz
+i= 2Sx
+iSz
+i, (16)
+Qyz
+i= 2Sy
+iSz
+i, (17)
+aresummedoverall latticesites i.
+Where spin rotational symmetry is not already broken by
+magnetic field, i.e., for h= 0, spins may select a common
+axis without selecting a direction on it. This is convention al
+nematic order, of the type exhibited by uniaxial molecules,
+andcanbedetectedusingtheorderparameter
+Q2= (Q3z2−r2)2+(Qx2−y2)2
++(Qxy)2+(Qxz)2+(Qyz)2(18)
+which is invariant under O(3)rotations. This order param-
+eter takes on its maximal value /an}bracketle{tQ2/an}bracketri}ht →4/3in a perfectly
+collinearstate, suchasthe2:2state for T→0.TABLE I: Classification of tensor order operators according to ro-
+tational symmetry about a zaxis defined by magnetic field: Each
+forms anirreptransforming like einφ, wherenis aninteger and φis
+the polar angle in the xyplane. Also indicated are the finite values
+of the order parameters inthe 2:2and 3:1canted states.
+order par. tensor operators 2:2 1:3
+e2iφ{Qx2−y2,Qxy} finite finite
+eiφ{Qxz,Qyz} 0 finite
+{Sx,Sy} 0 0
+1 Q3z2−r2finite finite
+Szfinite finite
+In what follows we will also make use of the correlation
+functionmeasuringcollinearity
+P(rij) =3
+2/bracketleftbigg
+(Si·Sj)2−1
+3/bracketrightbigg
+, (19)
+considered in Ref. [17]. As defined, −1/2≤ /an}bracketle{tP(rij)/an}bracketri}ht ≤1,
+taking on the value /an}bracketle{tP(rij)/an}bracketri}ht= 0for uncorrelated spins. In
+factP(rij)canalso beexpressedintermsofquadrupolarop-
+eratorsas
+P(rij) =3
+4/summationdisplay
+αQα
+iQα
+j, (20)
+andit followsthat
+Q2=1
+N2/summationdisplay
+ij4
+3P(rij). (21)
+At finiteh,theO(3)invariantcorrelationfunctionEq.(19)
+still providesausefulmeasureofcollinearity,butdoesno tby
+itself signal a broken symmetry. In this case it is convenien t
+to group quadrupoles according to way in which they trans-
+form under the remaining O(2)rotations about the direction
+of magnetic field — conventionally the zaxis. We therefore
+consider
+Q⊥,2=/braceleftbig
+Qx2−y2,Qxy/bracerightbig
+, (22)
+Q⊥,1=/braceleftbig
+Qxz,Qyz/bracerightbig
+, (23)
+Q⊥,0=Q3z2−r2, (24)
+where the magnetic field is assumed to be parallel to the
+zaxis. Each of the separate irreps Q⊥,ntransforms like
+{cosnφ,sinnφ}— or equivalently, einφ— where φis the
+polar angle in the plane perpendicular to the magnetic field.
+They can therefore be used as order parameters to detect the
+n–fold breaking of rotational symmetry in the xyplane. The
+conventionalnematicorderparameterwith full O(3)symme-
+try,Eq.(18),isgivenbythesumofsquares
+Q2= (Q⊥,0)2+|Q⊥,1|2+|Q⊥,2|2.(25)
+In finite magnetic field, the one–dimensional irrep Q⊥,0
+does not contain any information about broken symme-6
+tries and can generally be discarded. However the two–
+dimensional irreps Q⊥,1andQ⊥,2distinguish different or-
+dered phases. In the 2:2 canted phase the mean square value
+ofQ⊥,2takesonafinite value
+/an}bracketle{t|Q⊥,2|2/an}bracketri}ht=/an}bracketle{t(Qx2−y2)2+(Qxy)2/an}bracketri}ht>0.(26)
+This is another form of nematic order of the transverse spin
+moments—onetransforminglike ei2φ—andreflectsthefact
+thatspinsselectacommonplaneinwhichtocant. Atthesame
+time mean square value of Q⊥,1— which transforms as eiφ,
+i.e.,a vectorinthe xyplane—vanishes.
+Similarly, /an}bracketle{t|Q⊥,2|2/an}bracketri}httakesonafinitevalueinthe3:1canted
+phase. However in this case the 3:1 asymmetry of the canted
+spinconfigurationdefinesadirectioninthe xyplane,and
+/an}bracketle{t|Q⊥,1|2/an}bracketri}ht=/an}bracketle{t(Qxz)2+(Qyz)2/an}bracketri}ht>0(27)
+isalsofinite. The3:1cantedphasethereforepossessaformo f
+vector-multipoleorder. ThesefactsaresummarizedinTabl eI.
+In what follows we concentrate almost exclusively on
+phases which do not exhibit conventional magnetic order, as
+definedby D(r)inEq.(5),andcharacterizethesestatesusing
+the rank–two tensor order parameters listed in Table I. For
+further details of conventional N´ eel phases, and comparis on
+with experiment, we refer the interested reader to Ref. [15] .
+Rank–three tensors which also occur as order parameters in
+thepresentmodelarediscussedinAppendixB.
+D. General considerations
+Many frustrated systems with disordered ground states
+managenonethelesstoorderatfinitetemperature. Thiseffe ct
+isknownas“orderfromdisorder”andoccurswherethereisa
+netentropygaininselectingoneparticularstateoutofthe dis-
+ordered manifold. Entropy is gained where a given spin con-
+figuration (typically, collinear or coplanar) has a higher d en-
+sity of low–energy excitations than its peers. However this
+entropy gain must be sufficient to offset the entropy lost by
+choosing one state out of the manifold. Where the ground
+state manifold has an extensive degeneracy, this is a very
+strong constraint. Order–from–disordereffects are known to
+select one particular N´ eel ordered ground state in e.g., th e
+frustrated square lattice22, but fail to do so in the case of the
+morefrustratedkagomelattice23,24.
+EvenwherefluctuationsfailtostabilizeoneparticularN´ e el
+ground state, they can still select a subset of states from th e
+groundstate manifoldwith a smaller — but none the less ex-
+tensive — degeneracy. This subset (submanifold) of states
+will not exhibit the long range spin–spin correlations whic h
+are the hallmark of conventional N´ eel–type magnetic order .
+However this does not necessarily mean that the system is
+truly disordered — it may well exhibit long range order of
+a morecomplextype.
+Agoodexampleofthissecondtypeoforder–from–disorder
+effect is provided by the nearest–neighbor classical XY
+model on the pyrochlore lattice, where thermal fluctuationslead to nematic order with broken spin–rotational symmetry ,
+butpower–lawdecayofspin–spincorrelations17.
+In what follows we use the order parameters defined in
+Sec. IIC to identify phases of Eq. (1) which exhibit nematic
+order in the absence of N´ eel order. We focus chiefly on dif-
+ferentformsof unconventionalorderfoundin magneticfield .
+Closely related studies in magnetic field have been made of
+the classical Heisenberg model on a kagome lattice23, and
+classicalXYmodelonacheckerboardlattice25. Inboththese
+cases unconventional order is stabilized by thermal fluctua -
+tions. Anothertypeofunconventionalorderforthepyrochl ore
+lattice with FM second–neighbor interactions J2<0and
+h= 0was recentlystudied in Ref. [26]. In our case the main
+drivingforce is not fluctuationsbut finite biquadratic inte rac-
+tionb; results for order stabilized by thermal fluctuations at
+finitehandJ3butb≡0will bepresentedelsewhere27.
+III. PARTIAL LIFTINGOFDEGENERACYIN FINITE
+MAGNETICFIELD
+A. Collinearnematic phasefor h= 0
+Intheabsenceofmagneticfield,thegroundstateofEq.(1)
+isdeterminedbytheconditionsthat(i)thetotalmagnetiza tion
+ofeachtetrahedronbezero,tominimizetheantiferromagne tic
+exchangeinteraction J1,and(ii)allspinsbecollinear,tomin-
+imizethebiquadraticinteraction b. Theseconditionsselectan
+extensivemanifoldof
+Ω0≈1.5N/2≈1.22N(28)
+states with exactly two–“up” and two–“down” spins ( uudd)
+ineachtetrahedron. Thedegeneracyofthisgroundstateman -
+ifoldisofthesameformasthatencounteredin Pauling’sthe -
+ory of water ice28, and we therefore refer to it as the “ice”
+manifold below. Since each spin is shared by two neighbor-
+ing tetrahedra, “up” and “down” spins form unbroken loops
+asshowninFig. 4. We returntothispointbelow.
+The fact that the direction along which “up” and “down”
+spinspoint is not determinedby the Hamiltonianimplies tha t
+spin rotational symmetry must be broken spontaneously (for
+simplicity, we none-the-less to use “up” and “‘down” to de-
+note the oppositely oriented spins). This can be seen in the
+spin collinearity Eq. (19), which takes on the maximal value
+P(r) = 1for all states in the ice manifold, implying that
+the ground state manifold has nematic (i.e., quadrupolar) o r-
+der. (This is explicitly confirmed by MC simulations below.)
+However,asalreadystated,thegroundstatemanifolddoesn ot
+possessN´ eelorderof anyform.
+In fact it is possible to calculate the asymptotic form
+of spin–spin correlations in the ice manifold by mapping
+them onto configurations of a notional electric (or magnetic )
+field29–31. The condition that every tetrahedron has exactly
+two–“up”andexactlytwo–“down”spinstranslatesintoazer o
+divergence condition for the electric (magnetic) field, and
+spin–spincorrelationstakeonadipolarform
+/an}bracketle{tSi·Sj/an}bracketri}ht ∼1
+|ri−rj|3, (29)7
+FIG. 4: (Color online) (a) Illustration of collinear E–symmetry ne-
+maticstateat h= 0,showingloop–likecoordinationofparallelspins
+associated with the “ice” manifold. (b) Four–sublattice lo ng range
+order with Esymmetry induced by FM J3. (c) Canted nematic state
+withpartialmagnetization under applied field.
+dictated by this effective electrodynamics. This power–la w
+decay of spin correlations is a signal property of the “ice”
+manifold. However it should not be taken to imply that “up”
+and“down”spinsareentirelyuncorrelated. Withineach uudd
+tetrahedron, each spin has twice as many AF aligned neigh-
+bors as FM aligned ones, and the net correlation on nearest
+neighborbondsis
+/an}bracketle{tSi.·Sj/an}bracketri}htn.n.=−1
+3. (30)
+Locally,orderiswellformed. Moreformally,wecanstateth at
+theseuuddtetrahedra belong to the two–dimensional EirrepFIG. 5: (Color online) Spin collinearity P[Eq. (19)] at h= 0for
+b= 0.1, showing the onset of nematic order at TQ≈0.13.Pis
+measured for the farthest spin pair along the /an}bracketle{t110/an}bracketri}htchains in the py-
+rochlore lattice in each system size ranging from L= 4toL= 16,
+andaveraged over the /an}bracketle{t110/an}bracketri}htchains running indifferent directions.
+of the tetrahedral symmetry group Td, defined in Sec. IIB,
+andthatlocalfluctuationsoforder λlocal
+Etakeontheirmaximal
+valueλlocal
+E= 16/3.
+Thisconcludesoursurveyofsymmetrybreakingfor T= 0,
+but it leaves open the question, what happens at finite tem-
+perature? By analogy with ordered systems where “order
+from disorder” is effective, thermal fluctuations might be e x-
+pected to select a single configuration from the “ice” mani-
+fold, and so restore N´ eel order. To address this question, w e
+have performed extensive Monte Carlo simulations using a
+local–update Metropolis algorithm to sample spin configura -
+tions. We typically perform 106MC samplings for measure-
+mentsafter 105stepsforthermalization. Wehavecheckedthe
+convergence of the results by comparing those for different
+initial spin configurations. In particular, to minimize the hys-
+teresis associated with first order transitions, we used mix ed
+initial conditions in which different parts of the system ar e
+assigned different ordered or disordered states32. Where the
+acceptance rate in MC updates becomes extremely slow, we
+usedthe exchangeMC method33, to avoidlocal spin-freezing
+at low temperatures. Results are divided into five bins to es-
+timate statistical errors by variance of average values in t he
+bins. The system sizes in the present work are up to L= 16,
+whereLis the linear dimension of the system measured in
+the cubic unitsshownin Fig. 1, i.e., the total numberofspin s
+Nis given by 16L3. We show the results for b= 0.1and
+b= 0.6, case by case, both of which exhibit qualitatively the
+samebehavior.
+Intheabsenceofappliedmagneticfield,thespincollinear-
+ityP(r)grows sharply below a transition temperature
+TQ≈b, as illustrated in Fig. 5. As expected, for T→0,
+P(r)→1, implying that all spins have a single common
+axis. The nematic order parameter Q[Eq. (18)] is plotted
+in Fig. 6(a), together with the heat capacity in Fig. 6(b) for a
+range of system sizes from L= 4toL= 16. Here the heat
+capacityiscalculatedbythe fluctuationofinternalenergy as
+Cv=/an}bracketle{tH2/an}bracketri}ht−/an}bracketle{tH/an}bracketri}ht2
+T2N. (31)8
+1.6 2.4 3.2 4.0 0.0 0.2 0.4 0.6 0.8 1.0 0.08 0.10 0.12 0.14 0.16 0.18 
+4
+6
+8
+12 
+16 LQT
+(a) 
+0510 15 20 Cv(b) 
+0.00 0.02 0.04 0.06 0.08 0.10 λλ(c) 4
+6
+8
+12 
+16 L
+4
+6
+8
+12 
+16 L
+0200 400 600 800 
+0.08 0.10 0.12 0.14 0.16 0.18 
+T(d) 
+4
+6
+8
+12 
+16 Lχ
+FIG. 6: (Color online) (a) Nematic order parameter Q[Eq. (18)],
+showingtheonsetofnematicorder at TQ≈0.13; (b)Heatcapacity;
+(c) Absence of long range four–sublattice order λglobal
+Eis accompa-
+nied by well–formed local correlations λlocal
+E; (d) The associated lo-
+cal susceptibility shows a sharp jump at TQ, where spins in tetrahe-
+dra with preformed local order gain energy by selecting long range
+collinearity. All data are for h= 0,b= 0.1, and for system size
+ranging from L= 4toL= 16.
+The sharp onset of order and jump in heat capacity imply a
+first orderphasetransitionat TQ≈0.13.
+Treating this nematic order at the level of a Ginzburg-
+Landau theory, the free-energy terms allowed by lattice and
+spinrotationalsymmetriesare
+F=aQ2+bQ3+cQ4+... , (32)
+whereQ2is defined in Eq. (18), the third order invariant Q3FIG. 7: (Color online) (a) Spin correlations D[Eq. (5)] at h= 0
+forb= 0.6, forr≤Lmeasured along /an}bracketle{t110/an}bracketri}htchains in the py-
+rochlore lattice and averaged over different chain directi ons as in
+Fig. 5.ris in units of the distance between nearest neighbor spins.
+(b) Characteristic 1/r3power–law decay of spin correlations in the
+low–temperature nematic phase for T < T Q∼0.5associated with
+the ice manifold of 2:2 states, plotted on a log–log scale. (c ) Expo-
+nential decay of spin correlations in the high–temperature paramag-
+netic phase for T > T Q∼0.5, plotted on a log–linear scale. In (b)
+and (c), grey lines are guides for the eye, ∼1/r3and∼exp(−r),
+respectively. Inallfiguresthedataarefortemperatures ra ngingfrom
+T= 0.30toT= 0.84andthe system size L= 16.
+isgivenby
+Q3= 2(Q3z2−r2)3+3Q3z2−r2/bracketleftbig
+(Qxz)2+(Qyz)2/bracketrightbig
+−6Q3z2−r2/bracketleftig
+(Qx2−y2)2+(Qxy)2/bracketrightig
++3√
+3/parenleftig
+Qx2−y2/bracketleftbig
+(Qxz)2−(Qyz)2/bracketrightbig
++2QxyQxzQyz/parenrightig
+. (33)
+In a three–dimensional uniaxial nematic state, such as that9
+FIG. 8: (Color online) Spin correlations spanning the nemat ic state
+andhightemperatureparamagnetplottedasafunctionofthe rescaled
+distancer√
+Ton (a) log–log scale and (b) log–linear scales. The
+data are identical to those plotted in Figs. 7(b) and (c); gre y lines
+are guides for the eye, showing ∼1/r3and∼exp(−r), respec-
+tively. Temperatures should be compared with the nematic or dering
+temperature TQ≈0.5.
+realized here, all quadrupoles moments Qnare proportional
+to a simple scalar Q. The presence of a cubic term in the
+freeenergyEq.(32)thereforeimpliesthatthephasetransi tion
+from nematic phase to paramagnet as a function of tempera-
+turemustbefirst order— asobservedinthe MCresults.
+Inprinciple,thermalfluctuationsmightselectasingleN´ e el
+statefromtheicemanifold,inwhichcase TQwouldmarkthe
+onsetofdipolaras wellas quadrupolarorder. Howeverthisi s
+not the case. Spin–spin correlations D(r), defined in Eq. (5),
+remain short ranged [Fig. 7(a)]. At the distances accessibl e
+to simulation, they rapidly cross over from the power–law
+decay characteristic of the ice manifold at low temperature s
+[Fig. 7(b)] to the exponential decay expected for a paramag-
+net[Fig.7(c)].
+The reason that the usual order–from–disordermechanism
+is ineffective in selecting dipolar order is the massive deg en-
+eracy of the ice manifold — the entropy gain of fluctuations
+aboutthefavoredstate(relativetotheaverage)wouldhave tocompensateforthelossofanextensiveentropyof
+lnΩ0/N∼0.5ln1.5≈0.20
+perspin. We returntothispointbelow.
+At finite temperature, the algebraic decay of spin corre-
+lationsD(r)∼1/r3in the Coulomb phase is expected
+to crossover to exponential decay D(r)∼exp(−r/ξc)for
+r/greaterorsimilarξc, where the characteristic length scale ξcdiverges
+forT→0. For Heisenberg spins in three dimensions,
+ξc∼1/√
+T31,34,35. This is the only length scale in the sim-
+plest Coulomb theory, and it is therefore interesting to plo t
+thespincorrelations |D(r)|forarescaleddistance r√
+T. This
+is done in Fig. 8. At low temperatures T < T Q, the data
+appear to collapse onto a single power–law behavior in this
+range,whiletheycollapseontoanexponentialbehaviorabo ve
+TQ: Thereis a rapid changebetweenthese behaviors,associ-
+atedwith the discontinuoustransitionat T=TQ. Theresults
+suggest that the Coulomb–phasetheoryappliesto the presen t
+bilinear–biquadraticmodel,andinaddition,thatthechar acter-
+istic length ξcsuddenlychangesfromseveral lattice spacings
+in the nematic phase for T < T Qto one comparable to the
+latticespacingintheparamagneticphasefor T > T Q.
+Once again, these results have a simple interpretation in
+terms of local, preformed order. In Fig. 6(c) we plot the ex-
+pectationvalueoftheorderparameterforthesimplestkind of
+four–sublattice order, λglobal
+E[Eq. (8)]. This clearly scales to
+zerowith systemsize. Howeverthereisa sharpfeaturein the
+susceptibility associated with λlocal
+E[Eq. (10)] at TQ, shown
+in Fig. 6(d), where tetrahedra with localEsymmetry collec-
+tively choose collinear configurations. Indeed, as T→0,
+λlocal
+Etakes on its maximum allowed value of λlocal
+E→16/3
+[Fig.6(c)],asrequiredforloopsofperfectlycollinearsp ins.
+B. Nematic phasewithlocal Esymmetry
+In applied magnetic field, the “up” and “down” spins of
+thecollinearnematicphase immediately“flop”into theplan e
+parallel to h, and transform into the 2:2 canted coplanar con-
+figurations shown in Fig. 3. Such canting is entirely com-
+patible with the ice manifold, as is illustrated in Fig. 4(c) —
+entire loops of spins cant simultaneously, to give a state wi th
+smoothlyevolvingmagnetization,but noN´ eel order.
+The correlation function P(r)retains a finite (reduced)
+value in this new canted manifold of states. However spin
+rotationalsymmetryisnowexplicitlybrokenbythemagneti c
+field, so this does not of itself imply nematic order. Nematic
+order is none the less present, in the selection of a common
+planewithinwhichthespinscant. Thisisequivalenttothes e-
+lection of a direction (but not an orientation)in the xyplane,
+and long range order can now be observed in the transverse
+momentQ⊥,2definedbyEq.(22), asdiscussedin TableI.
+Since this director breaks the residual O(2)symmetry,
+the resulting nematic state must possess a branch of gapless
+(Goldstone) modes associated with rotations of the plane of
+cantingaboutthe zaxis. It is worth notingthat a cantedN´ eel
+statewith E–typesymmetrywouldbreakrotationalsymmetry
+inthesameway15. Howeverfor HLRO→0,simulationsshow10
+0.0 0.2 0.4 0.6 0.8 0.06 0.08 0.10 0.12 0.14 0.16 
+4
+6
+8
+12 
+16 LT
+(a) 
+h=1.2 
+h=2.4 
+04812 16 
+0.06 0.08 0.10 0.12 0.14 0.16 Cv
+T4
+6
+8
+12 
+16 L(b) 
+h=1.2 h=2.4 Q,2 
+FIG. 9: (Color online) (a) Nematic order parameter Q⊥,2=|Q⊥,2|
+[Eq.(22)],signalingthe nematic long–range order infinite magnetic
+field; (b) Heat capacity [Eq. (31)]. Data are at h= 1.2andh= 2.4
+forb= 0.1, andfor system size ranging from L= 4toL= 16.
+thatspin–spincorrelationsretaintheirpower–lawcharac terat
+low temperatures, implying the absence of long–range N´ eel
+order.
+The transition from local– Esymmetry nematic state to
+collinear state as h→0is completely smooth, and the finite
+Tpropertiesofnematicstateatfinite harequalitativelyiden-
+ticaltothoseshowninFigs.6and7,withtheobviouscaveats
+thatP(r)<1forT→0, and the collinear order parameter
+Qmust be replaced by Q⊥,2=|Q⊥,2|. Once again the on-
+set of nematic order at TQis associated with a sharp peak in
+heatcapacity,ariseinthelocalfluctuationswith Esymmetry,
+λlocal
+E,andtheabsenceoflongrangeorderoftheform λglobal
+E.
+A suitablefreeenergytodescribethisnematicstate is
+F=a2|Q⊥,2|2+c22|Q⊥,2|4+e222|Q⊥,2|6,(34)
+which permits both first and second order phase transitions
+into a paramagnetic phase as a function of temperature, de-
+pending on the sign of c22. However MC simulations sug-
+gest that the transition remains first order. Figure 9 shows
+the temperature dependences of the nematic order parameter
+Q⊥,2=|Q⊥,2|[Eq.(22)]andtheheatcapacity[Eq.(31)]for
+h= 1.2andh= 2.4. For both cases, the order parameter
+exhibitsa sharponsetandtheheatcapacityshowsajump,in-
+dicatingthat the E–symmetrynematictransition isof the first
+order, as for h= 0in Fig. 6. As noted by comparing the
+results for h= 1.2andh= 2.4, the discontinuity becomes
+cleareras hincreases.FIG. 10: (Color online) Dependence of the magnetization m
+[Eq. (6)] on magnetic field hforb= 0.1andJ3= 0, showing the
+existence of the magnetization plateau in the absence of lon g–range
+N´ eel order. Symbols show the result of Monte Carlosimulati ons for
+temperatures ranging from T= 0.04toT= 0.24and the system
+sizeL= 8. Thedashed lineistheresultobtainedbyminimizingthe
+energyfor T= 0.
+In principle the E–symmetry nematic state could interpo-
+late to saturation, simply by canting all spins until they ar e
+alignedwith the magneticfield. Howeverthis is not energeti -
+callyfavorableatthelevelofasingletetrahedron(Fig.3) ,and
+fora magneticfield h≈3, the system undergoesa first order
+transition into a state with magnetization m= 1/2, seen as
+the plateau in Fig. 10. This state is discussed in detail in th e
+sectionbelow.
+C. Plateauliquidwithlocal T2symmetry
+Thehalf–magnetizationplateauxobservedinCrspinelsare
+associated with collinear states with three–up and one–dow n
+spinpertetrahedron. Thereareinfactan extensivenumber
+Ω0≈1.7N/4≈1.14N(35)
+of suchuuudstates — a manifold isomorphic to hard–core
+dimer coveringsof the diamond lattice formed by joining the
+centersof tetrahedra36. (Dimerson bondsof the diamondlat-
+tice correspond to the down spins in uuudstates on the py-
+rochlorelattice.) We thereforerefer to it as the “dimer” ma n-
+ifold below. Collinear uuudstates with and without simple
+N´ eelorderareillustratedin Fig.11.
+It is possible to construct a field theory for the dimer man-
+ifold atT= 0by exact analogy with the treatment of the
+ice manifold above. The conditionthat every tetrahedronha s
+exactly three–up and exactly one–down spins translates int o
+a zero divergence condition for an electric (magnetic) field ,
+modifiedtoincludea sourceterm13.
+Once again, thermal fluctuations are ineffective in restor-
+ing long–range N´ eel order. Reduced spin–spin correlation s
+Eq. (5) at finite distance exhibit a crossover between a dipo-
+lar form [cf. Eq. (29)] for low temperatures, and exponentia l11
+FIG. 11: (Color online) Half–magnetization plateau states (uuud
+states) on a pyrochlore lattice with exactly three–up and on e–down
+spins per tetrahedron. (a) Schematic picture of an uuudstate with
+nolongrangeorder,associatedwiththe“dimer”manifold; ( b)uuud
+state with long–range four–sublattice order with T2symmetry in-
+duced by FM J3, as considered in Refs. [2] and [15]; (c) 16–
+sublattice order. See alsoFig.15.
+decay for high temperatures — see Fig. 12. While spins are
+perfectlycollinearatlowtemperatures[Fig.14(a)],the zaxis
+isnowsingledoutbymagneticfield,and Q3z2−r2contributes
+toP(r).
+This means that there is no symmetry breaking associated
+with the smoothrise in collinearityfor T∗≈b, which should
+be regardedasa crossoverratherthana phasetransition. Si n-FIG.12: (Coloronline)Absenceoflong–rangemagneticorde rinthe
+plateauliquidstate for b= 0.6,J3= 0, andh= 4. (a) The reduced
+spin correlation function D(r), defined by Eq. (5), measured along
+the/an}bracketle{t110/an}bracketri}htchains, as in Fig. 7(a). (b) Characteristic 1/r3power–law
+decay of spincorrelations at low temperatures T < T∗∼0.6, asso-
+ciated with the dimer manifold of uuudstates, plotted on a log–log
+scale. (c)Exponentialdecayofspincorrelationsathighte mperatures
+T > T∗∼0.6, plotted on a log–linear scale. In (b) and (c), grey
+lines show guides for the eye, ∼1/r3and∼exp(−r),respectively.
+Inallfigures thedataarefortemperatures rangingfrom T= 0.30to
+T= 0.84andthe system size L= 16.
+gular features are similarly absent from the heat capacity,
+shown in Fig. 14(b). This smooth change is also seen in the
+rescaled plot of the spin correlations shown in Fig. 13. The
+crossoverfromthe low– Tpower–lawbehaviorto the high– T
+exponentialdecayismuchmoresmoothcomparedtothecase
+for the nematic transition at h= 0in Fig. 8. This suggests a
+smooth growth of the characteristic length scale ξc∼1/√
+T
+atT∼T∗≈b. We therefore conclude that the magnetiza-
+tionplateauisatruespin–liquidstate,continuouslyconn ected
+with the high– Tparamagnet. We refer to this as the plateau
+liquidbelow.12
+FIG.13: (Coloronline)Spincorrelationsspanningtheplat eauliquid
+and high temperature paramagnetic phases, plotted as a func tion of
+the rescaled distance r√
+Ton (a) log–log scale and (b) log–linear
+scales. The data are identical to those plotted in Figs. 12(b ) and (c);
+grey lines are guides for the eye, showing ∼1/r3and∼exp(−r),
+respectively. Temperatures should be compared with the cro ssover
+scaleT∗≈0.6.
+Itisinterestingtonotethat,despitetheabsenceof anykind
+oflongrangeorder,thedefiningpropertyoftheplateauliqu id
+— its magnetization (Fig. 10) — is almost indistinguishable
+fromthoseofthefour–sublatticeorderedstate15. Long–range
+four–sublatticeorder λglobal
+T2isexplicitly absent— the plateau
+liquidpossess the full A1symmetryof the paramagnet. None
+the less there is a marked rise in local T2order of individual
+tetrahedra λlocal
+T2atT∗≈baccompaniedbyabroadpeakinits
+susceptibility, as shown in Figs. 14(c) and (d). This curiou s
+spin–liquidstate clearlydeservesfurtherstudy.
+To this end, we have performed low– Texpansions of the
+free energy of many different ordered and disordered uuud
+states. Thesearecontrolledexpansionsaboutthegroundst ate
+inpowersof Tforaspinoflength S= 1,wherewewritethe
+energy
+H=E0+1
+2/summationdisplay
+i,jδSiMijδSj+... , (36)20 60 100 
+0.10 0.12 0.14 0.16 0.18 0.20 0.22 
+TCv
+1.0 1.5 2.0 2.5 4
+6
+8
+12 
+16 L(a) 
+(b) λλ(c) 4
+6
+8
+12 
+16 L
+4
+6
+8
+12 
+16 L
+(d) 
+4
+6
+8
+12 
+16 Lχ0.0 0.2 0.4 0.6 0.10 0.12 0.14 0.16 0.18 0.20 0.22 PT
+12345
+0.00 0.02 0.04 0.06 0.08 0.10 
+FIG. 14: (Color online) Temperature dependence of (a) colli near-
+ity [Eq. (19), cf. Fig.5], (b) heat capacity [Eq.(31)], (c) t he related
+measureof localcorrelation λlocal
+T2definedbyEq. (10),andthe global
+order parameter λglobal
+T2defined by Eq.(8), and (d) the associated lo-
+cal susceptibility. Simulations were performed for h= 4,b= 0.6,
+inclusters with L= 4toL= 16.
+intermsofthefluctuations
+δS= (δSx
+1,δSx
+2,...δSx
+N,δSy
+1,δSy
+2,...δSy
+N)(37)
+about a given uuudconfiguration. The leading fluctuation
+contributiontothefreeenergycanthenbecalculatedinter ms
+ofthetraceovereigenvaluesofthe 2N×2NmatrixMinthe13
+TABLEII:Fluctuationentropyper sitecalculatedforrando mly gen-
+erateduuudstatesinan N= 1024cluster. Here smin
+f,smax
+f,and/an}bracketle{tsf/an}bracketri}ht
+are the lowest, highest, and mean value of the entropy, respe ctively,
+and∆sf=smax
+f− /an}bracketle{tsf/an}bracketri}htmeasures the deviation of the highest value
+of entropy from the mean. Statistical errors on all numbers a re less
+than10−6.
+b smin
+f smax
+f /an}bracketle{tsf/an}bracketri}ht ∆sf
+0.05 -0.79931 -0.79608 -0.79837 0.00228
+0.1 -1.63451 -1.63191 -1.63374 0.00183
+0.2 -2.54944 -2.54758 -2.54888 0.00130
+0.3 -3.12864 -3.12725 -3.12823 0.00098
+0.4 -3.56055 -3.55948 -3.56025 0.00077
+0.5 -3.90757 -3.90672 -3.90734 0.00062
+0.6 -4.19874 -4.19806 -4.19857 0.00051
+form
+F
+N=E0
+N−TlnT+T
+2N/an}bracketle{tlndetM/an}bracketri}htΩ0
+−T
+NlnΩ0+O(T2), (38)
+whereE0is the ground state energy, Ω0its degeneracy, and
+/an}bracketle{t.../an}bracketri}htΩ0the averageoverall degenerategroundstates.
+For a generic ordered phase, Ω0is finite, and detMtakes
+onthesamevalueforall (symmetryrelated)groundstates. I n
+this caselnΩ0/N→0forN→ ∞. However for the dimer
+manifold, Ω0≈1.14N,whichmeansthatthegroundstatehas
+afiniteentropypersite
+S0
+N≈ln1.14≈0.13. (39)
+In this case, different ground states are not related by simp le
+lattice symmetriesandthefluctuationentropypersite
+sf=−lndetM
+2N(40)
+takesona rangeofvalues.
+We have studied the distribution of values of sfwithin
+the dimer manifold for a range of values of b, by numeri-
+cally calculating detMfor10000randomlygenerated uuud
+states in a cluster of N= 1024 sites (L= 4), using a
+Monte Carlo algorithm based on loop updates of spins. We
+found that the highest value of sfis achieved by an eight–
+fold degenerate, 16–sublattice “ R-state”13, in which the four
+A–sublattice tetrahedra within the 16–site cubic unit cell of
+the pyrochlore lattice take on all four possible uuudconfig-
+urations [Fig. 11(c)]. This state has overall cubic symmetr y,
+and is actually observed in the plateau phase of HgCd 2O46.
+The lowest value of sfis achieved by the four–sublattice or-
+der shown in Fig. 11(b). The calculated values of the maxi-
+mumandminimumvalues smax
+fandsmin
+farelisted inTableII
+togetherwith the mean value /an}bracketle{tsf/an}bracketri}htand the differencebetween
+smax
+fandthemean /an}bracketle{tsf/an}bracketri}ht,∆sf.
+Fromtheseresultsitisimmediatelyclear whythermalfluc-
+tuationsalonefailtoselectauniquegroundstateforanyva lueP (a.u.) 0 50 100 150 200 250
+Nflip-4.1988-4.1986-4.1984-4.1982-4.1980sf0300600P (a.u.)
+16 subl.
+4 subl.
+b=0.6N=1024(a)
+(b) (c)
+FIG. 15: (Color online) (a) Probability distribution of the flippable
+hexagons withinthedimer manifold. (b)The fluctuationentr opy per
+sitesf[Eq.(40)]asafunctionofthenumberof“flippable”hexagons .
+The lower bound sf=−4.19874is set by the four–sublattice state
+shown in Fig.11(b), which has no flippable hexagons. The uppe r
+boundsf=−4.19806is set by the 16–sublattice state with the
+maximum number of flippable hexagons [see Fig.11(c)]. The bl ue
+dots represent a sample of 10000 random configurations. (c) P rob-
+ability distribution of the fluctuation entropy per site sfwithin the
+dimer manifold. All results are for a cluster of N= 1024sites with
+b= 0.6.
+ofbconsideredin this paper. The fluctuation entropyper site
+gained by choosing the cubic 16–sublattice state is miserly ,
+forexample, ∆sf= 0.00183forb= 0.1and∆sf= 0.00051
+forb= 0.6. These numbers must be compared with the ex-
+tensiveentropy S0/N≈0.13oftheliquidphase, allofwhich
+is lost if the system orders. So for the values of bconsidered
+here,thermalfluctuationscannotdrivethesystemto order.
+Howeveritisamusingtonotethattheentropygain ∆sfin-
+creasesasbdecreases,scalingapproximatelyas lnb,asshown
+inTableII. Thisraisestheintriguingpossibilitythat bactsasa
+singular perturbation,and that for sufficiently small b, fluctu-
+ationsmightovercometheextensiveentropy S0/N≈0.13of
+the dimer manifold, driving the system order — even though
+it isdisordered forb= 0. Such an order-from disorder ef-
+fect would presumably favor the cubic 16–sublattice R-state,
+which is also believedto be selected by quantumfluctuations
+atT= 012,13,37. Howeverinthepresentmodel,itwouldoccur
+only for vanishingly low temperatures, and would therefore
+be extremely difficult to access in simulation. This questio n
+remainsforfuturestudy.
+The result above explains why the system does not order
+at finite temperature, but not whythe fluctuation entropy fa-
+vors the 16–sublattice state? We can answer this question
+by looking at the distribution of the fluctuation entropies sf
+within the dimer manifold of uuudstates. Figure 15 shows
+the distribution for b= 0.6. Theuuudstates can be broken
+up into classes of states with a different net flux of an effec-
+tivemagnetic(or,equivalently,electric)field12,13,29–31,37. This14
+flux is conservedby all cyclic exchangesof spins on loopsof
+alternating uanddspins, motivating a loop expansion of the
+fluctuation contribution to the free energy38–42. The leading
+term in suchan expansioncountsthe number Nflipof six–site
+hexagonalringsina“flippable” u–d–u–d–u–dconfiguration.
+In Fig. 15(b) we plot the fluctuation entropy sfas a func-
+tion ofNflip. The highest (lowest) values are achievedfor the
+16–sublattice (four–sublattice) ordered states with the m ost
+(least) flippable hexagons. The fact that randomly generate d
+uuudstates lie extremely close to the line connecting these
+twostatessuggeststhatloopsofmorethansixsitescontrib ute
+little tothe fluctuationentropy.
+The remaining question is why the overall difference in
+fluctuationentropybetweendifferent uuudstatesissosmall?
+We can express the fluctuation entropy in terms of the 2N
+eigenvalues {εn}ofMas
+sf=−1
+2N/summationdisplay
+nlnεn. (41)
+The eigenvalue spectrum {εn}associated with the simplest
+q= 0four–sublattice uuudstate can easily be calculatedan-
+alytically; working with a four–site unit cell, there are fo ur
+bands, one of which is nondispersive. The associated densit y
+of states (DOS) for b= 0.6is shown in Fig. 16(a), where
+the flat band appears as a sharp peak at ε= 16b+h−4.
+InFigs.16(b)–(d)wecomparetheintegratedDOSfromthese
+fourbandswithnumericalresultsforthe integratedDOS ofa
+1024–sitecluster.
+TheintegratedDOS,averagedwithinthedimermanifoldof
+uuudstates[Fig.16(d)],isindistinguishablebyeyefromthat
+of the four–sublattice state [Fig. 16(b)]. The step associa ted
+withtheflatbandsurvivesasasetof N/4localizedexcitations
+atε= 16b. And,critically,thegap
+∆ = 8b+4−/radicalbig
+(8b+4)(8b+4−h)+h2(42)
+to the lowest lying excitation is set by a nodeless eigenvec-
+tor, whose components δSα
+idependonly on whetherthe spin
+Sipoints up or down. All uuudstates can be made formally
+equivalent to four–sublattice order by renumbering the sit es
+ineachindividualtetrahedron,andtheenergyofthisnodel ess
+excitationisalsounchangedbythisrenumberingofsites. I tis
+thereforecompletely insensitive to whether or not the syst em
+is ordered. From the results it is clear why the thermody-
+namic properties of the plateau liquid state, and in particu lar
+the entropy associated with fluctuations about it, are so clo se
+tothoseoftheorderedplateaustate.
+Fromtheseresults,itisalsopossibletounderstandwhythe
+numericallydeterminedentropygain ∆sfincreases as b→0
+forh= 4(cf. Table II). This singular behaviorcan be traced
+backtoabandofexcitationsabovethespin-wavegap ∆≈4b,
+withbandwidth ∆ε∼b,whichcollapsestobecomeastrictset
+ofzeromodesfor b→0. Sincezeromodesareexcludedfrom
+thesumwhichdetermines ∆sf,whilethecollapsingbandcon-
+tributes as ∼lnb,bacts as a singular perturbation, and in-
+finitesimal bmaydrivethesystemtoorder. Thisisdespitethe
+fact that it is disorderedfor b= 0, and for the relativelylarge
+busedin oursimulations.FIG. 16: (Color online) (a) Density of states (DOS) for eigen values
+ofMfor four–sublattice uuudstate in thermodynamic limit,show-
+ingfinitegap ∆[Eq.(42)]andflatband at ε= 16b. Integrated DOS
+(IDOS) (b) of four–sublattice uuudstate (points), (c) of a typical
+disordered uuudstate (points), and (d) averaged within disordered
+uuudstates (dashed line). In all cases J3= 0,h= 4, andb= 0.6,
+and a cluster size is N= 1024. The integrated DOS corresponding
+to(a)is shown inshading on(b)–(d) for comparison.
+D. vector–multipole phasewithlocal T2symmetry
+At the uppercritical field of the magnetizationplateau, the
+collinearspinsofthe uuudconfigurationscantawayfromthe
+zaxis. This instability occurs at the level of a single tetrah e-
+dron(Fig.3),whereit iscontinuous. Ona lattice, it is asso ci-
+ated with theclosing ofthe gap ∆[Eq.(42)] in the excitation
+spectrum of the plateau liquid. Because of the special struc -
+ture of this excitation, discussed above, the gap closes at t he
+samevalueof hc= 4+8bforalluuudstates, andthetransi-
+tionisonceagaincontinuous— at least for T= 0. However,
+sincethespinconfigurationsinquestionaresimply3:1cant ed
+versionsofthe uuudstates,withlocal T2symmetry,allofthe
+entropicargumentspresentedabovefortheplateauliquids till
+hold. Thermalfluctuationsalonecannotrestore(canted)N´ eel
+order, and spin–spin correlations exhibit a power–law deca y
+of1/r3forT→0.
+The resulting state does however exhibit long range order
+inboththe rank–twotensororderparameters Q⊥,1andQ⊥,2
+[Eqs.(22)and(23),andTableI]. The3:1cantingofthe uuud
+spinsselectsadirectioninthe xyplane,andtheprimaryorder
+parameter is therefore the lower–symmetry irrep, Q⊥,1. The
+finite value of the nematic order parameter Q⊥,2reflects the
+fact that this canting is coplanar. Since Q⊥,1transforms like
+avectorunderrotationsaboutthe zaxis,weclassifythisstate
+asavector–multipolephasewith local T2symmetry.
+Within the framework of a Ginzburg–Landau theory, the
+contributionto the free energyfrom this pair of order param -
+etersis
+F=a1|Q⊥,1|2+a2|Q⊥,2|2
++b12/parenleftig
+Q⊥,2
+1/bracketleftig
+(Q⊥,1
+1)2−(Q⊥,1
+2)2/bracketrightig
++2Q⊥,2
+2Q⊥,1
+1Q⊥,1
+2/parenrightig
++c11|Q⊥,1|4+2c12|Q⊥,1|2|Q⊥,2|2+c22|Q⊥,2|4,
+(43)15
+FIG. 17: (Color online) Magnetic field dependence of (a) prim ary
+order parameter Q⊥,1=|Q⊥,1|[Eq. (23)] and (b) secondary order
+parameter Q⊥,2=|Q⊥,2|[Eq. (22)] in the vector–multipole phase
+forb= 0.1. The continuous transition from the plateau liquid state
+into the vector–multipole phase at lower magnetic fields, an d the di-
+rect transition from the vector–multipole phase into the (s aturated)
+paramagnet at higher fields are clearly visible. Dashed line s show
+behaviorat T= 0inasingle-tetrahedrontheory. Pointsshowresults
+of MC simulations for the system size N= 8fromT= 0.02to
+T= 0.08.
+where, following Eqs. (22) and (23), Q⊥,2
+1=Qx2−y2,
+Q⊥,2
+2=Qxy,Q⊥,1
+1=Qxz,andQ⊥,1
+2=Qyz.
+Eq. (43) should be contrasted with the form of free en-
+ergy in the absence of magnetic field, Eq. (32). The cubic
+invariantQ3survivesasaninteraction b12betweenQ⊥,1and
+Q⊥,2, which transforms like e2iφe−iφe−iφ∼1under rota-
+tions about the zaxis. This means that components of Q⊥,2
+couplelinearlyto a quadraticcombinationofthe component s
+ofQ⊥,1. Because of this, a finite valueof the (lower symme-
+try)orderparameter Q⊥,1,immediatelyinducesafinitevalue
+ofthe(highersymmetry)orderparameter Q⊥,2.
+In principle Eq. (43) permits both first and second order
+phase transitions into the vector–multipolephase from dis or-
+dered (paramagnetic) or pure nematic phases, depending on
+the signofthe coefficients c11,c12, andc22. Thefull solution
+forQ⊥,1andQ⊥,2isfurthercomplicatedbythefactthatthese
+order parameters also couple to octupolar spin moments (see
+Appendix B for details). However the relationship between
+Q⊥,1andQ⊥,2is clear at the level of a single-tetrahedron
+theory(cf. Ref.[2]).
+Within the theory for a single, embedded tetrahedron —0.0 0.5 1.0 1.5 2.0 2.5 
+0200 400 600 
+0.016 0.032 0.048 0.064 
+T04812 0.0 0.1 0.2 0.3 0.4 0.5 0.016 0.032 0.048 0.064 TCv4
+6
+8
+12 
+16 L(a) 
+(b) λλ(c) 4
+6
+8
+12 
+16 L
+4
+6
+8
+12 
+16 L
+(d) 
+4
+6
+8
+12 
+16 Lχ Q,1 
+0.00 0.02 0.04 0.06 0.08 0.10 
+FIG. 18: (Color online) Temperature dependence of (a) the pr imary
+order parameter Q⊥,1=|Q⊥,1|defined by Eq. (23), showing the
+onset of the vector–multipole order at T=TV≈0.042, (b) heat
+capacity Eq.(31), (c) the related measure of localcorrelation λlocal
+T2
+definedbyEq.(10),andthe globalorderparameter λglobal
+T2definedby
+Eq.(8), and (d) the associated local susceptibility Eq.(11 ). Simula-
+tions were performed for h= 6,b= 0.1, in clusters with L= 4to
+L= 16.
+which is exactforT= 0— the primary order parameter
+Q⊥,1=|Q⊥,1|growsas
+Q⊥,1(h/greaterorsimilarhc)|∼=3/radicalbig
+2(3−2b)/radicalbig
+h−hc,(44)
+while the secondary order parameter Q⊥,2=|Q⊥,2|grows16
+moreslowlyas
+Q⊥,2(h/greaterorsimilarhc)∼=3
+2(3−2b)(h−hc). (45)
+The results of this theory for the T2vector–multipole phase
+are shown by the dashed lines in Fig. 17. For the value of
+bused in the present study, the zero temperature transition
+fromT2vector–multipole phase to paramagnet at high field
+isstronglyfirst order,evenat thelevelofa single-tetrahe dron
+theory— cf. Fig.3— andremainsso throughout.
+The nature of the finite temperature transition from the
+vector–multipole phase into the paramagnet is harder to de-
+termine. However, as shown in Fig. 17, it appears to be first
+order for all h > h∗, where(T∗,h∗)≈(0.1,5.2)marks the
+pointatwhichthecrossoverline T∗joinstheboundaryofthe
+vector–multipole phase, TV, as shown in Fig. 2(b). On the
+basisofourresults,weconsiderthatthereisatricritical point
+at(T∗,h∗)where the nature of the phase transition into the
+T2vector–multipole phase changes from continuous to first
+order.
+In Fig. 18 we present MC simulation results for the finite–
+temperature transition into the vector–multipole phase fo r
+h= 6.0> h∗. The primary order parameter Q⊥,1be-
+comes nonzero with a sharp jump at a transition temperature
+TV≃0.042[Fig. 18(a)]. Both heat capacity and local T2
+susceptibility show a jump at the transition TV[Figs. 18(b)
+and (d)], but no sign of long range order in the bond–order
+parametergivenbyEq.(7) [Fig.18(c)].
+The finite temperature transition from the plateau liquid to
+T2vector–multipolephasefor h < h∗deservesspecial atten-
+tion, since the plateau liquid exhibits algebraic decay of c or-
+relationsfor intermediatedistances. MonteCarlo simulat ions
+suggestthatthetransitionhasacontinuouscharacterwith (ap-
+proximately) mean–field exponents. We return to this below
+inSec. IVD.
+E. Global structureof the h–Tphasediagram
+Ourresultsforthe h–Tphasediagramoftheantiferromag-
+netic nearest–neighbor Heisenberg model with additional b i-
+quadraticinteractions b[Eq.(1)]aresummarizedinFig.2(b).
+There are two ordered phases, a nematic phase with local E
+symmetry and a vector–multipole phase with local T2sym-
+metry, both of which break spin rotational symmetry about
+the direction of the magnetic field. These are separated by a
+plateau–liquid state with all the symmetries of a paramagne t
+inmagneticfield.
+Thisphasediagrambearsaverystrongresemblancetothat
+ofthecorrespondingmodelwithweakFM third–neighborin-
+teraction J3=−0.05, which enforces four–sublattice order,
+as shown in Fig. 2(a) (cf. Ref. [15]). So far as the topology
+of the phase diagram is concerned the onlychange is the re-
+placementofalineoffirstorderphasetransitionsterminat ing
+thefour–sublatticeplateaustate(whichbreakslatticesy mme-
+tries), by a crossover in the case of the plateau liquid (whic h
+doesnot).Throughout this paper, we have argued that preformed lo-
+cal order at the level of a single tetrahedron exists in allof
+thesephases. Moreover,inthecaseofthehalf–magnetizati on
+plateau,wehaveseeninSec.IIICthatconventionalmagneti c
+order has very little impact on the excitation spectrum, and
+thereforeonthethermodynamicpropertiesofthesystem.
+Viewed in this way, the correspondence between the two
+h–Tphasediagramsisnotatall surprising—theroleofsec-
+ondaryinteractions like J3is merely to select between an in-
+finite set of different ordered ground states. Precisely how
+enforcementoflongrangeorderworksatfinitetemperaturei s
+acomplexandveryinterestingquestion,towhichwe provide
+onlya partialanswerbelow.
+IV. THERMALTRANSITIONSBETWEENDIFFERENT
+ORDEREDAND DISORDEREDSTATES
+A. General context
+None of the phases described above possess conventional
+magnetic order of the form /an}bracketle{tSi/an}bracketri}ht /ne}ationslash= 0. However they all con-
+tain the seeds of such order in the form of well formed local
+ordersλlocal
+Eandλlocal
+T2. Long range order can easily be re-
+stored by addingadditional terms to the HamiltonianEq. (1) .
+The simplest possible choice is a FM third–neighborinterac -
+tionJ3<0in Eq. (2), leading to four–sublatticeorder of the
+form considered in Refs. [2] and [15]. In what follows we
+study how FM |J3| ≪bprecipitates an ordered uuudstate
+from the plateau liquid for h= 4, and contrast this with the
+wayinwhichN´ eelorderemergesfromthenematicphasewith
+localEsymmetry for h= 0. We also discuss the continuous
+transition from plateau liquid to vector–multipole phase f or
+h < h∗,T < T∗.
+We study these phase transitions as a function of tempera-
+tureTwhichalsogivesusaccesstothehightemperaturepara-
+magnetic phase. This is interesting because, for intermedi ate
+distances r < ξc∼√
+Tthe nematic phases exhibit the alge-
+braic decay of spin correlations characteristic of a Coulom b
+phase, rather than the exponentialdecay of correlationsmo re
+usually associated with a paramagnet. Transitions between a
+disordered phase subject to an ice–rule type constraint and a
+phase with conventionalorderhave beendiscussed for a long
+time in the context of hydrogen–bonded ferroelectrics43–46.
+More recently such questions have arisen again in the con-
+text of experimentsof many highly frustrated magnets47, and
+in the past few yearsthere has been a theoretical effort to un -
+derstandhowordercanemergefromaCoulombphaseinclas-
+sical dimer48–52andspinmodels53,54.
+A strong motivation for this work has been the possibility
+of observingan unusualcontinuousphase transitions, incl ud-
+ing transitions lying outside the Landau–Ginzburg–Wilson
+paradigm55. Indeed the (classical) dimer model on cubic
+lattice does exhibit a continuous transition from a Coulomb
+phase at high temperatures to a simple crystalline ordered
+phase as a function of temperature48. This transition has
+unusual scaling properties49, and does not naively admit a
+Landau–Ginzburg–Wilsondescription,sincethehightempe r-17
+ature phase cannot be described using an expansion in terms
+of the low–temperature order parameter. A recent very de-
+tailedsimulationstudyofafamilyofthree–dimensionaldi mer
+models with ordered ground states and high–temperature
+Coulomb phases found a rich variety of continuous and dis-
+continuous phase transitions, including double phase tran -
+sitions where monopole excitations condense out of the
+Coulomb phase to give a conventional paramagnet at inter-
+mediatetemperatures52.
+The present understanding of these phenomena is that the
+gaugefieldassociatedwithCoulombphaseisminimallycou-
+pled to a matter field which condenses in the ordered phase,
+following an Anderson–Higgs mechanism50–52. In fact it is
+alsopossibletostudyzerotemperature(quantum)phasetra n-
+sitionsfromCoulombtoorderedphasesinthree–dimensiona l
+quantum dimer models56. These can in principle be continu-
+ous, occurring throughthe condensation of monopole excita -
+tionsin the Coulombphase57, but numericalsimulationssug-
+gest thatthetransitionis firstorder37.
+Less is known about transitions in spin models, but one
+interesting scenario exists for a continuous transition in an
+extended Heisenberg model on a pyrochlore lattice from a
+Coulombphasetoafour–sublatticeorderedstate53. Thistran-
+sition is foundto be in the same universalityclass as a uniax -
+ial ferroelectricwith dipolarinteractions,forwhichthe upper
+critical dimension is three58. This makes possible to continu-
+ous transitions with mean-field exponents (up to log correc-
+tions) — a scenario which closely resembles the transition
+from plateau-liquid into vector-quadrupole phase discuss ed
+below. Generically, however, transitions from Coulomb liq -
+uids into ordered states seem to be first order53, a fact which
+may be explained by interactions between fluctuations of as-
+sociatedgaugefield54.
+We conclude by noting that the complex forms of order
+which can occur in Heisenbergmodels on the pyrochlorelat-
+tice as a result of the interplay between farther–neighbor i n-
+teractions and thermal fluctuations are also a topic of cur-
+rent interest26. In finite magnetic field, these lead to a half–
+magnetizationplateauwhichcanbetunedatwillbetweendif -
+ferent forms of order27. A similar fluctuation driven plateau,
+but with a uniquelydefined form of order,is also expected to
+occurfortheedgesharingtetrahedraoftheFCC lattice59.
+We nowreturntothe modelinquestion.
+B. Transitionfrom plateau–liquidtoordered uuudstate
+Forh≃4,T/lessorsimilarb, Eq. (1) exhibits the plateau–liquid
+state described in Sec. IIIC [cf. Fig. 2(b)]. Inclusion of a
+FM third–neighbor interaction J3[Eq. (2)] causes it to or-
+der at low temperatures. We consider first the conventional
+limit where both |J3|andbare “large”, choosing parameters
+J3=−0.06andb= 0.6. Inthiscasethereisstronglyfirstor-
+dertransitionfromparamagnettofour–sublatticeplateau state
+forTN≈0.70. Thiscanbeseenveryclearlyinsimulationre-
+sultsfortheheatcapacityandtheorderparameter λglobal
+T2,and
+its susceptibility χglobal
+T2, presented in Fig. 19. If we now de-
+crease|J3|,thetransitiontemperature TNmustalsodecrease,0.2 0.4 0.6 0.8 1.0 
+T0246810 0.2 0.4 0.6 0.8 1.0 CvT
+012345010 20 30 40 12345
+0100 200 300 λT2localλglobal
+T2χT2local χglobal
+T24
+6
+8
+12 
+16 L(a) 
+(b) 
+(c) 4
+6
+8
+12 
+16 L
+4
+6
+8
+12 
+16 L
+4
+6
+8
+12 
+16 L(d) 
+(e) 
+4
+6
+8
+12 
+16 L
+FIG.19: (Coloronline)Temperaturedependence of(a)heatc apacity
+defined by Eqs.(31), (b) and (c) the related measure of localcorre-
+lationλlocal
+T2and its susceptibility defined by Eqs.(10)and (11), and
+(d)and(e)the globalorderparameter λglobal
+T2anditssusceptibilityde-
+finedbyEqs. (8)and (9) for b= 0.6,h= 4anda range of values of
+system size. The results are for J3=−0.06, showing a single first
+order transition into the ordered phase at TN= 0.70(1)(indicated
+bythe vertical dashed line).
+and for sufficiently small |J3|it will become smaller than the
+crossovertemperature T∗≈bassociatedwiththeplateauliq-
+uid.
+In this case, there are anomaliesin thermodynamicquanti-
+ties at two distinct temperatures as demonstrated in Fig. 20 .
+There is a broadmaximum in χlocal
+T2atT∗≈0.6[Fig. 20(c)],
+signalingtheonsetoftheplateauliquidstate,accompanie dby
+abroadpeakintheheatcapacity Cvataslightlylowertemper-18
+0510 15 20 25 12345
+0.2 0.4 0.6 0.8 1.0 
+T012345Cv
+01234560.2 0.4 0.6 0.8 1.0 T
+0100 200 300 λT2localλglobal
+T2χT2local χglobal
+T24
+6
+8
+12 
+16 L(a) 
+(b) 
+(c) 4
+6
+8
+12 
+16 L
+4
+6
+8
+12 
+16 L
+4
+6
+8
+12 
+16 L(d) 
+(e) 
+4
+6
+8
+12 
+16 L
+FIG. 20: (Color online) The same plots as Fig. 19 but for
+J3=−0.02, showing both a crossover in to the plateau liquid for
+T∗≈0.6(bold grey line) and a transition into the ordered phase at
+TN= 0.46(1)(dashed line).
+ature [Fig. 20(a)]. And, at T=TN≈0.46< T∗, there is a
+smalljumpin Cv,accompaniedbyaclearsingularityinglobal
+orderparametersusceptibility χglobal
+T2[Fig.20(e)]. Whilethere
+isnotruephasetransitionat T∗,it isclearthatthebulkofthe
+entropyoftheparamagnetislostinthesmoothcrossoverint o
+the plateau liquid, and not in the first ordertransition into the
+orderedphase.
+So what happensfor J3→0? Unfortunatelythis question
+is hard to answer by Monte Carlo simulation, as the massive
+degeneracyofthe uuudstatestranslatesintomanycompeting
+local minima in the free energy. However TNis strictly zero
+forJ3= 0, and there are two obviousscenarios for how this0123456
+0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.015 
+0.020 
+0.025 
+0.03 
+0.04 
+0.05 
+0.06 
+0.07 
+0.08 -J 3λglobal
+T2
+T
+FIG.21: (Coloronline)Temperaturedependence oftheorder param-
+eterλglobal
+T2forfour-sublattice uuudorder[asdefinedbyEq.(8)],for
+a range of values of J3<0. The transition temperature becomes
+smaller as |J3| →0. At the same time the transition becomes more
+stronglyfirst-order. Allresultsarefor h= 4andb= 0.6,inacluster
+withL= 8. The lines are guides forthe eye.
+long-range
+ordered phaseT2paramagnet
+spin-liquid
+plateau
+J -30.00.20.40.60.81.0
+0.00 0.02 0.04 0.06 0.08TT*(χ local)T2
+T*(Cv)
+TN
+FIG. 22: (Color online) Phase diagram for the classical Heis enberg
+antiferromagnet on a pyrochlore lattice in applied magneti c field
+h= 4, with additional biquadratic interactions b= 0.6[Eq. (1)].
+The transition temperature TNassociated with the gapped, ordered,
+half–magnetization plateau state vanishes as the strength of ferro-
+magnetic third–neighbor interactions J3→0, as determined by
+Monte Carlo simulation. A state exhibiting a half–magnetiz ation
+plateau but no long–range magnetic order exists above TNup to a
+crossover temperature T∗. Estimates of the crossover temperature
+are takenfrom peaks inthe local susceptibilityand heat cap acity.
+canbeachieved.
+The first is that the first–order transition into the ordered
+phase becomes weaker as TN→0, terminating in a critical
+end point for J3= 0,TN= 0. This end point would in fact
+bemulticritical ,sincemanydifferentordered uuudstatescan
+be formed out of the dimer manifold for different choice of
+long range interactions. Within this scenario, the power–l aw
+correlations between spins in the plateau liquid for T→0
+could be viewed as evidenceof critical fluctuations. The sec -
+ond scenario is that the first order transition into the order ed
+phase persists down to TN= 0. Since an infinite number of19
+different ordered phases branch out from the point J3= 0,
+TN= 0,it canprobablybest betermed “multifurcative” .
+First–order phase transitions between different ordered
+phases with an infinite degeneracy at the transition occur in
+a numberof models. Such phase transitions are first order, in
+the sense that neither ordered parameter collapses approac h-
+ing the critical point. However they also exhibit one of the
+characteristic features of a second order transition, name ly a
+soft excitation or set of soft excitationsconnectingthe di ffer-
+entorderedgroundstates.
+Asfaraswecantellfromourpresentresults,itseemsmost
+likely that the classical pyrochlore AF with biquadratic in -
+teractions exists at a multifurcative point in parameter sp ace,
+with an infinite ground–statedegeneracy, notat a critical end
+point. As shown in Fig. 21, the low temperature value of the
+order parameter is broadly independent of TN. This implies
+that the phase transition in fact becomes more strongly first
+orderatTN→0,andappearsto ruleouta (multi)criticalend
+point. Our collected simulation results for J3→0are sum-
+marizedin theformofthephasediagraminFig.22.
+It is amusing to note that this phase diagram bears a su-
+perficial resemblance to the phenomenology of a second or-
+der(quantum)criticalpoint—atransitiontemperaturewhi ch
+collapsestoaspecialpointwithalgebraicdecayofcorrela tion
+functions, which in turn controls a broad region of the phase
+diagram up to a characteristic crossover temperature T∗. All
+of this despite the fact that the only phase transition prese nt
+is first order, which means that the length scale associated
+with fluctuationsremainsfinite. Some of the generic feature s
+seen in our model — power law decay of correlations over
+a large, but finite, length scale — have been previously dis-
+cussed in the context of other models with strong local con-
+straints, where they were dubbed “high temperature critica l-
+ity”60.
+Thetransitionfromacritical“Coulombic”phasedescribed
+by aU(1)gaugetheory into a simple orderedstate as a func-
+tion of temperature can be studied much more cleanly in the
+(classical) dimer model on cubic lattice, where the constra int
+enforcingthedimermanifoldisinfinite. Inthiscase,theph ase
+transition is continuous, and exhibits interesting and unu sual
+scaling properties48,49. We have made a preliminary study of
+the“stiffness” Kassociatedwithfluctuationsina U(1)gauge
+theoryfor temperaturesspanningthe paramagnetand platea u
+liquid phasesin our model (see Fig. 22), but find no clear ev-
+idence of a phase transition. However the way in which the
+dimer and loop manifolds break down at finite temperature
+in a model with a finite constraint is an interesting problem,
+andonewhichdeservesfurtherstudy. We noteinpassingthat
+interesting, related, problems arising the context of quan tum
+loopmodels61.
+C. Transitions from paramagnet to E–symmetry nematic
+phaseand N ´eelordered state
+Itisinterestingtocontrastthefinitetemperaturephasetr an-
+sitions associated with the plateau states for h≈4, with
+the transitions into E–symmetry N´ eel and nematic orderedstates for h= 0. Once again, for “large” |J3|there is a
+strongly first order transition from the paramagnet into the
+N´ eel phase at a unique temperature TN. Meanwhile, for
+“small”|J3| ≪b, there is double transition, first from para-
+magnet to E–symmetry nematic phase at TQ∼b, and then
+into the four–sublattice N´ eel order at a much lower tempera -
+tureTN, as demonstratedin Fig. 23. Within the limits of our
+simulation,bothofthesetransitionsappeartobefirst orde rin
+character64. The results for varying J3are summarizedin the
+phasediagraminFig.24. Thefirstordertransitionfrompara -
+magnet to nematic phase at TQath= 0should be compared
+with the crossoverfrom paramagnetto plateau liquid T∗∼b
+observedfor h= 4in Fig22.
+D. Transitionfrom plateau–liquidtovector–multipole pha se
+Perhapsthemostinterestingofthefinitetemperaturetrans i-
+tionsobservedin our modelis the one fromplateau–liquidto
+vector–multipole phase, already described in Sec. IIID. At
+one level this is the most exotic phase transition we study
+— a continuous phase transition from a “Coulombic” state
+with algebraicdecay of correlationfunctions(the plateau liq-
+uid) to a phase with long–rangemultipolarorder (the vector –
+multipole state). But at the same time it has the simplest
+phenomenology of any of the phase transitions in this pa-
+per, with the order parameter exhibiting a simple mean–field
+like behavior Q⊥,1(T)∼√TV−TwithTV≈0.09, as
+shown in Fig. 25(a). The secondary order parameter Q⊥,2
+grows more slowly as expected [Fig. 25(b)]. The heat ca-
+pacity does not show a noticeable singularity at T=TVin
+Fig. 25(c), which is also consistent with the mean–field be-
+haviorCv∼(T−TV)αwithα= 0. (The broad peak at
+T∼0.112again corresponds to the crossover temperature
+T∗forthe plateau-liquidstate.)
+Ataqualitativelevel,andinthespiritofthispaper,itise asy
+toseehowacontinuoustransitioncanarisebetweenthesetw o
+states. Both are built of tetrahedra with a local T2character,
+with three “up” and one “down” spin, joined at the corners.
+Both states will exhibit algebraic decay spin correlations at
+low temperatures, as a result of the infinite number of ways
+that these tetrahedra can be assembled to form a pyrochlore
+lattice. Theonlydifferenceisthatthree“up”andone“down ”
+spinsare canted in the vector–multipolephase, givinga fini te
+valueofQ⊥,1andQ⊥,2[Figs. 25(a)and (b)]. As longas this
+cantingcaninterpolatesmoothlytozerointhecollinear uuud
+state, the transition will be continuous. And at the level of a
+Ginzburg–Landautheory,nothingpreventsthis from happen -
+ing—cf. Eq.(43).
+However in principle it should also be possible to tran-
+scribeeachofthese phasesintermsofthemoresophisticate d
+“solenoidalfield”theoryusedtodescribeN´ eelorderinasp in
+modelwithahightemperatureCoulombphase(cf.Ref.[53]).
+To the best of our knowledge, nobody has yet attempted to
+extend the gauge–field description of a Heisenberg type spin
+model to treat multipolar order. But it is interesting to not e
+that the transition from Coulombic phase to simple N´ eel or-
+der was found to be continuous, and in a universality class20
+withuppercriticaldimensionthree,i.e.,onewherethecri tical
+behaviorismean–fieldlike, upto logcorrections53,58.
+V. SUMMARYAND CONCLUSIONS
+We have studied the ordered and disordered phases of the
+classical, bilinear–biquadratic Heisenberg model on the p y-
+rochlore lattice at finite temperature and in applied magnet ic
+field. Wefindarichcollectionofunconventionalstates—ne-
+matic and vector–multipolephases with distinct and differ ent
+local symmetries, separated by a half–magnetization plate au
+withspin–liquidcharacter. Allofthesephasesshowanunde r-
+lying“Coulombic”characterwithalgebraicdecayofspinco r-
+relation functions over distances r/lessorsimilarξc∼1/√
+T. Interest-
+ingly, the transition from plateau–liquid to vector–multi pole
+phase is continuous, and appears to be well–described by
+meanfieldtheory.
+While this behavioris undeniablyexotic,all of these state s
+can be understood — and even anticipated — from a proper
+understanding of the geometry of the pyrochlore lattice, an d
+the properties of a single tetrahedron. Strong localfluctua-
+tions of N´ eel order are present in all of these phases, and
+the zero temperature phase diagram can be understood sim-
+ply from the “self assembly” of these ordered tetrahedra int o
+complexstates withhighersymmetry.
+It is therefore unsurprising that conventional N´ eel order
+(with four–sublattice structure) is immediately restored by
+the introduction of a ferromagnetic third–neighbor coupli ng
+J3. Howeverforsmall |J3|, the unconventionalstates survive
+above the N´ eel transition temperature TN. In particular, the
+spin–liquidplateausurvivesabove TN,uptoacrossovertem-
+peratureT∗≈b. The transition between liquid and ordered
+plateaux is first order in nature, and remains so for TN→0.
+Forsmall |J3|,thesystemalsoexhibitsafirst–ordertransition
+betweentheN´ eel andnematicphases,in additiontothe first –
+ordertransitionfromthenematicphaseintohigh-temperat ure
+paramagnet.
+So far as experimentis concerned, our main finding is that
+the physicsof a pyrochloreantiferromagnetin magnetic fiel d
+can be largely determined by the properties of a single tetra -
+hedron. In the simple models which we have considered it is
+possible to tune between states with entirely different poi nt
+group symmetries at will, simply by changing the form of
+(weak)longrangeinteractionspresent. Thisisanoversimp li-
+fication, in the sense that magnetostriction in real systems is
+likely single out a particular phonon (or family of phonons)
+with definite symmetry, which will then drive the system
+towards collinearity ( b, in our model) andselect the low–
+temperatureorderingpattern(longrangeinteractions,e. g.,J3,
+in our model). However,as long as there is a strong coupling
+to phononswithinindividualtetrahedra,the formof the mag -
+netizationplateauandassociated phasesmaybe largelyind e-
+pendentofthese(systemdependent)details.Acknowledgments
+We are pleased to acknowledge stimulating discussions
+with F. Alet, J. Chalker, G. Kriza, G. Misguich, T. Momoi,
+H. Shiba, H. Takagi, O. Tchernyshyov, S. Trebst, H. Tsunet-
+sugu,andH.Ueda. We areparticularlyindebtedtoR.Moess-
+nerandM.E.Zhitomirskyforvaluablecommentsaboutmul-
+ticriticalpointsandtheclassificationofmultipolarorde r.
+This work was supported under EPSRC Grants No.
+EP/C539974/1and EP/G031460/1, and SFB 463 of the DFG
+(NS); Hungarian OTKA T049607 and K62280 (KP); Grant–
+in–Aid for Scientific Research No. 16GS50219, 17740244,
+and19052008fromMEXT,Japan;GlobalCOEProgram“the
+Physical Sciences Frontier”, MEXT, Japan, and Next Gener-
+ation Super ComputingProject, NanoscienceProgram(YM).
+PartofthisworkwasdonewhileKP andYMwere visitorsat
+KITP Santa Barbara. KP and NS also acknowledge the hos-
+pitality of MPI–PKS Dresden, where a part of this work was
+completed.
+AppendixA:Classification of symmetry breakingat thelevel of
+asingle site
+Inordertoidentifythedifferentpossibleformsofmagneti c
+orderwhichcansurvivewhereconventionalN´ eelorderbrea ks
+down,it ishelpfultoclassifythedifferentformsofsymmet ry
+breakingwhichexistatthelevelofasinglesite. Thisanaly sis
+is in the spirit of the detailed classification for the nemati cs
+in liquid crystals undertaken in Ref. [62], and motivates th e
+rank–two and rank–three tensor order parameters introduce d
+in Sec. IIC and Appendix B. In order to keep contact with
+quantum spins, which are axial rather than polar vectors, we
+mustkeeptrackoftime reversalsymmetry.
+In Table III we show the transformation rules for the spins
+under selected symmetry operations, including time revers al
+ΘS=−S. In contrast to the usual polar vectors, inver-
+sion leaves the axial vectors invariant – as a consequence, a ll
+the usual (reflection, rotation, and inversion) symmetry op er-
+ationcanberepresentedbyanorthogonalmatrixbelongingt o
+SO(3), with determinant equal to +1. The role of inversion
+in the case of polar vectors is taken over by the time reversal
+operatorΘ.
+All of the symmetryoperations,extendedwith the time re-
+versal, can be representedby orthogonalmatrices with dete r-
+minant -1. In the Table III we also check if the collinear and
+coplanarstatesareinvariantunderthosesymmetryoperati ons.
+Since we are interested in the symmetry breaking which can
+occurin theabsenceof brokentranslationalsymmetry,we do
+notapplythesymmetryelementstothelattice points(i.e., we
+treat all the spins as they were at the origin). We find that the
+invariant operations of the 2:2 state include a C2(z)rotation
+inadditiontothe symmetryoperationsofthe3:1state.
+In Table IV we show the symmetry group of each of the
+spin states. In order to facilitate comparison with Ref. [62 ],
+we also show the symmetry group of the states if the spins
+were polar vectors. We can see that as the time reversal does
+not play a role for the 2:2 collinear state, its symmetry grou p21
+being the grey–group D∞h+ΘD∞h. The magnetic(the 3:1
+collinear and both canted states) states have a magnetic poi nt
+groupasa symmetrygroup.
+When studying which symmetry group is broken for the
+magnetic states, we need to note that the external magnetic
+fields lowers the O(3)symmetry of the space to C∞×
+{E,I}+ΘσvC∞×{E,I},wheretheaxisofthe C∞ispar-
+allel to the magnetic field, and σvis a reflection to a plane
+that includes the axis of the magnetic field. The symmetry
+of the space with magnetic field is actually identical to the
+symmetryofthe3:1collinearstate. Thus,withinaGinzburg –
+Landauframeworkwedonotexpectacontinuous(secondor-
+der)phasetransitionbetweenthe T= 0liquidplateauandthe
+hightemperaturedisorderedphase. The Z2loweredsymmetry
+ofthe3:1statewithrespectto2:2cantedstateismanifeste din
+theZ2symmetryloweringofthevectorto thenematicphase.
+AppendixB:Higher order multipoles
+Inthispaper,wehaveclassifiedstatesaccordingtothe low-
+estmoment of spins which breaks spin rotational symmetry.
+Accordingtothisconventional,“commonsense”prescripti on,
+astatewhichlacksconventionaldipolar(e.g.,N´ eel)orde r,but
+exhibitsacommonplaneforthecantingofspins,isautomati -
+callyclassifiedasanematicorvector–multipolephase. Whi le
+this classification scheme is unambiguous,it is not complet e,
+and in some cases may give the wrong answer, so far as the
+primaryorderparameterisconcerned.
+This point was recently discussed at length for the copla-
+narground–statemanifold of the classical Heisenberg model
+on a kagome lattice, where the primary order parameter was
+convincingly argued to be octupolar, and notquadrupolar, in
+nature24. Incorrect assignment of the primary order parame-
+ter does not affect our ability to detect a bulk ordered phase ,
+butcanleadtofalseconclusionsaboutphasetransitions. T his
+is particularly true of two–dimensional systems at finite te m-
+perature,wherethehomotopygroupassociatedwiththeorde r
+TABLE III: The transformation of spins under different symm etry
+operations. Eis the identity element, Iis the inversion, Θis the
+time reversal operation, σαβis a reflection with a mirror plane αβ,
+andC2(α)is a two–fold rotation around the αaxis. In the last two
+columns we indicate if the 2:2 and 3:1 canted states (with mag netic
+moment along the zaxis and spins are in the xzplane) are invariant
+withrespect tothe particular operation.
+symmetryelements SxSySz2:2 3:1
+E,I SxSySzyes yes
+σyz,C2(x) Sx−Sy−Szno no
+σxz,C2(y) −SxSy−Szno no
+σxy,C2(z) −Sx−SySzyes no
+Θσyz,ΘC2(x)−SxSySzyes no
+Θσxz,ΘC2(y) Sx−SySzyes yes
+Θσxy,ΘC2(z) SxSy−Szno no
+Θ,ΘI −Sx−Sy−Szno noparameter determines the form of topological defect enteri ng
+intoBerezinsky–Kosterlitz–Thoulesstypephasetransiti ons.
+Infactthestateswhichweclassifyas“nematic”or“vector–
+multipole” in Sec. III also posses higher order multipole mo -
+ments which, under some circumstances, couple to the rank-
+two tensor order parameters used in this paper. We illustrat e
+this below for the specific case of the rank–threetensor asso -
+ciatedwithoctupolarorder.
+Thisisoddundertime reversal,andhassevencomponents
+Tα=1
+N/summationdisplay
+iTα
+i (B1)
+givenby
+Tx3−3xy2
+i = (Sx
+i)3−3Sx
+i(Sy
+i)2, (B2)
+Ty3−3yx2
+i = (Sy
+i)3−3Sy
+i(Sx
+i)2, (B3)
+Tz(x2−y2)
+i =√
+6/bracketleftbig
+(Sx
+i)2−(Sy
+i)2/bracketrightbig
+Sz
+i, (B4)
+Txyz
+i= 2√
+6Sx
+iSy
+iSz
+i, (B5)
+Tx(r2−5z2)
+i =/radicalbigg
+3
+5Sx
+i/bracketleftbig
+(Sx
+i)2+(Sy
+i)2−4(Sz
+i)2/bracketrightbig
+,(B6)
+Ty(r2−5z2)
+i =/radicalbigg
+3
+5Sy
+i/bracketleftbig
+(Sx
+i)2+(Sy
+i)2−4(Sz
+i)2/bracketrightbig
+,(B7)
+Tz(3r2−5z2)
+i =/radicalbigg
+2
+5Sz
+i/bracketleftbig
+3(Sx
+i)2+3(Sy
+i)2−2(Sz
+i)2/bracketrightbig
+.(B8)
+Intheabsenceofmagneticfield,quadrupolarordercancou-
+ple to (fluctuations of) octupolar order through terms of the
+form
+δF ∼/summationdisplay
+αβγδQαβTαγδTβγδ(B9)
+in the free energy, which respect the full O(3)symmetry of
+the Hamiltonian, and time reversal invariance62. Therefore,a
+finiteoctupolarorderparameterusuallyinducesaquadrupo lar
+one, while the opposite is not always true. When they occur
+together, some care must then be taken to assign the correct
+primaryorderparameter.
+In finite magnetic field we again classify these octupoles
+accordingto the way in which they transformunder rotations
+aboutdirectionofmagneticfield(the zaxis). Weobtainasin-
+gleone–dimensionalirrepandthreetwo–dimensionalirrep s,
+T⊥,3={Tx3−3xy2,Ty3−3yx2}, (B10)
+T⊥,2={Tz(x2−y2),Txyz}, (B11)
+T⊥,1={Tx(r2−5z2),Ty(r2−5z2)},(B12)
+T⊥,0=Tx3−3xy2, (B13)
+which take on finite values in the different ordered states.
+These results are summarized in Table V. As the magnetic
+field breakstime-reversalinvariance,the quadrupolarand oc-
+tupolar order parametersmay mix linearly in the free energy .
+For example, where Szis singled out by magnetic field, the
+newtermsthatenterthefreeenergyareoftheform
+δF ∼Sz/bracketleftig
+Qx2−y2Tz(x2−y2)+QxyTxyz/bracketrightig
+,(B14)22
+TABLE IV: The symmetry of the different configurations, trea ting the arrows as polar vectors, or as axial vectors with and without inclusion
+of the time reversal symmetry. Inthe lastcolumn we show the b roken symmetry (we assume nomagnetic fieldinthe case of the 2 :2 collinear
+state and magnetic fieldalong the zdirection for 3:1 collinear and for the twocanted states). T he notation is the same as inthe Table III,with
+theadditionoftwoelements: σvisareflectiontoaplane perpendicular tothe C∞axis(σxzisalsoaσv),whileC′
+2isatwo–foldrotationwith
+axis perpendicular tothe C∞axis.
+state polar vectors axial vectors spins (axial vectors + tim e reversal) symmetry broken
+2:2collinear D∞h=C∞⊗{1,C′
+2,σv,I} D∞h D∞h+ΘD∞h O(3)/(O(2)×O(1)) =RP2
+3:1collinear C∞v=C∞×{E,σv} C∞×{E,I}C∞×{E,I}+ΘσvC∞×{E,I} 1
+2:2canted C2v={E,C2(z),σxz,σyz}C2h={E,I,C 2(z),σxy} C2h+ΘσxzC2h C∞/C2
+3:1canted C1h={E,σxz} S2={E,I} S2+ΘσxzS2 C∞23
+TABLEV:Classificationofrank–threetensoroperatorsacco rdingto
+rotational symmetry about a zaxis defined by magnetic field. Also
+indicated are the finite values of the order parameters in the 2:2 and
+3:1canted states.
+order par. tensor operators 2:2 1:3
+e3iφ{Tx3−3xy2,Ty3−3yx2} 0 finite
+e2iφ{Tz(x2−y2),Txyz} finite finite
+eiφ{Tx(r2−5z2),Ty(r2−5z2)} 0 finite
+1 Tz(3r2−5z2)finite finite
+and
+δF ∼Sz/bracketleftig
+QxzTx(r2−5z2)+QyzTy(r2−5z2)/bracketrightig
+,(B15)
+whichrespecttheremaining O(2)rotationalsymmetry(moreprecisely, they can mix if the Szorder parameter is finite, ir-
+respectivelyofthepresenceofexternalmagneticfield). Ma g-
+netic field can therefore strongly modify the symmetry of a
+(primary) multipolar order parameter. For a related discus -
+sion,seeRef. [25].
+It is not our intention to give a definitive treatment of this
+complex set of coupled order parameters in this paper. How-
+ever we have made a preliminary study of the behavior of
+the rank–three and rank-four tensor order parameters in the
+present model, using the T= 0theory for a single tetrahe-
+dronembeddedinthelattice,andclassicalMonteCarlosimu -
+lation. Wehavebeenunabletoidentifyanyhigher-ordermul -
+tipole which grows faster at a continuous transition than th e
+rank-two tensor order parameters given in Section IIC, and
+sothese retaintheirtentativeassignmentasprimaryorder pa-
+rameters.
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+63Thepyrochlorelatticecanalsobethoughtofasbi-simplex, where
+iseach tetrahedron is asimplex.
+64We observe substantial hystereses and very slow relaxation pro-
+cess in MC calculations in the first order transitions at h= 0.0
+b= 0.6; We here adopt mixed initial configurations in which a
+halfofthesystemisnematicorderedandtheresthalfispara mag-
+netic disordered.25
+0123412340510 15 CvT
+020 40 60 80 λT2localλglobal
+T2χT2local χglobal
+T24
+6
+8
+12 
+16 L(a) 
+(b) 
+(c) 
+4
+6
+8
+12 
+16 L
+4
+6
+8
+12 
+16 L
+4
+6
+8
+12 
+16 L(d) 
+(e) 
+4
+6
+8
+12 
+16 L(f) 4
+6
+8
+12 
+16 L
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Q
+0100 200 300 400 
+0.2 0.3 0.4 0.5 0.6 0.7 0.8 
+T
+FIG. 23: (Color online) Temperature dependence of (a) nemat ic or-
+der parameter [Eq. (18)], (b) heat capacity [Eq. (31)], (b) a nd (c)
+the related measure of localcorrelation λlocal
+Eand its susceptibility
+[Eqs.(10)and(11)],and(d)and(e)the globalorderparameter λglobal
+E
+anditssusceptibility[Eqs.(8)and(9)]for b= 0.6,h= 0andarange
+of values of system size. The results are for J3=−0.02, showing
+bothafirstorder transitionintothenematic phase at TQ= 0.518(6)
+(indicatedbytheverticaldottedline)andatransitionint otheordered
+phase atTN= 0.42(2)(dashed line).26
+FIG. 24: (Color online) Phase diagram for the classical Heis enberg
+antiferromagnet on a pyrochlore lattice, Eq. (1), inapplie d magnetic
+fieldh= 0, with additional biquadratic interactions b= 0.6. The
+transition temperature TNassociated with the E–symmetry long–
+range ordervanishes asthestrengthofferromagnetic third –neighbor
+interactions J3→0, as determined by Monte Carlo simulation. A
+phase exhibitingnematic order exists above TNuptoaTQ∼b.27
+0.0 0.1 0.2 0.3 0.4 0.06 0.07 0.08 0.09 0.10 0.11 0.12 
+4
+6
+8
+12 
+16 LT
+0.00 0.02 0.04 0.06 (a) 
+(b) 
+(c) Q,1 Q,2 
+0.06 0.07 0.08 0.09 0.10 0.11 0.12 Cv
+T4
+6
+8
+12 
+16 L
+4
+6
+8
+12 
+16 L
+1.0 1.4 1.8 2.2 2.6 
+FIG. 25: (Color online) (a) Temperature dependence of the pr imary
+order parameter Q⊥,1in the vector–multipole phase. In the limit
+L→ ∞these results extrapolate to a mean field–like behavior
+Q⊥,1∼√TV−TwithTV≃0.091(solid black points and grey
+line). (b) The secondary order parameter Q⊥,2also takes on a finite
+value for T < T V, but grows more slowly at the transition. (c) Heat
+capacity,showingnomeasurablesingularityat T=TV. Alldataare
+forh= 5,b= 0.1,J3= 0, with system sizes ranging from L= 4
+toL= 16.
\ No newline at end of file
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@@ -0,0 +1,3 @@
+   Liquid-Crystal Transitions: A First Principles Multiscale Approach   Z. Shreif1, S. Pankavich1,2, and P. Ortoleva1   1Center for Cell and Virus Theory Department of Chemistry Indiana University Bloomington IN 47405  2Department of Mathematics  University of Texas at Arlington Arlington, TX 76019 USA   Abstract A rigorous theory of liquid-crystal transitions is developed starting from the Liouville equation.  The starting point is an all-atom description and a set of order parameter field variables that are shown to evolve slowly via Newton’s equations. The separation of timescales between that of atomic collision/vibrations and the order parameter fields enables the derivation of rigorous equations for stochastic order parameter field dynamics. When the fields provide a measure of the spatial profile of the probability of molecular position, orientation, and internal structure, a theory of liquid-crystal transitions emerges. The theory uses the all-atom/continuum approach developed earlier to obtain a functional generalization of the Smoluchowski equation wherein key atomic details are embedded. The equivalent non-local Langevin equations are derived and the computational aspects are discussed. The theory enables simulations that are much less computationally intensive than molecular dynamics and thus does not require oversimplification of the system’s constituent components. The equations obtained do not include factors that require calibration and can thus be applicable to various phase transitions which overcomes the limitations of phenomenological field models. The relation of the theory to phenomenological descriptions of Nematic and Smectic phase transitions, and the possible existence of other types of transitions involving intermolecular structural parameters are discussed.   Keywords: liquid-crystals, multiscale analysis, all-atom continuum theory, phase transitions, Liouville equation, probability functionals  I Introduction Condensed media composed of phospholipids or other elongated molecules can support liquid crystal phases [1-3]. In these phases, there is preferred molecular orientational order but, unlike crystals, there is no long-lived nearest-neighbor connectivity. The objective of this study is to place these phenomena in the framework of all-atom multiscale statistical mechanics. Liquid crystals are key elements of many engineered and natural systems. Examples of engineered systems include liquid crystal displays (such as TVs, LCD projectors, and computer monitors), liquid crystal thermometers, and sensors. Lyotropic (concentration-dependent) liquid crystals exist abundantly in biological systems; examples include cell membranes, enveloped viruses, and nanocapsules for therapeutic delivery. It has been suggested that the liquid-crystal properties of these biological systems play a key role in their function [3]. For instance, because of their liquid-crystal properties, membrane phospholipids can easily exchange position while keeping long-lived orientational order on the average. Thus, a predictive theory of the onset and stability of liquid-crystal behavior is of great fundamental and applied interest. There is an extensive literature on the theory of liquid-crystal phenomena [4-8]. There has been a general consensus that there must be a tradeoff between the level of detail and efficiency in simulating liquid crystals in order to achieve long time and length-scale results. At the highest level of detail, there are the atomistic models [5-6]. These yield insight into nematic and smectic phases of liquid crystals; however, they are unable to simulate long timescales (over 50 ns) or large liquid-crystal systems (such as polymers). Alternatively, different levels of coarse-graining were attempted in order to achieve longer time and length scales. These include molecular simulations using lattice and off-lattice models [7]. These models do not take into consideration the atomistic details and are still not efficient enough to simulate large systems over long time periods (over 1 ms). At the next level of coarse graining are mesoscopic models such as the continuum and Lattice Boltzmann models [8]. In summary, each of the approaches mentioned above has been successful in shedding light on specific aspects of liquid-crystal behavior at the various phases. However, if the aim is to simulate the full behavior of a large system displaying liquid-crystal states among other properties (such as the case of an enveloped virus), then it is necessary to develop a theory that can bridge the gap between the various methods to yield a model that takes into account both atomistic and continuum scales. Here we present a first-principle theory of liquid-crystals. Starting with Newton’s equations for the N-atom system, we derive equations for coarse-grained field variables (order parameter fields). The N-atom description is necessary to account for key atomic details; however, the liquid-crystal is understood in terms of more coarse-grained variables quantifying the preferred orientation of the elongated constituent molecules. Due to the lack of nearest-neighbor connectivity in such systems, field variables (i.e. variables that change continuously across the system) are needed to keep track of the constantly changing structures. The above prescription is achieved via an integrative all-atom/continuum multiscale (ACM) framework [9-10]. Other methods have been developed wherein an N-atom description of the system is used to obtain the coarse-grained model. This is usually done via “decoupled” coarse-graining wherein atoms are lumped together and effective forces on the lumped elements are set forth before simulation begins [11-12]. The shortcoming of such models is that they ignore the feedback loop of Fig.1. Obtaining the forces driving the dynamics of the coarse-grained variables (i.e. the order parameters) only at the beginning of the simulation is insufficient; while the atomic configuration mediates the forces and diffusion coefficients needed to drive the dynamics of the order parameters, the latter affect the ensemble of atomistic configurations. Thus, the former should be co-evolved with the order parameters during a simulation. 
+ Multiscale techniques and their application to Brownian motion have been discussed extensively in the literature [14-24], and date back to the first mathematical explanation of Brownian motion by Einstein and Smoluchowski independently. Most relevant to the work presented here are recent studies [13,25-33] wherein the N-atom Liouville equation is used as a starting point to derive reduced equations for the stochastic dynamics of the order parameters. The N-atom probability density is written as a function of the all-atom state both directly and, through a set of order parameters, indirectly. This formulation does not constitute an over counting of the number of degrees of freedom ; rather, it was shown [13,29-30] that when there is a separation of timescales between the rapidly fluctuating atomic state and the order parameters, both dependencies can be reconstructed. These studies led to our recent development of an ACM (All-atom/Continuum Multiscale) approach [9-10] where the N-atom probability Fig.1: Order parameter fields characterizing nanoscale features affect the statistics of the atomistic configurations which, in turn, mediate the forces driving the order parameter fields’ dynamics. This feedback loop is central to a complete multiscale understanding of nanosystems and the true nature of their dynamics [13]. density is postulated to be a function of the 6N atomic positions and momenta and a functional of a set of order parameter fields. ACM was demonstrated via an application to enveloped viruses [9] and nanocapsule therapeutic delivery [10]. However, that formulation of ACM does not account for liquid-crystals, i.e. does not describe preferred molecular orientation. In this study, we introduce order parameter field variables that describe the local state of preferred elongated molecule orientational order (Sect II). Starting with the Liouville equation for the N-atom probability density , we generalize our deductive multiscale procedure [9-10,13] and derive a stochastic equation for the reduced probability density as a functional of the order parameter field profiles (Sect III). In Sect IV, we set forth the equivalent Langevin equations needed for practical simulation of liquid-crystal transitions. In Sect V, we discuss how to proceed to develop a computer simulator for stochastic field dynamics and draw conclusions in Sect VI.  II Order Parameter Fields Counters for an incremental range of molecular structural variables are set forth as a first element of a theory of liquid-crystal transitions as follows. Let  be a structural parameter describing the state of the  molecule of type , where  is the set of positions of the atoms constituting the  molecule of type , and  labels the type of structural parameter (such as center-of-mass position and orientation [13,28]). The notation  represents the set of structural parameters with the number of components in the set depending on  (e.g. a large complex flexible molecule requires more structural variables to describe it than does a small rigid one). In other words, each type  has a specific number of distinct   values, each labeling a distinct type of structural variable. For example, we take  to label the center-of-mass (CM) and  to measure compression/extension along the X-axis. By convention, we take  to indicate the CM. With this,  is the CM of the  molecule of type .  To obtain the structural parameters, orthogonal basis functions  are introduced [9]. Atom  is moved via a deformation wherein its position in space  is considered a displacement from a reference point . With this,  is written    , (II.1) where  is the number of molecules of type ;   is the number of molecular types;  is the residual displacement of atom ; and  is one when atom  belongs to the  molecule of type  and zero otherwise. The ideal values of the structural parameters are those that minimize the information content (see below), with proper choice of basis functions and number of terms in the  sum. We start by defining a mass-weighted mean square residual  via    , (II.2) where  is the total number of atoms in the system and  is the mass of atom . The relationship between the structural parameters  and the set of atomic positions  can be found via minimizing  with respect to  keeping  constant. With this, one obtains   , (II.3)   . (II.4) By convention, we take  and thus  is the total mass of a molecule of type . We construct the  for  to be mass-weighted orthogonal to . With this,  for . This convention with equation (II.3) implies that  is the CM of the  molecule of type . This procedure was applied in another context (notable for modeling bionanostructures) and it was found that the mass-weighted residual method yields structural parameters similar to generalized CMs [13]. Let  be a given value of a structural parameter, i.e.  does not depend on the set of atomic positions . The subscript  indicates that the  value only includes structural variables for a type  molecule. The notation  corresponds to the set of  values, i.e. for all  relevant for type  molecules.  Let  be a window function centered about zero such that it is one inside the window and zero otherwise. The index  on  is introduced because the dimension of the window is the number of structural variables for a molecule of type  which, henceforth, does not include the CM. Therefore,  indicates that the  molecule of type  has internal structural variables  in a window about . Order parameter fields are conceived of as densities of subpopulations of molecules of type  with structural variables in a window. If a variable has a narrow window, then fluctuations of molecules in and out of the window will be large and occur on a short timescale; therefore the associated subpopulation density is not a viable order parameter. Let the length scale over which the profile of the CM distribution be broad, i.e. we seek a smoothly varying spatial profile of subpopulation CM distribution. Thus, we introduce a set of order parameter fields   for type  molecules such that   , (II.5) where  is the Dirac delta function centered at  (which, for formal development, we take to be a narrow unit-normalized, gaussian); the parameter  is the ratio of the average nearest-neighbor molecule CM distance to the characteristic size of features of interest;  labels a point in space and is scaled such that it undergoes a displacement of  when one average nearest-neighbor distance is traversed; and  times the mass of a type  molecule is the mass density of type  molecules with internal structure in the  window and CM near . The first step in our multiscale development is to show that the densities  are slowly varying in time and thus, can be used as order parameters [13]. Newton’s equations imply  for Liouville operator     (II.6) where  is the momentum of atom  and  is the force exerted on it. This yields   (II.7) . (II.8) Multiplying both sides of (II.7) by a sampling function , for a given point in space , and integrating over a volume in -space, we find that  is slowly varying due to the large number of molecules in such a volume; by examining the LHS of (II.7), one obtains    (II.9) which is of the order of the number of molecules of type  in a given structural window in the sampling volume. If the sampling volume is large enough, the number of molecules would also be large. Choosing the sampling volume such that the typical number of molecules of a given type in a structural window is , then (II.9) is of . By examining the RHS of (II.7), and using integration by parts, one obtains     (II.10) where   (II.11) The integral in the RHS of (II.11) is of the order of the number of molecules in the sampling volume, that is . However, (II.10) is a sum of terms the number of which is on the order of the number of molecules. Therefore, (II.10) is a sum of fluctuating terms of  each of order ; this makes the overall order of (II.10) of . This suggests introducing a scaled order parameter . With this, we obtain  , (II.12) where  is the set of  atomic positions and momenta. The field variables  and  are labeled by  and  and vary continuously across the system. Thus, with the dependency on the atomistic configurations being understood, we use the notation  and  to label them as the -associated parameters. In the following sections, we use the notation  to represent the infinite set composed of  for all values of  and .  III Multiscale Analysis: A Functional Smoluchowski Equation for Liquid Crystals To develop a theory for liquid crystals using the field variables introduced in the previous section, we derive an equation for the reduced probability density  that is a functional of the distribution of field variables in space  and over the continuous set of structural parameters . To address the continuum limit of the field theory we present, we divide the system into a large but finite number of boxes of volume  (each of which is labeled by its center point ). As indicated previously [10],  is taken to be much smaller than the characteristic length of the phenomena of interest, but large enough to contain a statistically significant number of molecules. By definition,  , (III.1) where   ; (III.2)  is the N-atom state over which integration is taken; ; a superscript * for any variable indicates evaluation at ;  is the N-atom probability density which satisfies the Liouville equation ; and  is a continuum product of  for all positions  and set of structural parameters , and is represented by . With these definitions and the chain rule for functionals,  is found to satisfy the conservation equation  , (III.3)  Given the ansatz that  depends on  both directly and, through the order parameter fields, indirectly,  is rewritten as a functional of  and a function of , i.e.    . (III.4) The set of scaled times ,  is introduced to account for processes taking place on increasingly long timescales; and  is the set of times tracking the slow processes, i.e. much slower than the atomic processes tracked by . With this and the chain rule, we obtain the multiscale Liouville equation    (III.5)    (III.6)   . (III.7) In  the  are taken at constant , while in  the functional derivatives are taken at constant . Following our earlier procedure [9-10], our strategy is to solve the Liouville equation perturbatively in the small  limit, and then use the solution to obtain a closed equation for the stochastic dynamics of the field variables. The development continues with the perturbation expansion   , (III.8) and examining the multiscale Liouville equation at each order in . We hypothesize that the lowest order solution  is slowly varying in time and is, thus, independent of .   To , the above assumptions imply . Since no further information is known,  is determined by adopting an entropy maximization procedure with canonical constraint of fixed average energy [13]. With this, we obtain     (III.9)    (III.10) where  is inverse temperature,  is the Hamiltonian     (III.11) for N-atom potential , and  is the  partition functional given by   . (III.12)  To , the multiscale Liouville equation implies  . (III.13) This admits the solution  , (III.14) where the initial first order distribution was taken to be zero (i.e. the system is in a quasi-equilibrium state at the initial instant) [29-30]. Inserting (III.7) and (III.9) in (III.14), we obtain , (III.15) where the thermal-average  is defined via  , (III.16) and  is the free energy, related to  via  . (III.17) Using the Gibbs hypothesis, imposing that  be finite as , and using the fact that the thermal-average of  is zero (since the weighing factor  is even in  while  is odd in it), we conclude that  is independent of . With this, (III.15) becomes  . (III.18) Inserting (III.9) and (III.18) in the conservation equation (III.3) and using the fact that  as  yields , (III.19) where the diffusion coefficients  are defined as  . (III.20) This constitutes a Smoluchowski equation for the stochastic dynamics of the field variables and can thus provide a theory of liquid-crystal phenomena. The set of Langevin equations equivalent to (III.19) allow a practical simulation as outlined in the next section.  IV Langevin Equations of Liquid-Crystal Transition For practical simulation of Liquid-Crystal transition, we derive the Langevin equations equivalent to the Smoluchowski equation (III.19). We suggest the Langevin equations take the form   (IV.1) where  are random forces and  are kernels which we will show to be related to  of Sect. III. In what follows, we demonstrate that this implies the Smoluchowski equation (III.19). For simplicity, we henceforth drop the subscript 2 on .  Starting with the Markov assumption and assuming the set of random forces  changes much more rapidly than the set of field order parameters ,  is postulated to evolve via    (IV.2) where  is a functional integral,  is the transition probability functional for a scenario of  during the time interval  to ; ; ;  is the set of  for all  and ;  is the set of  for all  and ; and  is the solution to (IV.1) at time  given that the state was  at time .  Assuming that  changes very little over , we get   (IV.3) Inserting (IV.3) in the LHS of (IV.2) and expanding the -function on the RHS in  truncated at second order (following similar procedure as earlier [10]), we obtain    (IV.4) where  is the T-weighted average for any  variable . In what follows, we assume that  and that the process generating  is stationary. With this, integrating (IV.1) and neglecting terms of order  and higher yields   (IV.5)   (IV.6) where   (IV.7)  . (IV.8) Inserting (IV.5) and (IV.6) in (IV.4) yields  . (IV.9) Comparing (IV.9) with the functional Smoluchowski equation (III.19), we find that the  kernels, the diffusion coefficients , and the statistics of the random forces are related via   (IV.10)  . (IV.11) The Langevin equations presented in this section provide a practical way to simulate stochastic liquid crystal dynamics wherein atomic details are retained via the use of the diffusion coefficients and the thermal average forces. The latter are to be numerically calculated via molecular dynamics (MD) simulations and an efficient algorithm for developing the ensembles. In addition, the statistics of the random forces used in the Langevin equations are restricted via (IV.11).  V Simulating Stochastic Field Dynamics The all-atom/continuum multiscale theory holds great promise for the efficient simulation of liquid-crystal transitions. Providing simulations that use the equations derived here is beyond the scope of this study. However, to illustrate the feasibility of a multiscale computational approach, we suggest the following algorithm.   As the set order parameters are field variables, the first step is to divide the space of  and that of each  into finite elements. The order parameter fields  are then defined in each finite volume. The space of  is discretized into a cubic grid while each  is discretized in a way that depends on . The size of each element/structural window should be smaller than a critical characteristic size but large enough to contain a statistically significant number of molecules. For example, consider a system consisting of two types of molecules. For the first type, only the center of mass position is of interest. For the second type, information on both center of mass position and orientation is needed. In this case,  and . The space of  is then divided into cones and a domain number  is assigned to each cone while a domain number  is assigned to each cube in the space of . With this,  corresponds to the number of type 1 molecules in the cube , while  corresponds to the number of type 2 molecules in cube  with an orientation in the cone . In general,  corresponds to the number of type  molecules in the  cube with characteristics in the set of domains  (i.e. in a structural window defined by the set of domains ). For example, while  corresponds to an orientational window,  can correspond to overall size,  to degree of coiling, etc, and which of these windows is appropriate will depend on the molecular type . Note that the size of each cube  should be large enough to contain several molecules as  represents a coarse-grained location and is not intended to capture small scale structure.  The next step is to calculate the thermal-average forces and diffusion coefficients. The thermal-average forces are computed via a method for generating an ensemble of atomistic configurations, followed by a Monte Carlo integration to obtain the thermal-average of the order parameter force calculated from each configuration. The friction factors are computed using MD-generated time courses used to compute correlation functions for the rates of change of the order parameters (see III.20). Even when using a highly efficient parallelized algorithm, the calculations might not be tractable unless the kernels  are short range; in this case, only interactions between nearest-neighbor elements are needed. The thermal-average forces and diffusion coefficients are then used to drive the dynamics of the order parameters.   at time  for timestep  will then be computed using   (V.1) where  is, for example, a sequence of Gaussian time pulses with random spacing, width, and time where the random numbers are chosen according to probability distributions consistent with the MD-correlation functions. The thermal-average forces and diffusion coefficients are then updated and the cycle is repeated (see Fig. 2).  
+ Fig.2 An efficient algorithm involves starting with an initial configuration to obtain the initial values of the thermal average forces  and diffusion coefficients , in addition to the structural parameters, and thus the initial values of the order parameter fields. The former are used to drive the dynamics of the order parameter fields. The thermal average forces and diffusion coefficients are then updated “on the fly” and the cycle is repeated. This scheme is consistent with the schematic NanoX workflow provided in Ref [9]. VI Conclusions An all-atom/continuum multiscale (ACM) approach for simulating liquid-crystal transitions is presented. Structural parameters describing the state of each type of molecule are introduced (e.g. center-of-mass, orientation, and compression/extension) and order parameter fields are defined as the densities of subpopulations of molecules whose structural variables reside in a specified window. A functional Smoluchowski equation for the stochastic dynamics of the order parameter fields and its equivalent Langevin equations are derived. An algorithm for simulating these Langevin equations is provided.  Our approach allows for simulation of liquid-crystal transitions that retains key atomic detail and does not need calibration, in contrast to phenomenological models. The theory is built on our ACM [9-10] approach which is tailored to systems without long-lasting nearest-neighbor molecular connectivity but which can support long-lasting organization on the average. ACM was originally developed to investigate the behavior of enveloped viruses [9] and nanocapsules for the delivery of therapeutic agents [10]. However, the liquid-crystal behavior of these two systems was not taken into consideration. With the present approach, a more complete study of both systems can now be conducted.  Computational approaches for liquid crystal transitions have been based on either semi-phenomenological molecular [34] or phenomenological field [4,35-36] models. The former describe the interactions at the molecular levels; however, they simplify the system to make the model computationally feasible (e.g. rigid rod or bead models). Field models, on the other hand, describe the system via order parameter profiles and use a phenomenological expression for the free-energy functional. Take, for example, the widely used de Gennes models [4]. Two different models were introduced, one for the isotropic-nematic transitions and another for the nematic-smecticA one; the reason for having distinct models is that the model that works for one doesn’t work for the other. Thus, there was a need for an approach that subsumes multiple types of transitions and possible more complex ones. Phenomenological models that extend the Landau-de Gennes free energy functional to simultaneously describe more than one transition were presented [35-36]. As the free energy expression becomes more complex, more parameters are introduced that need calibration with each new application. This implies that it is possible to introduce a unified phenomenological theory describing the various known transitions; however, such a model does not allow the discovery of new phase transitions as a first principle theory would. Also, the increased complexity and number of calibrated parameters would prove to be restrictive for a computational model and one could question the true predictive character of such a theory.  To illustrate our approach, consider a system that exhibits 3 phases: isotropic-nematic-smecticA. Keeping in mind that one can include as many structural parameters as needed in our theory, at least four are required for this case: center-of-mass  and orientations  for 3  values, for  and for each type  requiring these parameters. Let  be the probability that a type  molecule in a sampling volume  has orientations  specified by the sampling function  (see appendix for the special case of rigid axisymmetric molecules). When the system is in the isotropic liquid phase,  is a constant. As the system transitions to the nematic phase,  becomes non-constant and indicates preferred orientation. For the transition from the nematic to the smectic phases, one needs to examine the atomistic probability density to extract the short-scale order which is computed via correlation functions using the conditional probability  (of III.10) and Monte Carlo integration. With the inclusion of other internal molecular structural parameters, new types of transitions can be discovered. For example, the constituent molecules might have a tendency to assume a specific structure (e.g a helix) versus another one.  Our approach has the advantages of the molecular models in the sense that key small scale information is taken into consideration and can thus be applied to multiple types of phase transitions without the above cited difficulties of these models (i.e. without the need to simplify the components composing the system and for recalibration with each new application).   Appendix Consider the case of a single type of molecule so that the subscript  can be dropped for all parameters. If the molecules are rigid, then the residuals  are zero and the position of atom  is given by   , (A.1) where  (i.e. we consider the displacements from the CM of  as they are more convenient),  is a reference point,  is the relative position after rotation from the reference configuration, and the superscript “rot” on the sum indicates that the three essential basis functions are , , and  where , ,  are the components of  along the , , and -axes, respectively. With this, we obtain   . (A.2) Thus, the set of  constitutes a rotation matrix  such that    (A.3)    (A.4) Integrating  with respect to  over a volume  yields    (A.5) where  is the number of molecules in  with internal structural variables in a window about .  and  are the set of elements in  for any molecule and molecule , respectively. For axisymmetric molecules,  can be related to the set  of two polar angles. Letting  be the two-dimensional polar angle volume element, the orientational probability distribution  takes the form   . (A.6) For isotropic states,  is independent of , simply being a measure of the size of the window. In a liquid crystal, it takes structure, i.e., it indicates preferred orientation.  To relate our approach to existing models, consider, for example, the Maier-Saupe model for axisymmetric molecules. The proposed Smoluchowski equations [37-38] to study the dynamics of such systems describe the evolution of the orientational probability distribution. Here, we derive a Langevin equation describing the evolution of . If exchange of molecules between  and its surroundings are ignored, then we may choose  to be normalized  ; in this case, one obtains  . (A.7) Assuming spatial symmetry such that the thermal average force is independent of  and inserting the Langevin equation of Sect. IV in (A.7) yields   (A.8) where  , (A.9) and the rotational diffusion coefficient  is given by   . (A.10)  We suggest that the Langevin equation described here is more appropriate. For example, a Smoluchowski equation for  as presented in [37-38] implies a Langevin equation for the evolution of the set of polar angles. However, the latter does not qualify as an order parameter because its dynamics is on a similar timescale to that of atomistic fluctuations which violates the arguments needed to derive such an equation. Furthermore, our approach is calibration-free and can be readily generalized to more complex cases supporting a rich array of liquid-crystal phenomena.  Acknowledgements  This project was supported in part by the National Institute of Health (NIBIB), the National Science Foundation (CRC Program), and Indiana University’s college of Arts and Sciences through the Center for Cell and Virus Theory.   Literature Cited [1] SJ Woltman, GD Jay, and GP Crawford, Liquid Crystals: Frontiers in Biomedical Applications (World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ, 2007) [2] GT Stewart, Liquid crystals in biology I: Historical, biological and medical aspects, Liquid Crystals, 30 (5), 541-557 (2003)  [3] JW Goodby, V Görtz, SJ Cowling, G Mackenzie, P Martin, D Plusquellec, T Benvegnu, P Boullanger, D Lafont, Y Queneau, S Chambert, and J Fitremann, Thermotropic liquid crystalline glycolipids, Chemical Society Reviews, 36 (12), 1971-2128 (2007) [4] PG de Gennes and J Prost, The Physics of Liquid Crystals (2nd Ed. Oxford University Press, New York, 1993) [5] C McBride, MR Wilson, and JAK Howard, Molecular dynamics simulations of liquid crystal phases using atomistic potentials, Molecular Physics, 93 (6), 955-964 (1998) [6] AJ McDonald and S Hanna, Atomistic simulation of a model liquid crystal, Journal of Chemical Physics, 124 (16), 164906 (2006) [7] MR Wilson, Molecular simulation of liquid crystals: progress towards a better understanding of bulk structure and the prediction of material properties, Chemical Society Reviews, 36 (12), 1881-1888 (2007) [8] CM Care and DJ Cleaver, Computer simulation of liquid crystals, Reports on Progress in Physics, 68 (11), 2665-2700 (2005) [9] Z Shreif, P Adhangale, S Cheluvaraja, R Perera, R Kuhn, and P Ortoleva, Enveloped Viruses Understood via Multiscale Simulation: Computer-aided vaccine design, Scientific Modeling and Simulation, 15 (1-3), 363-380 (2008) [10] Z Shreif and P Ortoleva, All-atom/Continuum Multiscale Theory: Application to Nanocapsule Therapeutic Delivery, SIAM Multiscale Modeling and Simulation, 2009, (submitted) [11] S Izvekov and GA Voth, A Multiscale Coarse-Graining Method for Biomolecular Systems, Journal of Physical Chemistry B, 109 (7), 2469-2473 (2005) [12] J Zhou, IF Thorpe, S Izvekov, and GA Voth, Coarse-Grained Peptide Modeling Using a Systematic Multiscale Approach, Biophysical Journal, 92 (12), 4289-4303 (2007) [13] S Pankavich, Y Miao, J Ortoleva, Z Shreif, and P. Ortoleva, Stochastic Dynamics of Bionanosystems: Multiscale Analysis and Specialized Ensembles, Journal of Chemical Physics, 128 (23), 234908 (2008) [14] S Chandrasekhar, Stochastic Problems in Physics and Astronomy, Reviews of Modern Physics, 15 (1), 1-89 (1943) [15] RI Cukier and JM Deutch, Microscopic Theory of Brownian Motion: The Multiple-Time-Scale Point of View, Physical Review, 177 (1), 240-244 (1969) [16] S Bose and P Ortoleva, Reacting hard sphere dynamics: Liouville equation for condensed media, Journal of Chemical Physics, 70 (6), 3041-3056 (1979) [17] S Bose and P Ortoleva, A hard sphere model of chemical reaction in condensed media, Physics Letters A, 69 (5), 367-369 (1979) [18] JM Deutch and I Oppenheim, The Concept of Brownian Motion in Modern Statistical Mechanics, Faraday Discussions of the Chemical Society, 83 (83), 1-20 (1987) [19] JE Shea and I Oppenheim, Fokker-Planck equation and Langevin equation for one Brownian particle in a nonequilibrium bath, Journal of Physical Chemistry, 100 (49), 19035-19042 (1996) [20] Shea JE and I Oppenheim, Fokker-Planck equation and non-linear hydrodynamic equations of a system of several Brownian particles in a non-equilibrium bath, Physica A, 247 (1-4), 417-443 (1997) [21] MH Peters, Fokker-Planck equation and the grand molecular friction tensor for combined translational and rotational motions of structured Brownian particles near structures surface, Journal of Chemical Physics, 110 (1), 528-538 (1999) [22] MH Peters, Fokker-Planck equation, molecular friction, and molecular dynamics for Brownian particle transport near external solid surfaces, Journal of Statistical Physics, 94 (3-4), 557-586 (1999) [23] MH Peters, The Smoluchowski diffusion equation for structured macromolecules near structured surfaces, Journal of Chemical Physics, 112 (12), 5488-5498 (2000) [24] P Ortoleva, Nanoparticle Dynamics: A Multiscale Analysis of the Liouville Equation, Journal of Physical Chemistry B, 109 (45), 21258-21266 (2005) [25] Y Miao and P Ortoleva, All-atom multiscaling and new ensembles for dynamical nanoparticles, Journal of Chemical Physics, 125 (4), 044901 (2006) [26] Y Miao and P Ortoleva, Viral structural transitions: An all-atom multiscale theory, Journal of Chemical Physics, 125 (21), 214901 (2006) [27] Z Shreif and P Ortoleva, Curvilinear All-Atom Multiscale (CAM) Theory of Macromolecular Dynamics, Journal of Statistical Physics, 130 (4), 669-685 (2008) [28] Y Miao and P Ortoleva, Molecular Dynamics/Order Parameter Extrapolation for Bionanosystem Simulations, Journal of Computational Chemistry, 30 (3), 423-437 (2009) [29] S Pankavich, Z Shreif, and P Ortoleva, Multiscaling for Classical Nanosystems: Derivation of Smoluchowski and Fokker-Planck Equations, Physica A, 387 (16-17), 4053-4069 (2008) [30] Z Shreif and P Ortoleva, Multiscale Derivation of an Augmented Smoluchowski, Physica A, 388 (5), 593-600 (2009) [31] Z Shreif and P Ortoleva, Computer-aided design of nanocapsules for therapeutic delivery, Computational and Mathematical Methods in Medicine (ISSN: 1748-670X), 10 (1), 49-70 (2009) [32] S Pankavich, Z Shreif, Y Miao, and P Ortoleva, Self-Assembly of Nanocomponents into Composite Structures: Derivation and Simulation of Langevin Equation, Journal of Chemical Physics, 130 (19), 194115 (2009) [33] S Pankavich and P Ortoleva, Self-Assembly of Nanocomponents into Composite Structures: Multiscale Derivation of Stochastic Chemical Kinetic Models, in preparation [34] JM Ilnytskyi and MR Wilson, Molecular models in computer simulation of liquid crystals, Journal of Molecular Liquids, 92 (1-2), 21-28 (2001) [35] I Lelidis and G Durand, Landau model of electric field induced smectic phases in thermotropic liquid crystals, Journal de Physique II, 6 (10), 1359-1387 (1996) [36] P Biscari, MC Calderer, EM Terentjev, Landau-de Gennes theory of isotropic-nematic-smectic liquid crystal transitions, Physical Review E, 75 (5), 051707 (2007) [37] S Hess, Fokker-Planck-Equation Approach to Flow Alignment in Liquid Crystals, Zeitschrift Fur Naturforschung Section A- A Journal of Physical Sciences, 31, 1034-1037 (1976) [38] M Doi and SF Edwards, The Theory of Polymer Dynamics (Clarendon Press, Oxford, 1988) 
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+arXiv:1001.0272v1  [hep-ex]  2 Jan 2010A systematicstudyofthestronginteractionwith
+¯PANDA
+JohanMesschendorp∗forthe¯PANDA collaboration
+KVI, University ofGroningen,TheNetherlands
+E-mail:messchendorp@kvi.nl
+The theoryof QuantumChromoDynamics(QCD) reproducesthe s tronginteractionat distances
+much shorter than the size of the nucleon. At larger distance scales, the generation of hadron
+massesandconfinementcannotyetbederivedfromfirstprinci plesonbasisofQCD.The ¯PANDA
+experiment at FAIR will address the origin of these phenomen a in controlled environments.
+Beams of antiprotons together with a multi-purpose and comp act detection system will provide
+uniquetoolstoperformstudiesofthestronginteraction. T hiswillbeachievedviaprecisionspec-
+troscopy of charmonium and open-charm states, an extensive search for exotic objects such as
+glueballsandhybrids,in-mediumandhypernucleispectros copy,andmore. Anoverviewisgiven
+ofthephysicsprogramofthe ¯PANDAcollaboration.
+EuropeanPhysicalSocietyEurophysicsConferenceonHighE nergyPhysics,EPS-HEP2009,
+July 16- 222009
+Krakow, Poland
+∗Speaker.
+c/circlecopyrtCopyrightownedbytheauthor(s)underthetermsoftheCreat ive CommonsAttribution-NonCommercial-ShareAlikeLicen ce. http://pos.sissa.it/Stronginteractionstudieswith ¯PANDA JohanMesschendorp
+1. Introduction
+The fundamental building blocks of QCD are the quarks which i nteract with each other by
+exchanging gluons. QCD is well understood at short-distanc e scales, much shorter than the size
+of a nucleon ( <10−15m). In this regime, the basic quark-gluon interaction is suf ficiently weak.
+In fact, many processes at high energies can quantitatively be described by perturbative QCD.
+Perturbation theory fails when the distance among quarks be comes comparable to the size of the
+nucleon. Asaconsequenceofthestrongcoupling,weobserve therelativelyheavymassofhadrons,
+such as protons and neutrons, which is two orders of magnitud e larger than the sum of the masses
+of the individual quarks. This quantitatively yet-unexpla ined behavior is related to the effect of
+chiral symmetry breaking.
+The physics program of the ¯PANDA (anti-Proton ANnihilation at DArmstadt) collaborat ion
+will address various questions related to the strong intera ctions by employing a multi-purpose de-
+tector system [1, 2, 3] at the High Energy Storage Ring for ant i-protons (HESR) of the upcoming
+Facility for Anti-proton and Ion Research (FAIR) [4]. The ¯PANDA collaboration aims to connect
+the perturbative and the non-perturbative QCD regions, the reby providing insight in the mecha-
+nisms of mass generation and confinement. For this purpose, a large part of the program will be
+devoted to charmonium spectroscopy; gluonic excitations, e.g. hybrids and glueballs; open and
+hiddencharminnuclei. Inaddition, variousotherphysicst opicswillbestudiedwith ¯PANDAsuch
+as the hyperon-nucleon and hyperon-hyperon interactions v iaγ-ray spectroscopy of hypernuclei;
+CP violation studies exploiting rare decays in the D and/or Λsectors; studies of the structure of
+the proton by measuring Generalized Parton Distributions ( Drell-Yan and Virtual-Compton Scat-
+tering), "spin" structure functions using polarized anti- protons, and electro-magnetic form factors
+in the time-like region.
+2. The experimental facility
+The key ingredient for the ¯PANDA physics program is a high-intensity and a high-resolu tion
+beam of antiprotons in the momentum range of 1.5 to 15 GeV/c. S uch a beam gives access to
+a center-of-mass energy range from 2.2 to 5.5 GeV/c2in ¯ppannihilations. In this range, a rich
+spectrum of hadrons with various quark configurations can be studied. In particular, hadronic
+states which contain charmed quarks and gluon-rich matter b ecome experimentally accessible.
+The¯PANDA detector will be installed at the High Energy Storage R ing, HESR, at the fu-
+ture Facility for Antiproton and Ion Research, FAIR. FAIR pr ovides a storage ring for beams of
+phase-space cooled antiprotons withunprecedented qualit y and intensity [15]. Antiprotons will be
+transferred to the HESR where internal-target experiments in the beam momentum range of 1.5 –
+15GeV/ccanbeperformed. Electronandstochastic phase spa cecooling willbeavailable toallow
+for experiments with either high momentum resolution of abo ut∼10−5at reduced luminosity or
+with high luminosity up to2 ×1032cm−1s−1with anenlarged momentum spread of ∼10−4.
+The¯PANDA detector is designed as a large acceptance multi-purp ose setup. The experiment
+will use internal targets. It is conceived to use either pell ets of frozen H 2or cluster jet targets for
+the ¯ppreactions, and wiretargets for the ¯ pAreactions.
+2Stronginteractionstudieswith ¯PANDA JohanMesschendorp
+Toaddress thedifferent physics topics, the detector needs tocope withavariety of final states
+and a large range of particle momenta and emission angles. At present, the detector is being de-
+signedtohandlehighratesof107annihilations/s ,withgoodparticle identification andmom entum
+resolution for γ,e,µ,π,K, andpwith the ability to measure D,K0
+S, andΛwhich decay at
+displaced vertices. Furthermore, the detector will have an almost 4πdetection coverage both for
+charged particles and photons. This is an essential require ment for an unambiguous partial wave
+analysis of resonance states. Various design and physics pe rformance studies [16] are ongoing,
+partly making use of adedicated computing framework for sim ulations and data analysis.
+3.¯PANDAphysicstopics
+Thelevelschemeoflower-lyingbound ¯ ccstates,charmonium,isverysimilartothatofpositro-
+nium. Thesecharmoniumstatescanbedescribedfairlywelli ntermsofheavy-quarkpotentialmod-
+els. Precision measurements of the mass and width of the char monium spectrum give, therefore,
+access to the confinement potential in QCD.Extensive measur ements of the masses and widths of
+the 1−Ψstates have been performed at e+e−machines where they can be formed directly via a
+virtual-photon exchange. Otherstates,whichdonotcarryt hesamequantumnumberasthephoton,
+cannot be populated directly, but only via indirect product ion mechanisms. This is in contrast to
+the ¯ppreaction, which can form directly excited charmonium state s of all quantum numbers. Asa
+result, theresolution inthemassandwidthof charmonium st ates isdetermined bytheprecision of
+thephase-space cooled beam momentum distribution andnot b ythe(significantly poorer) detector
+resolution. Theneed for such atool becomes evident by revie wing the many open questions inthe
+charmonium sector. For example, our understanding of the st ates above the D¯Dthreshold is very
+poor and needs to be explored in more detail. Recent experime ntal evidences (see review [5]) hint
+at awhole series of surprisingly narrow states with masses a nd properties which, so-far, cannot be
+interpreted consistently by theory. Besides the spectrosc opy of charmonium states, ¯PANDA will
+alsoprovidethecapabilitytoperformopen-charmspectros copy astheanalogofthehydrogenatom
+inQED(heavy-light system). Strikingdiscrepancies ofrec entlydiscovered DsJstatesbyBaBar[6]
+andCLEO[7]withmodel calculations havebeenobserved. Pre cisionmeasurements ofthemasses
+and widths of these states using antiprotons and by performi ng near-threshold scans are needed to
+shed light on these open problems.
+The self-coupling of gluons in strong QCD has an important co nsequence, namely that QCD
+predictshadronicsystemsconsistingofonlygluons,glueb alls,orboundsystemsofquark-antiquark
+pairs with a strong gluon component, hybrids. These systems cannot be categorized as "ordinary"
+hadrons containing valence q¯qorqqq. The additional degrees of freedom carried by gluons allow
+glueballs and hybrids to have spin-exotic quantum numbers, JPC, that are forbidden for normal
+mesons and other fermion-antifermion systems. States with exotic quantum numbers provide the
+best opportunity to distinguish between gluonic hadrons an dq¯qstates. Exotic states with con-
+ventional quantum numbers can be identified by measuring an o verpopulation of the meson spec-
+trum and by comparing properties, like masses, quantum numb ers, and decay channels, with - for
+instance - predictions from Lattice Quantum Chromodynamic s (LQCD) calculations. The most
+promising energy range to discover unambiguously hybrid st ates and glueballs is in the region of
+3-5 GeV/c2,inwhichnarrow states areexpected tobesuperimposed onas tructureless continuum.
+3Stronginteractionstudieswith ¯PANDA JohanMesschendorp
+In this region, LQCD predicts an exotic 1−+¯cc-hybrid state with a mass of 4.2-4.5 GeV/c2and
+a glueball state around 4.5 GeV/c2with an exotic quantum number of JPC=0+−[8, 9]. The ¯ pp
+production cross section of these exotic states are similar to conventional states and in the order
+of 100 pb. All other states with ordinary quantum numbers are expected to have cross sections of
+about 1µb.
+One of the challenges in nuclear physics is to study the prope rties of hadrons and the modi-
+fication of these properties when the hadron is embedded in a n uclear many-body system. Only
+recently it became experimentally evident that the propert ies of mesons, such as masses of π,K,
+andωmesons, change in adense environment [10, 11, 12, 13]. The ¯PANDAexperiment provides
+a unique possibility to extend these studies towards the hea vy-quark sector by exploiting the ¯ pA
+reaction. For instance, an in-medium modification of the mas s of theDmeson would imply a
+modification of the energy threshold for the production of Dmesons, compared to a free mass. In
+addition, a lowering of the D-meson mass could cause charmonium states which lie just bel ow the
+D¯Dthreshold for the ¯ ppchannel to reside above the threshold for the ¯ pAreaction. In such a case,
+the width of the charmonium state will drastically increase , which can experimentally be verified.
+Although this is intuitively a simple picture, in practice t he situation is more complicated since
+the mass of various charmonium states might also change insi de the nuclear medium. Besides the
+indirect in-medium studies as described above, ¯PANDA will be capable to directly measure the
+in-medium spectral shape of charmonium states. This can be a chieved by measuring the invariant
+mass of thedi-lepton decay products. For the Ψ(3770), for instance, models predict mass shifts of
+the order of –100 MeV[14],which are experimentally feasibl e to observe.
+So far, this paper has concentrated on only a few of the topics which will be addressed by the
+¯PANDA collaboration. There exists, however, a large variet y of other physics topics which can
+ideally be studied with the ¯PANDA setup at the antiproton facility at FAIR. For example, there is
+growinginterest withinthe ¯PANDAcollaboration tomakeuseofelectro-magnetic probes , photons
+and leptons, in antiproton-proton annihilation. These pro bes will be used to study the structure
+of the proton by measuring Generalized Parton Distribution s (GPDs), to determine quark distribu-
+tion functions via Drell-Yan processes, and to obtain time- like electro-magnetic form factors by
+exploiting the ¯ pp→e+e−reaction with an intermediate massive virtual photon. Furt hermore, the
+¯PANDAcollaboration hastheambitiontoperformhypernucle i experiments, whichenablesastudy
+of the hyperon-nucleon and hyperon-hyperon interactions. Finally, plans for symmetry violation
+experiments with ¯PANDAwillopenawindowontophysicsbeyondtheStandardMod elofparticle
+physics.
+4. Summary
+The¯PANDA experiment at FAIR will address a wide range of topics i n the field of QCD, of
+whichonlyasmallpartcouldbepresentedinthispaper. Thep hysicsprogramwillbeconductedby
+using beams of antiprotons together with a multi-purpose de tection system, which enables experi-
+mentswithhighluminositiesandprecisionresolution. Thi scombinationprovidesuniquepossibili-
+tiestostudyhadronmatterviaprecisionspectroscopy ofth echarmonium systemandthediscovery
+of new hadronic matter, such as charmed hybrids or glueballs , as well as by measuring the prop-
+erties of hadronic particles in dense environments. New ins ights in the structure of the proton will
+4Stronginteractionstudieswith ¯PANDA JohanMesschendorp
+beobtained byexploiting electromagnetic probes. Further more, thenextgeneration ofhypernuclei
+spectroscopy will be conducted by the ¯PANDA collaboration and at the J-PARC facility in Japan.
+Tosummarize, ¯PANDAhastheambitiontoprovidevaluableandnewinsightsi nthefieldofhadron
+physics which would bridge our present knowledge obtained i nthe fieldof perturbative QCDwith
+that of non-perturbative QCDand nuclear structure.
+Acknowledgments
+Theauthor acknowledges the financial support from the Unive rsity of Groningen and the GSI
+Helmholtzzentrum für Schwerionenforschung GmbH,Darmsta dt.
+References
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+http://www-panda.gsi.de
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+Magnets, arXiv:0907.0169 .
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+arXiv:0810.1216 .
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+[5] T.Barnes, hep-ph/0608103 .
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+[11] R. Barth et al.,Phys.Rev.Lett. 78,4007(1997).
+[12] F. Laue etal.,Phys.Rev.Lett. 82,1640(1999).
+[13] D.Trnka etal.,Phys.Rev.Lett. 94,192303(2005).
+[14] S.H.Lee, nucl-th/0310080 .
+[15] TheHighEnergyStorageRing,http://www.fz-juelich. de/ikp/hesr/
+[16] PANDA Collaboration,PhysicsPerformanceReportforP ANDA,arXiv:0903.3905 .
+5
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+arXiv:1001.0274v4  [hep-th]  9 Sep 2010Bosons, fermions and anyons in the plane,
+and supersymmetry
+Peter A. Horv´ athya∗,Mikhail S. Plyushchayb†,Mauricio Valenzuelaa‡
+aLaboratoire de Math´ ematiques et de Physique Th´ eorique,
+Unit´ e Mixte de Recherche 6083du CNRS, F´ ed´ eration Denis Poisson, Universit´ e de Tours,
+Parc de Grandmont, F-37200 Tours, France
+bDepartamento de F´ ısica, Universidad de Santiago de Chile
+Casilla 307, Santiago 2, Chile
+Abstract
+Universal vector wave equations allowing for a unified description of anyons, and also
+of usual bosons and fermions in the plane are proposed. The existe nce of two essentially
+different types of anyons, based on unitary and also on non-unitar y infinite-dimensional
+half-bounded representations of the (2+1)D Lorentz algebra is r evealed. Those associated
+with non-unitary representations interpolate between bosons an d fermions. The extended
+formulation of the theory includes the previously known Jackiw-Nair (JN) and Majorana-
+Dirac (MD) descriptions of anyons as particular cases, and allows us to compose bosons
+and fermions from entangled anyons. The theory admits a simple sup ersymmetric general-
+ization, in which the JN and MD systems are unified in N= 1 andN= 2 supermultiplets.
+Two different non-relativistic limits of the theory are investigated. T he usual one general-
+izes L´ evy-Leblond’s spin 1 /2 theory to arbitrary spin, as well as to anyons. The second,
+“Jackiw-Nair” limit (that corresponds to In¨ on¨ u-Wigner contrac tion with both anyon spin
+and light velocity going to infinity), is generalized to boson/fermion fie lds and interpolating
+anyons. The resulting exotic Galilei symmetry is studied in both the no n-supersymmetric
+and supersymmetric cases.
+∗e-mails: horvathy@univ-tours.fr
+†mplyushc@lauca.usach.cl
+‡valenzue@lmpt.univ-tours.fr
+12Contents
+1 Introduction 4
+2 Wave equations 6
+2.1 Examples: Dirac and Deser-Jackiw-Templeton fields . . . . . . . . . . . . . . . 8
+2.2 Example: Majorana-Dirac anyon field . . . . . . . . . . . . . . . . . . . . . . . 9
+2.3 Example: Jackiw-Nair anyon field . . . . . . . . . . . . . . . . . . . . . . . . . . 9
+3so(2,1)lowest/highest weight representations 10
+3.1 Irreducible representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
+3.2 Representations with zero norm states . . . . . . . . . . . . . . . . . . . . . . . 13
+4 Solutions of the wave equations. Anyon interpolation between bosons and
+fermions 13
+5 Extended formulation. Bosons and fermions from anyons 15
+6 Deformed oscillator representation of osp(1|2) 19
+7N= 1supersymmetric generalization 22
+7.1 Dirac/Jackiw-Deser-Templeton supermultiplet . . . . . . . . . . . . . . . . . . . 24
+7.2 Jackiw-Nair/Majorana-Dirac supersymmetric system . . . . . . . . . . . . . . . 27
+8N= 2supersymmetry 28
+9 Nonrelativistic limit 32
+9.1 Usual non-relativistic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
+9.2 Exotic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
+10 Non-relativistic supersymmetry 38
+10.1 Usual N= 1 non-relativistic supermultiplet . . . . . . . . . . . . . . . . . . . . 38
+10.2 Exotic N= 1 non-relativistic supermultiplet . . . . . . . . . . . . . . . . . . . . 40
+10.3 Nonrelativistic Dirac/DJT supermultiplet . . . . . . . . . . . . . . . . . . . . . 41
+11 Extended N= 2non-relativistic supersymmetry 43
+11.1 Usual non-relativistic extended supersymmetry . . . . . . . . . . . . . . . . . . 43
+11.2 Exotic nonrelativistic extended supersymmetry . . . . . . . . . . . . . . . . . . 44
+12 Discussion and outlook 45
+A Finite-dimensional representations of osp(1|2) 47
+A.1osp(1|2)-supermultiplet ( α+=−1/2,α−= 0) . . . . . . . . . . . . . . . . . . . 47
+A.2osp(1|2)-supermultiplet ( α+=−1,α−=−1/2) . . . . . . . . . . . . . . . . . . 48
+31 Introduction
+According to Wigner [1], elementary particles in the plane c orrespond to irreducible represen-
+tations of the planar Poincar´ e group [2]. These are labeled by two Casimir invariants, namely1
+(PµPµ+m2)ψ= 0, (1.1)
+(PµJµ−sm)ψ= 0, (1.2)
+wheremis the mass, sthe spin, andJµis the “spin part” of the total angular momentum
+operator
+Mµ=−ǫµνλxνPλ+Jµ. (1.3)
+Mµ, together with Pµ, generate the (2+1)D Poincar´ e group2.
+For aspinlessparticle, s= 0, thesecond equation dropsout and theKlein-Gordon equat ion,
+(1.1), is sufficient in itself.
+For a Dirac particle, or for a topologically massive vector g auge field of Deser, Jackiw and
+Templeton (DJT) [3], i.e. for Poincar´ e spins s=±1/2 ors=±1, the Pauli-Lubanski condition,
+(1.2), implies (1.1) and provides, therefore, a satisfacto ry description. In the plane the spin can
+take arbitrary real values, however, and in the early nineti es two, slightly different descriptions
+have been proposed for fractional-spin particles (called a nyons [4, 5, 6, 7, 8, 9, 10, 11]). Both
+of them combine the half-bounded, infinite-dimensional, un itary, spin-αrepresentations, D±
+α,
+of the Lorentz group [12] with a finite dimensional, non-unit ary, “internal” representation.
+Choose, for example, D+
+α, which carries an infinite-dimensional, bounded-from-bel ow repre-
+sentation of the so(2,1) algebra of the planar Lorentz group, with generators Jµ. Diagonalizing
+J0,J0|α,n∝an}bracketri}ht= (α+n)|α,n∝an}bracketri}ht,withα>0 andn= 0,...,∞.
+Jackiw and Nair (JN) [13] combine this with the spin-1 repres entation of topologically
+massive gauge theory [3]. In fact, they consider the dual to t he topologically massive gauge
+field strength vectorFµ(x), which transforms under the spin-1 matrices ( jµ)νλ=−iǫµ
+ν λ.A
+“Jackiw-Nair” wave function is, hence,
+Fµ=∞/summationdisplay
+n=0Fµ
+n(x)|α,n∝an}bracketri}ht. (1.4)
+Their wave equation is the Pauli-Lubanski condition (1.2), withJµdenoting the total spin
+[13],
+Pλ(Jλ)nµ n′νFν
+n′−smFnµ= 0,Jµ=Jµ+jµ, s=α−1. (1.5)
+This description is redundant, however, necessitating sub sidiary conditions, which reduce the
+number of components from 3 to 1, and generate the mass shell c ondition [13].
+An alternative description, proposed by one of us [14] and re ferred to below as the
+“Majorana-Dirac” (MD) approach, also uses the infinite dime nsional discrete type representa-
+tionsD±
+α, but the finite dimensional part is, rather, a spinor,
+ψ=∞/summationdisplay
+n=0ψn(x)|α,n∝an}bracketri}ht. (1.6)
+1We tookc= 1, so that our metric is ( −1,+1,+1).
+2The irreducible representations of the universal covering group ofSO(2,1) are, for finite-dimensional non-
+unitary representations, fixed uniquely by the values of the so(2,1) Casimir operator JµJµ. Infinite-dimensional
+(unitary and non-unitary) representations require, in add ition, to specify also the concrete series we are con-
+sidering. On the other hand, the representations of the Ponc ar´ e group are labeled, in the massive case, by the
+Poincar´ e spin that corresponds to the value of the Casimir o peratorPµMµ/m=PµJµ/m.
+4where theψn(x) are two-component spinors. The posited field equations are the planar Majo-
+rana equation [15] supplemented by the planar Dirac equatio n,
+(PµJµ−αm)ψ= 0, (1.7)
+(Pµγµ−m)ψ= 0, (1.8)
+where theγµare the planar Dirac (in fact Pauli) matrices. The total spin is stillJµ=Jµ+jµ
+as in (1.5), except for jµ=−γµ/2, which carry a spin-1 /2 representation, so that s=α−1/2.
+Note that in (1.7) the total spin, Jµ, has been replaced by the D+
+αgeneratorJµ.
+The system of two equations eliminates the unphysical degre es of freedom, so that the
+system (1.7)-(1.8) does not necessitate any further subsid iary condition.
+Both the JN and MD systems are shown to imply the Klein-Gordon equation, (1.1), and
+carry, therefore, an irreducible representation of the pla nar Poincar´ e group.
+•Our first aim is to unify the two descriptions by replacing bot h of them by a suitable
+vector equation.
+To this end we use the vector set of equations proposed earlie r in [16]. By analyzing the
+lowest/highest weight representations of so(2,1), we show that the vector set is appropriate
+not only to describe anyons and usual boson and fermion fields , but also provides us with an
+anyon interpolation between bosons and fermions of arbitrary spin values shifted by one half.
+Considering an extended formulation of the vector set of equ ations, we reveal that the Jackiw-
+Nair [13] and Majorana-Dirac [14] descriptions of anyons ar e incorporated in it as particular
+cases. By a kind of fine tuning, which results in a destructive interference, the extended
+formulation allows us to compose boson and fermion fields fro mentangled anyons. Both this
+possibility and the interpolation mechanism mentioned abo ve rely on infinite dimensional non-
+unitary half-bounded representations of so(2,1).
+•Our second objective is to construct a supersymmetric exten sion of the theory.
+We show that our basic vector set of equations admits a simple and natural supersymmetric
+generalization. The construction is based on osp(1|2) representations realized in terms of a
+special deformation of the Heisenberg algebra [17, 18, 19, 2 0]. As interesting examples, we
+consider an N= 1 supermultiplet that includes the Dirac spinor and topolo gically massive
+DJT vector gauge fields, and unify the JN and MD anyon fields int oN= 1 andN= 2
+supermultiplets.
+•Finally, two different non-relativistic limits are investig ated both in the supersymmetric
+and the non-supersymmetric cases.
+The first, usual, limit yields the L´ evy-Leblond theory for b oson/fermion fields of arbitrary
+spin [21, 22], as well as its supersymmetric and anyon genera lizations. Among particular
+examples, we consider the non-relativistic limit of a topol ogically massive vector gauge field.
+In the second, “Jackiw-Nair” non-relativistic limit [23, 2 4], spin and light speed both tend
+simultaneously to infinity in a special way such that the (sup er) Poincar´ e symmetry is reduced
+to the exotic (super) Galilei symmetry [25, 26, 27, 28].
+52 Wave equations
+Our starting point is to posit the vector set of equations
+Vµψ= 0, V µ=ßPµ−iǫµνλPνJλ+mJµ, (2.1)
+wherePµ=−i∂µandJµgenerates (2+1)D Lorentz transformations, [ Jµ,Jν] =−iǫµνλJλ.
+Such a system of wave equations was proposed before in [16] to describe anyons of mass m>0
+and spin ß.
+Contracting (2.1) with Pµ,Jµand−iǫµνλPνJλproduces three equations, namely
+/parenleftbig
+ßP2+mPJ/parenrightbig
+ψ= 0, (2.2)/parenleftbig
+(ß−1)PJ+mJ2/parenrightbig
+ψ= 0, (2.3)/parenleftbig
+P2J2+(PJ)(m−PJ)/parenrightbig
+ψ= 0. (2.4)
+Ifß= 0, these equations do not to fix the value of the (2+1)D Poinca r´ e Casimir operator P2.
+We suppose therefore that ß∝ne}ationslash= 0. Then the system (2.2)–(2.4) is equivalent to
+P2/parenleftbig
+P2+m2/parenrightbig
+ψ= 0, (2.5)/parenleftbig
+ßP2+mPJ/parenrightbig
+ψ= 0, (2.6)/parenleftbig
+m2J2+ß(1−ß)P2/parenrightbig
+ψ= 0. (2.7)
+It follows that the field ψsatisfies either P2ψ= 0, PJψ= 0,J2ψ= 0, or
+/parenleftbig
+P2+m2/parenrightbig
+ψ= 0, (2.8)
+(PJ−ßm)ψ= 0, (2.9)/parenleftbig
+J2+ß(ß−1)/parenrightbig
+ψ= 0. (2.10)
+The first case corresponds to the trivial representation of t he (2+1)D Poincar´ e group with
+Pµ= 0 andJµ= 0 as seen in the frame Pµ= (p,p,0). Excluding this case, we assume
+henceforth that ψsatisfies Eqns. (2.8)–(2.10) with Jµ∝ne}ationslash= 0. Our vector system of equations
+implies, hence, the Klein-Gordon (2.8) andthe Pauli-Luban ski condition (2.9). Theconsistency
+of our vector equations (2.1) then follows from
+[Vµ,Vν] =−iǫµνλ/parenleftig
+mVλ+Pλ(PJ−ßm)/parenrightig
+. (2.11)
+When nontrivial solutions of (2.1) do exist, it follows that they describe an irreducible repre-
+sentation of the (2+1)D Poincar´ e group with nonzero mass mand spins=ß3.
+Our vector equations determine the representation of the sp in part of the Lorentz transfor-
+mation; eq. (2.10) fixes the value of the Lorentz Casimir.
+Taking into account Eq. (2.8) and passing to the rest frame Pµ= (εm,0,0) whereε= +1
+or−1, we find that our covariant vector equations, (2.1), are equ ivalent to two independent
+equations, namely to
+(J0−εß)ψ= 0, (2.12)
+(J1−iεJ2)ψ= 0. (2.13)
+3The spin zero case can also be incorporated into the theory, s ee below.
+6From Eq. (2.13) we infer that for ε= 1,Jµshould belong to a representation bounded from
+below, and for ε=−1 a representation has to be bounded from above4. (It can also be
+bounded from both sides.)
+The universal covering group of the (2+1)D Lorentz group adm its half-bounded unitary
+infinite-dimensional representations, namely, the discre te type series D±
+α[12, 16]. They are
+characterized by a diagonal operator and a highest (lowest) vector,
+D+
+α:J0|α,n∝an}bracketri}ht= (α+n)|α,n∝an}bracketri}htand (J1−iJ2)|α,0∝an}bracketri}ht= 0, (2.14)
+D−
+α:J0|α,n∝an}bracketri}ht=−(α+n)|α,n∝an}bracketri}htand (J1+iJ2)|α,0∝an}bracketri}ht= 0, (2.15)
+wheren= 0,1,.... The representations D+
+αandD−
+αare related via the Lorentz algebra
+automorphismJ0→−J0,J1→J1,J2→−J2. The Casimir operator takes here the value,
+J2=−α(α−1). (2.16)
+These unitary half-bounded infinite-dimensional represen tationsD±
+αappear in most earlier
+works on wave equations for relativistic [13, 14, 15, 16, 27, 29, 30, 31, 32, 33, 34] and non-
+relativistic [22, 27] anyons.
+The universal covering group of the (2+1)D Lorentz group adm its also unbounded repre-
+sentations, namely, the principal and the supplementary co ntinuous series. They only allow
+trivial solutions of the vector set of equations (2.1), and w ill therefore be discarded.
+The non-unitary (2 j+ 1)-dimensional representations ˜Dj,j=1
+2,1,..., characterized by
+(2.16) with α=−ji.e.,J2=−j(j+1)5, admits both highest- and lowest-spin states,
+˜Dj:J0|j,ℓ∝an}bracketri}ht=ℓ|j,ℓ∝an}bracketri}ht,(J1−iJ2)|j,0∝an}bracketri}ht= 0 and ( J1+iJ2)|j,j∝an}bracketri}ht= 0,(2.17)
+whereℓ=−j,−j+ 1,...,j.Hence, they also can be used in our equation. In this case, the
+vector equation describes usual bosons and fermions of arbi trary spin.
+Up to anunitary transformation, they are obtained from thes pin-jrepresentations of su(2),
+J0=ˆJ3,Jk=−iˆJk,k= 1,2, where [ ˆJa,ˆJb] =iǫabcˆJc,a,b,c= 1,2,3.
+There is a third type of representation of so(2,1) which plays an important role in our
+equations, namely half-bounded, infinite-dimensional non-unitary represen tations. These rep-
+resentations are characterized by relations of the form (2. 16) and (2.14) or (2.15), with the
+parameterαin (2.16) taking non-half-integer negative values,−α∝ne}ationslash=j= 1/2,1,.... They are
+denoted here by ˜D±
+α,
+˜D±
+α:J0|α,n∝an}bracketri}ht=±(α+n)|α,n∝an}bracketri}ht,(J1∓iJ2)|α,0∝an}bracketri}ht= 0,−j <α<−j+1/2.(2.18)
+These representations describe anyons whose fractional sp in interpolates between bosons
+and fermions.
+Occasionally, we will also use the unifying notation,
+D±
+α=
+
+D±
+αforα>0,
+˜D±
+αforα<0, α∝ne}ationslash=−j,
+˜Djforα=−j.(2.19)
+4Changingm→ −min (2.1) together with intertwining representation bounde d from below and above yields
+solutions of the same energy but of the opposite sign of spin .
+5We reserve the notation jto denote the (half-) integer values, 1 /2,1,3/2,....
+7ChoosingJµin one of irreducible representations (2.19) and putting
+ß=α, (2.20)
+Eq. (2.10) is identically satisfied for all real α∝ne}ationslash= 0.
+Matrix representations of D±
+α,˜Djand˜D±
+αare considered, in a unified way, in Section 3
+below; in Section 6 they are treated as Fock-like representa tions of a certain deformation of the
+Heisenberg algebra.
+In Section 4 we describe explicit solutions of the vector set of equations with Jµchosen in
+any of the three irreducible representations (2.19), and wi th parameter ßcoherently fixed by
+(2.20). In what follows we refer to such a realization of (2.1 ) as aminimal one.
+Anextended formulation, which combines several irreducible represen tations, will be con-
+sidered in Section 5.
+2.1 Examples: Dirac and Deser-Jackiw-Templeton fields
+The two simplest finite-dimensional representations ˜Djare,
+•j=−α=1
+2,Jµ=−1
+2γµ, when the linear differential equation (2.9) reduces to the Di rac
+equation,
+(Pγ−m)ψ= 0. (2.21)
+Conversely, our vector system (2.1) is the Dirac equation (2 .21), multiplied by1
+2γµ.
+•j=−α= 1, (Jµ)λν=−iǫλµν, when (2.9) reduces to the equation for a topologically
+massive vector gauge field [3],
+/parenleftig
+−iǫλµνPµ+mδλ
+ν/parenrightig
+ψν= 0, (2.22)
+where we switched to the notation ψµinstead ofFµ, used in (1.4). Eq. (2.22) implies
+the transversality condition
+Pµψµ= 0, (2.23)
+that allows us to view the vector field ψµas dual to the U(1) gauge invariant tensor,
+ψµ=1
+2ǫµνλFνλ,Fµν=∂µAν−∂νAµ.
+Conversely, the vectorial form (2.1) is obtained from (2.22 ) by multiplication by −iǫρσλ
+and addition to Eq. (2.23), multiplied by −δρ
+σ.
+Thevaluesj=1
+2andj= 1areexceptional inthatinthesecases(only)thelineardi fferential
+equation (2.9), i .e (2.21) and (2.22), does already imply th e Klein-Gordon equation, (2.8). This
+explains why, for |s|∝ne}ationslash=1
+2,1, getting an irreducible representation of the Poincar´ e g roup requires
+positing a vector set of equations for anyons, and for for usu al fields of integer and half-integer
+spin with|s|=j,j >16.
+6Any two of three equations (2.1) generate the third one as con sistency condition, see Section 4 below; the
+complete set of three equations provides us with an explicit ly covariant formulation.
+82.2 Example: Majorana-Dirac anyon field
+Both the Dirac and DJT models are associated with finite dimen sional representations. We
+can also combine them with fractional spin, however. Consid er, in fact, the representation of
+the spin part of the Lorentz generators as7,
+Jµ=Jµ−1
+2γµ, J µ∈D+
+α,−1
+2γµ∈˜D1/2, α∝ne}ationslash= 1/2, (2.24)
+and put
+ß=α−1/2. (2.25)
+Eq. (2.10) allows us to infer,
+Πψ= 0,where Π = Jγ+α. (2.26)
+The operator Π is a projector : Π2= (2α−1)Π. Multiplying (2.26) from the left by the
+[on-shell invertible, cf. (2.8)] operator Pγ, we find that the field also has to satisfy
+Υψ= 0,where Υ = PJ−αPγ−Λ,Λ =iǫµνλPµJνγλ. (2.27)
+Multiplication of (2.26) by −Λ yields ((α−1)Υ−Λ)ψ= 0, and we find, hence, that ψalso
+has to satisfy the equation
+Λψ= 0. (2.28)
+In the present case, Eq. (2.9) reads/parenleftbig
+PJ−αm)−1
+2(Pγ−m)/parenrightbig
+ψ= 0. Combining it with Eq.
+(2.27) and taking into account Eq. (2.28), we find that our fiel dψn
+b, that carries an index
+n= 0,1,...of the representation D+
+αas well as a spinor index b, has to satisfy, separately, the
+(2+1)DMajorana and Dirac equations,
+(PJ−αm)ψ= 0,(Pγ−m)ψ= 0. (2.29)
+Thissystemoftwoequationswasproposedin[14]todescribe (2+1)Danyons; thecorresponding
+field was called a “Majorana-Dirac field”.
+It is easy to check that these two equations imply (2.26) and ( 2.28) as consistency (inte-
+grability) conditions. Moreover, we have the following rem arkable property: the full set of four
+equations (2.29), (2.26) and (2.28) is generated by positin g any two equations out of the four,
+(one of which should involve the mass parameter). It follows that the vector system (2.1), with
+(2.24) and (2.25) describes an irreducible representation of the (2+1) DPoincar´ e group of mass
+mand spins=α−1
+2∝ne}ationslash= 0.
+2.3 Example: Jackiw-Nair anyon field
+Similarly, consider now,
+Jµ=Jµ+jµ, J µ∈D+
+α,(jµ)λν=−iǫλµν∈˜D1, α∝ne}ationslash= 1, (2.30)
+and
+ß=α−1, (2.31)
+for which the Pauli-Lubanski condition (2.9) becomes,
+/parenleftig
+PµJµδλ
+ν−iǫλµνPµ−(α−1)mδλ
+ν/parenrightig
+ψν= 0, (2.32)
+7The assumed tensor product, Jµ=Jµ⊗1−1⊗1
+2γµ, is not indicated for the sake of notational simplicity.
+9that is the anyon equation put forward by Jackiw and Nair [13]8. Eq. (2.10) reads in turn
+/parenleftig
+−iǫλµνJµ−αδλ
+ν/parenrightig
+ψν= 0. (2.33)
+Now we turn to the subsidiary conditions in [13]. Contractin g (2.33) from the left with Jλ
+yields
+Jµψµ= 0. (2.34)
+Next, contraction with the vector Pλproduces/parenleftbig
+−iǫµνλPνJλ−αPµ/parenrightbig
+ψµ= 0, while contraction
+with−iǫλρσPρJσproduces an equation of the same form, but with the sign befor eαreversed.
+In addition to (2.34), the field also has to satisfy therefore ,
+Pµψµ= 0,andǫµνλPµJνψλ= 0, (2.35)
+which are precisely the subsidiary conditions of Jackiw and Nair in Ref. [13]9.
+Contracting Eq. (2.33) with iǫρσλPσand taking into account Eq. (2.35), yields
+/parenleftig
+(PJ−αm)δλ
+ν+α(−iǫλµνPµ+mδλ
+ν)/parenrightig
+ψν= 0. (2.36)
+Then, using Eq. (2.32) shows that Eq. (2.33) splits the JN equ ation (2.32) into a Majorana
+equation, supplemented with the DJT equation of a topologic ally massive gauge vector field.
+Note however that, unlike in the Majorana-Dirac case (2.29) , imposing the pair of Majorana
+and DJT equations alone on ψn
+µdoes not imply the condition (2.33) that guarantees the irre -
+ducibility equation (2.10) as well as the other necessary su bsidiary conditions in (2.35).
+We shall see in Section 5 that Majorana-Dirac and Jackiw-Nai r descriptions of anyon fields
+are included as particular cases into a broader, extended realization of our basic set of equations
+(2.1).
+3so(2,1)lowest/highest weight representations
+Now we discuss some aspects of the representation theory of so(2,1), which will be used in the
+next two Sections to derive explicit solutions. This will al low us to see, in particular, how the
+non-unitary infinite dimensional representations ˜D+
+α(or˜D−
+α) produce an anyon interpolation
+between bosons and fermions, and how bosons and fermions can be composed from anyons.
+LetJµgenerate the so(2,1) algebra, [Jµ,Jν] =−iǫµνλJλ. In terms of the ladder operators
+J±=J1±iJ2,
+[J−,J+] = 2J0,[J0,J±] =±J±. (3.1)
+Eq. (3.1) implies
+[J−,Jn
++] = 2nJn−1
++/parenleftbigg1
+2(n−1)+J0/parenrightbigg
+, n= 0,1,2,.... (3.2)
+From now on, for the sake of definiteness, we consider the lowe st weight representations
+built over the state |0∝an}bracketri}ht,
+J0|0∝an}bracketri}ht=α|0∝an}bracketri}ht,J−|0∝an}bracketri}ht= 0,∝an}bracketle{t0|0∝an}bracketri}ht= 1, (3.3)
+8Both in (2.32) and in (2.33) the index nofψn
+µhas been suppressed.
+9The subsidiary condition (4.12a) of [13] is a linear combina tion of (2.34) and of the first equation from (2.35),
+(PµJν−JµPν)ψν= 0.
+10whereαis real10.
+The non-normalized states
+/tildewider|n∝an}bracketri}ht=Jn
++|0∝an}bracketri}ht, n= 0,1,..., (3.4)
+are eigenvectors of J0,
+J0/tildewider|n∝an}bracketri}ht= (α+n)/tildewider|n∝an}bracketri}ht, (3.5)
+on which the so(2,1) CasimirCso(2,1)=−J2
+0+1
+2(J+J−+J−J+) takes the value
+Cso(2,1)/tildewider|n∝an}bracketri}ht=−α(α−1)/tildewider|n∝an}bracketri}ht. (3.6)
+From relations (3.2)–(3.4) it follows that the vectors/tildewider|n∝an}bracketri}htform an orthogonal basis,
+/tildewider∝an}bracketle{tn|/tildewidern′∝an}bracketri}ht=δnn′C2
+α,n,Cα,0= 1,C2
+α,n=n!n−1/productdisplay
+k=0(2α+k), (3.7)
+whereC2
+α,nis a squared norm of/tildewider|n∝an}bracketri}ht.
+Forα∝ne}ationslash=−j,j= 0,1/2,1,..., the numbersC2
+α,ntake nonzero values for any n= 0,1,...;
+the corresponding representations are infinite-dimension al and irreducible.
+Forα=−j, only the first 2 j+1 numbersC2
+α,nwithn= 0,...,2jtake nonzero values, while
+C2
+α,n= 0 forn= 2j+1,2j+2,.... This happens if/tildewider|n∝an}bracketri}ht= 0 forn= 2j+1,..., and then we
+have a (2j+1)-dimensional irreducible representation.
+There is also another possibility, though : the states/tildewider|n∝an}bracketri}htwithn= 2j+1,2j+2,...may be
+nonzero. Curiously, they have zero norm.
+We first consider irreducible representations, where no zero norm states of the form (3.4)
+are assumed to appear, and then discuss how to treat represen tations with zero norm states.
+3.1 Irreducible representations
+To analyze the solutions of the wave equations, it is conveni ent to work with normalized states,
+denoted by|n),
+|0) =|0∝an}bracketri}ht,(J+)n|0) =/parenleftbig
+Πn−1
+k=0Cα
+k/parenrightbig
+|n) =/tildewider|n∝an}bracketri}ht, (3.8)
+Cα
+n=/radicalbig
+(2α+n)(n+1). (3.9)
+Note thatC2
+α,n=/producttextn−1
+k=0Cα
+k, cf. (3.7), and that
+J0|n) = (α+n)|n),J+|n) =Cα
+n|n+1),J−|n) =Cα
+n−1|n−1).(3.10)
+Due to the structure of the squared norm C2
+α,n, the following cases have to be distinguished.
+•Forα= 0, we have C0
+0= 1 [C0
+n= 0,n= 1,...], yielding the trivialone-dimensional
+spin-0 representation, Jµ|0) = 0.
+•D+
+αwhereα >0. The coefficients C2
+α,nare positive for any n. The|n) = (Cα,n)−1/tildewider|n∝an}bracketri}ht
+are just those|α,n∝an}bracketri}htin (2.14), and satisfy the orthonormality relation ( n|n′) =δnn′. This
+corresponds to the infinite-dimensional unitary half-bounded representatio ns.
+10Highest weight representations can be analyzed in the same w ay and can also be obtained directly from the
+lowest weight representations via the Lorentz algebra auto morphism J0→ −J 0,J±→ J∓.
+11•˜Djwithα=−j, a negative (half)integer. We have both a lowest weight vect or,J−|0) =
+0, and a highest weight vector, J+|2j) = 0; the coefficient C−j
+2jin (3.10) vanishes. This
+is theusual(2j+1)-dimensional non-unitary representation .
+By (3.9), the coefficients C−j
+n,n= 1,...,2j−1, are imaginary. Put
+C−j
+n=iC′−j
+n, C′−j
+n=/radicalbig
+(2j−n)(n+1). (3.11)
+Making use of the so(2,1) commutation relations, one finds that the states |n),n=
+0,1,...,2j, satisfy (n|n′) = (−1)nδnn′. The corresponding metric in the vector space ˜Dj,
+ηnn′=diag(1,−1,...,(−1)2j−1,(−1)2j), (3.12)
+is indefinite, consistently with the non-unitary character of such representations. Re-
+defining the generators as J±→±iJ±yields, instead of (3.10),
+J+|n) =C′−j
+n|n+1),J−|n) =−C′−j
+n−1|n−1), (3.13)
+whereC′−j
+nis given by (3.11). Such redefined operators J+andJ−are mutually conju-
+gate, (Ψ|J+Φ)∗= (Φ|J−Ψ), w.r.t. the indefinite scalar product (Ψ ,Φ) =¯ΨnΦn,¯Ψn=
+Ψ∗kˆηkn, where Φn= (n|Φ), and matrix ˆ ηis given by (3.12), while (Ψ |J0Φ)∗= (Φ|J0Ψ).
+•˜D+
+αwithαnegative such that α∝ne}ationslash=−j. In this case the C2
+α,nalternate. This corresponds
+to theinfinite-dimensional half-bounded non-unitary represent ations.
+Let us discuss this case in some detail. For
+−j <α<−j+1
+2, j= 1/2,1,3/2,..., (3.14)
+all coefficients (3.9) are nonzero. Those with index n <2jhave an imaginary factor i,
+while those with n≥2jare real. The metric is indefinite,
+ηnn′=diag(1,−1,...,(−1)2j−1,(−1)2j,(−1)2j,(−1)2j...). (3.15)
+In the first (2 j+1) positions the metric alternates as in the finite-dimensi onal represen-
+tation˜Dj, and after that it takes the same (constant) value ( −1)2jup to infinity.
+Both the finite and infinite non-unitary representations can be considered in a unified
+way by putting α=−(j+ǫ) in (3.10). Redefining J±→±iJ±, we get
+J0|n) = (−j−ǫ+n)|n),J+|n) =C′−(j+ǫ)
+n|n+1),J−|n) =−C′−(j+ǫ)
+n−1|n−1),(3.16)
+where,
+C′−(j+ǫ)
+n=/radicalbig
+(2(j+ǫ)−n)(n+1). (3.17)
+Here the parameter −1/2≤ǫ≤0 interpolates between −1/2 and 0, which correspond to
+the finite dimensional representations Dj−1/2andDj, respectively.
+In all the described cases, the structure of the metric is con sistent with the properties of
+theC2
+α,nin (3.7).
+123.2 Representations with zero norm states
+Consider now the infinite-dimensional representation with α=−j, which we denote here by
+/tildewidestD+
+−j. The necessity to treat such peculiar representations come s from their appearance in the
+extended realization which will be considered in Section 5.
+The representation/tildewidestD+
+−jis characterized by the presence of an infinite number of zero norm
+states of the form (3.4) with n= 2j+1+k,k= 0,1,.... In such a representation, there is a
+peculiar zero norm state/tildewider|2j+1∝an}bracketri}ht=J2j+1
++|0∝an}bracketri}ht, which by (3.2) and (3.5) satisfies
+J−/tildewider|2j+1∝an}bracketri}ht= 0,J0/tildewider|2j+1∝an}bracketri}ht= (j+1)/tildewider|2j+1∝an}bracketri}ht. (3.18)
+Due to (3.18), the infinite set of zero norm states /tildewider|2j+1+k∝an}bracketri}ht=Jk
++/tildewider|2j+1∝an}bracketri}ht,k= 0,1...,
+span a space invariant under the action of so(2,1), i.e., these states form an irreducible infinite-
+dimensional null subspacewhich wedenoteby D+0
+j+1. All theinfinite-dimensional representation
+/tildewidestD+
+−jspanned by the states (3.4) with n= 0,1,..., is, therefore, reducible.
+Notice that the invariant null subspace D+0
+j+1may appear here because the value
+−(−j)((−j)−1) =−j2−jof the Casimir operator (3.6) at α=−jadmits, by (3.18),
+also the alternative factorization, −j2−j=−(j+1)((j+1)−1).
+Since the zero norm states /tildewider|2j+1+k∝an}bracketri}ht,k= 0,1,...are orthogonal to all states/tildewider|n∝an}bracketri}ht,n=
+0,1,..., we can work with equivalence classes, viewing |Ψ∝an}bracketri}htand|Ψ∝an}bracketri}ht+|ϕ∝an}bracketri}htwhere|Ψ∝an}bracketri}ht∈/tildewidestD+
+−j
+and|ϕ∝an}bracketri}ht∈D+0
+j+1as equivalent11. In particular, we consider any state |ϕ∝an}bracketri}htequivalent to the
+zero state. In such a way, we reduce the peculiar infinite-dim ensional representation/tildewidestD+
+−jto
+the (2j+1)-dimensional non-unitary irreducible representation ˜Djdescribed above, i.e., we get
+/tildewidestD+
+−j/D+0
+j+1=˜Dj.
+4 Solutions of the wave equations. Anyon interpolation be-
+tween bosons and fermions
+Having in mind the three essentially different types of irredu cible representations, we analyze
+first the solutions of our vector system in the minimal realiz ationß=α, assuming that the
+wave function ψcarries irreducible representation D+
+α, whose type is defined by the value of
+the parameter α.
+The wave function in (2.1) can be decomposed into partial wav es,
+ψ(x) =r/summationdisplay
+n=0ψn(x)|n), (4.1)
+wherer=∞for the representations D+
+αand˜D+
+α, andr= 2jfor the finite-dimensional
+representation ˜Dj(α=−j).
+Having chosen an irreducible representation, (2.10) is sat isfied identically. The three equa-
+tions in (2.1) are not independent: choosing any two of them, the third one follows as a
+consequence.
+11Mathematicians call/tildewidestD+
+−ja “Verma module”.
+13It is convenient to choose equations (2.1) with µ= 1,2. Their complex linear combinations
+V±ψ= 0,V±=V1±iV2, give then the system of two equations
+[(α−J0)P++(m+P0)J+]ψ= 0,[(α+J0)P−+(m−P0)J−]ψ= 0,(4.2)
+whereP±=P1±iP2. Consistently with (2.11), these two equations generate th e one with
+µ= 0 as an integrability condition,
+/bracketleftbigg
+αP0+mJ0+1
+2(P−J+−P+J−)/bracketrightbigg
+ψ= 0. (4.3)
+With (4.1), (3.9) and (3.10), from (4.2) we get the equivalen t system
+√
+n+2α(m−P0)ψn−√
+n+1P+ψn+1= 0, (4.4)
+√
+n+2αP−ψn+√
+n+1(m+P0)ψn+1= 0. (4.5)
+Then some algebraic manipulations yield the component form of Eq. (4.3),
+(αP0+m(α+n))ψn+1
+2/parenleftig/radicalbig
+(2α+n−1)nP−ψn−1−/radicalbig
+(2α+n)(n+1)P+ψn+1/parenrightig
+= 0.(4.6)
+Eqns (4.4), (4.5) are conveniently analyzed in the momentum representation. For positive
+energy,P0>0, the operator P0+m∝ne}ationslash= 0 is invertible. Then (4.5) allows us to write all partial
+wave components ψnin terms of ψ0. From (4.5) one obtains, first,
+ψn+1=−√n+2α√n+1P−
+P0+mψn, P0>0, (4.7)
+and then iteratively,
+ψn=/radicalbig
+Cn/parenleftbigg−P−
+P0+m/parenrightbiggn
+ψ0, (4.8)
+Cn=2α(2α+1)...(2α+n−1)
+n!= (nB(2α,n))−1, (4.9)
+whereB(x,y) = Γ(x)Γ(y)/Γ(x+y) is Euler’s beta function.
+Note for further reference that if 2 αis a negative integer, then the coefficients Cnvanish
+whenn≥1−2α: theinfinitetower of the ψ’s terminates, andtherepresentation spacebecomes
+finite dimensional . Alternatively, the beta function in (4.9) has poles at nega tive integers.
+The mass-shell condition follows from Eqns. (4.2), and so th e component ψ0has the form
+ψ0=Cδ(P0−/radicalbig
+/vector p2+m2)δ(/vectorP−/vector p), (4.10)
+whereCis a (normalization) constant and /vector pis the momentum of the state.
+Similarly, for negative energy, P0−mis invertible, allowing us to express ψnin terms of
+ψn+1,
+ψn=√n+1√n+2αP+
+m−P0ψn+1, P0<0. (4.11)
+All components can be expressed in terms of the highest spin s tate component. The highest
+spin state exists, however, onlyfor the finite dimensional representations ˜Dj, in which case
+β=α=−j, and we get
+ψ2j−n=i−n/radicalbigg
+2j(2j−1)...(2j−n+1)
+n!/parenleftbiggP+
+P0−m/parenrightbiggn
+ψ2j, (4.12)
+14where
+ψ2j=Cδ(P0+/radicalbig
+/vector p2+m2)δ(/vectorP−/vector p). (4.13)
+Let us now analyze the positive–energy solutions given by Eq ns. (4.8), (4.9), (4.10). In
+the rest frame /vector p= 0, only the lowest component ψ0is nontrivial. For /vector p∝ne}ationslash= 0, every subsequent
+component includes an additional kinematical factor P−/(P0+m).
+For the infinite dimensional representations D+
+α(and in the generic case of non-unitary
+representations ˜D+
+α), the numerical coefficient Cnis of order 1 when ntends to infinity.
+However, in the case of non-unitary representations ˜D+
+αwithαclose to a negative
+(half)integer j, the nature of coefficients (4.9) is essentially different. Whe nα, supposed to
+satisfy the relation (3.14), tends to −jfrom above, the first coefficients with n= 1,...,2j
+tend to the values of those that correspond to the finite dimen sional case ˜Dj. In particular,
+C2j∼j+α→0 whenα→−j. In such a case, all higher components are suppressed by the
+same factor ( j+α)1/2→0.
+With this picture, having also in mind the metric described i n Section 3.1, we conclude
+thatthe infinite-dimensional half-bounded non-unitary repres entations ˜D+
+αinterpolate between
+boson and fermion states of positive energy. This is what corresponds, intuitively, to ananyon.
+It is worth noting that when αis between−jand−j+1
+2, j= 1/2,1,3/2 cf. (3.14) and
+approaches−j+1
+2from below, the structure with Poincar´ e spin s=−j+1
+2is recovered : it
+is described by the finite-dimensional non-unitary represe ntation˜Dj−1/2and by the indefinite
+metric (3.12), with jchanged into j−1/2.
+The spin zero case ß=α= 0, excluded so far, can be incorporated as the limit α→0 of
+Jµ∈˜Dα,ß=α<0.
+5 Extended formulation. Bosons and fermions from anyons
+In the general case Jµcan be taken as the sum of the generators of the (2 + 1) DLorentz
+group in representations that correspond to the series D±
+α,˜D±
+α, and to ˜Dj. The examples
+of Majorana-Dirac and Jackiw-Nair anyon fields considered i n Sections 2.2 and 2.3 are just
+particular cases of such an extended formulation (or, realization) of the vector set of equation s.
+Taking, for simplicity, just two representations,
+Jµ=Jµ+J′
+µ, J2=−α(α−1), J′2=−α′(α′−1), (5.1)
+and put
+ß=α+α′(5.2)
+where, for a finite-dimensional representation, the parame terαand/orα′is−j, and (5.2) such
+thatß∝ne}ationslash= 0. Then Eq. (2.10) transforms into
+/parenleftbig
+JJ′+αα′/parenrightbig
+ψ= 0. (5.3)
+Eq. (5.3) is a non-dynamical, subsidiary equation for the fie ldψ[which carries two indices not
+shown explicitly]. It, in turn, implies that our composite s ystem (5.1), (5.2) carries an irre-
+duciblerepresentation of the so(2,1)-spinß=α+α′, consistently with Eq. (2.10). This means
+that the extended formulation, in fact, recovers the previo usly considered minimal realization.
+It provides us, however, with a new possibility: in this form ulation usual boson and fermion
+fields of arbitrary spin can be composed from anyons .
+Before considering concrete examples, some general commen ts are in order.
+15•The operator of the non-dynamical equation (5.3) commutes w ith the total vector spin
+generatorJµ. Solutions of (5.3) then can be given by the eingestates/tildewider|k∝an}bracketri}htof the compact so(2,1)
+generatorJ0=J0+J′
+0,J0/tildewider|k∝an}bracketri}ht= (ß+k)/tildewider|k∝an}bracketri}ht,k= 0,..., presented as a certain linear combinations
+of the simultaneous J0andJ′
+0eigenstates,|n)|n′). Particularly, the state |0∝an}bracketri}ht=|0)|0′) is a
+solution that satisfies relations of the form (3.3) with αchanged into ß=α+α′[ for the sake
+of definiteness we assume here that Jµ∈D+
+α,J′
+µ∈D+
+α′].Jµcommutes with the scalar operator
+JJ′+αα′=−J0J0+1
+2(J+J′
+−+J−J′
++)+αα′; all other solutions/tildewider|k∝an}bracketri}htof Eq. (5.3) are obtained
+therefore by subsequent application of J+to the lowest weight state |0∝an}bracketri}htcf. Section 3.
+ForJµ∈˜DjandJ′
+µ∈˜Dj′, Eq. (5.3)singlesoutirreduciblefinitedimensionalrepre sentation
+˜Dj+j′.
+Forß∝ne}ationslash=−j, Eq. (5.3) separates an irreducible bounded from below infin ite dimensional
+representation of the unitary or non-unitary type dependin g onß>0 orß<0.
+Whenß=−jand at least one of the so(2,1) spins is in an infinite dimensional represen-
+tation, Eq. (5.3) gives a reducible representation/tildewidestD+
+−jwith invariant infinite dimensional null
+subspaceD+0
+j+1. As it is explained in Section 3.2, working modulo zero norm s tates|ϕ∝an}bracketri}ht∈D+0
+j+1,
+solutions of (5.3) gives rise to the irreducible finite-dime nsional non-unitary representation
+˜Dj=/tildewidestD+
+−j/D+0
+j+1.
+Asaresult, thesolution ofthevector systemofequationsin theextendedformulation, forall
+possible choices of the so(2,1) representations in (5.1) is reduced to the minimal realiz ation12.
+•WhenJµis composed as in (5.1), the vector set of equations (2.1) spl its, for each Lorentz
+index, into two vector sets.
+Consider, for instance, Jµ∈D+
+α,J′
+µ∈˜Dj, assuming, as in the Majorana-Dirac and Jackiw-
+Nair cases, (2.24) and (2.30), that ß∝ne}ationslash= 0. In the rest frame the system (2.12) – (2.13) becomes
+/parenleftbig
+(J0−εα)+(J′
+0+εj)/parenrightbig
+ψ= 0, (5.4)
+/parenleftbig
+(J1−iεJ2)+(J′
+1−iεJ′
+2)/parenrightbig
+ψ= 0. (5.5)
+Eq. (5.5) has nontrivial solutions if and only if ( J1−iεJ2)ψ= 0. Such a solution, proportional
+to the lowest state |α,0∝an}bracketri}htfor the representation D+
+α, is nontrivial only for ε= +1. Then Eq.
+(5.5) splits into
+(J1−iJ2)ψ= 0,and/parenleftbig
+J′
+1−iJ′
+2/parenrightbig
+ψ= 0,
+splitting Eq. (5.4) into
+(J0−α)ψ= 0,and/parenleftbig
+J′
+0+j/parenrightbig
+ψ= 0.
+Then one findsthat a solution of (2.1) has to be a simultaneous solution of two sets of equations
+of the form (2.1), in one of which JµisJµ∈D+
+α, and in the other JµisJ′
+µ∈˜Dj. The
+corresponding solution has positive energy, mass mand spins=α−j∝ne}ationslash= 0. This means that
+the system of equations,
+V(α)
+µψ= 0, Vα′
+µψ= 0, JJ′+αα′= 0, (5.6)
+is equivalent to (2.1) with ß=α+α′. In other words, spins simply add. This additive property
+allows us to create particles of spin ßbuilt from one of spin αand another one of spin α′.
+Intuitively, the non-dynamical constraint (5.3) “entangl es” the component systems.
+12The spin zero case s= 0 is incorporated into the extended formulation via a limit ß→0 if representations
+in (5.1) are chosen in such a way that −1/2<ß<0.
+16In the Majorana-Dirac and Jackiw-Nair examples discussed i n Section 2, the entangling
+equation (5.3) takes the form (2.26) and (2.33), respective ly.
+•ChoosingJµin a unitary infinite-dimensional representation D+
+αwith non-(half)integer
+α, andJ′
+µin a finite-dimensional non-unitary representation ˜Djin such a way that ß=α−
+j <0, the resulting irreducible representation is an infinite- dimensional half-bounded non-
+unitaryrepresentation of so(2,1). Suchinfinite-dimensional non-unitaryrepresentation s belong
+to the Majorana-Dirac an Jackiw-Nair descriptions, no atte ntion was paid to the corresponding
+interpolating anyons, in the original References [13, 14] t hough.
+•The choices Jµ∈˜D+
+α,J′
+µ∈˜D+
+α′orJ′
+µ∈D+
+α′, provide us with another interesting
+case : if the parameters αandα′add up to a negative (half)-integer ß=−j, our extended
+formulation, based on two half-bounded infinite-dimension al (i.e. anyon-like) representations,
+describes a usual relativistic finite-component field of spi nj. In particular, from two anyon-
+like representations given by α=α′=−1/4 [considered below], the extended formulation
+reproduces the theory of the Dirac particle with spin s=−1/2.
+Examples
+Consider some representative particular examples.
+•LetJµ, J′
+µ∈˜D1/2,α=α′=−1/2,ß=−1. The solutions of Eq. (5.3) are
+|0) =|0)|0′),|1) =1√
+2/bracketleftbig
+|0)|1′)+|1)|0′)/bracketrightbig
+,|2) =|1)|1′). (5.7)
+They are eigenstates of J0with eigenvalues−1, 0 and +1, respectively, and satisfy the relations
+J−|0) =J+|2) = 0,J+|0) =i√
+2|1),J+|1) =i√
+2|2), (5.8)
+(n|n′) = (−1)nδnn′, n, n′= 0,1,2, (5.9)
+[where we have used Eq. (3.9)] which correspond to the repres entation ˜D1. The vector set of
+equations (2.1) describes, in this case, a topologically ma ssive vector gauge field of mass mand
+spins=−1, which can be treated as composed of two massive particles o f spins=−1/2 each.
+•Let us take ( Jµ)λν=−iǫλµν∈˜D1,α=−1,J′
+µ=−1
+2γµ∈˜D1/2,α′=−1/2, and
+ß=−3/2. This corresponds to the Rarita-Schwinger field of spin s=−3/213. Having in mind
+the splitting of our vector equations, we find that the vector -spinor field ψµ[spinor indices are
+not display explicitly] satisfies the Dirac and DJT equation s, supplemented with the entangling
+equation (5.3),
+(Pγ−m)ψµ= 0, (5.10)/parenleftbig
+−iǫµνλPν+mδµ
+λ/parenrightbig
+ψλ= 0, (5.11)/parenleftig
+iǫµνλγν+δµ
+λ/parenrightig
+ψλ= 0. (5.12)
+Taking into account the identity γµγν=−ηµν+iǫµνλγλ, simple algebraic manipulations show
+that the system (5.10)–(5.12) is equivalent to the Rarita-S chwinger equations
+(Pγ−m)ψµ= 0, γ µψµ= 0. (5.13)
+13The chosen representation for α=−1 differs from the matrix representation of Section 3 by a unit ary
+transformation, see Eqns. (A.14), (A.17) and (A.16) in Appe ndix A.2.
+17Thus the system (5.13) generates, in itself, the DJT, and als o the entangling, (5.12), equations.
+Notice a similarity between the second equation in (5.13) an d Eq. (2.34).
+•Consider now the case Jµ∈˜D+
+−1/4,J′
+µ∈˜D+
+−1/4,α=α′=−1/4,ß=−1/2. The two
+lowestnormalizedeigenstates of J0aregivenformallybythesamerelationsasin(5.7). Itshoul d
+be remembered, though, that the states which appear on the ri ght hand sides of the relations
+belong to two copies of the infinite dimensional representat ion˜D+
+−1/4. The normalization
+is given by the indefinite metric described in Section 3. Thes e states|0) and|1) haveJ0-
+eigenvalues−1/2 and +1/2, respectively, and satisfy the relations
+(n|n′) = (−1)nδnn′, n,n′= 1,2,J+|0) =i|1). (5.14)
+Application of the ladder operator J+to|1) does not annihilate it, but produces a zero-norm
+state,
+J+|1) =1√
+2/bracketleftbig
+|0)|2′)+|2)|0′)/bracketrightbig
++i|1)|1′) =/tildewider|2∝an}bracketri}ht,/tildewide∝an}bracketle{t2|/tildewide2∝an}bracketri}ht= 0,
+which is orthogonal to |0) and|1), and is annihilated by J−.
+Other solutions of Eq. (5.3) are obtained by applying, subse quently,J+to the state/tildewider|2∝an}bracketri}ht,
+which, together with the latter, span the null space D+0
+1/2. Factoring out the states |ϕ∝an}bracketri}ht∈D+0
+1/2,
+we get the two-dimensional non-unitary representation ˜D1/2. In such a way two anyons of
+identical so(2,1) spin−1/4, yield, in our extended realization, a fermion field of spin −1/2.
+By this reason, in our picture, these anyons canbe called semions.
+The picture we have just described can be reinterpreted in th e following way. Consider
+a pair of positive energy states (particles) with Poincar´ e spins−1/4. Separately, each such
+particle can be described within the minimal realization wi thJµ∈˜D+
+−1/4,ß=α=−1/4.
+The wave function of each particle picks up a phase, e−iπ/2=−i, under a 2πspatial rotation
+generated by J0, and the system can be interpreted as a pair of semions.
+When switching to the extended formulation with Jµ∈˜D+
+−1/4,J′
+µ∈˜D+
+−1/4,α=α′=
+−1/4,ß=−1/2, the vector system of equations splits, as discussed above , into two vector
+systems ofequations, each describingasemion field. Theset wo fieldsarenot moreindependent,
+however. The vector system implies the non-dynamical equat ion (5.3) that guarantees the
+irreducibility of the Poincar´ e representation realized o n our composed, two-semion system.
+This equation introduces a kind of entanglement between the two semion states, that provides
+us, as a result, with a spin s=−1/2 fermion. Each independent semion is described by
+an infinite-component field in a frame different from the rest fr ame [with higher components
+suppressed by a kinematical factor, see Eq. (4.8)]. Equatio n (5.3) then introduces a kind
+of destructive interference between the semion wave functi ons, with an infinite number of
+components associated with the null subspace, with the exce ption of two surviving independent
+wave components, which, due to the entanglement, describe a two-component positive-energy
+fermion state.
+Notice here that if we choose instead two unitary infinite dim ensional representations, Jµ∈
+D+
++1/4,J′
+µ∈D+
++1/4,α=α′= +1/4,ß= +1/2, our vector set of wave equations would describe
+a positive-energy particle of mass mand spin +1 /2. Although its wave function picks up a
+phase−1 undera 2πrotation, we do not get a usual fermion, since its wave functi on has infinite
+number of components when /vector p∝ne}ationslash= 0, cf. the previous Section. Anyons described by the unitar y
+representations D+
++1/4can not be referred to, therefore, as semions14.
+14The name ‘semions’ cannot be applied therefore to the supers ymmetric system constructed in [30], based on
+18In the same way, if we add two spins corresponding to infinite d imensional anyon repre-
+sentations such that α+α′=ß=−j, we can treat the resulting boson or fermion field as
+composed of two anyons.
+•Finally, we consider shortly another particular interesti ng case,Jµ∈D+
+1/2,J′
+µ∈˜D1,
+α= 1/2,α′=−1,ß=−1/2. In our framework, it describes a usual fermion field15.
+According to Eq. (3.9), the nonzero coefficients are C−1
+0=C−1
+1=i√
+2, andC1/2
+n=n+1.
+The−1/2 and +1/2 eigenstates ofJ0are
+|0) =|0)|0′) and|1) =J+|0) =|1)|0′)+i√
+2|0)|1′), (5.15)
+which satisfy the relations (5.14). The lowest zero-norm st ate is given by
+J+|1) =/tildewider|2∝an}bracketri}htwhere/tildewider|2∝an}bracketri}ht= 2/bracketleftig
+|2)|0′)+i√
+2|1)|1′)−|0)|2′)/bracketrightig
+. (5.16)
+J−/tildewider|2∝an}bracketri}ht= 0. Higher zero norm states of the null subspace D+0
+1/2are produced by applying the
+ladder operatorJ+to/tildewider|2∝an}bracketri}ht.
+6 Deformed oscillator representation of osp(1|2)
+To get a concrete realization of the supersymmetric extensi ons of our model we shall discuss
+later, we use the reflection-deformed Heisenberg algebra (R DHA). The latter was (implicitly)
+introduced by Wigner [17] and led subsequently to parastati stics [18], that condensed, finally,
+into QCD color [19]. To make our discussion self-contained, we outline below those represen-
+tations of the RDHA which are necessary for our purposes. See Ref. [20]16for more details.
+Let us consider the reflection-deformed Heisenberg algebra of the harmonic oscillator
+/bracketleftbig
+a−,a+/bracketrightbig
+= 1+νR, R2= 1,{a±,R}= 0, (6.1)
+whereνis a real deformation parameter and Ris the reflection operator, R= (−1)N=
+exp(iπN). The number operator
+N=1
+2{a+,a−}−1
+2(ν+1),[N,a±] =±a±, (6.2)
+defines the Fock space,
+F={|n∝an}bracketri}ht,n= 0,1,2,...},N|n∝an}bracketri}ht=n|n∝an}bracketri}ht, (6.3)
+where|n∝an}bracketri}ht=Cn(a+)n|0∝an}bracketri}ht,n= 0,1,..., a−|0∝an}bracketri}ht= 0,∝an}bracketle{t0|0∝an}bracketri}ht= 1,
+Cn= ([n]ν!)−1/2,[0]ν! = 1,[n]ν! =n/productdisplay
+l=1[l]ν, n≥1,[l]ν=l+1
+2(1−(−1)l)ν.(6.4)
+The explicit form of the coefficients (6.4) indicates that for ν >−1 the algebra has infinite-
+dimensional, parabosonic-type unitary irreducible repre sentations; for negative-odd values
+the two infinite dimensional representations D+
++1/4andD+
++3/4; the term ‘quartions’ used alternatively seems to
+be more appropriate.
+15This case formally corresponds to the Jackiw-Nair scheme, n o attention was paid to it in [13], though.
+16In the context of quantum mechanical supersymmetry this alg ebra was used in [32, 35].
+19ν=−(2r+ 1),r= 1,2,..., it has (2r+ 1)-dimensional, non-unitary parafermionic-type rep-
+resentations, see below17. Forν <−1,ν∝ne}ationslash=−(2r+1), it has infinite dimensional non-unitary
+representations of the ˜D±
+αtype.
+The quadratic operators
+J0=1
+4{a+,a−},J±=J1±iJ2=1
+2(a±)2, (6.5)
+together with linear operators
+La=/parenleftbigga++a−
+√
+2, ia+−a−
+√
+2/parenrightbigg
+, a= 1,2, (6.6)
+generate the osp(1|2) superalgebra,
+[Jµ,Jν] =−iǫµνλJλ, (6.7)
+{La,Lb}= 4i(Jγ)ab,[Jµ,La] =1
+2(γµ)b
+aLb. (6.8)
+The Lorentz operators Jµare even, andLaare odd supercharges with respect to a Z2-grading
+provided by the reflection operator, [ R,Jµ] = 0,{R,La}= 0. The (2+1) dimensional Dirac
+matrices are in the Majorana representation,
+(γ0)b
+a=−(σ2)b
+a,(γ1)b
+a=i(σ1)b
+a,(γ2)b
+a=i(σ3)b
+a, (6.9)
+which satisfy,
+(γµ)aρ(γν)ρb=−ηµνǫab+iǫµνλ(γλ)ab, γµ
+ab=γµ
+ba, 㵆
+ab=−γµ
+ab.(6.10)
+The antisymmetric tensor, ǫab=−ǫba(ǫ12= 1), plays the role of a spinor metric, rising and
+lowering indices, Aa=ǫabAb,Aa=Abǫba. Thescalar producthastheproperty AaBa=−AaBa
+for any spinors AaandBa. Then the Lorentz generators can be written as
+Jµ=i
+4La(γµ)abLb, µ= 0,1,2. (6.11)
+The representation of osp(1|2) built in this way is irreducible, characterized by the sup er-
+Casimir operator
+C=JµJµ−i
+8LaLa=1
+16(1−ν2). (6.12)
+The representation of the Lorentz subalgebra is, however, r educible, due to supersymmetry.
+This is reflected by the eigenvalues of the Casimir of the Lore ntz subalgebra,
+JµJµ=−ˆα(ˆα−1) with ˆ α=1
+4(2+ν)−1
+4R. (6.13)
+The irreducible components are obtained by projecting to th e eigenspaces of R,
+RF±=±F±, (6.14)
+F+=/braceleftbig
+|n∝an}bracketri}ht+=|2n∝an}bracketri}ht,n= 0,1,2,.../bracerightbig
+,F−=/braceleftbig
+|n∝an}bracketri}ht−=|2n+1∝an}bracketri}ht,n= 0,1,2,.../bracerightbig
+.(6.15)
+17The Fock space construction here is similar to that in Sectio n 3. We refrain from repeating similar technical
+details related to the appearance of null subspaces at ν=−(2r+1).
+20On these subspaces, the so(2,1) Casimir (6.13) takes the values
+JµJµF±=−α±(α±−1)F±,whereα+=1+ν
+4, α−=α++1
+2.(6.16)
+The irreducible representations of the Lorentz algebra are extracted by the projectors, F±=
+Π±F,
+J(±)
+µ=JµΠ±,[J(±)
+µ,J(±)
+ν] =−iǫµνλJ(±)
+λ,[J(+)
+µ,J(−)
+ν] = 0, (6.17)
+J(±)
+µJ(±)µ=−α±(α±−1)Π±, (6.18)
+Π±=1
+2(1±R),Π2
+±= Π±,Π+Π−= 0,Π++Π−= 1. (6.19)
+The representation of Jµin (6.5) is therefore a direct sum, Jµ=J(+)
+µ+J(−)
+µ. Consistently
+with (6.5) and (6.2), J0has eigenvalues n+α±,
+J0|n∝an}bracketri}ht±= (n+α±)|n∝an}bracketri}ht±,|n∝an}bracketri}ht±∈F±. (6.20)
+For future use, it is convenient to introduce the shifted num ber operator,
+N=J0−ˆα,N|n∝an}bracketri}ht±=n|n∝an}bracketri}ht±,ˆα|n∝an}bracketri}ht±=α±|n∝an}bracketri}ht±. (6.21)
+The ladder operators J±=J1±iJ2act as
+J+|n∝an}bracketri}ht±=Cα±n|n+1∝an}bracketri}ht±,J−|n∝an}bracketri}ht±=Cα±
+n−1|n−1∝an}bracketri}ht±,|n∝an}bracketri}ht±∈F±, (6.22)
+Cα±n=/radicalbig
+(2α±+n)(n+1), (6.23)
+cf. (3.9). The rising/lowering operators interchange the F+andF−subspaces,a±:F+↔F−,
+a+|n∝an}bracketri}ht+=/radicalbig
+2(n+2α+)|n∝an}bracketri}ht−, a+|n∝an}bracketri}ht−=/radicalbig
+2(n+1)|n+1∝an}bracketri}ht+, (6.24)
+a−|n∝an}bracketri}ht+=√
+2n|n−1∝an}bracketri}ht−, a−|n∝an}bracketri}ht−=/radicalbig
+2(n+2α+)|n∝an}bracketri}ht+. (6.25)
+On the subspaces F±theso(2,1) spin is shifted by
+α−−α+= 1/2; (6.26)
+it is therefore legitimate to interpret the a±as odd supercharges.
+The deformed oscillator provides us with infinite-, and also finite, dimensional representa-
+tions ofosp(1|2). Observe that, in (6.24) and (6.25), the coefficient n+2α+vanishes when n
+takes the value n=−2α+=−(1+ν)/2. This happens for odd-integer negative values of the
+deformation parameter, ν=−(2r+ 1), r= 1,2,...,for whichn=r. Then the Fock space
+becomes finite,
+F={|0∝an}bracketri}ht,|1∝an}bracketri}ht,...,|2r∝an}bracketri}ht}=F++F−, (6.27)
+F+={|0∝an}bracketri}ht+=|0∝an}bracketri}ht,...,|r∝an}bracketri}ht+=|2r∝an}bracketri}ht},F−={|0∝an}bracketri}ht−=|1∝an}bracketri}ht,...,|r−1∝an}bracketri}ht−=|2r−1∝an}bracketri}ht}.
+These spaces are characterized by the equations
+a−(2r+1)=a+(2r+1)= 0, a−|0∝an}bracketri}ht= 0, a+|2r∝an}bracketri}ht= 0, dim(F) =|ν|= 2r+1.
+21The spins carried by the subspaces F+andF−areα+=−r/2 andα−=−r/2+1/2, respec-
+tively.
+Forν >−1, the spectrum of J0is positive definite and bounded from below, and the spin
+α±takes only positive values. This produces the discrete seri esD+
+α±ofso(2,1). Every series
+is generated by the operators (6.17). The discrete series bo unded from above, D−
+α±, of the
+Lorentz algebra are obtained by the external automorphism ( J0,J1,J2)→(−J0,J1,−J2),
+and projection to the corresponding subspaces F±.
+Forν <−1,ν∝ne}ationslash=−(2r+ 1), the RDHA provides us also with infinite dimensional half -
+bounded non-unitary irreducible representations ˜D±
+α±ofso(2,1).
+For the finite dimensional representation, the operators a+anda−are conjugate,
+(Ψ,a−Φ)∗= (Φ,a+Ψ), w.r.t. the scalar product,
+(Ψ,Φ) =¯ΨnΦn,¯Ψn= Ψ∗kˆηkn, (6.28)
+provided by the matrix ˆ η=diag(+1,−1,−1,+1,+1,...,(−1)r,(−1)r), where Φn=∝an}bracketle{tn|Φ∝an}bracketri}ht.
+The two simplest examples of finite-dimensional representa tions of the reflection-deformed
+Heisenberg algebra (6.1), namely those of α+=−1/2, α−= 0, and of α+=−1, α−=−1/2,
+are summarized in the Appendix.
+Note that for ν=−1, the algebra (6.1) has a one-dimensional trivial represen tation, in
+whichR= 1 anda±= 0. This corresponds to the spin-0 representation of so(2,1).
+The algebra (6.1) admits a nonlinear realization [36],
+a+=/radicalbigg
+1+ν
+˜N˜a+˜Π++˜a+˜Π−, a−=/radicalbigg
+1+ν
+˜N+1˜a−˜Π−+˜a−˜Π+, (6.29)
+in terms of the usual, non-deformed Heisenberg algebra of a b osonic oscillator,
+[˜a−,˜a+] = 1,[˜N,˜a±] =±˜a±. (6.30)
+Here˜N= ˜a+˜a−is the number operator, and ˜Π±=1
+2/parenleftig
+1±˜R/parenrightig
+,˜R= (−1)˜N, are projectors,
+[˜R,˜a±] = 0. Consistently with (6.29) and (6.2), the number operato rs of the deformed, (6.1),
+and the non-deformed, (6.30), algebras coincide, N=˜N.
+7N= 1supersymmetric generalization
+Having written the various spin cases in a unified form, now we proceed to unify them into a
+supersymmetric theory. Consider in fact the following modi fication of Eq. (2.1),
+V(ˆα)
+µψ(x) = 0, V(ˆα)
+µ= ˆαPµ−iǫµνλPνJλ+mJµ. (7.1)
+HereJµis the direct sum of irreducible representations of so(2,1), and ˆαis not more a c-
+number, but a (diagonal) operator, which takes the corresponding so(2,1)-spin value, α, on
+eachso(2,1)-irreducible subspace,
+JµJµ=−ˆα(ˆα−1), (7.2)
+cf. Eq. (2.16). Thevector system (7.1) yields equations of t heform (2.8)–(2.10) with ßreplaced
+by ˆα. Note that the analog of Eq. (2.10) is automatically satisfie d here, due to (7.2). Eq. (7.1)
+describes a multiplet of particles of the same mass, and the v alues of the Poincar´ e spins are
+given by the diagonal elements of ˆ α.
+22As found before (but in a less general framework) [27], choos ing the direct sum of just two
+so(2,1) representations shifted by one half,
+ˆα=diag(α,α+1/2), (7.3)
+providesuswithan N= 1supersymmetricsystem. Toidentify thesupersymmetrics tructureof
+(7.1), (7.3), we realize the so(2,1)-generators quadratically in terms of the creation-anni hilation
+operatorsoftheRDHA, see(6.5). Consistently with(7.2), t heysatisfytherelation (6.13). They
+commute with the reflection operator R, identified as the Z2-grading operator of the super-
+Poincar´ e structure we are looking for. The Fock space repre sentation of the RDHA depends
+on the deformation parameter, and provides us with infinite- (ν∝ne}ationslash=−(2r+ 1)), and (2 r+ 1)-
+dimensional ( ν=−(2r+1),r= 1,2,...) irreducible representations of the osp(1|2) superalge-
+bra. Each such representation is the direct sum of two irredu cibleso(2,1) representations with
+spin values shifted by 1 /2,
+α+=1
+4(1+ν), α −=α++1
+2, (7.4)
+cf. (6.26), and consistently also with (7.3). These are just the values taken by the operator ˆ α
+in (6.13), when restricted to the subspaces F±. The latter are invariant under the action of
+Jµ, and are eigensubspaces of the reflection operator, see (6.1 4), (6.15). The wave function is
+expanded as
+ψ(x) =ψ+(x)+ψ−(x), ψ±(x) =/summationdisplay
+n=0ψ±
+n(x)|n∝an}bracketri}ht±,|n∝an}bracketri}ht±∈F±,(7.5)
+and Eq. (7.1) is reduced to two independent equations,
+V(α+)
+µψ+(x) = 0 and V(α−)
+µψ−(x) = 0, (7.6)
+whose solutions form a supermultiplet with spins α+andα++ 1/2 =α−, respectively. The
+spinor supercharge of such an N= 1-supersymmetric system is given by
+Qa=i/radicalbig
+2m(1+ν)/parenleftig
+(Pµγµ)ab−mRδab/parenrightig
+Lb, a,b = 1,2, (7.7)
+where theγµ’s are the Dirac matrices (6.9) in the Majorana representati on. It is an operator-
+valued linear combination of the odd osp(1|2) generators (6.6). To see that the (7.7) is indeed
+the supercharge, we calculate the commutator
+[V(ˆα)
+µ,Qa] =i/radicalbig
+2m(1+ν)/parenleftbigg
+(Dµ)abDbΠ+−1
+2(γµ)abLb(P2+m2)/parenrightbigg
+, (7.8)
+where
+(Dµ)ab=Pµδb
+a−m(γµ)ab,Db= (Pµ(γµ)ab−mδab)Lb, (7.9)
+and Π+is the projector on the even subspace F+, see (6.19). The action of (7.8) on the wave
+functions that satisfy Eq. (7.1) produces zero if Daψ+= 0. Such a spinor set of equations was
+studied in [32], where it was shown that its solutions descri be a particle of mass mand spin
+α+.
+Moreover, V(α+)
+µψ+(x) =1
+4La(γµDµ)abDbψ+(x),and any solution of the spinor set of
+equations in the even subspace is also a solution of the vecto r set.
+23We conclude, therefore, that (7.7) is indeed a supercharge. Together with the Poincar´ e
+generators, it yields the off-shell nonlinear superalgebra
+[Pµ,Pν] = 0,[Mµ,Pν] =−iǫµνλPλ,[Mµ,Mν] =−iǫµνλMλ,(7.10)
+[Pµ,Qa] = 0,[Mµ,Qa] =1
+2(γµ)abQb, (7.11)
+{Qa,Qb}=−2i(Pγ)ab (7.12)
++2i
+m(1+ν)/bracketleftbig
+(Jγ)ab(P2+m2)−2(Pγ)ab(PJ−mˆα)/bracketrightbig
+. (7.13)
+The last term (7.13) vanishes on-shell, and (7.10)–(7.12) y ield the usual N= 1 planar super-
+Poincar´ e algebra.
+The deformed oscillator representation of the Lorentz alge bra produces both the infinite
+half-bounded series, and the usual finite-dimensional seri es with integer or half-integer spin.
+Hence, the system of equations (7.1), (7.3) describes, univ ersally, a supersymmetric system of
+massive particles of anyonic spin when ν∝ne}ationslash=−(2r+1),r= 1,2,..., and of usual integer/half-
+integer spin for ν=−3,−5,−7,.... This is summarized in Table 1, where the notation (2.19)
+was used. Note that for −3< ν <−1, the supermultiplet includes anyons with spins of
+opposite signs,−1/2< α+<0 andα−=α++ 1/2>0, which are described in terms of
+infinite-dimensional half-bounded non-unitary and unitar y representations, respectively.
+Table 1: Poincar´ e supermultiplets
+Deformation parameter Poincar´ e spin supermultiplet Lorentz rep. Supermultiplet
+ν∝ne}ationslash=−(2r+1) (α+,α−)D+
+α+⊕D+
+α−anyons
+ν=−(2r+1),r= 1,2,... (−r/2,−r/2+1/2)D+
+−r/2⊕D+
+−r/2+1/2boson/fermion
+Some examples of interest are listed in Table 2. Notice that t he supermultiplet (a) was
+studied in a superfield approach in [30], see also Footnote 14 . For the supermultiplet (c), the
+vector operator V(ˆα)
+µvanishes identically on the spin-0 subspace. In this sector the Klein-
+Gordon equation should therefore be imposed, assigning to i t|0∝an}bracketri}ht−in the Fock space. Then the
+superpartner-fields are mapped into each other by the superc harge (7.7). Another possibility
+consists in taking ν <−3, and then considering the limit ν→−3.
+The supermultiplet (e) is studied in the next Section; the su permultiplets (f) and (g) can
+be considered in the same way. The analog of the DJT formulati on [3] for the massive graviton
+was considered in [37, 38]; here we have the linearized form o f the correspondingfield equations.
+7.1 Dirac/Jackiw-Deser-Templeton supermultiplet
+To illustrate the general theory of the previous Section, we consider a supermultiplet composed
+of aDirac field , paired with the topological massive gauge vector field of Deser, Jackiw and
+Templeton [3]. It is described by the vector system (7.1), wh ere the spin operator ˆ αhas now
+eigenvalues−1and−1/2, respectively. Thedeformedalgebra (6.1) withdeformati onparameter
+ν=−5providesuswithan osp(1|2) representation, whoseirreducibleLorentzcomponents h ave
+24Table 2: Examples
+Deformation parameter Spin supermultiplet Supermultiplet
+ν= 0 (1/4,3/4) (a) Quartions
+ν=−2 (−1/4,1/4) (b) Semion/quartion
+ν=−3 (−1/2,0) (c) Dirac/scalar
+ν=−4 (−3/4,−1/4) (d) Semions
+ν=−5 (−1,−1/2) (e) DJT/Dirac
+ν=−7 (−3/2,−1) (f) Rarita-Schwinger/DJT
+ν=−9 (−2,−3/2) (g) Massive graviton/gravitino
+spinj=−α+= 1 andj=−α−= 1/2, respectively. The superfield ψ(x) is expanded in the
+|ν|= 5-dimensional basis (A.7),
+ψ(x) =
+ψ+
+0(x)
+ψ−
+0(x)
+ψ+
+1(x)
+ψ−
+1(x)
+ψ+
+2(x)
+. (7.14)
+Theosp(1|2) generators are 5 ×5 matrices given by Eqns. (A.14), (A.10) and (6.6). Using the
+projectors (A.9) and taking into account (A.13) and (A.15), we extract the components with
+spins 1 and 1 /2,
+j=−α+= 1 :ψ+(x) = Π+ψ(x) =
+ψ+
+0(x)
+ψ+
+1(x)
+ψ+
+2(x)
+
++, (7.15)
+j=−α−= 1/2 :ψ−(x) = Π−ψ(x) =/parenleftbiggψ−
+0(x)
+ψ−
+1(x)/parenrightbigg
+−. (7.16)
+The operator V(ˆα)
+µin Eq. (7.1), projected to these components,
+V(ˆα)
+µΠ±ψ(x) =V(α±)
+µψ±(x), (7.17)
+reduces to
+V(α±)
+µ=α±Pµ+mJ(±)
+µ−iǫµνλPνJ(±)λ, (7.18)
+where,
+J(−)
+0=−1
+2σ3, J(−)
+1=i
+2σ1, J(−)
+2=−i
+2σ2, (7.19)
+J(+)
+0=
+−1 0 0
+0 0 0
+0 0 1
+, J(+)
+1=i√
+2
+0 1 0
+1 0 1
+0 1 0
+, J(+)
+2=1√
+2
+0−1 0
+1 0−1
+0 1 0
+,(7.20)
+J(−)
+µJ(−)µ=−3
+4, J(+)
+µJ(+)µ=−2. (7.21)
+25The Pauli-Lubanski condition (2.9) implied by the equation s (7.1) becomes, on one of the
+sectors, the Dirac equation18
+(Pγ′−m)ab(ψ−(x))b= 0, γ′
+µ=−2J(−)
+µ, (7.22)
+while, on the other sector, it becomes equivalent to the topo logically massive gauge vector field
+equation19,
+(PµJ(+)
+µ+m)ψ+(x) = 0.
+Then we obtain (2.22), with
+/parenleftig
+−iǫλµνPµ+mδλ
+ν/parenrightig
+ψν= 0, ψν= (U†ψ+(x))ν,−iǫνµλ= (U†J(+)
+µU)νλ.(7.23)
+To see how the supercharge acts on this supermultiplet, it is convenient to consider the
+linear combinations
+Q±=Q1∓iQ2, Q ±=i/radicalbig
+m(1+ν)/bracketleftbig
+±a∓P±+a±(mR∓P0)/bracketrightbig
+,(7.24)
+whereP±=P1±iP2. Using the matrix representation (A.10) of the rising/lowe ring oper-
+atorsa±, and the representation (A.9) of the reflection operator, th e explicit action of the
+supercharges (7.24) on the spin-1 ( ψ+) and spin-1/2 ( ψ−) components of the supermultiplet is
+found,
+Q+ψ−(x) =1√m
+iP+ψ−
+0(x)
+i√
+2P+ψ−
+1(x)−1√
+2(m+P0)ψ−
+0(x)
+−(m+P0)ψ−
+1(x)
+
++, (7.25)
+Q−ψ−(x) =1√m
+i(P0−m)ψ−
+0(x)
+−1√
+2P−ψ−
+0(x)+i√
+2(P0−m)ψ−
+1(x)
+−P−ψ−
+1(x)
+
++, (7.26)
+Q+ψ+(x) =1√m/parenleftigg1√
+2P+ψ+
+1(x)+i(m−P0)ψ+
+0(x)
+P+ψ+
+2(x)+i√
+2(m−P0)ψ+
+1(x)/parenrightigg
+−, (7.27)
+Q−ψ+(x) =1√m/parenleftigg
+−iP−ψ+
+0(x)+1√
+2(m+P0)ψ+
+1(x)
+−i√
+2P−ψ+
+1(x)+(m+P0)ψ+
+2(x)/parenrightigg
+−. (7.28)
+Here, it is explicitly shown how the components of the transf ormed spin-1 field depend on the
+untransformed spin-1 /2 components (7.16) of the superfield (7.14).
+Conversely, the components of the transformed spin-1 /2 field are those of the spin 1 field
+(7.15).
+18Note that the representation of Dirac matrices, γ′
+µin (7.22), is related to that in (6.9) by the unitary
+transformation (A.6).
+19The standard (adjoint) representation of the spin-1 Lorent z algebra, ( Jµ)ν
+λ=−iǫν
+µλ, is obtained by the
+unitary transformation (A.17).
+267.2 Jackiw-Nair/Majorana-Dirac supersymmetric system
+Now, extending the Dirac–DJT correspondence established i n the previous Section, we con-
+struct anN= 1supersymmetric unification of the Jackiw-Nair and Majorana-Dirac anyonic
+fields. Thiscan bedoneby combiningtheextended realizatio n (5.1) and(5.2) [that corresponds
+to the sum of the so(2,1)-representations] with the N= 1 supersymmetric scheme of Section
+7. Taking the particular realization of the latter from Sect ion 7.1 [i.e. taking the Dirac/DJT
+supermultiplet], we get, as a result, a particular, N= 1 supermultiplet of Majorana-Dirac and
+JN fields.
+The generator Jµin (5.1) is taken in an irreducible representation D+
+α, whileJ′
+µis realized
+as a direct sum of two finite-dimensional so(2,1) representations with Lorentz spins j= 1 and
+j= 1/2. Finite-dimensional representations can be obtained by m eans of the RDHA with
+deformation parameter ν=−5. The parameter α′is promoted to a diagonal operator with
+eigenvalues−1 and−1/2. This immediately provides us with a supermultiplet of spi nsα−1
+andα−1/2. On the corresponding subspaces with β=α−1 andβ=α−1/2, our vector
+equations will take the form which corresponds to the Jackiw -Nair and Majorana-Dirac fields,
+respectively.
+Explicitly, the N= 1 JN/MD supersymmetric system is described by the vector sy stem of
+equations
+V(α+ˆα′)
+µψ(x) = 0, V(α+ˆα′)
+µ= (α+ ˆα′)Pµ−iǫµνλPνJλ+mJµ, (7.29)
+where
+Jµ=Jµ+J′
+µ, J µ∈D+
+α,J′
+µ∈D1⊕D1/2,ˆα′=diag(−1,−1/2).(7.30)
+Thefive-dimensional irreduciblerepresentation of osp(1|2) that correspondstothechosendirect
+sumforJ′
+µwas describedin thepreviousSection interms of theRDHA wit hparameter ν=−5.
+A solution of the system (7.29) is given by the JN field realize d onD+
+α⊕˜D1, and by the MD
+field realized on D+
+α⊕˜D1/2. Here ˜D1corresponds to the three-dimensional (vector) even
+subspace of five-dimensional Fock space and ˜D1/2corresponds to its two-dimensional (spinor)
+odd subspace.
+The spinor supercharge of such a supersymmetric system beco mes
+Qa=1
+2√
+2m/parenleftig
+(Pµγµ)ab−mRδab/parenrightig
+Lb, a,b = 1,2, (7.31)
+cf. (7.7). It is implied that (7.31) acts identically on the i ndexn[not shown explicitly and
+corresponding to the infinite-dimensional representation D+
+α] of the wave function ψ(x), while
+the matrix operator Lbacts in the 5-dimensional irreducible representation of th eosp(1|2)
+superalgebra. The action of this supercharge is given by Eqn s. (7.24), (7.25)–(7.28), where, in
+this case, it is implied that the components of the wave funct ions carry also [an not displayed]
+indexn. Together withMµandPµ, this supercharge generates a nonlinear superalgebra of th e
+form (7.10)–(7.13), with Jµin the last term (7.13) changed to J′
+µ, and ˆαchanged to ˆ α′.
+It was shown in Section 2 that the vector sets of equations for JN and MD fields decouple
+into Majorana and DJT equations, and Majorana and Dirac equa tions, respectively. This
+means that the factor ( PJ′−mˆα′) disappears on-shell like (7.13). Therefore, on-shell, th e
+standardN= 1 superalgebra given by Eqns. (7.10)–(7.12) is obtained.
+278N= 2supersymmetry
+In Section 7 we have shown that promoting the parameter αin (2.20) to a (diagonal) operator,
+ˆαin (7.3), and realizing the Lorentz generator Jµin terms of the RDHA creation-annihilation
+operators yields an N= 1 supersymmetric system of anyons, or of usual fields of arbi trary
+integer and half-integer spin.
+In the previous Section we showed in turn that N= 1 supersymmetry can be obtained,
+alternatively, if, in the extended realization of Section 5 , the second parameter α′is promoted
+to adiagonal operator, while αis left afixednumerical parameter. Thealternative constru ction
+allowed us to to unify the Majorana-Dirac and JN fields in one a nyon supermultiplet, and to
+get supermultiplets with partner anyon fields described by i nfinite-dimensional representations.
+The construction of the previous section was asymmetric, an d now we argue that the same
+prescription, applied to (5.2) symmetrically in the parameters αandα′, i.e., promoting them
+bothto operators, provides us with an N= 2-supersymmetric system of anyons, or of bosons
+and fermions. Insuch away we unify, in particular, theMajor ana-Dirac and Jackiw-Nair anyon
+systems into a single extended N= 2 supermultiplet.
+Let us take
+Jµ=Jµ
+1+Jµ
+2,JAµJµ
+A=−ˆαA(ˆαA−1), A= 1,2, (8.1)
+[with no summation in Aimplied], and postulate the vector system of equations
+V(ˆß)
+µψ(x) = 0, V(ˆß)
+µ=ˆßPµ+mJµ−iǫµνλPνJλ, (8.2)
+ˆß= ˆα1+ ˆα2. (8.3)
+Here we assume that each Lorentz generator JAµ,A= 1,2, is realized quadratically in terms
+of the creation-annihilation operators of two independent RDHA’s,
+/bracketleftbig
+a−
+A,a+
+A/bracketrightbig
+= (1+νARA),{a±
+A,RA}= 0, R2
+A= 1, A= 1,2. (8.4)
+So, here
+ˆαA=1
+4(2+νA−RA), (8.5)
+JA0=1
+2NA+αA+, α A+=1
+4(1+νA). (8.6)
+In internal space we have two copies of osp(1|2) superalgebra,
+osp1(1|2)⊕osp2(1|2),ospA(1|2) ={LAa,Jµ
+A}, A= 1,2, (8.7)
+which provide us with a representation of the Lorentz genera tors (8.1) with four irreducible
+components [two for each ospA(1|2)]. TheJµact on the tensor product of the Fock spaces,
+F=F1⊗F2,F=/braceleftbig
+|n1,n2∝an}bracketri}ht=|n1∝an}bracketri}ht⊗|n2∝an}bracketri}ht/vextendsingle/vextendsingle|n1∝an}bracketri}ht∈F 1,|n2∝an}bracketri}ht∈F 2/bracerightbig
+, (8.8)
+whereFAis the Fock space of the corresponding RDHA. The wave functio n is expanded on F,
+ψ(x) =/summationdisplay
+n1,n2=0ψn1,n2(x)|n1,n2∝an}bracketri}ht. (8.9)
+28The subdivision of the Fock spaces FAin even and odd subspaces,
+FA+=/braceleftbig
+|nA∝an}bracketri}ht+=|2nA∝an}bracketri}ht/bracerightbig
+,FA−=/braceleftbig
+|nA∝an}bracketri}ht−=|2nA+1∝an}bracketri}ht/bracerightbig
+,
+induces the decomposition of F,
+F=F1+⊗F2++F1+⊗F2−+F1−⊗F2++F1−⊗F2−. (8.10)
+The spin operator (8.3) takes, on the four corresponding sub spaces in (8.10), the values
+ˆß=diag/parenleftbigg
+χ,χ+1
+2,χ+1
+2,χ+1/parenrightbigg
+, χ=α1++α2+=1+ν1
+4+1+ν2
+4.(8.11)
+Observe that the spin eigenvalue χ+1
+2is doubly degenerated. The wave function (8.9) has
+therefore four components,
+ψ(x) =ψ++(x)+ψ+−(x)+ψ−+(x)+ψ−−(x), (8.12)
+ψ±±(x) =/summationdisplay
+n1,n2=0ψ±±
+n1,n2(x)|n1,n2∝an}bracketri}ht±±, (8.13)
+ψ±∓(x) =/summationdisplay
+n1,n2=0ψ±∓
+n1,n2(x)|n1,n2∝an}bracketri}ht±∓, (8.14)
+|n1,n2∝an}bracketri}ht±±∈F1±⊗F2±,|n1,n2∝an}bracketri}ht±∓∈F1∓⊗F2∓. (8.15)
+The wave function (8.12) solves (8.2) if each of its componen t solves, independently,
+V(χ)
+µψ++(x) = 0 ,
+V(χ+1/2)
+µψ+−(x) = 0,
+V(χ+1/2)
+µψ−+(x) = 0,
+V(χ+1)
+µψ−−(x) = 0.(8.16)
+Each component carries thereforean irreduciblerepresent ation of thePoincar´ e groupwith mass
+mand spin indicated by the upper index of the vector operator.
+Fields which differ by one half in their spins can be connected b y supercharges analogous
+to (7.7). They are given by
+Q1a=i/radicalbig
+2m(1+ν1)/parenleftig
+(Pµγµ)ab−mR1δab/parenrightig
+L1b, (8.17)
+Q2a=i/radicalbig
+2m(1+ν2)R1/parenleftig
+(Pµγµ)ab−mR2δab/parenrightig
+L2b. (8.18)
+The additional factor R1is included into the second supercharge in order to make (8.1 7) and
+(8.18) anti-commute, see (8.21) below. The action of the sup ercharges on the supermultiplet
+components is illustrated on Fig. 1.
+To see that (8.17) and (8.18) are physical operators in that t hey preserve the physical
+subspace, we pass to the rest frame20,Pµ= (m,0,0), in which the vector system is equivalent
+to (J0−ˆß)ψ(x) = 0,J−ψ(x) = 0, cf. (2.14). Observe that
+J−|0,0∝an}bracketri}ht±±= 0,J−|0,0∝an}bracketri}ht±∓= 0,
+20It is implied that at least one of the parameters νA,A= 1,2,corresponds to an infinite-dimensional
+representation of RDHA ( ν >−1).
+29[ψ+−(x);s=χ+1/2]←−Q1−→[ψ−−(x);s=χ+1]
+↑ ↑
+Q2 Q2
+↓ ↓
+[ψ++(x);s=χ]←−Q1−→[ψ−+(x);s=χ+1/2]
+Figure 1: The action of supercharges on the supermultiplet components . Each supercharge, Q1
+andQ2, changes the spin by 1/2; the first (second) supercharge acts nontrivially in the firs t
+(second) index of the Fock space decomposition (8.9).
+and that|0,0∝an}bracketri}ht++=|0,0∝an}bracketri}ht,|0,0∝an}bracketri}ht+−=|0,1∝an}bracketri}ht,|0,0∝an}bracketri}ht−+=|1,0∝an}bracketri}ht,|0,0∝an}bracketri}ht−−=|1,1∝an}bracketri}ht. The same set of
+kets is annihilated also by the operator J0−ˆß. In the rest frame, the general solution of (8.2)
+is, therefore, an arbitrary linear combination of the four l ower ket states,
+ψ(x)∝ψ++
+0,0|0,0∝an}bracketri}ht+++ψ+−
+0,0|0,0∝an}bracketri}ht+−+ψ−+
+0,0|0,0∝an}bracketri}ht−++ψ−−
+0,0|0,0∝an}bracketri}ht−−,
+multiplied by a time-dependent phase [not displayed here]. The coefficients are arbitrary con-
+stants. In the rest frame, the operators (8.17) and (8.18) ar e proportional to a+
+AΠA+anda−
+AΠA−
+with values of index A= 1and 2, respectively.
+|0,1∝an}bracketri}ht−→a+
+1−→
+←−a−
+1←−|1,1∝an}bracketri}ht (8.19)
+a+
+2↑ ↓a−
+2 a+
+2↑ ↓a−
+2
+|0,0∝an}bracketri}ht−→a+
+1−→
+←−a−
+1←−|1,0∝an}bracketri}ht
+Figure 2: In the rest frame Pµ= (m,0,0), the action of the supercharges, shown in Fig. 1, is
+reduced to that of creation-annihilation operators.
+Their action on solutions is reducedto that of thecreation a nd annihilation operators, as shown
+on Figure 2. Therefore, the space of solutions is invariant u nder the action of (8.17) and (8.18).
+They are odd operators with respect to Γ = R1R2, identified as the Z2-grading operator of the
+N= 2 Poincar´ e superalgebra. The even part of this superalgeb ra is given by Eq. (7.10). The
+part that involves the supercharges is
+[Pµ,QAa] = 0,[Jµ,QAa] =1
+2(γµ)abQAb, (8.20)
+{QAa,QBb}=−2iδAB(Pγ)ab, (8.21)
+where the anti-commutator of the supercharges is shown in on -shell form.
+30When both parameters νA,A= 1,2,correspond to finite-dimensional representations of
+RDHA, theequations (8.2), (8.3)provideuswithausual, bos on/fermion N= 2super-multiplet
+(8.11), with states of both signs of the energy.
+Jackiw-Nair/Majorana-Dirac N= 2 supermultiplet
+The Lorentz generator (8.1), Jµ=Jµ
+1+Jµ
+2, is constructed in terms of two RDHA algebras,
+one forJµ
+1and another for Jµ
+2. We have the freedom to choose the deformation parameters
+ν1andν2independently. As a result, the system of equations (8.2) ca n describe a variety of
+N= 2 supermultiplets (8.11), based on different representatio ns of the so(2,1) algebra, by
+mixing “anyonic” and/or usual integer/half-integer spin s eries.
+As an example, we consider here the Jackiw-Nair/Majorana-D iracN= 2 supermultiplet.
+It is realized by choosing ν1>−1 andν2=−5. This implies,
+α1+>0,−α2+=j2= 1⇒χ=α1+−1. (8.22)
+Hence we get the spectrum described in Table 3.
+Table 3:N= 2 JN/MD supermultiplet
+Field component Fine spin structure SpinsInternal space Field type
+ψ++(x) α1+−1α1+−1F(∞)
+1+⊗F(3)
+2+ JN
+ψ+−(x) α1+−1/2α1+−1/2F(∞)
+1+⊗F(2)
+2− MD
+ψ−+(x) α1−−1α1+−1/2F(∞)
+1−⊗F(3)
+2+ JN
+ψ−−(x) α1−−1/2 α1+F(∞)
+1−⊗F(2)
+2− MD
+It is interesting to note that, from the viewpoint of the Poin car´ e spin value s[see the third
+column in the Table], the field component ψ−−looks like the field we got in Section 2 using
+the irreducible representation D+
+αwithα=α1+>0. The components ψ−+andψ+−look like
+anyonic fields produced by the Majorana-Dirac system. Final ly, the component ψ++looks like
+that described by the Jackiw-Nair equations. Having in mind , however, that the equations for
+each field component have a structure that corresponds to the scheme (5.1), (5.2) based on the
+direct sum of two irreducible representations of so(2,1) (see the second and fourth columns),
+each component is interpreted as a field of the type indicated in the fifth column.
+The wave functions have two Lorentz indices, associated to F(∞)
+1=F(∞)
+1−+F(∞)
+1+(infinite
+31dimensional) and F(5)
+2=F(2)
+2−+F(3)
+2+(5-dimensional) representations of RDHA. Explicitly,
+ψ++(x) =∞/summationdisplay
+n=0/summationdisplay
+λ′=1,2,3(ψ++
+n(x))λ′|n,λ′∝an}bracketri}ht++,|n,λ′∝an}bracketri}ht++∈F(∞)
+1+⊗F(3)
+2+,(8.23)
+ψ+−(x) =∞/summationdisplay
+n=0/summationdisplay
+a=1,2(ψ+−
+n(x))a|n,a∝an}bracketri}ht+−,|n,a∝an}bracketri}ht+−∈F(∞)
+1+⊗F(2)
+2−, (8.24)
+ψ−+(x) =∞/summationdisplay
+n=0/summationdisplay
+λ′=1,2,3(ψ−+
+n(x))λ′|n,λ′∝an}bracketri}ht−+,|n,λ′∝an}bracketri}ht−+∈F(∞)
+1−⊗F(3)
+2+,(8.25)
+ψ−−(x) =∞/summationdisplay
+n=0/summationdisplay
+a=1,2(ψ−−
+n(x))a|n,a∝an}bracketri}ht−−,|n,a∝an}bracketri}ht−−∈F(∞)
+1−⊗F(2)
+2−. (8.26)
+Here,a= 1,2,is a spinor index in the basis (A.13), in which the so(2,1) generators are
+given by Eq. (A.5). The vector index λ′corresponds to the basis (A.15), and transforms
+under the so(2,1) representation (A.14). Index ntransforms under the infinite dimensional
+so(2,1) representation ( D+
+α1±), and endows a wave function with the anyonic part of the spin .
+Supercharges (8.17), (8.18) transform these fields as shown on Fig. 1.
+9 Nonrelativistic limit
+In this Section we study the non-relativistic limit of our ve ctor system of equations (2.1). This
+will provide us with a universal non-relativistic descript ion of either fermion/boson fields of
+arbitrary half-integer/integer spin, or of non-unitary an yons interpolating between them, or
+of anyons based on unitary representations. As shown below, the system of linear differen-
+tial equations we get implies, for each component, the Schr¨ odinger equation as integrability
+condition – just like the Klein-Gordon equation is implied i n the relativistic case. Also, the
+equations guarantee that the system carries an irreducible representation of the corresponding
+Galilei symmetry.
+We analyze two different types of non-relativistic limits [22 ].
+•The first one is the usual limit, in which the velocity of light ,c, tends to infinity, while
+the remaining parameters are kept constant. This generaliz es theL´ evy-Leblond equations valid
+for spin 1/2 [21] to arbitrary spin.
+•In the second, “exotic” type of limit introduced by Jackiw and Nair [23] (see also [24 , 22])
+the spin,s, also goes to infinity in such a way that a ratio
+c→∞, s→∞, s/c2=κ (9.1)
+remains constant. Its “raison d’ˆ etre” is that it yields exotic Galilei symmetry [39], which
+admits, besides the mass, m, also a second central charge, κ, associated with the non-
+commutativity of Galilean boosts [39, 40].
+Both non-relativistic limitsareconveniently derivedfro m(2.1), afterreinstatingthevelocity
+of light,c, and substituting m→mcandψ→e−imc2tψ, which gives
+P0=1
+c/parenleftbigg
+i∂
+∂t+mc2/parenrightbigg
+. (9.2)
+32Then Eqns. (4.4) and (4.5) take the equivalent form,
+1
+c√
+n+2α/parenleftbigg
+i∂
+∂t/parenrightbigg
+ψn+√
+n+1P+ψn+1= 0, (9.3)
+√
+n+2αP−ψn+√
+n+1/parenleftbigg
+2mc+1
+c/parenleftbigg
+i∂
+∂t/parenrightbigg/parenrightbigg
+ψn+1= 0, (9.4)
+while Eq. (4.6) reduces to
+/parenleftbigg
+−α1
+ci∂
+∂t+mcn)/parenrightbigg
+ψn+1
+2/parenleftig/radicalbig
+(2α+n−1)nP−ψn−1−/radicalbig
+(2α+n)(n+1)P+ψn+1/parenrightig
+= 0.
+(9.5)
+The Klein-Gordon equation, a consequence of Eqns. (9.3)–(9 .5), takes here the equivalent form
+/parenleftbigg/parenleftbigg
+1+i
+2mc2∂t/parenrightbigg
+i∂t−1
+2m/vectorP2/parenrightbigg
+ψ= 0. (9.6)
+We put
+Ki=−1
+cǫijMj=−tPi+mxi−ǫijJj+1
+c2xi·i∂t, (9.7)
+J =M0=ǫijxiPj+J0, (9.8)
+and get, for the commutator of the boosts,
+[Ki,Kj] =−i1
+c2ǫijJ, (9.9)
+whereǫij=−ǫji,ǫ12= 1.
+Let us insist that all these formulas are still relativistic ; we simply wrote them in a form
+where the role of the speed of light is highlighted.
+9.1 Usual non-relativistic limit
+From Eq. (9.4) we infer that, as c→ ∞, every subsequent component is O(1
+c)-times the
+previous one. The case of a boson/fermion of nonzero spin j=n/2>0 is described by a
+(2j+1)-component field. Considering the non-relativistic lim it, one could try to preserve all
+terms up to order 1 /cn,n >1. Though such an approximation would be rather natural for
+boson/fermionfieldsofcorrespondingspin,itwillnotposs essapropertyofuniversality. Indeed,
+particularly, as it follows from (9.6), the Hamiltonian ope rator in the Schr¨ odinger equation then
+will contain corrections with terms of the form ( /vectorP2/mc2)k, for which the highest value of kwill
+depend on the value of spin. Also, according to (9.9), the boo st generators will commute for
+j= 1/2, but will be non-commuting for j >1/2. The question is then: what kind of symmetry
+will underlie the resulting theory?
+Investigating the general question goes beyond our scope he re; we consider, instead, a
+certain non-relativistic limit which is characterized by t he following properties:
+•It has a universal structure; in particular, the dynamics is described by the Schr¨ odinger
+equation with a Hamiltonian operator having the usual non-r elativistic form;
+•ThecorrespondingLie-algebraic symmetryisproducedbyIn ¨ on¨ u-Wignercontraction from
+Poincar´ e symmetry;
+33•Forj= 1/2, it reproduces the L´ evy-Leblond theory for a non-relativ istic spin one-half
+field.
+Let us denote the multicomponent field by Ψ, and consider the ( invertible) similarity trans-
+formation Ψ→Φ,
+Φ =MΨ,M=diag(1,c1,c2,...). (9.10)
+The dimension of the field-column
+Φ =
+φ0
+φ1
+...
+, (9.11)
+and of diagonal matrix Mdepends on the chosen representation of the Lorentz algebra ; in the
+anyon case, it is infinite. The components of the transformed field Φ, unlike those of Ψ, are of
+orderc0= 1 cf. Eq. (9.4).
+Thetransformation (9.10) inducesasimilarity transforma tion of operators : to any operator
+Oacting on a state Ψ, there corresponds an operator
+ˇO=MOM−1(9.12)
+that acts on Φ. For the translational invariant (spin) part o f the Lorentz generators, O=J0,
+J+,J−, we get then
+ˇJ0=J0,ˇJ+=cJ+,ˇJ−=1
+cJ−. (9.13)
+The components of the vector operator Vµof our basic equations are transformed into
+ˇV0=−1
+c/parenleftbigg
+αi∂t+1
+2P+J−/parenrightbigg
++c/parenleftbigg
+m(J0−α)+1
+2P−J+/parenrightbigg
+, (9.14)
+ˇV+=−(J0−α)P+−J+i∂t, (9.15)
+ˇV−= (J0+α)P−+2mJ−+1
+c2i∂t. (9.16)
+So far we merely deformed the theory, which is still relativi stic. To obtain the non-relativistic
+limit of the equations presented in terms of the rescaled fiel d Φ, it is necessary to ‘renormalize’
+ˇV0and consider
+V0= lim
+c→∞1
+cˇV0=m(J0−α)+1
+2J+P−, (9.17)
+V+=−lim
+c→∞ˇV+= (J0−α)P++J+i∂t, (9.18)
+V−= lim
+c→∞ˇV−= (J0+α)P−+2mJ−. (9.19)
+Then we infer our new vector equations
+V0Φ = 0,V+Φ = 0,V−Φ = 0. (9.20)
+The dynamics of the field Φ is given by the Schr¨ odinger equati on,i∂tΦ =1
+2m/vectorP2Φ. The latter
+appears in fact as consistency (integrability) condition f or the system (9.20).
+34In component form the last two equations read
+√
+n+2α/parenleftbigg
+i∂
+∂t/parenrightbigg
+φn+√
+n+1P+φn+1= 0, (9.21)
+√
+n+2αP−φn+2m√
+n+1φn+1= 0, (9.22)
+wheren= 0,1,...,∞for anyons ( α >0, or 0> α∝ne}ationslash=−j), andn= 0,1,...,2jfor
+bosons/fermions ( α=−j). The component form of the first equation from (9.20) is redu ced
+here to Eq. (9.22). Expressing φn+1from Eq. (9.21),
+φn+1=−1
+2m/radicalbigg
+n+2α
+n+1P−φn. (9.23)
+Note that the tower of states is automatically finite as it sho uld if 2αis a negative inte-
+ger. Inserting (9.23) into (9.22) shows explicitly that eac h component satisfies, separately, the
+Schr¨ odinger equation,/parenleftigg
+i∂
+∂t−/vectorP2
+2m/parenrightigg
+φn= 0. (9.24)
+Iterating (9.23) allows us to express all components φnin terms of the lowest one,
+φn= (nB(2α,n))−1/2/parenleftbigg−P−
+2m/parenrightbiggn
+φ0, (9.25)
+cf. (4.8). By Eq. (9.24), φ0is a (superposition of) plane waves,
+φ0= exp/braceleftbigg
+−it/vector p2
+2m+i/vector x·/vector p/bracerightbigg
+. (9.26)
+Note that for integer/half-integer spin j=−α= 1/2,1,3/2,..., Eqns. (9.23)-(9.25) yield
+φn= 0 forn>2j, consistently with our expectation that spin- jparticle should be described
+by a (2j+1)-component field, cf. Section 3.
+As in the relativistic case, the infinite-component anyon fie lds of spin s=αgiven in Eq.
+(3.14), interpolate between the finite-component non-rela tivistic fields of spins j−1/2 andj,
+respectively. Here, however, each higher component φnis suppressed by the additional hidden
+factor1
+cnw.r.t. the leading component φ0. Whenαtends to either of the boundary values
+−j+1/2 or−j, the components φnwithn>2j−1 orn>2j+1 are suppressed, in addition,
+by the numerical factor ( j−1
+2+α)1/2or (j+α)1/2, see Section 3. In the case of anyons based
+on the unitary representations D+
+αwithα>0, no such additional suppression arises.
+Now we show that the system (9.21), (9.22) has a 1-parameter c entrally extended Galilei
+symmetry, obtained by In¨ on¨ u-Wigner contraction from the (2+1)D Poincar´ e algebra,
+[Pµ,Pν] = 0,[Mµ,Pν] =−iǫµνλPλ,[Mµ,Mν] =−iǫµνλMλ.(9.27)
+The Galilean boost generators are defined by
+Ki=−lim
+c→∞ǫijˇMj/c. (9.28)
+With taking into account Eqns. (9.7) and (9.13), we get
+K±=K1±iK2=−tP±+mx±+∆±,∆−= 0,∆+=iJ+. (9.29)
+35Note here the presence of the spin-dependent part ∆ +.
+The generator of rotations has, by (9.7), (9.13), the same fo rm as in relativistic case,
+J =ǫijxiPj+J0. (9.30)
+Defining also
+H=cP0−mc2, (9.31)
+we find that Pi,Ki, J andH=i∂
+∂tare symmetry operators of the system (9.21), (9.22), and
+generate the extended Galilei (“Bargmann”) algebra with a u nique central charge m,
+[Ki,Pj] =imδij,[Pi,Pj] = 0,[H,Pi] = 0,[Ki,Kj] = 0, (9.32)
+[Ki,H] =iPi,[J,Pi] =iǫijPj,[J,Ki] =iǫijKj. (9.33)
+Note that by (9.9) boosts commute.
+The Casimir operators of the algebra (9.32), (9.33) are
+C1=P2
+i−2mH,C2=mJ−ǫijKiPj. (9.34)
+On-shell, i.e. on the surface of Eqns. (9.21), (9.22), they t ake the values
+C1= 0,C2=αm. (9.35)
+The non-relativistic equations (9.21), (9.22) describe th erefore a massive non-relativistic parti-
+cle of spinα.
+The particular cases of spin 1 /2 and spin 1 will be discussed in Section 10.3.
+9.2 Exotic limit
+For the infinite-dimensional unitary representations D+
+α,α>0, we define the operators,
+N=J0−α, b±=J±√
+2α, (9.36)
+and putα→∞.In this limit, the operators (9.36) generate the harmonic os cillator algebra,
+[N,b±] =b±,[b−,b+] = 1, (9.37)
+and theso(2,1) representation (3.10) transforms into a Fock space repre sentation,
+N|n) =n|n), b−|n) =√
+n+1|n+1), b+|n) =√n|n−1), n= 0,1,2,... .(9.38)
+The exotic non-relativistic limit c→∞, α→∞, α/c2=κ=const,cf. (9.1), applied to the
+equations (4.2)–(4.3) with the substitution (9.2), produc es
+Λ0ψ= 0,Λ+ψ= 0,Λ−ψ= 0, (9.39)
+Λ0=i∂
+∂t−v−P+
+2+v+
+2Λ−,Λ+=κv+/parenleftbigg
+i∂
+∂t−1
+2v−P+/parenrightbigg
+,Λ−=P−−mv−.(9.40)
+Here, Λ 0, Λ−and Λ+correspond to the limit (9.1) of the operators −cV0/α,V−/2αandV+,
+respectively, and we have introduced the notation
+v±=v1±iv2=−/radicalbigg
+2
+κb±. (9.41)
+36Taking the combination Λ 0−v+Λ−, we get a minimal set of equations,
+/parenleftbigg
+i∂
+∂t−H/parenrightbigg
+ψ= 0,Λ−ψ= 0, (9.42)
+where
+H=/vectorP/vector v−m
+2v+v− (9.43)
+is identified as the [linear-in- P]Hamiltonian operator .
+Taking into account the second equation from (9.42), the firs t one becomes the Schr¨ odinger
+equation,/parenleftbigg
+i∂
+∂t−1
+2m/vectorP2/parenrightbigg
+ψ= 0. (9.44)
+On the other hand, decomposing as ψ=/summationtext
+nψn|n), the equations Λ ±ψ= 0 read
+√
+2κ/parenleftbigg
+i∂
+∂t/parenrightbigg
+ψn+√
+n+1P+ψn+1= 0, (9.45)
+√
+2κP−ψn+2m√
+n+1ψn+1= 0. (9.46)
+These equations can also be obtained, alternatively, by app lying the exotic limit (9.1) to Eqns.
+(9.3), (9.4). Using the second equation from (9.46), the hig her components can be expressed
+in terms of the lowest one, namely as
+ψn=1√
+n!/parenleftbigg
+−/radicalbiggκ
+2P−
+m/parenrightbiggn
+ψ0,
+while, according to the first equation, the dynamics of each c omponent is governed by the
+Schr¨ odinger equation, (9.44).
+The exotic limit with the additional ‘renormalization’
+M0→M0−α=M0−κc2, (9.47)
+cf. (9.31), provides us with rotation and Galilean boosts ge nerators,
+J =ǫijxiPj+1
+2κv+v−,Ki=mxi−tPi+κǫijvj. (9.48)
+Together withH=i∂
+∂tand momentum Pi, they generate the exotic Galilei symmetry,
+[Ki,Pj] =imδij,[Pi,Pj] = 0,[H,Pi] = 0,[Ki,Kj] =−iκǫij, (9.49)
+[Ki,H] =iPi,[J,Pi] =iǫijPj,[J,Ki] =iǫijKj. (9.50)
+In particular, the parameter κ, which measures the non-commutativity of boosts, becomes
+the second central charge, highlighting the exotic Galilea n symmetry [39, 40]. The Casimir
+operators are
+C1=P2
+i−2mH,C2=mJ−ǫijKiPj+κH. (9.51)
+On-shell, they take the values C1=C2= 0. The system of equations (9.42) describes an exotic
+(2+1)D non-relativistic particle with mass mand the exotic parameter κ. The system may be
+reinterpreted as a free massive particle on the non-commuta tive plane, see [22].
+37The exotic limit can be carried out also by letting α→−∞, that involves the non-unitary
+representations ˜Djand˜D+
+α. We reproduce here the same results as for α→+∞. Putting
+α=−j,j→∞, the limit can be applied, in particular, to usual boson/fer mion fields.
+Starting from Eqns. (3.16), (3.17), we now consider the exot ic non-relativistic limit (9.1),
+i.e.,c→∞, j→∞, j/c2=κ=const.The equations (9.39), with corresponding operators
+(9.40), are obtained by applying this limit to
+(κc)−1V0ψ= 0, V +ψ= 0,−κ(2κc2)−1V−ψ= 0,
+respectively. The oscillator number and creation-annihil ation operators are found by applying
+the limitj→∞to
+N=J0+j+ǫ, b±=±J±/radicalbig
+2(j+ǫ). (9.52)
+Note that while the nature of the operators (3.16) requires a n indefinite metric, due to the sign
+in the definition of b±in (9.52), we have now N†=N, (b+)†=b−with respect to the usual,
+positive definite metric, and reproduce the relations (9.38 ).
+10 Non-relativistic supersymmetry
+Studying the non-relativistic limit of the vector equation (2.1), we derived first order equations
+for bosons and fermions, and for anyons. The symmetry of thes e systems is the mass-extended
+Galilei symmetry, or the exotic-Galilei symmetry, dependi ng on the type of limit. They corre-
+spond to different In¨ on¨ u-Wigner contractions of the Poinca r´ e group.
+In Sections 7 and 8, the relativistic wave equation (2.1) was generalized to (7.1), or (8.2),
+respectively. They describe N= 1 orN= 2 Poincar´ e supermultiplets. It is natural to
+expect, therefore, that the non-relativistic limits of the se equations will describe(exotic) Galilei
+supermultiplets. It is also expected that the symmetries of these systems will be the In¨ on¨ u-
+Wigner contraction of the super-Poincar´ e algebra.
+In the case of N= 1 supersymmetry, for instance, the supermultiplet is desc ribed by (7.1).
+Projectedtothecorrespondingsubspaces, ityieldsEqns. ( 7.6),V(α+)
+µψ+= 0andV(α−)
+µψ−= 0.
+The fieldsψ+andψ−have spinα+andα−, respectively. The procedure described in Section 9
+to obtain the nonrelativistic limit can be repeated in each s ubspace. Having in mind that the
+fieldsψ±are expanded on the even (odd) Fock subspaces of the RDHA, see Eqns. (7.5) and
+(6.14),ψ±(x) =/summationtext
+n=0ψ±
+n(x)|n∝an}bracketri}ht±, we first obtain (9.3), (9.4). Then, taking the limit, yields
+Eqns. (9.21) and (9.22) with parameters α+andα−=α++ 1/2, respectively. Each set of
+non-relativistic fields will be Galilei-symmetric, and the fields are interchanged by the non-
+relativistic supercharge, obtained as the non-relativist ic limit of the relativistic supercharge
+(7.7).
+The extended N= 2 supersymmetry is derived by a similar procedure, startin g from equa-
+tions (8.2) and applying In¨ on¨ u-Wigner contraction. We pr esent this in detail in the following
+subsections.
+10.1 Usual N= 1non-relativistic supermultiplet
+In this case we get two copies of the non-relativistic equati ons (9.21), (9.22),
+/radicalbig
+n+2α±/parenleftbigg
+i∂
+∂t/parenrightbigg
+φ±
+n+√
+n+1P+φ±
+n+1= 0, (10.1)
+/radicalbig
+n+2α±P−φ±
+n+2m√
+n+1φ±
+n+1= 0. (10.2)
+38Definingφ±
+n=cnψ±
+nand taking the limit c→∞as in (9.21), (9.22), we get, cf. (9.25), (9.26),
+φ±
+n= (nB(2α±,n))−1/2/parenleftbigg−P−
+2m/parenrightbiggn
+φ0, φ±
+0=A±exp/braceleftbigg
+−it/vector p2
+2m+i/vector p/vector x/bracerightbigg
+,(10.3)
+where theA±are constants. The Galilean generators are defined as in (9.2 9), (9.30), (9.31).
+Augmented with the translation generators Pi, they span the algebra (9.32), (9.33). Here the
+reducible representations (6.5) of the Lorentz algebra (se e also (6.17)) are used for J+andJ0
+in the boost and rotation operators, K+and J, respectively. The operator C2, defined as in
+(9.34), is therefore multi-valued. On-shell, we have
+C2=mˆα,C2Φ±=C±
+2Φ±,C±
+2=mα±. (10.4)
+The states Φ±are vectors of the form (9.11), with components φ±
+n. The operatorC1vanishes
+in both subspaces.
+The supercharges that interchange the fields Φ±are the nonrelativistic limits of those in
+(7.7). To identify them, we note that the multi-component fie ld Ψ of our N= 1 supermultiplet
+has the structure ΨT= (ψ+
+0,ψ−
+0,ψ+
+1,ψ−
+1,...), whereTdenotes transposition. The rescaled
+field Φ has analogous structure. They are related by the trans formation of the form (9.10) with
+a matrix M
+M=diag(1,1,c,c,c2,c2,...) =M++M−,M±=MΠ±, (10.5)
+where Π +and Π−are the projectors (6.19). The similarity transformation ( 9.12), applied to
+the RDHA creation and annihilation operators, gives
+ˇa+=ca+Π−+a+Π+,ˇa−=c−1a−Π++a−Π−. (10.6)
+From (7.24) and (10.6) we infer the non-relativistic superc harges,
+Q±= limc→∞ˇQ±
+c, (10.7)
+Q+= 2i/radicalbiggm
+1+νa+Π+,Q−=−2i/radicalbiggm
+1+ν/parenleftbigg
+a−+1
+2mP−a+/parenrightbigg
+Π−,(10.8)
+whereν∝ne}ationslash=−1. The only nontrivial anticommutator is
+{Q+,Q−}=8
+1+νS, (10.9)
+where
+S=C2−mˆα+m1+ν
+2=V0+1
+2m(1+ν), (10.10)
+withV0defined in (9.17). The operator Scommutes with all Galilei generators. Unlike C2,S
+commutes also with the supercharges Q±. It can be identified [up to the factor m], therefore,
+with the superspin Casimir operator. Taking into account (1 0.4) [or the first equation from
+(9.20)], weget on-shell S=m1+ν
+2.Thesupercharges Q±extendtheGalilei algebra(9.32)-(9.33)
+with on-shell (anti)commutation relations,
+[J,Q±] =±1
+2Q±,{Q+,Q−}= 4m, (10.11)
+[Ki,Q±] = [Pi,Q±] = [H,Q±] = 0,Q2
+±= 0, (10.12)
+giving rise to the N= 1 super-extension of the Galilei algebra [41]21.
+21TheN= 1 super-Galilei algebra [41] has, besides the Q±, one more supercharge, namely the square root
+3910.2 Exotic N= 1non-relativistic supermultiplet
+Based on the deformed-oscillator representation of the Lor entz algebra, we define now the
+operators,
+N=J0−ˆα, b±=/radicalbigg
+2
+νJ±, (10.13)
+and take the limit ν→∞.In this limit we reproduce the usual harmonic oscillator alg ebraic
+relations of the form (9.37),
+[N,b±] =b±,[b−,b+] = 1. (10.14)
+However, herewegetthedirectsumoftwoirreduciblerepres entationsoftheHeisenbergalgebra,
+realized in the subspaces F±of the total Fock space Fof the RDHA (see (6.14), (6.15)),
+N|n∝an}bracketri}ht±=n|n∝an}bracketri}ht±, b−|n∝an}bracketri}ht±=√
+n+1|n+1∝an}bracketri}ht±, b+|n∝an}bracketri}ht±=√n|n−1∝an}bracketri}ht±, n= 0,1,2,... .
+Generalizing the exotic limit (9.1), we define, cf. (9.1),
+c→∞, ν→∞,ν/4
+c2=κ=const. (10.15)
+This limit produces J±/c→√
+2κb±and,
+ˆα
+c2→κ. (10.16)
+It follows that applying (10.15) to the supersymmetric equa tions (7.1) with the substitution
+(9.2) yields the sameequations (9.39) and (9.42) as in the non-supersymmetric ca se. There,
+thev±operators are defined as in (9.41), in terms of the operators ( 10.13).
+The Poincar´ e reducibility of the system is revealed by a non trivial integral, namely the
+reflection operator R. We have
+/parenleftbigg
+i∂
+∂t−H/parenrightbigg
+ψ±= 0,Λ−ψ±= 0, Rψ±=±ψ±, (10.17)
+wheretheoperators HandΛ−, whichcommutewith R, areformally givenbythesamerelations
+as in (9.43), (9.40).
+The finite part of the operator M0is now
+J = (ǫijxiPj+J0)−ν
+4=ǫijxiPj+N+1
+2−1
+4R. (10.18)
+The exotic Galilean generators are given by
+J =ǫijxiPj+1
+2κv+v−+1
+2−R
+4,Ki=mxi−tPi+κǫijvj. (10.19)
+Together with time and space translations, H=i∂
+∂t,Pi, they generate on-shell the algebra
+(9.49), (9.50). The Casimir operator C1, defined in (9.51), vanishes on-shell. In turn, the
+operatorC2, (9.51), takes different eigenvalues when acts on the ψ±wave functions,
+C2=m2−R
+4,C2ψ±=C±
+2ψ±,C±
+2=m/parenleftbigg1
+2±1
+4/parenrightbigg
+. (10.20)
+ofH. It is, however, notobtained as the non-relativistic limit of relativistic SUS Y. The possibility of extending
+the Galilei symmetry conformally, yielding a generalized S chr¨ odinger symmetry from s= 1/2 [42] to any spin, as
+well as its supersymmetric extensions, [43], are, in genera l, open questions. (See, however, [44]). The extension
+of exotic symmetry appears to be non-linear [28, 45].
+40The supercharges associated with the system (10.17) are obt ained from the exotic limit of
+(7.24),
+Q±= limc,ν→∞1√cQ±=±2i√m/parenleftbigg
+limν→∞Π∓a±
+√ν/parenrightbigg
+. (10.21)
+On account of the realization (6.29) of the RDHA generators i n terms of the bosonic oscillator
+operators, we get
+limν→∞a+
+√ν=1/radicalbig
+˜N˜a+˜Π+≡c+,limν→∞a−
+√ν=1/radicalbig
+1+˜N˜a−˜Π−≡c−.(10.22)
+The operators c±and˜R=Rsatisfy the same algebra as the Pauli matrices σ±=1
+2(σ1±iσ2)
+andσ3,
+{c+, c−}= 1, c±2= 0,[c+, c−] =R,{c±,R}= 0, R2= 1. (10.23)
+We obtain therefore
+Q±=±2i√mc±. (10.24)
+The supercharges Q±extend the exotic Galilei algebra (9.49)-(9.50) with (anti )commutation
+relations identical tothosein(10.11)-(10.12). Theonlyd ifferencewiththeusualnon-relativistic
+limit of the previous subsection is the non-commutativity o f Galilean boosts.
+The Casimir operator operator (10.20) is generalized to
+S=C2+1
+16[Q+,Q−],
+that commutes, unlike C2, with all superalgebra generators, including the supercha rgesQ±,
+and on-shell takes the value1
+2m. It is identified [up to the factor m] as the superspin operator.
+10.3 Nonrelativistic Dirac/DJT supermultiplet
+•For spin one-half ( −α= 1/2), putting
+φ0=ψ0, φ1=cψ1, (10.25)
+the general theory of Section 9 reduces, for n= 0, to the non-relativistic analogs of the (2+1)D
+Dirac equation due to the L´ evy-Leblond [21, 42], which take here a form
+i∂
+∂tφ0+−iP+φ1= 0,
+P−φ0−2imφ1= 0.(10.26)
+Eliminating one component yields the Schr¨ odinger equatio n for the other one. Eqns. (9.21)–
+(9.22) forn= 1 and then for any n>1 yieldφn= 0.
+The boosts operators (9.29) involve nontrivial spin [21, 42 ],
+K±=K1±iK2=−tP±+mx±+∆±,∆−= 0,∆+=/parenleftbigg0 0
+−1 0/parenrightbigg
+.(10.27)
+The angular momentum, (9.30), reads in turn,
+J =ǫijxiPj+J0,J0=/parenleftbigg−1/2 0
+0 1/2/parenrightbigg
+. (10.28)
+41•For−α=j= 1 one gets instead the non-relativistic analog of the DJT ve ctor equation
+for a topological massive gauge field. Introducing the redefi ned fields
+ϕ0=ψ0, ϕ 1=cψ1, ϕ 2=c2ψ2, (10.29)
+yields, putting n= 0 into (9.21)-(9.22) and multiplying by −i, the pair of equations
+√
+2i∂
+∂tϕ0−iP+ϕ1= 0,
+√
+2P−ϕ0−2imϕ1= 0.
+Similarly, for n= 1 we get
+P−ϕ1−i2√
+2mϕ2= 0,
+i∂
+∂tϕ1−i√
+2P+ϕ2= 0.
+The last equation here is readily seen to follow from the first three, however, and can therefore
+be dropped. For n= 2 and then for any n >2, the system (9.21)-(9.22) yields φn= 0. In
+conclusion, the non-relativistic limit of the Deser-Jacki w-Templeton equations (2.22) reads,
+i∂
+∂tϕ0−i√
+2P+ϕ1= 0,
+i√
+2mϕ1−P−ϕ0= 0,
+2√
+2mϕ2+iP−ϕ1= 0.(10.30)
+ϕ1andϕ2play the role of auxiliary fields, and may be expressed in term s ofϕ0. The dynamics
+of each of these free fields is governed by the Schr¨ odinger eq uation.
+The spin contribution to the boost, (9.29), and to the angula r momentum, (9.30), read
+∆+=
+0 0 0
+−√
+2 0 0
+0−√
+2 0
+,J0=
+−1 0 0
+0 0 0
+0 0 1
+. (10.31)
+The L´ evy-Leblond (spin s=−1/2) and non-relativistic DJT (spin s=−1) systems can
+now be unified into an N= 1 supermultiplet following the general recipe of Section 1 0. The
+non-relativistic spin one half, (10.25), and spin one, (10. 29), multiplets are thus unified into
+Φ=
+ϕ0
+φ0
+ϕ1
+φ1
+ϕ2
+. (10.32)
+The Galilei generators Piand (9.30) with J0=diag(−1,−1/2,0,1/2,1) [cf. (A.11)] act
+diagonally on the vectors (10.32). The corresponding super charges are (10.8) with ν=−5, i.e.
+Q+=√ma+Π+,Q−=−√m/parenleftbigg
+a−+1
+2mP−a+/parenrightbigg
+Π−, (10.33)
+where the odd osp(1|2) generators a±and projectors Π ±are those matrices (A.10) and (A.9).
+Note the asymmetry between Q+andQ−, which is consistent with the one between K+and
+K−in Eqns. (10.27), (10.31).
+The Dirac – DJT supermultiplet, in particular its superGali lei symmetry, are further dis-
+cussed in [44]. See also [37].
+4211 Extended N= 2non-relativistic supersymmetry
+Here we investigate the non-relativistic limit of the relat ivistic equations (8.16) with N= 2
+supersymmetry. The symmetries of the corresponding four-p article non-relativistic supermul-
+tiplet are obtained by In¨ on¨ u-Wigner contraction of the ex tended superPoincar´ e algebra (7.10),
+(8.20)-(8.21). The limit can be usual, or exotic, or a mixtur e of both kinds, since there are two
+independent spin parameters.
+11.1 Usual non-relativistic extended supersymmetry
+First we consider the usual non-relativistic limit of the fo ur-anyon system in Eqns.(8.16),
+V(χ)
+µψ++(x) = 0 ,
+V(χ+1/2)
+µψ+−(x) = 0,
+V(χ+1/2)
+µψ−+(x) = 0,
+V(χ+1)
+µψ−−(x) = 0.
+Forψ++(x), for instance, the simple limit yields
+/radicalbig
+n+2χ/parenleftbigg
+i∂
+∂t/parenrightbigg
+Φ++
+n+√
+n+1P+Φ++
+n+1= 0, (11.1)
+/radicalbig
+n+2χP−Φ++
+n+2m√
+n+1Φ++
+n+1= 0, (11.2)
+obtained as in (10.1), (10.2). Identical equations can be de rived for the three remaining fields,
+taking into account that the spin parameter in V(ˆß)
+µtakes the values,
+ˆß=diag/parenleftbigg
+χ,χ+1
+2,χ+1
+2,χ+1/parenrightbigg
+, χ=α1++α2+=1
+4(1+ν1)+1
+4(1+ν2),(11.3)
+as in (8.11). The index nof Φ refers to the lowest weight representations of the Loren tz algebra
+of spinß, from which the nonrelativistic limit was taken. These lowe st weight representations
+correspond to linear combinations of the Fock basis (8.14) ( see also (8.12) and (8.13)). Hence,
+ˆß= ˆα1++ ˆα2+=χ+1
+2(Π1−+Π2−), χ=α1++α2+=1
+2+1
+4(ν1+ν2).(11.4)
+The nonrelativistic limit of time translations, Galilei bo osts and rotation generators is
+obtained as in Section 9.
+This representation is, of course, reducible, since that of the Poincar´ e algebra was reducible
+too. In particular, rotations also involve the number opera tors of the lowest-weight represen-
+tations of the Lorentz algebra as well as ˆß,22
+J =ǫijxiPj+N1+N2+ˆß. (11.5)
+22Here Π A±= (1±RA)/2 are the projection operators on Fock space of every oscilla tor labeled by A. They
+satisfy the relations (6.19) for every index, and commute. T he numbers operators satisfy,
+NA|n1,n2/angbracketright±±=nA|n1,n2/angbracketright±±,NA|n1,n2/angbracketright±∓=nA|n1,n2/angbracketright±∓, A= 1,2.
+43The operatorC1in (9.35) is still zero, while C2becomesC2=mˆß,cf.Eq. (9.35). Their
+eigenvalues are,
+C2Φ++=χΦ++,C2Φ+−= (χ+1/2)Φ+−,
+C2Φ−+= (χ+1/2)Φ−+,C2Φ−−= (χ+1)Φ−−.(11.6)
+Each field Φ carries therefore a representation of the Galile i algebra.
+The supercharges associated to this supermultiplet are the nonrelativistic limits of the
+superchages (8.17) and (8.18), namely,
+Q1+= 2i/radicalbiggm
+1+ν1a+
+1Π1+,Q1−=−2i/radicalbiggm
+1+ν1/parenleftbigg
+a−
+1+1
+2mP−a+
+1/parenrightbigg
+Π1−,(11.7)
+Q2+= 2i/radicalbiggm
+1+ν2R1a+
+2Π2+,Q2−=−2i/radicalbiggm
+1+ν2R1/parenleftbigg
+a−
+2+1
+2mP−a+
+2/parenrightbigg
+Π2−,(11.8)
+cf. (10.8). The algebra generated by the supercharges (11.7 ), (11.8) augmented with the Galilei
+generators is analogous to (10.11), (10.12), with the addit ional relations
+{QA+,QB−}=δAB4m,{QA±,QB±}= 0. (11.9)
+The supercharges with different indices anticommute, due to t he presence of the reflection
+operator,R1.
+11.2 Exotic nonrelativistic extended supersymmetry
+The exotic limit (9.1) is generalized to
+c→∞, ν→∞,ˆß
+c2=χ
+c2=κ=const, (11.10)
+cf. (10.16). Since χdepends on both of the deformation parameters (see (11.5)), the exotic
+limit can be carried out varying only one, or both, parameter s simultaneously.
+Simple exotic limit
+Here we take the limit
+c→∞, ν 1→∞, ν 2=const,ν1/4
+c2=κ=const. (11.11)
+The rotation generator is obtained by removing the divergen tc-number term ν1/4 fromˆßin
+(11.5),
+J =ǫijxiPj+N1+N2+α2++1
+4+1
+2(Π1−+Π2−). (11.12)
+Galilei boosts are obtained from the non-relativistic limi t of Sec. 9,
+K+=mx+−tP++i(J2+−κv+),K−=mx−−tP−+iκv−, (11.13)
+v±=v1±iv2=−/radicalbigg
+2
+κb±, b±= lim
+ν1→∞/radicaligg
+2
+ν1J1±. (11.14)
+44Observe that vjandb±only involve the oscillator with label 1 .
+Galilei boosts satisfy the exotic commutation relation (9. 49). Theb±andN1operators
+satisfy relations like in (9.37), i.e.N1=b+b−. Due to the asymmetry of the limit (11.11), the
+supercharges are of different types,
+Q1±= limc,ν→∞1√cQ1±,Q2±= limc→∞1
+cˇQ2±, (11.15)
+Q1±=±2i√m/parenleftbigg
+limν1→∞Π1∓a±
+1√ν1/parenrightbigg
+=±2i√mc±
+1, (11.16)
+Q2+= 2i/radicalbiggm
+1+ν2R1a+
+2Π2+,Q2−=−2i/radicalbiggm
+1+ν2R1/parenleftbigg
+a−
+2+1
+2mP−a+
+2/parenrightbigg
+Π2−,(11.17)
+which are those obtained as in Eqns. (10.21) (“exotic superc harges”) and (10.8) (“usual super-
+charges”) respectively. These supercharges yield commuta tion relation as in (11.9).
+Double exotic limit
+Here we take the limit,
+c→∞, ν 1→∞, ν 2→∞,ν1/2
+c2=ν2/2
+c2=κ=const. (11.18)
+The finite part of angular momentum is now,
+J =ǫijxiPj+N1+N2+1
+2+1
+2(Π1−+Π2−). (11.19)
+The Galilei boosts are,
+Ki=mxi−tPi+κ
+2ǫijv1j+κ
+2ǫijv2j (11.20)
+where thevoperators are defined analogously to (11.2), and the labels Aindicate the depen-
+dence on the corresponding oscillator. In particular, here we haveb±
+AandNA=b+
+Ab−
+A, as in
+(9.37). Note that this double exotic limit is symmetric in bo th oscillators. Hence, we obtain,
+Q1±= limc,ν1→∞1√cQ1±=±2i√mc±
+1, (11.21)
+Q2±= limc,ν2→∞1√cQ2±=±2i√mR1c±
+2, (11.22)
+which are those in Eqns. (10.21) and (10.8) respectively.
+12 Discussion and outlook
+In conclusion, we review our main results and hint at some ope n problems which could be
+worth of further study.
+Our universal covariant vector equations describe massive spinning particles in 2 + 1 di-
+mensions. The Poincar´ e spin, a pseudoscalar, can take arbi trary real values. The equations fix
+themselves the type of the representation of the (2+1)D Lore ntz algebra: only so(2,1) repre-
+sentations bounded from below or from above (or from both sid es) are allowed. Depending on
+the spin parameter, α, our equations describe three classes of particles, namely
+45a) Bosons/Fermions : α=−j, 2j= 1,2,...,
+b) “Unitary” anyons : α>0,
+c) “Non-unitary” anyons : −j <α<−j+1/2.
+Case a), based on finite-dimensional non-unitary represent ations of the so(2,1), reproduces,
+forα=−1/2,−1,−3/2 the Dirac, Deser-Jackiw-Templeton, and Rarita-Schwinge r theories,
+respectively. For α=−2, our theory provides us with the linearized equations of ma ssive
+gravity. More generally, complete correspondence with the 2+1 dimensional Dirac-Fierz-Pauli
+equations [46] is expected.
+Then, we investigated two types of anyons. Both involve infin ite-dimensional half-bounded
+representations of the (2+1)D Lorentz algebra. The first typ e, which corresponds to unitary
+representations [case b) above], has been widely used in the anyon context since the beginning
+[13, 14, 15]. The second type [case c)] is associated with non-unitary representations. No at-
+tention was paid to them in earlier investigations on anyons . We argue, however, that precisely
+these new- [and not the conventional] types of fractional sp in particles do correspond to the
+intuitive picture, in which anyons interpolate between bos ons and fermions.
+Theessential differenceisthat whentheparameter αtendstoa(half) integer −j, thehigher
+field components ψn(x) with index n>2jare, for the new type, suppressed by the universal
+numerical factor ( α+j)1/2. Note that no such a suppression happens in the conventional ,
+unitary case. As a result, in the limiting case α=−j, such an anyonic field reduces to a
+usual (2j+ 1)-component field of spin modulus |s|=j. The spin zero case then also can be
+incorporated into the theory letting α→0 for non-unitary anyons.
+The physical difference between the two types of anyons could, perhaps, be revealed when
+interactions are switched on.
+The extended formulation of the theory, presented in Sectio n 5, generalizes the Jackiw-Nair
+[13] and the Majorana-Dirac [14] descriptions of anyons. Ou r framework allows us to combine
+several irreducible representations of the Lorentz algebr a, “entangled” by Eqn. (5.3), whereas
+spins behave additively. Usual bosons and fermions can be ob tained, in particular, as entangled
+anyons. The topologically massive vector gauge field of Jack iw-Deser-Templeton [3] can be, for
+example, viewed as a pair of entangled Dirac particles.
+It would be interesting to apply such a picture to quantum com puting [47].
+Promoting the parameter αto a diagonal operator ˆ α=diag(α,α+ 1/2) and taking the
+direct sum of two irreducible representations of so(2,1) provided us with an N= 1 supersym-
+metric system. It is described, on-shell, by the usual super-Poincar´ e algebra.
+The natural question is, then: what kind of symmetry will app ear if the elements of ˆ α
+are shifted by n/2 withninteger greater than 1 ? One can expect that such a theory will
+be characterized by a kind of generalized supersymmetry, wi th spin-tensorial supercharges
+[48, 49, 50], which, on-shell, would generate a nonlinear generalization of the usual N= 1
+super-Poincar´ e algebra [35, 51].
+In the simplest form of the extended realization two Lorentz spin vector generators are
+added; this also allows us to generalize the N= 1 supersymmetry to N= 2, see Section 8 for
+details.
+The extended formulation can further be generalized by addi ng an arbitrary number of the
+vector spin generators, Jµ=/summationtext
+iJ(i)
+µ, with each J(i)
+µbelonging to one of the representations
+D+
+α(i)(orD−
+α(i)) from (2.19). Choosing ß=/summationtext
+iα(i), one finds then that the basic vector set of
+equations splits into a set of vector equations, while the ir reducibility equation (2.10) produces
+46an entangling equation which generalizes (5.3),
+V(α(i))
+µψ= 0,/summationdisplay
+i>j(J(i)J(j)+α(i)α(j))ψ= 0. (12.1)
+EachJ(i)
+µcan be promoted to an even generator of the osp(1|2) superalgebra, realized in terms
+of an independent reflection-deformed oscillator with appr opriately chosen parameter ν(i). In
+this way, extended supermultiplets of anyons, or of bosons a nd fermions could be obtained.
+Considering two types of non-relativistic limits to the rel ativistic theory, we observed how
+either theusualcentrally-mass-extended, orthedoublyce ntrally extended“exotic” Galilei sym-
+metries appear. Their Schr¨ odinger-type conformal extens ions for arbitrary (including anyonic)
+spin, and also for supersymmetric generalizations could al so be studied.
+It would also be interesting to derive our basic vector equat ions from an action princi-
+ple. A nontrivial point is that the three equations of our sys tem are dependent: any two of
+them generate the third one as consistency (integrability) equation. This indicates that the
+corresponding action should have a kind of gauge symmetry, a nalogous to the Chern-Simons
+formulation of topologically massive gauge fields [3]. An ad ditional indication in this direc-
+tion is the appearance of an infinite-dimensional null subsp ace in the extended formulation.
+Analogous null subspaces also appear, in fact, in covariant quantization schemes of relativistic
+strings, and are associated with gauge (diffeomorphism) symm etries.
+In our unified scheme, all irreducible massive representati ons of the 2+1 D Poincar´ e group
+have been obtained. Does this theory have an analog in higher dimensions? This question
+directs us towards theories with arbitrary spin fields. For i nstance, to strings, or to higher spin
+theories [52].
+Acknowledgements . MP is indebted to the Laboratoire de Math´ ematiques et de Physique
+Th´ eorique ofToursUniversity, andPAHisindebtedtothe Departamento de F´ ısica, Universidad
+de Santiago de Chile , respectively, for hospitality. Partial support by the FON DECYT (Chile)
+under the grant 1095027 and by DICYT (USACH) is acknowledged . MV was supported by
+CNRS postdoctoral grant (contract number 87366).
+A Finite-dimensional representations of osp(1|2)
+A.1osp(1|2)-supermultiplet ( α+=−1/2,α−= 0)
+Hereν=−3 andr= 1. The finite-Fock space, F=F+⊕F−,is
+F+=
+
+|0∝an}bracketri}ht+=
+1
+0
+0
+,|1∝an}bracketri}ht+=
+0
+0
+1
+
+
+,F−=
+
+|0∝an}bracketri}ht−=
+0
+1
+0
+
+
+.(A.1)
+The reflection operator and projectors take the form,
+R=diag(1,−1,1),Π+=diag(1,0,1),Π−=diag(0,1,0). (A.2)
+The matrix form of the osp(1|2) algebra is obtained from (6.22)–(6.25), (6.15) and (6.16 ),
+a+=
+0 0 0
+i√
+2 0 0
+0√
+2 0
+, a−=
+0i√
+2 0
+0 0√
+2
+0 0 0
+, (A.3)
+47J0=
+−1/2 0 0
+0 0 0
+0 0 1/2
+,J+=
+0 0 0
+0 0 0
+i0 0
+,J−=
+0 0i
+0 0 0
+0 0 0
+.(A.4)
+The representation (A.4) of the Lorentz algebra is reducibl e. The spin−α+=j= 1/2 part is
+projected from (A.4) to
+J(+)
+0=−1
+2σ3, J(+)
+1=i
+2σ1, J(+)
+2=−i
+2σ2, J(+)
+µJ(+)µ=−3
+4,(A.5)
+whereσa,a= 1,2,3,are the Pauli matrices. These operators act on the vector spa ceF+
+formed by
+|0∝an}bracketri}ht+=/parenleftbigg1
+0/parenrightbigg
++,|1∝an}bracketri}ht+=/parenleftbigg0
+1/parenrightbigg
++.
+−2J(+)
+µ=γ′
+µare Dirac matrices that satisfy the Clifford algebra (6.10). T hey are related to
+the Majorana representation (6.9) by the unitary transform ation,
+γ′
+µ=UγµU†, U=1√
+2(σ2+σ3). (A.6)
+The Lorentz algebra (A.4) projected to the spin α−= 0 subspace is trivial, J(−)
+µ=JµΠ−=
+0. It leaves invariant the scalar, one-dimensional subspac eF−={|0∝an}bracketri}ht−}.
+A.2osp(1|2)-supermultiplet ( α+=−1,α−=−1/2)
+Hereν=−5 andr= 2. The finite-Fock space is F=F+⊕F−,
+F+=
+
+|0∝an}bracketri}ht+=
+1
+0
+0
+0
+0
+,|1∝an}bracketri}ht+=
+0
+0
+1
+0
+0
+,|2∝an}bracketri}ht+=
+0
+0
+0
+0
+1
+
+
+, (A.7)
+F−=
+
+|0∝an}bracketri}ht−=
+0
+1
+0
+0
+0
+,|1∝an}bracketri}ht−=
+0
+0
+0
+1
+0
+
+
+. (A.8)
+The reflection operator and projectors take the form
+R=diag(1,−1,1,−1,1),Π+=diag(1,0,1,0,1),Π−=diag(0,1,0,1,0).(A.9)
+Theosp(1|2) matrix algebra is generated by
+a+=
+0 0 0 0 0
+2i0 0 0 0
+0√
+2 0 0 0
+0 0i√
+2 0 0
+0 0 0 2 0
+, a−=
+0 2i0 0 0
+0 0√
+2 0 0
+0 0 0 i√
+2 0
+0 0 0 0 2
+0 0 0 0 0
+, (A.10)
+48J0=diag(−1,−1/2,0,1/2,1), (A.11)
+J+=
+0 0 0 0 0
+0 0 0 0 0
+i√
+2 0 0 0 0
+0i0 0 0
+0 0i√
+2 0 0
+,J−=
+0 0i√
+2 0 0
+0 0 0 i0
+0 0 0 0 i√
+2
+0 0 0 0 0
+0 0 0 0 0
+. (A.12)
+Projection to the odd subspace of spin −α−=j= 1/2 produces a representation with the
+same matrix elements as in (A.5) but now acting on F−,
+|0∝an}bracketri}ht−=/parenleftbigg1
+0/parenrightbigg
+−,|1∝an}bracketri}ht−=/parenleftbigg0
+1/parenrightbigg
+−. (A.13)
+The spin−α+=j= 1 part is projected to
+J(+)
+0=
+−1 0 0
+0 0 0
+0 0 1
+, J(+)
+1=1√
+2
+0i0
+i0i
+0i0
+, J(+)
+2=1√
+2
+0−1 0
+1 0−1
+0 1 0
+,(A.14)
+which act on the vector space F+,
+|0∝an}bracketri}ht+=
+1
+0
+0
+
++,|1∝an}bracketri}ht+=
+0
+1
+0
+
++,|2∝an}bracketri}ht+=
+0
+0
+1
+
++. (A.15)
+(A.14) is related to the adjoint representation of so(2,1),
+(Jµ)νλ=−iǫνµλ, µ,ν,λ = 0,1,2, ǫ012= 1, (A.16)
+by means of the unitary transformation
+J(+)
+µ=UJµU†, U=1√
+2
+0 1i
+−i√
+2 0 0
+0−1i
+. (A.17)
+The explicit matrix form of (A.16) is
+(J0)νλ=
+0 0 0
+0 0−i
+0i0
+,(J1)νλ=
+0 0−i
+0 0 0
+−i0 0
+,(J2)νλ=
+0i0
+i0 0
+0 0 0
+.(A.18)
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+54
\ No newline at end of file
diff --git a/1001.0275.txt b/1001.0275.txt
new file mode 100644
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--- /dev/null
+++ b/1001.0275.txt
@@ -0,0 +1,900 @@
+arXiv:1001.0275v1  [math.AP]  2 Jan 2010DIRAC–SOBOLEV SPACES AND SOBOLEV SPACES
+TAKASHI ICHINOSE AND YOSHIMI SAIT ¯O
+Abstract. TheaimofthisworkistostudythefirstorderDirac-Sobolevspaces in
+Lpnorm on an open subset of R3to clarify its relationship with the corresponding
+Sobolev spaces. It is shown that for 1 < p <∞, they coincide, while for p= 1,
+the latter spaces are proper subspaces of the former.
+Keywords : Sobolev spaces, Dirac oprerator.
+2000 Mathematics Subject Classification : 46E35, 46E40.
+1.Introduction
+In the recent work [2], Balinski-Evans-Sait¯ o introduced an Lp-seminorm /ba∇dbl(α·
+p)f/ba∇dblp,Ωof aC4-valued function fin an open subset Ω of R3, relevant to a massless
+Dirac operator
+(1.1) α·p=3/summationdisplay
+j=1αj(−i∂j) (∂j=∂/∂xj),
+wherep=−i∇, andα= (α1, α2, α3) is the triple of 4 ×4 Dirac matrices
+(1.2) αj=/parenleftbigg
+02σj
+σj02/parenrightbigg
+(j= 1,2,3)
+with the 2 ×2 zero matrix 0 2and the triple of 2 ×2 Pauli matrices
+σ1=/parenleftbigg
+0 1
+1 0/parenrightbigg
+, σ2=/parenleftbigg
+0−i
+i0/parenrightbigg
+, σ3=/parenleftbigg
+1 0
+0−1/parenrightbigg
+.
+Theyusedthisseminormtogiveagroupofinequalities called Dirac–Sobolev inequal-
+itiesin order to obtain Lp-estimates of the zero modes , i.e. eigenfunctions associated
+with the eigenvalue λ= 0, of the Dirac operator ( α·p)+Q, whereQ(x) is a 4×4
+Hermitian matrix-valued potential decaying at infinity. We believe tha t our above
+notation “ p” for the differential operator −i∇will not be confused with another
+“p” which appears as the superscript 1 ≤p<∞of the space Lp.
+12 TAKASHI ICHINOSE AND YOSHIMI SAIT ¯O
+Let Ω be an open subset of R3and let the first order Dirac–Sobolev space
+H1,p
+0(Ω), 1≤p<∞,be the completion of [ C∞
+0(Ω)]4with respect to the norm
+(1.3)
+/ba∇dblf/ba∇dblD,1,p,Ω:=/braceleftbigg/integraldisplay
+Ω(|f(x)|p
+p+|(α·p)f(x)|p
+p)dx/bracerightbigg1/p
+=/parenleftbig
+/ba∇dblf/ba∇dblp
+p,Ω+/ba∇dbl(α·p)f/ba∇dblp
+p,Ω/parenrightbig1/p,
+wheref(x) =t(f1(x),f2(x),f3(x),f4(x)), the norm of a vector a=t(a1,a2,a3,a4)∈
+C4being denoted by
+(1.4) |a|p=/bracketleftBig4/summationdisplay
+k=1|ak|p/bracketrightBig1/p
+.
+AsoneofthesimplestDirac–Sobolevinequalities([2],Corollary2),the yshowed:
+If Ω is a bounded open subset of R3andf∈H1,p
+0(Ω) with 1 ≤p <∞, then for
+1≤k<p(p+3)/3 there exists a positive constant Csuch that
+(1.5) /ba∇dblf/ba∇dblk,Ω≤C/ba∇dbl(α·p)f/ba∇dblp,Ω,
+where/ba∇dblg/ba∇dblp,Ωstands for the norm of g=t(g1,g2,g3,g4)∈[Lp(Ω)]4given by
+(1.6) /ba∇dblg/ba∇dblp,Ω=/braceleftbigg/integraldisplay
+Ω|g(x)|p
+pdx/bracerightbigg1/p
+=/braceleftBigg/integraldisplay
+Ω4/summationdisplay
+j=1|gj(x)|pdx/bracerightBigg1/p
+.
+Now letβbe the fourth Dirac matrix given by
+(1.7) β=/parenleftbigg
+1202
+02−12/parenrightbigg
+,
+where 1 2is the 2×2 unit matrix. It has been known that the free massless Dirac
+operatorα·por the free Dirac operator α·p+mβwith positive mass mand
+the relativistic Schr¨ odinger operator√
+m2−∆ may bring similar properties in L2
+sometimes but not necessarily in Lpwithp/\e}atio\slash= 2. On the other hand, the following
+two norms are equivalent: for 1 <p<∞,
+/braceleftBigg
+(/ba∇dblψ/ba∇dblp
+p+/ba∇dbl∇ψ/ba∇dblp
+p)1/p,
+/ba∇dbl√
+1−∆ψ/ba∇dblp.
+whereψis a scalar-valued function in R3(Stein [6], p.135, Theorem 3 or p.136,
+Lemma 3). However, for p= 1 orp=∞, these two norms are not equivalent, in
+fact, the one does not dominate the other ([6], p.160, 6.6).
+For an open subset Ω of R3and 1≤p<∞, letAp(Ω) be allC∞functionsψon
+Ω such that ψand∇ψbelong toLp(Ω) and let H1,p(Ω) be the completion of Ap(Ω)DIRAC–SOBOLEV SPACES 3
+with respect to the norm given by
+(1.8) /ba∇dblψ/ba∇dbl1,p,Ω=/braceleftbigg/integraldisplay
+Ω(|ψ(x)|p+|∇ψ(x)|p)dx/bracerightbigg1/p
+={/ba∇dblψ/ba∇dblp
+p,Ω+/ba∇dbl∇ψ/ba∇dblp
+p,Ω}1/p,
+where|∇ψ(x)|p=/summationtext3
+j=1|∂jψ(x)|p. LetC∞
+0(Ω) be the space of all C∞functionsφ
+on Ω such that support of φis contained in Ω and let H1,p
+0(Ω) be the completion
+ofC∞
+0(Ω) with respect to the norm (1.8). The space H1,p
+0(Ω) is a closed subspace
+ofH1,p(Ω). The norm /ba∇dbl·/ba∇dblS,1,p,Ωof the Banach space [ H1,p(Ω)]4(and [H1,p
+0(Ω)]4) is
+given by
+(1.9) /ba∇dblf/ba∇dblS,1,p,Ω=/braceleftbigg/integraldisplay
+Ω/parenleftbig
+|f(x)|p
+p+|∇f(x)|p
+p/parenrightbig
+dx/bracerightbigg1/p
+,
+wheref=t(f1,f2,f3,f4)∈[H1,p(Ω)]4and
+(1.10)
+
+|f(x)|p
+p=4/summationdisplay
+k=1|fk(x)|p,
+|∇f(x)|p
+p=3/summationdisplay
+j=1|∂jf(x)|p
+p=3/summationdisplay
+j=14/summationdisplay
+k=1|∂jfk(x)|p
+(cf. (1.4)).
+Definition 1.1. LetAp,D(Ω) be all [ C∞(Ω)]4functionsfon Ω such that fand
+(α·p)fbelong to [Lp(Ω)]4. Then the Dirac-Sobolev spaces H1,p(Ω) andH1,p
+0(Ω) are
+the completion of Ap,D(Ω) and [C∞
+0(Ω)]4with respect to the norm
+(1.11) /ba∇dblf/ba∇dblD,1,p,Ω=/braceleftbigg/integraldisplay
+Ω(|f(x)|p
+p+|(α·p)f(x)|p
+p)dx/bracerightbigg1/p
+,
+respectively, where f(x) =t(f1(x),f2(x),f3(x),f4(x)), and
+(1.12)
+
+|f(x)|p
+p=4/summationdisplay
+k=1|fk(x)|p,
+|(α·p)f(x)|p
+p=|3/summationdisplay
+j=1αjpjf(x)|p
+p=4/summationdisplay
+k=1/vextendsingle/vextendsingle/vextendsingle/parenleftBig3/summationdisplay
+j=1−iαj∂jf/parenrightBig
+k(x)/vextendsingle/vextendsingle/vextendsinglep
+p
+It should be noted that in the paper [2] the space H1,1
+0(Ω) of our paper was
+denoted by H1,1(Ω) without subscript ‘0’. We have adopted this notation, following
+the usual Sobolev space convention.
+Remark 1.2. (i) As in the case of Sobolev spaces, we have H1,p(R3) =H1,p
+0(R3)
+since [C∞
+0(R3)]4is dense in Ap,D(R3) with respect to the norm (1.11).4 TAKASHI ICHINOSE AND YOSHIMI SAIT ¯O
+(ii) LetW1,p(Ω) be defined by
+W1,p(Ω) ={f∈[Lp(Ω)]4; (α·p)f∈[Lp(Ω)]4},
+where (α·p)fis taken in the sense of distributions. As in the case of Sobolev space s
+(see, e.g.,Adams-Fournier[1],Theorem 3.17),byapproximating elem ents inW1,p(Ω)
+using the molifier, we have W1,p(Ω) =H1,p(Ω), where 1 ≤p<∞and Ω is an open
+subset of R3.
+In this work we are going to investigate the relationship of the Dirac– Sobolev
+spacesH1,p(Ω) and the ordinary Sobolev spaces [ H1,p(Ω)]4as well as the relationship
+of the Dirac–Sobolev spaces H1,p
+0(Ω) and the ordinary Sobolev spaces [ H1,p
+0(Ω)]4.
+To proceed, we note the inclusions [ H1,p(Ω)]4⊂H1,p(Ω) and [H1,p
+0(Ω)]4⊂
+H1,p
+0(Ω) to hold, which we shall see precisely later in the next section, Pro position
+2.2. So for an open subset Ω of R3, define the linear map JΩandJ0,Ωby
+(1.13)/braceleftBigg
+JΩ: [H1,p(Ω)]4∋f/ma√sto→JΩf=f∈H1,p(Ω),
+J0,Ω: [H1,p
+0(Ω)]4∋f/ma√sto→J0,Ωf=f∈H1,p
+0(Ω).
+Our main result is as follows:
+Theorem 1.3. LetΩbe an open subset of R3and letJΩandJ0,Ωbe as above.
+(i)Then, for 1≤p<∞, the map both JΩandJ0,Ωare one-to-one and contin-
+uous. The Sobolev space [H1,p
+0(Ω)]4is a dense subspace of the Dirac–Sobolev spaces
+H1,p
+0(Ω).
+(ii)Let1< p <∞. Then[H1,p
+0(Ω)]4=H1,p
+0(Ω), i.e., the map J0,Ωis not only
+one-to-one and continuous, but also they are onto with conti nuous inverse map J−1
+0,Ω.
+(iii)Forp= 1the map neither JΩnorJ0,Ωare onto, or we have [H1,1(Ω)]4/subsetornotdbleql
+H1,1(Ω)and[H1,1
+0(Ω)]4/subsetornotdbleql H1,1
+0(Ω). Forp= 1, the norms /ba∇dbl · /ba∇dblD,1,1,Ωand/ba∇dbl · /ba∇dblS,1,1,Ω
+of these two spaces are not equivalent; /ba∇dbl·/ba∇dblD,1,1,Ωis dominated by /ba∇dbl·/ba∇dblS,1,1,Ω, but not
+conversely.
+Remark 1.4. We don’t know in(ii) whether or not it holds for a proper open subset
+Ω that [H1,p(Ω)]4=H1,p(Ω), i.e., that the map JΩis onto with continuous inverse
+mapJ−1
+Ωwhen Ω /subsetornotdbleql R3, although it holds by (ii) for Ω = R3that [H1,p(R3)]4=
+H1,p(R3), because this space coincides with [ H1,p
+0(R3)]4=H1,p
+0(R3).
+For the proof we shall use a method of classical analysis rather tha n a subtle
+pseudo-differentical calculus, in particular, in the case p= 1.DIRAC–SOBOLEV SPACES 5
+In Section 2 we shall prove Theorem 1.3, (i) (Proposition 2.2). In Sec tion 3 we
+are going to give the proof of Theorem 1.3, (ii) by first dealing with JΩand then
+J0,Ω. Theorem 1.3, (iii), the case that p= 1, will be discussed and proved in Section
+4.
+2.Continuity of the map J Ω
+In this section we are going to prove Theorem 1.3, (i). Let αj,j= 1,2,3, be
+the Dirac matrices given in (1.2). Then we have
+Lemma 2.1. Let a=t(a1,a2,a3,a4)∈C4. Then, for p∈[1,∞),
+(2.1) |αja|p=|a|p(j= 1,2,3),
+where the norm |·|pis given by (1.4).
+Proof.By the definition of α1we have
+α1a=t(a4,a3,a2,a1),
+and hence
+|α1a|p
+p=|a4|p+|a3|p+|a2|p+|a1|p=|a|p
+p.
+In quite a similar manner (2.1) can be proved for j= 2,3. /square
+Now we are in a position to show the continuity of the map JΩandJ0,Ωgiven
+by (1.13).
+Proposition 2.2. LetΩbe an open subset of R3and let f ∈[H1,p(Ω)]4. Then,
+forp∈[1,∞), we have f ∈H1,p(Ω)and there exists a positive constant C=Cp,
+depending only on p, not on Ω, such that
+(2.2) /ba∇dblf/ba∇dblD,1,p,Ω≤C/ba∇dblf/ba∇dblS,1,p,Ω,
+where the norms /ba∇dblf/ba∇dblD,1,p,Ωand/ba∇dblf/ba∇dblS,1,p,Ωare given in (1.11)and(1.9), respectively.
+Thus the identity maps JΩon[H1,p(Ω)]4andJ0,Ωon[H1,p
+0(Ω)]4are continuous,
+one-to-one maps from [H1,p(Ω)]4intoH1,p(Ω), and[H1,p
+0(Ω)]4intoH1,p
+0(Ω). Further
+[H1,p
+0(Ω)]4is a dense subset of H1,p
+0(Ω).6 TAKASHI ICHINOSE AND YOSHIMI SAIT ¯O
+Proof.Letf=t(f1,f2,f3,f4)∈[H1,p(Ω)]4. By using Lemma 2.1 and H¨ older’s in-
+equality for p>1 or the triangle inequality for p= 1, we have
+|(α·p)f|p
+p=|(3/summationdisplay
+l=1−αli∂l)f|p
+p≤/parenleftbig3/summationdisplay
+l=1|αli∂lf|p/parenrightbigp=/parenleftbig3/summationdisplay
+ℓ=1|∂lf|p/parenrightbigp
+≤/bracketleftbig/parenleftbig3/summationdisplay
+l=11q/parenrightbig1/q/parenleftbig3/summationdisplay
+ℓ=1|∂lf|p
+p/parenrightbig1/p/bracketrightbigp= 3p−1|∇f|p
+p,
+wherep−1+q−1= 1 and see (1.10) for the definition of |∇f|p. It follows that
+(2.3)/ba∇dbl(α·p)f/ba∇dblp,Ω=/parenleftBig/integraldisplay
+Ω|(α·p)f|p
+pdx/parenrightBig1/p
+≤3p−1
+p/parenleftBig/integraldisplay
+Ω|∇f|p
+pdx/parenrightBig1/p
+= 3p−1
+p/ba∇dbl∇f/ba∇dblp,Ω,
+where/ba∇dbl(α·p)f/ba∇dblp,Ωis the norm of ( α·p)f∈[Lp(Ω)]4given by (1.6), and /ba∇dbl∇f/ba∇dblp,Ωis
+given by
+/ba∇dbl∇f/ba∇dblp,Ω=/braceleftbigg/integraldisplay
+Ω|∇f|p
+pdx/bracerightbigg1/p
+.
+Then it is easy to see that (2.3) implies (2.2). The map Jis one-to-one since, for
+fj∈[H1,p(Ω)]4,j= 1,2, we have
+(2.4)JΩf1=JΩf2inH1,p(Ω) =⇒f1=f2in [Lp(Ω)]4=⇒f1=f2in [H1,p(Ω)]4.
+Using(2.2)andproceedingasin(2.4),weseethattheidentitymap J0,Ωon[H1,p
+0(Ω)]4
+is also continuous and one-to-one on [ H1,p
+0(Ω)]4. Since [C∞
+0(Ω)]4is dense in both
+[H1,p
+0(Ω)]4andH1,p
+0(Ω), [H1,p
+0(Ω)]4is a dense subset of H1,p
+0(Ω). This completes the
+proof. /square
+3.Range of the map J 0,Ω
+In this section we are going to prove Theorem 1.3, (ii).
+Proposition 3.1. Let1<p<∞. Then the map J0,R3is ontoH1,p
+0(R3).
+The proof will be given after the following two lemmas.
+Lemma 3.2. Let1<q <∞. Then∆(C∞
+0(R3))is dense in Lq(R3).
+Proof of Lemma 3.2. Suppose that f∈Lr(R3) with 1/q+1/r= 1 satisfies
+/a\}b∇acketle{tf,∆φ/a\}b∇acket∇i}ht=/integraldisplay
+f(x)∆φ(x)dx= 0,for allφ∈C∞
+0(R3).DIRAC–SOBOLEV SPACES 7
+Then we obtain in the sense of distributions ∆ f= 0, namely, ∆ annihilates f. By
+elliptic regularity, we see that fmust beC∞, and hence f(x) is a polynomial of x.
+Sincef(x) should belong to Lr(R3), we havef= 0. This proves Lemma 3.2. /square
+Remark 3.3. Lemma 3.2 does not hold for q= 1, since, in this case, the Laplacian
+∆ always annihilates a constant C/\e}atio\slash= 0 which is a nonzero element of L∞(R3) =
+L1(R3)∗.
+Lemma 3.4. Let1<q<∞. LetΩ⊂R3be an open set. Then, for each pair (j,k),
+j,k= 1,2,3, there exists a positive constant C=Cjksuch that
+(3.1) /ba∇dbl∂j∂kφ/ba∇dblq,Ω≤C/ba∇dbl∆φ/ba∇dblq,Ω(φ∈C∞
+0(Ω)).
+Proof.By Stein [6], p.59, Proposition 3, there exists a positive constant C=Cjk
+such that
+/ba∇dbl∂j∂kφ/ba∇dblq≤C/ba∇dbl∆φ/ba∇dblq, φ∈C∞
+0(R3).
+Of course, this holds for φ∈C∞
+0(R3). /square
+Proof of Proposition 3.1. The proof will be divided into four steps.
+(I) Letf=t(f1,f2,f3,f4)∈H1,p
+0(R3). Thenf∈[Lp(R3)]4, and
+g:= (α·p)f=−i[α1∂1f+α2∂2f+α3∂3f]
+belongs to [ Lp(R3)]4. Using the definition (1.2) of the Dirac matrices αj,j= 1,2,3,
+we can rewrite this with g=t(g1,g2,g3.g4) as
+(3.2)
+
+ig1= (∂1−i∂2)f4+∂3f3,
+ig2= (∂1+i∂2)f3−∂3f4,
+ig3= (∂1−i∂2)f2+∂3f1,
+ig4= (∂1+i∂2)f1−∂3f2.
+Then from the first and second equations of (3.2) we have
+(∂1+i∂2)ig1= (∂2
+1+∂2
+2)f4+∂3(∂1+i∂2)f3
+= (∂2
+1+∂2
+2)f4+∂3(ig2+∂3f4),
+so that
+∆f4= (∂2
+1+∂2
+2+∂3
+3)f4= (∂1+i∂2)(ig1)−∂3(ig2),
+andhence,byapplying ∂jtobothsidesoftheaboveequation,wehave,for j= 1,2,3,
+(3.3) ∆ ∂jf4= (∂j∂1+i∂j∂2)(ig1)−∂j∂3(ig2).8 TAKASHI ICHINOSE AND YOSHIMI SAIT ¯O
+Similarly we have from (3.2)
+(3.4)
+
+∆∂jf3= (∂j∂1−i∂j∂2)(ig2)−∂j∂3(ig1),
+∆∂jf2= (∂j∂1+i∂j∂2)(ig3)−∂j∂3(ig4),
+∆∂jf1= (∂j∂1−i∂j∂2)(ig4)−∂j∂3(ig3).
+The equalities in (3.3) and (3.4) should be interpreted as equalities in th e space
+D′(R3) of distributions on R3.
+(II) Our first goal is to show that each distribution ∂jfkactually belongs to
+Lp(R3), wherej= 1,2,3 andk= 1,2,3,4, namely, for each jandkthere exists
+Fjk∈Lp(R3) such that
+(3.5) /a\}b∇acketle{t∂jfk, φ/a\}b∇acket∇i}ht=/integraldisplay
+R3Fjk(x)φ(x)dx
+for anyφ∈C∞
+0(R3), where the left-hand side is a bilinear form on D′(R3)×C∞
+0(R3).
+This will show that fbelongs to [ H1,p(R3)]4. We shall prove (3.5) for k= 4 and
+j= 1,2,3 since other cases can be proved in a similar manner. After that, fin ally
+we show that fbelongs to [ H1,p
+0(R3)]4to complete our proof.
+(III) Letqbe the conjugate of por letqsatisfyp−1+q−1= 1. We see from
+(3.3) that for φ∈C∞
+0(Ω),
+/a\}b∇acketle{t∂jf4,∆φ/a\}b∇acket∇i}ht=/a\}b∇acketle{t∆∂jf4,φ/a\}b∇acket∇i}ht=/a\}b∇acketle{t(∂j∂1+i∂j∂2)(ig1)−∂j∂3(ig2),φ/a\}b∇acket∇i}ht
+=/a\}b∇acketle{tig1,(∂j∂1+i∂j∂2)φ/a\}b∇acket∇i}ht−/a\}b∇acketle{tig2,∂j∂3φ/a\}b∇acket∇i}ht.
+Hence by Lemma 3.5 we have
+(3.6) |/a\}b∇acketle{t∂jf4,∆φ/a\}b∇acket∇i}ht| ≤ /ba∇dblg1/ba∇dblp/ba∇dbl(∂j∂1+i∂j∂2)φ/ba∇dblq+/ba∇dblg2/ba∇dblp/ba∇dbl∂j∂3φ/ba∇dblq
+≤(Cj1+Cj2)/ba∇dblg1/ba∇dblp/ba∇dbl∆φ/ba∇dblq+Cj3/ba∇dblg2/ba∇dblp/ba∇dbl∆φ/ba∇dblq
+= [(Cj1+Cj2)/ba∇dblg1/ba∇dblp+Cj3/ba∇dblg2/ba∇dblp]/ba∇dbl∆φ/ba∇dblq.
+Since ∆C∞
+0(R3)) is dense in Lq(R3) as has been shown in Lemma 3.2, the inequality
+(3.6) is extended uniquely to a continuous linear form on Lq(R3). SinceLp(R3) is
+the dual space of Lq(R3), there exists a function Fj4∈Lp(R3) such that /a\}b∇acketle{t∂jf4, ψ/a\}b∇acket∇i}ht=/integraltext
+R3Fj4(x)ψ(x)dx, ψ∈Lq(R3), which implies (3.5) with k= 4 andj= 1,2,3. In
+particular, we have also shown that
+(3.7) /ba∇dbl∂jfk/ba∇dblp≤C0{Σ4
+k=1/ba∇dblgk/ba∇dblp
+p}1/p=C0/ba∇dblg/ba∇dblp,
+with a positive constant C0for allj= 1,2,3 andk= 1,2,3,4.DIRAC–SOBOLEV SPACES 9
+(IV) Finally, since fisH1,p
+0(R3), by definition there exist a sequence {fn}∞
+n=1in
+C∞
+0(R3) withfn= (fn,1,fn,2,fn,3,fn,4) such that with gn= (gn,1,gn,2,gn,3,gn,4) :=
+(α·p)fn,
+/ba∇dblfn−f/ba∇dblp
+D,1,p,R3=/ba∇dblfn−f/ba∇dblp
+p+/ba∇dbl(α·p)(fn−f)/ba∇dblp
+p
+=/ba∇dblfn−f/ba∇dblp
+p+/ba∇dblgn−g/ba∇dblp
+p→0, n→ ∞.
+Since by the same argument used to get (3.7) we have /ba∇dbl∂j(fn,k−fk)/ba∇dblp≤C0/ba∇dblgn−g/ba∇dblp
+for allj= 1,2,3 andk= 1,2,3,4, it follows that /ba∇dblfn−f/ba∇dblp
+S,1,p,R3→0 asn→ ∞, so
+thatf∈[H1,p
+0(R3)]4. This completes the proof of Proposition 3.1. /square
+Proposition 3.5. Let1< p <∞. LetΩ⊂R3be an open set and J0,Ωbe given
+in(1.13). Then the map J0,Ωis ontoH1,p
+0(Ω). Further, the inverse map J−1
+0,Ωis well-
+defined as a bounded linear operator.
+Proof.(I) We have seen in Propositions 3.1 that [ H1,p
+0(R3)]4=H1,p
+0(R3) as sets and
+there exist positive constants C1andC2such that
+(3.8) C1/ba∇dblf/ba∇dblD,1,p≤ /ba∇dblf/ba∇dblS,1,p≤C2/ba∇dblf/ba∇dblD,1,p
+forf∈[H1,p
+0(R3)]4=H1,p
+0(R3).
+(II) Letf∈[H1,p
+0(Ω)]4. Then there exists a sequence {φn}∞
+n=1⊂[C∞
+0(Ω)]4such
+that
+(3.9) /ba∇dblf−φn/ba∇dblS,1,p,Ω→0 (n→ ∞).
+Since each φncan be naturally extended to be an element of [ C∞
+0(R3)]4by setting
+φn(x) = 0 forx∈R3\Ω, we have
+(3.10) /ba∇dblf−φn/ba∇dblS,1,p→0 (n→ ∞),
+wherefis also extended to be a function on R3by setting 0 outside Ω, and hence
+f∈[H1,p(R3)]4with support in the closure of Ω. Therefore f∈H1,p(R3) andf
+satisfies (3.8). Then, (3.10) is combined with (3.8) to yield
+(3.11) /ba∇dblf−φn/ba∇dblD,1,p→0 (n→ ∞),
+which implies, together with the fact that and φnhave support in Ω, that
+(3.12) /ba∇dblf−φn/ba∇dblD,1,p,Ω→0 (n→ ∞).
+Thus we have f∈H1,p
+0(Ω), and we obtain from (3.8)
+(3.13) C1/ba∇dblf/ba∇dblD,1,p,Ω≤ /ba∇dblf/ba∇dblS,1,p,Ω≤C2/ba∇dblf/ba∇dblD,1,p,Ω.10 TAKASHI ICHINOSE AND YOSHIMI SAIT ¯O
+(III) Let f∈H1,p
+0(Ω). Then, starting with f∈H1,p
+0(Ω) proceeding as in (II), we
+can show that f∈[H1,p
+0(Ω)]4and the estimates (3.13) are satisfied, which completes
+the proof. /square
+Proof of Theorem 1.3, (ii). Theorem 1.3, (ii) follows from Propositions 3.1 and 3.5.
+/square
+4.The case p=1
+The goalof thissection isto prove Theorem 1.3, (iii), that is, toprov e [H1,1(Ω)]4
+and [H1,1
+0(Ω)]4are proper subspaces of H1,1(Ω) andH1,1
+0(Ω), respectively. First, we
+are going to show, for Ω = R3, that [H1,1
+0(R3)]4= [H1,1(R3)]4is a proper subspace
+ofH1,1
+0(R3) =H1,1(R3) (Proposition 4.4). Then all other statements in Theorem 1.3,
+(iii) will follow from Proposition 4.4. In the following, when speaking of H1,1
+0(R3) or
+H1,1(R3), and [H1,1
+0(R3)]4or [H1,1(R3)]4, we shall use the latter, namely, H1,1(R3)
+and [H1,1(R3)]4. As easily seen, H1,p(R3) is also the subspace of [ Lp(R3)]4consisting
+of allf∈[Lp(R3)]4such that (α·p+β)f∈[Lp(R3)]4instead of ( α·p)f∈[Lp(R3)]4,
+whereβis the fourth Dirac matrix βgiven by (1.7).
+Lemma 4.1. The map
+(α·p)+β:H1,1(R3)∋f/ma√sto→(α·p+β)f∈[L1(R3)]4
+mapsH1,1(R3)one-to-one and onto [L1(R3)]4.
+Proof.(I) We define the Dirac operator H0= (α·p) +βas a linear operator in
+[L1(R3)]4with domain D(H0) =H1,1(R3). It is easy to see that H0is a closed oper-
+ator in [L1(R3)]4. Let the operator H= (α·p)+βbe defined as a pseudodifferential
+operator acting on [ S′(R3)]4, the dual space of [ S(R3)]4, with 4×4 matrix symbol
+(4.1) σH(ξ) =α·ξ+β=3/summationdisplay
+j=1ξjαj+β.
+Then the operator H0can beviewed asthe restriction of the operator HtoH1,1(R3).
+LetBbe a pseudodifferential operator acting on [ S′(R3)]4with symbol
+σB(ξ) = (1+ |ξ|2)−1σH(ξ) =σH(ξ)[(1+|ξ|2)−1I4].
+By the anti-commutative relation
+αjαk+αkαj= 2δjkI4(j,k= 1,2,3,4,α4=β),DIRAC–SOBOLEV SPACES 11
+whereI4is the 4×4 unit matrix, we see that
+σH(ξ)σB(ξ) =σB(ξ)σH(ξ) =I4,
+which implies that
+(4.2) HBf=BHf=f(f∈[S′(R3)]4),
+i.e.,the operator Bis the inverse operator of Hon [S′(R3)]4.
+(II) Note that
+(4.3) B=HB(1),
+where the symbol σB(1)(ξ) ofB(1)is given by σ(1)
+B(ξ) = (|ξ|2+1)−1I4. Letσ0(ξ) =
+(|ξ|2+1)−1and letσ0(p) be the pseudodifferential operator on S′(R3) with symbol
+σ0(ξ). The symbol σ0(ξ) is aC∞function on R3
+ξand bounded together with all their
+derivatives. Then, by noting that the integrand ( |ξ|2+1)−1(Fφ)(ξ) is a function in
+S(R3
+ξ) forφ∈ S(R3), we have
+(σ0(p)φ)(x) = lim
+R→∞(2π)−3/2/integraldisplay
+|ξ|<Reix·ξ(|ξ|2+1)−1(Fφ)(ξ)dξ
+= (2π)−3lim
+R→∞/integraldisplay
+R3/braceleftBig/integraldisplay
+|ξ|<Rei(x−y)·ξ(|ξ|2+1)−1dξ/bracerightBig
+φ(y)dy.
+= (2π)−3lim
+R→∞/integraldisplay
+R3/braceleftBig/integraldisplay
+|ξ|<Reiy·ξ(|ξ|2+1)−1dξ/bracerightBig
+φ(x−y)dy.
+forφ∈ S(R3). Since ( |ξ|2+ 1)−1∈L2(R3
+ξ), the integral/integraltext
+|ξ|<Reiy·ξ(|ξ|2+ 1)−1dξ
+converges in L2(R3
+y) asR→ ∞. At the same time it is known that the limit
+lim
+R→∞(2π)−3/integraldisplay
+|ξ|<Reiy·ξ(|ξ|2+1)−1dξ=G(y)
+exist fory/\e}atio\slash= 0 with
+G(y) =e−|y|
+4π|y|,
+which is the Green function of the operator 1 −∆. Thus we have
+(σ0(p)φ)(x) = (G∗φ)(x) (φ∈ S(R3)),
+whereG∗φdenotes the convolution of Gandφ, which implies that
+(4.4)σ(1)
+B(p)φ(x) = (G∗φ)(x) =t((G∗φ1)(x),(G∗φ2)(x),(G∗φ3)(x),(G∗φ4)(x))
+forφ=t(φ1,φ2,φ3,φ4)∈[S(R3)]4.12 TAKASHI ICHINOSE AND YOSHIMI SAIT ¯O
+(III) LetBbe the pseudodifferential operator as in (I). It follows from (4.4)
+that
+(4.5) Bφ(x) =σH(p)σ(1)
+B(p)φ(x) =/parenleftBig3/summationdisplay
+j=1−iαj∂j+β/parenrightBig
+(G∗φ)(x)
+forφ(x)∈[S(R3)]4, where∂j=∂/∂xj. Define a 4 ×4 matrix-valued function
+K(x) = (Kkℓ(x))1≤k,ℓ≤4onR3by
+K(x) =/parenleftBig3/summationdisplay
+j=1−iαj∂j+β/parenrightBig
+G(x) (G(x) =t(G(x),G(x),G(x),G(x)).
+Therefore we have
+(4.6) K(x) =1
+4π/bracketleftBig3/summationdisplay
+j=1iαj/parenleftBigxj
+|x|3+xj
+|x|2/parenrightBig
++β1
+|x|/bracketrightBig
+e−|x|,
+and each element Kkℓ(x) ofK(x) belongs to L1(R3). Thus the k-th component
+(Bφ)k,k= 1,2,3,4, ofBφis expressed as
+(Bφ)k(x) =4/summationdisplay
+ℓ=1(Kkℓ∗φℓ)(x),
+which allows us to apply Young’s inequality to see that
+/ba∇dblBφ/ba∇dbl1≤C/ba∇dblφ/ba∇dbl1(φ∈[S(R3)]4)
+with a positive constant C. Therefore Brestricted on [ S(R3)]4is uniquely extended
+toa bounded linear operatoron[ L1(R3)]4which will bedenoted by B0. The operator
+B0is actually the restriction of Bto [L1(R3)]4.
+(IV)Let g∈[L1(R3)]4. Let{gm}∞
+m=1⊂[S(R3)]4beasequence such that gm→g
+in [L1(R3)]4asm→ ∞. It follows from (4.2) that
+H0B0gm= (α·p+β)B0gm=gm(m= 1,2,···).
+Therefore, recalling that B0is a bounded operator on [ L1(R3)]4, we have
+(4.7)/braceleftBigg
+B0gm→B0g,
+(α·p+β)B0gm=gm→g
+in [L1(R3)]4asm→ ∞. Since the operator H0=α·p+βdefined on H1,1(R3) is
+a closed operator, we see from (4.7) that B0g∈H1,1(R3) andH0B0g=g, which
+implies that H0=α·p+βis onto [L1(R3)]4.DIRAC–SOBOLEV SPACES 13
+(V) Suppose that f∈H1,1(R3) such that H0f= (α·p+β)f= 0. Let {fm}∞
+m=1⊂
+[C∞
+0(R3)]4such that fm→finH1,1(R3) asm→ ∞. Thus we have
+(4.8) fm→f,(α·p+β)fm→(α·p+β)fin [L1(R3)]4.
+On the other hand, we have from (4.2)
+(4.9) B0(α·p+β)fm= (α·p+β)B0fm=fm(m= 1,2,···).
+Lettingm→ ∞in (4.9), noting that B0is a bounded operator and using (4.8), we
+see that
+0 =B0(α·p+β)f= lim
+m→∞(α·p+β)B0fm= lim
+m→∞fm=f
+in[L1(R3)]4,whichimplies that H0isone-to-one.Thiscompletes theproofofLemma
+4.1. /square
+Remark 4.2. As has been seen, the key element of the proof of the above Lemma
+4.1 is to show that the pseudodifferential operator Bgiven by (4.3) is a bounded
+linear operator on [ L1(R3)]4. Actually it can be shown that Bis a bounded linear
+operator on [ Lp(R3)]4for 1≤p <∞. Thus we can prove that Lemma 4.1 holds
+for any 1 ≤p <∞. In fact, for 1 < p <∞, a theorem in Fefferman[3] (Theorem,
+a), p.414) can be applied to show that Bis a bounded linear operator on [ Lp(Rn)]4
+withn= 3. Letσ(x,p) be a pseudodifferential operator in Rnwhose symbol σ(x,ξ)
+belongs to the H¨ ormander class S−b
+1−a,δ(Rn) with 0≤δ <1−a<1. Then it follows
+from the above theorem by Fefferman that σ(x,p) is a bounded operator on Lp(Rn)
+ifb<na/2 and
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
+p−1
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤b
+n/bracketleftBign
+2+λ
+b+λ/bracketrightBig /parenleftBig
+λ=na
+2−b
+1−a/parenrightBig
+.
+By takingn= 3, b= 1, δ= 0, the above two condition becomes
+(4.10) 3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
+p−1
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1
+a<3
+2.
+Forp >1 there exists a∈(0,1) which satisfies (4.10). For p= 1, however, there
+is noawhich satisfies (4.10) since both sides of (4.10) become 3/2. Indeed , our
+pseudodifferential operator Bhas symbol σB(ξ) belonging to the H¨ ormander class
+S−1
+1−a,0(R3). To prove Lemma 4.1, which is the case p= 1, we have discussed the
+integral kernel of the Dirac operator.
+To proceed, we need some facts on the local Hardy space h1(R3), which is
+introduced in Goldberg[5] in connection with the Hardy space H1(R3). The Hardy14 TAKASHI ICHINOSE AND YOSHIMI SAIT ¯O
+space is (see e.g. Fefferman-Stein [4]) the proper subspace of L1(R3) consisting of
+the functions f∈L1(R3) such that Rjf∈L1(R3) forj= 1,2,3, whereRj:=
+∂j·(−∆)−1/2are the Riesz transforms, having symbols iξj/|ξ|. Letϕbe a fixed
+function in the Schwartz space S(R3) such that ϕ= 1 in a neighborhood of the
+origin. By definition a distribution fbelongs toh1(R3) if and only if f∈L1(R3) and
+rjf∈L1(R3) forj= 1,2,3, whererj, j= 1,2,3, are pseudodifferential operators
+with symbol σrj(ξ) = (1−ϕ(ξ))(iξj/|ξ|) ([5], Theorem 2 (p.33)). The definition is
+independent of the choice of ϕ. It is a Banach space with norm /ba∇dblf/ba∇dblh1=/ba∇dblf/ba∇dblL1+/summationtext3
+j=1/ba∇dblrjf/ba∇dblL1. The space h1(R3) is a proper subspace of L1(R3), which is strictly
+larger than the Hardy space H1(R3) (see e.g. [5], p.33, just after Theorem 3).
+Now, we are introducing the following operator
+(4.11) r′
+j=∂j(1−∆)−1/2=∂j
+(−∆)1/2(−∆)1/2
+(1−∆)1/2=Rj·(−∆)1/2
+(1−∆)1/2,
+where we note that the pseudodifferential operator ( −∆)1/2/(1−∆)1/2is a bounded
+operator on L1(R3) (see Stein[6], p.133, Eq.(31)).
+The proof of the lemma below was inspired by the proof of [5], Theorem 2
+(p.33).
+Lemma 4.3. A distribution finR3belongs toh1(R3)if and only if f∈L1(R3)
+andr′
+jf∈L1(R3)forj= 1,2,3.
+Proof.(I) Itis sufficient to showthat rj−r′
+j,j= 1,2,3,areboundedlinear operators
+onL1(R3) (or, more exactly, the pseudodifferential operator rj−r′
+jdefined on S(R3)
+can be uniquely extended to a bounded linear operator on L1(R3)). Note that the
+operatorsrjandr′
+jhave symbols
+(4.12) σrj(ξ) =(1−ϕ(ξ))iξj
+|ξ|, σr′
+j(ξ) =iξj
+(1+|ξ|2)1/2,
+and both symbols are C∞functions in R3
+ξand bounded together with all their
+derivatives, and we have
+σrj(ξ)−σr′
+j(ξ) =(1−ϕ(ξ))iξj
+|ξ|/parenleftBig
+1−|ξ|
+(1+|ξ|2)1/2/parenrightBig
+−ϕ(ξ)iξj
+(1+|ξ|2)1/2
+=:σ1j(ξ)+σ2j(ξ).
+As in the proof of Lemma 4.1, we are going to show that, for each j= 1,2,3, the
+pseudodifferential operator with symbols σ1jandσ2jhave integral kernels belonging
+toL1(R3), inother words, thattheir inverse Fouriertransforms Fσ1j(x) andFσ2j(x)DIRAC–SOBOLEV SPACES 15
+belong toL1(R3), where Fis given by
+Fφ(x) = (2π)−3/2/integraldisplay
+R3eix·ξφ(ξ)dξ.
+It is easy to see that Fσ2j∈L1(R3), becauseσ2jbelongs to S(R3), so that Fσ2j
+belongs to S(R3) and hence it belongs to L1(R3). In the rest of the proof we are
+going to show that Fσ1j∈L1(R3).
+(II) By definition we have
+σ1j(ξ) =i(1−ϕ(ξ))ξj
+|ξ|(1+|ξ|2)1/2(|ξ|+(1+|ξ|2)1/2),
+and hence, σ1j(ξ) =O(|ξ|−2) as|ξ| → ∞. Thus, by noting that 1 −ϕ(ξ) is 0 around
+the originξ= 0, we see that σ1j∈L2(R3). Therefore Ij(x) = (Fσ1j)(x) exists as a
+function in L2(R3
+x). Letρ(t) be a real-valued C∞function on [0 ,∞) such that
+ρ(t) = 1 (0 ≤t≤1),= 0 (t≥2).
+Then, since σ1j(ξ)ρ(ǫ|ξ|),ǫ>0, converges to σ1j(ξ) inL2(R3
+ξ) asǫ↓0, we have, by
+setting
+Ij(x,ǫ) :=F(σ1j(ξ)ρ(ǫ|ξ|)),
+Ij(x,ǫ) converges to Ij(x) inL2(R3
+x) asǫ↓0, and hence there exists a decreasing
+sequence
+1≥ǫ2>ǫ2>···>ǫm>→0
+such thatIj(ǫm,x)→Ij(x) a.e.xasm→ ∞. For the sake of the simplicity of
+notations, we shall use ǫ≤1 instead of ǫm.
+(III) Letα= (α1,α2,α3) be a multi-index. Then we have
+(4.13) |∂α
+ξσ1j(ξ)| ≤Cj,α(1+|ξ|)−2−|α|(ξ∈R3
+ξ)
+with a constant Cj,α>0, where we should note that 1 −ϕ(ξ) is bounded and
+∂α
+ξ(1−ϕ(ξ)) =−∂α
+ξϕ(ξ)∈ S(R3
+ξ)
+forα/\e}atio\slash= 0. Letℓbe a positive integer and k= 1,2,3. Then, by integration by parts,
+(4.14) (2 π)3/2xℓ
+kIj(x,ǫ) =/integraldisplay
+R3/braceleftbig
+(−i∂ℓ
+ξk)eix·ξ/bracerightbig
+σ1j(ξ)ρ(ǫ|ξ|)dξ
+= (−i)ℓ(−1)ℓ/integraldisplay
+R3eix·ξ∂ℓ
+ξk/braceleftbig
+σ1j(ξ)ρ(ǫ|ξ|)/bracerightbig
+dξ.16 TAKASHI ICHINOSE AND YOSHIMI SAIT ¯O
+Here, by the Leibniz formula, we have
+∂ℓ
+ξk/braceleftbig
+σ1j(ξ)ρ(ǫ|ξ|)/bracerightbig
+=/parenleftbig
+∂ℓ
+ξkσ1j(ξ)/parenrightbig
+ρ(ǫ|ξ|)+ℓ/summationdisplay
+m=1ℓCm/parenleftbig
+∂ℓ−m
+ξkσ1j(ξ)/parenrightbig/parenleftbig
+∂m
+ξkρ(ǫ|ξ|)/parenrightbig
+=:J0(ξ,ǫ)+J1(ξ,ǫ)
+withℓCm=ℓ/(m!(ℓ−m)!). Form= 1,2,···,ℓ, we have
+ξ∈supp(∂m
+ξkρ(ǫ|ξ|)) =⇒1≤ǫ|ξ| ≤2 =⇒ǫ≤2
+|ξ|,
+where supp( f) denotes thesupport of f. Thus we can replace ǫin∂m
+ξkρ(ǫ|ξ|)by 2|ξ|−1
+when we evaluate |∂m
+ξkρ(ǫ|ξ|)|. Therefore it follows that
+(4.15) |∂m
+ξkρ(ǫ|ξ|)| ≤c(1+|ξ|)−mχǫ(ξ),
+wherec=cj,k,mis a positive constant and χǫ(ξ) is the characteristic function of
+the setAǫ={ξ:ǫ−1≤ |ξ| ≤2ǫ−1}. Since it is supposed that ǫ≤1, we have
+Aǫ⊂ {ξ:|ξ| ≥1}. The inequalities (4.13) and (4.15) are combined to give
+|J1(ξ,ǫ)| ≤C(1+|ξ|)−2−ℓχǫ(ξ) (ξ∈R3
+ξ)
+with positive constant C=Cj,k,ℓ. Letℓ≥2. Then it is seen that |J1(ξ,ǫ)|is
+dominated by C(1+|ξ|)−2−ℓ, which is in L1(R3
+ξ), andJ1(ξ,ǫ)→0 for eachξ∈R3
+ξ
+asǫ→0, and hence, by the Lebesgue convergence theorem, we have
+/integraldisplay
+R3eix·ξJ1(ξ,ǫ)dξ→0 (ǫ→0).
+Similarly, since |J0(ξ,ǫ)|is dominated by |∂ℓ
+ξkσ1j(ξ)|which is in L1(R3
+ξ), andJ0(ξ,ǫ)
+converges to ∂ℓ
+ξkσ1j(ξ) for eachξ∈R3
+ξasǫ→0, we have
+/integraldisplay
+R3eix·ξJ0(ξ,ǫ)dξ→/integraldisplay
+R3eix·ξ∂ℓ
+ξkσ1j(ξ)dξ(ǫ→0).
+Therefore, by letting ǫ→0 in (4.14), we obtain
+(4.16) xℓ
+kIj(x) =xℓ
+k(Fσ1j)(x) =iℓ(2π)−3/2/integraldisplay
+R3eix·ξ∂ℓ
+ξkσ1j(ξ)dξ
+a.e.x∈R3forℓ≥2andj,k= 1,2,3.Heretheright-handside isuniformlybounded
+forx∈R3. Thus, by considering the case ℓ= 2 andℓ= 4, it follows that
+|(Fσ1j)(x)| ≤Cjmin(|x|−2,|x|−4) (j= 1,2,3)
+with a positive constant Cj, which implies that Fσ1j∈L1(R3). This completes the
+proof of Lemma 4.3. /squareDIRAC–SOBOLEV SPACES 17
+Now we are going to prove that [ H1,1(R3)]4is a proper subspace of H1,1(R3),
+which is the most crucial part of Theorem 1.3, (iii) in the following strat egy: Let
+g∈[L1(R3)]4\[h1(R3)]4, whereh1(R3) is the local Hardy space which can be
+defined, by Lemma 4.3, as the space of all distributions fsuch thatf∈L1(R3)
+andr′
+jf∈L1(R3) forj= 1,2,3, wherer′
+jis given by (4.11). Set f= (α·p+β)−1g.
+Then, by using Lemma 4.1, we have f∈H1,1(R3). Then we shall be able to show
+thatf/∈[H1,1(R3)]4.
+Proposition 4.4. H1,1(R3)is strictly larger than [H1,1(R3)]4.
+Proof.(I) It follows from Lemma 4.1 that, for every g∈[L1(R3)]4there exists a
+uniquef∈H1,1(R3) such that g= (α·p+β)f. The equation g= (α·p+β)fcan
+be written as
+
+g1=−i(∂1−i∂2)f4−i∂3f3+f1,
+g2=−i(∂1+i∂2)f3+i∂3f4+f2,
+g3=−i(∂1−i∂2)f2−i∂3f1−f3,
+g4=−i(∂1+i∂2)f1+i∂3f2−f4.
+Solving the above equation for f1,f2,f3andf4, we obtain
+(4.17)
+
+(1−∆)f1=−i(∂1−i∂2)g4−i∂3g3+g1,
+(1−∆)f2=−i(∂1+i∂2)g3−i∂3g4+g2,
+(1−∆)f3=−i(∂1−i∂2)g2−i∂3g1−g3,
+(1−∆)f4=−i(∂1+i∂2)g1−i∂3g2−g4,
+where∂j=∂/∂xj. Here each equation in (4.17) should be viewed as equations in
+S′(R3). As has been shown in the proof of Lemma 4.1, the differential oper ator 1−∆
+has the inverse (1 −∆)−1as a pseudodifferential operator with symbol (1+ |ξ|2)−1,
+and hence, by applying (1 −∆)−1and∂jto each of the equations in (4.17), it follows
+that
+(4.18)
+
+∂jf1=−i(∂j∂1−i∂j∂2)(1−∆)−1g4−i∂j∂3(1−∆)−1g3+∂j(1−∆)−1g1,
+∂jf2=−i(∂j∂1+i∂j∂2)(1−∆)−1g3+i∂j∂3(1−∆)−1g4+∂j(1−∆)−1g2,
+∂jf3=−i(∂j∂1−i∂j∂2)(1−∆)−1g2−i∂j∂3(1−∆)−1g1−∂j(1−∆)−1g3,
+∂jf4=−i(∂j∂1+i∂j∂2)(1−∆)−1g1+i∂j∂3(1−∆)−1g2−∂j(1−∆)−1g4.
+(II) By Lemma 4.3 we can choose g0∈L1(R3)\h1(R3) such that r′
+3g0=
+∂3(1−∆)−1/2g0/∈L1(R3). Then define g=t(g1,g2,g3,g4)∈[L1(R3)]4by
+g1(x) =g3(x) =g4(x) = 0 and g2(x) =g0(x).18 TAKASHI ICHINOSE AND YOSHIMI SAIT ¯O
+Then we have from (4.18)
+(4.19) ∂jf4=i∂j∂3(1−∆)−1g2(j= 1,2,3).
+Sincer′
+3g2/∈L1(R3),wehavenecessarily r′
+3g2/∈h1.Then/parenleftbig
+r′
+1(r′
+3g2),r′
+2(r′
+3g2),r′
+3(r′
+3g2)/parenrightbig
+does not belong to [ L1(R3)]3. It follows from (4.11) that the symbol sj3(ξ) ofr′
+jr′
+3is
+given by
+sj3(ξ) =iξj
+(1+|ξ|2)1/2iξ3
+(1+|ξ|2)1/2= (iξj)(iξ3)(1+|ξ|2)−1,
+and hence by using (4.11) again, we see that
+(4.20) r′
+jr′
+3g2= (∂j(1−∆)−1/2)(∂3(1−∆)−1/2)g2=∂j∂3(1−∆)−1g2
+forj= 1,2,3. Thus we have from (4.19) and (4.20)
+(∂1f4,∂2f4,∂3f4) =i(r′
+1r′
+3g2,r′
+2r′
+3g2,r′
+3r′
+3g2)/∈[L1(R3)]3,
+which implies that f/∈[H1,1(R3)]4. This completes the proof of Proposition 4.4. /square
+Proposition 4.5. LetΩbe an open subset of R3. Then
+(i) [H1,1
+0(Ω)]4is a proper subspace of H1,1
+0(Ω).
+(ii) [H1,1(Ω)]4is a proper subspace of H1,1(Ω).
+Proof.(I) We are going to show the norms /ba∇dblf/ba∇dblS,1,1,Ωof [H1,1
+0(Ω)]4and/ba∇dblf/ba∇dblD,1,1,Ωof
+H1,1
+0(Ω) are not equivalent on [ C∞
+0(Ω)]4(see (1.9) and (1.11) for the definition of
+these norms). To this end we use Proposition 4.4. Without loss of gen erality, we may
+assume that Ω contains the unit ball {x:|x| ≤1}with center at the origin. As in
+(1.6) (with p= 1), we denote the norm of [ L1(Ω)]4by/ba∇dblf/ba∇dbl1,Ω,i.e.,
+/ba∇dblf/ba∇dbl1,Ω=/integraldisplay
+Ω4/summationdisplay
+j=1|fj(x)|dx(f(x) =t(f1(x),f2(x),f3(x),f4(x)).
+By Proposition 4.4 and the fact that [ C∞
+0(R3)]4is dense in both [ H1,1(R3)]4and
+H1,1(R3) (Theorem 1.3, (i)), the norms /ba∇dblf/ba∇dblS,1,1,R3and/ba∇dblf/ba∇dblD,1,1,R3are not equivalent
+on[C∞
+0(R3)]4.Therefore, bytaking noteofProposition2.2,(2.2)withΩ = R3,which
+says that the norm /ba∇dblf/ba∇dblD,1,1,R3is dominated by the norm /ba∇dblf/ba∇dblS,1,1,R3, there exists a
+sequence {fn}∞
+n=1of functions in [ C∞
+0(R3)]4such that fn/\e}atio\slash= 0 and /ba∇dblfn/ba∇dblS,1,1,R3≥
+(n+1)/ba∇dblfn/ba∇dblD,1,1,R3, or
+(4.21) /ba∇dblfn/ba∇dbl1,R3+/ba∇dbl∇fn/ba∇dbl1,R3≥(n+1)[/ba∇dblfn/ba∇dbl1,R3+/ba∇dbl(α·p)fn/ba∇dbl1,R3]
+for eachn= 1,2,···. Eachfnhas support in some ball {x:|x| ≤Rn}with radius
+Rn>0 and center at the origin. We may assume with no loss of generality th atDIRAC–SOBOLEV SPACES 19
+Rn≥1 (n= 1,2,···). Putgn(x) =R3
+nfn(Rnx). Thengnhas support in the unit ball
+{x:|x| ≤1}, and hence in Ω, so that {gn} ⊂[C∞
+0(Ω)]4for eachn. We have
+/ba∇dblgn/ba∇dbl1,Ω=/ba∇dblfn/ba∇dbl1,R3and/ba∇dbl∂jgn/ba∇dbl1,Ω=Rn/ba∇dbl∂jfn/ba∇dbl1,R3
+forj= 1,2,3. Then by (4.21) we have
+/ba∇dblgn/ba∇dbl1,Ω+1
+Rn/ba∇dbl∇gn/ba∇dbl1,Ω≥(n+1)[/ba∇dblgn/ba∇dbl1,Ω+1
+Rn/ba∇dbl(α·p)gn/ba∇dbl1,Ω],
+and hence, by noting Rn≥1
+1
+Rn/ba∇dbl∇gn/ba∇dbl1,Ω≥n/ba∇dblgn/ba∇dbl1,Ω+n+1
+Rn/ba∇dbl(α·p)gn/ba∇dbl1,Ω
+≥n
+Rn[/ba∇dblgn/ba∇dbl1,Ω+/ba∇dbl(α·p)gn/ba∇dbl1,Ω/ba∇dbl].
+Therefore
+/ba∇dbl∇gn/ba∇dbl1,Ω≥n[/ba∇dblgn/ba∇dbl1,Ω+/ba∇dbl(α·p)gn/ba∇dbl1,Ω],
+which implies that /ba∇dblgn/ba∇dblS,1,1,Ω≥n/ba∇dblgn/ba∇dblD,1,1,Ωforn= 1,2,···. This proves that the
+norms/ba∇dblf/ba∇dblS,1,1,Ωand/ba∇dblf/ba∇dblD,1,1Ωare not equivalent on [ C∞
+0(Ω)]4, showing (i).
+(II) As we have seen in the proof of (i), the norms of [ H1,1
+0(Ω)]4andH1,1
+0(Ω) are
+not equivalent. In fact, there exists a sequence {gn} ⊂[C∞
+0(Ω)]4such that
+/ba∇dblgn/ba∇dblS,1,1,Ω≥n/ba∇dblgn/ba∇dblD,1,1,Ω, n= 1,2, ....
+Sinceclearlythissequence isalso containedin[ H1,1(Ω)]4,thisimpliesthatthenorms
+of [H1,1(Ω)]4andH1,1(Ω) are not equivalent. This shows (ii), completing the proof
+of Proposition 4.5. /square
+Proof of Theorem 1.3, (iii). Theorem 1.3, (iii) follows from Propositions 4.5. /square
+Acknowledgement. Being deeply grieved that our respected friend, Professor
+Tetsuro Miyakawa, suddenly passed away in February 2009, we gra tefully remember
+nice useful discussion with him about the Riesz transform at the ear ly stage of the
+present work. Thanks are also due to Professor Shuichi Sato for valuable discussion
+about the Hardy space and local Hardy space. The research of T.I . is supported in
+part by JSPS Grant-in-Aid for Sientific Research No. 17654030. We wish to thank
+the referee for his kindly pointing out a flaw in an assertion in Theorem 1.3(ii) of
+our original version of the paper.20 TAKASHI ICHINOSE AND YOSHIMI SAIT ¯O
+References
+[1] R. Adams and J. Fournier, Sobolev Spaces, 2nd edition , Academic Press, 2003.
+[2] A. Balinsky, W. D. Evans and Y. Sait¯ o, Dirac–Sobolev inequalities and estimates for the zero
+modes of massless Dirac operators , J. Math. Physics 49, (2008), 043524-1 – 043524-10.
+[3] C. Fefferman, Lpbounds for pseudo-differential operators , Israel J. Math. 14 (1973), 413–417.
+[4] C. Fefferman and E. Stein, Hpspaces of several variables , Acta Math. 129(1972), no. 3-4,
+137–193.
+[5] D. Goldberg, A local version of real Hardy spaces , Duke Math. J., 46(1979), 27–42.
+[6] E. M. Stein, Singular Integrals and Differntiability Propoerties of Fun ctions, Princeton Uni-
+versity Press, 1970.
+Department of Mathematics, Kanazawa University, Kanazawa , 920-1192, Japan
+E-mail address :ichinose@kenroku.kanazawa-u.ac.jp
+Department of Mathematics, University of Alabama at Birmin gham, Birming-
+ham, AL 35294, USA
+E-mail address :saito@math.uab.edu
\ No newline at end of file
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+arXiv:1001.0276v1  [astro-ph.CO]  2 Jan 2010Accepted by ApJ Letters 24 December 2009
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+THE GALAXY-WIDE DISTRIBUTIONS OF MEAN ELECTRON
+DENSITY IN THE HII REGIONS OF M51 AND NGC 4449.
+Leonel Guti ´errez1,2and John E. Beckman2,3
+Accepted by ApJ Letters 24 December 2009
+ABSTRACT
+Using ACS-HST images to yield continuum subtracted photometric ma ps in Hαof the Sbc galaxy
+M51 and the dwarf irregular galaxy NGC 4449, we produced extensiv e (over 2000 regions for M51,
+over 200 regions for NGC4449) catalogues of parameters of their Hiiregions: their H αluminosities,
+equivalent radii and coordinates with respect to the galaxy center s. From these data we derived, for
+each region, its mean luminosity weighted electron density, /angbracketleftne/angbracketright, determined from the H αluminosity
+and the radius, R, of the region. Plotting these densities against the radii of the reg ions we find
+excellent fits for /angbracketleftne/angbracketrightvarying as R−1/2. This relatively simple relation has not, as far as we know,
+been predictedfrommodelsofH iiregionstructure, andshouldbe useful inconstrainingfuture mod els.
+Plotting the densities against the galactocentric radii, r, of the regions we find good exponential fits,
+with scalelengths ofcloseto 10kpc for both galaxies. These valuesa recomparableto the scalelengths
+of the Hicolumn densities for both galaxies, although their optical structur es, related to their stellar
+components are very different. This result indicates that to a first approximation the H iiregions can
+be considered in pressure equilibrium with their surroundings. We also plot the electron density of
+the Hiiregions across the spiral arms of M51, showing an envelope which pe aks along the ridge lines
+of the arms.
+Subject headings: ISM: general — HII regions — galaxies: structure
+1.INTRODUCTION
+Electron densities for H iiregions can be measured in
+two different ways, which respectively yield two differ-
+ent ranges of values. Density sensitive emission line
+ratios, such as the [SII] λλ6731/6716 ˚A doublet ratio,
+or the [O II]λλ3729/3726 ˚A doublet ratio give values in
+therangeofhundreds(Zaritsky et al.1994;McCall et al.
+1985). However measurements using H αsurface bright-
+ness to derive the emission measure, together with the
+estimatedradiusofanH iiregion,givevaluesinthe range
+1-10 (see e.g. Rozas et al. 1996). This type of dichotomy
+was recognized long ago by Osterbrock & Flather (1959)
+who used the emission line ratio of the [O II] doublet,
+λλ3729/3726 ˚A, as their version of the first technique,
+and cm wavelength radio emission (from the literature)
+to estimate the emission measure, i.e. for the second
+technique. They reached the conclusion that the best
+way to reconcile the differences, of more than an order of
+magnitude between the two techniques, is to assume that
+the emission nebula is clumpy, with dense clouds embed-
+dedinamuchmoretenuoussubstrate. Thelineratiosare
+strongly weighted by density, and yield values reflecting
+the densities of the dense clumps, while the electron den-
+sity derived using the emission measure and the overall
+size of the H iiregion is a geometrical average, whose bal-
+ancebetweenthe contributionofthe clumps andthesub-
+strate depends on the fractional volume occupied by the
+former. In models for H iiregions based on recognition
+of this inhomogeneity, the fractional volume occupied by
+leonel@astrosen.unam.mx, jeb@iac.es
+1Universidad Nacional Aut´ onoma de M´ exico, Instituto de As -
+tronom´ ıa, Ensenada, B. C. M´ exico
+2Instituto de Astrof´ ısica de Canarias, C/ Via L´ actea s/n,
+38200 La Laguna, Tenerife, Spain
+3Consejo Superior de Investigaciones Cient´ ıficas, Spainthe clumps is termed the filling factor, and a first or-
+der approximationcan be made that the interclump sub-
+strate makes a negligible contribution to the line emis-
+sion, so that the interclump volume can be considered to
+have negligible electron density (see Osterbrock (1989)
+for a standard treatment). In the present article we are
+using the measured luminosities of H iiregions in H αto
+compute an electron density using the second method,
+so our values are to be taken as the means produced by
+the second technique. We will use them to derive com-
+parative electron densities of selected H iiregions in M51
+and NGC 4449, to establish functional relations between
+them, the sizes of the H iiregions, and their positions
+within the discs of their respective galaxies, and hence
+to provide an overview of the global behavior of the elec-
+tron density within the discs of the two galaxies.
+2.THE OBSERVATIONAL DATA AND ITS
+TREATMENT
+M51 was observed in January 2005 with the ACS
+on HST using the broad band filters F435W (close to
+the standard B band) F555W (close to V) and F814W
+(close to I) and the narrow band filter F658N (H α)
+within observational program 10452 (P.I. S. Beckwith,
+Hubble Heritage Team). The data used here were
+corrected, calibrated, and combined into a mosaic of
+the galaxy by Mutchler et al. (2005). The images of
+NGC 4449 were also taken with the ACS, through the
+same filters as those used for M51, in November 2005,
+within program GO 10585 (P.I. A. Aloisi). The im-
+ages were taken at two different positions along the ma-
+jor axis of the galaxy, with four dithered pointings on
+each position (Annibali et al. 2008) to help cosmic ray
+removal. We combined them using MULTIDRIZZLE
+(Koekemoer et al. 2002), yielding overall exposure times2 Guti´ errez & Beckman
+of 3600s, 2400s, 2000s and 360s respectively for the four
+named filters. The total resulting field is sized 345 ×200
+arcsec2; we produced mosaics containing complete im-
+ages of the galaxy in all bands.
+For both galaxies standard correction and calibration
+procedures were used (bias, flat-field, dark current, and
+distortion corrections) using the ACS pipeline calibra-
+tion suite CALACS (Hack et al 2000). Use of the Mul-
+tidrizzle process allowed us to correct for detector de-
+fects and distortions, though we retained the original
+pixel scales (0.05 arcsec per pixel). After this process
+we converted the images from e-/s to flux in erg cm−2
+s−1˚A−1using the standard PHOTOFLAM conversion
+factor (Sirianni et al. 2005). We aligned the continuum
+and Hαimages and produced a continuum subtracted
+Hαimage, after deriving the factor of proportionality
+between the continuum in the on-band and off-band fil-
+ters, followingthe method ofB¨ oker et al.(1999). Amore
+detailed description of our image treatment can be found
+in Guti´ errez et al. (2010, in preparation). The method
+for measuring the H αflux of an H iiregionrequires defin-
+ing its boundaries in the presence of any local diffuse H α
+emission and any overlapping neighbour regions, using
+circular or rectangular defining apertures. The bound-
+ary of an HII region is defined as containing only pixels
+whose brightness is greater than 3 σ, measured on the
+background. For NGC 4449 the diffuse background sub-
+traction presented special difficulties due to its strength,
+and to perform this accurately we used a selective fil-
+ter, as defined in Guti´ errez et al. (2010, in preparation).
+Once a local background brightness was determined, it
+was multiplied by the number of pixels within the H ii
+region and subtracted from the flux directly measured
+within the region, to yield the measured flux from the
+region. To allow for the overlap of regions, if the bright-
+ness between the two maxima falls below 2/3ofthe value
+of the fainter maximum we considered them to be two
+separate regions, with the boundary in the depression of
+the brightness distribution. Where this criterion was not
+met we classified the region as single.
+To allow for the contribution of the [N II] doublet
+through the F658N filter, for M51 we used the mean
+of the observed ratios of these lines to H αfor 10 H ii
+regions in that galaxy by Bresolin et al. (2004), which
+show very limited variations from region to region, to
+calculate the factor by which the observed flux should
+be reduced, using the filter transmission curve and the
+observedredshift of M51. For NGC 4449we used the ob-
+served ratio from Kennicutt (1992). The factor for M51
+was 0.67, and the corresponding factor for NGC 4449 is
+0.86. We understand that radial decline in metallicity
+(especially in M51), will give rise to errors in using these
+blanket numbers for each object, but we can show that
+these errors are of second order (Guti´ errez et al. 2010,
+in preparation). Also, we estimated the possible system-
+atic error due to the presence of the [OIII] emisison line
+atλ5007˚A in the F555W filter and found this to be less
+than 1%.
+Finally, in deriving absolute fluxes we took the dis-
+tance to M51 as 8.39 Mpc, following Feldmeier et al.
+(1997) and the distance to NGC 4449 as 3.82 Mpc, fol-
+lowing Annibali et al. (2008). We measuredthe luminos-
+ity of each region as outlined above, and also, from thecontinuum- and background-subtracted H αimage, de-
+termined the area subtended by the image, and derived
+a value for an equivalent radius, obtained dividing the
+area byπand taking the square root. With these mea-
+surements we could determine a mean electron density
+for a each region, as described below. A catalogue of
+2657 regions of M51, and 273 regions of NGC4449 was
+produced, containing positions, measured fluxes and lu-
+minosities in H α, and equivalent radii (see Guti´ errez et
+al. 2010 for details).
+3.THE MEAN ELECTRON DENSITY OF THE H II
+REGIONS
+3.1.Electron density as a function of H II region radius
+We described in the introduction the inhomogeneity of
+Hiiregions, but with the set of observations described
+here we confine our attention to the mean electron den-
+sity (/angbracketleftne/angbracketright), and use the “Case B” formula (Osterbrock
+1989)
+/angbracketleftne/angbracketright=/radicalBigg
+2.2L(Hα)
+4
+3παBhνHαR3(1)
+to derive this, where L(H α) is the H αluminosity, αBis
+the case B recombination coefficient, hνHαis the energy
+of an Hαphoton, and Ris the equivalent radius of the
+Hiiregion. If we measure Rin pc and L(H α) in erg
+s−1, and taking a canonical value for the temperature of
+10000 K we can rewrite equation 1 as
+/angbracketleftne/angbracketright= 1.5×10−16/radicalbigg
+L(Hα)
+R3cm−3.(2)
+In Fig. 1 we show the mean electron densities of all the
+regions observed in M51 against their equivalent radii.
+There is considerable scatter, which we will investigate
+further below. A linear fit to the points gives the result:
+/angbracketleftne/angbracketright=28.9
+R0.56; (3)
+in qualitative terms, the larger regions have lower mean
+electron densities. The functional trend is quite well re-
+produced by the H iiregions in NGC 4449 as we can see
+in Fig. 2. With far fewer points, the best linear fit is
+given by
+/angbracketleftne/angbracketright=44.2
+R0.45. (4)
+These are only two galaxies, but the trends can at this
+stage be usefully summarized by the simple expression
+/angbracketleftne/angbracketright ∼1
+R0.5. (5)
+This relation is obeyed more closely for both galaxies
+if we correct the relationship between /angbracketleftne/angbracketrightandRfor
+the dependence of /angbracketleftne/angbracketrighton the galactocentric radius, r,
+as described in section 3.2 below. Making this correction
+givesadjusted valuesof-0.52,and-0.47fortheexponents
+in M51 and NGC 4449, respectively.Distributions of mean electron density in HII regions. 3
+Fig. 1.— Mean electron densities of all the H iiregions measured
+in M51 as a function of equivalent radius. In the lower panel t he
+densities have been corrected by a factor which takes into ac count
+their galactocentric distances (plotted in Fig. 3).
+3.2.Electron density as a function of galactocentric
+distance
+In the upper panel of Fig. 3 we have plotted the mean
+electron density, /angbracketleftne/angbracketright, against galactocentric radius, r,
+for M51, including all the regions, but distinguishing by
+point styles among those found in different large scale
+features of the galaxy: arms, interarm zones, and the
+central kpc. The outstanding feature of this plot is the
+single straight line fit (which is an exponential, as the
+electrondensityscaleislogarithmic), whichhastheform:
+/angbracketleftne/angbracketright=/angbracketleftne/angbracketrightoe−r/h, (6)
+where/angbracketleftne/angbracketrighto, with value 10 ±1 cm−3is a central value of
+electron density and his a scale length, which takes the
+value 10±1.0 kpc. We have limited the fit to galactocen-
+tric radiiwithin r= 10.4kpc, because beyond this radius
+there is an abrupt local rise and fall in /angbracketleftne/angbracketright, attributable
+to the effect of the interacting neighbour galaxy NGC
+5195. We can make separate estimates for the constants
+/angbracketleftne/angbracketrightoandhin the arms and in the interarm zones. For
+the arms this gives us /angbracketleftne/angbracketrighto= 10 cm−3andh= 11 kpc,
+while for the interarm zones we find /angbracketleftne/angbracketrighto= 8 cm−3, and
+h= 11 kpc .
+Since we know that in the upper panel of Fig. 3 much
+ofthe apparentscatterisdue tothe (inverse)dependence
+of/angbracketleftne/angbracketrighton the radii of the individual H iiregions, we next
+opted to apply a normalized correction factor for this ef-
+fect to all the data, and replot the figure, yielding the
+central panel of Fig. 3, in which the (exponential) linear
+fit is clearly much improved. To make the fit we have
+chosen to leave out the zones from 1.4 to 4.6 kpc from
+the center, and from 10.4 to 11.5 kpc from the center,
+and now find values of /angbracketleftne/angbracketrightoandhof 12 cm−3and 9 kpc
+respectively. It is notable that the scale length found
+for the neutral atomic hydrogen component of M51 byFig. 2.— Mean electron densities of the H iregions measured in
+NGC 4449 as a function of equivalent radius. In the lower pane l the
+densities have been corrected by a factor which takes into ac count
+their galactocentric distances (plotted in Fig. 4).
+Tilanus & Allen (1991) is 9.1 kpc, on which we will com-
+ment further below. We should point out that the zone
+between the center of the galaxy and a radius of 1.4 kpc
+does fit the exponential for the disc as a whole.
+To take a look at the global behavior we have taken
+averages of all the regions within annuli of width 200 pc
+and have plotted, in the lower panel of Fig. 3 these val-
+ues against galactocentric radius. This shows the overall
+trend in /angbracketleftne/angbracketright, and we can pick out the general decline
+with radius, but note a plateau between 1.4 kpc and 4.6
+kpc. If we omit the points in this interval from those
+entering the exponential fit, we obtain values for /angbracketleftne/angbracketrighto
+andh, of 11 cm−3and 9 kpc respectively.
+To compute the effect of the temperature gradient
+on the determination of /angbracketleftne/angbracketrightand of its gradient, we
+used data from Bresolin et al. (2004) giving the O abun-
+dance gradient, and the widely used CLOUDY model
+(Ferland et al. 1998) to derive the effective temperature
+gradient, which we then incorporated into the expression
+/angbracketleftne/angbracketright= 2.3×10−18/radicalbigg
+L(Hα)T0.91
+R3cm−3,(7)
+and recalculated the electron densities. This introduced
+a change in the exponent of the relation between /angbracketleftne/angbracketright
+and radius from -0.56 to -0.55, and in the scale length
+from 9.2 to 9.9 kpc. These are the values represented in
+Fig. 3.
+Here we should point out the similarity of the lower
+panel of Fig. 3 with the radial plot of the azimuthally
+averaged H icolumn density in Tilanus & Allen (1991).
+General expressions for /angbracketleftne/angbracketrightare then:4 Guti´ errez & Beckman
+Fig. 3.— Mean electron density ( /angbracketleftne/angbracketright) vs galactocentric radius,
+r, for M51, distinguishing by the point styles those found in t he
+arms, in the interarmzones, and within the central kpc. The p oints
+in the centre panel were found by applying a normalized corre ction
+factor to those in the upper panel, in order to take into accou nt the
+variation of /angbracketleftne/angbracketrightwith region radius, R. In the lower panel we show
+the result of taking the median value of /angbracketleftne/angbracketrightfor the full sample of
+regiones within annuli of width 200 pc.
+Fig. 4.— Mean electron densities ( /angbracketleftne/angbracketright) for the measured H i
+regions NGC 4449. In the upper panel /angbracketleftne/angbracketrightis plotted against
+galactocentric radius, in the centre panel /angbracketleftne/angbracketrighthas been modified
+by a correction factor to take into account the variation wit h the
+radiiRof the individual regions (a 1 /R0.5dependence), and in the
+lower panel we plot the median values of all the regions sampl ed
+within annuli of 100 pc width.
+/angbracketleftne/angbracketright=
+
+45.8e−r/10R−0.55cm−3;
+r <1.4 kpc &r >4.6 kpc
+28.9R−0.55cm−3;
+r∈[1.4,4.6] kpc.(8)
+The results for NGC 4449 are summarized in Fig. 4.
+In the upper panel of Fig. 4 we have plotted /angbracketleftne/angbracketrightagainst
+galactocentric radius for all our measured regions, in the
+central panel we present the data normalized to take into
+account the 1 /R0.5relation explained above, while in thelower panel we have averaged the points in the central
+panel in annuli of width 200 pc. In the latter two panels
+thelinearfitsshowtheexponentialradialdeclineinmean
+electron density, which is fitted by the whole galaxy out-
+side the central 120 pc. Using the best fit in the lower
+panel of Fig. 4 and normalizing, /angbracketleftne/angbracketrightis given by:
+/angbracketleftne/angbracketright= 45.7e−r/11R−0.45. (9)
+Thisscalelengthhereis11kpcwhichismuchbiggerthan
+the optical radius of the galaxy. However, Hunter et al
+(1998) observed an H idisc with a central dense compo-
+nent some 4.5 kpc in radius embedded in a much larger
+elliptical component with a major axis limiting radius
+of 18 kpc. Taken our cue from M51, which has an H i
+scale length of 9 kpc and a limiting H iradius of 15 kpc
+(Meijerink et al. 2005), we can claim here that the H ii
+region scale length and the H iscale length of NGC 4449
+are comparable, consistent with the scenario where the
+electron densities in the H iiregions follow the column
+density of the H igas, as in M51.
+3.3.The mean electron density as a function of
+position in the spiral arms of M51
+Finally in this section we show in Fig. 5 the electron
+density in the spiral arms of M51 as a function of the po-
+sition relative to the central ridge line of the arms. The
+ridge lines of the arms, used to define this relative posi-
+tion, were measured using the continuum image through
+the F814W filter. The abscissa in these plots is the dis-
+tance from the ridge line to the regions. We can see
+that the mean electron density reaches peak values along
+the ridge line, which is not surprising, but taking the
+complete populations of H iiregions we can see that the
+systematic increase of density towards the ridge line is a
+property of the upper envelopes of the plots for the two
+arms. There aremany regionsclose to the ridge line with
+relatively low densities, though no high density regions
+towards the edges of the arms.
+4.CONCLUSIONS
+We have used the H αluminosities and measured radii
+of the populations of H iiregions in M51 and NGC 4449,
+from HST images, to derive their mean electron densi-
+ties, aware that their inhomogeneous structures imply
+that the mean electron density does not describe either
+the in situ densities of their strongly emitting clumps, or
+the densities of their more tenuous interclump gas, but
+is a luminosity weighted mean over the H iiregion vol-
+ume. We have found two functional dependences of this
+mean electron density parameter, valid for both galax-
+ies: it varies inversely proportionally to the square root
+of the radius of an H iiregion, and it falls exponentially
+with galactocentric radius. The former relation must de-
+pend on the effects of the energy outflow from the mas-
+sive ionizing stars on their environment. As far as we
+are aware it does not correspond to the predictions of
+any previously published specific model (Scoville et al.
+(2001) show a graph for M51 which could be used to
+infer a similar result, but did not derive an explicit rela-
+tion for these variables), and sets interesting constraints
+on the physics of OB cluster formation and the subse-
+quent interaction with the cluster environment. The lat-
+ter relation, with the scale length for the electron densityDistributions of mean electron density in HII regions. 5
+Fig. 5.— The mean electron densities of the HII regions in M51 plotted against distance from the central ridge line of the arms in M5 1
+(squares). Also shown is the variation of the average value i n 200 pc bins of these densities as a function of this distance (thick lines) for
+both arms.
+comparablewith that for the H icolumn density in galax-
+ies of very different sizes and types (M51 is a Sbc while
+NGC 4449 is an IBm), demonstrates that, globally, an
+Hiiregion is in quasi-pressure equilibrium with its sur-
+rounding gas. The quantitative differences between the
+density of arm and interarm regions are not great, but
+also reflect this quasi-equilibrium, since the values are
+systematically somewhat lower for the interarm regions.Finally the distribution of mean electron densities across
+a spiral arm has an upper envelope peaking along the
+center, the ridge line of the arm, but values well below
+the envelope are distributed fairly uniformly across the
+arm. These relations should be of value for improving
+physical descriptions of H iiregions as zones of interac-
+tion between hot stars and their surrounding gas.
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\ No newline at end of file
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+arXiv:1001.0277v1  [astro-ph.GA]  4 Jan 2010Mon. Not. R. Astron. Soc. 000, 1–??() Printed 30 October 2018 (MN L ATEX style file v2.2)
+Thermomagnetic instability in hot discs
+Edward Liverts,⋆Michael Mond, and Vadim Urpin
+Department of Mechanical Engineering, Ben-Gurion Univers ity of the Negev,
+P.O. Box 653, Beer-Sheva 84105, Israel
+A.F. Ioffe Institute of Physics and Technology and Isaac Newt on Institute of Chile, Branch in St.Petersburg, 194021 St. P etersburg, Russia
+Accepted —. Received —-; in original form —-
+ABSTRACT
+A linear stability analysis of ionized discs with a temperature gradient a nd an external
+axial magnetic field is presented. It is shown that both hydromagne tic and thermo-
+magnetic effects can lead to the amplification of waves and make discs unstable. The
+conditions under which the instabilities grow are found and the chara cteristic growth
+rate is calculated. The regimes at which both the thermomagnetic an d magnetorota-
+tional instabilities can operate are discussed.
+Key words: accretion, accretion discs, MHD, MRI.
+1 INTRODUCTION
+The origin of turbulence in astrophysical discs is often att ributed to hydrodynamic and hydromagnetic instabilities t hat can
+occur in differentially rotating stratified gas. The magneto rotational instability (MRI), first investigated by Velikh ov (1959)
+and Chandrasekhar (1960) and later fully recognized by Balb us & Hawley (1991), is usually considered as one of the possib le
+candidates to generate such turbulence and is thought to pla y an important role in the evolution and dynamics of astrophy sical
+accretion discs. The growth of the instability in weakly ion ized magnetized discs is of interest for models of star forma tion and
+thesubsequentevolution ofprotostellar discs. Numerical simulations of the MRI in accretion discs (Hawley, Gammie & B albus
+1995; Brandenburg et al. 1995; Matsumoto & Tajima 1995; Tork elsson et al. 1996; Arlt & R¨ udiger 2001) show that turbu-
+lence generated can enhance essentially the angular moment um transport. The MRI has been studied in detail for both
+stellar and accretion disc conditions (see, e.g., Fricke (1 969); Safronov (1969); Acheson (1978); Balbus & Hawley (199 1);
+Kaisig, Tajima & Lovelace (1992); Zhang, Diamond & Vishniac (1994)). In fact, the MRI can occur only in a relatively weak
+magnetic field, but a sufficiently strong field can suppress the instability completely (Balbus & Hawley 1991). This is rela ted
+to the fact that the MRI is basically a long wavelength phenom ena in the sense that it is stabilized for wavelengths shorte r
+thanλcr∼2πcA/Ω where cAis the Alfven velocity and Ω is the angular velocity. Defining β=c2
+s/c2
+Awherecsis the sound
+speed, we can estimate that the critical wavelength λcris longer than the half-thickness of the disc, H∼cs/Ω forβ >1.
+More rigorous calculations for high and low values of the rad ial wave number krmay be found in Coppi & Keyes (2003), and
+Liverts & Mond (2009), respectively. In particular, it has b een shown by Liverts & Mond (2009) that the number of unstable
+MRI modes is decreasing with β. For example, it was found that there exist only three unstab le modes in the disc for β∼103.
+It is the goal of the present paper to study the effect of the tem perature gradient on stability of magnetized discs in the ca se
+of high beta. It will be shown that the range of unstable wavel engths is significantly widened due to the vertical temperat ure
+gradients and can extend to values shorter than the disc thic kness even in the case β >1.
+In discs, the centrifugal force almost is balanced by the gra vitational force, while the vertical structure approximat ely
+is determined by hydrostatic equilibrium. The asymptotic a nalysis [see, e.g.,Regev (1983), Klu` zniak & Kita (2000), O gilvie
+(1997), Shtemler, Mond, & R¨ udiger (2009)] reveals that the vertical temperature gradient is comparable to that of the d ensity
+in the case of a small aspect ratio. The destabilizing effect o f a temperature gradient on Alfv´ en waves has been studies fir st
+by Gurevich (1963); Gurevich & Gel’mont (1967) and later on b y Dolginov & Urpin (1979); Urpin (1981) for plasma in
+hydrostatic equilibrium. It has been shown that the thermom agnetic instability (TMI) can arise that transforms a fract ion of
+the thermal energy into magnetic one. Also, Coppi (2008) has shown that the vertical temperature gradient in discs combi ned
+with the radial rotation shear give rise to vertically local ized ballooning instabilities, the growth rate of which inc reases with
+⋆E-mail: eliverts@bgu.ac.il (EL); mond@bgu.ac.il (MM); Va dim.Urpin@uv.es (VU)
+c/circlecopyrtRAS2Edward Liverts, Michael Mond, and Vadim Urpin
+the temperature gradient. Thus, for a super adiabatic tempe rature gradient that can be obtained due to a strong heating
+source around the equatorial plane, the growth rate of the en suing ballooning modes is found to be comparable to that of th e
+”cylindrical” MRIs.
+In this paper, we consider the stability of hot magnetized di scs taking into account the temperature gradient. We will
+take a look at the MRI and TMI in various ranges of the waveleng th and determine the domains in a parameter space where
+each of those instabilities is dominant.
+2 BASIC EQUATIONS
+The dynamics of a magnetized disc is governed by the momentum , continuity, and induction equations:
+ρ(∂/vector v
+∂t+/vector v·/vector∇/vector v) =−/vector∇p+/vectorJ×/vectorB+ρ/vectorG, (1)
+∂ρ
+∂t+/vector∇·(ρ/vector v) = 0, (2)
+∂/vectorB
+∂t=−/vector∇×/vectorE, (3)
+where/vectorGis gravity.
+The current density that is given by Amp´ ere’s law:
+/vector∇×/vectorB=µ0/vectorJ, (4)
+whereµ0is magnetic constant and /vectorBsatisfies a divergence-free condition
+/vector∇·/vectorB= 0, (5)
+These set of equations should be complemented by the general ized Ohm’s law that expresses the electric field /vectorEin terms
+of the electric current /vectorJand gradients of the thermodynamic quantities. It should be noted that due to quasi-neutrality of
+plasma the independent thermodynamic variables are only th e temperature and pressure. Then, the Ohm’s law reads (see,
+e.g., Braginskii (1963))
+/vectorE=−/vector v×/vectorB−1
+qne/vector∇Pe+ ˆη·/vectorj+ˆΛ·/vector∇T, (6)
+where the last two terms represent the galvanomagnetic and t hermomagnetic effects, correspondingly. The tensors ˆ ηandˆΛ
+depend on the magnetic field /vectorBand, as a result, the transport properties of plasma are anis otropic with substantially different
+properties along and across the magnetic field if the field is s ufficiently strong. The effect of the magnetic field on the trans port
+properties is characterized by the magnetization paramete rae=ωBτwhereωB=qB/m eis the gyrofrequency of the electrons
+andτis their relaxation time (see, e.g., Spitzer (1978)). Even p oorly conducting protostellar discs can be strongly magnet ized
+if the electrons are the main charge carriers (Wardle 1999).
+The tensor terms in Eq.(6) can be written as follows
+ˆη/vectorJ=η/vectorJ+η1(/vectorJ×/vectorB)+η2/vectorB(/vectorB·/vectorJ), (7)
+ˆΛ/vector∇T= Λ/vector∇T+Λ1(/vector∇T×/vectorB)+Λ2/vectorB(/vectorB·/vector∇T), (8)
+where the second terms in both expressions represent the Hal l and the Nernst effects, correspondingly. The coefficients of
+these expressions can be obtained from the relations given b y Braginskii (1963); Lifshitz & Pitaevskii (1981). It is con venient
+to represent the effects of the temperature gradients by intr oducing two quantities that have the dimension of velocity:
+/vector u1T= Λ 1/vector∇T (9)
+u2T= Λ 2(/vectorB·/vector∇T) (10)
+both of which are of the order of u1T∼u2T∼kBτ/me|/vector∇T|for moderate values of the magnetization parameter aewithkB
+being the Boltzmann constant.
+The basic state of the considered rotating discs is assumed t o be characterized by an angular velocity that depends
+on the radial coordinate alone, Ω( r), and by hydrostatic equilibrium along the rotation axis z; (r;ϕ;z) are the cylindrical
+coordinates. Thus, assuming polytropic equation of state, asymptotic analysis of the steady-state properties of the d iscs
+demonstrates that the vertical temperature gradients are m uch larger than the radial ones (Regev 1983; Klu` zniak & Kita
+2000; Shtemler, Mond, & R¨ udiger 2009). Consequently, as a m odel problem, in the current calculations the steady-state
+temperature is assumed to vary only vertically. In addition , the background magnetic field is assumed to be a constant
+parallel to rotation axis z.
+c/circlecopyrtRAS, MNRAS 000, 1–??Thermomagnetic instability in hot discs 3
+As a first step toward studying the TMI it is instructive to est imate the relative importance of the thermomagnetic
+and galvanomagnetic terms contribution to Ohm’s law. For mo derate values of the magnetization parameter and wavelengt h
+of perturbations measured by cA/Ω (typical axial wave number for the MRI spectrum), assuming the disc thickness as a
+characteristic length scale for the temperature gradient o ne finds that the thermomagnetic terms in the Ohm’s law and the
+induction equation are much bigger than dissipative ones ar ising due to resistivity terms if
+kBT
+mec2ω2
+pτ2≫/radicalbig
+β (11)
+whereωp=/radicalbig
+neq2/ε0meis the plasma frequency with nebeing the number density of electrons and ε0being the electric
+constant. This condition can be satisfied for a very hot plasm a with arbitrary values of magnetic field and small or moderat e
+number density. Thus for example, Boettcher, Liang, & Smith (1998) have investigated Galactic black hole candidates. A p-
+plying their model to GX-339-4 the relevant data is that the d isc temperature is of the order of 106K with number density of
+the order of 1024m−3, and a corona with temperature of the order of 108K. Such range of parameters, with somewhat higher
+temperatures, has also been obtained by Artemova et al. (200 6). According to (11) such conditions are indeed favorable f or
+the onset of the TMI. The latter is consequently not suppress ed by resistivity so that the galvanomagnetic terms can be
+dropped from Ohm’s law.
+Even though condition (11) may be fulfilled in optically thic k as well as in optically thin environments (as was exemplifie d
+in the previous paragraph) we restrict the current discussi on to media such that radiative heat exchange is efficient in th at
+medium, thus any thermal transport due to the electron condu ction may be neglected and the energy equation may be
+dropped out. This indeed simplifies the subsequent calculat ions. To this end one may refer to optically thick medium wher e
+the radiative transport can be described as diffusive heat tr ansport with a thermal diffusivity that depends on the temper ature
+and therefore a temperature gradient can be well exist. It sh ould be emphasized though, that the TMI may also be excited
+in optically thin discs, however in such environments, the e nergy equation should be taken into account.
+Consider a linear stability of the discs under axisymmetric short wavelength perturbations that propagate in z-direction.
+The perturbed electric field /vectorE, magnetic field /vectorb, current density /vectorjand hydrodynamic velocity /vector vvary in time and space
+according to exp( ikz−iωt) so local approximation is used. Retaining the thermomagne tic terms in Ohm’s law (6), invoking
+incompressibility that results in zero pressure perturbat ions (the latter also means that the perturbations are restr icted to
+Alfv´ en modes) and assuming Keplerian angular velocity Ω( r) one can reduce linearized Eqs.(1)-(6) to the following sys tem of
+equations:
+/bracketleftbigg/parenleftbigg
+ω2+3Ω2−ω2
+A−2iΩω
+2iΩω ω2−ω2
+A/parenrightbigg
+−/parenleftbigg
+−ωω1T−2iΩω2T−ωω2T+2iΩω1T
+ωω2T−1
+2iΩω1T−ωω1T−1
+2iΩω2T/parenrightbigg/bracketrightbigg
+/vectorb= 0. (12)
+Here the characteristic frequencies are the Alfv´ en ωA=kcAand the thermomagnetic ω1,2T=ku1,2Tfrequencies.
+Before turning to the stability analysis it is instructive t o notice that for long wave lengths perturbations all the mat rix
+elements of second matrix in square brackets of the last syst em of equations are small and consequently the thermomagnet ic
+effects can be neglected. In that case the MRI is the dominant m ode of instability. We will elaborate on that point in the nex t
+section. The main interest of the current work however, is th e case for which the matrix elements of second matrix in squar e
+brackets of the equations (12) is comparable to the first one. This happens when: ω1,2T∼Ω, that indeed could occur in hot
+discs.
+3 DISPERSION EQUATION FOR THERMOMAGNETIC WAVES
+The dispersion relation that is obtained from Eqs.(12) read s
+ω4+a3ω3+a2ω2+a1ω+a0= 0, (13)
+where
+a3= 2ω1T (14)
+a2=−Ω2−2ω2
+A+ω2
+1T+ω2
+2T−i3
+2Ωω2T (15)
+a1=−2ω1T(Ω2+ω2
+A) (16)
+a0=−3Ω2ω2
+A+ω4
+A−Ω2(ω2
+1T+ω2
+2T)+i
+2Ωω2T(3Ω2−5ω2
+A). (17)
+If Ω = 0, Eq. (13) reduces to the dispersion relation for Alfv´ en waves modified by the temperature gradients (see Gurevich
+(1963); Gurevich & Gel’mont (1967); Dolginov & Urpin (1979) ; Urpin (1981)). If Ω /ne}ationslash= 0 but ∇T= 0, Eq.(13) reduces to the
+dispersion relation for the MRI obtained by Velikhov (1959) , and Balbus & Hawley (1991), which describes the combined
+effect of inertial and Alfv´ en waves propagating in rotating discs.
+c/circlecopyrtRAS, MNRAS 000, 1–??4Edward Liverts, Michael Mond, and Vadim Urpin
+01234567800.10.20.30.40.50.60.70.80.91
+kcA/ΩΓ=ℑω/Ω
+Figure 1. The growth rate, obtained by numerical solutions of eq.(13) foruT/cA= 0.2.
+It is instructive to examine a regime when the both MRI and TMI may operate. This implies the limit of thick disc
+(β≫1) andω1,2T/lessorequalslantΩ. For typical wave number of the MRI spectrum (Ω /cA) such regime reveals itself if ζ≡u2T/cA/lessorequalslant1.
+Furthermore one should note that in hot discs with the values of temperatures and numberdensity quoted above the inequal ity
+(11) can be hold for such regime and thus magnetic diffusivity may be neglected. A solution of the dispersion equation for
+ζ≡u2T/cA= 0.2 is depicted in Fig. 1. It is seen from this figure that there ar e two modes that become unstable. The
+upper curve corresponds to the MRI modified by the temperatur e gradient whereas the lower one describes a new instability
+that occurs due to the temperature gradient. It can be also ob tained from the analytical consideration that properties o f the
+instability are qualitatively different in the limiting cas es ofωA/lessorequalslantΩ (relatively long wavelengths) and ωA/greaterorequalslantΩ (relatively
+short wavelengths). In the long wavelength limit ( ωA/lessorequalslantΩ), the instability is predominantly the MRI with small corr ections
+caused by the temperature gradient. In particular, the temp erature gradient is responsible for a finite phase velocity w hich is
+vanishing in the isothermal case. In this limit, the eigenva lue is given by
+ω=±(Ω
+4u2T
+cA+i√
+3ωA). (18)
+The corresponding eigenvectors are
+/vectorb={2ωA√
+3Ω,∓1}Tb, /vector v=i{∓2ωA
+Ω,−4ω2
+A√
+3Ω2}Tb√µ0ρ, (19)
+These eigenvectors represent the pair of linearly polarize d waves that propagate in the opposite directions along the z-axis.
+The wave propagating in the positive direction (this direct ion has been assumed conventionally by setting a positive va lue for
+Ω) is exponentially growing but the other one is decaying.
+The another branch of modes in the limit of relatively long wa velengths is characterized by ω∼Ω and small growth rate
+that is given by
+Γ =Imω≈ ±5
+4ω2Tω2
+A
+Ω2(20)
+The eigenvalue and eigenvector for that branch are
+ω=±(Ω+i5
+4ω2Tω2
+A
+Ω2),/vectorb={1,∓2i}Tb, /vector v={1,i
+2}Tb√µ0ρ. (21)
+These eigenvectors describe the pair of left elliptically p olarized waves that propagate in the opposite directions. I n the absence
+of temperature gradients both waves are stable and represen t inertial waves of the pure hydrodynamic nature.
+In the limit of relatively short wavelengths (at ω∼ωA/greaterorequalslantΩ) and for large temperature gradients such as uT/cA/lessorequalslant1
+the thermomagnetic effects play an important role in the deve lopment of instability. The Alfv´ en waves can be significant ly
+c/circlecopyrtRAS, MNRAS 000, 1–??Thermomagnetic instability in hot discs 5
+modified by the temperature gradient and become unstable bey ond the stability limit of the classical MRIs. The growth rat e
+in this regime is given by
+Γ =ℑω≈ω2T
+2(1±2ω1T+Ω
+4ωA) (22)
+Finally, in the short wavelength limit and under the conditi onuT/cA/lessorequalslant1 the rotation can be neglected and the problem
+reduces to that described by Gurevich & Gel’mont (1967).
+In order to gain deeper understanding into the nature of the T MI in the short wavelength limit, the dispersion equation
+is rewritten in terms of b±≡bx±ibyin the limit Ω →0. Then, the dispersion equation reads
+/bracketleftbig
+ω2+ω(ω1T∓iω2T)−ω2
+A/bracketrightbig
+b±= 0 (23)
+This equation describes four circularly polarized waves. O ne pair of waves propagates along the magnetic field, and its
+frequency is
+ω=ωA−ω1T
+2±iω2T
+2. (24)
+The corresponding eigenvectors are
+b±=b, b∓= 0,v±
+cA=/bracketleftBig
+−1−ω1T
+2ωA±iω2T
+2ωA/bracketrightBigb
+B0, v∓= 0. (25)
+The unstable wave (upper sign) is right circularly polarize d while the damped wave (lower sign) is left circularly polar ized.
+The wave for which the phase shift between perturbations of t he magnetic field and velocity is smaller than π(πis the
+appropriate value for pure Alfv´ en waves) grows exponentia lly. Another wave for which the phase shift between the magne tic
+and velocity perturbations is larger than πis damped. The other pair also comprises of two right and left polarized waves
+that propagate in the opposite direction to the magnetic fiel d. One of these waves grows exponentially whereas the other i s
+damped in the same manner as in the first pair. Summarizing, ri ght circularly polarized waves propagating in the opposite
+directions along the magnetic field grow exponentially with the growth rate
+Γ =ω2T
+2(26)
+The picture of circularly polarized unstable waves allows a n additional interpretation of the instability. To that end it
+should be noted that j∗±=∓kb∗±/µ0wherej∗is the complex conjugate amplitude of the current density of the circularly
+polarized waves. The thermomagnetic contribution to the pe rturbed electric field is anti Hermitian as it can be seen from
+Eq.(25). Then, it is easy to show that the average work over th e wave period is given by
+/an}b∇acketle{t/vectorE·/vectorj/an}b∇acket∇i}ht=−ω2T
+2µ0|b|2. (27)
+The time derivative of the average energy /an}b∇acketle{tW/an}b∇acket∇i}htfor the wave propagating along the external magnetic field is given by
+∂/an}b∇acketle{tW/an}b∇acket∇i}ht
+∂t=∂
+∂t/integraldisplay
+/an}b∇acketle{tρv2
+2+b2
+2µ0/an}b∇acket∇i}htd3r=−/integraldisplay
+/an}b∇acketle{t/vectorj·/vectorE/an}b∇acket∇i}htd3r. (28)
+This equation together with Eq. (27) clearly shows that the w ave energy grows exponentially with the rate 2Γ.
+4 SUMMARY
+We have considered the thermomagnetic instability in ioniz ed non-isothermal Keplerian discs that are threaded by an ex -
+ternal magnetic field. It has been shown that, in the short wav elength limit, the temperature gradient allows to develop t he
+thermomagnetorotational instability for wavelengths sat isfying the condition ωA≫Ω for which the MRI does not occurs.
+The growth rate of this instability can be comparable to that of the MRI. Physically, the thermomagnetorotational insta bility
+transforms a fraction of the thermal energy carried out by th e heat flux into the magnetic energy of Alfv´ en waves (modified by
+the temperature gradient, see Gurevich (1963); Gurevich & G el’mont (1967); Dolginov & Urpin (1979); Urpin (1981)). The
+presence of shear, however, changes substantially the grow th rate of instability. The considered combined magnetorot ational
+and thermomagnetic instability can play an important role i n dynamics of astrophysical discs.
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diff --git a/1001.0278.txt b/1001.0278.txt
new file mode 100644
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--- /dev/null
+++ b/1001.0278.txt
@@ -0,0 +1,334 @@
+arXiv:1001.0278v1  [cs.CR]  2 Jan 2010How to retrieve priced data
+Li Xi
+Abstract
+Databases are an indispensable resourcefor retrieving up-to-da te information. How-
+ever, curious database operators may be able to find out the user s’ interests when the
+users buy something from the database. For these cases, if the d igital goods have the
+identical prices, then a k-out-of-noblivious transfer protocol could help the users to
+hide their choices, but when the goods have different prices, this wo uld not work. In
+this paper, we propose a scheme to help users to keep their choices secret when buying
+priced digital goods from databases.
+1. Introduction
+I am quite surethat all readers are familiar with digital lib raries, such as the digital libraries
+ofACM, IEEEandSIAM.Theselibrariesprovideresearchersw ithacomprehensiveresource
+of published papers, and users can easily retrieve their des ired papers by visiting these
+libraries. Recall how we retrieve data from digital librari es: we log in the system, select our
+desired ones and download them. If one does not own a membersh ipof a database, hewould
+have to pay for the papershereads, perhapsaccording to the l ength of thepublication. This
+process is convenient, but we undertake the risk of revealin g our private research interests
+to the database operators.
+If every paper has the same price, this problem can be resolve d perfectly: suppose that
+there are npublications in the library and we are interested in kof them, then by k-out-of-n
+oblivious transfer, we pay for some kpublications whilerevealing nothingabout our choices.
+In other word, the operator learns nothing but k, and could get the payment by adding up
+the prices for the ksold items.
+However, it is naive to assume that all publications have the same price. Nowadays,
+most papers are priced according to their lengths, perhaps o ne dollar per page. A more
+scientific way (although not perfect) is to price the data acc ording to the number of bits it
+contains. More formally, here we may view the database as a bi nary string x=x1x2···xn
+of length n, and every bit has thesame weight. Thenthe we could still use oblivious transfer
+to buy our desired bits from the library, leaking nothing but the number of bits we pay for.
+1How to retrieve priced data
+This scheme is not efficient(since the number of bits can be ver y large), and a more
+serious problem is that we should not assume every bit to have the same value. Instead
+of finding a method to assign prices for different goods, we woul d rather let the database
+operator to assign the value herself himself: a two-page com munication may cost you 100
+dollars, while a 200-page review may cost only 1 dollar. Afte r all, we let the operator to
+assign the prices himself.
+Nowweconsiderthegeneralproblem: thedatabasehas nitems, namely m1,m2,···,mn.
+Each item mi(1≤i≤n) has its own weight pi. Letσ1,σ2,···,σk, a subset of {1,2,···,n},
+be the choices of a user, the goal is to leak mσj(1≤j≤k) to the user while revealing
+nothing but Σ jpσjto the database operator. This is a special case of oblivious transfer, we
+denote it weighted oblivious transfer.
+Organization. In the rest of this section we discuss in more detail traditio nal oblivious
+transfer and weighted oblivious transfer. Section 2 presen ts two protocols for weighted
+oblivious transfer. Section 3 concludes the paper.
+1.1. A short review of oblivious transfer
+Oblivious Transfer ( OT) refers to a kind of two-party protocols where at the beginni ng
+of the protocol one party, the sender, has an input, and at the end of the protocol the
+other party, the receiver, learns some information about th is input in a way that does not
+allow the sender to figure out what it has learned [ 1]. Oblivious transfer is one of the key
+components of many cryptographic protocols and a fundament al primitive for cryptogra-
+phy and secure distributed computation [ 2,3,4]. The concept of oblivious transfer was
+proposed by Rabin [ 5], since then, many flavors of oblivious transfer were introd uced and
+analyzed [ 5,6,7,8,9]. Now oblivious transfer is one of the most remarkable achie vements
+in foundation of cryptography. The main flavors of oblivious transfer are as follows:
+•Original oblivious transfer ( OT)[5].ForOT, the sender has only one secret, m, and
+would like to have the receiver obtain mwith probability 0 .5. On the other hand, the
+receiver does not want the sender to know whether it gets mor not.
+•1-out-of-2 oblivious transfer( OT1
+2)[6].ForOT1
+2, the sender has two secrets, m1and
+m2, and would like to give the receiver one of them at the receive r’s choice. Again,
+the receiver does not want the sender to know which secret it c hooses.
+•1-out-of-noblivious transfer( OT1
+n)[7]. OT1
+nis a natural extension of OT1
+2to the case of
+nsecrets, in which the sender has nsecretsm1,m2,...,m nand is willing to disclose
+exactly one of them to the receiver at its choice.
+•k-out-of-noblivious transfer( OTk
+n)[10].ForOTk
+n, the receiver can receive only kmes-
+sages out of nmessages sent by the sender. In general, one thinks that OTk
+nis exten-
+2How to retrieve priced data
+sion ofOT1
+n. It is obvious that a trivial OTk
+nprotocol can be obtained by performing
+OT1
+nprotocol ktimes.
+Essentially, all these flavors are equivalent in the informa tion theoretic sense [ 11], but
+their functions vary, intuitively, we may use the following relation to describe the relation
+among all four flavors:
+OT⊆OT1
+2⊆OT1
+n⊆OTk
+n. (1)
+There are many ways to construct an efficient oblivious transf er protocol. Classical
+oblivious transfer protocols are based on discrete logarit hm [12,13], the hardness of the
+decisional Diffie-Hellman problem [ 14]etc.
+1.2. Definition of weighted oblivious transfer
+To define the requirements of a weighted oblivious transfer p rotocol, we simply apply the
+requirements of general OT1
+nprotocol (with minor revisions) [ 15] to it: for convenience, let
+m1,m2,···,mnto be items and pibe the weight of mi.
+DEFINITION 1A(k-out-of-n)weighted oblivious transfer should meet the following re-
+quirements:
+•Correctness. The protocol achieves its goal if both the receiver and the sen der behave
+properly. That is, if both the receiver and the sender follow t he protocol step by step,
+the receiver gets mσi’s after executing the protocol with the sender, where σi’s are the
+receiver’s choices, and the sender learns/summationtextk
+i=1pσi(i.e., the whole price of the goods).
+•Receivers’ Privacy-indistinguishability. The transcripts corresponding to the receiver’s
+different choices {σai}and{σbi},{σai} /ne}ationslash={σbi}, are computationally indistinguishable
+to the sender if the following equation is satisfied:
+Σpσai= Σpσbi. (2)
+If the transcripts are identically distributed, the choice o f the receiver is uncondition-
+ally secure.
+•Sender’s Privacy-compared with ideal model. We say that the sender’s privacy is guar-
+anteed if, for every possible malicious Rwhich interacts with S, there is a simulator R′
+(a probabilistic polynomial time machine) which interacts withTsuch that the output
+ofR′is computationally indistinguishable from the output of R.
+Remark. The weighted oblivious transfer also relies on the intracta bility of subset sub
+problem. Theprotocolimpliesthatbythetotal priceofthes olditems, thesendercannottell
+3How to retrieve priced data
+which items the receiver bought. Although subset sum proble m is know to be NP-complete ,
+sometimes it is still solvable (consider the case where the p rices are 1 ,2,4,8,···,2n−1, then
+the binary representation of the total price would betray th e receiver’s choice). However,
+for this case, even if we apply a trusted third party T, the problem still exists. This problem
+is solvable when the database is stored by more than one serve rs (recall PIR), but this is
+out of the scope of this paper.
+1.3. Comparing to priced oblivious transfer
+Perhaps the idea that of weighted oblivious transfer is simi lar to [16]. In [16], the notion
+of “priced oblivious transfer” is proposed. Informally, as sume that a buyer first deposits
+a pre-payment at the hands of a vendor. The buyer should then b e able to engage in a
+virtually unlimited number of interactions with the vendor in order to obtain digital goods
+(also referred to as items) at a total cost which does not exce ed its initial deposit amount.
+After spending all of its initial credit, the buyer should be unable to obtain any additional
+items before depositing an additional pre-payment. For pri ced oblivious transfer, unlimited
+number of interactions and prepayment is required, while th ese requirements relax in this
+paper. However, for weighted oblivious transfer, the recei ver would disclose how much,
+whento the sender, since we do not assume that the receiver intera cts with the sender
+many times. Comparing to priced oblivious transfer, weight ed oblivious transfer is used
+when the receiver would like to buy the desired items once at t he same time.
+2. Weighted oblivious transfer
+The idea of our first protocol is straightforward. The sender locks the ever item mi
+withpidifferent locks. In this way, only with all pilocks can the receiver get mi. This
+implies that the sender needs to generate/summationtextn
+i=1pikeys, and with/summationtextk
+i=1pσilocks and keys,
+the receiver could unlock the locks for mσi’s. By using a/summationtextk
+i=1pσi-out-of-/summationtextn
+i=1pioblivious
+transfer, the sender could leak the corresponding keys to th e receiver without knowing the
+chosen ones, thus unable to figure out the receiver’s choices . For simplicity, the best(i.e.,
+most efficient) lock should be symmetric key encryption schem e.
+PROTOCOL 1
+In this protocol, the sender has/summationtextn
+i=1pipairs of (different) keys, denoted by Kij(i∈
+{1,2,···,n},j∈ {1,2,···,pi}). Intuitively, the sender locks miwithKij’s.
+4How to retrieve priced data
+Input: The receiver’s input is composed of knumbers σ1,σ2,...,σ k, which is a subset
+of{1,2,...,n}, and the sender’s input is composed of npriced items m1,m2,...,m n, the
+weight (price) of item miispi.
+Output: Thereceiver’soutputsare mσ1,mσ2,...,m σk, andthesender’soutputisΣk
+i=1pσi.
+•Step 1The sender encrypts the items m1,m2,···,mnwith the encryption keys. For
+mi, thisisdonebycomputing EKi1(EKi2(···(EKipi(mi))···). Thatis, miisencrypted
+bypilocks:Ki1,···,Kipirespectively.
+•Step 2The sender sends all the ciphertexts to the receiver.
+•Step 3By/summationtextk
+i=1pσi-out-of-/summationtextn
+i=1pioblivious transfer, the sender reveals the keys for
+allmσi’s, while learning nothing about the receiver’s choices.
+•Step 4With the keys, the receiver easily decrypts and learns all mσi’s (and nothing
+else).
+This protocol is straightforward, and we do not prove that it is actually a weighted
+oblivious transfer in a formal way. Informally, assume the s ecurity of/summationtextk
+i=1pσi-out-of-
+/summationtextn
+i=1pioblivious transfer protocol used in step 3, the sender leaks nothing but/summationtextk
+i=1pσi
+during step 3, the only communication from the receiver to th e sender. Also, the receiver
+could unlock no more than/summationtextk
+i=1pσilocks, thus leans no more than what “costs”/summationtextk
+i=1pσi.
+Although the protocol is not efficient enough, it is the corner stone of the next protocol.
+Remark. When the protocol is applied by databases, the firsttwo steps are donebefore
+transactions. That is, the database publishes the encrypte d items online and everyone could
+download them. When interested in some of the items, the user interacts with the database
+operator and completes thelast two steps. Inthis way, they w ould not need to communicate
+the whole encrypted data, which turns out to be huge. Also, th e items are encrypted only
+once for all users.
+2.1. Making our protocol efficient
+It is not hard to show that protocol 1 needs O(/summationtextn
+i=1pi) encryptions, this number is
+clearly impractical, at least sometimes. In this subsectio n, we propose a very efficient
+protocol which only needs O(n) encryptions.
+If for any mi, if there is way to divide miinto pieces such that it is easily reconstructable
+frompipieces, butevencompleteknowledgeof pi−1piecesreveals absolutelynoinformation
+aboutmi, then we can propose a new protocol. This is really easy: let
+mi=pi/circleplusdisplay
+j=1mij=mi1⊕mi2⊕···⊕mipi (3)
+5How to retrieve priced data
+Thenmicould only be recovered with all pimij’s. However, this division scheme is not
+very efficient since mican be very long. Also, for this case, instead of downloading the
+encrypted data from the website, the receiver has to learn al l he needs from the sender. So
+we slightly revise our idea and comes up with protocol 2:
+PROTOCOL 2
+In this protocol, the sender has npairs of different keys, denoted by Ki(i∈ {1,2,···,n}).
+Intuitively, she intends to encrypt miwithKi.
+Input: The receiver’s input is composed of knumbers σ1,σ2,...,σ k∈ {1,2,...,n}, and
+the sender’s input is composed of nitemsm1,m2,...,m n, the weight of item miispi.
+Output: The receiver’s outputs are mσ1,mσ2,...,m σk.
+•Step 1The sender encrypts the items m1,m2,···,mnwith the encryption keys and
+obtainsEK1(m1),EK2(m2),···,EKn(mn).
+•Step 2The sender sends all the ciphertexts to the receiver.
+•Step 3For every i∈ {1,2,···,n}, the sender divides Kiintopiparts. This is done
+by finding Ki1,Ki2,···,Kipisuch that/circleplustextj=pi
+j=1Kij=Ki.
+•Step 4Using oblivious transfer, the sender leaks Kσ1,···,Kσkto the receiver while
+learning nothing about σi’s. This is done by revealing/summationtextk
+i=1pσiparts (i.e., the parts
+of keyKσi’s) out of all/summationtextn
+i=1piparts(i.e., for every σi, revealKσi1,···,Kσipσi).
+•Step 5The receiver recovers the keys for all mσi’s by exclusive-oring mσijand de-
+crypts them.
+Similarly, when the protocol is applied by databases, the fir stthree steps are donebefore
+transactions. That is, the database publishes the encrypte d items online and everyone could
+download them. When interested in some of the items, the user interacts with the database
+operator and completes the last two steps.
+Now we show that protocol 2 is indeed a weighted oblivious tra nsfer: If both parties
+behave properly, then the receiver would learn all parts of Kσi,i∈ {1,···,k}, and by XOR
+operations he learns Kσi, thus able to learn mσi.
+The scheme takes only three rounds. This is almost optimal si nce at least the receiver
+has to choose {σ1,···,σk}’s and let the sender know and the sender has to respond to the
+receiver’s request.
+For computation, the receiver needs k < ndecryptions and/summationtextk
+i=1pσiXOR operations.
+The sender needs nencryptions, nXOR operations and choosing/summationtextn
+i=1(pi−1) random
+numbers.
+6How to retrieve priced data
+LEMMA1For protocol 2, the receiver’s choice is unconditional secu re, assuming that
+the oblivious transfer used in step 4 is secure.
+Proof. For any choices {σai}, if/summationtextpσai=/summationtextpσiholds, in step 4 the receiver and
+the sender still perform/summationtextpσi-out-of-/summationtextpioblivious transfer, then the security of oblivious
+transfer used in step 4 shows that the sender cannot learn any thing-it cannot tell {σai}
+from{σi}. Since the receiver sends nothing else, the sender cannot te ll what the receiver’s
+choice is. ✷
+LEMMA2For protocol 2, if the receiver is semi-honest, it gets no inf ormation about
+mi,i /∈ {σ1,···,σk}, assuming the security of the encryption scheme and oblivio us transfer.
+Proof.If the receiver is semi-honest, due to the security of oblivi ous transfer, it learns
+nothing about Ki,i /∈ {σ1,···,σk}. The security of the encryption scheme promises that
+EKi(mi),i /∈ {σ1,···,σk}is computational indistinguishable from EKi(r),i /∈ {σ1,···,σk},
+whereris a randomly chosen sequence. ✷
+LEMMA3Protocol 2 meets the requirement of sender’s privacy assumi ng the security
+of oblivious transfer used in step 4.
+Proof.For each malicious receiver Rin the real run, we construct a simulator R′in
+the Ideal Model such that the outputs of RandR′are computationally indistinguishable.
+As the oblivious transfer used in step 4 is secure, there exis ts a simulator R′′in the Ideal
+Model such that the outputs of RandR′′are computationally indistinguishable. Now let
+R′acts the same as R′′, thenRandR′are computationally indistinguishable. Since there
+is no other iteration with T, we prove the theorem. ✷
+With these preparations, we come up with:
+THEOREM 1Protocol 2 is indeed weighted oblivious transfer.
+Since the fact that OTk
+ncan be achieved from weighted oblivious transfer is trivial , we
+see the equivalence between the two flavors. Moreover, all (e xisting) flavors of oblivious
+transfer are equivalent in the information theoretic sense .
+2.2. Reducing the computation complexity
+The computational complexity of the protocol is much more ex pensive than a tradi-
+tionalk-out-of-nscheme where every item has the same weight. This deficiency m ay, to
+some extent, affect the application of the protocol, but there are some ways to reduce the
+computational complexity.
+7How to retrieve priced data
+Ifp1,···,pnshare a greatest common divisor q >1 , then by dividing qand paying q
+times the money for each decryption, the weight of the i-th element becomes pi/q, where
+piis the original weight, and the times of encryptions and decr yptions can be decreased
+to 1/q(of the original one). In the cases that the greatest common d ivisor of p1,···,pnis
+1(q= 1), we introduce two additional methods:
+Method 1 Arrange the items into some certain categories, and the item s of each
+category share the same weight. This is practical in everyda y life. Assume that there are
+three categories, with weight one, two and three, then the co mputation is around 3 n.
+Method 2 Generally speaking, methods onecould save most computatio n. In thecases
+where it is difficult to arrange the items into certain categor ies, we have following method:
+First consider an example. Assume that there are four items w ith weights 105, 190, 307,
+689. Then the greatest common divisor of the weights is q= 1, and we could do nothing
+with them. However, if the weights change a little into 100, 2 00, 300, 700, then p= 100,
+and the complexity can be greatly reduced by changing the wei ghts into 1, 2, 3, 7.
+More generally, suppose that the weight of each item is pi(i= 1,2,...,n). The two
+parties could decide a “greatest common divisor” q, and calculate the new weights-the
+closest integer of pi/q. In this way, the complexity could also be greatly reduced.
+3. Concluding remarks
+This paper discusses weighted oblivious transfer, which ca n be used for selling priced
+digital goods. Two implementations of it was proposed and an alyzed. The protocol is
+especially useful when the prices of the items are not very la rge, or the prices of digital
+goods fall in very limited categories. In this way, the compu tation can be done most
+efficiently.
+However, weighted oblivious transfer also suffers from short comings. We assume that
+subset problem is hard to compute, but sometimes it is possib le (recall the example that
+the prices are 1 ,2,···,2n−1). And this shortcoming is unsolvable even when we apply the
+trusted third party. In addition, sometimes the whole price of the digital goods itself can
+leak part (not all) of the choices, which would also not be sec ure. Similarly, the problem
+exists even a trusted third party is employed. This addition al asks the sender to be careful
+when assigning the prices for the goods.
+I think that the above problems are unsolvable in current set tings. However, the idea
+can be used for SPIR, when there are more than one servers. Fur ther we may consider the
+implementation of adaptive queries in the future.
+8How to retrieve priced data
+References
+[1] M. Naor. Computationally Secure Oblivious Transfer. J. Cryptol. , 18(1), 1-35.
+[2] Y. Ishai, M. Prabhakaran, A. Sahai. Founding Cryptograp hy on Oblivious Transfer-
+Efficiently. Advances in Cryptology-CRYPTO 2008, LNCS 5157 , 572-591, 2008.
+[3] O. Goldreich, R. Vainish. How to Solve Any Protocol Probl em: An Efficient Improve-
+ment.Advances in Cryptology-CRYPTO 1987, LNCS 293 , 73-86, 1988.
+[4] J. Kilian. Founding Cryptography on Oblivious Transfer .Proceedings of the 20th ACM
+Symposium on Theory of Computing , 20-31, 1988.
+[5] M. Rabin. How to Exchange Secrets by Oblivious Transfer. Technical Report TR- 81,
+Aiken Computation Laboratory, Harvard University, 1981.
+[6] S. Even, O. Goldreich, A. Lempel. A Randomized Protocol f or Signing Contracts.
+Commun. ACM , 28(6), 637-647.
+[7] G. Brassard, C. Crepeau, J.M. Robert. All-or-Nothing Di sclosure of Secrets. Advances
+in Cryptology-CRYPTO 1986, LNCS 263 , 234-238, 1987.
+[8] J.Garay, P. MacKenzie. ConcurrentObliviousTransfer. Proceedings of the 41th Annual
+Symposium on Foudations of Computer Science , 314-324, 2000.
+[9] I. Haitner. Semi-honest to Malicious Oblivious Transfe r-The Black-Box Way. Proceed-
+ings of the Fifth Theory of Cryptography Conference, LNCS 4948 , 412-426, 2008.
+[10] J. Zhang, Y. Wang. Two provably secure k-out-of-noblivious transfer schemes. Appl.
+Math. Comput. , 169(2), 1211-1220, 2005.
+[11] C. Cachin. On the Foundations of Oblivious Transfer. Advances in Cryptology-
+EUROCRYPT 1998, LNCS 1403 , 361-374, 1998.
+[12] K. Bruce, L. Cardelli, B. Pierce. Comparing object enco dings.Inform. Comput. , 155(1-
+2), 108-133.
+[13] M. Naor, B. Pinkas. Efficient oblivious transfer protoco ls.Proceedings of the twelfth
+annual ACM-SIAM symposium on discrete algorithms , 448-457, 2001.
+[14] WG Tzeng. Efficient 1-Out-of-n Oblivious Transfer Schem es with Universally Usable
+Parameters. IEEE Trans. Comput. , 53(2), 232-240.
+[15] M. Naor, B. Pinkas. Oblivious Transfer and Polynomial E valuation. Proceedings of the
+31st ACM Symposium on Theory of Computing , 145-254, 1999.
+9How to retrieve priced data
+[16] B. Aiello, Y. Ishai and O. Reingold. Priced Oblivious Tr ansfer: How to Sell Digital
+Goods.Advances in cryptology-EUROCRYPT 2001, LNCS 2045 , 119-135, 2001.
+10
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+arXiv:1001.0279v1  [stat.ML]  2 Jan 2010Regularization for Matrix Completion
+Raghunandan H. Keshavan∗and Andrea Montanari∗†
+Departments of Electrical Engineering∗and Statistics†, Stanford University
+Abstract—We consider the problem of reconstructing a low
+rank matrix from noisy observations of a subset of its entrie s.
+This task has applications in statistical learning, comput er vision,
+and signal processing. In these contexts, ‘noise’ generica lly refers
+to any contribution to the data that is not captured by the
+low-rank model. In most applications, the noise level is lar ge
+compared to the underlying signal and it is important to avoi d
+overfitting.Inordertotacklethisproblem,wedefinea regularized
+cost function well suited for spectral reconstruction meth ods.
+Within a random noise model, and in the large system limit, we
+prove that the resulting accuracy undergoes a phase transit ion
+depending on the noise level and on the fraction of observed
+entries. The cost function can be minimized using OPTSPACE
+(a manifold gradient descent algorithm). Numerical simula tions
+show that this approach is competitive with state-of-the-a rt
+alternatives.
+I. INTRODUCTION
+LetNbe anm×nmatrix which is ‘approximately’ low
+rank, that is
+N=M+W=UΣVT+W . (1)
+whereUhas dimensions m×r,Vhas dimensions n×r, and
+Σis a diagonal r×rmatrix. Thus Mhas rank randWcan
+be thought of as noise, or ‘unexplained contributions’ to N.
+Throughout the paper we assume the normalization UTU=
+mIr×randVTV=nIr×r(Id×dbeing the d×didentity).
+Out of the m×nentries of N, a subset E⊆[m]×[n]is
+observed. We let PE(N)be them×nmatrix that contains
+the observed entries of N, and is filled with 0’s in the other
+positions
+PE(N)ij=/braceleftbiggNijif(i,j)∈E,
+0otherwise.(2)
+Thenoisy matrix completion problem requires to reconstruct
+the low rank matrix Mfrom the observations PE(N). In the
+following we will also write NE=PE(N)for the sparsified
+matrix. Over the last year, matrix completion has attracted
+significant attention because of its relevance –among other
+applications– to colaborative filtering. In this case, the m atrix
+Ncontains evaluations of a group of customers on a group of
+products, and one is interested in exploiting a sparsely fill ed
+matrix to provide personalized recommendations [1].
+In suchapplications,the noise Wis not a small perturbation
+and it is crucial to avoid overfitting. For instance, in the li mit
+M→0, the estimate of /hatwiderMrisks to be a low-rank approxima-
+tion of the noise W, which would be grossly incorrect.
+In order to overcomethis problem, we proposein this paper
+an algorithm based on minimizing the following cost functio n
+FE(X,Y;S)≡1
+2||PE(N−XSYT)||2
+F+1
+2λ||S||2
+F.(3)Here the minimization variables are S∈Rr×r, andX∈
+Rm×r,Y∈Rn×rwithXTX=YTY=Ir×r. Finally, λ >0
+is a regularization parameter.
+A. Algorithm and main results
+The algorithm is an adaptation of the O PTSPACEalgorithm
+developed in [2]. A key observation is that the following
+modified cost function can be minimized by singular value
+decomposition (see Section I.1):
+/hatwideFE(X,Y;S)≡1
+2||PE(N)−XSYT||2
+F+1
+2λ||S||2
+F.(4)
+As emphasized in [2], [3], which analyzed the case λ= 0,
+this minimization can yield poor results unless the set of
+observations Eis ‘well balanced’. This problem can be
+bypassedby ‘trimming’the set E, and constructinga balanced
+set/tildewideE. The O PTSPACEalgorithm is given as follows.
+OPTSPACE( setE, matrixNE)
+1: Trim E, and let /tildewideEbe the output;
+2: Minimize /hatwideF/tildewideE(X,Y;S)via SVD,
+letX0,Y0,S0be the output;
+3: Minimize FE(X,Y;S)by gradient descent
+usingX0,Y0,S0as initial condition.
+In this paper we will study this algorithm under a model
+for which step 1 (trimming) is never called, i.e. /tildewideE=Ewith
+high probability. We will therefore not discuss it any furth er.
+Section II compares the behavior of the present approach
+with alternative schemes. Our main analytical result is a sh arp
+characterization of the mean square error after step 2. Here
+and below the limit n→ ∞is understood to be taken with
+m/n→α∈(0,∞).
+Theorem I.1. Assume|Mij| ≤Mmax,Wijto be i.i.d. random
+variables with mean 0variance√mnσ2andE{W4
+ij} ≤Cn2,
+and that for each entry (i,j),Nijis observed (i.e. (i,j)∈E)
+independently with probability p. Finally let /hatwiderM=X0S0YT
+0
+be the rank rmatrix reconstructed by step 2ofOPTSPACE,
+for the optimal choice of λ. Then, almost surely for n→ ∞
+1
+||M||2
+F||/hatwiderM−M||2
+F= 1−
+−/braceleftBig/summationtextr
+k=1Σ2
+k/parenleftBig
+1−σ4
+p2Σ4
+k/parenrightBig
++/bracerightBig2
+||Σ||2
+F/braceleftBig/summationtextr
+k=1Σ2
+k/parenleftBig
+1+√ασ2
+pΣ2
+k/parenrightBig/parenleftBig
+1+σ2
+pΣ2
+k√α/parenrightBig/bracerightBig+on(1).
+This theorem focuses on a high-noise regime, and predicts
+a sharp phase transition: if σ2/p <Σ1, we can successfully
+extract information on M, from the observations NE. If onthe other hand σ2/p≥Σ1, the observations are essentialy
+useless in reconstructing M. It is possible to prove [4] that the
+resulting tradeoff between noise and observed entries is ti ght:
+no algorithm can obtain relative mean square error smaller
+than one for σ2/p≥Σ1, under a simple random model for
+M. To the best of our knowledge, this is the first sharp phase
+transition result for low rank matrix completion.
+For the proof of Theorem I.1, we refer to Section III. An
+important byproduct of the proof is that it provides a rule fo r
+choosing the regularization parameter λ, in the large system
+limit.
+B. Related work
+The importance of regularization in matrix completion is
+well known to practitioners. For instance, one important co m-
+ponentofmanyalgorithmscompetingfortheNetflixchalleng e
+[1], consisted in minimizing the cost function HE(X,Y;S)≡
+1
+2||PE(N−/tildewideX/tildewideYT)||2
+F+1
+2λ||/tildewideX||2
+F+1
+2λ||/tildewideY||2
+F(this is also
+knownas maximummargin matrix factorization [5], [6]). Here
+the minimization variables are /tildewideX∈Rm×r,/tildewideY∈Rn×r.
+Unlike in O PTSPACE, these matrices are not constrained to
+be orthogonal, and as a consequence the problem becomes
+significantly more degenerate. Notice that, in our approach ,
+the orthogonality constraint fixes the norms ||X||F,||Y||F.
+This motivates the use of ||S||2
+Fas a regularization term.
+Convex relaxations of the matrix completion problem were
+recently studied in [7], [8]. As emphasized by Mazumder,
+Hastie and Tibshirani [9], such nuclear norms relaxations
+can be viewed as spectral regularizations of a least square
+problem.Finally,thephasetransitionphenomenoninTheor em
+I.1, generalizes a result of Johnstone and Lu on principal
+component analysis [10], and similar random matrix models
+were studied in [11].
+II. NUMERICAL SIMULATIONS
+In this section, we present the results of numerical sim-
+ulations on synthetically generated matrices. The data are
+generated following the recipe of [9]: sample U∈Rn×r
+andV∈Rm×rby choosing UijandVijindependently and
+indentically as N(0,1). Sample independently W∈Rm×n
+by choosing Wijiid with distribution N(0,σ2√mn). Set
+N=UVT+W. We also use the parameters chosen in [9]
+and define
+SNR =/radicalBigg
+Var((UVT)ij)
+Var(Wij),
+TestError =||P⊥
+E(UVT−/hatwideN)||2
+F
+||P⊥
+E(UVT)||2
+F,
+TrainError =||PE(N−/hatwideN)||2
+F
+||PE(N)||2
+F,
+whereP⊥
+E(A)≡A−PE(A).
+In Figure 1, we plot the train error and test error for the
+OPTSPACEalgorithm on matrices generated as above with
+n= 100,r= 10, SNR=1andp= 0.5. For comparison, we 0.5 0.6 0.7 0.8 0.9 1
+ 10  20  30  40  50  60
+ 0 0.2 0.4 0.6 0.8 1
+ 10  20  30  40  50  60 
+ 
+L1
+L1-L0
+L1-U
+COPTSPACE(λ∗)OPTSPACE(0)
+Fig. 1. Test (top) and train (bottom) error vs. rank for O PTSPACE, SOFT-
+IMPUTE, HARD-IMPUTEand SVT. Here m=n= 100,r= 10,p=
+0.5,SNR = 1 .
+also plot the corresponding curves for S OFT-IMPUTE,HARD-
+IMPUTEand SVT taken from [9]. In Figures 2 and 3, we
+plot the same curves for different values of r,ǫ,SNR. In these
+plots, O PTSPACE(λ)corresponds to the algorithm that min-
+imizes the cost (3). In particular O PTSPACE(0)corresponds
+to the algorithm described in [2]. Further, λ∗=λ∗(ρ)is the
+value of the regularization parameter that minimizes the te st
+error while using rank ρ(this can be estimated on a subset of
+the data, not used for training).
+It is clear that regularization greatly improves the perfor -
+mance of O PTSPACEand makes it competitive with the best
+alternative methods.
+III. PROOF OF THEOREM I.1
+The proof of Theorem 1 is based on the following three
+steps:(i)Obtain an explicit expression for the root mean
+square error in terms of right and left singular vectors of N;
+(ii)Estimate the effect of the noise Won the right and left
+singular vectors; (iii)Estimate the effect of missing entries.
+Step(ii)buildsonrecentestimatesontheeigenvectorsoflarge
+covariancematrices[12].Instep (iii)we usethe resultsof[2].
+Step(i)is based on the following linear algebra calculation,
+whose proof we omit due to space constraints(here and below
+/a\}bracketle{tA,B/a\}bracketri}ht ≡Tr(ABT)). 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
+ 10  20  30  40  50  60
+ 0 0.2 0.4 0.6 0.8 1
+ 10  20  30  40  50  60 
+ 
+L1
+L1-L0
+L1-U
+COPTSPACE(λ∗)OPTSPACE(0)
+Fig. 2. Test (top) and train (bottom) error vs. rank for O PTSPACE, SOFT-
+IMPUTE, HARD-IMPUTEand SVT. Here m=n= 100,r= 6,p=
+0.5,SNR = 1 .
+Proposition III.1. LetX0∈Rm×randY0∈Rm×rbe the
+matrices whose columns are the first r, right and left, singular
+vectors of NE. Then the rank- rmatrix reconstructed by step
+2of ofOPTSPACE, with regularization parameter λ, has the
+form/hatwiderM(λ) =X0S0(λ)YT
+0Further, there exists λ∗>0such
+that
+1
+mn||M−/hatwiderM(λ∗)||2
+F=||Σ||2
+F−/parenleftbigg/a\}bracketle{tXT
+0MY0, XT
+0NEY0/a\}bracketri}ht√mn||X0NEY0||F/parenrightbigg2
+.
+(5)
+A. The effect of noise
+Inorderto isolate theeffectof noise,we considerthematri x
+/hatwideN=pUΣVT+WE. Throughoutthis section we assume that
+the hypotheses of Theorem I.1 hold.
+Lemma III.2. Let(nz1,n,...,nz r,n)be therlargest singular
+valuesof /hatwideN. Then,as n→ ∞,zi,n→zialmost surely, where,
+forΣ2
+i> σ2/p,
+zi=pΣi/braceleftbigg
+α/parenleftbiggσ2
+pΣ2
+i+1√α/parenrightbigg/parenleftbiggσ2
+pΣ2
+i+√α/parenrightbigg/bracerightbigg1/2
+,(6)
+andzi=σ/radicalbig
+pα1/2(1+√α)forΣ2
+i≤σ2/p.
+Further, let X∈Rm×randY∈Rn×rbe the matrices
+whose columns are the first r, right and left, singular vectors 0 0.2 0.4 0.6 0.8 1
+ 5 10 15 20 25 30
+ 0 0.2 0.4 0.6 0.8 1
+ 5 10 15 20 25 30 
+ 
+L1
+L1-L0
+L1-U
+COPTSPACE(λ∗)OPTSPACE(0)
+Fig. 3. Test (top) and train (bottom) error vs. rank for O PTSPACE, SOFT-
+IMPUTE, HARD-IMPUTEand SVT. Here m=n= 100,r= 5,p=
+0.2,SNR = 10 .
+of/hatwideN.Thenthereexistsasequenceof r×rorthogonalmatrices
+Qnsuch that, almost surely ||1√mUTX−AQn||F→0,
+||1√nVTY−BQn||F→0withA= diag( a1,...,a r),
+B= diag(b1,...,b r)and
+a2
+i=/parenleftBig
+1−σ4
+p2Σ4
+i/parenrightBig/parenleftBig
+1+√ασ2
+pΣ2
+i/parenrightBig−1
+,
+b2
+i=/parenleftBig
+1−σ4
+p2Σ4
+i/parenrightBig/parenleftBig
+1+σ2
+p√αΣ2
+i/parenrightBig−1
+,(7)
+forΣ2
+i> σ2/p, whileai=bi= 0otherwise.
+Proof:Dueto space limitations,we will focushere onthe
+caseΣ1,...,Σr> σ2/p. The general proof proceeds along
+the same lines, and we defer it to [4].
+Notice that WEis anm×nmatrix with i.i.d. entries with
+variance√mnσ2pand fourth moment bounded by Cn2. It
+is therefore sufficient to prove our claim for p= 1and then
+rescaleΣbypandσby√p. We will also assume that,without
+loss of generality, m≥n.
+Let/hatwideZbeanr×rdiagonalmatrixcontainingtheeigenvalues
+(nzn,1,...,nz n,r). The eigenvalue equations read
+Uˆβy+WY−X/hatwideZ= 0, (8)
+Vˆβx+WTX−Y/hatwideZ= 0. (9)where we defined ˆβx≡ΣUTX,ˆβy≡ΣVTY∈
+Rr×r. By singular value decomposition we can write W=
+Ldiag(w1,w2,...wn)RT, withLTL=Im×m,RTR=
+In×n.
+LetuT
+i,xT
+i,vT
+i,yT
+i∈Rrbe thei-th row of -respectively-
+LTU,LTX,RTV,RTY. In this basis equations (8) and (9)
+read
+uT
+iˆβy+wiyT
+i−xT
+i/hatwideZ= 0, i∈[n],
+uT
+iˆβy−xT
+i/hatwideZ= 0, i∈[m]\[n],
+vT
+iˆβx+wixT
+i−yT
+i/hatwideZ= 0, i∈[n].
+These can be solved to get
+xT
+i= (uT
+iˆβy/hatwideZ+wivT
+iˆβx)(Z2−w2
+i)−1, i∈[n],
+xT
+i=uT
+iˆβy/hatwideZ−1, i ∈[m]\[n],
+yT
+i= (vT
+iˆβx/hatwideZ+wiuT
+iˆβy)(/hatwideZ2−w2
+i)−1, i∈[n].(10)
+By definition Σ−1ˆβx=/summationtextm
+i=1uixT
+i, andΣ−1ˆβy=/summationtextn
+i=1viyT
+i, whence
+Σ−1ˆβx=n/summationdisplay
+i=1ui(uT
+iˆβy/hatwideZ+wivT
+iˆβx)(/hatwideZ2−w2
+i)−1
++m/summationdisplay
+i=n+1uiuT
+iβy/hatwideZ−1, (11)
+Σ−1ˆβy=n/summationdisplay
+i=1vi(vT
+iˆβx/hatwideZ+wiuT
+iˆβy)(/hatwideZ2−w2
+i)−1.(12)
+Letλ=w2
+iα1/2/(m2σ2). Then, it is a well known fact
+[13] that as n→ ∞the empirical law of the λi’s converges
+weaklyalmost surelyto the Marcenko-Pasturlaw,with densi ty
+ρ(λ) =α/radicalBig
+(λ−c2
+−)(c2
++−λ)/(2πλ), withc±= 1±α−1/2.
+Letβx=ˆβx/√m,βy=ˆβx/√n,Z=/hatwideZ/n. A priori, it
+is not clear that the sequence (βx,βy,Z)–dependent on n–
+converges.However,it is immediate to show that the sequenc e
+is tight, and hence we can restrict ourselves to a subsequenc e
+Ξ≡ {ni}i∈Nalong which a limit exists. Eventually we will
+show that the limit doesnot depend on the subsequence,apart ,
+possibly, from the rotation Qn. Hence we shall denote the
+subsequential limit, by an abuse of notation, as (βx,βy,Z).
+Consider now a such a convergent subsequence. It is possi-
+ble to show that Σ2
+i> σ2/pimpliesZ2
+ii> α3/2σ2c+(α)2+δ
+for some positive δ. Since almost surely as n→ ∞,w2
+i<
+α3/2σ2c+(α)2+δ/2for alli, for all purposes the summands
+on the rhs of Eqs. (11), (12) can be replaced by uniformly
+continuous, bounded functions of the limiting eigenvalues λi.
+Further, each entry of ui(resp.vi) is just a single coordinate
+of the left (right) singular vectors of the random matrix W.
+Using Theorem 1in [12], it follows that any subsequential
+limit satisfies the equations
+βx= Σβy/braceleftBig
+Z/integraldisplay
+(Z2−α3/2σ2λ)−1ρ(λ)dλ+(α−1)Z−1/bracerightBig
+,
+(13)
+βy= Σβx/braceleftBig
+Z/integraldisplay
+(Z2−α3/2σ2λ)−1ρ(λ)dλ/bracerightBig
+,. (14)Solving for βy, we get an equation of the form
+Σ−2βy=βyf(Z) (15)
+wheref(·)is a function that can be given explicitely using
+the Stieltjis transform of the measure ρ(λ)dλ. Equation (15)
+implies that βyis block diagonal according to the degeneracy
+pattern of Σ. Considering each block, either βyvanishes in
+the block (a case that can be excluded using Σ2
+i> σ2/p)
+orΣ−2
+i=f(Zii)in the block. Solving for Ziishows that
+the eigenvalues are uniquely determined (independent of th e
+subsequence) and given by Eq. (6).
+In order to determine βxandβyfirst observe that, since
+Ir×r=YTY=/summationtextn
+i=1yiyT
+i, we have, using Eq. (10)
+Ir×r=n/summationdisplay
+i=1(/hatwideZ2−w2
+i)−1(/hatwideZˆβT
+xvi+wiˆβT
+yui)
+(vT
+iˆβx/hatwideZ+wiuT
+iˆβy)(/hatwideZ2−w2
+i)−1.
+In the limit n→ ∞, and assuming a convergent subsequence
+for(Z,βx,βy), this sum can be computed as above. After
+Ir×r=/braceleftBig/integraldisplayZ2
+(Z2−α3/2σ2λ)2ρ(λ)dλ/bracerightBig
+Cx
++/braceleftBig/integraldisplayα3/2σ2λ
+(Z2−α3/2σ2λ)2ρ(λ)dλ/bracerightBig
+Cy,
+whereCx=βT
+xβx,Cy=βT
+yβyand the functions of Zon
+the rhs are defined as standard analyic functions of matrices .
+Using Eqs. (13), (14) and solving the above, we get Cx=
+diag(Σ2
+1a2
+1,...Σ2
+ra2
+r), andBy= diag(Σ2
+1b2
+1,...Σ2
+rb2
+r). We
+already concluded that βxandβyare block diagonals with
+blocks in correspondence with the degeneracy pattern of Σ.
+SinceβT
+xβx=CxandβT
+yβy=Cyarediagonal,withthesame
+degeneracypattern, it follows that, inside each block of si zed,
+each ofβxandβyis proportionalto a d×dorthogonalmatrix.
+Therefore βx= ΣAQs,βy= ΣBQ′
+s, for some othogonal
+matriced Qs,Q′
+s. Also, using equation (13) one can prove
+thatQs=Q′
+s.
+Notice, by the above argument A,Bare uniquely fixed by
+our construction. On the other hand Qsmight depend on the
+subsequence Ξ. Since our statmement allows for a seqence
+of rotations Qn, that depend on n, the eventual subsequence
+dependence of Qscan be factored out.
+It is useful to point out a straightforward consequence of
+the above.
+Corollary III.3. There exists a sequence of orthogonal ma-
+tricesQn∈Rr×rsuch that, almost surely,
+lim
+n→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1√mnXTUΣVTY−QnDQT
+n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+F= 0,(16)
+withD= diag(Σ 1a1b1,...,Σrarbr).
+B. The effect of missing entries
+The proof of Theorem I.1 is completed by estabilishing a
+relation between the singular vectors X0,Y0ofNEand the
+singular vectors XandYof/hatwideN.Lemma III.4. Letk≤rbe the largest integer such that
+Σ1≥ ··· ≥Σk> σ2/p, and denote by X(k)
+0,Y(k)
+0,X(k), and
+Y(k)the matrices containing the first kcolumns of X0,Y0,
+X, andY, respectively. Let X(k)
+0=X(k)Sx+X(k)
+⊥,Y(k)
+0=
+Y(k)Sy+Y(k)
+⊥where(X(k)
+⊥)TX(k)= 0,(Y(k)
+⊥)TY(k)= 0
+andSx,Sy∈Rr×r. Then there exists a numerical constant
+C=C(Σi,σ2,α,Mmax), such that, with high probability,
+||X(k)
+⊥||2
+F,||Y(k)
+⊥||2
+F≤Cr/radicalbigg
+1
+n, (17)
+with probability approaching 1asn→ ∞.
+Proof:We will prove our claim for the right singular
+vectorY, since the left case is completely analogous. Further
+we will drop the superscript kto lighten the notation.
+We start by noticing that ||NEY0||2
+F=/summationtextk
+a=1(n˜za,n)2,
+wheren˜za,nare the singular values of NE. Using Lemma
+3.2 in [2] which bounds ||ME−pM||2=||NE−/hatwideN||2, we
+get
+||NEY0||2
+F≥k/summationdisplay
+a=1(nza,n−CMmax√pn)2.(18)
+On the other hand ||NEY0||F≤ ||/hatwideNY0||F+||NE−
+/hatwideN||2||Y0||F. Further by letting Sy=LyΘyRT
+y, forLy,Ry
+orthogonal matrices, we get ||/hatwideNY0||2
+F=||/hatwideNYLyΘy||2
+F+
+||/hatwideNY⊥||2
+F. Since YT
+0Y0=Ik×k, we have Ik×k=
+RyΘT
+yΘyRT
+y+YT
+⊥Y⊥, and therefore
+||/hatwideNY0||2
+F=||/hatwideNYLy||2
+F−||/hatwideNYLyRT
+yYT
+⊥||2
+F+||/hatwideNY⊥||2
+F
+≤n2k/summationdisplay
+a=1z2
+a,n−n2z2
+k,n||Y⊥||2
+F
++n2pσ2α(c+(α)+δ)||Y⊥||2
+F
+=n2k/summationdisplay
+a=1z2
+a,n−n2ey||Y⊥||2
+F,
+whereey≡z2
+k,n−pσ2α(c+(α)+δ), and used the inequality
+||/hatwideNY⊥||2
+F≤n2pσ2α(c+(α) +δ)||Y⊥||2
+Fwhich holds for all
+δ >0asymptotically almost surely as n→ ∞(by an
+immediate generalization of Lemma III.2). It is simple to
+check that Σk≥σ2/pimpliesey>0.
+Using triangular inequality, Lemma 3.2 in [2], we get
+||NY0||2
+F≤n2r/summationdisplay
+a=1z2
+a,n−n2ey||Y⊥||2
+F+Cnpα3/2M2
+maxr
++2Cn√npα3/4Mmax√r||z||,
+which, combined with equation (18), implies the thesis.
+Proof of Theorem I.1: We now turn to upper bounding
+the right hand side of Eq. (5). Let kbe defined as in the
+last lemma. Notice that by Lemma III.2, XT(UΣVT)Yis
+well approximated by (X(k))T(UΣVT)Y(k). Analogously, it
+can be proved that XT
+0(UΣVT)Y0is well approximated by
+(X(k)
+0)T(UΣVT)Y(k)
+0. Due to space limitations, we will omit
+this technical step and thus focus here on the case k=r(equivalently, neglect the error incurred by this approxim a-
+tion).
+Using Lemma III.4 to bound the contribution of X⊥,Y⊥,
+we have
+/a\}bracketle{tXT
+0(UΣVT)Y0, XT
+0NEY0/a\}bracketri}ht
+=/a\}bracketle{tST
+xXT(UΣVT)YSy, XT
+0NEY0/a\}bracketri}ht(1+on(1))
+=/a\}bracketle{tXT(UΣVT)Y , ST
+xXT
+0NEY0Sy/a\}bracketri}ht(1+on(1)).(19)
+FurtherXT
+0NEY0=XT
+0/hatwideNY0+XT
+0(NE−/hatwideN)Y0and, using
+once more the bound in Lemma 3.2 of [2], that implies
+|XT
+0(NE−/hatwideN)Y0| ≤Cr√nrp, we get
+ST
+xXT
+0NEY0Sy=LxΘ2
+xLT
+xXT/hatwideNYRyΘ2
+yRT
+y+E1
+=Z+E2,
+where we recall that Zis the diagonal matrix with entries
+given by the singular values of /hatwideN, and||E1||2
+F,||E2||2
+F≤
+C(p,r)√n. Using this estimate in Eq. (19), together with the
+result in Lemma III.2, we finally get
+/a\}bracketle{tXT
+0(UΣVT)Y0, XT
+0NEY0/a\}bracketri}ht√mn||XT
+0NEY0||2
+F≥/summationtextr
+k=1Σkakbkzk√α||z||−on(1),
+which implies the thesis after simple algebraic manipulati ons
+ACKNOWLEDGEMENTS
+We aregratefulto T.Hastie, R.MazumderandR. Tibshirani
+for stimulating discussions, and for making available thei r
+data. This work was supported by a Terman fellowship, and
+the NSF grants CCF-0743978 and DMS-0806211.
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+arXiv:1001.0280v2  [math.CO]  5 Apr 2010Symmetric chains, Gelfand-Tsetlin chains, and the Terwill iger
+algebra of the binary Hamming scheme
+Murali K. Srinivasan
+Department of Mathematics
+Indian Institute of Technology, Bombay
+Powai, Mumbai 400076, INDIA
+mks@math.iitb.ac.in
+Mathematics Subject Classifications: 05A15, 05A30.
+To the memory of Mobi
+Abstract
+The de Bruijn-Tengbergen-Kruyswijk (BTK) construction is a simp le algorithm that
+produces an explicit symmetric chain decomposition of a product of c hains. We linearize
+the BTK algorithm and show that it produces an explicit symmetric Jor dan basis (SJB).
+In the special case of a Boolean algebra the resulting SJB is orthogo nal with respect to the
+standard inner product and, moreover, we can write down an explic it formula for the ratio
+of the lengths of the successive vectors in these chains (i.e., the sin gular values). This yields
+a new, constructive proof of the explicit block diagonalization of the Terwilliger algebra of
+the binary Hamming scheme. We also give a representation theoretic characterization of
+this basis that explains its orthogonality, namely, that it is the canon ically defined (upto
+scalars) symmetric Gelfand-Tsetlin basis.
+1 Introduction
+The de Bruijn-Tengbergen-Kruyswijk (BTK) construction is a simple visual algorithm in
+matching theory that produces an explicit symmetric chain d ecomposition of a (finite) product
+of (finite) linear orders. We show that the BTK algorithm admi ts a simple and natural linear
+analog. The main purpose of this paper is to study the linear B TK algorithm as an object in
+itself. It enables us to explicitly study the up operator (i. e., the nilpotent operator taking an
+element to the sum of the elements covering it) on a product of linear orders by producing an
+explicit symmetric Jordan basis (SJB). In the special case o f a Boolean algebra the resulting
+SJB is orthogonal with respect to (wrt) the standard inner pr oduct and, moreover, we can
+write down an explicit formula for the ratio of the lengths of the successive vectors in these
+chains (i.e., the singular values). This yields a new constr uctive proof of the explicit block
+1diagonalization of the Terwilliger algebra of the binary Ha mming scheme, recently achieved
+by Schrijver. We also give a representation theoretic chara cterization of this basis that ex-
+plains its orthogonality, namely, that it is the canonicall y defined (upto scalars) symmetric
+Gelfand-Tsetlin basis (wrt the up operator on the Boolean al gebra).
+A (finite) graded poset is a (finite) poset Ptogether with a rank function r:P→Nsuch
+that ifqcoverspinPthenr(q) =r(p) + 1. The rankofPisr(P) = max {r(p) :p∈P}
+and, for i= 0,1,...,r(P),Pidenotes the set of elements of Pof ranki. Asymmetric chain
+in a graded poset Pis a sequence ( p1,...,ph) of elements of Psuch that picoverspi−1, for
+i= 2,...h, andr(p1)+r(ph) =r(P), ifh≥2, or else 2 r(p1) =r(P), ifh= 1. Asymmetric
+chain decomposition (SCD) of a graded poset Pis a decomposition of Pinto pairwise disjoint
+symmetric chains.
+We now define the linear analog of a SCD. For a finite set S, letV(S) denote the complex
+vector space with Sas basis. Let Pbe a graded poset with n=r(P). Then we have
+V(P) =V(P0)⊕V(P1)⊕ ··· ⊕V(Pn) (vector space direct sum). An element v∈V(P) is
+homogeneous ifv∈V(Pi) for some i, and we extend the notion of rank to homogeneous
+elements by writing r(v) =i. A linear map T:V(P)→V(P) is said to be order raising if,
+for allp∈P,T(p) is a linear combination of the elements covering p(note that this implies
+thatT(p) = 0 for all maximal elements of Pand that T(v) is homogeneous for homogeneous
+v). Theup operator U:V(P)→V(P) is defined, for p∈P, byU(p) =/summationtext
+qq, where the sum
+is over all qcovering p. LetTbe an order raising map on a graded poset P. Agraded Jordan
+chaininV(P) with respect to T(wrtTfor short) is a sequence v= (v1,...,vh) of nonzero
+homogeneous elements of V(P) such that T(vi−1) =vi, fori= 2,...h, andT(vh) = 0 (note
+that the elements of this sequence are linearly independent , being nonzero and of different
+ranks). We say that vstartsat rank r(v1) andendsat rank r(vh). If, in addition, vis
+symmetric, i.e., r(v1)+r(vh) =r(P), ifh≥2, or else 2 r(v1) =r(P), ifh= 1, we say that vis
+asymmetric Jordan chain . Agraded Jordan basis ofV(P) wrtTis a basis of V(P) consisting
+of a disjoint union of graded Jordan chains in V(P) wrtT. If every chain in a graded Jordan
+basis is symmetric we speak of a symmetric Jordan basis (SJB) of V(P) wrtT. WhenTis the
+up mapU, we drop the “wrt U” from the notation and speak of a graded Jordan basis/SJB
+ofV(P).
+Letnbe a positive integer and let k1,...,knbe nonnegative integers. Define
+M(n,k1,...,kn) ={(x1,...,x n)∈Nn: 0≤xi≤ki,for alli},
+and partially order it by componentwise ≤. The cardinality of M(n,k1,...,kn) is (k1+
+1)···(kn+1) and the rank of ( x1,...,x n) isx1+···+xn, sor(M(n,k1,...,kn)) =k1+···+kn.
+It is easily seen that M(n,k1,...,kn) is (order) isomorphic to a product of nchains of lengths
+k1,...,kn, respectively. Two special cases of M(n,k1,...,kn) are of interest: the uniformcase,
+wherek1=···=kn=kand we write M(n,k) forM(n,k,...,k ) and the Boolean algebra or
+setcase, where k1=···=kn= 1 and we write B(n) forM(n,1).
+Analgorithm toconstructanexplicitSCDof M(n,k1,...,kn)wasgiven bydeBruijn,Tengber-
+gen, and Kruyswijk [BTK, A, E] . We call this the BTK algorithm. Canfield [C], Proctor [P],
+and Proctor, Saks, and Sturtevant [PSS]proved the existence of a SJBof V(M(n,k1,...,kn)).
+2These authors work in the more general context of Sperner the ory which we do not need here.
+An overview of this area is given in Chapter 6 of Engel’s book [E]. In Section 3 we present a
+linear analog of the BTK algorithm and prove the following re sult.
+Theorem 1.1 For a positive integer nand nonnegative integers k1,...,kn, the linear BTK
+algorithm constructs an explicit SJB of V(M(n,k1,...,kn)). The vectors in this basis have
+integral coefficients when expressed in the standard basis M(n,k1,...,kn).
+When applied to the Boolean algebra B(n) the linear BTK algorithm has properties that go
+well beyond producing an SJB of V(B(n)). We now discuss this. Perhaps the linear BTK
+algorithm (or a variant) has interesting properties also in the general case but we are unable
+to say anything on this point here.
+When the linear BTK algorithm is run on B(n), the resulting SJB’s are all orthogonal wrt the
+standard inner product on B(n). Moreover, any two symmetric Jordan chains starting at ran k
+kand ending at rank n−k“look alike” in the sense made precise in the following resul t. Let
+/an}⌊ra⌋ketle{t,/an}⌊ra⌋ketri}htdenote the standard inner product on V(B(n)), i.e.,/an}⌊ra⌋ketle{tX,Y/an}⌊ra⌋ketri}ht=δ(X,Y), (Kronecker delta)
+forX,Y∈B(n). Thelength/radicalbig
+/an}⌊ra⌋ketle{tv,v/an}⌊ra⌋ketri}htofv∈V(B(n)) is denoted /⌊ard⌊lv/⌊ard⌊l. The following result is
+proved in Section 3. (In the formulation below item (ii) is cl early implied by item (iii) but for
+convenience of later reference we have spelt out item (ii) ex plicitly.)
+Theorem 1.2 LetO(n)be the SJB produced by the linear BTK algorithm when applied to
+B(n).
+(i) The elements of O(n)are orthogonal with respect to /an}⌊ra⌋ketle{t,/an}⌊ra⌋ketri}ht.
+(ii) Let0≤k≤ ⌊n/2⌋and let(xk,...,x n−k)and(yk,...,yn−k)be any two symmetric Jordan
+chains in O(n)starting at rank kand ending at rank n−k. Then
+/⌊ard⌊lxu+1/⌊ard⌊l
+/⌊ard⌊lxu/⌊ard⌊l=/⌊ard⌊lyu+1/⌊ard⌊l
+/⌊ard⌊lyu/⌊ard⌊l, k≤u < n−k.
+(iii) In the notation of part (ii) we have, for k≤u < n−k,
+/⌊ard⌊lxu+1/⌊ard⌊l
+/⌊ard⌊lxu/⌊ard⌊l=/radicalbig
+(u+1−k)(n−k−u) (1)
+= (n−k−u)/parenleftbiggn−2k
+u−k/parenrightbigg1
+2/parenleftbiggn−2k
+u+1−k/parenrightbigg−1
+2
+. (2)
+In a recent breakthrough, Schrijver [S]obtained new polynomial time computable upper
+bounds on binary code size using semidefinite programming an d the Terwilliger algebra (this
+approach was later extended to nonbinary codes in [GST]). There are two main steps involved
+here. The first is to (upper) bound binary code size by the opti mal value of an exponential
+size semidefinite program and the second is to reduce the semi definite program to polynomial
+size by explicitly block diagonalizing the Terwilliger alg ebra. For background on coding the-
+ory we refer to [GST, S] . In this paper we consider the second step. Theorem 1.2 conta ins
+3most of the information necessary to explicitly block diago nalize the Terwilliger algebra of the
+binary Hamming scheme. We show this in Section 2. Here we woul d like to add a few remarks
+about the present proof of explicit block diagonalization. There are two proofs available for
+this result: the linear algebraic proof of Schrijver and the representation theoretic proof of
+Vallentin [V]based on the work of Dunkl [D1, D2] . Our proof can be seen as a constructive
+version of Schrijver’s proof that also has representation t heoretic meaning (see Theorem 1.3
+below). The basic pattern of the proof is the same as in [S]. Given Theorem 1.2, the rest of
+the proof is a binomial inversion argument from [S]. On the other hand, though not explicitly
+stated in this form in [S], the existence of a SJB of V(B(n)) satisfying (i), (ii), and (iii) of
+Theorem 1.2 easily follows from the results in [S]. So the new ingredient here is the explicit
+construction of the SJB O(n) and its representation theoretic characterization in The orem 1.3
+below. We remark here that this explicit construction is pri marily of mathematical interest
+and is not important from the complexity point of view since e ven to write down O(n) takes
+exponential time. We rewrite equation (1) as equation (2) so that the final formula for the
+block diagonalization turns out to equal Schrijver’s which is in a very convenient form with
+respect to the location of square roots (see Theorem 2.2).
+Our proof of Theorem 1.2 is self-contained and elementary bu t more insight into the result is
+obtained by using a bit of representation theory. We first giv e a short proof of existence of a
+SJB ofV(B(n)) satisfying (i), (ii), and (iii) of Theorem 1.2 by putting t ogether standard and
+well-known facts from representation theory.
+Define the down operator DonV(B(n)) analogous to the up operator and define the operator
+HonV(B(n)) byH(vi) = (2i−n)vi, vi∈V(B(n)i), i= 0,1,...,n. It is easy to check that
+[H,U] = 2U, [H,D] =−2D, and [U,D] =H. Thus the linear map sl(2 ,C)→gl(V(B(n)))
+given by/parenleftbigg0 1
+0 0/parenrightbigg
+/mapsto→U,/parenleftbigg0 0
+1 0/parenrightbigg
+/mapsto→D,/parenleftbigg1 0
+0−1/parenrightbigg
+/mapsto→H
+is a representation of sl(2 ,C). Decompose V(B(n)) into irreducible sl(2 ,C)-submodules and
+letWbe an irreducible in this decomposition with dimension l+1. It follows from the repre-
+sentation theory of sl(2 ,C) (see Section 2.3.1 in [GW]) that there exists a basis {v0,v1,...,vl}
+ofWsuch that, for i= 0,1,...,l, we have (below we take v−1=vl+1= 0)
+U(vi) =vi+1, D(vi) =i(l−i+1)vi−1, H(vi) = (2i−l)vi (3)
+So the eigenvalues of Honv0,v1,...,vlare, respectively, −l,−l+2,...,l−2,l. It now follows
+fromthedefinitionof Hthateach viishomogeneousandthat( v0,...,vl)isasymmetricJordan
+chain in V(B(n)). Thus there exists a SJB of V(B(n)). Note also that the basis {v0,...,vl}
+ofWis canonically determined (upto a common scalar multiple) s ince the eigenvalues of the
+vionHare distinct. Let r(v0) =kand put xj=vj−k, k≤j≤n−k. The symmetric Jordan
+chain (v0,...,vl) now gets rewritten as ( xk,...,x n−k) and (3) becomes, for k≤u≤n−k,
+U(xu) =xu+1, D(xu) = (u−k)(n−k−u+1)xu−1. (4)
+Now we use the substitution action of the symmetric group SnonB(n). As is easily seen the
+existence of a SJB of V(B(n)) satisfying (i) and (ii) of Theorem 1.2 follows from the fol lowing
+facts by an application of Schur’s lemma:
+4(a) Existence of some SJB of V(B(n)).
+(b)UisSn-linear.
+(c) For 0 ≤k≤n,V(B(n)k) is the sum of min {k,n−k}+1 distinct irreducible Sn-modules
+(this result is well known).
+(d) For a finite group G, aG-invariant inner product on an irreducible G-module is unique
+upto scalars.
+We now again use the sl(2 ,C) action to prove Theorem 1.2(iii). Let J(n) be a SJB of V(B(n))
+satisfying parts (i) and (ii) of Theorem 1.2. Normalize J(n) to get an orthonormal basis J′(n)
+ofV(B(n)). Let ( xk,...,x n−k),xu∈V(B(n)u)) for all u, be a symmetric Jordan chain in
+J(n). Putx′
+u=xu
+/bardblxu/bardblandαu=/bardblxu+1/bardbl
+/bardblxu/bardbl, k≤u≤n−k(we take xk−1=xn−k+1= 0). We have,
+fork≤u≤n−k,
+U(x′
+u) =U(xu)
+/⌊ard⌊lxu/⌊ard⌊l=xu+1
+/⌊ard⌊lxu/⌊ard⌊l=αux′
+u+1. (5)
+Now observe that the matrices, in the standard basis, of UandDare real and transposes of
+each other. Since J′(n) is orthonormal wrt the standard inner product, it follows t hat the
+matrices of UandD, in the basis J′(n), must be adjoints of each other. Thus we must have,
+using (5), D(x′
+u+1) =αux′
+u. So the subspace spanned by {xk,...,x n−k}is an irreducible
+sl(2,C)-module and formula (4) applies. We have
+DU(x′
+u) =α2
+ux′
+u= (u+1−k)(n−k−u)x′
+u
+and thus αu=/radicalbig
+(u+1−k)(n−k−u).
+Using a little bit more representation theory we can give a ch aracterization of O(n) among all
+SJB’s satisfying Theorem 1.2. Consideran irreducible Sn-module V. By the branchingrulethe
+decomposition of Vintoirreducible Sn−1-modulesis multiplicity freeandisthereforecanonical.
+Each ofthesemodules, inturn, decomposecanonically intoi rreducible Sn−2-modules. Iterating
+this construction we get a canonical decomposition of Vinto irreducible S1-modules, i.e., one
+dimensional subspaces. Thus, there is a canonical basis of V, determined upto scalars, and
+called the Gelfand-Tsetlin or Young basis (GZ-basis) (see[VO]). Note that the GZ-basis is
+orthogonal wrt the (unique upto scalars) Sn-invariant inner product on V. We now observe
+the following:
+(i) Iff:V→Wis aSn-linear isomorphism between irreducibles V,Wthen the GZ-basis of
+Vgoes to the GZ-basis of W.
+(ii) LetVbe aSn-module whose decomposition into irreducibles is multipli city free. By the
+GZ-basis of Vwe mean the union of the GZ-bases of the various irreducibles occuring in the
+(canonical) decomposition of Vinto irreducibles. Then the GZ-basis of Vis orthogonal wrt
+anySn-invariant inner product on V.
+Now consider the Snaction on V(B(n)). Since UisSn-linear, the action is multiplicity free
+onV(B(n)k), for all k, and there exists a SJB of V(B(n)), it follows from points (i) and
+(ii) above that there is a canonically defined (upto scalars) orthogonal SJB of V(B(n)) that
+5consists of the union of the GZ-bases of V(B(n)k), 0≤k≤n. We call this basis the symmetric
+Gelfand-Tsetlin basis ofV(B(n)). We prove the following result in Section 4.
+Theorem 1.3 The SJB O(n), produced by the linear BTK algorithm when applied to the
+Boolean algebra B(n), is the symmetric Gelfand-Tsetlin basis of V(B(n)).
+In Example 3.4 in Section 3 we write down the symmetric Gelfan d-Tsetlin bases of V(B(n))
+forn= 1,2,3,4.
+2 Terwilliger algebra of the binary Hamming scheme
+TheTerwilliger algebra of the binary Hamming scheme , denoted Tn, is defined to be the
+commutant of the Snaction on B(n), i.e.,Tn= End Sn(V(B(n))) (in this definition the order
+structure on B(n) is irrelevant). In this section we explicitly block diagon alizeTn. It is
+convenient to think of B(n) as the poset of subsets of the set {1,2,...,n}.
+Being the commutant of a finite group action Tnis aC∗-algebra. Note that Tnis noncom-
+mutative (since, for example, the trivial representation o ccurs at every rank and thus more
+than once). Let us first describe Tnin matrix terms. We represent elements of End( V(B(n)))
+(in the standard basis) as B(n)×B(n) matrices (we think of elements of V(B(n)) as column
+vectors with coordinates indexed by B(n)). ForX,Y∈B(n), the entry in row X, column Y
+of a matrix Mwill be denoted M(X,Y). The matrix corresponding to f∈End(V(B(n))) is
+denoted Mf.
+Lemma 2.1 Letf:V(B(n))→V(B(n))be a linear map. Then fisSn-linear if and only if
+Mf(X,Y) =Mf(π(X),π(Y)),for allX,Y∈B(n), π∈Sn.
+Proof(only if) For Y∈B(n) we have f(Y) =/summationtext
+X∈B(n)Mf(X,Y)X. SincefisSn-linear we
+now have, for π∈Sn,
+/summationdisplay
+X∈B(n)Mf(X,π(Y))X=f(π(Y)) =π(f(Y)) =/summationdisplay
+X∈B(n)Mf(X,Y)π(X).
+It follows that Mf(X,π(Y)) =Mf(π−1(X),Y). Thus we have Mf(π(X),π(Y)) =Mf(X,Y).
+(if) Similar to the only if part. ✷
+DefineAnto be the set of all B(n)×B(n) complex matrices Msatisfying M(X,Y) =
+M(π(X),π(Y)), for all X,Y∈B(n), π∈Sn. It follows from Lemma 2.1 that Anis aC∗-
+algebra of matrices isomorphic to Tn. We can easily determine its dimension. For nonnegative
+integersi,j,tletMt
+i,jbe theB(n)×B(n) matrix given by
+Mt
+i,j(X,Y) =/braceleftbigg1 if|X|=i,|Y|=j,|X∩Y|=t
+0 otherwise
+6Given (X,Y),(X′,Y′)∈B(n)×B(n), there exists π∈Snwithπ(X) =X′, π(Y) =Y′if and
+only if|X|=|X′|,|Y|=|Y′|, and|X∩Y|=|X′∩Y′|. It follows that
+{Mt
+i,j|i−t+t+j−t≤n, i−t,t,j−t≥0}
+is a basis of Anand its cardinality is/parenleftbign+3
+3/parenrightbig
+.
+It follows from general C∗-algebra theory that there exists a block diagonalization ofAn, i.e.,
+there exists a B(n)×Sunitary matrix N(n), for some index set Sof cardinality 2n, and
+positive integers p0,q0,...,pm,qmsuch that N(n)∗AnN(n) is equal to the set of all S×S
+block-diagonal matrices
+
+C00...0
+0C1...0
+............
+0 0 ... C m
+(6)
+where each Ckis a block-diagonal matrix with qkrepeated, identical blocks of order pk
+Ck=
+Bk0...0
+0Bk...0
+............
+0 0 ... B k
+(7)
+Thusp2
+0+···+p2
+m= dim(An) andp0q0+···+pmqm= 2n. The numbers p0,q0,...,pm,qm
+andmare uniquely determined (upto permutation of the indices) b yAn.
+By dropping duplicate blocks we get a positive semidefiniten ess preserving C∗-algebra isomor-
+phism (below Mat( n×n) denotes the algebra of complex n×nmatrices)
+Φ :An∼=m/circleplusdisplay
+k=0Mat(pk×pk).
+In anexplicit block diagonalization we need to know this isomorphism explicitly, i.e., we need to
+know the entries in the image of Mt
+i,j. In[S], an explicit block diagonalization was determined.
+We now show that this result follows from Theorem 1.2. The pre sent proof also yields an
+explicit, canonical (real) unitary matrix N(n) achieving the isomorphism Φ.
+The first step is a binomial inversion argument. Fix i,j∈ {0,...,n}. Then we have
+Mt
+i,tMt
+t,j=n/summationdisplay
+u=0/parenleftbiggu
+t/parenrightbigg
+Mu
+i,j, t= 0,...,n,
+since the entry of the lhs in row X, colYwith|X|=i,|Y|=jis equal to the number of
+common subsets of XandYof sizet. Apply binomial inversion to get
+Mt
+i,j=n/summationdisplay
+u=0(−1)u−t/parenleftbiggu
+t/parenrightbigg
+Mu
+i,uMu
+u,j, t= 0,...,n. (8)
+7SinceMu
+u,j= (Mu
+j,u)tand it will turn out that N(n) can be taken to be real (see the definition
+below) it follows that
+Φ(Mt
+i,j) =n/summationdisplay
+u=0(−1)u−t/parenleftbiggu
+t/parenrightbigg
+Φ(Mu
+i,u)Φ(Mu
+j,u)t, t= 0,...,n, (9)
+and hence all the images under Φ can be calculated by knowing t he images Φ( Mu
+i,u).
+For the second step we useTheorem 1.2 whosenotation we prese rve. For therest of this section
+setm=⌊n/2⌋, andpk=n−2k+1, qk=/parenleftbign
+k/parenrightbig
+−/parenleftbign
+k−1/parenrightbig
+, k= 0,...,m. Note that
+m/summationdisplay
+k=0p2
+k=/parenleftbiggn+3
+3/parenrightbigg
+, (10)
+since both sides are polynomials in l(treating the cases n= 2landn= 2l+1 separately) of
+degree 3 and agree for l= 0,1,2,3.
+For 0≤k≤m,O(n) will contain qksymmetric Jordan chains, each containing pkvectors,
+starting at rank kand ending at rank n−k. We can formalize this as follows: define the finite
+setS={(k,b,i)|0≤k≤m,1≤b≤qk, k≤i≤n−k}. For each 0 ≤k≤m, fix some linear
+ordering of the qkJordan chains of O(n) going from rank kto rankn−k. Then there is a
+bijection B:O(n)→Sdefined as follows: let v∈O(n). Then B(v) = (k,b,i), where i=r(v)
+andvoccurs on the bth symmetric Jordan chain going from rank kto rankn−k(there are
+unique such k,b). Linearly order Sas follows: ( k,b,i)<ℓ(k′,b′,i′) iffk < k′ork=k′,b < b′
+ork=k′,b=b′,i < i′. Form a B(n)×SmatrixN(n) as follows: the columns of N(n) are the
+normalized imagesB−1(s)
+/bardblB−1(s)/bardbl,s∈Slisted in increasingorder (of <ℓ). By Theorem 1.2(i), N(n) is
+unitary. Since the action of Mu
+i,uonV(B(n)u) is1
+(i−u)!times the action of Ui−uonV(B(n)u),
+it follows by Theorem 1.2(ii) and identities (8), (10) above that conjugating by N(n) provides
+a block diagonalization of Anof the form (6), (7) above. Set Φ equal to conjugation by N(n)
+followed by dropping duplicate blocks. To calculate the ima ges under Φ we shall now use part
+(iii) of Theorem 1.2.
+Fori,j,k,t∈ {0,...,n}define
+βt
+i,j,k=n/summationdisplay
+u=0(−1)u−t/parenleftbiggu
+t/parenrightbigg/parenleftbiggn−2k
+u−k/parenrightbigg/parenleftbiggn−k−u
+i−u/parenrightbigg/parenleftbiggn−k−u
+j−u/parenrightbigg
+.
+For 0≤k≤mandk≤i,j≤n−k, defineEi,j,kto be the pk×pkmatrix, with rows and
+columns indexed by {k,k+1,...,n−k}, and with entry in row iand column jequal to 1 and
+all other entries 0.
+Theorem 2.2 (Schrijver [S])Leti,j,t∈ {0,...,n}. Write
+Φ(Mt
+i,j) = (N0,...,N m),
+where, for k= 0,...,m, the rows and columns of Nkare indexed by {k,k+ 1,...,n−k}.
+Then, for 0≤k≤m,
+Nk=/braceleftigg/parenleftbign−2k
+i−k/parenrightbig−1
+2/parenleftbign−2k
+j−k/parenrightbig−1
+2βt
+i,j,kEi,j,kifk≤i,j≤n−k
+0 otherwise
+8ProofFix 0≤k≤m. If both i,jare not elements of {k,...,n−k}then clearly Nk= 0. So
+we may assume k≤i,j≤n−k. Clearly, Nk=λEi,j,kfor some λ. We now find λ=Nk(i,j).
+Letu∈ {0,...,n}. Write Φ( Mu
+i,u) = (Au
+0,...,Au
+m). We claim that
+Au
+k=/braceleftigg/parenleftbign−k−u
+i−u/parenrightbig/parenleftbign−2k
+u−k/parenrightbig1
+2/parenleftbign−2k
+i−k/parenrightbig−1
+2Ei,u,kifk≤u≤n−k
+0 otherwise
+The otherwise part of the claim is clear. If k≤u≤n−kandi < uthen we have Au
+k= 0. This
+also follows from the rhs since the binomial coefficient/parenleftbiga
+b/parenrightbig
+is 0 forb <0. So we may assume
+thatk≤u≤n−kandi≥u. Clearly, in this case we have Au
+k=αEi,u,k, for some α. We now
+determine α=Au
+k(i,u). We have using Theorem 1.2(iii)
+Au
+k(i,u) =/producttexti−1
+w=u/braceleftbigg
+(n−k−w)/parenleftbign−2k
+w−k/parenrightbig1
+2/parenleftbign−2k
+w+1−k/parenrightbig−1
+2/bracerightbigg
+(i−u)!=/parenleftbiggn−k−u
+i−u/parenrightbigg/parenleftbiggn−2k
+u−k/parenrightbigg1
+2/parenleftbiggn−2k
+i−k/parenrightbigg−1
+2
+Similarly, if we write Φ( Mu
+u,j) = (Bu
+0,...,Bu
+m) then, since Mu
+u,j= (Mu
+j,u)t, we have
+Bu
+k=/braceleftigg/parenleftbign−k−u
+j−u/parenrightbig/parenleftbign−2k
+u−k/parenrightbig1
+2/parenleftbign−2k
+j−k/parenrightbig−1
+2Eu,j,kifk≤u≤n−k
+0 otherwise
+It now follows from (8) that Nk=/summationtextn
+u=0(−1)u−t/parenleftbigu
+t/parenrightbig
+Au
+kBu
+k=/summationtextn−k
+u=k(−1)u−t/parenleftbigu
+t/parenrightbig
+Au
+kBu
+k.Thus
+Nk(i,j)
+=n−k/summationdisplay
+u=k(−1)u−t/parenleftbiggu
+t/parenrightbigg/braceleftiggn−k/summationdisplay
+l=kAu
+k(i,l)Bu
+k(l,j)/bracerightigg
+=n−k/summationdisplay
+u=k(−1)u−t/parenleftbiggu
+t/parenrightbigg
+Au
+k(i,u)Bu
+k(u,j)
+=n−k/summationdisplay
+u=k(−1)u−t/parenleftbiggu
+t/parenrightbigg/parenleftbiggn−k−u
+i−u/parenrightbigg/parenleftbiggn−2k
+u−k/parenrightbigg1
+2/parenleftbiggn−2k
+i−k/parenrightbigg−1
+2/parenleftbiggn−k−u
+j−u/parenrightbigg/parenleftbiggn−2k
+u−k/parenrightbigg1
+2/parenleftbiggn−2k
+j−k/parenrightbigg−1
+2
+=/parenleftbiggn−2k
+i−k/parenrightbigg−1
+2/parenleftbiggn−2k
+j−k/parenrightbigg−1
+2/braceleftiggn/summationdisplay
+u=0(−1)u−t/parenleftbiggu
+t/parenrightbigg/parenleftbiggn−k−u
+i−u/parenrightbigg/parenleftbiggn−k−u
+j−u/parenrightbigg/parenleftbiggn−2k
+u−k/parenrightbigg/bracerightigg
+.
+3 The linear BTK algorithm
+In this section we present the linear analog of the BTK algori thm for constructing a SJB of
+V(M(n,k1,...,kn)). Though we do not recall here the BTK algorithm for constru cting a SCD
+9ofM(n,k1,...,kn) (see[BTK, A, E] ), readers familiar with that method will easily recognize
+the present algorithm as its linear analog.
+The basic building block of the linear BTK algorithm is an ind uctive method for constructing
+a SJB of V(M(2,p,q)). Ifporq= 0, then M(2,p,q) is order isomorphic to a chain and the
+characteristic vectors of the elements of the chain form a SJ B . For positive p,qwe shall now
+reduce the problem of constructing a SJB of V(M(2,p,q)) to that of constructing a SJB of
+V(M(2,p−1,q−1)). We begin with the following elementary lemma on determi nants.
+Lemma 3.1 LetN= (ai,j)be an×nreal matrix, n≥2. Suppose that
+(i)ai,1>0, fori∈ {1,...,n}, i.e., the first column contains positive entries.
+(ii) Forj∈ {2,...,n},aj,j>0,aj−1,j<0, and all other entries in column jare 0.
+Then det (N)>0.
+ProofBy induction on n. The assertion is clear for n= 2. Now assume that n >2. Let
+N[i,j] denote the ( n−1)×(n−1) matrix obtained from Nby deleting row iand column j.
+Expanding det( N) by the first row we get
+det(N) =a1,1det(N[1,1])−a1,2det(N[1,2])
+Nowa1,1>0,a1,2<0,N[1,1] is upper triangular with positive diagonal entries, and
+det(N[1,2])>0 (by induction hypothesis). The result follows. ✷
+Letp,qbe positive and set P=M(2,p,q),W=V(P) with up operator U. Letrdenote the
+rank function of P. We have dim W= (p+1)(q+1). The action of Uon the standard basis
+ofWis given as follows: for 0 ≤i≤p, 0≤j≤q
+U((i,j)) =
+
+(i+1,j)+(i,j+1) ifi < p,j < q
+(i+1,j) if i < p,j=q
+(i,j+1) if i=p,j < q
+0 if i=p,j=q
+Consider the following symmetric Jordan chain in Wgenerated by v(0) = (0,0):
+(v(0),v(1),v(2),...,v(p+q)),
+where, for 0 ≤k≤p+q,
+v(k) =Uk((0,0)) =/summationdisplay
+i,j/parenleftbiggk
+i/parenrightbigg
+(i,j), (11)
+the sum being over all 0 ≤i≤p, 0≤j≤qwithi+j=k.
+The following result is basic to our inductive approach.
+Theorem 3.2 Define homogeneous vectors in Was follows:
+v(i,j) = (p−i)(i,j)−(q−j+1)(i+1,j−1),0≤i≤p−1,1≤j≤q. (12)
+10Then
+(i)v(i,j)is nonzero and r(v(i,j)) =i+j,0≤i≤p−1,1≤j≤q.
+(ii){v(k)|0≤k≤p+q}∪{v(i,j)|0≤i≤p−1,1≤j≤q}is a basis of W.
+(iii) For 0≤i≤p−1,1≤j≤qwe have
+U(v(i,j)) =
+
+v(i+1,j)+v(i,j+1)ifi < p−1,j < q
+v(i+1,j) ifi < p−1,j=q
+v(i,j+1) ifi=p−1,j < q
+0 ifi=p−1,j=q
+Thus, the action of Uon thev(i,j)is isomorphic to the action of the up operator on the
+standard basis of V(M(2,p−1,q−1)), except that the map (i,j)/mapsto→v(i,j+ 1),(i,j)∈
+M(2,p−1,q−1)shifts ranks by one (sincer(v(i,j+1)) =i+j+1).
+Proof(i) This is clear.
+(ii) For 0 ≤k≤p+qdefine
+Xk={v(k)}∪{v(i,j) : 0≤i≤p−1,1≤j≤q, i+j=k}
+The map φk:Xk→M(2,p,q)kgiven by φk(v(i,j)) = (i,j) and
+φk(v(k)) =/braceleftbigg(k,0) if k < p
+(p,k−p) ifk≥p
+is clearly a bijection. It is enough to show that Xkis a basis of V(M(2,p,q)k). Linearly order
+the elements of M(2,p,q)kusing reverse lexicographic order <r: (i,j)<r(i′,j′) if and only if
+i > i′. Transfer this order to Xkviaφ−1
+k. Consider the M(2,p,q)k×XkmatrixN, with rows
+and columns listed in the order <r, and whose columns are the coordinate vectors of elements
+ofXkin the standard basis M(2,p,q)kofV(M(2,p,q)k). Equation (11) shows that hypothesis
+(i) of Lemma 3.1 is satisfied and equation (12) shows that hypo thesis (ii) of Lemma 3.1 is
+satisfied. The result now follows from Lemma 3.1.
+(iii) We check the first case. The other cases are similar. Let i < p−1,j < q. Then
+U(v(i,j))
+=U((p−i)(i,j)−(q−j+1)(i+1,j−1))
+= (p−i)((i+1,j)+(i,j+1))−(q−j+1)((i+2,j−1)+(i+1,j))
+= (p−i−1)(i+1,j)−(q−j+1)(i+2,j−1)+(p−i)(i,j+1)−(q−j)(i+1,j)
+=v(i+1,j)+v(i,j+1),
+completing the proof. ✷
+Theorem3.2, whosenotation wepreserve, gives thefollowin ginductivemethodforconstructing
+a SJB of V(M(2,p,q)): ifporqequals 0, the chain M(2,p,q) itself gives a SJB. Now suppose
+p,q >1. Setv(0) = (0,0) and form the symmetric Jordan chain C= (v(0),v(1),...,v(p+q)),
+11wherev(k) is given by (11). Take the (inductively constructed) SJB of V(M(2,p−1,q−1))
+and (using parts (ii) and (iii) of Theorem 3.2) transfer each Jordan chain in this SJB to a
+Jordan chain in V(M(2,p,q)) via the map ( i,j)/mapsto→v(i,j+ 1). Since r(v(0,1)) = 1 and
+r(v(p−1,q)) =p+q−1, each such transferred Jordan chain is symmetric in V(M(2,p,q)) and
+part (ii) of Theorem 3.2 now shows that the collection of thes e chains together with Cgives a
+SJB ofV(M(2,p,q)).
+Example 3.3 (i) HereweworkouttheSJBof V(M(2,2,2)) producedbythealgorithm above.
+The symmetric Jordan chain generated by (0 ,0) is given by
+((0,0),(1,0) +(0,1),(2,0) +2(1,1) +(0,2),3(2,1)+3(1,2),6(2,2)). (13)
+Thev(i,j) are given by
+v(0,1) = 2(0,1)−2(1,0)v(1,1) = (1,1)−2(2,0)
+v(0,2) = 2(0,2)−(1,1)v(1,2) = (1,2)−(2,1)
+The SJB of V(M(2,1,1)) is given by the following two chains
+((0,0),(1,0)+(0,1),2(1,1))
+((0,1)−(1,0))
+Transferring these chains to V(M(2,2,2)) via the map ( i,j)/mapsto→v(i,j+1) gives the following
+two chains
+(v(0,1), v(1,1) +v(0,2),2v(1,2)) (14)
+(v(0,2)−v(1,1)) (15)
+Chains (13, 14, 15) give a SJB of V(M(2,2,2)).
+(ii) Theprocedureabove is especially simplewhen, say q= 1, as in this case the recursion stops
+right at the first stage. For later reference we spell out this case in detail. Consider M(2,n,1).
+Define a 1-1 linear map V(M(2,n,0))→V(M(2,n,1)) by (i,0)/mapsto→(i,1),(i,0)∈M(2,n,0).
+Forv∈V(M(2,n,0)), we denote the image of vunder this map by v.
+Let (x0,x1,...,x n), where xi= (i,0) be the SJB of V(M(2,n,0)). Set x−1=xn+1= 0. We
+now consider two cases:
+(a)n= 0 : In this case
+(x0,x0). (16)
+is the SJB of V(M(2,n,1)) produced by Theorem 3.2.
+(b)n≥1 : The symmetric Jordan chain in V(M(2,n,1)) generated by (0 ,0) can be written
+as (using (11))
+(y0,y1,...,yn+1),whereyl=xl+lxl−1,0≤l≤n+1. (17)
+The SJB of V(M(2,n,1)) produced by Theorem 3.2 is given by (17) and the following sym-
+metric Jordan chain:
+(z1,...,zn),wherezl= (n−l+1)xl−1−xl,1≤l≤n. (18)
+12We can now give the full linear BTK algorithm which reduces th e general case to the n= 2
+case.
+Proof of Theorem 1.1 The proof is by induction on n, the case n= 1 being clear and the
+casen= 2 established above. Let P=M(n,k1,...,kn),n≥3 and set V=V(P). Denote the
+rank function of Pbyr. Define induced subposets P(j) ofPby
+P(j) ={(a1,...,a n)∈P:an=j},0≤j≤kn,
+and setV(j) =V(P(j)). LetUdenote the up operator on VandUjdenote the up operator
+onV(j), 0≤j≤kn. Note that all the P(j) are order isomorphic. We have
+V=V(0)⊕V(1)⊕···⊕V(kn). (19)
+For 0≤j < kndefine linear isomorphisms Rj:V(j)→V(j+1) by
+Rj((a1,...,an−1,j)) = (a1,...,an−1,j+1),(a1,...,an−1,j)∈P(j)
+PutV(kn+1) ={0}and define Rknto be the zero map.
+Forv∈V(j),0≤j≤knwe have
+U(v) =Uj(v)+Rj(v) (20)
+Uj+1Rj(v) =RjUj(v) (21)
+By induction there is a SJB of V(0) (wrt U0). Lettdenote the number of symmetric Jordan
+chains in this SJB, with the mthchain denoted by
+S(0,m) = (v(0,0,m),v(1,0,m),...,v(lm,0,m)),1≤m≤t,
+wherelm≥0 is the length of the mthsymmetric Jordan chain. Thus
+r(v(0,0,m)) +r(v(lm,0,m)) =k1+···+kn−1iflm>0
+2r(v(0,0,m)) =k1+···+kn−1 iflm= 0
+Define subspaces X(0,m)⊆V(0) by
+X(0,m) = Span S(0,m) = Span {v(0,0,m),...,v(lm,0,m)},1≤m≤t
+We have V(0) =X(0,1)⊕···⊕X(0,t). Note that dim X(0,m) =lm+1.
+For 1≤m≤t,1≤j≤kn,0≤i≤lmdefine, by induction on j(starting with j= 1),
+v(i,j,m) =Rj−1(v(i,j−1,m)) (22)
+S(j,m) = (v(0,j,m),v(1,j,m),...,v(lm,j,m))
+X(j,m) = Span S(j,m) = Span {v(0,j,m),...,v(lm,j,m)} (23)
+Since the Rj, 0≤j < knare isomorphisms we have
+{v(0,j,m),...,v(lm,j,m)}is independent ,0≤j≤kn,1≤m≤t. (24)
+V(j) =X(j,1)⊕X(j,2)⊕···⊕X(j,t),0≤j≤kn. (25)
+13For 1≤m≤t,1≤j≤knwe see from (24) and the following inductive calculation on j(using
+(21)) that each S(j,m) is a graded Jordan chain in V(j) (below we take v(lm+1,0,m) = 0,
+for allm)
+Uj(v(i,j,m)) =UjRj−1(v(i,j−1,m))
+=Rj−1Uj−1(v(i,j−1,m))
+=Rj−1(v(i+1,j−1,m))
+=v(i+1,j,m) (26)
+Form= 1,...,tdefine subspaces Y(m)⊆Vby
+Y(m) =X(0,m)⊕X(1,m)⊕···⊕X(kn,m). (27)
+We have from (19) and (25) that
+V=Y(1)⊕···⊕Y(t). (28)
+It follows from (23), (24), and (27) that the set
+B(m) ={v(i,j,m) : 0≤i≤lm,0≤j≤kn},1≤m≤t
+is a basis of Y(m). Note that
+r(v(i,j,m)) =r(v(0,0,m)) +i+j,1≤m≤t,0≤i≤lm,0≤j≤kn.(29)
+Fix 1≤m≤t. Using (20), (22), and (26) we see that the action of Uon the basis B(m) is
+given by
+U(v(i,j,m)) =
+
+v(i+1,j,m)+v(i,j+1,m) ifi < lm,j < k n
+v(i+1,j,m) if i < lm,j=kn
+v(i,j+1,m) if i=lm,j < k n
+0 if i=lm,j=kn
+Sothisactionisisomorphictotheactionoftheupoperatoro nthestandardbasisof V(M(2,lm,kn)),
+except for the shift of rank given by (29). We can now use the al gorithm of Theorem 3.2 to
+construct an SJB of Y(m) wrtU. Since
+r(v(0,0,m)) +r(v(lm,kn,m)) =r(v(0,0,m)) +r(v(lm,0,m))+kn=k1+···+kn,
+each graded Jordan chain in this SJB is symmetric in V. From (28) it follows that the union
+of the SJB’s of Y(m) gives an SJB of V. That completes the proof. ✷
+Proof of Theorem 1.2 The proof is by induction of n. The result is clear for n= 1. We
+writeB(n) asM(n,k1,...,kn), where k1=···=kn= 1.
+Consider V=V(M(n+1,k1,...,kn+1)), withki= 1 for all i. We preserve the notation of the
+proof of Theorem 1.1. Let there be tsymmetric Jordan chains in the SJB of V(0) =V(B(n)).
+Forv∈V(0), we denote R0(v) =v(agreeing with the notation in Example 3.3(ii)).
+14We have
+V=V(0)⊕V(1)
+V(0) = X(0,1)⊕X(0,2)⊕···⊕X(0,t)
+V(1) = X(1,1)⊕X(1,2)⊕···⊕X(1,t)
+Since the inner product is standard we have
+/an}⌊ra⌋ketle{tu,v/an}⌊ra⌋ketri}ht=/an}⌊ra⌋ketle{tu,v/an}⌊ra⌋ketri}ht,/an}⌊ra⌋ketle{tu,v/an}⌊ra⌋ketri}ht= 0, u,v∈V(0). (30)
+It follows that V(0) is orthogonal to V(1). The subspaces X(0,1),...,X(0,t) are mutually
+orthogonal, by the induction hypothesis, and thus
+V=Y(1)⊕···⊕Y(t)
+is an orthogonal decomposition of V, where
+Y(m) =X(0,m)⊕X(1,m),1≤m≤t.
+Fix 1≤m≤t. The subspaces X(0,m) andX(1,m) are orthogonal but Theorem 3.2 will
+produce new symmetric Jordan chains from linear combinatio ns of vectors in X(0,m) and
+X(1,m). We now show that these too are orthogonal and satisfy (1). T his will prove the
+theorem.
+Write the mthsymmetric chain in V(0) as
+(xk,...,x n−k),0≤k≤ ⌊n/2⌋, (31)
+wherer(xk) =k. By induction hypothesis
+/an}⌊ra⌋ketle{txu+1,xu+1/an}⌊ra⌋ketri}ht
+/an}⌊ra⌋ketle{txu,xu/an}⌊ra⌋ketri}ht= (u+1−k)(n−k−u), k≤u < n−k. (32)
+We now consider two cases.
+(a)k=n−k: It follows from (16) (after changing nton−2kand shifting the rank by k)
+that the SJB of Y(m) will consist of the single symmetric Jordan chain
+(xk,xk). (33)
+We have
+/an}⌊ra⌋ketle{txk,xk/an}⌊ra⌋ketri}ht
+/an}⌊ra⌋ketle{txk,xk/an}⌊ra⌋ketri}ht=/an}⌊ra⌋ketle{txk,xk/an}⌊ra⌋ketri}ht
+/an}⌊ra⌋ketle{txk,xk/an}⌊ra⌋ketri}ht= 1 = (k+1−k)(n+1−k−k).
+(b)k < n−k: By (17, 18) the SJB of Y(m) will consist of the following two symmetric Jordan
+chains (after changing nton−2kand shifting the rank by k):
+(yk,...,yn+1−k),and (zk+1,...,zn−k), (34)
+15where
+yl=xl+(l−k)xl−1, k≤l≤n+1−k. (35)
+zl= (n−k−l+1)xl−1−xl, k+1≤l≤n−k. (36)
+andxk−1=xn+1−k= 0.
+Fork+1≤l≤n−kwe have from (30)
+/an}⌊ra⌋ketle{tyl,zl/an}⌊ra⌋ketri}ht= (n−k−l+1)(l−k)/an}⌊ra⌋ketle{txl−1,xl−1/an}⌊ra⌋ketri}ht−/an}⌊ra⌋ketle{txl,xl/an}⌊ra⌋ketri}ht= 0,
+where the last step follows from (32) upon substituting u=l−1. Thus
+{yk,...,yn+1−k,zk+1,...,zn−k}
+is an orthogonal basis of Y(m).
+We now check (1) for the case n+1. Let k≤u < n+1−k. Note that (32) holds for u=n−k
+also. Also note that in the following computation (in the las t but one step) we have used (32)
+whenu=k−1 (in which case the rhs is 0 and the lhs is ∞). This is permissible here because
+of the presence of the factor ( u−k)2.
+We have, using (30) and (32),
+/an}⌊ra⌋ketle{tyu+1,yu+1/an}⌊ra⌋ketri}ht
+/an}⌊ra⌋ketle{tyu,yu/an}⌊ra⌋ketri}ht=/an}⌊ra⌋ketle{txu+1+(u+1−k)xu, xu+1+(u+1−k)xu/an}⌊ra⌋ketri}ht
+/an}⌊ra⌋ketle{txu+(u−k)xu−1, xu+(u−k)xu−1/an}⌊ra⌋ketri}ht
+=/an}⌊ra⌋ketle{txu+1,xu+1/an}⌊ra⌋ketri}ht+(u+1−k)2/an}⌊ra⌋ketle{txu,xu/an}⌊ra⌋ketri}ht
+/an}⌊ra⌋ketle{txu,xu/an}⌊ra⌋ketri}ht+(u−k)2/an}⌊ra⌋ketle{txu−1,xu−1/an}⌊ra⌋ketri}ht
+=/angbracketleftxu+1,xu+1/angbracketright
+/angbracketleftxu,xu/angbracketright+(u+1−k)2
+1+/angbracketleftxu−1,xu−1/angbracketright
+/angbracketleftxu,xu/angbracketright(u−k)2
+=(u+1−k)(n−k−u)+(u+1−k)2
+1+(u−k)2
+(u−k)(n−k−u+1)
+= (u+1−k)(n+1−k−u)
+The calculation for/angbracketleftzu+1,zu+1/angbracketright
+/angbracketleftzu,zu/angbracketrightis similar and is omitted. ✷
+Example 3.4 In this example we work out the SJB’s of V(B(n)), forn= 2,3,4, starting with
+the SJB of V(B(1)), using the formulas (31, 33, 34, 35, 36) given in the proo f of Theorem 1.2.
+We write elements of B(n) as subsets of [ n] ={1,2,...,n}rather than as their characteristic
+vectors. Thus, for X⊆[n], we have R0(X) =X=X∪{n+1}.
+(i) The SJB of V(B(1)) is given by
+(∅,{1})
+(ii) The SJB of V(B(2)) consists of
+(∅,{1}+{2},2{1,2})
+({2}−{1})
+16(iii) The SJB of V(B(3)) consists of
+(∅,{1}+{2}+{3},2({1,2}+{1,3}+{2,3}),6{1,2,3})
+( 2{3}−{1}−{2},{1,3}+{2,3}−2{1,2})
+({2}−{1},{2,3}−{1,3})
+(iv) The SJB of V(B(4)) consists of (some of the chains are split across two line s)
+(∅,{1}+{2}+{3}+{4},2({1,2}+{1,3}+{1,4}+{2,3}+{2,4}+{3,4}),
+6({1,2,3}+{1,2,4}+{1,3,4}+{2,3,4}),24{1,2,3,4})
+( 3{4}−({1}+{2}+{3}),2({1,4}+{2,4}+{3,4})−2({1,2}+{1,3}+{2,3}),
+2({1,2,4}+{1,3,4}+{2,3,4})−6{1,2,3})
+( 2{3}−({1}+{2}),{1,3}+{2,3}−2{1,2}+2{3,4}−({1,4}+{2,4}),
+2({1,3,4}+{2,3,4})−4{1,2,4})
+({2}−{1},{2,3}−{1,3}+{2,4}−{1,4},2({2,3,4} −{1,3,4}) )
+({2,4}−{1,4}−{2,3}+{1,3})
+( 2({3,4}+{1,2})−({1,4}+{2,4}+{1,3}+{2,3}) )
+It may be verified that each of the SJB’s are orthogonal and sat isfy (1).
+4 Symmetric Gelfand-Tsetlin basis
+In this section we prove Theorem 1.3. The proof is an applicat ion of the Vershik-Okounkov
+[VO]theory of (complex) irreducible representations of the sym metric group. We first recall
+briefly (without proofs) those points of the theory which we n eed:
+(A) A direct elementary argument is given to show that branch ing from SntoSn−1is simple,
+i.e., multiplicity free. Once this is done we have the canoni cally defined (upto scalars) Gelfand-
+Tsetlin basis (or GZ-basis) of an irreducible Sn-module, as inthe introduction. As stated there,
+the GZ-basis of an irreducible representation Vis orthogonal wrt the unique (upto scalars)
+Sn-invariant inner product on V.
+(B) Denote by S∧
+nthe set of equivalence classes of finite dimensional complex irreducible
+representations of Sn. Denote by Lλthe irreducible Sn-module corresponding to λ∈S∧
+n.
+17We have identified a canonical basis, namely the GZ-basis, in each irreducible representation
+ofSn. A natural question at this point is to identify those elemen ts ofC[Sn] that act diago-
+nally in this basis (in every irreducible representation). In other words, consider the algebra
+isomorphism
+C[Sn]∼=/circleplusdisplay
+λ∈S∧nEnd(Lλ), (37)
+given by
+π/mapsto→(Lλπ→Lλ:λ∈S∧
+n), π∈Sn.
+Let D(Lλ) consist of all operators on Lλdiagonal in the GZ-basis of Lλ. The question above
+can now be stated as: what is the image under the isomorphism ( 37) of the subalgebra/circleplustext
+λ∈S∧nD(Lλ) of/circleplustext
+λ∈S∧nEnd(Lλ).
+LetZndenote the center of the algebra C[Sn] and set GZnequal to the subalgebra of C[Sn]
+generated by Z1∪Z2∪···∪Zn(where we have the natural inclusions S1⊆S2⊆ ···). It is easy
+to see that GZnis a commutative subalgebra of C[Sn]. It is called the Gelfand-Tsetlin algebra
+(GZ-algebra) of the inductive family of group algebras C[Sn]. It is proved that GZnis the
+image of/circleplustext
+λ∈S∧nD(Lλ) under the isomorphism (37) above, i.e., GZnconsists of all elements of
+C[Sn] that act diagonally in the GZ-basis in every irreducible re presentation of Sn. ThusGZn
+is a maximal commutative subalgebra of C[Sn] and its dimension is equal to/summationtext
+λ∈S∧ndimLλ.
+By a GZ-vector vwe mean an element of the GZ-basis of Lλ, for some λ∈S∧
+n. It follows that
+the GZ-vectors are the only vectors that are eigenvectors fo r the action of every element of
+GZn. Moreover, any GZ-vector is uniquely determined by the eige nvalues of the elements of
+GZnon this vector.
+(C) Fori= 1,2,...,ndefineXi= (1,i)+(2,i)+···+(i−1,i)∈C[Sn]. TheXi’s are called the
+Young-Jucys-Murphy elements (YJM-elements) and it is show n that they generate GZn. To a
+GZ-vector vwe associate the tuple α(v) = (a1,a2,...,a n), where ai= eigenvalue of Xionv.
+We callα(v) theweightofvand we set
+spec(n) ={α(v) :vis a GZ-vector }.
+It follows from step (B) above that, for GZ-vectors uandv,u=viffα(u) =α(v) and thus
+#spec(n) =/summationtext
+λ∈S∧ndimLλ. Given α∈spec(n) we denote by vα(∈Lλfor some unique
+λ∈S∧
+n) the GZ-vector with weight α.
+There is a natural equivalence relation ∼on spec(n): forα,β∈spec(n),
+α∼β⇔vαandvβbelong to the same irreducible Sn-module Lλfor some λ∈S∧
+n
+Clearly we have #(spec( n)/∼) = #S∧
+n= number of partitions of n.
+(D) In the final step we construct a bijection between spec( n) and tab( n) (= set of all standard
+Young tableaux on the letters {1,2,...,n}) sending weights in spec( n) to content vectors of
+standard Young tableaux showing, in particular, that the we ights are integral. Weights in
+spec(n) that are related by ∼go to standard Young tableaux of the same shape. We shall not
+use this step.
+18LetVbe aSn-module, not necessarily multiplicity free. For α= (a1,a2,...,a n)∈spec(n)
+define the weight space
+V(α) ={v∈V:Xi(v) =aiv, i= 1,...,n}.
+Note that, if Vis multiplicity free, then it follows from items (B), (C) abo ve that every weight
+space is either zero or one dimensional.
+Forλ∈S∧
+n, letV(λ)⊆Vdenote the isotypical component of Lλand let spec( n,λ) denote the
+set of all weights α∈spec(n) withvα∈Lλ. Basic properties of the weight spaces are given
+below.
+Lemma 4.1 LetV,WbeSn-modules. We have
+(i) Forλ∈S∧
+n,V(λ) =⊕α∈spec(n,λ)V(α)is an orthogonal decomposition of V(λ)under any
+Sn-invariant inner product on V.
+(ii)V=⊕α∈spec(n)V(α)is an orthogonal decomposition of Vunder any Sn-invariant inner
+product on V.
+(iii) Let λ∈S∧
+n, andα,β∈spec(n,λ). There is a canonical linear isomorphism fα,β:
+V(α)→V(β), unique upto scalars, satisfying the following property: f orv∈V(α), the subspace
+C[Sn]v∩V(β)is generated by fα,β(v).
+(iv) Letλ∈S∧
+n,α∈spec(n,λ), andf:V(λ)→W(λ)aSn-linear map. Then f(V(α))⊆W(α)
+and any linear map g:V(α)→W(α)has a unique Sn-linear extension g:V(λ)→W(λ).
+Proof(i) Fix a Sn-invariant inner producton Vand letV(λ) =W1⊕···⊕Wtbean orthogonal
+decomposition of V(λ) into irreducible submodules, with all Wiisomorphic to Lλ. Forα∈
+spec(n,λ) it follows from (B), (C) above that V(α)∩Wiis one dimensional for all iand that
+the span of these one dimensional subspaces is V(α). Since the GZ-bases of Wiare orthogonal
+the result follows.
+(ii)Follows frompart(i), sincethedecompositionof Vintoisotypical componentsisorthogonal.
+(iii) Let v∈V(α). Consider the irreducible submodule C[Sn]vofVgenerated by v. By item
+(B) above the subspace C[Sn]v∩V(β) is one dimensional, say generated by u. There is an
+elementa∈C[Sn] withav=u. The linear map
+V(α)→V(β), x/mapsto→ax (38)
+has the required properties. It is also clear that such a map i s unique upto scalars.
+(iv) The first part of the assertion is clear. For the second pa rt note that (38) implies that
+ghas atmost one Sn-linear extension. Dimension considerations now show that ghas exactly
+oneSn-linear extension. ✷
+Consider the poset M(n,k), on which the symmetric group Snacts by substitution. Set
+V=V(M(n,k)) andVi=V(M(n,k)i), 0≤i≤kn. Note that each Viis aSn-submodule of
+Vand, for α∈spec(n), we have V(α) =V0(α)⊕ ··· ⊕Vkn(α). Since the up operator Uis
+Sn-linear it follows from Lemma 4.1(iv) that
+U(Vi(α))⊆Vi+1(α),0≤i < kn, α ∈spec(n). (39)
+19LetC= (vi,vi+1,...,vkn−i), withr(vl) =lfor alll, be a symmetric Jordan chain in V. We
+say that Cis asymmetric Gelfand-Tsetlin chain if eachvlis a simultaneous eigenvector for
+the action of X1,X2,...,X n. It follows that vl∈Vl(α),i≤l≤kn−i, for some α∈spec(n).
+Since an SJB of Vexists, it follows from Lemma 4.1(ii) and (39) that there exi sts a SJB of V
+consisting of symmetric Gelfand-Tsetlin chains. However, in such a SJB the Gelfand-Tsetlin
+chains belonging to different weight spaces (of the same isoty pical component) need not be
+related to each other. Note that, for α∼β, α,β∈spec(n), ifCis a Gelfand-Tsetlin chain
+inV(α) thenfα,β(C) is a Gelfand-Tsetlin chain in V(β) (from (38) and the fact that Uis
+Sn-linear). We now add this condition to the definition.
+Asymmetric Gelfand-Tsetlin basis (SGZB) of Vis a SJB BofVsatisfying the following
+conditions:
+(a) Each vector in Bis a simultaneous eigenvector for the action of X1,...,X n.
+(b) Forα∼β, α,β∈spec(n), ifv∈B∩V(α), then some multiple of fα,β(v) is also in B.
+Clearly, aSGZBof Vexists (foreachisotypical componentchooseaSJBofanyone weightspace
+and transfer it to the other weightspaces via fα,β). In the special case of Boolean algebras this
+definition of SGZB coincides with the one given in the introdu ction (see proof of Theorem 1.3
+below).
+At this point two natural questions arise:
+(i) Is it possible to characterize in some way the SJB produce d by the linear BTK algorithm?
+(ii) Is there an explicit construction (inductive or direct ) of a SGZB of V(M(n,k))?
+Theorem 1.3 answers both these questions in the special case of Boolean algebras. We now
+prove this result.
+Lemma 4.2 For0≤i≤n,V(B(n)i)is a multiplicity free Sn-module with min {i,n−i}+1
+irreducible summands.
+For the proof of this lemma see Theorem 29.13 in [JL]. To actually identify the irreducibles (as
+corresponding to two part partitions) see Example 7.18.8 in [S1]. To identify the irreducibles,
+along with their multiplicity (given by the number of semist andard Young tableaux), in the
+Sn-module V(M(n,k)i) see Exercise 7.75 in [S1].
+Proof of Theorem 1.3 We shall show inductively that each element of O(n) is a simultaneous
+eigenvector of X1,...,X n, the case n= 1 being clear. By Lemma 4.2, condition (b) in the
+defnition of SGZB is then automatically satisfied. It also fo llows from Lemma 4.2 that in the
+case ofV(B(n)) the definition of SGZB given in the introduction agrees wit h the definition
+given above.
+Assume that each element of O(n) is an eigenvector for the action of X1,...,X n. Note that if
+v∈V(B(n)k) is an eigenvector for Xi, for some 1 ≤i≤n, thenv∈V(B(n+1)k+1) is also an
+eigenvector for Xiwith the same eigenvalue. Thus it follows from (31, 33, 34, 35 , 36) that each
+element of O(n+1) is an eigenvector for X1,...,X n. It remains to show that each element of
+O(n+1) is an eigenvector for Xn+1.
+20For 0≤i≤n+1
+2and 0≤k≤idefine a subset R(k,i)⊆O(n+1) consisting of all v∈O(n+1)
+satisfying: r(v) =iand the symmetric Jordan chain in which vlies starts at rank kand ends
+at rankn+1−k. PutW(k,i) = Span R(k,i). Clearly V(B(n+1)i) =W(0,i)⊕W(1,i)⊕
+··· ⊕W(i,i). We claim that each W(k,i) is aSn+1-submodule. We prove this by induction
+oni, the case W(0,0) being clear. Assume inductively that W(0,i−1),...,W(i−1,i−1) are
+submodules, where i≤n+1
+2. SinceUisSn+1-linear,U(W(j,i−1)) =W(j,i), 0≤j≤i−1
+are submodules. Now consider W(i,i). Letu∈W(i,i) andπ∈Sn+1. SinceUisSn+1-linear
+we have Un+2−2i(πu) =πUn+2−2i(u) = 0. It follows that πu∈W(i,i).
+We now have from Lemma 4.2 that, for 0 ≤i≤n+1
+2,W(0,i),...,W(i,i) are mutually noni-
+somorphic irreducibles. Consider the Sn+1-linear map f:V(B(n+1)i)→V(B(n+1)i) given
+byf(v) =av, where
+a= sum of all transpositions in Sn+1=X1+···+Xn+1.
+It follows by Schur’s lemma that there exist scalars λ0,...,λ isuch that f(u) =λku, for
+u∈W(k,i). Thus each element of R(k,i) is an eigenvector for X1+···+Xn+1(and also for
+X1,...,X n). It follows that each element of R(k,i) is an eigenvector for Xn+1.
+The paragraph above has shown that the bottom element of each symmetric Jordan chain in
+O(n+1) is a simultaneous eigenvector for X1,...,X n+1. It now follows from Lemma 4.1(iv)
+that each element of O(n+1) is a simultaneous eigenvector for X1,...,X n+1. That completes
+the proof. ✷
+Acknowledgement
+I am grateful to Sivaramakrishnan Sivasubramanian for seve ral helpful discussions and to
+Professor Alexander Schrijver for an encouraging e-mail.
+References
+[A] I. Anderson, Combinatorics of finite sets, Clarendon Press, Oxford , 1987.
+[BTK] N. G. de Bruijn, C. A. v. E. Tengbergen and D. Kruyswijk, On the set of divisors of
+a number , Nieuw Arch. Wiskunde, 23: 191-193 (1951).
+[C] E. R. Canfield, A Sperner property preserved by product , Linear and Multilinear
+Algebra, 9: 151-157 (1980).
+[E] K. Engel, Sperner theory, Cambridge University Press , 1997.
+[D1] C. F. Dunkl, A Krawtchouk polynomial addition theorem and wreath produc t of sym-
+metric groups , Indiana University Mathematics Journal, 26: 335-358 (197 6).
+[D2] C. F. Dunkl, Spherical functions on compact groups and applications to s pecial func-
+tions, Symposia Mathematica, 22: 145-161, Academic Press, Londo n - 1977.
+21[GST] D. Gijswijt, A. Schrijver, and H. Tanaka, New upper bounds for nonbinary codes based
+on the Terwilliger algebra and semidefinite programming , Journal of Combinatorial
+Theory, Series A, 113: 1719-1731 (2006).
+[GW] R.Goodman, andN.R.Wallach, Symmetry, Representatio ns, andInvariants, Graduate
+Texts in Mathematics, Springer , 2009.
+[JL] G. James, and M.Liebeck, Representations and Characte rs of Groups, Cambridge
+University Press , 2001.
+[P] R. A. Proctor, Representations of sl(2,C)on posets and the Sperner property , SIAM
+Journal of Algebraic and Discrete Methods, 3:275-280 (1982 ).
+[PSS] R. A. Proctor, M. E. Saks and P. G. Sturtevant, Product partial orders with the
+Sperner property , Discrete Mathematics, 30: 173-180 (1980).
+[S] A. Schrijver, New code upper bounds from the Terwilliger algebra and semid efinite
+programming , IEEE Transactions on Information Theory, 51: 2859-2866 (2 005).
+[S1] R. P. Stanley, Enumerative Combinatorics - Volume 2, Cambridge University Press ,
+1999.
+[V] F. Vallentin, Symmetry in semidefinite programs , Linear Algebra and its Applications,
+430: 360-369 (2009).
+[VO] A. M. Vershik and A. Okounkov, A new approach to the representation theory of the
+symmetric groups - II , Journal of Mathematical Sciences (New York), 131: 5471-54 94
+(2005).
+22
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+arXiv:1001.0281v1  [cond-mat.stat-mech]  2 Jan 2010Proof of the determinantal form of the spontaneous
+magnetization of the superintegrable chiral Potts model
+R.J. Baxter
+Mathematical Sciences Institute, The Australian National
+University, Canberra, A.C.T. 0200, Australia
+2 January 2010
+Abstract
+The superintegrable chiral Potts model has many resemblanc es to the Ising model,
+so it is natural to look for algebraic properties similar to t hose found for the Ising
+model by Onsager, Kaufman and Yang. The spontaneous magneti zationMrcan be
+written in terms of a sum over the elements of a matrix Sr. The author conjectured the
+form of the elements, and this conjecture has been verified by Iorgovet al. The author
+also conjectured in 2008 that this sum could be expressed as a determinant, and has
+recently evaluated the determinant to obtain the known resu lt forMr. Here we prove
+that the sum and the determinant are indeed identical expres sions.
+1 Introduction
+There has been considerable progress recently towards the g oal of finding an algebraic
+method of calculating the spontaneous magnetization Mr(the order parameter) of the
+superintegrable N-state chiral Potts model. By considering the square lattic e of finite
+widthL, the author[1]-[3] showed that it could be expressed in term s of a sum over the
+elements of a 2mby 2m′matrixSr. In [3] he conjectured a formula for these elements
+as simple products. He also conjectured in [2, 3] that the sum could be written as the
+determinant DPQof anm-dimensional (or m′-dimensional) matrix. These P,Q,rare
+related by Q=P+r(modN), and (for 0 ≤P,Q < N )
+m=/bracketleftbigg
+(N−1)L−P
+N/bracketrightbigg
+, m′=/bracketleftbigg
+(N−1)L−Q
+N/bracketrightbigg
+(1)
+where [x] is the integer part of x. Note that these defintions imply
+|m−m′| ≤1. (2)
+In a recent paper[4], the author has shown that this determin ant can be written the
+product (or ratio) of Cauchy-like determinants, so is also a simple product. Taking the
+limitL→ ∞, one does indeed regain the known result[5, 6, 7]
+Mr= (1−k′2)r(N−r)/2N2. (3)
+This still left open the two conjectures, the first being the p roduct expression for
+the elements of Sr. The matrix Srsatisfies two commutation relations. In unpublished
+1work, the author had been able to prove that the conjectured f orm did in fact satisfy
+these relations, and from numerical studies for small value s ofN,Lit appeared that
+there was only one such solution to these linear equations, b ut this fell short of a proof.
+The problem of calculating such matrix elements has been stu died more directly by
+Au-Yang and Perk.[8, 9] Iorgov et alhave now given a proof, and have gone on to
+calculate Mrdirectly.[10]
+The second conjecture was that thesumoverthe elements of Srwas thedeterminant
+DPQ. This is the problem we address here. It is a very self-contai ned problem, being
+just an algebraic identity between rational functions of ma ny variables. It completes
+the algebraic proof of the formula (3) via the determinant DPQ.
+We shall refer to papers [1] - [4] as papers I, II, III, IV, resp ectively, and quote their
+equations accordingly.
+2 Definitions
+Letc1,...,c m,y1,...,y mandc′
+1,...,c′
+n,y′
+1,...,y′
+nbe sets of variables, where
+n=m′
+andm,m′are the integers mentioned above. In this paper we do notuse the above
+definitions (1), nor the integers N,L,P,Q . We can take m,m′to both be arbitrary
+positive integers, and the ci,yi,c′
+i,y′
+ito be arbitrary variables. In this paper we can
+and do allow mandm′to be arbitrary. We ignorethe restriction (2).
+Lets={s1,...,s m}be a set of mintegers with values
+si= 0 or 1 for 1 ≤i≤m. (4)
+Similarly, let s′={s′
+1,...,s′
+n}, where each s′
+i= 0 or 1. Set
+κs=s1+···+sm, κs′=s′
+1+···+s′
+n. (5)
+For a given set s, letVbe the set of integers isuch that si= 0 and Wthe set such
+thatsi= 1. Hence, from (5), Vhasm−κselements, while Whasκs. Define V′,W′
+similarly for the set s′, soV′hasn−κs′elements, while W′hasκs′.
+Define
+As,s′=/productdisplay
+i∈W/productdisplay
+j∈V′(ci−c′
+j), Bs,s′=/productdisplay
+i∈V/productdisplay
+j∈W′(ci−c′
+j),
+Cs=/productdisplay
+i∈W/productdisplay
+j∈V(cj−ci), Ds′=/productdisplay
+i∈V′/productdisplay
+j∈W′(c′
+j−c′
+i).
+Then the afore-mentioned matrix Srhas elements ( Sr)s,s′which are proportional
+toAs,s′Bs,s′/(CsDs′) and the sum over its elements is given in III.3.48 as
+R=/summationdisplay
+s/summationdisplay
+s′ys1
+1ys2
+2···ysm
+m/parenleftbigg
+As,s′Bs,s′
+CsDs′/parenrightbigg
+y′
+1s′
+1y′
+2s′
+2···y′
+ns′
+n, (6)
+the sum being restricted to s,s′such that
+κs=κs′. (7)
+Now define ai,...,a m,b1,...bmanda′
+i,...,a′
+n,b′
+1,...b′
+nby
+ai=n/productdisplay
+j=1(ci−c′
+j), a′
+i=m/productdisplay
+j=1(c′
+i−cj), (8)
+2bi=m/productdisplay
+j=1,j/negationslash=i(ci−cj), b′
+i=n/productdisplay
+j=1,j/negationslash=i(c′
+i−c′
+j), (9)
+and letBbe anmbynmatrix, and B′annbymmatrix, with elements
+Bij=ai
+bi(ci−c′
+j),B′
+ij=a′
+i
+b′
+i(c′
+i−cj). (10)
+Also, define an mbymdiagonal matrix Yand annbyndiagonal matrix Y′by
+Yi,j=yiδij, Y′
+i,j=y′
+iδij. (11)
+Then the determinant mentioned above is
+D=DPQ= det[Im+YBY′B′], (12)
+whereImis the identity matrix of dimension m.
+(The definition (12) is the same as the (II.7.2), (III.4.9), ( IV.1.18). If f,f′
+iare
+defined by IV.2.31 and BPQby IV.2.29, and F,F′are the diagonal matrices with
+elements Fi,i=fi,F′
+i,i=f′
+i, thenB=ǫFBPQF′−1,B′=−ǫF′BQPF−1andǫ2= 1.
+We can then observe that (12) is the same as IV.1.18.)
+We can also write (12) as
+D= det[In+Y′B′YB], (13)
+so bothRandDare unaltered by simultaneously interchanging mwithn, theciwith
+thec′
+iand the yiwith the y′
+i. It follows that without loss of generality, we can here
+choose
+n≥m. (14)
+Theexpressions R,Darefunctionsof m,n,c 1,...cm,y1,...,y m,c′
+1,...c′
+n,y′
+1,...,y′
+n.
+We shall write them as Rmn,Dmn.
+3 Proof that Rmn=Dmn
+BothRmnandDmnare rational functions of c1,...cm,c′
+1,...c′
+n. They are symmet-
+ric, being unchanged by simultaneously permuting the ciand the yi, as well as by
+simultaneously permuting the c′
+iand the y′
+i. We find that they are identical, for all
+ci,yi,c′
+i,y′
+i.
+The proof proceeds by recurrence, in the following four step s.
+1. The case m= 1
+Calculation of R1n
+Ifm= 1 then s={s1}and either s1= 0 ors1= 1. In the first case, since κs=κs′,
+all thes′
+imust be zero and the sets W,W′are both empty, so we get a contribution to
+(6) of unity.
+In the second case, s′={0,...,0,1,0,...,0}, with the 1 in position r, forr=
+1,...,n. ThenVis empty, so Bs,s′=Cs= 1, while
+As,s′=n/productdisplay
+j=1,j/negationslash=r(c1−c′
+j), Ds′=n/productdisplay
+j=1,j/negationslash=r(c′
+r−c′
+j). (15)
+From (6) it follows that
+R1n= 1+n/summationdisplay
+r=1y1y′
+rn/productdisplay
+j=1,j/negationslash=rc1−c′
+j
+c′r−c′
+j. (16)
+3Calculation of D1n
+The RHS of (12) is a determinant of dimension 1, so
+D1n= 1+Y1,1n/summationdisplay
+r=1B1,rY′
+r,rB′
+r,1
+= 1−a1y1
+b1n/summationdisplay
+r=1a′
+ry′
+r
+b′r(c1−c′r)2, (17)
+whereai,a′
+i,bi,b′
+iare defined by (8), (9). Note that here b1= 1 .
+We therefore obtain
+D1n= 1+n/summationdisplay
+r=1y1y′
+rn/productdisplay
+j=1,j/negationslash=rc1−c′
+j
+c′r−c′
+j(18)
+and we see explicitly that
+R1n=D1n. (19)
+2: Degree of the numerator polynomials
+Consider RmnandDmnas functions of cm. They are both rational functions. We show
+here that they are both of the form
+polynomial of degree ( n−1)
+bm(20)
+Degree for Rmn
+First considerthesum Rmnin(6)as afunctionof cm. Eachtermisplainly apolynomial
+divided by bm. Ifm∈W, thensm= 1 and the numerator is proportional to As,s′.
+The degree of the numerator is the number of elements of V′. The condition (7) means
+thatW′must have at least one element, so V′must have at most n−1. The degree of
+the numerator is therefore at most n−1.
+Ifm∈V, then the numerator is proportional to Bs,s′and the degree of the nu-
+merator is the number of elements of W′, which from (7) is the same as the number of
+elements of W. SinceVhas at least one element, Wcan have at most m−1. From
+(14), this is at most n−1
+The sum of all the terms in (6) is therefore a polynomial in cmof degree at most
+n−1, divided by bm, as in (20).
+Degree for Dmn
+Now consider the determinant Dmnin (12) as a function of cm. At first sight there
+appear to be poles at cm=c′
+j, coming from Bmj. However, they are cancelled by the
+factoram. Similarly, the ones in the element B′
+jmof the matrix B′are cancelled by the
+factora′
+j. So there are no poles at cm=c′
+j, for any j.
+There are poles at cm=ci(for 1≤i < m) coming from the bi,bmfactors in
+Bij,Bmj, respectively, so there are simple poles in each of the rows iandm. This
+threatens to create a double pole in the determinant Dmn. However, if cm=ci, the
+rowsiandmof the matrix ( cm−ci)Bare equal and opposite. By replacing row iby
+the sum of the two rows (corresponding to pre-multiplying Bby an elementary matrix),
+we can eliminate the poles in row i. Hence there is only a single pole at cm=ci. The
+determinant is therefore a polynomial in cm, divided by bm.
+4To determine the degree of this polynomial, consider the beh aviour of Dmnwhen
+cm→ ∞. Then, writing cmsimply as c,
+Bij∼c−1ifi < m, Bij∼cn−mifi=m,
+B′
+ij∼cifj < m, B′
+i,j∼1 ifj=m.
+and hence the orders of the elements of the matrix product in ( 12) are given by
+YBY′B′∼
+1 1 ...1 c−1
+1 1 ...1 c−1
+.. .. ... .. ..
+1 1 ...1 c−1
+cn−m+1cn−m+1... cn−m+1cn−m
+.(21)
+Sincen≥m, it follows that Dmngrows at most as
+Dmn∼cn−m(22)
+asc→ ∞. The numerator polynomial in (20) is therefore of degree at m ostn−1. This
+completes the proof of (20 ).
+3. The case cm=c′
+n
+The sum Rmn
+Consider the case when cm=c′
+n. Ifm∈Wandn∈V′, then from (6) As,s′= 0.
+Similarly, If m∈Vandn∈W′, then from Bs,s′= 0. So the summand in (6) is zero
+unless either m∈V,n∈V′, orm∈W,n∈W′.
+In the first instance, sm=s′
+n= 0. The AB/CD factor in (6) is the same as if we
+take replace m,nbym−1,n−1, respectively, except for a factor
+/productdisplay
+i∈Wci−c′
+n
+cm−ci/productdisplay
+j∈W′cm−c′
+j
+c′
+j−c′n. (23)
+Sincecm=c′
+n, the factors in the product cancel, except for a sign. From (7 ), there as
+many elements in Was inW′, so the sign products also cancel , leaving unity. Thus
+this contribution to (6) is exactly that obtained by replaci ngm,nbym−1,n−1.
+In the second instance, sm=s′
+n= 1. This time AB/CD has an extra factor
+/productdisplay
+j∈V′cm−c′
+j
+c′n−c′
+j/productdisplay
+i∈Vci−c′
+n
+ci−cm= 1, (24)
+so this contribution to (6) is again that obtained by replaci ngm,nbym−1,n−1,
+except that now there is an extra factor ymy′
+n. Adding the two contributions, we see
+that
+Rmn= (1+ymy′
+n)Rm−1,n−1. (25)
+The determinant Dmn
+Now look at the determinant (12) when cm=c′
+n. Sinceamanda′
+nboth contain the
+factorcm−c′
+n, themth row of Bvanishes except for the element m,n, which is
+Bm,n=n−1/productdisplay
+j=1(cm−c′
+j)/slashiggm−1/productdisplay
+j=1(cm−cj). (26)
+5Similarly, the nth row of B′vanishes except for
+B′
+n,m=m−1/productdisplay
+j=1(c′
+n−cj)/slashiggn−1/productdisplay
+j=1(c′
+n−c′
+j). (27)
+Sincecm=c′
+n, we see that Bm,nB′
+n,m= 1.
+It follows that the matrix in (12) has a block-triangular str ucture:
+Im+YBY′B′=/parenleftigg
+1+yby′b′···
+0 1+ymy′
+n/parenrightigg
+, (28)
+where1,y,b,y′,b′are the matrices Im,Y,B,Y′,B′with their last rows and columns
+omitted. Hence
+Dm,n= (1+ymy′
+n)Dm−1,n−1. (29)
+4. Proof by recurrence
+The proof now proceeds by recurrence. Suppose D(m−1,n−1) =R(m−1,n−1)
+for allci,c′
+i. Then from (25) and (29) it is true that Dm,n=Rm,nwhencm=c′
+n. By
+symmetry it is also true for cm=c′
+jforj= 1,...,n. ThusDm,n−Rm,nis zero for all
+thesenvalues. But from point 2 above, this difference (times bm) is a polynomial in
+cmof degree n−1. The polynomial must therefore vanish identically, so Rm,n=Dm,n
+forarbitrary values of cm.
+Since it is true for m= 1, it follows that
+Rm,n=Dm,n (30)
+forallm,n.
+This proves the second conjecture of [3].
+4 Summary
+We have proved that the sum Rover the elements of the matrix Sris identical to the
+determinant D. In paper IV we have calculated Dand hence obtained the spontaneous
+magnetization Mr.
+The recent work by Irgov et al[10] proves that the elements of the matrix Srare
+indeed given by III.3.45, being proportional to As,s′Bs,s′/(CsDs′), so the algebraic
+calculation of Mris now complete.
+Further, Irgov et algo on to calculate R, and hence Mr, directly, thereby avoiding
+the determinantal formulation altogether. While this last step is efficient, it by-passes
+the author’s original motivation for this work, which was to obtain a derivation of Mr
+that more closely resembled the algebraic and combinatoria l determinantal calculations
+for the Ising model of Yang, Kac and Ward, Hurst and Green, and Montroll, Potts and
+Ward, [11] – [14] all of whom write the partition function in t erms of a determinant or
+pfaffian (the square root of an anti-symmetric determinant). Indeed, the D=DPQof
+thispaperis theimmediategeneralization oftheIsingmode ldeterminant, asformulated
+in I.7.7 in the first paper of this series.
+In fact, it would still be illuminating to obtain a simple and direct derivation of
+DPQparallelling Kaufman’s spinor operators (Clifford algebra ) method for the Ising
+model.[15]
+6References
+[1] Baxter R. J. Algebraic reduction of the Ising model J. Stat. Phys. 132959 –
+982 (2008)
+[2] Baxter, R. J.: A conjecture for the superintegrable chir al Potts model. J. Stat.
+Phys.132, 983 – 1000 (2008)
+[3] Baxter R. J. Some remarks on a generalization of the super integrable chiral
+Potts model J. Stat. Phys. October 2009 in Springerlink “Online First”, also at
+arXiv: 0906.3551 (2009)
+[4] Baxter, R. J.: Spontaneous magnetization of the superin tegrable chiral Potts
+model: calculation of the determinant DPQ.arXiv: 0912.45.49 (2009)
+[5] G. Albertini, B. M. McCoy, J. H. H. Perk and S. Tang, Excita tion spectrum
+and order parameter for the integrable N-state chiral Potts model. Nucl. Phys.
+B314:741–763 (1989).
+[6] Baxter, R. J. Derivation of the order parameter of the chi ral Potts model. Phys.
+Rev. Lett. 94, 130602 (2005)
+[7] Baxter, R. J., The order parameter of the chiral Potts mod el.J. Stat. Phys.
+120:1–36 (2005).
+[8] Au-Yang H. and Perk J. H. H. Identities in the superintegr able chiral Potts
+modelJ. Phys. A: Math. Theor. in press, also at arXiv: 0906.3153 (2009)
+[9] Au-Yang H. and Perk J. H. H. Quantum loop subalgebra and ei genvectors of
+the superintegrable chiral Potts transfer matrices arXiv: 0907.0362 (2009)
+[10] Iorgov, N., Shadura, V., Tykhyy, Yu., Pakuliak, S. and v on Gehlen, G. Spin
+operator matrix elements in the superintegrable chiral Pot ts quantum chain.
+arXiv: 0912.5027 (2009)
+[11] Yang, C. N.: The spontaneous magnetization of a two-dim ensional Ising model.
+Phys. Rev. 85, 808–816 (1952)
+[12] Kac, M. and Ward, J. C. A combinatorial solution of the th e two-dimensional
+Ising model, Phys. Rev. 88, 1332 – 1337 (1952)
+[13] Hurst,C. A. and Green, H. S. Newsolution of the Ising pro blem for a rectangular
+lattice, J. Chem ˙Phys.33, 1059 – 1062 (1960)
+[14] Montroll, E. W., Potts, R. B. and Ward, J. C. Correlation s and spontaneous
+magnetization of the two-dimensional Ising model. J. Math. Phys.4308– 322
+(1963)
+[15] Kaufman, B.: Crystal statistics. II. Partition functi on evaluated by spinor anal-
+ysis. Phys. Rev 76, 1232–1243 (1949)
+7
\ No newline at end of file
diff --git a/1001.0282.txt b/1001.0282.txt
new file mode 100644
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--- /dev/null
+++ b/1001.0282.txt
@@ -0,0 +1,326 @@
+Robust Image Watermarking in the Wavelet Domain for 
+ Copyright Protection 
+ 
+Hamed Dehghan, S. Ebrahim Safavi 
+Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran. 
+Email : {dehghan, safavi}@ee.sharif.edu  
+ 
+ 
+Abstract:  In this paper a new approach to image water-
+marking in wavelet domain is presented. The idea is to hide 
+the watermark data in blocks of the block segmented image. 
+Two schemes are presented based on this idea by embedding the watermark data in the low pass wavelet coefficients of each block. Due to low computational complexity of the 
+proposed approach, this algorithm can be implemented in 
+real time. Experimental results demonstrate the impercepti-bility of the proposed method and its high robustness 
+against various attacks such as filtering, JPEG compres-
+sion, cropping, noise addition and geometric distortions.  Keywords: Watermarking, Wavelet Domain, Low Pass Co-
+efficients, Robustness.   
+1. INTRODUCTION 
+Due to the increasing use and ease of manipulation of digital 
+media such as audio, image, and video over the internet, the 
+problem of ownership protection has become increasingly important. Embedding a signature, or watermark, into the 
+multimedia signal, is a solution which has been developed in 
+recent years. Digital watermarking allows owners or provid-ers to hide an invisible and robust message inside multime-
+dia content, which is able to be retrieved later [1].  
+A watermarking scheme should balance between three re-
+quirements: robustness, capacity and imperceptibility of 
+embedded data [2]. But, the main challenge in ownership verification system for copyright protection is the robustness 
+of the watermarking techniques against various attacks [3]. 
+The attacks that are usually encountered in image water-marking methods are filtering, JPEG compression, cropping, 
+noise addition and geometric distortions such as rotation and 
+scaling. Although for the first two types of attacks many 
+robust methods were proposed [4-14], but they are still weak against other types of attacks such as geometric distortions 
+because of their high computational complexity, implemen-
+tation difficulties, and poor performance.  
+There are two major classes of detection in a watermark-
+ing system: blind detection, which does not need the original 
+signal for detecting the mark and non-blind detection that 
+uses the original signal in the detection process.  
+In [15] a new non-blind watermarking scheme is proposed 
+for audio signals. In this paper we extend this method to two 
+wavelet-based image watermarking scheme. In these meth-ods we segment the original image to smaller blocks, apply wavelet transform to these blocks and embed the binary 
+watermark data in the lowpass scale of each block.  
+2. BACKGROUND 
+In [15] a novel method is presented for audio watermarking. 
+In this method the host signal in transform domains is modi-
+fied with respect to the binary watermark signal (0 or 1).  
+The embedding  processes used in this method can be sum-
+marized a s follows:  
+ Windowing the host signal. 
+ Applying the wavelet transform to each frame. 
+ Embedding the watermark bit of 1 or 0 to a number 
+of wavelet coefficients, W(i),  of each frame based on 
+the following equations: 
+).( )( iWiW          For embedding 1          (1) 
+/)( )( iWiW         For embedding 0          (2) 
+whereis he strength factor. 
+ Inverse wavelet transforming the resulted coeffi-
+cients of each frame. 
+In the detection process, the embedded data is detected 
+by the following process: 
+ Windowing the original and received signal and Ap-
+plying the wavelet transform to each frame of them. 
+ Calculating a compare vector by dividing the wavelet 
+coefficients of the received signal to the original one. 
+ Detecting the embedded bit by comparing the vector 
+which is calculated in the second step with a thresh-old level. If the majority of the vector components is 
+larger than the threshold level, the embedded bit 
+would be 1, otherwise 0 is detected  
+It is proved in [15] that as the embedding process is 
+symmetrical, the best threshold  value is 
+
+
+1
+21 .  
+3. PROPOSED METHODS 
+Now we describe our proposed image watermarking scheme. 
+We extend the audio watermarking method shown in [15] to 
+two image watermarking techniques. 
+3.1 Method 1  
+In the first approach we propose a simple watermarking 
+scheme.  3.1.1 Watermark embedding 
+The embedding processes used in this method can be 
+summarized as: 
+ Segmenting the original image to small non-
+overlapping blocks and then applying the wavelet transforming to each block. 
+ In each block, the wavelet coefficients in the last 
+lowpass scale are modified for embedding 1 or 0 based on (1) and (2). 
+ Applying the inverse wavelet transform to the ob-
+tained wavelet coefficients.  
+3.1.2 Watermark detection 
+The detection process can be described as follows: 
+ Block segmenting of the received and the original 
+image and applying the wavelet transform to each 
+block for the both images. 
+ Calculating the comparing matrix by dividing the 
+wavelet coefficients of the received image to the 
+original one. 
+ Detecting the embedded bit by comparing the matrix 
+which is calculated in the second step with a thresh-
+old level. If the majority of the matrix components is 
+larger than the threshold level, the embedded bit would be 1, otherwise 0 is detected. 
+We use the threshold level of  
+
+
+1
+21 , same as [15]. 
+  
+3.2 Method 2  
+In the second approach we propose a modified version 
+of the first method.  
+3.2.1 Watermark embedding 
+The embedding process in this method is the same as 
+method one except that we perform an additional task after block segmentation of the original image: 
+ The variance of each block is calculated and the first 
+N blocks with higher variances are selected for wa-
+termark embedding.  
+Then the same tasks are performed to embed watermark 
+data in these selected blocks. 
+3.2.2 Watermark detection 
+In the detection step we first select N blocks of the 
+original image with higher variances and the same blocks in 
+the received image. Then we perform the detection process 
+same as before on these selected blocks. Using this approach 
+we can choose a larger strength factor α, which increases the 
+robustness while maintains the imperceptibility. 
+4. EXPERIMENTAL RESULTS 
+In this section we perform se veral experiments to test the 
+proposed algorithms and evaluated its performance against 
+various kinds of attacks. Throughout our experiments we 
+choose the strength factor of 
+01.1 for Method 1 and 
+025.1 for Method 2. These factors are selected to 
+ 
+  
+  
+  
+(a) Original images. ( Goldhill, Baboon, Barbara  and Boat) 
+ 
+  
+  
+  
+(b) Watermarked images using Method 1 
+ 
+  
+  
+  
+(c) Watermarked images using Method 2  
+Figure 1 – Original and Watermarked images . maximize the robustness of this approach, while the modifi-
+cations introduced by the watermarking process are imper-ceptible. Also we use the Daubechies length-8 symlet filters 
+with maximum levels of decomposition to compute the 2-D 
+DWT. It means that for N×N  block size we use log
+2N levels 
+of decomposition.  All the results are obtained by averaging 
+on five runs. We also use a pseudorandom binary sequence 
+as the watermarking signal. Throughout the experiments we mention Method 1 as M1 and Method 2 as M2. 
+We use BER (bit-error-rate) to measure the performance 
+of the watermarking scheme against attacks. A set of four common images were tested for our experiments. The im-
+ages are illustrated in Figure 1.a. They are all 512×512 stan-
+dard images including: Goldhill, Baboon, Barbara  and Boat. 
+Their watermarked versions using Method 1 and Method 2 
+are shown in Figures 1.b and 1.c. As we see the impercepti-
+bility of the watermarked images are satisfied for both methods. The mean PSNR (peak-signal-to-noise-ratio) of 
+the watermarked images ar e 46.45dB, 45.93dB, 46.26dB 
+and 45.60dB respectively for method1 and 47.34dB, 47.65dB, 48.04dB and 47.79dB respectively for method2.  
+In the second experiment, we test the robustness of the 
+proposed watermarking methods against JPEG compression attacks. Figure 2 shows the resulted BER for different im-
+ages. As we see, both of our proposed methods, especially 
+Method 2 are highly robust against JPEG attacks. 
+In the third experiment, the watermarks are tested against 
+additive white Gaussian noise attack with different noise 
+levels. The BER results are shown in Figure 3. Again, we see that the proposed watermarking schemes are robust 
+against even high variance noise attack. 
+Robustness against rotation is the concept of the fourth 
+experiment. Using template matching the rotation attack can be compensated by identifying the rotation angle and then rotating the image back [8, 16 and 17]. Therefore, the intro-
+duced distortion only comes from the interpolation due to 
+image rotation. The BER results are shown in Table 1. As 
+we see even for large rotation angles our watermarking 
+methods are still robust. 
+In the next experiment we test scaling attack. In this 
+case we should first restore the image to its original size and 
+then perform the watermark detection process. Most of the watermarking schemes are less robust against this attack. 
+The results of our watermarking schemes for scaling attack 
+with different scaling factors are shown in Table 2. As we see this attack could cause much distortion in watermark 
+signal but our method is still robust and could detect the 
+watermark successfully. Also we can see that in this case the Method 1 has a better performance than the Method 2, 
+which is due to the larger block size in Method 1. 
+We investigate the robustness of the proposed methods 
+against filtering attack in the sixth experiment. In this ex-
+periment mean filter as a linear filter and median filter as a 
+nonlinear filter with different window sizes are tested. The results can be seen in Table 3. We see that although these 
+filters highly distort the image and degrade its quality, but 
+our watermarking techniques, especially the second one, are 
+still robust against these attacks and could detect the water-mark signal successfully. 
+As the last part of this section, we compare the proposed 
+method with several established image watermarking tech-niques. These methods include: Dugad's wavelet based 
+ 
+(a)                                               (b) 
+Figure 2 – BER(%) results after JPEG  attack with various qualities. 
+ 
+ 
+(a)                                               (b) 
+Figure 3 – BER(%) results after Noise attack for different noise standard  
+deviations σn Table 1. BER(%) resulted for different rotation angles. 
+Image Method  0.5º -.05º 1º -1º 5º -5º 10º 30º 
+M1 0.78 0.78 5.70 6.33 6.95 6.64 8.59 9.77  Goldhill M2 0 0 0.31 1.56 0.94 1.56 0.94 2.19  
+M1 5.16 5.31 4.53 5.00 5.86 6.25 7.62 11.91  Baboon M2 0 0 3.75 5.63 5.94 7.81 9.69 12.03  
+M1 5.23 5.47 5.47 5.70 6.25 6.45 8.98 12.89  Barbara M2 0 0 3.59 3.13 5.78 5.00 5.00 13.13  
+M1 5.39 5.55 5.23 5.78 5.86 6.05 9.96 12.30  Boat M2 0 0 0 1.56 0.47 2.5 0.16 0 
+ 
+Table 2. BER(%) resulted for different scaling factors (SF). 
+Image Method  SF=0.9 SF=0.8 SF=0.7 SF=0.6 SF=0.5 
+M1 11.72  9.18  7.03  7.91  8.98  Goldhill M2 15.47  18.59  12.19  18.59  13.44  
+M1 6.25  9.38  12.03  13.48  10.74  Baboon M2 16.41  15.63  14.69  19.22  16.72  
+M1 14.06  14.84  11.91  7.62  14.45  Barbara M2 18.44  20.16  19.38  17.97  25.16  
+M1 10.55  10.74  6.45  8.79  12.11  Boat M2 27.34  30.78  18.12  20.16  25.62  
+ 
+Table 3. BER(%) resulted for mean and median filtering attacks 
+with different window sizes. 
+Mean Filter Median Filter  Image Method  3×3 5×5 7×7 3×3 5×5 7×7 
+M1 7.42 12.30  14.45 2.93 12.11 18.95 Goldhill  M2 0 2.61 6.51 0.52 5.21 17.19 
+M1 8.20 11.33  14.65 13.48 20.51 26.17 Baboon  M2 2.61 9.11 14.06 4.69 19.01 22.14 
+M1 11.91  15.82  15.82 1.95 11.91 15.23 Barbara  M2 5.73 8.33 11.46 3.91 14.84 23.44 
+M1 6.45 12.11  15.43 3.52 10.35 19.73 Boat M2 0.78 2.61 11.46 3.91 16.41 17.44 
+ spread spectrum method [9], the improved SS (ISS) method 
+[10], the holographic method [11], EPCM [8], the multistage 
+VQ (MVQ) method [12], P&Z method [13] and K&R method [14]. We test these algorithms against JPEG attack using Lena  image. In some of these papers correlation coeffi-
+cient is used to measure the watermark robustness against 
+attacks. Correlation coefficient is defines as: 
+
+
+
+2 2)( )()()(
+),(
+nw nwnwnw
+ww          (3) 
+where w is the watermark signal which is a string of {+1 and 
+-1}and w' is the extracted signal. Table 5 shows the result of 
+these methods along with our proposed methods. As we see 
+both of the proposed methods outperforms all of these estab-
+lished techniques. Also we see that the Method 2 is more robust than Method 1. 
+As the last part of this section we should mention the 
+low computational complexity of the proposed methods. The CPU time for watermarking a 512×512 image using a 3 GHz 
+Pentium IV machine is about 6.17 sec and 2.66 sec for 
+Method 1 and Method 2, respectively. Also the CPU time for watermark detection of the same image is about 3.92 sec and 
+1.49 sec, respectively. 
+5. CONCLUSION 
+In this paper we presented a new wavelet-based image wa-
+termarking technique which is suitable for image copyright 
+protection. In this method the host image was segmented to 
+small blocks and the watermark data was embedded in the lowpass wavelet coefficients of each block with one of two 
+methods mentioned in Section 3. The simulation results on 
+several images confirm the imperceptibility of the water-marked image. We also showed the robustness of the pro-
+posed method against several attacks. Experiments demon-
+strate that Method 2 outperforms Method 1 against most of 
+attacks. Due to the low computational complexity of the pro-posed algorithm it can be impl emented in real time purposes. 
+Future work may be done by trying to develop a blind wa-
+termarking scheme considering the idea of the proposed method. Moreover, we want to improve the idea of Method 2 
+for selecting high variance blocks for watermark embedding. 
+  
+ 
+  
+ REFERENCES 
+[1] J. L., Dugelay, S. Roche, C. Rey, G. Doërr, "Still-image water-
+marking robust to local geometric distortions,"  IEEE Trans. on 
+Image Proc. , vol. 15, no. 9, pp. 2831-2842, 2006. 
+[2] S. Katzenbeisser and F. A. P. Petitcolas, Information Hiding 
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+Artech House, 2000. 
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+hiding for ownership verification," in Proceeding of ICASSP2004,  
+vol. 3, pp. 401-404, 2004.  
+[4] Wolfgang R.B., Podilchuk C.I., and E.J. Delp, "Perceptual Wa-
+termarks for Digital Images and Video," Proceedings of the IEEE , 
+vol. 87, no. 7, pp. 1108-1126, 1999. [5] E. Koch and J. Zhao, "Towards robust hidden image copyright labeling," in Proc. IEEE Workshop Nonlinear Signal and Image 
+Processing , pp. 452–455, 1995.  
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+age Process. , vol. 6, no. 12, pp. 1673–1687, 1997. 
+[7] M.D. Swanson, M. Kobayashi, and A.H. Tewfik, "Multimedia 
+data embedding and watermarking technologies," Proceed.  of the 
+IEEE , vol. 86, no. 6, pp. 1064–1087,  1998. 
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+with application to digital watermarking,"  IEEE Trans. on Image 
+Proc ., vol. 14, no. 10, pp. 1590-1602. 2005. 
+[9] R. Dugad, K. Ratakunda and N. Ahuja, "A new wavelet-based 
+scheme for watermarking images," in Proc. IEEE Int. Conf. Image. 
+Proc. (ICIP98),  vol. 2, pp. 419-423,1998.  
+[10] H.S. Malvar, D. A. F. Flore ̂ncio, "Improved spread spectrum: 
+A new modulation technique fo r robust watermarkimg ," IEEE 
+Transactions on Signal Processing , vol. 51, no. 4, pp. 898-905, 
+2003. 
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+domain image watermarking method ," Circuits, System and Sig. 
+Process.,  vol. 17, no. 3, pp. 361-389, 1998. 
+[12] Z. M. Lu, D. G. Xu, S. H. Sun, "Multipurpose image water-
+marking algorithm based on multistage vector quantization," IEEE 
+Trans. on Image Process ., vol. 14, no. 6, pp. 822-831, 2005. 
+[13] C. I. Podilchuk and W. Zeng, "Image-adaptive watermarking using visual models," IEEE journal on Selected Areas in Commu-
+nications , vol. 16, no. 4, pp. 525-540, 1998. 
+[14] N. Kaewkamnerd and K. R. Rao, "Wavelet based image adap-
+tive watermarking scheme," IEE Electronic Letters , vol. 36, pp. 
+312-313, 2000. 
+[15] M. A. Akhaee, S. GhaemMaghami, and N. Khademi, "A 
+Novel Technique for Audio Signals Watermarking in the Wavelet 
+and Walsh Transform Domains," in International Symposium on 
+Intelligent Signal Processing and Communication Systems 
+(ISPACS),  Japan, Dec, 2006. 
+[16] X. Qi, J. Qi, "Improved affine resistant watermarking by using 
+robust templates," in Proc. Of ICASSP2004 , vol. 3, pp. 405-408, 
+2004.  
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+tation-, scale-, and translation resilient public watermarking of images,"  IEEE Trans. on Image Process. , vol. 10, no. 5, pp. 767-
+782, 2001.  Table 4. Robustness comparison of several methods against JPEG 
+compression attack for Lena  image.  (CC: Correlation coefficient) 
+JPEG Quality (%) Method Measure  10 20 30 40 50 60 70 80 90 
+Dugads CC 0.78  .86 1 1 1 1 1 1 1 
+P&Z CC 0.1 .18 .25 .34 .42 .49 .64 .80 .94  
+KR CC 0.05  .16 .18 .26 .32 .45 .54 .68 .88  
+M1 CC 0.82 .99 .99 1 1 1 1 1 1 
+M2 CC 0.92 1 1 1 1 1 1 1 1 
+ISS BER(%)  8 1.8 1.8 0 0 0 0 0 0 
+holo BER(%)  14 14 14 9.7 7.9 6.1 6.1 - - 
+ECPM BER(%)  1.8 1.8 0 0 0 0 0 0 0 
+MVQ BER(%)  - - 6.30  - 3.20 - - 1.20 - 
+M1 BER(%)  9.29 0.39 0.16 0 0 0 0 0 0 
+M2 BER(%)  3.91 0 0 0 0 0 0 0 0 
+ 
\ No newline at end of file
diff --git a/1001.0283.txt b/1001.0283.txt
new file mode 100644
index 0000000000000000000000000000000000000000..567933bbe4df5929a0feca5c1eaf6c39501b386d
--- /dev/null
+++ b/1001.0283.txt
@@ -0,0 +1,109 @@
+arXiv:1001.0283v1  [physics.gen-ph]  2 Jan 2010Reply to the comments of Karami on the
+Interacting holographic dark energy model
+and generalized second law of
+thermodynamics in a non-flat universe
+M.R. Setare∗
+Department of Physics, University of Kurdistan, Pasdaran A ve., Sanandaj, Iran
+Abstract
+In [2] K. Karami commented on my publication [1] and claimed t hat my result
+would be invalid. In fact he found an error in calculation of t he entropy of cold dark
+matter in section 3 of my article, which is, however, a rather trivial one. It is shown
+that my discussion in section 3 of my paper works correctly if we take the value of
+free parameter b2>0.264.
+∗E-mail: rezakord@ipm.ir
+11 Reply
+The comment of author [2] is based onan error in the sign of first ter m in Eq.(35) of paper
+[1]. Due to this error the result of equations (37) and (38) also will ch ange. The correct
+form is given by Eq.(4) in [2]. But this correction do not change my resu lt in paper [1].
+In the paper [1] I have claimed that the generalized second law (GSL) of thermodynamics
+can be satisfy in non-flat universe enclosed by the event horizon me asured fromthe sphere
+of the event horizon L. In my formula (41), or in Eq.(6) of [2] which is correct version of
+(41) due the correction on eq.(35), there are some parameters, Ωk, ΩΛ, cosy,c, andb2.
+The holographic dark energy model has been tested and constrain ed by various astronom-
+ical observations, in both flat and non-flat cases. These observa tional data include type
+Ia supernovae, cosmic microwave background, baryon acoustic o scillation, and the X-ray
+gas mass fraction of galaxy clusters. According to the analysis of t he observational data
+for the holographic dark energy model, we find that generally c <1. Here we summarized
+the main constraint results as follows:
+1. For flat universe, using only the SN data to constrain the hologra phic dark energy
+model, we get the fit results: c= 0.21+0.41
+−0.12, Ωm0= 0.47+0.06
+−0.15, with the minimal
+chi-square corresponding to the best fit χ2
+min= 173.44 [3]. When combining the
+information from SN Ia [5], CMB [4] and BAO [6], the fitting for the hologr aphic
+dark energy model gives the parameter constraints in 1 σ:c= 0.81+0.23
+−0.16, Ωm0=
+0.28±0.03, with χ2
+min= 176.67 [3].
+2. Also for the flat case, the X-ray gas mass fraction of rich cluste rs, as a function
+of redshift, has also been used to constrain the holographic dark e nergy model [7].
+The main results, i.e. the 1 σfit values for cand Ω m0are:c= 0.61+0.45
+−0.21and
+Ωm0= 0.24+0.06
+−0.05, with the best-fit chi-square χ2
+min= 25.00 [7].
+3. For the non-flat universe, the authors of [8] used the data com ing from the SN and
+CMB to constrain the holographic dark energy model, and got the 1 σfit results:
+c= 0.84+0.16
+−0.03, Ωm0= 0.29+0.06
+0.08, and Ω k0= 0.02±0.10, with the best-fit chi-square
+χ2
+min= 176.12.
+Due to these result, and such as [1] we can assume the value of Ω k, ΩΛ, cosy, andcin
+present time respectively as, 0 .01, 0.73, 0.99, and 1. In the other hand bis a dimensionless
+coupling constant, which is a free parameter, where appear in Eq.(2 1) in [1]. If we
+substitute thementioned values forΩ k, ΩΛ, cosy,c= 1inEq.(6) of[2]we obtainfollowing
+relation
+d
+dx(SΛ+Sm+SL) =π2M2
+p
+H2(3.38−2.86−10.732+38.63b2) (1)
+easily one can see if we take the value of b2>0.264, then we obtaind
+dx(SΛ+Sm+SL)>0.
+It should be stressed that evidence was recently provided by the A bell Cluster A586 in
+support of the interaction between dark energy and dark matter [9]. However, despite the
+fact that numerous works have been performed till now, there ar e no strong observational
+bounds on the strength of this interaction [10]. This weakness to se t stringent (observa-
+tional or theoretical) constraints on the strength of the coupling between dark energy and
+dark matter stems from our unawareness of the nature and origin of dark components of
+2the Universe. It is therefore more than obvious that further wor k is needed to this direc-
+tion. In any case may be we can said that for the validity of GSL in the f ramework of
+our model, the coupling constant between dark energy and cold dar k matter must satisfy
+b2>0.264 in present time.
+So in contrast to the claim of author [2], the essential conclusion of [1 ] is true and the
+GSL can be valid for the present time for the interacting holographic dark energy with
+cold dark matter in a non-flat universe enveloped by the event horiz on measured from the
+sphere of the horizon named L.
+Finally it should be mentioned that recently we have investigated the v alidity of the gen-
+eralized second law of thermodynamics, in the cosmological scenario where dark energy
+interacts with both dark matter and radiation [11]. Then we have sho wn that the gen-
+eralized second law is always and generally valid, as long as one consider s the apparent
+horizon as the universe “radius” (the use of other choices, such is the future event horizon,
+leads to conditional validity only), independently of the specific inter action form, of the
+fluids equation-of-state parameters and of the background geo metry.
+References
+[1] M. R. Setare, J. Cosmol. Astropart. Phys. 01, 023 (2007).
+[2] K. Karami, arXiv:0911.4808 [physics.gen-ph].
+[3] X. Zhang and F. Q. Wu, Phys. Rev. D 72, 043524 (2005) [astro-ph/0506310].
+[4] D. N. Spergel et al.[WMAP Collaboration], Astrophys. J. Suppl. 148, 175 (2003)
+[astro-ph/0302209];
+D. N. Spergel et al., astro-ph/0603449.
+[5] A. G. Riess et al.[Supernova Search Team Collaboration], Astrophys. J. 607, 665
+(2004) [astro-ph/0402512].
+[6] D. J. Eisenstein et al.[SDSS Collaboration], Astrophys. J. 633, 560 (2005) [astro-
+ph/0501171].
+[7] Z. Chang, F. Q. Wu and X. Zhang, Phys. Lett. B 633, 14 (2006) [astro-ph/0509531].
+[8] Y. G. Gong, B. Wang and Y. Z. Zhang, Phys. Rev. D 72, 043510 (2005) [hep-
+th/0412218].
+[9] O. Bertolami, F. Gil Pedro and M. Le Delliou, Phys. Lett. B 654, 165 (2007)
+[arXiv:astro-ph/0703462]; M. Le Delliou, O. Bertolami and F. Gil Pedr o, AIP Conf.
+Proc. 957, 421 (2007) [arXiv:0709.2505 [astro-ph]].
+[10] L.Amendola, G.CamargoCamposandR.Rosenfeld, Phys. Rev. D75, 083506(2007)
+[arXiv:astro-ph/0610806]; Z. K. Guo, N. Ohta and S. Tsujikawa, Ph ys. Rev. D 76,
+023508 (2007) [arXiv:astroph/ 0702015]; C. Feng, B. Wang, Y. Gon g and R. K. Su,
+JCAP 0709, 005 (2007) [arXiv:0706.4033 [astro-ph]]; E. Abdalla, L. R. W. Abramo,
+L. . J. Sodre and B. Wang, arXiv:0710.1198 [astro-ph].
+3[11] M. Jamil, E. N. Saridakis, M. R. Setare, arXiv:0910.0822v1 [hep-t h], accepted for
+publication in Phys. Rev. D (2010).
+4
\ No newline at end of file
diff --git a/1001.0284.txt b/1001.0284.txt
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+arXiv:1001.0284v2  [physics.bio-ph]  6 Jan 2010Evidence of Conformational Changes in Adsorbed Lysozyme
+Molecule on Silver Colloids
+Goutam Chandra1, Kalyan S. Ghosh2, Swagata Dasgupta2and Anushree Roy1
+1Department of Physics, Indian Institute of Technology, Kha ragpur 721302, India
+2Department of Chemistry, Indian Institute of Technology, K haragpur 721302, India
+Abstract
+In this article, we discuss metal-protein interactions in t he Ag-lysozyme complex by spectro-
+scopic measurements. The analysis of the variation in relat ive intensities of SERS bands reveal the
+orientation and the change in conformation of the protein mo lecules on the Ag surface with time.
+The interaction kinetics of metal-protein complexes has be en analyzed over a period of three hours
+via both Raman and absorption measurements. Our analysis in dicates that the Ag nanoparticles
+most likely interact with Trp-123 which is in close proximit y to Phe-34 of the lysozyme molecule.
+Keywords : Lysozyme, Ag colloids, SERS, optical absorption
+1I. INTRODUCTION
+Metal nanoparticles are finding increasing use in nanomedicine for su rface interactions
+with an array of proteins and/or small molecules. Metal nanopartic les tagged to biomacro-
+molecules are oftentimes used to identify specific antibody-antigen binding sites in cells
+and tissues [1, 2]. Surface Enhanced Raman scattering (SERS) is an extremely sensitive
+technique for monitoring the adsorption of species of very low conc entration and for char-
+acterizing the structure and orientation of the adsorbed species on the rough metal surface
+or metal colloidal particles [3]. Nonresonance SERS is primarily sensitiv e to the species at-
+tached to the metal surface or present within a few of the metal-d ielectric interface as well
+as to the orientation of the ad-molecule [4]. Thus, it is possible to char acterize molecular
+rearrangement on the metal surface by SERS mechanism. This implie s that conformational
+changes occurring in the protein structure or due to denaturatio n through possible interac-
+tions with metal particles can be detected from a detailed analysis of the SERS spectrum.
+Modification of the specific interaction sites by covalent or non-cov alent methods makes
+the possibilities of study enormous. In this regard, the interaction s of silver nanoparticles
+with several proteins have been investigated. Some of the protein s studied with silver (Ag)
+colloids are hemoglobin [5], antirabbit IgG [6], albumin [7] and lysozyme [8–1 0].
+Lysozyme is a small monomeric globular protein consisting of 129 amino acids that has
+the ability to disrupt many bacterial functions including their membra ne structure [11–14].
+The protein has α-helix andβ-sheet components with four disulfide bonds. It contains six
+tryptophan (Trp) residues, three of them are located at the sub strate binding sites, two in
+the hydrophobic matrix box, while one is separated from the others [15]. Among them Trp
+62 and Trp108 are the most dominant fluorophores, both being loca ted at the substrate
+binding sites [16]. Lysozyme action on bacteria has been the focus of a lot of research, but
+interaction studies with Ag colloids have not previously revealed the s pecific interaction sites
+on the protein. We have further probed the interactions of lysozy me with Ag nanoparticles
+to obtain an insight into the specific residues involved in the interactio n. This has been
+investigated by spectral methods and new insights have been obta ined in the understanding
+of how this protein is involved in the interaction with Ag colloids.
+Earlier studies on the SERS spectrum of lysozyme molecules adsorbe d on Ag electrodes
+and hydrosols indicate the proximity of the S-S bond and aromatic am ino acid residues of
+2lysozyme molecule on the Ag surface [8–10]. These articles report th e orientation of the
+aromatic amino acid side-chains located on the surface of the metal electrodes. The role
+of anα-helical component of the lysozyme molecule for binding on the metal surface has
+been shown [8, 9] and a recent study has reported the conformat ional changes in lysozyme
+molecule with temperature, while adsorbed on a gold nano-patterne d SERS substrate [17].
+Thus previous studies indicate the possible regions of interaction of the lysozyme molecule
+while interacting on the surface of metal colloids. It is necessary to probe the specific region
+of binding of the protein molecule to the Ag colloidal surface to permit an understanding
+of which regions of the protein would be free to interact with ligands a nd/or drugs. For
+example, the antimicrobial macromolecule formed by a lysozyme-poly phenol complex can
+be used as a carrier for phenolic antioxidants. Epigallocatechin gallat e (EGCG) [18] and
+Triclosan [19] molecules are known to form complexes with lysozyme via Trp-62 and Trp-63
+residues. The knowledge of the specific binding site in Ag-lysozyme co mplex can provide
+guidelines to develop nanoparticle tagged lysozyme molecule for drug delivery. It may be
+noted that the lysozyme is a potential carrier molecule as a renal-se lective drug carrier for
+delivery of the angiotensin-converting enzyme (ACE) inhibitor capt opril [20].
+II. MATERIALS AND METHODS
+Synthesis route for preparing stabilized particles: Ag nanoparticles have been syn-
+thesized by chemical routes, using sodium borohydride (NaBH 4) as reducing agents. Silver
+nitrate (AgNO 3), sodium borohydride (NaBH 4) of analytical reagent grade (SRL, India),
+were used to prepare the Ag sol. A colloidal silver solution was prepar ed in deionized wa-
+ter following the method described by Creighton et al. [21]. Essentially , in this chemical
+route, AgNO 3is reduced by an excess amount of NaBH 4. 2.2×10−3M AgNO 3was added
+dropwise to 1 mM NaBH 4at 4◦C. Stirring for 20 minutes was necessary to stabilize the
+colloidal solution. Later, it was left at room temperature for appro ximately 1 hour. The
+excess NaBH 4evaporated and the remaining solution became transparent yellow in color.
+Colloids generated by above method were stable over a considerable period of time (4-6
+weeks). However, all our reported results are either with freshly prepared Ag sol or aged
+only for a day.
+Chicken egg-white lysozyme was obtained from Sigma-Aldrich. In solu tion, depending on
+3the pH values, the majority of lysozyme molecules assume different s tructural forms. The
+protein has a zero net charge at the isoelectric point (pI). Intera ctions of neutral protein
+molecules with water is less favorable than when they are in the charg ed state. At the pI,
+the solubility of proteins in water is low and the repulsion between prot ein molecules is also
+a minimum. These two factors help in adsorption of the molecules on me tal surfaces for pH
+of the solution close to pI. For lysozyme the value of pI is 10.5. At pH 1 0.5 the protein is not
+soluble in water, which would cause problem in comparing the SERS spec trum and normal
+Raman spectrum of lysozyme molecule. At pH 11 the molecule undergo es rapid hydrolysis
+[22]. Moreover, thepHofuntreatedmetal sol wasbetween 7.0-7.5 . Forthesereasons wehave
+maintained thepHofthelysozyme solutioninslightly acidic state, such thattheeffective pH
+in the final sol (lysozyme+Ag sol) is close to 7.0-7.5 in all our experimen ts. 4 mM solution
+of lysozyme was prepared in deionized water. For SERS measuremen t the volume ratio of
+Ag sol to protein solution was maintained at 1:1.
+SERS spectra were measured using a 488 nm Argon ion laser as an exc itation source
+using a microRaman spectrometer with 100X objective lens. The spe ctrometer is equipped
+with 1200 g /mm holographic grating, a holographic super-notch filter, and a Pelt ier cooled
+CCD detector. The laser power on the samples was 5 mW. The unchan ged circular dichro-
+ism spectra of laser irradiated and un-irradiated samples eliminate th e possibility of dam-
+age/conformational changes in the protein structure by laser irr adiation under the given
+experimental conditions. The data acquisition time for each Raman a nd SERS spectrum
+was 120 sec.
+UV-Visible spectra were measured by Spectrascan UV 2600 (Chemit o). To study inter-
+action kinetics, measurements were carried out immediately after a ddition of lysozyme in
+Ag sol and subsequently after 15 mins, 30 mins, 1 hr, 2hrs and 3 hrs .
+Foropticalabsorptionmeasurements thesampleswerekept inamic rocuvettewithoptical
+pathlength 1 cm. For SERS measurements a drop of solution was tak en on a glass slide
+and allowed to dry. It took around 10-15 minutes to dry the samples . During kinetic
+measurements, to investigate the interaction between metal collo ids and protein molecules
+in the solution, drops were deposited after the specific time interva ls as given above. Thus,
+we carried out all measurements after the completion of metal-mole cule interaction for the
+required time, in ambient conditions. All measurements (SERS and op tical absorption) and
+analysis have been carried out in triplicate.
+4PyMol [23] was used for visualization of the protein conformation. T he accessible surface
+area (ASA) of all the amino acid residues in uncomplexed lysozyme are estimated using
+NACCESS [24].
+III. EFFECT OF ADDED LYSOZYME MOLECULES ON METAL COLLOIDS
+A. Analysis of Plasmon Band of Silver Colloids
+All biomolecules do not interact with metal colloids in the same way. Som e of them result
+in agglomeration of the colloidal particles, others do not cause aggr egation but modify the
+surface charge of the particle [7]. To understand the effect of add ed lysozyme molecules
+on Ag colloids, we study the plasmon band of Ag particles in the sol on a dsorption of the
+protein molecules. Plasmon band of Ag colloids is shown by black dotted line in Fig. 1
+(a). Ag sols absorb light at 398 nm, due to the dipolar surface plasmo n of small spherical
+particles of Ag. The average particle size of Ag has been estimated t o be∼100˚A.
+Time-dependance of the plasmon band of Ag colloids in the sols is shown in Fig. 1 (a).
+Spectra, recorded just after the addition of protein molecules in t he metal sols, are shown
+by black solid line. Subsequent changes in the spectra with time (afte r 15 mins., 30 mins, 1
+hr, 2 hrs and 3 hrs.) are shown by colored solid lines in the Fig. 1 (a). Ad dition of lysozyme
+to the sol results in (i) decrease in intensity and (ii) broadening with a (iii) red shift of the
+plasmon resonance band of Ag particles in the sol. The decrease in re lative intensity of
+the plasmon absorption band in solutions (ratio of maximum intensity o f the plasmon band
+of Ag colloids after addition of protein to that of the Ag colloids), is sh own in Fig. 1(b).
+We observe that the intensity of the plasmon band decays exponen tially. The decay time
+constant of intensity is estimated to be 60 min in the sol. The observe d broadening and
+the red shift of the plasmon band indicate formation of bigger partic les in the sol by added
+lysozyme molecules. Furthermore, the decrease in the intensity of the peak can be explained
+by precipitation of metal particles from the sol followed by immediate aggregation.
+IV. EFFECT OF METAL COLLOIDS ON ADDED LYSOZYME MOLECULES
+Analysis of SERS Spectra
+5Next, we focus on the effect of metal particles on added lysozyme m olecules in the sol.
+Raman spectra and SERS spectra of 4 mM lysozyme in water and in the Ag sol (just after
+addition) are shown by solid lines and dotted lines respectively in Fig. 2. Appearance of the
+characteristic feature of the Ag-N vibrational mode at 233 cm−1[8] in the SERS spectrum
+(shown in the inset of Fig. 2) is a clear signature of metal-protein inte raction in the sol
+via N atom of the ad-molecule. Time variation of SERS spectra of lysoz yme in Ag sol
+over the spectral range 400–900 cm−1, 900–1100 cm−1and 1100–1700 cm−1are shown in
+Fig. 3(a)–(c) The assignments of the prominent bands (indicated b y⋆marks in Fig. 2)
+are summarized in Table I. To study the interaction of the Ag-lysozy me complex, we have
+estimated the relative intensity of each feature in the range over 4 00–1100 cm−1[Fig. 3(a)
+and (b)] with respect to the intensity of the peak at 877 cm−1(vibration of N 1H site of Trp
+amino acid residues in lysozyme). The methylene δ(CH2) vibration at 1442 cm−1has been
+used as reference for the spectral features in the range 1100– 1700 cm−1[Fig. 3(c)]. Both
+these peaks (at 877 and 1442 cm−1) maintained a constant frequency and were of quite
+strong intensities. Invariant Raman shift of a band indicates uncha nged chemical bonding
+responsible for these particular vibrational modes over experimen tal duration.
+Itwasinterestingtonotethattherelativeintensityofallvibration albandsoftheadsorbed
+lysozyme molecule do not follow the same trend. A possible explanation for the time-
+evolution of the spectra can be the conformational changes of th e ad-molecules with time,
+which we discuss in detail below.
+A. Vibrational modes of amide bands
+Vibrations of peptide backbone in proteins, associated with amide I, mainly involve the
+C=O stretch with a small contribution of C-N stretching and N-H ben ding. The same for
+the amide III region arises predominantly by an in-plane N-H deforma tion coupled with a
+Cα-N stretch. In lysozyme, amide bands consist of mainly three struc tural components —
+three stretches of α-helix, antiparallel pleated βsheet, in which the polypeptide is hydrogen-
+bondedbetweenoneregiontoanotherviaahairpinturnofthechain andrandomcoils, which
+are folded in an irregular way. Lysozyme is a globular protein with distin ctly demarcated
+α-helix and β-sheet regions. These secondary structural units are connect ed by random
+coils. The positions and intensities of the vibrational modes of amides can be used for an
+empirical estimation of these secondary structures in proteins.
+SERS spectrum of amide I band appears between 1610–1700 cm−1. The contributions of
+6the three secondary structures— α-helix,β-sheet and random coil— are expected to occur
+over the spectral windows 1665-1680 cm−1, 1610–1632 cm−1and 1636–1644 cm−1, respec-
+tively [28]. The amide III band appears over the spectral range bet ween 1230 and 1310
+cm−1— the signatures of α-helix,β-sheet and random coils are characterized by spectral
+peaks over the range 1280–1320 cm−1, 1235–1242 cm−1and 1250–1260 cm−1, respectively
+[29]. The region 1254–1260 cm−1has been assigned to β-turn [30].
+In the measured SERS spectra of lysozyme in Ag sol, the vibrational modes of amide
+III appear at 1260 cm−1, as a shoulder of the strong peak at 1331 cm−1of Trp (Fig. 3).
+Thus, it is difficult to extract the subtle spectral features of amide III band from spectral
+analysis. To study the correlation between three secondary stru ctural components of the
+adsorbed protein molecule, the broad envelope of the amide I band o f each spectrum needs
+to be deconvoluted into the contributions from each of them. Such analysis may also provide
+information on conformational changes in lysozyme with time upon ad sorption on colloidal
+surface. The time-variation of the vibrational spectra of the amid e I band are shown in Fig.
+4 by + signs. The deconvoluted and net fitted spectra are shown in F ig. 4 by dotted and
+solid lines. From the analysis of the spectral profile of amide I band we obtain the following:
+(i) the spectral feature over the spectral range between 1610 and 1632 cm−1forβ-sheet
+is missing in all spectra and (ii) the relative vibrational intensities of ra ndom coils (I R) to
+α-helix (I α) increases with time (in the inset of Fig. 4).
+B. Vibrational modes of amino acid residues
+As mentioned before, the sharp SERS spectral line at 877 cm−1corresponds to the vibra-
+tion of N 1H site of Trp residues [31]. The frequency of this band depends on th e strength
+of the H-bond between Trp and other side chain molecule. The same f eature is expected to
+shift by 8 cm−1for non-H-bonding Trp derivatives and lowered by 7 cm−1for H-bonded Trp
+derivatives. The single sharp peak at 877 cm−1in the Raman and SERS spectra indicates
+that in the metal-protein complex the Trp-residues do not form any new chemical bond
+directly with the Ag surface or with other residues. The variation in r elative intensities
+of other vibrational modes of Trp are shown in Fig. 5(a). The intens ity of ring breathing
+mode of Phe-residue at 1004 cm−1first increases and then decreases with time (Fig. 5(b)).
+The change in δ(CH2) mode of glutamic acid at 1449 cm−1(just next to δ(CH2) methylene
+vibrational mode at 1442 cm−1) is shown in Fig. 5(c). In each plot, the solid line is the
+guide to the eye.
+7We did not find any clear signature of Asp and His (histidine), two othe r amino residues
+of lysozyme, in the SERS spectra of the molecule.
+C. Disulfide bond
+There are four pairs of S-S bond in lysozyme. Among these four pair s, two pairs of S-S
+bonds are near the α-helix region and two others are in the β-sheet region. The strong SERS
+spectrum for vibration of S-S bond appears at 505 cm−1just after addition of lysozyme in
+the Ag sol. However, the intensity of this band decreases with time ( Fig. 5(d)). To check
+the effect of the reducing environment on the disulfide bond, an add itional experiment with
+lysozyme and NaBH 4was conducted. The trend in the change of relative intensity of S-S
+bond (Fig. 5(e)) indicates that reduction of the disulphide linkage by BH−
+4(note that the
+Ag colloidal particles are stabilized by BH−
+4in NaBH 4stabilized sol) is possible.
+Discussion
+Here we report the specific amino acid residues of lysozyme molecule w hich are adsorbed
+onAg colloids. Appearance ofstrong vibrationalmodes ofTrpandP he residues intheSERS
+spectra of lysozyme molecule and the similar trend in the change in rela tive intensities of
+these modes (Fig. 5(a) and Fig. 5(b)) signify that both these resid ues are close to the
+Ag surface and occupying adjacent sites in the adsorbed molecule. A lysozyme molecule
+contains six Trp residues (Fig. 6). The accessible surface area (AS A) available for these
+residues are shown in Table II. Three (Trp-62, Trp-63 and Trp-12 3) of the six Trp residues
+of a free lysozyme molecule have ASA more than 50 ˚A2and, hence, are likely to be free to
+bind with a substrate. Trp-62 and Trp-63 are exposed at the edge of the active site cleft and
+knowntointeract specifically withasubstrate[29]. Trp-123isclose tothec-endoftheamino
+acid chain of the molecule. Trp-28, -108, -111 have relatively less AS A and are expected to
+be burried. Of all three Phe residues (Phe-3, Phe-34 and Phe-38) of lysozyme molecule the
+ring of Phe-34 is very close to Trp-123 and its ASA is relatively large co mpared to that of
+the other two [Fig. 6 and Table II]. Thus, it is most likely that the lysozy me molecules are
+adsorbed on Ag surface via the red marked region which includes bot h Trp-123 and Phe-34
+residues of the molecule [Fig. 6].
+Next we discuss the observed SERS spectra of amide I band. It is to be noted that the
+ratio of the secondary structure contents i.e αhelix:βsheet:random coil in free lysozyme is
+expected to be 0.54:0.15:1 [32]. Dominant spectral features of α-helical component in Fig. 4
+indicate that lysozyme is adsorbed on the colloidal Ag surface either in anα-helical region
+8or in a loop with an α-helical element [33]. The increase in the intensity ratio (I R: Iα) in the
+inset of Fig. 4 indicates unfolding of the α-helix or the α-helical turn element to a random
+coil of adsorbed protein molecule with time. It is interesting to note t hat theφ,ψangles of
+Trp-123 in lysozyme are -124.9,-2.5 [red arrow in Fig. 6], that is close to theα-helical region
+of the Ramachandran plot.
+Furthermore, the disulfide bond is expected to be affected by the c hange in protein
+backbone conformation [34]. Of all four S-S bonds, two of them, (S 6-S127) and (S30-S115)
+are very close to Trp-123. As the turn element, at the arrow in Fig. 6, gets affected due
+to the interaction with the Ag surface, S6-S127/S30-S115 bond/ bonds may break down
+and this explains the decrease in the intensity of S-S vibrational mod e with time (Fig.
+5(d)). Furthermore, Glu-7 of lysozyme lies very close to the S6-S1 27 bond with an ASA
+corresponding to 82.4 ˚A2. If the S-S bond breaks down (Fig. 5(d)), the vibrational mode
+of Glu-7 is expected to be affected in a similar fashion (Fig. 5(c)). Mor eover, it is to be
+noted that this particular region of the molecule corresponds to th e C-terminal end of the
+protein. Hence, this region of the molecule is more flexible and suscep tible to changes in the
+environment.
+Addition of lysozyme in Ag sol results in an immediate agglomeration of t he Ag particles.
+The initial increase in the relative intensity of vibrational bands of am ino-acid residues of
+the molecule with time in Fig. 5(a) and Fig. 5(b) suggest re-orientatio n of the residues to
+obtain a low energy configuration from an initial high energy state du e to initial favorable
+electrostatic interactions during change in conformation. The time constant for the unfold-
+ing of theα-helix is also estimated to be 45 min (inset of Fig. 4). If we assume only t he local
+electrostatic interaction between the adsorbates and metal collo ids, all our results suggest
+that lysozyme molecules interact with Ag colloids instantaneously; ho wever, it takes longer
+time for the protein molecule to be stabilized on the metal surface. I n general, irreversible
+laws of thermodynamics are involved in adsorption of proteins on met al surfaces [35]. Var-
+ious time scales are involved in metal-molecule interactions. The fast s teps are sometimes
+reversible (can be probed only by time-resolved experimental meas urements) and the rela-
+tively slow steps (of time-scale varies from seconds to minutes) res ult in rearrangements of
+protein structure by the surface environment and in many cases, can be irreversible [35]. We
+believe that the initial metal-protein interaction via agglomeration of the particles involves
+the fast process mentioned above and the conformational chang es of the lysozyme molecule
+9with time corresponds to the latter process. All above conjectur es, obtained from spec-
+troscopic measurements, can be further confirmed by studying in teraction of Ag-lysozyme
+complexes with other ligands that are currently underway, for whic h the interaction sites of
+lysozyme are already established.
+As mentioned earlier that there are few reports in which SERS spect ra of lysozyme
+molecules on Ag colloidal surface have been discussed. Prior knowled ge available on the
+interaction of the Ag-nanoparticles with lysozyme indicates the α-helical and random coil
+components [9]. Our experimental approach identifies the particula r Trp residue involved in
+the interaction. Besides this, the effect of the interaction on the in tensities of SERS bands
+of Phe and disulfide bonds confirm the specific site of the protein invo lved in the complexa-
+tion. Conformational changes in the protein molecule on interaction with Ag nanoparticles
+indicate partial unfolding of the α-helix constituted of residues 25-35 of which Phe-34 is a
+member. Interestingly, it is to be noted that the parts of the prot ein molecule with β-sheet
+comprising of Trp-62 and Trp-63, known to have higher bio-activity , remain unaffected by
+metal-protein complexation.
+Acknowledgements
+AR and SDG thank DRDO, India for financial assistance
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+12Figure Captions
+Figure 1. (a)Kinetic study of the plasmon band of Ag colloids and (b)Time variat ion in
+relative intensity of the plasmon band of the Ag upon addition of lysoz yme in the sol.
+Figure 2. Comparison between normal Raman spectrum of 4 ×10−3M lysozyme in water
+(solid lines) and SERS spectrum of the same in the Ag sol (dotted lines ). Inset of the figure
+shows the characteristic feature of Ag-N vibrational mode.
+Figure 3. SERS spectra of Ag-lysozyme complex taken at different time over t he range
+(a)450-900 cm−1, (b) 900-1100 cm−1and (c) 1100-1700 cm−1.
+Figure 4. SERS spectrum (+ signs) and deconvoluted spectra (dashed lines) of the amide
+I band. Inset of the figure shows the variation in intensity ratio of r andom coil to α-helix
+components of the protein with time.
+Figure 5. Variation in intensity of vibrational bands of (a)Trp residues (b)Ph e residues
+(c)glutamic acid (d)S-S bonds of adsorbed lysozyme in Ag sol and (e )S-S bonds of lysozyme
+in BH 4environment (without metal colloid) with time respectively.
+Figure 6. Cartoon of lysozyme molecule and its site of interaction with Ag surfa ce (the
+region within the red mark. The α-helix element which unfolds upon interaction with metal
+surface is shown by arrow.
+13Table Captions
+Table I. Assignment of vibrational bands for lysozyme SERS spectra.
+Table II. Calculated accessible surface area (ASA) for Trp, Phe and Glu resid ues in a
+lysozyme molecule.
+14/s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53
+/s32/s32
+/s32/s65 /s103/s95/s78/s97/s66/s72
+/s52
+/s32/s65 /s103/s32/s43/s32/s76/s121/s115/s32/s40/s48/s32/s109/s105/s110/s41
+/s32/s65 /s102/s116/s101/s114/s32/s49/s53/s32/s109/s105/s110/s115
+/s32/s65 /s102/s116/s101/s114/s32/s51/s48/s32/s109/s105/s110/s115
+/s32/s65 /s102/s116/s101/s114/s32/s49/s32/s104/s114
+/s32/s65 /s102/s116/s101/s114/s32/s50/s32/s104/s114/s115
+/s32/s65 /s102/s116/s101/s114/s32/s51/s32/s104/s114/s115/s65/s98/s115/s111/s114/s98/s97/s110/s99/s101/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
+/s87/s97/s118/s101/s108/s101/s110/s103/s116/s104/s32/s40/s110/s109/s41/s40/s97/s41
+/s48 /s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48 /s50/s48/s48/s48/s46/s52/s48/s48/s46/s52/s53/s48/s46/s53/s48/s48/s46/s53/s53/s48/s46/s54/s48/s48/s46/s54/s53/s48/s46/s55/s48/s48/s46/s55/s53
+/s32/s32/s82/s101/s108/s97/s116/s105/s118/s101/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121
+/s84/s105/s109/s101/s32/s40/s109/s105/s110/s41/s40/s98/s41
+FIG. 1: Chandra et al
+/s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s50/s48/s48 /s49/s54/s48/s48/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48/s50/s53/s48/s48/s51/s48/s48/s48
+/s50/s48/s48 /s50/s52/s48 /s50/s56/s48/s48/s52/s48/s48/s56/s48/s48/s49/s50/s48/s48/s49/s54/s48/s48
+/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
+/s82/s97/s109/s97/s110/s32/s115/s104/s105/s102/s116/s32 /s40/s99/s109/s45/s49
+/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
+/s82/s97/s109/s97/s110/s32/s115/s104/s105/s102/s116/s32/s40/s99/s109/s45/s49
+/s41
+FIG. 2: Chandra et al
+15/s53/s48/s48 /s54/s48/s48 /s55/s48/s48 /s56/s48/s48 /s57/s48/s48/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48/s50/s53/s48/s48/s51/s48/s48/s48
+/s32/s65 /s103/s43/s76/s121/s115/s40/s106/s117/s115/s116/s32/s97/s102/s116/s101/s114/s32/s109/s105/s120/s105/s110/s103/s41
+/s32/s97/s102/s116/s101/s114/s32/s49/s53/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s51/s48/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s54/s48/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s49/s50/s48/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s49/s56/s48/s32/s109/s105/s110/s40/s97/s41
+/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
+/s82/s97/s109/s97/s110/s32/s115/s104/s105/s102/s116/s32 /s40/s99/s109/s45/s49
+/s41/s57/s48/s48 /s57/s53/s48 /s49/s48/s48/s48 /s49/s48/s53/s48 /s49/s49/s48/s48/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48/s50/s53/s48/s48/s51/s48/s48/s48
+/s32/s65 /s103/s43/s76/s121/s115/s40/s106/s117/s115/s116/s32/s97/s102/s116/s101/s114/s32/s109/s105/s120/s105/s110/s103/s41
+/s32/s97/s102/s116/s101/s114/s32/s49/s53/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s51/s48/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s54/s48/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s49/s50/s48/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s49/s56/s48/s32/s109/s105/s110/s40/s98/s41
+/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
+/s82/s97/s109/s97/s110/s32/s115/s104/s105/s102/s116/s32 /s40/s99/s109/s45/s49
+/s41
+/s49/s49/s48/s48 /s49/s50/s48/s48 /s49/s51/s48/s48 /s49/s52/s48/s48 /s49/s53/s48/s48 /s49/s54/s48/s48 /s49/s55/s48/s48/s48/s51/s48/s48/s48/s54/s48/s48/s48/s57/s48/s48/s48/s49/s50/s48/s48/s48
+/s32/s65 /s103/s43/s76/s121/s115/s40/s106/s117/s115/s116/s32/s97/s102/s116/s101/s114/s32/s109/s105/s120/s105/s110/s103/s41
+/s32/s97/s102/s116/s101/s114/s32/s49/s53/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s51/s48/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s54/s48/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s49/s50/s48/s32/s109/s105/s110
+/s32/s97/s102/s116/s101/s114/s32/s49/s56/s48/s32/s109/s105/s110/s40/s99/s41
+/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
+/s82/s97/s109/s97/s110/s32/s115/s104/s105/s102/s116/s32 /s40/s99/s109/s45/s49
+/s41
+FIG. 3: Chandra et al
+16/s48/s49/s48/s48/s48/s50/s48/s48/s48
+/s48/s49/s48/s48/s48/s48/s49/s48/s48/s48/s48/s49/s48/s48/s48
+/s49/s54/s48/s48 /s49/s54/s50/s48 /s49/s54/s52/s48 /s49/s54/s54/s48 /s49/s54/s56/s48 /s49/s55/s48/s48/s48/s49/s48/s48/s48/s48/s49/s48/s48/s48/s48 /s54/s48 /s49/s50/s48 /s49/s56/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54
+/s32/s32/s40/s49/s56/s48/s32/s109/s105/s110/s41/s32
+/s32/s40/s49/s53/s32/s109/s105/s110/s41/s32
+/s32/s40/s54/s48/s32/s109/s105/s110/s41
+/s82/s97/s109/s97/s110/s32/s115/s104/s105/s102/s116/s32/s40/s99/s109/s45/s49
+/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32/s40/s49/s50/s48/s32/s109/s105/s110/s41/s32
+/s32/s40/s48/s32/s109/s105/s110/s41/s32
+/s32/s40/s51/s48/s32/s109/s105/s110/s41
+/s32/s32/s32
+/s84/s105/s109/s101/s32/s40/s109/s105/s110/s41/s82
+FIG. 4: Chandra et al
+17/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50
+/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52
+/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56
+/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50
+/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56
+/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s32
+/s32/s55/s54/s48/s32/s99/s109/s45/s49/s84/s114/s112/s32/s40/s97/s41/s32
+/s32/s49/s48/s49/s53/s32/s99/s109/s45/s49
+/s32/s82/s101/s108/s97/s116/s105/s118/s101/s32/s114/s97/s116/s105/s111/s49/s51/s51/s48/s32/s99/s109/s45/s49
+/s32
+/s32/s49/s51/s53/s54/s32/s99/s109/s45/s49/s32
+/s84/s105/s109/s101/s32/s40/s109/s105/s110/s41/s49/s53/s52/s50/s32/s99/s109/s45/s49
+/s32
+/s32/s49/s53/s54/s56/s32/s99/s109/s45/s49
+/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50
+/s32/s32
+/s49/s48/s48/s52/s32/s99/s109/s45/s49/s82/s101/s108/s97/s116/s105/s118/s101/s32/s114/s97/s116/s105/s111
+/s84/s105/s109/s101/s32/s40/s109/s105/s110/s41/s80/s104/s101/s32/s40/s98/s41
+/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s52/s53/s48/s32/s99/s109/s45/s49
+/s32/s32/s82/s101/s108/s97/s116/s105/s118/s101/s32/s114/s97/s116/s105/s111
+/s84/s105/s109/s101/s32/s40/s109/s105/s110/s41/s71/s108/s117/s32/s40/s99/s41
+/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50
+/s32/s32/s82/s101/s108/s97/s116/s105/s118/s101/s32/s114/s97/s116/s105/s111
+/s84/s105/s109/s101/s32/s40/s109/s105/s110/s41/s53/s48/s53/s32/s99/s109/s45/s49/s83/s45/s83/s32/s98/s97/s110/s100/s32/s40/s100/s41
+/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50
+/s32/s32
+/s53/s48/s53/s32/s99/s109/s45/s49/s82/s101/s108/s97/s116/s105/s118/s101/s32/s114/s97/s116/s105/s111
+/s84/s105/s109/s101/s32/s40/s109/s105/s110/s41/s83/s45/s83/s32/s98/s97/s110/s100/s32/s40/s101/s41
+FIG. 5: Chandra et al
+18FIG. 6: Chandra et al
+19TABLE I: Chandra et al
+Assignment peak (cm−1)Reference
+S-S 505 [8, 22]
+Phe 620 [25]
+Tyr 644 [25]
+Trp - indole breathing (W18) 763 [26]
+Trp- N 1H site vibration (W17) 877 [26]
+Phe - ring breathing 1004 [8]
+Trp - ring breathing 1015 [8]
+Trp- indole ring (C-H deformation) 1338 [26]
+Trp- W7 wagging 1356 [26]
+δ(CH2) 1442 [25]
+Glu-δ(CH2) 1449 [27]
+Trp -indole ring stretching 1540 [26]
+Trp -W2 mode 1568 [26]
+Phe/Tyr 1598 [8]
+Amide - I 1644 [8]
+20TABLE II: Chandra et al
+Trp sequence ASA (˚A2)Phe-sequence ASA (˚A2)
+Trp-28 0.12 Phe-3 17.82
+Trp-62 120.57 Phe-34 64.02
+Trp-63 48.01 Phe-38 15.33
+Trp-108 10.59 Glu-7 82.44
+Trp-111 12.69 Glu-35 34.16
+Trp-123 55.5
+21
\ No newline at end of file
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--- /dev/null
+++ b/1001.0285.txt
@@ -0,0 +1,414 @@
+ 1Interaction  of (-)-epigallocatechin gallate with silver nanoparticles 
+Goutam Kumar Chandra1, Debi Ranjan Tripathy2, Swagata Dasgupta2 and Anushree Roy1* 
+1Department of Physics, Indian Institu te of Technology Kharagpur, 721302, India 
+2Department of Chemistry, Indian Institute of Technology Kharagpur, 721302, India 
+ 
+Abstract 
+Interactions between silver nanoparticles and (-)-epigallocatechin gallate (EGCG) have 
+been investigated. Prior to the addition of  EGCG molecules the silver particles are 
+stabilized by borate ions. Studies on the surface plasmon resonance band of silver particles suggest that the EGCG  molecules remove the borate ions from the surface of the 
+metal particles due to the chelating propert y of the ions. The complex formation by 
+EGCG and borate ions has b een confirmed by NMR studies and pH titration. A possible 
+scheme of interaction between the two has been proposed.  Key words:  Silver colloid, EGCG, UV-Vis, NMR 
+ PACS codes:  87.64.k, 87.14.ej, 82.70.Dd, 68.43.Mn 
+ 
+ 
+     
+ 
+Electronic mail: anushree@phy.iitkgp.ernet.in                         21. Introduction 
+Polyphenols are natural substances, presen t in beverages obtained from plants, 
+fruits and vegetables, such as red wine and tea. The dominant and most important 
+catechin in green tea is (-)-E pigallocatechin gallate (EGCG) , a potent antioxidant, which 
+has been reported to show an ti-cancer activity and is pres ently under investigation in 
+clinical studies [1-3]. Silver ions (Ag+) and silver-based com pounds are also known to 
+exhibit strong antimicrobial effects [4, 5]. Th e active ingredients in silver-based drugs are 
+“oligodynamic” silver ions [6]. In medical litera ture, an interesting case-history reveals 
+the oncolytic role of Ag+ [7]. This case history also reports the role of EGCG in anti-
+cancer dietary therapy. However, Ag+ can have only limited usefulness as an 
+antimicrobial agent due to the interfering effect s of their salts [8]. Su ch adverse effects of 
+Ag+ can be avoided by using the corresponding me tal nanoparticles. Use of silver (Ag) 
+nanoparticles is an efficient and reliable tool  for improving the biocompatibility of Ag in 
+different biological processes. It is common knowledge that the biol ogical activities of 
+several polyphenols are affected by transition metal ions [9-11] . With increased use of Ag 
+nanoparticles as drug carriers it would be interesting to inve stigate its inte raction with 
+EGCG.  
+2. Experimental 
+Silver colloids are prepared by using sodium borohydride (NaBH 4) as a reducing 
+agent. Silver nitrate (AgNO 3), sodium borohydride (NaBH 4) of analytical reagent grade 
+(SRL, India), were used to prepare the silver sol.  A colloidal silver solution was prepared 
+in deionized water following the method desc ribed by Creighton et al [12]. In this 
+chemical route, AgNO 3 is reduced by an excess amount of NaBH 4. 2.2 mM AgNO 3 was  3added dropwise to 1 mM NaBH 4 at 4oC. Vigorous stirring for 20 minutes was necessary 
+to stabilize the colloid al solution. Later, it was left at room temperature for approximately 
+1 hour till the solution became transparen t yellow. EGCG was obtained from Sigma-
+Aldrich (USA).  
+UV-Visible (UV-Vis) spectra were recorded usi ng a spectrophotometer, Model 
+Lambda-45 (Make Perkin-Elmer). For optical absorption measurements the samples were 
+kept in a microcuvette with an  optical pathlength of 1 cm. 1H NMR spectra were recorded 
+on a Bruker 400 MHz spectrometer at 22.5 ºC. 
+3. Results and Discussion 
+3.1 Formation of borate esters of EGCG 
+The pH of the Ag sol is measured to be 8.43, which remains constant during the 
+time course of the experiment. Just afte r addition of 0.5 mM of EGCG, the pH of 
+colloidal solution decreases to 7.72. As time pr ogresses the pH of the solution gradually 
+decreases further. After 8 hrs the pH of the solution reaches a value ~ 7.30 and remains 
+constant with time as shown in Fig. 1.  The acid dissociation constants of EGCG 
+molecules (inset of Fig. 1) [13] indicate th at in a basic medium deprotonation of the 
+aromatic OH groups of the molecule occurs.  The decrease in pH of the colloidal solution 
+after addition of EG CG, thus, indicates the release of H+ due to such an ad-molecule in 
+the sol. We believe that the pH of the solution decreases further due the formation of borate esters of  EGCG [14]. 
+The complexation of EGCG with borat e ion has been demonstrated by NMR 
+spectroscopic measurements. The 
+1H NMR spectrum of the gallate and pyrogallol moiety 
+o f  E G C G  i n  D 2O in water suppression mode are ( δ6.883(s)2H) and  (δ6.474(s)2H)  4(shown in Fig. 2(a)). In presence of NaOH the 1H NMR spectrum of gallate and 
+pyrogallol moiety of EGCG in D 2O in water suppression mode are ( δ6.869(s)2H) and  
+(δ6.469(s)2H) (shown in Fig. 2(b)). However, in presence of NaBH 4 the 1H NMR 
+spectrum of gallate and pyrogallol moiety of EGCG in D 2O in water suppression mode 
+are (δ6.878(s) and δ6.748(s) 2H) and ( δ6.467(S) and δ6.345(s) 2H) (shown in Fig. 2(c)). 
+Due to the formation of a chelate involving the ortho phenolic group with borate, there is 
+a split in the NMR spectrum along with broa dening [15-16]. We have seen that two 
+protons show more shielding which is only po ssible if boron is tetra coordinated and with 
+a negative charge. Since the two aromatic pr otons of ring A are deuterated they do not 
+appear here.  The possible path of the reaction of EGCG and borate ions is proposed in 
+Scheme 1. 
+3.2 Stability of meta l colloids with EGCG in solution 
+Furthermore, studies on the surface pl asmon resonance (SPR) band of the Ag 
+particle provide direct evid ence of the interaction between  Ag colloids and EGCG. The 
+SPR band of Ag colloids (shown in Fig. 3) appears at 397 nm. The measurements were 
+carried out immediately after di rect addition of EGCG molecu les (0.5 mM in the sol) in 
+the metal sol and subsequently at half hour in tervals over a duration of 3 hrs. The broken 
+lines in Fig. 3 demonstrate the change in spectral profile of the plasmon band of Ag 
+colloids at different time intervals on additi on of EGCG (for clarity, we have shown only 
+a few characteristic spectra taken just after mi xing (0 hr), after 1 hr, 2 hrs and 3 hrs). We 
+observe (i) an increase in inte nsity of the absorption maximu m and (ii) a red shift (by 8 
+nm) of the absorption maximum of plasmon ba nd on addition of EGCG in the beginning, 
+which remains constant at a later time.  5The nanosized particles possess a very large surface-to-volume ratio, and 
+consequently their properties are mostly governed by the surface states. It is well known that the SPR band of Ag colloids in solutio n is strongly influenced by any chemical 
+modification of the surface, depending on wh ether the ad-molecule is nucleophilic and 
+therefore donates electron density into the pa rticle surface (blue-shift of the SPR band), 
+or is electrophilic and withdr aws electron density from the particle surface (red-shift of 
+the SPR band). In the present case, the Ag part icles in the sol are stabilized by the anions 
+(borate ions). Thus the red-shift of the SPR band of the Ag colloids (shown in Fig. 3) by 
+addition of EGCG indicates a decrease of el ectron density (anions) from the colloidal 
+surface that indicates a desorption or removal of anions (borate ions) from the surface of 
+the metal particles.  
+As the Ag colloids in our experiments ar e suspended in an aqueous solution and 
+excess NaBH
+4 was used in the chemical synthesis, the following experiments were 
+performed to obtain a clearer view on the me tal-molecule interaction. The absorption 
+band of an aqueous solution of EGCG  appe ars at 275 nm (solid line in Fig. 4(a)) 
+whereas, a double peak absorption band of  0.5 mM of EGCG in 1 mM NaBH 4 (same 
+concentration as added during synthesis of Ag sol)   solution  appears at 279 nm and 316 
+nm (shown by dotted line in Fig. 4(a)). This feature of the absorp tion band of EGCG in 
+NaBH 4 solution, most likely arises from the deprotonated EGCG molecule and the 
+corresponding borate esters of EGCG. The absorption spectra obtained on successive 
+addition of 0.5 mM aqueous solution of EGCG and NaBH 4 solution of EGCG to Ag 
+colloids is given in Fig. 4(b) and Fig. 4(c) , respectively. On addition of an aqueous 
+solution of EGCG to Ag colloids, the char acteristic absorption ba nd at 275 nm of EGCG  6remains unchanged and another weak shoul der originating from the EGCG-borate 
+complex appears at a higher wavelength (indicate d by an arrow in Fig. 4(b)). Initially we 
+observe an increase in intensity and a red sh ift of the plasmon band of the metal colloid 
+by ~ 8 nm. Subsequently, there is a further red shift of the plasmon band (as shown in 
+Fig. 4(b)) for the higher conc entration of the added molecu le. However, on addition of 
+NaBH 4 solution of EGCG in Ag sol , along with the appearance  of the characteristic 
+double peak structure of deproton ated EGCG and borate ester in  the spectra, there is an 
+increase in intensity and red-shift by ~ 4 nm  of the absorption maximum of plasmon band 
+of silver colloids (Fig. 4(c)). The relative shift of the plasmon band in above two cases 
+has been compared in Fig. 5. The smaller shif t in plasmon band in Fig. 4(c) than what 
+was observed in Fig. 4(b) (shown by filled and open circles, respectively, in Fig. 5), can be explained by the fact that in case of the aqueous solution of EGCG in Ag colloids, the 
+ad-molecules are free to strip off the borate ions present at the surface of the Ag colloids 
+and can form a complex. However, for EGCG dissolved in NaBH
+4 solution the molecules 
+are deprotonated and form a complex with borat e ions (prior to addition to the colloidal 
+solution of Ag). Thus, the probability of the EGCG molecules to interact with the borate ions present on the surface of th e Ag colloids is lower in comparison to the previous case 
+where the interaction of EGCG occurs directly with the sol.  The above results, once 
+again, indicate that EGCG molecules interact  with the borate ions on the surface of Ag 
+particles.  
+3.3 Change in surface charge of Ag colloids in presence of EGCG 
+Next we estimate the change in surf ace charge and surface potential of Ag 
+colloidal particles due to the interaction wi th the EGCG molecules. The shift in the  7wavelength of absorption maximum of the plasmon band before ( λ0) and after ( λ) 
+desorption of the anions from the partic le surface is given by the equation [17], 
+ 
+                                             …………………… (1)  
+where [Ag] is the net silver concentration in th e colloid (distributed in all particles of the 
+colloidal solution) and [anion] is the concentration of anions removed from the particle 
+surface. The spectral shift ( λ-λ0) in Fig. 3 is 61 meV. Hence, we estimate the fractional 
+change in charge density  upon desorption of anions from the surface of th e Ag particles to 
+be ∆[]
+[]Aganionq=  = 0.04, which is equivalent to ~ 0.5×10-4 M concentration of borate ions 
+(anion)  in solution for [Ag]=1.2 ×10-3 M (used during sample preparation).  
+To estimate the change in surface potential  of the Ag particles due to desorption 
+of anions from the surface by EGCG, we have studied the change in maximum wavelength of the SPR band of Ag  colloids with different concentrations of EGCG in the 
+solution (shown in Fig. 6). From the shift in  the plasmon band and using the expression in 
+Eq. 1, we estimate the fractional change in char ge density of metal particles with increase 
+in concentration of EGCG in solution. The change in surface potential arising from the 
+change in charge ∆q on the surface of particle of ra dius R as estimated using the 
+expression ∆V = ∆q/4πεR [18] is shown in inset of Fig. 6. It is clear that as we increase 
+the concentration of EGCG there is an initial increase in ∆V. At around 0.2 to 0.3 mM 
+concentration of EGCG, ∆V reaches a maximum value and then decreases with further 
+increase in the concentration of the ad-mol ecule. A possible explanation to the observed 
+change in surface potential of the Ag collo id on addition of EGCG is given. With []
+[]2/1
+0Aganion1⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ =λλ 8increase in concentration, as  more EGCG molecules are av ailable in the solution, higher 
+amounts of borate ions are removed from the surface of Ag colloids and thus, the surface 
+potential of the colloids increases as shown in  Fig. 6. However, with further increase in 
+concentration of EGCG, the ex cess ad-molecules (after removing the negative ions layers 
+from the colloidal surface) or the complex start to interact directly w ith the surface of the 
+colloids (by neutralizing the particles) wh ich decreases the surface potential of the 
+particle. 
+ 
+Spectroscopic measurements thus indica te that the added EGCG molecules are 
+capable of stripping off borate ions from the surface of the Ag colloidal particles in the 
+sol with the formation of borate esters. This study has far reaching implications in terms 
+of the usage of metal nanoparticles as drug carriers. Changes in chemical composition and surface charge for such drug-nanoparticle combinations require further investigation 
+to be able to use them ef fectively and beneficially. 
+ 
+ 
+Acknowledgments 
+A. Roy and S. Dasgupta thank DRDO, India fo r financial assistance. D. R. Tripathy is 
+thankful to CSIR, India for a Junior Research Fellowship.  
+ 
+         9References  
+ 
+[1] S. Valcic, B. N. Timmermann, D. S. Alberts, G. A. Wächter, M.  Krutzsch, J. Wymer, 
+J. M. Guillén, Anti-Cancer Drugs 7(4), 461 (1996).  [2] H. L. Gensler, B. N. Timmermann, S. Valc ic, G. A. Wächter, R. Dorr, K. Dvorakova, 
+D. S. Alberts, Nutr. Cancer 26(3), 325 (1996).  [3] M. Barthelman, W. B. Ba ir III, K. K. Stickland, W. Chen, B. N. Timmermann, S. 
+Valcic, Z. Dong, G. T. Bowden, Carcinogenesis 19, 2201 (1998).  [4] F. Furno, K. S. Morley, B. Wong, B. L.  Sharp, P. L. Arnold, S. M. Howdle et al., J 
+Antimicrob Chemother 54, 1019 (2004).  [5] S. Abuskhuna, J. Briody, M. McCann, M. De vereux, K. Kavanagh, J. B. Fontecha et 
+al., Polyhedron 23, 1249 (2004).  [6] Von Nageli C., Naturforsch Ges, Denkschriften der Schweiz. 33, 1 (1893).  [7] http://www.imref.org/articles/pdfs/Townsend_III.pdf
+. 
+ [8] J. S. Kim, E. Kuk, K. N. Yu, J. H. Kim, S.  J. Park, H. J. Lee, S. H. Kim, Y. K. Park, 
+Y. H. Park, C. Y. Hwang, Y. K. Kim, Y. S. Lee, D. H. Jeong, M. H. Cho, Nanomedicine 
+3, 95 (2007). 
+ 
+[9] S. A. Jovanovic, M. G. Simic,  S. Steenken, Y. Hara, J. Chem . Soc., Perkin Trans. 2,    
+2365 (1998).  [10] A. Furukawa, S. Oikawa, M. Mura ta, Y. Hiraku, S. Kawanishi, Biochem. 
+Pharmacol. 66, 1769 (2003).  [11] S. Azam, N. Hadi, N. U. Khan, S. M. Hadi, Toxicol. Vitro 18, 55 (2004).
+ 
+ 
+[12] J.A. Creighton, C.G. Blatchford, M. Gr ant Albrecht, J. Chem. Soc. Faraday Trans.  
+75, 790 (1979).  [13] M. Kumamoto, T. Sonda , K. Nagayama, M. Tabata, Biosci. Biotechnol. Biochem. 
+65(1), 126 (2001). 
+ 
+ [14] J. Böesken,  N. Vermaas, J. Phys. Chem. 35(5), 1477 (1931).  
+ [15] J. Knoeck, J. K. Taylor, Anal ytical Chemistry 41(13), 1730 (1969).  
+ 
+[16] R. Pizer, P. J. Ricatto, Inorg. Chem. 33,  4985 (1994).  
+ [17] A. Henglein, Chem. Mater. 10, 444 (1998).   10 
+[18] P. C. Hiemenz, in Principles of Colloid and Surface Chemistry  edited Marcel 
+Dekker, Inc. New York (1986), p.746.      
+ 
+    
+ 
+  
+ 
+  
+ 
+  
+ 
+   
+ 
+  
+ 
+  
+ 
+  
+ 
+  
+ 
+    11Figure Captions 
+ 
+Figure 1. Change in pH of Ag colloids on addition of  EGCG in the sol.  Inset presnts the 
+structure of the EGCG molecule and th e pKa value of two protons are shown. 
+ Figure2.  (a) 1H NMR spectrum of EGCG in D
+2O in water suppression mode. (b) 1H 
+NMR spectrum of EGCG and NaOH in D 2O in water suppression mode, (c) 1H NMR 
+spectrum of EGCG and NaBH 4 in D 2O in water suppression mode.  
+ 
+Figure 3.  Time dependent study of the plasmon band of Ag colloids on addition of 
+EGCG in the sol.  Figure 4.  (a) The solid and dash lines represent the absorption spectra of EGCG in 
+aqueous solution and EGCG in NaBH
+4 solution respectively. (b) Titration of Ag colloids 
+with 0.5 mM EGCG in aqueous solution. (c) Titration of Ag  colloids with 0.5 mM EGCG 
+in NaBH 4 solution.  The red arrows indicate the order of the spectra taken with increase 
+in volume ratio of EGCG.  Figure 5. Comparison between the relative shift of  the plasmon band in case of aqueous 
+solution of EGCG (filled circles)  and NaBH
+4 solution of EGCG (open circles) in Ag 
+colloids.  Figure 6.  Concentration dependent study of the plasmon band of Ag colloids on addition 
+of different concentration of EG CG in the Ag sol. Inset of the figure shows the change in 
+surface potential of the Ag colloids ( ∆V) with different concentration (C-EGCG) of 
+EGCG.  
+ Scheme 1.  Complex formation of EGCG with NaBH
+4.  
+    
+  
+         
+  120.0 2.5 5.0 7.5 24.07.27.47.67.8
+  pH of the solution
+Time (hr)(S)(S)O
+O
+OHO
+OHOH
+OH
+OH
+OH
+OH
+OHAB
+C
+pKa 7.59pKa 7.87
+(c) (a
+(b
+)Figures 
+ 
+ 
+  
+ 
+   
+ 
+  
+ 
+  
+ 
+  
+Figure 1 
+ 
+ 
+ 
+ 
+   
+ 
+  
+ 
+  
+ 
+  
+ 
+  
+ 
+ 
+Figure 2   13BNaB(OH) 4
+(S)(S)O
+O
+OOH
+OH
+OH
+OH
+O
+OHO
+OHA
+CB
+B(OH)4(S)(S)O
+O
+OOH
+OH
+OH
+OH
+OH
+OHHO
+OH
+CA
+B
+(S)(S)O
+O
+OO
+O
+O
+O
+O
+OHO
+OHB
+BOH
+OHA
+CNaBH4 + H2O  NaBO2+4 H2
+ 
+ 
+  
+Scheme 1 
+   14400 5000.00.20.40.60.81.01.21.4
+  
+Ag Coll
+T=0 min
+T=1 hr
+T=2 hr
+T=3 hrAbsorbance (Arb. units)
+Wavelength (nm)
+200 250 300 350 400 4500.00.20.40.60.81.0(a)
+  Absorbance (Arb. units)
+Wavelength (nm) 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 3  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+Figure 4
+ 
+      
+ 
+  200 300 400 500 6000.00.51.0
+  Absorbance (a.u.)
+Wavelength (nm)200 300 400 500 6000.00.20.40.60.81.0 (c)
+  Absorbance (a.u.)
+Wavelength (nm)(b)  150.00 0.02 0.04 0.06 0.085678910111213
+  Relative shift of plasmon band (nm)Concentration of EGCG (mM)
+300 400 500 6000.00.51.01.52.02.5 Bare Ag colloids
+ 0.02 mM EGCG
+ 0.05 mM EGCG
+ 0.1 mM EGCG
+ 0.2 mM EGCG
+ 0.3 mM EGCG
+ 0.5 mM EGCG
+ 0.6 mM EGCG
+ 0.7 mM EGCG
+  Absorbance (Arb. units)
+Wavelength (nm)0.00 0.25 0.50 0.756.57.07.58.08.59.09.510.0
+  ∆V (mV)
+C-EGCG(mM) 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+    
+ 
+ 
+ 
+ 
+ 
+Figure 5
+ 
+ 
+ 
+ 
+  
+ 
+   
+ 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 6  
+ 
+  
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+++ b/1001.0286.txt
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+arXiv:1001.0286v1  [astro-ph.CO]  2 Jan 2010IL NUOVO CIMENTO Vol. ?, N. ? ?
+Model Independent Methods of Describing GRB Spectra Using
+BATSE MER Data
+P. Veres (1)(2), I. Horv ´ath(3), Z. Bagoly (1), L.G. Bal ´azs(4), A. M´esz´aros(5)(7),
+G. Tusn ´ady(6) andF. Ryde (7)(8)
+(1)E¨ otv¨ os University, H-1117 Budapest, P´ azm´ any P. s. 1./A , Hungary.
+(2)Department of Physics, University College Cork, Ireland.
+(3)Dept. of Physics, Bolyai Military University, Budapest, Bo x-12, H-1456, Hungary.
+(4)Konkoly Observatory, H-1525 Budapest, POB 67, Hungary.
+(5)Astronomical Institute of the Charles University, Prague, Czech Republic.
+(6)R´ enyi Institute of Mathematics, H-1364 Budapest, POB 127, Hungary.
+(7)Department of Physics, The Royal Institute of Technology, 1 0691 Stockholm, Sweden.
+(8)Stockholm Observatory, AlbaNova, SE-106 91 Stockholm, Swe den.
+Summary. — The Gamma Ray Inverse Problem is discussed. Four methods of
+spectral deconvolution are presented here and applied to th e BATSE’s MER data
+type. We compare these to the Band spectra.
+PACS 00.00 – By the way, which PACS is it, the 00.00? GOK..
+PACS —.— – ....
+1. – Introduction
+The Gamma Ray Inverse Problem ([1]) is as follows: from a given number of observed
+parameters we have to reconstruct the intrinsic spectra. This will be estimated at some
+other number of points. The relation between the measured param eters and the intrinsic
+spectra is given by the detector response matrix (DRM). In the ca se of the BATSE MER
+data, the spectra has to be approximated at 62 intervals from 16 m easured count rates.
+The inverse problem corresponds to the discretized version of the Fredholm integral
+equation of the first kind.
+d(y) =/integraldisplay
+R(x,y)f(x)dx. (1)
+HereR(x,y) is the so-called kernel function, in our case the detector’s respo nse matrix.
+d(y) is the measured count rate, f(x) is the quantity to be determined. This is known
+to be an ill-posed and underdetermined problem. In the case of BATS E MERdis a 16
+element vector, fa 62 element vector and Ris a 16×62 matrix.
+c/circlecopyrtSociet` a Italiana di Fisica 12 P. VERES ETC.
+2. – Data
+Data were taken from the BATSE database (ftp://cossc.gsfc.nas a.gov). We only used
+the Medium Energy Resolution data type. Background fits were mad e using intervals
+by Norris (http:// cossc.gsfc.nasa.gov/ docs/ cgro/ batse/ bats eburst/ sixtyfour ms/
+batfiles.revamp join) for the cat64data. We have subtracted 3rddegree polynomials
+from all the 16 channels of MER data.
+3. – Methods
+Throughoutthe literaturetherearearticleswhichdiscussindetail theinverseproblem
+in astronomy ([1, 2]) and other sciences ([3]). Gamma-ray bursts’ s pectra are most
+commonly reconstructed by the method parameter fitting also kno ws as forward folding
+[4]. This is done by postulating spectral forms with some adjustable p arameters. Once
+multiplied with the DRM we get a solution for the count rates and we the n compare
+them to the measured set of data. We accept the set of paramete rs which minimizes the
+χ2of the solution and the measured counts. Although widely used and a well established
+method, still it is obliging when it comes to the shape of the spectra an d it consists of
+estimating the parameters and not the spectra itself [1].
+3.1.Singular Value Decomposition . – We decompose the response matrix using this
+well known mathematical method. We solve the GRIP by multiplying with the gener-
+alised inverse obtained through SVD. Spectra obtained from simply m ultiplying with the
+general inverse tend to be noisy. To avoid this we can use filters.
+3.2.Maximum entropy method . – This method introduces some a priori information
+to reconstructthe spectra ( m). This is done by fitting a Band GRB spectral form [5]. We
+propose to maximize the expression: L(f,λ) =λS(f,m)−1
+2χ2whereλis a parameter,
+mis thea priori spectra (from model fitting or SVD). It is with respect to this that
+we measure the entropy of the intrinsic spectra ( f). The expression of the entropy is
+as follows: S(f,m) =/summationtextN(=62)
+j=1fj−mj−fjlog/parenleftBig
+fj
+mj/parenrightBig
+. When Lhas a maximum then
+∇L(f,λ) = 0.. From this we derive: λlog/parenleftBig
+f
+m/parenrightBig
+=R⊤[σ−2](d−Rf), where [ σ2] is a
+diagonal matrix with elements of the σ2vector and Ris the response matrix. Since
+log/parenleftBig
+f
+m/parenrightBig
+results from the linear combination of the R⊤’s columns, we can search for the
+solution in the form of: f=meR⊤w. Herewis a 16 element vector. This ensures that
+fwill always be positive, and since whas a smaller scale, it is easier to find it as a
+result of a minimizing process. The choice of λcan be made with the so-called cross-
+validation technique or another method based on a Bayesian approa ch. We have applied
+the method to BATSE MER data. First results suggest that the Ban d GRB function
+provides a good enough fit.
+3.3.Backus Gilbert method . – Its main advantage is that it is a linear method([6]),
+data in the intrinsic spectra is reconstructed as a linear combination of the measured
+count rates: f(Ej) =/summationtextM
+i=1ai(Ej)di=/summationtextN
+k=1∆k(Ej)fkand ∆ k(Ej) =/summationtextM
+i=1ai(Ej)Ri,k..
+We callR(E) the resolution function. Our aim is to approximate the resolution fu nction
+with Dirac-delta function as well as possible. By doing this we minimize fo rai. The
+other issue at hand is the variance of the spectra. This is dealt with b y minimizing
+σ2
+j(Ej) =/summationtextM
+i=1a2
+i(Ej)σ2
+ialso forai. These two expressions are combined into a singleMODEL INDEPENDENT METHODS OF DESCRIBING GRB SPECTRA USING B ATSE MER DATA 3
+Fig. 1. – Sample Backus Gilbert inverse.
+one depending on the parameter θin the following manner: w(Ej,θ) =R(Ej)cosθ+
+vσ2
+j(Ej)sinθ. By minimizing wwith respect to aiwe get the Backus Gilbert inverse
+(Fig.1). ( vis merely a factor to get the two components to the same order of m agnitude)
+3.4.Phillips-Towmey method .–Wewillprovideonlyabriefdescriptionofthemethod.
+Like at the maximum entropy method, we are also in need here of an a priorispectrum.
+This information is used to reduce the characteristic length scale of the problem. We
+introduce the sum-of-squares of the spectrum’s k-th derivative ( Ck).
+We propose to minimize some linear combination of the χ2and theCk. As ana priori
+spectrum we fit a Band GRB function. We find that there is only insignifi cant gain in
+the function to be minimized with respect to the Band GRB function.
+4. – Conclusion
+HavingappliedseveraloftheabovepresentedmethodstoinvertB ATSEMERspectra,
+we conclude that the methods give a consistent estimation of the inc ident spectrum each
+with regard to the minimizing function they use. The SVD method is the fastest, the
+Backus Gilbert requires the most time, but it is less sensitive to noise. We have also
+found that the Band GRB function provides a good fit to the spectr a we analyzed.
+∗ ∗ ∗
+This research was supported by OTKA grant T048870 and by a gran t from Swedish
+Wenner-Gren Foundations (A.M.).
+REFERENCES
+[1]Loredo T. andEpstein R. ,ApJ,336(1989) 896;
+[2]Bouchet L. ,AandASS,113(1995) 167;
+[3]Parker R.L. ,Annual Review of Earth and Planetary Sciences ,5(1977) 35;
+[4]Preece R.D. andal.,ApJ,496(1998) 849;
+[5]Band D. andal.,ApJ,413(1993) 281;
+[6]Backus G.E. andGilbert J. F. ,Royal Society of London Philosophical Transactions
+Series A ,266(1970) 123;
\ No newline at end of file
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+arXiv:1001.0287v1  [math.CO]  2 Jan 2010Upper bounds for the rainbow connection
+numbers of line graphs∗
+Xueliang Li, Yuefang Sun
+Center for Combinatorics and LPMC-TJKLC
+Nankai University, Tianjin 300071, P.R. China
+E-mail: lxl@nankai.edu.cn; syf@cfc.nankai.edu.cn
+Abstract
+A path in an edge-colored graph G, where adjacent edges may be colored
+the same, is called a rainbow path if no two edges of it are colo red the same.
+A nontrivial connected graph Gis rainbow connected if for any two vertices of
+Gthere is a rainbow path connecting them. The rainbow connect ion number
+ofG, denoted by rc(G), is defined as the smallest number of colors by using
+which there is a coloring such that Gis rainbow connected. In this paper,
+we mainly study the rainbow connection number of the line gra ph of a graph
+which contains triangles and get two sharp upper bounds for rc(L(G)), in
+terms of the number of edge-disjoint triangles of GwhereL(G) is the line
+graph of G. We also give results on the iterated line graphs.
+Keywords: rainbow path, rainbow connection number, (iterated) line g raph,
+edge-disjoint triangles.
+AMS Subject Classification 2000: 05C15, 05C40
+1 Introduction
+All graphs in this paper are simple, finite and undirected. Let Gbe a nontrivial
+connected graph with an edge coloring c:E(G)→ {1,2,···,k},k∈N, where
+adjacent edges may be colored the same. A path of Gis called rainbow if no two
+edges of it are colored the same. An edge-colored graph Gisrainbow connected if
+for any two vertices there is a rainbow path connecting them. Clear ly, if a graph
+is rainbow connected, it must be connected. Conversely, any conn ected graph has
+a trivial edge coloring that makes it rainbow connected, i.e., the color ing such that
+each edge has a distinct color. Thus, we define the rainbow connection number
+∗Supported by NSFC, PCSIRT and the “973” program.
+1of a connected graph G, denoted by rc(G), as the smallest number of colors for
+which there is an edge coloring of Gsuch that Gis rainbow connected. An easy
+observation is that if Ghasnvertices then rc(G)≤n−1, since one may color the
+edges of a spanning tree with distinct colors, and color the remaining edges with
+one of the colors already used. Generally, if G1is a connected spanning subgraph of
+G, thenrc(G)≤rc(G1). We notice the trivial fact that rc(G) = 1 if and only if G
+is complete, and the fact that rc(G) =n−1 if and only if Gis a tree, as well as the
+easy observation that a cycle with k >3 vertices has rainbow connection number
+⌈k
+2⌉([2]). Since a Hamiltonian graph Ghas a Hamiltonian cycle which contains all
+nvertices, then Ghas rainbow connection number at most ⌈n
+2⌉. Also notice that,
+clearly,rc(G)≥diam(G) wherediam(G) denotes the diameter of G.
+Chartrand et al. in [2] determined that the rainbow connection numb ers of
+some graphs including trees, cycles, wheels, complete bipartite gra phs and complete
+multipartite graphs. Caro et al. [3] gave some results on general gr aphs in terms
+of some graph parameters, such as the order or the minimum degre e of a graph.
+They observed that rc(G) can be bounded by a function of δ(G), the minimum
+degree of G. They proved that if δ(G)≥3 thenrc(G)≤αnwhereα <1 is
+a constant and n=|V(G)|. They conjectured that α= 3/4 suffices and proved
+thatα <5/6. Specifically, it was proved in [3] that if δ=δ(G) thenrc(G)≤
+min{lnδ
+δn(1 +oδ(1)),n4lnδ+3
+δ}. Some special graph classes, such as line graphs,
+have many special properties, and by these properties, we can ge t some interesting
+results ontheir rainbowconnection numbers interms ofsome graph parameters. For
+example, in [3] the authors got a very good upper bound for the rain bow connection
+number of a 2-connected graph according to their ear-decompos ition. And in [6],
+we studied the rainbow connection numbers of line graphs of triangle -free graphs
+in the light of particular properties of line graphs of triangle-free gr aphs shown
+in [4], and particularly, of 2-connected triangle-free graphs accor ding to their ear
+decompositions. However, wedidnot getboundsoftherainbowcon nection numbers
+for line graphs that do contain triangles. In this paper, we aim to inve stigate the
+remaining case, i.e., line graphs that do contain triangles, and give two sharp upper
+bounds in terms of the number of edge-disjoint triangles.
+We useV(G),E(G) for the sets of vertices and edges of G, respectively. For
+any subset XofV(G), letG[X] denote the subgraph induced by X, andE[X]
+the edge set of G[X]; similarly, for any subset E1ofE(G), letG[E1] denote the
+subgraph induced by E1. LetGbe a set of graphs, then V(G) =/uniontext
+G∈GV(G),
+E(G) =/uniontext
+G∈GE(G). We define a cliquein a graph Gto be a complete subgraph
+ofG, and amaximal clique is a clique that is not contained in any larger clique.
+Theclique graph K (G) ofGis the intersection graph of the maximal cliques of
+G–that is, the vertices of K(G) correspond to the maximal cliques of G, and two of
+these vertices are joined by an edge if and only if the corresponding maximal cliques
+intersect. Let [ n] ={1,···,n}denote the set of the first nnatural numbers. For a
+setS,|S|denotes the cardinality of S. We follow the notations and terminology of
+[1] for those not defined here.
+22 Some basic observations
+We first list two observations which were given in [6] and will be used in th e
+sequel.
+Observation 2.1 ([6]) IfGis a connected graph and {Ei}i∈[t]is a partition of the
+edge set of Ginto connected subgraphs Gi=G[Ei]andrc(Gi) =ci, then
+rc(G)≤t/summationdisplay
+i=1ci.
+LetGbe a connected graph, and Xa proper subset of V(G). Toshrink X is to
+delete all the edges between vertices of Xand then identify the vertices of Xinto a
+single vertex, namely w. We denote the resulting graph by G/X.
+Observation 2.2 ([6]) Let G′andGbe two connected graphs, where G′is obtained
+fromGby shrinking a proper subset XofV(G), that is, G′=G/X, such that any
+two vertices of Xhave no common adjacent vertex in V\X. Then
+rc(G′)≤rc(G).
+Now we introduce two graph operations and two corresponding res ults which
+will be used later.
+Operation 1: As shown in Figure 2.1, for any edge e=uv∈Gwithmin{degG(u),
+degG(v)} ≥2, wefirst subdivide e, thenreplacethenewvertex withtwo newvertices
+ue,vewithdegG′(ue) =degG′(ve) = 1 where G′is the new graph.
+Operation 1 
+uve
+u v
+ueve
+G G′
+Figure 2.1 G′is obtained from Gby doing Operation 1 to edge e.
+SincedegG′(ue) =degG′(ve) = 1 inG′, and by the definition of a line graph, it is
+easy to show that L(G) can be obtained from L(G′) by shrinking a vertex set of two
+3nonadjacent vertices (these two vertices correspond to edges uue,vve, and belong to
+cliques/a\}brack⌉tl⌉{tS(u)/a\}brack⌉tri}ht,/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht, respectively, in L(G′)). Recall that the line graph of a graph
+Gis the graph L(G) whose vertex set V(L(G)) =E(G) and two vertices e1,e2of
+L(G) are adjacent if and only if they are adjacent in G. So by Observation 2.2, we
+have
+Observation 2.3 If graphG′is obtained from a connected graph Gby doing Oper-
+ation 1 at some edge e∈G, then
+rc(L(G))≤rc(L(G′)).
+Operation 2 . As shown in Figure 2.2, vis a common vertex of a set of edge-disjoint
+triangles in G. We replace vby two nonadjacent vertices v′andv′′such that v′is the
+common vertex of some triangles, and v′′is the common vertex of the rest triangles.
+Gvv′′ v′
+G′Operation 2 
+Figure 2.2 Figure of Operation 2.
+Since during this procedure, the number of edges does not change , the order of
+the line graph L(G) is equal to that of L(G′). Furthermore, by the definition of a
+line graph, L(G′) is a spanning subgraph of L(G). So we have
+Observation 2.4 If a connected graph G′is obtained from a connected graph Gby
+doing Operation 2 at some vertex v∈G, then
+rc(L(G))≤rc(L(G′)).
+3 Main results
+3.1 A sharp upper bound
+Recall that the star, S(v), at a vertex vof graph G, is the set of all edges incident
+tov. Aclique decomposition ofGis a collection Cof cliques such that each edge
+ofGoccurs in exactly one clique in C.
+4We now introduce a new terminology. For a connected graph G, we call Ga
+clique-tree-structure , if it satisfies the following condition:
+T1.Each block is a maximal clique.
+We call a graph Haclique-forest-structure , ifHis a disjoint union of some
+clique-tree-structures, thatis, eachcomponent ofaclique-for est-structureisaclique-
+tree-structure. By condition T1, we know that any two maximal cliques of Ghave
+at most one common vertex. Furthermore, Gis formed by its maximal cliques. The
+sizeof the clique-tree(forest)-structure is the number of its maxima l cliques. An
+example of clique-forest-structure is shown in Figure 3.1.
+Figure 3.1 A clique-forest-structure with size 6 and 2 components.
+If each block of a clique-tree-structure is a triangle, we call it a triangle-tree-
+structure . Letℓbe the size of a triangle-tree-structure. Then, by definition, it
+is easy to show that there are 2 ℓ+ 1 vertices in it. Similarly, we can give the
+definition of triangle-forest-structure . A clique-tree-structure Gis called a clique-
+path-structure if the clique graph K(G) is a path.
+For a connected graph G, we call Gaclique-cycle-structure , if it satisfies the
+following three conditions:
+C1.Ghas at least three maximal cliques;
+C2.Each edge belongs to exactly one maximal clique;
+C3.The clique graph is a cycle. (In particular, if each maximal clique is a
+triangle,thenitisa triangle-cycle-structure . Anexampleoftriangle-cycle-structure
+is shown in Figure 3.2.)
+Figure 3.2 An example of triangle-cycle-structure.
+Aninner vertex of a graph is a vertex with degree at least two. For a graph G,
+5we useV2to denote the set of all inner vertices of G. Letn1=|{v:degG(v) = 1}|,
+n2=|V2|./a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}htis the subgraph of L(G) induced by S(v), clearly it is a clique
+ofL(G). Let K0={/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht:v∈V(G)},K={/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht:v∈V2}. It is easy to
+show that K0is a clique decomposition of L(G) ([5]) and each vertex of the line
+graph belongs to at most two elements of K0. We know that each element /a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht
+ofK0\K, a single vertex of L(G), is contained in the clique induced by uthat is
+adjacent to vinG. SoKis a clique decomposition of L(G).
+LetXandYbe sets of vertices of a graph G. We denote by E[X,Y] the set of
+the edges of Gwith one end in Xand the other end in Y. IfY=X, we simply
+writeE(X) forE[X,X]. When Y=V\X, the set E[X,Y] is called the edge cut of
+Gassociated with X, and is denoted by ∂(X).
+Theorem 3.1 For any set Toftedge-disjoint triangles of a connected graph G, if
+the subgraph induced by the edge set E(T)is a triangle-forest-structure, then
+rc(L(G))≤n2−t.
+Moreover, the bound is sharp.
+Proof.LetT=/uniontextc
+i=1Ti=/uniontextc
+i=1{Ti,jiis a triangle of G: 1≤ji≤ti}(/summationtextc
+i=1ti=t)
+be a set of tedge-disjoint triangles of Gsuch that the subgraph of G,G[E(Ti)],
+induced by each E(Ti) is a component of the subgraph G[E(T)], that is, a triangle-
+tree-structure of size ti.
+InG, for each 1 ≤i≤c, letGi=G[E(Ti)],Vi=V(Gi),Ei=E(Ti);E0
+i=
+E(Vi)∪∂(Vi)⊇Ei, andG0
+i=G[E0
+i]. We obtain a new graph G′fromGby doing
+Operation 1 at each edge e∈E(Vi)\Eifor 1≤i≤c, and we denote by G′
+ithe
+new subgraph (of G′) corresponding to G0
+i. Applying Observation 2.3 repeatedly,
+we haverc(L(G))≤rc(L(G′)).
+Next we will show rc(L(G′))≤n2−t. By previous discussion, we know that
+K={/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht:v∈V2}=c/uniondisplay
+i=1{/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht:v∈Vi}/uniondisplay
+{/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht:v∈V2\c/uniondisplay
+i=1Vi}
+is a clique partition of L(G). So
+{E(/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht) :v∈Vi}c
+i=1/uniondisplay
+{E(/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht) :v∈V2\c/uniondisplay
+i=1Vi},
+that is,
+{E(L(G0
+i))}c
+i=1/uniondisplay
+{E(/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht) :v∈V2\c/uniondisplay
+i=1Vi}
+is an edge partition of L(G). So
+{E(L(G′
+i))}c
+i=1/uniondisplay
+{E(/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht) :v∈V2\c/uniondisplay
+i=1Vi}
+6is an edge partition of L(G′). By Observation 2.1, we have
+rc(L(G′))≤c/summationdisplay
+i=1rc(L(G′
+i))+/summationdisplay
+v∈V2\/uniontextc
+i=1Virc(/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht).
+We know |Vi|= 2ti+1, since the triangle-tree-structure Gihas sizeti. So
+rc(L(G′))≤c/summationdisplay
+i=1rc(L(G′
+i))+(n2−2t−c).
+In order to get rc(L(G′))≤n2−t, we need to show rc(L(G′
+i))≤ti+1.
+Claim.rc(L(G′
+i))≤ti+1.
+Proof of the Claim . Sincethegraph G′
+iisobtainedfrom GibydoingOperation
+1 at each edge e∈E(Vi)\Ei,G′
+icontains exactly titriangles: {Ti,ji∈ Ti: 1≤ji≤
+ti}. We will show that there is a ( ti+ 1)-rainbow coloring of L(G′
+i) by induction
+onti. Forti= 1,G′
+icontains exactly one triangle, and we give its line graph a
+2-rainbow coloring as shown in Figure 3.3. We give color 1 to the edges o f/a\}brack⌉tl⌉{tS(u)/a\}brack⌉tri}ht
+incident with vertex e1, edges of /a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}htincident with vertex e2, and edges of /a\}brack⌉tl⌉{tS(w)/a\}brack⌉tri}ht
+incident with vertex e3; We then give color 2 to the edges of /a\}brack⌉tl⌉{tS(u)/a\}brack⌉tri}htincident with
+vertexe3, edges of /a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}htincident with vertex e1, and edges of /a\}brack⌉tl⌉{tS(w)/a\}brack⌉tri}htincident with
+vertexe2; Finally, give color 2 to the rest of the edges. It is easy to show that this is
+a rainbow coloring. So, the above conclusion holds for the case ti= 1. We assume
+u
+vwe1
+e2e31
+11 2
+22
+Figure 3.3 2-rainbow coloring of line graph of graph with exactly one tr iangle.
+that the conclusion holds for the case ti=h−1, and now show that it holds for
+the case ti=h. By the definition of triangle-tree-structure, there must exist o ne
+triangle: T={u,v,w}, which has exactly one common vertex, namely u, with other
+triangles and v,wdo not belong to any other triangle. We now obtain a new graph
+G′
+ifromG′
+iby doing Operation 1 at edges e1ande2. It is clear that L(G′
+i) can
+be obtained from L(G′
+i) by subdividing vertex e1into two vertices {e′
+1,e′′
+1}ande2
+into two vertices {e′
+2,e′′
+2}. SinceH1hash−1 (edge-disjoint) triangles, by induction
+hypothesis, rc(L(H1))≤h. So the subgraph L(G′
+i)\{/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht,/a\}brack⌉tl⌉{tS(w)/a\}brack⌉tri}ht}∼=L(H1) has
+a rainbow h-coloring, and we now color the edges of /a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}htand/a\}brack⌉tl⌉{tS(w)/a\}brack⌉tri}htin the graph
+L(G′
+i) asfollows: give a new color to the edges of /a\}brack⌉tl⌉{tS(w)/a\}brack⌉tri}htincident with vertex e1, and
+7edges of/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}htincident with vertex e2. Lete3=vw, we then give any one color, say
+c1, of the former hcolors to the edges of /a\}brack⌉tl⌉{tS(w)/a\}brack⌉tri}htincident with vertex e3, and give
+a distinct color, say c2, of the former hcolors to the edges of /a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}htincident with
+vertexe3. It is easy to show that, with above coloring, L(G′
+i) is rainbow connected,
+and soL(G′
+i)≤ti+1 holds for ti=h.
+u
+vwe1e2u
+vw
+G′
+i G′
+iH1
+e′
+1e′′ 
+1
+e′
+2
+e′′ 
+2 H2
+Figure 3.4 G′
+iis obtained from Giby doing Operation 1 to edges e1ande2.
+For the sharpness of the upper bound, see Example 3.3.
+Wecallasetoftriangles independent ifanytwoofthemarevertex-disjoint. Since
+each single triangle is a triangle-tree-structure, applying Theorem 3.1, we have the
+following corollary, and for the sharpness of the upper bound, see Example 3.3:
+Corollary 3.2 IfGis a connected graph with t′independent triangles, then
+rc(L(G))≤n2−t′.
+Moreover, the bound is sharp.
+Example 3.3 As shown in Figure 3.5, Gconsists of t(independent) triangles and
+t−1edges which do not belong to any triangles. Since Ghas3tinner vertices, by
+Theorem 3.1(Corollary 3.2), we know rc(L(G))≤2t; on the other hand, it is easy to
+show that the diameter of the line graph L(G)is2t, and so we have rc(L(G)) = 2t.
+3.2 Another upper bound from Theorem 3.1
+Now we use the notation similar to that of Theorem 3.1. Let T=/uniontextc
+i=1Ti=/uniontextc
+i=1{Ti,jiis a triangle of G: 1≤ji≤ti}(/summationtextc
+i=1ti=t) be a set of tedge-disjoint
+triangles of G; the subgraph of G,G[E(Ti)], induced by each E(Ti) is a connected
+8G
+Figure 3.5 Figure of Example 3.3.
+component of the subgraph G[E(T)] which may not be a triangle-tree-structure,
+that is, it may contain a triangle-cycle-structure as a subgraph.
+InG, for each 1 ≤i≤c, letGi=G[E(Ti)],Vi=V(Gi),Ei=E(Ti);E0
+i=
+E(Vi)∪∂(Vi)⊇Ei, andG0
+i=G[E0
+i]. We obtain a new graph G′fromGby doing
+Operation 1 at each edge e∈E(Vi)\Eifor 1≤i≤c, and we denote by G′
+ithe
+new subgraph (of G′) corresponding to G0
+i. Applying Observation 2.3 repeatedly,
+we have rc(L(G))≤rc(L(G′)). Now each G′
+icontains exactly titriangles: Ti,ji
+where 1≤ji≤ti. We now obtain a new graph G′′fromG′by doing Operation 2 to
+thoseG′
+iswhichcontaintriangle-cycle-structures suchthateachsubgra ph(ofG′′)G′′
+i
+correspondingto G′
+icontainsnotriangle-cycle-structure. Let op(G′)betheminimum
+times of doing Operation 2 needed during above procedure. Clearly, op(G′) =
+op(G[E(T)]) (minimum times of doing Operation 2 needed for G[E(T)] such that
+the resulting graph contains no triangle-cycle-structure). Since there are op(G′)
+new inner vertices totally produced, and by Observation 2.4 and the discussion of
+Theorem 3.1, we have
+Lemma 3.4 For any set Toftedge-disjoint triangles of a connected graph Gwith
+n2inner vertices, we have
+rc(L(G))≤n2+op(G[E(T)])−t.
+We know that a triangle-forest-structure of size ℓcontains 2 ℓ+c(inner) vertices
+wherecis the number of components of it. Operation 2 does not change the number
+of edge-disjoint triangles, but the number of inner vertices increa ses 1 after we did
+Operation 2 once. Then it is easy to show that after doing Operation 2op(G[E(T)])
+times, the number of inner vertices of the new graph is op(G[E(T)]) +n2=2t+1+n′
+2
+wheren′
+2denotes the number of inner vertices not covered by the original tedge-
+disjoint triangles. So, by Lemma 3.4, we have the following result and f or the
+sharpness of the bound see Example 3.6.
+Theorem 3.5 IfGis a connected graph, Tis a set of tedge-disjoint triangles
+that cover all but n′
+2inner vertices of Gandcis the number of components of the
+subgraph G[E(T)], then
+rc(L(G))≤t+n′
+2+c.
+Moreover, the bound is sharp.
+9Example 3.6 LetGbe a graph shown in Figure 3.6. The set T={ui,vi,ui+1}k−1
+i=1
+is a set of k−1edge-disjoint triangles, n′
+2= 1andc= 1. By Theorem 3.5, we have
+rc(L(G))≤k+1; on the other hand, it is easy to show that the diameter of L(G)
+isk+1, and so rc(L(G)) =k+1.
+x
+yu1u2u3 uk
+v1v2v3 vk
+G
+Figure 3.6 Figure of Example 3.6.
+4 Bounds for iterated line graphs
+Recall that the iterated line graph of a graph G, denoted by L2(G), is the line
+graph of the graph L(G). The following corollary deduced from Theorem 3.4 is an
+upper bound of the rainbow connection numbers of iterated line gra phs of connected
+cubic graphs.
+Corollary 4.1 IfGis a connected cubic graph with nvertices, then
+rc(L2(G))≤n+1.
+Proof.SinceGis a connected cubic graph, each vertex is an inner vertex and the
+clique/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}htinL(G) corresponding to each vertex vis a triangle. We know that
+K={/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht:v∈V2}={/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht:v∈V}is a clique decomposition of L(G). Let
+T=K. ThenTis a set of nedge-disjoint triangles that cover all vertices of L(G)
+andL(G) =L(G)[E(T)]. Son′
+2= 0 andc= 1, and by Theorem 3.5, the conclusion
+holds.
+In graph G, we call a path of length kapendent k -length path if one of its end
+vertex has degree 1 and all inner vertices has degree 2. By definitio n, a pendent
+k-length path contains a pendent ℓ-length path(1 ≤ℓ≤k). A pendent 1-length
+path is a pendent edge .
+Theorem 4.2 LetGbe a connected graph with medges and m1pendent 2-length
+paths. Then rc(L2(G))≤m−m1, the equality holds if and only if Gis a path of
+length at least 3.
+Proof. NowL(G) is a graph with mvertices and m1pendent edges. Then it
+hasm−m1inner vertices. By the discussion before Theorem 3.1, we give each
+10clique/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}htinL2(G) a fresh color, where vis an inner vertex of L(G). It is
+easy to show that this gives a rainbow ( m−m1)-edge-coloring of L2(G), and so
+rc(L2(G))≤m−m1.
+IfGis a path of length at least 3, then the equality clearly holds.
+IfGcontains a cycle, then L(G) clearly contains a minimal cycle C:v1,···,vℓ.
+SoL2(G) contains a clique-cycle-structure of size ℓwhich is formed by cliques
+{/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht}ℓ
+i=1. By a theorem in [6], we know that this structure needs at most ⌈ℓ+1
+2⌉
+colorstomake sure that it israinbowconnected. The rest part of L2(G) isformedby
+cliques{/a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}ht:v∈V2\{vi}ℓ
+i=1}whereV2is the set of inner vertices of L(G). We
+giveeach /a\}brack⌉tl⌉{tS(v)/a\}brack⌉tri}htafresh color, byObservation 2.1, rc(L2(G)≤ ⌈ℓ+1
+2⌉+(m−m1−ℓ)<
+m−m1.
+IfGis a tree with a vertex of degree at least 3, then L(G) contains a cycle, a
+similar argument will show rc(L2(G))< m−m1.
+SoGis a tree with maximum degree 2, and hence it must be a path of length a t
+least 3.
+References
+[1] J.A. Bondy, U.S.R. Murty, Graph Theory , GTM 244, Springer, 2008.
+[2] G. Chartrand, G.L. Johns, K.A. McKeon, P. Zhang, Rainbow connection in graphs ,
+Math. Bohem. 133(2008), 85-98.
+[3] Y. Caro, A. Lev, Y. Roditty, Z. Tuza, R. Yuster, On rainbow connection , Electron.
+J. Combin. 15(2008), R57.
+[4] S.T.Hedetniemi, P.J. Slater, Linegraphs of trianglelessgraphs anditerated clique
+graphs, inGraph Theory and Applications , Lecture Notes in Mathematics 303(ed.
+Y. Alavi et al.), Springer-Verlag, Berlin, Heidelberg, New York, 1972, p p. 139-147;
+MR49#151.
+[5] Bo-Jr Li, G.J. Chang, Clique coverings and partitions of line graphs , Discrete
+Math.308(2008), 2075-2079.
+[6] X. Li, Y.Sun, Rainbow connectionnumbersof linegraphs , accepted forpublication
+in Ars Combin.
+11
\ No newline at end of file
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+arXiv:1001.0288v2  [gr-qc]  28 Jun 2010Dynamical Horizon Entropy Bound Conjecture in Loop Quantum Cosmology
+Li-Fang Li and Jian-Yang Zhu∗
+Department of Physics, Beijing Normal University, Beijing 100875, China
+The covariant entropy bound conjecture is an important hint for the quantum gravity, with
+several versions available in the literature. For cosmolog y, Ashtekar and Wilson-Ewing ever show
+the consistence between the loop gravity theory and one vers ion of this conjecture. Recently, S.
+He and H. Zhang proposed a version for the dynamical horizon o f the universe, which validates
+the entropy bound conjecture for the cosmology filled with pe rfect fluid in the classical scenario
+when the universe is far away from the big bang singularity. H owever, their conjecture breaks down
+near big bang region. We examine this conjecture in the conte xt of the loop quantum cosmology.
+With the example of photon gas, this conjecture is protected by the quantum geometry effects as
+expected.
+PACS numbers: 04.60.Pp,04.60.-m, 65.40.gd
+I. INTRODUCTION
+The thermodynamical property of spacetime is an im-
+portant hint for the quantization of gravity. Start-
+ing from Hawking’s discovery of black hole’s radiation
+[1], a theory of thermodynamics of spacetime is be-
+ing constructed gradually. Recently, the second law of
+this thermodynamics was generalized to the covariant
+entropy bound conjecture [3]. It states that the en-
+tropy flux Sthrough any null hypersurface generated
+by geodesics with non-positive expansion, emanating or-
+thogonallyfrom atwo-dimensional(2D) spacelikesurface
+of areaA, must satisfy
+S
+A≤1
+4l2p, (1)
+wherelp=√
+/planckover2pi1is the Planck length. Here and in what
+follows, we adopt the units c=G=kB= 1. Soon,
+Flanagan, Marolfand Wald [4] proposed a new versionof
+the entropybound conjecture. Ifone allowsthe geodesics
+generating the null hypersurface from a 2D spacelike sur-
+face of area Ato terminate at another 2D spacelike sur-
+face of area A′before coming to a caustic, boundary or
+singularity of spacetime, one can replace the above con-
+jecture with
+S
+A′−A≤1
+4l2p. (2)
+More recently, He and Zhang related these conjectures
+to dynamical horizon and proposed a covariant entropy
+bound conjecture on the cosmological dynamical horizon
+[5]: Let A(t) be the area of the cosmological dynami-
+cal horizon at cosmological time t, then the entropy flux
+Sthrough the cosmological dynamical horizon between
+∗Author to whom correspondence should be addressed; Electro nic
+address: zhujy@bnu.edu.cntimetandt′(t′> t) must satisfy
+S
+A(t′)−A(t)≤1
+4l2p, (3)
+if the dominant energy condition holds for matter.
+As a non-perturbative and background-independent
+quantization of gravity, the loop quantum gravity (LQG)
+developed rapidly in recent years. Its cosmological ver-
+sion, the loop quantum cosmology (LQC) achieved many
+successes, including resolution of the classical singularity
+[6, 7], quantum suppression of classical chaotic behavior
+near singularities [8, 9], generic phase of inflation [10–
+12] and more (for example [13]). Since it has been sug-
+gested that the holographic principle is a powerful hint
+and should be used as an essential building block for any
+quantum gravitytheory [2], it is important and tempting
+to investigate the covariant entropy bound conjecture in
+the framework of the LQC, which is a successful applica-
+tion of the non-perturbative quantum gravity scheme—
+the LQG. The authors of [17] investigated the Bousso’s
+covariant entropy bound [2, 3] with a cosmology filled
+with photon gas and found that the conjecture is vio-
+lated near the big bang in the classical scenario. But
+they found the LQC can protect this conjecture even in
+the deep quantum region. In [5], He and Zhang proposed
+a new version of the entropy bound conjecture for the
+dynamical horizon in cosmology and validated it through
+a cosmology filled with adiabatic perfect fluid, governed
+by the classical Einstein equation when the universe is
+far away from the big bang singularity. But when the
+universe approaches the big bang singularity, the strong
+quantum fluctuation does break down their conjecture.
+In analogy to Ashtekar and Wilson-Ewing’s result [17],
+one may wonder if He and Zhang’sconjecture can also be
+protected by the quantum geometry effect of the LQG.
+Following [17], we use photon gas as an example to
+investigate this problem. As expected, we find that the
+loop quantum effects can indeed protect the conjecture.
+Besidestheresultof[17], ourresultpresentsonemoreev-
+idence for the consistence between the loop gravity and
+the covariantentropyconjecture. Thispaper isorganized2
+as follows. In Sec. II, we briefly review the framework
+of the effective LQC and describe the covariant entropy
+bound conjecture proposed by He and Zhang [5]. Then
+in Sec. III, we test this conjecture with cosmology filled
+with photon gas, and show that the LQC is able to pro-
+tect the conjecture in all. We conclude the paper in Sec.
+IV and discuss the implications.
+II. THE EFFECTIVE FRAMEWORK OF LQC
+AND COVARIANT ENTROPY BOUND
+CONJECTURE
+To a good degree of approximation, the universe can
+be described by the well-known Friedmann-Robertson-
+Walker (FRW) metric,
+ds2=−dt2+a2(t)/parenleftbiggdr2
+1−kr2+r2dΩ2/parenrightbigg
+,(4)
+which describes a homogeneous and isotropic universe.
+Here the spatial curvature k=−1,0,1 correspond to
+open, flat and closed universes respectively, and ais the
+scale factor of the universe. For simplicity we focus on a
+flat universe in this paper. In the LQC, the phase space
+for a flat universe is spanned by the coordinates c=γ˙a,
+being the gravitational gauge connection, and p=a2,
+being the densitized triad. γ= 0.2375 is the Barbero-
+Immirzi parameter [14]. Then the effective Hamiltonian
+in the LQC is given by [15, 16]
+Heff=−3
+8πγ2¯µ2√psin2(¯µc)+HM.(5)
+The variable ¯ µcorresponds to the dimensionless length
+of the edge of the elementary loop and is given by
+¯µ=ξpλ, (6)
+whereξ >0 andλdepend on the particular scheme in
+the holonomy corrections. In this paper we take the ¯ µ-
+scheme, which gives
+ξ2= 2√
+3πγl2
+p (7)
+andλ=−1/2. With this effective Hamiltonian, we have
+the canonical equation
+˙p={p,Heff}=−8πγ
+3∂Heff
+∂c, (8)
+or,
+˙a=sin(¯µc)cos(¯µc)
+γ¯µ. (9)
+Combining with the constraint on the Hamiltonian,
+Heff= 0, we obtain the modified Friedmann equation,
+H2=8π
+3ρ/parenleftbigg
+1−ρ
+ρc/parenrightbigg
+, (10)in terms of the Hubble rate H≡˙a
+a, whereρ:=HM
+p3/2and
+ρc=3
+8πγ2¯µ2p=√
+3
+16π2γ2/planckover2pi1.ρcis the critical energy density
+coming from the quantum effect in the LQC. It is a large
+quantity and when it goes to infinity all the quantum
+effects disappear.
+According to [5], the cosmological dynamical horizon
+[2] is defined geometrically as a three-dimensional hyper-
+surface foliated by spheres, where at least one orthogonal
+null congruence with vanishing expansion exists. For a
+sphere characterized by any value of ( t,r), there are two
+future directed null directions
+ka
+±=1
+a/parenleftbigg∂
+∂t/parenrightbigga
+±1
+a2/parenleftbigg∂
+∂r/parenrightbigga
+, (11)
+satisfying geodesic equation kb∇bka= 0. The expansion
+of these null directions is
+θ:=▽aka
+±=2
+a2/parenleftbigg
+˙a±1
+r/parenrightbigg
+, (12)
+where the dot denotes differential with respect to t, and
+the sign +( −) represents the null direction pointing to
+larger (smaller) values of r. For an expanding universe,
+i.e. ˙a >0,θ= 0determinesthelocationofthedynamical
+horizon, rH= 1/˙a, bythedefinitionofdynamicalhorizon
+givenabove. The LQC replacesthe big bang with the big
+bounce, so the universe is symmetric with respect to the
+point of the bounce, expanding on one side of the bounce
+andcontractingontheotherside. Thedynamicalhorizon
+in the contracting stage of the LQC corresponds to rH=
+−1/˙a, andalloftherelationsaresimilartothe onesgiven
+here. In this paper we only consider the expanding stage
+for the LQC, but note that the contracting stage is the
+same.
+Since the area of the dynamical horizon is A=
+4πa2r2
+H= 4πH−2, the covariant entropy bound conjec-
+ture in our question becomes
+l2
+pS≤π/bracketleftbig
+H−2(t′)−H−2(t)/bracketrightbig
+, (13)
+whereSis the entropy flux through the dynamical hori-
+zon between cosmological time tandt′(t′> t), andHis
+the Hubble parameter. Considering that the cosmology
+model discussed here is isotropic and homogeneous, we
+can write the entropy current vector as
+sa=s
+a3/parenleftbigg∂
+∂t/parenrightbigga
+, (14)
+wheresis the ordinary comoving entropy density, inde-
+pendent of space. If the entropy current of the perfect
+fluid is conserved, i.e., ∇asa= 0,swill be independent
+oftas well. For simplicity we restrict ourselves to this
+special case. The entropy flux through the dynamical
+horizon (shown in Fig.1) is given by
+S=/integraldisplay
+CDHsaǫabcd=4πs
+3/parenleftbig
+r3
+H(t′)−r3
+H(t)/parenrightbig
+(15)3
+whereǫabcd=a3r2sinθ(dt∧dr∧dθ∧dφ)abcdis the
+spacetime volume 4-form. So the conjecture is reduced
+to
+H−2(t′)−4
+3l2
+ps˙a−3(t′)≥H−2(t)−4
+3l2
+ps˙a−3(t), t′> t.
+(16)
+/s65/s40/s116/s39/s41
+/s67/s68/s72/s67/s68/s72
+/s65/s40/s116/s41/s115/s97
+FIG. 1: A schematic of the entropy current flowing across the
+cosmological dynamical horizon. The thick solid line marke d
+by “CDH” is the cosmological dynamical horizon. The thin
+solid line is the region enclosed by the CDH at time tandt′
+respectively. The dashed lines are the entropy current.
+III. CONJECTURE TEST FOR A COSMOLOGY
+FILLED WITH PERFECT FLUID
+Given that the FRW universe is filled with photon gas,
+the energy momentum tensor can be expressed as
+Tab=ρ(t)(dt)a(dt)b+P(t)a2(t){(dr)a(dr)b
++r2[(dθ)a(dθ)b+sin2θ(dφ)a(dφ)b]/bracerightbig
+.(17)
+The pressure Pand the energy density ρsatisfy a fixed
+equation of state
+P=ωρ, (18)
+where the constant ω=1
+3. From∇aTab= 0, we have the
+conservation equation
+˙ρ+3H(ρ+P) = 0. (19)
+The comoving entropy density sis given by
+s=a3ρ+P
+T=a3(1+ω)ρ
+T, (20)andρdepends only on the temperature T,
+ρ=Kol−2−1+ω
+ωpT1+ω
+ω, (21)
+whereKois a dimensionless constant depending on
+the density of energy state of the perfect fluid. For
+photon gas Ko=π2
+15. Plugging above thermody-
+namics relation into equation (20) we get s= (1 +
+ω)Kω
+1+ωol−1−2ω
+1+ωpρ1
+1+ωa3. Written the above conservation
+equation as
+˙ρ+3(1+ω)ρ˙a
+a= 0, (22)
+we have an integration constant C=ρ1
+1+ωa3. Thens=
+(1+ω)Kω
+1+ωol−1−2ω
+1+ωpC. Combining our equation of state
+Eq. (18) with the above conservation equation, we get
+the relationship between ρand the Hubble parameter,
+H=−1
+3(1+ω)˙ρ
+ρ. (23)
+Substituting the above relation (23) into the modified
+Friedmann equation (10), we can get
+ρ=1
+6π(t+C1)2(1+ω)2+1
+ρc(24)
+whereC1is an integration constant without direct phys-
+ical significance, and we can always drop it by resetting
+the time coordinate. Setting C1= 0 gives
+H=4πt(1+ω)
+6πt2(1+ω)2+1
+ρc. (25)
+With the definition of the Hubble parameter, we can in-
+tegrate once again to get
+a(t) =C1/3/bracketleftbigg
+6πt2(1+ω)2+1
+ρc/bracketrightbigg1
+3(1+ω)
+.(26)
+Whenρcgoes to infinity, all of the above solutions be-
+come the same as the classical ones [18] presented in [5].
+In the classical scenario,
+H−2−4
+3l2
+ps˙a−3
+=9
+4t2(1+ω)2−9Kω
+1+ωol1−2ω
+1+ωp
+2(6π)1/(1+ω)t3−2
+1+ω(1+ω)4−2
+1+ω.
+(27)
+Whent≪1,H−2−4
+3l2
+ps˙a−3∼ −t3−2
+1+ω=−t3/2whichis
+a decreasing function of t, so the conjecture breaks down
+when the universe approaches the big bang singularity.
+We introduce a new variable τ=√2πρc(1 +ω)tfor
+the LQC to simplify the above expressions to
+H=/radicalbig
+2πρc2τ
+3τ2+1, (28)
+a=C1/3ρ−1
+3(1+ω)
+c/parenleftbig
+3τ2+1/parenrightbig1
+3(1+ω), (29)
+˙a=aH= 2τC1/3ρ−1
+3(1+ω)
+c/radicalbig
+2πρc/parenleftbig
+3τ2+1/parenrightbig1
+3(1+ω)−1.
+(30)4
+/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s49/s50/s51/s52/s53/s54/s55/s56
+/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48/s49/s48/s48
+/s32/s100/s101/s114/s105/s118/s97/s116/s105/s118/s101
+/s32/s32/s108/s111/s103/s97/s114/s105/s116/s104/s109/s32/s111/s102/s32/s100/s101/s114/s105/s118/s97/s116/s105/s118/s101/s32/s102/s117/s110/s99/s116/s105/s111/s110/s32/s105/s116/s115/s101/s108/s102
+/s32/s102/s117/s110/s99/s116/s105/s111/s110/s32/s105/s116/s115/s101/s108/s102/s32
+FIG. 2: Function H−2−4
+3l2
+ps˙a−3and its derivative respect to
+τfor photon gas.
+Then
+H−2−4
+3l2
+ps˙a−3
+=1
+2πρc/bracketleftBigg/parenleftbigg3
+2τ+1
+2τ/parenrightbigg2
+−(1+ω)
+6√
+2πKω
+1+ωo(√
+3
+16π2γ2)1
+1+ω−1
+2/parenleftbig
+3τ2+1/parenrightbig3−1
+(1+ω)
+τ3
+.
+(31)
+It is obvious that the necessary and sufficient condition
+for meeting the covariant entropy bound conjecture is
+that the above expression increases with τ. In order to
+investigate the monotone property of above function, we
+plotH−2−4
+3l2
+ps˙a−3itself and its derivative respect to τin Fig.2. The minimal value of the derivative is about
+1.16>0. The covariant entropy bound conjecture for
+dynamicalhorizonin cosmologyisfully protectedbyloop
+quantum effect.
+IV. CONCLUSION AND DISCUSSION
+The covariant entropy bound conjecture comes from
+the holographic principle and is an important hint for
+the quantum gravity theory. In the recent years we have
+witnessed more and more success of the loop quantum
+gravity, especially for the problem of the big bang singu-
+larity in cosmology. The entropy bound conjecture usu-
+ally breaks down in the strong gravity region of space-
+time where the quantum fluctuation is strong, and one
+would expect the loop quantum correction to protect the
+conjecture from the quantum fluctuation. And Ashtekar
+and Wilson-Ewing do find a result in [17] which is consis-
+tent with above expectation. In this paper, we general-
+izedthecovariantentropyconjectureforthe cosmological
+dynamical horizon proposed in [5] to the loop quantum
+cosmology scenario. We found that the quantum geom-
+etry effects of the loop quantum gravity can also protect
+the conjecture. Our result gives out one more evidence
+for the consistence of covariant entropy conjecture and
+loop quantum gravity theory. This adds one more en-
+couraging result of loop quantum gravity theory besides
+previous ones.
+Acknowledgments
+The work was supported by the National Natural Sci-
+ence of China (No.10875012) and the Scientific Research
+Foundation of Beijing Normal University.
+[1] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).
+[2] R. Bousso, Rev. Mod. Phys. 74, 825 (2002).
+[3] R. Bousso, JHEP 07(1999)004.
+[4] E. Flanagan, D. Marolf and R. Wald, Phys. Rev. D 62,
+084035 (2000).
+[5] S. He and H. Zhang, JHEP 10(2007)077.
+[6] M. Bojowald, Phys. Rev. Lett. 86, 5227 (2001).
+[7] M. Bojowald, Class. Quantum Grav. 20, 2595 (2003).
+[8] M. Bojowald, and G. Date, Phys. Rev. Lett. 92, 071302
+(2004).
+[9] M. Bojowald, Class. Quantum Grav. 21, 3541 (2004).
+[10] M. Bojowald, Phys. Rev. Lett. 89, 261301 (2002).
+[11] G. Date and G. Hossain, Phys. Rev. Lett. 94, 011301
+(2005).
+[12] Hua-Hui Xiong and Jian-Yang Zhu, Phys. Rev. D 75,
+084023 (2007).[13] Li-Fang Li and Jian-Yang Zhu, Phys. Rev. D 79, 044011
+(2009); Phys. Rev. D 79, 124011 (2009); Advances in
+High Energy Physics, doi:10.1155/2009/905705.
+[14] A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Phys.
+Rev. Lett. 80, 904 (1998).
+[15] A. Ashtekar, T. Pawlowski, and P. Singh, Phys. Rev. D
+73, 124038 (2006); 74, 084003 (2006).
+[16] J. Mielczarek, T. Stachowiak, M. Szydlowski, Phys. Rev .
+D77, 123506 (2008).
+[17] A. Ashtekar and E. Wilson-Ewing, Phys. Rev. D 78,
+064047 (2008).
+[18] Note that the original result in [5] used conformal time
+η, while we use universe time tin this paper. ηcan be
+negative which divides the discussion into two cases. tis
+always positive and makes the discussion simpler.
\ No newline at end of file
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+arXiv:1001.0289v6  [math.AT]  9 May 2010ON THE CONSTRUCTIONS OF FREE AND LOCALLY
+STANDARD Z2-TORUS ACTIONS ON MANIFOLDS
+LI YU
+Abstract. We introduce an elementary way of constructing principal ( Z2)m-
+bundles over compact smooth manifolds. In addition, we will define a g eneral
+notion of locally standard ( Z2)m-actions on closed manifolds for all m≥1,
+and then give a general way to construct all such ( Z2)m-actions from the orbit
+space. Some related topology problems are also studied.
+1.Introduction
+If the group ( Z2)macts freely and smoothly on a closed manifold Mn, the orbit
+spaceQnis also a closed manifold. We can think of Mneither as a principal
+(Z2)m-bundle over Qnor as a regular covering over Qnwith deck transformation
+group (Z2)m. In algebraic topology, we have a standard way to recover Mnfrom
+Qnusing the universal covering space of Qnand the monodromy map of the
+covering (see [1]). However, it is not very easy to visualize the total space of
+the covering using that construction. Considering the speciality of (Z2)m, it is
+desirable to have a new way of constructing such regular coverings from the orbit
+spaces which can really help us to visualize the total space more easily . In this
+paper, such a construction will be given with the name glue-back construction .
+Another source of nice Z2-torus actions on manifolds are locally standard ac-
+tions (see [2]). Suppose Mnis a closed manifold with a smooth locally standard
+(Z2)n-action, let Xn=Mn/(Z2)nbe the orbit space and π:Mn→Xnbe the
+orbit map. It is well known that Xnis a nicen-manifold with corners, and if
+the action is not free, Xnwill have boundary. The ( Z2)n-action determines a
+characteristic function νπ(taking values in ( Z2)n) on the facets of Xn, which
+encodes the information of isotropy subgroups of the non-free o rbits. In partic-
+ular, when Xnis a convex simple polytope, there is a standard construction to
+recoverMn(up to equivariant homeomorphism) from the characteristic funct ion
+2000Mathematics Subject Classification. 57R22, 57S10, 57S17, 52B70.
+Key words and phrases. Z2-torus, regular covering, locally standard action, involutive panel
+structure, glue-back construction.
+This work is partially supported by a grant from NSFC (No.10826040) and by the Scientific
+Research Foundation for the Returned Overseas Chinese Scholar s, State Education Ministry.
+12 LI YU
+νπonXn(see [2]). But in general, if H1(Xn,Z2) is not trivial, we need an addi-
+tional piece of data to recover Mn— a principal ( Z2)n-bundleξπoverXnwhich
+encodes the information of the free orbits of the action (see [3]). H owever, the
+bundle information ξπhas a quite different flavor from the characteristic function
+νπand is not so easy to be visualized in the orbit space Xn. In this paper, we will
+combine the characteristic function νπand the ( Z2)n-bundleξπonXninto a com-
+posite (Z2)n-valued colorings ( λπ,µπ) on a new manifold Un(called a Z2-coreof
+Xn), which is a nice manifold with corners obtained from Xn(but not uniquely).
+And up to equivariant homeomorphisms, we can recover Mnfrom the composite
+(Z2)n-coloring (λπ,µπ) onUnfrom a generalized glue-back construction.
+Moreover, we can define a general notion of locally standard ( Z2)m-action on
+n-dimensional manifolds for all m≥1, which includes all free ( Z2)m-actions on n-
+dimensional manifolds. The glue-back construction can be applied in t his general
+setting as well. So actually we do not assume m=nat all in this paper.
+The paper is organized as following. In section 2, we will explain how to g et a
+Z2-coreVnfromaclosedmanifold Qnandintroduceanimportantstructureon Vn
+calledinvolutive panel structure . We will introduce several definitions concerning
+this structure to make our subsequent discussions precise and co nvenient. Some
+explicit examples will be analyzed to illustrate these definitions. In sec tion 3,
+we will introduce the glue-back construction from a Z2-coreVnofQnwith a
+(Z2)m-colorings. And we will show that any principal ( Z2)m-bundles over Qncan
+be obtained in this way. Also the glue-back construction makes sens e for any
+nice manifold with corners equipped with an involutive panel structur e. Some
+properties of this construction will be studied along with some explicit examples.
+InSection4, wewillgeneralizethenotionof Z2-coreandglue-backconstructionto
+compact manifolds with boundary as well. Then in Section 5 we define a g eneral
+notion of locally standard ( Z2)m-actions on closed n-manifolds for any m≥1 and
+apply the glue-back construction to this general setting. Especia lly, the notion
+of involutive panel structure is used to unify all our constructions . In addition,
+we will state some classification theorems of locally standard ( Z2)m-actions on
+closedn-manifolds up to (weak) equivariant homeomorphisms. In Section 6,
+we will discuss how to get some topological information (e.g. the numb er of
+connected components and orientability) of the glue-back constr uction of locally
+standard ( Z2)m-actions from the ( Z2)m-colorings. In the end, we will propose
+some problem for the further study.
+The main idea of the paper is inspired by the description of locally stand ard
+Z2-torus manifolds in [3]. An aim of this paper is to establish a framework f or
+studying general locally standard ( Z2)m-actions on n-manifolds in the future. In3
+particular, the author will use the glue-back construction to stud y the Halperin-
+Carlsson conjecture for free ( Z2)m-actions on compact manifolds in a sequel pa-
+per. Also, the involutive panel structure defined in this paper might have some
+independent value.
+In this paper, we denote the quotient group Z/2ZbyZ2and always think of
+(Z2)mas an additive group. In addition, we will use the following conventions :
+(1) any manifold and submanifold in this paper is smooth;
+(2) we always identify an embedded submanifold with its image;
+(3) any ( Z2)m-actions on manifolds in this paper are smooth and effective.
+2.Z2-core of a closed manifold and Involutive panel structure
+SupposeMnis ann-dimensional closed manifold with a free ( Z2)m-action (m
+is an arbitrary positive integer), let Qn=Mn/(Z2)mbe the orbit space and
+π:Mn→Qnbe the orbit map. Then Qnis also a closed manifold. In addition,
+we always assume Qnis connected in this paper.
+We can consider the orbit map π:Mn→Qneither as a regular ( Z2)mcovering
+or as a principal ( Z2)m-bundle map. Note that Mnmay not be connected in
+general.
+It is well-known that the up to bundle isomorphism, principal ( Z2)m-bundles
+overQnare one-to-one correspondent with elements of H1(Qn,(Z2)m). Then
+π:Mn→Qndetermines an element
+Λπ∈H1(Qn,(Z2)m)∼=Hom(H1(Qn,Z2),(Z2)m).
+From another viewpoint, as a regular covering space, π:Mn→Qnis deter-
+mined by its monodromy map Hπ:π1(Qn)→(Z2)m. Since (Z2)mis an abelian
+group, wegetaninducegrouphomomorphism Hab
+π:H1(Qn,Z2)→(Z2)mwhichis
+exactlytheΛ πabove. Moreover, bythePoincar´ eduality, wehave Hn−1(Qn,Z2)∼=
+H1(Qn,Z2). So we obtain a group homomorphism Λ∗
+π:Hn−1(Qn,Z2)→(Z2)m.
+The above analysis suggests us to construct a new geometric obje ct fromQn
+which can carry all the information of Λ π(or Λ∗
+π). First, we recall a well-known
+theorem in algebraic topology.
+Theorem 2.1 (Hopf).Letf:Mm→Nnbe a smooth map between closed
+oriented manifolds and Ln−1⊂Nna closed, oriented submanifold of codimension
+psuch thatfis transverse to L. Writeu∈Hp(N)for the Poincar´ e dual of [L]N,
+that is,u∩[N] = [L]N. Then[f−1(L)]M=f∗(u)∩[M]. In other words: If u
+is Poincar´ e dual to [L]N, thenf∗(u)∈Hp(M)is Poincar´ e dual to [f−1(L)]M. If
+usingZ2coefficient, we do not need to assume that MmandNnare orientable.4 LI YU
+Proof.Use the naturality of the Thom class of the tangent bundle. /square
+IfH1(Qn,Z2) = 0, then any principal ( Z2)m-bundle over Qnis trivial. So in
+the rest of this paper, we always assume H1(Qn,Z2)/\e}atio\slash= 0. let {ϕ1,···,ϕk}be
+a basis ofH1(Qn,Z2), and let {α1,···,αk}be the basis of Hn−1(Qn,Z2) that is
+dual to{ϕ1,···,ϕk}under the Poincar´ e duality.
+Lemma 2.2. α1,···,αkcan be represented by codimension one connected em-
+bedded submanifolds of Qn.
+Proof.SinceH1(Qn,Z2)∼=[Qn,K(Z2,1)] = [Qn,RP∞] = [Qn,RPn+1], an ele-
+mentϕ∈H1(Qn,Z2) corresponds to a homotopy class of maps [ f] :Qn→RPn+1
+such thatϕ=f∗(Φ) where Φ is a generator of H1(RPn+1,Z2). Then the Z2-
+homology class represented by a canonically embedded RPn⊂RPn+1is the
+Poincar´ e dual of Φ. We can always assume that fis smooth and transverse to
+RPn. Then by above theorem, Σ = f−1(RPn) is a codimension 1 embedded
+submanifold in Qnwhich is Poincar´ e dual to ϕ. So we can find codimension one
+embedded submanifolds Σ 1,···,Σkwhich arePoincar´ e dual to ϕ1,···,ϕkrespec-
+tively. In addition, we can always choose Σ 1,···,Σkto be connected. Indeed, for
+n= 1,2, this is obviously true. And when n≥3, if some Σ iis not connected, we
+can connect all its components via thin tubes in Qn, which will not change the
+homology class of Σ iinHn−1(Qn,Z2). /square
+A collection of codimension-one embedded closed submanifolds {Σ1,···,Σk}
+is called a Z2-cut system ofQnif they satisfy the following conditions:
+(1) thehomologyclasses [Σ 1],···,[Σk] formaZ2-linearbasisof Hn−1(Qn,Z2).
+(2) Σ1,···,Σkare in general position in Qnwhich means that:
+(a) Σ1,···,Σkintersect transversely with each other and,
+(b) if Σ i1∩···∩Σisis not empty, then it is an embedded submanifold of
+Qnwith codimension s.
+Now we choose a small tubular neighborhood N(Σi) of each Σ iinQn, and then
+remove the interior of each N(Σi) fromQn. The manifold that we get is:
+Vn=Qn−k/uniondisplay
+i=1int(N(Σi))
+which is called a Z2-coreofQnfrom cutting Qnopen along Σ 1,···,Σk. The
+boundary of Vnis∂(/uniontext
+iN(Σi)). We call ∂N(Σi) thecut section of ΣiinQn.
+Notice that the projection ηi:∂N(Σi)→Σiis a double cover, either trivial or
+nontrivial. Let τibe the generator of the deck transformation of ηi. Thenτiis5
  
+  
+τ
+	τ
+
+ Σ
+Σ
+ΣΣΣ
+Figure 1. Local deformation of τi’s
+a free involution on ∂N(Σi), i.e.τiis a homeomorphism with no fixed point and
+τ2
+i=id.
+The boundary of Vnis tessellated by ( n−1)-dimensional compact connected
+manifolds (with boundary) called facetsofVn. Any connected component of the
+intersection of some facets is called a (closed) face ofVn. Since Σ 1,···,Σkare
+in general position in Qn, soVnis anice manifold with corners , which means
+that each codimension lface ofVnis the intersection of exactly lfacets. For
+a comprehensive introduction of manifolds with corners and related concepts,
+see [4] and [5].
+Remark 2.3. Vnmightnothave vertices(0-dimensionalstrata)ontheboundary.
+for example, if Qn=Sn−1×S1(n≥3), cutting QnalongSn−1× {1}gives a
+Z2-coreVn=Sn−1×[0,1] ofQnwhose boundary consists of two disjoint Sn−1.
+In addition, we call the union of facets of Vnthat belong to ∂N(Σi) apanel,
+denoted by Pi(see Figure 2). So {P1,...,P k}forms a panel structure on Vn.
+Recall that a panel structure on a topological space Yis a locally finite family of
+closed subspaces {Yα}α∈Aindexed by some set A. EachYαis called a panelofY
+(see [5]).
+Notice that the involution τimay not map a Pi⊂∂N(Σi) intoPi. This is
+because that there might be some N(Σj) so thatτiandτjdo not commute
+at the intersections ∂N(Σi)∩∂N(Σj) (see the left picture in Figure 1). But
+the following lemma shows that we can always deform the τiandτjlocally by
+isotopies to make them commute at ∂N(Σi)∩∂N(Σj).
+Lemma 2.4. We can deform τi’s around the intersections of ∂N(Σi)’s so that
+after the deformations, we have:
+(i)eachτiis still a free involution ∂N(Σi)→∂N(Σi)and the quotient
+∂N(Σi)//a\}bracketle{tx∼τi(x)/a\}bracketri}ht∼=Σi;6 LI YU
+(ii)for any1≤i,j≤k,∂N(Σi)∩∂N(Σj)becomes an invariant set of both
+τiandτj;
+(iii)for any point x∈∂N(Σi)∩∂N(Σj),τi(τj(x)) =τj(τi(x)).
+Proof.For∀p∈Σi, letTpΣibe the tangent plane of Σ iatpinQn. Suppose
+Σi1∩ ··· ∩Σisis nonempty. For any p∈Σi1∩ ··· ∩Σis, there exists an open
+neighborhood Uofpand a homeomorphism φ:U→Rnsuch thatφ(p) = 0 and
+for anyi∈ {i1,···,is}, we have
+•φ(Σi∩U) is the coordinate hyperplane Hi={(x1,···,xn)∈Rn|xi= 0}.
+Soφ(Σi1∩···∩Σis∩U) =Hi1∩···∩His.
+•φ(N(Σi)∩U) =Hi×[−1,1]⊂Rn.
+•inthechart( U,φ),τidefinesahomeomorphism fi:Hi×{1} →Hi×{−1}.
+Then we can deform these τivia some isotopies in Usuch that they satisfy
+our requirements (i) – (iii) in U. Indeed, let ribe the reflection of Rnabout the
+hyperplane Hi. Then we can isotope fisuch that for any j∈ {i1,···,is}with
+j/\e}atio\slash=i, we have
+fi(x) =ri(x),for anyx∈Hj×{±1}∩Hi×{1}. (1)
+Then these τi’s obviously meet our requirements. Moreover, since the (i) – (iii)
+arecoordinate-independent properties, wecancarryoutthede formationsofthese
+τi’s chart by chart around Σ i1∩ ···∩Σisuntil the (i) – (iii) are satisfied at all
+places. In addition, we should do the deformation of the τi’s in the charts around
+the higher degree intersection points first, then extend to the ch arts around lower
+degreeintersectionpoints. Intheend, wewill get τi’swhichsatisfyalltherequire-
+ments (i) – (iii). We remark that doing the isotopy of these τiin a chart might
+alter what we have previous done in another chart, but since the (i) – (iii) are
+coordinate-independent properties, the altering will not cause an y inconsistency
+in our construction. /square
+After the local deformations of τi’s described in the preceding lemma, the
+restriction of each τionPi⊂∂N(Σi) defines a free involution on Pi, denoted
+byτi. Because of the existence of these τi, we call the set of panels of Vnan
+involutive panel structure . We will always assume that Vnhas this involutive
+panel structure in the rest of the paper. Note that {τi:Pi→Pi}1≤i≤ksatisfy:
+•τimaps a face fofPito a facef′ofPi(it is possible that f′=fthough);
+•τi(Pi∩Pj)⊂Pi∩Pjfor all 1≤i,j≤k;
+•τi◦τj=τj◦τi:Pi∩Pj→Pi∩Pjfor all 1≤i,j≤k.
+For anyI={i1,···,is} ⊂ {1,···,k}with|I|=s≥1, we define:
+PI:=Pi1∩···∩Pis⊂Vn,ΣI:= Σi1∩···∩Σis⊂Qn. (2)7

+Σ
+Σ
+.. .
+..
+Figure 2. AZ2-core of torus        !" # $% & '..(
+)* +,- . / 0 1 2 34 567 8 9 :; <= P >
+Figure 3. AZ2-core of real projective plane
+WhenI=∅, we define P∅=Vnand Σ ∅=Qn. IfPIwith|I| ≥2 is nonempty,
+it is called a subpanel ofVn. Notice that the PIis empty whenever |I|> n.
+AlthoughQnis assumed to be connected, the Σ Imay not be connected.
+For any point x∈Pi, letx∗
+Pi=τi(x)∈Pi. We callx∗
+Pithetwin point ofxin
+Pi. Obviously, x∗
+Pi/\e}atio\slash=xsinceτihere is free.
+Generally, for a face f⊂Pi, if the face f∗
+Pi=τi(f) is disjoint from f, it is called
+thetwin face offinPi. Otherwise, fis called self-involutive inPiin the sense
+thatτi(f) =f. In particular, if a facet FofVnis not self-involutive, it has a
+unique twin facet F∗which belongs to the same panel as F. We call /hatwideF:=F∪F∗
+afacet pair .
+As an embedded submanifold of Qn, Σicould be two-sided or one-sided, which
+is determined by the orientability of the normal bundle of Σ iinQn. If Σiis two-
+sided, any facet FinPihas a twin facet F∗(see Figure 2). But if Σ iis one-sided,
+some facet in Pimight be self-involutive (see Figure 3).
+Remark 2.5. The facets in the same panel of Vnare pairwise disjoint since each
+Σiin theZ2-cut system has no self-intersections. But a panel of Vnmay consist
+of more than one facet pair (see the Example 1 below).
+If we identify any points of Vnwith all their duplicate points in Vn, we will
+get a manifold denoted by /hatwideQn. Let̺:Vn→/hatwideQnbe the quotient map.
+Lemma 2.6. There exists a homeomorphism h:/hatwideQn→Qnwithh(̺(PI)) = Σ I
+for anyI⊂ {1,···,k}.8 LI YU
+Proof.Byourconstructionof τi, itiseasytoseethat ̺(Pi)∼=Σiforany1 ≤i≤k.
+In addition, there exists a neighborhood N(∂Vn) of∂VninVnwithN(∂Vn)∼=
+∂Vn×[0,ε] so that̺(N(∂Vn))⊂/hatwideQnis homeomorphic tok/uniontext
+i=1N(Σi)⊂Qn. Let
+Un=Qn\int(k/uniontext
+i=1N(Σi)). Then we can think of /hatwideQn(orQn) as the gluing of
+N(∂Vn)/parenleftbigg
+ork/uniontext
+i=1N(Σi)/parenrightbigg
+withUnalong their boundary, that is:
+/hatwideQn=̺(N(∂Vn))/uniontext
+ϕ1Un, Qn=/parenleftbiggk/uniontext
+i=1N(Σi)/parenrightbigg/uniontext
+ϕ2Un,
+where the ϕ1:∂(̺(N(∂Vn)))→∂Unandϕ2:∂/parenleftbiggk/uniontext
+i=1N(Σi)/parenrightbigg
+→∂Unare
+homeomorphisms. If we identify ∂(̺(N(∂Vn))) with∂/parenleftbiggk/uniontext
+i=1N(Σi)/parenrightbigg
+,ϕ1andϕ2
+are actually isotopic because the local deformations we make on the τi’s are all
+isotopies of homeomorphisms. So we can construct a homeomorphis mh:/hatwideQn→
+Qnfrom an isotopy between ϕ1andϕ2, which satisfies our requirement. /square
+We callρ=h◦̺:Vn→Qntherestoring map ofVn. ThenPI=ρ−1(ΣI)
+for anyI⊂ {1,···,k}. Obviously, for any point xin the relative interior of
+Pi1∩···∩Pis, we have:
+ρ−1(ρ(x)) ={τεs
+is◦···◦τε1
+i1(x);εj∈ {0,1},1≤j≤s},
+whereτ0
+ij:=id. It is easy to see that ρ−1(ρ(x)) consists of exactly 2sdifferent
+points inVn. Any point x′∈ρ−1(ρ(x)) (including xitself) is called a duplicate
+pointofxinVn.
+Example 1. SupposeQnis a small cover over some simple polytope. It is well
+known that the Z2-homology classes of Qncan all be represented by some special
+embedded submanifolds of Qn, calledfacial submanifolds (see [2] and [6]). And
+cuttingQnopen along a collection of facial submanifolds of Qnwill give us a
+connected Z2-coreVnofQn. Figure 4 shows such an example in dimension 2
+whereQ2=RP2#RP2#RP2is a small cover over a pentagon. A Z2-core ofQ2
+is an octagon where the four edges marked by“ A”belong to the same panel.
+Remark 2.7. A closed connected manifold Qnmay have a Z2-coreVnwith
+H1(Vn,Z2)/\e}atio\slash= 0. For example, the Z2-core ofQ2=T2#T2shown in Figure 5 is
+homeomorphic to an annulus.9?@A
+B
+CDE
+F G
+H I
+JK
+Figure 4. AZ2-core ofRP2#RP2#RP2LM
+N..O
+Figure 5. AZ2-core ofT2#T2
+Next, we define a general notion of involutive panel structure amo ng all nice
+manifolds with corners. The involutive panel structure on a Z2-coreVncon-
+structed above is just a special case of this general notion.
+Definition 2.8 (Involutive Panel Structure) .SupposeWnisanicemanifoldwith
+corners (may not be connected). Suppose the boundary of Wnis the union of
+several panels P1,···,Pkwhich satisfy the following conditions:
+(a) eachpanel Piisadisjoint unionoffacetsof Wnandeachfacetiscontained
+in exactly one panel;
+(b) there is an involution τion eachPi(i.e.τiis a homeomorphism with
+τ2
+i=idPi) which sends a face f⊂Pito a facef′⊂Pi(it is possible that
+f′=f);
+(c) for alli/\e}atio\slash=j,τi(Pi∩Pj)⊂Pi∩Pjandτi◦τj=τj◦τi:Pi∩Pj→Pi∩Pj.
+Then we say that Wnhas aninvolutive panel structure defined by {Pi,τi}1≤i≤k
+on the boundary. Note here, we do not require that the involution τionPiis free.
+Similar to the Z2-coreVn, for anyx∈Pi⊂Wn, we callτi(x) the twin point x
+inPi. Moreover, if xis in the relative interior of Pi1∩···∩Pis, anyτεs
+is◦···◦τε1
+i1(x)
+whereεi∈ {0,1}is called a duplicate point of xinWn. But in this case, it is not
+necessarily that xhas exactly 2sduplicate points (even if each τionPiis free,
+see Figure 10). Also we can define subpanels for Wnas in (3).
+Remark 2.9. A nice manifold with corners Wnmay admit many different invo-
+lutive panel structures on the boundary (for example see Figure 6 ).10 LI YU. .
+... .
+... .
+. .
+Figure 6. Three different involutive panel structures on a square
+Example 2. SupposeXnis a nice manifold with corners, and let F1,···,Fl
+be all the facets of Xn. We can think of Xnhaving a trivial involutive panel
+structure which is defined by: for any 1 ≤i≤l,Pi=Fiand the involution
+τi=idFi:Fi→Fi. In this case, we call each Piareflexive panel ofXn.
+Obviously, in the trivial involutive panel structure, any point of Xnhas only one
+duplicate point — itself.
+In general, suppose {Pi,τi}1≤i≤kis an involutive panel structure on a nice man-
+ifoldwithcorners Wn. Iftheτi:Pi→Piistheidentity map, we call Piareflexive
+panelofWn.
+Example 3. SupposeVnis aZ2-core ofQn. Use the notations above, for any
+panelPiofVn,Piitself is a nice manifold with corners and has an involutive
+panel structure on the boundary induced from Vn. Indeed, the panels of Piare:
+{Pj∩Pi;τj|Pj∩Pi:Pj∩Pi→Pj∩Pifor∀1≤j≤k,j/\e}atio\slash=i}.
+Moregenerally, foran I={i1,···,is} ⊂ {1,···,k}, thesubpanel PIisan(n−s)-
+dimensional nice manifold with corners (may not be connected), and PIhas an
+involutive panel structure on its boundary induced from Vnwhich is given by:
+{Pj∩PI/\e}atio\slash=∅;τj|Pj∩PI:Pj∩PI→Pj∩PIfor∀1≤j≤k,j /∈I}.
+Obviously, the Z2-coreof a closed manifold Qnis far fromunique. The topolog-
+ical type of a Z2-core depends on the corresponding Z2-cut system in Qn. For an
+arbitrary Z2-cut system of Qn, it is fairly possible that the corresponding Z2-core
+ofQnis not connected. But we can prove the following statement (which is not
+used in any other place in the paper).
+Theorem 2.10. For any closed connected manifold Qn, there always exists a
+connected Z2-core forQn.
+Proof.Forn= 1 and 2, the statement is obviously true. So assume n≥3 in the
+rest of the proof.
+First, let us choose a Z2-cut system Σ 1,···,ΣkofQnwith each Σ ibeing con-
+nected. We claim that each Σ iis non-separating in Qn. let{[Γ1],···,[Γk]} ⊂11
+H1(Qn,Z2) the dual basis of the {[Σ1],···,[Σk]} ⊂Hn−1(Qn,Z2) under Z2-
+intersection form of Qn, i.e.
+#(Γi∩Σj) =δijmod 2
+So the curve Γ imust intersect Σ iodd number of times. Let N(Σi) be a small
+tubular neighborhood of Σ iinQn. Since Σ iis connected, Qn−int(N(Σi)) is
+either connected or has exactly two connected-component. but the later case
+contradicts #(Γ i∩Σi) = 1 mod 2. So Σ imust be non-separating in Qn.
+LetQn
+jbe the manifold we get by cutting Qnopen along {Σ1,···,Σj}, i.e.
+Qn
+j=Qn−j/uniondisplay
+i=1int(N(Σi)).
+In addition, for any j+1≤i≤k, let Σ(j)
+i:= Σi∩Qn
+jand Γ(j)
+i:= Γi∩Qn
+j.
+AssumeQn
+jis connected and we cut Qn
+jopen along Σ(j)
+j+1. Since the relative
+intersection number of Σ(j)
+j+1and Γ(j)
+j+1inH∗(Qn
+j,∂Qn
+j,Z2) is 1 (mod 2), if Σ(j)
+j+1is
+connected in Qn
+j, then Σ(j)
+j+1must be non-separating in Qn
+jfor the same reason as
+above. If Σ(j)
+j+1is not connected in Qn
+j, we can connect all the components of Σ(j)
+j+1
+via some thin tubes in Qn
+jwhich are transverse to other Σ(j)
+i’s. This operation
+will change the original Σ j+1inQnsimultaneously, but it will not change the
+homology class of Σ j+1inHn−1(Qn,Z2). Now, cutting Qn
+jopen along the new
+Σ(j)
+j+1, we get a nice manifold with corners Qn
+j+1which remains connected.
+By iterating the above argument from j= 1 toj=k, we will get a connected
+nicemanifoldwithcorners Vn. Bydefinition, VnistheZ2-coreofQnfromcutting
+Qnopen along a Z2-cut system {Σ′
+1,···,Σ′
+k}, which is obtained from the original
+Z2-cut system by some homology preserving operations. /square
+3.Construction of free (Z2)m-actions on closed manifolds
+Supposeπ:Mn→Qnis a principal ( Z2)m-bundle over a closed connected
+manifoldQn. LetVnbe aZ2-core ofQnfrom cutting Qnalong aZ2-cut system
+{Σ1,···,Σk}inQn(we do not assume that Vnis connected). We have shown
+that the principal ( Z2)m-bundleπis classified by an element
+Λπ∈H1(Qn,(Z2)m)∼=Hom(H1(Qn,Z2),(Z2)m) (3)12 LI YU
+BythePoincar´ eduality, thereisanisomorphism ψ:Hn−1(Qn,Z2)∼=H1(Qn,Z2).
+So we get an element Λ∗
+π∈Hom(Hn−1(Qn,Z2),(Z2)m) defined by:
+Λ∗
+π:{[Σ1],···,[Σk]} −→(Z2)m
+[Σi]/mapsto→Λπ(ψ([Σi]))
+LetPi⊂∂Vnbe the panel corresponding to Σ i. So we have a map
+λπ:{P1,···,Pk} −→(Z2)m
+Pi/mapsto→Λ∗
+π([Σi]) = Λ π(ψ([Σi]))
+λπis called the associated (Z2)m-coloring ofπ:Mn→QnonVn. In general,
+any mapλ:{P1,···,Pk} →(Z2)mis called a ( Z2)m-coloring on Vn, and any
+element in ( Z2)mis called a color.
+Inaddition, if weconsider π:Mn→Qnasaregular covering, theΛ πisjust the
+abelianization of the monodromy map Hπ:π1(Qn,q0)→(Z2)m, whereq0∈Qn
+is a base point and ( Z2)mis identified with the deck transformation group of
+this covering π. Indeed, let {[Γ1],···,[Γk]} ⊂H1(Qn,Z2) be the dual basis of
+{[Σ1],···,[Σk]}under the Z2-intersection form of Qnwhere each Γ iis a closed
+curve that intersects all Σ j’s transversely. If we fix a point x0∈π−1(q0), and let
+/tildewideΓi: [0,1]→Mnbe a lifting of Γ iwith/tildewideΓi(0) =x0, then
+/tildewideΓi(1) =Hπ(Γi)·x0= Λπ([Γi])·x0. (4)
+Conversely, given an arbitrary ( Z2)m-coloringλonVn, we can construct a
+principal ( Z2)m-bundle over Qnby the following rule:
+M(Vn,{Pi,τi},λ) :=Vn×(Z2)m/∼ (5)
+Where(x,g)∼(x′,g′)whenever x′=τi(x)forsomePiandg′=g+λ(Pi)∈(Z2)m.
+It is easy to see that if xis in the interior of Pi1∩···∩Pis, (x,g)∼(x′,g′) if and
+only if (x′,g′) = (τεs
+is◦···◦τε1
+i1(x),g+ε1λ(P1)+···+εsλ(Ps)) whereεj∈ {0,1}
+for∀1≤j≤s.
+We callM(Vn,{Pi,τi},λ) theglue-back construction from (Vn,λ). Also, we
+useM(Vn,λ) to denote M(Vn,{Pi,τi},λ) if there is no ambivalence about the
+involutive panel structure on Vnin the context.
+Let [(x,g)]∈M(Vn,λ) denote the equivalence class of ( x,g) defined in (5).
+Then we can define a natural ( Z2)m-action onM(Vn,λ) by:
+g·[(x,g0)] := [(x,g+g0)],∀x∈Vn,∀g,g0∈(Z2)m. (6)
+It is easy to check that the ( Z2)m-action is well defined. And for any element
+g/\e}atio\slash= 0∈(Z2)m,g·[(x,g0)] = [(x,g+g0)]/\e}atio\slash= [(x,g0)]. This is because:13
+(i) whenxis in the interior of Vn, (x,g+g0) and (x,g0) are not equivalent
+under∼for anyg/\e}atio\slash= 0;
+(ii) whenxis in the relative interior of Pi1∩ ··· ∩Pis, (x,g+g0)∼(x,g0)
+would force ( x,g+g0) = (τεs
+is◦···◦τε1
+i1(x),g0+ε1λ(P1)+···+εsλ(Ps)).
+Notice that g/\e}atio\slash= 0 implies that at least one of the ε1,···,εsis not 0. But
+sincexhas exactly 2sduplicate points in Vn,τεs
+is◦···◦τε1
+i1(x)/\e}atio\slash=xas long
+as someεj/\e}atio\slash= 0.
+So the action of ( Z2)monM(Vn,λ) defined by (6) is always a free group action.
+In the rest of this paper, we will always assume that M(Vn,λ) is equipped with
+this free ( Z2)m-action.
+Remark 3.1. SinceQnis smooth, the M(Vn,λ) is naturally a smooth manifold
+and the natural ( Z2)m-action onM(Vn,λ) defined in (6) is smooth.
+Remark 3.2. A similar idea to the glue-back construction was used to construct
+cyclicandinfinitecycliccovering spacesofthecomplement ofknotsin S3(see[7]).
+Theorem 3.3. M(Vn,λ)is a closed n-manifold and the orbit space of the free
+(Z2)m-action onM(Vn,λ)defined in (6)is homeomorphic to Qn.
+Proof.Observe that each orbit of this ( Z2)m-action has some representative in
+Vn×{0}. And for any point xin the relative interior of Pi1∩···∩Pis, a point
+(x′,0)∈Vn× {0}is in the same orbit as ( x,0) under the above ( Z2)m-action
+if and only if x′=τεs
+is◦ ···◦τε1
+i1(x) for some ε1,···εs∈ {0,1}(in other words,
+x′is a duplicate point of xinVn). So the orbit space is homeomorphic to the
+space of gluing all points of Vnwith their duplicate points together, which is
+homeomorphic to Qnby Lemma 2.6. And since Qnis a closed manifold, so is
+M(Vn,λ). /square
+Following are some explicit examples of free ( Z2)m-actions on manifolds from
+the glue-back construction.
+Example 4. A meridian and a longitude of the torus T2forms aZ2-cut system
+ofT2. The corresponding Z2-core ofT2is a square V2. Given a coloring of the
+edges ofV2by elements in ( Z2)2=/a\}bracketle{te1/a\}bracketri}ht⊕/a\}bracketle{te2/a\}bracketri}htsuch that opposite edges of V2are
+colored by the same element of ( Z2)2, we can construct all principal ( Z2)2-bundles
+overT2(see Figure 7 and Figure 8 for such examples).
+Example 5. LetM2be a disjoint union of two S2. Figure 9 shows a free ( Z2)2-
+action onM2whose orbit space is RP2. AZ2-coreV2ofRP2is a disk with only
+one panelP=∂V2. Let{e1,e2}be a basis of ( Z2)2. ThenM2∼=M(V2,λ) where
+λis a (Z2)2-coloring on V2given byλ(P) =e1(ore2).14 LI YUPQ. .
+. ..R
+S
+...TU
+VW
+XY
+Z
+[ \ ]
+. .
+. .....
+Figure 7. Examples of the glue-back construction
+∪....^
+_`
+a
+b
+cd
+e
+....
+f
+gh
+ij
+k
+l
+m. .
+. .
+. .
+. .
+Figure 8. Examples of the glue-back constructionno
+p=q
+r=
+st
+u=
+v w
+x
+yz
+{|
+}∪
+~ €
+P 
+Figure 9.
+More generally, for any nice manifold with corners Wnwith an involutive panel
+structure {τi:Pi→Pi}1≤i≤k, any map µ:{P1,···,Pk} →(Z2)mis called a
+(Z2)m-coloring on Wn. We can define the glue-back construction M(Wn,µ) by
+thesame ruleas in(5). Also we have anatural ( Z2)m-actiononM(Wn,µ)defined
+by (6). But this ( Z2)m-action onM(Wn,µ) may not be free. Indeed, suppose
+θµ:Wn×(Z2)m→M(Wn,µ) is the quotient map. For a point xin the relative
+interior of a codimension sface ofWn, it is possible that xhas less than 2s
+duplicate points in Wn. In that case, θµ(x×(Z2)m) would consist of less than 2m15
+.
+.....
+..
+‚ƒ„…
+†‡
+ˆ‰Š
+‹
+Figure 10.
+points, which implies that θµ(x×(Z2)m) can not a free orbit under the natural
+(Z2)m-action onM(Wn,µ) defined in (6) (see the examples below).
+Example 6. For a simple polytope Vn, consider each facet of Vnas a panel and
+Vnhas the trivial involutive panel structure (see Example 2). Then a s mall cover
+overVncan be thought of as the glue-back construction M(Vn,λ) whereλis
+acharacteristic function onVnwith value in ( Z2)n(see [2]). But the natural
+action of ( Z2)ndefined by (6) on a small cover is exactly the locally standard
+action defined in [2], which is not free.
+Remark 3.4. From the Example 6, we can see that the significance of intro-
+ducing the general notion of involutive panel structure in Definition 2.8 is that:
+it allows us to unify the way of constructing free ( Z2)m-actions and non-free lo-
+cally standard ( Z2)m-actions on manifolds from the orbit spaces (see Section 5
+for details).
+Example 7. Suppose a square [0 ,1]2is equipped with an involutive panel struc-
+ture as indicated by the arrows in Figure 10. For a ( Z2)2-coloringλdefined by
+λ(P1) =e1,λ(P2) =e2where{e1,e2}is a basis of ( Z2)2, the glue-back construc-
+tionM([0,1]2,λ) is homeomorphic to T2. But the natural ( Z2)2-action on T2
+defined by (6) is not free.
+For a nice manifold with corners Wnequipped with an involutive panel struc-
+ture, if for any s≥0, any point in the relative interior of any codimension sface
+ofWnhas exactly 2sduplicate points in Wn, the involutive panel structure is
+calledperfect. For example: the involutive panel structure on any Z2-coreVnof
+Qnconstructed from Lemma 2.4 above is perfect.
+We can easily show that if the involutive panel structure on Wnis perfect, the
+natural ( Z2)m-action on M(Wn,λ) defined by (6) is free for all ( Z2)m-coloring
+λonWn. However, even if the involutive panel structure on Wnis not perfect,
+it is still possible that there exists some nontrivial ( Z2)m-coloringλonWnso
+that the natural ( Z2)m-action onM(Wn,λ) is free. For example, although the
+involutive panel structure on the square in Example 7 is not perfect , if we color
+the two panels of the square both by e1∈(Z2)2, we will obtain a disjoint union16 LI YU
+of two spheres S2∪S2from the glue-back construction. Obviously, the natural
+(Z2)2-action on this S2∪S2is free.
+LetVnbe aZ2-core of a closed manifold Qndescribed as above. As in Exam-
+ple 3, we can think of a panel Pi⊂Vnitself as a nice manifold with corners with
+an involutive panel structure defined by {Pj∩Pi; 1≤j≤k,j/\e}atio\slash=i}. Then we
+have an induced ( Z2)m-coloringλin
+PiofPidefined by
+λin
+Pi(Pj∩Pi) :=λ(Pj),∀j/\e}atio\slash=i,Pj∩Pi/\e}atio\slash=∅
+Furthermore, for I={i1,···,is} ⊂ {1,···,k}, thesubpanel PI=Pi1∩···∩Pis
+has an involutive panel structure on its boundary defined by {Pj∩PI/\e}atio\slash=∅; 1≤
+j≤k,j /∈I}. Theinduced(Z2)m-coloringλin
+PIofPIis
+λin
+PI(Pj∩PI) =λ(Pj)∈(Z2)m,1≤j≤k, j /∈I, Pj∩PI/\e}atio\slash=∅(7)
+If we apply the glue-back construction (5) to ( PI,λin
+PI), we will get a closed
+manifoldM(PI,λin
+PI). Notice that when |I| ≥1, by the definition of M(PI,λin
+PI),
+the relative interior points of the 2mcopies ofPIare not glued together like they
+are inM(Vn,λ). In fact, it is easy to see that M(PI,λin
+PI) is homeomorphic to a
+disjoint union of 2scopies ofθ−1
+λ(ΣI), whereθλ:Vn×(Z2)m→M(Vn,λ) is the
+quotient map defined in (5).
+Theorem 3.5. For any principal (Z2)m-bundleπ:Mn→Qn, letλbe the asso-
+ciated(Z2)m-coloring of πonVn. Then there is an equivariant homeomorphism
+fromM(Vn,λ)toMnwhich covers the identity of Qn.
+Proof.Letθλ:Vn×(Z2)m→M(Vn,λ)bethequotientmapand ξλ:M(Vn,λ)→
+Qnbe the orbit map of the natural ( Z2)m-action defined by (6). It suffice to show
+thatξλandπdefines the same monodromy map as regular coverings over Qn. So
+let us first compute the monodromy Hξλ([Γ]) for any closed curve Γ : [0 ,1]→Qn.
+Suppose Γ( t) meets Σ i1,···,Σirconsecutively in Qnas the time tgoes from
+0 to 1. When cutting Qnopen along {Σ1,···,Σk}, the cut-open image of Γ is
+Γ∩Vn:=γ. Note that the curve γmight be disconnected in Vn. When the time
+parameterincreases, γwillmeet thepanels Pi1,···,PirofVnconsecutively. Then
+when we glue the 2mcopies ofVntogether in the glue-back construction, the γin
+different copies of Vnare fit together which gives all the liftings of Γ in M(Vn,λ).
+Indeed, if we choose the start point of a lifting of Γ in θλ(Vn×g0),g0∈(Z2)m,
+the end point of the lifting would be in θλ(Vn×(g0+λ(Pi1)+···+λ(Pir))). So
+the monodromy Hξλofξλis:
+Hξλ([Γ]) =λ(Pi1)+···+λ(Pir)∈(Z2)m. (8)17
+Now let{[Γ1],···,[Γk]} ⊂H1(Qn,Z2) the dual basis of the {[Σ1],···,[Σk]} ⊂
+Hn−1(Qn,Z2) underZ2-intersection form of Qn. Then
+#(Γi∩Σj) =δijmod 2 (9)
+We can assume that each Γ i: [0,1]→Qnis a closed curve which intersects all
+Σj’s transversely and starts at the same base point q0∈Qn. Supposeγi= Γi∩Vn
+is the cut-open image of Γ iinVn. Then by (9), γiwill meetPiodd number of
+times and meet all other Pj(j/\e}atio\slash=i) even number of times. So by (8), we have:
+Hξλ([Γi]) =λ(Pi) =Hπ([Γi]),1≤i≤k. (10)
+This implies that Hξλ=Hπ. So the theorem is prove. /square
+Remark 3.6. For aZ2-coreVnofQnwithH1(Vn,Z2)/\e}atio\slash= 0, a principal ( Z2)m-
+bundle over Vnis not necessarily trivial. If we apply the gluing rule (5) to an
+arbitrary principal ( Z2)m-bundle over Vn, we may get a principal ( Z2)m-bundle
+overQntoo. ThesignificanceofTheorem3.5isthatwecanactuallyusethetr ivial
+(Z2)m-bundle over Vn(i.e.Vn×(Z2)m) and the gluing rule (5) to construct all
+principal ( Z2)m-bundles over Qn, which is enough for our purpose in this paper.
+For a (Z2)m-coloringλon the panels P1,···,PkofVn, define:
+Lλ:= the subgroup of ( Z2)mgenerated by {λ(P1),···,λ(Pk)},(11)
+rank(λ) := dim Z2Lλ. (12)
+Sinceλencodes all the structural information of M(Vn,λ), so any topological
+invariant of M(Vn,λ) (e.g. homology groups) should be completely determined
+by (Vn,λ). But if we try to compute the Z2-homology groups of M(Vn,λ) via
+the Serre-spectral sequence, the problem of twisted local coeffi cients could occur
+when the orbit space is not simply-connected. This problem is hard to get around
+in general. However, we can at least compute H0(M(Vn,λ),Z2), i.e. the number
+of connected components of M(Vn,λ), from the ( Z2)m-coloringλ.
+Theorem 3.7. For any (Z2)m-coloringλofVn,M(Vn,λ)has2m−rank(λ)con-
+nected components which are pairwise homeomorphic. Let θλ:Vn×(Z2)m→
+M(Vn,λ)be the quotient map. Then each connected component of M(Vn,λ)is
+homeomorphic to θλ(Vn×Lλ). And there is a free action of Lλ∼=(Z2)rank(λ)on
+each connected component of M(Vn,λ)whose orbit space is Qn.
+Proof.Letθλ:Vn×(Z2)m→M(Vn,λ) be the quotient map defined by (5). Here
+we do not assume Vnis connected. Then by the definition of M(Vn,λ), for two
+arbitrary connected components K,K′ofVnand∀g,g′∈(Z2)m, we have:18 LI YU
+(a)θλ(K×g) andθλ(K′×g′) are in the same connected component of
+M(Vn,λ) if and only if there is a sequence
+(K,g) = (K0,g0)↔(K1,g1)··· ↔(Kr−1,gr−1)↔(Kr,gr) = (K′,g′)
+whereeach Kiisaconnectedcomponent of Vn,gi∈(Z2)m, andθλ(Ki×gi)
+andθλ(Ki+1×gi+1) share an ( n−1)-dimensional face in M(Vn,λ).
+(b)θλ(K×g) andθλ(K′×g′) share an ( n−1)-dimensional face in M(Vn,λ)
+if and only if there is a facet FofKwith its twin facet F∗⊂K′and
+g′−g=λ(F).
+So ifθλ(K×g) andθλ(K′×g′) are inthe same connected component of M(Vn,λ),
+it is necessary that g′∈g+Lλ.
+Conversely, we claim: for any g′∈g+Lλ,θλ(K×g) andθλ(K′×g′) are always
+in the same connected component of M(Vn,λ) for any connected components K
+andK′ofVn.
+Indeed, since Qnis connected, for KandK′, there always exists a sequence,
+K=K0,K1,···,Kr−1,Kr=K′, suchthatsomefacet Fai⊂KiwhileF∗
+ai⊂Ki+1.
+So by the above argument, θλ(K×g) lies in the same connected component as
+θλ(K′×g∗) inM(Vn,λ) for some g∗∈g+Lλ. Then it remains to show that
+θλ(K′×g∗) andθλ(K′×g′) are always in the same connected component of
+M(Vn,λ) whenever g′−g∗∈Lλ.
+To see this, let Γ : [0 ,1]→Qnbe an arbitrary closed curves based at a point
+q0∈K′⊂Vn. For anyg∗∈(Z2)m, there is a lifting of Γ in M(Vn,λ) which
+goes from a point in θλ(K′×g∗) to a point in θλ(K′×(g∗+Hξλ([Γ]))), where
+Hξλ([Γ]) is the monodromy of Γ with respect to the covering ξλ:M(Vn,λ)→Qn
+(see (4)). So θλ(K′×g∗) andθλ(K′×(g∗+Hξλ([Γ]))) are in the same connected
+component of M(Vn,λ).
+LetΓ1,···,Γkbesomeclosedcurvesin Qnbasedatq0sothat{[Γ1],···,[Γk]} ⊂
+H1(Qn,Z2) is the dual basis of the {[Σ1],···,[Σk]} ⊂Hn−1(Qn,Z2) underZ2-
+intersection form of Qn. Then by (10), we have
+{Hξλ([Γ])| ∀Γ : [0,1]→Qn}= Im(Hξλ)
+=/a\}bracketle{tHξλ([Γ1]),···,Hξλ([Γk])/a\}bracketri}ht ⊂(Z2)m
+=/a\}bracketle{tλ(P1),···,λ(Pk)/a\}bracketri}ht=Lλ. (13)
+So any element of Lλcan be realized by Hξλ([Γ]) for some closed curve Γ. Then
+the above claim is proved.
+Soθλ(K×g) andθλ(K′×g′) belong to the same connected component of
+M(Vn,λ)⇐⇒g′∈g+Lλ. Since dim Z2Lλ= rank(λ), each connected component19
+ofM(Vn,λ) is made up of 2rank(λ)copies ofVnfromVn×(Z2)m. Indeed, suppose
+(Z2)m=Lλ⊕ /a\}bracketle{tω1/a\}bracketri}ht ⊕ ··· ⊕ /a\}bracketle{tωq/a\}bracketri}htwhereq=m−rank(λ). ThenM(Vn,λ) has
+2m−rank(λ)connected components which are given by:
+θλ(Vn×(Lλ+t1ω1+···tqωq)), ti∈ {0,1},1≤i≤q,
+each of which is equipped with a natural free action by Lλ∼=(Z2)rank(λ)defined
+in (6) whose orbit space is Qn. /square
+Remark 3.8. In the above proof, let κ: (Z2)m→Lλ∼=(Z2)rank(λ)be a quotient
+homomorphism. Then for any ( Z2)m-coloringλonVn, we can think of κ◦λas
+a (Z2)rank(λ)-coloring on Vn. It is easy to see that each connected component K
+ofM(Vn,λ) is homeomorphic to M(Vn,κ◦λ).
+In general, for I={i1,···,is} ⊂ {1,···,k}, the submanifold Σ I⊂Qnmay
+not be connected. Suppose Sis a connected component of Σ I. ThenSis an
+(n−s)-dimensional embedded submanifold of Qn. We also want to compute the
+number of components in ξ−1
+λ(S), whereξλ:M(Vn,λ)→Qnis the quotient map
+defined by (5).
+Remark 3.9. Two connected components SandS′of ΣImay not be homeo-
+morphic to each other, and the ξ−1
+λ(S) andξ−1
+λ(S′) may not have equal number
+of connected components either.
+If we cutSopen along the transversely intersected embedded submanifolds
+{Σj∩S/\e}atio\slash=∅|j /∈I}, we will get a nice manifold with corners, denoted by
+VS. Similar to the Z2-core ofQn, we can construct a (perfect) involutive panel
+structure on the boundary of VSfrom the cut sections of {Σj∩S/\e}atio\slash=∅|j /∈I}.
+And any ( Z2)m-coloringλonVninduces a ( Z2)m-coloringλin
+Son the panels of
+VSby: ifEis the panel of VScorresponding to Σ j∩S,
+λin
+S(E) :=λ(Pj).
+It is easy to see that the glue-back construction M(VS,λin
+S) is homeomorphic
+toξ−1
+λ(S). ButVSmay not be a Z2-core ofS, since the homology classes of
+{Σj∩S/\e}atio\slash=∅|j /∈I}may not form a basis of Hn−s−1(S,Z2). So we can not
+directly apply the formula in Theorem 3.7 to compute the number of co mponents
+ofM(VS,λS). In fact, the number of components of M(VS,λS) also depends on
+what homology classes are represented by {Σj∩S/\e}atio\slash=∅|j /∈I}inHn−s−1(S,Z2).
+So we need to modify the proof of Theorem 3.7 to deal with this case. In the
+following, we will treat this problem in a very general setting on Qn.
+Suppose {N1,···,Nr}is an arbitrary collection of codimension one embedded
+submanifolds of a closed manifold Qnwhich lie in general position. Cutting Qn20 LI YU
+open along N1,···,Nrgives us a nice manifold with corners Wn. As before, we
+can construct a (perfect) involutive panel structure /hatwideP1,···,/hatwidePron the boundary
+ofWnfrom the cut sections of N1,···,Nr. In addition, suppose Γ 1,···,Γkare
+simple closed curves in Qnwhose homology classes form a basis of H1(Qn,Z2),
+and each Γ iintersects {N1,···,Nr}transversely. Let aij∈Z2be the mod 2
+intersection number between Γ iandNj. Note that for any fixed j, the{aij}is
+completely determined by the homology class of NjinHn−1(Qn,Z2). Indeed, if
+{Σ1,···,Σk}is aZ2-cut system of Qnwhose homology classes is a dual basis
+of{[Γ1],···,[Γk]}under the Z2-intersection form, then for each 1 ≤j≤r, the
+homology class [ Nj] =/summationtext
+iaij[Σi]∈Hn−1(Qn).
+For any ( Z2)m-coloringλ:{/hatwideP1,···,/hatwidePr} →(Z2)monWn, we can show that
+M(Wn,λ) is a closed manifold with a natural free ( Z2)m-action defined by (6)
+whoseorbitspaceis Qn. TheproofofthisfactisexactlythesameasTheorem 3.3,
+hence omitted. In addition, by a similar argument as in the proof Theo rem 3.5,
+the monodromy of each Γ iwith respect to the covering M(Wn,λ)→Qnis given
+by:
+/hatwideHλ(Γi) :=r/summationdisplay
+j=1aijλ(/hatwidePj)∈(Z2)m. (14)
+Now, we associate a new ( Z2)m-coloring /hatwideλtoλon the panels of Wnby:
+/hatwideλ(/hatwidePi) :=/hatwideHλ(Γi),1≤i≤r.
+LetL/hatwideλ= the subgroup of ( Z2)mgenerated by {/hatwideλ(/hatwideP1),···,/hatwideλ(/hatwidePr)}.
+Theorem 3.10. For any (Z2)m-coloringλon the panels of Wnhere, the num-
+ber of connected components of M(Wn,λ)equals2m−l, wherel= dim Z2L/hatwideλ. In
+addition, all the connected components of M(Wn,λ)are homeomorphic to each
+other, and there is free L/hatwideλ∼=(Z2)l-action on each component of M(Wn,λ)whose
+orbit space is Qn.
+Proof.The argument here is parallel to that in Theorem 3.7 except that in (1 3),
+the monodromy of Γ ishould be replaced by /hatwideHλ(Γi) in (14). So the proof is left
+to the reader. /square
+4.Generalize to compact manifolds with boundary
+We can generalize the notion of Z2-core and glue-back construction to any
+compact manifold with boundary. Suppose Xnis ann-dimensional compact
+connected nice manifold with corners and H1(Xn,Z2)/\e}atio\slash= 0. Let {F1,...,F l}be
+thesetoffacetsof Xn. AZ2-cut system ofXnisacollectionof( n−1)-dimensional
+embedded submanifoldsΣ 1,···,Σk(possiblewithboundary)of Xnwhichsatisfy:21
+(i) Σ1,···,Σkare in general position in Xn; and
+(ii) the (relative) homology classes [Σ 1],···,[Σk] form a Z2-linear basis of
+Hn−1(Xn,∂Xn,Z2)∼=H1(Xn,Z2)/\e}atio\slash= 0.
+Moreover, we can choose each Σ ito be connected. If we cut Xnopen along
+{Σ1,...,Σk}, we get a nice manifold with corners Un, called a Z2-coreofXn.
+Similar to Theorem 2.10, we can show that there always exists a conne ctZ2-core
+forXn. Note that the boundary stratification of Unis a mixture of the facets in
+the cut section ofΣ iandthe facets from ∂Xn. So we define thepanel structure on
+Unby{P1,···,Pk,P′
+1,...,P′
+l}, wherePiconsists of the facets in the cut section
+of ΣiandP′
+jconsists of the facets in the cut open image of Fj.
+Bya similar argument asinLemma 2.4, we canconstruct a freeinvolutio nτion
+eachPi(1≤i≤k) which satisfies the conditions (a), (b)and (c) in Definition 2.8.
+If we do not define any involution on P′
+j, we say that {τi:Pi→Pi}1≤i≤kalong
+with{P′
+1,...,P′
+l}is apartial involutive panel structure on the boundary of Un.
+LetP(Un) ={P1,···,Pk}be the set of all panels in Unthat are equipped with
+involutions. Any map from P(Un) to (Z2)mis called a ( Z2)m-coloring on Un. It
+is easy to see that the glue-back construction M(Un,λ) makes perfect sense for
+Unwith a (Z2)m-coloringλ. Indeed,M(Un,λ) is got by glue 2mcopies ofUn
+only along those panels equipped with involutions according to the rule in (5).
+By a parallel argument as Theorem 3.3, we can show that (6) defines a natural
+free (Z2)m-action onM(Un,λ) whose orbit space is homeomorphic to Xn. And
+similarly, we can prove the following.
+Theorem 4.1. SupposeXnis a compact connected nice manifold with corners
+andUnis aZ2-core ofXn. Then we have:
+(1)any principal (Z2)m-bundleπ:Mn→Xndetermines a (Z2)m-coloringλπ
+onUn.
+(2)there is an equivariant homeomorphism from the M(Un,λπ)toMnwhich
+covers the identity of Xn.
+In addition, we can similarly define Lλand rank(λ) for any ( Z2)m-coloringλ
+onUn(see (11) and (12)) and extend the results in Theorem 3.7 to M(Un,λ) as
+well. Here we only give the statement below and leave the proof to the reader.
+Theorem 4.2. For any (Z2)m-coloringλonUn, theM(Un,λ)has2m−rank(λ)
+connected components which are pairwise homeomorphic. Let θλ:Un×(Z2)m→
+M(Un,λ)be the quotient map. Then each connected component of M(Un,λ)is
+homeomorphic to θλ(Un×Lλ). And there is a free action of Lλ∼=(Z2)rank(λ)on
+each connected component of M(Un,λ)whose orbit space is Xn.22 LI YU
+5.Locally Standard (Z2)m-action on closed n-manifolds
+First, let us define the meaning of standard action of (Z2)min dimension nfor
+anym≥1. Suppose g= (g1,···,gm) is an arbitrary element of ( Z2)m.
+(1) Ifm≤n, the standard ( Z2)m-action on Rnis:
+(x1,···,xn)/mapsto−→((−1)g1x1,···,(−1)gmxm,xm+1,···,xn),
+whose orbit space is Rn,m
++:={(x1,...,x n);xi≥0 for∀1≤i≤m}.
+(2) Form>n, the standard ( Z2)m-action on Rn×(Z2)m−nis:
+((x1,···,xn),(h1,···,hm−n))/mapsto−→
+(((−1)g1x1,···,(−1)gnxn),(gn+1+h1,···,gm+hm−n)),
+whose orbit space is Rn,n
++.
+Suppose ( Z2)macts effectively and smoothly on a closed n-manifoldMn. A
+local isomorphism ofMnwith the standard action (defined above) consists of:
+(1) a group automorphism σ: (Z2)m→(Z2)m;
+(2) a (Z2)m-stable open set VinMnandUinRn(ifm≤n) orRn×(Z2)m−n
+(ifm>n);
+(3) aσ-equivariant homeomorphism f:V→U, i.e.f(g·v) =σ(g)·f(v) for
+anyg∈(Z2)mandv∈V.
+A (Z2)m-action onMnis called locally standard if each point of Mnis in the
+domain of some local isomorphism. Then Mnis called a locally standard (Z2)m-
+manifold overMn/(Z2)m. Note that here we generalize the notion of locally
+standard 2-torus manifold defined in [3] where mis required to be equal to n.
+Now, suppose we have a locally standard ( Z2)m-action on a closed manifold
+Mn. Then the orbit space Xn=Mn/(Z2)mis a nice manifold with corners (in
+the rest of the paper, we always assume Xnis connected). Let π:Mn→Xnbe
+the orbit map. Suppose the set of all facets of XnisF(Xn) ={F1,···,Fl}. The
+characteristic function νπ:F(Xn)→(Z2)mof the action is defined by:
+νπ(Fj) = the element of ( Z2)mthat fixesπ−1(Fj) pointwise.
+Observe that whenever Fj1∩ ··· ∩Fjs/\e}atio\slash=∅,{νπ(Fj1),···,νπ(Fjs)}should be
+linearly independent vectors in ( Z2)moverZ2. And the isotropy group of the set
+π−1(Fj1∩···∩Fjs) is the subgroup of ( Z2)mgenerated by {νπ(Fj1),···,νπ(Fjs)}.
+In addition, Mndetermines a principal ( Z2)m-bundle over Xn, denoted by ξπ.
+IfH1(Xn,Z2) = 0,ξπis always trivial. So we assume H1(Xn,Z2)/\e}atio\slash= 0 in the rest
+of this section.
+Remark 5.1. Ifm<n, the dimension of any face of Xnis at leastn−m.23
+We will see that the characteristic function νπand the principal ξπencode all
+the structural information of the ( Z2)m-action, and we can classify locally stan-
+dard (Z2)m-manifoldsπ:Mn→Xnbyνπandξπup to some natural equivalence
+relations. The following discussions are parallel to those in [3].
+First of all, we say that two locally standard ( Z2)m-manifoldsMnandNnover
+Xnareequivalent if there is a homeomorphism f:Mn→Nntogether with an
+elementσ∈GL(m,Z2) such that
+(1)f(g·x) =σ(g)·f(x) for allg∈(Z2)mandx∈Mn, and
+(2)finduces the identity map on the orbit space.
+In addition, we call two locally standard ( Z2)m-manifoldsMnandNnoverXn
+equivariantly homeomorphic if there is a homeomorphism f:Mn→Nnsuch
+thatf(g·x) =g·f(x) for allg∈(Z2)mandx∈Mn. Such an fis called
+anequivariant homeomorphism betweenMnandNn. Notice that fwill induce
+a homeomorphism hf:Xn→Xnwhich preserves the manifold with corners
+structure of Xn. Buthfmay not be the identity map of Xn.
+LetUnbe aZ2-core ofXnfrom cutting Xnopen along a Z2-cut system
+{Σ1,···,Σk}inXnas described in Section 4. The panel structure of Unis
+{P1,···,Pk,P′
+1,···,P′
+l}wherePicorresponds to the cut section of Σ iandP′
+j
+is the cut open image of the facet Fj, and there is a partial involutive panel
+structureτi:Pi→PionUn. Moreover, if we define τ′
+j=id:P′
+j→P′
+jfor
+any 1≤j≤l, then{Pi,τi}1≤i≤kand{P′
+j,τ′
+j}1≤j≤ltogether define a complete
+involutive panel structure on Un. We call {P1,···,Pk}theprincipal panels of
+Unand call {P′
+1,···,P′
+l}thereflexive panels ofUn. And we assume Unhaving
+this involutive panel structure in the rest of this paper.
+By Theorem 4.1, the principal bundle ξπdetermines a ( Z2)m-coloringλπon the
+set of principal panels {P1,···,Pk}, and the characteristic function νπinduces
+a (Z2)m-coloringµπon the reflexive panels {P′
+1,···,P′
+l}byµπ(P′
+j) =νπ(Fj),
+1≤j≤l. So for any P′
+j1∩···∩P′
+js/\e}atio\slash=∅, we should have:
+µπ(P′
+j1),···,µπ(P′
+js) is linearly independent vectors in ( Z2)moverZ2.(15)
+The glue-back construction of Unwith respect to the composite ( Z2)m-coloring
+(λπ,µπ) onthepanels of Ungives usa closedmanifold, denotedby M(Un,λπ,µπ).
+The natural ( Z2)m-action onM(Un,λπ,µπ) is also defined by:
+g·[(x,g0)] := [(x,g+g0)],∀x∈Un,∀g,g0∈(Z2)m. (16)
+The following theorem is parallel to that in [3].24 LI YU
+Theorem 5.2. The action (Z2)m/archrightdownM(Un,λπ,µπ)defined in (16)is locally
+standard and there is an equivariant homeomorphism from M(Un,λπ,µπ)toMn
+which covers the identity of Xn.
+Proof.It is easy to check the action is locally standard. And by a parallel arg u-
+ment as the proof of Theorem 3.3, the orbit space of ( Z2)m/archrightdownM(Un,λπ,µπ) is
+Xn. Moreover, ( Z2)m/archrightdownM(Un,λπ,µπ) defines the same principal ( Z2)m-bundle
+overXnandthesamecharacteristicfunctionon Xnasthelocallystandard( Z2)m-
+action onMn. Then it is easy to construct an equivariant homeomorphism from
+M(Un,λπ,µπ) toMnwhich covers the identity of Xn. /square
+Denote by Ξ( Un,(Z2)m) the set of all eligible composite ( Z2)m-colorings on Un,
+Ξ(Un,(Z2)m) :={(λ,µ)|λis a (Z2)m-coloring on the principal panels of Un;
+µis a (Z2)m-coloring on the reflexive panels of Un
+which satisfies the condition (15) }.
+ThenbyTheorem5.2, anylocallystandard( Z2)m-actiononaclosed n-manifold
+withXnas the orbit space can be obtained from Unand some composite ( Z2)m-
+coloring (λ,µ)∈Ξ(Un,(Z2)m).
+By the definition of Un, it is easy to see that we can identify Ξ( Un,(Z2)m)
+withH1(Xn,(Z2)m)×V(Xn,(Z2)m) as a set, where V(Xn,(Z2)m) is the set of all
+characteristic functions on Xn, i.e.
+V(Xn,(Z2)m) :={ν:F(Xn)→(Z2)m;ν(Fj1),···,ν(Fjs) are linearly
+independent vectors in ( Z2)mwheneverFj1∩···∩Fjs/\e}atio\slash=∅}.
+Example 8. SupposePnisaconvex simplepolytopewith mfacets{F1,...,F m}.
+Let{e1,...,e m}be a basis of ( Z2)m. If we color Fibyei, the glue-back construc-
+tion forPnwith the trivial involutive panel structure (see Example 2) gives us a
+manifold ZPn, called the real moment-angle manifold overPn(see [2] and [6]).
+TheZ2-coefficient equivariant cohomology ring of ZPnwith respect to the nat-
+ural (Z2)maction is isomorphic to the face ring of Pn(see [2]). The ordinary
+Z2-cohomology groups of ZPnwere calculated by XiangYu Cao and Zhi L¨ u in [8].
+Example 9. In Figure 11, we have three different ( Z2)3-colorings of a pentagon
+which is equipped with the trivial involutive panel structure (see Exa mple 2).
+The glue-back construction for the left picture gives T2#T2(connected sum of
+two tori). For the other two pictures, the glue-back constructio ns both give the
+connected sum of two Klein bottles (these examples are taken from [9]).
+Example 10. LetX2=T2−P2whereP2is a polygon on T2. In Figure 12, we
+have three different Z2-cores ofX2. Then any locally standard ( Z2)m-action on
+a closed manifold with X2as the orbit space can be obtained by the glue-back25
+1e
+2e2e3e
+3e1e2e
+2e3eŒ1 2e   e1e2e
+2e3e
+1 2e   e1 2e   eŽ 
+Figure 11.
+‘’ “
+”• – — ˜ ™ š › œ
+ž Ÿ   ¡ ¢ £ ¤ ¥¦
+§¨
+Figure 12.
+construction from any one of the three Z2-cores with some suitable composite
+(Z2)m-coloring.
+Next, let us classify the locally standard ( Z2)m-manifolds over Xnup to the
+equivalence from the viewpoint of glue-back construction. By a simila r argument
+as in [3], we can prove:
+Theorem 5.3. The set of equivalence classes in locally standard (Z2)m-manifolds
+overXnbijectively corresponds to the coset Ξ(Un,(Z2)m)/GL(m,Z2), where the
+GL(m,Z2)acts onΞ(Un,(Z2)m)viaautomorphismsof the coefficientgroup (Z2)m.
+Moreover, similar to [3], we can classify locally standard ( Z2)m-manifolds over
+Xnupto equivariant homeomorphisms asfollowing. Let Aut( Xn)bethegroupof
+self-homeomorphisms of Xnwhich preserve the manifold with corners structure
+ofXn. An element h∈Aut(Xn) will induce a permutation on F(Xn) denoted
+by Φ(h) :F(Xn)→ F(Xn). Sohnaturally acts on V(Xn,(Z2)m) by sending any
+ν∈ V(Xn,(Z2)m) toν◦Φ(h).
+Theorem 5.4. Supposeπ:Mn→Xnandπ′:Nn→Xnare two locally standard
+(Z2)m-manifolds over Xn.MnandNnare equivariantly homeomorphic if and
+only if there is an h∈Aut(Xn)such thatνπ′=νπ◦Φ(h)andh∗(ξπ′) =ξπwhere
+h∗(ξπ′)is the pull-back bundle by h.
+Theorem 5.5. The setofequivarianthomeomorphismclassesof all n-dimensional
+locally standard (Z2)m-manifolds over Xnbijectively corresponds to the coset
+/parenleftbig
+H1(Xn,(Z2)m)×V(Xn,(Z2)m)/parenrightbig
+/Aut(Xn)
+whereAut(Xn)acts diagonally on the two factors.26 LI YU
+In addition, we say that two locally standard ( Z2)m-manifolds MnandNn
+overXnareweakly equivariantly homeomorphic if there is a homeomorphism
+f:Mn→Nnand an element σ∈GL(m,Z2) such that f(g·x) =σ(g)·f(x) for
+allg∈(Z2)mandx∈Mn.
+Theorem 5.6. The set of weakly equivariant homeomorphism classes in all n-
+dimensional locally standard (Z2)m-manifolds over Xnbijectively corresponds to
+the double coset
+GL(m,Z2)\/parenleftbig
+H1(Xn,(Z2)m)×V(Xn,(Z2)m)/parenrightbig
+/Aut(Xn)
+where both GL(m,Z2)andAut(Xn)act diagonally on the two factors.
+6.Some Topological Information of Locally Standard
+(Z2)m-Manifolds from the (Z2)m-colorings
+SupposeUnis aZ2-core of a connected nice manifold with corners Xn, and
+the principal panels and reflexive panels of Unare{P1,···,Pk}and{P′
+1,···,P′
+l}
+as described in the preceding section. Similar to Theorem 3.7, we can c ompute
+the number of connected components in any M(Un,λ,µ) from the composite
+(Z2)m-coloring (λ,µ) onUnas following.
+Theorem 6.1. For any(λ,µ)∈Ξ(Un,(Z2)m), the number of connected compo-
+nents inM(Un,λ,µ)is2m−rank(λ,µ)where
+rank(λ,µ) = dim Z2/a\}bracketle{tλ(P1),···,λ(Pk),µ(P′
+1),···,µ(P′
+l)/a\}bracketri}ht.
+The connected components of M(Un,λ,µ)are pairwise homeomorphic, and there
+is aninducedlocallystandard (Z2)rank(λ,µ)-actionon eachcomponentof M(Un,λ,µ).
+Proof.Suppose we glue the 2mcopies ofUnonly along the principal panels first
+according to the coloring λ, we will get a manifold with boundary denoted by
+M(Un,λ). By the same argument as in the proof of Theorem 3.7, M(Un,λ) has
+2m−rank(λ)connected components which are pairwise homeomorphic. Let Lλ:=
+/a\}bracketle{tλ(P1),···,λ(Pk)/a\}bracketri}ht ⊂(Z2)mand letθλ:Un×(Z2)m→M(Un,λ) be the quotient
+map. Then an arbitrary connected component of M(Un,λ) is of the following
+form
+Ng=/uniondisplay
+g′∈g+Lλθλ(Un×g′) for some fixed g∈(Z2)m
+SoM(Un,λ) =Ng1∪ ··· ∪Ngrwherer= 2m−rank(λ)andgi′−gi/∈Lλfor
+any 1≤i,i′≤r. The boundary of M(Un,λ) consists of those facets from the
+reflexive panels of Un×g′s.27
+Next, we glue the facets in the boundary of M(Un,λ) together according to (5)
+and the coloring µon the reflexive panels, which will give us the M(Un,λ,µ).
+Letθµ:M(Un,λ)→M(Un,λ,µ) denote the corresponding quotient map. In
+addition, let Lµ=/a\}bracketle{tµ(P′
+1),···,µ(P′
+l)/a\}bracketri}ht ⊂(Z2)m. It is easy to see that θµ(Ngi)
+andθµ(Ngi′) are in the same connected component of M(Un,λ,µ) if and only
+ifgi′−gi∈Lµ. So for any g,g′∈(Z2)m, the two blocks θµ(θλ(Un×g)) and
+θµ(θλ(Un×g′)) are in the same connected component of M(Un,λ,µ) if and only
+ifg′−g∈Lλ+Lµ. Since the Z2-dimension of Lλ+Lµis rank(λ,µ), so each
+connected component of M(Un,λ,µ) is the gluing of 2rank(λ,µ)copies ofUnfrom
+the glue-back construction. Then M(Un,λ,µ) has exactly 2m−rank(λ,µ)connected
+components which are pairwise homeomorphic. Obviously, the restr icted action
+of (Z2)mtoLλ+Lµon each component of M(Un,λ,µ) is locally standard. So
+our theorem is proved. /square
+Inaddition, if Xnisorientable(inintegralcoefficient), usingthesameargument
+as the Theorem1.7 in [10], we can prove the following.
+Theorem 6.2. Fora basis {e1,···,em}of(Z2)m, there is a group homomorphism
+ǫ: (Z2)m→Z2defined byǫ(ei) = 1for alli. SupposeXnis orientable. Then
+M(Un,λ,µ)is orientable if and only if there exists a basis {e1,···,em}of(Z2)m
+such thatǫ(µ(P′
+1)) =···=ǫ(µ(P′
+l)) = 1. So in this case, the orientablity of
+M(Un,λ,µ)is determined only by the coloring µon the reflexive panels.
+It was shown in [2] that the Z2-coefficient cohomology ring of any small cover
+canbecomputedfromthecombinatorial structureoftheorbitsp ace andtheasso-
+ciated characteristic function. But for a general locally standard (Z2)m-manifold
+MnoverXn, it is not clear how to compute the Z2-homology group or cohomol-
+ogy ring of MnfromXn. From our preceding discussions, the simplest case in
+this problem would be when Xnhas aZ2-coreUnwhose faces are all contractible.
+So we propose the following problem.
+Problem: for a locally standard ( Z2)m-manifoldM(Un,λ,µ) where (λ,µ)∈
+Ξ(Un,(Z2)m), if each face of Unis contractible (or a Z2-homology ball), find some
+way to compute the Z2-homologygroupandcohomologyring of M(Un,λ,µ)from
+the data (Un,λ,µ).
+Acknowledgement: I want to thank professor Zhi L¨ u for showing me the
+paper [3] and a lot of patient explanation.28 LI YU
+References
+[1] A. Hatcher, Algebraic topology , Cambridge University Press, Cambridge, 2002.
+[2] M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions ,
+Duke Math. J. 62(1991), no. 2, 417–451.
+[3] Z. L¨ u and M. Masuda, Equivariant classification of 2-torus manifolds , Colloq. Math. 115
+(2009), 171–188.
+[4] K. J¨ anich, On the classification of O(n)-manifolds , Math. Ann. 176(1968) 53–76.
+[5] M. Davis, Groups generated by reflections and aspherical manifolds no t covered by Eu-
+clidean space , Ann. of Math. 117(1983), 293–324.
+[6] V. M. Buchstaber and T.E. Panov, Torus actions and their applications in topology and
+combinatorics , University Lecture Series, 24. American Mathematical Society, P rovidence,
+RI, 2002.
+[7] D. Rolfsen, Knots and links , Mathematics Lecture Series, No. 7. Publish or Perish, Inc.,
+Berkeley, Calif., 1976.
+[8] X. Cao and Z. L¨ u, M¨ obius transform, moment-angle complex and Halperin-Car lsson con-
+jecture, Preprint (2009); arXiv:0908.3174
+[9] Z. L¨ u and L. Yu, Topological types of 3-dimensional small covers , math/arXiv:0710.4496,
+to appear in Forum. Math.
+[10] H. Nakayama and Y. Nishimura, The orientability of small covers and coloring simple
+polytopes , Osaka J. Math. 42 (2005), 243-256.
+Department of Mathematics and IMS, Nanjing University, Nan jing, 210093,
+P.R.China
+E-mail address :yuli@nju.edu.cn
\ No newline at end of file
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+arXiv:1001.0290v1  [math.CV]  2 Jan 2010QUASI-CONFORMAL DEFORMATIONS OF NONLINEARIZABLE
+GERMS
+KINGSHOOK BISWAS
+Abstract. Letf(z) =e2πiαz+O(z2),α∈Rbe a germ of holomorphic diffeo-
+morphism in C. Forαrational and fof infinite order, the space of conformal
+conjugacy classes of germs topologically conjugate to fis parametrized by the
+Ecalle-Voronin invariants (and in particular is infinite-d imensional). When α
+is irrational and fis nonlinearizable it is not known whether fadmits quasi-
+conformal deformations. We show that if fhas a sequence of repelling periodic
+orbits converging to the fixed point then fembeds into an infinite-dimensional
+family of quasi-conformally conjugate germs no two of which are conformally
+conjugate.
+AMS Subject Classification: 37F50
+Contents
+1. Introduction 1
+2. Deformations. 2
+2.1. Deforming repelling periodic orbits. 3
+2.2. Deforming germs with small cycles. 4
+References 4
+1.Introduction
+Letf(z) =e2πiαz+O(z2),α∈R/Zbe a germ of holomorphic diffeomorphism
+fixing the origin in C. We consider the question of when fadmits quasi-conformal
+deformations, i.e. when do there exist germs gwhich are quasi-conformally but
+not conformally conjugate to f? Iffislinearizable (i.e. analytically conjugate
+to the rigid rotation Rα(z) =e2πiαz) then any germ topologically conjugate to f
+is linearizable. In the nondegenerate parabolic case (i.e.α=p/q∈Q,fq/ne}ationslash=id),
+the quasi-conformal conjugacy class of fcontains an infinite dimensional family of
+conformalconjugacyclassesparametrizedbytheEcalle-Voronin invariants([Eca75],
+[Vor81]). In the irrationally indifferent nonlinearizable case (αirrational,fnot
+linearizable), it seems to be unknown whether quasi-conformal def ormations are
+possible. We show the following:
+Theorem 1.1. Letfbe an irrationally indifferent nonlinearizable germ with a
+sequence of repelling periodic orbits accumulating the ori gin. Then there is a family
+of quasi-conformal maps {hΛ}Λ∈ Mparametrized by sequences M={(λn)n≥0:
+Research partly supported by Department of Science and Tech nology research project grant
+DyNo. 100/IFD/8347/2008-2009.
+12 KINGSHOOK BISWAS
+λn∈C,|λn|>1}such that all conjugates of gΛ=hΛ◦f◦h−1
+Λare holomorphic,
+andgΛ1,gΛ2are conformally conjugate if and only if all but finitely many terms of
+Λ1andΛ2agree. For fixed z,hΛ(z)depends holomorphically on each λn.
+The proof proceeds as follows: given Λ = ( λn) the germgΛis obtained by quasi-
+conformally deforming fnear its periodic orbits. Each periodic orbit is attracting
+forf−1, and it is possible to construct an f-invariant Beltrami differential on each
+basin of attraction such that the quasi-conformal map hΛrectifying the Beltrami
+differential conjugates fto a holomorphic germ gΛhaving multipliers λnat the
+periodic orbits. The multipliers at periodic orbits being invariant under conformal
+conjugacies, the conclusion of the theorem follows.
+Examples of germs satisfying the hypothesis of the theorem are ge rms of rational
+mapsofdegree dwithαsatisfyingthe Cremerconditionofdegree d(seeforexample
+Milnor [Mil99], Ch. 8)
+limsuploglogqn+1
+qn>logd
+(where (pn/qn) are the continued fraction convergents of α) which ensures that the
+fixed point is accumulated by periodic orbits (only finitely many of which can be
+repelling for a rational map).
+Perez-Marcohas shown ([PM97]) for any nonlinearizable germ fthe existence of
+a unique monotone one-parameter family ( Kt)t>0of full, totally invariant continua
+calledhedgehogs containing the fixed point. In [Bis09] it is proved that any confor-
+mal mapping in a neighbourhood of a hedgehog Kof a germf1mappingKto a
+hedgehog of a germ f2necessarily conjugates f1tof2. As a corollary of Theorem
+1.1 we have
+Corollary 1.2. There exists a holomorphic motion φ:D∗׈C→ˆCofˆCover
+D∗and a hedgehog Ksuch that all the sets φ(t,K)are hedgehogs, all of which are
+quasi-conformal images of K, but fors/ne}ationslash=t,φ(s,K)cannot be conformally mapped
+toφ(t,K).
+Acknowledgements. The author thanks Ricardo Perez-Marco for many helpful
+discussions and comments. This article was written in part during a vis it to the
+Chennai Mathematical Institute. The author is very grateful to the organizers for
+their warmth and hospitality.
+2.Deformations.
+We fix a germ f(z) =e2πiαz+O(z2),α∈R−Qand a neighbourhood Uof the
+origin such that fandf−1are univalent on a neighbourhood of U. By aperiodic
+orbitorcycleoffof orderq≥1 we mean a finite set O={z1,...,z q} ⊂Usuch
+thatf(zi) =zi+1,1≤i≤q−1 andf(zq) =z1. Themultiplier at the periodic
+orbit is defined to be λ=f′(z1)f′(z2)...f′(zq) = (fq)′(zi). The periodic orbit is
+calledattracting ,indifferent andrepelling according as |λ|<1,|λ|= 1 and |λ|>1
+respectively. A periodic orbit for fof multiplier λis a periodic orbit for f−1of
+multiplierλ−1. The basin of attraction of an attracting periodic cycle is defined by
+A(O,f) :={z∈U:fn(z)→ Oasn→+∞}.QUASI-CONFORMAL DEFORMATIONS OF NONLINEARIZABLE GERMS 3
+We observe that fcan have only finitely many cycles of a given order qinU
+(by the uniqueness principle). Since fis asymptotic to an irrational rotation, i.e.
+f(z)/z→e2πiαasz→0, it follows that if fhas small cycles then the orders of the
+cycles must go to infinity.
+2.1.Deforming repelling periodic orbits. Given a repelling periodic orbit O
+offwith multiplier λand|λ′|>1 we deform the multiplier of fquasi-conformally
+fromλtoλ′by constructing a corresponding f-invariant Beltrami differential µ=
+µ(O,λ,λ′) on the basin of attraction A(O,f−1) as follows:
+By K¨ oenigs linearization theorem (see [Mil99], Ch. 6) for any repelling p eriodic
+orbitOoffwith multiplier λthere exists a unique holomorphic map φdefined on
+a neighbourhood of Osuch thatφ(zi) = 0,φ′(zi) = 1,i= 1,...,nandφ(fq(z)) =
+λφλ(z). The conformal isomorphism L:C∗→C/Z,w/mapsto→ξ=1
+2πilogwconjugates
+the linear map w/mapsto→λwonC∗to the translation ξ/mapsto→ξ+τonC/Zwhereτ=L(λ)
+and Imτ <0.
+Letτ′=L(λ′)andletKbethereallinearmapon CdefinedbyK(1) = 1,K(τ) =
+τ′. ThenKcommuteswiththetranslationbyoneandhencegivesaquasi-confo rmal
+orientation preserving (since Im τ,Imτ′<0) homeomorphism ˜K:C/Z→C/Z.
+The Beltrami differential of ˜Kis constant and invariant under translations of C/Z,
+and˜Kconjugates the translation ξ/mapsto→ξ+τonC/Ztoξ/mapsto→ξ+τ′.
+We letµbe the Beltrami differential of ˜K◦L◦φλrestricted to a small neigh-
+bourhoodDofz1∈ O. The map ˜K◦φλconjugatesfqto the translation ξ/mapsto→ξ+τ′.
+So at points z,z′=fq(z) inD,µsatisfies the invariance condition
+µ(z′)(fq)′(z) =µ(z)(fq)′(z)
+Soµextended to the neighbourhood V=D∪f(D)∪···∪fq−1(D) ofOby putting
+µ(fj(z)) =µ(z)(fj)′(z)
+(fj)′(z)
+forz∈D,j= 1,...,q−1isanf-invariantBeltramidifferential. Similarlytheabove
+equation allows us to extend µto anf-invariant Beltrami differential µ(O,λ,λ′) on
+A(O,f−1) =∪n≥1fn(V).
+Lemma 2.1. Any quasi-conformal homeomorphism hwith Beltrami coefficient
+equal toµ=µ(O,λ,λ′)on a neighbourhood of Oconjugates fto a mapg=
+h◦f◦h−1with a periodic orbit h(O)of multiplier λ′. The dependence of µ(O,λ,λ′)
+onλ′is holomorphic.
+Proof: Fori= 1,...,qwe letψibe the branch of φ−1sending 0 to zi. By construc-
+tion the map kdefined byk=ψi◦L−1◦˜K◦L◦φ(z) forzin a neighbourhood of
+zihas Beltrami coefficient equal to µand conjugates fon a neighbourhood of O
+to a holomorphic map f1=k◦f◦k−1with periodic orbit k(O) and multiplier λ′.
+Sincehandkhave the same Beltrami coefficient the map h◦k−1is holomorphic,
+and conjugates f1tog, so the multipliers of f1andgare equal. The Beltrami dif-
+ferential of ˜Kis constant equal to−log(λ′/λ)
+2log|λ|+log(λ′/λ)which depends holomorphically
+onλ′, so the Beltrami differential µ(O,λ,λ′) of˜K◦L◦φλdepends holomorphically
+onλ′.⋄4 KINGSHOOK BISWAS
+2.2.Deforming germs with small cycles. We use the Beltrami differentials
+µ(O,λ,λ′) to deform a germ with small cycles as follows:
+Proof of Theorem 1.1: Let fbe a germ with a sequence of repelling small cycles
+of orders (qn) (which we may assume to be strictly increasing). The basins of
+attraction A(O,f−1) of distinct repelling cycles of fare disjoint, so for any Λ =
+(λn)∈ Mwe can define an f-invariant Beltrami differential µΛonUby putting
+µΛ(z) =µ(O,λ,λn)(z) whenzbelongs to A(O,f−1) for a cycle Oof orderqn
+and multiplier λ, andµΛ(z) = 0 otherwise. Let hΛbe the unique quasi-conformal
+homeomorphismwith Beltramicoefficient µΛfixing0,1,∞givenby the Measurable
+Riemann Mapping Theorem. Then gΛ=hΛ◦f◦h−1
+Λis a holomorphic germ fixing
+the origin with small cycles. By Naishul’s theorem [Nai83] the multiplier at an
+indifferent fixed point is a topological conjugacy invariant so g′
+Λ(0) =e2πiα. By
+Lemma 2.1, the multiplier of gΛatallits repelling cycles of order qnis equal toλn.
+Ifφis a holomorphic germ conjugating two such germs gΛ1,gΛ2,φmust take
+all repelling cycles of gΛ1of orderqn(in its domain) to repelling cycles of gΛ2
+of orderqnand preserve multipliers, so the sequences of multipliers Λ 1,Λ2must
+agree for all but finitely many terms. It follows from Lemma 2.1 that µΛdepends
+holomorphically on each λn, hence by the Measurable Riemann Mapping Theorem
+so doeshΛ(z) for fixedz.⋄
+ProofofCorollary1.2: Let fbe asaboveand K⊂Ube ahedgehogof f. Fort∈D∗
+we let Λ tbe the constant sequence ( λn= 1/t). ThenµΛtdepends holomorphically
+ont, andφ: (t,z)/mapsto→hΛt(z) gives the required holomorphic motion. ⋄
+References
+[Bis09] K. Biswas. Complete conjugacy invariants of nonlin earizable holomorphic dynamics. To
+appear in DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS , 2009.
+[Eca75] J. Ecalle. Th´ eorie it´ erative: introduction ` a la th´ eorie des invariantes holomorphes. J.
+Math pures et appliqu´ es 54 , pages 183–258, 1975.
+[Mil99] J. Milnor. Dynamics in one complex variable. Friedr. Vieweg and Sohn, Braunschweig ,
+1999.
+[Nai83] V. I. Naishul. Topological invariants of analytic a nd area-preserving mappings and their
+applications to analytic differential equations in C2andCP2.Trans. Moscow Math. Soc.
+42, pages 239–250, 1983.
+[PM97] R. Perez-Marco. Fixed points and circle maps. Acta Mathematica 179:2 , pages 243–294,
+1997.
+[Vor81] S. M. Voronin. Analytic classification of germs of co nformal mappings ( C,0)→(C,0)
+with identity linear part. Funktsional. Anal. i Prilozhen 15 , pages 1–17, 1981.
+Ramakrishna Mission Vivekananda University, Belur Math, WB-71120 2, India
+email: kingshook@rkmvu.ac.in
\ No newline at end of file
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+1ASimple Form for the Ground State Rotational Band of 
+even-even Actinide nuclei 
+ 
+Mohamed E. Kelabi, Khaled A. Mazuz, Eman O. Farhat,  
+Howida K. Elgowiry, and Samira E. Abushnag  
+ 
+Department of Physics, Faculty of Science, Al-Fatah University, 
+Tripoli, LIBYA 
+P.O. Box 13220, Tripoli, LIBYA 
+ 
+Abstract 
+F r o m  B o h r - M o t t e l s o n  m o d e l ,  a  t h r e e  p a r a m e t r i c  s i m p l e  e x p r e s s i o n  f o r  t h e  g r o u n d  s t a t e  r o t a t i o n a l  
+band of deformed even-even nuclei is deduced by incorporating the variable moment of inertia and 
+the softness parameter. Our obtained results show good agreement with data in comparison with other 
+existing models. 
+ 
+Introduction 
+T h e  g r o u n d  s t a t e  r o t a t i o n a l  b a n d  o f  d e f o r m e d  n u c l e i  w a s  e a r l y  d e s c r i b e d  b y  t h e  s e m i - c l a s s i c a l  
+expression [1] 
+ 
+2
+() ( 1 )2EI II=+

+J. (1) 
+ 
+where Jis the nuclear moment of inertia, and Iis the nuclear spin state angular momentum, follows 
+t h e  s e q u e n c e  0 ,  2 ,  4 ,  6 , …  w i t h  e v e n  p a r i t y .  F o r  r e l a t i v e l y  h i g h  s p i n  s t a t e s ,  E q .  ( 1 )  s h o w s  a  h i g h e r  
+s y s t e m a t i c  d e v i a t i o n  f r o m  t h e  e n e r g y  s p e c t r u m  [ 2 ] .  T h i s  d e v i a t i o n  c a n  b e  m o d e r a t e d  b y  a d d i n g  a  
+correction term to Eq. (1), expressive of rotation-vibration interaction [3], [4] 
+ 
+22() ( 1 ) ( 1 )EI A II B I I=+  + (2) 
+ 
+where 2(/ 2 )A=
 Jand Bneed to be determined from experiment1.Leaving aside a few very rigid 
+n u c l e i ,  E q  ( 2 )  s h o w s  a  n o t i c e a b l e  d e v i a t i o n  f r o m  d a t a  a n d  t h o u g h t  a s  i n a p p r o p r i a t e  t o  d e s c r i b e  
+r o t a t i o n a l  e n e r g y  s p e c t r u m  [ 5 ] .  R  K  G u p t a  [ 6 ]  a t t r i b u t e d  t h i s  e f f e c t  t o  t h e  v a r i a t i o n  o f  m o m e n t  o f  
+inertia Jwith nuclear angular momentum I,giving  
+ 
+2
+(1 )2I
+IEI I=+

+J. (3) 
+ 
+1The parameter B is related to the head energies of 
-and -vibrational bands [4]. 2B y  u s i n g  T a y l o r  e x p a n s i o n  o f  IJa b o u t  t h e  g r o u n d  s t a t e  v a l u e0J,c o r r e s p o n d i n g  t o  I=0,and 
+incorporating Morinaga’s softness parameter [7] 
+ 
+0 011
+!n
+I
+n n
+JnIJ
+J

+==,n=1,2,3 , …    ( 4 )  
+ 
+Gupta could extract from Eq. (3), by keeping the first and the second order of 
n,the following two 
+expressions: 
+ 
+1(1 )
+1IAI IEI+=+(5) 
+ 
+2
+12(1 )
+1IAI IE
+II+=
+++(6) 
+ 
+these are known as NS2 and NS3 models, respectively, where the parameters A,
1and 
2can be 
+obtained by fitting with data.  
+ 
+Approach and Formalism 
+I n  t h i s  w o r k ,  w e  c o m b i n e  t h e  e f f e c t  o f  v a r i a t i o n  o f  t h e  m o m e n t  o f  i n e r t i a Jw i t h  n u c l e a r  a n g u l a r  
+momentum I,and the effect of nuclear rotation-vibration interaction in a simple expression 
+ 
+22
+1(1 )() ( 1 )1AI IEI B I II+=  ++. (7) 
+ 
+W e  f u r t h e r  s i m p l i f y  E q .  ( 7 )  b y  e x p a n d i n g  t h e  d e n o m i n a t o r ,  u s i n g  t h e  f o l l o w i n g  f o r m  o f  g e o m e t r i c  
+series [8] 
+ 
+1
+1 01()1n
+nII
+== + ,1|| 1 I <.
+The higher powers of nbear less contribution, we therefore keep only the first two terms for n<2, 
+ 
+22 2
+1 () ( 1 ) ( 1 ) ( 1 )EI A II A I I B I I  =+  +  + . (8) 
+ 
+By setting 1 aA = we obtain an elegant linear expression 
+ 
+22 2() ( 1 ) ( 1 ) ( 1 )EI A II a I I B I I=+  +  + (9) 
+ 
+contains three parameters A,a,and Bwhich can be determined straight forward, using linear method 
+of least squares fitting. 
+ 
+Eq. (9) is our fundamental expression and will be used to calculate the energies of the ground state 
+rotational band of deformed even-even Actinide nuclei.  3Results and Comparison 
+In this section we compare our results with other available calculations, namely: 
+1)The Exponential model (Expo1) [9] 
+ 
+1/ 22
+0
+0() ( 1 ) e x p 12cIEI III 	
+ 
  
 =+   
  
 

+2)The Exponential model (Expo2) [10] 
+ 
+1/2
+0
+0() ( 1 ) e x p 12cIEI III
+	
+ 
  
 =+   
  
 

+3)The Nuclear Softness model (NS3) [6] 
+ 
+2
+12(1 )
+1IAI IE
+II+=
+++
+4) The Variable Moment of Inertia model (VMI) [11] 
+ 
+2
+0(1 ) 1()22II
+IIIEC +=+ 
+In Table 1, we present the result of our calculations tabulated as Linear form of nuclear Rotational and 
+Vibrational interaction (LRV), along with experimental data in comparison other existing models. The 
+corresponding fits of our calculations are given in Table 2. We show in Figure 1, the error of each model relative to experimental data by means of the chi squared per degrees of freedom 
+ 
+( )22 1 exp calc
+mm
+mEEmp=  ,
+where mis the number of data points, and pis the number of free parameters. 
+ 
+Table 1 .Energy levels of various calculations in the units of [MeV]. Data taken from [12]. 
+ E(I) Data Expo1 
+p=3Expo2 
+p=4NS3 
+p=3VMI 
+p=2LRV 
+p=3
+224Th  E( 2 )  0 . 0 9 8 1  0 . 0 7 7 9 8 6  0 . 0 9 3 4 7 8  0 . 0 9 6 6 7 2  0 . 0 9 3 1 4 9  0 . 0 9 0 1 3  
+E( 4 )  0 . 2 8 4 1  0 . 2 4 8 1 3 3  0 . 2 8 0 2 5 5  0 . 2 8 3 6 6  0 . 2 7 9 5 2  0 . 2 7 5 3 2 2  
+ E( 6 )  0 . 5 3 4 7  0 . 4 9 6 4 0 1  0 . 5 3 3 6 3 2  0 . 5 3 4 4 6 3  0 . 5 3 1 3 2 6  0 . 5 3 0 5 5 8  
+ E( 8 )  0 . 8 3 3 9  0 . 8 0 8 8 4 3  0 . 8 3 6 2 0 7  0 . 8 3 4 3 4 7  0 . 8 3 3 2 6 8  0 . 8 3 6 6 6 2  
+ E( 1 0 )  1 . 1 7 3 8  1 . 1 7 1 4 9  1 . 1 7 7 3 4  1 . 1 7 4 6 9  1 . 1 7 6 2  1 . 1 8 0 3 0 1  
+ E( 1 2 )  1 . 5 4 9 8  1 . 5 7 0 2  1 . 5 5 1 3 6  1 . 5 5 0 3 5  1 . 5 5 4 0 7  1 . 5 5 3 9 8 7  
+ E( 1 4 )  1 . 9 5 8 9  1 . 9 9 0 4 6  1 . 9 5 6 4 5  1 . 9 5 8 2 4  1 . 9 6 2 5 6  1 . 9 5 6 0 7 2  
+ E( 1 6 )  2 . 3 9 8  2 . 4 1 6 9 3  2 . 3 9 3 7 4  2 . 3 9 6 6 8  2 . 3 9 8 4 6  2 . 3 9 0 7 5 6  
+ E( 1 8 )  2 . 8 6 4  2 . 8 3 2 8 8  2 . 8 6 6 9 8  2 . 8 6 4 8 9  2 . 8 5 9 2 2  2 . 8 6 8 0 7 8  4Table 1,  continued… 
+ E(I) Data Expo1 
+p=3Expo2 
+p=4NS3 
+p=3VMI 
+p=2LRV 
+p=3
+226Th E( 2 )  0 . 0 7 2 2  0 . 0 6 4 8 8  0 . 0 7 2 4 6 9  0 . 0 7 3 5 6 2  0 . 0 7 3 0 5 3  0 . 0 7 1 8 3 2  
+E( 4 )  0 . 2 2 6 4 3  0 . 2 0 8 9 3 6  0 . 2 2 6 5 1 4  0 . 2 2 8 1 0 3  0 . 2 2 9 2 7 4  0 . 2 2 5 4 8 7  
+ E( 6 )  0 . 4 4 7 3  0 . 4 2 3 2 9 3  0 . 4 4 6 9 5 1  0 . 4 4 7 9 1  0 . 4 5 0 4 9 7  0 . 4 4 6 1 8 9  
+ E( 8 )  0 . 7 2 1 9  0 . 6 9 8 8 8 6  0 . 7 2 1 4 7  0 . 7 2 1 2 1 8  0 . 7 2 3 6 4 7  0 . 7 2 1 4 0 2  
+ E( 1 0 )  1 . 0 4 0 3  1 . 0 2 6 3 8  1 . 0 4 0 2 6  1 . 0 3 9 0 4  1 . 0 3 9 9 2  1 . 0 4 0 8 3 3  
+ E( 1 2 )  1 . 3 9 5 2  1 . 3 9 6 0 5  1 . 3 9 5 6 9  1 . 3 9 4 3 9  1 . 3 9 3 1 7  1 . 3 9 6 4 3 5  
+ E( 1 4 )  1 . 7 8 1 5  1 . 7 9 7 5 6  1 . 7 8 2 1  1 . 7 8 1 7 8  1 . 7 7 8 9  1 . 7 8 2 4 0 1  
+ E( 1 6 )  2 . 1 9 5 8  2 . 2 1 9 6 4  2 . 1 9 5 6 3  2 . 1 9 6 8 1  2 . 1 9 3 6 9  2 . 1 9 5 1 6 8  
+ E( 1 8 )  2 . 6 3 5 1  2 . 6 4 9 5 8  2 . 6 3 4 2  2 . 6 3 5 9 2  2 . 6 3 4 8 6  2 . 6 3 3 4 1 9  
+ E( 2 0 )  3 . 0 9 7 1  3 . 0 7 2 1  3 . 0 9 7 6  3 . 0 9 6 2  3 . 1 0 0 2  3 . 0 9 8 0 7 6  
+ 
+228Th  E( 2 )  0 . 0 5 7 7 5 9  0 . 0 5 6 3 6 6  0 . 0 5 9 1 8 9  0 . 0 5 9 2 1 7  0 . 0 5 8 4 5 8  0 . 0 5 9 1 0 8  
+ E( 4 )  0 . 1 8 6 8 2 3  0 . 1 8 2 3 0 3  0 . 1 8 8 6 4 8  0 . 1 8 8 6 2 4  0 . 1 8 8 5 7 6  0 . 1 8 8 5 4 5  
+ E( 6 )  0 . 3 7 8 1 7 9  0 . 3 7 0 9 7 8  0 . 3 7 9 0 3  0 . 3 7 8 8 8 4  0 . 3 8 0 0 5 3  0 . 3 7 9 0 0 3  
+ E( 8 )  0 . 6 2 2 5  0 . 6 1 5 3 6 2  0 . 6 2 2 0 8 4  0 . 6 2 1 8 4 6  0 . 6 2 3 3 8 4  0 . 6 2 2 1 6 4  
+ E( 1 0 )  0 . 9 1 1 8  0 . 9 0 8 1  0 . 9 1 0 5 8  0 . 9 1 0 3 8  0 . 9 1 1 0 7  0 . 9 1 0 7 0 3  
+ E( 1 2 )  1 . 2 3 9 4  1 . 2 4 1 6 2  1 . 2 3 8 2 3  1 . 2 3 8 2 5  1 . 2 3 7 4  1 . 2 3 8 2 8 9  
+ E( 1 4 )  1 . 5 9 9 5  1 . 6 0 7 5 3  1 . 5 9 9 6 6  1 . 5 9 9 9 8  1 . 5 9 7 9 6  1 . 5 9 9 5 7 8  
+ E( 1 6 )  1 . 9 8 8 1  1 . 9 9 6 7 5  1 . 9 9 0 3 8  1 . 9 9 0 7 6  1 . 9 8 9 2 7  1 . 9 9 0 2 2 4  
+ E( 1 8 )  2 . 4 0 7 9  2 . 3 9 8 8 7  2 . 4 0 6 7 6  2 . 4 0 6 3 9  2 . 4 0 8 5 8  2 . 4 0 6 8 6 8  
+ 
+230Th E( 2 )  0 . 0 5 3 2  0 . 0 5 1  0 . 0 5 4 5  0 . 0 5 4 3  0 . 0 5 4 7 3  0 . 0 5 . 4 4 2 7  
+ E( 4 )  0 . 1 7 4 1  0 . 1 6 6 7 7 1  0 . 1 7 5 9  0 . 1 7 5 5 7 7  0 . 1 7 8 2  0 . 1 7 5 8 1 3  
+ E( 6 )  0 . 3 5 6 6  0 . 3 4 3 2 8 2  0 . 3 5 7 8  0 . 3 5 7 4 6 5  0 . 3 6 2 7 6  0 . 3 5 7 7 0 7  
+ E( 8 )  0 . 5 9 4 1  0 . 5 7 6 2 9  0 . 5 9 4  0 . 5 9 3 8 2 7  0 . 6 0 0 5 3  0 . 5 9 3 9 8  
+ E( 1 0 )  0 . 8 7 9 7  0 . 8 6 1 2 8  0 . 8 7 8 8  0 . 8 7 8 8 0 4  0 . 8 8 4 7 2 8  0 . 8 7 8 8 2 5  
+ E( 1 2 )  1 . 2 0 7 8  1 . 1 9 3 3 5  1 . 2 0 6 6 5  1 . 2 0 6 8 5  1 . 2 0 9 8 6  1 . 2 0 6 7 5 5  
+ E( 1 4 )  1 . 5 7 2 9  1 . 5 6 7 1 1  1 . 5 7 2 4 8  1 . 5 7 2 7 5  1 . 5 7 1 5 3  1 . 5 7 2 6 0 9  
+ E( 1 6 )  1 . 9 7 1 5  1 . 9 7 6 4 6  1 . 9 7 1 4 7  1 . 9 7 1 6 6  1 . 9 6 6 2  1 . 9 7 1 5 4 3  
+ E( 1 8 )  2 . 3 9 7 8  2 . 4 1 4 2  2 . 3 9 9 0 7  2 . 3 9 9 0 6  2 . 3 9 0 9 7  2 . 3 9 9 0 3 9  
+ E( 2 0 )  2 . 8 5  2 . 8 7 1  2 . 8 5  2 . 8 5 0 8  2 . 8 4 3 4  2 . 8 5 0 8 9 9  
+ E( 2 2 )  3 . 3 2 5  3 . 3 3 5 8  3 . 3 2 3  3 . 3 2 3  3 . 3 2 1 6  3 . 3 2 3 2 4 6  
+ E( 2 4 )  3 . 8 1 2  3 . 7 8 8 6  3 . 8 1 2  3 . 8 1 2 6  3 . 8 2 3 8  3 . 8 1 2 5 2 6  5Table 1,  continued… 
+ E(I) Data Expo1 
+p=3Expo2 
+p=4NS3 
+p=3VMI 
+p=2LRV 
+p=3
+232Th E( 2 )  0 . 0 4 9 3 6 9  0 . 0 4 6 9 8 6  0 . 0 5 1 5 5 4  0 . 0 5 2 1 1 1  0 . 0 5 1 2 9 6  0 . 0 5 1 5 6 6  
+E( 4 )  0 . 1 6 2 1 2  0 . 1 5 3 7  0 . 1 6 6 1 4  0 . 1 6 7 3 4 6  0 . 1 6 7 1 9 6  0 . 1 6 6 2 2 2  
+ E( 6 )  0 . 3 3 3 2  0 . 3 1 6 5 7 6  0 . 3 3 7 4  0 . 3 3 8 9 0 2  0 . 3 4 0 7 8 5  0 . 3 3 7 5 9 8  
+ E( 8 )  0 . 5 5 6 9  0 . 5 3 1 9 5 9  0 . 5 5 9 5  0 . 5 6 0 7 9 6  0 . 5 6 4 8 2 7  0 . 5 5 9 8 0 9  
+ E( 1 0 )  0 . 8 2 7  0 . 7 9 6 0 8 7  0 . 8 2 7  0 . 8 2 7 7 4 7  0 . 8 3 3 0 2 2  0 . 8 2 7 4 5 4  
+ E( 1 2 )  1 . 1 3 7 1  1 . 1 0 5 0 7  1 . 1 3 5 3  1 . 1 3 5 0 8  1 . 1 4 0 2 2  1 . 1 3 5 6 2  
+ E( 1 4 )  1 . 4 8 2 8  1 . 4 5 4 8 3  1 . 4 7 9 7  1 . 4 7 8 6 4  1 . 4 8 2 2 8  1 . 4 7 9 8 7 8  
+ E( 1 6 )  1 . 8 5 8 6  1 . 8 4 1 1 1  1 . 8 5 6 4  1 . 8 5 4 7 3  1 . 8 5 5 8 4  1 . 8 5 6 2 8 4  
+ E( 1 8 )  2 . 2 6 2 9  2 . 2 5 9 3 6  2 . 2 6 1 8  2 . 2 6 0 0 4  2 . 2 5 8 1 5  2 . 2 6 1 3 8 1  
+ E( 2 0 )  2 . 6 9 1 5  2 . 7 0 4 7  2 . 6 9 2 8  2 . 6 9 1 6  2 . 6 8 7  2 . 6 9 2 1 9 8  
+ E( 2 2 )  3 . 1 4 4 2  3 . 1 7 1 7  3 . 1 4 6 7  3 . 1 4 6 8  3 . 1 4 0 3  3 . 1 4 6 2 4 7  
+ E( 2 4 )  3 . 6 1 9 6  3 . 6 5 4 3  3 . 6 2 1 6  3 . 6 2 3 2  3 . 6 1 6 6  3 . 6 2 1 5 2 8  
+ E( 2 6 )  4 . 1 1 6 2  4 . 1 4 5 4  4 . 1 1 5 9  4 . 1 1 8 7  4 . 1 1 4 4  4 . 1 1 6 5 2 5  
+ E( 2 8 )  4 . 6 3 1 8  4 . 6 3 6 6  4 . 6 2 9 1  4 . 6 3 1 4  4 . 6 3 2 5  4 . 6 3 0 2 0 9  
+ E( 3 0 )  5 . 1 6 2  5 . 1 1 7  5 . 1 6 3  5 . 1 5 9  5 . 1 7  5 . 1 6 2 0 3 5  
+ 
+234Th  E( 2 )  0 . 0 4 9 5 5  0 . 0 5 0 0 8 7  0 . 0 4 9 6 4 7  0 . 0 4 9 3 2 9  0 . 0 4 9 8 4 2  0 . 0 4 9 4 3 5  
+ E( 4 )  0 . 1 6 3  0 . 1 6 3 7 8 9  0 . 1 6 2 9 6 4  0 . 1 6 2 7 5 2  0 . 1 6 3 8 3  0 . 1 6 2 8 1 2  
+ E( 6 )  0 . 3 3 6 5  0 . 3 3 6 9 4 6  0 . 3 3 6 3 1 8  0 . 3 3 6 5 5 5  0 . 3 3 7 3 3 2  0 . 3 3 6 4 7 2  
+ E( 8 )  0 . 5 6 4 8  0 . 5 6 4 8 3 2  0 . 5 6 5 0 6 3  0 . 5 6 5 3 4 7  0 . 5 6 4 8 6 3  0 . 5 6 5 2 7 1  
+ E( 1 0 )  0 . 8 4 3  0 . 8 4 1 8 2 8  0 . 8 4 2 8 6 3  0 . 8 4 2 4 5 2  0 . 8 4 1 0 7 5  0 . 8 4 2 5 8 6  
+ E( 1 2 )  1 . 1 6 0 2  1 . 1 6 0 7 7  1 . 1 6 0 2 3  1 . 1 6 0 3 6  1 . 1 6 1 1 9  1 . 1 6 0 3 1 1  
+ 
+230U E( 2 )  0 . 0 5 1 7 2  0 . 0 5 1 4 5 7  0 . 0 5 2 7 1  0 . 0 5 2 5 0 5  0 . 0 5 1 9 9  0 . 0 5 2 1 3 6  
+ E( 4 )  0 . 1 6 9 5  0 . 1 6 7 7 5 4  0 . 1 7 0 5 2 1  0 . 1 7 0 2 0 7  0 . 1 6 9 9 9 5  0 . 1 6 9 5 9 1  
+ E( 6 )  0 . 3 4 7 1  0 . 3 4 4 1 0 4  0 . 3 4 7 4 2 9  0 . 3 4 7 2 2 8  0 . 3 4 7 7 6 9  0 . 3 4 6 8 9 2  
+ E( 8 )  0 . 5 7 8 2  0 . 5 7 5 3 2  0 . 5 7 7 6 4 4  0 . 5 7 7 6 7 5  0 . 5 7 8 4 8 8  0 . 5 7 8 0 6 5  
+ E( 1 0 )  0 . 8 5 6 4  0 . 8 5 5 6 5 4  0 . 8 5 5 5 5 4  0 . 8 5 5 7 4 6  0 . 8 5 5 9 7 8  0 . 8 5 6 6 3 5  
+ E( 1 2 )  1 . 1 7 5 7  1 . 1 7 8 5 2  1 . 1 7 5 6 8  1 . 1 7 5 8 2  1 . 1 7 5 0 5  1 . 1 7 5 6 2 5  
+ 
+232U E( 2 )  0 . 0 4 7 5 7 2  0 . 0 4 7 6 9 5  0 . 0 4 8 7 5 4  0 . 0 4 8 3 9 9  0 . 0 4 8 3 5 5  0 . 0 4 8 6 3 8  
+ E( 4 )  0 . 1 5 6 5 7  0 . 1 5 5 8 0 3  0 . 1 5 8 3 9 5  0 . 1 5 7 7 4 6  0 . 1 5 8 5 6 5  0 . 1 5 8 1 6 6  
+ E( 6 )  0 . 3 2 2 6  0 . 3 2 0 3 6 8  0 . 3 2 4 0 8  0 . 3 2 3 4 6 9  0 . 3 2 5 5 2 1  0 . 3 2 3 8 4 2  
+ E( 8 )  0 . 5 4 1  0 . 5 3 7 2 3 1  0 . 5 4 1 0 6 1  0 . 5 4 0 7 9 1  0 . 5 4 3 3 9 4  0 . 5 4 0 9 2 8  
+ E( 1 0 )  0 . 8 0 5 8  0 . 8 0 1 9 7  0 . 8 0 4 6 6 7  0 . 8 0 4 8 3 5  0 . 8 0 6 7 0 4  0 . 8 0 4 6 9 9  
+ E( 1 2 )  1 . 1 1 1 5  1 . 1 0 9 8 1  1 . 1 1 0 2 7  1 . 1 1 0 7 2  1 . 1 1 0 7  1 . 1 1 0 4 3 7  
+ E( 1 4 )  1 . 4 5 3 7  1 . 4 5 5 5  1 . 4 5 3 2 6  1 . 4 5 3 6 4  1 . 4 5 1 3 9  1 . 4 5 3 4 3 2  
+ E( 1 6 )  1 . 8 2 8 1  1 . 8 3 3 1  1 . 8 2 8 9 6  1 . 8 2 8 9 4  1 . 8 2 5 4 6  1 . 8 2 8 9 8 5  
+ E( 1 8 )  2 . 2 3 1 5  2 . 2 3 5 7 2  2 . 2 3 2 5 7  2 . 2 3 2 1 6  2 . 2 3 0 1 1  2 . 2 3 2 4 0 4  
+ E( 2 0 )  2 . 6 5 9 7  2 . 6 5 4 6  2 . 6 5 8 9 8  2 . 6 5 9 0 7  2 . 6 6 3 0 1  2 . 6 5 9 0 0 5  6Table 1,  continued… 
+ E(I) Data Expo1 
+p=3Expo2 
+p=4NS3 
+p=3VMI 
+p=2LRV 
+p=3
+234U E( 2 )  0 . 0 4 3 4 9 8  0 . 0 4 1 0 3 9  0 . 0 4 5 3 4 5  0 . 0 4 5 4 9 5  0 . 0 4 4 3 9 9  0 . 0 4 4 5 9 9  
+E( 4 )  0 . 1 4 3 3 5 1  0 . 1 3 4 9 9 5  0 . 1 4 6 8 5 4  0 . 1 4 7 1 5  0 . 1 4 5 7 0 5  0 . 1 4 5 1 2 4  
+ E( 6 )  0 . 2 9 6 0 7 1  0 . 2 7 9 6 0 9  0 . 2 9 9 6 7 4  0 . 2 9 9 9 8  0 . 2 9 9 4 0 9  0 . 2 9 7 3 5 3  
+ E( 8 )  0 . 4 9 7 0 4  0 . 4 7 2 4 9 1  0 . 4 9 9 2 8 1  0 . 4 9 9 4 3 2  0 . 5 0 0 3 0 5  0 . 4 9 7 1 0 3  
+ E( 1 0 )  0 . 7 4 1 2  0 . 7 1 1 0 8 6  0 . 7 4 1 4 6 9  0 . 7 4 1 3 4 4  0 . 7 4 3 4 3 9  0 . 7 4 0 2 1 8  
+ E( 1 2 )  1 . 0 2 3 8  0 . 9 9 2 6 3  1 . 0 2 2 3 4  1 . 0 2 1 9 1  1 . 0 2 4 4 8  1 . 0 2 2 5 8  
+ E( 1 4 )  1 . 3 4 0 8  1 . 3 1 4 0 8  1 . 3 3 8 2 9  1 . 3 3 7 6 3  1 . 3 3 9 7 7  1 . 3 4 0 1 0 1  
+ E( 1 6 )  1 . 6 8 7 8  1 . 6 7 2 0 2  1 . 6 8 6 0 3  1 . 6 8 5 3 3  1 . 6 8 6 2 3  1 . 6 8 8 7 2 7  
+ E( 1 8 )  2 . 0 6 3  2 . 0 6 2 4 9  2 . 0 6 2 5 3  2 . 0 6 2 0 7  2 . 0 6 1 2 9  2 . 0 6 4 4 3 9  
+ E( 2 0 )  2 . 4 6 4 2  2 . 4 8 0 7 6  2 . 4 6 5 1 1  2 . 4 6 5 1 7  2 . 4 6 2 7 7  2 . 4 6 3 2 4 9  
+ 
+236U E( 2 )  0 . 0 4 5 2 4 2  0 . 0 4 7 3 1 5  0 . 0 4 6 8 8 9  0 . 0 4 6 2 3 4  0 . 0 4 7 3 1 8  0 . 0 4 6 4 0 4  
+ E( 4 )  0 . 1 4 9 4 7 6  0 . 1 5 3 9 4 3  0 . 1 5 2 8 4 7  0 . 1 5 1 4 0 5  0 . 1 5 5 2 6 3  0 . 1 5 1 6 5 4  
+ E( 6 )  0 . 3 0 9 7 8 4  0 . 3 1 5 4 4 5  0 . 3 1 3 7 3 9  0 . 3 1 1 8 8 7  0 . 3 1 8 9 8 9  0 . 3 1 1 9 9 9  
+ E( 8 )  0 . 5 2 2 2 4  0 . 5 2 7 4 8  0 . 5 2 5 4 3 6  0 . 5 2 3 7 1 8  0 . 5 3 2 9 1 8  0 . 5 2 3 5 5 3  
+ E( 1 0 )  0 . 7 8 2 3  0 . 7 8 5 8 0 4  0 . 7 8 3 8 1 6  0 . 7 8 2 6 9 5  0 . 7 9 1 7 5 4  0 . 7 8 2 2 9 4  
+ E( 1 2 )  1 . 0 8 5 3  1 . 0 8 6 2 7  1 . 0 8 4 7 6  1 . 0 8 4 4 7  1 . 0 9 0 8 7  1 . 0 8 4 0 6 3  
+ E( 1 4 )  1 . 4 2 6 3  1 . 4 2 4 8 1  1 . 4 2 4 1 4  1 . 4 2 4 6 5  1 . 4 2 6 3 7  1 . 4 2 4 5 6 5  
+ E( 1 6 )  1 . 8 0 0 9  1 . 7 9 7 4 7  1 . 7 9 7 8 4  1 . 7 9 8 8 6  1 . 7 9 4 9 8  1 . 7 9 9 3 6 9  
+ E( 1 8 )  2 . 2 0 3 9  2 . 2 0 0 3 5  2 . 2 0 1 7 3  2 . 2 0 2 8 1  2 . 1 9 3 9 7  2 . 2 0 3 9 1  
+ E( 2 0 )  2 . 6 3 1 7  2 . 6 2 9 6 5  2 . 6 3 1 7  2 . 6 3 2 3 5  2 . 6 2 1 0 1  2 . 6 3 3 4 8 4  
+ E( 2 2 )  3 . 0 8 1 2  3 . 0 8 1 6 7  3 . 0 8 3 6  3 . 0 8 3 5 5  3 . 0 7 4 1 3  3 . 0 8 3 2 5 3  
+ E( 2 4 )  3 . 5 5  3 . 5 5 2 7 4  3 . 5 5 3 2 8  3 . 5 5 2 6 7  3 . 5 5 1 6 5  3 . 5 4 8 2 4 2  
+ 
+238U E( 2 )  0 . 0 4 4 9 1 6  0 . 0 4 6 9 4 1  0 . 0 4 6 3 5 5  0 . 0 4 5 7 2  0 . 0 4 7 3 4 5  0 . 0 4 6 3 2 9  
+ E( 4 )  0 . 1 4 8 3 8  0 . 1 5 2 7 6 6  0 . 1 5 1 2 2 1  0 . 1 4 9 8 3 9  0 . 1 5 5 2 3 2  0 . 1 5 1 1 6  
+ E( 6 )  0 . 3 0 7 1 8  0 . 3 1 3 1 1 3  0 . 3 1 0 6 3 6  0 . 3 0 8 8 9 3  0 . 3 1 8 6 2 6  0 . 3 1 0 5 4 9  
+ E( 8 )  0 . 5 1 8 1  0 . 5 2 3 7 1 8  0 . 5 2 0 6 2 2  0 . 5 1 9 0 6 1  0 . 5 3 1 7 9 7  0 . 5 2 0 5 3  
+ E( 1 0 )  0 . 7 7 5 9  0 . 7 8 0 4 0 6  0 . 7 7 7 1 9  0 . 7 7 6 2 6 6  0 . 7 8 9 3 6 4  0 . 7 7 7 1 1 4  
+ E( 1 2 )  1 . 0 7 6 7  1 . 0 7 9 1  1 . 0 7 6 3 3  1 . 0 7 6 2 7  1 . 0 8 6 6 7  1 . 0 7 6 2 8 7  
+ E( 1 4 )  1 . 4 1 5 5  1 . 4 1 5 7 9  1 . 4 1 4 0 2  1 . 4 1 4 7 5  1 . 4 1 9 8 1  1 . 4 1 4 0 1 4  
+ E( 1 6 )  1 . 7 8 8 4  1 . 7 8 6 5 8  1 . 7 8 6 2 1  1 . 7 8 7 4  1 . 7 8 5 5 4  1 . 7 8 6 2 3 8  
+ E( 1 8 )  2 . 1 9 1 1  2 . 1 8 7 6 4  2 . 1 8 8 8 3  2 . 1 8 9 9 8  2 . 1 8 1 1 3  2 . 1 8 8 8 7 8  
+ E( 2 0 )  2 . 6 1 9 1  2 . 6 1 5 2 2  2 . 6 1 7 7 8  2 . 6 1 8 3 7  2 . 6 0 4 2 9  2 . 6 1 7 8 2 9  
+ E( 2 2 )  3 . 0 6 8 1  3 . 0 6 5 6 4  3 . 0 6 8 9 4  3 . 0 6 8 6 5  3 . 0 5 3 0 8  3 . 0 6 8 9 6 4  
+ E( 2 4 )  3 . 5 3 5 3  3 . 5 3 5 3 1  3 . 5 3 8 1 4  3 . 5 3 7 0 8  3 . 5 2 5 8 1  3 . 5 3 8 1 3 4  
+ E( 2 6 )  4 . 0 1 8 1  4 . 0 2 0 6 9  4 . 0 2 1 2  4 . 0 2 0 1 9  4 . 0 2 1 0 6  4 . 0 2 1 1 6 6  
+ E( 2 8 )  4 . 5 1 7  4 . 5 1 8 3 2  4 . 5 1 3 8 7  4 . 5 1 4 7 3  4 . 5 3 7 5 3  4 . 5 1 3 8 6 5  7Table 1 ,continued… 
+ E(I) Data Expo1 
+p=3Expo2 
+p=4NS3 
+p=3VMI 
+p=2LRV 
+p=3
+236Pu E( 2 )  0 . 0 4 4 6 3  0 . 0 4 5 4 7 7  0 . 0 4 5 0 5 9  0 . 0 4 4 7 5 1  0 . 0 4 4 9 6  0 . 0 4 4 8 9 8  
+ E( 4 )  0 . 1 4 7 4 5  0 . 1 4 8 9 2 1  0 . 1 4 7 9 9 4  0 . 1 4 7 5 5 2  0 . 1 4 8 3 3 9  0 . 1 4 7 7 5  
+ E( 6 )  0 . 3 0 5 8  0 . 3 0 7 0 1 4  0 . 3 0 5 9  0 . 3 0 5 6 9 5  0 . 3 0 6 9 2 9  0 . 3 0 5 7 6 5  
+ E( 8 )  0 . 5 1 5 7  0 . 5 1 6 2 7 6  0 . 5 1 5 5 1  0 . 5 1 5 6 9 9  0 . 5 1 6 6 8 2  0 . 5 1 5 5 8 6  
+ E( 1 0 )  0 . 7 7 3 5  0 . 7 7 3 0 2 5  0 . 7 7 3 0 9 3  0 . 7 7 3 4 5 4  0 . 7 7 3 3 8 3  0 . 7 7 3 2 8 7  
+ E( 1 2 )  1 . 0 7 4 3  1 . 0 7 3 3 2  1 . 0 7 4 3 1  1 . 0 7 4 3 8  1 . 0 7 3 0 5  1 . 0 7 4 3 7 6  
+ E( 1 4 )  1 . 4 1 3 6  1 . 4 1 2 8 5  1 . 4 1 4  1 . 4 1 3 5 8  1 . 4 1 2 1 1  1 . 4 1 3 7 9  
+ E( 1 6 )  1 . 7 8 6  1 . 7 8 6 8 7  1 . 7 8 5 8 4  1 . 7 8 6  1 . 7 8 7 4 2  1 . 7 8 5 9 0 3  
+ 
+238Pu  E( 2 )  0 . 0 4 4 0 7 6  0 . 0 4 5 2 1 2  0 . 0 4 4 8 8 3  0 . 0 4 4 6 4 4  0 . 0 4 4 9 6  0 . 0 4 4 8 4 6  
+ E( 4 )  0 . 1 4 5 9 5 2  0 . 1 4 8 2 5 1  0 . 1 4 7 3 9 4  0 . 1 4 6 8 9 2  0 . 1 4 8 3 3 9  0 . 1 4 7 3 1 2  
+ E( 6 )  0 . 3 0 3 3 8  0 . 3 0 6 1 8 8  0 . 3 0 4 8 3 6  0 . 3 0 4 2 3 7  0 . 3 0 6 9 2 9  0 . 3 0 4 7 2 9  
+ E ( 8 )  0 . 5 1 3 5 8  0 . 5 1 6 1 0 4  0 . 5 1 4 4 5 9  0 . 5 1 3 9 7 9  0 . 5 1 6 6 8 2  0 . 5 1 4 3 6 1  
+ E( 1 0 )  0 . 7 7 3 4 8  0 . 7 7 5 0 9 5  0 . 7 7 3 4 6 1  0 . 7 7 3 2 6 4  0 . 7 7 3 3 8 3  0 . 7 7 3 4 0 1  
+ E( 1 2 )  1 . 0 8 0 1  1 . 0 8 0 2 7  1 . 0 7 8 9 8  1 . 0 7 9 1 2  1 . 0 7 3 0 5  1 . 0 7 8 9 7 7  
+ E( 1 4 )  1 . 4 2 9 1  1 . 4 2 8 7 3  1 . 4 2 8 1  1 . 4 2 8 4 9  1 . 4 1 2 1 1  1 . 4 2 8 1 4 4  
+ E( 1 6 )  1 . 8 1 8 5  1 . 8 1 7 6 1  1 . 8 1 7 8 2  1 . 8 1 8 2 7  1 . 7 8 7 4 2  1 . 8 1 7 8 9 2  
+ E( 1 8 )  2 . 2 4 4 9  2 . 2 4 4 0 2  2 . 2 4 5 0 8  2 . 2 4 5 3 7  2 . 1 9 6 2 4  2 . 2 4 5 1 4 2  
+ E( 2 0 )  2 . 7 0 5 7  2 . 7 0 5 0 8  2 . 7 0 6 7 2  2 . 7 0 6 6 6  2 . 6 3 6 2 1  2 . 7 0 6 7 4 4  
+ E( 2 2 )  3 . 1 9 8 8  3 . 1 9 7 9  3 . 1 9 9 5  3 . 1 9 9 1  3 . 1 0 5 2 7  3 . 1 9 9 4 8 4  
+ E( 2 4 )  3 . 7 2 0 8  3 . 7 1 9 6  3 . 7 2 0 1 4  3 . 7 1 9 7  3 . 6 0 1 6 1  3 . 7 2 0 0 7 5  
+ E( 2 6 )  4 . 2 6 5 2  4 . 2 6 7 3  4 . 2 6 5 1  4 . 2 6 5 5 2  4 . 1 2 3 6 4  4 . 2 6 5 1 6 4  
+ 
+240Pu E( 2 )  0 . 0 4 2 8 2 4  0 . 0 4 3 6 2 5  0 . 0 4 4 0 8 1  0 . 0 4 4 0 1 3  0 . 0 4 2 8 1 2  0 . 0 4 4 0 6 8  
+ E( 4 )  0 . 1 4 1 6 9  0 . 1 4 2 9 4 7  0 . 1 4 4 1 2  0 . 1 4 3 9 6 5  0 . 1 4 1 6 1 4  0 . 1 4 4 0 9 6  
+ E( 6 )  0 . 2 9 4 3 1 9  0 . 2 9 5 0 2 2  0 . 2 9 6 8 4 3  0 . 2 9 6 6 3 7  0 . 2 9 4 0 4 3  0 . 2 9 6 8 2 5  
+ E( 8 )  0 . 4 9 7 5 2  0 . 4 9 6 9 2 9  0 . 4 9 9 1 1  0 . 4 9 8 9 0 3  0 . 4 9 6 9 6 6  0 . 4 9 9 1 1 2  
+ E( 1 0 )  0 . 7 4 7 8  0 . 7 4 5 7 6 9  0 . 7 4 8  0 . 7 4 7 7 3 7  0 . 7 4 6 9 4 7  0 . 7 4 7 9 2 3  
+ E( 1 2 )  1 . 0 4 1 8  1 . 0 3 8 6 6  1 . 0 4 0 3  1 . 0 4 0 2 1  1 . 0 4 0 5 8  1 . 0 4 0 3 3 9  
+ E( 1 4 )  1 . 3 7 5 6  1 . 3 7 2 7 5  1 . 3 7 3 5  1 . 3 7 3 5 1  1 . 3 7 4 6 6  1 . 3 7 3 5 5 3  
+ E( 1 6 )  1 . 7 4 5 6  1 . 7 4 5 1 8  1 . 7 4 4 8  1 . 7 4 4 9 3  1 . 7 4 6 2 8  1 . 7 4 4 8 7 1  
+ E( 1 8 )  2 . 1 5 2  2 . 1 5 3 1 3  2 . 1 5 2  2 . 1 5 1 8 5  2 . 1 5 2 8 6  2 . 1 5 1 7 1 3  
+ E( 2 0 )  2 . 5 9 1  2 . 5 9 3 7 7  2 . 5 9 2  2 . 5 9 1 7 9  2 . 5 9 2 0 8  2 . 5 9 1 6 0 9  
+ E( 2 2 )  3 . 0 6 1  3 . 0 6 4 3  3 . 0 6 2  3 . 0 6 2 3 6  3 . 0 6 1 9  3 . 0 6 2 2 0 5  
+ E( 2 4 )  3 . 5 6  3 . 5 6 1 9  3 . 5 6  3 . 5 6 1 2 9  3 . 5 6 0 5 2  3 . 5 6 1 2 5 9  
+ E( 2 6 )  4 . 0 8 8  4 . 0 8 3 7 8  4 . 0 8 7  4 . 0 8 6 4  4 . 0 8 6 3 2  4 . 0 8 6 6 3 9  8Table 1 ,continued… 
+ E(I) Data Expo1 
+p=3Expo2 
+p=4NS3 
+p=3VMI 
+p=2LRV 
+p=3
+242Pu E( 2 )  0 . 0 4 4 5 4  0 . 0 4 6 0 4 6  0 . 0 4 5 3 6  0 . 0 4 4 8 0 9  0 . 0 4 5 6 8 4  0 . 0 4 5 2 2 9  
+E( 4 )  0 . 1 4 7 3  0 . 1 5 0 6 4 9  0 . 1 4 8 8  0 . 1 4 7 6 9  0 . 1 5 0 7 2  0 . 1 4 8 5 5 3  
+ E( 6 )  0 . 3 0 6 4  0 . 3 1 0 3 9 9  0 . 3 0 7 5  0 . 3 0 6 1 6 9  0 . 3 1 1 8 3 7  0 . 3 0 7 1 5 5  
+ E( 8 )  0 . 5 1 8 1  0 . 5 2 1 8 8 1  0 . 5 1 8 3  0 . 5 1 7 3 0 4  0 . 5 2 4 9 0 8  0 . 5 1 8 0 1 9  
+ E( 1 0 )  0 . 7 7 8 6  0 . 7 8 1 6 6 7  0 . 7 7 8 1  0 . 7 7 7 7 6 8  0 . 7 8 5 6 4 2  0 . 7 7 7 9 3 3  
+ E( 1 2 )  1 . 0 8 4 4  1 . 0 8 6 3 1  1 . 0 8 3 4  1 . 0 8 3 9 3  1 . 0 8 9 9 9  1 . 0 8 3 4 8 9  
+ E( 1 4 )  1 . 4 3 1 7  1 . 4 3 2 3 5  1 . 4 3 0 8  1 . 4 3 1 9 2  1 . 4 3 4 3 1  1 . 4 3 1 0 8 5  
+ E( 1 6 )  1 . 8 1 6 7  1 . 8 1 6 2 6  1 . 8 1 6 6  1 . 8 1 7 7 7  1 . 8 1 5 4 1  1 . 8 1 6 9 2 2  
+ E( 1 8 )  2 . 2 3 6  2 . 2 3 4 5 2  2 . 2 3 7  2 . 2 3 7 4  2 . 2 3 0 5 2  2 . 2 3 7 0 0 5  
+ E( 2 0 )  2 . 6 8 6  2 . 6 8 3 5 1  2 . 6 8 7  2 . 6 8 6 7 7  2 . 6 7 7 2 3  2 . 6 8 7 1 4 2  
+ E( 2 2 )  3 . 1 6 3  3 . 1 5 9 5 6  3 . 1 6 3  3 . 1 6 1 9  3 . 1 5 3 4 5  3 . 1 6 2 9 4 8  
+ E( 2 4 )  3 . 6 6 2  3 . 6 5 8 9 1  3 . 6 6  3 . 6 5 8 9 2  3 . 6 5 7 3 4  3 . 6 5 9 8 3 9  
+ E( 2 6 )  4 . 1 7 2  4 . 1 7 7 6 8  4 . 1 7 3  4 . 1 7 4 1 2  4 . 1 8 7 2 9  4 . 1 7 3 0 3 8  
+ 
+244Pu E( 2 )  0 . 0 4 4 2  0 . 0 4 8 4 4 8  0 . 0 4 6 6 3  0 . 0 4 3 9 6 8  0 . 0 5 0 9 1 3  0 . 0 4 5 5 0 9  
+ E( 4 )  0 . 1 5 5  0 . 1 5 8 1 7 4  0 . 1 5 3 1 5 7  0 . 1 4 7 1 3  0 . 1 6 6 2 5 8  0 . 1 5 0 5 5  
+ E( 6 )  0 . 3 1 7 9  0 . 3 2 5 1 4 1  0 . 3 1 6 6 6 1  0 . 3 0 8 8 2 9  0 . 3 3 9 6 0 2  0 . 3 1 3 1 2 3  
+ E( 8 )  0 . 5 3 5  0 . 5 4 5 2 2 6  0 . 5 3 3 9 6 9  0 . 5 2 6 8 8 1  0 . 5 6 4 0 4 6  0 . 5 3 0 5 1 3  
+ E( 1 0 )  0 . 8 0 2 4  0 . 8 1 4 2 0 4  0 . 8 0 1 6 0 9  0 . 7 9 7 6 6 9  0 . 8 3 3 4 4 8  0 . 7 9 9 2 9 3  
+ E( 1 2 )  1 . 1 1 5 9  1 . 1 2 7 7 2  1 . 1 1 5 7 4  1 . 1 1 6 3 4  1 . 1 4 2 7 1  1 . 1 1 5 3 2 1  
+ E( 1 4 )  1 . 4 7 1  1 . 4 8 1 2 5  1 . 4 7 2 0 8  1 . 4 7 7 0 4  1 . 4 8 7 6 8  1 . 4 7 3 7 4 1  
+ E( 1 6 )  1 . 8 6 3 5  1 . 8 7 0 0 7  1 . 8 6 5 7 7  1 . 8 7 3 2 5  1 . 8 6 4 9 6  1 . 8 6 8 9 8 2  
+ E( 1 8 )  2 . 2 8 9  2 . 2 8 9 1 5  2 . 2 9 1 2 3  2 . 2 9 8 0 6  2 . 2 7 1 7 8  2 . 2 9 4 7 6 1  
+ E( 2 0 )  2 . 7 4 2  2 . 7 3 3 1 3  2 . 7 4 1 9 5  2 . 7 4 4 5 2  2 . 7 0 5 8 1  2 . 7 4 4 0 8  
+ E( 2 2 )  3 . 2 1 5  3 . 1 9 6 1 2  3 . 2 1 0 0 9  3 . 2 0 5 8 6  3 . 1 6 5 0 9  3 . 2 0 9 2 2 6  
+ E( 2 4 )  3 . 6 9  3 . 6 7 1 5 3  3 . 6 8 5 9 9  3 . 6 7 5 7 4  3 . 6 4 7 9 5  3 . 6 8 1 7 7 5  
+ E( 2 6 )  4 . 1 4 9  4 . 1 5 1 7 7  4 . 1 5 7 2 7  4 . 1 4 8 4 2  4 . 1 5 2 9 5  4 . 1 5 2 5 8 6  
+ E( 2 8 )  4 . 6 1  4 . 6 2 7 6 2  4 . 6 0 7 2 4  4 . 6 1 8 8 2  4 . 6 7 8 8 1  4 . 6 1 1 8 0 6  9Table 2 .The corresponding fits of our calculation are given in the units of [MeV]. 
+Nucleus A310× a510× B710×
+2 2 4 T h  1 6 . 3 9 8 8 3  7 3 . 4 2 6 8 5  152.1694 
+2 2 6 T h  1 2 . 7 1 6 1 9  3 8 . 9 6 6 4 2  58.41588 
+2 2 8 T h  1 0 . 2 9 5 9 9  2 3 . 0 1 0 3 3  25.83317 
+2 3 0 T h  9 . 3 5 8 4 2 5  1 4 . 6 1 3 1 6  8.382393 
+2 3 2 T h  8 . 8 8 7 5 1 7  1 5 . 0 4 3  12.6447 
+2 3 4 T h  8 . 3 0 6 7 8 2  2 . 2 2 5 1 3  3 8 . 5 8 1 7 1  
+230U 8.888611 9.574225 13.05427
+232U 8.304480 9.915433 0.2278621 
+234U 7.610906 8.911491 0.8635747 
+236U 7.882479 7.317457 3.542552
+238U 7.884489 8.132648 0.5960606
+2 3 6 P u  7 . 5 6 6 7 7 1  3 . 7 4 2 6 2 4  1 4 . 7 8 4 1 8  2 3 8 P u  7 . 5 8 1 5 4 4  5 . 3 1 0 0 3 1  1 . 7 8 3 5 3 9  
+2 4 0 P u  7 . 4 8 6 8 6 0  7 . 1 9 8 7 9 7  
+2.938037 
+2 4 2 P u  7 . 6 4 4 2 6 7  5 . 1 6 0 4 8 3  5 . 1 0 0 3 1 3  
+2 4 4 P u  7 . 6 2 7 2 3 0  1 . 5 6 4 0 6 7  1 8 . 5 9 2 7 0  
+Expo1    Expo2    NS3   VMI    LRV224Th
+226Th
+228Th
+230Th
+232Th
+234Th
+230U
+232U
+234U
+236U
+238U
+236Pu
+238Pu
+240Pu
+242Pu
+244Pu0.0000.0010.0020.0030.0040.0052[MeV]2
+Figure 1 .Deviation from data of various mentioned calculations. 
+ (The zero level is shifted upward for convenience) 
+ 10Conclusion 
+In this work, a simple linear expression is deduced to calculate the energy levels of the ground state 
+r o t a t i o n a l  b a n d  o f  d e f o r m e d  e v e n - e v e n  A c t i n i d e  n u c l e i .  I t  i n c l u d e s  t h r e e  p a r a m e t e r s  A,a,and B
+which are determined straight forward using linear least squares fitting. The results of our calculation 
+show good agreement with data in comparison with other calculations. 
+ 
+We observe that the correction parameter Bbears small values and could be neglected in some cases 
+without affecting the results. This is an indication for which the 
-and -vibrations do not contribute 
+much to the ground state band. On the other hand, the moment of inertia and its variation at high spin 
+states show considerable contributions. Our sim ple expression can also be em ployed to study som e 
+nuclear effects in deformed even-even nuclei, including the backbending effects in some nuclei and 
+possibly the variation of moment of inertia with high energy spin states.  
+[1]  A. Bohr and B. R. Mottelson, Kgl. Danske Videnskab. Sclskab, Mat. Fys. Medd. 26, no 14 
+ ( 1 9 5 2 ) .  
+[2]  Nathan, O., and S. G. Nelsson, Collective Nuclear Motion and the Unified Model, Alpha-Beta 
+ and Gamma-Ray Spectroscopy, Vol I, North Holland Publishing Co. Amsterdam (1965). [3] A. Bohr and B. R. Mottelson, Kgl. Danske Videnskab. Sclskab, Mat. Fys. Medd. 27, no 16 
+ ( 1 9 5 3 ) .  
+[4]  Raymond K. Sheline, Rev. Mod. Phys., 32-1(1960)1. 
+[5]  P. C. Sood, Phys. Rev. 161 (1967) 1063.   
+[6]  Raj K. Gupta, ICTP internal report, IC/71/32, Trieste (1971). [7] H. Morinaga, Nucl. Phys. 75 (1966) 385. 
+[8]  Arfen and Weber, Mathematical Methods for Physicists, 6th Ed., Elsevier Inc. (2007). 
+[9] P. C. Sood and A. K. Jain, Phys. Rev. C18 (1983) 1522. 
+[10] H. A. Alhendi, H. H. Alharbi, S. U. El-Kameesy, ”Improved Exponential model with pairing 
+ attenuation and the backbending phenomenon”, nucl-th/0409065. 
+[11] M. A. J. Mariscotti, G. Schorff-Goldhaber, and B. Buck, Phys. Rev. 178 (1969) 1864. 
+[12] H. A. Alhendi, H. H. Alharbi, S. U. El-Kameesy, “Nuclear Structure Study of Some Actinides 
+ N u c l e i ” ,  n u c l - t h / 0 5 0 2 0 1 7 .  
+ References 
\ No newline at end of file
diff --git a/1001.0293.txt b/1001.0293.txt
new file mode 100644
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@@ -0,0 +1,281 @@
+arXiv:1001.0293v2  [hep-ph]  13 Jan 2010The Aligned Two-Higgs-Doublet Model∗
+Paula Tuz ´on
+and
+Antonio Pich
+Departament de F´ ısica Te` orica, IFIC, Universitat de Val` encia - CSIC
+Apt. Correus 22085, E-46071 Val` encia, Spain
+In the two-Higgs-Doublet model the alignment of the Yukawa flavou r
+matrices of the two scalar doublets guarantees the absence of tr ee-level
+flavour-changing neutral couplings. The resulting fermion-scalar interac-
+tions are parameterized in terms of three complex parameters ζf, leading
+to a generic Yukawa structure which contains as particular cases a ll known
+specific implementationsofthemodelbasedon Z2symmetries. Thesethree
+complex parameters are potential new sources of CP violation.
+PACS numbers: 12.60.Fr, 12.15.-y, 12.15.Ji, 14.80.Cp
+1. The generic two-Higgs-doublet model
+The two-Higgs-doublet model (THDM) [1, 2] is one of the most s im-
+plest extensions of the Standard Model (SM) and brings many i nterest-
+ing phenomenological features (new sources of CP violation , dark mat-
+ter candidates, axion phenomenology ...). It is a gauge theo ry with the
+same fermion content as the SM and two Higgs doublets φa(a= 1,2)
+with hypercharge Y=1
+2. The neutral components of the scalar dou-
+blets acquire vacuum expectation values that are, in genera l, complex:
+<0|φT
+a(x)|0>=1√
+2(0,vaeiθa). One can fix θ1= 0 through an appropriate
+U(1)Ytransformation, leaving the relative phase θ≡θ2−θ1.
+A global SU(2) transformation in the scalar space ( φ1,φ2) allows us to
+define the so-called Higgs basis (Φ 1,Φ2) where only one doublet acquires a
+∗Presented at the Flavianet topical workshop on Low energy constraints on extensions
+of the Standard Model , Kazimierz, Poland, 23-27 July 20092
+vacuum expectation value v≡/radicalBig
+v2
+1+v2
+2:
+/parenleftbiggΦ1
+−Φ2/parenrightbigg
+≡1
+v/bracketleftbiggv1v2
+v2−v1/bracketrightbigg /parenleftbiggφ1
+e−iθφ2/parenrightbigg
+, (1)
+Φ1=/bracketleftBigg
+G+
+1√
+2(v+S1+iG0)/bracketrightBigg
+, Φ2=/bracketleftBigg
+H+
+1√
+2(S2+iS3)/bracketrightBigg
+.(2)
+In addition to the three Goldstone fields G±(x) andG0(x), there are five
+physical degrees of freedom in the scalar sector: two charge d fieldsH±(x)
+and three neutrals ϕ0
+i(x) ={h(x),H(x),A(x)}, which are related to the Si
+fields through an orthogonal transformation ϕ0
+i(x) =RijSj(x). The form
+ofRijdepends on the scalar potential, which could violate CP in it s most
+general version; in this case the resulting mass eigenstate s will not have a
+definite CP parity. If it is CP symmetric, the CP admixture dis appears.
+1.1. Yukawa sector
+The most general Yukawa Lagrangian is given by
+LY=−¯Q′
+L(Γ1φ1+Γ2φ2)d′
+R−¯Q′
+L(∆1/tildewideφ1+∆2/tildewideφ2)u′
+R
+−¯L′
+L(Π1˜φ1+Π2˜φ2)l′
+R+ h.c. , (3)
+where¯Q′
+Land¯L′
+Lare the left-handed quark and lepton doublets, respec-
+tively, and ˜φa(x)≡iτ2φ∗
+athe charge-conjugated scalar fields. All fermionic
+fields are written as NG–dimensional vectors and the couplings Γ a, ∆aand
+ΠaareNG×NGcomplex matrices in flavour space. Moving to the Higgs
+basis, the Lagrangian reads
+LY=−√
+2
+v/braceleftBig¯Q′
+L(M′
+dΦ1+Y′
+dΦ2)d′
+R+¯Q′
+L(M′
+u˜Φ1+Y′
+u˜Φ2)u′
+R
++¯L′
+L(M′
+l˜Φ1+Y′
+l˜Φ2)l′
+R+ h.c./bracerightBig
+. (4)
+Here,M′
+f(f=u,d,l) are the non-diagonal fermion mass matrices and Y′
+f
+contain the Yukawa couplings to the scalar doublet that does n’t acquire a
+vacuum expectation value. Each right-handed fermion coupl es to two differ-
+ent matrices that in general cannot be diagonalized simulta neously. In the
+fermion mass-eigenstate basis f(x), theM′
+fmatrices become the diagonal
+mass matrices Mf, whileY′
+fbecomeYfwhich remain non-diagonal and un-
+related to Mf. This originates dangerous flavour-changing neutral-curr ent
+(FCNC) interactions, which are tightly constrained phenom enologically.3
+To avoid this problem one can assume that the non-diagonal Yu kawa
+couplings are proportional to thegeometric mean of thetwo f ermion masses,
+gij∝√mimj[3]. This kind of phenomenologically viable models, known
+as Type III THDM [4], can be obtained assuming particular tex tures of
+the Yukawa matrices [3]. Another possibility is to make the s calars heavy
+enough to suppress low-energy FCNC effects. However, this ass umption
+leads to a phenomenologically non relevant THDM.
+A more elegant solution is to impose a discrete Z2symmetry to enforce
+that only one scalar doublet couples to a given right-handed fermion [5]; one
+of the two Yukawa matrices has to be zero. One requires the Lag rangian
+to remain invariant under the change φ1→φ1,φ2→ −φ2,QL→QL,
+LL→LL, and appropriate transformation properties for the right- handed
+fermions. Therearefournon-equivalent choices: theTypeI (onlyφ2couples
+to fermions) [6, 7, 8], Type II ( φ1couples to dandlandφ2tou) [7, 9, 8],
+Type X or leptophilic ( φ1couples to fermions and φ2to quarks) and Type
+Y (φ1couples to dandφ2touandl) models [8, 10, 11, 12, 13]. The Z2
+symmetry can be also implemented in the Higgs basis, forcing all fermions
+to couple to Φ 1; thisinertdoublet model is a natural frame for dark matter
+[14, 15, 16].
+2. Aligned THDM
+Amoregeneral waytoavoid tree-level FCNCsistorequirethe alignment
+in flavour space of the Yukawa coupling matrices [17]:
+Γ2=ξde−iθΓ1,∆2=ξ∗
+ueiθ∆1,Π2=ξle−iθΠ1, (5)
+withξfarbitrary complex numbers. The alignment guarantees that t he ma-
+tricesY′
+fandM′
+fare proportional and can be simultaneously diagonalized:
+Yd,l=ζd,lMd,l, Y u=ζuMu, ζ f≡ξf−tanβ
+1+ξftanβ,(6)
+where tan β≡v2/v1. In terms of mass-eigenstate fields, the Yukawa La-
+grangian takes then the form:
+LY=−√
+2
+vH+(x){¯u(x)[ζdVMdPR−ζuMuVPL]d(x)+ζl¯ν(x)MlPRl(x)}
+−1
+v/summationdisplay
+ϕ,fϕ0
+i(x)yϕ0
+i
+f¯f(x)MfPRf(x) + h.c. , (7)
+whereVis the CKM matrix, PR,L≡1±γ5
+2andyϕ0
+i
+fare neutral couplings of
+the physical scalar fields [17]. Therefore, within this appr oach:4
+Table 1. Choices of ξfassociated with Z2models and corresponding values of ζf.
+Model (ξd,ξu,ξl) ζdζuζl
+Type I (∞,∞,∞) cotβcotβcotβ
+Type II (0,∞,0) −tanβcotβ−tanβ
+Type X (∞,∞,0) cotβcotβ−tanβ
+Type Y (0,∞,∞)−tanβcotβcotβ
+Inert (tanβ,tanβ,tanβ)0 00
+– All scalar-fermion couplings are proportional to the corr esponding
+fermion masses.
+– The neutral Yukawas are diagonal in flavour.
+– The only source of flavour-changing interactions is the qua rk-mixing
+matrix V in the charged sector.
+– All leptonic couplings are diagonal in flavour, since there are no right-
+handed neutrinos in the theory.
+– There are only three new parameters ζf, which encode all possible
+freedom allowed by the alignment conditions. These couplin gs satisfy
+universality among the different generations, i.e. there is t he same
+universal coupling for all fermions with a given electric ch arge. They
+are also invariant under global SU(2) transformations of th e scalar
+fieldsφa→φ′
+a=Uabφb[18], i.e. they are scalar basis independent.
+– The usual models based on Z2symmetries are recovered taking the
+appropriated limits shown at Table 1.
+– Theζfcan be arbitrary complex numbers. Their phases introduce
+new sources of CP violation without tree-level FCNCs.
+Lepton-flavour-violating neutral couplings are identical ly zero to all or-
+ders in perturbation theory, because of the absence of right -handed neu-
+trinos. So, the usually adopted Z2symmetries are not necessary in the
+lepton sector, making the Type X and Y models less compelling . From a
+phenomenological point of view, the coupling ζlcould take any value.
+Quantum corrections could introduce some misalignment of t he quark
+Yukawa coupling matrices, generating small FCNCs suppress ed by the cor-
+responding loop factors. However, the flavour mixing induce d by loop cor-
+rections has a very characteristic structure [17] which res embles the popular
+Minimal Flavour Violation scenarios [19]. This restrictio n in the structure
+of the local FCNC terms is due to the following symmetry of the aligned
+THDM: the Lagrangian remains invariant under flavour-depen dent phase5
+transformations of thedifferent fermionmass eigenstates, fi(x)→eiαf
+ifi(x),
+provided the quark mixing matrix is transformed as Vij→eiαu
+iVije−αd
+j[17].
+The only possible FCNC terms are then of the type ¯ ui[V(Md)nV†]ijujand
+¯di[V†(Mu)mV]jdj(n,m >0), or similar structures with additional factors
+ofVandV†[17]. Structures of this type have been recently discussed i n
+Ref. [20].
+3. Phenomenology
+One of the most distinctive features of the THDM is the presen ce of a
+charged scalar H±. Assuming that Br( H+→c¯s) +Br(H+→τ+ντ) = 1,
+the combined LEP data on e+e−→H+H−constrain MH±>78.6 GeV
+(95% CL) [21]. The CDF [22] and D0 [23] collaborations have se arched for
+t→H+bdecays with negative results. CDF assumes Br( H+→c¯s) = 1
+(ζl<< ζu,d), while D0 adopts the opposite hypothesis Br( H+→τντ) = 1
+(ζl>> ζu,d). Both experiments find upper bounds on Br( t→H+b) around
+0.2 (95% CL) for charged scalar masses between 60 and 155 GeV. This
+implies|ζu|<0.3−2.5, where the exact number depends on MH±.
+Onecan also extract information from thesemileptonic deca ys of a pseu-
+doscalar meson P+→l+νl, which are sensitive to H+exchange due to
+the helicity suppression of the SM amplitude: Γ( P+
+ij→l+νl)/Γ(P+
+ij→
+l+νl)SM=|1−∆ij|2, where the new physics information is encoded in
+∆ij= (mP±
+ij/MH±)2ζ∗
+l(ζumui+ζdmdj)/(mui+mdj). The correction ∆ ij
+is in general complex and its real part can have either sign. T o determine
+its size one needs to know Vijand a theoretical determination of the me-
+son decay constant. From the present knowledge of B+→τ+ντ[24], we
+obtain|1−∆ub|= 1.32±0.20 which implies MH±//radicalBig
+|Re(ζ+
+lζd)|>5.7 GeV
+(90% CL). With fDs= 242±6 MeV [25], the recent CLEO measurement
+Br(D+
+s→τ+ντ) = (5.62±0.64)% [26] implies |1−∆cs|= 1.07±0.05 and
+MH±//radicalBig
+|Re(ζ+
+lζu)|>4.4 GeV (90% CL).
+FCNC processes such as b→sγorP0−¯P0mixing are also very sen-
+sitive to scalar contributions and give strong constraints on the parameter
+space (MH±,ζu,d). A detailed phenomenological analysis will be presented
+somewhere else.
+Acknowledgements
+This work has been supported in part by the EU Contract MRTN-C T-
+2006-035482 (FLAVIAnet), by MICINN, Spain [grants FPA2007 -60323 and
+Consolider-Ingenio 2010 CSD2007-00042 CPAN] and by Genera litat Valen-6
+ciana (PROMETEO/2008/069). P.T is indebted to the Spanish M ICINN
+for a FPU fellowship.
+REFERENCES
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+for Higgs Boson Searches, LHWG note 2001-05 [arXiv:hep-ex/0107 031].
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\ No newline at end of file
diff --git a/1001.0294.txt b/1001.0294.txt
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@@ -0,0 +1,443 @@
+arXiv:1001.0294v1  [astro-ph.IM]  2 Jan 20100
+UV and EUV Instruments
+January 2010
+to appear in:
+Landolt-B¨ ornstein, New Series VI/4A, Astronomy, Astroph ysics, and Cosmology; In-
+struments and Methods, ed. J.E. Tr¨ umper, Springer-Verlag , Berlin, 2010
+K. Werner
+Institut for Astronomy and Astrophysics
+Kepler Center for Astro and Particle Physics
+University of T¨ ubingen
+Sand 1, 72076 T¨ ubingen, Germany
+Landolt-B¨ ornstein
+New Series VI/4A1
+1.4 UV and EUV instruments
+We describe telescopes and instruments that were developed and u sed for astronomical research
+in the ultraviolet (UV) and extreme ultraviolet (EUV) regions of the e lectromagnetic spectrum.
+The wavelength ranges covered by these bands are not uniquely de fined. We use the following
+convention here: The EUV and UV span the regions λλ100–912˚A and 912–3000 ˚A, respectively.
+The limitation between both ranges is a natural choice, because the hydrogen Lyman absorption
+edge is located at λ912˚A. At smaller wavelengths, astronomical sources are strongly abs orbed
+by the interstellar medium. It also marks a technical limit, because te lescopes and instruments
+are of different design. In the EUV range, the technology is strong ly related to that utilized in
+X-ray astronomy, while in the UV range the instruments in many case s have their roots in optical
+astronomy. We will, therefore, describe the UV and EUV instrument s in appropriate conciseness
+and refer to the respective chapters of this volume for more tech nical details.
+Theλ100˚A limit of the EUV range to the soft X-ray range is fuzzy, because it is covered by
+both, specific X-ray and EUV instrumentation, and because there is no scientific motivation for
+a sharp distinction between these regions. As a consequence, the re will be some minor overlap
+with the X-ray chapter of this volume concerning the presentation of specific instruments. In
+contrast, the long-wavelength limit of the UV range to the optical r egion is an obvious choice. The
+λ >3000˚A region is accessible with ground-based telescopes, whereas inves tigation shortwards of
+this limit requires observations from above Earth’s atmosphere.
+All facts presented here are mainly taken from original papers ref erred to in Table 1. A number
+of reviews are useful sources, e.g.: [69Wil] and [72Ble] on UV astrono my, [71Car] on electronic
+imaging devices, [91Bob] on the cosmic far-UV ( λλ1000–2000 ˚A) background, and [00Bow] on
+EUV astronomy. Significant books on the state-of-the-art in the relatively concise field of EUV
+astronomy (in comparison to the UV) are two conference proceed ings [91Mal], [96Bow], and a very
+useful, exhaustive presentation by [03Bar].
+We restrict our description here to space-based instruments, i.e., to those facilities that were
+carried by satellites or by interplanetary and lunar probes. For con ciseness, we do not address
+the issue of sounding rocket experiments, although they have bee n essential for the development
+of UV and EUV instruments and obtaining early scientific results. Like wise, we will not describe
+stratospheric balloon experiments, which exploit the fact that Ear th’s atmosphere beyond a height
+of≈40km above sea level is sufficiently transparent in the λλ2000–3000 ˚A region to perform
+astronomical observations.
+For UV and EUV instrumentation used to observe the Sun, we refer to another chapter of this
+volume. Solar observatories (e.g. SOHO, STEREO) occasionally perf ormed stellar observations
+for calibration purposes [07Val]. We also disregard here UV/EUV instr uments that were used by
+space probes to observe exclusively solar-system objects (Moon , planets, interplanetary matter,
+etc.). But we will point out UV/EUV instruments of such probes that were also used to perform
+astronomical observations of Galactic and extragalactic objects .
+In the following, we describe EUV and UV instruments in two separate sections. In the sense
+just outlined, Table 1 represents a complete compilation of all EUV an d UV instruments that flew
+on space probes, in chronological order of their launch date. For d etails on the instrumentation
+references are made to Table 2.
+Most instruments were carried on Earth-orbiting satellites , the first few of them were unstabi-
+lized. Importantmeasurementswerecontributedfrominstrumen tsthat flewonmissionstoplanets.
+Duringinterplanetary cruise phases, astronomical observations were performed with instrum ents
+primarily devoted to the study of planetary atmospheres: Venera 2 & 3 to Venus, Mariner 9 to
+Mars (and in Mars orbit), and Voyager 1 & 2 to the outer solar syste m. Other instruments were
+operated on manned space vehicles: Gemini (during extravehicular stand-up activities by astro-
+nauts), Apollo (on the lunar surface and during translunar coast) , Soyuz, the MIR and Skylab
+space stations, and quite frequently on several Space Shuttle flig hts.
+Landolt-B¨ ornstein
+New Series VI/4A2
+1.4.1 EUV instruments
+An excellent summary about the current state of technology for E UV space-instrumentation is
+given in [03Bar].
+1.4.1.1 EUV detectors
+EUV detector developments were determined by two approaches ( photon counting devices):
+•Proportional counters (as used in X-ray astronomy), and
+•Photomultipliers (as used in UV range).
+The problem with proportional counters is, that detector windows should not absorb photons
+(this is amuch smallerproblemin the X-rayrange). Therefore, thinn er windowsorplastic windows
+are used. But significant gas losses occur, hence, “thin window, ga sflow” proportional counters
+were developed. But these are limited to λ <200˚A, therefore other approaches are more common
+in EUVastronomy: CEM (channel electronmultiplier) and MCP (microc hannelplates). The CEM
+is a compact, windowless version of a photomultiplier. Therefore, ph otons at all EUV wavelengths
+can be detected.
+Two-dimensional (2-d) position-sensitive detectors are required for observations with focussing
+instruments and for an efficient recording of spectra. Three tech niques have been advanced:
+•MCP detectors (e.g. used in Chandra/LETG, EUVE, ROSAT/WFC, EX OSAT/CMA, Ein-
+stein/HRI). Essentially, MCPs are arrays of single CEMs. They have no energy resolution.
+•IPCs (imaging proportional counters; e.g. ROSAT/PSPC). These a re position sensitive pro-
+portional counters (PSPC). They have rather modest energy re solution.
+•CCDs (charge coupled devices); solid-state semiconductor detec tors developed for X-ray in-
+struments (XMM-Newton). They are sensitive up to λ≈100˚A and have rather modest
+energy resolution in the EUV. They are different from “usual” CCDs ( i.e. those record-
+ing optical photons) which were often used in UV/EUV experiments in combination with
+UV–optical converters (so-called intensified CCDs, see below).
+In EUV experiments, MCPs are mostly used because of the limited wav elength range of pro-
+portional counters. Quantum efficiencies of MCPs are relatively low ( ≈5%). They can be in-
+creased significantly by depositing a photocathode on their front f ace. The materials are mostly
+alkali-halides (e.g. MgF 2, CsI, KBr). Different read-out systems for 2-d MCPs are used, e .g., a
+wedge-and-strip anode (WSA), the mostly used design (e.g. EUVE) . MCPs are solar blind.
+1.4.1.2 EUV telescopes
+The first orbiting EUV telescopes (after Apollo-Soyuz/EUVT) were grazing-incidence telescopes
+of Wolter types I and II (EUVE, ROSAT-WFC). Nested mirrors incre ase the collecting area (type
+I Wolter). Grazing-incidence is necessary for wavelengths below λ≈300˚A. However, the incidence
+angles can be rather large (10◦) compared to X-ray telescopes (1 −2◦), because the reflectivity
+is still quite high in EUV. Likewise, the tolerances on surface roughne ss are not so severe as for
+X-rays. Materials are usually glass or metal shells coated with gold.
+At wavelengths larger than λ≈500˚A normal-incidence telescopes with relatively high reflec-
+tivity by SiC are possible. Aluminum coated with LiF or MgF 2is preferred above λλ1050˚A and
+1150˚A, respectively (e.g. ORFEUS, HUT, HST, FUSE).
+Landolt-B¨ ornstein
+New Series VI/4A3
+More recently, normal-incidence telescopes are used even at shor t wavelengths, i.e., λ <300˚A.
+Theyuseinterferencetechniquesforreflectionwithmultilayercoa tingsofopticalsurfaces(ALEXIS,
+6 different telescopes with large collecting area). The price to pay is a narrow bandpass. Its width
+is about 10% of the peak wavelength.
+Simple mechanicalcollimatorsinstead ofimaging telescopeswere emplo yedin a number ofEUV
+missions that particularly aimed at measuring the Galactic EUV backgr ound radiation and diffuse
+emission from the interstellar medium (e.g. CHIPS, EURD), but also fo r stellar observations (e.g.
+Voyager/UVS).
+1.4.1.3 EUV photometry and spectroscopy
+Thinfilmfiltersandopticalcoatingmaterialsdefinebandpassesforp hotometryandblockunwanted
+background radiation. There are two categories of filters:
+•Plastic films; material: hydrocarbon polymers (polypropylene, pary lene, Lexan), and
+•Metal foils (often used with mechanical support grids).
+Spectroscopic instruments in the EUV utilize reflection (EUVE, CHIP S) and transmissiongrat-
+ings (Chandra/LETG, EXOSAT/TGS). Filters can suppress higher- order spectra.
+1.4.1.4 Notes on specific EUV missions
+For a long time, it was thought that observations in the EUV band wou ld be impossible, because
+of the presumed high opacity of the interstellar medium [59All]. Encour aged by new evidence on
+the average density of neutral hydrogen and probable inhomogen eities, a first successful search for
+EUV sourceswas performed in 1975with an instrument on the Apollo- SoyuzTest Project [77Bow].
+Further detections were made in the soft-energy range of X-ray observatories(Einstein, EXOSAT).
+Important milestones were the photometric all-sky surveys condu cted with the wide-field camera
+(WFC) aboard ROSAT and with the EUVE satellite. For the first (and h itherto the only) time,
+EUVE offered to a wide astronomical community access to a spectro scopic instrument covering
+the entire EUV range. ALEXIS is remarkable and probably pathbrea king, because it utilized for
+the first time a normal-incidence telescope in the far-EUV range.
+A few UV instruments were also capable of performing near-EUV spe ctroscopy, e.g., HUT,
+ORFEUS/BEFS, and the UVS instruments on the Voyager probes. A t the time of writing (2009),
+the UVS on Voyager 1 is still operating, more than 30 years after lau nch. It stares at a point in
+anti-sun direction and returns spectra of the interplanetary med ium. These are used to study the
+Lyαemission from interstellar gas penetrating the solar system [09Hol]. C urrently, there is no
+EUV instrument in operation, and there are no accepted proposals for future EUV missions. The
+XMM-Newton and Chandra X-ray observatories provide access to the very-far EUV ( λ <150˚A),
+only.
+Landolt-B¨ ornstein
+New Series VI/4A4
+1.4.2 UV instruments
+AconciseoverviewaboutthecurrentstateoftechnologyforUVs pace-instrumentationandpossible
+future developments is given in a special section of [99Mor].
+1.4.2.1 UV detectors
+In the first instruments, and up to the mid-1980s, photomultipliers were used for photometric
+work. Before the availability of 2-d detectors, spectra were scan ned with photomultipliers, too
+(e.g. OAO-2, Copernicus, Apollo 17, Astron).
+In some instruments (OAO-2, Copernicus, IUE) television (TV) cam era tubes were used as
+2-d detectors. The secondary electron conduction (SEC) vidicon detectors aboard IUE integrated
+spectrographic images on a potassium chloride SEC target and were then read out by scanning
+the SEC target with an electron beam. The photocathode of the TV camera tubes, designed
+for visible light response, required a converter (with a CsTe photoc athode) to transform UV into
+visible radiation.
+Image recording on photographic film was an important detection te chnique for a long time.
+Two primary methods were used: one is electronography, in which hig h-energy electrons from
+the photocathode cause the blackening of the grains directly; the other makes use of a phosphor
+screen to convert the electron energy into visible light, which is then recorded on ordinary, light-
+sensitive photographic film. In the latter method the electronic imag ing device serves mainly to
+intensify the image, so as to increase the speed of the ordinary pho tographic process, and, in some
+cases, to extend its spectral range into wavelengthranges to wh ich the film is not directly sensitive.
+Hence, these latter type devicesarecommonlycalled imageintensifie rsorimageconverters[71Car].
+Film detectors were exclusively used in short-term, American and So viet manned space missions,
+commencing with Gemini flights in 1966, and ending with several instru ments on Space Shuttles,
+up to 1995 (UIT, FUVIS).
+Today, MCPs as well as CCDs (the latter at wavelengths longer than λ≈1500˚A) are utilized.
+The UV response of CCDs is sometimes enhanced by addition of a phos phor, such as Lumogen
+on the HST/WFPC-2 CCDs. CCDs with improved UV response are unde r active development.
+Intensified CCDs (ICCD) are image intensifier tubes or MCPs in front of optically sensitive CCDs,
+e.g., MCP-intensified CCDs in the Optical Monitor (OM) of XMM-Newton . The first use of an
+MCP detector was on the Voyager UV-Spectrographs; they serv ed as image intensifiers for a newly
+developed self-scanned linear anode array (so-called Ssanacon) [7 7Bro].
+Another detector type are intensified photodiode arrays. One re alisation was an MCP with a
+phosphor screen anode put in front of a Reticon, that is, a visual-p hoton recording photodiode
+array (HUT). Digicons were used in the GHRS and FOS spectrograph s aboard HST. They consist
+ofaphotocathode, magneticandelectrostaticfocussingcoilsand electrodes, andadiodearray. The
+photoelectronsareaccelerateddirectlyontothediodeswithnoco nversionintovisible-lightphotons.
+Digiconsgavewayafterthefirst-generationofHST instrumentst oeithermulti-anodemicrochannel
+arrays (MAMAs, HST/STIS instrument) which achieve small pixels wit h high photon-count rates,
+or electron-bombarded CCDs.
+1.4.2.2 UV telescopes
+Normal-incidence mirrors are used. As for the coatings of mirrors ( and optical elements), different
+materials are necessary for different wavelength bands in order to achieve high performance. For
+example, the HST mirror coating (Al+MgF 2) is “blind” for λ <1150˚A, hence, other materials
+are needed for high reflectivity when going to the far-UV (towards the hydrogen Lyman edge).
+Landolt-B¨ ornstein
+New Series VI/4A5
+For example, the FUSE instrument consists of four coaligned telesc opes and spectrographs. Two
+channels with SiC coatings coverthe range λλ905–1100 ˚A, and two channels with Al+LiF coatings
+cover the range λλ1000–1195 ˚A (04Moo).
+In earlier instruments refractive optics cameras were employed fo r imaging (Gemini), and later
+on Schmidt cameras. Open collimators, i.e., non-imaging instruments, were also used for UV-
+background studies since the earliest mission (Kosmos 51) until mor e recently (EURD). Also,
+point source detection and spectroscopy with non-imaging instrum ents were performed (Apollo 17,
+Voyager 1 & 2, IMAPS).
+1.4.2.3 UV photometry, spectroscopy, and polarimetry
+In analogy to instruments working in the optical wavelength band, fi lters, coating materials and
+detector efficiency characteristics define bandpasses for photo metry and block unwanted back-
+ground radiation. Likewise, the principles of employed spectroscop ic (prisms, grisms, gratings)
+and polarimetric devices are in principle the same as in optical astrono my.
+1.4.2.4 Notes on specific UV missions
+The early-phase history of UV space instruments culminated in 1968 with the launch of OAO-2,
+the first observatory-type mission. A similar success was OAO-3 (C opernicus), which obtained
+first high-resolution UV spectra. TD-1A and ANS were important UV -survey instruments. IUE
+was an outstanding success. For almost two decades it was operat ed as a true observatory with
+real-timeaccessby guestobserversat groundstations, and deliv eredmorethan 100,000UV spectra
+of all kinds of sources. FUSE was seminal because it took high-reso lution far-UV spectra ( λλ912–
+1180˚A) for many years. GALEX made a deep UV all-sky survey.
+The Hubble Space Telescope (HST) carries a diverse suite of imaging a nd spectroscopic in-
+struments working in a broad wavelength range spanning from the U V to the infrared. It is the
+only orbiting space telescope that was frequently maintained by Spa ce Shuttle crews repairing
+instruments and installing new ones. In 2004, HST lost its UV spectro scopic capability when the
+STIS spectrograph failed. Hence, at the time of writing (2009), no UV spectroscopic instrument
+is available at all (except for the small UV/optical monitors aboard t he XMM-Newton and Swift
+X-ray observatories). For 2009, the last of HST servicing missions is planned, during which STIS
+shall be repaired and a new UV spectrograph (COS) shall be installed . If successful, then HST
+will again provide access to the UV for another ≈5 years.
+There are no accepted proposals for spectroscopic UV missions be yond HST, except for a
+Russian-led multi-national initiative, the World Space Observatory U ltraviolet (WSO/UV). It
+shall perform imaging and spectroscopy in the λλ1020–3200 ˚A band (resolving power R=1000 and
+50000) and is planned for launch in 2014. Two Indian-led multiple-band UV-imaging missions
+(Astrosat/UVIT and GSat4/TAUVEX) are scheduled for launch in 2 009.
+Landolt-B¨ ornstein
+New Series VI/4A6
+Table 1. List of space-based UV and EUV instruments. Abbreviations: ph.=photometry, im.=imaging, sp.=spectroscopy. The numb ers in the column
+“Instr.” refer to the description of telescopes, instrumen ts, and detectors in Table2. ∆ λand R denote spectral resolution and resolving power, respe ctively.
+Mission/Instrument Country Period Instr. Wavelength Remarks Ref.
+or Agency Range [ ˚A]
+Kosmos 51 USSR 1964 1,7 2300–7500 2 bands, UV background 70Di1,7 2Dim
+1964 83C USA 1964 2,7 1300–1650 1 band, stars 67Smi
+Venera 2, 3 USSR 1965 1,4 1050–1340 2 bands, UV background 67Ku r,68Kur
+Gemini 10–12/S-013 USA 1966 3,6,16 2500–4400 ∆ λ=7−20˚A, stars 70Kon,74Spe
+Kosmos 213 USSR 1968 1,7 2000–6500 2 bands, UV background 70Di1 ,72Dim
+Kosmos 215 USSR 1968 1,7 1250–2800 several telescopes/bands, stars 70Di2,76Dim
+OAO-2 USA 1968-73 3,8,16 1000–4250 multiband im., sp.: ∆ λ=12 and 22 ˚A 72Cod
+Mariner 9/UVS USA 1971 2,7,17 1150–3400 ∆ λ=7.5 and 15 ˚A 71Pea,75Mol
+Salyut1/Orion1 USSR 1971 3,6,16 2000–3800 R=500 72Gur
+STP 72-1 USA 1972 1,10 912–1050 1 band, UV background 84OpaApollo16/S-201 USA 1972 3,6,14 1050–1600 2 bands, operated on lun ar surface 73Car,83Car
+Apollo17/S-169 USA 1972 1,7,17 1180–1680 ∆ λ=10˚A 73Fas,75Hry
+Copernicus (OAO-3) USA 1972-81 3,7,17 912–3275 ∆ λ=0.05−0.4˚A 73Rog
+TD-1A ESRO 1972-74 3,7,17 1350–2550 S2/68 instr. sp. ∆ λ=35−40˚A, ph. 2750 ±310˚A 73Bok
+3,7,17 2000–2900 S59 instr. ∆ λ=1.7˚A 74Jag
+Soyuz13/Orion2 USSR 1973 3,6,15 2000–4000 ∆ λ=8-28˚A 76Gur
+Skylab 2-4 USA 1973-74 3,6,15 1300–5000 S-019: ∆ λ=2−42˚A 75Hen,77Oca
+3,6,18 2200–3000 S-183: one band, French instrument 77Lag,88Vu i
+3,6,14 1100–1500 S-201: one band 74Car
+ANS NL 1974-76 3,7,17 1500–3250 multiple bands 75Dui,82WesD2B-Aura/ELZ F 1975 3,7,16 1100–3300 multiple bands, UV backgrou nd 78Mau
+Apollo-Soyuz/EUVT USA,USSR 1975 3,10 40–1550 multiple bands, 1st E UV source detections 77Bow,80Par
+Solrad-11B USA 1976 1,7 1220–1500 1 band, UV background 83WelPrognoz-6/Galaktika USSR 1977-78 1,7,17 1100–1850 ∆ λ=100˚A, UV background 82Zve
+Voyager 1+2/UVS USA 1977- 1,12,16 500–1700 ∆ λ=18˚A, 1st EUV star spectra 77Bro
+Einstein/HRI,OGS USA 1978-81 3,9,18 3–120 HRI: multiband im., with OGS : sp. R=10 −50 79Gia
+IUE Intl. 1978-96 3,8,17 1150–3300 USA, ESA, UK; ∆ λ=0.1 and 6 ˚A 78Bog,97Har
+Spacelab1/VWFC USA 1983 3,6,14 1250–3000 Shuttle flight, 3 bands, French instrument 94Tob
+Astron USSR 1983-89 3,7,17 1100–3500 ∆ λ=0.4 and 28 ˚A 84Boy,86Boy
+Landolt-B¨ ornstein
+New Series VI/4A7Table 1. (continued)
+Mission/Instrument Country Period Instr. Wavelength Remarks Ref.
+or Agency Range [ ˚A]
+EXOSAT/CMA,TGS ESA 1983-86 3,9,18 6–400 CMA: multib.im., with TGS: sp . ∆λ=1−4˚A 81Tay,91Whi
+Dynamics Explorer 1 USA 1984-87 1,7 1350–2000 1 band, UV backgro und 89Fix
+UVX USA 1986 3,9,17 1200–3100 Shuttle flight, ∆ λ=15−27˚A, UV background 89Mur,91Hur
+Mir/GLAZAR USSR 1987-90 3,6,14 1500–1800 1 band 88Tov,09Tov
+HUT (Astro1+2) USA 1990,95 3,11,14,17 415–1850 2 Shuttle flights, ∆ λ=1.5−3˚A 92Dav,95Kru
+UIT (Astro1+2) USA 1990,95 3,6,14,18 1200–3400 2 Shuttle flights, m ultiband im., sp.: ∆ λ=19˚A 97Ste
+WUPPE (Astro1+2) USA 1990,95 3,11,14,17,20 1350–3300 2 Shuttle flig hts, ∆λ=8˚A 94Nor
+ROSAT/WFC,PSPC D,UK 1990-99 3,5,9 5–600 WFCmultib.im.60 −600˚A;PSPC5 −120˚A,R≈1 03Bar,08Pfe
+HST USA,ESA 1990- see text 1150–IR several instruments, see te xt 09Hst
+FUVCam USA 1991 3,6,14 1220–2000 Shuttle flight, 2 bands 95SchFAUST USA 1992 3,9 1400–1800 Shuttle flight, 1 band 93BowDUVE USA 1992 1,9,17 950–1080 ∆ λ=3˚A, diffuse emission from hot ISM 98Kor
+EUVE USA 1992-01 2,9,17 60–800 multiband im., sp.: R=200 91BoaALEXIS USA 1993-05 3,9 130–186 3 narrow bands, normal incidence optics 96Blo
+ORFEUS D,USA 1993,96 3,9,17 380–1400 2 Shuttle flights, BEFS: 380 −1220˚A, R=4600 96Hur
+TUES: 912-1400 ˚A, R=11000 99Bar
+IMAPS USA 1993,96 1,13,14,16 950–1150 2 Shuttle flights with ORFEUS, R=120000 96Jen
+FUVIS USA 1995 3,6,14,17? 970–2000 Shuttle flight, ∆ λ=5 and 30 ˚A, diffuse sources 90Car
+UVSTAR Italy,USA 1995-98 3,13,14,17 535–1250 3 Shuttle flights (199 5,97,98), ∆ λ=1−12˚A 93Sta,02Gre
+MSX/UVISI USA 1996-97 3,13,14 1240–8270 multiple bands, primarily m ilitary mission 01Mur,04New
+Minisat-01/EURD Spain,USA 1997-01 1,9,16 350–1100 ∆ λ=5˚A, spectral im. 97Bow,01Mor
+FUSE USA 1999-07 3,9,17 912–1180 R=20000 00Moo,00Sah
+Chandra/LETG USA 1999- 3,9,13,18 6–150 R=100 −1000 02Wei
+XMM/OM,pn-CCD ESA 1999- 3,13,14,19 1700–6000 OM: multiband im., sp. R=250 01Mas
+3,13 1–100 pn-CCD: R=1 01Str
+CHIPS USA 2003-08 1,9,17 90–260 R=50 −150, EUV background 03Hur,05Hur
+GALEX USA 2003- 3,9,19 1350–2800 2 bands im., grism sp.: R=90 −200 05Mar,05Mor
+SPEAR (=FIMS) Korea,USA 2003-05 2,9,17 900–1750 R=550, spectr al im. 06Ede
+Swift/UVOT USA 2004- 3,13,14,19 1700–6500 multiband im., sp.: R=150 0 4Geh
+Landolt-B¨ ornstein
+New Series VI/4A8
+Table 2. Instrumentation reference for UV and EUV space missions lis ted in Table 1.
+# Description
+Telescope type 1 mechanical collimator
+2 collecting mirror
+3 imaging mirror
+Detector 4 Geiger counter
+5 gas proportional counter
+6 photographic film
+7 photomultiplier
+8 television tube (including UV–visible photon converter)
+9 microchannel plate (MCP)
+10 channel electron multiplier (CEM)
+11 photodiode detector (Reticon, Digicon)
+12 MCP-intensified self-scanned linear anode array (Ssanacon)
+13 charge-coupled device (CCD)
+14 image intensifier (e.g. MCP); acts as UV–visible photon converter
+in front of optical photon detectors (film, CCDs, photodiodes)
+Dispersive element 15 objective prism
+16 objective grating
+17 reflection grating
+18 transmission grating
+19 grism
+Polarimetry 20 polarizing beamsplitter
+1.4.3 References for 1.4
+59All Aller, L.H.: Publ. Astron. Soc. Pacific 71(1959) 324.
+67SmiSmith, A.M.: Astrophys. J. 147(1967) 158.
+67KurKurt, V.G., Sunyaev, R.A.: Soviet J. Exp. Theoret. Ph. Lette rs5(1967) 246.
+68KurKurt, V.G., Sunyaev, R.A.: Soviet Astron. 11(1968) 928.
+69Wil Wilson, R., Boksenberg, A.: Annu. Rev. Astron. Astrophys. 7(1969) 421.
+70Di1 Dimov,N.A., Severny,A.B., Zvereva,A.M.: inUltravioletStellarSpe ctraandGround-Based
+Observations, eds. L. Houziaux, Butler, H.-E., IAU Symp. 36(1970), 325.
+70Di2 Dimov, N.A.: in Ultraviolet Stellar Spectra and Ground-Based Obs ervations, eds. L. Houzi-
+aux, Butler, H.-E., IAU Symp. 36(1970), 138.
+70KonKondo, Y., Henize, K.G., Kotila, C.L.: Astrophys. J. 159(1970) 927.
+71CarCarruthers, G.R.: Astrophys. Sp. Science 14(1971) 332.
+71PeaPearce, J.B., Gause, K.A., Mackey, E.F., Kelly, K.K., Fastie, W.G., B arth, C.A.: Appl.
+Optics10(1971) 805.
+72Ble Bless, R.C., Code, A.D.: Annu. Rev. Astron. Astrophys. 10(1972) 197.
+72CodCode, A.D. (ed.).: The Scientific Results from the Orbiting Astr onomical Observatory
+(OAO-2), NASA SP-310 (1972).
+72DimDimov, N.A., Zvereva, A.M., Severny, A.B.: Izv. Krymskoj Astro fiz. Obs. 45(1972) 67.
+72GurGurzadyan, G.A., Ohanesyan, J.B.: Space Sci. Rev. 13(1972) 647.
+73BokBoksenberg, A., Evans, R.G., Fowler, R.G., et al.: Mon. Not. R. A str. Soc. 163(1973)
+291.
+73CarCarruthers, G.R.: Appl. Opt. 12(1973) 2501.
+73Fas Fastie, W.G.: The Moon 7(1973) 49.
+Landolt-B¨ ornstein
+New Series VI/4A9
+73RogRogerson, J.B., Spitzer, L., Drake, J.F., et al.: Astrophys. J. Lett.181(1973) L97.
+74CarCarruthers, G.R., Opal, C.B., Page, T.L., Meier, R.R., Prinz, D.K.: I carus23(1974) 526.
+74Jag de Jager, C., Hoekstra, R., van der Hucht, K.A.: Astrophys. Sp. Science 26(1974) 207.
+74Spe Spear, G.G., Kondo, Y., Henize, K.G.: Astrophys. J. 192(1974) 615.
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+arXiv:1001.0295v1  [quant-ph]  2 Jan 2010Distinguishing quantum channels via magic squares game
+M. Ramzan and M. K. Khan
+Department of Physics Quaid-i-Azam University
+Islamabad 45320, Pakistan
+(Dated: October 29, 2018)
+We study the effect of quantum memory in magic squares game when p layed in quantum
+domain. We consider different noisy quantum channels and analyze th eir influence on the
+magic squares quantum pseudo-telepathy game. We show that the probability of success can
+be used to distinguish the quantum channels. It is seen that the mea n success probability
+decreases with increase of quantum noise. Where as the mean succ ess probability increases
+with increase of quantum memory. It is also seen that the behaviour of amplitude damping
+and phase damping channels is similar. On the other hand, the behavio ur of depolarizing
+channel is similar to the flipping channels. Therefore, the probability of success of the game
+can be used to distinguish the quantum channels.
+Keywords: Quantum magic squares game; quantum memory; succ ess probability
+I. INTRODUCTION
+Applications of game theory [1] can be found in several resea rch areas, such as economics,
+biology, physics and computer sciences. In the recent past, rapid interest has been developed in
+the discipline of quantum information [2] that has led to the creation of quantum game theory
+[3]. During last few years, number of authors have contribut ed to the development of quantum
+game theory [4-9]. In this direction, much work has been done on quantum prisoners’ dilemma
+game [10-12] and other games have been converted to the quant um realm including the battle of
+the sexes [8, 13], the Monty Hall problem [14, 15], the rock-s cissors-paper [16], and the others
+[17-20]. Quantum pseudo-telepathy game [21] is something w hich can not be won in the classical
+world without communication but can be won in the quantum wor ld using entangled state without
+any use of classical communication. Brassard et al. [21] hav e shown that how to win the magic2
+square game for n= 3 with certainty by sharing a two qubit entanglement betwee n Alice and Bob.
+Gawron et al. [19] have studied noise effects in quantum magic s quares game. They have shown
+that the probability of success can be used to determine the c haracteristics of quantum channels.
+Recently, Sousa et al. [22] have proposed a quantum game whic h can be used to control the access
+of processes to the CPU in a quantum computer. More recently, James et al. [23] have analyzed
+the quantum penny flip game using geometric algebra.
+Decoherence is an integral part of the theory of quantum comp utation and communication.
+A major problem of quantum communication is to faithfully tr ansmit unknown quantum states
+through a noisy quantum channel. When quantum information i s sent through a channel (optical
+fiber), the carriers of the information interact with the cha nnel and get entangled with its many
+degrees of freedom. This gives rise to the phenomenon of deco herence on the state space of the
+information carriers. To deal with the problem of decoheren ce, two methods have been devel-
+oped, known as quantum error correction [24] and entangleme nt purification [25]. When quantum
+information is processed in the real-world, the decoherenc e caused by the external environment
+is inevitable. Implementation of decoherence in quantum ga mes have been studied by different
+authors [6, 26]. Recently, interest has been developed in im plementing quantum memory in the
+field of quantum game theory [8].
+Quantum channels are implemented by suitable quantum devic es consisting of intrinsic degrees
+of freedom associated with the environment and acting on the system via particular interactions
+between the system and the environment. The assumption that noise is uncorrelated between
+successive uses of a channel is not realistic. Hence memory e ffects need to be taken into account.
+Quantum channels with memory [27-29] are the natural theore tical framework for the study of
+any noisy quantum communication system where correlation t ime is longer than the time between
+consecutive uses of the channel. A more general model of a qua ntum channel with memory was
+introduced by Bowen and Mancini [30] and also studied by Kret schmann and Werner [31].
+In this paper, we study the quantum magic squares game influen ced by different memory chan-
+nels, suchasamplitudedamping, depolarizing, phasedampi ng, bit-flip, phase-flipandbit-phase-flip
+channels, parameterized by a quantum noiseparameter αand memory parameter µ. Hereα∈[0,1]
+andµ∈[0,1] represent the lower and upper limits of quantum noise para meter and memory pa-
+rameter respectively. It is seen that the behaviour of the da mping (amplitude and phase damping)
+channels is different as compared tothedepolarizingand flipp ingchannels. Therefore, thequantum
+channels can easily be distinguished using the success prob ability of the game.3
+II. THE MAGIC SQUARES GAME
+The magic squares game is a two-player game presented by Arav ind [32] which was primarily
+built on the work done by Mermin [33]. The magic square is a 3 ×3 matrix filled with numbers
+0 or 1,such that the sum of entries in each row is even and the sum of en tries in each column is
+odd. However, it is impossible to have such a matrix with all t he rows having even parity and all
+the columns having odd parity. Therefore, there can be no cla ssical strategy that always wins. In
+the magic squares game, there are two players, say, Alice and Bob. Alice is given the number of
+the row and Bob is given the number of the column. Alice gives t he entries for a row and Bob
+gives entries for a column. The winning condition is that the parity of the row must be even, the
+parity of the column must be odd, and the intersection of the g iven row and column must be same.
+During the game, Alice and Bob are not allowed to communicate with each other. There exists
+a (classical) strategy that leads to winning probability of 8/9. If parties are allowed to share a
+quantum state they can achieve probability 1. An interestin g feature of this game is that it does
+not require a promise.
+In the quantum version of this game [21], Alice and Bob share a n entangled quantum state of
+the form
+|Ψin/an}bracketri}ht=1
+2(|0011/an}bracketri}ht −|1100/an}bracketri}ht−|0110/an}bracketri}ht+|1001/an}bracketri}ht) (1)
+In equation (1) the first two qubits correspond to Alice, wher eas the last two qubits correspond
+to Bob. Depending upon their inputs (the specific row and colu mn to be filled in) Alice and Bob4
+apply unitary operators Ai⊗IandI⊗Bj, respectively,
+A1=1√
+2
+i0 0 1
+0−i1 0
+0i1 0
+1 0 0 i
+, B 1=1
+2
+i−i1 1
+−i−i1−1
+1 1−i i
+−i i1 1
+
+A2=1
+2
+i1 1i
+−i1−1i
+i1−1−i
+−i1 1−i
+, B 2=1
+2
+−1i1i
+1i1−i
+1−i1i
+−1−i1−i
+
+A3=1
+2
+−1−1−1 1
+1 1−1 1
+1−1 1 1
+1−1−1−1
+, B 3=1√
+2
+1 0 0 1
+−1 0 0 1
+0 1 1 0
+0 1−1 0
+(2)
+whereiandjdenote the corresponding input. Finally, Alice and Bob meas ure their qubits in the
+computational basis. Further steps to calculate the succes s probability of the game are given in
+the next section.
+A. Quantum memory channels
+A major hurdle in the path of efficient information transmissi on is the presence of noise, in both
+classical and quantum channels. This noise causes a distort ion of the information sent through
+the channel. Error correcting codes are used to overcome thi s problem. Messages are encoded
+into code-words, which are then sent through the channel. In formation transmission is said to be
+reliable if the probability of error, in decoding the output of the channel, vanishes asymptotically
+in the number of uses of the channel. A basic question of infor mation theory is whether there
+is any advantage in using entangled states as input states. T hat is, whether or not encoding the
+classical data into entangled rather than separable states increases the mutual information. For the
+case when multiple uses of the channel are not correlated, th ere is no advantage in using entangled
+states. Correlated noise, also referred as memory in the lit erature, acts on consecutive uses of
+the channels. However, in general, one may want to encode cla ssical data into entangled strings
+or consecutive uses of the channel may be correlated to each o ther. Hence, we are dealing with
+a strongly correlated quantum system, the correlation of wh ich results from the memory of the5
+channel itself.
+In ref. [26] a Pauli channel with partial memory was studied. The action of the channel on two
+consecutive qubits is given by the Kraus operators
+Aij=/radicalBig
+αi[(1−µ)αj+µδij]σi⊗σj (3)
+whereσi(σj) are usual Pauli matrices, αi(αj) represent the quantum noise and indices iandj
+runs from 0 to 3 .The above expression means that with probability µthe channel acts on the
+second qubit with the same error operator as on the first qubit , and with probability (1 −µ) it acts
+on the second qubit independently. Physically the paramete rµis determined by the relaxation
+time of the channel when a qubit passes through it. In order to remove correlations, one can wait
+until the channel has relaxed to its original state before se nding the next qubit, however this lowers
+the rate of information transfer. Thus it is necessary to con sider the performance of the channel for
+arbitrary values of µto reach a compromise between various factors which determi ne the final rate
+of information transfer. Thus in passing through the channe l any two consecutive qubits undergo
+random independent (uncorrelated) errors with probabilit y (1−µ) and identical (correlated) errors
+with probability µ. Thisshouldbethecase if thechannel hasamemory depending onits relaxation
+time and if we stream the qubits through it. The action of the P auli channels on n-qubits can be
+generalized in Kraus operator form as given below
+Ai1.....in=/radicaltp/radicalvertex/radicalvertex/radicalbtαinn−1/productdisplay
+m=1[(1−µ)αim+µδim,im+1]σi1⊗.....⊗σin (4)
+As stated above, with probability (1 −µ) the noise is uncorrelated and can be completely specified
+by the Kraus operators
+Du
+ij=√αiαjσi⊗σj, (5)
+and with probability µthe noise is correlated (i.e. the channel has memory) which c an be specified
+by the Kraus operators
+Dc
+kk=√αkσk⊗σk, (6)
+A detailed list of single qubit Kraus operators for different q uantum channels with uncorrelated
+noise is given in table 1. The action of such a channel if nqubits are streamed through it, can be
+described in operator sum representation as [2]
+ρf=n−1/summationdisplay
+k1,....,.kn=0(Akn⊗.....Ak1)ρin(A†
+k1⊗.....A†
+kn) (7)6
+whereρinrepresents the initial density matrix for quantum state and Aknare the Kraus operators
+expressed in equation (4). The Kraus operators satisfy the c ompleteness relation
+n−1/summationdisplay
+kn=0A†
+knAkn= 1 (8)
+However, the Kraus operators for a quantum amplitude dampin g channel with correlated noise are
+given by Yeo and Skeen [27] as given as
+Ac
+00=
+cosχ0 0 0
+0 1 0 0
+0 0 1 0
+0 0 0 1
+, Ac
+11=
+0 0 0 0
+0 0 0 0
+0 0 0 0
+sinχ0 0 0
+(9)
+where, 0 ≤χ≤π/2 and is related to the quantum noise parameter as
+sinχ=√α (10)
+It is clear that Ac
+00cannot be written as a tensor product of two two-by-two matri ces. This gives
+rise to the typical spooky action of the channel: |01/an}bracketri}htand|10/an}bracketri}ht, and any linear combination of
+them, and |11/an}bracketri}htwill go through the channel undisturbed, but not |00/an}bracketri}ht.The action of this non-unital
+channel is given by
+π→ρ= Φ(π) = (1−µ)1/summationdisplay
+i,j=0Au
+ijπAu†
+ij+µ1/summationdisplay
+k=0Ac
+kkπAc†
+kk(11)
+The action of the super-operators provides a way of describi ng the evolution of quantum states
+in a noisy environment. In our scheme, the Kraus operators ar e of the dimension 23. They
+are constructed from single qubit Kraus operators by taking their tensor product over all n3
+combinations
+Ak=⊗
+knAkn (12)
+wherenis the number of Kraus operator for a single qubit channel. Th e final state of the game
+after the action of the channel can be obtained as
+ρf= Φα,µ(|Ψ/an}bracketri}ht/an}bracketle{tΨ|) (13)
+where Φ α,µis the super-operator realizing the quantum channel parame trized by real numbers α
+andµ. After the action of the players unitary operations, the gam e’s final state transforms to
+ρ´f= (Ai⊗Bj)(Φα,µ(|Ψ/an}bracketri}ht/an}bracketle{tΨ|)(A†
+i⊗Bj†) (14)7
+The probability of success Pi,j(α,µ) can be computed as the probability of measuring ρ´fin the
+state indicating success
+Pi,j(α,µ) = Tr(ρ´f/summationdisplay
+m|ξm/an}bracketri}ht/an}bracketle{tξm|) (15)
+where|ξm/an}bracketri}htare the states that imply success and Tr represents the trace of the matrix. The mean
+probability of success of the game is calculated from
+¯Pi,j(α,µ) =/summationdisplay
+i,j∈{1,2,3}Pi,j(α,µ). (16)
+It can be easily checked that the results of ref. [19] can be re produced if we put µ= 0 in tables 2
+and 3 for success probability and mean success probability r espectively.
+III. RESULTS AND DISCUSSIONS
+We compute the success probability Pi,j(α,µ) for different inputs ( i,j∈ {1,2,3}) in the magic
+squaresgamefordifferentquantumchannelssuchasdepolariz ing, amplitudedamping,phasedamp-
+ingandflippingchannels. Itis seen that themeanprobabilit y of success, ¯Pi,j(α,µ),heavily depends
+on the quantum noise parameter αand memory parameter µ(see table 3). The game results for
+each combination of operators Ai,Bjfor depolarizing, amplitude damping, phase damping, bit
+flip, phase flip and bit-phase flip channels are listed in table 2. In table 3, we present the results
+for the mean success probability ¯Pi,j(α,µ). It is seen that in the case of depolarizing channel,
+the success probability as the function of quantum noise par ameterαand memory parameter µ
+is the same for all the possible inputs. For amplitude dampin g and phase damping channels, one
+can observe three different types of functions. These functio ns are increasing for these channels
+under the effect of memory. The depolarizing function reaches its minimum for α=µ= 1/2 and
+is symmetrical. It is seen that in case of input (1 ,3) the phase-flip channel does not influence the
+probability of success. The same is true for input (2 ,3) for bit flip channel and also for input (3 ,3)
+for bit-phase flip channel. Hence, it is possible to distingu ish these channels by looking at the
+success probability of magic-squares game.
+In figures 1 and 2, we plot mean success probability ¯P(α,µ) as a function of quantum noise
+parameter αfor memory parameter µ= 0.5 andµ= 1 respectively for amplitude damping,
+depolarizing, phase damping and flipping channels. It is see n that the amplitude damping, phase
+damping channels cause monotonic decrease of mean success p robability as a function of quantum
+noise parameter for µ <1 (see figure 1). On the other hand, depolarizing and flippingc hannels give8
+symmetrical function. However, for µ= 1,the amplitude damping and phase damping channels
+causemonotonic decrease of mean success probability. Wher eas incase of depolarizingandflipping
+channels, thesuccess functionsattain their maximumproba bility ofsuccess (seefigure2). Infigures
+3 and 4, we plot mean success probability ¯P(α,µ) as a function of memory parameter µforα= 0.5
+andα= 1 respectively, for amplitude damping, depolarizing, pha se damping and flipping channels.
+It is seen that the mean success probability increases linea rly as a function of memory parameter
+µ,forα <1 (see figure 3). However, for depolarizing and flipping chann els it reaches its maximum
+atα= 1 for the entire range of the memory parameter µ.
+In figures 5-8, we present the 3D graphs of mean success probab ility as a function of αandµ
+for amplitude damping, depolarizing, phase damping and flip ping channels respectively. One can
+easily see that the mean success probability is dependent on the quantum noise parameter αand
+memory parameter µ.It is seen that the mean success probability decreases with αand increases
+withµ,which indicates that both the parameters influence the proba bility differently. It is also
+seen that the behaviour of amplitude damping and phase dampi ng channels is same. On the other
+hand the behaviour of depolarizing channel is similar to the flipping channels. Therefore, it is
+possible to distinguish these channels by looking at the suc cess probability of the game.
+IV. CONCLUSIONS
+We study quantum magic squares game under the influence of qua ntum memory. We analyze
+that how the quantum noise and memory of quantum channels can influence the magic squares
+quantum pseudo-telepathy game. It is seen that the mean succ ess probability decreases with
+quantum noise parameter α, forµ <1.Where as it increases with µfor damping channels for
+α <1. On the other hand, for depolarizing and flipping channels t he success functions attain
+their maximum probability of success for entire range of µ. It is further seen that the behaviour of
+depolarizing and flipping channels is rather different from th e damping channels (amplitude and
+phase damping). Therefore, we can conclude with the comment that, it is possible to distinguish
+these channels by looking at the probability of success of th e game. In other words, the probability
+of success can be used to distinguish the quantum channels.
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+[2] Nielson, M. A., Chuang, I. L.: Quantum Computation and Quantum I nformation, Cambridge, Cam-
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+Figures captions
+Figure 1 . Mean success probability ¯P(α,µ) plotted as a function of quantum noise parameter α
+for memory parameter µ= 0.5 for amplitude damping, depolarizing, phase damping and fli pping
+(phase flip, bit flip or bit-phase flip) channels.
+Figure 2 . Mean success probability ¯P(α,µ) plotted as a function of quantum noise parameter α
+for memory parameter µ= 1 for amplitude damping, depolarizing, phase damping and fl ipping
+channels.
+Figure 3 . Mean success probability ¯P(α,µ) plotted as a function of memory parameter µfor
+quantum noise parameter α= 0.5 for amplitude damping, depolarizing, phase damping and fli p-
+ping channels.
+Figure 4 . Mean success probability ¯P(α,µ) plotted as a function of memory parameter µfor
+quantum noise parameter α= 1 for amplitude damping, depolarizing, phase damping and fl ipping
+channels.
+Figure 5 . Mean success probability ¯P(α,µ) plotted as a function of memory parameter µand
+quantum noise parameter αfor amplitude damping channel.
+Figure 6 . Mean success probability ¯P(α,µ) plotted as a function of memory parameter µand
+quantum noise parameter αfor depolarizing channel.
+Figure 7 . Mean success probability ¯P(α,µ) plotted as a function of memory parameter µand
+quantum noise parameter αfor phase flip channel.
+Figure 8 . Mean success probability ¯P(α,µ) plotted as a function of memory parameter µand
+quantum noise parameter αfor phase damping channel.
+Tables Captions
+Table 1. Single qubit Kraus operators for typical noise channels su ch as depolarizing, amplitude
+damping, phase damping, phase flip, bit flip and bit-phase flip whereαrepresents the quantum
+noise parameter.
+Table 2. Success probability P(α,µ) for all combinations of magic squares game inputs for depo-
+larizing, amplitude damping, phase damping, phase flip, bit flip and bit-phase flip channels in the
+presence of memory.
+Table 3. Mean success probability ¯P(α,µ) for depolarizing, amplitude damping, phase damping
+and flipping (phase flip, bit flip or bit-phase flip) channels in the presence of memory.11
+0.0 0.2 0.4 0.6 0.8 1.00.50.60.70.80.91.01.11.2Mean Success Probability
+α amplitude damping channel
+ depolarizing channel
+ flipping channels
+ phase damping channel
+ classical threshold
+FIG. 1: Mean success probability ¯P(α,µ) plotted as a function of quantum noise parameter αfor memory
+parameter µ= 0.5 for amplitude damping, depolarizing, phase damping and flipping (pha se flip, bit flip or
+bit-phase flip) channels.12
+0.0 0.2 0.4 0.6 0.8 1.00.70.80.91.01.11.2Mean Success Probability
+α amplitude damping channel
+ depolarizing channel
+ flipping channels
+ phase damping channel
+ classical threshold
+FIG. 2: Mean success probability ¯P(α,µ) plotted as a function of quantum noise parameter αfor memory
+parameter µ= 1 for amplitude damping, depolarizing, phase damping and flipping cha nnels.13
+0.0 0.2 0.4 0.6 0.8 1.00.50.60.70.80.91.01.11.2Mean Success Probability
+µ amplitude damping channel
+ depolarizing channel
+ flipping channels
+ phase damping channel
+ classical threshold
+FIG. 3: Mean success probability ¯P(α,µ) plotted as a function of memory parameter µfor quantum noise
+parameter α= 0.5 for amplitude damping, depolarizing, phase damping and flipping chan nels.14
+0.0 0.2 0.4 0.6 0.8 1.00.50.60.70.80.91.01.11.2Mean Success Probability
+µ amplitude damping channel
+ depolarizing channel
+ flipping channels
+ phase damping channel
+ classical threshold
+FIG. 4: Mean success probability ¯P(α,µ) plotted as a function of memory parameter µfor quantum noise
+parameter α= 1 for amplitude damping, depolarizing, phase damping and flipping cha nnels.15
+0.0
+0.2
+0.4
+0.6
+0.8
+1.00.60.81.0
+0.00.20.40.60.81.0Me
+an
+Success P 
+robability
+α
+µ
+FIG. 5: Mean success probability ¯P(α,µ) plotted as a function of memory parameter µand quantum noise
+parameter αfor amplitude damping channel.16
+0.0
+0.2
+0.4
+0.6
+0.8
+1.00.60.81.0
+0.00.20.40.60.81.0Me
+an
+Success P 
+robability
+α
+µ
+FIG. 6: Mean success probability ¯P(α,µ) plotted as a function of memory parameter µand quantum noise
+parameter αfor depolarizing channel.17
+0.0
+0.2
+0.4
+0.6
+0.8
+1.00.60.81.0
+0.00.20.40.60.81.0Me
+an
+Success P 
+robability
+α
+µ
+FIG. 7: Mean success probability ¯P(α,µ) plotted as a function of memory parameter µand quantum noise
+parameter αfor phase flip channel.18
+0.0
+0.2
+0.4
+0.6
+0.8
+1.00.60.81.0
+0.00.20.40.60.81.0Me
+an
+SuccessProbability
+α
+µ
+FIG. 8: Mean success probability ¯P(α,µ) plotted as a function of memory parameter µand quantum noise
+parameter αfor phase damping channel.19
+TABLE I: Single qubit Kraus operators for typical noise channels su ch as depolarizing, amplitude damping,
+phase damping, phase flip, bit flip and bit-phase flip where αrepresents the quantum noise parameter.
+Depolarizing channelE0=/radicalBig
+1−3α
+4I, E 1=/radicalbigα
+4σx
+E2=/radicalbigα
+4σy, E 3=/radicalbigα
+4σz
+Amplitude damping channel E0=
+1 0
+0√1−α
+, E1=
+0√α
+0 0
+
+Phase damping channel E0=
+1 0
+0√1−α
+,E1=
+0 0
+0√α
+
+Phase flip channel E0=√1−αI, E 1=√ασz
+Bit flip channel E0=√1−αI, E 1=√ασx
+Bit-phase flip channel E0=√1−αI, E 1=√ασy20
+TABLE II: Success probability P(α,µ) for all combinations of magic squares game inputs for depolarizing,
+amplitude damping, phasedamping, phaseflip, bit flipand bit-phaseflip channelsin thepresenceofmemory.
+Depolarizing channel:
+{(i,j)|i,j= 1,2,3}Pij(α,µ) =1
+2(2−4α+3µα+µ3α+6α2−9µα2
++3µ2α2−4α3+9µα3−6µ2α3+µ3α3
++α4−3µα4+3µ2α4−µ3α4)
+Amplitude damping channel:
+i,j∈ {(1,1),(1,2),(2,3),(3,3)}
+i,j∈(1,3)
+i,j∈ {(2,1),(2,2),(3,1),(3,2)}Pij(α,µ) =1
+4(4−4α+3µα+2α2−2µα2)
+Pij(α,µ) = 1−2α+2µα+2α2−2µα2
+Pij(α,µ) =1
+2(2−µ+µ√1−α−3α+3µα+2α2−2µα2)
+Phase damping channel:
+i,j∈ {(1,1),(1,2),(2,3),(3,3)}
+i,j∈(1,3)
+i,j∈ {(2,1),(2,2),(3,1),(3,2)}Pij(α,µ) =1
+4(4−4α+3µα+2α2−2µα2)
+Pij(α,µ) = 1
+Pij(α,µ) =1
+2(2−µ+µ√1−α−α+µα)
+Phase Flip channel:
+i,j∈ {(1,1),(1,2),(2,3),(3,3)}
+i,j∈(1,3)
+i,j∈ {(2,1),(2,2),(3,1),(3,2)}Pij(α,µ) = 1−4α+6µα−4µ2α+2µ3α+12α2+24µ2α4
+−30µα2+28µ2α2−10µ3α2−16α3+48µα3
+−48µ2α3+16µ3α3+8α4−24µα4−8µ3α4
+Pij(α,µ) = 1
+Pij(α,µ) = 1−2α+2µ2α+2α2−2µ2α2
+Bit flip channel:
+i,j∈ {(1,1),(1,2),(3,1),(3,2)}
+i,j∈(2,3)
+i,j∈ {(1,3),(2,1),(2,2),(3,3)}Pij(α,µ) = 1−2α+2µ2α+2α2−2µ2α2
+Pij(α,µ) = 1
+Pij(α,µ) = 1−4α+6µα−4µ2α+2µ3α+12α2+24µ2α4
+−30µα2+28µ2α2−10µ3α2−16α3+48µα3
+−48µ2α3+16µ3α3+8α4−24µα4−8µ3α4
+Bit-phase flip channel:
+i,j∈ {(1,1),(1,2),(2,1),(2,2)}
+i,j∈(3,3)
+i,j∈ {(1,3),(2,3),(3,1),(3,2)}Pij(α,µ) = 1−2α+2µ2α+2α2−2µ2α2
+Pij(α,µ) = 1
+Pij(α,µ) = 1−4α+6µα−4µ2α+2µ3α+12α2+24µ2α4
+−30µα2+28µ2α2−10µ3α2−16α3+48µα3
+−48µ2α3+16µ3α3+8α4−24µα4−8µ3α421
+TABLE III: Mean success probability ¯P(α,µ) for depolarizing, amplitude damping, phase damping and
+flipping (phase flip, bit flip or bit-phase flip) channels in the presence o f memory.
+Depolarizing channel:
+¯P(α,µ) =1
+2(2−4α+3µα+µ3α+6α2−9µα2+3µ2α2−4α3
++9µα3−6µ2α3+µ3α3+α4−3µα4+3µ2α4−µ3α4)
+Amplitude damping channel:
+¯P(α,µ) = 1/9(9−2µ+2µ√1−α−12α+11µα+8α2−8µα2)
+Phase damping channel:
+¯P(α,µ) = 1/9(9−2µ+2µ√1−α−6α+5µα+2α2−2µα2)
+Flipping channels:
+¯P(α,µ) = 1/9(9−24α+24µα−8µ2α+8µ3α+56α2−120µα2
++104µ2α2−40µ3α2−64α3+192µα3−192µ2α3
++64µ3α3+32α4−96µα4+96µ2α4−32µ3α4)arXiv:1001.0295v1  [quant-ph]  2 Jan 2010Table 1: Single qubit Kraus operators for typical noise channels suc h as depo-
+larizing, amplitude damping, phase damping, phase flip, bit flip and bit-p hase
+flip.
+Depolarizing channelE0=/radicalBig
+1−3α
+4I, E 1=/radicalbigα
+4σx
+E2=/radicalbigα
+4σy, E 3=/radicalbigα
+4σz
+Amplitude damping channel E0=/bracketleftbigg1 0
+0√1−α/bracketrightbigg
+, E1=/bracketleftbigg0√α
+0 0/bracketrightbigg
+Phase damping channel E0=/bracketleftbigg1 0
+0√1−α/bracketrightbigg
+,E1=/bracketleftbigg0 0
+0√α/bracketrightbigg
+Phase flip channel E0=√1−αI, E 1=√ασz
+Bit flip channel E0=√1−αI, E 1=√ασx
+Bit-phase flip channel E0=√1−αI, E 1=√ασy
\ No newline at end of file
diff --git a/1001.0296.txt b/1001.0296.txt
new file mode 100644
index 0000000000000000000000000000000000000000..71d8d548eef76f3d59fbb01dd8faf8d4ca1098c0
--- /dev/null
+++ b/1001.0296.txt
@@ -0,0 +1,691 @@
+arXiv:1001.0296v3  [math.PR]  22 Mar 2013Certain Periodically Correlated Multi-component Locally
+Stationary Processes
+N. Modaresi and S. Rezakhah∗
+Abstract
+By introducing Xls(t) as a random mixture of two stationary processes where the
+time dependent random weights have exponentially convex covarian ce, we show that
+this process has a multi-component locally stationary covariance fu nction in Silver-
+man’s sense. We also define Xp(t) as a certain continuous time periodically correlated
+(PC) process where its covariance function is generated by the co variance function of
+a discrete time through defining some simple random measure on real line. We also
+impose a bi-periodic correlation for this PC process with Xls(t). The existence of
+such random measure is proved. Then by defining X(t) =Xls(t) +Xp(t) as a cer-
+tain periodicallycorrelatedmulti-component locallystationaryproce ss, the covariance
+structure and time varying spectral representationof such pro cesses are characterized.
+Keywords: Periodically correlated; Spectral representation; Multi-compone nt lo-
+cally stationary processes; exponentially convex covariance.
+1 Introduction
+Recently a large amount of work has been devoted to time serie s analysis, with the focus
+placed on locally stationary (LS) and periodically correla ted (PC) processes which give
+plausible description of real world. Silverman [18] presen ted a definition of LS process to
+model systems which behave as a function of time. He presente d a relation between the
+covariance and the spectral density of LS processes which co nstitutes a natural general-
+ization of the Wiener-Khintchine relation. A general class of LS processes where their
+spectral structure varies smoothly over time is introduced by Dahlhaus [4]. A compu-
+tationally efficient state space method for estimating, pred icting and making statistical
+inferences about these non-stationary models are proposed in [15]. Exponentially con-
+vex stochastic processes, their covariance functions and s pectral representation have been
+studied by Loeve [12] among others [6].
+Gladyshev [8] introduced PC sequences and showed that all ar e harmonizable in the
+senseof Loeve [12]. He providedthestructureof thecovaria nce function andan interesting
+spectral representation. At the most basic level, the conne ction of the PC structure to a
+group of shift and unitary operators was mentioned as a motiv ation for the development of
+spectral representations that are much like those of Gladys hev. One of the classical results
+of unitary operator theory is the spectral theorem, the noti on of spectral measures and the
+time varying spectral representation of a PC process [9]. Th e contributions to the analysis
+of PC processes in the literature are [5], [9] and [16], where various properties of PC
+∗Faculty of Mathematics and Computer Science, Amirkabir Uni versity of Technology, 424 Hafez Avenue, Tehran
+15914, Iran. E-mail: namomath@aut.ac.ir(N. Modaresi), re zakhah@aut.ac.ir(S. Rezakhah).
+1processes are presented. Harmonic series representation, coherent and component method
+for mean and covariance function estimation by using linear filtration of PC processes are
+studiedin [10] and[11]. A class of bilinear processes withp eriodictime-varying coefficients
+of periodic ARMA and periodic GARCH models has been applied i n [2].
+We consider a certain non-stationary process {X(t),t∈R+}, say periodically corre-
+lated multi-component locally stationary (PC-LS) process . For introducing the structure
+of this process we consider a partition of the positive real l ine.This paper can be con-
+sidered as an extension of previous works in three stages. First by presenting
+spectral representation of certain continuous PC process. This is established
+by introducing some random measure on subsets of some proper partition of
+real line where the increments of the process on such partition, provides a
+discrete time PC process. We specify the covariance structure of such random
+measure through the covariance structure of the discrete time PC process, and
+provide the spectral representation of such continuous time PC process. The
+intuition of this process comes from the aggregated traffic on various networks.
+Second we present a locally stationary process in the Silverman sense as a time
+varying random mixture of stationary processes where the coefficients have
+exponentially convex covariances and are independent of the stationary ones.
+This consideration provides smooth transition in the correlation structure of
+the process which is in effect of two or more stationary resources at the nodes
+of the mentioned partitions. This makes a convenient way for analyzing traffic
+systems like the mobile cellular networks. Third we introduce the process as
+sum of such PC and LS processes which have bi-periodic correlations and have
+potential to model periodic and local variation in network traffics. The idea
+of introducing such process comes from the behavior of traffic flow in different
+networks, like telecommunication, internet or transportation networks. This
+process is determined by aggregation of the effects of two main components. The first
+component often has periodic behavior of the flow in time, Xp(t), which shows over all
+usage of the network. We assume that Xp(t) in turn can be approximated via a periodic
+sequenceXp
+j, which represents the increment of the flow on some proper par tition of time,
+sayBj,j∈N. Type and amount of the usage of the network by different users a t a time
+have some interaction which effects the flow and provide the oth er component of the pro-
+cess and can be approximated as a multi-component locally st ationary process Xls(t). We
+assume that Xls(t) is a random mixture, with some exponentially convex random weight
+process, of two stationary processes at time points inside e achBj. We consider some fixed
+T∈Nand a partition of the positive real line as 0 = s0< s1< ...,Bj= (sj−1,sj]
+forj∈Nand|Bj|=|Bj+kT|,k∈N, thenS=/summationtextT
+i=1|Bi|. LetX(t) =Xls(t) +Xp(t),
+t∈R+represents a stochastic process, where Xls(t) =/summationtext∞
+j=1Xls
+j(t)IBj(t) andXls
+j(t) is a
+multi-component LS process which is an exponentially conve x mixture of two stationary
+processes. Also let {Xp
+j},j∈Nbe the discrete time PC process, Mj,j∈Nthe sequence
+of random measure on BjwhereMj(Bj) :=Xp
+jandXp
+j(t) :=Mj(sj−1,t] fort∈Bj, j∈N
+with some special correlation. As the amount of the process o n a setA, say traffic, we
+consider simple random measure M(A) =/summationtextm
+j=1Mij(A∩Bij) whereA=∪m
+j=1(A∩Bij).
+We prove the existence of such measures by introducing some m easurable map and some
+T-variate random measure. Then we study Xp(t) =/summationtext∞
+j=1∆p
+j(t)IBj(t), where ∆p
+jis a
+linear combination of Xp
+j−1andXp
+j(t), which describes elimination of the effect of traffic
+onBj−1as a decreasing fraction of XP
+j−1and aggregate of the process on BjasXp
+j(t),
+2and study its spectral representation.
+In section 2, we study the general framework and preliminari es of the harmonizable
+representation of stationary an PC processes and also, we pr esent exponentially convex
+processes. The main result as the covariance structure and s pectral representation of such
+continuous time multi-component PC-LS processes are given in section 3.
+2 Preliminaries
+We review the harmonizable representations of PC processes based on unitary operators
+and also the definition of exponentially convex processes in this section.
+2.1 Spectral representation of PC processes
+We review harmonizable representation of PC processes by un itary operators. For a
+comprehensive review of these one can refer to Hurd and Miame e [9]. Every wide sense
+stationary process X(t) has harmonizable representation
+X(t) =/integraldisplay∞
+−∞eiλtdZ(λ) (2.1)
+whereZ(λ) has orthogonal increments.
+A unitary operator on a Hilbert space His a linear operator from HtoHwhich
+preserve inner product as/angbracketleftbig
+Ux,Uy/angbracketrightbig
+=/angbracketleftbig
+x,y/angbracketrightbig
+= cov(x,y) for every x,y∈ H.
+Theorem 2.1 For any unitary operator Uon a Hilbert space H, there exists a unique
+spectral measure Qon the Borel subsets of [0,2π)such thatU=/integraltext2π
+0eiλQ(dλ), andUt=/integraltext2π
+0eiλtQ(dλ)for any integer t.
+Existence of unitary operator for a PC sequence, characteri ze its spectral representation.
+Proposition 2.1 A second order stochastic sequence Xjis PC with period Tif and only
+if for every j∈Z, there exists a unitary operator U=VTand a periodic sequence
+(process)Pjwith period Ttaking values in HX=sp{Xj,j∈Z}for whichXj=VjPj
+whereV=/integraltext2π
+0eiλ/TQ(dλ)andQis the spectral measure defined in Theorem 2.1, so
+Xj=/integraltext2π
+0eiλj/TQ(dλ)Pj.
+2.2 Exponentially convex process
+We give a brief description of exponentially convex process and its covariance function,
+for more details see [6], [7].
+Definition 2.1 The covariance function of a second order zero mean process {Z(t),t∈R}
+with finite variance,/angbracketleftbig
+Z(ti),Z(tj)/angbracketrightbig
+=ψ(ti+tj), is called exponentially convex if and only
+if/summationtextn
+i=1/summationtextn
+j=1aiajψ(ti+tj)/greaterorequalslant0, for all finite sets of complex coefficients a1,...,anand
+pointst1,...,tn∈R.
+A stochastic process with such covariance function is calle d exponentially convex. The
+result of Berg et al. [1] implies that a continuous function i s exponentially convex if and
+only if it is the Laplace transform of a non-negative finite me asure.
+33 Main results: A continuous time PC-LS model
+We introduce a new class of certain non-stationary process, say multi-component period-
+ically correlated locally stationary (PC-LS) process as
+X(t) =Xls(t)+Xp(t), t> 0 (3.1)
+whereXp(·) is a continuous time PC process and Xls(·) is a multi-component LS process
+which have bi-periodic correlation. We introduce their spe cial structures in subsections
+3.1 and 3.2 respectively. Their covariance and cross covari ance structures are studied in
+subsection 3.3. Finally the spectral representation of the process is characterized in 3.4.
+3.1 Continuous PC process
+Let{Xp
+j,j∈N}be a positive second order discrete time PC process with peri odT,Mja
+random measure on Borel field of subsets of Bj, whereMj(Bj) :=Xp
+j, and forA,B⊂Bj,
+D⊂Bk, we define E[Mj(A)] =|A|
+|Bj|E[Xp
+j] and covariance functions
+γj,j(A,B)=λ|A||B|+(1−λ)aj|A∩B|
+a2
+jγp
+jj, γj,k(A,D):=|A||D|
+ajakγp
+jk(3.2)
+whereγj,j(A,B) =/angbracketleftbig
+Mj(A),Mj(B)/angbracketrightbig
+,γj,k(A,D) =/angbracketleftbig
+Mj(A),Mk(D)/angbracketrightbig
+,γp
+jk=/angbracketleftbig
+Xp
+j,Xp
+k/angbracketrightbig
+,
+γp
+jj= Var(Xp
+j), 0≤λ≤1 andaj=|Bj|.
+IfB=Bkthen forj/ne}ationslash=k,/angbracketleftbig
+Mj(A),Mk(Bk)/angbracketrightbig
+=|A|
+ajγp
+jk.This inner product is well defined.
+LetXp
+j(t) :=Mj(sj−1,t],t∈Bj,j∈N. So fort,u∈Bj,t≤u, andv∈Bk
+/angbracketleftbig
+Xp
+j(t),Xp
+j(u)/angbracketrightbig
+=λat
+jau
+j+(1−λ)ajat
+j
+a2
+jγp
+jj,/angbracketleftbig
+Xp
+j(t),Xp
+k(v)/angbracketrightbig
+=at
+jav
+k
+ajakγp
+jk(3.3)
+whereat
+j=t−sj−1. Thus by defining
+Nj(t−sj−1) :=Mj(sj−1,t] =Xp
+j(t), (3.4)
+we find that Nj(y) is a discrete time PC process with period Twith respect to jfor fixed
+0<y/lessorequalslantajandXp
+j(t) is a bi-periodic process with period Tinjand period S=/summationtextT
+i=1|Bi|
+int, that is<Xp
+j(t),Xp
+k(u)>=<Xp
+j+T(t+S),Xp
+k+T(u+S)>. Also fort>0
+Xp(t) =∞/summationdisplay
+j=1∆p
+j(t)IBj(t);∆p
+j(t) =aj−at
+j
+ajXp
+j−1+Xp
+j(t), (3.5)
+is a continuous time PC process with respect to twith period S.Using a similar
+method such as the one described by Soltani and Parvardeh [17], we prove the
+existence of such random measures by the following.
+Let(D,D)be a measurable space and L2(Ω,F,P)the Hilbert space of real
+random variables on the probability space (Ω,F,P)with finite second moment.
+A mapping Φ :D → L2(Ω,F,P)is a second order random measure on DifΦ(∅) =
+0andE/vextendsingle/vextendsingleΦ/parenleftbig
+∪∞
+i=1Ai/parenrightbig
+−/summationtextn
+i=1Φ(Ai)/vextendsingle/vextendsingle2→0asn→ ∞, for disjoint sets A1,A2,...∈ D.
+We introduce simple random measure M(A)forA∈ DbyM(A) =M1(A∩B1)+
+M2(A∩B2) +...+MT(A∩BT)whereBjis the support of Mj, j= 1,...,Tand
+Bi∩Bj=∅fori/ne}ationslash=j. For the random measure M,E|M(dx)|2=γj,j(dx)forx∈Bj.
+4Also forx∈Bjandy∈Biwe haveE/parenleftbig
+M(dx)M(dy)/parenrightbig
+=E/parenleftbig
+Mj(dx),Mi(dy)/parenrightbig
+=
+γj,k(dx,dy). To show the existence of such random measures, let Ψ = (Ψ 1,...ΨT)
+be aT-variate random measure on (X,X), whereX=B1×...×BTandXis
+the corresponding class of finite disjoint union of measurable rectangles. In
+this case for A∈ D,A=∪m
+j=1(A∩Bij)and{i1,i2,...,im}aremdistinct integer of
+{1,2,...,T}. LetX∗=Bi1×...×BimandX∗the class of finite disjoint rectangles
+F∗= (F∩Bi1)×...×(F∩Bim)whereF⊂D∗=∪m
+j=1BijandS:X∗→D∗be a
+measurable map that S(A∗) =∪m
+i=1{xi:x= (x1,...,x m)∈A∗}forA∗∈ X∗. Then
+S−1(F) ={x= (x1,···,xm) :xj∈F∩Bij,j= 1,...,m}forF⊂D∗. Also let
+Mij(F) = ΨijS−1(F), andΨ∗=/parenleftbig
+Ψi1,...,Ψim/parenrightbig
+be anm-variate random measure
+on(X∗,X∗)with covariance matrix µ= [µj,k], then for the resulting Mand for
+1≤j,k≤m
+µj,k/parenleftbig
+S−1(F)/parenrightbig
+=E/parenleftbig
+Ψij(S−1(F))Ψik(S−1(F))/parenrightbig
+=γij,ik(F∩Bij,F∩Bik)
+µj,j/parenleftbig
+S−1(F)/parenrightbig
+=E/parenleftbig
+Ψij(S−1(F))Ψij(S−1(F))/parenrightbig
+=γij,ij(F∩Bij,F∩Bij)
+Interestingly
+ν(A×C) =/summationdisplay
+j,kγj,k(A∩Bj,C∩Bk)
+defines a product measure on (D∗×D∗,Y), where Yis the class of finite disjoint
+union of corresponding measurable rectangles. Also for Ej,k∈ YwhereEj,k⊂
+Bij×Bik
+ν(Ej,k) =µj,k/braceleftbig
+x:x∈X∗,(xj,xk)∈Ej,k/bracerightbig
+Thus forE∈ YandEj,k=E∩(Bij×Bik)
+ν(E) =/summationdisplay
+j,kν(Ej,k) =m−1/summationdisplay
+l=−m+1min{m−l,m}/summationdisplay
+j=max{1−l,1}ν(Ej,l+j)
+=m/summationdisplay
+j=1ν(Ej,j)+m−1/summationdisplay
+l=1/braceleftbiggm−l/summationdisplay
+j=1ν(Ej,l+j)+min{2m−l,m}/summationdisplay
+j=m−l+1ν(Ej,l+j−m)/bracerightbigg
+.
+Now define µ0,...,µ m−1onD∗through
+µ0(A) =m/summationdisplay
+j=1µj,j/parenleftbig
+S−1(A)/parenrightbig
+,
+µl(A) =min{2m−l,m}/summationdisplay
+j=m−l+1µj,j+l−m/parenleftbig
+S−1(A)/parenrightbig
++m−l/summationdisplay
+j=1µj,j+l/parenleftbig
+S−1(A))/parenrightbig
+forl= 1,...,m−1; then for E∈ Ywe have that ν(E) =/summationtextm−1
+l=0µl{x∈D∗;(x,y)∈
+Efor some y}.Therefore the product measure νis specified by mset functions
+µ0,...,µ m−1onD∗that are determined by µ= [µj,k], the covariance matrix of
+m-variate random vector M= (M1,...,M m). The covariance matrix µalso can
+be specified from µ0,...,µ m−1, namely for G∈ X∗
+µj,k(G) =µm+k−j/parenleftbig
+{xk:x∈G}∪{xj:x∈G}/parenrightbig
+, k −j <0,
+µj,k(G) =µk−j/parenleftbig
+{xk:x∈G}∪{xj:x∈G}/parenrightbig
+, k −j≥0,
+Therefore νandµuniquely specify each other.
+53.2 Multi-component LS process
+To provide the smoothing transition and LS behavior of the process between
+inside each partition we introduce a random mixture of two or more sta-
+tionary processes or resources at nodes of such partition with exponentially
+convex coefficients which provides an LS processes in the Silverman sense. Let/braceleftbig
+Ys
+j(t),t∈Bj∪Bj+1/bracerightbig
+,Ys
+j(t) =/integraltext∞
+−∞eitληj(dλ),j∈Nbe a sequence of independent
+stationary processes, ηj(·)d=ηj+T(·) and{Ys
+j(t)}d={Ys
+j+kT(t+kS)}fork∈N. We define
+Xls(t) =∞/summationdisplay
+j=1Xls
+j(t)IBj(t), Xls
+j(t)=Uj−1(t)Ys
+j−1(t)+Uj(t)Ys
+j(t) (3.6)
+fort∈Bj,j= 2,···,TandXls
+1(t) =U1(t)Ys
+1(t) fort∈B1, in which {Uj(t),j∈N}is a
+random weight process with exponentially convex covarianc e and are independent of the
+processYs
+j(·). We callXls
+j(·), a multi-component LS process motivated from its covarian ce
+function, which is represented in Lemma 3.1.
+3.3 Covariance function of PC-LS process
+We present some lemmas in this section which we use to provide the covariance function
+of the process X(t) in Theorem 3.1.
+Lemma 3.1 LetBj,j∈Nbe the introduced partition of positive real line. The covari ance
+function of Xls(t),t∈R+satisfies
+γls(t,u)≡/angbracketleftbig
+Xls(t),Xls(u)/angbracketrightbig
+=∞/summationdisplay
+m=1m+1/summationdisplay
+n=m−1γls
+mn(t,u)IBm(t)IBn(u) (3.7)
+whereγls
+mm(t,u)≡/angbracketleftbig
+Xls
+m(t),Xls
+m(u)/angbracketrightbig
+=ψm(t+u)γm(t−u) +ψm−1(t+u)γm−1(t−u),
+γls
+mn(t,u)≡/angbracketleftbig
+Xls
+m(t),Xls
+n(u)/angbracketrightbig
+=ψk(t+u)γk(t−u),k= min{m,n}and|n−m|= 1in
+whichψm(t+u) =/angbracketleftbig
+Um(t),Um(u)/angbracketrightbig
+,γm(t−u) =/angbracketleftbig
+Ys
+m(t),Ys
+m(u)/angbracketrightbig
+form∈N.
+Proof:According to (3 .6) and the fact that Uj(·) andYs
+j(·) are independent processes,
+soγls
+mm(t,u) andγls
+mn(t,u) for|n−m|= 1 are as presented by the lemma. Thus Xls
+j(·) is
+a multi-component LS process in the Silverman sense [18]. Al so by (3.6)
+γls(t,u) =∞/summationdisplay
+m=1/angbracketleftbig
+Xls
+m(t),Xls(u)/angbracketrightbig
+IBm(t) =∞/summationdisplay
+m=1m+1/summationdisplay
+n=m−1γls
+mn(t,u)IBm(t)IBn(u).
+Lemma 3.2 LetBj,j∈Nbe a partition of positive real line and γp
+mn=/angbracketleftbig
+Xp
+m,Xp
+n/angbracketrightbig
+. The
+covariance function of Xp(t)fort∈Bmandu∈Bn,t≤uis
+γp(t,u)≡/angbracketleftbig
+Xp(t),Xp(u)/angbracketrightbig
+= (3.8)
+/braceleftBigg
+At,m/parenleftbig
+Au,mγm−1,m−1+au
+m
+amγm−1,m/parenrightbig
++Au,mat
+m
+amγm−1,m+Bm(t,u)γp
+mmn=m,
+At,m/parenleftbig
+Au,nγp
+m−1.n−1+au
+n
+anγp
+m−1,n/parenrightbig
++Au,nat
+m
+amγp
+m,n−1+at
+mau
+n
+amanγp
+m,n|m−n| ≥1
+whereAt,m= (1−at
+m/am),Au,n= (1−au
+n/an)andBm(t,u) =λat
+mau
+m+(1−λ)amat
+m
+a2m.
+6Proof: By (3.5) the covariance function of Xp(·) fort∈Bm,u∈Bnisγp(t,u) =/angbracketleftbig
+Xp
+m(t),Xp
+n(u)/angbracketrightbig
+and by (3.3) we have the result.
+Letaj=|Bj|,a=E[Uj(t)] fort∈R+andθr(j,u) =/angbracketleftbig
+Xp
+j,Ys
+r(u)/angbracketrightbig
+. We also define the
+cross covariance function of Xp
+j(t) =Mj(sj−1,t] andYs
+r(u) fort∈Bj, u∈Bras
+/angbracketleftbig
+Xp
+j(t),Ys
+r(u)/angbracketrightbig
+=at
+j
+ajθr(j,u),/angbracketleftbig
+Xp
+j(t),Xls
+r(u)/angbracketrightbig
+=aat
+j
+aj/bracketleftbig
+θr−1(j,u)+θr(j,u)/bracketrightbig
+.(3.9)
+Remark 3.1 The cross covariance function of Xls(t)andXp(t)can be written as
+γp,ls(t,u)≡/angbracketleftbig
+Xp(t),Xls(u)/angbracketrightbig
+=a∞/summationdisplay
+m=1∞/summationdisplay
+n=1/bracketleftbigat
+m
+amDm,n(u)+At,mDm−1,n(u)/bracketrightbig
+IBm(t)IBn(u)
+(3.10)
+whereDm,n(u) =θn−1(m,u)+θn(m,u),At,m= 1−at
+m/amandθr(m,u)is defined by
+(3.9).
+Theorem 3.1 The covariance function of the multi-component PC-LSprocessX(t) =
+Xls(t) +Xp(t), whereXls(t)andXp(t)are dependent and defined by (3.6)and(3.5)
+respectively, is γ(t,u) = cov/parenleftbig
+X(t),X(u)/parenrightbig
+=γls(t,u)+γp(t,u)+γp,ls(t,u)whereγls(t,u),
+γp(t,u)andγp,ls(t,u)are as in Lemma 3.1, Lemma 3.2 and Remark 3.1 respectively.
+3.4 Spectral representation
+In this section we obtain spectral representations of PC pro cess{Xp(t),t∈R+}, and
+locally stationary process {Xls(t),t∈R+}. Then by imposing bi-periodic property for
+the cross covariance function of Xls(t) andXp(t) their cross spectrum is characterized.
+Usingtheseresultsthespectral representation ofthemult i-component PC-LSprocess X(·)
+and its spectral measure is provided by Theorem 3.2.
+Let{Xp
+j},j∈N}be a PC process with period T. By using theGladyshev result for
+PC processes, it has been shown in [9] that for j,k∈N
+Xp
+j=/integraldisplay2π
+0eiλjdϑ(λ), (3.11)
+wheredϑ(λ) =/summationtextT−1
+k=0Q(Tdλ−2kπ)/tildewidePkI∆k(λ), ∆k= [2kπ/T,2(k+1)π/T],/tildewidePkareFourier
+coefficients of the periodic sequence Pj,Q(·) andPjare defined by Proposition 2.1, that
+Pj=/summationtextT−1
+k=0/tildewidePkei2πkj/T. The support of the spectral density ˜θ(dλ,dω) =/angbracketleftbig
+ϑ(dλ),ϑ(dω)/angbracketrightbig
+is
+the intersection of the set Ψ = {(λ,ω) :λ=ω−2πk/T,k∈[−(T−1),T−1]}and the
+square [0,2π)×[0,2π).
+Remark 3.2 Let{Xp
+j}be a sequence of PCprocess with period T, by proposition (2.1)
+Xp
+j=/integraltext2π
+0eiλj/Tξ(dλ,j)whereξ(dλ,j) =Q(dλ)Pj,Q(dλ)is an orthogonally scattered
+random measure and Pja periodic sequence with period T. Letaj=|Bj|,at
+j=t−sj−1
+fort∈Bj. As by (3.4), Xp
+j(t)≡Nj(at
+j)andNj(y)is a discrete PC process with period T
+for fixedy, so by Proposition 2.1Xp
+j(t) =/integraltext2π
+0eiλj/Tζj(dλ,t)fort∈Bj, whereζj(dλ,t) =
+Q(dλ)/hatwidePj,at
+jand/hatwidePj,yis a periodic function in jwith period Tfor fixedy. Using the
+relation (3.4) we have Xp
+j≡Xp
+j(sj), so/hatwidePj,aj≡Pj. By a similar method as in (3.11)
+7Xp
+j(t) =/integraltext2π
+0eiλjΥ(dλ,at
+j)fort∈Bj, whereΥ(dλ,at
+j)has the same structure as ϑ(dλ)in
+(3.12), by replacing /tildewidePkwith/tildewidePk,at
+kwhere/hatwidePj,at
+j=/summationtextT−1
+k=0/tildewidePk,at
+kei2πjk/T. So(3.3)implies that
+Θj,k(dλ,dω,at
+j,au
+k) =/angbracketleftbig
+Υ(dλ,at
+j),Υ(dω,au
+k)/angbracketrightbig
+=
+
+λat
+jau
+j+(1−λ)ajat
+j
+a2
+j˜θ(dλ,dω)j=k
+at
+jau
+k
+ajak˜θ(dλ,dω)j/ne}ationslash=k
+which is bi-periodic with period Tinjandk, and period Sintandu(t≤u) where as in
+(3.11),˜θ(dλ,dω) =/angbracketleftbig
+ϑ(dλ),ϑ(dω)/angbracketrightbig
+haveΨas its support.
+We represent the spectral representation for Xls
+j(·)by the following remark.
+Remark 3.3 Spectral representation of the process Xls
+j(·),j∈Ndefined by (3.6)is
+Xls
+j(t) =/integraldisplay∞
+−∞eiλtΦj(dλ,t), t∈Bj (3.12)
+whereΦj(dλ,t) =Uj−1(t)ηj−1(dλ) +Uj(t)ηj(dλ), andηjis the orthogonally scattered
+random measure in the spectral representation of the statio nary process Ys
+j(·). Also by
+the independence of {Uj(·)}and{Ys
+j(·)}, the cross spectral covariance Fj,k(dλ,dλ,t,u ) =/angbracketleftbig
+Φj(dλ,t),Φk(dλ,u)/angbracketrightbig
+forλ/ne}ationslash=ωcan be written as
+Fj,k(dλ,dλ,t,u ) =ψj−1(t+u)Gj−1(dλ)I{k,k+1}(j)+ψj(t+u)Gj(dλ)I{k,k−1}(j),
+whereψj(t+u) =/angbracketleftbig
+Uj(t),Uj(u)/angbracketrightbig
+, Gj(dλ) =E|ηj(dλ)|2, andFj,k(dλ,dω,t,u ) = 0.
+Lemma 3.3 Under the assumptions of Remark 3.2-3.3, the cross covarian ce of the pro-
+cessesXp
+j(t)andYs
+r(u)is bi-periodic, with period Sintanduand period Tinjandr,/angbracketleftbig
+Xp
+j(t),Ys
+r(u)/angbracketrightbig
+=/angbracketleftbig
+Xp
+j+T(t),Ys
+r+T(u+S)/angbracketrightbig
+,wheret∈Bjandu∈Brif and only if
+/angbracketleftbig
+Xp
+j(t),Ys
+r(u)/angbracketrightbig
+=at
+j
+aj/integraldisplay2π
+0/integraldisplay∞
+−∞ei(jλ−uω)∆r(dλ,dω) (3.13)
+where support of ∆r(dλ,dω)=/angbracketleftbig
+ϑ(dλ),ηr(dω)/angbracketrightbig
+isΓ={(λ,ω) :λT=Sω−2πl,l∈Z}.
+Proof:By (3.9)< Xp
+j(t),Ys
+r(u)>=at
+j
+aj< Xp
+j,Ys
+r(u)>, by (3.11) Xp
+j=/integraltext2π
+0eiλjϑ(dλ),
+and by (3.6) Ys
+r(u) =/integraltext∞
+−∞eiωuηr(dω), whereηr+T(·) =ηr(·) . So we have (3.13), and
+one can easily verify that if Γ is the support of the cross spec tral covariance then the
+covariance function has bi-periodic property. On the other hand by assuming that the
+cross covariance is bi-periodic, using (3.9), equality of θr(j,u) =θr+T(j+T,u+S) implies
+that for any natural number N,
+θr(j,u) =1
+2N+1N/summationdisplay
+k=−Nθr+kT(j+kT,u+kS) =/integraldisplay2π
+0/integraldisplay∞
+−∞ei(jλ−uω)DN(Tλ−Sω)∆r(dλ,dω),
+whereDN(2πl) = 1, l∈Z, andDN(σ) =sin(N+1/2)σ
+2(N+1/2)sin(σ/2)and converges to zero for
+σ/ne}ationslash= 2πZ. This implies that the support of cross spectral density mus t be contained in Γ.
+8Theorem 3.2 The spectral representation of the multi-component PC-LSprocessX(t) =
+Xls(t)+Xp(t), whereXp(t)andXls(t)are defined by (3.11),(3.5)and(3.6)can be written
+as
+X(t) =/integraldisplay∞
+−∞eiλtdZm,t(λ), t∈Bm, (3.14)
+wheredZm,t(λ)=Φm(dλ,t)+I[0,2π)(λ)eiλ(m−t)/bracketleftbigg
+Υ(dλ,at
+m)+At,me−iλdϑ/bracketrightbigg
+, in whichϑ,
+ΦmandΥare defined by (3.12)and Remark 3.2. The cross spectral covariance for t∈Bm,
+u∈Bn
+/angbracketleftbig
+dZm,t(λ),dZn,u(ω)/angbracketrightbig
+=Fm,n(dλ,dω,t,u)+K(m,n,λ,ω)/bracketleftbigg
+e−i(λ−ω)At,mAu,n˜θ(dλ,dω)
++Θm,n(dλ,dω,at
+m,au
+n)/bracketrightbigg
++aau
+n
+anΛm,n(dω,dλ,t)+aat
+m
+amΛn,m(dλ,dω,u)
++Au,naΛm,n−1(dω,dλ,t)e−iω+At,maΛn,m−1(dλ,dω,u)e−iλ
+whereAt,m= 1−at
+m/am,Λl,k(dθ,dη,v) =ei(θk−vη)/parenleftbig
+∆l(dθ,dη)+∆l−1(dθ,dη)/parenrightbig
+I[0,2π)(dθ), l,k=
+m,n, andΘ, F,and∆are defined in Remark 3.2 ,Remark 3.3 andLemma 3.3 respec-
+tively, and K(m,n,λ,ω ) =eiλ(m−t)−iω(n−u)I[0,2π)(λ)I[0,2π)(ω).
+Proof:By the result of Remarks 3.2-3.3 and relations (3.5),(3.6) w e have that {X(t),t∈
+R+}has the time varying spectral representation (3.14). Also b y (3.9), Lemma 3.3, and
+Remark 3.2-3.3, the last assertion of the theorem can be easi ly obtained as the cross
+spectral covariance.
+References
+[1] C. Berg, J. P. Christensen, P. Ressel (1984) Harmonic ana lysis on semigroups, New
+York: Springer-Verlag.
+[2] A. Bibi, I. Lescheb (2012) On general periodic time-vary ing bilinear processes, Eco-
+nomics Letters , Vol.114, pp.353-357.
+[3] D. R. Cox, H. D. Miller (1994) The Theory of Stochastic Pro cesses, Chapman Hall.
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+arXiv:1001.0297v2  [gr-qc]  8 Feb 2010Towards a more exact value of deflection of
+light due to static gravitational mass
+A. K. Sen
+Department of Physics, Assam University, Silchar 788011,
+India
+Email: asokesen@bsnl.com
+Abstract The deflection of a ray of light passing close to a gravitational
+mass, is generally calculated from the null geodesic which the light ray (
+photon) follows. However, there is an alternate approach, where the effect
+of gravitation on the ray of light is estimated by considering the ray t o be
+passing through a material medium . Calculations have been done in this
+paper, following the later approach, to estimate the amount of defl ection
+due to a static non-rotating mass. The refractive index of such a m aterial
+medium, has been calculated in a more rigorous manner in the present work
+andthefinalexpressionfortheamountofdeflectioncalculatedhe reisclaimed
+to be more accurate than all other expressions derived so far usin gmaterial
+mediumapproach. Based on this expression, the amount of deflection for a
+sun grazing ray has been also calculated.
+1 Introduction
+The gravitational deflection of light is one of the important predictio ns of the
+General Theory of Relativity (GTR) proposed by Einstein, which play s a key
+role in understanding problems related to Astronomy, Cosmology, G ravita-
+tional Physics and other related branches.
+Newtons theory of universal gravitation had already predicted th at the
+path of any material particle moving at a finite speed is affected by th e pull
+of gravity. By the late 18th century, it was possible to apply Newton s law
+to compute the deflection of light by gravity. Cavendish commented briefly
+on the gravitational deflection of light in the late 1700s and Soldner g ave a
+detailed derivation in 1801.
+1Theidea ofbending oflightwasrevived byEinstein in1911andthequan -
+titative prediction for the amount of deflection of light passing near a large
+mass (M) was identical to the old Newtonian prediction, d= 2GM/(c2r/circledottext),
+wherer/circledottextis the closet distance of approach and in this case approximately
+the solar radius. It wasn’t until late in 1915, as Einstein completed th e
+general theory, he realized his earlier prediction was incorrect and the angu-
+lar deflection should actually be twice the size he predicted in 1911. Th is
+was subsequently confirmed by Eddington in 1919 through an exper iment
+performed during the solar eclipse.
+The exact amount of deflection for a ray of light passing close to a gr avi-
+tational mass can be worked out from the null geodesic, which a ray of light
+follows [1,2,3].
+The deflection of a light ray passing close to a gravitational mass can be
+alternately calculated by following an approach, where the effect of gravita-
+tion on the light ray is estimated by considering the light ray to be pass ing
+through a material medium with a value of refractive index decided by the
+value of gravitational field [4].
+The concept of this equivalent material medium was discussed by Balazs
+[5] as early as in 1958, to calculate the effect of a rotating body, on t he
+polarization of an electromagnetic wave passing close to it. Plebansk i [6]
+had also utilized this concept in 1960, to study the scattering of a pla ne
+electromagnetic wave by gravitational field, where theauthor men tioned that
+this concept of equivalent material medium was first pointed out by Tamm
+[7] in 1924. A general procedure for utilizing this concept, for defle ction
+calculation has been worked out by Felice [8]. Later this concept was a lso
+usedbyMashoon[9,10],tocalculatethedeflectionandpolarizationdu etothe
+Schwarzschild and Kerr black holes. Fischbach and Freeman [11], der ived the
+effective refractive index of the material medium and calculated the second
+order contribution to the gravitational deflection. In a similar way S ereno
+[12] has used this idea, for gravitational lensing calculation by drawin g the
+trajectory of the ray by Fermat’s principle. More recently Ye and L in [13],
+emphasized the simplicity of this approach and calculated the gravita tional
+time delay and the effect of lensing.
+Ontheotherhand, thecalculationofhigher orderdeflectionterms , dueto
+Schwarzschild Blackhole, fromthenullgeodesic, hasbeenperform edrecently
+by Iyer and Petters ( [14] and references their in). Using null geod esics,
+gravitational lensing calculations have been done by a number of aut hors in
+2past [15,16].
+With the above background, in the present work, we follow the mate-
+rial medium approach, to calculate a more accurate value for the deflection
+term due to a non-rotating sphere ( Schwarzschild geometry ). It is claimed
+that the present calculated value will be more accurate than all oth er values
+calculated in past, using material medium concept.
+2 The effective refractive index and the tra-
+jectory of light ray
+As discussed earlier, the gravitational field influences the propaga tion of elec-
+tromagnetic radiation by imparting to the space an effective index of refrac-
+tionn(r)[4].
+For a static and spherically symmetric gravitational field, the solutio n of
+Einstein’s Field Equation was given by K. Schwarzschild in 1961, which is a s
+follows[4]:
+ds2= (1−rg
+r)c2dt2−r2(sin2θdφ2+dθ2)−dr2
+(1−rg
+r)(1)
+whererg=2km
+c2calledSchwarzschild Radius, whichcompletely definesthe
+gravitational field in vacuum produced by any centrally-symmetric d istribu-
+tion of masses. The above equation can be expressed in an isotropic form by
+introducing a new radius co-ordinate ( ρ) with the following transformation
+equation [4]
+ρ=1
+2[(r−rg
+2)+r1/2(r−rg)1/2] (2)
+OR
+r=ρ(1+rg
+4ρ)2(3)
+The resulting isotropic form of Schwarzschild equation will be now:
+ds2= (1−rg/(4ρ)
+1+rg/(4ρ))2c2dt2−(1+rg
+4ρ)4(dρ2+ρ2(sin2θdφ2+dθ2)) (4)
+3Now in spherical co-ordinate system the quantity ( dρ2+ρ2(sin2θdφ2+
+dθ2)) has the dimension of square of infinitesimal length vector d− →ρ.
+By setting ds= 0, the velocity of light can be identified from the ex-
+pression of the form ds2=f(ρ)dt2−d− →ρ2, asv(ρ) =/radicalBig
+f(ρ). Therefore the
+velocity of light in the present case ( characterized by Schwarzsch ild radius
+rg) can be expressed as :
+v(ρ) =(1−rg
+4ρ)c
+(1+rg
+4ρ)3(5)
+But this above expression of velocity of light is in the unit of length ρper
+unit time. We therefore write
+v(r) =v(ρ)dr
+dρ
+=v(ρ)[(1+rg
+4ρ)2−rg
+2ρ(1+rg
+4ρ)]
+= (rg−4ρ
+rg+4ρ)2c (6)
+Substituting the value of ρfrom Eqn (2) in Eqn.(6), we get:
+v(r) = (rg/2−2ρ
+rg/2+2ρ)2c
+= (rg/2−((r−rg
+2)+r1/2(r−rg)1/2)
+rg/2+((r−rg
+2)+r1/2(r−rg)1/2))2c
+= (rg−r−r1/2(r−rg)1/2
+r+r1/2(r−rg)1/2)2c
+=c(r−rg)
+r(7)
+Therefore the refractive index n(r) at a point with spherical polar co-
+ordinate ( r), can be expressed by the relation:
+n(r) =c
+v(r)=r
+r−rg(8)
+4At this stage the entire problem, can become a problem of geometric al
+optics, where we have to find the trajectory of a light ray travelling in a
+medium, whose refractive index has spherical symmetry. The traj ectory of
+the light ray and the center of mass ( source of gravitational pote ntial) will
+together define a plane. The equation of such a ray inplane polar co- ordinate
+system (r,θ) can be written as [17]:
+θ=A./integraldisplay∞
+r/circledottextdr
+r√
+n2r2−A2(9)
+The trajectory is such that n(r).dalways remains a constant, wheredis
+the perpendicular distance between the trajectory of the light ra y from the
+origin and the constant is taken here as A[17]. In our present problem the
+lightiscomingfrominfinity( r=−∞)anditisapproachingthegravitational
+mass, which is placed at the originand characterized by Schwarzsch ild radius
+rg. The closest distance of approach, for the approaching ray is band the
+ray goes to r=∞, after undergoing certain amount of deflection ( △φ).
+Here, the parameter bcan be replaced by solar radius r/circledottext. When the
+light ray passes through the closest distance of approach (ie r=borr/circledottext),
+the tangent to the trajectory becomes perpendicular to the vec tor− →r( which
+is− →r/circledottext). Therefore, we can write A=n(r/circledottext)r/circledottext. The trajectory of the light
+ray had been already constructed before like this, by Ye and Lin [13] and the
+value of deflection ( △φ), can be written as :
+△φ= 2/integraldisplay∞
+r/circledottextdr
+r/radicalbigg
+(n(r).r
+n(r/circledottext).r/circledottext)2−1−π (10)
+However, Ye and Lin [13], had in our opinion used a value of refractive
+indexn(r) which was approximated and somewhat ad hoc. Fischbach and
+Freeman [11] also in their attempt to calculate a more accurate value of
+deflection, considered terms only up to second order in the expres sion for
+refractive index. However, in our attempt to do so we shall avoid ma king
+any such approximation in the following. We denote the above integra l in
+Eqn. (10) by Iand write
+5I=/integraldisplay∞
+r/circledottextdr
+r/radicalbigg
+(n(r).r
+n(r/circledottext).r/circledottext)2−1
+=n(r/circledottext)r/circledottext/integraldisplay∞
+r/circledottextdr
+r/radicalBig
+(n(r).r)2−(n(r/circledottext).r/circledottext)2
+=n(r/circledottext)r/circledottext/integraldisplay∞
+r/circledottextdr
+r/radicaltp/radicalvertex/radicalvertex/radicalbtr4
+(r−rg)2−r4/circledottext
+(r/circledottext−rg)2
+=n(r/circledottext)r/circledottext/integraldisplay∞
+r/circledottextdr
+r2/radicaltp/radicalvertex/radicalvertex/radicalbt1
+(1−rg
+r)2−r2/circledottextr−2
+(1−rg
+r/circledottext)2(11)
+Now we change the variable to x=rg
+rand introduce a quantity a=rg
+r/circledottext.
+We also denote n(r/circledottext) byn/circledottext. Accordingly we write:
+I=n/circledottextr/circledottext/integraldisplay0
+a−x−2rgdx
+r2/radicalBig
+1
+(1−x)2−x2
+(a(1−a))2
+=n/circledottextr/circledottext/integraldisplay0
+a−x−2rgdx
+xr2/radicalBig1
+(x(1−x))2−1
+(a(1−a))2
+=n/circledottextr/circledottext
+rg/integraldisplaya
+0dx
+x/radicalBig1
+(x(1−x))2−1
+(a(1−a))2
+=n/circledottextr/circledottext
+rg/integraldisplaya
+0(1−x)dx/radicalbigg
+1−(x(1−x))2
+(a(1−a))2(12)
+For our convenience we can denote the quantity 1 /(a(1−a)) byD. This
+also implies
+D=r2/circledottext
+rg(r/circledottext−rg)(13)
+6However, the Integral Ican not be solved as a standard integral at this
+stage. We split the above Integral, as a sum of two Integrals and pr oceed as
+follows:
+I= (n/circledottextr/circledottext
+rg)[/integraldisplaya
+0(1−2x)dx/radicalBig
+1−D2x2(1−x)2+/integraldisplaya
+0xdx/radicalBig
+1−D2x2(1−x)2]
+= (n/circledottextr/circledottext
+rg)/integraldisplaya
+0(1−2x)dx/radicalBig
+1−D2x2(1−x)2+(n/circledottextr/circledottext
+rg)/integraldisplaya
+0xdx/radicalBig
+1−D2x2(1−x)2
+= (n/circledottextr/circledottext
+rg)I1+(n/circledottextr/circledottext
+rg)I2 (14)
+whereI1andI2are used to denote the above two integrals. Now we can
+identify
+n/circledottextr/circledottext
+rg=1
+1−a.1
+a=1
+a(1−a)=D
+Changing the variable from xtoy=Dx(1−x), we can write D(1−2x)dx=
+dy. Accordingly the upper and lower limits x= 0 andx=achange to y= 0
+andy=Da(1−rg
+r/circledottext) =1
+a(1−a)a(1−a) = 1. Therefore for the first part in
+Eqn (14) we can write :
+(n/circledottextr/circledottext
+rg)I1=/integraldisplaya
+0D(1−2x)dx/radicalBig
+1−D2x2(1−x)2
+=/integraldisplay1
+0dy√1−y2
+= [sin−1y]1
+0
+=π/2 (15)
+Therefore, from Eqn (10), one may write the amount of deflection as:
+△φ= 2/integraldisplay∞
+r/circledottextdr
+r/radicalbigg
+(n(r).r
+n(r/circledottext).r/circledottext)2−1−π
+7= 2(n/circledottextr/circledottext
+rg)I1+2(n/circledottextr/circledottext
+rg)I2−π
+=π+2(n/circledottextr/circledottext
+rg)I2−π
+= (2n/circledottextr/circledottext
+rg)/integraldisplaya
+0xdx/radicalBig
+1−D2x2(1−x)2(16)
+Thus the gravitational bending for a ray of light grazing the static g rav-
+itational mass ( with Schwarzschild radius rg) with the closest distance of
+approach r/circledottextcan be expressed as:
+△φ= 2D/integraldisplaya
+0xdx/radicalBig
+1−D2x2(1−x)2(17)
+The above expression for gravitational deflection has been obtain ed from
+the Schwarzschild Equation ( Eqn(1)), with out applying any approx imation
+at any stage. Owing to this, it is claimed that this expression of bendin g
+is more exact as compared to all other expressions derived till toda y, using
+equivalentmaterialmedium concept. However, theintegrationofthequantity
+in Eqn (17)), involves some complicated algebraic expressions conta ining
+Elliptical functions. Using Mathematica, we obtain the following expre ssion
+after integration :
+/integraldisplayxdx/radicalBig
+1−D2x2(1−x)2= 2(√
+D+√D−4)E−(2√D−4)F
+D(√
+D+4−√
+D−4)
+(18)
+whereE≡E(p,q2)istheEllipticIntegraloffirstkindand F≡F(−q,p,q2)
+is Incomplete Elliptic Integral of Third kind. The arguments p, q2,-q,p,q2are
+expressed by the following mathematical relations:
+p= arcsin/radicaltp/radicalvertex/radicalvertex/radicalbt(√
+D−4−√
+D+4)(√
+D−4+(2x−1)√
+D)
+(√
+D−4+√
+D+4)(√
+D−4−(2x−1)√
+D)
+(19)
+8q=(√
+D−4+√
+D+4)
+(√
+D−4−√
+D+4)
+(20)
+Finally we can write the expression for gravitational deflection ( △φ) of
+the light ray, due to a static mass ( rg) with the closest distance of approach
+r/circledottextas :
+△φ= 4/braceleftBigg(√
+D+√
+D−4)E−(2√
+D−4)F
+(√
+D+4−√
+D−4)/bracerightBiggx=a
+x=0(21)
+where the value of Dis given by Eqn.(13) as D=r2/circledottext
+rg(r/circledottext−rg)anda=
+rg/r/circledottext. Eqn. (21) is a general expression for bending of light, where r/circledottext
+can be replaced by the closest distance of approach of the light ray . This
+mathematical expression for deflection, derived here is claimed to b e more
+accurate than all other expressions derived so far using material medium
+approach and it is equally valid for strong field. For a Sun grazing ray,
+we can take the closest distance of approach as equal to solar rad ius which
+isr/circledottext= 695,500 km and Schwarzschild radius corresponding to the mass
+of Sun as rg= 3 km. We, therefore, get a= (rg/r/circledottext) = 1/231,833 and
+D= 231,834. Finally, we get a value of △φ= 8.62690E10−6radians or
+1.77943arc sec. This value of gravitational deflection suffered by a Sun
+grazing ray, is claimed to be more accurate than all other values obt ained in
+past.
+Acknowledgments: Some of the calculations reported in this paper
+were done using Mathematica, at IUCAA, Pune, India. I sincerely ac knowl-
+edge this help and support.
+9References
+[1] C W Misner, K S Thorne and J A Wheeler, Graviation ,(W.
+H. Freeman and Company, Newyork 1972)
+[2] S. Weinberg, Principles and Applications of the General
+Theory of Relativity ,(John Wiley & Sons Inc. 1972)
+[3] P. Schneider, P. Ehlers, E. E. Falco, Gravitational Lenses ,
+(Springer, Berlin 1999)
+[4]The Classical Theory of Fields volume 2; J. L. D. Landau and
+E. M. Lifshitz (1st edition Pergamon Press, 1951; 4 th editio n’
+Butterworth-Heinemann 1980)
+[5] N. L. Balazs , Phys. Rev.110, No. 1, 236 (1958)
+[6] J. Plebanski, Phys. Rev. 118, No.5, 1396 (1960)
+[7] J. E. Tamm, J. Russ. Phys.-Chem. Soc. 56,2-3, 284 (1924)
+[8]F de Felice, Gen. Relativ. Gravit. 2, No. 4, 347 (1971)
+[9] B Mashhoon, Phys. Rev. D 7, No. 10, 2807 (1973)
+[10]B Mashhoon, Phys. Rev. D 11, No. 10, 2679 (1975)
+[11] E. Fischbach and B. S. Freeman, Phys. Rev. D 67, (1980)
+[12]M Sereno, Phys. Rev. D 67, (2003)
+[13] X-H. Ye and Q. Lin , Journal Mod. Opt. 55, Issue 7, 1119
+(2007)
+[14] S. V. Iyer and A. O. Petters, Gen. Relativ. Gravit. 39,
+1563 (2007)
+[15] K. S. Virbhadra and George F. R. Ellis, Phys. Rev. D 62,
+084003 (2000)
+[16] S. Frittelli, T.P. Kiling, and E.T. Newman, Phys. Rev. D .
+62, 064021 (2000)
+[17]M. Born and E wolf 1947, Principles of Optics (7th Edition,
+Cambridge University Press, Cambridge,1999)p121
+10
\ No newline at end of file
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+ 1Surface magnetic states of Ni nanoch ains modified by using different 
+organic surfactants  
+Weimeng Chen1, Wei Zhou2, Lin He1, Chinping Chen1, and Lin Guo2  
+1Department of Physics, Peking Univer sity, Beijing, 100871, People’s Republic of 
+China 
+2School of Chemistry and Environment Scie nce, Beijing University of Aeronautics 
+and Astronautics, Beijing, 100191, People’s Republic of China  
+E-mail: cpchen@pku.edu.cn  and guolin@buaa.edu.cn   
+PACS: 75.70.Rf, 75.50.Lk   Keywords: Ni nanochain, core-shell struct ure, surface magnetic property, Ni@Ni
+3C. 
+ 
+Abstract  
+Three powder samples of Ni nanochains fo rmed of polycrystalline Ni nanoparticles 
+with an estimated diameter of about 30 nm  have been synthesized by a wet chemical 
+method using different organic surfactants. These samples, having magnetically/structurally core-shell structur es, all with a ferromagnetic Ni core, are 
+Ni@Ni
+3C nanochains, Ni@NiSG nanochains with a spin glass (SG) surface layer, and 
+Ni@NiNM nanochains with a nonmagnetic (NM) surface layer. The average thickness 
+of the shell for these three samples is de termined as about 2 nm. Magnetic properties 
+tailored by the different surface magnetism ar e studied. In particular, suppression in  2saturation magnetization usually observed with  magnetic nanoparticles is revealed to 
+arise from the surface magnetic states with the present samples.                        31. Introduction  
+Surface magnetic states have l ong been a subject of intens e studies in the past few 
+decades. It becomes increasingly important w ith the emerging field of nanomagnetics. 
+For biomedical applications, magnetic nanopa rticles play an important role for the 
+bounding of antibiotics, nucleotides, vitamins, peptides, etc. [1, 2, 3]. These depend 
+much on the magnetic properties of the nanoparticles, especially, the surface 
+magnetism. In information storage, the surface anisotropy plays an important role. It offers an extra degree of freedom to tune  the magnetic anisotropy energy, making it 
+attractive for basic investigati on [4, 5]. The fundamental role  of exchange bias, widely 
+counted on in spin valve and tunneling devices , is directly related to the interaction 
+from the interface or surface [6]. Therefor e, the investigations of surface magnetic 
+states are interesting a nd of great importance. 
+Theoretical and experimental works reveal  that surface atoms may have either 
+enhanced or quenched moments, depending on their chemical en vironment [7, 8]. 
+Enhancement of saturation magnetization, M
+S, has been reported in small-sized, 
+elemental metallic clusters [9]. However,  many more thin films and nanoparticles 
+demonstrate otherwise  [10]. It is reported th at the coating of organic molecules [11], 
+CO chemisorption [12] or carbonyl liga tion [13] on the surface of magnetic 
+nanoparticles dramatically affects the magnetic properties. The low temperature 
+magnetization reaches only 75% of the bul k value at low temperature for NiFe 2O4 
+nanoparticles coated by organic molecule s [11]. The chemisorption of CO on the 
+metallic Ni surface leads to the quenching of  Ni magnetic moments because electrons  4of the carbonyl ligation drive the Ni 4 s electrons to fill up the 3 d shell by repulsive 
+interaction. The Ni atoms in the surface la yer, therefore, become nonmagnetic (NM), 
+leaving the magnetism of the inner core unaffected [12, 13]. In many cases, surface 
+magnetic effects are usually featured with  a surface spin glass (SG) state and an 
+appreciable reduction of saturation magneti zation. For instance, surface SG behaviour 
+has often been observed with ferromagnetic (F M) or ferrimagnetic nanoparticles, such 
+as 6.5 nm NiFe 2O4 [14], 9-10 nm γ-Fe 2O3 [15], 12 nm Fe nanoparticles [16] and Ni 
+nanochains [17]. Even, the surface SG prope rties are reported w ith antiferromagnetic 
+(AFM) Co 3O4 nanowires having a magneti c core-shell structure of 
+Co3O4AFM@Co 3O4SG [18]. Surface magnetism has al so been observed with the 
+partially-oxidized compos ite nanoparticles, Cu@(Cu 2ONM+CuOAFM) [19]. 
+Nevertheless, there are other origins lead ing to surface “dead layer” of magnetic 
+nanoparticles and resulting in the suppressi on of saturation magnetization [20]. These 
+indicate that the surface magnetic state is one  of the most important factors dictating 
+the magnetic properties of nanoparticles.     Ni nanochains with dendritic morphol ogy have been fabricated previously using 
+polyvinyl pyrrolidone (PVP) as a surface modi fier [21, 22]. These samples show a SG 
+behavior in addition to a FM phase. The SG behaviors are an alyzed from the results of 
+magnetization measurements and are revealed to come from the surface layer [17]. It 
+has been suggested that the surface modi fier (PVP) actually tailors the surface 
+magnetic state. In order to further investig ate magnetic properties such as the finite 
+size effect of the ferromagnetic (FM) tran sition point with the Ni nanochains, the  5diamater-dependent magnetization reversal behavior with the quasi-one dimensional 
+chain-like nanostructure, and the possible transport properties with the nanochain 
+structures, it is important to understa nd the surface magnetic properties and the 
+thickness of the surface layer. We investig ate three powder samples of Ni nanochains 
+with magnetically/structurally core-shell structure, synthesized by a wet chemical method using different surface modifying agen ts. The estimated outer diameter of the 
+three samples is about 30 nm. They include Ni@Ni
+3C synthesized using 
+trioctylphosphine oxide (TOPO), and Ni na nochains of two kinds synthesized using 
+PVP and hexadecyltrimethyl ammonium bromide (CTAB), forming the magnetic core-shell structure of Ni@Ni
+SG and Ni@NiNM, respectively. They are thus labeled as 
+S-TOPO, S-PVP and S-CTAB. For the three samples, the cores are all of FM Ni, 
+however, with a surface shell layer of different magnetic pr operties, i.e., a NM surface 
+shell of Ni 3C for S-TOPO, a magnetically dead layer of Ni (also NM) for S-CTAB, 
+and a SG surface shell for S-PVP. 
+2. Sample preparations and characterizations  
+The detailed processes of synthesis ar e reported elsewhere. The process and 
+characterizations for S-TOPO are reported in  reference [23], S-PVP is Sample B in 
+reference [22]. The process and the chemical reagents used to prepare S-CTAB is the 
+same as that reported in reference [24], how ever, with a different reaction temperature 
+at 197 ºC. The chemical reagen ts used for the preparation of S-CTAB are analytical 
+grade without being further purified. In a typical experiment, 0.5 mmol NiCl 2·6H2O 
+and 3 mmol CTAB were dissolved in the so lvent of 60 ml glycol. Afterwards, the  6solution of hydrazine hydrate (50% v/v) by 1 ml was dropped into the mixture. When 
+the solution mixed homogeneously, it was h eated to the boiling poi nt (~197 ºC), and 
+kept for 5 hours. The as-obtained sample was washed with ethanol and deionized 
+water. 
+The crystal structures were character ized by powder X-ray diffraction (XRD) 
+using a Rigaku Dmax 2200 X-ray diffractometer with Cu K α radiation ( λ=0.1542 nm). 
+Transmission electron microscopy (TEM ) and high-resolution TEM (HRTEM) 
+investigations were carr ied out by a JEOL JEM-2100F microscope, equipped with 
+EDS (energy dispersive X-ray spectroscopy ). The thermogravimetric (TG) analysis 
+was performed using a Diamond therm ogravimetric analyzer (Perkin–Elmer 
+instruments) under a stream of air. The pr oduct was heated from 50 °C to 600 °C at a 
+scan rate of 10 °C/min. 
+Figure 1 shows the XRD patterns for S-TOPO , S-PVP and S-CTAB. It reveals that 
+S-PVP and S-CTAB are of pure Ni phase w ith a face-centered cubic (fcc) structure. 
+The diffraction peaks correspond to the planes  of (111), (200) and (220) of fcc Ni 
+(JCPDS 04-0850). Besides the nickel phase, S-TOPO contains the Ni 3C phase also, 
+which has been characterized in detail and reported in [23]. Th e peaks marked with 
+open triangles could be assigned to Ni 3C (JCPDS 77-0194), corresponding to the 
+planes of (110), (006), (113), (116), (300), and (119) of Ni 3C. It is noteworthy that the 
+(111) peak of Ni overlaps the (113) peak of Ni 3C. 
+Figures 2a, 2b, and 2c show the TEM im ages of S-TOPO, S-PVP, and S-CTAB, 
+respectively. The morphology is similar, showing chain-like shape with dendritic  7structure. Their average diameter is estim ated about 30 nm. The corresponding insets 
+show the magnified images. In the inse t of figure 2a, a HRTEM image of Ni@Ni 3C 
+with a core-shell structure is presented. Th ere is an almost invi sible and very thin 
+capped layer outside the Ni 3C shell. It is known that a residual nonmagnetic, organic 
+capped layer, TOPO in this case, is inev itable even after se veral times of thorough 
+rinsing. About 10% mass ratio of the organic capped layer is determined by the TG measurement described below. The Ni
+3C shell is estimated from  the inset about 2 to 4 
+nm in thickness, which is slightly larger  than the average th ickness of about 2 nm 
+determined by the magnetic measurements. In the inset of figur e 2b for S-PVP, the 
+HRTEM image reveals that the Ni chains ar e covered with a vague  layer of organic 
+remnants. A layer of almost invisible orga nic remnants is also observed for S-CTAB, 
+as shown in the inset of fi gure 2c. To further confirm the composition of the samples, 
+EDS measurements were conducted, showi ng pure Ni element without any other 
+magnetic elements.   The mass ratio of the nonmagnetic or ganic capped layer is determined by TG 
+measurements, as shown in figure 3. The ma ss of the as-prepared samples at room 
+temperature is denoted as m
+0. By heating up the samples gradually, the mass first 
+decrease slightly due to the burning of th e organic remnants. Then, the mass increases 
+dramatically arising from the oxidati on of Ni, forming NiO, denoted as m(NiO). The 
+mass of Ni is then calculated according to  the relative ratio of  the formula weight, 
+m(Ni) = (59/75) m(NiO). The correction factor, m(Ni)/ m0, is then determined as 90%, 
+87%, and 93% for S-TOPO, S-PVP, and S-CTAB, respectively. The mass correction  8factor has been accounted fo r in the normalization of the measured magnetization. It 
+is noted that the mass of C atoms in the Ni 3C shell layer is not taken into account in 
+the above estimation. The NM shell of Ni 3C is treated as if it were a magnetically 
+dead layer of Ni. This introduces  about 2% error in mass for Ni 3C being treated as Ni. 
+3. Magnetic measurements and analysis  
+The magnetization measurements were performed by a Quantum Design SQUID 
+magnetometer. The measurements include, a) temperature dependent saturation 
+magnetization, MS(T), recorded in the field of 20 kOe from T = 380 to 5 K, b) 
+zero-field-cooling (ZFC) and fiel d-cooling (FC) measurements,  MZFC(T) and MFC(T) 
+from T = 5 to 380 K, and c) field dependent hysteresis loops, M(H), at a series of 
+fixed temperature from T = 5 to 380 K. 
+Figure 4 shows the temperature depe ndence of saturation magnetization, MS(T), of 
+the three samples recorded in the applied field of Happ = 20 kOe. At T > 80 K, the 
+three MS(T) curves nearly collapse. The values of MS(T) at T = 300 K are determined 
+as 37.7 emu/g (S-TOPO), 38.2 emu/g (S-PVP ), and 38.3 emu/g (S-CTAB). They 
+account for about 70% of the corresponding bulk value, ~54.2 emu/g, at 300 K [25]. 
+The reduction of the saturation magnetization is attributed to the magnetically inert property of the surface shell. It is noted th at the mass effect of the NM organic capped 
+layer has already been corrected for by the TG measurements. Otherwise, the saturation magnetization per unit mass would have been further reduced by 10 to 15%. For S-TOPO, the reduction in the satura tion magnetization is obvious because the 
+Ni
+3C shell is NM [26]. The average diameter of the FM cores is then estimated as  9Dcore = 26.6 nm with the Ni 3C shell thickness of about 1.7 nm. It is consistent with the 
+result of shell thickness,  1-4 nm, observed by the HRTEM investigation [23]. For 
+S-PVP and S-CTAB, although there is no obvious shell structure like the Ni 3C shell of 
+S-TOPO, a FM core of Ni with roughly the same diameter, Dcore ~ 26 nm, is also 
+concluded. At T < 80 K, MS(T) of S-PVP increases dram atically, reaching the bulk 
+value of Ni, ~ 55 emu/g. It is attributed to  the contribution from the surface SG shell 
+[17]. On the other hand, th e shell layer of S-CTAB, although of pure Ni, does not 
+show any magnetism at all, very much like S-TOPO with a NM Ni 3C shell layer. 
+Therefore, S-CTAB exhibits properties of a magnetically core-shell structure, 
+Ni@NiNM, with a magnetically “dead shell” of NiNM. The presence of a magnetically 
+“dead layer” has been reported in the early days with the surface of Ni films [27]. In 
+addition, the magnetic moments of the surface  Ni are reported to be quenched, e.g., by 
+the carbonyl ligation on the surface [13].    
+Figure 5 shows the MZFC(T) and MFC(T) curves. The ZFC curves were recorded in 
+the applied field, Happ = 90 Oe, in the warming process after the sample was cooled 
+down to T = 5 K under zero applied fi eld. For the FC measurement, the procedure of 
+data collection was the same except that th e sample was cooled down to 5 K in the 
+applied magnetic field of 20 kOe. For all of the three samples, the MFC(T) and MZFC(T) 
+curves separate from each other with the temperature going up to 380 K. It indicates 
+that the blocking temperature is higher th an 380 K. The inset shows the blown-up 
+MZFC(T) curves in the low temperature region. A freezing peak appears at TF ~ 8 K 
+with S-PVP, similar to those at about 13 K observed for 50, 75 and 150 nm Ni  10nanochains also synthesized usin g the surfactant of PVP [17] . It further confirms that 
+S-PVP exhibits a magnetically co re-shell structure of Ni@NiSG. For the other two 
+samples, S-TOPO and S-CTAB, there is no such  characteristic featur e, as is expected 
+for the samples with a NM shell layer.      T h e  M(H) measurements for the three samples are performed at various 
+temperature from 5 to 380 K. Figure 6a shows the M(H) curves measured at T = 5 K. 
+The magnetization determined in the high field region at H = 10 kOe, and the 
+coercivity, H
+C, are 45.0 emu/g and 600 Oe for S-TOPO, 52.3 emu/g and 568 Oe for 
+S-PVP, and 43.1 emu/g and 634 Oe for S-CTAB , respectively. The magnetization of 
+S-PVP is higher than that of S-TOPO, or S-CTAB by about 20%, attributed to the 
+contribution from the SG shell. The M(H) curves at T = 300 K is shown in Figure 6b. 
+The difference in saturation magnetization is  more or less reduced. The coercivity 
+determined from the M(H) curves at various temperatur es is shown in figure 6c. The 
+magnetization reversal can be described by the fanning mode based on the chain of 
+sphere model proposed by Jacobs and Bean [28],  as discussed in our previous work 
+for S-TOPO [23].  
+4. Conclusion  
+We have investigated the magnetic properties of three samples of Ni nanochains, with 
+an estimated outer diameter of 30 nm, synthesized by different surface modifying 
+agents, including CTAB, TOPO and PVP.  The surface magnetic properties are 
+modified significantly and differently by th e surfactants in the synthesis process, 
+forming different magnetic shell structures . The CTAB leads to the nanochains of  11Ni@NiNM with a magnetically dead layer of Ni  surface shell, while  the TOPO makes 
+the final products of Ni@Ni 3C nanochains with a shell of NM Ni 3C. With PVP, a SG 
+shell layer of Ni is formed, resulti ng in the magnetic structure of Ni@NiSG. The 
+saturation magnetization of these samples acco unts for only 70% of the bulk value at 
+300 K attributed to the presence of the magne tically inert layer. This value would be 
+further reduced by more than 10% if the mass of the outermost capped layer of 
+organic remnants is not corrected for. The th ickness of the shell layer is determined as 
+about 2 nm for all of the three samples.  
+Acknowledgements 
+This work is supported by the NSFC (Grant No.10874006 and 50725208), State Key 
+Project of Fundamental Research for Nanoscience and Nanotechnology 
+(2006CB932301), National Basic Research Program of China (Nos. 2009CB939901 
+and 2010CB934601), and Specialized Research  Fund for the Doctoral Program of 
+Higher Education (20060006005) as well as  by the Innovation Foundation of BUAA 
+for PhD Graduates.          12References 
+[1] Pankhurst Q A, Connolly J, Jones S K and Dobson J 2003 J. Phys. D: Appl. Phys.  
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+[2] Tartaj P, Morales M P, Veintemilla s-Verdaguer S and Gonzalez-Carreno T 2003 J. 
+Phys. D: Appl. Phys.  36, R182  
+[3] Lu A, Salabas E L and Schuth F 2007 Angew. Chem. Int. Ed  46, 1222 
+[4] Gambardella P, Rusponi S, Veronese M,  Dhesi S S, Grazioli C, Dallmeyer A, 
+Cabria I, Zeller R, Dederichs P H, Kern K, Carbone C and Brune H 2003 Science  
+300, 1130 
+[5] Kaneyoshi T, 1991 J. Phys.: Condens. Matter 3, 4497 
+[6] Nogues J, Sort J, Langlais V , Skumryev  V , Surinach S, Munoz J S and Baro M D 
+2005 Physics Reports  422, 65 
+[7] Bader S D 2002 Surf. Sci.  500 172  
+[8] Liu F, Press M R, Khanna S N and Jena P 1989 Phys. Rev. B  39 6914  
+[9] Jensen P J and Bennemann K H 1995 Phys Z. D 35 273  
+[10] Dormann J L, Fiorani D and Tronc E 1997 Adv. Chem. Phys.  XCVIII 283  
+[11] Berkowitz A E, Lahut J A, Jacobs I S, Levinson L M and Forester D W 1975 
+Phys. Rev. Lett.  34 594  
+[12] Feigerle C S, Seiler A, Pena  J L, Celotta R J and Pierce D T 1986 Phys. Rev. Lett.    
+56 2207 
+[13] van Leeuwen D A, van Ruitenbeek J M,  de Jongh L J, Cerio tti A, Pacchioni G, 
+Haberlen O D and Rosch N 1994 Phys. Rev. Lett.  73 1432   13[14] Kodama R H, Berkowitz A E,  McNiff, Jr. E J and Foner S 1996 Phys. Rev. Lett.  
+77 394  
+[15] Martinez B, Obradors X, Balc ells Ll, Rouanet A and Monty C 1997 Phys. Rev. 
+Lett. 80 181  
+[16] Bonetti E, Bianco L D, Fiorani D, Ri naldi D, Caciuffo R and Hernando A 1999    
+Phys. Rev. Lett.  83 2829  
+[17] He L, Zheng W Z, Zhou W, Du H L, Chen C P and Guo L 2007 J. Phys.: 
+Condens. Matter  19 036216   
+[18] Benitez M J, Petracic O, Salabas E L, Radu E, Tuysuz H, Schuth F and Zabel H 
+2008 Phys. Rev. Lett.  101 097206    
+[19] Li Q, Zhang S W, Zhang Y and Chen C P 2006 Nanotechnology  17 4981 
+[20] Westman C, Jang S, Kim C, He S, Harmon G, Miller N, Graves B, Poudyal N, 
+Sabirianov R, Zeng H, DeMarco M and Liu J P 2008 J. Phys. D: Appl. Phys. 41 
+225003 
+[21] Liu C M, Guo L, Wang R M, Deng Y , Xu H B and Yang S H 2004 Chem. 
+Commun.  23 2726 
+[22] Zhou W, He L, Cheng R, Guo L, Chen C P and Wang J L. 2009 J. Phys. Chem. C  
+113 17355 
+[23] Zhou W, Zheng K, He L, Wang R M, Guo L, Chen C P, Han X D and Zhang Z 
+2008 Nano Lett . 8 1147 
+[24] Zhou W, Guo L, He L, Chen C P 2008 Physica Status Solidi (a)  205 1109 
+[25] Sellmyer D J, Zheng M and Skomski R 2001 J. Phys.: Condens. Matter  13 R433   14[26] Yue L P, Sabiryanov R, Kir kpatrick E M, Leslie-Pelecky D L 2000 Phys. Rev. B 
+62 8969 
+[27] Liebermann L, Clinton J, Edwards D M and Mathon J 1970 Phys. Rev. Lett.  25 
+232  
+[28] Jacobs L S and Bean C P 1955 Phys. Rev . 100 1060  
+ 
+                 15Figure captions 
+Figure 1. XRD patterns for samples modifi ed with TOPO (S-TOPO), PVP (S-PVP) 
+and CTAB (S-CTAB).  Figure 2. Morphology revealed by TEM images  for (a) S-TOPO, (b) S-PVP and (c) 
+S-CTAB. The insets are the corresponding HRTEM images.   Figure 3. Thermogravimetric analyses fo r the samples of S-TOPO, S-PVP and 
+S-CTAB. The mass of the as-pre pared sample is denoted as m
+0, and the final NiO is 
+denoted as mNiO. The calculated Ni contents, mNi, is 93%, 90% and 87% for TOPO, 
+CTAB and PVP, respectively.  
+ Figure 4. Saturation magnetization, M
+S(T), recorded in the a pplied field of 20 kOe 
+from 380 to 5 K. The magnetization of  S-PVP increases dramatically at T < 80 K. 
+ Figure 5. ZFC and FC M(T) curves for S-TOPO, S-PVP a nd S-CTAB measured in the 
+applied field of 90 Oe from 5 K to 380 K. The inset shows the ZFC curves in the low 
+temperature region. The peak ar ound 8 K in the ZFC curve with  S-PVP is attributed to 
+the freezing of the su rface SG shell.  
+ Figure 6. M(H) curves for S-TOPO, S-PVP and S-CTAB at (a) T = 5 K, (b) T = 300 K, 
+and (c) coercivity de termined from the M(H) curves at different temperatures.   16Figures   
+              
+Figure 1 
+      
+  17 
+         
+Figure 2a 
+          
+Figure 2b 
+ 
+  18 
+            
+Figure 2c 
+        
+  19 
+            
+Figure 3 
+        
+  20 
+             
+Figure 4  
+       
+ 21 
+             
+Figure 5 
+       
+ 22 
+        
+Figure 6a 
+          
+Figure 6b 
+ 
+ 
+  23 
+             
+Figure 6c  
+  
+ 
\ No newline at end of file
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+arXiv:1001.0299v1  [math.CO]  2 Jan 2010The solutions of four q-functional equations
+Jun-Ming Zhu∗
+Abstract In this note we obtain the solutions of four q-functional equations and express
+the solutions in q-operator forms. These equations give sufficient conditions forq-operator
+methods
+Key words: Basic hypergeometric series, q-Series,q-Functional equation, q-Difference
+equation, q-Exponential operator
+1 Introduction
+We follow the notation and terminology in [6] , and for conven ience, we always assume
+that 0<|q|<1. Theq-shifted factorials are defined by
+(a;q)n=/braceleftBigg
+1, n = 0,
+(1−a)(1−aq)···(1−aqn−1), n= 1,2,3,······,
+Theq-derivative operator is defined by
+Dq{f(a)}=f(a)−f(aq)
+a, Dn
+q{f(a)}=Dq{Dn−1
+q{f(a)}}.
+The operator θis defined by
+θ=η−1Dq, θn{f(a)}=θ{θn−1{f(a)}},
+whereη−1{f(a)}=f(q−1a).BothDqandθare obviously linear transforms, and by
+convention, D0
+qandθ0are both understood as the identity.
+The two q-exponential operators (see, [2, 3, 7, 8, 10]) are defined by
+T(bDq) =∞/summationdisplay
+n=0(bDq)n
+(q;q)n,
+and
+E(bθ) =∞/summationdisplay
+n=0qn(n−1)/2(bθ)n
+(q;q)n,
+respectively.
+∗E-mail address: jm zh@sohu.com, junming zhu@163.com
+1The following operator was first introduced by Fang [5]. But w e follow Chen and Gu’s
+notation [1]. It seems to be more convenient.
+T(a,b;Dq) =∞/summationdisplay
+n=0(a,q)n
+(q;q)n(bDq)n.
+Chen and Gu [1] named T(a,b;Dq) as Cauchy operator. But compared with the following
+operator, we called T(a,b;Dq) the first Cauchy operator in this paper.
+The operator T(a,b;θ), introduced by Fang [5], is defined by
+T(a,b;θ) =∞/summationdisplay
+n=0(a,q)n
+(q;q)n(bθ)n.
+We called T(a,b;θ) the second Cauchy operator.
+All the papers using the q-operator methods imply the following definition but do not
+state it explicitly. Unless otherwise stated, all the opera tors are applied with respect to
+the parameter a.
+Definition 1.1
+T(bDq){f(a)}=∞/summationdisplay
+n=0bn
+(q;q)n{Dqn{f(a)}},
+and
+E(bθ){f(a)}=∞/summationdisplay
+n=0qn(n−1)/2bn
+(q;q)n{θn{f(a)}}.
+The first and the second Cauchy operators and the operators T(bθ) andE(bDq) below
+also act in this way.
+When using the methods of operators, rearrangememts of seri es are often emploied.
+We know that rearrangememts of series must be under the condi tion that the series are
+absolutely convergent, which may benot very easy to check so metimes. Inhis paper [9] ,
+Liu obtained the following two theorems using the q-functional equation method.
+Theorem 1.1Letf(a,b)be a two variables analytic function in a neighborhood of
+(a,b) = (0,0)∈C2, satisfying the q-difference equation
+bf(aq,b)−af(a,bq) = (b−a)f(a,b).
+Then we have
+f(a,b) =T(bDq){f(a,0)}.
+Theorem 1.2Letf(a,b)be a two variables analytic function in a neighborhood of
+(a,b) = (0,0)∈C2, satisfying the q-difference equation
+af(aq,b)−bf(a,bq) = (a−b)f(aq,bq).
+Then we have
+f(a,b) =E(bθ){f(a,0)}.
+2These two theorems give us a method to use operator T(bDq) andE(bθ) without having
+to check the absolute convergence of the series.
+Originated by Liu’s work, we give the above two theorems more general forms in the
+following section.
+2 Four q-functional equations and the solutions
+Theorem 2.1Letf(a,b,c)be a three variables analytic function in a neighborhood
+of(a,b,c) = (0,0,0)∈C3, satisfying the q-functional equation
+c(f(a,b,c)−f(a,bq,c)) =b(f(a,b,c)−f(a,b,cq)−af(a,bq,c)+af(a,bq,cq)).(1)
+Then we have
+f(a,b,c) =T(a,b;Dq){f(a,0,c)},
+whereT(a,b;Dq)is applied with respect to the parameter c.
+Theorem 2.2Letf(a,b,c)be a three variables analytic function in a neighborhood
+of(a,b,c) = (0,0,0)∈C3, satisfying the q-functional equation
+c(f(a,bq,cq)−f(a,b,cq)) =b(f(a,bq,cq)−f(a,bq,c)−af(a,b,c)+af(a,b,cq)).(2)
+Then we have
+f(a,b,c) =T(−1
+a,ab;θ){f(a,0,c)},
+whereT(−1
+a,ab;θ)is also applied with respect to the parameter c.
+When letting a→0 in theorem 2.1 and 2.2, we get theorem 1.1 and 1.2, respectiv ely.
+The proof of theorem 2.2 is similar to that of theorem 2.1 and s o is omitted.
+Proof of Theorem 2.1 We now begin to solve this equation. From the theory of
+several complex variables, we assume that
+f(a,b,c) =+∞/summationdisplay
+n=0An(a,c)bn. (3)
+We substitute the above equation into (1) to get
+c+∞/summationdisplay
+n=0(1−qn)An(a,c)bn= (+∞/summationdisplay
+n=0An(a,c)−+∞/summationdisplay
+n=0An(a,cq)−a+∞/summationdisplay
+n=0An(a,c)qn
++a+∞/summationdisplay
+n=0An(a,cq)qn)bn+1.
+This is
+c+∞/summationdisplay
+n=1(1−qn)An(a,c)bn=+∞/summationdisplay
+n=1(1−aqn−1)(An−1(a,c)−An−1(a,cq))bn.
+3Comparing the coefficients of bngives
+c(1−qn)An(a,c) = (1−aqn−1)(An−1(a,c)−An−1(a,cq)).
+Then we have
+An(a,c) =(1−aqn−1)
+(1−qn)(An−1(a,c)−An−1(a,cq))
+c
+=(1−aqn−1)
+(1−qn)Dq{An−1(a,c)},
+whereDqis applied with respect to the parameter c. Iterate the above equation to get
+An(a,c) =(a;q)n
+(q,q)nDq{A0(a,c)}. (4)
+It remains to calculate A0(a,c). Putting b= 0 in (3), we immediately deduce that
+A0(a,c) =f(a,0,c). Substituting (4) back into (3) gives
+f(a,b,c) =∞/summationdisplay
+n=0(a,q)n
+(q;q)n(bDq)n{f(a,0,c)}=T(a,b;Dq){f(a,0,c)}.
+This completes the proof.
+Using the q-functional equation method we easily obtain the following two theorems.
+Theorem 2.3Letf(a,b)be a two variables analytic function in a neighborhood of
+(a,b) = (0,0)∈C2, satisfying the q-functional equation
+af(a,b)+bf(aq,bq) = (a+b)f(a,bq).
+Then we have
+f(a,b) =E(bDq){f(a,0)},
+where the operator E(bDq)is defined by
+E(bDq) =∞/summationdisplay
+n=0qn(n−1)/2(bDq)n
+(q;q)n.
+Theorem 2.4Letf(a,b)be a two variables analytic function in a neighborhood of
+(a,b) = (0,0)∈C2, satisfying the q-functional equation
+af(aq,bq)+bf(a,b) = (a+b)f(aq,b).
+Then we have
+f(a,b) =T(−bθ){f(a,0)},
+where the operator T(bθ)is defined by
+T(bθ) =∞/summationdisplay
+n=0(bθ)n
+(q;q)n.
+From all the theorems in this little paper, we can look at the q -operators from a different
+standpoint.
+4References
+[1] V. Y. B. Chen, N. S. S. Gu, The Cauchy operator for basic hypergeometric series , Adv.
+in Appl. Math, 41(2008), 177–196.
+[2] W. Y. C. Chen, Z.-G. Liu, Parameter augmentation for basic hypergeometric series,
+II, J. Combin. Theory Ser. A 80(1997), 175–195.
+[3] W. Y. C. Chen, Z.-G. Liu, Parameter augmentation for basic hypergeometric series,
+I, in: B. E. Sagan, R. P. Stanley (Eds.), Mathematical Essays in honor of Gian-Carlo
+Rota, Birk¨ auser, Basel, 1998, pp. 111–129.
+[4] J.-P. Fang, q-Differential operator identities and applications , J. Math. Anal. Appl.
+332(2007), 1393–1407.
+[5] J.-P. Fang, q-Operator identities and its applications , Journal of East China Normal
+University, 1(2008), 20–24 (in Chinese).
+[6] G. Gasper, M. Rahman, Basic Hypergeometric Series , 2nd ed., Cambridge Univ. Press,
+Cambridge, MA, 2003.
+[7] Z.-G. Liu, A new proof of the Nassrallah-Rahman integral , Acta Mathematica Sinica
+41(1998), 405–410 (in Chinese).
+[8] Z.-G. Liu, Some operator identities and q-series transformation formulas , Discrete
+Math.,265(2003), 119–139.
+[9] Z.-G. Liu, Twoq-difference equations and q-operator indetities , Journal of Difference
+Equations and Applications (in press).
+[10] L. J. Rogers, On the expansion of some infinite products , Proc. London Math. Soc.,
+24(1894), 337-352.
+5
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+ 1 Stability of Multipole-mode So litons in Thermal Nonlinear 
+Media 
+Liangwei Dong1*, Fangwei Ye 2 
+1Institue of Information Optics of Zhe jiang Normal University, Jinhua, China, 321004 
+2Department of Physics, Centre for Nonlinear Studies,  
+and The Beijing-Hong Kong-Singapore Joint Centre 
+ for Nonlinear and Complex System(Hong Kong), 
+Hong Kong Baptist University, Kowloon Tong, China 
+*Email: donglw@zjnu.cn 
+We study the stability of multipole-mode solitons in one-dimensional thermal nonlinear media. We show how the sample  geometry impacts the stability of 
+mutlipole-mode solitons and reveal that th e tripole and quadrupole can be made stable 
+in their whole domain of existence, provided that the sample width exceeds a critical 
+value. In spite of such geometry-dependent  soliton stability, we find that the maximal 
+number of peaks in stable multipole-mode solitons in thermal media is the same as 
+that in nonlinear materials with finite-range nonlocality. 
+PACS number(s): 42.65.Tg, 42.65.Jx, 42.65.Wi  
+Solitons in nonlinear media can take a va riety of forms and shapes. In addition 
+to the node-less amplitude distributions fe atured by fundamental solitons, solitons 
+could also display complex spatial shapes. In one-dimensional system, these are dipole-modes which are composed of two out-of-phase fundamental solitons, or  2 multipole-mode solitons which are composed of several fundamental solitons with 
+alternating phases. Due to the repulsive fo rces between the peak s of them, multipole 
+soliton can not exist in uniform media with  local nonlinear responses, except in the 
+form of vector solitons under suitable c onditions. However, such situation changes 
+significantly in nonlocal nonlinear media, where,  in sharp contrast to the case in local 
+media, out-of-phase bright solitons can attr act each other and may even, form scalar 
+bound states [1].    
+The nonlocality of the nonlinear response is a generic property of nonlinear 
+materials. Nonlocality arises when the light-matter interaction involves mechanism 
+such as diffusion of carriers, reorientation of molecules, heat transfer, etc [2, 3]. 
+Recent interest in the study of nonlocal  solitons has been stimulated by the 
+experiments in nematic liquid crystals (NLCs) [4-7] as well as in thermal materials such as lead glasses [8, 9]. The nonlocal nonlinear responses of these two materials 
+could be quite different. The nonlinear respons e of a NLC is charac terized by a finite 
+nonlocality degree, thus when the size of the sample is large enough relative to the 
+spatial extent of the light, the boundary of a NLC does not affect the light dynamics or 
+soliton properties. In contrast, in a thermal medium, the nonlinear response is always greatly determined by the details of heat  diffusion at sample boundaries and thus the 
+nonlocality degree is naturally  “infinite” as the afar boun daries could significantly 
+change the light dynamics and soliton proper ties [12-18], even when spatial extent of 
+the light beam is significant small compared  with that of the sample. Especially, 
+numerical simulations [14] ha ve demonstrated that the so liton stability may depend  3 crucially on the sample-geometry and a recta ngular sample with a proper aspect ratio 
+may stabilize the otherwise unstable dipole mode s in a square sample. Note that such 
+strongly sample-geometry-dependent sol iton properties are specific to thermal 
+materials and are usually abse nt in both local and finite -range nonlocal materials. 
+The present work is motivat ed by the study of Z. Xu et al. in Ref. [11], where 
+the authors have revealed that, in a nonlinea r material with a finite-range nonlocality 
+such as in a NLC, the maximal number of peaks in stable multipole-mode solitons is 
+four and the modes containing more than four peaks are always unstable. As the 
+authors in Ref. [11] deal with a material  with a finite-range nonlocality, no boundary 
+effects have been took into consideration. In  contrast, in nonlocal thermal materials, 
+boundary is always crucial, and therefor e, one may naturally ask, how a thermal 
+nonlinear material affects the stability of multipoles? Especially, one might ask whether a nonlinear response of an infi nite-range nonlocality supports stable 
+multipoles composed of more peaks than th at supported by a fin ite-range nonlocality? 
+It is worthy to mention that, the physical model describing light propagation in a NLC 
+without the static bias electri c field is mathematically equi valent to that in a thermal 
+nonlinear medium, thus, boundary effects may also come into play in a NLC, as numerically analyzed and experimentally observed in [24,25]. Therefore our present 
+study may also find potential applications in liquid crystals, although we following 
+put our research in the context of  thermal nonlocal materials.   
+The purposes of this paper are tw ofold. First, we study whether the 
+sample-geometry-dependent soliton stability in  thermal materials, which was revealed  4 in two-dimensional settings where the contro lling parameter being the sample aspect 
+ratio, also persists in one-dimensional ge ometry when the controlling parameter 
+simply being the sample width. Indeed, we find that the stability of multipole-mode solitons depends crucially on the physical dimension of the thermal material, and complete stability is achieved for tri pole- and quadrupole-mode solitons when the 
+width of the sample exceeds some crucial value. We are able to build this important finding on a rigorous linear stability analysis , which is absent in higher-dimensional 
+settings [14] due to the requirement of huge computational resources. The second 
+purpose of this paper is to reveal the maxi mum number of peaks in stable multipoles 
+that hold in a thermal nonlinear medium. This purpose stems from the first one, as the stability characteristics of multipoles now cr ucially depends on sample width and thus 
+the stability analysis for NLCs [11] does not hold in thermal nonlinear media. 
+Surprisingly, we found that a nonlin ear thermal medium possesses the same 
+restriction on the maximum number of peaks in stable multipoles as in the case of a 
+finite-range nonlocal medium. In another word, the maximal number of peaks in 
+stable multipoles supported by thermal materi als is also four and all modes composed 
+of five peaks or more are unstable. 
+We consider the propagation of a laser beam along the 
+z axis of a thermal 
+medium occupying the region /2 /2Lx L-£ £ + ( L is the sample width), which 
+can be described by the system of equati ons for the dimensionless field amplitude q 
+and the nonlinear contribution to the refractive index n that is proportional to the 
+temperature variation, given by  5 2
+2
+22
+21,2
+,qqiq nz x
+nq
+x¶¶=- -¶ ¶
+¶=-
+¶  (1) 
+Here the transverse and longitudinal coordinate ,xz are scaled to the beam width 
+and to the diffraction length, respectively.  We assume that the opposite boundaries 
+of the thermal medium are thermostabilized, thus,/2,| 0xLqn==. 
+  We search for soliton solutions of Eqs. (1) in the form () e x p ( ) qw x i b z= , 
+where w is a real function and b is the propagation constant . Such solitons can be 
+characterized by their energy flow/22
+/2||L
+LUq d x
+-=ò, which is a conserved quantity 
+of Eqs.(1). The refractive index induced by the soliton is given 
+by/22
+/2(, ' ) | ( ' ) | 'L
+LnG x x q x d x
+-=-ò, where the response function 
+1(, ' ) ( / 2 ) ( ' / 2 )Gxx L x L x L-=+ - for ' xx£ and 
+1( , ' ) ( ' /2 ) ( /2 )Gxx L x L x L-=+ - for ' xx³ . To elucidate the linear stability of 
+solitons we searched for perturbed solution in the form of 
+**( , ) [ ( ) ( )exp( ) ( )exp( )]exp( )qxz wx ux iz v x i z i b z dd =+ + - , where the perturbation 
+modes u and v can grow with a complex rate d upon propagation(“ *” stands for 
+complex conjugate). Linearization of  Eqs. (1) around stationary solution ()wx  
+yields, 
+/2
+/2
+/2
+/21( ,' )(' ) [(' ) (' ) ] ' ,2
+1(, ' )( ' ) [( ' ) ( ' ) ] ' ,2L
+xx
+L
+L
+xx
+Luu b u u n w G x x w x u x v x d x
+v v bv vn w G x x w x u x v x dxd
+d-
+-=- + + +
+=- + - - +ò
+ò   (2)  6 Figure 1 (a) presents an example for funda mental solitons. As shown in the figure, 
+the induced refractive index ()nx is nonzero across the whole sample, which is due to 
+the infinite range of nonlo cality. Note also that the ()nx  produced by the dipole 
+displays a maximum platform near 0 x=[Fig. 1 (b)], rather a small dip encountered 
+in a NLC [11].  Such index platform expa nds for lower-power dipoles [Fig. 1(c)].  
+The dependence of energy flow Uversus propagation constant bis shown in Fig.1 
+(d). We find that fundamental and dipole so litons are stable in their whole domain of 
+existence. 
+Thermal nonlinear media in principle s upport bound states with arbitrary number 
+of peaks. This can be explained from their associated index profiles, as ()nx always 
+features a parabolic-like shape across the w hole light regions [Fi g. 2(a), Fig. 2(d) and 
+Fig. 3(a)]. The stability analysis for tr ipole and quadrupole-mode solitons is 
+summarized in Fig.2. An important result is th at the stability of solitons is strongly 
+dependent on the sample width, and inst ability domain shrinks  quickly with the 
+increase of the sample width [Fi g. 2(c) and (f)]. For example, when 20 L= , tripole 
+modes are stable in the region 0.7crbb>» . However, when Lis increased to 
+40 L=  the stable region expands to 0.15crbb>» [Fig. 2 (b)]. Further, 
+completely stable are achieved if 60 L> . Such sample-width-dependent stability 
+characteristics are also observed for quadrup ole solitons [Fig. 2(e)]. However, it 
+should be mentioned that the instability region, defined by [0, ]crbbÌ , for 
+quadrupoles is wider than that for tripol e. Also the instab ility growth rate 
+id(imaginary part of d) for quadrupole is higher than th at for tripoles. Note that, as  7 mentioned above, such geometry-dependent st ability have been reported recently in 
+two-dimensional thermal media for dipole so litons [14], where the dipole modes with 
+correct orientation with respect to sample ge ometry are found to be stable provided 
+that the sample is cut into a suitable geom etrical aspect ratio. However, the finding in 
+[14] is based on the propagation simulations wh ile a linear stability analysis is absent 
+as the latter requires huge computational re sources. In contrast, here we work on a 
+most fundamental system, thus allowing us to build the im portant result of 
+geometry-dependent-stability on a ri gorous linear stability analysis.  
+Another important finding of our stability  analysis is that, although the soliton 
+stability in thermal materials has a pr ofound dependence on the sample-width as 
+discussed above, the maximum number of p eaks in stable multipoles is still four, 
+which is the same as in a nonlinear medi um with a finite-ra nge nonlocality. Thus, 
+bound states incorporating five [Fig.3 (a)] or more than five peaks are always 
+oscillatory unstable. Such global instability holds irrespective of variation of energy 
+flow [Fig.3 (b)] or sample wi dth, although the specific value of id quantitatively 
+depends on both. The same re striction on the maximal numb er of peaks in stable 
+multipoles for a thermal medium and a NLC might counter one’s intuition, as in NLC 
+the instability domain decreases with the in crease of nonlocality degree [11], thus for 
+a thermal medium with an infinite range  of nonlocality and a strong dependence of 
+stability on sample-geometry, one might expect  that the maximal number of peaks in 
+stable multipoles might be larger than four or at least different with that. However,  8 our results here reveal that thermal non linear media actually possess the same upper 
+threshold for stability as that in NLCs.  All results reported above are confirmed by the outcome of the direct numerical 
+integration of Eqs. (1) with the input condition 
+(, 0 ) () [ 1 () ]qxz wx x r == + , where  
+()wx  is the profile of th e stationary wave and ()xr stands for the broadband input 
+noise with the variance 2
+noise0.01s= . As expected, the multipole-mode that belong 
+to the stable region predicted by linear stability analysis retain their shapes over 
+indefinitely long propagation di stance, as clearly seen in Fig. 4(a) for a dipole mode, 
+and Fig. 4(d) for a quadrupole mode. Note th at, Figure 4(c) shows propagation result 
+for another quadrupole mode at a smaller sa mple width (all the other parameters are 
+the same as those in Fig. 4(d)). In contra st with that in wider sample, this quadrupole 
+experiences oscillatory instability and destr oys itself. Finally, all fifth-order solitons 
+experience the similar instab ility scenario [Fig.4 (b)]. 
+In conclusion, we studied the stability charact eristics of bound states composed of 
+several field peaks in thermal nonlinear medi a with an infinite-range nonlocality. We 
+found that the stability of tr ipole- and quadrupole-mode so liton depends crucially on 
+the sample width, and they can be completely stable when the sample width exceeds a critical value. We based the result of sa mple-width-dependent soliton stability on a 
+rigorous linear stability analysis with co llaboration by direct propagation simulations. 
+Thermal nonlinear media cannot support stab le bound modes containing more than 
+four peaks, thus they posse ss the same restriction on the maximal number of peaks in  9 sable multipoles as that in nonlinear media with finite-range of nonlocality, although 
+their nonlocal nonlinear responses  are essentially different.  
+This work was supported by the Nationa l Natural Science Foundation of China 
+(Grant No. 10704067).   10 References  
+1. M. Peccianti, K. A. Brzdkiewicz, and G. Assanto , Opt. Lett.  27, 1460(2002). 
+2. W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, 
+and D. Edmundson,  J. Opt. B 6, S288 (2004). 
+3. M. Peccianti, C. Conti, and G. Assanto, Opt. Lett.  30, 415(2005). 
+4. C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett.  91, 073901(2003). 
+5. C. Conti, M. Peccianti, and G. Assanto , Phys. Rev. Lett.  92,113902(2004). 
+6. M. Peccianti, C. Conti, and G. Assanto, “Optical multisoliton generation in 
+nematic liquid crystals,” Opt. Lett.  28, 2231(2003).  
+7. C. Conti, M. Peccianti, and G. Assanto, Opt. Lett.  31, 2030(2006).  
+8. C. Rotschild, O. Cohen, O. Manela, M.  Segev, and T. Carmon, Phys. Rev. Lett. 
+95, 213904 (2005);  
+9. C. Rotschild, B. Alfassi, O. Cohen and M. Segev, Nature Physics 2, 769(2006). 
+10. F. Ye, Y. V. Kartashov, and L. Torner, Phys. Rev. A  76, 033812(2007). 
+11. Z. Xu, Y. V. Kartashov, and L. Toner, Opt. Lett. 30, 3171(2005). 
+12. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. 
+Lett. 32, 154 (2007); Q. Shou, Q. Liang, Q. Jian g, Y. Zheng, W. Hu, and Q. Guo, 
+Opt. Lett . 34, 3523 (2009). 
+13. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides,   Phys. 
+Rev. Lett. 98, 213901(2007). 
+14. F. Ye, Y. V. Kartashov, and L. Torner , Phys. Rev. A  77, 043821(2008).   11 15. Y. V. Kartashov, F. Ye, V. A. Vysloukh, and L. Torner , Opt. Lett.  32, 
+2260(2007).  
+16. Y. V. Kartashov, V. A. Vysloukh, and L. Torner , Opt. Lett. 33, 1774(2008).  
+17. Y. Kartahov, V. A. Vysloukh, and L. Torner , Opt. Lett. 34, 283(2009).  
+18. F. Ye, Y. V. Kartashov, and L. Torner , Phys. Rev. A 77, 033829(2008).  
+19. F. Ye, L. Dong, and B. Hu, Opt. Lett . 34, 584 (2009). 
+20. I. Kaminer, C. Rotschild, O. Manela, and M. Segev, Opt. Lett . 32, 3209 (2007). 
+21. N. K. Efremidis, Phys. Rev. A  77, 063824(2008). 
+22. D. Buccoliero, A. S. Desyatnikov, and W. Krolikowski, and Y. Kivshar, J. Opt. A: 
+Pure Appl. Opt.  11, 094014(2009). 
+23. D. Buccoliero, and A. S. Desyatnikov, Opt. Express  17, 9608(2009).  
+24. A. Alberucci, M. Peccianti, and G. Assanto , Opt. Lett. 32, 2795 (2007). 
+25. A. Alberucci, G. Assanto, D. Buccoliero, A.  S. Desyatnikov, T. R. Marchant, and 
+N. F. Smyth, Phys. Rev. A 79, 043816(2009).  12 Figure captions 
+FIG. 1. (Color online) (a) Prof ile of a fundamental soliton at 0.5b . Profiles of dipole 
+solitons at (b) 0.5b  and (c) 5b, respectively. The thick line stands for the light 
+field ( w) while the light line sta nds for refractive index ( n). (d) the energy flow 
+versus propagation constant diagram for f undamental(dashed line), dipole(light line) 
+and tripole(thick line) solutions. The two points marked by circle s corresponds to the 
+soliton solutions in (b) and (c ) respectively. In all plots 60L . All quantities are 
+plotted in arbitrary dimensi onless units.         
+FIG. 2. (Color online) (a) Profile of a tripole soliton at 1band 40L . (b) 
+Instability growth rate for tripole soli tons versus propaga tion constant at 
+20,  30 L and40 . (c) Critical value of propagatio n constant (above which solitons 
+are stable) for tripole solitons  versus sample width. (d) Profile of a quadrupole soliton 
+at 1band 40L . (e) Instability growth rate for quadrupole solitons versus 
+propagation constant at 20,  40 L and60. (f) Critical value of propagation constant 
+for quadrupole solitons versus sample width.  All quantities are plotted in arbitrary 
+dimensionless units.   
+FIG. 3 (a) Profile of a fifth-order soliton at 1b. (b) Instability growth rate for the 
+fifth-order soliton. 60L . All quantities are pl otted in arbitrary dimensionless units.   
+FIG. 4.  (Color online) Propagation of a perturbed dipole soliton (a) and a fifth-order 
+mode (b) in a sample with 60L .  Propagation of a pertur bed quadrupole mode at 
+(c) 20L  and (d) 40L . For all the plots, 1b. All quantities are plotted in 
+arbitrary dimensionless units.    13 
+ 
+FIG. 1. (Color online) (a) Prof ile of a fundamental soliton at 0.5b . Profiles of dipole 
+solitons at (b) 5b and (c) 0.5b , respectively. The thick line stands for the 
+light field ( w) while the light line stands for refractive index ( n). (d) Energy 
+flow versus propagation constant diagra m for fundamental (dashed line), dipole 
+(light line) and tripole(thick line) solutions. The two points marked by circles corresponds to the soliton solutions in (b ) and (c) respectively. In all plots 
+60L . All quantities are plotted in arbitrary dimensionless units.            14           
+ 
+FIG. 2. (Color online) (a) Profile of a tripole soliton at 1band 40L . (b) 
+Instability growth rate for tripole soli tons versus propaga tion constant at 
+20,  30 L and40 . (c) Critical value of propagatio n constant (above which solitons 
+are stable) for tripole solitons  versus sample width. (d) Profile of a quadrupole soliton 
+at 1band 40L . (e) Instability growth rate for quadrupole solitons versus 
+propagation constant at 20,  40 L and60. (f) Critical value of propagation constant 
+for quadrupole solitons versus sample width.  All quantities are plotted in arbitrary 
+dimensionless units.   15                  
+  
+FIG. 3 (a) Profile of a fifth-order soliton at 1b. (b) Instability growth rate for the 
+fifth-order soliton. 60L . All quantities are plotted in arbitrary dimensionless units.   16  
+  
+FIG. 4.  (Color online) Propagation of a perturbed dipole soliton (a) and a fifth-order 
+mode (b) in a sample with 60L .  Propagation of a pertur bed quadrupole mode at 
+(c) 20L  and (d) 40L . For all the plots, 1b. All quantities are plotted in 
+arbitrary dimensionless units.   
+ 
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+arXiv:1001.0302v1  [cond-mat.supr-con]  3 Jan 2010Interaction of linear waves and moving Josephson vortex lat tices in layered
+superconductors
+A. V. Chiginev∗and V. V. Kurin
+Institute for Physics of Microstructure of the Russian Acad emy of Science, GSP-105, 603950 Nizhny Novgorod, Russia
+(Dated: May 13, 2017)
+A general phenomenological theory describing dynamics of J osephson vortices coupled to wide
+class of linear waves in layered high-T csuperconductors is developed. The theory is based on
+hydrodynamic long wave approximation and describes intera ction of vortices with electromagnetic,
+electronic and phonon degrees of freedom on an equal footing . In the limiting cases the proposed
+theory degenerates to simple models considered earlier. In the framework of the suggested model we
+undertook the numerical simulation of resistive state in la yered superconductors placed in external
+magnetic field and demonstrate excitation of linear waves of various origin by a moving vortex
+lattice, manifesting in existence of resonances on current -voltage characteristics.
+I. INTRODUCTION
+High-T csuperconductors (HTSCs) with strong anisotropy has been the s ubject of intensive investigation during
+many years. The layered structure, intrinsic Josephson effect, a nd complex chemical composition provide a variety
+of physical properties of such materials. A great deal of attentio n has been paid to Josephson dynamics of layered
+superconductors.
+The application of an external magnetic field parallel to the layers to a layered HTSC leads to formation of a
+Josephson vortex lattice (JVL) which can be moved under the influe nce of an external current applied perpendicular
+to the layers. When this lattice collides with the edge of the structur e the electromagnetic radiation is generated.
+This principle may be used for construction of small and efficient elect romagnetic oscillators with the frequency
+being limited from above by the energy gap in HTSC which is of the order of 10 THz. Recently, the radiation with
+the frequency of 0.85 THz and power of 0.5 µW has been obtained from the structure based on Bi 2Sr2CaCu2O81.
+The small and compact sources of electromagnetic radiation of ter ahertz band are very interesting for applications
+in astrophysics, medicine, biology and many other branches of scien ce. Therefore, the investigation of Josephson
+dynamics in layered HTSCs is highly important.
+Layered HTSCs represent complex physical system with wide spect rum of eigenwaves, which includes electromag-
+netic, plasma, different phonon modes and Carlson-Goldman mode2. All these modes are coupled to vortices and can
+be excited by the moving JVL. The reverse action of the excited wav es on JVL resulting in changes of vortex shape
+and mutual arrangement of vortices will affect the current-volta ge characteristics (CVCs) of layered superconductor
+and intensity of electromagnetic radiation. Thus coupling of linear mo des and moving vortices can be practically used
+for controlling the dispersion properties of eigenwaves by the exte rnal magnetic field, for diagnostics of eigenwaves by
+methods of Josephson spectroscopy3, and for controlling electromagnetic radiation from layered superc onductor.
+The model adequately describing vortex dynamics in layered HTSCs s hould take into account the interaction of
+vortices with all weakly damping linear modes. However, the developm ent of the theory of such comprehensive kind is
+not a simple task and a main obstacle is a nonlocality of the constitutive equations of a layered superconductor or, in
+other words, spatial dispersion. Even in hydrodynamic approximat ion when the dynamical equations are differential,
+the consistent approach to description of layered media requires s olution of differential equations for field and matter
+in the domains of homogeneity, and subsequent joining of the solutio ns found for adjacent domains on interfaces.
+If the number of modes in the system is rather high then the exact d escription leads to sophisticated problem for
+eigenvectorsand eigenvalues ofhigh-dimension transfermatrix. E xample ofcalculations of this kind applied to normal
+nonsuperconducting layered media can be found in Ref. 4. The desc ription of layered systems can be considerably
+simplified by employing the long-wavelength limit which deals with smoothe d dynamical variables averaged over the
+spatial period of the layered structure. The goal of the present paper is the development of such a theory for a layered
+superconductor with intrinsic Josephson effect. We propose an av eraged, sufficiently simple, hydrodynamic theory
+accounting for a wide spectrum of linear waves and vortex degrees of freedom. The model is easily extendable for
+including additional linear modes and mechanisms of their exitation.
+By present, a number of models describing Josephson dynamics of la yered HTSCs in some special cases has been
+proposed. The first and most known one is the local model account ing only for the magnetic (inductive) coupling
+between adjacent junctions of the stack which was applied to desc ription of the artificial multilayer Josephson struc-
+tures5and laterto layeredsuperconductorswith intrinsicJosephsoneffe ct6. The chargecouplingbetween the adjacent
+junctions in layered superconductors has first been investigated in Ref. 7 for the case of spatially uniform distributions
+of Josephsonphase. There have been also some attempts to comb ine magnetic and chargecouplings into one model8,9.2
+The ”global” coupling of the junctions via the external waveguide co nnected parallel to the long Josephson junction
+stack has been considered in our previous paper10. The influence of nonequilibrium effects, such as quasiparticle
+imbalance, to the Josephson dynamics of intrinsic junctions in layere d HTSCs, has been investigated in Refs. 11–14.
+The effects of the in-plane dissipation to the Josephson vortex mot ion in layered HTSCs has been studied in Refs. 15–
+17. The role of phonons in Josephon dynamics has been considered in Refs. 18,19 for the case of direct excitation of
+phonons by the electric field, in Refs. 20,21 for phonon excitation du e to phonon-assisted tunneling. All these models
+may be unified into one theory describing the linear waves and Joseph son vortices in layered HTSCs.
+In the present paper we formulate the comprehensive theory acc ounting for vortex interaction with linear waves of
+different physical nature which unifies the models mentioned above. The prescription to design a theory of such kind
+is the following. Instead of exact consideration of high-T csuperconductor as a layered medium we consider it as a
+continuous anisotropic medium which consists of superconducting a nd normal electrons and of several sort of ions. To
+describe the dynamics of such a media coupled to electromagnetic fie ld we use Maxwell equations jointly with two-
+liquid anisotropic hydrodynamic equations for electrons and equatio ns of adiabatic Born-Oppenheimer approximation
+for ions. But this set of equations describe only linear properties of layered superconductor in long-wave limit. To
+introduce vortex degrees of freedom to the model and allow finite j ump of Josephson phase difference on dielectric
+layers we ”discretize” the set of equations simultaneously restorin g the nonlinear term with sinusoidal dependence on
+Josephson phase difference in interlayer current. In various part icular cases our model is reduced to the known models
+considered earlier. In the framework of the suggested model we u ndertook the numerical simulation of resistive state
+in layered superconductors placed in external magnetic field and de monstrate excitation of linear waves of various
+origin by a moving vortex lattice, manifesting in existence of resonan ces on current-voltage characteristics.
+The paper is organized as follows. The next section is devoted to for mulation of effective averaged model describing
+linearwavesandvortexdegreesoffreedomin layeredsupercondu ctorswith intrinsicJosephsoneffect andtoestimation
+of effective parameters of the model. The third section is devoted t o numerical experiment showing the effects of
+excitation of linear waves by Josephson vortex lattice moving in a laye red HTSC. In the conclusion we summarize the
+main results obtained in the paper.
+II. THE PHENOMENOLOGICAL DESCRIPTION OF JOSEPHSON DYNAMIC S OF A LAYERED
+HTSC — HYDRODYNAMIC APPROACH.
+In this section we derive the set of equations describing Josephson dynamics of a layered superconductor treating it
+as a continuous but anisotropic medium. First, we write down the Max well equations and anisotropic hydrodynamic
+equations describing superconducting condensate, electronic qu asiparticles and phonon degrees of freedom of super-
+conductor. As a result of this stage we obtain a system describing lin ear wavesin a layered HTSC in a long wavelength
+limit. Then, substituting the derivatives over the coordinate perpe ndicular to the layers by finite differences and, si-
+multaneously, restoring the nonlinear expression for the interlaye r supercurrent in Josephson form js=jcsinθn, we
+obtain the desired set of equations.
+FIG. 1: Side view of the multilayer Josephson structure. The numeration of superconducting and insulator layers, and th e
+coordinate system, are shown.
+In this paper we adopt the widely used concept that high-T csuperconductor can be treated as a sequence of
+superconducting layers coupled through dielectric layers by tunne l effect providing the existence of Josephson effect
+and quasiparticle conductivity in c-axis direction. Such a layered superconductor with chosen coord inate system is3
+shown in Fig. 1. The xaxis is chosen laying in abplane of a superconductor which is assumed to be isotropic in this
+plane, direction of the yaxis is chosen coinciding with the direction of the magnetic field which is a ssumed to have
+only one component, both static and alternating, and the zaxis is directed perpendicular to the layers along caxis of
+layered superconductor. For such magnetic field the electric field a nd other physical variables would not depend on y
+and the problem becomes two-dimensional one. The superconduct ing and adjacent dielectric layers are enumerated
+by number n, as shown in Fig. 1.
+We start the derivation from the assumption that, first, the char acteristic scale of the solutions is much greater than
+the layer structure period, and, second, the module of the Josep hson phase difference over dielectric layer is much
+smaller than unity. These assumptions allow us to treat layered supe rconductor as a continuous medium, and, thus,
+to formulate the desired model in the continual limit. Then, after ”d iscretizing” the continuous model, we will restore
+possibility of finite jump of the phase of the order parameter and fin ally obtain the set of finite-difference equations
+representing the desired model allowing vortex solutions.
+As the basic equations we use Maxwell equations for electromagnet ic field together with hydrodynamic equations
+describing contributions from superconducting electrons, norma l electrons, and phonons. The next subsections are
+devoted to derivation of constitutive equations needed to constr uct the desired model.
+A. Field equations and superconducting electrons
+In this subsection we derive the set of equations phenomenologically describing Josephson dynamics of a layered
+superconductorinacontinuallimit. Todothis, firstweuseMaxwelle quationstogetherwithanisotropichydrodynamic
+equations for supercondicting electrons. Aftewards we suppleme nt the set of obtained equations with constitutive
+equationsdescribingcontributionsfromnormalelectronsand pho nons, which areto be derivedin the next subsections.
+As the starting point of the derivation of the equations describing t he superconducting electron subsystem we use
+the expression of the superconducting momentum which we denote aspvia the phase χof superconducting order
+parameter
+p=/planckover2pi1∇χ+2e
+cA, (1)
+hereAis the vector potential of the electromagnetic field and the negativ e electron charge is explicitly taken into
+account. On this stage we suppose that there are no vortices in th e material, so that [ ∇×∇χ] = 0. Accounting for
+this, we obtain
+B=c
+2e[∇×p]. (2)
+Substituting this expression into Maxwell equation [ ∇×B] = 4πc−1jtot+c−1˙Ewe get the evolutional equation for E
+1
+c˙E=c
+2e[∇×[∇×p]]−4π
+cjtot. (3)
+The total current jtotin (3) is the sum of the supercurrent js, current of normal electrons jn, and ion current ji.
+The supercurrent is related to velocity by usual expression js=−en0vbut in anisotropic Ginzburg-Landau model
+the momentum is connected to velocity by relation p= 2ˆmsvvia tensor ˆ msof effective mass. We write down this
+tensor in the form ˆ ms= diag(m/Γs,mΓs), where m= (mxxmzz)1/2is the geometric average of the effective masses
+of superconducting electrons in the in-plane and interlayer directio ns, factor Γ s= (mxx/mzz)1/2is a measure of
+anisotropy of superconducting properties of the material. Using t he expression for supercurrent, one rewrites (3) in
+components
+1
+c˙Ex=c
+2e/parenleftbigg∂2pz
+∂x∂z−∂2px
+∂z2/parenrightbigg
+−4π
+c/parenleftbigg
+−en0Γs
+2mpx+jnx+jix/parenrightbigg
+, (4a)
+1
+c˙Ez=c
+2e/parenleftbigg
+−∂2pz
+∂x2+∂2px
+∂x∂z/parenrightbigg
+−4π
+c/parenleftbigg
+−en0
+2mΓspz+jnz+jiz−jext/parenrightbigg
+. (4b)
+Though we assume that nothing depends on y-coordinate in our system, in the Eq. (4b) we take into account tha t
+thez-component of [ ∇×B] actually consists of two terms
+[∇×B] =∂By
+∂x−∂Bx
+∂y≡∂By
+∂x+4π
+cjext;4
+with the last term in the r. h. s. of this equation being proportional t o the bias current jext. The Eqs. (4a), (4b) are
+to be supplemented with evolutional equations for p.
+If we differentiate Eq. (1) with respect to time and introduce chemic al potential of superconducting electrons
+2µ=/planckover2pi1˙χ−2eϕthen we come to equation of hydrodynamic type
+ˆm˙v+∇µ=−eE. (5)
+Further we make a model assumption that chemical potential of su perconducting electrons is defined by formula
+of degenerated Fermi gas µ=εFwhereεF=/planckover2pi12(3π2n)2/3/2mis the Fermi energy and nis a concentration of
+superconducting electrons. Assuming that deviation of the conce ntration nsfrom its equilibrium value n0is small we
+can linearize equation (5) what yields
+ˆms˙ vs+2
+3εF,sns
+n0=−eE, (6)
+where we denote as εF,s=εF(n=n0) a Fermi energy for unperturbed superconducting concentrat ion. Further,
+excluding the concentration deviation nsfrom hydrodynamic equations using Maxwell equation which we write in the
+form (∇·D) = 4πρs≡ −4πens, we come to evolutional equation for p
+1
+2e˙p=E−r2
+d∇(∇·D), (7)
+or, in components
+−1
+2e˙px=Ex−r2
+d/parenleftbigg∂2Dx
+∂x2+∂2Dz
+∂x∂z/parenrightbigg
+, (8a)
+−1
+2e˙pz=Ez−r2
+d/parenleftbigg∂2Dx
+∂x∂z+∂2Dz
+∂z2/parenrightbigg
+. (8b)
+Hererd= (2/3)1/2vF/ωpis a screening length of longitudinal electric field, v2
+F= 2εF/m, andω2
+p= 4πe2n0m−1are
+averaged Fermi velocity and plasma frequency respectively which a re defined as v2
+F= (v2
+Fxv2
+Fz)1/2,ω2
+p= (ω2
+pxω2
+pz)1/2.
+Vector of electric displacement field Dis defined so that it includes charges of normal electrons and ions as bound
+charges, i.e.
+D=E+4πPn+4πPi≡(ˆI+4πˆχn+4πˆχi)E, (9)
+wherePnandPiare the polarizations associated with normal electrons and phonon s, respectively, so that
+(∇·Pn,i) =−ρn,i. Tensors ˆ χn,ˆχiare normal electrons and phonons susceptibility respectively.
+A set of equations Eqs. (4a), (4b), (8a), (8b) describes linear dy namics of electromagnetic field and superconducting
+condensate in layered superconductor in the long wave limit. Other d egrees of freedom as normal quasiparticles and
+phonons of different kind come in the Eqs (4a), (4b) via extra curre nt densities jn,jiand in the Eqs. (8a), (8b)
+via diplacement vector D. In the next two subsection we find contributions from quasipartic les and phonons to
+susceptibility and make the set of equations complete.
+B. Contribution from normal electrons
+To describe normal electrons we assume that they form degenera te Fermi gas and apply hydrodynamical Thomas-
+Fermi approach what gives the following equations
+˙nn+n0n(∇·vn) = 0,
+ˆmn(˙ vn+ ˆνvn)+2
+3εF
+n0n∇nn=eE. (10)
+In the Eqs. (10) ˆ mn= diag(m/Γn,mΓn) is the mass tensor of normal electrons, Γ nis the anisotropy parameter, n0n
+is the unperturbed electron concentration, nnis the deviation of electron concentration from its unperturbed va lue,
+vnis the electron velocity, εF,n=εF(n=n0,n) is the Fermi energy of normal electrons and tensor ˆ ν= diag(νx,νz)
+characterizes the electron collision frequencies. For supercondu ctor the condition ω≪νx,zis usually satisfied and
+equations can be somewhat simplified by neglecting the term ˙ vin comparison with ˆ νv. Then such hydrodynamic5
+equations can be solved and the relation between normal current jnand electric field Ecan be easily obtained. In
+(ω,k)-representation one gets
+jn=ω2
+pn
+4π∆
+3iωνzΓn−2v2
+Fnk2
+z2v2
+Fnkxkz
+2v2
+Fnkxkz3iωνx/Γn−2v2
+Fnk2
+x
+E≡ −iωˆχnE. (11)
+Hereω2
+pn= 4πe2n0n/mis the plasma frequency, vFn= (εF/m)1/2is the Fermi velocity, both related to normal
+electrons, ∆ = 3 iωνxνz−2νxΓ−1
+nv2
+Fnk2
+z−2νzΓnv2
+Fnk2
+x. Without spatial dispersion of normal electrons, i. e. at
+vFn= 0, this equation became the simple anisotropic Ohm law
+jn= ˆσE,ˆσ= diag(σxx,σzz), (12)
+whereσxx=ω2
+pnΓn/(4πνx),σzz=ω2
+pn/(4πνzΓn) are respectively the in-plane and interlayer normal conductivities .
+Here it is worth to mention that such local model of normal conduct ivity has been used in works15–17devoted to
+study the influence of the in-plane normal conductivity to the dyna mics and stability of moving JVLs in layered
+HTSCs. Our approach naturally introduces into consideration the s patial dispersion originated from nonzero pressure
+of normal electrons.
+C. Phonon contribution
+This subsectionis devoted toaccountingforphonons in ourphenom enologicalmodel describingJosephsondynamics
+of layered HTSCs. In the present paper we take into account only t he direct excitation of infrared-active phonons by
+the electric field of Josephson oscillations. In fact here we will actua lly generalize the approach used in Refs. 18,19
+to the case of non-uniform solutions in distributed Josephson syst ems. The nonlinear interaction between Josephson
+oscillations and phonons due to effects of phonon assisted tunneling considered in Refs. 20,21 would not be considered
+in this subsection, but later, when we formulate general nonlinear e quation, we will show how this mechanism can be
+introduced in our model.
+In order to find the phonon contribution to the dielectric permittivit y we use standard approach (see, for example,
+Ref.22). Let us write the classical equation of motion for ions derive d in Born-Oppenheimer adiabatic approximation,
+Mν¨zν
+N=−/summationdisplay
+µ,MˆGνµ
+N−Mzµ
+M+qνEν
+N. (13)
+In this formula zν
+Nis the ion displacement from the equilibrium position, νis the ion index in the unit cell, N=
+n1a1+n2a2+n3a3is the unit cell index, a1,2,3are lattice periods, ˆGνµ
+N−Mis the ”seed” force tensor resulting from
+the interaction between ions via valence electrons, qνis the ion charge, Mνis the ion mass, Eν
+Nis the microscopic
+electric field at the point of the ion. The interaction between ions and conductivity electrons is carried out via the
+last term in the r. h. s. of (13). In the equation (13) and further t he product of a tensor and a vector is written in
+components as ( ˆAx)i=Aijxj. The relation between microscopic field and average macroscopic fie ld can be written
+via so called Lorentz tensor ˆLνµ
+N−M
+Eν
+N=EN+4π/summationdisplay
+µ,MˆLνµ
+N−MPµ
+M, (14)
+wherePµ
+M=qµzµ
+Mis the dipole momentum of the µ-th ion in the M-th cell. With the account for the difference
+between microscopic electric field from macroscopic field the equatio n forν-th ion motion takes the form
+Mν¨zν
+N=−/summationdisplay
+µ,MˆFνµ
+N−Mzµ
+M+qνEN, (15)
+whereˆFνµ
+N−M=ˆGνµ
+N−M−4πqνˆLνµ
+N−Mqµis renormalized force tensor.
+The contribution of ions into dielectric permittivity εmay be found from the expression for ion current density
+jN=1
+V/summationdisplay
+νqν˙zν
+N, (16)6
+whereVis the unit cell volume. After some algebra we get the ion current den sity
+ji=−iω
+V3L/summationdisplay
+a=11
+−ω2+ω2
+ph(k,a)/summationtext
+ν,µqνqµeν(e∗
+µ,E)/summationtext
+νMνe∗νeν≡ −iωˆχiE, (17)
+whereais the phonon mode index, Lis the number of ions in the unit cell, ωph(k,a) is the phonon frequency, eν(k,a)
+is the polarization vector. ωph(k,a) andeν(k,a) are yielded from the equation for eigenvalues and eigenvectors fo r
+the matrix ˆFνµ(k)
+/summationdisplay
+µˆFνµ(k)eµ(k,a) =Mνω2
+ph(k,a)eν(k,a), (18)
+whereˆFνµ(k) is the Fourier image of the tensor ˆFνµ
+N−M
+ˆFνµ(k) =/summationdisplay
+Ne−ikNˆFνµ
+N. (19)
+The expression (17) is the well-known formula describing phonon con tribution to dielectric permittivity, which may
+be found in many solid state physics courses. Of course, the concr ete definition of phonon frequencies and polarization
+vectors for layered superconductors require detailed spectros copic or numerical investigation. Example of calculations
+of phonon properties of typical layered superconductor with intr insic Josephson effect Bi 2Sr2CaCu2O8can be found
+in Ref. 23.
+So now we have found the contributions from normal electrons and phonons to the dielectric permittivities and
+are able to define electric displacement vector D= (ˆI+ 4πˆχn+ 4πˆχi)E≡ˆεE, using expressions (11) and (17) for
+susceptibilities ˆ χnand ˆχi.
+The Eqs. (4a), (4b), (8a), (8b) together with the relations (11) and (17) represent complete system describing, in
+principle, all linear wavesin long wavelength limit. However, this system does not describe Josephson vortex degree of
+freedom in layered HTSCs because it is linear and does not allow the finit e jumps of the phase of order parameter. In
+the next subsection we will show how Josephson vorticescan be inclu ded in the model and formulate general nonlinear
+model describing both Josephson vortex dynamics and linear waves in layered HTSCs. We will compare this model
+with several known ones, and reveal the relations between the pa rameters of continual model and the parameters of
+layered superconductors.
+D. Discretization of the model
+The aim of the present subsection is the transformation of the set of equations in the continual limit, to the form
+admitting solutions in the form of vortices, which are able to move alon g the layers of the HTSC. To do this, we
+split the continious medium into the series of layers of thickness sequal to the period of the layers of the HTSC,
+and oriented perpendicular to the caxis of superconductor. We introduce new variables describing the corresponding
+fields at the moment t, coordinate x, and in some point within the n-th layer. It is possible provided field distributions
+are smooth inside the layer. After such discretization we allow the ph ase of order parameter to make a finite jump
+between layers and restore the nonlinear expression for the inter layer Josephson current.
+First of all, let us decompose all dynamical variables to two sets defin ed on two different lattices shifted with respect
+to each other. The nodes of these two lattices can be thought to b e placed in the middle of superconducting and
+dielectric layers, respectively. The nodes of these sublattices we w ill denote by the number of period n. We attribute
+the variables Ex,Dx,jn,s,i
+x,vn,s,i
+xto the first set and Ez,Dz,jn,s,i
+z,vn,s,i
+zto the second one. After such a decomposition
+we replace the derivatives of the dynamical variables over the coor dinate perpendicular to the layers in the set of
+Eqs. (4a), (4b), (8a), (8b), (11), (17), by finite differences, u sing the following rule :
+s∂V
+∂z→ ±Vn±1∓Vn, s2∂2V
+∂z2→Vn+1−2Vn+Vn−1≡∆nVn, (20)
+whereVnstands for the one of the dynamical variables of the problem, uppe r sign should be used for variables from
+the first set and lower sign — for the second one. Such a rule of discr etisation follows from integral form of Maxwell
+and hydrodynamic equationsand providesnecessarysymmetry of finite-difference equations. To demonstratehow this
+procedure works let us apply it to well known telegraph equations de scribing electromagnetic waves in transmission
+lines
+L˙I+Ux= 0, C˙U+Ix= 0, (21)7
+hereU,Iare voltage and current in the line, L,Care linear densities of inductance and capacity, respectively.
+Attributing voltage and current to different sets and replacing spa tial derivatives by finite differences using our rule
+we come to equations
+sL˙In+Un−Un−1= 0, sC˙Un+In+1−In= 0, (22)
+expressing two Kirchhoff laws for discrete L,Cchain. Now let us return to our problem. Applying the formulated
+procedure to our set of differential the substitution (20) we obta in the following system
+1
+c∂
+∂tExn=c
+2es∂
+∂x(pzn−pzn−1)−c
+2es2∆npxn−4π
+c/parenleftbigg
+−en0Γs
+2mpxn+jn
+xn+ji
+xn/parenrightbigg
+, (23)
+1
+c∂
+∂tEzn=−c
+2e∂2
+∂x2pzn−c
+2es∂
+∂x(pxn+1−pxn)+4π
+c/parenleftbigg
+−en0
+2mΓspzn+jn
+zn+ji
+zn−jext/parenrightbigg
+, (24)
+−1
+2e∂
+∂tpxn=Exn−r2
+d/parenleftbigg∂2
+∂x2Dxn+1
+s∂
+∂x(Dzn−Dzn−1)/parenrightbigg
+, (25)
+−1
+2e∂
+∂tpzn=Ezn−r2
+d/parenleftbigg1
+s∂
+∂x(Dxn+1−Dxn)+1
+s2∆nDzn/parenrightbigg
+, (26)
+D= ˆεE,jn,i=∂
+∂t(ˆχn,iE),ˆε=ˆI+4πˆχn+4πˆχi. (27)
+The Eq. (27) of this system combines the contributions from norma l electrons and phonons which are given by the
+formulas (11) and (17) for corresponding tensors of susceptibilit ies. In a common case, when spacial dispersion
+takes place, these tensors contain wavevectors in the interlayer direction. Therefore, the procedure of discretization
+should be applied also to Eq. (27) accounting for specific expression s for normal electron and phonon contributions
+to dielectric permittivity.
+Now this system allows finite jumps of physical variables but it still rem ains linear and, therefore, does not describe
+Josephson vortices. To describe them we need to introduce Josep hson nonlinearity in the expression for interlayer
+supercurrent. This procedure is made in the following way. Integra ting the expression for z-component of the
+superconducting momentum over zfromn-th ton+1-th layer, we get Josephson phase difference
+θn=−1
+/planckover2pi1/integraldisplay(n+1)s+ds
+2
+ns+ds
+2pzdz=χn−χn+1−2π
+Φ0/integraldisplay(n+1)s+ds
+2
+ns+ds
+2Azdz. (28)
+Then we need to substitute θninto the set of equations in a correct way. As the interlayer superc urrent in layered
+HTSCs has Josephson nature, we need to substitute pzn→ −/planckover2pi1s−1sinθnin the last term of the Eq. (24). The
+remaining terms in Eqs. (23) and (24) having pznare in fact the components of [ ∇,B], that is why in these terms
+pzn→ −/planckover2pi1s−1θn. The Eq. (26) at rd= 0 is actually the Josephson relation /planckover2pi1˙θ= 2eU, then in this expression also
+pzn→ −/planckover2pi1s−1θn. Atrd/negationslash= 0 the Eq. (26) describes violation of the Josephson relation due to charge effects; the
+influence of these effects on Josephson dynamics is considered late r, in the section devoted to numerical experiment.
+Let us rewrite the system (23), (24), (25), (26), (27) in the not ations used when describing bulk superconductors:
+jc=Φ0
+8π2csω2
+ps
+Γs, λ2
+ab=cΦ0
+8π2jcsΓ2s=c2
+Γsω2ps, r2
+d=v2
+Fs
+ω2psds
+s.
+As a result, we get the following set of equations:
+1
+c∂
+∂tExn=−Φ0
+2πs2∂
+∂x(θn−θn−1)+4π
+cλ2
+ab
+s2∆njs
+xn−4π
+c(js
+xn+jn
+xn+ji
+xn), (29)
+1
+c∂
+∂tEzn=Φ0
+2πs∂2
+∂x2θn−4πλ2
+ab
+cs∂
+∂x(js
+xn+1−js
+xn)−4π
+c(jcsinθn+jn
+zn+ji
+zn−jext), (30)
+4πλ2
+ab
+c2∂
+∂tjs
+xn=Exn−sr2
+d
+ds/parenleftbigg∂2
+∂x2Dxn+1
+s∂
+∂x(Dzn−Dzn−1)/parenrightbigg
+, (31)
+Φ0
+2πcs∂
+∂tθn=Ezn−sr2
+d
+ds/parenleftbigg1
+s∂
+∂x(Dxn+1−Dxn)+1
+s2∆nDzn/parenrightbigg
+. (32)
+D= ˆεE,jn,i=∂
+∂t(ˆχn,iE),ˆε=ˆI+4πˆχn+4πˆχi. (33)8
+The model represented by these equations (29), (30), (31), (3 2), (33) describes interaction of Josephson vortices
+withelectromagneticwaves,plasmons, andphononsonanequalfo oting. Thismodelpossessesthe necessarysymmetry
+with respect to permutation of x,zcoordinates and describes the spatial dispersion resulted from ele ctron and ion
+degree of freedom in the system. Besides, our system allows to acc ount for the influence of linear modes of any nature
+to the dynamics of Josephson vortices in a layered HTSC, by inclusion of the corresponding terms into the dielectric
+permittivity ˆ ε. Say, in recently discovered FeAs based superconductors24–26it can turn out that magnetic degrees of
+freedom play an important role. Our approach allows to include them in this general scheme.
+Of course, our general model contain the particular theories con sidered in earlier works as a limiting cases. Among
+these special casesare the model with the magnetic coupling betwe en the layers5,6, the model with the chargecoupling
+between the layers7, the model taking into account the in-plane quasiparticle current15, the model accounting for the
+infrared-active phonons polarized perpendicular to the layers18,19, etc. Results of two works8,9, where the attempt to
+unify magnetic and charge coupling have been undertaken also can b e reproduced by our model. But our model have
+extra terms in Eqs. (31) and (32) containing ∂/∂xwhich are absent in Refs 8,9. Although small far from resonances,
+these terms become significant when Josephson frequency get clo se to phonon and plasmon frequencies.
+The set of Eqs. (29), (30), (31), (32), (33) contains the only no nlinear term, which describes Josephson nonlinearity.
+While deriving this system we have neglected all hydrodynamic nonlinea rities such as nonlinear dependence of a
+pressure on concentration, difference between Lagrange and Eu ler varyables, nonlinear expression for the current
+density and so on. The account for these terms would lead to the ex istence of combination processes, such as, for
+exaple, Raman and Brillouin scattering. Due to strong anisotropy of the layered HTSCs, the most significant effects
+of this type are the ones providing the dependence on concentrat ion of the parameters determining the transport and
+elastic properties of the material in the direction of c-axis, such as Josephson critical current density jc, interlayer
+normal conductivity σzz, etc. The accountingfor these dependencies leads to such effect s as phonon-assistedtunneling
+due to the interaction of the electrons with Raman-active phonons . This effect has been considered in Ref. 21 for
+acoustic phonons in long Josephson junction and later in Ref. 20 for intrinsic junctions. The similar effect should be
+also expected for plasmons. In the present work we restrict ours elves to the case when these effects are unimportant.
+Let us consider now the examples how some known models arise from o ur equations in the limiting cases. For
+instance, if we set the phonon current, the in-plane displacement a nd normal currents to be equal to zero ( jx,zph=
+∂Exn/∂t=jn
+xn= 0), propose the interlayer normal current to have the purely Oh mic character ( jzn=σzzEzn), and
+neglect the nonzero screening length of the longitudinal electric fie ld (rd= 0), we obtain
+∂2θn
+∂x2=/parenleftigg
+1
+λ2
+j∆n+1
+λ2c/parenrightigg/parenleftbigg
+ω2
+j∂2θn
+∂t2+σzzΦ0
+2πcs∂θn
+∂t+sinθn−jext
+jc/parenrightbigg
+, (34)
+whereλj= Γss,λc= Γsλab,ω2
+j= (8π2jccs)/Φ0. First this set of equations has been derived in Refs. 5,6 and describ es
+the Josephson dynamics of stacked distributed Josephson junct ions and layered superconductors with the magnetic
+coupling between the layers. In another case, when we suppose th e lateral dimensions of the system to be smaller
+than Josephson length, so that ∂/∂x= 0, neglect phonons, and set jzn=σzzEzn, we obtain
+1
+c2∂2θn
+∂t2=−1
+λ2c(1−η∆n)/parenleftbigg
+sinθn−jext
+jc/parenrightbigg
+−4πσzz
+c2∂θn
+∂t, (35)
+whereη=r2
+d/(sds) is the parameter of charge coupling. This set of equations has bee n first considered in Ref. 7 and
+describes the dynamics of the layered superconductor with the ch arge coupling between the layers. Other particular
+examples of models considered earlier may be obtained from our mode l in a similar way.
+III. DISPERSION CHARACTERISTICS OF LINEAR WAVES.
+The layered HTSCs has a wide spectrum of eigenwaves which, in princip le, may be radiated by a moving JVL. The
+linear modes excited by JVL, in turn, may affect the shape and mutua l arrangement of vortices in the lattice. The
+excitation of linear waves by the moving JVL leads to appearance of r esonant steps on CVCs with the frequencies
+being equal to the ones of the radiated modes. In order to identify the resonances on CVC it is necessary to know
+the dispersion characteristics of linear waves in layered HTSCs, whic h can give much information about eigenmodes
+in the material and conditions of their excitation.
+In order to build the dispersion characteristics we first linearize the set of Eqs. (29), (30), (31), (32), (33) assuming
+|θn| ≪1, so that sin θn≈θn, and set the bias current to zero. We also neglect dissipation in the s ystem (σxx=
+σzz= 0) keeping in mind that it may be accounted for by perturbation the ory. Taking all dynamic variables9
+∼exp(ikx+iqn−iωt) and setting the determinant of the obtained linear system to zero , we get the dispersion
+equation of the linear modes. It has the standard form
+det/vextenddouble/vextenddouble/vextenddouble/vextenddoubleω2
+c2εtotalij(ω,k)+kikj−k2δij/vextenddouble/vextenddouble/vextenddouble/vextenddouble= 0, (36)
+where ˆεtotalcontains linear contributions from superconducting and normal ele ctrons, and phonons. For short, we do
+not write this equation in an explicit form. We also assume the simple mod el expression of phonon susceptibility
+ˆχph=−1
+4π
+Ω2
+x
+ω2+iωγphx−ω2
+0x0
+0Ω2
+z
+ω2+iωγphz−ω2
+0z
+. (37)
+This expression describes two optical phonon modes: one polarized along the layers ( x-phonon) and other polarized
+perpendicular to the layers ( z-phonon). Here ω0xandω0zare the frequencies of the x-phonon and the z-phonon,
+respectively, γphxareγphzthe damping coefficients of these modes, Ω xand Ω zare the oscillator strengths. For
+simplicity we assume ω0x,z,γphx,z, and Ω x,zto be independent of the quasimomentum, i. e. the bare phonon mod es
+has no spatial dispersion. Moreover, in this section we neglect phon on damping ( γphx,z= 0), assuming that it may
+be accounted for by perturbation theory. We will return to the no nzeroγphx,zlater, in numerical experiment.
+In our previous work we have considered wave vector surfaces fo r the analysis of the dispersion characteristics of
+the layered superconductor with phonons in the continual limit27. Now we build the dispersion curves ω(k,q) for the
+periodic system.
+The dispersion curves of linear modes in a layered HTSC are schematic ally shown in Fig. 2. The electromagnetic,
+plasma, and phonon modes are shown. We build the dispersion charac teristics for certain directions in the Brillouin
+zone. The curves in the right part of the plot are for the Γ −X direction (growing k), the curves in the central part of
+the plot are for the Γ −X direction (growing q), and the curves in the left part are for the X −W direction (growing
+k), along the edge of the Brillouin zone. We use standard notations Γ ,X,W for the points of the Brillouin zone. The
+dispersion curves for the case of absence of the interaction betw een electromagnetic and plasma modes with phonons,
+i. e. at zero oscillator strengths (Ω x,z= 0), are plotted by dashed lines. It is seen that at nonzero oscillato r strengths
+the splitting of dispersion curves appears in the places of intersect ion between bare curves.
+We note that due to anisotropy the separation of modes into the ele ctromagnetic and plasma mode is relative,
+depending on the direction of the wave. For example, consider the lo west branch of the dispersion characteristics.
+The wave having q= 0 is the electromagnetic mode and the one having k= 0 is the plasma mode. And vice versa,
+for highest branch, the wave with q= 0 is the plasma mode and the one with k= 0 is the electromagnetic mode. For
+the arbitrary direction of the wave vector is it impossible to say whet her it is an electromagnetic or plasma wave.
+The frequency of the lowest branch of the dispersion characteris tics corresponding to the point Γ ( k= 0,q= 0)
+isωj—the frequency of the Josephson plasma resonance. In the nota tions of the hydrodynamic model it is equal to
+ωps/√Γs. The typical values of ωjfor layered HTSCs are of the order 100 GHz. At the values of the pa rameters
+corresponding to layered HTSCs, the frequencies of the highest b ranch of the dispersion characteristics are in the
+optical range. Therefore, this mode cannot affect the JVL motion in the layered superconductors.
+The important particular case of the modes in a layered HTSC is the Sw ihart wave, i. e. the mode belonging to
+the lowest branch of the dispersion characteristics, which propag ating along the layers and having the standing wave
+structure in the direction perpendicular to the layers. The dispers ion curve ω(k) of this wave has the form of the
+hyperbole. The slope of the asymptotes of this curve determines t he Swihart velocity
+¯c2(q) =µε−1
+zz+2η(1−cosq)
+1+2µ(1−cosq), (38)
+whereqis the transverse wave number of the Swihart wave. It is seen that the antisymmetric ( q=π) Swihart mode
+is the slowest one, and the symmetric ( q= 0) mode is the fastest one. We will use the formula (38) for the ana lyzing
+the results of the numerical experiment.
+IV. NUMERICAL EXPERIMENT
+Now we apply the derived system (29), (30), (31), (32), (33) to t he numerical investigation of the dynamics of
+Josephson vortices in layered superconductors with the account for their interaction with various linear modes. The
+motion ofJosephson vortexlattice (JVL) leads to excitation of linea r modes of a layeredHTSC which, generally, affect10
+FIG. 2: (Color online) The schematic drawing of the dispersi on characteristics of linear waves in a layered HTSC. The das hed
+lines show the dispersion without the interaction with phon on modes. The inset shows the fragment of the Brillouin zone o f
+the layered structure.
+the moving lattice, leading to distortions in vortex shape and change s in their mutual arrangement. The excited linear
+modes may lead to resonances on CVCs of layered HTSCs with moving J VL. In the present section we numerically
+investigate the excitation of linear waves by the moving JVL, and the ir influence on CVCs, accounting for possible
+changes in vortex shape and their rearrangement.
+In the numerical experiment we use periodic boundary conditions in b oth in-plane and interlayer directions for all
+variables. However, for Josephson phase difference θnthe boundary conditions in x-direction is modified so that
+θn(L) =θn(0)+2πRn, (39)
+whereRnis the number of vortices trapped in the n-th junction of the stack, and Lis the length of the system in
+the longitudinal direction. Instead of θnwe introduce θ′
+n=θn−2πRnx/Lsatisfying the periodic boundary condition
+in the longitudinal direction. Accounting for this, we write the expre ssion for the interlayer Josephson current
+asjcsin(θ′
+n+ 2πRnx/L). The choice of boundary conditions described above provides the simplicity of numerical
+solution of the system; however, when using such conditions one ta kes into account only volume effects, neglecting
+the influence of boundaries on Josephson vortex dynamics.
+In orderto solvethe system (29), (30), (31), (32), (33) numer icallywetransformit tothe set ofevolutionalequations
+in ordinary derivatives in time. To do this, we use the exponential Fou rier transform by standard 2D-FFT algorithm.
+The obtained set of ordinary time-dependent differential equation s is solved by Krank-Nikolson scheme. The similar
+approach to the numerical experiment has been used in Ref. 10.
+The periodic boundary conditions along the layers imply that the numb er of vortices captured in each junction of
+the system is a constant. In our calculation we assume the number o f vortices to be the same for all layers and denote
+it asR.
+In the numerical experiment we use the typical values of the param eters of Bi 2Sr2CaCu2O8:jc= 150 A/cm2,
+λab= 1700˚A,s= 15˚A,σab= 5·104(Ω·cm)−1,σc= 2·10−3(Ω·cm)−1,εzz= 12. The parameters of phonons have
+been taken from the experimental data obtained by spectral ellips ometry28. In the following we assume the length of
+the system Lto be measured in the units of Josephson length λj= Γssand the bias current density to be measured
+in the units of Josephson critical current jc. In the calculations we use values of Lcorresponding to the ones of the
+patterns used in experiments. The number of vortices in a layer is se t so that it corresponds to the dense JVL (at a
+givenL). We also assume the frequencies and the voltages to be measured in the units of ωpΓ−1/2
+s.
+A. Results
+This subsection is devoted to the results of the numerical modeling o f the dynamics of the JVL in a layered HTSC
+performed basing on the derived model. First, we investigate the sim plest case, when the spatial dispersion, phonons,
+and all interlayer couplings except the magnetic one are neglected. Then, we complicate the problem step-by-step,
+introducing phonon susceptibility, charge coupling, and study the e ffects caused by these complications.11
+1. The simplest case—the magnetic coupling without phonons and spatial dispersion.
+To check the efficiency of the used numerical scheme we perform th e calculation of the CVC of the layered HTSC
+with the moving JVL in the case of absence of spatial dispersion, pho nons, and in-plane dissipation. The interlayer
+normal current is assumed to be purely ohmic. This approximation co rresponds to the case of magnetic coupling
+between the layers, which has been considered in Refs. 5,6.
+Consider the CVC built for the case R= 6 which corresponds to typical values of the external magnetic fi eld
+(Fig. 3). At jext= 0 a triangular JVL is established in the structure (Fig. 4). As the bia s current increases, the
+JVL moves with the increasing velocity, keeping the triangular vorte x arrangement. When the bias current reaches
+0.18, the JVL velocity stops growing, so that the voltage on the stru cture is established at the value 0 .18. This step
+denoted by 1 appears due to coincidence of the JVL velocity with the characteristic velocity of the antisymmetric
+(q=π) Swihart mode of a layered HTSC.
+The JVL remains triangular on the whole step 1; the vortex shape ex hibit specific distortions due to the resonant
+increase of the harmonics with q=πwhich grow as the bias current increases. The distribution of the ma gnetic field
+in this regime does not qualitatively differ from the static one (Fig. 4).
+FIG. 3: A CVC of the layered HTSC with the moving JVL. The digit s enumerate the steps of the CVC in the order of resonance
+frequencies increase. The insets schematically show the mu tual arangement of vortices on different resonance steps.
+FIG. 4: (Color online) The magnetic field distribution corre sponding to the static vortex lattice ( jext= 0). For the sake of
+convenience, here and in the subsequent figures we add consta nts to the distributions of the magnetic field in different jun ctions.
+The actual value of the constant component of the magnetic fie ld is as for the junction 1.
+At the value jext= 0.44the CVC jumps from the step 1 to the Ohmic branch. The amplitude of the electromagnetic
+field on the Ohmic branch is small and the vortices form rectangular la ttice.
+When the bias current reaches the value jext≈1.0 the second step on the CVC appears. This step is due to the
+resonance with the symmetric ( q= 0) Swihart mode. In this regime, the moving vortices still form the r ectangular12
+FIG. 5: (Color online) The magnetic field distribution corre sponding to the end of the step 2 in Fig. 3. Here jext= 1.4. Giant
+amplitude of the electromagnetic field.
+lattice, but the amplitude of the electromagnetic field sharply increa ses. This regime of vortex motion is characterized
+by large amplitude of the electromagnetic wave which accompanies th e moving JVL, the amplitude of the wave grows
+with the bias current increase (Fig. 5).
+With further bias current increase the CVC again jumps to the Ohmic branch. The magnetic field distribution in
+this regime does not qualitatively differ from the one between the ste ps 1 and 2.
+The CVC obtained in our calculations is similar to the one obtained for tw o-stacked long Josephson junctions29.
+The only difference is that in our CVC the voltages of the steps differ b y two orders from each other. This is due to
+the fact that the magnetic coupling in layered HTSCs is usually much st ronger than the one of artificial multilayer
+structures.
+Consider now the CVC of the layered HTSC with the moving JVL at smalle r external magnetic field ( R= 4)
+(Fig. 6). One can see that, in addition to steps 1 and 3 correspondin g to the resonances with the antisymmetric
+Swihart mode ( u= 0.14) and the symmetric mode ( u= 32), respectively, there is a step 2 corresponding to half a
+frequency of the resonance with the symmetric Swihart mode. The magnetic field distribution on this step is shown
+in Fig. 7. It is seen that there are two oscillations of the magnetic field per spatial period of the system, though four
+magnetic flux quanta are captured in each junction of the stack. T his regime also shows a large amplitude of the
+electromagnetic field. The distributions of the magnetic field and the mutual arrangement of vortices at the steps 1
+and 3 and between the resonances are similar to the ones for the ca se considered earlier (see Figs. 3, 4, 5).
+FIG. 6: CVC of the layered HTSC with moving JVL for weaker exte rnal magnetic field.13
+FIG. 7: (Color online) The magnetic field distribution corre sponding to the end of the step 2 in Fig. 6. Here jext= 0.58.
+2. Excitation of a phonon by moving vortex lattice.
+The complex chemical composition of layered HTSCs provide large amo unt of phonon modes in such materials.
+Among these modes, there are ”soft” phonons having frequencie s of the order of several THz, which are smaller than
+the frequency of the energy gap in HTSCs. This makes possible the e xcitation of such phonon modes by a JVL moving
+in a layered HTSC. In this subsection we investigate the phonon excit ation by a moving JVL.
+For the calculations we slightly complicate the model used in the previo us subsection, introducing the simplified
+expression (37) for the phonon susceptibility. The phonon parame ters used in calculations are taken from Ref. 28.
+FIG. 8: (Color online) The amplitudes of alternate componen ts of the electric field and the polarization vs. external cur rent in
+the vicinity of the phonon frequency.
+The Figs. 8 and 9 illustrate the excitation of the phonon z-phonon by the moving JVL. The Fig. 8 shows the
+dependence of complex amplitude modules of PzandEzharmonics with q= 0 and k=klattice, on the bias current.
+HerePzis thez-componentofthe phononpolarizationand klattice= 2πRL−1isthe wavenumberofthe mainharmonic
+of the JVL. The peak on the dependence Pz(jext) is due to the excitation of the phonon mode and appears at the
+value ofjextwhich corresponds to the JVL motion with the frequency of the pho non. The Fig. 9 shows the peak on
+the contribution to the CVC due to the extra energy needed to exc ite the phonon. The similar phonon peaks on the
+CVC has been obtained earlier18,19for the case of spatially uniform Josephson junctions and junction chains. Our
+model actually generalizes the one used in Ref. 19 to the case of dist ributed Josephson juctions and junction stacks.
+In the present simulations we do not consider the excitation of the x-phonons. The reason is that the JVL is
+rectangular at low enough external magnetic field and at the frequ encies close to the phonon one. As one can see from
+the Eqs. (29), (30), (31), (32), x-phonon is not excited by a rectangular JVL, as Ex= 0 in such a lattice. However,
+if the tensor of phonon susceptibility contains non-diagonal compo nents, the excitation of x-phonons by Ezof the14
+FIG. 9: Contribution to CVC due to excitation of the phonon mo de in a HTSC.
+moving rectangular JVL is possible.
+In this subsection we have considered the excitation of a phonon by a moving JVL provided the condition that the
+phonon frequency is far from the frequencies of Swihart modes at a givenklattice. The next subsection is devoted to
+the situation when the phonon frequency coincides with the freque ncy of one of the Swihart modes.
+3. Excitation of the hybrid phonon+Swihart mode by a moving J osephson vortex lattice.
+By choosing the external magnetic field applied to the structure, it is possible to make the frequency of the Swihart
+mode at k=klatticeequal to the phonon frequency. The Fig. 11 shows the contributio n to the CVC in the vicinity
+of the bias current value corresponding to the phonon frequency . Two peaks located close to each other and having
+nearly equal height appear due to excitation of two modes with close frequencies. To explain this effect, consider the
+fragment of the dispersion characteristic of linear waves in a layere d HTSC in the vicinity of the point where the
+Swihart mode and the phonon mode interact (Fig. 10). In the absen ce of the interaction the dispersion characteristic
+would have the shape shown by dotted line, here the slanted line show s the dispersion of the Swihart mode and the
+horizontal line shows the phonon dispersion.
+In the presence of the interaction between two modes (Ω x,Ωz/negationslash= 0) the dispersion characteristic takes the form
+shown by solid line in the Fig. 10. The magnitude of the dispersion curve splitting is determined by the oscillator
+strength of the phonon mode (Ω xor Ωz, depending on the polarization of the mode interacting with the Swiha rt
+wave). The intersection points of the vertical line corresponding t oklattice, and dispersion curves, give the resonance
+frequencies.
+The Fig. 11 shows the contribution to the CVC from the excited hybr id phonon+Swihart modes. The distance
+between two peaks is equal to the difference between the resonan ce frequencies obtained from the dispersion charac-
+teristic. We note that the height of the peaks in Fig. 11 is of two orde rs higher that the height of the peak caused by
+the excitation of the pure phonon mode (Fig. 9).
+4. The violation of the Josephson relation — separation of no rmal electron charges by a moving JVL.
+In this subsection we investigate the influence of the charge effect s on the JVL motion in layered HTSCs. Starting
+again from the simple model with the magnetic coupling, we now assume the parameter of the charge coupling to be
+nonzeroη/negationslash= 0 and find out the differences in JVL dynamics compared to the case η= 0 considered above.
+Consider CVC of the layered superconductor, calculated in the abs ence of phonons and in the presence of the charge
+coupling (Fig. 12, dashed line). As it is seen from this figure, there ar e two steps of this CVC. As in the absence of the
+charge coupling, they appear due to resonance with the ancisymme tric and symmetric Swihart modes. Let us consider
+the differences between CVCs in Figs. 3 and 12. The first one is that t he frequency of the first step is shifted towards
+higher frequencies, while the frequency of the second step is not c hanged. As in the case of zero charge coupling, the
+step positions correspond to the formula ωres= ¯c(q)2πRL−1, where ¯c(q) is determined by the expression (38).
+The second feature of the regime with nonzero charge coupling is th at the amplitude of the second step is much
+smaller than in the case of zero charge coupling. To explain this effect , consider the dependence of the Josephson15
+FIG. 10: (Color online) Dispersion characteristic of the sy mmetric Swihart mode and phonon mode in the vicinity of their
+interaction point.
+FIG. 11: Contribution to CVC due to the excitation of the hybr id phonon+Swihart modes.
+phase growth rate on the bias current (Fig. 12, solid lines). It is see n that the phase growth rates vary from one
+junction to another or, the same, the vortex chains in different ju nction have different velocities. At the same time,
+the voltages on each junction are the same. According to the Eq. ( 32), this is the demonstration of the violation of
+the Josephson relation. As it is also seen from Fig. 12, only two or thr ee junctions of possible four ones are locked to
+the second resonance of CVC. Therefore, the range of bias curr ents where the system remains on the resonant step,
+is smaller than in the case of zero charge coupling, when all junctions are locked.
+V. CONCLUSION
+We propose the comprehensive phenomenological model which desc ribes the dynamics of the non-uniform distri-
+butions of Josephson phase difference in layered HTSCs, e. g. movin g Josephson vortices and linear waves of any
+nature. Basing on this system we numerically build CVCs of a layered su perconductor with the moving JVL and
+demonstrate the excitation of linear modes by moving vortices. The proposed model is shown to cover many effects
+which have been studied in previous works; in addition, we observe so me new effects such as excitation of a phonon
+and hybrid modes by a moving JVL in layered superconductors.16
+FIG. 12: (Color online) Dependence of phase growth rate in Jo sephson junctions of the structure, on the bias current. Das hed
+line shows the constant component of the electric field Ezin each junction.
+VI. ACKNOWLEDGEMENTS
+This work has been supported by the Russian Foundation for Basic R esearch (Grant # 09-02-01358-a), and by
+the following programs of the Russian Academy of Science: ”Nonlinea r Dynamics”, ”Quantum Macrophysics”, and
+”Problems of Radiophysics”.
+∗Electronic address: chig@ipm.sci-nnov.ru
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+arXiv:1001.0303v3  [math.RA]  11 Sep 2010THE IDEAL INTERSECTION PROPERTY FOR GROUPOID
+GRADED RINGS
+JOHAN ÖINERT AND PATRIK LUNDSTRÖM
+Abstract. We show that if a groupoid graded ring has a grading satis-
+fying a certain nondegeneracy property, then the commutant of the
+center of the principal component of the ring has the ideal in tersec-
+tion property, that is it intersects nontrivially every non zero ideal of
+the ring. Furthermore, we show that for skew groupoid algebr as with
+commutative principal component, the principal component is maximal
+commutative if and only if it has the ideal intersection prop erty.
+1.Introduction
+LetRbe a ring. By this we always mean that Ris an additive group
+equipped with a multiplication which is associative and uni tal. IfXandY
+are nonempty subsets of R, thenXYdenotes the set of all finite sums of
+elements of the form xywithx∈Xandy∈Y. The identity element of R
+is denoted 1Rand is always assumed to be nonzero. We always assume that
+ring homomorphisms respect the multiplicative identities . By the commutant
+of a subset Sof a ring R, denoted CR(S), we mean the set of elements of R
+that commute with each element of S; the commutant CR(R)is called the
+center ofRand will be denoted by Z(R).
+Suppose that R′is a subring of R, i.e. there is an injective ring homor-
+phismR′→R. By abuse of notation, we will, in that case, often identify R′
+with its image in R. We say that R′has the ideal intersection property inR
+ifR′∩I/\e}atio\slash={0}for each nonzero ideal IofR. An important ring theoretic
+impetus for studying the ideal intersection property is tha t it can be charac-
+terized by saying that any ring homomorphism R→Sis injective whenever
+the induced restriction R′→Sis injective (see Section 4 for details). Recall
+that ifR′is commutative, then it is called a maximal commutative subring
+ofRifR′=CR(R′), i.e.R′coincides with its commutant in R. A lot of
+2000Mathematics Subject Classification. primary 16W50; secondary 16D25, 16S99.
+Key words and phrases. graded rings, skew groupoid rings, commutants, ideals, max i-
+mal commutative subrings.
+The authors wish to thank Jonas T. Hartwig for stimulating an d useful discussions on
+the topic of this paper. The first author was partially suppor ted by The Swedish Research
+Council, The Swedish Foundation for International Coopera tion in Research and Higher
+Education (STINT), The Crafoord Foundation, The Royal Phys iographic Society in Lund,
+The Swedish Royal Academy of Sciences and "LieGrits", a Mari e Curie Research Training
+Network funded by the European Community as project MRTN-CT 2003-505078.
+12 JOHAN ÖINERT AND PATRIK LUNDSTRÖM
+work has been devoted to the investigation of the connection between maxi-
+mal commutativity and the ideal intersection property (see [2], [5], [6], [9],
+[10], [11], [15] and [25]). Recently (see [20], [21], [22], [ 23] and [24]) such a
+connection was established for the commutant of the neutral component of
+strongly group graded rings and crystalline graded rings (s ee Theorem 1 and
+Theorem 2 below).
+LetGbe a group with neutral element e. Recall that Ris said to be
+graded by the group Gif there is a set of additive subgroups, Rs, fors∈G,
+ofRsuch that R=/circleplustext
+s∈GRsandRsRt⊆Rst, fors,t∈G. The subring Re
+ofRis referred to as the neutral component ofR. LetRbe a ring graded by
+the group G. IfRsRt=Rst, fors,t∈G, thenRis called strongly graded.
+For more details concerning group graded rings, see e.g. [16 ].
+Theorem 1. If a strongly group graded ring has a commutative neutral com -
+ponent, then the commutant of the neutral component has the i deal intersec-
+tion property.
+For a proof of Theorem 1, see [23, Theorem 4]. Recall from [17] that a
+ringRgraded by the group Gis called crystalline graded if for each s∈G
+there is a nonzero element in Rswith the property that it freely generates
+Rsboth as a left and a right Re-module.
+Theorem 2. If a crystalline graded ring has a commutative neutral compo -
+nent, then the commutant of the neutral component has the ide al intersection
+property.
+For a proof of Theorem 2, see [23, Theorem 5]. The main purpose of this
+article is to simultaneously generalize Theorem 1 and Theor em 2 as well as
+extending the grading from groups to groupoids in the follow ing way.
+Theorem 3. If a groupoid graded ring has a right (or left) nondegenerate
+grading, then the commutant of the center of the principal co mponent of the
+ring has the ideal intersection property.
+For the definitions of principal component , groupoid graded rings and right
+(and left) nondegeneracy of a grading, see Definition 1 and Definition 2. For a
+proof of Theorem 3, see Section 3. In general, the ideal inter section property
+alone is of course not a sufficient condition for the principal component in a
+graded ring to be maximal commutative (see e.g. [21, Example 2.3]). The
+secondary purpose of this article is to show that this is howe ver true for a
+large class of skew category algebras.
+Theorem 4. Suppose that Ris a skew category algebra with a commutative
+principal component Aand that the grading of Ris defined by a category
+with finitely many objects. If Ahas the ideal intersection property in R, then
+Ais maximal commutative in R.
+For the definition of skew category algebras, see Example 3 in Section 2.
+For a proof of Theorem 4, see the end of Section 3. By Theorem 3 an d
+Theorem 4 the succeeding result follows immediately.THE IDEAL INTERSECTION PROPERTY FOR GROUPOID GRADED RINGS 3
+Theorem 5. Suppose that Ris a skew groupoid algebra with a commutative
+principal component Aand that the grading of Ris defined by a groupoid
+with finitely many objects. Then Ais maximal commutative in Rif and only
+ifAhas the ideal intersection property in R.
+Note that Theorem 5 simultaneously generalizes [18, Theore m 3.4], [19,
+Corollary 6] and [19, Proposition 10].
+2.Graded Rings
+In this section, we recall the definition of category graded r ings from [13]
+(see Definition 1) and we give some examples of such rings (see Example 1–
+5). In the end of this section, we show a result (see Propositi on 1) concerning
+identity elements in category graded rings and category cro ssed products for
+use in Section 3.
+Suppose that Gis a category. The family of objects of Gis denoted ob(G);
+we will often identify an object in Gwith its associated identity morphism.
+The family of morphisms in Gis denoted ob(G); by abuse of notation, we will
+often write s∈Gwhen we mean s∈mor(G). The domain and codomain
+of a morphism sinGis denoted d(s)andc(s)respectively. We let G(2)
+denote the collection of composable pairs of morphisms in G, i.e. all(s,t)in
+mor(G)×mor(G)satisfying d(s) =c(t). A category is called cancellable (a
+groupoid) if all its morphisms are both monomorphisms and ep imorphisms
+(isomorphisms). A category is called a monoid if it only has o ne object.
+Definition 1. LetRbe a ring and Ga category. A set of additive subgroups,
+Rs, fors∈G, ofRis said to be a G-filter inRif for all s,t∈G, we have
+RsRt⊆Rstif(s,t)∈G(2)andRsRt={0}otherwise. The ring Ris said to
+begraded by the category Gif there is a G-filter,Rs, fors∈G, inRsuch
+thatR=/circleplustext
+s∈GRs. IfRis graded by GandRsRt=Rst, for(s,t)∈G(2),
+thenRis called strongly graded . By the principal component ofRwe mean
+the setR0:=/circleplustext
+e∈ob(G)Re. IfXis a subset of Rwe putXs=X∩Rs, for
+s∈G; the set Xis said to be homogeneous ifX=/summationtext
+s∈GXs. A subring of
+Ris said to be graded if it is homogeneous as a subset of R.
+Remark 1. We always assume that filters and gradings are defined over
+small categories, that is categories where the collection o f morphisms is a
+set.
+Remark 2. Filtrations of rings, in the sense of e.g. [7], are examples o f
+G-filters with G= (N,+).
+Remark 3. Groupoid graded rings have recently arisen as natural objec ts
+of study in Galois theory for weak Hopf algebras (see e.g. [1] ).
+Category graded rings are very general mathematical object s. To give
+some flavor of this we now show some examples of such rings.4 JOHAN ÖINERT AND PATRIK LUNDSTRÖM
+Example 1. Recall from [19] that category crossed products are defined b y
+first specifying a crossed system i.e. a quadruple {A,G,σ,α }whereAis
+the direct sum of rings Ae,e∈ob(G),σs:Ad(s)→Ac(s), fors∈G, are
+ring homomorphisms and αis a map from G(2)to the disjoint union of the
+setsAe, fore∈ob(G), withα(s,t)∈Ac(s), for(s,t)∈G(2), satisfying the
+following five conditions:
+(1) σe= idAe
+(2) α(s,d(s)) = 1Ac(s)
+(3) α(c(t),t) = 1Ac(t)
+(4) α(s,t)α(st,r) =σs(α(t,r))α(s,tr)
+(5) σs(σt(a))α(s,t) =α(s,t)σst(a)
+for alle∈ob(G), all(s,t,r)∈G(3)and alla∈Ad(t). LetA⋊σ
+αGdenote the
+collection of formal sums/summationtext
+s∈Gasus, whereas∈Ac(s),s∈G, are chosen so
+that all but finitely many of them are zero. Define addition on A⋊σ
+αGby
+(6)/summationdisplay
+s∈Gasus+/summationdisplay
+s∈Gbsus=/summationdisplay
+s∈G(as+bs)us
+and define multiplication on A⋊σ
+αGby the bilinear extension of the relation
+(7) (asus)(btut) =asσs(bt)α(s,t)ust
+if(s,t)∈G(2)and(asus)(btut) = 0otherwise where as∈Ac(s)andbt∈Ac(t).
+By (4) the multiplication on A⋊σ
+αGis associative. We will often identify
+Awith/circleplustext
+e∈ob(G)Aeue; this ring will be referred to as the coefficient ring of
+A⋊σ
+αG. It is clear that A⋊σ
+αGis a category graded ring and it is strongly
+graded by Gif and only if each α(s,t), for(s,t)∈G(2), has a left inverse in
+Ac(s).
+Example 2. We say that A⋊σ
+αGis a twisted category algebra if each σs,
+s∈G, withd(s) =c(s)equals the identity map on Ad(s)=Ac(s); in that case
+the category crossed product is denoted A⋊αG. Now we consider two well
+known special cases of this construction when all the rings Ae, fore∈ob(G),
+coincide with a fixed ring D.
+IfGis a group, then D⋊αGcoincides with the usual construction of a
+twisted group algebra.
+Ifnis a positive integer and we define Gto be the groupoid with the n
+first positive integers as objects and as arrows all ordered p airs(i,j), for
+1≤i,j≤n, equipped with the partial binary operation defined by letti ng
+(i,j)(k,l)be defined and equal to (i,l)precisely when j=k, thenD⋊αG
+is the ring of twisted square matrices over Dof sizen.THE IDEAL INTERSECTION PROPERTY FOR GROUPOID GRADED RINGS 5
+Example 3. We say that A⋊σ
+αGis a skew category algebra if α(s,t) = 1Ac(s)
+for all(s,t)∈G(2); in that case the category crossed product is denoted
+A⋊σG. Now we consider two well known special cases of this constru ction
+when all the rings Ae, fore∈ob(G), coincide with a fixed ring D.
+IfGis a group, then D⋊σGis the usual skew group ring of GoverD.
+Even in the case when σs= idDfor alls∈G, we get interesting examples
+of algebras. Indeed, if Gis a a quiver, i.e. a directed graph where loops and
+multiple arrows between two vertices are allowed, then the c ategory crossed
+product coincides with the quiver algebra DG. If, on the other hand, G
+equals the groupoid in the end of Example 2, then the category crossed
+product coincides with the ring of n×nmatrices over D.
+Example 4. Recall from [12] the construction of crossed product algebr as
+F⋊σ
+αGdefined by finite separable (not necessarily normal) field ext ensions
+L/K. LetNdenote a normal closure of L/K and letGal(N/K)denote
+the Galois group of N/K. Furthermore, let Fdenote the direct sum of the
+conjugate fields Li, fori= 1,...,n ; putL1=L. If1≤i,j≤n, then let
+Gijdenote the set of field isomorphisms from LjtoLi. Ifs∈Gij, then we
+indicate this by writing d(s) =jandc(s) =i. If we let Gbe the union of
+theGij, for1≤i,j≤n, thenGis a groupoid. For each s∈G, letσs=s.
+Suppose that αis a map G(2)→/unionsqtextn
+i=1Liwithα(s,t)∈Lc(s), for(s,t)∈G(2)
+satisfying (2), (3) and (4) for all (s,t,r)∈G(3)and alla∈Ld(t). IfL/K
+is normal, then G= Gal(N/K)and hence F⋊σ
+αGcoincides with the usual
+crossed product defined by the Galois extension L/K.
+Example 5. Suppose that Kis a commutative ring and that Mn(K)denotes
+the ring of square matrices of size noverK. Furthermore, let eijdenote the
+matrix in Mn(K)with 1 in the ij:th position and 0 elsewhere. By an example
+of Dade [4] the decomposition R:=M3(K) =R0⊕R1, where
+R0=Ke11+Ke22+Ke23+Ke32+Ke33
+and
+R1=Ke21+Ke31+Ke12+Ke13,
+is an example of a strongly Z2-graded ring which is not a group crossed
+product with this grading. In fact, since dimK(R0) = 5/\e}atio\slash= 4 = dim K(R1),
+it follows that Ris not even crystalline graded. Inspired by this example,
+we now show that there are nontrivial strongly groupoid grad ed rings which
+are not, in a natural way, groupoid crossed products in the se nse defined in
+Example 1. Indeed, let Gbe the unique thin connected groupoid with two
+objects. Explicitly this means that the morphisms of Garee,f,sandt;
+multiplication is defined by the relations e2=e,f2=f,es=s,te=t,
+sf=s,ft=t,st=eandts=f. Define a 13-dimensional K-subalgebra R
+ofM5(K)byR=Re⊕Rf⊕Rs⊕Rtwhere
+Re=Ke11+Ke33 Rf=Ke22+Ke44+Ke45+Ke54+Ke55
+Rs=Ke12+Ke34+Ke35Rt=Ke21+Ke43+Ke536 JOHAN ÖINERT AND PATRIK LUNDSTRÖM
+A straightforward calculation shows that
+ReRe=Re, RfRf=Rf, ReRs=Rs, RtRe=Rt, RsRf=Rs, RfRt=Rt.
+Therefore Ris strongly graded by G. However, since dimK(Rf) = 5/\e}atio\slash= 3 =
+dimK(Rt), the left Rf-module Rtcan not be free on one generator.
+For use in the next section, we gather some facts concerning i dentity
+elements in category graded rings.
+Proposition 1. Suppose that Ris a ring graded by a category G.
+(a)IfGis cancellable, then 1R∈R0. In particular, the same conclusion
+holds ifGis a groupoid.
+(b)IfR=A⋊α
+σGis a category crossed product algebra, then Rhas an
+identity element if and only if Ghas finitely many objects; in that case
+1R=/summationtext
+e∈ob(G)ue.
+Proof. (a) Let1R=/summationtext
+s∈G1swhere1s∈Rsfors∈G. Ift∈G, then
+1t= 1R1t=/summationtext
+s∈G1s1t. SinceGis cancellable, this implies that 1s1t= 0
+whenever s∈G\ob(G). Therefore, if s∈G\ob(G), then1s= 1s1R=/summationtext
+t∈G1s1t= 0. The last part follows from the fact that groupoids are
+cancellable categories.
+(b) First we show the ”if” statement. If Ghas finitely many objects,
+then, by equations (2) and (3) from Example 1, it immediately follows that/summationtext
+e∈ob(G)ueis an identity element of R. Now we show the ”only if” statement.
+Suppose that 1R=/summationtext
+s∈Gasusfor some as∈Ac(s),s∈G, satisfying as= 0
+for all but finitely many s∈G. Sinceue= 1Rue=ue1Rholds for all e∈
+ob(G), it follows that as= 0whenever c(s)/\e}atio\slash=d(s). Sinceue=ue1Rueholds
+for alle∈ob(G), it follows that as= 0whenever s /∈ob(G)andae= 1Ae/\e}atio\slash= 0
+for alle∈ob(G). Therefore ob(G)is finite and 1R=/summationtext
+e∈ob(G)ue./square
+3.Commutants and Ideals
+In this section, we prove Theorem 3 and Theorem 4. To this end, we first
+gather some facts concerning commutants in graded rings.
+Proposition 2. Suppose that Ris a ring graded by a category G.
+(a)IfXis a subset of R, then{CR(X)s|s∈G}is aG-filter in R.
+(b)IfAis a homogeneous additive subgroup of R, then
+CR(A)s=/intersectiondisplay
+u∈GCRs(Au)
+for alls∈G.
+(c)The setCR(R0)is a graded subring of Rwith
+CR(R0)s=/braceleftbiggCRs(Rd(s)),ifc(s) =d(s),/braceleftbig
+rs∈Rs|Rc(s)rs=rsRd(s)={0}/bracerightbig
+,otherwise .THE IDEAL INTERSECTION PROPERTY FOR GROUPOID GRADED RINGS 7
+(d)If1R∈R0, thenCR(R0)is a graded subring of Rwith
+CR(R0)s=/braceleftbiggCRs(Rd(s)),ifc(s) =d(s),
+{0},otherwise .
+In particular, if Gis cancellable or Ris a category crossed product and
+Ghas finitely many objects, then the same conclusion holds.
+Proof. (a) Take s,t∈G. Ifd(s)/\e}atio\slash=c(t), thenCR(X)sCR(X)t⊆RsRt={0}.
+Now suppose that d(s) =c(t). SinceCR(X)is a subring of Rit follows that
+CR(X)sCR(X)t= (CR(X)∩Rs)(CR(X)∩Rt)⊆CR(X)CR(X)⊆CR(X).
+SinceRis graded it follows that
+CR(X)sCR(X)t= (CR(X)∩Rs)(CR(X)∩Rt)⊆RsRt⊆Rst.
+Therefore CR(X)sCR(X)t⊆CR(X)∩Rst=CR(X)st.
+(b) This is a consequence of the following chain of equalitie s
+CR(A)s=CR(A)∩Rs=CRs(A) =CRs/parenleftBigg/circleplusdisplay
+u∈GAu/parenrightBigg
+=/intersectiondisplay
+u∈GCRs(Au).
+(c) It is clear that CR(R0)⊇/circleplustext
+s∈GCR(R0)s. Now we show the re-
+versed inclusion. Take x∈CR(R0),e∈ob(G)andae∈Re. Then/summationtext
+s∈Gxsae=/summationtext
+s∈Gaexs. By comparing terms of the same degree, we can
+conclude that xsae=aexsfor alls∈G. Sincee∈ob(G)andae∈Ae
+were arbitrarily chosen this implies that xs∈CR(R0)sfor alls∈G.
+Now we show the second part of (c). Take e∈ob(G). Suppose that
+c(s) =d(s). Ifd(s)/\e}atio\slash=e, thenCRs(Re) =Rs. Hence, by (b), we get that
+CR(R0)s=/intersectiontext
+e∈ob(G)CRs(Re) =CRs(Rd(s)). Now suppose that c(s)/\e}atio\slash=d(s).
+Ifc(s)/\e}atio\slash=e/\e}atio\slash=d(s), thenCRs(Re) =Rs. Therefore, by (b), we get that
+CR(R0)s=/intersectiontext
+e∈ob(G)CRs(Re) =CRs(Rc(s))/intersectiontextCRs(Rd(s));CRs(Rc(s))equals
+the set of rs∈Rssuch that ars=rsafor alla∈Rc(s). Sinced(s)/\e}atio\slash=c(s),
+we get that rsae= 0;CRs(Rd(s))is treated similarly.
+(d) The claim follows immediately from (c). In fact, suppose thatc(s)/\e}atio\slash=
+d(s). Takers∈Rssuch that Rc(s)rs={0}. Thenrs= 1Rrs= 1c(s)rs= 0.
+The last part follows from Proposition 1. /square
+The essence of the following definition stems from the paper [ 3] by Cohen
+and Rowen.
+Definition 2. LetRbe a ring graded by a category G. TheG-grading
+ofRis said to be right nondegenerate (resp. left nondegenerate ) if to each
+isomorphism s∈Gand each nonzero x∈Rs, the right (resp. left) R0-ideal
+xRs−1(resp.Rs−1x) is nonzero.
+As the following example shows, a grading which is right nond egenerate
+is not necessarily left nondegenerate, and vice versa.8 JOHAN ÖINERT AND PATRIK LUNDSTRÖM
+Example 6. Letube a symbol and put R=R0⊕R1whereR0=Z2[x]
+(x2)and
+R1=R0u. We define multiplication in the obvious way and put u2= 0and
+ux= 0. It is clear that Ris aZ2-graded ring and xR1/\e}atio\slash={0}butR1x={0}.
+Thus, the grading is right nondegenerate, but not left nonde generate.
+Proof of Theorem 3. We prove the contrapositive statement. Let Cde-
+note the commutant of Z(R0)inRand suppose that Iis a twosided ideal
+ofRwith the property that I∩C={0}. We wish to show that I={0}.
+Takex∈I. Ifx∈C, then by the assumption x= 0. Therefore we now
+assume that x=/summationtext
+s∈Gxs∈I,xs∈Rs,s∈G, and that xis chosen so
+thatx /∈Cwith the set S:={s∈G|xs/\e}atio\slash= 0}of least possible cardi-
+nalityN. Seeking a contradiction, suppose that Nis positive. First note
+that there is e∈ob(G)with1ex∈I\C. In fact, if 1ex∈Cfor all
+e∈ob(G), thenx= 1Rx=/summationtext
+e∈ob(G)1ex∈Cwhich is a contradiction.
+Note that, by the proof of Proposition 1(a), the sum/summationtext
+e∈ob(G)1e, and hence
+the sum/summationtext
+e∈ob(G)1ex, is finite. By minimality of N, we can assume that
+c(s) =e,s∈S, for some fixed e∈ob(G). Taket∈S. By the right
+(or left) nondegeneracy of the grading there is y∈Rt−1withxty/\e}atio\slash= 0(or
+yxt/\e}atio\slash= 0). By minimality of N, we can therefore from now on assume that
+e∈Sandd(s) =c(s) =efor alls∈S. Taked=/summationtext
+f∈ob(G)df∈Z(R0)
+wheredf∈Rf,f∈ob(G)and note that Z(R0) =/circleplustext
+f∈ob(G)Z(Rf). Then
+I∋dx−xd=/summationtext
+s∈S/summationtext
+f∈ob(G)(dfxs−xsdf) =/summationtext
+s∈Sdexs−xsde. In theRe
+component of this sum we have dexe−xede= 0sincede∈Z(Re). Thus, the
+summand vanishes for s=e, and hence, by minimality of N, we get that
+dx−xd= 0. Sinced∈Z(R0)was arbitrarily chosen, we get that x∈C
+which is a contradiction. Therefore N= 0and hence S=∅which in turn
+implies that x= 0. Sincex∈Iwas arbitrarily chosen, we finally get that
+I={0}. /square
+The next two results show that Theorem 3 is a simultaneous gen eralization
+of Theorem 1 and Theorem 2.
+Proposition 3. The grading of any strongly groupoid graded ring is right
+(and left) nondegenerate. In particular, the same conclusi on holds for the
+grading of any strongly group graded ring.
+Proof. Suppose that Ris a ring strongly graded by the groupoid Gand that
+we have chosen s∈Gand a nonzero x∈Rs. By Proposition 1(a) it follows
+that1R∈R0. Hence 0/\e}atio\slash=x=x1R=x1d(s)∈xRs−1Rs. Therefore the
+rightR0-idealxRs−1is nonzero. The proof of left nondegeneracy is done
+analogously. The last part follows since a group is a groupoi d. /square
+Proposition 4. Suppose that R=A⋊σ
+αGis crossed groupoid product such
+thatGhas finitely many objects. If for each s∈G, the element α(s,s−1)is
+not a zero divisor in Ac(s), then the standard grading of Ris right (and left)THE IDEAL INTERSECTION PROPERTY FOR GROUPOID GRADED RINGS 9
+nondegenerate. In particular, the same conclusion holds fo r the standard
+gradings of crystalline graded rings.
+Proof. Takes∈Gandx=aus∈Rsfor some nonzero a∈Ac(s). By
+Proposition 1(b) it follows that 1R∈R0. Hence, since α(s,s−1)is not a zero
+divisor in Ac(s), we get that 0/\e}atio\slash=aα(s,s−1)1R=aα(s,s−1)uc(s)=xus−1∈
+xRs−1. Therefore, the right R0-idealxRs−1is nonzero. Similarly one can
+show that the left R0-idealRs−1xis nonzero.
+Now suppose that Gis a group with identity element e. It follows from [17,
+Corollary 1.7] and the discussion preceding it that every cr ystalline G-graded
+ringRmay be presented as A⋊σ
+αGwhereA=Re, the maps σs:A→A,
+fors∈G, are ring automorphisms and for all s,t∈Gthe element α(s,t)is
+not a zero divisor in Resatisfying equations (1)–(5) from Example 1 (note
+thatd(s) =c(t) =esinceGis a group). /square
+Remark 4. (a) Even if we restrict ourselves to the group graded situati on,
+there exist rings that are neither strongly graded nor cryst alline graded but
+still have a grading which is right nondegenerate. In fact, l etKbe any
+integral domain which is not a field; take a nonzero element πinKwhich
+is not a unit. Let R=R0⊕R1be the decomposition as additive groups
+given in Example 5 but with a new multiplication defined by eijejk=eik,
+ifi=jorj=k, andeijejk=πeik, otherwise. Then Ris aZ2-graded
+ring with a right nondegenerate grading. Indeed, suppose th at0/\e}atio\slash=x=
+ae21+be31+ce12+de13∈R1for some a,b,c,d∈K. Ifa/\e}atio\slash= 0orb/\e}atio\slash= 0, then
+xR1∋πae22+πbe32/\e}atio\slash= 0and hence xR1/\e}atio\slash= 0. The case when c/\e}atio\slash= 0ord/\e}atio\slash= 0
+is treated in a similar way. By the equality R1R1=πR0it is clear that Ris
+not strongly graded. By an argument similar to the one used in E xample 5
+it follows that Ris not crystalline graded.
+(b) The groupoid graded ring in the end of Example 5 can, in a si milar
+way, be used to construct a ring which has a right nondegenera te grading,
+but which is neither strongly graded nor a groupoid crossed p roduct. We
+leave the details in this construction to the reader.
+Remark 5. The ideal intersection property does not hold for the commu-
+tant of the principal component in arbitrary category grade d rings. In the
+following example we construct a group graded ring R, whereReis com-
+mutative, for which CR(Re)does not have the ideal intersection property.
+This is made possible through relation (i) below, which ensu res us that the
+grading of Ris not right nondegenerate nor left nondegenerate.
+Example 7. LetC[T]denote the polynomial ring in the indeterminate T.
+Choose some nonzero q∈Cwhich is not a root of unity, i.e. qn/\e}atio\slash= 1for all
+n∈Z\{0}. Consider the automorphism σofC[T]defined by
+σ:C[T]→C[T], p(T)/mapsto→p(qT).
+LetRdenote the algebra generated by C[T]and two symbols XandY,
+subject to the following relations:10 JOHAN ÖINERT AND PATRIK LUNDSTRÖM
+(i)XY= 0andYX= 0.
+(ii)Xp(T) =σ(p(T))Xforp(T)∈C[T].
+(iii)Yp(T) =σ−1(p(T))Yforp(T)∈C[T].
+We can endow Rwith aZ-gradation by putting R0=C[T],Rn=C[T]Xnfor
+n >0andRn=C[T]Y−nforn <0. Furthermore, it is clear that R0=C[T]
+is maximal commutative in Rsinceqis not a root of unity. It follows from
+(i) that the principal ideal I=/a\}bracketle{tX/a\}bracketri}htinRhas trivial intersection with R0and
+henceR0=CR(R0)does not have the ideal intersection property.
+Proof of Theorem 4. We show the contrapositive statement. Suppose
+thatAis not maximal commutative in A⋊σG. Then, by Proposition 2(d),
+there exists some e∈ob(G), somes∈G\ob(G), withd(s) =c(s) =e,
+and some nonzero a∈Ae, such that auscommutes with all of A. LetI
+be the nonzero ideal in A⋊σGgenerated by the element aue−ausand
+define the homomorphism of abelian groups ϕ:A⋊σG→Aby the additive
+extension of the relation ϕ(xut) =x, fort∈Gandx∈Ac(t). We claim that
+I⊆ker(ϕ). If we assume that the claim holds, then, since ϕ|A= idA, it
+follows that A∩I=ϕ|A(A∩I)⊆ϕ(I) ={0}. Now we show the claim. By
+the definition of Iit follows that it is enough to show that ϕmaps elements of
+the form xur(aue−aus)yutto zero, where x∈Ac(r),y∈Ac(t)andr,t∈G
+satisfyd(r) =e=c(t). However, since auscommutes with all of A, we
+get that xur(aue−aus)yut=xur(ayut−ayusut) =xur(ayut−ayust) =
+xσr(ay)urt−xσr(ay)urstwhich, obviously, is mapped to zero by ϕ./square
+4.Applications: Injectivity of ring morphisms
+The ideal intersection property is characterized by the fol lowing important
+proposition, whose proof is easy and therefore omitted.
+Proposition 5. LetR′be a subring of a ring Rsuch that R′has the ideal
+intersection property, then for any ring Sand any ring morphism ϕ:R→S,
+the following assertions are equivalent:
+(i)ϕ:R→Sis injective.
+(ii)ϕ|R′:R′→Sis injective. (The restriction of ϕtoR′.)
+Conversely, if R′is a subring of a ring Rsuch that for any ring Sand any
+ring morphism ϕ:R→S, properties (i)and(ii)above are equivalent, then
+R′has the ideal intersection property.
+We now obtain the following results as corollaries to Theore m 3 respec-
+tively Theorem 5.
+Corollary 1. If the grading of a groupoid graded ring R=/circleplustext
+g∈GRgis right
+(or left) nondegenerate, then for any ring Sand any ring morphism ϕ:R→
+S, the following assertions are equivalent (where R0=/circleplustext
+e∈ob(G)Re):
+(i)ϕ:R→Sis injective.
+(ii)ϕ|CR(Z(R0)):CR(Z(R0))→Sis injective.THE IDEAL INTERSECTION PROPERTY FOR GROUPOID GRADED RINGS 1 1
+Corollary 2. Suppose that Ris a skew groupoid algebra with a commutative
+principal component Aand that the grading of Ris defined by a groupoid
+with finitely many objects. Then the following assertions ar e equivalent:
+(i)Ais maximal commutative in R.
+(ii)For any ring Sand any ring morphism ϕ:R→S, the following
+assertions are equivalent:
+(a)ϕ:R→Sis injective.
+(b)ϕ|A:A→Sis injective.
+References
+[1] Caenepeel, S., De Groot, E. (2004). Galois theory for wea k Hopf algebras.
+arXiv:math/0406186v1 [math.RA]
+[2] Cohen, M., Montgomery, S. (1984). Group-Graded Rings, S mash Products and Group
+Actions. Trans. Amer. Math. Soc. 282(1):237–258.
+[3] Cohen, M., Rowen, L. H. (1983). Group graded rings. Comm. Algebra 11(11):1253–
+1270.
+[4] Dade, E. C. (1980). Group Graded Rings and Modules. Math. Z. 174:241–262.
+[5] Fisher, J. W., Montgomery, S. (1978). Semiprime Skew Gro up Rings. J. Algebra
+52(1):241–247.
+[6] Formanek, E., Lichtman, A. I. (1978). Ideals in Group Rin gs of Free Products. Israel
+J. Math. 31(1):101–104.
+[7] Lang, S. (2002). Algebra . Graduate Texts in Mathematics, 211. New York: Springer-
+Verlag.
+[8] Le Bruyn, L., Van den Bergh, M., Van Oystaeyen, F. (1988). Graded orders . Boston,
+MA: Birkhäuser.
+[9] Lorenz, M., Passman, D. S. (1979). Centers and Prime Idea ls in Group Algebras of
+Polycyclic-by-Finite Groups. J. Algebra 57(2):355–386.
+[10] Lorenz, M., Passman, D. S. (1979). Prime Ideals in Cross ed Products of Finite
+Groups. Israel J. Math. 33(2):89–132.
+[11] Lorenz, M., Passman, D. S. (1980). Addendum - Prime Idea ls in Crossed Products
+of Finite Groups. Israel J. Math. 35(4):311–322.
+[12] Lundström, P. (2005). Crossed Product Algebras Defined by Separable Extensions.
+J. Algebra 283:723–737.
+[13] Lundström, P. (2006). Separable Groupoid Rings. Comm. Algebra 34:3029–3041.
+[14] Lundström, P. (2006). The Picard Groupoid and Strongly Groupoid Graded Modules.
+Colloq. Math. 106:1–13.
+[15] Montgomery, S., Passman, D. S. (1978). Crossed Product s over Prime Rings. Israel
+J. Math. 31(3–4):224–256.
+[16] Nˇ astˇ asescu, C., Van Oystaeyen, F. (1982). Graded Ring Theory . Amsterdam-New
+York: North-Holland Publishing Co.
+[17] Nauwelaerts, E., Van Oystaeyen, F. (2008). Introducin g Crystalline Graded Algebras.
+Algebr. Represent. Theory 11:133–148.
+[18] Öinert, J. (2009). Simple group graded rings and maxima l commutativity. Contemp.
+Math., Amer. Math. Soc. 503:159–175.
+[19] Öinert, J., Lundström, P. (2008). Commutativity and Id eals in Category Crossed
+Products. To appear in the Proceedings of the Estonian Academy of Sciences .
+arXiv:0812.1468v1 [math.RA]
+[20] Öinert, J., Silvestrov, S. D. (2008). Commutativity an d Ideals in Algebraic Crossed
+Products. J. Gen. Lie T. Appl. 2(4):287–302.
+[21] Öinert, J., Silvestrov, S. D. (2008). On a Corresponden ce Between Ideals and Com-
+mutativity in Algebraic Crossed Products. J. Gen. Lie T. Appl. 2(3):216–220.12 JOHAN ÖINERT AND PATRIK LUNDSTRÖM
+[22] Öinert, J., Silvestrov, S. D. (2009). Crossed Product- Like and Pre-Crystalline Graded
+Rings. In: Silvestrov, S., et al., Generalized Lie Theory in Mathematics, Physics and
+Beyond. Springer, pp. 281–296.
+[23] Öinert, J., Silvestrov, S., Theohari-Apostolidi, T., Vavatsoulas, H. (2009). Commu-
+tativity and Ideals in Strongly Graded Rings. Acta Appl. Math. 108(3):585–602.
+[24] Öinert, J., Silvestrov, S. (2009). Commutativity and I deals in Pre-Crystalline Graded
+Rings. Acta Appl. Math. 108(3):603–615.
+[25] Passman, D. S. (1977). The Algebraic Structure of Group Rings . Pure and Applied
+Mathematics. New York-London-Sydney: Wiley-Interscienc e (John Wiley & Sons).
+Johan Öinert, Department of Mathematical Sciences, Univer sity of Copen-
+hagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmar k
+E-mail address :oinert@math.ku.dk
+Patrik Lundström, University West, Department of Engineer ing Science,
+SE-46186 Trollhättan, Sweden
+E-mail address :Patrik.Lundstrom@hv.se
\ No newline at end of file
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+arXiv:1001.0304v1  [cs.SC]  2 Jan 2010A Complete Method for Checking Hurwitz Stability of
+a Polytope of Matrices∗
+Junwei Shao, Xiaorong Hou†
+School of Automation Engineering, University of Electroni c Science and Technology of China, Sichuan, PRC
+E-mail:junweishao@gmail.com ,houxr@uestc.edu.cn
+Abstract: We present a novel method for checking the Hurwitz stability of a po lytope of
+matrices. First we prove that the polytope matrix is stable if and only if two homogenous
+polynomials are positive on a simplex, then through a newly proposed m ethod, i.e., the
+weighted difference substitution method, the latter can be checke d in finite steps. Exam-
+ples show the efficiency of our method.
+Key words: polytope of matrices; Hurwitz stability
+1 Introduction
+Given a linear time-invariant system ˙ x(t) =Ax(t), it is asymptotically stable if the
+system matrix Ais Hurwitz stable, i.e., all eigenvalues of A have negative real parts.
+Sometimes, we need to consider a type of system uncertainty in whic h case the family of
+system matrices forms a polytope, i.e., Ais varying in
+A=/braceleftiggm/summationdisplay
+k=1qkAk:m/summationdisplay
+k=1qk= 1,qk≥0 for all k/bracerightigg
+, (1)
+whereA1,...,A m∈Rn×nare constant matrices. We say that Ais robustly Hurwitz stable
+if each matrix in Ais Hurwitz stable. The stability of a polytope of matrices cannot be
+derived from the stability of all its edges [ 1], which is the case of the stability of a polytope
+of polynomials [ 2]. In fact, [ 3] proved that for a polytope of n×nmatrices, the stability of
+all 2n−4 dimensional faces can guarantee the stability of the polytope, an d the number
+2n−4 is minimal. But checking the stability of 2 n−4 dimensional faces of a polytope is
+also a difficult task. For a matrix polytope with normal vertex matrice s, [4] proved that
+∗Partially supported by a National Key Basic Research Project of Ch ina (2004CB318000) and by
+National Natural Science Foundation of China (10571095)
+†The author to whom all correspondence should be sent.
+1the stability of vertex matrices is necessary and sufficient for the s tability of the whole
+polytope. Meanwhile, serval sufficient criterions [ 5–11] are provided to check stability of
+matrix polytopes.
+Based on the newly proposed results on checking positivity of forms (i.e., homogenous
+polynomials), we present a method for checking the stability of a poly tope of matrices in
+this paper, this method is complete, moreover, it only has a power ex ponential complexity.
+2 Notations
+•R: the field of real numbers.
+•Z: the set of all integers.
+•N: the set of all nonnegative integers.
+•Rm×n: the space of m×nreal matrices.
+•In: the identity matrix of order n.
+•detA: the determinant of a square matrix A.
+•∆f: the Hurwitz matrix of the polynomial f(x) =anxn+an−1xn−1...+a0, it is an
+n×nmatrix defined as
+∆f=
+an−1an−3an−5···0
+anan−2an−4···0
+0an−1an−3···0
+0anan−2···0
+.........................
+.........................
+.
+The successive principal minors of ∆ fare denoted by ∆ k,k= 1,2,...,n.
+•(k1k2...km): a permutation of {1,2,...,m}, which changes itoki,i= 1,...,m.
+•Θm: the set of all m! permutations of {1,2,...,m}.
+•deg(f): the degree of a polynomial.
+•AT: the transpose of a matrix or vector A.
+•Sm: them−1 dimensional simplex in Rm, i.e.,
+Sm={(x1,...,x m) :m/summationdisplay
+i=1xi= 1,xi≥0,i= 1,...,m}.
+•[x]: the largest integer not exceeding the number x.
+23 Main Results
+Suppose
+A=m/summationdisplay
+k=1qkAk
+is a matrix in A, denote its characteristic polynomial by
+fA(s)/definesdet(sIn−A) =sn+an−1sn−1+...+a1s+a0. (2)
+It is well known that, for all 1 ≤i≤n, (−1)ian−iequals the sum of all principle minors
+of orderiofA, hencean−iis a form of degree ionq1,q2,...,q m. Denote by bijthe (i,j)th
+entry of the Hurwitz matrix of fA(s), thenbij= 0, orbijis a form of degree 2 j−ion
+q1,q2,...,q m. Since
+∆k=/summationdisplay
+(j1j2...jk)∈Θk±b1j1b2j2···bkjk,
+and
+b1j1b2j2···bkjk= 0,
+or
+deg(b1j1b2j2···bkjk) =k/summationdisplay
+i=1(2ji−i) =k/summationdisplay
+i=1i=k(k+1)
+2,
+we can see that ∆ kis a form of degree k(k+1)/2 onq1,q2,...,q m.
+Theorem 1. There exists a Hurwitz stable matrix in the matrix polytope A, thenAis
+Hurwitz stable if and only if
+∆n−1(q1,...,q m)>0anda0(q1,...,q m)>0,(q1,...,q m)∈Sm. (3)
+Proof.The necessity is directly from Routh-Hurwitz criterion. If Ais not Hurwitz stable,
+then by continuity there must exist a matrix AinAwhich has eigenvalues lying on the
+imaginary axis. Suppose eigenvalues of Aares1,...,s n. If some siequals zero, then
+a0= (−1)ns1s2···sn= 0,
+which contradicts the hypothesis a0>0. If some siandsjare a pair of conjugate eigen-
+values of Aon the imaginary axis, then from Orlando’s formula [ 12],
+∆n−1= (−1)n(n−1)
+2/productdisplay
+1≤i<j≤n(si+sj) = 0,
+which contradicts the hypothesis ∆ n−1>0.
+3Remark: [ 5] also proved that a polytope matrix Ais robustly stable if and only if an
+associated form γdet/tildewideAis positive on a simplex, where /tildewideAis the Kronecker sum of the
+matrixAinAwith itself, and γis the sign of det /tildewideA./tildewideAis ann2×n2matrix, hence the
+degree of γdet/tildewideAis much larger than ∆ n−1anda0in (3).
+A newly proposed method, i.e. the difference substitution method [ 13,14], can be used
+to check positivity of forms efficiently. [ 15] proved that a form is positive if and only if we
+can get forms with positive coefficients after finite steps of varied f orms of substitutions,
+i.e., weighted difference substitutions(WDS). [ 16] further gave a bound for the number of
+steps required, and pointed out that the WDS method is complete in c hecking positivity or
+nonnegativity of integral forms. We will introduce this method more detailedly in Section
+4. Based on this method, we have
+Theorem 2. Suppose entries of Ak,k= 1,...,min(1)are all rational, the magnitudes
+of coefficients of ∆n−1anda0in(3)are bounded by M, then the Hurwitz stability of A
+can be checked by an algorithm with complexity
+O/parenleftig
+mm+1n2m2(n2mlnM+n2(m+1)lnm+2(m2+mn2m)lnn)/parenrightig
+.
+4 Positivity of forms on simplices
+In this section, we will introduce the WDS method for checking positiv ity of forms [ 16],
+and analyze the complexity of checking Hurwitz stability of matrix poly topes through this
+method.
+Suppose θ= (k1k2...km)∈Θm, letPθ= (pij)m×mbe the permutation matrix corre-
+sponding θ, that is
+pij=/braceleftbigg
+1, j=ki
+0, j/ne}ationslash=ki.
+GivenTm∈Rm×m, where
+Tm=
+11
+2...1
+m
+01
+2...1
+m............
+0...01
+m
+, (4)
+let
+Aθ=PθTm,
+and call it the WDS matrix determined by the permutation θ. The variable substi-
+tutionx=Aθycorresponding θis called a WDS, where x= (x1,x2,...,x m)T,y=
+(y1,y2,...,y m)T.
+Letf(x)∈R[x1,x2,...,x m] be a form, we call
+WDS(f) =/uniondisplay
+θ∈Θm{f(Aθx)} (5)
+4the WDS set of f,
+WDS(k)(f) =/uniondisplay
+θk∈Θm···/uniondisplay
+θ1∈Θm{f(Aθk···Aθ1x)} (6)
+thekth WDS set of ffor positive integer k, and set WDS(0)(f) ={f}.
+Letα= (α1,α2,...,α m)∈Nm,|α|=α1+α2+···+αm. For a form of degree d
+f(x1,x2,...,x m) =/summationdisplay
+|α|=dcαxα1
+1xα2
+2···xαm
+m,
+if all coefficients cαare nonzero, we say fhas complete monomials.
+It is obvious that if there exists k∈N, such that forms in WDS(k)(f) all have complete
+monomials, and their coefficients are all positive, then fis positive on Sm. In fact, the
+reverse is also true, and for integral forms, the upper bound for kcan also be estimated.
+Theorem 3 ([16]).Supposef∈Z[x1,x2,...,x m]is a form of degree d, and the magnitudes
+of its coefficients are bounded by M, thenfis positive on Sm, if and only if there exists
+k≤Cp(M,m,d), such that each form in WDS(k)(f)has complete monomials, and its
+coefficients are all positive, where
+Cp(M,m,d) =
+ln/parenleftig
+2dmMdm+1mdm+1+dd(m+1)d+mdm(d+1)(m−1)(m+2)/parenrightig
+lnm−ln(m−1)
++2 (7)
+Remark: The Cp(M,m,d) in (7) provides a theoretical upper bound of the number of
+steps of substitutions required to check positivity of an integral f orm. In practice, numbers
+of steps used are generally much smaller than this bound [ 15].
+proof of Theorem 2.A formf(q1,...,q m) of degree dhas at most
+/parenleftbiggd+m−1
+m−1/parenrightbigg
+≤(d+1)m−1
+monomials thus the number of arithmetic operations of computing WD S(f) is bounded by
+m!(d+1)m−1(d+1)m(m−1)≤mm(d+1)m2. (8)
+Moreover
+Cp(M,m,d) =O/parenleftbig
+m(dmlnM+dm+1lnm+(m2+mdm)lnd)/parenrightbig
+,
+and deg(∆ n−1) =n(n−1)/2,deg(a0) =n, therefore the complexity of our method for
+checking the robust Hurwitz stability of a polytope of n×nmatrices is
+O/parenleftig
+mm+1n2m2(n2mlnM+n2(m+1)lnm+2(m2+mn2m)lnn)/parenrightig
+.
+55 Examples
+First we will illustrate our method through an example from [ 1]. Suppose Ais a
+polytope of following matrices
+A1=
+−1 0 1
+0−1 0
+−1 0 0 .1
+,
+A2=
+−1 0 0
+0−1 1
+0−1 0.1
+,
+A3=
+−1 0−1
+0−1−1
+1 1 0 .1
+,
+letA=q1A1+q2A2+q3A3, and denote by fA(s) the characteristic polynomial of A. The
+second successive principle minor of the Hurwitz matrix of fA(s) is
+∆2=63
+25q13+99
+25q12q3+243
+50q32q1+144
+25q1q22+153
+25q1q2q3
++144
+25q12q2+243
+50q2q32+63
+25q23+99
+25q22q3+171
+50q33,
+which is obviously positive on S3. The constant term of fA(s) is
+a0=9
+10q13+9
+10q23−23
+5q1q2q3−13
+10q12q3+7
+10q12q2
+−3/10q2q32−3/10q32q1+7
+10q1q22−13
+10q22q3+19
+10q33.(9)
+a0is not positive on S3since the following form (with a difference of a positive constant
+factor) belongs to WDS(3)(a0) and its coefficients are all negative:
+−6516q1q2q3−1296q13−891q23−3888q12q3−3888q12q2−1568q2q32
+−2828q32q1−3483q1q22−2223q22q3−236q33,
+therefore the polytope Ais not Hurwitz stable.
+Furthermore, we have checked 900 polytopes of matrices for n= 2,3,4 andm= 2,3,4,
+i.e., 100 polytopes for each pair ( n,m). The vertexes of these polytopes are generated
+following a similar method as was described in [ 11]: their entries are real numbers with 4
+significant numbers anduniformly distributed intheinterval [ −1,1], moreover themaximal
+real parts of their eigenvalues equal −0.0001 (if not so, a shift should be performed). Table
+6Table 1: Time used to check robust Hurwitz stability of 100 polytopes for each pair ( n,m)
+nmnumber of stable / unstable polytopes total time (in seconds)
+22 67 / 33 0.125
+3 29 / 71 1.578
+4 11 / 89 2537.360
+32 58 / 42 0.123
+3 28 / 72 2.009
+4 9 / 91 665.129
+42 54 / 46 0.090
+3 21 / 79 2.608
+4 4 / 96 1894.602
+1shows the time used to check the stability of these polytopes on a co mputer equipped
+with Intel Core 2 Duo E4500 CPU at 2.2 GHz and 4.0 GB of RAM memory, o ur program
+have been implemented in the computer algebra system Maple.
+Remark: Generally, the time gets longer with the increase of norm, but there are
+some extreme examples that very long time may be spent even for sm allnandm. For
+example, in our experiment, it takes 1129.656 and 1172.234 seconds respectively to check
+the Hurwitz stability of two matrix polytopes generated for ( n,m) = (2,4), this makes the
+time corresponding to ( n,m) = (2,4) much longer than that corresponding to ( n,m) =
+(3,4) or (n,m) = (4,4) in Table 1.
+References
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+[12] F. R. Gantmacher. The Theory of Matrices. Chelsea Publishing C ompany, 1960. 3
+[13] Lu Yang, Difference substitution and automated inequality prov ing. Journal of
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+Asian Technology Conference in Mathematics, ATCM Inc, 2005, 37- 46.4
+[15] Yong Yao. Termination of the Sequence of SDS Sets and Machine Decision for Positive
+Semi-definite Forms. arXiv: 0904.4030. 4,5
+[16] X. Hou, J. Shao. Completeness of the WDS method in Checking Po sitivity of Integral
+Forms. arXiv:0912.1649v1. 4,5
+8
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+arXiv:1001.0305v1  [astro-ph.IM]  2 Jan 2010Preliminary Astrometric Results from Kepler
+David G. Monet
+U. S. Naval Observatory, Flagstaff, AZ 86001
+Jon M. Jenkins
+SETI Institute, Mountain View, CA 94043
+Edward Dunham
+Lowell Observatory, Flagstaff, AZ 86001
+Stephen T. Bryson
+NASA/Ames Research Center, Moffett Field, CA 94035
+Ronald L. Gilliland
+Space Telescope Science Institute, Baltimore, MD 21218
+David W. Latham
+Harvard-Smithsonian Center for Astrophysics, Cambridge MA 021 38
+William J. Borucki
+NASA/Ames Research Center, Moffett Field, CA 94035
+and
+David G. Koch
+NASA/Ames Research Center, Moffett Field, CA 94035
+Received ; accepted– 2 –
+ABSTRACT
+Although not designed as an astrometric instrument, Kepleris expected to
+produce astrometric results of a quality appropriate to support m any of the
+astrophysical investigations enabled by its photometric results. O n the basis of
+data collected during the first few months of operation, the astro metric precision
+for a single 30 minute measure appears to be better than 4 milliarcsec onds (0.001
+pixel). Solutions for stellar parallax andproper motions await more ob servations,
+but the analysis of the astrometric residuals from a local solution in t he vicinity
+of a star have already proved to be an important tool in the proces s of confirming
+the hypothesis of a planetary transit.
+Subject headings: astrometry — stars: fundamental parameters– 3 –
+1. Introduction
+The measurement of astrometric parameters, particularly the pa rallax, for Keplerstars
+is a critical component of computing the physical values for various stellar parameters
+using the relative values that are computed from the photometric a nalysis. If the Kepler
+data can be shown to have the necessary astrometric accuracy, then such a conversion can
+be included in the processing for most if not all Keplerstars. Although the discussion of
+Keplerastrometric accuracy must await more data and modeling, the very high precision
+ofKeplerpositions is already a powerful tool for understanding the photom etric variations
+of stars and the possible presence of planetary companions. Deta iled discussions of the
+Keplerspacecraft and mission are presented by Borucki et al. (2010) an d Koch et al.
+(2010). A short overview for the astrometric discussion is the follo wing. The Schmidt
+telescope has a 1.4-m primary and a 0.95-m corrector, and the phot ometer is a mosaic of
+42 charge coupled devices (CCDs). The boresight of the telescope remains constant for
+the mission, but the spacecraft rolls 90 degrees every three mont hs. Due to restrictions in
+memory and bandwidth, only the pixels associated with the target st ars are sent to the
+ground for processing. Target stars are defined by software, a nd pixels not associated with
+targets are saved only infrequently. The basic integration time is 6.0 2 seconds, and the long
+cadence (LC) sequence (Jenkins et al. 2010) co-adds the target pixels for 29.4 minutes. The
+co-added pixel data are sent to the ground every month for proc essing and analysis. This
+strategy enables the extremely high signal-to-noise (SNR) observ ations needed to achieve
+the planetary detection mission.
+To obtain the large field of view, the image sampling is very coarse comp ared to
+other astrometric assets. The Pixel Response Function (PRF) is d escribed in great detail
+by Bryson et al. (2010), but a quick summary is as follows. The images contain three
+components, a sharp spike in the middle that comes from optical diffr action (about 0.1– 4 –
+arcsecond), a wider component with a characteristic size of 5 or 6 a rcseconds set primarily
+by mechanical alignment tolerances of the CCD mosaic, and a much br oader scattering
+profile. The intermediate component of the image profile produces t he majority of the
+astrometric signal, and it contains about 70% of the light. The 43 day s of data available so
+far1have shown a remarkable astrometric precision, but the demonstr ation of astrometric
+accuracy is still a work in progress. Even if the centroiding process was fully understood,
+the short interval of available data precludes the lifting of the dege neracies between effects
+of proper motion, parallax, and velocity aberration. No measured a strometric parameters
+are given here. Indeed, the entire range of observed image motion is about 0.2 pixels, and
+this is dominated by the spacecraft guiding precision.
+Various authors, including King (1983) and Kaiser et al. (2000), hav e developed
+theoretical expectations for the astrometric precision of an imag e. A simple approximation
+is
+precision =FWHM/ (2∗SNR) (1)
+where FWHM is the image full width at half maximum, and SNR is the photo metric
+signal-to-noise ratio of that star image. The differences in the theo retical derivations concern
+the exact value for which the approximate value of 2 is used above. T he observational
+confirmation of this relationship has yet to be done, but essentially a ll ground- and
+space-based astrometric studies have demonstrated the validity of the scaling of this
+relationship. Improved astrometric precision is obtained for smaller image FWHM, higher
+SNR, or both assuming that adequate image sampling is available.
+Kepleroperates in a heretofore unstudied astrometric domain. The pixels are very
+1As described more fully by Caldwell at al. (2010), the spacecraft da ta are grouped by
+observing quarters as defined by the mandatory rolls of the space craft itself. So far, data
+from Quarters 0 and 1 are available on the ground.– 5 –
+large, 3.98 arcseconds, as compared to other ground- and space -based astrometric assets,
+and the observed FWHM is approximately 5 to 6 arcseconds and depe nds on the location
+in the field of view. (See Bryson et al. (2010) for further discussion and examples.) The
+effects of undersampled image components are not captured by Eq . 1. However, Kepler
+was designed for extremely high SNR observations. The well capacit y of the 27-micron
+CCD pixels is more than a million electrons, and most of the stars are br ight. As more
+fully discussed by Caldwell at al. (2010), the onset of saturation in t he basic 6.02 second
+integration cycle is near the magnitude Kp= 11.3.2A single LC co-addition produces a
+SNR of about 10,000 for an bright, unsaturated, uncrowded star . There is no atmosphere to
+degrade the image quality, and the flux is so large that effects such a s sensor readout noise
+and dark current are unimportant for the brightest 2-3 magnitud es of unsaturated stars.
+2. Preliminary Astrometric Investigations
+The astrometric processing of Keplerdata is conceptually no different than the
+traditional differential astrometric process of data from other g round- and space-based
+assets. There is no need to worry about the actual coordinates ( i.e., J2000 RA and Dec)
+of the stars. Essentially all Keplerstars are in the 2MASS catalog (Skrutskie et al. 2006),
+and most are in the UCAC-2 catalog (Zacharias, et al. 2004). Rathe r, the goal of the
+analysis is to measure the small changes in position associated with pr oper motion, parallax,
+perturbations from unseen companions, and blending with photome trically variable stars.
+The current astrometric pipeline involves three distinct steps: cen troids are computed from
+2TheKeplermagnitude Kpincludes a very wide passband and is similar to an astronom-
+icalRmagnitude in central wavelength.– 6 –
+the pixel data, transformations are computed from each channe l3of each LC co-addition
+into an intermediate coordinate system, and solutions for each sta r are computed using the
+intermediate coordinates and terms such as time, parallax factor, etc. Steps two and three
+are iterated a few times, and convergence is quite rapid. Details of e ach of these steps are
+presented in the following subsections.
+2.1. Centroids
+Most modern centroiding algorithms fall into three classes: moment analysis, fits
+to analytic functions, and fits to the instrumental point spread fu nction (PSF). So far,
+various algorithms from the first two classes have been implemented and tested. The data
+processing pipeline of the Science Operations Center (SOC; Jenkins et al. (2010)) compute
+flux-weighted means for all stars and Gaussian fits and PSF fits for a few stars. PSF fitting
+of all stars remains a task for the future. In a separate effort, s everal other centroiding
+algorithms based on fitting the images to analytic functions have bee n evaluated, and
+centroids for all stars have been computed for many of these. Th e choice of centroiding
+algorithm requires special attention because of the properties of theKeplerimages discussed
+above. The effect of the undersampled components of the images h as not been fully
+evaluated, and the number of pixels for each star has been minimized by the Optimal
+Aperture algorithm (Bryson et al. 2010) so that the maximum numbe r of stars can be
+transmitted in the fixed spacecraft bandwidth. Because the best astrometric centroiding
+algorithm has not been identified yet, analysis is proceeding with para llel tracks to evaluate
+3As described more fully by Jenkins et al. (2010), each CCD is split into t wo channels by
+the flight electronics. The astrometric verification of the stability b etween the two channels
+of a single CCD has yet to be performed.– 7 –
+a few of the most promising algorithms.
+2.2. Transformation Coefficients
+The current astrometric pipeline supports two different coordinat e systems. For some
+investigations, working in a sky-based system seems appropriate. The tangent point is
+taken to be the nominal Keplerboresight ( α= 19h22m40s,δ= +44◦30′, J2000) and a simple
+tangent plane projection is computed from the nominal positions of the stars listed in the
+KeplerInput Catalog (KIC).4In this coordinate system, effects such as differential velocity
+aberration and parallax are easy to visualize. For other investigatio ns, a channel-based
+coordinate system seems appropriate, and effects arising from th e structure and behavior of
+CCD pixels are easier to visualize. Of particular importance are effect s based on where the
+star falls with respect to the pixel grid, a term called “pixel phase”. In either coordinate
+system, a separate set of transformation coefficients is compute d for the measures from each
+channel for each cadence. A sample of 1000 known distant giant st ars was included in the
+Keplerstar list, and are used during the first iteration to generate a tran sformed coordinate
+system that is as close to inertial as possible.
+2.3. Astrometric Coefficients
+The tasks of computing the centroids and the transformation coe fficients for the Kepler
+data are similar to their counterparts for traditional ground- and space-based assets. It
+is the modeling of the measured positions for each star that involves special attention,
+and this flows from the extremely high SNR of the data. The simplest s olutions based on
+4http://archive.stsci.edu/kepler/kepler fov/search.php– 8 –
+computing only the mean positions from the data currently available s how an astrometric
+precision near 20 milliarcseconds (about 0.005 pixels). This value and t hose presented below
+refer to the uncertainty in a single measure of a single axis (row or co lumn) for a single star
+from a single LC co-addition. Because these solutions are local and c ontain only a small
+number of measures, they should be construed as estimators of t he astrometric precision
+and not of the overall astrometric accuracy of the spacecraft a nd photometer. Adding terms
+that model the differential velocity aberration across the Keplerfield reduces this error to
+about 4 milliarcseconds (about 0.001 pixels), and adding terms arising from the pixel phase
+reduce the errors to about 2 milliarcseconds (about 0.0005 pixels). These pixel phase terms
+are empirical fits, and are not derived from detailed modeling of the im age formation and
+sampling processes.
+During the first 33.5 days of science operations that followed the en d of commissioning,
+the number of stars was increased to 156,000. The volume of data f or this number of stars
+observed every 29.4 minutes is large, and the data from the 84 chan nels are diverse. Thus
+it is difficult to characterize the entirety of the astrometric solution with a single number.
+Again, much development in the astrometric processing is needed be cause every Kepler
+star is important. Although only preliminary measures of astrometr ic precision have been
+obtained, it is reassuring to see that the observed errors follow th e prediction based on the
+flux of the stars involved. Fig. 1 shows the errors for stars in a sing le channel as a function
+of the measured Kp, and demonstrates that the error rises as the SNR decreases.
+3. Local Astrometric Solutions and the “Rain” Plots
+A special case of the astrometric solutions described above can be computed in the
+vicinity of individual stars. Under the assumptions of small parallax a nd adequate removal
+of differential velocity aberration, the equations for the apparen t place of a star can be– 9 –
+linearized. Simple trend analysis produces a robust estimator for th e mean position of
+a star, and residuals from each measurement are computed. This e nables astrometric
+processing to contribute to the understanding of the Keplerstars. As more fully discussed
+by Batalha et al. (2010), what appears to be a single object can be t wo or more stars,
+and each can have photometric variability. Such astrations can mimic the photometric
+properties of a transiting planet, and adding astrometry to the ve tting procedure can assist
+in the confirmation or denial process. Fig. 2 shows astrometric res iduals for two stars.
+The upper star is a blend of a variable star and one or more constant stars while the lower
+shows residuals that are typical for a bright, constant star. The astrometric amplitude of
+the variable star is huge - almost 0.02 pixels.
+The astrometric behavior of an image composed of an unknown numb er of stars each
+of which having unknown photometric variations is complicated. Howe ver, the simple case
+of two stars, a small photometric variation, and centroids comput ed from flux-weighted
+means provides much insight. Where ∆ sis the true separation of the stars, δsis the small
+measured astrometric shift, Fis the small relative brightness of the fainter B component
+compared to the brighter A component, and fis the small relative change in total brightness
+due to a transit or stellar variability, the observed astrometric shif t assuming that the A
+component is variable is given by
+δs=Ff∆s (2)
+If the B component has the photometric variation, the observed s hift is
+δs=f∆s (3)
+When the Keplerobservations of δsandfare combined with high resolution imagery that
+can measure ∆ s,F, and other characteristics such as stellar colors, then the model of the
+composite image can be improved.
+On the basis of its photometric signature alone, the KeplerObject of Interest (KOI-)– 10 –
+15 might involve a transiting planet. The analysis of the combined phot ometric and
+astrometric residuals denies this hypothesis. Fig. 3. shows the time series astrometric and
+photometric residuals for KOI-15 after they have been high-pass filtered so as to emphasize
+signals with shorter timescales. A different visualization of the same d ata is called a “Rain
+Plot”, and is shown in Fig. 4. Clearly, the astrometric and photometr ic residuals for
+KOI-15 are strongly correlated, and these correlations are the s ignature of an astration that
+includes one or more relatively constant stars and a background ec lipsing binary. Indeed,
+the secondary eclipse is more apparent in the astrometric residuals than in the photometric
+residuals. A true transiting planet system should not show these co rrelations. As suggested
+by the Rain Plot, the pixel data were re-examined and the offending v ariable star was
+identified as being about 11 arcseconds away from and about 4.8 Kpmagnitudes fainter
+than the brighter star.
+4. Conclusions
+Although based on just a small fraction of the data expected from the entire mission,
+the following conclusions can be drawn.
+a) Both the preliminary version of the full astrometric solution and t he locally
+linearized astrometric solution indicate that the precision of a single m easure of a typical
+star is about 0.004 arcsecond (about 0.001 pixel). Results such as t hose shown in Figs. 2
+and 3 suggest that the precision for some stars might be substant ially better.
+b) On the basis of the limited data available so far, many astrometric e ffects cannot be
+separated. The astrometric precision should enable the measurem ent of the proper motions
+of the known large motion stars in the star list, but as yet proper mo tion is indistinguishable
+from differential velocity aberration, pixel phase, and similar effect s.– 11 –
+c) Because of the large pixel size, Keplerimages can often be blends of multiple stars. If
+one or more components of such a blend is a photometrically variable s tar, the astrometric
+position of the image can have a significant motion. Tools such as the R ain Plot have
+already demonstrated the utility of combining astrometric and phot ometric processing in
+the evaluation of planetary transit candidates.
+d) Whereas the astrometric precision of Keplerdata has been demonstrated, the
+astrometric accuracy has yet to be evaluated.
+In summary, it is the extremely high SNR of the Keplerphotometer that enables
+the astrometric analysis of Keplerdata. Even with just the preliminary data, astrometric
+analysis is providing an important tool for the physical understand ing of the observations.
+Should the astrometric results continue to follow the theoretical e xpectation of improving
+with the SNR, then solutions for the parallax and proper motions of a ll stars will be
+computed.
+Funding for this Discovery mission is provided by NASA’s Science Mission Directorate.
+Many people have contributed to the success of the KeplerMission, and the authors wish
+to express their profound thanks to all.
+Facilities: The Kepler Mission.– 12 –
+Fig. 1.— Astrometric error as a function of Kpmagnitude for stars on Channel 2 in the
+Quarter 1 data collection.
+Fig. 2.— Astrometric residuals from a blended variable (top) and a non -variable star (bot-
+tom) taken from a solution for a single channel. Red symbols are from the columns and blue
+symbols are from the rows.– 13 –
+−1.5−1−0.500.51Relative Flux Residuals, pptA
+−14−12−10−8−6−4−202Row Residuals, mpixB
+4965 4970 4975 4980 4985 4990 4995−50510152025Column Residuals, mpix
+Time, JD−2450000C
+Fig. 3.— Time series residuals for the relative photometric flux (A) in pa rts per thousand,
+and the relative astrometric row (B) and column (C) residuals in millipixe ls (1 pixel = 3.98
+arcseconds) for KOI-15. The observations have been filtered to remove long-period trends.
+The strong correlation between these residuals is taken as evidenc e that this Keplerstar is
+actually an astration of one or more relatively constant stars and a background eclipsing
+binary.– 14 –
+−0.02 −0.01 00.01 0.02 0.03−1.5−1−0.500.51
+Centroid Shift, pixelsNormalized Flux Residuals, ppt  
+Fig. 4.— The Rain Plot for KOI-15. This visualization shows the correlat ion between
+the photometric and astrometric row (red X) and column (blue +) re siduals. The strong
+correlation between the position and brightness is evidence that th is is an astration of one
+or more relatively constant stars and a background eclipsing binary .– 15 –
+REFERENCES
+Batalha, N. M., et al. 2010, this issue
+Borucki, W. J. et al. 2010, Science, submitted
+Bryson, S. T., et al. 2010, this issue
+Caldwell, D. A., et al. 2010, this issue
+Jenkins, J. M., et al. 2010, this issue
+Kaiser, N., Tonry, J. L., Luppino, J. L. 2000, PASP112, 768
+King, I. R. 1983, PASP95, 163
+Koch, D. G., et al. 2010, this issue
+Skrutskie, M. J., et al. 2006, AJ131, 1163
+Zacharias N., et al. 2004, AJ127, 3043
+This manuscript was prepared with the AAS L ATEX macros v5.2.
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+arXiv:1001.0306v2  [gr-qc]  28 Jun 2010Thermodynamics properties of the dark energy in loop quantu m cosmology
+Kui Xiao∗and Jian-Yang Zhu†
+Department of Physics, Beijing Normal University, Beijing 100875, China
+(Dated: June 27, 2018)
+Considering an arbitrary, varying equation of the state par ameter, the thermodynamic properties
+of the dark energy fluid in a semiclassical loop quantum cosmo logy scenario, which we consider the
+inverse volume modification, is studied. The equation of the state parameters are corrected as a
+semiclassical one during considering the effective behavio r. Assuming that the apparent horizon
+has Hawking temperature, the modified entropy-area relatio n is obtained, we find that this relation
+is different from the one which is obtained by considering the holonomy correction. Considering
+the dark energy is a thermal equilibrium fluid, we get the expr essions for modified temperature,
+chemical potential and entropy. The temperature, chemical potential and entropy are well-defined
+in the semiclassical regions.
+PACS numbers: 04.60.Pp, 04.60.Kz, 98.80.Qc
+I. INTRODUCTION
+More and more evidences [1] show that our universe
+may be dominated by dark energy (for more recent re-
+view, please read [2] and references therein.). Although
+the cosmology constant Λ maybe responsible for the ob-
+served data, but it is also the possible that there are
+exiting the dynamical mechanism is at work. There are
+some candidates to describe those dynamical systems,
+e.g., a canonical scalar field (quintessence) [3], a scalar
+field with negative sign of kinetic term (phantom) [4], or
+thecombinationofquintessenceandphantominaunified
+model named quintom [5] and so on [2]. Many authors
+have analysed the characters of dark energy in the clas-
+sical cosmology, e.g., the inflation caused by dark energy
+[6], attractor behavior and solution [7], the relationship
+between geometry and dark energy [8], the fate of uni-
+verse with dark energy [9], the singularity of universe
+with dark energy [10], the thermodynamical properties
+of dark energy [11] and so on. And some authors stud-
+ied the quantum properties of it [12]. But we still have
+very little knowledge about it, and there is not any direct
+evidence shows whether the dark energy exists. So it is
+necessary to study the properties of it.
+The universe can be considered as a thermodynamical
+system. The thermodynamical properties of the universe
+was studied by many authors [13, 14]. The apparent
+horizon and event horizon of universe are different, the
+first and second law of thermodynamics hold on the ap-
+parent horizon, but they break down while we consider
+the event horizon [14]. According to [15], the thermo-
+dynamical properties of the universe in loop quantum
+cosmology (LQC) is still held. But the corrected Fired-
+mann equation will modify the entropy-area relation of
+apparent horizon in LQC [16]. Due to this subtlety, we
+will focus our attention on the thermodynamical prop-
+∗Electronic address: 87xiaokui@mail.bnu.edu.cn
+†Electronic address: zhujy@bnu.edu.cnerties of the apparent horizon first. And then we will
+derive thermodynamical variables of dark energy fluid in
+the inner side ofthe apparent horizonin the semiclassical
+LQC scenario.
+Loop quantum cosmology (for more recent review,
+please read [17, 18]) is a canonical quantization of homo-
+geneous spactimes based upon techniques used in loop
+quantum gravity (LQG). Due to the homogeneous and
+isotropic spacetime, the phase space of LQC is simpler
+than LQG. The connection is determined by a single pa-
+rametercand the triadis determined by p. The variables
+candpare canonically conjugate with Poisson bracket
+{c,p}=8πG
+3γ, in which γis the Barbero-Immirzi pa-
+rameter. In the LQC scenario, the initial singularity is
+instead by a bounce. So, thanks for the quantum effect,
+the universe is initially contracting phase with the min-
+imal but not zero volume, and then the quantum effect
+drivesit to the expanding phase. If one wants to consider
+the physical effect near the minimal area, e.g., very small
+scalefactor a < ai=√γlpl, it isnecessarytoconsiderthe
+difference equation [19]. But if we consider the interval
+ai< a < a ⋆=/radicalbig
+γj/3lpl, with the quantum parameter j,
+we can consider the effective dynamics of LQC. And the
+classical cosmology can be recovered if a > a ⋆. In this
+paper, we just consider the thermodynamical of the dark
+energy fluid in the semiclassical regions.
+In the classicalregions, the darkenergythermodynam-
+ics has been studied by many authors [11]. The thermo-
+dynamics properties of dark energy fluid described by
+the equation of state parameter ωcl=Pcl/ρcl, in which
+Pcl,ρclare the pressure and energy density of dark en-
+ergy fluids respective. For the thermodynamical proper-
+ties of quintessence filed, with a constant ωcl>−1, it
+is nature to use the results of conventional perfect fluid
+thermodynamics [20]. But, in the case of phantom fields,
+it is more complex if we consider constant ωcl<−1.
+Considering the chemical potential µ= 0, some authors
+showed that the energy density and entropy are positive,
+but the temperature is negative [21, 22], and they ar-
+gued that this negative temperature due to the quantum
+properties of phantom fields. And some other authors2
+showed that, if the temperature, energy density and en-
+tropy is positive, one must get a negative chemical po-
+tential [23]. But considering the varying ωcl(a), one can
+obtain more physics contents, well-defined temperature,
+energy density, chemical potential, and entropy on the
+case ofωcl(a) =−1, but the temperature will divergence
+whena= 0 in the classical universe [20]. In addition,
+if considering that the temperature of dark energy fluids
+have the same temperature of the Hawking temperature
+on the apparent horizon, one maybe get negative entropy
+in condition of ωcl(a)<−1 [24].
+Based on those confused properties of the thermody-
+namics of dark energy fluids, it is necessary to discuss
+it in the semiclassical regions, even quantum regions. In
+this paper, we will focus our interested on the semiclas-
+sical one based on the effective LQC. Just as the above
+discussion, the scale factor should be in the regions of
+ai< a < a ⋆. The aim of this paper is tripartite. The
+first is to get the modified equation of state parameters
+ωscin the semiclassicalLQC. The second is to discuss the
+modified entropy-arearelation of apparent horizon which
+is caused by the corrected Firedmann equation. And the
+last is to obtainthe expressionsofthermodynamicalvari-
+ables of dark energy fluids which is in the inner side of
+the apparent horizon.
+The organization of this paper is as follows. In Sec.
+II, we will introduce the basic concepts of semiclassical
+LQC and get the modified varying ωsc(a). And then
+in Section III, we will obtain the modified entropy-area
+relation of the apparent horizon and the expressions of
+thermodynamicalvariablesofthedarkenergyfluidinside
+of the apparent horizon. In the last section IV, we will
+get some conclusions and discussions.
+II. MODIFIED EFFECTIVE DYNAMICS IN
+LQC
+The basic variables of phase space of LQC are the con-
+nectioncand the triad p. In our interested background
+spacetime, the connection and triad can be related to the
+scale factor, c=γ˙a,p=a2=V2/3, with the Barbero-
+Immirziparameter γ≈0.2375. The connectionandtriad
+satisfy the Possion bracket
+{c,p}=8πG
+3γ. (1)
+When the scale factor lies in the regions ai< a < a ⋆, we
+consider the effective dynamics of LQC, with ai=√γlpl,
+a⋆=/radicalbig
+γj/3lpl, andjis the quantum parameter and
+must be taken half integer values. If a < ai, the quantum
+effect dominates, we must consider the difference equa-
+tion. And a > a⋆, we turn to the classical effect. In this
+paper, we just consider the semiclassical effect of LQC.
+The effective Hamiltonian can be written as [25, 26]
+Heff=−3
+γ28πGSa[(c−1)2+γ2]+Hm,(2)with the dark energy Hamiltonian Hm. AndSis
+S=1
+4/bracketleftbig
+2((q+1)3−|q−1|3)
+−3q((q+1)2−sgn(q−1)|q−1|2) ] (3)
+withq=a2/a2
+⋆.
+One can get the equations of motion of connection and
+triad from the Hamiltonian (2)
+˙p={p,Heff}=−8πGγ
+3∂Heff
+∂c=2
+γSa(c−1),(4)
+˙c={c,Heff}=8πGγ
+3∂Heff
+∂p
+=−(c−1)2+γ2
+2γ/parenleftBigg
+S
+a+˙S
+˙a/parenrightBigg
++8πGγ
+3∂Hm
+∂p,(5)
+wheredot denotesthe derivation with respect to the time
+t. Considering Eq.(4) and remembering the Hamiltonian
+constrain satisfies Heff≈0, one can obtain the modified
+Friedmann equation
+H2=/parenleftbigg˙a
+a/parenrightbigg2
+=8πG
+3Sρsc−S2
+a2, (6)
+with the Hubble parameter H= ˙a/aand the modified
+dark energy density ρsc=Em/a3.Emis the eigenvalues
+for the dark energy Hamiltonian operator that includes
+the appropriate modifications to the inverse volume [26].
+We will discuss the definition of ρsclater on.
+Considering the modified pressure can be written as
+Psc=−∂Hm
+∂V=−2
+3p−1/2∂Hm
+∂p, and using the equations
+of motion of connection and triad (4) and (5), one can
+obtain the modified Raychaudhuri equation
+¨a
+a=−4πG
+3S(ρsc+3Psc)+1
+2˙S
+a/parenleftbigg˙a
+S−S
+˙a/parenrightbigg
+.(7)
+Combining the Friedmann equation (6) and the Ray-
+chaudhuri equation (7), one can get
+˙H=¨a
+a−˙a2
+a2=−8πG
+2S(ρsc+Psc)
++1
+2˙S
+a/bracketleftbigg˙a
+S−S
+˙a/bracketrightbigg
++S2
+a2, (8)
+and it is easy to verify the conservation equation
+˙ρsc+3H(ρsc+Psc) = 0. (9)
+Defining the equationof state (EoS) parameters ωsc(a) =
+Psc(a)/ρsc(a), in which ωsc(a) is a function of scale factor
+a. We can rewrite the above equation as
+˙ρsc+3˙a
+aρsc[1+ωsc(a)] = 0. (10)3
+Integrating the above equation, it is easy to get the so-
+lution for ρsc
+ρsc=ρ0/braceleftBigg
+a3[ω0+1]
+0
+a3[ωsc(a)+1]/bracerightBigg
+exp/bracketleftbigg
+−3/integraldisplaya0
+adaω′
+sc(a)lna/bracketrightbigg
+,
+(11)
+withρ0,a0,ω0denote today’s energy density, scale factor
+and EoS parameter respectively. For a0≫a⋆, it’s not
+necessary to consider the quantum effect of those present
+values (For conservation, we will define the subscript ’0’
+denotes the present value of a physical quantity.). And
+the prime means the derivation with respect to the scale
+factora.
+In the above discussion, we defined the semiclassical
+energy density as ρsc=Em/a3, but this is not the only
+definition. Singh has showed us that there are two ways
+to obtain the semiclassical density [27]. One way is just
+like the definition we discussed above. The other one is
+to define a density operator ˆ ρq=/hatwiderHm/a3and then take
+their eigenvalues. The eigenvalues for ˆ ρqcan be obtained
+by considering ˆHmand/hatwidest1/a3[27]
+ρq=dj,l(a)Em(a,φ) =Dl(q)a−3Em(a,φ) =Dl(q)ρsc,
+(12)
+in which dj,l=Dl(q)a−3is the eigenvalue for/hatwidest1/a3for
+the large j. AndDlis given as [27]
+Dl(q) =/braceleftbigg27|q|1−2l/3
+8l/bracketleftbigg1
+l+3/parenleftBig
+(q+1)2(l+3)/3
+−|q−1|2(l+3)/3/parenrightBig
+−2q
+2l+3/parenleftBig
+(q+1)2(l+3)/3
+−sgn(q−1)|q−1|2(l+3)/3/parenrightBig/bracerightBig 3
+2−2l,(13)
+whereq=a2/a2
+⋆, andlis the quantum ambiguity pa-
+rameters with 0 < l <1 [26]. For a/lessorsimilara⋆,Dl/lessorsimilar1, this
+meansρsc/lessorsimilarρq.
+Now, we turn to discuss the relationship between the
+semiclassical energy density and the classical one. The
+classical energy conservation is
+˙ρcl+3H(ρcl+Pcl) = 0. (14)
+Considering the varying EoS parameter ωcl(a) =ρcl/Pcl,
+the above equation can be written as
+˙ρcl+3Hρcl[1+ωcl(a)] = 0. (15)
+It is easy to obtain the expression for ρclfrom the above
+equation
+ρcl=ρ0/braceleftBigg
+a3[ω0+1]
+0
+a3[ωcl(a)+1]/bracerightBigg
+exp/bracketleftbigg
+−3/integraldisplaya0
+adaω′
+cl(a)lna/bracketrightbigg
+.
+(16)
+Considering the definition of ρq, we can get the eigenval-
+ues for/hatwideρqusing the eigenvalues for/hatwidest1/a3,dj,l, instead ofa−3in Eq.(16). So, the semiclassical expression for ρq
+can be written as
+ρq=ρ0/braceleftBig
+a3[ω0+1]
+0d[ωcl(a)+1]
+j,l/bracerightBig
+exp/bracketleftbigg
+−3/integraldisplaya0
+adaω′
+cl(a)lna/bracketrightbigg
+=Dωcl(a)+1
+lρcl(a). (17)
+Remembering the relationship between ρqandρsc, which
+is described by Eq.(12), we can get the relationship be-
+tweenρscandρcl
+ρsc=Dωcl(a)
+lρcl. (18)
+Now we want to get the relationship between ωsc(a) and
+ωcl(a). Differentiating ln ρscwith respect to ln a, one can
+obtain
+dlnρsc
+dlna= lnDldωcl(a)
+dlna+ωcl(a)dlnDl
+dlna+dlnρcl(a)
+dlna.(19)
+Note thatdlnρsc(a)
+dlna=dρsc(a)
+Hρsc(a)dt=−3[1 +ωsc(a)], and
+dlnρcl(a)
+dlna=dρcl(a)
+Hρcl(a)dt=−3[1+ωcl(a)], it is easy to get
+ωsc(a) =ωcl(a)/bracketleftbigg
+1−1
+3lnDldln|ωcl(a)|
+dlna−1
+3dlnDl
+dlna/bracketrightbigg
+.
+(20)
+Remembering that ωcl(a) may be a negative value for
+dark energy fluid, so we need choose the absolute value
+when we calculate its logarithm, but it dose not change
+the answer. Considering Eq.(20), it is easy to find that
+the effective EoS in loop quantum cosmology is possible
+”-1” crossing. If ωcl= const., Eq.(20) will still hold,
+but the second term in the square brackets is zero. So
+the constant classical EoS will be a varying term in the
+effective region.
+In this section, we get the equation of the effective EoS
+of dark energy with a varying EoS in loop quantum cos-
+mology. Now, we can turn to considerthe thermodynam-
+icalpropertiesofdarkenergyin loop quantumcosmology
+scenario.
+III. THERMODYNAMICS PROPERTIES OF
+DARK ENERGY
+In this section, we will discuss the thermodynamics
+properties of dark energy in semiclassical regions. This
+section includes two subsections. In the first subsection,
+wewilldiscussthe entropy-arearelationofapparenthori-
+zon in semiclassical scenario. In the next subsection, we
+will show the temperature, chemical potential and en-
+tropy of dark energy fluid inner of the apparent horizon.
+A. Modified entropy-area relation
+According to the fact that the Einstein equation can
+be rewritten as the form dEm=TdS+WmdV[28–30],4
+with the total energy Em=ρVand the work density
+Wm= (ρ−p)/2, the authors of [16] showed that the
+entropy-area relationship of apparent horizon would be
+corrected by the modified Friedmann equation in loop
+quantum cosmology. In this subsection, we will discuss
+the modified entropy-arearelation which is caused by the
+corrected Friedmann equation (6). Just as [16] stated,
+the temperature of apparent horizon is just determined
+by the spacetime metric, so it is convenient to assume
+the temperature of apparent horizon is TA= 1/(2π˜rA),
+with the radius of the apparent horizon ˜ rA. Noticed that
+the LQC is just correcting the Hamiltonian, not the La-
+grangian. And the spacetime metric is described by the
+Lagrangian, so it is natural to believe that the definition
+oftemperatureofthe apparenthorizonin LQCisassame
+as in the classical one.
+Our interested spacetime is the closed Friedmann-
+Robertson-Walker (FRW) universe which is described by
+the metric
+ds2=−dt2+a2/parenleftbiggdr2
+1−r2+r2dΩ2
+2/parenrightbigg
+=habdxadxb+ ˜r2dΩ2
+2, (21)
+withhab= diag(−1,a2/(1−r2)),˜r=a(t)r, andkdenotes
+the spatial curvature. If we define the Missner-Sharpe
+mass [24, 31, 32]
+hab∂a˜r∂b˜r≡f, (22)
+the apparent horizon is determined from f= 0. So, we
+can get the radius of the apparent horizon
+˜rA=1/radicalbig
+H2+1/a2. (23)
+with the Hubble parameter Hdescribed by Eq.(6).
+In this paper, we consider the dark energy fluid is a
+perfect fluid satisfies Tµν= (ρsc+Psc)UµUν+Pscgµν,
+whereρscandPscare the modified energy density and
+pressure in the semiclassical LQC, respectively. And the
+energy conservation law which is described by Eq.(10)
+still satisfied.
+The Einstein equation is written as
+dE=AΨ+WdV, (24)
+whereA= 4π˜r2
+Ais the area of the apparent horizon and
+V=4π
+3˜r3
+Ais the volume of 3-dimensionalsphere. E=M
+is the Misser-Sharp energy which is given by Eq.(22). W
+is the work density and Ψ is the energy-supply vector,
+are defined as
+W=−1
+2Tabhab, (25)
+Ψa=Tb
+a∂b˜r+W∂a˜r, (26)
+whereTabis the projection of the (3 + 1)-dimensional
+energy-momentum tensor Tµνof dark energy fluid in the
+FRW universe in the normal direction of 2-sphere[16].Considering our interested metric (21), we can get the
+obvious expressions for W,Ψa[16]:
+W=1
+2(ρsc−Psc), (27)
+Ψa=−1
+2(ρsc+Psc)H˜rdt+1
+2(ρsc+Psc)adr,(28)
+In those equations, we have considered the quantal cor-
+rection.
+The energy thrill through the apparent horizon at the
+interval time dtis given as [16]
+δQ=−dE=−AΨ =A(ρsc+Psc)H˜rAdt,(29)
+with the area of the apparent horizon A= 4π˜r2
+A. If the
+modified entropy of the apparent horizon is denoted as
+Ssc
+A, considering δQ=TdSsc
+AandEq.(29), onecanobtain
+dSsc
+A= 8π21
+H3(ρsc+Psc)dt
+=1
+8πG/bracketleftBigg
+2π˙A
+S+A2H2
+2˙S
+S2
+−1
+2˙SA2
+a2−1−S2
+a2HA2
+S/bracketrightBigg
+dt(30)
+in which we have used the temperature of the apparent
+horizonTA= 1/(2π˜rA) with the radius of the apparent
+horizon ˜rA= 1//radicalbig
+H2+1/a2and Eq.(8). Expanding the
+aboveequation and integratingit, Eq.(30) can be written
+as
+Ssc
+A=Ssc
+Aa+Aeff
+4G+1
+4G/integraldisplay
+AeffdlnS
++1
+8πG/integraldisplayA2
+effH2
+2dS −1
+16πG/integraldisplayA2
+eff
+a2S2dS
+−1
+8πG/integraldisplay(1−S2)S
+a2HA2
+effdt, (31)
+in which Aeff=A/Sdenotes the effective area of the ap-
+parent horizon in the semiclassical LQC. in which Ssc
+Aais
+a constant and the value depends on the specific physics.
+In a short conclusion, considering the Einstein equa-
+tion (24) and assuming that the apparent horizon has
+Hawkingtemperature, wegetthe expressionforthemod-
+ified entropy-area relation in the semiclassical LQC. The
+semiclassical theory not only modify the entropy of ap-
+parent horizon, but also the area of it. And it is worth
+to notice that this modified entropy-area relation is not
+as same as the correction of black hole in loop quantum
+gravity [33]
+Sbh=SBH+∞/summationdisplay
+n=0CnA−n
+H, (32)
+with some constants Cn. And it is different from the
+modified entropy-area relationship, which has the same5
+form as Eq.(32), when one considers the effective LQC
+with holonomycorrections[16]. This difference is reason-
+able. Because we consider the inverse volume correction
+in the semiclassical regions, but the authors of [16] con-
+sidered the holonomy correction. Notice that there are
+many differences between inverse volume correction and
+holonomy correction. For example, the inverse volume
+correction modifies the Klein-Gorden equation [34], but
+holonomy correction does not [35]; and the semiclassical
+energy density and effective pressure which caused by
+the inverse volume correction are different from the ones
+caused by the holonomy correction [36].
+B. The thermodynamical variables of dark energy
+fluid
+In this section, we will discuss the thermodynamics
+properties of the dark energy in the inner side of the ap-
+parent horizon. Some authors advised that the tempera-
+ture of dark energy fluid has the same temperature of the
+apparent horizon which the temperature is described by
+Hawking temperature, but this may be get a ill-defined
+entropy when ωcl<−1 [24]. And other authors find the
+dark energy either have negative entropy or temperature
+in the classical regions [21–23]. In this subsection, we
+will consider the temperature and entropy of the dark
+energy in the inner side of the apparent horizon. And we
+will assume the dark energy is a ideal fluid and with the
+chemical potential. This subsection, we will follow the
+method of [20] to discuss the temperature and entropy of
+the dark energy in semiclassical LQC scenario.
+The entropy fluid and particle fluid are described by
+Sµ
+sc=sscuµandNµ=nuµ, with the particle number
+densitynand the entropy density ssc, respectively. Con-
+sidering they satisfy Sµ
+;µ= 0,Nµ
+;µ= 0, it is easy to obtain
+˙ssc+3Hssc= 0, (33)
+˙n+3Hn= 0. (34)
+The solutions for above equations are
+n(a) =n0/parenleftbigga
+a0/parenrightbigg3
+, s sc(a) =s0/parenleftbigga
+a0/parenrightbigg3
+.(35)
+Now we consider the thermodynamics. We assume the
+thermodynamic properties is described by the particle
+number density nand the temperature Tsc. We have
+assumed the quantum effect will modify the tempera-
+ture. This is reasonable. If assuming that the modified
+energy density and the temperature is still satisfy the
+Stenfan-Boltzmann law ρsc∝f(Tsc), the temperature
+Tscshould be dependent on a, for the energy density is
+the function of the scale factor a, and then the temper-
+ature should be modified by the semiclassical quantum
+effect. We will see that the Stenfan-Boltzmann law still
+is tenable in our model later on. We consider the Gibbs
+law,Tsc/parenleftBig
+∂Psc
+∂Tsc/parenrightBig
+n=ρsc+Psc−n/bracketleftBig
+∂ρsc(a)
+∂n/bracketrightBig
+Tsc, then theprefect fluid can be described by
+Tsc(a)/bracketleftbigg∂ωsc(a)ρsc
+∂Tsc(a)/bracketrightbigg
+n= (ωsc(a)+1)ρsc−n/bracketleftbigg∂ρsc(a)
+∂n/bracketrightbigg
+Tsc.
+(36)
+Considering ˙ ρsc(a) = ˙n/parenleftBig
+∂ρsc(a)
+∂n/parenrightBig
+Tsc(a)+˙Tsc(a)/parenleftBig
+∂ρsc(a)
+∂Tsc(a)/parenrightBig
+n
+and Eqs.(10) and (36), we can get the evolution of mod-
+ified temperature with respect to the semiclassical EoS
+parameters
+/bracketleftbigg3
+aTsc(a)ωsc(a)+T′
+sc(a)/bracketrightbigg
+(ωsc(a)+1) = Tsc(a)ω′
+sc(a).
+(37)
+Just as the mention in Sec.II, the modified EoS param-
+etersωsc(a) are possible -1 crossing. It is obvious that
+the modified temperature which is described by above
+equation is zero when ωsc(a) =−1 forω′
+sc(a)∝ne}ationslash= 0. In-
+tegrating the above equation when ωsc(a)∝ne}ationslash=−1, we can
+get the expression for Tsc(a)
+Tsc(a) =T0/parenleftbiggωsc(a)+1
+ω0+1/parenrightbigg/parenleftbigg1
+a3ωsc(a)/parenrightbigg
+×exp/bracketleftbigg
+−3/integraldisplay1
+adaω′
+sc(a)lna/bracketrightbigg
+,(38)
+with the present temperature T0. We have considered
+that the present scale factor a0= 1. There are four
+points should be noticed. First, if ω0=−1 andT0∝ne}ationslash= 0,
+Tsc(a) will be divergent. But if we consider ωsc(a) de-
+scribes today’s dark energy, ωsc(a) should not be modi-
+fied by the quantum effect for a0≫a⋆and is substituted
+byωcl(a). In this case, Eq.(37) describes the temper-
+ature for the dark energy in the classical regions. The
+temperature of dark energy with ω(a0) =−1 should be
+zero, for ω′
+cl(a)|a=a0∝ne}ationslash= 0 in Eq.(37). So, T0= 0 when
+ω0=−1 if the EoS ωclis a function of the scale fac-
+tor. Second, the temperature is zero when ωsc(a) =−1.
+But this ”-1” EoS parameter is caused by the effective
+behavior. Third, for T0andω0+ 1 have the same sign,
+ifωsc(a)>−1 the temperature is positive, the tempera-
+ture is negative when ωsc(a)<−1, andTsc(a) = 0 when
+ωsc(a) =−1, this is as same as the behavior of temper-
+ature in the classical regions [20]. But notice that ωsc
+are modified EoS parameters and includes the quantum
+geometry effect. Fourth, the scale factor is in the regions
+ai< a < a ⋆, so1
+ain Eq.(38) will not divergent, this is
+different from [20] in which the authors considered the
+thermodynamical properties of dark energy in the classi-
+cal regions.
+Now we have the expressions for modified temperature
+and modified energy density. It is easy to obtain the6
+generalized Stefan-Boltzmann law
+ρsc(a) =ρ0/bracketleftbiggTsc(a)
+T0ω0+1
+ωsc(a)+1/bracketrightbiggωsc(a)+1
+ωsc(a)
+×exp/bracketleftbigg3
+ωsc(a)/integraldisplay1
+adaω′
+sc(a)lna/bracketrightbigg
+,
+forωsc(a)∝ne}ationslash= 0;
+ρsc(a) =ρ0/bracketleftbiggTsc(a)
+T0ω0+1
+ωsc(a)+1/bracketrightbigg/parenleftbigg1
+a/parenrightbigg3
+,
+forωsc(a) = 0.(39)
+It is worth to notice that the above equations are still
+valid for ωsc(a) =−1, forTsc(a) = 0 in this moment.
+Now we turn to consider the chemical potential and
+entropy. Consider the Euler’s relation of Tsc(a)ssc(a) =
+[1+ωsc(a)]ρsc(a)−µsc(a)n, with the chemical potential
+µsc(a), the chemical potential can be given as [20]
+µsc(a) =1
+n[(1+ωsc(a))ρsc(a)−Tsc(a)ssc(a)].(40)
+Using the Gibbs law (36) and the modified temperature
+(38), it is easy to get the modified chemical potential
+µsc(a) =µ0/bracketleftbiggωsc(a)+1
+ω0+1/bracketrightbigg/bracketleftbigg1
+a3ωsc(a)/bracketrightbigg
+×exp/bracketleftbigg
+−3/integraldisplay1
+adaω′
+sc(a)lna/bracketrightbigg
+,(41)
+with chemical potential at present µ0=1
+n0[ρ0(ω0+1)−
+T0s0]. As before, the aboveequation is well-defined when
+ω0=−1, forT0= 0. But the sign of µsc(a) is indepen-
+dent onωsc(a), for the sign of µ0can be general. And the
+chemical potential can be wrote as a function of Tsc(a),
+µsc(a) =µ0Tsc(a)
+T0. (42)
+It is easy to get the entropy of the dark energy
+fluid by considering the Eq.(40). Remembering S(a) =
+s(a)V(a) =s(a)a3, and using Eqs.(35),(39) and (42), we
+can get the expression for entropy.
+Ssc(a) =s0V(a)/bracketleftbiggTsc(a)
+T0ω0+1
+ωsc(a)+1/bracketrightbigg1
+ωsc(a)
+×exp/bracketleftbigg3
+ωsc(a)/integraldisplay1
+adaω′
+sc(a)lna/bracketrightbigg
+,(43)
+in which we have considered the present scale factor a0=
+1. When ωsc(a) = 0, combining Eq.(39) and Eq.(42), we
+find it is a trivial result. Note that, above equation is
+regular at ωsc(a) =−1, because if ωsc(a) =−1,Tsc= 0.
+And ifω0=−1, thenT0= 0, so the above equation is
+still well-defined. It is easy to get s0V0=ssc(a)V(a).
+In a short conclusion, in this subsection, we has dis-
+cussed the thermodynamics properties of dark energy in
+the inside of apparent horizon, and get the expressionsfortemperature, chemicalpotential and entropy. We find
+that those variablesare correcteddue to the semiclassical
+quantum effect, and those corrections are connected with
+the modified EoS parameters. Our method is following
+[20], in which the temperature is regular for the vacuum
+when they discuss the thermodynamics properties in the
+classical regions but the temperature will be divergent
+ata= 0. But due to the modification of semiclassical
+quantum effect, a= 0 doesn’t lie in the regions we inter-
+ested, so the expression for temperature always holds for
+a varying classical EoS ωcl. The expressions for chemical
+potential and entropy are obtain in this subsection.
+IV. CONCLUSION AND DISCUSSION
+In this paper, we have discussed the thermodynamics
+properties of the dark energy in the semiclassical LQC
+scenario. The common feature of the all models of dark
+energy is the EoS parameters ωof dark energy fluid is
+included by the field evolution. So, at the first, we con-
+sider the correction of EoS parameters in the regions we
+interested. We find that the corrected EoS parameters
+depends on the changing rate of inverse volume factor
+Dland classical EoS parameters ωcl(a).
+We discussed the area-entropy relation of the appar-
+ent horizon of universe which is full of dark energy fluid.
+We found that the effective LQC not only modify the en-
+tropyofapparenthorizon,butalsotheareaofit. Andthe
+modified area-entropy relation is different from the one
+considered by [16], in which the authors considered the
+holonomy correction. And then, we discussed the ther-
+modynamical properties of dark energy fluid in the inner
+side of apparent horizon. The expressions for tempera-
+ture, chemical potential and entropy are obtained. We
+find that, if the classical EoS parameter is a function of
+the scale factor, the thermodynamical variables are func-
+tions of the modified EoS parameters and well-defined
+whenω(a0) =−1. The sign of temperature is dependent
+onthe modifiedEoS ωsc, whenωsc>−1,Tsc>0,Tsc<0
+whenωsc<−1, andTsc= 0 ifωsc=−1, but the sign
+of the modified temperature is independent on the sign
+of potential, which is determined by the sign of µ0. This
+is as same as the discussion in the classical region [20].
+Tsc(a) =Ssc(a) = 0 when ωsc(a) =−1, this is like the be-
+havior of massless degenerate Fermi gas [37, 38]. Just as
+we discussed above, when ωsc<−1, the temperature is
+negative. But this negative temperature is independent
+on the sign of potential, for the sign of potential is just
+dependent on µ0. The negative temperature can only be
+explain in the quantum framework. The quantum field
+theory for the field with EoS parameters ωcl<−1 is still
+an open question. In the semiclassical regions, the EoS
+parametersshould be modified bythe quantumgeometry
+effect, so the negative temperature in the semiclassical
+regions partial is cased by the quantum geometry effect.
+In this paper, we just consider the thermodynamics
+properties of dark energy fluid, it is easy to verify that7
+our method can apply to the other kinds of matter, e.g.,
+stiff matter, radiation and so on, for the EoS of this mat-
+ter will be modified in the effective region in loop quan-
+tum cosmology, just as Singh shown [27]. But this model
+can not describe the thermodynamics properties of cos-
+mological constant with the constant EoS -1. Although
+the EoS of cosmological constant will be modified in the
+effective region, just as Eq.(20) described. The temper-
+ature of cosmological constant today is still unclear, so
+the term T0/(ω0+1) in Eq.(38) will be divergent for ωscdoes not always equal to -1.
+Acknowledgments
+The work was supported by the National Natural Sci-
+ence ofChina (No. 10875012)and the Scientific Research
+Foundation of Beijing Normal University.
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+arXiv:1001.0308v2  [quant-ph]  28 Nov 2010How to prepare quantum states that follow classical paths
+Pouria Pedram∗
+Plasma Physics Research Center, Science and Research Branc h, Islamic Azad University, Tehran, Iran
+November 5, 2018
+Abstract
+We present an alternative quantization procedure for the on e-dimensional non-relativistic quan-
+tum mechanics. We show that, for the case of a free particle an d a particle in a box, the complete
+classical and quantum correspondence can be obtained using this formulation. The resulting wave
+packets do not disperse and strongly peak on the classical pa ths. Moreover, for the case of the free
+particle, they satisfy minimum uncertainty relation.
+Keywords : Quantum mechanics; Classical-quantum correspondence; Wave p ackets.
+Pacs: 03.65.-w
+1 Introduction
+Schr¨ odinger equation Hψ=i¯h˙ψis the main equation of non-relativistic quantum physics which can be
+obtained upon the quantization procedure /vector p→ −i¯h/vector∇andH→i¯h∂tin the Hamiltonian formulation of
+quantum mechanics [1, 2]. In fact, this equation determines the time evolution of the particle’s wave
+function. Often, weareinterestedtofind solutionsin suchawayth at theyfollowtheclassicaltrajectories
+without dispersion and peak on them. But, the construction of suc h kind of wave packets, except for the
+case of the simple harmonic oscillator, is not an easy task. For instan ce, for the case of a free particle,
+the initial wave function disperses quickly as the particle moves.
+The problem of classical and quantum correspondence has attrac ted much attention in the literature
+[3]. These efforts have begun by Schr¨ odinger [4] and followed by oth ers in the context of the coherent
+states. These wavepacketsare specific kind of quantum states t hat describe a maximal kind ofcoherence
+∗pouria.pedram@gmail.com
+1and a classical kind of behavior [5]. In quantum physics, we can const ruct such kind of quantum states
+by the superposition of the energy eigenstates which would peak ar ound the classical trajectories. But,
+in most cases, the wave packets do not maintain their shape and eve ntually disperse.
+One possible way to solve this problem is using an alternative quantizat ion method. In fact, we need
+to change the differential structure of the Schr¨ odinger equatio n which has a parabolic form. It has been
+shown that in the context of quantum cosmology, the hyperbolic na ture of its main equation gives us
+the possibility of complete classical and quantum correspondence [6 , 7]. So, the resulting wave packets
+strongly peak on the classical trajectories and never disperse.
+Here, we first consider a one-dimensional model in the presence of an external general potential.
+Then, we use an alternative classical picture which, after quantiza tion, results in a hyperbolic differential
+equation. For the case of a free particle and a particle in a box, we so lve this equation and construct
+wave packets using appropriate initial conditions. We show that the se wave packets follow the classical
+paths and strongly peak on them in the whole configuration space.
+2 The model
+Let us consider a harmonic oscillator as a simple example. In the classic al domain, this model has a
+well-known solution u(t) =Acos(ωt+δ1). Note that, time explicitly appears in this solution which
+parameterize the temporal behavior of u. To eliminate the explicit presence of t, we can use an another
+general solution with the same total energy but different phase i.e.v(t) =Acos(ωt+δ2). Now, we can
+write the variable uin terms of vinstead oft, namely
+u= cos(∆)v±sin(∆)/radicalbig
+A2−v2, (1)
+where ∆ = δ1−δ2and−A≤u,v≤A. So, in general, the trajectory is an ellipse which for ∆ = π/2
+represents a circle. This example shows that we can alwaysparamet erizethe solution in terms of another
+one which has the same energy but with arbitrary phase shift. More over, for each model, the trajectories
+are unique due to the specific form of the potential.
+2Since the both solutions have a same energy and obey a same equatio n of motion we have
+
+
+p2
+u2m+V(u) =E,
+p2
+v2m+V(v) =E.(2)
+Since the right hand sides are equal, we obtain
+F ≡p2
+u
+2m−p2
+v
+2m+V(u)−V(v) = 0. (3)
+Inthequantummechanicaldomain, weneedtoquantizeaboveequa tion. Theoperator Fcanbeobtained
+upon quantization procedure pu→ −i¯h∂
+∂uandpv→ −i¯h∂
+∂v. So, we demand that Fannihilate the
+wave function i.e.
+FΨ(u,v) = 0, (4)
+or, equivalently
+/braceleftbigg
+−¯h2
+2m∂2
+∂u2+¯h2
+2m∂2
+∂v2+V(u)−V(v)/bracerightbigg
+Ψ(u,v) = 0. (5)
+Therefore, Eq. (5) is an alternative quantum mechanical equation which has a different structure with
+respect to the Schr¨ odinger equation, but both explain the same p hysics.
+Although there is no intrinsic preference between these two pictur es, the later has some additional
+advantages. This is due to the hyperbolic form of Eq. (5) which gives us freedom for choosing the initial
+wave function and its initial slope. On the other hand, the solutions o f a hyperbolic equation are usually
+highly oscillatory. Since the oscillation around the classical paths is no t acceptable, we need to choose
+appropriate initial conditions to guarantee the classical–quantum c orrespondence.
+In the next section, to show the method, we consider the case of a free particle. For this case, the
+trajectories in both u−tandu−vplanes are straight lines. Since Eq. (5) admits solutions which never
+disperseandstronglypeakontheclassicalpaths, weshowthatth e classicalandquantumcorrespondence
+arises naturally from this formulation.
+33 Free particle
+For the case of the free particle ( V= 0), we have
+
+
+u(t) =βut+u0,
+v(t) =βvt+v0,(6)
+whereβis the particle’s velocity. Since both solutions have a same total ener gy, we also have |βu|=|βv|
+which results in u=±v+u0∓v0. So, the trajectories are straight lines with unit absolute slope.
+It is straightforward to check that, for V= 0, cos(ku)cos(kv) and sin(ku)sin(kv) are the eigenfunc-
+tions of Eq. (5). Therefore, the general solution is
+Ψ(u,v) =1√
+2π/integraldisplay∞
+−∞[A(k)cos(ku)cos(kv)
++iB(k)sin(ku)sin(kv)]dk. (7)
+To find the complete form of the wave packet, we need to specify th e coefficients A(k) andB(k). These
+coefficients can be determined from the initial form of the wave pack et atv= 0
+Ψ(u,0) =1√
+2π/integraldisplay∞
+−∞A(k)cos(ku)dk, (8)
+∂Ψ(u,v)
+∂v/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+v=0=i√
+2π/integraldisplay∞
+−∞B(k)sin(ku)kdk. (9)
+It is obvious that the presence of B(k) dose not have any effect on the form of the initial wave function
+but it is responsible for the slope of the wave function at v= 0, and vice versa for A(k). Moreover,
+a complete description of the problem would include the specification o f both of these quantities. On
+the other hand, since we are interested to construct wave packe ts with classical properties, we need to
+assume a specific relationship between these coefficients. The pres cription is that these coefficients have
+the same functional form [6, 7] i.e.
+A(k) =B(k), (10)
+which results in the following wave packet
+Ψ(u,v) =1√
+2π/integraldisplay∞
+−∞A(k)[cos(ku)cos(kv)
++isin(ku)sin(kv)]dk. (11)
+4To completely determine the wave packet, we should specify A(k). We can find A(k) by choosing an
+appropriate initial wave function such as two Gaussians at u=±d
+Ψ(u,0) =e−α(u−d)2+e−α(u+d)2. (12)
+This choice of initial condition, using inverse fourier transform, is re lated to
+A(k) =/radicalbigg
+2
+αe−k2
+4αcos(kd), (13)
+which gives
+Ψ(u,v) =1−i
+2/parenleftBig
+e−α(u+v+d)2+e−α(u+v−d)2
++ie−α(u−v+d)2+ie−α(u−v−d)2/parenrightBig
+. (14)
+Figure 1 shows the initial wave function and its initial slope for d= 4 andα= 1. Since, from Bohmian
+interpretation of the quantum mechanics, the initial derivative of t he imaginary part of the wave packet
+corresponds to the initial classical velocity, the presence of a non zero and appropriate form of B(k) can
+be justified. The plot of the initial slope of the wave packet contains a negative and a positive peaks
+which correspond to a incoming or outgoing particle, respectively. F igure 2 shows the resulting wave
+packet ford= 2 andα= 1. As it can be seen from the figure, this wave packet never disper ses and
+strongly peaks on the classical trajectory. Moreover, the heigh t of the crest of the wave packet, which
+from WKB approximation corresponds to the inverse velocity of the particle, is constant along the
+classical path. This behavior is in complete agreement with the classic al picture. However, the square
+of the wave packet increases at the intersection points of the tra jectories which correctly corresponds
+to the probability of two possible particle’s directions of motion. Note that our choice of the expansion
+coefficients (13) corresponds to a free particle with a positive or ne gative momentum which is located
+initially at u=±d.
+5-6 -4 -2 2 4 6u0.20.40.60.81/CapPsi/LParen1u,0/RParen1
+-6 -4 -2 2 4 6u
+-0.75-0.5-0.250.250.50.75/Minus/Imagi/PartialDiff/CapPsi/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/PartialDiffv/VertBar1v/Equal0
+Figure 1: The initial wave function (left) and the initial derivative of t he wave function (right) for d= 4
+andα= 1.
+-5
+0
+5u
+0246
+v00.250.50.751
+/VertBar1/CapPsi/VertBar12
+-5
+0
+5u-5
+0
+5u
+0246
+v00.250.50.751
+/VertBar1/CapPsi1/VertBar12
+-5
+0
+5u-5
+0
+5u
+0246
+v00.20.4
+/VertBar1/CapPsi2/VertBar12
+-5
+0
+5u
+-5 0 5
+u0246
+v
+00.250.50.751/VertBar1/CapPsi/VertBar12046
+-5 0 5
+u0246
+v
+00.250.50.751/VertBar1/CapPsi1/VertBar12046
+-5 0 5
+u0246
+v
+00.20.4
+/VertBar1/CapPsi2/VertBar12046
+Figure 2: Up: the square of the wave packet |Ψ(u,v)|2(left), [Re[Ψ( u,v)]]2(middle), and [Im[Ψ( u,v)]]2
+(right) ford= 2 andα= 1, Down: the respective upper view.
+64 Particle in a box
+For a particle in a box, we have
+V(q) =
+
+0 −L
+2<q<L
+2,
+∞ otherwise,(15)
+whereqstands foruorv. For this case, the trajectories also are straight lines with unit abs olute slope.
+Moreover, this model has well-known orthonormal even and odd eig enfunctions
+ψn(q) =
+
+/radicalBig
+2
+Lcos(nπq
+L), n = 1,3,5,...,
+/radicalBig
+2
+Lsin(nπq
+L), n = 2,4,6,....(16)
+Now, using the exact form of the eigenstates, we find the following w ave packet
+Ψ(u,v) =/summationdisplay
+n=1,3,5,...A(n)cos(nπu
+L)cos(nπv
+L)
++i/summationdisplay
+n=2,4,6,...A(n)sin(nπu
+L)sin(nπv
+L). (17)
+If we demand that this solution satisfies the initial condition of Eq. (1 2), we obtain the following form
+of the expansion coefficients
+A(n) =−ienπ(nπ+4iαdL)
+4αL2
+L/radicalbig
+α/π/bracketleftbigg
+−Erfi(nπ+2iαL(d−L)
+2√αL)
++e2idnπ
+L/parenleftbigg
+Erfi(nπ−2iαL(d−L)
+2√αL)
+−Erfi(nπ−2iαL(d+L)
+2√αL)/parenrightbigg
++ Erfi(nπ+2iαL(d+L)
+2√αL)/bracketrightbigg
+, (18)
+where Erfi( x) is the imaginary error function Erfi( x)≡ −iErf(ix). Figure 3 shows the wave packet
+ford= 1.5 andα= 5. Classically, the particle is free inside the well and moves with positiv e or
+negative initial velocity from any arbitrary position in the range −L/2< u,v < L/ 2. As it can seen
+from the figure, the wave packet follows the classical path and str ongly peaks on it which is in complete
+agreement with the classical scenario. In fact, the value of ddetermines the initial classical position and
+the presence of two rectangles with opposite directions indicates t he two possible directions of motion.
+7-4
+-2
+0
+2
+4-4-2024
+01234
+-4
+-2
+0
+2
+4 -4 -2 0 2 4-4-2024
+01234-4-2024
+Figure 3: The square of the wave packet |Ψ(u,v)|2(left), and the respective upper view (right) for
+d= 1.5 andα= 5.
+Since the height of the crest of the wave packet is constant along t he classical trajectory, the probability
+of finding the particle is constant along the classical path. On the ot her hand, since the classical velocity
+of the particle for this case is a constant of motion, the classical pr obability of finding the particle is
+also constant along its trajectory. So, the classical probability co mpletely coincides with the quantum
+mechanical probability.
+5 Classical limit, notion of time, uncertainty principle, a nd the
+conservation law
+Now, we can define the classical wave packet which is nearly zero bey ond the classical path as follows:
+Ψcl(u,v)≡lim
+α→∞Ψ(u,v). (19)
+Thiswavepackethasthe desiredpropertyΨ cl(u,v)/ne}ationslash= 0at{u=ucl,v=vcl}andΨ cl(u,v)≃0elsewhere.
+In fact, the wavepacket does not disperse even for largevalues o fα. This is in contrastwith the solutions
+of the Schr¨ odinger equation which usually disperses quickly as the p article moves.
+The notion of time does not appear explicitly in our main equation (5). H owever, as we have shown,
+the results depend on time in an implicit manner. We also encounter this phenomenon in the context of
+quantum cosmology where its main equation, which is the Wheeler-DeW itt equation, does not contain
+time. Moreover, it is a hyperbolic differential equation and in particula r conditions can be written in
+8the form of Eq. (5) [6]. Therefore, our approach can be considere d as a bridge between relativistic
+and non–relativistic quantum mechanics. On the other hand, we can find the notion of time using the
+Bohmian interpretation of quantum mechanics, namely
+pµ=∂µS, (20)
+where Ψ = Rexp(iS). So, we can obtain the time evolution of each variable using this inter pretation.
+Although this definition of time is not genuine, at the classical limit ( α→ ∞) it will coincide with the
+classical time as desired.
+At this point, we encounter an important question: do these wave p ackets satisfy the Heisenberg
+uncertainty relation? For the case of a free particle, using the exp licit form of the wave packet (14), we
+can check the uncertainty principle, for instance at v= 0. At this point, the uncertainties in uandpu
+take the following form
+(∆u)2=1
+4α+d2
+2/parenleftbig
+1+tanh(d2α)/parenrightbig
+−
+/radicalBig
+2
+πα+d2e2d2αErf(√
+2αd2)
+1+e2d2α
+2
+, (21)
+(∆pu)2=α¯h2/bracketleftbigg
+1−4α
+1+e2d2α/parenleftbigg
+d2α−2/π
+1+e2d2α/parenrightbigg/bracketrightbigg
+, (22)
+where the integration is over {0,∞}. Since the initial wave function contains two well-separated Gaus-
+sians located at ±d(αd2>1), we have (∆ u)2≃1
+4αand (∆pu)2≃α¯h2which results in (∆ u)2(∆pu)2≃
+¯h2/4. So, similar to the coherent states, the initial form of the wave pa cket satisfies the minimized
+uncertainty relation for all values of dandαsubjected to αd2>1. Since the wave packet preserves its
+shape during the motion, we expect that this result also holds for ot her values of v.
+Note that, in this formulation, the total probability at fixed uorvis not a conserved quantity. In
+fact, the wave packets which are solutions of the Eq. (5) satisfy t he following conservation law
+∂µJµ= 0, (23)
+9where the probability current is defined as
+Jµ=¯h2
+2m(Ψ∗∂µΨ−Ψ∂µΨ∗). (24)
+Therefore, the probability interpretation of Eq. (5) is similar to the Klein–Gordon equation.
+6 Conclusions
+In this paper, we have presented an alternative quantization proc edure which results in a hyperbolic dif-
+ferential equation. We showed that, for the case of a free partic le and a particle in a box, the structure of
+the underlyingquantum mechanicalequationandappropriateinitial conditionsled tocompleteclassical–
+quantum correspondence. The wave packets never dispersed an d followed the classical trajectories in
+the whole configuration space and strongly peaked on them.
+References
+[1] E. Schr¨ odinger, Ann. der Physik, 79, 361 (1926).
+[2] E. Schr¨ odinger, Ann. der Physik, 80, 437 (1926); 81, 109 (1926).
+[3] A. O. Bolivar, Quantum-classical correspondence: dynamical quantizati on and the classical limit
+(Springer, New York, 2004).
+[4] E. Schr¨ odinger, Naturwissenschaften. 14, 664 (1926).
+[5] R. J. Glauber, Phys. Rev. Lett. 10, 84 (1963); A. M. Perelomov, Generalized coherent states and
+their applications (Springer, Berlin, 1986).
+[6] P. Pedram, Phys. Lett. B 671, 1 (2009).
+[7] P. Pedram, JCAP 07, 006 (2008).
+10
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+arXiv:1001.0309v1  [physics.soc-ph]  2 Jan 2010Ternary Social Networks: Dynamic Balance and Self-Organiz ed Criticality
+Qing-Kuan Meng,1,∗Wei Liu,2,†and Jian-Yang Zhu1,‡
+1Department of Physics, Beijing Normal University, Beijing 100875, China
+2School of Science, Xi’an University of Science and Technolo gy, Xi’an 710054, China
+(Dated: October 30, 2018)
+Antalet al.[Phys. Rev. E 72, 036121 (2005)] have studied the balance dynamics on the
+social networks. In this paper, based on the model proposed b y Antal et al., we improve it and
+generalize the binary social networks to the ternary social networks. When the social networks get
+dynamically balanced, we obtain the distributions of each r elation and the time needed for dynamic
+balance. Besides, we study the self-organized criticality on the ternary social networks based on our
+model. For the ternary social networks evolving to the sensi tive state, any small disturbance may
+result in an avalanche. The occurrence of the avalanche sati sfies the power-law form both spatially
+and temporally. Numerical results verify our theoretical e xpectations.
+PACS numbers: 05.65.+b, 87.23.Ge, 89.75.-k
+I. INTRODUCTION
+Antalet al.[1] have studied the balance dynamics on
+the social networks based on the notion of social balance
+[2, 3]. In the networks, each node is connected to all the
+others, representing each person knows all the others in
+the society. Each edge in the networks has two values,
++1 and -1. If the edge is +1, it means the two persons
+are friendly with each other. If the edge is -1, it means
+the two persons are hostile towards each other. At every
+step, they choose a triangle randomly from the network.
+If the product of the three edges is +1, the triangle is
+stable. Otherwise, if the product is -1, the triangle is un-
+stable. For the stable triangle, it satisfies (i) the friend
+of my friend being my friend; (ii) the enemy of my friend
+being my enemy; (iii) the friend of my enemy being my
+enemy; and (iv) the enemy of my enemy being my friend.
+The unstable triangles always try to be stable, but the
+final state of the network depends on the edge flipping
+probability p, which is set manually. If p≥1
+2, the net-
+work will reach the state of ”paradise”, with all relations
+being friendly. Several studies around the balance dy-
+namics have been carried out, including the studies of
+the university class of triad dynamics [4] and the satisfi-
+ability problem of computer science [5], etc.
+In the first half of this paper, at first based on Ref.
+[1], we map the edge relations to the node relations with
+somerelaxation,sotherearetwoopposingopinionsinthe
+network. We suppose for each triad relation in the social
+networks, it will changefrom stable to unstable or the re-
+verse, caused by the change of some persons’ opinions in
+it. While for the change, the person’s opinion depending
+both on the other opinions in the triad relation and the
+opinions all around him. Next, considering in the social
+∗Electronic address: qkmeng@mail.bnu.edu.cn
+†Electronic address: wliuphys@gmail.com
+‡Author to whom correspondence should be addressed; Electro nic
+address: zhujy@bnu.edu.cnnetworks, there being two opposing opinions is extreme,
+we introduce the neutral opinion. So the values of each
+node are generalized to +1, 0 and -1, and the metastable
+triad relationsappear. We find the densities ofeach triad
+relationassociatedwiththeHamiltonianandthegeomet-
+rical temperature which is determined by the structure
+of the network without manual setting. Based on our
+model, we obtain the distributions of each triad relation
+and the time needed for dynamic balance both for the
+binary and ternary cases.
+In the latter half of this paper, we study the self-
+organized criticality (SOC) [6–8] on the ternary social
+networks. The previous studies of the SOC on complex
+networks are mainly based on the BTW model [6, 9–14].
+In this paper, we establish our model with some differ-
+ences. As we know, in the society, if one relation changes
+from one state into anther, it may affect other relations
+associating with it. The effect may be big or small, de-
+pending on many internal factors. So in this paper, we
+simplify this phenomenon, with an eye to the triad bal-
+ance dynamics, and make two modifications compared to
+Sec. II. We find under specific conditions the SOC to be
+observable. Besides, for the small-world network [15, 16],
+we find the way of construction have an influence on the
+occurrence of the avalanche. We analyze it theoretically.
+Numerical results verify our theoretical expectations.
+This paper is organized as follows. In Sec. II, we map
+theedgerelationstothenoderelationsandgeneralizethe
+binary networks to the ternary ones. For both the binary
+and the ternary networks, we obtain the distributions
+of each triad relation and the time needed for dynamic
+balance. In Sec. III, we study the SOC on the regular
+network and the small-world network respectively, and
+find out the small-world effect on the occurrence of the
+avalanche. We end this paper with conclusions in Sec.
+IV.2
+FIG. 1:a1−a4stand for each type of the triad relations, and
+the corresponding densities in addition. a1anda4are stable
+relations, while a2anda3are unstable relations.
+II. DYNAMIC BALANCE
+A. The Node Model
+First we define the node relations in analogy with Ref.
+[1] with some relaxation. We propose each node in the
+network has two values, +1 and -1, standing for each
+person’s two opposite standpoints in the society, or two
+opposing opinions. For a triad relation, if all nodes have
+the same value, i.e., all persons have the same opinion,
+the relation is stable. If there are different opinions in
+a triad relation, conflicts are easily to occur, so the re-
+lation is unstable. The stable and unstable triad rela-
+tions are given in Fig. 1. In our model, the stable and
+unstable triad relations have the probabilities to change
+into the other, which is more realistic for the social net-
+works. As we know in the society, stable relations may
+become unstable because of conflicts and the surround-
+ings, and unstable relations may become stable because
+of the same interest and the surroundings. Considering
+each node has two values which have the probabilities to
+change into the other, we study the social networks in
+association with the spin systems. In the following con-
+text, we establish our model in association with the Ising
+model and the generalized Glauber dynamics [17, 18].
+When associated with the spin systems, each node in
+the network standing for a spin. We propose the Hamil-
+tonian of each triangle to be
+H∆=−α(σiσj+σjσk+σiσk), (1)
+whereσi,σjandσkstand for the three spins of the tri-
+angle. In the binary case, σi,σjandσkcan take either
+of the two values, +1 and -1. Besides, α >0 is a real
+parameter, standing for the strength of coupling. So the
+Hamiltonian is the form of the Ising model. The evolving
+rules are as follows. At every step we choose a triangle
+from the network randomly, and choose one of the threenodes from the selected triangle at random, then let the
+selected node do spin flipping. The spin flipping proba-
+bility is defined as
+p(σi→σ′
+i) =e−βH′
+∆
+Z, (2)
+whereH′
+∆=−α(σ′
+iσj+σjσk+σ′
+iσk) stands for the fi-
+nal Hamiltonian after the spin flipping, with σ′
+ithe final
+value of the selected spin, and σj,σkthe values of the
+other spins unchanged. It is noted that the spin flipping
+probability depends on the final value, independent of
+the initial one [18]. In addition, Z=/summationtext
+σ′
+i=+1,−1e−βH′
+∆
+is the normalizing factor and βis a geometrical temper-
+ature determined by the structure of the network. Since
+βis not a real temperature, and in order to embody the
+majority principle, we propose
+β= 1/|/summationdisplay
+j/negationslash=iσj/(N−1)−σ′
+i| ≃1/|M−σ′
+i|,(3)
+whereNis the size of the network, and M=/summationtext
+i/angbracketleftσi(t)/angbracketright/Nis the average value of all spins. So the
+spin flipping probability depends both on the triad rela-
+tion and the surroundings [18, 19].
+The numbers ofpositive and negative nodes in the net-
+work are as follows,
+N+=C3
+N(3a1+2a2+a3)
+C2
+N=N
+3(3a1+2a2+a3),
+N−=C3
+N(a2+2a3+3a4)
+C2
+N=N
+3(a2+2a3+3a4).(4)
+whereNis the number of total nodes and aistands for
+the triangle density, satisfying/summationtext4
+i=1ai= 1. The densi-
+ties of each triangle attached to a positive node are
+n+
+ai=N
+3Ciai
+N+=Ciai
+3a1+2a2+a3, (5)
+wherei= 1,2,3, andCiis the number of positive nodes
+inai. In a similar way, the densities of each triangle
+attached to a negative node are
+n−
+aj=N
+3Djaj
+N−=Djaj
+a2+2a3+3a4, (6)
+wherej= 2,3,4, withDjthe number of negtive nodes
+inaj.
+For each node, the spin flipping probabilities from one
+value to the other are given by Eq. (2), i.e.,
+p(σi→σ′
+i) =eα
+|M−σ′
+i|(σ′
+iσj+σjσk+σ′
+iσk)
+e−α
+(1+M)(−σj+σjσk−σk)+eα
+(1−M)(σj+σjσk+σk).
+(7)
+So the total flipping probabilities for each node from one
+value to the other are as follows,3
+p(1→ −1) =4/summationdisplay
+i=1Ciai
+3p(1→ −1)
+=a1e−α/(1+M)
+e−α/(1+M)+e3α/(1−M)+2a2
+3e−α/(1+M)
+e−α/(1+M)+e−α/(1−M)+a3
+3e3α/(1+M)
+e3α/(1+M)+e−α/(1−M),(8)
+p(−1→1) =4/summationdisplay
+i=1Diai
+3p(−1→1)
+=a2
+3e3α/(1−M)
+e3α/(1−M)+e−α/(1+M)+2a3
+3e−α/(1−M)
+e−α/(1−M)+e−α/(1+M)+a4e−α/(1−M)
+e−α/(1−M)+e3α/(1+M),(9)
+/s49/s48/s52/s49/s48/s49
+/s32/s32/s84
+/s78
+/s78/s32 /s32 /s61/s48/s46/s49
+/s32 /s32 /s61/s50/s46/s48
+FIG. 2: A plot of TNwithNin the log-log coordinate.
+M(0) =−1,Nis the size of the network, and TNis the
+time needed for dynamic balance. The numerical results are
+obtained by taking the average of 5 ×102simulations on the
+networks with size from 2 ×103to 2×104.
+wherea1toa4stand for each triangle density, and Ci
+andDithe number of positive and negative nodes in ai
+respectively. When the network gets dynamically bal-
+anced, the densities of each triad relation are as follows,
+a1=1
+8, a2=3
+8, a3=3
+8, a4=1
+8.(10)
+We note the network reaches an explicit state, indepen-
+dent of the parameter α. And in the dynamically bal-
+anced state n+=n−, wheren+andn−stand for the
+densities of positive and negative nodes respectively.
+The time needed for dynamic balance is
+TN∼NC(α), (11)
+whereNis the size of the network, and C(α)>0, being
+a function of α. The numerical results are given in Fig.
+2. More details are given in Appendix A1./s32/s32/s32/s32/s32/s32/s32/s32/s32/s48/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s48/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s48
+/s43/s32/s32/s32/s32/s32/s32/s32/s97
+/s53/s32/s32/s32/s32/s32/s43/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s43/s32/s32/s32/s32/s32/s32/s32/s97
+/s54/s32/s32/s32/s32/s32/s32/s45/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s45/s32/s32/s32/s32/s32/s32/s32/s97
+/s55/s32/s32/s32/s32/s32/s32/s45
+/s32/s32/s32/s32/s32/s32/s32/s32/s48/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s48/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s48
+/s48/s32/s32/s32/s32/s32/s32/s32/s97
+/s56/s32/s32/s32/s32/s32/s43/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s48/s32/s32/s32/s32/s32/s32/s32/s32/s97
+/s57/s32/s32/s32/s32/s32/s45/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s48/s32/s32/s32/s32/s32/s32/s32/s32/s97
+/s49/s48/s32/s32/s32/s32/s48
+FIG. 3: a5anda7represent the stable relations, a6the un-
+stable relation, while a8−a10the metastable relations.
+B. The Generalized Node Model
+There being only two opposing opinions in the social
+networks is extreme. We propose the neutral opinion
+shouldexist, whichmeansthepersonholdingthisopinion
+is indifferent to other opinions. So we propose each node
+should have three values, +1, 0 and -1, standing for the
+three opinions. If there are the same opinion in the triad
+relation except the neutral opinion, the relation is stable.
+If there are different opinions in the triad relation except
+the neutral opinion, the relation is unstable. Otherwise
+therelationismetastable. Thesimplestwayisbyjudging
+the Hamiltonian (see Eq. (1)). If H∆<0, it is stable.
+IfH∆>0, it is unstable. If H∆= 0, it is metastable.
+There are four stable relations, three unstable relations,
+and three metastable relations in the network, as shown
+by Fig. 1 and Fig. 3.
+Compared with Eq. (4), the numbers of positive, neu-4
+tral and negative nodes in the network are as follows,
+N+=N
+3(3a1+2a2+a3+2a5+a6+a8),
+N0=N
+3(a5+a6+a7+2a8+2a9+3a10),
+N−=N
+3(a2+2a3+3a4+a6+2a7+a9),(12)
+wherea1-a10stand for the densities of each triangle. The
+densities of each triangle attached to a positive node are
+as follows,
+n+
+ai=Eiai
+3a1+2a2+a3+2a5+a6+a8,(13)
+wherei= 1,2,3,5,6,8, andEiis the number of positive
+nodesin ai. Inasimilarway,thedensitiesofeachtriangle
+attached to a neutral node are as follows,
+n0
+aj=Fjaj
+a5+a6+a7+2a8+2a9+3a10,(14)wherej= 5,6,7,8,9,10, andFjis the number of neutral
+nodes in aj. The densities of each triangle attached to a
+negative node are as follows,
+n−
+ak=Gkak
+a2+2a3+3a4+a6+2a7+a9,(15)
+wherek= 2,3,4,6,7,9, withGkbeing the the number
+of negtive nodes in ak.
+We study the ternary network in association with the
+Potts spin systems. Since the values of each spin are
+more than two, the spin flipping mechanism changes into
+the spin transition mechanism. For the spin transition
+mechanism, the spin takes one of the three values, +1, 0
+and -1, depending both on the triad relation and the sur-
+roundings [18, 19]. The Hamiltonian, the spin transition
+probabilities and the geometrical temperature are illus-
+trated by Eqs. (1), (2) and (3) respectively. The total
+probabilities from one value to the others for each node
+are as follows,
+p(−1→0) =a2
+3eα
+|M|
+eα
+|M|+e−α
+1+M+e3α
+1−M+2a3
+3e−α
+|M|
+e−α
+|M|+e−α
+1+M+e−α
+1−M+a4eα
+|M|
+eα
+|M|+e3α
+1+M+e−α
+1−M
++a6
+31
+1+e−α
+1+M+eα
+1−M+2a7
+31
+1+eα
+1+M+e−α
+1−M+a9
+9,
+p(−1→1) =a2
+3e3α
+1−M
+eα
+|M|+e−α
+1+M+e3α
+1−M+2a3
+3e−α
+1−M
+e−α
+|M|+e−α
+1+M+e−α
+1−M+a4e−α
+1−M
+eα
+|M|+e3α
+1+M+e−α
+1−M
++a6
+3eα
+1−M
+1+e−α
+1+M+eα
+1−M+2a7
+3e−α
+1−M
+1+eα
+1+M+e−α
+1−M+a9
+9,
+p(0→ −1) =a5
+3e−α
+1+M
+e−α
+1+M+eα
+|M|+e3α
+1−M+a6
+3e−α
+1+M
+e−α
+1+M+e−α
+|M|+e−α
+1−M+a7
+3e3α
+1+M
+e3α
+1+M+eα
+|M|+e−α
+1−M
++2a8
+3e−α
+1+M
+e−α
+1+M+1+eα
+1−M+2a9
+3eα
+1+M
+eα
+1+M+1+e−α
+1−M+a10
+3,
+p(0→1) =a5
+3e3α
+1−M
+e−α
+1+M+eα
+|M|+e3α
+1−M+a6
+3e−α
+1−M
+e−α
+1+M+e−α
+|M|+e−α
+1−M+a7
+3e−α
+1−M
+e3α
+1+M+eα
+|M|+e−α
+1−M
++2a8
+3eα
+1−M
+e−α
+1+M+1+eα
+1−M+2a9
+3e−α
+1−M
+eα
+1+M+1+e−α
+1−M+a10
+3,
+p(1→ −1) =a1e−α
+1+M
+e−α
+1+M+eα
+|M|+e3α
+1−M+2a2
+3e−α
+1+M
+e−α
+1+M+e−α
+|M|+e−α
+1−M+a3
+3e3α
+1+M
+e3α
+1+M+eα
+|M|+e−α
+1−M
++2a5
+3e−α
+1+M
+e−α
+1+M+1+eα
+1−M+a6
+3eα
+1+M
+eα
+1+M+1+e−α
+1−M+a8
+9,
+p(1→0) =a1eα
+|M|
+e−α
+1+M+eα
+|M|+e3α
+1−M+2a2
+3e−α
+|M|
+e−α
+1+M+e−α
+|M|+e−α
+1−M+a3
+3eα
+|M|
+e3α
+1+M+eα
+|M|+e−α
+1−M
++2a5
+31
+e−α
+1+M+1+eα
+1−M+a6
+31
+eα
+1+M+1+e−α
+1−M+a8
+9. (16)
+But Eq. (16) is correct only for M/negationslash= 0. When M→0, the transition probabilities are defined as (the explana-5
+/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53
+/s32/s32/s32
+/s32 /s32/s110/s43
+/s32 /s32/s110/s45
+/s32 /s32/s110/s48
+FIG. 4: The numerical results are obtained by taking the
+average of 20 simulations on the network with 104nodes. Ini-
+tiallyM(0) =−1/2, y(0) = 1/2, and the network takes a
+long enough time to get dynamically balanced.
+tions are given in Appendix A1)
+p(−1→0) = 0,p(1→0) = 0,
+p(0→1) =1
+2/parenleftbigga5
+3+a6
+3+a7
+3+2a8
+3+2a9
+3+a10/parenrightbigg
+,
+p(0→ −1) =1
+2/parenleftbigga5
+3+a6
+3+a7
+3+2a8
+3+2a9
+3+a10/parenrightbigg
+,
+p(1→ −1) =1
+2/parenleftbigg
+a1+2a2
+3+a3
+3+2a5
+3+a6
+3+a8
+3/parenrightbigg
+,
+p(−1→1) =1
+2/parenleftbigga2
+3+2a3
+3+a4+a6
+3+2a7
+3+a9
+3/parenrightbigg
+.
+(17)
+Since the transition probabilities are not continued at
+M= 0,wecannotgivetheanalyticalresultsof n+,n−,n0
+withα, so we turn to the numerical method. The results
+are shown by Fig. 4. We note when α→0,n+=n−=
+n0= 1/3. And when α→+∞,n0= 0, which means
+the neutral opinion disappears, and the ternary network
+comes back to the binary network.
+When the network gets dynamically balanced, the dis-
+tributions of each triangle are as follows,
+ai≃Clmnxl+nym, (18)
+wheren+≃n−=xandn0=y. Besides, i=
+1,2,3,...,10, and l,m,nare the respective numbers of
+positive, neutral, negative nodes in ai, along with Clmn
+the corresponding combinatorial number. For example
+C201= 3.
+For the time needed for dynamic balance, we expect it
+to satisfy
+TN∼ND(α), (19)
+whereNis the size of the network and D(α)>0, being
+a function of α./s49/s48/s52/s49/s48/s49
+/s32/s32/s84
+/s78
+/s78/s32 /s32 /s61/s48/s46/s49
+/s32 /s32 /s61/s50/s46/s48
+FIG. 5: A plot of TNwithNin the log-log coordinate.
+M(0) =−1/2, y(0) = 1/2.Nis the size of the network.
+TNis the time needed for dynamic balance. The numerical
+results are obtained by taking the average of 5 ×102simula-
+tions on the networks with size from 2 ×103to 2×104.
+The numerical results of the time needed for dynamic
+balance are given in Fig. 5. More details are given in
+Appendix A2.
+III. SELF-ORGANIZED CRITICALITY
+As is well known, SOC [6–8] is studied on many com-
+plex systems, including on the random graph [9], the
+scale-free network [10–12] and the small-world network
+[13], etc, which are mainly based on the BTW model [6].
+Besides, there arethe studies ofthe SOCon complexnet-
+works based on other models, which we refer the readers
+to Ref. [14] for details. In this paper, we study the SOC
+on the ternary social networks based on our model and
+the evolving rules, along with two modifications com-
+pared to Sec. II. The first modification is the struc-
+ture of the network. Because for a completely connected
+network, any disturbance is globe, with no propagation,
+so the occurrence of the avalanche will not follow the
+power-law form either spatially or temporally. Besides,
+compared to the completely connected network, the reg-
+ular network or the small-world network [15, 16] is closer
+to the real structure of the society. So we take the latter
+forms for study. The second modification is the judge-
+ment ofatriad relation. In orderto embody the majority
+principle, we take the judgement as the gauge invariance
+or the gauge variance [19–22], which is explained as fol-
+lows. Under this judgement, the triad relation evolves
+from one state to the other by changing node’s value.
+If the value of a node being +1, it means the person
+shows the friendly face in the triad relation, and always
+tires to keep the relation stable. If the value being -1, it
+means the person shows the hostile face, and always tries
+to destroy the stability of the triad relation. While the
+value being 0, it means the person does not care whether
+the triad relation is stable or not. So we propose if the6
+friendly faces are in the majority, the triad relation is in
+one state, or the gauge invariance state. While the hos-
+tile faces are in the majority, the triad relation is in the
+other state, or the gauge variance state. The explicit ex-
+pressions of the gauge invariance and the gauge variance
+states [19–22] are as follows.
+For each triad relation, we propose if
+σ1+σ2+σ3≥0or σ1=σ2=σ3= 0,(20)
+it satisfies the gauge invariance. If
+σ1+σ2+σ3<0or σ1/negationslash=σ2/negationslash=σ3= 0,(21)
+the triad relation does not satisfy the gauge invariance.
+Hereσ1,σ2andσ3arethe valuesofthe threenodesofthe
+triangle. So a1,a2,a5,a8,a10are gauge invariance states,
+whilea3,a4,a6,a7,a9are gauge variance ones, as shown
+by Fig. 1 and Fig. 3.
+We study the ternary network in association with the
+Potts spin systems. Since each node has three values,
+the way of spin changing is the spin transition [18]. Be-
+sides, we propose the spin transition depends both on
+itself and the surroundings [18, 19]. Considering the
+SOC is determined by the structure of the system, we
+set the geometric temperature as β=1
+T>0, where
+T=|/summationtext
+<i,j>σj/ni−σ′
+i|, withσ′
+ithe final value of i, and/summationtext
+<i,j>σj/nithe average value of all neighboring spins
+ofi. Based on these considerations, we give the spin
+transition probability of each node as follows,
+W(σi→σ′
+i) =1
+Ze−βH(σ′
+i,Σ<i,j>σj).(22)
+In Eq. (22), H(σ′
+i,/summationtext
+<i,j>σj) =−σ′
+i/summationtext
+<i,j>σj, where/summationtext
+<i,j>σjis the sum of values of all neighboring spins
+ofi, andZ=/summationtext
+σ′
+i=−1,0,+1eβσ′
+iΣ<i,j>σjis the normaliz-
+ing factor. In order to avoid the denominator of βbeing
+zero, we propose if the final value of the spin results in
+T= 0, it happens with probability zero, while the other
+values happen with probability one. In this way, what-
+ever the initial condition is, the network will evolve to
+the sensitive state.
+The evolving rules are as follows. We set all triad rela-
+tions in the network to satisfy gauge invariance initially.
+The simplest way is to set all spins to be zero, which
+stands for all persons in the society being neutral with
+each other initially. After a long enough time of evolu-
+tion, the network will reach the sensitive state. Then any
+small disturbance may result in an avalanche.
+For any small disturbance:
+(1)Chooseatriangleatrandom. Ifitisgaugevariance,
+nothing happens. If it is gauge invariance, we choose
+one of the three nodes at random, and let the node do
+spin transition according to Eq. (22). If after the spin
+transition, the triangle is still gauge invariance, nothing
+happens. Otherwise, we store all the gauge invariance
+triangles attached to the selected triangle, and goto step
+(2).FIG. 6: There are 9 ×104nodes in the two-dimensional lattice
+with equal length and width. It is one of the simplest two-
+dimensional regular networks satisfying the triad relatio ns.
+(2) Choose all the stored triangles successively at ran-
+dom. Because of the interaction, first we judge whether
+the triangle is gauge invariance. If it is gauge variance,
+nothing happens. Otherwise, let one of the three nodes
+do spin transition. If the final triangle is gauge invari-
+ance, nothing happens, otherwise we find out all the
+gauge invariance triangles attached to the selected tri-
+angle.
+(3) Find out all the gauge invariance triangles by the
+combined action of the stored triangles in step (2). If the
+number of gauge invariance triangles to store is nonzero,
+goto step (2), otherwise goto step (1).
+For each avalanche, we define the size as the number of
+trianglesstoredinstep(2)changedfromthegaugeinvari-
+ance state to the other state, and plus the one changed
+in step (1). Besides, we define the time of avalanche as
+follows. If the number of stored triangles in step (2)
+changed from the gauge invariance state to the other
+state is nonzero, the time is increased by one, and plus
+the one changed in step (1).
+Atfirst, westudytheSOContheregularnetwork. The
+structure of the regular network is given in Fig. 6. The
+size distribution and the time distribution are given in
+Fig. 7(a)and7(b)respectively. Inthefollowingcontent,
+we keep the number of nodes in both the regular network
+andthe small-worldnetworkconstantas9 ×104. Besides,
+the numerical results are obtained by taking the average
+of 100 simulations. For each simulation, we execute 9 ×
+106time of disturbance.
+In Fig. 7 (a) and Fig. 7 (b), the diagrams are curved
+whensandtare small. Because we set all spins to be
+zero initially. It takes a long time for the network to
+reach the sensitive state. While the diagrams curved at
+the ends is caused by the finite size effect.
+Next, we study the SOC on the small-world network
+[15, 16]. The structure of the small-world network is
+given in Fig. 8.
+When a triangle changes from gauge invariance to
+gauge variance, the value of the selected node is de-7
+/s49/s48/s48
+/s49/s48/s49
+/s49/s48/s50/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51
+/s49/s48/s48
+/s49/s48/s49
+/s49/s48/s50/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52
+/s40/s97/s41
+/s32/s32/s80/s40/s115/s41
+/s115/s32/s78/s61/s57/s42/s49/s48/s52
+/s32/s80/s40/s115/s41/s126/s115/s45/s49/s46/s55/s54/s40/s98/s41
+/s32/s32/s80/s40/s116/s41
+/s116/s32/s78/s61/s57/s42/s49/s48/s52
+/s32/s80/s40/s116/s41/s126/s116/s32/s45/s50/s46/s51/s56
+FIG. 7: Plots of P(s)versussandP(t) versuston the regular
+network. (a) sis thesize ofavalanche. P(s)is thedistribution
+function, satisfying the power-law form, with exponent -1. 76.
+(b)tisthetimeofavalanche. P(t)isthedistributionfunction,
+satisfying the power-law form, with exponent -2.38.
+/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s98/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s99
+/s97
+FIG. 8: From a two-dimensional lattice with equal length
+and width, for each node in the network, it sends out an
+edge with probability q, and attaches to another node at ran-
+dom, then attaches to one neighbor of the selected node at
+random. Multi-connection and self-connection are avoided .
+Thus, by adding edges, more triangles are created. The man-
+ner of adding random edges is shown by nodes a,b,c.
+creased. Because of the small-world effect, it may result
+in the triangles far away compared to the regular net-
+work becoming of gauge variance. If the triangles far
+away compared to the regular network become of gauge
+variance, it will blockthe spread. Soweexpect the sizeof
+avalancheto decrease, and the exponent ofthe power-law
+distributiontodecreaseaccordingly. Basedontheseanal-
+ysis, we set the edge adding probability qof the small-
+worldfrom q= 0.05 toq= 0.3, and observethe exponent
+of the size distribution decrease from -1.91 to -2.79, as
+given in Fig. 9 (a) and Fig. 9 (b) respectively.
+As is well known, for the small-world network[15, 16],/s49/s48/s48
+/s49/s48/s49
+/s49/s48/s50
+/s49/s48/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52
+/s49/s48/s48
+/s49/s48/s49
+/s49/s48/s50/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s32/s32/s80/s40/s115/s41
+/s115/s32/s78/s61/s57/s42/s49/s48/s52
+/s44/s32/s113/s61/s48/s46/s48/s53
+/s32/s80/s40/s115/s41/s126/s115/s45/s49/s46/s57/s49/s40/s97/s41
+/s32/s32/s80/s40/s115/s41
+/s115/s32/s78/s61/s57/s42/s49/s48/s52
+/s44/s32/s113/s61/s48/s46/s51
+/s32/s80/s40/s115/s41/s126/s115/s45/s50/s46/s55/s57/s40/s98/s41
+FIG. 9: A plot of P(s) versusson the small-world network. q
+is the edge adding probability. sis the size of the avalanche.
+P(s) is the distribution function. (a) q= 0.05. The exponent
+of the power-law distribution is -1.91. (b) q= 0.3. The
+exponent of the power-law distribution is -2.79
+/s49/s48/s48
+/s49/s48/s49
+/s49/s48/s50/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52
+/s49/s48/s48
+/s49/s48/s49
+/s49/s48/s50/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52/s32/s32/s80/s40/s116/s41
+/s116/s32/s78/s61/s57/s42/s49/s48/s52
+/s44/s32/s113/s61/s48/s46/s48/s53
+/s32/s80/s40/s116/s41/s126/s116/s45/s50/s46/s54/s53/s40/s97/s41
+/s32/s32/s80/s40/s116/s41
+/s116/s32/s78/s61/s57/s42/s49/s48/s52
+/s44/s32/s113/s61/s48/s46/s51
+/s32/s40/s98/s41
+FIG. 10: A plot of P(t) versus ton the small-world network.
+qis the edge adding probability. tis the time of avalanche.
+P(t) is the distribution function. (a) q= 0.05. The exponent
+of the power-law distribution is -2.65. (b) q= 0.3.
+when the edge adding probability qis big enough, the
+structure of the network becomes random. We note the
+diagrams are curved in Fig. 9 (a) and Fig. 9 (b) when s
+is small, but they take different forms. It means that the
+random networks are easier for the spread in the initial
+conditions.
+Then there is the time distribution of the avalanche
+on the small-world network, which is similar to the size
+distribution of the avalanche, as given in Fig. 10 (a)
+and Fig. 10 (b) for comparison. But we note, when q
+is big enough, with the structure of the network becom-
+ing random, the time distribution no longer satisfies the
+power-law form. Besides, there is condensation at the
+end of the diagram, which we want to study in the later
+work.
+At last, for the construction of the small-world net-8
+/s49/s48/s48
+/s49/s48/s49
+/s49/s48/s50
+/s49/s48/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52
+/s49/s48/s48
+/s49/s48/s49
+/s49/s48/s50
+/s49/s48/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52/s32/s32/s80/s40/s115/s41
+/s115/s32/s78/s61/s57/s42/s49/s48/s52
+/s44/s32/s113/s61/s48/s46/s48/s53
+/s32δ /s61/s49/s46/s48/s44/s32/s80/s40/s115/s41/s126/s115/s45/s49/s46/s56/s53/s40/s97/s41
+/s32/s32/s80/s40/s115/s41
+/s115/s32/s78/s61/s57/s42/s49/s48/s52
+/s44/s32/s113/s61/s48/s46/s48/s53
+/s32δ /s61/s52/s46/s48/s44/s32/s80/s40/s115/s41/s126/s115/s45/s49/s46/s54/s57/s40/s98/s41
+FIG. 11: A plot of P(s) versus son the small-world network
+considering the distance effect. qis the edge adding prob-
+ability. sis the size of avalanche. P(s) is the distribution
+function. (a) q= 0.05,δ= 1.0, and the exponent of the
+power-law distribution is -1.85. (b) q= 0.05,δ= 4.0, and the
+exponent of the power-law distribution is -1.69.
+work, when considering the factor of distance, the small-
+world effect on the triangles far away compared to the
+regular network is weakened, which may be easier for the
+spread. Besides, the number of local triangles are in-
+creased. So we expect the size of avalanche to increase,
+and the exponent of the size distribution to increase ac-
+cordingly. Forthesmall-worldnetwork,whenconsidering
+the distance effect, it is constructed as follows. From a
+regular network, as shown by Fig. 6, for each node in
+the network, it sends out an edge with probability qand
+attachesto anothernode in the network. The probability
+of attachment between nodes iandjis
+pij=ℓ−δ
+i,j/summationtext
+m<nℓ−δm,n. (23)
+Hereδ >0, andℓi,jstands for the smallest distance
+between iandjon the regular network. Then it at-
+taches to one neighbor of the selected node at random.
+Multi-connection and self-connection are forbidden. In
+this manner, more triangles are created. For the small-
+worldnetwork,wekeep q= 0.05constant,andset δ= 1.0
+andδ= 4.0 for comparison. For the size distribution of
+the avalanche, the exponent increasesfrom -1.85to -1.69,
+as shown by Fig. 11 (a) and Fig. 11 (b). Meanwhile, for
+the time distribution of the avalanche, the exponent in-
+creases from -2.56 to -2.22, as shown by Fig. 12 (a) and
+Fig. 12 (b).
+In a word, the size distribution of the avalanchealways
+satisfies the power-law form, no matter on the regular
+network, the small-world network or the random net-
+work. But for the time distribution of the avalanche,
+it deviates from the power-law form on the random net-
+work, which we want to do further study in the later
+work./s49/s48/s48
+/s49/s48/s49
+/s49/s48/s50/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52
+/s49/s48/s48
+/s49/s48/s49
+/s49/s48/s50/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52/s32/s32/s80/s40/s116/s41
+/s116/s32/s78/s61/s57/s42/s49/s48/s52
+/s44/s32/s113/s61/s48/s46/s48/s53
+/s32δ /s61/s49/s46/s48/s44/s32/s80/s40/s116/s41/s126/s116/s45/s50/s46/s53/s54/s40/s97/s41
+/s32/s32/s80/s40/s116/s41
+/s116/s32/s78/s61/s57/s42/s49/s48/s52
+/s44/s32/s113/s61/s48/s46/s48/s53
+/s32δ /s61/s52/s46/s48/s44/s32/s80/s40/s116/s41/s126/s116/s45/s50/s46/s50/s50/s40/s98/s41
+FIG. 12: A plot of P(t) versus ton the small-world network
+considering the distance effect. qis the edge adding prob-
+ability. tis the time of avalanche. P(t) is the distribution
+function. (a) q= 0.05,δ= 1.0, and the exponent of the
+power-law distribution is -2.56. (b) q= 0.05,δ= 4.0, and the
+exponent of the power-law distribution is -2.22.
+IV. CONCLUSIONS
+We have studied the social networks based on our
+model and obtained meaningful results. First, in Sec.
+II of this paper, based on Ref. [1], we map the edge re-
+lations to the node relations with some relaxation, along
+with introducing the neutral relation. So the social net-
+works change from the binary networks into the ternary
+ones. We suppose the change of each triad relation de-
+pending both on itself and the surroundings. So basedon
+ourmodel, whenthe socialnetworksgetdynamicallybal-
+anced, we obtain the distributions of each triad relation
+and the time needed for dynamic balance. Besides, we
+find under the extreme condition α→+∞, the ternary
+networks come back to the binary ones. Second, in Sec.
+III of this paper, we study the SOC on the ternary social
+networksbasedon ourmodel. Becausewesuppose, if one
+triad relation changes, it will affect other triad relations
+associating with it. The effect may be big or small, de-
+pending on many internal factors. While compared with
+Sec. II of this paper, we make two modifications for our
+node model as follows. One is the structure of the social
+networks, which takes the form of regular or small-world
+network. The other is the judgement of a triad relation,
+which takesthe formof the gaugeinvarianceorthe gauge
+variance. When the ternary social networks evolving to
+thesensitivestate,anysmalldisturbancemayresultinan
+avalanche. Sincethe numberoftriadrelationschangedin
+the avalanche being no scale preferred, it should satisfy
+the power-law distributions both spatially and tempo-
+rally. Third, for the small-world network, we find out
+the small-world effect on the occurrence of the avalanche
+both theoretically and numerically.9
+Acknowledgments
+WethankH. Zhuforhelpful discussions. Theworkwas
+supportedby the NationalNaturalScienceFoundationof
+China (No. 10875012).
+Appendix A: Distributions of each triad relation and
+the time needed for dynamic balance
+1. The node model
+For the node model, each node has two spin values, +1
+and -1. We define one step of time as executing Nspin
+flipping successively at random [1]. Because each spin
+flipping is linked up with ( N−1)(N−2)/2 variations of
+the triangles, the variations of the densities of each triad
+relation with time are as follows,
+da1
+dt=p(−1→1)n−
+a2−p(1→ −1)n+
+a1,
+da2
+dt=p(1→ −1)n+
+a1+p(−1→1)n−
+a3
+−p(1→ −1)n+
+a2−p(−1→1)n−
+a2,
+da3
+dt=p(1→ −1)n+
+a2+p(−1→1)n−
+a4
+−p(1→ −1)n+
+a3−p(−1→1)n−
+a3,
+da4
+dt=p(1→ −1)n+
+a3−p(−1→1)n−
+a4.(A1)
+When the network gets dynamically balanced, we have
+da1
+dt=da2
+dt=da3
+dt=da4
+dt= 0, (A2)
+and
+p(−1→1) =p(1→ −1). (A3)From Eqs. (A1), (A2) and (A3), along with the normal-
+ization/summationtext4
+i=1ai= 1, we obtain
+a1=1
+8,a2=3
+8,a3=3
+8,a4=1
+8.(A4)
+We note the distributions of each triad relation indepen-
+dentoftheparameter α, andn+=n−inthedynamically
+balanced state.
+Defining at any time the density of positive nodes to
+beρ, we obtain
+ai=Cj
+3ρj(1−ρ)3−j,
+M=n+−n−= 2ρ−1,
+ρ= (1+M)/2, ρ(∞) =1
+2, (A5)
+wherejis the number of positive nodes in aiandCj
+3is
+the combinatorial number. We have the new equations
+as follows,
+dn+
+dt=p(−1→1)−p(1→ −1),
+dn−
+dt=p(1→ −1)−p(−1→1),
+dM
+dt=dn+
+dt−dn−
+dt
+= 2[p(−1→1)−p(1→ −1)].(A6)
+Substitute Eqs. (8), (9) and (A5) into Eq. (A6), we
+obtain
+dM
+dt= 2/bracketleftBigg/parenleftbigg1+M
+2/parenrightbigg2/parenleftbigg1−M
+2/parenrightbigge3α/(1−M)
+e3α/(1−M)+e−α/(1+M)+2/parenleftbigg1+M
+2/parenrightbigg/parenleftbigg1−M
+2/parenrightbigg2e−α/(1−M)
+e−α/(1−M)+e−α/(1+M)
++/parenleftbigg1−M
+2/parenrightbigg3e−α/(1−M)
+e−α/(1−M)+e3α/(1+M)−/parenleftbigg1+M
+2/parenrightbigg3e−α/(1+M)
+e−α/(1+M)+e3α/(1−M)
+−2/parenleftbigg1+M
+2/parenrightbigg2/parenleftbigg1−M
+2/parenrightbigge−α/(1+M)
+e−α/(1+M)+e−α/(1−M)−/parenleftbigg1+M
+2/parenrightbigg/parenleftbigg1−M
+2/parenrightbigg2e3α/(1+M)
+e3α/(1+M)+e−α/(1−M)/bracketrightBigg
+(A7)
+For Eq. (A7), when M(t)∼0, after some calculations,
+we obtain the approximate result,
+M(t)∼t−1/C(α), (A8)
+whereC(α)>0, being a function of α.For the dynamic balance of the network, we propose
+|M(TN)| ∼O/parenleftbigg1
+N/parenrightbigg
+. (A9)10
+So the time needed for dynamic balance is
+TN∼NC(α). (A10)
+In the simulations, both for the binary and the ternary
+cases, if the final value σ′
+iresults in the denominator
+ofβ(see Eq. (3)) being zero, we propose it happens
+with probabilityzero, while the other values happen with
+probability one (see Eq. (17) for example). In this man-
+ner, no matter what M(0) (orM(0) and ρ(0)) is, thenetwork will reach the dynamically balanced state.
+2. The generalized model
+When each node has three spin values, the variations
+of the densities of each triad relation with time are as
+follows,
+da1
+dt=p(−1→1)n−
+a2+p(0→1)n0
+a5−[p(1→0)+p(1→ −1)]n+
+a1,
+da2
+dt=p(1→ −1)n+
+a1+p(−1→1)n−
+a3+p(0→ −1)n0
+a5+p(0→1)n0
+a6−[p(1→0)+p(1→ −1)]n+
+a2
+−[p(−1→0)+p(−1→1)]n−
+a2,
+...
+da9
+dt=p(1→0)n+
+a6+p(−1→0)n−
+a7+p(1→ −1)n+
+a8+p(0→ −1)n0
+a10−[p(0→ −1)+p(0→1)]n0
+a9
+−[p(−1→0)+p(−1→1)]n−
+a9,
+da10
+dt=p(1→0)n+
+a8+p(−1→0)n−
+a9−[p(0→ −1)+p(0→1)]n0
+a10. (A11)
+When the network gets dynamically balanced, it satisfies
+dai
+dt= 0, (A12)
+wherei= 1,2,3,...,10, along with the normalization
+10/summationdisplay
+i=1ai= 1. (A13)
+Although Eqs. (A11) are very hard to solve, from ap-
+pendixA1andthesymmetricspintransitionmechanism,we expect n+≃n−andM(∞)≃0 in the dynamically
+balanced state. Defining n+≃n−=xandn0=yin the
+dynamically balanced state, the densities of each type of
+the triangles are
+ai≃Clmnxl+nym, (A14)
+wherel,mandnare the numbers of positive, neutral
+and negative nodes in airespectively. Clmnis the corre-
+sponding combinatorial number. For example C201= 3.
+Because near M= 0, there are fluctuations. The spin
+transition probabilities are ( M/negationslash= 0)
+p(−1→0)≃x3+x3+2x2y
+1+eα+e−α+2x2y
+1+eα+e−α+xy2
+3,
+p(−1→1)≃x3+2x2yeα
+1+eα+e−α+2x2ye−α
+1+eα+e−α+xy2
+3,
+p(0→ −1)≃x2y+2xy2e−α
+1+eα+e−α+2xy2eα
+1+eα+e−α+y3
+3,
+p(0→1)≃x2y+2xy2e−α
+1+eα+e−α+2xy2eα
+1+eα+e−α+y3
+3,
+p(1→ −1)≃x3+2x2ye−α
+1+eα+e−α+2x2yeα
+1+eα+e−α+xy2
+3,
+p(1→0)≃x3+x3+2x2y
+1+eα+e−α+2x2y
+1+eα+e−α+xy2
+3,
+(A15)11
+and (M→0)
+p(−1→0) = 0,p(1→0) = 0,
+p(0→1) = 2x2y+2xy2+y3
+2,
+p(0→ −1) = 2x2y+2xy2+y3
+2,
+p(1→ −1) = 2x3+2x2y+xy2
+2,
+p(−1→1) = 2x3+2x2y+xy2
+2.(A16)
+The densities of each triangle attached to a positive
+node are
+n+
+ai≃c1x2+2c2x2+2c3xy+c4y2
+(2x+y)2,(A17)
+wherei= 1,2,3,5,6,8, and/summationtext4
+j=1cj= 1,cj= 0,1. That
+is, only one of the coefficient cjis nonzero. In a similar
+way, the densities of each triangle attached to a neutral
+node are
+n0
+aj≃c1x2+2c2x2+2c3xy+c4y2
+(2x+y)2,(A18)wherej= 5,6,7,8,9,10. And the densities of each tri-
+angle attached to a negative node are
+n−
+ak≃c1x2+2c2x2+2c3xy+c4y2
+(2x+y)2,(A19)
+wherek= 2,3,4,6,7,9.
+Because the transition probabilities (see Eq. (16) and
+Eq. (17)) are not continued at M= 0, we cannot obtain
+the analytical results for the distributions of n+,n−and
+n0withα, so we turn to the numerical method, as shown
+by Fig. 4. We note when α→+∞, the neutral opinion
+disappears, and the ternary network comes back to the
+binary network. That can be explained as follows. In the
+triad relation, the more people try to keep it stable, i.e.,
+to show the same opinion with the others, the less likely
+the neutral opinion will exist.
+[1] T. Antal, P. L. Krapivsky, and S. Redner, Phys. Rev. E
+72, 036121 (2005).
+[2] F. Heider, Psychol. Rev. 51, 358 (1944); F. Heider, J.
+Psychol. 21, 107 (1946); F. Heider, The Psychology of
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+[3] S. Wasserman and K. Faust, Social Network Analysis:
+Methods and Applications (Cambridge University Press,
+New York, 1994).
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+Rev. E75, 021118 (2007).
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+[9] E. Bonabeau, J. Phys. Soc. Jpn. 64, 327 (1995).
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+Lett.91, 148701 (2003).
+[11] D. S. Lee, K. I. Goh, B. Kahng, and D. Kim, Physica A338, 84 (2004).
+[12] D. S. Lee, K. I. Goh, B. Kahng, and D. Kim, J. Korean
+Phys. Soc. 44, 633 (2004).
+[13] L. de Arcangelis and H. Herrmann, Physca A 308, 545
+(2002).
+[14] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes,
+Rev. Mod. Phys. 80, 1275 (2008).
+[15] D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998).
+[16] M. E. J. Newman and D. J. Watts, Phys. lett. A 263,
+341 (1999).
+[17] R. J. Glauber, J. Math. Phys. 4, 294 (1963).
+[18] J. Y. Zhu and Z. R. Yang, Phys. Rev. E 59, 1551 (1999);
+H. Zhu, J. Y. Zhu, and Y. Zhou, Phys. Rev. E 66, 036106
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+[20] C. Rovelli and L. Smolin, Phys. Rev. D 52, 5743 (1995).
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+095016 (2008).
\ No newline at end of file
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+arXiv:1001.0310v1  [astro-ph.SR]  2 Jan 2010Astronomy & Astrophysics manuscript no. mfried c/circlecopyrtESO 2018
+November 4, 2018
+UV Emission line shifts of symbiotic binaries
+M. Friedjung1, J. Miko/suppress lajewska2, A. Zajczyk3, and M. Eriksson4
+1Institut d’Astrophysique de Paris -UMR 7095, CNRS/Univers it´ e Pierre et Marie Curie, 98 bis Boulevard Arago,
+75014 Paris France
+2Nicolaus Copernicus Astronomical Center, Bartycka 18, 00- 716 Warsaw, Poland
+3Nicolaus Copernicus Astronomical Center, Rabia´ nska 8, 87 -100 Toru´ n, Poland
+4University College of Kalmar, 391 82 Kalmar, Sweden
+Received , Accepted...
+ABSTRACT
+Context. Relative and absolute emission line shifts have been previo usly found for symbiotic binaries, but their cause
+was not clear.
+Aims.This work aims to better understand the emission line shifts .
+Methods. Positions of strong emission lines were measured on archiva l UV spectra of Z And, AG Dra, RW Hya, SY
+Mus and AX Per and relative shifts between the lines of differe nt ions compared. Profiles of lines of RW Hya and Z
+And were also examined.
+Results. The reality of the relative shift between resonance and inte rcombination lines of several times ionised atoms
+was clearly shown except for AG Dra. This redshift shows a wel l defined variation with orbital phase for Z And and RW
+Hya. In addition the intercombination lines from more ionis ed atoms and especially O ivare redshifted with respect to
+those from less ionised atoms. Other effects are seen in the pr ofiles
+Conclusions. The resonance-intercombination line shift variation can b e explained in quiescence by P Cygni shorter
+wavelength component absorption, due to the wind of the cool component, which is specially strong in inferior conjunc-
+tion of this cool giant. The velocity stratification permits absorption of line emission. The relative intercombinatio n
+line shifts may be connected with varying occultation of lin e emission near an accretion disk, which is optically thick
+in the continuum.
+Key words. Stars: binaries: symbiotic – Stars: mass loss Stars: - accre tion – Stars: individual:
+Z And - CI Cyg – AG Dra – RW Hya – SY Mus – AX Per
+1. Introduction
+The variations of the radial velocities of the emission lines
+formed in symbiotic systems are not easy to interpret. The
+lines are emitted in moving plasma, due to winds and/or
+gas streams, which do not need to have the same motion
+as either stellar component. In addition the lines can be
+affected by absorption of overlying material such as the
+absorption of other lines of the iron forest as well as by ra-
+diative transfer effects. The line profiles are as a result not
+necessarily simple, affecting radial velocity measurements.
+The present work was undertaken in order to better under-
+stand both radial velocities and in some cases line profiles.
+Among the effects previously found, there is a system-
+atic redshift of the wavelengths of resonance emission lines
+of highly ionized atoms with respect to those of intercombi-
+nation emission lines in the ultraviolet spectra of a number
+of symbiotic binaries (Friedjung, Stencel and Viotti 1983).
+The former lines might be expected to be optically thick, so
+it appeared that such a redshift might be explainable in an
+expanding medium, either by the presence of absorption at
+the short wavelength edge of the resonance emission lines
+or by a radiative transfer effect, associated with the scat-
+tering of radiation in this expanding medium. The latter
+can occur only if the optical thickness is very large.
+Send offprint requests to : M. Friedjung
+Correspondence to : fried@iap.frMore recently a GHRS/HST spectrum and a number of
+IUE spectra of the symbiotic binary CI Cyg were studied
+by Miko/suppress lajewska, Friedjung and Quiroga (2006). The red-
+shift was confirmed for that binary. The resonance lines had
+an almost constant radial velocity during an orbital cycle,
+interpreted as being most probably due to the presence of
+a blueshifted absorption component, produced in a circum-
+binary region. According to the interpretation proposed,
+the circum-binary region appeared to be mostly expand-
+ing, while in addition, a part appeared to be contracting.
+Other radial velocity effects in the emission lines were also
+investigated and interpreted in that paper.
+In the present work we examine the radial velocities
+of other symbiotic binaries, comparing mean radial veloci-
+ties of emission lines of different ions, in order to look for
+optical thickness and ionisation potential dependent strat-
+ification effects. We can in this connection note that ac-
+cording to Nussbaumer et al (1988), among the emission
+lines, which interest us C iii, Civ, Niii,Nivand Oiiiare
+expected to be formed in the same region for a temperature
+of the source of ionising radiation of not less than 60 000K
+and an electron density of around 109cm−3. In addition
+the critical densities, where most emission of the intercom-
+bination lines is produced, are of the order of 109cm−3.
+This paper together with our previous paper on CI Cyg
+(Miko/suppress lajewska, Friedjung, and Quiroga) (2006), completes
+the study of radial velocity shifts in all symbiotic binaries2 Friedjung et al.: Symbiotic line shifts
+which have strong emission lines and for which high reso-
+lution ultraviolet spectra exist.
+2. The data
+We have used the MAST archive, containing IUE and HST
+spectra. Among the IUE spectra only those, having a high
+resolution, taken with the large aperture and the SWP cam-
+era, were analysed. HST spectra were available for AG Dra
+and RW Hya. This left 41 spectra of Z And, 44 spectra of
+AG Dra, 11 spectra of RW Hya, only 7 spectra for SY Mus
+and even fewer (3) for AX Per.
+Z And had two outbursts between JD 45797 and JD
+46596, while AG Dra had more than one outburst dur-
+ing the period studied. AX Per, SY Mus and RW Hya are
+eclipsing systems. Near infrared photometry indicates elip-
+soidal variations for SY Mus and RW Hya with Roche lobe
+filling factors near unity of 0.83 and 0.91 respectively ac-
+cording to Rutkowski, Miko/suppress lajewska and Whitelock (2007),
+while Miko/suppress lajewska (2007) gives reasons for believing that
+the cool components of most S type systems fill their Roche
+lobes.
+The radial velocities were measured by Gaussian fits to
+the whole profile, using the SPLOT programme of IRAF,
+This averages over any real or instrumental assymmetry.
+Saturated lines were measured by fitting to the wings. We
+have checked that the radial velocities determined in this
+way closely agree with the line centroid radial velocities. It
+is because of our method that we were able to use emission
+lines, whose centres are saturated. The nature of our IUE
+spectra does not enable us to do anything more precise.
+Let us note however that the HST/STIS spectra of AG
+Dra suggest some assymmetry of the resonance emission
+line profiles, with line centres being shifted by apparent
+absorption on the low wavelength side (Young 2006, private
+communication as well as the MAST archive).
+The errors, indicated in the radial veocity tables, corre-
+spond to 1 σerrors of the respective mean values. They do
+not indicate systematic errors, such as for instance those of
+errors in the absolute wavelength scale. However the mean
+values are based on data of different transitions, so their σ
+values include that due to uncertainty of laboratory wave-
+lengths. The other systematic errors due to the absolute
+wavelength scale, have been estimated by comparing our
+radial velocity estimates from different spectra of the same
+target, taken at the same epoch. In particular we found 14
+pairs of IUE spectra (4 for Z And, 10 for AG Dra) taken on
+the same day or one day later. The offset in radial velocity
+for these pairs is between ∆ v∼0−20kms−1, with an off-
+set for 10 pairs of 2-8 km s−1, a median value of around 5
+km s−1and a mean of ∼6.7±1.3kms−1. There is no dif-
+ference in the offset between saturated and non-saturated
+lines, when we compare a spectrum on which a particular
+line is saturated and another spectrum on which it is not
+saturated. The corresponding offset for 2 HST/STIS spec-
+tra of AG Dra taken on the same day is only 1.7 km s−1.
+3. Results
+Our measured radial velocity means and their 1 σerrors are
+tabulated in table 1 for AX Per, SY Mus and RW Hya, in
+table 2 for Z And and in table 3 for AG Dra. The phases of
+the tables are from the ephemeris of Miko/suppress lajewska (2003), 0 10 20 30 40
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ CIV
+ 0 10 20 30 40
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ NV
+-10 0 10 20 30
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+φHeII
+Fig.1. Radial velocities for SY Mus. The dashed line shows
+the orbital velocity of the cool giant. The symbols are ex-
+plained in the text.
+with respect to inferior conjunction of the giant. The last 8
+lines in table 1 for RW Hya and the last 2 lines in table 3 for
+AG Dra are based on HST/STIS spectra. The wavelength of
+the Heii1640 ˚A line used, is a mean wavelength of the fine
+structure, given by Clegg et al (1999). Let us note that this
+mean is fairly insensitive to the exact physical conditions of
+line formation; ranges of electron density from 104to 109
+cm−3and temperatures from 10 000 to 30 000K give for
+case B line formation a shift of 0.007 ˚A or 1.3 km s−1.
+Some radial velocities in the tables, including means and
+the radial velocity of the He ii1640 ˚A line, plotted against
+orbital phase, are shown in fig 1, for SY Mus, while radial
+velocity differences with much less scatter are shown in figs.
+2 and 3 for the same symbiotic. Similar figures shown, are
+4, 5 and 6 for Z And and 7, 8 and 9 for AG Dra. The
+difference graphs are with respect to the resonance doublet
+of Civ, as measurements of the strong lines belonging to
+this doublet could be made for all the spectra studied by us.
+Shifts, such as those of the intercombination lines relative
+to Civ, depend on the shift of C ivitself. The values for
+Siivare not plotted, but their behaviour can be seen from
+the tables.Friedjung et al.: Symbiotic line shifts 3
+Table 1. Radial velocities of the UV emission lines (in units of km s−1) for AX Per, SY Mus and RW Hya.
+MJD φSiiii] Ciii] Niii] O iii] Niv] Oiv] Si iv Civ NvHeii
+IP[eV] 16.3 24.4 29.6 35.1 47.4 54.1 33.5 47.5 77.5 54.4
+AX Per
+44156 0.000 -123 -121 -120(2) -124(2) -106 -105(2) -113
+45642 0.183 -137 -131-134(1) -136(5) -135 -135 -107(4) -111(1) -137
+48226 0.979 -111 -110 -116(2) -111(2) -111 -104(2) -103(2) - 113
+SY Mus
+44767 0.340 7 15 7 9(1) 8 13 19(1) 22(2) 16(3) 3
+45020 0.746 -6 -10 20(1) 2
+48451 0.240 9 10 2(1) 7 23(2) -6
+48473 0.275 14 19 14 13(1) 10 26 27(1) 28 3
+48629 0.525 16 16 16(1) 16(1) 13 21(1) 23(1) 22(4) 12
+48623 0.675 11 14 9 11(1) 13 18(2) 23(1) 21(1) 8
+50187 0.018 0 5 -15 -2 -5 23(1) -3
+RW Hya
+44118 0.437 10 11 3 2(1) -8-3(2) 17(1) 26(3) 19(1) -26
+44256 0.809 7 6 7 6(2) 832(2) 24 36(1) 32(7) 34
+44692 0.987 14 23 17 18(1) 24 47(1) 45(3) 55
+52012∗0.749 11.9(0.5) 17.7 35.3(0.1) 29.0(1.0) 40.0(1.2) 46.5 (0 .4) 40.4
+52016∗0.760 8.7(0.3) 16.1 34.0(0.5) 29.4(0.4) 38.0(1.3) 48.1(0. 4) 39.0
+52020∗0.771 9.5(0.8) 17.1 35.4(0.5) 30.3(0.2) 39.0(0.9) 50.1(0. 5) 40.2
+52024∗0.782 11.9(1.1) 20.0 39.2(0.5) 33.9(0.5) 42.6(0.8) 52.9(0 .9) 43.7
+52028∗0.793 10.3(1.3) 19.8 39.9(0.5) 32.9(0.6) 42.1(0.5) 53.3(1 .3) 44.3
+52032∗0.803 12.8(1.4) 23.2 43.5(0.6) 34.4(0.7) 45.3(0.4) 57.6(1 .5) 48.1
+52036∗0.814 13.8(1.3) 25.0 45.0(0.9) 36.6(0.4) 46.4(0.2) 61.9(0 .9) 49.7
+52040∗0.825 14.7(1.5) 26.2 46.0(0.7) 34.6(1.4) 46.7(0.2) 63.3(1 .1) 51.0
+For saturated lines the radial velocities (itallic) were ob tained by fitting their wings.
+1-σerrors of the mean values are in brackets.
+∗– HST STIS spectra.
+-10 0 10 20 30 40
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ CIV-NV
+ 0 10 20 30 40 50
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+φCIV-HeII
+Fig.2. Nvand Heiiradial velocity differences for SY Mus.
+The dashed line shows the orbital velocity of the cool giant.
+The symbols are explained in the text.
+In the graphs mentioned, the open circles are of fits of
+the line centres to a Gaussian for IUE observations in quies-
+cence, while in the cases of IUE observations of Z And andAG Dra, the filled circles are for IUE observations in out-
+burst. Open and filled triangles denote corresponding fits of
+the line wings in quiescence and outburst, when the centres
+are saturated. Finally an X is used to plot HST/STIS data
+of AG Dra. The dotted curve in these figures is the radial
+velocity curve of the cool component from Belczy´ nski et al
+(2000), taking phase 0 as that of inferior conjuction of that
+component.
+We mainly show and study the difference graphs and
+values, which are much less affected by the systematic er-
+rors, as shown above. In the case of SY Mus, Fig. 1 shows
+that the C ivradial velocity is redshifted and almost con-
+stant, while He iitends to follow the hot compoment with
+a blueshift. One can see from Fig. 2 that the mean C iv
+and Nvredshifts are almost the same without systematic
+variations with orbital phase, while in addition the table 1
+Civ- Siivdifferences are small and slightly positive. The
+Fig. 2 Civ- Heii1640 ˚A radial velocity graph shows a rel-
+ative redshift of C ivwith a minimum perhaps near phase
+0.5, this being a reflection of the radial velocity variation of
+Heiiby itself. The redshift of the C ivresonance with re-
+spect to the intercombination lines of SY Mus varies fairly
+smoothly with orbital phase, with a maximum approaching
+30 km s−1somewhat before phase 0 and a much smaller
+minimum (Fig.3).
+There are far more measurements for Z And which
+shows comparable effects. The O iv] and He ii1640 ˚A radial
+velocities show a maximum which seems to be in phase with
+the radial velocity of the hot component at least during qui-
+escence (Fig.4). The He iiline may have a small tendency
+to have a blueshift. The difference graphs of Z And give
+clear results. Shifts of N vrelative to C ivin Fig. 5 are at
+least during quiescence almost always small and indepen-4 Friedjung et al.: Symbiotic line shifts
+Table 2. Radial velocities of the UV emission lines (in units of km s−1) for Z And.
+MJD φSiiii] Ciii] Niii] O iii] Niv] O iv] Si iv Civ NvHeii
+IP[eV] 16.3 24.4 29.6 35.1 47.4 54.1 33.5 47.5 77.5 54.4
+43922 0.655 -6 -7 -6 -2.6(0.1) -8 5.3(1.1) 7.3(2.6) 10.1(1.3) 0.4(2.0) -6
+44075 0.856 -13 -14 -14 -13.4(1.0) -11 -4.4(1.4) 2 7.1(1.6) - 0.2(2.0) -12
+44402 0.287 0 -1 -9 -1.8(1.1) -7 10.1(2.5) 11.6(2.6) 14.2(0. 1) 8.1(1.1) -12
+44480 0.391 -5 1 -4 -1.7(4.0) 0.8 7.9(1.0) 6.9(6.0) 13.1(0.9 ) 5.9(0.6) -12
+44719 0.706 9 10 4 8 17.9(0.3) 15 1
+45797∗∗0.126 -24 -18 -21 -27.1(5.1) -29 -21.8(0.9) 18.0(4.5) 17.7( 1.7) 12.3(4.6) -4
+45863∗∗0.213 -9 -11 -11.8(0.5) -10.6(2.0) -11 -1.9(2.1) 9.0(0.1) 30.7(4.3) 12.5(3.5) 1
+45946 0.323 -8 -17 -14 -10.0(1.0) -16 -1.5(1.5) 7.6(3.8) 16.5(5.4) 2.7(0.8) -14
+45948 0.325 6 10 5 4.1(0.2) 2 11.1(1.6) 17.6(1.3) 21.7(0.4) 1 6.3(3.5) -3
+46232 0.700 -6 -1 -0.5(2.8) -1.9(1.5) -7 6.1(1.0) 11.4(2.8) 11.2(0.8) 11.8(1.7) 5
+46233 0.701 5 4 -4 12.1(4.0) 6 17 23.1(2.7) 24.1(1.3) 22.7(1.6) 1
+46283 0.767 10 3 1 6 14 26.2(3.9) 25.9(3.1) 21.5(0.9) 3
+46595∗∗0.178 -13 -20 -24.6(1.7) -19.4(1.6) -23 -10.9(0.1) 5.4(5.0) 4.4(3.8) 12.1(4.8) -20
+46596∗∗0.179 -7 -15-14.7(1.4) -11.4(1.2) -11 -3.5(1.6) 10.7(1.3) 18.5(2.4) -11
+46715 0.336 -21 -27 -19.4(1.9) -19.4(1.9) -28-12.8(1.3) -4.3(1.7) -2.3(1.4) -2.5(0.1) -21
+46778 0.419 -8 -16 -6.3(2.1) -6.2(0.1) -8 -1.1=(1.1) 5.0(0.8) 8.1(0.4) 3.7(2.8) -20
+46962 0.662 7 11 11 10.4(1.7) 7 16.8(4.3) 22.5(0.5) 24.0(0.7) 19.2(2.3) -8
+47081 0.819 0 0 -2 -1.0(3.1) -5 10 19.0(1.8) 16.3(4.7) 6
+47101 0.845 -10 -16 -13 -8.3(1.5) -13 -0.1(3.1) 8 11.1(1.0) 9 .4(5.3) -3
+47195 0.969 -23 -18 -15 4.4(0.1) -16
+47327 0.143 -33 -37 -34 -31 -39 -30.3(2.1) -13.5(1.2) -6.6(0 .8) -13 -33
+47713 0.650 -2 -2 -11 -3.3(2.5) -5 2.5(1.2) 4.6(3.5) 11.4(1. 2) 7.4(1.1) -8
+47845 0.825 -8 -8 -8 -6.8(1.90) -12 -2.7(0.8) 7.1(0.1) 11.1(2.6) 6.4(1.3) -7
+47885 0.878 -16 -18 -15 -26 -21 -17(3.7) -1.5(1.5) -3 -20
+48092 0.150 -1 3 -4 -8 2 25.1(0.5) -4
+48929 0.254 -9 -8 -7 -7 0 10.8(4.6) 10.9 -20
+48997 0.343 -7 7 -4 -3.9(0.3) -1 10.5(0.5) 9.4(0.9) 12.2(1.0 ) 7.6(3.0) -9
+49163 0.563 6 8 0 3.7(2.3) 0 8.4(0.1) 10.3(2.8) 15.4(0.5) 9.8 (0.4) -8
+49229 0.650 7 10 5 11.0(0.4) 9 23.1(1.0) 20.5(0.5) 20.4(0.3) 18.8(1.2) 5
+49617 0.161 -30 -18 -21 -27 -15 2.2(1.1) -24
+49678 0.241 -7 -7 -14.9(0.1) -18 -10.3(2.3) -4 7.0(0.2) 6.8 - 18
+49728 0.307 -6 0 -9.3(1.5) -2 -0.2(1.9) 4.5(0.2) 7.8(0.5) 1. 0(4.0) -17
+49898 0.523 -21 -14 -20 -18.3(1.3) -20 -9 -8.3(4.0) 0.2(1.7) -8.8(1.1) -30
+49963 0.617 -3 3 -15 -5.1(0.6) -13 6 7.9(1.0) 2 -12
+50104 0.802 -16 -7 -13.4(2.6) -11 4.6(0.2) -16
+50104 0.802 -21 -25 -19 -17.0(0.6) -18 -11.3(4.2) -3.0(0.2) -1.4(1.0) 3.0(0.6) -18
+For saturated lines the radial velocities (itallic) were ob tained by fitting their wings.
+1-σerrors of the mean values are in brackets.
+∗∗– spectra taken during outbursts. The profiles on spectra tak en during the outburst on MJD46353–46462 are very complex an d
+they cannot be characterised by single radial velocity.
+dent of orbital phase. We may note that the C iv- Siiv
+differences from the table have a certain amount of scatter,
+but are usually small (less than 10 km s−1and positive).
+Fig. 5 shows that the C iv- Heiiradial velocity is in phase
+with the cool component, or that the He iiradial velocity
+relative to that of C ivis in phase with the hot component.
+The situation is somewhat less certain for O iv], which may
+reach a maximum radial velocity relative to that of C iv
+earlier. C ivis however redshifted with respect to O iv] at
+most orbital phases. Fig. 6 shows radial velocity differences
+between C ivand other intercombination lines, which have
+a similar orbital phase dependence, but which are generally
+more positive than the O iv] difference. Let us note that the
+phase dependence would appear to be somewhat different
+than that of SY Mus. In connection with these results let us
+note that IUE wavelengths of O iv] are usually less accurate
+than those of He ii.
+AG Dra shows fewer clear effects. Unlike in quiescence
+there are enough He iiand Oiv] measurements during out-
+burst, to indicate that they appear to follow the hot com-
+ponent at least at such times (Fig. 7). The difference graphs
+of Fig. 8 show constant differences at almost all times, C ivhaving a small redshift relative to He ii. This constancy is
+violated during outburst between JD 49538 and 49613 at
+phases of 0.7-0.9 (Table 1). Fig. 9 shows the radial velocity
+shift between the C ivand intercombimation lines exclud-
+ing Oiv]; except perhaps for C iii] in outburst, no clear
+sign of any orbital variation is seen, while there might be
+a significant redshift for line centre meaurements of C iv
+relative to C iii] in quiescence. As can be seen in Table 3
+the higher quality HST/STIS spectra show a clear redshift
+of Oiv] relative to the other intercombination lines, due to
+ions with a lower ionization potential.
+Figures are not shown for the other symbiotics for which
+the observations have poor phase coverage. However most
+RW Hya measurements, based on HST/STIS spectra are
+very accurate. The radial velocity C iv- Siivdifferences
+are small and positive, with very little variation between
+8.7 and 12.1 km s−1. The Civ- Nvdifference shows how-
+ever a more complex behaviour. The first 2 values, based on
+IUE spectra are similarly small (7 and 4 kms−1), while the
+later values from HST/STIS spectra as well as being closely
+spaced in time at phases not very far from that of the second
+IUE spectrum, but taken two decades later, are curiouslyFriedjung et al.: Symbiotic line shifts 5
+Table 3. Radial velocities of the UV emission lines (in units of km s−1) for AG Dra.
+MJD φSiiii] Ciii] O iii] N iv] O iv] Si iv Civ NvHeii
+IP[eV] 16.3 24.4 35.1 47.4 54.1 33.5 47.5 77.5 54.4
+44418 0.421 -138 -141.8(0.9) -137 -133.9(2.0) -132 -131.9( 0.6) -145.3(5.0) -149
+44418 0.421 -142 -137 -143.1(1.3) -142 -137.0(1.5) -142 -13 8.2(1.0) -145.0(2.9) -155
+44698∗∗0.932 -145 -156.3(4.8) -159 -151.4(0.7) -156.7(1.8) -149. 9(0.1) -149.6(1.7) -162
+44700∗∗0.934 -148 -154 -153.0(1.9) -154-143.8(1.2) -144.6(0.6) -144.3(0.1) -147.9(3.0) -152
+44719∗∗0.969 -141-145 -153 -156-146.1(4.6) -148.7(0.4) -147.5(1.1) -146.3 (0.6) -158
+44719∗∗0.969 -146-144 -153.5(1.4.0) -155 -144.2(0.8) -145.4(1.2) -144.7(1 .4) -142.9(0.1) -159
+44820∗∗0.154 -157 -145 -156.5(1.3) -161 -149.6(1.6) -154.7(2.3) - 152.5(0.2) -158.0(2.6) -168
+44944∗∗0.379 -160-179 -166.3(0.3) -168 -162.1(0.6) -163.3(1.5) -170.6(0.1) -169.7(1.0) -185
+44950∗∗0.390 -153 -161 -160.4(1.7) -162-158.3(3.0) -159.3(1.8) -160.6(1.5) -168.3(3.9) -188
+44950∗∗0.390 -154 -157 -166 -160 -150.7(2.7) -159.7(1.0) -160.0(4.6) -179
+45492 0.378 -143-140 -146.6(0.5) -145-137.1(1.7) -145.8(0.1) -141.7(0.7) -145 -158
+46138 0.554 -144 -151.3(0.7) -152 -142.2(3.2) -149.9(0.8) -148.0(0.7) -153 -165
+46155 0.585 -153 -148 -148 -150.7(1.4) -153.1(3.8) -168
+46374 0.984 -152 -157 -154 -136 -137.0(0.3) -139 -149
+47827 0.631 -156 -160.7(0.8) -163 -148.9(5.9) -152 -150.5( 0.8) -150 -164
+47828 0.632 -152 -148.1(7.7) -157 -159
+48225 0.355 -134 -145.7(1.3) -140 -140 -135.9(0.4) -132.1( 3.1) -151
+48812 0.424 -149 -137 -149.6(7.1) -157 -157
+48878 0.545 -137.3(2.9) -128 -140
+48898 0.581 -124 -125.4(3.4) -126 -139
+48960 0.695 -142 -146.2 -148 -138.2(1.4) -143 -139.4(2.2) - 138.9(1.0) -154
+49058 0.873 -161 -165.7(3.6) -163 -152.2(4.4) -157 -150.9( 2.0) -149 -166
+49533∗∗0.737 -140 -134 -145 -144 -140.8(2.1) -146.7(4.6) -142.7(0 .6) 139.2(0.7) -143
+49533∗∗0.737 -154 -153 -142 -148 -148.0(1.3) -150 -153
+49538∗∗0.748 -133 -143 -150 -153 -138.2(2.9) -142.9(1.6) -144.9(0 .1) -134.9(1.3) -152
+49540∗∗0.750 -164 -172.4(0.2) -167 -153.4(5.3) -163.3(1.1) -164. 3(1.6) -154.6(2.9) -173
+49543∗∗0.755 -170 -144 -125 -143 -136.5(0.8) -110.5(1.8) -120
+49543∗∗0.755 -151 -167 -148 -140 -133.7(1.6) -143.0(0.7) -130 -130
+49546∗∗0.762 -168 -164.6(6.0) -143 -165
+49561∗∗0.789 -144 -143 -147.2(3.5) -132 -158
+49562∗∗0.791 -151 -146 -164 -160 -154 -153.4(4.5) -154.8(0.9) -148 .0(5.6) -169
+49613∗∗0.884 -132 -146 -145.6(4.7) -145 -145.3(3.9) -146.6(5.3) - 137.9(0.1) -138 -154
+49693∗∗0.029 -140 -141.9(0.4) -143 -134.9(0.5) -130 -134.6(1.0) - 133.5(0.8) -154
+49693∗∗0.029 -147 -149.0(1.5) -149 -141(3.1) -144.7(0.2) -140.8( 0.6) -140.9(2.1) -157
+49846∗∗0.307 -153 -154 -154 -144.0(0.5) -145.5(1.2) -151 -162
+49846∗∗0.307 -154 -142 -158.1(2.0) -154 -147.7(0.9) -151.8(0.7) - 152.5(0.9) -154 -169
+49927∗∗0.456 -157 -147 -162.6(1.4) -161 -150.5(1.8) -151.9(0.4) -148.2(1.2) -153.0(1.2) -155
+49928∗∗0.456 -146 -138 -154 -156 -143 -143 -141.0(0.3) -145 -161
+49928∗∗0.456 -152 -161 -138.5(1.1) -161
+49975∗∗0.541 -154 -137 -160.5(0.4) -160 -153.0(1.8) -154.9(3.4) - 150.9(0.5) -152.8(2.6) -159
+50016∗∗0.618 -151 -150 -157.6(1.5) -160 -151.8(1.4) -151.1(1.3) - 150.3(0.7) -152 -170
+50064∗∗0.705 -140 -128 -143.1 -143 -134.7(0.3) -134.2(4.5) -131.4 (0.9) -138.4(1.2) -155
+50128∗∗0.823 -147 -151 -145 -135 -136 -130.8(0.9) -133 -147
+52756∗0.608 -148.2(0.1) -148.4 -138.9(0.6) -141.8(1.0) -138.4( 1.5) -134.9(0.6) -151
+52756∗0.608 -150.0(0.1) -150.0 -140.1(0.9) -143.4(0.6) -140.1( 1.6) -136.1(0.6) -152
+For saturated lines the radial velocities (itallic) were ob tained by fitting their wings.
+1-σerrors of the mean values are in brackets.
+∗– HST STIS spectrum.
+∗∗– spectra taken during outbursts.
+negative, decreasing between the phases 0.749 and 0.825
+from -6.5 to -16.6 km s−1. In this way N vappears to be
+more redshifted with respect to the intercombination lines
+than Civ. The Civredshift relative to the intercombina-
+tion lines, deduced from the HST/STIS spectra, decreases
+with the ionization potential of the corresponding ions, be-
+ing much less for O iv] than for O iii] and Niv]. The He ii
+1640 ˚A line appears to be for all but the first observation
+strongly redshifted with respect to the systemic radial ve-
+locity of 12.4 or 12.9 km s−1, but its radial velocity may
+be compatible with that of the compact component at the
+phases of the observations, if this component is much less
+massive than the cool one. However the radial velocity vari-ations of He iiand the intercombination lines, deduced from
+the HST/STIS spectra, would appear to be in phase with
+the orbital radial velocity variations of the cool one.
+The AX Per spectra are all for phases very close to con-
+junction. The C iv, Nvand Siivlines have similar radial
+velocities, which are redshifted with respect to the inter-
+combination lines, including O iv.
+A few line profiles of Z And and RW Hya, obtained
+from the spectra, are shown in figs. 10 and 11. They will be
+considered in the discussion of our results in order to better
+understand the reasons for the radial velocity shifts.6 Friedjung et al.: Symbiotic line shifts
+ 0 15 30 45
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ CIV-CIII]
+ 0 15 30 45
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ CIV-OIII]
+ 0 15 30 45
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+φCIV-NIV]
+Fig.3. Intercombination line radial velocity differences for
+SY Mus. The dashed line shows the orbital velocity of the
+cool giant. The symbols are explained in the text.
+4. Discussion
+Let it first be emphasized, that the differences between the
+Civmean resonance line radial velocity and the means of
+other resonance doublets, are generally much smaller than
+the difference between the C ivresonance line mean and
+the intercombonation line means in many orbital phases
+for SY Mus, Z And and AX Per. This clearly indicates that
+any explanation of the relative shift between the resonance
+and the intercombination line radial velocities, which is not
+the same for the different resonance line doublets, cannot
+work. The shift is in many orbital phases much larger than
+the 1σerrors given in the tables, which shows in addition
+that shifts between either the different resonance or the
+intercombination lines of the same ion are small.
+We can immediately draw the conclusion that it may be
+difficult to explain the relative radial velocity shift between
+the optically thick resonance lines and the intercombination
+lines by a resonance line radiative transfer effect, because
+the different optical thicknesses of the resonance line might
+be expected at first sight to produce different shifts.
+Line radial velocities can be influenced by various ef-
+fects. We need to first examine such effects, before dis--40-20 0 20 40
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ OIV]
+-40-25-10 5 20
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+φHeII
+Fig.4. Intercombination O iv]and Heiiradial velocities
+for Z And. The dashed line gives the orbital velocity of the
+cool giant. The symbols are expained in the text.
+cussing what is observed. Among the former we shall in
+the first three subsections consider the following.
+4.1. Effects due to the presence of interstellar lines on shift ed
+resonance line
+Interstellar absorption components can be superposed on
+the resonance emission lines and can distort the line pro-
+files. Such an effect should not be present for the symbiotic
+systems with large systemic velocities (-116.5 and -148.3
+km s−1of AX Per and AG Dra respectively). In the case
+of the others we can look for narrow at least partly inter-
+stellar absorption lines in stars, which are close in the sky,
+but which do not have smaller linear distances from the
+observer. Let us note that this test for the effect of inter-
+stellar absorption is not completely watertight. Radiation
+from the hot component of a symbiotic binary could ionize
+the nearby interstellar medium, so producing interstellar
+absorption from highly ionized atoms, without such lines
+being seen in the spectra of other stars in similar lines of
+sight.
+The estimated distance of SY Mus is 850 pc according to
+Schmutz et al (1994), while TU Mus, a βLyr type eclipsing
+variable with a O star component, has a distance of 2.1 kpc
+according to Penny et al (2008) and is only 0.3oin the sky
+from SY Mus. The stronger N v1238 ˚A doublet resonance
+line absorption of TU Mus is weak as is also the stronger
+Siiv1393 ˚A doublet resonance line. The shifts of N vand
+Siivrelative to C ivof SY Mus are both near zero however,
+suggesting hardly any effect of interstellar absorption in the
+direction of TU Mus and SY Mus.
+High resolution HST/STIS spectra are available for RW
+Hya and we can directly inspect the spectra for the pres-Friedjung et al.: Symbiotic line shifts 7
+-20 0 20 40 60
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ CIV-OIV]
+-20 0 20 40
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ CIV-NV
+-20 0 20 40 60
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+φCIV-HeII
+Fig.5. Intercombination O iv]as well as N vand Heiira-
+dial velocity differences for Z And. The dashed line shows
+the orbital velocity of the cool giant. The symbols are ex-
+plained in the text.
+ence of narrow absorption lines. In fact extremely narrow
+absorption lines are seen, superposed on N v1242 ˚A emis-
+sion as well as in other places, though they may rather be
+circumstellar (see discussion below on the iron forest).
+Z And has a distance of 1.5 kpc according to
+Miko/suppress lajewska and Kenyon (1996), while HD218195, sep-
+arated 10.3oin the sky, has a distance of 2.9 kpc (2001).
+There is neither any sign of N vabsorption nor of that of
+Civwithin the noise. In any case the relative shifts of N v,
+Civand Siivin the Z And spectra, which might be ex-
+pected not to be affected in the same way by interstellar
+absorption, are usually small at many orbital phases com-
+pared with the shift between C ivand the intercombination
+lines.
+4.2. Effects due to the absorption component of a P Cygni
+profile of shifted resonance lines
+It should first be noted that any explanation involving a P
+Cygni absorption component of a line at a shorter wave-
+length than the line emission studied has a difficulty when-20 0 20 40 60
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ CIV-CIII]
+-20 0 20 40 60
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ CIV-OIII]
+-20 0 20 40 60
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+φCIV-NIV]
+Fig.6. Intercombination line radial velocity differences of
+less ionized atoms for Z And. The dashed line shows the
+orbital velocity of the cool giant. The symbols are explained
+in the text.
+the continuum is weak. In that case, line absorption can
+however still be produced by an overlying layer with the
+same radial velocity as regions from which part of the line
+emission comes. It is this kind of explanation, which we
+shall use to understand many observations of the resonance
+lines. In this connection, we can mention that Young et al
+(2005) suggest that the O viP Cygni profile of AG Dra is
+affected by absorption of a false continuum, that is by a
+continuum enhanced by the presence of electron scattering
+wings of the line. However the 1/e width of electron scat-
+tering wings is 550 km s−1at 10 000 K for one scattering. If
+much narrower line emission is only seen, with no evidence
+of wings with at least this width, such an explanation will
+not work for any geometry of an electron scattering region
+relative to that of line formation. In particular in the case
+of CI Cyg, previously studied by Miko/suppress lajewska, Friedjung
+and Quiroga (2006), there is no evidence of broad wings
+for the C ivresonance lines, extending to much more than
+100 km s−1in the HST/GHRS spectrum (see Fig. 1 of that
+paper).8 Friedjung et al.: Symbiotic line shifts
+-180-160-140-120-100
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ OIV]
+-200-180-160-140-120-100
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+φHeII
+Fig.7. Oiv]intercombination line and He iiradial veloci-
+ties for AG Dra. The dashed line shows the orbital velocity
+of the cool giant. The symbols are explained in the text
+4.3. Analysis of iron forest absorption and plausible line sh ifts
+of lines producing pumping by accidental resonance
+(PAR)
+The importance of iron forest absorption by the low ve-
+locity wind of the cool component, leading to pumping by
+accidental resonance (PAR), needs to be checked. The ef-
+fect of the iron forest is well known since the work of Shore
+and Aufdenberg (1993). In principle such an effect could
+be quite large, though in fact we shall find usually no large
+influence of this kind of effect on the radial velocities of
+the lines studied by us. A major problem of the Shore
+and Aufdenberg calculations is moreover that the oscilla-
+tor strengths, used for the lines, are not always reliable.
+This is for instance the case for the 4s-4p∗transitions in-
+volving high parent terms (sometimes refered to as 4s∗and
+4p∗transitions) superposed on short wavelength region IUE
+emission lines. We can in such a way explain why for in-
+stance lines excited or “pumped” as a result of absorption
+of emission by the C iv1548 ˚A line are generally observed
+unlike those from the C iv1550 ˚A line (Erikson, Johansson
+and Wahlgren 2006).
+We can, in order to test for effects of the iron forest,
+firstly look for the presence of emission lines, emitted in
+the wind of the cool component by levels “pumped” as
+a result of absorption of the emission of lines, studied in
+this paper. The amount of pumping clearly depends on the
+widths of the lines which can pump. Pumping is summa-
+rized for different symbiotic binaries by Eriksson, Johansson
+and Wahlgren (2006).
+In the case of Z And, C iv1548 ˚A pumps two channels,
+while Siiv1393 ˚A probably pumps another channel. The
+3d7a4F9/2- 3d6(3G) 4p y4H11/2channel pumps 10 ob-
+served Fe iiemission lines at 2772, 2493, 2481, 2459, 2436,-30-15 0 15 30
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ CIV-OIV]
+-30-15 0 15 30
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+  CIV-NV
+-30-15 0 15 30 45Vrad   [ km s-1 ]
+ CIV-HeII
+-160-140
+-0.5  0  0.5  1  1.5
+φ
+Fig.8. Intercombination line O iv], Nv, Heiiradial veloc-
+ity differences with the orbital radial velocity of the cool
+giant for AG Dra. The symbols are explained in the text.
+2228, 2220, 2211, 2168 and 1975 ˚A. The 3d7a4P1/2- 4p
+w2D3/2channel, pumped by the same C ivline produces
+the observed 2979.95, 2483, 2479.98 and 1965 ˚A lines. The
+3 Feiilines probably pumped by Si ivthrough the 3d6
+(5D) 4s a6D7/2- 3d6(1G) 4p x2H9/2channel are at 2588,
+25492 and 1793 ˚A. Correcting the fluxes for the radiation
+absorbed from the pumping lines, before being re-emitted
+in the pumped lines, gives nearly ”optically thin” flux ra-
+tios near 2 for the C ivand Siivdoublets, except for the
+Civdoublet near phase zero in outburst, when probably
+blueshifted P Cygni absorption (see Fig. 10) is visible. In
+addition pumping by the O iii] intercombination line at
+1660 ˚A may be responsable for weak emission features in
+the spectrum of Z And at 2784.5 and 2728.2 ˚A . HoweverFriedjung et al.: Symbiotic line shifts 9
+-40-20 0 20 40
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ CIV-CIII]
+-20 0 20 40 60
+-0.5  0  0.5  1  1.5Vrad   [ km s-1 ]
+ CIV-OIII]
+-15 0 15 30 45Vrad   [ km s-1 ]
+ CIV-NIV]
+-160-140
+-0.5  0  0.5  1  1.5
+φ
+Fig.9. Intercombination line radial velocity differences of
+less ionized atoms with the orbital radial velocity of the
+cool giant for AG Dra. The symbols are explained in the
+text.
+we do not see any systematic velocity shift of the O iii] line
+with respect to the other intercombination lines.
+Pumping by the first 1548 ˚A Civline channel produced
+4 observed Fe iiemission lines in the spectrum of SY Mus.
+The Civflux ratio, correcting for pumping again gives an
+”optically thin” value of 2.
+Obtaining a clear result is more difficult for AG Dra.
+Emission lines, which could be produced by pumping, are
+also observed sometimes in absorption. However in a pri-
+vate communication P.R. Young states that he sees lines
+pumped by C iv, Heii1640 ˚A and 1085 ˚A, N IV] 1486 ˚A
+and O III] 1660 ˚A. We have however seen no sign of Fe ii
+lines pumped by two possible He ii1640 ˚A channels for AG
+Dra so any effect of pumping on the He iiprofile is small. Inany case, let us note that pumping may be less important
+for this metal underabundant symbiotic system.
+The Civ1548 ˚A line pumps the Fe iiy4H11/2chan-
+nel in the spectrum of RW Hya, but lines pumped by this
+mechanism are not observed in the spectrum of AX Per,
+suggesting no strong iron forest absorption for the latter
+symbiotic. We must in the former RW Hya case note in
+addition, that the HST/STIS spectra show strong narrow
+absorption components, superposed especially on the N v
+1242 ˚A profile and on the last date on the Si ivprofile.
+In any case we can note, that the already mentioned
+small 1σvalues for the radial velocities of the two lines of
+the same resonance doublet and also of the intercombina-
+tion lines of the same ion as well as the usually small values
+of the C ivradial velocity mean - mean resonance doublet
+radial velocities of other ions, would be hard to explain, if
+effects on radial velocities of the iron forest, which should
+not be the same for differnt lines, were important.
+4.4. Clues about the nature of the observed radial velocity
+shifts for different systems
+More clues concerning the nature of the shifts can be ob-
+tained from the time variations and line profiles, for which
+we have information especialy from the observations of Z
+And and RW Hya.
+The periodic variation of the Z And C ivresonance dou-
+blet -intercombination line radial velocity means suggests
+a maximum at phases not long after conjunctions when
+the cool component is nearer the observer. This might
+be understood if the increased redshift of Z And at such
+phases is due to larger cool component wind line absorp-
+tion on the short wavelength side of resonance line emis-
+sion for strongly ionised atoms, with perhaps additional
+effects behind the cool giant at that phase. Among such
+effects there is wind focussing towards the orbital plane,
+with a three dimensional spiral stream, which occurs when
+the cool component does not quite fill its Roche lobe
+(Gawryszczak, Miko/suppress lajewska, R´ o˙ zyczka 2003). This inter-
+pretation is not contradicted by the C ivand Nvfluxes
+found by Fernandez-Castro et al (1988) including a mini-
+mum near phase zero, confirmed by later more numerous
+observations. However, Fernandez-Castro et al (1988) also
+find simlar minima for the intercombnation lines, so some-
+thing else than line absorption is also involved.
+There has been a certain amount of disagreement and
+uncertainty about the causes of the outbursts of Z And
+and similar symbiotic systems. Our ultraviolet line study
+may be relevant to this question. According to Sokoloski et
+al (2006), based on later observations after the end of the
+life of IUE, outbursts of Z And can be understood as disk
+instabilites, which in the case of major outbursts are fol-
+lowed by thermonuclear burning. The V magnitude of one
+outburst, studied by these authors, became brighter than
+about 9.7, when according to them thermonuclear burning
+started. Bisikalo et al (2006) also suggested an increase in
+thermo-nuclear burning during outbursts as well as effects
+of colliding winds. They made fairly detailed calculations.
+Fig 10 shows the IUE profiles of the C ivresonance dou-
+blet of Z And at three epochs. The first in the upper panel
+is at phase 0.961 during a major outburst, when according
+to Sokoloski et al (2006) thermonuclear burning could have
+occured. On this date the line profiles were faint and com-
+plex so we do not attempt to give radial velocities in table10 Friedjung et al.: Symbiotic line shifts
+Fig.10. Civdoublet profiles of Z And for 3 dates. The
+upper panel is for phase 0.960 during a major outburst, the
+middle panel for phase 0.178 at a fainter stage of outburst
+and the lower panel for phase 0.523 in quiescence.
+2. The middle panel shows the profiles at phase 0.171, in a
+fainter stage of outburst, while the bottom panel shows the
+profiles at phase 0.523 in quiescence. The greater width of
+lines in outburst, previously detected by Fernandez-Castro
+et al (1995), is clearly seen even through the presence of
+very weak wings in the fainter stage of outburst and is gi-
+gantic during what could be a thermo-nuclear burning stage
+according to Sokoloski et al. Fig. 10 shows particularly a
+large quantity of redshifted emission. However we can note
+in addition to faint red wings, the presence of what looks
+like blue shifted absorption superposed on weak line emis-
+sion in the fainter stage of the outburst. Fernandez-Castro
+et al (1995) suggested that envelopes were ejected during
+outburst; another interpretation is discussed below.
+We may try to understand the changing profiles, shown
+in Fig. 10, as due to a decrease in ionising radiation near
+the maximum of outburst, corrsponding to a lower photo-
+spheric temperature of an expanded white dwarf. In addi-
+tion we expect the optical thickness of the C ivresonance
+lines to be probably very large; because taking an elec-
+tron density of the order of 1010cm−3from Altamore et
+al (1981) and Fernandez-Castro et al (1988), a solar abun-
+dance of carbon, and assuming 10 percent of carbon three
+times ionised, would lead to the weaker line becoming op-
+tically thick for distances of only 2 109cm, which is small
+compared wth the binary separation times the sine of the
+inclination of 7 1012cm of Mikolajewwska (2003). The
+fractional abundance of carbon in the form of C+++used,
+is of the order of that calcuated for different conditions by
+Nussbaumer (1982). Therefore the enlargement of what is
+still visible of the C ivion doublet lines towards the red, in
+very probable conditions of large optical thicknesses near
+the cool component at that time, might be due to radiative
+transfer in an expanding medium such as the cool compo-
+nent wind or a region of collision between the winds from
+the components rather than an optically thinner ejected
+shell. Iron forest absorption can be expected to also playa role in the profile. An explanation only involving line
+emission due to the wind from the hot component is un-
+likely, because the escape velocity from the photosphere of
+the outbursting component would for a largest measured
+radius of 0.36 solar radii and a white dwarf mass of 0.65
+solar masses (taking the estimates of Sokoloski et al 2006)
+be then around 830 km s−1. A wind from that component
+might be expected to be probably faster, while in outburst
+the red sides of those resonance line profiles, which are un-
+affected by any classical P Cygni absorption, do not extend
+to more than about 300 km s−1.
+We may note in this connection that Lamers et al
+(1995), in their study of wind terminal velocities of hot
+stars near the main sequence, found ratios of the terminal
+to escape velocity, decreasing from around 2.7 at an effec-
+tive temperature of more than 40 000K and highly ionized
+winds to 0.7 at an effective temperature of 8000K and much
+less ionized winds, which produce P Cygni spectral line pro-
+files of C ii, Aliiiand Mgii. In addition Dumm et al (2000)
+state that the terminal velocities of central stars of plane-
+tary nebulae are typically 2.5 times the escape velocity. Let
+it be noted however that the hot component wind could
+be concentrated towards the polar axis by the white dwarf
+magnetic field and/or the accretion disk, so the observed ra-
+dial velocity would be then the above expected value times
+the cosine of the inclination and so smaller.
+We may also note that Sokoloski et al (2006) observed
+P Cygni profiles for Z And in outburst in the far ultravio-
+let, using the FUSE satellite. An enhanced continuum was
+seen by them during outburst, classical P Cygni blueshifted
+absorption being strong at phases 0.116 to 0.155, when the
+continuum was also strong. P v1117 ˚A absorption, indicat-
+ing the presence of a not very fast wind, is clearly visible
+to about -270 km s−1at phase 0.116 in their Fig. 6.
+We can compare our AG Dra results with results given
+in previous papers for that system. Let us note that Viotti
+et al (1984) found that the N vlines were broader when
+the luminosity of AG Dra was higher like for Z And. These
+authors found in addition P Cygni profiles for all epochs
+of Nv1238 ˚A with a terminal velocity of 170 km s−1for
+the stellar wind supposed to produce it and line assymme-
+try suggesting P Cygni absorption of line emission, which
+was also present when the continuum was weak. Young et
+al (2005) observed AG Dra with FUSE and saw a redshift
+of the O viresonance doublet (which had a P Cygni pro-
+file) relative to the intercombination Ne vand Nevilines.
+However their measurements indicated no redshift for the
+Sivand Svizero energy lines. These last mentioned au-
+thors suggest that the O viP Cygni absorption is only that
+of the resonance doublet, superposed on a false continuum,
+produced by electron scattering wings. We may finally note
+that the geometry of this system with a fairly small pre-
+ferred orbital inclintion of 30-45oaccording to Miko/suppress lajewska
+et al (1995) and a low metal abundance, may play a role in
+reducing shifts between the resonance and intercombination
+lines.
+We shall here just emphasize one of the results for SY
+Mus. The constancy of the C ivradial velocity mean at dif-
+ferent orbital phases reminds us of the resonance line mean
+constant velocity, found by Miko/suppress lajewska, Friedjung and
+Quiroga (2006) for CI Cyg, which like SY Mus is a high
+inclination eclipsing system. Unlike in the case of CI Cyg,
+we do not have very high resolution HST spectra for SY
+Mus, but we may be able to invoke a similar explanationFriedjung et al.: Symbiotic line shifts 11
+Fig.11. HST/STIS profiles for RW Hya on MJD 52040
+(phase 0.825). The full line shows the C iv1551 ˚A profile,
+the dotted line the N iv] 1486 ˚A profile and the dashed line
+the Oiv1401 ˚A profile
+to that given by Miko/suppress lajewska, Friedjung and Quiroga, ex-
+plaing the redshift of the resonance doublet emission lines
+by the presence of a circum-binary region. Such a region
+could be near the plane of the orbit.
+As far as RW Hya is concerned, the observations avail-
+able, do not enable us to say anything certain about
+the variation of redshift with orbital phase. According to
+Dumm et al (2000 low resolution UV spectra indicate a
+flux decrease in the continuum between 1250 ˚A and 1290
+˚A at phase 0.78 relative to phase 0.71, which was almost
+over at phase 0.81. They interpreted this result as due to
+additional apparent extinction byRayleigh scattering of
+neutral hydrogen and due to iron forest absorption in an
+accretion wake produced by wind accretion. Our exami-
+nation indicates no large variation in the values of C iv-
+intercombination line radial velocity between phases in the
+range 0.749 to 0.825 on the high resolution HST/STIS spec-
+tra. In any case, there is no accretion wake, if accretion is
+by Roche lobe overflow, as suggested by the observations
+of ellipsoidal light variability in the near infrared accord-
+ing to Rutkowski, Miko/suppress lajewska and Whitelock (2007) and
+another explanation is required for the observations of ul-
+traviolet fluxes in such a case than that of Dumm et al
+(2000). Another possibility is absorption due to impact of
+the stream. In addition Dumm et al (1999) suggest that
+the density distribution around the M giant of SY Mus
+is assymetric with additional extinction than that due to
+Rayleigh scattering, which they suggest is produced by the
+iron forest.
+Unlike most of the spectra studied here, those of RW
+Hya did have a significant continuum, with P Cygni ab-
+sorption components superposed on it, which could have
+contributed to some measured velocities of the N vand Siiv
+lines. The edge velocities relative to the systemic velocity
+are lower than 200 km−1. Let us note also the presence
+of broad wings of more than 20 ˚A wide around the C iv
+resonance lines. It remains to be seen whether that can be
+explained by electron scattering.The high resolution HST/STIS spectra of RW Hya can
+be used to look at the line profiles, in order to better under-
+stand the radial velocity shifts. Fig. 11 shows profiles of the
+Civ1551 ˚A , Niv1586 ˚A and O iv] 1401 ˚A lines. The weak
+continuum fluxes of less than 2% the line centre fluxes have
+been subtracted, the profiles being insensitive to the contin-
+uum and unaffected by the iron forest absorption. The C iv
+profile was divided by 3 and the O iv] profile by 1.7, so as
+to make their red wings coincide approximately. One sees
+that the blue wings of the C ivand Oiv] lines are reduced
+with respect to N iv]. The Civline is optically thick, so the
+reduction appears to be due to P Cygni absorption of line
+emission, starting at radial velocities which are positive rel-
+ative to the systemic velocity of 12.4 or 12.9 km s−1(both
+values being given as alternatives by Belczynski et al (2000)
+and Mikolajeswska (2003)) and a cool giant radial velocity
+7.8 km s−1smaller than those values at phase 0.825. This
+could suggest the presence of a line absorbing wind from the
+cool component absorbing some of of the line emission com-
+ing from regions rotating not very rapidly around the white
+dwarf with perhaps a contribution to the absorption at cer-
+tain orbital phases by a stream from the inner Lagrangian
+point. The latter is possible for high inclination eclipsing
+systems like RW Hya if the cool component fills its Roche
+lobe (Rutkowski, Mikolajewska and Whitelock (2007). Let
+us note that the wind from the cool component can be de-
+viated by the gravitational field of the compact component.
+The Oiv] line is optically thin, so no explanation, invovlv-
+ing absorption can work for this line. The effect may be ex-
+plainable by a large part of the intercombination emission
+line flux coming from the already mentioned wind from the
+cool component in front of an accretion disk, with the latter
+being opticaly thick in the continuum, so occulting emis-
+sion behind it, plus a possible contribution to line emission
+due to the presence of the already mentioned stream from
+the inner Lagrangian point in front of the disk. Such effects
+might produce other emission line shifts. Let us finally note
+that it is not quite clear to what extent the small absolute
+redshift of the N iv] line centre relative to the systemic and
+cool giant velocities is real. In any case the very incomplete
+phasing of the HST/STIS spectra makes it hard to draw
+more conclusions.
+5. Conclusions
+A number of new results have been obtained, from a more
+detailed analysis of relative emission line shifts, which con-
+clude the study of suitable ultraviolet spectra, which are
+available.
+The relative shift between the radial velocities of the
+resonance lines and the intercombination lines of multiply
+ionised atoms is confirmed for most systems and appears
+to be not due to various artifacts. It is rather probably
+due to the absorption component of P Cygni profiles of
+the cool component’s wind, which must in many cases be
+able to absorb emission from the emission part of line pro-
+files and not only radiation from the continuous spectrum.
+That places strong constraints on the geometry, which we
+might try to interpret as for instance involving absorption
+of line radiation from near the outer edge of an accretion
+disk. The variation of the shifts with orbital phase, espe-
+cialy for Z And, can be understood as due to a greater
+optical thickness of regions connected with the cool com-
+ponent’s wind on its far side with respect to the compact12 Friedjung et al.: Symbiotic line shifts
+component. However the wide redshifted profile of C ivof
+Z And near phase zero during outburst may still be due to
+a radiative transfer effect.
+Ionisation potential dependent stratification of radial
+velocity is present for the intercombination lines, O iv]be-
+ing much more redshifted than intercombination lines of
+less ionised atoms. This could be connected with occulta-
+tion by an accretion disk, which is optically thick in the
+continuum. However let us note that our data do not en-
+able us to be certain about any difference for Z And and
+AG Dra in outburst, if a comparison is made with quies-
+cence We might expect that as a change in the properties
+of an accretion disk might be expected.
+Differences can be due to different geometries and metal
+abundances. AG Dra in particular has a strong metal un-
+derabundance. Streams and cool component winds affected
+by the presence of the hot component, may play a major
+role. However the sample of symbiotic systems with high
+spectral resolution ultraviolet observations at many differ-
+ent orbital phases is very small, making it difficult to look
+for correlations with the properties of these systems. It is
+therefore probably not convenient to make more detailed
+speculations at the present time.
+Acknowledgements. This research has been partly supported by
+Polish research grants 1P03D 017 27 and N203 395539, and by th e
+European Associated Laboratory “Astrophysics-Poland-Fr ance”. It
+also made use of the NASA Astrophysics Data System and SIMBAD
+database. We thank Peter Young for giving information on lin e pro-
+files and other possible channels of possible pumping of exci ted Fe
+II levels in AG Dra. J. Zorec must also be thanked for supplyin g a
+computer programme, while a friend helped in the preparatio n of one
+figure.
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diff --git a/1001.0311.txt b/1001.0311.txt
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+arXiv:1001.0311v2  [quant-ph]  13 Mar 2010Exact solutions of a particle in a box with a delta function
+potential: The factorization method
+Pouria Pedram∗
+Plasma Physics Research Center, Science and Research Campu s,
+Islamic Azad University, P.O. Box 14665–678, Tehran, Iran
+M. Vahabi
+Department of Physics, Shahid Beheshti University, G. C., E vin, Tehran 19839, Iran
+Abstract
+We use the factorization method to find the exact eigenvalues and eigenfunctions for a particle
+in a box with the delta function potential V(x) =λδ(x−x0). We show that the presence of the
+potential results in the discontinuity of the correspondin g ladder operators. The presence of the
+delta function potential allows us to obtain the full spectr um in the first step of the factorization
+procedure even in the weak coupling limit λ→0.
+1I. INTRODUCTION
+The time-independent form of Schr¨ odinger’s equation in the prese nce of a potential V
+can be written as1
+−¯h2
+2m∇2ψ+Vψ=Eψ, (1)
+whereψis the wave function, mis the mass of the particle, and Eis the eigenenergy.
+Equation (1) can be solved exactly only for a few potentials. A partic le in a box and a
+particle in a delta function potential are two well-known and instruct ive examples.1The
+former can be used to describe quantum dots and quantum wells at lo w temperatures,2,3
+and the latter can be used as a model for atoms and molecules.4
+The solution for a particle in a box with a delta function potential has b een investigated
+using a perturbative expansion in the strength of the delta functio n potential λ.5Exact
+solutions have been obtained for the weak ( λ→0) and the strong (1 /λ→0) coupling
+limits.6
+In this paper we discuss the solution for a particle in a box with a delta f unction potential
+using the factorization method and show that the presence of the delta function simplifies
+the factorization procedure. In this way we find the full spectrum of the Hamiltonian in
+the first step of the factorization method. We also show that this r esult applies in the weak
+coupling limit λ→0. Note that if we put λ= 0 from the beginning, we need to continue
+the factorization procedure to find each eigenvalue in each step.
+II. PARTICLE IN A BOX WITH A DELTA FUNCTION POTENTIAL
+Consider a particle in a one-dimensional box of size awith the delta function potential
+V(x) =λδ(x−x0) =λδ(x−pa), where 0<p<1. In this case Eq. (1) takes the form
+−¯h2
+2md2ψn(x)
+dx2+λδ(x−pa)ψn(x) =Enψn(x), (2)
+whereψn(x) andEnare the corresponding eigenfunctions and eigenvalues, respectiv ely.
+Because of the boundary conditions, ψn(x) = 0 for x≤0 orx≥a, the form of the
+eigenfunctions inside the box is
+ψn(x) =
+
+Asin(knx) (0 ≤x≤pa)
+Bsin[kn(x−a)] (pa≤x≤a),(3)
+20.5 1 1.5 2 2.5 3x
+-1.5-1-0.50.511.5Ψ/LParen1x/RParen1
+Ψ1
+Ψ2Ψ3
+FIG. 1: First three stationary states ψn(x) for 2mλ/¯h2= 8,a= 3,A= 1, andp= 1/2.
+wherekn=√2mEn/¯h. The continuity condition of the wave function at x=pagives
+A/B= sin[(p−1)kna]/sin(pkna). Because the delta function is infinite at x=pa, the
+first derivative of the wave function is not continuous and the relat ion between the left and
+right derivatives of the wave function can be obtained by integratin g Eq. (2) over the small
+interval (x0−ǫ,x0+ǫ)
+dψn(x)
+dx|pa+ǫ−dψn(x)
+dx|pa−ǫ=2m
+¯h2/integraldisplaypa+ǫ
+pa−ǫV(x)ψn(x)dx (4a)
+=2mλ
+¯h2ψn(pa). (4b)
+We substitute the eigenfunctions in Eq. (3) in Eq. (4b) and obtain th e quantization
+condition5,6
+knsin(kna) =2mλ
+¯h2sin(pkna)sin[(p−1)kna]. (5)
+The solutions to Eq. (5) give the energy spectrum of the Hamiltonian ,En= ¯h2k2
+n/2m.
+Figure 1 shows the first three stationary states for 2 mλ/¯h2= 8,a= 3,A= 1, andp= 1/2.
+III. THE FACTORIZATION METHOD
+To calculate the eigenvalues and eigenfunctions of a Hamiltonian oper atorH, we can
+use a general operational procedure called the factorization met hod. In this method the
+Hamiltonian is written as the product of two ladder operators plus a c onstant,H=a†a+E.
+These operators are used to obtain the eigenfunctions of the Ham iltonian. In contrast to
+the simple harmonic oscillator, one ladder operator is usually not suffic ient to form all the
+Hamiltonian’s eigenfunctions and another ladder operator is needed for each eigenfunction.
+3ThismethodwasfirstintroducedbySchr¨ odinger7–9andDirac10andwasfurtherdeveloped
+by Infeld and Hull11and Green.12The spirit of the factorization method is to write the
+second-order differential operator Has the product of two first-order differential operators
+aanda†, plus a real constant E. The form of these operators depends on the form of the
+potentialV(x) and the factorization energy.
+The procedure for finding the ladder operators and the eigenfunc tions is as follows.13We
+find operators a1,a2,a3,...and real constants E1,E2,E3,...from the recursive relations13
+a†
+1a1+E1=H, (6a)
+a†
+2a2+E2=a1a†
+1+E1, (6b)
+a†
+3a3+E3=a2a†
+2+E2,... . (6c)
+More generally
+a†
+n+1an+1+En+1=ana†
+n+En,(n= 1,2,...). (7)
+Also assume that there exists a null eigenfunction (root function) |ξn/an}b∇acket∇i}htwith zero eigenvalue
+for eachan, namely,
+an|ξn/an}b∇acket∇i}ht= 0. (8)
+Hence,Enis thenth eigenvalue of the Hamiltonian with the corresponding eigenfunctio n13
+(up to a normalization coefficient)
+|En/an}b∇acket∇i}ht=a†
+1a†
+2...a†
+n−1|ξn/an}b∇acket∇i}ht. (9)
+Figure 2 shows a schematic diagram describing the relation between r oot functions, the
+stationary states, and the ladder operators. As can be seen fro m Eq. (6),a†
+1a1is equal toH
+except for a constant. Thus, a1must have a linear momentum term to be consistent with
+the kinetic energy part of the Hamiltonian. According to Eq. (7), ea ch of the annihilation
+operatorsanshould also have a a linear momentum term. Thus, ancan be written as
+an=1√
+2m[P+ifn(x)], (10)
+wherePis the momentum operator and fn(x) is a real function of x. Although these
+operators are not hermitian ( a†
+n=1√
+2m(P−ifn(x))/ne}ationslash=an), their product is hermitian
+a†
+nan=1
+2mP2+1
+2mf2
+n+¯h
+2mdfn
+dx. (11)
+4/1/2/1/3/1/4
+/5/2
+/5/3
+/5/4/1/6
+/5/6/7/8/7/8/7/8/7
+/7/8/7/8/7/8/7
+/7/8/7/8/7/8/7
+/7/8/7/8/7/8/7/7/8/7/8/7/8/7/9/10
+/6
+/9/10
+/4/11/6
+/9/10
+/4/9/10
+/6/9/10
+/6
+/9/10
+/2
+/9/2/9/6
+/9/3
+/9/4
+FIG. 2: A schematic diagram describing the relation between the root functions ξn, the stationary
+states Ψ n, and the ladder operators an,a†
+n.
+We are now ready to find the ladder operators and eigenenergies fo r our problem. First,
+we consider Eq. (6) for n= 1
+a†
+1a1+E1=H. (12)
+From the form of the Hamiltonian (2) and Eq. (11) we can rewrite Eq. (12) as
+P2
+2m+1
+2mf2
+1+¯h
+2mdf1
+dx+E1=P2
+2m+λδ(x−pa), (13)
+or equivalently
+1
+2mf2
+1+¯h
+2mdf1
+dx+E1=λδ(x−pa). (14)
+Note that in contrast to Eq. (2), Eq. (14) is a nonlinear first-orde r differential equation.
+To solve Eq. (14), we consider the left-hand and right-hand sides o f the delta function
+potential separately. For these regions, Eq. (14) reduces to
+1
+2mf2
+1+¯h
+2mdf1
+dx+E1= 0 (x/ne}ationslash=pa), (15)
+which has the solution
+f1(x) =/radicalbig
+2mE1cot[√2mE1
+¯h(x−b)], (16)
+5wherebis a constant of integration. We require that f1(x) be finite in the range 0 <x<a,
+where it is the solution for our problem. Because the singularities of t he cotangent function
+areπradian apart, we choose the points x= 0 andx=aas the singularity points of f1(x).
+At these points thepotential andhence the Hamiltonianareinfinite a ndf1(x) canbeinfinite
+at the boundaries. Hence, we have
+f1(x) =
+
+√2mE1cot[√2mE1
+¯hx],(x<pa)
+√2mE1cot[√2mE1
+¯h(x−b)],(x>pa),(17)
+where (√2mE1/¯h)(a−b) =π. To fix the value of E1, we need to use the discontinuity
+relation of the ladder operators, which can be obtained by integrat ing Eq. (14) over a small
+interval (pa−ǫ,pa+ǫ)
+f1(pa+ǫ)−f1(pa−ǫ) =2mλ
+¯h. (18)
+Equation (18) shows that the presence of the delta function resu lts in the discontinuity of
+f1(x) atx=pa. We use Eq. (17) and b=a−π¯h√2mE1, to rewrite Eq. (18) as
+/radicalbig
+2mE1/braceleftbigg
+cot/bracketleftBig√2mE1
+¯h(pa+π¯h√2mE1−a)/bracketrightBig
+−cot/bracketleftBig√2mE1
+¯h(pa)/bracketrightBig/bracerightbigg
+=2mλ
+¯h.(19)
+Because cot( π+α) = cotα, Eq. (19) can be written in the form
+k1{cot[(p−1)k1a]−cot(pk1a)}=2mλ
+¯h2, (20)
+or
+k1sin(k1a) =2mλ
+¯h2sin(pk1a)sin[(p−1)k1a], (21)
+wherek1=√2mE1/¯h. Equation(20)issimilartoEq.(5). Notethat, althoughwecalculate d
+the ground state energy of the Hamiltonian in the first step of the f actorization method, we
+found the full spectrum. This conclusion applies to the weak coupling limitλ→0. In this
+limit sin(k1a) = 0 from Eq. (21), which gives the full spectrum of a particle in a box with
+k1=nπ/a. This result has an interesting consequence. If we use the factor ization method
+to calculate the energy levels of a particle in a box with λ= 0, we cannot find all of them
+in the first step of the procedure. In fact, we should use the recu rsive relations to find each
+eigenvalue in each step.13Thus, contrary to the Schr¨ odinger equation, the presence of t he
+delta function in the factorization method simplifies the problem and r epeated application
+of the recursion relations is not necessary. We need only to replace the index 1 with nin
+Eqs. (17) and (21).
+6To obtain the root functions, we rewrite Eq. (8) as
+/parenleftbigg¯h
+id
+dx+i¯hkncot(knx)/parenrightbigg
+ξn(x) = 0,(x<pa), (22a)
+/parenleftbigg¯h
+id
+dx+i¯hkncot[kn(x−b)]/parenrightbigg
+ξn(x) = 0,(x>pa), (22b)
+whereξn(x)≡ /an}b∇acketle{tx|ξn/an}b∇acket∇i}ht. It can be easily checked that the following is a solution to Eq. (22)
+ξn(x) =
+
+Asin(knx), (x<pa),
+−Bsin[kn(x−b)],(x>pa).(23)
+Because in the first step of the factorization procedure the root functionξ1(x) is equal
+to the first eigenfunction ψ1(x) (see Eq. (9) and Fig. 2) and we have replaced the index
+1 byn, all the root functions are equal to the eigenfunctions of the Ham iltonian, that is,
+ξn(x) =ψn(x). This result is similar to Eq. (3) for x<pa. Moreover, because knb=kna−π,
+we have sin[ kn(x−b)] = sin[kn(x−a)+π] =−sin[kn(x−a)]. So, these eigenfunctions are
+equivalent to the solutions in Eq. (3).
+IV. SUGGESTED PROBLEMS
+Problem 1. Usetherecursive relations(6)toshowthat |En/an}b∇acket∇i}htinEq.(9)istheeigenfunction
+of the Hamiltonian with Enas its eigenvalue.
+Problem 2. Show that the eigenvalues E1,E2,E3,...form a monotonic increasing se-
+quence, that is, E1≤E2≤E3≤....
+Problem 3. Apply the factorization method to a particle in a box with no delta func tion
+potential. In the first step show that k1=π/a. Continue the factorization procedure to
+find all the eigenenergies and eigenfunctions. Hint: consider an=1√
+2m[P+icncot(dnx)],
+wherecnanddnare real constants.
+∗Electronic address: pouria.pedram@gmail.com
+1D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, Upper Saddle River, NJ,
+2004).
+2B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors (Springer, New
+York, 1984), Chap. 1.
+73P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, New York, 2005), Chap.
+9.
+4I. R. Lapidus, “Relativistic one-dimensional hydrogen ato m,” Am. J. Phys. 51, 1036–1038
+(1983).
+5N. Bera, K. Bhattacharya, and J. K. Bhattacharjee, “Perturb ative and non-perturbative studies
+with the delta function potential,” Am. J. Phys. 76, 250–257 (2008).
+6Y. N. Joglekara, “Particle in a box with a δ-function potential: Strong and weak coupling
+limits,” Am. J. Phys. 77, 734–736 (2009).
+7E. Schr¨ odinger, “A method of determining quantum-mechani cal eigenvalues and eigenfunc-
+tions,” Proc. Roy. Irish Acad. Sect. A 46, 9–16 (1941).
+8E. Schr¨ odinger, “Further studies on solving eigenvalue pr oblems by factorization,” Proc. Roy.
+Irish Acad. Sect. A 46, 183–206 (1941).
+9E. Schr¨ odinger, “The factorization of the hypergeometric equation,” Proc. Roy. Irish Acad.
+Sect. A47, 53–54 (1941) or arXiv:physics/9910003.
+10P. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, Oxford, 1958).
+11I. Infeld and T. E. Hull, “The factorization method,” Rev. Mo d. Phys. 23, 21–68 (1951).
+12H. S. Green, Matrix Methods in Quantum Mechanics (Barnes & Noble, New York, 1968).
+13H. C. Ohanian, Principles of Quantum Mechanics (Prentice Hall, Englewood Cliffs, NJ, 1990).
+8
\ No newline at end of file
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+arXiv:1001.0312v2  [math.CO]  5 Sep 2010The Method of Combinatorial Telescoping
+William Y.C. Chen1, Qing-Hu Hou2and Lisa H. Sun3
+Center for Combinatorics, LPMC-TJKLC
+Nankai University
+Tianjin 300071, P.R. China
+1chen@nankai.edu.cn,2hou@nankai.edu.cn,3sunhui@nankai.edu.cn
+Abstract. We present a method for proving q-series identities by combinatorial telescoping,
+in the sense that one can transform a bijection or a classifica tion of combinatorial objects
+into a telescoping relation. We shall illustrate this metho d by giving a combinatorial proof
+of Watson’s identity which implies the Rogers-Ramanujan id entities.
+Keywords. Watson’s identity, Sylvester’s identity, Rogers-Ramanuj an identities, combina-
+torial telescoping
+AMS Subject Classification. 05A17; 11P83
+1 Introduction
+The main objective of this paper is to present the method of co mbinatorial telescoping for
+provingq-series identities. A benchmark of this approach is the clas sical identity of Watson
+which implies the Rogers-Ramanujan identities.
+There have been many combinatorial proofs of the Rogers-Ram anujan identities. Schur
+[13] provided an involution for the following identity whic h is equivalent to the first Rogers-
+Ramanujan identity:
+∞/productdisplay
+k=1(1−qk)/parenleftBigg
+1+∞/summationdisplay
+k=1qk2
+(1−q)(1−q2)···(1−qk)/parenrightBigg
+=∞/summationdisplay
+k=−∞(−1)kqk(5k−1)/2.
+Andrews[1]provedtheRogers-Ramanujanidentitiesbyintr oducingthenotionof k-partitions.
+Garsia and Milne [9] gave a bijection by using the involution principle. Bressoud and Zeil-
+berger [5,6] provided a different involution principle proof based on an algebraic proof due to
+Bressoud [4]. Boulet and Pak [3] found a combinatorial proof which relies on the symmetry
+properties of a generalization of Dyson’s rank.
+Let us consider a summation of the following form
+∞/summationdisplay
+k=0(−1)kf(k). (1.1)
+1Suppose that f(k) is a weighted count of a set Ak, that is,
+f(k) =/summationdisplay
+α∈Akw(α).
+Motivated by the idea of creative telescoping of Zeilberger [16], we aim to find sets Bkand
+Hkwith a weight assignment wsuch that there is a weight preserving bijection
+φk:Ak−→Bk∪Hk∪Hk+1, (1.2)
+where∪stands for disjoint union. Since φkandφk+1are weight preserving, both φ−1
+k(Hk+1)
+andφ−1
+k+1(Hk+1) have the same weight as Hk+1. Realizing that φ−1
+k(Hk+1)⊆Akand
+φ−1
+k+1(Hk+1)⊆Ak+1, they cancel each other in the sum (1.1). More precisely, if w e set
+g(k) =/summationdisplay
+α∈Bkw(α) andh(k) =/summationdisplay
+α∈Hkw(α),
+then the bijection (1.2) implies that
+f(k) =g(k)+h(k)+h(k+1). (1.3)
+To see that the above equation is indeed a telescoping relati on with respect to the sum (1.1),
+let
+f′(k) = (−1)kf(k), g′(k) = (−1)kg(k), h′(k) = (−1)kh(k).
+Thus we have
+f′(k) =g′(k)+h′(k)−h′(k+1). (1.4)
+Just like the conditions for the creative telescoping, we su ppose that H0=∅andHkvanishes
+for sufficiently large k. Summing (1.4) over k, we deduce the following relation
+∞/summationdisplay
+k=0(−1)kf(k) =∞/summationdisplay
+k=0(−1)kg(k), (1.5)
+which is often an identity we wish to establish.
+The above approach to proving an identity like (1.5) is calle d combinatorial telescoping.
+It can be seen that the bijections φklead to a correspondence between A=∞/uniontext
+k=0Akand
+B=∞/uniontext
+k=0Bkafter the cancelations of Hk’s. To be more specific, we can derive a bijection
+φ:A\∞/uniondisplay
+k=0φ−1
+k(Hk∪Hk+1)−→B
+and an involution
+ψ:∞/uniondisplay
+k=0φ−1
+k(Hk∪Hk+1)−→∞/uniondisplay
+k=0φ−1
+k(Hk∪Hk+1),
+2given byφ(α) =φk(α) ifα∈Akand
+ψ(α) =
+
+φ−1
+k−1φk(α),ifα∈φ−1
+k(Hk),
+φ−1
+k+1φk(α),ifα∈φ−1
+k(Hk+1).
+In the examples of this paper, the set Akis of the following form
+Ak=∞/uniondisplay
+n=0An,k.
+Fix an integer n, for any nonnegative integer k, we can establish a bijection φn,ksuch that
+the corresponding set Bn,kis related to An,k,An−1,k,...,A n−r,kfor an integer r. Let
+Fn,k=/summationdisplay
+α∈An,kw(α)
+be a weighted count of the set An,k, and let
+Fn=∞/summationdisplay
+k=0(−1)kFn,k.
+By (1.5), the bijections {φn,k}∞
+k=0imply a recurrence relation of Fn, which leads to an explicit
+expressionu(n) forFnby iteration. Finally, we deduce the following identity
+∞/summationdisplay
+k=0(−1)kf(k) =∞/summationdisplay
+k=0(−1)k∞/summationdisplay
+n=0Fn,k=∞/summationdisplay
+n=0Fn=∞/summationdisplay
+n=0u(n). (1.6)
+As a simple example, one can easily give a combinatorial tele scoping proof of the classical
+identity of Gauss, see also, [7,11,12]:
+n/summationdisplay
+k=0(−1)k/bracketleftbiggn
+k/bracketrightbigg
+=/braceleftBigg
+0, n odd,
+(1−q)(1−q3)···(1−qn−1), neven.
+Let us consider the following reformulation
+n/summationdisplay
+k=0(−1)k1
+(q;q)k(q;q)n−k=
+
+0, n odd,
+1
+(1−q2)(1−q4)···(1−qn), neven.(1.7)
+Let
+Pn,k={(λ,µ):λ1≤k, µ1≤n−k},
+whereλandµare partitions, and let
+Hn,k={(λ,µ)∈Pn,k:mk(λ)<mn−k(µ)},
+wheremk(λ)denotesthenumberofoccurrencesofthepart kinλandweadopttheconvention
+thatm0(λ) = +∞. By definition, Hn,k=∅fork= 0 ork >n. For any integers n≥1 and
+k≥0, we shall construct a bijection
+φn,k:Pn,k−→ {0,n,2n,...}×Pn−2,k∪Hn,k∪Hn,k+1.
+3Let (λ,µ)∈Pn,k. Ifmk(λ)< mn−k(µ), then (λ,µ)∈Hn,k. In this case, φn,k/parenleftbig
+(λ,µ)/parenrightbig
+=
+(λ,µ). Ifmk(λ)≥mn−k(µ), we letmn−k(µ) =t. In this case, if µt+1=n−1−k, we
+increase each of the first tparts ofλby one and decrease each of the first tparts ofµby
+one. It is easily seen that the resulting pair of partitions ( λ′,µ′) belongs to Hn,k+1and we
+setφn,k/parenleftbig
+(λ,µ)/parenrightbig
+= (λ′,µ′). Finally, if µt+1≤n−2−k, then we set
+φn,k/parenleftbig
+(λ,µ)/parenrightbig
+=/parenleftbig
+tn,(ˆλ,ˆµ)/parenrightbig
+∈ {0,n,2n,...}×Pn−2,k,
+whereˆλ= (λt+1,λt+2,...) and ˆµ= (µt+1,µt+2,...) are the partitions obtained from λandµ
+by removing the first tparts. Define the weight function wonPn,kand{0,n,2n,...}×Pn−2,k
+as follows
+w(λ,µ) =q|λ|+|µ|,andw(tn,(λ,µ)) =qtn+|λ|+|µ|,
+where|λ|=λ1+λ2+···. It can be checked that φn,kis weight preserving. Hence we obtain
+the following recurrence relation
+Fn(q) =1
+1−qnFn−2(q), (1.8)
+whereFn(q) denotes the sum on the left hand side of (1.7). By iteration o f (1.8), we arrive
+at (1.7).
+It should be noted that the bijections φn,klead to an involution on Pn,k, which can be
+considered as a variation of the involution given by Chen, Ho u and Lascoux [7].
+In Section 2, we use the idea of combinatorial telescoping to give a proof of Watson’s
+identity [15] in the following form, see also [10, Section 2. 7],
+∞/summationdisplay
+k=0(−1)k1−aq2k
+(q;q)k(aqk;q)∞a2kqk(5k−1)/2=∞/summationdisplay
+n=0anqn2
+(q;q)n, (1.9)
+where
+(a;q)k= (1−a)(1−aq)···(1−aqk−1),and (a;q)∞=∞/productdisplay
+i=0(1−aqi).
+Settinga= 1, Watson’s identity reduces to Schur’s identity [3]
+1
+(q;q)∞∞/summationdisplay
+k=−∞(−1)kqk(5k−1)/2=∞/summationdisplay
+n=0qn2
+(q;q)n.
+Applying Jacobi’s triple product identity to the left hand s ide, we are led to the first Rogers-
+Ramanujan identity. Similarly, setting a=qin Watson’s identity yields the second Rogers-
+Ramanujan identity.
+Here is a sketch of the proof. Assume that the k-th summand regardless of the sign on
+the left hand side of (1.9) is the weight of a set Pk. We further divide Pkinto a disjoint union
+of subsetsPn,k,n= 0,1,..., by considering the expansion of the summand in the paramete r
+a. For a positive integer nand a nonnegative integer k, we can construct a bijection
+φn,k:Pn,k→ {n}×Pn,k∪{2n−1}×Pn−1,k∪Hn,k∪Hn,k+1. (1.10)
+4Let
+Fn(a,q) =∞/summationdisplay
+k=0(−1)k/summationdisplay
+α∈Pn,kw(α).
+The bijections φn,kyield a recurrence relation
+Fn(a,q) =qnFn(a,q)+aq2n−1Fn−1(a,q), n≥1.
+By iteration, we find that Fn(a,q) =anqn2/(q;q)n, and hence (1.9) holds.
+As another example, it can be seen that the method of combinat orial telescoping also
+applies to Sylvester’s identity [14]
+∞/summationdisplay
+k=0(−1)kqk(3k+1)/2xk1−xq2k+1
+(q;q)k(xqk+1;q)∞= 1. (1.11)
+This identity has been investigated by Andrews [1,2].
+2 Watson’s identity
+In this section, we shall use Watson’s identity as an example to illustrate the idea of com-
+binatorial telescoping. Let us recall some definitions conc erning partitions. A partition is a
+non-increasing finite sequence of positive integers λ= (λ1,...,λ ℓ). The integers λiare called
+thepartsofλ. The sum of parts and the number of parts are denoted by |λ|=λ1+···+λℓ
+andℓ(λ) =ℓ, respectively. The number of k-parts inλis denoted by mk(λ). The special
+partition with no parts is denoted by ∅. We shall use diagrams to represent partitions and
+use columns instead of rows to represent parts.
+Set
+Pk={(τ,λ,µ):τ= (k2k,k−1,...,2,1), λℓ(λ)≥k, λi/\e}atio\slash= 2k, µ1≤k},(2.1)
+wherek2kdenotes 2koccurrences of a part k. In other words, τis a trapezoid partition
+with|τ|=k(5k−1)/2,λis a partition with parts at least kbut not equal to 2 k, andµis a
+partition with parts at most k. In particular, we have P0={(∅,λ,∅)}. It is clear that the
+k-th summand of the left hand side of (1.9) without sign can be v iewed as the weight of Pk,
+that is,/summationdisplay
+(τ,λ,µ)∈Pkaℓ(λ)+2kq|τ|+|λ|+|µ|.
+According to the exponent of ain the above definition, we divide Pkinto a disjoint union
+of subsets
+Pn,k={(τ,λ,µ)∈Pk:ℓ(λ) =n−2k}, (2.2)
+withPn,0={(∅,λ,∅)∈P0:ℓ(λ) =n}andPn,k=∅forn <2k. The elements of Pn,kare
+illustrated in Figure 2.1.
+We have the following combinatorial telescoping relation f orPn,k.
+5n
+k λ
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+τ ≥kbut without 2 kµ
+Figure 2.1: The diagram ( τ,λ,µ)∈Pn,k
+Theorem 2.1 Let
+Hn,k={(τ,λ,µ)∈Pn,k:mk(λ)+2>mk(µ)}. (2.3)
+Then, for any positive integer nand any nonnegative integer k, there is a bijection
+φn,k:Pn,k−→ {n}×Pn,k∪{2n−1}×Pn−1,k∪Hn,k∪Hn,k+1. (2.4)
+Proof.The bijection is essentially a classification of Pn,kinto four cases. Let ( τ,λ,µ) be a
+3-tuple of partitions in Pn,k.
+Case 1.mk(λ)+2>mk(µ). In this case, ( τ,λ,µ)∈Hn,kand the image of ( τ,λ,µ) is defined
+to be itself.
+Case 2.mk(λ) + 2≤mk(µ) andm2k+1(λ) = 0. Denote the set of 3-tuples ( τ,λ,µ) in this
+case byUn,k. Note that
+Un,0={(∅,λ,∅)∈Pn,0:m1(λ) = 0}.
+Sincemk(µ)≥mk(λ)+2, we can remove ( mk(λ)+2)k-parts from µto generate a partition
+µ′. In the meantime, we change each k-part ofλinto a 2k-part in order to obtain a partition
+λ′whose minimal part is strictly greater than k.
+Next, we decrease each part of λ′by one in order to producea partition λ′′whose minimal
+part is greater than or equal to k. Sinceλcontains no parts equal to 2 k+1, we see that λ′′
+contains no parts equal to 2 k. Thus we obtain a bijection ϕ1:Un,k→ {n}×Pn,kdefined by
+(τ,λ,µ)/mapsto→(n,(τ,λ′′,µ′)). This case is illustrated by Figure 2.2.
+k λ′′µ′k k n −2k
+Figure 2.2: The resulting partition under the bijection ϕ1.
+Case 3.mk(λ)+2≤mk(µ),m2k+1(λ)>0 andmk+1(λ)+m2k+2(λ) = 0. Denote the set of
+3-tuples (τ,λ,µ) in this case by Vn,k. We remark that when k= 0, one 1-part is regarded as
+a (2k+1)-part and the other 1-parts are regarded as ( k+1)-parts so that
+Vn,0={(∅,λ,∅)∈Pn,0:m1(λ) = 1 andm2(λ) = 0}.
+6Letλ′,µ′be given as in Case 2. We can remove one (2 k+1)-part from λ′and decrease each
+of the remaining parts by two in order to obtain λ′′. This leads to a bijection ϕ2:Vn,k→
+{2n−1}×Pn−1,kas given by ( τ,λ,µ)/mapsto→(2n−1,(τ,λ′′,µ′)). See Figure 2.3 for an illustration.
+k λ′′µ′n−2k−1 2k+1
+Figure 2.3: The resulting partition under the bijection ϕ2.
+Case 4.mk(λ)+2≤mk(µ),m2k+1(λ)>0 andmk+1(λ)+m2k+2(λ)>0. Denote the set of
+3-tuples (τ,λ,µ) in this case by Wn,k. As in Case 3, we have
+Wn,0={(∅,λ,∅)∈Pn,0:m1(λ)>0 andm1(λ)+m2(λ)>1}.
+Letλ′,µ′be given as in Case 2. We can change each (2 k+2)-part of λ′to a (k+1)-part and
+addm2k+2(λ′) (k+1)-parts to µ′. Denote the resulting partitions by λ′′andµ′′. Then we
+have
+mk+1(λ′′) =mk+1(λ)+m2k+2(λ)>0, mk+1(µ′′) =m2k+2(λ). (2.5)
+Now remove one ( k+1)-part and one (2 k+1)-part from λ′′to obtainλ′′′. By (2.5), we find
+mk+1(λ′′′) =mk+1(λ′′)−1≥mk+1(µ′′)−1.
+Moreover, it is clear that
+|λ|+|µ|= 2k+(k+1)+(2k+1)+|λ′′′|+|µ′′|.
+Letτ′be the trapezoid partition of size k+1. So we obtain a bijection ϕ3:Wn,k→Hn,k+1
+defined by ( τ,λ,µ)/mapsto→(τ′,λ′′′,µ′′). This case is illustrated in Figure 2.4.
+n
+k+1 λ′′′
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+τ′≥k+1 and without 2 k+2µ′′
+Figure 2.4: The resulting partition under the bijection ϕ3.
+Assign a weight function wonPn,k,{n}×Pn,kand{2n−1}×Pn−1,kas follows:
+w/parenleftbig
+τ,λ,µ/parenrightbig
+=anq|τ|+|λ|+|µ|,
+w/parenleftbig
+n,(τ,λ,µ)/parenrightbig
+=qn·anq|τ|+|λ|+|µ|,
+w/parenleftbig
+2n−1,(τ,λ,µ)/parenrightbig
+=aq2n−1·an−1q|τ|+|λ|+|µ|.
+7Observe that the bijections ϕ1,ϕ2andϕ3are weight preserving. In addition, Hn,0=∅and
+Hn,k=∅fork >n
+2. Thus the bijections φn,kimmediately lead to a recurrence relation of
+Fn(a,q) defined as follows.
+Corollary 2.2 Let
+Fn(a,q) =∞/summationdisplay
+k=0(−1)k/summationdisplay
+(τ,λ,µ)∈Pn,kanq|τ|+|λ|+|µ|. (2.6)
+Then, for any positive integer n, we have
+Fn(a,q) =qnFn(a,q)+aq2n−1Fn−1(a,q). (2.7)
+SinceF0(a,q) = 1, by iteration we find that
+Fn(a,q) =aq2n−1
+1−qnFn−1(a,q) =a2q4n−4
+(1−qn)(1−qn−1)Fn−2(a,q) =···=anqn2
+(q;q)n.
+Summing over n, we arrive at Watson’s identity (1.9).
+3 Sylvester’s identity
+In this section, we describe the approach of combinatorial t elescoping for Sylvester’s identity
+(1.11). Define
+Qn,k={(τ,λ):τ= (kk+1,k−1,...,2,1),λi/\e}atio\slash= 2k+1,m>k(λ) =n−k},
+wherem>k(λ) denotes the number of parts of λwhich are greater than k. See Figure 3.1 for
+an illustration. In particular, we have
+Qn,0={(∅,λ):λi/\e}atio\slash= 1,ℓ(λ) =n}.
+n
+k λ
+τ/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright
+Figure 3.1: The diagram of ( τ,λ)∈Qn,k.
+Let
+Hn,k={(τ,λ)∈Qn,k:mk+1(λ)≥mk(λ)}.
+Then, for each positive integer nand each nonnegative integer k, we have a bijection
+φn,k:Qn,k−→ {n}×Qn,k∪Hn,k∪Hn,k+1,
+which is a classification of Qn,kinto three cases. Let ( τ,λ)∈Qn,k.
+8Case 1.mk+1(λ)≥mk(λ). In this case, ( τ,λ)∈Hn,kand the image of ( τ,λ) underφn,kis
+defined to be itself.
+Case 2.mk+1(λ)< mk(λ) andm2k+2(λ) = 0. Denote the set of pairs ( τ,λ) in this case
+byUn,k. We remove one k-part from λ. Then, for each ( k+1)-part of λ, we can add it to
+ak-part to form a (2 k+1)-part. Finally, we decrease each part greater than k+1 by one
+to generate a partition λ′. Sincem2k+2(λ) = 0, we see that ( τ,λ′)∈Qn,k. So we obtain a
+bijectionϕ1:Un,k→ {n}×Qn,kgiven by (τ,λ)/mapsto→(n,(τ,λ′)).
+Case 3.mk+1(λ)< mk(λ) andm2k+2(λ)>0. Denote the set of pairs ( τ,λ) in this case by
+Vn,k. We first remove one k-part and one (2 k+2)-part from λand add them to τto form a
+partitionτ′. Hereτ′is a trapezoid partition of size k+1. Then for each ( k+1)-part of λwe
+combine it with a k-part to form a (2 k+1)-part. Finally we decompose each (2 k+3)-part
+ofλinto a (k+1)-part and a ( k+2)-part to form a partition λ′. Sincem2k+3(λ′) = 0, we
+obtain a bijection ϕ2:Vn,k→Hn,k+1defined by ( τ,λ)/mapsto→(τ′,λ′).
+It is not difficult to see that Sylvester’s identity follows fr om the bijections φn,k. Let
+In(q) =∞/summationdisplay
+k=0(−1)k/summationdisplay
+(τ,λ)∈Qn,kq|τ|+|λ|.
+Noting that Hn,0=∅because of the definition m0(λ) = +∞, the bijections φn,klead to the
+recurrence relation
+In(q) =qnIn(q),
+which implies that In(q) = 0 forn≥1. Clearly I0(q) = 1, and hence Sylvester’s identity
+holds.
+To conclude this paper, we notice that both Watson’s identit y and Sylvester’s identity can
+be verified by employing the q-Zeilberger algorithm for infinite q-series developed by Chen,
+Hou and Mu [8]. Let
+f(a) =∞/summationdisplay
+k=0(−1)k(1−aq2k)
+(q;q)k(aqk;q)∞a2kqk(5k−1)/2.
+Denote the k-th summand of f(a) byFk(a). Theq-Zeilberger algorithm gives that
+Fk(a)−Fk(aq)−aqFk(aq2) =Hk+1(a)−Hk(a), (3.1)
+where
+Hk(a) = (−1)k(−1−qk+aq2k)
+(q;q)k−1(aqk;q)∞a2kqk(5k−1)/2.
+Summing (3.1) over k, we find that
+f(a) =f(aq)+aqf(aq2).
+Extracting the coefficients of anleads to the same recurrence relation as (2.7). It is easily
+checked that the right hand side of (1.9) satisfies the same re cursion. By Theorem 3.1 of
+Chen, Hou and Mu [8], one sees that (1.9) holds for any aprovided that it is valid for the
+trivial case a= 0. Similarly, let
+f(x) =∞/summationdisplay
+k=0(−1)kqk(3k+1)/2xk1−xq2k+1
+(q;q)k(xqk+1;q)∞.
+9Theq-Zeilberger algorithm gives that
+Fk(x)−Fk(xq) =Hk+1(x)−Hk(x), (3.2)
+whereFk(x) is thek-th summand of f(x) and
+Hk(x) = (−1)k+1qk(3k+1)/2xk
+(q;q)k−1(xqk+1;q)∞. (3.3)
+Summing (3.2) over k, we deduce that f(x) =f(xq), which implies f(x) = 1.
+Acknowledgments. We wish to thank the referees for helpful comments. This work was supported
+by the National Science Foundation, the PCSIRT project, the Pro ject NCET-09-0479, and the Fun-
+damental Research Funds for Universities of the Ministry of Educa tion of China.
+References
+[1] G.E. Andrews, Partially ordered sets and the Rogers-Ram anujan identities, Aequationes
+Math. 12 (1975) 94–107.
+[2] G.E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Appli-
+cations, Vol. 2, Addison-Wesley Publishing Co., Reading, M ass., 1976.
+[3] C. Boulet and I. Pak, A combinatorial proof of the Rogers- Ramanujan and Schur iden-
+tities, J. Combin. Theory, Ser. A 113 (2006) 1019–1030.
+[4] D.M. Bressoud, An easy proof of the Rogers-Ramanujan ide ntities, J. Number Theory
+16 (1983) 235–241.
+[5] D.M. Bressoud and D. Zeilberger, A short Rogers-Ramanuj an bijection, Discrete Math.
+38 (1982) 313–315.
+[6] D.M.BressoudandD.Zeilberger, GeneralizedRogers-Ra manujanbijections, Adv.Math.
+78 (1989) 42–75.
+[7] W.Y.C. Chen, Q.-H. Hou and A. Lascoux, An involution for t he Gauss identity, J.
+Combin. Theory, Ser. A 102 (2003) 309–320.
+[8] W.Y.C. Chen, Q.-H. Hou and Y.-P. Mu, Nonterminating basi c hypergeometric series
+and theq-Zeilberger algorithm, Proc. Edinb. Math. Soc. 51 (2008) 60 9–633.
+[9] A.M. Garsia and S.C. Milne, A Rogers-Ramanujan bijectio n, J. Combin. Theory, Ser. A
+31 (1981) 289–339.
+[10] G.GasperandM.Rahman, BasicHypergeometricSeries, 2 ndEd., CambridgeUniversity
+Press, Cambridge, MA, 2004.
+[11] J.Y. Lee, A combinatorial proof for an identity of Gauss , Ramanujan J. 21 (2010) 65–69.
+[12] I. Pak, The nature of partition bijections I. Involutio ns, Adv. Appl. Math. 33 (2004)
+263–289.
+10[13] I. Pak, Partition bijections, a survey, Ramanujan J. 12 (2006) 5–75.
+[14] J.J. Sylvester, A constructive theory of partitions, a rranged in three acts, an interact,
+and an exodion, Amer. J. Math. 5 (1882) 251–330.
+[15] G.N. Watson, A new proof of the Rogers-Ramanujan identi ties, J. London Math. Soc. 4
+(1929) 4–9.
+[16] D. Zeilberger, The method of creatvie telescoping, J. S ymbolic Comput. 11 (1991) 195–
+204.
+11
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+arXiv:1001.0313v3  [math.CO]  29 Nov 2010ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL
+COMPLEXES
+RUSS WOODROOFE
+Abstract. A recent framework for generalizing the Erd˝ os-Ko-
+Rado Theorem, due to Holroyd, Spencer, and Talbot, defines the
+Erd˝ os-Ko-Rado property for a graph in terms of the graph’s inde -
+pendent sets. Since the family of all independent sets of a graph
+forms a simplicial complex, it is natural to further generalize the
+Erd˝ os-Ko-Rado property to an arbitrary simplicial complex. An
+advantage of working in simplicial complexes is the availability
+of algebraic shifting, a powerful shifting (compression) technique ,
+which we use to verify a conjecture of Holroyd and Talbot in the
+case of sequentially Cohen-Macaulay near-cones.
+1.Introduction
+A family Aof sets is intersecting if every pair of sets in Ahas non-
+emptyintersection, andisan r-familyifeverysetin Ahascardinality r.
+A well-known theorem of Erd˝ os, Ko, and Rado bounds the cardinalit y
+of an intersecting r-family:
+Theorem 1.1. (Erd˝ os-Ko-Rado [11]) Letr≤n
+2andAbe an inter-
+sectingr-family of subsets of [n]. Then|A| ≤/parenleftbign−1
+r−1/parenrightbig
+.
+Given a simplicial complex ∆ (defined in Section 2) and a face σof
+∆, we define the linkofσin ∆ to be
+link∆σ={τ:τ∪σis a face of ∆ , τ∩σ=∅}.
+Anr-faceof∆isafaceofcardinality r. Wefurtherlet fr(∆)bedefined
+asthenumberof r-facesin∆, andthetuple( f0(∆),f1(∆),...,f d+1(∆))
+(wheredis the dimension of ∆) is called the f-vectorof ∆.
+Note1.2.WefollowSwartz[28]inourdefinitionof r-faceand fr. Other
+sources define an r-face to be a face with dimension r(rather than
+cardinality r) which shifts the indices of the f-vector by 1.
+We restate Theorem 1.1 using this language:
+2000Mathematics Subject Classification. 05E45, 05D05.
+1ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 2
+Theorem 1.3. Letr≤n
+2andAbe an intersecting r-family of faces
+of the simplex with nvertices. Then |A| ≤fr−1(link∆v1).
+LetGbe a graph with edge set E(G) and vertex set V(G). The
+independence complex ofG, denoted I(G), is the simplicial complex
+consisting of all independent sets of G. For a vertex v∈V(G), the
+closed neighborhood N[v] consists of vand all its neighbors. We notice
+that link I(G)v=I(G\N[v]).
+Following Holroyd, Spencer, and Talbot [17, Section 1], we define:
+Definition 1.4. A simplicial complex ∆ is r-EKRif every intersect-
+ingr-familyAof faces of ∆ satisfies |A| ≤maxv∈V(∆)fr−1(link∆v).
+Equivalently, ∆ is r-EKR if the set of all r-faces containing some vhas
+maximal cardinality among all intersecting families of r-faces.
+Holroyd and Talbot [18] further investigated the problem of when
+graphs are r-EKR, and made the following conjecture:
+Conjecture 1.5. (Holroyd-Talbot [18, Conjecture 7]) IfGis a graph
+where the minimal facet cardinality of I(G)isk, thenI(G)isr-EKR
+forr≤k
+2.
+The natural extension was made by Borg [5]:
+Conjecture1.6. (Borg[5,Conjecture1.7]) If∆isa simplicialcomplex
+having minimal facet cardinality k, then∆isr-EKR for r≤k
+2.
+Remark1.7.Adifferent version ofanEKR propertyfor simplicial com-
+plexes was studied by Chv´ atal [8], who conjectured that if Ais an
+intersecting family of faces (of possibly differing dimensions) then
+|A| ≤max
+v∈V(∆)/parenleftBigg/summationdisplay
+rfr(link∆v)/parenrightBigg
+,
+i.e., that the set of all faces containing some vhas maximal cardinality
+among all intersecting families of faces. We notice that Conjecture 1.6
+isananalogueofChv´ atal’sConjectureforuniformintersecting fa milies.
+We refer the reader to [6] for additional background on the r-EKR
+property in graphs, and to [5] for further relationships with more g en-
+eral intersection problems.
+This paper is organized as follows. In Section 2 we review the nec-
+essary background on shifted complexes, algebraic shifting, the C ohen-
+Macaulay property, and near-cones. We also characterize the gr aphsG
+such that Shift I(G) is the independence complex of some other graph,
+and the graphs such that I(G) is a near-cone. In Section 3 we presentERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 3
+and prove our main theorem, Theorem 3.3. In Section 4 we give appli-
+cations of Theorem 3.3 to Conjecture 1.5. In particular, we recove r the
+main result of [19], and many of the results of [17]. We close in Section
+5 with further questions regarding the strict r-EKR property.
+Acknowledgements
+The article at Gil Kalai’s blog (at http://gilkalai.wordpress.com/ )
+on the use of algebraic shifting in intersection theorems was very he lp-
+ful, especially inproving Lemma3.1. ArtDuvalandIsabellaNovik also
+answered some of my questions about algebraic shifting. Glenn Hurl-
+bert and Vikram Kamat explained the difficulties in extending their
+proof of Corollary 4.2 to all vertex decomposable graphs.
+2.Shifting
+We will need some basic simplicial complex language: An (abstract)
+simplicial complex ∆ is a system of sets (called faces) on base set V(∆)
+(calledvertices) such that if σis a face then every subset of σis also a
+face. We assume that every vertex is contained in some face. A facet
+of ∆ is a face that is maximal under inclusion. It is well-known that
+anyabstractsimplicialcomplexhasa geometric realization ,ageometric
+simplicial complex with the same face incidences; so we can use terms
+from geometry such as dimension to describe a simplicial complex.
+IfFis some family of sets, then the simplicial complex ∆( F)gen-
+erated by Fhas faces consisting of all subsets of all sets in F. For a
+simplicial complex ∆, the r-skeleton ∆(r)consists of all faces of ∆ hav-
+ing dimension at most r, while the purer-skeleton is the subcomplex
+generated by all faces of ∆ having dimension exactly r. Thejoinof
+disjoint simplicial complexes ∆ and Σ is the simplicial complex ∆ ∗Σ
+with faces τ∪σ, whereτis a face of ∆ and σa face of Σ.
+A simplicial complex ∆ with ordered vertex set {v1,...,v n}isshifted
+if whenever σis a face of ∆ containing vertex vi, then (σ\{vi})∪{vj}
+is a face of ∆ for every j < i. Anr-familyFof subsets of {v1,...,v n}
+isshiftedif it generates a shifted complex.
+A general approach to proving theorems similar to Theorem 1.3 is to
+define a shifting operation orcompression operation which replaces a
+non-shifted set system with a shifted system obeying some of the s ame
+combinatorial properties. Erd˝ os, Ko, and Rado pioneered this te ch-
+nique in [11], and their operation is now called combinatorial shifting.
+Combinatorial shifting is discussed in the survey article [13], particu-
+larly as applied to intersection theorems.ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 4
+2.1.Algebraic shifting. The specific shifting operation we will use
+is called exterior algebraic shifting (with respect to a field F), and we
+denote the (exterior) algebraic shift of ∆ by Shift∆, or Shift F∆ if we
+want to emphasize the field. Algebraic shifting was first studied by
+Kalai, and unless otherwise stated the facts we present here were first
+proved by him.
+The precise definition of Shift∆will not be important forus, but can
+be found in Kalai’s survey article [22]. Rather than working with the
+definition, we examine Shift∆using a series oflemmas collected in[22].
+The following basic properties we will use without further mention:
+Lemma 2.1. [22]Let∆be a simplicial complex with nvertices. Then:
+(1) Shift∆ is a shifted simplicial complex on an ordered vertex set
+{v1,...,v n}.
+(2)If∆is shifted, then Shift∆∼=∆.
+(3)fi(Shift∆) = fi(∆)for alli.
+Shifting respects subcomplexes, at least in a weak sense:
+Lemma 2.2. [22, Theorem 2.2] IfΣ⊂∆are simplicial complexes,
+thenShiftΣ⊂Shift∆.
+Example 2.3. Let ∆ be the simplicial complex with facets {1,2}and
+{3,4}. Then it is easy to see that the unique shifted complex with
+the same f-vector has facets {1,2},{2,3}, and{4}, hence that this
+complex is Shift∆. Then Shift {1,2}= Shift{3,4}={1,2} ⊂Shift∆.
+Corollary 2.4. If∆isa simplicialcomplex, then Shift(∆(r)) = (Shift∆)(r).
+Proof.Lemma 2.2 gives that Shift(∆(r))⊆(Shift∆)(r), and by Lemma
+2.1 part (3) the f-vectors are equal. /square
+Notice that if ∆ is a shifted complex, then it is immediate that ∆(r)
+is shifted for every r.
+IfAis somer-family of sets, then Shift Ais the pure r-skeleton of
+Shift∆(A). (Kalai’s equivalent definition actually defines Shift∆ as a
+union of the shift of its r-faces [22, Section 2.1].) Kalai proves:
+Lemma 2.5. [22, Corollary 6.3] IfAis an intersecting r-family, then
+ShiftAis an intersecting r-family.
+2.2.Near-cones. A simplicial complex ∆ is a near-cone with respect
+to anapex vertex vif for every face σ, the set ( σ\ {w})∪{v}is also
+a face for each vertex w∈σ. Equivalently, the boundary of every
+facet of ∆ is contained in v∗link∆v; another equivalent condition is
+that ∆ consists of v∗link∆vunion some set of facets not containing vERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 5
+(but whose boundary is contained in link ∆v). If ∆ is a cone with apex
+vertexv, then obviously ∆ = v∗link∆v, thus every cone is a near-cone.
+Because the apex vertex is always the vertex with the largest link,
+near-conesarerelativelyeasytoworkwithinthecontextofinters ection
+theorems, as has been previously noticed in e.g. [27, 4]. In particular :
+Lemma 2.6. If∆is a near-conewith apex vertex v, thenfr(link∆w)≤
+fr(link∆v)for any vertex wand allr.
+Proof.For every ( r+1)-face σcontaining w, eitherv∈σor else (σ\
+w)∪vis anr-face containing vbut notw. /square
+Notice that ∆ being a near-cone with apex vertex vessentially says
+that ∆ is“shifted with respect to v”. In particular, any shifted complex
+is a near-cone with apex vertex v1. Nevo examined the algebraic shift
+of a near-cone, showing:
+Lemma 2.7. (Nevo [25, Theorems 5.2 and 5.3]) If∆is a near-cone
+with apex v, let us consider Shift(link ∆v)as having ordered vertex set
+{v2,...,v n}. Then
+(1) link Shift∆v1= Shift(link ∆v).
+(2) Shift∆ = ( v1∗Shift(link ∆v))∪ B, whereBis a set of facets
+not containing v1.
+Corollary 2.8. If∆is a near-cone with apex v, thenfr(link∆v) =
+fr(linkShift∆v1)for allr.
+2.3.Pure complexes, Cohen-Macaulay complexes, and depth.
+A simplicial complex ∆ is pureif all facets of ∆ have the same dimen-
+sion. Graphs with a pure independence complex are sometimes called
+well-covered .
+LetFbeeitheranyfield, ortheringofintegers. Asimplicialcomplex
+∆ isCohen-Macaulay over F if its homology satisfies ˜Hi(link∆σ;F) =
+0 for all i <dim(link ∆σ) and all faces σof ∆ (including σ=∅). It
+is well-known that every Cohen-Macaulay complex is pure, and that
+every skeleton of a Cohen-Macaulay complex is Cohen-Macaulay.
+A simplicial complex is sequentially Cohen-Macaulay over Fif the
+purer-skeleton of ∆ is Cohen-Macaulay (over F) for all r. Thus, a
+pure sequentially Cohen-Macaulay complex is Cohen-Macaulay.
+When we simply say that a simplicial complex ∆ is (sequentially)
+Cohen-Macaulay, with no mention of F, then we mean that ∆ is (se-
+quentially) Cohen-Macaulay over all F. For example, every“shellable”
+or“vertex decomposable”complex issequentially Cohen-Macaulay o ver
+anyF[2, 3].ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 6
+The main relationships between the Cohen-Macaulay property and
+shifting are the following:
+Lemma 2.9. [3, Theorem 11.3] If∆is shifted, then ∆is “vertex de-
+composable”, hence sequentially Cohen-Macaulay (over any F).
+Lemma 2.10. ([22,Theorem4.1],seealso[1,Proposition8.4]) Shift F∆
+is pure if and only if ∆is Cohen-Macaulay (over F).
+Duval [9] also examined the algebraic shift of a sequentially Cohen-
+Macaulay complex, and more generally of Cohen-Macaulay skeletons .
+A result of his that will be of particular interest to us is:
+Corollary 2.11. (Duval [9, Corollary 4.5]) The minimum facet dimen-
+sion ofShiftF∆is≥dif and only if ∆(d)is Cohen-Macaulay (over F).
+Corollary 2.11 suggests the definition of the depth of ∆overFas
+depthF∆ = max {d: ∆(d)is Cohen-Macaulay over F}.
+Thus, depthF∆ is the minimum facet dimension of Shift F∆. We
+note that depthF∆ is one less than the ring-theoretic depth of the
+“Stanley-Reisner ring” F[∆] [26, Theorem 3.7]. Thus, just as in the
+ring-theoretic situation, ∆ is Cohen-Macaulay over Fif and only if
+depthF∆ = dim∆. If ∆ is sequentially Cohen-Macaulay over Fthen
+depthF∆ is the minimum facet dimension of ∆.
+By the definition of simplicial homology we have ˜Hd(∆(d+1);F) =
+˜Hd(∆;F), hence the easy equivalent characterization:
+depthF∆ = max {d:˜Hi(link∆σ;F) = 0 for all σ∈∆ andi < d−|σ|}.
+In particular, we notice that depthF∆ is at most the minimal facet
+dimension, since if σis a facet then ˜H−1(link∆σ;F) =˜H−1(∅;F) =F.
+The following result about the depth of the join of complexes will be
+especially useful in Section 4:
+Lemma 2.12. LetFbe a field, and ∆1and∆2be simplicial complexes.
+ThendepthF(∆1∗∆2) = depthF∆1+depthF∆2+1.
+Proof.Faces of ∆ 1∗∆2have the form σ=σ1˙∪σ2whereσiis a face
+of ∆i, hence link ∆1∗∆2σ= link ∆1σ1∗link∆2σ2. The result then fol-
+lows from the standard algebraic topology fact [20, Corollary 4.23] t hat
+˜Hn+1(∆1∗∆2;F) =/circleplustextn+1
+i=−1˜Hi(∆1;F)⊗˜Hn−i(∆2;F). /square
+The reader is referred to [20] for additional background on depthF∆
+and the (sequentially) Cohen-Macaulay property. We will hencefor th
+take the field Fto be understood, and drop it from our notation.ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 7
+2.4.Shifting independence complexes. A graph Gischordalif
+every induced subgraph of Gwhich is a cyclic graph has length 3. A
+graph isco-chordal if its complement graph is chordal.
+Since the original question of Holroyd, Spencer and Talbot was re-
+stricted to the independence complexes of graphs, one might ask w hen
+ShiftI(G)isisomorphicto I(G′)forsomegraph G′. Theanswer iseasy,
+given the necessary machinery. The Alexander dual of ∆, denoted ∆∨,
+is the complex with facets {σ:V\σis a minimal non-face of ∆ }. See
+e.g. [21, Section 6] for more information and background on Alexand er
+duality.
+Theorem 2.13. LetGbe a graph. Then ShiftI(G)is the independence
+complexI(G′)of some other graph G′if and only if Gis co-chordal.
+Proof.We need the following three facts about Alexander duality: 1)
+It is clear from the definition that ∆ is the independence complex of
+a graph if and only if ∆∨is pure (n−2)-dimensional, where nis the
+number of vertices of ∆. 2) Alexander duality and shifting commute,
+i.e. Shift∆ = (Shift(∆∨))∨[22, Section 3.5.6]. 3) If Gis a graph, then
+I(G)∨is Cohen-Macaulay if and only if Gis co-chordal [10, Proposition
+8]. The result is then immediate from Lemma 2.10. /square
+Remark2.14.Thefamilyofindependencecomplexesofgraphshasbeen
+extensively studied in the literature under the name of flag complexes .
+The shifted flag complexes were classified by Klivans, as follows:
+Givenagraph G, letD(G)beG˙∪{v}foranewvertex v(Dfor“disjoint
+union”). Let S(G) be the graph on vertex set V(G)∪ {v}for a new
+vertexvand with edge set E(G)∪ {wv:w∈V(G)}(Sfor “star”).
+In the independence complex, we have I(D(G)) as the cone over I(G),
+andI(S(G)) asI(G)˙∪{v}, thus ifGis sequentially Cohen-Macaulay
+then both D(G) andS(G) are.
+Agraphis threshold [24] if it isobtainedfroma singlevertex by some
+sequence of DandSoperations. Every threshold graph is bothchordal
+and co-chordal. Since a Doperation adds a cone vertex, which can be
+takenastheinitialvertexinashiftedcomplex; andan Soperationadds
+a disjoint vertex, which can be taken as the final vertex in a shifted
+complex, we have proved inductively that the independence complex of
+any threshold graph is shifted. Klivans [23] showed the converse r esult
+that all graphs with shifted independence complex are threshold.
+We prove the following generalization of [23, Theorem 1]:
+Proposition 2.15. IfGis a graph such that I(G)is a near-cone, then
+Gis obtained from some graph G0by a sequence of DandSoperations,
+including at least one Doperation.ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 8
+Proof.Letvbe the apex vertex of I(G), and suppose that wv∈E(G).
+Thenwxis also an edge for every x∈V(G), since if wxwere a face of
+I(G) thenwvwould also be independent, a contradiction. We see that
+G=SkD(G\N[v]), where kis the number of neighbors of v./square
+Remark2.16.Since the independence complex of S(G) has an isolated
+vertex, its minimum facet dimension is 0. Hence Proposition 2.15 tells
+us that for a graph Gwe have that I(G) is a near-cone with non-trivial
+minimum facet dimension if and only if Ghas an isolated vertex.
+3.Main theorem
+The following lemma followsfromBorg’smoregeneral result [5, The-
+orem 2.7]. We use algebraic shifting to give a short new proof of the
+specific result.
+Lemma 3.1. (Borg [5, Theorem 2.7]) If∆is a shifted complex having
+minimal facet cardinality k, then∆isr-EKR for r≤k
+2.
+Proof.Let ∆ have ordered vertex set {v1,...,v n}, and let Abe an
+intersecting r-family of faces of ∆. We proceed by induction: our base
+cases arewhen ∆ isa simplex (Theorem 1.3), andthetrivial case wher e
+r= 1.
+If ∆isnota simplex and r >1, thenby Lemmas2.5and2.2, we have
+that Shift Ais a shifted intersecting r-family of faces of ∆ = Shift∆
+with|ShiftA|=|A|. We decompose Shift Ainto the subfamilies C
+consisting of all σ∈ShiftAwithvn∈σ, andD= (ShiftA)\C, so that
+|A|=|ShiftA|=|C|+|D|.
+We first consider C. LetC0={σ\{vn}:σ∈ C}, so that |C|=|C0|.
+Suppose that C0is not intersecting. Then there are σ,τ∈ Csuch that
+σ∩τ={vn}, and|σ∪τ|< r+r≤k < n. It follows that there
+is avℓ/∈σ∪τ. But then τ′= (τ\ {vn})∪ {vℓ}is in Shift Aby
+the definition of shiftedness, and σ∩τ′=∅, which contradicts that
+ShiftAis intersecting. We conclude that C0is an intersecting ( r−1)-
+family of faces of link ∆vn. Since link ∆vnis a shifted complex with
+minimum facet cardinality at least k−1, we get that |C|=|C0| ≤
+fr−2(link∆{v1,vn}) by induction and Lemma 2.6.
+We now consider D. It is obvious that Dis an intersecting r-family
+contained in the shifted complex ∆ \vn. Since ∆ is not a simplex, it
+follows easily from the definition of shiftedness that the minimum face t
+cardinality of ∆ \vnis at least k. By induction and Lemma 2.6 we
+have|D| ≤fr−1(link∆\vnv1).ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 9
+Putting our two parts together, we have
+|A|=|C|+|D| ≤fr−2(link∆{v1,vn})+fr−1(link∆\vnv1) =fr−1(link∆v1).
+/square
+Remark 3.2.Our requirement for Lemma 3.1 on the minimum facet
+cardinality seems much stronger than necessary. Our essential n eed is
+for a parameter kwhich we can control in both ∆ \vnand link ∆vn,
+and such that r≤k
+2forcesr <n
+2. Use of another such parameter
+might give a stronger result version of Lemma 3.1. Any strengthenin g
+of Lemma 3.1 would likely also strengthen Theorem 3.3.
+Holroyd and Talbot [18, Section 3] construct several examples of
+independence complexes that are not r-EKR for various r, which may
+give some intuition about what parameters are tractable. (They in
+particularconstruct anexample withmaximum facet cardinality ℓsuch
+that ∆ is not/floorleftbigℓ
+2/floorrightbig
+-EKR.)
+By applying algebraic shifting to an arbitrary complex, we prove:
+Theorem 3.3. If∆is a near-cone and Fis an arbitrary field, then ∆
+isr-EKR for r≤depthF∆+1
+2.
+Proof.LetAbe an intersecting r-family of faces of ∆. By Lemma 2.6,
+we need to show that |A| ≤fr−1(link∆v) for the apex vertex v.
+Apply algebraic shifting. Shift FAis an intersecting r-family of faces
+ofShift∆with |ShiftA|=|A|by Lemmas2.5and2.2. ByLemma2.11
+andthe following discussion, the minimum facet cardinality of Shift F∆
+is depthF∆ + 1, hence |A| ≤fr−1(linkShift∆v1) =fr−1(link∆v) by
+Lemma 3.1 and Corollary 2.8. /square
+To the best of my knowledge, Theorem 3.3 is the first ‘new’ intersec-
+tion theorem to be proved by algebraic shifting.
+Corollary 3.4. If∆is a sequentially Cohen-Macaulay near-cone with
+minimum facet cardinality k, then∆isr-EKR for r≤k
+2.
+Remark3.5.Borg’s aforementioned result [5, Theorem 2.7] generalizes
+Lemma 3.1 to include non-uniform families (i.e., sets of different sizes),
+and tot-intersecting families (i.e., to families where |A∩B| ≥t).
+By Kalai [22, Corollary 6.3 and following], algebraic shifting preserves
+thet-intersecting property for any r-family, hence a reduction to [5,
+Theorem 2.7] similar to that in Theorem 3.3 will show that if ∆ is a
+t-fold near-cone (i.e., shifted with respect to its first telements) with
+depth equal to its minimum facet dimension, then [5, Conjecture 2.7]
+holds for uniform r-families of faces in ∆. In particular, [5, Conjecture
+2.7] holds for sequentially Cohen-Macaulay t-fold near-cones.ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 10
+4.Applications
+It is immediate from the definitions that if G=G1˙∪G2, thenI(G)
+decomposes as the join I(G1)∗I(G2). In particular, if Ghas an iso-
+lated vertex vthenI(G) is a cone over v, as discussed in detail in
+Section 2.2. We will call a graph Gsequentially Cohen-Macaulay if
+its independence complex I(G) is sequentially Cohen-Macaulay. An
+immediate consequence of Corollary 3.4 is:
+Theorem 4.1. IfGis a sequentially Cohen-Macaulay graph with an
+isolated vertex, then Gsatisfies Conjecture 1.5.
+The family of sequentially Cohen-Macaulay graphs includes:
+(1) Chordal graphs. [12]
+(2) Graphs with no induced cycle of length other than 3 or 5. [31]
+(3) Bipartite graphs containing a vertex vof degree 1 such that
+G\N[v] andG\N[w] (where wis the unique neighbor of v)
+recursively satisfy the same condition. [30, Corollary 3.11]
+(4) Incomparability graphs of shellable posets. [2]
+(5) The minimal set of non-faces of the isocahedron, or any other
+polytope where the set of minimal non-faces forms a graph. [7]
+(6) Disjoint unions of sequentially Cohen-Macaulay graphs, since
+I(G1˙∪G2) =I(G1)∗I(G2).
+In particular, we recover the following theorem of Hurlbert and Kam at:
+Corollary 4.2. (Hurlbert-Kamat[19, Theorem 1.22]) IfGis a chordal
+graph with an isolated vertex, then Gsatisfies Conjecture 1.5.
+Obviously we also have that if Gis e.g. a threshold graph, then G
+satisfies Conjecture 1.5. But Remark 2.16 tells us that this result is
+not an interesting improvement on Corollary 4.2, since in this case the
+minimum facet cardinality of I(G) is 1 unless Ghas an isolated vertex.
+We apply Lemma 2.12 for a result in a slightly different direction:
+Proposition 4.3. IfG=G1˙∪...˙∪Gnis the disjoint union of n≥2r
+nonempty graphs, including at least one isolated vertex, th enI(G)is
+r-EKR.
+Proof.The 0-skeleton of any non-empty complex is Cohen-Macaulay,
+hence depth I(Gi)≥0, and by repeated application of Lemma 2.12 we
+get depth I(G) = depth I(G1)∗ ···∗I(Gn)≥n−1. The result then
+follows by Theorem 3.3. /square
+Proposition 4.3 significantly improves [17, Theorem 8], which proves
+the result in the special case where each Giis a complete graph, path,
+or cycle. By considering graphs of depth 1, we can do slightly better .ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 11
+Lemma 4.4. The independence complex of a graph Ghas depth ≥1
+if and only if |G|>1and the complement graph ¯Gis connected.
+Proof.The complement graph ¯Gforms the 1-skeleton of I(G) under
+the hypothesis, and a 1-dimensional complex is Cohen-Macaulay if an d
+only if it is connected. /square
+Example 4.5. LetCnbe the cyclic graph on nvertices. If n≥5, then
+Cnsatisfies the conditions of Lemma 4.4, hence depth I(Cn)≥1. But
+the cyclic graph C4on 4 vertices has disconnected complement graph,
+hence depth I(C4) = 0.
+Proposition 4.6. LetG=G1˙∪...˙∪Gnbe the disjoint union of n
+graphs, including at least one isolated vertex. Suppose tha tmof theGi
+satisfy the conditions of Lemma 4.4. Then I(G)isr-EKR for r≤n+m
+2.
+The proof is exactly as in Proposition 4.3.
+5.Further questions
+As we have discussed, for ∆ to be r-EKR means that every maximal
+intersecting r-family of faces Ahas|A| ≤maxv∈V(∆)fr−1(link∆v). We
+say that ∆ is strictlyr-EKRif every maximum cardinality intersecting
+r-family of faces Aconsists of the r-faces of v∗(link∆v)(r−1)for some
+v. That is to say, every maximum intersecting r-familyAsatisfies/intersectiontext
+A∈AA/\e}atio\slash=∅. Hilton and Milner [15] improved Theorem 1.3 to:
+Theorem 5.1. (Hilton-Milner [15]) If∆is the simplex with nvertices,
+then∆is strictly r-EKR for 2≤r <n
+2.
+Holroyd and Talbot, and later Borg, actually conjectured slightly
+more than we stated in Conjectures 1.5 and 1.6:
+Conjecture 5.2. (Holroyd-Talbot [18, Conjecture 7]; Borg [5, Conjec-
+ture 1.7]) If∆is a simplicial complex having minimal facet cardinality
+k, then∆isr-EKR for r≤k
+2, and strictly r-EKR for r <k
+2.
+Can an algebraic shifting (or some other) argument be adapted to
+prove Conjecture 5.2, optionally restricted to the case of a seque ntially
+Cohen-Macaulay complex?
+Of course, even the more restricted Conjecture 1.6 remains open
+for general complexes. Theorem 3.3 suggests that a counterexa mple to
+Conjecture1.6, ifoneexists, shouldbeacomplexwhichbadlyfailstob e
+Cohen-Macaulay. We discuss briefly some examples of such complexe s:ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 12
+Example 5.3. The facets of the boundary ∆ 0of a simplex with n+1
+vertices is intersecting, hence not n-EKR. One can increase the dimen-
+sion by coning kpoints over each facet to obtain a pure complex ∆
+with (k+ 1)·(n+ 1) points. Since ˜Hn(∆)/\e}atio\slash= 0, the complex is not
+Cohen-Macaulay, and the hope for a counterexample would be: for
+somek > nandn≤r≤ ⌊k+n
+2⌋, that the family Aconsisting of all
+r-faces that contain a facet of ∆ 0would be larger than fr−1(link∆v).
+But we count: if v∈∆0, thenfr−1(link∆v) =n·/parenleftbign+k
+r−1/parenrightbig
+, while|A|=
+(n+ 1)·/parenleftbigk
+r−n/parenrightbig
+. A straightforward computation (cancel, then match
+terms) yields that fr−1(link∆v)/|A|>1. Hence |A|< fr−1(link∆v),
+and thus Aand ∆ are not a counterexample to Conjecture 1.6.
+Example 5.4. Cyclic graphs Cnare not sequentially Cohen-Macaulay
+forn/\e}atio\slash= 3,5 [12, Proposition 4.1]. For example, I(C4) consists of two
+disjoint 2-faces, while I(C7) is a triangulation of the M ¨obius strip.
+Nonetheless, Talbot [29] showed that the independence complex of ev-
+ery cyclic graph is r-EKR for all r. More recently, the independence
+complex of the disjoint union of two cycles [14], and of the disjoint
+union of an arbitrary number of cycles and a path [16] were shown t o
+ber-EKR for all r. Conjecture 1.5 also holds [6] for the disjoint union
+of an isolated vertex and a somewhat wider class of non-sequentially
+Cohen-Macaulay graphs, including cycles and complete multipartite
+graphs.
+References
+[1] AnnettaAramovaandJ ¨urgenHerzog, Almost regular sequences and Betti num-
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+1974, pp. 61–66. Lecture Notes in Math., Vol. 411.ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 13
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+analogue of the Erd˝ os-Ko-Rado theorem , Graph theory in Paris, Trends Math.,
+Birkh¨auser, Basel, 2007, pp. 225–242.
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+Rado graphs , Discrete Math. 293(2005), no. 1-3, 155–164.
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+1995.
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+arXiv:math/0512086.ERD˝OS-KO-RADO THEOREMS FOR SIMPLICIAL COMPLEXES 14
+[29] John Talbot, Intersecting families of separated sets , J. London Math. Soc. (2)
+68(2003), no. 1, 37–51.
+[30] Adam Van Tuyl and Rafael H. Villarreal, Shellable graphs and sequentially
+Cohen-Macaulay bipartite graphs , J. Combin. Theory Ser. A 115(2008), no. 5,
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+Proc. Amer. Math. Soc. 137(2009), no. 10, 3235–3246, arXiv:0810.0311.
+E-mail address :russw@math.wustl.edu
+Department of Mathematics, Washington University in St. Lo uis,
+St. Louis, MO, 63130
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+arXiv:1001.0314v1  [math.DS]  4 Jan 2010ORBITS FOR PRODUCTS OF MAPS
+APISIT PAKAPONGPUN AND THOMAS WARD
+Abstract. We study the behaviour of the dynamical zeta func-
+tion and the orbit Dirichlet series for products of maps. The be-
+haviour under products of the radius of convergence for the zet a
+function, and the abscissa of convergence for the orbit Dirichlet s e-
+ries, are discussed. The orbit Dirichlet series of the cartesian cube
+of a map with one orbit of each length is shown to have a natural
+boundary.
+1.Introduction
+A fundamental topological invariant of a dynamical system – here
+thought of as a continuous map T:X→Xof a compact metric space
+– is its orbit-counting data. Analytic properties of functions captu r-
+ing this data have been widely exploited in dynamics. Recently the
+authors [12] studied functorial properties of orbit-counting fun ctions,
+relating disjoint unions, Cartesian products, and iterates of maps to
+corresponding operations on the orbit-counting functions. Here we fo-
+cus on some analytic questions in the same spirit, a simple example be-
+ingthis: Whatistherelationshipbetween theanalyticpropertiesoft he
+dynamical zeta functions ζT1,ζT2andζT1×T2? Similar questions arise
+forthe orbit Dirichlet series introduced in [7], where analyticpropert ies
+are directly related by Tauberian theorems to the usual orbit-gro wth
+function πT.
+In order to highlight the underlying combinatorial questions, we
+take a cavalier attitude to maps in the following sense. For any se-
+quencea= (an)n/greaterorequalslant1of non-negative integers, there is (manifestly) a
+map on Nwithanclosed orbits of length nfor each n/greaterorequalslant1; via a
+compactification there is a continuous map on a compact metric spac e
+withthesameproperty; finally, viaabeautifultheoremofWindsor[1 7],
+there is a C∞diffeomorphism of the two-torus with the same property.
+Thus all our remarks below may be seen as being about abstract com -
+binatorial maps or about (unspecified) smoothexamples. Inthe fo rmer
+setting, the paradigmatic examples are those for which the sequen cea
+Date: Draft - November 2, 2018.
+2010Mathematics Subject Classification. 37P35.
+12 APISIT PAKAPONGPUN AND THOMAS WARD
+is arithmetically simple, the prototype being a map with exactly one
+orbit of each length, with the natural analytic tool being the orbit
+Dirichlet series. In the latter setting, the paradigmatic example migh t
+be an Axiom A diffeomorphism of the torus, with the natural tool be-
+ing the dynamical zeta function. Thus two examples of the argumen ts
+below are the following. Firstly, if Thas one orbit of each length,
+then the orbit Dirichlet series dTis the Riemann zeta function, and a
+calculation shows that dT×T(s) =ζ(s)2ζ(s−1)
+ζ(2s), with abscissa of conver-
+gence at 2 and a meromorphic extension to the plane; more surprisin g
+is the fact that for the Cartesian cube we find that dT×T×T(s) has
+abscissa of convergence at 3, a meromorphic extension to ℜ(s)>1,
+and a natural boundary at ℜ(s) = 1. This is a striking instance of
+a naturally-occurring Dirichlet series with a natural boundary. Sec -
+ondly, if T1andT2are maps with rational dynamical zeta functions,
+what relates the discs of convergence of ζT1andζT2to that of ζT1×T2?
+2.Products and Iterates
+LetT(orT1,T2,...) be maps. A closed orbit τof length |τ|is a set
+of the form {x,Tx,...,T|τ|x=x}with cardinality |τ|; writeOT(n) for
+the number of closed orbits of length nunderT. We always assume
+thatOT(n)<∞for alln/greaterorequalslant1.
+The number of points of period n(that is, the number of points
+fixed by the iterate Tn) isFT(n) =/summationtext
+d|ndOT(d). The dynamical zeta
+function associated to Tis the function
+ζT(z) = exp∞/summationdisplay
+n=1FT(n)zn
+n,
+with radius of convergence ̺(ζT) = 1/limsupn→∞FT(n)1/n(which may
+be zero), and the orbit Dirichlet series associated to Tis
+dT(s) =∞/summationdisplay
+n=1OT(n)
+ns,
+convergent ona(possiblyempty) half-plane ℜ(s)> σ(dT), whereσ(dT)
+is the abscissa of convergence. Analytic properties of ζTanddTmay
+be used in several ways, the most immediate being that asymptotics
+for
+πT(N) =|{τ| |τ|/lessorequalslantN}|3
+may be found via Tauberian theorems. The usual M¨ obius relation
+between the sequences ( OT(n)) and (FT(n)) means that
+dT(s) =1
+ζ(s+1)∞/summationdisplay
+n=1FT(n)
+ns+1, (1)
+and, viewed via the Euler transform, the same relation means that
+ζT(z) =/productdisplay
+τ(1−z|τ|)−1=∞/productdisplay
+n=1(1−z)−OT(n).
+ClearlyFT1×T2(n) =FT1(n)FT2(n) for all n/greaterorequalslant1, and as pointed out
+in [12, Lem. 1], it follows that
+OT1×T2(n) =/summationdisplay
+lcm(d1,d2)=ngcd(d1,d2)OT1(d1)OT2(d2) (2)
+(this may be seen using (1) or by pure thought). The arithmetic pro p-
+erties of the operation (2) are rather subtle.
+Turning now to iterates of a single map (rather than products of
+pairs of maps), write D(n) for the set of prime divisors of n∈N,
+and for a prime decomposition n=pa=pa1
+1···parrand a subset J⊂
+D(n), writepaJ
+Jfortherestricted product/producttext
+pj∈Jpaj
+j. The basicformula
+for orbit-counting under iteration is found in [12, Th. 4]: if m=pa
+andJ=J(n) =D(m)\D(n), then
+OTm(n) =/summationdisplay
+d|paJ
+Jm
+dOT(mn
+d). (3)
+In this expression Jdepends on n, so it involves a splitting into cases
+depending on the set of primes dividing n. The corresponding formula
+for fixed points is once again trivial: FTk(n) =FT(kn) for alln,k/greaterorequalslant1.
+Example 2.1. The quadratic map T:x/ma√sto→1−cx2on the inter-
+val [−1,1] at the Feigenbaum value c= 1.401155···has exactly one
+orbit of length 2kfor eachk/greaterorequalslant0, so (as pointed out by Ruelle [16])
+ζT(z) =∞/productdisplay
+n=0/parenleftbig
+1−z2n/parenrightbig−1=∞/productdisplay
+n=0/parenleftbig
+1+z2n/parenrightbign+1,
+satisfying the functional equation ζT(z2) = (1−z)ζT(z). More enlight-
+ening from an analytic point of view is to note that
+dT(s) =1
+1−2−s, (4)
+withσ(dT) = 0. It is clear that πT(N) =logN
+log2+O(1); this toy case may
+also be found by applying Perron’s theorem [13] or Agmon’s Tauberian4 APISIT PAKAPONGPUN AND THOMAS WARD
+theorem[1]to(4). Even inthissimplecase somecareisneeded asthe re
+are infinitely many poles on the critical line ℜ(s) = 0, and the corre-
+sponding residue sums areonly conditionally convergent. A calculatio n
+using (3) (see [12] for the details) shows that
+dTk(s) =|k|−1
+2−1+|k|−1
+2dT(s),
+so in this case σ(dTk) =σ(dT) for allk/greaterorequalslant1. Similarly,
+dT×T(s) =3
+1−2−(s−1)−2
+1−2−s,
+soσ(dT×T) =σ(dT)+σ(dT)+1 in this case.
+In pursuit of the behaviour of the abscissa of convergence for pr od-
+ucts, Ramanujan’s formula [15] for the Dirichlet series with coeffi-
+cientsσa(n)σb(n) may be used together with (2) to give the following
+(the detailed calculation is in the first author’s thesis [11]).
+Example 2.2. LetT1be map with naorbits of length nand letT2be
+a map with nborbits of length n, so that dT1(s) =ζ(s−a) anddT2(s) =
+ζ(s−b). Then
+dT1×T2(s) =ζ(s−a)ζ(s−b)ζ(s−a−b−1)
+ζ(2s−a−b).
+Thusσ(dT1×T2) =σ(dT1)+σ(dT2) in this case. Perron’s theorem applies
+to show that
+πT1×T2(N)∼Res(dT1×T2(s)Ns/s)s=a+b+2
+=ζ(a+2)ζ(b+2)
+2ζ(a+b+4)+(a+b)ζ(a+b+4)Na+b+2.
+Example 2.3. LetT1be the full shift on asymbols, and T2the full
+shift onbsymbols, so that ζT1(z) = 1/(1−az) andζT2(z) = 1/(1−bz).
+Clearly in this case ζT1×T2(z) = 1/(1−abz), so̺(ζT1×T2) =̺(ζT1)̺(ζT2).
+Our first result is that the phenomena in Example 2.3 holds for
+rational zeta functions. Recall that a linear recurrence sequenc e is said
+to be non-degenerate if among the non-trivial ratios of zeros of t he
+characteristic polynomial no unit roots are found (see [9, Sect. 1 .1.9]),
+and we say that a rational zeta function ζTis non-degenerate if the
+linear recurrence sequence satisfied by the sequence ( FT(n)) is non-
+degenerate.
+Theorem 2.4. IfζT1andζT2are non-degenerate rational functions,
+then̺(ζTk
+1) =̺(ζT1)k, and̺(ζT1×T2) =̺(ζT1)̺(ζT2).5
+Proof.The first assertion is immediate: if ζT1is rational, then by [2]
+there are algebraic numbers β1,...,β randα1,...,α swith
+FT1(n) =r/summationdisplay
+i=1βn
+i−s/summationdisplay
+i=1αn
+i, (5)
+giving the statement at once.
+The second statement is more delicate. If
+ζTj(s) =r(j)/productdisplay
+i=1/parenleftBig
+1−α(j)
+iz/parenrightBigs(j)/productdisplay
+i=1/parenleftBig
+1−β(j)
+iz/parenrightBig−1
+forj= 1,2 then
+ζT1×T2(z) =r(1)/productdisplay
+i=1s(2)/productdisplay
+j=1/parenleftBig
+1−α(1)
+iβ(2)
+jz/parenrightBigr(2)/productdisplay
+i=1s(1)/productdisplay
+j=1/parenleftBig
+1−α(2)
+iβ(1)
+jz/parenrightBig
+r(1)/productdisplay
+i=1r(2)/productdisplay
+j=1/parenleftBig
+1−α(1)
+iα(2)
+jz/parenrightBigs(2)/productdisplay
+i=1s(1)/productdisplay
+j=1/parenleftBig
+1−β(1)
+iβ(2)
+jz/parenrightBig.
+Thus̺(ζT1×T2) is the reciprocal of
+max{α(1)
+iα(2)
+j,β(1)
+kβ(2)
+ℓ|1/lessorequalslanti/lessorequalslantr(1),1/lessorequalslantj/lessorequalslantr(2),1/lessorequalslantk/lessorequalslants(1),1/lessorequalslantℓ/lessorequalslants(2)},
+(6)
+and we claim that the reciprocal of (6) is equal to
+max{β(1)
+iβ(2)
+j|1/lessorequalslanti/lessorequalslants(1),1/lessorequalslantj/lessorequalslants(2)}−1.
+That is, the exponential growth due to the poles of the zeta funct ion
+dominatesthegrowthduetothezeros. InsimplecaseslikeExample2 .3
+this is obvious, but in general account needs to be taken of possible
+cancellation among terms of equal modulus in (5).
+Lemma 2.5. IfζTis a non-degenerate zeta function with (5), then
+max{|βi| |1/lessorequalslanti/lessorequalslantr}/greaterorequalslantmax{|αi| |1/lessorequalslanti/lessorequalslants}.
+Proof.If max{|αi|,|βj|}<1 thenFT(n)→0 asn→ ∞, soFT(n) = 0
+foralllarge n, andthereforethefunctionisdegenerate(see[9, Th.2.1]).
+It follows that max {|αi|,|βj|}/greaterorequalslant1. If max {|αi|}= 1, then max {|βj|}/greaterorequalslant
+1 sinceFT(n)/greaterorequalslant0 for all n/greaterorequalslant1 and we are done. Assume therefore
+that max {|αi|}>1, and for the purposes of a contradiction assume
+that
+1/lessorequalslantmax{|βj|}<max{|αi|},
+and choose ǫ >0 so that
+max{|βj|}<(max{|αi|})1−ǫ. (7)6 APISIT PAKAPONGPUN AND THOMAS WARD
+By [2, Prop. 1] the numbers αiandβjare algebraic numbers (in-
+deed, are reciprocals of algebraic integers), so that the estimate s of
+Evertse [10] or van der Poorten and Schlickewei [14] may be applied t o
+see that there is an N(T,ǫ) with
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingler/summationdisplay
+i=1αn
+i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/greaterorequalslants(max{|αi|})n(1−ǫ)
+forn/greaterorequalslantN(T,ǫ). Then, by (7),
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingler/summationdisplay
+i=1αn
+i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle> smax{|βj|}n(for all large n)
+/greaterorequalslants/summationdisplay
+j=1|βj|n(for all large n),
+which would make FT(n) negative for large n, an impossibility. /square
+This completes the proof, since Lemma 2.5 shows that ̺(ζTj) =
+max{|β(j)
+i|}forj= 1,2 and that ̺(ζT1×T2) is the product. /square
+The next two examples show that the relationships found in Theo-
+rem 2.4 do not hold in general.
+Example 2.6. LetP={p1,p2,...}be the set of primes written in
+order, and let P1={p2,p4,...},P2={p1,p3,...}be the primes of
+even and of odd index respectively. Let Tjbe a map with
+OTj(n) =/braceleftBigg
+An
+jifn∈ Pj;
+0 if not ,
+forj= 1,2, whereA1= 2andA2= 3. Then FTj(n) =/summationtext
+p∈Pj;|n|p<1pAp
+j,
+and so
+Fp2k+j(Tj)1/p2k+j→Aj (8)
+ask→ ∞for each j= 1,2. On the other hand, a simple induction
+argument shows that
+/parenleftbig
+a1Aa1
+j+a2Aa2
+j+···+arAar
+j/parenrightbig1/a1a2···ar/lessorequalslantAj
+for distinct a1,...,a r/greaterorequalslant1, soAjis in fact the upper limit in (8),
+and̺(ζTj) = 1/Ajforj= 1,2. Turning to the product, let n=n1n2,
+where
+nj=u(j)/productdisplay
+i=1qai,j
+i,j7
+withqi,j∈ Pjandai,j>0. Then
+FT1×T2(n) =
+u(1)/summationdisplay
+i=1si,12si,1
+
+u(1)/summationdisplay
+i=1si,23si,2
+,
+and straightforward estimates show that
+limsup
+n→∞FT1×T2(n)1/n<6.
+Thus, for this example, ̺(ζT1×T2)< ̺(ζT1)̺(ζT2).
+Example 2.7. The map T1from Example 2.6 has 2norbits of length n
+ifn∈ P1, and none otherwise, and we have seen in (8) that ̺(ζT1) =1
+2.
+On the other hand, FT2
+1(n) =FT1(2n) =/summationtext
+p∈P1;|n|p<1p2p, so̺(ζT2
+1) =1
+2
+also.
+Example 2.1 has σ(dT1)+σ(dT2)−σ(dT1×T2) = 1; some simple esti-
+mates show that this discrepancy cannot be any larger.
+Proposition 2.8. σ(dT1×T2)/lessorequalslantσ(dT1)+σ(dT2)+1.
+Proof.Letσj=σ(Tj)forj= 1,2. Then, forany ǫ >0,dTj(σj+ǫ)<∞
+and so∞/summationdisplay
+n=1FTj(n)
+n1+σj+ǫ<∞
+forj= 1,2 by (1). Thus
+∞/summationdisplay
+n=1FT1×T2(n)
+n2+σ1+σ2+2ǫ<∞,
+and therefore dT1×T2(1+σ1+σ2+2ǫ)<∞by (1) again. /square
+3.Higher products
+Even in the simplest of situations, higher products have quite subtle
+combinatorial and analytic properties, and for simplicity we restrict
+attention to the case of a map with a single orbit of each length. Sim-
+ilar methods will apply to maps for which the sequence ( OT(n)) is
+multiplicative.
+Proposition 3.1. LetTbe a map with OT(n) =naforn/greaterorequalslant1.
+ThenOT×···×T(n)is equal to
+/productdisplay
+p|n1
+(pa+1−1)m−1
+m−1/summationdisplay
+r=0(−1)r/parenleftbiggm
+m−r/parenrightbigg
+p((m−r)(a+1)−1)ordp(n)(m−r)(a+1)−1/summationdisplay
+j=0pj
+
+where there are mterms in the Cartesian product.8 APISIT PAKAPONGPUN AND THOMAS WARD
+Proof.We have FT(n) =/summationtext
+d|nda+1=σa+1(d) and fixed points for iter-
+atessimply multiply forCartesianproducts so, foraprime pandk/greaterorequalslant1,
+by (1),
+OT×···×T(pk) =1
+pk/summationdisplay
+d|pkµ(pk/d)(σa+1(d))m
+=1
+pk/parenleftBig/parenleftBig
+p(a+1)(k+1)−1
+pa+1−1/parenrightBigm
+−/parenleftBig
+p(a+1)k−1
+pa+1−1/parenrightBigm/parenrightBig
+.
+Clearlyn/ma√sto→OT×···×T(n) is multiplicative, so this proves the proposi-
+tion. /square
+Proposition 3.1 allows the orbit Dirichlet series for higher powers to
+be computed (in the case dT(s) =ζ(s), with trivial changes for OT(s)
+polynomial). To this end, assume that dT(s) =ζ(s−a), letf(n) =
+OT×···×T(n), write
+dT×···×T(s) =/productdisplay
+p∈P/parenleftbig
+1+f(p)p−s+f(p2)p−2s+···/parenrightbig
+=/productdisplay
+p∈PEp(s),
+and define θby
+f(pk) =1
+(pa+1−1)m−1θ(pk).
+Then
+Ep(s) = 1+1
+(pa+1−1)m−1m−1/summationdisplay
+b=0Abp(b+1)a+b−s1
+1−p(b+1)a+b−s,
+whereAb= (−1)r/parenleftbigm
+m−r/parenrightbig/summationtext(b+1)a+b
+j=0pj, andm−r−1 =b, so by rear-
+ranging
+Ep(s) =Mp(s)
+(1−pa−s)(1−p2a+1−s)(1−p3a+2−s)···(1−pma+(m−1)−s)
+withMp(s)/\e}atio\slash= 0. Thus dT×···×T(s) is given by
+ζ(s−a)ζ(s−(2a+1))ζ(s−(3a+2))···ζ(s−(ma+m−1))/productdisplay
+p∈PMp(s),
+whereMp(s) is (in principle) explicitly computable, and so
+σ(dT×···×T) =ma+m.
+Example 3.2. By Perron’s theorem [13] we deduce that if dT(s) =
+ζ(s), then
+πT×···×T(N)∼Cmζ(m)ζ(m−1)···ζ(2)Nm
+m9
+whereCm=/producttext
+pMp(m) is an explicit constant. Thus, for exam-
+ple,πT(N)∼NandπT×T(N)∼π2
+12N2, while
+πT×T×T(N)∼C3π2ζ(3)
+18N3,
+where
+C3=/productdisplay
+p(1+p−5+2p−2+2p−3) = 2.835979....
+Example 3.3. Example 2.2 with a=b= 0 gives
+dT×T(s) =ζ(s)2ζ(s−1)
+ζ(2s),
+and the calculation above gives
+dT×T×T(s) =ζ(s)ζ(s−1)ζ(s−2)/productdisplay
+p∈P/parenleftbig
+1+(2p+2)p−s+p1−2s/parenrightbig
+.(9)
+Remark 3.4. The Euler product/producttext
+p(1+p1−2s+2p1−s+2p−s)issugges-
+tive, but deceptively so. Under the Hecke correspondence, the m odular
+form with Fourier series f(τ) =c(0)+/summationtext∞
+n=1c(n)e2πinτhas associated
+Dirichlet series
+φ(s) =∞/summationdisplay
+n=1c(n)
+ns=/productdisplay
+p/parenleftbig
+1−c(p)p−s+p2k−1p−2s/parenrightbig−1.
+However, there is no real connection because the choice of param e-
+ters needed violates the (known) Weil bounds that |r2|=|r2|=√p
+where 1−c(p)x+p2k−1x2= (1−r1x)(1−r2x). (Equivalently, the seem-
+ing relationship with an L-function of an elliptic curve is meaningless
+because the choice of parameters violates the Hasse bounds).
+4.Natural boundaries
+Natural boundaries for Dirichlet series arise in several contexts. Es-
+terman’s theorem [6] gives a large class of Euler products of the for m
+/productdisplay
+ph(p−s)
+with natural boundaries. The example below is more closely related
+to the work of Grunewald, du Sautoy and Woodward [4], [5] on zeta
+functions for subgroup growth, where products of ‘ghost’ polyn omials
+are used to exhibit natural boundaries for products of the form
+/productdisplay
+ph(p−s,p).10 APISIT PAKAPONGPUN AND THOMAS WARD
+Natural boundaries also arise for dynamical zeta functions in seve ral
+natural dynamical settings, including certain random maps [3] and a u-
+tomorphisms of certain solenoids [8].
+Weexhibit anaturalboundaryforaspecificcase, buttheappeara nce
+of a natural boundary for triple (and higher) products of system s with
+polynomial orbit growth is a widespread phenomena.
+Theorem 4.1. IfdT(s) =ζ(s), thendT×T×T(s)has abscissa of con-
+vergence at 3, a meromorphic extension to ℜ(s)>1, and a natural
+boundary at ℜ(s) = 1.
+Proof.By (9) we know that
+dT×T×T(s) =ζ(s)ζ(s−1)ζ(s−2)/productdisplay
+p∈Pf(p−s,p)
+wheref(p−s,p) = (1 + (2 p+ 2)p−s+p1−2s). The term/producttext
+pf(p−s,p)
+converges for ℜ(s)>2, so the abscissa of convergence is determined
+by the term ζ(s−2).
+To show that ℜ(s) = 1 is a natural boundary, we show that each
+pointswithℜ(s) = 1isalimitofasequence( sn)ofzerosof/producttext
+pf(p−s,p)
+withℜ(sn)>1. Solving the quadratic f(x,p) = 0 for xgives the solu-
+tions
+α±
+p=−/parenleftBig
+1+1
+p/parenrightBig
+±/radicalbigg/parenleftBig
+1+1
+p/parenrightBig2
+−1
+p,
+and the zero α+
+p=p−shas solutions
+sn,p=−log|α+
+p|
+logp+πi+2kπi
+logp
+fork∈Z. Notice that−3+√
+7
+2< α+
+p<0 for all p,α+
+p→0 asp→ ∞,
+and by the binomial theorem α+
+p∼ −1
+2pfor large primes p. It follows
+thatℜ(sn,p)>1 andℜ(sn,p)→1 asp→ ∞. Thus given any s
+withℜ(s) = 1 we may choose a sequence ( snk,pk) with the properties
+that
+(i)ℜ(snk,pk)>1;
+(ii)snk,pk→sask→ ∞;
+(iii)/producttext
+pf(p−snk,pk,p) = 0 for all k/greaterorequalslant1.
+This shows that ℜ(s) = 1 is a natural boundary.
+It remains to show that there is a meromorphic extension to the
+half-plane ℜ(s)>1, and here we follow the methods of [5]. Using the
+lexicographic ordering on N2to eliminate terms in ascending powers11
+ofxand then y, there is a unique decomposition
+f(x,y) =/productdisplay
+(m,n)∈N2(1−xmyn)c(m,n)
+withc(m,n)∈Z, wheref(x,y) = 1 + 2 x+ 2xy+yx2. This may
+be constructed using factors of the shape (1 −xayb)eto eliminate a
+term−exayb(e >0) and factors of the shape (1 −x2ay2b)e(1−xayb)−e
+to eliminate a term exayb(e >0), obtaining an approximation valid to
+larger and larger powers of xby induction. Thus, for example, we find
+that
+f(x,y) = (1−x2)3(1−x)−2(1−x2y2)2(1−xy)−2(1−x2y)3(1−x2y2)+O(x3).
+By construction, if
+f(x,y) =/productdisplay
+m/lessorequalslantM(1−xmyn)c(m,n)+/summationdisplay
+m>Me(m,n)xmyn
+then ifc(m,n) ande(m,n) are non-zero we must have n/lessorequalslantm. Thus
+for eachMandℜ(s)>max{(n+1)/m|e(m,n)/\e}atio\slash= 0}the product
+fM(s) =/productdisplay
+p/parenleftBigg
+1+/summationtext
+m>Me(m,n)pn−ms
+/producttext
+p(1−pn−ms)c(m,n)/parenrightBigg
+converges absolutely, allowing/producttext
+pf(p−s,p) to be defined there by
+/productdisplay
+(m,n)∈N2,m/lessorequalslantMζ(ms−n)−c(m,n)fM(s).
+LettingM→ ∞gives a meromorphic extension to ℜ(s)>1. /square
+References
+[1] S. Agmon. Complex variable Tauberians. Trans. Amer. Math. Soc. , 74:444–
+481, 1953.
+[2] R. Bowen and O. E. Lanford, III. Zeta functions of restriction s of the shift
+transformation. In Global Analysis (Proc. Sympos. Pure Math., Vol. XIV,
+Berkeley, Calif., 1968) , pages43–49.Amer.Math.Soc., Providence,R.I., 1970.
+[3] J. Buzzi. Some remarks on random zeta functions. Ergodic Theory Dynam.
+Systems, 22(4):1031–1040, 2002.
+[4] M. du Sautoy and F. Grunewald. Zeta functions of groups: zero s and friendly
+ghosts.Amer. J. Math. , 124(1):1–48, 2002.
+[5] M. du Sautoy and L. Woodward. Zeta functions of groups and rings , volume
+1925 ofLecture Notes in Mathematics . Springer-Verlag, Berlin, 2008.
+[6] T. Estermann. On certain functions represented by Dirichlet se ries.Proc. Lon-
+don Math. Soc. , 27:435–448, 1928.
+[7] G. Everest, R. Miles, S. Stevens, and T. Ward. Dirichlet series fo r finite com-
+binatorial rank dynamics. Trans. Amer. Math. Soc. , 362(7):199–227, 2010.12 APISIT PAKAPONGPUN AND THOMAS WARD
+[8] G. Everest, V. Stangoe, and T. Ward. Orbit counting with an isom etric direc-
+tion. InAlgebraic and topological dynamics , volume 385 of Contemp. Math. ,
+pages 293–302. Amer. Math. Soc., Providence, RI, 2005.
+[9] G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward. Recurrence
+sequences , volume 104 of Mathematical Surveys and Monographs . American
+Mathematical Society, Providence, RI, 2003.
+[10] J.-H. Evertse. On sums of S-units and linear recurrences. Compositio Math. ,
+53(2):225–244, 1984.
+[11] A. Pakapongpun. Functorial orbit counting . PhD thesis, Univ. East Anglia,
+2010.
+[12] A. Pakapongpun and T. Ward. Functorial orbit counting. J. Integer Seq. ,
+12(2):Article 09.2.4, 20, 2009.
+[13] O. Perron. Zur Theorie der Dirichletschen Reihen. J. Reine Angew. Math. ,
+134:95–143, 1908.
+[14] A. J. van der Poorten and H. P. Schlickewei. Additive relations in fi elds.J.
+Austral. Math. Soc. Ser. A , 51(1):154–170, 1991.
+[15] S. Ramanujan. Some formulae in the analytic theory of numbers .Messenger ,
+45:81–84, 1915.
+[16] D. Ruelle. Dynamical zeta functions and transfer operators. Preprint
+IHES/M/02/66.
+[17] A. J. Windsor. Smoothness is not an obstruction to realizability. Ergodic The-
+ory Dynam. Systems , 28(3):1037–1041, 2008.
+School of Mathematics, University of East Anglia, Norwich, NR4
+7TJ, UK
+E-mail address :t.ward@uea.ac.uk
\ No newline at end of file
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+arXiv:1001.0315v2  [hep-ph]  12 Mar 2010TUHEP-TH-09170
+Computation of the coefficients for the order p6anomalous chiral Lagrangian
+Shao-Zhou Jiang1,2∗, and Qing Wang1,2†‡
+1Center for High Energy Physics, Tsinghua University, Beiji ng 100084, P.R. China
+2Department of Physics, Tsinghua University, Beijing 10008 4, P.R. China§
+We present the results of computing the order p6low energy constants in the anomalous part of
+the chiral Lagrangian for both two and three flavor pseudosca lar mesons. This is a generalization of
+ourprevious work on calculating theorder p6coefficients for thenormal partof thechiral Lagrangian
+in terms of the quark self energy Σ( p2). We show that most of our results are consistent with those
+we have found in the literature.
+PACS numbers: 12.39.Fe, 11.30.Rd, 12.38.Aw, 12.38.Lg
+I. INTRODUCTION AND BACKGROUND
+It is well known that the chiral symmetry in quantum chromodynamic s (QCD) suffers anomalies due to the non-
+invariance of the path integral measure of the quark fields under t he chiral symmetry transformation. The anomaly
+reflectsthe fact that the classicalchiralsymmetry maybe violate dby quantum corrections. At the levelofthe effective
+chiralLagrangianforthepseudoscalarmesonfield U, anomalynolongercomesfromthepathintegralmeasure. Instea d
+it is due to the non-invariance of the effective chiral Lagrangian. If we denote by Γ eff[U,J] the effective action for the
+pseudoscalar meson field Uand the external source J, then this non-invariance can be expressed as
+Γeff[U,J]−Γeff[UΩ,JΩ] = Γ[Ω,J], (1)
+whereUΩ≡Ω†UΩ†andJΩ≡[ΩPR+Ω†PL][J+/∂][ΩPR+Ω†PL]. Γ[Ω,J] is the anomaly from the light quark path
+integral measure D¯ψΩDψΩ=D¯ψDψ eΓor the well known Wess-Zumino-Witten term. We can formally expres s it as
+Γ[Ω,J] =−lnDet[(ΩPR+Ω†PL)(ΩPR+Ω†PL)] =−Trln[/∂+JΩ]+Trln[/∂+J], (2)
+Because for Nflight quarks, each generator of the chiral symmetry SU(Nf)L⊗SU(Nf)R/SU(Nf)Vcorresponds to
+a Goldstone boson, which is treated phenomenologically as the physic al pseudoscalar meson field, the phase angle of
+the chiral rotation group element Ω can be treated as the pseudos calar meson field, i.e. U= Ω2. Then comparing (1)
+and (2), we can rewrite the effective action Γ effas
+Γeff[U,J] =−Trln[/∂+JΩ]+Trln[/∂+J]+F[U,J] F[U,J] =F[UΩ,JΩ] (3)
+TheUandJdependence for F[U,J] is not fixed by (1), but F[U,J] is invariant on U→UΩandJ→JΩ. Hence
+F[U,J] represents those chiral invariant terms. In fact UΩ= Ω†Ω2Ω†= 1, and Γ eff[UΩ,JΩ] = Γeff[1,JΩ] =−Trln[/∂+
+JΩ]+Trln[/∂+JΩ]+F[UΩ,JΩ] =F[U,J]. Note that the effective action is the path integration result for Seff[U,J],
+the action of the effective chiral Lagrangian for the pseudoscalar meson field Uand the external source J,
+e−Γeff[Ucl,J]=/integraldisplay
+DU e−Seff[U,J]Ucl(x)≡/integraldisplay
+DU U(x)e−Seff[U,J], (4)
+where the second equation gives the definition of Uclwhich fixes Uclas the functional of the external source J. With
+(3), (4) becomes
+eTrln[/∂+JΩ]−Trln[/∂+J]−F(Ucl,J)=/integraldisplay
+DU e−Seff[U,J]. (5)
+Ref.[1],[2],[3] choose as an approximation
+Seff,0[U,J] =−Trln[/∂+JΩ]+Trln[/∂+J], (6)
+∗Email:jsz@mails.tsinghua.edu.cn.
+†Email: wangq@mail.tsinghua.edu.cn.
+‡corresponding author
+§mailing addresswhere subscript 0is used to denote the approximation. From (1), (3) and (5), we find that under the chiral symmetry
+transformation, Seff,0[U,J], defined in (6), is not invariant. Substituting (6) back into (5) and u sing standard loop
+expansion as developed in Ref.[4], we find F[Ucl,J] is the pure loop correction from the action Seff,0[U,J]. From the
+action (6), one can calculate various low energy constants (LECs) of the effective chiral Lagrangian for pseudoscalar
+mesons. In Ref.[5], we call (6) the anomaly approach. In our previou s paper [6], we have shown that the finite
+orderp4LECs of the normal part of Seff,0[U,J] are exactly canceled by the summation of all the p6and higher order
+terms. Eq.(2) further shows that even for the anomalous part, Seff,0[U,J] only contributes the Wess-Zumino-Witten
+term; it cannot produce the p6and higher order anomaly terms. This absence of the normal part a nd thep6and
+more higher order anomalous part reflects the fact that the choic e of (6) is not correct, although it offers the correct
+Wess-Zumino-Witten term. Further, (6) is independent of the str ong interaction dynamics, i.e., even we switch off the
+quark-gluon interaction by deleting the strong interaction coupling constant, (6) is not changed. These facts imply
+that we need to add some strong dynamics dependent correction t erm ∆Seff[U,J] toSeff,0[U,J] as given in (6),
+Seff[U,J] =Seff,0[U,J]+∆Seff[U,J]. (7)
+From(5)and(6), wefindthat∆ Seff[U,J], introducedin(7), mustbeinvariantunderchiralsymmetrytrans formations.
+In Refs.[7] and [8], ∆ Seff[U,J] is taken to be
+∆Seff[U,J] = Trln[/∂+JΩ+Σ(−¯∇2)] (8)
+with Σbeing the quarkselfenergysatisfyingtheSchwinger-Dysone quation(SDE) and ¯∇µis definedas ¯∇µ≡∂µ−ivµ
+Ω.
+Thisexpressionfor∆ Seff[U,J]encodesthedynamicsoftheunderlyingQCDthroughquarkselfen ergyΣandinRef.[9],
+we have shown that (8) does not produce the Wess-Zumino-Witten term ensuring the correctness of (1).
+In Ref.[7], we have calculated the orders p2andp4normal part LECs in terms of the action (7) and (8). The
+importance of knowledge of LECs of the chiral Lagrangian, especia lly for order p6LECs was emphasized in Ref.[10].
+Recently, in Ref.[6], we improved the computation procedure and gen eralized the calculations up to the order p6
+normal part LECs. In Ref.[9], we have calculated the p4order anomalous part and shown that the Σ dependent
+coefficient generates the correct coefficient Ncfor the Wess-Zumino-Witten term. It is the purpose of this paper t o
+calculate all order p6LECs for the anomalous part of the chiral Lagrangian (7). In fact the general structure of the
+p6order anomalous part chiral Lagrangian was first given by Refs.[11] and [12] and later clarified by Refs.[13] and
+[14]. Ref.[15] estimates the values of several of the order p6LECs for the anomalous part of the chiral Lagrangian.
+Although order p6LECs for the normal part of the chiral Lagrangian seem attract m ore attentions in the literature
+(see references given in [6]), they are the next next to leading orde r terms. The order p6LECs for the anomalous part
+of the chiral Lagrangian are belong to the next leading order terms .
+This paper is organized as follows: in Sec.II, we review the calculation o f the order p4anomalous part of the chiral
+Lagrangian in terms of the action (7). With the method used in sectio n II, in Sec.III, we compute the order p6LECs
+for the anomalous part of the chiral Lagrangian, and obtain the an alytical expression for the LECs in terms of quark
+self energy Σ. We further compute the numerical values for these LECs. We compare our results with those obtained
+in literature. Sec.IV is the summary and future directions of our wor k. We list some necessary tables and formulae
+in appendices.
+II. REVIEW THE ORDER p4ANOMALOUS PART OF THE CHIRAL LAGRANGIAN
+For the anomalous part of the chiral Lagrangian, the leading nontr ivial order is p4and it is the well known Wess-
+Zumino-Witten term. In Ref.[9], we have calculated the action (7) by s everal different methods and all obtain the
+same Wess-Zumino-Witten term. If we naively apply these methods t o the next to leading order p6computations,
+we will find that they are too complex to be achieved even with the help of the computer. In this section, we build
+a method which is suitable to be generalized to the order p6calculations. The order p4of the anomalous chiral
+Lagrangian is here only to be used to explain our method. Ref.[9] only e xpresses the Wess-Zumino-Witten term in
+terms of a parameter integration. In this section, we will explicitly fin ish this parameter integration and show that it
+does recover the Wess-Zumino-Witten term.
+Since we are only interested in the Ufield dependent part of the anomalous part of the chiral Lagrangia n, we can
+drop out the pure source terms. Then our choice of ∆ Seff[U,J] in (8) gives the result that only Σ dependent terms in
+∆Seff[U,J] contribute to the chiral Lagrangian, while the Σ independent term s in ∆Seff[U,J] are completely canceled
+by the term −Trln[/∂+JΩ] inSeff,0[U,J], leaving a pure Ufield independent term Trln[ /∂+J]. So what we need to
+compute is
+Seff[U,J] =/bracketleftbigg
+Trln[/∂+JΩ+Σ(−¯∇2)]−Trln[/∂+J+Σ(−∇2)]/bracketrightbigg
+Σdependent, (9)
+2in which we have added in Seff[U,J] an extra pure source term −Trln[/∂+J+Σ(−∇2)]/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+Σdependentfor later use, and we
+define∇µ≡∂µ−ivµ. Now we write Ω as Ω = e−iβand further introduce a parameter tdependent rotation element
+Ω(t) =e−itβ. With the help of the relation Ω(1) = Ω and Ω(0) = 1, (9) becomes
+Seff[U,J] = Trln[/∂+JΩ(t)+Σ(−∇2
+t)]/vextendsingle/vextendsingle/vextendsingle/vextendsinglet=1
+t=0,Σdependent∇µ
+t=∇µ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+Ω→Ω(t)(10)
+with∇µ
+t=∂µ−ivµ
+Ω(t).JΩ(t)isJΩwith Ω replaced by Ω( t). We decompose JΩasJΩ=−i/vΩ−i/aΩγ5−sΩ+ipΩγ5,
+so we can also decompose JΩ(t)asJΩ(t)=−i/vt−i/atγ5−st+iptγ5. Result (10) implies that our chiral Lagrangian
+can be expressed as the difference of Trln( ···) attdependent chiral rotation between t= 1 andt= 0. Since the t
+dependent rotated source JΩ(t)satisfies
+∂JΩ(t)
+∂t=1
+2[∂Ut
+∂tU†
+tγ5,/∂+JΩ(t)]+ Ut= Ω2(t), (11)
+we can further proceed to express the chiral Lagrangian in terms of integration over the parameter t:
+Seff[U,J] =/integraldisplay1
+0dtd
+dtTrln[i/∂+JΩ(t)+Σ(−∇2
+t)]/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+Σdependent
+=/integraldisplay1
+0dtTr/bracketleftbigg
+[∂JΩ(t)
+∂t+∂Σ(−∇2
+t)
+∂t][i/∂+JΩ(t)+Σ(−∇2
+t)]−1/bracketrightbigg
+Σdependent
+=/integraldisplay1
+0dtTr/bracketleftbigg/parenleftbigg1
+2[∂Ut
+∂tU†
+tγ5,/∂+JΩ(t)]++∂Σ(−∇2
+t)
+∂t/parenrightbigg
+[i/∂+JΩ(t)+Σ(−∇2
+t)]−1/bracketrightbigg
+Σdependent.(12)
+(12) is the main formula we rely on to calculate LECs. Ref.[9] explicitly ca lculates the order p4anomalous part of the
+r.h.s. of (12) and finds the result
+Seff[U,J]/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+anomalous p4=−2Ncǫµναβ/integraldisplay
+d4x/integraldisplay1
+0dt/integraldisplayd4k
+(2π)4trf/bracketleftbigg∂Ut
+∂tU†
+t/parenleftbiggΣ(k2)[Σ2(k2)−k2][Σ(k2)−2k2Σ′(k2)]
+[Σ2(k2)+k2]4
+×(2∇µ
+t∇ν
+t∇α
+t∇β
+t+2aµ
+taν
+t∇α
+t∇β
+t−2∇µ
+taν
+t∇α
+taβ
+t+2∇µ
+taν
+taα
+t∇β
+t+2aµ∇ν
+t∇α
+taβ
+t−2aµ
+t∇ν
+taα
+t∇β
+t
++2∇µ
+t∇ν
+taα
+taβ
+t+2aµ
+taν
+taα
+taβ
+t)+k2Σ(k2)[Σ(k2)−2k2Σ′(k2)]
+[Σ2(k2)+k2]4(4∇µ
+t∇ν
+t∇α
+t∇β
+t+2aµ
+taν
+t∇α
+t∇β
+t
+−2∇µ
+taν
+t∇α
+taβ
+t+4aµ
+t∇ν
+t∇α
+taβ
+t−2aµ
+t∇ν
+taα
+t∇β
+t+2∇µ
+t∇ν
+taα
+taβ
+t)/parenrightbigg/bracketrightbigg
+. (13)
+The momentum integration can be calculated analytically, because th e integrand is a total derivative. The result is
+Seff[U,J]/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+anomalous p4=1
+32π2ǫµναβ/integraldisplay
+d4x/integraldisplay1
+0dttrf/bracketleftbigg∂Ut
+∂tU†
+t/parenleftbigg
+Vµν
+tVαβ
+t+2i
+3[aµ
+taν
+t,Vαβ
+t]++4
+3dµ
+taν
+tdα
+taβ
+t
++8i
+3aµ
+tVνα
+taβ
+t+4
+3aµ
+taν
+taα
+taβ
+t/parenrightbigg/bracketrightbigg
+, (14)
+whereVµν
+t=∂µvν
+t−∂νvµ
+t−i[vµ
+t,vν
+t] anddµ
+taν
+t=∂µaν
+t−i[vµ
+t,aν
+t]. Ref.[9] only gives the above result (14) without
+finishing the integration over parameter t. Now we continue to achieve this integration, with the help of following
+relations:
+∂aµ
+t
+∂t=i
+2[∇µ
+t∂Ut
+∂tU†
+t−∂Ut
+∂tU†
+t∇µ
+t]∂vµ
+t
+∂t=1
+2[aµ
+t∂Ut
+∂tU†
+t−∂Ut
+∂tU†
+taµ
+t]
+∂st
+∂t=−i
+2[pt∂Ut
+∂tU†
+t+∂Ut
+∂tU†
+tpt]∂pt
+∂t=i
+2[st∂Ut
+∂tU†
+t+∂Ut
+∂tU†
+tst] (15)
+∂dµaν
+t
+∂t=i
+2[(∇µ
+t∇ν
+t+aν
+taµ
+t)∂Ut
+∂tU†
+t−∇ν
+t∂Ut
+∂tU†
+t∇µ
+t−∇µ
+t∂Ut
+∂tU†
+t∇ν
+t−aν
+t∂Ut
+∂tU†
+taµ
+t−aµ
+t∂Ut
+∂tU†
+taν
+t+∂Ut
+∂tU†
+t(∇ν
+t∇µ
+t+aµ
+taν
+t)]
+∂Vµν
+t
+∂t=1
+2[−∇µ
+t∂Ut
+∂tU†
+taν
+t+∂Ut
+∂tU†
+t∇µ
+taν
+t−∂Ut
+∂tU†
+tdµ
+taν
+t+dµ
+taν
+t∂Ut
+∂tU†
+t+aν
+t∇µ
+t∂Ut
+∂tU†
+t−aν
+t∂Ut
+∂tU†
+t∇µ
+t
+3+∇ν
+t∂Ut
+∂tU†
+taµ
+t−∂Ut
+∂tU†
+t∇ν
+taµ
+t+∂Ut
+∂tU†
+tdν
+taµ
+t−dν
+taµ
+t∂Ut
+∂tU†
+t−aµ
+t∇ν
+t∂Ut
+∂tU†
+t+aµ
+t∂Ut
+∂tU†
+t∇ν
+t]
+and by lengthy calculations, we can rewrite (14) as
+Seff[U,J]/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+anomalous p4=−Nc
+48π2/integraldisplay
+d4x/integraldisplay1
+0dt ǫµνλρtrf/bracketleftbigg∂Ut
+∂tU†
+tRµ
+tRν
+tRλ
+tRρ
+t+d
+dtWµνλρ(Ut,l,r)/bracketrightbigg
+(16)
+withlµ=vµ−aµ, rµ=vµ+aµ,Rµ
+t=U†
+t∂µUt,Lµ
+t= (∂µUt)U†
+tand
+Wµνλρ(Ut,l,r) =iRµ
+tRν
+tRλ
+tlρ+lµ∂νlλRρ
+t+∂µlνlλRρ
+t−1
+2Rµ
+tlνRλ
+tlρ+rµUtlνRλ
+tRρU†
+t+iRµ
+tlνlλlρ+iU†
+trµUt∂νlλlρ
++iU†
+trµ∂νrλUtlρ−ilµU†
+trνUtlλRρ
+t−Rµ
+tU†
+t∂νrλUtlρ+U†
+trµUtlνlλlρ+1
+4U†
+trµUtlνU†
+trλUtlρ
+−(Ut↔U†
+t,lµ↔rµ,Lµ
+t↔ −Rµ
+t). (17)
+In Ref.[9], we already show that the first term of the r.h.s. of Eq.(16) is just the Wess-Zumino-Witten term of the
+form defined on a four dimensional boundary disc Qin five dimensional space-time
+−Nc
+48π2/integraldisplay
+d4x/integraldisplay1
+0dt ǫµνλρtrf/bracketleftbigg∂Ut
+∂tU†
+tRµ
+tRν
+tRλ
+tRρ
+t/bracketrightbigg
+=−Nc
+240π2/integraldisplay
+QdΣijklmtrf[RiRjRkRlRm] (18)
+withRi≡U†∂iU. For the second term of the r.h.s. of Eq.(16), the integration over parametertcan be calculated
+explicitly,
+−Nc
+48π2/integraldisplay
+d4x/integraldisplay1
+0dt ǫµνλρtrf/bracketleftbiggd
+dtWµνλρ(Ut,l,r)/bracketrightbigg
+=−Nc
+48π2/integraldisplay
+d4x ǫµνλρtrf/bracketleftbigg
+Wµνλρ(U,l,r)−Wµνλρ(1,l,r)/bracketrightbigg
+,(19)
+which the just the gauge part of the Wess-Zumino-Witten term give n by Ref.[13] and [16]. This finishes the explicit
+calculation of the order p4anomalous part of the chiral Lagrangian starting from formula (12 ). We leave the order
+p6part to the next section.
+III. CALCULATION OF THE ORDER p6ANOMALOUS PART OF THE CHIRAL LAGRANGIAN
+In this section, we start from Eq.(12) to calculate its order p6anomalous part of the chiral Lagrangian. For
+convenience, we change to the Minkowski space to perform our ca lculations. Direct computation gives the result
+Seff[U,J]/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+anomalous p6=210/summationdisplay
+m=1/integraldisplay
+d4x¯KW
+m/integraldisplay1
+0dttrf[¯OW
+m(x,t)], (20)
+where¯KW
+mis the coefficient in front of the operator ¯OW
+m(x,t), which depends on quark self energy Σ( k2). The 210
+parametertdependent operators ¯OW
+m(x,t) allhavethe structureof ¯OW
+m(x,t) =ǫµνλρ∂Ut
+∂tU†
+t¯Oµνλρ
+m(x,t) and¯Oµνλρ
+m(x,t)
+are orderp6operators consisting of multiplications of various compositions of aµ
+t,∇ν
+t,standpt. In Appendix A we
+list all these operators in Table V. In obtaining (20), we have applied t he Schouten identity, which reduces the original
+total 294 operators to the present 210 operators. In the litera ture, the general p6order anomalous part of the chiral
+Lagrangian given in Ref.[13] has only 24 independent operators. For Nf= 3,2 this number reduces to 23 and five
+respectively. Specially for the case of Nf= 2, to incorporate the electro-magnetic field into the external so urcevµ,
+the original traceless property of vµmust be dropped, this changes the original five independent p6order anomalous
+operators into thirteen. If we denote the independent operator s byOW
+n(x) (oW
+n(x) forNf= 2) and corresponding
+coefficients in front of the operators by CW
+n(cW
+n(x) forNf= 2) respectively, then (20) becomes
+Seff[U,J]/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+anomalous p6=24/summationdisplay
+n=1/integraldisplay
+d4x CW
+nOW
+n(x)Nf=2=====13/summationdisplay
+n=1/integraldisplay
+d4x cW
+noW
+n(x). (21)
+Note that our starting chiral Lagrangian(7) only involves one trac e for flavor indices. If we further apply the equation
+of motion to (21), there will appear some operators with two flavor traces. Our result prohibits the appearance
+of three operators OW
+3,OW
+18,OW
+24, leaving 21 independent operators. This implies that our formulation givesCW
+3=
+4CW
+18=CW
+24= 0. If we do not apply the equation of motion, there will be more indep endent operators and now this
+number is 23. To make our calculation more convenient, we denote th ese operators before applying the equation of
+motion by ˜OW
+n(x) and the corresponding coefficients in front of the operators by ˜KW
+n. We list all possible ˜OW
+n(x) in
+the Table VI of Appendix A. With these operators, (21) can also be w ritten as
+Seff[U,J]/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+anomalous p6=23/summationdisplay
+n=1/integraldisplay
+d4x˜KW
+n˜OW
+n(x). (22)
+Through using the equation of motion, we can obtain the relations am ong the two sets of operators ˜OW
+n(x) andOW
+n(x)
+as follows
+˜OW
+1=OW
+1/B0˜OW
+2=OW
+2/B0˜OW
+3=OW
+4/B0˜OW
+4=OW
+5/B0˜OW
+5=OW
+7/B0˜OW
+6=OW
+9/B0
+˜OW
+7=OW
+11/B0˜OW
+8=OW
+12˜OW
+9=OW
+1˜OW
+10=OW
+16˜OW
+11=OW
+17˜OW
+12=OW
+13˜OW
+13=OW
+14
+˜OW
+14=OW
+15˜OW
+15=−OW
+4+2
+NfOW
+6˜OW
+16=−OW
+5−1
+NfOW
+6˜OW
+17=OW
+19˜OW
+18=OW
+20˜OW
+19=OW
+21
+˜OW
+20=OW
+22˜OW
+21=OW
+23˜OW
+22=OW
+7−1
+NfOW
+8˜OW
+23=OW
+9−1
+NfOW
+10, (23)
+whereB0is the order p2LEC in the normal part of the chiral Lagrangian. Here we divide OW
+1,···,OW
+7byB0,
+making the matrices Amnintroduced later in Eq.(27) independent of B0. ForNf= 2, (23) is changed to
+˜OW
+1= 0 ˜OW
+2=oW
+1/B0˜OW
+3=oW
+2/B0˜OW
+4=−oW
+2/(2B0)+oW
+6/B0˜OW
+5=oW
+3/B0˜OW
+6=oW
+4/B0
+˜OW
+7=oW
+5/B0˜OW
+8=˜OW
+9=˜OW
+10=˜OW
+11= 0 ˜OW
+12=−oW
+9˜OW
+13=˜OW
+14=−1
+2oW
+6+oW
+9˜OW
+15=−oW
+6
+˜OW
+16=−1
+2oW
+6˜OW
+17=oW
+10˜OW
+18=˜OW
+19=−oW
+10˜OW
+20=1
+4oW
+7−1
+8oW
+8−oW
+10+oW
+11−2oW
+13˜OW
+21= 0
+˜OW
+22=oW
+7−1
+2oW
+8˜OW
+23= 0. (24)
+Direct comparison between (20) and (22) is difficult, since in (20) we h ave an extra integration over parameter tand
+the number of operators in (20) is much larger than it is in (22). Inst ead of finishing the integration over parameter
+tin (20), we introduce an integration of parameter tin (22). Since we are only interested in the Udependent part of
+the chiral Lagrangian, adding some Ufield independent pure source terms in (22) will not change our resu lt; therefore
+we can rewrite (22) as
+Seff[U,J]/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+anomalous p6=23/summationdisplay
+n=1/integraldisplay
+d4x˜KW
+n[˜OW
+n(x)−˜OW
+n(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+U=1] =23/summationdisplay
+n=1/integraldisplay
+d4x˜KW
+n[˜OW
+n(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+U→Ut=1−˜OW
+n(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+U→Ut=0]
+=23/summationdisplay
+n=1/integraldisplay
+d4x˜KW
+n[˜OW
+n(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+U→Ut]/vextendsingle/vextendsingle/vextendsingle/vextendsinglet=1
+t=0=23/summationdisplay
+n=1/integraldisplay
+d4x˜KW
+n/integraldisplay1
+0dtd
+dt[˜OW
+n(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+U→Ut]. (25)
+In expression (25), integration of parameter tis already present in the formula, then the only remaining problem is
+that in (25) there are only 23 independent terms acted on by the diff erential oft, while in (20) there are 210 terms.
+comparing (25) and (20), we obtain
+23/summationdisplay
+n=1˜KW
+nd
+dt[˜OW
+n(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+U→Ut] =210/summationdisplay
+m=1¯KW
+m¯OW
+m(x,t). (26)
+Note that with the help of relation (15),d
+dt[˜OW
+n(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+U→Ut] appearing in the above equation can be reduced to linear
+composition of ¯OW
+m(x,t), i.e.
+d
+dt[˜OW
+n(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+U→Ut] =210/summationdisplay
+m=1Anm¯OW
+m(x,t) (27)
+5with the 23 ×210 matrix Anmgiven by Table VII in Appendix B, Then we rearrange (27) by multiplying both sides
+of the equation by some 23 ×23 matrix elements Cn′n,
+23/summationdisplay
+n=1Cn′nd
+dt[˜OW
+n(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+U→Ut] =23/summationdisplay
+n=1210/summationdisplay
+m=1Cn′nAnm¯OW
+m(x,t) =210/summationdisplay
+m=1Rn′m¯OW
+m(x,t)Rn′m≡23/summationdisplay
+n=1Cn′nAnm(28)
+and tuneCn′nin such a way that a 23 ×23 submatrix R′is a unit matrix, i.e. R′
+n′m′=δn′m′withn′,m′=
+1,3,4,5,6,7,20,43,44,49,50,51,52, 54,57,59,62,63,64,127,128,133,134. TheCmatrix is found to be of the form/parenleftbigg¯C7×707×15
+015×7˜C15×15/parenrightbigg
+where¯Cand˜Care 7×7 and 15 ×15 matrices respectively. The off diagonal parts are two matrices
+with null matrix elements and the dimensions are 7 ×15 and 15 ×7. We label the dimension of the sub-matrices as
+their subscripts. ¯Cand˜Cmatrices are given in Table VIII and Table IX in Appendix B. We call the r emaining part
+ofRn′mthe matrix Rn′m′′m′′/negationslash=m′. Then (28) is changed to
+23/summationdisplay
+n=1Cm′nd
+dt[˜OW
+n(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+U→Ut] =¯OW
+m′(x,t)+/summationdisplay
+m′′Rm′m′′¯OW
+m′′(x,t). (29)
+Multiplying both sides of the above equation by ¯KW
+m′,
+/summationdisplay
+m′23/summationdisplay
+n=1¯KW
+m′Cm′nd
+dt[˜OW
+n(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+U→Ut] =/summationdisplay
+m′¯KW
+m′¯OW
+m′(x,t)+/summationdisplay
+m′/summationdisplay
+m′′¯KW
+m′Rm′m′′¯OW
+m′′(x,t). (30)
+Comparing (30) and (26), to make these two equations consistent with each other, we must have conditions,
+˜KW
+n=/summationdisplay
+m′¯KW
+m′Cm′n¯KW
+m′′=/summationdisplay
+m′¯KW
+m′Rm′m′′, (31)
+in which the second equation is a consistency check for the coefficien ts¯KW
+m′′of the dependent operators ¯OW
+m′′(x,t). We
+have checked analytically that these constraints are all automatic ally satisfied and this can be seen as a consistency
+check of our formulation. The first equation gives ˜KW
+nin terms of ¯KW
+m′andCm′n. Substituting it in the expressions
+obtained for ¯KW
+m′andCm′n, we finally obtain the 23 order p6LECs for the three and more flavors anomalous part of
+chiral Lagrangian.
+The resulting analytical expressions for ˜KW
+nas functions of quark self energy Σ are given in Appendix C. With ˜KW
+n
+given in Appendix C, we can choose a suitable running coupling constan tαs(p2), solve the Schwinger-Dyson equation
+numerically, obtaining the quark self energy Σ, and then calculate th e numerical values of all order p6anomalous
+LECs. To obtain the final numerical result, we have assumed F0= 87MeV as input to fix the dimensional parameter
+ΛQCDappearing in the running coupling constant αs(p2) and taken momentum cutoff Λ = 1 .00+0.10
+−0.10GeV. Because
+of the appearance of the divergent order p2LECB0in Eqs.(23) and (24), we need a momentum cutoff Λ to make
+B0finite as we did previously in Ref.[6]. In Table I, we give the numerical valu es for all 21 nonzero LECs for three
+flavors(CW
+3=CW
+18= 0 in our formulation).
+Combined with our numerical result, we also list the numerical estimat es for some of the LECs from five different
+models and different processes given in Ref.[15],[17],[18],[19] and [20]. In Re f.[15], model I and III are all from
+direct chiral perturbation(ChPT) computations, except that mo del I is the full ChPT result, while in model III, low
+energy experiment data are extrapolated to the high energy regio n; model II is the vector meson dominance model
+(VMD); model IV and V are the chiral constituent quark model (CQ M) with some extrapolations included in model
+V. For a fixed model, different processes may give different results. For example, in model I for CW
+7and models I and
+IV forCW
+22, we all obtain two results from two different processes. Further, Ref.[15],[19] and [20] also give estimations
+on some combinations or ratios of LECs. We list our and their results in Table II. For Nf= 2, in Table III, we give
+the numerical values of all 12 nonzero LECs ( cW
+12= 0 in our formulation) which are actually of the very same structure
+as that given by [13].
+TABLE I. The nonzero values of the order p6anomalous LECs CW
+1,CW
+2,CW
+4,...,CW
+17,CW
+19,...,CW
+23for three flavors.
+The LECs are in units of 10−3GeV−2. The 2nd column is our result LECs with the values at Λ = 1GeV wi th superscript
+the difference caused at Λ = 1 .1GeV (i.e. CW
+i/vextendsingle/vextendsingle
+Λ=1.1GeV−CW
+i/vextendsingle/vextendsingle
+Λ=1GeV) and subscript the difference caused at Λ = 0 .9GeV
+(i.e.CW
+i/vextendsingle/vextendsingle
+Λ=0.9GeV−CW
+i/vextendsingle/vextendsingle
+Λ=1GeV). The 3rd to 7th columns are results given in Ref.[15]: (I)–ChPT, (II )–VMD,
+(III)–ChPT(extrapolation), (IV)–CQM, (V)–CQM(extrapol ation). The 8th column shows results from Ref.[17],[18],[1 9],[20].
+6nCW
+nours [15](I) [15](II) [15](III) [15](IV) [15](V) [17], [18], [19], [20]
+14.97+0.55
+−0.79
+2−1.43+0.10
+−0.12−0.32±10.4 0.78±12.74.96±9.70−0.074±13.3
+4−0.96+0.22
+−0.290.28±9.19 0.67±10.96.32±6.09−0.55±9.05
+53.26+0.34
+−0.4928.50±28.83 9.38±152.233.05±28.6634.51±41.13
+60.91+0.03
+−0.04
+71.68−0.24
++0.310.013±1.17 0.51±0.06 0.1±1.2
+20.3±18.7 0.1∗
+80.41+0.01
+−0.020.76±0.18 0.58±0.20
+0.5∗
+91.15−0.03
++0.03
+10−0.18−0.01
++0.01
+11−1.15+0.08
+−0.10−6.37±4.54 −0.00143±0.03 0.68±0.21
+12−5.13−0.15
++0.25
+13−6.37−0.18
++0.31−74.09±55.89−20.00−8.44±69.914.15±15.22−7.46±19.62
+14−2.00−0.06
++0.1029.99±11.14−6.010.72±15.310.23±7.56−0.58±9.77
+154.17+0.12
+−0.20−25.30±23.932.00−3.10±28.619.70±7.498.89±9.72
+163.58+0.10
+−0.17
+171.98+0.06
+−0.10
+190.29+0.01
+−0.01
+201.83+0.05
+−0.09
+212.48+0.07
+−0.12
+225.01+0.14
+−0.246.52±0.788.01 3.94±0.43 5.4±0.8
+5.07±0.71 3.94±0.43
+232.74+0.08
+−0.13
+∗This result is just the absolute value given in Ref.[19].
+TABLE II. Some combinations or ratios of LECs in units of 10−3GeV−2. The 2nd column is our result LECs
+with the values at Λ = 1GeV, and with superscript the differenc e caused at Λ = 1 .1GeV and subscript the difference
+caused at Λ = 0 .9GeV. The 3rd to 5th columns are results given in Ref.[15]: (I )–ChPT, (II)–VMD, (III)–ChPT
+(extrapolation), (IV)–CQM, (V)–CQM (extrapolation). The 6th and 7th columns are results given in Ref.[19]
+and [20] respectively.
+ours [15] [15] [15] [19][20]
+CW
+3−CW
+6−0.91−0.03
++0.0421.67±17.41 (I)5.07±5.07 (IV) −2.14±6.54 (V)
+2CW
+15−4CW
+14+CW
+139.95+0.29
+−0.48−244.7±148.4 (I) ≈8.0 (II) −17.52±188.3 (III)
+2CW
+14−CW
+132.38+0.07
+−0.12134.1±78.17 (I) ≈8.0 (II) 9.88±100.5 (III)
+|CW
+7|/|CW
+8|4.12−0.69
++1.01 0.2<0.1
+TABLE III. The nonzero values of the p6order anomalous LECs cW
+1,...,cW
+11,cW
+13for two flavor in units of 10−3GeV−2.
+cW
+1cW
+2cW
+3cW
+4cW
+5cW
+6cW
+7cW
+8cW
+9cW
+10cW
+11cW
+13
+−1.46+0.10
+−0.12−1.25+0.09
+−0.112.96−0.20
++0.250.63−0.04
++0.05−1.17+0.08
+−0.100.77+0.26
+−0.36−0.04−0.00
++0.000.02+0.00
+−0.008.19+0.23
+−0.38−8.73−0.24
++0.414.85+0.13
+−0.23−9.70−0.27
++0.45
+We see that most of our results are consistent with those we have f ound in the literature.
+As a phenomenological check for two flavor anomalous LECs, we disc uss theπ0→γγprocess. Ref.[20] gives the
+amplitude of this process by
+TLO+NLO =1
+F/braceleftbigg1
+4π2+16
+3m2
+π(−4cWr
+3−4cWr
+7+cWr
+11)+64
+9B(md−mu)(5cWr
+3+cWr
+r+2cWr
+8)/bracerightbigg
+.(32)
+In our calculation, we choose the center value B(md−mu) = 0.32m2
+π0given in Ref.[20]. Experimentally, the π0→γγ
+process dominates the life time of π0to 98.79%, and if we ignore that small fraction from other processes, th en the
+7life time of π0can be expressed in terms of amplitude Tas 1/τ=παm3
+πT2/4. In Table IV we give our result for
+τLOup to the leading order p4, which corresponds to the first term of the r.h.s of Eq.(32), and τNLOup to the next
+leading order p6of the low energy expansion. Experimental result from particle dat a group[21] is also included in the
+table for comparison.
+TABLE IV. π0life time in units of 10−17s.
+τLOτNLO
+F= 87MeV 7.567.59−0.03
++0.04
+F= 93MeV 8.638.67−0.03
++0.04
+Exp.[21] 8.4±0.6
+Our result roughly matches the experimental value and we see that the orderp6results have less effect on the life
+time ofπ0.
+IV. SUMMARY AND FUTURE WORK
+In this work, we review the general anomaly structure of the effec tive chiral Lagrangian and then generalize our
+orderp6calculation in Ref.[6] from the normal part to the anomalous part of t he chiral Lagrangian for pseudoscalar
+mesons. The result is obtained by computing the imaginary Σ depende nt part of Trln[ /∂+JΩ+ Σ(−¯∇2)]. To
+match the calculation of the order p4anomalous part, in practice we calculate the integration of paramet ertover
+d
+dtTrln[i/∂+JΩ(t)+ Σ(−∇2
+t)]. The conventional chiral Lagrangian is also reformulated to an int egration of tand
+through comparison of it with our result, we read out all order p6anomalous LECs expressed in terms of quark self
+energy Σ. Inputting the SDE solution of Σ( k2), we obtain numerical values and compare them with those we can fin d
+in literature. Some of them are consistent, some are not. We leave t hose inconsistent results to future investigations.
+Combined with the previous result on the order p6normal LECs given in Ref.[6], we have now completed all the
+orderp6LECs computations. Based on them, one direction of the further r esearch is to apply the order p6chiral
+Lagrangian to various pseudoscalar meson processes and discuss the corresponding physics. Another direction is to
+improve the precision of (7) and our ladder approximation SDE. With t hese improvements, we expect a more precise
+estimation on all LECs in future.
+Acknowledgments
+This work was supported by National Science Foundation of China (N SFC) under Grant No.10875065.
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+9Appendix A: List of All Operators ¯Oµνλρ
+nand˜OW
+n
+In this appendix, we first explicitly write down all 210 ¯Oµνλρ
+noperators. To save the space, we use some simplified
+symbols to represent the original symbols in the text. Our ¯Oµνλρ
+ns are constructed in such a way that they are
+invariant under charge conjugation transformation. This causes the result that most of ¯Oµνλρ
+ns consist of two terms
+which are charge conjugates to each other.
+TABLE V. µ≡ ∇µ
+t, ν≡ ∇ν
+t, λ≡ ∇λ
+t, ρ≡ ∇ρ
+t,¯µ≡aµ
+t,¯ν≡aν
+t,¯λ≡aλ
+t,¯ρ≡aρ
+t, s≡st, p≡pt
+···σ···σ≡ ···∇σ
+t···∇t,σ,···¯σ···¯σ≡ ···aσ
+t···at,σ,···σ···¯σ≡ ···∇σ
+t···at,σ,···¯σ···σ≡ ···aσ
+t···∇t,σ
+n ¯Oµνλρ
+n n ¯Oµνλρ
+n n ¯Oµνλρ
+n n ¯Oµνλρ
+n n ¯Oµνλρ
+n
+1sµνλρ+µνλρs 43µνλρσσ +σσµνλρ 85µσ¯νλρ¯σ+ ¯σµν¯λσρ127µν¯λ¯ρ¯σ¯σ+ ¯σ¯σ¯µ¯νλρ169µ¯ν¯λ¯σ¯σρ+µ¯σ¯σ¯ν¯λρ
+2µsνλρ+µνλsρ 44µνλσρσ +σµσνλρ 86µσ¯νλσ¯ρ+ ¯µσν¯λσρ128µν¯λ¯σ¯ρ¯σ+ ¯σ¯µ¯σ¯νλρ170µ¯ν¯σ¯λ¯ρσ+σ¯µ¯ν¯σ¯λρ
+3 µνsλρ 45µνλσσρ +µσσνλρ 87µσ¯νσλ¯ρ+ ¯µνσ¯λσρ129µν¯λ¯σ¯σ¯ρ+ ¯µ¯σ¯σ¯νλρ171µ¯ν¯σ¯λ¯σρ+µ¯σ¯ν¯σ¯λρ
+4sµν¯λ¯ρ+ ¯µ¯νλρs46µνσλρσ +σµνσλρ 88µσ¯σνλ¯ρ+ ¯µνλ¯σσρ130µν¯σ¯λ¯ρ¯σ+ ¯σ¯µ¯ν¯σλρ172µ¯σ¯ν¯λ¯ρσ+σ¯µ¯ν¯λ¯σρ
+5sµ¯νλ¯ρ+ ¯µν¯λρs47µνσλσρ +µσνσλρ 89µν¯λρ¯σσ+σ¯σµ¯νλρ131µν¯σ¯λ¯σ¯ρ+ ¯µ¯σ¯ν¯σλρ173¯µνλ¯ρ¯σ¯σ+ ¯σ¯σ¯µνλ¯ρ
+6sµ¯ν¯λρ+µ¯ν¯λρs48µσνλρσ +σµνλσρ 90µν¯λσ¯ρσ+σ¯µσ¯νλρ132µν¯σ¯σ¯λ¯ρ+ ¯µ¯ν¯σ¯σλρ174¯µνλ¯σ¯ρ¯σ+ ¯σ¯µ¯σνλ¯ρ
+7s¯µνλ¯ρ+ ¯µνλ¯ρs49µνλρ¯σ¯σ+ ¯σ¯σµνλρ 91µν¯λσ¯σρ+µ¯σσ¯νλρ133µσ¯ν¯λ¯ρ¯σ+ ¯σ¯µ¯ν¯λσρ175¯µνλ¯σ¯σ¯ρ+ ¯µ¯σ¯σνλ¯ρ
+8s¯µν¯λρ+µ¯νλ¯ρs50µνλσ¯ρ¯σ+ ¯σ¯µσνλρ 92µν¯σλ¯ρσ+σ¯µν¯σλρ134µσ¯ν¯λ¯σ¯ρ+ ¯µ¯σ¯ν¯λσρ176¯µνσ¯λ¯ρ¯σ+ ¯σ¯µ¯νσλ¯ρ
+9s¯µ¯νλρ+µν¯λ¯ρs51µνλσ¯σ¯ρ+ ¯µ¯σσνλρ 93µν¯σλ¯σρ+µ¯σν¯σλρ135µσ¯ν¯σ¯λ¯ρ+ ¯µ¯ν¯σ¯λσρ177¯µνσ¯λ¯σ¯ρ+ ¯µ¯σ¯νσλ¯ρ
+10µsν¯λ¯ρ+ ¯µ¯νλsρ52µνσλ¯ρ¯σ+ ¯σ¯µνσλρ 94µν¯σσ¯λρ+µ¯νσ¯σλρ136µσ¯σ¯ν¯λ¯ρ+ ¯µ¯ν¯λ¯σσρ178¯µνσ¯σ¯λ¯ρ+ ¯µ¯ν¯σσλ¯ρ
+11µs¯νλ¯ρ+ ¯µν¯λsρ53µνσλ¯σ¯ρ+ ¯µ¯σνσλρ 95µσ¯νλ¯ρσ+σ¯µν¯λσρ137µ¯νλ¯ρ¯σ¯σ+ ¯σ¯σ¯µν¯λρ179¯µσν¯λ¯ρ¯σ+ ¯σ¯µ¯νλσ¯ρ
+12µs¯ν¯λρ+µ¯ν¯λsρ54µνσσ¯λ¯ρ+ ¯µ¯νσσλρ 96µσ¯νλ¯σρ+µ¯σν¯λσρ138µ¯νλ¯σ¯ρ¯σ+ ¯σ¯µ¯σν¯λρ180¯µσν¯λ¯σ¯ρ+ ¯µ¯σ¯νλσ¯ρ
+13¯µsνλ¯ρ+ ¯µνλs¯ρ55µσνλ¯ρ¯σ+ ¯σ¯µνλσρ 97µσ¯νσ¯λρ+µ¯νσ¯λσρ139µ¯νλ¯σ¯σ¯ρ+ ¯µ¯σ¯σν¯λρ181¯µσν¯σ¯λ¯ρ+ ¯µ¯ν¯σλσ¯ρ
+14¯µsν¯λρ+µ¯νλs¯ρ56µσνλ¯σ¯ρ+ ¯µ¯σνλσρ 98µσ¯σν¯λρ+µ¯νλ¯σσρ140µ¯νσ¯λ¯ρ¯σ+ ¯σ¯µ¯νσ¯λρ182¯µσσ¯ν¯λ¯ρ+ ¯µ¯ν¯λσσ¯ρ
+15¯µs¯νλρ+µν¯λs¯ρ57µσνσ¯λ¯ρ+ ¯µ¯νσλσρ 99µν¯λ¯ρσσ+σσ¯µ¯νλρ141µ¯νσ¯λ¯σ¯ρ+ ¯µ¯σ¯νσ¯λρ183¯µν¯λρ¯σ¯σ+ ¯σ¯σµ¯νλ¯ρ
+16µνs¯λ¯ρ+ ¯µ¯νsλρ58µσσν¯λ¯ρ+ ¯µ¯νλσσρ 100µν¯λ¯σρσ+σµ¯σ¯νλρ142µ¯νσ¯σ¯λ¯ρ+ ¯µ¯ν¯σσ¯λρ184¯µν¯λσ¯ρ¯σ+ ¯σ¯µσ¯νλ¯ρ
+17µ¯νsλ¯ρ+ ¯µνs¯λρ59µνλ¯ρσ¯σ+ ¯σσ¯µνλρ101µν¯λ¯σσρ+µσ¯σ¯νλρ143µ¯σν¯λ¯ρ¯σ+ ¯σ¯µ¯νλ¯σρ185¯µν¯λσ¯σ¯ρ+ ¯µ¯σσ¯νλ¯ρ
+18 µ¯νs¯λρ 60µνλ¯σρ¯σ+ ¯σµ¯σνλρ102µν¯σ¯λρσ+σµ¯ν¯σλρ144µ¯σν¯λ¯σ¯ρ+ ¯µ¯σ¯νλ¯σρ186¯µν¯σλ¯ρ¯σ+ ¯σ¯µν¯σλ¯ρ
+19 ¯µνsλ¯ρ 61µνλ¯σσ¯ρ+ ¯µσ¯σνλρ103µν¯σ¯λσρ+µσ¯ν¯σλρ145µ¯σν¯σ¯λ¯ρ+ ¯µ¯ν¯σλ¯σρ187¯µν¯σλ¯σ¯ρ+ ¯µ¯σν¯σλ¯ρ
+20s¯µ¯ν¯λ¯ρ+ ¯µ¯ν¯λ¯ρs62µνσ¯λρ¯σ+ ¯σµ¯νσλρ104µσ¯ν¯λρσ+σµ¯ν¯λσρ146µ¯σσ¯ν¯λ¯ρ+ ¯µ¯ν¯λσ¯σρ188¯µν¯σσ¯λ¯ρ+ ¯µ¯νσ¯σλ¯ρ
+21¯µs¯ν¯λ¯ρ+ ¯µ¯ν¯λs¯ρ63µνσ¯λσ¯ρ+ ¯µσ¯νσλρ105µ¯νλρσ¯σ+ ¯σσµν¯λρ147µ¯ν¯λρ¯σ¯σ+ ¯σ¯σµ¯ν¯λρ189¯µσ¯νλ¯ρ¯σ+ ¯σ¯µν¯λσ¯ρ
+22 ¯µ¯νs¯λ¯ρ 64µνσ¯σλ¯ρ+ ¯µν¯σσλρ106µ¯νλσρ¯σ+ ¯σµσν¯λρ148µ¯ν¯λσ¯ρ¯σ+ ¯σ¯µσ¯ν¯λρ190¯µσ¯νλ¯σ¯ρ+ ¯µ¯σν¯λσ¯ρ
+23pµνλ¯ρ−¯µνλρp 65µσν¯λρ¯σ+ ¯σµ¯νλσρ107µ¯νλσσ¯ρ+ ¯µσσν¯λρ149µ¯ν¯λσ¯σ¯ρ+ ¯µ¯σσ¯ν¯λρ191¯µσ¯νσ¯λ¯ρ+ ¯µ¯νσ¯λσ¯ρ
+24pµν¯λρ−µ¯νλρp66µσν¯λσ¯ρ+ ¯µσ¯νλσρ108µ¯νσλρ¯σ+ ¯σµνσ¯λρ150µ¯ν¯σλ¯ρ¯σ+ ¯σ¯µν¯σ¯λρ192¯µσ¯σν¯λ¯ρ+ ¯µ¯νλ¯σσ¯ρ
+25pµ¯νλρ−µν¯λρp67µσν¯σλ¯ρ+ ¯µν¯σλσρ109µ¯νσλσ¯ρ+ ¯µσνσ¯λρ151µ¯ν¯σλ¯σ¯ρ+ ¯µ¯σν¯σ¯λρ193¯µν¯λ¯ρσ¯σ+ ¯σσ¯µ¯νλ¯ρ
+26p¯µνλρ−µνλ¯ρp68µσσ¯νλ¯ρ+ ¯µν¯λσσρ110µ¯νσσλ¯ρ+ ¯µνσσ¯λρ152µ¯ν¯σσ¯λ¯ρ+ ¯µ¯νσ¯σ¯λρ194¯µν¯λ¯σρ¯σ+ ¯σµ¯σ¯νλ¯ρ
+27µpνλ¯ρ−¯µνλpρ 69µνλ¯ρ¯σσ+σ¯σ¯µνλρ111µ¯σνλρ¯σ+ ¯σµνλ¯σρ153µ¯σ¯νλ¯ρ¯σ+ ¯σ¯µν¯λ¯σρ195¯µν¯λ¯σσ¯ρ+ ¯µσ¯σ¯νλ¯ρ
+28µpν¯λρ−µ¯νλpρ70µνλ¯σ¯ρσ+σ¯µ¯σνλρ112µ¯σνλσ¯ρ+ ¯µσνλ¯σρ154µ¯σ¯νλ¯σ¯ρ+ ¯µ¯σν¯λ¯σρ196¯µν¯σ¯λρ¯σ+ ¯σµ¯ν¯σλ¯ρ
+29µp¯νλρ−µν¯λpρ71µνλ¯σ¯σρ+µ¯σ¯σνλρ113µ¯σνσλ¯ρ+ ¯µνσλ¯σρ155µ¯σ¯νσ¯λ¯ρ+ ¯µ¯νσ¯λ¯σρ197¯µν¯σ¯λσ¯ρ+ ¯µσ¯ν¯σλ¯ρ
+30¯µpνλρ−µνλp¯ρ72µνσ¯λ¯ρσ+σ¯µ¯νσλρ114µ¯σσνλ¯ρ+ ¯µνλσ¯σρ156µ¯σ¯σν¯λ¯ρ+ ¯µ¯νλ¯σ¯σρ198¯µσ¯ν¯λρ¯σ+ ¯σµ¯ν¯λσ¯ρ
+31µνpλ¯ρ−¯µνpλρ 73µνσ¯λ¯σρ+µ¯σ¯νσλρ115µ¯νλρ¯σσ+σ¯σµν¯λρ157µ¯ν¯λ¯ρσ¯σ+ ¯σσ¯µ¯ν¯λρ199¯µ¯νλρ¯σ¯σ+ ¯σ¯σµν¯λ¯ρ
+32µνp¯λρ−µ¯νpλρ74µνσ¯σ¯λρ+µ¯ν¯σσλρ116µ¯νλσ¯ρσ+σ¯µσν¯λρ158µ¯ν¯λ¯σρ¯σ+ ¯σµ¯σ¯ν¯λρ200¯µ¯νλσ¯ρ¯σ+ ¯σ¯µσν¯λ¯ρ
+33pµ¯ν¯λ¯ρ−¯µ¯ν¯λρp75µσν¯λ¯ρσ+σ¯µ¯νλσρ117µ¯νλσ¯σρ+µ¯σσν¯λρ159µ¯ν¯λ¯σσ¯ρ+ ¯µσ¯σ¯ν¯λρ201¯µ¯νλσ¯σ¯ρ+ ¯µ¯σσν¯λ¯ρ
+34p¯µν¯λ¯ρ−¯µ¯νλ¯ρp76µσν¯λ¯σρ+µ¯σ¯νλσρ118µ¯νσλ¯ρσ+σ¯µνσ¯λρ160µ¯ν¯σ¯λρ¯σ+ ¯σµ¯ν¯σ¯λρ202¯µ¯νσλ¯ρ¯σ+ ¯σ¯µνσ¯λ¯ρ
+35p¯µ¯νλ¯ρ−¯µν¯λ¯ρp77µσν¯σ¯λρ+µ¯ν¯σλσρ119µ¯νσλ¯σρ+µ¯σνσ¯λρ161µ¯ν¯σ¯λσ¯ρ+ ¯µσ¯ν¯σ¯λρ203¯µ¯νσλ¯σ¯ρ+ ¯µ¯σνσ¯λ¯ρ
+36p¯µ¯ν¯λρ−µ¯ν¯λ¯ρp78µσσ¯ν¯λρ+µ¯ν¯λσσρ120µ¯σνλ¯ρσ+σ¯µνλ¯σρ162µ¯ν¯σ¯σλ¯ρ+ ¯µν¯σ¯σ¯λρ204¯µ¯σνλ¯ρ¯σ+ ¯σ¯µνλ¯σ¯ρ
+37µp¯ν¯λ¯ρ−¯µ¯ν¯λpρ79µν¯λρσ¯σ+ ¯σσµ¯νλρ121¯µνλρσ¯σ+ ¯σσµνλ¯ρ163µ¯σ¯ν¯λρ¯σ+ ¯σµ¯ν¯λ¯σρ205¯µ¯ν¯λ¯ρ¯σ¯σ+ ¯σ¯σ¯µ¯ν¯λ¯ρ
+38¯µpν¯λ¯ρ−¯µ¯νλp¯ρ80µν¯λσρ¯σ+ ¯σµσ¯νλρ122¯µνλσρ¯σ+ ¯σµσνλ¯ρ164µ¯σ¯ν¯λσ¯ρ+ ¯µσ¯ν¯λ¯σρ206¯µ¯ν¯λ¯σ¯ρ¯σ+ ¯σ¯µ¯σ¯ν¯λ¯ρ
+39¯µp¯νλ¯ρ−¯µν¯λp¯ρ81µν¯λσσ¯ρ+ ¯µσσ¯νλρ123¯µνλσσ¯ρ+ ¯µσσνλ¯ρ165µ¯σ¯ν¯σλ¯ρ+ ¯µν¯σ¯λ¯σρ207¯µ¯ν¯λ¯σ¯σ¯ρ+ ¯µ¯σ¯σ¯ν¯λ¯ρ
+40¯µp¯ν¯λρ−µ¯ν¯λp¯ρ82µν¯σλρ¯σ+ ¯σµν¯σλρ124¯µνσλρ¯σ+ ¯σµνσλ¯ρ166µ¯σ¯σ¯νλ¯ρ+ ¯µν¯λ¯σ¯σρ208¯µ¯ν¯σ¯λ¯ρ¯σ+ ¯σ¯µ¯ν¯σ¯λ¯ρ
+41µ¯νp¯λ¯ρ−¯µ¯νp¯λρ83µν¯σλσ¯ρ+ ¯µσν¯σλρ125¯µνσλσ¯ρ+ ¯µσνσλ¯ρ167µ¯ν¯λ¯ρ¯σσ+σ¯σ¯µ¯ν¯λρ209¯µ¯ν¯σ¯λ¯σ¯ρ+ ¯µ¯σ¯ν¯σ¯λ¯ρ
+42¯µνp¯λ¯ρ−¯µ¯νpλ¯ρ84µν¯σσλ¯ρ+ ¯µνσ¯σλρ126¯µσνλρ¯σ+ ¯σµνλσ¯ρ168µ¯ν¯λ¯σ¯ρσ+σ¯µ¯σ¯ν¯λρ210¯µ¯σ¯ν¯λ¯ρ¯σ+ ¯σ¯µ¯ν¯λ¯σ¯ρ
+10Next, we list all 23 /tildewideOW
+noperators.
+TABLE VI. List of/tildewideOW
+noperators, where we divide ˜OW
+1,...,˜OW
+7byB0making the matrices Amnintroduced
+in Eq.(36) independent of B0. The symbols are introduced in Ref.[22]. The comparisons between
+the symbols introduced in Ref.[22] and ours are given in Tabl e XV. of Ref.[6].
+n /tildewideOW
+n n /tildewideOW
+n
+1 /angb∇acketleftiuµuνuλuρχ−/angb∇acket∇ightǫµνλρ/B0 13−i/angb∇acketleftfµν
++uσuλhρσ/angb∇acket∇ightǫµνλρ−i/angb∇acketleftfµν
++hλσuρuσ/angb∇acket∇ightǫµνλρ
+2/angb∇acketleftuµuνχ+fλρ
+−/angb∇acket∇ightǫµνλρ/B0−/angb∇acketleftuµuνfλρ
+−χ+/angb∇acket∇ightǫµνλρ/B014−i/angb∇acketleftfµν
++uλhρσuσ/angb∇acket∇ightǫµνλρ−i/angb∇acketleftfµν
++uσhλσuρ/angb∇acket∇ightǫµνλρ
+3/angb∇acketleftfµν
++uλuρχ−/angb∇acket∇ightǫµνλρ/B0+/angb∇acketleftfµν
++χ−uλuρ/angb∇acket∇ightǫµνλρ/B015i/angb∇acketleftfµν
++uλuρhσ
+σ/angb∇acket∇ightǫµνλρ+i/angb∇acketleftfµν
++hσ
+σuλuρ/angb∇acket∇ightǫµνλρ
+4 −/angb∇acketleftfµν
++uλχ−uρ/angb∇acket∇ightǫµνλρ/B0 16 −i/angb∇acketleftfµν
++uλhσ
+σuρ/angb∇acket∇ightǫµνλρ
+5 i/angb∇acketleftfµν
++fλρ
++χ−/angb∇acket∇ightǫµνλρ/B0 17i/angb∇acketleftfµ
++σuνuλfρσ
+−/angb∇acket∇ightǫµνλρ+i/angb∇acketleftfµ
++σfνσ
+−uλuρ/angb∇acket∇ightǫµνλρ
+6 i/angb∇acketleftfµν
+−fλρ
+−χ−/angb∇acket∇ightǫµνλρ/B0 18i/angb∇acketleftfµσ
++uνuσfλρ
+−/angb∇acket∇ightǫµνλρ−i/angb∇acketleftfµσ
++fνλ
+−uσuρ/angb∇acket∇ightǫµνλρ
+7i/angb∇acketleftfµν
++fλρ
+−χ+/angb∇acket∇ightǫµνλρ/B0−i/angb∇acketleftfµν
++χ+fλρ
+−/angb∇acket∇ightǫµνλρ/B019−i/angb∇acketleftfµσ
++uνfλρ
+−uσ/angb∇acket∇ightǫµνλρ+i/angb∇acketleftfµσ
++uσfνλ
+−uρ/angb∇acket∇ightǫµνλρ
+8−/angb∇acketleftuσuµuνuλhρσ/angb∇acket∇ightǫµνλρ+/angb∇acketleftuµuνuλuσhρσ/angb∇acket∇ightǫµνλρ20−/angb∇acketleftfµν
++uλ∇σfρσ
++/angb∇acket∇ightǫµνλρ+/angb∇acketleftfµν
++∇σfλσ
++uρ/angb∇acket∇ightǫµνλρ
+9 /angb∇acketleftuµuνuλuρhσ
+σ/angb∇acket∇ightǫµνλρ 21−/angb∇acketleftuµ∇σfνσ
+−fλρ
+−/angb∇acket∇ightǫµνλρ−/angb∇acketleftuµfνλ
+−∇σfρσ
+−/angb∇acket∇ightǫµνλρ
+10/angb∇acketleftuσuµuνuλfρσ
+−/angb∇acket∇ightǫµνλρ−/angb∇acketleftuµuνuλuσfρσ
+−/angb∇acket∇ightǫµνλρ22 /angb∇acketleftfµν
++fλρ
++hσ
+σ/angb∇acket∇ightǫµνλρ
+11/angb∇acketleftu2uµuνfλρ
+−/angb∇acket∇ightǫµνλρ−/angb∇acketleftuµuνu2fλρ
+−/angb∇acket∇ightǫµνλρ 23 /angb∇acketlefthσ
+σfµν
+−fλρ
+−/angb∇acket∇ightǫµνλρ
+12i/angb∇acketleftfµσ
++uνuλhρ
+σ/angb∇acket∇ightǫµνλρ+i/angb∇acketleftfµσ
++hν
+σuλuρ/angb∇acket∇ightǫµνλρ
+11Appendix B: A,CandRmatrices
+In this appendix, we give matrix Anm,Cm′nandRm′m. For convenience of writing, in practice, we do not write A
+matrix, but its transverse ATmultiplied by −i.
+TABLE VII. −i(AT)mnmatrix
+m,n 1234567891011121314151617181920212223
+10000320320000000000000000
+200000000000000000000000
+3000000−640000000000000000
+400−320−320−320000000000000000
+500000−32320000000000000000
+60−32000−32320000000000000000
+7000−320−32320000000000000000
+80−32000−32320000000000000000
+9032−320−320−320000000000000000
+100−32000000000000000000000
+110−32000000000000000000000
+1200000000000000000000000
+1300000000000000000000000
+14032000000000000000000000
+15032000000000000000000000
+160−320000640000000000000000
+170320000−640000000000000000
+180640000−640000000000000000
+19000000−640000000000000000
+20−32−326432320320000000000000000
+2100000000000000000000000
+220640000−640000000000000000
+230000−320−320000000000000000
+2400−320−320−320000000000000000
+2500000−32320000000000000000
+26000−320−32320000000000000000
+2700−3200000000000000000000
+2800000000000000000000000
+29000−320000000000000000000
+30003200000000000000000000
+310000−323200000000000000000
+3200−320−323200000000000000000
+3303200032−320000000000000000
+3403200032−320000000000000000
+350−32320320320000000000000000
+36−32−326432320320000000000000000
+37−32032320000000000000000000
+3800000000000000000000000
+39000−320000000000000000000
+40320−32−320000000000000000000
+41320−640−323200000000000000000
+4200−320−323200000000000000000
+43000000000000000000080−640
+440000000000000000000−240320
+45000000000000000000024000
+460000000000000000000−80−320
+470000000000000000000−8000
+48000000000000000000000320
+490000000000003200000080−320
+5000000000000−8160008−1600000
+51000000000008−480−320−800−8000
+520000000000080000−8008800
+5300000000000−163200016000−1600
+5400000000000−816000−800−81600
+5500000000000000320016000−320
+12TABLE VII (continued). −i(AT)mnmatrix
+m,n1234567891011121314151617181920212223
+56000000000008−480−320−80000320
+570000000000024−160320800160−320
+5800000000000−1600−320000−80320
+59000000000000−1632032000000−32
+60000000000000−160000008000
+61000000000000320320000−8800
+62000000000000016000−16−16−241600
+63000000000008−161600801624−3200
+64000000000008−16000−800−162432−32
+650000000000000−1600016161600−32
+6600000000000−816−1600−80−16−160032
+6700000000000−816000800880−32
+6800000000000000000000−8032
+690000000000000−32−64−320−160−8−86432
+7000000000000−8800640800168−640
+71000000000008−32000−8160−8000
+7200000000000−4016−16−640−80−16−400640
+7300000000000240−16008−161648−800
+7400000000000−32320−320000−40832−32
+7500000000000320166400016240−64−32
+7600000000000−160160000−16−240032
+770000000000016−320000001680−32
+7800000000000000000000−8032
+790000000000000−1600016161600−32
+80000000000000016000−16−16−241600
+8100000000000000000008−800
+82000000000000−160000008000
+830000000000080−1600−816160−800
+8400000000000−8016008−16−160−8−3232
+85000000000000−1632032000000−32
+86000000000008000−328−16000032
+8700000000000−800032−81600160−32
+880000000000000−32−32−320000−8032
+89000000000000016000−16−160−32096
+90000000000008−1616008016−8480−64
+9100000000000−800032−81608−1600
+92000000000008−16000−8000−24064
+9300000000000−8016008−16−160800
+94000000000001601632000160−16−3232
+950000000000000000000080−96
+9600000000000000000000−8032
+9700000000000000000000240−32
+9800000000000000000000−8032
+9900000000000−160−16−32000−160−243232
+1000000000000080000−8000240−32
+1010000000000080−160−328016−8000
+10200000000000−81616008−16−160−8032
+10300000000000−816−3200−81608000
+1040000000000000000000000−32
+10500000000000000320016000−320
+1060000000000080000−8008800
+10700000000000800008−1600−800
+10800000000000−8160008−1600000
+10900000000000−160000−16320−16000
+11000000000000800008−16016−800
+1110000000000003200000080−320
+11200000000000−8160008−160−80320
+1130000000000080000−80008−320
+11400000000000000320016000320
+13TABLE VII (continued). −i(AT)mnmatrix
+m,n1234567891011121314151617181920212223
+115000000000000−16−32−32−32000−8−83232
+11600000000000−800032−81608240−32
+11700000000000−8160008000−1600
+118000000000008000−328−160−88032
+119000000000008−16000−8000000
+120000000000000−3232032000000−32
+121000000000000000000080−640
+1220000000000000000000−240320
+123000000000000000000024000
+1240000000000000000000−80−320
+1250000000000000000000−8000
+126000000000000000000000320
+1270000000160160−16−1616−3200−16−16−240320
+1280000000−160−16−32816−16008161624−800
+1290000000000−3280000−81600800
+1300000000−160−1632−832−1600−8016−161600
+1310000000320320−160320016−32−320−1600
+1320000000−160−1632816−1600−80160800
+1330000000−48−3216−32−1616−48−64−3200−160−16320
+13400000001632−160832166432−81616016−320
+13500000000−3200−8−320−64−328−160−8−8320
+136000000064640016064966400080−320
+137000000000064000000−3200−8032
+1380000000000−320000001600−800
+1390000000−160−16−32016000016001600
+1400000000160−16320−1616000016−162400
+1410000000−320320816−16008−32−1624−3200
+14200000000−3200−24−320−640−8160−241632−32
+14300000001601600−1616000−16−160−24032
+1440000000−160−16−3280−1600−832160240−32
+1450000000−160−1632−8160008−1600−8032
+1460000000−48−3216−32016−32−32−32000000−32
+14700000001601600−1616000−16−160−24032
+1480000000160−16320−1616000016−162400
+14900000001632−16001603232016016000
+1500000000000−320000001600−800
+15100000003203208−321600−8−16−160−800
+15200000001632−160241616640801680−3232
+153000000000064000000−3200−8032
+1540000000000−32−8160008160080−32
+1550000000160−1632−80000−800−80032
+15600000001601600−16000000000−32
+1570000000−48−3216−32−1616−48−64−3200−160−16320
+1580000000−160−1632−832−1600−8016−161600
+15900000000−32008−480−32−328008000
+1600000000−160−16−32816−16008161624−800
+1610000000−320320−1632−3200−160−32−16000
+1620000000−160−16−32−832−1600816168000
+1630000000160160−16−1616−3200−16−16−240320
+1640000000160−163224−32163208016240−320
+1650000000000−328−480−320−8160−80320
+1660000000000640003200−16000−320
+16700000004864−1632321648128640016320−640
+16800000000−32032−16−320−64−320−160−320320
+169000000016016−320−16160000−160000
+17000000001632−163216161664320016160−320
+1710000000000−320000001600000
+1720000000−64−3200−1632−64−64−3200000320
+1730000000000000−32−64−320−160−8−86432
+14TABLE VII (continued). −i(AT)mnmatrix
+m,n1234567891011121314151617181920212223
+17400000000000−8800640800168−640
+175000000000008−32000−8160−8000
+17600000000000−4016−16−640−80−16−400640
+17700000000000240−16008−161648−800
+17800000000000−32320−320000−40832−32
+17900000000000320166400016240−64−32
+18000000000000−160160000−16−240032
+1810000000000016−320000001680−32
+18200000000000000000000−8032
+183000000000000016000−16−160−32096
+184000000000008−1616008016−8480−64
+18500000000000−800032−81608−1600
+186000000000008−16000−8000−24064
+18700000000000−8016008−16−160800
+188000000000001601632000160−16−3232
+1890000000000000000000080−96
+19000000000000000000000−8032
+19100000000000000000000240−32
+19200000000000000000000−8032
+19300000000000−160−16−32000−160−243232
+1940000000000080000−8000240−32
+1950000000000080−160−328016−8000
+19600000000000−81616008−16−160−8032
+19700000000000−816−3200−81608000
+1980000000000000000000000−32
+199000000000000−16−32−32−32000−8−83232
+20000000000000−800032−81608240−32
+20100000000000−8160008000−1600
+202000000000008000−328−160−88032
+203000000000008−16000−8000000
+204000000000000−3232032000000−32
+20500000004864−1632321648128640016320−640
+20600000000−32032−16−320−64−320−160−320320
+207000000016016−320−16160000−160000
+20800000001632−163216161664320016160−320
+2090000000000−320000001600000
+2100000000−64−3200−1632−64−64−3200000320
+From (28), Cmatrix is consist of two sub-matrices ¯Cand˜C.¯Cmatrix is
+TABLE. VIII ¯Cm′nmatrix
+m′,n1234567
+1−i
+320−i
+320i
+3200
+30000i
+64−i
+64−i
+64
+4−i
+160−i
+320000
+50i
+320i
+320−i
+320
+6i
+32−i
+3200000
+7−i
+3200−i
+32000
+20−i
+32000000
+15˜Cmatrix is
+TABLE. IX ˜Cm′nmatrix
+m′,n891011121314151617181920212223
+4319i
+160−11i
+80−19i
+160−13i
+160−3i
+80−i
+80−7i
+1609i
+160−i
+160−13i
+80−3i
+40−i
+32−i
+40−i
+10−3i
+160−7i
+160
+4431i
+320−3i
+20−21i
+320−17i
+3203i
+160i
+160−13i
+32011i
+32011i
+320−17i
+160−9i
+160i
+64−i
+20−3i
+40−i
+160−3i
+320
+49−i
+163i
+323i
+32i
+16i
+16i
+32i
+32−i
+160i
+8i
+16i
+320i
+160i
+64
+50000000−i
+320i
+320−i
+16i
+320000
+51−5i
+643i
+325i
+643i
+64i
+320i
+64−3i
+64i
+643i
+32i
+323i
+640i
+160i
+32
+525i
+32−3i
+16−5i
+32−3i
+3200−i
+16i
+16i
+32−i
+4−i
+8−i
+160−i
+80−i
+32
+54−3i
+32i
+325i
+32i
+3200i
+16−i
+320i
+8i
+16i
+80i
+80i
+16
+57−3i
+64i
+327i
+64i
+64i
+320i
+64−i
+64i
+643i
+32i
+323i
+640i
+160i
+32
+590−i
+32000000i
+320000000
+62−i
+32i
+32−i
+32000i
+320−i
+3200−i
+320000
+63−i
+32i
+32i
+32000i
+320−i
+3200i
+320000
+640i
+32000000−i
+32000000−i
+32
+127i
+320i
+32−i
+32000000000000
+128i
+32−i
+32−i
+32−i
+32000000000000
+133−i
+32i
+32i
+320000000000000
+134−i
+32i
+16i
+320000000000000
+Appendix C: Final Analytical Result on ˜KW
+n
+In this appendix, we list our analytical result on 23 LECs for p6order anomalous part of the chiral Lagrangian,
+˜KW
+1=/integraldisplayd4k
+(2π)4/bracketleftbigg
+−1
+2k2Σ3
+kX5−1
+2Σ5
+kX5+3
+4k4Σ2
+kΣ′
+kX5+3
+4k2Σ4
+kΣ′
+kX5/bracketrightbigg
+˜KW
+2=/integraldisplayd4k
+(2π)4/bracketleftbigg
+−1
+4Σ5
+kX5+1
+8k4Σ2
+kΣ′
+kX5+5
+8k2Σ4
+kΣ′
+kX5/bracketrightbigg
+˜KW
+3=/integraldisplayd4k
+(2π)4/bracketleftbigg
+−1
+4k2Σ3
+kX5−1
+4Σ5
+kX5+1
+2k4Σ2
+kΣ′
+kX5+1
+2k2Σ4
+kΣ′
+kX5/bracketrightbigg
+˜KW
+4=/integraldisplayd4k
+(2π)4/bracketleftbigg
+−1
+4k2Σ3
+kX5−1
+4Σ5
+kX5+1
+2k4Σ2
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+(2π)4/bracketleftbigg1
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+(9
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++7
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++13
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++893
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+(2π)4/bracketleftbigg
+(−7
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+(2π)4/bracketleftbigg
+(−33
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+(2π)4/bracketleftbigg
+(−1
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++1
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+(1
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++1
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++(−7
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++13
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+14=/integraldisplayd4k
+(2π)4/bracketleftbigg
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++5
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++197
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+(11
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+−1
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+(2π)4/bracketleftbigg
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++109
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+(2π)4/bracketleftbigg
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++29
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++43
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++11
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+k−29
+8k6Σ3
+kΣ′3
+k+3
+16k4Σ5
+kΣ′3
+k)X6/bracketrightbigg
+˜KW
+20=/integraldisplayd4k
+(2π)4/bracketleftbigg
+(1
+40k2Σ′′
+k+1
+40Σ2
+kΣ′′
+k−1
+180k4Σ′′′
+k−1
+180k2Σ2
+kΣ′′′
+k)k2ΣkX4+(1
+20k4ΣkΣ′
+k+1
+10k2Σ3
+kΣ′
+k+1
+20Σ5
+kΣ′
+k
+−1
+8k6Σ′2
+k−1
+4k4Σ2
+kΣ′2
+k−1
+8k2Σ4
+kΣ′2
+k+11
+120k6ΣkΣ′′
+k+7
+60k4Σ3
+kΣ′′
+k+1
+40k2Σ5
+kΣ′′
+k−13
+120k8Σ′
+kΣ′′
+k
+−1
+12k6Σ2
+kΣ′
+kΣ′′
+k+1
+40k4Σ4
+kΣ′
+kΣ′′
+k)X5+(3
+20k4Σ2
+k+1
+4k2Σ4
+k+1
+10Σ6
+k−2
+5k6ΣkΣ′
+k−1
+2k4Σ3
+kΣ′
+k−1
+10k2Σ5
+kΣ′
+k
++1
+5k8Σ′2
+k−1
+10k6Σ2
+kΣ′2
+k−1
+5k4Σ4
+kΣ′2
+k+1
+10k2Σ6
+kΣ′2
+k+1
+5k8ΣkΣ′3
+k−1
+5k4Σ5
+kΣ′3
+k)X6/bracketrightbigg
+˜KW
+21=/integraldisplayd4k
+(2π)4/bracketleftbigg
+(−11
+40k2Σ′′
+k+9
+40Σ2
+kΣ′′
+k−11
+120k4Σ′′′
+k+3
+40k2Σ2
+kΣ′′′
+k)k2ΣkX4+(13
+40k4ΣkΣ′
+k−19
+40k2Σ3
+kΣ′
+k+1
+5Σ5
+kΣ′
+k
+−1
+4k6Σ′2
+k+5
+4k4Σ2
+kΣ′2
+k−1
+2k2Σ4
+kΣ′2
+k+13
+40k6ΣkΣ′′
+k−23
+40k4Σ3
+kΣ′′
+k+1
+10k2Σ5
+kΣ′′
+k−11
+60k8Σ′
+kΣ′′
+k+5
+4k6Σ2
+kΣ′
+kΣ′′
+k
+−17
+30k4Σ4
+kΣ′
+kΣ′′
+k)X5+(1
+10k4Σ2
+k−1
+4k2Σ4
+k+3
+20Σ6
+k−31
+60k6ΣkΣ′
+k+4
+3k4Σ3
+kΣ′
+k−23
+20k2Σ5
+kΣ′
+k+3
+10k8Σ′2
+k
+−169
+60k6Σ2
+kΣ′2
+k+38
+15k4Σ4
+kΣ′2
+k−7
+20k2Σ6
+kΣ′2
+k+29
+30k8ΣkΣ′3
+k−7
+3k6Σ3
+kΣ′3
+k+7
+10k4Σ5
+kΣ′3
+k)X6/bracketrightbigg
+18˜KW
+22=/integraldisplayd4k
+(2π)4/bracketleftbigg
+(−1
+80k2Σ′′
+k−1
+80Σ2
+kΣ′′
+k−1
+240k4Σ′′′
+k−1
+240k2Σ2
+kΣ′′′
+k)k2ΣkX4+(3
+80k4ΣkΣ′
+k+1
+80k2Σ3
+kΣ′
+k−1
+40Σ5
+kΣ′
+k
+−1
+16k6Σ′2
+k+1
+16k2Σ4
+kΣ′2
+k+3
+80k6ΣkΣ′′
+k+1
+40k4Σ3
+kΣ′′
+k−1
+80k2Σ5
+kΣ′′
+k−19
+480k8Σ′
+kΣ′′
+k+1
+48k6Σ2
+kΣ′
+kΣ′′
+k
++29
+480k4Σ4
+kΣ′
+kΣ′′
+k)X5+(−1
+80k4Σ2
+k−1
+16k2Σ4
+k−1
+20Σ6
+k+1
+80k6ΣkΣ′
+k+1
+4k4Σ3
+kΣ′
+k+19
+80k2Σ5
+kΣ′
+k+1
+40k8Σ′2
+k
+−13
+40k6Σ2
+kΣ′2
+k−11
+40k4Σ4
+kΣ′2
+k+3
+40k2Σ6
+kΣ′2
+k+3
+20k8ΣkΣ′3
+k−3
+20k4Σ5
+kΣ′3
+k)X6/bracketrightbigg
+˜KW
+23=/integraldisplayd4k
+(2π)4/bracketleftbigg
+(−23
+160k2Σ′′
+k+17
+160Σ2
+kΣ′′
+k−23
+480k4Σ′′′
+k+17
+480k2Σ2
+kΣ′′′
+k)k2ΣkX4+(19
+160k4ΣkΣ′
+k−47
+160k2Σ3
+kΣ′
+k+7
+80Σ5
+kΣ′
+k
+−1
+32k6Σ′2
+k+3
+4k4Σ2
+kΣ′2
+k−7
+32k2Σ4
+kΣ′2
+k+31
+240k6ΣkΣ′′
+k−157
+480k4Σ3
+kΣ′′
+k+7
+160k2Σ5
+kΣ′′
+k−11
+480k8Σ′
+kΣ′′
+k+19
+24k6Σ2
+kΣ′
+kΣ′′
+k
+−89
+480k4Σ4
+kΣ′
+kΣ′′
+k)X5+(−9
+80k4Σ2
+k−3
+16k2Σ4
+k+7
+40Σ6
+k+1
+120k6ΣkΣ′
+k+7
+12k4Σ3
+kΣ′
+k−37
+40k2Σ5
+kΣ′
+k+1
+60k8Σ′2
+k
+−223
+160k6Σ2
+kΣ′2
+k+371
+240k4Σ4
+kΣ′2
+k−7
+160k2Σ6
+kΣ′2
+k+23
+80k8ΣkΣ′3
+k−13
+8k6Σ3
+kΣ′3
+k+7
+80k4Σ5
+kΣ′3
+k)X6/bracketrightbigg
+(C1)
+whereB0is the LEC appear in p2order normal part of the chiral Lagrangian. Σ k≡Σ(k2) andX≡1
+k2+Σ2
+k.
+19
\ No newline at end of file
diff --git a/1001.0316.txt b/1001.0316.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9f92add94a77ae0efc5d87fefb3b857fa5af73e1
--- /dev/null
+++ b/1001.0316.txt
@@ -0,0 +1,1261 @@
+Dark Matter: The evidence from astronomy,
+astrophysics and cosmology
+Matts Roos
+Department of Physics, FI-00014 University of Helsinki, Finland
+E-mail: matts.roos@helsinki.fi
+Abstract. Dark matter has been introduced to explain many independent gravitational
+effects at different astronomical scales, in galaxies, groups of galaxies, clusters, superclusters
+and even across the full horizon. This review describes the accumulated astronomical,
+astrophysical, and cosmological evidence for dark matter. It is written at a non-specialist
+level and intended for an audience with little or only partial knowledge of astrophysics or
+cosmology.
+PACS numbers: 95.35.+d, 98.65.-r, 98.62.-g, 98.80.-k, 98.90.+sarXiv:1001.0316v2  [astro-ph.CO]  18 Oct 2010Dark Matter: The evidence from astronomy, astrophysics and cosmology 2
+Contents
+1. Introduction
+2. Stars near the Galactic disk
+3. Virially bound systems
+4. Rotation curves of spiral galaxies
+5. Small galaxy groups emitting X-rays
+6. Mass to luminosity ratios
+7. Mass autocorrelation functions
+8. Strong and weak lensing
+9. Cosmic Microwave Background
+10. Baryonic acoustic oscillations
+11. Galaxy formation in purely baryonic matter
+12. Large Scale Structures simulated
+13. Dark matter from overall fits
+14. Merging galaxy clusters
+15. Comments and conclusions
+1. Introduction
+According to the standard “Cold Dark Matter” (CDM) paradigm, dark matter originated
+from quantum fluctuations in the almost uniform, early Universe during a very short period
+of cosmic inflation, and emerged with negligible thermal velocities and a Gaussian and scale-
+free distribution of density fluctuations. It is one of the most important tasks of our time to
+establish whether this paradigm is supported by observations.
+In spite of the importance of this task, there is no review of the accumulated astronomical,
+astrophysical, and cosmological observational testimony on the CDM paradigm. The literature
+summing such evidence usually refers only to the rotation curves of spiral galaxies, to the
+observations of strongly lensing galaxies and clusters, and to the determination of parameters
+in fits combining the Cosmic Microwave Background (CMB) with some other constraints.
+Quite recently the “Bullet cluster” gained fame due to the combination of lensing observations
+of its gravitational field and X-ray observations of the heated Intra-Cluster Medium (ICM).
+But there exists much more evidence.
+The purpose of this review is to summarize all the gravitational evidence of dark matter
+without entering into speculation on its true nature. Whether the correct explanation implies
+conventional matter, unconventional particles, or modifications to gravitational theory, all the
+observed gravitational effects described below have to be explained. Thus the use of the term
+“dark matter” is generic and does not imply that we are taking sides.
+Beginning from the first startling discoveries of Jan Hendrik Oort in 1932 [1] of missing
+matter in the Galactic disk which has not been confirmed by modern observations (Sec. 2),
+and that of Fritz Zwicky in 1933 [2] of missing matter in the Coma cluster (Sec. 3), much
+later understood to be “dark matter”, we describe the kinematics of virially bound systems
+(Sec. 3) and rotating spiral galaxies (Sec. 4). Different testimonies of missing mass in groups
+and clusters are offered by the comparison of visible light and X-rays (Sec. 5). The massDark Matter: The evidence from astronomy, astrophysics and cosmology 3
+to luminosity ratios of objects at all scales can be compared to what would be required to
+close the Universe (Sec. 6), and mass autocorrelation functions relate galaxy masses to dark
+halo masses (Sec. 7). An increasingly important method to determine the weights of galaxies,
+clusters and gravitational fields at large, independently of electromagnetic radiation, is lensing,
+strong as well as weak (Sec. 8).
+In radiation the most important tools are the temperature and polarization anisotropies
+in the Cosmic Microwave Background (Sec. 9), which give information on the mean density of
+both dark matter and baryonic matter as well as on the geometry of the Universe. The large
+scale structures of matter exhibit similar fluctuations with similar information in the Baryonic
+Acoustic Oscillations (BAO) (Sec. 10). The amplitude of the temperature variations in the
+CMB prove, that galaxies could not have formed in a purely baryonic Universe (Sec. 11).
+Simulations of large scale structures also show that dark matter must be present (Sec. 12).
+The best quantitative estimates of the density of dark matter come from overall parametric
+fits (to various cosmological models) of CMB data, BAO data, and redshifts of supernovae of
+type Ia (SNe Ia) (Sec. 13). A particularly impressive testimony comes from combining lensing
+information on pairs of colliding clusters with X-ray maps (Sec. 14). We conclude (Sec. 15)
+with a few general comments on the nature of dark matter.
+2. Stars near the Galactic disk
+In 1932 the Dutch astronomer Jan Hendrik Oort analyzed the vertical motions of all known
+stars near the Galactic plane and used these data to calculate the acceleration of matter [1].
+This amounts to treating the stars as members of a ”star atmosphere”, a statistical ensemble
+in which the density of stars and their velocity dispersion defines a ”temperature” from which
+one obtains the gravitational potential. This is analogous to how one obtains the gravitational
+potential of the Earth from a study of the atmosphere. The result contradicted grossly the
+expectations: the potential provided by the known stars was not sufficient to keep the stars
+bound to the Galactic disk, the Galaxy should rapidly be losing stars. Since the Galaxy
+appeared to be stable there had to be some missing matter near the Galactic plane, Oort
+thought, exerting gravitational attraction. This used to be counted as the first indication for
+the possible presence of dark matter in our Galaxy.
+The possibility that this missing matter would be non-baryonic could not even be thought
+of at that time. Note that the first neutral baryon, the neutron, was discovered by James
+Chadwick in the same year, in 1932.
+However, it is nowadays considered, that this does not prove the existence of dark matter
+in the disk. The potential in which the stars are moving is not only due to the disk, but rather
+to the totality of matter in the Galaxy which is dominated by the Galactic halo. The advent of
+much more precise data in 1998 led Holmberg and Flynn [3] to conclude that no dark matter
+was present in the disk.
+Oort determined the mass of the Galaxy to be 1011Msun, and thought that the
+nonluminous component was mainly gas. Still in 1969 he thought that intergalactic gas made
+up a large fraction of the mass of the universe [4]. The general recognition of the missing
+matter as a possibly new type of non-baryonic dark matter dates to the early eighties.Dark Matter: The evidence from astronomy, astrophysics and cosmology 4
+3. Virially bound systems
+Stars move in galaxies and galaxies in clusters along their orbits; the orbital velocities are
+balanced by the total gravity of the system, similar to the orbital velocities of planets moving
+around the Sun in its gravitational field. In the simplest dynamical framework one treats
+clusters of galaxies as statistically steady, spherical, self-gravitating systems of Nobjects of
+average mass mand average orbital velocity v. The total kinetic energy Eof such a system is
+then
+E=1
+2Nmv2. (1)
+If the average separation is r, the potential energy of N(N−1)/2 pairings is
+U=−1
+2N(N−1)Gm2
+r. (2)
+The virial theorem states that for such a system
+E=−U/2. (3)
+The total dynamic mass Mcan then be estimated from v, andrfrom the cluster volume
+M=Nm=2rv2
+G. (4)
+An empirical formula often used in simulations of the density distribution of dark matter
+halos in clusters is
+ρDM(r) =ρ0
+(r/rs)α(1 +r/rs)3−α, (5)
+whereρ0is a normalization constant and 0 ≤α≤3/2. Standard choices are α= 1 for the
+Navarro-Frenk-White profile [5] and the α= 3/2 profile of Moore & al. [6], both cusped at
+r= 0.
+However, Gentile & al. [7] have shown that cusped profiles are in clear conflict with data on
+spiral galaxies. Central densities are rather flat, scaling approximately as ρ0∝r−2/3
+luminous . The
+best-fit disk + NFW halo mass model fits the rotation curves poorly, it implies an implausibly
+low stellar mass-to-light ratio and an unphysically high halo mass.
+3.1. The Coma cluster
+Historically, the second possible indication of dark matter, the first time in an object at a
+cosmological distance, was found by Fritz Zwicky in 1933 [2]. While measuring radial velocities
+of member galaxies in the Coma cluster (that contains some 1000 galaxies), and the cluster
+radius from the volume they occupy, Zwicky was the first to use the virial theorem to infer
+the existence of unseen matter. He was able to infer the average mass of galaxies within the
+cluster, and obtained a value about 160 times greater than expected from their luminosity (a
+value revised today), and proposed that most of the missing matter was dark. His suggestion
+was not taken seriously at first by the astronomical community, that Zwicky felt as hostile and
+prejudicial. Clearly, there was no candidate for the dark matter because gas radiating X-rays
+and dust radiating in the infrared could not yet be observed, and non-baryonic matter was
+unthinkable. Only some forty years later when studies of motions of stars within galaxies alsoDark Matter: The evidence from astronomy, astrophysics and cosmology 5
+Figure 1. Density profile of matter components enclosed within a given radius rin the Coma
+cluster versus r/rvirial . From E. L. Lokas & G. A. Mamon [8].
+implied the presence of a large halo of unseen matter extending beyond the visible stars, dark
+matter became a serious possibility.
+Zwicky found to his surprise that the orbital velocities were almost a factor of ten larger
+than expected from the summed mass of all galaxies belonging to the Coma cluster. He then
+concluded that in order to hold galaxies together the cluster must contain huge amounts of some
+non-luminous matter. Since that time modern observations have revised our understanding of
+the composition of clusters. Luminous stars represent a very small fraction of a cluster mass;
+in addition there is a baryonic, hot intracluster medium (ICM) visible in the X-ray spectrum.
+Rich clusters typically have more mass in hot gas than in stars; in the largest virial systems
+like the Coma the composition is about 85% dark matter, 14% ICM, and only 1% stars [8].
+In modern applications of the virial theorem one also needs to model and parametrize
+the radial distributions of the ICM and the dark matter densities. In the outskirts of galaxy
+clusters the virial radius roughly separates bound galaxies from galaxies which may either be
+infalling or unbound. The virial radius rviris conventionally defined as the radius within which
+the mean density is 200 times the background density.
+Matter accretion is in general quite well described within the approximation of the
+Spherical Collapse Model. According to this model, the velocity of the infall motion and
+the matter overdensity are related. Mass profile estimation is thus possible once the infall
+pattern of galaxies is known [9].
+In Fig. 1 the Coma profile is fitted [8] with Eq. (5) with α= 0 which describes a centrally
+finite profile which is almost flat. The separation of different components in the core is notDark Matter: The evidence from astronomy, astrophysics and cosmology 6
+well done with Eq. (5) because the Coma has a binary center like many other clusters [10].
+The physical size of clusters is expected to depend on the mass, characterized by the mass
+concentration index cs≡rvir/rs, roughlycs∝M−0.1. This can be tested by strong lensing of
+a sequence of halos.
+Figure 2. Density profile of matter components in the cluster AC 114, enclosed within a
+given projected radius. From M. Sereno & al.[14].
+3.2. The AC 114 cluster
+Dark matter is usually dissected from baryons in lensing analyses by first fitting the lensing
+features to obtain a map of the total matter distribution and then subtracting the gas mass
+fraction as inferred from X-ray observations [11, 12]. The total mass map can then be
+obtained with parametric models in which the contribution from cluster-sized DM haloes can
+be considered together with the main galactic DM haloes [13]. Mass in stars and stellar
+remnants is estimated converting galaxy luminosity to mass assuming suitable stellar mass to
+light ratios.
+One may go one step further by exploiting a parametric model which has three kinds of
+components: cluster-sized dark matter haloes, galaxy-sized (dark plus stellar) matter haloes,
+and a cluster-sized gas distribution [10, 14]. The results of such an analysis of the dynamically
+active cluster AC 114 is shown in Fig. 2.
+3.3. The Local Group
+The Local Group is a very small virial system, dominated by two large galaxies, the M31 or
+Andromeda galaxy, and the Milky Way. The M31 exhibits blueshift, falling in towards us.Dark Matter: The evidence from astronomy, astrophysics and cosmology 7
+Evidently the Galaxy and M31, as well as all the minor galaxies in the Local Group, form a
+bound system which is oscillating.
+In this virial system the two large galaxies dominate the dynamics, so that it is not
+meaningful to define, as for the Coma, an average pairwise separation between galaxies, an
+average mass or an average orbital velocity. The total kinetic energy Eis still given by the
+sum over all the group members, and the potential energy Uby the sum over all the galaxy
+pairs, but here the pair formed by the M31 and the Milky Way dominates, and the pairings
+of the small members with each other are negligible.
+An interesting recent claim is, that the mass estimate of the Local Group is also affected
+by the accelerated expansion, the “dark energy”. A. D. Chernin & al. [15] have shown that the
+the potential energy Uis reduced in the force field of dark energy, so that the virial theorem
+forNmassesmiwith baryocentric radius vectors ritakes the form
+E=−1
+2U+U2, (6)
+whereUis defined as in Eq. (3), and
+U2=−4πρv
+3/summationdisplay
+mir2
+i (7)
+is a correction which reduces the potential energy due to the background dark energy density
+ρv. In the Local Group this correction to the mass appears to be quite substantial, of the order
+of 30%−50%.
+The dynamical mass of the local group comes out to be 3 .2−3.7 1012solar masses whereas
+the total visible mass of the Galaxy + M31 is only 2 1011solar masses. Thus there is a large
+amount of dark matter.
+4. Rotation curves of spiral galaxies
+Spiral galaxies are stable gravitationally bound systems in which visible matter is composed
+of stars and interstellar gas. Most of the observable matter is in a relatively thin disc, where
+stars and gas rotate around the galactic center on nearly circular orbits. If the circular velocity
+at radiusrisvin a galaxy with mass M(r) insider, the condition for stability is that the
+centrifugal acceleration v/rshould equal the gravitational pull GM(r)/r2, and the radial
+dependence of vwould then be expected to follow Kepler’s law
+v=/radicalbigg
+GM(r)
+r. (8)
+The surprising result from measurements of galaxy-rotation curves is that the velocity does
+not follow this 1 /√rlaw, but stays constant after attaining a maximum at about 5 kpc. The
+most obvious solution to this is that the galaxies are embedded in extensive, diffuse haloes of
+dark matter. If the enclosed mass inside the radius r, M(r), is proportional to r it follows that
+v(r)≈constant.
+The rotation curve of most galaxies can be fitted by the superposition of contributions
+from the stellar and gaseous disks, the luminous components, both modeled by thin exponential
+disks, sometimes a bulge, and the dark halo, modeled by a quasi-isothermal sphere. The
+inner part is difficult to model because the density of stars is high, rendering observations ofDark Matter: The evidence from astronomy, astrophysics and cosmology 8
+Figure 3. Best disk-halo fits to the Universal Rotation Curve (dotted/dashed line: disc/halo).
+Each object is identified by the halo virial mass [16].
+individual star velocities difficult. Thus the fits are not unique, the relative contributions of
+disk and dark matter is model-dependent, and it is not even sure whether galactic disks do
+contain dark matter. Typically the dark matter is about half of the total mass.
+In Fig. 3 we show the rotation curves fitted for eleven well-measured galaxies [16]. One
+notes, that at very small radii the dark halo component is indeed much smaller than the
+luminous disk component. At large radii, however, the need for a halo of dark matter is
+obvious. On galactic scales, the contribution of dark matter generally dominates the total mass.
+Note the contribution of the baryonic component, negligible for small masses but increasingly
+important in the larger structures.Dark Matter: The evidence from astronomy, astrophysics and cosmology 9
+Our Galaxy is complicated because of what appears to be a density dip at 9 kpc and a
+small dip at 3 kpc, as is seen in Fig. 4 [17]. To fit the measured rotation curve one needs at
+least three contributing components: a central bulge, the star disk + gas, and a dark matter
+halo [17, 18, 19]. For small radii there is a choice of empirical rotation curves, and no dark
+matter component appears to be needed until radii beyond 15 kpc.
+Figure 4. Decomposition of the rotation curve of the Milky Way into the components bulge,
+stellar disk + interstellar gas, dark matter halo (the red curves from left to right). From Y.
+Sofue et al. [17].Dark Matter: The evidence from astronomy, astrophysics and cosmology 10
+5. Small galaxy groups emitting X-rays
+There are examples of groups formed by a small number of galaxies which are enveloped in a
+large cloud of hot gas (ICM), visible by its X-ray emission. One may assume that the electron
+density distribution associated with the X-ray brightness is in hydrostatic equilibrium, and
+one can extract the ICM radial density profiles by fits.
+The amount of matter in the form of hot gas can be deduced from the intensity of this
+radiation. Adding the gas mass to the observed luminous matter, the total amount of baryonic
+matter,Mb, can be estimated [20, 21]. In clusters studied, the gas fraction increases with the
+distance from the center; the dark matter is more concentrated than the visible matter.
+The temperature of the gas depends on the strength of the gravitational field, from which
+the total amount of gravitating matter, Mgrav, in the system can be deduced. In many such
+small galaxy groups one finds Mgrav/Mb≥3. Thus a dark halo must be present. An accurate
+estimate of Mgravrequires that also dark energy is taken into account, because it reduces
+the strength of the gravitational potential. There are sometimes doubts whether all galaxies
+apparently in these groups are physical members. If not, they will artificially increase the
+velocity scatter and thus lead to larger virial masses.
+On the scale of large clusters of galaxies like the Coma, it is generally observed that dark
+matter represents about 85% of the total mass and that the visible matter is mostly in the
+form of a hot ICM.
+6. Mass to luminosity ratios
+Let us define the mass/luminosity ratio of an astronomical object as Υ ≡M/L . Stellar
+populations exhibit values Υ = 1 −10 in solar units, in the solar neighborhood Υ = 2 .5−7,
+in the Galactic disk Υ = 1 .0−1.7 [22]. In the Coma the value of Υ depends strongly on the
+halo profile for r <0.2 Mpc; beyond that the value is 400 and slowly decreasing with radius
+[8]. Many dwarf spheroidal galaxies exhibit very high ratios: in Draco Υ = 330 ±125, in
+Andromeda IX Υ = 93+120
+−50. The cluster AC 114 discussed in Sec. 2.2 is very underluminous
+having Υ = 700±100 [14].
+A special case is the ultra-faint dwarf disk galaxy Segue 1 [23] which has a baryon mass
+of only about 1000 solar masses. One interpretation is that it is a thin non-rotating stellar
+disk not accompanied by a gas disk, embedded in an axisymmetric DM halo and with an
+f=Mhalo/Mb≈200. But if the disk rotates, fcould be as high as 2000. If Segue 1 also has a
+magnetized gas disk, the DM halo has to confine the effective pressure in the stellar disk and
+the magnetic Lorentz force in the gas disk as well as possible rotation. Then fcould be very
+large [23].
+These mass/luminosity ratios are however local, characteristic for isolated objects but not
+for the whole observable Universe. The value for LUniverse is roughly known. One can then
+deduce that the critical density to close the Universe would require Υ ≈990. However, there
+is not sufficiently much luminous matter to achieve closure, therefore a large amount of DM is
+needed.Dark Matter: The evidence from astronomy, astrophysics and cosmology 11
+Figure 5. Dark matter halo mass Mhaloas a function of stellar mass M∗. The thick black curve
+is the prediction from abundance matching assuming no dispersion in the relation between the
+two masses. Red and green dashed curves assume some dispersion in log M∗. The dashed
+black curve is the satellite fraction as a function of stellar mass, as labeled on the axis at the
+right-hand side of the plot. From Qi Guo & al. [24].
+7. Mass autocorrelation functions
+If galaxy formation is a local process, then on large scales galaxies must trace mass. This
+requires the study of how galaxies populate dark matter haloes. In simulations one attempts
+to track galaxy and dark matter halo evolution across cosmic time in a physically consistent
+way, providing positions, velocities, star formation histories and other physical properties for
+the galaxy populations of interest.
+Guo & al.[24] use abundance matching arguments to derive an accurate relation between
+galaxy stellar mass and dark matter halo mass. They combine a stellar mass function
+based on spectroscopic observations with a precise halo/subhalo mass function obtained from
+simulations. Assuming this stellar mass halo mass relation to be unique and monotonic, they
+compare it with direct observational estimates of the mean mass of haloes surrounding galaxies
+of given stellar mass inferred from gravitational lensing and satellite galaxy dynamics data,
+and use it to populate haloes in simulations. The stellar mass halo mass relation is shown
+in Fig. 5. The implied spatial clustering of stellar mass turns out to be in remarkably good
+agreement with a direct and precise measurement. By comparing the galaxy autocorrelation
+function with the total mass autocorrelation function, as averaged over the Local SuperclusterDark Matter: The evidence from astronomy, astrophysics and cosmology 12
+Figure 6. Dark matter column density vs. dark matter halo mass in solar units. From A.
+Boyarsky & al.[25]
+(LSC) volume, one concludes that a large amount of matter in the LSC is dark.
+A similar study is that of Boyarsky & al. [25] who find a universal relation between DM
+column density and DM halo mass, satisfied by matter distributions at all observable scales,
+in halo sizes from 108to 1016solar masses, as shown in Fig. 6. Such a universal property is
+difficult to explain without dark matter.
+8. Strong and weak lensing
+A consequence of the Strong Equivalence Principle (SEP) is that a photon in a gravitational
+field moves as if it possessed mass, and light rays therefore bend around gravitating masses.
+Thus celestial bodies can serve as gravitational lenses probing the gravitational field, whether
+baryonic or dark without distinction.
+Since photons are neither emitted nor absorbed in the process of gravitational light
+deflection, the surface brightness of lensed sources remains unchanged. Changing the size of
+the cross-section of a light bundle only changes the flux observed from a source and magnifies it
+at fixed surface-brightness level. If the mass of the lensing object is very small, one will merely
+observe a magnification of the brightness of the lensed object an effect called microlensing .
+Microlensing of distant quasars by compact lensing objects (stars, planets) has also been
+observed and used for estimating the mass distribution of the lens–quasar systems.
+Weak Lensing refers to deflection through a small angle when the light ray can be treated
+as a straight line (Fig. 7), and the deflection as if it occurred discontinuously at the point ofDark Matter: The evidence from astronomy, astrophysics and cosmology 13
+Figure 7. Wave fronts and light rays in the presence of a cluster perturbation. From N.
+Straumann [26].
+closest approach (the thin-lens approximation in optics). One then only invokes SEP which
+accounts for the distortion of clock rates.
+Figure 8. This image resulted from color-subtracteion of a lensing singular isothermal
+elliptical galaxy. The strongly lensed object forms two prominent arcs A, B and a less extended
+third image C. From R.J. Smith & al.[27]Dark Matter: The evidence from astronomy, astrophysics and cosmology 14
+Figure 9. Map of the dark matter distribution in the 2-square degree COSMOS field [28]: the
+linear blue scale on top shows the gravitational lensing magnification κ, which is proportional
+to the projected mass along the line of sight. Contours begin at 0.4% and are spaced by 0.5%
+inκ.
+InStrong Lensing the photons move along geodesics in a strong gravitational potential
+which distorts space as well as time, causing larger deflection angles and requiring the full
+theory of GR. The images in the observer plane can then become quite complicated because
+there may be more than one null geodesic connecting source and observer; it may not even be
+possible to find a unique mapping onto the source plane c.f.Fig.7 . Strong lensing is a tool
+for testing the distribution of mass in the lens rather than purely a tool for testing GR. An
+illustration is seen in Fig. 8 where the lens is an elliptical galaxy [27].
+At cosmological distances one may observe lensing by composed objects such as galaxy
+groups which are ensembles of “point-like”, individual galaxies. Lensing effects are very model-
+dependent, so to learn the true magnification effect one needs very detailed information on the
+structure of the lens.
+The large-scale distribution of matter in the Universe is inhomogeneous in every direction,
+so one can expect that everything we observe is displaced and distorted by weak lensing. Since
+the tidal gravitational field and the deflection angles depend neither on the nature of the matter
+nor on its physical state, light deflection probes the total projected mass distribution. Lensing
+in infrared light offers an additional advantage of being able to sense distant background
+galaxies, since their number density is higher than in the optical range.
+Background galaxies would be ideal tracers of distortions if they were intrinsically circular,Dark Matter: The evidence from astronomy, astrophysics and cosmology 15
+Figure 10. Strong lensing analysis of the galaxy cluster Abell 370 (z=0.375) shown with
+the locations of the identified multiple systems within the critical line at z=1.2 in white (the
+redshift for the majority of the mutiple images). The red line delimits the region of multiple
+images for very high redshift sources (here assuming z = 6). From J. Richard & al.[29]
+because lensing transforms circular sources into ellipses. Any measured ellipticity would then
+directly reflect the action of the gravitational tidal field of the interposed lensing matter,
+and the statistical properties of the distortions would reflect the properties of the matter
+distribution. But many galaxies are actually intrinsically elliptical, and the ellipses are
+randomly oriented. This introduces noise into the inference of the tidal field from observed
+ellipticities. A useful feature in the sky is a fine-grained pattern of faint and distant blue
+galaxies appearing as a ‘wall paper’. This makes statistical weak-lensing studies possible,
+because it allows the detection of the coherent distortions imprinted by gravitational lensing
+on the images of the galaxy population. Thus weak lensing has become an important technique
+to map non-luminous matter. A reconstruction of the largest and most detailed weak lensing
+survey ever undertaken with the Hubble Space Telescope is shown in Fig. 9 [28]. This map
+covers a large enough area to see extended filamentary structures.
+In Fig. 10 we see a first detailed example of the strong lensing of a cluster [29].
+9. Cosmic Microwave Background (CMB)
+The tight coupling between radiation and matter density before decoupling caused the
+primordial adiabatic perturbations to oscillate in phase. Beginning from the time of last
+scattering, the receding horizon has been revealing these frozen density perturbations, settingDark Matter: The evidence from astronomy, astrophysics and cosmology 16
+Figure 11. The maps of CMB radiation temperature (TT) and temperature-polarization
+correlations (TE) from WMAP show anisotropies which can be analyzed by power spectra as
+functions of multipole moments. From G. Hinshaw & al.[30]
+up a pattern of standing acoustic waves in the baryon–photon fluid. After decoupling, this
+pattern is visible today as temperature anisotropies with a certain regularity across the sky.
+The primordial photons are polarized by the anisotropic Thomson scattering process, but
+as long as the photons continue to meet free electrons their polarization is washed out, and no
+net polarization is produced. At a photon’s last scattering, however, the induced polarization
+remains and the subsequently free-streaming photon possesses a quadrupole moment.
+Temperature and polarization fluctuations are analyzed in terms of multipole components
+or powers. The resulting distribution of powers versus multipole /lscript, or multipole moment
+k= 2π//lscript, is the power spectrum which exhibits conspicuous Doppler peaks . In Fig. 11
+we display the TT and TE power spectra from the five-year data of WMAP as functionsDark Matter: The evidence from astronomy, astrophysics and cosmology 17
+of multipole moments [30]. The spectra can then be compared to theory, and theoretical
+parameters determined. Many experiments have determined the power spectra, so a wealth of
+data exists.
+Figure 12. The E-mode polarization power spectrum (EE) from the CMB observations of
+the QUaD collaboration & al.[32]
+Baryonic matter feels attractive self-gravity and is pressure-supported, whereas dark
+matter only feels attractive self-gravity, but is pressureless. Thus the Doppler peaks in the
+CMBR power spectrum testify about baryonic and dark matter, whereas the troughs testify
+about rarefaction caused by the baryonic pressure. The position of the first peak determines
+Ωmh2. Combining the 5-year WMAP measurements of the temperature power spectrum (TT)
+with determinations of the Hubble constant h, the WMAP team finds the total mass density
+parameter Ω m≈0.26 [31]. The ratio of amplitudes of the second-to-first Doppler peaks
+determines the baryonic density parameter Ω b≈0.04; the dark matter component is then
+Ωdm≈0.22.
+Information on the temperature – E-mode polarization correlations (TE) comes from
+several measurements, among them WMAP [30, 31], and on the E-mode polarization power
+spectrum alone (EE) from the QUAD collaboration [32], Fig. 12.
+The results show two surprises: Firstly, since Ω m/lessmuch1, a large component Ω Λ≈0.74 is
+missing, of unknown nature, and termed dark energy . The second surprise is that ordinary
+baryonic matter is only a small fraction of the total matter budget. The remainder is then
+dark matter, of unknown composition. Of the 4.2% of baryons in the Universe only about 1%
+is stars.
+10. Baryonic acoustic oscillations (BAO)
+A cornerstone of cosmology is the Copernican principle, that matter in the Universe is
+distributed homogeneously, if only on the largest scales of superclusters separated by voids. On
+smaller scales we observe inhomogeneities in the forms of galaxies, galaxy groups, and clusters.Dark Matter: The evidence from astronomy, astrophysics and cosmology 18
+The common approach to this situation is to turn to non-relativistic hydrodynamics and treat
+matter in the Universe as an adiabatic, viscous, non-static fluid, in which random fluctuations
+around the mean density appear, manifested by compressions in some regions and rarefactions
+in other. The origin of these density fluctuations was the tight coupling established before
+decoupling between radiation and charged matter density, causing them to oscillate in phase.
+An ordinary fluid is dominated by the material pressure, but in the fluid of our Universe three
+effects are competing: gravitational attraction, density dilution due to the Hubble flow, and
+radiation pressure felt by charged particles only.
+The inflationary fluctuations crossed the post-inflationary Hubble radius, to come back
+into vision with a wavelength corresponding to the size of the Hubble radius at that moment.
+At timeteqthe overdensities began to amplify and grow into larger inhomogeneities. In
+overdense regions where the gravitational forces dominate, matter contracts locally and
+attracts surrounding matter, becoming increasingly unstable until it eventually collapses into
+a gravitationally bound object. In regions where the pressure forces dominate, the fluctuations
+move with constant amplitude as sound waves in the fluid, transporting energy from one region
+of space to another.
+Inflationary models predict that the primordial mass density fluctuations should be
+adiabatic, Gaussian, and exhibit the same scale invariance as the CMB fluctuations. The
+baryonic acoustic oscillations can be treated similarly to CMB, they are specified by the
+dimensionless mass autocorrelation function which is the Fourier transform of the power
+spectrum of a spherical harmonic expansion. The power spectrum is shown in Fig. 13 [33].
+As the Universe approached decoupling, the photon mean free path increases and
+radiation can diffused from overdense regions into underdense ones, thereby smoothing out
+any inhomogeneities in the plasma. The situation changed dramatically at recombination, at
+time 380 000 yr after Big Bang, when all the free electrons suddenly disappeared, captured
+into atomic Bohr orbits, and the radiation pressure almost vanished. Now the baryon acoustic
+waves and the CMB continued to oscillate independently, but adiabatically, and the density
+perturbations which have entered the Hubble radius since then can grow with full vigor into
+baryonic structures.
+The scale of BAO depends on Ω mand on the Hubble constant, h, so one needs information
+onhto break the degeneracy. The result is then Ω m≈0.26. In the ratio Ω b/Ωmtheh-
+dependence cancels out, so one can also quantify the amount of dark matter on very large
+scales by Ω b/Ωm= 0.18±0.04.
+11. Galaxy formation in purely baryonic matter
+We have seen in Sec. 9 that the baryonic density parameter, Ω b, is very small. The critical
+density Ω critical is determined by the expansion speed of the Universe, and the mean baryonic
+density of the Universe (stars, interstellar & intergalactic gas) is only Ω b= 0.046×Ωcritical
+[31].
+The question arises whether the galaxies could have formed from primordial density
+fluctuations in a purely baryonic medium. We have also noted, that the fluctuations in CMB
+and BAO maintain adiabaticity. The amplitude of the primordial baryon density fluctuationsDark Matter: The evidence from astronomy, astrophysics and cosmology 19
+Figure 13. BAO in power spectra calculated from (a) the combined SDSS and 2dFGRS
+main galaxies, (b) the SDSS DR5 LRG sample, and (c) the combination of these two samples
+(solid symbols with 1 σerrors). The data are correlated and the errors are calculated from the
+diagonal terms in the covariance matrix. A Standard ΛCDM distance-redshift relation was
+assumed to calculate the power spectra with Ω m= 0.25,ΩΛ= 0.75. From Percival & al.[33].
+would have needed to be very large in order to form the observed number of galaxies. But then
+the amplitude of the CMB fluctuations would also have been very large, leading to intolerably
+large CMB anisotropies today. Thus galaxy formation in purely baryonic matter is ruled out
+by this argument alone.
+Thus one concludes, that the galaxies could only have been formed in the presence of
+gravitating dark matter which started to fluctuate early, unhindered by radiation pressure.
+This conclusion is further strengthened in the next Section.
+12. Large Scale Structures simulated
+In the ΛCDM paradigm, the nonlinear growth of dark matter structure is a well-posed problem
+where both the initial conditions and the evolution equations are known (at least when the
+effects of the baryons can be neglected).
+The Aquarius Project [34] is a Virgo Consortium programme to carry out high-resolution
+dark matter simulations of Milky-Way-sized halos in the ΛCDM cosmology. This project seeksDark Matter: The evidence from astronomy, astrophysics and cosmology 20
+clues to the formation of galaxies and to the nature of the dark matter by designing strategies
+for exploring the formation of our Galaxy and its luminous and dark satellites. The galaxy
+Figure 14. The left panel shows the projected dark matter density at z = 0 in a slice of
+thickness 13.7 Mpc through the full box (137 Mpc on a side) of the 9003parent simulation.
+The right panel show this halo resimulated at a different numerical resolution. The image
+brightness is proportional to the logarithm of the squared dark matter density projected along
+the line-of-sight. The circles mark r50, the radius within which the mean density is 50 times
+the background density. From V. Springel & al. [34]
+population on scales from 50 kpc to the size of the observable Universe has been predicted by
+hierarchical ΛCDM scenarios, and compared directly with a wide array of observations. So far,
+the ΛCDM paradigm has passed these tests successfully, particularly those that consider the
+large-scale matter distribution and has led to the discovery of a universal internal structure for
+dark matter halos As was noted already in Sec. 11, the observed structure of galaxies, clusters
+and superclusters, as illustrated by Fig. 14, could not have formed in a baryonic medium
+devoid of dark matter.
+Given this success, it is important to test ΛCDM predictions also on smaller scales, not
+least because these are sensitive to the nature of the dark matter. Indeed, a number of serious
+challenges to the paradigm have emerged on the scale of individual galaxies and their central
+structure. In particular, the abundance of small dark matter subhalos predicted within CDM
+halos is much larger than the number of known satellite galaxies surrounding the Milky Way.
+13. Dark matter from overall fits
+In Sec. 9 we have seen that the WMAP-5 year CMB data together with the Hubble constant
+value testify about the existence of dark matter [30, 31]. In Sec. 10 we addressed the BAO
+data [33] with the same conclusion. In overall fits one combines these with supernova data
+(SN Ia) which offer a constraint nearly orthogonal to that of CMB in the Ω Λ−Ωm-plane.
+The Union compilation of 414 SN Ia, which reduces to 307 SNe after selection cuts, includes
+the recent large samples of SNe Ia from the Supernova Legacy Survey and ESSENCE Survey,Dark Matter: The evidence from astronomy, astrophysics and cosmology 21
+Figure 15. 68.3 %, 95.4 % and 99.7% confidence level contours on Ω Λand Ω mobtained from
+CMB, BAO and the Union SN set, as well as their combination (assuming w = -1). Note the
+straight line corresponding to a flat Universe with Ω Λ+Ω m= 1. From M. Kowalski & al. [35].Dark Matter: The evidence from astronomy, astrophysics and cosmology 22
+the older data sets, as well as the recently extended data set of distant supernovae observed
+with HST. M. Kowalski & al. [35] present the latest results from this Union compilation and
+discuss the cosmological constraints and its combination with CMB and BAO measurements.
+The CMB constraint is close to the line Ω Λ+ Ωm= 1, whereas the supernova constraint is
+close to the line Ω Λ−1.6×Ωm= 0.2. The BAO data constrain Ω m, but hardly at all Ω Λ. This
+is shown in Fig. 15.
+Defining the vacuum energy density parameter by
+Ωk= 1−ΩΛ−Ωm, (9)
+a flat Universe corresponds to Ω k= 0. For a Λ CDM Universe with a cosmological constant
+responsible for dark energy, a simultaneous fit to the data sets gives
+Ωm= 0.285+0.020
+−0.019±0.011,Ωk=−0.009+0.009+0.002
+−0.010−0.003, (10)
+where the first error is statistical and the second error systematic. This is the most precise
+determination at present of the total mass density parameter Ω m. At the same time one notes
+that the Universe is consistent with being flat. Subtracting Ω b= 0.046 from Ω m= 0.285 one
+obtains the density parameter for dark matter, Ω dm≈0.24. Assuming flatness, M. Kowalski
+& al. [35] find Ω m= 0.274±0.016±0.013.
+Figure 16. The merging cluster 1E0657-558. On the right is the smaller ’bullet’ cluster which
+has traversed the larger cluster. The colors indicate the X-ray temperature of the plasma: blue
+is coolest and white is hottest. The green contours are the weak lensing reconstruction of the
+gravitational potential of the cluster. From D. Clowe & al. [36]
+14. Merging galaxy clusters
+A direct empirical proof of the existence of dark matter is furnished by observations of 1E0657-
+558, a unique cluster merger [36]. Due to the collision of two clusters, the dissipationless stellarDark Matter: The evidence from astronomy, astrophysics and cosmology 23
+component and the fluid-like X-ray emitting plasma are spatially segregated. The gravitational
+potential observed by weak and strong lensing does not trace the plasma distribution which is
+the dominant baryonic mass component, but rather approximately traces the distribution of
+galaxies, cf. Fig. 16. The center of the total mass is offset from the center of the baryonic mass
+peaks, proving that the majority of the matter in the system is unseen. In front of the smaller
+’bullet’ cluster which has traversed the larger one, a bow shock is evident in the X-rays.
+Two other merging systems with similar characteristics have been seen, although with
+lower spatial resolution and less clear-cut cluster geometry. In Fig.17 we show the post-merging
+galaxy cluster MACS J0025.4-1222 [11].
+Figure 17. The F555W-F814W color composite of the cluster MACS J0025.4.1222. Overlaid
+in red contours is the surface mass density κfrom the combined weak and strong lensing mass
+reconstruction. The contour levels are linearly spaced with ∆ κ= 0.1, starting at ∆ κ= 0.5,
+for a fiducial source at a redshift of zs→∞ . The X-ray brightness contours (also linearly
+spaced) are overlaid in yellow and the I-band light is overlaid in white. The measured peak
+position and error bars for the total mass are given as a cyan cross [11].
+15. Comments and conclusions
+What we have termed “dark matter” is generic for observed gravitational effects on all scales:
+galaxies, small and large galaxy groups, clusters and superclusters, CMB anisotropies over the
+full horizon, baryonic oscillations over large scales, and cosmic shear in the large-scale matter
+distribution. The correct explanation or nature of dark matter is not known, whether it implies
+conventional matter, unconventional particles, or modifications to gravitational theory. but
+gravitational effects prove its existence in some form.Dark Matter: The evidence from astronomy, astrophysics and cosmology 24
+Only a few per cent of the total mass of the Universe is found in stars and hydrogen
+clouds, and this amount of baryonic matter is well accounted for by nucleosynthesis. If there
+exist particles which were very slow at time teqwhen galaxy formation started, they could be
+candidates for cold dark matter. If dark matter were composed of such new particles, they
+must have become non-relativistic much earlier than the leptons, and then decoupled from the
+hot plasma.
+Whenever laboratory searches discover a new particle, it must pass several tests in order
+to be considered a viable DM candidate: it must be neutral, compatible with constraints
+on self-interactions (essentially collisionless), consistent with Big Bang nucleosynthesis, and
+match the appropriate relic density. It must be consistent with direct DM searches and
+gamma-ray constraints, it must leave stellar evolution unchanged, and be compatible with
+other astrophysical bounds.
+Theories attempting to explain dark matter without new particles, with new interactions
+or modified gravity, likewise have the burden to explain all the observed gravitational effects
+described in here. Thus there remains much to be done.
+Acknowledgements
+It is a pleasure to thank K. Mattila at the University of Helsinki, M. Valtonen at the University
+of Turku, and P. Salucci at SISSA, Trieste, for valuable comments.
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+astrophysics and cosmology
+Matts Roos
+Department of Physics, FI-00014 University of Helsinki, Finland
+E-mail: matts.roos@helsinki.fi
+Abstract. Dark matter has been introduced to explain many independent gravitational
+effects at different astronomical scales, in galaxies, groups of galaxies, clusters, superclusters
+and even across the full horizon. This review describes the accumulated astronomical,
+astrophysical, and cosmological evidence for dark matter. It is written at a non-specialist
+level and intended for an audience with little or only partial knowledge of astrophysics or
+cosmology.
+PACS numbers: 95.35.+d, 98.65.-r, 98.62.-g, 98.80.-k, 98.90.+sarXiv:1001.0316v2  [astro-ph.CO]  18 Oct 2010Dark Matter: The evidence from astronomy, astrophysics and cosmology 2
+Contents
+1. Introduction
+2. Stars near the Galactic disk
+3. Virially bound systems
+4. Rotation curves of spiral galaxies
+5. Small galaxy groups emitting X-rays
+6. Mass to luminosity ratios
+7. Mass autocorrelation functions
+8. Strong and weak lensing
+9. Cosmic Microwave Background
+10. Baryonic acoustic oscillations
+11. Galaxy formation in purely baryonic matter
+12. Large Scale Structures simulated
+13. Dark matter from overall fits
+14. Merging galaxy clusters
+15. Comments and conclusions
+1. Introduction
+According to the standard “Cold Dark Matter” (CDM) paradigm, dark matter originated
+from quantum fluctuations in the almost uniform, early Universe during a very short period
+of cosmic inflation, and emerged with negligible thermal velocities and a Gaussian and scale-
+free distribution of density fluctuations. It is one of the most important tasks of our time to
+establish whether this paradigm is supported by observations.
+In spite of the importance of this task, there is no review of the accumulated astronomical,
+astrophysical, and cosmological observational testimony on the CDM paradigm. The literature
+summing such evidence usually refers only to the rotation curves of spiral galaxies, to the
+observations of strongly lensing galaxies and clusters, and to the determination of parameters
+in fits combining the Cosmic Microwave Background (CMB) with some other constraints.
+Quite recently the “Bullet cluster” gained fame due to the combination of lensing observations
+of its gravitational field and X-ray observations of the heated Intra-Cluster Medium (ICM).
+But there exists much more evidence.
+The purpose of this review is to summarize all the gravitational evidence of dark matter
+without entering into speculation on its true nature. Whether the correct explanation implies
+conventional matter, unconventional particles, or modifications to gravitational theory, all the
+observed gravitational effects described below have to be explained. Thus the use of the term
+“dark matter” is generic and does not imply that we are taking sides.
+Beginning from the first startling discoveries of Jan Hendrik Oort in 1932 [1] of missing
+matter in the Galactic disk which has not been confirmed by modern observations (Sec. 2),
+and that of Fritz Zwicky in 1933 [2] of missing matter in the Coma cluster (Sec. 3), much
+later understood to be “dark matter”, we describe the kinematics of virially bound systems
+(Sec. 3) and rotating spiral galaxies (Sec. 4). Different testimonies of missing mass in groups
+and clusters are offered by the comparison of visible light and X-rays (Sec. 5). The massDark Matter: The evidence from astronomy, astrophysics and cosmology 3
+to luminosity ratios of objects at all scales can be compared to what would be required to
+close the Universe (Sec. 6), and mass autocorrelation functions relate galaxy masses to dark
+halo masses (Sec. 7). An increasingly important method to determine the weights of galaxies,
+clusters and gravitational fields at large, independently of electromagnetic radiation, is lensing,
+strong as well as weak (Sec. 8).
+In radiation the most important tools are the temperature and polarization anisotropies
+in the Cosmic Microwave Background (Sec. 9), which give information on the mean density of
+both dark matter and baryonic matter as well as on the geometry of the Universe. The large
+scale structures of matter exhibit similar fluctuations with similar information in the Baryonic
+Acoustic Oscillations (BAO) (Sec. 10). The amplitude of the temperature variations in the
+CMB prove, that galaxies could not have formed in a purely baryonic Universe (Sec. 11).
+Simulations of large scale structures also show that dark matter must be present (Sec. 12).
+The best quantitative estimates of the density of dark matter come from overall parametric
+fits (to various cosmological models) of CMB data, BAO data, and redshifts of supernovae of
+type Ia (SNe Ia) (Sec. 13). A particularly impressive testimony comes from combining lensing
+information on pairs of colliding clusters with X-ray maps (Sec. 14). We conclude (Sec. 15)
+with a few general comments on the nature of dark matter.
+2. Stars near the Galactic disk
+In 1932 the Dutch astronomer Jan Hendrik Oort analyzed the vertical motions of all known
+stars near the Galactic plane and used these data to calculate the acceleration of matter [1].
+This amounts to treating the stars as members of a ”star atmosphere”, a statistical ensemble
+in which the density of stars and their velocity dispersion defines a ”temperature” from which
+one obtains the gravitational potential. This is analogous to how one obtains the gravitational
+potential of the Earth from a study of the atmosphere. The result contradicted grossly the
+expectations: the potential provided by the known stars was not sufficient to keep the stars
+bound to the Galactic disk, the Galaxy should rapidly be losing stars. Since the Galaxy
+appeared to be stable there had to be some missing matter near the Galactic plane, Oort
+thought, exerting gravitational attraction. This used to be counted as the first indication for
+the possible presence of dark matter in our Galaxy.
+The possibility that this missing matter would be non-baryonic could not even be thought
+of at that time. Note that the first neutral baryon, the neutron, was discovered by James
+Chadwick in the same year, in 1932.
+However, it is nowadays considered, that this does not prove the existence of dark matter
+in the disk. The potential in which the stars are moving is not only due to the disk, but rather
+to the totality of matter in the Galaxy which is dominated by the Galactic halo. The advent of
+much more precise data in 1998 led Holmberg and Flynn [3] to conclude that no dark matter
+was present in the disk.
+Oort determined the mass of the Galaxy to be 1011Msun, and thought that the
+nonluminous component was mainly gas. Still in 1969 he thought that intergalactic gas made
+up a large fraction of the mass of the universe [4]. The general recognition of the missing
+matter as a possibly new type of non-baryonic dark matter dates to the early eighties.Dark Matter: The evidence from astronomy, astrophysics and cosmology 4
+3. Virially bound systems
+Stars move in galaxies and galaxies in clusters along their orbits; the orbital velocities are
+balanced by the total gravity of the system, similar to the orbital velocities of planets moving
+around the Sun in its gravitational field. In the simplest dynamical framework one treats
+clusters of galaxies as statistically steady, spherical, self-gravitating systems of Nobjects of
+average mass mand average orbital velocity v. The total kinetic energy Eof such a system is
+then
+E=1
+2Nmv2. (1)
+If the average separation is r, the potential energy of N(N−1)/2 pairings is
+U=−1
+2N(N−1)Gm2
+r. (2)
+The virial theorem states that for such a system
+E=−U/2. (3)
+The total dynamic mass Mcan then be estimated from v, andrfrom the cluster volume
+M=Nm=2rv2
+G. (4)
+A generally used empirical formula to describe the density distribution of dark matter
+halos in clusters is
+ρDM(r) =ρ0
+(r/rs)α(1 +r/rs)3−α, (5)
+whereρ0is a normalization constant and 0 ≤α≤3/2. Standard choices are α= 1 for the
+Navarro-Frenk-White profile [5] and the α= 3/2 profile of Moore & al. [6], both cusped at
+r= 0.
+3.1. The Coma cluster
+Historically, the second possible indication of dark matter, the first time in an object at a
+cosmological distance, was found by Fritz Zwicky in 1933 [2]. While measuring radial velocities
+of member galaxies in the Coma cluster (that contains some 1000 galaxies), and the cluster
+radius from the volume they occupy, Zwicky was the first to use the virial theorem to infer
+the existence of unseen matter. He was able to infer the average mass of galaxies within the
+cluster, and obtained a value about 160 times greater than expected from their luminosity (a
+value revised today), and proposed that most of the missing matter was dark. His suggestion
+was not taken seriously at first by the astronomical community, that Zwicky felt as hostile and
+prejudicial. Clearly, there was no candidate for the dark matter because gas radiating X-rays
+and dust radiating in the infrared could not yet be observed, and non-baryonic matter was
+unthinkable. Only some forty years later when studies of motions of stars within galaxies also
+implied the presence of a large halo of unseen matter extending beyond the visible stars, dark
+matter became a serious possibility.
+Zwicky found to his surprise that the orbital velocities were almost a factor of ten larger
+than expected from the summed mass of all galaxies belonging to the Coma cluster. He thenDark Matter: The evidence from astronomy, astrophysics and cosmology 5
+Figure 1. Density profile of matter components enclosed within a given radius rin the Coma
+cluster versus r/rvirial . From E. L. Lokas & G. A. Mamon [7].
+concluded that in order to hold galaxies together the cluster must contain huge amounts of some
+non-luminous matter. Since that time modern observations have revised our understanding of
+the composition of clusters. Luminous stars represent a very small fraction of a cluster mass;
+in addition there is a baryonic, hot intracluster medium (ICM) visible in the X-ray spectrum.
+Rich clusters typically have more mass in hot gas than in stars; in the largest virial systems
+like the Coma the composition is about 85% dark matter, 14% ICM, and only 1% stars [7].
+In modern applications of the virial theorem one also needs to model and parametrize
+the radial distributions of the ICM and the dark matter densities. In the outskirts of galaxy
+clusters the virial radius roughly separates bound galaxies from galaxies which may either be
+infalling or unbound. The virial radius rviris conventionally defined as the radius within which
+the mean density is 200 times the background density.
+Matter accretion is in general quite well described within the approximation of the
+Spherical Collapse Model. According to this model, the velocity of the infall motion and
+the matter overdensity are related. Mass profile estimation is thus possible once the infall
+pattern of galaxies is known [8].
+In Fig. 1 the Coma profile is fitted [7] with Eq. (5) with α= 0 which describes a centrally
+finite profile which is almost flat. The separation of different components in the core is not
+well done with Eq. (5) because the Coma has a binary center like many other clusters [9].
+The physical size of clusters is expected to depend on the mass, characterized by the mass
+concentration index cs≡rvir/rs, roughlycs∝M−0.1. This can be tested by strong lensing of
+a sequence of halos.Dark Matter: The evidence from astronomy, astrophysics and cosmology 6
+Figure 2. Density profile of matter components in the cluster AC 114, enclosed within a
+given projected radius. From M. Sereno & al.[13].
+3.2. The AC 114 cluster
+Dark matter is usually dissected from baryons in lensing analyses by first fitting the lensing
+features to obtain a map of the total matter distribution and then subtracting the gas mass
+fraction as inferred from X-ray observations [10, 11]. The total mass map can then be
+obtained with parametric models in which the contribution from cluster-sized DM haloes can
+be considered together with the main galactic DM haloes [12]. Mass in stars and stellar
+remnants is estimated converting galaxy luminosity to mass assuming suitable stellar mass to
+light ratios.
+One may go one step further by exploiting a parametric model which has three kinds of
+components: cluster-sized dark matter haloes, galaxy-sized (dark plus stellar) matter haloes,
+and a cluster-sized gas distribution [9, 13]. The results of such an analysis of the dynamically
+active cluster AC 114 is shown in Fig. 2.
+3.3. The Local Group
+The Local Group is a very small virial system, dominated by two large galaxies, the M31 or
+Andromeda galaxy, and the Milky Way. The M31 exhibits blueshift, falling in towards us.
+Evidently the Galaxy and M31, as well as all the minor galaxies in the Local Group, form a
+bound system which is oscillating.
+In this virial system the two large galaxies dominate the dynamics, so that it is not
+meaningful to define, as for the Coma, an average pairwise separation between galaxies, an
+average mass or an average orbital velocity. The total kinetic energy Eis still given by the
+sum over all the group members, and the potential energy Uby the sum over all the galaxyDark Matter: The evidence from astronomy, astrophysics and cosmology 7
+pairs, but here the pair formed by the M31 and the Milky Way dominates, and the pairings
+of the small members with each other are negligible.
+An interesting recent claim is, that the mass estimate of the Local Group is also affected
+by the accelerated expansion, the “dark energy”. A. D. Chernin & al. [14] have shown that the
+the potential energy Uis reduced in the force field of dark energy, so that the virial theorem
+forNmassesmiwith baryocentric radius vectors ritakes the form
+E=−1
+2U+U2, (6)
+whereUis defined as in Eq. (3), and
+U2=−4πρv
+3/summationdisplay
+mir2
+i (7)
+is a correction which reduces the potential energy due to the background dark energy density
+ρv. In the Local Group this correction to the mass appears to be quite substantial, of the order
+of 30%−50%.
+The dynamical mass of the local group comes out to be 3 .2−3.7 1012solar masses whereas
+the total visible mass of the Galaxy + M31 is only 2 1011solar masses. Thus there is a large
+amount of dark matter.
+4. Rotation curves of spiral galaxies
+Spiral galaxies are stable gravitationally bound systems in which visible matter is composed
+of stars and interstellar gas. Most of the observable matter is in a relatively thin disc, where
+stars and gas rotate around the galactic center on nearly circular orbits. If the circular velocity
+at radiusrisvin a galaxy with mass M(r) insider, the condition for stability is that the
+centrifugal acceleration v/rshould equal the gravitational pull GM(r)/r2, and the radial
+dependence of vwould then be expected to follow Kepler’s law
+v=/radicalbigg
+GM(r)
+r. (8)
+The surprising result from measurements of galaxy-rotation curves is that the velocity does
+not follow this 1 /√rlaw, but stays constant after attaining a maximum at about 5 kpc. The
+most obvious solution to this is that the galaxies are embedded in extensive, diffuse haloes of
+dark matter. If the enclosed mass inside the radius r, M(r), is proportional to r it follows that
+v(r)≈constant.
+The rotation curve of most galaxies can be fitted by the superposition of contributions
+from the stellar and gaseous disks, both modeled by thin exponential disks, sometimes a
+bulge, and the dark halo, modeled by a quasi-isothermal sphere. The inner part is difficult to
+model because the density of stars is high, rendering observations of individual star velocities
+difficult. Thus the fits are not unique, the relative contributions of disk and dark matter
+is model-dependent, and it is not even sure whether galactic disks do contain dark matter.
+Typically the dark matter is about half of the total mass.
+In Figs. 3 and 4 we show the rotation curves fitted for two well-measured galaxies, NGC
+6503 and NGC 3198 [15]. One notes, that at very small radii the dark halo component is
+indeed much smaller than the stellar disk component. At large radii, however, the need for aDark Matter: The evidence from astronomy, astrophysics and cosmology 8
+halo of dark matter is obvious. On galactic scales, the contribution of dark matter generally
+dominates the total mass.
+Our Galaxy is complicated because of what appears to be a density dip at 9 kpc and a
+small dip at 3 kpc, as is seen in Fig. 5 [16]. To fit the measured rotation curve one needs at
+least three contributing components: a central bulge, the star disk + gas, and a dark matter
+halo [16, 17, 18]. For small radii there is a choice of empirical rotation curves, and no dark
+matter component appears to be needed until radii beyond 15 kpc.
+Figure 3. Decomposition of the rotation curve of NGC3198 with a submaximal stellar disk.
+From D. Fuchs [15].Dark Matter: The evidence from astronomy, astrophysics and cosmology 9
+Figure 4. Decomposition of the rotation curve of NGC6503 with a submaximal stellar disk.
+From D. Fuchs [15].
+Figure 5. Decomposition of the rotation curve of the Milky Way into the components bulge,
+stellar disk + interstellar gas, dark matter halo (the red curves from left to right). From Y.
+Sofue et al. [16].Dark Matter: The evidence from astronomy, astrophysics and cosmology 10
+5. Small galaxy groups emitting X-rays
+There are examples of groups formed by a small number of galaxies which are enveloped in a
+large cloud of hot gas (ICM), visible by its X-ray emission. One may assume that the electron
+density distribution associated with the X-ray brightness is in hydrostatic equilibrium, and
+one can extract the ICM radial density profiles by fits.
+The amount of matter in the form of hot gas can be deduced from the intensity of this
+radiation. Adding the gas mass to the observed luminous matter, the total amount of baryonic
+matter,Mb, can be estimated [19, 20]. In clusters studied, the gas fraction increases with the
+distance from the center; the dark matter is more concentrated than the visible matter.
+The temperature of the gas depends on the strength of the gravitational field, from which
+the total amount of gravitating matter, Mgrav, in the system can be deduced. In many such
+small galaxy groups one finds Mgrav/Mb≥3. Thus a dark halo must be present. An accurate
+estimate of Mgravrequires that also dark energy is taken into account, because it reduces
+the strength of the gravitational potential. There are sometimes doubts whether all galaxies
+apparently in these groups are physical members. If not, they will artificially increase the
+velocity scatter and thus lead to larger virial masses.
+On the scale of large clusters of galaxies like the Coma, it is generally observed that dark
+matter represents about 85% of the total mass and that the visible matter is mostly in the
+form of a hot ICM.
+6. Mass to luminosity ratios
+Let us define the mass/luminosity ratio of an astronomical object as Υ ≡M/L . Stellar
+populations exhibit values Υ = 1 −10 in solar units, in the solar neighborhood Υ = 2 .5−7,
+in the Galactic disk Υ = 1 .0−1.7 [21]. In the Coma the value of Υ depends strongly on the
+halo profile for r <0.2 Mpc; beyond that the value is 400 and slowly decreasing with radius
+[7]. Many dwarf spheroidal galaxies exhibit very high ratios: in Draco Υ = 330 ±125, in
+Andromeda IX Υ = 93+120
+−50. The cluster AC 114 discussed in Sec. 2.2 is very underluminous
+having Υ = 700±100 [13].
+A special case is the ultra-faint dwarf disk galaxy Segue 1 [22] which has a baryon mass
+of only about 1000 solar masses. One interpretation is that it is a thin non-rotating stellar
+disk not accompanied by a gas disk, embedded in an axisymmetric DM halo and with an
+f=Mhalo/Mb≈200. But if the disk rotates, fcould be as high as 2000. If Segue 1 also has a
+magnetized gas disk, the DM halo has to confine the effective pressure in the stellar disk and
+the magnetic Lorentz force in the gas disk as well as possible rotation. Then fcould be very
+large [22].
+These mass/luminosity ratios are however local, characteristic for isolated objects but not
+for the whole observable Universe. The value for LUniverse is roughly known. One can then
+deduce that the critical density to close the Universe would require Υ ≈990. However, there
+is not sufficiently much luminous matter to achieve closure, therefore a large amount of DM is
+needed.Dark Matter: The evidence from astronomy, astrophysics and cosmology 11
+Figure 6. Dark matter halo mass Mhaloas a function of stellar mass M∗. The thick black curve
+is the prediction from abundance matching assuming no dispersion in the relation between the
+two masses. Red and green dashed curves assume some dispersion in log M∗. The dashed
+black curve is the satellite fraction as a function of stellar mass, as labeled on the axis at the
+right-hand side of the plot. From Qi Guo & al. [23].
+7. Mass autocorrelation functions
+If galaxy formation is a local process, then on large scales galaxies must trace mass. This
+requires the study of how galaxies populate dark matter haloes. In simulations one attempts
+to track galaxy and dark matter halo evolution across cosmic time in a physically consistent
+way, providing positions, velocities, star formation histories and other physical properties for
+the galaxy populations of interest.
+Guo & al.[23] use abundance matching arguments to derive an accurate relation between
+galaxy stellar mass and dark matter halo mass. They combine a stellar mass function
+based on spectroscopic observations with a precise halo/subhalo mass function obtained from
+simulations. Assuming this stellar mass – halo mass relation to be unique and monotonic, they
+compare it with direct observational estimates of the mean mass of haloes surrounding galaxies
+of given stellar mass inferred from gravitational lensing and satellite galaxy dynamics data,
+and use it to populate haloes in simulations. The stellar mass – halo mass relation is shown
+in Fig. 6. The implied spatial clustering of stellar mass turns out to be in remarkably good
+agreement with a direct and precise measurement. By comparing the galaxy autocorrelation
+function with the total mass autocorrelation function, as averaged over the Local SuperclusterDark Matter: The evidence from astronomy, astrophysics and cosmology 12
+Figure 7. Dark matter column density vs. dark matter halo mass in solar units. From A.
+Boyarsky & al.[24]
+(LSC) volume, one concludes that a large amount of matter in the LSC is dark.
+A similar study is that of Boyarsky & al. [24] who find a universal relation between DM
+column density and DM halo mass, satisfied by matter distributions at all observable scales,
+in halo sizes from 108to 1016solar masses, as shown in Fig. 7. Such a universal property is
+difficult to explain without dark matter.
+8. Strong and weak lensing
+A consequence of the Strong Equivalence Principle (SEP) is that a photon in a gravitational
+field moves as if it possessed mass, and light rays therefore bend around gravitating masses.
+Thus celestial bodies can serve as gravitational lenses probing the gravitational field, whether
+baryonic or dark without distinction.
+Since photons are neither emitted nor absorbed in the process of gravitational light
+deflection, the surface brightness of lensed sources remains unchanged. Changing the size of
+the cross-section of a light bundle only changes the flux observed from a source and magnifies it
+at fixed surface-brightness level. If the mass of the lensing object is very small, one will merely
+observe a magnification of the brightness of the lensed object an effect called microlensing .
+Microlensing of distant quasars by compact lensing objects (stars, planets) has also been
+observed and used for estimating the mass distribution of the lens–quasar systems.
+Weak Lensing refers to deflection through a small angle when the light ray can be treated
+as a straight line (Fig. 8), and the deflection as if it occurred discontinuously at the point ofDark Matter: The evidence from astronomy, astrophysics and cosmology 13
+Figure 8. Wave fronts and light rays in the presence of a cluster perturbation. From N.
+Straumann [25].
+closest approach (the thin-lens approximation in optics). One then only invokes SEP which
+accounts for the distortion of clock rates.
+Figure 9. This image resulted from color-subtracteion of a lensing singular isothermal
+elliptical galaxy. The strongly lensed object forms two prominent arcs A, B and a less extended
+third image C. From R.J. Smith & al.[26]Dark Matter: The evidence from astronomy, astrophysics and cosmology 14
+Figure 10. Map of the dark matter distribution in the 2-square degree COSMOS field [27]: the
+linear blue scale on top shows the gravitational lensing magnification κ, which is proportional
+to the projected mass along the line of sight. Contours begin at 0.4% and are spaced by 0.5%
+inκ.
+InStrong Lensing the photons move along geodesics in a strong gravitational potential
+which distorts space as well as time, causing larger deflection angles and requiring the full
+theory of GR. The images in the observer plane can then become quite complicated because
+there may be more than one null geodesic connecting source and observer; it may not even be
+possible to find a unique mapping onto the source plane c.f.Fig.8 . Strong lensing is a tool
+for testing the distribution of mass in the lens rather than purely a tool for testing GR. An
+illustration is seen in Fig. 9 where the lens is an elliptical galaxy [26].
+At cosmological distances one may observe lensing by composed objects such as galaxy
+groups which are ensembles of “point-like”, individual galaxies. Lensing effects are very model-
+dependent, so to learn the true magnification effect one needs very detailed information on the
+structure of the lens.
+The large-scale distribution of matter in the Universe is inhomogeneous in every direction,
+so one can expect that everything we observe is displaced and distorted by weak lensing. Since
+the tidal gravitational field and the deflection angles depend neither on the nature of the matter
+nor on its physical state, light deflection probes the total projected mass distribution. Lensing
+in infrared light offers an additional advantage of being able to sense distant background
+galaxies, since their number density is higher than in the optical range.
+Background galaxies would be ideal tracers of distortions if they were intrinsically circular,Dark Matter: The evidence from astronomy, astrophysics and cosmology 15
+Figure 11. Strong lensing analysis of the galaxy cluster Abell 370 (z=0.375) shown with
+the locations of the identified multiple systems within the critical line at z=1.2 in white (the
+redshift for the majority of the mutiple images). The red line delimits the region of multiple
+images for very high redshift sources (here assuming z = 6). From J. Richard & al.[28]
+because lensing transforms circular sources into ellipses. Any measured ellipticity would then
+directly reflect the action of the gravitational tidal field of the interposed lensing matter,
+and the statistical properties of the distortions would reflect the properties of the matter
+distribution. But many galaxies are actually intrinsically elliptical, and the ellipses are
+randomly oriented. This introduces noise into the inference of the tidal field from observed
+ellipticities. A useful feature in the sky is a fine-grained pattern of faint and distant blue
+galaxies appearing as a ‘wall paper’. This makes statistical weak-lensing studies possible,
+because it allows the detection of the coherent distortions imprinted by gravitational lensing
+on the images of the galaxy population. Thus weak lensing has become an important technique
+to map non-luminous matter. A reconstruction of the largest and most detailed weak lensing
+survey ever undertaken with the Hubble Space Telescope is shown in Fig. 10 [27]. This map
+covers a large enough area to see extended filamentary structures.
+In Fig. 11 we see a first detailed example of the strong lensing of a cluster [28].
+9. Cosmic Microwave Background (CMB)
+The tight coupling between radiation and matter density before decoupling caused the
+primordial adiabatic perturbations to oscillate in phase. Beginning from the time of last
+scattering, the receding horizon has been revealing these frozen density perturbations, settingDark Matter: The evidence from astronomy, astrophysics and cosmology 16
+Figure 12. The maps of CMB radiation temperature (TT) and temperature-polarization
+correlations (TE) from WMAP show anisotropies which can be analyzed by power spectra as
+functions of multipole moments. From G. Hinshaw & al.[29]
+up a pattern of standing acoustic waves in the baryon–photon fluid. After decoupling, this
+pattern is visible today as temperature anisotropies with a certain regularity across the sky.
+The primordial photons are polarized by the anisotropic Thomson scattering process, but
+as long as the photons continue to meet free electrons their polarization is washed out, and no
+net polarization is produced. At a photon’s last scattering, however, the induced polarization
+remains and the subsequently free-streaming photon possesses a quadrupole moment.
+Temperature and polarization fluctuations are analyzed in terms of multipole components
+or powers. The resulting distribution of powers versus multipole /lscript, or multipole moment
+k= 2π//lscript, is the power spectrum which exhibits conspicuous Doppler peaks . In Fig. 12
+we display the TT and TE power spectra from the 5 year data of WMAP as functionsDark Matter: The evidence from astronomy, astrophysics and cosmology 17
+of multipole moments [29]. The spectra can then be compared to theory, and theoretical
+parameters determined. Many experiments have determined the power spectra, so a wealth of
+data exists.
+Figure 13. The E-mode polarization power spectrum (EE) from the CMB observations of
+the QUaD collaboration & al.[31]
+Baryonic matter feels attractive self-gravity and is pressure-supported, whereas dark
+matter only feels attractive self-gravity, but is pressureless. Thus the Doppler peaks in the
+CMBR power spectrum testify about baryonic and dark matter, whereas the troughs testify
+about rarefaction caused by the baryonic pressure. The position of the first peak determines
+Ωmh2. Combining the 5-year WMAP measurements of the temperature power spectrum (TT)
+with determinations of the Hubble constant h, the WMAP team finds the total mass density
+parameter Ω m≈0.26 [30]. The ratio of amplitudes of the second-to-first Doppler peaks
+determines the baryonic density parameter Ω b≈0.04; the dark matter component is then
+Ωdm≈0.22.
+Information on the temperature – E-mode polarization correlations (TE) comes from
+several measurements, among them WMAP [29, 30], and on the E-mode polarization power
+spectrum alone (EE) from the QUAD collaboration [31], Fig. 13.
+The results show two surprises: Firstly, since Ω m/lessmuch1, a large component Ω Λ≈0.74 is
+missing, of unknown nature, and termed dark energy . The second surprise is that ordinary
+baryonic matter is only a small fraction of the total matter budget. The remainder is then
+dark matter, of unknown composition. Of the 4.2% of baryons in the Universe only about 1%
+is stars.
+10. Baryonic acoustic oscillations (BAO)
+A cornerstone of cosmology is the Copernican principle, that matter in the Universe is
+distributed homogeneously, if only on the largest scales of superclusters separated by voids. On
+smaller scales we observe inhomogeneities in the forms of galaxies, galaxy groups, and clusters.Dark Matter: The evidence from astronomy, astrophysics and cosmology 18
+The common approach to this situation is to turn to non-relativistic hydrodynamics and treat
+matter in the Universe as an adiabatic, viscous, non-static fluid, in which random fluctuations
+around the mean density appear, manifested by compressions in some regions and rarefactions
+in other. The origin of these density fluctuations was the tight coupling established before
+decoupling between radiation and charged matter density, causing them to oscillate in phase.
+An ordinary fluid is dominated by the material pressure, but in the fluid of our Universe three
+effects are competing: gravitational attraction, density dilution due to the Hubble flow, and
+radiation pressure felt by charged particles only.
+The inflationary fluctuations crossed the post-inflationary Hubble radius, to come back
+into vision with a wavelength corresponding to the size of the Hubble radius at that moment.
+At timeteqthe overdensities began to amplify and grow into larger inhomogeneities. In
+overdense regions where the gravitational forces dominate, matter contracts locally and
+attracts surrounding matter, becoming increasingly unstable until it eventually collapses into
+a gravitationally bound object. In regions where the pressure forces dominate, the fluctuations
+move with constant amplitude as sound waves in the fluid, transporting energy from one region
+of space to another.
+Inflationary models predict that the primordial mass density fluctuations should be
+adiabatic, Gaussian, and exhibit the same scale invariance as the CMB fluctuations. The
+baryonic acoustic oscillations can be treated similarly to CMB, they are specified by the
+dimensionless mass autocorrelation function which is the Fourier transform of the power
+spectrum of a spherical harmonic expansion. The power spectrum is shown in Fig. 14 [32].
+As the Universe approached decoupling, the photon mean free path increases and
+radiation can diffused from overdense regions into underdense ones, thereby smoothing out
+any inhomogeneities in the plasma. The situation changed dramatically at recombination, at
+time 380 000 yr after Big Bang, when all the free electrons suddenly disappeared, captured
+into atomic Bohr orbits, and the radiation pressure almost vanished. Now the baryon acoustic
+waves and the CMB continued to oscillate independently, but adiabatically, and the density
+perturbations which have entered the Hubble radius since then can grow with full vigor into
+baryonic structures.
+The scale of BAO depends on Ω mand on the Hubble constant, h, so one needs information
+onhto break the degeneracy. The result is then Ω m≈0.26. In the ratio Ω b/Ωmtheh-
+dependence cancels out, so one can also quantify the amount of dark matter on very large
+scales by Ω b/Ωm= 0.18±0.04.
+11. Galaxy formation in purely baryonic matter
+We have seen in Sec. 9 that the baryonic density parameter, Ω b, is very small. The critical
+density Ω critical is determined by the expansion speed of the Universe, and the mean baryonic
+density of the Universe (stars, interstellar & intergalactic gas) is only Ω b= 0.046×Ωcritical
+[30].
+The question arises whether the galaxies could have formed from primordial density
+fluctuations in a purely baryonic medium. We have also noted, that the fluctuations in CMB
+and BAO maintain adiabaticity. The amplitude of the primordial baryon density fluctuationsDark Matter: The evidence from astronomy, astrophysics and cosmology 19
+Figure 14. BAO in power spectra calculated from (a) the combined SDSS and 2dFGRS
+main galaxies, (b) the SDSS DR5 LRG sample, and (c) the combination of these two samples
+(solid symbols with 1 σerrors). The data are correlated and the errors are calculated from the
+diagonal terms in the covariance matrix. A Standard ΛCDM distance – redshift relation was
+assumed to calculate the power spectra with Ω m= 0.25,ΩΛ= 0.75. From Percival & al.[32].
+would have needed to be very large in order to form the observed number of galaxies. But then
+the amplitude of the CMB fluctuations would also have been very large, leading to intolerably
+large CMB anisotropies today. Thus galaxy formation in purely baryonic matter is ruled out
+by this argument alone.
+Thus one concludes, that the galaxies could only have been formed in the presence of
+gravitating dark matter which started to fluctuate early, unhindered by radiation pressure.
+This conclusion is further strengthened in the next Section.
+12. Large Scale Structures simulated
+In the ΛCDM paradigm, the nonlinear growth of dark matter structure is a well-posed problem
+where both the initial conditions and the evolution equations are known (at least when the
+effects of the baryons can be neglected).
+The Aquarius Project [33] is a Virgo Consortium programme to carry out high-resolution
+dark matter simulations of Milky-Way-sized halos in the ΛCDM cosmology. This project seeksDark Matter: The evidence from astronomy, astrophysics and cosmology 20
+clues to the formation of galaxies and to the nature of the dark matter by designing strategies
+for exploring the formation of our Galaxy and its luminous and dark satellites. The galaxy
+Figure 15. The left panel shows the projected dark matter density at z = 0 in a slice of
+thickness 13.7 Mpc through the full box (137 Mpc on a side) of the 9003parent simulation.
+The right panel show this halo resimulated at a different numerical resolution. The image
+brightness is proportional to the logarithm of the squared dark matter density projected along
+the line-of-sight. The circles mark r50, the radius within which the mean density is 50 times
+the background density. From V. Springel & al. [33]
+population on scales from 50 kpc to the size of the observable Universe has been predicted by
+hierarchical ΛCDM scenarios, and compared directly with a wide array of observations. So far,
+the ΛCDM paradigm has passed these tests successfully, particularly those that consider the
+large-scale matter distribution and has led to the discovery of a universal internal structure for
+dark matter halos As was noted already in Sec. 11, the observed structure of galaxies, clusters
+and superclusters, as illustrated by Fig. 15, could not have formed in a baryonic medium
+devoid of dark matter.
+Given this success, it is important to test ΛCDM predictions also on smaller scales, not
+least because these are sensitive to the nature of the dark matter. Indeed, a number of serious
+challenges to the paradigm have emerged on the scale of individual galaxies and their central
+structure. In particular, the abundance of small dark matter subhalos predicted within CDM
+halos is much larger than the number of known satellite galaxies surrounding the Milky Way.
+13. Dark matter from overall fits
+In Sec. 9 we have seen that the WMAP-5 year CMB data together with the Hubble constant
+value testify about the existence of dark matter [29, 30]. In Sec. 10 we addressed the BAO
+data [32] with the same conclusion. In overall fits one combines these with supernova data
+(SN Ia) which offer a constraint nearly orthogonal to that of CMB in the Ω Λ−Ωm-plane.
+The Union compilation of 414 SN Ia, which reduces to 307 SNe after selection cuts, includes
+the recent large samples of SNe Ia from the Supernova Legacy Survey and ESSENCE Survey,Dark Matter: The evidence from astronomy, astrophysics and cosmology 21
+Figure 16. 68.3 %, 95.4 % and 99.7% confidence level contours on Ω Λand Ω mobtained from
+CMB, BAO and the Union SN set, as well as their combination (assuming w = -1). Note the
+straight line corresponding to a flat Universe with Ω Λ+Ω m= 1. From M. Kowalski & al. [34].Dark Matter: The evidence from astronomy, astrophysics and cosmology 22
+the older data sets, as well as the recently extended data set of distant supernovae observed
+with HST. M. Kowalski & al. [34] present the latest results from this Union compilation and
+discuss the cosmological constraints and its combination with CMB and BAO measurements.
+The CMB constraint is close to the line Ω Λ+ Ωm= 1, whereas the supernova constraint is
+close to the line Ω Λ−1.6×Ωm= 0.2. The BAO data constrain Ω m, but hardly at all Ω Λ. This
+is shown in Fig. 16.
+Defining the vacuum energy density parameter by
+Ωk= 1−ΩΛ−Ωm, (9)
+a flat Universe corresponds to Ω k= 0. For a Λ CDM Universe with a cosmological constant
+responsible for dark energy, a simultaneous fit to the data sets gives
+Ωm= 0.285+0.020
+−0.019±0.011,Ωk=−0.009+0.009+0.002
+−0.010−0.003, (10)
+where the first error is statistical and the second error systematic. This is the most precise
+determination at present of the total mass density parameter Ω m. At the same time one notes
+that the Universe is consistent with being flat. Subtracting Ω b= 0.046 from Ω m= 0.285 one
+obtains the density parameter for dark matter, Ω dm≈0.24. Assuming flatness, M. Kowalski
+& al. [34] find Ω m= 0.274±0.016±0.013.
+Figure 17. The merging cluster 1E0657-558. On the right is the smaller ’bullet’ cluster which
+has traversed the larger cluster. The colors indicate the X-ray temperature of the plasma: blue
+is coolest and white is hottest. The green contours are the weak lensing reconstruction of the
+gravitational potential of the cluster. From D. Clowe & al. [35]
+14. Merging galaxy clusters
+A direct empirical proof of the existence of dark matter is furnished by observations of 1E0657-
+558, a unique cluster merger [35]. Due to the collision of two clusters, the dissipationless stellarDark Matter: The evidence from astronomy, astrophysics and cosmology 23
+component and the fluid-like X-ray emitting plasma are spatially segregated. The gravitational
+potential observed by weak and strong lensing does not trace the plasma distribution which is
+the dominant baryonic mass component, but rather approximately traces the distribution of
+galaxies, cf. Fig. 17. The center of the total mass is offset from the center of the baryonic mass
+peaks, proving that the majority of the matter in the system is unseen. In front of the smaller
+’bullet’ cluster which has traversed the larger one, a bow shock is evident in the X-rays.
+Two other merging systems with similar characteristics have been seen, although with
+lower spatial resolution and less clear-cut cluster geometry. In Fig.18 we show the post-merging
+galaxy cluster MACS J0025.4-1222 [10].
+Figure 18. The F555W-F814W color composite of the cluster MACS J0025.4.1222. Overlaid
+in red contours is the surface mass density κfrom the combined weak and strong lensing mass
+reconstruction. The contour levels are linearly spaced with ∆ κ= 0.1, starting at ∆ κ= 0.5,
+for a fiducial source at a redshift of zs→∞ . The X-ray brightness contours (also linearly
+spaced) are overlaid in yellow and the I-band light is overlaid in white. The measured peak
+position and error bars for the total mass are given as a cyan cross [10].
+15. Comments and conclusions
+What we have termed “dark matter” is generic for observed gravitational effects on all scales:
+galaxies, small and large galaxy groups, clusters and superclusters, CMB anisotropies over the
+full horizon, baryonic oscillations over large scales, and cosmic shear in the large-scale matter
+distribution. The correct explanation or nature of dark matter is not known, whether it implies
+conventional matter, unconventional particles, or modifications to gravitational theory. but
+gravitational effects prove its existence in some form.Dark Matter: The evidence from astronomy, astrophysics and cosmology 24
+Only a few per cent of the total mass of the Universe is found in stars and hydrogen
+clouds, and this amount of baryonic matter is well accounted for by nucleosynthesis. If there
+exist particles which were very slow at time teqwhen galaxy formation started, they could be
+candidates for cold dark matter. If dark matter were composed of such new particles, they
+must have become non-relativistic much earlier than the leptons, and then decoupled from the
+hot plasma.
+Whenever laboratory searches discover a new particle, it must pass several tests in order
+to be considered a viable DM candidate: it must be neutral, compatible with constraints
+on self-interactions (essentially collisionless), consistent with Big Bang nucleosynthesis, and
+match the appropriate relic density. It must be consistent with direct DM searches and
+gamma-ray constraints, it must leave stellar evolution unchanged, and be compatible with
+other astrophysical bounds.
+Theories attempting to explain dark matter without new particles, with new interactions
+or modified gravity, likewise have the burden to explain all the observed gravitational effects
+described in here. Thus there remains much to be done.
+Acknowledgements
+It is a pleasure to thank the professors K. Mattila at the University of Helsinki and M. Valtonen
+at the University of Turku for valuable comments.
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+relaxed galaxy clusters , Mon. Not. Roy. Astr. Soc. , 383, 879 (2008)
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+Vaucouleurs Bulge, Disk, Dark Halo, and the 9-kpc Rotation Dip , Publ. Astron. Soc. Japan, 61, 227
+(2009) and arXiv:0811.0859[astro-ph] (2008)
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+[25] N. Straumann, Matter in the Universe , Space Science Series of ISSI, vol. 14. Kluwer (2002) (Reprinted
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+arXiv:0910.5553[astro-ph.CO] (2009)
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\ No newline at end of file
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+1 of 92 Pseudorandomness in Central Force Optimization 
+ 
+Richard A. Formato 1 
+ 
+Abstract:   Central Force Optimization is a deterministic metahe uristic for an evolutionary 
+algorithm that searches a decision space by flying probes w hose trajectories are computed 
+using a gravitational metaphor.  CFO benefits substantial ly from the inclusion of a 
+pseudorandom component (a numerical sequence that is precise ly known by specification or 
+calculation but otherwise arbitrary).  The essential requ irement is that the sequence is 
+uncorrelated with the decision space topology, so that its effect is to pseudorandomly 
+distribute probes throughout the landscape.  While this pro cess may appear to be similar to 
+the randomness in an inherently stochastic algorithm, it is in fact fundamentally different 
+because CFO remains deterministic at every step.  Three  pseudorandom methods are 
+discussed (initial probe distribution, repositioning factor , and decision space adaptation).  A 
+sample problem is presented in detail and summary data incl uded for a 23-function 
+benchmark suite.  CFO’s performance is quite good compared to other highly developed, 
+state-of-the-art algorithms. 
+ 
+Ver. 2  (typographical errors in Step (c) of Fig. 1 and discussion of Fig.  8 corrected; clarification of definition 
+of best Rr
+; error in Table 2 test function f 12  corrected; f 12  source code in Appendix 3, page 60 corrected).  
+3 February 2010 
+Saint Augustine, Florida  2 of 92 Pseudorandomness in Central Force Optimization 
+ 
+Richard A. Formato 1 
+ 
+1.  Introduction 
+This note examines the role of pseudorandomness in Central  Force Optimization.  CFO is a 
+deterministic Nature-inspired search and optimization met aheuristic for an evolutionary 
+algorithm (EA) based on gravitational kinematics [1-10].  CF O analogizes Newton’s 
+mathematically precise laws of motion and gravity, so t hat its underlying equations are 
+equally precise.  The algorithm locates the global maxima  of an objective function  defined 
+on a decision space  Ω with unknown topology (“landscape”).  CFO searches Ω by “flying” a 
+group of “probes” whose trajectories are computed from t wo deterministic equations of 
+motion  at a series of discrete “time” steps (iterations).  Details of the CFO metaheuristic are 
+in Appendix 1. 
+ CFO is fundamentally different from Nature-inspired E As whose underlying 
+equations are inherently stochastic.  Particle Swarm Opt imization (PSO) and Ant Colony 
+Optimization (ACO) are examples.  Their equations are formulated in terms of random 
+variables, and removing randomness causes these algorithms to fail completely.  By 
+contrast, CFO’s equations are inherently deterministic.  Every CFO run with the same setup 
+returns precisely the same values step-by-step throughout t he entire run.  Nevertheless, 
+effective implementations substantially benefit from  a “pseudorandom” component that 
+enters the algorithm indirectly, not through its basic equa tions.  Although pseudorandomness 
+is not required in CFO, numerical experiments show that it is an important feature in 
+effective implementations. 
+ A pseudorandom variable is defined here as one whose value is precisely k nown but 
+arbitrarily assigned.  The value can be specified in adva nce (for example, an arbitrary 
+sequence of numbers) or it can be calculated in a prescr ibed manner.  The variable’s 
+“randomness” derives from the fact that its value is a rbitrary and uncorrelated with Ω’s 
+topology, not that it is uncertain in the sense of a true random variable.  A random variable’s 
+value is calculated from a probability distribution with successive calculations yielding 
+different values that cannot be known in advance.  This type of randomness is fundamentally 
+different from CFO’s pseudorandomness.  Even when CFO includes pseudorandomness, it 
+remains deterministic, always yielding the same result f or runs with the same setup.  Once a 
+pseudorandom variable is specified, either explicitly or by  calculation, its value is known 
+with absolute precision, so that CFO’s trajectory calc ulations are deterministic even in the 
+presence of pseudorandomness.  There are many ways pseudoran domness can be added to 
+CFO; three simple methods are described here. 
+2.  CFO Implementation 
+This note discusses a CFO implementation with pseudorando mness injected in the following 
+ways: (1) the initial probe distribution; (2) the “reposit ioning factor;” and (3) changing the 
+decision space boundaries.  The algorithm is referred to a s “CFO-PR.”  Every CFO run 
+starts with a user-specified initial probe distribution ( total number and locations in Ω at the 
+beginning of the run, step 0).  An arbitrary, variable initia l probe distribution is a convenient 
+way to inject pseudorandomness, the effect of which is to  provide better sampling of Ω’s 3 of 92 topology than a static distribution.  Each initial p robe distribution in a set of distributions 
+provides different information about Ω’s landscape.  As the results below show, certain on es 
+perform much better than others. 
+ The second method of injecting pseudorandomness is the us e of a step-by-step 
+variable repositioning factor 1≤ ≤ ∆rep rep F F  where rep F∆  is the step increment.  
+“Repositioning” refers to the process of retrieving a probe that has flown outside the 
+decision space (discussed in detail in Appendix 1).  A var iable rep F has the effect of 
+pseudorandomly distributing probes throughout Ω, which provides better sampling of the 
+decision space landscape as a run progresses. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+                                                       Fig. 1.  CFO-PR Pseudocode with Variable Initial 
+                                                                     Probes and rep Fand DS Adaptation For  2=d pNN  to 8 by 2: 
+For  start γγ=  to stop γ  by γ∆: 
+(a.1) Compute initial probe distribution. 
+(a.2) Compute initial fitness matrix. 
+(a.3) Assign initial probe accelerations. 
+(a.4) Set initial rep F. 
+For  0=j  to tN (or earlier termination criterion): 
+(b) Compute probe position vectors, 
+ pp
+j Np R ≤≤1 ,r
+ [eq.(2)]. 
+(c) Retrieve errant probes ( pNp≤≤1 ): 
+ If ∴ <⋅min ˆi ip
+j xeRr
+ 
+ ) ˆ ( ˆmin 
+1min 
+i ip
+j rep i ip
+j xe RF xeR −⋅ + =⋅−r r
+ 
+ If ∴ >⋅max ˆi ip
+j xeRr
+ 
+ )ˆ ( ˆ1max max 
+ip
+j i rep i ip
+j e R xF xeR ⋅ − − =⋅−r r
+ 
+ 
+(d) Compute fitness matrix for current probe 
+ distribution. 
+ 
+(e) Compute accelerations using current probe 
+ distribution and fitnesses [eq. (1)]. 
+ 
+(f) Increment rep F by rep F∆ : 
+ If rep rep rep F F F ∆= ∴>1 . 
+ 
+(g) If ∴=0 20 MOD j  
+ Shrink Ω around best Rr
+. 
+Next  j 
+Next  γ 
+Next  d pNN  4 of 92  The third way pseudorandomness is injected is by shrinking t he decision space 
+around the best probe’s location.  This process coupled wi th variable rep F redistributes 
+probes in the smaller Ω in an arbitrary but precise, hence pseudorandom, manner.  The effect 
+is to speed CFO’s convergence, but at the risk of premature  convergence [on an empirical 
+basis, this issue does not appear to be significant using t he procedures described below]. 
+ Pseudocode for CFO-PR appears in Fig. 1 and source code in A ppendix 3.  The inner 
+time step loop (“ j” loop) is common to all CFO implementations, but the  two outer loops 
+inject initial probe pseudorandomness.  The γ loop controls where initial probes are 
+deployed, and the d pNN  loop determines their number.  These two parameters defi ne the 
+initial probe distribution, and the two loops together cre ate a variable, pseudorandom 
+distribution (see Appendix 1 for notation and equations).  The variable rep F procedure 
+appears in step (f) of the pseudocode.  And the pseudorandom dec ision space adaptation is 
+in step (g).  Ω’s boundaries shrink around the then best probe position vec tor every 20 th  step 
+as discussed in detail below.  A two-dimensional exampl e is used to illustrate these 
+techniques because it provides a concrete visualization of th e different methods.  In the 
+actual CFO-PR implementation, of course, these technique s are generalized to the dN-
+dimensional case as shown in the pseudocode and source co de. 
+ Initial Probes:  The manner in which initial probes are deployed using γ is shown 
+schematically in Fig. 2, which provides a 2-dimensional (2D) s chematic representation of a 
+variable initial probe distribution comprising an orthogo nal array of 
+dp
+NN probes per axis 
+deployed uniformly on “probe lines” parallel to the coordin ate axes that intersect at a point 
+along Ω’s principal diagonal.  dN is Ω’s dimensionality (here 2 =dN ), and max min 
+i i i xx x ≤≤ , 
+dNi≤≤1  define Ω’s domain. 
+ For illustrative purposes, Fig. 2 shows nine probes on each  probe line.  The lines, 
+which are parallel to the 1x and 2x axes, intersect at a point on Ω’s principal diagonal 
+marked by position vector ) (min max min X X XDrrrr
+− + = γ , where iN
+iiex Xd
+ˆ
+1min 
+min ∑
+==r
+ and 
+iN
+iiex Xd
+ˆ
+1max 
+max ∑
+==r
+ are the diagonal’s endpoint vectors.  Parameter 1 0 ≤≤γ  specifies where 
+along the diagonal the orthogonal probe array is pl aced by locating the probe lines’ 
+intersection point. 
+ While Fig. 2 shows an equal number of probes on ea ch line, a different number of 
+probes per axis can be used instead.  For example, if equal probe spacing were desired in a 
+decision space with unequal boundaries, or if overl apping probes were to be excluded in a 
+symmetrical space, then unequal numbers would be us ed.  Unequal numbers also might be 
+appropriate if a priori  knowledge of Ω’s landscape, however obtained, suggests denser 
+sampling in one area.  The initial probe distributi on in Fig. 2 with variable 
+dp
+NN was used 
+for the CFO-PR runs reported here, but any number o f other variable initial probe 
+distributions could be used instead.  The key idea is that the initial probe distribution must 5 of 92 be pseudorandom, that is, arbitrary and therefore u ncorrelated with the decision space 
+landscape. 
+ Repositioning Factor:   A variable value for rep F also adds pseudorandomness.  rep F 
+starts at some arbitrary initial value that is incr emented at each iteration by an arbitrary 
+amount rep F∆  so that 1≤ ≤ ∆rep rep F F .  This errant probe retrieval scheme is an example  of 
+an arbitrary sequence of calculated numbers determi nistically assigned to a CFO parameter.  
+CFO’s ability to search the decision space depends on where errant probes are reinserted in 
+Ω, and this process is pseudorandom in nature becaus e rep F is deterministic but arbitrary and 
+uncorrelated with Ω’s topology.  By placing errant probes pseudorandom ly throughout the 
+decision space, more information is developed about  its topology as the run progresses.  The 
+scheme used here was empirically determined, and ap pears to work well across a wide range 
+of objective functions.  But, of course, there are many other procedures for setting rep F’s 
+value, some no doubt better than others. 
+ 
+                          Fig. 2.  Variable 2-D Ini tial Probe Distribution used for CFO Runs Reported Here 
+ 
+ Decision Space Adaptation:   CFO-PR also includes adaptive reconfiguration of the 
+decision space in order to improve convergence spee d.  This feature also is pseudorandom in 
+nature because the way Ω’s boundaries are changed is arbitrary and uncorrel ated with the 
+landscape.  Fig. 3 illustrates in 2D how Ω’s size is adaptively reduced in this case every 20th  
+step around the probe’s location with the then best  fitness throughout the run up to the 
+current iteration, best Rr
+.  Ω’s boundary coordinates are reduced by one-half the  distance from 6 of 92 the best probe’s position to the each boundary on a  coordinate-by-coordinate basis, that is, 
+2ˆmin 
+min min i i best 
+i ixe Rx x−⋅+ =′r
+ and 2ˆmax 
+max max i best i
+i ie R xx x⋅ −− =′r
+, where the primed coordinate 
+is the new decision space boundary, and the dot den otes vector inner product.  For clarity, 
+Fig. 3 shows best Rr
+ as fixed, whereas generally it varies throughout a  run.  Changing Ω’s 
+boundary every twenty steps instead of some other i nterval was chosen arbitrarily (another, 
+probably better approach, might be a reactive adapt ation based on performance measures 
+such as convergence speed or fitness saturation). 
+ 
+                                    Fig. 3.  Schema tic 2-D Decision Space Adaptation (with constant best Rr
+) 
+ 
+ 
+3.  A Sample Problem 
+The effectiveness of injecting pseudorandomness int o CFO-PR will be illustrated with the 
+2D Goldstein-Price function (“GP”) plotted in Fig. 4.  GP is defined as 
+                   
+)] 27 36 48 12 32 18 ()3 2 ( 30 [)] 3 6 14 3 14 19 () 1 (1 [ ),(
+2
+2 21 22
+1 12
+2 12
+2 21 22
+1 12
+2 1 2 1
+x xx x x x x xx xx x x x xx xxf
++ − + + −⋅ − + ×+ + − + −⋅ ++ +−=   , 7 of 92 where Ω: 100 , 100 2 1≤ ≤ − xx  [note that in most published reports Ω is much smaller, viz. , 
+2 , 22 1≤ ≤− xx ].  GP’s global maximum is /uni20123 at (0, /uni20121).  This function is multimodal with few 
+local maxima, and it varies over nearly nineteen or ders of magnitude as shown in Fig. 4. 
+ 
+ 
+Fig. 4.  Goldstein-Price Function 
+ 
+ The following parameter values were used for all r uns reported in this note: 2=α , 
+2=β , 2=G , 1=∆t, initial acceleration of zero, initial 5 . 0=rep F , 05 . 0= ∆rep F , 0 =start γ , 
+1=stop γ  with 1 . 0 =∆γ  (eleven runs).  For GP, 2by 14 to 4=d pNN , with different ranges 
+of this parameter for the other test functions as d escribed below.  In all cases, a run was 
+terminated early if the average best fitness over 5 0 steps (including the current step) and the 
+current best fitness differed by less than 10 -6.  The pseudorandom initial probes distributions 
+computed using the procedure illustrated in Fig. 2 are plotted in Fig. 5. 
+ Table 1 shows a summary of the results for the GP function.  A total of sixty-six 
+optimization runs were made in six groups of eleven  runs each [data for “Run #0” are 
+starting values].  The column headings are for the most part self-explanatory.  Each run 
+began with 500 =tN , but, as the #Steps  column shows, in no case were 500 iterations used 
+because every run terminated early.  eval N is the number of function evaluations performed 
+for the shortened run, and the total number of eval uations over all runs appears at the bottom 
+of this column.  The rep F column lists rep F’s value at the end of the run, and the 8 of 92 
+ 
+ 
+  
+ 
+ 
+Fig. 5.  GP Best Run Initial Probe Distributions 9 of 92                               Table 1.  Summary Res ults for GP Function using Pseudorandom CFO. 
+ 
+“V” denotes that rep F was variable as discussed above.  Fitness  tabulates the best fitness 
+returned during the run.  The Initial Probes  column shows the type of initial probe 
+distribution, in this case probes uniformly spaced along probe lines parallel to Ω’s axes 
+(notated “ /uni01C1-AXIS ”) as described above and shown in Fig. 5. 
+ The best fitness ranged from a low of ... 6643 .236 −  in run 12 to the global maximum 
+of 3− at ) 1 , 0 (− returned in run #54 (best results highlighted in blue ).  Parameters for run 
+#54 were 12 =d pNN , 24 =pN , and 9 . 0 =γ .  The total number of function evaluations 
+over all runs was 472 ,180  while eval N for the best run was 464 , 1  (60 iterations).  Fig. 6 plots Run ID: 12-26-2009, 14:02:08 
+ 
+FUNCTION: GP 
+ 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+ 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500      2      8     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000   UNIFORM /uni01C1-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500      2      8     2.0    1. 0    2.00     2.00       78       632   0.60000V        -5.48169471   UNIFORM /uni01C1-AXIS 
+   2       0.100    500      2      8     2.0    1. 0    2.00     2.00      134      1080   0.55000V       -84.78003234   UNIFORM /uni01C1-AXIS 
+   3       0.200    500      2      8     2.0    1. 0    2.00     2.00      191      1536   0.55000V        -3.19822531   UNIFORM /uni01C1-AXIS 
+   4       0.300    500      2      8     2.0    1. 0    2.00     2.00       79       640   0.65000V        -8.62285433   UNIFORM /uni01C1-AXIS 
+   5       0.400    500      2      8     2.0    1. 0    2.00     2.00      403      3232   0.70000V       -84.47139703   UNIFORM /uni01C1-AXIS 
+   6       0.500    500      2      8     2.0    1. 0    2.00     2.00      261      2096   0.25000V       -10.56054743   UNIFORM /uni01C1-AXIS 
+   7       0.600    500      2      8     2.0    1. 0    2.00     2.00      135      1088   0.60000V        -4.98190912   UNIFORM /uni01C1-AXIS 
+   8       0.700    500      2      8     2.0    1. 0    2.00     2.00      229      1840   0.55000V        -3.00130536   UNIFORM /uni01C1-AXIS 
+   9       0.800    500      2      8     2.0    1. 0    2.00     2.00      133      1072   0.50000V       -89.42718008   UNIFORM /uni01C1-AXIS 
+  10       0.900    500      2      8     2.0    1. 0    2.00     2.00      116       936   0.60000V        -8.57642421   UNIFORM /uni01C1-AXIS 
+  11       1.000    500      2      8     2.0    1. 0    2.00     2.00      193      1552   0.65000V        -3.08573605   UNIFORM /uni01C1-AXIS 
+  12       0.000    500      2     12     2.0    1. 0    2.00     2.00       78       948   0.60000V      -236.66437724   UNIFORM /uni01C1-AXIS 
+  13       0.100    500      2     12     2.0    1. 0    2.00     2.00      263      3168   0.35000V        -3.00574349   UNIFORM /uni01C1-AXIS 
+  14       0.200    500      2     12     2.0    1. 0    2.00     2.00       78       948   0.60000V       -57.61728445   UNIFORM /uni01C1-AXIS 
+  15       0.300    500      2     12     2.0    1. 0    2.00     2.00      304      3660   0.50000V        -3.00017384   UNIFORM /uni01C1-AXIS 
+  16       0.400    500      2     12     2.0    1. 0    2.00     2.00      154      1860   0.60000V        -3.55376552   UNIFORM /uni01C1-AXIS 
+  17       0.500    500      2     12     2.0    1. 0    2.00     2.00      196      2364   0.80000V        -3.00014046   UNIFORM /uni01C1-AXIS 
+  18       0.600    500      2     12     2.0    1. 0    2.00     2.00      154      1860   0.60000V        -6.35080630   UNIFORM /uni01C1-AXIS 
+  19       0.700    500      2     12     2.0    1. 0    2.00     2.00      212      2556   0.65000V        -3.03478155   UNIFORM /uni01C1-AXIS 
+  20       0.800    500      2     12     2.0    1. 0    2.00     2.00       78       948   0.60000V       -16.66444362   UNIFORM /uni01C1-AXIS 
+  21       0.900    500      2     12     2.0    1. 0    2.00     2.00       99      1200   0.70000V        -6.52063755   UNIFORM /uni01C1-AXIS 
+  22       1.000    500      2     12     2.0    1. 0    2.00     2.00      250      3012   0.65000V        -3.01375068   UNIFORM /uni01C1-AXIS 
+  23       0.000    500      2     16     2.0    1. 0    2.00     2.00       78      1264   0.60000V       -22.18779159   UNIFORM /uni01C1-AXIS 
+  24       0.100    500      2     16     2.0    1. 0    2.00     2.00       60       976   0.65000V       -16.89097168   UNIFORM /uni01C1-AXIS 
+  25       0.200    500      2     16     2.0    1. 0    2.00     2.00      173      2784   0.60000V        -3.01613943   UNIFORM /uni01C1-AXIS 
+  26       0.300    500      2     16     2.0    1. 0    2.00     2.00       79      1280   0.65000V       -76.69765275   UNIFORM /uni01C1-AXIS 
+  27       0.400    500      2     16     2.0    1. 0    2.00     2.00      189      3040   0.45000V        -3.00972155   UNIFORM /uni01C1-AXIS 
+  28       0.500    500      2     16     2.0    1. 0    2.00     2.00      191      3072   0.55000V        -6.65915188   UNIFORM /uni01C1-AXIS 
+  29       0.600    500      2     16     2.0    1. 0    2.00     2.00      122      1968   0.90000V       -15.75050525   UNIFORM /uni01C1-AXIS 
+  30       0.700    500      2     16     2.0    1. 0    2.00     2.00       60       976   0.65000V       -31.95935579   UNIFORM /uni01C1-AXIS 
+  31       0.800    500      2     16     2.0    1. 0    2.00     2.00       95      1536   0.50000V       -49.88324658   UNIFORM /uni01C1-AXIS 
+  32       0.900    500      2     16     2.0    1. 0    2.00     2.00      304      4880   0.50000V        -3.00045306   UNIFORM /uni01C1-AXIS 
+  33       1.000    500      2     16     2.0    1. 0    2.00     2.00      318      5104   0.25000V        -3.00010350   UNIFORM /uni01C1-AXIS 
+  34       0.000    500      2     20     2.0    1. 0    2.00     2.00       78      1580   0.60000V       -76.94725945   UNIFORM /uni01C1-AXIS 
+  35       0.100    500      2     20     2.0    1. 0    2.00     2.00      116      2340   0.60000V        -3.12194540   UNIFORM /uni01C1-AXIS 
+  36       0.200    500      2     20     2.0    1. 0    2.00     2.00      191      3840   0.55000V        -3.03175136   UNIFORM /uni01C1-AXIS 
+  37       0.300    500      2     20     2.0    1. 0    2.00     2.00       78      1580   0.60000V      -182.65850920   UNIFORM /uni01C1-AXIS 
+  38       0.400    500      2     20     2.0    1. 0    2.00     2.00      193      3880   0.65000V        -3.14860260   UNIFORM /uni01C1-AXIS 
+  39       0.500    500      2     20     2.0    1. 0    2.00     2.00       98      1980   0.65000V        -3.62879802   UNIFORM /uni01C1-AXIS 
+  40       0.600    500      2     20     2.0    1. 0    2.00     2.00      192      3860   0.60000V        -3.04067143   UNIFORM /uni01C1-AXIS 
+  41       0.700    500      2     20     2.0    1. 0    2.00     2.00      139      2800   0.80000V        -3.54277590   UNIFORM /uni01C1-AXIS 
+  42       0.800    500      2     20     2.0    1. 0    2.00     2.00      171      3440   0.50000V        -3.00072472   UNIFORM /uni01C1-AXIS 
+  43       0.900    500      2     20     2.0    1. 0    2.00     2.00      101      2040   0.80000V        -4.37891524   UNIFORM /uni01C1-AXIS 
+  44       1.000    500      2     20     2.0    1. 0    2.00     2.00       79      1600   0.65000V       -35.88632429   UNIFORM /uni01C1-AXIS 
+  45       0.000    500      2     24     2.0    1. 0    2.00     2.00      226      5448   0.40000V        -3.00031923   UNIFORM /uni01C1-AXIS 
+  46       0.100    500      2     24     2.0    1. 0    2.00     2.00       78      1896   0.60000V        -9.06529259   UNIFORM /uni01C1-AXIS 
+  47       0.200    500      2     24     2.0    1. 0    2.00     2.00      209      5040   0.50000V        -3.03088087   UNIFORM /uni01C1-AXIS 
+  48       0.300    500      2     24     2.0    1. 0    2.00     2.00       79      1920   0.65000V       -30.56252660   UNIFORM /uni01C1-AXIS 
+  49       0.400    500      2     24     2.0    1. 0    2.00     2.00      306      7368   0.60000V        -3.00075380   UNIFORM /uni01C1-AXIS 
+  50       0.500    500      2     24     2.0    1. 0    2.00     2.00      248      5976   0.55000V        -3.00968878   UNIFORM /uni01C1-AXIS 
+  51       0.600    500      2     24     2.0    1. 0    2.00     2.00       97      2352   0.60000V        -3.34890591   UNIFORM /uni01C1-AXIS 
+  52       0.700    500      2     24     2.0    1. 0    2.00     2.00      246      5928   0.45000V        -3.01530923   UNIFORM /uni01C1-AXIS 
+  53       0.800    500      2     24     2.0    1. 0    2.00     2.00       78      1896   0.60000V       -31.34561763   UNIFORM /uni01C1-AXIS 
+  54       0.900    500      2     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V        -3.00000000   UNIFORM /uni01C1-AXIS 
+  55       1.000    500      2     24     2.0    1. 0    2.00     2.00      101      2448   0.80000V       -15.41042957   UNIFORM /uni01C1-AXIS 
+  56       0.000    500      2     28     2.0    1. 0    2.00     2.00      266      7476   0.50000V        -3.00178783   UNIFORM /uni01C1-AXIS 
+  57       0.100    500      2     28     2.0    1. 0    2.00     2.00      257      7224   0.05000V        -3.02198109   UNIFORM /uni01C1-AXIS 
+  58       0.200    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V      -102.95525789   UNIFORM /uni01C1-AXIS 
+  59       0.300    500      2     28     2.0    1. 0    2.00     2.00      154      4340   0.60000V        -3.40567728   UNIFORM /uni01C1-AXIS 
+  60       0.400    500      2     28     2.0    1. 0    2.00     2.00       79      2240   0.65000V       -40.28712313   UNIFORM /uni01C1-AXIS 
+  61       0.500    500      2     28     2.0    1. 0    2.00     2.00      210      5908   0.55000V        -3.00034670   UNIFORM /uni01C1-AXIS 
+  62       0.600    500      2     28     2.0    1. 0    2.00     2.00       79      2240   0.65000V       -32.25789818   UNIFORM /uni01C1-AXIS 
+  63       0.700    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V       -36.68532659   UNIFORM /uni01C1-AXIS 
+  64       0.800    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V       -14.65270155   UNIFORM /uni01C1-AXIS 
+  65       0.900    500      2     28     2.0    1. 0    2.00     2.00       78      2212   0.60000V       -31.47896038   UNIFORM /uni01C1-AXIS 
+  66       1.000    500      2     28     2.0    1. 0    2.00     2.00      282      7924   0.35000V        -3.00151179   UNIFORM /uni01C1-AXIS 
+ 
+                                                 To tal Function Evaluations:      180472 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  54       0.900    500      2     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V        -3.00000000   UNIFORM /uni01C1-AXIS 10 of 92 the evolution of GP’s best fitness, which in only t wo steps increases from 
+12 10 x72 9922682476 . 2−−  to GP’s actual global maximum of 3−.  This seems to be quite 
+remarkable in view of the initial probe distributio n for 9 . 0 =γ  (Fig. 5) in which all probes 
+are far removed from the maximum’s location at ) 1 , 0 (−. As the data in Table 1 clearly 
+show, some sets of parameters are much better than others.  Without pseudorandom initial 
+probes, one run would be made with, in this example , only 13.6% probability of locating the 
+global maximum with a fractional accuracy of 0.03% (9 of 66 runs).  This statistic highlights 
+the importance of pseudorandomness in CFO. 
+ 
+                                                            Fig. 6.  Evolution of GP Function Best Fit ness 
+ Fig. 7 plots CFO’s “ avg D curve” for GP.  avg D is the normalized average distance 
+between the probe with the best fitness and all oth er probes at each time step, viz. , 
+∑ ∑
+= =−− ⋅=p dN
+pN
+ijp
+ijp
+i
+pavg x xNLD
+1 12 *, ,) () 1 (1 where *p is the number of the probe with the best 
+fitness, and ∑
+=− =dN
+ii i x x L
+12 min max ) (  is the length of Ω’s principal diagonal (see Appendix 1 for 
+definitions).  avg D decreases monotonically through step 10 to 0.04969 77, then increases 
+very quickly to a peak of 0.4767301 at step 11, fol lowed by another quasi-monotonic 
+decrease through step 29 to 0.0274355.  This cycle repeats through step 48 where avg D is 
+0.0169657, followed by a jump to 0.1194779 at step 49.  After a slight dip through step 53, 
+avg D flattens out around a value of 0.11…   The quasi-oscillatory behavior in avg D usually 
+correlates with local trapping, which in this case happens to be at the global maximum.  
+Oscillation in avg D may be a similar phenomenon to oscillation seen in  “ V∆” curves for 
+gravitationally trapped Near Earth Objects (NEOs), strongly suggesting that NEO theory 11 of 92 may hold the key to analytical mitigation or elimin ation of local trapping at local maxima 
+and possibly a proof of convergence for CFO [7-9]. 
+ Because CFO-PR converges so quickly on GP’s global  maximum, the number of the 
+probe with the best fitness is constant after step #1 as seen in the best probe plot in Fig. 8.  
+The best probe number (#14) is the same for steps 0  and 1, but it switches to probe #2 at step 
+2.  Neither the number of the best probe nor the fi tness change after step 2.  Of course, for 
+most functions the best probe number varies through out a run, often quite erratically. 
+ 
+                                                                  Fig. 7.  Evolution of GP Function avg D  
+ 
+                                                               Fig. 8.  GP Function Best Probe Number 12 of 92  Figs. 9 and 10, respectively, plot the probe traje ctories for the probes with the best 
+ten fitnesses ordered by fitness and for the first sixteen individual probes ordered by probe 
+number [number of trajectories plotted chosen as a matter of convenience].  Both plots are 
+very chaotic with no obvious sign of regularity in how probes gravitate to the global 
+maximum.  Nevertheless, there is some measure of re gularity as reflected in the avg D curve 
+because its appearance is not nearly as chaotic as the trajectory plots.  In fact, in many cases 
+avg D exhibits a mathematically precise oscillation even  when the probe trajectories look like 
+Figs. 9 and 10 (see in particular [7,9]). 
+ 
+ 
+                                               Fig.  9.  GP Function Trajectories of Probes with Best F itness 13 of 92 
+ 
+                                               Fig.  10.  GP Function Probe Trajectories by Probe Numbe r 
+ 
+4.  A Benchmark Suite 
+CFO-PR was tested against the twenty-three function  benchmark suite in [11], and the 
+results compared to the other algorithms’ reported in that paper.  The benchmarks, their 
+decision spaces, and the three other algorithms (“G SO,” “PSO,” “GA”) are discussed in 
+detail in [11].  GSO (Group Search Optimizer) is a novel, highly refined Nature-inspired 
+metaheuristic that mimics animal searching behavior  based on a “producer-scrounger” 
+model.  PSO was implemented using “PSOt,” a MATLAB- based toolbox that includes 
+standard and variant PSO algorithms.  The genetic a lgorithm in [11] was implemented using 
+the GAOT toolbox (genetic algorithm optimization to olbox).  Recommended default 
+parameter values were used for the PSOt and GA algo rithms as described in [11]. 
+ Table 2 summarizes CFO-PR’s results using the same  function numbering as [11] 
+(more detailed data appear in Appendix 2).  max f is the known global maximum  (note that 
+the negative of each benchmark in [11] is used beca use, unlike the other algorithms, CFO 
+locates maxima, not minima).  >⋅<  denotes average value.  Because GSO, PSO and GA ar e 
+inherently stochastic, evaluating their performance  requires a statistical assessment.  
+Statistical data for those algorithms in Table 2 ar e reproduced from [11].  CFO’s results, of 14 of 92 course, are repeatable over runs with the same para meters because it is deterministic; no 
+statistical description is necessary. 
+ The CFO-PR data correspond to the best fitness ret urned by the single best run in the 
+set of runs with variable d pNN  and variable γ.  For functions f14  - f23  eleven runs were 
+made with 1 0 ≤≤γ  in increments of 0.1 and 2by 14 4 ≤ ≤d pNN  (66 runs total).  For f1 - 
+f13  the same procedure was used, but with 2by 6 2 ≤ ≤d pNN  (33 runs total) in order to 
+avoid excessive runtimes.  Table 2 shows the γ value corresponding to the best fitness, 
+best γ, and the corresponding best value of d pNN .  eval N is the number of function 
+evaluations, and it is tabulated for the single bes t run and for the group of runs used to 
+determine best γ and the best number of probes per axis. 
+ In the first group of high dimensionality unimodal  functions, f1 - f7, CFO-PR returned 
+the best fitness on five of the seven functions ( f3 - f7).  PSO performed best on the first two.  
+In the second set of high dimensionality multimodal  functions with many local maxima, f8 – 
+f13 , CFO-PR performed best on two ( f9, f10 ) and essentially the same as the best other 
+algorithm (GSO) on f8.  In last group of ten multimodal functions with f ew local maxima, f14  
+- f23 , CFO-PR returned the best fitness on four ( f20  - f23 ), equal fitnesses on three ( f14, f17, f18 ), 
+and very slightly lower fitnesses on the remaining three. 
+ Even though it is in its infancy, CFO-PR performed  very well against three other 
+highly sophisticated algorithms.  It returned the b est, equal, essentially equal or very slightly 
+lower fitnesses on eighteen of the twenty three tes t functions.  It is reasonable to conclude 
+that, overall, CFO-PR performed as well or better t han GSO, which in turn performed better 
+than PSO or GA. 
+ 
+5.  Conclusion 
+This note suggests that pseudorandomness is an impo rtant, indeed perhaps essential, aspect 
+of effective CFO implementations.  A pseudorandom v ariable has an arbitrary but precisely 
+known value that may be assigned or calculated.  It s essential characteristic is that the value 
+is uncorrelated with the decision space’s topology,  so that it has the effect of distributing 
+probes pseudorandomly throughout the landscape. 
+ While in a general sense this process may appear t o be similar to the randomness in 
+an inherently stochastic algorithm, it is in fact f undamentally different.  The equations 
+underlying stochastic algorithms are formulated in terms of true random variables whose 
+values are computed from probability distributions and consequently unknowable until the 
+calculation is made.  Therefore successive calculat ions yield different values, and every 
+optimization run has a different outcome. 
+ By contrast, a pseudorandom variable in the contex t of CFO is known with absolute 
+precision because of how its value is determined (a ssignment or deterministic calculation).  
+This property allows CFO to compute probe trajector ies precisely because it is inherently 
+deterministic.  Every CFO with the same setup, with  or without a pseudorandom component, 
+yields exactly the same results step-by-step throug hout an entire run.  Importantly, CFO’s 
+reproducibility lends itself well to “reactive” imp lementations in which run parameters are 15 of 92 “tuned” in response to performance metrics such as rate of convergence or fitness saturation, 
+for example.  Reactive stochastic algorithms, on th e other hand, are not easily implemented. 
+ 
+                               Table 2.  CFP-PR Com parative Results for 23 Benchmark Functions 
+                                                                        ( dN = Function Dimension, max f = Known Global Maximum) 
+ 
+- - - - - - - - - - - - - CFO  - - - - - - - - - - - - - 
+  
+ 
+Test 
+Func-
+tion *  
+ 
+ 
+dN  
+ 
+ 
+max f*  
+<Best 
+Fitness>/ 
+Other 
+Algorithm  Best 
+Fitness best γ
+ Best 
+d pNN
+ eval N 
+Best 
+Run eval N 
+Total 
+Unimodal Functions (other algorithms: average of 1000 runs) 
+f1 30 0 -3.6927x10 -37  
+/ PSO -4.8438x10 -4 0.1 4 20,640 507,060 
+f2 30 0 -2.9168x10 -24  
+/ PSO -4x10 -8 0.5 2 5,040 716,400 
+f3 30 0 -1.1979x10 -3 / 
+PSO -6x10 -8 0.5 2 10,260 1,534,260 
+f4 30 0 -0.1078 / 
+GSO -4.2x10 -7 0.5 2 5,160 332,340 
+f5 30 0 -37.3582 / 
+PSO -1.09289x10 -3 0.9 6 34,560 845,640 
+f6 30 0 -1.6000x10 -2 / 
+GSO 0 1.0 6 10,980 350,280 
+f7 30 0 -9.9024x10 -3 / 
+PSO -4.249x10 -5 0.1 4 60,120 1,983,960 
+Multimodal Functions, Many Local Maxima (other algorithms: a vg 1000 runs) 
+f8 30 12,569.5 12,569.4882 / 
+GSO 12,569.4866 0.5 4 12,720 448,800 
+f9 30 0 -0.6509 / GA -2.05x10 -6 0.7 4 16,440 680,640 
+f10  30 0 -2.6548x10 -5 / 
+GSO -1.5x10 -7 0.5 2 5,100 904,980 
+f11  30 0 -3.0792x10 -2 / 
+GSO -9.97293x10 -2 0.1 6 42,660 489,060 
+f12  30 0 -2.7648x10 -11  
+/ GSO -2.067x10 -5 0.5 2 3,660 341,400 
+f13  30 0 -4.6948x10 -5 / 
+GSO -3.2853x10 -3 0.6 6 16,920 679,620 
+Multimodal Functions, Few Local Maxima (other algorithms: a vg 50 runs) 
+f14  2 -1 -0.9980  / 
+GSO -0.9980 0.2 12 5,952 141,076 
+f15  4 -0.0003075 -3.7713x10 -4 / 
+GSO -4.889x10 -4 0 12 3,360 304,664 
+f16  2 1.0316285 1.031628 /  
+GSO  1.031626 0.4 12 6,288 124,340 
+f17  2 -0.398 -0.3979  / 
+GSO -0.3979 0 8 1,872 108,340 
+f18  2 -3 -3 / GSO  -3 0.9 12 1,464 180,472 
+f19  3 3.86 3.8628 /  GSO  3.8627 0.2 14 3,150 200,268 
+f20  6 3.32 3.2697 / GSO 3.32173 0.3 12 18,072 730,212 
+f21  4 10 7.5439 / PSO 10.1532 0.4 6 1,896 336,712 
+f22  4 10 8.3553 / PSO 10.4029 0.8 6 2,208 386,176 
+f23  4 10 8.9439 / PSO 10.5363 0.8 6 2,256 394,320 
+                           *  Note: negative of the functions in [11] are compute d by CFO because CFO searches for maxima 
+                   instead of minima.  
+ 16 of 92  This note provides examples of how pseudorandomnes s can improve CFO’s 
+performance.  Three different approaches are used ( initial probe distribution, repositioning 
+factor, and decision space adaptation), and each wa s discussed in detail.  A sample CFO 
+problem was presented in detail, and summary data i ncluded for two versions of the 
+algorithm tested against a 23-function benchmark su ite.  CFO’s performance is quite good 
+compared to other highly developed, state-of-the-ar t algorithms. 
+ Hopefully these results will encourage further wor k on improved methodologies for 
+injecting pseudorandomness into CFO.  Of course, an y or all of CFO’s run parameters can 
+be pseudorandomized, not only the three considered here.  But even with respect to those 
+parameters, different approaches to how they are ps eudorandomized may yield better results 
+or faster runtimes.  There are many fruitful areas of research on CFO, and it is the author’s 
+hope that this and the other CFO papers will provid e the foundation and catalyst for that 
+work. 
+2 January 2010 
+Brewster, Massachusetts 
+ 
+ 
+Ver. 2  (typographical errors in Step (c) of Fig. 1 and di scussion of Fig. 8 corrected; clarification of defi nition 
+of best Rr
+; error in Table 2 test function f 12  corrected; f 12  source code in Appendix 3, page 60 corrected).  
+3 February 2010 
+Saint Augustine, Florida  
+ 
+ 
+1Richard A. Formato, JD, PhD 
+ Registered Patent Attorney & Consulting Engineer 
+ P.O. Box 1714, Harwich, MA  02645  USA 
+ rf2@ieee.org 
+ 
+ 
+ 
+© 2010 Richard A. Formato 17 of 92 Appendix 1.  CFO Metaheuristic 
+CFO searches an dN-dimensional decision space Ω for the global maxima  of an objective 
+function  ) ,..., ,(2 1dNx xxf  defined on Ω: max min 
+i i i xx x ≤≤ , dNi≤≤1 .  The ix are the decision 
+variables , and i the coordinate number.  The term fitness  refers to the value of )(xfr at point 
+xr in Ω.  There is no a priori  information about the objective function’s maxima,  that is, 
+)(xfr’s topology or “landscape” is unknown. 
+ CFO searches Ω by flying “probes” through the space at discrete “ time” steps 
+(iterations).  Each probe’s location is specified b y its position vector computed from two 
+equations of motion  that analogize their real-world counterparts for m aterial objects moving 
+through physical space under the influence of gravi ty without energy dissipation. 
+ Probe p’s position vector at step j is kN
+kjp
+kp
+j ex Rd
+ˆ
+1,∑
+==r
+, where the jp
+kx, are its 
+coordinates and keˆ the unit vector along the kx-axis.  The indices p, pNp≤≤1 , and j, 
+tNj≤≤0 , respectively, are the probe number and iteration number, where pN and tN are 
+the corresponding total  number of probes and total  number of time steps. 
+ Equations of Motion.  In metaphorical “CFO space” each of the pN probes 
+experiences an acceleration created by the “gravita tional pull” of “masses” in Ω.  Probe p’s 
+acceleration at step 1−j is given by 
+βα
+p
+jk
+jp
+jk
+jN
+pkkp
+jk
+jp
+jk
+jp
+j
+R RR RM M M MU G ap
+1 11 1
+11 1 1 1 1) () () (
+− −− −
+≠=− − − − −
+−−× − ⋅ − =∑ rrrr
+r
+,     (1) 
+which is the first of CFO’s two equations of motion .  In equation (1), 
+) ,..., , (1, 1,
+21,
+1 1− − −
+−=jp
+Njp jp p
+jdx x xf M  is the objective function’s fitness at probe p’s location at 
+time step 1−j.  Each of the other probes at that step (iteration ) has associated with it fitness 
+pk
+j N p p k M ,..., 1, 1 ,..., 1 ,1 + − =− .  G is CFO’s gravitational constant , and ) (⋅U  is the Unit 
+Step function, 
+
+ ≥=otherwise zzU, 00 , 1)( . 
+ The acceleration p
+ja1−r causes probe p to move from position p
+jR1−r
+ at step 1−j to 
+p
+jRr
+ at step j according to the trajectory equation 
+1 ,21 2
+1 1 ≥ ∆ + =− − jt a R Rp
+jp
+jp
+jrrr
+,          (2) 
+which is CFO’s second equation of motion.  
+ The CFO equations of motion, (1) and (2), combine to compute a new probe 
+distribution at each time step using “masses” disco vered by the probe distribution at the 
+previous step.  t∆ is the “time” interval between steps during which the acceleration is 18 of 92 constant.  Note that CFO’s terminology has no signi ficance beyond being a reflection of 
+CFO’s kinematic roots, as is the factor ½ in eq. (2).  The gravitational constant, G, and 
+time increment, t∆, have direct analogues in Newton’s equations of mo tion for real masses 
+moving under real gravity through three-dimensional  physical space.  The CFO exponents 
+α and β, by contrast, have no analogues in Nature.  They p rovide added flexibility to the 
+algorithm designer who, in metaphorical “CFO space, ” is free to change how “gravity” 
+varies with distance, or mass, or both, if doing so  creates a more effective algorithm. 
+ “Mass.”  The concept of “mass” in CFO space is very importan t and quite different 
+than it is in real space.  Mass in the physical Uni verse is an inherent, immutable property of 
+matter, whereas in CFO space it is a positive-defin ite user-defined function  of the objective 
+function’s fitness [not (necessarily) the fitness i tself].  For example, in equation (1) mass is 
+defined as α) () (1 1 1 1p
+jk
+jp
+jk
+j CFO M M M MU MASS − − − − − ⋅ − =  [difference in fitness values raised to 
+the α power multiplied by the Unit Step].  A different f unction can be used if it results in a 
+better performing CFO algorithm.  In this specific implementation the Unit Step is a critical 
+element because it prevents negative mass.  Without  the Unit Step CFO mass could be 
+negative depending on which fitness is greater.  Bu t mass in the real Universe always is 
+positive, and as a consequence the force of gravity  always attractive.  By contrast, mass can 
+be positive or negative in metaphorical CFO space, depending on how it is defined, and 
+undesirable effects may result from the wrong defin ition.  Negative mass creates a repulsive  
+gravitational force that flies probes away from max ima instead of toward them, thus 
+defeating the very purpose of the algorithm. 
+ Errant Probes.  At any iteration in a CFO run, it is possible that a probe’s 
+acceleration computed from eq. (2) may be too great  to keep it inside Ω.  If any coordinate 
+min 
+i ixx< or max 
+i ixx> , the probe enters a region of unfeasible  solutions that are not valid for 
+the problem at hand.  The question is what to do wi th an errant probe, and it arises in many 
+algorithms.  There are many approaches s.  While ma ny schemes are possible, a simple, 
+empirically determined one is used here.  On a coor dinate-by-coordinate basis, probes flying 
+out of the decision space are placed a fraction 1≤ ≤ ∆rep rep F F  of the distance between the 
+probe’s starting coordinate and the corresponding b oundary coordinate.  rep F is the variable 
+“repositioning factor” introduced in [2].  Its valu e, as well as those of all the CFO 
+parameters, were determined empirically. 19 of 92 References 
+ 
+[1] Formato, R. A., “Central Force Optimization: A New Metaheuristic with 
+ Applications in Applied Electromagnetics,” Prog. Electromagnetics Research , PIER 
+ 77 , pp. 425-491 (2007), online at http://ceta.mit.edu /PIER/pier.php?volume=77 . 
+ (DOI:10.2528/PIER07082403) 
+[2] Formato, R. A., “Central Force Optimization: A New Computational Framework for 
+ Multidimensional Search and Optimization,” in Nature Inspired Cooperative 
+ Strategies for Optimization (NICSO 2007) , Studies in Computational Intelligence 
+ 129  (N. Krasnogor, G. Nicosia, M. Pavone, and D. Pelta , Eds.), vol. 129, pp. 221-
+ 238, Springer-Verlag, Heidelberg (2008). 
+[3] Formato, R. A., “Central Force Optimisation: a New Gradient-Like Metaheuristic for 
+ Multidimensional Search and Optimisation,” Int. J. Bio-Inspired Computation , 1, no. 
+ 4, pp. 217-238 (2009). (DOI: 10.1504/IJBIC.2009.02 4721). 
+[4] Formato, R. A., “Central Force Optimization: A New Deterministic Gradient-Like 
+ Optimization Metaheuristic,” OPSEARCH, Jour. of the Operations Research Society 
+ of India , 46 , no. 1, pp. 25-51 (2009). (DOI: 10.1007/s12597-009 -0003-4). 
+[5] Qubati, G. M., Formato, R. A., and Dib, N. I., “Antenna Benchmark Performance 
+ and Array Synthesis using Central Force Optimisati on,”  IET (U.K.) Microwaves, 
+ Antennas & Propagation  (in press). (DOI: 10.1049/iet-map.2009.0147). 
+[6] Formato, R. A., “Improved CFO Algorithm for Ant enna Optimization,” Prog. 
+ Electromagnetics Research (in revision). 
+[7] Formato, R. A., “Are Near Earth Objects the Key  to Optimization Theory?,” 
+ arXiv:0912.1394v1  [astro-ph.EP], online at www.arXiv.org . 
+[8] Formato, R. A., “Central Force Optimization and  NEOs – First Cousins?,”  Journal of 
+ Multiple-Valued Logic and Soft Computing (in press). 
+[9] Formato, R. A., “NEOs – A Physicomimetic Framew ork for Central Force 
+ Optimization?,” Applied Mathematics and Computation  (review). 
+[10] Formato, R. A., “Central Force Optimization wi th Variable Initial Probes and 
+Adaptive Decision Space,” Applied Mathematics and Computation  (review). 
+[11] He, S., Wu, Q. H., Saunders, J. R., “Group Sea rch Optimizer: An Optimization 
+Algorithm Inspired by Animal Searching Behavior,” IEEE Trans. Evol. 
+Computation , vol. 13, no. 5, pp. 973-990, Oct. 2009. 20 of 92 Appendix 2.  CFO Summary Data for the GSO Benchmark Suite  [11] 
+ 
+Note:  In the tables below, “ UNIFORM P-AXIS” means  “ /uni01C1-AXIS” as discussed above. 
+ 
+F1  
+Run ID: 12-27-2009, 13:28:55 
+ 
+FUNCTION: F1 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00       85      5160   0.95000V        -0.06315972    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00      232     13980   0.70000V      -135.67004943    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00      238     14340   0.05000V      -345.75281239    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00      162      9780   0.05000V       -85.84498630    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00       97      5880   0.60000V      -163.14259732    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00      115      6960   0.55000V        -0.02659219    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00       72      4380   0.30000V      -212.60250000    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00      222     13380   0.20000V      -559.30076908    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00      160      9660   0.90000V       -20.19453374    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00      197     11880   0.85000V        -2.43693893    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00      220     13260   0.10000V        -2.41967196    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00      110     13320   0.30000V        -1.23728667    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00      171     20640   0.50000V        -0.00048438    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00      128     15480   0.25000V        -0.00954054    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00      136     16440   0.65000V        -0.00386787    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00       98     11880   0.65000V        -0.03513774    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00      117     14160   0.65000V        -2.48228919    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00       98     11880   0.65000V        -0.03513774    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00      134     16200   0.55000V        -0.31826953    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00      126     15240   0.15000V        -0.00992779    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00       98     11880   0.65000V        -0.00915096    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00      110     13320   0.30000V        -1.23728667    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00      130     23580   0.35000V        -0.00413390    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00      150     27180   0.40000V        -2.11477561    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V        -0.33188771    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V        -2.12513165    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00      150     27180   0.40000V        -0.00219997    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00      136     24660   0.65000V        -0.06292008    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00      153     27720   0.55000V        -0.06039108    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V        -2.12513165    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V        -0.33188771    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00      150     27180   0.40000V        -2.11477561    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00      130     23580   0.35000V        -0.00413390    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      507060 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00      171     20640   0.50000V        -0.00048438    UNIFORM P-AXIS 
+ 
+F2 
+Run ID: 12-27-2009, 13:29:23 
+ 
+FUNCTION: F2 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00      175     10560   0.70000V        -6.60617435    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00      370     22260   0.95000V        -3.81990738    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00      324     19500   0.55000V        -2.06811449    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00      190     11460   0.50000V        -2.09728545    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00      126      7620   0.15000V        -2.73598457    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00       83      5040   0.85000V        -0.00000004    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00      174     10500   0.65000V        -1.24666633    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00      157      9480   0.75000V        -0.70580201    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00      123      7440   0.95000V        -3.00749022    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00      461     27720   0.75000V        -3.47243586    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.19552788    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00      173     20880   0.60000V        -0.50882324    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00       97     11760   0.60000V        -0.13817869    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00       80      9720   0.70000V        -0.42701634    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00       79      9600   0.65000V        -0.31587227    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00      118     14280   0.70000V        -0.18736453    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -0.06929315    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00      353     42480   0.10000V        -0.06374008    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00      134     16200   0.55000V        -0.06659569    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00      380     45720   0.50000V        -0.11070981    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00      184     22200   0.20000V        -0.13620894    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00       98     11880   0.65000V        -0.09823314    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00       96     17460   0.55000V        -0.01628443    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00      499     90000   0.75000V        -0.00051367    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00      216     39060   0.85000V        -0.02264350    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00       81     14760   0.75000V        -0.08109305    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V        -0.16869854    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -0.03716355    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V        -0.16869854    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00       81     14760   0.75000V        -0.08109305    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00      210     37980   0.55000V        -0.03264013    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00      398     71820   0.45000V        -0.00042562    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00       96     17460   0.55000V        -0.22930127    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      716400 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00       83      5040   0.85000V        -0.00000004    UNIFORM P-AXIS 21 of 92 F3 
+Run ID: 12-27-2009, 14:55:32 
+ 
+FUNCTION: F3 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+ 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00      362     21780   0.55000V      -981.62841988    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00      383     23040   0.65000V       -91.16096360    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00      354     21300   0.15000V       -43.94966422    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00      358     21540   0.35000V       -88.81550993    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00      348     20940   0.80000V       -66.55740578    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00      170     10260   0.45000V        -0.00000006    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00      456     27420   0.50000V       -66.39589627    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00      331     19920   0.90000V       -86.26322188    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00      383     23040   0.65000V       -42.03391601    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00      333     20040   0.05000V       -75.23554855    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V      -979.00439553    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00      388     46680   0.90000V       -74.16162186    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00      420     50520   0.60000V       -31.07754473    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00      363     43680   0.60000V       -44.64307545    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00      390     46920   0.05000V      -134.76969539    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00      407     48960   0.90000V       -47.09603099    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00      370     44520   0.95000V        -1.54893769    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00      446     53640   0.95000V       -36.81437550    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00      389     46800   0.95000V      -136.56793953    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00      380     45720   0.50000V       -43.06679509    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00      345     41520   0.65000V       -25.74778799    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00      417     50160   0.45000V       -76.55528086    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00      364     65700   0.65000V       -39.37989070    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00      419     75600   0.55000V       -10.05898847    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00      425     76680   0.85000V        -6.83905075    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00      364     65700   0.65000V       -92.48498686    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00      383     69120   0.65000V       -55.05339898    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00      403     72720   0.70000V        -0.22231755    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00      366     66060   0.75000V       -55.86579942    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00      402     72540   0.65000V      -101.88858589    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00      389     70200   0.95000V        -7.12140022    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00      397     71640   0.40000V        -9.52128172    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00      387     69840   0.85000V       -54.11204028    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:     1534260 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00      170     10260   0.45000V        -0.00000006    UNIFORM P-AXIS 
+ 
+F4 
+Run ID: 12-27-2009, 15:30:00 
+ 
+FUNCTION: F4 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00       60      3660   0.65000V      -100.00000000    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00       71      4320   0.25000V       -78.34901257    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00       71      4320   0.25000V       -54.46679317    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00      100      6060   0.75000V       -37.42713572    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00      121      7320   0.85000V       -12.05143023    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00       85      5160   0.95000V        -0.00000042    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00      110      6660   0.30000V       -12.81211639    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00      155      9360   0.65000V       -29.63311265    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00       71      4320   0.25000V       -54.46679317    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00       71      4320   0.25000V       -77.68060588    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00       60      3660   0.65000V       -10.00000000    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V      -100.00000000    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V       -69.33580190    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V       -27.05570292    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00       75      9120   0.45000V        -5.57949024    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -7.97580490    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00      101     12240   0.80000V        -0.05938789    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -7.97580490    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00       75      9120   0.45000V        -5.57949024    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V       -27.05570292    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V       -69.33580190    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -4.10445402    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V      -100.00000000    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V       -53.09339766    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00       75     13680   0.45000V        -2.16245242    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00       98     17820   0.65000V        -0.02337879    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00      108     19620   0.20000V        -0.70650669    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00       77     14040   0.55000V        -0.12574949    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00      190     34380   0.50000V        -0.01553213    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00       98     17820   0.65000V        -0.02337879    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00      114     20700   0.50000V        -0.00451056    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V       -53.09339766    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00       72     13140   0.30000V        -1.23280952    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      332340 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00       85      5160   0.95000V        -0.00000042    UNIFORM P-AXIS 
+ 
+ 
+ 22 of 92 F5 
+Run ID: 12-27-2009, 16:32:40 
+ 
+FUNCTION: F5 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+ 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00       98      5940   0.65000V        -3.70063828    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00      191     11520   0.55000V       -13.15019089    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00      359     21600   0.40000V        -0.02171873    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00      320     19260   0.35000V        -0.06060966    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00      347     20880   0.75000V        -0.99902885    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00      334     20100   0.10000V        -1.16157214    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00       97      5880   0.60000V    -19026.84403442    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00       79      4800   0.65000V      -495.91491702    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00      441     26520   0.70000V        -2.54248267    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00      399     24000   0.50000V        -0.58852057    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00       60      3660   0.65000V    -44159.15850962    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00      264     31800   0.40000V       -48.73428954    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00       96     11640   0.55000V        -4.72035934    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00      208     25080   0.45000V        -0.00163122    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00       99     12000   0.70000V        -1.16925621    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00      365     43920   0.70000V        -1.06926420    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00       79      9600   0.65000V        -4.80953659    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00      392     47160   0.15000V        -2.19512324    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00      343     41280   0.55000V        -0.02220271    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00      400     48120   0.55000V        -0.48415875    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00      192     23160   0.60000V        -0.33830043    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00      173     20880   0.60000V     -2586.52615560    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00      338     61020   0.30000V        -0.11427428    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V        -0.86881815    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00      191     34560   0.55000V        -0.06891125    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00       79     14400   0.65000V        -1.47748671    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00       80     14580   0.70000V       -13.38325143    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00       98     17820   0.65000V        -2.14030613    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00      192     34740   0.60000V      -294.28499320    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00      349     63000   0.85000V        -0.53537670    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00       77     14040   0.55000V        -2.50318155    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00      191     34560   0.55000V        -0.00109289    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00      354     63900   0.15000V        -5.02357451    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      845640 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00      191     34560   0.55000V        -0.00109289    UNIFORM P-AXIS 
+ 
+ 
+F6 
+Run ID: 12-31-2009, 11:30:35 
+ 
+FUNCTION: F6 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00       82      4980   0.80000V         0.00000000    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00      117      7080   0.65000V      -150.00000000    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00      119      7200   0.75000V      -324.00000000    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00      123      7440   0.95000V       -81.00000000    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00      109      6600   0.25000V      -150.00000000    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00      111      6720   0.35000V      -256.00000000    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00      137      8280   0.70000V      -169.00000000    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00       92      5580   0.35000V      -870.00000000    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00       92      5580   0.35000V      -390.00000000    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00      117      7080   0.65000V      -150.00000000    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00      117      7080   0.65000V      -480.00000000    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00      110     13320   0.30000V        -1.00000000    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00      130     15720   0.35000V         0.00000000    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00      116     14040   0.60000V         0.00000000    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00       78      9480   0.60000V        -1.00000000    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00       79      9600   0.65000V         0.00000000    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00       78      9480   0.60000V         0.00000000    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00       79      9600   0.65000V         0.00000000    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00       78      9480   0.60000V         0.00000000    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00       78      9480   0.60000V        -1.00000000    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00       79      9600   0.65000V        -1.00000000    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00      110     13320   0.30000V        -1.00000000    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V         0.00000000    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V        -4.00000000    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V         0.00000000    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00       97     17640   0.60000V        -1.00000000    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V         0.00000000    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V        -1.00000000    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V         0.00000000    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00       97     17640   0.60000V        -1.00000000    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V         0.00000000    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V        -4.00000000    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V         0.00000000    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      350280 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V         0.00000000    UNIFORM P-AXIS 
+ 
+ 23 of 92 F7 
+Run ID: 12-31-2009, 11:31:18 
+ 
+FUNCTION: F7 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.01097246    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.00474254    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.00105431    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.00191198    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.00207141    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.00075131    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.00308822    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.00122170    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.00161902    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.00311318    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V        -0.00373317    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00332989    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00004249    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00135109    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00196345    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00169604    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00094857    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00065703    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00084987    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00099223    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00033298    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00078422    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.00150250    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.00021166    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.00057275    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.00024470    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.00202352    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.00025176    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.00179181    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.00033893    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.00100772    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.00060779    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.00118826    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:     1983960 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.00004249    UNIFORM P-AXIS 
+ 
+NOTE: THIS FILE HAS BEEN EDITED TO SHOW THE CORRECT  BEST RUN ON THE BOTTOM LINE BECAUSE FUNCTION F7 IS  RANDOM. 
+ 
+F8 
+Run ID: 12-31-2009, 16:01:36 
+ 
+FUNCTION: F8 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00      110      6660   0.30000V      5537.61608593    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00       85      5160   0.95000V     12568.82823317    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00       86      5220   0.05000V     12569.28760395    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00       88      5340   0.15000V     12569.48093048    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00       86      5220   0.05000V     12569.43736670    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00       95      5760   0.50000V     12561.34875424    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00      137      8280   0.70000V     12559.51991759    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00       97      5880   0.60000V      9016.10961344    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00      132      7980   0.45000V     12569.48515193    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00      105      6360   0.05000V     12569.46303573    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00      130      7860   0.35000V      8652.91904178    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00      110     13320   0.30000V      5536.79333863    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00      124     15000   0.05000V     12569.48174634    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00       87     10560   0.10000V     12569.47740007    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00      106     12840   0.10000V     12569.48333599    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00       86     10440   0.05000V     12569.40587582    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00      105     12720   0.05000V     12569.48661484    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00      124     15000   0.05000V     12569.43670675    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00       87     10560   0.10000V     12569.47329569    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00       73      8880   0.35000V      8995.25108485    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00      105     12720   0.05000V     12569.48652992    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00      194     23400   0.70000V     12568.87307882    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00       95     17280   0.50000V     12569.35060751    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00      144     26100   0.10000V     12569.43649774    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00      162     29340   0.05000V     12351.27895503    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00       87     15840   0.10000V     12569.45150324    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V     12174.51472557    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00      125     22680   0.10000V     12569.47757504    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00      134     24300   0.55000V     12569.20312794    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00      148     26820   0.30000V     12569.39892325    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00      110     19980   0.30000V     12569.45496162    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00       86     15660   0.05000V     12569.48420468    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00      118     21420   0.70000V     12569.46181648    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      448800 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00      105     12720   0.05000V     12569.48661484    UNIFORM P-AXIS 
+ 
+ 
+ 24 of 92 F9 
+Run ID: 12-31-2009, 13:36:56 
+ 
+FUNCTION: F9 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00       97      5880   0.60000V       -31.70098539    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00       99      6000   0.70000V       -34.19310095    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00      311     18720   0.85000V      -123.74637495    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00      239     14400   0.10000V      -108.91030539    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00      103      6240   0.90000V       -31.69341889    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00      113      6840   0.45000V        -0.99112119    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V       -41.36476630    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00      172     10380   0.55000V       -44.82618586    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00      116      7020   0.60000V      -112.60669341    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00       78      4740   0.60000V      -483.01739630    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00      113      6840   0.45000V        -0.00000352    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V       -40.51140239    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00      247     29760   0.50000V       -29.69982488    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00      116     14040   0.60000V        -0.00383322    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00      151     18240   0.45000V        -0.00133000    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00       96     11640   0.55000V        -0.00003333    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00      165     19920   0.20000V        -0.99011355    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00       96     11640   0.55000V        -0.00003333    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00      136     16440   0.65000V        -0.00000205    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00      116     14040   0.60000V        -0.00383322    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00      155     18720   0.65000V       -30.01506187    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00      500     60120   0.80000V        -0.22468142    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00      116     21060   0.60000V        -0.00005257    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00      116     21060   0.60000V        -0.00035429    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00      182     32940   0.10000V        -0.99080917    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00      176     31860   0.75000V        -0.99013929    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00      155     28080   0.65000V        -0.00054147    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00       96     17460   0.55000V        -0.00014569    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V        -0.28712215    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00      116     21060   0.60000V        -0.00015616    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00      143     25920   0.05000V       -15.84039530    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00      171     30960   0.50000V        -0.00051308    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00      116     21060   0.60000V        -0.00005257    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      680640 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00      136     16440   0.65000V        -0.00000205    UNIFORM P-AXIS 
+ 
+F10 
+Run ID: 12-31-2009, 17:26:18 
+ 
+FUNCTION: F10 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+ 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00      171     10320   0.50000V       -19.95554498    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00      196     11820   0.80000V       -21.94028909    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00      179     10800   0.90000V       -18.77780599    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00      273     16440   0.85000V       -14.99021318    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00      164      9900   0.15000V       -12.68499222    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00       84      5100   0.90000V        -0.00000015    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00      120      7260   0.80000V       -16.10492237    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00      156      9420   0.70000V       -17.47681504    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00      171     10320   0.50000V       -19.67837086    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00      196     11820   0.80000V       -21.97779101    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00      140      8460   0.85000V        -0.65731364    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00      163     19680   0.10000V       -19.95577490    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00      225     27120   0.35000V       -21.88619837    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00      229     27600   0.55000V       -18.91894002    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00      207     24960   0.40000V        -4.27574675    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00      155     18720   0.65000V       -12.54005560    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00      155     18720   0.65000V        -0.01693374    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00      332     39960   0.95000V        -6.87736931    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00      290     34920   0.75000V       -15.81273556    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00      283     34080   0.40000V       -19.29625344    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00      226     27240   0.40000V       -21.87441323    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00      426     51240   0.90000V        -0.00004949    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00      138     25020   0.75000V       -19.95643855    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00      206     37260   0.35000V       -21.74579561    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00      203     36720   0.20000V       -19.62244329    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00      500     90180   0.80000V       -17.80088045    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00      232     41940   0.70000V       -10.88650930    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00       78     14220   0.60000V        -0.39610444    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00      125     22680   0.10000V        -0.01473085    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00      194     35100   0.70000V        -6.47287042    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00      393     70920   0.20000V        -0.01545910    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00      386     69660   0.80000V       -19.96912371    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00      140     25380   0.85000V        -0.00944870    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      904980 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00       84      5100   0.90000V        -0.00000015    UNIFORM P-AXIS 
+ 
+ 
+ 25 of 92 F11 
+Run ID: 12-31-2009, 19:27:23 
+ 
+FUNCTION: F11 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+ 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00      161      9720   0.95000V     -3349.89947839    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00      109      6600   0.25000V      -682.73550403    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00      121      7320   0.85000V      -787.80355520    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00      242     14580   0.25000V      -779.13514096    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V      -328.25766449    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00      500     30060   0.80000V       -64.46413940    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00      473     28440   0.40000V        -4.00165096    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00      122      7380   0.90000V      -142.36479694    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00      103      6240   0.90000V      -487.25760030    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00      104      6300   0.95000V     -1024.52777584    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00      232     13980   0.70000V        -2.00124049    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00      174     21000   0.65000V     -1042.16470301    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -3.03975464    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -2.57331594    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -1.39086441    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -5.67090092    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00      140     16920   0.85000V       -64.20929684    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -6.40000000    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00      178     21480   0.85000V       -97.02717321    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V      -445.43143407    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V       -40.34010306    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -4.05519280    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00      111     20160   0.35000V     -1127.36887599    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00      236     42660   0.90000V        -0.09972929    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00      251     45360   0.70000V        -1.89829119    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -4.42021222    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -8.01901276    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V       -68.05499564    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -3.84611240    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00      140     25380   0.85000V      -121.65998851    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V       -12.45943847    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -3.38969420    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -3.56479100    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      489060 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00      236     42660   0.90000V        -0.09972929    UNIFORM P-AXIS 
+ 
+F12 
+Run ID: 02-03-2010, 14:55:15 [CORRECTED 02-03-2010;  SEE F12 SOURCE LISTING PAGE 60] 
+ 
+FUNCTION: F12 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000   UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00       78      4740   0.60000V        -0.79539561   UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00       99      6000   0.70000V        -0.74543651   UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00       92      5580   0.35000V        -0.42992625   UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00       96      5820   0.55000V        -0.43773493   UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00      156      9420   0.70000V        -0.25675666   UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00       60      3660   0.65000V        -0.00002067   UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00       60      3660   0.65000V        -0.24925810   UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00       60      3660   0.65000V        -0.72371775   UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00       60      3660   0.65000V        -2.19256987   UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00      115      6960   0.55000V        -0.13285456   UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00       82      4980   0.80000V        -1.60665754   UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00      135     16320   0.60000V        -0.39097490   UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -0.00368253   UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00       62      7560   0.75000V        -0.25600092   UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00      133     16080   0.50000V        -0.02016903   UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00       78      9480   0.60000V        -0.25920627   UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00       96     11640   0.55000V        -0.01298711   UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00      135     16320   0.60000V        -0.26677253   UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00      115     13920   0.55000V        -0.36764888   UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -0.02887759   UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00       97     11760   0.60000V        -0.03222183   UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00       78      9480   0.60000V        -0.06522233   UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00      117     21240   0.65000V        -0.14434782   UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -0.00696541   UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -0.07738490   UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00      116     21060   0.60000V        -0.16057613   UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -0.03082085   UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00       97     17640   0.60000V        -0.00446426   UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00       96     17460   0.55000V        -0.11569986   UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -0.00606971   UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -0.01177648   UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -0.01952312   UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00       70     12780   0.20000V        -0.26244867   UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      341400 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00       60      3660   0.65000V        -0.00002067   UNIFORM P-AXIS 
+ 
+ 
+ 26 of 92 F13 
+Run ID: 12-31-2009, 21:12:44 
+ 
+FUNCTION: F13 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+ 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500     30     60     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500     30     60     2.0    1. 0    2.00     2.00       91      5520   0.30000V        -1.98285152    UNIFORM P-AXIS 
+   2       0.100    500     30     60     2.0    1. 0    2.00     2.00      312     18780   0.90000V        -0.23909156    UNIFORM P-AXIS 
+   3       0.200    500     30     60     2.0    1. 0    2.00     2.00       90      5460   0.25000V        -0.15479827    UNIFORM P-AXIS 
+   4       0.300    500     30     60     2.0    1. 0    2.00     2.00       81      4920   0.75000V        -0.28720755    UNIFORM P-AXIS 
+   5       0.400    500     30     60     2.0    1. 0    2.00     2.00      233     14040   0.75000V        -0.14883921    UNIFORM P-AXIS 
+   6       0.500    500     30     60     2.0    1. 0    2.00     2.00      184     11100   0.20000V        -0.06596601    UNIFORM P-AXIS 
+   7       0.600    500     30     60     2.0    1. 0    2.00     2.00      226     13620   0.40000V        -0.28878418    UNIFORM P-AXIS 
+   8       0.700    500     30     60     2.0    1. 0    2.00     2.00      122      7380   0.90000V        -0.76742123    UNIFORM P-AXIS 
+   9       0.800    500     30     60     2.0    1. 0    2.00     2.00       90      5460   0.25000V        -0.60350146    UNIFORM P-AXIS 
+  10       0.900    500     30     60     2.0    1. 0    2.00     2.00      134      8100   0.55000V        -0.50142794    UNIFORM P-AXIS 
+  11       1.000    500     30     60     2.0    1. 0    2.00     2.00      148      8940   0.30000V        -0.26195645    UNIFORM P-AXIS 
+  12       0.000    500     30    120     2.0    1. 0    2.00     2.00      331     39840   0.90000V        -0.17132369    UNIFORM P-AXIS 
+  13       0.100    500     30    120     2.0    1. 0    2.00     2.00      130     15720   0.35000V        -0.63280216    UNIFORM P-AXIS 
+  14       0.200    500     30    120     2.0    1. 0    2.00     2.00      312     37560   0.90000V        -0.33885975    UNIFORM P-AXIS 
+  15       0.300    500     30    120     2.0    1. 0    2.00     2.00      204     24600   0.25000V        -0.18849309    UNIFORM P-AXIS 
+  16       0.400    500     30    120     2.0    1. 0    2.00     2.00       60      7320   0.65000V        -0.41319188    UNIFORM P-AXIS 
+  17       0.500    500     30    120     2.0    1. 0    2.00     2.00       98     11880   0.65000V        -0.03399272    UNIFORM P-AXIS 
+  18       0.600    500     30    120     2.0    1. 0    2.00     2.00      165     19920   0.20000V        -0.25527275    UNIFORM P-AXIS 
+  19       0.700    500     30    120     2.0    1. 0    2.00     2.00      383     46080   0.65000V        -0.27622507    UNIFORM P-AXIS 
+  20       0.800    500     30    120     2.0    1. 0    2.00     2.00      172     20760   0.55000V        -0.17442617    UNIFORM P-AXIS 
+  21       0.900    500     30    120     2.0    1. 0    2.00     2.00      212     25560   0.65000V        -0.02638101    UNIFORM P-AXIS 
+  22       1.000    500     30    120     2.0    1. 0    2.00     2.00      215     25920   0.80000V        -0.27534519    UNIFORM P-AXIS 
+  23       0.000    500     30    180     2.0    1. 0    2.00     2.00       97     17640   0.60000V        -0.07220597    UNIFORM P-AXIS 
+  24       0.100    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -0.07174276    UNIFORM P-AXIS 
+  25       0.200    500     30    180     2.0    1. 0    2.00     2.00      370     66780   0.95000V        -0.15772989    UNIFORM P-AXIS 
+  26       0.300    500     30    180     2.0    1. 0    2.00     2.00      151     27360   0.45000V        -0.08917428    UNIFORM P-AXIS 
+  27       0.400    500     30    180     2.0    1. 0    2.00     2.00      195     35280   0.75000V        -0.06522780    UNIFORM P-AXIS 
+  28       0.500    500     30    180     2.0    1. 0    2.00     2.00      134     24300   0.55000V        -0.00713353    UNIFORM P-AXIS 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00       93     16920   0.40000V        -0.00328529    UNIFORM P-AXIS 
+  30       0.700    500     30    180     2.0    1. 0    2.00     2.00      186     33660   0.30000V        -0.19308067    UNIFORM P-AXIS 
+  31       0.800    500     30    180     2.0    1. 0    2.00     2.00      170     30780   0.45000V        -0.07304213    UNIFORM P-AXIS 
+  32       0.900    500     30    180     2.0    1. 0    2.00     2.00       60     10980   0.65000V        -0.24282441    UNIFORM P-AXIS 
+  33       1.000    500     30    180     2.0    1. 0    2.00     2.00      146     26460   0.20000V        -0.02567155    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      679620 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  29       0.600    500     30    180     2.0    1. 0    2.00     2.00       93     16920   0.40000V        -0.00328529    UNIFORM P-AXIS 
+ 
+F14 
+Run ID: 12-26-2009, 23:38:50 
+ 
+FUNCTION: F14 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500      2      8     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500      2      8     2.0    1. 0    2.00     2.00      213      1712   0.70000V        -5.93292124    UNIFORM P-AXIS 
+   2       0.100    500      2      8     2.0    1. 0    2.00     2.00      500      4008   0.80000V        -2.08127846    UNIFORM P-AXIS 
+   3       0.200    500      2      8     2.0    1. 0    2.00     2.00       97       784   0.60000V        -2.98238078    UNIFORM P-AXIS 
+   4       0.300    500      2      8     2.0    1. 0    2.00     2.00      114       920   0.50000V        -1.00528367    UNIFORM P-AXIS 
+   5       0.400    500      2      8     2.0    1. 0    2.00     2.00       78       632   0.60000V        -6.90336530    UNIFORM P-AXIS 
+   6       0.500    500      2      8     2.0    1. 0    2.00     2.00      123       992   0.95000V        -1.99203466    UNIFORM P-AXIS 
+   7       0.600    500      2      8     2.0    1. 0    2.00     2.00       70       568   0.20000V       -18.31709957    UNIFORM P-AXIS 
+   8       0.700    500      2      8     2.0    1. 0    2.00     2.00      118       952   0.70000V        -2.98345327    UNIFORM P-AXIS 
+   9       0.800    500      2      8     2.0    1. 0    2.00     2.00      108       872   0.20000V        -7.87400969    UNIFORM P-AXIS 
+  10       0.900    500      2      8     2.0    1. 0    2.00     2.00       95       768   0.50000V        -9.80389837    UNIFORM P-AXIS 
+  11       1.000    500      2      8     2.0    1. 0    2.00     2.00       76       616   0.50000V       -10.76397606    UNIFORM P-AXIS 
+  12       0.000    500      2     12     2.0    1. 0    2.00     2.00      213      2568   0.70000V        -1.99245403    UNIFORM P-AXIS 
+  13       0.100    500      2     12     2.0    1. 0    2.00     2.00      139      1680   0.80000V        -1.00776304    UNIFORM P-AXIS 
+  14       0.200    500      2     12     2.0    1. 0    2.00     2.00      114      1380   0.50000V       -10.76325056    UNIFORM P-AXIS 
+  15       0.300    500      2     12     2.0    1. 0    2.00     2.00       92      1116   0.35000V        -1.00708019    UNIFORM P-AXIS 
+  16       0.400    500      2     12     2.0    1. 0    2.00     2.00       98      1188   0.65000V        -5.92895444    UNIFORM P-AXIS 
+  17       0.500    500      2     12     2.0    1. 0    2.00     2.00       72       876   0.30000V        -3.03091277    UNIFORM P-AXIS 
+  18       0.600    500      2     12     2.0    1. 0    2.00     2.00       82       996   0.80000V        -6.90589693    UNIFORM P-AXIS 
+  19       0.700    500      2     12     2.0    1. 0    2.00     2.00      110      1332   0.30000V        -8.84107275    UNIFORM P-AXIS 
+  20       0.800    500      2     12     2.0    1. 0    2.00     2.00      136      1644   0.65000V        -1.08554446    UNIFORM P-AXIS 
+  21       0.900    500      2     12     2.0    1. 0    2.00     2.00       87      1056   0.10000V        -2.98244726    UNIFORM P-AXIS 
+  22       1.000    500      2     12     2.0    1. 0    2.00     2.00       76       924   0.50000V       -10.76397605    UNIFORM P-AXIS 
+  23       0.000    500      2     16     2.0    1. 0    2.00     2.00      130      2096   0.35000V        -1.99739732    UNIFORM P-AXIS 
+  24       0.100    500      2     16     2.0    1. 0    2.00     2.00      115      1856   0.55000V        -1.99209759    UNIFORM P-AXIS 
+  25       0.200    500      2     16     2.0    1. 0    2.00     2.00       98      1584   0.65000V        -0.99806031    UNIFORM P-AXIS 
+  26       0.300    500      2     16     2.0    1. 0    2.00     2.00       79      1280   0.65000V        -3.96825029    UNIFORM P-AXIS 
+  27       0.400    500      2     16     2.0    1. 0    2.00     2.00       60       976   0.65000V       -13.61860920    UNIFORM P-AXIS 
+  28       0.500    500      2     16     2.0    1. 0    2.00     2.00       93      1504   0.40000V        -1.00064288    UNIFORM P-AXIS 
+  29       0.600    500      2     16     2.0    1. 0    2.00     2.00       81      1312   0.75000V        -4.95050169    UNIFORM P-AXIS 
+  30       0.700    500      2     16     2.0    1. 0    2.00     2.00       96      1552   0.55000V        -4.95057264    UNIFORM P-AXIS 
+  31       0.800    500      2     16     2.0    1. 0    2.00     2.00       95      1536   0.50000V       -13.61905412    UNIFORM P-AXIS 
+  32       0.900    500      2     16     2.0    1. 0    2.00     2.00       82      1328   0.80000V        -2.98333672    UNIFORM P-AXIS 
+  33       1.000    500      2     16     2.0    1. 0    2.00     2.00      153      2464   0.55000V        -0.99800512    UNIFORM P-AXIS 
+  34       0.000    500      2     20     2.0    1. 0    2.00     2.00      128      2580   0.25000V        -6.90577771    UNIFORM P-AXIS 
+  35       0.100    500      2     20     2.0    1. 0    2.00     2.00      155      3120   0.65000V       -11.71875390    UNIFORM P-AXIS 
+  36       0.200    500      2     20     2.0    1. 0    2.00     2.00      124      2500   0.05000V        -5.92888566    UNIFORM P-AXIS 
+  37       0.300    500      2     20     2.0    1. 0    2.00     2.00       60      1220   0.65000V        -2.00372774    UNIFORM P-AXIS 
+  38       0.400    500      2     20     2.0    1. 0    2.00     2.00      110      2220   0.30000V        -6.90334175    UNIFORM P-AXIS 
+  39       0.500    500      2     20     2.0    1. 0    2.00     2.00       77      1560   0.55000V        -1.99362004    UNIFORM P-AXIS 
+  40       0.600    500      2     20     2.0    1. 0    2.00     2.00       94      1900   0.45000V        -4.95052579    UNIFORM P-AXIS 
+  41       0.700    500      2     20     2.0    1. 0    2.00     2.00      122      2460   0.90000V        -1.00012042    UNIFORM P-AXIS 
+  42       0.800    500      2     20     2.0    1. 0    2.00     2.00       97      1960   0.60000V       -12.67050646    UNIFORM P-AXIS 
+  43       0.900    500      2     20     2.0    1. 0    2.00     2.00       95      1920   0.50000V        -2.98355029    UNIFORM P-AXIS 
+  44       1.000    500      2     20     2.0    1. 0    2.00     2.00      116      2340   0.60000V        -6.90750161    UNIFORM P-AXIS 
+  45       0.000    500      2     24     2.0    1. 0    2.00     2.00      103      2496   0.90000V       -13.62018424    UNIFORM P-AXIS 
+  46       0.100    500      2     24     2.0    1. 0    2.00     2.00      469     11280   0.20000V        -1.01402257    UNIFORM P-AXIS 
+  47       0.200    500      2     24     2.0    1. 0    2.00     2.00      247      5952   0.50000V        -0.99800389    UNIFORM P-AXIS 
+  48       0.300    500      2     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V       -12.67050583    UNIFORM P-AXIS 27 of 92   49       0.400    500      2     24     2.0    1. 0    2.00     2.00      114      2760   0.50000V        -1.99259236    UNIFORM P-AXIS 
+  50       0.500    500      2     24     2.0    1. 0    2.00     2.00      135      3264   0.60000V        -2.98494679    UNIFORM P-AXIS 
+  51       0.600    500      2     24     2.0    1. 0    2.00     2.00      112      2712   0.40000V        -2.98305135    UNIFORM P-AXIS 
+  52       0.700    500      2     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V       -12.67050581    UNIFORM P-AXIS 
+  53       0.800    500      2     24     2.0    1. 0    2.00     2.00       78      1896   0.60000V        -8.84121987    UNIFORM P-AXIS 
+  54       0.900    500      2     24     2.0    1. 0    2.00     2.00      129      3120   0.30000V        -9.80393440    UNIFORM P-AXIS 
+  55       1.000    500      2     24     2.0    1. 0    2.00     2.00       75      1824   0.45000V        -6.90339260    UNIFORM P-AXIS 
+  56       0.000    500      2     28     2.0    1. 0    2.00     2.00      105      2968   0.05000V       -12.72309266    UNIFORM P-AXIS 
+  57       0.100    500      2     28     2.0    1. 0    2.00     2.00      291      8176   0.80000V        -2.05112505    UNIFORM P-AXIS 
+  58       0.200    500      2     28     2.0    1. 0    2.00     2.00       77      2184   0.55000V        -0.99800500    UNIFORM P-AXIS 
+  59       0.300    500      2     28     2.0    1. 0    2.00     2.00       78      2212   0.60000V       -12.68239387    UNIFORM P-AXIS 
+  60       0.400    500      2     28     2.0    1. 0    2.00     2.00       77      2184   0.55000V        -5.92884534    UNIFORM P-AXIS 
+  61       0.500    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V        -7.87399305    UNIFORM P-AXIS 
+  62       0.600    500      2     28     2.0    1. 0    2.00     2.00       79      2240   0.65000V        -7.87399303    UNIFORM P-AXIS 
+  63       0.700    500      2     28     2.0    1. 0    2.00     2.00      123      3472   0.95000V        -7.89231597    UNIFORM P-AXIS 
+  64       0.800    500      2     28     2.0    1. 0    2.00     2.00       92      2604   0.35000V        -1.00752170    UNIFORM P-AXIS 
+  65       0.900    500      2     28     2.0    1. 0    2.00     2.00      110      3108   0.30000V        -8.84100679    UNIFORM P-AXIS 
+  66       1.000    500      2     28     2.0    1. 0    2.00     2.00      111      3136   0.35000V        -0.99828689    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      141076 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  47       0.200    500      2     24     2.0    1. 0    2.00     2.00      247      5952   0.50000V        -0.99800389    UNIFORM P-AXIS 
+ 
+F15 
+Run ID: 12-26-2009, 23:44:46 
+ 
+FUNCTION: F15 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500      4     16     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V        -0.00969087    UNIFORM P-AXIS 
+   2       0.100    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V        -0.04147696    UNIFORM P-AXIS 
+   3       0.200    500      4     16     2.0    1. 0    2.00     2.00      293      4704   0.90000V        -0.01107829    UNIFORM P-AXIS 
+   4       0.300    500      4     16     2.0    1. 0    2.00     2.00      174      2800   0.65000V        -0.01340605    UNIFORM P-AXIS 
+   5       0.400    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V        -0.04063055    UNIFORM P-AXIS 
+   6       0.500    500      4     16     2.0    1. 0    2.00     2.00       98      1584   0.65000V        -0.00241631    UNIFORM P-AXIS 
+   7       0.600    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V        -0.12870474    UNIFORM P-AXIS 
+   8       0.700    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V        -0.11830207    UNIFORM P-AXIS 
+   9       0.800    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V        -0.06996965    UNIFORM P-AXIS 
+  10       0.900    500      4     16     2.0    1. 0    2.00     2.00      135      2176   0.60000V        -0.00326219    UNIFORM P-AXIS 
+  11       1.000    500      4     16     2.0    1. 0    2.00     2.00      174      2800   0.65000V        -0.01224564    UNIFORM P-AXIS 
+  12       0.000    500      4     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V        -0.01602030    UNIFORM P-AXIS 
+  13       0.100    500      4     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V        -0.00973842    UNIFORM P-AXIS 
+  14       0.200    500      4     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V        -0.01306855    UNIFORM P-AXIS 
+  15       0.300    500      4     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V        -0.00676815    UNIFORM P-AXIS 
+  16       0.400    500      4     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V        -0.04798806    UNIFORM P-AXIS 
+  17       0.500    500      4     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V        -0.02226235    UNIFORM P-AXIS 
+  18       0.600    500      4     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V        -0.09216877    UNIFORM P-AXIS 
+  19       0.700    500      4     24     2.0    1. 0    2.00     2.00      174      4200   0.65000V        -0.00277113    UNIFORM P-AXIS 
+  20       0.800    500      4     24     2.0    1. 0    2.00     2.00      256      6168   0.95000V        -0.04058785    UNIFORM P-AXIS 
+  21       0.900    500      4     24     2.0    1. 0    2.00     2.00      128      3096   0.25000V        -0.00937831    UNIFORM P-AXIS 
+  22       1.000    500      4     24     2.0    1. 0    2.00     2.00      193      4656   0.65000V        -0.01337984    UNIFORM P-AXIS 
+  23       0.000    500      4     32     2.0    1. 0    2.00     2.00       60      1952   0.65000V        -0.02197297    UNIFORM P-AXIS 
+  24       0.100    500      4     32     2.0    1. 0    2.00     2.00       60      1952   0.65000V        -0.00559270    UNIFORM P-AXIS 
+  25       0.200    500      4     32     2.0    1. 0    2.00     2.00      311      9984   0.85000V        -0.00272084    UNIFORM P-AXIS 
+  26       0.300    500      4     32     2.0    1. 0    2.00     2.00      350     11232   0.90000V        -0.00190484    UNIFORM P-AXIS 
+  27       0.400    500      4     32     2.0    1. 0    2.00     2.00       60      1952   0.65000V        -0.02345614    UNIFORM P-AXIS 
+  28       0.500    500      4     32     2.0    1. 0    2.00     2.00       92      2976   0.35000V        -0.00265986    UNIFORM P-AXIS 
+  29       0.600    500      4     32     2.0    1. 0    2.00     2.00       98      3168   0.65000V        -0.01874733    UNIFORM P-AXIS 
+  30       0.700    500      4     32     2.0    1. 0    2.00     2.00      231      7424   0.65000V        -0.01974054    UNIFORM P-AXIS 
+  31       0.800    500      4     32     2.0    1. 0    2.00     2.00      249      8000   0.60000V        -0.03410112    UNIFORM P-AXIS 
+  32       0.900    500      4     32     2.0    1. 0    2.00     2.00       60      1952   0.65000V        -0.09252196    UNIFORM P-AXIS 
+  33       1.000    500      4     32     2.0    1. 0    2.00     2.00      271      8704   0.75000V        -0.01358199    UNIFORM P-AXIS 
+  34       0.000    500      4     40     2.0    1. 0    2.00     2.00       60      2440   0.65000V        -0.00760705    UNIFORM P-AXIS 
+  35       0.100    500      4     40     2.0    1. 0    2.00     2.00      285     11440   0.50000V        -0.00496634    UNIFORM P-AXIS 
+  36       0.200    500      4     40     2.0    1. 0    2.00     2.00       60      2440   0.65000V        -0.00822103    UNIFORM P-AXIS 
+  37       0.300    500      4     40     2.0    1. 0    2.00     2.00      231      9280   0.65000V        -0.00222432    UNIFORM P-AXIS 
+  38       0.400    500      4     40     2.0    1. 0    2.00     2.00       60      2440   0.65000V        -0.05848026    UNIFORM P-AXIS 
+  39       0.500    500      4     40     2.0    1. 0    2.00     2.00      117      4720   0.65000V        -0.00290803    UNIFORM P-AXIS 
+  40       0.600    500      4     40     2.0    1. 0    2.00     2.00       60      2440   0.65000V        -0.04223306    UNIFORM P-AXIS 
+  41       0.700    500      4     40     2.0    1. 0    2.00     2.00       60      2440   0.65000V        -0.03074483    UNIFORM P-AXIS 
+  42       0.800    500      4     40     2.0    1. 0    2.00     2.00      175      7040   0.70000V        -0.01799522    UNIFORM P-AXIS 
+  43       0.900    500      4     40     2.0    1. 0    2.00     2.00      270     10840   0.70000V        -0.00111803    UNIFORM P-AXIS 
+  44       1.000    500      4     40     2.0    1. 0    2.00     2.00      210      8440   0.55000V        -0.00191557    UNIFORM P-AXIS 
+  45       0.000    500      4     48     2.0    1. 0    2.00     2.00       69      3360   0.15000V        -0.00048890    UNIFORM P-AXIS 
+  46       0.100    500      4     48     2.0    1. 0    2.00     2.00      174      8400   0.65000V        -0.00194328    UNIFORM P-AXIS 
+  47       0.200    500      4     48     2.0    1. 0    2.00     2.00       70      3408   0.20000V        -0.02907792    UNIFORM P-AXIS 
+  48       0.300    500      4     48     2.0    1. 0    2.00     2.00      121      5856   0.85000V        -0.00352392    UNIFORM P-AXIS 
+  49       0.400    500      4     48     2.0    1. 0    2.00     2.00       60      2928   0.65000V        -0.05265765    UNIFORM P-AXIS 
+  50       0.500    500      4     48     2.0    1. 0    2.00     2.00       74      3600   0.40000V        -0.00340201    UNIFORM P-AXIS 
+  51       0.600    500      4     48     2.0    1. 0    2.00     2.00       60      2928   0.65000V        -0.03430604    UNIFORM P-AXIS 
+  52       0.700    500      4     48     2.0    1. 0    2.00     2.00       99      4800   0.70000V        -0.01095233    UNIFORM P-AXIS 
+  53       0.800    500      4     48     2.0    1. 0    2.00     2.00      117      5664   0.65000V        -0.02417579    UNIFORM P-AXIS 
+  54       0.900    500      4     48     2.0    1. 0    2.00     2.00      110      5328   0.30000V        -0.00077444    UNIFORM P-AXIS 
+  55       1.000    500      4     48     2.0    1. 0    2.00     2.00      231     11136   0.65000V        -0.01333495    UNIFORM P-AXIS 
+  56       0.000    500      4     56     2.0    1. 0    2.00     2.00       82      4648   0.80000V        -0.00177343    UNIFORM P-AXIS 
+  57       0.100    500      4     56     2.0    1. 0    2.00     2.00      136      7672   0.65000V        -0.00246125    UNIFORM P-AXIS 
+  58       0.200    500      4     56     2.0    1. 0    2.00     2.00       60      3416   0.65000V        -0.01356890    UNIFORM P-AXIS 
+  59       0.300    500      4     56     2.0    1. 0    2.00     2.00       60      3416   0.65000V        -0.03136321    UNIFORM P-AXIS 
+  60       0.400    500      4     56     2.0    1. 0    2.00     2.00      175      9856   0.70000V        -0.00734608    UNIFORM P-AXIS 
+  61       0.500    500      4     56     2.0    1. 0    2.00     2.00       78      4424   0.60000V        -0.00702747    UNIFORM P-AXIS 
+  62       0.600    500      4     56     2.0    1. 0    2.00     2.00      279     15680   0.20000V        -0.00472297    UNIFORM P-AXIS 
+  63       0.700    500      4     56     2.0    1. 0    2.00     2.00      156      8792   0.70000V        -0.01750298    UNIFORM P-AXIS 
+  64       0.800    500      4     56     2.0    1. 0    2.00     2.00       99      5600   0.70000V        -0.01215245    UNIFORM P-AXIS 
+  65       0.900    500      4     56     2.0    1. 0    2.00     2.00       98      5544   0.65000V        -0.00050140    UNIFORM P-AXIS 
+  66       1.000    500      4     56     2.0    1. 0    2.00     2.00      118      6664   0.70000V        -0.00970616    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      304664 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  45       0.000    500      4     48     2.0    1. 0    2.00     2.00       69      3360   0.15000V        -0.00048890    UNIFORM P-AXIS 
+ 28 of 92 F16 
+Run ID: 12-26-2009, 23:48:35 
+ 
+FUNCTION: F16 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500      2      8     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500      2      8     2.0    1. 0    2.00     2.00       78       632   0.60000V         1.02587776    UNIFORM P-AXIS 
+   2       0.100    500      2      8     2.0    1. 0    2.00     2.00      111       896   0.35000V         1.03160668    UNIFORM P-AXIS 
+   3       0.200    500      2      8     2.0    1. 0    2.00     2.00       60       488   0.65000V         0.99305889    UNIFORM P-AXIS 
+   4       0.300    500      2      8     2.0    1. 0    2.00     2.00      137      1104   0.70000V         1.03148086    UNIFORM P-AXIS 
+   5       0.400    500      2      8     2.0    1. 0    2.00     2.00      115       928   0.55000V         1.03125790    UNIFORM P-AXIS 
+   6       0.500    500      2      8     2.0    1. 0    2.00     2.00      144      1160   0.10000V         1.03070958    UNIFORM P-AXIS 
+   7       0.600    500      2      8     2.0    1. 0    2.00     2.00      115       928   0.55000V         1.03125790    UNIFORM P-AXIS 
+   8       0.700    500      2      8     2.0    1. 0    2.00     2.00      137      1104   0.70000V         1.03148086    UNIFORM P-AXIS 
+   9       0.800    500      2      8     2.0    1. 0    2.00     2.00       60       488   0.65000V         0.99305889    UNIFORM P-AXIS 
+  10       0.900    500      2      8     2.0    1. 0    2.00     2.00      111       896   0.35000V         1.03160668    UNIFORM P-AXIS 
+  11       1.000    500      2      8     2.0    1. 0    2.00     2.00       78       632   0.60000V         1.02587776    UNIFORM P-AXIS 
+  12       0.000    500      2     12     2.0    1. 0    2.00     2.00       96      1164   0.55000V         1.03013810    UNIFORM P-AXIS 
+  13       0.100    500      2     12     2.0    1. 0    2.00     2.00       73       888   0.35000V         1.02492533    UNIFORM P-AXIS 
+  14       0.200    500      2     12     2.0    1. 0    2.00     2.00       77       936   0.55000V         1.03161663    UNIFORM P-AXIS 
+  15       0.300    500      2     12     2.0    1. 0    2.00     2.00      111      1344   0.35000V         1.02941460    UNIFORM P-AXIS 
+  16       0.400    500      2     12     2.0    1. 0    2.00     2.00      114      1380   0.50000V         1.03130657    UNIFORM P-AXIS 
+  17       0.500    500      2     12     2.0    1. 0    2.00     2.00       60       732   0.65000V         0.99998652    UNIFORM P-AXIS 
+  18       0.600    500      2     12     2.0    1. 0    2.00     2.00      114      1380   0.50000V         1.03130657    UNIFORM P-AXIS 
+  19       0.700    500      2     12     2.0    1. 0    2.00     2.00      111      1344   0.35000V         1.02941460    UNIFORM P-AXIS 
+  20       0.800    500      2     12     2.0    1. 0    2.00     2.00       60       732   0.65000V         1.03031188    UNIFORM P-AXIS 
+  21       0.900    500      2     12     2.0    1. 0    2.00     2.00       73       888   0.35000V         1.02492533    UNIFORM P-AXIS 
+  22       1.000    500      2     12     2.0    1. 0    2.00     2.00       96      1164   0.55000V         1.03013810    UNIFORM P-AXIS 
+  23       0.000    500      2     16     2.0    1. 0    2.00     2.00      148      2384   0.30000V         1.03018603    UNIFORM P-AXIS 
+  24       0.100    500      2     16     2.0    1. 0    2.00     2.00      131      2112   0.40000V         1.02973102    UNIFORM P-AXIS 
+  25       0.200    500      2     16     2.0    1. 0    2.00     2.00      130      2096   0.35000V         1.02998274    UNIFORM P-AXIS 
+  26       0.300    500      2     16     2.0    1. 0    2.00     2.00      122      1968   0.90000V         1.02837012    UNIFORM P-AXIS 
+  27       0.400    500      2     16     2.0    1. 0    2.00     2.00       97      1568   0.60000V         1.01423368    UNIFORM P-AXIS 
+  28       0.500    500      2     16     2.0    1. 0    2.00     2.00      132      2128   0.45000V         1.02874055    UNIFORM P-AXIS 
+  29       0.600    500      2     16     2.0    1. 0    2.00     2.00       97      1568   0.60000V         1.01423368    UNIFORM P-AXIS 
+  30       0.700    500      2     16     2.0    1. 0    2.00     2.00      122      1968   0.90000V         1.02837016    UNIFORM P-AXIS 
+  31       0.800    500      2     16     2.0    1. 0    2.00     2.00      130      2096   0.35000V         1.02998274    UNIFORM P-AXIS 
+  32       0.900    500      2     16     2.0    1. 0    2.00     2.00      131      2112   0.40000V         1.02973102    UNIFORM P-AXIS 
+  33       1.000    500      2     16     2.0    1. 0    2.00     2.00       96      1552   0.55000V         1.02313447    UNIFORM P-AXIS 
+  34       0.000    500      2     20     2.0    1. 0    2.00     2.00      111      2240   0.35000V         1.01250976    UNIFORM P-AXIS 
+  35       0.100    500      2     20     2.0    1. 0    2.00     2.00       60      1220   0.65000V         1.02425714    UNIFORM P-AXIS 
+  36       0.200    500      2     20     2.0    1. 0    2.00     2.00      151      3040   0.45000V         1.03113113    UNIFORM P-AXIS 
+  37       0.300    500      2     20     2.0    1. 0    2.00     2.00       77      1560   0.55000V         1.02996462    UNIFORM P-AXIS 
+  38       0.400    500      2     20     2.0    1. 0    2.00     2.00       78      1580   0.60000V         1.03120923    UNIFORM P-AXIS 
+  39       0.500    500      2     20     2.0    1. 0    2.00     2.00      155      3120   0.65000V         1.02700912    UNIFORM P-AXIS 
+  40       0.600    500      2     20     2.0    1. 0    2.00     2.00       78      1580   0.60000V         1.03120923    UNIFORM P-AXIS 
+  41       0.700    500      2     20     2.0    1. 0    2.00     2.00       77      1560   0.55000V         1.02996462    UNIFORM P-AXIS 
+  42       0.800    500      2     20     2.0    1. 0    2.00     2.00      151      3040   0.45000V         1.03114606    UNIFORM P-AXIS 
+  43       0.900    500      2     20     2.0    1. 0    2.00     2.00       60      1220   0.65000V         1.02425714    UNIFORM P-AXIS 
+  44       1.000    500      2     20     2.0    1. 0    2.00     2.00       60      1220   0.65000V         1.02062108    UNIFORM P-AXIS 
+  45       0.000    500      2     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V         1.02305918    UNIFORM P-AXIS 
+  46       0.100    500      2     24     2.0    1. 0    2.00     2.00      113      2736   0.45000V         1.02572173    UNIFORM P-AXIS 
+  47       0.200    500      2     24     2.0    1. 0    2.00     2.00      130      3144   0.35000V         1.02370473    UNIFORM P-AXIS 
+  48       0.300    500      2     24     2.0    1. 0    2.00     2.00       96      2328   0.55000V         1.02984598    UNIFORM P-AXIS 
+  49       0.400    500      2     24     2.0    1. 0    2.00     2.00      261      6288   0.25000V         1.03162595    UNIFORM P-AXIS 
+  50       0.500    500      2     24     2.0    1. 0    2.00     2.00       97      2352   0.60000V         1.03047576    UNIFORM P-AXIS 
+  51       0.600    500      2     24     2.0    1. 0    2.00     2.00       95      2304   0.50000V         1.03079194    UNIFORM P-AXIS 
+  52       0.700    500      2     24     2.0    1. 0    2.00     2.00      115      2784   0.55000V         1.02989343    UNIFORM P-AXIS 
+  53       0.800    500      2     24     2.0    1. 0    2.00     2.00      129      3120   0.30000V         1.02930539    UNIFORM P-AXIS 
+  54       0.900    500      2     24     2.0    1. 0    2.00     2.00      303      7296   0.45000V         1.03149759    UNIFORM P-AXIS 
+  55       1.000    500      2     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V         1.01912590    UNIFORM P-AXIS 
+  56       0.000    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V         1.02614496    UNIFORM P-AXIS 
+  57       0.100    500      2     28     2.0    1. 0    2.00     2.00       74      2100   0.40000V         1.03161154    UNIFORM P-AXIS 
+  58       0.200    500      2     28     2.0    1. 0    2.00     2.00       80      2268   0.70000V         1.03032871    UNIFORM P-AXIS 
+  59       0.300    500      2     28     2.0    1. 0    2.00     2.00      126      3556   0.15000V         1.03155024    UNIFORM P-AXIS 
+  60       0.400    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V         1.02949203    UNIFORM P-AXIS 
+  61       0.500    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V         0.99911920    UNIFORM P-AXIS 
+  62       0.600    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V         1.02949203    UNIFORM P-AXIS 
+  63       0.700    500      2     28     2.0    1. 0    2.00     2.00      126      3556   0.15000V         1.03155024    UNIFORM P-AXIS 
+  64       0.800    500      2     28     2.0    1. 0    2.00     2.00       80      2268   0.70000V         1.03032871    UNIFORM P-AXIS 
+  65       0.900    500      2     28     2.0    1. 0    2.00     2.00       74      2100   0.40000V         1.03161154    UNIFORM P-AXIS 
+  66       1.000    500      2     28     2.0    1. 0    2.00     2.00       79      2240   0.65000V         1.02808103    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      124340 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  49       0.400    500      2     24     2.0    1. 0    2.00     2.00      261      6288   0.25000V         1.03162595    UNIFORM P-AXIS 
+ 
+F17 
+Run ID: 12-26-2009, 23:52:43 
+ 
+FUNCTION: F17 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500      2      8     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500      2      8     2.0    1. 0    2.00     2.00      100       808   0.75000V        -0.46439597    UNIFORM P-AXIS 
+   2       0.100    500      2      8     2.0    1. 0    2.00     2.00       82       664   0.80000V        -0.39953210    UNIFORM P-AXIS 
+   3       0.200    500      2      8     2.0    1. 0    2.00     2.00       96       776   0.55000V        -0.42672486    UNIFORM P-AXIS 
+   4       0.300    500      2      8     2.0    1. 0    2.00     2.00       82       664   0.80000V        -0.46602832    UNIFORM P-AXIS 
+   5       0.400    500      2      8     2.0    1. 0    2.00     2.00      270      2168   0.70000V        -0.42249455    UNIFORM P-AXIS 
+   6       0.500    500      2      8     2.0    1. 0    2.00     2.00       72       584   0.30000V        -0.41429073    UNIFORM P-AXIS 
+   7       0.600    500      2      8     2.0    1. 0    2.00     2.00       76       616   0.50000V        -0.39795354    UNIFORM P-AXIS 
+   8       0.700    500      2      8     2.0    1. 0    2.00     2.00       60       488   0.65000V        -0.44406252    UNIFORM P-AXIS 
+   9       0.800    500      2      8     2.0    1. 0    2.00     2.00       89       720   0.20000V        -0.41551430    UNIFORM P-AXIS 
+  10       0.900    500      2      8     2.0    1. 0    2.00     2.00       95       768   0.50000V        -0.44055628    UNIFORM P-AXIS 
+  11       1.000    500      2      8     2.0    1. 0    2.00     2.00       70       568   0.20000V        -0.42345094    UNIFORM P-AXIS 
+  12       0.000    500      2     12     2.0    1. 0    2.00     2.00       92      1116   0.35000V        -0.41691841    UNIFORM P-AXIS 
+  13       0.100    500      2     12     2.0    1. 0    2.00     2.00       99      1200   0.70000V        -0.50113512    UNIFORM P-AXIS 
+  14       0.200    500      2     12     2.0    1. 0    2.00     2.00      103      1248   0.90000V        -0.40440758    UNIFORM P-AXIS 
+  15       0.300    500      2     12     2.0    1. 0    2.00     2.00       60       732   0.65000V        -0.40024733    UNIFORM P-AXIS 
+  16       0.400    500      2     12     2.0    1. 0    2.00     2.00       60       732   0.65000V        -0.44377180    UNIFORM P-AXIS 29 of 92   17       0.500    500      2     12     2.0    1. 0    2.00     2.00      114      1380   0.50000V        -0.45485690    UNIFORM P-AXIS 
+  18       0.600    500      2     12     2.0    1. 0    2.00     2.00       60       732   0.65000V        -0.42645784    UNIFORM P-AXIS 
+  19       0.700    500      2     12     2.0    1. 0    2.00     2.00       72       876   0.30000V        -0.39934165    UNIFORM P-AXIS 
+  20       0.800    500      2     12     2.0    1. 0    2.00     2.00       60       732   0.65000V        -0.46538381    UNIFORM P-AXIS 
+  21       0.900    500      2     12     2.0    1. 0    2.00     2.00       82       996   0.80000V        -0.41912809    UNIFORM P-AXIS 
+  22       1.000    500      2     12     2.0    1. 0    2.00     2.00       95      1152   0.50000V        -0.40267002    UNIFORM P-AXIS 
+  23       0.000    500      2     16     2.0    1. 0    2.00     2.00      116      1872   0.60000V        -0.39790238    UNIFORM P-AXIS 
+  24       0.100    500      2     16     2.0    1. 0    2.00     2.00      137      2208   0.70000V        -0.39963599    UNIFORM P-AXIS 
+  25       0.200    500      2     16     2.0    1. 0    2.00     2.00       77      1248   0.55000V        -0.40617340    UNIFORM P-AXIS 
+  26       0.300    500      2     16     2.0    1. 0    2.00     2.00       85      1376   0.95000V        -0.39852121    UNIFORM P-AXIS 
+  27       0.400    500      2     16     2.0    1. 0    2.00     2.00      112      1808   0.40000V        -0.39840760    UNIFORM P-AXIS 
+  28       0.500    500      2     16     2.0    1. 0    2.00     2.00       79      1280   0.65000V        -0.40838689    UNIFORM P-AXIS 
+  29       0.600    500      2     16     2.0    1. 0    2.00     2.00      133      2144   0.50000V        -0.41414181    UNIFORM P-AXIS 
+  30       0.700    500      2     16     2.0    1. 0    2.00     2.00       76      1232   0.50000V        -0.39932652    UNIFORM P-AXIS 
+  31       0.800    500      2     16     2.0    1. 0    2.00     2.00       60       976   0.65000V        -0.42335713    UNIFORM P-AXIS 
+  32       0.900    500      2     16     2.0    1. 0    2.00     2.00       60       976   0.65000V        -0.40134318    UNIFORM P-AXIS 
+  33       1.000    500      2     16     2.0    1. 0    2.00     2.00      309      4960   0.75000V        -0.39989906    UNIFORM P-AXIS 
+  34       0.000    500      2     20     2.0    1. 0    2.00     2.00       78      1580   0.60000V        -0.40090871    UNIFORM P-AXIS 
+  35       0.100    500      2     20     2.0    1. 0    2.00     2.00       60      1220   0.65000V        -0.40240145    UNIFORM P-AXIS 
+  36       0.200    500      2     20     2.0    1. 0    2.00     2.00       76      1540   0.50000V        -0.40601199    UNIFORM P-AXIS 
+  37       0.300    500      2     20     2.0    1. 0    2.00     2.00       95      1920   0.50000V        -0.40997353    UNIFORM P-AXIS 
+  38       0.400    500      2     20     2.0    1. 0    2.00     2.00       76      1540   0.50000V        -0.40543885    UNIFORM P-AXIS 
+  39       0.500    500      2     20     2.0    1. 0    2.00     2.00       76      1540   0.50000V        -0.40212945    UNIFORM P-AXIS 
+  40       0.600    500      2     20     2.0    1. 0    2.00     2.00       80      1620   0.70000V        -0.39803955    UNIFORM P-AXIS 
+  41       0.700    500      2     20     2.0    1. 0    2.00     2.00       85      1720   0.95000V        -0.47106855    UNIFORM P-AXIS 
+  42       0.800    500      2     20     2.0    1. 0    2.00     2.00       60      1220   0.65000V        -0.40920220    UNIFORM P-AXIS 
+  43       0.900    500      2     20     2.0    1. 0    2.00     2.00       74      1500   0.40000V        -0.40471949    UNIFORM P-AXIS 
+  44       1.000    500      2     20     2.0    1. 0    2.00     2.00       74      1500   0.40000V        -0.40068500    UNIFORM P-AXIS 
+  45       0.000    500      2     24     2.0    1. 0    2.00     2.00       93      2256   0.40000V        -0.39872669    UNIFORM P-AXIS 
+  46       0.100    500      2     24     2.0    1. 0    2.00     2.00       75      1824   0.45000V        -0.39898701    UNIFORM P-AXIS 
+  47       0.200    500      2     24     2.0    1. 0    2.00     2.00       78      1896   0.60000V        -0.40251366    UNIFORM P-AXIS 
+  48       0.300    500      2     24     2.0    1. 0    2.00     2.00      114      2760   0.50000V        -0.41446304    UNIFORM P-AXIS 
+  49       0.400    500      2     24     2.0    1. 0    2.00     2.00       97      2352   0.60000V        -0.40190883    UNIFORM P-AXIS 
+  50       0.500    500      2     24     2.0    1. 0    2.00     2.00       79      1920   0.65000V        -0.40045620    UNIFORM P-AXIS 
+  51       0.600    500      2     24     2.0    1. 0    2.00     2.00      135      3264   0.60000V        -0.42255103    UNIFORM P-AXIS 
+  52       0.700    500      2     24     2.0    1. 0    2.00     2.00       74      1800   0.40000V        -0.41716267    UNIFORM P-AXIS 
+  53       0.800    500      2     24     2.0    1. 0    2.00     2.00       92      2232   0.35000V        -0.40648412    UNIFORM P-AXIS 
+  54       0.900    500      2     24     2.0    1. 0    2.00     2.00       77      1872   0.55000V        -0.40305464    UNIFORM P-AXIS 
+  55       1.000    500      2     24     2.0    1. 0    2.00     2.00       74      1800   0.40000V        -0.39811469    UNIFORM P-AXIS 
+  56       0.000    500      2     28     2.0    1. 0    2.00     2.00      111      3136   0.35000V        -0.40172094    UNIFORM P-AXIS 
+  57       0.100    500      2     28     2.0    1. 0    2.00     2.00       90      2548   0.25000V        -0.40193575    UNIFORM P-AXIS 
+  58       0.200    500      2     28     2.0    1. 0    2.00     2.00      101      2856   0.80000V        -0.39848716    UNIFORM P-AXIS 
+  59       0.300    500      2     28     2.0    1. 0    2.00     2.00       73      2072   0.35000V        -0.41375115    UNIFORM P-AXIS 
+  60       0.400    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V        -0.40399916    UNIFORM P-AXIS 
+  61       0.500    500      2     28     2.0    1. 0    2.00     2.00       96      2716   0.55000V        -0.40467652    UNIFORM P-AXIS 
+  62       0.600    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V        -0.40740570    UNIFORM P-AXIS 
+  63       0.700    500      2     28     2.0    1. 0    2.00     2.00      110      3108   0.30000V        -0.39803164    UNIFORM P-AXIS 
+  64       0.800    500      2     28     2.0    1. 0    2.00     2.00      106      2996   0.10000V        -0.39793899    UNIFORM P-AXIS 
+  65       0.900    500      2     28     2.0    1. 0    2.00     2.00       77      2184   0.55000V        -0.39935405    UNIFORM P-AXIS 
+  66       1.000    500      2     28     2.0    1. 0    2.00     2.00       93      2632   0.40000V        -0.39935800    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      108340 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  23       0.000    500      2     16     2.0    1. 0    2.00     2.00      116      1872   0.60000V        -0.39790238    UNIFORM P-AXIS 
+ 
+F18 
+Run ID: 12-26-2009, 22:19:28 
+ 
+FUNCTION: F18 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500      2      8     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500      2      8     2.0    1. 0    2.00     2.00       78       632   0.60000V        -5.48169471    UNIFORM P-AXIS 
+   2       0.100    500      2      8     2.0    1. 0    2.00     2.00      134      1080   0.55000V       -84.78003234    UNIFORM P-AXIS 
+   3       0.200    500      2      8     2.0    1. 0    2.00     2.00      191      1536   0.55000V        -3.19822531    UNIFORM P-AXIS 
+   4       0.300    500      2      8     2.0    1. 0    2.00     2.00       79       640   0.65000V        -8.62285433    UNIFORM P-AXIS 
+   5       0.400    500      2      8     2.0    1. 0    2.00     2.00      403      3232   0.70000V       -84.47139703    UNIFORM P-AXIS 
+   6       0.500    500      2      8     2.0    1. 0    2.00     2.00      261      2096   0.25000V       -10.56054743    UNIFORM P-AXIS 
+   7       0.600    500      2      8     2.0    1. 0    2.00     2.00      135      1088   0.60000V        -4.98190912    UNIFORM P-AXIS 
+   8       0.700    500      2      8     2.0    1. 0    2.00     2.00      229      1840   0.55000V        -3.00130536    UNIFORM P-AXIS 
+   9       0.800    500      2      8     2.0    1. 0    2.00     2.00      133      1072   0.50000V       -89.42718008    UNIFORM P-AXIS 
+  10       0.900    500      2      8     2.0    1. 0    2.00     2.00      116       936   0.60000V        -8.57642421    UNIFORM P-AXIS 
+  11       1.000    500      2      8     2.0    1. 0    2.00     2.00      193      1552   0.65000V        -3.08573605    UNIFORM P-AXIS 
+  12       0.000    500      2     12     2.0    1. 0    2.00     2.00       78       948   0.60000V      -236.66437724    UNIFORM P-AXIS 
+  13       0.100    500      2     12     2.0    1. 0    2.00     2.00      263      3168   0.35000V        -3.00574349    UNIFORM P-AXIS 
+  14       0.200    500      2     12     2.0    1. 0    2.00     2.00       78       948   0.60000V       -57.61728445    UNIFORM P-AXIS 
+  15       0.300    500      2     12     2.0    1. 0    2.00     2.00      304      3660   0.50000V        -3.00017384    UNIFORM P-AXIS 
+  16       0.400    500      2     12     2.0    1. 0    2.00     2.00      154      1860   0.60000V        -3.55376552    UNIFORM P-AXIS 
+  17       0.500    500      2     12     2.0    1. 0    2.00     2.00      196      2364   0.80000V        -3.00014046    UNIFORM P-AXIS 
+  18       0.600    500      2     12     2.0    1. 0    2.00     2.00      154      1860   0.60000V        -6.35080630    UNIFORM P-AXIS 
+  19       0.700    500      2     12     2.0    1. 0    2.00     2.00      212      2556   0.65000V        -3.03478155    UNIFORM P-AXIS 
+  20       0.800    500      2     12     2.0    1. 0    2.00     2.00       78       948   0.60000V       -16.66444362    UNIFORM P-AXIS 
+  21       0.900    500      2     12     2.0    1. 0    2.00     2.00       99      1200   0.70000V        -6.52063755    UNIFORM P-AXIS 
+  22       1.000    500      2     12     2.0    1. 0    2.00     2.00      250      3012   0.65000V        -3.01375068    UNIFORM P-AXIS 
+  23       0.000    500      2     16     2.0    1. 0    2.00     2.00       78      1264   0.60000V       -22.18779159    UNIFORM P-AXIS 
+  24       0.100    500      2     16     2.0    1. 0    2.00     2.00       60       976   0.65000V       -16.89097168    UNIFORM P-AXIS 
+  25       0.200    500      2     16     2.0    1. 0    2.00     2.00      173      2784   0.60000V        -3.01613943    UNIFORM P-AXIS 
+  26       0.300    500      2     16     2.0    1. 0    2.00     2.00       79      1280   0.65000V       -76.69765275    UNIFORM P-AXIS 
+  27       0.400    500      2     16     2.0    1. 0    2.00     2.00      189      3040   0.45000V        -3.00972155    UNIFORM P-AXIS 
+  28       0.500    500      2     16     2.0    1. 0    2.00     2.00      191      3072   0.55000V        -6.65915188    UNIFORM P-AXIS 
+  29       0.600    500      2     16     2.0    1. 0    2.00     2.00      122      1968   0.90000V       -15.75050525    UNIFORM P-AXIS 
+  30       0.700    500      2     16     2.0    1. 0    2.00     2.00       60       976   0.65000V       -31.95935579    UNIFORM P-AXIS 
+  31       0.800    500      2     16     2.0    1. 0    2.00     2.00       95      1536   0.50000V       -49.88324658    UNIFORM P-AXIS 
+  32       0.900    500      2     16     2.0    1. 0    2.00     2.00      304      4880   0.50000V        -3.00045306    UNIFORM P-AXIS 
+  33       1.000    500      2     16     2.0    1. 0    2.00     2.00      318      5104   0.25000V        -3.00010350    UNIFORM P-AXIS 
+  34       0.000    500      2     20     2.0    1. 0    2.00     2.00       78      1580   0.60000V       -76.94725945    UNIFORM P-AXIS 
+  35       0.100    500      2     20     2.0    1. 0    2.00     2.00      116      2340   0.60000V        -3.12194540    UNIFORM P-AXIS 
+  36       0.200    500      2     20     2.0    1. 0    2.00     2.00      191      3840   0.55000V        -3.03175136    UNIFORM P-AXIS 
+  37       0.300    500      2     20     2.0    1. 0    2.00     2.00       78      1580   0.60000V      -182.65850920    UNIFORM P-AXIS 
+  38       0.400    500      2     20     2.0    1. 0    2.00     2.00      193      3880   0.65000V        -3.14860260    UNIFORM P-AXIS 
+  39       0.500    500      2     20     2.0    1. 0    2.00     2.00       98      1980   0.65000V        -3.62879802    UNIFORM P-AXIS 
+  40       0.600    500      2     20     2.0    1. 0    2.00     2.00      192      3860   0.60000V        -3.04067143    UNIFORM P-AXIS 
+  41       0.700    500      2     20     2.0    1. 0    2.00     2.00      139      2800   0.80000V        -3.54277590    UNIFORM P-AXIS 
+  42       0.800    500      2     20     2.0    1. 0    2.00     2.00      171      3440   0.50000V        -3.00072472    UNIFORM P-AXIS 
+  43       0.900    500      2     20     2.0    1. 0    2.00     2.00      101      2040   0.80000V        -4.37891524    UNIFORM P-AXIS 
+  44       1.000    500      2     20     2.0    1. 0    2.00     2.00       79      1600   0.65000V       -35.88632429    UNIFORM P-AXIS 30 of 92   45       0.000    500      2     24     2.0    1. 0    2.00     2.00      226      5448   0.40000V        -3.00031923    UNIFORM P-AXIS 
+  46       0.100    500      2     24     2.0    1. 0    2.00     2.00       78      1896   0.60000V        -9.06529259    UNIFORM P-AXIS 
+  47       0.200    500      2     24     2.0    1. 0    2.00     2.00      209      5040   0.50000V        -3.03088087    UNIFORM P-AXIS 
+  48       0.300    500      2     24     2.0    1. 0    2.00     2.00       79      1920   0.65000V       -30.56252660    UNIFORM P-AXIS 
+  49       0.400    500      2     24     2.0    1. 0    2.00     2.00      306      7368   0.60000V        -3.00075380    UNIFORM P-AXIS 
+  50       0.500    500      2     24     2.0    1. 0    2.00     2.00      248      5976   0.55000V        -3.00968878    UNIFORM P-AXIS 
+  51       0.600    500      2     24     2.0    1. 0    2.00     2.00       97      2352   0.60000V        -3.34890591    UNIFORM P-AXIS 
+  52       0.700    500      2     24     2.0    1. 0    2.00     2.00      246      5928   0.45000V        -3.01530923    UNIFORM P-AXIS 
+  53       0.800    500      2     24     2.0    1. 0    2.00     2.00       78      1896   0.60000V       -31.34561763    UNIFORM P-AXIS 
+  54       0.900    500      2     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V        -3.00000000    UNIFORM P-AXIS 
+  55       1.000    500      2     24     2.0    1. 0    2.00     2.00      101      2448   0.80000V       -15.41042957    UNIFORM P-AXIS 
+  56       0.000    500      2     28     2.0    1. 0    2.00     2.00      266      7476   0.50000V        -3.00178783    UNIFORM P-AXIS 
+  57       0.100    500      2     28     2.0    1. 0    2.00     2.00      257      7224   0.05000V        -3.02198109    UNIFORM P-AXIS 
+  58       0.200    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V      -102.95525789    UNIFORM P-AXIS 
+  59       0.300    500      2     28     2.0    1. 0    2.00     2.00      154      4340   0.60000V        -3.40567728    UNIFORM P-AXIS 
+  60       0.400    500      2     28     2.0    1. 0    2.00     2.00       79      2240   0.65000V       -40.28712313    UNIFORM P-AXIS 
+  61       0.500    500      2     28     2.0    1. 0    2.00     2.00      210      5908   0.55000V        -3.00034670    UNIFORM P-AXIS 
+  62       0.600    500      2     28     2.0    1. 0    2.00     2.00       79      2240   0.65000V       -32.25789818    UNIFORM P-AXIS 
+  63       0.700    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V       -36.68532659    UNIFORM P-AXIS 
+  64       0.800    500      2     28     2.0    1. 0    2.00     2.00       60      1708   0.65000V       -14.65270155    UNIFORM P-AXIS 
+  65       0.900    500      2     28     2.0    1. 0    2.00     2.00       78      2212   0.60000V       -31.47896038    UNIFORM P-AXIS 
+  66       1.000    500      2     28     2.0    1. 0    2.00     2.00      282      7924   0.35000V        -3.00151179    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      180472 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  54       0.900    500      2     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V        -3.00000000    UNIFORM P-AXIS 
+ 
+F19 
+Run ID: 12-26-2009, 23:57:27 
+ 
+FUNCTION: F19 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500      3     12     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500      3     12     2.0    1. 0    2.00     2.00       92      1116   0.35000V         3.81087000    UNIFORM P-AXIS 
+   2       0.100    500      3     12     2.0    1. 0    2.00     2.00      156      1884   0.70000V         3.80252096    UNIFORM P-AXIS 
+   3       0.200    500      3     12     2.0    1. 0    2.00     2.00       92      1116   0.35000V         3.85625317    UNIFORM P-AXIS 
+   4       0.300    500      3     12     2.0    1. 0    2.00     2.00       90      1092   0.25000V         3.82032439    UNIFORM P-AXIS 
+   5       0.400    500      3     12     2.0    1. 0    2.00     2.00       76       924   0.50000V         3.86157246    UNIFORM P-AXIS 
+   6       0.500    500      3     12     2.0    1. 0    2.00     2.00       77       936   0.55000V         3.84618140    UNIFORM P-AXIS 
+   7       0.600    500      3     12     2.0    1. 0    2.00     2.00       74       900   0.40000V         3.83838507    UNIFORM P-AXIS 
+   8       0.700    500      3     12     2.0    1. 0    2.00     2.00      112      1356   0.40000V         3.83554332    UNIFORM P-AXIS 
+   9       0.800    500      3     12     2.0    1. 0    2.00     2.00       60       732   0.65000V         3.84711642    UNIFORM P-AXIS 
+  10       0.900    500      3     12     2.0    1. 0    2.00     2.00      137      1656   0.70000V         3.85599159    UNIFORM P-AXIS 
+  11       1.000    500      3     12     2.0    1. 0    2.00     2.00       74       900   0.40000V         3.70557750    UNIFORM P-AXIS 
+  12       0.000    500      3     18     2.0    1. 0    2.00     2.00      149      2700   0.35000V         3.78695126    UNIFORM P-AXIS 
+  13       0.100    500      3     18     2.0    1. 0    2.00     2.00       91      1656   0.30000V         3.85424146    UNIFORM P-AXIS 
+  14       0.200    500      3     18     2.0    1. 0    2.00     2.00      249      4500   0.60000V         3.85002422    UNIFORM P-AXIS 
+  15       0.300    500      3     18     2.0    1. 0    2.00     2.00       93      1692   0.40000V         3.85668197    UNIFORM P-AXIS 
+  16       0.400    500      3     18     2.0    1. 0    2.00     2.00       90      1638   0.25000V         3.84336439    UNIFORM P-AXIS 
+  17       0.500    500      3     18     2.0    1. 0    2.00     2.00       77      1404   0.55000V         3.85075229    UNIFORM P-AXIS 
+  18       0.600    500      3     18     2.0    1. 0    2.00     2.00      356      6426   0.25000V         3.84476719    UNIFORM P-AXIS 
+  19       0.700    500      3     18     2.0    1. 0    2.00     2.00      114      2070   0.50000V         3.84820906    UNIFORM P-AXIS 
+  20       0.800    500      3     18     2.0    1. 0    2.00     2.00      114      2070   0.50000V         3.83793620    UNIFORM P-AXIS 
+  21       0.900    500      3     18     2.0    1. 0    2.00     2.00       93      1692   0.40000V         3.83173651    UNIFORM P-AXIS 
+  22       1.000    500      3     18     2.0    1. 0    2.00     2.00      116      2106   0.60000V         3.79007044    UNIFORM P-AXIS 
+  23       0.000    500      3     24     2.0    1. 0    2.00     2.00      103      2496   0.90000V         3.77571774    UNIFORM P-AXIS 
+  24       0.100    500      3     24     2.0    1. 0    2.00     2.00      193      4656   0.65000V         3.85998185    UNIFORM P-AXIS 
+  25       0.200    500      3     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V         3.85827634    UNIFORM P-AXIS 
+  26       0.300    500      3     24     2.0    1. 0    2.00     2.00      112      2712   0.40000V         3.85863108    UNIFORM P-AXIS 
+  27       0.400    500      3     24     2.0    1. 0    2.00     2.00       61      1488   0.70000V         3.80982656    UNIFORM P-AXIS 
+  28       0.500    500      3     24     2.0    1. 0    2.00     2.00      275      6624   0.95000V         3.86090728    UNIFORM P-AXIS 
+  29       0.600    500      3     24     2.0    1. 0    2.00     2.00       94      2280   0.45000V         3.81276278    UNIFORM P-AXIS 
+  30       0.700    500      3     24     2.0    1. 0    2.00     2.00       75      1824   0.45000V         3.84748287    UNIFORM P-AXIS 
+  31       0.800    500      3     24     2.0    1. 0    2.00     2.00       77      1872   0.55000V         3.84487382    UNIFORM P-AXIS 
+  32       0.900    500      3     24     2.0    1. 0    2.00     2.00      117      2832   0.65000V         3.85440945    UNIFORM P-AXIS 
+  33       1.000    500      3     24     2.0    1. 0    2.00     2.00      111      2688   0.35000V         3.85064729    UNIFORM P-AXIS 
+  34       0.000    500      3     30     2.0    1. 0    2.00     2.00       74      2250   0.40000V         3.83941990    UNIFORM P-AXIS 
+  35       0.100    500      3     30     2.0    1. 0    2.00     2.00      113      3420   0.45000V         3.84742951    UNIFORM P-AXIS 
+  36       0.200    500      3     30     2.0    1. 0    2.00     2.00      231      6960   0.65000V         3.82489621    UNIFORM P-AXIS 
+  37       0.300    500      3     30     2.0    1. 0    2.00     2.00       77      2340   0.55000V         3.85474673    UNIFORM P-AXIS 
+  38       0.400    500      3     30     2.0    1. 0    2.00     2.00       60      1830   0.65000V         3.86095502    UNIFORM P-AXIS 
+  39       0.500    500      3     30     2.0    1. 0    2.00     2.00      174      5250   0.65000V         3.84955030    UNIFORM P-AXIS 
+  40       0.600    500      3     30     2.0    1. 0    2.00     2.00       96      2910   0.55000V         3.83954812    UNIFORM P-AXIS 
+  41       0.700    500      3     30     2.0    1. 0    2.00     2.00       96      2910   0.55000V         3.83746687    UNIFORM P-AXIS 
+  42       0.800    500      3     30     2.0    1. 0    2.00     2.00      272      8190   0.80000V         3.83739135    UNIFORM P-AXIS 
+  43       0.900    500      3     30     2.0    1. 0    2.00     2.00       60      1830   0.65000V         3.85079537    UNIFORM P-AXIS 
+  44       1.000    500      3     30     2.0    1. 0    2.00     2.00      231      6960   0.65000V         3.79786366    UNIFORM P-AXIS 
+  45       0.000    500      3     36     2.0    1. 0    2.00     2.00       60      2196   0.65000V         3.84685849    UNIFORM P-AXIS 
+  46       0.100    500      3     36     2.0    1. 0    2.00     2.00       60      2196   0.65000V         3.85060158    UNIFORM P-AXIS 
+  47       0.200    500      3     36     2.0    1. 0    2.00     2.00      108      3924   0.20000V         3.84686078    UNIFORM P-AXIS 
+  48       0.300    500      3     36     2.0    1. 0    2.00     2.00       76      2772   0.50000V         3.84532644    UNIFORM P-AXIS 
+  49       0.400    500      3     36     2.0    1. 0    2.00     2.00       79      2880   0.65000V         3.85460096    UNIFORM P-AXIS 
+  50       0.500    500      3     36     2.0    1. 0    2.00     2.00      113      4104   0.45000V         3.84320651    UNIFORM P-AXIS 
+  51       0.600    500      3     36     2.0    1. 0    2.00     2.00       74      2700   0.40000V         3.84383985    UNIFORM P-AXIS 
+  52       0.700    500      3     36     2.0    1. 0    2.00     2.00      286     10332   0.55000V         3.84332469    UNIFORM P-AXIS 
+  53       0.800    500      3     36     2.0    1. 0    2.00     2.00       96      3492   0.55000V         3.85185414    UNIFORM P-AXIS 
+  54       0.900    500      3     36     2.0    1. 0    2.00     2.00       94      3420   0.45000V         3.84805588    UNIFORM P-AXIS 
+  55       1.000    500      3     36     2.0    1. 0    2.00     2.00      126      4572   0.15000V         3.80228240    UNIFORM P-AXIS 
+  56       0.000    500      3     42     2.0    1. 0    2.00     2.00       60      2562   0.65000V         3.83661732    UNIFORM P-AXIS 
+  57       0.100    500      3     42     2.0    1. 0    2.00     2.00       79      3360   0.65000V         3.81761193    UNIFORM P-AXIS 
+  58       0.200    500      3     42     2.0    1. 0    2.00     2.00       74      3150   0.40000V         3.86268376    UNIFORM P-AXIS 
+  59       0.300    500      3     42     2.0    1. 0    2.00     2.00      101      4284   0.80000V         3.85655395    UNIFORM P-AXIS 
+  60       0.400    500      3     42     2.0    1. 0    2.00     2.00      111      4704   0.35000V         3.84157656    UNIFORM P-AXIS 
+  61       0.500    500      3     42     2.0    1. 0    2.00     2.00       60      2562   0.65000V         3.85432960    UNIFORM P-AXIS 
+  62       0.600    500      3     42     2.0    1. 0    2.00     2.00       93      3948   0.40000V         3.83483193    UNIFORM P-AXIS 
+  63       0.700    500      3     42     2.0    1. 0    2.00     2.00       60      2562   0.65000V         3.85431079    UNIFORM P-AXIS 
+  64       0.800    500      3     42     2.0    1. 0    2.00     2.00      112      4746   0.40000V         3.84765623    UNIFORM P-AXIS 
+  65       0.900    500      3     42     2.0    1. 0    2.00     2.00       77      3276   0.55000V         3.85815731    UNIFORM P-AXIS 
+  66       1.000    500      3     42     2.0    1. 0    2.00     2.00      146      6174   0.20000V         3.84384847    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      200268 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  58       0.200    500      3     42     2.0    1. 0    2.00     2.00       74      3150   0.40000V         3.86268376    UNIFORM P-AXIS 31 of 92 F20 
+Run ID: 12-27-2009, 00:00:52 
+ 
+FUNCTION: F20 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500      6     24     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500      6     24     2.0    1. 0    2.00     2.00      154      3720   0.60000V         1.79481098    UNIFORM P-AXIS 
+   2       0.100    500      6     24     2.0    1. 0    2.00     2.00      168      4056   0.35000V         3.29593022    UNIFORM P-AXIS 
+   3       0.200    500      6     24     2.0    1. 0    2.00     2.00      124      3000   0.05000V         3.31976377    UNIFORM P-AXIS 
+   4       0.300    500      6     24     2.0    1. 0    2.00     2.00      250      6024   0.65000V         3.29485054    UNIFORM P-AXIS 
+   5       0.400    500      6     24     2.0    1. 0    2.00     2.00      284      6840   0.45000V         3.31728832    UNIFORM P-AXIS 
+   6       0.500    500      6     24     2.0    1. 0    2.00     2.00      129      3120   0.30000V         3.31388945    UNIFORM P-AXIS 
+   7       0.600    500      6     24     2.0    1. 0    2.00     2.00      250      6024   0.65000V         3.28047153    UNIFORM P-AXIS 
+   8       0.700    500      6     24     2.0    1. 0    2.00     2.00      284      6840   0.45000V         3.15797322    UNIFORM P-AXIS 
+   9       0.800    500      6     24     2.0    1. 0    2.00     2.00      162      3912   0.05000V         3.25303766    UNIFORM P-AXIS 
+  10       0.900    500      6     24     2.0    1. 0    2.00     2.00      142      3432   0.95000V         1.44877557    UNIFORM P-AXIS 
+  11       1.000    500      6     24     2.0    1. 0    2.00     2.00      214      5160   0.75000V         3.21996273    UNIFORM P-AXIS 
+  12       0.000    500      6     36     2.0    1. 0    2.00     2.00      250      9036   0.65000V         3.30877277    UNIFORM P-AXIS 
+  13       0.100    500      6     36     2.0    1. 0    2.00     2.00      194      7020   0.70000V         3.31923851    UNIFORM P-AXIS 
+  14       0.200    500      6     36     2.0    1. 0    2.00     2.00      175      6336   0.70000V         3.31659640    UNIFORM P-AXIS 
+  15       0.300    500      6     36     2.0    1. 0    2.00     2.00      191      6912   0.55000V         3.30506311    UNIFORM P-AXIS 
+  16       0.400    500      6     36     2.0    1. 0    2.00     2.00      175      6336   0.70000V         3.28854629    UNIFORM P-AXIS 
+  17       0.500    500      6     36     2.0    1. 0    2.00     2.00      231      8352   0.65000V         3.31583530    UNIFORM P-AXIS 
+  18       0.600    500      6     36     2.0    1. 0    2.00     2.00      155      5616   0.65000V         3.11598206    UNIFORM P-AXIS 
+  19       0.700    500      6     36     2.0    1. 0    2.00     2.00      232      8388   0.70000V         3.13277320    UNIFORM P-AXIS 
+  20       0.800    500      6     36     2.0    1. 0    2.00     2.00      193      6984   0.65000V         3.31407408    UNIFORM P-AXIS 
+  21       0.900    500      6     36     2.0    1. 0    2.00     2.00      109      3960   0.25000V         0.63932000    UNIFORM P-AXIS 
+  22       1.000    500      6     36     2.0    1. 0    2.00     2.00      179      6480   0.90000V         3.26629451    UNIFORM P-AXIS 
+  23       0.000    500      6     48     2.0    1. 0    2.00     2.00      250     12048   0.65000V         3.32048934    UNIFORM P-AXIS 
+  24       0.100    500      6     48     2.0    1. 0    2.00     2.00      213     10272   0.70000V         3.31842289    UNIFORM P-AXIS 
+  25       0.200    500      6     48     2.0    1. 0    2.00     2.00      193      9312   0.65000V         3.31716803    UNIFORM P-AXIS 
+  26       0.300    500      6     48     2.0    1. 0    2.00     2.00      277     13344   0.10000V         3.31881259    UNIFORM P-AXIS 
+  27       0.400    500      6     48     2.0    1. 0    2.00     2.00      231     11136   0.65000V         3.31690037    UNIFORM P-AXIS 
+  28       0.500    500      6     48     2.0    1. 0    2.00     2.00      286     13776   0.55000V         3.31917080    UNIFORM P-AXIS 
+  29       0.600    500      6     48     2.0    1. 0    2.00     2.00      151      7296   0.45000V         3.12577992    UNIFORM P-AXIS 
+  30       0.700    500      6     48     2.0    1. 0    2.00     2.00      168      8112   0.35000V         3.13969728    UNIFORM P-AXIS 
+  31       0.800    500      6     48     2.0    1. 0    2.00     2.00      144      6960   0.10000V         3.31609910    UNIFORM P-AXIS 
+  32       0.900    500      6     48     2.0    1. 0    2.00     2.00      180      8688   0.95000V         0.21817480    UNIFORM P-AXIS 
+  33       1.000    500      6     48     2.0    1. 0    2.00     2.00      250     12048   0.65000V         3.16479410    UNIFORM P-AXIS 
+  34       0.000    500      6     60     2.0    1. 0    2.00     2.00      134      8100   0.55000V         3.31699991    UNIFORM P-AXIS 
+  35       0.100    500      6     60     2.0    1. 0    2.00     2.00      193     11640   0.65000V         3.31372407    UNIFORM P-AXIS 
+  36       0.200    500      6     60     2.0    1. 0    2.00     2.00      126      7620   0.15000V         3.31638228    UNIFORM P-AXIS 
+  37       0.300    500      6     60     2.0    1. 0    2.00     2.00      139      8400   0.80000V         3.31640953    UNIFORM P-AXIS 
+  38       0.400    500      6     60     2.0    1. 0    2.00     2.00      145      8760   0.15000V         3.31306608    UNIFORM P-AXIS 
+  39       0.500    500      6     60     2.0    1. 0    2.00     2.00      251     15120   0.70000V         3.31365808    UNIFORM P-AXIS 
+  40       0.600    500      6     60     2.0    1. 0    2.00     2.00      249     15000   0.60000V         3.31607817    UNIFORM P-AXIS 
+  41       0.700    500      6     60     2.0    1. 0    2.00     2.00      158      9540   0.80000V         3.17449059    UNIFORM P-AXIS 
+  42       0.800    500      6     60     2.0    1. 0    2.00     2.00      193     11640   0.65000V         3.32110070    UNIFORM P-AXIS 
+  43       0.900    500      6     60     2.0    1. 0    2.00     2.00      167     10080   0.30000V         0.23623635    UNIFORM P-AXIS 
+  44       1.000    500      6     60     2.0    1. 0    2.00     2.00      282     16980   0.35000V         3.26645604    UNIFORM P-AXIS 
+  45       0.000    500      6     72     2.0    1. 0    2.00     2.00      193     13968   0.65000V         3.31934262    UNIFORM P-AXIS 
+  46       0.100    500      6     72     2.0    1. 0    2.00     2.00      157     11376   0.75000V         3.31520000    UNIFORM P-AXIS 
+  47       0.200    500      6     72     2.0    1. 0    2.00     2.00      250     18072   0.65000V         3.31085989    UNIFORM P-AXIS 
+  48       0.300    500      6     72     2.0    1. 0    2.00     2.00      250     18072   0.65000V         3.32173273    UNIFORM P-AXIS 
+  49       0.400    500      6     72     2.0    1. 0    2.00     2.00      250     18072   0.65000V         3.30166220    UNIFORM P-AXIS 
+  50       0.500    500      6     72     2.0    1. 0    2.00     2.00      230     16632   0.60000V         3.31879682    UNIFORM P-AXIS 
+  51       0.600    500      6     72     2.0    1. 0    2.00     2.00      267     19296   0.55000V         3.31773563    UNIFORM P-AXIS 
+  52       0.700    500      6     72     2.0    1. 0    2.00     2.00      160     11592   0.90000V         3.17735937    UNIFORM P-AXIS 
+  53       0.800    500      6     72     2.0    1. 0    2.00     2.00      227     16416   0.45000V         3.31701910    UNIFORM P-AXIS 
+  54       0.900    500      6     72     2.0    1. 0    2.00     2.00      174     12600   0.65000V         3.25654442    UNIFORM P-AXIS 
+  55       1.000    500      6     72     2.0    1. 0    2.00     2.00      270     19512   0.70000V         3.31105409    UNIFORM P-AXIS 
+  56       0.000    500      6     84     2.0    1. 0    2.00     2.00      283     23856   0.40000V         3.31541041    UNIFORM P-AXIS 
+  57       0.100    500      6     84     2.0    1. 0    2.00     2.00      160     13524   0.90000V         3.31043298    UNIFORM P-AXIS 
+  58       0.200    500      6     84     2.0    1. 0    2.00     2.00      206     17388   0.35000V         3.31145519    UNIFORM P-AXIS 
+  59       0.300    500      6     84     2.0    1. 0    2.00     2.00      103      8736   0.90000V         3.32074934    UNIFORM P-AXIS 
+  60       0.400    500      6     84     2.0    1. 0    2.00     2.00      231     19488   0.65000V         3.32093772    UNIFORM P-AXIS 
+  61       0.500    500      6     84     2.0    1. 0    2.00     2.00      249     21000   0.60000V         3.29768846    UNIFORM P-AXIS 
+  62       0.600    500      6     84     2.0    1. 0    2.00     2.00      250     21084   0.65000V         3.31956844    UNIFORM P-AXIS 
+  63       0.700    500      6     84     2.0    1. 0    2.00     2.00       99      8400   0.70000V         3.18238200    UNIFORM P-AXIS 
+  64       0.800    500      6     84     2.0    1. 0    2.00     2.00      165     13944   0.20000V         3.31971372    UNIFORM P-AXIS 
+  65       0.900    500      6     84     2.0    1. 0    2.00     2.00      268     22596   0.60000V         3.30768144    UNIFORM P-AXIS 
+  66       1.000    500      6     84     2.0    1. 0    2.00     2.00      251     21168   0.70000V         3.30198521    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      730212 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  48       0.300    500      6     72     2.0    1. 0    2.00     2.00      250     18072   0.65000V         3.32173273    UNIFORM P-AXIS 
+ 
+F21 
+Run ID: 12-27-2009, 00:08:36 
+ 
+FUNCTION: F21 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500      4     16     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V         0.27311534    UNIFORM P-AXIS 
+   2       0.100    500      4     16     2.0    1. 0    2.00     2.00       96      1552   0.55000V         5.04133435    UNIFORM P-AXIS 
+   3       0.200    500      4     16     2.0    1. 0    2.00     2.00      138      2224   0.75000V         0.45919710    UNIFORM P-AXIS 
+   4       0.300    500      4     16     2.0    1. 0    2.00     2.00      138      2224   0.75000V         0.67224136    UNIFORM P-AXIS 
+   5       0.400    500      4     16     2.0    1. 0    2.00     2.00      172      2768   0.55000V        10.14144490    UNIFORM P-AXIS 
+   6       0.500    500      4     16     2.0    1. 0    2.00     2.00      156      2512   0.70000V         0.61614905    UNIFORM P-AXIS 
+   7       0.600    500      4     16     2.0    1. 0    2.00     2.00      272      4368   0.80000V        10.14663737    UNIFORM P-AXIS 
+   8       0.700    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V         0.53087283    UNIFORM P-AXIS 
+   9       0.800    500      4     16     2.0    1. 0    2.00     2.00       96      1552   0.55000V         4.85279463    UNIFORM P-AXIS 
+  10       0.900    500      4     16     2.0    1. 0    2.00     2.00      196      3152   0.80000V         1.25937046    UNIFORM P-AXIS 
+  11       1.000    500      4     16     2.0    1. 0    2.00     2.00      191      3072   0.55000V        10.14038367    UNIFORM P-AXIS 
+  12       0.000    500      4     24     2.0    1. 0    2.00     2.00      171      4128   0.50000V         5.05394549    UNIFORM P-AXIS 
+  13       0.100    500      4     24     2.0    1. 0    2.00     2.00      127      3072   0.20000V        10.15293741    UNIFORM P-AXIS 
+  14       0.200    500      4     24     2.0    1. 0    2.00     2.00      173      4176   0.60000V        10.15309558    UNIFORM P-AXIS 
+  15       0.300    500      4     24     2.0    1. 0    2.00     2.00      153      3696   0.55000V         1.97500752    UNIFORM P-AXIS 
+  16       0.400    500      4     24     2.0    1. 0    2.00     2.00       78      1896   0.60000V        10.15319585    UNIFORM P-AXIS 32 of 92   17       0.500    500      4     24     2.0    1. 0    2.00     2.00       60      1464   0.65000V         0.57297331    UNIFORM P-AXIS 
+  18       0.600    500      4     24     2.0    1. 0    2.00     2.00      500     12024   0.80000V         8.82793293    UNIFORM P-AXIS 
+  19       0.700    500      4     24     2.0    1. 0    2.00     2.00      174      4200   0.65000V         0.53158719    UNIFORM P-AXIS 
+  20       0.800    500      4     24     2.0    1. 0    2.00     2.00      117      2832   0.65000V        10.14658660    UNIFORM P-AXIS 
+  21       0.900    500      4     24     2.0    1. 0    2.00     2.00      208      5016   0.45000V         2.05090467    UNIFORM P-AXIS 
+  22       1.000    500      4     24     2.0    1. 0    2.00     2.00      204      4920   0.25000V        10.15142802    UNIFORM P-AXIS 
+  23       0.000    500      4     32     2.0    1. 0    2.00     2.00      134      4320   0.55000V         5.04863711    UNIFORM P-AXIS 
+  24       0.100    500      4     32     2.0    1. 0    2.00     2.00       96      3104   0.55000V         5.05472330    UNIFORM P-AXIS 
+  25       0.200    500      4     32     2.0    1. 0    2.00     2.00      138      4448   0.75000V         0.42839721    UNIFORM P-AXIS 
+  26       0.300    500      4     32     2.0    1. 0    2.00     2.00      212      6816   0.65000V         0.85458904    UNIFORM P-AXIS 
+  27       0.400    500      4     32     2.0    1. 0    2.00     2.00      114      3680   0.50000V        10.15296968    UNIFORM P-AXIS 
+  28       0.500    500      4     32     2.0    1. 0    2.00     2.00       60      1952   0.65000V         0.58456509    UNIFORM P-AXIS 
+  29       0.600    500      4     32     2.0    1. 0    2.00     2.00       98      3168   0.65000V         5.02590739    UNIFORM P-AXIS 
+  30       0.700    500      4     32     2.0    1. 0    2.00     2.00       60      1952   0.65000V         0.53137912    UNIFORM P-AXIS 
+  31       0.800    500      4     32     2.0    1. 0    2.00     2.00       75      2432   0.45000V        10.02571078    UNIFORM P-AXIS 
+  32       0.900    500      4     32     2.0    1. 0    2.00     2.00      176      5664   0.75000V         1.02391414    UNIFORM P-AXIS 
+  33       1.000    500      4     32     2.0    1. 0    2.00     2.00       74      2400   0.40000V        10.14076773    UNIFORM P-AXIS 
+  34       0.000    500      4     40     2.0    1. 0    2.00     2.00      151      6080   0.45000V         5.05281229    UNIFORM P-AXIS 
+  35       0.100    500      4     40     2.0    1. 0    2.00     2.00       99      4000   0.70000V         5.05441601    UNIFORM P-AXIS 
+  36       0.200    500      4     40     2.0    1. 0    2.00     2.00      194      7800   0.70000V         0.46731350    UNIFORM P-AXIS 
+  37       0.300    500      4     40     2.0    1. 0    2.00     2.00      229      9200   0.55000V         7.93894966    UNIFORM P-AXIS 
+  38       0.400    500      4     40     2.0    1. 0    2.00     2.00      153      6160   0.55000V        10.15163190    UNIFORM P-AXIS 
+  39       0.500    500      4     40     2.0    1. 0    2.00     2.00       60      2440   0.65000V         0.58586398    UNIFORM P-AXIS 
+  40       0.600    500      4     40     2.0    1. 0    2.00     2.00       94      3800   0.45000V        10.15158968    UNIFORM P-AXIS 
+  41       0.700    500      4     40     2.0    1. 0    2.00     2.00       60      2440   0.65000V         0.53087283    UNIFORM P-AXIS 
+  42       0.800    500      4     40     2.0    1. 0    2.00     2.00      273     10960   0.85000V        10.12002071    UNIFORM P-AXIS 
+  43       0.900    500      4     40     2.0    1. 0    2.00     2.00      196      7880   0.80000V         4.11351704    UNIFORM P-AXIS 
+  44       1.000    500      4     40     2.0    1. 0    2.00     2.00      135      5440   0.60000V        10.15035466    UNIFORM P-AXIS 
+  45       0.000    500      4     48     2.0    1. 0    2.00     2.00      249     12000   0.60000V         5.05432308    UNIFORM P-AXIS 
+  46       0.100    500      4     48     2.0    1. 0    2.00     2.00      190      9168   0.50000V        10.15273852    UNIFORM P-AXIS 
+  47       0.200    500      4     48     2.0    1. 0    2.00     2.00      193      9312   0.65000V         0.47276129    UNIFORM P-AXIS 
+  48       0.300    500      4     48     2.0    1. 0    2.00     2.00      158      7632   0.80000V         1.95052494    UNIFORM P-AXIS 
+  49       0.400    500      4     48     2.0    1. 0    2.00     2.00      252     12144   0.75000V        10.15205928    UNIFORM P-AXIS 
+  50       0.500    500      4     48     2.0    1. 0    2.00     2.00       60      2928   0.65000V         0.58526200    UNIFORM P-AXIS 
+  51       0.600    500      4     48     2.0    1. 0    2.00     2.00      118      5712   0.70000V        10.14685194    UNIFORM P-AXIS 
+  52       0.700    500      4     48     2.0    1. 0    2.00     2.00       60      2928   0.65000V         0.53140202    UNIFORM P-AXIS 
+  53       0.800    500      4     48     2.0    1. 0    2.00     2.00       80      3888   0.70000V         5.09974140    UNIFORM P-AXIS 
+  54       0.900    500      4     48     2.0    1. 0    2.00     2.00      174      8400   0.65000V         1.94704742    UNIFORM P-AXIS 
+  55       1.000    500      4     48     2.0    1. 0    2.00     2.00      167      8064   0.30000V        10.15311224    UNIFORM P-AXIS 
+  56       0.000    500      4     56     2.0    1. 0    2.00     2.00      246     13832   0.45000V         5.05171296    UNIFORM P-AXIS 
+  57       0.100    500      4     56     2.0    1. 0    2.00     2.00       78      4424   0.60000V         5.05389514    UNIFORM P-AXIS 
+  58       0.200    500      4     56     2.0    1. 0    2.00     2.00      176      9912   0.75000V         0.47849329    UNIFORM P-AXIS 
+  59       0.300    500      4     56     2.0    1. 0    2.00     2.00      161      9072   0.95000V         1.97272685    UNIFORM P-AXIS 
+  60       0.400    500      4     56     2.0    1. 0    2.00     2.00       60      3416   0.65000V        10.11776011    UNIFORM P-AXIS 
+  61       0.500    500      4     56     2.0    1. 0    2.00     2.00       60      3416   0.65000V         0.58434919    UNIFORM P-AXIS 
+  62       0.600    500      4     56     2.0    1. 0    2.00     2.00       80      4536   0.70000V        10.09567420    UNIFORM P-AXIS 
+  63       0.700    500      4     56     2.0    1. 0    2.00     2.00       60      3416   0.65000V         0.53103979    UNIFORM P-AXIS 
+  64       0.800    500      4     56     2.0    1. 0    2.00     2.00      106      5992   0.10000V        10.14160544    UNIFORM P-AXIS 
+  65       0.900    500      4     56     2.0    1. 0    2.00     2.00      209     11760   0.50000V         5.08969804    UNIFORM P-AXIS 
+  66       1.000    500      4     56     2.0    1. 0    2.00     2.00      103      5824   0.90000V        10.15317755    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      336712 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  16       0.400    500      4     24     2.0    1. 0    2.00     2.00       78      1896   0.60000V        10.15319585    UNIFORM P-AXIS 
+ 
+F22  
+Run ID: 12-27-2009, 00:12:14 
+ 
+FUNCTION: F22 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500      4     16     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V         0.29361829    UNIFORM P-AXIS 
+   2       0.100    500      4     16     2.0    1. 0    2.00     2.00      203      3264   0.20000V        10.26261276    UNIFORM P-AXIS 
+   3       0.200    500      4     16     2.0    1. 0    2.00     2.00      138      2224   0.75000V         0.50537915    UNIFORM P-AXIS 
+   4       0.300    500      4     16     2.0    1. 0    2.00     2.00      157      2528   0.75000V         0.95497293    UNIFORM P-AXIS 
+   5       0.400    500      4     16     2.0    1. 0    2.00     2.00      291      4672   0.80000V        10.40284412    UNIFORM P-AXIS 
+   6       0.500    500      4     16     2.0    1. 0    2.00     2.00      218      3504   0.95000V         1.00484226    UNIFORM P-AXIS 
+   7       0.600    500      4     16     2.0    1. 0    2.00     2.00      117      1888   0.65000V         5.12591666    UNIFORM P-AXIS 
+   8       0.700    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V         0.57542458    UNIFORM P-AXIS 
+   9       0.800    500      4     16     2.0    1. 0    2.00     2.00      177      2848   0.80000V         5.11317026    UNIFORM P-AXIS 
+  10       0.900    500      4     16     2.0    1. 0    2.00     2.00      175      2816   0.70000V         1.19582715    UNIFORM P-AXIS 
+  11       1.000    500      4     16     2.0    1. 0    2.00     2.00      216      3472   0.85000V        10.39966301    UNIFORM P-AXIS 
+  12       0.000    500      4     24     2.0    1. 0    2.00     2.00      182      4392   0.10000V         5.07724293    UNIFORM P-AXIS 
+  13       0.100    500      4     24     2.0    1. 0    2.00     2.00       96      2328   0.55000V         5.08627190    UNIFORM P-AXIS 
+  14       0.200    500      4     24     2.0    1. 0    2.00     2.00      144      3480   0.10000V        10.40264736    UNIFORM P-AXIS 
+  15       0.300    500      4     24     2.0    1. 0    2.00     2.00      207      4992   0.40000V         1.24076642    UNIFORM P-AXIS 
+  16       0.400    500      4     24     2.0    1. 0    2.00     2.00      130      3144   0.35000V        10.40281884    UNIFORM P-AXIS 
+  17       0.500    500      4     24     2.0    1. 0    2.00     2.00      194      4680   0.70000V         0.98671234    UNIFORM P-AXIS 
+  18       0.600    500      4     24     2.0    1. 0    2.00     2.00      153      3696   0.55000V        10.39418334    UNIFORM P-AXIS 
+  19       0.700    500      4     24     2.0    1. 0    2.00     2.00      194      4680   0.70000V         0.59488646    UNIFORM P-AXIS 
+  20       0.800    500      4     24     2.0    1. 0    2.00     2.00       91      2208   0.30000V        10.40286059    UNIFORM P-AXIS 
+  21       0.900    500      4     24     2.0    1. 0    2.00     2.00      193      4656   0.65000V         1.91215458    UNIFORM P-AXIS 
+  22       1.000    500      4     24     2.0    1. 0    2.00     2.00      116      2808   0.60000V        10.35044377    UNIFORM P-AXIS 
+  23       0.000    500      4     32     2.0    1. 0    2.00     2.00      180      5792   0.95000V         5.08311858    UNIFORM P-AXIS 
+  24       0.100    500      4     32     2.0    1. 0    2.00     2.00      329     10560   0.80000V        10.40175889    UNIFORM P-AXIS 
+  25       0.200    500      4     32     2.0    1. 0    2.00     2.00      138      4448   0.75000V         0.48151911    UNIFORM P-AXIS 
+  26       0.300    500      4     32     2.0    1. 0    2.00     2.00      139      4480   0.80000V         0.87836043    UNIFORM P-AXIS 
+  27       0.400    500      4     32     2.0    1. 0    2.00     2.00       60      1952   0.65000V        10.35599982    UNIFORM P-AXIS 
+  28       0.500    500      4     32     2.0    1. 0    2.00     2.00      176      5664   0.75000V         0.89227405    UNIFORM P-AXIS 
+  29       0.600    500      4     32     2.0    1. 0    2.00     2.00      110      3552   0.30000V         5.12666016    UNIFORM P-AXIS 
+  30       0.700    500      4     32     2.0    1. 0    2.00     2.00       60      1952   0.65000V         0.57306071    UNIFORM P-AXIS 
+  31       0.800    500      4     32     2.0    1. 0    2.00     2.00      178      5728   0.85000V        10.40070098    UNIFORM P-AXIS 
+  32       0.900    500      4     32     2.0    1. 0    2.00     2.00      177      5696   0.80000V         0.84664455    UNIFORM P-AXIS 
+  33       1.000    500      4     32     2.0    1. 0    2.00     2.00      226      7264   0.40000V        10.40258245    UNIFORM P-AXIS 
+  34       0.000    500      4     40     2.0    1. 0    2.00     2.00      134      5400   0.55000V         5.08672034    UNIFORM P-AXIS 
+  35       0.100    500      4     40     2.0    1. 0    2.00     2.00      123      4960   0.95000V        10.40267265    UNIFORM P-AXIS 
+  36       0.200    500      4     40     2.0    1. 0    2.00     2.00      138      5560   0.75000V         0.48686581    UNIFORM P-AXIS 
+  37       0.300    500      4     40     2.0    1. 0    2.00     2.00      176      7080   0.75000V         1.51013313    UNIFORM P-AXIS 
+  38       0.400    500      4     40     2.0    1. 0    2.00     2.00      167      6720   0.30000V        10.39925385    UNIFORM P-AXIS 
+  39       0.500    500      4     40     2.0    1. 0    2.00     2.00      176      7080   0.75000V         1.00062130    UNIFORM P-AXIS 
+  40       0.600    500      4     40     2.0    1. 0    2.00     2.00      131      5280   0.40000V        10.39124772    UNIFORM P-AXIS 
+  41       0.700    500      4     40     2.0    1. 0    2.00     2.00       60      2440   0.65000V         0.57542458    UNIFORM P-AXIS 
+  42       0.800    500      4     40     2.0    1. 0    2.00     2.00       75      3040   0.45000V        10.40200744    UNIFORM P-AXIS 
+  43       0.900    500      4     40     2.0    1. 0    2.00     2.00      176      7080   0.75000V         4.00941138    UNIFORM P-AXIS 
+  44       1.000    500      4     40     2.0    1. 0    2.00     2.00      254     10200   0.85000V        10.39754781    UNIFORM P-AXIS 33 of 92   45       0.000    500      4     48     2.0    1. 0    2.00     2.00      176      8496   0.75000V         5.08636835    UNIFORM P-AXIS 
+  46       0.100    500      4     48     2.0    1. 0    2.00     2.00      131      6336   0.40000V         5.08612144    UNIFORM P-AXIS 
+  47       0.200    500      4     48     2.0    1. 0    2.00     2.00      143      6912   0.05000V         0.48654928    UNIFORM P-AXIS 
+  48       0.300    500      4     48     2.0    1. 0    2.00     2.00      165      7968   0.20000V         4.44538707    UNIFORM P-AXIS 
+  49       0.400    500      4     48     2.0    1. 0    2.00     2.00      124      6000   0.05000V        10.40044163    UNIFORM P-AXIS 
+  50       0.500    500      4     48     2.0    1. 0    2.00     2.00      155      7488   0.65000V         1.39721189    UNIFORM P-AXIS 
+  51       0.600    500      4     48     2.0    1. 0    2.00     2.00      207      9984   0.40000V        10.39959118    UNIFORM P-AXIS 
+  52       0.700    500      4     48     2.0    1. 0    2.00     2.00       60      2928   0.65000V         0.57429457    UNIFORM P-AXIS 
+  53       0.800    500      4     48     2.0    1. 0    2.00     2.00       86      4176   0.05000V        10.39514139    UNIFORM P-AXIS 
+  54       0.900    500      4     48     2.0    1. 0    2.00     2.00      178      8592   0.85000V         2.14526622    UNIFORM P-AXIS 
+  55       1.000    500      4     48     2.0    1. 0    2.00     2.00      163      7872   0.10000V        10.40265915    UNIFORM P-AXIS 
+  56       0.000    500      4     56     2.0    1. 0    2.00     2.00      231     12992   0.65000V         5.07298037    UNIFORM P-AXIS 
+  57       0.100    500      4     56     2.0    1. 0    2.00     2.00      344     19320   0.60000V        10.40232156    UNIFORM P-AXIS 
+  58       0.200    500      4     56     2.0    1. 0    2.00     2.00      213     11984   0.70000V         0.50205717    UNIFORM P-AXIS 
+  59       0.300    500      4     56     2.0    1. 0    2.00     2.00      240     13496   0.15000V         6.54548019    UNIFORM P-AXIS 
+  60       0.400    500      4     56     2.0    1. 0    2.00     2.00      130      7336   0.35000V        10.40282272    UNIFORM P-AXIS 
+  61       0.500    500      4     56     2.0    1. 0    2.00     2.00      154      8680   0.60000V         1.40129961    UNIFORM P-AXIS 
+  62       0.600    500      4     56     2.0    1. 0    2.00     2.00      209     11760   0.50000V        10.40172732    UNIFORM P-AXIS 
+  63       0.700    500      4     56     2.0    1. 0    2.00     2.00       60      3416   0.65000V         0.57352478    UNIFORM P-AXIS 
+  64       0.800    500      4     56     2.0    1. 0    2.00     2.00      111      6272   0.35000V         5.12730501    UNIFORM P-AXIS 
+  65       0.900    500      4     56     2.0    1. 0    2.00     2.00      225     12656   0.35000V         5.12155298    UNIFORM P-AXIS 
+  66       1.000    500      4     56     2.0    1. 0    2.00     2.00      166      9352   0.25000V        10.40279968    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      386176 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  20       0.800    500      4     24     2.0    1. 0    2.00     2.00       91      2208   0.30000V        10.40286059    UNIFORM P-AXIS 
+ 
+F23  
+Run ID: 12-27-2009, 00:16:12 
+ 
+FUNCTION: F23 
+ 
+Run #     Gamma     Nt     Nd     Np      G     Del T   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes 
+-----    -------   ----   ----   ----   -----   --- -   -----   ------   ------   -------   --------    ---------------   --------------- 
+   0       0.000    500      4     16     2.0    1. 0    2.00     2.00        0         0   0.50000V     -9999.00000000    UNIFORM P-AXIS 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+   1       0.000    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V         0.32172905    UNIFORM P-AXIS 
+   2       0.100    500      4     16     2.0    1. 0    2.00     2.00      137      2208   0.70000V        10.51946328    UNIFORM P-AXIS 
+   3       0.200    500      4     16     2.0    1. 0    2.00     2.00      138      2224   0.75000V         0.55435306    UNIFORM P-AXIS 
+   4       0.300    500      4     16     2.0    1. 0    2.00     2.00      157      2528   0.75000V         1.07283233    UNIFORM P-AXIS 
+   5       0.400    500      4     16     2.0    1. 0    2.00     2.00      140      2256   0.85000V        10.53615023    UNIFORM P-AXIS 
+   6       0.500    500      4     16     2.0    1. 0    2.00     2.00      231      3712   0.65000V         1.14397265    UNIFORM P-AXIS 
+   7       0.600    500      4     16     2.0    1. 0    2.00     2.00      111      1792   0.35000V         2.86919827    UNIFORM P-AXIS 
+   8       0.700    500      4     16     2.0    1. 0    2.00     2.00       60       976   0.65000V         0.65227664    UNIFORM P-AXIS 
+   9       0.800    500      4     16     2.0    1. 0    2.00     2.00      132      2128   0.45000V         5.17558057    UNIFORM P-AXIS 
+  10       0.900    500      4     16     2.0    1. 0    2.00     2.00      241      3872   0.20000V         1.33159419    UNIFORM P-AXIS 
+  11       1.000    500      4     16     2.0    1. 0    2.00     2.00      131      2112   0.40000V        10.53532965    UNIFORM P-AXIS 
+  12       0.000    500      4     24     2.0    1. 0    2.00     2.00      194      4680   0.70000V         5.10962539    UNIFORM P-AXIS 
+  13       0.100    500      4     24     2.0    1. 0    2.00     2.00      310      7464   0.80000V        10.53629238    UNIFORM P-AXIS 
+  14       0.200    500      4     24     2.0    1. 0    2.00     2.00       69      1680   0.15000V         5.10013673    UNIFORM P-AXIS 
+  15       0.300    500      4     24     2.0    1. 0    2.00     2.00      162      3912   0.05000V         1.51972338    UNIFORM P-AXIS 
+  16       0.400    500      4     24     2.0    1. 0    2.00     2.00      135      3264   0.60000V        10.53628373    UNIFORM P-AXIS 
+  17       0.500    500      4     24     2.0    1. 0    2.00     2.00      182      4392   0.10000V         1.13540741    UNIFORM P-AXIS 
+  18       0.600    500      4     24     2.0    1. 0    2.00     2.00      413      9936   0.25000V        10.53581352    UNIFORM P-AXIS 
+  19       0.700    500      4     24     2.0    1. 0    2.00     2.00      176      4248   0.75000V         0.72542885    UNIFORM P-AXIS 
+  20       0.800    500      4     24     2.0    1. 0    2.00     2.00       93      2256   0.40000V        10.53632019    UNIFORM P-AXIS 
+  21       0.900    500      4     24     2.0    1. 0    2.00     2.00      174      4200   0.65000V         2.06446347    UNIFORM P-AXIS 
+  22       1.000    500      4     24     2.0    1. 0    2.00     2.00      190      4584   0.50000V        10.53152157    UNIFORM P-AXIS 
+  23       0.000    500      4     32     2.0    1. 0    2.00     2.00      182      5856   0.10000V         5.11195183    UNIFORM P-AXIS 
+  24       0.100    500      4     32     2.0    1. 0    2.00     2.00      191      6144   0.55000V        10.53233518    UNIFORM P-AXIS 
+  25       0.200    500      4     32     2.0    1. 0    2.00     2.00      138      4448   0.75000V         0.52798028    UNIFORM P-AXIS 
+  26       0.300    500      4     32     2.0    1. 0    2.00     2.00      147      4736   0.25000V         1.00075402    UNIFORM P-AXIS 
+  27       0.400    500      4     32     2.0    1. 0    2.00     2.00       60      1952   0.65000V        10.47302654    UNIFORM P-AXIS 
+  28       0.500    500      4     32     2.0    1. 0    2.00     2.00      176      5664   0.75000V         1.03683311    UNIFORM P-AXIS 
+  29       0.600    500      4     32     2.0    1. 0    2.00     2.00       76      2464   0.50000V         5.11476720    UNIFORM P-AXIS 
+  30       0.700    500      4     32     2.0    1. 0    2.00     2.00       60      1952   0.65000V         0.64610891    UNIFORM P-AXIS 
+  31       0.800    500      4     32     2.0    1. 0    2.00     2.00      208      6688   0.45000V        10.53519865    UNIFORM P-AXIS 
+  32       0.900    500      4     32     2.0    1. 0    2.00     2.00      183      5888   0.15000V         1.98038727    UNIFORM P-AXIS 
+  33       1.000    500      4     32     2.0    1. 0    2.00     2.00      232      7456   0.70000V        10.53612921    UNIFORM P-AXIS 
+  34       0.000    500      4     40     2.0    1. 0    2.00     2.00      339     13600   0.35000V         5.12842148    UNIFORM P-AXIS 
+  35       0.100    500      4     40     2.0    1. 0    2.00     2.00      171      6880   0.50000V        10.51343276    UNIFORM P-AXIS 
+  36       0.200    500      4     40     2.0    1. 0    2.00     2.00      192      7720   0.60000V         0.56506609    UNIFORM P-AXIS 
+  37       0.300    500      4     40     2.0    1. 0    2.00     2.00      171      6880   0.50000V         2.46078960    UNIFORM P-AXIS 
+  38       0.400    500      4     40     2.0    1. 0    2.00     2.00      226      9080   0.40000V        10.53511786    UNIFORM P-AXIS 
+  39       0.500    500      4     40     2.0    1. 0    2.00     2.00      269     10800   0.65000V         1.05309681    UNIFORM P-AXIS 
+  40       0.600    500      4     40     2.0    1. 0    2.00     2.00      117      4720   0.65000V        10.16796476    UNIFORM P-AXIS 
+  41       0.700    500      4     40     2.0    1. 0    2.00     2.00       60      2440   0.65000V         0.65227664    UNIFORM P-AXIS 
+  42       0.800    500      4     40     2.0    1. 0    2.00     2.00       81      3280   0.75000V        10.52269113    UNIFORM P-AXIS 
+  43       0.900    500      4     40     2.0    1. 0    2.00     2.00      176      7080   0.75000V         4.37301283    UNIFORM P-AXIS 
+  44       1.000    500      4     40     2.0    1. 0    2.00     2.00      129      5200   0.30000V        10.48509321    UNIFORM P-AXIS 
+  45       0.000    500      4     48     2.0    1. 0    2.00     2.00      152      7344   0.50000V         5.12815383    UNIFORM P-AXIS 
+  46       0.100    500      4     48     2.0    1. 0    2.00     2.00       74      3600   0.40000V        10.52851278    UNIFORM P-AXIS 
+  47       0.200    500      4     48     2.0    1. 0    2.00     2.00      138      6672   0.75000V         0.53841817    UNIFORM P-AXIS 
+  48       0.300    500      4     48     2.0    1. 0    2.00     2.00      193      9312   0.65000V         3.30001202    UNIFORM P-AXIS 
+  49       0.400    500      4     48     2.0    1. 0    2.00     2.00      112      5424   0.40000V        10.53439920    UNIFORM P-AXIS 
+  50       0.500    500      4     48     2.0    1. 0    2.00     2.00      157      7584   0.75000V         1.46710892    UNIFORM P-AXIS 
+  51       0.600    500      4     48     2.0    1. 0    2.00     2.00       97      4704   0.60000V         5.10176153    UNIFORM P-AXIS 
+  52       0.700    500      4     48     2.0    1. 0    2.00     2.00       60      2928   0.65000V         0.65302072    UNIFORM P-AXIS 
+  53       0.800    500      4     48     2.0    1. 0    2.00     2.00      322     15504   0.45000V        10.53096318    UNIFORM P-AXIS 
+  54       0.900    500      4     48     2.0    1. 0    2.00     2.00      176      8496   0.75000V         1.85678248    UNIFORM P-AXIS 
+  55       1.000    500      4     48     2.0    1. 0    2.00     2.00      251     12096   0.70000V        10.53165877    UNIFORM P-AXIS 
+  56       0.000    500      4     56     2.0    1. 0    2.00     2.00      217     12208   0.90000V         5.11016962    UNIFORM P-AXIS 
+  57       0.100    500      4     56     2.0    1. 0    2.00     2.00      208     11704   0.45000V        10.53571364    UNIFORM P-AXIS 
+  58       0.200    500      4     56     2.0    1. 0    2.00     2.00       74      4200   0.40000V         0.51919113    UNIFORM P-AXIS 
+  59       0.300    500      4     56     2.0    1. 0    2.00     2.00      229     12880   0.55000V         7.31576150    UNIFORM P-AXIS 
+  60       0.400    500      4     56     2.0    1. 0    2.00     2.00      145      8176   0.15000V        10.53546357    UNIFORM P-AXIS 
+  61       0.500    500      4     56     2.0    1. 0    2.00     2.00      157      8848   0.75000V         1.52874747    UNIFORM P-AXIS 
+  62       0.600    500      4     56     2.0    1. 0    2.00     2.00      179     10080   0.90000V        10.53625126    UNIFORM P-AXIS 
+  63       0.700    500      4     56     2.0    1. 0    2.00     2.00       60      3416   0.65000V         0.64916466    UNIFORM P-AXIS 
+  64       0.800    500      4     56     2.0    1. 0    2.00     2.00      141      7952   0.90000V         5.17550217    UNIFORM P-AXIS 
+  65       0.900    500      4     56     2.0    1. 0    2.00     2.00      329     18480   0.80000V         5.17556929    UNIFORM P-AXIS 
+  66       1.000    500      4     56     2.0    1. 0    2.00     2.00      113      6384   0.45000V        10.52136164    UNIFORM P-AXIS 
+ 
+                                                 To tal Function Evaluations:      394320 
+ 
+--------------------------------------------------- --------------------------------------------------- ---------------------------------- 
+  20       0.800    500      4     24     2.0    1. 0    2.00     2.00       93      2256   0.40000V        10.53632019    UNIFORM P-AXIS 34 of 92 Appendix 3:  CFO Source Code 
+'Program 'CFO_12-25-09.BAS' compiled with 
+'Power Basic/Windows Compiler 9.0 
+ 
+'LAST MOD 12-26-2009 ~1349 HRS EST 
+ 
+'CHANGES MADE TO "CFO_11-26-09.BAS" TO CREATE THIS VERSION 
+'WHICH IS USED FOR arXiv SUBMISSION #2: 
+'   (1).  OUTER LOOP Np/Nd = 2 to MaxProbesPerAxis by 2 ADDED 
+'   (2).  Np IGNORED & ARRAYS REDIMENSIONED AS REQU IRED BY (1). 
+ 
+'NOTE: ALL PBM FUNCTIONS HAVE WIRE RADIUS SET TO 0. 00001LAMBDA 
+'================================================== =========== 
+ 
+'THIS PROGRAM IMPLEMENTS A SIMPLE VERSION OF "CENTR AL 
+'FORCE OPTIMIZATION."  IT IS DISTRIBUTED FREE OF CH ARGE 
+'TO INCREASE AWARENESS OF CFO AND TO ENCOURAGE EXPE RI- 
+'MENTATION WITH THE ALGORITHM. 
+ 
+'CFO IS A MULTIDIMENSIONAL SEARCH AND OPTIMIZATION 
+'ALGORITHM THAT LOCATES THE GLOBAL MAXIMA OF A FUNC TION. 
+'UNLIKE MOST OTHER ALGORITHMS, CFO IS COMPLETELY DE TERMIN- 
+'ISTIC, SO THAT EVERY RUN WITH THE SAME SETUP PRODU CES 
+'THE SAME RESULTS. 
+ 
+'Please email questions, comments, and significant 
+'results to: CFO_questions@yahoo.com.  Thanks! 
+ 
+'(c) 2006-2009 Richard A. Formato 
+ 
+'ALL RIGHTS RESERVED WORLDWIDE 
+ 
+'THIS PROGRAM IS FREEWARE.  IT MAY BE COPIED AND 
+'DISTRIBUTED WITHOUT LIMITATION AS LONG AS THIS 
+'COPYRIGHT NOTICE AND THE GNUPLOT AND REFERENCE 
+'INFORMATION BELOW ARE INCLUDED WITHOUT MODIFICATIO N, 
+'AND AS LONG AS NO FEE OR COMPENSATION IS CHARGED, 
+'INCLUDING "TIE-IN" OR "BUNDLING" FEES CHARGED FOR 
+'OTHER PRODUCTS. 
+ 
+'================================================== ====== 
+ 
+'THIS PROGRAM REQUIRES wgnuplot.exe TO DISPLAY PLOT S. 
+'Gnuplot is a copyrighted freeware plotting program  
+'available at http://www.gnuplot.info/index.html. 
+ 
+'================================================== ====== 
+ 
+'REFERENCES 
+'---------- 
+'1. "Central Force Optimization: A New Metaheuristi c with Applications in Applied Electromagnetics," 
+'    Progress in Electromagnetics Research, PIER 77 , 425-491, 2007 (online). 
+'Abstract – Central Force Optimization (CFO) is a n ew deterministic multi-dimensional search 
+'metaheuristic based on the metaphor of gravitation al kinematics.  It models “probes” that “fly” 
+'through the decision space by analogy to masses mo ving under the influence of gravity.  Equations 
+'are developed for the probes’ positions and accele rations using the analogy of particle motion 
+'in a gravitational field.  In the physical univers e, objects traveling through three-dimensional 
+'space become trapped in close orbits around highly  gravitating masses, which is analogous to 
+'locating the maximum value of an objective functio n.  In the CFO metaphor, “mass” is a user-defined 
+'function of the value of the objective function to  be maximized.  CFO is readily implemented in a 
+'compact computer program, and sample pseudocode is  presented.  As tests of CFO’s effectiveness, 
+'an equalizer is designed for the well-known Fano l oad, and a 32-element linear array is synthesized. 
+'CFO results are compared to several other optimiza tion methods. 
+ 
+'2. "Central Force Optimization: A New Nature Inspi red Computational Framework for Multidimensional 
+'    Search and Optimization," Studies in Computati onal Intelligence (SCI), vol. 129, 221-238 (2008), 
+'    www.springerlink.com, Springer-Verlag Berlin H eidelberg 2008. 
+'Abstract – This paper presents Central Force Optim ization, a novel, nature inspired, deterministic 
+'search metaheuristic for constrained multi-dimensi onal optimization.  CFO is based on the metaphor 
+'of gravitational kinematics.  Equations are presen ted for the positions and accelerations experienced  
+'by “probes” that “fly” through the decision space by analogy to masses moving under the influence of 
+'gravity.  In the physical universe, probe satellit es become trapped in close orbits around highly 
+'gravitating masses.  In the CFO analogy, “mass” co rresponds to a user-defined function of the value 
+'of an objective function to be maximized.  CFO is a simple algorithm that is easily implemented in 
+'a compact computer program.  A typical CFO impleme ntation is applied to several test functions. 
+'CFO exhibits very good performance, suggesting tha t it merits further study. 
+ 
+'3. "Central Force Optimisation: A New Gradient-Lik e Metaheuristic for Multidimensional Search and 
+'    Optimisation," International Journal of Bio-In spired Computation, vol. 1, no. 4, 217-238 (2009), 
+'Abstract: This paper introduces Central Force Opti mization, a novel, Nature-inspired, deterministic 
+'search metaheuristic for constrained multidimensio nal optimization in highly multimodal, smooth, or 
+'discontinuous decision spaces.  CFO is based on th e metaphor of gravitational kinematics.  The 
+'algorithm searches a decision space by “flying” it s “probes” through the space by analogy to masses 
+'moving through physical space under the influence of gravity.  Equations are developed for the probes ’ 
+'positions and accelerations using the gravitationa l metaphor.  Small objects in our Universe can beco me 
+'trapped in close orbits around highly gravitating masses.  In “CFO space” probes are attracted to “ma sses” 
+'created by a user-defined function of the value of  an objective function to be maximized.  CFO may be  
+'thought of in terms of a vector “force field” or, loosely, as a “generalized gradient” methodology 
+'because the force of gravity can be computed as th e gradient of a scalar potential.  The CFO algorith m 
+'is simple and easily implemented in a compact comp uter program.  Its effectiveness is demonstrated by  
+'running CFO against several widely used benchmark functions.  The algorithm exhibits very good perfor mance, 
+'suggesting that it merits further study. 
+ 
+'4. "Central Force Optimization: A New Gradient-Lik e Optimization Metaheuristic," OPSEARCH, Journal 
+'    of the Operational Research Society of India, vol. 46, no. 1, 25-51, 2009, Springer India. 
+'Abstract:   This paper introduces Central Force Op timization as a new, Nature-inspired metaheuristic 
+'for multidimensional search and optimization based  on the metaphor of gravitational kinematics.  CFO is a 
+'“gradient-like” deterministic algorithm that explo res a decision space by “flying” a group of “probes ” 
+'whose trajectories are governed by equations analo gous to the equations of gravitational motion in th e 
+'physical Universe.  This paper suggests the possib ility of creating a new “hyperspace directional der ivative” 
+'using the Unit Step function t7o create positive-d efinite “masses” in “CFO space.”  A simple CFO impl ementation 
+'is tested against several recognized benchmark fun ctions with excellent results, suggesting that CFO merits 
+'further investigation. 
+ 
+'5. "Synthesis of Antenna Arrays Using Central Forc e Optimization," Mosharaka International Conference  
+'    on Communications, Computers and Applications,  MIC-CPE 2009. 
+'Abstract: Central force optimization (CFO) techniq ue is a new deterministic multi-dimensional 
+'search metaheuristic based on an analogy to classi cal particle kinematics in a gravitational 
+'field. CFO is a simple technique and is still in i ts infancy. To enhance its global search 
+'ability while keeping its simplicity, a new select ion part is introduced in this paper. CFO 
+'is briefly presented and applied in the design of linear antenna array and the modified CFO 35 of 92 'is applied to the design of circular array. The re sults are compared with those obtained using 
+'other evolutionary optimization techniques. 
+ 
+'6. "Central Force Optimization and NEOs - First Co usins?" Journal of Multiple Valued Logic and Soft 
+'    Computing (to be published) 
+'Abstract: Central Force Optimization is a new dete rministic multidimensional search and optimization algorithm 
+'based on the metaphor of gravitational kinematics.   This paper describes CFO and suggests some possib le 
+'directions for its future development.  Because CF O is deterministic, it is more computationally effi cient 
+'than stochastic algorithms and may lend itself wel l to “parameter tuning” implementations.  But, like  
+'all deterministic algorithms, CFO is prone to loca l trapping.  Oscillation in CFO’s Davg curve appear s to 
+'be a reliable harbinger of trapping.  And there se ems to be a reasonable basis for believing that 
+'trapping can be handled deterministically using th e theory of gravitationally trapped Near Earth Obje cts. 
+'Deterministic mitigation of local trapping would b e a major step forward in optimization theory.  Fin ally, 
+'CFO may be thought of as a “gradient-like” algorit hm utilizing the Unit Step function as a critical e lement, 
+'and it is suggested that a useful, new derivative- like mathematical construct might be defined based on the Unit 
+'Step. 
+ 
+'7. "Array Synthesis and Antenna Benchmark Performa nce Using Central Force Optimization," IET (UK) 
+'    (in review) 
+'Abstract: Central force optimization (CFO) is a ne w deterministic multi-dimensional search metaheuris tic 
+'based on an analogy to classical particle kinemati cs in a gravitational field. CFO is a simple techni que 
+'that is still in its infancy.  This paper evaluate s CFO’s performance and provides further examples o f its 
+'effectiveness by applying it to a set of “real wor ld” antenna benchmarks and to pattern synthesis for  linear 
+'and circular array antennas.  A new selection sche me is introduced that enhances CFO’s global search ability 
+'while maintaining its simplicity.  The improved CF O algorithm is applied to the design of a circular array 
+'with very good results.  CFO’s performance on the antenna benchmarks and the synthesis problems is co mpared 
+'to that of other evolutionary optimization techniq ues. 
+'================================================== ===================================================  
+ 
+#COMPILE EXE 
+ 
+#DIM ALL 
+ 
+%USEMACROS = 1 
+ 
+#INCLUDE "Win32API.inc" 
+ 
+DEFEXT A-Z 
+ 
+'------ EQUATES ----- 
+ 
+%IDC_FRAME1     = 101 
+%IDC_FRAME2     = 102 
+ 
+%IDC_Function_Number1  = 121 
+%IDC_Function_Number2  = 122 
+%IDC_Function_Number3  = 123 
+%IDC_Function_Number4  = 124 
+%IDC_Function_Number5  = 125 
+%IDC_Function_Number6  = 126 
+%IDC_Function_Number7  = 127 
+%IDC_Function_Number8  = 128 
+%IDC_Function_Number9  = 129 
+%IDC_Function_Number10 = 130 
+%IDC_Function_Number11 = 131 
+%IDC_Function_Number12 = 132 
+%IDC_Function_Number13 = 133 
+%IDC_Function_Number14 = 134 
+%IDC_Function_Number15 = 135 
+%IDC_Function_Number16 = 136 
+%IDC_Function_Number17 = 137 
+%IDC_Function_Number18 = 138 
+%IDC_Function_Number19 = 139 
+%IDC_Function_Number20 = 140 
+%IDC_Function_Number21 = 141 
+%IDC_Function_Number22 = 142 
+%IDC_Function_Number23 = 143 
+%IDC_Function_Number24 = 144 
+%IDC_Function_Number25 = 145 
+%IDC_Function_Number26 = 146 
+%IDC_Function_Number27 = 147 
+%IDC_Function_Number28 = 148 
+%IDC_Function_Number29 = 149 
+%IDC_Function_Number30 = 150 
+%IDC_Function_Number31 = 151 
+%IDC_Function_Number32 = 152 
+%IDC_Function_Number33 = 153 
+%IDC_Function_Number34 = 154 
+%IDC_Function_Number35 = 155 
+%IDC_Function_Number36 = 156 
+%IDC_Function_Number37 = 157 
+%IDC_Function_Number38 = 158 
+%IDC_Function_Number39 = 159 
+%IDC_Function_Number40 = 160 
+%IDC_Function_Number41 = 161 
+%IDC_Function_Number42 = 162 
+%IDC_Function_Number43 = 163 
+%IDC_Function_Number44 = 164 
+%IDC_Function_Number45 = 165 
+%IDC_Function_Number46 = 166 
+%IDC_Function_Number47 = 167 
+%IDC_Function_Number48 = 168 
+%IDC_Function_Number49 = 169 
+%IDC_Function_Number50 = 170 
+ 
+'------------------------------- GLOBAL CONSTANTS &  SYMBOLS --------------------------- 
+ 
+GLOBAL Aij() AS EXT 'array for Shekel's Foxholes fu nction 
+ 
+GLOBAL EulerConst, Pi, Pi2, Pi4, TwoPi, FourPi, e, Root2 AS EXT 'mathematical constants 
+ 
+GLOBAL Alphabet$, Digits$, RunID$  'upper/lower cas e alphabet, digits 0-9 & Run ID 
+ 
+GLOBAL Quote$, SpecialCharacters$  'quotation mark & special symbols 
+ 
+GLOBAL Mu0, Eps0, c, eta0 AS EXT   'E&M constants 
+ 
+GLOBAL Rad2Deg, Deg2Rad, Feet2Meters, Meters2Feet, Inches2Meters, Meters2Inches AS EXT 'conversion fac tors 
+ 
+GLOBAL Miles2Meters, Meters2Miles, NautMi2Meters, M eters2NautMi AS EXT                 'conversion fac tors 
+ 
+GLOBAL ScreenWidth&, ScreenHeight& 'screen width & height 
+ 
+GLOBAL xOffset&, yOffset&          'offsets for pro be plot windows 36 of 92  
+GLOBAL FunctionNumber% 
+ 
+GLOBAL AddNoiseToPBM2$ 
+ 
+'------------------------------ TEST FUNCTION DECLA RATIONS -------------------------------- 
+ 
+DECLARE FUNCTION F1(R(),Nd%,p%,j&)              'F1  (n-D) 
+ 
+DECLARE FUNCTION F2(R(),Nd%,p%,j&)              'F2 (n-D) 
+ 
+DECLARE FUNCTION F3(R(),Nd%,p%,j&)              'F3  (n-D) 
+ 
+DECLARE FUNCTION F4(R(),Nd%,p%,j&)              'F4  (n-D) 
+ 
+DECLARE FUNCTION F5(R(),Nd%,p%,j&)              'F5  (n-D) 
+ 
+DECLARE FUNCTION F6(R(),Nd%,p%,j&)              'F6  (n-D) 
+ 
+DECLARE FUNCTION F7(R(),Nd%,p%,j&)              'F7  (n-D) 
+ 
+DECLARE FUNCTION F8(R(),Nd%,p%,j&)              'F8  (n-D) 
+ 
+DECLARE FUNCTION F9(R(),Nd%,p%,j&)              'F9  (n-D) 
+ 
+DECLARE FUNCTION F10(R(),Nd%,p%,j&)             'F1 0 (n-D) 
+ 
+DECLARE FUNCTION F11(R(),Nd%,p%,j&)             'F1 1 (n-D) 
+ 
+DECLARE FUNCTION F12(R(),Nd%,p%,j&)             'F1 2 (n-D) 
+ 
+DECLARE FUNCTION F13(R(),Nd%,p%,j&)             'F1 3 (n-D) 
+ 
+DECLARE FUNCTION F14(R(),Nd%,p%,j&)             'F1 4 (n-D) 
+ 
+DECLARE FUNCTION F15(R(),Nd%,p%,j&)             'F1 5 (n-D) 
+ 
+DECLARE FUNCTION F16(R(),Nd%,p%,j&)             'F1 6 (n-D) 
+ 
+DECLARE FUNCTION F17(R(),Nd%,p%,j&)             'F1 7 (n-D) 
+ 
+DECLARE FUNCTION F18(R(),Nd%,p%,j&)             'F1 8 (n-D) 
+ 
+DECLARE FUNCTION F19(R(),Nd%,p%,j&)             'F1 9 (n-D) 
+ 
+DECLARE FUNCTION F20(R(),Nd%,p%,j&)             'F2 0 (n-D) 
+ 
+DECLARE FUNCTION F21(R(),Nd%,p%,j&)             'F2 1 (n-D) 
+ 
+DECLARE FUNCTION F22(R(),Nd%,p%,j&)             'F2 2 (n-D) 
+ 
+DECLARE FUNCTION F23(R(),Nd%,p%,j&)             'F2 3 (n-D) 
+ 
+DECLARE FUNCTION F24(R(),Nd%,p%,j&)             'F2 4 (n-D) 
+ 
+DECLARE FUNCTION F25(R(),Nd%,p%,j&)             'F2 5 (n-D) 
+ 
+DECLARE FUNCTION F26(R(),Nd%,p%,j&)             'F2 6 (n-D) 
+ 
+DECLARE FUNCTION F27(R(),Nd%,p%,j&)             'F2 7 (n-D) 
+ 
+DECLARE FUNCTION ParrottF4(R(),Nd%,p%,j&)       'Pa rrott F4 (1-D) 
+ 
+DECLARE FUNCTION SGO(R(),Nd%,p%,j&)             'SG O Function (2-D) 
+ 
+DECLARE FUNCTION GoldsteinPrice(R(),Nd%,p%,j&)  'Go ldstein-Price Function (2-D) 
+ 
+DECLARE FUNCTION StepFunction(R(),Nd%,p%,j&)    'St ep Function (n-D) 
+ 
+DECLARE FUNCTION Schwefel226(R(),Nd%,p%,j&)     'Sc hwefel Prob. 2.26 (n-D) 
+ 
+DECLARE FUNCTION Colville(R(),Nd%,p%,j&)        'Co lville Function (4-D) 
+ 
+DECLARE FUNCTION Griewank(R(),Nd%,p%,j&)        'Gr iewank (n-D) 
+ 
+DECLARE FUNCTION Himmelblau(R(),Nd%,p%,j&)      'Hi mmelblau (2-D) 
+ 
+DECLARE FUNCTION PBM_1(R(),Nd%,p%,j&)           'PB M Benchmark #1 
+ 
+DECLARE FUNCTION PBM_2(R(),Nd%,p%,j&)           'PB M Benchmark #2 
+ 
+DECLARE FUNCTION PBM_3(R(),Nd%,p%,j&)           'PB M Benchmark #3 
+ 
+DECLARE FUNCTION PBM_4(R(),Nd%,p%,j&)           'PB M Benchmark #4 
+ 
+DECLARE FUNCTION PBM_5(R(),Nd%,p%,j&)           'PB M Benchmark #5 
+ 
+'----------------------------------- SUB DECLARATIO NS ------------------------------------- 
+ 
+DECLARE SUB GetTestFunctionNumber(FunctionName$) 
+ 
+DECLARE SUB FillArrayAij 
+ 
+DECLARE SUB ResetDecisionSpaceBoundaries(Nd%,XiMin( ),XiMax(),StartingXiMin(),StartingXiMax()) 
+ 
+DECLARE SUB Plot3DbestProbeTrajectories(NumTrajecto ries%,M(),R(),XiMin(),XiMax(),Np%,Nd%,LastStep&,Fun ctionName$) 
+ 
+DECLARE SUB Plot2DbestProbeTrajectories(NumTrajecto ries%,M(),R(),XiMin(),XiMax(),Np%,Nd%,LastStep&,Fun ctionName$) 
+ 
+DECLARE SUB Plot2DindividualProbeTrajectories(NumTr ajectories%,M(),R(),XiMin(),XiMax(),Np%,Nd%,LastSte p&,FunctionName$) 
+ 
+DECLARE SUB SelectTestFunction(FunctionName$) 
+ 
+DECLARE SUB Show2Dprobes(R(),Np%,Nt&,j&,XiMin(),XiM ax(),Frep,BestFitness,BestProbeNumber%,BestTimeStep &,FunctionName$,RepositionFactor$,Gamma) 
+ 
+DECLARE SUB StatusWindow(FunctionName$,StatusWindow Handle???) 
+ 
+DECLARE SUB 
+DisplayRunParameters(FunctionName$,Nd%,Np%,Nt&,G,De ltaT,Alpha,Beta,Frep,R(),A(),M(),PlaceInitialProbes $,InitialAcceleration$,RepositionFactor$,RunCF 
+O$,ShrinkDS$,CheckForEarlyTermination$) 
+ 
+DECLARE SUB GetBestFitness(M(),Np%,StepNumber&,Best Fitness,BestProbeNumber%,BestTimeStep&) 
+ 37 of 92 DECLARE SUB 
+Tabulate1DprobeCoordinates(Max1DprobesPlotted%,Nd%, Np%,LastStep&,G,DeltaT,Alpha,Beta,Frep,R(),M(),Plac eInitialProbes$,InitialAcceleration$,Repositio 
+nFactor$,FunctionName$,Gamma) 
+ 
+DECLARE SUB 
+GetPlotAnnotation(PlotAnnotation$,Nd%,Np%,Nt&,G,Del taT,Alpha,Beta,Frep,M(),PlaceInitialProbes$,Initial Acceleration$,RepositionFactor$,FunctionName$, 
+Gamma) 
+ 
+DECLARE SUB 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+DECLARE SUB CLEANUP 
+ 
+DECLARE SUB DisplayBestFitness(Np%,Nd%,LastStep&,M( ),R(),BestFitnessProbeNumber%,BestFitnessTimeStep&, FunctionName$) 
+ 
+DECLARE SUB 
+Plot1DprobePositions(Max1DprobesPlotted%,Nd%,Np%,La stStep&,G,DeltaT,Alpha,Beta,Frep,R(),M(),PlaceIniti alProbes$,InitialAcceleration$,RepositionFacto 
+r$,FunctionName$,Gamma) 
+ 
+DECLARE SUB DisplayMmatrix(Np%,Nt&,M()) 
+ 
+DECLARE SUB DisplayMmatrixThisTimeStep(Np%,j&,M()) 
+ 
+DECLARE SUB DisplayAmatrix(Np%,Nd%,Nt&,A()) 
+ 
+DECLARE SUB DisplayAmatrixThisTimeStep(Np%,Nd%,j&,A ()) 
+ 
+DECLARE SUB DisplayRmatrix(Np%,Nd%,Nt&,R()) 
+ 
+DECLARE SUB DisplayRmatrixThisTimeStep(Np%,Nd%,j&,R ()) 
+ 
+DECLARE SUB DisplayXiMinMax(Nd%,XiMin(),XiMax()) 
+ 
+DECLARE SUB DisplayRunParameters2(FunctionName$,Nd% ,Np%,Nt&,G,DeltaT,Alpha,Beta,Frep,PlaceInitialProbe s$,InitialAcceleration$,RepositionFactor$) 
+ 
+DECLARE SUB 
+PlotBestProbeVsTimeStep(Nd%,Np%,LastStep&,G,DeltaT, Alpha,Beta,Frep,M(),PlaceInitialProbes$,InitialAcce leration$,RepositionFactor$,FunctionName$,Gamm 
+a) 
+ 
+DECLARE SUB 
+PlotBestFitnessEvolution(Nd%,Np%,LastStep&,G,DeltaT ,Alpha,Beta,Frep,M(),PlaceInitialProbes$,InitialAcc eleration$,RepositionFactor$,FunctionName$,Gam 
+ma) 
+ 
+DECLARE SUB 
+PlotAverageDistance(Nd%,Np%,LastStep&,G,DeltaT,Alph a,Beta,Frep,M(),PlaceInitialProbes$,InitialAccelera tion$,RepositionFactor$,FunctionName$,R(),Diag 
+Length,Gamma) 
+ 
+DECLARE SUB Plot2Dfunction(FunctionName$,XiMin(),Xi Max(),R()) 
+ 
+DECLARE SUB Plot1Dfunction(FunctionName$,XiMin(),Xi Max(),R()) 
+ 
+DECLARE SUB 
+GetFunctionRunParameters(FunctionName$,Nd%,Np%,Nt&, G,DeltaT,Alpha,Beta,Frep,R(),A(),M(),XiMin(),XiMax( ),StartingXiMin(),StartingXiMax(),_ 
+                                     DiagLength,Pla ceInitialProbes$,InitialAcceleration$,RepositionFac tor$) 
+ 
+DECLARE SUB InitialProbeDistribution(Np%,Nd%,Nt&,Xi Min(),XiMax(),R(),PlaceInitialProbes$,Gamma) 
+ 
+DECLARE SUB InitialProbeAccelerations(Np%,Nd%,A(),I nitialAcceleration$,MaxInitialRandomAcceleration,Ma xInitialFixedAcceleration) 
+ 
+DECLARE SUB RetrieveErrantProbes(Np%,Nd%,j&,XiMin() ,XiMax(),R(),M(),RepositionFactor$,Frep) 
+ 
+DECLARE SUB CFO(Nd%,Np%,Nt&,G,DeltaT,Alpha,Beta,Fre p,R(),A(),M(),XiMin(),XiMax(),DiagLength,PlaceIniti alProbes$,InitialAcceleration$,_ 
+                
+RepositionFactor$,FunctionName$,LastStep&,CheckForE arlyTermination$,BestFitnessThisRun,RunNumber%,NumR uns%,Gamma,BestOverallFitness,ShrinkDS$,MaxPro 
+besPerAxis%,MinProbesPerAxis%) 
+ 
+DECLARE SUB ThreeDplot(PlotFileName$,PlotTitle$,Ann otation$,xCoord$,yCoord$,zCoord$, _ 
+                       XaxisLabel$,YaxisLabel$,Zaxi sLabel$,zMin$,zMax$,GnuPlotEXE$,A$) 
+ 
+DECLARE SUB ThreeDplot2(PlotFileName$,PlotTitle$,An notation$,xCoord$,yCoord$,zCoord$,XaxisLabel$,_ 
+                        YaxisLabel$,ZaxisLabel$,zMi n$,zMax$,GnuPlotEXE$,A$,xStart$,xStop$,yStart$,ySto p$) 
+ 
+DECLARE SUB TwoDplot(PlotFileName$,PlotTitle$,xCoor d$,yCoord$,XaxisLabel$,YaxisLabel$, _ 
+                     LogXaxis$,LogYaxis$,xMin$,xMax $,yMin$,yMax$,xTics$,yTics$,GnuPlotEXE$,LineType$,A nnotation$) 
+ 
+DECLARE SUB TwoDplot2Curves(PlotFileName1$,PlotFile Name2$,PlotTitle$,Annotation$,xCoord$,yCoord$,Xaxis Label$,YaxisLabel$, _ 
+                            LogXaxis$,LogYaxis$,xMi n$,xMax$,yMin$,yMax$,xTics$,yTics$,GnuPlotEXE$,Line Size) 
+ 
+DECLARE SUB TwoDplot3curves(NumCurves%,PlotFileName 1$,PlotFileName2$,PlotFileName3$,PlotTitle$,Annotat ion$,xCoord$,yCoord$,XaxisLabel$,YaxisLabel$, 
+_ 
+                            LogXaxis$,LogYaxis$,xMi n$,xMax$,yMin$,yMax$,xTics$,yTics$,GnuPlotEXE$) 
+ 
+DECLARE SUB CreateGNUplotINIfile(PlotWindowULC_X%,P lotWindowULC_Y%,PlotWindowWidth%,PlotWindowHeight%)  
+ 
+DECLARE SUB Delay(NumSecs) 
+ 
+DECLARE SUB MathematicalConstants 
+ 
+DECLARE SUB AlphabetAndDigits 
+ 
+DECLARE SUB SpecialSymbols 
+ 
+DECLARE SUB EMconstants 
+ 
+DECLARE SUB ConversionFactors 
+ 
+DECLARE SUB ShowConstants 
+ 
+'------ FUNCTION DECLARATIONS ------- 
+ 
+DECLARE CALLBACK FUNCTION DlgProc 
+ 
+DECLARE FUNCTION HasFITNESSsaturated$(Nsteps&,j&,Np %,Nd%,M(),R(),DiagLength) 
+ 
+DECLARE FUNCTION HasDAVGsaturated$(Nsteps&,j&,Np%,N d%,M(),R(),DiagLength) 
+ 
+DECLARE FUNCTION OscillationInDavg$(j&,Np%,Nd%,M(), R(),DiagLength) 
+ 
+DECLARE FUNCTION DavgThisStep(j&,Np%,Nd%,M(),R(),Di agLength) 
+ 
+DECLARE FUNCTION NoSpaces$(X,NumDigits%) 
+ 
+DECLARE FUNCTION FormatFP$(X,Ndigits%) 38 of 92  
+DECLARE FUNCTION FormatInteger$(M%) 
+ 
+DECLARE FUNCTION TerminateNowForSaturation$(j&,Nd%, Np%,Nt&,G,DeltaT,Alpha,Beta,R(),A(),M()) 
+ 
+DECLARE FUNCTION MagVector(V(),N%) 
+ 
+DECLARE FUNCTION UniformDeviate(u&&) 
+ 
+DECLARE FUNCTION RandomNum(a,b) 
+ 
+DECLARE FUNCTION GaussianDeviate(Mu,Sigma) 
+ 
+DECLARE FUNCTION UnitStep(X) 
+ 
+DECLARE FUNCTION Fibonacci&&(N%) 
+ 
+DECLARE FUNCTION ObjectiveFunction(R(),Nd%,p%,j&,Fu nctionName$) 
+ 
+DECLARE FUNCTION UnitStep(X) 
+ 
+'================================================== ================================================== 
+ 
+'----- MAIN PROGRAM ------ 
+ 
+FUNCTION PBMAIN () AS LONG 
+ 
+'   ------ CFO Parameters ----- 
+ 
+    LOCAL Nd%, Np%, Nt& 
+ 
+    LOCAL G, DeltaT, Alpha, Beta, Frep AS EXT 
+ 
+    LOCAL PlaceInitialProbes$, InitialAcceleration$ , RepositionFactor$ 
+ 
+    LOCAL R(), A(), M() AS EXT    'position, accele ration & fitness matrices 
+ 
+    LOCAL XiMin(), XiMax(), StartingXiMin(), Starti ngXiMax() AS EXT 'decision space boundaries 
+ 
+    LOCAL FunctionName$           'name of objectiv e function 
+ 
+    LOCAL DiagLength AS EXT 
+ 
+'   ------------ Miscellaneuous Setup Parameters -- ------------ 
+ 
+    LOCAL StartingG, StartingDeltaT, StartingAlpha,  StartingBeta, StartingFrep AS EXT 
+ 
+    LOCAL N%, i%, YN& 
+ 
+    LOCAL A$ 
+ 
+    LOCAL NumRuns%, RunNumber%, BestRunNumber%, Tot alFunctionEvaluations&&, AbsoluteRunNumber% 
+ 
+    LOCAL Gamma, StartingGamma, StoppingGamma, Best FitnessThisRun, BestOverallFitness AS EXT 
+ 
+    LOCAL NumTrajectories% 
+ 
+    LOCAL Max1DprobesPlotted% 
+ 
+    LOCAL BestFitnessProbeNumber%, BestFitnessTimeS tep&, BestNumProbes%, MaxProbesPerAxis%, MinProbesP erAxis%, NumProbesPerAxis% 
+ 
+    LOCAL RunCFO$ 
+ 
+    LOCAL StatusWindowHandle??? 
+ 
+    LOCAL LastStep& 
+ 
+    LOCAL CheckForEarlyTermination$ 'early terminat ion checking? (YES/NO) 
+ 
+    LOCAL ShrinkDS$ 'adaptively shrink DS? (YES/NO)  
+ 
+'   -------------------- Global Constants --------- ------------ 
+ 
+    REDIM Aij(1 TO 2, 1 TO 25) '(GLOBAL array for S hekel's Foxholes function) 
+ 
+    CALL FillArrayAij 
+ 
+    CALL MathematicalConstants 'NOTE: Calling order  is important!! 
+ 
+    CALL AlphabetAndDigits 
+ 
+    CALL SpecialSymbols 
+ 
+    CALL EMconstants 
+ 
+    CALL ConversionFactors        ': CALL ShowConst ants 'to verify constants have been set 
+ 
+    xOffset& = 20 : yOffset& = 20 'offsets for succ essive probe position plots 
+ 
+'   --------------------------- General Setup ----- ----------------------- 
+ 
+    RANDOMIZE TIMER  'seed random number generator with program start time 
+ 
+    DESKTOP GET SIZE TO ScreenWidth&, ScreenHeight&   'get screen size (global variables) 
+ 
+    IF DIR$("wgnuplot.exe") = "" THEN 
+ 
+        MSGBOX("WARNING!  'wgnuplot.exe' not found.   Run terminated.") : EXIT FUNCTION 
+ 
+    END IF 
+ 
+'   ------------------- NEC Files Required for PBM Antenna Benchmarks ------------------- 
+ 
+    IF DIR$("n41_2k1.exe") = "" THEN 
+ 
+            MSGBOX("WARNING!  'n41_2k1.exe' not fou nd.  Run terminated.") : EXIT FUNCTION 
+ 
+    END IF 
+ 
+    N% = FREEFILE : OPEN "FILE_MSG.DAT" FOR OUTPUT AS #N% : PRINT #N%, "YES"     : CLOSE #N% 
+ 
+    N% = FREEFILE : OPEN "FILE_MSG.DAT" FOR OUTPUT AS #N% : PRINT #N%, "NO"      : CLOSE #N% 
+ 
+    N% = FREEFILE : OPEN "NHSCALE.DAT"  FOR OUTPUT AS #N% : PRINT #N%, "0.00001" : CLOSE #N% 39 of 92  
+    IF DIR$("ProbeCoordinates.DAT") <> "" THEN KILL  "ProbeCoordinates.DAT" 'get rid of this file to av oid confusion 
+ 
+    IF DIR$("Frep.DAT") <> "" THEN KILL "Frep.DAT" 'ditto 
+ 
+'   ----------------------------------------------- ---------------------------- CFO RUN PARAMETERS --- ---------------------------------------------- 
+--------------------------- 
+ 
+    Max1DprobesPlotted% = 15 : Max1DprobesPlotted% = MIN(15,Max1DprobesPlotted%) '15 MAX! 
+ 
+   ' CALL SelectTestFunction(FunctionName$) 
+ 
+    CALL GetTestFunctionNumber(FunctionName$)' : ex it function 'DEBUG 
+ 
+    'MSGBOX("Test Function is #"+STR$(FunctionNumbe r%)+", "+FunctionName$) 
+ 
+    CALL GetFunctionRunParameters(FunctionName$,Nd% ,Np%,Nt&,G,DeltaT,Alpha,Beta,Frep,R(),A(),M(),XiMin (),XiMax(),StartingXiMin(),StartingXiMax(),_ 
+                                  DiagLength,PlaceI nitialProbes$,InitialAcceleration$,RepositionFactor $) 
+ 
+' !! NOTE !!  IN THIS VERSION Np% IS IGNORED BECAUS E THE PROGRAM LOOPS ON Np/Nd (= 2-8 by 2) AND COMPU TES Np FROM Np = (Np/Nd)*Nd. 
+'             BUT THE PROGRAM STILL RETURNS Np FROM  THIS SUBROUTINE (TOO MUCH OF A BOTHER TO TAKE IT O UT ...). 
+ 
+    StartingG = G : StartingDeltaT = DeltaT : Start ingAlpha = Alpha : StartingBeta = Beta : StartingFr ep = Frep 
+ 
+'   IMPORTANT NOTE: Arrays XiMin() & XiMax() are di mensioned (1 TO Nd%) in SUB GetFunctionRunParameter s 
+ 
+    REDIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&), A(1 TO N p%, 1 TO Nd%, 0 TO Nt&), M(1 TO Np%, 0 TO Nt&) 'pos ition, acceleration & fitness matrices 
+ 
+'   ------------------------ PLOT 1D and 2D FUNCTIO NS ON-SCREEN FOR VISUALIZATION -------------------- ---- 
+ 
+    IF Nd% = 2 AND INSTR(FunctionName$,"PBM_") > 0 THEN MSGBOX("Begin computing plot of function "+Fun ctionName$+"?  May take a while - be 
+patient...") 
+ 
+    SELECT CASE Nd% 
+        CASE 1 : CALL Plot1Dfunction(FunctionName$, XiMin(),XiMax(),R()) : REDIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'erases coordinate data in R()used 
+to plot function 
+        CASE 2 : CALL Plot2Dfunction(FunctionName$, XiMin(),XiMax(),R()) : REDIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'ditto 
+    END SELECT 
+ 
+'   ----------------------------------------------- ------------------ DISPLAY PARAMETERS & RUN CFO --- ---------------------------------------------- 
+----------------------- 
+ 
+    CALL 
+DisplayRunParameters(FunctionName$,Nd%,Np%,Nt&,G,De ltaT,Alpha,Beta,Frep,R(),A(),M(),PlaceInitialProbes $,InitialAcceleration$,RepositionFactor$,RunCF 
+O$,ShrinkDS$,CheckForEarlyTermination$) 
+ 
+    IF RunCFO$ = "YES" THEN 'run CFO 
+ 
+        CALL StatusWindow(FunctionName$,StatusWindo wHandle???) 
+ 
+        StartingGamma = 0## : StoppingGamma = 1## 
+ 
+        NumRuns% = 11 'For Initial Probe Dist'n GAM MA LOOP 
+ 
+        MinProbesPerAxis% = 4 : MaxProbesPerAxis% =  14 'For Initial Probe Dist'n Np/Nd LOOP 
+ 
+'       ---------------- Output Data File & Header ---------------- 
+ 
+        N% = FREEFILE : OPEN FunctionName$+".DAT" F OR OUTPUT AS #N% 
+ 
+        PRINT #N%, "Run ID: "+RunID$+CHR$(13)+CHR$( 13)+"FUNCTION: "+UCASE$(FunctionName$)+CHR$(13)+CHR $(13) 
+ 
+        PRINT #N%, _ 
+        "Run #     Gamma     Nt     Nd     Np      G     DelT   Alpha    Beta    #Steps    Neval       Frep         Fitness       Initial Probes" + 
+CHR$(13) +_ 
+        "-----    -------   ----   ----   ----   -- ---   ----   -----   ------   ------   -------   -- ------    ---------------   ---------------" 
+        A$ = _ 
+        "####     ###.###   ####   ####   ####   ## #.#   ##.#   ##.##    ##.##   ######   #######   #. #####\\ ########.########   \             \" 
+'                                                                                                                UNIFORM ON-AXIS 
+        PRINT #N%, 
+USING$(A$,0,StartingGamma,Nt&,Nd%,Np%,StartingG,Sta rtingDeltaT,StartingAlpha,StartingBeta,0,0,Starting Frep,LEFT$(RepositionFactor$,1),-
+9999,PlaceInitialProbes$) 'header 
+ 
+        PRINT #N%,_ 
+        "------------------------------------------ --------------------------------------------------- -------------------------------------------" 
+ 
+'       -------------------------- Np/Nd LOOP ----- ---------------- 
+ 
+        BestOverallFitness = -1E4200 'very large ne gative number... 
+ 
+        TotalFunctionEvaluations&& = 0 
+ 
+        FOR NumProbesPerAxis% = MinProbesPerAxis% T O MaxProbesPerAxis% STEP 2 '4/6/.../MaxProbesPerAxi s% probes per axis 
+ 
+        Np% = NumProbesPerAxis%*Nd% 'sets Np% regar dless of what Sub GetFunctionRunParameters() return s. 
+ 
+        REDIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&), A(1 TO Np%, 1 TO Nd%, 0 TO Nt&), M(1 TO Np%, 0 TO Nt&) 'must redim these matrices because Np has 
+changed 
+ 
+'       -------------------------- GAMMA LOOP ----- ---------------- 
+ 
+        FOR RunNumber% = 1 TO NumRuns% 
+ 
+            Gamma = StartingGamma + (RunNumber%-1)* (StoppingGamma-StartingGamma)/(NumRuns%-1) 
+ 
+            G = StartingG : DeltaT = StartingDeltaT  : Alpha = StartingAlpha : Beta = StartingBeta : Fr ep = StartingFrep 
+ 
+            CALL ResetDecisionSpaceBoundaries(Nd%,X iMin(),XiMax(),StartingXiMin(),StartingXiMax()) 
+ 
+            CALL CFO(Nd%,Np%,Nt&,G,DeltaT,Alpha,Bet a,Frep,R(),A(),M(),XiMin(),XiMax(),DiagLength,Place InitialProbes$,InitialAcceleration$,_ 
+                     
+RepositionFactor$,FunctionName$,LastStep&,CheckForE arlyTermination$,BestFitnessThisRun,RunNumber%,NumR uns%,Gamma,BestOverallFitness,ShrinkDS$,MaxPro 
+besPerAxis%,MinProbesPerAxis%) 
+ 
+            IF BestFitnessThisRun >= BestOverallFit ness THEN 
+ 
+                BestOverallFitness = BestFitnessThi sRun : BestRunNumber% = RunNumber% : BestNumProbes%  = Np% 
+ 
+'MSGBOX("Best Fitness="+STR$(BestOverallFitness)+"  Best Run#="+STR$(BestRunNumber%)) 
+ 
+            END IF 
+ 
+            TotalFunctionEvaluations&& = TotalFunct ionEvaluations&& + (LastStep&+1)*Np% 
+ 40 of 92             AbsoluteRunNumber% = RunNumber% + ((Np% \Nd%)\2-MinProbesPerAxis%\2)*NumRuns% 
+ 
+'msgbox("Run#="+STR$(RunNumber%)+"    Abs Run #="+S TR$(AbsoluteRunNumber%)) 
+ 
+            PRINT #N%, USING$(A$,AbsoluteRunNumber% ,Gamma,Nt&,Nd%,Np%,G,DeltaT,Alpha,Beta,LastStep&,(L aststep&+1)*Np%,Frep,_ 
+                              LEFT$(RepositionFacto r$,1),BestFitnessThisRun,PlaceInitialProbes$) 
+ 
+        NEXT RunNumber% 'GAMMA LOOP 
+ 
+        NEXT NumProbesPerAxis% 'Np/Nd LOOP 
+ 
+        PRINT #N%, CHR$(13)+USING$("                                                 Total Function Ev aluations: 
+###########",TotalFunctionEvaluations&&)+CHR$(13) 
+ 
+'       ------------------------------------------- - Re-Run Best Run for Final Results --------------- --------------------------------- 
+ 
+        Gamma = StartingGamma + (BestRunNumber%-1)* (StoppingGamma-StartingGamma)/(NumRuns%-1) 
+ 
+MSGBOX("  Best Run#="+STR$(BestRunNumber%)+"   Gamm a="+STR$(Gamma)) 
+ 
+        G = StartingG : DeltaT = StartingDeltaT : A lpha = StartingAlpha : Beta = StartingBeta : Frep =  StartingFrep : Np% = BestNumProbes% 
+ 
+        REDIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&), A(1 TO Np%, 1 TO Nd%, 0 TO Nt&), M(1 TO Np%, 0 TO Nt&) 'must redim these matrices because Np has 
+changed 
+ 
+        CALL ResetDecisionSpaceBoundaries(Nd%,XiMin (),XiMax(),StartingXiMin(),StartingXiMax()) 
+ 
+        CALL CFO(Nd%,Np%,Nt&,G,DeltaT,Alpha,Beta,Fr ep,R(),A(),M(),XiMin(),XiMax(),DiagLength,PlaceInit ialProbes$,InitialAcceleration$,_ 
+                 
+RepositionFactor$,FunctionName$,LastStep&,CheckForE arlyTermination$,BestFitnessThisRun,BestRunNumber%, NumRuns%,Gamma,BestOverallFitness,ShrinkDS$,Ma 
+xProbesPerAxis%,MinProbesPerAxis%) 
+ 
+        CALL ResetDecisionSpaceBoundaries(Nd%,XiMin (),XiMax(),StartingXiMin(),StartingXiMax()) 
+ 
+        PRINT #N%,"-------------------------------- --------------------------------------------------- ---------------------------------------------- 
+-------" 
+ 
+        AbsoluteRunNumber% = BestRunNumber% + ((Np% \Nd%)\2-MinProbesPerAxis%\2)*NumRuns% 
+ 
+        PRINT #N%, USING$(A$,AbsoluteRunNumber%,Gam ma,Nt&,Nd%,BestNumProbes%,G,DeltaT,Alpha,Beta,LastS tep&,(Laststep&+1)*Np%,Frep,_ 
+                          LEFT$(RepositionFactor$,1 ),BestFitnessThisRun,PlaceInitialProbes$) 
+        CLOSE #N% 
+ 
+    ELSE 
+ 
+        GOTO ExitPBMAIN 
+ 
+    END IF 
+ 
+'   -------------------------------- Display Best F itness ----------------------------------------- 
+ 
+    CALL DisplayBestFitness(Np%,Nd%,LastStep&,M(),R (),BestFitnessProbeNumber%,BestFitnessTimeStep&,Fun ctionName$) 
+ 
+'   -----------------------------------------------  PLOT EVOLUTION OF BEST FITNESS, AVG DISTANCE & BES T PROBE # ------------------------------------ 
+-------------------- 
+ 
+    CALL 
+PlotBestFitnessEvolution(Nd%,Np%,LastStep&,G,DeltaT ,Alpha,Beta,Frep,M(),PlaceInitialProbes$,InitialAcc eleration$,RepositionFactor$,FunctionName$,Gam 
+ma) 
+ 
+    CALL 
+PlotAverageDistance(Nd%,Np%,LastStep&,G,DeltaT,Alph a,Beta,Frep,M(),PlaceInitialProbes$,InitialAccelera tion$,RepositionFactor$,FunctionName$,R(),Diag 
+Length,Gamma) 
+ 
+    CALL 
+PlotBestProbeVsTimeStep(Nd%,Np%,LastStep&,G,DeltaT, Alpha,Beta,Frep,M(),PlaceInitialProbes$,InitialAcce leration$,RepositionFactor$,FunctionName$,Gamm 
+a) 
+ 
+'   ---------------------------------------------- PLOT TRAJECTORIES OF BEST PROBES FOR 2/3-D FUNCTION S -------------------------------------------- 
+------- 
+ 
+    IF Nd% = 2 THEN 
+ 
+        NumTrajectories% = 10 : CALL Plot2DbestProb eTrajectories(NumTrajectories%,M(),R(),XiMin(),XiMa x(),Np%,Nd%,LastStep&,FunctionName$) 
+ 
+        NumTrajectories% = 16 : CALL Plot2Dindividu alProbeTrajectories(NumTrajectories%,M(),R(),XiMin( ),XiMax(),Np%,Nd%,LastStep&,FunctionName$) 
+ 
+    END IF 
+ 
+    IF Nd% = 3 THEN 
+ 
+        NumTrajectories% = 4 : CALL Plot3DbestProbe Trajectories(NumTrajectories%,M(),R(),XiMin(),XiMax (),Np%,Nd%,LastStep&,FunctionName$) 
+ 
+    END IF 
+ 
+'   ----------- For 1-D Objective Functions, Tabula te Probe Coordinates & if Np% =< Max1DprobesPlotted % Plot Evolution of Probe Positions ---------- 
+-- 
+ 
+    IF Nd% = 1 THEN 
+ 
+        CALL 
+Tabulate1DprobeCoordinates(Max1DprobesPlotted%,Nd%, Np%,LastStep&,G,DeltaT,Alpha,Beta,Frep,R(),M(),Plac eInitialProbes$,InitialAcceleration$,Repositio 
+nFactor$,FunctionName$,Gamma) 
+ 
+        IF Np% =< Max1DprobesPlotted% THEN CALL 
+Plot1DprobePositions(Max1DprobesPlotted%,Nd%,Np%,La stStep&,G,DeltaT,Alpha,Beta,Frep,R(),M(),PlaceIniti alProbes$,InitialAcceleration$,RepositionFacto 
+r$,FunctionName$,Gamma) 
+ 
+        CALL CLEANUP 'delete probe coordinate plot files, if any 
+ 
+    END IF 
+ 
+ExitPBMAIN: 
+ 
+END FUNCTION 'PBMAIN() 
+ 
+'================================================== ===================================  CFO SUBROUTINE   
+=================================================== ====================================== 
+ 
+SUB CFO(Nd%,Np%,Nt&,G,DeltaT,Alpha,Beta,Frep,R(),A( ),M(),XiMin(),XiMax(),DiagLength,PlaceInitialProbes $,InitialAcceleration$,_ 
+        
+RepositionFactor$,FunctionName$,LastStep&,CheckForE arlyTermination$,BestFitnessThisRun,RunNumber%,NumR uns%,Gamma,BestOverallFitness,ShrinkDS$,MaxPro 
+besPerAxis%,MinProbesPerAxis%) 
+ 41 of 92 LOCAL p%, i%, j& 'Standard Indices: Probe #, Coordi nate #, Time Step # 
+ 
+LOCAL k%, L%     'Dummy summation indices 
+ 
+LOCAL SumSQ, Denom, Numerator, MaxInitialRandomAcce leration, MaxInitialFixedAcceleration AS EXT 
+ 
+LOCAL StepNumber&, BestProbeNumber%, BestTimeStep& 
+ 
+LOCAL BestFitness AS EXT 
+ 
+LOCAL DavgOscillation$, DavgSaturation$, FitnessSat uration$ 
+ 
+LOCAL Nsteps& 
+ 
+'STEP (A1) ------------- Compute Initial Probe Dist ribution (Step 0)---------------- 
+ 
+    CALL InitialProbeDistribution(Np%,Nd%,Nt&,XiMin (),XiMax(),R(),PlaceInitialProbes$,Gamma) 
+ 
+    IF Nd% = 2 THEN 'plot 2-D initial probe distrib ution at Step 0 
+ 
+        CALL CreateGNUplotINIfile(0.1##*ScreenWidth &,0.25##*ScreenHeight&,0.6##*ScreenHeight&,0.6##*Sc reenHeight&) 
+ 
+        CALL Show2Dprobes(R(),Np%,Nt&,0,XiMin(),XiM ax(),Frep,0##,1,0,FunctionName$,RepositionFactor$,G amma) 'show initial probes for 2-D functions 
+ 
+'       msgbox("Press ENTER to continue.") 
+ 
+    END IF 
+ 
+'STEP (A2) ------------- Compute Initial Fitness Ma trix (Step 0) ------------- 
+ 
+    FOR p% = 1 TO Np% : M(p%,0) = ObjectiveFunction (R(),Nd%,p%,0,FunctionName$) : NEXT p% 
+ 
+'STEP (A3) ------------- Compute Initial Probe Acce lerations (Step 0)--------------- 
+ 
+    MaxInitialRandomAcceleration =  2## 'maximum va lue for random initial acceleration: Random[0-MAX] 
+ 
+    MaxInitialFixedAcceleration  = 10## 'maximum va lue for fixed initial acceleration: [0.001-MAX] 
+ 
+    CALL InitialProbeAccelerations(Np%,Nd%,A(),Init ialAcceleration$,MaxInitialRandomAcceleration,MaxIn itialFixedAcceleration) 
+ 
+'   ======================================= LOOP ON  TIME STEPS STARTING AT STEP #1 =================== ======================== 
+ 
+    LastStep& = Nt& 'unless run is terminated earli er 
+ 
+    BestFitnessThisRun = M(1,0) 
+ 
+    FOR j& = 1 TO Nt& 
+ 
+'STEP (B) ----------- Compute Probe Position Vector s for this Time Step -------- 
+ 
+        FOR p% = 1 TO Np% : FOR i% = 1 TO Nd% : R(p %,i%,j&) = R(p%,i%,j&-1) + 0.5##*A(p%,i%,j&-1)*Delt aT^2 : NEXT i% : NEXT p% 
+ 
+'STEP (C) ----------- Retrieve Errant Probes ------ --------- 
+ 
+        CALL RetrieveErrantProbes(Np%,Nd%,j&,XiMin( ),XiMax(),R(),M(),RepositionFactor$,Frep) 
+ 
+'STEP (D) ----------- Compute Fitness Matrix for Cu rrent Probe Distribution --------- 
+ 
+        FOR p% = 1 TO Np% : M(p%,j&) = ObjectiveFun ction(R(),Nd%,p%,j&,FunctionName$) : NEXT p% 
+ 
+'STEP (E) ----------- Compute Accelerations Based o n Current Probe Distribution & Fitnesses ---------- ------ 
+ 
+        FOR p% = 1 TO Np% 
+ 
+            FOR i% = 1 TO Nd% 
+ 
+                A(p%,i%,j&) = 0 
+ 
+                FOR k% = 1 TO Np% 
+ 
+                    IF k% <> p% THEN 
+ 
+                        SumSQ = 0## 
+ 
+                        FOR L% = 1 TO Nd%  : SumSQ = SumSQ + (R(k%,L%,j&)-R(p%,L%,j&))^2 : NEXT L% 'du mmy index 
+ 
+                        Denom = SQR(SumSQ) : Numera tor = UnitStep((M(k%,j&)-M(p%,j&)))*(M(k%,j&)-M(p%, j&)) 
+ 
+                        A(p%,i%,j&) = A(p%,i%,j&) +  G*(R(k%,i%,j&)-R(p%,i%,j&))*Numerator^Alpha/Denom^ Beta 
+ 
+                    END IF 
+ 
+                NEXT k% 'dummy index 
+ 
+            NEXT i% 'coord (dimension) # 
+ 
+        NEXT p% 'probe # 
+ 
+'   --------- Get Best Fitness & Corresponding Prob e # and Time Step --------- 
+ 
+    CALL GetBestFitness(M(),Np%,j&,BestFitness,Best ProbeNumber%,BestTimeStep&) 
+ 
+    IF BestFitness >= BestFitnessThisRun THEN BestF itnessThisRun = BestFitness 
+ 
+'   ----------------------------------------- Displ ay Best Fitness/Probe#/Time Step and Plot Probe Pos itions for 2-D Functions --------------------- 
+---------------------- 
+ 
+'    IF Nd% = 2 AND ((j& MOD MAX(Nt&\16,4) = 0 OR ( j& =< 16 AND j& MOD 2 = 0) OR j& = Nt&)) THEN 'plot  2D probes at selected time steps 
+ 
+'        xOffset& = xOffset& - 20 : yOffset& = yOff set& + 10 
+ 
+'        CALL CreateGNUplotINIfile(0.55##*ScreenWid th&+xOffset&,0.01##*ScreenHeight&+yOffset&,0.6##*Sc reenHeight&,0.6##*ScreenHeight&) 
+ 
+'        CALL Show2Dprobes(R(),Np%,Nt&,j&,XiMin(),X iMax(),Frep,BestFitness,BestProbeNumber%,BestTimeSt ep&,FunctionName$,RepositionFactor$,Gamma) 
+'plot positions of probes in 2-D 
+ 
+'    END IF 
+ 
+'   ---------------------- Check for Davg OSC and F itness/Davg SAT &  Display Run Status ------------- ------------- 
+ 
+    DavgOscillation$   = OscillationInDavg$(j&,Np%, Nd%,M(),R(),DiagLength)           'check for oscill ation in Davg 
+ 
+    Nsteps& = 50 '25 'for 100D tests 'used 50 for n ew paper '15 '# steps for averaging Davg & Best Fit ness to test for saturation 
+ 42 of 92     DavgSaturation$    = HasDAVGsaturated$(Nsteps&, j&,Np%,Nd%,M(),R(),DiagLength)    'check for satura tion of Davg 
+ 
+    FitnessSaturation$ = HasFITNESSsaturated$(Nstep s&,j&,Np%,Nd%,M(),R(),DiagLength) 'check for satura tion of best fitness 
+ 
+    GRAPHIC SET PIXEL (35,15) : GRAPHIC PRINT "Run #"    + REMOVE$(STR$(RunNumber%+((Np%\Nd%)\2-MinPro besPerAxis%\2)*NumRuns%),ANY" ")     + "/" 
++REMOVE$(STR$(((MaxProbesPerAxis%-MinProbesPerAxis% )\2+1)*NumRuns%),ANY" ") +_ 
+                                              ", Np /Nd=" + REMOVE$(STR$(Np%\Nd%),ANY" ") +_ 
+                                              ", St ep #" + REMOVE$(STR$(j&),ANY" ")             + "/" + REMOVE$(STR$(Nt&),ANY" ")     +_ 
+                                              ", Ga mma=" + REMOVE$(STR$(ROUND(Gamma,4)),ANY" ") +_ 
+                                              ", Fr ep="  + REMOVE$(STR$(Frep,4),ANY" ")         + STRI NG$(10," ") 
+ 
+    GRAPHIC SET PIXEL (35,35) : GRAPHIC PRINT "Best  Fitness This Run = " + REMOVE$(STR$(ROUND(BestFitn ess,8)),ANY" ") +_ 
+                                              " @ P robe #" + REMOVE$(STR$(BestProbeNumber%),ANY" ") + ", Step #" + REMOVE$(STR$(BestTimeStep&),ANY" 
+") + STRING$(15," ") 
+ 
+    IF RunNumber% = 1 AND Np%\Nd% = 2 THEN 
+ 
+        GRAPHIC SET PIXEL (35,55) : GRAPHIC PRINT " Best Fitness Overall = N/A @ Run #1" + STRING$(15,"  ") 
+ 
+    ELSE 
+ 
+        GRAPHIC SET PIXEL (35,55) : GRAPHIC PRINT " Best Fitness Overall = " + REMOVE$(STR$(ROUND(BestO verallFitness,8)),ANY" ") + STRING$(15," ") 
+ 
+    END IF 
+ 
+    GRAPHIC SET PIXEL (35,75) : GRAPHIC PRINT "Osc Davg? " +  DavgOscillation$ + ",  Sat Davg? " +  Da vgSaturation$ +",  Sat Fitness? " + 
+FitnessSaturation$ + STRING$(10," "): GRAPHIC REDRA W 
+ 
+'   -------------------------------------- If Varia ble Frep, Adjust Value----------------------------- -------------- 
+ 
+    IF RepositionFactor$ = "VARIABLE" THEN 
+ 
+        Frep = Frep + 0.05## 
+ 
+        IF Frep > 1## THEN Frep = 0.05## 'keep Frep  in range [0.05,1] 
+ 
+    END IF 
+ 
+    IF RepositionFactor$ = "RANDOM" THEN Frep = Ran domNum(0##,1##) 'set new Frep value randomly 
+ 
+'   ------------------------------- If DavgOSC => R eposition Probes ------------------------------ 
+ 
+'   ---------------------------- If DavgSauration = > Reposition Probes ------------------------------ 
+ 
+'   IF DavgSaturation$ = "YES" and j& < Nt& THEN 
+ 
+'   IF DavgOscillation$ = "YES" THEN 
+ 
+'        FOR p% = 1 TO Np% 
+ 
+'            If p% = BestProbeNumber% then 
+ 
+'                FOR i% = 1 TO Nd% 
+ 
+'                    R(p%,i%,j&) = 0.5##*R(p%,i%,j& ) 'adjust each coordinate by 10% 
+ 
+'                    R(p%,i%,j&) = R(p%,i%,j&)*(1## +GaussianDeviate(0##,0.1##)) 'adjust each coordinat e randomly (mu,sigma) 
+ 
+'                   IF R(p%,i%,j&) < XiMin(i%) THEN  R(p%,i%,j&) = XiMin(i%) 
+'                   IF R(p%,i%,j&) > XiMax(i%) THEN  R(p%,i%,j&) = XiMax(i%) 
+ 
+'                NEXT i% 
+ 
+'            end if 
+ 
+'        NEXT p% 
+ 
+'        CALL RetrieveErrantProbes(Np%,Nd%,j&,XiMin (),XiMax(),R(),M(),RepositionFactor$,Frep) 
+ 
+'    END IF 
+ 
+'   ------------------------------------ Every 20 S teps Shrink Decision Space Around Best Probe ------ ------------------------------ 
+ 
+    IF j& MOD 20 = 0 AND ShrinkDS$ = "YES" THEN 
+'    IF j& MOD 30 = 0 THEN 
+ 
+        FOR i% = 1 TO Nd% 
+ 
+            XiMin(i%) = XiMin(i%)+(R(BestProbeNumbe r%,i%,BestTimeStep&)-XiMin(i%))/2## : XiMax(i%) = X iMax(i%)-(XiMax(i%)-
+R(BestProbeNumber%,i%,BestTimeStep&))/2## 
+ 
+        NEXT i% 
+ 
+'       LastStep& = j& : EXIT FOR  'time step loop 
+ 
+    END IF 
+ 
+'   ------------------------------------ If DavgOSC  => Increase All Probes' Acceleration to Break Away  from Trapping(?)  --------------------------- 
+------ 
+ 
+'    IF DavgOscillation$ = "YES" THEN 
+ 
+ '       For p% = 1 to Np% 
+ 
+  '          For i% = 1 to Nd% 
+ 
+   '             A(p%,i%,j&) = 1##*A(p%,i%,j&) 'fac tor of XX increase 
+ 
+    '        next i% 
+ 
+     '   next p% 
+ 
+    'END IF 
+ 
+'STEP (F) -------------------- Check for Early Run Termination --------------------- 
+ 
+    IF CheckForEarlyTermination$ = "YES" THEN 'inse rt termination test 
+ 
+        IF FitnessSaturation = "YES" THEN 'terminat e run 
+ 
+            LastStep& = j& : EXIT FOR  'time step l oop 
+ 
+        END IF 
+ 43 of 92     END IF 
+ 
+    NEXT j& 'END OF TIME STEP LOOP 
+ 
+END SUB 'CFO() 
+ 
+'================================================== =================================================== ============================================ 
+ 
+SUB GetBestFitness(M(),Np%,StepNumber&,BestFitness, BestProbeNumber%,BestTimeStep&) 
+ 
+LOCAL p%, i&, A$ 
+ 
+    BestFitness = M(1,0) 
+ 
+    FOR i& = 0 TO StepNumber& 
+ 
+        FOR p% = 1 TO Np% 
+ 
+            IF M(p%,i&) >= BestFitness THEN 
+ 
+                BestFitness = M(p%,i&) : BestProbeN umber% = p% : BestTimeStep& = i& 
+ 
+            END IF 
+ 
+        NEXT p% 
+ 
+    NEXT i& 
+ 
+END SUB 
+ 
+'================================================== =========== FUNCTION DEFINITIONS ================== ============================================ 
+ 
+SUB SelectTestFunction(FunctionName$) 
+ 
+LOCAL A$ 
+ 
+    A$ = INPUTBOX$("Which Function?"+CHR$(13)+"1 - Parrott F4 (1-D)"+CHR$(13)+"2 - SGO (2-D)"+CHR$(13) +"3 - GP (2-D)"+CHR$(13)+"4 - Step (n-
+D)"+CHR$(13)+ _ 
+                   "5 - Schwefel 2.26 (n-D)"+CHR$(1 3)+"6 - Colville (4-D)"+CHR$(13)+"7 - RESERVED"+CHR $(13)+"8 - more","SELECT OBJECTIVE 
+FUNCTION","1") 
+ 
+    IF VAL(A$) < 1 OR VAL(A$) > 8 THEN A$ = "1" 
+ 
+    SELECT CASE VAL(A$) 
+ 
+        CASE 1 : FunctionName$ = "ParrottF4" 
+        CASE 2 : FunctionName$ = "SGO" 
+        CASE 3 : FunctionName$ = "GP" 
+        CASE 4 : FunctionName$ = "STEP" 
+        CASE 5 : FunctionName$ = "SCHWEFEL_226" 
+        CASE 6 : FunctionName$ = "COLVILLE" 
+        CASE 7 : FunctionName$ = "GRIEWANK" 
+        CASE 8 : FunctionName$ = "more" 
+ 
+    END SELECT 
+ 
+    IF FunctionName$ = "more" THEN 
+ 
+        A$ = INPUTBOX$("Which Function?"+CHR$(13)+" 1 - F1"+CHR$(13)+"2 - F2"+CHR$(13)+"3 - F3"+CHR$(13 )+"4 - F4"+CHR$(13)+ _ 
+                   "5 - F5"+CHR$(13)+"6 - F6"+CHR$( 13)+"7 - F7"+CHR$(13)+"8 - more","SELECT OBJECTIVE FUNCTION","1") 
+ 
+        IF VAL(A$) < 1 OR VAL(A$) > 8 THEN A$ = "1"  
+ 
+        SELECT CASE VAL(A$) 
+ 
+            CASE 1 : FunctionName$ = "F1" 
+            CASE 2 : FunctionName$ = "F2" 
+            CASE 3 : FunctionName$ = "F3" 
+            CASE 4 : FunctionName$ = "F4" 
+            CASE 5 : FunctionName$ = "F5" 
+            CASE 6 : FunctionName$ = "F6" 
+            CASE 7 : FunctionName$ = "F7" 
+            CASE 8 : FunctionName$ = "more" 
+ 
+        END SELECT 
+ 
+    END IF 
+ 
+        IF FunctionName$ = "more" THEN 
+ 
+        A$ = INPUTBOX$("Which Function?"+CHR$(13)+" 1 - F8"+CHR$(13)+"2 - F9"+CHR$(13)+"3 - F10"+CHR$(1 3)+"4 - F11"+CHR$(13)+ _ 
+                   "5 - F12"+CHR$(13)+"6 - F13"+CHR $(13)+"7 - F14"+CHR$(13)+"8 - more","SELECT OBJECTI VE FUNCTION","1") 
+ 
+        IF VAL(A$) < 1 OR VAL(A$) > 8 THEN A$ = "1"  
+ 
+        SELECT CASE VAL(A$) 
+ 
+            CASE 1 : FunctionName$ = "F8" 
+            CASE 2 : FunctionName$ = "F9" 
+            CASE 3 : FunctionName$ = "F10" 
+            CASE 4 : FunctionName$ = "F11" 
+            CASE 5 : FunctionName$ = "F12" 
+            CASE 6 : FunctionName$ = "F13" 
+            CASE 7 : FunctionName$ = "F14" 
+            CASE 8 : FunctionName$ = "more" 
+ 
+        END SELECT 
+ 
+    END IF 
+ 
+    IF FunctionName$ = "more" THEN 
+ 
+        A$ = INPUTBOX$("Which Function?"+CHR$(13)+" 1 - F15"+CHR$(13)+"2 - F16"+CHR$(13)+"3 - F17"+CHR$ (13)+"4 - F18"+CHR$(13)+ _ 
+                   "5 - F19"+CHR$(13)+"6 - F20"+CHR $(13)+"7 - F21"+CHR$(13)+"8 - more","SELECT OBJECTI VE FUNCTION","1") 
+ 
+        IF VAL(A$) < 1 OR VAL(A$) > 8 THEN A$ = "1"  
+ 
+        SELECT CASE VAL(A$) 
+ 
+            CASE 1 : FunctionName$ = "F15" 
+            CASE 2 : FunctionName$ = "F16" 
+            CASE 3 : FunctionName$ = "F17" 
+            CASE 4 : FunctionName$ = "F18" 
+            CASE 5 : FunctionName$ = "F19" 44 of 92             CASE 6 : FunctionName$ = "F20" 
+            CASE 7 : FunctionName$ = "F21" 
+            CASE 8 : FunctionName$ = "more" 
+ 
+        END SELECT 
+ 
+    END IF 
+ 
+    IF FunctionName$ = "more" THEN 
+ 
+        A$ = INPUTBOX$("Which Function?"+CHR$(13)+" 1 - F22"+CHR$(13)+"2 - F23"+CHR$(13)+"3 - F24"+CHR$ (13)+"4 - F25"+CHR$(13)+ _ 
+                   "5 - F26"+CHR$(13)+"6 - F27"+CHR $(13)+"7 - F28"+CHR$(13)+"8 - more","SELECT OBJECTI VE FUNCTION","1") 
+ 
+        IF VAL(A$) < 1 OR VAL(A$) > 2 THEN A$ = "1"  
+ 
+        SELECT CASE VAL(A$) 
+ 
+            CASE 1 : FunctionName$ = "F22" 
+            CASE 2 : FunctionName$ = "F23" 
+            CASE 3 : FunctionName$ = "F24" 
+            CASE 4 : FunctionName$ = "F25" 
+            CASE 5 : FunctionName$ = "F26" 
+            CASE 6 : FunctionName$ = "F27" 
+            CASE 7 : FunctionName$ = "F28" 
+            CASE 8 : FunctionName$ = "more" 
+ 
+        END SELECT 
+ 
+    END IF 
+ 
+END SUB 'SelectTestFunction() 
+ 
+'------ 
+ 
+FUNCTION ObjectiveFunction(R(),Nd%,p%,j&,FunctionNa me$) 'Objective function to be MAXIMIZED is defined  here 
+ 
+    SELECT CASE FunctionName$ 
+ 
+        CASE "ParrottF4"     : ObjectiveFunction = ParrottF4(R(),Nd%,p%,j&)      'Parrott F4 (1-D) 
+ 
+        CASE "SGO"           : ObjectiveFunction = SGO(R(),Nd%,p%,j&)            'SGO Function (2-D) 
+ 
+        CASE "GP"            : ObjectiveFunction = GoldsteinPrice(R(),Nd%,p%,j&) 'Goldstein-Price Func tion (2-D) 
+ 
+        CASE "STEP"          : ObjectiveFunction = StepFunction(R(),Nd%,p%,j&)   'Step Function (n-D) 
+ 
+        CASE "SCHWEFEL_226"  : ObjectiveFunction = Schwefel226(R(),Nd%,p%,j&)    'Schwefel Prob. 2.26 (n-D) 
+ 
+        CASE "COLVILLE"      : ObjectiveFunction = Colville(R(),Nd%,p%,j&)       'Colville Function (4 -D) 
+ 
+        CASE "GRIEWANK"      : ObjectiveFunction = Griewank(R(),Nd%,p%,j&)       'Griewank Function (n -D) 
+ 
+        CASE "HIMMELBLAU"    : ObjectiveFunction = Himmelblau(R(),Nd%,p%,j&)     'Himmelblau Function (2-D) 
+ 
+'       ------------------------- GSO Paper Benchma rk Functions --------------------------- 
+        CASE  "F1"           : ObjectiveFunction = F1(R(),Nd%,p%,j&)             'F1  (n-D) 
+        CASE  "F2"           : ObjectiveFunction = F2(R(),Nd%,p%,j&)             'F2  (n-D) 
+        CASE  "F3"           : ObjectiveFunction = F3(R(),Nd%,p%,j&)             'F3  (n-D) 
+        CASE  "F4"           : ObjectiveFunction = F4(R(),Nd%,p%,j&)             'F4  (n-D) 
+        CASE  "F5"           : ObjectiveFunction = F5(R(),Nd%,p%,j&)             'F5  (n-D) 
+        CASE  "F6"           : ObjectiveFunction = F6(R(),Nd%,p%,j&)             'F6  (n-D) 
+        CASE  "F7"           : ObjectiveFunction = F7(R(),Nd%,p%,j&)             'F7  (n-D) 
+        CASE  "F8"           : ObjectiveFunction = F8(R(),Nd%,p%,j&)             'F8  (n-D) 
+        CASE  "F9"           : ObjectiveFunction = F9(R(),Nd%,p%,j&)             'F9  (n-D) 
+        CASE "F10"           : ObjectiveFunction = F10(R(),Nd%,p%,j&)            'F10 (n-D) 
+        CASE "F11"           : ObjectiveFunction = F11(R(),Nd%,p%,j&)            'F11 (n-D) 
+        CASE "F12"           : ObjectiveFunction = F12(R(),Nd%,p%,j&)            'F12 (n-D) 
+        CASE "F13"           : ObjectiveFunction = F13(R(),Nd%,p%,j&)            'F13 (n-D) 
+        CASE "F14"           : ObjectiveFunction = F14(R(),Nd%,p%,j&)            'F14 (2-D) 
+        CASE "F15"           : ObjectiveFunction = F15(R(),Nd%,p%,j&)            'F15 (4-D) 
+        CASE "F16"           : ObjectiveFunction = F16(R(),Nd%,p%,j&)            'F16 (2-D) 
+        CASE "F17"           : ObjectiveFunction = F17(R(),Nd%,p%,j&)            'F17 (2-D) 
+        CASE "F18"           : ObjectiveFunction = F18(R(),Nd%,p%,j&)            'F18 (2-D) 
+        CASE "F19"           : ObjectiveFunction = F19(R(),Nd%,p%,j&)            'F19 (3-D) 
+        CASE "F20"           : ObjectiveFunction = F20(R(),Nd%,p%,j&)            'F20 (6-D) 
+        CASE "F21"           : ObjectiveFunction = F21(R(),Nd%,p%,j&)            'F21 (4-D) 
+        CASE "F22"           : ObjectiveFunction = F22(R(),Nd%,p%,j&)            'F22 (4-D) 
+        CASE "F23"           : ObjectiveFunction = F23(R(),Nd%,p%,j&)            'F23 (4-D) 
+ 
+'       ------------------------------ PBM Antenna Benchmarks ------------------------------ 
+ 
+        CASE "PBM_1"         : ObjectiveFunction = PBM_1(R(),Nd%,p%,j&)          'PBM_1 (2-D) 
+        CASE "PBM_2"         : ObjectiveFunction = PBM_2(R(),Nd%,p%,j&)          'PBM_2 (2-D) 
+        CASE "PBM_3"         : ObjectiveFunction = PBM_3(R(),Nd%,p%,j&)          'PBM_3 (2-D) 
+        CASE "PBM_4"         : ObjectiveFunction = PBM_4(R(),Nd%,p%,j&)          'PBM_4 (2-D) 
+        CASE "PBM_5"         : ObjectiveFunction = PBM_5(R(),Nd%,p%,j&)          'PBM_5 (2-D) 
+ 
+    END SELECT 
+ 
+END FUNCTION 'ObjectiveFunction() 
+ 
+'----------- 
+ 
+SUB ResetDecisionSpaceBoundaries(Nd%,XiMin(),XiMax( ),StartingXiMin(),StartingXiMax()) 
+ 
+LOCAL i% 
+ 
+    FOR i% = 1 TO Nd% : XiMin(i%) = StartingXiMin(i %) : XiMax(i%) = StartingXiMax(i%) : NEXT i% 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB GetFunctionRunParameters(FunctionName$,Nd%,Np%, Nt&,G,DeltaT,Alpha,Beta,Frep,R(),A(),M(),XiMin(),Xi Max(),StartingXiMin(),StartingXiMax(),_ 
+                             DiagLength,PlaceInitia lProbes$,InitialAcceleration$,RepositionFactor$) 
+ 
+LOCAL i%, NumProbesPerDimension%, NN%, NumCollinear Elements% 
+ 
+    SELECT CASE FunctionName$ 
+ 
+        CASE "ParrottF4" 
+ 
+            Nd%                    = 1 
+            NumProbesPerDimension% = 3 45 of 92             Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.9## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "FIXED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 1 'cannot change dimensionality o f Parrott F4 function! 
+ 
+            NumProbesPerDimension% = MAX(NumProbesP erDimension%,3) 'at least three for 1-D functions 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : XiMin(1) = 0## : XiMax(1) = 1## 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+        CASE "SGO" 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '10 '4 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " '"2D GRID" 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 2 'cannot change dimensionality o f SGO function! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -50## : XiMax(i%)  = 50## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 
+ 
+        CASE "GP" 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '10 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## '0.2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## '0.8## '0.9## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " '"2D GRID" 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 2 'cannot change dimensionality o f GP function! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -100## : XiMax(i% ) = 100## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 
+ 
+        CASE "STEP" 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 46 of 92             InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -100## : XiMax(i% ) = 100## : NEXT i% 
+ 
+'           REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : XiMin(1) = 72## : XiMax(1) = 78## : XiMin(2) = 27 ## : XiMax(2) = 33## 'use this to plot STEP 
+detail 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'STEP 
+ 
+        CASE "SCHWEFEL_226" 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 4 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -500## : XiMax(i% ) = 500## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 
+ 
+        CASE "COLVILLE" 
+ 
+            Nd%                    = 4 
+            NumProbesPerDimension% = 4 '14 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 4 'cannot change dimensionality o f Colville function! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -10## : XiMax(i%)  = 10## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 
+ 
+        CASE "GRIEWANK" 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '14 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 47 of 92             Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -600## : XiMax(i% ) = 600## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 
+ 
+       CASE "HIMMELBLAU" 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '14 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 2 'cannot change dimensionality o f Himmelblau function! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -6## : XiMax(i%) = 6## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 
+ 
+        CASE "F1" '(n-D) 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -100## : XiMax(i% ) = 100## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 
+ 
+       CASE "F2" '(n-D) 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -10## : XiMax(i%)  = 10## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 48 of 92  
+            END IF 
+ 
+      CASE "F3" '(n-D) 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -100## : XiMax(i% ) = 100## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 
+ 
+      CASE "F4" '(n-D) 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -100## : XiMax(i% ) = 100## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 
+ 
+      CASE "F5" '(n-D) 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -30## : XiMax(i%)  = 30## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 
+ 
+       CASE "F6" '(n-D) STEP 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 49 of 92             Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -100## : XiMax(i% ) = 100## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 
+ 
+       CASE "F7" '(n-D) 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 100          'BECAUSE THIS F UNCTION HAS A RANDOM COMPONENT !! 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -1.28## : XiMax(i %) = 1.28## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'F7 
+ 
+       CASE "F8" '(n-D) 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 '4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -500## : XiMax(i% ) = 500## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'F8 
+ 
+       CASE "F9" '(n-D) 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 '4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 50 of 92             Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -5.12## : XiMax(i %) = 5.12## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'F9 
+ 
+       CASE "F10" '(n-D) Ackley's Function 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 '4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -32## : XiMax(i%)  = 32## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'F10 
+ 
+       CASE "F11" '(n-D) 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 '4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -600## : XiMax(i% ) = 600## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'F11 
+ 
+       CASE "F12" '(n-D) Penalized #1 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 '4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -50## : XiMax(i%)  = 50## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'F12 51 of 92  
+       CASE "F13" '(n-D) Penalized #2 
+ 
+            Nd%                    = 30 
+            NumProbesPerDimension% = 2 '4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -50## : XiMax(i%)  = 50## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'F13 
+ 
+       CASE "F14" '(2-D) Shekel's Foxholes 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 2 'cannot change dimensionality o f Shekel's Foxholes function! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -65.536## : XiMax (i%) = 65.536## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'F14 
+ 
+       CASE "F15" '(4-D) Kowalik's Function 
+ 
+            Nd%                    = 4 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 4 'cannot change dimensionality o f Kowalik's Function! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -5## : XiMax(i%) = 5## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+       CASE "F16" '(2-D) Camel Back 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 52 of 92             PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 2 'cannot change dimensionality o f Camel Back function! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = -5## : XiMax(i%) = 5## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'F16 
+ 
+       CASE "F17" '(2-D) Branin 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 2 'cannot change dimensionality o f Branin function! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : XiMin(1) = -5## : XiMax(1) = 10## : XiMin(2) = 0# # : XiMax(2) = 15## 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'F17 
+ 
+       CASE "F18" '(2-D) Goldstein-Price 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 2 'cannot change dimensionality o f Branin function! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : XiMin(1) = -2## : XiMax(1) = 2## : XiMin(2) = -2# # : XiMax(2) = 2## 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            IF PlaceInitialProbes$ = "2D GRID" THEN  
+ 
+                Np% = NumProbesPerDimension%^2 : RE DIM R(1 TO Np%, 1 TO Nd%, 0 TO Nt&) 'to create (Np/ Nd) x (Np/Nd) grid 
+ 
+            END IF 'F18 
+ 
+       CASE "F19" '(3-D) Hartman's Family #1 
+ 
+            Nd%                    = 3 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 53 of 92             CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 3 'cannot change dimensionality o f Hartman's Family! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = 0## : XiMax(i%) =  1## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+       CASE "F20" '(6-D) Hartman's Family #2 
+ 
+            Nd%                    = 6 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 6 'cannot change dimensionality o f Hartman's Family! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = 0## : XiMax(i%) =  1## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+       CASE "F21" '(4-D) Shekel's Family m=5 
+ 
+            Nd%                    = 4 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 4 'cannot change dimensionality o f Shekel's Family! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = 0## : XiMax(i%) =  10## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+       CASE "F22" '(4-D) Shekel's Family m=7 
+ 
+            Nd%                    = 4 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 4 'cannot change dimensionality o f Shekel's Family! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = 0## : XiMax(i%) =  10## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+       CASE "F23" '(4-D) Shekel's Family m=10 
+ 
+            Nd%                    = 4 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 500 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 54 of 92             Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 4 'cannot change dimensionality o f Shekel's Family! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = 0## : XiMax(i%) =  10## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+         CASE "PBM_1" '2-D 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 2 '4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 100 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 2 'cannot change dimensionality o f PBM_1! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) 
+ 
+            XiMin(1) = 0.5## : XiMax(1) = 3## 'dipo le length, L, in Wavelengths 
+            XiMin(2) = 0##   : XiMax(2) = Pi2 'pola r angle, Theta, in Radians 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            NN% = FREEFILE : OPEN "INFILE.DAT" FOR OUTPUT AS #NN% : PRINT #NN%,"PBM1.NEC" : PRINT #NN% ,"PBM1.OUT" : CLOSE #NN% 'NEC Input/Output 
+Files 
+ 
+        CASE "PBM_2" '2-D 
+ 
+            AddNoiseToPBM2$ = "NO" '"YES" '"NO" '"Y ES" 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 100 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 2 'cannot change dimensionality o f PBM_2! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) 
+ 
+            XiMin(1) = 5## : XiMax(1) = 15## 'dipol e separation, D, in Wavelengths 
+            XiMin(2) = 0## : XiMax(2) = Pi   'polar  angle, Theta, in Radians 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            NN% = FREEFILE : OPEN "INFILE.DAT" FOR OUTPUT AS #NN% : PRINT #NN%,"PBM2.NEC" : PRINT #NN% ,"PBM2.OUT" : CLOSE #NN% 
+ 
+        CASE "PBM_3" '2-D 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 100 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 2 'cannot change dimensionality o f PBM_3! 55 of 92  
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) 
+ 
+            XiMin(1) = 0## : XiMax(1) = 4## 'Phase Parameter, Beta (0-4) 
+            XiMin(2) = 0## : XiMax(2) = Pi  'polar angle, Theta, in Radians 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            NN% = FREEFILE : OPEN "INFILE.DAT" FOR OUTPUT AS #NN% : PRINT #NN%,"PBM3.NEC" : PRINT #NN% ,"PBM3.OUT" : CLOSE #NN% 
+ 
+        CASE "PBM_4" '2-D 
+ 
+            Nd%                    = 2 
+            NumProbesPerDimension% = 4 '6 '2 '4 '20  
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 100 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = 2 'cannot change dimensionality o f PBM_4! 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) 
+ 
+            XiMin(1) = 0.5##   : XiMax(1) = 1.5##  'ARM LENGTH (NOT Total Length), wavelengths (0.5-1. 5) 
+            XiMin(2) = Pi/18## : XiMax(2) = Pi/2## 'Inner angle, Alpha, in Radians (Pi/18-Pi/2) 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            NN% = FREEFILE : OPEN "INFILE.DAT" FOR OUTPUT AS #NN% : PRINT #NN%,"PBM4.NEC" : PRINT #NN% ,"PBM4.OUT" : CLOSE #NN% 
+ 
+        CASE "PBM_5" 
+ 
+            NumCollinearElements%  = 6 '30 'EVEN or  ODD: 6,7,10,13,16,24 used by PBM 
+ 
+            Nd%                    = NumCollinearEl ements% - 1 
+            NumProbesPerDimension% = 4 '20 
+            Np%                    = NumProbesPerDi mension%*Nd% 
+ 
+            Nt&      = 100 
+            G        = 2## 
+            Alpha    = 2## 
+            Beta     = 2## 
+            DeltaT   = 1## 
+            Frep     = 0.5## 
+ 
+            PlaceInitialProbes$  = "UNIFORM ON-AXIS " 
+            InitialAcceleration$ = "ZERO" 
+            RepositionFactor$    = "VARIABLE" '"FIX ED" 
+ 
+            CALL 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 
+            Nd% = NumCollinearElements% - 1 
+ 
+            Np% = NumProbesPerDimension%*Nd% 
+ 
+            REDIM XiMin(1 TO Nd%), XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : XiMin(i%) = 0.5## : XiMax(i%)  = 1.5## : NEXT i% 
+ 
+            REDIM StartingXiMin(1 TO Nd%), Starting XiMax(1 TO Nd%) : FOR i% = 1 TO Nd% : StartingXiMin (i%) = XiMin(i%) : StartingXiMax(i%) = 
+XiMax(i%) : NEXT i% 
+ 
+            NN% = FREEFILE : OPEN "INFILE.DAT" FOR OUTPUT AS #NN% : PRINT #NN%,"PBM5.NEC" : PRINT #NN% ,"PBM5.OUT" : CLOSE #NN% 
+ 
+'   =============================================== ======================================= 
+'   NOTE - DON'T FORGET TO ADD NEW TEST FUNCTIONS T O FUNCTION ObjectiveFunction() ABOVE !! 
+'   =============================================== ======================================= 
+ 
+    END SELECT 
+ 
+    IF Nd% > 100 THEN Nt& = MIN(Nt&,200) 'to avoid array dimensioning problems 
+ 
+    DiagLength = 0## : FOR i% = 1 TO Nd% : DiagLeng th = DiagLength + (XiMax(i%)-XiMin(i%))^2 : NEXT i%  : DiagLength = SQR(DiagLength) 'compute 
+length of decision space principal diagonal 
+ 
+END SUB 'GetFunctionRunParameters() 
+ 
+'-------------------------------- 
+ 
+FUNCTION ParrottF4(R(),Nd%,p%,j&) 'Parrott F4 (1-D)  
+ 
+'MAXIMUM = 1 AT ~0.0796875... WITH ZERO OFFSET. 
+ 
+'References: 
+ 
+'Beasley, D., D. R. Bull, and R. R. Martin, “A Sequ ential Niche Technique for Multimodal 
+'Function Optimization,” Evol. Comp. (MIT Press), v ol. 1, no. 2, 1993, pp. 101-125 
+'(online at http://citeseer.ist.psu.edu/beasley93se quential.html). 
+ 
+'Parrott, D., and X. Li, “Locating and Tracking Mul tiple Dynamic Optima by a Particle Swarm 
+'Model Using Speciation,” IEEE Trans. Evol. Computa tion, vol. 10, no. 4, Aug. 2006, pp. 440-458. 
+ 
+ 
+LOCAL Z, x, offset AS EXT 
+ 
+    offset = 0## 56 of 92  
+    x = R(p%,1,j&) 
+ 
+    Z = EXP(-2##*LOG(2##)*((x-0.08##-offset)/0.854# #)^2)*(SIN(5##*Pi*((x-offset)^0.75##-0.05##)))^6 'W ARNING! This is a NATURAL LOG, NOT Log10!!! 
+ 
+    ParrottF4 = Z 
+ 
+END FUNCTION 'ParrottF4() 
+ 
+'----------------------------- 
+ 
+FUNCTION SGO(R(),Nd%,p%,j&) 'SGO Function (2-D) 
+ 
+'MAXIMUM = ~130.8323226... @ ~(-2.8362075...,-2.836 2075...) WITH ZERO OFFSET. 
+ 
+'Reference: 
+ 
+'Hsiao, Y., Chuang, C., Jiang, J., and Chien, C., " A Novel Optimization Algorithm: Space 
+'Gravitational Optimization," Proc. of 2005 IEEE In ternational Conference on Systems, Man, 
+'and Cybernetics, 3, 2323-2328. (2005) 
+ 
+    LOCAL x1, x2, Z, t1, t2, SGOx1offset, SGOx2offs et AS EXT 
+ 
+    SGOx1offset = 0## : SGOx2offset = 0## 
+ 
+'   SGOx1offset = 40## : SGOx2offset = 10## 
+ 
+    x1 = R(p%,1,j&) - SGOx1offset : x2 = R(p%,2,j&)  - SGOx2offset 
+ 
+    t1 = x1^4 - 16##*x1^2 + 0.5##*x1 : t2 = x2^4 - 16##*x2^2 + 0.5##*x2 
+ 
+    Z = t1 + t2 
+ 
+    SGO = -Z 
+ 
+END FUNCTION 'SGO() 
+ 
+'------------------ 
+ 
+FUNCTION GoldsteinPrice(R(),Nd%,p%,j&) 'Goldstein-P rice Function (2-D) 
+ 
+'MAXIMUM = -3 @ (0,-1) WITH ZERO OFFSET. 
+ 
+'Reference: 
+ 
+'Cui, Z., Zeng, J., and Sun, G. (2006) ‘A Fast Part icle Swarm Optimization,’ Int’l. J. 
+'Innovative Computing, Information and Control, vol . 2, no. 6, December, pp. 1365-1380. 
+ 
+    LOCAL Z, x1, x2, offset1, offset2, t1, t2 AS EX T 
+ 
+    offset1 = 0## : offset2 = 0## 
+ 
+'    offset1 = 20## : offset2 = -10## 
+ 
+    x1 = R(p%,1,j&)-offset1 : x2 = R(p%,2,j&)-offse t2 
+ 
+    t1 = 1##+(x1+x2+1##)^2*(19##-14##*x1+3##*x1^2-1 4##*x2+6##*x1*x2+3##*x2^2) 
+ 
+    t2 = 30##+(2##*x1-3##*x2)^2*(18##-32##*x1+12##* x1^2+48##*x2-36##*x1*x2+27##*x2^2) 
+ 
+    Z  = t1*t2 
+ 
+    GoldsteinPrice = -Z 
+ 
+END FUNCTION 'GoldesteinPrice() 
+ 
+'----------- 
+ 
+FUNCTION StepFunction(R(),Nd%,p%,j&) 'Step Function  (n-D) 
+ 
+'MAXIMUM VALUE = 0 @ [Offset]^n. 
+ 
+'Reference: 
+ 
+'Yao, X., Liu, Y., and Lin, G., “Evolutionary Progr amming Made Faster,” 
+'IEEE Trans. Evolutionary Computation, Vol. 3, No. 2, 82-102, Jul. 1999. 
+ 
+ 
+    LOCAL Offset, Z AS EXT 
+ 
+    LOCAL i% 
+ 
+    Z = 0## : Offset = 0## '75.123## '0## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        IF Nd% = 2 AND i% = 1 THEN Offset = 75 '75# # 
+ 
+        IF Nd% = 2 AND i% = 2 THEN Offset = 35 '30 '35## 
+ 
+        Z = Z + INT((R(p%,i%,j&)-Offset) + 0.5##)^2  
+ 
+    NEXT i% 
+ 
+    StepFunction = -Z 
+ 
+END FUNCTION 'StepFunction() 
+ 
+'----------- 
+ 
+FUNCTION Schwefel226(R(),Nd%,p%,j&) 'Schwefel Probl em 2.26 (n-D) 
+ 
+'MAXIMUM = 12,569.5 @ [420.8687]^30 (30-D CASE). 
+ 
+'Reference: 
+ 
+'Yao, X., Liu, Y., and Lin, G., “Evolutionary Progr amming Made Faster,” 
+'IEEE Trans. Evolutionary Computation, Vol. 3, No. 2, 82-102, Jul. 1999. 
+ 
+    LOCAL Z, Xi AS EXT 
+ 
+    LOCAL i% 
+ 
+    Z = 0## 57 of 92  
+    FOR i% = 1 TO Nd% 
+ 
+        Xi = R(p%,i%,j&) 
+ 
+        Z = Z + Xi*SIN(SQR(ABS(Xi))) 
+ 
+    NEXT i% 
+ 
+    Schwefel226 = Z 
+ 
+END FUNCTION 'SCHWEFEL226() 
+ 
+'----------- 
+ 
+FUNCTION Colville(R(),Nd%,p%,j&) 'Colville Function  (4-D) 
+ 
+'MAXIMUM = 0 @ (1,1,1,1) WITH ZERO OFFSET. 
+ 
+'Reference: 
+ 
+'Doo-Hyun, and Se-Young, O., “A New Mutation Rule f or Evolutionary Programming Motivated from 
+'Backpropagation Learning,” IEEE Trans. Evolutionar y Computation, Vol. 4, No. 2, pp. 188-190, 
+'July 2000. 
+ 
+    LOCAL Z, x1, x2, x3, x4, offset AS EXT 
+ 
+    offset = 0## '7.123## 
+ 
+    x1 = R(p%,1,j&)-offset : x2 = R(p%,2,j&)-offset  : x3 = R(p%,3,j&)-offset : x4 = R(p%,4,j&)-offset 
+ 
+    Z =  100##*(x2-x1^2)^2 + (1##-x1)^2  + _ 
+          90##*(x4-x3^2)^2 + (1##-x3)^2  + _ 
+        10.1##*((x2-1##)^2 + (x4-1##)^2) + _ 
+        19.8##*(x2-1##)*(x4-1##) 
+ 
+    Colville = -Z 
+ 
+END FUNCTION 'Colville() 
+ 
+'----------- 
+ 
+'Max of zero at (0,...,0) 
+ 
+FUNCTION Griewank(R(),Nd%,p%,j&) 'Griewank (n-D) 
+ 
+    LOCAL Offset, Sum, Prod, Z, Xi AS EXT 
+ 
+    LOCAL i% 
+ 
+    Sum = 0## : Prod = 1## 
+ 
+    Offset = 75.123## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Xi = R(p%,i%,j&) - Offset 
+ 
+        Sum = Sum + Xi^2 
+ 
+        Prod = Prod*COS(Xi/SQR(i%)) 
+ 
+    NEXT i% 
+ 
+    Z = Sum/4000## - Prod + 1## 
+ 
+    Griewank = -Z 
+ 
+END FUNCTION 'Griewank() 
+ 
+'----------- 
+ 
+FUNCTION Himmelblau(R(),Nd%,p%,j&) 'Himmelblau (2-D ) 
+ 
+    LOCAL Z, x1, x2, offset AS EXT 
+ 
+    offset = 0## 
+ 
+    x1 = R(p%,1,j&)-offset : x2 = R(p%,2,j&)-offset  
+ 
+    Z = 200## - (x1^2 + x2 -11##)^2 - (x1+x2^2-7##) ^2 
+ 
+    Himmelblau = Z 
+ 
+END FUNCTION 'Himmelblau() 
+ 
+'----------- 
+ 
+FUNCTION F1(R(),Nd%,p%,j&) 'F1 (n-D) 
+ 
+'MAXIMUM = ZERO (n-D CASE). 
+ 
+'Reference: 
+ 
+    LOCAL Z, Xi AS EXT 
+ 
+    LOCAL i% 
+ 
+    Z = 0## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Xi = R(p%,i%,j&) 
+ 
+        Z = Z + Xi^2 
+ 
+    NEXT i% 
+ 
+    F1 = -Z 
+ 
+END FUNCTION 'F1 
+ 
+'----------- 
+ 58 of 92 FUNCTION F2(R(),Nd%,p%,j&) 'F2 (n-D) 
+ 
+'MAXIMUM = ZERO (n-D CASE). 
+ 
+'Reference: 
+ 
+    LOCAL Sum, prod, Z, Xi AS EXT 
+ 
+    LOCAL i% 
+ 
+    Z = 0## : Sum = 0## : Prod = 1## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Xi = R(p%,i%,j&) 
+ 
+        Sum  = Sum+ ABS(Xi) 
+ 
+        Prod = Prod*ABS(Xi) 
+ 
+    NEXT i% 
+ 
+    Z = Sum + Prod 
+ 
+    F2 = -Z 
+ 
+END FUNCTION 'F2 
+ 
+'----------- 
+ 
+FUNCTION F3(R(),Nd%,p%,j&) 'F3 (n-D) 
+ 
+'MAXIMUM = ZERO (n-D CASE). 
+ 
+'Reference: 
+ 
+    LOCAL Z, Xk, Sum AS EXT 
+ 
+    LOCAL i%, k% 
+ 
+    Z = 0## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Sum = 0## 
+ 
+        FOR k% = 1 TO i% 
+ 
+            Xk = R(p%,k%,j&) 
+ 
+            Sum = Sum + Xk 
+ 
+        NEXT k% 
+ 
+        Z = Z + Sum^2 
+ 
+    NEXT i% 
+ 
+    F3 = -Z 
+ 
+END FUNCTION 'F3 
+ 
+ 
+'----------- 
+ 
+FUNCTION F4(R(),Nd%,p%,j&) 'F4 (n-D) 
+ 
+'MAXIMUM = ZERO (n-D CASE). 
+ 
+'Reference: 
+ 
+    LOCAL Z, Xi, MaxXi AS EXT 
+ 
+    LOCAL i% 
+ 
+    MaxXi = -1E4200 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Xi = R(p%,i%,j&) 
+ 
+        IF ABS(Xi) >= MaxXi THEN MaxXi = ABS(Xi) 
+ 
+    NEXT i% 
+ 
+    F4 = -MaxXi 
+ 
+END FUNCTION 'F4 
+ 
+'----------- 
+ 
+FUNCTION F5(R(),Nd%,p%,j&) 'F5 (n-D) 
+ 
+'MAXIMUM = ZERO (n-D CASE). 
+ 
+'Reference: 
+ 
+    LOCAL Z, Xi, XiPlus1 AS EXT 
+ 
+    LOCAL i% 
+ 
+    Z = 0## 
+ 
+    FOR i% = 1 TO Nd%-1 
+ 
+        Xi      = R(p%,i%,j&) 
+ 
+        XiPlus1 = R(p%,i%+1,j&) 
+ 
+        Z = Z + (100##*(XiPlus1-Xi^2)^2+(Xi-1##))^2  
+ 
+    NEXT i% 
+ 
+    F5 = -Z 59 of 92  
+END FUNCTION 'F5 
+ 
+'----------- 
+ 
+FUNCTION F6(R(),Nd%,p%,j&) 'F6 
+ 
+'MAXIMUM VALUE = 0 @ [Offset]^n. 
+ 
+'Reference: 
+ 
+'Yao, X., Liu, Y., and Lin, G., “Evolutionary Progr amming Made Faster,” 
+'IEEE Trans. Evolutionary Computation, Vol. 3, No. 2, 82-102, Jul. 1999. 
+ 
+ 
+    LOCAL Z AS EXT 
+ 
+    LOCAL i% 
+ 
+    Z = 0## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Z = Z + INT(R(p%,i%,j&) + 0.5##)^2 
+ 
+    NEXT i% 
+ 
+    F6 = -Z 
+ 
+END FUNCTION 'F6 
+ 
+'----------- 
+ 
+FUNCTION F7(R(),Nd%,p%,j&) 'F7 
+ 
+'MAXIMUM VALUE = 0 @ [Offset]^n. 
+ 
+'Reference: 
+ 
+'Yao, X., Liu, Y., and Lin, G., “Evolutionary Progr amming Made Faster,” 
+'IEEE Trans. Evolutionary Computation, Vol. 3, No. 2, 82-102, Jul. 1999. 
+ 
+ 
+    LOCAL Z, Xi AS EXT 
+ 
+    LOCAL i% 
+ 
+    Z = 0## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Xi = R(p%,i%,j&) 
+ 
+        Z = Z + i%*Xi^4 
+ 
+    NEXT i% 
+ 
+    F7 = -Z - RandomNum(0##,1##) 
+ 
+END FUNCTION 'F7 
+ 
+'----------- 
+ 
+FUNCTION F8(R(),Nd%,p%,j&) '(n-D) F8 [Schwefel Prob lem 2.26] 
+ 
+'MAXIMUM = 12,569.5 @ [420.8687]^30 (30-D CASE). 
+ 
+'Reference: 
+ 
+'Yao, X., Liu, Y., and Lin, G., “Evolutionary Progr amming Made Faster,” 
+'IEEE Trans. Evolutionary Computation, Vol. 3, No. 2, 82-102, Jul. 1999. 
+ 
+    LOCAL Z, Xi AS EXT 
+ 
+    LOCAL i% 
+ 
+    Z = 0## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Xi = R(p%,i%,j&) 
+ 
+        Z  = Z - Xi*SIN(SQR(ABS(Xi))) 
+ 
+    NEXT i% 
+ 
+    F8 = -Z 
+ 
+END FUNCTION 'F8 
+ 
+'----------- 
+ 
+FUNCTION F9(R(),Nd%,p%,j&) '(n-D) F9 [Rastrigin] 
+ 
+'MAXIMUM = ZERO (n-D CASE). 
+ 
+'Reference: 
+ 
+'Yao, X., Liu, Y., and Lin, G., “Evolutionary Progr amming Made Faster,” 
+'IEEE Trans. Evolutionary Computation, Vol. 3, No. 2, 82-102, Jul. 1999. 
+ 
+    LOCAL Z, Xi AS EXT 
+ 
+    LOCAL i% 
+ 
+    Z = 0## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Xi = R(p%,i%,j&) 
+ 
+        Z  = Z + (Xi^2 - 10##*COS(TwoPi*Xi) + 10##) ^2 
+ 
+    NEXT i% 60 of 92  
+    F9 = -Z 
+ 
+END FUNCTION 'F9 
+ 
+'----------- 
+ 
+FUNCTION F10(R(),Nd%,p%,j&) '(n-D) F10 [Ackley's Fu nction] 
+ 
+'MAXIMUM = ZERO (n-D CASE). 
+ 
+'Reference: 
+ 
+'Yao, X., Liu, Y., and Lin, G., “Evolutionary Progr amming Made Faster,” 
+'IEEE Trans. Evolutionary Computation, Vol. 3, No. 2, 82-102, Jul. 1999. 
+ 
+    LOCAL Z, Xi, Sum1, Sum2 AS EXT 
+ 
+    LOCAL i% 
+ 
+    Z = 0## : Sum1 = 0## : Sum2 = 0## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Xi   = R(p%,i%,j&) 
+ 
+        Sum1 = Sum1 + Xi^2 
+ 
+        Sum2 = Sum2 + COS(TwoPi*Xi) 
+ 
+    NEXT i% 
+ 
+    Z = -20##*EXP(-0.2##*SQR(Sum1/Nd%)) - EXP(Sum2/ Nd%) + 20## + e 
+ 
+    F10 = -Z 
+ 
+END FUNCTION 'F10 
+ 
+'----------- 
+ 
+FUNCTION F11(R(),Nd%,p%,j&) '(n-D) F11 
+ 
+'MAXIMUM = ZERO (n-D CASE). 
+ 
+'Reference: 
+ 
+'Yao, X., Liu, Y., and Lin, G., “Evolutionary Progr amming Made Faster,” 
+'IEEE Trans. Evolutionary Computation, Vol. 3, No. 2, 82-102, Jul. 1999. 
+ 
+    LOCAL Z, Xi, Sum, Prod AS EXT 
+ 
+    LOCAL i% 
+ 
+    Z = 0## : Sum = 0## : Prod = 1## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Xi   = R(p%,i%,j&) 
+ 
+        Sum  = Sum + (Xi-100##)^2 
+ 
+        Prod = Prod*COS((Xi-100##)/SQR(i%)) 
+ 
+    NEXT i% 
+ 
+    Z = Sum/4000## - Prod + 1## 
+ 
+    F11 = -Z 
+ 
+END FUNCTION 'F11 
+ 
+'----- 
+ 
+FUNCTION F12(R(),Nd%,p%,j&) '(n-D) F12, Penalized # 1 
+ 
+    LOCAL Offset, Sum1, Sum2, Z, u, Xi, Xn, XiPlus1 , Yi, YiPlus1, Yn, X1, Y1, a, k AS EXT 
+ 
+    LOCAL i%, m% 
+ 
+    a = 10## : k = 100## : m% = 4  '[NOTE IN CORREC TION 02-03-2010: Coefficient “a” was incorrectly se t to 5 in the original version!] 
+ 
+    X1 = R(p%,1,j&) : Y1 = 1## + (X1+1##)/4## 
+ 
+    Sum1 = 10##*SIN(Pi*Y1)^2 + (Yn-1##)^2 
+ 
+    FOR i% = 1 TO Nd%-1 
+ 
+        Xi = R(p%,i%,j&) : XiPlus1 = R(p%,i%+1,j&) : Xn = R(p%,Nd%,j&) 
+ 
+        Yi = 1## + (Xi+1##)/4## : YiPlus1 = 1## + ( XiPlus1+1##)/4## : Yn = 1## + (Xn+1##)/4## 
+ 
+        Sum1 = Sum1 + (Yi-1##)^2*(1##+10##*SIN(Pi*Y iPlus1)^2) 
+ 
+    NEXT i% 
+ 
+    Sum1 = Pi*Sum1/Nd% 
+ 
+    Sum2 = 0## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Xi = R(p%,i%,j&) 
+ 
+        u = 0## 
+ 
+        IF Xi >  a THEN u = k*(Xi-a)^m% 
+ 
+        IF Xi < -a THEN u = k*(-Xi-a)^m% 
+ 
+        Sum2 = Sum2 + u 
+ 
+    NEXT i% 
+ 
+    Z = Sum1 + Sum2 61 of 92  
+    F12 = -Z 
+ 
+END FUNCTION 'F12() 
+ 
+'----- 
+ 
+FUNCTION F13(R(),Nd%,p%,j&) '(n-D) F13, Penalized # 2 
+ 
+    LOCAL Offset, Sum1, Sum2, Z, u, Xi, Xn, Xi1, X1 , a, k AS EXT 
+ 
+    LOCAL i%, m% 
+ 
+    a = 5## : k = 100## : m% = 4 
+ 
+    X1 = R(p%,1,j&) 
+ 
+    Xn = R(p%,Nd%,j&) 
+ 
+    Sum1 = 0## 
+ 
+    FOR i% = 1 TO Nd%-1 
+ 
+        Xi  = R(p%,i%,j&) 
+ 
+        Xi1 = R(p%,i%+1,j&) 
+ 
+        Sum1 = Sum1 + (Xi-1##)^2*(1##+(SIN(3##*Pi*X i1))^2)+(Xn-1##)^2*(1##+(SIN(TwoPi*Xn))^2) 
+ 
+    NEXT i% 
+ 
+    Sum2 = 0## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Xi = R(p%,i%,j&) 
+        u = 0## 
+        IF Xi >  a THEN u = k*(Xi-a)^m% 
+        IF Xi < -a THEN u = k*(-Xi-a)^m% 
+        Sum2 = Sum2 + u 
+ 
+    NEXT i% 
+ 
+    Z = ((SIN(3##*Pi*X1))^2+Sum1)/10## + Sum2 
+ 
+    F13 = -Z 
+ 
+END FUNCTION 'F13() 
+ 
+'----- 
+ 
+SUB FillArrayAij  'needed for function F14, Shekel' s Foxholes 
+ 
+    Aij(1,1)=-32##  : Aij(1,2)=-16##  : Aij(1,3)=0# #  : Aij(1,4)=16##  : Aij(1,5)=32## 
+    Aij(1,6)=-32##  : Aij(1,7)=-16##  : Aij(1,8)=0# #  : Aij(1,9)=16##  : Aij(1,10)=32## 
+    Aij(1,11)=-32## : Aij(1,12)=-16## : Aij(1,13)=0 ## : Aij(1,14)=16## : Aij(1,15)=32## 
+    Aij(1,16)=-32## : Aij(1,17)=-16## : Aij(1,18)=0 ## : Aij(1,19)=16## : Aij(1,20)=32## 
+    Aij(1,21)=-32## : Aij(1,22)=-16## : Aij(1,23)=0 ## : Aij(1,24)=16## : Aij(1,25)=32## 
+ 
+    Aij(2,1)=-32##  : Aij(2,2)=-32## : Aij(2,3)=-32 ## : Aij(2,4)=-32## : Aij(2,5)=-32## 
+    Aij(2,6)=-16##  : Aij(2,7)=-16## : Aij(2,8)=-16 ## : Aij(2,9)=-16## : Aij(2,10)=-16## 
+    Aij(2,11)=0##   : Aij(2,12)=0##  : Aij(2,13)=0# #  : Aij(2,14)=0##  : Aij(2,15)=0## 
+    Aij(2,16)=16##  : Aij(2,17)=16## : Aij(2,18)=16 ## : Aij(2,19)=16## : Aij(2,20)=16## 
+    Aij(2,21)=32##  : Aij(2,22)=32## : Aij(2,23)=32 ## : Aij(2,24)=32## : Aij(2,25)=32# 
+ 
+END SUB 
+ 
+'----- 
+ 
+FUNCTION F14(R(),Nd%,p%,j&) 'F14 (2-D) Shekel's Fox holes (INVERTED...) 
+ 
+    LOCAL Sum1, Sum2, Z, Xi AS EXT 
+ 
+    LOCAL i%, jj% 
+ 
+    Sum1 = 0## 
+ 
+    FOR jj% = 1 TO 25 
+ 
+        Sum2 = 0## 
+ 
+        FOR i% = 1 TO 2 
+ 
+            Xi = R(p%,i%,j&) 
+ 
+            Sum2 = Sum2 + (Xi-Aij(i%,jj%))^6 
+ 
+        NEXT i% 
+ 
+        Sum1 = Sum1 + 1##/(jj%+Sum2) 
+ 
+    NEXT j% 
+ 
+    Z = 1##/(0.002##+Sum1) 
+ 
+    F14 = -Z 
+ 
+END FUNCTION 'F14 
+ 
+'----------- 
+ 
+FUNCTION F16(R(),Nd%,p%,j&) 'F16 (2-D) 6-Hump Camel -Back 
+ 
+    LOCAL x1, x2, Z AS EXT 
+ 
+    x1 = R(p%,1,j&) : x2 = R(p%,2,j&) 
+ 
+    Z = 4##*x1^2 - 2.1##*x1^4 + x1^6/3## + x1*x2 - 4*x2^2 + 4*x2^4 
+ 
+    F16 = -Z 
+ 
+END FUNCTION 'F16 
+ 
+'----------- 62 of 92  
+FUNCTION F15(R(),Nd%,p%,j&) 'F15 (4-D) Kowalik's Fu nction 
+ 
+'Global maximum = -0.0003075 @ (0.1928,0.1908,0.123 1,0.1358) 
+ 
+    LOCAL x1, x2, x3, x4, Num, Denom, Z, Aj(), Bj()  AS EXT 
+ 
+    LOCAL jj% 
+ 
+    REDIM Aj(1 TO 11), Bj(1 TO 11) 
+ 
+    Aj(1)  = 0.1957## : Bj(1)  = 1##/0.25## 
+    Aj(2)  = 0.1947## : Bj(2)  = 1##/0.50## 
+    Aj(3)  = 0.1735## : Bj(3)  = 1##/1.00## 
+    Aj(4)  = 0.1600## : Bj(4)  = 1##/2.00## 
+    Aj(5)  = 0.0844## : Bj(5)  = 1##/4.00## 
+    Aj(6)  = 0.0627## : Bj(6)  = 1##/6.00## 
+    Aj(7)  = 0.0456## : Bj(7)  = 1##/8.00## 
+    Aj(8)  = 0.0342## : Bj(8)  = 1##/10.0## 
+    Aj(9)  = 0.0323## : Bj(9)  = 1##/12.0## 
+    Aj(10) = 0.0235## : Bj(10) = 1##/14.0## 
+    Aj(11) = 0.0246## : Bj(11) = 1##/16.0## 
+ 
+    Z = 0## 
+ 
+    x1 = R(p%,1,j&) : x2 = R(p%,2,j&) : x3 = R(p%,3 ,j&) : x4 = R(p%,4,j&) 
+ 
+    FOR jj% = 1 TO 11 
+ 
+       Num   = x1*(Bj(jj%)^2+Bj(jj%)*x2) 
+ 
+       Denom = Bj(jj%)^2+Bj(jj%)*x3+x4 
+ 
+       Z = Z + (Aj(jj%)-Num/Denom)^2 
+ 
+    NEXT jj% 
+ 
+    F15 = -Z 
+ 
+END FUNCTION 'F15 
+ 
+'----------- 
+ 
+FUNCTION F17(R(),Nd%,p%,j&) 'F17, (2-D) Branin 
+ 
+'Global maximum = -0.398 @ (-3.142.12.275), (3.142, 2.275), (9.425,2.425) 
+ 
+    LOCAL x1, x2, Z AS EXT 
+ 
+    x1 = R(p%,1,j&) : x2 = R(p%,2,j&) 
+ 
+    Z = (x2-5.1##*x1^2/(4##*Pi^2)+5##*x1/Pi-6##)^2 + 10##*(1##-1##/(8##*Pi))*COS(x1) + 10## 
+ 
+    F17 = -Z 
+ 
+END FUNCTION 'F17 
+ 
+'----------- 
+ 
+FUNCTION F18(R(),Nd%,p%,j&) 'Goldstein-Price 2-D Te st Function 
+ 
+'Global maximum = -3 @ (0,-1) 
+ 
+    LOCAL Z, x1, x2, t1, t2 AS EXT 
+ 
+    x1 = R(p%,1,j&) : x2 = R(p%,2,j&) 
+ 
+    t1 = 1##+(x1+x2+1##)^2*(19##-14##*x1+3##*x1^2-1 4##*x2+6##*x1*x2+3##*x2^2) 
+ 
+    t2 = 30##+(2##*x1-3##*x2)^2*(18##-32##*x1+12##* x1^2+48##*x2-36##*x1*x2+27##*x2^2) 
+ 
+    Z  = t1*t2 
+ 
+    F18 = -Z 
+ 
+END FUNCTION 'F18() 
+ 
+'----------- 
+ 
+FUNCTION F19(R(),Nd%,p%,j&) 'F19 (3-D) Hartman's Fa mily #1 
+ 
+'Global maximum = 3.86 @ (0.114,0.556,0.852) 
+ 
+    LOCAL Xi, Z, Sum, Aji(), Cj(), Pji() AS EXT 
+ 
+    LOCAL i%, jj%, m% 
+ 
+    REDIM Aji(1 TO 4, 1 TO 3), Cj(1 TO 4), Pji(1 TO  4, 1 TO 3) 
+ 
+    Aji(1,1) = 3.0## : Aji(1,2) = 10## : Aji(1,3) =  30## : Cj(1) = 1.0## 
+    Aji(2,1) = 0.1## : Aji(2,2) = 10## : Aji(2,3) =  35## : Cj(2) = 1.2## 
+    Aji(3,1) = 3.0## : Aji(3,2) = 10## : Aji(3,3) =  30## : Cj(3) = 3.0## 
+    Aji(4,1) = 0.1## : Aji(4,2) = 10## : Aji(4,3) =  35## : Cj(4) = 3.2## 
+ 
+    Pji(1,1) = 0.36890## : Pji(1,2) = 0.1170## : Pj i(1,3) = 0.2673## 
+    Pji(2,1) = 0.46990## : Pji(2,2) = 0.4387## : Pj i(2,3) = 0.7470## 
+    Pji(3,1) = 0.10910## : Pji(3,2) = 0.8732## : Pj i(3,3) = 0.5547## 
+    Pji(4,1) = 0.03815## : Pji(4,2) = 0.5743## : Pj i(4,3) = 0.8828## 
+ 
+    Z = 0## 
+ 
+    FOR jj% = 1 TO 4 
+ 
+        Sum = 0## 
+ 
+        FOR i% = 1 TO 3 
+ 
+            Xi = R(p%,i%,j&) 
+ 
+            Sum = Sum + Aji(jj%,i%)*(Xi-Pji(jj%,i%) )^2 
+ 
+        NEXT i% 
+ 
+        Z = Z + Cj(jj%)*EXP(-Sum) 63 of 92  
+    NEXT jj% 
+ 
+    F19 = Z 
+ 
+END FUNCTION 'F19 
+ 
+'----------- 
+ 
+FUNCTION F20(R(),Nd%,p%,j&) 'F20 (6-D) Hartman's Fa mily #2 
+ 
+'Global maximum = 3.32 @ (0.201,0.150,0.477,0.275,0 .311,0.657) 
+ 
+    LOCAL Xi, Z, Sum, Aji(), Cj(), Pji() AS EXT 
+ 
+    LOCAL i%, jj%, m% 
+ 
+    REDIM Aji(1 TO 4, 1 TO 6), Cj(1 TO 4), Pji(1 TO  4, 1 TO 6) 
+ 
+    Aji(1,1) = 10.0## : Aji(1,2) = 3.00## : Aji(1,3 ) = 17.0## : Cj(1) = 1.0## 
+    Aji(2,1) = 0.05## : Aji(2,2) = 10.0## : Aji(2,3 ) = 17.0## : Cj(2) = 1.2## 
+    Aji(3,1) = 3.00## : Aji(3,2) = 3.50## : Aji(3,3 ) = 1.70## : Cj(3) = 3.0## 
+    Aji(4,1) = 17.0## : Aji(4,2) = 8.00## : Aji(4,3 ) = 0.05## : Cj(4) = 3.2## 
+ 
+    Aji(1,4) = 3.5## : Aji(1,5) = 1.7## : Aji(1,6) =  8## 
+    Aji(2,4) = 0.1## : Aji(2,5) =   8## : Aji(2,6) = 14## 
+    Aji(3,4) =  10## : Aji(3,5) =  17## : Aji(3,6) =  8## 
+    Aji(4,4) =  10## : Aji(4,5) = 0.1## : Aji(4,6) = 14## 
+ 
+    Pji(1,1) = 0.13120## : Pji(1,2) = 0.1696## : Pj i(1,3) = 0.5569## 
+    Pji(2,1) = 0.23290## : Pji(2,2) = 0.4135## : Pj i(2,3) = 0.8307## 
+    Pji(3,1) = 0.23480## : Pji(3,2) = 0.1415## : Pj i(3,3) = 0.3522## 
+    Pji(4,1) = 0.40470## : Pji(4,2) = 0.8828## : Pj i(4,3) = 0.8732## 
+ 
+    Pji(1,4) = 0.01240## : Pji(1,5) = 0.8283## : Pj i(1,6) = 0.5886## 
+    Pji(2,4) = 0.37360## : Pji(2,5) = 0.1004## : Pj i(2,6) = 0.9991## 
+    Pji(3,4) = 0.28830## : Pji(3,5) = 0.3047## : Pj i(3,6) = 0.6650## 
+    Pji(4,4) = 0.57430## : Pji(4,5) = 0.1091## : Pj i(4,6) = 0.0381## 
+ 
+    Z = 0## 
+ 
+    FOR jj% = 1 TO 4 
+ 
+        Sum = 0## 
+ 
+        FOR i% = 1 TO 6 
+ 
+            Xi = R(p%,i%,j&) 
+ 
+            Sum = Sum + Aji(jj%,i%)*(Xi-Pji(jj%,i%) )^2 
+ 
+        NEXT i% 
+ 
+        Z = Z + Cj(jj%)*EXP(-Sum) 
+ 
+    NEXT jj% 
+ 
+    F20 = Z 
+ 
+END FUNCTION 'F20 
+ 
+'----------- 
+ 
+FUNCTION F21(R(),Nd%,p%,j&) 'F21 (4-D) Shekel's Fam ily m=5 
+ 
+'Global maximum = 10 
+ 
+    LOCAL Xi, Z, Sum, Aji(), Cj() AS EXT 
+ 
+    LOCAL i%, jj%, m% 
+ 
+    m% = 5 : REDIM Aji(1 TO m%, 1 TO 4), Cj(1 TO m% ) 
+ 
+    Aji(1,1)  = 4## : Aji(1,2)  =   4## : Aji(1,3)  = 4## : Aji(1,4)  =   4## : Cj(1)  = 0.1## 
+    Aji(2,1)  = 1## : Aji(2,2)  =   1## : Aji(2,3)  = 1## : Aji(2,4)  =   1## : Cj(2)  = 0.2## 
+    Aji(3,1)  = 8## : Aji(3,2)  =   8## : Aji(3,3)  = 8## : Aji(3,4)  =   8## : Cj(3)  = 0.2## 
+    Aji(4,1)  = 6## : Aji(4,2)  =   6## : Aji(4,3)  = 6## : Aji(4,4)  =   6## : Cj(4)  = 0.4## 
+    Aji(5,1)  = 3## : Aji(5,2)  =   7## : Aji(5,3)  = 3## : Aji(5,4)  =   7## : Cj(5)  = 0.4## 
+ 
+    Z = 0## 
+ 
+    FOR jj% = 1 TO m%  'NOTE:  Index jj% is used to  avoid same variable name as j& 
+ 
+        Sum = 0## 
+ 
+        FOR i% = 1 TO 4 'Shekel's family is 4-D onl y 
+ 
+            Xi = R(p%,i%,j&) 
+ 
+            Sum = Sum + (Xi-Aji(jj%,i%))^2 
+ 
+        NEXT i% 
+ 
+        Z = Z + 1##/(Sum + Cj(jj%)) 
+ 
+    NEXT jj% 
+ 
+    F21 = Z 
+ 
+END FUNCTION 'F21 
+ 
+'----------- 
+ 
+FUNCTION F22(R(),Nd%,p%,j&) 'F22 (4-D) Shekel's Fam ily m=7 
+ 
+'Global maximum = 10 
+ 
+    LOCAL Xi, Z, Sum, Aji(), Cj() AS EXT 
+ 
+    LOCAL i%, jj%, m% 
+ 
+    m% = 7 : REDIM Aji(1 TO m%, 1 TO 4), Cj(1 TO m% ) 
+ 
+    Aji(1,1)  = 4## : Aji(1,2)  =   4## : Aji(1,3)  = 4## : Aji(1,4)  =   4## : Cj(1)  = 0.1## 64 of 92     Aji(2,1)  = 1## : Aji(2,2)  =   1## : Aji(2,3)  = 1## : Aji(2,4)  =   1## : Cj(2)  = 0.2## 
+    Aji(3,1)  = 8## : Aji(3,2)  =   8## : Aji(3,3)  = 8## : Aji(3,4)  =   8## : Cj(3)  = 0.2## 
+    Aji(4,1)  = 6## : Aji(4,2)  =   6## : Aji(4,3)  = 6## : Aji(4,4)  =   6## : Cj(4)  = 0.4## 
+    Aji(5,1)  = 3## : Aji(5,2)  =   7## : Aji(5,3)  = 3## : Aji(5,4)  =   7## : Cj(5)  = 0.4## 
+    Aji(6,1)  = 2## : Aji(6,2)  =   9## : Aji(6,3)  = 2## : Aji(6,4)  =   9## : Cj(6)  = 0.6## 
+    Aji(7,1)  = 5## : Aji(7,2)  =   5## : Aji(7,3)  = 3## : Aji(7,4)  =   3## : Cj(7)  = 0.3## 
+ 
+    Z = 0## 
+ 
+    FOR jj% = 1 TO m%  'NOTE:  Index jj% is used to  avoid same variable name as j& 
+ 
+        Sum = 0## 
+ 
+        FOR i% = 1 TO 4 'Shekel's family is 4-D onl y 
+ 
+            Xi = R(p%,i%,j&) 
+ 
+            Sum = Sum + (Xi-Aji(jj%,i%))^2 
+ 
+        NEXT i% 
+ 
+        Z = Z + 1##/(Sum + Cj(jj%)) 
+ 
+    NEXT jj% 
+ 
+    F22 = Z 
+ 
+END FUNCTION 'F22 
+ 
+'----------- 
+ 
+FUNCTION F23(R(),Nd%,p%,j&) 'F23 (4-D) Shekel's Fam ily m=10 
+ 
+'Global maximum = 10 
+ 
+    LOCAL Xi, Z, Sum, Aji(), Cj() AS EXT 
+ 
+    LOCAL i%, jj%, m% 
+ 
+    m% = 10 : REDIM Aji(1 TO m%, 1 TO 4), Cj(1 TO m %) 
+ 
+    Aji(1,1)  = 4## : Aji(1,2)  =   4## : Aji(1,3)  = 4## : Aji(1,4)  =   4## : Cj(1)  = 0.1## 
+    Aji(2,1)  = 1## : Aji(2,2)  =   1## : Aji(2,3)  = 1## : Aji(2,4)  =   1## : Cj(2)  = 0.2## 
+    Aji(3,1)  = 8## : Aji(3,2)  =   8## : Aji(3,3)  = 8## : Aji(3,4)  =   8## : Cj(3)  = 0.2## 
+    Aji(4,1)  = 6## : Aji(4,2)  =   6## : Aji(4,3)  = 6## : Aji(4,4)  =   6## : Cj(4)  = 0.4## 
+    Aji(5,1)  = 3## : Aji(5,2)  =   7## : Aji(5,3)  = 3## : Aji(5,4)  =   7## : Cj(5)  = 0.4## 
+    Aji(6,1)  = 2## : Aji(6,2)  =   9## : Aji(6,3)  = 2## : Aji(6,4)  =   9## : Cj(6)  = 0.6## 
+    Aji(7,1)  = 5## : Aji(7,2)  =   5## : Aji(7,3)  = 3## : Aji(7,4)  =   3## : Cj(7)  = 0.3## 
+    Aji(8,1)  = 8## : Aji(8,2)  =   1## : Aji(8,3)  = 8## : Aji(8,4)  =   1## : Cj(8)  = 0.7## 
+    Aji(9,1)  = 6## : Aji(9,2)  =   2## : Aji(9,3)  = 6## : Aji(9,4)  =   2## : Cj(9)  = 0.5## 
+    Aji(10,1) = 7## : Aji(10,2) = 3.6## : Aji(10,3)  = 7## : Aji(10,4) = 3.6## : Cj(10) = 0.5## 
+ 
+    Z = 0## 
+ 
+    FOR jj% = 1 TO m%  'NOTE:  Index jj% is used to  avoid same variable name as j& 
+ 
+        Sum = 0## 
+ 
+        FOR i% = 1 TO 4 'Shekel's family is 4-D onl y 
+ 
+            Xi = R(p%,i%,j&) 
+ 
+            Sum = Sum + (Xi-Aji(jj%,i%))^2 
+ 
+        NEXT i% 
+ 
+        Z = Z + 1##/(Sum + Cj(jj%)) 
+ 
+    NEXT jj% 
+ 
+    F23 = Z 
+ 
+END FUNCTION 'F23 
+ 
+'================================================== ====== END FUNCTION DEFINITIONS =================== ============================================ 
+ 
+SUB Plot2DbestProbeTrajectories(NumTrajectories%,M( ),R(),XiMin(),XiMax(),Np%,Nd%,LastStep&,FunctionNam e$) 
+ 
+LOCAL TrajectoryNumber%, ProbeNumber%, StepNumber&,  N%, M%, ProcID??? 
+ 
+LOCAL MaximumFitness, MinimumFitness AS EXT 
+ 
+LOCAL BestProbeThisStep%() 
+ 
+LOCAL BestFitnessThisStep(), TempFitness() AS EXT 
+ 
+LOCAL Annotation$, xCoord$, yCoord$, GnuPlotEXE$, P lotWithLines$ 
+ 
+    Annotation$    = "" 
+ 
+    PlotWithLines$ = "YES" '"NO" 
+ 
+    NumTrajectories% = MIN(Np%,NumTrajectories%) 
+ 
+    GnuPlotEXE$ = "wgnuplot.exe" 
+ 
+'   ---------------- Get Min/Max Fitnesses -------- --------- 
+ 
+    MaximumFitness = M(1,0) : MinimumFitness = M(1, 0)  'Note:  M(p%,j&) 
+ 
+    FOR StepNumber& = 0 TO LastStep& 
+ 
+        FOR ProbeNumber% = 1 TO Np% 
+ 
+            IF M(ProbeNumber%,StepNumber&) >= Maxim umFitness THEN MaximumFitness = M(ProbeNumber%,Step Number&) 
+ 
+            IF M(ProbeNumber%,StepNumber&) =< Minim umFitness THEN MinimumFitness = M(ProbeNumber%,Step Number&) 
+ 
+        NEXT ProbeNumber% 
+ 
+    NEXT StepNumber% 
+ 
+'   ------------- Copy Fitness Array M() into TempF itness to Preserve M() ---------------- 
+ 65 of 92     REDIM TempFitness(1 TO Np%, 0 TO LastStep&) 
+ 
+    FOR StepNumber& = 0 TO LastStep& 
+ 
+        FOR ProbeNumber% = 1 TO Np% 
+ 
+            TempFitness(ProbeNumber%,StepNumber&) =  M(ProbeNumber%,StepNumber&) 
+ 
+        NEXT ProbeNumber% 
+ 
+    NEXT StepNumber% 
+ 
+'   ------------ LOOP ON TRAJECTORIES ----------- 
+ 
+    FOR TrajectoryNumber% = 1 TO NumTrajectories% 
+ 
+'       --------------- Get Trajectory Coordinate D ata ----------------- 
+ 
+        REDIM BestFitnessThisStep(0 TO LastStep&), BestProbeThisStep%(0 TO LastStep&) 
+ 
+        FOR StepNumber& = 0 TO LastStep& 
+ 
+            BestFitnessThisStep(StepNumber&) = Temp Fitness(1,StepNumber&) 
+ 
+            FOR ProbeNumber% = 1 TO Np% 
+ 
+                IF TempFitness(ProbeNumber%,StepNum ber&) >= BestFitnessThisStep(StepNumber&) THEN 
+ 
+                    BestFitnessThisStep(StepNumber& ) = TempFitness(ProbeNumber%,StepNumber&) 
+ 
+                    BestProbeThisStep%(StepNumber&)   = ProbeNumber% 
+ 
+                END IF 
+ 
+            NEXT ProbeNumber% 
+ 
+        NEXT StepNumber& 
+ 
+'   ----- Create Plot Data File ----- 
+ 
+    N% = FREEFILE 
+ 
+    SELECT CASE TrajectoryNumber% 
+ 
+        CASE 1  : OPEN "t1"  FOR OUTPUT AS #N% 
+        CASE 2  : OPEN "t2"  FOR OUTPUT AS #N% 
+        CASE 3  : OPEN "t3"  FOR OUTPUT AS #N% 
+        CASE 4  : OPEN "t4"  FOR OUTPUT AS #N% 
+        CASE 5  : OPEN "t5"  FOR OUTPUT AS #N% 
+        CASE 6  : OPEN "t6"  FOR OUTPUT AS #N% 
+        CASE 7  : OPEN "t7"  FOR OUTPUT AS #N% 
+        CASE 8  : OPEN "t8"  FOR OUTPUT AS #N% 
+        CASE 9  : OPEN "t9"  FOR OUTPUT AS #N% 
+        CASE 10 : OPEN "t10" FOR OUTPUT AS #N% 
+ 
+    END SELECT 
+ 
+'   ------------ Write Plot File Data ------------ 
+ 
+'   M% = freefile : open "BestProbebData" for outpu t as #M% 'debug 
+ 
+'   print #M%, "  Step #  BestProbe#           x1                  x2" 
+ 
+    FOR StepNumber& = 0 TO LastStep& 
+ 
+        PRINT #N%, USING$("######.######## 
+######.########",R(BestProbeThisStep%(StepNumber&), 1,StepNumber&),R(BestProbeThisStep%(StepNumber&),2, StepNumber&)) 
+ 
+'       PRINT #M%, USING$("#####     #####      ### ###.########      
+######.########",StepNumber&,BestProbeThisStep%(Ste pNumber&),R(BestProbeThisStep%(StepNumber&),1,StepN umber&),R(BestProbeThisStep%(StepNumber&),2,St 
+epNumber&)) 
+ 
+        TempFitness(BestProbeThisStep%(StepNumber&) ,StepNumber&) = MinimumFitness 'so that same max wi ll not be found for next trajectory 
+ 
+    NEXT StepNumber% 
+ 
+    CLOSE #N% 
+ 
+'   Close #M% 
+ 
+    NEXT TrajectoryNumber% 
+ 
+'   ------------------------- Plot Trajectories --- ----------------------- 
+ 
+    CALL CreateGNUplotINIfile(0.13##*ScreenWidth&,0 .18##*ScreenHeight&,0.7##*ScreenHeight&,0.7##*Scree nHeight&) 
+ 
+    Annotation$ = "" 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "cmd2d.gp" FOR OUTPUT AS #N% 
+ 
+        PRINT #N%, "set xrange ["+REMOVE$(STR$(XiMi n(1)),ANY"" )+":"+REMOVE$(STR$(XiMax(1)),ANY" ")+"] " 
+        PRINT #N%, "set yrange ["+REMOVE$(STR$(XiMi n(2)),ANY" ")+":"+REMOVE$(STR$(XiMax(2)),ANY" ")+"] " 
+ 
+        'PRINT #N%, "set label "      + Quote$ + An notation$ + Quote$ + " at graph " + xCoord$ + "," +  yCoord$ 
+        PRINT #N%, "set grid xtics " + "10" 
+        PRINT #N%, "set grid ytics " + "10" 
+        PRINT #N%, "set grid mxtics" 
+        PRINT #N%, "set grid mytics" 
+        PRINT #N%, "show grid" 
+        PRINT #N%, "set title "  + Quote$ + "2D "+ FunctionName$+" TRAJECTORIES OF PROBES WITH BEST\nF ITNESSES (ORDERED BY FITNESS)" + "\n" + RunID$ 
++ Quote$ 
+        PRINT #N%, "set xlabel " + Quote$ + "x1\n\n "                                   + Quote$ 
+        PRINT #N%, "set ylabel " + Quote$ + "\nx2"                                     + Quote$ 
+ 
+        IF PlotWithLines$ = "YES" THEN 
+ 
+            SELECT CASE NumTrajectories% 
+ 
+                CASE 1  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" w l lw 3" 
+                CASE 2  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l"  
+                CASE 3  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l, "+Quote$+"t3"+Quote$+" w l" 66 of 92                 CASE 4  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l, "+Quote$+"t3"+Quote$+" w 
+l,"+Quote$+"t4"+Quote$+" w l" 
+                CASE 5  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l, "+Quote$+"t3"+Quote$+" w 
+l,"+Quote$+"t4"+Quote$+" w l,"+Quote$+"t5"+Quote$+"  w l" 
+                CASE 6  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l, "+Quote$+"t3"+Quote$+" w 
+l,"+Quote$+"t4"+Quote$+" w l,"+Quote$+"t5"+Quote$+"  w l,"+Quote$+"t6"+Quote$+" w l" 
+                CASE 7  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l, "+Quote$+"t3"+Quote$+" w 
+l,"+Quote$+"t4"+Quote$+" w l,"+Quote$+"t5"+Quote$+"  w l,"+Quote$+"t6"+Quote$+" w l,"+_ 
+                                             Quote$ +"t7"+Quote$+" w l" 
+                CASE 8  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l, "+Quote$+"t3"+Quote$+" w 
+l,"+Quote$+"t4"+Quote$+" w l,"+Quote$+"t5"+Quote$+"  w l,"+Quote$+"t6"+Quote$+" w l,"+_ 
+                                             Quote$ +"t7"+Quote$+" w l,"     +Quote$+"t8"+Quote$+" w l"  
+                CASE 9  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l, "+Quote$+"t3"+Quote$+" w 
+l,"+Quote$+"t4"+Quote$+" w l,"+Quote$+"t5"+Quote$+"  w l,"+Quote$+"t6"+Quote$+" w l,"+_ 
+                                             Quote$ +"t7"+Quote$+" w l,"     +Quote$+"t8"+Quote$+" w l, "+Quote$+"t9"+Quote$+" w l" 
+                CASE 10 : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l, "+Quote$+"t3"+Quote$+" w 
+l,"+Quote$+"t4"+Quote$+" w l,"+Quote$+"t5"+Quote$+"  w l,"+Quote$+"t6"+Quote$+" w l,"+_ 
+                                             Quote$ +"t7"+Quote$+" w l,"     +Quote$+"t8"+Quote$+" w l, "+Quote$+"t9"+Quote$+" w 
+l,"+Quote$+"t10"+Quote$+" w l" 
+            END SELECT 
+ 
+        ELSE 
+ 
+            SELECT CASE NumTrajectories% 
+ 
+                CASE 1  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" lw 2" 
+                CASE 2  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$ 
+                CASE 3  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote $+"t3"+Quote$ 
+                CASE 4  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote $+"t3"+Quote$+" ,"+Quote$+"t4"+Quote$ 
+                CASE 5  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote $+"t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$ 
+                CASE 6  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote $+"t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$+" ,"+Quote$+"t6"+Quote$ 
+                CASE 7  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote $+"t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$+" ,"+Quote$+"t6"+Quote$+" ,"+ _ 
+                                             Quote$ +"t7"+Quote$ 
+                CASE 8  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote $+"t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$+" ,"+Quote$+"t6"+Quote$+" ,"+ _ 
+                                             Quote$ +"t7"+Quote$+" ,"    +Quote$+"t8"+Quote$ 
+                CASE 9  : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote $+"t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$+" ,"+Quote$+"t6"+Quote$+" ,"+ _ 
+                                             Quote$ +"t7"+Quote$+" ,"    +Quote$+"t8"+Quote$+" ,"+Quote $+"t9"+Quote$ 
+                CASE 10 : PRINT #N%, "plot "+Quote$ +"t1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote $+"t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$+" ,"+Quote$+"t6"+Quote$+" ,"+ _ 
+                                             Quote$ +"t7"+Quote$+" ,"    +Quote$+"t8"+Quote$+" ,"+Quote $+"t9"+Quote$+" ,"+Quote$+"t10"+Quote$ 
+            END SELECT 
+ 
+        END IF 
+ 
+    CLOSE #N% 
+ 
+    ProcID??? = SHELL(GnuPlotEXE$+" cmd2d.gp -") : CALL Delay(0.5##) 
+ 
+END SUB 'Plot2DbestProbeTrajectories() 
+ 
+'---- 
+ 
+SUB Plot2DindividualProbeTrajectories(NumTrajectori es%,M(),R(),XiMin(),XiMax(),Np%,Nd%,LastStep&,Funct ionName$) 
+ 
+LOCAL ProbeNumber%, StepNumber&, N%, ProcID??? 
+ 
+LOCAL Annotation$, xCoord$, yCoord$, GnuPlotEXE$, P lotWithLines$ 
+ 
+    NumTrajectories% = MIN(Np%,NumTrajectories%) 
+ 
+    Annotation$    = "" 
+ 
+    PlotWithLines$ = "YES" '"NO" 
+ 
+    GnuPlotEXE$ = "wgnuplot.exe" 
+ 
+'   -------------- LOOP ON PROBES --------------- 
+ 
+    FOR ProbeNumber% = 1 TO MIN(NumTrajectories%,Np %) 
+ 
+'   ----- Create Plot Data File ----- 
+ 
+    N% = FREEFILE 
+ 
+    SELECT CASE ProbeNumber% 
+ 
+        CASE 1  : OPEN "p1"  FOR OUTPUT AS #N% 
+        CASE 2  : OPEN "p2"  FOR OUTPUT AS #N% 
+        CASE 3  : OPEN "p3"  FOR OUTPUT AS #N% 
+        CASE 4  : OPEN "p4"  FOR OUTPUT AS #N% 
+        CASE 5  : OPEN "p5"  FOR OUTPUT AS #N% 
+        CASE 6  : OPEN "p6"  FOR OUTPUT AS #N% 
+        CASE 7  : OPEN "p7"  FOR OUTPUT AS #N% 
+        CASE 8  : OPEN "p8"  FOR OUTPUT AS #N% 
+        CASE 9  : OPEN "p9"  FOR OUTPUT AS #N% 
+        CASE 10 : OPEN "p10" FOR OUTPUT AS #N% 
+        CASE 11 : OPEN "p11" FOR OUTPUT AS #N% 
+        CASE 12 : OPEN "p12" FOR OUTPUT AS #N% 
+        CASE 13 : OPEN "p13" FOR OUTPUT AS #N% 
+        CASE 14 : OPEN "p14" FOR OUTPUT AS #N% 
+        CASE 15 : OPEN "p15" FOR OUTPUT AS #N% 
+        CASE 16 : OPEN "p16" FOR OUTPUT AS #N% 
+ 
+    END SELECT 
+ 
+'   ------------ Write Plot File Data ------------ 
+ 
+    FOR StepNumber& = 0 TO LastStep& 
+ 
+        PRINT #N%, USING$("######.######## ######.# #######",R(ProbeNumber%,1,StepNumber&),R(ProbeNumbe r%,2,StepNumber&)) 
+ 
+    NEXT StepNumber% 
+ 
+    CLOSE #N% 
+ 
+    NEXT ProbeNumber% 
+ 
+'   ------------------------------------------- Plo t Trajectories ------------------------------------ --------- 
+ 67 of 92 'usage:  CALL CreateGNUplotINIfile(PlotWindowULC_X% ,PlotWindowULC_Y%,PlotWindowWidth%,PlotWindowHeight %) 
+ 
+    CALL CreateGNUplotINIfile(0.17##*ScreenWidth&,0 .22##*ScreenHeight&,0.7##*ScreenHeight&,0.7##*Scree nHeight&) 
+ 
+    Annotation$ = "" 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "cmd2d.gp" FOR OUTPUT AS #N% 
+ 
+        PRINT #N%, "set xrange ["+REMOVE$(STR$(XiMi n(1)),ANY"" )+":"+REMOVE$(STR$(XiMax(1)),ANY" ")+"] " 
+        PRINT #N%, "set yrange ["+REMOVE$(STR$(XiMi n(2)),ANY" ")+":"+REMOVE$(STR$(XiMax(2)),ANY" ")+"] " 
+ 
+        'PRINT #N%, "set label "      + Quote$ + An notation$ + Quote$ + " at graph " + xCoord$ + "," +  yCoord$ 
+        PRINT #N%, "set grid xtics " + "10" 
+        PRINT #N%, "set grid ytics " + "10" 
+        PRINT #N%, "set grid mxtics" 
+        PRINT #N%, "set grid mytics" 
+        PRINT #N%, "show grid" 
+        PRINT #N%, "set title "  + Quote$ + "2D "+ FunctionName$+" INDIVIDUAL PROBE TRAJECTORIES\n(ORD ERED BY PROBE #)" + "\n" + RunID$ + Quote$ 
+        PRINT #N%, "set xlabel " + Quote$ + "x1\n\n "                                   + Quote$ 
+        PRINT #N%, "set ylabel " + Quote$ + "\nx2"                                     + Quote$ 
+ 
+        IF PlotWithLines$ = "YES" THEN 
+ 
+            SELECT CASE NumTrajectories% 
+ 
+                CASE 1  : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1" 
+                CASE 2  : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2"+Quote$+" w l" 
+                CASE 3  : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2"+Quote$+" w l,"+Quote$+"p3"+Quote$+" w l" 
+                CASE 4  : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2"+Quote$+" w l,"+Quote$+"p3"+Quote$+" w 
+l,"+Quote$+"p4"+Quote$+" w l" 
+                CASE 5  : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2"+Quote$+" w l,"+Quote$+"p3"+Quote$+" w 
+l,"+Quote$+"p4"+Quote$+" w l,"+Quote$+"p5"+Quote$+"  w l" 
+                CASE 6  : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2"+Quote$+" w l,"+Quote$+"p3"+Quote$+" w 
+l,"+Quote$+"p4"+Quote$+" w l,"+Quote$+"p5"+Quote$+"  w l,"+Quote$+"p6"+Quote$+" w l" 
+                CASE 7  : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2"+Quote$+" w l,"+Quote$+"p3"+Quote$+" w 
+l,"+Quote$+"p4"+Quote$+" w l,"+Quote$+"p5"+Quote$+"  w l,"+Quote$+"p6"+Quote$+" w l,"+_ 
+                                             Quote$ +"p7"  +Quote$+" w l" 
+                CASE 8  : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2"+Quote$+" w l,"+Quote$+"p3"+Quote$+" w 
+l,"+Quote$+"p4"+Quote$+" w l,"+Quote$+"p5"+Quote$+"  w l,"+Quote$+"p6"+Quote$+" w l,"+_ 
+                                             Quote$ +"p7"  +Quote$+" w l,"     +Quote$+"p8"+Quote$+" w l" 
+                CASE 9  : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2"+Quote$+" w l,"+Quote$+"p3"+Quote$+" w 
+l,"+Quote$+"p4"+Quote$+" w l,"+Quote$+"p5"+Quote$+"  w l,"+Quote$+"p6"+Quote$+" w l,"+_ 
+                                             Quote$ +"p7"  +Quote$+" w l,"     +Quote$+"p8"+Quote$+" w l,"+Quote$+"p9"+Quote$+" w l" 
+ 
+                CASE 10 : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2" +Quote$+" w  l,"+Quote$+"p3" +Quote$+" w l,"+Quote$+"p4" 
++Quote$+" w l,"+Quote$+"p5"+Quote$+" w l,"+Quote$+" p6"+Quote$+" w l,"+_ 
+                                             Quote$ +"p7"  +Quote$+" w l,"     +Quote$+"p8" +Quote$+" w  l,"+Quote$+"p9" +Quote$+" w 
+l,"+Quote$+"p10"+Quote$+" w l" 
+ 
+                CASE 11 : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2" +Quote$+" w  l,"+Quote$+"p3" +Quote$+" w l,"+Quote$+"p4" 
++Quote$+" w l,"+Quote$+"p5" +Quote$+" w l,"+Quote$+ "p6" +Quote$+" w l,"+_ 
+                                             Quote$ +"p7"  +Quote$+" w l,"     +Quote$+"p8" +Quote$+" w  l,"+Quote$+"p9" +Quote$+" w 
+l,"+Quote$+"p10"+Quote$+" w l,"+Quote$+"p11"+Quote$ +" w l" 
+ 
+                CASE 12 : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2" +Quote$+" w  l,"+Quote$+"p3" +Quote$+" w l,"+Quote$+"p4" 
++Quote$+" w l,"+Quote$+"p5" +Quote$+" w l,"+Quote$+ "p6" +Quote$+" w l,"+_ 
+                                             Quote$ +"p7"  +Quote$+" w l,"     +Quote$+"p8" +Quote$+" w  l,"+Quote$+"p9" +Quote$+" w 
+l,"+Quote$+"p10"+Quote$+" w l,"+Quote$+"p11"+Quote$ +" w l,"+Quote$+"p12"+Quote$+" w l" 
+ 
+                CASE 13 : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2" +Quote$+" w  l,"+Quote$+"p3" +Quote$+" w l,"+Quote$+"p4" 
++Quote$+" w l,"+Quote$+"p5" +Quote$+" w l,"+Quote$+ "p6" +Quote$+" w l,"+_ 
+                                             Quote$ +"p7"  +Quote$+" w l,"     +Quote$+"p8" +Quote$+" w  l,"+Quote$+"p9" +Quote$+" w 
+l,"+Quote$+"p10"+Quote$+" w l,"+Quote$+"p11"+Quote$ +" w l,"+Quote$+"p12"+Quote$+" w l,"+_ 
+                                             Quote$ +"p13 "+Quote$+" w l" 
+ 
+                CASE 14 : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2" +Quote$+" w  l,"+Quote$+"p3" +Quote$+" w l,"+Quote$+"p4" 
++Quote$+" w l,"+Quote$+"p5" +Quote$+" w l,"+Quote$+ "p6" +Quote$+" w l,"+_ 
+                                             Quote$ +"p7"  +Quote$+" w l,"     +Quote$+"p8" +Quote$+" w  l,"+Quote$+"p9" +Quote$+" w 
+l,"+Quote$+"p10"+Quote$+" w l,"+Quote$+"p11"+Quote$ +" w l,"+Quote$+"p12"+Quote$+" w l,"+_ 
+                                             Quote$ +"p13" +Quote$+" w l,"     +Quote$+"p14"+Quote$+" w  l" 
+ 
+                CASE 15 : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2" +Quote$+" w  l,"+Quote$+"p3" +Quote$+" w l,"+Quote$+"p4" 
++Quote$+" w l,"+Quote$+"p5" +Quote$+" w l,"+Quote$+ "p6" +Quote$+" w l,"+_ 
+                                             Quote$ +"p7"  +Quote$+" w l,"     +Quote$+"p8" +Quote$+" w  l,"+Quote$+"p9" +Quote$+" w 
+l,"+Quote$+"p10"+Quote$+" w l,"+Quote$+"p11"+Quote$ +" w l,"+Quote$+"p12"+Quote$+" w l,"+_ 
+                                             Quote$ +"p13" +Quote$+" w l,"     +Quote$+"p14"+Quote$+" w  l,"+Quote$+"p15"+Quote$+" w l" 
+ 
+                CASE 16 : PRINT #N%, "plot "+Quote$ +"p1"  +Quote$+" w l lw 1,"+Quote$+"p2" +Quote$+" w  l,"+Quote$+"p3" +Quote$+" w l,"+Quote$+"p4" 
++Quote$+" w l,"+Quote$+"p5" +Quote$+" w l,"+Quote$+ "p6" +Quote$+" w l,"+_ 
+                                             Quote$ +"p7"  +Quote$+" w l,"     +Quote$+"p8" +Quote$+" w  l,"+Quote$+"p9" +Quote$+" w 
+l,"+Quote$+"p10"+Quote$+" w l,"+Quote$+"p11"+Quote$ +" w l,"+Quote$+"p12"+Quote$+" w l,"+_ 
+                                             Quote$ +"p13" +Quote$+" w l,"     +Quote$+"p14"+Quote$+" w  l,"+Quote$+"p15"+Quote$+" w 
+l,"+Quote$+"p16"+Quote$+" w l" 
+            END SELECT 
+ 
+        ELSE 
+ 
+            SELECT CASE NumTrajectories% 
+ 
+                CASE 1  : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1" 
+                CASE 2  : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1,"+Quote$+"p2"+Quote$ 
+                CASE 3  : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1,"+Quote$+"p2"+Quote$+" ,"+Quote $+"p3"+Quote$ 
+                CASE 4  : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1,"+Quote$+"p2"+Quote$+" ,"+Quote $+"p3"+Quote$+" ,"+Quote$+"p4"+Quote$ 
+                CASE 5  : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1,"+Quote$+"p2"+Quote$+" ,"+Quote $+"p3"+Quote$+" ,"+Quote$+"p4"+Quote$+" 
+,"+Quote$+"p5"+Quote$ 
+                CASE 6  : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1,"+Quote$+"p2"+Quote$+" ,"+Quote $+"p3"+Quote$+" ,"+Quote$+"p4"+Quote$+" 
+,"+Quote$+"p5"+Quote$+" ,"+Quote$+"p6"+Quote$ 
+                CASE 7  : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1,"+Quote$+"p2"+Quote$+" ,"+Quote $+"p3"+Quote$+" ,"+Quote$+"p4"+Quote$+" 
+,"+Quote$+"p5"+Quote$+" ,"+Quote$+"p6"+Quote$+" ,"+ _ 
+                                             Quote$ +"p7"+Quote$ 
+                CASE 8  : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1,"+Quote$+"p2"+Quote$+" ,"+Quote $+"p3"+Quote$+" ,"+Quote$+"p4"+Quote$+" 
+,"+Quote$+"p5"+Quote$+" ,"+Quote$+"p6"+Quote$+" ,"+ _ 
+                                             Quote$ +"p7"+Quote$+" ,"    +Quote$+"p8"+Quote$ 
+                CASE 9  : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1,"+Quote$+"p2"+Quote$+" ,"+Quote $+"p3"+Quote$+" ,"+Quote$+"p4"+Quote$+" 
+,"+Quote$+"p5"+Quote$+" ,"+Quote$+"p6"+Quote$+" ,"+ _ 
+                                             Quote$ +"p7"+Quote$+" ,"    +Quote$+"p8"+Quote$+" ,"+Quote $+"p9"+Quote$ 
+ 
+                CASE 10 : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1," +Quote$+"p2" +Quote$+" ,"+Quo te$+"p3"+Quote$+" ,"+Quote$+"p4"  +Quote$+" 
+,"+Quote$+"p5"+Quote$+" ,"+Quote$+"p6"+Quote$+" ,"+ _ 
+                                             Quote$ +"p7"+Quote$+" ,"     +Quote$+"p8" +Quote$+" ,"+Quo te$+"p9"+Quote$+" ,"+Quote$+"p10" +Quote$ 
+ 68 of 92                 CASE 11 : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1," +Quote$+"p2" +Quote$+" ,"+Quo te$+"p3"+Quote$+" ,"+Quote$+"p4"  +Quote$+" 
+,"+Quote$+"p5" +Quote$+" ,"+Quote$+"p6"+Quote$+" ," +_ 
+                                             Quote$ +"p7"+Quote$+" ,"     +Quote$+"p8" +Quote$+" ,"+Quo te$+"p9"+Quote$+" ,"+Quote$+"p10" +Quote$+" 
+,"+Quote$+"p11"+Quote$ 
+ 
+                CASE 12 : PRINT #N%, "plot "+Quote$ +"p1"+Quote$+" lw 1," +Quote$+"p2" +Quote$+" ,"+Quo te$+"p3"+Quote$+" ,"+Quote$+"p4"  +Quote$+" 
+,"+Quote$+"p5" +Quote$+" ,"+Quote$+"p6" +Quote$+" , "+_ 
+                                             Quote$ +"p7"+Quote$+" ,"     +Quote$+"p8" +Quote$+" ,"+Quo te$+"p9"+Quote$+" ,"+Quote$+"p10" +Quote$+" 
+,"+Quote$+"p11"+Quote$+" ,"+Quote$+"p12"+Quote$ 
+ 
+                CASE 13 : PRINT #N%, "plot "+Quote$ +"p1" +Quote$+" lw 1,"+Quote$+"p2" +Quote$+" ,"+Quo te$+"p3"+Quote$+" ,"+Quote$+"p4"  +Quote$+" 
+,"+Quote$+"p5" +Quote$+" ," +Quote$+"p6" +Quote$+" ,"+_ 
+                                             Quote$ +"p7" +Quote$+" ,"    +Quote$+"p8" +Quote$+" ,"+Quo te$+"p9"+Quote$+" ,"+Quote$+"p10" +Quote$+" 
+,"+Quote$+"p11"+Quote$+" ," +Quote$+"p12"+Quote$+" ,"+_ 
+                                             Quote$ +"p13"+Quote$ 
+ 
+                CASE 14 : PRINT #N%, "plot "+Quote$ +"p1" +Quote$+" lw 1,"+Quote$+"p2" +Quote$+" ,"+Quo te$+"p3"+Quote$+" ,"+Quote$+"p4"  +Quote$+" 
+,"+Quote$+"p5" +Quote$+" ," +Quote$+"p6" +Quote$+" ,"+_ 
+                                             Quote$ +"p7" +Quote$+" ,"    +Quote$+"p8" +Quote$+" ,"+Quo te$+"p9"+Quote$+" ,"+Quote$+"p10" +Quote$+" 
+,"+Quote$+"p11"+Quote$+" ," +Quote$+"p12"+Quote$+" ,"+_ 
+                                             Quote$ +"p13"+Quote$+" ,"    +Quote$+"p14"+Quote$ 
+ 
+                CASE 15 : PRINT #N%, "plot "+Quote$ +"p1" +Quote$+" lw 1,"+Quote$+"p2" +Quote$+" ,"+Quo te$+"p3" +Quote$+" ,"+Quote$+"p4" +Quote$+" 
+,"+Quote$+"p5" +Quote$+" ," +Quote$+"p6" +Quote$+" ,"+_ 
+                                             Quote$ +"p7" +Quote$+" ,"    +Quote$+"p8" +Quote$+" ,"+Quo te$+"p9" +Quote$+" ,"+Quote$+"p10"+Quote$+" 
+,"+Quote$+"p11"+Quote$+" ," +Quote$+"p12"+Quote$+" ,"+_ 
+                                             Quote$ +"p13"+Quote$+" ,"    +Quote$+"p14"+Quote$+" ,"+Quo te$+"p15"+Quote$ 
+ 
+                CASE 16 : PRINT #N%, "plot "+Quote$ +"p1" +Quote$+" lw 1,"+Quote$+"p2" +Quote$+" ,"+Quo te$+"p3" +Quote$+" ,"+Quote$+"p4" +Quote$+" 
+,"+Quote$+"p5" +Quote$+" ," +Quote$+"p6" +Quote$+" ,"+_ 
+                                             Quote$ +"p7" +Quote$+" ,"    +Quote$+"p8" +Quote$+" ,"+Quo te$+"p9" +Quote$+" ,"+Quote$+"p10"+Quote$+" 
+,"+Quote$+"p11"+Quote$+" ," +Quote$+"p12"+Quote$+" ,"+_ 
+                                             Quote$ +"p13"+Quote$+" ,"    +Quote$+"p14"+Quote$+" ,"+Quo te$+"p15"+Quote$+" ,"+Quote$+"p16"+Quote$ 
+            END SELECT 
+ 
+        END IF 
+ 
+    CLOSE #N% 
+ 
+    ProcID??? = SHELL(GnuPlotEXE$+" cmd2d.gp -") : CALL Delay(0.5##) 
+ 
+END SUB 'Plot2DindividualProbeTrajectories() 
+ 
+'---- 
+ 
+SUB Plot3DbestProbeTrajectories(NumTrajectories%,M( ),R(),XiMin(),XiMax(),Np%,Nd%,LastStep&,FunctionNam e$) 'XYZZY 
+ 
+LOCAL TrajectoryNumber%, ProbeNumber%, StepNumber&,  N%, M%, ProcID??? 
+ 
+LOCAL MaximumFitness, MinimumFitness AS EXT 
+ 
+LOCAL BestProbeThisStep%() 
+ 
+LOCAL BestFitnessThisStep(), TempFitness() AS EXT 
+ 
+LOCAL Annotation$, xCoord$, yCoord$, zCoord$, GnuPl otEXE$, PlotWithLines$ 
+ 
+    Annotation$      = "" 
+ 
+    PlotWithLines$   = "NO" '"YES" '"NO" 
+ 
+    NumTrajectories% = MIN(Np%,NumTrajectories%) 
+ 
+    GnuPlotEXE$ = "wgnuplot.exe" 
+ 
+'   ---------------- Get Min/Max Fitnesses -------- --------- 
+ 
+    MaximumFitness = M(1,0) : MinimumFitness = M(1, 0)  'Note:  M(p%,j&) 
+ 
+    FOR StepNumber& = 0 TO LastStep& 
+ 
+        FOR ProbeNumber% = 1 TO Np% 
+ 
+            IF M(ProbeNumber%,StepNumber&) >= Maxim umFitness THEN MaximumFitness = M(ProbeNumber%,Step Number&) 
+ 
+            IF M(ProbeNumber%,StepNumber&) =< Minim umFitness THEN MinimumFitness = M(ProbeNumber%,Step Number&) 
+ 
+        NEXT ProbeNumber% 
+ 
+    NEXT StepNumber% 
+ 
+'   ------------- Copy Fitness Array M() into TempF itness to Preserve M() ---------------- 
+ 
+    REDIM TempFitness(1 TO Np%, 0 TO LastStep&) 
+ 
+    FOR StepNumber& = 0 TO LastStep& 
+ 
+        FOR ProbeNumber% = 1 TO Np% 
+ 
+            TempFitness(ProbeNumber%,StepNumber&) =  M(ProbeNumber%,StepNumber&) 
+ 
+        NEXT ProbeNumber% 
+ 
+    NEXT StepNumber% 
+ 
+'   ------------ LOOP ON TRAJECTORIES ----------- 
+ 
+    FOR TrajectoryNumber% = 1 TO NumTrajectories% 
+ 
+'       --------------- Get Trajectory Coordinate D ata ----------------- 
+ 
+        REDIM BestFitnessThisStep(0 TO LastStep&), BestProbeThisStep%(0 TO LastStep&) 
+ 
+        FOR StepNumber& = 0 TO LastStep& 
+ 
+            BestFitnessThisStep(StepNumber&) = Temp Fitness(1,StepNumber&) 
+ 
+            FOR ProbeNumber% = 1 TO Np% 
+ 
+                IF TempFitness(ProbeNumber%,StepNum ber&) >= BestFitnessThisStep(StepNumber&) THEN 
+ 
+                    BestFitnessThisStep(StepNumber& ) = TempFitness(ProbeNumber%,StepNumber&) 
+ 
+                    BestProbeThisStep%(StepNumber&)   = ProbeNumber% 69 of 92  
+                END IF 
+ 
+            NEXT ProbeNumber% 
+ 
+        NEXT StepNumber& 
+ 
+'   ----- Create Plot Data File ----- 
+ 
+    N% = FREEFILE 
+ 
+    SELECT CASE TrajectoryNumber% 
+ 
+        CASE 1  : OPEN "t1"  FOR OUTPUT AS #N% 
+        CASE 2  : OPEN "t2"  FOR OUTPUT AS #N% 
+        CASE 3  : OPEN "t3"  FOR OUTPUT AS #N% 
+        CASE 4  : OPEN "t4"  FOR OUTPUT AS #N% 
+        CASE 5  : OPEN "t5"  FOR OUTPUT AS #N% 
+        CASE 6  : OPEN "t6"  FOR OUTPUT AS #N% 
+        CASE 7  : OPEN "t7"  FOR OUTPUT AS #N% 
+        CASE 8  : OPEN "t8"  FOR OUTPUT AS #N% 
+        CASE 9  : OPEN "t9"  FOR OUTPUT AS #N% 
+        CASE 10 : OPEN "t10" FOR OUTPUT AS #N% 
+ 
+    END SELECT 
+ 
+'   ------------ Write Plot File Data ------------ 
+ 
+    FOR StepNumber& = 0 TO LastStep& 
+ 
+        PRINT #N%, USING$("######.######## ######.# ####### 
+######.########",R(BestProbeThisStep%(StepNumber&), 1,StepNumber&),R(BestProbeThisStep%(StepNumber&),2, StepNumber&),R(BestProbeThisStep%(StepNumber&) 
+,3,StepNumber&))+CHR$(13) 
+ 
+        TempFitness(BestProbeThisStep%(StepNumber&) ,StepNumber&) = MinimumFitness 'so that same max wi ll not be found for next trajectory 
+ 
+    NEXT StepNumber% 
+ 
+    CLOSE #N% 
+ 
+    NEXT TrajectoryNumber% 
+ 
+'   ------------------------- Plot Trajectories --- ----------------------- 
+ 
+   'CALL CreateGNUplotINIfile(0.1##*ScreenWidth&,0. 25##*ScreenHeight&,0.6##*ScreenHeight&,0.6##*Screen Height&) 
+ 
+    Annotation$ = "" 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "cmd3d.gp" FOR OUTPUT AS #N% 
+ 
+    PRINT #N%, "set pm3d" 
+    PRINT #N%, "show pm3d" 
+    PRINT #N%, "set hidden3d" 
+    PRINT #N%, "set view 45, 45, 1, 1" 
+ 
+    PRINT #N%, "unset colorbox" 
+ 
+    PRINT #N%, "set xrange [" + REMOVE$(STR$(XiMin( 1)),ANY"" ) + ":" + REMOVE$(STR$(XiMax(1)),ANY"" ) + "]" 
+    PRINT #N%, "set yrange [" + REMOVE$(STR$(XiMin( 2)),ANY"" ) + ":" + REMOVE$(STR$(XiMax(2)),ANY"" ) + "]" 
+    PRINT #N%, "set zrange [" + REMOVE$(STR$(XiMin( 3)),ANY"" ) + ":" + REMOVE$(STR$(XiMax(3)),ANY"" ) + "]" 
+ 
+'   PRINT #N%, "set label "   + Quote$  + Annotatio n$ + Quote$+" at graph "+xCoord$+","+yCoord$+","+zC oord$ 
+'   PRINT #N%, "show label" 
+    PRINT #N%, "set grid xtics ytics ztics" 
+    PRINT #N%, "show grid" 
+    PRINT #N%, "set title "  + Quote$ + "3D " + Fun ctionName$ + " PROBE TRAJECTORIES" + "\n" + RunID$ + Quote$ 
+    PRINT #N%, "set xlabel " + Quote$ + "x1"                                          + Quote$ 
+    PRINT #N%, "set ylabel " + Quote$ + "x2"                                          + Quote$ 
+    PRINT #N%, "set zlabel " + Quote$ + "x3"                                          + Quote$ 
+ 
+    IF PlotWithLines$ = "YES" THEN 
+ 
+        SELECT CASE NumTrajectories% 
+ 
+            CASE 1  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" w l lw 3" 
+            CASE 2  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l" 
+            CASE 3  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l,"+Q uote$+"t3"+Quote$+" w l" 
+            CASE 4  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l,"+Q uote$+"t3"+Quote$+" w l,"+Quote$+"t4"+Quote$+" 
+w l" 
+            CASE 5  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l,"+Q uote$+"t3"+Quote$+" w l,"+Quote$+"t4"+Quote$+" 
+w l,"+Quote$+"t5"+Quote$+" w l" 
+            CASE 6  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l,"+Q uote$+"t3"+Quote$+" w l,"+Quote$+"t4"+Quote$+" 
+w l,"+Quote$+"t5"+Quote$+" w l,"+Quote$+"t6"+Quote$ +" w l" 
+            CASE 7  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l,"+Q uote$+"t3"+Quote$+" w l,"+Quote$+"t4"+Quote$+" 
+w l,"+Quote$+"t5"+Quote$+" w l,"+Quote$+"t6"+Quote$ +" w l,"+_ 
+                                          Quote$+"t 7"+Quote$+" w l" 
+            CASE 8  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l,"+Q uote$+"t3"+Quote$+" w l,"+Quote$+"t4"+Quote$+" 
+w l,"+Quote$+"t5"+Quote$+" w l,"+Quote$+"t6"+Quote$ +" w l,"+_ 
+                                          Quote$+"t 7"+Quote$+" w l,"     +Quote$+"t8"+Quote$+" w l" 
+            CASE 9  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l,"+Q uote$+"t3"+Quote$+" w l,"+Quote$+"t4"+Quote$+" 
+w l,"+Quote$+"t5"+Quote$+" w l,"+Quote$+"t6"+Quote$ +" w l,"+_ 
+                                          Quote$+"t 7"+Quote$+" w l,"     +Quote$+"t8"+Quote$+" w l,"+Q uote$+"t9"+Quote$+" w l" 
+            CASE 10 : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" w l lw 3,"+Quote$+"t2"+Quote$+" w l,"+Q uote$+"t3"+Quote$+" w l,"+Quote$+"t4"+Quote$+" 
+w l,"+Quote$+"t5"+Quote$+" w l,"+Quote$+"t6"+Quote$ +" w l,"+_ 
+                                          Quote$+"t 7"+Quote$+" w l,"     +Quote$+"t8"+Quote$+" w l,"+Q uote$+"t9"+Quote$+" w 
+l,"+Quote$+"t10"+Quote$+" w l" 
+        END SELECT 
+ 
+    ELSE 
+ 
+        SELECT CASE NumTrajectories% 
+ 
+            CASE 1  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" lw 2" 
+            CASE 2  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$ 
+            CASE 3  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote$+" t3"+Quote$ 
+            CASE 4  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote$+" t3"+Quote$+" ,"+Quote$+"t4"+Quote$ 
+            CASE 5  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote$+" t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$ 
+            CASE 6  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote$+" t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$+" ,"+Quote$+"t6"+Quote$ 
+            CASE 7  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote$+" t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$+" ,"+Quote$+"t6"+Quote$+" ,"+ _ 70 of 92                                           Quote$+"t 7"+Quote$ 
+            CASE 8  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote$+" t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$+" ,"+Quote$+"t6"+Quote$+" ,"+ _ 
+                                          Quote$+"t 7"+Quote$+" ,"    +Quote$+"t8"+Quote$ 
+            CASE 9  : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote$+" t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$+" ,"+Quote$+"t6"+Quote$+" ,"+ _ 
+                                          Quote$+"t 7"+Quote$+" ,"    +Quote$+"t8"+Quote$+" ,"+Quote$+" t9"+Quote$ 
+            CASE 10 : PRINT #N%, "splot "+Quote$+"t 1"+Quote$+" lw 2,"+Quote$+"t2"+Quote$+" ,"+Quote$+" t3"+Quote$+" ,"+Quote$+"t4"+Quote$+" 
+,"+Quote$+"t5"+Quote$+" ,"+Quote$+"t6"+Quote$+" ,"+ _ 
+                                          Quote$+"t 7"+Quote$+" ,"    +Quote$+"t8"+Quote$+" ,"+Quote$+" t9"+Quote$+" ,"+Quote$+"t10"+Quote$ 
+        END SELECT 
+ 
+    END IF 
+ 
+    CLOSE #N% 
+ 
+    ProcID??? = SHELL(GnuPlotEXE$+" cmd3d.gp -") : CALL Delay(0.5##) 
+ 
+END SUB 'Plot3DbestProbeTrajectories() 
+ 
+'---- 
+ 
+FUNCTION HasFITNESSsaturated$(Nsteps&,j&,Np%,Nd%,M( ),R(),DiagLength) 
+ 
+LOCAL A$ 
+ 
+LOCAL k&, p% 
+ 
+LOCAL BestFitness, SumOfBestFitnesses, BestFitnessS tepJ, FitnessSatTOL AS EXT 
+ 
+    A$ = "NO" 
+ 
+    FitnessSatTOL = 0.000001## 'tolerance for FITNE SS saturation 
+ 
+    IF j& < Nsteps& + 10 THEN GOTO ExitHasFITNESSsa turated 'execute at least 10 steps after averaging interval before performing this check 
+ 
+    SumOfBestFitnesses = 0## 
+ 
+    FOR k& = j&-Nsteps&+1 TO j& 
+ 
+        BestFitness = M(k&,1) 
+ 
+        FOR p% = 1 TO Np% 
+ 
+            IF M(p%,k&) >= BestFitness THEN BestFit ness = M(p%,k&) 
+ 
+        NEXT p% 
+ 
+        IF k& = j& THEN BestFitnessStepJ = BestFitn ess 
+ 
+        SumOfBestFitnesses = SumOfBestFitnesses + B estFitness 
+ 
+    NEXT k& 
+ 
+    IF ABS(SumOfBestFitnesses/Nsteps&-BestFitnessSt epJ) =< FitnessSatTOL THEN A$ = "YES" 'saturation i f (avg value - last value) are within TOL 
+ 
+ExitHasFITNESSsaturated: 
+ 
+    HasFITNESSsaturated$ = A$ 
+ 
+END FUNCTION 'HasFITNESSsaturated$() 
+ 
+'----------- 
+ 
+FUNCTION HasDAVGsaturated$(Nsteps&,j&,Np%,Nd%,M(),R (),DiagLength) 
+ 
+LOCAL A$ 
+ 
+LOCAL k& 
+ 
+LOCAL SumOfDavg, DavgStepJ AS EXT 
+ 
+LOCAL DavgSatTOL AS EXT 
+ 
+    A$ = "NO" 
+ 
+    DavgSatTOL = 0.0005## 'tolerance for DAVG satur ation 
+ 
+    IF j& < Nsteps& + 10 THEN GOTO ExitHasDAVGsatur ated 'execute at least 10 steps after averaging int erval before performing this check 
+ 
+    DavgStepJ = DavgThisStep(j&,Np%,Nd%,M(),R(),Dia gLength) 
+ 
+    SumOfDavg = 0## 
+ 
+    FOR k& = j&-Nsteps&+1 TO j& 'check this step an d previous (Nsteps&-1) steps 
+ 
+        SumOfDavg = SumOfDavg + DavgThisStep(k&,Np% ,Nd%,M(),R(),DiagLength) 
+ 
+    NEXT k& 
+ 
+    IF ABS(SumOfDavg/Nsteps&-DavgStepJ) =< DavgSatT OL THEN A$ = "YES" 'saturation if (avg value - last  value) are within TOL 
+ 
+ExitHasDAVGsaturated: 
+ 
+    HasDAVGsaturated$ = A$ 
+ 
+END FUNCTION 'HasDAVGsaturated$() 
+ 
+'----------- 
+ 
+FUNCTION OscillationInDavg$(j&,Np%,Nd%,M(),R(),Diag Length) 
+ 
+LOCAL A$ 
+ 
+LOCAL k&, NumSlopeChanges% 
+ 
+    A$ = "NO" 
+ 
+    NumSlopeChanges% = 0 
+ 
+    IF j& < 15 THEN GOTO ExitDavgOscillation 'wait at least 15 steps 
+ 
+    FOR k& = j&-10 TO j&-1 'check previous ten step s 
+ 71 of 92         IF (DavgThisStep(k&,Np%,Nd%,M(),R(),DiagLen gth)-DavgThisStep(k&-1,Np%,Nd%,M(),R(),DiagLength)) * _ 
+           (DavgThisStep(k&+1,Np%,Nd%,M(),R(),DiagL ength)-DavgThisStep(k&,Np%,Nd%,M(),R(),DiagLength))  < 0## THEN INCR NumSlopeChanges% 
+ 
+    NEXT j& 
+ 
+    IF NumSlopeChanges% >= 3 THEN A$ = "YES" 
+ 
+ExitDavgOscillation: 
+ 
+    OscillationInDavg$ = A$ 
+ 
+END FUNCTION 'OscillationInDavg() 
+ 
+'------ 
+ 
+FUNCTION DavgThisStep(j&,Np%,Nd%,M(),R(),DiagLength ) 
+ 
+LOCAL BestFitness, TotalDistanceAllProbes, SumSQ AS  EXT 
+ 
+LOCAL p%, k&, N%, i%, BestProbeNumber%, BestTimeSte p& 
+ 
+'   ----------- Best Probe #, etc. ----------- 
+ 
+    FOR k& = 0 TO j& 
+ 
+        BestFitness = M(1,k&) 
+ 
+        FOR p% = 1 TO Np% 
+ 
+            IF M(p%,k&) >= BestFitness THEN 
+ 
+                BestFitness = M(p%,k&) : BestProbeN umber% = p% : BestTimeStep& = k& 
+ 
+            END IF 
+ 
+        NEXT p% 'probe # 
+ 
+    NEXT k& 'time step 
+ 
+'   --------- Average Distance to Best Probe ------ ----- 
+ 
+    TotalDistanceAllProbes = 0## 
+ 
+    FOR p% = 1 TO Np% 
+ 
+        SumSQ = 0## 
+ 
+        FOR i% = 1 TO Nd% 
+ 
+            SumSQ = SumSQ + (R(BestProbeNumber%,i%, BestTimeStep&)-R(p%,i%,j&))^2 'do not exclude p%=Be stProbeNumber%(j&) from sum because it adds 
+zero 
+ 
+         NEXT i% 
+ 
+        TotalDistanceAllProbes = TotalDistanceAllPr obes + SQR(SumSQ) 
+ 
+    NEXT p% 
+ 
+    DavgThisStep = TotalDistanceAllProbes/(DiagLeng th*(Np%-1)) 'but exclude best prove from average 
+ 
+END FUNCTION 'DavgThisStep() 
+ 
+'----------- 
+ 
+SUB 
+PlotBestFitnessEvolution(Nd%,Np%,LastStep&,G,DeltaT ,Alpha,Beta,Frep,M(),PlaceInitialProbes$,InitialAcc eleration$,RepositionFactor$,FunctionName$,Gam 
+ma) 
+ 
+LOCAL BestFitness(), GlobalBestFitness AS EXT 
+ 
+LOCAL PlotAnnotation$, PlotTitle$ 
+ 
+LOCAL p%, j&, N% 
+ 
+    REDIM BestFitness(0 TO LastStep&) 
+ 
+    CALL 
+GetPlotAnnotation(PlotAnnotation$,Nd%,Np%,LastStep& ,G,DeltaT,Alpha,Beta,Frep,M(),PlaceInitialProbes$,I nitialAcceleration$,RepositionFactor$,Function 
+Name$,Gamma) 
+ 
+    GlobalBestFitness = M(1,0) 
+ 
+    FOR j& = 0 TO LastStep& 
+ 
+        BestFitness(j&) = M(1,j&) 
+ 
+        FOR p% = 1 TO Np% 
+ 
+            IF M(p%,j&) >= BestFitness(j&)   THEN B estFitness(j&)   = M(p%,j&) 
+ 
+            IF M(p%,j&) >= GlobalBestFitness THEN G lobalBestFitness = M(p%,j&) 
+ 
+        NEXT p% 'probe # 
+ 
+    NEXT j& 'time step 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "Fitness" FOR OUTPUT AS #N% 
+ 
+        FOR j& = 0 TO LastStep& 
+ 
+            PRINT #N%, USING$("###### #######.##### ##",j&,BestFitness(j&)) 
+ 
+        NEXT j& 
+ 
+ 
+    CLOSE #N% 
+ 
+    PlotAnnotation$ = PlotAnnotation$ + "Best Fitne ss = " + REMOVE$(STR$(ROUND(GlobalBestFitness,8)),A NY" ") 
+ 
+    PlotTitle$ = "Best Fitness vs Time Step\n" + "[ " + REMOVE$(STR$(Np%),ANY" ") + " probes, "+REMOVE$ (STR$(LastStep&),ANY" ")+" time steps]" 
+ 
+    CALL CreateGNUplotINIfile(0.1##*ScreenWidth&,0. 1##*ScreenHeight&,0.6##*ScreenWidth&,0.6##*ScreenHe ight&) 72 of 92  
+    CALL TwoDplot("Fitness","Best Fitness","0.7","0 .7","Time Step\n\n.",".\n\nBest Fitness(X)", _ 
+                  "","","","","","","","","wgnuplot .exe"," with lines linewidth 2",PlotAnnotation$) 
+ 
+END SUB 'PlotBestFitnessEvolution() 
+ 
+'------ 
+ 
+SUB 
+PlotAverageDistance(Nd%,Np%,LastStep&,G,DeltaT,Alph a,Beta,Frep,M(),PlaceInitialProbes$,InitialAccelera tion$,RepositionFactor$,FunctionName$,R(),Diag 
+Length,Gamma) 
+ 
+LOCAL Davg(), BestFitness(), TotalDistanceAllProbes , SumSQ AS EXT 
+ 
+LOCAL PlotAnnotation$, PlotTitle$ 
+ 
+LOCAL p%, j&, N%, i%, BestProbeNumber%(), BestTimeS tep&() 
+ 
+    REDIM Davg(0 TO LastStep&), BestFitness(0 TO La stStep&), BestProbeNumber%(0 TO LastStep&), BestTim eStep&(0 TO LastStep&) 
+ 
+    CALL 
+GetPlotAnnotation(PlotAnnotation$,Nd%,Np%,LastStep& ,G,DeltaT,Alpha,Beta,Frep,M(),PlaceInitialProbes$,I nitialAcceleration$,RepositionFactor$,Function 
+Name$,Gamma) 
+ 
+'   ----------- Best Probe #, etc. ----------- 
+ 
+    FOR j& = 0 TO LastStep& 
+ 
+        BestFitness(j&) = M(1,j&) 
+ 
+        FOR p% = 1 TO Np% 
+ 
+            IF M(p%,j&) >= BestFitness(j&) THEN 
+ 
+                BestFitness(j&) = M(p%,j&) : BestPr obeNumber%(j&) = p% : BestTimeStep&(j&) = j& 'only probe number is used at this time, but other 
+data are computed for possible future use. 
+ 
+            END IF 
+ 
+        NEXT p% 'probe # 
+ 
+    NEXT j& 'time step 
+ 
+    N% = FREEFILE 
+ 
+'   --------- Average Distance to Best Probe ------ ----- 
+ 
+    FOR j& = 0 TO LastStep& 
+ 
+        TotalDistanceAllProbes = 0## 
+ 
+        FOR p% = 1 TO Np% 
+ 
+            SumSQ = 0## 
+ 
+            FOR i% = 1 TO Nd% 
+ 
+                SumSQ = SumSQ + (R(BestProbeNumber% (j&),i%,j&)-R(p%,i%,j&))^2 'do not exclude p%=BestP robeNumber%(j&) from sum because it adds zero 
+ 
+            NEXT i% 
+ 
+            TotalDistanceAllProbes = TotalDistanceA llProbes + SQR(SumSQ) 
+ 
+        NEXT p% 
+ 
+        Davg(j&) = TotalDistanceAllProbes/(DiagLeng th*(Np%-1)) 'but exclude best prove from average 
+ 
+    NEXT j& 
+ 
+'   ------------ Create Plot Data File ----------- 
+ 
+    OPEN "Davg" FOR OUTPUT AS #N% 
+ 
+        FOR j& = 0 TO LastStep& 
+ 
+            PRINT #N%, USING$("###### #######.##### ##",j&,Davg(j&)) 
+ 
+        NEXT j& 
+ 
+    CLOSE #N% 
+ 
+    PlotTitle$ = "Average Distance of " + REMOVE$(S TR$(Np%-1),ANY" ") + " Probes to Best Probe\nNormal ized to Size of Decision Space\n" + _ 
+                 "[" + REMOVE$(STR$(Np%),ANY" ") + " probes, " + REMOVE$(STR$(LastStep&),ANY" ") + " t ime steps]" 
+ 
+    CALL CreateGNUplotINIfile(0.2##*ScreenWidth&,0. 2##*ScreenHeight&,0.6##*ScreenWidth&,0.6##*ScreenHe ight&) 
+ 
+    CALL TwoDplot("Davg",PlotTitle$,"0.7","0.9","Ti me Step\n\n.",".\n\n<D>/Ldiag", _ 
+                  "","","","","","","","","wgnuplot .exe"," with lines linewidth 2",PlotAnnotation$) 
+ 
+END SUB 'PlotAverageDistance() 
+ 
+'------ 
+ 
+SUB 
+GetPlotAnnotation(PlotAnnotation$,Nd%,Np%,LastStep& ,G,DeltaT,Alpha,Beta,Frep,M(),PlaceInitialProbes$,I nitialAcceleration$,RepositionFactor$,Function 
+Name$,Gamma) 
+ 
+LOCAL A$ 
+ 
+    A$ = "" : IF PlaceInitialProbes$ = "UNIFORM ON- AXIS" AND Nd% > 1 THEN A$ = " ("+REMOVE$(STR$(Np%/N d%),ANY" ") + "/axis)" 
+ 
+    PlotAnnotation$ = RunID$ + "\n" + _ 
+                      FunctionName$ + " Function" +  " ("+ FormatInteger$(Nd%) + "-D) \n"    +_ 
+                      FormatInteger$(Np%) + " probe s"       + A$ + "\n" +_ 
+                      "G = " + FormatFP$(G,2)               + "\n" +_ 
+                      "Alpha = "     + FormatFP$(Al pha,1)   + "\n" +_ 
+                      "Beta = "      + FormatFP$(Be ta,1)    + "\n" +_ 
+                      "DelT = "      + FormatFP$(De ltaT,1)  + "\n" +_ 
+                      "Gamma = "     + FormatFP$(Ga mma,3)   + "\n" +_ 
+                      "Init Probes " + PlaceInitial Probes$  + "\n" +_ 
+                      "Init Accel "  + InitialAccel eration$ + "\n" +_ 
+                      "Frep = "      + FormatFP$(Fr ep,3)    + " (" + RepositionFactor$ + ")\n" 
+ 
+'    SELECT CASE RepositionFactor$ 73 of 92  
+ '       CASE "FIXED" : PlotAnnotation$ = PlotAnnot ation$ + "Frep = " + FormatFP$(Frep,3) + " fixed" +  "\n" 
+ 
+  '  END SELECT 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB 
+PlotBestProbeVsTimeStep(Nd%,Np%,LastStep&,G,DeltaT, Alpha,Beta,Frep,M(),PlaceInitialProbes$,InitialAcce leration$,RepositionFactor$,FunctionName$,Gamm 
+a) 
+ 
+LOCAL BestFitness AS EXT 
+ 
+LOCAL PlotAnnotation$, PlotTitle$ 
+ 
+LOCAL p%, j&, N%, BestProbeNumber%() 
+ 
+    REDIM BestProbeNumber%(0 TO LastStep&) 
+ 
+    CALL 
+GetPlotAnnotation(PlotAnnotation$,Nd%,Np%,LastStep& ,G,DeltaT,Alpha,Beta,Frep,M(),PlaceInitialProbes$,I nitialAcceleration$,RepositionFactor$,Function 
+Name$,Gamma) 
+ 
+    FOR j& = 0 TO LastStep& 
+ 
+        Bestfitness = M(1,j&) 
+ 
+        FOR p% = 1 TO Np% 
+ 
+            IF M(p%,j&) >= BestFitness THEN 
+ 
+                BestFitness = M(p%,j&) : BestProbeN umber%(j&) = p% 
+ 
+            END IF 
+ 
+        NEXT p% 'probe # 
+ 
+    NEXT j& 'time step 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "Best Probe" FOR OUTPUT AS #N% 
+ 
+        FOR j& = 0 TO LastStep& 
+ 
+            PRINT #N%, USING$("###### #####",j&,Bes tProbeNumber%(j&)) 
+ 
+        NEXT j& 
+ 
+ 
+    CLOSE #N% 
+ 
+    PlotTitle$ = "Best Probe Number vs Time Step\n"  + "[" +REMOVE$(STR$(Np%),ANY" ") + " probes, " + R EMOVE$(STR$(LastStep&),ANY" ") + " time 
+steps]" 
+ 
+    CALL CreateGNUplotINIfile(0.15#*ScreenWidth&,0. 15##*ScreenHeight&,0.6##*ScreenWidth&,0.6##*ScreenH eight&) 
+ 
+'USAGE: CALL 
+TwoDplot(PlotFileName$,PlotTitle$,xCoord$,yCoord$,X axisLabel$,YaxisLabel$,LogXaxis$,LogYaxis$,xMin$,xM ax$,yMin$,yMax$,xTics$,yTics$,GnuPlotEXE$,Line 
+Type$,Annotation$) 
+ 
+    CALL TwoDplot("Best Probe",PlotTitle$,"0.7","0. 7","Time Step\n\n.",".\n\nBest Probe #","","","","" ,"0",NoSpaces$(Np%+1,0),"","","wgnuplot.exe"," 
+pt 8 ps .5 lw 1",PlotAnnotation$) 'pt, pointtype; p s, pointsize; lw, linewidth 
+ 
+END SUB 'PlotBestProbeVsTimeStep() 
+ 
+'------ 
+ 
+SUB DisplayBestFitness(Np%,Nd%,LastStep&,M(),R(),Be stFitnessProbeNumber%,BestFitnessTimeStep&,Function Name$) 
+ 
+LOCAL A$, B$, p%, i%, j& 
+ 
+LOCAL BestFitness AS EXT 
+ 
+    B$ = "" : IF Nd% > 1 THEN B$ = "s" 
+ 
+    BestFitness = M(1,0) 
+ 
+    FOR j& = 0 TO LastStep& 
+ 
+        FOR p% = 1 TO Np% 
+ 
+            IF M(p%,j&) >= BestFitness THEN 
+ 
+                BestFitness = M(p%,j&) : BestFitnes sProbeNumber% = p% : BestFitnessTimeStep& = j& 
+ 
+            END IF 
+ 
+        NEXT p% 
+ 
+    NEXT j& 
+ 
+    A$ = FunctionName$ + CHR$(13) +_ 
+         "Best Fitness = " + REMOVE$(STR$(ROUND(Bes tFitness,8)),ANY" ") + " returned by" + CHR$(13)        +_ 
+         "Probe # "        + REMOVE$(STR$(BestFitne ssProbeNumber%),ANY" ") +_ 
+         " at Time Step "  + REMOVE$(STR$(BestFitne ssTimeStep&),ANY" ") + CHR$(13) + CHR$(13) + "P" + REMOVE$(STR$(BestFitnessProbeNumber%),ANY" ") 
++ " coordinate" + B$ + ":" + CHR$(13) 
+ 
+    FOR i% = 1 TO Nd% : A$ = A$ + STR$(i%)+"    "+R EMOVE$(STR$(ROUND(R(BestFitnessProbeNumber%,i%,Best FitnessTimeStep&),8)),ANY" ")+CHR$(13) : NEXT 
+i% 
+ 
+    MSGBOX(A$) 
+ 
+END SUB 'DisplayBestFitness() 
+ 
+'------ 
+ 
+FUNCTION FormatInteger$(M%) : FormatInteger$ = REMO VE$(STR$(M%),ANY" ") : END FUNCTION 
+ 
+'------ 
+ 
+FUNCTION FormatFP$(X,Ndigits%) 74 of 92  
+LOCAL A$ 
+ 
+    IF X = 0## THEN 
+ 
+        A$ = "0." : GOTO ExitFormatFP 
+ 
+    END IF 
+ 
+    A$ = REMOVE$(STR$(ROUND(ABS(X),Ndigits%)),ANY" ") 
+ 
+    IF ABS(X) < 1## THEN 
+ 
+        IF X > 0## THEN 
+ 
+            A$ = "0" + A$ 
+ 
+        ELSE 
+ 
+            A$ = "-0" + A$ 
+ 
+        END IF 
+ 
+    ELSE 
+ 
+        IF X < 0## THEN A$ = "-" + A$ 
+ 
+    END IF 
+ 
+ExitFormatFP: 
+ 
+    FormatFP$ = A$ 
+ 
+END FUNCTION 
+ 
+'----------- 
+ 
+SUB InitialProbeDistribution(Np%,Nd%,Nt&,XiMin(),Xi Max(),R(),PlaceInitialProbes$,Gamma) 
+ 
+LOCAL DeltaXi, DelX1, DelX2, Di AS EXT 
+ 
+LOCAL NumProbesPerAxis%, p%, i%, k%, NumX1points%, NumX2points%, x1pointNum%, x2pointNum%, A$ 
+ 
+    SELECT CASE PlaceInitialProbes$ 
+ 
+        CASE "UNIFORM ON-AXIS" 
+ 
+            IF Nd% > 1 THEN 
+ 
+                NumProbesPerAxis% = Np%\Nd% 'even #  
+ 
+            ELSE 
+ 
+                NumProbesPerAxis% = Np% 
+ 
+            END IF 
+ 
+            FOR i% = 1 TO Nd% 
+ 
+                FOR p% = 1 TO Np% 
+ 
+                    R(p%,i%,0) = XiMin(i%) + Gamma* (XiMax(i%)-XiMin(i%)) 
+ 
+                NEXT Np% 
+ 
+            NEXT i% 
+ 
+            FOR i% = 1 TO Nd% 'place probes axis-by -axis (i% is axis [dimension] number) 
+ 
+                DeltaXi = (XiMax(i%)-XiMin(i%))/(Nu mProbesPerAxis%-1) 
+ 
+                FOR k% = 1 TO NumProbesPerAxis% 
+ 
+                    p% = k% + NumProbesPerAxis%*(i% -1) 'probe # 
+ 
+                    R(p%,i%,0) = XiMin(i%) + (k%-1) *DeltaXi 
+ 
+                NEXT k% 
+ 
+                'DeltaXi = (XiMax(i%)-XiMin(i%))/(N umProbesPerAxis%+1) 
+ 
+                'FOR k% = 1 TO NumProbesPerAxis% 
+ 
+                '    p% = k% + NumProbesPerAxis%*(i %-1) 'probe # 
+ 
+                '    R(p%,i%,0) = XiMin(i%) + k%*De ltaXi 
+ 
+                'NEXT k% 
+ 
+            NEXT i% 
+ 
+        CASE "UNIFORM ON-DIAGONAL" 
+ 
+            FOR p% = 1 TO Np% 
+ 
+                FOR i% = 1 TO Nd% 
+ 
+                    DeltaXi = (XiMax(i%)-XiMin(i%)) /(Np%-1) 
+ 
+                    R(p%,i%,0) = XiMin(i%) + (p%-1) *DeltaXi 
+ 
+                NEXT i% 
+ 
+            NEXT p% 
+ 
+        CASE "2D GRID" 
+ 
+            NumProbesPerAxis% = SQR(Np%) : NumX1poi nts% = NumProbesPerAxis% : NumX2points% = NumX1poin ts% 'broken down for possible future use 
+ 
+            DelX1 = (XiMax(1)-XiMin(1))/(NumX1point s%-1) 
+ 
+            DelX2 = (XiMax(2)-XiMin(2))/(NumX2point s%-1) 
+ 
+            FOR x1pointNum% = 1 TO NumX1points% 75 of 92  
+                FOR x2pointNum% = 1 TO NumX2points%  
+ 
+                    p% = NumX1points%*(x1pointNum%- 1)+x2pointNum% 'probe # 
+ 
+                    R(p%,1,0) = XiMin(1) + DelX1*(x 1pointNum%-1) 'x1 coord 
+                    R(p%,2,0) = XiMin(2) + DelX2*(x 2pointNum%-1) 'x2 coord 
+ 
+                NEXT x2pointNum% 
+ 
+            NEXT x1pointNum% 
+ 
+        CASE "RANDOM" 
+ 
+            FOR p% = 1 TO Np% 
+ 
+                FOR i% = 1 TO Nd% 
+ 
+                    R(p%,i%,0) = XiMin(i%) + Random Num(0##,1##)*(XiMax(i%)-XiMin(i%)) 
+ 
+                NEXT i% 
+ 
+            NEXT p% 
+ 
+    END SELECT 
+ 
+END SUB 'InitialProbeDistribution() 
+ 
+'------ 
+ 
+SUB InitialProbeAccelerations(Np%,Nd%,A(),InitialAc celeration$,MaxInitialRandomAcceleration,MaxInitial FixedAcceleration) 
+ 
+LOCAL p%, i% 
+ 
+LOCAL A$ 
+ 
+LOCAL FixedInitialAcceleration AS EXT 
+ 
+    SELECT CASE InitialAcceleration$ 
+ 
+        CASE "ZERO" 
+ 
+            FOR p% = 1 TO Np% 
+ 
+                FOR i% = 1 TO Nd% 
+ 
+                    A(p%,i%,0) = 0## 
+ 
+                NEXT i% 'coordinate # 
+ 
+            NEXT p% 'probe # 
+ 
+        CASE "FIXED" 
+ 
+            A$ = INPUTBOX$("Fixed Initial Probe"+CH R$(13)+"Acceleration? [0.001-"+REMOVE$(STR$(MaxInit ialFixedAcceleration),ANY" ")+"]","PROBES' 
+INITIAL ACCELERATION","0.5") 
+ 
+            FixedInitialAcceleration = VAL(A$) 
+ 
+            IF FixedInitialAcceleration < 0.001## O R FixedInitialAcceleration > MaxInitialFixedAcceler ation THEN FixedInitialAcceleration = 0.5## 
+ 
+            FOR p% = 1 TO Np% 
+ 
+                FOR i% = 1 TO Nd% 
+ 
+                    A(p%,i%,0) = FixedInitialAccele ration 
+ 
+                NEXT i% 'coordinate 
+ 
+            NEXT p% 'probe # 
+ 
+        CASE "RANDOM" 
+ 
+            FOR p% = 1 TO Np% 
+ 
+                FOR i% = 1 TO Nd% 
+ 
+                    A(p%,i%,0) = RandomNum(0##,1##) *MaxInitialRandomAcceleration 
+ 
+                NEXT i% 'coordinate 
+ 
+            NEXT p% 'probe # 
+ 
+    END SELECT 
+ 
+END SUB 'InitialProbeAccelerations() 
+ 
+'------ 
+ 
+SUB RetrieveErrantProbes(Np%,Nd%,j&,XiMin(),XiMax() ,R(),M(),RepositionFactor$,Frep) 'note: M(), Reposi tionFcator$ passed but not used in thsi 
+version 
+ 
+LOCAL p%, i% 
+ 
+    FOR p% = 1 TO Np% 
+ 
+        FOR i% = 1 TO Nd% 
+ 
+            IF R(p%,i%,j&) < XiMin(i%) THEN R(p%,i% ,j&) = XiMin(i%) + Frep*(R(p%,i%,j&-1)-XiMin(i%)) 
+ 
+            IF R(p%,i%,j&) > XiMax(i%) THEN R(p%,i% ,j&) = XiMax(i%) - Frep*(XiMax(i%)-R(p%,i%,j&-1)) 
+ 
+        NEXT i% 
+ 
+    NEXT p% 
+ 
+END SUB 'RetrieveErrantProbes() 
+ 
+'------ 
+ 
+SUB 
+ChangeRunParameters(NumProbesPerDimension%,Np%,Nd%, Nt&,G,Alpha,Beta,DeltaT,Frep,PlaceInitialProbes$,In itialAcceleration$,RepositionFactor$,FunctionN 
+ame$) 
+ 76 of 92 LOCAL A$, DefaultValue$ 
+ 
+    A$ = INPUTBOX$("# dimensions?","Change # Dimens ions ("+FunctionName$+")",NoSpaces$(Nd%+0,0)) : Nd%     = VAL(A$) : IF Nd% < 1 OR Nd% > 500 THEN 
+Nd% = 2 
+ 
+'    A$ = INPUTBOX$("# probes/dimension?","Change #  Probes/Axis ("+FunctionName$+")",NoSpaces$(NumProb esPerDimension%+0,0)) : NumProbesPerDimension% 
+= VAL(A$) 
+ '   IF NumProbesPerDimension% < 2 OR NumProbesPerD imension% > 500 THEN NumProbesPerDimension% = 10 'r estrict values to reasonable (?) ranges 
+ 
+    IF Nd% > 1 THEN NumProbesPerDimension% = 2*((Nu mProbesPerDimension%+1)\2) 'require an even # probe s on each axis to avoid overlapping at origin 
+(in symmetrical spaces at least...) 
+ 
+    IF Nd% = 1 THEN NumProbesPerDimension% = MAX(Nu mProbesPerDimension%,3)    'at least 3 probes on x- axis for 1-D functions 
+ 
+    Np% = NumProbesPerDimension%*Nd% 
+ 
+    A$ = INPUTBOX$("# time steps?","Change # Steps ("+FunctionName$+")",NoSpaces$(Nt&+0,0)) : Nt&    =  VAL(A$) : IF Nt& < 3                       
+THEN Nt& = 50 
+ 
+    A$ = INPUTBOX$("Grav Const G?","Change G ("+Fun ctionName$+")",NoSpaces$(G,2))           : G      =  VAL(A$) : IF G < -100##    OR G > 100##    
+THEN G   = 2## 
+ 
+    A$ = INPUTBOX$("Alpha?","Change Alpha ("+Functi onName$+")",NoSpaces$(Alpha,2))          : Alpha  =  VAL(A$) : IF Alpha < -50## OR Alpha > 50## 
+THEN Alpha = 2## 
+ 
+    A$ = INPUTBOX$("Beta?","Change Beta ("+Function Name$+")",NoSpaces$(Beta,2))             : Beta   =  VAL(A$) : IF Beta  < -50## OR Beta  > 50## 
+THEN Beta  = 2##' 
+ 
+    A$ = INPUTBOX$("Delta T","Change Delta-T ("+Fun ctionName$+")",NoSpaces$(DeltaT,2))      : DeltaT =  VAL(A$) : IF DeltaT =< 0##                 
+THEN DeltaT = 1## 
+ 
+    A$ = INPUTBOX$("Frep [0-1]?","Change Frep ("+Fu nctionName$+")",NoSpaces$(Frep,3))       : Frep   =  VAL(A$) : IF Frep < 0##    OR Frep > 1##   
+THEN Frep = 0.5## 
+ 
+'   ------------ Initial Probe Distribution ------- ----- 
+ 
+    SELECT CASE PlaceInitialProbes$ 
+        CASE "UNIFORM ON-AXIS"     : DefaultValue$ = "1" 
+        CASE "UNIFORM ON-DIAGONAL" : DefaultValue$ = "2" 
+        CASE "2D GRID"             : DefaultValue$ = "3" 
+        CASE "RANDOM"              : DefaultValue$ = "4" 
+    END SELECT 
+ 
+    A$ = INPUTBOX$("Initial Probes?"+CHR$(13)+"1 - UNIFORM ON-AXIS"+CHR$(13)+"2 - UNIFORM ON-DIAGONAL" +CHR$(13)+"3 - 2D GRID"+CHR$(13)+"4 - 
+RANDOM","Initial Probe Distribution ("+FunctionName $+")",DefaultValue$) 
+ 
+    IF VAL(A$) < 1 OR VAL(A$) > 4 THEN A$ = "1" 
+ 
+    SELECT CASE VAL(A$) 
+        CASE 1 : PlaceInitialProbes$ = "UNIFORM ON- AXIS" 
+        CASE 2 : PlaceInitialProbes$ = "UNIFORM ON- DIAGONAL" 
+        CASE 3 : PlaceInitialProbes$ = "2D GRID" 
+        CASE 4 : PlaceInitialProbes$ = "RANDOM" 
+    END SELECT 
+ 
+    IF Nd% = 1  AND PlaceInitialProbes$ = "UNIFORM ON-DIAGONAL" THEN PlaceInitialProbes$ = "UNIFORM ON -AXIS" 'cannot do diagonal in 1-D space 
+ 
+    IF Nd% <> 2 AND PlaceInitialProbes$ = "2D GRID"  THEN PlaceInitialProbes$ = "UNIFORM ON-AXIS" '2D g rid is available only in 2 dimensions! 
+ 
+'   ------------- Initial Acceleration ------------ ----- 
+ 
+    SELECT CASE InitialAcceleration$ 
+        CASE "ZERO"   : DefaultValue$ = "1" 
+        CASE "FIXED"  : DefaultValue$ = "2" 
+        CASE "RANDOM" : DefaultValue$ = "3" 
+    END SELECT 
+ 
+    A$ = INPUTBOX$("Initial Acceleration?"+CHR$(13) +"1 - ZERO"+CHR$(13)+"2 - FIXED"+CHR$(13)+"3 - RAND OM","Initial Acceleration 
+("+FunctionName$+")",DefaultValue$) 
+ 
+   IF VAL(A$) < 1 OR VAL(A$) > 3 THEN A$ = "1" 
+ 
+    SELECT CASE VAL(A$) 
+        CASE 1 : InitialAcceleration$ = "ZERO" 
+        CASE 2 : InitialAcceleration$ = "FIXED" 
+        CASE 3 : InitialAcceleration$ = "RANDOM" 
+    END SELECT 
+ 
+'   ----------- Reposition Factor --------------- 
+ 
+    SELECT CASE RepositionFactor$ 
+        CASE "FIXED"    : DefaultValue$ = "1" 
+        CASE "VARIABLE" : DefaultValue$ = "2" 
+        CASE "RANDOM"   : DefaultValue$ = "3" 
+    END SELECT 
+ 
+    A$ = INPUTBOX$("Reposition Factor?"+CHR$(13)+"1  - FIXED"+CHR$(13)+"2 - VARIABLE"+CHR$(13)+"3 - RAN DOM","Retrieve Probes 
+("+FunctionName$+")",DefaultValue$) 
+ 
+    IF VAL(A$) < 1 OR VAL(A$) > 3 THEN A$ = "1" 
+ 
+    SELECT CASE VAL(A$) 
+        CASE 1 : RepositionFactor$ = "FIXED" 
+        CASE 2 : RepositionFactor$ = "VARIABLE" 
+        CASE 3 : RepositionFactor$ = "RANDOM" 
+    END SELECT 
+ 
+END SUB 'ChangeRunParameters() 
+ 
+'------ 
+ 
+FUNCTION NoSpaces$(X,NumDigits%) :  NoSpaces$ = REM OVE$(STR$(X,NumDigits%),ANY" ") : END FUNCTION 
+ 
+'----------- 
+ 
+FUNCTION TerminateNowForSaturation$(j&,Nd%,Np%,Nt&, G,DeltaT,Alpha,Beta,R(),A(),M()) 
+ 
+LOCAL A$, i&, p%, NumStepsForAveraging& 
+ 
+LOCAL BestFitness, AvgFitness, FitnessTOL AS EXT 't erminate if avg fitness does not change over NumSte psForAveraging& time steps 
+ 
+    FitnessTOL = 0.00001## : NumStepsForAveraging& = 10 
+ 
+    A$ = "NO" 77 of 92  
+    IF j& >= NumStepsForAveraging+10 THEN 'wait unt il step 10 to start checking for fitness saturation  
+ 
+        AvgFitness = 0## 
+ 
+        FOR i& = j&-NumStepsForAveraging&+1 TO j& ' avg fitness over current step & previous NumStepsFo rAveraging&-1 steps 
+ 
+            BestFitness = M(1,i&) 
+ 
+            FOR p% = 1 TO Np% 
+ 
+                IF M(p%,i&) >= BestFitness THEN Bes tFitness = M(p%,i&) 
+ 
+            NEXT p% 
+ 
+            AvgFitness = AvgFitness + BestFitness 
+ 
+        NEXT i& 
+ 
+        AvgFitness = AvgFitness/NumStepsForAveragin g& 
+ 
+        IF ABS(AvgFitness-BestFitness) < FitnessTOL  THEN A$ = "YES" 'compare avg fitness to best fitne ss at this step 
+ 
+    END IF 
+ 
+    TerminateNowForSaturation$ = A$ 
+ 
+END FUNCTION 'TerminateNowForSaturation$() 
+ 
+'----------- 
+ 
+FUNCTION MagVector(V(),N%) 'returns magnitude of Nx 1 column vector V 
+ 
+LOCAL SumSq AS EXT 
+ 
+LOCAL i% 
+ 
+    SumSQ = 0## : FOR i% = 1 TO N% : SumSQ = SumSQ + V(i%)^2 : NEXT i% : MagVector = SQR(SumSQ) 
+ 
+END FUNCTION 'MagVector() 
+ 
+'--- 
+ 
+FUNCTION UnitStep(X) 
+ 
+LOCAL Z AS EXT 
+ 
+    IF X < 0## THEN 
+ 
+        Z = 0## 
+ 
+    ELSE 
+ 
+        Z = 1## 
+ 
+    END IF 
+ 
+    UnitStep = Z 
+ 
+END FUNCTION 'UnitStep() 
+ 
+'--- 
+ 
+SUB Plot1Dfunction(FunctionName$,XiMin(),XiMax(),R( )) 'plots 1D function on-screen 
+ 
+LOCAL NumPoints%, i%, N% 
+ 
+LOCAL DeltaX, X AS EXT 
+ 
+    NumPoints% = 32001 
+ 
+    DeltaX = (XiMax(1)-XiMin(1))/(NumPoints%-1) 
+ 
+    N% = FREEFILE 
+ 
+    SELECT CASE FunctionName$ 
+ 
+        CASE "ParrottF4" 'PARROTT F4 FUNCTION 
+ 
+            OPEN "ParrottF4" FOR OUTPUT AS #N% 
+ 
+                FOR i% = 1 TO NumPoints% 
+ 
+                    R(1,1,0) = XiMin(1) + (i%-1)*De ltaX 
+ 
+                    PRINT #N%, USING$("#.######  #. ######",R(1,1,0),ParrottF4(R(),1,1,0)) 
+ 
+                NEXT i% 
+ 
+            CLOSE #N% 
+ 
+            CALL CreateGNUplotINIfile(0.2##*ScreenW idth&,0.2##*ScreenHeight&,0.6##*ScreenWidth&,0.6##* ScreenHeight&) 
+ 
+            CALL TwoDplot("ParrottF4","Parrott F4 F unction","0.7","0.7","X\n\n.",".\n\nParrott F4(X)", "","","0","1","0","1","","","wgnuplot.exe"," 
+with lines linewidth 2","") 
+ 
+    END SELECT 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB CLEANUP 'probe coordinate plot files 
+ 
+    IF DIR$("P1")  <> "" THEN KILL "P1" 
+    IF DIR$("P2")  <> "" THEN KILL "P2" 
+    IF DIR$("P3")  <> "" THEN KILL "P3" 
+    IF DIR$("P4")  <> "" THEN KILL "P4" 
+    IF DIR$("P5")  <> "" THEN KILL "P5" 
+    IF DIR$("P6")  <> "" THEN KILL "P6" 
+    IF DIR$("P7")  <> "" THEN KILL "P7" 
+    IF DIR$("P8")  <> "" THEN KILL "P8" 
+    IF DIR$("P9")  <> "" THEN KILL "P9" 
+    IF DIR$("P10") <> "" THEN KILL "P10" 78 of 92     IF DIR$("P11") <> "" THEN KILL "P11" 
+    IF DIR$("P12") <> "" THEN KILL "P12" 
+    IF DIR$("P13") <> "" THEN KILL "P13" 
+    IF DIR$("P14") <> "" THEN KILL "P14" 
+    IF DIR$("P15") <> "" THEN KILL "P15" 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB Plot2Dfunction(FunctionName$,XiMin(),XiMax(),R( )) 
+ 
+LOCAL A$ 
+ 
+LOCAL NumPoints%, i%, k%, N% 
+ 
+LOCAL DelX1, DelX2, Z AS EXT 
+ 
+    SELECT CASE FunctionName$ 
+ 
+        CASE "PBM_1","PBM_2","PBM_3","PBM_4","PBM_5 " : NumPoints% = 25 
+ 
+        CASE ELSE : NumPoints% = 100 
+ 
+    END SELECT 
+ 
+    N% = FREEFILE : OPEN "TwoDplot.DAT" FOR OUTPUT AS #N% 
+ 
+    DelX1 = (XiMax(1)-XiMin(1))/(NumPoints%-1) : De lX2 = (XiMax(2)-XiMin(2))/(NumPoints%-1) 
+ 
+    FOR i% = 1 TO NumPoints% 
+ 
+        R(1,1,0) = XiMin(1) + (i%-1)*DelX1 'x1 valu e 
+ 
+        FOR k% = 1 TO NumPoints% 
+ 
+            R(1,2,0) = XiMin(2) + (k%-1)*DelX2 'x2 value 
+ 
+            Z = ObjectiveFunction(R(),2,1,0,Functio nName$) 
+ 
+            PRINT #N%, USING$("######.###### ###### .###### #######.######^^^^",R(1,1,0),R(1,2,0),Z) 
+ 
+        NEXT k% 
+ 
+        PRINT #N%, "" 
+ 
+    NEXT i% 
+ 
+    CLOSE #N% 
+ 
+    CALL CreateGNUplotINIfile(0.1##*ScreenWidth&,0. 1##*ScreenHeight&,0.6##*ScreenWidth&,0.6##*ScreenHe ight&) 
+ 
+    A$ = "" : IF INSTR(FunctionName$,"PBM_") > 0 TH EN A$ = "Coarse " 
+ 
+    CALL ThreeDplot2("TwoDplot.DAT",A$+"Plot of "+F unctionName$+" Function","","0.6","0.6","1.2", _ 
+                     "x1","x2","z=F(x1,x2)","",""," wgnuplot.exe","","","","","") 
+ 
+END SUB 
+ 
+'------ 
+ 
+    SUB TwoDplot3curves(NumCurves%,PlotFileName1$,P lotFileName2$,PlotFileName3$,PlotTitle$,Annotation$ ,xCoord$,yCoord$,XaxisLabel$,YaxisLabel$, _ 
+                        LogXaxis$,LogYaxis$,xMin$,x Max$,yMin$,yMax$,xTics$,yTics$,GnuPlotEXE$) 
+ 
+        LOCAL N% 
+ 
+        LOCAL LineSize$ 
+ 
+        LineSize$ = "2" 
+ 
+        N% = FREEFILE 
+ 
+        OPEN "cmd2d.gp" FOR OUTPUT AS #N% 
+ 
+            IF LogXaxis$ = "YES" AND LogYaxis$ = "N O"  THEN PRINT #N%, "set logscale x" 
+            IF LogXaxis$ = "NO"  AND LogYaxis$ = "Y ES" THEN PRINT #N%, "set logscale y" 
+            IF LogXaxis$ = "YES" AND LogYaxis$ = "Y ES" THEN PRINT #N%, "set logscale xy" 
+ 
+            IF xMin$ <> "" AND xMax$ <> "" THEN  PR INT #N%, "set xrange ["+xMin$+":"+xMax$+"]" 
+ 
+            IF yMin$ <> "" AND yMax$ <> "" THEN  PR INT #N%, "set yrange ["+yMin$+":"+yMax$+"]" 
+ 
+            PRINT #N%, "set label "+Quote$+AnnoTati on$+Quote$+" at graph "+xCoord$+","+yCoord$ 
+            PRINT #N%, "set grid xtics" 
+            PRINT #N%, "set grid ytics" 
+            PRINT #N%, "set xtics "+xTics$ 
+            PRINT #N%, "set ytics "+yTics$ 
+            PRINT #N%, "set grid mxtics" 
+            PRINT #N%, "set grid mytics" 
+            PRINT #N%, "set title " +Quote$+PlotTit le$+Quote$ 
+            PRINT #N%, "set xlabel "+Quote$+XaxisLa bel$+Quote$ 
+            PRINT #N%, "set ylabel "+Quote$+YaxisLa bel$+Quote$ 
+ 
+            SELECT CASE NumCurves% 
+ 
+            CASE 1 
+            PRINT #N%, "plot " + Quote$ + PlotFileN ame1$ + Quote$ + " with lines linewidth " + LineSiz e$ 
+ 
+            CASE 2 
+            PRINT #N%, "plot " + Quote$ + PlotFileN ame1$ + Quote$ + " with lines linewidth " + LineSiz e$+", " + _ 
+                                 Quote$ + PlotFileN ame2$ + Quote$ + " with lines linewidth " + LineSiz e$ 
+            CASE 3 
+            PRINT #N%, "plot " + Quote$ + PlotFileN ame1$ + Quote$ + " with lines linewidth " + LineSiz e$+", " + _ 
+                                 Quote$ + PlotFileN ame2$ + Quote$ + " with lines linewidth " + LineSiz e$+", " + _ 
+                                 Quote$ + PlotFileN ame3$ + Quote$ + " with lines linewidth " + LineSiz e$ 
+            END SELECT 
+ 
+        CLOSE #N% 
+ 
+        SHELL(GnuPlotEXE$+" cmd2d.gp -") 
+ 
+        CALL Delay(0.3) 
+ 
+    END SUB 'TwoDplot3Curves() 79 of 92  
+'--- 
+ 
+FUNCTION Fibonacci&&(N%) 'RETURNS Nth FIBONACCI NUM BER 
+ 
+LOCAL i%, Fn&&, Fn1&&, Fn2&& 
+ 
+LOCAL A$ 
+ 
+    IF N% > 91 OR N% < 0 THEN 
+ 
+        MSGBOX("ERROR!  Fibonacci argument"+STR$(N% )+" > 91.  Out of range or < 0...") : EXIT FUNCTION  
+ 
+    END IF 
+ 
+    SELECT CASE N% 
+ 
+        CASE 0: Fn&& = 1 
+ 
+        CASE ELSE 
+ 
+            Fn&& = 0 : Fn2&& = 1 : i% = 0 
+ 
+            FOR i% = 1 TO N% 
+ 
+                Fn&& = Fn1&& + Fn2&& 
+ 
+                Fn1&& = Fn2&& 
+ 
+                Fn2&& = Fn&& 
+ 
+            NEXT i% 'LOOP 
+ 
+    END SELECT 
+ 
+    Fibonacci&& = Fn&& 
+ 
+END FUNCTION 'Fibonacci&&() 
+ 
+'----------- 
+ 
+FUNCTION RandomNum(a,b) 'Returns random number X, a =< X < b. 
+ 
+    RandomNum = a + (b-a)*RND 
+ 
+END FUNCTION 'RandomNum() 
+ 
+'----------- 
+ 
+FUNCTION GaussianDeviate(Mu,Sigma) 'returns NORMAL (Gaussian) random deviate with mean Mu and standard  deviation Sigma (variance = Sigma^2) 
+ 
+'Refs: (1) Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T., "Numerical Recipes: The  Art of Scientific Computing," 
+'          §7.2, Cambridge University Press, Cambri dge, UK, 1986. 
+'      (2) Shinzato, T., "Box Muller Method," 2007,  http://www.sp.dis.titech.ac.jp/~shinzato/boxmuller .pdf 
+ 
+LOCAL s, t, Z AS EXT 
+ 
+    s = RND : t = RND 
+ 
+    Z = Mu + Sigma*SQR(-2##*LOG(s))*COS(TwoPi*t) 
+ 
+    GaussianDeviate = Z 
+ 
+END FUNCTION 'GaussianDeviate() 
+ 
+'----------- 
+ 
+    SUB ContourPlot(PlotFileName$,PlotTitle$,Annota tion$,xCoord$,yCoord$,zCoord$, _ 
+                    XaxisLabel$,YaxisLabel$,ZaxisLa bel$,zMin$,zMax$,GnuPlotEXE$,A$) 
+        LOCAL N% 
+ 
+        N% = FREEFILE 
+ 
+        OPEN "cmd3d.gp" FOR OUTPUT AS #N% 
+ 
+            PRINT #N%, "show surface" 
+            PRINT #N%, "set hidden3d" 
+            IF zMin$ <> "" AND zMax$ <> "" THEN  PR INT #N%, "set zrange ["+zMin$+":"+zMax$+"]" 
+            PRINT #N%, "set label "+Quote$+AnnoTati on$+Quote$+" at graph "+xCoord$+","+yCoord$+","+zCo ord$ 
+            PRINT #N%, "show label" 
+            PRINT #N%, "set grid xtics ytics ztics"  
+            PRINT #N%, "show grid" 
+            PRINT #N%, "set title "+Quote$+PlotTitl e$+Quote$ 
+            PRINT #N%, "set xlabel "+Quote$+XaxisLa bel$+Quote$ 
+            PRINT #N%, "set ylabel "+Quote$+YaxisLa bel$+Quote$ 
+            PRINT #N%, "set zlabel "+Quote$+ZaxisLa bel$+Quote$ 
+            PRINT #N%, "splot "+Quote$+PlotFileName $+Quote$+A$  '" notitle with linespoints" 'A$'" not itle with lines" 
+        CLOSE #N% 
+ 
+        SHELL(GnuPlotEXE$+" cmd3d.gp -") 
+ 
+    END SUB 'ContourPlot() 
+ 
+'--- 
+ 
+    SUB ThreeDplot(PlotFileName$,PlotTitle$,Annotat ion$,xCoord$,yCoord$,zCoord$, _ 
+                   XaxisLabel$,YaxisLabel$,ZaxisLab el$,zMin$,zMax$,GnuPlotEXE$,A$) 
+ 
+        LOCAL N%, ProcessID??? 
+ 
+        N% = FREEFILE 
+ 
+        OPEN "cmd3d.gp" FOR OUTPUT AS #N% 
+ 
+            PRINT #N%, "set pm3d" 
+            PRINT #N%, "show pm3d" 
+            IF zMin$ <> "" AND zMax$ <> "" THEN  PR INT #N%, "set zrange ["+zMin$+":"+zMax$+"]" 
+            PRINT #N%, "set label "+Quote$+AnnoTati on$+Quote$+" at graph "+xCoord$+","+yCoord$+","+zCo ord$ 
+            PRINT #N%, "show label" 
+            PRINT #N%, "set grid xtics ytics ztics"  
+            PRINT #N%, "show grid" 
+            PRINT #N%, "set title "+Quote$+PlotTitl e$+Quote$ 
+            PRINT #N%, "set xlabel "+Quote$+XaxisLa bel$+Quote$ 
+            PRINT #N%, "set ylabel "+Quote$+YaxisLa bel$+Quote$ 80 of 92             PRINT #N%, "set zlabel "+Quote$+ZaxisLa bel$+Quote$ 
+            PRINT #N%, "splot "+Quote$+PlotFileName $+Quote$+A$+" notitle"' with lines" 
+        CLOSE #N% 
+ 
+        SHELL(GnuPlotEXE$+" cmd3d.gp -") : CALL Del ay(0.5##) 
+ 
+    END SUB 'ThreeDplot() 
+ 
+'--- 
+ 
+    SUB ThreeDplot2(PlotFileName$,PlotTitle$,Annota tion$,xCoord$,yCoord$,zCoord$, _ 
+                    XaxisLabel$,YaxisLabel$,ZaxisLa bel$,zMin$,zMax$,GnuPlotEXE$,A$,xStart$,xStop$,ySta rt$,yStop$) 
+ 
+        LOCAL N% 
+ 
+        N% = FREEFILE 
+ 
+        OPEN "cmd3d.gp" FOR OUTPUT AS #N% 
+ 
+            PRINT #N%, "set pm3d" 
+            PRINT #N%, "show pm3d" 
+            PRINT #N%, "set hidden3d" 
+            PRINT #N%, "set view 45, 45, 1, 1" 
+ 
+            IF zMin$ <> "" AND zMax$ <> "" THEN  PR INT #N%, "set zrange ["+zMin$+":"+zMax$+"]" 
+ 
+            PRINT #N%, "set xrange [" + xStart$ + " :" + xStop$ + "]" 
+            PRINT #N%, "set yrange [" + yStart$ + " :" + yStop$ + "]" 
+            PRINT #N%, "set label "   + Quote$  + A nnoTation$ + Quote$+" at graph "+xCoord$+","+yCoord $+","+zCoord$ 
+            PRINT #N%, "show label" 
+            PRINT #N%, "set grid xtics ytics ztics"  
+            PRINT #N%, "show grid" 
+            PRINT #N%, "set title "  + Quote$+PlotT itle$    + Quote$ 
+            PRINT #N%, "set xlabel " + Quote$+Xaxis Label$   + Quote$ 
+            PRINT #N%, "set ylabel " + Quote$+Yaxis Label$   + Quote$ 
+            PRINT #N%, "set zlabel " + Quote$+Zaxis Label$   + Quote$ 
+            PRINT #N%, "splot "      + Quote$+PlotF ileName$ + Quote$ + A$ + " notitle with lines" 
+        CLOSE #N% 
+ 
+        SHELL(GnuPlotEXE$+" cmd3d.gp -") 
+ 
+    END SUB 'ThreeDplot2() 
+ 
+'--- 
+ 
+     SUB TwoDplot2Curves(PlotFileName1$,PlotFileNam e2$,PlotTitle$,Annotation$,xCoord$,yCoord$,XaxisLab el$,YaxisLabel$, _ 
+                       LogXaxis$,LogYaxis$,xMin$,xM ax$,yMin$,yMax$,xTics$,yTics$,GnuPlotEXE$,LineSize)  
+ 
+        LOCAL N%, ProcessID??? 
+ 
+        N% = FREEFILE 
+ 
+        OPEN "cmd2d.gp" FOR OUTPUT AS #N% 
+            'print #N%, "set output "+Quote$+"test. plt"+Quote$ 'tried this 3/11/06, didn't work... 
+ 
+            IF LogXaxis$ = "YES" AND LogYaxis$ = "N O"  THEN PRINT #N%, "set logscale x" 
+            IF LogXaxis$ = "NO"  AND LogYaxis$ = "Y ES" THEN PRINT #N%, "set logscale y" 
+            IF LogXaxis$ = "YES" AND LogYaxis$ = "Y ES" THEN PRINT #N%, "set logscale xy" 
+ 
+            IF xMin$ <> "" AND xMax$ <> "" THEN  PR INT #N%, "set xrange ["+xMin$+":"+xMax$+"]" 
+ 
+            IF yMin$ <> "" AND yMax$ <> "" THEN  PR INT #N%, "set yrange ["+yMin$+":"+yMax$+"]" 
+ 
+            PRINT #N%, "set label "+Quote$+AnnoTati on$+Quote$+" at graph "+xCoord$+","+yCoord$ 
+            PRINT #N%, "set grid xtics" 
+            PRINT #N%, "set grid ytics" 
+            PRINT #N%, "set xtics "+xTics$ 
+            PRINT #N%, "set ytics "+yTics$ 
+            PRINT #N%, "set grid mxtics" 
+            PRINT #N%, "set grid mytics" 
+            PRINT #N%, "set title "+Quote$+PlotTitl e$+Quote$ 
+            PRINT #N%, "set xlabel "+Quote$+XaxisLa bel$+Quote$ 
+            PRINT #N%, "set ylabel "+Quote$+YaxisLa bel$+Quote$ 
+ 
+            PRINT #N%, "plot "+Quote$+PlotFileName1 $+Quote$+" with lines linewidth "+REMOVE$(STR$(Line Size),ANY" ")+","+ _ 
+                               Quote$+PlotFileName2 $+Quote$+" with points pointsize 0.05"'+REMOVE$(STR $(LineSize),ANY" ") 
+ 
+        CLOSE #N% 
+ 
+        ProcessID??? = SHELL(GnuPlotEXE$+" cmd2d.gp  -") : CALL Delay(0.5##) 
+ 
+    END SUB 'TwoDplot2Curves() 
+ 
+'--- 
+ 
+    SUB Probe2Dplots(ProbePlotsFileList$,PlotTitle$ ,Annotation$,xCoord$,yCoord$,XaxisLabel$,YaxisLabel $, _ 
+                     LogXaxis$,LogYaxis$,xMin$,xMax $,yMin$,yMax$,xTics$,yTics$,GnuPlotEXE$) 
+ 
+        LOCAL N%, ProcessID??? 
+ 
+        N% = FREEFILE 
+ 
+        OPEN "cmd2d.gp" FOR OUTPUT AS #N% 
+ 
+            IF LogXaxis$ = "YES" AND LogYaxis$ = "N O"  THEN PRINT #N%, "set logscale x" 
+            IF LogXaxis$ = "NO"  AND LogYaxis$ = "Y ES" THEN PRINT #N%, "set logscale y" 
+            IF LogXaxis$ = "YES" AND LogYaxis$ = "Y ES" THEN PRINT #N%, "set logscale xy" 
+ 
+            IF xMin$ <> "" AND xMax$ <> "" THEN  PR INT #N%, "set xrange ["+xMin$+":"+xMax$+"]" 
+ 
+            IF yMin$ <> "" AND yMax$ <> "" THEN  PR INT #N%, "set yrange ["+yMin$+":"+yMax$+"]" 
+ 
+            PRINT #N%, "set label "+Quote$+AnnoTati on$+Quote$+" at graph "+xCoord$+","+yCoord$ 
+            PRINT #N%, "set grid xtics" 
+            PRINT #N%, "set grid ytics" 
+            PRINT #N%, "set xtics "+xTics$ 
+            PRINT #N%, "set ytics "+yTics$ 
+            PRINT #N%, "set grid mxtics" 
+            PRINT #N%, "set grid mytics" 
+            PRINT #N%, "set title "+Quote$+PlotTitl e$+Quote$ 
+            PRINT #N%, "set xlabel "+Quote$+XaxisLa bel$+Quote$ 
+            PRINT #N%, "set ylabel "+Quote$+YaxisLa bel$+Quote$ 
+ 
+            PRINT #N%, ProbePlotsFileList$ 81 of 92  
+        CLOSE #N% 
+ 
+        ProcessID??? = SHELL(GnuPlotEXE$+" cmd2d.gp  -") : CALL Delay(0.5##) 
+ 
+    END SUB 'Probe2Dplots() 
+ 
+'--- 
+ 
+SUB Show2Dprobes(R(),Np%,Nt&,j&,XiMin(),XiMax(),Fre p,BestFitness,BestProbeNumber%,BestTimeStep&,Functi onName$,RepositionFactor$,Gamma) 
+ 
+    LOCAL N%, p% 
+ 
+    LOCAL A$, PlotFileName$, PlotTitle$, Symbols$ 
+ 
+    LOCAL xMin$, xMax$, yMin$, yMax$ 
+ 
+    LOCAL s1, s2, s3, s4 AS EXT 
+ 
+    PlotFileName$ = "Probes("+REMOVE$(STR$(j&),ANY"  ")+")" 
+ 
+    IF j& > 0 THEN 'PLOT PROBES AT THIS TIME STEP 
+ 
+        PlotTitle$ = "\nLOCATIONS OF "+REMOVE$(STR$ (Np%),ANY" ") + " PROBES AT TIME STEP" + STR$(j&) +  " / " + REMOVE$(STR$(Nt&),ANY" ") + "\n" + _ 
+                     "Fitness = "+REMOVE$(STR$(ROUN D(BestFitness,3)),ANY" ") + ", Probe #" + REMOVE$(S TR$(BestProbeNumber%),ANY" ") + " at Step #" + 
+REMOVE$(STR$(BestTimeStep&),ANY" ") + _ 
+                     "  [Frep = "+REMOVE$(STR$(Frep ,4),ANY" ") + " " + RepositionFactor$ + "]\n" 
+ 
+    ELSE 'PLOT INITIAL PROBE DISTRIBUTION 
+ 
+        PlotTitle$ = "\nLOCATIONS OF "+REMOVE$(STR$ (Np%),ANY" ") + " INITIAL PROBES FOR " + FunctionNa me$ + " FUNCTION\n[gamma = 
+"+STR$(ROUND(Gamma,3))+"]\n" 
+ 
+    END IF 
+ 
+    N% = FREEFILE : OPEN PlotFileName$ FOR OUTPUT A S #N% 
+ 
+        FOR p% = 1 TO Np% : PRINT #N%, USING$("#### ##.#####    ######.#####",R(p%,1,j&),R(p%,2,j&)) : NEXT p% 
+ 
+    CLOSE #N% 
+ 
+    s1 = 1.1## : s2 = 1.1## : s3 = 1.1## : s4 = 1.1 ## 'expand plots axes by 10% 
+ 
+    IF XiMin(1) > 0## THEN s1 = 0.9## 
+    IF XiMax(1) < 0## THEN s2 = 0.9## 
+    IF XiMin(2) > 0## THEN s3 = 0.9## 
+    IF XiMax(2) < 0## THEN s4 = 0.9## 
+ 
+    xMin$ = REMOVE$(STR$(s1*XiMin(1),2),ANY" ") 
+    xMax$ = REMOVE$(STR$(s2*XiMax(1),2),ANY" ") 
+    yMin$ = REMOVE$(STR$(s3*XiMin(2),2),ANY" ") 
+    yMax$ = REMOVE$(STR$(s4*XiMax(2),2),ANY" ") 
+ 
+    CALL TwoDplot(PlotFileName$,PlotTitle$,"0.6","0 .7","x1\n\n","\nx2","NO","NO",xMin$,xMax$,yMin$,yMa x$,"5","5","wgnuplot.exe"," pointsize 1 
+linewidth 2","") 
+ 
+    KILL PlotFileName$ 'erase plot data file after probes have been displayed 
+ 
+END SUB 'ShowProbes() 
+ 
+'---- 
+ 
+    SUB TwoDplot(PlotFileName$,PlotTitle$,xCoord$,y Coord$,XaxisLabel$,YaxisLabel$, _ 
+                 LogXaxis$,LogYaxis$,xMin$,xMax$,yM in$,yMax$,xTics$,yTics$,GnuPlotEXE$,LineType$,Annot ation$) 
+ 
+        LOCAL N%, ProcessID??? 
+ 
+        N% = FREEFILE 
+ 
+        OPEN "cmd2d.gp" FOR OUTPUT AS #N% 
+ 
+            IF LogXaxis$ = "YES" AND LogYaxis$ = "N O"  THEN PRINT #N%, "set logscale x" 
+            IF LogXaxis$ = "NO"  AND LogYaxis$ = "Y ES" THEN PRINT #N%, "set logscale y" 
+            IF LogXaxis$ = "YES" AND LogYaxis$ = "Y ES" THEN PRINT #N%, "set logscale xy" 
+ 
+            IF xMin$ <> "" AND xMax$ <> "" THEN  PR INT #N%, "set xrange ["+xMin$+":"+xMax$+"]" 
+            IF yMin$ <> "" AND yMax$ <> "" THEN  PR INT #N%, "set yrange ["+yMin$+":"+yMax$+"]" 
+ 
+            PRINT #N%, "set label "      + Quote$ +  Annotation$ + Quote$ + " at graph " + xCoord$ + ", " + yCoord$ 
+            PRINT #N%, "set grid xtics " + XTics$ 
+            PRINT #N%, "set grid ytics " + yTics$ 
+            PRINT #N%, "set grid mxtics" 
+            PRINT #N%, "set grid mytics" 
+            PRINT #N%, "show grid" 
+            PRINT #N%, "set title "  + Quote$+PlotT itle$+Quote$ 
+            PRINT #N%, "set xlabel " + Quote$+Xaxis Label$+Quote$ 
+            PRINT #N%, "set ylabel " + Quote$+Yaxis Label$+Quote$ 
+ 
+            PRINT #N%, "plot "+Quote$+PlotFileName$ +Quote$+" notitle"+LineType$ 
+ 
+        CLOSE #N% 
+ 
+        ProcessID??? = SHELL(GnuPlotEXE$+" cmd2d.gp  -") : CALL Delay(0.5##) 
+ 
+    END SUB 'TwoDplot() 
+ 
+'----- 
+ 
+    SUB CreateGNUplotINIfile(PlotWindowULC_X%,PlotW indowULC_Y%,PlotWindowWidth%,PlotWindowHeight%) 
+ 
+    LOCAL N%, WinPath$, A$, B$, WindowsDirectory$ 
+ 
+    WinPath$ = UCASE$(ENVIRON$("Path"))'DIR$("C:\WI NDOWS",23) 
+ 
+    DO 
+        B$ = A$ 
+ 
+        A$ = EXTRACT$(WinPath$,";") 
+ 
+        WinPath$ = REMOVE$(WinPath$,A$+";") 
+ 
+        IF RIGHT$(A$,7) = "WINDOWS" OR A$ = B$ THEN  EXIT LOOP 
+ 
+        IF RIGHT$(A$,5) = "WINNT"   OR A$ = B$ THEN  EXIT LOOP 82 of 92  
+    LOOP 
+ 
+    WindowsDirectory$ = A$ 
+ 
+    N% = FREEFILE 
+ 
+'   ----------- WGNUPLOT.INPUT FILE ----------- 
+    OPEN WindowsDirectory$+"\wgnuplot.ini" FOR OUTP UT AS #N% 
+ 
+        PRINT #N%,"[WGNUPLOT]" 
+        PRINT #N%,"TextOrigin=0 0" 
+        PRINT #N%,"TextSize=640 150" 
+        PRINT #N%,"TextFont=Terminal,9" 
+        PRINT #N%,"GraphOrigin="+REMOVE$(STR$(PlotW indowULC_X%),ANY" ")+" "+REMOVE$(STR$(PlotWindowULC _Y%),ANY" ") 
+        PRINT #N%,"GraphSize="  +REMOVE$(STR$(PlotW indowWidth%),ANY" ")+" "+REMOVE$(STR$(PlotWindowHei ght%),ANY" ") 
+        PRINT #N%,"GraphFont=Arial,10" 
+        PRINT #N%,"GraphColor=1" 
+        PRINT #N%,"GraphToTop=1" 
+        PRINT #N%,"GraphBackground=255 255 255" 
+        PRINT #N%,"Border=0 0 0 0 0" 
+        PRINT #N%,"Axis=192 192 192 2 2" 
+        PRINT #N%,"Line1=0 0 255 0 0" 
+        PRINT #N%,"Line2=0 255 0 0 1" 
+        PRINT #N%,"Line3=255 0 0 0 2" 
+        PRINT #N%,"Line4=255 0 255 0 3" 
+        PRINT #N%,"Line5=0 0 128 0 4" 
+ 
+    CLOSE #N% 
+ 
+    END SUB 'CreateGNUplotINIfile() 
+ 
+'------ 
+ 
+    SUB Delay(NumSecs) 
+ 
+        LOCAL StartTime, StopTime AS EXT 
+ 
+        StartTime = TIMER 
+ 
+        DO UNTIL (StopTime-StartTime) >= NumSecs 
+ 
+            StopTime = TIMER 
+ 
+        LOOP 
+ 
+    END SUB 'Delay() 
+ 
+'----- 
+ 
+SUB MathematicalConstants 
+    EulerConst    = 0.577215664901532860606512## 
+    Pi            = 3.141592653589793238462643## 
+    Pi2           = Pi/2## 
+    Pi4           = Pi/4## 
+    TwoPi         = 2##*Pi 
+    FourPi        = 4##*Pi 
+    e             = 2.718281828459045235360287## 
+    Root2         = 1.4142135623730950488## 
+END SUB 
+ 
+'----- 
+ 
+SUB AlphabetAndDigits 
+    Alphabet$   = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdef ghijklmnopqrstuvwxyz" 
+    Digits$     = "0123456789" 
+    RunID$      = DATE$ + ", " + TIME$ 
+END SUB 
+ 
+'------ 
+ 
+SUB SpecialSymbols 
+    Quote$             = CHR$(34) 'Quotation mark "  
+    SpecialCharacters$ = "'(),#:;/_" 
+END SUB 
+ 
+'----- 
+ 
+SUB EMconstants 
+    Mu0  = 4E-7##*Pi     'hy/meter 
+    Eps0 = 8.854##*1E-12 'fd/meter 
+    c    = 2.998E8##     'velocity of light, 1##/SQ R(Mu0*Eps0) 'meters/sec 
+    eta0 = SQR(Mu0/Eps0) 'impedance of free space, ohms 
+END SUB 
+ 
+'------ 
+ 
+SUB ConversionFactors 
+    Rad2Deg       = 180##/Pi 
+    Deg2Rad       = 1##/Rad2Deg 
+    Feet2Meters   = 0.3048## 
+    Meters2Feet   = 1##/Feet2Meters 
+    Inches2Meters = 0.0254## 
+    Meters2Inches = 1##/Inches2Meters 
+    Miles2Meters  = 1609.344## 
+    Meters2Miles  = 1##/Miles2Meters 
+    NautMi2Meters = 1852## 
+    Meters2NautMi = 1##/NautMi2Meters 
+END SUB 
+ 
+'------ 
+ 
+SUB ShowConstants 'puts up msgbox showing all const ants 
+ 
+LOCAL A$ 
+ 
+A$ = _ 
+"Mathematical Constants:"+CHR$(13)+_ 
+"Euler const="+STR$(EulerConst)+CHR$(13)+_ 
+"Pi="+STR$(Pi)+CHR$(13)+_ 
+"Pi/2="+STR$(Pi2)+CHR$(13)+_ 
+"Pi/4="+STR$(Pi4)+CHR$(13)+_ 
+"2Pi="+STR$(TwoPi)+CHR$(13)+_ 
+"4Pi="+STR$(FourPi)+CHR$(13)+_ 83 of 92 "e="+STR$(e)+CHR$(13)+_ 
+"Sqr2="+STR$(Root2)+CHR$(13)+CHR$(13)+_ 
+"Alphabet, Digits & Special Characters:"+CHR$(13)+_  
+"Alphabet="+Alphabet$+CHR$(13)+_ 
+"Digits="+Digits$+CHR$(13)+_ 
+"quote="+Quote$+CHR$(13)+_ 
+"Spec chars="+SpecialCharacters$+CHR$(13)+CHR$(13)+ _ 
+"E&M Constants:"+CHR$(13)+_ 
+"Mu0="+STR$(Mu0)+CHR$(13)+_ 
+"Eps0="+STR$(Eps0)+CHR$(13)+_ 
+"c="+STR$(c)+CHR$(13)+_ 
+"Eta0="+STR$(eta0)+CHR$(13)+CHR$(13)+_ 
+"Conversion Factors:"+CHR$(13)+_ 
+"Rad2Deg="+STR$(Rad2Deg)+CHR$(13)+_ 
+"Deg2Rad="+STR$(Deg2Rad)+CHR$(13)+_ 
+"Ft2meters="+STR$(Feet2Meters)+CHR$(13)+_ 
+"Meters2Ft="+STR$(Meters2Feet)+CHR$(13)+_ 
+"Inches2Meters="+STR$(Inches2Meters)+CHR$(13)+_ 
+"Meters2Inches="+STR$(Meters2Inches)+CHR$(13)+_ 
+"Miles2Meters="+STR$(Miles2Meters)+CHR$(13)+_ 
+"Meters2Miles="+STR$(Meters2Miles)+CHR$(13)+_ 
+"NautMi2Meters="+STR$(NautMi2Meters)+CHR$(13)+_ 
+"Meters2NautMi="+STR$(Meters2NautMi)+CHR$(13)+CHR$( 13) 
+ 
+MSGBOX(A$) 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB DisplayRmatrix(Np%,Nd%,Nt&,R()) 
+ 
+LOCAL p%, i%, j&, A$ 
+ 
+    A$ = "Position Vector Matrix R()"+CHR$(13) 
+ 
+    FOR p% = 1 TO Np% 
+ 
+        FOR i% = 1 TO Nd% 
+ 
+            FOR j& = 0 TO Nt& 
+ 
+                A$ = A$ + "R("+STR$(p%)+", "+STR$(i %)+", "+STR$(j&)+ ") ="+STR$(R(p%,i%,j&)) + CHR$(13 ) 
+ 
+            NEXT j& 
+ 
+        NEXT i% 
+ 
+    NEXT p% 
+ 
+    MSGBOX(A$) 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB DisplayRmatrixThisTimeStep(Np%,Nd%,j&,R()) 
+ 
+LOCAL p%, i%, A$ 
+ 
+    A$ = "Position Vector Matrix R() at step "+STR$ (j&)+":"+CHR$(13) 
+ 
+    FOR p% = 1 TO Np% 
+ 
+        FOR i% = 1 TO Nd% 
+ 
+            A$ = A$ + "R("+STR$(p%)+", "+STR$(i%)+" , "+STR$(j&)+ ") ="+STR$(R(p%,i%,j&)) + CHR$(13) 
+ 
+        NEXT i% 
+ 
+    NEXT p% 
+ 
+    MSGBOX(A$) 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB DisplayAmatrix(Np%,Nd%,Nt&,A()) 
+ 
+LOCAL p%, i%, j&, A$ 
+ 
+    A$ = "Acceleration Vector Matrix A()"+CHR$(13) 
+ 
+    FOR p% = 1 TO Np% 
+ 
+        FOR i% = 1 TO Nd% 
+ 
+            FOR j& = 0 TO Nt& 
+ 
+                A$ = A$ + "A("+STR$(p%)+", "+STR$(i %)+", "+STR$(j&)+ ") ="+STR$(A(p%,i%,j&)) + CHR$(13 ) 
+ 
+            NEXT j& 
+ 
+        NEXT i% 
+ 
+    NEXT p% 
+ 
+    MSGBOX(A$) 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB DisplayAmatrixThisTimeStep(Np%,Nd%,j&,A()) 
+ 
+LOCAL p%, i%, A$ 
+ 
+    A$ = "Acceleration matrix A() at step "+STR$(j& )+":"+CHR$(13) 
+ 
+    FOR p% = 1 TO Np% 
+ 
+        FOR i% = 1 TO Nd% 
+ 84 of 92             A$ = A$ + "A("+STR$(p%)+", "+STR$(i%)+" , "+STR$(j&)+ ") ="+STR$(A(p%,i%,j&)) + CHR$(13) 
+ 
+        NEXT i% 
+ 
+    NEXT p% 
+ 
+    MSGBOX(A$) 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB DisplayMmatrix(Np%,Nt&,M()) 
+ 
+LOCAL p%, j&, A$ 
+ 
+    A$ = "Fitness Matrix M()"+CHR$(13) 
+ 
+    FOR p% = 1 TO Np% 
+ 
+       FOR j& = 0 TO Nt& 
+ 
+          A$ = A$ + "M("+STR$(p%)+", "+STR$(j&)+ ")  ="+STR$(M(p%,j&)) + CHR$(13) 
+ 
+       NEXT j& 
+ 
+    NEXT p% 
+ 
+    MSGBOX(A$) 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB DisplayMmatrixThisTimeStep(Np%,j&,M()) 
+ 
+LOCAL p%, A$ 
+ 
+    A$ = "Fitness matrix M() at step "+STR$(j&)+":" +CHR$(13) 
+ 
+    FOR p% = 1 TO Np% 
+ 
+          A$ = A$ + "M("+STR$(p%)+", "+STR$(j&)+ ")  ="+STR$(M(p%,j&)) + CHR$(13) 
+ 
+    NEXT p% 
+ 
+    MSGBOX(A$) 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB DisplayXiMinMax(Nd%,XiMin(),XiMax()) 
+ 
+LOCAL i%, A$ 
+ 
+    A$ = "" 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        A$ = A$ + "XiMin("+STR$(i%)+" ) = "+STR$(Xi Min(i%))+"   XiMax("+STR$(i%)+" ) = "+STR$(XiMax(i% )) + CHR$(13) 
+ 
+    NEXT i% 
+ 
+    MSGBOX(A$) 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB DisplayRunParameters2(FunctionName$,Nd%,Np%,Nt& ,G,DeltaT,Alpha,Beta,Frep,PlaceInitialProbes$,Initi alAcceleration$,RepositionFactor$) 
+ 
+LOCAL A$ 
+ 
+    A$ = "Function = "+ FunctionName$+CHR$(13)+_ 
+         "Nd = "+STR$(Nd%)+CHR$(13)+_ 
+         "Np = "+STR$(Np%)+CHR$(13)+_ 
+         "Nt = "+STR$(Nt&)+CHR$(13)+_ 
+         "G  = "+STR$(G)+CHR$(13)+_ 
+         "DeltaT = "+STR$(DeltaT)+CHR$(13)+_ 
+         "Alpha = "+STR$(Alpha)+CHR$(13)+_ 
+         "Beta  = "+STR$(Beta)+CHR$(13)+_ 
+         "Frep  = "+STR$(Frep)+CHR$(13)+_ 
+         "Init Probes: "+PlaceInitialProbes$+CHR$(1 3)+_ 
+         "Init Accel:  "+InitialAcceleration$+CHR$( 13)+_ 
+         "Retrive Method: "+RepositionFactor$+CHR$( 13) 
+ 
+    MSGBOX(A$) 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB 
+Tabulate1DprobeCoordinates(Max1DprobesPlotted%,Nd%, Np%,LastStep&,G,DeltaT,Alpha,Beta,Frep,R(),M(),Plac eInitialProbes$,InitialAcceleration$,Repositio 
+nFactor$,FunctionName$,Gamma) 
+ 
+LOCAL N%, ProbeNum%, FileHeader$, A$, B$, C$, D$, E $, F$, H$, StepNum&, FieldNumber% 'kludgy, yes, but  it accomplishes its purpose... 
+ 
+        CALL 
+GetPlotAnnotation(FileHeader$,Nd%,Np%,LastStep&,G,D eltaT,Alpha,Beta,Frep,M(),PlaceInitialProbes$,Initi alAcceleration$,RepositionFactor$,FunctionName 
+$,Gamma) 
+ 
+        REPLACE "\n" WITH ", " IN FileHeader$ 
+ 
+        FileHeader$ = LEFT$(FileHeader$,LEN(FileHea der$)-2) 
+ 
+        FileHeader$ = "PROBE COORDINATES" + CHR$(13 ) +_ 
+                      "-----------------" + CHR$(13 ) + FileHeader$ 
+ 
+        N% = FREEFILE : OPEN "ProbeCoordinates.DAT"  FOR OUTPUT AS #N% 
+ 
+            A$ = "   Step #     " : B$ = "   ------      " : C$ = "" 
+ 85 of 92             FOR ProbeNum% = 1 TO Np% 'create out da ta file header 
+ 
+                SELECT CASE ProbeNum% 
+                    CASE   1 TO   9 : E$ = ""   : F $ = "            " : H$ = "            " 
+                    CASE  10 TO  99 : E$ = "-"  : F $ = "           "  : H$ = "           " 
+                    CASE 100 TO 999 : E$ = "--" : F $ = "          "   : H$ = "          " 
+                END SELECT 
+ 
+                A$ = A$ + "P" + NoSpaces$(ProbeNum% +0,0) + F$ 'note: adding zero to ProbeNum% necessar y to convert to floating point... 
+ 
+                B$ = B$ + E$ + "--" + H$ 
+ 
+                C$ = C$ + "######.###    " 
+ 
+'                C$ = C$ + "##.#######" 
+ 
+            NEXT ProbeNum% 
+ 
+            PRINT #N%, FileHeader$ + CHR$(13) : PRI NT #N%, A$ : PRINT #N%, B$ 
+ 
+            FOR StepNum& = 0 TO LastStep& 
+ 
+                D$ = USING$("######   ",StepNum&) 
+ 
+                FOR ProbeNum% = 1 TO Np% : D$ = D$ + USING$(C$,R(ProbeNum%,1,StepNum&)) : NEXT ProbeNu m% 
+ 
+                PRINT #N%, D$ 
+ 
+            NEXT StepNum& 
+ 
+        CLOSE #N% 
+ 
+END SUB 'Tabulate1DprobeCoordinates() 
+ 
+'------ 
+ 
+SUB 
+Plot1DprobePositions(Max1DprobesPlotted%,Nd%,Np%,La stStep&,G,DeltaT,Alpha,Beta,Frep,R(),M(),PlaceIniti alProbes$,InitialAcceleration$,RepositionFacto 
+r$,FunctionName$,Gamma) 
+   'plots on-screen 1D function probe positions vs time step if Np =< 10 
+ 
+LOCAL ProcessID???, N%, n1%, n2%, n3%, n4%, n5%, n6 %, n7%, n8%, n9%, n10%, n11%, n12%, n13%, n14%, n15 %, ProbeNum%, StepNum&, A$ 
+ 
+LOCAL PlotAnnotation$ 
+ 
+    IF Np% > Max1DprobesPlotted% THEN EXIT SUB 
+ 
+    CALL CLEANUP 'delete old "Px" plot files, if an y 
+ 
+    ProbeNum% = 0 
+ 
+    DO 'create output data files, probe-by-probe 
+        INCR ProbeNum% : n1%  = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n1%   : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n2%  = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n2%   : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n3%  = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n3%   : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n4%  = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n4%   : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n5%  = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n5%   : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n6%  = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n6%   : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n7%  = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n7%   : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n8%  = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n8%   : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n9%  = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n9%   : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n10% = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n10 % : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n11% = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n11 % : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n12% = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n12 % : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n13% = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n13 % : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n14% = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n14   : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : n15% = FREEFILE : OPEN "P" +REMOVE$(STR$(ProbeNum%),ANY" ") FOR OUTPUT AS #n15 % : IF ProbeNum% = Np% THEN EXIT LOOP 
+    LOOP 
+ 
+    ProbeNum% = 0 
+ 
+    DO 'output probe positions as a function of tim e step 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n1%,  USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n2%,  USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n3%,  USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n4%,  USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n5%,  USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n6%,  USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n7%,  USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n8%,  USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n9%,  USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n10%, USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n11,  USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n12%, USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n13%, USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n14%, USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : FOR StepNum& = 0 TO LastSt ep& : PRINT #n15%, USING$("######  ######.########" ,StepNum&,R(ProbeNum%,1,StepNum&)) : NEXT 
+StepNum& : IF ProbeNum% = Np% THEN EXIT LOOP 
+    LOOP 
+ 
+    ProbeNum% = 0 
+ 
+    DO 'close output data files 
+        INCR ProbeNum% : CLOSE #n1%  : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n2%  : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n3%  : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n4%  : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n5%  : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n6%  : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n7%  : IF ProbeNum%  = Np% THEN EXIT LOOP 86 of 92         INCR ProbeNum% : CLOSE #n8%  : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n9%  : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n10% : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n11% : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n12% : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n13% : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n14% : IF ProbeNum%  = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : CLOSE #n15% : IF ProbeNum%  = Np% THEN EXIT LOOP 
+    LOOP 
+ 
+    ProbeNum% = 0 : A$ = "" 
+ 
+    DO 'create file string for plot command file 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+        INCR ProbeNum% : A$ = A$ + Quote$ + "P"+REM OVE$(STR$(ProbeNum%),ANY" ") + Quote$ + " w l lw 2,  " : IF ProbeNum% = Np% THEN EXIT LOOP 
+    LOOP 
+ 
+    A$ = LEFT$(A$,LEN(A$)-2) 
+ 
+    CALL 
+GetPlotAnnotation(PlotAnnotation$,Nd%,Np%,LastStep& ,G,DeltaT,Alpha,Beta,Frep,M(),PlaceInitialProbes$,I nitialAcceleration$,RepositionFactor$,Function 
+Name$,Gamma) 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "cmd2d.gp" FOR OUTPUT AS #N% 
+ 
+        PRINT #N%, "set label "      + Quote$ + Plo tAnnotation$ + Quote$ + " at graph 0.5,0.95" 
+        PRINT #N%, "set grid xtics" 
+        PRINT #N%, "set grid ytics" 
+        PRINT #N%, "set title "  + Quote$ + "Evolut ion of "    + FunctionName$ + " Probe Positions"+ " \n" + RunID$ + Quote$ 
+        PRINT #N%, "set xlabel " + Quote$ + "Time S tep"        + Quote$ 
+        PRINT #N%, "set ylabel " + Quote$ + "Probe Coordinate" + Quote$ 
+        PRINT #N%, "plot "       + A$ 
+ 
+    CLOSE #N% 
+ 
+    CALL CreateGNUplotINIfile(0.2##*ScreenWidth&,0. 2##*ScreenHeight&,0.6##*ScreenWidth&,0.6##*ScreenHe ight&) 'USAGE: CALL 
+CreateGNUplotINIfile(PlotWindowULC_X%,PlotWindowULC _Y%,PlotWindowWidth%,PlotWindowHeight%) 
+ 
+    ProcessID??? = SHELL("wgnuplot.exe"+" cmd2d.gp -") : CALL Delay(5##) 'before SUB Cleanup is called  
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB 
+DisplayRunParameters(FunctionName$,Nd%,Np%,Nt&,G,De ltaT,Alpha,Beta,Frep,R(),A(),M(),PlaceInitialProbes $,InitialAcceleration$,RepositionFactor$,RunCF 
+O$,ShrinkDS$,CheckForEarlyTermination$) 
+ 
+LOCAL A$, B$, YN& 
+ 
+    ShrinkDS$ = "NO"                 : YN& = MSGBOX ("Adaptively Shrink DS?",%MB_YESNO,"ADAPTIVE DS?")              : IF YN& = %IDYES THEN ShrinkDS$ 
+= "YES" 
+ 
+    CheckForEarlyTermination$ = "NO" : YN& = MSGBOX ("Check for Early Termination?",%MB_YESNO,"EARLY TE RMINATION?") : IF YN& = %IDYES THEN 
+CheckForEarlyTermination$ = "YES" 
+ 
+    B$ = "" : IF PlaceInitialProbes$ = "UNIFORM ON- AXIS" AND Nd% > 1 THEN B$ = "  ["+REMOVE$(STR$(Np%/ Nd%),ANY" ") + "/axis]" 
+ 
+    RunCFO$ = "NO" 
+ 
+'    A$ = "RUN CFO WITH THE" + CHR$(13) +_ 
+ '        "FOLLOWING PARAMETERS?"                             + CHR$(13) + CHR$(13) +_ 
+  '       "Function "        + FunctionName$                  + " (" + REMOVE$(STR$(Nd%),ANY" ") + "-D )" + CHR$(13) + CHR$(13) +_ 
+   '      "# probes = "      + REMOVE$(STR$(Np%),AN Y" ")      + B$ + CHR$(13) + _ 
+    '     "# time steps = "  + REMOVE$(STR$(Nt&),AN Y" ")      + CHR$(13) + _ 
+     '    "Grav Const G = "  + REMOVE$(STR$(G,2),AN Y" ")      + CHR$(13) + _ 
+      '   "Delta-T = "       + REMOVE$(STR$(DeltaT, 3),ANY" ") + CHR$(13) + _ 
+       '  "Exp Alpha = "     + REMOVE$(STR$(Alpha,3 ),ANY" ")  + CHR$(13) + _ 
+        ' "Exp Beta = "      + REMOVE$(STR$(Beta,3) ,ANY" ")   + CHR$(13) + _ 
+         '"Frep = "          + REMOVE$(STR$(Frep,4) ,ANY" ")   + " ("+RepositionFactor$ + ")" + CHR$(13 ) + _ 
+         '"Initial Probes: " + PlaceInitialProbes$            + CHR$(13) + _ 
+         '"Initial Accel: "  + InitialAcceleration$            + CHR$(13) + CHR$(13) 
+ 
+ A$ = "RUN CFO WITH THE" + CHR$(13) +_ 
+         "FOLLOWING PARAMETERS?"                             + CHR$(13) + CHR$(13) +_ 
+         "Function "        + FunctionName$                  + " (" + REMOVE$(STR$(Nd%),ANY" ") + "-D) " + CHR$(13) +_ 
+         "# time steps = "  + REMOVE$(STR$(Nt&),ANY " ")      + CHR$(13) + _ 
+         "Grav Const G = "  + REMOVE$(STR$(G,2),ANY " ")      + CHR$(13) + _ 
+         "Delta-T = "       + REMOVE$(STR$(DeltaT,3 ),ANY" ") + CHR$(13) + _ 
+         "Exp Alpha = "     + REMOVE$(STR$(Alpha,3) ,ANY" ")  + CHR$(13) + _ 
+         "Exp Beta = "      + REMOVE$(STR$(Beta,3), ANY" ")   + CHR$(13) + _ 
+         "Frep = "          + REMOVE$(STR$(Frep,4), ANY" ")   + " ("+RepositionFactor$ + ")" + CHR$(13)  + _ 
+         "Initial Probes: " + PlaceInitialProbes$            + CHR$(13) + _ 
+         "Initial Accel: "  + InitialAcceleration$           + CHR$(13) + _ 
+         "Check for Early Termination? " + CheckFor EarlyTermination$ + CHR$(13) + _ 
+         "Shrink Decision Space? "       + ShrinkDS $ + CHR$(13) +CHR$(13) 
+ 
+'   lResult& = MSGBOX(txt$ [, [style&], title$]) 
+ 
+    YN& = MSGBOX(A$,%MB_YESNO,"CONFIRM RUN") 
+ 
+    IF YN& = %IDYES THEN RunCFO$ = "YES" 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB StatusWindow(FunctionName$,StatusWindowHandle?? ?) 
+ 87 of 92     GRAPHIC WINDOW "Run Progress, "+FunctionName$,0 .08##*ScreenWidth&,0.08##*ScreenHeight&,0.25##*Scre enWidth&,0.1##*ScreenHeight& TO 
+StatusWindowHandle??? 
+ 
+    GRAPHIC ATTACH StatusWindowHandle???,0,REDRAW 
+ 
+    GRAPHIC FONT "Lucida Console",8,0 '"Courier New ",8,0 'Fixed width fonts 
+ 
+    GRAPHIC SET PIXEL (35,15) : GRAPHIC PRINT "  In itializing...      " : GRAPHIC REDRAW 
+ 
+END SUB 
+ 
+'------ 
+ 
+SUB GetTestFunctionNumber(FunctionName$) 
+ 
+  LOCAL hDlg AS DWORD 
+ 
+  LOCAL N%, M% 
+ 
+  LOCAL FrameWidth&, FrameHeight&, BoxWidth&, BoxHe ight& 
+ 
+  BoxWidth& = 276 : BoxHeight& = 300 : FrameWidth& = 80 : FrameHeight& = BoxHeight&-5 
+ 
+  DIALOG NEW 0, "CENTRAL FORCE OPTIMIZATION TEST FU NCTIONS",,, BoxWidth&, BoxHeight&, %WS_CAPTION OR % WS_SYSMENU, 0 TO hDlg 
+ 
+'-------------------------------------------------- ---------------- 
+ 
+  CONTROL ADD FRAME,  hDlg, %IDC_FRAME1,  "Test Fun ctions",      5,  2, FrameWidth&, FrameHeight& 
+  CONTROL ADD FRAME,  hDlg, %IDC_FRAME2,  "GSO Test  Functions",  95, 2, FrameWidth&, 255 
+ 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number1,  "Parrott F4",10,  14, 60, 10, %WS_GROUP OR %WS_TAB STOP 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number2,  "SGO"      , 10,  24, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number3,  "Goldstein-Price", 10,  34, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number4,  "Step"     , 10,  44, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number5,  "Schwefel 2.26", 10,  54, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number6,  "Colville", 10,  64, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number7,  "Griewank", 10,  74, 60, 10 
+ 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number31,  "PBM #1",    10,  84, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number32,  "PBM #2",    10,  94, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number33,  "PBM #3",    10, 104, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number34,  "PBM #4",    10, 114, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number35,  "PBM #5",    10, 124, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number36,  "Himmelblau",10, 134, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number37,  "Reserved",  10, 144, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number38,  "Reserved",  10, 154, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number39,  "Reserved",  10, 164, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number40,  "Reserved",  10, 174, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number41,  "Reserved",  10, 184, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number42,  "Reserved",  10, 194, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number43,  "Reserved",  10, 204, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number44,  "Reserved",  10, 214, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number45,  "Reserved",  10, 224, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number46,  "Reserved",  10, 234, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number47,  "Reserved",  10, 244, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number48,  "Reserved",  10, 254, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number49,  "Reserved",  10, 264, 60, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number50,  "Reserved",  10, 274, 60, 10 
+ 
+' --------------------- Test Functions from GSO Pap er --------------------- 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number8,  "f1" , 120,  14, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number9,  "f2" , 120,  24, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number10,  "f3" , 120,  34, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number11,  "f4" , 120,  44, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number12,  "f5" , 120,  54, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number13,  "f6" , 120,  64, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number14,  "f7" , 120,  74, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number15,  "f8" , 120,  84, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number16,  "f9" , 120,  94, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number17,  "f10", 120, 104, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number18,  "f11", 120, 114, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number19,  "f12", 120, 124, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number20,  "f13", 120, 134, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number21,  "f14", 120, 144, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number22,  "f15", 120, 154, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number23,  "f16", 120, 164, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number24,  "f17", 120, 174, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number25,  "f18", 120, 184, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number26,  "f19", 120, 194, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number27,  "f20", 120, 204, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number28,  "f21", 120, 214, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number29,  "f22", 120, 224, 40, 10 
+  CONTROL ADD OPTION, hDlg, %IDC_Function_Number30,  "f23", 120, 234, 40, 10 
+ 
+  CONTROL SET OPTION  hDlg, %IDC_Function_Number1, %IDC_Function_Number1, %IDC_Function_Number3 'defau lt to Parrott F4 
+ 
+'-------------------------------------------------- --------------------- 
+ 
+  CONTROL ADD BUTTON, hDlg, %IDOK, "&OK", 200, 0.45 ##*BoxHeight&, 50, 14 
+ 
+'-------------------------------------------------- --------------------- 
+ 
+  DIALOG SHOW MODAL hDlg CALL DlgProc 
+ 
+  CALL Delay(0.5##) 
+ 
+  IF FunctionNumber% < 1 OR FunctionNumber% > 36 TH EN 
+ 
+    FunctionNumber% = 1 : MSGBOX("Error in function  number...") 
+ 
+  END IF 
+ 
+' MSGBOX("Test Function is #"+STR$(FunctionNumber%) ) 
+ 
+    SELECT CASE FunctionNumber% 
+ 
+        CASE 1 : FunctionName$ = "ParrottF4" 
+        CASE 2 : FunctionName$ = "SGO" 
+        CASE 3 : FunctionName$ = "GP" 
+        CASE 4 : FunctionName$ = "STEP" 
+        CASE 5 : FunctionName$ = "SCHWEFEL_226" 
+        CASE 6 : FunctionName$ = "COLVILLE" 
+        CASE 7 : FunctionName$ = "GRIEWANK" 
+        CASE 8 : FunctionName$ = "F1" 88 of 92         CASE 9 : FunctionName$ = "F2" 
+        CASE 10: FunctionName$ = "F3" 
+        CASE 11: FunctionName$ = "F4" 
+        CASE 12: FunctionName$ = "F5" 
+        CASE 13: FunctionName$ = "F6" 
+        CASE 14: FunctionName$ = "F7" 
+        CASE 15: FunctionName$ = "F8" 
+        CASE 16: FunctionName$ = "F9" 
+        CASE 17: FunctionName$ = "F10" 
+        CASE 18: FunctionName$ = "F11" 
+        CASE 19: FunctionName$ = "F12" 
+        CASE 20: FunctionName$ = "F13" 
+        CASE 21: FunctionName$ = "F14" 
+        CASE 22: FunctionName$ = "F15" 
+        CASE 23: FunctionName$ = "F16" 
+        CASE 24: FunctionName$ = "F17" 
+        CASE 25: FunctionName$ = "F18" 
+        CASE 26: FunctionName$ = "F19" 
+        CASE 27: FunctionName$ = "F20" 
+        CASE 28: FunctionName$ = "F21" 
+        CASE 29: FunctionName$ = "F22" 
+        CASE 30: FunctionName$ = "F23" 
+        CASE 31: FunctionName$ = "PBM_1" 
+        CASE 32: FunctionName$ = "PBM_2" 
+        CASE 33: FunctionName$ = "PBM_3" 
+        CASE 34: FunctionName$ = "PBM_4" 
+        CASE 35: FunctionName$ = "PBM_5" 
+        CASE 36: FunctionName$ = "HIMMELBLAU" 
+ 
+    END SELECT 
+ 
+END SUB 
+ 
+'----------- 
+ 
+CALLBACK FUNCTION DlgProc() AS LONG 
+ 
+  '------------------------------------------------ -------------------- 
+  ' Callback procedure for the main dialog 
+  '------------------------------------------------ ------------------ 
+  LOCAL c, lRes AS LONG, sText AS STRING 
+ 
+  SELECT CASE AS LONG CBMSG 
+ 
+  CASE %WM_INITDIALOG' %WM_INITDIALOG is sent right  before the dialog is shown. 
+ 
+  CASE %WM_COMMAND              ' <- a control is c alling 
+ 
+     SELECT CASE AS LONG CBCTL  ' <- look at contro l's id 
+ 
+     CASE %IDOK                 ' <- OK button or E nter key was pressed 
+ 
+         IF CBCTLMSG = %BN_CLICKED THEN 
+             '------------------------------------- --- 
+             ' Loop through the Function_Number con trols 
+             ' to see which one is selected 
+             '------------------------------------- --- 
+             FOR c = %IDC_Function_Number1 TO %IDC_ Function_Number50 
+ 
+                 CONTROL GET CHECK CBHNDL, c TO lRe s 
+ 
+                 IF lRes THEN EXIT FOR 
+ 
+             NEXT 'c holds the id for selected test  function. 
+ 
+             FunctionNumber% = c-120 
+ 
+'DEBUG       sText = FORMAT$(c - %IDC_Function_Numb er1 + 1) + $CRLF 
+'            MSGBOX sText, %MB_TASKMODAL, "Selected  Function" 
+ 
+            DIALOG END CBHNDL 
+ 
+         END IF 
+ 
+     END SELECT 
+ 
+  END SELECT 
+ 
+END FUNCTION 
+ 
+'----------------------------- PBM ANTENNA BENCHMAR K FUNCTIONS --------------------------- 
+ 
+'Reference for benchmarks PBM_1 through PBM_5: 
+ 
+'Pantoja, M F., Bretones, A. R., Martin, R. G., "Be nchmark Antenna Problems for Evolutionary 
+'Optimization Algorithms," IEEE Trans. Antennas & P ropagation, vol. 55, no. 4, April 2007, 
+'pp. 1111-1121 
+ 
+FUNCTION PBM_1(R(),Nd%,p%,j&) 'PBM Benchmark #1: Ma x D for Variable-Length CF Dipole 
+ 
+    LOCAL Z, LengthWaves, ThetaRadians AS EXT 
+ 
+    LOCAL N%, Nsegs%, FeedSegNum% 
+ 
+    LOCAL NumSegs$, FeedSeg$, HalfLength$, Radius$,  ThetaDeg$, Lyne$, GainDB$ 
+ 
+    LengthWaves  = R(p%,1,j&) 
+ 
+    ThetaRadians = R(p%,2,j&) 
+ 
+    ThetaDeg$ = REMOVE$(STR$(ROUND(ThetaRadians*Rad 2Deg,2)),ANY" ") 
+ 
+    IF TALLY(ThetaDeg$,".") = 0 THEN ThetaDeg$ = Th etaDeg$+"." 
+ 
+    Nsegs% = 2*(INT(100*LengthWaves)\2)+1 '100 segs  per wavelength, must be an odd #, VOLTAGE SOURCE 
+ 
+    FeedSegNum% = Nsegs%\2 + 1 'center segment numb er, VOLTAGE SOURCE 
+ 
+    'Nsegs% = 2*(INT(100*LengthWaves)\2) '100 segs per wavelength, must be an even # for BICONE SOURCE  
+ 
+    'FeedSegNum% = Nsegs%\2 'center segment number for BICONE SOURCE 
+ 
+    NumSegs$    = REMOVE$(STR$(Nsegs%),ANY" ") 
+ 89 of 92     FeedSeg$    = REMOVE$(STR$(FeedSegNum%),ANY" ")  
+ 
+    HalfLength$ = REMOVE$(STR$(ROUND(LengthWaves/2# #,6)),ANY" ") 
+ 
+    IF TALLY(HalfLength$,".") = 0 THEN HalfLength$ = HalfLength$+"." 
+ 
+    Radius$     = "0.00001" 'REMOVE$(STR$(ROUND(Len gthWaves/1000##,6)),ANY" ") 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "PBM1.NEC" FOR OUTPUT AS #N% 
+ 
+        PRINT #N%,"CM File: PBM1.NEC" 
+        PRINT #N%,"CM Run ID "+DATE$+" "+TIME$ 
+        PRINT #N%,"CM Nd="+STR$(Nd%)+", p="STR$(p%) +", j="+STR$(j&) 
+        PRINT #N%,"CM R(p,1,j)="+STR$(R(p%,1,j&))+" , R(p,2,j)="+STR$(R(p%,2,j&)) 
+        PRINT #N%,"CE" 
+        PRINT #N%,"GW 1,"+NumSegs$+",0.,0.,-"+HalfL ength$+",0.,0.,"+HalfLength$+","+Radius$ 
+        PRINT #N%,"GE" 
+        'PRINT #N%,"EX 5,1,"+FeedSeg$+",0,1.,0." 'B ICONE SOURCE 
+        PRINT #N%,"EX 0,1,"+FeedSeg$+",0,1.,0." 'VO LTAGE SOURCE 
+        PRINT #N%,"FR 0,1,0,0,299.79564,0." 
+        PRINT #N%,"RP 0,1,1,1001,"+ThetaDeg$+",0.,0 .,0.,1000." 'gain at 1000 wavelengths range 
+        PRINT #N%,"XQ" 
+        PRINT #N%,"EN" 
+ 
+    CLOSE #N% 
+ 
+'      - - ANGLES - -          - POWER GAINS -        - - - POLARIZATION - - -    - - - E(THETA) - - -     - - - E(PHI) - - - 
+'  THETA     PHI        VERT.   HOR.    TOTAL      AXIAL     TILT   SENSE     MAGNITUDE    PHASE      MAGNITUDE    PHASE 
+' DEGREES  DEGREES       DB      DB      DB        RATIO     DEG.              VOLTS/M    DEGREES      VOLTS/M    DEGREES 
+'   90.00     0.00       3.91 -999.99    3.91    0. 00000     0.00  LINEAR    1.29504E-04     5.37    0 .00000E+00    -5.24 
+'123456789x123456789x123456789x123456789x123456789x 123456789x123456789x123456789x123456789x123456789x1 23456789x123456789x 
+'        10        20        30        40        50         60        70        80        90        100        110       120 
+ 
+    SHELL "n41_2k1.exe",0 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "PBM1.OUT" FOR INPUT AS #N% 
+ 
+        WHILE NOT EOF(N%) 
+ 
+            LINE INPUT #N%, Lyne$ 
+ 
+            IF INSTR(Lyne$,"DEGREES  DEGREES") > 0 THEN EXIT LOOP 
+ 
+        WEND 'position at next data line 
+ 
+        LINE INPUT #N%, Lyne$ 
+ 
+    CLOSE #N% 
+ 
+    GainDB$ = REMOVE$(MID$(Lyne$,37,8),ANY" ") 
+ 
+    PBM_1 = 10^(VAL(GainDB$)/10##) 'Directivity 
+ 
+END FUNCTION 'PBM_1() 
+ 
+'---- 
+ 
+FUNCTION PBM_2(R(),Nd%,p%,j&) 'PBM Benchmark #2: Ma x D for Variable-Separation Array of CF Dipoles 
+ 
+    LOCAL Z, DipoleSeparationWaves, ThetaRadians AS  EXT 
+ 
+    LOCAL N%, i% 
+ 
+    LOCAL NumSegs$, FeedSeg$, Radius$, ThetaDeg$, L yne$, GainDB$, Xcoord$, WireNum$ 
+ 
+    DipoleSeparationWaves = R(p%,1,j&) 
+ 
+    ThetaRadians          = R(p%,2,j&) 
+ 
+    ThetaDeg$ = REMOVE$(STR$(ROUND(ThetaRadians*Rad 2Deg,2)),ANY" ") 
+ 
+    IF TALLY(ThetaDeg$,".") = 0 THEN ThetaDeg$ = Th etaDeg$+"." 
+ 
+    NumSegs$ = "49" 
+ 
+    FeedSeg$ = "25" 
+ 
+    Radius$  = "0.00001" 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "PBM2.NEC" FOR OUTPUT AS #N% 
+ 
+        PRINT #N%,"CM File: PBM2.NEC" 
+        PRINT #N%,"CM Run ID "+DATE$+" "+TIME$ 
+        PRINT #N%,"CM Nd="+STR$(Nd%)+", p="STR$(p%) +", j="+STR$(j&) 
+        PRINT #N%,"CM R(p,1,j)="+STR$(R(p%,1,j&))+" , R(p,2,j)="+STR$(R(p%,2,j&)) 
+        PRINT #N%,"CE" 
+ 
+        FOR i% = -9 TO 9 STEP 2 
+            WireNum$ = REMOVE$(STR$((i%+11)\2),ANY"  ") 
+            Xcoord$  = REMOVE$(STR$(i%*DipoleSepara tionWaves/2##),ANY" ") 
+            PRINT #N%,"GW "+WireNum$+","+NumSegs$+" ,"+Xcoord$+",0.,-0.25,"+Xcoord$+",0.,0.25,"+Radius$  
+        NEXT i% 
+ 
+        PRINT #N%,"GE" 
+ 
+        FOR i% = 1 TO 10 
+            PRINT #N%,"EX 0,"+REMOVE$(STR$(i%),ANY"  ")+","+FeedSeg$+",0,1.,0." 'VOLTAGE SOURCE 
+        NEXT i% 
+        PRINT #N%,"FR 0,1,0,0,299.79564,0." 
+        PRINT #N%,"RP 0,1,1,1001,"+ThetaDeg$+",90., 0.,0.,1000." 'gain at 1000 wavelengths range 
+        PRINT #N%,"XQ" 
+        PRINT #N%,"EN" 
+ 
+    CLOSE #N% 
+ 
+'      - - ANGLES - -          - POWER GAINS -        - - - POLARIZATION - - -    - - - E(THETA) - - -     - - - E(PHI) - - - 
+'  THETA     PHI        VERT.   HOR.    TOTAL      AXIAL     TILT   SENSE     MAGNITUDE    PHASE      MAGNITUDE    PHASE 
+' DEGREES  DEGREES       DB      DB      DB        RATIO     DEG.              VOLTS/M    DEGREES      VOLTS/M    DEGREES 90 of 92 '   90.00     0.00       3.91 -999.99    3.91    0. 00000     0.00  LINEAR    1.29504E-04     5.37    0 .00000E+00    -5.24 
+'123456789x123456789x123456789x123456789x123456789x 123456789x123456789x123456789x123456789x123456789x1 23456789x123456789x 
+'        10        20        30        40        50         60        70        80        90        100        110       120 
+ 
+    SHELL "n41_2k1.exe",0 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "PBM2.OUT" FOR INPUT AS #N% 
+ 
+        WHILE NOT EOF(N%) 
+ 
+            LINE INPUT #N%, Lyne$ 
+ 
+            IF INSTR(Lyne$,"DEGREES  DEGREES") > 0 THEN EXIT LOOP 
+ 
+        WEND 'position at next data line 
+ 
+        LINE INPUT #N%, Lyne$ 
+ 
+    CLOSE #N% 
+ 
+    GainDB$ = REMOVE$(MID$(Lyne$,37,8),ANY" ") 
+ 
+    IF AddNoiseToPBM2$ = "YES" THEN 
+ 
+        Z = 10^(VAL(GainDB$)/10##) + GaussianDeviat e(0##,0.4472##) 'Directivity with Gaussian noise (z ero mean, 0.2 variance) 
+ 
+    ELSE 
+ 
+        Z = 10^(VAL(GainDB$)/10##) 'Directivity wit hout noise 
+ 
+    END IF 
+ 
+    PBM_2 = Z 
+ 
+END FUNCTION 'PBM_2() 
+ 
+'---- 
+ 
+FUNCTION PBM_3(R(),Nd%,p%,j&) 'PBM Benchmark #3: Ma x D for Circular Dipole Array 
+ 
+    LOCAL Beta, ThetaRadians, Alpha, ReV, ImV AS EX T 
+ 
+    LOCAL N%, i% 
+ 
+    LOCAL NumSegs$, FeedSeg$, Radius$, ThetaDeg$, L yne$, GainDB$, Xcoord$, Ycoord$, WireNum$, ReEX$, I mEX$ 
+ 
+    Beta         = R(p%,1,j&) 
+ 
+    ThetaRadians = R(p%,2,j&) 
+ 
+    ThetaDeg$ = REMOVE$(STR$(ROUND(ThetaRadians*Rad 2Deg,2)),ANY" ") 
+ 
+    IF TALLY(ThetaDeg$,".") = 0 THEN ThetaDeg$ = Th etaDeg$+"." 
+ 
+    NumSegs$ = "49" 
+ 
+    FeedSeg$ = "25" 
+ 
+    Radius$  = "0.00001" 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "PBM3.NEC" FOR OUTPUT AS #N% 
+ 
+        PRINT #N%,"CM File: PBM3.NEC" 
+        PRINT #N%,"CM Run ID "+DATE$+" "+TIME$ 
+        PRINT #N%,"CM Nd="+STR$(Nd%)+", p="STR$(p%) +", j="+STR$(j&) 
+        PRINT #N%,"CM R(p,1,j)="+STR$(R(p%,1,j&))+" , R(p,2,j)="+STR$(R(p%,2,j&)) 
+        PRINT #N%,"CE" 
+ 
+        FOR i% = 1 TO 8 
+            WireNum$ = REMOVE$(STR$(i%),ANY" ") 
+ 
+            SELECT CASE i% 
+                CASE 1 : Xcoord$ = "1"        : Yco ord$ = "0" 
+                CASE 2 : Xcoord$ = "0.70711"  : Yco ord$ = "0.70711" 
+                CASE 3 : Xcoord$ = "0"        : Yco ord$ = "1" 
+                CASE 4 : Xcoord$ = "-0.70711" : Yco ord$ = "0.70711" 
+                CASE 5 : Xcoord$ = "-1"       : Yco ord$ = "0" 
+                CASE 6 : Xcoord$ = "-0.70711" : Yco ord$ = "-0.70711" 
+                CASE 7 : Xcoord$ = "0"        : Yco ord$ = "-1" 
+                CASE 8 : Xcoord$ = "0.70711"  : Yco ord$ = "-0.70711" 
+            END SELECT 
+ 
+            PRINT #N%,"GW "+WireNum$+","+NumSegs$+" ,"+Xcoord$+","+Ycoord$+",-0.25,"+Xcoord$+","Ycoord$ +",0.25,"+Radius$ 
+        NEXT i% 
+ 
+        PRINT #N%,"GE" 
+ 
+        FOR i% = 1 TO 8 
+            Alpha = -COS(TwoPi*Beta*(i%-1)) 
+ 
+            ReV   = COS(Alpha) 
+            ImV   = SIN(Alpha) 
+ 
+            ReEX$ = REMOVE$(STR$(ROUND(ReV,6)),ANY"  ") 
+            ImEX$ = REMOVE$(STR$(ROUND(ImV,6)),ANY"  ") 
+ 
+            IF TALLY(ReEX$,".") = 0 THEN ReEX$ = Re EX$+"." 
+            IF TALLY(ImEX$,".") = 0 THEN ImEX$ = Im EX$+"." 
+ 
+            PRINT #N%,"EX 0,"+REMOVE$(STR$(i%),ANY"  ")+","+FeedSeg$+",0,"+ReEX$+","+ImEX$ 'VOLTAGE SOU RCE 
+        NEXT i% 
+ 
+        PRINT #N%,"FR 0,1,0,0,299.79564,0." 
+        PRINT #N%,"RP 0,1,1,1001,"+ThetaDeg$+",0.,0 .,0.,1000." 'gain at 1000 wavelengths range 
+        PRINT #N%,"XQ" 
+        PRINT #N%,"EN" 
+ 
+    CLOSE #N% 
+ 
+'      - - ANGLES - -          - POWER GAINS -        - - - POLARIZATION - - -    - - - E(THETA) - - -     - - - E(PHI) - - - 91 of 92 '  THETA     PHI        VERT.   HOR.    TOTAL      AXIAL     TILT   SENSE     MAGNITUDE    PHASE      MAGNITUDE    PHASE 
+' DEGREES  DEGREES       DB      DB      DB        RATIO     DEG.              VOLTS/M    DEGREES      VOLTS/M    DEGREES 
+'   90.00     0.00       3.91 -999.99    3.91    0. 00000     0.00  LINEAR    1.29504E-04     5.37    0 .00000E+00    -5.24 
+'123456789x123456789x123456789x123456789x123456789x 123456789x123456789x123456789x123456789x123456789x1 23456789x123456789x 
+'        10        20        30        40        50         60        70        80        90        100        110       120 
+ 
+    SHELL "n41_2k1.exe",0 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "PBM3.OUT" FOR INPUT AS #N% 
+ 
+        WHILE NOT EOF(N%) 
+ 
+            LINE INPUT #N%, Lyne$ 
+ 
+            IF INSTR(Lyne$,"DEGREES  DEGREES") > 0 THEN EXIT LOOP 
+ 
+        WEND 'position at next data line 
+ 
+        LINE INPUT #N%, Lyne$ 
+ 
+    CLOSE #N% 
+ 
+    GainDB$ = REMOVE$(MID$(Lyne$,37,8),ANY" ") 
+ 
+    PBM_3 = 10^(VAL(GainDB$)/10##) 'Directivity 
+ 
+END FUNCTION 'PBM_3() 
+ 
+'---- 
+ 
+FUNCTION PBM_4(R(),Nd%,p%,j&) 'PBM Benchmark #4: Ma x D for Vee Dipole 
+ 
+    LOCAL TotalLengthWaves, AlphaRadians, ArmLength , Xlength, Zlength, Lfeed AS EXT 
+ 
+    LOCAL N%, i%, Nsegs%, FeedZcoord$ 
+ 
+    LOCAL NumSegs$, Lyne$, GainDB$, Xcoord$, Zcoord $ 
+ 
+    TotalLengthWaves = 2##*R(p%,1,j&) 
+ 
+    AlphaRadians     = R(p%,2,j&) 
+ 
+    Lfeed            = 0.01## 
+ 
+    FeedZcoord$      = REMOVE$(STR$(Lfeed),ANY" ") 
+ 
+    ArmLength = (TotalLengthWaves-2##*Lfeed)/2## 
+ 
+    Xlength  = ROUND(ArmLength*COS(AlphaRadians),6)  
+ 
+    Xcoord$  = REMOVE$(STR$(Xlength),ANY" ") : IF T ALLY(Xcoord$,".") = 0 THEN Xcoord$ = Xcoord$+"." 
+ 
+    Zlength  = ROUND(ArmLength*SIN(AlphaRadians),6)  
+ 
+    Zcoord$  = REMOVE$(STR$(Zlength+Lfeed),ANY" ") : IF TALLY(Zcoord$,".") = 0 THEN Zcoord$ = Zcoord$+ "." 
+ 
+    Nsegs%   = 2*(INT(TotalLengthWaves*100)\2) 'eve n number, total # segs 
+ 
+    NumSegs$ = REMOVE$(STR$(Nsegs%\2),ANY" ") '# se gs per arm 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "PBM4.NEC" FOR OUTPUT AS #N% 
+ 
+        PRINT #N%,"CM File: PBM4.NEC" 
+        PRINT #N%,"CM Run ID "+DATE$+" "+TIME$ 
+        PRINT #N%,"CM Nd="+STR$(Nd%)+", p="STR$(p%) +", j="+STR$(j&) 
+        PRINT #N%,"CM R(p,1,j)="+STR$(R(p%,1,j&))+" , R(p,2,j)="+STR$(R(p%,2,j&)) 
+        PRINT #N%,"CE" 
+ 
+        PRINT #N%,"GW 1,5,0.,0.,-"+FeedZcoord$+",0. ,0.,"+FeedZcoord$+",0.00001" 'feed wire, 1 segment,  0.01 wvln 
+ 
+        PRINT #N%,"GW 2,"+NumSegs$+",0.,0.,"+FeedZc oord$+","+Xcoord$+",0.,"+Zcoord$+",0.00001" 'upper arm 
+ 
+        PRINT #N%,"GW 3,"+NumSegs$+",0.,0.,-"+FeedZ coord$+","+Xcoord$+",0.,-"+Zcoord$+",0.00001" 'lowe r arm 
+ 
+        PRINT #N%,"GE" 
+ 
+        PRINT #N%,"EX 0,1,3,0,1.,0." 'VOLTAGE SOURC E 
+ 
+        PRINT #N%,"FR 0,1,0,0,299.79564,0." 
+        PRINT #N%,"RP 0,1,1,1001,90.,0.,0.,0.,1000. " 'ENDFIRE gain at 1000 wavelengths range 
+        PRINT #N%,"XQ" 
+        PRINT #N%,"EN" 
+ 
+    CLOSE #N% 
+ 
+'      - - ANGLES - -          - POWER GAINS -        - - - POLARIZATION - - -    - - - E(THETA) - - -     - - - E(PHI) - - - 
+'  THETA     PHI        VERT.   HOR.    TOTAL      AXIAL     TILT   SENSE     MAGNITUDE    PHASE      MAGNITUDE    PHASE 
+' DEGREES  DEGREES       DB      DB      DB        RATIO     DEG.              VOLTS/M    DEGREES      VOLTS/M    DEGREES 
+'   90.00     0.00       3.91 -999.99    3.91    0. 00000     0.00  LINEAR    1.29504E-04     5.37    0 .00000E+00    -5.24 
+'123456789x123456789x123456789x123456789x123456789x 123456789x123456789x123456789x123456789x123456789x1 23456789x123456789x 
+'        10        20        30        40        50         60        70        80        90        100        110       120 
+ 
+    SHELL "n41_2k1.exe",0 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "PBM4.OUT" FOR INPUT AS #N% 
+ 
+        WHILE NOT EOF(N%) 
+ 
+            LINE INPUT #N%, Lyne$ 
+ 
+            IF INSTR(Lyne$,"DEGREES  DEGREES") > 0 THEN EXIT LOOP 
+ 
+        WEND 'position at next data line 
+ 
+        LINE INPUT #N%, Lyne$ 
+ 
+    CLOSE #N% 
+ 92 of 92     GainDB$ = REMOVE$(MID$(Lyne$,37,8),ANY" ") 
+ 
+    PBM_4 = 10^(VAL(GainDB$)/10##) 'Directivity 
+ 
+END FUNCTION 'PBM_4() 
+ 
+'---- 
+ 
+FUNCTION PBM_5(R(),Nd%,p%,j&) 'PBM Benchmark #5: N- element collinear array (Nd=N-1) 
+ 
+    LOCAL TotalLengthWaves, Di(), Ystart, Y1, Y2, S umDi AS EXT 
+ 
+    LOCAL N%, i%, q% 
+ 
+    LOCAL Lyne$, GainDB$ 
+ 
+    REDIM Di(1 TO Nd%) 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        Di(i%) = R(p%,i%,j&) 'dipole separation, wa velengths 
+ 
+        'MSGBOX("R="+STR$(R(p%,i%,j&))+"  p="+STR$( p%)+"   i="+STR$(i%)+"    j="+STR$(j&)+"    Nd="+ST R$(Nd%)) 
+ 
+    NEXT i% 
+ 
+    TotalLengthWaves = 0## 
+ 
+    FOR i% = 1 TO Nd% 
+ 
+        TotalLengthwaves = TotalLengthWaves + Di(i% ) 
+ 
+    NEXT i% 
+ 
+    TotalLengthWaves = TotalLengthWaves + 0.5## 'ad d half-wavelength of 1 meter at 299.8 MHz 
+ 
+    Ystart = -TotalLengthWaves/2## 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "PBM5.NEC" FOR OUTPUT AS #N% 
+ 
+        PRINT #N%,"CM File: PBM5.NEC" 
+        PRINT #N%,"CM Run ID "+DATE$+" "+TIME$ 
+        PRINT #N%,"CM Nd="+STR$(Nd%)+", p="STR$(p%) +", j="+STR$(j&) 
+        PRINT #N%,"CM R(p,1,j)="+STR$(R(p%,1,j&))+" , R(p,2,j)="+STR$(R(p%,2,j&)) 
+        PRINT #N%,"CE" 
+ 
+        FOR i% = 1 TO Nd%+1 
+ 
+            SumDi = 0## 
+ 
+            FOR q% = 1 TO i%-1 
+ 
+                SumDi = SumDi + Di(q%) 
+ 
+            NEXT q% 
+ 
+            Y1 = ROUND(Ystart + SumDi,6) 
+ 
+            Y2 = ROUND(Y1+0.5##,6) 'add one-half wa velength for other end of dipole 
+ 
+            PRINT #N%,"GW "+REMOVE$(STR$(i%),ANY" " )+",49,0.,"+REMOVE$(STR$(Y1),ANY" ")+",0.,0.,"+REMO VE$(STR$(Y2),ANY" ")+",0.,0.00001" 
+ 
+        NEXT i% 
+ 
+        PRINT #N%,"GE" 
+ 
+        FOR i% = 1 TO Nd%+1 
+            PRINT #N%,"EX 0,"+REMOVE$(STR$(i%),ANY"  ")+",25,0,1.,0." 'VOLTAGE SOURCES 
+        NEXT i% 
+ 
+        PRINT #N%,"FR 0,1,0,0,299.79564,0." 
+        PRINT #N%,"RP 0,1,1,1001,90.,0.,0.,0.,1000. " 'gain at 1000 wavelengths range 
+        PRINT #N%,"XQ" 
+        PRINT #N%,"EN" 
+ 
+    CLOSE #N% 
+ 
+'      - - ANGLES - -          - POWER GAINS -        - - - POLARIZATION - - -    - - - E(THETA) - - -     - - - E(PHI) - - - 
+'  THETA     PHI        VERT.   HOR.    TOTAL      AXIAL     TILT   SENSE     MAGNITUDE    PHASE      MAGNITUDE    PHASE 
+' DEGREES  DEGREES       DB      DB      DB        RATIO     DEG.              VOLTS/M    DEGREES      VOLTS/M    DEGREES 
+'   90.00     0.00       3.91 -999.99    3.91    0. 00000     0.00  LINEAR    1.29504E-04     5.37    0 .00000E+00    -5.24 
+'123456789x123456789x123456789x123456789x123456789x 123456789x123456789x123456789x123456789x123456789x1 23456789x123456789x 
+'        10        20        30        40        50         60        70        80        90        100        110       120 
+ 
+    SHELL "n41_2k1.exe",0 
+ 
+    N% = FREEFILE 
+ 
+    OPEN "PBM5.OUT" FOR INPUT AS #N% 
+ 
+        WHILE NOT EOF(N%) 
+ 
+            LINE INPUT #N%, Lyne$ 
+ 
+            IF INSTR(Lyne$,"DEGREES  DEGREES") > 0 THEN EXIT LOOP 
+ 
+        WEND 'position at next data line 
+ 
+        LINE INPUT #N%, Lyne$ 
+ 
+    CLOSE #N% 
+ 
+    GainDB$ = REMOVE$(MID$(Lyne$,37,8),ANY" ") 
+ 
+    PBM_5 = 10^(VAL(GainDB$)/10##) 'Directivity 
+ 
+END FUNCTION 'PBM_5() 
+ 
+'*************************************************  END PROGRAM 'CFO_12-25-09.BAS'  ****************** **************************************** 
\ No newline at end of file
diff --git a/1001.0318.txt b/1001.0318.txt
new file mode 100644
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--- /dev/null
+++ b/1001.0318.txt
@@ -0,0 +1,672 @@
+arXiv:1001.0318v3  [math.DS]  21 Sep 2018SCHMIDT’S GAME, FRACTALS, AND
+ORBITS OF TORAL ENDOMORPHISMS
+RYAN BRODERICK, LIOR FISHMAN AND DMITRY KLEINBOCK
+Abstract. Given an integer matrix M∈GLn(R) and a point y∈Rn/Zn,
+consider the set
+˜E(M,y)def=/braceleftBig
+x∈Rn:y /∈{MkxmodZn:k∈N}/bracerightBig
+.
+S.G. Dani showed in 1988 that whenever Mis semisimple and y∈Qn/Zn,
+the set˜E(M,y) has full Hausdorff dimension. In this paper we strengthen
+this result, extending it to arbitrary M∈GLn(R)∩Mn×n(Z) andy∈
+Rn/Zn, and in fact replacing the sequence of powers of Mby any lacunary
+sequence of (not necessarily integer) m×nmatrices. Furthermore, we show
+that sets of the form ˜E(M,y) and their generalizations always intersect
+with ‘sufficiently regular’ fractal subsets of Rn. As an application we give
+an alternative proof of a recent result of [7] on badly approximable systems
+of affine forms.
+1.Introduction
+LetTndef=Rn/Znbe then-dimensional torus. Any non-singular n×nmatrix
+Mwith integer entries defines a continuous surjective endomorphism fMofTn
+given by
+fM(x+Zn)def=Mx+Zn∀x∈Rn,
+and any continuous surjective endomoprhism fofTncan be obtained this
+way. Criteria for ergodicity of f(with respect to Haar measure on Tn) are
+well known, and ergodicity implies that f-orbits of almost all points are dense
+inTn. Also in many cases it is known that exceptional sets of points with
+non-dense orbits are rather big. For example, following the notatio n used in
+[14], let us define
+E(f,y)def=/braceleftBig
+x∈Tn:y /∈{fk(x) :k∈N}/bracerightBig
+(1.1)
+for a fixed y∈Tnand a self-map fofTn. In 1988 Dani proved
+Theorem 1.1. [5, Theorem 2.1] For any semisimple M∈GLn(R)∩Mn×n(Z)
+and any y∈Qn/Zn, the set E(fM,y)is1
+2-winning.
+The above winning property is based on a game, introduced by Schmid t in
+[26], which is usually referred to as Schmidt’s game. This property implie s
+density and full Hausdorff dimension and is stable with respect to cou ntable
+intersections; see §2 for more detail.
+Date: May 2010.
+12 BRODERICK, FISHMAN, KLEINBOCK
+Oneofthegoalsofthepresent paperistoproveafar-reachingge neralization
+of Theorem 1.1. Namely, we remove the assumptions of Mbeing semisimple
+andybeing rational. Also we are able to intersect sets E(f,y) with many
+‘sufficiently regular’ fractal subsets of Tn. In fact it will be more convenient
+to lift the problem to Rn: denote by πthe quotient map Rn→Tnand, for
+M∈Mn×n(R) andy∈Tn, consider
+˜E(M,y)def=/braceleftBig
+x∈Rn:y /∈{π(Mkx) :k∈N}/bracerightBig
+. (1.2)
+Clearly˜E(M,y) =π−1/parenleftbig
+E(fM,y)/parenrightbig
+whenM∈GLn(R)∩Mn×n(Z); however
+the definition (1.2) makes sense even when Mis singular or has non-integer
+entries.
+The ‘sufficient regularity’ of subsets of Rnwill be characterized by their
+ability to support so-called absolutely decaying measures; see [16] or §3 for a
+definition. Examples include Rnitself and limit sets of irreducible families of
+contracting similarities of Rnsatisfying the Open Set Condition, such as the
+Koch snowflake or the Sierpinski carpet. Other interesting examp les can be
+found in [16, 29, 32].
+It turns out, as was first observed in [12], that the absolute deca y property
+of a measure can be used for playing Schmidt’s game on its support. N amely,
+we will say, following [1], that a subset SofRnisα-winning on a subset Kof
+RnifS∩Kisα-winning for Schmidt’s game played on the metric space K
+with the metric induced from Rn. From [26] it immediately follows that the
+intersection of countably many sets α-winning on Kis alsoα-winning on K.
+We will say that Siswinning on Kif it isα-winning on Kfor some α >0.
+Precise definitions are given in §2. As a trivial consequence of Corollary 3.3,
+ifSis winning on K= suppµ, whereµis absolutely decaying, then S∩Kis
+not contained in a countable union of affine hyperplanes. Furthermo re, under
+some additional assumptions on µ, for example when K=Rnor one of the
+self-similar sets mentioned above, one can show that the Hausdorff dimension
+ofS∩Kis equal to dim( K) whenever Sis winning on K. See§3 for precise
+statements.
+In this paper we prove a generalization of Theorem 1.1:
+Theorem 1.2. For every K⊂Rnwhich supports an absolutely decaying mea-
+sure there exists α=α(K)>0such that for any M∈GLn(R)∩Mn×n(Z)and
+anyy∈Tn, the set ˜E(M,y)isα-winning on K.
+In particular, for any countable subsetYofTn, the set
+/intersectiondisplay
+y∈Y/intersectiondisplay
+M∈GLn(R)∩Mn×n(Z)˜E(M,y)
+is alsoα-winning on K. It immediately follows that sets E(fM,y) discussed in
+Theorem 1.1 and their countable intersections always intersect tho se subsets
+of the torus whose pullbacks to Rnsupport absolutely decaying measures. It
+can also be shown that α(Rn) = 1/2, recovering Dani’s result, see §5.1.SCHMIDT’S GAME, FRACTALS, AND ORBITS OF TORAL ENDOMORPHISM S 3
+The one-dimensional case of Theorem 1.2 appeared recently in [1], an d also,
+independently and for K=R, in [10]; see also [30]. In other words, the sets
+˜E(b,y)def=/braceleftBig
+x∈R:y /∈{π(bkx) :k∈N}/bracerightBig
+. (1.3)
+were shown to be winning on supp µfor any absolutely decaying measure µon
+R, any integer b >1 and any y∈T. However, the main result of [1] applies to
+much more general situations, recovering earlier work [6, 23, 24] b y Pollington
+and de Mathan. In particular, bin (1.3) does not have to be an integer, and
+one can replace the sequence of powers of bby an arbitrary lacunary sequence
+tkof real numbers (we recall that ( tk) is called lacunary if infk∈Ntk+1
+tk>1.)
+We now describe an analogous generalization of Theorem 1.2, which is t he
+main result of the present paper. We are going to fix m,n∈N, consider a
+sequence M= (Mk) ofm×nmatrices and a sequence Z= (Zk) of subsets of
+Rm, and define
+˜E(M,Z)def={x∈Rn: inf
+k∈Nd(Mkx,Zk)>0}. (1.4)
+(Hered(·,·)standsfortheEuclideandistanceon Rn.) Thesets ˜E(M,y)defined
+in (1.2) constitute a special case, with m=n,M= (Mk) andZk=π−1(y).
+Some assumptions on MandZare in order. We will say that a sequence
+Mof nonzero m×nmatrices is lacunary if so is the sequence ( /⌊ar⌈⌊lMk/⌊ar⌈⌊lop) of the
+values of their operator norms. A subset ZofRnwill be called δ-uniformly
+discreteif infx,y∈Z,x/ne}ationslash=yd(x,y)> δ. With some abuse of terminology, we say
+that a sequence Z= (Zk) isδ-uniformly discrete if Zkisδ-uniformly discrete
+for every k∈N, and that Zisuniformly discrete if it isδ-uniformly discrete
+for some δ >0. For example, for an arbitrary sequence ( yk) of points of Tm,
+the sequence of sets Zk=π−1(yk)⊂Rmis 1-uniformly discrete.
+We can now formulate our main result, which is proved in §4:
+Theorem 1.3. For every K⊂Rnwhich supports an absolutely decaying mea-
+sure there exists a positive α=α(K)such that if Zis a uniformly discrete
+sequence of subsets of RmandMis a lacunary sequence of m×nmatrices
+with real entries, then ˜E(M,Z)isα-winning on K.
+An important special case is m=nandM= (Mk), where Mis ann×n
+matrix with spectral radius strictly greater than 1 (not necessar ily invertible
+and not necessarily with integer entries); this is used to derive Theo rem 1.2
+from Theorem 1.3, see §4. Our main theorem also generalizes results from [2]
+and [22] dealing with a special case where
+Mis a lacunary sequence of 1 ×ninteger matrices
+andZ= (Zk),whereZk=Z=π−1(0)∀k∈N.(1.5)
+It was observed both in [2] and in [22] that the latter set-up can be u sed to
+prove the abundance of badly approximable systems of affine forms . Recall
+that a pair ( A,x), interpreted as a function q/mapsto→Aq−x,Rm→Rn(here
+A∈Mn×m(R) andx∈Rn) is said to be badly approximable if
+inf
+q∈Zm/integerdivide{0}/⌊ar⌈⌊lq/⌊ar⌈⌊lm/nd(Aq−x,Zn)>0.4 BRODERICK, FISHMAN, KLEINBOCK
+This is an inhomogeneous analog of the notion of badly approximable sy stems
+of linear forms, see [27, 28]. It was proved in [15] that the set Bad(n,m) of
+badly approximable pairs ( A,x) has full Hausdorff dimension. Then a much
+easier proof was found in [2], where, for fixed A∈Mn×m(R), the sets
+BadA(n,m) =/braceleftbig
+x∈Rn: (A,x)∈Bad(n,m)/bracerightbig
+were considered, and it was shown that dim/parenleftbig
+BadA(n,m)/parenrightbig
+=nfor anyA. The
+latter result was strengthened by Tseng in the case m=n= 1: he proved
+[31] that Bada(1,1)⊂Ris1
+8-winning for any a∈R. Shortly thereafter,
+Moshchevitin concluded [22] that the sets BadA(n,m) are1
+2-winning for any
+m,nand any A∈Mn×m(R). Our main theorem can be used to deduce
+Corollary 1.4. LeK⊂Rnbe the support of an absolutely decaying measure,
+and letαbe as in Theorem 1.3. Then for any A∈Mn×m(R),BadA(n,m)is
+α-winning on K.
+Independently, in a recent preprint [7] Einsiedler and Tseng provide d an-
+other proof of this result, with a smaller value of α. We derive Corollary 1.4
+in§4. At the end of the paper a remark is made explaining how all our resu lts
+can be strengthened to replace ‘winning’ with ‘strong winning’, a prop erty
+introduced recently in [10, 11, 21].
+Acknowledgements: The authors are grateful to Manfred Einsiedler,
+Nikolay Moshchevitin and Barak Weiss for helpful discussions, and to the
+referee for useful comments. D.K. was supported in part by NSF G rant DMS-
+0801064.
+2.Schmidt’s game
+In this section we describe the game, first introduced by Schmidt in [2 6].
+Let (X,d) be a complete metric space. Consider Ωdef=X×R+, and define a
+partial ordering
+(x2,ρ2)≤s(x1,ρ1) ifρ2+d(x1,x2)≤ρ1.
+We associate to each pair ( x,ρ) a ball in ( X,d) via
+B(x,ρ) ={x′∈X:d(x,x′)≤ρ}.
+Note that ( x2,ρ2)≤s(x1,ρ1) implies (but is not necessarily implied by)
+B(x2,ρ2)⊂B(x1,ρ1). However the two conditions are equivalent when X
+is a Euclidean space.
+Schmidt’s game is played by two players, whom we will call Alice and Bob,
+following a convention used previously in [18, 1]. The two players are eq uipped
+withparameters αandβrespectively, satisfying 0 < α,β < 1. Chooseasubset
+SofX(a target set). The game starts with Bob picking x1∈Xandρ >0,
+hence specifying a pair ω1= (x1,ρ). Alice and Bob then take turns choosing
+ω′
+k= (x′
+k,ρ′
+k)≤sωkandωk+1= (xk+1,ρk+1)≤sω′
+krespectively satisfying
+ρ′
+k=αρkandρk+1=βρ′
+k. (2.1)
+As the game is played on a complete metric space and the diameters of the
+nested ballsSCHMIDT’S GAME, FRACTALS, AND ORBITS OF TORAL ENDOMORPHISM S 5
+B(ω1)⊃B(ω′
+1)⊃...⊃B(ωk)⊃B(ω′
+k)⊃...
+tend to zero as k→ ∞, the intersection of these balls is a point x∞∈X. Call
+Alice the winner if x∞∈S. Otherwise Bob is declared the winner. A strategy
+consists of specifications for a player’s choices of centers for his o r her balls
+given the opponent’s previous moves.
+If for certain α,βand a target set SAlice has a winning strategy, i.e., a
+strategy for winning the game regardless of how well Bob plays, we s ay that
+Sis an (α,β)-winning set . IfSandαare such that Sis an (α,β)-winning set
+for all possible β’s, we say that Sis anα-winning set. Call a set winning if
+such anαexists.
+Intuitively one expects winning sets to be large. Indeed, every suc h set is
+clearlydensein X; moreover, undersomeadditionalassumptionsonthemetric
+space winning sets can be proved to have positive, and even full, Hau sdorff
+dimension. For example, the fact that a winning subset of Rnhas Hausdorff
+dimension nis due to Schmidt [26, Corollary 2]. Another useful result of
+Schmidt [26, Theorem 2] states that the intersection of countab ly many α-
+winning sets is α-winning.
+Schmidt himself used the machinery of the game he invented to prove that
+certain subsets of RorRnare winning, and hence have full Hausdorff dimen-
+sion. Now let Kbe a closed subset of X. Following an approach initially
+introduced in [12], we will say that a subset SofXis (α,β)-winning on K
+(resp.,α-winning on K,winning on K) ifS∩Kis (α,β)-winning (resp., α-
+winning, winning) for Schmidt’s game played on the metric space Kwith the
+metric induced from ( X,d). In the present paper we let X=Rnand take K
+to be the support of an absolutely decaying measure. In other wor ds, since the
+metric is induced, playing the game on Kamounts to choosing balls in Rnac-
+cording to the rules of a game played on Rn, but with an additional constraint
+that the centers of all the balls lie in K. Since the first appearance of this
+approach in [12], where it was used to show that sufficiently regular fr actals
+meet with a countable intersection of non-singular affine images of th e set of
+badly approximable vectors in Rn, it has been utilized in [13, 8], and most
+recently in [1], of which the present paper is a sequel and a generaliz ation.
+3.Absolutely decaying measures
+In this section we describe in detail the class of absolutely decaying m ea-
+sures and discuss other related properties and their applications. Following a
+terminology introduced in [16, 25], say that a locally finite Borel meas ureµ
+onRnis (C,γ)-absolutely decaying if there exists ρ0>0 such that
+µ/parenleftbig
+B(x,ρ)∩L(ε)/parenrightbig
+< C(ε/ρ)γµ/parenleftbig
+B(x,ρ)/parenrightbig
+for any affine hyperplane L ⊂Rn
+and any x∈suppµ,0< ρ < ρ 0, ε >0.(3.1)
+HereB(x,ρ) stands for the closed Euclidean ball in Rnof radius ρcentered
+atx, andL(ε)def={x∈Rn:d(x,L)≤ε}is the closed ε-neighborhood of L.
+We say that µisabsolutely decaying if it is (C,γ)-absolutely decaying for
+someC,γ >0. (This terminology differs slightly from the one introduced in6 BRODERICK, FISHMAN, KLEINBOCK
+[16], where a less uniform version was considered.) If µis (C,γ)-absolutely
+decaying, we will denote by ρC,γ(µ) the supremum of ρ0for which (3.1) holds.
+Another property, which often comes in a package with absolute de cay, is
+the so-called doubling, or Federer, condition. One says that µisD-Federer if
+there exists ρ0>0 such that
+µ/parenleftbig
+B(x,2ρ)/parenrightbig
+< Dµ/parenleftbig
+B(x,ρ)/parenrightbig
+∀x∈suppµ,∀0< ρ < ρ 0,(3.2)
+andFedererif it isD-Federer for some D >0. Measures which are both
+absolutely decaying and Federer are called absolutely friendly , a term coined
+in [25].
+Many examples of absolutely friendly measures can be found in [16, 17 , 32,
+29]. The Federer condition is very well studied; it obviously holds when µ
+satisfies a power law , i.e. there exist positive δ,c1,c2,ρ0such that
+c1ρδ≤µ/parenleftbig
+B(x,ρ)/parenrightbig
+≤c2ρδ∀x∈suppµ,∀0< ρ < ρ 0.(3.3)
+Such measures are often referred to as δ-Ahlfors regular . However it is not
+hard to construct absolutely friendly measures not satisfying a po wer law,
+see [17] for an example. Also, when n= 1 the Federer property is implied
+by the absolute decay, which in its turn is implied by a power law (see [1]
+for a thorough discussion of equivalent definitions of absolute frien dliness in
+the one-dimensional case). However these implications fail to hold in higher
+dimensions. In particular, the volume measures on smooth k-dimensional sub-
+manifolds of Rnobviously are k-Ahlfors regular but not absolutely decaying
+unlessk=n.
+The goal of the current work, as well as in several earlier papers [1 7, 19, 12,
+13, 8], is to use measures in order to construct points in their suppo rts with
+prescribed(dynamical orDiophantine)properties. Ourattention willtherefore
+be focused on closed subsets KofRnwhich support absolutely decaying and
+absolutely friendly measures. For example, this is the case when K=Rn, or
+whenKis the limit set of an irreducible family of contracting self-similar [16]
+orself-conformal [32] transformationsof Rnsatisfying theOpenSet Condition.
+More examples can be found in [17, 29]. Note that the paper [2] estab lished
+full Hausdorff dimension of ˜E(M,Z)∩KforM,Zas in (1.5) and under an
+assumption that K⊂Rnsupports an absolutely decaying, δ-Ahlfors regular
+measure with δ > n−1. It is not hard to show, using an elementary covering
+argument, that (3.3) with δ > n−1 implies (3.1) with γ=δ−n+1. Hence
+the sets considered in [2] support absolutely decaying measures.
+Recall that the lower pointwise dimension of a measure µatx∈suppµis
+defined as
+dµ(x)def= liminf
+ρ→0logµ(B(x,ρ))
+logρ.
+For an open Uwithµ(U)>0 let
+dµ(U)def= inf
+x∈suppµ∩Udµ(x). (3.4)
+It is well known, see e.g. [9, Proposition 4.9], that (3.4) constitutes a lower
+bound for the Hausdorff dimension of supp µ∩U(where this bound is sharp
+whenµsatisfies a power law). It is also easy to see that dµ(x)≥γfor everySCHMIDT’S GAME, FRACTALS, AND ORBITS OF TORAL ENDOMORPHISM S 7
+x∈suppµwhenever µis (C,γ)-absolutely decaying: indeed, take ρ < ρ0<
+ρC,γ(µ) andx∈suppµ; then, using (3.1) and noting that B(x,ρ)⊂ L(ρ)for
+some hyperplane L, one has
+µ/parenleftbig
+B(x,ρ)/parenrightbig
+< C/parenleftbiggρ
+ρ0/parenrightbiggγ
+µ/parenleftbig
+B(x,ρ0)/parenrightbig
+.
+Thus, for ρ <1,
+logµ/parenleftbig
+B(x,ρ)/parenrightbig
+logρ≥γ+logC−γlogρ0+logµ/parenleftbig
+B(x,ρ0)/parenrightbig
+logρ,
+and the claim follows.
+The following proposition [18, Proposition 5.1] makes it possible to estim ate
+the Hausdorff dimension of sets winning on supports of Federer mea sures:
+Proposition 3.1. LetKbe the support of a Federer measure µonRn, and
+letSbe winning on K. Then for any open U⊂Rnwithµ(U)>0one has
+dim(S∩K∩U)≥dµ(U).
+In particular, if in addition µis (C,γ)-absolutely decaying, in the above
+proposition one can replace dµ(U) withγ, and with dim( K) ifµsatisfies a
+power law. Note that this generalizes estimates for the Hausdorff d imension of
+winning sets due to Schmidt [26] for µbeing Lebesgue on Rn, and to Fishman
+[12,§5] for measures satisfying a power law.
+The next lemma exhibits a crucial feature of sets supporting absolu tely
+decaying measures, namely the fact that while playing Schmidt’s game on such
+aset, Alicecandistanceherself fromhyperplanes ‘efficiently’. Thisob servation
+is the cornerstone of the proof of our main theorem. The argumen t has been
+adapted from the one in [22], where the case K=Rnwas proved with α=1
+2
+(see§5.1 for more detail), and then refined using an observation from [7].
+Lemma 3.2. For every C,γ >0and
+α <1
+2C1/γ+1(3.5)
+there exists ε=ε(C,γ,α)∈(0,1)such that if Kis the support of a (C,γ)-
+absolutely decaying measure µonRn,0< ρ < ρ C,γ(µ),x1∈K,N∈N, and
+L1,...,LNare hyperplanes in Rn, there exists x2∈Kwith
+B(x2,αρ)⊂B(x1,ρ) (3.6)
+and
+d(B(x2,αρ),Li)> αρfor at least ⌈εN⌉of the hyperplanes Li.(3.7)
+Proof.LetAi=B/parenleftbig
+x1,(1−α)ρ/parenrightbig
+/integerdivideL(2αρ)
+i. By (3.1) and (3.5), for each 1 ≤i≤
+N,
+µ(Ai)
+µ/parenleftbig
+B(x1,(1−α)ρ)/parenrightbig>1−C/parenleftbigg2α
+1−α/parenrightbiggγ
+def=ε >0.8 BRODERICK, FISHMAN, KLEINBOCK
+We claim there exist j1,...,j k, wherek≥ ⌈εN⌉, such that K∩/intersectiontextk
+i=1Aji/n⌉}ationslash=∅.
+To see this, let f(x) =/summationtextN
+i=1χAi(x). Then/integraldisplay
+B(x1,(1−α)ρ)f(x)dµ(x)≥Nεµ/parenleftbig
+B(x1,(1−α)ρ)/parenrightbig
+,
+so clearly there exists some x2∈Kwithf(x2)≥Nǫ. Sincef(x2)∈Z, there
+must exist j1,...,j kas above. Hence, x2satisfies (3.6) and (3.7). /square
+We will also need the following corollary of the above lemma:
+Corollary 3.3. LetKbe the support of a (C,γ)-absolutely decaying measure
+onRn, letαbe as in(3.5), letS⊂Rnbeα-winning on K, and let S′⊂Sbe
+a countable union of hyperplanes. Then S/integerdivideS′is alsoα-winning on K.
+Proof.In view of the countable intersection property, it suffices to show t hat
+for any hyperplane L ⊂Rn, the set Rn/integerdivideLis (α,β)-winning on Kfor anyβ.
+Letµbe a (C,γ)-absolutely decaying measure with K= suppµ. We let Alice
+play arbitrarily until the radius of a ball chosen by Bob is less than ρC,γ(µ).
+Then apply Lemma 3.2 with N= 1 and L1=L, which yields a ball disjoint
+fromL. Afterwards she can keep playing arbitrarily, winning the game. /square
+4.Proofs
+Let us now state a more precise version of Theorem 1.3:
+Theorem 4.1. LetKbe the support of a (C,γ)-absolutely decaying measure
+onRn, and let αbe as in (3.5). Then for any uniformly discrete sequence Z
+of subsets of Rmand any lacunary sequence Mofm×nreal matrices, the set
+˜E(M,Z)isα-winning on K.
+Proof.WriteM= (Mk), lettkdef=/⌊ar⌈⌊lMk/⌊ar⌈⌊lopand letvkbe a unit vector satisfying
+/⌊ar⌈⌊lMkvk/⌊ar⌈⌊l=tk.
+Takeδ >0 such that Zisδ-uniformly discrete, and let
+inf
+ktk+1
+tk=Q >1. (4.1)
+Now pick an arbitrary 0 < β <1, takeεas in Lemma 3.2, and choose Nlarge
+enough that
+(αβ)−r≤QN, wherer=⌊log1
+1−εN⌋+1. (4.2)
+We will denote by M−1
+k(Z) the preimage of a set Z⊂RnunderMk. Notice
+thatforeach k∈N,M−1
+k(Zk)iscontainedinacountableunionof hyperplanes,
+so applying Corollary 3.3 a finite number of times, we may assume that t1≥1.
+By playing arbitrary moves if needed, we may assume without loss of g en-
+erality that B(ω1) has radius
+ρ <min/parenleftbiggαβδ
+4,ρC,γ/parenrightbigg
+. (4.3)
+Now let
+c= min/parenleftbigg
+ρ(αβ)2r−1,δ
+4/parenrightbigg
+. (4.4)SCHMIDT’S GAME, FRACTALS, AND ORBITS OF TORAL ENDOMORPHISM S 9
+WewilldescribeastrategyforAlicetoplaythe( α,β)-gameon Kandtoensure
+that for all j∈N, for allx∈B(ω′
+r(j+1)) and for all kwith 1≤tk<(αβ)−rj,
+one has d(Mkx,Zk)> c. This will imply that/intersectiontext
+kB(ω′
+k)∈˜E(M,Z)∩K,
+finishing the proof.
+To satisfy the above goal, Alice can choose ω′
+iarbitrarily for i < r. Now fix
+j∈N. By (4.1) and (4.2), there are at most Nindicesk∈Nfor which
+(αβ)−r(j−1)≤tk<(αβ)−rj. (4.5)
+Letkbe one of these indices. For any x∈Rn,/⌊ar⌈⌊lx/⌊ar⌈⌊l ≥1
+tk/⌊ar⌈⌊lMk(x)/⌊ar⌈⌊l. Thus, if
+y1,y2are two different points in Zk, then by (4.3) and (4.5)
+d/parenleftBig
+M−1
+k/parenleftbig
+B(y1,c)/parenrightbig
+,M−1
+k/parenleftbig
+B(y2,c)/parenrightbig/parenrightBig
+≥δ−2c
+tk≥δ
+2tk>δ
+2(αβ)rj≥2ρ(αβ)rj−1;
+(4.6)
+therefore B(ωrj) intersects with at most one set of the form M−1
+k/parenleftbig
+B(y,c)/parenrightbig
+,
+wherey∈Zk. Hence, for each ksatisfying (4.5),
+B(ωrj)∩M−1
+k(Z(c)
+k)⊂M−1
+k/parenleftbig
+B(y,c)/parenrightbig
+for some y∈Zk.(4.7)
+We will now show that the preimage of such a ball is contained in a ‘small
+enough’ neighborhood of some hyperplane, so that we can apply th e decay
+condition. Toward this end, let V⊂Rmbe the hyperplane perpendicular to
+Mkvkand passing through 0. Then
+Wdef=M−1
+k(V)
+is a hyperplane in Rnpassing through 0.
+Ifx/n⌉}ationslash∈W(c/tk), thenx=w+ηvkfor some η > c/t kandw∈W, thus
+/⌊ar⌈⌊lMkx/⌊ar⌈⌊l=/⌊ar⌈⌊lMkw+Mkηvk/⌊ar⌈⌊l ≥η/⌊ar⌈⌊lMkvk/⌊ar⌈⌊l=tkη > c.
+Hence,M−1
+k/parenleftbig
+B(0,c)/parenrightbig
+⊂W(c/tk), which clearly implies that for each y∈Zk,
+M−1
+k/parenleftbig
+B(y,c)/parenrightbig
+⊂ L(c/tk)for some hyperplane L ⊂Rn. By (4.4) and (4.5),
+c
+tk≤(αβ)r(j+1)−1ρdef=ζ.
+Therefore, by (4.7),
+/uniondisplay
+tksatisfies (4.5)B(ωrj)∩M−1
+k/parenleftbig
+Z(c)
+k/parenrightbig
+⊂N/uniondisplay
+i=1L(ζ)
+i, (4.8)
+whereLiare hyperplanes. Noticing that by (4.2) (1 −ε)rN <1, Alice can
+utilize Lemma 3.2 rtimes to distance herself by ζfromeach of the hyperplanes
+Liafterrturns. Thus for ksatisfying (4.5), it holds that
+B(ω′
+r(j+1))∩M−1
+k/parenleftbig
+Z(c)
+k/parenrightbig
+=∅.
+We conclude that d(Mkx,Zk)≥cfor anyx∈B(ω′
+r(j+1)), which implies the
+desired statement. /square
+Proof of Theorem 1.2. Recall that we are given M∈GLn(R)∩Mn(Z). If all
+the eigenvalues of Mhave modulus less than or equal to 1, then obviously
+every eigenvalue of Mmust have modulus 1. By a theorem of Kronecker [20],10 BRODERICK, FISHMAN, KLEINBOCK
+they must be roots of unity, so there exists an N∈Nsuch that the only
+eigenvalue of MNis 1. Let J=L−1MNLbe the Jordan normal form of MN,
+and letvi=Lei,i= 1,...,n, be the Jordan basis for MN. Then, since MN
+is an integer matrix, we have vi∈Qnfor each 1 ≤i≤n. Hence, letting
+V= span(v1,...,vn−1),V+Znis a union of positively separated parallel
+hyperplanes. Since Jfixes the last coordinate of any vector, if a1,...,a n∈R,
+then
+MN/parenleftBiggn/summationdisplay
+i=1aivi/parenrightBigg
+∈anvn+V.
+Therefore, for x,y∈Rnwithx−y/n⌉}ationslash∈V+Znand any k∈None has
+d(MNkx,y+Zn)≥c0d(x−y,V+Zn)>0,
+wherec0is a positive constant depending only on v1,...,vn. Hence, for any
+y∈Tn,
+˜E(MN,y)⊃Rn/integerdivide(π−1(y)+V) =Rn/integerdivide(y+V+Zn),
+whereyis an arbitrary vector in π−1(y). Thus ˜E(MN,y) isα-winning on K
+by Corollary 3.3. Hence ˜E(MN,z) isα-winning on Kwhenever z∈f−i
+M(y),
+where 0≤i < N. Thus the intersection
+˜E(M,y) =N−1/intersectiondisplay
+i=0/intersectiondisplay
+z∈f−i
+M(y)˜E/parenleftbig
+MN,f−i
+M(y)/parenrightbig
+is alsoα-winning on K.
+In the case where at least one of the eigenvalues is of absolute value strictly
+greater than 1, we will show that the sequence ( /⌊ar⌈⌊lMk/⌊ar⌈⌊lop) is a finite union of
+lacunary sequences, which will clearly imply that ˜E/parenleftbig
+(Mk),Z/parenrightbig
+isα-winning on
+K. LetJ=L−1MLbe the Jordan normal form of M. Since the operator
+norm ofMas a real transformation is equal to its operator norm as a complex
+transformation and
+/⌊ar⌈⌊lJk/⌊ar⌈⌊lop≤ /⌊ar⌈⌊lL/⌊ar⌈⌊lop/⌊ar⌈⌊lL−1/⌊ar⌈⌊lop/⌊ar⌈⌊lMk/⌊ar⌈⌊lopand/⌊ar⌈⌊lMk/⌊ar⌈⌊lop≤ /⌊ar⌈⌊lL/⌊ar⌈⌊lop/⌊ar⌈⌊lL−1/⌊ar⌈⌊lop/⌊ar⌈⌊lJk/⌊ar⌈⌊lop,
+lettingc=/⌊ar⌈⌊lL/⌊ar⌈⌊lop/⌊ar⌈⌊lL−1/⌊ar⌈⌊lop, wehave1
+c/⌊ar⌈⌊lMk/⌊ar⌈⌊lop≤ /⌊ar⌈⌊lJk/⌊ar⌈⌊lop≤c/⌊ar⌈⌊lMk/⌊ar⌈⌊lopforallk∈N.
+Hence, if ( /⌊ar⌈⌊lJk/⌊ar⌈⌊lop) is eventually lacunary, then there exists ℓ,N∈NandQ >1
+such that, for all k≥N,
+/⌊ar⌈⌊lMk+ℓ/⌊ar⌈⌊lop
+/⌊ar⌈⌊lMk/⌊ar⌈⌊lop≥1
+c2/⌊ar⌈⌊lJk+ℓ/⌊ar⌈⌊lop
+/⌊ar⌈⌊lJk/⌊ar⌈⌊lop≥Q.
+Thus it will suffice to show that ( /⌊ar⌈⌊lJk/⌊ar⌈⌊lop) is eventually lacunary.
+LetBbe anm×mblock of Jassociated to an eigenvalue λand write
+Bk=/parenleftbig
+bij(k)/parenrightbig
+. Direct computation shows that, for 0 ≤j−i≤k,
+bij(k) =/parenleftbiggk
+j−i/parenrightbigg
+λk−(j−i), (4.9)SCHMIDT’S GAME, FRACTALS, AND ORBITS OF TORAL ENDOMORPHISM S 11
+andbij(k) = 0 otherwise. Since |bij(k)|=o(|b1m(k)|) as functions of kfor all
+(i,j)/n⌉}ationslash= (1,m),
+lim
+k→∞/⌊ar⌈⌊lBk/⌊ar⌈⌊lop
+|b1m(k)|= 1. (4.10)
+Hence,
+lim
+k→∞/⌊ar⌈⌊lBk+1/⌊ar⌈⌊lop
+/⌊ar⌈⌊lBk/⌊ar⌈⌊lop=|λ|, (4.11)
+so clearly if |λ|>1 then (/⌊ar⌈⌊lBk/⌊ar⌈⌊lop) is eventually lacunary. Write J=B1⊕···⊕
+Bs, wheres∈NandBiare the Jordan blocks, with associated eigenvalues
+λi. Letλmax= max|λi|, and let Bmaxbe a block with associated eigenvalue
+having absolute value λmaxand of maximal dimension among such blocks. By
+(4.9) and (4.10), for any i,
+lim
+k→∞/⌊ar⌈⌊lBk
+max/⌊ar⌈⌊lop
+/⌊ar⌈⌊lBk
+i/⌊ar⌈⌊lop≥1.
+Hence, by (4.11),
+lim
+k→∞/⌊ar⌈⌊lJk+1/⌊ar⌈⌊lop
+/⌊ar⌈⌊lJk/⌊ar⌈⌊lop= lim
+k→∞/⌊ar⌈⌊lBk+1
+max/⌊ar⌈⌊lop
+/⌊ar⌈⌊lBkmax/⌊ar⌈⌊lop=λmax.
+Since by assumption M(and therefore J) has an eigenvalue with absolute
+value greater than 1, ( /⌊ar⌈⌊lJk/⌊ar⌈⌊lop) is eventually lacunary. /square
+In the remaining part of this section we apply Theorem 1.3 to badly ap-
+proximable systems of affine forms.
+Proof of Corollary 1.4. Recall that we need to fix A∈Mn×m(R) and study
+the set
+BadA(n,m) =/braceleftbigg
+x∈Rn: inf
+q∈Zm/integerdivide{0}/⌊ar⌈⌊lq/⌊ar⌈⌊lm/nd(Aq−x,Zn)>0/bracerightbigg
+.
+First observe that the above set is easy to understand in the ‘ratio nal’ case
+whenthereexists anonzero u∈Znsuchthat ATu∈Zm(orequivalently, when
+the rank of the group ATZn+Zmis strictly smaller than m+n). In this case,
+by a theorem of Kronecker, see [4, Ch. III, Theorem IV], inf q∈Zmd(Aq−x,Zn)
+is positive if and only if the value of u·xis not an integer. Therefore
+BadA(n,m)⊃ {x∈Rn:u·x/∈Z}.
+Since the right-hand side is the complement of a countable union of hy per-
+planes, in view of Corollary 3.3 BadA(n,m) isα-winning on Kwhenever K
+is absolutely decaying and αis as in Theorem 1.3.
+In the more interesting ‘irrational’ case when rank( ATZn+Zm) =m+n,
+one can utilize the theory of best approximations to Aas developed by Cassels
+[4, Ch. III] and recently made more precise by Bugeaud and Lauren t [3]. In [2,
+§§5–6], using results from [3], it is shown that if rank( ATZn+Zm) =m+n,
+then there exists a lacunary sequence of vectors yk∈Zn(a subsequence of the
+sequence of best approximations to A) such that whenever x∈Rnsatisfies
+inf
+k∈Nd(yk·x,Z)>0,12 BRODERICK, FISHMAN, KLEINBOCK
+it follows that x∈BadA(n,m). In other words,
+˜E(Y,Z)⊂BadA(n,m),
+whereYdef= (yk) andZ= (Zk) withZk=Zfor each k. (See also [22, §2] for
+an alternative exposition.) Therefore in this case BadA(n,m) isα-winning on
+Kby Theorem 1.3. /square
+5.Concluding remarks
+5.1.Playing on Rnwithα= 1/2.As was mentioned before, the special
+caseK=Rnof our main theorem is essentially contained in [22]. In fact,
+arguing as in §4 and using [22, Lemma 2] (the analogue of our Lemma 3.2)
+and [22, Lemma 3] (Schmidt’s escaping lemma, cf. [28, Ch. 3, Lemma 1B ]),
+one can show that for ZandMas in Theorem 1.3 and any α,β >0 with
+1 +αβ−2α >0, the sets ˜E(M,Z) are (α,β)-winning. In particular, this
+shows that one can take α(Rn) = 1/2 in Theorems 1.2 and 1.3.
+5.2.Strong winning. Recently in [10, 11] and independently in [21] a mod-
+ification of Schmidt’s game has been introduced, where condition (2.1 ) is re-
+placed by
+ρ′
+k≥αρkandρk+1≥βρ′
+k. (5.1)
+Following [21], a subset Sof a metric space Xis said to be ( α,β)-strong win-
+ningif Alice has a winning strategy in the game defined by (5.1). Analogously ,
+one defines α-strong winning and strong winning sets. It is not hard to verify
+thatstrongwinning implies winning (see[11]foraproof), andthatac ountable
+intersection of α-strong winning sets is α-strong winning. Furthermore, this
+class has stronger invariance properties, e.g. it is proved in [21] tha t strong
+winning subsets of Rnare preserved by quasisymmetric homeomorphisms.
+It is not hard to modify the proofs given above to show that in Theor em 1.3
+(and therefore in all its corollaries), α-winning may be replaced by α-strong
+winning. This is done by adding ‘dummy moves’ in order to accommodate
+the possibly slower decrease in radii of the chosen balls. Details will ap pear
+elsewhere.
+References
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+fractals, and numbers normal to no base , Math. Res Letters, 17(2010), no. 2, 309–
+323.
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+Dynam. Syst. 8(1988), 523–529.
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+Acad. Sci. Hungar. 36(1980), 237–241.
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+and Schmidt games , Preprint, arXiv:0912.2445 .SCHMIDT’S GAME, FRACTALS, AND ORBITS OF TORAL ENDOMORPHISM S 13
+[8] R. Esdahl-Schou and S. Kristensen, On badly approximable complex numbers ,
+Glasg. Math. J., to appear.
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+Wiley & Sons, Inc., Hoboken, NJ, 2003.
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+Preprint, arXiv:0904.4365v1 .
+[11] D.F¨ arm,T.PerssonandJ.Schmeling, DimensionofCountableIntersectionsofSome
+Sets Arising in Expansions in Non-Integer Bases , Fundamenta Math., to appear.
+[12] L. Fishman, Schmidt’s game on fractals , Israel J. Math. 171(2009), no. 1, 77–92.
+[13] ,Schmidt’s game, badly approximablematrices and fractals , J. Number The-
+ory129(2009), no. 9, 2133–2153.
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+Dynam. Systems 18(1998), 373–396.
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+no. 1, 83–102.
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+approximation , Selecta Math. 10(2004), 479–523.
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+149(2005), 137–170.
+[18] ,Modified Schmidt games and Diophantine approximation with weights , Ad-
+vances in Math. 223(2010), no. 4, 1276–1298.
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+approximable sets , Advances in Math. 203(2006), 132–169.
+[20] L. Kronecker, Zwei S¨ atse ¨ uber Gleichungen mit ganzzahligen Coefcienten , J. Reine
+Angew. Math. 53(1857), 173–175; see also Werke, Vol. 1, 103–108, Chelsea Pub-
+lishing Co., New York, 1968.
+[21] C. McMullen, Winning sets, quasiconformal maps and Diophantine approximation ,
+Geom. Funct. Anal., to appear.
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+Preprint, arXiv:0812.3998v2 .
+[23] A.D. Pollington, On nowhere dense ∂-sets, in: Groupe de travail d’analyse ul-
+tramtrique (1982/83), No. 2, Exp. No. 22, 2 pp., Inst. Henri Poin car´ e, Paris, 1984.
+[24] ,On the density of sequence {ηkξ}, Illinois J. Math. 23(1979), no. 4, 511–
+515.
+[25] A.D. Pollington and S.L. Velani, Metric Diophantine approximation and ‘absolutely
+friendly’ measures , Selecta Math. 11(2005) 297–307.
+[26] W.M. Schmidt, On badly approximable numbers and certain games , Trans. A.M.S.
+123(1966), 27–50.
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+139–154.
+[28] ,Diophantine approximation , Lecture Notes in Mathematics, vol. 785,
+Springer-Verlag, Berlin, 1980.
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+Math. Proc. Cambridge Phil. Soc. 140(2006), 297–304.
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+525–543.
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+Theory110(2005), 219–235.
+Department of Mathematics, Brandeis University, Waltham M A 02454-9110
+E-mail address :ryanb@brandeis.edu, lfishman@brandeis.edu, kleinboc@b randeis.edu
\ No newline at end of file
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+arXiv:1001.0319v1  [math.NA]  2 Jan 2010Efficient PML for the wave equation
+Marcus J. Grote1, and Imbo Sim2∗
+1Department of Mathematics, University of Basel, CH-4051 Ba sel, Switzerland
+2Institut d’Analyse et Calcul Scientifique, Ecole Polytechn ique F´ ed´ erale de Lausanne,
+CH-1015 Lausanne, Switzerland
+Abstract. In the last decade, the perfectly matched layer (PML) approa ch has proved
+a flexible and accurate method for the simulation of waves in u nbounded media.
+Most PML formulations, however, usually require wave equat ions stated in their stan-
+dard second-order form to be reformulated as first-order sys tems, thereby introducing
+many additional unknowns. To circumvent this cumbersome an d somewhat expen-
+sive step, we instead propose a simple PML formulation direc tly for the wave equation
+in its second-order form. Inside the absorbing layer, our fo rmulation requires only two
+auxiliary variables in two space dimensions and four auxili ary variables in three space
+dimensions; hence it is cheap to implement. Since our formul ation requires no higher
+derivatives, it is also easily coupled with standard finite d ifference or finite element
+methods. Strong stability is proved while numerical exampl es in two and three space
+dimensions illustrate the accuracy and long time stability of our PML formulation.
+AMS subject classifications : 35L05, 35L20, 65M06, 65M12
+Key words : PML, wave equation, second-order
+1 Introduction
+The accurate and reliable simulation of wave propagations i n unbounded media is of fun-
+damental importance in a wide range of applications. The per fectly matched layer (PML)
+approach [6] has proved a flexible and accurate method for the simulation of waves. It
+consists in surrounding the computational domain by an abso rbing layer, which gener-
+ates no reflections at its interface with the computational d omain; hence, it is perfectly
+matched. Inside the absorbing layer a damping term is added t o the wave equation,
+which acts only in the direction perpendicular to the layer. This approach is analogous to
+the physical treatment of the walls of an anechoic chamber an d provides an alternative
+to absorbing or nonrelfecting boundary conditions [11, 12, 14–16].
+The initial PML formulation of B´ erenger [6] was based on spli tting the electromagnetic
+∗Corresponding author. Email addresses: Marcus.Grote@unibas.ch (Marcus J. Grote), Imbo.Sim@epfl.ch
+(Imbo Sim)
+http://www.global-sci.com/ Global Science Preprint2
+fields into two parts, the first containing the tangential deriv atives and the second con-
+taining the normal derivatives; damping was then enforced o nly upon the normal com-
+ponent. Later Abarbanel and Gottlieb [1] showed that B´ eren ger’s approach was only
+weakly well-posed due to the unphysical splitting of the field variables. Several strongly
+well-posed approaches have been suggested since, some of wh ich were shown to be lin-
+early equivalent [2, 20].
+The PML approach has proved very successful in practice, beca use of its simplicity, ver-
+satility, and robust treatment of corners. Once discretize d and truncated at a finite thick-
+ness, the layer is no longer perfectly absorbing and the opti mal damping parameters need
+to be determined via numerical experiments. Stability prop erties of the PML approach
+has been analyzed in several works, such as in [1, 2, 7, 9] amon g others.
+The best implementation in the time domain is still under deb ate. Most PML formula-
+tions require wave equations stated in their standard secon d-order form to be reformu-
+lated as first-order hyperbolic systems, thereby introducin g many additional unknowns.
+Here we propose instead a simple PML formulation directly for the second-order wave
+equation both in two and in three space dimensions. Our formu lation also requires
+fewer auxiliary variables than previous formulations for t he second-order wave equa-
+tion – see [3, 5, 19], for instance.
+Our paper is organized as follows. In Section 2 we derive a PML f ormulation for the
+wave equation in its standard second-order form. By judicio usly choosing the auxiliary
+variables in the Laplace transformed domain, the resulting PML modified equations re-
+quire only two auxiliary variables in two dimensions and fou r auxiliary variables in three
+dimensions inside the absorbing layer. Next, in Section 3 we prove stability of our PML
+formulation by using standard theory from [18]. The finite dif ference discretization of the
+PML modified wave equation is shown in Section 4. In Section 5, ou r numerical results
+both in two and three space dimensions demonstrate the accur acy and long time stability
+of the PML formulation.
+2 PML formulation
+We consider a time dependent wave field upropagating through unbounded three di-
+mensional space and assume that all sources and initial dist urbances are confined to the
+rectangular domain Ω=[−a1,a1]×[−a2,a2]×[−a3,a3],a1,a2,a3>0. Outside Ω, we further
+assume the speed of propagation c>0 to be constant; hence, all waves are purely outgo-
+ing in the unbounded exterior R3\Ω. Inside Ω, the wave field u(x1,x2,x3,t)satisfies
+utt−∇·/parenleftbig
+c2∇u/parenrightbig=f t>0, (2.1)
+u=u0 t=0, (2.2)
+ut=v0 t=0. (2.3)
+We wish to truncate the unbounded exterior and thereby restr ict the computation to the
+finite computational domain Ω. In doing so, we need to ensure that all waves propagat-3
+ing outward leave Ωwithout spurious reflection. Thus we shall surround Ωby a per-
+fectly matched layer (PML) of thickness Li,i=1,2,3, in each coordinate which is designed
+to absorb the waves exiting Ω. Inside the absorbing layer, uthen satisfies a modified
+wave equation whose solutions decay exponentially fast wit h distance from the compu-
+tational domain.
+Following [1, 2], we let ˆudenote the Laplace transform of u, defined as
+ˆu=ˆu(x,s)=/integraldisplay∞
+0estu(x,t)dt, s∈C. (2.4)
+Outside Ω,ˆuthen satisfies the Helmholtz equation,
+s2ˆu=∂
+∂x1/parenleftbigg
+c2∂ˆu
+∂x1/parenrightbigg
++∂
+∂x2/parenleftbigg
+c2∂ˆu
+∂x2/parenrightbigg
++∂
+∂x3/parenleftbigg
+c2∂ˆu
+∂x3/parenrightbigg
+. (2.5)
+Next, we introduce the coordinate transformation
+xi/mapsto→˜xi:=xi+1
+s/integraldisplayxi
+0ζi(x)dx, i=1,2,3, (2.6)
+where the damping profile ζiis positive inside the absorbing layer, |xi|>ai,i=1,2,3, but
+vanishes inside Ω. If we now require ˆuto satisfy the modified Helmholtz equation in
+those stretched coordinates,
+s2ˆu=∂
+∂˜x1/parenleftbigg
+c2∂ˆu
+∂˜x1/parenrightbigg
++∂
+∂˜x2/parenleftbigg
+c2∂ˆu
+∂˜x2/parenrightbigg
++∂
+∂˜x3/parenleftbigg
+c2∂ˆu
+∂˜x3/parenrightbigg
+, (2.7)
+it is well-known that uwill remain unaltered inside Ω, but decay exponentially fast inside
+the layer; hence the absorbing layer will be perfectly match ed. In fact, the (unsplit) PML
+modified Helmholtz equation (2.7) in the Laplace transformed domain is standard [1, 2].
+The difficulty lies in transforming (2.7) back to the time doma in, without introducing
+high order derivatives or too many auxiliary variables.
+From(2.6), we observe that partial differentiation with respect to ˜xiis related to partial
+differentiation with respect to the physical coordinate, xi, through
+∂
+∂˜xi=s
+s+ζi∂
+∂xi. (2.8)
+We now let γi=γi(ζi; s),i=1,2,3 denote
+γi:=1+ζi
+s. (2.9)
+Then, by replacing partial derivatives according to (2.8) a nd multiplying the resulting
+expression by γ1γ2γ3, we rewrite (2.7) in physical coordinates as
+s2γ1γ2γ3ˆu=∂
+∂x1/parenleftbigg
+c2γ2γ3
+γ1∂ˆu
+∂x1/parenrightbigg
++∂
+∂x2/parenleftbigg
+c2γ3γ1
+γ2∂ˆu
+∂x2/parenrightbigg
++∂
+∂x3/parenleftbigg
+c2γ1γ2
+γ3∂ˆu
+∂x3/parenrightbigg
+. (2.10)4
+From (2.9) we derive after some algebra the following identi ties:
+γ2γ3
+sγ1=1+(ζ2+ζ3−ζ1)s+ζ2ζ3
+(s+ζ1)s,
+γ3γ1
+sγ2=1+(ζ3+ζ1−ζ2)s+ζ3ζ1
+(s+ζ2)s,
+γ1γ2
+sγ3=1+(ζ1+ζ2−ζ3)s+ζ1ζ2
+(s+ζ3)s.(2.11)
+By using (2.11) in (2.10) we find
+/parenleftbigg
+s2+s(ζ1+ζ2+ζ3)+(ζ1ζ2+ζ2ζ3+ζ3ζ1)+ζ1ζ2ζ3
+s/parenrightbigg
+ˆu
+=∂
+∂x1/parenleftbigg
+c2∂ˆu
+∂x1/parenrightbigg
++∂
+∂x2/parenleftbigg
+c2∂ˆu
+∂x2/parenrightbigg
++∂
+∂x3/parenleftbigg
+c2∂ˆu
+∂x3/parenrightbigg
++∂
+∂x1/parenleftbigg
+c2/parenleftBig(ζ2+ζ3−ζ1)s+ζ2ζ3
+(s+ζ1)s/parenrightBig∂ˆu
+∂x1/parenrightbigg
++∂
+∂x2/parenleftbigg
+c2/parenleftBig(ζ3+ζ1−ζ2)s+ζ3ζ1
+(s+ζ2)s/parenrightBig∂ˆu
+∂x2/parenrightbigg
++∂
+∂x3/parenleftbigg
+c2/parenleftBig(ζ1+ζ2−ζ3)s+ζ1ζ2
+(s+ζ3)s/parenrightBig∂ˆu
+∂x3/parenrightbigg
+.(2.12)
+Next, we introduce the auxiliary functions ψand φ=(φ1,φ2,φ3)⊤,
+/hatwideψ=1
+sˆu,
+/hatwideφ1=c2/parenleftbiggζ2+ζ3−ζ1
+s+ζ1+ζ2ζ3
+(s+ζ1)s/parenrightbigg∂ˆu
+∂x1,
+/hatwideφ2=c2/parenleftbiggζ3+ζ1−ζ2
+s+ζ2+ζ3ζ1
+(s+ζ2)s/parenrightbigg∂ˆu
+∂x2,
+/hatwideφ3=c2/parenleftbiggζ1+ζ2−ζ3
+s+ζ3+ζ1ζ2
+(s+ζ3)s/parenrightbigg∂ˆu
+∂x3,
+or equivalently
+s/hatwideψ=ˆu,
+(s+ζ1)/hatwideφ1=c2/parenleftbigg
+(ζ2+ζ3−ζ1)+ζ2ζ3
+s/parenrightbigg∂ˆu
+∂x1,
+(s+ζ2)/hatwideφ2=c2/parenleftbigg
+(ζ3+ζ1−ζ2)+ζ3ζ1
+s/parenrightbigg∂ˆu
+∂x2,
+and(s+ζ3)/hatwideφ3=c2/parenleftbigg
+(ζ1+ζ2−ζ3)+ζ1ζ2
+s/parenrightbigg∂ˆu
+∂x3.5
+Finally, we use the above relations in (2.12) and transform t he resulting equations back to
+the time domain, which yields the PML modified wave equation
+utt+(ζ1+ζ2+ζ3)ut+(ζ1ζ2+ζ2ζ3+ζ3ζ1)u=∇·/parenleftbig
+c2∇u/parenrightbig+∇·φ−ζ1ζ2ζ3ψ,
+φt=Γ1φ+c2Γ2∇u+c2Γ3∇ψ,
+ψt=u,(2.13)
+where
+Γ1=
+−ζ1 0 0
+0−ζ2 0
+0 0 −ζ3
+,Γ2=
+ζ2+ζ3−ζ1 0 0
+0 ζ3+ζ1−ζ2 0
+0 0 ζ1+ζ2−ζ3
+
+and Γ3=
+ζ2ζ3 0 0
+0 ζ3ζ1 0
+0 0 ζ1ζ2
+.
+In the interior of Ω, the damping profiles ζi,i=1,2,3 and the auxiliary variables φ,ψvan-
+ish; hence, (2.13) reduces to (2.1) in Ω. Because our PML formulation (2.13) requires only
+four auxiliary scalar variables φ1,φ2,φ3,ψinside the layer and no high order derivatives,
+its implementation is not only straightforward but also che ap to implement.
+In two space dimensions, ζ3and φ3and ψvanish and our PML formulation reduces to
+utt+(ζ1+ζ2)ut+ζ1ζ2u=∇·/parenleftbig
+c2∇u/parenrightbig+∇·φ,
+φt=Γ1φ+c2Γ2∇u,(2.14)
+where
+Γ1=/bracketleftbigg−ζ1 0
+0−ζ2/bracketrightbigg
+,Γ2=/bracketleftbiggζ2−ζ1 0
+0 ζ1−ζ2/bracketrightbigg
+.
+Remarkably only two auxiliary functions are needed here.
+The choice of the damping profiles ζi(x)≥0,i=1,2,3 is arbitrary; it can be constant,
+linear, or quadratic among others. In our computations, we a lways use
+ζi(xi)=
+
+0 for |xi|<ai,i=1,2,3
+¯ζi/parenleftBigg
+|xi−ai|
+Li−sin/parenleftBig2π|xi−ai|
+Li/parenrightBig
+2π/parenrightBigg
+forai≤|xi|≤ai+Li,i=1,2,3.(2.15)
+Because ζi(x)is twice continuously differentiable throughout the inter face at|xi|=ai,
+no special transmission conditions are needed there. The co nstant ¯ζidepends on the
+discretization and the thickness of the layer, which in prac tice is truncated by a homo-
+geneous Dirichlet (or Neumann) boundary condition. Then th e relative reflection, R, is
+given by
+¯ζi=c
+Lilog/parenleftBig1
+R/parenrightBig
+,i=1,2,3. (2.16)
+In Figure 1 we show damping profiles for different values of ¯ζi.6
+00.02 0.04 0.06 0.08 0.1010203040
+xζ(x)
+  
+ζ= 10
+ζ= 25
+ζ= 40
+Figure 1: The damping profile ζi(xi)given by (2.15) is shown for different values of ¯ζi, with c=1and Li=0.1.
+3 Stability
+We now establish the stability and well-posedness of our PML f ormulation, first in two
+and then in three space dimensions, where we assume that the a bsorbing layer extends
+to infinity. Here we follow standard stability theory for hype rbolic systems [18], which
+we briefly recall below.
+Consider a general Cauchy problem,
+Ut=P/parenleftbigg∂
+∂x/parenrightbigg
+U, 0≤t≤T,U∈Rp, (3.1)
+where P(∂x)denotes a linear differential operator, with initial condi tions
+U(x,0)=U0(x),x∈R3. (3.2)
+Following [18], the Cauchy Problem is weakly (resp. strongly) well-posed , if the solution
+U(·,t)satisfies
+/ba∇dblU(·,t)/ba∇dblL2≤Keαt/ba∇dblU(·,0)/ba∇dblHs (3.3)
+with s>0 (resp. s=0). The Cauchy Problem is weakly (resp. strongly) stable , if the solution
+U(·,t)satisfies
+/ba∇dblU(·,t)/ba∇dblL2≤K(1+t)s/ba∇dblU(·,0)/ba∇dblHs (3.4)
+with s>0 (resp. s=0). A necessary and sufficient condition for weak well-posedness (resp.
+stability) is that all eigenvalues λof the operator P(ik))satisfy
+ℜ{λ(P(ik))}≤C, k∈R, (3.5)
+with C>0 (resp. C=0) independent of k. For strong well-posedness (resp. stability) , the
+corresponding eigenvectors must also be complete.
+By rewriting the PML-modified wave equations (2.13), (2.14) as a first-order hyperbolic
+system and applying the stability theory from [18] delineat ed above, we can prove the
+following two stability results.7
+Theorem 3.1. The Cauchy problem for the PML formulation (2.14) in two space dimen-
+sions is strongly stable for ζ1,ζ2≥0.
+proof )
+For simplicity, we assume that ζ1,ζ2are constant; note, however, that the stability theory
+from [18] extends to smoothly varying coefficients. We introd uce the new variable vto
+rewrite the first equation in (2.14) equivalently as
+ut=−ζ2u+divv,vt=−ζ1v+c2∇u+φ. (3.6)
+By using (3.6), we now rewrite (2.14) as a first order hyperboli c system:
+Ut=AU x+BU y+C, (3.7)
+where
+Ut=(u,φ1,φ2,v1,v2)⊤, (3.8)
+A=
+0 0 0 1 0
+c2(ζ2−ζ1)0 0 0 0
+0 0 0 0 0
+c20 0 0 0
+0 0 0 0 0
+,B=
+0 0 0 0 1
+c2(ζ1−ζ2)0 0 0 0
+0 0 0 0 0
+0 0 0 0 0
+c20 0 0 0
+,
+and C=−
+ζ20 0 0 0
+0ζ10 0 0
+0 0 ζ20 0
+0 0 0 ζ10
+0 0 0 0 ζ1
+.(3.9)
+By using a symbolic algebra program we find that the eigenvalue s of the principal part
+ofP(ik)for (3.7) are
+λ(P(ik))=±ic(k2
+1+k2
+2)1/2. (3.10)
+Thus,
+ℜ{λ(P(ik))}=0, (3.11)
+while the corresponding eigenvectors are also complete for allζ1,ζ2≥0. Therefore, since
+Cis a diagonal matrix with negative entries for ζ1,ζ2≥0, we conclude that (2.14) is
+strongly stable.
+Theorem 3.2. The Cauchy problem for the PML formulation (2.13) in three spa ce dimen-
+sions is strongly stable, if at least two ζj=0,j=1,2,3, and weakly stable, otherwise.
+proof )
+We introduce the new variable vto rewrite the first equation in (2.13) as
+ut=−ζ2u+divv−ζ3ψ,vt=−ζ1v+c2∇u+φ. (3.12)8
+By using (3.12), we can rewrite (2.13) as a first order hyperbol ic system:
+Ut=AU x+BU y+CU z+D, (3.13)
+where
+Ut=(u,φ1,φ2,φ3,v1,v2,v3,ψ)⊤, (3.14)
+A=
+0 0 0 0 1 0 0 0
+c2(ζ2+ζ3−ζ1)0 0 0 0 0 0 ζ2ζ3
+0 0 0 0 0 0 0 0
+0 0 0 0 0 0 0 0
+c20 0 0 0 0 0 0
+0 0 0 0 0 0 0 0
+0 0 0 0 0 0 0 0
+0 0 0 0 0 0 0 0
+, (3.15)
+B=
+0 0 0 0 0 1 0 0
+0 0 0 0 0 0 0 0
+c2(ζ3+ζ1−ζ2)0 0 0 0 0 0 ζ3ζ1
+0 0 0 0 0 0 0 0
+0 0 0 0 0 0 0 0
+c20 0 0 0 0 0 0
+0 0 0 0 0 0 0 0
+0 0 0 0 0 0 0 0
+, (3.16)
+and
+C=
+0 0 0 0 0 0 1 0
+0 0 0 0 0 0 0 0
+0 0 0 0 0 0 0 0
+c2(ζ1+ζ2−ζ3)0 0 0 0 0 0 ζ1ζ2
+0 0 0 0 0 0 0 0
+0 0 0 0 0 0 0 0
+c20 0 0 0 0 0 0
+0 0 0 0 0 0 0 0
+. (3.17)
+By using a symbolic algebra program, we find that the eigenvalu esλofP(ik)for (3.13)
+are
+λ(P(ik))=±ic(k2
+1+k2
+2+k2
+3)1/2. (3.18)
+Thus,
+ℜ{λ(P(ik))}=0, (3.19)
+while the corresponding eigenvectors are also complete, if at least two ζj=0,j=1,2,3;
+else, they are not complete. Therefore, since Dis a diagonal matrix with negative entries
+forζ1,ζ2,ζ3≥0, we conclude that (2.13) is strongly stable, if at least two ζj=0, and weakly
+stable, otherwise.9
+4 Finite difference discretization
+Here we show how to discretize (2.13) with standard second-o rder finite differences on a
+uniform mesh at grid points xi,iℓ=x1,0+iℓ∆xi, with i=1,2,3 and iℓ=0,1,..., Mℓ. For the time
+discretization we use a constant step size ∆tand denote the time levels by tn=t0+n∆t,
+n=0,1,..., N. Inside the absorbing layer, we further introduce a space-t ime staggered grid
+at locations xi,iℓ+1
+2=xi,0+/parenleftbig
+iℓ+1
+2/parenrightbig
+∆xi,i=1,2,3 and times tn+1
+2=t0+/parenleftbig
+n+1
+2/parenrightbig
+∆t. Then the
+numerical solution un
+i,j,k, which approximates uat grid point (x1,i,x2,j,x3,k)and time tn,
+satisfies
+un+1
+i,j,k−2un
+i,j,k+un−1
+i,j,k
+∆t2+/parenleftbig
+ζ1i+ζ2j+ζ3k/parenrightbigun+1
+i,j,k−un−1
+i,j,k
+2∆t+/parenleftbig
+ζ1iζ2j+ζ2jζ3k+ζ3kζ1i/parenrightbig
+un
+i,j,k
+=c2
+i+1
+2,j,kun
+i+1,j,k−(c2
+i+1
+2,j,k+c2
+i−1
+2,j,k)un
+i,j,k+c2
+i−1
+2,j,kun
+i−1,j,k
+∆x12
++c2
+i,j+1
+2,kun
+i,j+1,k−(c2
+i,j+1
+2,k+c2
+i,j−1
+2,k)un
+i,j,k+c2
+i,j−1
+2,kun
+i,j−1,k
+∆x22
++c2
+i,j,k+1
+2un
+i,j,k+1−(c2
+i,j,k+1
+2+c2
+i,j,k−1
+2)un
+i,j,k+c2
+i,j,k−1
+2un
+i,j,k−1
+∆x32
++˜φn
+1i+1
+2,j,k−˜φn
+1i−1
+2,j,k
+∆x1+˜φn
+2i,j+1
+2,k−˜φn
+2i,j−1
+2,k
+∆x2+˜φn
+3i,j,k+1
+2−˜φn
+3i,j,k−1
+2
+∆x3−ζ1iζ2jζ3kψn+1
+2
+i,j,k+ψn−1
+2
+i,j,k
+2,
+where the cell averages of the auxiliary functions φ1,φ2and φ3are defined as
+˜φn
+1i+1
+2,j,k=1
+4/parenleftBig
+φn
+1i+1
+2,j−1
+2,k−1
+2+φn
+1i+1
+2,j−1
+2,k+1
+2+φn
+1i+1
+2,j+1
+2,k−1
+2+φn
+1i+1
+2,j+1
+2,k+1
+2/parenrightBig
+,
+˜φn
+2i,j+1
+2,k=1
+4/parenleftBig
+φn
+2i−1
+2,j+1
+2,k−1
+2+φn
+2i−1
+2,j+1
+2,k+1
+2+φn
+2i+1
+2,j+1
+2,k−1
+2+φn
+2i+1
+2,j+1
+2,k+1
+2/parenrightBig
+,
+˜φn
+3i,j,k+1
+2=1
+4/parenleftBig
+φn
+3i−1
+2,j−1
+2,k+1
+2+φn
+3i−1
+2,j+1
+2,k+1
+2+φn
+3i+1
+2,j−1
+2,k+1
+2+φn
+3i+1
+2,j+1
+2,k+1
+2/parenrightBig
+.
+Concurrently with the above discretized wave equation, we a lso advance the (scalar)
+auxiliary variables ψ,φj,j=1,2,3 inside the absorbing layer by using standard finite
+differences. For ψwe use
+ψn+1
+2
+i,j,k−ψn−1
+2
+i,j,k
+∆t=un
+i,j,k,10
+whereas for φ1we use
+φn+1
+1i+1
+2,j+1
+2,k+1
+2−φn
+1i+1
+2,j+1
+2,k+1
+2
+∆t
+=−ζ1i+1
+2φn+1
+1i+1
+2,j+1
+2,k+1
+2+φn
+1i+1
+2,j+1
+2,k+1
+2
+2+/parenleftBig
+ζ2j+1
+2+ζ3k+1
+2−ζ1i+1
+2/parenrightBig
+Dh
+x1un+1
+2
+i+1
+2,j+1
+2,k+1
+2
++ζ2j+1
+2ζ3k+1
+2Dh
+x1ψn+1
+2
+1i+1
+2,j+1
+2,k+1
+2,
+where
+Dh
+x1un+1
+2
+i+1
+2,j+1
+2,k+1
+2=1
+2
+˜un+1
+i+1,j+1
+2,k+1
+2−˜un+1
+i,j+1
+2,k+1
+2
+∆x1+˜un
+i+1,j+1
+2,k+1
+2−˜un
+i,j+1
+2,k+1
+2
+∆x1
+,
+Dh
+x1ψn+1
+2
+1i+1
+2,j+1
+2,k+1
+2=˜ψn+1
+2
+i+1,j+1
+2,k+1
+2−˜ψn+1
+2
+i,j+1
+2,k+1
+2
+∆x1.
+Here, the cell averages of uand ψare defined as
+˜un
+i,j+1
+2,k+1
+2=1
+4/parenleftBig
+un
+i,j,k+un
+i,j,k+1+un
+i,j+1,k+un
+i,j+1,k+1/parenrightBig
+,
+˜ψn+1
+2
+i,j+1
+2,k+1
+2=1
+4/parenleftbigg
+ψn+1
+2
+i,j,k+ψn+1
+2
+i,j,k+1+ψn+1
+2
+i,j+1,k+ψn+1
+2
+i,j+1,k+1/parenrightbigg
+.
+The finite difference approximations for φ2and φ3are analogous.
+5 Numerical experiments
+Here we present numerical experiments that illustrate the a ccuracy, versatility and long-
+time stability of our PML formulation discretized with stand ard finite differences as in
+Section 4. In all cases we choose ¯ζi=80 in the damping profile, which yields a relative
+reflection R≈10−3for the the typical values c=1 and Li=0.1. At the exterior boundary
+of the absorbing layer we impose homogeneous Dirichlet boun dary conditions.
+5.1 Point source in 2D
+First, we consider the wave equation (2.1) in two space dimen sions with constant speed
+of propagation c=1 and zero initial conditions, u0=v0=0. The source term fcorresponds
+to a truncated first derivative of a Gaussian:
+f(x,y,t)=δ(x)δ(y)h(t) (5.1)11
+with
+h(t)=d
+dt/parenleftbig
+e−π2(f0t−1)2/parenrightbig
+, f0=10Hz . (5.2)
+The grid spacing is uniform in x1and x2, with ∆x=0.002.
+In Figure 2 we display snapshots of the numerical solutions a t different times in Ω=
+[−0.5,0.5]2, surrounded by a PML of width L=0.1. We observe how the circular wave
+propagates outward essentially without spurious reflectio n from the PML. By time t=
+1 the wave has essentially left the computational domain. To assess the error in the
+numerical solution, we compute a reference solution in a muc h larger domain of size
+[−5.5,5.5]×[5.5,5.5], so that boundary effects are postponed to later times insid eΩ. In
+Figure 3, the time evolution of the L2–error is shown for different values of the damping
+coefficient ¯ζi. Until t=8 we observe a steady decrease of the error over seven orders o f
+magnitude, regardless of the value of ¯ζi, which demonstrates the long-time stability of
+our method. Moreover, our formulation appears robust with re spect to the parameter
+value ¯ζi.
+Figure 2: Point source in 2D: snapshots of the numerical solu tions at different times in Ω=[−0.5,0.5]2,
+surrounded by a PML of width L=0.1.
+5.2 Heterogeneous medium in 2D
+Next, to illustrate the versatility of our PML formulation, w e consider the homogeneous
+wave equation (2.1) in a heterogeneous medium with varying w ave speed c=c(x2), given12
+0 1 2 3 4 5 6 7 8 910−2510−2010−1510−10
+tlog10 (|| Uref  −  Upml || L2)
+  
+ζ= 20
+ζ= 40
+ζ= 60
+ζ= 80
+Figure 3: Point source in 2D: time evolution of the L2–error for different damping coefficients ¯ζi.
+by
+c(x1,x2)=
+
+0.5, if x2<−b
+1+y
+2b+1
+2πsin/parenleftbigπx2
+b/parenrightbig
+, if|x2|<b
+1.5, otherwise.(5.3)
+We set b=0.95 which yields the vertical velocity profile shown in Figur e 4. The initial
+conditions are
+u|t=0=u0(x1,x2)and ut|t=0=0, (5.4)
+where
+u0(x1,x2)=/braceleftBigg/parenleftbig
+4(x1+0.4)(0.4−x1)/parenrightbig3sin(3πx2), if−0.4<x1<0.4,−1<x2<1
+0, otherwise.
+Here Ωis the square domain [−1,1]×[−1,1], surrounded by a PML of width L=0.2. The
+−1 −0.5 0 0.5 10.50.7511.251.5
+x2c(x2)
+Figure 4: Heterogeneous medium in 2D: varying wave speed cgiven by (5.3).13
+finite difference grid is uniform with grid spacing ∆x=0.004. In Fig.5, we display snap-
+shots of the solution at different times, where again the las t frame is purposely chosen
+at a much later time. In spite of the varying wave speed and the glancing angle of in-
+cidence along the vertical artificial boundaries, the waves a re damped without spurious
+reflection. Even at much later times we do not observe any inst ability in the numerical
+scheme.
+Figure 5: Heterogeous medium in 2D: snapshots of the numeric al solution are shown at different times in
+Ω=[−1,1]2, surrounded by a PML of width L=0.2.
+5.3 Point source in 3D
+Finally, we consider the wave equation (2.1) in three space d imensions with zero initial
+conditions and the same point source fas in (5.1). The grid spacing is uniform in x1,x2
+and x3with ∆x=0.006. In Figure 6, we display snapshots of the numerical sol utions at
+different times in Ω=[−0.5,0.5]2, surrounded by a PML of width L=0.1. We observe
+how the spherical wave propagates outward essentially with out spurious reflection from
+the PML. By time t=1 the wave has essentially left the computational domain. Ag ain we
+observe no instabilities in the numerical solution even at m uch later times.
+6 Concluding remarks
+We have presented a PML formulation for the wave equation in it s standard second-order
+form. It distinguishes itself from known formulations by it s simplicity and the small14
+Figure 6: Point source in 3D: snapshots of the numerical solu tion are shown at different times in Ω=[−0.5,0.5]3,
+surrounded by a PML of width L=0.1.15
+number of auxiliary variables needed inside the absorbing l ayer. We have proved that
+the continuous Cauchy problem with the unbounded PML is stabl e and well-posed. Our
+numerical results in two and in three space dimensions with s tandard finite differences
+illustrate the accuracy, versatility and long-time stabil ity of our PML formulation.
+Because it involves no high space or time derivatives, our PML formulation easily fits
+continuous or discontinuous Galerkin formulation for use w ith finite element methods
+[10, 13]. It also immediately generalizes to Maxwell’s equat ions in second-order form.
+Current work involves the extension to second-order wave eq uations in complex elastic
+and poro-elastic media, and will be reported elsewhere in th e near future.
+References
+[1] S. Abarbanel, D. Gottlieb and J. S. Hesthaven : Long time behavior of the perfectly matched layer
+equations in computational electromagnetics , J. Sci. Comput. 17(1-4), pp. 405–422. 2002.
+[2] D. Appel¨ o, T. Hagstrom and G. Kreiss : Perfectly matched layers for hyperbolic systems: general
+formulation, well-posedness, and stability , SIAM J. Appl. Math. 67(1), pp. 1–23, 2006.
+[3] D. Appel¨ o and G. Kreiss : Application of a perfectly matched layer to the nonlinear wa ve equation ,
+Wave Motion 44, pp. 531–548, 2007
+[4] D. Appel¨ o and G. Kreiss : A new absorbing layer for elastic waves , J. Comput. Phys. 215(2), pp.
+642–660, 2006.
+[5] G. Cohen : Higher-order numerical methods for transient wave equatio ns, Springer, 2002.
+[6] J. P . B´ erenger : A perfectly matched layer for the absorption of electromagn etic waves , J. Comput.
+Phys., 114, pp. 185–200, 1994.
+[7] E. B´ ecache, S. Fauqueux and P . Joly : Stability of perfectly matched layers, group velocities an d
+anisotropic waves , J. Comput. Phys., 188, pp. 399–433, 2003.
+[8] E. B´ ecache, and P . Joly : On the analysis of B´ erengers perfectly matched layers for M axwells equa-
+tions , Mod´ elisation Math´ ematique et Analyse Num´ erique, 36(1), pp. 87–119, 2002.
+[9] J. Diaz and P . Joly : A time domain analysis of PML models in acoustics , Comput. Methods Appl.
+Mech. Engrg. 195(29-32), pp. 3820–3853, 2006.
+[10] J. Diaz and M. J. Grote : Energy conserving explicit local time stepping for second- order wave
+equations , SIAM J. Sci. Comput., in press.
+[11] M. J. Grote and J. B. Keller : Exact nonreflecting boundary conditions for the time Depend ent wave
+equation , SIAM J. Appl. Math. 55(2), pp. 280–297, 1995.
+[12] M. J. Grote and J. B. Keller : Nonreflecting boundary conditions for time-dependent scat tering , J.
+Comput. Phys. 127(1), pp. 52–65, 1996.
+[13] M. J. Grote, A. Schneebeli and D. Sch¨ otzau, Discontinuous galerkin finite element method for the
+wave Equation , SIAM J. Numer. Anal. 44, pp. 2408–2431, 2006.
+[14] T. Hagstrom : Radiation boundary conditions for the numerical simulatio n of waves , Acta Numer.
+8, pp. 47–106, 1999.
+[15] T. Hagstrom and S. I. Hariharan : A formulation of asymptotic and exact boundary conditions
+using local operators , Appl. Numer. Math. 27(4), pp. 403–416, 1998.
+[16] T. Hagstrom, A. Mar-Or and D. Givoli : High-order local absorbing conditions for the wave equa-
+tion: extensions and improvements , J. Comput. Phys. 227 no. 6, 3322–3357, 2008.
+[17] F. Ihlenburg : Finite element analysis of acoustic scattering , Springer-Verlag, New York, 1998.
+[18] H.-O. Kreiss and J. Lorenz : Initial-boundary value problems and the Navier-Stokes equ ations ,
+Academic Press, 1989.16
+[19] B. Sj¨ ogreen and N. A. Petersson : Perfectly matched layers for Maxwell’s equations in second order
+formulation , J. Comput. Phys., 209(1), pp. 19–46, 2005.
+[20] L. Zhao and A.C. Cangellaris : A general approach for the development of unsplit-field time -
+domain implementations of perfectly matched layers for FDT D grid truncation , IEEE Microwave
+and Guided Letters, 6(5), 1996.
\ No newline at end of file
diff --git a/1001.0320.txt b/1001.0320.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8481f6654d54ce219dd9432a893aa1f4cc21fe9e
--- /dev/null
+++ b/1001.0320.txt
@@ -0,0 +1,396 @@
+arXiv:1001.0320v1  [physics.atom-ph]  2 Jan 2010EPJ manuscript No.
+(will be inserted by the editor)
+Anomalous photon diffusion in atomic vapors
+M. Chevrollier1a, N. Mercadier2, W. Guerin2b, and R. Kaiser2
+1Laborat´ orio de Espectroscopia ´Otica, Departamento de F´ ısica, Universidade Federal da Pa ra´ ıba, Cx. Postal 5008
+58051-900 Jo˜ ao Pessoa, PB, Brazil
+2Institut Non Lin´ eaire de Nice, CNRS and Universit´ e Nice So phia-Antipolis, 1361 route des Lucioles, 06560 Valbonne, F rance
+This draft: July 16, 2018/ Received: / Revised version:
+Abstract. The multiple scattering of photons in a hot, resonant, atomi c vapor is investigated and shown to
+exhibit a L´ evy Flight-like behavior. Monte Carlo simulati ons give insights into the frequency redistribution
+process that originates the long steps characteristic of th is class of random walk phenomena.
+PACS.02.50.Ey Stochastic processes – 32.50.+d Fluorescence, ph osphorescence (including quenching)
+1 Introduction
+Random walk processes of particles, such as the Brown-
+ian motion, are usually described as Gaussian stochastic
+processes, where the distribution P(x) of the size xof the
+random steps has finite first and second moment, so that,
+according to the Central Limit Theorem (in the limit of
+a large number of steps), the total displacement, r, dis-
+tribution converges towards a Gaussian and the process
+is said to be normal or diffusive. Properties such as the
+square root dependence of the total displacement with
+time characterize this so-called normal diffusion ( γ= 1
+in Eq. 1):
+< r2>=Dtγ. (1)
+The 1990’s saw the emergence of a new class of sys-
+tems which do not follow the Central Limit Theorem, and
+whose dynamics is dominated by large but rare random
+events. Systems exhibiting long-tailed step distributions
+with diverging second moment give rise to superdiffusion,
+withγ >1 in Eq. 1. If this step size distribution asymp-
+totically scales like a power-law P(x)∝x−αwithα <3,
+the total displacement profile converges towards a L´ evy
+distribution after a large number of steps [1]. Note that, if
+α <2, even the first moment of the step size distribution
+diverges, i.e. the mean-free-path between two scattering
+events cannot be defined.
+The number of physical phenomena shown to follow
+L´ evyinsteadofGaussianstatisticshasbeensteadilygrow-
+ing in the last 20 years [2,3,4]. Recently, L´ evy processes
+involving light propagation have been demonstrated in an
+optical material engineered to induce photon superdiffu-
+sive behavior [5]. In such a system, long steps take their
+amartine@otica.ufpb.br
+bPresent address: Physikalisches Institut, Eberhard-Karl s-
+Universit¨ at T¨ ubingen, Auf der Morgenstelle 14, D-72076
+T¨ ubingen, Germanyorigin in inhomogeneities of spatial order artificially intro-
+duced in the material. As we shall show, another, much
+simpler, photonic system that exhibits L´ evy flights, is a
+hotatomicvapor.Inthatcasehowever,spatialL´ evyflights
+of resonant light take their origin in spectral inhomo-
+geneities.
+In this paper, we present our study of L´ evy flights
+of photons in atomic vapors. Contrary to most previous
+work, we measure directly the microscopic ingredient of
+the superdiffusive behavior, i.e. the single step-size distri-
+bution. In the next section, we present the physical origin
+of the L´ evy flights as well as our experimental methods
+and measurements. Then, in section 3, we detail our nu-
+merical simulations of such a system.
+2 Photon diffusion in atomic vapors
+2.1 Origin of long steps
+Radiation trapping in an atomic vapor is a well known
+phenomenon [6,7,8], whereby atoms in a vapor absorb res-
+onant radiation from an incident source and re-emit pho-
+tons that can be re-absorbed in the vapor, and so on.
+This photon multiple scattering happens in atomic media
+ranging from laboratory atomic vapors in optical cells to
+interstellar space. Under simple hypotheses, one can show
+that the distribution for the length covered by a photon
+between an emission and an absorption (jump size dis-
+tribution) asymptotically decays like a power law of di-
+verging second moment, characterizing resonant photons
+trajectoriesin atomic vapors as L´ evy flights [9]. The diffu-
+sion of photons exhibits an anomalous behavior through
+the inhomogeneity introduced in the scattering process by
+thefrequencydependenceoftheabsorptioncoefficientand
+the frequency redistribution due to the atoms velocity.2 M. Chevrollier et al.: Anomalous photon diffusion in atomic vapors
+In a dilute atomic vapor of two level atoms where the
+Dopplerbroadeningoftheatomiclineshapeisnegligiblein
+comparison with the natural width, as with laser-cooled
+atomic vapors [10,11,12,13], the scattering is quasi elas-
+tic, i.e. the frequency of the re-emitted photon is almost
+the same as the frequency of the absorbed one, so that
+frequency redistribution can be neglected. For light at a
+frequency ν, the jump size distribution then writes
+P(x,ν) =−∂T(x,ν)
+∂x=ka(ν)e−ka(ν)x,(2)
+whereka(ν) is the absorptionprofile and T(x,ν) the prob-
+abilityofaphotonoffrequency νgoingthroughadistance
+xwithout being absorbed,
+T(x,ν) =e−ka(ν)x. (3)
+In the general case where inelastic scattering cannot be
+discarded, as in normal laboratorytemperature condition,
+however, the jump size distribution function has to be
+computed by averagingover the spectrum of emitted light
+ke:
+P(x) =/integraldisplay+∞
+0ke(ν)ka(ν)e−ka(ν)xdν . (4)
+In presence of Doppler broadening alone, the absorp-
+tion profile is gaussian
+kD(ν) =Nσ0Γ
+4c√πv0ν0e−χ(ν)2, (5)
+while it is lorentzian for natural broadening
+kL(ν) =Nσ01
+1+[4π(ν−ν0)/Γ]2, (6)
+and Voigt if the two mechanisms are mixed,
+kV(ν) =Nσ0Γ
+4a
+πc√πv0ν0/integraldisplay+∞
+−∞exp(−y2)
+a2+(χ(ν)−y)2dy .
+(7)
+Here,Nis the atomic density, σ0the on-resonance atomic
+crosssection for atoms at rest, ν0the resonancefrequency,
+andΓthe natural width of the atomic transition. The
+mostprobablevelocityattemperature Tisv0=/radicalbig
+2kBT/m
+withkBthe Boltzman constant, cthe speed of light and
+mthe atomic mass. χ(ν) =c
+v0ν−ν0
+ν0is a reducedfrequency.
+Finally, to define the Voigt profile, we have introduced the
+Voigt parameter a=Γc/(4πv0ν0). The determination of
+the emission profiles, however, is not straightforward, as
+they depend on the light initial spectra, but Monte-Carlo
+simulations show that, in the multiple scattering regime,
+they are very close to the absorption lines (see Section 3).
+For Gaussian emission and absorption spectra ka(ν) =
+ke(ν) =kD(ν), the expected power of the step length dis-
+tribution asymptotic decay is α= 2, while for Lorentzian
+(Cauchy) spectra ka(ν) =ke(ν) =kL(ν),α= 1.5. As
+Voigtspectra kV(ν) showLorentzian-likefrequencywings,
+they also lead to a decay with α= 1.5 [9].2.2 Experimental methods and measurements
+While the measurement of the single step size distribution
+of a photon in large, hot atomic vapors such as in stars
+is not feasible, a specific multicell arrangement allowed
+for such an experimental observation[14]. The experimen-
+tal arrangement devised to observe individual scattering
+events in a hot vapor consists of an optical cell filled with
+a vapor of Rb atoms at a controlled temperature/density,
+from now on called the observation cell (part (a) of Figure
+1). These atoms are excited by a collimated beam of pho-
+tons, so as to define a direction of observation (the ballis-
+tic one). The atoms scatter the incident photons in all the
+directions. Photons detected by a CCD device show the
+position in the cell where they have been last scattered. A
+narrowstrip(2mm) paralleltothe ballisticdirectionisse-
+lectedformeasurements.Assumingthatphotonsscattered
+from this strip to outside the cell have undergone a sin-
+gle scattering event in the observation cell, the measured
+fluorescence intensity is directly proportional to the step-
+size distribution P(x). In fact, insuring a single scattering
+Source cell Observation cell 
+Laser beam CCD 
+Incident 
+photons Laser beam 
+(a) (b) 
+(c) 
+(d) First cell 
+(multiple scattering 
+regime) Laser 
+beam Second cell (single 
+scattering regime) 
+Fig. 1. Experimental setup for observing the spatial profile
+of fluorescence in a hot atomic vapor, i.e. the step-size dis-
+tribution function. (a) Observation cell and imaging syste m.
+This part is common to the three performed measurements.
+(b) The incident photons are directly provided by a monochro -
+matic laser, for calibration. (c) The incident photons have been
+scattered once, at 90◦, in the low-optical-thickness source-cell.
+(d)Theincidentphotonshavebeenfirstscatteredseveral ti mes
+(∼4) in a first, optically-thick, cell before being scattered o nce
+more in the second cell.M. Chevrollier et al.: Anomalous photon diffusion in atomic v apors 3
+regime in the observation cell is quite incompatible with
+having a sufficient dynamics of P(x) so that higher order
+scattering inevitably happens. As multiple scattering cor-
+rections might also mimic a power law intensity along Ox,
+it is crucial to correct for this effect. This can be achieved
+by subtracting a narrow strip measured slightly off-axis,
+as those photons are due to multiple scattering only [14].
+A reference measurement is first carried out with a
+monochromatic laser beam (frequency νL) as the incident
+photonsource(SeeFigure1(b)).Thestep-sizedistribution
+P(x) for this specific configuration is given by Eq. 2 with
+ν=νL,
+P(x)∝e−k(νL)x. (8)
+The measurement of this exponential decay enables the
+calibration of the mean-free-path ℓ(νL) = 1/k(νL) at res-
+onance, and consequently the atomic density n, which
+can be varied through the temperature of the Rb reser-
+voir in the cell. The temperature is varied between 20
+and≈47◦C, corresponding to an atomic density between
+9×109and 2×1011atoms/cm3and a subsequent resonant
+mean free path between 50 and 5 mm.
+A second experimental measurement of the jump size
+distribution P(x) is performed by using a two cell config-
+uration (Fig. 1(c)). Laser light is sent on a first rubidium
+cell of very low atomic density, so that photons are scat-
+tered at most once in this source cell. Light with a spectra
+broadened by Doppler effect is emitted in all directions in
+space,butacollimatedbeamorthogonaltotheinitiallaser
+propagation axis is selected out of it by two diaphragms
+and sent towards the observation cell. The knowledge of
+the position of the first scattering event, in the source cell,
+and the second one, in the observation cell, gives us access
+to the jump size distribution function P(x) (Fig. 2). We
+find that, for xlarge enough compared to the mean-free-
+path of resonant light, P(x) scales like a power law 1 /xα,
+withα= 2.41±0.12.
+Although the obtained distribution has clearly a di-
+verging second moment, this measurement alone does not
+provethatlighttransportinthemultiplescatteringregime
+is superdiffusive, because the spectrum of light scattered
+in the first cell, and hence the jump size distribution for
+the first step, depends on the laser frequency. The fre-
+quency redistribution is only partial [15]. To characterize
+the multiple scattering regime, we need to measure the
+jump size distribution function after several steps, once
+photons have lost memory of the initial laser frequency, so
+that the emission spectra has converged towards a limit
+spectra (see section 3 for a numerical demonstration of
+this convergence). To achieve this, we use a three cell con-
+figuration (Fig. 1(d)). Laser photons are sent in a first
+rubidium cell of high atomic density, where they are scat-
+tered several times ( ∼4), before reaching a second source
+cell, where they undergo one more scattering event with a
+well-known position before being sent to the observation
+cell. We thus measure the jump size distribution func-
+tion in the multiple scattering regime. Once again, we
+find a power law asymptotic behavior P(x)∼1/xα, with
+α= 2.09±0.15,whichhasaninfinitevarianceandisthere-
+fore characteristic of a superdiffusive regime. One should3 4 5 6101100P (x)
+x (cm)  
+Fig. 2.Experimental measurements of the step size distribu-
+tion function (log-log scale) in the configuration of Figure 1(a)
+and (b). With an incident laser beam (green crosses), the ob-
+served decay is well fitted by an exponential (red dotted line )
+as expected from Beer-Lambert law. In the two-cell configura -
+tion, the data (continuous blue line) are well fitted by a powe r
+law (black dotted line) P(x) = 1/xα, withα= 2.41±0.12.
+notice, however,that as α >2 the mean free path remains
+finite.
+3 Monte Carlo simulations
+A Monte Carlo (MC) code simulates the spectral and spa-
+tial properties of individual photons as they propagate in
+an atomic vapor at temperature T, in an optical cell hav-
+ing the dimensions of the experimental observation cell.
+Monte Carlo simulations may in general help in the sig-
+naldeconvolutionorin choosingsuitable experimentalpa-
+rameters. In this work, it proved indeed useful to validate
+the data analysis and for a deeper understanding of the
+third experimental configuration, in the multiple scatter-
+ing regime (Fig. 1(d)).
+3.1 Description
+The code consists essentially in an elementary procedure,
+corresponding to a scattering order, which is repeated as
+long as the simulated photon is inside the optical cell.
+When it gets out, it is either discarded or detected, de-
+pending on the direction it takes. If detected, its last scat-
+teringpositioninthecellisrecordedinafile,togetherwith
+the order of the last scattering process. The elementary
+procedure consists in taking one photon, of frequency ν1
+and wavenumber k1, and to determine the position where
+it will undergo a scattering process. The distance from
+thispositiontotheinitialoneisfrequency-dependent.The
+probability T(x,ν) that a photon of frequency νtravel a
+distance xwithout being absorbed in the vapor is ran-
+domly attributed a value between 0 and 1 and the step
+lengthxis then determined as
+x=−lnT
+k(ν), (9)4 M. Chevrollier et al.: Anomalous photon diffusion in atomic vapors
+withk(ν) the absorption spectrum. A new wavenumber
+k2(direction) is then drawn for the scattered photon and
+its frequency ν2is calculated from
+ν2=ν1+v·(k2−k1), (10)
+wherevis the velocity of the scattering atom. The fre-
+quencyν2will then determine the step length to the next
+scattering process.
+The photons incident on the optical cell are initially
+homogeneously distributed over a disk whose diameter is
+the one of the second diaphragm of the setup shown in
+Figure 1(c,d). Their direction is parallel to the longitudi-
+nal axis of the observation cell and their frequency can be
+drawn either from a laser’s lorentzian lineshape (calibra-
+tion experiment, see text) or from a Voigt distribution at
+the temperature of the source cell.
+From the equation (10), we can see that the relevant
+components of both the wavenumbers and of the atom
+velocity are the one parallel and the one in the plane per-
+pendicular to the incident photon’s direction. Regarding
+the direction of the scattered photon, an isotropic angular
+distribution follows from isoprobabilistically drawing the
+value of the polar angle’s cosine between −1 and +1 and
+the value of the azimuthal angle between 0 and 2 π.
+Finally, the velocity component vpof the scattering
+atom in the plane perpendicular to the direction of the
+incident photon can take any value, with a probability law
+given by the Gaussian Maxwell-Boltzmann distribution of
+standarddeviation σv=/radicalbig
+kBT/m:G(vp)∝e−(vp/√
+2σv)2.
+The velocity component vkof the scattering atom in the
+direction of the incident photon is drawn from a prob-
+ability law p(vk) =L(ν,vk)×G(vk), convolution of the
+atomic Lorentzian lineshape L(ν,vk)∝1/(1+4π(ν−ν0−
+vk/λ)2/Γ2) and of the Gaussian Maxwell-Boltzmann ve-
+locity distribution G(vk), where ν0,λandΓare respec-
+tively the frequency, the wavelength and the linewidth of
+the atomic transition considered.
+/s50 /s51 /s52 /s53 /s54/s49/s49/s48/s49/s48/s48/s49/s48/s48/s48
+/s80/s40/s120/s41
+/s120/s32/s40/s99/s109/s41
+Fig. 3. First-step size distribution function (log-log scale) in
+the configuration of Figure 1. Black solid line: Experimenta l
+measurements, as in Figure 2. Grey line: Monte Carlo simula-
+tion.Thesimulatedfirst-stepdistributionisshowninFigure
+3 to reproduce the observed one.
+3.2 Multiple scattering regime
+To characterize the multiple scattering regime from the
+measurement of a single step-size distribution in the mid-
+dle of the scattering sequence, it is necessary that this
+distribution have reached a stationnary state and do not
+depend any more of the photon frequency at the previous
+step. To know how many scattering events are required
+to fulfill this condition, we have used the MC simulations
+to follow the evolution of the light spectral distribution,
+from the monochromatic laser beam (lorentzian shape of
+width≈1 MHz) to the asymptotical Voigt distribution
+(Doppler width for the preparationcell temperature). The
+half width at half maximum (HWHM) of the spectra is
+shown as a function of the diffusion order in Figure 4(a).
+We can see that in the conditions of the experiment (300
+K) at least 5 diffusions are necessary for assuring a Voigt-
+(a)
+(b)
+Fig. 4. Evolution of (a) the HWHM of the photons spectral
+distribution and (b) the power-law exponent α, with the num-
+ber of scattering events in a Rb vapor at 300K. The original
+spectral distribution (photons absorbed in the zeroth orde r dif-
+fusion) is the one of a resonant, monochromatic laser.M. Chevrollier et al.: Anomalous photon diffusion in atomic v apors 5
+broadened distribution of the photons incident on the ob-
+servation cell and fulfill the Complete Frequency Redis-
+tribution condition. The second setup used in the exper-
+iments (see Figure 1(d)) takes this requirement into ac-
+count. The exponent of the power-law of the correspond-
+ing step-distribution is shown in Figure 4(b). It decreases
+fromα≈2.4 to≈2.05 as the diffusion order increases
+from 1 to ∼5.
+In our experimental conditions, the Doppler width is
+typically two orders of magnitude larger than the natural
+linewidth (Voigt parameter a≈10−2). In such a regime,
+thelorentziannatureofthenaturalbroadeningisexpected
+to dominate the spectral dynamics (asymptotical Lorentz
+power-law exponent α= 1.5) for optical densities nσ0x
+larger than 104[9], i.e. for jump sizes larger than 1 m in
+our experimental conditions ( σ0≈1.25×10−13m2,n≈
+5×1016m−3). Below 10 cm, the spatial range observed
+in our experiment, the spectral behavior (assuming CFR)
+is purely Doppler-driven, with an exponent α= 2 for the
+step distribution power law.
+4 Conclusion
+We have evidenced a L´ evy Flights behavior of photons
+propagation in a hot resonant atomic vapor by measur-
+ing the single step distribution in this multiple scattering
+process. Monte Carlo simulations help deconvoluating the
+effects of multiple scattering and analyzing the frequency
+redistribution process in a Doppler-broadened vapor ex-
+cited by a monochromatic light source.
+References
+1. P. L´ evy, Th´ eorie de l’addition des variables al´ eatoires
+(Gauthier-Villiers, 1937).
+2. J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 127–293
+(1990).
+3. M. Shlesinger, G. Zaslavsky and U. Frisch, L´ evy Flights
+and Related Topics in Physics (Springer, 1995).
+4. R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
+5. P. Barthelemy, J. Bertolotti and D. S. Wiersma, Nature
+453, 495 (2008).
+6. C. Kenty, Phys. Rev. 42, 823 (1932).
+7. T. Holstein, Phys. Rev. 721212 (1947).
+8. A. F. Molisch and B. P. Oehry, Radiation Trapping in
+Atomic Vapours (Oxford Univ., 1998).
+9. E. Pereira, J. M. G. Martinho and M. N. Berberan-Santos,
+Phys. Rev. Lett. 93, 120201 (2004).
+10. A. Fioretti et al., Opt. Commun. 149, 415 (1998).
+11. G. Labeyrie et al., Phys. Rev. Lett. 91, 223904 (2003).
+12. G. Labeyrie, R. Kaiser, D. Delande, Appl. Phys. B 81,
+1001 (2005).
+13. R. Pierrat, B. Gr´ emaud and D. Delande, Phys. Rev. A.
+80, 013831 (2009).
+14. N. Mercadier, W. Guerin, M. Chevrollier and R. Kaiser,
+Nature Phys. 5, 602 (2009).
+15. A. R. Alves-Pereira, E. J. Nunes-Pereira, J. M. G. Mart-
+inho, M. N. Berberan-Santos, J. Chem. Phys. 126, 154505
+(2007).
\ No newline at end of file
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+arXiv:1001.0321v2  [math.GR]  6 Sep 2010On the Cartan matrix of Mackey algebras
+Serge Bouc
+Abstract : Letkbe a field of characteristic p >0, andGbe a finite group. The first
+result of this paper is an explicit formula for the determinant of the C artan matrix of the
+Mackey algebra µk(G) ofGoverk. The second one is a formula for the rank of the Cartan
+matrix of the cohomological Mackey algebra coµk(G) ofGoverk, and a characterization
+of the groups Gfor which this matrix is non singular. The third result is a generalization
+of this rank formulaand characterizationto blocks of coµk(G) : in particular, if bis a block
+ofkG, the Cartan matrix of the corresponding block coµk(b) ofcoµk(G) is non singular if
+and only if bis nilpotent with cyclic defect groups.
+AMS Subject Classification : 18G05, 20C20, 20J06.
+Keywords : Cartan matrix, cohomological, Mackey functor, block.
+1 Introduction
+1.1.The theory of Mackey functors for a finite group Gover a commutative
+ringRis by many aspects very similar to the theory of RG-modules. It has
+been shown by J. Th´ evenaz and P. Webb ([13]), among many fundam ental
+other results, that the category of Mackey functors for GoverRis equivalent
+to the category of modules over the Mackey algebra µR(G). This algebra
+sharesmanypropertieswiththegroupalgebra RG: itisfreeasan R-module,
+and itsR-rank does not depend on R; ifKis a field of characteristic 0 or
+coprime to the order of G, the algebra µK(G) is semisimple ; when ( K,O,k)
+is ap-modular system, there is a decomposition theory from Mackey fun ctors
+forGoverKto Mackey functors for Goverk; the Cartan matrix of µk(G)
+is symmetric and non singular. This list of common properties between the
+Mackey algebra and the group algebra is far from exhaustive...
+1.2.However some well known results for group algebras are no longer
+true for Mackey algebras. It was observed on small examples in par ticular
+by M. Nicollerat in her thesis ([12] Chapitre 5) that the determinant o f the
+Cartan matrix of µk(G), whenkis a field of characteristic p, is generally not
+a power of p, even when Gitself is a p-group. Instead, some rather strange
+prime factors appear in this determinant. One of the motivations of this
+paperistogiveanexplanationforthesestrangefactors, bystat inganexplicit
+formula for the determinant of the Cartan matrix of µk(G) (Theorem 2.28).
+1.3.The other motivation is the similar problem for the cohomological
+Mackey algebra coµk(G), which was defined by Th´ evenaz and Webb as a
+specific quotient of µk(G), with the property that the modules over coµk(G)
+1are exactly the cohomological Mackey functors : the first major difference is
+that in general, the Cartan matrix of this algebra is singular. This rais es the
+question of characterizing those finite groups Gfor which the Cartan matrix
+ofcoµk(G) is non singular, and the answer is the second main result of this
+paper (Theorem 3.4) : these groups are exactly the p-nilpotent groups with
+cyclic Sylow p-subgroups. The other possibly interesting result is that for
+an arbitrary finite group G, the rank of the Cartan matrix of coµk(G) is
+equal to the number of conjugacy classes of pairs ( R,s), where Ris a cyclic
+p-subgroup of G, andsis ap′-element of the centralizer of RinG.
+The third result of this paper is a natural generalization of Theorem 3.4
+toblocks, suggested by the one to one correspondence b/mapsto→coµk(b) between
+blocks of kGand blocks of coµk(G) : Theorem 4.2 gives a formula for the
+rank of the Cartan matrix of coµk(b), in terms of b-Brauer pairs, and shows
+that this matrix is non singular if and only if the block bis nilpotent with
+cyclic defect groups.
+1.4.The paper is organized according to these results : Section 2 is devo ted
+to the case of µk(G), starting by recalling some standard notation, defini-
+tions and properties, and Section 3 deals with coµk(G). Finally, Section 4 is
+devoted to the case of blocks of cohomological Mackey functors.
+Acknowledgment : I wish to thank Jacques Th´ evenaz for very fruitful
+discussions andfriendlycollaborationonallthesequestionsalso, at theEPFL
+in June 2009.
+2 The Mackey algebra
+2.1.Throughout the paper when Gis a finite group and pis a prime
+number, the set of p′-elements of Gis denoted by Gp′, and the symbol [ Gp′]
+denotes a set of representatives of G-conjugacy classes in Gp′. The set of
+p-subgroups of Gis denoted by Sp(G), and [Sp(G)] denotes similarly a set of
+representatives of G-conjugacy classes in Sp(G).
+2.2.Fromnow on Gwill be a fixed finite group, and pa fixed prime number.
+Let (K,O,k)beap-modularsystem : thus Oisadiscrete valuationring with
+residue field kof characteristic p, and field of fractions Kof characteristic 0.
+Assume that Kandkare splitting fields for all the groups NG(Q)/Q, for
+Q∈ Sp(G) (e.g. assume that Kcontains the e-th roots of unity, where eis
+the exponent of G).
+2.3.WhenRis a commutative ring with identity element, the Mackey
+algebraµR(G) ofGoverRhas been defined by Th´ evenaz and Webb ([13]
+Section 3). It is an associative algebra with the property that the c ategory
+2µR(G)-Modof leftµR(G)-modules is equivalent to the category MackR(G) of
+Mackey functors for GoverR.
+2.4.Th´ evenaz and Webb have also shown ([13] Theorem 10.1) that ther e is
+an equivalence of abelian categories
+Mackk(G)∼=/productdisplay
+HMackk/parenleftbig
+NG(H)/H,1/parenrightbig
+,
+whereHruns through a set of representatives of conjugacy classes of p-
+perfect subgroups of G, andMackk/parenleftbig
+NG(H)/H,1/parenrightbig
+denotes the subcategory of
+Mackk/parenleftbig
+NG(H)/H/parenrightbig
+consisting of Mackey functors which are projective relative
+top-subgroups. The category Mackk(G,1) is equivalent to µk(G,1)-Mod,
+whereµk(G,1) is a direct summand of µk(G) of the form µk(G)f, for a
+specific central idempotent fofµk(G).
+2.5.([13] Theorem 12.7 and Corollary 12.8) The correspondence M/mapsto→
+M(1), that is evaluation at the trivial subgroup of G, induces a one to one
+correspondence between the set of isomorphism classes of indeco mposable
+projective µk(G,1)-modules and the set of isomorphism classes of indecom-
+posablep-permutation kG-modules (also called trivial source kG-modules).
+2.6.Letppk(G) denote the Green ring of p-permutation kG-modules : as a
+group, it is the Grothendieck group of the category of finitely gene ratedp-
+permutation kG-modules, for relations given by direct sum decompositions.
+The product on ppk(G) is induced by the tensor product of kG-modules
+overk.
+IfWis a finitely generated p-permutation kG-module, its dual W∗=
+Homk(W,k) is also a p-permutation kG-module, and this duality extends to
+a ring automorphism W/mapsto→W∗ofppk(G).
+WhenQisap-subgroupof G,theBrauer quotient W[Q]isap-permutation
+kNG(Q)-module, where NG(Q) =NG(Q)/Q(see [7]). This construction
+commutes with duality and tensor product of p-permutation modules : if V
+andWare (finitely generated) p-permutation kG-modules, there are isomor-
+phisms of kNG(Q)/Q-modules
+(2.7) W[Q]∗∼=W∗[Q], V[Q]⊗W[Q]∼=(V⊗W)[Q].
+Inparticular, theBrauerquotientinducesaringhomomorphismfro mppk(G)
+toppk/parenleftbig
+NG(Q)/parenrightbig
+, still denoted by W/mapsto→W[Q] (see e.g. [5] Proposition 2.11).
+Thereisa Z-valuedbilinearform /a\}bracketle{t,/a\}bracketri}htonppk(G)definedfor p-permutation
+kG-modules VandWby
+/a\}bracketle{tV,W/a\}bracketri}htG= dim kHomkG(V,W).
+3It is worth noticing that
+(2.8) /a\}bracketle{tV,W/a\}bracketri}htG= dim kHomkG(k,V∗⊗kW).
+Thebilinear form /a\}bracketle{t,/a\}bracketri}htGextends toa K-valuedbilinear formon K⊗Zppk(G),
+still denoted by /a\}bracketle{t,/a\}bracketri}htG.
+2.9.Let/a\}bracketle{t/a\}bracketle{t,/a\}bracketri}ht/a\}bracketri}htGdenotethebilinearformon ppk(G)definedfor p-permutation
+kG-modules VandWby
+/a\}bracketle{t/a\}bracketle{tV,W/a\}bracketri}ht/a\}bracketri}htG=/summationdisplay
+Q∈[Sp(G)]/a\}bracketle{tV[Q],W[Q]/a\}bracketri}htNG(Q)
+=/summationdisplay
+Q∈[Sp(G)]dimkHomkNG(Q)(V[Q],W[Q]).
+It follows from 2.7 that
+(2.10) /a\}bracketle{t/a\}bracketle{tV,W/a\}bracketri}ht/a\}bracketri}htG=/a\}bracketle{t/a\}bracketle{tk,V∗⊗kW/a\}bracketri}ht/a\}bracketri}htG.
+It was shown in [3] Proposition 5.11 that if LandMare projective Mackey
+functors in Mackk(G,1), then
+dimkHomMackk(G,1)(L,M) =/a\}bracketle{t/a\}bracketle{tL(1),M(1)/a\}bracketri}ht/a\}bracketri}htG.
+WhenLandMare indecomposable, this is equal to the coefficient cL,Mof
+the Cartan matrix of the algebra µk(G,1).
+2.11.LetQG,pdenote the set of pairs ( R,s) consisting of a p-subgroup R
+ofGand ap′-element sofNG(R). The group Gacts by conjugation on
+QG,p. Let [QG,p] denote a set of representatives of G-orbits on QG,p, and let
+NG(R,s) denote the stabilizer of ( R,s)∈ QG,pinG: it is the set of elements
+g∈NG(R) such that the image gofginNG(R) centralizes s. In other words,
+there is an exact sequence of groups
+(2.12) 1→R→NG(R,s)→CNG(R)(s)→1.
+It was shown in [5] that the primitive idempotents of the (commutative) ring
+K⊗Zppk(G) are indexed by the orbits of GonQG,p: the idempotent FG
+R,s
+associated to the orbit of the pair ( R,s) is equal to
+(2.13)
+FG
+R,s=1
+|R||s||CNG(R)(s)|/summationdisplay
+ϕ,L˜ϕ(s−1)|L|µ(L,<sR> )IndG
+LRes<sR>
+Lkϕ,
+where
+4•the group <sR>is the inverse image in NG(R) of the subgroup <s>
+ofNG(R) under the map x/mapsto→xR.
+•the morphism ϕruns through group homomorphisms <s>→k×, and
+˜ϕliftsϕtoK. The module kϕis the vector space kon which <sR>
+acts by<sR>→<s>ϕ→k×(thus ˜ϕis the Brauer character of the
+modulekϕ).
+•the group Lruns through the set of subgroups of <sR>such that
+LR=<sR>.
+The idempotents FG
+R,s, for (R,s)∈[QG,p], form a K-basis of K⊗Zppk(G).
+Any element WofK⊗Zppk(G) can be expressed in this basis as
+(2.14) W=/summationdisplay
+(R,s)∈[QG,p]FG
+R,s⊗kW=/summationdisplay
+(R,s)∈[QG,p]tG
+R,s(W)FG
+R,s,
+wheretG
+R,sis the extension to K⊗Zppk(G) of the species(i.e. the ring
+homomorphism, see [1] Lemma 2.2.1, page 26) ppk(G)→Kassociated to
+the pair ( R,s).
+2.15.Recall that the value of tG
+R,son ap-permutation kG-module Wis
+equal to the value at sof the Brauer character of the Brauer quotient W[R]
+(see [5] Notation 2.15 and Proposition 2.18). This will be also be called th e
+Brauer trace ofsonW[R], and denoted by BrTr( s|W[R]).
+2.16.The ring automorphism W/mapsto→W∗extends by K-linearity to an auto-
+morphism of the ring K⊗Zppk(G). This automorphism preserves the set of
+primitive idempotents. The bilinear form /a\}bracketle{t/a\}bracketle{t,/a\}bracketri}ht/a\}bracketri}htGalso extends to a K-valued
+bilinear form on K⊗Zppk(G). Since the dual of kϕis isomorphic to kϕ−1,
+and since /tildewider(ϕ−1)(s−1) = ˜ϕ(s), it follows from 2.13 that ( FG
+R,s)∗=FG
+R,s−1. Now
+Equation 2.10 shows that
+/a\}bracketle{t/a\}bracketle{tFG
+R′,s′,FG
+R,s/a\}bracketri}ht/a\}bracketri}htG=/braceleftbigg/a\}bracketle{t/a\}bracketle{tk,FG
+R,s/a\}bracketri}ht/a\}bracketri}htGif (R′,s′) =G(R,s−1)
+0 otherwise
+(where = GdenotesG-conjugacy). Thus, if VandWare any elements of
+K⊗Zppk(G), it follows from 2.14 that
+/a\}bracketle{t/a\}bracketle{tV,W/a\}bracketri}ht/a\}bracketri}htG=/summationdisplay
+(R,s)∈[QG,p]tG
+R,s−1(V)tG
+R,s(W)/a\}bracketle{t/a\}bracketle{tk,FG
+R,s/a\}bracketri}ht/a\}bracketri}htG.
+2.17.The indecomposable p-permutation kG-modules are indexed by con-
+5jugacy classes of pairs ( P,E), where Pis ap-subgroup of G, andEis an
+indecomposable projective kNG(P)-module ([7] Theorem 3.2). Let [ PG,p]
+denote a set of representatives of such conjugacy classes in G.
+The indecomposable module MP,Eindexed by the pair ( P,E) has ver-
+texP, and the projective module Eis isomorphic to the Brauer quotient
+MP,E[P]. In particular, if Qis a subgroup of G, thenMP,E[Q] is non zero
+only ifQ≤GP(and in fact, if and only if Q≤GP).
+2.18.The modules MP,E, for (P,E)∈[PG,p], form a basis of ppk(G), hence
+also a basis of K⊗Zppk(G). The Cartan matrix C/parenleftbig
+µk(G,1)/parenrightbig
+of the algebra
+µk(G,1) can be viewed as the square matrix indexed by [ PG,p], where the
+coefficient indexed by the elements ( P,E) and (Q,F) of [PG,p] is equal to
+c(P,E),(Q,F)=/a\}bracketle{t/a\}bracketle{tMP,E,MQ,F/a\}bracketri}ht/a\}bracketri}htG
+=/summationdisplay
+(R,s)∈[QG,p]tG
+R,s−1(MP,E)tG
+R,s(MQ,F)/a\}bracketle{t/a\}bracketle{tk,FG
+R,s/a\}bracketri}ht/a\}bracketri}htG.
+2.19.LetTandT′denotethematricesindexedbytheproduct[ PG,p]×[QG,p],
+defined by
+T(P,E),(R,s)=tG
+R,s(MP,E),
+T′
+(P,E),(R,s)=tG
+R,s−1(MP,E),
+respectively. Note that TandT′are square matrices. Let moreover Sbe the
+(square) diagonal matrix indexed by [ QG,p], where the diagonal term indexed
+by (R,s) is equal to /a\}bracketle{t/a\}bracketle{tFG
+R,s,k/a\}bracketri}ht/a\}bracketri}htG. It follows that
+C/parenleftbig
+µk(G,1)/parenrightbig
+=T′·S·tT.
+Thus
+(2.20) det C/parenleftbig
+µk(G,1)/parenrightbig
+= detT′detSdettT,
+and moreover, since Sis diagonal
+(2.21) det S=/productdisplay
+(R,s)∈[QG,p]/a\}bracketle{t/a\}bracketle{tk,FG
+R,s/a\}bracketri}ht/a\}bracketri}htG=/productdisplay
+R∈[Sp(G)]/productdisplay
+s∈[NG(R)p′]/a\}bracketle{t/a\}bracketle{tk,FG
+R,s/a\}bracketri}ht/a\}bracketri}htG
+2.22.Now since MP,E[R] ={0}unlessR≤GP, the matrices TandT′
+6are actually block triangular, with diagonal blocks ∆( P) indexed by the
+conjugacy classes of p-subgroups PofG. Thus
+detT=/productdisplay
+P∈[Sp(G)]det∆(P).
+The blockmatrix∆( P)isa(square) matrixwithrows indexed bytheisomor-
+phismclasses ofindecomposableprojective kNG(P)-modules E, andcolumns
+indexed by the conjugacy classes of p′-elements sofNG(P). The coefficient
+∆(P)E,sis equal to
+∆(P)E,s=tG
+P,s(MP,E)
+= Brauertrace( s|MP,E[P])
+= ΦE(s),
+sinceMP,E[P]∼=E, where Φ Eis the Brauer character of the projective
+kNG(P)-module E.
+Similarly,
+detT′=/productdisplay
+P∈[Sp(G)]det∆′(P),
+where the coefficients of the diagonal block ∆′(P) ofT′are given by
+∆′(P)E,s= ΦE(s−1).
+2.23.Let Σ(P) denote the (square) diagonal matrix indexed by the conju-
+gacy classes of p′-elements of NG(P), with diagonal coefficient Σ( P)s,sequal
+to the inverse of the order of the centralizer CNG(P)(s) ofsinNG(P), and
+letU(P) = ∆(P)·Σ(P)·t∆′(P). The coefficient of U(P) indexed by the
+indecomposable projective kNG(P)-modules EandFis equal to
+U(P)E,F=/summationdisplay
+s∈[NG(P)p′]ΦE(s)1
+|CNG(P)(s)|ΦF(s−1)
+=1
+|NG(P)|/summationdisplay
+s∈NG(P)p′ΦE(s)ΦF(s−1)
+=/a\}bracketle{tE,F/a\}bracketri}htNG(P).
+In other words, the matrix U(P) is equal to the Cartan matrix of the group
+algebrakNG(P). It is then well known (see [6] III.16 or [10]) that
+detU(P) =/productdisplay
+s∈[NG(P)p′]|CNG(P)(s)|p,
+7and it follows that
+det∆(P)dett∆′(P) =/productdisplay
+s∈[NG(P)p′]|CNG(P)(s)|p
+detΣ(P)
+=/productdisplay
+s∈[NG(P)p′]|CNG(P)(s)|p/productdisplay
+s∈[NG(P)p′]|CNG(P)(s)|.
+Together with 2.20 and 2.21, this gives
+(2.24)
+detC/parenleftbig
+µk(G,1)/parenrightbig
+=/productdisplay
+R∈[Sp(G)]/productdisplay
+s∈[NG(R)p′]/parenleftbig
+|CNG(R)(s)|p|CNG(R)(s)|/a\}bracketle{t/a\}bracketle{tk,FG
+R,s/a\}bracketri}ht/a\}bracketri}htG/parenrightbig
+.
+2.25. Lemma : LetQ∈ Sp(G)and(R,s)∈ QG,p. Then
+FG
+R,s[Q] =/summationdisplay
+x∈NG(Q)\G/NG(R,s)
+Q/triangleleftequalx<Rs>FNG(Q)
+xR/Q,xs,
+(wherexsdenotes the image in NG(Q,xR)/xRof thex-conjugate of s).
+Proof :The Brauer map W/mapsto→W[Q] is a ring homomorphism from ppk(G)
+toppk/parenleftbig
+NG(Q)/parenrightbig
+. ThusFG
+R,s[Q] is an idempotent of ppk/parenleftbig
+NG(Q)/parenrightbig
+, hence a sum
+of distinct primitive idempotents FNG(Q)
+X,uassociated to some pairs ( X,u)∈
+QNG(Q),p. In other words Xis ap-subgroup of NG(Q), of the form Y/Q,
+for some p-subgroup YofGsuch that Q/triangleleftequalY, anduis ap′-element of
+NNG(Q)(X)∼=NG(Q,Y)/Y.
+Theidempotent FNG(Q)
+X,uappearsin FG
+R,s[Q]ifandonlyifthespecies tNG(Q)
+X,u
+takes the value 1 when evaluated at FG
+R,s[Q]. ButtNG(Q)
+X,u(FG
+R,s[Q]) is equal to
+the Brauer trace (see 2.15) of uonFG
+R,s[Q][Y/Q] = ResNG(Y)/Y
+NG(Q,Y)/YFG
+R,s[Y]. In
+other words
+tNG(Q)
+X,u(FG
+R,s[Q]) =tG
+Y,u(FG
+R,s),
+and this is equal to 1 if and only if the pair ( Y,u) isG-conjugate to the pair
+(R,s). The lemma follows.
+82.26. Lemma : Let(R,s)∈ QG,p. Then
+/a\}bracketle{tk,FG
+R,s/a\}bracketri}htG=/braceleftBiggφ(|R|)
+|NG(R,s)|if<sR>is cyclic
+0otherwise,
+whereφis the Euler totient function.
+Proof :Recall from 2.13 that
+FG
+R,s=1
+|R||s||CNG(R)(s)|/summationdisplay
+ϕ,L˜ϕ(s−1)|L|µ(L,<sR> )IndG
+LRes<sR>
+Lkϕ,
+whereϕruns through homomorphisms <s>→k×(i.e. equivalently ho-
+momorphisms <sR>→k×), andLthrough subgroups of <sR>such that
+LR=<sR>. Now
+/a\}bracketle{tk,IndG
+LRes<sR>
+Lkϕ/a\}bracketri}htG=/a\}bracketle{tk,kϕ/a\}bracketri}htL
+is equal to zero unless the restriction of ϕtoLis trivial, i.e. if L≤Kerϕ.
+SinceR≤Kerϕ, this implies that <sR>≤Kerϕ, i.e. that ϕis trivial.
+Thus
+/a\}bracketle{tk,FG
+R,s/a\}bracketri}htG=1
+|R||s||CNG(R)(s)|/summationdisplay
+LR=<sR>|L|µ(L,<sR> ).
+Now a subgroup LofH=<sR>is such that LR=Hif and only if it
+contains some conjugate of sinH, i.e. if it is of the form Q<xs>, for some
+x∈Rand some subgroup QofRnormalized byxs. In this case, there
+are|Q:CQ(xs)|conjugates of scontained in L. Moreover, since LR=H
+andL∩R=Op(H) =Q, the map X/mapsto→X∩R, from the poset ] L,H[ of
+proper subgroups of Hstrictly containing L, to the poset ] Q,R[L=]Q,R[xsof
+L-invariant proper subgroups of Rstrictly containing Q, is an isomorphism :
+the inverse isomorphism is the map Y/mapsto→Y·L. Thusµ(L,<sR> ) is equal
+to the value µ/parenleftbig
+(Q,R)xs/parenrightbig
+of the M¨ obius function of the poset of subgroups
+normalized byxs, and
+/a\}bracketle{tk,FG
+R,s/a\}bracketri}htG=1
+|R||s||CNG(R)(s)|/summationdisplay
+x∈R/CR(s)/summationdisplay
+Q≤R
+Qxs=Q|Q||s|µ/parenleftbig
+(Q,R)xs/parenrightbig
+/|Q:CQ(xs)|
+=1
+|R||CNG(R)(s)|/summationdisplay
+x∈R/CR(s)/summationdisplay
+Q≤R
+Qxs=Q|CQ(xs)|µ/parenleftbig
+(Q,R)xs/parenrightbig
+.
+9NowQis normalized byxsif and only if Qxis normalized by s. Moreover
+µ/parenleftbig
+(Q,R)xs/parenrightbig
+=µ/parenleftbig
+(Qx,R)s/parenrightbig
+, and|CQ(xs)|=|CQx(s)|. Thus
+/a\}bracketle{tk,FG
+R,s/a\}bracketri}htG=1
+|R||CNG(R)(s)|/summationdisplay
+x∈R/CR(s)/summationdisplay
+Q≤R
+Qs=Q|CQ(s)|µ/parenleftbig
+(Q,R)s/parenrightbig
+=1
+|CR(s)||CNG(R)(s)|/summationdisplay
+Q≤R
+Qs=Q|CQ(s)|µ/parenleftbig
+(Q,R)s/parenrightbig
+.
+Now
+/summationdisplay
+Q≤R
+Qs=Q|CQ(s)|µ/parenleftbig
+(Q,R)s/parenrightbig
+=/summationdisplay
+y∈CR(s)/summationdisplay
+y∈Q≤R
+Qs=Qµ/parenleftbig
+(Q,R)s/parenrightbig
+=/summationdisplay
+y∈CR(s)δ<y>,R
+=|{y∈CR(s)|<y>=R}|.
+This is non zero if and only if Ris cyclic and centralized by s, i.e. if the
+group<sR>is cyclic. In this case CR(s) =R, and
+/a\}bracketle{tk,FG
+R,s/a\}bracketri}htG=1
+|R||CNG(R)(s)|φ(|R|) =φ(|R|)
+|NG(R,s)|,
+since|R||CNG(R)(s)|=|NG(R,s)|, in view of Exact Sequence 2.12. This
+completes the proof of the lemma.
+2.27. Lemma : Let(R,s)∈ QG,p. Then
+/a\}bracketle{t/a\}bracketle{tk,FG
+R,s/a\}bracketri}ht/a\}bracketri}htG=1
+|CNG(R)(s)|/summationdisplay
+x∈R/[<sR>,R ]1
+|x|.
+Proof :By Definition 2.9
+/a\}bracketle{t/a\}bracketle{tk,FG
+R,s/a\}bracketri}ht/a\}bracketri}htG=/summationdisplay
+Q∈[Sp(G)]/a\}bracketle{tk,FG
+R,s[Q]/a\}bracketri}htNG(Q).
+10Hence, by Lemma 2.25
+/a\}bracketle{t/a\}bracketle{tk,FG
+R,s/a\}bracketri}ht/a\}bracketri}htG=/summationdisplay
+Q∈Sp(G)|NG(Q)|
+|G|/summationdisplay
+x∈NG(Q)\G/NG(R,s)
+Q/triangleleftequalx<sR>/a\}bracketle{tk,FNG(Q)
+xR/Q,xs/a\}bracketri}htNG(Q)
+=/summationdisplay
+Q∈Sp(G)
+x∈G
+Qx/triangleleftequal<sR>|NG(Q)|
+|G||NG(Qx)∩NG(R,s)|
+|NG(Q)||NG(R,s)|/a\}bracketle{tk,FNG(Qx)
+R/Qx,s/a\}bracketri}htNG(Qx)
+=/summationdisplay
+Q≤R
+Q/triangleleftequal<sR>|NG(Q)∩NG(R,s)|
+|NG(R,s)|/a\}bracketle{tk,FNG(Q)
+R/Q,s/a\}bracketri}htNG(Q)
+Now/a\}bracketle{tk,FNG(Q)
+R/Q,s/a\}bracketri}htNG(Q)= 0 by Lemma 2.26, unless R/Qis cyclic and cen-
+tralized by s, i.e. ifR/Qis cyclic and Qcontains the commutator subgroup
+[<sR>,R ],inwhichcaseitisequaltoφ(|R/Q|)
+|M|,whereM=NNG(Q)(R/Q,s).
+Now the two exact sequences
+1→R→NG(Q)∩NG(R,s)→CNG(Q,R)/R(s)→1
+1→R/Q→NNG(Q)(R/Q,s)→CNG(Q,R)/R(s)→1
+show that
+|NG(Q)∩NG(R,s)|
+|NNG(Q)(R/Q,s)|=|Q|.
+It follows that
+/a\}bracketle{t/a\}bracketle{tk,FG
+R,s/a\}bracketri}ht/a\}bracketri}htG=/summationdisplay
+[<sR>,R ]≤Q≤R
+R/Qcyclic|Q|φ(|R/Q|)
+|NG(R,s)|
+=1
+|CNG(R)(s)|/summationdisplay
+[<sR>,R ]≤Q≤R
+R/Qcyclicφ(|R/Q|)
+|R/Q|.
+Now the summation is equal to the summation over all cyclic quotients X=
+R/Qof the abelian group A=R/[<sR>,R ] ofφ(|X|)
+|X|, i.e. by a classical
+dualityargument, tothesummationofthisquantityover the cyclic subgroups
+ofA, i.e. finally to/summationdisplay
+x∈A1
+|x|. Thus
+/a\}bracketle{t/a\}bracketle{tk,FG
+R,s/a\}bracketri}ht/a\}bracketri}htG=1
+|CNG(R)(s)|/summationdisplay
+x∈R/[<sR>,R ]1
+|x|,
+as was to be shown.
+112.28. Theorem : LetGbe a finite group, let pbe a prime number, and kbe
+a field of characteristic p, big enough to be a splitting field for all the groups
+NG(Q)/Q, forQ∈ Sp(G). Then the determinant of the Cartan matrix of
+the algebra µk(G,1)is equal to
+detC/parenleftbig
+µk(G,1)/parenrightbig
+=/productdisplay
+R∈[Sp(G)]/productdisplay
+s∈[NG(R)p′]/parenleftbig
+|CNG(R)(s)|p/summationdisplay
+x∈R/[<sR>,R ]1
+|x|/parenrightbig
+.
+Proof :The Cartan matrix of µk(G,1) is independent of the field k, as long
+as it is big enough. So one can assume e.g. that kis algebraically closed,
+and choose a corresponding p-modular system ( K,O,k), whereKis also big
+enough. Then the formula for det C/parenleftbig
+µk(G,1)/parenrightbig
+follows from Equation 2.24
+and Lemma 2.27.
+2.29. Examples : Recall that kis a (big enough) field of characteristic p.
+•LetGbe a cyclic group of order pn. Then
+detC/parenleftbig
+µk(G,1)/parenrightbig
+=p(n
+2)n/productdisplay
+i=1/parenleftbig
+p+i(p−1)/parenrightbig
+.
+•LetGbe an elementary abelian group of order p2. Then
+detC/parenleftbig
+µk(G,1)/parenrightbig
+=p(2p−1)p+1(p2+p−1).
+3 The cohomological Mackey algebra
+3.1.The cohomological Mackey algebra coµR(G) of a finite group Gover
+a commutative ring Rhas also been introduced by Th´ evenaz and Webb
+(see [13] Section 16). It is an associative algebra with the property that the
+categoryofleft coµR(G)-modulesisequivalent tothecategory coMack R(G)of
+cohomological Mackey functors . The algebra coµR(G) is defined as a quotient
+of the Mackey algebra µR(G).
+In the case where Ris a field kof characteristic p, the cohomological
+Mackey functors for Goverkare projective relative to p-subgroups. In other
+words, the algebra coµk(G) is a quotient of µk(G,1).
+3.2.Th´ evenaz and Webb have shown that the projective cohomologica l
+12Mackey functors for GoverRare exactly the fixed points functors FPW,
+whereWis a direct summand of some permutation RG-module. Moreover,
+for anyRG-modules VandW,
+HomcoMack R(G)(FPW,FPV)∼=HomRG(W,V).
+In particular, the indecomposable projective coµk(G)-modules are the func-
+torsFPW, whereWis an indecomposable p-permutation kG-module. Thus,
+the Cartan matrix of the algebra coµk(G) is the square matrix indexed by
+the set [PG,p], defined by
+c(P,E),(Q,F)= dim kHomcoMack k(G)(FPMP,E,FPMQ,F) =/a\}bracketle{tMP,E,MQ,F/a\}bracketri}htG.
+3.3.Thus by the same argument already used in Section 2, the Cartan
+matrix of the algebra coµk(G) is the matrix of the bilinear form /a\}bracketle{t,/a\}bracketri}hton
+ppk(G). This extends to a bilinear form on K⊗Zppk(G), and by 2.8, for
+(R,s) and (R′,s′) inQG,p
+/a\}bracketle{tFG
+R′,s′,FG
+R,s/a\}bracketri}ht=/braceleftbigg/a\}bracketle{tk,FG
+R,s/a\}bracketri}htGif (R′,s′) =G(R,s−1)
+0 otherwise
+By Lemma 2.26, this is non zero if and only if the group <sR>is cyclic.
+Theargument ofparagraph2.19shows thattheCartanmatrixof coµk(G)
+can be expressed as
+C/parenleftbig
+coµk(G)/parenrightbig
+=T′·S′·tT,
+whereS′is the diagonal matrix indexed by [ QG,p] with (R,s)-diagonal entry
+equal to /a\}bracketle{tFG
+R,s,k/a\}bracketri}htG. SinceTandT′are invertible, it follows that the rank
+ofC/parenleftbig
+coµk(G)/parenrightbig
+is equal to the number of elements ( R,s) of [QG,p] such that
+<sR>is cyclic.
+3.4. Theorem : LetGbe a finite group, let pbe a prime number, and
+kbe a field of characteristic p, big enough to be a splitting field for all the
+groupsNG(Q)/Q, forQ∈ Sp(G). Then :
+1. The rank of the Cartan matrix of the algebra coµk(G)is equal to the
+number of G-conjugacy classes of pairs (R,s), whereR∈ Sp(G)and
+s∈NG(R)p′, such that <sR>is cyclic. In other words
+rkC/parenleftbig
+coµk(G)/parenrightbig
+=/summationdisplay
+R∈[Cp(G)]|NG(R)\CG(R)p′|
+=/summationdisplay
+s∈[Gp′]cp/parenleftbig
+CG(s)/parenrightbig
+,
+13where[Cp(G)]denotes a set of representatives of conjugacy classes of
+cyclicp-subgroups of a group G, andcp(G)is the number of such con-
+jugacy classes. Moreover, in the first summation, |NG(R)\CG(R)p′|
+denotes the number of NG(R)-conjugacy classes of p-regular elements
+ofCG(R).
+2. The Cartan matrix of the algebra coµk(G)is non singular if and only if
+the group Gisp-nilpotent with cyclic Sylow p-subgroups. In this case,
+ifG=N⋊P, whereNis ap′-group and Pis a cyclic group of order pn
+detC/parenleftbig
+coµk(G)/parenrightbig
+=/productdisplay
+R≤P/parenleftbiggφ(|R|)
+|R|/parenrightbigglp/parenleftbig
+NG(R)/parenrightbig
+detC/parenleftbig
+kNG(R)/parenrightbig
+=/productdisplay
+R≤P/productdisplay
+x∈/bracketleftbig
+P\[CN(R)]/bracketrightbig/parenleftbiggφ(|R|)
+|R||CCN(R)⋊P/R(x)|p/parenrightbigg
+,
+wherelp/parenleftbig
+NG(R)/parenrightbig
+is the number of p-regular classes of NG(R), where
+C/parenleftbig
+kNG(R)/parenrightbig
+is the Cartan matrix of the group algebra kNG(R), and/bracketleftbig
+P\[CN(R)]/bracketrightbig
+is a set of representatives of CN(R)⋊P-conjugacy classes
+inCN(R).
+Proof :The first sentence of Assertion 1 follows from the above argument s,
+and the second one follows by counting the number of pairs ( R,s) of [QG,p]
+such that <sR>is cyclic, in two different ways. For Assertion 2, observe
+that the Cartan matrix C/parenleftbig
+coµk(G)/parenrightbig
+is non singular if and only if for any
+(R,s)∈ QG,p, the group <sR>is cyclic. This amounts to saying that the
+Sylowp-subgroups of Gare cyclic, and that whenever s∈Gp′normalizes
+ap-subgroup R, it centralizes it. In other words, the group NG(R)/CG(R)
+is ap-group, for any R∈ Sp(G). This is equivalent to saying that Gis
+p-nilpotent, by the theorem of Frobenius ([11] Theorem 4.5).
+In the case G=N⋊P, wherePis cyclic of order pn, then as in 2.24
+detC/parenleftbig
+coµk(G)/parenrightbig
+= detT′detS′dettT
+=/productdisplay
+R∈[Sp(G)]/productdisplay
+s∈[NG(R)p′]/parenleftbig
+|CNG(R)(s)|p|CNG(R)(s)|/a\}bracketle{tk,FG
+R,s/a\}bracketri}htG/parenrightbig
+=/productdisplay
+R∈[Sp(G)]
+detC/parenleftbig
+kNG(R)/parenrightbig/productdisplay
+s∈[NG(R)p′]/parenleftbig
+|CNG(R)(s)|/a\}bracketle{tk,FG
+R,s/a\}bracketri}htG/parenrightbig
+.
+14The set [ Sp(G)] can be chosen to be the set of subgroups of P. Moreover by
+Lemma 2.26, for R≤Pands∈NG(R)p′
+/a\}bracketle{tk,FG
+R,s/a\}bracketri}htG=φ(|R|)
+|NG(R,s)|=φ(|R|)
+|R||CNG(R)(s)|.
+It follows that
+detC/parenleftbig
+coµk(G)/parenrightbig
+=/productdisplay
+R≤P/parenleftbiggφ(|R|)
+|R|/parenrightbigglp/parenleftbig
+NG(R)/parenrightbig
+detC/parenleftbig
+kNG(R)/parenrightbig
+.
+Finally the p′-elements of the group NG(R)∼=CN(R)⋊(P/R) are the ele-
+ments of CN(R). The last formula of the theorem follows.
+3.5. Remark : Thus when it is non zero, the determinant of the Cartan
+matrix of coµk(G) is equal to ( p−1)npm, for suitable non negative integers
+nandm.
+4 Blocks of cohomological Mackey functors
+4.1.It was shown by Th´ evenaz and Webb (see [13] Theorem 17.1 and its
+proof, see also [4])) that the blocks of the algebra µk(G,1) and the blocks of
+the algebra coµk(G) are in one to one correspondence with the blocks of the
+group algebra kG. Ifbis a block of kG, denote by coµk(b) the corresponding
+block ofcoµk(G). When Ris ap-subgroup of G, let Br R: (kG)R→kCG(R)
+denote the Brauer morphism. If moreover R/triangleleftequalG, denote by u/mapsto→uthe
+projection map kG→k(G/R). When Ais ak-algebra, denote by Irr k(A)
+the set of isomorphism classes of simple A-modules.
+This section is devoted to the proof of the following block version of
+Theorem 3.4 for such blocks of cohomological Mackey functors :
+4.2. Theorem : LetGbe a finite group, let pbe a prime number, and k
+be an algebraically closed field of characteristic p. Let moreover bbe a block
+ofkG. Then :
+1. The rank of the Cartan matrix of the algebra coµk(b)is equal to
+rkC/parenleftbig
+coµk(b)/parenrightbig
+=/summationdisplay
+R∈[Cp(G)]|NG(R)\Irrk/parenleftbig
+kCG(R)BrR(b)/parenrightbig
+|
+=/summationdisplay
+(R,c)∈[Cp(b)]|NG(R,c)\Irrk/parenleftbig
+kCG(R)c/parenrightbig
+|,
+15where[Cp(b)]is a set of representatives of G-conjugacy classes of b-
+Brauer pairs (R,c)for which Ris cyclic.
+2. The Cartan matrix of the block coµk(b)is non singular if and only if
+bis a nilpotent block with cyclic defect groups.
+4.3.The Cartan matrix Cofcoµk(b) is non singular if and only if the rows
+of the decomposition matrix Dare linearly independent (since C=D·tD) :
+indeed, if a vector uis such that Cu= 0, thentuCu=t(tDu)·(tDu) = 0,
+thustDu= 0. Conversely, iftDu= 0, then obviously Cu= 0.
+4.4.Thedecompositionmatrix Dofcoµk(b)hasrowsindexedbythe(isomor-
+phism classes of) indecomposable p-permutation kG-modules in the block b,
+and columns indexed by the (isomorphism classes of) simple KG-modules
+in the block b. The coefficient DW,χcorresponding to the indecomposable
+p-permutation kGb-module Wand the simple KG-module χis equal to the
+multiplicity of χinK⊗O˜W, where˜Wis anOG-module lifting WtoO(i.e.
+such that k⊗O˜W∼=W). Such an OG-module is unique up to isomorphism,
+sinceWis ap-permutation module. The character of the module K⊗O˜W
+will be called the (ordinary) character ofW.
+It follows that the rank of the Cartan matrix of coµk(b) is equal to
+the dimension of the subspace of Q⊗ZRK(b) generated by characters of
+p-permutation modules in the block b.
+In particular, the Cartan matrix of coµk(b) is non singular if and only if
+the ordinary characters of the indecomposable p-permutation modules in b
+are linearly independent.
+4.5.Recall (see 2.15, and [8] Proposition 3.3) that the value of the ordina ry
+character of a p-permutation module Won an element sofGis equal to
+the Brauer trace BrTr( sp′|W[<sp>]), where spandsp′are thep-part and
+p′-part ofs, respectively.
+4.6. Notation : LetZp(G) denote the set of pairs ( P,E) consisting of
+a cyclic p-subgroup PofGand an indecomposable projective kCG(P)-
+moduleE, whereCG(P) =CG(P)/P, and let [ Zp(G)] be a set of repre-
+sentatives of G-orbits on Zp(G).
+LetZp(b) denote the subset of Zp(G) consisting of pairs ( P,E) such that
+BrP(b)InfCG(P)
+CG(P)E= InfCG(P)
+CG(P)E .
+Set moreover [ Zp(b)] =Zp(b)∩[Zp(G)].
+164.7. Lemma : The characters of the modules IndG
+CG(P)InfCG(P)
+CG(P)E, for
+(P,E)∈[Zp(G)], form a basis of the subspace of Q⊗ZRK(G)generated by
+the characters of the p-permutation modules.
+Proof : Since projective modules are p-permutation modules, and since in-
+duction and inflation preserves this class of modules, the module LP,E=
+IndG
+CG(P)InfCG(P)
+CG(P)E, for (P,E)∈ Zp(G), is ap-permutation module. Up to
+isomorphism, this module depends only on the G-orbit of ( P,E). Moreover
+the number of G-orbits on the set Zp(G) is equal to
+|G\Zp(G)|=/summationdisplay
+P∈[Cp(G)]|NG(P)\Irrk/parenleftbig
+kCG(P)/parenrightbig
+|.
+Indeed, the number of isomorphism classes of indecomposable proj ective
+kCG(P)-modules is equal to the number of isomorphism classes of sim-
+plekCG(P)-modules, i.e. to the number of isomorphism classes of simple
+kCG(P)-modules, since Pis a normal p-subgroup of CG(P).
+By Theorem 3.4, it follows that the cardinality of the set [ Zp(G)] is pre-
+cisely equal to the rank of the Cartan matrix of the algebra coµk(G), i.e. to
+the dimension of the subspace of Q⊗ZRK(G) generated by the characters of
+thep-permutation modules. Thus, to prove Lemma 4.7, it is enough to pro ve
+that the characters of the modules LP,E, for (P,E)∈[Zp(G)], are linearly
+independent.
+The character χP,Eof the module LP,Eis equal to IndG
+CG(P)InfCG(P)
+CG(P)ΦE,
+where Φ Eis the character of the module E. Suppose that some non trivial
+linearcombinationofthesecharactersisequalto0, i.e. thatthere areintegers
+nP,E∈Z, for (P,E)∈[Zp(G)], not all equal to 0, such that
+(4.8)/summationdisplay
+(P,E)∈[Zp(G)]nP,EχP,E= 0.
+LetQbe maximal such that there exists ( Q,F)∈[Zp(G)] withnQ,F/\e}atio\slash= 0,
+and letsbe a generator of the cyclic group Q. By 4.5, for any t∈CG(s)p′,
+the value of the linear combination 4.8 at the element stis equal to
+0 =/summationdisplay
+(P,E)∈[Zp(G)]nP,EBrTr(t|LP,E[Q]).
+ButLP,E[Q] = 0, unless some conjugate of Qis contained in P. By maxi-
+mality of Q, it follows that
+0 =/summationdisplay
+EnQ,EBrTr(t|LQ,E[Q]),
+17whereErunsthroughaset[ P]ofindecomposableprojective kCG(Q)-modules,
+up to isomorphism and conjugation by NG(Q). Since moreover
+LQ,E[Q]∼=IndNG(Q)
+CG(Q)InfCG(Q)
+CG(Q)E ,
+it follows that for any t∈CG(Q)p′
+/summationdisplay
+E∈[P]nQ,E/parenleftbig
+IndNG(Q)
+CG(Q)InfCG(Q)
+CG(Q)ΦE/parenrightbig
+(t) = 0.
+This is also equal to
+/summationdisplay
+E∈[P]nQ,E/parenleftbig
+ResNG(Q)
+CG(Q)IndNG(Q)
+CG(Q)InfCG(Q)
+CG(Q)ΦE/parenrightbig
+(t) =/summationdisplay
+E∈[P]
+x∈NG(Q)/CG(Q)nQ,EΦE(xt)
+=/summationdisplay
+E∈[P]
+x∈NG(Q)/CG(Q)nQ,EΦEx(t),
+wheretis the image of tinCG(Q), andExis the image of Eby conjugation
+byx∈NG(Q).
+Since the map t/mapsto→tis a surjection from CG(Q)p′toCG(Q)p′, it follows
+that the modular character/summationtext
+E∈[P]
+x∈NG(Q)/CG(Q)nQ,EΦExofCG(Q) is equal to zero.
+But the set of modules Ex, forE∈[P] andx∈NG(Q)/CG(Q), is exactly
+the set of projective indecomposable kCG(Q)-modules, up to isomorphism.
+Thus /summationdisplay
+F∈QnQ,FmFΦF= 0,
+whereQis a set of representatives of isomorphism classes of indecomposab le
+projective kCG(Q)-modules, where nQ,Fis defined as nQ,Eif (Q,E)∈[P]
+and if there exists x∈NG(Q) such that Ex∼=F, and where mFis the
+number of elements x∈NG(Q)/CG(Q) such that Fx∼=F.
+Now the characters Φ F, forF∈ Q, are linearly independent, and it
+follows that nQ,F= 0 for any F. This contradicts the definition of Q, and
+completes the proof of Lemma 4.7.
+4.9. Notation : LetP∈ Sp(G), andEbe any projective kNG(P)-module.
+ThenEsplits as a direct sum ⊕
+i∈IEiof indecomposable kNG(P)-modules Ei.
+In this situation, set MP,E=⊕
+i∈IMP,Ei.
+184.10. Lemma : The characters of the modules MP,IndNG(P)
+CG(P)E, for(P,E)∈
+[Zp(b)], form a basis of the subspace of Q⊗ZRK(b)generatedby the characters
+of thep-permutation modules in the block b.
+Proof : Let (P,E)∈ Zp(b). The module M= IndNG(P)
+CG(P)Esplits as a direct
+sum⊕
+i∈IEiof indecomposable kNG(P)-modules Ei. Since Br P(b) isNG(P)-
+invariant and belongs to kCG(P), it follows that BrP(b) acts as the identity
+onM, hence on every direct summand Ei. Recall that the indecompos-
+able module MP,Eibelongs to the block bif and only if BrP(b)Ei=Ei(see
+[2] Corollary 6.3.2). Hence all the indecomposable kG-modules MP,Eiare in
+the block b, and their direct sum MP,IndNG(P)
+CG(P)Eis also in b.
+To prove Lemma 4.10, it suffices to observe that the sets Zp(b), when
+bruns through the blocks of kG, form a partition of Zp(G), and to prove
+that the characters of the modules MP,IndNG(P)
+CG(P)E, for (P,E)∈[Zp(G)], form
+a basis of the subspace of Q⊗ZRK(G) generated by the characters of the
+p-permutation modules.
+For this, observe that
+IndG
+CG(P)InfCG(P)
+CG(P)E∼=IndG
+NG(P)InfNG(P)
+NG(P)IndNG(P)
+CG(P)E
+is a direct sum of MP,IndNG(P)
+CG(P)Eand of indecomposable modules with vertex
+strictly contained in Pup to conjugation. This yields a triangular transition
+matrix, withnonzero diagonalcoefficients, andsuch amatrixchang es abasis
+to another basis.
+4.11.The first equality in Assertion 1 of Theorem 4.2 follows trivially from
+Lemma 4.10. The second one follows from the fact that, for any b-Brauer
+pair (R,c)
+|NG(R)\Irrk/parenleftbig
+kCG(R)TrNG(R)
+NG(R,c)c/parenrightbig
+|=|NG(R,c)\Irrk/parenleftbig
+kCG(R)c/parenrightbig
+|.
+This is because the algebra kCG(R)TrNG(R)
+NG(R,c)cis isomorphic to the direct sum
+of the block algebras kCG(R)xc, forx∈[NG(R)/NG(R,c)], which are tran-
+sitively permuted by NG(R). Moreover the stabilizer in NG(R) ofkCG(R)c
+is equal to NG(R,c).
+4.12.For anyp-subgroup RofG, induction from CG(R) toNG(R) induces
+19an inequality
+(4.13)|NG(R)\Irrk/parenleftbig
+kCG(R)BrR(b)/parenrightbig
+| ≤ |Irrk/parenleftbig
+kNG(R)BrR(b)/parenrightbig
+|.
+Indeed, the left hand side is equal to the rank of the group
+Pk/parenleftbig
+kCG(R)BrR(b)/parenrightbig
+NG(R)
+ofNG(R)-coinvariants on the group of projective kCG(R)BrR(b)-modules. It
+is easy to see that IndNG(R)
+CG(R)induces an injective map from this group to
+the group Pk/parenleftbig
+kNG(R)BrR(b)/parenrightbig
+of projective kNG(R)BrR(b)-modules, and the
+rank of this group is equal to the right hand side of 4.13.
+The Cartan matrix of coµk(b) is non singular if and only if
+/summationdisplay
+R∈[Cp(G)]|NG(R)\Irrk/parenleftbig
+kCG(R)BrR(b)/parenrightbig
+|=/summationdisplay
+R∈[Sp(G)]|Irrk/parenleftbig
+kNG(R)BrR(b)/parenrightbig
+|.
+Indeed, thelefthandsideistherankoftheCartanmatrix, andthe righthand
+side the size of this matrix, i.e. the number of indecomposable p-permutation
+modules in the block b, up to isomorphism.
+By inequality 4.13, this is in turn equivalent to the following equality, for
+anyR∈ Sp(G) :
+|Irrk/parenleftbig
+kNG(R)BrR(b)/parenrightbig
+|=/braceleftbigg
+|NG(R)\Irrk/parenleftbig
+kCG(R)BrR(b)/parenrightbig
+|ifRis cyclic
+0 otherwise .
+4.14.Hence if the Cartan matrix of coµk(b) is non singular, and bhas
+defectD, then in particular Br D(b)/\e}atio\slash= 0, and |Irrk/parenleftbig
+kNG(D)BrD(b)/parenrightbig
+| /\e}atio\slash= 0,
+soDis cyclic. Let ( D,c) be a maximal b-Brauer pair. Then Br D(b) =
+TrNG(D)
+NG(D,c)c, and the algebra kNG(D)BrD(b) is isomorphic to a matrix algebra
+overkNG(D,c)c. In particular
+(4.15) |Irrk/parenleftbig
+kNG(D)BrD(b)/parenrightbig
+|=|Irrk/parenleftbig
+kNG(D,c)c/parenrightbig
+|.
+Moreover Br D(b) splits as a sum of blocks of CG(D), which are the distinct
+NG(D)-conjugates of c. Hence, the orbits of NG(D) on Irr k/parenleftbig
+kCG(D)BrD(b)/parenrightbig
+areinonetoonecorrespondencewiththeorbitsof NG(D,c)onIrr k/parenleftbig
+kCG(D)c/parenrightbig
+.
+It follows that
+|NG(D,c)\Irrk/parenleftbig
+kCG(D)c/parenrightbig
+|=|Irrk/parenleftbig
+kNG(D,c)c/parenrightbig
+|.
+Now the group NG(D,c)/CG(D) is the inertial quotient of b. It is a cyclic
+group of order edividingp−1. The block cis a nilpotent block of CG(D), so
+20in particular there is a unique simple kCG(D)c-module S, which is invariant
+byNG(D,c). This simple module can be extended to a simple kNG(D,c)c
+module in edifferent ways, by the following argument (see [2], proof of
+Proposition 6.5.4) : let gbe a generator of the group NG(Q)/CG(Q), and
+θ:S→Sganisomorphismof kCG(Q)-modules. Thenthemap ge(s)/mapsto→θe(s)
+is akCG(Q) automorphism of S, because ge∈CG(Q). Askis algebraically
+closed, there is a scalar µ∈ksuch that ge(s) =µθe(s), for any s∈S.
+Moreover there are edistinct elements λ∈ksuch that λe=µ. For each
+suchλ, onecanlet gactonSbyλθ, andthis gives emutually nonisomorphic
+extensions of Sto a simple kNG(Q,c)-module, which are all in the block c
+sincec∈kCG(Q).
+These modules are not isomorphic to each other (since their restric tions
+to<g>are not). Hence e= 1 by 4.15, since |Irrk/parenleftbig
+kCG(D)c/parenrightbig
+|= 1, and this
+is equivalent to saying that bis nilpotent, since bhas cyclic defect.
+4.16.Conversely, if bis a nilpotent block with a cyclic defect group D, then
+for each R∈ Sp(G), either Br R(b) = 0 ifRis not contained in Dup toG
+conjugation, or Br R(b) is a sum/summationtext
+i∈Ibiof blocks biofNG(R), ifRis contained
+inDup to conjugation, and in that case Ris cyclic since Dis. Each of
+the blocks biis equal to TrNG(R)
+NG(R,ci)ci, where ( R,ci) is some b-Brauer pair. In
+particular
+|Irrk/parenleftbig
+kNG(R)BrR(b)/parenrightbig
+|=/summationdisplay
+i∈I|Irrk/parenleftbig
+kNG(R)BrR(bi)/parenrightbig
+|
+=/summationdisplay
+i∈I|Irrk/parenleftbig
+kNG(R,ci)ci/parenrightbig
+|.
+Since the block bis nilpotent, all the groups NG(R,ci)/CG(R) arep-groups,
+so|Irrk/parenleftbig
+kNG(R,ci)ci/parenrightbig
+|=|NG(R,ci)\Irrk/parenleftbig
+kCG(R)ci/parenrightbig
+|, and this is equal to 1
+sinceciis a nilpotent block of CG(R) by Theorem 1.2 of [9]. It follows that
+|Irrk/parenleftbig
+kNG(R)BrR(b)/parenrightbig
+|=|I|=|NG(R)\Irrk/parenleftbig
+kCG(R)BrR(b)/parenrightbig
+|.
+This implies that the Cartanmatrix of coµk(b) is non singular, andcompletes
+the proof of Theorem 4.2.
+References
+[1] D. Benson. Modular representation theory: New trends and methods ,
+volume 1081 of Lecture Notes in Mathematics . Springer, 1984.
+21[2] D. Benson. Representations and cohomology I , volume 30 of Cambridge
+studies in advanced mathematics . Cambridge University Press, 1991.
+[3] S. Bouc. R´ esolutions de foncteurs de Mackey. In Group representations:
+Cohomology, Group Actions andTopology ,volume63,pages31–83.AMS
+Proceedings of Symposia in Pure Mathematics, 1998.
+[4] S. Bouc. The p-blocks of the Mackey algebra. Algebras and Represen-
+tation Theory , 6:515–543, 2003.
+[5] S. Bouc and J. Th´ evenaz. The primitive idempotents of the p-
+permutation ring. Journal of Algebra , 323:2905-2915, 2010.
+[6] R. Brauer and C. Nesbitt. On the modular characters of groups .Ann.
+of Maths, 42(2):556–590, 1941.
+[7] M. Brou´ e. On Scott modules and p-permutation modules: an approach
+through the Brauer morphism. Proc.AMS , 93(3):401–408, March 1985.
+[8] M. Brou´ e. Rickard equivalences and block theory , volume 211 of London.
+Math. Soc. Lecture Note , pages 58–79. Cambridge Univ. Press, 1995.
+[9] M. Brou´ e and L. Puig. A Frobenius theorem for blocks. Invent. Math. ,
+56:117–128, 1980.
+[10] W. Feit. The representation theory of finite groups . North Holland
+Publishing Company, 1982.
+[11] W. Gorenstein. Finite groups . Chelsea, 1968.
+[12] M. Nicollerat. Foncteurs de Mackey projectifs . PhD thesis, Ecole Poly-
+technique F´ ed´ erale de Lausanne, 2008.
+[13] J. Th´ evenaz and P. Webb. The structure of Mackey functor s.Trans.
+Amer. Math. Soc. , 347(6):1865–1961, June 1995.
+Serge Bouc
+LAMFA - CNRS UMR 6140
+Universit´ e de Picardie Jules Verne, 33, rue St Leu
+80039 - Amiens - FRANCE.
+email : serge.bouc@u-picardie.fr
+web : http://www.mathinfo.u-picardie.fr/bouc/
+22
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+arXiv:1001.0322v3  [math.CV]  24 Feb 2015A Brian¸ con-Skoda type result for a non-reduced analytic
+space
+Jacob Sznajdman
+Abstract. We present here an analogue of the Brian¸ con-Skoda
+theorem for a germ of an analytic space Zat 0, such that OZ,0is
+not necessarily reduced.
+1.Introduction
+The Brian¸ con-Skoda theorem, [14], states that for any ideal a⊂
+OCn,0generated by mgerms, we have the inclusion amin(m,n)+r−1⊂ar,
+whereIdenotes the integral closure of I. We will refer to this theorem
+as the classical Brian¸ con-Skoda theorem. The generalization to a n
+arbitrary regular Noetherian ring was proven algebraically in [21].
+Huneke, [18], showed that for a quite general Noetherian reduced
+local ringSthere is an integer Nsuch that aN+r−1⊂arfor all ideals
+a⊂Sandr≥1. In particular this applies when S=OV,0, the local
+ring of holomorphic functions of a germ of a reduced analytic space
+V. This case of Huneke’s theorem was recently reproven analytically,
+[7]. Assume that ais generated by some elements aiand let|a|2=/summationtextm
+1|ai|2; up to constants this does not depend on the choice of the
+generators. Since a function φinaMis characterized by the property
+that|φ| ≤C|a|M, [20], an equivalent formulation of the theorem is that
+φbelongs to arwhenever |φ| ≤C|a|N+r−1(onV).
+We will consider a germ of an analytic space, that is, a pair Z=
+(X,OZ,0) of a germ of a reduced analytic variety X⊂Cnat 0∈Cn
+and its local ring OZ,0=OCn,0/J, whereJ ⊂ O Cn,0is an ideal such
+thatZ(J) =X. We assume throughout this paper that Jhas pure
+dimension.
+The aim of this paper is to find an appropriate generalization of the
+Brian¸ con-Skoda theorem to this setting – when S=OZ,0. Now that we
+have dropped the assumption that Zis reduced, the situation becomes
+different; the integral closure of any ideal contains the nilradical√
+0 by
+definition, so aN⊂acan only hold if√
+0⊂a. In the following example
+we consider the most simple non-reduced space, which will help to
+illustrate some general notions and our main result Theorem 1.2.
+Date: November 17, 2018.
+2000Mathematics Subject Classification. 32C30, 13B22, 32A26.
+12 JACOB SZNAJDMAN
+Example 1.1. Consider the analytic space Zsuch that
+X={w= 0}=Cn−1⊂Cn
+OZ=C[[z1,...zn−1,w]]/(wk), k≥2.
+The nilradical is ( w), and is not contained in a= (w2) ifk>2. It may
+be helpful to think of the space ZasCn−1with an extra infinitesimal
+direction transversal to X, and its structure sheaf being the k:th order
+Taylor expansions in that direction. For each f∈ OZ,0we have
+f(z,w) =k−1/summationdisplay
+i=0∂if
+∂wi(z,0)wi
+i!,and
+OZ,0≃ O⊕k
+X,0.
+Although the function wis identically zero on X(so that|w| ≤C|a|M
+onXfor anyM), the element wdoes not belong to a= (w2). SinceZ
+is non-reduced, evaluating w, or any other element, as a function on X
+does not give enough information to determine ideal membership. We
+also have to take into account the transversal derivatives.
+A germ of a holomorphic differential operator Lis calledNoetherian
+with respect to an ideal J ⊂ O Cn,0ifLφ∈√
+Jfor allφ∈ J. We say
+thatL1,...,L Misadefiningset ofNoetherianoperatorsfor Jifφ∈ J
+if and only if L1φ,...,L Mφ∈√J. The existence of a defining set for
+any ideal Jis due to Ehrenpreis [15] and Palamodov [23], see also [12],
+[17] and [22]. As for the example above, 1 ,∂/∂w,...,∂k−1/∂wk−1is a
+defining set for ( wk).
+IfLis Noetherian with respect to J, thenLψis a well-defined func-
+tion onXfor anyψ∈ OZ,0, andLinduces a mapping L:OZ,0→ OX,0.
+LetN(Z) be the set of all such mappings; this set does not depend on
+the local representation of Zas a subscheme of Cn. If (Li) is a defin-
+ing set for J, then by definition any element φ∈ OZ,0is determined
+uniquely by the tuple of functions ( Liφ) onX, cf. Example 1.1. This
+fact indicates that it is natural to impose size conditions on the whole
+set (Liφ) to generalize the Brian¸ con-Skoda theorem:
+Theorem 1.2. LetZbe a germ of an analytic space such that OZ
+has pure dimension. Then there exists an integer Nand operators
+L1,...,L M∈ N(Z)such that for all ideals a⊂ OZand allr≥1,
+|Ljφ| ≤C|a|N+r−1onX,1≤j≤M, (1)
+implies that φ∈ar.
+In Section 5 we give a version of Theorem 1.2 for the case when X
+is smooth.
+Andersson and Wulcan gave in [10] a proof of a global version of the
+Brian¸ con-Skoda-Huneke theorem on a reduced singular variety, which
+is a version of the effective Nullstellensatz. The proof is based on the3
+corresponding local result in [7]. One can therefore hope that Theo -
+rem 1.2 can be globalized in a similar way, cf., Remark 4.3 below.
+Although the formulation of Theorem 1.2 is intrinsic, we will choose
+an embedding and work in the ambient space exclusively. For the
+remainder of this paper we fix a choice of functions aj∈ OCn,0, 1≤
+j≤m, so that theimages of ajinOZ,0generate a. We will also identify
+φ∈ OZ,0with an arbitrary representative in OCn,0, and each operator
+L∈ N(Z)withanoperatoron Cnthatrepresents it. Withthispointof
+view, we are to show that if the inequalities (1) hold on X⊂Cn, then
+(the representative of) φbelongs to ( a)r+J, where (a) = (a1,...,a m).
+Example 1.3. In this example we shall give a direct proof of Theo-
+rem 1.2, based on the classical Brian¸ con-Skoda theorem, for the ana-
+lytic space in Example 1.1. For simplicity, we will assume that r= 1.
+As we saw above, a set of defining differential operators for the ide al
+(wk) is formed by Lj=∂j/∂wj, 0≤j≤k−1. Let
+̺= min(n−1,m),and
+|a|2=m/summationdisplay
+1|ai|2.
+We will prove that if
+|Ljφ|/lessorsimilar|a|̺+k−1−j, (2)
+onX={w= 0}, thenφ∈(a) + (wk). Thus Theorem 1.2 follows
+in this case with N=̺+k−1. We will also show that none of the
+hypotheses (2) can be relaxed.
+In the proof, we will allow ourselves to abuse notation; for example,
+we will write simply ( a) when we actually are referring to some element
+thatbelongsto( a). From(2)andtheclassicalBrian¸ con-Skodatheorem
+forOX,0=OCn,0/(w), we get
+∂jφ
+∂wj= (a)k−j+kjw, k j∈ OCn,0. (3)
+We will show inductively that
+φ=p/summationdisplay
+i=0wi(a)k−i+gpwp+1, gp∈ OCn,0, (4)
+holds forp≤k−1. First assume that p= 0. Then (4) reduces to (3)
+withj= 0. Now assume that (4) holds for some p < k−1. Let us
+differentiate (4) p+1 times with respect to w, and compare the result
+with (3) for j=p+1. This gives
+gp∈(a)k−p−1+(w).4 JACOB SZNAJDMAN
+If we substitute this back into (4), we get
+φ∈p+1/summationdisplay
+i=0wi(a)k−i+(wp+2).
+This completes the proof of (4). Now note that for p=k−1, (4)
+implies that φ∈(a)+(wk), which completes the proof.
+Finally, we will show that if any of the hypotheses (2) are relaxed,
+it is possible that φ /∈(a)+(wk). To this end, we need to find φpfor
+each 0≤p≤k−1, such that φp/∈(a)+(wk) and
+|∂j
+wφp| ≤ |a|̺+k−1−j, j/\e}atio\slash=p (5)
+|∂p
+wφp| ≤ |a|̺+k−2−p. (6)
+Taken= 2 and (a) = (z+w). Then̺= 1, sincen−1 =m= 1. A
+suitable choice is now φp=wpzk−1−p. It is easy to verify (5)-(6). The
+functionφpdoes not belong to ( a) +(wk), because if it did we would
+have
+wpzk−1−p−(z+w)(a0(z)+···+ak−1(z)wk−1)∈(wk),
+which would give a0=···=ap−1= 0 andzap=zk−1−p,zap+1=−ap,
+zap+2=−ap+1, etc, soak−1=±1/z. This is a contradiction since ak−1
+is holomorphic at 0 ∈Cn.
+The idea of the proof of Theorem 1.2 is to use a certain residue cur-
+rent associated to ( a)randJ; ifφ∈ OCn,0annihilates this current,
+then by solving a sequence of ∂-equations it follows that φbelongs to
+(a)r+J, so that the image of φinOZ,0belongs to ar. Alternatively,
+one can also use a division formula to obtain an explicit integral repre-
+sentation of the membership φ∈(a)r+Jwheneverφannihilates the
+associated residue current. These ways of proving ideal members hip
+are used in [7] and go back to [1] and [5]. We will show in Section 4
+thatφannihilates the residue current mentioned above whenever (1)
+holds.
+2.Coleff-Herrera currents and Noetherian differential
+operators
+Assume that Xis a germ of an analytic set of pure codimension
+pat 0∈Cn. Letµbe a current of bidegree (0 ,p) with support on
+X. Throughout this paper we let χbe a smooth function such that
+χ≡0 on [0,x1] andχ≡1 on [x2,∞) for some 0 < x1< x2<
+∞. One says that µhas the standard extension property (SEP) if
+µ= lim ε→0χ(|h|2/ε2)µfor anyh∈ OCn,0that does not vanish iden-
+tically on any component of X.
+Definition 2.1. A current of bidegree (0 ,p) with support on Xis
+a Coleff-Herrera current on Xif it is∂-closed, has the SEP and is
+annihilated by φfor allφ∈ OCn,0that vanish on X.5
+The set of all Coleff-Herrera currents on Xis anOCn,0-module which
+wedenoteby CHX. Foranyµ∈ CHX, annµisapure-dimensionalideal
+whose associated primes correspond to the irreducible component s of
+X. InTheorem 2.2below, which isduetoBj¨ ork, [12], wehave principal
+value integrals of the form
+/integraldisplay
+Xξ
+hk:= lim
+ε→0/integraldisplay
+Xχ(|h|2/ε2)ξ
+hk, (7)
+whereh∈ OX,0andξis a test form. For details about the existence of
+such integrals, see e.g. [13]. The right hand side of (7) is independent
+of the choice of χas long as it has the properties mentioned above. We
+will use the symbol /rightanglesefor interior multiplication of a differential form
+by a section of/logicalandtextpT(Cn).
+Theorem 2.2 (Bj¨ ork, [12]) .Givenµ∈ CHX, there is a multi-index
+M, two finite sets of holomorphic differential operators {Lα}and{Kα}
+forα≤M, an integer N0, an analytic function h, and a smooth section
+Sof/logicalandtextpT(Cn), such that {Lα}is a defining set for the ideal annµand
+φµ.ξ=/integraldisplay
+X/summationdisplay
+α≤M1
+hN0Lα(φ)Kα(S/rightangleseξ), (8)
+for any holomorphic function φand test form ξ.
+Proof.The proof is essentially that of Bj¨ ork, [12], but we include it for
+thereader’sconvenience since ourformulationofthetheoremisslig htly
+different. It follows from the local parametrization theorem that o ne
+can find holomorphic functions f1,...,f pforming a complete intersec-
+tion, such that Xis a union of a number of irreducible components of
+Vf={f1=···=fp= 0}anddf1∧...∧dfp/\e}atio\slash= 0 generically on X. Let
+the firstn−pcoordinates in Cnbe denoted by ζand the last pones
+byη. We then get local holomorphic coordinates
+z=ζ
+w=f(ζ,η) (9)
+outside the hypersurface Wdefined by
+h:= det∂f
+∂η. (10)
+By possibly rotating the coordinates ( ζ,η), we can make sure that h
+will not vanish identically on any irreducible component of X.
+We will first show that (8) holds for some operators Lαwhenξhas
+support outside of {h= 0}. We will then see that it follows from this
+special case of (8) that the operators Lαindeed form a defining set for
+annµ. Finally, we will show that (8) holds for general ξ.6 JACOB SZNAJDMAN
+Sinceµis a Coleff-Herrera current on the complete intersection Vf,
+we get by by Theorem 4.2 in [3] that
+µ=A/bracketleftBigg
+∂1
+f1+M1
+1∧···∧∂1
+f1+Mp
+p/bracketrightBigg
+(11)
+for some integers Mjand holomorphic function A. A basic fact is that
+for an (n,n−p) test form ξ,
+/bracketleftBigg
+∂1
+w1+M1
+1∧···∧∂1
+w1+Mp
+p/bracketrightBigg
+.ξ=M!(2πi)p/integraldisplay
+w=0∂M1
+w1...∂Mp
+wp/parenleftbigg∂
+∂w/rightangleseξ/parenrightbigg
+,
+where the derivative symbols refer to Lie derivatives and ∂/∂w=
+∂/∂w1∧...∧∂/∂wp. Using first Leibniz’ rule and (11) and then that
+∂/∂w/rightangleseξ=h−1∂/∂η/rightangleseξ, we get
+µ.ξ=/integraldisplay
+w=0/summationdisplay
+α≤Mcα∂M−α
+w(A)∂α
+w(S/rightangleseξ/h). (12)
+whereM= (M1,...,M p) andS=∂/∂η. We now want to express
+φµ.ξin terms of derivatives with respect to the variables ηiinstead of
+wi. By inverting the matrix ∂f/∂ηwe get
+∂
+∂wj=1
+h/summationdisplay
+kγjk∂
+∂ηk, (13)
+whereγjkare holomorphic. Multiplying the test form in (12) by φ, we
+thus get operators Qαso that
+φµ.ξ=/integraldisplay
+w=01
+h/summationtextMj/summationdisplay
+α≤MQα(φ)∂α
+w(S/rightangleseξ) =
+=/integraldisplay
+X/summationdisplay
+α≤M1
+hN0Lα(φ)Kα(S/rightangleseξ), (14)
+whereN0andN1are integers such that N0= 2N1+/summationtext
+jMjand
+Lα=hN1QαandKα=hN1∂α
+ware differential operators with respect to
+the original variables ( ζ,η). It follows from (13) that LαandKαare
+holomorphic across WifN1is chosen sufficiently large.
+Clearly, the values of ∂α
+w(S/rightangleseξ) can be prescribed on {w= 0}.
+Thereforeφµ= 0 onX\Wif and only if Lα(φ) = 0 onX\Wfor
+allα≤M, but by continuity and the SEP, these relations hold if and
+only if they hold across W. Thus{Lα}is a defining set for ann µ.
+Let nowξbe an arbitrary test form. For the sake of simplicity, we
+assume that φ= 1, but the same idea works for general φ. We thus
+want to show that
+µ.ξ= lim
+ε→0/integraldisplay
+Xχ(|h|2/ε2)L(S/rightangleseξ)
+hN0, (15)
+whereL=/summationtext
+α≤MLα(1)Kα.7
+The right hand side of (15) defines a current τ. Outside of {h= 0},
+τandµare equal, and the latter current has the SEP. It is therefore
+enough to show that τhas the SEP. By expanding
+χ(|h|2/δ2)τ.ξ=/integraldisplay
+XL(χ(|h|2/δ2)S/rightangleseξ)
+hN0(16)
+we get one term when all derivatives of LhitS/rightangleseξ, and clearly this
+term is precisely τ.ξin the limit. We will now explain why all other
+contributions vanish; all these terms contain derivatives of χ(|h|2/δ2)
+as a factor, and such a factor can be written as a sum of terms
+χ(k)/parenleftBigg
+|h|2
+δ2/parenrightBigg
+·/parenleftBigg
+|h|2
+δ2/parenrightBiggk
+σ
+hκ, (17)
+for some integers kandκand a smooth function σ. Let us define
+˜χ(x) =χ(x)−xkχ(k)(x). We note that the current σ/hκcan be defined
+both as lim δ→0χ(|h|2/δ2)σ/hκand lim δ→0˜χ(|h|2/δ2)σ/hκ, so it follows
+that (17) must be zero in the limit. /square
+3.Residue currents associated to ideals
+We will give a summary of the machinery of residue currents needed
+to prove Theorem 1.2. In [8] was presented a method for construc ting
+currentsRandUassociated to any generically exact complex
+··· →E2f2→E1f1→E0 (18)
+of hermitian vector bundles over an open set in Cn. The total bundle
+is then (E,f), whereE=/circleplustextEk,f=⊕fj. The currents UandRtake
+values in the bundle End E. Letσk:Ek−1→Ekbe the mapping of
+pointwise minimal norm that is the inverse of fkon the image of fk
+and extend it by zero on the orthogonal complement of the image. L et
+Wbe the analytic set where (18) is not exact. Outside of W, we define
+u=n+1/summationdisplay
+j=1uj (19)
+uj=∂σj∧...∧(∂σ2)σ1,
+so thatujis the (0,j−1)-bidegree component of uwhich takes values
+in Hom(E0,Ej). One can then extend uto a current UacrossWby
+setting
+U= lim
+ǫ→0χ(|F|2/ǫ2)∧u, (20)
+whereFis a holomorphic tuple vanishing on W. We define anoperator
+∇facting on currents with values in Eby
+∇f=f−∂. (21)8 JACOB SZNAJDMAN
+The residue current Ris defined by
+∇f◦U= 1−R. (22)
+One can check that ∇f◦U= 1 onX\W, soRhas support on W.
+The (0,k)-bidegree component of Ris denoted by Rk. Since∇2
+f= 0,
+(22) gives that ∇fR= 0, which is equivalent to
+f1R1= 0 (23)
+fk+1Rk+1=∂Rk, k≥1.
+An easy calculation shows that
+R= lim
+ǫ→0R0,ǫ+R1,ǫ+...+Rn,ǫ, (24)
+where
+R0,ǫ= (1−χ(|F|2/ǫ2)) (25)
+Rj,ǫ=∂χ(|F|2/ǫ2)∧uj,1≤j≤n.
+We restrict our attention to the case when rank E0= 1. For any
+idealJ ⊂ O Cn,0we can choose (18) (in various ways) so that J=
+Im(O(E1)→ O(E0)). Then ann R⊂ J; indeed, if φR= 0, then by
+(22)∇fUφ=φ. By solving a sequence of ∂-equations one can then
+show that φ∈ J, see [1]. The converse inclusion does not hold in
+general. A main result of [8] is that if (18) is chosen so that O(Ek)
+with maps ( fk) is a resolution of OCn/J, that is, an exact complex of
+sheaves, then ann R=J. Another choice for (18) is the Koszul com-
+plex. This has the advantage that the residue has an explicit form, a nd
+ifJhappens to be a complete intersection, then the Koszul complex
+is a resolution, so ann R=J.
+Now letJ ⊂ O Cn,0be the ideal that defines the analytic space Zof
+Theorem 1.2. Thus we assume that Jhas pure codimension p. We can
+thenchoose(18)sothatthecorrespondingsheafcomplexisares olution
+ofOCn,0/J. LetRZbe the current associated to Jas above so that
+annRZ=J. It has support on W=X.
+We also need a current Rarwith the property ann Rar⊂(a)r. We
+obtain such a current by the procedure above, where the complex (18)
+should be chosen so that ( a)r= Im(O(E1)→ O(E0)). The form u
+and the current Uwill be denoted by uarandUarrespectively. For the
+actual choice of (18), we follow [2], where uarandRarare described
+explicitly. Define
+σi=m/summationdisplay
+j=1ajei
+j
+|a|2(26)
+outside ofW=Z(a), where {ei
+j}jare frames of some trivial vector
+bundles. Andersson shows, see eq. (2.3) and the beginning of Sect ion 39
+in [2], that
+uar=/summationdisplay
+/summationtextji≤̺−1σ1∧/parenleftbig
+∂σ1/parenrightbig∧j1∧···∧σr∧/parenleftbig
+∂σr/parenrightbig∧jr, (27)
+where̺= min(n−p,m). We then obtain Rarby (24) and (25) with
+F=|a|, that is,
+Rar= lim
+ǫ→0(1−χa
+ǫ)+∂χa
+ǫ∧uar, (28)
+whereχa
+ǫ=χ(|a|2/ǫ2).
+3.1.Almost semi-meromorphic currents. In order to describe the
+structure of the current RZwe need to make a digression into almost
+semi-meromorphic currents. Recall thata current is semi-meromorphic
+if is a principal value current of the form α/f, wherefis a holomor-
+phic section of a line bundle and αis a smooth section of the same
+bundle. We say that a current ain an open subset V⊂Cnis al-
+most semi-meromorphic if there is a modification π:V′→Vand a
+semi-meromorphic current ˜ aonV′such thata=π∗˜a.
+Notice that if ais almost semi-meromorphic and ξis smooth, then
+ξ∧ais almost semi-meromorphic. In fact, ξ∧a=π∗(π∗ξ∧˜a).
+We will have use for the following result which is a part of Theo-
+rem 5.1 in [11].
+Lemma 3.1. Ifais an almost semi-meromorphic (0,∗)-current in an
+open subset of Cn, then also (∂/∂zℓ)ais almost semi-meromorphic.
+By [11, Theorem 4.6] one can define the product a∧µ, whereais
+almost semi-meromorphic and µis either an almost semi-meromorphic
+current or a Coleff-Herrera current, in the following way; notice fir st
+thatais smooth outside of an exceptional analytic set. If µis almost
+semi-meromorphic, let {h= 0}be any hypersurface containing the ex-
+ceptional set of aand leta∧µ:= lim ǫ→0χ(|h|2/ǫ2)a∧µ; the resulting
+current is independent of the choices of χandhand it is almost semi-
+meromorphic. If µis a Coleff-Herrera current on the analytic set Wwe
+will assume that the exceptional set of aintersectsWproperly since
+this is the only case of interest for us. Then let {h= 0}be a hyper-
+surface containing the exceptional set of asuch thathis generically
+non-vanishing on Wand as before set a∧µ:= lim ǫ→0χ(|h|2/ǫ2)a∧µ.
+Again the resulting current is independent of the choices made and,
+by [11, Corollary 4.7], a∧µis supported on Wand has the SEP with
+respect toW.
+We also need a result which is a slight variation of (part of) Propo-
+sition 3.3 in [6]. We include here the needed part of the proof.
+Proposition 3.2. There is a vector-valued Coleff-Herrera current µ
+and an endomorphism-valued almost semi-meromorphic curre ntbin a10 JACOB SZNAJDMAN
+neighborhood of 0inCnsuch thatbis smooth outside of Xsingand such
+thatRZ=bµ.
+Proof.Decompose RZ=/summationtextn
+k=0RZ
+kso thatRZ
+khas bidegree (0 ,k).
+Proposition 2.2 in [8] gives that the conjugate of I(Z) =√
+Janni-
+hilatesRZand thatRZ
+k= 0 fork < p, soRZ=RZ
+p+RZ
+p+1+....
+Consider the dual complex of (18)
+0→ O(E∗
+0)f∗
+1→ O(E∗
+1)→ ··· → O (E∗
+p)f∗
+p+1→ O(E∗
+p+1)→...,
+wheref∗
+jis the matrix transpose of fj; we now assume that the sheaf
+complex associated with (18) is a resolution of O/J. Since the sheaf
+Kerf∗
+p+1is coherent, we have an exact sequence
+O(F∗)g∗
+→ O(E∗
+p)f∗
+p+1→ O(E∗
+p+1),
+whereFisavectorbundle. Since ∇fRZ= 0, wehave ∂RZ
+p=fp+1RZ
+p+1,
+which gives that
+∂(gRZ
+p) =g∂RZ
+p=gfp+1RZ
+p+1= 0,
+sincegfp+1= 0. Since, by Corollary 2.4 in [9] and Proposition 2.2
+in [8],RZ
+phas the SEP with respect to Xand is annihilated by anti-
+holomorphic functions vanishing on X, it follows that gRZ
+pis a Coleff-
+Herrera current on X.
+LetZp+1be the complement of the set where fp+1has optimal rank.
+This is an analytic subset of Xsingthat is intrinsic to Z. Outside of
+Zp+1, themapping g:Ep→Fhasconstant rank. Wedefineamapping
+σF:F→Epon the complement of Zp+1by
+(σF)|(Img)⊥= 0
+σFg|(kerg)⊥= 1(kerg)⊥.
+This means that σFis the minimal norm inverse of gon the image
+ofg, and it is extended by zero to the orthogonal complement of this
+image; we here choose an auxiliary Hermitian metric on F. It is shown
+in Section 2 of [8] that σFhas an almost semi-meromorphic exten-
+sion across Zp+1; the extension is denoted σFas well. From equation
+(19), we see that outside of Zp+1,uZ
+ptakes values in the subbundle
+(Imfp+1)⊥= (kerg)⊥⊂Epsince this is true for σZ
+p. It follows that
+RZ
+p=σFgRZ
+poutside ofZp+1⊂Xsingand since both RZ
+pandσFgRZ
+p
+have the SEP with respect to Xwe see that RZ
+p=σFgRZ
+pholds cross
+Zp+1.
+Now, letZp+lbe the set where fp+ldoes not have optimal rank; this
+is again an analytic subset of Xsingthat is intrinsic to Zand since
+Zhas pure dimension we also have that codim Zp+l≥p+l+ 1, see
+Corollary 20.14 in [16]. By Theorem 4.4 in [8] there are almost semi-
+meromorphic endomorphism valued forms αlsuch thatRZ
+p+l=αlRZ
+p11
+outside ofZp+l. As in the proof of Proposition 3.3 in [6] this equality
+extends across Zp+land we are done. /square
+4.Proof of Theorem 1.2
+Wewillusetheproductofthecurrents RarandRZwhichwasdefined
+in [7]:
+Rar∧RZ= lim
+ǫ→0/bracketleftbig
+Rar
+0,ǫ+∂χa
+ǫ∧uar/bracketrightbig
+∧RZ. (29)
+Thiscurrenttakesvaluesinthetensorproductofthetwocorres ponding
+complexes (18) for RarandRZ.
+It follows by Proposition 2.2 in [7] (which holds in the non-reduced
+setting with the same proof), that φRar∧RZ= 0 forφ∈ OCn,0implies
+thatφ∈ J+ (a)r, that is, the image of φinOZ,0belongs to ar.
+AlthoughRar∧RZis a current in Cn,φRar∧RZdepends only on
+the image of φinOCn,0/J; in fact, if φ∈ J, thenφRZ= 0 and so
+φRar∧RZ= limǫ→0φRar
+ǫ∧RZ= 0. Moreover, Rar∧RZonly depends
+on the images of the generators ajinOCn,0/J. This can be deduced
+for example from Proposition 2.2 in [8]. Hence the proof of Theorem
+1.2 is reduced to showing that φRar∧RZ= 0 ifφsatisfies (1) with a
+suitable constant N.
+By Proposition 3.2, each component of RZis a sum of terms that
+can be factored as bµwherebis almost semi-meromorphic and µis
+a Coleff-Herrera current. According to Theorem 2.2, the annihilato r
+of each such µhas a defining set Mµ={Lµ
+1,...,Lµ
+Mµ}(and these
+operators satisfy (8)). As the operators L1,...,L Min Theorem 1.2, we
+take the union of Mµover all Coleff-Herrera currents µthat arise in
+the way we described.
+Although
+φRar∧RZ= lim
+ǫ→0φ(Rar
+0,ǫ+∂χa
+ǫ∧uar)∧RZ, (30)
+we will only prove that lim ǫ→0φ∂χa
+ǫ∧uar∧RZ= 0 as the proof for the
+remaining term is similar but easier. It suffices to show that
+φ∂χa
+ǫ∧uar∧bµ.ω (31)
+has limit 0 as ǫgoes to 0, where ωis a test form. Let hbe the holo-
+morphic function defined in (10) and set χb
+δ=χ(|h|2/δ2). We apply
+(8) withξ=χb
+δb∂χa
+ǫ∧uar∧ωto compute (31). This gives
+φ∂χa
+ǫ∧uar∧bµ.ω= lim
+δ→0φµ.∂χa
+ǫ∧uar∧χb
+δbω=
+= lim
+δ→0/integraldisplay
+X/summationdisplay
+α≤M1
+hN0Lα(φ)Kα(S/rightanglese∂χa
+ǫ∧uar∧χb
+δbω). (32)12 JACOB SZNAJDMAN
+Proposition4.1. Therearedifferentialoperators ˜Kαwith almostsemi-
+meromorphic coefficients and holomorphic derivatives so tha t
+φ∂χa
+ǫ∧uar∧bµ.ω=/integraldisplay
+X/summationdisplay
+α≤MLα(φ)˜Kα(∂χa
+ǫ∧uar∧ω). (33)
+This integral should be interpreted as a principal value, that is, as
+the limit of
+/integraldisplay
+X/summationdisplay
+α≤MLα(φ)χb
+δ˜Kα(∂χa
+ǫ∧uar∧ω)
+asδ→0.
+Proof.We apply Leibniz’ rule to see that the integral in (32) is a sum
+of terms of the form
+/integraldisplay
+Xψ∧∂α/parenleftbig
+∂χa
+ǫ∧uar∧ω/parenrightbig
+∧∂βχb
+δ∧∂γb
+hN0, (34)
+whereψis smooth. By Lemma 3.1 and the discussion following it we
+have that∂γb/hN0is almost semi-meromorphic. Moreover, it is smooth
+outside{h= 0}and since [X] is a Coleff-Herrera current on X(with
+values in the ( p,0)-forms) it follows that
+∂γb
+hN0∧[X] = lim
+δ→0χb
+δ∂γb
+hN0∧[X].
+Hence, ifβ= 0 then (34) goes to an expression of the form (33) as
+δ→0. Assume now that β/\e}atio\slash= 0. Then ∂βχb
+δis a sum of terms of the
+form (17) with k≥1. It follows in the same way as in (the end of)
+Section 2 that, in this case, (34) goes to 0 as δ→0.
+/square
+We now choose a resolution X′π→Xsuch thatX′is smooth and
+π∗his locally a monomial and the coefficients of ˜Kαare push forwards
+of semi-meromorphic forms whose denominators have normal cros sing
+zero sets. It suffices to show that
+Iǫ=/integraldisplay
+X′ξ′
+s1+n1
+1·...·s1+nn−p
+n−p∧π∗(Lαφ)π∗/parenleftbig
+∂β
+η(∂χa
+ǫ∧uar)/parenrightbig
+→0, (35)
+whereβ≤M,ξ′is a smooth form with compact support and ηis a
+local coordinate system on the ambient space CnofX. This integral
+is defined as a principal value as before.
+We want to integrate by parts in (35). The principal value current
+/bracketleftBig
+1/s1+n1
+1·...·s1+nn−p
+n−p/bracketrightBig13
+is a tensor product of one variable currents/bracketleftBig
+1/s1+nj
+j/bracketrightBig
+. Furthermore,
+one has for m≥1,
+∂
+∂sj/bracketleftbigg1
+sm
+j/bracketrightbigg
+=−m/bracketleftBigg
+1
+sm+1
+j/bracketrightBigg
+.
+This yields indeed that
+Iǫ=/integraldisplay
+X′ds
+s∧∂(n1,...,nn−p)
+s/parenleftbig
+ξ′∧π∗(Lαφ)π∗/parenleftbig
+∂β
+η(∂χa
+ǫ∧uar)/parenrightbig/parenrightbig
+, (36)
+where
+ds
+s=ds1∧···∧dsn−p
+s1·····sn−p.
+As the constant Nin the formulation of Theorem 1.2 we set
+N= max/bracketleftBigg
+2̺+|M|+n−p/summationdisplay
+j=1nj/bracketrightBigg
+, (37)
+where̺= min(m,dimX) = min(m,n−p) and the quantities |M|and/summationtextn−p
+1njare maximized over the components of RZand over all local
+charts of a (finite) covering of X′on whichhis a monomial. We claim
+that if (1) holds, then for any integers kj≤nj, 1≤j≤n−p,
+∂(k1,...,kn−p)
+s/parenleftbig
+π∗(Lαφ)π∗/parenleftbig
+∂β
+η(∂χa
+ǫ∧uar)/parenrightbig/parenrightbig
+(38)
+is bounded by a constant that is independent of ǫ. Note that the
+pointwise limit of any derivative of ∂χa
+ǫis zero almost everywhere. An
+application of dominated convergence in (36) thus gives that Iǫ→0,
+and thereby concludes the proof, given the claim.
+It remains to prove the claim above. We will assume that kj=nj
+for 1≤j≤n−p, as this is the worst case. We extend πto a resolution
+X′′π′
+→X′π→Xthat principalizes ( a), that isπ′∗π∗aj=a0a′
+jfor some
+non-vanishing tuple {a′
+j}. Note that the form ds/sbecomes a sum of
+similar forms when we pull it back to X′′. After pulling back (38) to
+X′′, we write it as a linear combination of forms
+π′∗/bracketleftbig
+∂(ˆn1,...,ˆnn−p)
+sπ∗(Lαφ)/bracketrightbig
+π′∗/bracketleftbig
+∂(˜n1,...,˜nn−p)
+sπ∗/parenleftbig
+∂β
+η(∂χa
+ǫ∧uar)/parenrightbig/bracketrightbig
+, (39)
+for integers ˆ njand ˜njsuch that ˆnj+ ˜nj=nj.
+By (1) we have that
+|π∗(Lαφ)|/lessorsimilar|π∗a|N+r−1. (40)
+The classical Brian¸ con-Skoda theorem therefore implies that loca lly
+π∗(Lαφ)∈(π∗a)N+r−̺. For the first factor of (39), we therefore locally
+have that
+π′∗/bracketleftbig
+∂(ˆn1,...,ˆnn−p)
+sπ∗(Lαφ)/bracketrightbig
+∈(a0)N+r−̺−ˆn, (41)14 JACOB SZNAJDMAN
+where ˆn= ˆn1+...+ ˆnn−p. As for the second factor of (39), we argue
+that
+π′∗/bracketleftbig
+∂(˜n1,...,˜nn−p)
+sπ∗/parenleftbig
+∂β
+η(∂χa
+ǫ∧uar)/parenrightbig/bracketrightbig
+=O(|a0|−(|β|+˜n+̺+r)), (42)
+where ˜n= ˜n1+...+ ˜nn−p. The chain rule gives that
+∂sjπ∗ω=n/summationdisplay
+j=1(∂sjπj)π∗∂ηjω,
+for any (0,q)-formω. Consequently, we can rewrite the left hand side
+of (42) as a sum of terms like
+ξ′′π′∗π∗/bracketleftBig
+∂˜β
+η(∂χa
+ǫ∧uar)/bracketrightBig
+, (43)
+whereξ′′is a smooth function and |˜β|=|β|+˜n. A slight reformulation
+of Lemma 4.2 in [24] states that
+Lemma 4.2. Let˜π:X′′→Xbe any principalization of the ideal (a)
+onXso that˜π∗(a) = (a0). Then for any multi-index β0
+˜π∗/bracketleftbig
+∂β0
+η(∂χa
+ǫ∧uar)/bracketrightbig
+=/summationdisplay
+j=0,1/parenleftbiggda0
+a0/parenrightbiggj
+∧O(|a0|−(|β0|+̺+r−1)).
+We apply this lemma to (43) with ˜ π=π◦π′andα0=˜β. We will
+use thatda0/a0is integrable in the following section, but here we can
+just estimate da0byO(1). We conclude by (41) and (42) that the form
+in (38) is indeed bounded if
+N≥ |β|+2̺+ ˆn+ ˜n=|β|+2̺+n−p/summationdisplay
+j=1nj, (44)
+and|β| ≤ |M|.
+Remark 4.3.LetZ⊂Pnbe a pure-dimensional projective variety, let
+Jbe the associated coherent ideal sheaf and let Xbe the associated
+reduced projective variety X. We believe that one can globalize Propo-
+sition 2.2 above and obtain Noetherian operators LαinCnsuch that
+an analogue of Theorem 1.2 holds: There is a constant Nsuch that if
+ais a polynomial ideal and (1) holds (with say r= 1), thenφbelongs
+to the ideal a.
+In general, the fact that φis inaonly implies that there is a rep-
+resentation φ=/summationtextajqj, where deg ajqjis like degφ+ 22d, ifdis the
+(maximal) degree of the generators of a. However, in the reduced
+case, i.e., as in [10], the condition (1) implies a degree estimate like
+degφ+Cdn−p. One could hope for a result of this kind even in the
+global non-reduced case.15
+Remark 4.4.Instead of using a log resolution X′→Xand integrate
+by parts on X′to see that the integral in (33) is a sum of terms of the
+form (36) it would be interesting to try to proceed as follows: Make
+successive normalized blow-ups along suitable ideals to get a normal
+modification X′→Xsuch that the almost semi-meromorphic coeffi-
+cients of all the operators ˜Kαare push-forwards of semi-meromorphic
+currents on X′. Then use Bernstein-Sato functional equations on X′to
+see that the integral in (33) is a sum of terms of a form very similar to
+(36). In cases where the orders of the differential operators ap pearing
+in the Bernstein-Sato equations are known one could thus possibly b e
+able to estimate the number Nin Theorem 1.2.
+5.The Cohen-Macaulay case
+In general it is hard to get a good estimate of the constant Nin
+Theorem 1.2 from the proof. In the case when Zis Cohen-Macaulay
+things are somewhat simpler. To begin with, then RZonly consists
+of the term RZ
+pwhich is a vector-valued Coleff-Herrera current on X,
+see [4]. That is, the factor bin Proposition 3.3 is just 1 and so one
+does not need the technical Proposition 4.1 to prove Theorem 1.2 in
+the Cohen-Macaulay case; it is sufficient to use Theorem 2.2.
+To have the representation (11), with µ=RZ
+p, it is precisely required
+thatf1+Mj
+jare inJ, and thusMjare governed by the constant in the
+local Nullstellensatz. If, for instance, the generators are polyno mials,
+then one can control Mjby the degree of the polynomials via Bezout
+estimates. Notice that Mjgive an upper bound of the degree of the
+resulting Noetherian operators Lj. It could be mentioned here that in
+the Cohen-Macaulay case also the function Ain (11) can be obtained
+more explicitly by means of a formula from [19]. The function hin
+the denominator in (8) is related to the structure form in [6], and the
+powerN0depends on the degree of the resulting Noetherian operators.
+We consider now the case where in addition Xis smooth. Choose
+the smallest possible multi-index M= (M1,...,M p) as above. For any
+component µj, there is a defining set of operators {Lj,α},α≤M, such
+that (8) holds. The union {Lj,α}of these sets is a defining set for J.
+Let us introduce the notation Lαfor the vector-valued operator Lj,α.
+Notice that, since Xis smooth, we can take w=ηin (9) so that
+h= 1 in (10).
+For each differential operator Lwe denote by ord( L) its order as
+a differential operator in the ambient space. One can also define the
+order of an element in N(Z) as the minimal order of any operator
+representing it.
+Letdbe the maximal distribution order all Coleff-Herrera currents
+that are annihilated by J. One can show that this number does not
+depend on the embedding of ZintoCn. By Example 1 in [4], the16 JACOB SZNAJDMAN
+componentsof µ=RZgeneratetheColeff-Herreracurrents annihilated
+byJ. Thusdcan also be expressed as the maximal distribution order
+of the components of RZ.
+Theorem 5.1. LetZbe a germ of a Cohen-Macaulay analytic space
+such thatXis smooth, and let {Lα},α≤M, be the vector-valued
+Noetherian operators obtained from µ=RZ
+pas above. If r≥1, and
+a⊂ OZ,0can be generated by melements, then
+|Lαφ| ≤C|a|min(m,dimX)+d−ord(Lα)+r−1onX, α≤M, (45)
+implies that φ∈ar.
+For the analytic space in Example 1.3, (45) coincides with the opti-
+mal hypotheses (2).
+Proof.We can assume that X=Cn−p⊂Cn, and we call the last p
+coordinates w1,...,w p. Clearly these functions form a complete inter-
+section, so if µ=RZ
+p, we have
+µ=A/bracketleftBigg
+∂1
+w1+M1
+1∧···∧∂1
+w1+Mp
+p/bracketrightBigg
+,
+for some (vector-valued) holomorphic function A, cf. (11). A basic
+computation rule for Coleff-Herrera products is that
+wα/bracketleftbigg
+∂1
+w1+M/bracketrightbigg
+=/bracketleftbigg
+∂1
+w1+M−α/bracketrightbigg
+, (46)
+where we have used multi-index notation. Using this, we get
+µ=/summationdisplay
+α≤MCαAα(z)/bracketleftbigg
+∂1
+w1+α/bracketrightbigg
+, (47)
+wherez= (z1,...,z n−p), andAαare holomorphic functions and Cα
+suitable constants so that
+µ.ξ=/integraldisplay
+w=0/summationdisplay
+α≤MAα(z)∂α
+w(∂/∂w/rightangleseξ). (48)
+Notice that since Mis minimal, AMmust be nonzero. Therefore, the
+distribution order of µis|M|.
+As in Section 2, we multiply (48) by φto obtain
+φµ.ξ=/integraldisplay
+w=0/summationdisplay
+α≤MLα(φ)∂α
+w(∂/∂w/rightangleseξ), (49)
+where
+Lα=/summationdisplay
+M≥γ≥α/parenleftbiggγ
+α/parenrightbigg
+Aγ(z)∂γ−α
+w.17
+Note that
+ord(Lα)≤ |M|−|α|. (50)
+We substitue ξ=∂χa
+ǫ∧uar∧ωinto (49). The analogue of (35) now
+becomes
+Iǫ=/integraldisplay
+w=0Lα(φ)∂α
+w(∂χa
+ǫ∧uar)∧ξ′. (51)
+Applying again Lemma 4.2, we get
+π′∗∂α
+w(∂χa
+ǫ∧uar) =/summationdisplay
+j=0,1/parenleftbiggda0
+a0/parenrightbiggj
+∧O(|a0|−(|α|+̺+r−1)). (52)
+By (50), we have d−ord(Lα)≥ |M|−ord(Lα)≥ |α|. Thus, assuming
+that|Lα(φ)| ≤C|a|̺+d−ord(Lα)+r−1, it follows from (52) and the fact
+thatda0/a0isintegrablethatdominatedconvergence maybeappliedin
+(51). Since ∂χa
+ǫand all of its derivatives go to zero almost everywhere,
+we see that lim ǫ→0Iǫ= 0, which was to be shown. /square
+Acknowledgements : I thank my advisors Mats Andersson and
+H˚ akan Samuelsson who gave insightful comments on the many prelim -
+inary versions.
+References
+1. M. Andersson, Residue currents and ideals of holomorphic functions , Bull. Sci.
+Math.128(2004), 481–512.
+2. ,Explicit versions of the Brian¸ con-Skoda theorem with vari ations, Michi-
+gan Math. J. 54(2006), 361–373.
+3. ,Uniqueness and factorization of Coleff-Herrera currents , Ann. Fac. Sci.
+Toulouse Math. (6)18, no. 4 (2009), 651–661.
+4. ,Coleff-Herrera currents, duality, and Noetherian operator s, Bull. Soc.
+Math. France. 139(2011), 535–554.
+5. ,A residue criterion for strong holomorphicity , Ark. Mat. 48, (2010),
+1–15.
+6. M. Andersson and H. Samuelsson, A Dolbeault-Grothendieck lemma on complex
+spaces via Koppelman formulas , Invent. Math., 190(2012), 261–297.
+7. M. Andersson, H. Samuelsson, and J. Sznajdman, On the Brian¸ con-Skoda the-
+orem on a singular variety , Ann. Inst. Fourier 60(2010), 417–432.
+8. M. Andersson and E. Wulcan, Residue currents with prescribed annihilator
+ideals, Ann. Sci. ´Ecole Norm. Sup. 40(2007), 985–1007.
+9. ,Decomposition of residue currents , J. Reine Angew. Math. 638(2010),
+103–118.
+10. ,Global effective versions of the Brian¸ con-Skoda-Huneke th eoremIn-
+vent. Math. (to appear)
+11. ,Direct images of semi-meromorphic currents , arXiv:1411.4832
+[math.CV].
+12. J.-E. Bj¨ ork, Residue currents and D-modules , The legacy of Niels Henrik Abel,
+Springer, Berlin, (2007), 605–651.
+13. J.-E. Bj¨ ork and H. Samuelsson, Regularizations of residue currents , J. Reine
+Angew. Math. 649(2010), 33–54.18 JACOB SZNAJDMAN
+14. J. Brian¸ con and H. Skoda, Sur la clˆ oture int´ egrale d’un id´ eal de germes de
+fonctions holomorphes en un point de Cn, C. R. Acad. Sci. Paris S´ er. A 278
+(1974), 949–951.
+15. L. Ehrenpreis, Fourier analysis in several complex variables , Pure and Applied
+Mathematics, vol. XXVII, Wiley-Interscience Publishers A Division of John
+Wiley & Sons, New York-London-Sydney, (1970).
+16. D. Eisenbud, Commutative Algebra. With a View Toward Algebraic Geometry .,
+Graduate Texts in Mathematics, vol. 150. Springer, New-York, (1 995).
+17. L. H¨ ormander, An introduction to complex analysis in several variables , North-
+Holland, (1990).
+18. C. Huneke, Uniform bounds in Noetherian rings , Invent. Math. 107(1992),
+203–223.
+19. R. L¨ ark¨ ang A comparison formula for residue currents arXiv:1207.1279
+20. M. Lejeune-Jalabert, B. Tessier, and J-J. Risler, Clˆ oture int´ egrale des id´ eaux et
+´ equisingularit´ e , Ann. Fac. Sci. Toulouse Math. (6)17, no. 4 (2008), 781–859.
+21. J. Lipman and A. Sathaye, Jacobian ideals and a theorem of Brian¸ con-Skoda ,
+Michigan Math. J. 28(1981), 199–222.
+22. U. Oberst, The construction of Noetherian operators , J. Algebra 222(1999),
+595–620.
+23. V.P. Palamodov, Linear differential operators with constant coefficients , Die
+Grundlehren der mathematischen Wissenschaften, Band 168, Spr inger-Verlag,
+New York-Berlin, (1970), Translated from the Russian by A.A. Brow n.
+24. J. Sznajdman, A residue calculus approach to the uniform Artin-Rees lemma ,
+Israel J. Math. 106(2013), 33–50.
+Jacob Sznajdman, Mathematical Sciences, Chalmers Univers ity and
+Gothenburg University, S-412 96 GOTHENBURG, SWEDEN
+E-mail address :sznajdma@chalmers.se
\ No newline at end of file
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+arXiv:1001.0323v4  [math.RT]  6 May 2014ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC
+REPRESENTATIONS
+SASCHA ORLIK AND MATTHIAS STRAUCH
+Abstract. LetGbe a split reductive p-adic group. This paper is about the Jordan-H¨ older series o f
+locally analytic G-representations which are induced from locally algebraic representations of a parabolic
+subgroup P⊂G. We construct for every representation Mof Lie(G) in the BGG-category O, which
+is equipped with an algebraic P-action, and for every smooth P-representation V, a locally analytic
+representation FG
+P(M,V) ofG. This gives rise to a bi-functor to the category of locally an alytic repre-
+sentations. We prove that it is exact and give a criterion for the topological irreducibility of FG
+P(M,V)
+in terms of MandV.
+Contents
+1. Introduction 1
+2. Preliminaries on locally analytic representations and g-modules 5
+3. FromOalgto locally analytic representations 9
+4. The functorFG
+Pand its properties 18
+5. Irreducibility results 25
+6. Composition series 40
+7. Applications to equivariant vector bundles on Drinfeld’s half space 42
+8. Appendix: some properties of highest weight modules in O 47
+References 60
+1.Introduction
+LetLbe a finite extension of Qpand letG=G(L) be the groupof L-valuedpoints ofa split connected
+reductive algebraic group GoverL. This paper is about the Jordan-H¨ older series of certain classes o f
+locally analytic G-representations. In particular, our treatment includes those r epresentations which are
+induced from locally algebraic representations of a parabolic subgro up.
+The theory of locally analytic representations was introduced by P. Schneider and J. Teitelbaum in
+[32]. Such representations appear in the study of vector bundles o np-adic period domains [33, 24] and
+in thep-adic Langlands program. In [31, 37] the authors pose the questio n of determining irreducible
+M. S. would like to acknowledge the support of the National Sc ience Foundation (award numbers DMS-0902103 and
+DMS-1202303).
+12 SASCHA ORLIK AND MATTHIAS STRAUCH
+parabolicallyinducedlocally-analyticrepresentations. Moregenera lly, onewouldliketoknowtheJordan-
+H¨ older series of such an object. Here we give a partial answer to t hese questions. As an application we
+consider the GL d+1-equivariant filtration constructed in [24] on the Fr´ echet space H0(X,L), whereLis a
+homogeneous line bundle on projective space Pd
+KandX⊂Pd
+Kis the Drinfeld half space. It is shown here
+that this filtration is either a Jordan-H¨ older series, or that it has a simple refinement (which we describe
+explicitly) that is a Jordan-H¨ older series.
+In order to explain our main construction and results, we introduce some notation. Fix a Borel
+subgroup B⊂G, a split maximal torus T⊂B, a standard parabolic subgroup P⊃B, and let
+b= Lie(B),t= Lie(T),p= Lie(P), andg= Lie(G) be their Lie algebras. The corresponding groups
+ofL-valued points are denoted by B=B(L),T=T(L), andP=P(L). Furthermore, we fix once
+and for all a finite extension KofLwhich will be our field of coefficients. The base change of an L-
+vector space (or scheme over L) toKwill always be denoted by the subscript K, e.g.,gK=g⊗LK,
+GK=G×Spec(L)Spec(K).
+We will study locally analytic representations which are in the image of c ertain bi-functors
+FG
+P(·,·) :Op
+alg×Rep∞,adm
+K(LP)−→Repℓa
+K(G)
+depending on the standard parabolic subgroup P⊂G. Here Rep∞,adm
+K(LP) is the category of smooth
+admissible representations of the Levi subgroup LP⊂PonK-vector spaces. Further Repℓa
+K(G) is the
+category of locally analytic representations of GonK-vector spaces. By O=Obwe denote the following
+category of U(gK)-modules1: objects are U(gK)-modulesMsatisfying the following properties:
+(1)Mis finitely generated as U(gK)-module.
+(2)Mdecomposes as a direct sum of one-dimensional U(tK)-modules.
+(3) The action of bKonMis locally finite, i.e. for every m∈M, the subspace U(bK)·m⊂Mis
+finite-dimensional over K.
+For a standard parabolic subalgebra p⊃bwe letOpbe the subcategory of those modules MinOon
+whichpKacts locally finitely. In Owe consider the subcategory Oalgof modules Mwhich have the
+additional property:
+(4) All weights of tKonMlie in the image of the natural map X∗(T) = Hom( T,Gm)→t∗
+K=
+HomL(t,K).
+We will show, cf. 2.8, that modules MinOalgon which pKacts locally finitely have the property that
+thepK-action lifts uniquely to a locally finite-dimensional algebraic PK-representation, and we denote
+this subcategory by Op
+alg, i.e.,Op
+alg=Oalg∩Op.
+Given a module MinOp
+alg, there is a finite-dimensional K-subspaceW⊂Mwhich is stable un-
+derU(pK) and which generates MasU(gK)-module. Let dbe the kernel of the canonical surjection
+U(gK)⊗U(pK)W→M, and letW′be the dual representation. By what we have said above, W′lifts
+uniquely toanalgebraicrepresentationof P. Then weconsiderthe locallyanalyticinduced representation
+IndG
+P(W′) ={f∈Can(G,W′)|∀g∈G,p∈P:f(gp) =p−1·f(g)},
+1this is an adaptation of the BGG-category Oto our setting in which the coefficient field is not algebraical ly closedON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 3
+whereCan(G,W′) =Can
+L(G,W′) is theK-vector space of L-locally analytic W′-valued functions on G.
+The action of Gon IndG
+P(W′) is given by left translation: ( g·f)(x) =f(g−1x). There is a Can(G,K)-
+valued pairing
+/a\}bracketle{t·,·/a\}bracketri}htCan(G,K):/parenleftbig
+U(gK)⊗U(pK)W/parenrightbig
+⊗KIndG
+P(W′)−→ Can(G,K)
+(y⊗w)⊗f/mapsto→/bracketleftBig
+g/mapsto→/parenleftbig
+(y·rf)(g)/parenrightbig
+(w)/bracketrightBig
+where, for x∈gwe have, by definition,
+(x·rf)(g) =d
+dtf(gexp(tx))|t=0.
+The common kernel of all /a\}bracketle{tz,·/a\}bracketri}htCan(G,K), withz∈d, is a closed subrepresentation
+FG
+P(M) = IndG
+P(W′)d={f∈IndG
+P(W′)|for allz∈d:/a\}bracketle{tz,f/a\}bracketri}htCan(G,K)= 0}.
+More generally, for a smooth admissible LP-representation Vwe define the representation
+FG
+P(M,V) = IndG
+P(W′⊗KV)d={f∈IndG
+P(W′⊗KV)|for allz∈d:/a\}bracketle{tz,f/a\}bracketri}htCan(G,V)= 0}
+similarly as before (here, the pairing /a\}bracketle{t·,·/a\}bracketri}htCan(G,V)takes values in Can(G,V)). The representation
+FG
+P(M,V) is an admissible locally analytic representation.
+LetMbe an object ofOp. We call the parabolic subalgebra pmaximal for M, ifMdoes not lie
+in a subcategory Oqwith a parabolic subalgebra qproperly containing p. It follows from [13, sec. 9.4]
+that for every object MofOpthere is unique parabolic subalgebra q⊃pwhich is maximal for M. The
+same definition applies for objects in the subcategory Op
+algin which case we also say that the parabolic
+subgroupPmaximal for M.
+The main result of this paper is the following:
+Theorem. (i)FG
+Pis functorial in both arguments: contravariant in Mand covariant in V.
+(ii)FG
+Pis exact in both arguments.
+(iii) IfQ⊃Pis a parabolic subgroup and Mis an object ofOq
+alg(and hence, in particular, an object of
+Op
+alg), then
+FG
+P(M,V) =FG
+Q(M,indLQ
+LP(LQ∩UP)(V))
+for all smooth admissible LP-representations V. Here,indLQ
+LP(LQ∩UP)(V) = indQ
+P(V)denotes the smooth
+induction of the representation V.
+(iv)FG
+P(M,V)is topologically irreducible if and only if the following co nditions are both satisfied:
+-Mis simple,
+- ifQ⊃Pis maximal for M, then the smooth representation indLQ
+LP(LQ∩UP)(V)is irreducible as an
+LQ-representation.4 SASCHA ORLIK AND MATTHIAS STRAUCH
+In (iv) we assume that the residue field characteristic pofLis odd, if the root system Φ = Φ(g,t)has
+irreducible components of type B,CorF4, and ifΦhas irreducible components of type G2we assume
+thatp>3.
+Remark. The assumptions on the residue field characteristic pofLare imposed because of our method of
+proof, but might not be necessary. That is, we do not know of any e xample where the fourth statement
+fails because p≤3.
+Our main result covers a particular case of the irreducibility result sh own in [25]. In loc.cit. we
+consider arbitrary reductive groups and arbitrary locally analytic fi nite-dimensional LP-representations
+W, but where, onthe otherhand, nodifferentialequationsappear. Usingasaninput Jordan-H¨ olderseries
+forMandV, and for possibly occurring smooth induced representations of th e form indLQ
+LP(LQ∩UP)(V),
+it is a rather formal exercise to determine a Jordan-H¨ older series forFG
+P(M,V), cf. section 6. Indeed, we
+concretize the determination of the composition factors for locally analytic representations of the form
+IndG
+P(W′)d. Finally, we study for a homogenous line bundle Lon projective space Pd
+kand the Drinfeld
+spaceX⊂Pd
+k, the space of sections H0(X,L) which is a Fr´ echet space equipped with a continuous action
+ofG= GL d+1(K). More precisely, we consider the filtration H0(X,L) =L(X)0⊃L(X)1⊃···⊃
+L(X)d−1⊃L(X)d=H0(Pd,L) constructed in [24] consisting of closed G-subspaces. We show that for
+i≥1,the dual space (L(X)i/L(X)i+1)′is a locally analytic G-representation which lies in the image of
+the functorFG
+P(i+1,d−i). HereP(i+1,d−i)is the maximal standard parabolic subgroup corresponding to the
+decomposition ( i+1,d−i) ofd+1. On the other hand, the dual of L(X)0/L(X)1sits inside an extension
+0→vG
+B⊗KHd(Pd
+K,L)′→(L(X)0/L(X)1)′→FG
+P(1,d)(M)→0
+for some simple object M∈Op(1,d).HerevG
+Bis the smooth Steinberg representation and Hd(Pd
+K,L) is
+a simple finite-dimensional algebraic G-module. By using our main result, we prove that all the locally
+analytic representations above are irreducible.
+Independently of our approach, O. Jones considers in [16] the rela tion between the Bernstein-Gelfand-
+Gelfand resolution and locally analytic principal series representatio ns ofG(and, more generally, of
+subgroups of Gpossessing an Iwahori decomposition). In [16, Thm. 26] he prove s the existence of an
+exact sequence which coincides with the exact sequence 4.11.3 in par agraph 4.11. The results about
+irreducibility in section 5 can then be used to decide if this sequence is a ctually a composition series, and,
+if not, how it can be refined to give such a series.
+Notation and conventions: We denote by pa prime number and consider fields L⊂Kwhich are both
+finite extensions of Qp. LetOLandOKbe the rings of integers of L, resp.K, and let|·|Kbe the
+absolute value on Ksuch that|p|K=p−1. The field Lis our ”base field”, whereas we consider Kas
+our ”coefficient field”. For a locally convex K-vector space Vwe denote by V′
+bits strong dual, i.e., the
+K-vector space of continuous linear forms equipped with the strong topology of bounded convergence.
+Sometimes, in particular when Vis finite-dimensional, we simplify notation and write V′instead ofV′
+b.
+All finite-dimensional K-vector spaces are equipped with the unique Hausdorff locally conve x topology.
+We letG0be a split reductive group scheme over OLandT0⊂B0⊂G0a maximal split torus and a
+Borel subgroup scheme, respectively. We denote by G,B,Tthe base change of G0,B0andT0toL. By
+G0=G0(OL),B0=B0(OL), etc., and G=G(L),B=B(L), etc., we denote the corresponding groups
+ofOL-valued points and L-valued points, respectively. Standard parabolic subgroups of G(resp.G) areON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 5
+those which contain B(resp.B). For each standard parabolic subgroup P(orP) we letLP(orLP) be
+the unique Levi subgroup which contains T(resp.T). Finally, Gothic letters g,p, etc., will denote the
+Lie algebras of G,P, etc.:g= Lie(G),t= Lie(T),b= Lie(B),p= Lie(P),lP= Lie(LP), etc.. Base
+change toKis usually denoted by the subscript K, for instance, gK=g⊗LK.
+We make the general convention that we denote by U(g),U(p), etc., the corresponding enveloping
+algebras, after base change to K, i.e., what would be usually denoted by U(g)⊗LK,U(p)⊗LKetc.
+Similarly, we use the abbreviations D(G) =D(G,K),D(P) =D(P,K) etc. for the locally L-analytic
+distributions with values in K.
+Acknowledgments. We thank O. Gabber, G. Henniart, V. Mazorchuk, T. Schmidt and B. Schraen for
+some helpful remarks. Furthermore, we thank the referees for their comments which helped us to improve
+the paper in a number of places.
+2.Preliminaries on locally analytic representations and g-modules
+2.1.Locally analytic representations. We start with recalling some basic facts on locally analytic
+representations as introduced by Schneider and Teitelbaum [32].
+Foralocally L-analyticgroup H, letCan(H,K) bethe locallyconvexvectorspaceoflocally L-analytic
+K-valued functions. More generally, if Vis a Hausdorff locally convex K-vector space, let Can(H,V)
+be theK-vector space consisting of locally analytic functions with values in V. It has the structure
+of a Hausdorff locally convex vector space, as well. The dual space D(H) =D(H,K) =Can(H,K)′
+is aK-algebra, and the multiplication (convolution) is separately continuo us. It has the structure of a
+Fr´ echet-Stein algebrawhen His compact [34]. Here the (convolution) product of δ1,δ2∈D(H) is defined
+by
+(2.1.1) δ1·δ2(f) = (δ1⊗δ2)((h1,h2)/mapsto→f(h1h2)).
+LetVbe a Hausdorff locally convex K-vector space. A homomorphism ρ:H→GLK(V) (or simply
+V) is called a locally analytic representation ofHif the topological K-vector space Vis barrelled, the
+action ofHonVis by continuous automorphisms, and the orbit maps ρv:H→V, h/mapsto→ρ(h)(v), are
+elements in Can(H,V) for allv∈V. We recall that a Hausdorff locally convex K-vector space Vis
+called of compact type , if it is an inductive limit of countably many Banach spaces with injective and
+compact transition maps, cf. [32, sec. 1]. In this case, the strong dualV′
+bis a nuclear Fr´ echet space (by
+[30, 16.10, 19.9]) and a separately continuous left D(H)-module. By [32, 3.3], the duality functor gives
+an anti-equivalence of categories
+(2.1.2)
+
+locally analytic H-represen-
+tations onK-vector spaces
+of compact type with
+continuous linear H-maps
+
+∼−→
+
+separately continuous D(H)-
+modules on nuclear Fr´ echet
+spaces with continuous
+D(H)-module maps
+
+.6 SASCHA ORLIK AND MATTHIAS STRAUCH
+In particular, Vis topologicallyirreducibleif V′
+bis a simpleD(H,K)-module. We also recallthat a locally
+analyticH-representation Vis called strongly admissible ifVis of compact type and if its strong dual
+V′
+bis a finitely generated D(H0)-module for any (equivalently, one) compact open subgroup H0⊂H, cf.
+[32, sec. 3].
+2.2.Induced locally analytic representations. For any closed subgroup H′ofHand any locally analytic
+representation VofH′, we denote by IndH
+H′(V) the induced locally analytic representation. It is defined
+by
+IndH
+H′(V) :=/braceleftBig
+f∈Can(H,V)|∀h′∈H′,∀h∈H:f(h·h′) = (h′)−1·f(h)/bracerightBig
+.
+The group Hacts on this vector space by ( h·f)(x) =f(h−1x). We will frequently consider the strong
+dual space of IndH
+H′(V). In this paper, Vwill always be of compact type and H/H′will be compact. As
+recalled above, cf. 2.1, V′
+bis then a nuclear Fr´ echet space.
+Lemma 2.3. Assume that HandH′are compact p-adic Lie groups. If Vis a finite-dimensional locally
+analyticH′-representation, then (IndH
+H′V)′
+bis canonically isomorphic, as D(H)-module, to D(H)⊗D(H′)
+V′.
+Proof.Consider the canonical map of D(H)-modules
+(2.3.1) D(H)⊗D(H′)V′→(IndH
+H′V)′
+b, δ⊗ϕ/mapsto→δ·ϕ,
+where (δ·ϕ)(f) =δ(g/mapsto→ϕ(f(g−1))). Because H′is a compact p-adic Lie group, it is topologically finitely
+generated (cf. [8, 8.34]). Because Vis finite-dimensional the same arguments as in [35], before Lemma
+6.1, show that 2.3.1 is an isomorphism of topological vector spaces, if we give the left side the quotient
+topology of the projective tensor product topology on D(H,K)⊗KV′. (We remark that the projective
+and inductive tensor product topologies coincide for tensor produ cts of Fr´ echet spaces, cf. [30, 17.6].)
+✷
+We recall that in this paper G=G(L) denotes the group of L-valued points of a split reductive
+algebraic group over L. We use the notation as fixed at the end of the introduction. In par ticular,
+G0=G0(OL) is the subgroup of OL-valued points of a smooth model of GoverOL.
+Furthermore, every rational G-representation and every smooth G(L)-representation may be consid-
+ered as a locally analytic representation, cf. [29, §2]. In the latter case the underlying vector space is the
+locally convex inductive limit of its finite-dimensional subspaces, and it is of compact type, if we assume
+that the smooth representation is admissible (because it is then of c ountable dimension).
+Lemma 2.4. LetP⊂Gbe a parabolic subgroup and put P0=P∩G0. LetVbe a smooth representation
+of the Levi subgroup LP, and letEbe a finite-dimensional algebraic P-representation. Both VandEare
+assumed to be K-vector spaces, and E′denotes the contragredient representation.
+(i) IfVis an admissible smooth representation, then IndG
+P(E′⊗KV)is an admissible locally analytic
+representation2.
+(ii) IfVis of finite length, then IndG
+P(E′⊗KV)is strongly admissible.
+2in the sense of [34, sec. 6]ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 7
+Proof.(i) Assume first that the P-representation Ecomes by inflation from an LP-representation (i.e.,
+the unipotent radical UPofPacts trivially on E). IfVis a smooth admissible representation, then it is
+admissible as a locally analytic representation by [34, 6.6]. This implies tha tE′⊗KVis an admissible
+LP-representation by [10, 6.1.5], and by [9, 2.1.2] we can conclude that I ndG
+P(E′⊗KV) is an admissible
+locally analytic representation. To treat the general case (when UPdoes not necessarily act trivially
+onE), we note first that we can find a descending filtration E=E0⊃E1⊃...⊃Er= 0 byP-
+stable subrepresentations Eisuch thatUPacts trivially on the associated graded vector space ¯E=/circleplustextr−1
+i=0Ei/Ei+1. We have already proved the assertion in the case r= 1. Arguing inductively, it suffices
+to treat the case r= 2. Consider the extension 0 →E1→E→E/E1→0, which gives 0→(E/E1)′→
+E′→E′
+1→0bypassingtodualspaces. Asthesearefinite-dimensional, thisse quencesplitsastopological
+vector spaces, and the same is true for the sequence that we get by tensoring with V:
+0−→(E/E1)′⊗KV−→E′⊗KV−→E′
+1⊗KV−→0.
+By [19, 5.1, 5.4] this gives an exact sequence of locally analytic repres entations
+0−→IndG
+P((E/E1)′⊗KV)−→IndG
+P(E′⊗KV)−→IndG
+P(E′
+1⊗KV)→0.
+As we pointed out above, the representations on the left and on th e right are admissible. Then the
+representation in the middle is admissible as well, because the categor y of admissible representations is
+closed under extensions, cf. [35, p. 304], [34, Remark 3.2].
+(ii) Any admissible representation that is an extension of two strong ly admissible representations is
+strongly admissible as well. Therefore, by the same d´ evissage as ab ove, we may assume that the action of
+PonEcomes by inflation froman action of LP. BecauseVis now assumedto be offinite length, the dual
+spaceV′is a finitely generated D(P0)-module, by [29, 2.2]. By [29, 3.3], this implies that also E⊗KV′is
+a finitely generated D(P0)-module. Therefore, E′⊗KVis a strongly admissible LP-representation. By
+[9, 2.1.2], the induced representation IndG
+P(E′⊗KV) is strongly admissible. ✷
+2.5.The categoryOand its parabolic variants Op.LetGbe a connected split reductive group over
+L. We use the notation as introduced at the end of the introduction. In particular, we take care to
+distinguish between vector spaces (and schemes) over our ”base field”Land our ”coefficient field” K.
+The subscript Kdenotes base change to K. However, when dealing with universal enveloping algebras,
+we make the generalconventionto write U(g),U(b) etc. to denote the correspondinguniversalenveloping
+algebras after base change to K, i.e., what is precisely U(gK),U(bK) and so on.
+The categoryOin the sense of Bernstein, Gelfand, Gelfand, cf. [2], [14], is defined fo r complex semi-
+simple Lie algebras. Here we consider the following variant for split red uctive Lie algebras over a field
+of characteristic zero. Thus we let Obe the full subcategory of all U(g)-modulesMwhich satisfy the
+following properties:
+(1)Mis finitely generated as a U(g)-module.
+(2)Mdecomposes as a direct sum of one-dimensional tK-representations.
+(3) The action of bKonMis locally finite, i.e. for every m∈M, the subspace U(b)·m⊂Mis
+finite-dimensional over K.
+As in the classical case one shows that Ois aK-linear, abelian, noetherian, artinian category which
+is closed under submodules and quotients, cf. [14, 1.1, 1.11]. In part icular, every object of Ohas a
+Jordan-H¨ older series.8 SASCHA ORLIK AND MATTHIAS STRAUCH
+In our paper we are mainly interested in a certain subcategory of O. The reason will become clear
+later on. Note that by property (2), we may write any object MinOas a direct sum
+(2.5.3) M=/circleplusdisplay
+λ∈t∗
+KMλ
+whereMλ={m∈M|∀x∈tK:x·m=λ(x)m}is theλ-eigenspace attached to λ∈t∗
+K= Hom K(tK,K).
+LetX∗(T) = Hom( T,Gm) be the group of characters of the torus Twhich we consider via the derivative
+as a subgroup of t∗
+K.
+Definition 2.6. We denote byOalgthe full subcategory of Owhose objects are U(g)-modules such that
+allλappearing in 2.5.3, for which Mλ/\e}atio\slash= 0, are contained in X∗(T)⊂t∗
+K.
+ThusM∈ Ois an object ofOalgif thetK-module structure on every Mλlifts to an algebraic
+action of T. Again,Oalgis an abelian noetherian, artinian category which is closed under subm odules
+and quotients. The Jordan-H¨ older series of a given U(g)-module lying in Oalgis the same as the one
+considered in the category O.
+Example 2.7. Forλ∈t∗
+K, letKλ=Kbe the 1-dimensional tK-module where the action is given by λ.
+ThenKλextends uniquely to a bK-module. Let
+M(λ) =U(g)⊗U(b)Kλ∈O
+be the corresponding Verma module. Denote by L(λ)∈Oits simple quotient. Then M(λ) resp.L(λ) is
+an object ofOalgif and only if λ∈X∗(T).
+LetPbe a standard parabolic subgroup of Gwith Levi decomposition P=LP·UPwhereT⊂LP.
+We putuP= Lie(UP) andlP= Lie(LP). We defineOpto be the category of U(g)-modulesMsatisfying
+the following properties:
+(1)Mis finitely generated as a U(g)-module.
+(2) Viewed as a lP,K-module,Mis the direct sum of finite-dimensional simple modules.
+(3) The action of uP,KonMis locally finite.
+This is analogous to the definition over an algebraically closed field, cf. [14, ch. 9]. Clearly, the
+categoryOpis a full subcategory of O. Furthermore, it is K-linear, abelian and closed under submodules
+and quotients, cf. [14, 9.3]. Hence the Jordan-H¨ older series of ev eryU(g)-module inOp⊂Olies inOp
+as well. IfQis a standard parabolic subgroup with Q⊃P, thenOq⊂Op.Finally, consider the extreme
+casep=g: the categoryOgconsists of all finite-dimensional semi-simple gK-modules. On the other
+hand,Ob=O.
+Similarly as before we define a subcategory Op
+algofOpas follows. Let Irr( lP,K)fdbe the set of
+isomorphism classes of finite-dimensional irreducible lP,K-modules. Again, any object in Ophas by
+property (2) a decomposition into lP,K-modules
+(2.7.4) M=/circleplusdisplay
+a∈Irr(lP,K)fdMa
+whereMa⊂Mis thea-isotypic part of the representation a. We denote byOp
+algthe full subcategory of
+Opconsisting of objects MofOpwith the following property: if Ma/\e}atio\slash= 0 (with the notation as in 2.7.4),
+thenais the Lie algebra representation induced by a finite-dimensional alge braicLP,K-representation,ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 9
+whereLP,K=LP×Spec(L)Spec(K). Again, the category Og
+algis contained inOalgand contains all
+finite-dimensional gK-modules which are induced by algebraic G-modules. Every object in Op
+alghas a
+Jordan-H¨ older series which coincides with the Jordan-H¨ older ser ies inOalg. The following lemma will
+later play a crucial role.
+Lemma 2.8. LetMbe an object ofOp. ThenMis inOp
+algif and only it is in Oalg.
+Proof.This follows from the discussion in section 1.20 of part II in [15]. It is sho wn there that a
+finite-dimensional U(lP)-module which is equipped with an algebraic T-action, whose induced U(t)-
+structure coincides with the one coming from the U(lP)-structure, lifts (uniquely) to an algebraic LP,K-
+representation. Note that the algebra Dist( LP,K) used in loc.cit. is in characteristic zero canonically
+isomorphic to the enveloping algebra U(lP,K), cf. [15, part I, 7.10]. Apart from the condition that the
+inducedU(t)-structure coincides with the one coming from the U(lP)-structure, Jantzen requires the
+following compatibility between the T-action and U(lP)-action:
+Ad(t)(x)·m=t·(x·(t−1·m)),
+for allm∈M,t∈Tandx∈lP,K. In our situation this is automatic: we may assume m∈Mλand
+x∈gα. This implies that x·m∈Mλ+α. Therefore Ad( t)(x)·m=α(t)(x·m), andt·(x·(t−1·m)) =
+(α+λ)(t)λ(t−1)·(x·m) =α(t)(x·m). ✷
+Remark 2.9. The preceding lemma can also be proved by using the description of irr educible algebraic
+representations of LPin terms of characters of T.
+Example 2.10. Let ∆ be the set of simple roots of Gwith respect to T⊂B. Letλ∈X(T)∗and
+setI={α∈∆|/a\}bracketle{tλ,α∨/a\}bracketri}ht∈Z≥0}.LetP=PIbe the standard parabolic subgroup of Gattached to I.
+Thenλis dominant with respect to the Levi subgroup LP.Denote byVI(λ) the correspondingirreducible
+finite-dimensional algebraic LP-representation, cf. [15, Pt. 2, 2.13]. We consider it as a P-module by
+letting act UPtrivially on it. The generalized parabolic Verma module (in the sense of L epowsky [21])
+attached to the weight λis given by
+MI(λ) =U(g)⊗U(pI)VI(λ).
+ThenMI(λ) is an object ofOp
+alg.Further, there is a surjective map
+M(λ)→MI(λ),
+where the kernel is given by the image of ⊕α∈IM(sα·λ)→M(λ). It follows that L(λ) is an object of
+Op
+alg, as well, cf. [14, sec. 9.4].
+3.FromOalgto locally analytic representations
+3.1.The representation associated to an object of Oalg.We fix as in the previous chapter a standard
+parabolic subgroup Pwith Levi decomposition P=LP·UPwhereT⊂LP. LetMbe an object ofOp
+alg.
+By the defining properties (1)-(3) for Op, we may choose a finite-dimensional representation W⊂Mof
+pKwhich generates MasU(g)-module. Thus we have a short exact sequence of U(g)-modules
+(3.1.1) 0 →d→U(g)⊗U(p)W→M→0,10 SASCHA ORLIK AND MATTHIAS STRAUCH
+where, by definition, dis the kernel of the canonical map U(g)⊗U(p)W→M. We denote the represen-
+tation of pKonWbyρ. It is induced by an algebraic PK-representation by the following lemma.
+Lemma 3.2. In the setting above, the representation ρlifts uniquely to an algebraic PK-representation
+onW(which we denote again by ρ).
+Proof.AsMis an object ofOp, theU(p)-moduleW, considered as a U(lp)-module, decomposes into
+a direct sum of isotypic modules Wa. Each module Walifts uniquely to an algebraic representation of
+LP,Kby 2.8. We denote this action of LP,KonW=/circleplustext
+aWabyρL. Moreover, the action of the Lie
+algebraup,Kintegrates uniquely to give an algebraic action of UPonWas follows. Given an element
+u= exp(x)∈UP(K), whereKdenotes an algebraic closure of K, we define ρ(u) :=/summationtext
+n≥0ρ(x)n
+n!, where
+ρ(x)n= 0 forn≫0. These two representations are compatible in the sense that ρL(h)◦ρ(u)◦ρL(h−1) =
+ρ(Ad(h)(u)), forh∈LP(K),u∈UP(K). This shows that ρlifts uniquely to an algebraic representation
+ofPKonW. ✷
+The induced locallyanalytic representationof Ponthe dual space W′= Hom K(W,K) will be denoted
+byρ′. We consider the locally analytic induced representation IndG
+P(W′). By 2.4 the canonical map of
+D(G)-modules
+D(G)⊗D(P)W−→/parenleftBig
+IndG
+P(W′)/parenrightBig′
+, δ⊗w/mapsto→[f/mapsto→δ(f)(w)],
+is an isomorphism of topological vector spaces. We thus have a pairin g
+(3.2.1) /a\}bracketle{t·,·/a\}bracketri}ht:/parenleftbig
+D(G)⊗D(P)W/parenrightbig
+⊗KIndG
+P(W′)−→K,
+which identifies the left hand side with the topological dual of the righ t hand side and vice versa. We
+remark that the Iwasawa decomposition G=G0·Bshows that as D(G0)-modules, the restriction of
+D(G)⊗D(P)WtoD(G0) is isomorphic to D(G0)⊗D(P0)W. The pairing 3.2.1 is obtained from the
+followingCan(G,K)-valued pairing by composition with the evaluation map Can(G,K)→K,f/mapsto→f(1).
+(3.2.2)/a\}bracketle{t·,·/a\}bracketri}htCan(G,K):/parenleftbig
+D(G)⊗D(P)W/parenrightbig
+⊗KIndG
+P(W′)−→ Can(G,K)
+(δ⊗w)⊗f/mapsto→/bracketleftBig
+g/mapsto→/parenleftbig
+δ·r(f(·)(w))/parenrightbig
+(g)/bracketrightBig
+where, by definition, we have/parenleftbig
+δ·r(f(·)(w))/parenrightbig
+(g) =δ(x/mapsto→f(gx)(w)). The same arguments as in [25, 3.4.1]
+show that the obvious map
+U(g)⊗U(p)W→D(G)⊗D(P)W
+is injective, so that we consider U(g)⊗U(p)Was being contained in the dual space of IndG
+P(W′). We
+then denote by IndG
+P(W′)dthe subspace of IndG
+P(W′) annihilated by dvia the pairing/a\}bracketle{t·,·/a\}bracketri}htCan(G,K):
+(3.2.3) IndG
+P(W′)d={f∈IndG
+P(W′)|for allδ∈d:/a\}bracketle{tδ,f/a\}bracketri}htCan(G,K)= 0Can(G,K)}.ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 11
+(This is exactly as in [33]; cf. the definition of the representation be fore Thm. 8.6. in [33].) Here
+IndG
+P(W′)dis clearly a closed subspace, since the action of U(g) is continuous on IndG
+P(W′). It isG-
+invariant because the pairing /a\}bracketle{t·,·/a\}bracketri}htCan(G,K)uses the right translation of Gon itself (in the argument of
+f), but theG-action on the induced representation is given by left translation.
+Proposition 3.3. (i) The representation IndG
+P(W′)dis a strongly admissible locally analytic G-represen-
+tation.
+(ii) The annihilator of IndG
+P(W′)dinD(G)⊗D(P)W, i.e., the set
+{ψ∈D(G)⊗D(P)W|for allf∈IndG
+P(W′)d:/a\}bracketle{tψ,f/a\}bracketri}ht= 0}.
+is equal toD(G)d. We therefore have a canonical isomorphism of coadmissible D(G)-modules
+/parenleftBig
+IndG
+P(W′)d/parenrightBig′∼=/parenleftbig
+D(G)⊗D(P)W/parenrightbig
+/D(G)d.
+Proof.(i) The representation IndG
+P(W′) is strongly admissible by 2.4. By [32, 3.5] closed invariant
+subspaces of strongly admissible representations are strongly ad missible again.
+(ii) By [34, 6.3], the admissible subrepresentations of IndG
+P(W′) are in bijection with the coadmissible
+quotients of D(G)⊗D(P)W. By [34, 3.6] these are given by the closed submodules of D(G)⊗D(P)W.
+The bijection is given by associating to a closed submodule J⊂D(G)⊗D(P)Wthe representation
+{f∈IndG
+P(W′)|for allψ∈J:/a\}bracketle{tψ,f/a\}bracketri}ht= 0}.
+By definition, for f∈IndG
+P(W′) to lie in IndG
+P(W′)dis equivalent to satisfy
+m/summationdisplay
+i=1/parenleftBig
+(yi·rf)(g)/parenrightBig
+(wi) = 0
+for allg∈Gand all/summationtext
+iyi⊗wi∈d. By the definition of ·rand the definition of the convolution product
+2.1.1 inD(G) this is equivalent to
+m/summationdisplay
+i=1/parenleftBig
+(δg·yi)(f)/parenrightBig
+(wi) = 0
+for allg∈Gand all/summationtext
+iyi⊗wi∈d. Hencefis in IndG
+P(W′)dif and only if/a\}bracketle{tδg·z,f/a\}bracketri}ht= 0, for all g∈G
+and allz∈d, where/a\}bracketle{t·,·/a\}bracketri}htis theK-valued pairing in 3.2.1. Because the subspace generated by the Dira c
+distributions δgis dense in D(G), this is equivalent to saying that /a\}bracketle{tδ,f/a\}bracketri}ht= 0, for all δ∈D(G)d. Asd
+is again an object of Op
+alg, it is finitely generated as U(g)-module. Hence D(G)dis finitely generated as
+D(G)-module and therefore closed by [34, Cor. 3.4, Lemma 3.6]. The asse rtion follows. ✷
+3.4.Another description. We proceed by giving another description of the dual space of IndG
+P(W′)d.
+LetMbe an object ofOp
+algas above. Then Mis the union of finite-dimensional pK-modules. Denote
+byXone of these finite-dimensional submodules. Xlifts by Lemma 3.2 uniquely to an algebraic PK-
+representation. In particular, we can consider Xas a locally analytic P-representation, and as such it has
+a unique structure as a D(P)-module, where the Dirac distributions act as group elements, cf. [32, Prop.
+3.2], and the sentence before Lemma 3.1 in [32]. (Note that we do not c onsider here the D(P)-module12 SASCHA ORLIK AND MATTHIAS STRAUCH
+structure on the dual space X′.) It thus follows that there is a unique D(P)-module structure on M
+which extends the pK-module structure and such that the Dirac distributions δgact as group elements
+g∈P.
+The next step is to consider the subring D(g,P) generated by U(g) andD(P) insideD(G). It follows
+from the proposition below that this ring is equal to U(g)D(P), i.e., every element of D(g,P) can be
+written as a finite sum of elements of the form z·δwithz∈U(g) andδ∈D(P).
+Proposition 3.5. LetH⊂Gbe a closed analytic subgroup, and let δ∈D(H). Thenδ·U(g)⊂
+U(g)·D(H). In particular, the smallest subring of D(G)containing U(g)andD(H)consists of finite
+sums/summationtext
+jzj·δjwithzj∈U(g)andδj∈D(H).
+Proof.Of course, it is enough to show that for any x∈gwe haveδ·x∈g·D(H). Letf∈Can(G,K). For
+g∈Gwe denote by g.fthe function defined by ( g.f)(x) =f(g−1x). It is easy to see that the convolution
+product 2.1.1 of two distributions λ1,λ2∈D(G) satisfies
+(λ1·λ2)(f) =λ1(g/mapsto→λ2(g−1.f)) =λ1(g/mapsto→λ2(h/mapsto→f(gh)) =λ2(h/mapsto→λ1(g/mapsto→f(gh))).
+The last equality may be called ”Fubini’s Theorem”, and this is used below in one instance. Furthermore,
+the image of xinD(G) is given by the formula
+x(f) = lim
+t→01
+t(f(exp(tx))−f(1)).
+Forh∈Handxas above write Ad( h)(x) =/summationtextd
+i=1ci(h)xi, where ( xi)iis some basis for g, and theciare
+locally analytic functions on H. Define distributions δi∈D(H) byδi(f) =δ(h/mapsto→ci(h)f(h)). Then we
+compute:
+(δ·x)(f) =δ(h/mapsto→x(h−1.f)) =δ(h/mapsto→Ad(h)(x)(g/mapsto→f(gh)))
+=δ/parenleftBig
+h/mapsto→(/summationtextd
+i=1ci(h)xi)(g/mapsto→f(gh))/parenrightBig
+=/summationtextd
+i=1δ/parenleftBig
+h/mapsto→(ci(h)xi)(g/mapsto→f(gh))/parenrightBig
+=/summationtextd
+i=1δ(h/mapsto→xi(g/mapsto→ci(h)f(gh)))Fubini=/summationtextd
+i=1xi(g/mapsto→δ(h/mapsto→ci(h)f(gh)))
+=/summationtextd
+i=1xi(g/mapsto→δ(h/mapsto→ci(h)(g−1.f)(h))) =/summationtextd
+i=1xi(g/mapsto→δi(g−1.f))
+=/summationtextd
+i=1(xi·δi)(f)
+And this shows that δ·x=/summationtextd
+i=1xi·δiis ing·D(H). ✷
+Corollary 3.6. There is on any object MofOp
+alga uniqueD(g,P)-module structure with the following
+properties:
+(i) The action of U(p), as a subring of U(g), coincides with the action of U(p)as a subring of D(P).
+(ii) The Dirac distributions δg∈D(P)act like group elements g∈P(the latter action being given by
+Lemma 3.2).
+Moreover, any morphism M1→M2inOp
+algis automatically a homomorphism of D(g,P)-modules.ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 13
+Proof.The first assertion about the existence and uniqueness of the D(g,P)-module structure follows
+from the discussion in the beginning of section 3.4 (which in turn uses [3 2, 3.1, 3.2]), together with the
+Prop. 3.5 just proved. The assertion about homomorphisms M1→M2follows from the uniqueness of
+theD(g,P)-module structure. ✷
+In the following we consider the tensor product D(G)⊗D(g,P)Mfor objects MofOp
+alg. Because
+Mis a finitely generated U(g)-module, hence a finitely generated D(g,P)-module, the tensor product
+D(G)⊗D(g,P)Mis a finitely generated D(G)-module. Let D(g,P0) be the subring of D(G0) generated
+byU(g) andD(P0). By 3.5 we have D(g,P0) =U(g)·D(P0) andD(G0)⊗D(g,P0)Mis likewise a finitely
+generatedD(G0)-module. Furthermore, it follows from the Iwasawa decomposition G=G0Pthat the
+canonical map
+D(G0)⊗D(g,P0)M−→D(G)⊗D(g,P)M
+is an isomorphism of D(G0)-modules, cf. [35, 6.1 (i)]. The next proposition shows that D(G0)⊗D(g,P0)M
+is a co-admissible D(G0)-module.
+Proposition 3.7. The canonical map
+ι:M=/parenleftbig
+U(g)⊗U(p)W/parenrightbig
+/d−→/parenleftbig
+D(G)⊗D(P)W/parenrightbig
+/D(G)d
+extends to an isomorphism of D(G)-modules
+D(G)⊗D(g,P)M∼=/parenleftbig
+D(G)⊗D(P)W/parenrightbig
+/D(G)d,
+Together with 3.3 we hence get an isomorphism of D(G)-modules
+D(G)⊗D(g,P)M∼=/parenleftBig
+IndG
+P(W′)d/parenrightBig′
+.
+As the canonical map D(G0)⊗D(g,P0)M→D(G)⊗D(g,P)Mis an isomorphism, we also have an iso-
+morphism of D(G0)-modules
+D(G0)⊗D(g,P0)M∼=/parenleftBig
+IndG
+P(W′)d/parenrightBig′
+.
+In particular, D(G0)⊗D(g,P0)Mis a coadmissible D(G0)-module.
+Proof.First we have to show that ιisD(g,P)-linear. But this follows from the D(g,P)-linearity of the
+natural maps q:U(g)⊗U(p)W→MandU(g)⊗U(p)W−→D(G)⊗D(P)Wtogether with the fact that
+dis aD(g,P)-submodule of U(g)⊗U(p)W.This shows that ιextends to a D(G)-module homomorphism
+Φ :D(G)⊗D(g,P)M−→/parenleftbig
+D(G)⊗D(P)W/parenrightbig
+/D(G)d
+by setting Φ( δ⊗v) =δι(v). In the other direction we define
+Ψ :/parenleftbig
+D(G)⊗D(P)W/parenrightbig
+/D(G)d−→D(G)⊗D(g,P)M14 SASCHA ORLIK AND MATTHIAS STRAUCH
+by Ψ((δ⊗w)+D(G)d) =δ⊗(w+d). It is immediate that Ψ is well-defined and easily checked that Φ
+and Ψ are inverse to each other. The last statement is now an immedia te consequence of 3.3 (i). ✷
+3.8.Yet another description. In this paragraph we present another approach to the represen tation
+IndG
+P(W′)dwhich appeared already in [24] in the case of G= GLn.Here we correct certain group actions
+which were formulated in loc.cit..
+LetG0be a split reductive group model of GoverOK.LetT0⊂G0be a maximal torus and fix a
+Borel subgroup B0⊂G0containing T0. We fix a standard parabolic subgroup P0ofG0. We denote
+byUP,0its unipotent radical, by U−
+P,0its opposite unipotent radical and by LP,0its Levi component
+containing T0.
+Letπ∈OLbe a uniformizer. For any positive integer n∈N, we consider the reduction map
+(3.8.1) pn:G0(OL)→G0(OL/(πn))
+Set
+Pn=p−1
+n/parenleftbig
+P0(OL/(πn))/parenrightbig
+, Un
+P=p−1
+n/parenleftbig
+UP,0(OL/(πn))/parenrightbig
+, Ln
+P=p−1
+n/parenleftbig
+LP,0(OL/(πn))/parenrightbig
+and
+U−,n
+P= ker/parenleftbig
+U−
+P,0(OL)→U−
+P,0(OL/(πn))/parenrightbig
+,U+,n
+P= ker/parenleftbig
+UP,0(OL)→UP,0(OL/(πn))/parenrightbig
+.
+These are compact open subgroups of G0=G(OL).The Levi decomposition on P0(OL/(πn)) induces a
+decomposition
+(3.8.2) Pn=Ln
+P·Un
+P.
+Further, we have equalities
+Pn=U−,n
+P·P0=P0·U−,n
+P,
+Un
+P=U−,n
+P·UP,0=UP,0·U−,n
+P, (3.8.2)
+Ln
+P=U+,n
+P·U−,n
+P·LP,0=LP,0·U+,n
+P·U−,n
+P.
+We may interpret U−,n
+P⊂U−
+P,0as theL-valued points of an open L-affinoid polydisc, since all non-
+diagonal entries xinU−,n
+Phave norm|x|≤|πn|.Then the ring of K-valued rigid-analytic functions
+O(U−,n
+P) is aK-Banach algebra equipped with the following norm. Let
+ΦU−
+P={β1,...,β r}
+be the set of roots appearing in u−
+P.We consider the ring O(U−
+P) ofK-valued algebraic functions on U−
+P
+as the polynomial K-algebra in the indeterminates Xβ1,...,X βr.Forn∈N, setǫn=|π|n.Then
+(3.8.3)/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+(i1,...,ir)∈Nr
+0ai1,...,irXi1
+β1···Xir
+βr/vextendsingle/vextendsingle/vextendsingle
+n:= sup(i1,...,ir)∈Nr
+0|ai1,...,ir|Kǫi1+···+irn.
+defines a norm on the K-algebraO(U−
+P) so thatO(U−,n
+P) becomes the completion of it. This K-Banach
+space is contained in the larger ring of bounded functions on U−,n
+P
+Ob(U−,n
+P) :=/braceleftBig/summationdisplay
+(i1,...,ir)ai1,...,ir·Xi1
+β1···Xir
+βr|ai1,...,ir∈K,sup(i1,...,ir)|ai1,...,ir|Kǫi1+···+ir
+n<∞/bracerightBig
+which is aK-Banach space with the same norm, as well.ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 15
+LetWbe a finite-dimensional locally analytic P-representation. For any n∈N,set
+Vn={rigid-analytic maps f:Pn→W′|f(gp) =p−1·f(g)}.
+This is a locally analytic Pn-representation where Pnacts via left translation. There is a natural identi-
+fication
+(3.8.4) Vn∼=O(U−,n
+P)⊗KW′.
+The action of Pn=U−,n
+P·LP,0·UP,0on the latter object translates as follows. The subgroup LP,0
+acts via conjugation on U−n
+Pon the first factor and on W′by the given one. The subgroup U−,n
+Pacts by
+translations on the first factor and trivially on W′. We omit the description of the action of UP,0since
+we will not use it explicitly.
+Lemma 3.9. There is a canonical isomorphism
+IndG0
+P0(W′)∼=lim−→n∈NIndG0
+Pn(Vn),
+whereIndG0
+Pndenotes the ordinary induction of group representations.
+Proof.Letf∈IndG0
+Pn(Vn) and denote by Rn⊂G0a set of representatives for G0/Pn.We may consider
+fas a function on G0=/uniontext
+g∈RgPnsuch that the restriction to each open subset gPnis rigid-analytic.
+Hence it gives rise to an locally analytic function on G0which lies in IndG0
+P0(W′).This construction is
+compatible with varying the integer n.
+On the other hand, if f∈IndG0
+P0(W′), then there is a disjoint union G0=/uniontext
+iUibyP0-stable subsets
+(with respect to the action from the right) such that frestricted to Uiis rigid-analytic. The open
+subgroupsPn⊂G,n∈N,form a cofinal system of P0-stable neighborhoods of the identity. Hence by
+choosing a refinement of the covering we may assume that there is s omensuch that it is of the shape
+G0=/uniontext
+g∈RgPnas above. Thus we get an element in IndG0
+Pn(Vn).
+Hence we have constructed two maps which are clearly inverse to ea ch other. It follows from the
+definition of the space of W′-valued locally analytic functions that these maps are also topologica l iso-
+morphisms. ✷
+Next we consider Was a Lie algebra representation of p.Since the universal enveloping algebra of
+gsplits (by the PBW-theorem) into a tensor product U(g) =U(u−
+P)⊗KU(p),we getU(g)⊗U(p)W′=
+U(u−
+P)⊗KW′.Similarlyasabove,thereisanactionof P0onthisspace. Onelements x⊗w∈U(g)⊗U(p)W′,
+a givenp∈P0acts by
+p·(x⊗w) = Ad(p)(x)⊗pw.
+Here Ad denotes the adjoint action. Under the identification U(g)⊗U(p)W′=U(u−)⊗KW′the action
+of the Levi factor LPis again via the adjoint action on U−,n
+Pand the natural one on W′.We stress that
+there is no natural action of U−,n
+Pon the above space since it is not complete in a suitable sense, cf. Ste p
+2 in the proof of Thm. 5.5.
+LetLβ1,...,L βrbe a basis of the vector space u−
+P. We consider the LP-equivariant pairing
+O(U−
+P)×U(u−
+P)→K (3.9.4)
+(f,z)/mapsto→(z·f)(1).16 SASCHA ORLIK AND MATTHIAS STRAUCH
+This is a non-degenerate pairing and induces therefore a K-linearLP-equivariant injection
+O(U−
+P)֒→HomK(U(u−
+P),K).
+given by
+Xi1
+β1···Xir
+βr/mapsto→(i1)!···(ir)!·(Li1
+β1···Lir
+βr)∗.
+Here/braceleftbig
+(Li1
+β1···Lir
+βr)∗|(i1,...,ir)∈Nr
+0/bracerightbig
+is the dual basis of {Li1
+β1···Lir
+βr|(i1,...,ir)∈Nr
+0}.The pairing
+3.9.4 extends by continuity to a non-degenerate ( LP)0-equivariant pairing
+(3.9.5) O(U−,n
+P)×U(u−
+P)→K.
+which induces a P0-equivariant pairing
+(3.9.6) ( ,) : (O(U−,n
+P)⊗KW′)×(U(u−
+P)⊗KW)→K
+(f⊗n,z⊗φ)/mapsto→φ(n)·(z·f)(1).
+Forǫ∈|K∗|,we consider the norm | |ǫonU(u−
+P) given by
+(3.9.7)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+(i1,...,ir)∈Nr
+0ai1,...,irLi1
+β1···Lir
+βr/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ǫ:= sup
+(i1,...,ir)∈Nr
+0|(i1)!···(ir)!·ai1,...,ir|ǫi1+···+ir.
+The completion of U(u−
+P) with respect to this norm yields the K-Banach space
+U(u−
+P)ǫ:=/braceleftBig/summationdisplay
+(i1,...,ir)∈Nr
+0ai1,...,irLi1
+β1···Lir
+βr|ai1,...,ir∈K,
+|(i1)!···(ir)!·ai1,...,ir|ǫi1+···+ir→0,i1+···+ir→∞/bracerightBig
+.
+We abbreviate U(u−
+P)n:=U(u−
+P)1
+ǫn.The pairing 3.9.5 extends to
+Ob(U−,n
+P)×U(u−
+P)n→K
+such thatOb(U−,n
+P) becomes the topological dual of U(u−
+P)n(cf. the Example in [30, ch. I, §3]). It follows
+thatOb(U−,n
+P)⊗KW′is the topological dual of U(u−
+P)n⊗KW. In particular, we get an action of U−,n
+P
+and hence of PnonU(u−
+P)n⊗KW.
+The following statement has been proved for G= GLnin [24].
+Proposition 3.10. There is an isomorphism of (Hausdorff) locally convex K-vector spaces
+lim−→n∈NO(U−,n
+P)⊗KW′∼−→/parenleftBig
+lim←−n∈NU(u−
+P)n⊗KW/parenrightBig′
+compatible with the action of lim←−nPn=P0.
+Proof.The proof is the same as in loc.cit. and proceeds as follows. There are identifications
+lim−→nOb(U−,n
+P) = lim−→nO(U−,n
+P)
+resp.
+lim−→nOb(U−,n
+P)⊗KW′= lim−→nO(U−,n
+P)⊗KW′ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 17
+of locally convex K-vector spaces. As we saw above, the space Ob(U−,n
+P)⊗KW′is the topological dual of
+U(u−
+P)n⊗KW. The claim follows now from [22, Thm. 3.4] respectively [30, Prop. 16.1 0] on the duality
+of projective limits of K-Fr´ echet spaces and injective limits of K-Banach spaces with compact transition
+maps. ✷
+Now we consider more generally a surjective map
+φ:U(g)⊗U(p)W→M
+ofU(g)-modules as in 3.1.1. Let d= kerφbe its kernel. We may consider dby PBW as a submodule of
+U(u−
+P)⊗KW. We denote by
+dn⊂U(u−
+P)n⊗KW
+its topological closure in U(u−
+P)n⊗KW. Finally, we put
+(O(U−,n
+P)⊗KW′)d={f∈O(U−,n
+P)⊗KW′|/a\}bracketle{tz,f/a\}bracketri}ht= 0∀z∈d}.
+ThenU(u−
+P)n⊗KW′/dnand (O(U−,n
+P)⊗KW′)dareK-Banachspaces equipped with a natural Pn-action,
+as well. The following statement generalizes Proposition 3.10.
+Proposition 3.11. There is an isomorphism of (Hausdorff) locally convex K-vector spaces
+lim−→n∈N(O(U−,n
+P)⊗KW′)d∼−→/parenleftBig
+lim←−n∈NU(u−
+P)n⊗KW/dn/parenrightBig′
+compatible with the action of lim←−nPn=P0.
+Proof.We define similarly ( O(U−,n
+P)⊗KW′)dn,(Ob(U−,n
+P)⊗KW′)dand (Ob(U−,n
+P)⊗KW′)dn.By the
+density of dindnwe have
+(O(U−,n
+P)⊗KW′)d= (O(U−,n
+P)⊗KW′)dn
+resp.
+(Ob(U−,n
+P)⊗KW′)d= (Ob(U−,n
+P)⊗KW′)dn.
+Now, theK-Banach space (Ob(U−,n
+P)⊗KW′)dnis the topological dual of U(u−
+P)n⊗KW/dn(here we use
+thatKis spherically complete, cf. [30, 9.4]). Then we proceed with the argu mentation as in the proof of
+Proposition 3.10. ✷
+Finally we arrive at the next alternative description of IndG
+P(W′)d.
+Corollary 3.12. There is an isomorphism of locally convex K-vector spaces compatible with the action
+ofG0:
+IndG0
+P0(W′)d= lim−→nIndG0
+Pn(O(U−,n
+P)⊗KW′)d∼=/parenleftBig
+lim←−nIndG0
+Pn(U(u−
+P)n⊗KW/dn)/parenrightBig′
+.
+Proof.Lemma 3.9 together with the identity (3.8.4) give rise to an isomorphism IndG0
+P0(W′)d=
+lim−→nIndG0
+Pn(O(U−,n
+P)⊗KW′)d.With the same reasoning as in the proof of the foregoing statement we
+see that IndG0
+Pn(Ob(U−,n
+P)⊗KW′)dis the topological dual of IndG0
+Pn(U(u−
+P)n⊗KW/dn). Passing to the
+limit yields the claim. ✷18 SASCHA ORLIK AND MATTHIAS STRAUCH
+4.The functorFG
+Pand its properties
+4.1.The functorFG
+P.Denote by Repℓa
+K(G) the category of locally analytic representations of Gon
+barrelled locally convex Hausdorff K-vector spaces. In the previous section we defined for any objec t
+M∈Op
+algtheG-representation IndG
+P(W′)d, cf. 3.2.3. By 3.3 it is strongly admissible. In particular, its
+underlying topological vector space is reflexive. Furthermore, th e continuous dual space, equipped with
+the strong topology, is isomorphic to D(G)⊗D(g,P)M, as aD(G)-module, cf. 3.7. This prompts us to
+consider the functor
+FG
+P:Op
+alg−→Repℓa
+K(G)
+defined by
+FG
+P(M) =/parenleftbig
+D(G)⊗D(g,P)M/parenrightbig′.
+Proposition 4.2. The functorFG
+Pis exact.
+Proof.Let 0→M1→M2→M3→0 be an exact sequence in the category Op
+alg.Then there are finite-
+dimensional algebraic generating P-representations Wi⊂Mi,i= 1,2,3, such that we have an induced
+exact sequence 0 →W1→W2→W3→0. Indeed, let W2be aP-submodule which generates M2. Then
+the imageW3ofW2inM3generatesM3. Consider X:= ker(W2→W3).If it does not generate M1,
+then we let Ybe any generating system and put W1=X+Yresp. replace W2byW2+Y. The resulting
+sequence satisfies the claim.
+We get a commutative diagram with exact rows:
+0 0 0
+↑ ↑ ↑
+0→M1→M2→M3→0
+↑ ↑ ↑
+0→U(g)⊗U(p)W1→U(g)⊗U(p)W2→U(g)⊗U(p)W3→0
+↑ ↑ ↑
+0→ d1→ d2→ d3→0
+↑ ↑ ↑
+0 0 0
+Note, that exactness of the middle row follows from the exactness of the functor U(g)⊗U(p)−.The
+exactness of the last row is a consequence of the snake resp. 3 ×3 lemma. If we apply our functor
+F=FG
+Pto the above diagram we get a new diagram:ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 19
+0 0 0
+↓ ↓ ↓
+0← F(M1)← F(M2)← F(M3)←0
+↓ ↓ ↓
+0←IndG
+P(W′
+1)←IndG
+P(W′
+2)←IndG
+P(W′
+3)←0
+↓ ↓ ↓
+0← F(d1)← F(d2)← F(d3)←0
+↓ ↓ ↓
+0 0 0
+The middle row is obviously exact. The first row coincides by definition w ith the sequence
+0←IndG
+P(W′
+1)d1←IndG
+P(W′
+2)d2←IndG
+P(W′
+3)d3←0.
+The following argumentation shows that it is exact on the right. Inde ed, the functor D(G)⊗U(g,P(·)
+is right exact and, by coadmissibility, yields strict K-linear maps. By the theorem of Hahn-Banach it
+follows that our functor FG
+Pis in particular left exact. So all we have to show by using the snake lem ma
+applied to the last two rows is to prove that the columns of the diagra m are exact. Exemplarily, we
+consider the middle column and abbreviate d=d2resp.W=W2. The submodule dis again an object
+of the categoryOp
+alg. Thus we have a short exact sequence
+0→¯d→U(g)⊗U(p)¯W→d→0
+for some finite-dimensional algebraic P-representation ¯W. HenceF(d) = IndG
+P(¯W′)¯d.In order to show
+the surjectivity of IndG
+P(W′)→F(d), it is enough to show the injectivity of its dual map since the image
+of IndG
+P(W′)→F(d) is closed by admissibility. Applying the description in Corollary 3.12 the d ual map
+is given by
+lim←−nIndG0
+Pn(U(u−)n⊗K¯W/¯dn)→lim←−nIndG0
+Pn(U(u−)n⊗KW).
+Now the functor lim←−nis left exact on projective systems of K-vector spaces. Hence it suffices to prove
+that for each n∈Nthe mapgn:U(u−)n⊗K¯W/¯dn→U(u−)n⊗KW, which is induced by the map
+g:U(u−)⊗K¯W/¯d∼=d֒→U(u−)⊗KWby passing to completions with respect to the norm 3.9.7, is
+injective. The left hand side U(u−)n⊗K¯W/¯dncarries the quotient norm. The map gnis continuous
+(this can been seen, e.g. by using the continuity criterion with respe ct to sequences). Thus its kernel is a
+closed subspace. On the other hand, by the same reasoning as in [2 5, Lemma 3.4.4] we deduce that the
+natural action of tonU(u−) is continuous with respect to the norm on U(u−)n. Hence all assumptions
+are satisfied for applying [11, Korollar 1.3.12]. The latter result says t hat the intersection of ker( gn) with
+U(u−)⊗K¯W/¯dis non-trivial, if ker( gn) is non-trivial. This is a contradiction since this intersection is
+nothing else but ker( g) ={0}. The claim follows. ✷
+Corollary 4.3. IfM/\e}atio\slash= 0thenFG
+P(M)/\e}atio\slash= 0.
+Proof.LetM→Nbe a simple quotient of M. By 4.2, this map induces an injective map FG
+P(N)→
+FG
+P(M). Wecanthereforeassumethat Missimple, and itis thereforeahighest weightmodule, generated
+by some vector v+with highest weight λ. LetW⊂Mbe a finite-dimensional p-module which contains
+v+, and hence generates MasU(g)-module. The weight space of U(g)⊗U(p)Wwith weight λis then the
+one-dimensional K-vector space generated by 1 ⊗v+. Putd= ker(U(g)⊗U(p)W→M). Then 1⊗v+is
+not contained in d, and all weights in dare distinct from λ. In the following we consider das a subspace
+ofU(u−
+P)⊗KW, which is tK-stable.20 SASCHA ORLIK AND MATTHIAS STRAUCH
+With the notation introduced in 3.8, U(u−
+P)n⊗KWis a Banach space completion of U(u−
+P)⊗KW.
+As above, let dnbe the closure of dinU(u−
+P)n⊗KW. By the last assertion in [11, 1.3.12], the weight
+spaces of dnwith respect to the tK-action are all contained in d, and we conclude that dndoes not contain
+1⊗v+, and the class of 1 ⊗v+in/parenleftbig
+U(u−
+P)n⊗KW/parenrightbig
+/dnis non-zero for all n. Therefore, the projective
+limit
+lim←−nIndG0
+Pn(U(u−
+P)n⊗KW/dn)
+is non-zero. By 3.12, this projective limit is reflexive, and its continuo us dual space isFG
+P(M), which is
+thus non-zero as well. ✷
+4.4.Extending the functor FG
+P.LetP⊂Gbe a standard parabolic subgroup. Let Vbe aK-vector
+space, equipped with a smooth admissible representation of the Lev i subgroup LP⊂P, and regard it
+via inflation as a representation of P. We recall that we always consider on Vthe finest locally convex
+K-vector space topology, i.e., the locally convex inductive limit of its finit e-dimensional subspaces. As
+suchVis of compact type and furnishes a locally analytic P-representation.
+LetMbe an object ofOp
+algand write it as a quotient of a generalized Verma module as in 3.1.1:
+0→d→U(g)⊗U(p)W→M→0.
+AsWis a finite-dimensional space, the injective tensor product W′⊗K,ιVcoincides with the projective
+tensor product W′⊗K,πV. Therefore we simply write W′⊗KVfor it. It is complete and we have
+W′⊗KV= lim−→HW′⊗KVH, whereHruns through all compact open subgroups of P, holds as locally
+convex vector spaces. Equipped with the diagonal action of P, cf. 3.2,W′⊗KVis a locally analytic
+representation. We set
+(4.4.1)FG
+P(M,W,V) = IndG
+P(W′⊗KV)d={f∈IndG
+P(W′⊗KV)|∀z∈d:/a\}bracketle{tz,f/a\}bracketri}htCan(G,V)= 0},
+where the pairing /a\}bracketle{t·,·/a\}bracketri}htCan(G,V)is defined as in 3.2.2. We are going to show that FG
+P(M,W,V) is
+independent of the chosen P-representation W, in the sense of 4.5 below. Later on we will therefore
+simplify notation by writing FG
+P(M,V) instead ofFG
+P(M,W,V).
+Proposition 4.5. LetG,P,M, andVbe as above. Let W1⊂W2be two finite-dimensional U(p)-
+submodules of Mwhich generate Mas aU(g)-module. Then the canonical map IndG
+P(W′
+2⊗KV)→
+IndG
+P(W′
+1⊗KV)ofG-representations, induced by the map W′
+2→W′
+1, restricts toa topological isomorphism
+FG
+P(M,W2,V)≃−→FG
+P(M,W1,V).
+Proof.Fori= 1,2, let 0→di→U(g)⊗U(p)Wi→M→0 be as in 3.1.1. It is immediate that under the
+(injective) map U(g)⊗U(p)W1→U(g)⊗U(p)W2induced by the inclusion W1⊂W2the submodule d1is
+mapped to d2. It follows that the canonical map IndG
+P(W′
+2⊗KV)→IndG
+P(W′
+1⊗KV) mapsFG
+P(M,W2,V)
+intoFG
+P(M,W1,V).
+We fix a locally L-analytic section s:G/P→Gof the projection G→G/Pand letH=s(G/P)⊂G
+be itsimage, sothat wehaveanisomorphismoflocally L-analyticmanifolds G≃H×P(this isomorphism
+depends on s). Using this isomorphism, we can identify the underlying topological v ector space of
+IndG
+P(W′
+i⊗KV) with
+Can
+L(H)ˆ⊗K(W′
+i⊗KV) =Can
+L(H)⊗KW′
+i⊗KV ,ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 21
+the completed tensor product on the left being the projective ten sor product, cf. [19, formula (56) and
+Remark 5.4]. This completed tensor product is equal to the ordinary tensor product, as W′⊗KVis
+an inductive limit of finite-dimensional vector spaces. Using this isomo rphism, which is natural in W′
+i
+andV, we identify f∈IndG
+P(W′
+i⊗KV) withψf=/summationtext
+kψk⊗φk⊗vk∈Can
+L(H)⊗KW′
+i⊗KV, where
+ψk∈Can
+L(H),φk∈W′
+i,vk∈V. Then, for fto be annihilated by ditranslates into a condition on ψf
+which is only a condition on/summationtext
+kψk⊗φk. With these identifications we find an isomorphism of K-vector
+spaces, natural in W′
+iandV,
+IndG
+P(W′
+i⊗KV)di≃−→/parenleftBig
+Can
+L(H)⊗KW′
+i/parenrightBigdi
+⊗KV .
+We get therefore a commutative diagram
+IndG
+P(W′
+2⊗KV)d2≃−→/parenleftBig
+Can
+L(H)⊗KW′
+2/parenrightBigd2
+⊗KV≃←−IndG
+P(W′
+2)d2⊗KV
+↓ ↓
+IndG
+P(W′
+1⊗KV)d1≃−→/parenleftBig
+Can
+L(H)⊗KW′
+1/parenrightBigd1
+⊗KV≃←−IndG
+P(W′
+1)d1⊗KV
+The vertical map on the right is an isomorphism because the map IndG
+P(W′
+2)d2→IndG
+P(W′
+1)d1is an
+isomorphism of topological vector spaces, by 3.7. The map on the lef t must hence be an isomorphism
+too. ✷
+4.6.LetM∈Op
+algandVbe as above. Given two finite-dimensional U(p)-submodules W1,W2⊂M,
+which both generate Mas aU(g)-module, we let W3⊂Mbe any finite-dimensional U(p)-submodule
+containing both W1andW2(e.g.,W3=W1+W2). By 4.5 the canonical maps
+(4.6.2) FG
+P(M,W2,V)←−FG
+P(M,W3,V)−→FG
+P(M,W1,V)
+are both isomorphisms of locally analytic G-representations. We have therefore a canonical isomorphism
+FG
+P(M,W2,V)−→ FG
+P(M,W1,V) which does not depend on the choice of W3. This is the unique
+isomorphism which is obtained from choosing any W3as above and inverting the map on the left of
+the resulting diagram 4.6.2. Via these uniquely specified isomorphisms w e can henceforth identify all
+representationsFG
+P(M,W,V) and writeFG
+P(M,V) for any one of them.
+Proposition 4.7. FG
+P(·,·)is a bi-functor
+Op
+alg×Rep∞,adm
+K(LP)−→Repℓa
+K(G),(M,V)/mapsto→FG
+P(M,V),
+which is contravariant in Mand covariant in V. HereRep∞,adm
+K(LP)is the category of smooth admissible
+representations of the Levi subgroup LP⊂PonK-vector spaces.
+Proof.Letα:M→Nbe a morphism in Op
+alg, and letβ:U→Vbe a morphism in Rep∞,adm
+K(LP).
+Choose a finite-dimensional U(p)-submodule WM⊂Mwhich generates Mas aU(g)-module. Then
+choose a finite-dimensional U(p)-submodule WN⊂Nwhich generates Nas aU(g)-module and which
+containsα(WM). With the notation as in 3.1.1 we get a commutative diagram
+0→dM→U(g)⊗U(p)WM→M→0
+↓ ↓
+0→dN→U(g)⊗U(p)WN→N→0
+It follows that the map in the middle maps dMintodN. This shows that the map22 SASCHA ORLIK AND MATTHIAS STRAUCH
+IndG
+P(W′
+N⊗KU)−→IndG
+P(W′
+M⊗KV)
+induced by α′⊗β:W′
+N⊗KU−→W′
+M⊗KVmapsFG
+P(N,U) intoFG
+P(M,V). ✷
+Proposition 4.8. (i) For all M∈Op
+alg, and for all smooth admissible LP-representations VtheG-
+representationFG
+P(M,V)is admissible.
+(ii) IfVis of finite length, then FG
+P(M,V)is strongly admissible for all M∈Op
+alg.
+Proof.(i) By 2.4 (i), the representation IndG
+P(W′⊗KV) is admissible. By [34, 6.4 iii.], the closed
+subrepresentation FG
+P(M,V) = IndG
+P(W′⊗KV)dis admissible as well.
+(ii) IfVis of finite length, then IndG
+P(W′⊗KV) is strongly admissible by 2.4 (ii). As a closed
+subrepresentation of IndG
+P(W′⊗KV), the representation FG
+P(M,V) is strongly admissible, cf. [32, 3.5].
+✷
+Proposition 4.9. a) The bi-functor FG
+Pis exact in both arguments.
+b) IfQ⊃Pis a parabolic subgroup, q= Lie(Q), andMan object ofOq
+alg, then
+FG
+P(M,V) =FG
+Q(M,iLQ
+LP(LQ∩UP)(V)),
+whereiLQ
+LP(LQ∩UP)(V) =iQ
+P(V)denotes the corresponding induced representation in the ca tegory of smooth
+representations.
+Proof.a) The arguments here are similar to the ones used in the proof of 4.5 . We fix a locally L-analytic
+sections:G/P→Gof the projection G→G/Pand letH=s(G/P)⊂Gbe its image. Then we obtain
+(as in the proof of 4.5) an isomorphism of K-vector spaces, which is natural in W′andV,
+IndG
+P(W′⊗KV)d∼=/parenleftBig
+Can
+L(H)⊗KW′/parenrightBigd
+⊗KV .
+This shows thatFG
+Pis exact in the second argument. That it is exact in the first argumen t is Prop. 4.2.
+b) LetMbe an object ofOq
+alg, and letW⊂Mbe a finite-dimensional qK-submodule which generates
+Mas aU(g)-module. Consider the canonical map U(g)⊗U(q)W→M, and denote by dits kernel. We
+have the tautological exact sequence
+(4.9.3) 0 −→d−→U(g)⊗U(q)W−→M−→0.
+Similarly, let 1the trivial one-dimensional representation (of qKorpK), letU(q)⊗U(p)1→1be the
+canonical map, dq
+pits kernel, and consider the corresponding exact sequence
+(4.9.4) 0 −→dq
+p−→U(q)⊗U(p)1−→1−→0.
+There is a canonicalisomorphism U(q)⊗U(p)W≃−→W⊗K(U(q)⊗U(p)1) which sends 1⊗wtow⊗(1⊗1)
+(cf. [17, Prop. 6.5]). This isomorphism, togetherwith 4.9.4showstha t there is acanonicalexact sequence
+(4.9.5) 0−→W⊗Kdq
+p−→W⊗K(U(q)⊗U(p)1)≃U(q)⊗U(p)W−→W−→0.ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 23
+Inducing 4.9.5 from U(q) toU(g) (which is an exact functor by the PBW theorem) gives the exact
+sequence
+(4.9.6) 0−→U(g)⊗U(q)(W⊗Kdq
+p)−→U(g)⊗U(p)W−→U(g)⊗U(q)W−→0.
+If we denote the kernel of the natural map U(g)⊗U(p)W→Mby˜d, we have the exact sequence
+(4.9.7) 0 −→˜d−→U(g)⊗U(p)W−→M−→0,
+and from 4.9.6 and 4.9.3 we deduce the exact sequence
+(4.9.8) 0 −→U(g)⊗U(q)(W⊗Kdq
+p)−→˜d−→d−→0.
+Furthermore, there is a canonical isomorphism
+(4.9.9) IndG
+P(W′⊗KV)≃IndG
+Q/parenleftBig
+IndQ
+P(W′⊗KV)/parenrightBig
+.
+This follows from [19, 5.3 and 2.6]. Let ( φ1,...,φ d) be a fixed basis of W′. Then, for every q∈Q, the
+family (q−1.φ1,...,q−1.φd) is a basis of W′, and given f∈IndQ
+P(W′⊗KV), we can write
+f(q) = (q−1.φ1)⊗ψ1(q)+...+(q−1.φd)⊗ψd(q)
+with uniquely determined ψi(q)∈V. It is straightforward to check that ψi∈IndQ
+P(V), and that
+IndQ
+P(W′⊗KV)≃−→W′⊗KIndQ
+P(V), f/mapsto→φ1⊗ψ1+...+φd⊗ψd,
+is an isomorphism of locally analytic Q-representations; the inverse map sends the element φ⊗ψ∈
+W′⊗KIndQ
+P(V) to [g/mapsto→(g−1.φ)⊗ψ(g)]∈IndQ
+P(W′⊗KV). From now on we identify IndG
+P(W′⊗KV)˜d
+withIndG
+Q/parenleftBig
+W′⊗KIndQ
+P(V)/parenrightBig˜d
+. Supposenowthat f:G→W′⊗KIndQ
+P(V)liesinIndG
+Q/parenleftBig
+W′⊗KIndQ
+P(V)/parenrightBig˜d
+.
+Because ˜dcontainsW⊗Kdq
+p(cf. 4.9.8), it follows that ftakes values in
+W′⊗KIndQ
+P(V)dq
+p=W′⊗KiQ
+P(V).
+This, together with 4.9.8, shows that
+FG
+P(M,V) = IndG
+Q/parenleftBig
+W′⊗KIndQ
+P(V)/parenrightBig˜d
+= IndG
+Q/parenleftBig
+W′⊗KiQ
+P(V)/parenrightBigd
+=FG
+Q/parenleftBig
+M,iLQ
+LP(LQ∩UP)(V)/parenrightBig
+.
+✷
+Remark 4.10. We remark that also the description in Cor. 3.12 has a counterpart in the ’enriched’
+version, i.e., including a smooth representation V. Indeed, let Vbe as before a smooth admissible LP-
+representation. By letting act UPtrivially on V, it has the structure of a P-representation. Then we get
+a topological isomorphism
+IndG0
+P0(W′⊗KV)d∼=/parenleftBig
+lim←−nIndG0
+Pn/parenleftbig
+(U(u−
+P)n⊗KW)ˆ⊗KV′/dn/parenrightbig/parenrightBig′
+.
+Indeed, for the proof we write again W′⊗KV= lim−→HW′⊗KVH.HereHruns through the set of
+all normal compact open subgroups of ( LP)0.Each subspace W′⊗KVHis stable by the action of P0.
+Moreover, we have ( W′⊗KVH)′∼=W⊗K(VH)′asVHis finite-dimensional (cf. [30, Prop. 20.13]). By24 SASCHA ORLIK AND MATTHIAS STRAUCH
+the very definition of Can(G,W′⊗V) [32] and again by the finite-dimensionality of VHwe see by using
+a variant of Prop. 3.11 that
+IndG0
+P0(W′⊗KV)d= lim−→nlim−→HIndG0
+Pn(O(U−,n
+P)⊗KW′⊗KVH)d
+=/parenleftBig
+lim←−nlim←−HIndG0
+Pn(U(u−
+P)n⊗KW⊗K(VH)′/dn)/parenrightBig′
+=/parenleftBig
+lim←−nIndG0
+Pn/parenleftbig
+((U(u−
+P)n⊗KW)ˆ⊗KV′/dn/parenrightbig/parenrightBig′
+.
+4.11.Locally analytic BGG-resolutions. Recall that for a character λ∈X∗(T), we denote by M(λ) =
+U(g)⊗U(b)Kλthe corresponding Verma module and by L(λ) its simple quotient. Let ∆ be the set of
+simple roots, Φ the set of roots and Φ+the set of positive roots of Gwith respect to our data T⊂B.
+Let
+X+={λ∈X∗(T)|∀α∈∆ : (λ,α∨)≥0}
+be the set of dominant weights in X∗(T).Ifλ∈X+, thenL(λ) is finite-dimensional and comes from an
+irreducible algebraic G-representation. In this situation, we also write V(λ) forL(λ).Finally, let Wbe
+the Weyl group of Gand denote by w0∈Wits longest element with respect to the Bruhat order.
+Denote by·the dot action of WonX∗(T) given by
+w·λ=w(λ+ρ)−ρ,
+whereρ=1
+2/summationtext
+α∈Φ+α. For a dominant weight λ∈X+, the BGG-resolution of the finite-dimensional
+G-moduleV(λ) has the following shape
+(4.11.1) 0→M(w0·λ)→/circleplusdisplay
+w∈W
+ℓ(w)=ℓ(w0)−1M(w·λ)→···→/circleplusdisplay
+w∈W
+ℓ(w)=1M(w·λ)→M(λ)→V(λ)→0.
+We refer to [20] for the definition of the differentials in this complex. B y applying our exact functor FG
+B,
+we get a resolution
+(4.11.2) 0←FG
+B(M(w0·λ))←/circleplusdisplay
+w∈W
+ℓ(w)=ℓ(w0)−1FG
+B(M(w·λ))←...
+···←/circleplusdisplay
+w∈W
+ℓ(w)=1FG
+B(M(w·λ))←FG
+B(M(λ))←FG
+B(V(λ))←0,
+which by Prop. 4.9 coincides with
+(4.11.3) 0←IndG
+B((w0·λ)−1)←/circleplusdisplay
+w∈W
+ℓ(w)=ℓ(w0)−1IndG
+B((w·λ)−1)←...
+···←/circleplusdisplay
+w∈W
+ℓ(w)=1IndG
+B((w·λ)−1)←IndG
+B(λ−1)←V(λ)⊗iG
+B(K)←0.ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 25
+There is also a parabolic versionof the BGG-resolutiondue to Lepows ky [21]. Let P=PI⊂G, I⊂∆
+be a standard parabolic subgroup and let WI⊂Wbe the subgroup generated by the reflections in I.
+Let
+X+
+I={λ∈X∗(T)|∀α∈I: (λ,α)≥0}
+be the set of LI-dominant weights. In particular X+⊂X+
+I. LetIW=WI\Wwhich we identify with
+the representatives of shortest length in W. LetIw0be the element of maximal length inIW. Ifλis in
+X+andw∈IWthenw·λ∈X+
+I, cf. [21, p. 502]. The I-parabolic BGG-resolution of V(λ) is given by
+a sequence
+0→MI(Iw0·λ)→/circleplusdisplay
+w∈IW
+ℓ(w)=ℓ(Iw0)−1MI(w·λ)→···→/circleplusdisplay
+w∈IW
+ℓ(w)=1MI(w·λ)→MI(λ)→V(λ)→0.
+Again, we refer to [20] for the definition of the differentials in this com plex. By applying our exact functor
+FG
+P, we get a resolution
+(4.11.4) 0←FG
+P(MI(Iw0·λ))←/circleplusdisplay
+w∈IW
+ℓ(w)=ℓ(Iw0)−1FG
+P(MI(w·λ))←...
+···←/circleplusdisplay
+w∈IW
+ℓ(w)=1FG
+P(MI(w·λ))←FG
+P(MI(λ))←FG
+P(V(λ))←0
+which coincides by Prop. 4.9 with
+(4.11.5) 0←IndG
+P(VI(Iw0·λ)′)←/circleplusdisplay
+w∈IW
+ℓ(w)=ℓ(Iw0)−1IndG
+P(VI(w·λ)′)←...
+···←/circleplusdisplay
+w∈IW
+ℓ(w)=1IndG
+P(VI(w·λ)′)←IndG
+P(VI(λ)′)←V(λ)⊗iG
+P(K)←0.
+Example 4.12. LetG= SL2,T⊂Gthe diagonal split torus, and λ∈X∗(T) the trivial character.
+ThusL(λ) =Kis the trivial representation. Put λ′=s·λ=s(ρ)−ρ=−2ρ∈X∗(T), wheres∈Wis
+the non-trivial element. The BGG-resolution of the trivial represe ntation is the short exact sequence
+0→M(λ′)→M(λ)→L(λ)→0.
+By applying the functor FG
+Bto this sequence, we obtain an exact sequence
+0→iG
+B(K)→IndG
+B(K)→IndG
+B((λ′)−1)→0
+The last map coincides with the derivative map, cf. [32, §6], [23].
+5.Irreducibility results
+The results in this section are valid under the following assumption on t he residue characteristic pof
+L:
+Assumption 5.1. If the root system Φ = Φ( g,t) has irreducible components of type B,CorF4, we
+assumep>2, and if Φ has irreducible components of type G2, we assume that p>3.26 SASCHA ORLIK AND MATTHIAS STRAUCH
+In section 8 we prove certain results on highest weight modules unde r the same assumption, and these
+results are used in this section, which is why this restriction is imposed here. It might be possible that a
+more refined analysis in section 8 shows that one can obtain results w hich are actually sufficient for our
+purposes here, without assuming 5.1.
+Definition 5.2. LetMbe an object of the category O. We call a standard parabolic subalgebra p
+maximal for MifM∈Opand ifM /∈Oqfor all parabolic subalgebras qstrictly containing p.
+It followsfrom[13, sec. 9.4]that foreveryobject MofOthere isuniquestandardparabolicsubalgebra
+pwhich is maximal for M. The same definition applies for objects in the subcategory Oalgin which case
+we also say that the standard parabolic subgroup Pcorresponding to pismaximal for M. In that case
+Mlies inOp
+alg, by 2.8.
+Theorem 5.3. Suppose 5.1 holds. Let M∈Op
+algbe simple and assume that Pis maximal for M. Then
+(i)D(G0)⊗D(g,P0)Mis simple as a D(G0)-module.
+(ii)FG
+P(M) = IndG
+P(W′)dis topologically irreducible as a G0-representation.
+(iii)FG
+P(M) = IndG
+P(W′)dis topologically irreducible as a G-representation.
+Proof.By the remark after 2.1.2, part (i) implies part (ii), and part (ii) implies o bviously part (iii).
+Assertion (i) is a special of Theorem 5.5 below if we take H=G0. ✷
+5.4.Generalization to open normal subgroups. It will be useful later (in the proof of 5.8), to have a
+similar criterion for the irreducibility of subrepresentations of FG
+P(M)|H, whereHis an arbitrary open
+normal subgroup of G0. For eachg∈G0,the subspace
+/braceleftBig
+f∈IndG0
+P0(W′)d|supp(f)⊂HgP0/bracerightBig
+of IndG0
+P0(W′)dis closed and stable under the action of H, and we therefore have a decomposition of
+H-representations
+IndG0
+P0(W′)d|H=/circleplusdisplay
+g∈H\G0/P0/braceleftBig
+f∈IndG0
+P0(W′)d|supp(f)⊂HgP0/bracerightBig
+.
+(TheH-representation IndG0
+P0(W′)d|His local in the sense that a function f, when restricted to a double
+cosetHgP0and extended by zero, call this function ( f|HgP0)!, still lies in that space. This is obviously
+the case for the induced representation IndG0
+P0(W′). The condition to be annihilated by the differential
+operators z·rwith inz∈dis a condition on the germ of fat any element in G0. Hence the function
+(f|HgP0)!belongs to IndG0
+P0(W′)das well.)
+We define the representation IndHP0
+P0(W′)das in 3.2.3. Extending functions on HP0by zero gives an
+isomorphism of H-representations
+IndHP0
+P0(W′)d∼=/braceleftBig
+f∈IndG0
+P0(W′)d|supp(f)⊂HP0/bracerightBig
+.ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 27
+Forg∈G0, we denote by IndHP0
+P0(W′)d,gthe space IndHP0
+P0(W′)dequipped with the H-action defined by
+(h·gf)(x) =f(g−1h−1gx).
+The map
+IndHP0
+P0(W′)d,g−→/braceleftBig
+f∈IndG0
+P0(W′)d|supp(f)⊂HgP0/bracerightBig
+,
+f/mapsto−→x/mapsto→/braceleftBigg
+f(g−1x), x∈HgP0
+0,else
+is then an isomorphism of H-representations. In the theorem below we use the following notat ion. For
+aD(H)-moduleNandg∈G0,we denote by δg⋆Nthe spaceN, equipped with the structure of a
+D(H)-module given by δ·gn= (δg−1δδg)n, wheren∈N,δ∈D(H), and the product δg−1δδg, is an
+element ofD(G0), and is an element of D(H).
+Theorem 5.5. Suppose 5.1 holds. Let M∈Op
+algbe simple and assume that pis maximal for M. LetH
+be an open normal subgroup of G0, and letg,g1,g2∈G0. Then
+(i) The dual space of IndHP0
+P0(W′)d,gis isomorphic to δg⋆/parenleftbig
+D(HP0)⊗D(g,P0)M/parenrightbig
+asD(H)-module.
+(ii) TheD(H)-moduleδg⋆/parenleftbig
+D(HP0)⊗D(g,P0)M/parenrightbig
+is simple.
+(iii) TheD(H)-modulesδg1⋆/parenleftbig
+D(HP0)⊗D(g,P0)M/parenrightbig
+andδg2⋆/parenleftbig
+D(HP0)⊗D(g,P0)M/parenrightbig
+are isomorphic
+if and only if g1HP0=g2HP0.
+(iv) TheH-representation IndHP0
+P0(W′)d,gis topologically irreducible.
+(v) The topological H-representations IndHP0
+P0(W′)d,g1andIndHP0
+P0(W′)d,g2are isomorphic if and only
+ifg1HP0=g2HP0.
+Proof.(i) The proofs of 3.7 and 3.3 only make use of the fact that G0is open (hence its Lie algebra
+is equal to g) and that it contains P0, and the corresponding statements carry over for any group wit h
+these properties. This shows statement (i) for the case when g= 1. From this the general case follows
+immediately.
+(iv) This follows from (ii), by 2.1.2.
+(v) This follows from (iii), by 2.1.2.
+We will now start with the proofs of (ii) and (iii). The first step is to red uce to the case of a suitable
+subgroupH0⊂Hwhich is normal in G0and uniform pro- p. Having this accomplished, we rename H0
+toH. The second step is to pass to modules over the Banach algebra Dr(H), cf. [34, sec. 4]. The third
+step consists in studying the restriction of these modules to the su bringUr(g) =Ur(g,H), which is the
+completion of U(g) with respect to the norm /bardbl·/bardblronDr(H).
+Step 1: reduction to H0.The standard parabolic subgroup P⊂Ghas a smooth model P0⊂G0,
+and the unipotent radical U−
+Pof the parabolic subgroup opposite to Phas a smooth model UP,0⊂G0
+as well.28 SASCHA ORLIK AND MATTHIAS STRAUCH
+Let Lie(G0), Lie(P0) and Lie( UP,0) be the Lie algebras of these (smooth) group schemes. These are
+OL-lattices in g= Lie(G),p= Lie(P) andu−
+P= Lie(U−
+P), respectively. Moreover, Lie( G0), Lie(P0) and
+Lie(UP,0) areZp-Lie algebras, and we have
+Lie(G0) = Lie(U−
+P,0)⊕Lie(P0).
+Form0≥1 (m0≥2 ifp= 2) theOL-latticespm0Lie(G0),pm0Lie(P0) andpm0Lie(UP,0) are powerful
+Zp-Lie algebras, cf. [8, sec. 9.4], and hence expG:g/axisshort/axisshort/arrowaxisrightGconverges on these OL-lattices. Therefore,
+(5.5.1) H0= expG/parenleftBig
+pm0Lie(G0)/parenrightBig
+, H+
+0= expG/parenleftBig
+pm0Lie(P0)/parenrightBig
+, H−
+0= expG/parenleftBig
+pm0Lie(U−
+P,0)/parenrightBig
+are uniform pro- pgroups. The adjoint action of G0=G0(OL) leaves Lie( G0) invariant, and hence
+pm0Lie(G0). This shows that H0is normal in G0, and analogous arguments show that H+
+0andH−
+0are
+normal inP0andU−
+P,0, respectively. It follows from the existence of ’coordinates of the second kind’ that
+H0=H−
+0H+
+0andH0∩P0=H+
+0andH0∩U−
+P,0=H−
+0, cf. [25, 2.2.4 (ii)].
+Moreover, for large enough m0the groupH0is contained H, andD(H) =/circleplustext
+h∈H/H0δhD(H0). This
+implies that, as D(H0)-modules,
+(5.5.2) δg⋆/parenleftbig
+D(HP0)⊗D(g,P0)M/parenrightbig
+=/circleplusdisplay
+h∈HP0/H0P0δgh⋆/parenleftbig
+D(H0P0)⊗D(g,P0)M/parenrightbig
+.
+Let us now assume that statements (ii) and (iii) hold for H0. Then the direct sum on the right of 5.5.2
+consistsofsimple D(H0)-modules which arepairwisenon-isomorphic. Asthey arepermuted by the action
+ofH, it follows that the left side is simple as D(H)-module. Furthermore, if δg1⋆/parenleftbig
+D(HP0)⊗D(g,P0)M/parenrightbig
+is isomorphic to δg2⋆/parenleftbig
+D(HP0)⊗D(g,P0)M/parenrightbig
+asD(H)-module, then also as D(H0)-module. But then we
+must haveg1h1H0P0=g2h2H0P0for someh1,h2∈H, in particular g1HP0=g2HP0.
+Thus we have shown that statements (ii) and (iii) for D(H) follow from the corresponding statements
+forD(H0).
+To simplify notation, we will from now on assume that H, and its subgroups H+andH−, are defined
+as in 5.5.1, and are in particular uniform pro- pgroups.
+Step 2: passage to Dr(H).We put
+κ=/braceleftBigg
+1, p>2
+2, p= 2
+(We point out that in [25], which we are going to use in the following, κis denoted by εp.) Letralways
+denote a real number in (0 ,1)∩pQwith the property:
+(5.5.3) there is m∈Z≥0such thats=rpmsatisfies1
+p<sandsκ<p−1/(p−1).
+For the definition of the canonical p-valuation on a uniform pro- pgroup we refer to [25, 2.2.3]. We let
+/bardbl·/bardblrdenote the norm on D(H) associated to rand the canonical p-valuation, cf. [25, 2.2.6] (where thisON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 29
+norm is denoted by ¯ qr).Dr(H) denotes the corresponding Banach space completion. This is a noe therian
+Banach algebra, and D(H) = lim←−r<1Dr(H) (cf. [34] for more information). If /tildewideH⊂G0is a compact
+open subgroup which contains H, then
+(5.5.4) D(/tildewideH) =/circleplusdisplay
+g∈/tildewideH/HδgD(H)
+and we let/bardbl·/bardblrbe the maximum norm on D(/tildewideH) given by
+(5.5.5) /bardbl/summationdisplay
+g∈/tildewideH/Hδgλg/bardblr= max{/bardblλg/bardblr|g∈/tildewideH/H}.
+Then the completion Dr(/tildewideH) ofD(/tildewideH) with respect to /bardbl·/bardblrhas an analogous decomposition:
+Dr(/tildewideH) =/circleplusdisplay
+g∈/tildewideH/HδgDr(H)
+and therefore Dr(/tildewideH) =Dr(H)⊗D(H)D(/tildewideH). We are going to use the preceding discussion in the case
+when/tildewideH=HP0. PutM=D(HP0)⊗D(g,P0)M. To show that Mis a simple D(H)-module it suffices,
+by [34, 3.9], to show that
+Mr:=Dr(H)⊗D(H)M=Dr(HP0)⊗D(g,P0)M
+is a simple Dr(H)-module for a sequence of r’s converging to 1. By 3.7 and 4.3
+FG0
+P0(M)′=D(G0)⊗D(HP0)M
+is non-zero, and thus M= lim←−r<1Mris non-zero. Hence we must have Mr/\e}atio\slash= 0 forrsufficiently close to
+1. As the image of MinMrgenerates MrasDr(HP0)-module, and because Mis simple, the canonical
+mapM→Mrmust be injective when Mr/\e}atio\slash= 0. From now on we assume that, in addition to 5.5.3, ris
+such that Mr/\e}atio\slash= 0, and we consider Mas being contained in Mr.
+Step 3: passage to Ur(g).LetUr(g) =Ur(g,H) be the topological closure of U(g) inDr(H). It is
+important for our approach to have a useful description of Ur(g). We will freely use the following fact,
+which follows from [27, 5.6]:
+(5.5.6) U(g) is dense in Dr(H) ifrκ<p−1
+p−1. In this case Ur(g) =Dr(H).
+Let (Pm(H))m≥1be the lower p-series ofH, cf. [8, 1.15]. Note that P1(H) =H. Form≥0 put
+Hm:=Pm+1(H) so thatH0=H. We refer to [8, sec. 4.5, sec. 9.4] for the notion of a Zp-Lie algebra
+of a uniform pro- pgroup. It follows from [8, 3.6] that Hmis a uniform pro- pgroup with Zp-Lie algebra
+equal topmLieZp(H). Lets=rpmbe as in 5.5.3. Denote by /bardbl·/bardbl(m)
+sthe norm on D(Hm) associated to s
+and the canonical p-valuation on Hm. Then, by [27, 6.2, 6.4], the restriction of /bardbl·/bardblronD(H) toD(Hm)
+is equivalent to/bardbl·/bardbl(m)
+s, andDr(H) is a finite and free Ds(Hm)-module a basis of which is given by any
+set of coset representatives for H/Hm. By 5.5.6 we can conclude
+(5.5.7) If s=rpmis as in 5.5.3, then U(g) is dense in Ds(Hm), henceUr(g,H) =Ds(Hm).30 SASCHA ORLIK AND MATTHIAS STRAUCH
+In particular, Ur(g)∩H=Hmis an open subgroup of H. Let
+mr:=Ur(g)M
+be theUr(g)-submodule of Mrgenerated by M. Because we assume Mr/\e}atio\slash= 0 we also have mr/\e}atio\slash= 0, and
+Mis contained in mr.
+Likewise, we equip D(H+) (D(H−), resp.) with the norm /bardbl·/bardblrassociated to the canonical p-valuation
+onH+(H−, resp.), and give D(P0) (D(U−
+P,0), resp.) the maximum norm as above (cf. 5.5.4 and 5.5.5).
+Again, we denote by Dr(P0) (Dr(U−
+P,0), resp.) the corresponding completions. As is easy to see, these
+norms onD(P0) (D(U−
+P,0), resp.) are equal to the restriction of the norms on D(HP0) (D(U−
+P,0H), resp.)
+coming from the norm /bardbl·/bardblronD(H) by the recipe explained above.
+LetUr(p) be the closure of U(p) inDr(H+). ThenUr(p) is the completion of U(p) with respect to the
+norm associated to the canonical p-valuation on H+. Because the canonical p-valuation of an element of
+h∈H(h∈H+, resp.) can be read off from logG(h)∈pm0Lie(G0) (logG(h)∈pm0Lie(P0), resp.), the
+canonicalp-valuation on H+is the restriction of the canonical p-valuation on H. HenceUr(p) is also the
+closure ofU(p) inUr(g).
+It follows from 5.5.7 (applied to H+andp= LieZp(H+)) thatDr(P0) is generated as a module
+overUr(p) by finitely many Dirac distributions δg1,...,δ gk, withgi∈P0. BecauseP0acts onM, the
+Ur(g)-module mris actually a module over the subring
+(5.5.8) Dr(g,P0) =Ur(g)Dr(P0) =k/summationdisplay
+i=1Ur(g)·δgi
+generated by Ur(g) andDr(P0) insideDr(G0). Moreover, mris a finitely generated Ur(g)-module (in
+fact, generated by a single vector of highest weight), hence carr ies a canonical Ur(g)-module topology, as
+Ur(g) is anoetherianBanachalgebra(cf. [18, 1.4.2]which applies herebe cause weassume r∈(1
+p,1)∩pQ).
+The module Mis clearly dense in mrwith respect to this topology. It follows from [11, 1.3.12] or [25,
+3.4.8] that mris a simple Ur(g)-module, and in particular a simple Dr(g,P0)-module.
+Sublemma 5.6. Letrandsbe as in 5.5.3. Put P0,r=HP0∩Dr(g,P0). Then
+(i)P0,r=HmP0=H−,mP0, whereH−,m=Pm(H−).
+(ii)Dr(HP0) =/circleplustext
+g∈HP0/P0,rδgDr(g,P0).
+Proof.By 5.5.7 (and the line following) we have Ur(g)∩H=Hm, and thusP0,r⊃HmP0. On the other
+hand, we have
+(5.6.1) Dr(HP0) =/circleplusdisplay
+g∈HP0/HδgDr(H)
+and
+(5.6.2) Dr(H) =/circleplusdisplay
+h∈H/HmδhUr(g).ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 31
+5.6.1 and 5.6.2 taken together give
+(5.6.3) Dr(HP0) =/circleplusdisplay
+g∈HP0/HmδgUr(g) =/circleplusdisplay
+h∈H−/H−,m/circleplusdisplay
+g∈P0/H+,mδhδgUr(g).
+5.6.3 and 5.5.8 taken together give
+(5.6.4) Dr(g,P0) =/circleplusdisplay
+g∈P0/H+,mδgUr(g).
+Ifnowh∈Hissuch that δh∈Dr(g,P0), then 5.6.2and 5.6.4togethershowthat δh=δgδwithg∈P0∩H
+andδ∈Ur(g). Therefore δ=δg−1h∈H∩Ur(g) =Hm, and soh∈HmP0. This shows (i). Because the
+inclusionH−⊂Hinduces a bijection H−/H−,m→(HP0)/(HmP0), assertion (ii) now follows from (i)
+and 5.6.3 and 5.6.4. ✷
+By 5.6 (ii) we have mr=Dr(g,P0)⊗D(g,P0)M⊂Dr(HP0)⊗D(g,P0)M=Mr. Using 5.6 (ii) we can
+conclude
+(5.6.5) Mr=Dr(HP0)⊗D(g,P0)M=Dr(HP0)⊗Dr(g,P0)mr=/circleplusdisplay
+g∈HP0/P0,rδgmr.
+Note that, as Ur(g)-modules, the submodule δgmron the right hand side is the same as δg⋆mr, where
+we use the notation as introduced before 5.5 (for Ur(g) instead of D(H)). By Theorem 5.7 below the
+modulesδg⋆mrare all simple, and there is no isomorphism of Ur(g)-modulesδg1⋆mr∼=δg2⋆mrif
+g1P0,r/\e}atio\slash=g2P0,r. This shows that Mris a simple Dr(H)-module.
+Now assume that δg1⋆Mandδg2⋆Mare isomorphic as D(H)-modules. Then, after tensoring
+withDr(H) we find that δg1⋆Mrandδg2⋆Mrare isomorphic as Dr(H)-modules, hence as Ur(g)-
+modules. Then 5.6.5, together with 5.7, implies that g1h1P0,r=g2h2P0,rfor someh1,h2∈H, and hence
+g1HP0=g2HP0.
+We have therefore reduced statements (ii) and (iii) of 5.5 to the ass ertions of the theorem below. ✷
+Theorem 5.7. Assume 5.1 holds. Let M∈Op
+algbe as in 5.5. Let H= expG(pm0Lie(G0))be uniform
+pro-p. Assume that r∈(1
+p,1)∩pQandsare as in 5.5.3. Assume that
+mr=Dr(g,P0)⊗D(g,P0)M/\e}atio\slash= 0.
+(This will be the case if ris close enough to 1.) Then, for every g∈G0theUr(g)-moduleδg⋆mris simple.
+For anyg1,g2∈G0withg1P0,r/\e}atio\slash=g2P0,rtheUr(g)-modulesδg1⋆mrandδg2⋆mrare not isomorphic.
+Proof.We continue to use the notation introduced in the proof of 5.5. We ha ve already seen that mris
+simple asUr(g)-module, and this implies that δg⋆mris simple as well.
+It is trivial to check that for x,y∈G0one hasδx⋆(δy⋆mr) =δxy⋆mr, and ifx∈P0,r, then
+δx⋆mr→mr,n/mapsto→δx·n, is an isomorphism of Ur(g)-modules.
+Now letφ:δg1⋆mr≃−→δg2⋆mrbe an isomorphism of Ur(g)-modules. A straightforward calculation
+shows that φis also aUr(g)-module isomorphism δg−1
+2g1⋆mr≃−→mr, so that we may assume g2= 1.32 SASCHA ORLIK AND MATTHIAS STRAUCH
+LetI0⊂G0be the Iwahori subgroup whose image in G(OL/(πL)) isB(OL/(πL)).3Using the Bruhat
+decomposition
+G0=/coproductdisplay
+w∈W/W PI0wP0
+we may write g=g1=h−1wp1withh∈I0,w∈Wandp1∈P0. By the Iwahori decomposition
+I0= (I0∩U−
+P,0)·(I0∩P0)
+we can write h=up2withp2∈I0∩P0andu∈I0∩U−
+P,0. The same reasoning as before then shows that
+an isomorphism δg⋆mr≃−→mrinduces an isomorphism of Ur(g)-modules which we again denote by φ:
+φ:δw⋆mr≃−→δu⋆mr.
+Step 1.We show first that this can only happen if w∈WP. Letλ∈X(T)∗be the highest weight of M,
+i.e.M=L(λ), andI={α∈∆|/a\}bracketle{tλ,α∨/a\}bracketri}ht∈Z≥0}, cf. Example 2.10.
+Then by Cor. 8.7 and the maximality condition with respect to p, the parabolic subalgebra pis
+pI, where the root system of the Levi subalgebra of pIhasIas a basis of simple roots. Suppose wis
+not contained in WI=WP. Then there is a positive root β∈Φ+\Φ+
+Isuch thatw−1β <0, hence
+w−1(−β)>0, cf. [5, 2.3]. Consider a non-zero element element y∈g−β, and letv+∈Mbe a weight
+vector of weight λ(uniquely determined up to a non-zero scalar). Then we have the fo llowing identity in
+δw⋆mr,
+y·wv+= Ad(w−1)(y)·v+= 0
+as Ad(w−1)(y)∈g−w−1βannihilates v+. We haveφ(v+) =vfor some nonzero v∈mr. And therefore
+0 =φ(y·wv+) =y·uφ(v+) =y·uv= Ad(u−1)(y)·v.
+On the other hand, y′:= Ad(u−1)(y) is an element of u−
+Pand as such acts injectively on M, cf. Cor.
+8.6. We show that y′actually acts injectively on mr, too. Indeed, let v∈mrwithy′·v= 0.Write
+vas a convergent sum v=/summationtext
+µ∈Λ(v)vλ−µwhere Λ(v) is a (non-empty) subset of Z≥0α1⊕...⊕Z≥0αℓ
+andvλ−µ∈Mλ−µ\{0}is a vector of weight λ−µ(cf. the fourth assertion in [11, 1.3.12]). Here
+∆ ={α1,...,α ℓ}is the set of simple roots. Write y′=/summationtext
+γ∈Byγ, whereBis a non-empty subset of
+Φ+\Φ+
+Iandyγis a non-zero element of g−γ⊂u−
+P. Then we have
+0 =y′·v=/summationdisplay
+ν∈Z≥0α1⊕...⊕Z≥0αℓ
+/summationdisplay
+µ∈Λ(v),γ∈B,ν=µ+γyγ·vλ−µ
+,
+where
+/summationdisplay
+µ∈Λ(v),γ∈B,ν=µ+γyγ·vλ−µ
+lies inMλ−ν. Because of the uniqueness of the expansions of elements of mras convergent series of weight
+vectors, cf. [11, 1.3.12], we can conclude that/summationtext
+µ∈Λ(v),γ∈B,ν=µ+γyγ·vλ−µvanishes.
+3We denote the Iwahori subgroup by I0because we use Ifor the subset of simple roots corresponding to p.ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 33
+Define on Qα1⊕...⊕Qαℓthe lexicographic ordering as in the proof of Prop. 8.5. Choose γ+∈B
+andµ+∈Λ(v), both minimal with respect to this ordering. With ν+=γ++µ+we then have
+0 =/summationdisplay
+µ∈Λ(v),γ∈B,ν+=µ+γyγ·vλ−µ=yγ+·vλ−µ+.
+This contradicts Cor. 8.6, and we can thus conclude that wmust be an element of WP.
+From now on we may even assume that w= 1 sinceWP⊂P0⊂P0,r.
+Step 2.Now letu∈U−
+P,0and letφ:mr≃−→δu⋆mrbe an isomorphism of Ur(g)-modules. We suppose
+thatuis not contained in P0,rand are seeking to derive a contradiction. Let v+∈Mλ\{0}be a vector
+of highest weight as above. Put φ(v+) =vwith some non-zero v∈mr. Then we have for any x∈t, the
+following identity in δu⋆mr:
+λ(x)v=φ(x·v+) =x·uφ(v+) = Ad(u−1)(x)·v.
+We have thus for all x∈t, the following identity in mr
+(5.7.1) λ(x)v= Ad(u−1)(x)·v .
+Note that this equation only involves the action of elements of u−
+P⊕t, because Ad( u−1)(x) is inu−
+P⊕t.
+Next we embed mrinto the ”formal completion” of M, i.e.,
+ˆM=/productdisplay
+µMµ
+by mapping the weight spaces Mµ⊂mrto the corresponding space Mµ⊂ˆM(cf. the fourth assertion in
+[11, 1.3.12]). Then ˆMis in an obvious way a module for U(u−
+P⊕t). This module structure extends to
+a representation of U−
+Pas follows. Since U−
+Pis a unipotent group, the exponential expU−
+P:u−
+P/axisshort/axisshort/arrowaxisrightU−
+P
+is actually defined on the whole Lie algebra, so that we have a bijective map expU−
+P:u−
+P→U−
+P. Let
+logU−
+P:U−
+P→u−
+Pbe the inverse map. Then we can define for u∈U−
+Pandv=/summationtext
+µvµ∈ˆM:
+δu·v=/summationdisplay
+µ/summationdisplay
+n≥01
+n!logU−
+P(u)n·vµ.
+Note that this sum is well-defined in ˆM, because logU−
+P(u) is inu−
+P, and there are hence only finitely
+many terms of given weight in this sum. (We continue to write the actio n of an element u∈U−
+PonˆM
+byδu.) Moreover, it gives an action of U−
+PonˆMbecause
+exp(logU−
+P(u1))·exp(logU−
+P(u2)) = exp/parenleftbig
+H(logU−
+P(u1),logU−
+P(u2))/parenrightbig
+= exp(logU−
+P(u1u2)),
+whereH(X,Y) = log(exp( X)exp(Y)) is the Baker-Campbell-Hausdorff series (which converges on u−
+P,
+asu−
+Pis nilpotent). It is then immediate that this action is compatible with the action ofU(u−
+P⊕t). The
+identity 5.7.1 implies then the following identity in ˆM34 SASCHA ORLIK AND MATTHIAS STRAUCH
+δu−1·(x·(δu·v)) = Ad(u−1)(x)·v=λ(x)v ,
+for allx∈t, and thus, multiplying both sides of the previous equation with δu,
+x·(δu·v) =λ(x)δu·v .
+Henceδu·v∈ˆMis a weight vector of weight λand must therefore be equal to a non-zero scalar multiple
+ofv+. After scaling vappropriately we have δu·v=v+orv=δu−1·v+with
+(5.7.2) δu−1·v+=/summationdisplay
+n≥01
+n!(−logU−
+P(u))n·v+=: Σ.
+Our goal is to show that the series Σ, which is an element of ˆM, does in fact not lie in the image of mr
+inˆM, ifu /∈P0,r, thus achieving a contradiction.
+Step 3.LetgZbe aZ-form ofg, i.e.gZ⊗ZL=g. We fix a Chevalley basis ( xγ,yγ,hα|γ∈Φ+,α∈∆)
+of [gZ,gZ]. We have xγ∈gγ,yγ∈g−γ, andhα= [xα,yα]∈t, forα∈∆. Then
+LieZp(H−) =pm0Lie(U−
+P,0) =/circleplusdisplay
+β∈Φ+\Φ+
+IOLy(0)
+β,
+wherey(0)
+β=pm0yβ. Moreover, the Zp-Lie algebra of H−,mis
+LieZp(H−,m) =pmLie(H−) =/circleplusdisplay
+β∈Φ+\Φ+
+IOLy(m)
+β,
+wherey(m)
+β=pm0+myβ. We assume that randsare as in 5.5.3. Then 5.5.7 shows that Ur(u−
+P) =
+Ur(u−
+P,H−) =Ds(H−,m). Elements in Ur(u−
+P) thus have a description as power series in ( y(m)
+β)β∈Φ+\Φ+
+I:
+Ur(u−
+P) =
+
+/summationdisplay
+n=(nβ)dn(y(m))n|lim
+|n|→∞|dn|sκ|n|= 0
+
+,
+where (y(m))nis the product of the ( y(m)
+β)nβover allβ∈Φ+\Φ+
+I, taken in some fixed order. Let
+/bardbl·/bardbl(m)
+sbe the norm on Ds(H−,m) induced by the canonical p-valuation on H−,m. Then we have for any
+generatory(m)
+β
+(5.7.3) /bardbly(m)
+β/bardbl(m)
+s=sκ,
+as follows from [27, 5.3, 5.8]. We recall that by 5.6 we have P0,r=H−,mP0. Write
+logU−
+P(u) =/summationdisplay
+β∈B(u)zβ,
+with a non-empty set B(u)⊂Φ+\Φ+
+Iand non-zero elements zβ∈g−β. PutON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 35
+B+(u) ={β∈B(u)|zβ/∈OLy(m)
+β}.
+This is a non-empty subset of B(u) sinceu /∈P0,r. PutB′(u) =B(u)\B+(u),
+z+=/summationdisplay
+β∈B+(u)zβandz′=/summationdisplay
+β∈B′(u)zβ= logU−
+P(u)−z+.
+Thenz′∈LieZp(H−,m), and thus exp( z′)∈H−,m. The element u1=uexp(z′)−1=uexp(−z′) does not
+lie inH−,m, andδu⋆mr∼=δu1⋆mr. We may hence replace ubyu1=uexp(−z′). Then we compute
+logU−(u1) = logU−(uexp(−z′)) by means of the Baker-Campbell-Hausdorff formula. Because of the
+commutators appearing in this formula we have
+(5.7.4) logU−(u1)∈z++/summationdisplay/parenleftbig
+iterated commutators of g−B+(u)withg−B′(u)/parenrightbig
+,
+whereg−B+(u)=/circleplustext
+β∈B+(u)g−βand analogous for g−B′(u). Recall that the heightht(β) of the root
+β=/summationtext
+α∈∆nααis defined to be the sum/summationtext
+α∈∆nα. Putht′(u) = min{ht(β)|β∈B′(u)}. It follows from
+the preceding formula 5.7.4 that if the right summand does not vanish we haveht′(u1)>ht′(u).
+The process of passing from utou1can be repeated finitely many times, but will finally produce an
+elementuN∈U−
+P,0\H−,mwhich has the property that B′(uN) =∅(and hence uN+1=uN). We will
+denote this element again by u.
+Next we chose an extremal element β+among the β∈B(u) =B+(u), i.e. one of the minimal
+generators of the cone/summationtext
+β∈B(u)R≥0βinside/circleplustext
+α∈∆Rα. Then we have
+R>0β+∩/summationdisplay
+β∈B(u),β/\egatio\slash=β+R≥0β=∅.
+This means in particular that no positive multiple of β+can be written as a linear combination/summationtext
+β∈B(u),β/\egatio\slash=β+cββwith non-negative integers cβ∈Z≥0. It follows that if nis a positive integer and
+(5.7.5)nβ+=γ1+...+γmwith not necessarily distinct γi∈B(u)
+⇒[m=nandγ1=...=γn=β+].
+After these intermediate considerations we return to the series
+Σ =/summationdisplay
+n≥01
+n!(−logU−
+P(u))n·v+=/summationdisplay
+n≥0(−1)n
+n!(zβ++zβ2+...+zβk)n·v+
+whereB(u) ={β+,β2,...,β k}. It follows from 5.7.5 and Cor. 8.6 that if we write Σ as a (formal) sum of
+weight vectors, there is for any n∈Z≥0a unique non-zero weight vector in this series which is of weight
+λ−nβ+, and this element is(−1)n
+n!zn
+β+·v+.36 SASCHA ORLIK AND MATTHIAS STRAUCH
+Step 4.The last part of the proof is to show that the formal sum of weight v ectors
+(5.7.6)/summationdisplay
+n≥0(−1)n
+n!zn
+β+·v+
+cannot be the corresponding sum of weight components of an eleme nt ofmr, when considered as an
+element of ˆMand written as a sum of weight vectors.
+Write Φ+={β1,...,β t}such that Φ+
+I={βτ+1,...,β t}. (Note: the roots βiare not necessarily the
+same as in Step 3.) Choose finitely many weight vectors vj∈Mof weightλ−µj,j= 1,...,k, generating
+MasU(u−
+P)-module. We can and will assume that vjis of the form/parenleftBig/producttextt
+η=τ+1yνj,η
+βk/parenrightBig
+·v+.
+Putyi=yβiandy(m)
+i=y(m)
+βifori= 1,...,τ. To show that 5.7.6 does not converge to an element in
+mr, we write
+(5.7.7)yn
+β+
+n!.v+=/summationdisplay
+1≤j≤k/summationdisplay
+ν∈In,jc(n)
+ν,jyν1
+1·...·yνττ.vj,
+whereIn,j⊂Zτ
+≥0consists of those ν= (ν1,...,ν τ) such that µj+ν1β1+...+ντβτ=nβ+. It follows
+from the Prop. 8.13 that there is at least one index ( ν,j) = (ν1,...,ν τ,j) with the property that
+(5.7.8) |c(n)
+ν,j|K≥/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
+n!/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+Kandν1+...+ντ+t/summationdisplay
+k=τ+1νj,k≥n.
+We will use this inequality to estimate the ||·||r- norm of any lift of1
+n!zn
+β+.v+to
+/circleplusdisplay
+1≤j≤kUr(u−
+P)⊗KKvj.
+Here the/bardbl·/bardblr- norm on this free Ur(u−
+P)-module is the supremum norm of the ||·||r- norms on each
+summand, defined by
+/bardblδ⊗vj/bardblr:=/bardblδ/bardblr,
+forδ∈Ur(u−
+P). Because B′(u) =∅(cf. Step 3), we have zβ+=p−ay(m)
+β+for somea >0. We then get
+from 5.7.7
+zn
+β+
+n!.v+=p−na(y(m)
+β+)n
+n!·v+=/summationdisplay
+1≤j≤k/summationdisplay
+ν∈In,jc(n)
+ν,jp−na+(m0+m)(n−ν1−...−ντ)(y(m)
+1)ν1·...·(y(m)
+τ)ντ.vj.
+It follows from [27, 6.2, 6.4] that the restriction of the norm /bardbl·/bardblronD(H,K) toD(Hm,K) is equivalent
+to the norm/bardbl·/bardbl(m)
+sonD(Hm,K) (cf. also [27, 7.4], where it is stated that the induced topologies ar e
+equivalent). Therefore, we are now going to estimate the /bardbl·/bardbl(m)
+s- norm of the term
+c(n)
+ν,jp−na+(m0+m)(n−ν1−...−νr)(y(m)
+1)ν1·...·(y(m)
+r)νr,ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 37
+where (ν,j) is such that 5.7.8 holds. By 5.7.3, the /bardbl·/bardbl(m)
+s-norm of this term is greater or equal to
+/vextendsingle/vextendsingle1
+n!/vextendsingle/vextendsingle
+Kpna+(m0+m)(ν1+...+νr−n)sν1+...+νr
+=/vextendsingle/vextendsingle1
+n!/vextendsingle/vextendsingle
+Kpna−n+(m0+m−1)(ν1+...+νr−n)(ps)ν1+...+νr.
+Note that 5.7.8 implies that n−(ν1+...+νr) is bounded from above by some constant C1. Hence we
+get
+pna−n+(m0+m−1)(ν1+...+νr−n)≥pn(a−1)·p−(m0+m−1)C1
+is bounded away from 0 (recall that a∈Z>0). And because s >1
+pwe haveps >1, and the term
+(ps)ν1+...+νris unbounded as n→∞(ν1+...+ντ≥n−C1). Altogether we get that any lift of1
+n!zn
+β+.v+
+to/circleplustext
+1≤j≤tUr(u−)⊗KKvihas an/bardbl·/bardblr- norm which exceeds C2(ps)n, whereC2>0 is some constant
+(cf. the relation between the norms /bardbl·/bardblrand/bardbl·/bardbl(m)
+smentioned above).
+The sum/summationtext
+n≥01
+n!zn
+β+.v+, which is an element of ˆMis therefore not contained in the image of mr
+(undertheinjection mr֒→ˆM). Andaswehaveseenbefore, thisinturnprovesthat/summationtext
+n≥0(−log(u))n
+n!.v+=
+δu−1.v+∈ˆMis not in the image of mr. ✷
+Theorem 5.8. Assume that Assumption 5.1 holds. Let M∈ Oalgbe simple and assume that pis
+maximal for M(cf. 5.2). Let Vbe a smooth and irreducible LP-representation. Then FG
+P(M,V) =
+IndG
+P(W′⊗KV)dis topologically irreducible as a G-representation.
+Proof.We start by giving an outline of the proof. Put X= (FG
+P(M,V),π). Then, to any G-
+subrepresentation U⊂Xthere is asmooth representation Usmand anatural G-morphism IndG
+P(W′)d⊗K
+Usm→U. HereUsmis a subrepresentation of Xsm, and we will show that Xsmis isomorphic to the
+smoothly induced representation indG
+P(V). We will prove that if Uis closed and non-zero, then Usmis
+non-zero as well, and the composite map
+IndG
+P(W′)d⊗KUsm֒→IndG
+P(W′)d⊗KindG
+P(V)−→X
+is surjective. As this map factors through Uwe necessarily have U=X.
+Step 1: the smooth representation Usm.For aG-subrepresentation U⊂X, we put
+Usm= lim−→HHomH(IndG
+P(W′)d|H,U|H),
+where the limit is taken overall compact open subgroups HofG, and the homomorphisms are continuous
+homomorphisms of topological vector spaces. Of course, it suffice s to take the limit over a set of compact
+open subgroups which is a neighborhood basis of 1 ∈G. There is an action of Gdefined on Usmas
+follows: for φ∈Usm,g∈Gput
+(gφ)(f) =π(g)(φ(π′(g−1)(f))),
+whereπ′is the representation on IndG
+P(W′)d.Note thatUsmis by construction a smooth representation.
+Taking for Uthe whole space Xgives us a smooth G-representation Xsm. Note that there is always a
+continuous map of G-representations38 SASCHA ORLIK AND MATTHIAS STRAUCH
+ΦU: IndG
+P(W′)d⊗KUsm−→U, f⊗φ/mapsto→φ(f).
+In order to analyze Usmwe will need to understand the restriction of IndG
+P(W′)dto compact-open
+subgroupsH.
+Step 2: restricting IndG
+P(W′)dto compact open subgroups. LetH⊂G0be an open normal subgroup. We
+have
+(5.8.9) IndG
+P(W′)d|H= IndG0
+P0(W′)d|H=/circleplusdisplay
+γ∈G0/HP0IndH
+H∩γP0γ−1((W′)γ)d.
+The notation on the right hand side has the following meaning. The und erlyingK-vector spaces of
+(W′)γandW′are the same. Further, if ( ρ′,W′) is the given representation of then (( ρ′)γ,(W′)γ) is the
+representation of H∩γP0γ−1defined by
+(ρ′)γ(h) =ρ′(γ−1hγ).
+It is now straightforward to check that, as H-representations, with the notation introduced before 5.5,
+IndH
+H∩γP0γ−1((W′)γ)d≃δγ⋆IndHP0
+P0(W′)d.
+By 5.5 (ii, iii), all H-representations on the right hand side of 5.8.9 are topologically irre ducible, and
+between any two such representations IndH
+H∩γ1P0γ−1
+1(W) and IndH
+H∩γ2P0γ−1
+2(W) withγ1HP0/\e}atio\slash=γ2HP0,
+there is no non-zero continuous H-equivariant morphism. Similarly, for the representation IndG
+P(W′⊗K
+V)dwe have
+IndG
+P(W′⊗KV)d|H= IndG0
+P0(W′⊗KV)d|H=/circleplusdisplay
+γ∈G0/HP0IndH
+H∩γP0γ−1((W′)γ⊗KVγ)d.
+Denote byVH∩P0⊂Vthe subspace on which H∩P0acts trivially. Then
+IndH
+H∩γP0γ−1((W′)γ⊗K(VH∩P0)γ)d
+is anH-subrepresentation of IndH
+H∩γP0γ−1((W′)γ⊗KVγ)d. And
+IndH
+H∩γP0γ−1((W′)γ⊗K(VH∩P0)γ)d
+is canonically isomorphic to
+/parenleftBig
+IndH
+H∩γP0γ−1((W′)γ)/parenrightBigd
+⊗KVH∩P0.
+Let
+φ: IndG0
+P0(W′)d|H→IndG0
+P0(W′⊗KV)d|H
+be a continuous homomorphism of H-representations. For each γ,letfγbe a non-zero vector in
+IndH
+H∩γP0γ−1((W′)γ)d.ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 39
+Because of the irreducibility of the representations IndH
+H∩γP0γ−1((W′)γ)d, the map φis uniquely de-
+termined by the images φ(fγ)∈IndG0
+P0(W′⊗KV)d. By the very definition of the representation
+IndG0
+P0(W′⊗KV)d, the function φ(fγ) :G0→W′⊗KVis rigid-analytic on the components of some
+covering of G0by ’polydiscs’ ∆ i, and takes values in a BH-subspace of W′⊗KV. But any such BH-
+subspace is finite-dimensional, hence contained in a subspace of the formW′⊗KVH′∩P0for some (small
+enough) open subgroup H′⊂Hwhich is normal in G0. Shrinking H′we may further suppose that each
+∆iisH′-stable (by multiplication from the left), and that each φ(fγ) takes values in W′⊗KVH′∩P0. For
+anyh∈H, we have
+φ(h.fγ)(g) = [h.φ(fγ)](g) =φ(fγ)(h−1g)∈W′⊗KVH′∩P0.
+Because the subspace of IndH
+H∩γP0γ−1((W′)γ)dgenerated by the functions h.fγ,h∈H, is dense in
+IndH
+H∩γP0γ−1((W′)γ)d, we haveφ(f)(g)∈W′⊗KVH′∩P0for allf∈IndH
+H∩γP0γ−1((W′)γ)d. It follows
+thatφinduces a continuous map of H′-representations
+φ: IndG0
+P0(W′)d|H′→IndG0
+P0(W′⊗KVH′∩P0)d|H′=/bracketleftBig
+IndG0
+P0(W′)d/bracketrightBig
+|H′⊗KVH′∩P0
+By 5.5 (ii, iii), the irreducible H′-representation IndH′
+H′∩γP0γ−1((W′)γ)dcontained in the left hand side
+is mapped by φto the corresponding summand/bracketleftBig
+IndH′
+H′∩γP0γ−1((W′)γ)d/bracketrightBig
+⊗KVH′∩P0on the right hand
+side. The map φ, restricted to this summand, is thus of the form f/mapsto→f⊗vfor somev∈VH′∩P0. This
+shows that any element in Xsmis ”locally” (i.e. on the subrepresentations IndH′
+H′∩γP0γ−1(Wγ)d) given
+by vectors in VH′∩P0(for sufficiently small subgroups H′). It is now an easy matter to verify that the
+G-action onXsmidentifiesXsmwith the smoothly induced representation indG
+P(V). Thus we have shown
+the
+Sublemma 5.9. The canonical map indG
+P(V)→Xsm,ϕ/mapsto→[f/mapsto→(g/mapsto→f(g)⊗ϕ(g))], is an isomorphism.
+Step 3:Usmis non-zero for non-zero U.LetU⊂Xbe a non-zero closed G-invariant subspace. Let
+f∈Ube a non-zeroelement. Then, aswe haveseen before, ftakes valuesin a vectorspace W′⊗KVH∩P0
+for a sufficiently small compact open subgroup H. Therefore, fis contained in
+/circleplusdisplay
+γ∈G0/HP0IndH
+H∩γP0γ−1/parenleftbig
+(W′)γ⊗K(VH∩P0)γ/parenrightbigd=/circleplusdisplay
+γ∈G0/HP0IndH
+H∩γP0γ−1((W′)γ)d⊗KVH∩P0
+It follows from this that Ucontains a non-zero H-invariant subspace of
+/circleplusdisplay
+γ∈G0/HP0IndH
+H∩γP0γ−1((W′)γ)d⊗KVH∩P0.
+Because the representations IndH
+H∩γP0γ−1(Wγ) are irreducible, and because there is no non-zero con-
+tinuous intertwiner between any two such (with Hγ1P0/\e}atio\slash=Hγ2P0), any such subspace is isomorphic
+to
+/circleplusdisplay
+γ∈G0/HP0IndH
+H∩γP0γ−1((W′)γ)d⊗KVγ,40 SASCHA ORLIK AND MATTHIAS STRAUCH
+with asubspace VγofVH∩P0. Thus, anynon-zerovectorin some Vγgivesrisetoanon-zero H-equivariant
+homomorphism from IndG0
+P0(W′)d|HtoU. Therefore, Usm/\e}atio\slash={0}.
+Step 4:IndG
+P(W′)d⊗KUsmsurjects onto X.Letφ0∈Usmbe a non-zero element which we identify with
+an element of the smoothly induced representation indG
+P(V). Without loss of generality we may assume
+φ0(1)/\e}atio\slash= 0. Consider the P-morphism Π : Usm→Vgiven byφ/mapsto→φ(1). Asφ0(1)/\e}atio\slash= 0 we deduce that the
+image of Π is a non-zero subrepresentation of V, and is hence equal to V(becauseVwas supposed to be
+irreducible). Therefore, for any v∈Vthere is some φ∈Usmwithφ(1) =v. AsUsmis aG-representation
+we find that for any g∈Gand anyv∈Vthere is some φ∈Usmwithφ(g) =v. Then, on a neighborhood
+Nofg, the mapφis constant with value von this neighborhood. Let S⊂G0be a compact locally
+analytic subset such that G=S·Pand henceG0=S·P0. (That is, every g∈Gis the product s·p
+with uniquely determined s∈Sandp∈P). Then there is a compact open subset S′⊂Sand a compact
+open neighborhood of the identity P′⊂P0such thatS′·P′⊂N. Letf:G0→W′be a function whose
+support is contained in S′·P0. Then the function S→W′⊗KV,s/mapsto→f(s)⊗φ(s), is equal to the function
+S→W′⊗KV,s/mapsto→f(s)⊗v.
+Letf∈IndG
+P(W′⊗KV)dbe any element. Then fis uniquely determined by its restriction to S.
+As we have seen above, the set f(G0) is contained in a finite-dimensional vector space W′⊗KV0with
+V0⊂V. Letv1,...,v rbe a basis of V0. Writef(s) =f1(s)⊗v1+...+fr(s)⊗vrwith locally analytic
+W′-valued functions fionS. Extend each function fitoGbyfi(s·p) =ρ′(p−1)(fi(s)), whereρ′is the
+representation of PonW′. Thenfi∈IndG
+P(W′) for alli. In fact, examining the action of the differential
+operators in dshows that fi∈IndG
+P(W′)dfor alli. For anys∈Schoose some φi,s∈Usmsuch that
+φi,s(x) =vifor allxin a compact open neighborhood Ni,s⊂Sofs. AsSis compact, finitely many
+of theNi,swill coverS. We can then choose a (finite) disjoint refinement ( Ni,j)jof the finite covering.
+Then we restrict fito each of these Ni,j·P, and extend it by 0 to ( S\Ni,j)·P. Denote the function
+thus obtained by fi,j. Again, all fi,jlie in IndG
+P(W′)d, andx/mapsto→fi,j(x)⊗φi,s(i,j)(x) =fi,j(x)⊗vi, for a
+suitably chosen s(i,j)∈S. Obviously, we have: f=/summationtext
+i,jfi,j⊗φi,s(i,j). Indeed, by construction, both
+f|Sand the restriction of the sum to Scoincide. And both are functions in IndG
+P(W′⊗KV)d; therefore,
+they are equal. This shows that fcan be written as a sum of simple tensors of functions in IndG
+P(W′)d
+and functions in Usmhence, the map IndG
+P(W′)d⊗KUsm→IndG
+P(W′⊗KV)d=Xis surjective. ✷
+Example 5.10. a) LetMbe a finite-dimensional irreducible algebraic G-module, so that in the above
+theorem we may choose P=G. Then we deduce that FG
+G(M,V) =M⊗KVis topologically irreducible
+for any smooth irreducible representation VofG. This result was already proved by D. Prasad [29, app.],
+and our proof is modeled after his approach.
+b) LetM=U(g)⊗U(p)W∈Op
+algbe an irreducible generalized Verma module for some irreducible
+LP-representation W. Then by Cor. 8.7 the parabolic subalgebra pis maximal in the above sense.
+It follows thatFG
+P(M) = IndG
+P(W′) is topologically irreducible. This result was proved in [25] for an
+arbitrary irreducible finite-dimensional locally analytic LP-representation W.
+6.Composition series
+6.1.Before we formulate our main result in this section, we recall the defi nition of (smooth) generalized
+Steinberg representations, cf. [3], [6]. Let Pbe a standard parabolic subgroup of Gand denote byON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 41
+iG
+P= indG
+P(1) =C∞(G/P,K) the smooth induced representation of locally constant functions onG/P.
+The generalized Steinberg representation to Pis the quotient
+vG
+P=iG
+P//summationdisplay
+P/subsetnoteqlQ⊂GiG
+Q.
+This is an irreducible G-representation and all irreducible subquotients of iG
+Pare of the form vG
+Qwith
+P⊂Q. Each such representation occurs with multiplicity one, cf. loc.cit.
+6.2.How to obtain a composition series for FG
+P(M,V).We are going to describe a method for finding
+a composition series of the representation FG
+P(M,V). LetMbe an object ofOp
+algandVan object of
+Rep∞,a
+K(LP) of finite length. Let (0) = V0/subset⋉oteqlV1/subset⋉oteql.../subset⋉oteqlVr=Vbe a composition series of smooth
+LP-representations for V. By the exactness of FG
+Pin the second argument, cf. Prop. 4.9, we get a chain
+of inclusions
+(0) =FG
+P(M,V0)/subset⋉oteqlFG
+P(M,V1)/subset⋉oteql.../subset⋉oteqlFG
+P(M,Vr) =FG
+P(M,V).
+Further the quotient FG
+P(M,Vi+1)/FG
+P(M,Vi) is isomorphic to FG
+P(M,Vi+1/Vi). We may hence assume
+thatVis irreducible.
+LetMbe an object of the category Op
+alg.Then it has a Jordan-H¨ older series
+M=M0/superset⋉oteqlM1/superset⋉oteqlM2/superset⋉oteql.../superset⋉oteqlMr/superset⋉oteqlMr+1= (0)
+in the same category. Thus we may apply our functor FG
+Pto the pair ( M,V). We get a sequence of
+surjections
+FG
+P(M,V) =FG
+P(M0,V)p0−→FG
+P(M1,V)p1−→FG
+P(M2,V)p2−→...pr−1−→FG
+P(Mr,V)pr−→(0).
+For any integer iwith 0≤i≤r,we put
+qi:=pi◦pi−1◦···◦p1◦p0.
+and
+Fi:= ker(qi)
+which is a closed subrepresentation of FG
+P(M,V). We obtain a filtration
+(6.2.1) F−1= (0)⊂F0⊂···⊂Fr−1⊂Fr=FG
+P(M,V)
+by closed subspaces with
+Fi/Fi−1∼=FG
+P(Mi/Mi+1,V).
+LetQi=Li·Ui⊃Pbe the standard parabolic subgroup which is maximal for Mi/Mi+1. Then
+FG
+P(Mi/Mi+1,V) =FG
+Qi(Mi/Mi+1,iLi
+LP·Li∩P(V)).
+Then we can choose a Jordan-H¨ older series for iLi
+LP·Li∩P(V) and we obtain a Jordan-H¨ older series for42 SASCHA ORLIK AND MATTHIAS STRAUCH
+FG
+Qi(Mi/Mi+1,iLi
+LP·Li∩P(V)).
+By refining the filtration 6.2.1 we get thus a Jordan-H¨ older series of FG
+P(M).
+Inthecasewhere Visactuallythetrivialrepresentationtheirreduciblesubquotientso fthesmoothrep-
+resentation iLi
+LP·Li∩P=iQi
+Pare the generalized Steinberg representations vQi
+QwithQi⊃Q⊃P, recalled
+above. Thusthe irreduciblesubquotientsof FG
+P(M,1) =FG
+P(M)areoftheformFG
+Qi(Mi/Mi+1,vQi
+Q), i=
+1,...,r, and where Qi⊃Q⊃P.
+Remark 6.3. Independently ofourwork, B. Schraen[36] determined the Jorda n-H¨ olderseries forcertain
+subquotients of parabolically induced locally analytic representation s for the group GL 3(Qp), including
+the locally analytic Steinberg representation.
+7.Applications to equivariant vector bundles on Drinfeld’s h alf space
+LetXbe Drinfeld’s half space of dimension d≥1 overK. This is a rigid-analytic variety over K
+given by the complement of all K-rational hyperplanes in projective space Pd
+K,i.e.,
+X=Pd
+K\/uniondisplay
+H/subsetornotdbleqlKd+1P(H),
+whereHruns over the set of K-rational hyperplanes in Kd+1.
+There is natural action of G= GLd+1(K) onXinduced by the algebraic action m:G×Pd
+K→Pd
+Kof
+G= GLd+1/Kdefined by
+g·[q0:···:qd] :=m(g,[q0:···:qd]) := [q0:···:qd]g−1.
+It is known thatXis a Stein space [28, §1, Prop. 4]. Moreover, for every homogeneous vector bundle E
+onPd
+Kthe space of sections E(X) =H0(X,E) has the structure of a K-Fr´ echet space equipped with a
+continuous G-action. Its topological dual E(X)′is a locally analytic G-representation, cf. [33, 24].
+In [24] one of us studied the structure of E(X) for homogeneous vector bundles EonPd
+K. The theorem
+below generalizes the result of Schneider and Teitelbaum [33] for the canonical bundle E= Ωdresp.
+Pohlkamp [26] for the structure sheaf E=O.
+Theorem 7.1. LetEbe a homogeneous vector bundle on Pd
+K.There is aG-equivariant filtration by closed
+K-subspaces ofE(X)
+(7.1.2) E(X) =E(X)0⊃E(X)1⊃···⊃E (X)d−1⊃E(X)d=H0(Pd,E),
+such that for j= 0,...,d−1,there are extensions of locally analytic G-representations
+(7.1.3) 0→vG
+P(j+1,1,...,1)(Hd−j(Pd
+K,E)′)→(E(X)j/E(X)j+1)′→IndG
+P(j+1,d−j)(U′
+j)dj→0.
+Here, for a decomposition ( n1,...,n s) ofd+1, the symbol P(n1,...,ns)denotes the corresponding lower
+standard parabolic subgroup of G. The module vG
+P(j+1,1,...,1)(Hd−j(Pd
+K,E)′) is a generalized Steinberg
+representation with coefficients in the finite-dimensional algebraic G-moduleHd−j(Pd
+K,E)′.ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 43
+TheP(j+1,d−j)-representation U′
+jis a tensor product N′
+j⊗KvGLd−j
+GLd−j∩Bof an algebraic representation
+N′
+jand the Steinberg representation vGLd−j
+GLd−j∩Bof the factor GL d−jof the Levi subgroup L(j+1,d−j)⊂
+P(j+1,d−j). The first one is characterized by the property that it generates the kernel ˜Hd−j
+Pj
+K(Pd
+K,E) of the
+natural homomorphism
+Hd−j
+Pj
+K(Pd
+K,E)→Hd−j(Pd
+K,E)
+as a module with respect to U(g). HerePj
+K=V(Xj+1,...,X d) is the linear subvariety of Pd
+Kdefined by
+the vanishing of the coordinates Xj+1,...,X d.The module djis just the kernel of the induced surjection
+U(g)⊗U(p(j+1,d−j))N′
+j→˜Hd−j
+Pj
+K(Pd
+K,E).
+In the case where Earises from an irreducible representation of the Levi subgroup L(1,d),we could
+make our result more precise, i.e., concerning the structure of Nj.Rather than recalling this result in full
+generality, we restrict our attention from now on to homogeneous line bundles on Pd
+K, where we will get
+even a more precise formula, cf. Prop. 7.5.
+Lets∈Zand denote by λ′= (s,...,s)∈Zdthe constant integral weight of GL d. Letr=λ0∈Z
+and set
+λ:= (r,s,...,s )∈Zd+1.
+We denote byLλthe homogeneousline bundle on Pd
+Ksuchthat its fibre in the basepoint is the irreducible
+algebraicL(1,d)-representation corresponding to λ. Then we obtain
+(7.1.4) Lλ=O(r−s)
+where theG-linearization is given by the tensor product of the natural one on O(r−s) with the character
+dets.
+Putwj:=sj···s1,wheresi∈Wis the (standard) simple reflection in the Weyl group W∼=Sd+1of
+G. Recall that·denotes the dot action of WonX∗(T)Q. Ifχ= (χ0,...,χ d)∈Zd+1we get
+wi·χ= (χ1−1,χ2−1,...,χ i−1,χ0+i,χi+1,...,χ d).
+Hence forχ=λ= (r,s,...,s ), we compute
+w0·λ=λ
+w1·λ= (s−1,r+1,s,...,s)
+...
+wi·λ= (s−1,s−1,...,s−1,r+i,s,...,s )
+...
+wd·λ= (s−1,s−1,...,s−1,r+d).
+In particular, there is at most one integer 0 ≤i0≤d, such that wi0·λis dominant with respect
+to the Borel subgroup B+of upper triangular matrices. In fact, this integer is characterize d by the
+non-vanishing of Hi0(Pd
+K,Lλ) (cf. [4, Thm. IV’] resp. [12, Thm. III.5.1]), which is i0= 0 forr≥sresp.44 SASCHA ORLIK AND MATTHIAS STRAUCH
+i0=dfors≥r+d+1. Otherwise, there is a unique integer i0<dwithwi0·λ=wi0+1·λ.This is the
+case for 0≤i0=s−r−1<d+1.We get
+(7.1.5) wi·λ≻wi+1·λ
+for alli≥i0(resp.i>i0ifwi0·λ=wi0+1·λ), and
+(7.1.6) wi·λ≺wi+1·λ
+for alli<i0, with respect to the dominance order ≻onX∗(T)Q. We put
+µi,λ:=/braceleftBigg
+wi−1·λ:i≤i0
+wi·λ:i>i0/bracerightBigg
+,i= 1,...,d.
+This is a L(i,d−i+1)-dominant weight (with respect to the Borel subgroup L(i,d−i+1)∩B+). LetVi,λbe
+the finite-dimensional irreducible L(i,d−i+1)-module with highest weight µi,λ.LetP+
+(i,d−i+1)be the upper
+triangular parabolic subgroup to the decomposition ( i,d+1−i) and denote by U+
+(i,d−i+1)its unipotent
+radical. By considering the trivial action of U+
+(i,d−i+1)onVi,λ, we may view it as a P+
+(i,d−i+1)-module.
+In the following we identify the linear subvarieties Pd−i
+K,0≤i≤d,with the closed subschemes
+V(X0,...,X i−1)⊂Pd
+Kdefined by the vanishing of the first icoordinate functions. Note that the
+stabilizer of this subvariety in Gis justP+
+(i,d+1−i).4In loc.cit. we saw that we can realize Vi,λas a
+submodule of ˜Hi
+Pd−i
+K(Pd
+K,Lλ).
+Infact,˜Hi
+Pd−i
+K(Pd
+K,Lλ)isnaturallyamodulefor P+
+(i,d−i+1)⋉U(g). Thissymbolicnotationsignifiesthat
+it is both a U(g)-module and an algebraic representation of P+
+(i,d−i+1), and that these two structures are
+compatible in the sense that h.(x.(h−1.v)) = Ad(h)(x).vfor allh∈P+
+(i,d−i+1),x∈g,v∈˜Hi
+Pd−i
+K(Pd
+K,Lλ).
+The operation of gis induced by the homogeneous line bundle. The second one is induced b y functo-
+riality since it is the stabilizer of Pd−i
+K.
+Proposition 7.2. For1≤i≤d,theP+
+(i,d−i+1)⋉U(g)-module ˜Hi
+Pd−i
+K(Pd
+K,Lλ)coincides with the module
+/circleplustext
+k0,...,ki−1≤0
+ki,...,kd≥0
+k0+···+kd=0K·Xk0
+0Xk1
+1···Xkd
+d·Vi,λ.
+Proof.This was proved in loc.cit. Prop. 1.4.2. ✷
+From this latter statement we deduce immediately the following corolla ry.
+Corollary 7.3. For1≤i≤d,theU(g)-module ˜Hi
+Pd−i
+K(Pd
+K,Lλ)lies in the category Op+
+(i,d−i+1)
+alg.
+Remark 7.4. It is well-known that the above objects lie in O,although, mostly the case of the full flag
+variety is considered, cf. e.g. [20], [1]. Indeed this can be seen by usin g the Grothendieck-Cousin complex
+which was used in loc.cit. to compute ˜Hi
+Pd−i
+K(Pd
+K,Lλ). Here all contributions of this complex are objects
+inO.Hence the cohomology of this complex which computes ˜Hi
+Pd−i
+K(Pd
+K,Lλ) lies inO, as well.
+Recall that we denote for a character µ∈X∗(T) byL(µ) the irreducible highest weight U(g)-module
+of weightµ.
+4We note that for formulating Thm. 7.1 we have used in loc.cit. the identification of Pd−i
+KwithV(Xd−i+1,...,X d).
+Therefore the standard parabolic subgroups used there are l ower (block) triangular. Afterwards we used the conjugacy o f
+V(X0,...,X i−1) andV(Xd−i+1,...,X d) within Pd
+Kvia the action of GonPd
+K.ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 45
+Proposition 7.5. For1≤i≤d,theP+
+(i,d−i+1)⋉U(g)-module ˜Hi
+Pd−i
+K(Pd
+K,Lλ)is isomorphic to L(µi,λ).
+Proof.In order to show the statement it suffices to check the following con ditions.
+1)˜Hi
+Pd−i
+K(Pd
+K,Lλ) is a highest weight module of weight µi,λ.
+2)˜Hi
+Pd−i
+K(Pd
+K,Lλ) is irreducible.
+Because of identity 7.1.4 it suffices to check the conditions in the case wheres= 0.ThenLλ=O(r) is the
+r-thSerretwistwithitsnatural G-linearization. Allcohomologygroupsare K-subspacesof/circleplustext
+k0,...,kd∈ZK·
+Xk0
+0Xk1
+1···Xkd
+d. Here the action of gis given as follows. For a root α=αi,j=ǫi−ǫj∈Φ,let
+Lα:=L(i,j)∈gα
+be the standard generator of the weight space gαing.Then the action of gis determined by
+(7.5.1) L(i,j)·Xk0
+0Xk1
+1···Xkd
+d=kj·Xi
+Xj·Xk0
+0Xk1
+1···Xkd
+d
+and
+t·Xk0
+0Xk1
+1···Xkd
+d= (/summationdisplay
+ikiti)·Xk0
+0Xk1
+1···Xkd
+dfort= (t0,...,td)∈t.
+We prove statements 1) and 2) case by case.
+a) Caser≥0. Thenµi,λ=wi·λand
+Vi,λ=/circleplusdisplay
+ki,...,kd≥0
+ki+···+kd=r+iK·X−1
+0X−1
+1···X−1
+i−1Xki
+i···Xkd
+d.
+Furthervi,λ=X−1
+0X−1
+1···X−1
+i−1Xr+i
+iis a maximal vector in ˜Hi
+Pd−i
+K(Pd
+K,Lλ) of weight µi,λ.
+It is easy to see by formula 7.5.1 that vi,λgenerates ˜Hi
+Pd−i
+K(Pd
+K,Lλ) as aU(g)-module, so condition 1)
+is satisfied. As for condition 2), let N⊂˜Hi
+Pd−i
+K(Pd
+K,Lλ) be a submodule, N/\e}atio\slash= (0). Then Nlies in the
+categoryOas a submodule, hence tacts semi-simply. Let Xk0
+0Xk1
+1···Xkd
+d∈N, wherek0,...ki−1<0,
+ki...,kd≥0 and/summationtextd
+j=0kj=r.Then by repeated multiplication with L(k,l)with 0≤k≤i−1 andl≥i,
+we can achieve that - up to scalar - k0=···=ki−1=−1.By further multiplying with L(k,l)wherek=i
+andl≥i+1,...,d, we see that vi,λ∈N.ThusN=˜Hi
+Pd−i
+K(Pd
+K,Lλ).
+b) Caser≤−d−1. Thenµi,λ=wi−1·λand
+Vi,λ=/circleplusdisplay
+k0,...,ki−1<0
+k0+···+ki−1=rK·Xk0
+0Xk1
+1···Xki−1
+i−1.
+FurtherX−1
+0X−1
+1···X−1
+i−2Xr+i−1
+i−1is a maximal vector in ˜Hi
+Pd−i
+K(Pd
+K,Lλ) of weight µi,λ.
+It is easyto see by formula7.5.1that vi,λgenerates ˜Hi
+Pd−i
+K(Pd
+K,Lλ) as aU(g)-module, so condition 1) is
+satisfied. As forcondition 2), let N⊂˜Hi
+Pd−i
+K(Pd
+K,Lλ) be a submodule, N/\e}atio\slash= (0). LetXk0
+0Xk1
+1···Xkd
+d∈N,
+wherek0,...ki−1<0,ki...,kd≥0 and/summationtextd
+j=0kj=r.Then by repeated multiplication with L(k,l)with
+0≤k≤i−1 andl≥i, we can achieve that - up to scalar - ki=···=kd= 0.By further multiplying
+withL(k,l)wherek<i−1 andl=i−1, we see that vi,λ∈N.ThusN=˜Hi
+Pd−i
+K(Pd
+K,Lλ).46 SASCHA ORLIK AND MATTHIAS STRAUCH
+c) Case 0>r>−d−1. Then for i≤i0=−r−1, we have µi,λ=wi−1·λand
+Vi,λ=/circleplusdisplay
+k0,...,ki−1<0
+k0+···+ki−1=rK·Xk0
+0Xk1
+1···Xki−1
+i−1.
+FurtherX−1
+0X−1
+1···X−1
+i−2Xr+i−1
+i−1is a maximal vector in ˜Hi
+Pd−i
+K(Pd
+K,Lλ) of weight µi,λ.
+Fori>i0=−r−1, we have µi,λ=wi·λand
+Vi,λ=/circleplusdisplay
+k0,...,ki−1<0
+k0+···+ki−1=rK·Xk0
+0Xk1
+1···Xki−1
+i−1.
+FurtherX−1
+0X−1
+1···X−1
+i−2Xr+i−1
+i−1is a highest weight vector in ˜Hi
+Pd−i
+K(Pd
+K,Lλ) of weight µi,λ.
+Here the reasoning is a mixture of the previous cases. ✷
+Now we use the above result for the computation of ˜Hi
+Pd−i
+K(Pd
+K,E) wherePd−i
+Kis identified with the
+closed subscheme Vi=V(Xd−i+1,...,X d) ofPd
+K.Consider the block matrix
+zi:=/parenleftBigg
+0Ii
+Id+1−i0/parenrightBigg
+∈G,
+whereIj∈GLj(K) denotes the j×j-identity matrix. Then V(X0,...Xi−1) is transformed into
+V(Xd−i+1,...,X d) under the action of zionPd
+K.We have
+(7.5.1) zi·P(d−i+1,i)·z−1
+i=P+
+(i,d+1−i)
+and on the Levi subgroups the conjugacy map is given by
+L(d−i+1,i)∋/parenleftBigg
+A0
+0B/parenrightBigg
+/mapsto→/parenleftBigg
+B0
+0A/parenrightBigg
+∈L(i,d−i+1).
+Hence the P(d−i+1,i)⋉U(g)-module ˜Hi
+Vi(Pd
+K,E),is given by ˜Hi
+Pd−i
+K(Pd
+K,E) twisted with the action of zi.
+In particular, we can choose Vi,λ- equipped with its action of P(d−i+1,i)via the isomorphism 7.5.1 - to
+be the representation Nd−iof Thm. 7.1. Its highest weight is z−1
+i·µi,λ.
+Corollary 7.6. IfE=Lλis a line bundle, then the contributions IndG
+P(j+1,d−j)(U′
+j)djin 7.1.3 are topo-
+logically irreducible, j= 0,...,d−1.
+Proof.By definition of our functor FG
+Pthere is for j= 0,...,d−1,the identity IndG
+P(j+1,d−j)(U′
+j)dj
+=FG
+P(j+1,d−j)(L(z−1
+j·µj,λ)).By Thm. 5.3 all these contributions are topologically irreducible since t he
+occuring standard parabolic subgroups are (proper) maximal and the simple modules L(z−1
+j·µj,λ) do
+not lie inOg(since they are not finite-dimensional). ✷
+Now we consider the remaining ingredients of the filtration 7.1.2. For t he line bundleLλ, we compute
+using [12, Thm. III.5.1]
+H∗(Pd
+K,Lλ) =Hi0(Pd
+K,Lλ)
+=
+
+Symr−s(Kd+1)⊗detsforr−s≥0 andi0= 0
+Sym−d−1−(r−s)(Kd+1)′⊗dets−1forr−s≤−d−1 andi0=d
+(0) otherwise.ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 47
+Hence we see that the filtration step Lλ(X)dvanishes or it is irreducible. Further we conclude that
+the contribution vG
+P(j+1,1,...,1)(Hd−j(Pd
+K,E)′) in 7.1.3 vanishes for j/\e}atio\slash= 0.In the case j= 0 this object
+coincides with vG
+B(K)⊗(Sym−d−1−(r−s)(Kd+1)⊗det1−s) which is irreducible by [29, app.] resp. Thm.
+5.8. Consequently, the filtration 7.1.2 is essentially a Jordan-H¨ older series ofH0(X,Lλ), i.e., by refining
+the filtration in the case r−s≤d−1 in the naive way we get a real Jordan-H¨ older series.
+8.Appendix: some properties of highest weight modules in O
+This section is about relations in a simple U(g)-moduleM∈Op
+algof the form
+yn
+γ·v+=/summationdisplay
+ν∈Incνyν1
+β1·...·yνt
+βt·v+,
+where
+- Φ+={β1,...,β t},
+-Inconsists of all t-tuples (ν1,...,ν t)∈Zt
+≥0satisfyingν1β1+...+νtβt=nγ,
+- the elements yγ∈g−γ,yβi∈g−βiare part of a Chevalley basis,
+-v+is a highest weight vector for M,
+- the standard parabolic subalgebra p=pIis maximal for M,
+-γ∈Φ+\Φ+
+I,
+- the coefficients cνare inK.
+Our aim (cf. 8.13) is to show that there is at least one νwith|cν|≥1 andν1+...+νt≥n.
+In particular, this result implies that yγacts injectively on M(takecν= 0 for all ν). In fact, we
+prove first this statement about the injectivity of the action of yγ(in 8.6) and later use it to prove the
+more general result about the absolute value of (at least one) cν.
+Thatyγacts injectively on M(assumingγ /∈ΦI) is also contained in a paper by I. Dimitrov, O.
+Mathieu and I. Penkov, as was kindly pointed out to us by V. Mazorch uk, cf. [7, 3.6]. Our proof is
+completely different from the proof given in loc.cit.
+We begin with a standard commutator relation valid in any associative u nital algebra A. Forx∈A
+we let ad(x) :A→Abe defined by ad( x)(z) = [x,z] =xz−zx. In the following we also write [ x(i),z]
+for ad(x)i(z), the value on zof thei-th iteration of ad( x).
+Lemma 8.1. Letx,z1,...,z n∈A. For allk∈Z≥0one has
+xk·z1z2...zn=/summationdisplay
+i1+...+in+1=k/parenleftbiggk
+i1...in+1/parenrightbigg
+[x(i1),z1]·...·[x(in),zn]xin+1
+where, as usual,
+/parenleftbiggk
+i1...in+1/parenrightbigg
+=k!
+(i1!)·...·(in+1!)
+Similarly,48 SASCHA ORLIK AND MATTHIAS STRAUCH
+[x(k),z1z2...zn] =/summationdisplay
+i1+...+in=k/parenleftbiggk
+i1...in/parenrightbigg
+[x(i1),z1]·...·[x(in),zn].
+Proof.This is straightforwardly proved by induction on k. ✷
+Lemma 8.2. Letx∈g, letMbe aU(g)-module and v∈M.
+(i) Ifxacts locally finitely on v(i.e., theK-vector space generated by (xi.v)i≥0is finite-dimensional),
+thenxacts locally finitely on U(g).v.
+(ii) Ifx.v= 0and[x,[x,y]] = 0for somey∈g, then
+xnyn.v=n![x,y]n.v.
+Proof.(i) It suffices to showthat xacts locally finitely on any element ofthe form z1...zn.v(for arbitrary
+nand arbitrary elements z1,...,z n∈g). Any element of U(g) of the form [ x(i1),z1]·...·[x(in),zn] is
+contained in g·...·g(nfactors), and g·...·gis a finite-dimensional K-subspace of U(g). As the elements
+xi.v,i≥0, are all contained in a finite-dimensional vector space, it follows fr om the formula in Lemma
+8.1 that all elements xi·z1...zn.vare contained in a finite-dimensional K-vector space.
+(ii) In the formula of Lemma 8.1 we let zi=y, 1≤i≤n, and get that the term
+[x(i1),y]·...·[x(in),y]xin+1.v
+vanishes as soon as in+1>0 or someij>1 for 1≤j≤n. Therefore, the only non-zero term corresponds
+to (i1,...,in+1) = (1,...,1,0). ✷
+As before, we let Φ = Φ( g,t) be the root system and
+Φ+={β1,...,β t}and ∆ ={α1,...,α ℓ}⊂Φ+
+be a set of positive roots and a corresponding basis.
+Lemma 8.3. Assumeγ∈Φ+is not simple and write γ=α+βwithα∈∆andβ∈Φ+. Suppose that
+iβ−jαis inΦ+for somei,j∈Z>0.
+(i) Then (iβ−jα)−γ= (i−1)β−(j+1)αiseithera positive root ornot inΦ∪{0}. Therefore: either
+[giβ−jα,g−γ]is equal to a root space gχwithχ=i′β−j′α∈Φ+for somei′,j′∈Z>0or[giβ−jα,g−γ] = 0.
+(ii) Moreover, (iβ−jα)−α=iβ−(j+1)αiseithera positive root ornot inΦ∪{0}. Therefore: either
+[giβ−jα,g−α]is equal to a root space gχwithχ=i′β−j′α∈Φ+for somei′,j′∈Z>0or[giβ−jα,g−α] = 0.
+(iii) LetMbe aU(g)-module and v∈Mbe annihilated by the radical uofb. Letx∈gβandy∈g−γ.
+Then, for any sequence of non-negative integers i1,...,inwe have
+[x(i1),y]·...·[x(in),y].v= 0ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 49
+if there is at least one ij>1.
+Proof.Assertions (i) and (ii) follow from the fact that βis not a multiple of α(the root system Φ is
+reduced), and βmust then contain a simple root α′/\e}atio\slash=α, i.e.β−α′∈/summationtext
+τ∈∆Z≥0τ.
+(iii) We may clearly assume that i1>1 and that all ij∈{0,1}forj >1. We will show by induction
+onnthat [x(i1),y]·...·[x(in),y] is contained in
+/summationdisplay
+i>0,j>0
+iβ−jα∈Φ+U(g)giβ−jα
+To begin with, notice that [ x(i1),y] is ingi1β−γ=g(i1−1)β−α, and asi1−1>0 this space is either zero
+or a root space gχwithχ∈Φ+. The assertion is hence true for n= 1. Now let n>1,in∈{0,1}, and
+fix an element z∈giβ−jαwithi>0,j >0, andiβ−jα∈Φ+. Consider the product z[x(in),y].
+(a)Ifin= 0then[x(in),y] =y∈g−γandz[x(in),y] =zy= [z,y]+yz. Ifiβ−jα−γ= (i−1)β−(j+1)α
+is a positive root, we are done, because then [ z,y] is ingi′α−j′βwithi′α−j′β∈Φ+. Otherwise, by (i),
+iβ−jα−γis not in Φ∪{0}, and then [ z,y] = 0.
+(b) Ifin= 1 then [x(in),y] = [x,y]∈g−αandz[x(in),y] =z[x,y] = [z,[x,y]]+[x,y]z. Ifiβ−jα−α=
+iβ−(j+1)αis a positive root, we are done, because then [ z,[x,y]] is ingi′α−j′βwithi′α−j′β∈Φ+.
+Otherwise, by (ii), iβ−jα−αis not in Φ∪{0}, and then [ z,[x,y]] = 0. ✷
+8.4.In the remainder of this section we will use the following lexicographic o rdering on
+Zα1⊕...⊕Zαℓ:
+ℓ/summationdisplay
+i=1niαi>ℓ/summationdisplay
+i=1n′
+iαi⇐⇒ ∃k≥1 :ni=n′
+ifor 1≤i≤k−1 andnk>n′
+k.
+Proposition 8.5. Letp=pIfor someI⊂∆. SupposeM∈Opis a highest weight module with highest
+weightλand
+I={α∈∆|/a\}bracketle{tλ,α∨/a\}bracketri}ht∈Z≥0}.
+Then no non-zero element of u−
+pacts locally finitely on M.
+Proof.Letv+be a weight vector with weight λ. Lety∈u−
+pbe a non-zero element which acts locally
+finitely on M. Writey=/summationtext
+γ∈Φ+\Φ+
+Iyγwith elements yγ∈g−γ. PutB={γ∈Φ+\Φ+
+I|yγ/\e}atio\slash= 0}
+(this set is non-empty) and choose γ+∈Bto be maximal among the elements in Bfor the lexicographic
+ordering 8.4. Write
+yn.v+=/summationdisplay
+1≤i1,...,in≤τyγi1·...·yγin.v+.
+where Φ+\Φ+
+I={γ1,...,γ τ}. Using the total ordering 8.4 it is easily seen that among the elements
+yγi1·...yγin.v+onlyyn
+γ+.v+can have weight λ−nγ+. But as all yi.v+,i≥0, are contained in a50 SASCHA ORLIK AND MATTHIAS STRAUCH
+finite-dimensional subspace, we must have yn
+γ+.v+= 0 for some n∈Z>0. It follows from Lemma 8.2 that
+yγ+acts then locally finitely on M.
+We can therefore assume that y=yγ∈g−γ\{0}is contained in a root space, where γ∈Φ+\ΦI.
+Writeγ=/summationtext
+α∈∆cαα(with non-negative integers cα). We show by induction on the height of γ,ht(γ) =/summationtext
+α∈∆cα, thatyγcan not act locally finitely. By Lemma 8.2 this is equivalent to the statem ent that
+yn
+γ.v+/\e}atio\slash= 0 for all positive integers n. (Note that the vectors yn
+γ·v+, if non-zero, have pairwise distinct
+weightsλ−n·γ, hence are linearly independent.) yn
+γ.v+. Ifht(γ) = 1, then γis an element of ∆ \I.
+Rescalingyγwe can choose xγ∈gγsuch that [xγ,yγ] =hγand [hγ,xγ] = 2xγand [hγ,yγ] =−2yγ. A
+well-known formula (which is easy to prove by in duction) gives
+(8.5.2) xn
+γyn
+γ.v+=n!n−1/productdisplay
+i=0(λ(hγ)−i).v+=n!n−1/productdisplay
+i=0(/a\}bracketle{tλ,γ∨/a\}bracketri}ht−i).v+.
+AsI={α∈∆|/a\}bracketle{tλ,α∨/a\}bracketri}ht∈Z≥0}, it follows that/a\}bracketle{tλ,γ∨/a\}bracketri}ht/∈Z≥0and the term on the right of 8.5.2 does not
+vanish. In particular, yn
+γ.v+/\e}atio\slash= 0 for alln≥0.
+Now suppose ht(γ)>1. Then we can write γ=α+βwithα∈∆ andβ∈Φ+. Clearly, not both α
+andβcan be contained in Φ I. We distinguish two cases.
+(a) Suppose β−αis not in Φ. As β/\e}atio\slash=α(the root system Φ is reduced) we have [ gα,g−β] = [g−α,gβ] =
+{0}. Then, ifα /∈I, we letxβbe a non-zero element of gβand have by Lemma 8.2:
+xn
+βyn
+γ.v+=n![xβ,yγ]n.v+.
+But as [xβ,yγ] is a non-zero element of g−αwe can conclude by induction that [ xβ,yγ]n.v+/\e}atio\slash= 0 for all
+n≥0. And thus yn
+γ.v+/\e}atio\slash= 0 for alln≥0.
+If, on the other hand, α∈I, thenβ /∈ΦI. Letxαbe a non-zero element of gα. Then we have by
+Lemma 8.2:
+xn
+αyn
+γ.v+=n![xα,yγ]n.v+.
+And as [xα,yγ] is a non-zero element of g−βwe can again conclude by induction.
+(b) Suppose β−αis in Φ. Then it must be in Φ+, and we have γ−kα∈Φ+for 0≤k≤k0(with
+k0≤3, cf. [14, 0.2]), and γ−kα /∈Φ∪{0}fork>k0. This implies [ x(i)
+α,yγ] = 0 fori>k0. By Lemma
+8.1 we have
+xnk0
+αyn
+γ.v+=/summationdisplay
+i1+...+in+1=nk0/parenleftbiggnk0
+i1...inin+1/parenrightbigg
+[x(i1)
+α,yγ]·...·[x(in)
+α,yγ]xin+1
+α.v+
+and asxαannihilates v+this reduces to
+/summationdisplay
+i1+...+in=nk0/parenleftbiggnk0
+i1...in/parenrightbigg
+[x(i1)
+α,yγ]·...·[x(in)
+α,yγ].v+
+By what we have just observed, the corresponding term vanishes if there is one ij>k0. Therefore, only
+the term with all ij=k0contributes, and this sum is hence equal toON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 51
+/parenleftbiggnk0
+k0...k0/parenrightbigg
+[x(k0)
+α,yγ]n.v+=(nk0)!
+(k0!)n[x(k0)
+α,yγ]n.v+
+Ifγ−k0αis not in Φ Iwe are done, because [ x(k0)
+α,yγ] is a non-zero element of g−(γ−k0α). Otherwise we
+necessarily have α /∈I. In this case, if we choose some xβ∈gβ\{0}, we have by Lemma 8.1 and Lemma
+8.3 (iii)
+xn
+βyn
+γ.v+=n![xβ,yγ]n.v+,
+and [xβ,yγ] is a non-zero element of g−α. And, as we are now in the case of height one, we can thus
+conclude again. ✷
+Corollary 8.6. Letp=pIfor someI⊂∆. SupposeM∈Opis a simple module of highest weight λ
+and
+I={α∈∆|/a\}bracketle{tλ,α∨/a\}bracketri}ht∈Z≥0}.
+Then the action of any non-zero element of u−
+ponMis injective.
+Proof.By Lemma 8.2 (i), the set Nof elements v∈Mon which a fixed element x∈u−
+p\{0}acts locally
+finitely is a U(g)-submodule. Clearly Ncontains ker( Mx·−→M). Because Mis assumed to be simple,
+and asxis not acting locally finitely on Mby Prop. 8.5, this submodule must be the zero module. ✷
+Corollary 8.7. LetM∈Obe a highest weight module. Then the set of elements in gwhich act locally
+finitely onMis a standard parabolic Lie subalgebra of g. IfMhas highest weight λ, then this standard
+parabolic subalgebra is pIwhere
+I={α∈∆|/a\}bracketle{tλ,α∨/a\}bracketri}ht∈Z≥0}.
+pIis maximal for Min the sense of 5.2.
+Proof.Letv+be a weight vector of weight λand defineIas above. Then λis in
+Λ+
+I={µ∈t∗|∀α∈I:/a\}bracketle{tµ,α∨/a\}bracketri}ht∈Z≥0}
+(cf. [14, sec. 9.2] for the notation) and Mis inOpwithp=pI, cf. [14, Thm. in sec. 9.4]. Suppose
+z∈g\pacts locally finitely on M. Then there is n∈Z>0andc1,...,c n∈Ksuch that
+(8.7.3) zn.v++c1zn−1.v++...+cn−1z.v++cn.v+= 0.
+Writez=x+/summationtext
+γ∈Φ+\ΦIyγwith elements yγ∈g−γandx∈p. Asz /∈pthere is at least one γ∈Φ+\ΦI
+withyγ/\e}atio\slash= 0. PutB={γ∈Φ+\ΦI|yγ/\e}atio\slash= 0}(this set is non-empty) and choose β∈Bto be maximal
+among the elements in Bfor the lexicographic ordering 8.4. By expanding znas a sum of products of x’s
+andyγ’s it is then easily seen that only yn
+β.v+can have weight λ−nβ. From equation 8.7.3 we deduce
+thatyn
+β.v+= 0. This however contradicts Prop. 8.5. ✷
+On certain relations in highest weight modules. For the rest of this section we fix a Chevalley basis
+(xβ,yβ,hα|β∈Φ+,α∈∆) ofg′= [g,g], cf. [13, Thm. in sec. 25.2]. Here we have xβ∈gβand52 SASCHA ORLIK AND MATTHIAS STRAUCH
+yβ∈g−β. We then let g′
+Zbe theZ-span of these basis elements. g′
+Zis a Lie algebra over Z. For later use
+we quote from [13, sec. 25] the following facts.
+Proposition 8.8. Letβ,β′be linearly independent roots. Let zβbexβ, ifβis positive, and let zβbe
+y−β, ifβis negative. Define zβ′andzβ+β′in the same manner, if β+β′is a root. Let β′−rβ,β′−(r−
+1)β,...,β′+qβ(r≥0,q≥0) be theβ-string through β′. Then:
+(i)r−q=/a\}bracketle{tβ,(β′)∨/a\}bracketri}ht.
+(ii)[zβ,zβ′] =±(r+1)zβ+β′ifβ+β′is a root.
+Proof.This is contained in [13, Prop. in sec. 25.1, Thm. in sec. 25.2]. ✷
+8.9.LetMbe a simple highest weight module in the category Op
+alg, and assume that p=pIis maximal
+forM. Denote by v+a vector of highest weight λ. We will be studying relations
+yn
+γ.v+=/summationdisplay
+ν∈Incνyν1
+β1·...·yνt
+βt.v+,
+whereInconsists of all tuples ν= (ν1,...,ν t)∈Zt
+≥0satisfyingν1β1+...+νtβt=nγ, andcνare
+coefficients in K. As before, Φ+={β1,...,β t}. Our aim is to show that there is at least one ν∈In
+having both of the following properties:
+i. ν1+...+νt≥n,
+ii.|cν|K≥1.
+We start by showing the existence of some ν∈Inwith the second property.
+Lemma 8.10. LetM∈Op
+algandp=pIbe as above 8.9. Suppose the residue characteristic of Kdoes
+not divide any of the non-zero numbers among /a\}bracketle{tβ,α∨/a\}bracketri}ht,α,β∈Φ,α/\e}atio\slash=±β. Letγ∈Φ+\Φ+
+I. Denote by
+Inthe set of all ν∈Zr
+≥0such thatν1β1+...+νrβr=nγ. Then, for any n∈Z≥0and any expression
+(8.10.1) yn
+γ.v+=/summationdisplay
+ν∈Incνyν1
+1·...·yνt
+t.v+,
+(whereyi=yβi,i= 1,...,t) there is at least one index ν∈Insuch that|cν|K≥1.
+Proof.We start by recalling that for any i≥0 andβ∈Φ+the endomorphism of g:
+1
+i![x(i)
+β,·] =1
+i!ad(xβ)i
+preserves the Z-formg′
+Zofg′, cf. [13, Prop. in sec. 25.5]. Denote by U(g′
+Z) the enveloping algebra of
+g′
+Z, i.e., the quotient of the tensor algebra TZ(g′
+Z) by the two-sided ideal generated by all elements of the
+formxy−yx−[x,y] withx,y∈g′
+Z. (We point out that U(g′
+Z) is, of course, notthe Kostant Z-form of
+the enveloping algebra, as in [13, sec. 26.4].) It follows from 8.1 that1
+i!ad(xβ)ialso preserves U(g′
+Z).
+The proof proceeds by induction on ht(γ). Suppose ht(γ) = 1 and let βi=γ. Then the setInconsists
+of a single element νwhich is the t-tuple that has the entry nin theithplace and zeros elsewhere. TheON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 53
+right hand side of 8.10.1 is thus cνyn
+γ.v+. By Cor. 8.6 the element yγacts injectively on M, and we
+therefore get cν= 1.
+Now we assume that ht(γ)>1. Writeγ=α+βwith a simple root α∈∆ and a positive root β.
+Clearly, not both αandβcan be contained in Φ I. We distinguish two cases.
+(a) Suppose β−αis not in Φ. As β/\e}atio\slash=α(the root system Φ is reduced) we have [ gα,g−β] = [g−α,gβ] =
+{0}. Then, ifα /∈I, we consider xβ, the element of the Chevalley basis which generates gβ. We have by
+Lemma 8.2:
+xn
+βyn
+γ.v+=n![xβ,yγ]n.v+.
+Consider the β-string through−γ:−γ−rβ,...,−γ+qβ. Then−α−(r+1)β,...,−α+(q−1)βis the
+β-string through−α, and because we assume here that −α+β /∈Φ, we deduce that q= 1. By 8.8 we
+then conclude r+1 =/a\}bracketle{t−α,β∨/a\}bracketri}htand [xβ,yγ] =±/a\}bracketle{tα,β∨/a\}bracketri}htyα. Hence
+xn
+βyn
+γ.v+=n!(±/a\}bracketle{tα,β∨/a\}bracketri}ht)nyn
+α.v+.
+As/a\}bracketle{tα,β∨/a\}bracketri}ht=−(r+1)/\e}atio\slash= 0 and because of our assumption on the residue characteristic of K, the integer
+/a\}bracketle{tα,β∨/a\}bracketri}htis invertible in OK.
+On the other hand, equation 8.10.1 gives
+xn
+β.yn
+γ.v+=/summationdisplay
+ν∈Incνxn
+β(yν1
+1·...·yνt
+t).v+=/summationdisplay
+ν∈Incνad(xβ)n(yν1
+1·...·yνt
+t).v+.
+But as ad(xβ)n(yν1
+1·...·yνt
+t) is inn!U(g′
+Z), and because hτ.v+=/a\}bracketle{tλ,τ∨/a\}bracketri}htv+∈Zv+(for allτ∈∆), we
+find thatxn
+β.yn
+γ.v+is of the form
+n!/summationdisplay
+ν′∈I′
+nc′
+ν′yν′
+1
+1·...·yν′
+t
+t.v+,
+whereI′
+nconsists of all ν′∈Zt
+≥0such thatν′
+1β1+...+ν′
+tβt=nα=n(γ−β), and the numbers c′
+ν′are
+linear combinations of the cνwith integral coefficients. We therefore get
+yn
+α.v+=1
+(±/a\}bracketle{tα,β∨/a\}bracketri}ht)n/summationdisplay
+ν′∈I′nc′
+ν′yν′
+1
+1·...·yν′
+t
+t.v+.
+The induction hypothesis shows that at least one of the coefficients c′
+ν′must be of absolute value at least
+1, and this implies that at least one of the coefficients cνis of absolute value at least 1.
+Now suppose α∈I. Thenβ /∈ΦIand we consider xα, the member of the Chevalley basis which
+generates gα. By Lemma 8.2:
+xn
+αyn
+γ.v+=n![xα,yγ]n.v+.
+The same arguments as above give [ xα,yγ] =±/a\}bracketle{tβ,α∨/a\}bracketri}htyβ, and/a\}bracketle{tβ,α∨/a\}bracketri}htis a unit in OK. As before, we
+then multiply the right hand side of 8.10.1 with xn
+αand find that
+yn
+β.v+=1
+(±/a\}bracketle{tβ,α∨/a\}bracketri}ht)n/summationdisplay
+ν′∈I′nc′
+ν′yν′
+1
+1·...·yν′
+t
+t.v+,54 SASCHA ORLIK AND MATTHIAS STRAUCH
+where the numbers c′
+ν′are linear combinations of the cνwith integral coefficients. And we conclude again
+by induction.
+(b) Suppose β−αis in Φ. Then it must be in Φ+, and we have γ−kα∈Φ+for 0≤k≤k0(with
+k0≤3, cf. [14, 0.2]), and γ−kα /∈Φ∪{0}fork>k0. This implies [ x(i)
+α,yγ] = 0 fori>k0. By Lemma
+8.1 we have
+xnk0
+αyn
+γ.v+=/summationdisplay
+i1+...+in+1=nk0/parenleftbiggnk0
+i1...inin+1/parenrightbigg
+[x(i1)
+α,yγ]·...·[x(in)
+α,yγ]xin+1
+α.v+
+and asxannihilates v+this reduces to
+/summationdisplay
+i1+...+in=nk0/parenleftbiggnk0
+i1...in/parenrightbigg
+[x(i1)
+α,yγ]·...·[x(in)
+α,yγ].v+
+By what we have just observed, the corresponding term vanishes if there is one ij>k0. Therefore, only
+the term with all ij=k0contributes, and this sum is hence equal to
+/parenleftbiggnk0
+k0...k0/parenrightbigg
+[x(k0)
+α,yγ]n.v+=(nk0)!
+(k0!)n[x(k0)
+α,yγ]n.v+
+Sublemma 8.11. [x(k0)
+α,yγ] =k0!·c·yγ−k0αwith an integer cwhich is a unit in OK.
+Proof.Supposek0= 2. Letγ−2α,...,γ+qαbe theα-string through γ. As such a string consists of
+at most four roots (cf. [14, 0.2]) we have q≤1. Theα-string through−γthen begins with −γ−qα.
+Assume first that q= 0. By 8.8 we have [ xα,yγ] =±yγ−αand then [xα,yγ−α] =±2yγ−2α, so indeed
+[x(2)
+α,yγ] =±2!·yγ−2α. Ifq= 1 then there is a string of length four, and Φ must contain an irredu cible
+component of type G2. By 5.1 we then assume that 3 is invertible in OK. Moreover, we have [ xα,yγ] =
+±2yγ−αand then [xα,yγ−α] =±3yγ−2α, so that indeed [ x(2)
+α,yγ] =±2!·3·yγ−2αwith 3∈O∗
+K.
+Supposek0= 3. Thenγ−3α,γ−2α,γ−α,γistheα-stringthrough γand−γ,−γ+α,−γ+2α,−γ+3α
+is theα-string through−γ. Using 8.8 we compute [ x(3)
+α,yγ] =±3!yγ−3α. ✷
+We conclude that
+(8.11.2) xnk0
+αyn
+γ.v+= (nk0)!·cn·(yγ−k0α)n.v+withc∈O∗
+K.
+Ifγ−k0αis not in Φ Iwe are done, by the same reasoning as in part (a). Otherwise we nec essarily have
+α /∈I. In this case we have by Lemma 8.1 and Lemma 8.3 (iii)
+xn
+βyn
+γ.v+=n![xβ,yγ]n.v+.
+Let−γ−rβ,...,−γ+qβbe theβ-string through−γ. We have q≥2 and thus r≤1. Ifr= 0 then
+[xβ,yγ] =±yα. Ifr= 1 then [xβ,yγ] =±2yα. But in this case the β-string through−γconsists of four
+roots, and Φ must have an irreducible component of type G2. By 5.1 we have p>3, and thus 2∈O∗
+K.
+Therefore, we always have [ xβ,yγ] =cyαwithc∈O∗
+K. With the same arguments as in part (a) we can
+thus deduce the claim. ✷ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 55
+Lemma 8.12. Suppose that none of the irreducible components of Φis of typeG2. Letγ∈Φ+. Consider
+a relation
+(8.12.1) nγ=ν1β1+...+νtβt
+with non-negative integers n,ν1,...,ν t∈Z≥0. Thenn≤ν1+...+νt.
+Proof.We may assume that Φ is irreducible and that nis positive (there is nothing to show when n= 0).
+It is known that for irreducible reduced roots systems other than G2the square of the ratio between the
+lengths of any two roots β,γis among{1
+2,1,2}. Root systems of type A,D,E(so-called simply laced
+root systems) have the property that its roots are all of equal le ngth, whereas for root systems of type
+B,C, andF4the ratio can also be1
+2or 2. The relations
+/a\}bracketle{tβ,γ/a\}bracketri}ht= 2(β,γ)
+(γ,γ)= 2/bardblβ/bardbl
+/bardblγ/bardblcos(θ) and/a\}bracketle{tβ,γ/a\}bracketri}ht/a\}bracketle{tγ,β/a\}bracketri}ht= 4cos(θ)2
+show that if/bardblβ/bardbl2
+/bardblγ/bardbl2∈{1
+2,1,2}then/a\}bracketle{tβ,γ/a\}bracketri}ht≤2, i.e., (β,γ)≤(γ,γ), cf. [13, 9.4]. Taking the scalar product
+of both sides of 8.12.1 with γand dividing by ( γ,γ) we get
+n=ν1(β1,γ)
+(γ,γ)+...+νt(βt,γ)
+(γ,γ)≤ν1+...+νt.
+And this is what we asserted. ✷
+Now we generalize the preceding lemma so as to assure the existence of someν∈Insatisfying both
+conditionsi.andii.of 8.9.
+Proposition 8.13. LetM∈Op
+algandp=pIbe as in 8.9. Suppose the residue characteristic of Kdoes
+not divide any of the non-zero numbers among /a\}bracketle{tβ,α∨/a\}bracketri}ht,α,β∈Φ,α/\e}atio\slash=±β. Letγ∈Φ+\Φ+
+I. Denote by
+Inthe set of all ν∈Zt
+≥0such thatν1β1+...+νtβt=nγ. Then, for any n∈Z≥0and any expression
+(8.13.1) yn
+γ.v+=/summationdisplay
+ν∈Incνyν1
+1·...·yνt
+t.v+,
+(whereyi=yβi,i= 1,...,t) there is at least one index ν∈Insuch thatν1+...+νt≥nand|cν|K≥1.
+Proof.If Φ does not have an irreducible component of type G2then 8.10 and 8.12 prove the assertion of
+8.13. Therefore we assume in the following that Φ is irreducible of type G2.
+The basic idea of the proof is as follows. There is nothing to show if ht(γ) = 1. Now suppose that
+ht(γ)>1. Then there is γ′∈Φ+andk0∈Z>0such thatγ−k0γ′∈Φ+\Φ+
+Iand
+1
+(nk0)!xnk0
+γ′·yn
+γ.v+=1
+(k0!)n·[x(k0)
+γ′,yγ]n.v+=c·(yγ−k0γ′)n.v+,
+withc∈O∗
+K, cf. 8.11.2. Considering the right hand side of 8.13.1, we aim to show th at for anyν∈In
+withν1+...+νt<nthe term
+(8.13.2) xnk0
+γ′·yν1
+1·...·yνt
+t.v+56 SASCHA ORLIK AND MATTHIAS STRAUCH
+in
+(8.13.3)1
+(nk0)!xnk0
+γ′·/parenleftBigg/summationdisplay
+ν∈Incνyν1
+1·...·yνt
+t.v+/parenrightBigg
+=/summationdisplay
+ν∈Incν·1
+(nk0)!xnk0
+γ′·yν1
+1·...·yνt
+t.v+
+vanishes. This means that the sum on the right hand side of 8.13.3 is eq ual to
+(8.13.4)/summationdisplay
+ν∈Jncν·1
+(nk0)!xnk0
+γ′·yν1
+1·...·yνt
+t.v+,
+whereJn⊂Inconsists only of those ν∈Infor whichν1+...+νt≥n. We can then rewrite 8.13.4 as
+/summationdisplay
+ν′∈I′
+nc′
+ν′·yν′
+1
+1·...·yν′
+t
+t.v+,
+whereI′
+nconsists of all ν′∈Zt
+≥0such thatν′
+1β1+...+ν′
+tν′
+t=n(γ−k0γ′), and the numbers c′
+ν′are
+linear combinations with integral coefficients of the numbers cν, withν∈Jn. (Recall that the operator
+1
+(nk0)!ad(xγ′)nk0preserves g′
+Z.) Applying Lemma 8.10, we find that there is ν′∈I′
+nsuch that|c′
+ν′|K≥1.
+Hence there is at least one ν∈Jnsuch that|cν|K≥1. Thus proving our assertion.
+We need to show that the term 8.13.2 actually vanishes if ν1+...+νt<n.
+Recall that Φ is of type G2here. Let ∆ ={α,β}withαbeing the short and βbeing the long root.
+We put
+β1=α, β 2=β, β3=α+β, β4= 2α+β, β5= 3α+β, β6= 3α+2β .
+Consider the equation nγ=ν1β1+...+ν6β6, i.e.,
+(8.13.5) nγ=ν1α+ν2β+ν3(α+β)+ν4(2α+β)+ν5(3α+β)+ν6(3α+2β)
+We consider all possible cases for γandI.
+Case when γ=αorγ=β.There is nothing to show in this case, as we can write a multiple of a
+simple root in one and only one way as a sum of positive roots.
+Case when γ=β5orγ=β6.Comparing the coefficients of α, the equation 8.13.5 implies in this case
+3n=ν1+ν3+2ν4+3ν5+3ν6.
+On the other hand, assuming ν1+...+ν6<nwe get that 3 ν1+...+3ν6<ν1+ν3+2ν4+3ν5+3ν6
+which implies 2 ν1+3ν2+2ν3+ν4<0 which is impossible. It remains to discuss the cases when γ=α+β
+orγ= 2α+β.
+Case when γ=α+β.In this case equation 8.13.5 implies
+n=ν1+ν3+2ν4+3ν5+3ν6andn=ν2+ν3+ν4+ν5+2ν6ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 57
+Subtracting the equation on the right from the equation on the left and adding ν2on both sides gives
+ν2=ν1+ν4+2ν5+ν6.
+(a) Suppose I=∅orI={α}. Then we let γ′=α. Consider
+xn
+α·yν1
+1·...·yν6
+6.v+= [x(n)
+α,yν1
+1·...·yν6
+6].v+
+=/summationtext
+i1+...+i6=n/parenleftbign
+i1...i6/parenrightbig
+[x(i1)
+α,yν1
+1]·...·[x(i6)
+α,yν6
+6].v+
+Moreover,
+(8.13.6) [ x(ij)
+α,yνj
+j] =/summationdisplay
+k1+...+kνj=ij/parenleftbiggij
+k1...kνj/parenrightbigg
+[x(k1)
+α,yj]·...·[x(kνj)
+α,yj].
+Therefore, if βj−αis not in Φ∪{0}, the term 8.13.6 vanishes if ij>0. Hencei2=i6= 0. Moreover,
+ifij> νj, then there must be at least one kt≥2 in the formula 8.13.6. However, if βj−2αis not in
+Φ∪{0}, the term 8.13.6 vanishes then. And because [ x(2)
+α,yα+β]∈g−β+α= 0, we get i3≤ν3. Similarly
+we see that i1≤2ν1. Now suppose n>ν1+...+ν6and consider the inequality
+n=i1+...+i6>ν1+...+ν6= 2ν1+ν3+2ν4+3ν5+2ν6.
+Here we have used that ν2=ν1+ν4+2ν5+ν6. Asi2=i6= 0,i3≤ν3andi1≤2ν1we get
+i4+i5>(2ν1−i1)+(ν3−i3)+2ν4+3ν5+2ν6≥2ν4+3ν5.
+This shows that either i4>2ν4ori5>3ν5. But in each of these cases the corresponding term [ x(i4)
+α,yν4
+4]
+or [x(i5)
+α,yν5
+5] vanishes. This proves our assertion in this case.
+(b) Suppose I={β}. Then we let γ′=β. Consider
+xn
+β·yν1
+1·...·yν6
+6.v+= [x(n)
+β,yν1
+1·...·yν6
+6].v+
+=/summationtext
+i1+...+i6=n/parenleftbign
+i1...i6/parenrightbig
+[x(i1)
+β,yν1
+1]·...·[x(i6)
+β,yν6
+6].v+
+Moreover,
+(8.13.7) [ x(ij)
+β,yνj
+j] =/summationdisplay
+k1+...+kνj=ij/parenleftbiggij
+k1...kνj/parenrightbigg
+[x(k1)
+β,yj]·...·[x(kνj)
+β,yj].
+Therefore, if βj−βis not in Φ∪{0}, the term 8.13.7 vanishes if ij>0. Hencei1=i4=i5= 0. Moreover,
+ifij> νj, then there must be at least one kt≥2 in the formula 8.13.7. However, if βj−2βis not in
+Φ∪{0}, the term 8.13.7 vanishes then. So we get i3≤ν3andi6≤ν6. Similarly we see that i2≤2ν2.
+Assumingn>ν1+...+ν6we get
+n=i1+...+i6>ν1+...+ν6,
+and hence58 SASCHA ORLIK AND MATTHIAS STRAUCH
+i2>ν1+ν2+(ν3−i3)+ν4+ν5+(ν6−i6)≥ν2.
+The left hand side of 8.13.7, for j= 2, is of weight ( i2−ν2)β, and [x(i2)
+β,yν2
+2].v+is therefore 0 or of
+weightλ+(i2−ν2)β, which shows that it must vanish. Assuming, as we may, that we had o rdered the
+positive roots such that βcomes last (i.e., β=β6instead ofβ=β2), we see that xn
+β·yν1
+1·...·yν6
+6.v+
+vanishes.
+Case when γ= 2α+β.In this case equation 8.13.5 implies
+(8.13.8) 2 n=ν1+ν3+2ν4+3ν5+3ν6
+(8.13.9) n=ν2+ν3+ν4+ν5+2ν6
+We assume that n>ν1+ν2+ν3+ν4+ν5+ν6. By 8.13.9, this implies
+ν2+ν3+ν4+ν5+2ν6>ν1+ν2+ν3+ν4+ν5+ν6,
+and therefore ν6>ν1. In particular,
+(8.13.10) ν6>0.
+Subtracting 8.13.9 from 8.13.8 we get n=ν1−ν2+ν4+2ν5+ν6, and using again our assumption that
+n>ν1+ν2+ν3+ν4+ν5+ν6we find
+ν1−ν2+ν4+2ν5+ν6>ν1+ν2+ν3+ν4+ν5+ν6,
+or, equivalently, ν5>2ν2+ν3. In particular
+(8.13.11) ν5>0.
+We distinguish two cases.
+(a) Suppose I=∅orI={α}. Then we let γ′=α. We arrange the positive roots in this order:
+β6,β1,β2,β3,β4,β5, and write elements of U(u−
+b) as sums of monomials of the form yν6
+6·yν1
+1·...·yν5
+5.
+Consider
+(8.13.12)x2n
+α·yν6
+6·yν1
+1·...·yν5
+5.v+
+= [x(2n)
+α,yν6
+6·yν1
+1·...·yν5
+5].v+
+=/summationtext
+i1+...+i6=2n/parenleftbig2n
+i1...i6/parenrightbig
+[x(i6)
+α,yν6
+6]·[x(i1)
+α,yν1
+1]·...·[x(i5)
+α,yν5
+5].v+
+Moreover,ON JORDAN-H ¨OLDER SERIES OF SOME LOCALLY ANALYTIC REPRESENTATIONS 59
+(8.13.13) [ x(ij)
+α,yνj
+j] =/summationdisplay
+k1+...+kνj=ij/parenleftbiggij
+k1...kνj/parenrightbigg
+[x(k1)
+α,yj]·...·[x(kνj)
+α,yj].
+Since [xα,y6] = 0, it follows from 8.13.13that we only need to consider tuples ( i1,...,i6) for which i6= 0.
+For those tuples we have 2 n=i1+i2+i3+i4+i5. On the other hand, 8.13.8 implies
+(8.13.14) ν1+ν3+2ν4+3ν5= 2n−3ν6<2n,
+becauseν6>0 (cf. 8.13.10). Now consider the term
+[x(i1)
+α,yν1
+1]·...·[x(i5)
+α,yν5
+5].v+
+in the third line of 8.13.12. It has weight
+(i1+i2+i3+i4+i5)α−ν1β1−ν2β2−ν3β3−ν4β4−ν5β5+λ
+= (2n−ν1+ν3+2ν4+3ν5)α+(2n−ν2+ν3+ν4+ν5)β+λ
+By 8.13.14, this weight is not of the form λ−(sum of positive roots), and must therefore vanish. Hence
+x2n
+α·yν6
+6·yν1
+1·...·yν5
+5.v+= 0
+for allνwithn>ν1+ν2+ν3+ν4+ν5+ν6.
+(b) Suppose I={β}. Then we let γ′=α+β. In this case we order the positive roots as follows:
+β5,β1,β2,β3,β4,β6, and write elements of U(u−
+b) as sums of monomials of the form yν5
+5·yν1
+1·...·yν4
+4·yν6
+6.
+Consider
+(8.13.15)xn
+α+β·yν5
+5·yν1
+1·...·yν4
+4·yν6
+6.v+= [x(n)
+α+β,yν5
+5·yν1
+1·...·yν4
+4·yν6
+6].v+
+=/summationtext
+i1+...+i6=n/parenleftbign
+i1...i6/parenrightbig
+[x(i5)
+α+β,yν5
+5]·[x(i1)
+α+β,yν1
+1]·...·[x(i4)
+α+β,yν4
+4]·[x(i6)
+α+β,yν6
+6].v+
+Moreover,
+(8.13.16) [ x(ij)
+α+β,yνj
+j] =/summationdisplay
+k1+...+kνj=ij/parenleftbiggij
+k1...kνj/parenrightbigg
+[x(k1)
+α+β,yj]·...·[x(kνj)
+α+β,yj].
+Since [xα+β,y5] = 0, it follows from 8.13.16 that we only need to consider tuples ( i1,...,i6) for which
+i5= 0. For those tuples we have n=i1+i2+i3+i4+i6. On the other hand, 8.13.9 implies
+(8.13.17) ν2+ν3+ν4+2ν6=n−ν5<n,
+becauseν5>0 (cf. 8.13.11). Now consider the term60 SASCHA ORLIK AND MATTHIAS STRAUCH
+[x(i1)
+α,yν1
+1]·...·[x(i4)
+α,yν4
+4]·[x(i6)
+α,yν6
+6].v+
+in the last line of 8.13.15. It has weight
+(i1+i2+i3+i4+i6)(α+β)−ν1β1−ν2β2−ν3β3−ν4β4−ν6β6+λ
+= (n−ν1+ν3+2ν4+3ν5)α+(n−ν2+ν3+ν4+2ν6)β+λ
+By 8.13.17, this weight is not of the form λ−(sum of positive roots), and must therefore vanish. Hence
+xn
+α+β·yν5
+5·yν1
+1·...·yν4
+4·yν6
+6.v+= 0
+for allνwithn>ν1+ν2+ν3+ν4+ν5+ν6. This completes the proof. ✷
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+Fachbereich C - Mathematik und Naturwissenschaften, Bergis che Universit ¨at Wuppertal, Gaußstraße 20,
+D-42119 Wuppertal, Germany
+E-mail address :orlik@math.uni-wuppertal.de
+Indiana University, Department of Mathematics, Rawles Hall, Bloomington, IN 47405, USA
+E-mail address :mstrauch@indiana.edu
\ No newline at end of file
diff --git a/1001.0324.txt b/1001.0324.txt
new file mode 100644
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@@ -0,0 +1,1053 @@
+arXiv:1001.0324v2  [math.AG]  14 May 2010Some Siegel threefolds with a Calabi-Yau model II
+Eberhard Freitag Riccardo Salvati Manni
+Mathematisches Institut Dipartimento di Matematica,
+Im Neuenheimer Feld 288 Piazzale Aldo Moro, 2
+D69120 Heidelberg I-00185 Roma, Italy.
+freitag@mathi.uni-heidelberg.de salvati@mat.uniroma1 .it
+2010
+Introduction
+In the paper [FSM] we described some Siegel modular threefol ds which admit a
+Calabi-Yau model. Using a different method we give in this pap er an enlarged
+list of such varieties. Basic for our method is the paper [GN] of van Geemen
+and Nygaard. In this paper they prove that the complete inter sectionXin
+P7(C), defined by the equations
+X:Y2
+0=X2
+0+X2
+1+X2
+2+X2
+3,
+Y2
+1=X2
+0−X2
+1+X2
+2−X2
+3,
+Y2
+2=X2
+0+X2
+1−X2
+2−X2
+3,
+Y2
+3=X2
+0−X2
+1−X2
+2+X2
+3
+is biholomorphic equivalent to the Satake compactification ofH2/Γ′for a cer-
+tainsubgroup Γ′⊂Sp(2,Z). Thisvarietyhas96singularitieswhichcorrespond
+to certain zero-dimensional cusps and these singularities are ordinary double
+points (nodes). In the paper [CM] it has been pointed out that the results
+of [GN] imply that a (projective) small resolution of this va riety is a rigid
+Calabi-Yau manifold ˜X.
+We describe the basic occurring groups: We use the standard n otations,
+M=/parenleftbigAB
+C D/parenrightbig
+:
+Γn[l] = kernel(Sp( n,Z)−→Sp(n,Z/lZ)),
+Γn[l,2l] =/braceleftbig
+M∈Γn[l];AtB/landCtD/lhave even diagonal/bracerightbig
+,
+Γn,0[l] ={M∈Γn;C≡0 modl},
+Γn,0,ϑ[l] ={M∈Γn;C≡0 modl, CtD/lhas even diagonal }.
+The group Γ n,0[l] can be extended by the Fricke involution
+J=Jl=/parenleftbigg
+0E/√
+l
+−√
+lE0/parenrightbigg/parenleftig
+JZ=−1
+lZ/parenrightig
+.2 Some Siegel threefolds with a Calabi-Yau model II
+We denote this extension (of index 2) by
+ˆΓn,0[l] = Γn,0[l]∪JΓn,0[l].
+The group Γ′, which belongs to van Geemen’s and Nygaard’s variety is a sub -
+group of index two of the group
+Γ2[2,4]∩Γ2,0,ϑ[4] ={M∈Γ2[2,4];C≡0 mod 4 , C/4 has even diagonal },
+namely
+Γ′={M∈Γ2[2,4]∩Γ2,0,ϑ[4]; det D≡ ±1 mod 8 }.
+In [FSM] we introduced a certain subgroup Γ 2,0[2]nof index two of Γ 2,0[2] as
+kernel of a certain character χn. This character is the product of the unique
+non-trivial character of the full Siegel modular group and t he character
+iα+β+γ, CtD=/parenleftbigg
+α β
+β γ/parenrightbigg
+,/parenleftig
+M=/parenleftbigg
+A B
+C D/parenrightbigg
+∈Γ2,0[2]/parenrightig
+.
+The group Γ′is contained in Γ 2,0[2]nas subgroup of index 6144.
+The character χnextends to a character ˆ χnofˆΓ2,0[2] by means of
+ˆχn(J) = 1.
+We denote its kernel by ˆΓ2,0[2]n. This is an extension of index two of Γ 2,0[2]n.
+The group Γ′is contained in ˆΓ2,0[2]nas subgroup of index 12288. The main
+result of this paper is:
+Theorem. The Siegel modular threefold, which belongs to a group betwe enΓ′
+andˆΓ2,0[2]n, admits a Calabi-Yau model in the following weak sense: Ther e
+exists a desingularization in the category of complex space s of the Satake com-
+pactification which admits a holomorphic three-form withou t zeros and whose
+first Betti number vanishes.
+It has been pointed out by van Geemen that it is not always poss ible to get
+a projective model. We want to come back to the question of pro jectivity in
+an extra paper. Here we just mention a result from [CFS] that c an be proved
+with the methods of [FSM]:
+Supplement. In the case of a group between Γ2,0[2]nandΓ2,0[4]∩Γ2[2]one
+can get a projective Calabi-Yau manifold.
+We will develop a method to compute the Picard number pic (ran k of the
+group of divisor classes) and the Euler number for the interm ediate groups. In
+the projective case they determine the Hodge numbers, the Pi card number is
+pic =h11and the Euler number is
+e= 2(h11−h12).§1. The variety of van Geemen and Nygaard 3
+For a small resolution ˜Xof the variety of van Geemen and Nygaard Xthey
+are known [GN], [CM], pic = 32, e= 64. To get it for other groups, one needs
+the action of the group ˆΓ2,0[2]non the Picard group of the regular locus of
+X. We will determine this action in section 6. The result of thi s section allow
+in principle to compute the numbers for all intermediate gro ups. There are
+thousands of conjugacy classes of intermediate groups.
+In section 7 we treat some simple examples, namely all subgro ups of order
+two ofˆΓ2,0[2]n/Γ′.
+In a subsequent paper we will treat some more significative ca ses.
+We are grateful for helpful disussions with Bert van Geemen a nd also for
+useful comments to a preliminary version of our paper from Ph ilip Candelas.
+He brought our attention to the paper [BH] of Borisov and Hua, where other
+examples of complete intersections of 4 quadrics in P7(C) that lead to Calabi-
+Yau manifolds with big fundamental groups are described.
+1. The variety of van Geemen and Nygaard
+We recall some basic facts about Siegel modular varieties. F or details we refer
+to [Fr1]. The symplectic group Sp( n,R) acts on the Siegel upper half plane
+Hn:={Z∈C(n,n);Z=tZ=X+iY, Y >0 (positive definite) }
+by means of the formula MZ= (AZ+B)(CZ+D)−1. For any subgroup
+Γ⊂Sp(n,R) that is commensurable with Sp( n,Z), the quotient Hn/Γ carries
+a natural structure as quasi projective algebraic variety. The Satake compacti-
+ficationHn/Γ is a projective variety which is closely related to Siegel m odular
+forms. The Satake compactification can be identified with the projective va-
+riety associated to a graded algebra of modular forms. We rec all briefly its
+definition. A modular form fof weight r/2,r∈Z, is a holomorphic function
+fonHnwith the transformation property
+f(MZ) =v(M)/radicalbig
+det(CZ+D)rf(Z)
+for allM∈Γ. In the case n= 1 a regularity condition at the cusps has to
+be added. Here v(M) is system of complex numbers of absolute valued one,
+called the multiplier system. It has to fulfil an obvious cocy cle condition. We
+denote this space by [Γ ,r/2,v]. Fixing some starting weight r0and a multiplier
+systemvfor it, we define the ring
+A(Γ) :=/circleplusdisplay
+r∈Z[Γ,rr0/2,vr].4 Some Siegel threefolds with a Calabi-Yau model II
+This turns out to be a finitely generated graded algebra and it s associated
+projectivevarietyproj A(Γ)canbeidentifiedwiththeSatakecompactifictation.
+The ring depends on the starting weight and the multiplier sy stem but the
+assiciated projective variety does not.
+Basic examples of modular forms are given be theta series wit h respect to
+characteristics. By definition, a theta characteristic is a n element m=/parenleftbiga
+b/parenrightbig
+from (Z/2Z)2n. Herea,b∈(Z/2Z)nare column vectors. The characteristic is
+called even iftab= 0 and odd otherwise. The group Sp( n,Z/2Z) acts on the
+set of characteristics by
+M{m}:=tM−1m+/parenleftbigg
+(AtB)0
+(CtD)0/parenrightbigg
+.
+HereS0denotes the column built of the diagonal of a square matrix S. It is
+well-known that Sp( n,Z/2Z) acts transitively on the subsets of even and odd
+characteristics. Recall that to any characteristic the the ta function
+ϑ[m] =/summationdisplay
+g∈Zneπi(Z[g+a/2]+tb(g+a/2))(Z[g] =tgZg)
+can be defined. Here we use the identification of Z/2Zwith the subset {0,1} ⊂
+Z. It vanishes if and only if mis odd. Recall also that the formula
+ϑ[M{m}](MZ) =v(M,m)/radicalbig
+det(CZ+D)ϑ[m](Z)
+holds for M∈Γn, wherev(M,m) is a rather delicate 8throot of unity which
+depends on the choice of the square root. Sometimes we will us e the notation
+ϑ[m] =ϑ/bracketleftiga1a2
+b1b2/bracketrightig
+form=
+a1
+a2
+b1
+b2
+.
+Following van Geemen and Nygaard, we consider the 8 function s
+ϑ/bracketleftig00
+00/bracketrightig
+(Z), ϑ/bracketleftig00
+10/bracketrightig
+(Z), ϑ/bracketleftig00
+01/bracketrightig
+(Z), ϑ/bracketleftig00
+11/bracketrightig
+(Z),
+ϑ/bracketleftig00
+00/bracketrightig
+(2Z), ϑ/bracketleftig10
+00/bracketrightig
+(2Z), ϑ/bracketleftig01
+00/bracketrightig
+(2Z), ϑ/bracketleftig11
+00/bracketrightig
+(2Z).
+If we denote them by Y0,...,Y 3,X0...,X3, then classical theta relations show
+that the relations listed at the beginning of the introducti on hold. The classical
+theta transformation formalism shows that the eight forms a re modular forms
+of weight 1 /2 for the group Γ′introduced in the introduction and that their
+multipliers on this group agree. Since this is standard, we o nly give a short
+sketch of the proof. The theta series Xiare the so-called theta series of second§1. The variety of van Geemen and Nygaard 5
+kind. One knows classically that they have the same multipli ersκ(M) on the
+group Γ 2[2,4] (s. for example [Ru]). By conjugation with the transforma tion
+Z/ma√s⊔o→2Zone shows that the Yihave the same multipliers ˜ κ(M) on the group
+Γ2,0,ϑ[2]. We have to find the subgroup Γ′of Γ2[2,4]∩Γ2,0,ϑ[2], where κand
+˜κagree. This just means that ϑ[0](Z)ϑ[0](2Z) andϑ[0](2Z)2have the same
+multipliers. But these are the standard thetas series with r espect to the binary
+forms/parenleftbig10
+02/parenrightbig
+and/parenleftbig20
+02/parenrightbig
+. The advantage of binary forms is that they have an even
+number of variables and standard formula can be used, for exa mple [Fr2], 7.1.
+⊔ ⊓
+We can express this in saying that the Y0,...,X 3are contained in the ring
+A(Γ′) :=/circleplusdisplay
+r∈Z[Γ′,r/2,κr].
+The precise result, slightly extending results of [GN], sta tes:
+RingvGN 1.1 Proposition. Let be
+Γ′={M∈Γ2[2,4];C≡0 mod 4,diagC≡0 mod 8,detD≡ ±1 mod 8}.
+The ring A(Γ′)is generated by the eight theta series above. The relations
+Y2
+0=X2
+0+X2
+1+X2
+2+X2
+3,
+Y2
+1=X2
+0−X2
+1+X2
+2−X2
+3,
+Y2
+2=X2
+0+X2
+1−X2
+2−X2
+3,
+Y2
+3=X2
+0−X2
+1−X2
+2+X2
+3
+are defining relations. They define a subvariety XofP7(C)which can be
+identified with the Satake compactification of H2/Γ′.
+Proof.Proposition2.5of[GN] saysthatthiscomplete intersectio n isthe Satake
+compatification for some subgroup of Sp(2 ,Z). Necessarily this must be a
+subgroup of what we called Γ′. An index computation gives that they agree.
+TheequalityofthecompleteintersectionandtheSatakecom pactificationshows
+thatA(Γ′) must be the normalization of the factor ring C[X0,...,Y 3] by the
+ideal generated by the above 4 relations. Using Serre’s crit erion for normality,
+it follows that this ring is normal. This proves 1.1. ⊔ ⊓
+In [GN] the modular form of weight 3
+T=ϑ/bracketleftig10
+00/bracketrightig
+(Z)ϑ/bracketleftig10
+01/bracketrightig
+(Z)ϑ/bracketleftig01
+00/bracketrightig
+(Z)ϑ/bracketleftig01
+10/bracketrightig
+(Z)ϑ/bracketleftig11
+00/bracketrightig
+(Z)ϑ/bracketleftig11
+11/bracketrightig
+(Z)
+has been introduced. The differential form
+ω=T dz0∧dz1∧dz2, Z=/parenleftbigg
+z0z1
+z1z2/parenrightbigg
+,
+is invariant under ˆΓ2,0[2]n(The invariance under Γ 2,0[2]nhas been proved in
+[FSM]. The behavior under the Fricke involution can be deriv ed from the fol-
+lowing explicit formula.)6 Some Siegel threefolds with a Calabi-Yau model II
+omegaXY 1.2 Lemma. In terms of the coordinates X0,...X3,Y0...,Y3we have that,
+up to a multiplicative constant, ωequals
+X4
+2
+Y0Y1Y2Y3d(X0/X2)∧d(X1/X2)∧d(X3/X2).
+Proof.This is essentially the form of ω, which has been derived in [FSM] (see
+Theorem 4.5 and the formulae before it). ⊔ ⊓
+The zero locus of Tconsists of the fixed point sets of all M∈Γ2,0[2]n,
+which are conjugate inside Sp(2 ,Z) to the diagonal matrix with diagonal
+(1,−1,1,−1). It can be checked that all these Mare contained in Γ′. Hence
+we obtain the following well-known result:
+TCalDif 1.3 Remark. The differential form ωdefines a holomorphic differential form
+without zeros on the smooth variety H2/Γ′. It is invariant under ˆΓ2,0[2]n.
+2. Automorphisms of the variety of van Geemen and
+Nygaard
+As we will see (2.1), the group Γ′is normal in ˆΓ2,0[2]. Hence this group acts
+on the variety of van Geemen and Nygaard. We want to describe t his action.
+We will see that this action can be described by a linear actio n on the variables
+Y0,...,X 3, more precisely by a finite subgroup of PGL(8 ,C). Recall that
+Γ2,0[2] is generated by the matrices of the form
+/parenleftbigg
+E S
+0E/parenrightbigg
+,/parenleftbigg
+U′0
+0U−1/parenrightbigg
+,/parenleftbigg
+E0
+2S E/parenrightbigg
+(S=S′integral).
+LetM∈ˆΓ2,0[2]. Forf∈[Γ′,1/2,vϑ] we set
+f|M(Z) =/radicalbig
+det(CZ+D)−1/2f(MZ).
+The map f/ma√s⊔o→f|Misanautomorphism ϕMofthe8-dimensional space spanned
+byY0,...,X 3. It depends on the choice of a square root of det( CZ+D). Hence
+±ϕMis well defined. Using standard theta transformation formul ae we can
+compute these automorphisms for the generators. It is suffici ent to take the
+following 4.§3. The stabilizer of a node 7
+matrix corresponding transformation
+/parenleftbiggtU0
+0U−1/parenrightbigg
+, U=/parenleftbigg
+1 1
+0 1/parenrightbigg
+(Y0,Y1,Y3,Y2,X0,X3,X2,X1)
+/parenleftbiggtU0
+0U−1/parenrightbigg
+, U=/parenleftbigg
+1 0
+1 1/parenrightbigg
+(Y0,Y3,Y2,Y1,X0,X1,X3,X2)
+/parenleftbigg
+E0
+S E/parenrightbigg
+, S=/parenleftbigg
+2 0
+0 0/parenrightbigg
+(Y0,−iY1,Y2,−iY3,X1,X0,X3,X2)
+J(Fricke involution)√
+2·(X0,X1,X2,X3,Y0/2,Y1/2,Y2/2,Y3/2)
+calGgen 2.1 Lemma. The group Γ′is normal in Γ2,0[2]. The group Ggenerated
+by the transformations ±ϕM,M∈ˆΓ2,0[2], is already generated by the above
+four transformations. It has order 98304. The map M/ma√s⊔o→ ±ϕMdefines a
+homomorphism
+ˆΓ2,0[2]−→ G/±.
+The group Gcontains the subgroup Zof order 4which is generated by multi-
+plication with i. The above homomorphism induces an isomorphism
+ˆΓ2,0[2]/Γ′∼−→ G/Z(order24576).
+Proof.Since the generators of Γ 2,0[2] act on X, the group Γ′must be a normal
+subgroup. The rest follows by comparing indices. ⊔ ⊓
+3. The stabilizer of a node
+Thevariety Xhas96singularities, whichallareordinarydoublepoints( nodes).
+They are zero dimensional boundary points, but not each zero dimensional
+boundary is singular. In coordinates the singularities can be described as fol-
+lows:
+One node is given by
+P= [√
+2,0,√
+2,0,1,1,0,0].
+Changing signs it produces 8 nodes, which are characterized by the property
+Y1=Y3=X2=X3= 0. Similarly, one gets 8 nodes with Y1=Y3=X0=
+X1= 0. So one has 16 nodes with Y1=Y3= 0. In the same way one gets
+16 nodes with the property Yi=Yj= 0 for each other pair 0 ≤i < j≤3.
+This gives 96 = 8 ·16 nodes. It is easy to check by hand that they exhaust all
+singular points. This description of the nodes also shows:8 Some Siegel threefolds with a Calabi-Yau model II
+ActNode 3.1 Lemma. The group ˆΓ2,0[2]acts on the 96nodes is transitive.
+The following matrices
+M1=
+1 0 0 0
+0 1 0 0
+0 2 1 0
+2 0 0 1
+, M 2=
+1 0 0 0
+0 1 0 0
+2 0 1 0
+0 0 0 1
+,
+M3=
+1 0 0 0
+0 1 0 1
+0 0 1 0
+0 0 0 1
+, M 4=
+1 0 0 0
+1 1 0 0
+0 0 1 −1
+0 0 0 1
+,
+M5=
+1 0 0 1
+0 1 1 0
+0 0 1 0
+0 0 0 1
+, M 6=
+0 1 0 0
+1 0 0 0
+0 0 0 1
+0 0 1 0
+◦J
+are contained in ˆΓ2,0[2] and stabilize the node P. We consider the group, which
+is generated by them and Γ′. One can check that the factor group mod Γ′has
+order 128. Together with 3.1 we obtain:
+StabGen 3.2 Proposition. The stabilizer ˆΓ2,0[2]Pof the node PinsideˆΓ2,0[2]is
+generated by Γ′and the matrices M1,M2,...,M 6.
+In a neighborhood of Pwe can use the affine coordinates
+η0=Y0/X1, η1=Y1/X1, η2=Y2/X1, η3=Y3/X1,
+ξ0=X0/X1, ξ2=X2/X1, ξ3=X3X1
+Then substituting the affine version of the third equation in t he fourth we get
+η2
+1−η2
+3= 2(ξ2
+2−ξ2
+3)
+Setting
+x1=η1−η3, x4=η1+η3, x2=√
+2(ξ2−ξ3), x3=√
+2(ξ2+ξ3),
+the relation gets the simple form
+x1x4=x2x3.
+So we have lead to consider the quadric
+Q:={(x1,x2,x3,x4)∈C4;x1x4=x2x3}.
+Thisisa three dimensional affinevariety witha unique singul arityat the origin.
+The above construction gives an ´ etale map of germs
+(X,P)−→(Q,0).§3. The stabilizer of a node 9
+Sometimes we write the elements of C4as matrices
+X=/parenleftbigg
+x1x2
+x3x4/parenrightbigg
+.
+ThenQis defined by det X= 0. The group GL(2 ,C)×GL(2,C) acts on the
+quadric by means of
+X/ma√s⊔o−→AXtB.
+Inthiswaywecanconsider (GL(2 ,C)×GL(2,C))/C∗assubgroup ofGL(4 ,C).
+Another transformation, which leaves the quadric invarian t, isX/ma√s⊔o→tX. We
+also can consider it as element of GL(4 ,C).
+We consider the transformations
+m1/parenleftbigg
+x1x2
+x3x4/parenrightbigg
+=/parenleftbigg
+x4x2
+x3x1/parenrightbigg
+, m 2/parenleftbigg
+x1x2
+x3x4/parenrightbigg
+=/parenleftbigg
+−ix1−x2
+x3−ix4/parenrightbigg
+,
+m3/parenleftbigg
+x1x2
+x3x4/parenrightbigg
+=/parenleftbigg
+−x1ix2
+ix3x4/parenrightbigg
+, m4/parenleftbigg
+x1x2
+x3x4/parenrightbigg
+=/parenleftbigg
+−x1−x2
+x3x4/parenrightbigg
+,
+m5/parenleftbigg
+x1x2
+x3x4/parenrightbigg
+=/parenleftbigg
+x1x3
+x2x4/parenrightbigg
+, m 6/parenleftbigg
+x1x2
+x3x4/parenrightbigg
+=/parenleftbigg
+x2x1
+x4x3/parenrightbigg
+.
+They are contained in the extension of index two of the image o f the group
+(GL(2,C)×GL(2,C)), which is generated by X/ma√s⊔o→tX. Hence this subgroup
+acts onQ.
+DefGH 3.3 Lemma. The transformations
+m1,m2,m3,m4,m5,m6
+generate a group Gof order 256.
+Since the proofs of this Lemma and the following Proposition can be given by
+computation, we omit them.
+IsoStabs 3.4 Proposition. The assignment
+Mi/ma√s⊔o−→mi(1≤i≤6)
+induces an isomorphism
+ˆΓ2,0[2]P/Γ′∼−→G.
+The described identification of germs (X(Γ′),P)and(Q,0)is equivariant.
+We are interested in the subgroup of index two ˆΓ2,0[2]n. We have to intersect
+it with the stabilizer. It is easy to check that the elements
+M2
+2, M2
+3, M2M1, M3M1, M4M1, M5M1, M6M1
+are contained in ˆΓ2,0[2]n. One also can check that the elements
+m2
+2, m2
+3, m2m1, m3m1, m4m1, m5m1, m6m1.
+generate a group Hof order 128. In this way one obtains:10 Some Siegel threefolds with a Calabi-Yau model II
+Isozstabs 3.5 Proposition. The stabilizer of PinˆΓ2,0[2]nis a subgroup of index two
+ofˆΓ2,0[2]P. The restriction of 3.4 induces an isomorphism
+(ˆΓ2,0[2]P∩ˆΓ2,0[2]n)/Γ′∼−→H.
+4. Some results about crepant resolutions
+Inthissectionwerecallsomeknownresultaboutprojective crepant resolutions.
+We need the notion of a crepant resolution for certain normal three dimen-
+sional varieties X.
+DefCrep 4.1 Definition. LetXbe a connected three dimensional normal complex
+space with singular locus S. Assume that for each point a∈Xthere exists an
+open neighborhood Uand holomorphic three form αonU−Swithout zeros. A
+crepant resolution f:˜X→Xis a holomorphic map of a connected (smooth)
+complex manifold ˜XontoX, such that ˜X−f−1(S)→X−Sis biholomorphic
+and such that αextends to a holomorphic three form without zeros on the
+inverse image ˜U=f−1(U).
+The existence ofa quasiprojective crepant desingularizationisonly a localques-
+tion (in the three-dimensional case):
+LocCrep 4.2 Lemma (Roan). Under the assumptions of 4.1 the following holds:
+The existence of a crepant desingularization is granted, if there exists an open
+covering Ui⊂X, such that each Uiadmits a crepant desingularization.
+We reproduce the argument of Roan: The singular locus is a cur veS. Over
+the generic point of Sthe crepant resolution is unique. For this reason one
+can choose the crepant resolutions over the finitely many sin gular points of S
+arbitrarily and glue them to a global resolution.
+We use a result of a general result about the existence of a res olutions of
+quotient singularities:
+Ito 4.3 Theorem. LetXbe a complex manifold of dimension three and Ga finite
+group of automorphisms of it. Assume that every point of X/Gadmits an open
+neighborhood (in the analytic topology) such that on its reg ular locus there exists
+a three-form without zeros. Then X/Gadmits a crepant desingularization.
+Because of 4.2 it is sufficient to prove this result in the sitat ionC3/Gwhere
+G⊂GL(3,C) is a finite subgroup. We can assume that Gdoesn’t contain
+quasi reflections. Then the assumption in 4.3 gives G⊂SL(3,C). We refer to
+[Re] (especially section 5) for historical comments and bas ic results.§5. Some quotients of an ordinary double point 11
+5. Some quotients of an ordinary double point
+We will study the node ( Q,0) and some of its quotients. This is related to work
+of Davis [Da], where also several quotients have been consid ered.
+Diffinv 5.1 Lemma. The restriction of
+α=1
+x2
+1−x2
+4(x1dx1+x4dx4)∧dx2∧dx3
+is a holomorphic differential form of degree three on Q−{0}without zeros. If
+one identifies a small neighborhood of the origin in Qwith a small neighborhood
+ofP∈X(Γ′), we have ω=hα, wherehis a holomorphic invertible function
+on this neighborhood.
+Proof.We cover Qby 4 charts corresponding to xi/ne}a⊔ionslash= 0. For example the part
+x1/ne}a⊔ionslash= 0 is the image of
+{(x1,x2,x3)∈C3;x1/ne}a⊔ionslash= 0} −→C4,(x1,x2,x3)/ma√s⊔o−→(x1,x2,x3,x2x3/x1).
+Pulling back αwe get
+1
+x2
+1−(x2x3/x1)2(x1dx1−x2
+2x2
+3/x3
+1dx1)∧dx2∧dx3=dx1∧dx2∧dx3
+x1
+This is holomorphic and without zeros on this chart. The othe r charts are
+treated in a similar way.
+Sinceαandωboth are 3-forms without zeros outside the singularity, we
+getω=hα, where his a holomorphic function without zeros outside the
+singularity. Since isolated singularities of analytic fun ctions in more than one
+variable cannot exist, handh−1are holomorphic also at the singularity.
+⊔ ⊓
+The following result can be found in [Fri], see also [Jo], 6.3 .
+Mdes 5.2 Proposition. The quadric Qadmits a small desingularization ˜Q→
+Q. This means that ˜Qis a smooth connected variety, the inverse image of
+the node 0is a curve and the map from the complement of this curve maps
+biholomorphically onto Q−{0}. Such a desingularization is crepant.
+A small resolution is not unique. Actually there exist two di fferent isomorphy
+classes of such small resolutions. They can be obtained by bl owing up the
+ideals (x1,x3) or (x1,x4) inC[x1,x2,x3,x4]/(x1x4−x2x3). From this explicit
+description one can derive:12 Some Siegel threefolds with a Calabi-Yau model II
+Fort 5.3 Lemma. The elements of the image of GL(2,C)×GL(2,C)extend to
+biholomorphic transformations of any small resolution, bu t the transformation
+X/ma√s⊔o−→tX,
+which is also an automorphism of Q, does not.
+The group G(see 3.4) is not contained in the image of GL(2 ,C)×GL(2,C).
+The intersection with this group defines a subgroup G0⊂Gof index two. It
+is generated by the elements m1m5,m2,m3,m4,m6. One checks that these
+elements have determinant 1 (considered in GL(4 ,C)). Hence G0also can be
+defined as intersection of Gwith SL(4 ,C). We denote by H0the intersection
+ofHandG0. This is a group of order 64.
+Cent 5.4 Lemma. The groups HandH0have the same center. It is the group of
+order2generated by the transformation x/ma√s⊔o→ −x.
+We omit the proof, since it can be done by simple computation.
+Difinv 5.5 Lemma. The differential form αonC4(s. 5.1) is invariant under H.
+Hence also the function hin 5.1 is H-invariant.
+Proof.The invariance can be checked directly for the generators. ⊔ ⊓
+Crep 5.6 Theorem. LetKbe any subgroup of H. Then the quotient Q0:=Q/K
+admits a crepant desingularization.
+For the proof we have to differ between 4 types of subgroups K.
+1)Kis contained in the subgroup H0.
+2)Kcontains the transformation x/ma√s⊔o→ −x.
+3)Kcontains one of the two transformations
+/parenleftbigg
+x1x2
+x3x4/parenrightbigg
+/ma√s⊔o−→/parenleftbigg
+−x1x2
+x3−x4/parenrightbigg
+or/parenleftbigg
+x1−x2
+−x3x4/parenrightbigg
+in its center.
+4)Kis a the cyclic subgroup of order 4 that is conjugated to one of the two
+following (given by a generator of order 4):
+/parenleftbigg
+x1x2
+x3x4/parenrightbigg
+/ma√s⊔o−→/parenleftbigg
+x2x4
+x1x3/parenrightbigg
+or/parenleftbigg
+x2ix4
+−ix1x3/parenrightbigg
+.
+These classes are not disjoint. But each subgroup of His contained in at least
+one of the 4 classes. This can be checked by hand or quicker by m eans of a
+computer.
+We are going to discuss the 4 cases. We begin with the§5. Some quotients of an ordinary double point 13
+First type. Because of 5.3 in this case the action of Kextends to a small resolu-
+tion˜Q→Q. The differential form αextends to a holomorphic differential form
+without zeros on ˜Q, since singularities or zeros can only occur in codimension
+one on a smooth variety. By 4.3 the variety ˜Q/Kadmits a crepant resolution.
+HenceQ/Kalso admits one.
+Second type. We start to blow up the origin of C4. The group Kstill acts
+biholomorphically on this blow up. A typical chart of the blo w up is the C4
+with the coordinates
+(t1,t2,t3,x4) = (x1/x4,x2/x4,x3/x4,x4).
+We consider in the blow up of C4the closed smooth subvariety ˜Q, which is
+defined in this chart by t1=t2t3. Its image in C4isQ. Hence ˜Q→Qis
+just a desingularization of Q. (Actually it is the blow-up of Qat the origin.)
+The chart of ˜Q, which we consider, is a C3with the coordinates t2,t3,x4. The
+differential form αin these coordinates can be computed. Up to a constant
+factor it is
+x4dt2∧dt3∧dx4.
+So it gets a zero of order one along x4= 0. The transformation x→ −xjust
+changes the sign of each variable xi. Hence it acts on t2,t3,x4as reflection,
+which changes the sign of the third variable only. The quotie nt is aC3with
+the coordinates
+(t2,t3,t4) = (x2/x4,x3/x4,x2
+4).
+Hence˜Q/±is a smooth variety, the affine piece in consideration a C3with
+coordinates t2,t3,t4. In these coordinates αappears as holomorphicdifferential
+form without zeros. By the general theorem 4.3, the quotient ˜Q/Kand hence
+Q/Kadmits a crepant resolution.
+Third type. The two cases are equivalent, hence we can assume that
+σ(x1,x2,x3,x4) = (x1,−x2,−x3,x4)
+is in the center of K. The ideal ( x2,x3) is invariant under K, since it describes
+the fixed point locus of σwhich is in the center of K. Hence the action of
+Kextends to an action by biholomorphic transformations on th e blow up C
+ofC4along this ideal. The manifold Ccan be covered by two C4using the
+coordinates ( x1,x2/x3,x3,x4) and (x1,x2,x3/x2,x4). We take the quotient of
+Cbyσ. The quotient C/σis covered by two C4with coordinates
+(u1,u2,u3,u4) = (x1,x2/x3,x2
+3,x4) and ( v1,v2,v3,v4) = (x1,x2
+2,x3/x2,x4).
+We consider the subvariety Q′⊂ C/σ, which in the two affine pieces is given
+byu1u4=u2u3andv1v4=v2v3. This variety has two singular points, which
+correspond to the origins of the affine pieces and which are ord inary double14 Some Siegel threefolds with a Calabi-Yau model II
+points. There is a natural projection Q′→Q/σ. The group Kacts onQ′. We
+need the stabilizers of the two singularities.
+Claim.The stabilizer of the singularity u1=...=u4= 0 acts by substitutions
+of the form
+U/ma√s⊔o−→CUtD,whereU=/parenleftbigg
+u1u2
+u3u4/parenrightbigg
+.
+Proof of the Claim. We consider an element of the stabilizer. It might be of
+the form
+X/ma√s⊔o−→X/ma√s⊔o−→AXtBorX/ma√s⊔o−→AtXtB.
+We have to use now that this transformation commutes with σ. In each case
+this means that AandBboth are of the form/parenleftbig∗0
+0∗/parenrightbig
+or both are of the form/parenleftbig0∗
+∗0/parenrightbig
+. Both cases are similar. For simplicity we take the case A=/parenleftbiga0
+0d/parenrightbig
+and
+B=/parenleftbigα0
+0δ/parenrightbig
+. Then the transformation AXtBcorresponds to CUtD, where
+C=/parenleftbigg
+1 0
+0d2α/a/parenrightbigg
+, D=/parenleftbigg
+aα0
+0aδ/dα/parenrightbigg
+.
+But the transformation AtXtBinterchanges the u- andv-chart and especially
+the two singularities are interchanged. Hence this transfo rmation is not con-
+tained in the stabilizer.
+We consider now the holomorphic map
+Q′/K −→Q/K,
+which is induced by Q′→Q/σ. The differential form αcan be considered
+as meromorphic differential form on C/σ. One checks that in the coordinates
+u1,u2,u3,u4it is given by the same equation as the original α, just replacing
+the letters xbyu. The same is true for the coordinates v1,v2,v3,v4. Henceα
+gives a holomorphic differential form without zeros on the re gular locus of Q′.
+The claim shows that Kextends to a suitable chosen small resolution ˜Q′ofQ′.
+To be concrete one can blow up the irreducible surface, which is defined in the
+u-coordinates by the ideal ( u2,u3) and in the v-coordinates by ( v2,v3). Then
+the group Kacts on ˜Q′. The differential form αextends to a holomorphic
+differential form on ˜Q′without zeros and is invariant under K.
+Fourth type. We consider the first case, σ(x1,x2,x3,x4) = (x2,x4,x1,x3), the
+second is similar. It is better then to use the coordinates
+y1=x1+x4, y2=x1−x4, y3=x2+x3, y4=x2−x3.
+The quadric then takes the equation y2
+1+y2
+4=y2
+2+y2
+3. We blow up the ideal
+(y2,y4). We denote by Cthe blow up of C4along this ideal. One chart of the
+blow up is ( y1,y2,y3,y4/y2). Taking quotient by σ2gives the chart
+(u1,u2,u3,u4) = (y1,y2
+2,y3,y4/y2).§5. Some quotients of an ordinary double point 15
+Now the the quadric y2
+1+y2
+4=y2
+2+y2
+3gets the form
+u2
+1+u2(u2
+4−1)−u2
+3= 0.
+This 3-fold has two singular points, (0 ,0,0,1) and (0 ,0,0,−1). Take the first
+one. For this point one can take as local parameter
+v4:=u2
+4−1 (= ( u4−1)(u4+1)).
+Now the 3-fold is given by u2
+1+u2v4−u2
+3= 0. Hence the singularity is an
+ordinary double point. This consideration shows that the tr ansform of the
+quadric appears in C/σ2as 3-fold with two singular points, which are nodes.
+The transformation σinterchanges the two nodes and hence acts without fixed
+points. The rest of the proof is analogously to the third case .
+Now the general result 4.3 shows that ˜Q′/Kadmits a crepant resolution.
+This gives a crepant resolution of Q/K. ⊔ ⊓
+We recall that we defined a local etale map
+(X(Γ′),P) = (X,P)−→(Q,0).
+If Γ is a group between Γ′andˆΓ2,0[2]nandKthe corresponding subgroup of
+H, we still get a local etale map
+(X(Γ),P)−→(Q/K,0).
+Pulling back a crepant resolution of Q/Kwe get a crepant resolution of some
+affine neighborhood of PinX(Γ).
+Now we can prove the a basic result of this paper, formulated i n the intro-
+duction, which states that the Siegel threefold for each gro up between Γ′and
+ˆΓ2,0[2]nadmits a (weak) Calabi-Yau model: By 4.3 there exists a crepa nt reso-
+lution of the complement of the (images of the) nodes in X(Γ). As we have just
+seen for each node there exists an open neighborhood which ad mits a crepant
+resolution. Hence we can apply 4.2 to obtain a (not necessari ly projective)
+crepant resolution of X(Γ).
+It is a natural question whether a group Γ extends to a group of biholomor-
+phic maps of a crepant resolution ˜X. Such a resolution is not unique. There
+exists a projective one but there also exist some which are no t projective. A
+necessary condition of Γ to extend is that the stabilizers of the nodes extend
+as described in 5.3. This is a condition which can be checked. Assume that
+it is satisfied. Then we can choose one node aand desingularization of this
+node. We can extend Γ ato this desingularization. Let now g∈Γ. The choice
+of the resolution of adictates us the choice of the resolution at g(a) and the
+assumption about Gamakes this choice independent of the choice of g. In this
+way we resolve the whole orbit of aand then in the same way the other orbits.
+In this way we obtain:16 Some Siegel threefolds with a Calabi-Yau model II
+ExtCom 5.7 Lemma. LetΓbe a group between Γ′andˆΓ2,0[2]n. Assume that the
+stabilizer at an arbitrary node satisfies the local conditio n 5.3. Then there
+exists a crepant resolution ˜X → Xin the category of complex spaces such that
+Γextends biholomorphicallly to ˜X.
+As van Geemen pointed out to the authors it is usually not poss ible to get ˜X
+as a projective variety. For example one can show that there a re freely acting
+groups of order #Γ /Γ′= 32. For them it is not possible to get a projective ˜X.
+6. The Picard group
+In the following we consider a group Γ between Γ′andˆΓ2,0[2]n. We denote by
+X(Γ) the Satake compactification of H2/Γ and by X(Γ)regits regular locus.
+We want to compute the Picard number of this variety. We denot e it by
+pic(Γ) := dimPic( X(Γ)reg)⊗ZQ.
+Since the map X(Γ)→X(Γ′) is unramified in codimension one, we have
+Pic(X(Γ)reg) = Pic(X(Γ′)reg)Γ.
+in [GN] the Picard number has been computed for Γ′:
+pic(Γ′) = 32.
+TheproofofvanGeemenandNygaardrestsoncountingnumbers ofpointswith
+values in some finite fields and making use of the Weil conjectu res. Therefore
+it may be difficult to get from these computations the action of ˆΓ2,0[2] on
+Pic(Xreg). For this reason we need some explicit description of gener ating
+divisors.
+We use Igusa’s cusp form χ35of weight 35 for the full Siegel modular group.
+picGN 6.1 Theorem. The group Pic(Xreg)is generated by the irreducible components
+of the zero divisor of χ35.
+The proof needs some computer calculation, which rests on th e following infor-
+mations about the structure of χ35. First we recall that χ35is the product
+χ35=χ5·χ30
+oftwo formsofweight 5and30, alsofor thefull modulargroup but withrespect
+to its non-trivial character. The form χ5can be defined as the product of the
+ten theta constants. They can be produced as follows: One sta rts with the
+most trivial theta constant
+ϑ[0](Z) =/summationdisplay
+g∈Z2eπiZ[g]
+and applies the full modular group to it. This gives 10 modula r forms and
+there product is χ5(up to a constant factor). The form χ30can be constructed
+in a similar way.§6. The Picard group 17
+chiDr 6.2 Lemma. If on applies the full modular group to X0=ϑ[0](2Z), on gets
+(up to constant factors) 60modular forms (living on Γ2[2,4]all with the same
+multipliers). There product up to a constant is χ30. Examples of forms in the
+orbit are
+ϑ[0](Z/2)andX0,X1,X2,X3.
+The60modular forms can be written as linear combinations of X0,...,X 3.
+The last statement is true, since the full modular group acts on the space
+generated by X0,...,X 3. This action has been studied in detail by Runge
+[Ru].
+So far we have seen that the zero locus of χ35considered on Xsplits into
+the sum of 70 pairwise different divisors. But these 70 diviso rs need not to be
+irreducible.
+Now computer algebra comes into the game. Since we know the eq uations
+of the 70 divisors in P7(C) we can decompose them into irreducibles by using
+the facility of computer algebra to compute the primary deco mposition of an
+ideal. We have to be a little careful, since computer algebra works well not
+overCbut only over a finitely generated field. Hence we have to use a n umber
+fieldK. We use K=Q(ζ8), where ζ8is a primitive 8throot of unity. We got
+the following result using MAGMA [BMP]:
+decIrr 6.3 Proposition. We consider Xas variety over K=Q(ζ8)(using the
+equations of van Geemen and Nygaard). The zero locus of χ35is defined over
+this field. It splits over Kinto132irreducible components. More precisely we
+have:
+1)The theta constants Y0,Y1,Y2,Y3have irreducible divisors. The other 6
+decompose into two irreducibles. Hence χ5contributes with 16 = 4 + 2 ·6
+irreducible components.
+2)The forms X0,X1,X2,X3have irreducible divisors. The other 56factors
+ofχ30decompose into pairs of irreducibles. Hence χ30contributes with 116 =
+4+2·56irreducible components.
+We conjecture that these 132 components are irreducible ove rC. But there is
+no need for us to check this.
+The proof of 6.1 now can be given as follows. Using Poincar` e d uality it
+is sufficient to construct a system of curves C1,...,C m, which are complete
+and contained in the regular locus of X(i.e. they don’t contain one of the
+96 nodes) and such that the intersection matrix between the 1 32 divisors and
+these curves has rank 32. The construction of these curves ca n be given (again
+by means of a computer) as follows: Take pairwise intersecti ons of the 132
+divisors, decompose them into irreducibles and single out t hose components,
+which don’t contain nodes.
+In this way 6.1 can be proved. This explicit description of th e Picard group
+allows us to describe the action of the group ˆΓ2,0[2] on it. The group Gacts18 Some Siegel threefolds with a Calabi-Yau model II
+in a natural way on the ring C[Y0,...,Y 3,X0,...,X 4] and on its ideals. Hence
+we can describe the action of Gon Pic(Xreg) explicitly. Of course Zacts
+trivially. Using the ismomorphism ˆΓ2,0[2]/Γ′∼=G/Zwe get the action of
+ˆΓ2,0[2] on Pic( Xreg). Using the character table for G/Zwhich can be produced
+by computer algebra we get the decomposition into irreducib les.
+DecIrr 6.4 Theorem. The space Pic(Xreg)⊗ZQdecomposes under ˆΓ2,0[2]into four
+irreducible components of dimensions 1,3,12,16.
+These numbers can be explained as follows:
+1) The 1-dimensional component comes from the divisor of a mo dular form.
+2) The 3-dimensional space comes from the components of the 6 theta con-
+stants, which are different form Y0,...,Y 3.
+3) The 56 factors of χ30, which are different from X0,...,X 3decompose under
+Γ2,0[2] into two orbits of 24 and 32 elements. Their irreducible c omponents
+produce the spaces of dimension 12 and 16.
+This explicit picture of the action of ˆΓ2,0[2] on Pic( Xreg) allows to compute the
+number pic(Γ) for every group Γ between Γ′andˆΓ2,0[2] and this can be done
+by means of a program.
+7. Involutions
+As we have seen, the group ˆΓ2,0[2]/Γ′∼=G/Zhas order 24576. We are in-
+terested in its subgroups of order two. One can compute that t here are 18
+conjugacy classes of such subgroups, and one can show that 10 of them are in
+the image of ˆΓ2,0[2]n. In the following we list them. There are two possibilities
+to define such a group. We could describe it by an element M∈ˆΓ2,0[2], such
+that the image of MinˆΓ2,0[2]/Γ′generates the group. In the case M∈Γ2,0[2]
+it is enough to consider its image in Sp(2 ,Z/8Z), since Γ′contains Γ 2[8]. The
+other possibility is to give a matrix g∈ Gsuch that its image in G/Zgenerates
+the subgroup of G/Z. Such a gis determined up to a power of i. We give the
+10 groups the names G2i, 1≤i≤10.
+GroupG21
+3 0 4 0
+0 1 0 0
+0 0 3 0
+0 0 0 1
+ (Y0,Y1,Y2,Y3,−X0,−X1,−X2,−X3)
+GroupG22
+5 2 6 2
+2 1 2 6
+4 4 1 6
+4 4 6 5
+ (Y0,−Y1,−Y2,Y3,X0,−X1,−X2,X3)§7. Involutions 19
+GroupG23
+1 0 2 6
+2 1 2 6
+0 0 1 6
+0 0 0 1
+ (Y0,Y1,Y2,Y3,X0,−X1,−X2,X3)
+GroupG24
+3 6 4 2
+4 7 6 2
+0 0 3 4
+0 4 2 7
+ (−Y0,−Y1,Y2,Y3,X0,X1,−X2,−X3)
+GroupG25
+3 2 6 7
+2 3 7 2
+4 2 1 2
+2 4 2 1
+ (−Y0,−Y1,−Y2,Y3,X0,X1,X2,−X3)
+GroupG26
+5 2 6 7
+0 3 1 0
+0 2 3 0
+6 4 6 5
+ (−Y0,−Y1,−Y2,Y3,−X0,−X1,−X2,X3)
+GroupG27
+7 4 7 6
+0 7 2 3
+6 4 5 0
+4 6 4 5
+ (−Y3,−iY2,−iY1,Y0,X3,iX2,iX1,−X0)
+GroupG28
+3 7 5 2
+6 1 2 6
+0 6 1 6
+2 0 7 3
+ (−Y1,Y0,−iY2,−iY3,X2,−iX1,−X0,iX3)
+GroupG29
+1 0 7 0
+0 7 0 0
+2 0 7 0
+0 0 0 7
+ (−iY1,Y0,−iY3,Y2,X1,−iX0,X3,−iX2)
+GroupG210 Fricke involution√
+2·(X0,X1,X2,X3,Y0/2,Y1/2,Y2/2,Y3/2)
+somPp 7.1 Proposition. The following table gives the numbers dimPic(X(Γ)reg)⊗Z
+Qfor the groups Γcorresponding to the groups G2i,1≤i≤10and the
+dimension of the fixed point locus of the generating involuti on in each case.
+(Dimension -1 means that the locus is empty.)
+Picard number 16 24 16 16 16 20 20 16 16 18
+Dimension -1 0 1 -1 -1 1 1 1 1 1
+We want to compute the Picard- and Euler number for a crepant r esolution.
+The numbers of a small resolution ˜XofXare [GN], [CM]
+e= 64,pic = 32.
+Hence˜Xis a rigid Calabi-Yau manifold.20 Some Siegel threefolds with a Calabi-Yau model II
+ActExt 7.2 Lemma. The action of the groups G2iextends to a suitable crepant
+desingularization ˜X.
+The proof rests on Lemma . We omit details. ⊔ ⊓
+Each of the groups G2iis generated by an involution σi. We need infor-
+mation by the fixed point locus. It can be checked that there is no component
+of dimension two. We also know that the Calabi-Yau 3-form is i nvariant under
+σi. This implies that the action of σion the tangent space of a fixed point can
+be diagonalized with diagonal ( −1,−1,1). From this follows:
+FixCur 7.3 Lemma. The fixed point locus of σion˜Xis the disjoint union of smooth
+curves. They are in one to one correspondence with the irredu cible components
+of the fixed point locus on X. The fixed point locus on Xconsists of curves and
+isolated points which are nodes. If a node occurs as isolated fixed point, then
+the exceptional line over it is in the fixed point locus on ˜X.
+Theimageofthefixedpointlocusof σiin˜X/σiiscurve, whichisinthesingular
+locus. We claim that in a crepant resolution over each of its c omponents there
+is only one exceptional divisor. Locally around a fixed point the involution
+can be described by ( w1,w1,w3)/ma√s⊔o→(−w1,−w2,w3). The crepant resolution of
+the quotient of C3by this involution is easy to describe (we did it in [FSM])
+and one sees from this description that the exceptional divi sor is smooth and
+connected.
+PicNumb 7.4 Lemma. The Picard number of a crepant resolution of X/G2iequals
+the sum of the Picard number of the regular locus of X/G2i(see 7.1) and the
+number of irreducible components of the fixed point locus of σi(considered in
+Xis enough).
+We also have to compute the Euler number of a crepant resoluti on ofX/G2i.
+This is given by the string theoretic Euler number e(˜X,G2i). We refer to [Re]
+for some comments about this. We recall the definition of e(M,G). Here G
+is a finite group acting on a compact differentiable manifold M. One has to
+consider the subset of G×Gof all commuting pairs ( g,h). Then the string
+theoretic Euler number is defined as
+e(M,G) =1
+#G/summationdisplay
+gh=hge(M/angbracketleftg,h/angbracketright).
+HereM/angbracketleftg,h/angbracketrightdenotes the common fixed point set of g,h. The string theoretic
+Euler number has the following basic property: Assume that Mis a Calabi Yau
+manifold and that Gacts by biholomorphic transformations, which leave the
+Calabi-Yau 3-form invariant. Assume that for a∈Mthe stabilizer Gaacts on
+the tangent space as subgroup of the special linear group. Th en there exists a
+crepant desingularization of M/Gand for each such desingularization its usual
+Euler number is e(M,G).
+We apply this formula in the case, where the order of Gis two. There are
+4 commuting pairs ( e,e),(σ,e),(e,σ),(σ,σ).§7. Involutions 21
+EulBer 7.5 Lemma. The Euler number of a crepant resolution of ˜X/G2iis
+e= 32+3
+2/summationdisplay
+Ce(C),
+whereCruns through the components of the fixed point locus of σiin˜X.
+The fixed point sets are easy to determine. The involution can be considered
+as a linear transformation A:C8→C8, whereA2=aE. We want to consider
+the fixed point locus of AonP7(C). It corresponds to the eigenspaces of A.
+The possible eigenvalues are the two square roots of a. We denote the two
+eigenspaces by V+andV−. Hence C8=V+⊕V−. The projective spaces
+P(V+) andP(V−) are two disjoint linear subspaces of P7(C) =P(C8). To
+get the fixed point set of Ainside the variety Xwe have to intersect this variety
+with the two linear subspaces. Hence the fixed point set insid eXis the disjoint
+union of two parts, where each of the parts can be empty of cour se.
+Following these lines one gets:
+FixSet 7.6 Proposition. The following table describes fixed point sets of the invo-
+lutionsσi,i≤i≤11, onXand the Hodge numbers of a Calabi-Yau model of
+the quotient X/G2i.
+fixed points pic e
+σ1:empty set 16 32
+σ2:16 nodes 40 80
+σ3:4 elliptic curves 20 32
+σ4:empty set 16 32
+σ5:empty set 16 32
+σ6:8 conics in planes ( ∼=P1) 28 56
+σ7:8 lines (∼=P1) 28 56
+σ8:2 elliptic curves 18 32
+σ9:2 elliptic curves 18 32
+σ10:4 conics in planes ( ∼=P1) 22 44
+The equations for the fixed point loci can be given explicitly . We just give as an
+example the 4 elliptic curves which describe the fixed point l ocus ofσ3: They
+are described by the following 4 ideals:
+(Y0+Y1, Y2+Y3, X1, X3, Y2
+1−X2
+0−X2
+2, Y2
+3−X2
+0+X2
+2),
+(Y0+Y1, Y2−Y3, X1, X3, Y2
+1−X2
+0−X2
+2, Y2
+3−X2
+0+X2
+2),
+(Y0−Y1, Y2+Y3, X1, X3, Y2
+1−X2
+0−X2
+2, Y2
+3−X2
+0+X2
+2),
+(Y0−Y1, Y2−Y3, X1, X3, Y2
+1−X2
+0−X2
+2, Y2
+3−X2
+0+X2
+2).22 Some Siegel threefolds with a Calabi-Yau model II
+References
+[BH] Borisov, L. Hua, Z.: On Calabi-Yau threefolds with large nonabelian funda-
+mental groups, Proc. Amer. Math. Soc. 136, 1549–1551 (2008)
+[BMP] Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The
+user language, J. Symbolic Comput. 24(3-4), 235–265 (1997)
+[CM] Cynk, S., Meyer, C.: Modular Calabi-Yau Threefolds of level eight, Inter-
+nat. J. Math. 18, no. 3, 331–347 (2007)
+[CFS] Cynk, S., Freitag, E., Salvati-Manni, R.: The geometry and arithmetic of a
+Calabi-Yau Siegel threefold, preprint 2010
+[Da] Davis, R.: Quotients of the conifold in compact Calabi-Yau threefolds , and
+new topological transitions, eprint arXiv: 0911.0708 [hep-th] (2009)
+[Fr1] Freitag, E.: Siegelsche Modulfunktionen, Grundlehren der mathematischen
+Wissenschaften, Bd. 254. Berlin Heidelberg New York: Springer (1983)
+[Fr2] Freitag, E.: Singular modular forms and theta relations, Lecture notes in
+Math.1487, Springer-Verlag, Berlin Heidelberg New York (1991)
+[Fri] Friedmann, R.: Simultaneous resolution of threefold double points, Mathe-
+matische Ann. 274, 671–689 (1986)
+[FSM] Freitag, E. Salvati Manni, R.: Some Siegel threefolds with a Calabi-Yau
+model,preprint 2009
+[GN] van Geemen, B., Nygaard, N.O.: On the geometry and arithmetic of some
+Siegel modular threefolds, Journal of Number Theory 53, 45–87 (1995)
+[Ig] Igusa, I.: A desingularization problem in the theory of Siegel modular func-
+tions,Math. Annalen 168228–260 (1967)
+[Re] Reid, M.: La correspondence de McKay, S´ eminaire Bourbaki 1999/2000.
+Ast´ erisque No. 276, 53–72 (2002)
+[SM] Salvati Manni, R.: Thetanullwerte and stable modular forms, Am. J. of
+Math.111, 435–455 (1989)
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+arXiv:1001.0325v2  [math.GR]  2 Feb 2011A Bers-like proof of the existence of train tracks for
+free group automorphisms
+Mladen Bestvina∗
+January 12, 2011
+To Mike Davis on the occasion of his 60th birthday
+Abstract
+Using Lipschitz distance on Outer space we give another proof of
+the train track theorem.
+1 Introduction
+An elegant proof of Thurston’s classification of surface hom eomorphisms
+[Thu88] was given by Bers [Ber78]. Given a surface homeomorp hism Φ
+the proof proceeds by studying the associated displacement function ˜Φ on
+Teichm¨ uller space Twith respect to Teichm¨ uller metric, i.e. the function
+˜Φ(x) =d(x,Φ(x))
+There are three possibilities:
+•(elliptic) inf ˜Φ = 0 and the infimum is realized. In this case ˜Φ has a
+fixed point and it is not hard to show that Φ is isotopic to a home o-
+morphism of finite order.
+•(hyperbolic) inf ˜Φ>0 and the infimum is realized. In this case Bers
+proceeds to show that Φ is isotopic to a pseudo-Anosov homeom or-
+phism.
+•(parabolic) The infimum is not realized. In this case Bers sho ws that
+Φ is reducible.
+∗The author gratefully acknowledges the support by the Natio nal Science Foundation.
+1In this paper we carry out the Bers’ argument in the context of Out(Fn)
+and Outer space. The role of Teichm¨ uller metric is played by the Lipschitz
+metric, a direct analog of Thurston’s Lipschitz metric on Te ichm¨ uller space
+[Thu]. This metric is not symmetric and one must carefully ch oose the order
+of the two points when measuring distance. The result is an al ternate proof
+of the train track theorem [BH92].
+Theorem. Every irreducible automorphism Φhas a topological representa-
+tive which is a train track map.
+Acknowledgements. The author is indebted to Ric Wade for reading the
+paper carefully and making helpful comments.
+2 Outer space and Lipschitz metric
+In this section we review basic definitions and set some notat ion.
+AgraphΓ is a finite cell complex of dimension ≤1. Acore graph is a
+graph Γ with all vertices of valence ≥2. Arosein ranknis the wedge Rn
+ofncircles. A marking of a graph Γ is a homotopy equivalence f:Rn→Γ.
+Ametricon Γ is an assignment ℓof positive lengths ℓ(e) to the edges eof Γ
+such that the sum is 1. If αis an immersed loop in Γ we define the lengthof
+αwith respect to the metric ℓas the sum ℓ(α) of the lengths of edges of Γ
+crossed by α, with multiplicity. If αis not immersed, we define ℓ(α) as the
+length of the immersed loop homotopic to α(or 0 ifαis nullhomotopic).
+We may view Γ as a geodesic metric space with each edge ehaving length
+ℓ(e). Adirection atx∈Γ is a germ of isometric embeddings d: [0,ǫ)→Γ
+withd(0) =x. Thus most points of Γ have two directions, and the number
+of directions at a vertex is the valence. Directions can be vi ewed as analogs
+of unit tangent vectors. If φ: Γ→Γ′is a map which is linear on edges and
+φ(x) =x′, thenφinduces a map φ∗from the set of directions at xto the
+set of directions at x′(unless the slope of φis 0 on an edge containing x).
+Recall that Culler-Vogtmann’s Outer space Xnin rankn[CV86] is the
+set of equivalence classes of triples (Γ ,f,ℓ) where
+•Γ is a core graph,
+•f:Rn→Γ is a marking, and
+•ℓis a metric on Γ.
+Two such triples (Γ ,f,ℓ) and (Γ′,f′,ℓ′) are equivalent if there is a home-
+omorphism φ: Γ→Γ′such that
+2•φf≃f′, and
+•for all loops αin Γ,ℓ(α) =ℓ′(φ(α)).
+If Γ and Γ′have no vertices of valence 2, then φmust induce a bijection
+between the edges of Γ and of Γ′andℓ(e) =ℓ′(φ(e)) for every edge e⊂Γ.
+Every triple (Γ ,f,ℓ) is equivalent (for n≥2) to some (Γ′,f′,ℓ′) so that Γ′
+has no vertices of valence 2, by “unsubdividing” and assigni ng the sum of
+the lengths of subdivision edges to the newly created edges.
+There are three ways of defining the topology on Xn, all yielding the
+same topology.
+• Xncan be decomposed into open simplices corresponding to fixin g the
+marking and varying lengths on Γ. This gives Xnthe structure of a
+“complex of simplices with missing faces”. Take the weak top ology
+with respect to the collection of these simplices with missi ng faces.
+•There is an embedding Xn֒→RSwhereSis the set of nontrivial
+conjugacy classes in Fn, or equivalently, the set of immersed loops in
+Rn. The embedding is given by
+[(Γ,f,ℓ)]/mapsto→(α/mapsto→ℓ(f(α)))
+Now take the subspace topology.
+•A neighborhood of [(Γ ,f,ℓ)] is determined by ǫ >0 and consists of
+classes [(Γ′,f′,ℓ′)] such that there is a map φ: Γ→Γ′withφf≃f′
+and such that φis<(1+ǫ)-Lipschitz.
+Out(Fn) acts on Xnon the right as follows. Let Φ ∈Out(Fn). We may
+view Φ as a homotopy equivalence (defined up to homotopy) Φ : Rn→Rn.
+Then
+[(Γ,f,ℓ)]Φ = [(Γ ,fΦ,ℓ)]
+The third definition of the topology on Xncan be promoted to a (non-
+symmetric) metric.
+Let [(Γ,f,ℓ)],[(Γ′,f′,ℓ′)]∈ Xn. Consider maps φ: Γ→Γ′so that
+•φf≃f′, and
+•φis linear on edges.
+3Call any map φthat satisfies these two conditions a difference of mark-
+ings. Letσ(φ) denote the maximal slope of φ.
+Observe that if αis any loop in Γ then
+ℓ′(φ(α))≤σ(φ)ℓ(α)
+The following resultis dueto Tad White (unpublished). A pro of appears
+in [FMb], but for completeness we include a proof.
+Proposition 1. Letφ0: Γ→Γ′be a difference of markings. Then
+inf{σ(φ)|φ≃φ0: Γ→Γ′is a difference of markings }= sup
+αℓ′(φ0(α))
+ℓ(α)
+and moreover both infandsupare realized.
+Proof.That inf ≥sup follows from the observation just before the proposi-
+tion. Arzela-Ascoli implies that inf is realized. Let φ: Γ→Γ′bea difference
+of markings that realizes inf. By ∆ = ∆( φ) denote the union of all edges
+of Γ on which φhas slope equal to σ(φ). We may also assume that ∆ is
+minimal possible. Therefore ∆ is a core graph (any edges with a valence 1
+vertex can be removed by a small homotopy of φ, by moving the image of
+the valence 1 vertex in the direction that lowers the slope on the edge of ∆
+containing it).
+Letvbe a vertex of ∆ and consider a turn, i.e. pair of distinct directions
+{d,d′}out ofvin ∆. We say that this turn is legalifφ∗(d)/ne}ationslash=φ∗(d′), and it is
+illegalifφ∗(d) =φ∗(d′). A path or a loop in ∆ is legalif it crosses only legal
+turns. Note that the loop φ(α) is immersed if and only if αis legal. Also
+observe that there is an equivalence relation on the set of di rections (within
+∆) out of each vertex v∈∆ where d∼d′if and only if φ∗(d) =φ∗(d′). A
+turn{d,d′}is legal if and only if d/ne}ationslash∼d′. We will call equivalence classes
+gates.
+If∆containsavertexwithonlyonegate, thenasmallhomotop yasabove
+would reduce ∆. Therefore there are at least two gates at each vertex. It
+follows that every legal edge path in ∆ can be extended in a leg al fashion.
+In particular, ∆ admits a legal loop α. By construction ℓ′(φ(α)) =σ(φ)ℓ(α),
+and thus αrealizes the sup and equality holds as claimed.
+Remark 2.The legal loop αcan be chosen to cross each edge of ∆ at most
+twice. Therefore, the sup can be calculated by taking the max imum over a
+finite collection of loops in Γ. Furthermore, ℓ′(φ0(α)) does not depend on
+φ0, only on the homotopy class of φ0, so the quantity in the statement can
+be easily calculated.
+4To simplify notation, we will replace [(Γ ,f,ℓ)] with Γ. Denote by
+σ(Γ,Γ′)
+the quantity in the statement of Proposition 1.
+Definition 3.Let Γ,Γ′∈ Xn. A difference of markings φ: Γ→Γ′isoptimal
+ifσ(φ) =σ(Γ,Γ′). Thetension (sub)graph ∆ = ∆ φ⊂Γ with respect to an
+optimal map φis the union of edges on which the slope of φequalsσ(φ).
+The tension graph is equippedwith a train track structure as in the proof
+of Proposition 1: a turn {d,d′}is legal if φ∗(d)/ne}ationslash=φ∗(d′) and otherwise it
+is illegal. Directions (in ∆) at a vertex break up into equiva lence classes,
+calledgates, so that {d,d′}is illegal if and only if d,d′are in the same gate.
+Definition 4.
+d(Γ,Γ′) = logσ(Γ,Γ′)
+Proposition 5. •d(Γ,Γ′)≥0with equality only when Γ = Γ′.
+•d(Γ,Γ′′)≤d(Γ,Γ′)+d(Γ′,Γ′′).
+•Out(Fn)acts onXnby isometries: d(Γ·Φ,Γ′·Φ) =d(Γ,Γ′).
+Proof.Ifσ(Γ,Γ′)<1 then the volume of the image (i.e. the sum of the
+lengths of images of edges) of an optimal map φ: Γ→Γ′is<1, contradict-
+ing the fact that φis surjective. Likewise, if σ(Γ,Γ′) = 1 then φmust be an
+isometry.
+The second statement follows by composing optimal maps and h omotop-
+ing rel vertices to a map linear on edges. Thethird statement is obvious.
+But note that in general d(Γ,Γ′)/ne}ationslash=d(Γ′,Γ). See [AKB]. It can also be
+shown that d:Xn×Xn→[0,∞) is continuous.
+3 Trichotomy
+Fix an automorphism Φ ∈Out(Fn) and consider the displacement function
+˜Φ :Xn→[0,∞)
+given by
+Γ/mapsto→d(Γ,Γ·Φ)
+There are three possibilities, as follows:
+•(elliptic) inf ˜Φ = 0 and it is realized.
+5•(hyperbolic) inf ˜Φ>0 and it is realized.
+•(parabolic) inf ˜Φ is not realized.
+We consider these cases separately.
+3.1Φis elliptic
+An example is pictured below.
+a bc
+Figure 1: Φ( a) =b, Φ(b) =c, Φ(c) =a. All 3 edges have length 1 /3, the
+tension graph is all of Γ and all nondegenerate turns are lega l. Φ has order
+6. Bar denotes the edge with opposite orientation.
+Here we are assuming that Φ has a fixed point, i.e. Γ ·Φ = Γ for some
+Γ∈ Xn. IffisthemarkingofΓ,wehave fΦ≃φfforanisometry φ: Γ→Γ.
+Since (for n≥2) isometries of Γ have finite order, it follows that for some
+k >0 we have φk=idand hence f=φkf≃fΦk, so Φkis (homotopic to)
+the identity, i.e. Φ has finite order.
+3.2Φis hyperbolic
+Suppose that inf ˜Φ is realized on Γ ∈ Xn. Let log λ=d(Γ,Γ·Φ) = inf ˜Φ>0,
+soλ >1. Let ∆ ⊆Γ be the tension graph with its train track structure,
+with respect to an optimal map φ.
+An example of a hyperbolic automorphism is given in Figure 2.
+6b
+a
+Figure 2: Φ( a) =ab, Φ(b) =bab. Lengths of edges and σ(φ) =λare
+computed from the equations ℓ(a) +ℓ(b) = 1,λℓ(a) =ℓ(a) +ℓ(b) and
+λℓ(b) =ℓ(a)+2ℓ(b), i.e.λ=3+√
+5
+2,ℓ(a) =3−√
+5
+2,ℓ(b) =√
+5−1
+2. The tension
+graph is all of Γ. The gates are indicated by the little triang le at the vertex:
+they are {a,b},{a},{b}, where we adopt the convention that arepresents
+the initial direction of the edge a, whilearepresents the terminal direction
+ofa, and similarly for b.
+Proposition 6. After an arbitrarily small perturbation of Γthat preserves
+the condition that d(Γ,Γ·Φ) = log λthere is an optimal map φ: Γ→Γ·Φ
+such that
+•φ(∆)⊆∆,
+•φsends edges of ∆to legal paths, and
+•φ∗sends legal turns to legal turns.
+A 2-gate vertex vis allowed to be mapped to a non-vertex, with the
+directions in each gate mapping to one of two directions out o fφ(v), i.e. the
+two directions out of a non-vertex are regarded as forming a l egal turn.
+Proof.Fix an optimal map φ: Γ→Γ·Φ. Thecomplexity of ∆ is the pair
+(rank(H1(∆)),−rank(H0(∆)). In the moves that follow the complexity is
+never increased, and is often decreased. Note that if ∆ is a co re graph and
+∆′⊂∆ is a proper core subgraph, then the complexity of ∆′is strictly
+smaller than that of ∆.
+7Suppose that all vertices of ∆ have ≥2 gates (in particular, ∆ is a core
+graph), that φ(∆)/ne}ationslash⊆∆ and let ebe an edge of ∆ with φ(e)/ne}ationslash⊂∆. Perturb
+the metric on Γ by scaling by µ >1 on the edges of ∆ and scaling down
+on the edges in the complement of ∆, maintaining volume 1. Den ote the
+new metric graph Γ′. The tension graph ∆′for the new pair Γ′→Γ′·Φ
+(using the same map made linear on edges) is contained in ∆ and does not
+containe. Note that the slope on some edge must be λby the minimality
+assumption. Continuing in this way we obtain a perturbation of Γ where
+φ(∆)⊆∆. If during the process we encounter ∆ with a 1-gate vertex, w e
+perturb the map as in the proof of Proposition 1 with the effect t hat ∆ is
+replaced by a smaller graph.
+Ifφmaps an edge eof ∆ over an illegal turn, first perturb by folding the
+illegal turn (this means identify initial segments of small lengthǫ >0 in the
+two edges and rescale the metric so the volume is 1; the map φnaturally
+induces a map on the quotient). After homotoping rel vertice s to a map
+linear on edges, we see that an edge (induced by e) drops out of ∆ and
+complexity decreases. Again it may be necessary to remove 1- gate vertices.
+Now suppose that φ∗maps a legal turn to an illegal turn. Perturb by
+folding the illegal turn. This converts the legal turn to an i llegal turn and
+it either lowers the number
+/summationdisplay
+v[max(0,G(v)−2)]
+whereG(v) is the number of gates at v, or else it introduces a 1-gate vertex.
+In the latter case the subsequent perturbation of φlowers the complexity of
+∆. At the end of the process we have φand ∆ satisfying the conclusion.
+Corollary 7. LetΓrealizeinf˜Φ = logλ. Thend(Γ,Γ·Φk) =klogλfor
+anyk= 1,2,3,···.
+Proof.After perturbing as in the proof of Proposition 6, the statem ent fol-
+lows by observing that for a legal loop αthe loop φ(α) is also legal, and
+iterating we find that the length of φk(α) is equal to λkℓ(α). By continuity,
+this is true before perturbing as well.
+Remark 8.It is easy to see that if Φ is any automorphism and φ: Γ→Γ·Φ
+an optimal map satisfying the conclusions of Proposition 6, then˜Φ achieves
+minimum at Γ. Indeed, as above we have d(Γ,Γ·Φk) =kd(Γ,Γ·Φ) fork >0
+and if Γ′∈ Xnthen
+d(Γ,Γ·Φk)≤d(Γ,Γ′)+d(Γ′,Γ′·Φk)+d(Γ′·Φk,Γ·Φk)≤
+d(Γ,Γ′)+kd(Γ′,Γ′·Φ)+d(Γ′,Γ)
+8The first and last terms are independent of kso by dividing by kand taking
+the limit as k→ ∞we see that d(Γ,Γ·Φ)≤d(Γ′,Γ′·Φ).
+For another example of an automorphism of this type consider Φ :F3→
+F3given by Φ( a) =ab, Φ(b) =baband Φ(c) =cwwherewis any word in
+aandb. Let Γ be the rose with the metric on a,bas in Figure 2 but scaled
+byt >0, and let the length of cbe 1−t. When t >0 is sufficiently small
+(depending on w) Γ realizes the minimal displacement log λwith the same
+λas in Figure 2.
+3.3Φis parabolic
+One example of a parabolic automorphism, with inf ˜Φ = 0, is shown in
+Figure 3.
+ab
+Figure 3: Φ( a) =a, Φ(b) =ab. Let the length of abet >0 and the length
+ofbis 1−t. Then the slope on ais 1 and the slope on bis1
+1−t. Thus as
+t→0 the displacement converges to 0. The tension graph is the lo opb.
+For another example, consider Φ( a) =ab, Φ(b) =bab, Φ(c) =cad,
+Φ(d) =dcad. OnaandbtakethemetricfromFigure2scaledbyaparameter
+t >0, and on canddtake the same metric (with ccorresponding to aand
+dtob) scaled by 1 −t. The tension graph consists of candd, with slope
+converging to λ=3+√
+5
+2ast→0. In this example inf ˜Φ = logλ >0 is not
+realized. (One way to see this is to show that the (combinator ial) length of
+Φk(c) grows like kλk.)
+Fix a sequence Γ ksuch that
+d(Γk,Γk·Φ)→D= inf˜Φ
+butDis not realized.
+Forθ >0 denote by Xn(θ) the subspace of Xnconsisting of marked
+metric graphs Γ where every nontrivial loop has length ≥θ(theθ-thick part
+9ofXn). ThenXn(θ) is anOut(Fn)-invariant closed subset on which Out(Fn)
+acts cocompactly.
+Proposition 9. For any θ >0there are only finitely many ΓkwithΓk∈
+Xn(θ).
+Proof.Suppose not, and after passing to a subsequence assume that Γ k∈
+Xn(θ) for every k. By cocompactness, there are Ψ k∈Out(Fn) such that
+Γk·Ψk→Γ∞(after a subsequence). Therefore, we have
+d(Γ∞·Ψ−1
+k,Γ∞·Ψ−1
+kΦ)
+≤d(Γ∞·Ψ−1
+k,Γk)+d(Γk,Γk·Φ)+d(Γk·Φ,Γ∞·Ψ−1
+kΦ)
+=d(Γ∞,Γk·Ψk)+d(Γk,Γk·Φ)+d(Γk·Ψk,Γ∞)
+and hence
+d(Γ∞·Ψ−1
+k,Γ∞·Ψ−1
+kΦ)→D
+(sinced(ΓkΨk,Γ∞)→0 andd(Γ∞,ΓkΨk)→0). In other words,
+d(Γ∞·Ψ−1
+kΦ−1Ψk,Γ∞)→D
+Note that Arzela-Ascoli implies that there are only finitely many Ψ ∈
+Out(Fn) such that d(Γ∞Ψ,Γ∞)≤D+ 1 (the set of eD+1-Lipschitz maps
+Γ∞→Γ∞is compact and nearby maps are homotopic, so such maps rep-
+resent only finitely many homotopy classes). It follows that , after a subse-
+quence, Ψ−1
+kΦ−1Ψkis a constant sequence and
+d(Γ∞·Ψ−1
+kΦ−1Ψk,Γ∞) =D
+i.e. thedisplacement of Γ ∞·Ψ−1
+kunderΦ is D, contradicting our assumption
+that inf is not realized.
+Forǫ >0 theǫ-small subspace Γǫof Γ is the union of all essential (not
+necessarily immersed) loops of length ≤ǫ(but note that Γǫmay not be a
+subgraph). There is ǫn>0 such that for any Γ ∈ Xnthe subspace Γǫnis
+always proper (i.e. not equal to Γ, e.g. there is always an edg e of length
+≥1/(3n−3), assuming no valence 2 vertices, so ǫn<1/(3n−3) works).
+Moreover, there is a bound Bnto the length of any chain of proper core
+subgraphs.
+Proposition 10. For large kany optimal map φ: Γk→Γk·Φleaves
+a nonempty proper core subgraph invariant up to homotopy (an d soφis
+homotopic to a possibly non-optimal map that maps this core s ubgraph to
+itself).
+10Proof.Letθ=ǫn/(eD+1)Bn. By Proposition 9 eventually Γ k/ne}ationslash∈ Xn(θ).
+Choosekso large that in addition the displacement of Γ kis< D+1. Set
+δi=ǫn/e(D+1)i,i= 0,1,2,···,Bn. Then
+Γδ0
+k⊃Γδ1
+k⊃ ··· ⊃ΓδBn
+k
+form a chain of homotopically nontrivial proper subspaces ( not necessarily
+subgraphs, but abstract graphs) of length Bn+ 1, so there must be some
+iso that Γδi
+kand Γδi+1
+khave the same core. By definition, an optimal map
+must send Γδi+1
+kinto Γδi
+k, so the common core is mapped to itself up to
+homotopy.
+4 Train tracks
+In this section we complete the proof of the train track theor em. We first
+make the standard definitions; the definition of a train track structure is as
+before, but does not require a metric on the graph.
+Definition 11.Let Γ be a marked graph with marking f:Rn→Γ. We say
+thatφ: Γ→Γ represents Φ ∈Out(Fn) ifφf≃fΦ. The automorphism
+Φ isreducible if there is some φ: Γ→Γ that represents Φ and leaves
+a homotopically nontrivial (i.e. not a forest) proper subgr aph invariant.
+Otherwise Φ is irreducible .
+Definition 12.Atrain track structure on a core graph Γ is an equivalence
+relation on the set of directions at every vertex of Γ with at l east two equiva-
+lence classes (called gates) at every vertex. As before, a tu rn{d,d′}isillegal
+ifd∼d′, and otherwise it is legal. An immersed loop or a path is legalif it
+takes only legal turns.
+Definition 13.Let Φ∈Out(Fn) be irreducible. A map φ: Γ→Γ represent-
+ing Φ is a train track map if it sends each edge to a nondegenerate immersed
+path and the following two equivalent conditions are satisfi ed.
+(i) There is a train track structure on Γ such that φsends edges to legal
+paths and if vis a vertex of Γ then either
+•φ(v) =wis a vertex and inequivalent directions at vmap to
+inequivalent directions at w, or
+•φ(v) is not a vertex, vhas two gates, and all directions in one
+gate fold to one direction at φ(v), while all directions in the other
+gate fold to the other direction at φ(v).
+11(ii) There is a loop αin Γ such that every iterate φk(α) is immersed,
+k= 1,2,3,···.
+To prove that (i) implies (ii) it suffices to choose any legal lo op forα,
+since (i) guarantees that legal loops map to legal loops. For the converse,
+first note that the iterates of αcross every edge of Γ (otherwise the union of
+the edges crossed by the iterates of αwould be a homotopically nontrivial
+properφ-invariant subgraph). Thus iterated images of edges are imm ersed.
+Define a train track structure on Γ by declaring d∼d′ifφk
+∗(d) =φk
+∗(d′)
+for some k≥1. The reader may now easily check that the conditions in (i)
+hold.
+For example, any simplicial isomorphism φ: Γ→Γ is a train track map.
+Remark14.For agiven φ: Γ→Γtheremay bemorethanoneinvariant (i.e.
+satisfying (i)) train track structure. For example, in Figu re 2 we could take
+the gates to be {a,b}and{a,b}. In the above paragraph we constructed the
+invariant train track structure with the minimal collection of illegal turns.
+The discussion in Section 3 proves the following theorem.
+Theorem. [BH92] Every irreducible automorphism Φis represented by a
+train track map φ: Γ→Γ.
+There is an additional caveat. In [BH92] train track maps alw ays send
+vertices to vertices. This is very useful, and luckily it is n ot hard to achieve.
+Letφt,t∈[0,1], be a homotopy of φ=φ0that moves each φ(v) that is
+not a vertex to an endpoint of the edge containing φ(v). We also insist that
+during the homotopy the order of the images of vertices in the same edge
+does not change (i.e. there are no collisions, nor “uncollis ions”) until the
+very last moment when several images of vertices may arrive a t the same
+vertex. It is not hard to see that the images of legal loops und erφtare
+unchanged (and they are still legal loops). There may be edge s that map
+to points under φ1; collapse all such edges iteratively (i.e. after a collapse
+there may be new such edges that are then collapsed). We obtai n a new
+mapφ′: Γ′→Γ′representing Φ. Any legal loop in Γ induces a loop in Γ′
+whoseφ′-iterates are immersed, so φ′is a train track map that sends vertices
+to vertices.
+Finally, as in [BH92], one can put a metric on Γ′by solving linear equa-
+tions e.g. as in Figure 2. The array of lengths is a positive le ft eigenvector of
+the transition matrix Mforφ′whoseij-entry is the number of times φ′(ej)
+crosseseiin either direction. The irreducibility of Φ implies the irr educibil-
+ity ofM, so by the Perron-Frobenius theory Mhas a unique (positive)
+eigenvalue with an associated positive eigenvector.
+12Remark 15.Whenφ: Γ→Γ sends vertices to vertices and edges to non-
+degenerate immersed paths, finding an invariant train track structure is
+algorithmic (when it exists). One forms a (finite) directed g raph whose ver-
+tices are the directions at the vertices of Γ and a directed ed ge fromdtod′
+whend′=φ∗(d). Thus each vertex of the directed graph has one outgoing
+edge and following outgoing edges eventually ends in a perio dic orbit (i.e.
+a cycle). Then define d∼d′ifdandd′are based at the same vertex and
+their forward iterates eventually coincide. For example, i n Figure 2 we have
+a/mapsto→a,b/mapsto→banda/mapsto→b/mapsto→b, soa∼bis the only nontrivial equivalence.
+Along the same lines one can give a proof of the existence of re lative
+train track maps [BH92] representing any given Φ ∈Out(Fn). One works in
+a relative Outer space, where all graphs contain a fixed subgr aph Γ0and on
+whichφ: Γ0→Γ0partially representing Φ has already been constructed.
+The edges of Γ 0are assigned length 0. The strategy of the absolute case
+applies here as well. The details may appear elsewhere.
+References
+[AKB] Yael Algom-Kfir and Mladen Bestvina. Asymmetry of Oute r space.
+to appear.
+[Ber78] Lipman Bers. An extremal problem for quasiconforma l mappings
+and a theorem by Thurston. Acta Math. , 141(1-2):73–98, 1978.
+[BH92] Mladen Bestvina and Michael Handel. Train tracks and automor-
+phisms of free groups. Ann. of Math. (2) , 135(1):1–51, 1992.
+[CV86] Marc Culler and Karen Vogtmann. Moduli of graphs and a utomor-
+phisms of free groups. Invent. Math. , 84(1):91–119, 1986.
+[FMa] Stefano Francaviglia and Armando Martino. The isomet ry group
+of Outer space. arXiv:0912.0299.
+[FMb] Stefano Francaviglia and Armando Martino. Metric pro perties of
+Outer space. arXiv:0803.0640.
+[Thu] William P. Thurston. Minimal stretch maps between hyp erbolic
+surfaces. math.GT/9801039.
+[Thu88] William P. Thurston. On the geometry and dynamics of diffeomor-
+phisms of surfaces. Bull. Amer. Math. Soc. (N.S.) , 19(2):417–431,
+1988.
+13
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+arXiv:1001.0326v1  [math.RT]  3 Jan 2010Herz–Schur Multipliers and Non-Uniformly
+Bounded Representations of Locally Compact
+Groups
+Troels Steenstrup∗
+Abstract
+LetGbe a second countable, locally compact group and let ϕbe a
+continuous Herz–Schur multiplier on G. Our main result gives the ex-
+istence of a (not necessarily uniformly bounded) strongly c ontinuous
+representation πofGon a Hilbert space H, together with vectors
+ξ, η∈H, such that ϕ(y−1x) =/a\}bracketle{tπ(x)ξ, π(y−1)∗η/a\}bracketri}htforx, y∈Gand
+supx∈G/bardblπ(x)ξ/bardbl ·supy∈G/bardblπ(y−1)∗η/bardbl=/bardblϕ/bardblM0A(G). Moreover, we ob-
+tain control over the growth of the representation in the sen se that
+/bardblπ(g)/bardbl ≤exp(c
+2d(g, e)) for g∈G, where e∈Gis the identity element,
+cis a constant and d is a metric on G.
+Introduction
+LetYbe a non-empty set. A function ψ:Y×Y→Cis called a Schur
+multiplier if for every operator A= (ax,y)x,y∈Y∈B(ℓ2(Y)) the matrix
+(ψ(x,y)ax,y)x,y∈Yagain represents an operator from B(ℓ2(Y)) (this opera-
+tor is denoted by MψA). Ifψis a Schur multiplier it follows easily from the
+closed graph theorem that Mψ∈B(B(ℓ2(Y))), and one refers to /bardblMψ/bardblas
+theSchur norm ofψand denotes it by /bardblψ/bardblS.
+LetGbe a locally compact group. In [Her74], Herz introduced a cla ss of
+functions on G, which was later denoted the class of Herz–Schur multipliers
+onG. By the introduction to [BF84], a continuous function ϕ:G→Cis a
+Herz–Schur multiplier if and only if the function
+(0.1) ˆ ϕ(x,y) =ϕ(y−1x) (x,y∈G)
+∗Partially supported by the Ph.D.-school OP–ALG–TOP–GEO.
+1is a Schur multiplier, and the Herz–Schur norm ofϕis given by
+/bardblϕ/bardblHS=/bardblˆϕ/bardblS.
+In [DCH85] De Cannière and Haagerup introduced the Banach al gebra
+MA(G) ofFourier multipliers ofG, consisting of functions ϕ:G→Csuch
+that
+ϕψ∈A(G) (ψ∈A(G)),
+whereA(G) is the Fourier algebra ofGas introduced by Eymard in [Eym64]
+(the Fourier–Stieltjes algebra B(G) ofGis also introduced in this paper).
+The norm of ϕ(denoted /bardblϕ/bardblMA(G)) is given by considering ϕas an operator
+onA(G). According to [DCH85, Proposition 1.2] a Fourier multipli er ofG
+can also be characterized as a continuous function ϕ:G→Csuch that
+λ(g)Mϕ/ma√sto→ϕ(g)λ(g) (g∈G)
+extends to a σ-weakly continuous operator (still denoted Mϕ) on the group
+von Neumann algebra ( λ:G→B(L2(G)) is the left regular representation
+and the group von Neumann algebra is the closure of the span of λ(G) in the
+weak operator topology). Moreover, one has /bardblϕ/bardblMA(G)=/bardblMϕ/bardbl. The Banach
+algebraM0A(G) ofcompletely bounded Fourier multipliers ofGconsists of
+the Fourier multipliers of G,ϕ, for which Mϕis completely bounded. In this
+case they put /bardblϕ/bardblM0A(G)=/bardblMϕ/bardblcb.
+In [BF84] Bożejko and Fendler show that the completely bound ed Fourier
+multipliers coincide isometrically with the continuous He rz–Schur multipli-
+ers. In [Jol92] Jolissaint gives a short and self-contained proof of the result
+from [BF84] in the form stated below.
+0.1 Proposition ([BF84], [Jol92]) .LetGbe a locally compact group and
+assume that ϕ:G→Candk≥0are given, then the following are equivalent:
+(i)ϕis a completely bounded Fourier multiplier of Gwith /bardblϕ/bardblM0A(G)≤k.
+(ii)ϕis a continuous Herz–Schur multiplier on Gwith /bardblϕ/bardblHS≤k.
+(iii) There exists a Hilbert space Hand two bounded, continuous maps
+P,Q :G→Hsuch that
+ϕ(y−1x) =/a\}bracketle{tP(x),Q(y)/a\}bracketri}ht (x,y∈G)
+and
+/bardblP/bardbl∞/bardblQ/bardbl∞≤k,
+where
+/bardblP/bardbl∞= sup
+x∈G/bardblP(x)/bardbland /bardblQ/bardbl∞= sup
+y∈G/bardblQ(y)/bardbl.
+2By a representation (π,H) of a locally compact group Gon a Hilbert
+space Hwe mean a homomorphism of Ginto the invertible elements of
+B(G). A representation ( π,H) ofGis said to be uniformly bounded if
+sup
+g∈G/bardblπ(g)/bardbl<∞
+and one usually writes /bardblπ/bardblfor supg∈G/bardblπ(g)/bardbl. Ifg/ma√sto→π(g) is continuous with
+respect to the strong operator topology on B(G) then we say that ( π,H) is
+strongly continuous . Let (π,H) be a strongly continuous, uniformly bounded
+representation of Gthen, according to [DCH85, Theorem 2.2], any coefficient
+of(π,H) is a continuous Herz–Schur multiplier, i.e.,
+gϕ/ma√sto→ /a\}bracketle{tπ(g)ξ,η/a\}bracketri}ht (g∈G)
+is a continuous Herz–Schur multiplier with
+/bardblϕ/bardblM0A(G)≤ /bardblπ/bardbl2/bardblξ/bardbl/bardblη/bardbl
+for anyξ,η∈H(note that this result also follows as a corollary to Pro-
+position 0.1). U. Haagerup has shown that on the non-abelian free groups
+there are Herz–Schur multipliers which can not be realized a s coefficients of
+uniformly bounded representations. The proof by Haagerup h as remained un-
+published, but Pisier has later given a different proof, cf. [ Pis05]. Haagerup’s
+proof can be modified to prove the corresponding result for th e connected,
+real rank one, simple Lie groups with finite center, cf. [Ste0 9, Theorem 3.6].
+Strictly speaking, the requirement that the above represen tations be uni-
+formly bounded is not fully needed in order to construct a con tinuous Herz–
+Schur multiplier. From Proposition 0.1 it follows that it is enough to require
+that
+(0.2) sup
+x∈G/bardblπ(x)ξ/bardbl<∞and sup
+y∈G/bardblπ(y−1)∗η/bardbl<∞.
+In [BF91, Theorem 1.1] Bożejko and Fendler shows that, for co untable dis-
+crete groups, all Herz–Schur multipliers can be realized as coefficients of
+representations satisfying a condition similar to (0.2). M ore specifically, they
+show that if ϕis a Hermitian Herz–Schur multiplier on a countable discret e
+group Γ, then there exists a representation ( π,H) and vectors ξ,η∈H
+such that
+ϕ(y−1x) =/a\}bracketle{tπ(x)ξ,π(y)η/a\}bracketri}ht (x,y∈Γ),
+with
+sup
+x∈Γ/bardblπ(x)ξ/bardbl ≤ /bardblϕ/bardbl1
+2
+M0A(Γ)and sup
+y∈Γ/bardblπ(y)η/bardbl ≤ /bardblϕ/bardbl1
+2
+M0A(Γ).
+3Furthermore, they give a quantitative bound on /bardblπ(g)/bardblforg∈Γ and note
+that the same holds for non-Hermitian Herz–Schur multiplie rs by including a
+square root two factor in the bound for supx∈Γ/bardblπ(x)ξ/bardbland supy∈Γ/bardblπ(y)η/bardbl.
+In section 1 we present a generalization of the result by Boże jko and Fendler
+to second countable, locally compact groups (cf. Theorem 1. 1 and Corol-
+lary 1.4). We also present the following theorem based on cri terion (0.2)
+(cf. Theorem 1.5):
+0.2 Theorem. LetGbe a second countable, locally compact group and let d
+be a proper, left invariant metric on Gwhich has at most exponential growth ,
+i.e.,
+µ(Bn(e))≤a·ebn(n∈N)
+for some constants a,b> 0, whereµis a left invariant Haar measure on G
+andBn(e) ={g∈G: d(g,e)< n}is the open ball of radius ncentered
+around the identity element e∈G. Then for any continuous Herz–Schur
+multiplierϕonGthere exists a strongly continuous representation (π,H)
+and vectors ξ,η∈Hsuch that
+ϕ(y−1x) =/a\}bracketle{tπ(x)ξ,π(y−1)∗η/a\}bracketri}ht (x,y∈G),
+with
+sup
+x∈G/bardblπ(x)ξ/bardbl=/bardblϕ/bardbl1
+2
+M0A(G)and sup
+y∈G/bardblπ(y−1)∗η/bardbl=/bardblϕ/bardbl1
+2
+M0A(G).
+Moreover, for every fixed c>b ,(π,H)can be chosen such that
+/bardblπ(g)/bardbl ≤ec
+2·d(g,e)(g∈G).
+Note that the existence of a proper, left invariant metric wi th at most ex-
+ponential growth on a second countable, locally compact gro up is guarantied
+by [HP06].
+1 Coefficients of non-uniformly bounded re-
+presentations
+Second countability guarantees the existence of a proper, l eft invariant met-
+ric, cf. [Str74, Theorem]. Actually, according to Haagerup and Przybyszew-
+ska (cf. [HP06]) one can choose this metric, d, to have at most exponential
+growth , i.e.,
+(1.1) µ(Bn(e))≤a·ebn(n∈N)
+4for some constants a,b > 0, whereµis a left invariant Haar measure on G
+andBn(e) ={g∈G: d(g,e)< n}is the open ball of radius ncentered
+around the identity element e∈G.
+Inspired by the proof of [BF91, Theorem 1.1], we state and pro ve Theo-
+rem 1.1 for Hermitian Herz–Schur multipliers, i.e., Herz–Schur multipliers ϕ
+for whichϕ∗=ϕ, where
+ϕ∗(g) =ϕ(g−1) (g∈G).
+The general (not necessarily Hermitian) case is treated in C orollary 1.4 and
+Theorem 1.5.
+1.1 Theorem. Ifϕis a continuous Hermitian Herz–Schur multiplier on a
+second countable, locally compact group G, and dis a proper, left invari-
+ant metric on Gsatisfying (1.1) for somea,b > 0(which exists according
+to [HP06]), then there exists a strongly continuous represe ntation (π,H)and
+vectorsξ,η∈Hsuch that
+ϕ(y−1x) =/a\}bracketle{tπ(x)ξ,π(y)η/a\}bracketri}ht (x,y∈G),
+with
+sup
+x∈G/bardblπ(x)ξ/bardbl=/bardblϕ/bardbl1
+2
+M0A(G)and sup
+y∈G/bardblπ(y)η/bardbl=/bardblϕ/bardbl1
+2
+M0A(G).
+Moreover, for every fixed c>b ,(π,H)can be chosen such that
+/bardblπ(g)/bardbl ≤ec
+2·d(g,e)(g∈G).
+Before we proceed with the proof of Theorem 1.1 we need the fol lowing
+application of the above mentioned result from [HP06], whic h was commu-
+nicated to us by Haagerup.
+1.2 Lemma. IfGis a second countable, locally compact group, then there
+exists a positive function h∈L1(G)with/bardblh/bardbl1= 1, and a positive function c
+onGsuch that
+1
+c(g)/integraldisplay
+Gf(z)h(z)dµ(z)≤/integraldisplay
+Gf(z)h(gz)dµ(z)≤c(g)/integraldisplay
+Gf(z)h(z)dµ(z)
+forg∈Gand any positive f∈L∞(G), whereµis the Haar measure on G.
+Moreover, we may use
+h(g) =e−c·d(g,e)
+/integraltext
+Ge−c·d(x,e)dµ(x)andc(g) =ec·d(g,e)(g∈G)
+forc>b when dis a proper, left invariant metric on Gsatisfying (1.1).
+5Proof. Letµbe a left invariant Haar measure on Gand let d be a proper,
+left invariant metric on Gsatisfying (1.1). We claim that
+0</integraldisplay
+Ge−c·d(g,e)dµ(g)<∞.
+PutE1=B1(e) and define inductively
+En=Bn(e)\Bn−1(e) (n≥2).
+ThenG=/unionsqtext∞
+n=1Enand
+e−cn≤e−c·d(g,e)≤e−c(n−1)(g∈En).
+Hence,
+/integraldisplay
+Ge−c·d(g,e)dµ(g) =∞/summationdisplay
+n=1/integraldisplay
+Ene−c·d(g,e)dµ(g)
+≤∞/summationdisplay
+n=1e−c(n−1)µ(En)
+≤ec∞/summationdisplay
+n=1e−cnµ(Bn(e))
+≤aec∞/summationdisplay
+n=1e(b−c)n
+<∞
+becausec>b .
+By the reverse triangle inequality we see that
+|d(z,g−1)−d(z,e)| ≤d(e,g−1) (g,z∈G).
+Using left invariance of the metric one finds that
+|d(gz,e)−d(z,e)| ≤d(g,e) (g,z∈G).
+This implies
+1
+c(g)e−c·d(z,e)≤e−c·d(gz,e)≤c(g)e−c·d(z,e)(g,z∈G),
+which is easily seen to complete the proof.
+61.3 Lemma. Assume that His a Hilbert space and that R:G→His
+bounded and continuous. Let R′:G→L2(G,H,µ)be given by
+R′(x)(z) =/radicalBig
+h(z)R(z−1x) (z∈G)
+forx∈G, thenR′is bounded and continuous, with /bardblR′(x)/bardbl2≤ /bardblR/bardbl∞for
+allx∈G. Also, let KR=span{R′(x) :x∈G}be a sub-Hilbert space of
+L2(G,H,µ). Then there exists a unique representation (πR,KR)such that
+πR(g)R′(x) =R′(gx) (g,x∈G).
+Moreover,
+/bardblπR(g)/bardbl ≤ec
+2·d(g,e)(g∈G)
+and the representation is strongly continuous.
+Proof. From Lebesgue’s dominated convergence theorem it easily fo llows that
+R′is continuous. To see that R′is bounded, note that
+/bardblR′(x)/bardbl2
+2=/integraldisplay
+Gh(z)/bardblR(z−1x)/bardbl2dµ(z)≤ /bardblR/bardbl2
+∞ (x∈G).
+Ifn∈N,x1,...,xn∈Gandc1,...,cn∈C, then Lemma 1.2 implies that
+/integraldisplay
+G/bardbln/summationdisplay
+i=1ciR(z−1xi)/bardbl2h(gz)dµ(z)≤c(g)/integraldisplay
+G/bardbln/summationdisplay
+i=1ciR(z−1xi)/bardbl2h(z)dµ(z)
+forg∈G, where
+c(g) =ec·d(g,e)(g∈G).
+It follows that
+/bardbln/summationdisplay
+i=1ciR′(gxi)/bardbl2
+2≤c(g)/bardbln/summationdisplay
+i=1ciR′(xi)/bardbl2
+2 (g∈G),
+from which we conclude that there exists a unique representa tion (πR,KR)
+ofGsuch that
+πR(g)R′(x) =R′(gx) (g,x∈G).
+Furthermore,
+/bardblπR(g)/bardbl ≤/radicalBig
+c(g) (g∈G).
+We proceed to show that the representation is strongly conti nuous. Since
+span{R′(x) :x∈G}is total in KRand/bardblπR(g)/bardbl ≤/radicalBig
+c(g), whereg/ma√sto→/radicalBig
+c(g)
+is a continuous function, it is enough to show that
+limn→∞πR(gn)R′(x) =R′(x)
+forx∈G, when (gn)n∈Nis a sequence converging to the identity e∈G(since
+Gis second countable we do not have to consider nets). But πR(gn)R′(x) =
+R′(gnx) so this follows simply from continuity of R′.
+7Proof of Theorem 1.1. Assume that ϕis a continuous Hermitian Herz–Schur
+multiplier and use Proposition 0.1 to find a Hilbert space Hand bounded,
+continuous maps P,Q :G→Hsuch that
+ϕ(y−1x) =/a\}bracketle{tP(x),Q(y)/a\}bracketri}ht (x,y∈G)
+and
+/bardblP/bardbl∞=/bardblQ/bardbl∞=/bardblϕ/bardbl1
+2
+M0A(G).
+Define
+a±(x,y) =1
+4/a\}bracketle{tP(x)±Q(x),P(y)±Q(y)/a\}bracketri}ht (x,y∈G).
+This gives rise to two positive definite, bounded kernels on G×Gsatisfying
+a+(x,y)−a−(x,y) =1
+2ϕ(y−1x) +1
+2ϕ∗(y−1x) =ϕ(y−1x) (x,y∈G)
+and
+a+(x,x) +a−(x,x) =1
+2/bardblP(x)/bardbl2+1
+2/bardblQ(x)/bardbl2≤ /bardblϕ/bardblM0A(G) (x∈G).
+Let
+h(g) =e−c·d(g,e)
+/integraltext
+Ge−c·d(x,e)dµ(x)(g∈G)
+for somec>b , when d is a proper, left invariant metric on Gsatisfying (1.1)
+(cf. Lemma 1.2). Define ( P±Q)′:G→L2(G,H,µ) by
+(P±Q)′(x)(z) =/radicalBig
+h(z)(P±Q)(z−1x) (z∈G)
+forx∈G. According to Lemma 1.3 there exist strongly continuous re-
+presentations ( πP±Q,KP±Q), where KP±Q=span{(P′±Q′)(x) :x∈G}
+andπP±Q(g)(P±Q)′(x) = (P±Q)′(gx) forg,x∈G. Furthermore, these
+representations satisfy
+(1.2) /bardblπP±Q(g)/bardbl ≤ec
+2·d(g,e)(g∈G).
+Put
+A±(x,y) =/a\}bracketle{t(P±Q)′(x),(P±Q)′(y)/a\}bracketri}htKP±Q (x,y∈G).
+ThenA±are positive definite, bounded kernels on G×Gsatisfying
+(1.3) A+(x,y)−A−(x,y) =ϕ(y−1x) (x,y∈G)
+8and
+(1.4) A+(x,x) +A−(x,x)≤ /bardblϕ/bardblM0A(G) (x∈G).
+To make the notation less cumbersome, let π±=πP±QandK±=KP±Qand
+defineξ±= (P±Q)′(e). Notice that
+/a\}bracketle{tπ±(x)ξ±,π±(y)ξ±/a\}bracketri}htK±=A±(x,y) (x,y∈G),
+and that (1.2) now reads
+/bardblπ±(g)/bardbl ≤ec
+2·d(g,e)(g∈G).
+Put
+K=K+⊕K−, ξ =ξ+⊕ξ−, η =ξ+⊕ −ξ−andπ=π+⊕π−.
+Observe that πis a strongly continuous representation such that
+(1.5) /bardblπ(g)/bardbl ≤ec
+2·d(g,e)(g∈G)
+and
+/a\}bracketle{tπ(x)ξ,π(y)η/a\}bracketri}htK=ϕ(y−1x) (x,y∈G).
+Finally, observe that
+/bardblπ(x)ξ/bardbl2=/bardblπ+(x)ξ+/bardbl2+/bardblπ−(x)ξ−/bardbl2=A+(x,x) +A−(x,x)≤ /bardblϕ/bardblM0A(G)
+forx∈G, and similarly
+/bardblπ(y)η/bardbl2≤ /bardblϕ/bardblM0A(G)
+fory∈G. This finishes the proof.
+1.4 Corollary. Ifϕis a continuous Herz–Schur multiplier on a second count-
+able, locally compact group G, and dis a proper, left invariant metric on
+Gsatisfying (1.1) for somea,b > 0(which exists according to [HP06]),
+then there exists a strongly continuous representation (π,H)and vectors
+ξ,η∈Hsuch that
+ϕ(y−1x) =/a\}bracketle{tπ(x)ξ,π(y)η/a\}bracketri}ht (x,y∈G),
+with
+sup
+x∈G/bardblπ(x)ξ/bardbl ≤√
+2/bardblϕ/bardbl1
+2
+M0A(G)and sup
+y∈G/bardblπ(y)η/bardbl ≤√
+2/bardblϕ/bardbl1
+2
+M0A(G).
+Moreover, for every fixed c>b ,(π,H)can be chosen such that
+/bardblπ(g)/bardbl ≤ec
+2·d(g,e)(g∈G).
+9Proof. This follows from Theorem 1.1 since
+ϕ= Re(ϕ) +iIm(ϕ),
+where
+Re(ϕ) =ϕ+ϕ∗
+2and Im(ϕ) =ϕ−ϕ∗
+2i
+are continuous Hermitian Herz–Schur multipliers with
+/bardblRe(ϕ)/bardblM0A(G)≤ /bardblϕ/bardblM0A(G)and /bardblIm(ϕ)/bardblM0A(G)≤ /bardblϕ/bardblM0A(G).
+1.5 Theorem. Ifϕis a continuous Herz–Schur multiplier on a second count-
+able, locally compact group G, and dis a proper, left invariant metric on
+Gsatisfying (1.1) for somea,b > 0(which exists according to [HP06]),
+then there exists a strongly continuous representation (π,H)and vectors
+ξ,η∈Hsuch that
+ϕ(y−1x) =/a\}bracketle{tπ(x)ξ,π(y−1)∗η/a\}bracketri}ht (x,y∈G),
+with
+sup
+x∈G/bardblπ(x)ξ/bardbl=/bardblϕ/bardbl1
+2
+M0A(G)and sup
+y∈G/bardblπ(y−1)∗η/bardbl=/bardblϕ/bardbl1
+2
+M0A(G).
+Moreover, for every fixed c>b ,(π,H)can be chosen such that
+/bardblπ(g)/bardbl ≤ec
+2·d(g,e)(g∈G).
+Proof. Assume that ϕis a continuous Herz–Schur multiplier and use Propo-
+sition 0.1 to find a Hilbert space Hand bounded, continuous maps P,Q :
+G→Hsuch that
+ϕ(y−1x) =/a\}bracketle{tP(x),Q(y)/a\}bracketri}ht (x,y∈G)
+and
+/bardblP/bardbl∞=/bardblQ/bardbl∞=/bardblϕ/bardbl1
+2
+M0A(G).
+Let
+h(g) =e−c·d(g,e)
+/integraltext
+Ge−c·d(x,e)dµ(x)(g∈G)
+for somec>b , when d is a proper, left invariant metric on Gsatisfying (1.1)
+(cf. Lemma 1.2). Define P′,Q′:G→L2(G,H,µ) by
+P′(x)(z) =/radicalBig
+h(z)P(z−1x) andQ′(y)(z) =/radicalBig
+h(z)Q(z−1y) (z∈G)
+10forx,y ∈G. According to Lemma 1.3 there exists a strongly contin-
+uous representation ( πP,KP), where KP=span{P′(x) :x∈G}and
+πP(g)P′(x) =P′(gx) forg,x∈G. Furthermore, this representation satisfies
+/bardblπP(g)/bardbl ≤ec
+2·d(g,e)(g∈G).
+Observe that
+/bardblP′(x)/bardbl2
+2,/bardblQ′(y)/bardbl2
+2≤ /bardblϕ/bardblM0A(G)
+and
+/a\}bracketle{tP′(x),Q′(y)/a\}bracketri}htL2(G,H,µ)=/integraldisplay
+Gh(z)/a\}bracketle{tP(z−1x),Q(z−1y)/a\}bracketri}htHdµ(z) =ϕ(y−1x)
+forx,y∈G. Putξ=P′(e) andη=PKPQ′(e), wherePKPis the orthogonal
+projection on KP. Note that ξ,η∈KPand
+ϕ(y−1x) =/a\}bracketle{tπP(y−1x)ξ,η/a\}bracketri}htKP=/a\}bracketle{tπP(x)ξ,πP(y−1)∗η/a\}bracketri}htKP (x,y∈G).
+It is clear that /bardblπP(x)ξ/bardbl2
+KP=/bardblP′(x)/bardbl2
+2≤ /bardblϕ/bardblM0A(G). The corresponding
+result for /bardblπP(y−1)∗η/bardbl2
+KPrequires more work. For x∈Garbitrary we find
+that
+/a\}bracketle{tπP(y−1)P′(x),PKPQ′(e)/a\}bracketri}htKP=/a\}bracketle{tP′(y−1x),PKPQ′(e)/a\}bracketri}htKP
+=/a\}bracketle{tP′(y−1x),Q′(e)/a\}bracketri}htH
+=ϕ(y−1x)
+=/a\}bracketle{tP′(x),Q′(y)/a\}bracketri}htH
+=/a\}bracketle{tP′(x),PKPQ′(y)/a\}bracketri}htKP,
+from which we conclude that πP(y−1)∗PKPQ′(e) =PKPQ′(y) and therefore
+/bardblπP(y−1)∗η/bardbl2
+KP=/bardblPKPQ′(y)/bardbl2
+KP≤ /bardblQ′(y)/bardbl2
+2≤ /bardblϕ/bardblM0A(G).
+1.6 Remark. For the free group on Ngenerators (2 ≤N < ∞) the constants
+a,bin (1.1) may be chosen as a=N
+(N−1)(2N−1)andb= ln(2N−1). This
+implies that for r>√2N−1, the representations ( π,H) from Theorem 1.1,
+Corollary 1.4 and Theorem 1.5 may be chosen to satisfy /bardblπ(g)/bardbl ≤rd(g,e)for
+allg∈G.
+Acknowledgments
+I would like to thank Ryszard Szwarc and Marek Bożejko for an i nspiring
+stay in Wrocław during March 2009.
+11References
+[BF84] Marek Bożejko and Gero Fendler. Herz–Schur multipli ers and com-
+pletely bounded multipliers of the Fourier algebra of a loca lly com-
+pact group. Boll. Un. Mat. Ital. A (6) , 3(2):297–302, 1984.
+[BF91] Marek Bożejko and Gero Fendler. Herz–Schur multipli ers and uni-
+formly bounded representations of discrete groups. Arch. Math.
+(Basel) , 57(3):290–298, 1991.
+[DCH85] Jean De Cannière and Uffe Haagerup. Multipliers of th e Fourier
+algebras of some simple Lie groups and their discrete subgro ups.
+Amer. J. Math. , 107(2):455–500, 1985.
+[Eym64] Pierre Eymard. L’algèbre de Fourier d’un groupe loc alement com-
+pact. Bull. Soc. Math. France , 92:181–236, 1964.
+[Her74] Carl Herz. Une généralisation de la notion de transf ormée de
+Fourier-Stieltjes. Ann. Inst. Fourier (Grenoble) , 24(3):xiii, 145–
+157, 1974.
+[HP06] Uffe Haagerup and Agata Przybyszewska. Proper metric s on locally
+compact groups, and proper affine isometric actions on Banach
+spaces. 2006, arXiv:math/0606794v1 [math.OA].
+[Jol92] Paul Jolissaint. A characterization of completely bounded multi-
+pliers of Fourier algebras. Colloq. Math. , 63(2):311–313, 1992.
+[Pis05] Gilles Pisier. Are unitarizable groups amenable? I nInfinite
+groups: geometric, combinatorial and dynamical aspects , volume
+248 of Progr. Math. , pages 323–362. Birkhäuser, Basel, 2005.
+[Ste09] Troels Steenstrup. Fourier multiplier norms of sph erical functions
+on the generalized Lorentz groups. 2009, arXiv:0911.4977v1
+[math.RT].
+[Str74] Raimond A. Struble. Metrics in locally compact grou ps.Compo-
+sitio Math. , 28:217–222, 1974.
+Troels Steenstrup <troelssj@imada.sdu.dk>
+Department of Mathematics and Computer Science, Universit y of Southern
+Denmark, Campusvej 55, DK–5230 Odense M, Denmark.
+12
\ No newline at end of file
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+arXiv:1001.0327v1  [nucl-th]  2 Jan 2010Is chiral symmetry manifested in nuclear structure?
+R.J. Furnstahl1and A. Schwenk2,3,4
+1Department of Physics, The Ohio State University, Columbus, OH 43 210, USA
+2TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3, Canada
+3ExtreMe Matter Institute EMMI, GSI, 64291 Darmstadt, German y
+4Institut f¨ ur Kernphysik, TU Darmstadt, 64289 Darmstadt, Ger many
+E-mail:furnstahl.1@osu.edu, schwenk@triumf.ca
+Abstract. Spontaneously broken chiral symmetry is an established property of low-
+energy quantum chromodynamics, but finding direct evidence for it from nuclear
+structure data is a difficult challenge. Indeed, phenomenologically su ccessful energy-
+density functional approaches do not even have explicit pions. Are there smoking guns
+for chiral symmetry in nuclei?
+1. Challenge
+Spontaneously broken chiral symmetry is a key property of quant um chromodynamics
+(QCD) at low energies. It gives rise to pseudo-scalar Goldstone bos ons, the pions, which
+because of non-zero light quark masses are not massless, but are much lighter than other
+hadrons. Fornuclear forces, theimmediate consequence istheon e-pion-exchange (OPE)
+potential with its long range and characteristic tensor structure . In more detail there
+are constraints on the form of interactions in effective low-energy Lagrangians [1, 2, 3].
+Since nuclear structure is ultimately determined by low-energy QCD a nd the nuclear
+Fermi momentum is comparable to 2 mπ, one might expect stark evidence from data for
+the fingerprint of chiral symmetry. Thus we pose the question: Ca n we find smoking
+guns for chiral symmetry in nuclei?
+Because calculations based on chiral symmetry consequences (un derstood broadly
+to range from including OPE to full consistency with chiral symmetry constraints in
+low-energy Lagrangians) are becoming increasingly able to describe nuclear structure,
+one might think the question is moot. But we are not asking here whet her theories
+or models consistent with chiral symmetry are sufficient to describe data, but whether
+the experimental data says this physics is necessary. This is a more difficult question
+and one with some counterexamples to the need for explicit chiral sy mmetry. At one
+extreme, pionless effective field theory (EFT) describes few-body scattering at very low
+energies and nuclear binding up to at least the alpha particle [1, 4]. At the other
+extreme, energy-density functional approaches to medium and h eavy nuclei (such as
+Skyrme) have wide phenomenological success without explicit pions [5 ].Is chiral symmetry manifested in nuclear structure? 2
+In trying to identify manifestations of chiral symmetry, it is importa nt to make the
+distinction between long-range chiral physics and short-range ph ysics, which may be
+free of chiral symmetry constraints. Nuclear structure will in gen eral have contributions
+from both. To find a smoking gun, one will likely need to identify observ ables (or
+combinations of observables) that are sensitive to the long-range parts.
+Because it is spontaneously broken, chiral symmetry most directly relates processes
+with different numbers of pions. Clear signatures of chiral symmetr y, even of the most
+basic pion properties (such as OPE), can become a more subtle ques tion for processes
+with no pions in the initial or final state, which include nuclear structu re properties
+such as binding energies and spectra. The signatures can be partic ularly obscured for
+bulk properties, where the pion effects at the mean-field level may b e washed out by
+spin and isospin averaging.
+2. Smoking guns for chiral symmetry in other contexts
+For our challenge to find smoking guns in nuclear structure we adopt a broad definition
+of evidence for chiral symmetry, which encompasses the quantita tive implications of a
+lightpionanditsquantumnumbers (such asOPE)aswell asthemores ubtle effectsthat
+constrain low-energy chiral Lagrangians. Such smoking guns for s pontaneously broken
+chiral symmetry are readily found in data from other strong intera ction contexts. The
+spectrum of mesons is the most obvious, where m2
+π≪m2
+ρmanifests the approximate
+Goldstone boson nature of the pion and would-be parity doublets (e .g., theρanda1) are
+split in mass. The physics of low-energy π–πandπ–Nscattering, which is governed by
+chiral physics, provide many other examples. Here the successes of chiral perturbation
+theory closely tie experiment and theory through chiral symmetry [6, 7].
+For nucleon-nucleon (NN) interactions, the situation is more subtle . For example,
+the unnaturally large NN scattering lengths are not related to chira l symmetry, but
+dominate the physics at very low energies. However, a compelling pro totype for the sort
+of direct evidence for chiral symmetry we seek from nuclear struc ture data can be found
+in the partial-wave analyses of NN scattering by the Nijmegen group and collaborators.
+This example also lets us define more concretely what we are looking fo r. Simply an
+improved χ2for a fit to NN data using interactions with an input one-pion exchang e
+potential (or two-pion exchange) is not the strong and direct evid ence we prefer. (After
+all, it is possible to fit this data accurately with inverse scattering pot entials with no
+pion exchange component.) Rather it is the unbiased extraction thr ough the fitting
+process of pion parameters that we call a smoking gun.
+As part of the analysis described in Ref. [8], the masses of the charg ed and neutral
+pions were used as fit parameters. The extracted values agreed w ith experiment within
+estimated one percent errors. At the same time the pion-nucleon c oupling was obtained
+from the fit and found consistent with extracted values from π–Nscattering [8, 9]. The
+data provides very clear signals for this coupling; indeed, the ppπ0coupling is said to
+be determined from each individual partial wave except for1S0[8].Is chiral symmetry manifested in nuclear structure? 3
+Subsequent studies looked at the partial wave analysis with a finer m icroscope and
+founddirectevidence oftwo-pionexchange(TPE) physics[10,11]. Fittingthepionmass
+again but now as a parameter in the TPE potential gave agreement w ith experiment
+at the ten percent level. Other evidence includes consistency of th ecicouplings of the
+sub-leading TPE with determinations from π–Nscattering. It is worth noting that
+the sub-leading TPE provides the long-range part of three-nucleo n (3N) forces. Finding
+smoking gunsforchiral symmetry innuclear structure maytheref orebetiedtoincluding
+3N interactions.
+In the example of NN phase shifts, the sensitivity to pion physics can be enhanced
+by concentrating on the peripheral partial waves (e.g., G-waves a nd higher are fully
+explained by pion exchanges). This isolates the long-distance physic s, which is chiral
+physics. Even so, a careful analysis of statistical and systematic errors together with a
+large database was required to cleanly extract the direct evidence for sub-leading chiral
+effects. This highlights the difficulty in meeting the challenge for nuclea r structure,
+where the chiral origin of mid-range physics from TPE may be difficult t o resolve.
+3. Opportunities
+There aremany opportunitiesto buildonexisting effortsinthesearc h for chiral smoking
+guns. The growing applications of chiral EFT to nuclear structure t hrough no-core
+shell model, coupled cluster, and lattice EFT calculations are building f oundations
+for the desired evidence from data and for investigations similar to t he NN partial-
+wave analyses. We emphasize again that while phenomenological succ ess starting from
+microscopic theories incorporating chiral symmetry is gratifying, t his type of indirect
+evidence is not what we are looking for. As the accuracy improves, h owever, we can
+ask whether distinctions that point to pions and then more subtle ch iral symmetry
+constraints will become apparent (as with TPE in the prototype par tial-wave analysis).
+For nuclear structure evidence, many-body force contributions from long-range TPE
+may be key.
+Other possibilities for smoking guns are in shell-model calculations tha t start
+from interactions including pion physics, with monopole (or more) adj ustments fit to
+reproduce spectra and binding energies in a major shell. Do the fits o f shell-model
+interactions favor pion exchanges for the long-range parts? If s o, is it possible to isolate
+data that are most sensitive to the long-range parts (e.g., to cons train their sub-leading
+contributions)?
+A third category may provide the greatest challenge to finding explic it chiral
+symmetry: energy-density functionals (EDF). Accurate calculat ions of binding energies
+and other properties are made across the mass table by EDF’s such as Skyrme or Gogny,
+which do not have explicit pions. Relativistic EDF’s in principle start with p ions but
+they do not contribute at the mean-field level used in relativistic phe nomenology. Naive
+dimensional analysis for fit parameters in EDF’s (Skyrme or covarian t) is suggestive of
+chiral signatures [12, 13] but is not quantitative enough to be conc lusive. We expectIs chiral symmetry manifested in nuclear structure? 4
+generalized EDF’s will be needed to find smoking guns.
+There have been many studies that hint at a special role for chiral s ymmetry in
+nuclear structure. These include a wide array of models that impleme nt a partial
+restoration of chiral symmetry at finite density, in-medium QCD sum rules, and Brown-
+Rho scaling (see for example Refs. [14, 15]). None of these rise as ye t to the level of a
+smokinggunfornuclearstructure; thechallengeistomakethemsu fficientlyquantitative
+and predictive based on data (rather than model input).
+Other direct possibilities for finding smoking guns of chiral symmetry may lie with
+in-medium chiral EFT, which has been under recent development (se e Refs. [3, 16]
+and references therein). There are open questions, such as the convergence of the
+perturbative approach, but it has the explicit chiral ingredients ne eded. We can ask: are
+there unique predictions from long-range pion physics? These may b e difficult to find,
+as illustrated by the subtleties in pinning down the origin of the spin-or bit splittings in
+nuclei. One might hope for the same type fitting as in NN phase shifts. A possibility
+already being considered is the fitting of energy-density functiona ls that have been
+supplemented with long-range chiral EFT physics. This would mean let ting the pion
+parameters float in the fits in the same way as in the Nijmegen phase s hift analyses.
+The answer is as yet unknown to even the most basic question: Do me dium and heavy
+nuclei know the pion mass and coupling?
+What can we say in general about where one should look for chiral sy mmetry
+signatures? Searching in bulk properties will be difficult because of th e averaging
+of pion contributions and the important role of short-range physic s. The task is to
+identify observables (analogous to peripheral partial waves) tha t isolate contributions
+from long-range pion physics. Past suggestions have included single -particle energies,
+where the pion tensor force may be isolated; isotopechains, where small differences from
+the isospin dependence due to pion exchanges including long-range 3 N forces may be
+amplified; collective excitations with pionic quantum numbers; electro weak axial-charge
+transitions and properties of pionic atoms (see for example Refs. [1 6, 17, 18]).
+4. Final comments
+Our challenge to nuclear structure theorists is to search for directmanifestations
+of chiral symmetry in nuclei. Success in this quest will help to unify des criptions
+of strong-interaction phenomena, identify sensitivities that can s uggest profitable
+new experiments, guide microscopic calculations of nuclear propert ies toward greater
+precision, and foreshadow what to expect under extreme conditio ns. It can also lead to
+ties to lattice QCD, whose growing successes give rise to dreams of d irect calculations
+of the lightest nuclei, enabling new insights to chiral symmetry in nucle ar structure.
+However, the challenge to find smoking guns can also be constructiv ely answered in a
+negative way, by showing that explicit chiral symmetry is notnecessary for particular
+nuclear structure observables. So an alternative question to pos e is: where are effective
+field theories of nuclei without pionic degrees of freedom applicable?Is chiral symmetry manifested in nuclear structure? 5
+Acknowledgments
+This work was supported in part by the National Science Foundation under Grant
+No. PHY–0653312, the UNEDF SciDAC Collaboration under DOE Grant DE-FC02-
+07ER41457, the Natural Sciences and Engineering Research Coun cil of Canada
+(NSERC), and by the Helmholtz Alliance Program of the Helmholtz Asso ciation,
+contract HA216/EMMI “Extremes of Density and Temperature: C osmic Matter in the
+Laboratory”. TRIUMF receives federal funding via a contribution agreement through
+the National Research Council of Canada.
+References
+[1] P. F. Bedaque and U. van Kolck. Effective field theory for few-nu cleon systems. Ann. Rev. Nucl.
+Part. Sci. , 52:339, 2002.
+[2] E. Epelbaum. Few-nucleon forces and systems in chiral effective field theory. Prog. Part. Nucl.
+Phys., 57:654, 2006.
+[3] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner. Modern theory of nuclear forces. Rev. Mod.
+Phys., 81:1773, 2009.
+[4] L. Platter. Few-body systems and the pionless effective field the ory. arXiv:0910.0031.
+[5] M. Bender, P.-H. Heenen, and P.-G. Reinhard. Self-consistent m ean-field models for nuclear
+structure. Rev. Mod. Phys. , 75:121, 2003.
+[6] J. Gasser and H. Leutwyler. Chiral perturbation theory to one loop.Ann. Phys. , 158:142, 1984.
+[7] V. Bernard and U.-G. Meißner. Chiral perturbation theory. Ann. Rev. Nucl. Part. Sci. , 57:33,
+2007.
+[8] V. G. J. Stoks, R. Timmermans, and J. J. de Swart. On the pion–n ucleon coupling constant.
+Phys. Rev. C , 47:512, 1993.
+[9] U. van Kolck, M. C. M. Rentmeester, J. L. Friar, J. T. Goldman, a nd J. J. de Swart.
+Electromagneticcorrectionsto the one-pion-exchangepotentia l.Phys. Rev. Lett. , 80:4386,1998.
+[10] M. C. M. Rentmeester, R. G. E. Timmermans, J. L. Friar, and J. J. de Swart. Chiral two-pion
+exchange and proton-proton partial-wave analysis. Phys. Rev. Lett. , 82:4992, 1999.
+[11] M. C. M. Rentmeester, R. G. E. Timmermans, and J. J. de Swart . Determination of the chiral
+coupling constants c3andc4in new proton-proton and neutron-proton partial-wave analyses .
+Phys. Rev. C , 67:044001, 2003.
+[12] J. L. Friar, D. G. Madland, and B. W. Lynn. QCD scales in finite nuc lei.Phys. Rev. C , 53:3085,
+1996.
+[13] R. J. Furnstahl. Neutron radii in mean-field models. Nucl. Phys. A , 706:85, 2002.
+[14] U. Mosel. Chiral symmetry in nuclei – Theoretical expectations and hard facts. Mod. Phys. Lett.
+A, 23:2371, 2008.
+[15] M. C. Birse. Chiral symmetry in nuclei: Partial restoration and it s consequences. J. Phys. G ,
+20:1537, 1994.
+[16] W. Weise. Yukawa’s pion, low-energy QCD and nuclear chiral dyna mics.Prog. Theor. Phys.
+Suppl., 170:161, 2007.
+[17] G. E. Brown and M. Rho. On the manifestation of chiral symmetr y in nuclei and dense nuclear
+matter. Phys. Rept. , 363:85, 2002.
+[18] T. Otsuka, T. Suzuki, J. D. Holt, A. Schwenk, and Y. Akaishi. Th ree-body forces and the limit
+of oxygen isotopes, arXiv:0908.2607.
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+arXiv:1001.0328v1  [nucl-th]  2 Jan 2010How should one formulate, extract, and interpret
+‘non-observables’ for nuclei?
+R.J. Furnstahl1and A. Schwenk2,3,4
+1Department of Physics, The Ohio State University, Columbus, OH 43 210, USA
+2TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3, Canada
+3ExtreMe Matter Institute EMMI, GSI, 64291 Darmstadt, German y
+4Institut f¨ ur Kernphysik, TU Darmstadt, 64289 Darmstadt, Ger many
+E-mail:furnstahl.1@osu.edu, schwenk@triumf.ca
+Abstract. Nuclear observables such as binding energies and cross sections ca n be
+directly measured. Other physically useful quantities, such as spe ctroscopic factors,
+are related to measured quantities by a convolution whose decompo sition is not
+unique. Can a framework for these nuclear structure ‘non-obser vables’ be formulated
+systematicallysothat they canbe extractedfromexperiment with knownuncertainties
+and calculated with consistent theory? Parton distribution functio ns in hadrons serve
+as an illustrative example of how this can be done. A systematic frame work is also
+needed to address questions of interpretation, such as whether short-rangecorrelations
+are important for nuclear structure.
+1. Nuclear observables and non-observables
+In quantum mechanics, an observable is typically defined as a physica l property
+that can be measured. The usual examples include energy, moment um, and angular
+momentum. Butwhataboutfamiliarnuclearquantitiessuchasmomen tumdistributions
+and spectroscopic factors? Although one often says that momen tum distributions are
+“measured”, in fact they must be extracted from data. The gene ral structure is that a
+measured quantity such as a cross section is decomposed as a conv olution of subsidiary
+pieces, usually based on a factorization principle. This decomposition is not unique, and
+sowereferheretotheextractedquantitiesas‘non-observables ’. Thequotesareintended
+to soften the implication that it is improper to talk about them; never theless, unless the
+conventions (e.g., scale and scheme dependence) are controlled an d specified, there will
+be ambiguities that will be entangled with the structure and reaction approximations.
+The challenge is to formulate and carry out experimental extractio ns and theoretical
+calculations of non-observables systematically and consistently.
+The theoretical ambiguities for non-observables that take the fo rm of a matrix
+element of an operator O(e.g., the momentum distribution O(k) =a†
+kak) are manifested
+by considering unitary transformations. We are free to apply a (sh ort-range) unitary‘Non-observables’ for nuclei 2
+transformation UtoOand to states |Ψn/angbracketrightwith the result that matrix elements are
+invariant:
+Omn≡ /angbracketleftΨm|O|Ψn/angbracketright= (/angbracketleftΨm|U†)UOU†(U|Ψn/angbracketright) =/angbracketleft/tildewideΨm|/tildewideO|/tildewideΨn/angbracketright. (1)
+But the matrix elements of Oitself between the transformed states are in general
+modified (unless Ois a conserved charge):
+/tildewideOmn≡ /angbracketleft/tildewideΨm|O|/tildewideΨn/angbracketright /negationslash=Omn. (2)
+In a low-energy effective theory, there is no preferred set of sta tes (equivalently, there
+is no preferred Hamiltonian) so transformations that modify short -range unresolved
+physics generate equally acceptable states. This is for example disc ussed in the context
+ofnuclearforcesinRef.[1]. Thustotheextentthatdifferencesbe tweenOmnand/tildewideOmnare
+important(whichwilldependontheprocessandkinematics), oneca nnotunambiguously
+refer to one as a measured quantity without specifying the basis st ates used and
+the corresponding Hamiltonian (including regularization and renorma lization scheme). ‡
+The same considerations apply to any sort of wave function overlap . A framework to
+specify the relevant conventions and relate different choices is well developed for parton
+distribution functions but its counterparts do not yet exist for nu clear non-observables.
+The challenge is to formulate such frameworks.
+These considerations are critically important for non-observables such as nuclear
+spectroscopic factors, which are extracted from experimental measurements and used to
+compare data to data and data to theory. In particular cases the extracted values may
+vary from essentially convention independence to highly scheme dep endent; precision
+analysis of nuclear data requires a rigorous framework to assess t hese uncertainties and
+allow robust comparisons. Interpreting the physics implications of n on-observables also
+requires knowing how they evolve with transformations of nuclear H amiltonians that
+change the resolution.
+2. Nuclear and other examples
+In nuclear physics, there is a wealth of physics involved in analyzing sp ectroscopic
+factors. Examples includecomparingtheratioofmeasuredtocalcu latedknock-outcross
+sections as a function of isospin; focusing on long-range characte rizations of states, such
+as the Hoyle state as a triple-alpha cluster state (in a different basis this state may look
+very complicated); or using spectroscopic factors as a concept t o gain understanding
+of the nature of single-particle dominated versus more complex sta tes in nuclei. For a
+proper comparison and rigorous interpretation of spectroscopic factors it is important
+to have control over their extraction and calculation.
+We can gain insight into the issues with spectroscopic factors by con sidering the
+simplest nucleus, the deuteron. The D-state probability of the deu teron is a non-
+observable; it depends on the short-range tensor strength tha t changes under unitary
+‡The field-theoretic counterpart to this argument in terms of field r edefinitions is presented in the
+context of momentum distributions and occupation numbers in Refs . [2, 3].‘Non-observables’ for nuclei 3
+transformationsof nuclear forcesandtheassociated transfor mationsofbasis states. The
+D-state probability is a spectroscopic factor for the D-state par t of the wave function,
+so this is an example of a general model dependence of spectrosco pic factors. The
+deuteron example, with the resolution dependence of the S- and D- state probabilities
+(shown, e.g., in Fig. 57 of Ref. [1]), demonstrates that there may be a large theoretical
+uncertainty(roughlyafactoroftwofortheD-statedeuteronp robability)associatedwith
+spectroscopic factors that are sensitive to short-range parts , and a significant theoretical
+uncertainty (of order 10% for the S-state deuteron probability) for spectroscopic factors
+that probe mainly the long-range parts of nuclear forces.
+In addition, spectroscopic factors rely on a convention for the lon g-range parts in
+nuclear Hamiltonians, such as pion exchanges or a pionless theory. B oth describe very
+low-energy scattering with similar accuracy, but the correspondin g long-range parts of
+wave functions will differ. We stress the importance of separating m idrange physics
+from short-range contributions in the discussion of spectroscop ic factors. For example,
+midrange will receive contributions from one- and multi-pion physics in chiral effective
+field theory (EFT).
+In systems with a large separation of scale, spectroscopic factor s are effectively
+measurable (that is, there are only small systematic uncertainties ) because the unitary
+transformations that change spectroscopic factors lead to shif ts of order ( kR)n, with
+kR≪1 (where kis a typical momentum scale and Rdenotes the range of interactions).
+An excellent example is a precision measurement of the closed-chann el fraction of
+cold atom pairs across a broad Feshbach resonance, obtained fro m radio-frequency
+spectroscopy where initial and final state interactions are negligib le [4]. Similarly, the
+momentum distribution for strongly interacting cold atom gases nea r the unitarity limit
+follows a universal power law 1 /k4ask→ ∞as long as kR≪1. As for cold atoms
+near unitarity, when there is a large separation between the energ y scales of interactions
+and the excitation of the composite particles, the impulse approxima tion will be good
+and the separation of the momentum distribution from experimenta l data will also be
+clean. But this is not the case for nuclei, which therefore require gr eater care.
+3. Parton distribution functions as a paradigm
+We propose the framework of parton distribution functions (PDF’s ) as a paradigm for
+nuclear non-observables. The general scenario is that experimen tal cross sections are
+expressed as a well-defined convolution. The PDF analysis is based on the expectation
+that part of the convolution can be calculated reliably for given expe rimental conditions
+so that the remaining part can be extracted as a universal quantit y, which can then
+be related to other processes and kinematic conditions. In the cas e of hard-scattering
+processes with a large momentum transfer scale Q, factorization allows a separation
+of the momentum and distance scales in the reaction. (In short, th e time scale for
+binding interactions in the rest frame are time dilated in the center-o f-mass frame, so
+the interaction of an electron with a hadron in deep-inelastic scatte ring is with single‘Non-observables’ for nuclei 4
+non-interacting partons.) The short-distance part can be calcula ted systematically in
+low-order perturbative QCD and the long-distance part identified a s PDF’s, which are
+basically momentum distributions for partons (i.e., quarks and gluons ) in hadrons.
+The PDF’s are typically extracted from global fits to experimental d ata. Related
+quantities to PDF’s are fragmentation functions, parton distribut ion amplitudes,
+generalized parton densities, which all summarize the universal non -perturbative parts
+of the physics. Since they are universal, the same PDF’s appear in all reactions, so once
+determined they can relate processes such as the deep inelastic sc attering of leptons,
+Drell-Yan, jet production, and more. Thus one can measure them in a limited set
+of reactions and then perturbative calculations of hard scatterin g and PDF evolution
+enable first principles predictions of cross sections for other proc esses [5].
+The rigorousframework developed for PDF’sand related functions offers important
+lessons for analogous low-energy nuclear quantities. The momentu m distribution for a
+given hadron is not unique: there is dependence on Q2, which serves as the resolution
+scale and can be changed by renormalization group (RG) evolution, a nd the PDF
+analysis at next-to-leading order must be performed in a specific re normalization and
+factorization scheme (typical choices are MS and DIS) [6]. To mainta in consistency,
+any hard-scattering cross section calculations that are used for the input processes or
+that use the extracted PDF’s have to be implemented with the samescheme. There is
+careful treatment of the uncertainties in the PDF’s. It is notconsidered sufficient to just
+compare different extractions. Instead, the Lagrange Multiplier a nd Hessian techniques
+have been developed to estimate PDF uncertainties (see Ref. [6] fo r brief explanations
+and original references). Finally, in expressing a PDF in terms of a qu ark correlation
+function there are non-trivial details that must be done correctly ; as emphasized by
+Collins [5], naive factorization is not adequate.
+Each of these features has an analog in calculating spectroscopic f actors and other
+nuclear non-observables. Can we formulate our theory to have th e same control as with
+PDF’s using factorization? An underlying question is whether the nec essary ingredients
+(corresponding to asymptotic freedom, infra-red safety, and f actorization for PDF’s, see
+Ref. [7]) are present. If not, there will be intrinsic limitations that ha ve to be quantified.
+The seeds for such a framework have been under development for some years. These
+include EFT methods to consistently calculate structure and opera tors [8] and RG
+methods that can change the resolution [1]. Extending these metho ds to the problems
+discussed here is an important open task for nuclear theory.
+4. Interpreting non-observables for nuclei
+A rigorous framework for extracting nuclear non-observables is n eeded to offer clear
+interpretations. A case in point are recent experiments that are in terpreted as
+providing definitive evidence for the effects of short-range corre lations in nuclei. This
+interpretation raises many questions, as these correlations are f eatures of the nuclear
+wave function at short distances, which are particularly dependen t on the choice of the‘Non-observables’ for nuclei 5
+Hamiltonian (e.g., the resolution). How is this physics reconciled with ap proaches using
+chiralEFTorlow-momentuminteractions, forwhichshort-rangec orrelationsaregreatly
+suppressed, or with “mean-field” energy-density functionals? Fo r low-momentum
+interactions, the unitary transformations leave observable cros s sections unchanged by
+construction. Doesonethensimply have different interpretations at different resolutions
+(e.g., simple operator and complicated wave function versus complica ted operator and
+simple wave function) or are there basic limitations to what can be con cluded? Is the
+extraction from experiment better controlled at certain resolutio ns? Does factorization
+work better for a range of resolutions in nuclear Hamiltonians? The c hallenge is to
+develop a systematic framework that can be applied at different res olutions, to address
+these questions and enable the theoretical ambiguities involved in no n-observables to be
+quantified.
+While implications from high-momentum components in nuclear wave fun ctions
+have been claimed for nuclear structure at both normal and higher densities (e.g.,
+neutron stars), one should be cautious about attributing too muc h to resolution-
+dependent quantities. After all, with parton distributions one would not talk about the
+results at a particular Q2as being “the” quark or gluon momentum distribution also for
+lower or higher Q2. In addition, we note the importance of distinguishing short-range
+versus long-range (or mid-range) correlations. The issue is wheth er one understands the
+physics at the corresponding resolution scale, which makes it possib le to attribute the
+resulttothecorrectphysics. Thisisrelevantforrecent experime ntsatJeffersonLabthat
+measure the differences in correlations between proton-proton a nd proton-neutron pairs
+bylookingattheratioofdifferential crosssections dσ(e,e′pn)/dσ(e,e′pp), which showsa
+pronounced preference for pnpairs over pppairs [9]. It is important to disentangle what
+is due to effects of long-range (low-momentum) pion-exchange ten sor forces as opposed
+to the high-momentum reaction physics or short-range effects us ually associated with a
+strong short-range repulsion in nucleon-nucleon (NN) interaction s.
+It is sometimes said (e.g., see Ref. [10]) that short-range correlatio ns are “hidden”
+in the parameters of low-energy effective theories (e.g., an EFT). P resumably “hidden”
+is in the sense of integrated out; that is, contributions from loop mo menta are shifted
+into coupling constants as the resolution is decreased. When is it nec essary (or at
+least desirable) to “unhide” this physics when doing low-energy nucle ar physics? In
+particular, is it relevant when calculating low-energy nuclear struct ure, such as binding
+energies and low-lying excitations? A systematic framework is neede d to disentangle
+what is hidden but known from what is unconstrained short-distanc e physics.
+In interpreting occupation numbers or momentum distributions ext racted from
+experiment, a comparison is often made to independent-particle mo dels, where
+occupation numbers are either zero or one. This in turn sometimes le ads to criticism of
+mean-field energy-density functionals (EDF) because of the appa rent contradiction of
+fractional occupation numbers extracted and those in the EDF be ing equal to zero or
+one(ignoringpairinghere). However, thetheoretical underpinnin g ofEDFapproachesis
+Kohn-Shamdensityfunctionaltheory(DFT),forwhich(withoutp airing)theoccupation‘Non-observables’ for nuclei 6
+numbers arezero or one, regardless of the degree of correlation. At the same time,
+there are issues with using Kohn-Sham single-particle wave function s for non-Kohn-
+Sham observables (although calculations of single-particle levels in DF T have shown
+promise, even if not fully justified by the theory). How can we resolv e these conflicting
+views? A key development would be a unified framework for DFT and ex periments
+such as (e,e′p). This would also help to distentangle the role of short-range versu s long-
+range correlations in the theoretical calculation of occupation num bers. Recent results
+for the quenching of spectroscopic factors suggest that long-r ange correlations may be
+more dominant than previously realized [11]. Being able to make robust comparisons
+at different resolutions will be essential in addressing these issues.
+Afinal example ofa non-observablethat requires careful treatm ent istheextraction
+of NN potentials from recent lattice QCD calculations. In this case it is computational
+rather than experimental data that is analyzed. We first note tha t the short-range
+NN interaction is a non-observable, which is immediately evident from t he experience
+with unitary transformations [1]. In the case of the lattice calculatio ns, a choice has to
+be made of quark fields that are to be used as interpolating operato r for the nucleon
+(which is chosen to have a non-zero overlap with a nucleon state tha t will dominate at
+large Euclidean times). Different choices will lead to different potentia ls so one must
+be cautious in drawing conclusions. Nevertheless, this could be a pow erful tool to gain
+physical insight, in particular given the many knobs to change QCD pa rameters on the
+lattice.
+5. Final comments
+Thesystematicformulationof‘non-observables’suchasspectro scopicfactorsandrelated
+quantities is an important open problem for nuclear structure theo rists. The analogy
+to parton distribution functions seems to us to be a promising avenu e to pursue.
+However, thedevelopment andapplicationofcorrespondingframe worksthatcanaddress
+questions such asthevalidityoffactorizationapproximations withd ifferent Hamiltonian
+resolutions will require progress on other open questions of nuclea r structure outlined
+in this volume. Finally, we note that resolution dependence for a given non-observable
+does not mean that a particular choice of conventions cannot be ad vantageous, just as
+in field theory contexts the choice of a particular gauge may be most illuminating or
+predictive. But having a rigorous framework will enable this choice to be made in a
+controlled way.
+Acknowledgments
+This work was supported in part by the National Science Foundation under Grant
+No. PHY–0653312, the UNEDF SciDAC Collaboration under DOE Grant DE-FC02-
+07ER41457, the Natural Sciences and Engineering Research Coun cil of Canada
+(NSERC), and by the Helmholtz Alliance Program of the Helmholtz Asso ciation,‘Non-observables’ for nuclei 7
+contract HA216/EMMI “Extremes of Density and Temperature: C osmic Matter in the
+Laboratory”. TRIUMF receives federal funding via a contribution agreement through
+the National Research Council of Canada.
+References
+[1] S. K. Bogner, R. J. Furnstahl, and A. Schwenk. From low-momen tum interactions to nuclear
+structure, arXiv:0912.3688.
+[2] R. J. Furnstahl, H. W. Hammer, and N. Tirfessa. Field redefinition s at finite density. Nucl. Phys.
+A, 689:846, 2001.
+[3] R. J. Furnstahl and H. W. Hammer. Are occupation numbers obs ervable? Phys. Lett. B , 531:203,
+2002.
+[4] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack, and R . G. Hulet. Molecular probe of
+pairing in the BEC-BCS crossover. Phys. Rev. Lett. , 95:020404, 2005.
+[5] J. C. Collins. What exactly is a parton density? Acta Phys. Polon. B , 34:3103, 2003.
+[6] J. M. Campbell, J. W. Huston, and W. J. Stirling. Hard interactions of quarks and gluons: A
+primer for LHC physics. Rept. Prog. Phys. , 70:89, 2007.
+[7] W. K. Tung, Perturbative QCD and the parton structure of the nucleon. At the frontier of particle
+physics, ed. M. Shifman, Vol. 2, p. 887, World Scientific.
+[8] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner. Modern theory of nuclear forces. Rev. Mod.
+Phys., 81:1773, 2009.
+[9] R. Subedi et al. Probing cold dense nuclear matter. Science, 320:1476, 2008.
+[10] Leonid Frankfurt, Misak Sargsian, and Mark Strikman. Future directions for probing two and
+three nucleon short- range correlations at high energies. AIP Conf. Proc. , 1056:322–329, 2008.
+[11] C. Barbieri. Role of long-range correlations on the quenching of spectroscopic factors. Phys. Rev.
+Lett., 103:202502, 2009.
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+arXiv:1001.0330v1  [math.CO]  2 Jan 2010A note on dilation coefficient, plane-width,
+and resolution coefficient of graphs
+Martin Milaniˇ c ,
+University of Primorska—FAMNIT, Glagoljaˇ ska 8, 6000 Kope r, Slovenia,
+martin.milanic@upr.si
+Tomaˇ z Pisanski ,
+University of Ljubljana, IMFM, Jadranska 19, 1111 Ljubljan a, Slovenia, and
+University of Primorska—PINT, Muzejski trg 2, 6000 Koper, S lovenia,
+Tomaz.Pisanski@fmf.uni-lj.si
+Arjana ˇZitnik
+University of Ljubljana, IMFM, Jadranska 19, 1111 Ljubljan a, Slovenia,
+Arjana.Zitnik@fmf.uni-lj.si
+Keywords: dilation coefficient; plane-width; resolution coefficient; g raph representation;
+graph drawing; chromatic number; circular chromatic numbe r; bandwidth.
+Math. Subj. Class. (2000) 05C62
+Abstract
+In this note we study and compare three graph invariants related t o the ‘compactness’
+of graph drawing in the plane: the dilation coefficient , defined as the smallest possible
+quotient between the longest and the shortest edge length; the plane-width , which is the
+smallest possible quotient between the largest distance between an y two points and the
+shortest length of an edge; and the resolution coefficient , the smallest possible quotient
+between the longest edge length and the smallest distance between any two points. These
+three invariants coincide for complete graphs.
+We show that graphs with large dilation coefficient or plane-width have a vertex with
+large valence but there exist cubic graphs with arbitrarily large reso lution coefficient. Sur-
+prisinglyenough, the one-dimensionalanaloguesofthese three inv ariantsallowus to revisit
+the three well known graph parameters: the circular chromatic nu mber, the chromatic
+number, and the bandwidth. We also examine the connection betwee n bounded resolution
+coefficient and minor-closed graph classes.
+1 Introduction
+Given a simple, undirected, finite graph G= (V,E), arepresentation ofGin thed-dimensional
+Euclidean space Rdis a function ρassigning to each vertex of Ga point in Rd. AnR2-
+representation is called planar, anR3-representation is called spatial.A representation of a
+graph in Rdcan beviewed as a drawing of the graph with straight edges. Th elength of an edge
+uv∈Eis the Euclidean distance between ρ(u) andρ(v), i.e.,/⌊ar⌈⌊lρ(u)−ρ(v)/⌊ar⌈⌊l2. A representation
+ρisnon-vertex-degenerate (NVD) if the function ρis one-to-one. A representation ρisnon-
+edge-degenerate (NED) if for all edges uv∈E(G), it holds that ρ(u)/n⌉}ationslash=ρ(v); in other words,
+if the minimal edge length is positive. Clearly, a non-verte x-degenerate representation is also
+non-edge-degenerate.
+1Inthispaperwestudythreegraphinvariants thatarereleva ntforrepresentationsofgraphs.
+We restrict ourselves to planar representations. To avoid t rivialities, we only consider graphs
+with at least one edge.
+Following [12], we define the dilation coefficient of a non-edge-degenerate representation of
+a graph as the ratio between the longest and the shortest edge length in the representation.
+The minimum of the set of dilation coefficients of all non-edge -degenerate representations of a
+graphGis called the dilation coefficient ofG, and is denoted by dc( G).
+Theplane-width of a graph G= (V,E), introduced in [10] and denoted by pwd( G), is
+the minimum diameter of the image of the graph’s vertex set, o ver all representations ρwith
+the property that d(ρ(u),ρ(v))≥1 for each edge uv∈E, wheredis the Euclidean distance.
+Equivalently, the plane-width of a graph can be defined as the minimum, over all non-edge-
+degenerate representations, of the ratio between the large st distance between two points and
+the shortest length of an edge.
+We define the resolution coefficient of a non-vertex-degenerate representation of a graph
+as the ratio between the longest length of an edge and the smal lest distance between the
+images of any two distinct vertices. The minimum of the set of resolution coefficients of all
+representations of a graph Gis called the resolution coefficient ofG, and is denoted by re( G).
+In formulae, denoting by V2(G) the set of all 2-element subsets of V(G), we have
+dc(G) = min/braceleftbiggmaxuv∈E(G)d(ρ(u),ρ(v))
+minuv∈E(G)d(ρ(u),ρ(v)):ρis an NED representation of G/bracerightbigg
+,(1)
+pwd(G) = min/braceleftbiggmaxuv∈V2(G)d(ρ(u),ρ(v))
+minuv∈E(G)d(ρ(u),ρ(v)):ρis an NED representation of G/bracerightbigg
+,(2)
+re(G) = min/braceleftbiggmaxuv∈E(G)d(ρ(u),ρ(v))
+minuv∈V2(G)d(ρ(u),ρ(v)):ρis an NVD representation of G/bracerightbigg
+.(3)
+Note that the above parameters are well-defined, as we only co nsider graphs with at least
+one edge, and each of the three parameters can be expressed as the minimum value of a
+continuous real-valued function over a compact subset of R2|V(G)|.
+Notice alsothat consideringthesmallest ratiobetween the largest andthesmallestdistance
+between any two distinct points in a non-vertex-degenerate representation does not lead to a
+meaningful graph parameter – indeed, this quantity depends only on the number of vertices of
+the graph and not on the graph itself. The problem of determin ing the minimum value of this
+parameter for a given number of points nhas previously appeared in the literature in different
+contexts such as finding the minimum diameter of a set of npoints in the plane such that each
+pair of points is at distance at least one [3], or packing non- overlapping unit discs in the plane
+so as to minimize the maximum distance between any two disc ce nters [13]. For a given n, we
+denote the value of this parameter by h(n):
+h(n) = min/braceleftbiggmaxuv∈V2(Kn)d(ρ(u),ρ(v))
+minuv∈V2(Kn)d(ρ(u),ρ(v)):ρis an NVD representation of Kn/bracerightbigg
+.
+2n2345 67 8
+h(n)11√
+21+√
+5
+22sin72◦2(2sin(π/14))−1
+approx. 111.4141.618 1.902 2 2.246
+Table 1: Known values of h(n) = dc(Kn) = pwd( Kn) = re(Kn).
+The exact values for h(n) are known only up to n= 8 [1, 3], see Table 1. It follows
+directly from the definitions that if Gis a complete graph, the three parameters dc, pwd and
+re coincide.
+Proposition 1. For every complete graph Kn,
+dc(Kn) = pwd( Kn) = re(Kn) =h(n).
+The parameter h(n) was recently studied by Horvat, Pisanski and ˇZitnik [8].
+In this paper we give some relationships between the three pa rameters. After giving some
+basic properties and examples in Section 3, we show in Sectio n 4 that the dilation coefficient
+and theplane-width areequivalent graphparameters inthe s ensethat they areboundedon the
+same sets of graphs. These two parameters are also equivalen t to the chromatic number, and
+are therefore bounded from above by a function of the maximum degree. On the other hand,
+there exist cubic graphs with arbitrarily large resolution coefficient. In particular, while the
+plane-width and the dilation coefficient are boundedfrom abo ve by a function of the resolution
+coefficient, the converse is not true.
+In Section 5 we examine the natural one-dimensional analogu es of these three parameters
+(denoted by dc 1(G), pwd1(G) and re 1(G)), and show that they coincide or almost coincide
+with three well studied graph parameters: the circular chro matic number χc(G), the chromatic
+numberχ(G), and the bandwidth bw( G). As a corollary of some relations among dc 1(G),
+pwd1(G)andre 1(G), weobtainindependentproofsofsomeofthewell-knownrel ationsbetween
+χc(G),χ(G) and bw( G).
+In Section 6 we show that Gis planar whenever re( G)<√
+2, while there exist graphs with
+re(G) =√
+2 that contain arbitrarily large clique minors.
+2 Preliminaries
+In this section we review some definitions and basic results. For terms left undefined, we refer
+the reader to [6]. As usual, let χ(G) denote the chromatic number , i.e., the least number
+of colors needed for a proper vertex coloring of G, andω(G) theclique number , i.e., the
+maximum size of a complete subgraph of G. Clearly, ω(G)≤χ(G) for any graph G. Graphs
+Gsuch that for all induced subgraphs HofGthe equality ω(H) =χ(H) holds are called
+perfect. Recently perfect graphs have been characterized by Chudno vsky, Robertson, Seymour
+and Thomas [4], who proved the famous Strong Perfect Graph Th eorem conjectured by Berge
+in 1961 [2].
+Some useful connections between the plane-width and the chr omatic number of a graph,
+for small values of these two parameters, are given in [10]:
+3Theorem 2. For all graphs G,
+(a)pwd(G) = 1if and only if χ(G)≤3,
+(b)pwd(G)∈(2/√
+3,√
+2]if and only if χ(G) = 4,
+(c)pwd(K4) =√
+2.
+The following property of the plane-width will also be used i n some of our proofs.
+Proposition 3 ([10]).LetGbe a graph such that χ(G) =ω(G). Then, pwd(G) =h(χ(G)).
+Obviously, the statement of the above proposition holds for perfect graphs.
+3 Basic properties and examples
+The dilation coefficient, the plane-width and the resolution coefficient of a graph are related
+by the following inequalities.
+Proposition 4. For every graph G,
+1≤dc(G)≤min{pwd(G),re(G)}.
+Proof.The inequality dc( G)≥1 is clear.
+Letρbe a non-edge-degenerate representation of G. Then, since E(G)⊆V2(G),
+maxuv∈E(G)d(ρ(u),ρ(v))
+minuv∈E(G)d(ρ(u),ρ(v))≤maxuv∈V2(G)d(ρ(u),ρ(v))
+minuv∈E(G)d(ρ(u),ρ(v)).
+Taking the minimum over all such representations, it follow s that dc( G)≤pwd(G).
+To see that dc( G)≤re(G), letρbe a non-vertex-degenerate representation of Gthat
+achieves the minimum in the definition of re( G). Then, since ρis non-edge-degenerate and
+E(G)⊆V2(G), we have
+dc(G)≤maxuv∈E(G)d(ρ(u),ρ(v))
+minuv∈E(G)d(ρ(u),ρ(v))≤maxuv∈E(G)d(ρ(u),ρ(v))
+minuv∈V2(G)d(ρ(u),ρ(v))= re(G).
+Clearly, the inequalities from Proposition 4 are tight. For instance, dc( G) = pwd( G) =
+re(G) = 1 for every graph with at most three vertices and at least on e edge. Moreover, the
+dilation coefficient of a graph Gis equal to 1 if and only if Gadmits a representation with
+all edges of the same length. Such graphs are known under the n ame ofunit distance graphs .
+There is a strong chemical motivation for exploring unit dis tance graphs and related concepts,
+since relevant chemical graphs tend to have all bond lengths of almost the same size. By
+Theorem 2, graphs of plane-width 1 are precisely the 3-color able graphs. To the best of our
+knowledge, graphs of unit resolution coefficient have not pre viously appeared in the literature.
+Thefollowing two examples, based on unit distance graphs, s how that dc( G) can bestrictly
+smaller than min {pwd(G),re(G)}, and that re( G) and pwd( G) are incomparable.
+4Example 1. LetGbe the Moser spindle, see Fig. 1. This is a 4-chromatic unit dis tance graph
+on 7 vertices. Since Gis unit distance, we have dc(G) = 1. On the other hand, since Gis
+4-chromatic, pwd(G)>2/√
+3by Theorem 2. Furthermore, it is easy to see that Gcannot be
+drawn in the plane with all edges of unit length and all the oth er pairs of points at least unit
+distance apart. Therefore, re(G)>1.
+Figure 1: A unit-distance representation of the Moser spind le
+Example 2. LetGbe the graph depicted on Fig. 2.
+Figure 2: A graph with re( G)<pwd(G) (all edge lengths are the same)
+Observe that Gis 4-chromatic, thus pwd(G)>2/√
+3by Theorem 2. The drawing from
+Fig. 2 gives a representation ρofGsuch that all edges are of unit length, and all the
+other pairs of points are at least unit distance apart. Theref ore,maxuv∈E(G)d(ρ(u),ρ(v)) =
+minuv∈V2(G)d(ρ(u),ρ(v)), which implies that re(G) = 1. This shows that in general, pwd(G)
+is not bounded from above by re(G).
+To see that also re(G)is not bounded from above by pwd(G), letGbe the 4-wheel, that
+is, the graph obtained from a 4-cycle by adding to it a dominat ing vertex. Then, since Gis
+3-colorable, pwd(G) = 1by Theorem 2. However, it is easy to see that Gcannot be drawn in
+the plane with all edges of the same length, which implies tha tre(G)>1.
+In Section 4 we will determine which of these three parameter s is bounded from above by
+a function of another one.
+Ahomomorphism of a graph Gto a graph His an adjacency-preserving mapping, that
+is a mapping φ:V(G)→V(H) such that φ(u)φ(v)∈E(H) whenever uv∈E(G). We say
+that a graph Gishomomorphic to a graph Hif there exists a homomorphism of GtoH. A
+graph invariant fishomomorphism monotone iff(G)≤f(H) whenever Gis homomorphic
+5toH. In [10], it was shown that the plane-width is homomorphism m onotone. We now show
+that the same property holds for the dilation coefficient, and that the resolution coefficient is
+monotone with respect to the subgraph relation.
+Proposition 5. LetGandHbe graphs with at least one edge.
+(i)IfGis homomorphic to Hthendc(G)≤dc(H).
+(ii)IfGis a subgraph of Hthenre(G)≤re(H).
+Proof.It follows from definitions that dc( G) is the minimum value of psuch that Gis homo-
+morphic to some graph Hwhose vertex set is a subset of R2and such that every edge of H
+connects two points at Euclidean distance at least 1 and at mo stp. This observation together
+with the transitivity of the homomorphism relation immedia tely implies ( i).
+Tosee (ii), letρbeanon-vertex-degenerate representation of Hthat achieves theminimum
+in the definition of re( H). Then, since the restriction of ρtoV(G) is a non-vertex-degenerate
+representation of G,E(G)⊆E(H) andV2(G)⊆V2(H), we have
+re(G)≤maxuv∈E(G)d(ρ(u),ρ(v))
+minuv∈V2(G)d(ρ(u),ρ(v))≤maxuv∈E(H)d(ρ(u),ρ(v))
+minuv∈V2(H)d(ρ(u),ρ(v))= re(H).
+4 More on relationship between dc,pwd, andre
+In this section, we examine more closely the relations betwe en these three parameters. First,
+we show in Theorem 7 that the plane-width of a graph is bounded from above by a function
+of its dilation coefficient. Our proof will make use of the foll owing result from [10] (combining
+Lemmas 2.2 and 3.7 therein).
+Lemma 6. ([10]) There exists a constant C >0such that for every graph G,
+pwd(G)≤/radicalBigg
+2√
+3
+πχ(G)+C.
+Theorem 7. There exist positive constants K,C >0such that for every graph G,
+pwd(G)≤K·dc(G)+C.
+Proof.Letρbe a representation of Gachieving the minimum in the definition of the di-
+lation coefficient (cf. Equation (1)). We may assume, without loss of generality, that
+minuv∈E(G)d(ρ(u),ρ(v)) = 1(otherwise, wescaletherepresentationaccordingly) . Hencedc( G)
+equals the maximum length of an edge of Gw.r.t. the representation ρ.
+Let us cover the set {ρ(v) :v∈V(G)}with pairwise disjoint translates of the half-open
+squareS= [0,t/√
+2)×[0,t/√
+2), arranged in a grid-like way, where t=⌈√
+2dc(G)⌉+ 1.
+Furthermore, we partition each copy S′ofSintot2pairwise disjoint translates Aij(S′) of
+the set [0 ,1/√
+2)×[0,1/√
+2), for each i,j∈ {1,...,t}. Here, the pair ( i,j) denotes the
+6“coordinates” of the square Aij(S′) within S′. We do the assignment of the coordinate pairs
+to the small squares in the same way for all copies S′ofS.
+Letc:V(G)→ {1,...,t}2be a coloring of the vertices of Gthat assigns to each v∈V(G)
+theunique( i,j)∈ {1,...,t}2suchthat thereexistsatranslate S′ofSsuchthat ρ(v)∈Aij(S′).
+By construction, cis a proper coloring of G. Therefore, χ(G)≤t2. By Lemma 6, pwd( G)≤/radicalBig
+2√
+3
+πχ(G)+C≤/radicalBig
+2√
+3
+πt+C. Combining this inequality with the inequality t≤√
+2dc(G)+2,
+the proposition follows: we may take K=/radicalBig
+4√
+3
+π≈1.4850.1
+We say that two graph parameters fandgareequivalent if they are bounded on precisely
+the same sets of graphs, that is, if for every set of graphs G, we have
+sup{f(G) :G∈ G}<∞
+if and only if
+sup{g(G) :G∈ G}<∞.
+It follows from Proposition 4 and Theorem 7 that the dilation coefficient and the plane-width
+are equivalent graph parameters, which, according to the fo llowing result from [10], are also
+equivalent to the chromatic number χ:
+Theorem 8 ([10]).For every ǫ >0there exists an integer ksuch that for all graphs Gof
+chromatic number at least k,
+/parenleftBigg√
+3
+2−ǫ/parenrightBigg
+/radicalbig
+χ(G)<pwd(G)<
+/radicalBigg
+2√
+3
+π+ǫ
+/radicalbig
+χ(G).
+On the other hand, it turns out that the resolution coefficient is not equivalent to the
+dilation coefficient or to the plane-width.
+Theorem 9. For every positive function fthere exists a graph Gsuchthat re(G)> f(pwd(G)).
+There exists a function fsuch that for every graph G,pwd(G)≤f(re(G)).
+Proof.TakeG=K1,n. Since these graphs are bipartite, pwd( K1,n) = 1 for all n. However, the
+values of re( K1,n) tend to infinity with increasing n. This follows by observing that thereexists
+a constant C >0 such that for every N >0, in any non-vertex-degenerate representation ρof
+K1,nwith min uv∈V2(G)d(ρ(u),ρ(v)) = 1, less than CN2vertices can be mapped to distance at
+mostNfrom the image of the center of the star. Thus, re( K1,n) = Ω(√n). This shows the
+first part of the theorem.
+Given a graph Gwith at least one edge, let ∆( G) denote its maximum vertex degree. Since
+K1,∆(G)is a subgraph of G, Proposition 5 implies that re( G)≥re(K1,∆(G)) = Ω(/radicalbig
+∆(G)).
+Therefore, themaximumdegreeofagraphisboundedfromabov ebyafunctionofitsresolution
+coefficient. In particular, since χ(G)≤∆(G)+1, the chromatic number of Gis also bounded
+from above by a function of re( G), and the second part of the theorem follows by Theorem 2.
+1A better constant K=/radicalBig
+8√
+3
+3π+ǫ≤1.2126 can be obtained by covering the plane with hexagons inst ead of
+the squares (similarly as was done in Lemma 3.6 in [10]).
+7In the above proof, large degree caused the resolution coeffic ient to be large. The next
+example shows that large maximum degree is only a sufficient bu t not a necessary condition
+for large resolution coefficient. Therefore, while the chrom atic number of a graph is bounded
+from above by a function of its maximum vertex degree, and the n the same is true for the
+dilation coefficient and the plane-width, this is not the case for the resolution coefficient.
+Theorem 10. For each R >0there exists a graph GRof maximum degree at most 3 such
+thatre(GR)> R.
+Proof.LetHkdenote a full cubic tree with klayers, rooted at vertex v0. LetVidenote the
+number of vertices of layer i. Then V0= 1,V1= 3,V2= 3·2, ...,Vk= 3·2k−1and
+|V(Hk)|=V0+V1+···+Vk= 3·2k−2. Letρbe a representation of Hk, where re( Hk) is
+achieved. Let r= min uv∈V2(Hk){d(ρ(u),ρ(v))}.There exists at least one vertex xofVksuch
+thatd(ρ(v0),ρ(x))> C·r√
+3·2k−2 =:R′. Since the graph distance d(V0,x) =kthere exists
+an edgepqon the path from v0toxsuch that
+d(ρ(p),ρ(q))≥R′/k=C·r√
+3·2k−2
+k.
+Therefore
+ρ(Hk)≥d(ρ(p),ρ(q))
+r≥C·√
+3·2k−2
+k.
+For a given Rdefinek0to be the smallest kfor which C·(3·2k−2)≥Rand define
+GR=Hk0. Clearly, GRhas maximum degree at most 3 and re( GR)> R.
+5 One-dimensional restrictions
+It is interesting to consider the restrictions of these thre e parameters to a line instead of the
+plane. That is, every vertex gets mapped to a point on the real line instead of being mapped
+to a point in R2and the definitions are analogous to those given by Equations (1)-(3). We
+denote the corresponding parameters by dc 1(G), pwd1(G) and re 1(G).
+It turns out that the one-dimensional variant of the dilatio n coefficient is strongly related
+to the circular chromatic number of the graph, the “line-wid th” is strongly related to the
+chromatic number of the graph, while the one-dimensional va riant of the resolution coefficient
+coincides with the graph’s bandwidth, defined as
+bw(G) := min
+π:V→{1,...,n},bij.max
+uv∈E|π(u)−π(v)|,
+wheren=|V(G)|.
+Given a graph G= (V,E) andc≥1, acircularc-coloring is a mapping f:V→[0,c) such
+that for every edge uv∈E, it holds that 1 ≤ |f(u)−f(v)| ≤c−1. Thecircular chromatic
+numberχc(G) ofGis defined as
+χc(G) = inf{c:Gadmits a circular c-coloring }.
+It is known that the infimum in the definition above is always at tained (see, e.g., [15]).
+8Theorem 11. For every graph G, the following holds:
+(i) dc1(G) =χc(G)−1
+(ii) pwd1(G) =χ(G)−1.
+(iii) re1(G) = bw(G).
+Proof.LetG= (V,E) be a graph.
+(i) Letc=χc(G) and let fbe a circular c-coloring of G. Then, fdefines a non-edge-
+degenerate one-dimensional representation of Gsuch that min uv∈E|f(u)−f(v)| ≥1 and
+maxuv∈E|f(u)−f(v)| ≤c−1. It follows thatmaxuv∈E|f(u)−f(v)|
+minuv∈E|f(u)−f(v)|≤c−1, implying dc 1(G)≤
+χc(G)−1.
+Conversely, let ρ∗:V→Rbe a non-edge-degenerate one-dimensional representation ofG
+that achieves the minimum in the definition of dc 1(G). Without loss of generality, we may
+assume that min uv∈E|ρ∗(u)−ρ∗(v)|= 1 and that min v∈Vρ∗(v) = 0. Let c= dc1(G)+1, and
+definef:V→Ras follows:
+For allv∈V,
+f(v) =ρ∗(v)−c·/floorleftbiggρ∗(v)
+c/floorrightbigg
+.
+To show that χc(G)≤dc1(G)+1 =c, it suffices to verify that fis a circular c-coloring of G.
+Since 0≤ρ∗(v)
+c−/floorleftBig
+ρ∗(v)
+c/floorrightBig
+<1,fmaps vertices of Gto the interval [0 ,c). It remains to show
+that for every edge uv∈E, it holds that 1 ≤ |f(u)−f(v)| ≤c−1. Notice that by the choice
+ofρ∗, we have 1 ≤ |ρ∗(u)−ρ∗(v)| ≤c−1 for every edge uv∈E.
+Letuv∈E. We may assume that ρ∗(u)≤ρ∗(v). Letk=/floorleftBig
+ρ∗(u)
+c/floorrightBig
+andℓ=/floorleftBig
+ρ∗(v)
+c/floorrightBig
+. The
+inequality ρ∗(u)≤ρ∗(v) implies that k≤ℓ. Moreover, the inequality ρ∗(v)≤ρ∗(u) +c−1
+impliesℓ≤k+1.
+Ifℓ=k, thenf(u)−f(v) =ρ∗(u)−ρ∗(v) and the inequalities 1 ≤ |f(u)−f(v)| ≤c−1
+follow.
+Suppose now that ℓ=k+1. Then f(u)−f(v) =ρ∗(u)−ρ∗(v)+c. Therefore, it follows
+fromρ∗(v)−ρ∗(u)≤c−1 thatf(u)−f(v)≥1, implying f(u)−f(v) =|f(u)−f(v)| ≥1.
+Similarly, it follows from ρ∗(v)−ρ∗(u)≥1 thatf(u)−f(v) =|f(u)−f(v)| ≤c−1.
+This shows that fis a circular c-coloring of G, which implies that χc(G)≤dc1(G)+1.
+(ii) Anyk-coloring of Gwith colors in the set {1,...,k} ⊆Rdefines a non-edge-degenerate
+one-dimensional representation ρofGsuch thatmaxuv∈V2(G)|ρ(u)−ρ(v)|
+minuv∈E|ρ(u)−ρ(v)|≤k−1. Therefore,
+pwd1(G)≤χ(G)−1.
+Conversely, let ρ∗:V→Rbe a non-edge-degenerate one-dimensional representation of
+Gthat achieves the minimum in the definition of pwd1(G). Without loss of generality, we
+may assume that min uv∈E|ρ∗(u)−ρ∗(v)|= 1 and that min v∈Vρ∗(v) = 1. Define f:V→R
+as follows: For all v∈V, letf(v) =⌊ρ∗(v)⌋. Then, fis a proper k-coloring of G, where
+k=⌊pwd1(G)⌋+1. Therefore, χ(G)≤ ⌊pwd1(G)⌋+1≤pwd1(G)+1.
+(iii) Any bijective mapping π:V→ {1,...,n} ⊆Rdefines a non-vertex-degenerate
+one-dimensional representation of Gsuch that
+maxuv∈E|π(u)−π(v)|
+minuv∈V2(G)|π(u)−π(v)|= max
+uv∈E|π(u)−π(v)|.
+9Therefore, re 1(G)≤bw(G).
+Conversely, let ρ∗:V→Rbe a non-vertex-degenerate one-dimensional representati on of
+Gthat achieves the minimum in the definition of re 1(G). Without loss of generality, we may
+assume that min uv∈V2(G)|ρ∗(u)−ρ∗(v)|= 1. Then
+re1(G) =maxuv∈E|ρ∗(u)−ρ∗(v)|
+minuv∈V2(G)|ρ∗(u)−ρ∗(v)|= max
+uv∈E|ρ∗(u)−ρ∗(v)|.
+Therefore, to show that bw( G)≤re1(G), it suffices to show that bw( G)≤maxuv∈E|ρ∗(u)−
+ρ∗(v)|.
+Order the vertices of V={v1,...,vn}according to the increasing values of their images:
+ρ∗(v1)< ρ∗(v2)<···< ρ∗(vn),
+and letπ(vi) =ifor alli∈ {1,...,n}. This defines a bijective mapping π:V→ {1,...,n}.
+Letuv∈E, and assume that i:=π(u)< j:=π(v). By the definition of πand since
+minxy∈V2(G)|ρ∗(x)−ρ∗(y)|= 1, we infer that |ρ∗(u)−ρ∗(v)| ≥j−i. Onthe other hand, |π(u)−
+π(v)|=j−i, which implies that |π(u)−π(v)| ≤ |ρ∗(u)−ρ∗(v)|. Therefore, max uv∈E|π(u)−
+π(v)| ≤maxuv∈E|ρ∗(u)−ρ∗(v)|, implying bw( G)≤maxuv∈E|π(u)−π(v)| ≤maxuv∈E|ρ∗(u)−
+ρ∗(v)|= re1(G).
+Proposition 12. For every graph G,
+⌈dc1(G)⌉= pwd1(G)≤re1(G).
+Proof.In the same way as the inequality dc( G)≤pwd(G) from Proposition 4, one can prove
+the inequality dc 1(G)≤pwd1(G). Since pwd1(G) =χ(G)−1 by Theorem 11, pwd1(G) is
+always integral, therefore ⌈dc1(G)⌉ ≤pwd1(G).
+To see that pwd1(G)≤ ⌈dc1(G)⌉, consider a non-edge-degenerate one-dimensional repre-
+sentation ρ∗:V→RofGthat achieves the minimum in the definition of dc 1(G). Without
+loss of generality, we may assume that min uv∈E|ρ∗(u)−ρ∗(v)|= 1 and that min v∈Vρ∗(v) = 0.
+Letk:=⌈dc1(G)⌉+1, and define ρ′:V→ {0,1,...,k−1} ⊆Ras follows: For all v∈V, let
+ρ′(v) =⌊ρ∗(v)⌋(modk).
+For every uv∈E, we have |ρ′(u)−ρ′(v)| ≥1. Indeed: if ρ′(u) =ρ′(v) then
+|ρ∗(u)−ρ∗(v)| ≥k >dc1(G), contrary to the choice of ρ∗and the definition of k. Therefore,
+ρ′is a non-edge-degenerate one-dimensional representation ofG, and since the image of ρ′is
+contained in the set {0,1,...,k−1}, we have max uv∈V2(G)|ρ′(u)−ρ′(v)| ≤k−1 =⌈dc1(G)⌉,
+implying that pwd1(G)≤ ⌈dc1(G)⌉.
+In the same way as the inequality dc( G)≤re(G) from Proposition 4, one can prove the
+inequality dc 1(G)≤re1(G). Since re 1(G) = bw(G) by Theorem 11, re 1(G) is always integral,
+and consequently ⌈dc1(G)⌉ ≤re1(G).
+Theorem 11 and Proposition 12 provide an alternative proof o f the following well known
+relations:
+•The equality χ(G) =⌈χc(G)⌉showing that the circular chromatic number is the refine-
+ment of the chromatic number [14].
+10•The inequality bw( G)≥χ(G)−1, proved in [5].
+Theorems 8 and 11 together with Proposition 12 also imply tha t both dilation coefficient
+and plane-width are equivalent to their one-dimensional co unterparts. On the other hand, the
+resolution coefficient is not equivalent to its one-dimensio nal variant, the bandwidth. There
+exist graphs of unit resolution and arbitrarily large bandw idth:
+Example 3. LetGbe the Cartesian product of two paths, G=Pm✷Pn. (For the definition of
+the Cartesian product, see e.g. [9].) By a result of Hare et al. [7], the bandwidth of Gis equal
+tomin{m,n}. Representing the vertices of Gas points {1,...,m}×{1,...,n}of the integer
+gridZ2shows that the resolution coefficient of Gis equal to 1.
+The local density of a graph is defined as maxv,r[|N(v,r)|/(2r)], whereN(v,r)denotes the
+set of all vertices at distance at most rfrom a vertex v. While the local density is a lower
+bound for the bandwidth, this example shows that the local de nsity (or any increasing function
+of it) is not a lower bound for the resolution coefficient.
+6 Minor-closed classes
+A graph Gis aminorof a graph HifGcan be obtained from Hby a sequence of vertex
+deletions, edge deletions, and edge contractions. We say th at a class of graphs is minor closed
+if it contains together with any graph Gall minors of G. In this section, we prove some results
+that link the resolution coefficient to minor-closed graph cl asses. First, we observe that all
+graphs of small enough resolution coefficient are planar.
+Proposition 13. Ifre(G)<√
+2thenGis planar.
+Proof.LetG= (V,E) such that re( G)<√
+2, and consider a representation ρ∗:V→R2
+achieving the minimum in the definition of the resolution coe fficient. As usual, we assume
+that min uv∈V2(G)d(ρ∗(u),ρ∗(v)) = 1.
+First, observe that since re( G)<2, for every vertex v∈Vand every edge e∈E, it
+holds that ρ∗(v) is not contained in the interior of the line segment ρ∗(e). Therefore, since ρ∗
+is non-vertex-degenerate and maps no vertex to an interior o f a line segment connecting the
+endpoints of an edge, it defines a drawing of Gin the plane with straight line segments. We
+claim that this is a planar drawing. Suppose not, and let uvandxybe two distinct edges of
+Gsuch that the interiors of line segments ρ∗(u)ρ∗(v) andρ∗(x)ρ∗(y) have a point in common.
+Basic geometric arguments show that a longest diagonal in a c onvex 4-gon in the plane is at
+least a factor of√
+2 longer than its shortest edge. Therefore, we conclude that at least one
+of the two line segments ρ∗(u)ρ∗(v) andρ∗(x)ρ∗(y) has length√
+2 or more, contrary to the
+assumption that re( G)<√
+2.
+Corollary 14. Every graph Gsuch that re(G) = dc(G) = 1<pwd(G)is a 4-chromatic planar
+unit distance graph.
+Proof.Gis unit distance since dc( G) = 1. By Proposition 13, Gis planar. Since pwd( G)>1,
+Theorem 2 implies that Gis not 3-colorable. Finally, since Gis planar, it must be 4-chromatic
+by the Four Color Theorem.
+11Notice that the inclusion is proper: there exist 4-chromati c planar unit distance graphs
+that are not of unit resolution coefficient, for example the Mo ser spindle (cf. Example 1).
+Itturnsoutthat thestatement of Proposition13 cannot bege neralized tosurfaces ofhigher
+genus, in the sense that the genus of a graph would be bounded f rom above by a function of
+its resolution coefficient. In fact, we show that:
+Theorem 15. For every nthere exists a graph Gwithre(G) =√
+2and such that Knis a
+minor of G.
+Proof.Letn≥5. We will construct a graph Gof maximum degree 3 with a Kn-minor and
+such that re( G) =√
+2.
+First, we replace every vertex vofKnwith a cycle on n−1 vertices and connect each
+vertex of the cycle to a different neighbor of vinKn. Repeating this procedure for all vertices
+ofG, we obtain a cubic graph G′which can be contracted to Kn.
+We now place the vertices of G′into the plane so that all point coordinates are integer and
+no two vertices are mapped to the same point. Moreover, we dra w the edges between them so
+that: (i) each edge is represented by a rectilinear curve con sisting of finitely many vertical and
+horizontal line segments, (ii) no two edges share a common li ne segment, (iii) at every point
+where two edges cross, they cross properly (i.e., they do not touch each other), and (iv) no
+vertex is contained in the interior of an edge. This can be don e since the graph is of maximum
+degree 3. Moreover, all the coordinates of the breakpoints ( points where a rectilinear curve
+bends) can be chosen to be rational-valued. By scaling the dr awing appropriately, we then
+obtain a drawing of G′such that all vertex coordinates as well as all coordinates o f the crossing
+points and breakpoints are integer multiples of 4.
+We now modify the obtained drawing by making a local modificat ion at each crossing
+point. We essentially rotate each cross by 45◦, as shown in Figure 3.
+Figure 3: The local modification at a crossing point; the grid represents a portion of the
+two-dimensional integer grid around the crossing point.
+Due to the assumption that in the previous drawing, horizont al and vertical lines connect
+points in (4 Z)2, performing such a local modification at each crossing point will not introduce
+any further crossings or touchings.
+The resolution coefficient of G′may be large. We now complete the proof by modifying
+G′into a graph Gby subdividing edges. For every edge eofG′, letℓ(e) denote the curve
+representing ein the above (modified) drawing. We insert a new vertex on each point in
+ℓ(e)∩Z2that is not yet occupied by a vertex. We repeat this procedure for every edge
+eofG′, and call the resulting graph G. Notice that the resulting drawing of Gdefines a
+non-vertex-degenerate representation ρofG.
+Sinceallvertexcoordinatesareinteger, wehavemin uv∈V2(G)d(ρ(u),ρ(v))≥1. Ontheother
+hand, dueto thenewly introduced vertices, it also holds tha t maxuv∈E(G){d(ρ(u),ρ(v))} ≤√
+2.
+12Therefore, the resolution coefficient of Gis at most√
+2. On the other hand, since Gcontains
+K5as a minor, it is not planar, and hence re( G)≥√
+2 by Proposition 13.
+Finally, notice that as no new vertices were introduced at cr ossing points, Gcan be con-
+tracted to G′and hence to Kn, which implies that Gcontains Knas a minor.
+7 Concluding remarks
+As we have shown, the relationship between the dilation coeffi cient, the plane-width and the
+resolution coefficient is non-trivial in many respects. Alth ough all these parameters somehow
+reflect the departure from unit-distance representations t hey do not behave in a uniform way.
+The dilation coefficient and the plane-width are both equival ent to each other and also to their
+one-dimensional counterparts (in the sense that these para meters are bounded on precisely the
+same sets of graphs), while the resolution coefficient is not e quivalent to either plane-width or
+dilation coefficient and also not to its one-dimensional anal ogue, the bandwidth.
+There are several possibilities for further investigation s. For instance, one could study the
+parameters dc, pwd and re in a more general framework where in stead of R2the host space
+is an arbitrary metric space; we have explored in this paper t he case when the metric space is
+the real line. A related notion of colorings in distance spac es was studied by Mohar [11].
+Thereareseveral families of graphsfor whichat least two ou t of thethreeabove parameters
+coincide. For example, the dilation coefficient and the plane -width coincide for all graphs
+Gsuch that χ(G)∈ {2,3,ω(G)}. All three parameters coincide for complete graphs, for
+subgraphs of the Cartesian product of two paths, and for subg raphs of the strong product of
+two paths that contain a K4. It would be interesting to get a complete classification of f amilies
+of graphs for which at least two out of the three parameters co incide.
+Last but not least, we feel it would also be interesting to con sider other measures of
+degeneracyofgraphrepresentations, suchasthesmallesta ngleorthesmallestdistancebetween
+an edge and vertices non-incident to it.
+Acknowledgements
+This work was supported in part by the research program P1-02 94 of the Slovenian Agency
+for Research.
+References
+[1] P. Bateman and P. Erd˝ os. Geometrical extrema suggested by a lemma of Besicovitch.
+American Math. Monthly , 58:306–314, 1951.
+[2] C. Berge. F¨ arbung von Graphen, deren s¨ amtliche bzw. de ren ungerade Kreise starr sind,
+Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natu r. Reihe 10:114, 1961.
+[3] A. Bezdek and F. Fodor. Minimal diameter of certain sets i n the plane. J. Comb. Theory,
+Ser. A, 85:105–111, 1999.
+13[4] M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, Th e strong perfect graph
+theorem. Ann. of Math. 164:51–229, 2006.
+[5] J. Chvatalova, A.K. Dewdney, N.E. Gibbs and R.R. Korfhag e. The bandwidth problem
+for graphs: a collection of recent results. Research Report #24, Department of Computer
+Science, UWO, London, Ontario (1975).
+[6] R. Diestel, Graph Theory. Third Edition. Springer-Verlag, Heidelberg, 2005.
+[7] W.R. Hare, E.O’M. Hare and S.T. Hedetniemi, Bandwidth of grid graphs. Proceedings
+of the Sundance conference on combinatorics and related top ics (Sundance, Utah, 1985).
+Congr. Numer. 50:67–76, 1985.
+[8] B. Horvat, T. Pisanski and A. ˇZitnik. The dilation coefficient of a complete graph. Croat.
+Chem. Acta , 82:771–779, 2009.
+[9] W. Imrich, S. Klavˇ zar. Product graphs. Wiley-Interscience, New York, 2000.
+[10] M. Kami´ nski, P. Medvedev and M. Milaniˇ c. The plane-wi dth of graphs. Submitted. 2009.
+[11] B. Mohar. Chromatic number of a nonnegative matrix, in p reparation IMFM Preprint,
+39:785, 2001. http://www.imfm.si/preprinti/PDF/00785.pdf
+[12] T. Pisanski and A. ˇZitnik. Representing graphs and maps, Chapter in Topics in Topolog-
+ical Graph Theory , Series: Encyclopedia of Mathematics and its Applications (No. 129).
+Cambridge University Press, 2009.
+[13] A. Sch¨ urmann. On extremal finite packings. Discrete Comput. Geom. , 28:389–403, 2002.
+[14] A. Vince. Star chromatic number. J. Graph Theory , 12:551–559, 1988.
+[15] X. Zhu. Circular chromatic number: a survey. Discrete Math. , 229:371–410, 2001.
+14
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+arXiv:1001.0331v2  [astro-ph.EP]  24 Mar 2010TheKepler Pixel Response Function
+Stephen T. Bryson1, Peter Tenenbaum2, Jon M. Jenkins2, Hema Chandrasekaran2, Todd
+Klaus3, Douglas A. Caldwell2, Ronald L. Gilliland4, Michael R. Haas1, Jessie L. Dotson1,
+David G. Koch1, William J. Borucki1
+ABSTRACT
+Keplerseeks to detect sequences of transits of Earth-size exoplanets orbiting
+Solar-like stars. Such transit signals are on the order of 100 ppm. T he high
+photometric precision demanded by Keplerrequires detailed knowledge of how
+theKeplerpixels respond to starlight during a nominal observation. This infor-
+mation is provided by the Keplerpixel response function (PRF), defined as the
+composite of Kepler’s optical point spread function, integrated spacecraft point-
+ing jitter during a nominal cadence and other systematic effects. T o provide
+sub-pixel resolution, the PRF is represented as a piecewise-contin uous polyno-
+mial on a sub-pixel mesh. This continuous representation allows the prediction
+of a star’s flux value on any pixel given the star’s pixel position. The a dvantages
+and difficulties of this polynomial representation are discussed, inclu ding char-
+acterization of spatial variation in the PRF and the smoothing of disc ontinuities
+between sub-pixel polynomial patches. On-orbit super-resolutio n measurements
+ofthePRFacrossthe Keplerfieldofviewaredescribed. Two usesofthePRFare
+presented: the selection of pixels for each star that maximizes the photometric
+signal to noise ratio for that star, and PRF-fitted centroids which provide robust
+and accurate stellar positions on the CCD, primarily used for attitud e and plate
+scale tracking. Good knowledge of the PRF has been a critical compo nent for
+the successful collection of high-precision photometry by Kepler.
+Subject headings: planetary systems — techniques: photometric
+1NASA Ames Research Center, Moffett Field, CA 94035
+2SETI Institute, Mountain View, CA 94043
+3Orbital Sciences Inc., Moffett Field, CA 94035
+4Space Sciences Telescope Institute, Baltimore, MD 21218– 2 –
+1. Introduction
+Thehighphotometricprecisiondemandedby Kepler(Borucki et.al. 2010; Koch et.al. 2010)
+requires detailedknowledge ofhowpixels respondto starlight during anominal observational
+interval. This is provided by the Keplerpixel response function (PRF), a super-resolution
+representation of the interaction of starlight with pixels that includ es modulation by point-
+ing jitter and other systematic effects during an observation (this usage of PRF differs from
+that in (Lauer 1999)). The PRF provides a continuous representa tion that allows the pre-
+diction of a star’s flux value on any pixel given the star’s pixel position : these pixel values
+are samples of the PRF at the star’s sub-pixel position.
+TheKeplerfocalplane(Argabright et.al.2008; Caldwell et.al. 2010)consistsof 421024×
+2200 CCDs, each of which is divided into two 1024 ×1100 output channels. A PRF model is
+defined on each of these 84 output channels, accounting for CCD t ip/tilt and other channel-
+level variations. Due to variation in the optical PSF within an output c hannel, the PRF
+model is pixel-position dependent ( §2.1). Data is collected at a 29.4 minute cadence; we refer
+to this sampling interval as a “long cadence”. Bandwidth limitations pr event the storage
+and downlinking of all 95 ×106pixels at each long cadence: at most 5 .4×106pixels are
+downlinked per long cadence.
+The PRF currently contributes to Kepler’s precision by supporting the selection of opti-
+mal pixels for aperture photometry ( §4.1), and providing high-precision PRF-fitted centroids
+(§4.2) for attitude determination, crucial for removing the effects o f pointing jitter from pho-
+tometry. These aresufficient to attain Kepler’s current highprecision (Jenkins et.al. 2010a).
+ThefutureuseofdifferentialimageanalysiswillusethePRFtoperfo rmtargetandpixellevel
+sensitivity corrections, including local image motion, further increa singKepler’s precision.
+A star’s position on the Keplerfocal plane is computed with the software package
+raDec2Pix , developed by the KeplerScience Operations Center (SOC). This package uses
+theKeplerFocal Plane Geometry (FPG) model ( §2.2), as well as Kepler optical, nominal
+spacecraft attitude, and velocity aberration models.
+2. The Pixel Response Function
+TheKeplerPRF on an output channel is represented as a continuous piece-wis e poly-
+nomial function of sub-pixel position on each pixel, providing a super -resolution image of the
+light of a star falling on the pixels. Pixel values are determined by evalu ating the PRF given
+the pixel and sub-pixel position of a star: the pixel position determ ines where the PRF is
+placed in the output channel’s pixel array, while the sub-pixel positio n determines the pixel– 3 –
+values. Figure 1 shows two example PRFs and corresponding pixel va lues.
+Computation of the PRF from stellar images requires knowing where t hose images fall
+on the focal plane. Each star’s coordinates are provided by the Ke pler Input Catalog1
+(KIC), and the physical location of the CCDs relative to each other provided by the focal
+plane geometry (FPG) model ( §2.2). Determination of the FPG model, however, requires
+PRF-fitted stellar centroids ( §4.2), so FPG and PRF are mutually dependent. Therefore we
+determine both FPG and PRF in an iterative process ( §3.3).
+2.1. Representation of the PRF
+AKeplerPRF is defined on a grid of n×npixels called the PRF pixel array , wherenis
+large enough to capture the full PRF. Each pixel in the PRF pixel arr ay is sub-divided into
+m×msub-pixel regions. Each sub-pixel region is assigned a two-dimensio nal polynomial
+PRFi,j,s,t(x,y), where ( i,j) are the pixels indices in the n×nPRF pixel array and ( s,t)
+are the sub-pixel indices in the m×msub-pixel grid on pixel ( i,j). Therefore there are
+n2×m2two-dimensional polynomials associated with a PRF on an output chan nel. For
+most channels n= 11 is sufficient to capture the PRF, while for some channels n= 15 is
+required. We find m= 6 provides sufficient sub-pixel detail. The order of each of these
+polynomials is determined by a maximal information criterion ( §3.2).
+Each pixel of the PRF pixel array gives the flux on that pixel from a s tar whose location
+is in the central pixel of the PRF pixel array. Thus the central pixe l represents the peak of
+the star’s PRF while pixels towards the edge represent the flux in the wings of that star’s
+PRF. The values of these fluxes depend on the sub-pixel position of the star.
+PRF polynomials are determined independently of each other, so the y will not match
+at sub-pixel boundaries. We handle the resulting discontinuities usin g two strategies: (1)
+the discontinuities are minimized by fitting the PRF using data that ext ends into adjacent
+sub-pixel regionsasdescribed in3.2and(2)theremaining discontinu ities aresmoothedwhen
+evaluating the PRF as described in §2.3.
+A position-dependent PRF is implemented by measuring five PRFs on ea ch output
+channel: one at each corner and one at the center. Linear interpo lation on the resulting
+triangles as described in §2.3 provides an interpolated PRF at any point in the output
+channel. This model neglects pixel-level variations, which are treat ed during calibration
+(Jenkins et.al. 2010b).
+1http://archive.stsci.edu/kepler/kepler fov/search.php– 4 –
+2.2. Focal Plane Geometry
+The focal plane geometry (FPG) model represents each of the 42 CCDs on the Kepler
+focal plane via the center position, rotation angle, and plate scale o f each CCD. Prior to
+launchtheCCDpositionswereknownwithanaccuracyof ±3pixels; theFPGmodelprovides
+the required accuracy of ±0.1 pixels. The input to the FPG model computation is a two
+dimensional polynomial which maps right ascension and declination to p ixel location. These
+motion polynomials are derived via a fit to the observed pixel positions of known stars v ia
+PRF centroiding ( §4.2), and serve as a smoothing filter, reducing the impact of observ ational
+errors of individual stars.
+FPG computation determines the CCD locations of a set of sky coord inates in two
+ways: via the raDec2Pix library, which uses the FPG model; and via the motion polyno-
+mials, which do not use the FPG model. The FPG model parameters are adjusted via
+a Levenberg-Marquardt non-linear least squares algorithm until t he differences between the
+model and observation are minimized. The FPG model computation als o provides the space-
+craft attitude.
+2.3. From PRF to Pixel Values
+The PRF provides expected pixel values for a given star with a given m agnitude and sky
+location. The output channel and pixel position of the star is deter mined via the raDec2Pix
+library. This pixel position determines which triangle contains the sta r’s central pixel, which
+determines which three (of five) PRFs are used to compute the sta r’s final image.
+Pixel values for the star are determined from each of the three PR Fs by evaluating the
+PRF polynomials at the sub-pixel position of the star. On each pixel in the PRF pixel array
+the sub-pixel region containing the star’s sub-pixel coordinates is chosen. The polynomial
+in that sub-pixel region is evaluated at the star’s sub-pixel location , providing the relative
+flux for that pixel. This results in an array of n×npixels reflecting the relative flux of the
+star based on that PRF.
+Whenthesub-pixel locationofastar isclosetotheboundarybetwe en sub-pixel patches,
+the following method smoothes out inter-patch discontinuities. The pixel value is evaluated
+using all nearby polynomial patches (two patches near an edge, fo ur patches near a corner).
+Adjacent polynomial patches can be evaluated at these positions b ecause patches are defined
+using data that extend into adjacent patches ( §3.2). For simplicity we consider the case of
+an edge, where we have two values v1andv2from the two polynomial patches adjacent at
+that edge. These values are smoothly interpolated using the distan cezof the star’s sub-pixel– 5 –
+position from the edge with the formula v=w(z)v1+(1−w(z))v2. Here the weight w(z)
+is given, assuming zis normalized so that it ranges from 0 on one side of the edge to 1 on
+the other side of the edge, by (Warner 1983)
+w(z) =f(z)
+f(z)+f(1−z), f(z) =/braceleftbiggexp(−1/za)z >0
+0 z≤0.(1)
+The scale factor adetermines the steepness of the exponential curve. w(z) has the property
+that itisequal to0 for z≤0, equal to 1for z≥1, andequal to0.5for z= 0.5. Further, w(z)
+has continuous derivatives of all order, even at z= 0 and z= 1. This approach smoothly
+eliminates the discontinuity at the boundary between the polynomial patches.
+The above process is repeated for all three PRFs defined on the tr iangle containing the
+star’s central pixel, giving three n×npixel arrays, one at the location of each PRF. These
+pixel arraysarelinearly interpolatedintheplaneofthetriangle, givin g thefinal relative pixel
+values for this star. These relative pixel values are multiplied by the t otal flux of the star in
+theKeplerbandpass via the Keplermagnitude given by the Kepler input catalog. The final
+pixel array is then placed in the output channel pixel array in the loc ation corresponding to
+the pixel position of the star. Repeating this process for every st ar on an output channel
+results in a synthetic image that should match the actual image obse rved in flight, assuming
+the KIC is accurate (Figure 2).
+3. Measurement of the PRF
+The PRF was measured during the commissioning phase of the Keplermission, after
+the photometer was brought into final focus. The PRF measureme nt strategy was to col-
+lect observations of bright, uncrowded, unsaturated stars, wh ich are used to fit the PRF
+polynomial patches.
+3.1. In-flight Observations
+The PRF measurement used 19,189 stars in the Keplerfield with magnitudes between
+12 and 13 that did not have significant ( >30%) flux from other stars in a 21 ×21 pixel
+aperture centered on the star. These stars were observed for 242 cadences of 14.7 minute
+duration, half a long cadence, to reduce the time required to perfo rm the observations. Data
+for the PRF measurement were obtained from each odd cadence, w hile the spacecraft slewed
+to a new dither location during the even cadences. These 121 PRF da ta cadences visited the
+achieved locationsshown inFigure3, which were used inthePRF compu tation. The orderof– 6 –
+visitationofthispatternwasrandomizedtominimizetheimpactoftime -varyingsystematics.
+The pixel values for each star are converted to relative pixel value s by normalizing by the
+total flux from that star. The pixels from each star measurement of the 121 data cadences
+are treated as independent data points, resulting in 2,321,869 obse rvations distributed on 84
+output channels. The distribution was uneven, with some channels h aving many more stars
+than others.
+3.2. The PRF Computation
+Two types of PRF arecomputed foreach output channel: asingle PR F using all targets,
+and a set of five PRFs to capture intra-channel PRF variation as de scribed in §2.3. When
+computing the five-PRF set, the data for each of the five PRFs are selected from a region
+near where that PRF is defined. For each corner PRF the data is sele cted from a square
+extending from that corner into the input channel. For the centra l PRF the data is selected
+from a square centered on the output channel. The size of each sq uare is set to include a
+specified minimum number of targets, so the squares on a crowded c hannel will be smaller
+than the squares on a channel with fewer stars. We find that ten s tars is sufficient to create
+PRFs that capture the cores with good quality.
+Once the data for a particular PRF is selected, the pixel data is grou ped in two ways:
+(1) The integer pixel coordinates of each pixel are registered ont o then×nPRF pixel
+array so that the central pixel of the star corresponds to the c entral pixel of the PRF pixel
+array. (2) Each star’s pixel is placed into a sub-pixel bin correspon ding to the sub-pixel
+region containing that star. The sub-pixel bin covers a larger area than the sub-pixel region,
+providing overlapping data used for the polynomial fit, minimizing disco ntinuities between
+adjacent polynomial patches. For each pixel in the PRF pixel array , and for each sub-pixel
+region in that pixel, the kth pixel value pkis collected with its sub-pixel coordinates ( xk,yk).
+The PRF polynomial patch PRF( xk,yk) for that sub-pixel region is the polynomial that
+minimizes
+χ2=N/summationdisplay
+k=1/bracketleftbigg1
+σk(pk−PRF(xk,yk))/bracketrightbigg2
+. (2)
+HereNis the number of pixel values in this sub-pixel region for this pixel, and σkis the
+uncertainty in the measurement of pk(Figure 4).
+The order of the polynomial is determined by the modified Akaike infor mation criterion
+(Akaike 1974). If
+µ(o) =1
+NN/summationdisplay
+k=1(pk−PRFo(xk,yk))2. (3)– 7 –
+is the mean square error of the polynomial fit PRF ofor a given order o, then Akaike’s
+modified information criterion selects the order that minimizes
+2c+Nlog(µ(o))+2c(c−1)
+N−c−1(4)
+wherecis the number of coefficients in the polynomial PRF ofor order o. The polynomial
+order will vary from sub-pixel region to sub-pixel region. The minima l solution to equation
+(2) is found using a fitting method that is robust against outliers.
+A difficulty was encountered when computing PRFs in a five-PRF set. W hile the data
+was sufficient to capture the core of the PRF, the far wings were hig hly vulnerable to back-
+ground pollution from dim background stars due to the small number of targets used. In
+contrast, the single PRFs computed on the same output channel u sing all the targets had,
+for the most part, well-behaved wings (we occasionally hand-edited the single PRF far wings
+to remove obvious background pollution). The solution was to replac e polynomials in the
+far wings of the PRFs in the 5-PRF set with those from the correspo nding pixel and sub-
+pixel regions from the single PRF. The region of replacement was det ermined by computing
+a contour of a manually determined background value that encloses the PRF core in each
+PRF of the five-PRF set. All polynomial patches outside that conto ur were replaced by
+polynomials from the single PRF. The resulting discontinuities were of t he same magnitude
+as the discontinuities that already existed between polynomial patc hes.
+The resulting PRFs do a good job of simulating pixel flux from the star s in the KIC, as
+shown in Figure 2. We find that central pixel energies vary across t heKeplerfocal plane,
+with 11% of the output channels having central pixel energies less t han 0.3, 20% between 0.3
+and 0.4, 37% between 0.4 and 0.5, 28% between 0.5 and 0.6, and 4% betw een 0.6 and 0.7.
+There are several possible sources of PRF error, including:
+•Changes in the statistics of spacecraft pointing jitter. This has re mained stationary
+over time but is being monitored.
+•Focus has changed slightly but measurably over time due to water de sorption and
+thermal variations (Jenkins et.al. 2010a).
+•KIC errors, including variable stars and blends, can distort the mea sured PRF and
+cause errors for individual stars.
+•CCD non-uniformities.
+•The PRF computation did not attempt to include effects due to star c olor. Pre-flight
+simulated PRFs based on Code V simulations of the optical PSF from Ba ll Aerospace– 8 –
+and Technologies Corporation indicate that color can have a non-tr ivial effect on the
+PRF.
+The impact and mitigation of these errors is an area of active investig ation.
+3.3. The FPG/PRF Iteration
+The computation described in §3.2 assumes that the pixel data have been calibrated,
+had their background removed, and that that Keplerattitude and FPG model are known.
+Pixel data calibration and background removal are performed by t he same software from the
+KeplerSOC pipeline that is used to process Keplerscience data (Jenkins et.al. 2010b). The
+Keplerattitude and FPG model, however, are based on PRF centroiding ( §4.2). Therefore
+an iterative approach is required.
+Two SOC modules are used: CAL, which performs pixel-level calibratio n, and PA which
+performs background characterization and removal as well as st ellar centroiding and compu-
+tation of motion polynomials (PA also performs aperture photometr y during Keplerscience
+processing, but PRFdoes not use this feature). Background-re moved pixels arenot persisted
+in the SOC pipeline, so the background removal function of PA must b e called with each
+iteration.
+PRF processing begins with a non-iterative bootstrap phase: Pixel data is calibrated by
+CAL, then PAiscalledwith apre-launch estimate ofthePRF andFPGmo dels thatareused
+for centroiding and motion polynomial creation. At this point the iter ative loop is entered:
+the FPG calculation provides an updated FPG model and attitude solu tion; PA performs
+background removal; the PRF computation described in §3.2 is performed; PA performs
+centroiding and motion polynomial computation using the updated PR F. The iteration is
+repeated until the FPG model is observed to converge.
+To obtain convergence, it was necessary to “pin” the centroid of t he PRF computed in
+§3.2 to a specific location because both PRF and FPG can move the locat ion of a star image
+on a CCD. We chose to constrain the PRF so that the value-weighted centroid of all input
+pixel values is at the center of the PRF array.– 9 –
+4. PRF Applications
+4.1. Optimal Pixel Selection
+There is a limit of 5 .4×106pixels per Keplerlong cadence. Among these pixels are
+those chosen for stellar Keplertargets (Bathala et.al. 2010). These pixels are selected to
+support high-precision aperture photometry of these targets: for each stellar target those
+pixels are selected which maximize the signal-to-noise ratio (SNR) for that target.
+The selection process relies on synthetic images generated using th e PRF, KIC and a
+zodiacal light model. For each target, two synthetic images are cre ated: 1) an image that
+contains the target star only (“target”) and 2) all stars except the target star, plus zodiacal
+light (“scene”). Both images include effects of saturation, charge transfer efficiency and
+smear induced by the lack of a shutter. The noise for each pixel p(in electrons) contains
+contributions from shot, read ( σread) and quantization noise ( σquant) (Caldwell et.al. 2010)
+and is computed as σ=/radicalBig
+ptarget+pscene+σ2
+read+σ2
+quant. The SNR of the pixel is ptarget/σ.
+The pixels are sorted in decreasing order of SNR. Starting with the fi rst pixel, as each
+pixel isaddedtheSNRofthecollectionofpixelsiscomputed, andthec ollectionofpixelsthat
+maximize the collection’s SNR defines the optimal aperture for this ta rget. Margin against
+PRF and KIC errors is provided by a ring of pixels placed around the op timal aperture,
+as well as a column of pixels on the left for undershoot correction (H aas et al. 2010). The
+result is the requested aperture for storage and downlinking.
+Software on the Keplerspacecraft supports 1024 arbitrarily shaped spacecraft apertures ,
+far fewer than the number of requested apertures. Therefore each requested aperture is
+assigned to the spacecraft aperture with the smallest number of p ixels that contains the
+requested aperture. Figure 5 shows example optimal and spacecr aft apertures.
+A synthetic image containing all stars in the KIC is used to determine b ackground
+pixels that are stored, downlinked and used to build and remove a bac kground model in
+SOC processing of the science targets.
+4.2. Stellar Centroids
+Centroid locations for unsaturated, sufficiently bright target sta rs are determined by
+PRF fitting: a non-linear fit is performed which varies the location and flux of the PRF,
+minimizing the χ2of the pixel-wise difference of the PRF-based pixel image ( §2.3) and the
+flight pixel values for that target. The resulting centroids show hig h stability, with changes– 10 –
+over time of about 0 .1 millipixels (after removing systematic effects). Errors in the PRF
+model induce local biases, which exhibit a median of about 1 millipixel and median absolute
+deviation of 22 millipixels pixels, across the focal plane.
+We thank the larger Keplerteam for their support and hard work. Funding for the
+Keplermission is provided by NASA, Science Mission Directorate.
+Facilities: Kepler.
+REFERENCES
+Akaike, H., 1974 IEEE Transactions on Automatic Control 19 (6): 7 16
+Argabright, V. S., et al., 2008 SPIE 7010
+Bathala, N., et al., 2010 ApJ, in press
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+Caldwell, D. A., et al., 2010 ApJ, in press
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+Jenkins, J. M., et al., 2010 ApJ, in press
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+Lauer, T. R., 1999 PASP, 111 (765):1434
+Warner, F. W., 1983, Foundations of Differentiable Manifolds and Lie G roups, (Springer)
+This preprint was prepared with the AAS L ATEX macros v5.2.– 11 –pixels
+24681012144681012
+pixelspixels
+246810121446810120 5 10246810
+pixels0 5 102
+4
+6
+8
+10
+Fig. 1.— Two example PRFs. Left column: a PRF near the edge of the fo cal plane. Right
+column: a PRF near the focal plane center. Top row: the PRF model contours. Bottom
+row: the PRF converted into pixel values for a star centered on a p ixel.– 12 –
+Log10 of flight image
+490500510520530540550540
+550
+560
+570
+580
+590
+600
+610
+620Ratio of flight image to synthetic image
+  
+490500510520530540550540
+550
+560
+570
+580
+590
+600
+610
+620
+00.20.40.60.811.21.41.61.8Log10 of synthetic image
+490500510520530540550540
+550
+560
+570
+580
+590
+600
+610
+620PRF error
+pattern
+error due
+to saturation
+Spurious stars
+in catalogCatalog magnitude
+error
+Fig. 2.— Left: Synthetic image created using a PRF and the Kepler inpu t catalog. Center:
+Ratio of the the synthetic image and the flight image, indicating variou s examples of error
+sources. Right: Flight image for the same region of sky.– 13 –
+−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.4−0.200.20.40.6column offset (pixels)
+row offset (pixels)
+Fig. 3.— Commanded (+) and achieved ( ◦) dither pattern used to measure the PRF. The
+discrepancy is due to the early state of calibration of the fine guidan ce trackers during
+commissioning.– 14 –
+−0.5−0.4−0.3−0.2−0.10
+−0.3−0.2−0.100.10.350.40.450.50.550.60.650.7
+sub−row coordinatepolynomial fit and fitted data for a sub−pixel region
+sub−column coordinateobserved relative flux
+Fig. 4.— An example of a PRF polynomial patch fitted to data. The doma in of the data is
+larger than the domain of the patch to reduce the size of discontinu ities between patches.– 15 –
+Fig. 5.— Optimal and spacecraft apertures for three Keplertargets. Left: the optimal
+apertures for the targets in black and the spacecraft aperture s in grey. Right: the synthetic
+image containing the targets and other stars.
\ No newline at end of file
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+arXiv:1001.0332v2  [gr-qc]  2 Jun 2010Fluctuating Twistor-Beam Solutions and Holographic
+Pre-Quantum Kerr-Schild Geometry.
+Alexander Burinskii
+Laboratory of Theoretical Physics, NSI Russian Academy of S ciences,
+B.Tulskaya 52, Moscow, 115191, Russia
+E-mail:bur@ibrae.ac.ru
+Abstract. Kerr-Schild (KS) geometry is based on a congruence of twisto r null lines which
+forms a holographic space-time determined by the Kerr theor em. We describe in details
+integration of the non-stationary Debney-Kerr-Schild equ ations for electromagnetic excitations
+of black-holes takingintoaccount theconsistent back-rea ction tometric. The exact KSsolutions
+havetheform ofsingular beam-likepulses supportedontwis tor nulllines oftheKerrcongruence.
+These twistor-beam pulses have very strong back reaction to metric and BH horizon and
+produce a fluctuating holographic KS geometry which takes an intermediate position between
+the Classical and Quantum gravity.
+1. Introduction
+Singular pp-waves [1], playing the role of plane waves in gra vity, differ drastically from plane
+waves in flat spacetime, which is one of the origin of incompat ibility of the Quantum theory
+and Gravity. Quantum theory works basically in momentum spa ce, while Gravity demands
+explicit representation in configuration space-time. Twis tor theory forms a bridge between
+them. Geometrically, twistor is a null line formed by a pair ( xµ, θα),whereθαis a two-
+component spinor joined to the point xµ∈M4.This spinor fixes a null direction σµ
+˙αα¯θ˙αθα
+corresponding to momentum pµof a massless particle. A plane wave in twistor coordinates
+TI={θα, µ˙α}, µ˙α=xνσν
+˙ααθαhas the form exp {ixµσµ
+˙αα˜θ˙αθα}and may be transformed to
+twistor space by a ‘twistor’ Fourier transform. The result, [2](p.2.5), corresponds to a singular
+beam in direction pµsupported on the twistor null line with coordinates TI={θα, µ˙α}.Such
+twistor-beams are the sources of singular pp-wave solution s in gravity, and similar twistor-beams
+are obtained for typical exact electromagnetic (em) excita tions of black-holes (BH) [3] presented
+in the KS class of the algebraically special solutions [4] wh ich cover a wide range of the rotating
+andnon-rotating BH’s andcosmological solutions. Appeara nceof thetwistor-beams intheexact
+solutions is not accidental, since twistors form a skeleton of the KS space-time.
+In this work we give a detailed description of integration of the Debney-Kerr-Schild equations
+[4] which describe the exact em excitations of the Kerr black -hole and corresponding back
+reaction to metric1consistent with the Einstein-Maxwell equations with avera ged stress-energy
+tensor.
+TheKSgeometryhastwosheeted holographictwistorstructu rewhichturnsouttobeperfectly
+adapted to quantum treatment suggested in [5, 6], and we show that the resulting set of the
+1In this treatment we neglect by the recoil of the excitations on the position of the BH.solutions forms a fluctuating pre-quantum geometry taking i ntermediate position between the
+Classical and Quantum gravity.
+2. Twistor structure of the KS geometry
+The KS solutions are based on the KS form of metric
+gµν=ηµν+2Hkµkν, (1)
+in whichηµνis metric of auxiliary Minkowski space-time M4andkµis a field of null directions,
+forming a principal null congruence (PNC) K.The BH solutions may be represented in two
+different forms g±
+µν=ηµν+2Hk±
+µk±
+ν,werekµ±(x),(x=xµ∈M4) correspond to two different
+vector fields tangent to different congruences K±.The directions k±
+µare determined in the
+Cartesian null coordinates u= (z−t)/√
+2, v= (z+t)/√
+2, ζ= (x+iy)/√
+2,¯ζ=
+(x−iy)/√
+2 by the form
+k(±)
+µdxµ=P−1(du+¯Y±dζ+Y±d¯ζ−Y±¯Y(±)dv) (2)
+which depends on the complex angular coordinate Y=eiφtanθ
+2.Two solutions Y±(x) are
+determined by the Kerr Theorem [1, 4, 7, 8] which states that the (necessary for solutions
+of type D) geodesic and shear-free null congruences in M4are generated by algebraic equation
+F= 0,whereF(Zp) is an arbitrary holomorphic function of the projective twi stor coordinates.2
+The Kerr congruence represents a vortex of null lines, which covers the spacetime twice: in
+the form of ingoing ( kµ−∈ K−) and outgoing vector fields ( kµ+∈ K+). It forms two sheets of
+the KS geometry with two different metrics g±
+µνon the same spacetime M4.
+−10−50510
+−10−50510−10−50510Z 
+Figure 1. The Kerr singular ring
+and the Kerr congruence of twistor
+null lines.0 2 4 6 8 10 12 14 16 18 20−8−6−4−202468
+in−photons out−    
+photons out−    
+photons 
+conformal diagram of
+Minkowski space−timeunfolded conformal 
+diagram of the Kerr
+        space−time I− I+ 
+I− I+ 
+Figure 2. Penrose conformal di-
+agrams. Unfolding of the auxil-
+iaryM4space of the Kerr space-
+time yields twosheeted structure of
+a pre-quantum BH spacetime.
+The em field has to be aligned with the rays of PNC and turns out t o be different on the
+in- and out- sheets, and these two fields should not be mixed, w hich is ignored usually in the
+perturbative approach, leading to drastic discrepancy in t he form of the fundamental solutions.
+The typical exact em solutions on the Kerr background have th e form of singular twistor-
+beams propagating along the twistor null rays of the Kerr PNC , contrary to the smooth angular
+dependence of the typical wave solutions used perturbative ly!
+2Projective twistor coordinates Zp, p= 1,2,3 are related with twistor coordinates {θα, µ˙α}as follows
+TI=θ1(1,Zp).3. Holographic KS structure
+For a long time this twosheetedness was a mystery of the Kerr B H (see refs in [9]). It was
+suggested by Israel (1968) to truncate the second sheet and r eplace it by a rotating disk-like
+bubble covering the Kerr singular ring. Alternatively, the Kerr singular ring was considered as a
+closed‘Alice’ stringformingagateto‘Alice’ mirrorworld oftheadvancedfields[10]. Holographic
+approach resolves this problem unifying the both points of v iew: the source of Kerr solution has
+tobeconsideredasamembraneseparatingthein- andout- par tsoftheKSspace. Itissupported
+by the membrane paradigm and by the obtained recently struct ure of the consistent Kerr-
+Newman source [11]. The both sheets of the KS space are necess ary for description of quantum
+fields in gravity. In particular, Gibbons states in [5] that t he curved spacetime Mshould be
+separated into two time-ordered regions M−andM+which are associated with ingoing and
+outgoing vacuum states |0−>and|0+> .If the source is absent, the KS basic solutions may
+be extended analytically from in- to out-sheet and vice vers a. Presence of the source breaks
+this analyticity, separating the retarded and advanced fiel ds, which allows one to consider the
+BH evaporation as a scattering of the in-vacuum on the BH-sou rce. Similarly, in [6] authors
+consider a pre-quantum BH spacetime with separated the in- a nd out- sheets which correspond
+to a holographic correspondence: the source forms a hologra phically dual boundary (membrane)
+separating the in- and out- regions.3The usual Penrose conformal diagram, containing the in-
+and out-fields on the same of M4,has to be unfolded in the holographic interpretation to spli t
+twosheetedness, as it is shown on the Fig.2. The null rays of K err congruence perform the
+lightlike projection of the past null infinity I−,so the Kerr source appears as a holographic
+image of the data on the past null infinity I−.
+Conformal structure of the KN solution is determined by comp lex function Y(x).Tetrad
+derivatives ∂a=eµ
+a∂µof thefunction Y(x) determineprincipal parameters of theKSholographic
+projection. In particular, function Y(x) for the shear-free and geodesic congruences of the Kerr
+theorem has to satisfy the conditions
+Y,2=Y,4= 0, (3)
+which show that the expression dY=Y,aea=Y,1e1+Y,3e3is gradient of the complex null
+surfacesY=const., which form a fiber bundle of the KS space-time with fibers span ned by the
+tetrad forms e1ande3,providing conformal properties of the KS holographic proje ction.
+4. Stationary twistor-beam KS solutions
+For the Kerr congruence function F(Y,x) of the Kerr theorem is to be quadratic in Y,and the
+eq.F= 0 may resolved in explicit form. The corresponding solutio nsY(x) determine the
+stationary Kerr congruence via (2) and the adapted (oblate s pheroidal) coordinate system r,θ,φ
+in which
+Y=eiφtanθ
+2. (4)
+FunctionF(Y,x) determinesalsothefunction Huptoanarbitraryfunction ψ(Y),andtherefore,
+the KS metric (1). In agreement with [4]
+H=1
+2[m(Z+¯Z)P−1−|ψ|2Z¯ZP−2], (5)
+whereZ=Y,1and (Z/P)−1=−dF/dY.Complex function Z=ρ+iωcharacterizes the
+conformal properties of the congruence: expansion ρand rotation ωof the projected image. For
+the Kerr-Newman BH solution at rest, Zis inversely proportional to a complex radial distance
+3In the model [11] it is a domain wall boundary of a rotating dis k-like source forming a superconducting bubble.Z=−P/(r+iacosθ) where function Pis an extra conformal factor determined by Killing
+vector of the solution [4].
+In particular, for the Kerr and Kerr-Newman (KN) geometries at rest
+H=mr−|ψ|2/2
+r2+a2cos2θ. (6)
+The KN electromagnetic field is determined by the vector pote ntial
+α=αµdxµ=−1
+2Re[(ψ
+r+iacosθ)e3+χd¯Y], (7)
+whereχ= 2/integraltext(1+Y¯Y)−2ψdY ,which obeys the alignment condition αµkµ= 0.
+For the Kerr solution mis mass of BH and ψ= 0.For the KN solution ψ=q=const..
+However, any nonconstant holomorphic function ψ(Y) yields also an exact KS solution, [4].
+On the other hand, any nonconstant holomorphic functions on sphere acquire at least one
+pole. A single pole at Y=Yi, ψi(Y) =qi/(Y−Yi) produces the beam in angular directions
+Yi=eiφitanθi
+2.The function ψ(Y) acts immediately on the function Hwhich determines the
+metric and position of the horizon. The given in [12] analysi s showed that electromagnetic
+beams have very strong back reaction to metric and deform top ologically the horizon, forming
+in the horizon the holes which allow matter to escape interio r. Forψ(Y) =/summationtext
+iqi
+Y−Yi,the exact
+solutions have several beams in angular directions Yi=eiφitanθi
+2,leading to the horizon with
+many holes. In far zone the twistor beams tend to the known exa ct singular pp-wave solutions
+[1].
+5. Pre-quantum fluctuating KS geometry
+The stationary KS beamlike solutions may be generalized to t ime-dependent wave pulses, [3].
+Since the horizon is extra sensitive to electromagnetic exc itations, it may also be sensitive to the
+vacuumelectromagnetic field, andthevacuum beampulses may producea fine-grainedstructure
+of fluctuating microholes in the horizon, allowing radiatio n to escape interior of black-hole.
+Figure 3. Twistor-beam pulses
+perforate the BH horizon forming
+its fluctuating fine-grained struc-
+ture.
+Twistor-beam pulses depend on a retarded time τand have to obey to the non-stationary
+Debney-Kerr-Schild(DKS) equations [4]. Thecorrespondin gsolutions acquirean extraradiative
+termγ(Y,τ).The long-term attack on the DKS equations [3], has led to the o btained time-
+dependentsolutionsofthefluctuatingKerr-Schildspace-t imes. Derivation ofthetime-dependent
+solutions will be given in the next section. Here we give a bri ef preliminary description of the
+general structure of the DKS solutions obtained by integrat ion in [4].The general em field is described by the self-dual tetrad comp onents,
+F12=AZ2,F31=γZ−(AZ),1, (8)
+whereFab=eµ
+aeν
+bFµν.We will assume that the mass of BH is much greater then the ener gy
+of the BH excitation. It allows us to neglect by recoil and con sider the Kerr congruence as
+stationary. Fluctuation of the metric will be related with fl uctuation of the em field. By these
+assumptions we obtain, [3], that the exact time-dependent s olutions describe the em radiation
+from BH which contains two components:
+a) a set of the singular beam pulses (determined by function ψ(Y,τ)) propagating along the
+Kerr PNC and breaking the topology and impenetrability of th e horizon; and
+b)the regularized radiative component (determined by γreg(Y,τ)) which is smooth and
+determines evaporation of the black-hole.
+The obtained solutions describe excitations of singular el ectromagnetic beams A=
+ψ(Y,τ)/P2on the Kerr-Schild background and the back reaction of the si ngular field ψ(Y,τ) on
+the metric leading to fine-grained fluctuations of the metric and black-hole horizon.
+The most essential difference from the stationary case is the a ppearance of an extra field
+γ(Y,τ).This term is absent in the expression for (6), and therefore, it does not have immediate
+action on the KS metric. One sees also from (8) that it generat es the null EM radiation along
+the Kerr congruence which fall off asymptotically as ∼Z=P/(r+iacosθ)∼1/r,and so, it is
+the leading term at infinity. The corresponding components o f the stress-energy tensor
+Tµν=T33e3
+µe3
+ν∼=1
+2¯γγkµkν (9)
+describe a flow of energy along the Kerr congruence and are kno wn as radiation of the Vaidya
+‘shining star’ solution [1]. This term appears also in the Ki nnersley ‘photon rocket’ solution [1],
+where it is related with acceleration of the source.
+It shows explicitly that the horizon turns out to be covered b y fluctuating micro-holes which
+make it penetrable for outgoing radiation. In the same time, the BH radiation is determined
+by the smooth field γreg(Y,τ),corresponding to the Vaidya “shining star” radiation. The
+holographic space-time forms a fluctuating twosheeted pre- geometry which reflects the dynamics
+of singular beam pulses. This pre-geometry is classical, bu t has to be still regularized to get the
+usual smooth classical space-time. In this sense, it takes a n intermediate position between the
+classical and quantum gravity.
+Note also that this structure may be generalized by using the Kerr theorem with the higher
+degree inYfunctionsF(Y),which leads to the multi-particle KS solutions [13] having t he
+complicate networks of twistor-beams.
+6. Time-dependent Kerr-Schild solutions
+The nearest time-dependent generalization of (4) is given b y the form
+ψ(Y,τ) =/summationdisplay
+ici(τ)(Y−Yi)−1, (10)
+where one assumes that an elementary beam has ci(τ) =qi(τ)eiωiτwhereqi(τ) is amplitude and
+ωiis a carrier frequency. Similar to stationary case, the non- stationary Kerr-Schild solutions
+contain the singular beams which change the structure of bla ck hole horizon, but the beams may
+acquire the form of time-dependent pulses. Since the mass of a black hole is much more than
+the energy of its excitation, we will neglect by recoil and ta ke the assumption that the black hole
+is at rest and the Kerr congruence remains undisturbed. It co rresponds to P= 2−1/2(1+Y¯Y)
+and˙P= 0.The obtained in [4] DKS general equations are the equations f or functions Aandγwhich
+determine electromagnetic field
+A,2−2Z−1¯ZY,3A= 0, A,4= 0, (11)
+DA+¯Z−1γ,2−Z−1Y,3γ= 0, (12)
+and the equations for function Mwhich determine gravitational field taking into account the
+back reaction caused by electromagnetic field
+M,2−3Z−1¯ZY,3M=A¯γ¯Z, (13)
+DM=1
+2γ¯γ. (14)
+The used here operator Dis
+D=∂3−Z−1Y,3∂1−¯Z−1¯Y,3∂2. (15)
+General equations for the nonstationary EM KS solutions wer e obtained by by Debney, Kerr
+and Schild [4] in 1968. However, in [4] the equations were ful ly integrated out only for the
+particular case γ= 0 which corresponds to stationary solutions.4
+The first non-stationary wave EM solutions on the KS backgrou nd were obtained in [14] and
+contained singular beams along the ±z−half-axis. There appeared the conjecture that there is
+no the exact regular time-dependent solutions at all, and fo r very seldom exclusions the exact
+solutions should have the singular beams.
+The DKS equations are simplified by using the DKS relation
+Y,3=−ZP¯Y/P. (16)
+The equation (11) for function Atakes the same form as in stationary case,
+(AP2),2= 0, A,4= 0, (17)
+where the general solution is A=ψ/P2,and function ψhas to obey
+ψ,2=ψ,4= 0. (18)
+To obtain the non-stationary KS solutions we introduce a com plex retarded-time parameter τ
+which satisfies the relations
+(τ),2= (τ),4= 0. (19)
+It allows us to represent the equation (11) in the form ( AP2),2= 0,and to get solution which
+has the same form
+A=ψ(Y,τ)/P2(20)
+4The degree of difficulty of this problem is described in the exc erptions from the given by Roy Kerr in [15] (see
+Part 3 on the page 2481) review on the history of the Kerr solut ion. In this review Roy calls the eqs. (8), (17) and
+(18) as “easy” ones which ”... showed that the Einstein-Maxwell field depends on three functions, M,A,andγ
+... restricted by “hard” field equations...” [The “hard” equations are the EM eq. (12) and gravitational e quation
+(13). AB.] ... ”I could not solve the later unless γ= 0so I temporarily took this as an additional assumption
+and continued. This led to the complete charged Kerr-Schild metric metrics including, of course, charged Kerr.
+The null congruence for the later is the same as the Kerr congr uence but the EM field depends on an arbitrary
+function of a complex variable...If it is a more complicated function then the EM field will probably be singular.
+[This is exactly the case of stationary beam-like KS solutio ns. AB.] At that point I turned the problem over to
+G.C. Debney ... to see if he could solve the problem when γ/ne}ationslash= 0. ... In the end it became clear that the tree of us
+could not solve the general problem.”with the exception of the extra dependence of function ψfrom the retarded-time parameter τ.
+The principal difference from the stationary case is containe d in the second electromagnetic
+DKS equation. Action of operator Don the variables Y,¯Yandτis
+DY=D¯Y= 0. (21)
+For the considered here case P= 2−1/2(1+Y¯Y), and˙P= 0,which yields
+Dτ=−1/P . (22)
+Using (16), (21) and (22), we reduce the equation (12) to the f orm
+˙A=−(γP),¯Y, (23)
+where˙( )≡∂τ. Integrating this equation we obtain
+γ=−P−1/integraldisplay
+˙Ad¯Y=21/2˙ψ
+P2Y+φ(Y,τ)/P, (24)
+whereφis an arbitrary analytic function of Yandτ.
+This solution shows that any non-stationarity in electromagnetic field ( ˙A/ne}ationslash= 0) generates an
+extra function γwhich, in accord with (8), generates also the lightlike elec tromagnetic radiation
+along the Kerr congruence. Such a radiation is well-known fo r the Vaidya ‘shining star’ solution
+[1], in which the field A=ψP−2is absent and γis incoherent, being related with the loss of total
+mass into radiation. The KS analog of this Vaidya relation is obtained from (14) by substitution
+M=m/P3, (25)
+and using (16), (21) and (22), and also the fact that t=1
+2(τ+ ¯τ),
+˙m=−1
+2P4γ(reg)¯γ(reg). (26)
+It is one of two gravitational equations determining self-c onsistency of the Kerr-Schild solution
+[4]. The most important consequence following from DKS equa tions in the non-stationary case
+is the fact that the field γappears inevitable, however it does not contribute to defor mation of
+the horizon, since it is absent in the function Hof (6). Its back reaction on the metric is smooth
+and circumstantial, acting only via the slowly decreasing m ass parameter m.Therefore, in the
+non-stationary Kerr-Schild case we obtain that the lightlike fields, determined by functions ψ
+andγ,have essentially different impact on the horizon.
+Functionψ=ψ(τ,Y) obeys the equation (18) which shows that the retarded time τhas to
+satisfy the conditions similar to (3), and therefore, gradi ent ofτis to be aligned to congruence,
+kµτ,µ= 0. (27)
+It was obtained in [14] that the corresponding retarded-tim e parameter has the form
+τ=t−r−iacosθ. (28)
+Since function φcontributes only to γit does not impact on the form of the horizon too.
+Important role of this function is obtained from the analysi s of the gravitational Kerr-Schild
+equation (13) which is reduced by using (16), (21) and (25) to the form
+m,¯Y=ψ¯γP. (29)If we note that γ∼˙ψ≈iωψ,we obtain that the r.h.s. of this equation tends to zero in the
+low-frequency limit, as well as the r.h.s of the equation for ˙m.So, the full solution will tend to
+consistent with gravity at least in the low-frequency limit [9]. For the simplicity we consider
+first the case of a single time-dependent beam generated by a s ingle pole at Y=Yi.In this case
+˙ψi= ˙ci(τ)/(Y−Yi) (30)
+and the solution (24) looks singular. However, we have still the free function φ,form of which
+and parameters (time-dependence and position of its pole) m ay be tuned to cancel the pole of
+function ˙ψi,and thus, regularize the solution. We define the analytic in Yfunction
+Pi=P(Y,¯Yi) = 2−1/2(1+Y¯Yi) (31)
+and set
+φ(tun)
+i(Y,τ) =−21/2˙ci(τ)
+Y(Y−Yi)Pi. (32)
+Onesees that the required analyticity of function φ(tun)
+i(Y,τ) inYis ensured. Using the equality
+(Pi−P)/(Yi−Y) =Y√
+2(¯Yi−¯Y)
+(Yi−Y), (33)
+we obtain that the regularized solution γ(reg)
+i=21/2˙ψi
+P2Y+φ(tun)
+i(Y,τ)/P ,takes the form
+γ(reg)
+i=1
+P2˙ci
+Pi[¯Yi−¯Y
+Yi−Y]. (34)
+The r.h.s. of the equation (29) for this regular solution tak es the form
+ψi¯γ(reg)
+iP=−ci˙¯ci
+P¯Pi(¯Y−¯Yi). (35)
+The equation (29) takes the form
+m=−/integraldisplay
+d¯Yci˙¯ci
+P¯Pi(¯Y−¯Yi)(36)
+and may now be integrated by using the Cauchy integral formul a,/contintegraltextf(z)dz
+z−z′= 2πif(z′),and we obtain the expression
+m=m0(Y)−2πici˙¯ci
+PiPii, (37)
+containing free function m0(Y),and also the depending on Ycontribution from the residue at
+singular point ¯Yi,in which
+Pii=1√
+2(1+Yi¯Yi) (38)
+is a constant. On the real slice Pis real,Pi→Pii,and the real part of mtakes the form
+ℜe m=m0−iπci˙¯ci−˙ci¯ci
+P2
+ii, (39)wherem0is a real constant. Imaginary part is
+ℑm m=−iπci˙¯ci+ ˙ci¯ci
+P2
+ii. (40)
+Ifci(τ) is expressed via slowly varying amplitude qi(τ) and the carrier frequency ωi,
+ci(τ) =qi(τ)e−iωiτ,the impact of the carrier frequency disappears and we obtain
+ℜe m=m0−2πωi|qi|2
+P2
+ii−iπqi˙¯qi−˙qi¯qi
+P2
+ii, (41)
+So, the result of integration yields the time-dependence of mcaused by the amplitude qi(τ)
+and the weak and slow imaginary contribution
+ℑm m=−iπqi˙¯qi+ ˙qi¯qi
+P2
+ii., (42)
+meaning of which is not clear at this stage. We have
+m=m0−2πωi|qi|2
+P2
+ii−2πiqi˙¯qi
+P2
+ii=m0−2πiqi
+P2
+ii(˙¯qi−iωi¯qi). (43)
+Influence of the extra contributions from ˙ qi(τ) falls off as 1 /T,whereTis effective time of
+action of the beam. This contribution is negligible in the li mit of the very long beams. Although
+these relations may be interpreted physically as a reaction of the mass parameter to single pulse,
+they cannot be considered literally, since the imaginary ma ss is inconsistent with the KS class of
+metrics. To provide consistency with gravitational sector we enforced to perform an averaging
+of the mass term over time, which is equivalent to averaging o f the stress-energy tensor.
+Further, it is known [1] that the gravitational equation (26 ) for radiative solutions is not
+equation really, but is only a definition of the loss of mass in radiation ˙m=−1
+2˙ci˙¯ci
+|Pi|2.On the
+real slicePi→Pii,and in terms of the amplitudes of beams we obtain
+˙m=−1
+2ω2
+i|qi|2+|˙qi|2
+P2
+ii=−1
+2P2
+ii(˙qi−iωiqi)(˙¯qi−iωi¯qi). (44)
+Similar to the Vaidya ‘shining star’ solution [1], for the ca se of the many stochastic pulses,
+we assume their frequencies as uncorrelated. There appears a double sum over the interacting
+beams and extra beating between them with fluctuations which drop out after averaging. The
+surviving radiation contains only slow fluctuations of the m ass term caused by amplitudes of
+the self-correlated beams, i.e. the sum of the partial solut ions pulses. Therefore, the obtained
+solutions turn out to be consistent with respect to the Einst ein equations with averaged r.h.s.,
+which for the DKS gravitational equations is equivalent to
+m,¯Y=<Pψ¯γ(reg)>,˙m=−1
+2<P2γ(reg)¯γ(reg)>. (45)
+The averaging removes the beating of the incoherent beams, h owever, it does not remove the
+sharp back-reaction of the beams to metric and horizon, caus ed by the poles in function ψ(Y,τ)
+in agreement with (6).
+Therefore, the obtained solutions are exact and consistent with the Einstein-Maxwell system
+of equations with averaged stress-energy tensor. Recall al so that we have used the restriction
+on the absence of recoil. The case of the non-zero recoil is ve ry hard and was considered earlier
+by Kinnersley only for the non-rotating case [1]. It may be ve ry important for the problem of
+scattering of spinning particles in twistor theory.Acknowledgments
+Author thanks the Organizers and especially Elias Vagenas f or invitation to this very interesting
+conference and and the RFBR (grants 07-08-00234-a, 09-02-0 8353-) for financial support.
+[1] D.Kramer, H.Stephani, E. Herlt, M.MacCallum 1980 Exact Solutions of Einstein’s Field Equations ,
+Cambridge Univ. Press.
+[2] E. Witten 2004 Comm. Math. Phys. 252189.
+[3] A. Burinskii, First Award of Gravity Research Foundatio n 2009Gen. Rel. Grav. 412281, arXiv: 0903.3162[gr-
+qc]; details in 0903.2365[hep-th].
+[4] G.C. Debney, R.P. Kerr and A.Schild 2009 J. Math. Phys. 10, 1842.
+[5] G. W. Gibbons 1975 Comm. Math. Phys. 45191.
+[6] C.R. Stephens, G. ’t Hooft and B.F. Whiting 994 Class. Quant. Grav. 11621.
+[7] R. Penrose 1967 J. Math. Phys. 8345;
+[8] A. Burinskii 2004 Phys. Rev. D 70, 086006.
+[9] A. Burinskii, E.Elizalde, S. R. Hildebrandt and G. Magli 2002Phys. Rev. D65064039, arXiv:gr-qc/0109085.
+[10] A. Burinskii 1974 Sov.Phys.JETP ,39193; 1974 Russian Phys.J. 17, 1068, DOI:10.1007/BF00901591; 2008
+Grav. Cosmol. 14109.
+[11] A. Burinskii, 2009 Superconducting Source of the Kerr- Newman Electron, arXiv: 0910.5388[hep-th].
+[12] A. Burinskii, E. Elizalde, S.R. Hildebrandt and G. Magl i 2006Phys. Rev. D 74021502; 2006 Grav. Cosm.
+12 119.
+[13] A. Burinskii 2005 Grav. Cosmol. 11, 301; 2007 IJGMMP ,4, n.3, 437.
+[14] A. Burinskii, Grav. Cosmol. 10, (2004) 50.
+[15] Roy P. Kerr, Gen Relativ Gravit 412479 (2009); DOI 10.1007/s10714-009-0856-0.
+[16] A. Burinskii, Phys.Rev. D 67, (2003) 124024.
\ No newline at end of file
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+arXiv:1001.0333v1  [astro-ph.SR]  2 Jan 2010Version 5.2 — 28 December 2009
+Kepler-6b: A Transiting Hot Jupiter Orbiting a Metal-Rich S tar†
+Edward W. Dunham1, William J. Borucki2, David G. Koch2, Natalie M. Batalha3,
+Lars A. Buchhave4,5, Timothy M. Brown6, Douglas A. Caldwell7, William D. Cochran8,
+Michael Endl8, Debra Fischer17, Gabor F˝ ur´ esz4, Thomas N. Gautier III9, John C. Geary4,
+Ronald L. Gilliland10, Alan Gould16, Steve B. Howell11, Jon M. Jenkins7, Hans Kjeldsen15,
+David W. Latham4, Jack J. Lissauer2, Geoffrey W. Marcy12, Soren Meibom4,
+David G. Monet13, Jason F. Rowe14,7, Dimitar D. Sasselov4
+ABSTRACT
+†Based in part on observations obtained at the W. M. Keck Observat ory, which is operated by the
+University of California and the California Institute of Technology.
+1Lowell Observatory, Flagstaff, AZ 86001
+2NASA Ames Research Center, Moffett Field, CA 94035
+3San Jose State University, San Jose, CA 95192
+4Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02 138
+5Niels Bohr Institute, Copenhagen University, DK-2100 Copenhage n, Denmark
+6Las Cumbres Observatory Global Telescope, Goleta, CA 93117
+7SETI Institute, Mountain View, CA 94043
+8University of Texas, Austin, TX 78712
+9Jet Propulsion Laboratory/California Institute of Technology, Pa sadena, CA 91109
+10Space Telescope Science Institute, Baltimore, MD 21218
+11National Optical Astronomy Observatory, Tucson, AZ 85719
+12University of California, Berkeley, Berkeley, CA 94720
+13US Naval Observatory, Flagstaff Station, Flagstaff, AZ 86001
+14NASA Postdoctoral Program Fellow
+15University of Aarhus, Aarhus, Denmark
+16Lawrence Hall of Science, Berkeley, CA 94720
+17Yale University, New Haven, CT 06510– 2 –
+We announce the discovery of Kepler-6b, a transiting hot Jupiter o rbiting a
+star with unusually high metallicity, [Fe /H] = +0.34±0.04. The planet’s mass is
+about 2/3 that of Jupiter, MP= 0.67MJ, and the radius is thirty percent larger
+than that of Jupiter, RP= 1.32RJ, resulting in a density of ρP= 0.35gcm−3, a
+fairly typical value for such a planet. The orbital period is P= 3.235 days. The
+host star is both more massive than the Sun, M⋆= 1.21M⊙, and larger than the
+Sun,R⋆= 1.39R⊙.
+Subjectheadings: planetarysystems—stars: individual(Kepler-6,KIC10874614,
+2MASS 19472094+4814238) — techniques: spectroscopic
+1. INTRODUCTION
+TheKeplermission was launched on March 6, 2009 to undertake a search for Ea rth-size
+planets orbiting in the habitable zones of stars similar to the Sun. Kepleruses the transit
+photometry approach for this task (Borucki et al. 2010a). Kepler’s commissioning process
+went very well and the system is providing data of exceptional phot ometric quality (Koch
+et al. 2010a). Indeed, the final commissioning data, 9.7 days of scie nce-like observations of
+50000 stars selected from the Kepler Input Catalog (KIC) (Koch et al. 2010a), were of such
+high quality that they are now commonly, if incorrectly, referred to as the “quarter 0” data.
+A number of KeplerObjects of Interest (KOIs) simply fell out of this dataset, includin g
+the present object, known as KOI-17. Its transit signal is huge b yKeplerstandards, and
+excellent light curve results were obtained quickly. The follow-up spe ctroscopic observations
+took somewhat longer since observations were not planned to be un dertaken so soon after
+commissioning. As a result the first data intended for science, 33.5 d ays of observations of
+150000 KIC stars, became available in the meantime and were folded in with the original
+test data for the analysis presented here. The 1.3 day gap betwee n these two sets of data
+contained one transit of Kepler-6b. Haas et al. (2010) provide add itional details regarding
+theKeplerdata used in this analysis.
+Because the pace of early Keplerextrasolar planet discoveries is limited by the radial-
+velocity follow-up observations, the first objects to be announce d tend to be those most
+easily confirmed spectroscopically. Kepler-6b (= KIC 10874614, α= 19h47m20s.94,δ=
++48◦14′23′′.8, J2000, KIC r= 13.253mag) is one of these objects.– 3 –
+2. KEPLER PHOTOMETRY
+TheKeplerscience data for the primary transit search mission are the long cad ence
+data (Jenkins et al. 2010a). These consist of sums very nearly 30 m inutes in length (with
+rigorously the same integration time) of each pixel in the aperture c ontaining the target star
+inquestion. These dataproceedthroughananalysispipelinetoprod ucecorrectedpixel data,
+thensimpleunweighted aperturephotometrysums areformedtop roduceaphotometrictime
+series for each object (Jenkins et al. 2010b). The many thousand s of photometric time series
+are then processed by the transiting planet search (TPS) pipeline e lement (Jenkins et al.
+2010b). ThecandidatetransiteventsidentifiedbyTPSwerethenv ettedbyvisual inspection.
+The light curves produced by the photometry pipeline tend to show d rifts due to an
+extremely small, slow focus change (Jenkins et al. 2010a), and ther e are also sometimes low
+frequency variations in the stellar signal that can make analysis of t he transit somewhat
+problematic. These low frequency effects can be removed by modes t filters that have only an
+insignificant effect on the transit signal (Koch et al. 2010b). The un folded and folded light
+curves for Kepler-6b produced in this manner are shown in Figure 1.1
+Figure 1 illustrates two of the ways we reject false positives based o n theKeplerlight
+curves alone. The lower part of the figure shows the planetary tra nsit (referred to the left
+ordinate). Thepointsmarkedwith+and ×symbolsrefertoalternatingevenandoddtransit
+events. This aids in detecting eclipsing binary stars with nearly, but n ot quite equal primary
+and secondary minima. No such deviation is observed for Kepler-6. T he upper curve shows
+the time of secondary minimum, assuming zero eccentricity. The ver tical scale, on the right,
+is magnified and measured in parts per million to help search for a small s econdary minimum
+that might be due to occultation of an M dwarf companion. Again, no s uch signature is
+visible.
+3. FOLLOW-UP OBSERVATIONS
+The crucial importance of ground-based follow-up observations w as recognized long ago
+(Latham 2003) and a well-established plan was in place by the time of lau nch. This has
+evolved somewhat and our current approach is described by Gautie r et al. (2010). The
+key features are: 1) carry out high resolution reconnaissance sp ectroscopy to determine the
+stellar properties and the star’s suitability for radial-velocity work, and to search for signs of
+1The time series photometry and radial velocity data, including bisect or spans, may be retrieved from
+the MAST/HLSP data archive at http://archive.stsci.edu/prepds/ keplerhlsp.– 4 –
+stellar companions; 2) search for faint blended eclipsing binaries thr ough the centroid shift
+approach (Batalha et al. 2010) and by means of high resolution imagin g; and 3) detect the
+stellar reflex motion with precise radial-velocity measurements. In t he future we hope to
+improve our constraints on stellar properties by including both aste roseismology results for
+at least the brighter host stars (Gilliland et al. 2010a) and parallaxes (Monet et al. 2010) for
+every target. These results are not available for Kepler-6 at this t ime.
+3.1. Reconnaissance Spectroscopy
+The reconnaissance spectra for Kepler-6 were carried out very e arly inthe follow-up pro-
+gram, so observations were obtained with several of our resourc es for comparison purposes:
+the TRES spectrograph on the Tillinghast telescope (by L. Buchhav e and D. Latham), the
+McDonald 2.7-m Coude spectrograph (by M. Endl and W. Cochran), and the FIES spec-
+trograph on the Nordic Optical Telescope (by L. Buchhave and G. F ˝ ur´ esz). The spectra
+indicated that Kepler-6 was well suited for precise radial-velocity wo rk and no evidence
+of double lines or a stellar companion was seen. The best data on stella r properties were
+determined later and are discussed below.
+The mean radial velocity of Kepler-6 was determined by C. Chubak wit h two HIRES
+spectrausingtelluriclinestosettheirzeropoint. Observationsoft heradialvelocitystandard
+HD182488 using the same approach produced a mean value of −21.50±0.18kms−1, in good
+agreement with the adopted value of -21.508kms−1. Observations with FIES and TRES
+using the same standard star to determine their zero points were in good agreement with
+the HIRES velocity. The average of these values is shown in Table 2.
+3.2. High-Resolution Imaging and Centroid Motion
+Kepler-6 has a companion 4 .1′′distant and 3.8 magnitudes fainter as seen in both a
+HIRES guide camera image and a PHARO Jband AO image obtained with the Palomar
+Hale telescope (Gautier and Ciardi). A WIYN speckle image (Howell) sho ws only a single
+star within its 2′′square box. There are apparently two additional components bot h ap-
+proximately 5.4 magnitudes fainter and 11 .5′′distant that appear in the KIC as KIC IDs
+10874615 and 10874616. These two objects refer to a single star and erroneously appear as
+two objects in the KIC. This star is very close to the edge of the ape rture containing Kepler-
+6. The observed shift in the Keplerintensity weighted mean image centroid during transit,
+(−0.1, +0.3) millipixels in the (column, row) directions is consistent with the expec ted shift– 5 –
+if thefaint companions arenot thesource of thetransit signature . If thefainter, moredistant
+star is fully in the aperture the expected shift is (0 .0, +0.3) millipixels and if it is completely
+out of the aperture the expected shift is ( −0.1, +0.4) millipixels, both being consistent with
+the observed centroid shift. On the other hand if the closer, brigh ter background star is the
+transit source the expected shift is 30 times larger, ruling out this p ossibility. The fainter
+object is too faint to produce the observed transit depth even if it disappears entirely during
+the transit, and if it could the resulting centroid shift would be nearly a factor of 100 larger
+than the observed shift. The typical sensitivity level for detectio n of centroid shifts is on the
+order of 0.1 millipixel or somewhat less. See Batalha et al. (2010) for a detailed discussion
+of this approach for candidate vetting.
+3.3. PRECISE RADIAL VELOCITIES
+Precise velocities were obtained by using HIRES onthe Keck 1 telesco pe with iodine cell
+radial-velocity reference. A modified version of the standard HIRE S iodine cell reduction
+pipeline that includes improved cosmic ray rejection and sky brightne ss suppression was
+applied to these spectra by J. Johnson. The reduction pipeline modifi cation was necessary
+because of the faintness of this star compared to the bright star s normally observed for radial
+velocity planet search work. The nearby companion mentioned earlie r is sufficiently far away
+on the sky that it does not interfere with the HIRES work. The phas ed velocities are shown
+in Figure 2 with a fit to a circular orbit whose phase and period are fixed by the observed
+transit times.
+The Monte Carlo analysis described below shows no evidence of orbita l eccentricity. In
+this analysis the eccentricity and longitude of periastron were para meterized as esinωand
+ecosω. The probability distribution for these parameters is symmetrical w ith zero mean
+and a standard deviation of 0.029.
+The rms of the velocity residuals is 5 .7ms−1with the point nearest transit omitted from
+the fit. This point is potentially affected by the Rossiter-McLaughlin e ffect. Its residual of
+12ms−1islargerthanalltheothervelocity pointsandisinthesenseofaprog radeorbit. The
+rotational velocity given in Table 2 would result in an R-M amplitude of 20 ms−1, certainly
+detectable with further careful spectroscopy. An analysis of th e line bisectors (Figure 2)
+shows an rms of 6 .8ms−1with no organized structure that might result from a triple system
+(Mandushev et al. 2005).– 6 –
+4. DISCUSSION
+A Monte Carlo bootstrap procedure developed by Rowe and Brown, outlined in Koch
+et al. (2010b) and Borucki et al. (2010b), that simultaneously fits the light curve and the
+radial velocities, and that is consistent with stellar evolution models, was carried out to
+produce most of the stellar and planetary properties presented in Table 2. The model light
+curve (Mandel & Agol 2002) was integrated over the long cadence integration period prior
+to fitting to the Keplerdata. We note that the nearby companion star alluded to earlier
+is only about one Keplerpixel away from Kepler-6, so its light is included in the Kepler
+aperture. The fainter, more distant star is partially included in the a perture as well. This
+has the effect of diluting the depth of the transit by 3 .3±0.4%. This effect has been taken
+into account in the modeling process including the dilution uncertainty due to the partial
+inclusion of the more distant star. The derived planetary density, 0 .352+0.018
+−0.022gcm−3, places
+it in a well-populated area of the mass-radius relationship for curren tly known extrasolar
+planets.
+Several spectra obtained with HIRES without the iodine cell were su bjected to an SME
+analysis (Valenti & Piskunov 1996) by D. Fischer to provide the rema ining stellar properties
+reported in Table 2. In this analysis log gwas frozen at the transit-derived value since it
+is more reliable than the SME value, which was 4.59 if allowed to float. The discrepancy
+between the transit value of log gand the SME value is significant. Gilliland et al. (2010b)
+and Nutzman et al. (2010) find a similar effect in their comparison of log gderived from
+transit fits, asteroseismology constraints, and SME spectral an alysis in the case of HD17156
+([Fe/H] = +0.24). In that case the discrepancy in log gis smaller, 0.12, but is in the same
+sense. Bakos et al. (2009) find a similar situation for HAT-P-13 ([Fe /H] = +0 .41) with
+a loggdiscrepancy of +0.14. This suggests that there is a systematic err or in the log g
+values derived from SME analysis for high metallicity stars. However, Burke et al. (2007)
+find only a small log gdiscrepancy of +0.02 in the case of XO-2 ([Fe /H] = +0.45) and the
+discrepancy for HD80606 ([Fe /H] = +0.4) is only +.01 (Naef et al. 2001; Winn et al. 2009).
+Further comparisons of spectroscopic log gvalues with precise values derived from transits
+and asteroseismology are likely to be fruitful.
+Funding for this Discovery mission is provided by NASA’s Science Mission Directorate.
+It is with pleasure that we acknowledge the outstanding work done b y the entire Kepler
+team during development, launch, commissioning, and the ongoing op erations of the Kepler
+mission. It is an honor and a privilege to work with such a talented and d edicated group of
+people.– 7 –
+We thank the anonymous referee for several excellent suggestio ns that improved the
+paper.
+Facilities: The Kepler Mission, Tillinghast (TRES), NOT (FIES), McDonald 2.7-m
+(Coude Spectrograph), Keck:I (HIRES), WIYN (Speckle), Paloma r Hale (PHARO/NGS)
+REFERENCES
+Bakos, G. ´A., et al. 2009, ApJ, in press (arXiv0907.3525B)
+Batalha, N., et al. 2010, ApJ, this issue
+Borucki, W. J., et al. 2010a, Science, Submitted
+Borucki, W. J., et al. 2010b, ApJ, this issue
+Burke, C. J., et al. 2007, ApJ, 671, 2115
+Gautier, T. N., et al. 2010, ApJ, this issue
+Gilliland, R. L., et al. 2010a, PASP, in press
+Gilliland, R. L., et al., 2010b, ApJ, Submitted
+Haas, M. R., et al. 2010, ApJ, this issue
+Jenkins, J. M., et al. 2010a, ApJ, this issue
+Jenkins, J. M., et al. 2010b, ApJ, this issue
+Koch, D. G., et al. 2010a, ApJ, this issue
+Koch, D. G., et al. 2010b, ApJ, this issue
+Latham, D. W. 2003. In Scientific Frontiers in Research on Extraso lar Planets, ASP Conf.
+Ser. 294, eds. D. Deming & S. Seager (San Francisco: ASP), 409
+Mandel, K., & Agol, E. 2002, ApJ, 580, L171
+Mandushev, G., et al. 2005, ApJ, 621, 1061
+Monet, D., et al. 2010, ApJ, this issue
+Naef, D., et al. 2001, A&A, 375, L27– 8 –
+Nutzman, P., et al. 2010, ApJ, Submitted
+Valenti, J. A., & Piskunov, N. 1996, A&AS, 118, 595
+Winn, J. N., et al. 2009, ApJ, 703, 2091
+This preprint was prepared with the AAS L ATEX macros v5.2.– 9 –
+Fig. 1.— The detrended light curve for Kepler-6. The time series for t he entire data set is
+plotted in the upper panel. The lower panel shows the photometry f olded by the 3 .234723-
+day period. The model fit to the primary transit is overplotted in red (vertical axis on the
+left), and our attempt to fit a corresponding secondary eclipse is s hown in green with the
+expanded and offset scale on the right.– 10 –
+     -50050100Radial velocity (m s-1)
+       -15015 Residual (m s-1)
+0.0 0.2 0.4 0.6 0.8 1.0
+Phase -15015 Bisector
+var. (m s-1)
+Fig. 2.— The orbital solution for Kepler-6. The observed radial veloc ities obtained with
+HIRES on the Keck 1 Telescope are plotted in the top panel togethe r with the velocity curve
+for a circular orbit with the period and time of transit fixed by the pho tometric ephemeris.
+The middle panel shows the residuals to the fit. The point potentially a ffected by the
+Rossiter-McLaughlin effect is the one at the smallest phase. With this point omitted from
+the fit the rms of the residuals is 5.7 ms−1. The lower panel shows the line bisector variation.
+No structure connected to the radial velocity curve is evident.– 11 –
+Table 1. Relative Radial-Velocity Measurements of Kepler-6
+BJD Phase RV σRV
+(ms−1) (ms−1)
+2454986.087289 0.769 +68 .42 4.48
+2454987.087348 0.078 −59.94 4.59
+2454988.077695 0.384 −67.64 4.75
+2454989.019649 0.676 +48 .64 4.64
+2454995.112273 0.559 +18 .56 4.79
+2455014.881155 0.671 +51 .75 4.54
+2455015.984052 0.012 −34.79 4.65
+2455017.067098 0.346 −75.84 4.94
+2455044.019956 0.679 +50 .59 5.42– 12 –
+Table 2. System Parameters for Kepler-6
+Parameter Value Notes
+Transit and orbital parameters
+Orbital period P(d) 3 .234723±0.000017 A
+Midtransit time E(HJD) 2454954 .48636±0.00014 A
+Scaled semimajor axis a/R⋆ 7.05+0.11
+−0.06A
+Scaled planet radius RP/R⋆ 0.09829+0.00014
+−0.00050A
+Impact parameter b≡acosi/R⋆ 0.398+0.020
+−0.039A
+Orbital inclination i 86.◦8±0.3 A
+Orbital semi-amplitude K(ms−1) 80 .9±2.6 A,B
+Orbital eccentricity e 0 (adopted) A,B
+Center-of-mass velocity γ(ms−1) −18.3±3.5 A,B
+Observed stellar parameters
+Effective temperature Teff(K) 5647 ±44 C
+Spectroscopic gravity log g(cgs) Fixed at transit value C
+Metallicity [Fe/H] +0 .34±0.04 C
+Projected rotation vsini(kms−1) 3 .0±1.0 C
+Mean radial velocity (kms−1) −49.14±0.10 B
+Derived stellar parameters
+MassM⋆(M⊙) 1 .209+0.044
+−0.038C,D
+RadiusR⋆(R⊙) 1 .391+0.017
+−0.034C,D
+Surface gravity log g⋆(cgs) 4 .236±0.011 C,D
+Luminosity L⋆(L⊙) 1 .99+0.24
+−0.21C,D
+Age (Gyr) 3.8±1.0 C,D
+Planetary parameters
+MassMP(MJ) 0 .669+0.025
+−0.030A,B,C,D
+RadiusRP(RJ, equatorial) 1 .323+0.026
+−0.029A,B,C,D
+Density ρP(gcm−3) 0 .352+0.018
+−0.022A,B,C,D
+Surface gravity log gP(cgs) 2 .974+0.016
+−0.022A,B,C
+Orbital semimajor axis a(AU) 0 .04567+0.00055
+−0.00046E
+Equilibrium temperature Teq(K) 1500 ±200 F
+Note. —
+A: Based on the photometry.
+B: Based on the radial velocities assuming an elliptical orb it.
+C: Based on an SME analysis of the HIRES spectra.
+D: Based on the Yale-Yonsei stellar evolution tracks.
+E: Based on Newton’s version of Kepler’s Third Law and total m ass.
+F: Assumes Bond albedo = 0.1 and complete redistribution. Th e uncertainty
+reflects the uncertainties in these assumptions.
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+arXiv:1001.0334v1  [astro-ph.GA]  2 Jan 2010THEASTROPHYSICAL JOURNAL 709(2010) 386
+Preprint typesetusingL ATEX styleemulateapjv. 11/10/09
+FORMALDEHYDEANTI-INVERSIONAT Z=0.68INTHEGRAVITATIONALLENS B0218 +357
+BENJAMIN ZEIGER1,2& JEREMYDARLING1,3,4
+TheAstrophysical Journal 709(2010) 386
+ABSTRACT
+We report new observations of the 1 10−111(6 cm) and 2 11−212(2 cm) transitions of ortho-formaldehyde
+(o-H2CO) in absorption at z= 0.68466 toward the gravitational lens system B0218 +357. Radiative transfer
+modeling indicates that both transitions are anti-inverte d relative to the 4.6 K cosmic microwave background
+(CMB), regardless of the source covering factor, with excit ation temperatures of ∼1 K and 1 .5−2 K for the
+110−111and211−212lines,respectively. Usingtheseobservationsandalargev elocitygradientradiativetransfer
+model that assumes a gradient of 1 kms−1pc−1, we obtain a molecular hydrogen number density of 2 ×
+103cm−3<n(H2)<1×104cm−3and a columndensity of 2 .5×1013cm−2<N(o-H2CO)<8.9×1013cm−2,
+wheretheallowedrangesconservativelyincludetherangeo fpossiblesourcecoveringfactorsinbothlines. The
+measurementssuggestthatH 2COexcitationintheabsorbingcloudsintheB0218 +357lensisanalogoustothat
+in Galactic molecular clouds: it would show H 2CO absorption against the CMB if it were not illuminated by
+thebackgroundquasarorif itwere viewedfromanotherdirec tion.
+Subject headings: galaxies: individual(B0218 +357)— galaxies: ISM — radiation mechanisms: non-thermal
+— radiolines: galaxies— quasars: absorptionlines
+1.INTRODUCTION
+Formaldehyde(H 2CO)centimetertransitionshavebeenob-
+served with excitation temperatures less than the tempera-
+ture of the cosmic microwave background (T ex<TCMB) in
+the Milky Way (Palmeretal. 1969). The “anti-inversion,” or
+non-thermal absorption, is pumped by collisions, coupling
+H2CO excitation to the number density n(H2) of molecu-
+lar hydrogen even at kinetic temperatures as low as ∼10 K
+(Townes& Cheung 1969; Evanset al. 1975; Garrisonetal.
+1975). Thus, H 2CO provides a density and temperature di-
+agnosticformoleculargas(Mühle,Seaquist,&Henkel2007;
+Mangumet al. 2008), and it often produces detectable lines
+before dense molecular clouds begin to form stars and warm
+sufficientlytoemitlinesfromtheusualmoleculartracerss uch
+asCO andHCO+.
+The rotation states of H 2CO are defined by the three quan-
+tum numbers JKaKc(total angular momentum and the angu-
+lar momentaabouttheaxeswith the smallest andlargestmo-
+ments of inertia, respectively). The lower energy transiti ons
+forKa<2 are shown in Figure 1. The spin symmetry of
+the hydrogen nuclei defines the ortho ( Ka= odd) and para
+(Ka= even) states. Since the spin symmetry is about the a
+axis, which is aligned with the electric dipole along the C–
+O double bond and thus unchanged by dipole interactions,
+the mean transition time from o-H 2CO to p-H 2CO states (or
+vice versa) is longer than the mean life of the molecule, and
+the ortho/para ratio preserves information about the domi-
+nant formation channel: a ratio of unity indicates catalysi s
+on the surface of cold (T kin/lessorsimilar15 K) dust grains, while a ra-
+tio of 3 indicates gas-phase formation (Kahaneet al. 1984;
+Dickens&Irvine1999).
+Themostcommonlydetectedo-H 2COlinesarethe ∆J=0,
+1Center for Astrophysics and Space Astronomy, Department of Astro-
+physical and Planetary Sciences, University of Colorado, 3 89 UCB, Boul-
+der, CO80309-0389, USA
+2benjamin.zeiger@colorado.edu
+3NASA Lunar Science Institute, NASA Ames Research Center, Mo f-
+fett Field, CA, USA
+4jdarling@colorado.edu∆Kc=±1“K-doublet”1 10−1114.8GHz(6cm)and2 11−212
+14.5 GHz (2 cm) centimeter transitions and the ∆J=±1,
+∆Kc=±1 millimeter transitions connecting the J= 1 states
+to theJ= 2 states at 140.8 and 150.5 GHz (2 12−111and
+211−110, respectively). The coupling of the centimeter and
+millimeter transitions allows the K-doublet transitions to be
+driven significantly out of local thermodynamic equilibriu m
+(LTE) and makes H 2CO anin situprobe of conditions in
+molecular clouds. The relative excitation of K-doublets is
+a powerful densitometer (Mangumet al. 2008), while the
+relative excitation of millimeter lines is a molecular ther -
+mometer (Mühle,Seaquist,&Henkel 2007). The 1 10−111
+and 211−212transitions have both been observed with pop-
+ulation anti-inversions, causing stimulated absorption o f the
+CMB,andthe1 10−111transitionhasbeenobservedasamaser
+(Haar& Pelling 1974; Evansetal. 1975; Arayaet al. 2008).
+We report observations of the 1 10−111and 211−212transi-
+tions in absorption toward B0218 +357, a gravitational lens
+system with two images separated by 0 .′′33. The lens lies at
+a redshift of z= 0.68466(4) (Browneet al. 1993), while the
+backgroundquasar is at z=0.944(2) (Cohenet al. 2003). At
+lowfrequencies,the“gravitationallenswiththesmallest sep-
+aration” of its images also displays the smallest known Ein-
+stein ring, with a diameter of 0 .′′335 (Patnaiketal. 1993).
+The two images, dubbed A and B in order of radio bright-
+ness,displaysignificantlydifferentabsorptioncharacte ristics,
+with B unobscured while A shows strong molecular absorp-
+tion lines and is heavily obscured by dust in optical obser-
+vations(Wiklind&Combes1995; Grundahl& Hjorth1995).
+Both A and B show the backgroundsource to have a core-jet
+morphology.
+Many molecules have been observed at the redshift of the
+lens in absorption against the strong continuum of image A,
+includingCO,HCO+,HCN, H 2O,NH 3,OH, andH 2CO; typ-
+ical line widths are 10 −15kms−1(Wiklind& Combes 1995;
+Combes& Wiklind 1997; Henkeletal. 2005; Kanekaret al.
+2003; Menten& Reid 1996). H Iobservations indicate
+the lensing galaxy to be gas-rich, with a column density2 Zeiger& Darling
+Table 1
+Journal of o-H 2CO Observations Toward B0218 +357
+Telescope Receiver UTDate Transition ν◦ νobs tint∆varms
+(GHz) (GHz) (s) (km s−1) (mJy)
+Arecibo S-low 2004 Aug 3–4 1 10−1114.829660(1) 2.8668455(6) 3600b1.3 0.8
+GBT X 2006 Apr26 2 11−21214.488479(1) 8.6002392(6) 3280 2.1 1.1
+a∆v is the rest-frame spectral velocity resolution of the final , smoothed spectrum.
+bTwo1800 ssessions fromconsecutive days were combined for t he final spectrum.
+Figure1. Formaldehyde (H 2CO) energy level diagram showing only the
+lower levels and Ka<2. Theindicated transitions are in GHz.
+N(HI) = 4×1018(Ts/f) cm−2across a line width (FWHM)
+of 43kms−1, whereTsis the spin temperature and fis the
+HIcovering factor (Carilli et al. 1993). Four spectrally re-
+solvedabsorptioncomponentscanbeidentifiedinfrontofim -
+age A in high-resolution interferometric HCO+observations
+(Mulleret al.2007).
+H2COwasfirstobservedinB0218 +357byMenten&Reid
+(1996) in the 2 11−212transition in absorption. Jethavaetal.
+(2007) followed these observations with a six-line study of
+H2CO in B0218 +357, adding observations of the lowest two
+∆J= 1 rotational transitions of both the ortho and the para
+species, as well as the 1 10−111transition of o-H 2CO. Us-
+ing the 55 K kinetic temperature derived from NH 3observa-
+tions(Henkelet al.2005)andalargevelocitygradient(LVG )
+model to determine the non-thermal H 2CO excitation due to
+collisions with H 2, Jethavaetal. (2007) determined a spatial
+number density of n(H2)<103cm−3and a column density
+ofN(H2CO)≈5.2×1013cm−2(no confidence interval is re-
+ported)in thestrongestabsorptioncomponent,witha weake r
+component adding ∼15% to the column density. They mea-
+sured an o-H 2CO/p-H 2CO ratio of 2 .0−3.0 with a best fit of
+2.8, giving N(o-H2CO) = 3.8×1013cm−2. They do not ad-
+dresstheissueofH 2COexcitationtemperatures,whichareofinterest because anti-inversion of the lines would imply th at
+thegasinthe B0218 +357lenswouldbe visible inabsorption
+againsttheCMBevenwithoutabackgroundquasartoillumi-
+nate it. Obtaining excitation temperatures of H 2CO lines in
+B0218+357will helptoset H 2CO observabilityconditionsin
+non-lensing(orotherwiseunilluminated)galaxies.
+The Mangumet al. (2008) surveyof local galaxiesdemon-
+strates the feasibility of determining both n(H2) and
+N(o-H2CO) strictly from the 1 10−111and 211−212transitions.
+The relative strengths of the transitions are set by the degr ee
+of collisional excitation, giving a precise densitometer t hat,
+in turn, yields the excitation temperatures of the transiti ons.
+In our analysis, we use the two-line densitometry method
+withnewdata,including1 10−111observationsofhigherspec-
+tral resolution and signal to noise ratio than that obtained by
+Jethavaetal.(2007)and2 11−212observationswithlowersig-
+nal to noise (4.7 versus 8.8) but comparable resolution to
+those of Menten&Reid (1996). Our analysis, while also
+yieldingN(o-H2CO) andn(H2) from an exhaustive search of
+the density space with our LVG model, focuses on line exci-
+tation temperatures in B0218 +357 to show that the o-H 2CO
+observed in absorption is anti-inverted relative to the 4.6 K
+TCMBatz=0.68, making it analogous to Galactic molecular
+clouds that show anti-inversion of the K-doublet transitions
+andthusCMB absorption.
+In Section 2, we describe our observational and data re-
+duction procedures used to acquire and reduce the new high
+resolutionspectra of the 1 10−111and 211−212o-H2CO transi-
+tionsinB0218 +357. Thepropertiesofbothlinesarereported
+in Section 3, and we use those results to determine n(H2),
+N(o-H2CO), and line excitation temperatures in Section 4.
+We show that the absorbing clouds in B0218 +357 are anal-
+ogous to Galactic molecular clouds that show anti-inversio n
+of o-H 2CO centimeter lines relative to the z= 0 TCMB, and
+that the anti-inversion of K-doublet transitions persists up to
+the615−616(101GHzrest)transition(Section5). Theconse-
+quencesofourresultsarediscussedinSection6.
+2.OBSERVATIONSAND DATAREDUCTION
+2.1.AreciboTelescope: 110−111Line
+Weobservedthe1 10−1114.829660(1)GHztransition5ofo-
+H2CO, redshifted to 2.8668455(6) GHz, toward B0218 +357
+attheAreciboradiotelescope6inAugust2004(Table1). Ob-
+servationswereconductedwitha6.25MHzbandcenteredon
+the redshifted line, using 5 minute position-switched scan s
+with a calibration diode fired after each position-switched
+pair and spectral records recorded every 6 s. Total integra-
+tion time was 3600 s. The autocorrelationspectrometer used
+5All line frequencies are from the JPL molecular line databas e
+(Pickett et al. 1998).
+6The Arecibo Observatory is part of the National Astronomy an d Iono-
+sphere Center, which is operated by Cornell University unde r a cooperative
+agreement with theNational Science Foundation.FormaldehydeAnti-InversioninB0218 +357 3
+Table 2
+Measured and Calculated Formaldehyde LineProperties
+Transition ν◦ Compt v obs−vsysDepth FWHM τapp/integraltext
+τappdv
+(GHz) (kms−1) (mJy) (kms−1) (kms−1)
+110−1114.829660(1) 1 +3.7(0.4) 9.6(0.4) 12.3(0.6)
+2−9.8(2.4) 1.8(0.3) 14.2(4.0)
+Total +3.4(0.4) 10.8(0.8) 12.6(0.6) 0.017(1) 0.239(8)
+211−21214.488479(1) 1 +4.2(0.8) 5.2(1.1) 8.5(1.5) 0.008(2) 0.083 (10)
+Note. — vobs−vsysis therest-frame centroid velocity offset from the assumed systemic redshif t,
+z=0.68466,FWHMistherest-framelinewidth,and τappisthemaximumapparentopticaldepthinthe
+line, assumingacovering factor ofunity andacontinuum in i mageAof650 ±16and 663 ±22mJyin
+the 110−111and 211−212lines, respectively (Section 3). Uncertainties are strict ly statistical uncertain-
+ties from the spectra and do not reflect the systematic uncert ainties associated with flux calibration or
+the radio continuum determinations. Reported quantities f or the 1 10−111“Total” and 2 11−212lines are
+measured directly from the data and are not dependent on profi le fitting; the individual components of
+the 110−111line are from Gaussian fits.
+nine-levelsampling in two (subsequentlyaveraged)polari za-
+tions. Bandpasseswere dividedinto 1024channelsand Han-
+ning smoothed to 512 channels. Rest-frame velocity resolu-
+tion was 1.3 km s−1(see Table 1). The observed band was
+interference-freeinthevicinityoftheobservedline.
+Records were individually calibrated and bandpasses were
+flattened using the calibration diode and the corresponding
+off-source records. Records and polarizations were subse-
+quently averaged, and a polynomial baseline with variation s
+much larger than the line was fit to and subtracted from the
+spectrum. Systematic flux calibration errors in these data
+are of order 10%. All data reduction was performed in the
+Astronomical Information Processing System ++, AIPS++.7
+Arecibo spectra often show spectral standing waves due to
+resonances of the strong continuum flux densities within the
+telescopesuperstructure. Thelinewidthwassignificantly less
+than the size of the standing wave features, so the spectrum
+wasnotsignificantlyaffected.
+2.2.GreenBankTelescope: 211−212Line
+We observed the 2 11−21214.488479(1) GHz transition of
+H2CO, redshifted to 8.6002392(6) GHz, toward B0218 +357
+withtheGreenBankTelescope8(GBT)on2006April26(Ta-
+ble1). Observationswereconductedwitha50MHzbandpass
+centered on the redshifted line in 5 minute position-switch ed
+scans with spectral data recorded every 3 s and a winking
+calibration diode firing during every other record. The to-
+tal on-source integration time was 3280 s. The autocorre-
+lation spectrometer used nine-level sampling in two (subse -
+quently averaged) linear polarizations. Bandpasses were d i-
+vided into 16384 channels, Hanning smoothed to 8192 in-
+dependent channels, and 10-channelGaussian smoothed to a
+rest-framevelocityresolutionof2.1kms−1. Recordswerein-
+dividuallycalibratedand bandpasseswere flattened using t he
+calibration diode and the corresponding off-source record s.
+The individual spectra were interference-free and fairly fl at,
+but the DC levels of the individual scans showed significant
+fluctuationsof order10%. Scans and polarizationswere sub-
+sequentlyaveraged,andafifth-orderpolynomialbaselinew as
+fit to the full 50 MHz bandwidth and subtracted. Systematic
+fluxcalibrationerrorsinthesedataareoforder10%. Alldat a
+7AIPS++isfreely available foruseundertheGnuPublicLicense. Fur ther
+information may beobtained fromhttp://aips2.nrao.edu.
+8The National Radio Astronomy Observatory is a facility of th e National
+Science Foundation operated under cooperative agreement b y Associated
+Universities, Inc.Figure2. Formaldehyde 1 10−111(6cm)and 2 11−212(2cm) absorption to-
+wardthegravitational lens systemB0218 +357. Therest-framevelocity scale
+assumes a heliocentric redshift of z= 0.684660, and the spectral resolutions
+are1.3 and 2.1 kms−1in the lower and upper lines, respectively. Thedashed
+line shows a two-component Gaussian fit to the 6 cm line profile . The 2 cm
+line spectrum, also detected by Menten &Reid (1996), is offs et by 4 mJy,
+with the zero point indicated by the dotted line.
+reductionwasperformedin GBTIDL.9
+3.RESULTS
+The Arecibo spectrum of the 1 01−111transition requires
+two Gaussian components for a good fit to the line profile
+(Figure 2). The rest frame velocity resolution is 1.3kms−1.
+Table 2 lists the two fit componentsand the properties of the
+total line. The total line propertiesare not based onGaussi an
+fits;theyarecomputeddirectlyfromthespectrum. Allveloc i-
+tiesareintherestframewithrespecttoaheliocentricreds hift
+9GBTIDL(http://gbtidl.nrao.edu/ )isthedatareductionpack-
+ageproduced by NRAO and written in theIDLlanguage for there duction of
+GBTdata.4 Zeiger& Darling
+ofz=0.68466. The integrated optical depth in this transition
+is0.239(8)kms−1over−20kms−1≤v≤+17kms−1assum-
+ing a continuumof 650 ±16 mJy and a unity coveringfactor
+(seeSections4.1and4.2).
+The GBT detection of the 2 11−212transition has a lower
+signal to noise ratio than the Arecibo 1 10−111transition. De-
+spite this and the lower spectral resolution of the GBT spec-
+trum (2.1 versus 1.3kms−1), both lines display similar pro-
+files (Figure 2), although the 2 11−212detection is insufficient
+to justify a second profile component as seen in the 1 10−111
+spectrum and in the data of Jethavaet al. (2007). The prop-
+erties listed in Table 2 for this line have been measured di-
+rectly from the data. The integrated optical depth, spannin g
+the same velocity range as the 4.8 GHz line, is 0 .083(10)km
+s−1, assuming a continuum of 663 ±22 mJy and a unity cov-
+eringfactor(seeSections4.1§4.2).
+The observed profiles are narrower than the 15kms−1ob-
+served in CO, HCO+, and HCN (Combes&Wiklind 1996),
+but broader than the NH 3lines that range from 6.6(1.6) to
+9.3(1.8)kms−1(Henkelet al. 2005). OH spectra (1.67 GHz
+rest) clearly show a two-component profile (Kanekaretal.
+2003), although at 20 and 27kms−1the line componentsare
+broader than the ∼10kms−1H2CO lines. Our detection of
+the211−212transitioniscomparabletothatofMenten&Reid
+(1996). Jethavaet al. (2007) report a much narrower 1 10−111
+line — with a FWHM of 5.5(5)kms−1— but our high sig-
+nalto noiseratioandresolutionmakeourmeasurementmore
+statistically robust. Both of our observations are coinci-
+dent in velocity space with previous molecular detections i n
+B0218+357.
+4.ANALYSIS
+4.1.SourceContinuum
+Ingeneral,whilethecontinuumdeterminationsmaybeun-
+certainandleadto errorsinthe absoluteopticaldepthandt o-
+tal columndensity,the computationof quantitiesthat depe nd
+on ratios of (low) optical depths, such as excitation temper a-
+tures and n(H2), will be significantly less affected, provided
+thatsystematicerrorsareconsistentacrossfrequency.
+The continuum emission at 2.9 GHz (the frequency of the
+redshifted4.8GHz1 10−111transition)isnearlyflatasafunc-
+tion of frequency for images A and B (O’Deaet al. 1992;
+Patnaiket al. 1993) but steeper for the Einstein ring and the
+extendedemission halo. At 5 GHz, the ratio of the two com-
+pact flat spectrum components A:B is ∼3.0 and the Ein-
+stein ring emission accounts for 10%–20% of the total emis-
+sion(O’Deaetal.1992;Patnaiket al.1993). Thetwosources
+showtimevariabilitywith a 10.5dayperiod,but thevariabi l-
+ity is constrainedto be <8% at low frequencies(Mittal etal.
+2006). Extendedemission is detected at 1.465 and 1.63 GHz
+at the∼10% level, and additional structure is seen at 5 GHz
+(O’Deaet al. 1992; Patnaiket al.1993).
+Comparison to other molecules confirms the association
+of H2CO absorption with image A (Menten&Reid 1996;
+Mulleretal. 2007). The appropriate continuum level to use
+in apparent optical depth calculations for the 1 10−111transi-
+tion is thus the total continuum at 2.9 GHz (8.6 GHz for the
+211−212line) less image B, the Einstein ring emission, and
+anyextendedsourcestructure. Interpolatingthefluxdensi ties
+for imagesA and B and the extendedemission from interfer-
+ometric observations allows measurement of the fraction of
+the continuum from image A, a fraction crucial to disentan-
+glingtheabsorptioninfrontof imageA fromtheunabsorbedcontinuumofimageBandotherextendedstructure.
+The flux density of image A at 2.9 GHz can be interpo-
+lated from a linear fit (in the log) to the flux density ver-
+sus frequency of image A. We use the very long baseline in-
+terferometry (VLBI) observations of Mittalet al. (2006) be -
+cause these straddle 2.9 GHz and because these observa-
+tions,of all the availableinterferometricmeasurements, were
+taken closest in time to our spectra (2002 January versus
+2004 August for the Arecibo 1 10−111observations and 2006
+April for the GBT 2 11−212observations). The continuum,
+interpolated from a fit to the image A data points spanning
+2.25–15.35 GHz (the 1.65 GHz datum is dropped due to a
+steep downturn below 2.25 GHz; see Mittalet al. 2006), is
+650±16 mJy, which neglects systematic and calibration er-
+rorstypicallyoforder10%. TheimageAcontinuumisnearly
+flat, and interpolation error is less of a concern than over-
+all flux calibration. Mittal etal. (2006) degrade their reso -
+lution to 50 ×50 mas, comparable in resolution to MER-
+LIN observations by Patnaiket al. (1993) to account for the
+loss of flux in VLBI observations, yielding similar flux den-
+sities to within 10%. Note that extrapolations from lower
+resolution (and much older) observations predict higher flu x
+densities at 2.9 GHz: O’Deaetal. (1992) predict 868 ±64
+mJy, and Patnaiket al. (1993) predict 791 ±68 mJy. We
+adoptSA(2.867GHz) = 650 ±16 mJy, bearing in mind that
+this estimate could potentially have large systematic erro rs.
+For the 14 GHz line redshifted to 8.6 GHz, we use the
+same VLBI data and continuum estimation method, adopt-
+ingSA(8.600GHz)= 663 ±22 mJy. The VLBI observations
+ofMittal etal. (2006) are degradedto 10mas at 8.4GHz and
+5masat 15GHz.
+4.2.SourceCoveringFactor
+Calculation of the true optical depth of each line requires
+knowledgeofthecoveringfactor fνdescribingthefractionof
+imageAobscuredbytheabsorbingcloudatthelinefrequency
+ν. The covering factor corrects the observed line intensity t o
+account for dilution of the line due to incomplete coverage
+of the continuum source by the absorbing cloud. Thus, the
+opticaldepth τlineofthelinetakestheform
+τline=−ln/parenleftbigg
+1−Sline
+fνSν/parenrightbigg
+, (1)
+whereSlineandSνare the line and continuum fluxes, re-
+spectively. Millimeter observations at ν >100 GHz show
+fν/greaterorsimilar0.77(Wiklind& Combes1995),consistentwithacover-
+ingfactorofunityinfrontofimageAand0infrontofimage
+B; the flux ratio (A/B) is close to 4 at 100 GHz (Mulleret al.
+2007).
+Due to the increased solid angle of image A at lower
+frequencies (Mittal etal. 2006), the covering factors for t he
+110−111and 211−212transitions could be significantly less
+than unity even after excluding flux from image B and ex-
+tended emission. Kanekaretal. (2003) estimate the cover-
+ing factor to be 0.4 at 990 MHz (redshifted 18 cm OH),
+and Jethavaetal. (2007) estimate the covering factor to be
+of the form fν= (ν/100)xforνin GHz, determining that
+0.0<x<0.5 by interpolatingbetween the millimeter obser-
+vations of Wiklind& Combes (1995) and the 990 MHz OH
+observations of Kanekaret al. (2003). (Note that, in arrivi ng
+at this range,theyreferto the OH linesas beingat 850MHz,
+the redshifted frequency of the H Iline.) For discussion,
+Jethavaetal.(2007)adoptthemostextremefrequencydepen -FormaldehydeAnti-InversioninB0218 +357 5
+denceofx=0.5,whichgivescoveringfactorsof0.17and0.29
+for the 1 10−111and 211−212lines; the results they report are
+determined significantly by this choice of covering factors .
+Since this paper is concerned with excitation temperatures ,
+which are determined via the ratio of the two lines, properly
+accountingforthefrequencydependenceofthecoveringfac -
+torisvitalto obtainingaccurateresults.
+VLBIobservationsofMittalet al.(2006)showanextended
+source at low frequencies that splits image A into two point-
+like imagesseparated at 15.4GHz by ∼1.4 mas, correspond-
+ing to a physical separation in the lensing galaxy of 9.6 pc
+(for a flat universe with H 0= 73kms−1Mpc−1,Ωm= 0.27,
+andΩv=0.73). Thesolidangle ΩAfollowsapowerlawofin-
+dexofα≈−2.7(whereΩA∝να)asfrequencyincreasesfrom
+1.65 to 15.65 GHz. (By comparing centimeter and millime-
+terobservations,Henkelet al.(2005)arguethat −2<α<−1.)
+Therelativelyflatspectrumofthesouth-westernsourceini m-
+age A suggests that it is the flat-spectrum core of the quasar
+andthelocationofcontinuumemissionseeninmillimeterob -
+servations, where the frequency-dependent jet would not be
+observable (Patnaiketal. 1995). Since the core is fully ob-
+scured(fν=1)inmillimeterobservations,wherethejetisnot
+visible,theabsorbingcloudliesalong—butisnotnecessar ily
+centeredupon—the lineofsighttoit.
+At8.6GHz,wherethejetandthecorearenotdistinctlysep-
+arated,the ∼15mas2regioncontainingthe core emits ∼60%
+of the flux of image A (Patnaiket al. (1995) measure 62(1)%
+with VLBI observations at 15 GHz). The total area of im-
+age A with flux density 5 times the rms background, found
+by extrapolating from the VLBI observations of Mittal etal.
+(2006), is ∼32mas2at this frequency. A spherical molecular
+cloud of 30 pc diameter subtends ∼15 mas2(∼47% ofΩAat
+8.6GHz)at z=0.68. Suchacloudgivesaminimumcovering
+factor of 0.3 if the core, which has fν= 1 at millimeter fre-
+quencies, is obscured only by a limb of the cloud; due to the
+comparable line widths observed at both centimeter and mil-
+limeter frequencies (Jethavaet al. 2007), it is likely that the
+absorbingcloudisfortuitouslyalignedwithimageAandtha t
+the coveringfactor is close to 0.6 for a 30 pc cloud. This is a
+strictly geometric argument, and weighting the covering fa c-
+torbythefractionoffluxthatisobscuredwillgive f14.5≥0.6
+evenforsomewhatsmallerclouds.
+The diameter of the cloud is likely to be significantly
+larger than 30 pc, since the optical obscuration of image
+A suggests the presence of large amounts of dust and pos-
+sibly a giant molecular cloud (Grundahl& Hjorth 1995);
+Wiklind& Combes (1999) report a visual extinction of Av=
+850infrontofimageA.Applyingtherelationbetweencloud
+sizeS(equivalenttodiameterfora sphericalcloud)in pcand
+velocity dispersion σvinkms−1measured empirically with
+CO forGalactic clouds(Solomonetal. 1987)of
+σv
+kms−1=1.0±0.1/parenleftbiggD
+pc/parenrightbigg0.50±0.05
+, (2)
+givesS≈225 pc (CO line widths are 15kms−1;
+Wiklind& Combes 1995), suggesting a cloud much larger
+than the 650 mas2solid angle subtended by image A at
+2.9 GHz. While Solomonet al. (1987) measure the relation
+for clouds with S<50 pc and line widths /lessorsimilar8kms−1, this
+is a strong indication that the cloud is quite large, increas ing
+theprobabilityofahighcoveringfactor. Mulleret al.(200 7),
+with high spectral resolution HCO+(2−1) observations, re-solve the line into four ∼4.5kms−1components, suggesting
+aclumpymediumofmoderatelysizedcloudsarrayedinfront
+of image A, allowing the covering factor to remain constant
+or even increase with decreasing observation frequency (an d
+increasing ΩA). While the HCO+observations reveal that it
+isnotasinglelargeabsorbingcloud,thepresenceofmultip le
+cloudsstill allowsforhighcoveringfactors.
+Agreement (1 σ) between the observed optical depths and
+ourLVGmodel(seeSection4.3)requires f4.8>0.2forf14.5=
+0.6 andf4.8>0.3 forf14.5= 1.0. Matching optical depths
+requires0 .5<f4.8<1.0forf14.5=0.6andf4.8=0.5forf14.5=
+1.0. Althoughlikely, f4.8<f14.5cannotbeset a prioridueto
+the clumpy absorbing medium. In subsequent analysis, we
+discussresultsspanningtherangeofviablecoveringfacto rs.
+4.3.RadiativeTransferModeling
+Formaldehydeissusceptibletonon-LTEexcitationineithe r
+direction(maserinversionoranti-inversion),soitisgen erally
+inappropriate to assume a single line excitation temperatu re
+for the molecule. It is generally also not valid to attribute
+a single excitation temperature to all of the centimeter ∆J=
+0 lines, since all line excitation temperatures can differ. We
+employ the still valid assumption of statistical equilibri umin
+theseH 2CO linesbutallowall lineexcitationtemperaturesto
+float.
+We used an LVG radiative transfer model to find rela-
+tive level populations up to the 40th rotational energy stat e
+(1037) of o-H 2CO and the 41st (7 43) of p-H 2CO. Assum-
+ingstatistical equilibrium,themodelcomputesopticalde pths
+and excitation temperatures for a given formaldehyde den-
+sityn(H2CO) and a hydrogen density n(H2). The model as-
+sumes an isothermal and constant density spherically sym-
+metric cloud with large-scale turbulence or gravitational col-
+lapse producing large velocity gradients. Following the ob -
+served velocity gradient of Solomonet al. (1987), we set the
+velocitygradientto1.0kms−1pc−1.N(o-H2CO) valuesscale
+inverselywiththe gradient dv/dr(Wangetal. 2004):
+N(o-H2CO)=[n(o−H2CO)/(dv/dr)]×3.08×1018×∆ν
+=[n(H2)·X/(dv/dr)]×3.08×1018×∆ν,(3)
+where∆νisthelinewidth(FWHM), n(o−H2CO)isthenum-
+ber density, and Xis the abundance of o-H 2CO (not total
+H2CO) relative to H 2. Since they are found via ratios of
+optical depths, which are proportional to N(o-H2CO),n(H2)
+and Texresults are independent of dv/dr. For a more thor-
+oughdiscussion of the LVG method, see Sobolev (1960) and
+Goldreich& Kwan(1974).
+Our model includes non-LTE collisional excitation of
+H2CO by H 2. We use the He collisional cross sectionscalcu-
+latedbyGreen(1991)asaproxyfortheasphericalH 2;Green
+notesthat the substitution of He for H 2could cause errorsup
+to 50% in individual collisional excitation rates between l ev-
+els, but that this is likely to translate to overall errorsin level
+populations of order 20%. Using pressure-broadened H 2–
+H2CO and He–H 2CO laboratory measurements of the mil-
+limeter transitions, Mengel&DeLucia (2000) find the He–
+H2CO collision rates to be “very good,” while the collisional
+cross sections of H 2–H2CO collisions are erroneousby up to
+afactorof2 fromthe scaledcollisionrates(/radicalbig
+µH2/µHe)with
+noapparentpatternto theerrors.
+We evaluated the effect on excitation temperatures of such
+anerrorin collisionratesbyrandomlyalteringcollisionr ates
+with Gaussian variation with a mean of zero and a standard6 Zeiger& Darling
+Figure3. Contours trace excitation temperatures of the 1 10−111(left) and 2 11−212(right) transitions as a function of n(H2) andN(o-H2CO)/(km s−1pc−1)−1as
+calculated with our LVG model. Overplotted white lines repr esent the observed peak optical depth of the 1 10−111line (∼horizontal) and the ratio of observed
+peak optical depths of the 1 10−111and 211−212lines (∼vertical) with 68% (1 σ) confidence intervals. TCMB=4.6 K atz=0.68, which we set as the background
+radiation temperature, andT kin=55K(Henkel etal.2005). Theblack lineisthe4.6Kcontour. CSobservations andmodelingfirmlylimit n(H2)<2×104cm−3
+(Henkel etal. 2005) — the range of these plots exceed that lim it to show the behavior of H 2CO as a function of n(H2) — andTex<TCMBfor all points on the
+map below that limit. Column density ( y-axis) is in units of cm−2(km s−1pc−1)−1,and number density ( x-axis) is in units ofcm−3. Itisassumed throughout this
+paper thatthevelocity gradient dv/dris1km s−1pc−1. Thewhiteathigh n(H2)represents aceiling tothecontours at12Kandnotaphysica lplateau. Theupper
+right corner of the 1 10−111plot reaches ∼55 K, the kinetic temperature of the gas. Overplotted white l ines on the 1 10−111plot represent the solution if covering
+factors are f4.8=0.5 andf14.5= 0.8; the central region is excluded, while a cloud with negligi ble collisions — log( n[H2]) =0 cm−3— but unrealistic Xis not
+excluded by the ratio of optical depths at the 68% confidence l evel if priors on Xare neglected. The similar lines on the 2 11−212plot represent the constraints
+placed byobservations ifcovering factors are f4.8=0.3andf14.5=1.0; thecentral region iswithin the68%confidence region alth ough there isnosolution inthe
+modelthat exactly matches the observed lines. Differences in solutions shown on the two plots are dueentirely to the sel ection of covering factors.
+deviation of 50%. This overestimates the “up to 50%” er-
+ror quoted by Green (1991) but is close to that measured
+by Mengel& DeLucia (2000). For each of 25 sets of “er-
+roneous” collisional rates, we generated the parameter spa ce
+shown in Figure 3. The effect of the errors on line excita-
+tion temperatures is small at low densities, where collisio ns
+areinfrequent(Section4.4); at moderatedensities,the rm sof
+the excitation temperatures of data sets is comparable to th e
+mean excitation temperatures. For the density measurement s
+reportedherein(seebelow),themeanexcitationtemperatu res
+of our erroneous data sets are 0.85 K for the 1 10−111transi-
+tion and 1.4–1.7 K for the 2 11−212transition; rms values are
+1.0and1.0–1.5,respectively. Takinganextremescenarioa nd
+uniformly doubling the collisional excitation rates of Gre en
+(1991)increasestheeffectofcollisionsatagivendensity ,ap-
+proximately halving the n(H2) values (0.3 dex) on the x-axis
+inFigure3.
+The model used to determine excitation temperatures was
+generated with an ambient radiation temperature of TCMB=
+2.73(1+z)=4.60 K and a kinetic temperature of 55 K. With
+a parameter space spanning 1 cm−3<n(H2)<107cm−3
+and3×1012cm−2(kms−1pc−1)−1<N(o-H2CO)/(dv/dr)<
+3×1016cm−2(kms−1pc−1)−1withasamplingintervalof0.05
+dex, we compared the observed peak optical depths to those
+generated with our LVG model. The model is weakly de-
+pendent on kinetic temperature: changes in modeled optical
+deptharelessthanobservationalerrorfor TK=55±20K,and
+we present results only for Tkin=55 K, as measured via NH 3
+observationsbyHenkeletal. (2005).
+Of the four spectral components Mulleret al. (2007) iden-
+tifyintheirhighresolutionHCO+spectrum,thestrongestab-
+sorptionoccursatthesamevelocityasboththeH 2CO211−212
+line and the strongest component of the 1 10−111profile. The
+ratio of the strongest 1 10−111absorption component and the
+211−212line allows in situdensitometric measurements: the
+optical depth of one transition gives N(o-H2CO), and the ra-
+tio of two K-doublets gives n(H2). Thus, with the two linesFigure4. Ratio of the optical depth of the 1 10−111transition to that of the
+211−212transition asafunction of n(H2). Themodeledopticaldepthsassume
+covering factors of unity and a constant H 2CO abundance relative to H 2of
+X=10−9. The two-line densitometry method cannot produce unique re sults
+in regions where a single value of the optical depth ratio cor responds to two
+values of n(H2) unless the density is constrained by priors or ancillary in for-
+mation. Thenon-uniqueregionisenlarged if Xisnotassumedtobeconstant,
+as can be seen in the two viable density regions plotted in the left image of
+Figure 3, where the two possible densities are n(H2)≈6×104cm−3(with
+X≈10−11)andn(H2)≈60cm−3(withX≈3×10−8). Ifpriorscan discrim-
+inatebetween n(H2)/lessorsimilar2.5×103cm−3andn(H2)/greaterorsimilar2.5×103cm−3,solutions
+areunique and H 2CO is apowerful densitometer.
+one can define a region of the modeled parameter space by
+matching observed optical depths to modeled ones. When
+n(H2)/lessorsimilar2×104,theratiooflinescanresultinbifurcatedsolu-
+tionsforn(H2) whenusingthetwo-linedensitometrymethod
+employed herein — one at moderate n(H2) and one at low
+n(H2),asisshowninFigure4.
+To break the degeneracy of solutions using observations
+of other molecules and abundance arguments, we divide theFormaldehydeAnti-InversioninB0218 +357 7
+Figure5. Diagonal dashed lines mark the abundance of o-H 2CO relative to
+H2(seeEquation 3);vertical dot-dashed lines separate there gions, labeled at
+top,discussed inSection 4.3and Table3. TypicalGalactic a bundances areof
+order10−9,whilemeasurements of N(H2CO)(Jethava et al.2007)and N(H2)
+(Combes & Wiklind1995;Gerin et al1997)placethetotalH 2COabundance
+between 2 .5×10−9and 10−10in B0218 +357; the ortho/para ratio, which
+Jethava etal. (2007) measure as ∼2.8, drops the abundance of o-H 2CO by
+∼25% relative to the total abundance of H 2CO.
+possible solutions into three regions separated at n(H2) =
+2×102cm−3andn(H2) = 2×104cm−3, as shown in Fig-
+ure 5 and described in Table 3. The Region I/II boundary at
+n(H2)=2×102isanarbitrarydivisionbetweenlowdensities,
+where T ex≈TCMBfor both the 1 10−111and 211−212transi-
+tions, and moderate densities, where T ex<TCMB. The Re-
+gionII/IIIboundaryat n(H2)=2×104cm−3isthefirmupper
+limit placed by CS observations and modeling (Henkeletal.
+2005). Ranges of abundances in each region are calculated
+for all covering factors for which equilibrium model solu-
+tions exist, determining N(o-H2CO) and n(H2) for all cover-
+ing factors within these ranges, and converting column den-
+sities to number densities of o-H 2CO with Equation 3; max-
+imum and minimum abundancevalues( X, Equation 3) listed
+in Table 3 are the maximum and minimum values of n(o-
+H2CO)/n(H2) in each region. Due to the high abundance
+(log[X]>−8.2),the low-densityRegionI isunlikely: typical
+Galactic abundances have a small dispersion around a few ×
+10−9(e.g., Evanset al. 1975; Dickel,Goss, &Rots 1987).
+Measurements of N(H2CO) (Jethavaetal. 2007) and N(H2)
+(Combes& Wiklind 1995; Gerinet al 1997) in B0218 +357
+place the total H 2CO abundance between 2 .5×10−9and
+10−10inB0218+357;theortho/pararatio,whichJethavaetal.
+(2007) measure as 2.8 in their best model of the lines they
+observed,dropstheabundanceofo-H 2COby∼25%fromthe
+totalH 2COabundance. Theseconstraintsagreewiththevalue
+ofn(H2)≈3.2×103cm−3estimatedfromNH 3(Henkeletal.
+2005,noconfidenceintervalgiven).
+Fitting observed optical depths to our LVG model within
+the above constraints and forcing f14.5>0.6 gives 2 ×
+103cm−3<n(H2)<1×104cm−3(1σ) for the covering fac-
+tors discussed above. Agreement between observations and
+modeling requires f4.8/f14.5/greaterorsimilar1/3, while the ratio of solid
+angles in image A is Ω4.8/Ω14.5≈20. Since the covering
+factors are poorly constrained, results for n(H2) and T excan-
+not have proper confidence intervals. However, our reported
+ranges include all reasonable covering factors ( f14.5>0.6,
+f4.8/f14.5/greaterorsimilar1/3) and are thus robust. We list n(H2) and T exTable3
+Formaldehyde Radiative Transfer Model Regions: H 2Number Density and
+H2CO Abundance
+Region Min. log( n(H2)) Max. log( n(H2)) Min. log( X) Max. log( X)
+I 0.0 2.3 −8.2 −5.6
+II 2.3 4.3 −10.2 −8.2
+III 4.3 5.5 −10.3 −10.2
+Note. —Molecularhydrogendensities n(H2),withunitsofcm−3,ando-H 2CO
+abundances X(Equation 3) at the boundaries between the regions discusse d in
+Section 4.3 and shown in Figure 5. The Region I/II boundary is an arbitrary di-
+vision between low densities, whereT ex≈TCMB,and moderate densities, where
+Tex≪TCMB. TheRegion II/III boundary is the upper limit placed by CS ob ser-
+vations and modeling Henkel et al. (2005). Ranges of abundan ces are calculated
+by determining N(o-H2CO) and n(H2) for all covering factors with equilibrium
+model solutions within these ranges, and converting column densities to number
+densities of o-H 2CO with Equation 3; maximum and minimum Xvalues are the
+maximum and minimum values of n(H2CO)/n(H2) in each region. The typical
+H2CO abundance relative to H 2in Galactic molecular clouds is ∼10−9, and the
+ortho/para ratio ranges from 1 to 3. In B0218 +357, the total H 2CO abundance is
+1−25×10−10, assuming an ortho/para ratio of 2.8.
+Figure6. Excitation temperatures ofthe lowestfive K-doublet transitions as
+afunction of n(H2)forN(o-H2CO)=1014cm−2,ascalculated with our LVG
+model. This figure is a horizontal slice through Figure 3 with additional K-
+doubletlinesadded. Thedivergence ofthe K-doublet excitation temperatures
+asafunction of n(H2)makes theratio ofthe anypair oflines asensitive den-
+sitometer. However, since the ratios converge to unity at lo wn(H2) (as Tex
+approaches T CMB)andcrossitagainat n(H2)≈106(asallexcitation temper-
+atures go to T kinat high densities), the ratios cannot provide a unique globa l
+solution unless the covering factor is known a priorior other information,
+such as observations of other molecules, is available to bre ak the degeneracy
+(Section 4.3).
+forvariouscoveringfactorsinTables4 −6;Tex<TCMBforall
+physicalpairsofcoveringfactors.
+4.4.ColumnDensityandExcitationTemperatures
+We find 2 .5×1013cm−2<N(o-H2CO)<8.9×1013cm−2
+(assuming a velocity gradient of 1 kms−1pc−1), with the
+range determined by the viable covering factors ( f14.5>0.6,
+f4.8/f14.5/greaterorsimilar1/3)andconstraintsonmolecularhydrogenden-
+sities(2×103cm−3<n(H2)<1×104cm−3). Sincethesere-
+sultscoverall reasonablecoveringfactors,thelimits are hard
+and conservative. The total H 2CO column density, assuming
+o-H2CO/p-H 2CO=2.8 fromthe best modelsof Jethavaet al.
+(2007),whichisdeterminedprimarilyfrommillimeterobse r-
+vations where covering factors are unity and not a concern,
+is 3.4×1013—1.2×1014cm−2. The column density scales
+inverselywiththe assumedvelocitygradient.8 Zeiger& Darling
+Table4
+Model-Derived Number and Column Densities For
+Various Covering Factors
+f4.8f14.5log(n(H2)) log(N(o−H2CO)/(dv/dr))
+(cm−3) (cm−2(km s−1pc−1)−1)
+0.50 0.60 5.05+0.15
+−0.1513.55+0.10
+−0.10
+0.03+1.47
+−0.0314.10+0.00
+−0.10
+0.50 0.80 4.80+0.20
+−0.2013.45+0.10
+−0.05
+1.80+0.45
+−1.8013.95+0.15
+−0.15
+0.30 1.00 3.53+0.37
+−0.3813.60+0.05
+−0.05
+1.00 1.00 5.15+0.15
+−0.1013.30+0.10
+−0.05
+Note. — Confidence intervals are set at 1 σ(68%)
+but are non-Gaussian. Where more than one solution ex-
+ists within 1 σobservational error, all solutions are given.
+An error of ‘0.00’ indicates that the modeled density
+did not change significantly across the region of density
+space defined by fitting observed optical depths to those
+derived with our LVG model. Number densities above
+2×104cm−3contradict thefirmupper limitobtained from
+LVG models of CS observations (Henkel etal. 2005).
+Regardlessofcoveringfactors(althoughtableshereinonl y
+list several, the entire range of reasonable covering facto rs
+was evaluated), Figure 3 shows that the excitation tempera-
+tures of both observed lines are below T CMBwithin the con-
+straints placed by CS, which require n(H2)/lessorsimilar2×104cm−3.
+Figures3 and 6 show clearlythat, at z=0.68,the H 2CO cen-
+timeter lines will be anti-inverted. The non-LTEexcitatio n is
+densitydependent:
+1.n(H2)/lessorsimilar102cm−3: H2CO excitation is dominated by
+CMB photons. All excitation temperatures are held
+near to the microwave background and the system is
+nearly in radiative equilibrium with the CMB because
+collisionswithH 2aretooinfrequenttopumptheH 2CO
+K-doubletpopulationintoanti-inversion.
+2. 102cm−3/lessorsimilarn(H2)/lessorsimilar3×105cm−3: Collisions with H 2
+anti-invert the H 2COK-doublet population. The mil-
+limetertransitions,however,haveT ex≥TCMB.
+3.n(H2)/greaterorsimilar3×105cm−3: Highdensitiesincreasecollision
+rates and drive H 2CO toward thermal equilibriumwith
+H2,witha kinetictemperatureof55K.
+Figure 6, a horizontal slice through Figure 3, shows a ‘U’
+shape of excitation temperatures. The two possible values
+of Texandn(H2) for a given ratio of observed optical depths
+cause the degeneracy discussed in Section 4.3 and shown in
+Figure4. The n(H2)-dependenceofcentimeterexcitationtem-
+peratures makes H 2CO a sensitive in situdensitometer if the
+degeneracycanbebrokenwith aprioriknowledgeofphysical
+conditionssuchasH 2CO abundanceorlimitson n(H2).
+5.DISCUSSION
+5.1.ComparisontoExcitationinGalacticMolecularClouds
+Galactic dark clouds show a decrement in the excitation
+temperature of the 1 10−111line relative to the CMB of T ex−
+TCMB≃−0.7 to−0.4 K (Evansetal. 1975). An LVG ra-
+diative transport model applied to Galactic giant molecula r
+clouds derives comparable decrements, ranging from −1 to
+0 K (Henkeletal. 1980). The cloud studied in this paper
+showsamuchlargerdecrement,withT ex−TCMB/lessorsimilar−3.6K(for2×103cm−3<n(H2)<1×104cm−3). Forthe2 11−212transi-
+tion,thedecrementis ∼−2.6to−3.1K.Themagnifieddecre-
+ment is due to the collisional dependence of excitation tem-
+peratures (Section 4.4): CMB excitation is only significant
+indiffusecloudswith n(H2)/lessorsimilar102cm−3,soforhigherdensity
+cloudstheexcitationtemperatureisnearlyconstantregar dless
+of the CMB temperature. Increasingthe backgroundtemper-
+aturethusmagnifiesthe T ex−TCMBdecrementproportionally
+to(1+z).
+Because T ex−TCMB<0 for both transitions, the cloud
+would be observable in absorption against the CMB in both
+lines without the background quasar – if the cloud were in a
+different place in the lensing galaxy rather than aligned wi th
+imageAorinagalaxywithnobackgroundcontinuumsource,
+for instance. In the case where the cloud is matched to the
+telescope beam (no beam dilution) at z= 0.68 and the cov-
+ering factor is unity, the observed 1 10−111line temperature
+wouldbe
+Tline=(1−e−τ)(Tex−TCMB)/(1+z)≈−36mK.(4)
+For a non-unity covering factor and small optical depth, as
+seen in B0218 +357,Tlineis approximately inversely propor-
+tional to the coveringfactor. Such a cloud illuminated by th e
+CMB will be observable with future instruments such as the
+EVLA. Since T exis small and nearly constant with redshift
+and TCMBscales as 1 +z, this raises the possibility of survey-
+ingmoleculargasingalaxies regardlessofdistance orchance
+alignmentwith sourcesofbackgroundillumination.
+Modelsindicatethattheanti-inversionpersistsinthe Ka=1
+K-doubletsatleastuptothe6 15−616o-H2COtransitionforall
+reasonable covering factors (which determine the measured
+molecular hydrogen density), and for some covering factors
+the anti-inversion persists into the 7 16−717transition (Table
+5). Although the highest energy lines may be difficult to de-
+tect, all of the anti-invertedtransitions should in princi ple be
+observableinabsorptionagainsttheCMB.
+K-doublet transitions with Ka= 2 (p-H 2CO) and Ka= 3
+(o-H2CO) are even more strongly anti-inverted than those in
+theKa= 1 ladder, but relative to the Ka= 1K-doublets the
+higher energy of the Ka>1 states (the lowest Ka= 2K-
+doublet level, 2 21at 57.6 K, has approximately the same en-
+ergy as the 5 15level at 62.5 K) generally makes them more
+optically thin and difficult to observe given the low temper-
+atures of molecular clouds. Our models indicate that the
+220−221transition (71.1 MHz rest frequency), the lowest K-
+doublet of the Ka= 2 ladder, has an excitation temperature
+less than 10 mK in B0218 +357. Due to the relatively warm
+temperature of the gas observed in B0218 +357 (55 K), the
+optical depth of the transition is predicted to be lower than
+that of the observed 1 10−111o-H2CO line by only a factor of
+∼5 and higher by a factor of ∼20 than would be expected
+in an otherwise equivalent cloud with a kinetic temperature
+of 15 K (assuming an ortho/para ratio of 2.8; Jethavaet al.
+2007). The next lowest K-doublet of the Ka= 2 ladder, the
+321−322transition (355.6 MHz rest), is predicted to have an
+optical depth that is an order of magnitude lower than that
+of the 2 20−221line. All other Ka>1K-doublets are weaker
+by yet another order of magnitude or more than the 3 21−322
+line. In the hypothetical case of a similar gas-rich galaxy i l-
+luminated only by the CMB at z= 0.68 (the flux from the
+quasar illuminating B0218 +357 is not well known at meter
+wavelengths), the observed line temperatures for the lowes t
+Ka= 2K-doublets are predicted to be between −4 and−11FormaldehydeAnti-InversioninB0218 +357 9
+Table 5
+Modeled K-doublet ( ∆J=0) o-H 2CO Line Excitation Temperatures
+f4.8f14.5log(n(H2)) 110−111211−212312−313413−414514−515615−616716−717
+(cm−3) 4.8 GHz 14.5 GHz 29.0 GHz 48.3GHz 72.4 GHz 101.3 GHz 135.0 GHz
+0.50 0.60 5.05+0.15
+−0.150.97+3.26
+−0.091.57+2.90
+−0.031.98+2.55
+−0.012.34+2.22
+−0.042.94+1.61
+−0.113.63+0.76
+−0.144.77+0.20
+−0.79
+0.03+1.47
+−0.034.57+0.00
+−1.234.59+0.00
+−0.514.59+0.00
+−0.284.60+0.00
+−0.194.60+0.00
+−0.204.58+0.00
+−0.654.41+0.00
+−0.48
+0.50 0.80 4.80+0.20
+−0.200.84+3.42
+−0.061.53+2.95
+−0.001.98+2.57
+−0.012.29+2.27
+−0.002.79+1.77
+−0.073.43+1.00
+−0.124.53+0.20
+−0.53
+1.80+0.45
+−1.803.24+1.33
+−0.994.03+0.56
+−0.634.30+0.29
+−0.434.40+0.20
+−0.304.39+0.21
+−0.313.92+0.66
+−0.443.93+0.48
+−0.03
+0.30 1.00 3.53+0.37
+−0.380.80+3.34
+−0.051.75+2.68
+−0.102.51+2.01
+−0.202.89+1.66
+−0.272.87+1.67
+−0.163.06+1.29
+−0.003.93+0.07
+−0.04
+1.00 1.00 5.15+0.15
+−0.101.04+3.31
+−0.081.60+2.91
+−0.032.01+2.55
+−0.032.39+2.18
+−0.053.03+1.54
+−0.103.76+0.71
+−0.144.92+0.25
+−0.90
+Note. — Excitation temperatures for selected pairs of covering f actorsf4.8andf14.5; boldfaced temperatures have
+Tex≥TCMB(TCMB= 4.60 K) within the listed 68% (non-Gaussian) confidence interv als. Frequencies are rest fre-
+quencies, and all excitation temperatures have units of K. R egions listed with errors of ‘0.00’ do not change excitation
+temperature significantly across the region of density spac e defined by fitting observed optical depths to those derived
+with our LVG model. Multiple excitation temperatures for a g iven pair of covering factors correspond to solutions in
+Table4whereapair ofoptical depths did notproduce aunique solution. Lowdensity solutions ( n(H2)<102cm−3)have
+Tex= TCMB(z= 0.68466) = 4 .60 K because collisions are insufficient to be a significant ex citation mechanism. Of the
+K-doublet transitions, only the1 10−111and 211−212lines havebeen observed in B0218 +357 (seeTable 2).
+Table 6
+Modeled ∆J=1H2COLine Excitation Temperatures
+f4.8f14.5log(n(H2)) 212−111(O) 2 11−110(O) 1 01−000(P) 2 02−101(P)
+(cm−3) 140.8 GHz 150.5 GHz 72.8 GHz 145.6 GHz
+0.50 0.60 5.05+0.15
+−0.157.73+1.45
+−3.136.70+0.89
+−2.1017.40+14.90
+−12.806.74+0.84
+−2.14
+0.03+1.47
+−0.034.60+0.00
+−0.004.60+0.00
+−0.004.60+0.00
+−0.004.60+0.00
+−0.00
+0.50 0.80 4.80+0.20
+−0.206.31+1.06
+−1.715.78+0.69
+−1.189.77+5.23
+−5.175.84+0.67
+−1.24
+1.80+0.45
+−1.804.60+0.00
+−0.004.60+0.00
+−0.004.60+0.01
+−0.004.60+0.00
+−0.00
+0.30 1.00 3.53+0.37
+−0.384.73+0.06
+−0.134.68+0.05
+−0.084.88+0.16
+−0.284.70+0.05
+−0.10
+1.00 1.00 5.15+0.15
+−0.108.63+1.97
+−4.037.26+1.14
+−2.6625.20+42.70
+−20.607.27+1.08
+−2.67
+Note. — Similar to Table 5 but for ∆J= 1 transitions of both o-H 2CO (O) and
+p-H2CO (P). All excitation temperatures have units of K. All tran sitions in this ta-
+ble have been observed in B0218 +357 (Jethava et al. 2007). Low density solutions
+(n(H2)<102cm−3) have T ex= TCMB(z= 0.68466) = 4 .60 K because collisions are
+insufficient to beasignificant excitation mechanism.
+mK for the 2 20−221line and−0.3 and−1 mK for the 3 21−322
+line (assuming N(H2CO) = 5×1013cm−2and an ortho/para
+ratio of 2.8, with the range set by our measured range of val-
+ues for the molecular hydrogen density in the cloud). While
+these low-frequency lines will be dwarfed by the foreground
+Galacticcontinuumanddifficulttoobservefortheirownsak e,
+this modeling indicates that H 2COKa=2 absorption lines in
+gas-rich galaxies present potential contaminants to Epoch of
+Reionizationstudies.
+5.2.ComparisontoPreviousResults
+Jethavaet al.(2007)measure N(o-H2CO)=3.8×1013cm−2
+andn(H2)<103cm−3(with a best fit of n(H2) = 200 cm−3)
+with their non-LTE LVG analysis of six H 2CO lines in
+B0218+357. Menten& Reid (1996), assuming LTE, mea-
+sureN(o-H2CO) = 1.2×1013cm−2from their observation
+the 211−212line, although they estimate that anti-inversion
+of theK-doublet would reduce their result by 50%, well be-
+low the column densities reported by Jethavaetal. (2007)
+and our range of 2 .5×1013−8.9×1013cm−2(fordv/dr=
+1kms−1pc−1).
+Our 110−111spectrum has a significantly higher signal to
+noise ratio and better spectral resolutionthan previousob ser-vations, and reliance solely on it and our new 2 11−212data
+to determine densities precludes high-frequencyatmosphe ric
+variability while magnifying the effect of uncertain cover -
+ing factors. Since the centimeter K-doublet lines measure
+n(H2)withgreaterprecisionthanthemillimeterlinescan,the
+trade-off makes the Jethavaetal. (2007) result less accura te
+for determining n(H2) and T ex(which they do not address)
+but more reliable for measuring N(o-H2CO), since they rely
+heavily on nearly thermal millimeter transitions which hav e
+covering factors of unity. The optimal combination of the
+twomethodsusesthecentimeterlinestoconstrain n(H2)(and
+Tex,sinceitiscloselyrelated)andthemillimeterlinestocon -
+strainN(o-H2CO). T exandn(H2) are nearly independent of
+N(o-H2CO) at the column densities observed in this source
+(Figure3).
+Our grid-search method is a more comprehensive analy-
+sis of the n(H2)-N(o-H2CO) density space than that used by
+Jethavaetal.(2007),whocomparedtheirobservationsonly to
+LVG models with n(H2)=0, 200, 500, 1000, and 3000 cm−3
+to determine a best-fit ( χ2) value. The small values of n(H2)
+sampled by Jethavaet al. (2007) also neglect the high- n(H2)
+solutions discussed in Section 4.3, essentially assuming t heir
+conclusionthattheabsorbingcloudisdiffuseandlow-dens ity10 Zeiger&Darling
+andnotconsideringthe alternativehigherdensityregime.
+6.CONCLUSIONS
+With new measurementsof the 1 10−111and 211−212transi-
+tionsofH 2COinB0218 +357,wehavemeasuredthemolecu-
+lar hydrogennumberdensity and H 2CO column density. Us-
+inganLVGmodelofthephysicalconditionsoftheabsorbing
+cloud in the gravitational lens and assuming a velocity gra-
+dient of 1kms−1pc−1, we measure N(o-H2CO) to be in the
+range 2.5×1013−8.9×1013cm−2, where the uncertainty in
+valuesis caused bythe uncertaintyin coveringfactors(in t he
+optically thin limit, the column density is inversely propo r-
+tional to the covering factor). Jethavaet al. (2007) measur e
+an ortho/para ratio of 2 .0−3.0 (with a best value of 2.8) and
+a second component to the profile that adds ∼15% to the to-
+tal column density; these two contributions increase the to -
+tal column N(H2CO) by∼55% above our measurement of
+N(o-H2CO) in the primaryabsorptioncomponent. Since col-
+lisionswithhydrogenarethedominantexcitationmechanis m,
+we are also able to measure n(H2) = 2×103−1×104cm−3,
+whichagainspanstherangeofviablecoveringfactors.
+Modeled excitation temperatures show both observed cen-
+timeter lines to be anti-invertedrelative to the 4.6 K CMB at
+z= 0.68, with T ex(110−111)≈1 K and T ex(211−212)/lessorsimilar2 K.
+Excitation temperatures of the millimeter lines are ≥4.60 K,
+as expected. These results show that the dominant absorbing
+cloudin the B0218 +357lens is analogousto Galactic molec-
+ular clouds that show anti-inversion of the K-doublet transi-
+tions, and the centimeter H 2CO lines could be observed in
+absorptionagainsttheCMB ifthebackgroundsourcewasre-
+moved or if the cloud was viewed along a different line of
+sight.
+Since T exis determined by collisions in dense clouds, the
+source-framedecrement T ex−TCMBis magnified when T CMB
+isincreased(n.b.,athighredshift). Theanti-invertedce ntime-
+ter transitions thus could provide absorptionlines agains t the
+CMB that would trace molecular gas independently of dis-
+tancewhen observed with a telescope capable of resolving
+the star-forming regions in galaxies. Future work targetin g
+additionalobjectswillquantifythechangeinT ex−TCMBwith
+redshiftandtest theseresults.
+We are indebted to C. Henkel for providing the LVG code
+used in this analysis and to the anonymousreferee who sug-
+gested that we model the excitation temperatures of Ka>1
+K-doubletsandcheckedourcalculationswithanindependent
+radiativetransfercode.
+SupportforthisworkwasprovidedbyNASAthroughHub-
+ble Fellowship grant no. HST-HF-01183.01-A awarded by
+the Space Telescope Science Institute, which is operated by
+the Association of Universities for Research in Astronomy,
+Incorporated, under NASA contract NAS5-26555. We also
+acknowledgethesupportoftheNSFthroughawardGSSP07-
+0015fromtheNRAO andNSF grantAST-0707713.
+The LUNAR consortium
+(http://lunar.colorado.edu ), headquartered at the
+University of Colorado, is funded by the NASA Lunar Sci-
+ence Institute (via Cooperative Agreement NNA09DB30A)to investigate concepts for astrophysical observatories o n the
+Moon.
+ThisresearchhasmadeuseoftheNASA/IPACExtragalac-
+tic Database (NED) which is operated by the Jet Propulsion
+Laboratory,CaliforniaInstituteofTechnology,undercon tract
+withtheNationalAeronauticsandSpaceAdministration.
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+arXiv:1001.0335v1  [quant-ph]  2 Jan 2010Wave communication across regular lattices
+Birgit Hein and Gregor Tanner
+School of Mathematical Sciences, University of Nottingham , University Park, Nottingham NG7 2RD, UK.
+(Dated: June 16, 2018)
+We propose a novel way to communicate signals in the form of wa ves across a d- dimensional
+lattice. The mechanism is based on quantum search algorithm s and makes it possible to both search
+for marked positions in a regular grid and to communicate bet ween two (or more) points on the
+lattice. Remarkably, neither the sender nor the receiver ne eds to know the position of each other
+despite the fact that the signal is only exchanged between th e contributing parties. This is an
+example of using wave interference as a resource by controll ing localisation phenomena effectively.
+Possible experimental realisations will be discussed.
+PACS numbers: 03.67.Hk, 03.65Sq, 03.67.Ac, 42.50.Ex
+Localization phenomena in linear wave systems are
+closely linked to wave interference effects. Anderson lo-
+calisation in disordered media is a prime example thereof
+still posing challenges to both theory and experiment
+[1] 60 years after its discovery [2]. Recently, a new re-
+search area has emerged focusing on interference as a
+resource and making use of localisation phenomena in a
+controlled way. Prominent examples are among others
+time reversal imaging [3] and reconstructing the Green
+function in terms of correlation functions [4], see also [5]
+. Here, information about the wave system is obtained
+by manipulating a seemingly ‘noisy’ signal using phase
+coherence. We will focus here on another class of wave
+localisation phenomena with counterintuitive properties,
+namely(quantum)searchalgorithmsand(quantum)ran-
+dom walks. Wave search algorithms gained prominence
+with Grover’s work [6] demonstrating a√
+Nspeed-up
+compared to a classical search within an unsorted data
+base ofNitems. Even though search algorithms became
+an inherent part of quantum information theory [7], the
+speed-up is in effect caused by wave interference as has
+already been pointed out by Grover [8] and has been im-
+plemented for a classical wave system in [9]. Based on
+ideas from quantum random walks [10–12], Grover’s al-
+gorithmhasbeen generalizedtospatialsearchalgorithms
+on networks such as on a hypercube [13, 14] and on reg-
+ular lattices [15]. Experimental realisations of quantum
+random walks have been achieved again both using clas-
+sical waves (optics) [16] and quantum devices [17].
+Starting from wave search algorithms, we will
+demonstrate that localisation can be used to establish
+communication channels across a regular lattice with
+surprising properties: (i) signals can be exchanged
+exclusively between a source and a receiver point, where
+neither the sender nor the receiver know the position of
+each other; (ii) the signal can track a moving receiver
+in the network; (iii) the algorithm can be used as a
+searching device without the necessity to know the time
+of measurement, (a typical requirement for Grover’s
+search algorithm); (iv) the protocol can act as a sensitive
+switching device for wave transport through a lattice;
+(v) the algorithm can be effectively implemented both
+on a quantum computer and using classical wavesx2−
+1+1−
+2+ aσ
+FIG.1: Regular gridwith d= 2,n= 8; local scatteringwithin
+a unit cell at vertex /vector xis described by the matrix σ.
+only. We will first describe the set-up of the search
+algorithm. We then introduce a simplified model for
+the searchand explainthe wavecommunicationprotocol.
+We consider wave propagation across d-dimensional
+periodic lattices of identical scatterers or periodic po-
+tentials with fixed lattice parameter a, see Fig. 1. It is
+important that the lattice has a finite number of sites n
+along each axis with a total number ndof lattice sites.
+To simplify the calculations we will restrict ourselves to
+models with nearest-neighbour interaction only and con-
+sider periodic boundary conditions. The wave dynamics
+within each unit cell is given by a local scattering matrix
+σmapping incoming channels onto outgoing channels,
+see Fig. 1. ( σis also denoted a coin matrix in the con-
+text of quantum walks). The overall wave dynamics is
+then given in terms of an operator U0mapping incom-
+ing onto outgoing wave coefficients between unit cells.
+We have dim U0=mndwheremdenotes the number of
+open scatteringchannels within a unit cell. Furthermore,
+U0is unitary when disregarding dissipation. Stationary
+solutions are obtained by the condition
+det/parenleftbig
+1−eıkaU0/parenrightbig
+≡0 (1)
+wherekis the wave length and exp(ı ka) is a phase
+shift between incoming and outgoing waves. We ne-
+glect any (in general weak) kdependence of σand
+thusU0. Note that the spectrum obtained from (1) is
+now periodic in kwith period 2 π/a; the eigenvalues are
+kj,l= (2πl−θj)/a, l∈N, whereθjare the eigenphases
+ofU0withj= 1,...,N. To start with, we will consider a2
+model consisting ofa singleopen channelbetween nearby
+lattice sites ( m= 2d) and we assume Kirchhoff bound-
+ary conditions at each vertex. The Hilbert space is then
+effectively N= 2dnddimensional. This physical model
+captures the essence behind the effect described below.
+We label incoming wave components from each unit
+cell as|i±∝an}bracketri}ht ⊗ |/vector x∝an}bracketri}ht=|i±,/vector x∝an}bracketri}ht, where/vector xspecifies the vertex
+in position space and i±, fori= 1,...,dgives the ±
+direction in dimension i. Incoming waves at vertex /vector x
+are mapped onto outgoing waves by a scattering matrix
+σ. For Kirchhoff boundary conditions, one obtains
+σ= 2|s∝an}bracketri}ht∝an}bracketle{ts| − /BD2dand|s∝an}bracketri}htis the uniform distribution
+|s∝an}bracketri}ht=1√
+2d/summationtextd
+i=1(|i+∝an}bracketri}ht+|i−∝an}bracketri}ht) [15]. Outgoing waves in
+direction i±are now identified with incoming waves
+at an adjacent vertex /vector x±a/vector eiwhere/vector eiis the unit
+vector in direction i. The local scattering processes
+is described in terms of a (global) scattering (or coin)
+matrixC=σ⊗ /BDnd. The full wave propagator (or
+quantum walk) U0is obtained from Cafter identifying
+incoming and outgoing waves of adjacent unit cells
+accordingly. The spectrum of the unperturbed walk U0
+exhibits a band structure as shown in Fig. 2 a), here for
+d= 2. (For a finite lattice, the quasi-momenta κx,κyare
+discretised accordingto κx,y= 2πj/(na);j= 0...n−1.)
+Following Ambainis, Kempe and Rivosh (AKR), the
+quantum walk U0acts as a search algorithm after mark-
+ing a target vertex |v∝an}bracketri}htby a modified scattering matrix
+σ′[15], that is, one considers C′=C−(σ−σ′)⊗|v∝an}bracketri}ht∝an}bracketle{tv|.
+The AKR search uses σ′=− /BD2d. Since|s∝an}bracketri}htis an eigen-
+vectorof σ, we may write U′=U0(1−2|sv∝an}bracketri}ht∝an}bracketle{tsv|), where
+|sv∝an}bracketri}ht=|s∝an}bracketri}ht ⊗ |v∝an}bracketri}ht. The search algorithm is initialised in
+the uniform state |Φ0∝an}bracketri}ht= 1/√
+N/summationtext
+/vector x|s/vector x∝an}bracketri}ht, and the walk
+(U′)T|Φ0∝an}bracketri}htlocalizes at |v∝an}bracketri}htafterT∝√
+Nsteps.
+The AKR search can be analysed by defining a one
+parameter family of unitary operators [14]
+Uλ=U0+/parenleftbig
+eıπλ−1/parenrightbig
+U0|sv∝an}bracketri}ht∝an}bracketle{tsv|; (2)
+one obtains U0forλ= 0 or 2 and the AKR search for
+λ= 1. The part ofthe eigenfrequency spectrum of Uλin-
+teractingwiththeperturbationisshowninFig.2b). The
+spectrum is periodic in kwith period 2 π/aindependent
+ofλ. When varying λ, a “perturber state” |νλ∝an}bracketri}htemerges
+whichcrossesthe kmod 2π/a= 0axisat λ= 1. There-
+sultingavoidedcrossingbetweentheinitialstate |Φ0∝an}bracketri}htand
+|νλ∝an}bracketri}htis shown in Fig. 2 c). Note that |Φ0∝an}bracketri}htis the fully sym-
+metric eigenstate of U0corresponding to a d-dimensional
+Bloch-vector /vector κ= 0 of the unperturbed spectrum with
+eigenvalue kmod 2π/a= 0. Like in Grover’s algorithm,
+the quantum search U′=Uλ=1rotates the initial state
+|Φ0∝an}bracketri}htinto a localised state |νλ∝an}bracketri}htwhich has here a strong
+overlap with the target state |sv∝an}bracketri}ht. The search time T0is
+inversely proportional to the gap at the avoided crossing
+∆, that is, T0≈π/∆.
+In order to obtain an estimate for the search time T0
+as well as the efficiency of the search, that is, the matrix
+element∝an}bracketle{tsv|νλ∝an}bracketri}ht, it is essential to find the approximately0 0.5 1 1.5 2
+λ-202ω
+±202
+kappa2±3±2±10123
+kappa1±3±2±10123
+κ−π
+−π−πκ1/a/aπ
+k mod 2   /aπ
+/aπ/a/aπ /a2
+νλ|>
+|Φo>ω|>−ω|>+∆a)
+c)
+b)π/a
+λ/a−π
+πk mod 2   /a
+FIG. 2: a) The band structure at λ= 0 for d= 2 and in
+the limit n→ ∞with/vector κ, the Bloch wave numbers; b) the
+eigenphases of Uλforn= 11,d= 2; c) the avoided crossing
+with spectral gap ∆ at λ= 1 and kmod 2π/a= 0 together
+with theapproximate eigenstates |νλ/angbracketrightand|Φ0/angbracketright(dashedlines).
+invariant two-level subspace near the crossing spanned
+by|Φ0∝an}bracketri}htand|νλ=1∝an}bracketri}ht. The technique developed in [14] for
+the hypercube hasbeen adaptedto regulargrids. We will
+only give the result here, further details will be presented
+elsewhere[18]. Onefindsforthenormalisedvector |νλ=1∝an}bracketri}ht
+|νλ=1∝an}bracketri}ht=−∝an}bracketle{tsv|vλ=1∝an}bracketri}ht/radicalbigg
+2
+N/summationdisplay
+/vector κ/negationslash=/vector0e−2πı
+n/vector κ/vector v× (3)
+/parenleftbiggeıθ/vector κ
+1−eıθ/vector κ/vextendsingle/vextendsingleΦ+
+/vector κ/angbracketrightbig
++e−ıθ/vector κ
+1−e−ıθ/vector κ/vextendsingle/vextendsingleΦ−
+/vector κ/angbracketrightbig/parenrightbigg
+,
+where/vector vis the position of the target vertex and |Φ±
+/vector κ∝an}bracketri}ht
+and±θ/vector κare the eigenvectors and eigenphases of the un-
+perturbed walk U0[15]. The d-dimensional label /vector κwith
+κi= 0,...,n−1 is equivalent to the (discretised) Bloch
+wave number, see Fig. 2 a). The eigenphases are ex-
+plicitly given as cos θ/vector κ=1
+d/summationtextd
+i=1cos2πκi
+n. The overlap-
+matrix element ∝an}bracketle{tsv|νλ=1∝an}bracketri}htcan be estimated [18]:
+∝an}bracketle{tsv|νλ=1∝an}bracketri}ht=/braceleftbigg
+O(1//radicalbig
+logN) ford= 2,
+O(1) for d >2.
+Detailed expressions for the leading order coefficients for
+d= 2 and 3 are given in [18]. We find that |νλ=1∝an}bracketri}htis ex-
+ponentially localised on the marked vertex /vector v; note, that
+the overlap of |sv∝an}bracketri}htwith a typical eigenstate of the unper-
+turbed spectrum |Φκ∝an}bracketri}htis of the order O(N−1
+2)≪ O(1).
+Near the avoided crossing, the level dynamics can be
+described in terms of the two-level sub-space spanned
+by the orthogonal vectors |νλ∝an}bracketri}htand|Φ0∝an}bracketri}ht. Writing the
+unitary operator UλasUλ= e−ıHλ, one obtains at λ= 1
+in the{|Φ0∝an}bracketri}ht,|νλ=1∝an}bracketri}ht}basis an effective two-dimensional
+Hamiltonian Hλ=1of the form
+H2×2=/parenleftbigg
+kla−ıǫ
+ıǫ kla/parenrightbigg
+(4)3
+    
+       
+    
+       
+    
+       
+    
+       01020 1020
+01020 1020
+01020 1020
+01020 102010−410−410−4
+10−410−210−2
+10−2 10−2T = 30T = 0 T = 15
+T = 451.0 1.0
+1.0 1.0
+FIG. 3: Probability distribution of the quantum walk on a
+31×31-grid up to T= 45 time steps.
+withkl= 2πl/a,l∈Nbeing the wave numbers corre-
+sponding to /vector κ= 0 states of the unperturbed lattice and
+ǫis a real and positive coupling parameter, that is,
+ǫ=∆
+2=∝an}bracketle{tνλ=1|U1|Φ0∝an}bracketri}ht ≈2|∝an}bracketle{tsv|νλ∝an}bracketri}ht|√
+N+O/parenleftbig
+N−1/parenrightbig
+(5)
+with ∆, the gap at the avoided crossing.
+The start vector (1 ,0)≡ |Φ0∝an}bracketri}htis rotated into the lo-
+calised state (0 ,1)≡ |νλ=1∝an}bracketri}htinT0=π/(2ǫ) steps leading
+to the√
+Nspeed-up [15]. The whole process is 2 T0-
+periodic , that is, one needs - like for Grover’s algorithm
+[6] - to know the period T0to perform the search. For a
+simulation of the search on a 31 ×31 grid, see Fig. 3.
+Interesting applications emerge when considering sev-
+eral target vertices,/vextendsingle/vextendsinglevi/angbracketrightbig
+,i= 1,...mwithm≪N. We
+now define a set of parameters λ= (λ1,...,λ m) and a
+search algorithm of the form
+Uλ=U+m/summationdisplay
+i=1/parenleftbig
+eıπλi−1/parenrightbig
+U/vextendsingle/vextendsinglesvi/angbracketrightbig/angbracketleftbig
+svi/vextendsingle/vextendsingle.(6)
+Atλ= (1,1,...,1) andkamod 2π= 0, one finds that
+there are m−1 degenerate eigenvalues and two further
+eigenvaluesforminganavoidedcrossingwiththedegener-
+atesubset. Thecorrespondingsetof m+1eigenstatesco-
+incidesingoodapproximationwiththesubspacespanned
+by the uniform distribution |Φ0∝an}bracketri}htand now mlocalised
+states/vextendsingle/vextendsingleνi
+λ/angbracketrightbig
+,i= 1,...,m. Each of the/vextendsingle/vextendsingleνi
+λ/angbracketrightbig
+is well de-
+scribed by the approximation (3) and ∝an}bracketle{tνi
+λ|νj
+λ∝an}bracketri}ht ≈δij. The
+localisedstatesinteractatthecrossingpredominantlyvia
+|Φ0∝an}bracketri}htwhich takes on the role of a carrier state . In analogy
+to (4), we write a model Hamiltonian at the crossing inthe basis {|Φ0∝an}bracketri}ht,|ν1
+λ=1∝an}bracketri}ht,...,|νm
+λ=1∝an}bracketri}ht}as
+H(m+1)×(m+1)=
+kla−ıǫ−ıǫ ...−ıǫ
+ıǫ kla0...0
+ıǫ0kla......
+............0
+ıǫ0...0kla
+.(7)
+Like for the full propagator Uλ=1, the spectrum of
+H(m+1)×(m+1)consists of ( m−1) eigenvalues equal to
+klaand two eigenvalues kla±√mǫwith eigenvectors
+|ω±∝an}bracketri}ht= ı/√
+2m(∓ı√m,1,...,1)t. The gap between the
+non-degenerate levels is now ∆ = 2√mǫwithǫgiven in
+(5). Starting the search in the totally symmetric state
+|Φ0∝an}bracketri}htatλ= (1,...,1), one finds that UT
+λ|Φ0∝an}bracketri}htlocalises
+on allmmarked vertices after T0∼π/2/radicalbig
+N/m(or, for
+d= 2,T0∼π/2/radicalbig
+N/mlogN) steps simultaneously.
+More interestingly, the quantum walk can also be used
+to transmit signals across the network. Such a sender-
+receiver configuration has to the best of our knowledge
+notbeendescribedbeforeandmayhaveinterestingappli-
+cations both in a quantum setting, but also for classical
+waves (such as microwaves, in optics or acoustics). In-
+stead of starting the walk in the uniform state |Φ0∝an}bracketri}ht, we
+propose to begin the walk at one of the localised states
+|νm
+λ∝an}bracketri}ht, say. At the avoided crossing, this state can approx-
+imately be described as
+|νm
+λ∝an}bracketri}ht=−ı√
+2m/parenleftig
+|ω+∝an}bracketri}ht+|ω−∝an}bracketri}ht−ı/radicalbig
+2(m−1)|ω0∝an}bracketri}ht/parenrightig
+(8)
+with|ω0∝an}bracketri}ht= (m(m−1))−1/2(0,1...,1−m)tin the basis
+spanning H(m+1)×(m+1);|ω0∝an}bracketri}htis a vector in the degener-
+ate eigenspace with eigenvalue kla. Applying the walk
+forTs=π/(ǫ√m) = 2T0steps leads to
+UTs|νm
+λ∝an}bracketri}ht=ıeıklaTs
+√
+2m/parenleftig
+|ω+∝an}bracketri}ht+|ω−∝an}bracketri}ht+ı/radicalbig
+2(m−1)|ω0∝an}bracketri}ht/parenrightig
+(9)
+= eıklaTs/parenleftbigg
+0,−2
+m,...,−2
+m,1−2
+m/parenrightbiggt
+.
+This implies that a signal of intensity 4 /m2is transmit-
+ted from the sender (located at the m-th marked ver-
+tex) to each of the m−1 other marked vertices. The
+intensity at the sender at time Tsis then of the order
+(1−2/m)2. Thesefindingshavebeenverifiednumerically
+in our model. Interestingly, in the case m= 2 with a sin-
+gle receiver, the signal is transmitted in full . This opens
+up the possibility of transferring signals directly between
+two points on a network where neither the sender nor
+the receiver know each other’s position. In addition, the
+sender has information about the number of receivers by
+recording the signal at time Ts.
+The effect persists also for a continuous source at the
+sender position. In Fig. 4, we recorded the signal both at
+the sender and at the receiver. (To keep the signal finite,4
+0 100 200 300 400
+time01020304050signal at sender and receiverssender
+receiver 1
+receiver 2
+FIG. 4: Sending continuously: signal at sender (full) and
+receivers (dashed/dotted). The response of the system to
+switching the position of the receiver at T= 200 is shown.
+(We introduced a damping of 1% at all vertices.)
+we added a small amount of absorption across the net-
+work). Again, without prior knowledge of the receiver’s
+position,thenetworklocalisesatthetwomarkedvertices,
+thus making it possible to actually exchange information
+continuouslybetweenthese twopoints. Changingthe po-
+sition of the receiver leads to a sudden drop of the signal
+at the old receiver position and a build-up at the new po-
+sition, see Fig. 4. The system is thus capable of tracking
+a moving receiver position! The signal speed is limited
+by the transfer time Ts∼√
+N(Ts∼√
+NlnNfor d=2)
+and thus is the speed at which the receiver can move.
+The continuous sender/receiver protocol can also be
+used to searchamarkeditem without knowingthe search
+timeT0; this is a serious complication of Grover-type
+search algorithms when the precise number of marked
+items is not know (as T0depends on m). Here, we findthe marked items as long as we wait for times T≥Ts.
+Furthermore, the system can act as a switching device.
+Wave transport between two points on the grid can
+only be achieved, if the system is tuned to the avoided
+crossing. Slight detuning by for example changing the
+parameter λin (2) will quickly cut-off the signal.
+The described effects open up completely new ways
+of transmitting signals across regular networks. While
+a construction of the map Uis certainly feasible in a
+quantum setting and can be implemented efficiently on a
+quantum computer [7], an implementation using classical
+waves may be even more promising. For dispersionless
+wave dynamics and on regular lattices, the unitary
+matrixUλis equivalent to a discretised version of the
+time dependent Green function (for times t=a/cwith
+cbeing the wave velocity; corresponding time scales in
+a quantum setting would be given by the group velocity
+[20]). Indeed, localised states due to a local perturbation
+in a regular lattice are well known; (for example in
+optical crystals, see [19]). We predict that the effect
+will occur if “defect states” created by local phase
+perturbations (equivalent to the perturbed coin C′) are
+pushed into the (discretised continuum) of the band
+close to the fully periodic state - the /vector κ= 0 state, see Fig.
+2 a. Furthermore, the interaction between the defect
+states and the lattice states must be small enough to
+lead to avoided crossings between these two states only.
+We expect that a signal (such as a laser or a microwave
+transmitter coupled into a periodic structure at a defect
+position) can be transmitted and focused onto another
+defect in the same lattice using the described effect. The
+totally symmetric state |Φ0∝an}bracketri}htacts then as a carrier state
+guiding the signal between the perturbations.
+We thank J Keating, U Kuhl, T Monteiro, U Peschel,
+H-JSt¨ ockmannandHSusantoforhelpfuldiscussionsand
+S Gnutzmann for carefully reading the manuscript.
+[1] H. Hu et al,Nature Physics 4, 945 (2008) and ref. therin.
+[2] A. Anderson, Phys. Rev. 109, 1492 (1958).
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+(2002).
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+(2007).
+[6] L. K. Grover, Phys. Rev. Lett. ,79325 (1997).
+[7] M. A. Nielsen and I. L. Chuang, Quantum Computation
+and Quantum Information , (Camb. Uni. Press 2000).
+[8] L. K. Grover and A. M. Sengupta, Phys. Rev. A 65,
+032319 (2002).
+[9] N. Bhattacharya, H. B. Linden van den Heuvell and R.
+J. C. Spreeuw, Phys. Rev. Lett. 88137901 (2002).
+[10] Y. Aharonov, L. Davidovich and N. Zagury, Phys. Rev.
+A48, 1687 (1993).
+[11] J. Kempe, Contemporary Physics ,44307 (2003)[12] M. Stefanak et al,Phys. Rev. Lett. 100, 020501 (2008).
+[13] N. Shenvi, J. Kempe and K. B. Whaley, Physical Review
+A,67052307 (2003).
+[14] B. Hein and G. Tanner, J. Phys. A 42085303 (2009).
+[15] A. Ambainis, J. Kempe and A. Rivosh, Proceedings of the
+16th ACM-SIAM SODA (SIAM, Philadelphia, 2005), p.
+1099.
+[16] D. Bouwmeester et al,Phys. Rev. A 61, 013410 (1999);
+P. L. Knight et al,Phys. Rev. A 68020301 (R) (2003).
+[17] P. Xue et al,Phys. Rev. A 78, 042334 (2008); H. Schmitz
+et al,Phys. Rev. Lett. 103, 090504.
+[18] B. Hein and G. Tanner, in preparation.
+[19] J. D. Joannopoulos et al,Photonic Crystals , (Princ. Uni.
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+(2009).
\ No newline at end of file
diff --git a/1001.0336.txt b/1001.0336.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5ab8d7dcfc86035aa06aa572d6ef89171de6b2c1
--- /dev/null
+++ b/1001.0336.txt
@@ -0,0 +1,1702 @@
+arXiv:1001.0336v3  [hep-ex]  11 Mar 2010FERMILAB-PUB-09-650-E, hep-ex 1001.0336
+Search for sterile neutrino mixing in the MINOS long-baseli ne experiment
+P. Adamson,8C. Andreopoulos,22D. J. Auty,26D. S. Ayres,1C. Backhouse,20P. D. Barnes Jr.,15G. Barr,20
+W. L. Barrett,31M. Bishai,4A. Blake,6G. J. Bock,8D. J. Boehnlein,8D. Bogert,8C. Bower,12S. Cavanaugh,9
+J. D. Chapman,6D. Cherdack,29S. Childress,8B. C. Choudhary,8,∗J. A. B. Coelho,7J. H. Cobb,20
+S. J. Coleman,32J. P. Cravens,28D. Cronin-Hennessy,17A. J. Culling,6I. Z. Danko,21J. K. de Jong,20,11
+N. E. Devenish,26M. V. Diwan,4M. Dorman,16,22A. R. Erwin,33C. O. Escobar,7J. J. Evans,16,20E. Falk,26
+G. J. Feldman,9M. V. Frohne,10,3H. R. Gallagher,29A. Godley,24M. C. Goodman,1P. Gouffon,23R. Gran,18
+E. W. Grashorn,17,†K. Grzelak,30,20A. Habig,18D. Harris,8P. G. Harris,26J. Hartnell,26,22R. Hatcher,8
+K. Heller,17A. Himmel,5A. Holin,16X. Huang,1J. Hylen,8G. M. Irwin,25Z. Isvan,21D. E. Jaffe,4C. James,8
+D. Jensen,8T. Kafka,29S. M. S. Kasahara,17G. Koizumi,8S. Kopp,28M. Kordosky,32,16D. J. Koskinen,16,‡
+Z. Krahn,17A. Kreymer,8K. Lang,28J. Ling,24P. J. Litchfield,17R. P. Litchfield,20L. Loiacono,28P. Lucas,8
+J. Ma,28W. A. Mann,29A. Marchionni,8M. L. Marshak,17J. S. Marshall,6N. Mayer,12A. M. McGowan,1,17,§
+R. Mehdiyev,28J. R. Meier,17M. D. Messier,12C. J. Metelko,22D. G. Michael,5,¶W. H. Miller,17
+S. R. Mishra,24J. Mitchell,6C. D. Moore,8J. Morf´ ın,8L. Mualem,5S. Mufson,12J. Musser,12D. Naples,21
+J. K. Nelson,32H. B. Newman,5R. J. Nichol,16T. C. Nicholls,22J. P. Ochoa-Ricoux,5,∗∗W. P. Oliver,29
+M. Orchanian,5T. Osiecki,28R. Ospanov,28,††J. Paley,12V. Paolone,21R. B. Patterson,5ˇZ. Pavlovi´ c,28,‡‡
+G. Pawloski,25G. F. Pearce,22R. Pittam,20R. K. Plunkett,8A. Rahaman,24R. A. Rameika,8T. M. Raufer,22
+B. Rebel,8P. A. Rodrigues,20C. Rosenfeld,24H. A. Rubin,11V. A. Ryabov,14M. C. Sanchez,13,1,9
+N. Saoulidou,8J. Schneps,29P. Schreiner,3P. Shanahan,8W. Smart,8C. Smith,16,26A. Sousa,9,20
+P. Stamoulis,2M. Strait,17N. Tagg,19,29R. L. Talaga,1J. Thomas,16M. A. Thomson,6G. Tinti,20R. Toner,6
+G. Tzanakos,2J. Urheim,12P. Vahle,32,16B. Viren,4M. Watabe,27A. Weber,20R. C. Webb,27N. West,20
+C. White,11L. Whitehead,4S. G. Wojcicki,25D. M. Wright,15T. Yang,25K. Zhang,4and R. Zwaska8
+(The MINOS Collaboration)
+1Argonne National Laboratory, Argonne, Illinois 60439, USA
+2Department of Physics, University of Athens, GR-15771 Athe ns, Greece
+3Physics Department, Benedictine University, Lisle, Illin ois 60532, USA
+4Brookhaven National Laboratory, Upton, New York 11973, USA
+5Lauritsen Laboratory, California Institute of Technology , Pasadena, California 91125, USA
+6Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, United Kingdom
+7Universidade Estadual de Campinas, IFGW-UNICAMP, CP 6165, 13083-970, Campinas, SP, Brazil
+8Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
+9Department of Physics, Harvard University, Cambridge, Mas sachusetts 02138, USA
+10Holy Cross College, Notre Dame, Indiana 46556, USA
+11Physics Division, Illinois Institute of Technology, Chica go, Illinois 60616, USA
+12Indiana University, Bloomington, Indiana 47405, USA
+13Department of Physics and Astronomy, Iowa State University , Ames, Iowa 50011 USA
+14Nuclear Physics Department, Lebedev Physical Institute, L eninsky Prospect 53, 119991 Moscow, Russia
+15Lawrence Livermore National Laboratory, Livermore, Calif ornia 94550, USA
+16Department of Physics and Astronomy, University College Lo ndon, Gower Street, London WC1E 6BT, United Kingdom
+17University of Minnesota, Minneapolis, Minnesota 55455, US A
+18Department of Physics, University of Minnesota – Duluth, Du luth, Minnesota 55812, USA
+19Otterbein College, Westerville, Ohio 43081, USA
+20Subdepartment of Particle Physics, University of Oxford, O xford OX1 3RH, United Kingdom
+21Department of Physics and Astronomy, University of Pittsbu rgh, Pittsburgh, Pennsylvania 15260, USA
+22Rutherford Appleton Laboratory, Science and Technologies Facilities Council, OX11 0QX, United Kingdom
+23Instituto de F´ ısica, Universidade de S˜ ao Paulo, CP 66318, 05315-970, S˜ ao Paulo, SP, Brazil
+24Department of Physics and Astronomy, University of South Ca rolina, Columbia, South Carolina 29208, USA
+25Department of Physics, Stanford University, Stanford, Cal ifornia 94305, USA
+26Department of Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
+27Physics Department, Texas A&M University, College Station , Texas 77843, USA
+28Department of Physics, University of Texas at Austin, 1 Univ ersity Station C1600, Austin, Texas 78712, USA
+29Physics Department, Tufts University, Medford, Massachus etts 02155, USA
+30Department of Physics, Warsaw University, Ho˙ za 69, PL-00- 681 Warsaw, Poland
+31Physics Department, Western Washington University, Belli ngham, Washington 98225, USA
+32Department of Physics, College of William & Mary, Williamsb urg, Virginia 23187, USA
+33Physics Department, University of Wisconsin, Madison, Wis consin 53706, USA
+(Dated: March 12, 2010)2
+A search for depletion of the combined flux of active neutrino species over a 735km baseline
+is reported using neutral-current interaction data record ed by the MINOS detectors in the NuMI
+neutrino beam. Such a depletion is not expected according to conventional interpretations of neu-
+trino oscillation data involving the three known neutrino fl avors. A depletion would be a signature
+of oscillations or decay to postulated noninteracting ster ile neutrinos, scenarios not ruled out by
+existing data. From an exposure of 3 .18×1020protons on target in which neutrinos of energies
+between ∼500MeV and 120GeV are produced predominantly as νµ, the visible energy spectrum of
+candidate neutral-current reactions in the MINOS far-dete ctor is reconstructed. Comparison of this
+spectrum to that inferred from a similarly selected near-de tector sample shows that of the portion
+of theνµflux observed to disappear in charged-current interaction d ata, the fraction that could be
+converting to a sterile state is less than 52% at 90% confidenc e level (C.L.). The hypothesis that
+active neutrinos mix with a single sterile neutrino via osci llations is tested by fitting the data to
+various models. In the particular four-neutrino models con sidered, the mixing angles θ24andθ34
+are constrained to be less than 11◦and 56◦at 90% C.L., respectively. The possibility that active
+neutrinos may decay to sterile neutrinos is also investigat ed. Pure neutrino decay without oscilla-
+tions is ruled out at 5.4 standard deviations. For the scenar io in which active neutrinos decay into
+sterile states concurrently with neutrino oscillations, a lower limit is established for the neutrino
+decay lifetime τ3/m3>2.1×10−12s/eV at 90% C.L.
+PACS numbers: 14.60.St, 12.15.Mm, 14.60.Pq, 14.60.Lm, 29. 27.-a, 29.30.-h
+I. INTRODUCTION
+Compelling evidence has been presented demonstrat-
+ing the disappearance of muon and electron neutrinos
+as they propagate from their production point. Dis-
+appearance of muon neutrinos has been observed from
+neutrino fluxes originating in the atmosphere [1, 2] and
+accelerator beams [3, 4]; disappearance of electron neu-
+trinos has been observed with neutrino fluxes from the
+Sun [5, 6] and terrestrial reactors [7]. Super-Kamiokande
+and other atmospheric neutrino experiments were the
+first to report significant deficits of νµcharged-current
+interactions from neutrinos propagating over baselines
+larger than several hundred kilometers. The K2K and
+MINOS experiments have observed the disappearance
+using accelerator-beam neutrinos propagating over fixed
+baselines of 250km and 735km, respectively. All experi-
+ments reporting muon-neutrinodisappearancefavorpure
+νµ→ντoscillations as the explanation for the observed
+disappearance of νµ[3, 8–10]. However, more exotic sce-
+narios in which one or more sterile neutrinos, νs, mix
+with the three active neutrino species remain as viable
+∗Now at Department of Physics & Astrophysics, University of
+Delhi, Delhi 110007, India.
+†Now at Center for Cosmology and Astro Particle Physics, Ohio
+State University, Columbus, Ohio 43210 USA.
+‡Now at Department of Physics, Pennsylvania State Universit y,
+State College, Pennsylvania 16802, USA.
+§Now at Physics Department, St. John Fisher College, Rochest er,
+New York 14618 USA.
+¶Deceased.
+∗∗Now at Lawrence Berkeley National Laboratory, Berkeley, Ca li-
+fornia 94720, USA.
+††Now at Department of Physics and Astronomy, University of
+Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
+‡‡Now at Los Alamos National Laboratory, Los Alamos, New Mex-
+ico 87545, USA.alternatives.
+Long-baseline experiments such as MINOS provide
+an opportunity to test alternative scenarios by com-
+paring the observed neutral-current interaction rates
+in near and far-detectors. Since all active neutrinos,
+νe, νµ, andντ, participate in the neutral-current inter-
+action, this comparison provides a sensitive probe to the
+existence of processes that deplete the flux of active neu-
+trinos between the two detectors. If neutrinos only oscil-
+late among the active flavors, the rate of neutral-current
+interactions at the far site of a long-baseline experiment
+remains unchanged from the null-oscillation prediction.
+However, if another process occurs concurrently with ac-
+tive neutrino oscillations, the rate of neutral-current in-
+teractions at the far site may be different. Two such pos-
+sibilities have attracted considerable attention and are
+the focus of the analysis reported here, (i) active neutri-
+nos oscillating to νs, and (ii) neutrino decay in conjunc-
+tion with oscillations.
+The possible existence of one or more sterile neutrinos
+that do not couple to the electroweak current but do mix
+with the active flavors has been discussed extensively in
+the literature [11–13]. The existence of sterile neutrinos
+would provide new degrees of freedom that could help
+clarify certain outstanding problems with the neutrino-
+mass spectrum [14] and with heavy element nucleosyn-
+thesis in supernovae [15]. A recent search for neutrino
+oscillations in a short baseline experiment provides no
+evidence for transitions that would imply existence of
+sterile neutrinos [16].
+The coupling between sterile neutrinos and the ac-
+tive neutrinos would likely involve the third mass eigen-
+state. Observations by the SNO experiment show the
+total flux of active neutrinos from the Sun to agree with
+expectations from solar models [6], thereby limiting the
+potential coupling of the first or second neutrino-mass
+eigenstates to a sterile neutrino. Additionally, the Super-
+Kamiokande experiment strongly disfavors pure νµ→νs3
+mixing [8, 9], but does not exclude an admixture of sub-
+dominant νµ→νsmixing with the dominant νµ→ντ
+mixing. MINOS has recently carried out the first ded-
+icated search at fixed long-baseline for νµoscillating to
+bothντandνs[17]. The analysis presented here uses a
+larger exposure and extends the earlier analysis by con-
+sidering specific models in which a sterile neutrino state
+is incorporated into the neutrino mixing matrix.
+The possibility that a neutrino may decayinto a sterile
+state [18] is also explored in this work. In this scenario,
+the mass eigenstate ν3is unstable and allows active neu-
+trinos to decay into undetectable final states. The decays
+would give rise to an anomalous depletion of the neutral-
+current event rate observed at the far-detector. The oc-
+currence of pure neutrino decay, without oscillation, has
+already been shown by MINOS and Super-Kamiokande
+to be highly disfavored [10, 19]. The analysis reported
+here represents the first direct test of the neutrino-decay-
+with-oscillations scenario in a long-baseline experiment.
+II. NUMI BEAM AND MINOS DETECTORS
+Neutrinos from the NuMI (Neutrinos from the Main
+Injector) beam [20] originate from decays of pions and
+kaons produced in the beamline target; a significantly
+smallercontributionarisesfromsubsequentmuondecays.
+The secondary mesons are created using 120GeV pro-
+tons extracted from the Fermilab Main Injector incident
+on a graphite target. The proton extraction occurs in
+10µs spills with a 2.2 s cycle. Positioned downstream of
+the target are two parabolic magnetic horns which fo-
+cusπ+andK+secondary particles. The focused mesons
+proceed into a 675m long evacuated decay pipe, where
+they may decay into muons and neutrinos. The remnant
+hadrons are stopped by a beam absorber placed at the
+end of the decay pipe. The tertiary muons are stopped
+by 240m of rock between the end of the decay volume
+and the near-detectorcavernso that only neutrinos reach
+the near-detector. The neutrino energy spectrum can be
+changed by adjusting either the horn current or the po-
+sition of the target relative to the horns. The data em-
+ployedinthisanalysiswereobtainedusingthelow-energy
+beam configuration, in which the peak neutrino energy
+is 3.3GeV [4], and correspond to a far-detector exposure
+of 3.18×1020protons on target, collected during the pe-
+riod of May 2005 to July 2007. In this configuration, ac-
+cording to Monte Carlo simulations, the neutrino flavor
+composition of the beam is 91.8% νµ, 6.9%νµ, and 1.3%
+νe+νe. For this analysis the neutrinos and antineutrinos
+are assumed to oscillate with the same parameters.
+The MINOS detectors are planar steel/scintillator
+tracking calorimeters [20]. The vertically oriented de-
+tector planes are composed of 2.54cm thick steel and
+1cm thick plastic scintillator. A scintillator layer is com-
+posed of 4.1cm wide strips. Each strip is coupled via
+a wavelength-shifting fiber to one pixel of a multianode
+photomultiplier tube (PMT) [21, 22].The MINOS near-detector is located 1.04km
+downstream of the target, has a total mass of
+980metric tonnes, and lies 103m underground at
+Fermilab. The detector consists of two sections, a
+calorimeter encompassing the upstream 121 planes and a
+spectrometer containing the downstream 161 planes. In
+both sections, one out of every five planes is fully covered
+with 96 scintillator strips attached to the steel plates. In
+the calorimeter section, the other four out of five planes
+are partially covered with 64 scintillator strips, whereas
+in the spectrometer section no scintillator is attached to
+the steel. The far-detector is 734km downstream of the
+near-detector, has a total mass of 5400metric tonnes,
+and is located in the Soudan Underground Laboratory,
+705m below the surface. It is composed of 484 fully
+instrumented planes organized in two supermodules [4].
+The fiducial masses used for the near and far-detectors
+are 27metric tonnes and 3800metric tonnes respectively.
+The near-detector steel is magnetized with an average
+field intensity of 1.3T whereas the far-detector has an
+average field of 1.4T in the steel.
+III. DATA SELECTION
+A neutrino interacting in one of the MINOS detectors
+produces either a charged-current event with a charged
+lepton plus hadrons emerging from the event vertex or a
+neutral-current event with hadrons but no charged lep-
+ton in the final state. In either case, the particles in
+the final state deposit energy in the scintillator strips,
+which is converted to light and collected by optical fibers
+and converted to electronic signals by PMTs. The MI-
+NOS reconstruction algorithms use event topology and
+the recorded time stamps of the strips where energy was
+deposited to identify neutrino events inside the detector.
+Events must have at least four strips with signal to be
+considered in the analysis. Individual scintillator strips
+are grouped into either reconstructed tracks or showers,
+and the tracks and showers are combined into events [4].
+The vertex of each event is required to be sufficiently far
+from any edge of the detector to ensure that the final-
+state hadronic showers are contained within the instru-
+mented regions of the detectors. On average, each GeV
+of energy deposition in a neutral-current event induces
+activity in 12 scintillator strips.
+The Monte Carlo simulation of the neutrino beam uti-
+lizes FLUKA05 [23] to model hadroproduction in the
+NuMI targetandaGEANT3[24] simulationofthe NuMI
+beam line to propagate the particles exiting the target.
+The neutrino interactions in the MINOS detectors are
+modeled by the NEUGEN-v3 [25] program. The simu-
+lated neutrino flux is constrained to agree with the neu-
+trino energy spectra measured in the near-detector for
+nine different beam configurations [4]. This procedure
+reduces the uncertainties due to the neutrino flux in the
+far-detector prediction.4
+A. Data integrity
+All of the data accepted for this analysis must pass a
+series of requirements on the beam and detector perfor-
+mance. The beam is required to strike within 2mm of
+the center of the upstream face of the NuMI target, a
+segmented rectangular graphite rod measuring 6.4mm
+in width, 15mm in height and 940mm in length [4].
+The full width at half-maximum of the beam at the
+target is required to be between 0.1mm and 2.0mm in
+the transverse horizontal direction and between 0.1mm
+and 1.5mm in the transverse vertical direction. The
+minimum allowed beam intensity during a beam spill is
+0.5×1012protons on target.
+For all data used in this analysis, all detector subsys-
+temsthataffectdataqualityarerequiredtobeinnormal,
+stable modes of operation. Checks are made to ensure
+the coil currents that energize the magnetic fields are
+at their nominal values in both detectors. The timing
+between the detectors is synchronized using the Global
+Positioning System to within 1 µs to ensure that correct
+beam extraction timing is provided to the far-detector
+electronics. The timing synchronization is not affected
+by the timing resolution of the detectors, which is 18.8ns
+and 1.9ns for the near and far-detectors, respectively.
+The high voltage supplied to each PMT is required to be
+at its nominal value.
+B. Fiducial requirements
+Only those events reconstructed in the fiducial volume
+are included in the analysis. In both detectors the re-
+constructed event vertex is required to be more than
+50cm from the edges of the instrumented regions and,
+in the case of the far-detector, more than 45cm from the
+center of the magnetic coil that runs through the mid-
+dle of each detector plane. In addition, a longitudinal
+veto region comprising either 30 planes at the front of
+the near-detector or four planes at the front of each of
+the far-detector supermodules eliminates events that en-
+ter through the first plane of a detector but may have
+originated outside the detector volume. To ensure good
+shower containment, the event vertex is required to be
+reconstructed more than 1m from the last plane of each
+far-detector supermodule and more than 1m from the
+last plane in the calorimeter region of the near-detector.
+The sparsely instrumented spectrometer region of the
+near-detector is not included in the fiducial volume. The
+primary vertex of an event is assigned according to the
+event’s reconstructed track vertex. However, for events
+without a reconstructed track, the vertex of the hadronic
+shower is used as the event vertex.C. Near-detector event selection
+The reconstruction algorithms are designed to han-
+dle the high rate of interactions occurring in the near-
+detector during running with typical intensities of 2 .2×
+1013protons on target per beam spill. At this intensity,
+an average of 16 neutrino interactions occur in the near-
+detector for each spill. For the majority of events the
+algorithms perform very well. However, for certain event
+subclassess, shortfalls have been identified and quantified
+using special studies including low intensity beam data
+and visual scanning. Monte Carlo studies show that 8%
+of all reconstructed events classified as neutral-current
+interactions are assigned a value of recontructed energy
+that is less than half of the actual deposited energy in
+the detector for the simulated interaction; the remaining
+energy has been reconstructed as a separate event, re-
+sulting in an overestimatein the number ofreconstructed
+events. In particular, the number of events with visible
+energy below 1GeV that are classified as neutral-current
+candidates is overestimated by 34%. Therefore, these
+poorlyreconstructedeventsaffect the neutral-currenten-
+ergyspectrum and carehas been taken in identifying and
+removing them from the analysis.
+Reconstruction failures are classified into three main
+categories: (i) split events , (ii)leakage events , and (iii)
+incomplete events . Split events occur when a single neu-
+trino interaction results in two or more reconstructed
+events. Leakage events are due to incorrectly assigned
+event vertices causing neutrino interactions outside the
+fiducialvolumetobe reconstructedwithin it. The incom-
+plete event category is a looser classification that refers
+to further types of failures in shower reconstruction to
+be described below. In all three categories, the visible
+energy of a neutrino candidate may be underestimated,
+resultinginabackgroundtoneutral-currenteventsatlow
+energies. As near-detector data are used to predict the
+expected spectra at the far-detector, reconstruction fail-
+ures specific to the near-detector must be minimized. A
+set ofselection requirementswasdeveloped[26] to reduce
+the occurrence of these failure modes. After applying the
+requirements, detailed below, simulations show that the
+background of poorly reconstructed events having visible
+energy below 1GeV has been reduced to 8%.
+Split events lead to double-counting of neutrino in-
+teractions and incorrect energy reconstruction. If two
+reconstructed events are caused by the same neutrino
+interaction, they are expected to appear close in time
+and in space. The time for each event has been taken
+to be the median of the recorded signal arrival times
+for event strips contained within five planes of the event
+vertex. To minimize the occurrence of split events, the
+time separation between an event and the closest other
+event in time is required to be |∆t|>40ns, as shown in
+Fig. 1. A requirement that the spatial separation be-
+tween events along the beam direction |∆z|>1m if
+40ns<|∆t|<120ns is also employed to further elimi-
+nate split events.5
+ t (ns)∆−400 −200 0 200 400 Events310
+050100150200
+Near Detector Data
+Monte Carlo
+Poor Reconstruction
+FIG.1: Thedistributionoftimeseparation betweenevents∆ t
+for data (solid points) and Monte Carlo simulated data (soli d
+histogram). A background component arising from poorly re-
+constructed events (hatched histogram) is confined to a nar-
+row region of low ∆ tvalues. Events accepted for further anal-
+ysis are indicated by the arrows.
+Leakage events are typically cosmic-ray muons causing
+steep showers with a high concentration of hit strips in
+a small number of detector planes. These events can be
+removed by selecting on this topological characteristic.
+Thus, a requirement is placed on the ratio of the average
+number of active strips per plane to the total number of
+planes with active strips in the event, represented more
+concisely as (active event strips)/(active event planes)2.
+Only events for which the ratio is less than 1.0 are ac-
+cepted by the analysis.
+Another type of leakage event is due to secondary par-
+ticles from interactions occurring outside of the fiducial
+volume. In the partially covered planes of the near-
+detector, the steel is instrumented with scintillator to
+within 16cm from the left-hand-side edges of the steel
+plate and 1.4m from the right-hand-side edges as viewed
+from along the beam direction. Consequently, secondary
+particles may enter the detector laterally due to the
+sparse instrumentation on the sides. For such cases the
+reconstruction algorithm is likely to fail in associating
+hits to events. Nevertheless, the extra activity at the
+edges of the fully covered planes is recorded and can be
+usedtovetoeventswithin atime window. Thevetocrite-
+rion uses the number of active strips and the pulse-height
+in the edge regions of the detector. An event is accepted
+if the number of active strips in the veto region recorded
+within a ±40ns window around the event vertex time is
+less than four and the pulse-height deposited in the vetoTotal Strips0 10 20 30 40 50 Events310
+0204060
+Near Detector Data
+Monte Carlo
+Poor Reconstruction
+FIG. 2: Section of the distribution of the number of strips
+with nonzero pulse-height, per event, after all other selec -
+tions are applied. The region displayed, in which the contri -
+bution from poorly reconstructed events (hatched histogra m)
+is significant, corresponds to low strip counts and represen ts a
+small fraction of the total number of events. The event range
+accepted for the analysis is identified by the arrow.
+regionis less than 2MIPa. These veto criteriaare applied
+to eventswith visible energyless than 5GeV in which the
+number of planes assigned to the reconstructed shower is
+greater than the number of planes assigned to the re-
+constructed track, as leakage events are reconstructed as
+low-energy showers without a clearly defined track.
+Incomplete events arise when the shower reconstruc-
+tion fails to assign all event-related strips to the shower.
+This type of reconstruction failure occurs if there are
+largegaps in a showeror if the showeris generally sparse.
+In a majority of these cases, events have a very low num-
+ber of reconstructed strips. Figure 2 shows the distribu-
+tion of the number of strips for the events, after applying
+the selection requirements. To minimize the number of
+incomplete events in the near-detector data sample, an
+event is required to have total number of strips greater
+than four.
+In summary, the selection criteria applied to the near-
+detector data are as follows: (i) the modulus of the time
+separation between events, |∆t|, must exceed 40ns; (ii)
+if 40ns<|∆t|<120ns, the modulus of the spatial
+separation between events, |∆z|, must exceed 1m; (iii)
+theratio(activeeventstrips)/(activeeventplanes)2must
+be less than unity; (iv) for events with less than 5GeV
+of reconstructed energy in which the number of planes
+is larger in the reconstructed shower than in the recon-
+aMinimum ionizing particle, equivalent to the response prod uced
+by a 1GeV muon traversing a detector plane at normalincidenc e.6
+structed track, the number of event strips reconstructed
+in the detector’svetoregionsshould be lessthan four and
+the total pulse-height in those regions must be less than
+2MIP; and (v) the total number of strips reconstructed
+in the event must be more than four. Only events that
+satisfy these criteria are used for further analysis.
+D. Far-detector event selection
+In contrast to the multiple events per beam spill ob-
+served in the near-detector, the rate measured in the
+far-detector is approximately two events per day within
+the beam spill times, so the appropriate requirements for
+event selection are necessarily different. Specifically, the
+probability that two or more neutrinos produced in the
+same 10µs beam spill will interact in the far-detector is
+negligible. Therefore, if two events are reconstructed in
+the same spill, the coincidence is due either to a recon-
+struction failure or else one of the events has a nonbeam
+origin. Effects of multiple event reconstruction are miti-
+gated by requiring an event to be used in the analysis to
+contain at least 75% of the total deposited energy during
+the beam spill.
+The main background in the far-detector results from
+detector noise arising from the electronics and PMTs or
+from spontaneous light emission from the scintillator and
+wavelength-shifting fibers [27]. The noise from the elec-
+tronicsandPMTsisremovedbysettinganenergythresh-
+old in the PMTs. The spontaneous light emission is re-
+moved by requiring accepted events to include at least
+nine strips or at least 10MIP deposited in the detector.
+Alternatively, events are also accepted if they include
+more than five strips and deposit more than 5MIP in
+the detector.
+Muons from cosmic raysare a potential source of back-
+ground events. Given the 0.2Hz cosmic-ray muon rate
+at the far-detector, the number of cosmic-ray-induced-
+muons that may potentially coincide with beam spills is
+comparable to the number of beam-induced neutrino in-
+teractions observed. Most cosmic-ray-induced muons are
+wellreconstructedandareefficientlyremovedbythefidu-
+cialrequirement. However,thereconstructionalgorithms
+are optimized to handle recorded energy flow in the gen-
+eral direction of the beam. Problem cases can thus arise
+with very steep cosmic muons, which are removed by re-
+quiring the absolute value of the muon direction cosine
+in the longitudinal direction, |pz|/E, to be higher than
+0.4. In some cases the events are so steep that they are
+reconstructed only as showers and may be removed by
+using selection variables that describe the transverse and
+longitudinal shower profiles. The transverse variable is
+defined by calculating the root-mean-square (rms) value
+of the shower strip positions, whereas the longitudinal
+variable is defined as the ratio of active strips per plane
+to the total number of active planes in the event. Only
+those showers with a transverse rms value lower than 0.5
+and (strips /plane)/(event planes) <1 are accepted forfurther analysis. Cosmic-ray muons that stop in the de-
+tector can mimic beam events if the end of the stopping
+muon track is interpreted as the vertex and the track
+is then reconstructed backwards. These events can be
+identified by performing a linear fit to the timing dis-
+tribution for strips on a track as a function of the strip
+longitudinal position. A fit resulting in a negative slope
+indicates that the event is a downward-going cosmic-ray
+muon and not a beam neutrino. The sample contamina-
+tion from cosmic-ray induced muons after these criteria
+are applied is estimated to be less than 0.1% [28].
+Another potential background arises from data
+recordedwhiletheLightInjectioncalibrationsystem(LI)
+is flashing during normal data taking. The light injec-
+tion events are removed with 99.99% efficiency by using
+information from a PMT directly connected to the light
+injection system. By applying additional requirements
+based on concentrated detector activity, it is estimated
+that much less than one LI event is accepted in the entire
+data sample [28]. Furthermore, application of the LI re-
+jection criteria results in no measurable loss of efficiency
+for beam-neutrino interactions.
+IV. EVENT CLASSIFICATION
+After the selection criteria described in the previous
+section are applied, the analysis proceeds by distinguish-
+ing charged-current events from neutral-current events.
+Distinct event classification procedures are employed for
+each sample, as described below. The reconstructed neu-
+trino energy spectra for both event classes are used in
+the fits described in Sec. VIII and Sec. IX.
+The goal of the event classification is to maximize
+the efficiency and purity of selected samples of neutral-
+current and charged-current events. Using Monte Carlo
+event samples, efficiency is defined as the number of true
+events of one type which are classified as that type, di-
+vided bythe total numberoftrueeventsofthat type that
+passthe criteriadescribed in Sec. III. Purityis defined as
+the ratioofthe number oftrue eventsofonetype selected
+to the total number of events selected as that type.
+To avoid biases, the methods for identifying neutral-
+currentcandidateeventsandproceduresemployedinpre-
+dicting the far-detector spectrum, described in Sec. V,
+were developed and tested using only the near-detector
+data and Monte Carlo simulation. All analysis proce-
+dures were finalized prior to examining data in the far-
+detector.
+The neutral-current event classification employs sev-
+eral criteria based on reconstruction variables displaying
+large differences between neutral-current and charged-
+current events [29]. Charged-current events with short
+or no apparent tracks and poorly reconstructed events
+are the two main sources of background. The latter is
+mitigated by employing the various selections described
+in Sec. III. The classification variables considered are:
+event length , expressed as the difference between the first7
+and last active plane in the event; number of tracks re-
+constructed in the event; and track extension , defined as
+the difference between track length and shower length.
+A sample of candidate neutral-current events is ob-
+tained by applying specific requirements on the classifi-
+cation variables. Since neutral-current events are typi-
+cally shorter than charged-current events, events cross-
+ing fewer than 60 planes and for which no track is re-
+constructed are classified as neutral-current. Because
+neutral-current events are expected to have short or no
+reconstructedtracks,eventscrossingfewerthan60planes
+that containatrackareclassifiedasneutral-currentifthe
+track extends fewer than 5 planes beyond the shower.
+The values chosen maximize sensitivity for detection of
+sterile-neutrino admixture. Finally, events that are not
+classified as neutral-current-like are labeled as charged-
+current-like if they pass the classification procedures de-
+scribed in a previous MINOS publicationb. These re-
+quirements are applied to both near and far-detectors to
+obtain neutral-current and charged-current event sam-
+ples.
+The 6% of near-detector events and 3.5% of far-
+detector events that are not classified as either neutral-
+current or charged-current are not used in further anal-
+ysis. Evaluation of these removed samples using simu-
+lated data shows that approximately 75% of the events
+in eachdetectorarecharged-currentinteractions, andthe
+reconstructedenergiesofthoseneutrinosdistribute in ac-
+cord with the expectation based on the simulation. The
+remaining events are neutral-current interactions whose
+distributions in reconstructed energy are very similar in
+the two detectors; the latter distributions span the full
+visibleenergyspectrum, but with modestaccentuationof
+the lower energy region 0GeV < Ereco<4GeV. There-
+fore, the removal of these events does not introduce anal-
+ysis biases.
+Distributions for the event length and track extension
+classification variables for data of the near-detector are
+shown in Figs. 3a and 3b. The data are plotted together
+with the prediction of the MINOS Monte Carlo simu-
+lation, which adequately reproduces the shapes of the
+classification-variable distributions. Comparisons of dis-
+tributions in the far-detector for the same event classi-
+fication variables are shown in Figs. 4a and 4b. Here,
+the Monte Carlo simulation uses oscillation parameter
+values obtained from the most recent MINOS charged-
+current measurement, |∆m2
+32|= 2.43×10−3eV2and
+sin22θ23= 1 [10].
+Efficiencies and purities for the neutral-current and
+bCandidate charged-current events are selected using a like lihood-
+based particle identification parameter constructed from t hree
+probability density functions. The functions represent di stribu-
+tions for the variables (i) event length, (ii) fraction of th e total
+event signal in the reconstructed track, and (iii) average s ignal
+per plane induced by the track. Further details are presente d in
+Ref. [4].Event Length (number of planes)0 20 40 60 80 100 Events510
+024681012
+Near Detector Data
+Monte Carlo
+CC BackgroundNear Detector Data
+Monte Carlo
+CC Backgrounda
+Track Extension (number of planes)−10 0 10 20 Events510
+00.511.52Near Detector Data
+Monte Carlo
+CC BackgroundNear Detector Data
+Monte Carlo
+CC Backgroundb
+FIG. 3: Comparisons of near-detector data with Monte Carlo
+predictions for distributions of the variables (a) event length
+and (b) track extension . The data quality requirements de-
+scribed in Sec. III are applied. Systematic uncertainties a re
+displayed as shaded bands on the Monte Carlo expectation.
+Events selected as neutral-current-like are indicated by t he
+arrows.
+charged-currentevent samples for both detectors are dis-
+played in Figs. 5 and 6. The classified neutral-current
+samples have nearly identical purities for the near and
+far-detectors. The far-detector purity curve is computed
+for the case of null neutrino oscillations. The neutral-
+current sample efficiencies have identical trends but ex-
+hibit a constant relative offset over the full range of re-
+constructedeventenergies, Ereco, dueto the specialnear-
+detector selection criteria. The minima observed in sam-
+ple purities for both detectors near the peak of the fo-
+cussed neutrino spectrum, Ereco≈4GeV, reflect the
+abundant rate and consequently increased background
+fromνµcharged-current events.
+The resulting reconstructed energy spectra for the
+neutral-current and charged-current samples in the near-
+detector are shown in Figs. 7 and 8 respectively. The
+figures show that the selected neutral-current sample in-
+cludes a sizable background contribution from misiden-
+tified charged-current events, in contrast to the se-8
+Event Length (number of planes)0 20 40 60 80 100Events
+050100150200 Far Detector Data
+Oscillated MC
+CC BackgroundFar Detector Data
+Oscillated MC
+CC Backgrounda
+Track Extension (number of planes)−10 0 10 20Events
+01020304050
+Far Detector Data
+Oscillated MC
+CC BackgroundFar Detector Data
+Oscillated MC
+CC Backgroundb
+FIG. 4: Far-detector data versus predictions from the Monte
+Carlo (MC) simulation including νµ→ντoscillations, for
+distributions of the classification variables (a) event length
+and (b) track extension . The data quality requirements of
+Sec. III are applied. The shaded bands show the systematic
+errors on the MC predictions. The arrows indicate events
+selected as neutral-current-like.
+lected charged-current sample which contains very little
+neutral-current background. For both samples, the data
+points are seen to fall within or near the 1 standard de-
+viation range of the systematic uncertainty of the Monte
+Carlo simulation.
+V. FAR-DETECTOR PREDICTION
+The predictions of the energy spectra of the neutral-
+current and charged-current samples at the far-detector
+are based on the observed near-detector data and make
+use of the expected relationship between the neutrino
+fluxes at the two sites. The process of making the pre-
+dictions is called “extrapolation” and may be viewed as
+making corrections to the simulation of interactions in
+the far-detector based on the energy spectrum measured
+in the near-detector.
+This analysis uses an extrapolation technique called (GeV)recoE0 10 20 30 40Purity, Efficiency
+00.20.40.60.81
+Near detector MC Efficiency
+Far detector MC Efficiency
+Near detector MC Purity
+Far detector MC Purity
+FIG. 5: Selection efficiency and sample purity for Monte
+Carlo (MC) events selected as neutral-current candidates i n
+the near and far-detectors, as a function of reconstructed
+event energy. Detector selection, fiducial volume and neutr al-
+current/charged-current separation requirements have be en
+applied. The shapes of both efficiency and purity distribu-
+tions are observed to be very similar for each of the two MI-
+NOS detectors.
+ (GeV)recoE0 10 20 30 40Purity, Efficiency
+00.20.40.60.81
+Near detector MC Efficiency
+Far detector MC Efficiency
+Near detector MC Purity
+Far detector MC Purity
+FIG. 6: Distributions ofselection efficiency (lower curves) and
+sample purity (upper curves) versus reconstructed event en -
+ergy, for MonteCarlo (MC) eventsselected as charged-curre nt
+candidate events in the near-detector and in the far-detect or.
+the “far over near” (F/N) method [4, 30]. This method
+makesthe prediction ofthe far-detectorspectrum by tak-
+ing the product of two quantities. The first quantity
+is the ratio of the expected number of events from the
+Monte Carlo simulation for each energy bin in the far-
+detector and near-detector spectra. The expected num-
+ber of events for each energy bin in the far-detector spec-
+tra depends on the composition of event types entering
+the samples and the corresponding oscillation probabili-
+ties for that energy. The second quantity is the number9
+ (GeV)recoE0 5 10 15 20 Events/GeV410
+010203040
+Near Detector Data
+Monte Carlo Expectation
+ CC Backgroundµν
+FIG. 7: Distribution of reconstructed visible energy for se -
+lected neutral-currenteventsinthenear-detector, for th edata
+(solid points) versus the Monte Carlo prediction (open his-
+togram). The systematic errors (1 σ) for the Monte Carlo are
+shown by the shaded band. Also shown is the Monte Carlo
+predictionforthebackgroundofmisidentifiedcharged-cur rent
+events in the near-detector sample (hatched histogram).
+ (GeV)recoE0 5 10 15 20 Events/GeV410
+0204060Near Detector Data
+Monte Carlo Expectation
+µν NC Background 
+FIG. 8: The reconstructed energy spectrum for selected
+charged-currenteventsofthenear-detector, for thedata( solid
+points) versus the Monte Carlo (open histogram). Systemati c
+errors at 1 σfor the Monte Carlo are indicated by the shaded
+band; the hatched histogram (lower left) shows the Monte
+Carlo expectation for the small background contribution by
+misidentified neutral-current events to this sample.
+of observed near-detector data events. The F/N method
+prediction is robust against distortions arising from dif-
+ferences between data and Monte Carlo simulation in the
+near-detector as these distortions are translated to the
+far-detector and do not affect the oscillation measure-
+ment [4].
+For example, the extrapolation for the neutral-current
+spectrum accounts for both the signal neutral-current
+events and the background charged-current events from
+each neutrino flavor. The extrapolation also addresses
+the ways in which the backgrounds change between thenear and far-detectors. Specifically, for the case of the
+νµcharged-currentcomponent of the neutral-currentand
+charged-current samples, the F/N extrapolation predicts
+the number of events at the far-detector for the i-th bin
+of reconstructed energy to be
+Fprediction
+i =Ndata
+i
+/summationdisplay
+x/summationdisplay
+jFMC
+ijPνµ→νx(Ej)
+NMC
+i
+,(1)
+whereNdata
+iis the number of selected events in the i-th
+reconstructed energy bin in the near-detector and NMC
+i
+is the number of events expected in that bin from the
+near-detector Monte Carlo simulation. The FMC
+ijrepre-
+sentsthenumberofeventsexpectedfromthefar-detector
+Monte Carlo simulation in the i-th bin of reconstructed
+energy and j-th bin of true neutrino energy. In the equa-
+tion,Ejis the trueneutrino energyand Pνµ→νxthe prob-
+ability of muon-neutrino transition to any other flavor.
+In particular, for the neutral-current spectrum, the
+extrapolation must take neutrino oscillations into ac-
+count to properly characterize the predominant back-
+ground arising from misidentified charged-current νµ,
+and it must include the small spectral distortion re-
+sulting from misidentified charged-current ντandνe
+events. Thus, there are five separate classes of events
+thatmustbeextrapolatedtothefar-detector: (i) genuine
+neutral-current interactions, (ii) νµcharged-current in-
+teractions, (iii) ντcharged-current interactions, (iv) pos-
+sibleνecharged-current interactions originating from
+νµoscillations, and (v) charged-current νeinteractions
+initiated by the intrinsic νebeam component. The muon
+neutrinos in the simulation include oscillations and are
+integrated in bins of reconstructed energy to account for
+the changing background. Oscillations of the intrinsic
+beamνeintoνµare not taken into account as those νe
+comprise only 1.3% of the neutrinos in the beam and
+mixing angle for such oscillations is so small as to make
+the contributions from that oscillation mode negligible.
+The five classes of simulated events are used to con-
+structindividualtwo-dimensionalhistogramsoftrueneu-
+trinoenergyversusreconstructedenergy. In eachofthese
+histograms, all the events in an individual bin of true en-
+ergy are multiplied by the same survival or transition
+probability. After applying oscillations the simulated
+events are converted to a reconstructed energy spectrum
+by integrating across all the true energy bins for each bin
+of reconstructed energy, producing the sum in Eq. (1).
+The reconstructed energy spectra for each separate data
+set, signal and backgrounds, are added together into one
+spectrum.
+VI. SYSTEMATIC UNCERTAINTIES
+The principal sources of systematic uncertainties in
+these analyses are: (i) absolute scale of the hadronic en-10
+ergy; (ii)relative calibration of the hadronic energy be-
+tween the two detectors; (iii) relative normalization be-
+tween the two detectors; (iv) charged-current background
+in selected neutral-current events; and (v) uncertainties
+due to the near-detector selection requirements in the
+near-detector event counts. Monte Carlo studies have
+been performed where, for each single uncertainty, the
+Monte Carlo spectrum is varied by ±1 standard devia-
+tion independently, in order to estimate the effect of each
+on the extrapolated spectrum. Beam and cross section
+uncertainties that are common to the two detectors ef-
+fectively cancel when using the F/N extrapolation.
+The absolute hadronic energy scale has an uncertainty
+of 12%. This value is a combination of the uncer-
+tainty in the hadronization model and intranuclear ef-
+fects (10%) and uncertainty of the detector response to
+singlehadrons(6%). Anuncertaintyof3%ontherelative
+energy scale between the two detectors was determined
+from cross-calibration studies using stopping muons [4].
+The relativenormalizationbetween the twodetectorshas
+an uncertainty of4%. This is a combination ofthe uncer-
+tainties due to fiducial mass, live time, and reconstruc-
+tion differences between the two detectors.
+To evaluate the uncertainties due to the near-detector
+selection, therequirementthatthetotalnumberofrecon-
+structed strips in an event is at least four was shifted by
+±1 strip. The effects on the reconstructed energy spec-
+trumweredeterminedforeachshift. Theuncertaintyhas
+been estimated to be 15.2% for Ereco<0.5 GeV, 2.9%
+for 0.5< Ereco<1 GeV, 0.4% for 1 < Ereco<1.5 GeV
+and is negligible for higher visible energies.
+The uncertainty in the number of charged-current
+background events is determined using near-detector
+data taken in several different beam configurations,
+namely, (i) horns-off , inwhichtherewasnocurrentinthe
+magnetic horns; (ii) medium energy , in which the target
+is moved upstream from the first horn by 100 cm; and
+(iii)high energy , in which the target is moved upstream
+from the first horn by 250 cm. For each of these beam
+configurations, described in further detail in Ref. [4], the
+charged-current background component has a consider-
+ably different spectrum from the one obtained in the low-
+energy (LE) configuration. The charged-current back-
+ground component in the neutral-current spectrum can
+thus be determined by using the observed differences in
+energyspectrumbetweenthelow-energybeamconfigura-
+tionandeachoftheotherbeamconfigurationsalongwith
+information from Monte Carlo simulation of each config-
+uration. In the low-energy configuration, the number
+of selected near-detector neutral-current events NLEcan
+be written as the sum of true neutral-current events and
+background charged-current events in that beam config-
+uration:
+NLE=NLE
+nc+NLE
+cc (2)
+For an alternate beam configuration “Alt”, the numberof selected neutral-current events may be written as
+NAlt=rAlt
+nc·NLE
+nc+rAlt
+cc·NLE
+cc, (3)
+whererAlt
+nc=NAlt
+nc/slashbig
+NLE
+ncandrAlt
+cc=NAlt
+cc/slashbig
+NLE
+ccare
+determined from the Monte Carlo simulation. Equa-
+tions (2) and (3) can be solved to yield the solutions:
+NLE
+cc=/parenleftbig
+NAlt−rAlt
+nc·NLE/parenrightbig/slashbig/parenleftbig
+rAlt
+cc−rAlt
+nc/parenrightbig
+NLE
+nc=/parenleftbig
+NAlt−rAlt
+cc·NLE/parenrightbig/slashbig/parenleftbig
+rAlt
+nc−rAlt
+cc/parenrightbig
+(4)
+The final estimate on the NLE
+ccbackground number re-
+sults from the weighted average of the three different so-
+lutions of Eq. (4) when the LE beam data is compared
+with each of the other beam configurations. The un-
+certainty on the NLE
+ccbackground number is 12% ±2%.
+Therefore, the uncertainty in the number of charged-
+current background events is conservatively taken to be
+15% at all energies at the near and far-detectors.
+VII. SEARCH FOR ACTIVE NEUTRINO
+DISAPPEARANCE
+The data collected in the near and far-detectors have
+been classified as either neutral-current or charged-
+current events using the selections described in the pre-
+vious sections. A total of 388 data events are selected as
+neutral-current in the far-detector. The measured and
+predicted Erecospectra at the far-detector are shown in
+Fig. 9. The prediction assumes that oscillations occur
+only among the three active flavors and uses the values
+of|∆m2
+32|andθ23previously measured by MINOS [10].
+Although the present analysis is not capable of isolat-
+ing an electron neutrino appearance signal, it must take
+νµ→νeoscillations into account because the classifica-
+tion criteria of this analysis include νecharged-current
+interactions in the neutral-current enriched sample with
+nearly 100% efficiency. This accounting is done by com-
+paring the observedneutral-currentspectrum to two pre-
+dictions, one that assumes null νeappearance, and an-
+other that assumes an upper limit for the νeappear-
+ance rate in the far-detector calculated with the normal
+neutrino-mass hierarchy, θ13= 12◦andδ= 3π/2. The
+choice of θ13corresponds to the 90% confidence level
+upper limit established by the CHOOZ reactor experi-
+ment [31] for the |∆m2
+32|value measured by MINOS [10];
+the choice of δmaximizes the νeappearance for the cho-
+sen value of θ13. As seen in Fig. 9, the observedspectrum
+matches the prediction based on oscillations among the
+threeactiveflavorsverywelloverthefull rangeofallowed
+values of θ13.
+The agreement between the observed and predicted
+neutral-current spectra is quantified using a statistic, R:
+R≡NData−BCC
+SNC, (5)
+where, within agivenenergyrange, NDataisthe observed
+event count, BCCis the extrapolated charged-current11
+ (GeV)recoE0 5 10 15 20Events/GeV
+0204060Far Detector Data
+° = 013θ
+/2π = 3δ  ° = 1213θ
+ CC Backgroundµν
+2 eV-310× | = 2.432
+32m∆| 
+ = 123θ22sin
+FIG. 9: Thereconstructed energyspectrumofneutral-curre nt
+selected events at the far-detector (points with statistic al un-
+certainties). The Monte Carlo prediction assuming standar d
+three-flavor oscillations is also shown, both with (dashed l ine)
+and without (solid line) νeappearance at the CHOOZ limit.
+The shaded region indicates the 1 standard deviation system -
+atic uncertainty on the prediction. The hatched region show s
+the Monte Carlo prediction for the background of misidenti-
+fied charged-current events in this sample.
+background from all flavors, and SNCis the extrapolated
+number of neutral-current interactions [17]. The values
+ofSNCand contributions to BCCare shown in Table I.
+The disappearance of νµoccurs mainly for true neu-
+trino energies <6 GeV [10]. While the true energy of
+a neutrino interacting through the neutral-current pro-
+cess cannot be measured, the data can be separated into
+twosampleswhosereconstructedenergyroughlydiscrim-
+inates between neutrinos with true energies greater than
+and less than 6 GeV. According to the Monte Carlo
+simulation, events with Ereco<3 GeV have a me-
+dian true neutrino energy of 3.1 GeV while events with
+3 GeV< Ereco<120 GeV have a median true neutrino
+energy of 7.9 GeV. The values of Rcalculated for these
+rangesin ErecoareshowninTableI. Foralldatasamples,
+Rdiffers from unity by less than 1 standard deviation.
+The effect of νeappearance on the value of Ris treated
+as an uncertainty in this analysis and is indicated by the
+last uncertainty in Table I. Because νe-charged-current
+eventsarealmostalwaysidentifiedasneutral-current,the
+effect of νeappearance at the far-detector is to decrease
+the value of R, since the predicted background will be
+larger than for the case of null νeappearance.
+The measured values of Rfor each energy range in-
+dicate that neutrino oscillations among the active fla-
+vors describes the observed data quite well. Over the
+full energy range, 0 −120 GeV, a value of R= 1.04±
+0.08(stat.) ±0.07(syst.) −0.10(νe) is measured, corre-
+sponding to a depletion of the total neutral-current eventrate assuming null (maximally-allowed) νeappearance of
+less than 8% (18%) at 90% confidence level. The follow-
+ing sections explore the extent to which the data allow
+Ereco(GeV)NDataSNCBνµ
+CCBντ
+CCBνe
+CC
+0−3 141 125.1 13.3 1.4 2.3 (12.4)
+3−120 247 130.4 84.0 4.9 16.0 (32.8)
+0−3 R= 0.99±0.09±0.07−0.08
+3−120 R= 1.09±0.12±0.10−0.13
+0−120 R= 1.04±0.08±0.07−0.10
+TABLE I: Values of R,NData,SNC,and the contributions to
+BCCfor various reconstructed energy ranges. The numbers
+in parentheses are calculated including νeappearance at the
+CHOOZ limit, as discussed in the text. The first uncertainty
+in the value of Rshown is the statistical uncertainty, the sec-
+ond is systematic uncertainty, and the third is due to possib le
+νeappearance.
+oscillationsbetweentheactiveflavorsandonesterileneu-
+trino or oscillations in combination with neutrino decay.
+VIII. OSCILLATIONS INCLUDING A STERILE
+NEUTRINO FLAVOR
+Mixing of the three active neutrino flavors with one
+sterile neutrino requires the addition of one mass eigen-
+state. Theprobabilityformixingbetweenanytwoflavors
+is described by the neutrino mixing matrix [32], U, which
+defines the rotation from the mass basis into the flavor
+basis. Thus, the mixing matrix needs to be expanded by
+one row and one column to accommodate the additional
+mass and flavor states. The expanded 4 ×4 mixing ma-
+trix contains six mixing angles and six phases, with three
+of the phases being Majorana phases that are not rele-
+vant to an oscillationexperiment [33]. The matrix can be
+written as a product of six independent rotation matri-
+ces about the Euler axes Rij, whereijrefers to the plane
+in which a particular rotation takes place. The ordering
+of the rotation matrices is arbitrary, reflecting that the
+mixing matrix can be parameterized in many ways. The
+orderingdescribedbelowwaschosentomakethe analysis
+more straightforward.
+MINOS is designed to precisely measure |∆m2
+32|but
+has no sensitivity to ∆ m2
+21. Consequently, the mass
+states 1 and 2 are treated as degenerate in this analy-
+sis. When two mass states are degenerate the rotation in
+theij-plane is unphysical and the corresponding mixing
+angle,θij, vanishes from the oscillation probabilities [12].
+Additionally, the matrix should be of the form such that
+theUe3component of the mixing matrix goes to zero
+whenθ13= 0◦to distinguish an electron component in
+the third mass eigenstate from effects of sterile neutrinos.
+For those reasons, the general form of the mixing matrix
+used by the current analysis is written as12
+U=R34(θ34)R24(θ24,δ2)R14(θ14)R23(θ23)R13(θ13,δ1)R12(θ12,δ3)
+=R34(θ34)R24(θ24,δ2)R14(θ14)R23(θ23)R13(θ13,δ1), (6)
+where the δkare the three CP-violatingDirac phases and
+the last equality reflects the assumption of degeneracy inmass states 1 and 2. Thus, the mixing matrix can be
+written as,
+U=
+Ue1Ue2 c14s13e−iδ1 s14
+Uµ1Uµ2 −s14s13e−iδ1s24e−iδ2+c13s23c24 c14s24e−iδ2
+Uτ1Uτ2−s14c24s34s13e−iδ1−c13s23s34s24eiδ2+c13c23c34c14c24s34
+Us1Us2−s14c24c34s13e−iδ1−c13s23c34s24eiδ2−c13c23s34c14c24c34
+. (7)
+Herecij= cosθijandsij= sinθij, and only the elements
+of the matrix that appear in the oscillation probabilities
+given below have been expressed in terms of the mixing
+angles and phases.
+A. Oscillation Probabilities
+The oscillation probabilities are derived following the
+example in Ref. [34]. The evolution of a neutrino with
+initial flavor state αis given by
+|να/angbracketright=/summationdisplay
+jU∗
+αje−im2
+jL/(2E)|νj,0/angbracketright, (8)
+where the sum is over the mass states, Eis the neutrino
+energy,Lis the distance traveled by the neutrino and mj
+is the mass of state j. As the NuMI beam is almost en-
+tirelyνµorνµ, only the amplitudes and probabilities for
+νµ→νx, wherexrepresents e,µ,τ,ors, are described.
+The amplitude for νµ→νxis given by
+A=4/summationdisplay
+j=1U∗
+µjUxje−im2
+jL/(2E). (9)In this equation Uµjis the element of the mixing matrix
+describing the coupling between the muon flavor state
+and mass state j. By squaring the amplitudes and using
+the unitarity of U, one obtains the oscillation probabili-
+ties for the different channels,
+P(να→νβ) =δαβ−4/summationdisplay
+j>iR(U∗
+αjUβjUαiU∗
+βi)sin2∆ji
++2/summationdisplay
+j>iI(U∗
+αjUβjUαiU∗
+βi)sin2∆ ji,(10)
+where ∆ ji≡(m2
+j−m2
+i)L/(4E) andR(),I() designate
+the real and imaginary parts of the amplitude products.
+Because ∆ 21≪∆31the first and second mass states are
+treated as degenerate and the factors of sin∆ 21can be
+set to 0. This degeneracy also implies that ∆ 42= ∆41
+and ∆ 32= ∆31. Equation (10) can be expanded for the
+different oscillation scenarios,
+Pνµ→νµ= 1−4/braceleftbigg
+|Uµ3|2(1−|Uµ3|2−|Uµ4|2)sin2∆31+|Uµ4|2|Uµ3|2sin2∆43+|Uµ4|2(1−|Uµ3|2−|Uµ4|2)sin2∆41/bracerightbigg
+,
+Pνµ→να= 4R/braceleftbigg
+|Uµ3|2|Uα3|2sin2∆31+|Uµ4|2|Uα4|2sin2∆41+U∗
+µ4Uα4Uµ3U∗
+α3(sin2∆31−sin2∆43+sin2∆41)/bracerightbigg
++2I/braceleftbigg
+U∗
+µ4Uα4Uµ3U∗
+α3(sin2∆ 31−sin2∆ 41+sin2∆ 43)/bracerightbigg
+, (11)
+whereα=e,τ,orsand the orthogonality constraint/summationtext
+iUaiU∗
+bi= 0 has been used to eliminate the matrix ele-ments corresponding to the first and second mass states.13
+1ν1ν1ν2ν2ν2ν3ν3ν3ν
+4ν4ν
+4ν
+1 = m4m3 >> m4m3 = m4mmass
+FIG. 10: Schematic representation of the possible mass spec -
+tra for models including four mass eigenstates. In the left
+panel, the first and fourth eigenstates are degenerate, the c en-
+ter panel has the fourth mass eigenstate much heavier than
+the third, and the right panel has the third and fourth eigen-
+states as degenerate. Only the scenarios illustrated in the left
+and center panels are considered in this analysis.
+As can be seen from Eqs. (7) and (11), mixing between
+three active and one sterile neutrino in the most general
+case involves ten parameters which are mostly unknown:
+fivemixingangles,threemass-squaredsplittings, andtwo
+CP-violating phases. The following simplifying assump-
+tions are made to reduce the number of possible parame-
+ters. First, θ13is eliminated as a free parameter by only
+considering two values, θ13= 0◦orθ13= 12◦, where the
+latter is the CHOOZ limit at the MINOS measured value
+of|∆m2
+32|. When θ13is set to the CHOOZ limit, δ1is
+taken to be 3 π/2 as that is the value of the CP-violating
+phasethatmaximizesthe νeappearanceprobability. Sec-
+ond, the additional CP-violatingphase isremovedby set-
+tingδ2= 0 as MINOS has no sensitivity to this phase.
+Finally, nonzero values of θ14do not measurably change
+the oscillation probabilities, so that angle is set to 0◦as
+well.
+In addition to the parameters represented in the mix-
+ing matrix and oscillation probability equations, there is
+one more free parameter, namely the mass of ν4rela-
+tive to the other mass states. To limit the number of
+free parameters in the models, this analysis only exam-
+ines possibilities for the relative mass of ν4that allow the
+oscillation probabilities to depend only on |∆m2
+31|. The
+possible hierarchies are illustrated in Fig. 10 and only
+the normal mass hierarchy is shown. An inverted mass
+hierarchy would change the relative locations of the pair
+ν1andν2toν3. The fourth mass state could either be
+degenerate with the first mass state, as seen in the left
+panel of the figure, much more massive than the third
+mass state, as seen in the middle panel, or degenerate
+with the third mass state, as shown in the right panel. In
+the scenario with the third and fourth eigenstates being
+nearly degenerate there would be no discernible mixingbetween the active and sterile components. This conse-
+quence comes from the SNO results [6] which indicate
+that any coupling between νsand the active neutrinos
+occurs in the third and fourth mass eigenstates. There-
+fore, in this case the mass difference in which oscillations
+between active and sterile neutrinos could arise is too
+small to be observed in terrestrial baselines and it is not
+explored further in this analysis.
+An alteration of the oscillation probability of Eq. (11)
+is predicted for neutrinos which traverse dense matter.
+By virtue of their neutral-current coherent forward scat-
+teringwith nucleons, activeneutrinosacquireaneffective
+matter potential whereas sterile neutrinos do not. The
+effective potentials are identical for the νµandντfla-
+vors so that matter effects vanish for νµ→ντmixing.
+Forνµ→νsoscillations, however, the overall nonzero
+matter potential yields MSW-type modifications to the
+mixing angle and oscillation length such that, in normal
+(inverted) hierarchy scenarios, the oscillation probability
+will be suppressed (enhanced) relative to νµ→ντ[35].
+Observations of neutrinos with energies above 12GeV
+that have traveled through several thousand kilometers
+of dense material, such as those described in Ref. [8],
+would be sensitive to this matter effect. By contrast, the
+beam-inducedneutrinosmonitoredattheMINOSfarsite
+originate from a Eνspectrum highly peaked between 2
+and 6GeV and travel 735km through the Earth’s crust.
+For the effective mixing angles and oscillation lengths of
+neutrinos in the NuMI beam, the νµ→νsmatter effect
+only gives rise to subpercent distortions of the neutrino
+oscillation probability that are mostly confined to neu-
+trinoenergiesbelow 2GeV. Distortionsofthis magnitude
+are of no consequence for the studies reported here, and
+so matter effects involving sterile neutrinos have been
+neglected.
+1. Active-sterile mixing when m4=m1
+In them4=m1case, the first and fourth mass eigen-
+states are assumed to be degenerate. Because the first
+andsecondeigenstatesarealsotreatedasdegenerate, the
+second and fourth states are degenerate as well. These
+degeneracies allow one to set θ14=θ24= 0◦in Eq. (7),
+which reduces the number of parameters in the model to
+four. There are no νeorνµcomponents in the fourth
+mass eigenstate, however there is a νscomponent in the
+third mass eigenstate. Using these simplifications, the
+oscillation probabilities become
+Pνµ→νµ= 1−4|Uµ3|2/parenleftbigg
+1−|Uµ3|2/parenrightbigg
+sin2∆31,
+Pνµ→να= 4|Uµ3|2|Uα3|2sin2∆31 (12)
+This model is equivalent to the phenomenological model
+presented by MINOS in Ref. [17].14
+2. Active-sterile mixing when m4≫m3
+In them4≫m3case, the fourth mass eigenstate is as-
+sumed tobe muchlargerthan the third; consequentlythe
+values of sin2∆41and sin2∆43average to1
+2. Addition-
+ally, sin2∆ 41and sin2∆ 43average to 0. In this model
+∆m2
+43is assumed to be O(eV2) such that the regime
+of rapid oscillations and thus the averages mentioned are
+validatthefar-detectorsite, whileensuringnodetectable
+depletion of νµoccurs at the near-detector. Such models
+have recently received attention in the literature [12, 13].
+Using the above simplifications reduces the number of
+parameters in this model by two, and allows Eq. (11) to
+be written as
+Pνµ→νµ= 1−4/braceleftbigg
+|Uµ3|2/parenleftbigg
+1−|Uµ3|2−|Uµ4|2/parenrightbigg
+sin2∆31
++|Uµ4|2
+2(1−|Uµ4|2)/bracerightbigg
+,
+Pνµ→να= 4R/braceleftbigg/parenleftbigg
+|Uµ3|2|Uα3|2+U∗
+µ4Uα4Uµ3U∗
+α3/parenrightbigg
+sin2∆31
++|Uµ4|2|Uα4|2
+2/bracerightbigg
+, (13)
+where the second term of Eq. (11) does not appear be-
+cause of the assumptions that θ14= 0◦andδ2= 0.
+B. Fitting active-sterile oscillations to the data
+The data are compared to Monte Carlo predictions
+based on the probabilities in Eqs. (12) and (13) using
+theχ2statistic appropriate to small sample sizes,
+χ2= 2N/summationdisplay
+i=1/bracketleftbigg
+ei−oi+oilnoi
+ei/bracketrightbigg
++5/summationdisplay
+j=1ǫ2
+j
+σ2
+j.(14)
+Hereeiis the expected number of events, assuming os-
+cillations among four flavors, in bin iof the energy spec-
+trum, and oiis the observed number of events in that
+bin. The second sum is the contribution to χ2from the
+parameters describing the systematic uncertainties; the
+nuisance parameter ǫjis the shift from the nominal fit
+value for the j-thsource of systematic uncertainty and
+σjis the uncertainty associated with that source. Both
+the neutral-current spectrum shown in Fig. 9 and the
+charged-current spectrum shown in Fig. 11 are used to
+obtain the oscillation parameters that best fit the data.
+The neutral-current spectrum provides information on
+the mixing angles for mixing between active and sterile
+neutrinos while the charged-current spectrum provides
+constraintsonthemixingangle θ23andthemasssplitting
+|∆m2
+31|. The charged-current-like spectrum of Fig. 11 is
+statistically consistent with that presented in Ref. [10]
+given the different fiducial volumes and event separation
+procedures used in the two analyses. The five system-
+atic uncertainties described in Sec. VI are included as (GeV)recoE0 5 10 15 20Events/GeV
+020406080100
+Far Detector Data
+Monte Carlo Expectation
+µν NC Background 
+2 eV-310× | = 2.432
+32m∆| 
+ = 123θ22sin
+FIG. 11: The reconstructed energy spectrum of charged-
+current selected events at the far-detector (points with st a-
+tistical uncertainties). The Monte Carlo prediction assum ing
+standard three-flavor oscillations (solid line) is shown wi th
+the 1 standard deviation systematic uncertainty on the pre-
+diction indicated by the shaded region. The prediction for
+the small background of misidentified neutral-current even ts
+in this sample (hatched region) is also shown.
+nuisance parameters in the far-detector fits. By fitting
+for the systematic parameters simultaneously using both
+near and far-detector data, the effect of the uncertainties
+is substantially reduced due to significant cancellations
+of uncertainties between the two detectors.
+The best-fit values for the mixing angles in the two
+models as well as the χ2for each are shown in Table II.
+A list of the systematic effects for the fit parameters
+is presented in Table III. The latter table shows that for
+each mixing angle evaluated in the four-neutrino models
+considered, the uncertainties introduced by the five most
+significant sources of systematic errorare relatively small
+compared to the total uncertainty ranges obtained from
+the fits, as summarized in Table II.
+The best-fit values obtained for each model for the
+mass splitting, |∆m2
+31|, agree well with the result found
+in Ref. [10]. The one-dimensional projections of the ∆ χ2
+Model θ13χ2/D.O.F. θ23θ24θ34fs
+m4=m1047.5/39 45.0+9.0
+−8.9 0.1+28.7
+−0.10.51
+1246.2/39 47.1+8.8
+−11.0 23.0+22.6
+−24.10.55
+m4≫m3047.5/38 45.0+9.0
+−8.90.0+7.2
+−0.00.1+28.7
+−0.10.52
+1246.2/38 47.1+8.8
+−11.00.0+7.2
+−0.023.0+22.6
+−24.10.55
+TABLE II: Best-fit points and uncertainty ranges obtained
+for the active-sterile oscillation models. Results are sho wn
+with and without νeappearance at the CHOOZ limit. All
+angles are given in degrees. The quantity fsis defined as the
+fraction of disappearing νµthat could transition to νsand is
+given at the 90% C.L. in this table. The values of fsin the
+m4≫m3model are evaluated for Eν= 1.4GeV.15
+Model ParameterShift due to Systematic Uncertainty
+Absolute E Had.Relative E Had.Normalization CC Background ND Selection Total
+m4=m1∆θ23 0.3 0.6 0.3 0.1 0.1 0.7
+∆θ34 3.6 9.9 12.6 9.9 9.9 21.6
+m4≫m3∆θ23 0.2 0.6 0.1 0.2 0.2 0.7
+∆θ24 1.5 2.1 5.1 0.3 0.3 5.7
+∆θ34 4.5 9.9 6.3 9.9 9.9 18.8
+Oscillations∆α(GeV/km) 2.54×10−40.70×10−46.25×10−41.23×10−41.15×10−46.99×10−4
+with decay ∆θ 2.6 3.7 0.9 4.0 3.9 7.2
+TABLE III: Summary of mixing-angle deviations introduced b y the major systematic uncertainties from best-fit results i n
+which systematic shifts have been neglected. Angular devia tions, shown in degrees, are displayed for each mixing angle fitted,
+for each of the neutrino models analyzed in this work.
+ (deg)23θ30 40 50 602χ ∆
+0123
+°=013θ
+°=1213θ
+ (deg)34θ20 40 6068%90%
+FIG. 12: Projections of∆ χ2as afunction ofthemixingangles
+for them4=m1model. The solid line contours are obtained
+with null νeappearance, whereas the dashed line contours
+includeνeappearance at the CHOOZ limit. The ranges of
+values allowed at 68% and 90% confidence levels lie within
+contours below the horizontal dashed lines.
+betweenthebest-fitpointandtheremainingpointsinthe
+space are shown in Fig. 12 for the m4=m1model. The
+two-dimensional 90% confidence level contours for that
+modelareshowninFig.13. Theprojectionsandcontours
+for them4≫m3model are shown in Figs. 14 and 15.
+As seen in these figures, θ34<38◦(56◦) at 90% con-
+fidence level for the m4=m1model. The number in
+parentheses represents the 90% confidence level limit ob-
+tained when maximal νeappearance is allowed. For the
+m4≫m3model,θ24<10◦(11◦) andθ34<38◦(56◦) at
+the 90% confidence level. These limits indicate that any
+coupling between the active neutrinos and a sterile neu-
+trino is submaximal. Furthermore, the χ2values indicate
+that these four-flavor models fit the data no better than
+oscillations among only the active neutrinos.
+A straightforwardmethod to quantify the coupling be-
+tween the active and sterile neutrinos is to determine the
+fraction of disappearing νµthat transition to νs. That
+fraction is expressed as
+fs≡Pνµ→νs
+1−Pνµ→νµ. (15) (deg)34θ0 20 40 60 80 100 (deg)23θ
+30405060 °=013θ
+Best fit point
+90% C.L.
+ 
+°=1213θ
+Best fit point
+90% C.L.
+FIG. 13: Contours representing 90% confidence level for the
+m4=m1model. The solid line and best-fit point (solid sym-
+bol) are obtained assuming null νeappearance, whereas the
+dashed line and corresponding best-fit point (open symbol)
+are obtained with νeappearance set at the CHOOZ limit.
+For them4=m1model, the disappearance fraction fs
+is energy independent, as can be seen upon inserting the
+expressions from Eq. (12) into Eq. (15). The 90% confi-
+dence level limit for fsis determined by selecting a large
+number oftest values of θ23andθ34fromGaussian distri-
+butions with mean and σgiven in Table II. The value of
+fsthat is larger than 90% of the test cases represents the
+limit. The value corresponding to the 90% confidence
+level isfs<0.51(0.55), with the value in parentheses
+indicating the value obtained for maximally-allowed νe
+appearance in the beam. This new limit on the value of
+fsrepresents a reduction of 33% compared to the previ-
+ous MINOS result without νeappearance [17].
+For the m4≫m3model, inserting the expressions
+from Eq. (13) into (15) shows that fsis energy depen-
+dent because of the constant terms in Eq. (13). For this
+reason the 90% confidence level value of fsfor this model
+is presented at Eν= 1.4GeV, the energy where the νµ16
+ (deg)23θ30 40 50 602χ ∆
+0123
+°=013θ
+°=1213θ
+ (deg)34θ20 40 60
+ (deg)24θ2 4 6 8 10 1268%90%
+FIG. 14: Projections of ∆ χ2as a function of the mixing angles for the m4≫m3model. The solid line is obtained for the
+case of null νeappearance whereas the dashed line represents solutions wi thνeappearance at the CHOOZ limit. The ranges
+of values allowed at 68% and 90% confidence levels lie within c ontours below the horizontal dashed lines.
+ (deg)23θ30 40 50 60051015
+ (deg)34θ0 20 40 60 (deg)23θ
+30405060 (deg)24θ
+051015
+°=013θ
+Best fit point
+90% C.L.
+ 
+°=1213θ
+Best fit point
+90% C.L.°=013θ
+Best fit point
+90% C.L.
+ 
+°=1213θ
+Best fit point
+90% C.L.°=013θ
+Best fit point
+90% C.L.
+ 
+°=1213θ
+Best fit point
+90% C.L.
+FIG. 15: Contours representing 90% confidence level for the
+m4≫m3model. The solid line and best-fit point (solid sym-
+bol) are obtained for the case of null νeappearance, whereas
+the dashed line and corresponding best-fit point (open sym-
+bol) is obtained with νeappearance included with θ13at the
+CHOOZ limit.
+disappearance probability is a maximum. The determi-
+nation ofthe limit followsthe proceduredescribed above,
+but with the addition of selecting a value of θ24for each
+test caseas well. At 90% confidencelevel fs<0.52(0.55)
+forEν= 1.4GeV in this model. Thus, in either model,
+approximately 50% of the disappearing νµcan convert to
+νsat 90%confidence level as longas the amount of νeap-
+pearance is less than the limit presented by the CHOOZ
+collaboration.IX. OSCILLATIONS WITH DECAY
+It was noted more than a decade ago that neutrino
+decay, as an alternative or companion process to neu-
+trino oscillations, offers some capability for reproducing
+neutrino disappearance trends [18]. The model investi-
+gated here [36] includes neutrino oscillations occurring in
+parallel with neutrino decay. Normal neutrino-mass or-
+dering is assumed, and the mass eigenstates ν1,ν2are
+approximately degenerate, so that m3≫m2≈m1. The
+heaviest neutrino-mass state ν3is allowed to decay into
+an invisible final state. With these assumptions, and ne-
+glecting the small contributions from νemixing, only the
+two neutrino flavor states νµandντ, and the correspond-
+ing mass states ν2andν3, are considered. The evolution
+of the neutrino flavor states is given by [36]:
+id/vector ν
+dx=/bracketleftbigg∆m2
+32
+4E/parenleftbigg
+−cos2θsin2θ
+sin2θcos2θ/parenrightbigg
+−im3
+4τ3E/parenleftbigg
+2sin2θsin2θ
+sin2θ2cos2θ/parenrightbigg/bracketrightbigg
+/vector ν,(16)
+whereτ3is the lifetime of the ν3mass state and θis the
+mixing angle governing oscillations between νµandντ.
+Solving Eq. (16) one obtains probabilities for νµsurvival
+or decay:
+Pµµ= cos4θ+sin4θe−m3L
+τ3E+
+2cos2θsin2θe−m3L
+2τ3Ecos/parenleftbigg∆m2
+32L
+2E/parenrightbigg
+(17)
+Pdecay=/parenleftig
+1−e−m3L
+τ3E/parenrightig
+sin2θ. (18)
+The limits τ3→ ∞and ∆m2
+32→0 correspond to a pure
+oscillations or a pure decay scenario, respectively.
+In a conventionalneutrino oscillationsscenario, the ra-
+tio of the predicted charged-current spectrum in the far-
+detectorwith thenull-oscillationexpectationdisplaysthe
+characteristic “dip” at the assumed ∆ m2
+32value that is17
+Model χ2/D.O.F. α(GeV/km) θ
+Osc. with Decay 47.5/39 0.00+0.90
+−0.0×10−345.0+10.83
+−8.96
+Pure Decay 76.4/40 4.6+3.1
+−2.3×10−350.9+39.1
+−11.27
+TABLE IV: Best-fit points and uncertainty ranges obtained
+for the relevant parameters of the oscillation with decay
+model. The result obtained for the pure decay scenario,
+∆m2
+32→0, is also presented. Angles are shown in degrees.
+ GeV/km)-3 (10α0 0.5 1 1.5 2 2.5 3 (deg)θ
+30405060
+Best fit point
+90% C.L.
+FIG. 16: The best-fit point and 90% C.L. contour for the two
+parameters of the neutrino oscillations-with-decay model , the
+neutrino mass-lifetime ratio αand the mixing angle θ.
+absentin the equivalent ratiocomputed forpure neutrino
+decay. Previously published results by MINOS using the
+charged-current far-detector spectrum support conven-
+tionaloscillationsanddisfavorascenarioofpureneutrino
+decay at 3.7 standard deviations [10]. In the present
+analysis, both the neutral-current and charged-current
+far-detector spectra shown in Figs. 9 and 11 are included
+in the fit. Consequently, additional sensitivity is gained
+with respect to previous analyses, since any neutrino de-
+cay into a noninteracting final state would also deplete
+the neutral-current spectrum according to Eq. (18). Ad-
+ditionally, the analysis is extended to the more general
+scenario combining oscillations and the decay model de-
+scribed above.
+The best-fit values extracted for θand the parameter
+α≡m3/τ3using this model are summarized in Table IV.
+Figure 16 shows the two-dimensional 90% confidence in-
+terval obtained by the fit. The results are consistent with
+maximal mixing ( θ= 45◦) and with no neutrino decay
+(α= 0). The best-fit value for |∆m2
+32|is consistent with
+Ref. [10].
+Figure 17 shows the one-dimensional ∆ χ2projections
+forαandθ, with other parameters marginalized. The
+90% confidence level limit found for the neutrino decay
+lifetime is τ3/m3>2.1×10−12s/eV.
+A ∆χ2of 28.9 is obtained for the pure decay scenario.
+Thus, a pure neutrino decay model with null oscillations,
+as considered in Ref. [10], is disfavored at the level of (deg)θ30 40 50 602χ ∆
+0123
+ GeV/km)-3 (10α0.5 1 1.5 268%90%
+FIG. 17: Projections in ∆ χ2for fit solutions for αandθ
+mixing angle for the oscillations-with-decay model. Param e-
+ter ranges allowed at 68% and 90% confidence levels lie below
+the corresponding dashed horizontal lines.
+5.4 standard deviations.
+X. SUMMARY
+Searches for depletion or distortion in rate and visi-
+ble energy spectra of neutral-current events recorded in
+the MINOS far-detector have been carried out for the
+purpose of detecting or constraining processes involving
+active-sterile neutrino mixing, as well as further restrict-
+ing models including neutrino decay. The data exposure
+analyzedcorrespondsto 3 .18×1020protonson targetcol-
+lected during the period of May 2005 to July 2007.
+A total number of 388 neutral-current events were
+observed in the far-detector, whereas the expecta-
+tion from standard three-flavor neutrino models is
+377±19.4(stat.) ±18.5(syst.) events. The value for the
+statistic Rthat gauges the agreement between the data
+and the expectation based on oscillations among the
+threeactiveflavorsis R= 1.04±0.08(stat.)±0.07(syst.) −
+0.10(νe), which is consistent with no depletion of the ac-
+tive neutrino flux.
+Joint fits to the observed neutral-current and charged-
+current energy spectra, assuming two different neutrino
+oscillation models that include an additional sterile neu-
+trino flavor, yield the following 90% confidence level
+limits on the oscillation parameters: θ34<38◦(56◦)
+for the model with m4=m1;θ24<10◦(11◦) and
+θ34<38◦(56◦) for the model with m4≫m3. The val-
+ues in parentheses represent the results for maximally
+allowedνeappearance. These limits for the mixing an-
+gles between active and sterile neutrinos show that mix-
+ing between the active flavors dominates oscillations. In
+fact, the fraction of active neutrinos that oscillate into
+a sterile species is constrained to be fs<0.51(0.55) at
+the 90% confidence level for the m4=m1model and
+fs<0.52(0.55) forEν= 1.4GeV in the case of the
+model with m4≫m3.
+Similar fits, assuming a two-flavor neutrino model in
+which oscillations may occur in parallel with decay into
+a sterile species, yield the best-fit values for the mass-18
+lifetime ratio α= 0.00+0.90
+−0.0×10−3GeV/km and mixing
+angleθ= 45.0◦+10.83
+−8.96. From these results, we extract a
+90% confidence level limit on the neutrino decay lifetime,
+τ3/m3>2.1×10−12s/eV. The pure decay scenario
+(∆m2
+32→0) is disfavored by 5.4 standard deviations in
+our data as an alternative explanation to neutrino oscil-
+lations.
+Acknowledgments
+We thank S. Parke and P. Huber for useful discussions
+concerning oscillations between active and sterile neutri-
+nos. This workwassupported by the U.S. Department ofEnergy, the U.S. National Science Foundation, the U.K.
+Science and Technology Facilities Council, the State and
+University of Minnesota, the Office of Special Accounts
+for Research Grants of the University of Athens, Greece,
+FAPESP (Funda¸ c˜ ao de Amparo a Pesquisa do Estado de
+S˜ aoPaulo)andCNPq (ConselhoNacionalde Desenvolvi-
+mento Cient´ ıfico e Tecnol´ ogico) in Brazil. We gratefully
+acknowledge the Minnesota Department of Natural Re-
+sources for their assistance and for allowing access to the
+facilities of the Soudan Underground Mine State Park.
+We thank the crew of the Soudan Underground Physics
+laboratory for their tireless work in building and operat-
+ing the MINOS far-detector.
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\ No newline at end of file
diff --git a/1001.0337.txt b/1001.0337.txt
new file mode 100644
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+arXiv:1001.0337v2  [astro-ph.CO]  27 Jan 2010Mem.S.A.It.Vol.75,282
+c/circlecopyrtSAIt 2008 Memorie della
+Thenatureofgasandstarsinthecircumnuclear
+regionsofAGN:achemicalapproach
+P. Rafanelli1,L. Vaona1,R. D’Abrusco1, S. Ciroi1, V. Cracco1
+Universit´ a di Padova, Dipartimento di Astronomia, vic. Os servatorio 3, 35122 Padova,
+Italia. e-mail:dabrusco@unipd.it
+Abstract. Aim of this communication is to describe the first results of a work-in-progress
+regarding the chemical properties of gas and stars in the cir cumnuclear regions of nearby
+galaxies. Different techniques have been employed to estimate the abundan ces of chemical
+elements inthe gaseous and stellarcomponents of nuclear su rroundings indifferent classes
+of galaxies according tothe levelof activityof thenucleus (normal orpassive, starforming
+galaxies andAGNs).
+1. Introduction
+A recent comparative study of the features
+of the stellar component of the circumnuclear
+(r∼5pc) regions, carried out on three dif-
+ferent samples of galaxies, namely 1302 star-
+forming galaxies (SFG), 1996 active galaxies
+and 2000 normal galaxies (passive galaxies)
+byRafanelliet al.(2009),hasshownthatSFGs
+are characterized by a spectral continuum dis-
+tributiondominatedbyO, B stars. Onthe con-
+trary, O and B stars are absent in active and
+normal galaxies which show a similar contin-
+uumintheopticalspectralrange,butaslightly
+different depth of the 4000Å break, deeper in
+the spectra of normal galaxies (NG) than in
+the spectra of active galaxies. This di fference
+in deepness of the 4000Å break is a clear in-
+dication that there are more stars of spectral
+type A, namely tracers of a recent star forma-
+tionhistory,inthecircumnuclearregionsofac-
+tivegalacticnuclei(AGN)thaninthesamere-
+gions of not-active galaxies. This finding rein-
+forces the scenario according to which galac-
+tic activity is correlated to stellar formation inthe proximity of the central engine of AGNs.
+Aim of this work is the characterizationof the
+chemical composition of the gaseous compo-
+nent ofthe circumnuclearregionsin bothSFG
+and AGN. We have focused our attention on
+two differentquestions:
+–Is gas in which star formation is occurring
+in SFG rich in heavy elements (taking the
+Sunchemicalcompositionasreference)?
+–Is the chemical composition of gas ion-
+izedbythecentralengineofAGNdi fferent
+fromthatofSFG?
+In our work, we have applied the CLOUDY
+photoionization code Ferlandet al.(1998) to
+the AGN and SFG samples on a grid of mod-
+els spanning a large area of the parameter
+spaceandobtainedthecorrespondingobserved
+emission lines. In particular, we have concen-
+trated our attention to Type 2 AGN, namely
+the Seyfert 2 galaxies in the redshift range
+[0.04,0.08], in order to avoid the broad emis-
+sion lines arising in the dense gaseous clouds
+closetothenucleusoftheAGNandhiddenby
+thetorusinSeyfert2galaxies(S2G).Ouranal-Rafanelli:The nature of gas andstars inthe circumnuclear r egions of AGN 283
+ysis allows us to derive the features of the gas
+componentsinS2andSFgalaxieswithdensity
+andopacitycomparabletothoseofthegaseous
+componentsin observedgalaxies.
+2. Thesamplesof galaxies
+The three different samples of galaxy spectra
+studied in this work have been extracted from
+the SDSS DR7 spectroscopic dataset and
+classified on the basis of a set of classical
+spectroscopic diagnostic diagrams which
+exploit the flux ratios of spectral emission
+lines to determine the presence of AGN
+activity in the spectrum of the nuclear region
+of the galaxies. These samples have all been
+selected from a larger dataset composed of all
+galaxies observed spectroscopically and with
+redshift comprised in the range [0.04, 0.08],
+with signal-to-noise S/N>5 in the spectral
+continuum atλ=5500Å with the additional
+requirementof a signal-to-noiseratio S/N>3
+in the strong emission lines [OIII] λ5007,
+[OI]λ6300, [OII]λ3727, Hβ, Hα, [NII]λ6584,
+[SII]λλ6717 6731, when observed. The selec-
+tion in redshift has been performedin order to
+minimize the effect on the spectra of the fixed
+apertureofspectroscopicfibersusedforSDSS
+observations (see Kewleyet al.(2006) for
+comparison) and allow the smallest possible
+contamination by stellar light emitted in the
+regions of the galaxies far outside from the
+nucleus. The first step of the selection process
+has been the separation of the sources without
+emissionlinesfromtheemissionlinegalaxies.
+The 2000 galaxies with highest S /N ratios of
+this class have been chosen as members of
+the passive galaxies sample. The remaining
+emission line galaxies have been split in two
+subsamples, the first corresponding to the
+sources showing the signs of the presence
+of an AGN and the second containing SFG,
+Liners and composite galaxies by applying
+the diagnostic diagram based on the Oxygen
+lines fluxratiosVaona(2009), usingthe fluxes
+measured by the SDSS and the empirical
+relation in the[OI] λ6300/[OIII]λ5007 vs
+[OII]λ3727/[OIII]λ5007 diagram derived by
+Vaona(2009). The spectra of the galaxies se-
+lected as AGN-hostingsourcesin the previousstep have been retrieved by the spectroscopic
+SDSS database and corrected for galactic
+reddening using the values of the extinction
+provided by the Nasa Extragalactic Database
+(NED) web-service1, based on the maps and
+toolsdiscussedinSchlegelet al.(1998).Then,
+they have been reduced to the rest-frame
+using the value of the redshift as measured by
+the SDSS spectroscopic pipeline. The stellar
+continuum of the spectra of these presumptive
+Seyfert galaxies has been evaluated using
+STARLIGHT (Bruzual& Charlot(2003)),
+and subtracted from the observed spectra in
+order to retain the emission and absorption
+features of the spectra. Afterthen, the spectral
+parametersof all emission lines with S/N>5
+have been re-measured using an original
+method performing a multi-gaussian fit of
+the the emission lines developed by L. Vaona
+during his PhD thesis (Vaona(2009)). The
+galaxies showing in their spectra H αand Hβ
+line profiles broader than the [OIII] λ5007
+profile have been rejected, and the remain-
+ing sample has been classified using the
+classical spectroscopic diagnostic diagrams
+known as Veilleux-Osterbrock diagrams
+Veilleux&Osterbrock(1987). The last step
+of the selection process of the S2G galax-
+ies has been the extraction of the galaxies
+which satisfy the empirical relations in
+the [NII]λ6584/Hαvs [OIII]λ5007/Hβ,
+[SII]λ6717/Hαvs [OIII]λ5007/Hβand
+[OI]λ6300/Hαvs [OIII]λ5007/Hβdiagrams
+as determined in (Kewleyet al.(2006)) for a
+sample of galaxies observed spectroscopically
+inthe SDSS DR4:
+0.61
+[log[NII]
+Hα−0.47]+1.19<log[OIII]
+Hβ(1)
+0.72
+[log[SII]
+Hα−0.32]+1.30<log[OIII]
+Hβ(2)
+0.73
+[log[OI]
+Hα−0.59]+1.33<log[OIII]
+Hβ(3)
+1At the website
+http://irsa.ipac.caltech.edu /applications/DUST/284 Rafanelli:The nature of gas andstars inthe circumnucle ar regions of AGN
+TheSFGgalaxieshavebeenextractedfrom
+the originalsample using the Oxygen diagram
+and then applying again the appropriate rela-
+tions in the Veilleux-Osterbrockdiagnostic di-
+agrams to the line fluxes as measured by the
+SDSS pipeline. To summarize, the final num-
+bers of members of the NG, SFG and S2G
+galaxysamplesusedforthefollowinganalysis
+are 2000,1302and1996respectively.
+3. Spectralcontinua
+The first step of the analysis has been the de-
+terminationofthepropertiesofthestellarpop-
+ulation in the close surroundingsof the galac-
+ticnucleus.Thespectraofthegalaxiesbelong-
+ing to each of the three samples described in
+the previous section have been corrected for
+galactic reddening using the extinctions pro-
+vided by the NED service and, similarly at
+what has been done for the selection of the
+S2G galaxies, reduced to the rest frame us-
+ing the spectroscopicredshiftmeasuredby the
+SDSS. Then, all spectra have been realigned
+to match the dispersion of 1 Å /pixel in the
+spectral range [3600Å ,9200Å]. At this point,
+STARLIGHT has been used to produce the
+templatespectraaccountingforthepurelystel-
+laremissioninthenuclearregionsofthegalax-
+ies. This code performs spectral synthesis on
+observed spectra, consisting in the decompo-
+sition of the spectrum in terms of a conve-
+nient superposition of a base of simple stel-
+lar populations of various ages and metallici-
+ties. The collection of stellar populations em-
+ployed to perform the fits of the input spec-
+tra in this work consists in a set of 92 stel-
+lar populations extracted from the Bruzual-
+Charlot library of evolutionarysynthesismod-
+els described Bruzual& Charlot(2003) with
+23 equally spaced distinct ages in the in-
+terval [106,1.3·1010] years and 4 distinct
+metallicity values (Z =4·10−3,8·10−3,2·
+10−2,5·10−2). The template spectra, obtained
+by STARLIGHT, of all galaxies of the three
+samples have been normalized to the spectral
+flux measured atλ=5500Å, since this region
+of the spectrum is devoid of emission and ab-
+sorption lines, thus being a good approxima-
+tion of the continuum component of the spec-Fig.1.Normalized average spectra for NGs
+(black curve), S2G galaxies (red curve) and
+SFGsgalaxies(bluecurve).
+trum. In figure 1, the comparison of the nor-
+malized average spectra for the three di fferent
+familiesisshown(blueisusedforSFG,redfor
+S2G galaxies and black for NGs). While the
+average spectrum of SFGs galaxies is signifi-
+cantly higher than others in the shorter wave-
+lengths region, the behaviour of the spectra of
+S2G and NGs galaxies is very similar. The ra-
+tio of the average normalized spectra of S2G
+galaxies and NGs is shown in figure 2 in or-
+der to highlight the di fferences between their
+shapes. The average spectrum of S2G galax-
+iesappearstobesystematicallyhigherthanthe
+average spectrum of the NGs at wavelengths
+smaller thanλ=5500Å, thus indicating the
+presence of a small fraction of young stellar
+populations in the nuclear regions of galaxies
+that harbor a Seyfert 2 nucleus which are not
+found in the surrounding of the nuclei of no-
+emissionlinesgalaxies.
+4. Metallicityof thethreesamplesof
+galaxies
+Three different methods have been used to es-
+timate the metallicity of the interstellar mat-
+ter and stellar populations in the central re-
+gionsofgalaxiesofthethreesamplesherecon-
+sidered. For Seyfert 2 galaxies, a photoioniza-
+tion model for the gas in the narrow line re-
+gionof the AGN (Vaona(2009)) hasbeenem-
+ployed, while the content of metals of central
+regions of SFs and normal galaxies has been
+evaluated using two empirical techniques, theRafanelli:The nature of gas andstars inthe circumnuclear r egions of AGN 285
+Fig.2.Ratioofthenormalizedaveragespectra
+ofNGsandS2Ggalaxies.
+P method (Pilyugin& Thuan(2005)) and the
+Mg2index method (Benderet al.(1993)) re-
+spectively. The last method, namely the em-
+pirical correlation between the Mg2index and
+the metallicity, has been employed also to es-
+timate the metallicity of S2 and SFs galaxies
+andcomparedwiththemetallicityprovidedby
+the other methods. In both cases, the circum-
+nuclear stellar populations have been recon-
+structed using STARLIGHT. In the following
+sections these methods and the results of their
+application to the galaxy samples considered
+will bedescribed.
+4.1. Photoionization models
+There are so many physical processes in-
+volved in the ionized gas emitting region of
+AGNs that building a model able to repro-
+duce the observed spectrum is a great chal-
+lenge.Following(Netzer(2008)),therearefive
+mainaspectstobetakenintoaccountforacor-
+rect modeling of the ionized gas physical pro-
+cesses: photoionizationand radiativerecombi-
+nation,thermalbalance,ionizingspectrum,gas
+chemical composition and clouds and /or fila-
+ments distribution. In the thermal balance we
+findthemechanicalheatingandtheroleofdust
+which is alwayspresent in the NLRs. We have
+excluded from our analysis the gas kinematics
+and winds and gas confinement mechanisms.
+AcodesuchasCLOUDYFerlandet al.(1998)
+is able to solve numerically the ionization and
+thermal structure of a single cloud so it canbe used to explore di fferent parameters val-
+ues,tobuildmoresophisticatedmodelssuchas
+compositemodelswithmorecloudsand /ordif-
+ferent geometriesand time-dependentmodels.
+CLOUDYworkswithtwodefaultsgeometries,
+open and close respectively. The first one is
+based on simple photoionization field through
+aplaneparallelslabwithadefinedionizedcol-
+umn density, while in the second case the gas
+is distributed all around the source. The mod-
+els were made using CLOUDY code version
+06.02 Ferlandetal. (1998). We have assumed
+open geometry for all models. Once the ion-
+izing spectrum is defined (no-thermal power-
+law in our case) the input parameters are the
+densityNe/cm−3, the ionized column density
+Nc(H+)/cm−2, the ionization parameter Uand
+themetalabundances, Z/Z⊙.Theionizationpa-
+rameter expresses the level of ionization of
+photoionizedgas and is proportionalto the ra-
+tio of the ionizing photon density over the gas
+densityaccordingtotherelation:
+U=Q(H)
+4πr2cNH(4)
+where c, the speed of light, is intro-
+duced to make U dimensionless. We built
+two kinds of photoionization models: single
+cloud models and composite models, the lat-
+ter being based on a combination of two
+different single-cloud models. In the litera-
+ture two fundamental approaches regarding
+composite models can be found: a dense
+cloud inside a medium with low density
+(Binetteet al. (1996), or combining separately
+two clouds (Komossa&Schulz(1997)). The
+evaluation of the model parameters on a grid
+of models is the only realistic way to compare
+agreatnumberofobservedspectratothemod-
+els. Since our aim is to define the mean phys-
+ical parameters which characterize the NLRs
+using spectra in the visible range, a particular
+attention is put on the metallicity because this
+parameter is important in the investigation of
+thenatureofgasanditsorigin.Wehavegener-
+atedagreatnumberofmodelsinordertofitall
+the observed lines. The well known and use-
+ful method of the line ratio diagrams employs
+few lines and so has limited separation power,
+so that if we reproduce all the measured lines286 Rafanelli:The nature of gas andstars inthe circumnucle ar regions of AGN
+the line ratio diagrams are automatically sat-
+isfied by a large number of models. In these
+cases, parameters values are obtained by av-
+eraging the different models. Another impor-
+tant result obtained by this method is the ex-
+tension toward the ultraviolet (UV) and near
+infrared(NIR) wavelengthsof the electromag-
+neticspectrum.Aχ2testhasbeenemployedto
+comparethemodelswiththeobservedfluxes.
+4.1.1. Onecloud modelsandtheinput
+parameters
+In order to explore the 5-dimensional param-
+eter space we built, a set of single cloud
+models with the values for the parameters re-
+ported in Table 1. The ionizing spectra were
+assumed to be power-laws with ναin the
+range 10µm- 50Kev, a cut-offat low en-
+ergies (ν5/2) and a cut-offat high energies
+(ν−2). This choice is in agreement with the
+results showed in (Komossa&Schulz(1997))
+and(Groveset al.(2004)),andsupportedbyX
+band observations (see Mainierietal. (2007)).
+Inordertoassureconsistency,thecolumnden-
+sity values were checked and the simulation
+was interrupted when the number of transmit-
+tedionizingphotonswasbelow1%;otherwise,
+the simulation stops when the final tempera-
+ture reaches 4000 K. The basic model is sum-
+marizedinFigure3:theionizingphotonscross
+a slabwithafixeddensity,columndensityand
+dust to gas ratio (D /G) (the distance of the
+sourceisnotrequiredasafreeparametersince
+the ionizationparameterisfixed).Thenthein-
+trinsic spectrumis calculatedusingCLOUDY:
+the calculation stops when either the fixed re-
+quiredcolumndensityhasbeenachievedorthe
+temperaturein the simulation has fallen below
+4000K. We assume that the intrinsic spectrum
+is reddened by the ISM, so when the models
+are comparedto the observedspectra it is nec-
+essary tocorrectforextinction.
+4.1.2. Setofchemicalabundances
+The metallicity in HII regions and in star-
+forming galaxies may be determined by
+direct methods (Alleret al.(1984)), measur-
+Fig.3.Basic scheme of the photoionization
+models.
+Table 1.Input parameter values. Note that U,
+NeandNc(H+)valuesarereportedinlogarith-
+micnotation.
+Z/Z⊙αU N ecm−3Nc(H+)cm−3
+0.5 -1.9 -3.6 1 19
+1 -1.6 -3.2 1.5 19.4
+1.5 -1.3 -2.8 2 19.8
+2 -1 -2.4 2.5 20.2
+2.5 -2 3 20.6
+3 -1.6 3.5 21
+3.5 4 21.4
+21.8
+22.2
+ing the line flux ratios, once the tempera-
+ture is determined, or through empirical but
+well tested metallicity sensitive calibrations
+(Pilyugin&Thuan(2005)),whenitisnotpos-
+sible to measure directly the abundances(high
+metallicityorlowexcitationcases).Inthestel-
+lar photoionization case, the ionization struc-
+tureisknownandsoitispossibletoobtainthe
+total abundancesbymeasuringthe ionic abun-
+dances using the ionization correction factors
+method (see (Mathis& Rosa(1991)) and ref-
+erences therein). In the AGN, the situation is
+more difficult to handle since, when the ion-
+izing spectrum is a no-thermal power-law, the
+ionizationstructureisverycomplex:thepartial
+ionized region is so large so that di fferent ion-
+izationstatesaremixedandtakingintoaccount
+theionicabundancesmakingupthetotalabun-
+dancesgets quitecomplicated.In thiscase, di-
+rect methods are not possible and the empiri-
+cal calibrations are obtained only by models.
+The choiceof the sets of chemical abundances
+a critical step in the creationof the modelsbe-
+cause they are critically linked to another fun-
+damentalaspectofthesimulations,i.e.therole
+of the dust and its composition. Once a ref-Rafanelli:The nature of gas andstars inthe circumnuclear r egions of AGN 287
+erence sample of di fferent abundances is as-
+sumed (usually, the solar set of values), a law
+describing the change in the chemical compo-
+sition and that will be used for the exploration
+of different metallicities, is required. The ob-
+servations of the ionized gas chemical abun-
+dances in HII regions and starburst galaxies
+show that all the elements, except nitrogen
+and helium, in first approximation are found
+in equal proportions. As usual, in the HII re-
+gions the metallicity is expressed in terms of
+oxygen abundance while in the visible spec-
+tra itispossibletomeasurethreedi fferention-
+ization states and oxygen is the most abun-
+dantelementafterhydrogenandhelium.Then,
+we can assume that all the elements scale di-
+rectly with the oxygen abundance, except for
+nitrogenand helium that must be treated sepa-
+rately. The nitrogen abundance, as well as he-
+lium, must be correctedfor secondaryproduc-
+tionVila Costas& Edmunds(1993).
+N
+H=O
+H[10−1.6+102.37+log(O/H)]. (5)
+He
+H=0.0737+0.0293Z
+Z⊙(6)
+The composition and quantity of dust are
+also sources of uncertainties. The set of abun-
+dancesneedstotakeintoaccountbothgasand
+dust compositionin orderto maintain the total
+assumedelementabundances.Asfortheabun-
+dances of the elements, whose reference val-
+ues are assumed to be similar to the solar val-
+ues, the default values of abundanceand com-
+position for dust are taken from the Galactic
+interstellar matter (ISM). Chosen the features
+of the dust, the abundances of the gas com-
+ponent are calculated by subtracting the dust
+abundances from the total. With the excep-
+tion of the noble gases, all abundances must
+be corrected for depletion into dust. This cor-
+rection correspondsto a variation in the abun-
+dances of elements in gaseous form. The de-
+terminationofthesecorrectionsisamainissue
+of modeling since it is only possible to make
+assumptions based on the observed properties
+of the Galactic ISM. It is reasonable to sup-
+pose an increase of dust with metallicity be-
+cause at a metallicity increase corresponds anFig.4.N/O vs O abundance, black dashed
+line total abundances, filled black circle the
+Sun abundances, colored lines: pale blue
+D/G=1.00, cyan D/G=0.75, blue D/G=0.50,
+greenD/G=0.25
+increment of the refracting elements. Another
+important assumption regards the dust to gas
+ratio value (D/G thereafter). By changing the
+relative internal dust content as a function of
+the metallicity, the depletion factors must also
+be changed, and a method to estimate the de-
+pletion factors with the variation of the dust
+content is presented in (Binetteet al. (1993)).
+In any case, some assumptions for the D /G
+and the variation of the dust with the metal-
+licity are necessary.It seemsthat thedepletion
+into grains could be constant in many direc-
+tions of the Galaxy (see (Vladilo(2002))), so
+we adopted the depletion factors reported in
+(Grovesetal. (2006))asreferencevaluesanda
+linearlawforthescalingofthedustabundance
+with the metallicity. Using the conventional
+logarithmic notation, the relation between de-
+pletion factor and gas abundance is given by
+theformula:
+depl(X)=log(X/H)gas−log(X/H)tot(7)
+In this work,it is assumed log(X/O)=cost
+for all chemical elements, with the exception
+of N and He. For these elements we follow
+(Grovesetal. (2004)) and ( ?) by employing288 Rafanelli:The nature of gas andstars inthe circumnucle ar regions of AGN
+a linear combination of the primary and
+secondary components of Nitrogen with the
+requirement of matching the adopted solar
+abundance patterns (Asplundet al. (2005))
+(Table 2). In the models, 4 D /G values were
+used in units of the ISM D /G=0.25, 0.50,
+0.75, 1.00. For each D/G value, we defined
+seven sets of chemical abundances, using
+the solar oxygen abundance as reference:
+Z/Z⊙=0.5; 1; 1.5; 2; 2.5; 3; 3.5. We focused
+our attention on the high metallicity levels
+because nuclear gas and Seyfert galaxies
+very rarely present sub-solar metallicities
+(Groveset al.(2006)). For each adopted D /G
+we changed the depletion factors set in order
+to retain the total abundances. The set of
+depletion factors is constant for each assumed
+D/G value with the exception of nitrogen
+because of the secondary production. The
+nitrogen depletion factor must change with
+the metallicity because, in agreement with our
+assumptions, if the dust increases with the
+metallicitytheremust beanexcessofnitrogen
+in gas form due to the formula 4.1.2. The
+Nitrogenmay be depletedin the carbonaceous
+grains: the involved functional groups are
+Amines (N-H, C-N), Amides (N-H), Nitriles
+(C,N triple bond stretch), Isocyanates (-
+N=C=O), Isothiocyanates (-N =C=S), Imines
+(R2C=N-R)andNitrogroups(-NO2aliphatic,
+aromatic)Kwok(2007). Excludingthe ammo-
+nia (NH 3), which cannot survive in an intense
+ionizing field such as in the AGN case, all
+the mentioned compounds need their partner
+elements linearly increasing with the metallic-
+ity. In Table 3 the depletion factors used for
+D/G=1.00 are reported. if this assumption
+is correct, a great increment of nitrogen gas
+abundancefor Z/Z⊙>1.5 shouldbeobserved,
+suggestingthat it is not necessaryto invokean
+excess of nitrogen in order to match the ob-
+served[NII]λ6584lineintensity.IntheFigure
+4 there are the relations between log(N/O)gas
+and 12+log(O/H)gasfor each couple of D /G
+andZ/Z⊙values. For the other elements the
+trend is constant at a given D /G value. It is
+important to stress that the depletion of the
+elements must be taken into consideration
+even if the nebula is without dust, such as
+in radiation-pressure dominated models of(Grovesetal. (2004)) according to which
+the dust is blown away by the wind. In our
+models, we assumed only two types of dust,
+graphite and silicates; their reference abun-
+dancesare [ C/H]=1.22E−04 when D/G=1
+in the graphite and [ O/H]=1.94E−04,
+[Mg/H]=3.15E−05, [Si/H]=2.82E−05,
+[Fe/H]=2.70E−05 for silicate respec-
+tively, while the distribution of their size is
+in the range [0.005,0.25]µm. We analyzed
+the abundances found in many HII regions
+with direct methods in order to check the
+relation between oxygen abundance and the
+other elements. In (VanZeeet al. (1998)),
+(VanZee&Haynes(2006)),
+(Izotov& Thuan(1999)), (Izotovet al.(2006)
+and (Continiet al.(2002)) a vast collection
+of chemical abundances were determined in
+186HIIregions spanning a range of radius
+in 13 spiral galaxies, 54 supergiant H II
+regions in 50 low-metallicity blue compact
+galaxies, 67 HIIregions in 21 dwarf irregular
+galaxies, 310 emission-line galaxies from
+the Data Release 3 of the SDSS and a sam-
+ple of 68 UV-selected intermediate-redshift
+(0≤z≤0.4)galaxies. Our set of modelsdoes
+not match very well with the trend seen in
+Figure 5, whereas there is a better fit with the
+samples used by (Groveset al.(2004))
+(Mouhcine& Contini(2002),
+Kennicuttet al.(2003)) but the log(N/O)
+trend is in any case overestimated when the
+abundances are greater than Z⊙. In order to
+compare our results with those obtained by
+the above mentioned authors, we adopted
+the same solar abundances, formalism and
+constraints. It is apparent that the UV-galaxies
+have a large spread in the log(N/O) values,
+so it could be possible that the HII regions
+and starburst galaxies have di fferent sets of
+chemical abundances. This hypothesis could
+be tested by using a sample of gas abun-
+dances in nuclei of spiral galaxies to ascertain
+whether the adopted functionis an upperlimit
+of the real one. The final number of models
+models is 10390, 9782, 9022 and 8207 for
+D/G=0.25, 0.50, 0.75, 1.00 respectively. The
+models will be rejected unless the resulting
+spectra are classified as Seyfert 2 spectra in
+the (Veilleux& Osterbrock(1987)) diagnosticRafanelli:The nature of gas andstars inthe circumnuclear r egions of AGN 289
+Fig.5.thelog(N/O)vs12+log(O/H):filledred
+circles UV galaxies, blue crosses HII regions
+in spiral galaxies, green crosses HII regionsin
+bluecompactgalaxies,blue***HIIregionsin
+dwarf irregular galaxies and green *** emis-
+sion line galaxies, black line the adopted N /O
+function, the black circles show the adopted
+metallicity values Z/Z⊙, respectively 0.5, 1
+(filledcircle),1.5,2,2.5,3,3.5.
+diagrams, following the Kewley’s conditions
+(Kewleyetal. (2006)),andthesyntheticfluxes
+are in the observed ranges, following Koski’s
+diagrams(Koski(1978)).
+4.1.3. Composite models
+The simple single cloud model described in
+the previous section is( only the first step
+of our modeling e ffort since in the ob-
+served Seyfert spectra two distinct c com-
+ponents are usually observed, yielding high
+and low ionization lines. A double compo-
+nent model with two clouds, each account-
+ing for either the high or the low ionization
+lines, can be created. We explored two kinds
+of composite models: Binette’s models and 2
+clouds models. The first one is based on the
+(Binette etal. (1996)) work, with the only im-
+provementsrepresentedbytheexplorationofa
+larger region of the parameter space; the sec-
+ond model is based on a combination of two
+different and independent single cloud models
+(Komossa&Schulz(1997)), as described pre-
+viously.We consideredall the possiblecombi-
+nations between two clouds with fixed power-
+law indexes and metallicities. The spectralTable 2. Adopted abundances and depletion
+factors (the depletion factor is defined as:
+depl=log(X/H)gas−log(X/H)tot).
+el.log(X/H)⊙depllog(X/H)⊙gasISM
+He -1.07 0.00 -1.07
+Li -10.95 0.00 -10.95
+Be -10.62 0.00 -10.62
+B -9.30 0.00 -9.30
+C -3.61 -0.30 -3.91 -3.91
+N -4.22 -0.30 -4.52 -4.52
+O -3.34 -0.24 -3.58 -3.71
+F -7.44 0.00 -7.44
+Ne -4.16 0.00 -4.16
+Na -5.83 -0.60 -6.43 -5.96
+Mg -4.47 -1.15 -5.62 -4.50
+Al -5.63 -1.44 -7.07 -5.65
+Si -4.49 -0.89 -5.38 -4.55
+P -6.64 0.00 -6.64
+S -4.86 -0.34 -5.20 -5.13
+Cl -6.50 -0.30 -6.80 -6.80
+Ar -5.82 0.00 -5.82
+K -6.92 0.00 -6.92
+Ca -5.68 -2.52 -8.20 -5.68
+Sc -8.95 0.00 -8.95
+Ti -7.10 0.00 -7.10
+V -8.00 0.00 -8.00
+Cr -6.36 0.00 -6.36
+Mn -6.61 0.00 -6.61
+Fe -4.55 -1.37 -5.92 -4.57
+Co -7.08 0.00 -7.08
+Ni -5.77 -1.40 -7.17 -5.79
+Cu -7.79 0.00 -7.79
+Zn -7.40 0.00 -7.40
+lines emitted by the first and second cloud
+are combined by a weighted mean, where the
+weightisgivenbytheproductof Hβluminosi-
+tiesratioandthesolidangleratio( ω1/ω2)sub-
+tended by the clouds as seen from the source
+oftheionizingradiation(thisratio,calledgeo-
+metrical factor, is indicated by GF). The fol-
+lowing steps lead to the relation which has
+been applied to the combined fluxes. The line
+luminosityisgivenby:
+L=4πr2Sω/4π (8)290 Rafanelli:The nature of gas andstars inthe circumnucle ar regions of AGN
+Table 3.Nitrogen depletion factors for the ra-
+tio D/G=1.00. Relative abundances of metals
+are inlogarithmicnotation.
+Z/Z⊙O/H N/H N/Hgasdepl
+0.5 -3.64 -4.75 -5.54 -0.79
+1 -3.34 -4.22 -4.52 -0.3
+1.5 -3.16 -3.89 -4.09 -0.19
+2 -3.04 -3.66 -3.8 -0.14
+2.5 -2.94 -3.48 -3.59 -0.11
+3 -2.86 -3.32 -3.41 -0.09
+3.5 -2.8 -3.19 -3.27 -0.08
+whereristhe innerradiusof thenebula, ω/4π
+is the covering factor, Sthe emission line in-
+tensity (erg s−1cm−2).Then:
+L=4πd2F=4πr2Sω/4π (9)
+whereFis the observedflux ( erg s−1cm−2),d
+is the distanceof the source.Definingthe den-
+sity ofionizingphotons( s−1cm−2)as:
+φ(H0)=Q(H0)/4πr2(10)
+whereQ(H0)isthenumberofionizingphotons
+emitted from the source in a second and the
+ionizationparameteras
+U=φ(H0)/nHc (11)
+Iftherearetwo clouds:
+U1/U2=r2
+2n2/r2
+1n1⇒r2
+2/r2
+1=U1n1/U2n2(12)
+and,asaconsequence:
+L2/L1=4πd2F2/4πd2F1=F2/F1(13)
+Then:
+F2/F1=U1n1S2ω2/U2n2S1ω1 (14)
+in termsof Hβthisratiocanbewrittenas:
+F2(Hβ)/F1(Hβ)=GF·L′
+2(Hβ)/L′
+1(Hβ) (15)
+whereL′
+2(Hβ)/L′
+1(Hβ) is theHβluminosity ra-
+tio ifω2=ω2. These quantities are calculated
+by CLOUDY for each model. If we want to
+Fig.6.2cloudsmodel,ω1andω2arethesolid
+anglessubtendedbytheclouds.
+findtheemissionlineintensity Iλrelativeto Hβ
+then:
+Iλ=(Fλ)2+(Fλ)1
+Ftot(Hβ)(16)
+whereFtot(Hβ) is the total observed flux, and
+rearrangingthetermswe obtain:
+Iλ=(Iλ)2+F(Hβ)1
+F(Hβ)2(Iλ)1
+1+F(Hβ)1
+F(Hβ)2(17)
+andfinally:
+Iλ=(Iλ)2+L′
+2(Hβ)
+L′
+1(Hβ)·GF·(Iλ)1
+1+L′
+2(Hβ)
+L′
+1(Hβ)·GF(18)
+IfGFgoes to zero only the second cloud is
+visible; on the contrary,if GFgoes to infinity,
+only the first cloud is visible. Five values for
+GFhave been used (0.25, 0.5, 1, 2 and 4), so
+that each couple of models provides five new
+syntheticalspectra.Thetotalnumberofmodels
+obtained with the constrains on D /G,α,Z/Z⊙
+fixed for each couple and after imposing the
+KewleyandKoskiconditions,isabout9 ·106.
+4.2. Comparison ofmodelswith
+observations
+Wehavecomparedallthemeasuredlinescom-
+ing from the observations with the syntheticRafanelli:The nature of gas andstars inthe circumnuclear r egions of AGN 291
+data provided by the models. Our goal is to
+reproduce as efficiently as possible the ob-
+served spectra with the aim of determining
+the most realistic set of input parameters. In
+(Olivaet al.(1999)), the authors proposed a
+new method for deriving abundances in the
+AGNnarrowlineregion(NLR),consistingina
+selection of ”good’”modelsfroma greatsam-
+ple(theyused27000models)whichiscapable
+of fairly reproducing the observed line spec-
+tra, and then slightly modifying the chemical
+abundancesin ordertodeterminethe best esti-
+mates of the metallicity of the AGN. The very
+large number of synthetical spectra we have
+produced has been employed to determine the
+set of models yielding the most similar syn-
+thetic spectral features to each observed spec-
+trumwithaχ2test.Inthiscase,the χ2isgiven
+by:
+χ2=/summationdisplay
+i(Fλobs−FλC)2
+i/(σi)2(19)
+whereFλobs−FλCisthedifferencebetweenthe
+observed flux and CLOUDY flux, and σithe
+errorassociatedtothei-thline.Themostprob-
+able set of input parameters can be therefore
+determined for each observed spectrum. The
+comparisonhasbeencarriedouttakingintoac-
+count the synthetical lines which are foreseen
+to be visible in the observed spectrum, since
+theselineswouldbedetectable( S/N>3ratio)
+intheobservedspectrum.Linesbelowthe S/N
+value of the observed spectrum or not visible
+areexcludedfromtheanalysis.Twofurtherpa-
+rameters consideredin the models are the red-
+dening(Av)andthescalefactor F(Hβ).Theχ2
+testisperformedusingthemeasuredfluxesnot
+corrected by reddening while the flux models
+are reddened in order to achieve the observed
+F(Hα)/F(Hβ)ratio.Sincewehavenotcorrected
+for the extinction using a fixed value, even if
+close to3.1,manymodelsshowasignificantly
+different value. Av is calculated applying the
+relationin(Cardelliet al.(1989)):
+Av=7.22·/bracketleftBigg
+log/parenleftBiggF(Hα)
+F(Hβ)/parenrightBigg
+o−log/parenleftBiggF(Hα)
+F(Hβ)/parenrightBigg
+C/bracketrightBigg
+(20)
+Nine different values of F(Hα)/F(Hβ), ob-
+tained combining F(Hα)±∆F(Hα) withF(Hβ)±∆F(Hβ) (writingin compactformthe
+fluxesFα/Fβ,(Fα+∆Fα)/Fβ),Fα/(Fβ−∆Fβ)
+and so on),have been adopted.Then the mean
+(/angbracketleftFα/Fβ/angbracketright) and the root mean square ( σ) of the
+ratiosarecalculated,takingintoaccountallthe
+values within the range −σ</angbracketleftFα/Fβ/angbracketright<σ,
+givingas a resulta positiveAvvalue.Foreach
+modelupto9differentsyntheticalspectrahave
+been calculated. The scale factor allows us to
+compare directly the reddened model with the
+observedspectrum.Relativefluxeswouldhave
+increased relative errors because of the prop-
+agation. The accepted model have to satisfy a
+predeterminedsignificancelevel,suggestedby
+the analysis of the reduced χ2distribution and
+givenby:
+χ2/dof=χ2/(nr−np) (21)
+where”dof”standsfordegreeoffreedom,”nr”
+is the number of measured lines and ”np” the
+numberof parametersused in the models.The
+minimal number of dof necessary in order to
+decide the acceptability level is given by the
+numberofmeasuredlinesminusthenumberof
+parametersusedinthemodels.Inordertocon-
+sider the model a good approximation of the
+observed spectrum, the χ2/dofdistribution is
+requiredtobesimilartoagaussiandistribution
+peaked on 1. If the peak is located at values
+<<1 there is an overestimation in the fitting,
+presumably caused by the fact that very large
+errorbarsfitverywellnotsignificantsetofpa-
+rameters of the model. On the other hand, if
+the peak is placed at values >>1, the models
+cannot reproduce the observed spectra or the
+errorsare too small. The significancelevel has
+beenfixed,afterseveraltests, between5%and
+10% and the assumed flux errors at 2 σlevel.
+The best fit models are associated to the min-
+imumχ2value, indicated with ( χ2
+min), with the
+assumption that if a model is accepted, all the
+models falling within the assumed confidence
+region are considered as acceptable as well.
+Thisconditioncanbeexpressedas:
+χ2−χ2
+min<A (22)
+where A is the value selected for a given
+probability Moll´ a&Hardy(2002) (25% level
+ofconfidenceisadoptedinthiswork).Foreach292 Rafanelli:The nature of gas andstars inthe circumnucle ar regions of AGN
+fitted spectrum there is a group of accepted
+models; the best values of the parameters are
+evaluated by averaging the single values over
+all theacceptedmodels.Inthisanalysis,7free
+parameters for the single cloud models (SC)
+have been considered. As a consequence, a
+minimum number of measured lines equal to
+7 and 11 to fit such models is required, while
+at least 12 measured lines and 7 parameters
+for Binette’s models. About 50% of the spec-
+tra shows less than 12 measured lines, mean-
+ing that the 2C models can be compared to
+only the 50% of the original sample of galax-
+iesbutwith thegreatadvantageofconsidering
+only the higher S/N spectra. A spectrum with
+a low number of measured lines has usually
+low S/N, i.e. then big errors on the measured
+spectral lines, so, even thoughit is easier to fit
+themodelswithdifferentparameters,thesefits
+havelittle ornoscientificvalidity.
+4.2.1. Thetest
+Inordertocomparethedi fferentkindsofmod-
+els, the spectra are splitted into two groups,
+namely spectra with nr≥12 and spectra with
+nr<12.SincetheD/Gratioisnotconsidereda
+parameterofthemodel,themodelswithdi ffer-
+entD/Gvalueswill betreatedseparately.Such
+classification leads to four families of models,
+one for each D/G value. The number of input
+parametersforeachkindofmodelaresumma-
+rizedinTable 4.
+4.2.2. Seyfert 2
+On a total of 2153,1141 spectra have nr<12
+and the remaining nr≥12. For this reason,
+the comparison between SC and 2C models
+will be done only over 1012 spectra. In Table
+5 the number of fitted spectra for each D /G
+value is reported;in more details, the first col-
+umn contains the D /G value, second to fourth
+columns contain the fitted spectra with SC
+models and the last column the fitted spectra
+with 2C models. The total percentageof spec-
+trawhichpassedthetestare ∼65%and∼50%
+for the SC and 2C models respectively. The
+percentage of spectra successfully fitted withFig.7.χ2/dofdistribution for SC models, up,
+and 2C models, down, and the di fferent D/G
+values,red0.25,blue0.50,green0.75andgrey
+1.00.
+nr<12is80%,while∼45%isthefractionof
+spectra fitted with nr≥12. Both two 2C mod-
+els accountsforonly50%of thespectra fitted,
+sincethisworkwascarriedoutwiththegoalof
+achievingthebestfittingandnotthemaximum
+numberofspectrafitted.
+Theχ2/dofdistributionsare shown in fig-
+ure 7. The distributions of the SC models are
+slightly skewed toward lower values for the
+D/G=1 and D/G=0.75 models, while the 2C
+modelsshowtailsforlargervalueseveniftheir
+overallshapesaremoreregular.Thesignificant
+differencebetweenthe modelsis evidentcom-
+paringthedistributionofabsolute χ2,sincethis
+statisticsrepresentsthegoodnessoftheoverall
+fitting procedure (the input model parameters,
+the choice of the significance level and errors)
+and it is possible to discriminate between two
+modelswhichareabletofitthesamespectrum
+bylookingat theχ2value.
+The cumulative distributions (Figure 8) of
+χ2/dofshowclearlythegreatdi fferenceinthe
+obtained fits between the two kinds of models
+used in this work. Within each type of mod-
+els, the D/G value yielding the minimum χ2
+values is 1 even if the di fference is not signi-
+ficative, while the larger overall number of fit-Rafanelli:The nature of gas andstars inthe circumnuclear r egions of AGN 293
+Table 4.The input parameters of the simulations for single cloud, do uble clouds and Binette’s
+models.
+Parameter SC2CBinette
+power-lawindex,α yesyes yes
+metallicity, Z/Z⊙ yesyes yes
+density, Ne yesyesfixed
+columndensity, Nc(H +) yesyes yes
+ionizationparameter, U yesyes yes
+density2nd cloud, Ne 2 noyesfixed
+columndensity 2nd cloud, Nc(H +)2noyescalculated
+ionizationparameter 2ndcloud, U 2noyescalculated
+geometrical factor, Gf noyes yes
+Reddening, Av yesyes yes
+Scalefactor, F(H β) yesyes yes
+number ofparameters 711 7
+Table 5.Numberoffitted spectra.
+D/GSC SCnr<12SCnr≥122C
+0.251249/2153 853/1141 396/1012 404/1012
+0.501375/2153 912/1141 463/1012 486/1012
+0.751445/2153 930/1141 515/1012 554/1012
+1.001426/2153 939/1141 487/1012 561/1012
+ted spectra is obtained with D /G=0.75÷1.
+More interesting is the comparison of χ2be-
+tween the matched fitted spectra with single
+and 2C models. For each D /G value (starting
+from0.25)therearerespectively289,356,410
+and396sharedspectra,whose χ2distributions
+are dramatically di fferent. The differences can
+beappreciatedlookingatthecumulativedistri-
+butions(Figure9):spectrahavinggoodS /Nare
+best fitted using 2C models. Many measured
+linesandmuchgreaternumberofspectracould
+befittedincreasingtheparameterresolution,in
+particularforthepower-lawindexandtheion-
+izationparameter,atthecostofamuchheavier
+computationalload.
+Table 6 contains the percentage of failures for
+each measured line of the observed spectra.
+The data show again that the 2C models fit
+more efficiently all the lines. Anyway, these
+results should be cautiously handled since the
+percentageof successful fits are calculated av-
+eragingoverfew spectra,so that a single mea-
+surecouldmodifydrasticallytheresult(inpar-
+ticular for the [ ArIV] and [FeVII] lines). TheFig.8.χ2cumulativedistribution for SC mod-
+els, thinlines,and2Cmodels,thicklines.
+lines mostly affected by this kind of issue are
+[NI]5200, [FeVII]5721,6087,HeI5876 and
+[ArIII]7135.
+4.2.3. Metallicities ofSeyfert 2galaxies
+In Table 7 are summarizedall the meanvalues
+of the parameterswith their root mean square,
+σ, respectively for the single cloud and 2C
+models. The power-law index αdecreases to
+−1.8withtheincreasingoftheD /Gratio,prob-294 Rafanelli:The nature of gas andstars inthe circumnucle ar regions of AGN
+Table 6.Percentageofline failuresforsinglecloudanddoubleclou dmodels.
+SC 2C
+line 0.250.500.751.000.250.500.751.00
+[OII]3727 66753221
+[NeIII]3869 22112003
+[NeIII]3968 11235211
+[SII]4074 81818250000
+Hγ4340 01110000
+[OIII]4363 100783668
+HeII4686 21130100
+[ArIV]4711 00000000
+[ArIV]4740 332525200000
+Hβ4861 00000000
+[OIII]4959 00000000
+[OIII]5007 23330000
+[NI]5200 6762595734293234
+[FeVII]5721 5060676714103033
+HeI5876 19171615810911
+[FeVII]6087 626166639151565
+[OI]6300 76762355
+[OI]6363 21110010
+[NII]6548 00000000
+Hα6563 00000000
+[NII]6584 23450000
+[SII]6716 23340011
+[SII]6731 44448641
+[ArIII]7135 7111622382627
+[OII]7325 000000010
+Fig.9.χ2cumulativedistributionforsharedfit-
+tedspectrawiththeusualmeaningofthesym-
+bols.
+ably because of the decreasing gas metallicity
+content (a large fraction of metals is depleted
+intograins).Infact,alowmetallicityofthegas
+implies a much slower cooling and a weaker
+ionizing spectrum is necessary to obtain the
+same degree of ionization. The total metallic-
+ityiswelldefinedinallmodelswithnegligiblescatter; 1.5÷2.5Z/Z⊙is the range of the fit-
+ted values with the SC models while for the
+2C model the estimate obtained is 1 .0÷2.0
+Z/Z⊙. The highmetallicity of the gasindicates
+that the AGN fuel is evolved material, which,
+in turn, supports the idea that the gas has a lo-
+cal origin in the vast majority of the observed
+Seyfert 2 galaxies. The densities are in agree-
+ment with the measured values: SC and SC
+models produce the same values of densities,
+both very similar to the measured sulfur den-
+sity.
+The column densities are very similar in
+the SC models and in the first cloud of the
+2C models, while the second cloud of the lat-
+ter model has a lower column density and a
+lower ionization parameter too, which allows
+us to treat such cloud as a ionization bounded
+case. Theionizationparameterin theSC mod-
+els isalmost constantfordi fferentD/G values:Rafanelli:The nature of gas andstars inthe circumnuclear r egions of AGN 295
+Table 7.Parametersoutput
+One cloud models 0.25 0.50 0.75 1.00
+Parameter meanσmeanσmeanσmeanσ
+α -1.50.3-1.60.3-1.70.3-1.80.3
+Z/Z⊙ 2.00.61.90.61.90.51.80.5
+log(Ne) 2.30.72.30.72.20.62.10.7
+log(Nc(H+)) 20.50.720.30.620.00.519.90.4
+U -3.10.2-3.10.2-3.00.2-3.00.3
+Av 1.40.61.50.61.50.61.60.6
+2C models
+α -1.30.3-1.40.3-1.60.3-1.80.3
+Z/Z⊙ 1.60.51.70.41.70.41.70.4
+log(Ne) 2.20.82.20.82.10.72.20.8
+log(Nc(H+)) 20.20.420.20.420.20.420.20.4
+U -2.70.4-2.60.4-2.60.4-2.50.4
+log(Ne)2 3.00.92.80.92.70.92.50.8
+log(Nc(H+))2 19.70.419.70.319.70.319.70.3
+U2 -3.30.2-3.30.2-3.30.2-3.30.3
+Av 1.20.41.20.41.30.41.40.4
+log(U)=−3. In the 2C the high ionization
+cloud shows a little variation in the ionization
+parameterwithrespecttoD /Gvalues,themean
+value being log(U)=−2.6, while a very simi-
+lar value is found for the low ionization cloud
+(log(U)=−3.3).Ingeneral,theamountofion-
+ization in the SC models is intermediate be-
+tween the values found in the first and second
+cloudsofthe2Cmodels.Anotherresult,which
+can be explained with an increasing dust con-
+tent, is the increase of the absorption Av with
+D/G.Thereisanaveragedi fferenceof0.2mag
+between the SC and 2C models. In Figure 10
+the values of the Av found by models for all
+thefittedspectraandobtainedwithallthemod-
+els are shown against the Av values estimated
+by Balmer decrement. The Av models appear
+to be slightly larger then the estimated values
+for both SC and 2C models. This e ffect can
+be explained by the fact that the Hα/Hβratio
+obtained by CLOUDY, in the vast majority of
+cases, is significantly lower than the canonical
+valueof3.1.
+The temperatures obtained by the models are
+shown in Table 8. The [ OII] and [SII] tem-
+peraturesvaluesareconsistentwiththosemea-
+suredbyRO2andRS2tratios,whilethe[ OIII]
+temperature is lower than the measured one
+Fig.10.Comparisonbetweentheobservedval-
+ues of the absorption Av and the values pro-
+ducedbyCLOUDY.
+but nonetheless consistent with the photoion-
+izationcase.
+4.3. Metallicity for star-forminggalaxies
+The determination of chemical abundances in
+the HII regions of SFs galaxies is compli-
+cated by the difficulty of reliable measures
+of the temperature of the HII clouds. The296 Rafanelli:The nature of gas andstars inthe circumnucle ar regions of AGN
+Table 8.Model temperatures for the observed and simulated spectral lines in single cloud and
+doublecloudsmodels.
+0.25 0.50 0.75 1.00
+SC TeσTeσTeσTeσ
+[OIII]82741421 85221437 88791387 93741379
+[OII]78131145 81421184 85681130 91361192
+[SII]70261038 72821163 76081220 80211527
+2C
+[OIII]10923 325711172 318911225 293711535 2853
+[OII]10020 335310390 316910361 280610736 2658
+[SII]92193715 95603567 93343294 97593170
+[OIII]292821118 92981085 95161103 96791140
+[OII]287881078 89061016 92221020 94641006
+[SII]279951418 80491353 83071450 84501480
+auroral lines, such as [OIII]4363, [SIII]6312
+and [NII]5755, used to evaluate the temper-
+ature are usually weak and in low-excitation
+metal-richHII regionsthey are often tooweak
+to be detected. For this reason, the direct
+method is almost useless in the vast majority
+ofthecases. Thepaperby(Pagelet al. (1979))
+suggested that some empirical calibrations,
+based on collection of observations, between
+the nebular line fluxes could be used to de-
+rive directly the oxygen abundance or the
+temperature of the HII regions. In more
+details, the parameter defined as R23=
+([OII]3727+[OIII]4959,5007)
+Hβwas used to directly
+measure the Oxygen abundances. This ratio
+was recalibrated using photoionization mod-
+els and confirmed through other observations.
+Nonetheless, the comparison between the re-
+sultsobtainedfromdi fferentsamplesofgalax-
+ies remains very di fficult because of the high
+heterogeneity of the model parameters and
+data. In all cases, the one-dimensionalcalibra-
+tions are systematically wrong as showed in
+(Pilyugin& Thuan(2005)). The only R23in-
+dexdoesnotremoveallthedegenerations,soa
+more general two-dimensional parametric cal-
+ibration, called the P-method, was proposed
+(Pilyugin& Thuan(2005)), introducing a new
+parameter, called the excitation parameter and
+definedas:
+P23=([OIII]4959,5007)
+([OII]3727+[OIII]4595,5007)(23)From a collection of high precision
+measurements of 700 HII regions, in
+(Pilyugin&Thuan(2005)) the authors re-
+vised the P-index calibration proposing two
+differentfunctionalformsfortheupperbranch
+and lower branch in the plane R23vs logO/H
+respectively.Fortheupperbranch:
+12+logO/H=
+(R23+726.1+842.2·P+337.5·P2)
+(85.96+82.76·P+43.98·P2+1.793·R23)
+These relations can be used to determine the
+Oxygen abundances of the HII regions in SF
+galaxies and remove the degeneracies, since
+the P-index increases from left to right in the
+figure11.
+4.3.1. Our sample
+Applying this procedure, we have found that
+our sample of SFs galaxies has P values in
+the range [0.1-0.9], consistent with the up-
+per branch sequence and that the sources are
+in the transition zone 12. At the same time,
+the estimated Oxygen abundance of our sam-
+ple is in the range [7.9-8.4]. Assuming the
+oxygen solar abundance ofN(O)
+N(H)=4.5710−4
+(Asplundet al.(2005)), the gas in this sam-
+ple of galaxies has a sub-solar abundance. If
+the metal abundances scale with the oxygen
+abundance, how it is usually assumed, the gasRafanelli:The nature of gas andstars inthe circumnuclear r egions of AGN 297
+Fig.11.Familyofcurvesinthe( R23,logO/H)
+plane labeled by di fferent values of the excita-
+tionparameter P,superimposedwithdatafrom
+different observations. The high-metallicity
+HII regions with 0 .0<P<0.3 are shown
+as opens squares, those with 0 .3<P<0.6
+as plus sign and those with 0 .6<P<0.9 as
+open circles. The low-metallicity HII regions
+with 0.5<P<0.7 are shown as filled trian-
+gles, those with 0.7<P<0.9 as crosses and
+thosewith 0.9<P<1.0asfilledcircles.
+metallicity is between 0.2 and 0.5 Z⊙in fig-
+ure 12. It needs to be stressed that these de-
+terminationsofthemetal abundancestakeinto
+account only the gas component and that, in
+orderto achievethe real metallicity,dust com-
+position has to be carefully modeled and con-
+sidered.AssumingthattheISMhassolarabun-
+dance, about 40% of the oxygenresides in sil-
+icate grains, implying thatN(O)
+N(H)=1.9410−4
+oxygen atoms per hydrogen atom are in dust
+form(Groveset al.(2006)). Evenif thereis an
+increase in the oxygen abundance, the metal-
+licity isin anycase sub-solarorsolar (theesti-
+mated value represents an upper limit because
+the dust content, usually, increases with the
+metallicity).
+5. Metallicityfromstellarpopulation
+synthesis
+The global relationship between the stel-
+lar populations and structural proper-
+ties of hot galaxies has been studied in
+Fig.12.The star-forming galaxies of the sam-
+ple used in this work in the ( R23,logO/H).
+These galaxies have excitation parameter Pin
+the interval [0.1,0.9], values that are consis-
+tent with their location in the upperbranch se-
+quenceoftheplot.
+Fig.13.Same plot as before,with the assump-
+tion that total metallicity scales linearly with
+the Oxygen abundance. In this case, the SFs
+galaxies of our sample have metallicity be-
+tween0.2·Z/summationtextand0.5·Z⊙
+(Benderet al.(1993))usingtwodi fferentmea-
+sures of the global stellar population, namely
+theMg2index and the ( B−V)0colour. The
+Mg2index,introducedin(Faberet al. (1977)),
+is defined as the the di fference in mag-
+nitude between the instrumental flux in a
+windowF2=[5156.0,5197.25]Å centered
+on the Magnesium spectral feature and the
+pseudo-continuum interpolated from two
+windows, F1=[4897.0,4958.25]Å and
+F3=[5303.0,5366.75]Å at the blue and red
+sidesrespectively:
+Mg2=F2
+F1+0.61(F3−F1)(24)
+Using population synthesis models of var-
+ious authors, the relation between the Mg2in-298 Rafanelli:The nature of gas andstars inthe circumnucle ar regions of AGN
+dex and the central velocity dispersion σ0can
+be translated in an approximate relation be-
+tween the same spectroscopic index and the
+ageandmetallicityofthestellar population:
+logMg2=0.41·logZ
+Z⊙+0.41·logt−1.00(25)
+whereZisthemetallicity, tistheagemea-
+sured in Gyr and the uncertainty on the coef-
+ficients is about±20%. This relation is rela-
+tively reliable for −0.5<log(Z/Z⊙)<0.3
+and 5<t<15 (always expressed in Gyr).
+The age and metallicities of the stellar pop-
+ulations of the normal galaxies considered in
+our work have been determined by exploiting
+this relation. In turn, one of the two param-
+eters has been derived from the fitted model
+of the continuum component of the spectrum
+obtained by STARLIGHT (whose output con-
+sist in a set of weights expressing the relative
+contribution of each single stellar population
+of given age and Zto the observed spectrum);
+this estimate of either Zorthas been used to
+calculate the other parameter as an unknown
+term in the 25 relation, using as approxima-
+tion of the measured values of the Mg2values
+theequivalentwidthoftheMgIlines,retrieved
+fromtheSDSSdatabase.Bothtemperatureand
+metallicityofthe stellarpopulationcouldhave
+been, in principle,derivedby the onlyspectral
+models obtained by STARLIGHT, but this ap-
+proach may prove to be essentially wrong due
+to the intrinsic limited coverage of the ( t,Z)
+plane of the library of template stellar popula-
+tionusedbySTARLIGHTtofitthecontinuum
+component of the observed spectra, since ev-
+ery possible final estimates of bothparameters
+is a weighted average of age and metallicities
+of the stellar population contained in the tem-
+plate library.
+The age tand metallicity Zdistributions
+for all galaxies belonging to the three di ffer-
+ent samples considered in this work and de-
+rived by STARLIGHT only are shown in fig-
+ures 14 and 15 as ”whisker and box” plots
+with, on a side, the histograms obtained from
+thesamedistributions.Asexpected,thenormal
+galaxies have significantly older stellar popu-
+lations than the other two samples of galax-
+ies, among which SFs are still younger than
+Fig.14.Distribution of mean weighted age t
+forthe three samplesofgalaxiesconsideredin
+this work, obtained using STARLIGHT. The
+”box and whisker” plots show the position of
+the first four quartiles of the age distribution
+andtheoutliers.
+Fig.15.”Box and whisker” plots of the distri-
+butionsofthe meanweightedmetallicity Zfor
+thethreesamplesofgalaxiesconsideredinthis
+work.
+Seyfert 2 galaxies. The metallicity distribu-
+tions show that normal galaxies and Seyfert
+galaxies, while almost consistent from a sta-
+tistical point of view, possess more evolved
+stellar populations than star-forming galaxies.
+On the other hand, the same distributions ob-
+tained using the measured values of the Mg2
+index and the relation 25 are shown in fig-
+ures??and 17 respectively. The comparison
+between these plots indicates that, though the
+relative relations between distributions of dif-
+ferent samples of galaxies remain unchanged,
+stellar populationsappear to be systematically
+youngerandless metal-richthanin thecase of
+theestimationbySTARLIGHTonly.Rafanelli:The nature of gas andstars inthe circumnuclear r egions of AGN 299
+Fig.16.”Box and whisker” plots of the distri-
+butionsofthemeanweightedage tforthethree
+samples of galaxies considered in this work.
+These estimates have been obtained using the
+relation 25, the mean metallicity provided by
+STARLIGHT and the observed Mg2 indexes
+ofthegalaxies.
+Fig.17.”Box and whisker” plots of the distri-
+butions of the mean weighted metallicities Z
+for the threesamplesof galaxiesconsideredin
+this work.These estimates have been obtained
+using the relation 25, the mean weighted age
+provided by STARLIGHT and the observed
+Mg2indexesofthegalaxies.
+6. Conclusions
+Inthisworkthemetalcontentofthestellarand
+gaseous components of the central regions of
+galaxies has been characterized with multiple
+methods. Three different samples of galaxies,
+namely normal galaxies, star-forming galax-
+ies and galaxies hosting an AGN (Seyfert 2),
+retrieved from the SDSS DR7 spectroscopic
+database and classified according to an origi-
+nalmethodbasedonspectroscopicdiagnostics
+(see Vaona(2009)), have been compared. For
+what the nuclear stellar populations are con-
+cerned, the presence of a strong blue contin-uum and the absence of any 4000Å break in
+thespectraoftheSF galaxiesareanindication
+ofthepresence,intheircircumnuclearregions,
+of an excess of younghot blue stars compared
+withNGandS2G.Onthecontrary,thespectra
+of the circumnuclear regions of S2G and NG
+indicate quite similar conditions: a dominat-
+ing stellar component of type F and a 4000Å
+break in NG deeper than in S2G of a factor
+∼1.5.Thisindicatesthatthepresenceofanex-
+cess of A stars in the central regions of S2G
+comparedtoNGisquitelikely,suggestingthat
+there have been in the recent past of S2 AGN
+slightlymorestarformationeventsthaninNG.
+Theanalysisofthegasofthecentralregionsof
+the Seyfert 2 galaxies has been performed us-
+inga veryrobustmodelingofthe photoioniza-
+tion processes occurring in gas clouds around
+the AGN and exploring a very large region
+of the model parameters space in order to re-
+producerealisticallytheobservedspectrafrom
+simulation.Bothsingleandtwocloudsmodels
+indicateforS2asignificantlylargermetallicity
+thansolar(1.5÷2.5Z/Z⊙with SCmodelsand
+1.0÷2.0Z/Z⊙for the 2C models). These re-
+sultssupporttheideathattheAGNfueliscom-
+posed byevolvedmaterial andthat the gashas
+a local origin in most of the observed Seyfert
+2 galaxies. The gas metal abundances for SF
+galaxieshavebeenestimatedusingthemethod
+of the P-index (Pilyugin&Thuan(2005)) and
+we have obtained results which clearly in-
+dicate sub-solar metal abundances (0.2 - 0.5
+Z⊙), confirming that young stars are fueled
+by relatively primordial and unprocessed ma-
+terial. The overall relations between the av-
+erage age and metallicity of central stellar
+populations of the three di fferent samples of
+galaxies here considered have been deter-
+mined using the Mg2absorption feature as
+proposed by(Faberet al.(1977)) and exploit-
+ing the spectral continuum componentsof ob-
+served spectra of the galaxies extracted using
+STARLIGHT (Rafanellietal. (2009)). This
+analysis qualitatively confirms that Seyfert 2
+galaxieshaveyounglow-metallicitycircumnu-
+clearstellarpopulationsfeedinginametal-rich
+environment where gas clouds are composed
+ofratherhighmetallicitymaterial;ontheother
+hand,SFgalaxieshaveveryyoungandprimor-300 Rafanelli:The nature of gas andstars inthe circumnucle ar regions of AGN
+dial central stellar populations. In conclusion,
+the comparative study of metallic content of
+stars and gas in this work indicate that Seyfert
+2 nuclei occur preferentially in a stellar envi-
+ronmentcomposedofyoungstarswhererecent
+circumnuclear star formation processes were
+active (as witnessed by the high metallicity of
+gas clouds), even if there is no compelling ev-
+idence that would connect present or past star
+formation activity to the nature of the central
+engine. The two phenomena,SF and AGN ac-
+tivity,maythereforebenotcausallyconnected
+to each other or, if so, through indirect or /and
+notclearphysicalmechanisms.
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+arXiv:1001.0339v1  [cs.IT]  2 Jan 2010Tight oracle bounds for low-rank matrix recovery
+from a minimal number of random measurements
+Emmanuel J. Cand` es1,2and Yaniv Plan3∗
+1Department of Statistics, Stanford University, Stanford, C A 94305
+2Department of Mathematics, Stanford University, Stanford, CA 94305
+3Applied and Computational Mathematics, Caltech, Pasadena, CA 91125
+December 2009
+Abstract
+This paper presents several novel theoretical results rega rding the recovery of a low-rank
+matrix from just a few measurements consisting of linear com binations of the matrix entries.
+Weshowthatproperlyconstrainednuclear-normminimizati onstablyrecoversalow-rankmatrix
+from a constant number of noisy measurements per degree of fr eedom; this seems to be the first
+result of this nature. Further, the recovery error from nois y data is within a constant of three
+targets: 1) the minimax risk, 2) an ‘oracle’ error that would be available if the column space of
+the matrix were known, and 3) a more adaptive ‘oracle’ error w hich would be available with the
+knowledge of the column space corresponding to the part of th e matrix that stands above the
+noise. Lastly, the error bounds regarding low-rank matrice s are extended to provide an error
+bound when the matrix has full rank with decaying singular va lues. The analysis in this paper
+is based on the restricted isometry property (RIP) introduc ed in [6] for vectors, and in [22] for
+matrices.
+Keywords. Matrix completion, The Dantzig selector, oracle inequalit ies, norm of random
+matrices, convex optimization and semidefinite programmin g.
+1 Introduction
+Low-rank matrix recovery is a burgeoning topic drawing the a ttention of many researchers in the
+closely related field of sparse approximation and compressi ve sensing. To draw an analogy, in the
+sparse approximation setup, the signal yis modeled as a sparse linear combination of elements from
+a dictionary Dso that
+y=Dx,
+wherexis a sparse coefficient vector. The goal is to recover x. In the matrix recovery problem, the
+signal to be recovered is a low-rank matrix M∈ /CAn1×n2, about which we have information supplied
+by means of a linear operator A: /CAn1×n2→ /CAm(typically, mis far less than n1n2),
+y=A(M), y∈ /CAm.
+∗Corresponding author: Yaniv Plan. Email: yanivplan@gmail.c om
+1In both cases, signal recovery appears to be an ill-posed pro blem because there are many more
+unknowns than equations. However, as has been shown extensi vely in the sparse-approximation
+literature, the assumption that the object of interest is sp arse makes this problem meaningful even
+when the linear system of equations is apparently underdete rmined. Further, when the measure-
+ments are corrupted by noise, we now know that by taking into a ccount the parsimony of the
+model, one can insure that the recovery error is within a log f actor of the error one would achieve
+by regressing yonto the low-dimensional subspace spanned by those columns withxi/ne}ationslash= 0; the
+squared error is adaptive, and proportional to the true dimension of the signal [3,7].
+In this paper, we derive similar results for matrix recovery . In contrast to results available in
+the literature on compressive sensing or sparse regression , we show that the error bound is within a
+constant factor (rather than a log factor) of an idealized ‘o racle’ error bound achieved by projecting
+the data onto a smaller subspace given by the ‘oracle’ (and al so within a constant of the minimax
+error bound). This error bound also applies to full-rank mat rices (which are well-approximated by
+low-rank matrices), and there appears to be no analogue of th is in the compressive sensing world.
+Another contribution of this paper is to lower the number of m easurements to stably recover
+a matrix of rank rby convex programming. It is not hard to see that we need at lea stm≥
+(n1+n2−r)rmeasurements to recover matrices of rank r, by any method whatsoever. To be sure,
+ifm <(n1+n2−r)r, we will always have two distinct matrices MandM′or rank at most r
+with the property A(M) =A(M′) no matter what Ais. To see this, fix two matrices U∈ /CAn1×r,
+V∈ /CAn2×rwith orthonormal columns, and consider the linear space of m atrices of the form
+T={UX∗−YV∗:X∈ /CAn2×r,Y∈ /CAn1×r}.
+Thedimensionof Tisr(n1+n2−r). Thus, if m <(n1+n2−r)r, thereexists M=UX∗−YV∗/ne}ationslash= 0in
+Tsuch that A(M) = 0. This proves the claim since A(UX∗) =A(YV∗) for two distinct matrices of
+rank at most r. Now a novel result of this paper is that, even without knowin g thatM∈T, one can
+stably recover Mfrom a constant times ( n1+n2)rmeasurements via nuclear-norm minimization.
+Once again, in contrast to similar results in compressive se nsing, the number of measurements
+required is within a constant of the theoretical lower limit – there is no extra log factor.
+1.1 A few applications
+Following a series of advances in the theory of low-rank matr ix recovery from undersampled linear
+measurements [5,8–10,16–18,20,22], a number of new applic ations have sprung up to join ranks
+with the already established ones. A quick survey shows that low-rank modeling is getting very
+popular in science and engineering, and we present a few ecle ctic examples to illustrate this point.
+•Quantum state tomography [15]. In quantum state tomography, a mixed quantum state
+is represented as a square positive semidefinite matrix, M(with trace 1). If Mis actually a
+pure state, then it has rank 1, and more generally, if it is app roximately pure then it will be
+well approximated by a low-rank matrix [15].
+•Face recognition [2,5]. Here the sequence of signals {yi}are images of the same face under
+varying illumination. In theory and under idealized circum stances (the images are assumed to
+be convex, Lambertian objects), these faces all reside near the same nine-dimensional linear
+subspace [2]. In practice, face-recognition techniques ba sed on the assumption that these
+images reside in a low-dimensional subspace are highly succ essful [2,5].
+2•Distance measurements. Letxi∈ /CAdbe a sequence of vectors representing several
+positions in ddimensional space, and let Mbe the matrix of squared distances between
+these vectors, Mi,j=/bardblxi−xj/bardbl2
+ℓ2. Then Mhas rank bounded by d+ 2. To see this, let
+X= [x1,x2,...,x n] be a concatenation of the positions vectors. Then, letting {ei}be the
+standard basis vectors,
+Mi,j= (ei−ej)∗X∗X(ei−ej) =−2e∗
+iX∗Xej+e∗
+i
+/BDq∗ej+e∗
+iq /BD∗ej,
+where /BDis a vector containing all ones, and qis a vector with qi=/an}bracketle{txi,xi/an}bracketri}ht. ThusM=
+X∗X+ /BDq∗+q /BD∗. The first matrix has rank bounded by dand the second two have rank
+bounded by 1. In fact, one can project out /BDq∗+q /BD∗in order to reduce the rank to d, which
+in usual applications will be 2 for positions constrained to lie in the plane, and 3 for positions
+constrained to lie somewhere in space.
+Quantum state tomography lends itself perfectly to the comp ressive sensing framework. On an
+abstract level, one sees measurements consisting of linear combinations of the unknown quantum
+stateM– inner products with certain observables which can be chose n with some flexibility by the
+physicist–andthegoalistorecoveragoodapproximationof M. Thesizeof Mgrowsexponentially
+with the number of particles in the system, so one would like t o use the structure of Mto reduce
+the number of measurements required, thus necessitating co mpressive sensing (see [15] for a more
+in depth discussion and a specific analysis of this problem).1Another more established example is
+sensor localization, in which one sees a subset of the entrie s of a distance matrix because the sensors
+have low power and can only sense reliably its distance to nea rby sensors. The goal is to fill in
+the missing entries (matrix completion). In some applicati ons of the face recognition example, one
+would see the entire set of faces (the sampling operator is th e identity), and the low-rank structure
+can be used to remove sparse errors, but otherwise arbitrari ly gross, from the data as described
+in [5] (we include this example to illustrate the different us es of the low-rank matrix model, but
+also note that it is quite different than the problem addresse d in our paper).
+1.2 Prior literature
+There has recently been an explosion of literature regardin g low-rank matrix recovery, with special
+attention given to the matrix completion subproblem (as mad e famous by the million dollar Netflix
+Prize). Several different algorithms have been proposed, wi th many drawing their roots from
+standard compressive sensing techniques [4,9,10,12,16,1 7,20–22]. For example, nuclear-norm
+minimization is highly analogous to ℓ1minimization (as a convex relaxation to an intractable
+problem), and the algorithms analyzed in this paper are anal ogous to the Dantzig Selector and the
+LASSO.
+The theory regarding the power of nuclear-norm minimizatio n in recovering low-rank matrices
+from undersampled measurements began with a paper by Recht e t al. [22], which sought to bridge
+compressive-sensing with low-rank matrix recovery via the RIP (to be defined in Section 2.1).
+Subsequently, several papers specialized the theory of nuc lear-norm minimization to the matrix
+1An interesting point about quantum state tomography is that i f one enforces the constraints trace( M) = 1 and
+M/followsequal0 then this ensures that ||M||∗= 1, and the scientist is left with a feasibility problem. In [ 15] the authors
+suggest to solve this feasibility problem by removing a const raint and then performing nuclear-norm minimization
+and they show that under certain conditions this is sufficient for exact recovery (and thus of course the solution obeys
+the unenforced constraint).
+3completion problem [5,8–10,14] which turns out to be ‘RIPle ss’; this literature is motivated by
+very clear applications such as recommender systems and net work localizations, and has required
+very sophisticated mathematical techniques.
+With the recent increase in attention given to the low-rank m atrix model, which the authors
+surmise is due to the spring of new theory, new applications a re being quickly discovered that
+deviate from the matrix completion setup (such as quantum st ate tomography [15]), and could
+benefit from a different analysis. Our paper returns to measur ement ensembles obeying the RIP as
+in [22], which are of a different nature than those involved in matrix completion. As in compressive
+sensing, theonlyknownmeasurementensembleswhichprovab lysatisfytheRIPatanearlyminimal
+sampling rate are random (such as the Gaussian measurement e nsemble in Section 2.1) Having said
+this, two comments are in order. First, our results provide a n absolute benchmark of what is
+achievable, thus allowing direct comparisons with other me thods and other sampling operators A.
+For instance, one can quantify how far the error bounds for th e RIPless matrix completion are
+from what is then known to be essentially unimprovable. Seco nd, since our results imply that
+the restricted isometry property aloneguarantees a near-optimal accuracy, we hope that this will
+encourage more applications with random ensembles, and als o encourage researchers to establish
+whether or not their measurements obey this desirable prope rty. Finally, we hope that our analysis
+offers insights for applications with nonrandom measuremen t ensembles.
+1.3 Problem setup
+We observe data yfrom the model
+y=A(M)+z, (1.1)
+whereMis an unknown n1×n2matrix,A: /CAn1×n2→ /CAmis a linear mapping, and zis anm-
+dimensional noise term. The synthesized versions of our err or bounds assume that zis a Gaussian
+vector with i.i.d. N(0,σ2) entries. The goal is to recover a good approximation of Mwhile requiring
+as few measurements as possible.
+We pause to demonstrate the form of A(X) explicitly: the ith entry of A(X) is [A(X)]i=
+/an}bracketle{tAi,X/an}bracketri}htforsomesequenceofmatrices {Ai}andwiththestandardinnerproduct /an}bracketle{tA,X/an}bracketri}ht= trace(A∗X).
+EachAican be likened to a row of a compressive sensing matrix, and in fact it can aid the intuition
+to think of Aas a large matrix, i.e. one could write A(X) as
+A(X) =
+vec(A1)
+vec(A2)
+...
+vec(Am)
+vec(X), (1.2)
+where vec( X) is a long vector obtained by stacking the columns of X. In the common matrix
+completion problem, each Aiis of the form eke∗
+jso that the ith component of A(X) is of the form
+/an}bracketle{teke∗
+j,M/an}bracketri}ht=e∗
+kMej=Mkjfor some ( j,k).
+41.4 Algorithms
+To recover M, we propose solving one of two nuclear-norm-minimization b ased algorithms. The
+first is an analogue to the Dantzig Selector from compressive sensing [7], defined as follows:
+minimize /bardblX/bardbl∗
+subject to /bardblA∗(r)/bardbl ≤λ
+r=y−A(X),(1.3)
+where the optimal solution is our estimate ˆM,/bardbl·/bardblis the operator norm and ||·||∗is its dual, i.e. the
+nuclear norm, and A∗is the adjoint of A. We call this convex program the matrix Dantzig selector .
+To pick a useful value for the parameter λin (1.3), we stipulate that the ‘true’ matrix M
+should be feasible (this is a necessary condition for our pro ofs). In other words, one should have
+/bardblA∗(z)/bardbl ≤λ; Section 2.2 provides further intuition about this require ment. In the case of Gaussian
+noise, this corresponds to λ=Cnσfor some numerical constant Cas in the following lemma.
+Lemma 1.1 Suppose zis a Gaussian vector with i.i.d. N(0,σ2)entries and let n= max(n1,n2).
+Then ifC0>4/radicalbig
+(1+δ1)log12
+/bardblA∗(z)/bardbl ≤C0√nσ, (1.4)
+with probability at least 1−2e−cnfor a fixed numerical constant c >0.
+This lemmais proved in Section 3using astandard coveringar gument. The scalar δ1isthe isometry
+constant at rank1, as defined in Section 2.1, but suffice for now that it is a very small constant
+bounded by√
+2−1 (with high probability) under the assumptions of all of our theorems.
+The optimization program (1.3) may be formulated as a semide finite program (SDP) and can
+thus be solved by any of the standard SDP solvers. To see this, we first recall that the nuclear
+norm admits an SDP characterization since /bardblX/bardbl∗is the optimal value of the SDP
+minimize/parenleftbig
+trace(W1)+trace( W2)/parenrightbig
+/2
+subject to/bracketleftbiggW1X
+X∗W2/bracketrightbigg
+/{ollowsequal0
+with optimization variables X,W1,W2∈ /CAn×n. Second, the constraint /bardblA∗(r)/bardbl ≤λis an SDP
+constraint since it can be expressed as the linear matrix ine quality (LMI)
+/bracketleftbiggλInA∗(r)
+[A∗(r)]∗λIn/bracketrightbigg
+/{ollowsequal0.
+This shows that (1.3) can be formulated as the SDP
+minimize/parenleftbig
+trace(W1)+trace( W2)/parenrightbig
+/2
+subject to
+W1X0 0
+X∗W20 0
+0 0 λInA∗(r)
+0 0 [ A∗(r)]∗λIn
+/{ollowsequal0
+r=y−A(X),
+5with optimization variables X,W1,W2∈ /CAn×n.
+However, a few algorithms have recently been developed to so lve similar nuclear-norm mini-
+mization problems without using interior-point methods wh ich work extremely efficiently in prac-
+tice [4,21]. The nuclear-norm minimization problem solved using fixed-point continuation in [21]
+is an analogue to the LASSO, and is defined as follows:
+minimize1
+2/bardblA(X)−y/bardbl2
+ℓ2+µ||X||∗. (1.5)
+Wecallthisconvexprogramthe matrix Lasso anditisthesecondconvexprogramwhosetheoretical
+properties are analyzed in this paper.
+1.5 Organization of the paper
+The results in this paper mostly concern random measurement s and random noise and so they
+hold with high probability. In Section 2.1, we show that cert ain classes of random measurements
+satisfy the RIP when only sampling a constant number of measu rements per degree of freedom. In
+Section 2.2 we present the simplest of our error bounds, demo nstrating that when the RIP holds,
+the solution to (1.3) is within a constant of the minimax risk . This error bound is refined in Section
+2.3 to provide a more adaptive error that holds improvements when the singular values of Mdecay
+below the noise level. It is shown that this error bound is wit hin a constant of the expected value
+of a certain ‘oracle’ error bound. In Section 2.4, we present an error bound handling the case when
+Mhas full rank but is well approximated by a low-rank matrix. S ection 3 contains the proofs and
+we finish with some concluding remarks in Section 4.
+1.6 Notation
+We review all notation used in this paper in order to ease read ability. We assume M∈ /CAn1×n2and
+letn= max(n1,n2). A variety of norms are used throughout this paper: ||X||∗is the nuclear norm
+(the sum of the singular values); /bardblX/bardblis the operator norm of X(the top singular value); /bardblX/bardblFis
+the Frobenius norm (the ℓ2-norm of the vector of singular values). The matrix X∗is the adjoint
+ofX, and for the linear operator A: /CAn1×n2→ /CAm,A∗: /CAm→ /CAn1×n2is the adjoint operator.
+Specifically, if [ A(X)]i=/an}bracketle{tAi,X/an}bracketri}htfor all matrices X∈ /CAn1×n2, then
+A∗(q) =m/summationdisplay
+i=1qiAi
+for all vectors q∈ /CAm.
+2 Main Results
+2.1 Matrix RIP
+The matrix version of the RIP is an integral tool in proving ou r theoretical results and we begin
+by defining the RIP in this setting and describing measuremen t ensembles that satisfy it. To
+characterize the RIP, we introduce the isometry constants o f a linear map A.
+6Definition 2.1 For each integer r= 1,2,...,n, the isometry constant δrofAis the smallest
+quantity such that
+(1−δr)/bardblX/bardbl2
+F≤ /bardblA(X)/bardbl2
+ℓ2≤(1+δr)/bardblX/bardbl2
+F (2.1)
+holds for all matrices of rank at most r.
+We say that Asatisfies the RIP at rank rifδris bounded by a sufficiently small constant between
+0 and 1, the value of which will become apparent in further sec tions (see e.g. Theorem 2.4).
+Which linear maps Asatisfy the RIP? As a quintessential example, we introduce t he Gaussian
+measurement ensemble.
+Definition 2.2 Ais a Gaussian measurement ensemble if each ‘row’ Ai,1≤i≤m, contains
+i.i.d.N(0,1/m)entries (and the Ai’s are independent from each other).
+This is of course highly analogous to the Gaussian random mat rices in compressive sensing. Our
+first result is that Gaussian measurement ensembles, along w ith many other random measurement
+ensembles, satisfy the RIP when m≥Cnr(with high probability) for some constant C >0.
+Theorem 2.3 Fix0≤δ <1and letAbe a random measurement ensemble obeying the following
+condition: for any given X∈ /CAn1×n2and any fixed 0< t <1,
+P(|/bardblA(X)/bardbl2
+ℓ2−/bardblX/bardbl2
+F|> t/bardblX/bardbl2
+F)≤Cexp(−cm) (2.2)
+for fixed constants C,c >0(which may depend on t). Then if m≥Dnr,Asatisfies the RIP with
+isometry constant δr≤δwith probability exceeding 1−Ce−dmfor fixed constants D,d >0.
+The many unspecified constants involved in the presentation of Theorem 2.3 are meant to allow
+for general use with many random measurement ensembles. How ever, to make the presentation
+more concrete we describe the constants involved in the conc entration bound (2.2) for a few special
+random measurement ensembles. If Ais a Gaussian random measurement ensemble, /bardblA(X)/bardbl2
+ℓ2is
+distributed as m−1/bardblX/bardbl2
+Ftimes a chi-squared random variable with mdegrees of freedom and (2.2)
+follows from standard concentration inequalities [19,22] . Specifically, we have
+P(|/bardblA(X)/bardbl2
+ℓ2−/bardblX/bardbl2
+F|> t/bardblX/bardbl2
+F)≤2exp/parenleftBig
+−m
+2(t2/2−t3/3)/parenrightBig
+. (2.3)
+Similarly, Asatisfies equation (2.3) in the case when each entry of each ‘r ow’Aihas i.i.d. entries
+that are equally likely to take the value 1 /√mor−1/√m, or ifAis a random projection [1,22].
+Further, Asatisfies (2.2) if the ‘rows’ Aicontain sub-Gaussian entries (properly normalized) [23],
+although in this case the constants involved depend on the pa rameters of the sub-Gaussian entries.
+In order to ascertain the strength of Theorem 2.3, note that t he number of degrees of freedom of
+ann1×n2matrixofrank risequalto r(n1+n2−r).2Thus, onemayexpectthatif m < r(n1+n2−r),
+there should be a rank- rmatrix in the null space of Aleading to a failure to achieve the lower
+bound in (2.1). In order to make this intuition rigorous (to w ithin a constant) assume without loss
+of generality that n2≥n1, and observe that the set of rank- rmatrices contains all those matrices
+restricted to have nonzero entries only in the first rrows. This is an n×rdimensional vector space
+and thus we must have m≥nror otherwise there will be a rank- rmatrix in the null space of A
+2This can be seen by counting the number of equations and unknown s in the singular value decomposition.
+7regardless of what measurements are used. (This is a similar alternative to the null-space argument
+posed in the introduction.)
+Theorem 2.3 is inspired by a similar theorem in [22][Theorem 4.2] and refines this result in two
+ways. First, it shows that one only needs a constant number of measurements per degree of freedom
+of the underlying rank- rmatrix in order to obtain the RIP at rank r(which improves on the result
+in [22] by a factor of log nand also achieves the theoretical lower bound to within a con stant).
+Second, it shows that one must only require a single concentr ation bound on A, removing another
+assumption required in [22]. A possible third benefit is that the proof follows simply and quickly
+fromaspecializedcoveringargument. Thenoveltyisinthem ethodusedtocoverlow-rankmatrices.
+2.2 The matrix Dantzig selector and the matrix Lasso are nearl y minimax
+In this section, we present our first and simplest error bound , which only requires that Asatisfies
+the RIP.
+Theorem 2.4 Assume that rank(M)≤rand let ˆMDSbe the solution to the matrix Dantzig
+selector(1.3)andˆMLbe the solution to the matrix Lasso (1.5). Ifδ4r<√
+2−1and/bardblA∗(z)/bardbl ≤λ
+then
+/bardblˆMDS−M/bardbl2
+F≤C0rλ2, (2.4)
+and ifδ4r<(3√
+2−1)/17and/bardblA∗(z)/bardbl ≤µ/2, then
+/bardblˆML−M/bardbl2
+F≤C1rµ2; (2.5)
+above,C0andC1are small constants depending only on the isometry constant δ4r. In particular,
+ifzis a Gaussian error and ˆMis either ˆMDSwithλ= 8nσ, orˆMLwithµ= 16nσ, we have
+/bardblˆM−M/bardbl2
+F≤C′
+0nrσ2(2.6)
+with probability at least 1−2e−cnfor a constant C′
+0(depending only on δ4r).
+Note that (2.6) follows from (2.4) and (2.5) simply by pluggi ng inλ,µ/2 = 8nσinto Lemma 1.1.
+In a nutshell, the error is proportional to the number of degr ees of freedom times the noise level.
+An important point is that one may expect the error to be reduc ed when further measurements
+are taken i.e. one may expect the error to be inversely propor tional to m. In fact, this is the case
+for the Gaussian measurement ensemble, but this extra facto r is absorbed into the definition in
+order to normalize the measurements so that they satisfy the RIP. If instead, each row ‘ Ai’ in the
+Gaussian measurement ensemble is defined to have i.i.d. stan dard normal entries, then by a simple
+rescaling argument (apply Theorem 2.4 to y/√m), the error bound reads
+/bardblˆM−M/bardbl2
+F≤C′
+0nrσ2/m.
+A second remark is that exploiting the low-rank structure he lps to denoise. For example, if we
+measured every entry of M(a measurement ensemble with isometry constant δr= 0), but with
+each measurement corrupted by a N(0,σ2) noise term, then taking the measurements as they are
+as the estimate of Mwould lead to an expected error equal to/BX/bardblˆM−M/bardbl2
+F=n2σ2.
+8Nuclear-norm minimization3reduces this error by a factor of about n/r.
+The strength of Theorem 2.4 is that the error bound (2.6) is ne arly optimal in the sense that
+no estimator can do essentially better without further assu mptions, as seen by lower-bounding the
+expected minimax error.
+Theorem 2.5 Ifzis a Gaussian error, then any estimator ˆM(y)obeys
+sup
+M:rank(M)≤r
+/BX/bardblˆM(y)−M/bardbl2
+F≥1
+1+δrnrσ2. (2.7)
+In other words, the minimax error over the class of matrices o f rank at most ris lower bounded by
+aboutnrσ2.
+Before continuing, it may be helpful to analyze the solution s to the matrix Dantzig selector
+and the matrix Lasso in a simple case in order to understand th e error bounds in Theorem 2.4
+intuitively, and also to understand our choice of λandµ. Suppose Ais the identity so that changing
+the notation a bit, the model is Y=M+Z, whereZis ann×nmatrix with i.i.d. Gaussian entries.
+We would like the unknown matrix Mto be a feasible point, which requires that /bardblZ/bardbl ≤λ(for
+example, if /bardblZ/bardbl> λ, we already have problems when M= 0). It is well known that the top singular
+value of a square n×nGaussian matrix, with per-entry variance σ2, is concentrated around√
+2nσ,
+and thus we require λ≥√
+2nσ(this provides a slightly sharper bound than Lemma 1.1). Let
+Tλ(X) denote the singular value thresholding operator given by
+Tλ(X) =/summationdisplay
+imax(σi(X)−λ,0)uiv∗
+i,
+whereX=/summationtext
+iσi(X)uiv∗
+iis any singular value decomposition. In this simple setting , the solution
+to (1.3) and (1.5) can be explicitly calculated, and for λ=µthey are both equal to Tλ(M+Z). If
+λis too large, then Tλ(M+Z) becomes strongly biased towards zero, and thus (loosely) λshould
+be as small as possible while still allowing Mto be feasible for the matrix Dantzig selector (1.3),
+leading to the choice λ≈√
+2nσ.
+Further, in this simple case we can calculate the error bound in a few lines. We have
+/bardblˆM−M/bardbl=/bardblTλ(Y)−Y+Z/bardbl
+≤ /bardblTλ(Y)−Y/bardbl+/bardblZ/bardbl
+≤2λ
+assuming that λ≥ /bardblZ/bardbl. Then
+/bardblˆM−M/bardbl2
+F≤ /bardblˆM−M/bardbl2rank(ˆM−M)
+≤4λ2rank(ˆM−M). (2.8)
+Once again, assuming that λ≥ /bardblZ/bardbl, we have rank( ˆM−M)≤rank(ˆM)+rank( M)≤2r. Plugging
+this in with λ=C√nσgives the error bound (2.6).
+3Of course if one sees all of the entries of the matrix plus noise , nuclear-norm minimization is unnecessary, and
+one can achieve minimax error bounds by truncating the singu lar values.
+92.3 Oracle inequalities
+Showing that an estimator achieves the minimax risk is reass uring but is sometimes not considered
+completely satisfactory. As is frequently discussed in the literature, the minimax approach focuses
+on the worst-case performance and it is quite reasonable to e xpect that for matrices of general inter-
+est, better performances are possible. In fact, a recent tre nd in statistical estimation is to compare
+the performance of an estimator with what is achievable with the help of an oracle that reveals
+extra information about the problem. A good match indicates an overall excellent performance.
+To develop an oracle bound, assume w.l.o.g. that n2≥n1so thatn=n2, and consider the
+family of estimators defined as follows: for each n1×rorthogonal matrix U, define
+ˆM[U] = argmin {/bardbly−A(ˆM)/bardblℓ2:ˆM=URfor some R}. (2.9)
+In other words, we fix the column space (the linear space spann ed by the columns of the matrix
+U), and then find the matrix with that column space which best fit s the data. Knowing the true
+matrixM, an oracle or a genie would then select the best column space t o use as to minimize the
+mean-squared error (MSE)
+inf
+U
+/BX/bardblM−ˆM[U]/bardbl2. (2.10)
+The question is whether it is possible to mimic the performan ce of the oracle and achieve a MSE
+close to (2.10) with a real estimator.
+Before giving a precise answer to this question, it is useful to determine how large the oracle
+risk is. To this end, consider a fixed orthogonal matrix U, and write the least-squares estimate
+(2.9) as
+ˆM[U] :=UHU(y),HU= (A∗
+UAU)−1A∗
+U,
+whereAUis the linear map
+AU: /CAr×n→ /CAm
+R/ma√sto→ A(UR),(2.11)
+and
+A∗
+U: /CAm→ /CAr×n
+y/ma√sto→U∗A∗(y).
+Then decompose the MSE as the sum of the squared bias and varia nce/BX/bardblM−ˆM[U]/bardbl2
+F=/bardblbias/bardbl2+variance
+=/bardbl /BXˆM[U]−M/bardbl2
+F+ /BX/bardblUHU(z)/bardbl2
+F.
+The variance term is classically equal to/BX/bardblUHU(z)/bardbl2
+F= /BX/bardblHU(z)/bardbl2
+F=σ2trace(H∗
+UHU) =σ2trace((A∗
+UAU)−1).
+Due to the restricted isometry property, all the eigenvalue s of the linear operator A∗
+UAUbelong to
+the interval [1 −δr,1+δr], see Lemma 3.12. Therefore, the variance term obeys
+σ2trace((A∗
+UAU)−1)≥1
+1+δrnrσ2.
+For the bias term, we have/BXˆM[U]−M=U(A∗
+UAU)−1A∗
+UA(M)−M,
+10which we rewrite as/BXˆM[U]−M=U(A∗
+UAU)−1A∗
+UA((I−UU∗+UU∗)M)−M
+=U(A∗
+UAU)−1A∗
+UA((I−UU∗)M)+U(A∗
+UAU)−1A∗
+UAU(U∗M)−M
+=U(A∗
+UAU)−1A∗
+UA((I−UU∗)M)−(I−UU∗)M.
+Hence, the bias is the sum of two matrices: the first has a colum n space included in the span of
+the columns of Uwhile the column space of the other is orthogonal to this span . PutPU⊥(M) =
+(I−UU∗)M; that is, PU⊥(M) is the (left) multiplication with the orthogonal projecti on matrix
+(I−UU∗). We have
+/bardbl /BXˆM[U]−M/bardbl2=/bardblU(A∗
+UAU)−1A∗
+UA(PU⊥(M))/bardbl2
+F+/bardblPU⊥(M)/bardbl2
+F
+≥ /bardblPU⊥(M)/bardbl2
+F.
+To summarize, the oracle bound obeys
+inf
+U
+/BX/bardblM−ˆM[U]/bardbl2≥inf
+U/bracketleftbigg
+/bardblPU⊥(M)/bardbl2
+2+nrσ2
+1+δr/bracketrightbigg
+.
+Now for a given dimension r, the best U– that minimizing the squared bias term or its proxy
+/bardblPU⊥(M)/bardbl2
+F– spans the top rsingular vectors of the matrix M. Denoting the singular values of
+Mbyσi(M), we obtain
+inf
+U
+/BX/bardblM−ˆM[U]/bardbl2≥inf
+r/bracketleftBigg/summationdisplay
+i>rσ2
+i(M) +1
+2nrσ2/bracketrightBigg
+,
+which for convenience we simplify to
+inf
+U
+/BX/bardblM−ˆM[U]/bardbl2≥1
+2/summationdisplay
+imin(σ2
+i(M),nσ2). (2.12)
+The right-hand side has a nice interpretation. Write the SVD ofMasM=/summationtextr
+i=1σi(M)uiv∗
+i.
+Then ifσ2
+i(M)> nσ2, one should try to estimate the rank-1 contribution σi(M)uiv∗
+iand pay the
+variance term (which is about nσ2) whereas if σ2
+i(M)≤nσ2, we should not try to estimate this
+component, and pay a squared bias term equal to σ2
+i(M). In other words, the right-hand side may
+be interpreted as an idealbias-variance trade-off.
+The main result of this section is that the matrix Dantzig Sel ector and matrix Lasso achieve
+this same ideal bias-variance trade-off to within a constant .
+Theorem 2.6 Assume that rank(M)≤rand let ˆMDSbe the solution to the matrix Dantzig
+selector(1.3)andˆMLbe the solution to the matrix Lasso (1.5). Suppose zis a Gaussian error and
+letλ= 16nσandµ= 32nσ2. Ifδ4r<√
+2−1, then
+/bardblˆMDS−M/bardbl2
+F≤C0/summationdisplay
+imin(σ2
+i(M),nσ2), (2.13)
+and ifδ4r<(3√
+2−1)/17, then
+/bardblˆML−M/bardbl2
+F≤C1/summationdisplay
+imin(σ2
+i(M),nσ2) (2.14)
+with probability at least 1−2e−cnfor constants C0andC1(depend only on δ4r).
+11In other words, not only does nuclear-norm minimization mim ic the performance that one would
+achieve with an oracle that gives the exact column space of M(as in Theorem 2.5), but in fact the
+error bound is within a constant of what one would achieve by p rojecting onto the optimal column
+space corresponding only to the significant singular values .
+While a similar result holds in the compressive sensing lite rature [7], we derive the result here
+using a novel technique. We use a middle estimate ¯Mwhich is the optimal solution to a certain
+rank-minimization problem (see Section 3) and is provably n earˆMandM. With this technique,
+the proof is a fairly simple extension of Theorem 2.4.
+2.4 Extension to full-rank matrices
+In some applications, such as sensor localization, Mhas exactly low rank, i.e. only the top few of
+its singular values are nonzero. However, in many applicati ons, such as quantum state tomography,
+Mhas full rank, but is well approximated by a low-rank matrix. In this section, we demonstrate
+an extension of the preceding error bound when Mhas full rank.
+First, suppose n1≤n2and note that a result of the form
+/bardblˆM−M/bardbl2
+F≤Cn1/summationdisplay
+i=1min(σ2
+i(M),nσ2) (2.15)
+would be impossible when undersampling Mbecause it would imply that as the noise level σ
+approaches zero, an arbitrary full-rank n×nmatrix could be exactly reconstructed from fewer
+thann2linear measurements. Instead, our result essentially spli tsMinto two parts,
+M=¯r/summationdisplay
+i=1σi(M)uiv∗
+i+n1/summationdisplay
+i=¯r+1σi(M)uiv∗
+i=M¯r+Mc
+where ¯r≈m/n, andM¯ris the best rank-¯ rapproximation to M. The error bound in the theorem
+reflects a near-optimal bias-variance trade-off in recoveri ngM¯r, but an inability to recover Mc(and
+indeed the proof essentially considers Mcas non-Gaussian noise). Note that ¯ r(n1+n2−¯r) is of
+the same order as mso that the part of the matrix which is well recovered has abou t as many
+degrees of freedom as the number of measurements. In other wo rds, even in the noiseless case
+this theorem demonstrates instance optimality i.e. the err or bound is proportional to the norm of
+the part of Mthat is irrecoverable given the number of measurements (see [25] for an analogous
+result in compressive sensing). In the noisy case there does not seem to be any current analogue
+to this error bound in compressive sensing, although the det ailed analysis can be translated to the
+compressive sensing problem and the authors are currently w riting a short paper containing this
+result.
+Theorem 2.7 FixM. Suppose that Ais sampled from the Gaussian measurement ensemble with
+m≤c0n2/log(m/n)and let¯r≤c1m/nfor some fixed numerical constants c0andc1. LetˆMbe the
+solution to the matrix Dantzig selector (1.3)withλ= 16√nσor the solution to the matrix Lasso
+(1.5)withµ= 32√nσ. Then
+/bardblˆM−M/bardbl2
+F≤C/parenleftBigg¯r/summationdisplay
+i=1min(σ2
+i(M),nσ2)+n/summationdisplay
+i=¯r+1σ2
+i(M)/parenrightBigg
+(2.16)
+12with probability greater than 1−De−dnfor fixed numerical constants C,D,d > 0. Roughly, the
+same conclusion extends to operators obeying the NNQ condit ion, see below.
+An interesting note is that in the noiseless case this error b ound provides a case of ‘instance
+optimality’
+First note that ¯ ris small enough so that the RIP holds with high probability (s ee Lemma
+2.3). However, the theorem requires more than just the RIP. T he other main requirement is a
+certain NNQ condition, which holds for Gaussian measuremen t ensembles and is introduced in
+Section 3. It is an analogous requirement to the LQ condition introduced by Wojtaszczyk [25] in
+compressive sensing. To keep the presentation of the Theore m simple, we defer the explanation of
+theNNQconditiontotheproofssectionandsimplystatethet heoremfortheGaussianmeasurement
+ensemble. However, the proof is not sensitive to the use of th is ensemble (for example sub-Gaussian
+measurements yield the same result). Many generalizations of this Theorem are available and the
+lemmas necessary to make such generalizations are spelled o ut in Section 3.
+The assumption that m≤cn2/log(m/n) seems to be an artifact of the proof technique. Indeed,
+one would not expect further measurements to negatively imp act performance. In fact, when
+m≥c′n2for a fixed constant c′, one can use Lemma 3.2 from Section 3 to derive the error
+bound (2.16) (with high probability), leaving the necessit y for a small ‘patch’ in the theory when
+cn2/log(m/n)≤n≤c′n2. However, our results intend to address the situation in whi chMis
+significantly undersampled, i.e. m≪n2, so the requirement that m≤cn2/log(m/n) should be
+intrinsic to the problem setup.
+3 Proofs
+The proofs of several of the theorems use ǫ-nets. For a set S, anǫ-netSǫwith respect to a norm
+/bardbl·/bardblsatisfies the following property: for any v∈S, there exists v0∈Sǫwith/bardblv0−v/bardbl ≤ǫ. In other
+words,Sǫapproximates Sto within distance ǫwith respect to the norm /bardbl · /bardbl. As shown in [24],
+there always exists an ǫ-netSǫsatisfying Sǫ⊂Sand
+|Sǫ| ≤Vol/parenleftbig
+S+1
+2D/parenrightbig
+Vol/parenleftbig1
+2D/parenrightbig
+where1
+2Dis anǫ/2 ball (with respect to the norm /bardbl·/bardbl) andS+1
+2D={x+y:x∈S,y∈1
+2D}. In
+particular, if Sis a unit ball in ndimensions (with respect to the norm /bardbl·/bardbl) or if it is the surface
+of the unit ball or any other subset of the unit ball, then S+1
+2Dis contained in the 1+ ǫ/2 ball,
+and the thus
+|Sǫ| ≤(1+ǫ/2)n
+(ǫ/2)n=/parenleftbigg2+ǫ
+ǫ/parenrightbiggn
+≤(3/ǫ)n
+where the last inequality follows because we always take ǫ≤1. See [24] for a more detailed
+argument. We will require in all of our proofs that Sǫ⊂S.
+3.1 Proof of Lemma 1.1
+We assume that σ= 1 without loss of generality. Put Z=A∗(z). The norm of Zis given by
+/bardblZ/bardbl= sup/an}bracketle{tw,Zv/an}bracketri}ht,
+13where the supremum is taken over all pairs of vectors on the un it sphere Sn−1. Consider a 1 /4-net
+N1/4ofSn−1with|N1/4| ≤12n. For each v,w∈Sn−1,
+/an}bracketle{tw,Zv/an}bracketri}ht=/an}bracketle{tw−w0,Zv/an}bracketri}ht+/an}bracketle{tw0,Z(v−v0)/an}bracketri}ht+/an}bracketle{tw0,Zv0/an}bracketri}ht
+≤ /bardblZ/bardbl/bardblw−w0/bardblℓ2+/bardblZ/bardbl/bardblv−v0/bardblℓ2+/an}bracketle{tw0,Zv0/an}bracketri}ht
+for some v0,w0∈ N1/4obeying/bardblv−v0/bardblℓ2≤1/4,/bardblw−w0/bardblℓ2≤1/4. Hence,
+/bardblZ/bardbl ≤2 sup
+v0,w0∈N1/4/an}bracketle{tw0,Zv0/an}bracketri}ht.
+Now for a fixed pair ( v0,w0),
+/an}bracketle{tw0,Zv0/an}bracketri}ht= trace(w∗
+0A∗(z)v0) = trace( v0w∗
+0A∗(z)) =/an}bracketle{tw0v∗
+0,A∗(z)/an}bracketri}ht=/an}bracketle{tA(w0v∗
+0),z/an}bracketri}ht.
+We deduce from this that /an}bracketle{tw0,Zv0/an}bracketri}ht ∼ N(0,/bardblA(w0v∗
+0)/bardbl2
+ℓ2). Now
+/bardblA(w0v∗
+0)/bardbl2
+ℓ2≤(1+δ1)/bardblw0v∗
+0/bardbl2
+F= (1+δ1)
+so that by a standard tail bound for Gaussian random variable s/C8(|/an}bracketle{tw0,Zv0/an}bracketri}ht| ≥λ)≤2e−1
+2λ2
+1+δ1.
+Therefore,/C8(max|/an}bracketle{tw0,Zv0/an}bracketri}ht| ≥γ/radicalbig
+(1+δ1)n)≤2|N1/4|2e−1
+2γ2n≤2e2nlog12−1
+2γ2n,
+which is bounded by 2 e−cnwithc=γ2/2−2log12 (we require γ >2√log12 so that c >0).
+3.2 Proof of Theorem 2.3
+The proof uses a covering argument, starting with the follow ing lemma.
+Lemma 3.1 (Covering number for low-rank matrices) LetSr={X∈ /CAn1×n2: rank(X)≤
+r,/bardblX/bardblF= 1}. Then there exists an ǫ-net¯Srfor the Frobenius norm obeying
+|¯Sr| ≤(9/ǫ)(n1+n2+1)r.
+ProofRecall the SVD X=UΣV∗of anyX∈Srobeying /bardblΣ/bardblF= 1. Our argument constructs
+anǫ-net forSrby covering the set of permissible U,Vand Σ. We work in the simpler case where
+n1=n2=nsince the general case is a straightforward modification.
+LetDbe the set of diagonal matrices with nonnegative diagonal en tries and Frobenius norm
+equal to one. We take ¯Dto be an ǫ/3-net for Dwith|¯D| ≤(9/ǫ)r. Next, let On,r={U∈ /CAn×r:
+U∗U=I}. To cover On,r, it is beneficial to use the ||·||1,2norm defined as
+||X||1,2= max
+i/bardblXi/bardblℓ2,
+whereXidenotes the ith column of X. LetQn,r={X∈ /CAn×r:||X||1,2≤1}. It is easy to see that
+On,r⊂Qn,rsince the columns of an orthogonal matrix are unit normed. We have seen that there is
+14anǫ/3-net¯On,rforOn,robeying|¯On,r| ≤(9/ǫ)nr. We now let ¯Sr={¯U¯Σ¯V∗:¯U,¯V∈On,r,¯Σ∈¯D},
+and remark that |¯Sr| ≤ |¯On,r|2|¯D| ≤(9/ǫ)(2n+1)r. It remains to show that for all X∈Srthere
+exists¯X∈¯Srwith/bardblX−¯X/bardblF≤ǫ.
+FixX∈Srand decompose XasX=UΣV∗as above. Then there exist ¯X=¯U¯Σ¯V∗∈¯Srwith
+¯U,¯V∈¯On,r,¯Σ∈¯Dobeying||U−¯U||1,2≤ǫ/3,||V−¯V||1,2≤ǫ/3, and/bardblΣ−¯Σ/bardblF≤ǫ/3. This gives
+/bardblX−¯X/bardblF=/bardblUΣV∗−¯U¯Σ¯V∗/bardblF
+=/bardblUΣV∗−¯UΣV∗+¯UΣV∗−¯U¯ΣV∗+¯U¯ΣV∗−¯U¯Σ¯V∗/bardblF
+≤ /bardbl(U−¯U)ΣV∗/bardblF+/bardbl¯U(Σ−¯Σ)V∗/bardblF+/bardbl¯U¯Σ(V−¯V)∗/bardblF. (3.1)
+For the first term, note that since Vis an orthogonal matrix, /bardbl(U−¯U)ΣV∗/bardblF=/bardbl(U−¯U)Σ/bardblF,
+and
+/bardbl(U−¯U)Σ/bardbl2
+F=/summationdisplay
+1≤i≤rΣ2
+i,i/bardbl¯Ui−Ui/bardbl2
+ℓ2
+≤ /bardblΣ/bardbl2
+F||U−¯U||2
+1,2
+≤(ǫ/3)2.
+Hence,/bardbl(U−¯U)ΣV∗/bardblF≤ǫ/3. The same argument gives /bardbl¯U¯Σ(V−¯V)∗/bardblF≤ǫ/3. To bound the
+middle term, observe that /bardbl¯U(Σ−¯Σ)V∗/bardblF=/bardblΣ−¯Σ/bardblF≤ǫ/3. This completes the proof.
+We now prove Theorem 2.3. It is a standard argument from this p oint, and is essentially the
+same as the proof of Lemma 4.3 in [22], but we repeat it here to k eep the paper self-contained.
+We begin by showing that Ais an approximate isometry on the covering set ¯Sr. Lemma 3.1 with
+ǫ=δ/(4√
+2) gives
+|¯Sr| ≤(36√
+2/δ)(n1+n2+1)r. (3.2)
+Then it follows from (2.2) together with the union bound that/C8/parenleftbigg
+max
+¯X∈¯Sr|/bardblA(¯X)/bardbl2
+ℓ2−/bardbl¯X/bardbl2
+F|> δ/2/parenrightbigg
+≤ |¯Sr|Ce−cm
+≤2(36√
+2/δ)(n1+n2+1)rCe−cm
+=Cexp/parenleftBig
+(n1+n2+1)rlog(36√
+2/δ)−cm/parenrightBig
+≤2exp(−dm)
+whered=c−log(36√
+2/δ)
+Cand we plugged in both requirements m≥C(n1+n2+ 1)randC >
+log(36√
+2/δ)/c.
+Now suppose that
+max
+¯X∈¯Sr|/bardblA(¯X)/bardbl2
+ℓ2−/bardbl¯X/bardbl2
+F| ≤δ/2
+(which occurs with probability at least 1 −Cexp(−dm)). We begin by showing that the upper
+bound in the RIP condition holds. Set
+κr= sup
+X∈Sr/bardblA(X)/bardblℓ2.
+15For anyX∈Sr, there exists ¯X∈¯Srwith/bardblX−¯X/bardblF≤δ/(4√
+2) and, therefore,
+/bardblA(X)/bardblℓ2≤ /bardblA(X−¯X)/bardblℓ2+/bardblA(¯X)/bardblℓ2≤ /bardblA(X−¯X)/bardblℓ2+1+δ/2. (3.3)
+Put ∆X=X−¯Xand note that rank(∆ X)≤2r. Write ∆ X= ∆X1+∆X2, where/an}bracketle{t∆X1,∆X2/an}bracketri}ht=
+0, and rank(∆ Xi)≤r,i= 1,2 (for example by splitting the SVD). Note that ∆ X1//bardbl∆X1/bardblF,
+∆X2//bardbl∆X2/bardblF∈Srand, thus,
+/bardblA(∆X)/bardblℓ2≤ /bardblA(∆X1)/bardblℓ2+/bardblA(∆X2)/bardblℓ2≤κr(/bardbl∆X1/bardblF+/bardbl∆X2/bardblF). (3.4)
+Now/bardbl∆X1/bardblF+/bardbl∆X2/bardblF≤√
+2/bardbl∆X/bardblFwhich follows from /bardbl∆X1/bardbl2
+F+/bardbl∆X2/bardbl2
+F=/bardbl∆X/bardbl2
+F. Also,
+/bardbl∆X/bardblF≤δ/(4√
+2) leading to /bardblA(∆X)/bardblℓ2≤δ/4. Plugging this into (3.3) gives
+/bardblA(X)/bardblℓ2≤κrδ/4+1+δ/2.
+Sincethisholdsforall X∈Sr, wehave κr≤κrδ/4+1+δ/2andthus κr≤(1+δ/2)/(1−δ/4)≤1+δ
+which essentially completes the upper bound. Now that this i s established, the lower bound now
+follows from
+/bardblA(X)/bardblℓ2≥ /bardblA(¯X)/bardblℓ2−/bardblA∆X/bardblℓ2≥1−δ/2−(1+δ)√
+2δ/(4√
+2)≥1−δ.
+Note that we have shown
+(1−δ)/bardblX/bardblF≤ /bardblA(X)/bardblℓ2≤(1+δ)/bardblX/bardblF,
+which can then be easily translated into the desired version of the RIP bound.
+3.3 Proof of Theorem 2.4
+We prove Theorems 2.4, 2.6, and 2.7 for the matrix Dantzig sel ector (1.3) and describe in Section
+3.7 how to extend these proofs to the matrix Lasso. We also ass ume that we are dealing with square
+matrices from this point forward ( n=n1=n2) for notational simplicity; the generalizations of the
+proofs to rectangular matrices are straightforward.
+We begin by a lemma, which applies to full-rank matrices, and contains Theorem 2.4 as a special
+case.4
+Lemma 3.2 Suppose δ4r<√
+2−1and letMrbe any rank-r matrix. Let Mc=M−Mr. Suppose
+λobeys/bardblA∗(z)/bardbl ≤λ. Then the solution ˆMto(1.3)obeys
+/bardblˆM−M/bardblF≤C0√rλ+C1/bardblMc/bardbl∗/r, (3.5)
+whereC0andC1are small constants depending only on the isometry constant δ4r.
+We shall use the fact that Amaps low-rank orthogonal matrices to approximately orthog onal
+vectors.
+Lemma 3.3 [6] For all X,X′obeying/an}bracketle{tX,X′/an}bracketri}ht= 0, andrank(X)≤r,rank(X′)≤r′,
+|/an}bracketle{tA(X),A(X′)/an}bracketri}ht| ≤δr+r′/bardblX/bardblF/bardblX′/bardblF.
+4We did not present this lemma in the main portion of the paper b ecause it does not seem to have an intuitive
+interpretation.
+16ProofThis is a simple application of the parallelogram identity. Suppose without loss of generality
+thatXandX′have unit Frobenius norms. Then
+(1−δr+r′)/bardblX±X′/bardbl2
+F≤ /bardblA(X±X′)/bardbl2
+F≤(1+δr+r′)/bardblX±X′/bardbl2
+F,
+since rank( X±X′)≤r+r′. We have /bardblX±X′/bardbl2
+F=/bardblX/bardbl2
+F+/bardblX′/bardbl2
+F= 2 and the parallelogram
+identity asserts that
+|/an}bracketle{tA(X),A(X′)/an}bracketri}ht|=1
+4/vextendsingle/vextendsingle/bardblA(X+X′)/bardbl2
+F−/bardblA(X−X′)/bardbl2
+F/vextendsingle/vextendsingle≤δr+r′,
+which concludes the proof.
+The proof of Lemma 3.2 parallels that of Cand` es and Tao about the recovery of nearly sparse
+vectors from a limited number of measurements [7]. It is also inspired by the work of Fazel, Recht,
+Cand` es and Parrilo [13,22]. Set H=ˆM−Mand observe that by the triangle inequality,
+/bardblA∗A(H)/bardbl ≤ /bardblA∗(A(ˆM)−y)/bardbl+/bardblA∗(y−A(M))/bardbl ≤2λ, (3.6)
+sinceMis feasible for the problem (1.3). Decompose Has
+H=H0+Hc,
+where rank( H0)≤2r,MrH∗
+c= 0 and M∗
+rHc= 0 (see [22]). We have
+/bardblM+H/bardbl∗≥ /bardblMr+Hc/bardbl∗−/bardblMc/bardbl∗−/bardblH0/bardbl∗
+=/bardblMr/bardbl∗+/bardblHc/bardbl∗−/bardblMc/bardbl∗−/bardblH0/bardbl∗.
+Since by definition, /bardblM+H/bardbl∗≤ /bardblM/bardbl∗≤ /bardblMr/bardbl∗+/bardblMc/bardbl∗, this gives
+/bardblHc/bardbl∗≤ /bardblH0/bardbl∗+2/bardblMc/bardbl∗. (3.7)
+Next, we use a classical estimate developed in [11] (see also [22]). Let Hc=Udiag(/vector σ)V∗be the
+SVD ofHc, where/vector σis the list of ordered singular values (not to be confused wit h the noise standard
+deviation). Decompose Hcinto a sum of matrices H1,H2,..., each of rank at most 2 ras follows.
+For each idefine the index set Ii={2r(i−1)+1,...,2ri}, and let Hi:=UIidiag(/vector σIi)V∗
+Ii; that is,
+H1is the part of Hccorresponding to the 2 rlargest singular values, H2is the part corresponding
+to the next 2 rlargest and so on. A now standard computation shows that
+/summationdisplay
+j≥2/bardblHj/bardblF≤1√
+2r/bardblHc/bardbl∗, (3.8)
+and thus/summationdisplay
+j≥2/bardblHj/bardblF≤ /bardblH0/bardblF+/radicalbigg
+2
+r/bardblMc/bardbl∗
+since/bardblH0/bardbl∗≤√
+2r/bardblH0/bardblFby Cauchy-Schwarz.
+Now the restricted isometry property gives
+(1−δ4r)/bardblH0+H1/bardbl2
+F≤ /bardblA(H0+H1)/bardbl2
+F, (3.9)
+17and observe that
+/bardblA(H0+H1)/bardbl2
+F=/an}bracketle{tA(H0+H1),A(H−/summationdisplay
+j≥2Hj)/an}bracketri}ht.
+We first argue that
+/an}bracketle{tA(H0+H1),A(H)/an}bracketri}ht ≤ /bardblH0+H1/bardblF√
+4r/bardblA∗A(H)/bardbl. (3.10)
+To see why this is true, let UΣV∗be the reduced SVD of H0+H1in which UandVaren×r′,
+and Σ is r′×r′withr′= rank(H0+H1)≤4r. We have
+/an}bracketle{tA(H0+H1),A(H)/an}bracketri}ht=/an}bracketle{tH0+H1,A∗A(H)/an}bracketri}ht
+=/an}bracketle{tΣ,U∗[A∗A(H)]V/an}bracketri}ht
+≤ /bardblΣ/bardblF/bardblU∗[A∗A(H)]V/bardblF
+=/bardblH0+H1/bardblF/bardblU∗[A∗A(H)]V/bardblF.
+The claim follows from /bardblU∗[A∗A(H)]V/bardblF≤√
+r′/bardblA∗A(H)/bardbl, which holds since U∗[A∗A(H)]Vis
+anr′×r′matrix with spectral norm bounded by /bardblA∗A(H)/bardbl. Second, Lemma 3.3 implies that for
+j≥2
+/an}bracketle{tA(H0),A(Hj)/an}bracketri}ht ≤δ4r/bardblH0/bardblF/bardblHj/bardblF, (3.11)
+and similarly with H1in place of H0. Note that because H0is orthogonal to H1, we have that
+/bardblH0+H1/bardbl2
+F=/bardblH0/bardbl2
+F+/bardblH1/bardbl2
+Fand thus /bardblH0/bardblF+/bardblH1/bardblF≤√
+2/bardblH0+H1/bardblF. This gives
+/an}bracketle{tA(H0+H1),A(Hj)/an}bracketri}ht ≤√
+2δ4r/bardblH0+H1/bardblF/bardblHj/bardblF. (3.12)
+Taken together, (3.9), (3.10) and (3.12) yield
+(1−δ4r)/bardblH0+H1/bardblF≤√
+4r/bardblA∗A(H)/bardbl+√
+2δ4r/summationdisplay
+j≥2/bardblHj/bardblF
+≤√
+4r/bardblA∗A(H)/bardbl+√
+2δ4r/bardblH0/bardblF+2δ4r√r/bardblMc/bardbl∗.
+To conclude, we have that
+/bardblH0+H1/bardblF≤C1√
+4r/bardblA∗A(H)/bardbl+C12δ4r√r/bardblMc/bardbl∗, C1= 1/[1−(√
+2+1)δ4r],
+provided that C1>0. Our claim (2.4) then follows from (3.6) together with
+/bardblH/bardblF≤ /bardblH0+H1/bardblF+/summationdisplay
+j≥2/bardblHj/bardblF≤2/bardblH0+H1/bardblF+/radicalbigg
+2
+r/bardblMc/bardbl∗.
+3.4 Proof of Theorem 2.4
+Theorem 2.4 follows by simply plugging Mr=Minto Theorem 3.2. To generalize the results, note
+that there are only two requirements on M,Aandyused in the proof.
+• /bardblA∗(A(M)−y)/bardbl ≤λ
+18•rank(M) =randδ4r<√
+2−1.
+Thus, the steps above also prove the following Lemma which is useful in proving Theorem 2.6.
+Lemma 3.4 Assume that Xis of rank at most rand that δ4r<√
+2−1. Suppose λobeys/bardblA∗(y−
+A(X))/bardbl ≤λ. Then the solution ˆMto(1.3)obeys
+/bardblˆM−X/bardbl2
+F≤C0rλ2. (3.13)
+whereC0is a small constant depending only on the isometry constant δ4r.
+3.5 Proof of Theorem 2.6
+In this section, λ= 16nσ2and we take as given that /bardblA∗(z)/bardbl ≤λ/2 (and thus, by Lemma 1.1, the
+end result holds with probability at least 1 −2e−cn). The novelty in this proof – the way it differs
+from analogous proofs in compressive sensing – is in the use o f a middle estimate ¯M. Define Kas
+K(X;M)≡γrank(X)+/bardblA(X)−A(M)/bardbl2
+ℓ2, γ=λ2
+4(1+δ1)(3.14)
+and let ¯M= argminXK(X,M). In words, ¯Machieves a compromise between goodness of fit
+and parsimony in the model with noiseless data. The factor γcould be replaced by λ2, but the
+derivations are cleanest in the present form. We begin by bou nding the distance between Mand
+¯Musing the RIP, and obtain
+/bardbl¯M−M/bardbl2
+ℓ2≤1
+1−δ2r/bardblA(¯M)−A(M)/bardbl2
+ℓ2(3.15)
+where the use of the isometry constant δ2rfollows from the fact that rank( ¯M)≤rank(M).
+We now develop a bound about /bardblˆM−¯M/bardbl2
+ℓ2. Lemma 3.5 gives
+/bardblA∗(y−A(¯M))/bardbl ≤ /bardblA∗(z)/bardbl+/bardblA∗A(M−¯M)/bardbl ≤λ,
+i.e.¯Mis feasible for (1.3). Also, rank( ¯M)≤rank(M) and, thus, plugging ¯Minto Lemma 3.4 gives
+/bardblˆM−¯M/bardbl2
+F≤Cλ2rank(¯M).
+Combining this with (3.15) gives
+/bardblˆM−M/bardbl2
+F≤2/bardblˆM−¯M/bardbl2
+F+2/bardbl¯M−M/bardbl2
+F
+≤2Cλ2rank(¯M)+2
+1−δ2r/bardblA(¯M)−A(M)/bardbl2
+ℓ2
+≤C′K(¯M;M) (3.16)
+whereC′= max(8 C(1+δ1),2/(1−δ2r)).
+Now¯Mis the minimizer of K(·;M), and so K(¯M;M)≤K(M0;M), where
+M0=/summationdisplay
+iσi(M)1{σi(M)>λ}uiv∗
+i. (3.17)
+19We have
+K(M0;M)≤γr/summationdisplay
+i=11{σi(M)>λ}+/bardblA(M−M0)/bardbl2
+ℓ2
+≤γr/summationdisplay
+i=11{σi(M)>λ}+(1+δr)/bardblM−M0/bardbl2
+F
+≤(1+δr)r/summationdisplay
+i=1min(λ2,σ2
+i(M)).
+In conclusion, the proof follows from λ= 16nσ2since
+/bardblˆM−M/bardbl2
+F≤C′r/summationdisplay
+i=1min(λ2,σ2
+i(M)).
+Lemma 3.5 The minimizer ¯Mobeys
+/bardblA∗A(¯M−M)/bardbl ≤λ/2.
+ProofSuppose not. Then there are unit-normed vectors u,v∈ /CAnobeying
+/an}bracketle{tuv∗,A∗A(¯M−M)/an}bracketri}ht> λ/2.
+We construct the rank-1 perturbation M′=¯M−αuv∗,α=/an}bracketle{tuv∗,A∗A(¯M−M)/an}bracketri}ht//bardblA(uv∗)/bardbl2
+ℓ2, and
+claim that K(M′:M)< K(¯M;M) thus providing the contradiction. We have
+/bardblA(M′−M)/bardbl2
+ℓ2=/bardblA(¯M−M)/bardbl2
+ℓ2−2α/an}bracketle{tA(uv∗),A(¯M−M)/an}bracketri}ht+α2/bardblA(uv∗)/bardbl2
+ℓ2
+=/bardblA(¯M−M)/bardbl2
+ℓ2−α2/bardblA(uv∗)/bardbl2
+ℓ2.
+It then follows that
+K(M′;M)≤γ(rank(M)+1)+/bardblA(¯M−M)/bardbl2
+ℓ2−α2/bardblA(uv∗)/bardbl2
+ℓ2
+=K(¯M;M)+γ−α2/bardblA(uv∗)/bardbl2
+ℓ2.
+However, /bardblA(uv∗)/bardbl2
+ℓ2≤(1+δ1)/bardbluv∗/bardbl2
+F= 1+δ1and,therefore, α2/bardblA(uv∗)/bardbl2
+ℓ2> γsince/an}bracketle{tuv∗,A∗A(¯M−
+M)/an}bracketri}ht> λ/2.
+3.6 Proof of Theorem 2.7
+Three useful lemmas are established in the course of the proo f of this more involved result, and we
+would like to point out that these can be used as powerful erro r bounds themselves. Throughout
+the proof, Cis a constant that may depend on δ4ronly, and whose value may change from line to
+line. An important fact to keep in mind is that under the assum ptions of the theorem, δ4¯rcan be
+bounded, with high probability, by an arbitrarily small con stant depending on the size of the scalar
+c1appearing in the condition ¯ r≤c1m/n. This is a consequence of Theorem 2.3. In particular,
+δ4¯r≤(√
+2−1)/2 with probability at least 1 −De−dm.
+20Lemma 3.6 Let¯MandM0be defined via (3.14)and(3.17), and set
+r= max(rank( ¯M),rank(M0)).
+Suppose that δ4r<√
+2−1and that λobeys/bardblA∗(z)/bardbl ≤λ/2. Then the solution ˆMto(1.3)obeys
+/bardblˆM−M/bardbl2
+F≤C0/parenleftBiggn/summationdisplay
+i=1min(λ2,σ2
+i(M))+/bardblA(M−M0)/bardbl2
+ℓ2/parenrightBigg
+, (3.18)
+whereC0is a small constant depending only on the isometry constant δ4r.
+ProofThe proof is essentially the same as that of Theorem 2.6, and s o we quickly go through the
+main steps. Set Mc=M−M0so thatMconly contains the singular values below the noise level.
+First,
+/bardbl¯M−M/bardbl2
+F≤2/bardbl¯M−M0/bardbl2
+F+2/bardblMc/bardbl2
+F
+≤2
+1−δ2r/bardblA(¯M−M0)/bardbl2
+ℓ2+2/bardblMc/bardbl2
+F
+≤4
+1−δ2r/bardblA(¯M−M)/bardbl2
+ℓ2+4
+1−δ2r/bardblA(Mc)/bardbl2
+ℓ2+2/bardblMc/bardbl2
+F.
+Second, we bound /bardblˆM−¯M/bardblFusing the exact same steps as in the proof of Theorem 2.6, and
+obtain
+/bardblˆM−¯M/bardbl2
+F≤Crλ2.
+Hence,
+/bardblˆM−M/bardbl2
+F≤C(K(¯M;M)+/bardblA(Mc)/bardbl2
+ℓ2+/bardblMc/bardbl2
+F).
+Finally, use K(¯M;M)≤K(M0;M) as before, and simplify to attain (3.18).
+The factor /bardblA(Mc)/bardbl2
+Fin (3.18) prevents us from stating the bound as the near-idea l bias-
+variance-trade-off (2.15). However, many random measureme nt ensembles obeying the RIP are
+also unlikely to drastically change the norm of anyfixed matrix (see (2.2)). Thus, we expect that
+/bardblA(Mc)/bardblℓ2≈ /bardblMc/bardblℓ2with high probability. Specifically, if Aobeys (2.2), then
+/bardblA(Mc)/bardbl2
+ℓ2≤1.5/bardblMc/bardbl2
+F (3.19)
+with probability at least 1 −De−cmfor fixed constants D,c. An important point here is that
+this inequality only holds (with high probability) when Mcis fixed, and Ais chosen randomly
+(independently). In the worst-case-scenario, one could ha ve
+/bardblA(Mc)/bardblℓ2=/bardblA/bardbl/bardblMc/bardblF
+where/bardblA/bardblis the operator norm of A. Thus we emphasize that the bound holds with high proba-
+bility for a given Mverifying our conditions, but may not hold uniformly over al l suchM’s.
+Returning to the proof, (3.24) together with
+/bardblMc/bardbl2
+F=n/summationdisplay
+i=1σ2
+i(M)1{σi(M)<λ}
+give the following lemma:
+21Lemma 3.7 FixMand suppose Aobeys(2.2). Then under the assumptions of Lemma 3.6, the
+solution ˆMto(1.3)obeys
+/bardblˆM−M/bardbl2
+F≤C0n/summationdisplay
+i=1min(λ2,σ2
+i(M)) (3.20)
+with probability at least 1−De−cnwhereC0is a small constant depending only on the isometry
+constant δ4r, andc,Dare fixed constants.
+The above two lemmas require a bound on the rank of M0. However, as the noise level ap-
+proaches zero, the rank of M0approaches the rank of M, which can be as large as the dimension.
+This requires further analysis, and in order to provide theo retical error bounds when the noise level
+is low (and Mhas full rank, say), a certain property of many measurement o perators is useful. We
+call it the NNQ property, and is inspired by a similar propert y from compressive sensing, see [25].
+Definition 3.8 (NNQ) LetBn×n
+∗be the set of n×nmatrices with nuclear norm bounded 1. Let
+Bm
+ℓ2be the standard ℓ2unit ball for vectors in /CAm. We say that Asatisfies NNQ( α) if
+A(Bn×n
+∗)⊇αBm
+ℓ2. (3.21)
+This condition may appear cryptic at the moment. To give a tas te of why it may be useful, note
+that Lemma 3.2 includes ||M−Mr||∗as part of the error bound. The point is that using the NNQ
+condition, we can find a proxy for M−Mr, which we call ˜M, satisfying A(˜M) =A(M−Mr), but
+also||˜M||∗≤ /bardblA(M−Mr)/bardblℓ2/α. Before continuing this line of thought, we prove that Gauss ian
+measurement ensembles satisfy NNQ( µ/radicalbig
+n/m) with high probability for some fixed constant µ >0.
+Theorem 3.9 (NNQ for Gaussian measurements) Suppose Ais a Gaussian measurement
+ensemble and m≤Cn2/log(m/n)for some fixed constant C >0. ThenAsatisfies NNQ (µ/radicalbig
+n/m)
+with probability at least 1−3e−cnfor fixed constants candµ.
+ProofPutα=µ/radicalbig
+n/mand suppose Adoes not satisfy NNQ( α). Then there exists a vector
+x∈ /CAmwith/bardblx/bardblℓ2= 1 such that
+/an}bracketle{tA(M),x/an}bracketri}ht ≤αfor allM∈Bn×n
+∗.
+In particular,
+/bardblA∗(x)/bardbl ≤α.
+Let¯Bm
+ℓ2⊂Bm
+ℓ2be anα-net forBm
+ℓ2with|¯Bm
+ℓ2| ≤(3/α)m. Then there exists ¯ x∈¯Bm
+ℓ2with/bardbl¯x−x/bardblℓ2≤
+αsatisfying
+/bardblA∗(¯x)/bardbl ≤ /bardblA∗(¯x−x)/bardbl+/bardblA(x)/bardbl ≤ /an}bracketle{tuv∗,A∗(¯x−x)/an}bracketri}ht+α,
+whereu,vare the left and right singular vectors of A∗(¯x−x) corresponding to the top singular
+value. Then
+/an}bracketle{tuv∗,A∗(¯x−x)/an}bracketri}ht=/an}bracketle{tA(uv∗),¯x−x/an}bracketri}ht ≤ /bardblA(uv∗)/bardblℓ2/bardbl¯x−x/bardblℓ2≤/radicalbig
+1+δ1α
+and, therefore,
+/bardblA∗(¯x)/bardbl ≤3α
+22assuming δ1≤1 (this occurs with probability at least 1 −2e−cnwhenm≥Cnfor fixed constants
+c,C).
+We will provide the contradiction by showing that with high p robability, /bardblA∗(¯x)/bardbl>3α, for
+all ¯x∈¯Bn×n
+∗. For each ¯ x,A∗(¯x) is equal in distribution to1√mZ, where Zis a matrix with
+i.i.d. standard normal entries. Let Zibe theith column of Z. Then/C8(/bardblA∗(¯x)/bardbl ≤3α)≤ /C8(/bardblZ/bardbl ≤3√mα)
+≤ /C8( max
+i=1,...,n/bardblZi/bardblℓ2≤3√mα);
+the second step uses the fact that the operator norm of Zis always larger or equal to the ℓ2norm
+of any column. With α=µ/radicalbig
+n/mand using the fact that the columns are independent, this yie lds/C8(/bardblA∗(¯x)/bardbl ≤3α)≤ /C8(/bardblZ1/bardbl2
+ℓ2≤9µ2n)n.
+However, /bardblZ1/bardbl2
+ℓ2is a chi-squared random variable with ndegrees of freedom, and can be bounded
+using a standard concentration of measure result [19]:/C8(/bardblZ1/bardbl2
+ℓ2−n≤ −t√
+2n)≤e−t2/2.
+Hence,/C8(/bardblA∗(¯x)/bardbl ≤3α)≤e−cn2,
+wherec= (1−9µ2)2/4 (we require µ <1/3 here). Thus, by the union bound,/C8/parenleftBigg
+min
+¯x∈¯Bm
+ℓ2/bardblA∗(¯x)/bardbl ≤3α/parenrightBigg
+≤(3/α)me−cn2= exp/parenleftbigg
+mlog/parenleftbigg3√m
+µ√n/parenrightbigg
+−cn2/parenrightbigg
+≤e−c′n2
+provided that m≤Cn2/log(m/n) for fixed constants, C,c′. The theorem is established.
+Notethattheprecedingproofcanberepeatedwhen Aisasub-Gaussianmeasurementensemble;
+the only difference is that Zabove will contain sub-Gaussian entries, rather than Gauss ian entries.
+Using the NNQ property, we can now bound the error when the noi se level is low; this does not
+involve any condition on the rank of M0, and does not involve a term in the bound depending on
+||M−M0||∗.
+Lemma 3.10 Suppose that Asatisfies NNQ( µ/radicalbig
+n/m) for a fixed constant µand that /bardblA∗(z)/bardbl ≤λ.
+Let¯r≥cm/nfor some fixed numerical constant c, and suppose that that δ4¯r≤1
+2(√
+2−1). Let
+M¯r=¯r/summationdisplay
+i=1σi(M)uiv∗
+i.
+LetˆMbe the solution to (1.3). Then
+/bardblˆM−M/bardblF≤C(λ√
+¯r+/bardblA(M−M¯r)/bardblℓ2)+/bardblM−M¯r/bardblF. (3.22)
+ProofSetMc=M−M¯r=/summationtextn
+i=¯r+1σi(M)uiv∗
+i. The NNQ( α) property with α=µ/radicalbig
+n/mgives
+A(Mc) =A(˜M)
+23for some ˜Mobeying||˜M||∗≤ /bardblA(Mc)/bardblℓ2/α. We also take note of the identity A(M¯r+˜M) =A(M).
+It follows from Lemma 3.2 that
+/bardblˆM−(M¯r+˜M)/bardblF≤C(λ√
+¯r+||˜M||∗/√
+¯r).
+Plugging in ||˜M||∗≤ /bardblA(Mc)/bardblℓ2/α, along with ¯ r≥cm/n, we obtain
+/bardblˆM−(M¯r+˜M)/bardblF≤C(λ√
+¯r+/bardblA(Mc)/bardblℓ2).
+Therefore,
+/bardblˆM−M/bardblF≤C(λ√
+¯r+/bardblA(Mc)/bardblℓ2)+/bardbl˜M/bardblF+/bardblMc/bardblF. (3.23)
+It remains to bound /bardbl˜M/bardblF. As in the proof of Lemma 3.2, decompose ˜Mas˜M=˜M1+˜M2+...
+sothat˜M1correspondstothelargest ¯ rsingularvaluesof ˜M,˜M2correspondswiththenext ¯ rlargest,
+and so on. Just as before,
+/bardbl˜M/bardblF≤/summationdisplay
+i/bardbl˜Mi/bardblF≤ /bardbl˜M1/bardblF+||˜M||∗/√
+¯r.
+We now bound /bardbl˜M1/bardblF. By the RIP,
+/bardbl˜M1/bardblF≤1√1−δ¯r/bardblA(˜M1)/bardblℓ2
+=1√1−δ¯r/parenleftBig
+/bardblA(˜M)−/summationdisplay
+i≥2A(˜Mi)/bardblℓ2/parenrightBig
+≤1√1−δ¯r/parenleftBig
+/bardblA(˜M)/bardblℓ2+/summationdisplay
+i≥2/bardblA(˜Mi)/bardblℓ2/parenrightBig
+.
+By the RIP again, /bardblA(˜Mi)/bardblℓ2≤√1+δ¯r/bardbl˜Mi/bardblF, and so
+/summationdisplay
+i≥2/bardblA(˜Mi)/bardblℓ2≤/radicalbig
+1+δ¯r/summationdisplay
+i≥2/bardbl˜Mi/bardblF≤/radicalbig
+1+δ¯r||˜M||∗√¯r.
+This together with A(˜M) =A(Mc) give
+/bardbl˜M/bardblF≤/radicalbigg
+1+δr
+1−δr/parenleftBig
+/bardblA(Mc)/bardblℓ2+||˜M||∗√¯r/parenrightBig
+.
+However, ||˜M||∗≤ /bardblA(M)/bardblℓ2/α≤√¯r/bardblA(M)/bardblℓ2/(µ√c) and, therefore,
+/bardbl˜M/bardblF≤C/bardblA(Mc)/bardblℓ2.
+Inserting this into (3.23) completes the proof of the lemma.
+We are now in position to prove our main theorem concerning th e recovery of matrices with
+decaying singular values (Theorem 2.7). There are three cas es to consider depending on the number
+of singular values of Mstanding above the noise level. In each case, we need the ineq uality
+/bardblA(Mc)/bardbl2
+F≤1.5/bardblMc/bardbl2
+F (3.24)
+24which holds with probability at least 1 −De−cnfor any measurement ensemble satisfying (2.2)
+(including the Gaussian measurement ensemble). Put λ= 16nσ2and recall the definition of M0:
+M0=n/summationdisplay
+i=1σi(M)1{σi(M)≥λ}uiv∗
+i
+whose rank is exactly the number of singular values of Mabove the noise level. There are three
+cases to consider depending mostly on the interplay between the singular values of Mand the noise
+level.
+Case 1: high noise level
+Suppose K(M0;M)≤λ2
+4(1+δ1)¯r. Then rank( M0)≤¯rand rank( ¯M)≤¯rby definition of ¯M. Hence,
+Lemma 3.7 gives
+/bardblˆM−M/bardbl2
+F≤Cn/summationdisplay
+i=1min(nσ2,σ2
+i(M))
+with probability at least 1 −2e−cn.
+Case 2: low noise level
+Suppose K(M0;M)>λ2
+4(1+δ1)¯rand rank( M0)≥¯r. It follows from (2.2) that
+/bardblA(M−M¯r)/bardbl2
+ℓ2≤√
+1.5/bardblM−M¯r/bardbl2
+F (3.25)
+withprobabilityatleast1 −De−cn. Now, fortheGaussianmeasurementensemble, therequireme nts
+of Lemma 3.10 are met with probability at least 1 −Ce−cn. Combining (3.25) with Lemma 3.10
+yields
+/bardblˆM−M/bardblF≤C(λ√
+¯r+/bardblM−M¯r/bardblF)
+and thus
+/bardblˆM−M/bardbl2
+F≤2C2(λ2¯r+/bardblM−M¯r/bardbl2
+F) = 2C2/parenleftBigg¯r/summationdisplay
+i=1min(λ2,σ2
+i(M))+n/summationdisplay
+i=¯r+1σ2
+i(M)/parenrightBigg
+.
+Sinceλ= 16nσ2, this is (3.23).
+Case 3: medium noise level
+Suppose K(M0;M)>λ2
+4(1+δ1)¯rand rank( M0)<¯r. As in Case 2, we have
+/bardblˆM−M/bardbl2
+F≤2C2(λ2¯r+/bardblM−M¯r/bardbl2
+F).
+Fromλ2¯r <4(1+δ1)K(M0;M), it follows that
+/bardblˆM−M/bardbl2
+F≤2C2(λ2rank(M0)+4(1+ δ1)/bardblA(M−M0)/bardbl2
+ℓ2+/bardblM−M¯r/bardbl2
+F).
+We also have /bardblA(M−M0)/bardbl2
+ℓ2≤1.5/bardblM−M0/bardbl2
+Fwith probability at least 1 −De−cn. Inserting
+this bound into the previous equation, along with /bardblM−M¯r/bardblF≤ /bardblM−M0/bardblF, gives the desired
+conclusion.
+These three cases comprise all possibilities. In short, the proof of Theorem 2.7 is complete.
+253.7 Extension of proofs to the solution to the Lasso (1.5)
+In the sparse regression setup, Bickel et al. [3] showed that the Dantzig Selector and the Lasso have
+analogous properties, leading to analogous error bounds. T he analogies still hold in the low-rank
+matrix recovery problem (for similar reasons). In fact, all of the theorems above also hold for the
+solution to (1.5) aside from a shift in those constants appea ring in the assumptions, and those
+appearing in the error bounds. To see this, note that our proo fs only used two crucial properties
+aboutˆM:
+1.||ˆM||∗≤ ||M||∗,
+2./bardblA∗(A(ˆM)−y)/bardbl ≤λ.
+The second property automatically holds for the solution to (1.5) (but with λreplaced by µ). This
+follows from the optimality conditions which states that A∗(y−A(ˆM))∈∂/bardblˆM/bardbl∗where/bardblˆM/bardbl∗is
+the family of subgradients to the nuclear norm at the minimiz er. Formally, let UΣV∗be the SVD
+ofˆM, then
+A∗(y−A(ˆM)) =λ(UV∗+W)
+for some Wobeying/bardblW/bardbl ≤1 andU∗W= 0,WV= 0 (see e.g. [10]). Hence, the second property
+follows from /bardblUV∗+W/bardbl ≤1.
+The first property does not necessarily hold for the matrix La sso, but a close enough approx-
+imation is verified (this is analogous to an argument made in [ 3]). Suppose that /bardblA∗(z)/bardbl ≤c0µ
+for a small constant c0(which, by Lemma 1.1, holds with high probability for Gaussia n noise if
+µ=Cnσ2). Then since ˆMminimizes (1.5), we have
+1
+2/bardblA(ˆM)−y/bardbl2
+ℓ2+µ||ˆM||∗≤1
+2/bardblA(M)−y/bardbl2
+ℓ2+µ||M||∗.
+Plug iny=A(M)+zand rearrange terms to give
+µ||ˆM||∗≤1
+2/bardblA(ˆM−M)/bardbl2
+ℓ2+µ||ˆM||∗≤ /an}bracketle{tˆM−M,A∗(z)/an}bracketri}ht+µ||M||∗.
+Since the nuclear norm and the operator norm are dual to each o ther, we have /an}bracketle{tˆM−M,A∗(z)/an}bracketri}ht ≤
+||ˆM−M||∗·/bardblA∗(z)/bardbl ≤c0µ||H||∗, where we use the notation H=ˆM−Mas in the proof of Lemma
+3.2. This gives
+||ˆM||∗≤c0||H||∗+||M||∗,
+which nearly is the first property. When c0is chosen to be a small constant, this factor has no
+essential detrimental effects on the proof. In particular, ( 3.7) in the proof of Lemma 3.2 is replaced
+by
+(1−c0)||Hc||∗≤(1+c0)||H0||∗+2||Mc||∗.
+In particular, for c0= 1/2,
+||Hc||∗≤3||H0||∗+4||Mc||∗.
+The rest of the proofs follow.
+263.8 Proof of Theorem 2.5
+We begin with a well-known lemma which gives the minimax risk for estimating the vector x∈ /CAn
+from the data y∈ /CAmand the linear model
+y=Ax+z, (3.26)
+whereA∈ /CAm×nand thezi’s are i.i.d. N(0,σ2).
+Lemma 3.11 Letλi(A∗A)be the eigenvalues of the matrix A∗A. Then
+inf
+ˆxsup
+x∈ /CAn
+/BX/bardblˆx−x/bardbl2=σ2trace((A∗A)−1) =/summationdisplay
+iσ2
+λi(A∗A). (3.27)
+In particular, if one of the eigenvalues vanishes (as in the c ase in which m < n), then the minimax
+risk is unbounded.
+ProofSuppose first that Ais the identity matrix. Then it is well known that the minimax risk is
+nσ2and is achieved by ˆ x=y. To see this, recall that for any prior on x, the minimax risk is lower
+bounded by the Bayes risk. Consider then the prior which assu mes that all the components of x
+are i.i.d. N(0,τ2). Then the Bayes’ estimator for this prior is the shrinkage e stimate given by
+ˆxi= /BX(xi|yi) =τ2
+τ2+σ2yi,
+and the Bayes risk is
+n/summationdisplay
+i=1
+/BX(ˆxi−xi)2=nσ2τ2
+τ2+σ2.
+Clearly, as τ→ ∞, the lower bound on the minimax risk goes to nσ2. Since this quantity is the
+risk of the maximum-likelihood estimate y, this proves the claim. Note that by a simple rescaling
+argument, this also proves that the minimax risk for estimat ingxfromyi=dixi+ziis/summationtextn
+i=11/d2
+i.
+We can now prove (3.27). We will assume that m≥nfor simplicity since for m < n, the
+minimax risk is unbounded. Let UΣV∗be the SVD of A, whereUism×n, Σ isn×nandVis
+n×n. All the information about xis inU∗y, and so we may just assume that the data is given by
+y′=U∗y= ΣV∗x+U∗z.
+Nowz′=U∗zis a Gaussian vector with i.i.d. N(0,σ2) components. Further, set x′=V∗x. Since
+Vis an orthogonal matrix, the minimax risk for estimating xorx′is the same and, therefore, our
+problem is that of computing the minimax risk for estimating x′from
+y′= Σx′+z′.
+Since Σ is a diagonal matrix with diagonal elements/radicalbig
+λi(A∗A), our previous result applies and
+establishes (3.27).
+We are now in position to prove Theorem 2.5. The set of rank- rmatrices is (much) larger than
+the set of matrices of the form
+M=UR,
+27whereUis afixedorthogonal n×rmatrix with orthonormal columns (note that the matrices of
+this form have a fixed r-dimensional column space). Thus,
+inf
+ˆMsup
+M:rank(M)=r
+/BX/bardblˆM−M/bardbl2
+F≥inf
+ˆMsup
+M:M=UR
+/BX/bardblˆM−M/bardbl2
+F.
+Knowing that M=URfor some unknown r×nmatrixR, one can of course limit ourselves to
+estimators of the form ˆM=UˆR, and since/BX/bardblˆM−M/bardbl2
+F= /BX/bardblUˆR−UR/bardbl2
+F= /BX/bardblˆR−R/bardbl2
+F,
+the minimax risk is lower bounded that by of estimating Rfrom the data
+y=AU(R)+z,
+whereAUis the linear map (2.11). We then apply Lemma 3.11 to conclude that the minimax rate
+is lower bounded by
+/summationdisplay
+iσ2
+λi(A∗
+UAU).
+The claim follows from the simple lemma below.
+Lemma 3.12 LetUbe ann×rmatrix with orthonormal columns. Then all the eigenvalues o f
+A∗
+UAUbelong to the interval [1−δr,1+δr].
+ProofBy definition,
+λmin(A∗
+UAU) = inf
+/bardblR/bardblF≤1/an}bracketle{tR,A∗
+UAU(R)/an}bracketri}ht
+and similarly for λmax(A∗
+UAU) with a sup in place of inf. Since
+/an}bracketle{tR,A∗
+UAU(R)/an}bracketri}ht=/bardblAU(R)/bardbl2
+ℓ2=/bardblA(UR)/bardbl2,
+the claim follows from
+(1−δr)/bardblUR/bardbl2
+F≤ /bardblA(UR)/bardbl2≤(1+δr)/bardblUR/bardbl2
+F,
+which is valid since rank( UR)≤rtogether with /bardblUR/bardbl2
+F=/bardblR/bardbl2
+F.
+4 Discussion
+Using RIP-based analysis, this paper has shown that low-ran k matrices can be stably recovered via
+nuclear-norm minimization from nearly the minimal possibl e number of linear samples. Further,
+the error bound is within a constant of the expected minimax e rror, and of an expected oracle
+error, and extends to the case when Mhas full rank.
+Thisworkdiffersfromthemainthrustoftherecentliteratur eonlow-rankmatrixrecovery,which
+has concentrated on the ‘RIPless’ matrix completion proble m. An interesting observation regarding
+matrix completion is that when the measurements are randoml y chosen entries of M, one requires
+at least about nrlognmeasurements to recover Mby any method when rank( M) =O(1) [8,10].
+28In contrast, this paper shows that on the order of nrmeasurements are enough provided these are
+sufficiently random.
+The popularity of the matrix completion model stems from the fact that this setup currently
+dominates the applications of low-rank matrix recovery. Th ere are far fewer applications in which
+the measurements are random linear combinations of many ent ries ofM(quantum-state tomog-
+raphy is a notable application though). As a great deal of att ention is given to low-rank matrix
+modeling these days, with new applications being discovere d all the time, this may change rapidly.
+We hope that our theory encourages further applications and research in this direction.
+References
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+30
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diff --git a/1001.0340.txt b/1001.0340.txt
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+arXiv:1001.0340v3  [cs.NA]  16 Mar 2010Computing the Least Fixed Point
+of Positive Polynomial Systems⋆
+Javier Esparza, Stefan Kiefer, and Michael Luttenberger
+Institut f¨ ur Informatik, Technische Universit¨ at M¨ unch en, 85748 Garching, Germany
+{esparza,kiefer,luttenbe }@model.in.tum.de
+Abstract. We consider equation systems of the form X1=f1(X1,...,X n),...,X n=
+fn(X1,...,X n) wheref1,...,f nare polynomials with positivereal coefficients. In vector
+form we denote such an equation system by X=f(X) and call fa system of positive
+polynomials, short SPP. Equation systems of this kind appea r naturally in the analysis
+of stochastic models like stochastic context-free grammar s (with numerous applications
+to natural language processing and computational biology) , probabilistic programs with
+procedures, web-surfing models with back buttons, and branc hing processes. The least
+nonnegative solution µfof an SPP equation X=f(X) is of central interest for these
+models. Etessami and Yannakakis [EY09] have suggested a par ticular version of Newton’s
+method to approximate µf.
+We extend a result of Etessami and Yannakakis and show that Ne wton’s method starting
+at0always converges to µf. We obtain lower bounds on the convergence speed of the
+method. For so-called strongly connected SPPs we prove the existence of a threshold
+kf∈Nsuch that for every i≥0 the (kf+i)-th iteration of Newton’s method has at least
+ivalid bits of µf. The proof yields an explicit bound for kfdepending only on syntactic
+parameters of f. We further show that for arbitrary SPP equations Newton’s m ethod still
+converges linearly: there exists a threshold kfand anαf>0 such that for every i≥0
+the (kf+αf·i)-th iteration of Newton’s method has at least ivalid bits of µf. The proof
+yields an explicit bound for αf; the bound is exponential in the number of equations in
+X=f(X), but we also show that it is essentially optimal. The proof d oes not yield
+any bound for kf, it only proves its existence. Constructing a bound for kfis still an
+open problem. Finally, we also provide a geometric interpre tation of Newton’s method for
+SPPs.
+1 Introduction
+We consider equation systems of the form
+X1=f1(X1,...,X n)
+...
+Xn=fn(X1,...,X n)
+wheref1,...,f nare polynomials with positivereal coefficients. In vector form we denote such
+an equation system by X=f(X). The vector fof polynomials is called a system of positive
+polynomials , orSPPfor short.Figure 1 showsthe graph ofa 2-dimensional SPP equatio n system
+X=f(X).
+Equation systems of this kind appear naturally in the analysis of stoc hastic context-free
+grammars (with numerous applications to natural language proces sing [MS99,GJ02] and com-
+putational biology [SBH+94,DEKM98,DE04,KH03]), probabilistic programs with procedures
+⋆This work was partially supported by the DFG project Algorithms for Software Model Checking .PSfrag replacements
+X1=f1(X1,X2)
+X2=f2(X1,X2)
+µf0
+0.20.4
+0.50.60.8
+1−0.4
+−0.2
+0.2
+0.4
+0.6
+0.2
+X1X2
+Fig.1.Graphs of the equations X1=f1(X1,X2) andX2=f2(X1,X2) withf1(X1,X2) =X1X2+1
+4
+andf2(X1,X2) =1
+6X2
+1+1
+9X1X2+2
+9X2
+2+3
+8. There are two real solutions in R2, the least one is labelled
+withµf.
+[EKM04,BKS05,EY09,EY05a,EKM05,EY05b,EY05c], and web-surfing mod els with back but-
+tons [FKK+00,FKK+01]. More generally, they play an important rˆ ole in the theory of branch-
+ing processes [Har63,AN72], stochastic processes describing the evolution of a po pulation whose
+individuals can die and reproduce. The probability of extinction of the population is the least
+solution of such a system, a result whose history goes back to [WG74 ].
+Since SPPs have positive coefficients, x≤x′impliesf(x)≤f(x′) forx,x′∈Rn
+≥0, i.e., the
+functions f1,...,f nare monotonic. This allows us to apply Kleene’s theorem (see for insta nce
+[Kui97]), and conclude that a feasible system X=f(X), i.e., one having at least one nonnega-
+tive solution, has a smallest solution µf. It follows easily from standard Galois theory that µf
+can be irrational and non-expressible by radicals. The problem of de ciding, given an SPP and a
+rational vector vencoded in binary, whether µf≤vholds, is known to be in PSPACE, and to
+be at least as hard as two relevant problems: SQUARE-ROOT-SUM an d PosSLP. SQUARE-
+ROOT-SUM is a well-known problem of computational geometry, whos e membership in NP is a
+long standing open question. PosSLP is the problem of deciding, giving a division-free straight-
+line program, whether it produces a positive integer (see [EY09] for more details). PosSLP has
+been recently shown to play a central rˆ ole in understanding the Blu m-Shub-Smale model of
+computation, where each single arithmetic operation over the reals can be carried out exactly
+and in constant time [ABKPM09].
+For the practical applications mentioned above the complexity of de termining if µfexceeds
+a given bound is less relevant than the complexity of, given i∈N, computing ivalid bits ofµf,
+i.e., computing a vector vsuch that/vextendsingle/vextendsingleµfj−vj/vextendsingle/vextendsingle//vextendsingle/vextendsingleµfj/vextendsingle/vextendsingle≤2−ifor every 1 ≤j≤n. Given an SPP
+fandi∈N, deciding whether the first ibits of a component of µf, sayµf1, are 0, remains
+as hard as SQUARE-ROOT-SUM and PosSLP. The reason is that in [EY0 9] both problems
+are reduced to the following one: given ǫ >0 and an SPP ffor which it is known that either
+µf1= 1 orµf1≤ǫ, decide which of the two is the case. So it suffices to take ǫ= 2−i.
+In this paper we study the problem of computing ivalid bits in the Blum-Shub-Smale model.
+Since the least fixed point of a feasible SPP fis a solution of F(X) =0forF(X) =f(X)−X,
+we can try to apply (the multivariate version of) Newton’s method [OR70]: starting at some
+x(0)∈Rn(we use uppercase to denote variables and lowercase to denote va lues), compute the
+2sequence
+x(k+1):=x(k)−(F′(x(k)))−1F(x(k))
+whereF′(X) is the Jacobian matrix of partial derivatives. A first difficulty is that the method
+mightnotevenbewell-defined,because F′(x(k))couldbesingularforsome k.However,Etessami
+and Yannakakis have recently studied SPPs derived from probabilist ic pushdown automata
+(actually, from an equivalent model called recursive Markov chains) [EY09], and shown that a
+particular version of Newton’s method always converges, namely a v ersion which decomposes
+the SPP into strongly connected components (SCCs)1and applies Newton’s method to them
+in a bottom-up fashion. Our first result generalizes Etessami and Y annakakis’: the ordinary
+Newton method converges for arbitrary SPPs, provided that the y are clean (which can be easily
+achieved).
+While these results show that Newton’s method can be an adequate a lgorithm for solving
+SPP equations, they provide no information on the number of iterat ions needed to compute i
+valid bits. To the best of our knowledge (and perhaps surprisingly), the rest of the literature
+does not contain relevant information either: it has not considered SPPs explicitly, and the
+existing results have very limited interest for SPPs, since they do no t apply even for very simple
+and relevant SPP cases (see Related work below). In this paper we obtain upper bounds on
+the number of iterations that Newton’s method needs to produce ivalid bits, first for strongly
+connected and then for arbitrary SPP equations.
+For strongly connected SPP equations X=f(X) we prove the existence of a threshold kf
+such that for every i≥0 the (kf+i)-th iteration of Newton’s method has at least ivalid bits
+ofµf. So, loosely speaking, after kfiterations Newton’s method is guaranteed to compute at
+least 1 new bit of the solution per iteration; we say that Newton’s met hod converges at least
+linearlywithrate 1. Moreover, we show that the threshold kfcan be chosen as
+kf=⌈4mn+3nmax{0,−logµmin}⌉
+wherenis the number of polynomials of the strongly connected SPP, mis such that all coeffi-
+cients of the SPP can be given as ratios of m-bit integers, and µminis the minimal component
+of the least fixed point µf.
+Notice that kfdepends on µf, which is what Newton’s method should compute. For this
+reason we also obtain bounds on kfdepending only on mandn. We show that for arbitrary
+strongly connected SPP equations kf= 4mn2nis also a valid threshold. For SPP equations
+coming from stochastic models, such as the ones listed above, we do far better. First, we show
+that if every procedure has a non-zero probability of terminating ( a condition that always holds
+for back-button processes [FKK+00,FKK+01]), then a valid threshold is kf= 2m(n+1). Since
+one iteration requires O(n3) arithmetic operations in a system of nequations, we immediately
+obtain an upper bound on the time complexity of Newton’s method in th e Blum-Shub-Smale
+model: for back-button processes, ivalid bits can be computed in time O(mn4+in3). Second,
+we observe that, since x(k)≤x(k+1)≤µfholds for every k≥0, as Newton’s method proceeds
+it provides better and better lower bounds for µminand thus for kf. We exhibit an SPP for
+which, using this fact and our theorem, we can prove that no compo nent of the solution reaches
+the value 1. This cannot be proved by just computing more iteration s, no matter how many.
+For general SPP equations, not necessarily strongly connected, we show that Newton’s
+method still converges linearly. Formally, we show the existence of a threshold kfand a real
+number 0 < αfsuch that for every i≥0 the (kf+αf·i)-th iteration of Newton’s method has at
+leastivalid bits of µf. So, loosely speaking, after the first kfiterations Newton’s method com-
+putes new bits of µfat a rate of at least 1 /αfbits per iteration. Unlike the strongly connected
+1Loosely speaking, a subset of variables and their associate d equations form an SCC, if the value of
+any variable in the subset influences the value of all variabl es in the subset, see §2 for details.
+3case, the proof does not provide any bound on the threshold kf: with respect to the threshold
+the proof is non-constructive, and finding a bound on kfis still an open problem. However, the
+proof does provide a bound for αf, it shows αf≤n·2nfor an SPP with npolynomials. We also
+exhibit a family of SPPs for which more than i·2n−1iterations are needed to compute ibits.
+Soαf≤n·2nfor every system f, and there exists a family of systems for which 2n−1≤αf.
+Finally,the lastresultofthe paperconcernsthe geometricinterpr etationofNewton’smethod
+for SPP equations. We show that, loosely speaking, the Newton app roximants stay within the
+hypervolumelimitedbythehypersurfacescorrespondingtoeachin dividualequation.Thismeans
+that a simple geometric intuition of how Newton’s method works, extr acted from the case of
+2-dimensional SPPs, is also correct for arbitrary dimensions. As a b yproduct we also obtain a
+new variant of Newton’s method.
+Related work. There is a large body of literature on the convergence speed of New ton’s method
+for arbitrarysystems ofdifferentiable functions. A comprehensiv e referenceis Ortegaand Rhein-
+boldt’s book [OR70] (see also Chapter 8 of Ortega’s course [Ort72] o r Chapter 5 of [Kel95] for a
+brief summary). Several theorems (for instance Theorem 8.1.10 o f [Ort72]) prove that the num-
+ber of valid bits grows linearly, superlinearly, or even exponentially in t he number of iterations,
+but only under the hypothesis that F′(x) is non-singular everywhere, in a neighborhood of µf,
+or at least at the point µfitself. However, the matrix F′(µf) can be singular for an SPP, even
+for the 1-dimensional SPP f(X) = 1/2X2+1/2.
+The general case in which F′(µf) may be singular for the solution µfthat the method
+converges to has been thoroughly studied. In a seminal paper [Red 78], Reddien shows that
+under certain conditions, the main ones being that the kernel of F′(µf) has dimension 1 and
+that the initial point is close enough to the solution, Newton’s method gains 1 bit per iteration.
+Decker and Kelly obtain results for kernels of arbitrary dimension, b ut they require a certain
+linear map B(X) to be non-singularfor all x/\⌉}a⊔io\slash=0[DK80]. Griewank observes in [GO81] that the
+non-singularity of B(X) is in fact a strong condition which, in particular, can only be satisfied
+by kernels of even dimension. He presents a weaker sufficient condit ion for linear convergence
+requiring B(X)tobe non-singularonlyatthe initialpoint x(0),i.e., itonlyrequirestomake“the
+right guess” for x(0). Unfortunately, none of these results can be directly applied to ar bitrary
+SPPs. The possible dimensions of the kernel of F′(µf) for an SPP f(X) are to the best of our
+knowledgeunknown,anddecidingthisquestionseemsashardastho serelatedtotheconvergence
+rate2. Griewank’s result does not apply to the decomposed Newton’s meth od either because the
+mapping B(x(0)) is always singular for x(0)=0.
+Kantorovich’s famous theorem (see e.g. Theorem 8.2.6 of [OR70] and [PP80] for an im-
+provement) guarantees global convergence and only requires F′to be non-singular at x(0).
+However, it also requires to find a Lipschitz constant for F′on a suitable region and some
+other bounds on F′. These latter conditions are far too restrictive for the application s men-
+tioned above. For instance, the stochastic context-free gramm ars whose associated SPPs satisfy
+Kantorovich’s conditions cannot exhibit two productions X→aYZandW→εsuch that
+Prob(X→aYZ)·Prob(W→ε)≥1/4. This class of grammars is too contrived to be of use.
+Summarizing, while the convergence of Newton’s method for system s of differentiable func-
+tions has been intensely studied, the case of SPPs does not seem to have been considered yet.
+The results obtained for other classes have very limited applicability t o SPPs: either they do
+not apply at all, or only apply to contrived SPP subclasses. Moreover , these results only provide
+information about the growth rate of the number of accurate bits , but not about the number
+itself. For the class of strongly connected SPPs, our thresholds le ad toexplicitlower bounds for
+the number of accurate bits depending only on syntactical parame ters: the number of equations
+2More precisely, SPPs with kernels of arbitrary dimension ex ist, but the cases we know of can be
+trivially reduced to SPPs with kernels of dimension 1.
+4and the size of the coefficients. For arbitrary SPPs we prove the ex istence of a threshold, while
+finding explicit lower bounds remains an open problem.
+Structure of the paper. §2 defines SPPs and briefly describes their applications to stochastic
+systems. §3 presents a short summary of our main theorems. §4 proves some fundamental
+properties of Newton’s method for SPP equations. §5 and§6 contain our results on the con-
+vergence speed for strongly connected and general SPP equatio ns, respectively. §7 shows that
+the bounds are essentially tight. §8 presents our results about the geometrical interpretation of
+Newton’s method, and §9 contains conclusions.
+2 Preliminaries
+In this section we introduce our notation used in the following and for malize the concepts
+mentioned in the introduction.
+2.1 Notation
+As usual, RandNdenote the set of real, respectively natural numbers. We assume 0∈N.Rn
+denotes the set of n-dimensional real valued column vectors and Rn
+≥0the subset of vectors with
+nonnegative components. We use bold letters for vectors, e.g. x∈Rn, where we assume that
+xhas the components x1,...,x n. Similarly, the i-th component of a function f:Rn→Rnis
+denoted by fi. We define 0:= (0,...,0)⊤and1:= (1,...,1)⊤where the superscript⊤indicates
+the transpose of a vector or a matrix. Let /ba∇⌈bl·/ba∇⌈bldenote some norm on Rn. Sometimes we use
+explicitly the maximum norm /ba∇⌈bl·/ba∇⌈bl∞with/ba∇⌈blx/ba∇⌈bl∞:= max 1≤i≤n|xi|.
+The partial order ≤onRnis defined as usual by setting x≤yifxi≤yifor all 1 ≤i≤n.
+Similarly, x<yifx≤yandx/\⌉}a⊔io\slash=y. Finally, we write x≺yifxi< yifor all 1≤i≤n, i.e., if
+every component of xis smaller than the corresponding component of y.
+WeuseX1,...,X nasvariableidentifiersandarrangethemintothevector X.Inthefollowing
+nalways denotes the number of variables, i.e., the dimension of X. While x,y,...denote
+arbitrary elements in Rn, we write Xif we want to emphasize that a function is given w.r.t.
+these variables. Hence, f(X) represents the function itself, whereas f(x) denotes its value for
+somex∈Rn.
+IfS⊆ {1,...,n}is a set of components and xa vector, then by xSwe mean the vector
+obtained by restricting xto the components in S.
+LetS⊆ {1,...,n}andS={1,...,n}\S. Given a function f(X) and a vector xS, then
+f[S/xS] is obtained by replacing, for each s∈S, each occurrence of Xsbyxsand removing
+thes-component. In other words, if f(X) =f(XS,XS) thenf[S/xS](yS) =fS(xS,yS). For
+instance, if f(X1,X2) = (X1X2+1
+2,X2
+2+1
+5)⊤, thenf[{2}/1
+2] :R→R,X1/maps⊔o→1
+2X1+1
+2.
+Rm×ndenotes the set of matrices having mrows and ncolumns. The transpose of a vector
+or matrix is indicated by the superscript⊤. The identity matrix of Rn×nis denoted by Id.
+Theformal Neumann series ofA∈Rn×nis defined by A∗=/summationtext
+k∈NAk. It is well-known
+thatA∗exists if and only if the spectral radius of Ais less than 1, i.e. max {|λ| |C∋
+λis an eigenvalue of A}<1. IfA∗exists then A∗= (Id−A)−1.
+The partial derivative of a function f(X) :Rn→Rw.r.t. the variable Xiis denoted by
+∂Xif. The gradient ∇foff(X) is then defined to be the (row) vector
+∇f:= (∂X1f,...,∂ Xnf).
+5TheJacobian of a function f(X) withf:Rn→Rmis the matrix f′(X) defined by
+f′(X) =
+∂X1f1... ∂Xnf1
+......
+∂X1fm... ∂Xnfm
+,
+i.e., thei-th row of f′is the gradient of fi.
+2.2 Systems of Positive Polynomials
+Definition 2.1. A function f(X)withf:Rn
+≥0→Rn
+≥0is asystem of positive polynomials
+(SPP), if every component fi(X)is a polynomial in the variables X1,...,X nwith coefficients
+inR≥0. We call an SPP f(X) feasible ify=f(y)for some y∈Rn
+≥0. An SPP is called linear
+(resp.quadratic ) if all polynomials have degree at most 1(resp.2).
+Fact 2.2. Every SPP fis monotone on Rn
+≥0, i.e. for 0≤x≤ywe have f(x)≤f(y).
+We will need the following lemma, a version of Taylor’s theorem.
+Lemma 2.3 (Taylor). Letfbe an SPP and x,u≥0. Then
+f(x)+f′(x)u≤f(x+u)≤f(x)+f′(x+u)u.
+Proof.It suffices to show this for a multivariate polynomial f(X) with nonnegative coefficients.
+Consider g(t) =f(x+tu). We then have
+f(x+u) =g(1) =g(0)+/integraldisplay1
+0g′(s)ds=f(x)+/integraldisplay1
+0f′(x+su)uds.
+The result follows as f′(x)≤f′(x+su)≤f′(x+u) fors∈[0,1]. ⊓ ⊔
+Since every SPP is continuous, Kleene’s fixed-point theorem (see e.g . [Kui97]) applies.
+Theorem 2.4 (Kleene’s fixed-point theorem). Every feasible SPP fhas a least fixed point
+µfinRn
+≥0i.e.,µf=f(µf)and, in addition, y=f(y)impliesµf≤y. Moreover, the sequence
+(κ(k)
+f)k∈Nwithκ(k)
+f=fk(0)(wherefkdenotes the k-fold iteration of f) is monotonically
+increasing with respect to ≤(i.e.κ(k)
+f≤κ(k+1)
+f)and converges to µf.
+In the following we call ( κ(k)
+f)k∈NtheKleene sequence off(X), and drop the subscript
+whenever fis clear from the context. Similarly, we sometimes write µinstead of µf.
+An SPP f(X) iscleanif for all variables Xithere is a k∈Nsuch that κ(k)
+i>0. It is easy
+to see that we have κ(k)
+i= 0 for all k∈Nifκ(n)
+i= 0. So we can “clean” an SPP f(X) in time
+linear in the size of fby determining the components iwithκ(n)
+i= 0 and removing them.
+We will also need the notion of dependence between variables.
+Definition 2.5. A polynomial f(X) contains a variable Xiif∂Xif(X)is not the zero-
+polynomial.
+Definition 2.6. Letf(X)be an SPP. A component idepends directly on a component kif
+fi(X)contains Xk. A component idepends onkif either idepends directly on kor there is
+a component jsuch that idepends on jandjdepends on k. The components {1,...,n}can
+be partitioned into strongly connected components (SCCs) w here an SCC Sis a maximal set of
+components such that each component in Sdepends on each other component in S. An SCC
+is called trivialif it consists of a single component that does not depend on it self. An SPP is
+strongly connected (short: an scSPP) if{1,...,n}is a non-trivial SCC.
+62.3 Convergence Speed
+We will analyze the convergence speed of Newton’s method. To this e nd we need the notion of
+valid bits .
+Definition 2.7. Letfbe a feasible SPP. A vector xhasivalid bits of the least fixed point µf
+if/vextendsingle/vextendsingleµfj−xj/vextendsingle/vextendsingle
+/vextendsingle/vextendsingleµfj/vextendsingle/vextendsingle≤2−i
+for every 1≤j≤n. Let(x(k))k∈Nbe a sequence with 0≤x(k)≤µf. Then the convergence
+orderβ:N→Nof the sequence (x(k))k∈Nis defined as follows: β(k)is the greatest natural
+numberisuch that x(k)hasivalid bits (or ∞if such a greatest number does not exist). We will
+always mean the convergence order of the Newton sequence (ν(k))k∈N, unless explicitly stated
+otherwise.
+We say that a sequence has linear, exponential, logarithmic, etc. co nvergence order if the
+function β(k) grows linearly, exponentially, or logarithmically in k, respectively.
+Remark 2.8. Our definition of convergence order differs from the one commonly u sed in numer-
+ical analysis (see e.g. [OR70]), where “quadratic convergence” or “ Q-quadratic convergence”
+means that the error e′of the new approximant (its distance to the least fixed point accord ing
+to some norm) is bounded by c·e2, whereeis the error of the old approximant and c >0 is
+some constant. We consider our notion more natural from a compu tational point of view, since
+it directly relates the number of iterations to the accuracy of the a pproximation. Notice that
+“quadratic convergence” implies exponential convergence order in the sense of Definition 2.7. In
+the following we avoid the notion of “quadratic convergence”.
+2.4 Stochastic Models
+As mentioned in the introduction, severalproblems concerningsto chastic models can be reduced
+to problems about the least fixed point µfof an SPP f. In these cases, µfis a vector of
+probabilities, and so µf≤1.
+Probabilistic Pushdown Automata Our study of SPPs was initially motivated by the ver-
+ification of probabilistic pushdown automata. A probabilistic pushdown automaton (pPDA) is
+a tupleP= (Q,Γ,δ,Prob) where Qis a finite set of control states ,Γis a finite stack al-
+phabet,δ⊆Q×Γ×Q×Γ∗is a finite transition relation (we write pX ֒− →qαinstead of
+(p,X,q,α )∈δ), andProbis a function which to each transition pX ֒− →qαassigns its probability
+Prob(pX ֒− →qα)∈(0,1]sothat forall p∈QandX∈Γwehave/summationtext
+pX֒− →qαProb(pX ֒− →qα) = 1.
+We write pXx֒− →qαinstead of Prob(pX ֒− →qα) =x. Aconfiguration ofPis a pair qw, whereq
+is a control state and w∈Γ∗is astack content . A pPDA Pnaturally induces a possibly infinite
+Markov chain with the configurations as states and transitions give n by:pXβx֒− →qαβfor every
+β∈Γ∗iffpXx֒− →qα. We assume w.l.o.g. that if pXx֒− →qαis a transition then |α| ≤2.
+pPDAs and the equivalent model of recursive Markov chains have be en very thoroughly
+studied [EKM04,BKS05,EY09,EY05a,EKM05,EY05b,EY05c]. This work has shown that the
+key to the analysis of pPDAs are the termination probabilities [pXq], where pandqare states,
+andXis a stack letter, defined as follows (see e.g. [EKM04] for a more form al definition):
+[pXq] is the probability that, starting at the configuration pX, the pPDA eventually reaches the
+7configuration qε(empty stack). It is not difficult to show that the vector of these p robabilities
+is the least solution of the SPP equation system containing the equat ion
+/a\}b∇ack⌉⊔l⌉{⊔pXq/a\}b∇ack⌉⊔∇i}h⊔=/summationdisplay
+pXx֒− →rYZx·/summationdisplay
+t∈Q/a\}b∇ack⌉⊔l⌉{⊔rYt/a\}b∇ack⌉⊔∇i}h⊔·/a\}b∇ack⌉⊔l⌉{⊔tZq/a\}b∇ack⌉⊔∇i}h⊔+/summationdisplay
+pXx֒− →rYx·/a\}b∇ack⌉⊔l⌉{⊔rYq/a\}b∇ack⌉⊔∇i}h⊔+/summationdisplay
+pXx֒− →qεx
+for each triple ( p,X,q). Call this quadratic SPP the termination SPP of the pPDA (we assume
+that termination SPPs are clean, and it is easy to see that they are a lways feasible).
+Strict pPDAs and Back-Button Processes A pPDA is strictif for all pX∈Q×Γand all
+q∈Qthe transition relation contains a pop-rule pXx֒− →qǫfor some x >0. Essentially, strict
+pPDAs model programs in which every procedure has at least one te rminating execution that
+does not call any other procedure. The termination SPP of a strict pPDA satisfies f(0)≻0.
+In [FKK+00,FKK+01] a class of stochastic processes is introduced to model the beh avior
+of web-surfers who from the current webpage Acan decide either to follow a link to another
+page, say B, with probability ℓAB, or to press the “back button” with nonzero probability bA.
+These back-button processes correspondto a veryspecial clas s ofstrict pPDAs havingone single
+control state (which in the following we omit), and rules of the form AbA֒− →ε(press the back
+button from A) orAℓAB֒− − →BA(follow the link from AtoB, remembering Aas destination of
+pressing the back button at B). The termination probabilities are given by an SPP equation
+system containing the equation
+/a\}b∇ack⌉⊔l⌉{⊔A/a\}b∇ack⌉⊔∇i}h⊔=bA+/summationdisplay
+AℓAB֒− − →BAℓAB/a\}b∇ack⌉⊔l⌉{⊔B/a\}b∇ack⌉⊔∇i}h⊔/a\}b∇ack⌉⊔l⌉{⊔A/a\}b∇ack⌉⊔∇i}h⊔=bA+/a\}b∇ack⌉⊔l⌉{⊔A/a\}b∇ack⌉⊔∇i}h⊔/summationdisplay
+AℓAB֒− − →BAℓAB/a\}b∇ack⌉⊔l⌉{⊔B/a\}b∇ack⌉⊔∇i}h⊔
+foreverywebpage A. In [FKK+00,FKK+01] thosetermination probabilitiesarecalled revocation
+probabilities. The revocation probability of a page Ais the probability that, when currently
+visiting webpage Aand having H0H1...Hn−1Hnas the browser history of previously visited
+pages, then during subsequent surfing from Athe random user eventually returns to webpage
+HnwithH0H1...Hn−1as the remaining browser history.
+Example 2.9. Consider the following equation system.
+
+X1
+X2
+X3
+=
+0.4X2X1+0.6
+0.3X1X2+0.4X3X2+0.3
+0.3X1X3+0.7
+
+The least solution of the system gives the revocation probabilities of a back-button process with
+three web-pages. For instance, if the surfer is at page 2 it can cho ose between following links
+to pages 1 and 3 with probabilities 0.3 and 0.4, respectively, or pressin g the back button with
+probability 0.3.
+3 Newton’s Method and an Overview of Our Results
+In order to approximate the least fixed point µfof an SPP fwe employ Newton’s method:
+Definition 3.1. Letfbe a clean and feasible SPP. The Newton operator Nfis defined as
+follows:
+Nf(X) :=X+/parenleftbig
+Id−f′(X)/parenrightbig−1(f(X)−X)
+The sequence (ν(k)
+f)k∈Nwithν(k)
+f=Nk
+f(0)(whereNk
+fdenotes the k-fold iteration of Nf) is
+calledNewton sequence . We drop the subscript of Nfandν(k)
+fwhenfis understood.
+8The main results of this paper concern the application of Newton’s me thod to SPPs. We
+summarize them in this section.
+Theorem 4.1 states that the Newton sequence ( ν(k))k∈Nis well-defined (i.e., the inverse
+matrices/parenleftbig
+Id−f′(ν(k))/parenrightbig−1exist for every k∈N), monotonically increasing and bounded from
+aboveby µf(i.e.ν(k)≤f(ν(k))≤ν(k+1)≤µf), and convergesto µf. This theorem generalizes
+the result of Etessami and Yannakakis in [EY09] to arbitrary clean a nd feasible SPPs and to
+the ordinary Newton’s method.
+For more quantitative results on the convergence speed it is conve nient to focus on quadratic
+SPPs.Theorem 4.13 shows that any clean and feasible SPP can be syntactically transfor med
+into a quadratic SPP without changing the least fixed point and withou t accelerating Newton’s
+method. This means, one canperform Newton’s method on the origin al(possibly non-quadratic)
+SPP and convergence will be at least as fast as for the correspond ing quadratic SPP.
+For quadratic n-dimensional SPPs, one iteration of Newton’s method involves O(n3) arith-
+metical operations and O(n3) operations in the Blum-Shub-Smale model. Hence, a bound on
+the number of iterations needed to compute a given number of valid b its immediately leads
+to a bound on the number of operations. In §5 we obtain such bounds for strongly connected
+quadratic SPPs. We give different thresholds for the number of iter ations, and show that when
+any of these thresholds is reached, Newton’s method gains at least one valid bit for each it-
+eration. More precisely, Theorem 5.12 states the following. Let fbe a quadratic, clean and
+feasible scSPP, let µminandµmaxbe the minimal and maximal component of µf, respectively,
+and let the coefficients of fbe given as ratios of m-bit integers. Then β(kf+i)≥iholds for all
+i∈Nand for any of the following choices of kf:
+1. 4mn+⌈3nmax{0,−logµmin}⌉;
+2. 4mn2n;
+3. 7mniffsatisfiesf(0)≻0;
+4. 2m(n+1) iffsatisfies both f(0)≻0andµmax≤1.
+We further show that Newton iteration can also be used to obtain a s equence of upper
+approximations of µf. Those upper approximations converge to µf, asymptotically as fast as
+the Newton sequence. More precisely, Theorem 5.15 states the following: Let fbe a quadratic,
+clean and feasible scSPP, let cminbe the smallest nonzero coefficient of f, and let µminbe the
+minimal component of µf. Further, for all Newton approximants ν(k)withν(k)≻0, letν(k)
+min
+be the smallest coefficient of ν(k). Then
+ν(k)≤µf≤ν(k)+
+/vextenddouble/vextenddoubleν(k)−ν(k−1)/vextenddouble/vextenddouble
+∞/parenleftig
+cmin·min{ν(k)
+min,1}/parenrightign
+
+where [s] denotes the vector xwithxj=sfor all 1≤j≤n.
+In§6 we turn to general (not necessarily strongly connected) clean a nd feasible SPPs. We
+show in Theorem 6.5 that Newton’s method still converges linearly. Formally, the theore m
+proves that for every quadratic, clean and feasible SPP f, there is a threshold kf∈Nand
+αf>0 such that β(kf+αf·i)≥ifor alli∈N. With respect to the threshold our proof is
+purely existential and does not provide any bound for kf. Forαfwe show an upper bound of
+n·2n, i.e., asymptotically at most n·2nextra iterations are needed in order to get one new
+valid bit. §7 exhibits a family of SPPs in which one new bit requires at least 2n−1iterations,
+implying that the bound on αfis essentially tight.
+Finally,§8 gives a geometrical interpretation of Newton’s method on quadra tic SPP equa-
+tions. Let Rbe the region bounded by the coordinate axes and by the quadrics c orresponding
+to the individual equations. Theorem 8.10 shows that all Kleene and Newton approximations
+lie within R, i.e.:ν(i),κ(i)∈Rfor every i∈N.
+94 Fundamental Properties of Newton’s Method
+4.1 Effectiveness
+Etessami and Yannakakis [EY09] suggested to use Newton’s metho d for SPPs. More precisely,
+they showed that the sequence obtained by applying Newton’s meth od to the equation system
+X=f(X) converges to µfas long as fis strongly connected. We extend their result to
+arbitrary SPPs, thereby reusing and extending several proofs o f [EY09].
+In Definition 3.1 we defined the Newton operator Nfand the associated Newton sequence
+(ν(k))k∈N. In this section we prove the following fundamental theorem on the Newton sequence.
+Theorem 4.1. Letfbe a clean and feasible SPP. Let the Newton operator Nfbe defined as in
+Definition 3.1:
+Nf(X) :=X+(Id−f′(X))−1(f(X)−X)
+1. Then the Newton sequence (ν(k))k∈Nwithν(k)=Nk
+f(0)is well-defined (i.e., the matrix
+inverses exist), monotonically increasing, bounded from a bove byµf(i.e.ν(k)≤f(ν(k))≤
+ν(k+1)≤µf), and converges to µf.
+2. We have (Id−f′(ν(k)))−1=f′(ν(k))∗for allk∈N.
+We also have (Id−f′(x))−1=f′(x)∗for allx≺µf.
+The proof of Theorem 4.1 consists of three steps. In the first pro of step we study a sequence
+generated by a somewhat weaker version of the Newton operator and obtain the following:
+Proposition 4.2. Letfbe a feasible SPP. Let the operator /hatwideNfbe defined as follows:
+/hatwideNf(X) :=X+∞/summationdisplay
+d=0/parenleftbig
+f′(X)d(f(X)−X)/parenrightbig
+.
+Then the sequence (ν(k))k∈Nwithν(k):=/hatwideNk
+f(0)is monotonically increasing, bounded from
+above by µf(i.e.ν(k)≤f(ν(k))≤ν(k+1)≤µf) and converges to µf.
+In a second proof step, we show another intermediary proposition , namely that the star of
+the Jacobian matrix f′converges for all Newton approximants:
+Proposition 4.3. Letfbe clean and feasible. Then the matrix series f′(ν(k))∗:= Id +
+f′(ν(k))+f′(ν(k))2+···converges in R≥0for all Newton approximants ν(k), i.e., there are no
+∞entries.
+In the third and final step we show that Propositions 4.2 and 4.3 imply T heorem 4.1.
+First Step. For the first proof step (i.e., the proof of Proposition 4.2) we will nee d the following
+generalization of Taylor’s theorem.
+Lemma 4.4. Letfbe an SPP, d∈N, and0≤u, and0≤x≤f(x). Then
+fd(x+u)≥fd(x)+f′(x)du.
+In particular, by setting u:=f(x)−xwe get
+fd+1(x)−fd(x)≥f′(x)d(f(x)−x).
+10Proof.By induction on d. Ford= 0 the statement is trivial. Let d≥0. Then, by Taylor’s
+theorem (Lemma 2.3), we have:
+fd+1(x+u) =f(fd(x+u))
+≥f(fd(x)+f′(x)du) (induction hypothesis)
+≥fd+1(x)+f′(fd(x))f′(x)du (Lemma 2.3)
+≥fd+1(x)+f′(x)d+1u (fd(x)≥x)
+Lemma 4.4 can be used to prove the following.
+Lemma 4.5. Letfbe a feasible SPP. Let 0≤x≤µfandx≤f(x). Then
+x+∞/summationdisplay
+d=0/parenleftbig
+f′(x)d(f(x)−x)/parenrightbig
+≤µf.
+Proof.Observe that
+lim
+d→∞fd(x) =µf (1)
+because0≤x≤µfimpliesfd(0)≤fd(x)≤µfand and as ( fd(0))d∈Nconverges to µfby
+Theorem 2.4, so does ( fd(x))d∈N. We have:
+x+∞/summationdisplay
+d=0/parenleftbig
+f′(x)d(f(x)−x)/parenrightbig
+≤x+∞/summationdisplay
+d=0/parenleftig
+fd+1(x)−fd(x)/parenrightig
+(Lemma 4.4)
+= lim
+d→∞fd(x)
+=µf (by (1))
+⊓ ⊔
+Now we can prove Proposition 4.2.
+Proof (of Proposition 4.2). First we prove the following inequality by induction on k:
+ν(k)≤f(ν(k))
+The induction base ( k= 0) is easy. For the step, let k≥0. Then
+ν(k+1)=ν(k)+∞/summationdisplay
+d=0/parenleftig
+f′(ν(k))d(f(ν(k))−ν(k))/parenrightig
+=f(ν(k))+∞/summationdisplay
+d=1/parenleftig
+f′(ν(k))d(f(ν(k))−ν(k))/parenrightig
+≤f(ν(k))+f′(ν(k))∞/summationdisplay
+d=0/parenleftig
+f′(ν(k))d(f(ν(k))−ν(k))/parenrightig
+≤f/parenleftigg
+ν(k)+∞/summationdisplay
+d=0/parenleftig
+f′(ν(k))d(f(ν(k))−ν(k))/parenrightig/parenrightigg
+(Lemma 2.3)
+=f(ν(k+1)).
+11Now, the inequality ν(k)≤µffollows from Lemma 4.5 by means of a straightforward
+induction proof. Hence, it follows f(ν(k))≤f(µf) =µf. Further we have
+f(ν(k)) =ν(k)+(f(ν(k))−ν(k))
+≤ν(k)+∞/summationdisplay
+d=0/parenleftig
+f′(ν(k))d(f(ν(k))−ν(k))/parenrightig
+=ν(k+1).(2)
+So it remains to show that ( ν(k))k∈Nconverges to µf. As we have already shown ν(k)≤µf
+it suffices to show that κ(k)≤ν(k)because ( κ(k))k∈Nconverges to µfby Theorem 2.4. We
+proceed by induction on k. The induction base ( k= 0) is easy. For the step, let k≥0. Then
+κ(k+1)=f(κ(k))
+≤f(ν(k)) (induction hypothesis)
+≤ν(k+1)(by (2))
+This completes the proof of Proposition 4.2 and, hence, the first st ep towards the proof of
+Theorem 4.1. ⊓ ⊔
+Second Step. For the second proof step (i.e., the proof of Proposition 4.3) it is con venient
+to move to the extended reals R[0,∞], i.e., we extend R≥0by an element ∞such that addition
+satisfies a+∞=∞+a=∞for alla∈R≥0and multiplication satisfies 0 · ∞=∞ ·0 =
+0 anda· ∞=∞ ·a=∞for alla∈R≥0. InR[0,∞], one can rewrite /hatwideN(ν(k)) =ν(k)+/summationtext∞
+d=0/parenleftbig
+f′(ν(k))d(f(ν(k))−ν(k))/parenrightbig
+asν(k)+f′(ν(k))∗(f(ν(k))−ν(k)). Notice that Proposition4.3
+does not follow trivially from Proposition 4.2, because ∞entries of f′(ν(k))∗could be cancelled
+out by matching 0 entries of f(ν(k))−ν(k).
+For the proof of Proposition 4.3 we need several lemmata. The follow ing lemma assures that
+a starred matrix has an ∞entry if and only if it has an ∞entry on the diagonal.
+Lemma 4.6. LetA= (aij)∈Rn×n
+≥0. LetA∗have an ∞entry. Then A∗also has an ∞entry
+on the diagonal, i.e., [A∗]ii=∞for some 1≤i≤n.
+Proof.By induction on n. The base case n= 1 is clear. For n >1 assume w.l.o.g. that [ A∗]1n=
+∞. We have
+[A∗]1n= [A∗]11n/summationdisplay
+j=2a1j/bracketleftig
+A∗
+[2..n,2..n]/bracketrightig
+jn, (3)
+where by A[2..n,2..n]we mean the square matrix obtained from Aby erasing the first row and
+the first column. To see why (3) holds, think of [ A∗]1nas the sum of weights of paths from 1 to
+nin the complete graph over the vertices {1,...,n}. The weight of a path Pis the product of
+the weight of P’s edges, and ai1i2is the weight of the edge from i1toi2. Each path Pfrom 1
+toncan be divided into two subpaths P1,P2as follows. The second subpath P2is the suffix of
+Pleading from 1 to nand not returning to 1. The first subpath P1, possibly empty, is chosen
+such that P=P1P2. Now, the sum of weights of all possible P1equals [A∗]11, and the sum of
+weights of all possible P2equals/summationtextn
+j=2a1j/bracketleftbig
+(A[2..n,2..n])∗/bracketrightbig
+jn. So (3) holds.
+As [A∗]1n=∞, it follows that either [ A∗]11or some/bracketleftbig
+(A[2..n,2..n])∗/bracketrightbig
+jnequals∞. In the first
+case, we are done. In the second case, by induction, there is an isuch that/bracketleftbig
+(A[2..n,2..n])∗/bracketrightbig
+ii=∞.
+But then also [ A∗]ii=∞, because every entry of/bracketleftbig
+(A[2..n,2..n])/bracketrightbig∗is less than or equal to the
+corresponding entry of A∗. ⊓ ⊔
+The following lemma treats the case that fis strongly connected (cf. [EY09]).
+12Lemma 4.7. Letfbe clean, feasible and non-trivially strongly connected. L et0≤x≺µf.
+Thenf′(x)∗does not have ∞as an entry.
+Proof.By Theorem 2.4 the Kleene sequence ( κ(i))i∈Nconverges to µf. Furthermore, κ(i)≺µf
+holds for all i, because, as every component depends non-trivially on itself, any increase in any
+component results in an increase of the same component in a later Kle ene approximant. So, we
+can choose a Kleene approximant y=κ(i)such that x≤y≺µf. Notice that y≤f(y). By
+monotonicity of f′it suffices to show that f′(y)∗does not have ∞as an entry. By Lemma 4.4
+(takingx:=yandu:=µf−y) we have
+f′(y)d(µf−y)≤µf−fd(y).
+Asd→ ∞, the right hand side converges to 0, because, by Kleene’s theorem, fd(y) converges
+toµf. So the left hand side also converges to 0. Sinceµf−y≻0, every entry of f′(y)dmust
+converge to 0. Then, by standard facts about matrices (see e.g. [LT85]), the sp ectral radius
+off′(y) is less than 1, i.e., |λ|<1 for all eigenvalues λoff′(y). This, in turn, implies that the
+seriesf′(y)∗= Id+f′(y)+f′(y)2+···converges in R≥0, see [LT85], page 531. In other words,
+f′(y)∗and hence f′(x)∗do not have ∞as an entry. ⊓ ⊔
+The following lemma states that Newton’s method can only terminate in a component s
+after certain other components ℓhave reached µfℓ.
+Lemma 4.8. Let1≤s,ℓ≤n. Let the term/bracketleftbig
+f′(X)∗/bracketrightbig
+sscontain the variable Xℓ. Let0≤x≤
+f(x)≤µfandxs< µfsandxℓ< µfℓ. Then/hatwideN(x)s< µfs.
+Proof.This proof follows closely a proof of [EY09]. Let d≥0 such that/bracketleftbig
+f′(X)d/bracketrightbig
+sscontains Xℓ.
+Letm′≥0 such that fm′(x)≻0andfm′(x)ℓ> xℓ. Such an m′exists because with Kleene’s
+theorem the sequence ( fk(x))k∈Nconverges to µf. Notice that our choice of m′guarantees/bracketleftig
+f′(fm′(x))d/bracketrightig
+ss>/bracketleftbig
+f′(x)d/bracketrightbig
+ss.
+Now choose m≥m′such that fm+1(x)s>fm(x)s. Such an mexists because the sequence
+(fk(x)s)k∈Nnever reaches µfs. This is because sdepends on itself (since/bracketleftbig
+f′(X)∗/bracketrightbig
+ssis not
+constant 0), and so every increase of the s-component results in an increase of the s-component
+in some later iteration of the Kleene sequence.
+Now we have
+fd+m+1(x)−fd+m(x)
+≥f′(fm(x))d(fm+1(x)−fm(x)) (Lemma 4.4)
+≥∗f′(x)d(fm+1(x)−fm(x))
+≥f′(x)df′(x)m(f(x)−x) (Lemma 4.4)
+=f′(x)d+m(f(x)−x).
+The inequality marked with ∗is strict in the s-component, due to the choice of dandm
+above. So, with b=d+mwe have:
+(fb+1(x)−fb(x))s>(f′(x)b(f(x)−x))s (4)
+Again by Lemma 4.4, inequality (4) holds for all b∈N, but with ≥instead of >. Therefore:
+µfs=/parenleftbig
+x+/summationtext∞
+i=0(fi+1(x)−fi(x))/parenrightbig
+s(Kleene)
+>/parenleftbig
+x+f′(x)∗(f(x)−x)/parenrightbig
+s(inequality (4))
+=/parenleftbig/hatwideN(x)/parenrightbig
+s
+⊓ ⊔
+13Now we are ready to prove Proposition 4.3.
+Proof (of Proposition 4.3). Using Lemma 4.6 it is enough to show that/bracketleftbig
+f′(ν(k))∗/bracketrightbig
+ss/\⌉}a⊔io\slash=∞for
+alls. If thes-component constitutes a trivial SCC, then/bracketleftbig
+f′(ν(k))∗/bracketrightbig
+ss= 0/\⌉}a⊔io\slash=∞. So we can
+assume in the following that the s-component belongs to a non-trivial SCC, say S. LetXLbe
+the set of variables contained by the term/bracketleftbig
+f′(X)∗/bracketrightbig
+ss. For any t∈Swe have/bracketleftbig
+f′(X)∗/bracketrightbig
+ss≥/bracketleftbig
+f′(X)∗/bracketrightbig
+st/bracketleftbig
+f′(X)∗/bracketrightbig
+tt/bracketleftbig
+f′(X)∗/bracketrightbig
+ts. Neither/bracketleftbig
+f′(X)∗/bracketrightbig
+stnor/bracketleftbig
+f′(X)∗/bracketrightbig
+tsis constant zero, because
+Sis non-trivial. Therefore,/bracketleftbig
+f′(X)∗/bracketrightbig
+sscontains all variables that/bracketleftbig
+f′(X)∗/bracketrightbig
+ttcontains, and vice
+versa, for all t∈S. So,XLis, for all t∈S, exactly the set of variables contained by/bracketleftbig
+f′(X)∗/bracketrightbig
+tt.
+We distinguish two cases.
+Case 1: There is a component ℓ∈Lsuch that the sequence ( ν(k)
+ℓ)k∈Ndoes not terminate, i.e.,
+ν(k)
+ℓ< µfℓholds for all k. Then, by Lemma 4.8, the sequence ( ν(k)
+s)k∈Ncannot reach µfseither.
+In fact, we have ν(k)
+S≺µfS. LetMdenote the set of those components that the S-components
+depend on,but donot depend on S. Inotherwords, Mcontainsthe componentsthatare“lower”
+in the DAG of SCCs than S. Defineg(XS) :=fS(X)[M/µfM]. Theng(XS) is an scSPP with
+µg=µfS. Asν(k)
+S≺µg, Lemma 4.7 is applicable, so g′(ν(k)
+S)∗does not have ∞as an entry.
+With/bracketleftbig
+f′(ν(k))∗/bracketrightbig
+SS≤g′(ν(k)
+S)∗, we get/bracketleftbig
+f′(ν(k))∗/bracketrightbig
+ss<∞, as desired.
+Case 2: For all components ℓ∈Lthe sequence ( ν(k)
+ℓ)k∈Nterminates. Let i∈Nbe the
+least number such that ν(i)
+ℓ=µfℓholds for all ℓ∈L. By Lemma 4.8 we have ν(i)
+s<
+µfs. But as, according to Proposition 4.2, ( ν(k)
+s)k∈Nconverges to µfs, there must exist a
+j≥isuch that 0 </parenleftbig
+f′(ν(j))∗(f(ν(j))−ν(j))/parenrightbig
+s<∞. So there is a component uwith
+0</bracketleftbig
+f′(ν(j))∗/bracketrightbig
+su(f(ν(j))−ν(j))u<∞. This implies 0 </bracketleftbig
+f′(ν(j))∗/bracketrightbig
+su<∞, therefore also/bracketleftbig
+f′(ν(j))∗/bracketrightbig
+ss<∞. By monotonicity of f′, we have/bracketleftbig
+f′(ν(k))∗/bracketrightbig
+ss≤/bracketleftbig
+f′(ν(j))∗/bracketrightbig
+ss<∞for all
+k≤j. On the other hand, since/bracketleftbig
+f′(X)∗/bracketrightbig
+sscontains only L-variables and ν(k)
+L=µfLholds for
+allk≥j, we also have/bracketleftbig
+f′(ν(k))∗/bracketrightbig
+ss=/bracketleftbig
+f′(ν(j))∗/bracketrightbig
+ss<∞for allk≥j. ⊓ ⊔
+This completes the second intermediary step towards the proof of Theorem 4.1.
+Third and Final Step. Now we can use Proposition 4.2 and Proposition 4.3 to complete the
+proof of Theorem 4.1.
+Proof (of Theorem 4.1). By Proposition 4.3 the matrix f′(ν(k))∗has no∞entries. Then we
+clearly have f′(ν(k))∗(Id−f′(ν(k))) = Id, so (Id −f′(ν(k)))−1=f′(ν(k))∗, which is the first
+claim of part 2. of the theorem. Hence, we also have
+/hatwideN(ν(k)) =ν(k)+∞/summationdisplay
+d=0/parenleftig
+f′(ν(k))d(f(ν(k))−ν(k))/parenrightig
+=ν(k)+f′(ν(k))∗(f(ν(k))−ν(k))
+=ν(k)+(Id−f′(ν(k)))−1(f(ν(k))−ν(k))
+=N(ν(k)),
+so we can replace /hatwideNbyN. Therefore, part 1. of the theorem is implied by Proposition 4.2. It
+remains to show (Id −f′(x))−1=f′(x)∗for allx≺µf. It suffices to show that f′(x)∗has
+no∞entries. By part 1. the sequence ( ν(k))k∈Nconverges to µf. So there is a k′such that
+x≤ν(k′). By Proposition 4.3, f′(ν(k′))∗has no∞entries, so, by monotonicity, f′(x)∗has no
+∞entries either. ⊓ ⊔
+144.2 Monotonicity
+Lemma 4.9 (Monotonicity of the Newton operator). Letfbe a clean and feasible SPP.
+Let0≤x≤y≤f(y)≤µfand letNf(y)exist. Then
+Nf(x)≤ Nf(y).
+Proof.Forx≤ywe have f′(x)≤f′(y) as every entry of f′(X) is a monotone polynomial.
+Hence,f′(x)∗≤f′(y)∗. With this at hand we get:
+Nf(y) =y+f′(y)∗(f(y)−y) (Theorem 4.1)
+≥y+f′(x)∗(f(y)−y) ( f′(y)∗≥f′(x)∗)
+≥y+f′(x)∗(f(x)+f′(x)(y−x)−y) (Lemma 2.3)
+=y+f′(x)∗((f(x)−x)−(Id−f′(x))(y−x))
+=y+f′(x)∗(f(x)−x)−(y−x) ( f′(x)∗=
+(Id−f′(x))−1)
+=Nf(x) (Theorem 4.1)
+⊓ ⊔
+4.3 Exponential Convergence Order in the Nonsingular Case
+If the matrix Id −f′(µf) is nonsingular, Newton’s method has exponential convergence or der
+in the sense of Definition 2.7. This is, in fact, a well known general pro perty of Newton’s
+method, see, e.g., Theorem 4.4 of [SM03]. For completeness, we show that Newton’s method for
+“nonsingular” SPPs has exponential convergence order, see The orem 4.12 below.
+Lemma 4.10. Letfbe a clean and feasible SPP. Let 0≤x≤µfsuch that f′(x)∗exists.
+Then there is a bilinear function B:Rn
+≥0×Rn
+≥0→Rn
+≥0with
+µf−N(x)≤f′(x)∗B(µf−x,µf−x).
+Proof.Writed:=µf−x. By Taylor’s theorem (cf. Lemma 2.3) we obtain
+f(x+d)≤f(x)+f′(x)d+B(d,d) (5)
+for the bilinear map B(X) :=f′′(µf)(X,X), where f′′(µf) denotes the rank-3 tensor of the
+second partial derivatives evaluated at µf[OR70]. We have
+µf−N(x) =d−f′(x)∗(f(x)−x)
+=d−f′(x)∗(d+f(x)−(x+d))
+=d−f′(x)∗(d+f(x)−f(x+d)) ( x+d=µf=f(µf))
+≤d−f′(x)∗/parenleftbig
+d−f′(x)d−B(d,d)/parenrightbig
+(by (5))
+=d−f′(x)∗/parenleftbig
+(Id−f′(x))d−B(d,d)/parenrightbig
+=d−d+f′(x)∗B(d,d) ( f′(x)∗= (Id−f′(x))−1)
+=f′(x)∗B(d,d)
+⊓ ⊔
+15Define for the following lemmata ∆(k):=µf−ν(k), i.e.,∆(k)is the error after kNew-
+ton iterations. The following lemma bounds/vextenddouble/vextenddouble/vextenddouble∆(k+1)/vextenddouble/vextenddouble/vextenddoublein terms of/vextenddouble/vextenddouble/vextenddouble∆(k)/vextenddouble/vextenddouble/vextenddouble2
+if Id−f′(µf) is
+nonsingular.
+Lemma 4.11. Letfbe a clean and feasible SPP such that Id−f′(µf)is nonsingular. Then
+there is a constant c >0such that
+/vextenddouble/vextenddouble/vextenddouble∆(k+1)/vextenddouble/vextenddouble/vextenddouble
+∞≤c·/vextenddouble/vextenddouble/vextenddouble∆(k)/vextenddouble/vextenddouble/vextenddouble2
+∞for allk∈N.
+Proof.As Id−f′(µf) is nonsingular, we have, by Theorem 4.1, (Id −f′(x))−1=f′(x)∗for
+all0≤x≤µf. By continuity, there is a c1>0 such that/vextenddouble/vextenddoublef′(x)∗/vextenddouble/vextenddouble≤c1for all0≤x≤µf.
+Similarly, there is a c2>0 such that /ba∇⌈blB(x,x)/ba∇⌈bl ≤c2/ba∇⌈blx/ba∇⌈bl2for all0≤x≤µf, because Bis
+bilinear. So it follows from Lemma 4.10 that/vextenddouble/vextenddouble/vextenddouble∆(k+1)/vextenddouble/vextenddouble/vextenddouble≤c1c2/vextenddouble/vextenddouble/vextenddouble∆(k)/vextenddouble/vextenddouble/vextenddouble2
+. ⊓ ⊔
+Lemma 4.11 implies that Newton’s method has an exponential converg ence order in the
+nonsingular case. More precisely:
+Theorem 4.12. Letfbe a clean and feasible SPP such that Id−f′(µf)is nonsingular. Then
+there is a constant kf∈Nsuch that
+β(kf+i)≥2ifor alli∈N.
+Proof.We first show that there is a constant /tildewidekf∈Nsuch that
+/vextenddouble/vextenddouble/vextenddouble∆(/tildewidekf+i)/vextenddouble/vextenddouble/vextenddouble
+∞≤2−2ifor alli∈N. (6)
+We can assume w.l.o.g. that c≥1 for the cfrom Lemma 4.11. As the ∆(k)converge to 0, we
+can choose/tildewidekf∈Nlarge enough such that d:=−log/vextenddouble/vextenddouble/vextenddouble∆(/tildewidekf)/vextenddouble/vextenddouble/vextenddouble−logc≥1. Asc,d≥1, it suffices
+to show the following inequality:
+/vextenddouble/vextenddouble/vextenddouble∆(/tildewidekf+i)/vextenddouble/vextenddouble/vextenddouble≤2−d·2i
+c.
+We proceed by induction on i. Fori= 0, the inequality above follows from the definition of d.
+Leti≥0. Then
+/vextenddouble/vextenddouble/vextenddouble∆(/tildewidekf+i+1)/vextenddouble/vextenddouble/vextenddouble≤c·/vextenddouble/vextenddouble/vextenddouble∆(/tildewidekf+i)/vextenddouble/vextenddouble/vextenddouble2
+(Lemma 4.11)
+≤c·2−d·2i·2
+c2(induction hypothesis)
+=2−d·2i+1
+c.
+Hence, (6) is proved.
+Choosem∈Nlarge enough such that 2m+i+log(µfj)≥2iholds for all components j. Thus
+∆(/tildewidekf+m+i)
+j /µfj≤2−2m+i/µfj (by (6))
+= 2−(2m+i+log(µfj))
+≤2−2i(choice of m).
+So, with kf:=/tildewidekf+m, the approximant ν(kf+i)has at least 2ivalid bits of µf. ⊓ ⊔
+16This type of analysis has serious shortcomings. In particular, Theo rem 4.12 excludes the case
+where Id −f′(µf) is singular. We will include this case in our convergence analysis in §5 and
+§6. Furthermore, and maybe more severely, Theorem 4.12 does not give any bound on kf. We
+solve this problem for strongly connected SPPs in §5.
+4.4 Reduction to the Quadratic Case
+In this section we reduce SPPs to quadratic SPPs, i.e., to SPPsin which everypolynomial fi(X)
+has degree at most 2, and show that the convergence on the quad ratic SPP is no faster than
+on the original SPP. In the following sections we will obtain convergen ce speed guarantees of
+Newton’s method on quadratic SPPs. Hence, one can perform Newt on’s method on the original
+SPP and, using the results of this section, convergence is at least a s fast as on the corresponding
+quadratic SPP.
+The idea to reduce the degree of our SPP fis to introduce auxiliary variables that express
+quadraticsubterms.Thiscanbedonerepeatedlyuntilallpolynomia lsinthesystemhavereached
+degree at most 2. The construction is very similar to the one that tr ansforms a context-free
+grammar into another grammar in Chomsky normal form. The followin g theorem shows that
+the transformation does not accelerate the convergence of New ton’s method.
+Theorem 4.13. Letf(X)be a clean and feasible SPP such that fs(X) =g(X)+h(X)XiXj
+for some 1≤i,j,s≤n, whereg(X)andh(X)are polynomials with nonnegative coefficients.
+Let/tildewidef(X,Y)be the SPP given by
+/tildewidefℓ(X,Y) =fℓ(X) for every ℓ∈ {1,...,s−1}
+/tildewidefs(X,Y) =g(X)+h(X)Y
+/tildewidefℓ(X,Y) =fℓ(X) for every ℓ∈ {s+1,...,n}
+/tildewidefn+1(X,Y) =XiXj.
+Then the function b:Rn→Rn+1given by b(X) = (X1,...,X n,XiXj)⊤is a bijection between
+the set of fixed points of f(X)and/tildewidef(X,Y). Moreover,/tildewideν(k)≤(ν(k)
+1,...,ν(k)
+n,ν(k)
+iν(k)
+j)⊤for all
+k∈N, where/tildewideν(k)andν(k)are the Newton approximants of /tildewidefandf, respectively.
+Proof.We first show the claim regarding b: ifxis a fixed point of f, thenb(x) = (x,xixj) is a
+fixed point of /tildewidef. Conversely, if ( x,y) is a fixed point of /tildewidef, then we have y=xixjimplying that
+xis a fixed point of f. Therefore, the least fixed point µfoffdetermines µ/tildewidef, and vice versa.
+NowweshowthattheNewtonsequenceof fconvergesatleastasfastastheNewtonsequence
+of/tildewidef. In the followingwe write Yfor the (n+1)-dimensionalvectorofvariables( X1,...,X n,Y)⊤
+and, as usual, Xfor (X1,...,X n)⊤. For an ( n+1)-dimensional vector x, we letx[1,n]denote
+its restriction to the nfirst components, i.e., x[1,n]:= (x1,...,x n)⊤. Note that Y[1,n]=X. Let
+esdenote the unit vector (0 ,...,0,1,0...0)⊤, where the “1” is on the s-th place. We have:
+/tildewidef(Y) =/parenleftbiggf(X)+esh(X)(Y−XiXj)
+XiXj/parenrightbigg
+and
+/tildewidef′(Y) =/parenleftbiggf′(X)+es∂Xh(X)(Y−XiXj)esh(X)
+∂XXiXj 0/parenrightbigg
+We need the following lemma.
+17Lemma 4.14. Letz∈Rn
+≥0,δ=/parenleftbig
+Id−f′(z)/parenrightbig−1(f(z)−z)and/tildewideδ=/parenleftig
+Id−/tildewidef′(z,zizj)/parenrightig−1
+(/tildewidef(z,zizj)−(z,zizj)⊤). Thenδ=/tildewideδ[1,n].
+Proof of the lemma.
+/tildewidef′(z,zizj) =/parenleftbiggf′(z)+esh(z)∂X(Y−XiXj)|Y=(z,zizj)esh(z)
+∂XXiXj|Y=(z,zizj) 0/parenrightbigg
+=/parenleftbiggf′(z)−esh(z)∂X(XiXj)|X=zesh(z)
+∂XXiXj|X=z 0/parenrightbigg
+We have (Id −/tildewidef′(z,zizj))/tildewideδ= (/tildewidef(z,zizj)−(z,zizj)⊤), or equivalently:
+/parenleftbigg
+Id−f′(z)+esh(z)∂X(XiXj)|X=z−esh(z)
+−∂XXiXj|X=z 1/parenrightbigg
+·/parenleftigg
+/tildewideδ[1,n]
+/tildewideδn+1/parenrightigg
+=/parenleftbigg
+f(z)−z
+0/parenrightbigg
+Multiplying the last row by esh(z) and adding to the first nrows yields:
+/parenleftbig
+Id−f′(z)/parenrightbig/tildewideδ[1,n]=f(z)−z
+So we have/tildewideδ[1,n]=/parenleftbig
+Id−f′(z)/parenrightbig−1(f(z)−z) =δ, which proves the lemma. ⊓ ⊔
+Now we proceed by induction on kto show/tildewideν(k)
+[1,n]≤ν(k), where/tildewideν(k)is the Newton sequence
+for/tildewidef. By definition of the Newton sequence this is true for k= 0. For the step, let k≥0 and
+defineu:= (/tildewideν(k)
+[1,n],/tildewideν(k)
+i·/tildewideν(k)
+j)⊤. Then we have:
+/tildewideν(k+1)
+[1,n]=N/tildewidef(/tildewideν(k))[1,n]
+(∗)
+≤ N/tildewidef(u)[1,n] (see below)
+=/tildewideν(k)
+[1,n]+/parenleftig
+(Id−/tildewidef′(u))−1(/tildewidef(u)−u)/parenrightig
+[1,n]
+=/tildewideν(k)
+[1,n]+(Id−f′(/tildewideν(k)
+[1,n]))−1(f(/tildewideν(k)
+[1,n])−/tildewideν(k)
+[1,n]) (Lemma 4.14)
+=Nf(/tildewideν(k)
+[1,n])
+≤ Nf(ν(k)) (induction)
+=ν(k+1)
+At the inequality marked with ( ∗) we used the monotonicity of N/tildewidef(Lemma 4.9) combined
+with Theorem 4.1, which states /tildewideν(k)≤/tildewidef(/tildewideν(k)), hence in particular /tildewideν(k)
+n+1≤/tildewideν(k)
+i/tildewideν(k)
+j. This
+concludes the proof of Theorem 4.13. ⊓ ⊔
+5 Strongly Connected SPPs
+In this section we study the convergencespeed of Newton’s metho d on stronglyconnected SPPs,
+short scSPPs, see Definition 2.6.
+5.1 Cone Vectors
+Our convergence speed analysis makes crucial use of the existenc e ofcone vectors .
+18Definition 5.1. Letfbe an SPP. A vector d∈Rn
+≥0is acone vector ifd≻0andf′(µf)d≤d.
+We will show that any scSPP has a cone vector, see Proposition 5.4 be low. As a first step,
+we show the following lemma.
+Lemma 5.2. Any clean and feasible scSPP fhas a vector d>0withf′(µf)d≤d.
+Proof.Consider the Kleene sequence ( κ(k))k∈N. Sincefis strongly connected, we have 0≤
+κ(k)≺µffor allk∈N. By Theorem 4.1.2., the matrices (Id −f′(κ(k)))−1=f′(κ(k))∗exist for
+allk. Let/ba∇⌈bl·/ba∇⌈blbe any norm. Define the vectors
+d(k):=f′(κ(k))∗1/vextenddouble/vextenddoublef′(κ(k))∗1/vextenddouble/vextenddouble.
+Notice that for all k∈Nwe have (Id −f′(κ(k)))d(k)=1
+/ba∇⌈blf′(κ(k))∗1/ba∇⌈bl·1≥0. Furthermore we have
+d(k)∈C, whereC:={x≥0| /ba∇⌈blx/ba∇⌈bl= 1}is compact. So the sequence ( d(k))k∈Nhas a convergent
+subsequence, whose limit, say d, is also in C. In particular d>0. As (κ(k))k∈Nconverges to µf
+and (Id−f′(κ(k)))d(k)≥0, it follows by continuity (Id −f′(µf))d≥0. ⊓ ⊔
+Lemma 5.3. Letfbe a clean and feasible scSPP and let d>0withf′(µf)d≤d. Thendis
+a cone vector, i.e., d≻0.
+Proof.Sincefis an SPP, every component of f′(µf) is nonnegative. So,
+0≤f′(µf)nd≤f′(µf)n−1d≤...≤f′(µf)d≤d.
+Let w.l.o.g. d1>0. Asfis strongly connected, there is for all jwith 1≤j≤nanrj≤nsuch
+that (f′(µf)rj)j1>0. Hence, ( f′(µf)rjd)j>0 for allj. With above inequality chain, it follows
+thatdj≥(f′(µf)rjd)j>0. So,d≻0. ⊓ ⊔
+The following proposition follows immediately by combining Lemmata 5.2 an d 5.3.
+Proposition 5.4. Any clean and feasible scSPP has a cone vector.
+We remark that using Perron-Frobenius theory [BP79] there is a sim pler proof for Proposi-
+tion 5.4: By Theorem 4.1 f′(x)∗exists for all x≺µf. So, by fundamental matrix facts [BP79],
+the spectral radius of f′(x) is less than 1 for all x≺µf. As the eigenvalues of a matrix de-
+pend continuously on the matrix, the spectral radius of f′(µf), sayρ, is at most 1. Since fis
+strongly connected, f′(µf) is irreducible, and so Perron-Frobenius theory guarantees the e xis-
+tence of an eigenvector d≻0off′(µf) with eigenvalue ρ. So we have f′(µf)d=ρd≤d, i.e.,
+the eigenvector dis a cone vector.
+5.2 Convergence Speed in Terms of Cone Vectors
+Now we show that cone vectors play a fundamental role for the con vergence speed of Newton’s
+method. The following lemma gives a lower bound of the Newton approx imantν(1)in terms of
+a cone vector.
+Lemma 5.5. Letfbe a feasible (not necessarily clean) SPP such that f′(0)∗exists. Let dbe
+a cone vector of f. Let0≥µf−λdfor some λ≥0. Then
+N(0)≥µf−1
+2λd.
+19Proof.We write f(X) as a sum f(X) =c+/summationtextD
+k=1T(k)(X,...,X), where Dis the degree of f
+and, for all k∈ {1,...,D}and alli∈ {1,...,n}, the component T(k)
+iofT(k)is the symmetric
+k-linear form associated to the degree- kterms of fi. LetL(k): (Rn)k−1→Rn×nsuch that
+T(k)(X(1),...,X(k)) =L(k)(X(1),...,X(k−1))·X(k). Now we can write
+f(X) =c+D/summationdisplay
+k=1L(k)(X,...,X)Xandf′(X) =D/summationdisplay
+k=1k·L(k)(X,...,X).
+We write LforL(1), andh(X) forf(X)−LX−c. We have:
+λ
+2d=λ
+2(L∗d−L∗Ld) ( L∗= Id+L∗L)
+≥λ
+2(L∗f′(µf)d−L∗Ld) ( f′(µf)d≤d)
+=λ
+2L∗h′(µf)d (f′(x) =h′(x)+L)
+=L∗1
+2h′(µf)λd
+≥L∗1
+2h′(µf)µf (λd≥µf)
+=L∗1
+2D/summationdisplay
+k=2k·L(k)(µf,...,µf)µf
+≥L∗D/summationdisplay
+k=2L(k)(µf,...,µf)µf
+=L∗h(µf)
+=L∗(f(µf)−Lµf−c) ( f(x) =h(x)+Lx+c)
+=L∗µf−L∗Lµf−L∗c (f(µf) =µf)
+=µf−L∗c (L∗= Id+L∗L)
+=µf−N(0) ( N(0) =f′(0)∗f(0) =L∗c) ⊓ ⊔
+We extend Lemma 5.5 to arbitrary vectors xas follows.
+Lemma 5.6. Letfbe a feasible (not necessarily clean) SPP. Let 0≤x≤µfandx≤f(x)
+such that f′(x)∗exists. Let dbe a cone vector of f. Letx≥µf−λdfor some λ≥0. Then
+N(x)≥µf−1
+2λd.
+Proof.Defineg(X) :=f(X+x)−x. We first show that gis an SPP (not necessarily clean).
+The only coefficients of gthat could be negative are those of degree 0. But we have g(0) =
+f(x)−x≥0, and so these coefficients are also nonnegative.
+It follows immediately from the definition that µf−x≥0is the least fixed point of g.
+Moreover, gsatisfies g′(µf−x)d≤d, and so dis also a cone vector of g. Finally, we have
+0≥µf−x−λd=µg−λd. So, Lemma 5.5 can be applied as follows.
+Nf(x) =x+f′(x)∗(f(x)−x)
+=x+g′(0)∗(g(0)−0)
+=x+Ng(0)
+20≥x+µg−1
+2λd (Lemma 5.5)
+=µf−1
+2λd
+⊓ ⊔
+By induction we can extend this lemma to the whole Newton sequence:
+Lemma 5.7. Letdbe a cone vector of a clean and feasible SPP fand letλmax= max j{µfj
+dj}.
+Then
+ν(k)≥µf−2−kλmaxd.
+PSfrag replacements
+X1=f1(X)X2=f2(X)µf=r(0)
+0
+−0.4−0.20.2 0 .4 0 .60.2
+X1X2
+r(λmax)
+Fig.2.Illustration of Lemma 5.7: The points (shape: +) on the ray ralong a cone vector are lower
+bounds on the Newton approximants (shape: ×).
+Before proving the lemma we illustrate it by a picture. The dashed line in Figure 2 is the ray
+r(t) =µf−tdalong a cone vector d. Notice that r(0) equals µfandr(λmax) is the greatest
+point on the ray that is below 0. The figure also shows the Newton iterates ν(k)for 0≤k≤2
+(shape:×) and the corresponding points r(2−kλmax) (shape: +) located on the ray r. Observe
+thatν(k)≥r(2−kλmax), as claimed by Lemma 5.7.
+Proof (of Lemma 5.7). By induction on k. For the induction base ( k= 0) we have for all
+components i:
+(µf−λmaxd)i=/parenleftbigg
+µf−max
+j/braceleftbiggµfj
+dj/bracerightbigg
+d/parenrightbigg
+i≤µfi−µfi
+didi= 0,
+soν(0)=0≥µf−λmaxd.
+For the induction step, let k≥0. By induction hypothesis we have ν(k)≥µf−2−kλmaxd.
+So we can apply Lemma 5.6 to get
+ν(k+1)=N(ν(k))≥µf−1
+22−kλmaxd=µf−2−(k+1)λmaxd.
+21⊓ ⊔
+The following proposition guarantees a convergence order of the N ewton sequence in terms
+of a cone vector.
+Proposition 5.8. Letdbe a cone vector of a clean and feasible SPP fand letλmax=
+maxj/braceleftigµfj
+dj/bracerightig
+andλmin= min j/braceleftigµfj
+dj/bracerightig
+. Letkf,d=/ceilingleftig
+logλmax
+λmin/ceilingrightig
+. Thenβ(kf,d+i)≥ifor all
+i∈N.
+Proof.For all 1 ≤j≤nthe following holds.
+/parenleftbig
+µf−ν(kf,d+i)/parenrightbig
+j≤2−(kf,d+i)λmaxdj (Lemma 5.7)
+≤λmin
+λmax2−iλmaxdj (def. ofkf,d)
+=λmindj·2−i
+≤µfj·2−i(def. ofλmin)
+Hence,ν(kf,d+i)hasivalid bits of µf. ⊓ ⊔
+5.3 Convergence Speed Independent from Cone Vectors
+The convergence order provided by Proposition 5.8 depends on a co ne vector d. While Proposi-
+tion 5.4 guarantees the existence of a cone vector for scSPPs, it d oes not give any information
+on the magnitude of its components. So we do not have any bound ye t on the “threshold” kf,d
+from Proposition 5.8. The following theorem solves this problem.
+Theorem 5.9. Letfbe a quadratic, clean and feasible scSPP. Let cminbe the smallest nonzero
+coefficient of fand letµminandµmaxbe the minimal and maximal component of µf, respec-
+tively. Let
+kf=/ceilingleftbigg
+logµmax
+µmin·(cmin·min{µmin,1})n/ceilingrightbigg
+.
+Then
+β(kf+i)≥ifor alli∈N.
+Before we prove Theorem 5.9 we give an example.
+Example 5.10. As an example of application of Theorem 5.9 consider the scSPP equat ion of the
+back button process of Example 2.9.
+
+X1
+X2
+X3
+=
+0.4X2X1+0.6
+0.3X1X2+0.4X3X2+0.3
+0.3X1X3+0.7
+
+We wish to know if there is a component s∈ {1,2,3}withµfs= 1. Notice that f(1) =1,
+soµf≤1. Performing 14 Newton steps (e.g. with Maple) yields an approximatio nν(14)toµf
+with 
+0.98
+0.97
+0.992
+≤ν(14)≤
+0.99
+0.98
+0.993
+.
+We have cmin= 0.3. In addition, since Newton’s method converges to µffrom below, we
+knowµmin≥0.97. Moreover, µmax≤1, as1=f(1) and so µf≤1. Hence kf≤
+22/ceilingleftbigg
+log1
+0.97·(0.3·0.97)3/ceilingrightbigg
+= 6. Theorem 5.9 then implies that ν(14)has 8 valid bits of µf. As
+µf≤1, the absolute errors are bounded by the relative errors, and sinc e 2−8≤0.004 we know:
+µf≤ν(14)+
+2−8
+2−8
+2−8
+≤
+0.994
+0.984
+0.997
+≺
+1
+1
+1
+
+So Theorem 5.9 yields a proof that µfs<1 for all three components s.
+Notice also that the Newton sequence converges much faster tha n the Kleene sequence
+(κ(k))k∈N. We have κ(14)≺/parenleftbig
+0.89,0.83,0.96/parenrightbig⊤, soκ(14)has no more than 4 valid bits in any
+component, whereas ν(14)has, in fact, more than 30 valid bits in each component. ⊓ ⊔
+For the proof of Theorem 5.9 we need the following lemma.
+Lemma 5.11. Letdbe a cone vector of a quadratic, clean and feasible scSPP f. Letcminbe
+the smallest nonzero coefficient of fandµminthe minimal component of µf. Letdminanddmax
+be the smallest and the largest component of d, respectively. Then
+dmin
+dmax≥(cmin·min{µmin,1})n.
+Proof.In what follows we shorten µftoµ. Let w.l.o.g. d1=dmaxanddn=dmin. We claim
+the existence of indices s,twith 1≤s,t≤nsuch that f′
+st(µ)/\⌉}a⊔io\slash= 0 and
+dmin
+dmax≥/parenleftbiggds
+dt/parenrightbiggn
+. (7)
+To prove that such s,texist, we use the fact that fis strongly connected, i.e., that there is a
+sequence 1 = r1,r2,...,r q=nwithq≤nsuch that f′
+rj+1rj(X) is not constant zero. As µ≻0,
+we have f′
+rj+1rj(µ)/\⌉}a⊔io\slash= 0. Furthermore
+d1
+dn=dr1
+dr2···drq−1
+drq, and so
+logd1
+dn= logdr1
+dr2+···+logdrq−1
+drq.
+So there must exist a jsuch that
+logd1
+dn≤(q−1)logdrj
+drj+1≤nlogdrj
+drj+1, and so
+dn
+d1≥/parenleftbiggdrj+1
+drj/parenrightbiggn
+.
+Hence one can choose s=rj+1andt=rj.
+Asdis a cone vector we have f′(µ)d≤dand thus f′
+st(µ)dt≤ds. Hence
+f′
+st(µ)≤ds
+dt. (8)
+On the other hand, since fis quadratic, f′is a linear mapping such that
+f′
+st(µ) = 2(b1·µ1+···+bn·µn)+ℓ
+23whereb1,...,b nandℓare coefficients of quadratic, respectively linear, monomials of f. As
+f′
+st(µ)/\⌉}a⊔io\slash= 0, at least one of these coefficients must be nonzero and so great er than or equal to
+cmin. It follows f′
+st(µ)≥cmin·min{µmin,1}. So we have
+(cmin·min{µmin,1})n≤/parenleftbig
+f′
+st(µ)/parenrightbign
+≤/parenleftbiggds
+dt/parenrightbiggn
+(by (8))
+≤dmin
+dmax(by (7)).
+⊓ ⊔
+Now we can prove Theorem 5.9.
+Proof (of Theorem 5.9). By Proposition 5.4, fhas a cone vector d. Letdmax= max j{dj}and
+dmin= min j{dj}andλmax= max j/braceleftigµfj
+dj/bracerightig
+andλmin= min j/braceleftigµfj
+dj/bracerightig
+. We have:
+λmax
+λmin≤µmax·dmax
+µmin·dmin(asλmax≤dmax
+µminandλmin≥dmin
+µmax)
+≤µmax
+µmin·(cmin·min{µmin,1})n(Lemma 5.11) .
+So the statement follows with Proposition 5.8. ⊓ ⊔
+The following consequence of Theorem 5.9 removes some of the para meters on which the kf
+from Theorem 5.9 depends.
+Theorem 5.12. Letfbe a quadratic, clean and feasible scSPP, let µminandµmaxbe the
+minimal and maximal component of µf, respectively, and let the coefficients of fbe given as
+ratios of m-bit integers. Then
+β(kf+i)≥ifor alli∈N
+holds for any of the following choices of kf.
+1.⌈4mn+3nmax{0,−logµmin}⌉;
+2.4mn2n;
+3.7mnwhenever f(0)≻0;
+4.2mn+mwhenever both f(0)≻0andµmax≤1.
+Items 3. and 4. of Theorem 5.12 apply in particular to termination SPP s of strict pPDAs ( §2.4),
+i.e., they satisfy f(0)≻0andµmax≤1.
+To prove Theorem 5.12 we need some relations between the paramet ers off. We collect
+them in the following lemma.
+Lemma 5.13. Letfbe a quadratic, clean and feasible scSPP. With the terminolo gy of Theo-
+rem 5.9 and Theorem 5.12 the following relations hold.
+1.cmin≥2−m.
+2. Iff(0)≻0thenµmin≥cmin.
+3. Ifcmin>1thenµmin>1.
+4. Ifcmin≤1thenµmin≥c2n−1
+min.
+5. Iffis strictly quadratic, i.e. nonlinear, then the following i nequalities hold: cmin≤1and
+µmax·c3n−2
+min·min{µ2n−2
+min,1} ≤1.
+Proof.We show the relations in turn.
+241. The smallest nonzero coefficient representable as a ratio of m-bit numbers is1
+2m.
+2. Asf(0)≻0, in all components ithere is a nonzero coefficient cisuch that fi(0) =ci. We
+haveµf≥f(0), soµfi≥fi(0) =ci≥cmin>0 holds for all i. Henceµmin>0.
+3. Letcmin>1. Recall the Kleene sequence ( κ(k))k∈Nwithκ(k)=fk(0). We first show by
+induction on kthat for all k∈Nand all components ieitherκ(k)
+i= 0 holds or κ(k)
+i>1. For
+the induction base we have κ(0)=0. Letk≥0. Thenκ(k+1)
+i=fi(κ(k)) is a sum of products
+of numbers which are either coefficients of f(and hence by assumption greater than 1) or
+which are equal to κ(k)
+jfor some j. By induction, κ(k)
+jis either 0 or greaterthan 1. So, κ(k+1)
+i
+must be 0 or greater than 1.
+By Theorem 2.4, the Kleene sequence converges to µf. Asfis clean, we have µf≻0, and
+so there is a k∈Nsuch that κ(k)≻1. The statement follows with µf≥κ(k).
+4. Letcmin≤1. We prove the following stronger statement by induction on k: For every kwith
+0≤k≤nthere isa set Sk⊆ {1,...,n},|Sk|=k, suchthat µfs≥c2k−1
+minholdsfor all s∈Sk.
+The induction base ( k= 0) is trivial. Let k≥0. Consider the SPP /hatwidef(X{1,...,n}\Sk) that is
+obtained from f(X) by removingthe Sk-componentsfrom fand replacingevery Sk-variable
+in the polynomials by the corresponding component of µf. Clearly, µ/hatwidef= (µf){1,...,n}\Sk.
+By induction, the smallest nonzero coefficient /hatwidecminof/hatwidefsatisfies/hatwidecmin≥cmin(c2k−1
+min)2=
+c2k+1−1
+min. Pick a component iwith/hatwidefi(0)>0. Then µ/hatwidefi≥/hatwidefi(0)≥/hatwidecmin≥c2k+1−1
+min. So set
+Sk+1:=Sk∪{i}.
+5. Let w.l.o.g. µmax=µf1. The proofis basedon the ideathat X1indirectlydepends quadrati-
+cally on itself. More precisely, as fis stronglyconnected and strictly quadratic,component 1
+depends (indirectly) onsome component, say ir, such that fircontainsadegree-2-monomial.
+The variables in that monomial, in turn, depend on X1. This gives an inequality of the form
+µf1≥C·µf12, implying µf1·C≤1.
+We give the details in the following. As fis strongly connected and strictly quadratic
+there exists a sequence of variables Xi1,...,X irand a sequence of monomials mi1,...,m ir
+(1≤r≤n) with the following properties:
+–Xi1=X1,
+–miuis a monomial appearing in fiu (1≤u≤r),
+–miu=ciu·Xiu+1 (1≤u≤r),
+–mir=cir·Xj1·Xk1for some variables Xj1,Xk1.
+Notice that
+µmax=µf1≥ci1·...·cir·µfj1·µfk1
+≥min(cn
+min,1)·µfj1·µfk1.(9)
+Again using that fis strongly connected, there exists a sequence of variables Xj1,...,X js
+and a sequence of monomials mj1,...,m js−1(1≤s≤n) with the following properties:
+–Xjs=X1,
+–mjuis a monomial appearing in fju(1≤u≤s−1),
+–mju=cju·Xju+1ormju=cju·Xju+1·Xj′
+u+1
+for some variable Xj′
+u+1(1≤u≤s−1).
+Notice that
+µfj1≥cj1·...·cjs−1·min(µs−1
+min,1)·µf1
+≥min(cn−1
+min,1)·min(µn−1
+min,1)·µf1.(10)
+25Similarly, there exists a sequence of variables Xk1,...,X kt(1≤t≤n) withXkt=X1
+showing
+µfk1≥min(cn−1
+min,1)·min(µn−1
+min,1)·µf1. (11)
+Combining (9) with (10) and (11) yields
+µmax≥min(c3n−2
+min,1)·min(µ2n−2
+min,1)·µ2
+max,
+or
+µmax·min(c3n−2
+min,1)·min(µ2n−2
+min,1)≤1. (12)
+Nowitsufficestoshow cmin≤1.Assumeforacontradiction cmin>1.Then,bystatement3.,
+µmin>1. Plugging this into (12) yields µmax≤1. This implies µmax< µmin, contradicting
+the definition of µmaxandµmin.
+⊓ ⊔
+Now we are ready to prove Theorem 5.12.
+Proof (of Theorem 5.12).
+1. Firstwecheckthecasewhere fislinear,i.e., allpolynomials fihavedegreeatmost 1.Inthis
+case, Newton’s method reaches µfafter one iteration, so the statement holds. Consequently,
+we can assume in the following that fis strictly quadratic, meaning that fis quadratic and
+there is a polynomial in fof degree 2.
+By Theorem 5.9 it suffices to show
+logµmax
+µmin·cn
+min·min{µn
+min,1}≤4mn+3nmax{0,−logµmin}.
+We have
+logµmax
+µmin·cn
+min·min{µn
+min,1}
+≤log1
+c4n−2
+min·min{µ3n−1
+min,1}(Lemma 5.13.5.)
+≤4n·log1
+cmin−log(min{µ3n−1
+min,1}) (Lemma 5.13.5.: cmin≤1)
+≤4mn−log(min{µ3n−1
+min,1}) (Lemma 5.13.1.) .
+Ifµmin≥1 we have −log(min{µ3n−1
+min,1})≤0, so we are done in this case. If µmin≤1 we
+have−log(min{µ3n−1
+min,1}) =−(3n−1)logµmin≤3n·(−logµmin).
+2. By statement 1. of this theorem, it suffices to show that 4 mn+ 3nmax{0,−logµmin} ≤
+4mn2n. This inequality obviously holds if µmin≥1. So let µmin≤1. Then, by
+Lemma 5.13.3., cmin≤1. Hence, by Lemma 5.13 parts 4. and 1., µmin≥c2n−1
+min≥2−m(2n−1).
+So we have an upper bound on −logµminwith−logµmin≤m(2n−1) and get:
+4mn+3nmax{0,−logµmin} ≤4mn+3nm(2n−1)
+≤4mn+4nm(2n−1) = 4mn2n
+3. Letf(0)≻0. By statement 1. of this theorem it suffices to show that 4 mn+
+3nmax{0,−logµmin} ≤7mnholds. By Lemma 5.13 parts 2. and 1., we have µmin≥
+cmin≥2−m, so−logµmin≤m. Hence, 4 mn+3nmax{0,−logµmin} ≤4mn+3nm= 7mn.
+264. Letf(0)≻0andµmax≤1. By Theorem 5.9 it suffices to show that
+logµmax
+µmin·cn
+min·min{µn
+min,1}≤2mn+m. We have:
+logµmax
+µmin·cn
+min·min{µn
+min,1}
+≤ −nlogcmin−(n+1)logµmin (asµmin≤µmax≤1)
+≤ −(2n+1)logcmin (Lemma 5.13.2.)
+≤2mn+m (Lemma 5.13.1.)
+⊓ ⊔
+5.4 Upper Bounds on the Least Fixed Point Via Newton Approxim ants
+By Theorem 4.1 each Newton approximant ν(k)is a lower bound on µf. Theorem 5.9 and
+Theorem 5.12 give us upper bounds on the error ∆(k):=µf−ν(k). Those bounds can directly
+transformed into upper bounds on µf, asµf=ν(k)+∆(k), cf. Example 5.10.
+Theorem 5.9 and Theorem 5.12 allow to compute bounds on ∆(k)even before the Newton
+iteration has been started. However, this may be more than we act ually need. In practice, we
+may wish to use an iterative method that yields guaranteed lower and upper bounds on µfthat
+improve during the iteration. The following theorem and its corollary c an be used to this end.
+Theorem 5.14. Letfbe a quadratic, clean and feasible scSPP. Let 0≤x≤µfandx≤f(x)
+such that f′(x)∗exists. Let cminbe the smallest nonzero coefficient of fandµminthe minimal
+component of µf. Then
+/ba∇⌈blN(x)−x/ba∇⌈bl∞
+/ba∇⌈blµf−N(x)/ba∇⌈bl∞≥(cmin·min{µmin,1})n.
+We prove Theorem 5.14 at the end of the section. The theorem can b e applied to the Newton
+approximants:
+Theorem 5.15. Letfbe a quadratic, clean and feasible scSPP. Let cminbe the smallest nonzero
+coefficient of fandµminthe minimal component of µf. For all Newton approximants ν(k)with
+ν(k)≻0, letν(k)
+minbe the smallest coefficient of ν(k). Then
+ν(k)≤µf≤ν(k)+
+/vextenddouble/vextenddoubleν(k)−ν(k−1)/vextenddouble/vextenddouble
+∞/parenleftig
+cmin·min{ν(k)
+min,1}/parenrightign
+
+where[s]denotes the vector xwithxj=sfor all1≤j≤n.
+Proof (of Theorem 5.15). Theorem 5.14 applies, due to Theorem 4.1, to the Newton approxi-
+mants with x=ν(k−1). So we get
+/vextenddouble/vextenddouble/vextenddoubleµf−ν(k)/vextenddouble/vextenddouble/vextenddouble
+∞≤/vextenddouble/vextenddoubleν(k)−ν(k−1)/vextenddouble/vextenddouble
+∞
+(cmin·min{µmin,1})n
+≤/vextenddouble/vextenddoubleν(k)−ν(k−1)/vextenddouble/vextenddouble
+∞/parenleftig
+cmin·min{ν(k)
+min,1}/parenrightign (asν(k)≤µf).
+Hence the statement follows from ν(k)≤µf. ⊓ ⊔
+27Example 5.16. Consider again the equation X=f(X) from Examples 2.9 and 5.10:
+
+X1
+X2
+X3
+=
+0.4X2X1+0.6
+0.3X1X2+0.4X3X2+0.3
+0.3X1X3+0.7
+
+Again we wish to verify that there is no component s∈ {1,2,3}withµfs= 1. Performing 10
+Newton steps yields an approximation ν(10)toµfwith
+
+0.9828
+0.9738
+0.9926
+≺ν(10)≺
+0.9829
+0.9739
+0.9927
+.
+Further, it holds/vextenddouble/vextenddoubleν(10)−ν(9)/vextenddouble/vextenddouble
+∞≤2·10−6. So we have
+/vextenddouble/vextenddoubleν(10)−ν(9)/vextenddouble/vextenddouble
+∞/parenleftig
+cmin·min{ν(10)
+min,1}/parenrightig3≤2·10−6
+(0.3·0.97)3≤0.00009
+and hence by Theorem 5.15
+ν(10)≤µf≤ν(10)+[0.00009]≤
+0.983
+0.974
+0.993
+
+In particular we know that µfs<1 for all three components s. ⊓ ⊔
+Example 5.17. Consider again the SPP ffrom Example 5.16. Setting
+u(k):=ν(k)+
+/vextenddouble/vextenddoubleν(k)−ν(k−1)/vextenddouble/vextenddouble
+∞/parenleftig
+0.3·ν(k)
+min/parenrightig3
+,
+Theorem 5.15 guarantees
+ν(k)≤µf≤u(k).
+Let us measure the tightness of the bounds ν(k)andu(k)onµfin the first component. Let
+plower(k) :=−log2(µf1−ν(k)
+1) and
+pupper(k) :=−log2(u(k)
+1−µf1).
+Roughly speaking, ν(k)
+1andu(k)
+1haveplower(k) andpupper(k) valid bits of µf1, respectively.
+Figure 3 shows plower(k) andpupper(k) fork∈ {1,...,11}.
+It canbe seenthatthe slopeof plower(k) isapproximately1for k= 2,...,6.Thiscorresponds
+to the linear convergence of Newton’s method according to Theore m 5.9. Since Id −f′(µf) is
+non-singular3, Newton’s method actually has, asymptotically, an exponential con vergenceorder,
+cf. Theorem 4.12. This behavior can be observed in Figure 3 for k≥7. Forpupper, we roughly
+have (using ν(k)≈µf):
+pupper(k)≈plower(k−1)+log/parenleftig
+0.3·ν(k)
+min/parenrightig3
+≈plower(k−1)−5.
+⊓ ⊔
+3In fact, the matrix is “almost” singular, with det(Id −f′(µf))≈0.006.
+28PSfrag replacements
+−100
+1 2 3 4 5 6 7 8 9 1010
+11 122030405060
+plower(k)
+pupper(k)
+k
+Fig.3.Number of valid bits of the lower (shape: ×) and upper (shape: +) bounds on µf1, see Exam-
+ple 5.17.
+The proof of Theorem 5.14 uses techniques similar to those of the pr oof of Theorem 5.9, in
+particular Lemma 5.11.
+Proof (of Theorem 5.14). By Proposition 5.4, fhas a cone vector d. Letdminanddmaxbe
+the smallest and the largest component of d, respectively. Let λmax:= max j{µfj−xj
+dj}, and let
+w.l.o.g.λmax=µf1−x1
+d1. We have x≥µf−λmaxd, so we can apply Lemma 5.6 to obtain
+N(x)≥µf−1
+2λmaxd. Thus
+/ba∇⌈blN(x)−x/ba∇⌈bl∞≥(N(x)−x)1≥µf1−1
+2λmaxd1−x1=1
+2λmaxd1≥1
+2λmaxdmin.
+Ontheotherhand,with Lemma4.5wehave 0≤µf−N(x)≤1
+2λmaxdandso/ba∇⌈blµf−N(x)/ba∇⌈bl∞≤
+1
+2λmaxdmax. Combining those inequalities we obtain
+/ba∇⌈blN(x)−x/ba∇⌈bl∞
+/ba∇⌈blµf−N(x)/ba∇⌈bl∞≥dmin
+dmax.
+Now the statement follows from Lemma 5.11. ⊓ ⊔
+6 General SPPs
+In§5 we considered strongly connected SPPs, see Definition 2.6. However, it is not always
+guaranteed that the SPP fis strongly connected. In this section we analyze the convergence
+speed of two variants of Newton’s method that both compute appr oximations of µf, wheref
+is a clean and feasible SPP that is not necessarily strongly connected (“general SPPs”).
+29The first one was suggested by Etessami and Yannakakis [EY09] an d is called Decomposed
+Newton’s Method (DNM) . It works by running Newton’s method separately on each SCC, see
+§6.1. The second one is the regular Newton’s method from §4. We will analyze its convergence
+speed in §6.2.
+The reason why we first analyze DNM is that our convergence speed results about Newton’s
+method for general SPPs (Theorem 6.5) build on our results about D NM (Theorem 6.2). From
+an efficiency point of view it actually may be advantageous to run Newt on’s method separately
+on each SCC. For those reasons DNM deserves a separate treatm ent.
+6.1 Convergence Speed of the Decomposed Newton’s Method (DN M)
+DNM, originally suggested in [EY09], works as follows. It starts by usin g Newton’s method for
+each bottom SCC, say S, of the SPP f. Then the corresponding variables XSare substituted
+for the obtained approximation for µfS, and the corresponding equations XS=fS(X) are
+removed. The same procedure is then applied to the new bottom SCC s, until all SCCs have
+been processed.
+Etessami and Yannakakis did not provide a particular criterion for t he number of Newton
+iterations to be applied in each SCC. Consequently, they did not analy ze the convergence speed
+of DNM. We will treat those issues in this section, thereby taking adv antage of our previous
+analysis of scSPPs.
+We fix aquadratic,cleanand feasibleSPP fforthis section.We assumethat wehavealready
+computed the DAG (directed acyclic graph) of SCCs. This can be don e in linear time in the size
+off. To each SCC Swe can associate its deptht: it is the longest path in the DAG of SCCs from
+Sto a top SCC. Notice that 0 ≤t≤n−1. We write SCC(t) for the set of SCCs of depth t. We
+define the height h(f) as the largest depth of an SCC and the width w(f) := max t|SCC(t)|as
+the largestnumber of SCCs ofthe same depth. Notice that fhas at most ( h(f)+1)·w(f) SCCs.
+Further we define the component sets [ t] :=/uniontext
+S∈SCC(t)Sand [>t] :=/uniontext
+t′>t[t′] and similarly[ < t].
+function DNM (f,i) /* The parameter icontrols the precision. */
+fortfromh(f)downto 0
+forallS∈ SCC(t) /* for all SCCs Sof depth t*/
+ρ(i)
+S:=Ni·2t
+fS(0) /* perform i·2tNewton iterations */
+f[<t]:=f[<t][S/ρ(i)
+S] /* applyρ(i)
+Sin the upper SCCs */
+returnρ(i)
+Fig.4.Decomposed Newton’s Method (DNM) for computing an approxim ationρ(i)ofµf.
+Figure 4 showsour versionof DNM. We suggest to run Newton’s meth od in eachSCC Sfor a
+number of steps that depends (exponentially) on the depth of Sand (linearly) on a parameter i
+that controls the precision.
+Proposition 6.1. The function DNM(f,i)of Figure 4 runs at most
+i·w(f)·2h(f)+1≤i·n·2niterations of Newton’s method.
+Proof.The number of iterations is/summationtexth(f)
+t=0|SCC(t)|·i·2t. This can be estimated as follows.
+h(f)/summationdisplay
+t=0|SCC(t)|·i·2t≤w(f)·i·h(f)/summationdisplay
+t=02t
+30≤w(f)·i·2h(f)+1
+≤i·n·2n(asw(f)≤nandh(f)< n)
+⊓ ⊔
+The following theorem states that DNM has linear convergence orde r.
+Theorem 6.2. Letfbe a quadratic, clean and feasible SPP. Let ρ(i)denote the result of calling
+DNM(f,i)(see Figure 4). Let βρdenote the convergence order of (ρ(i))i∈N. Then there is a
+kf∈Nsuch that βρ(kf+i)≥ifor alli∈N.
+Theorem 6.2 can be interpreted as follows: Increasing iby one yields asymptotically at least
+one additional bit in each component and, by Proposition 6.1, costs a t mostn·2nadditional
+Newton iterations. Notice that for simplicity we do not take into acco unt here that the cost of
+performing a Newton step on a single SCC is not uniform, but rather d epends on the size of the
+SCC (e.g. cubically if Gaussian elimination is used for solving the linear sys tems).
+For the proofofTheorem6.2, let ∆(i)denote the errorwhen running DNM with parameter i,
+i.e.,∆(i):=µf−ρ(i). Observe that the error ∆(i)can be understood as the sum of two errors:
+∆(i):=µf−ρ(i)= (µ−/tildewideµ(i))+(/tildewideµ(i)−ρ(i)),
+where/tildewideµ(i)
+[t]:=µ/parenleftbig
+f[t][[>t]/ρ(i)
+[>t]]/parenrightbig
+, i.e.,/tildewideµ(i)
+[t]is the least fixed point of f[t]after the approximations
+from the lower SCCs have been applied. So, ∆(i)
+[t]consists of the propagation error (µf[t]−/tildewideµ(i)
+[t])
+(resulting from the error at lower SCCs) and the approximation error (/tildewideµ(i)
+[t]−ρ(i)
+[t]) (resulting
+from the newly added error of Newton’s method on level t).
+The following lemma gives a bound on the propagation error.
+Lemma 6.3 (Propagation error). There is a constant Cf>0such that
+/vextenddouble/vextenddouble/vextenddoubleµf[t]−/tildewideµ[t]/vextenddouble/vextenddouble/vextenddouble≤Cf·/radicalbigg/vextenddouble/vextenddouble/vextenddoubleµf[>t]−ρ[>t]/vextenddouble/vextenddouble/vextenddouble
+holds for all ρ[>t]with0≤ρ[>t]≤µf[>t], where/tildewideµ[t]=µ/parenleftbig
+f[t][[>t]/ρ[>t]]/parenrightbig
+.
+Roughly speaking, Lemma 6.3 states that if ρ(i)
+[>t]haskvalid bits of µf[>t], then/tildewideµ(i)
+[t]has at
+least about k/2 valid bits of µf[t]. In other words, (at most) one half of the valid bits are lost
+on each level of the DAG due to the propagation error. The proof o f Lemma 6.3 is technically
+involved and, unfortunately, not constructive in that we know not hing about Cfexcept for its
+existence. Therefore, the statements in this section are indepen dent of a particular norm. The
+proof of Lemma 6.3 can be found in Appendix A.
+The following lemma gives a bound on the error/vextenddouble/vextenddouble/vextenddouble∆(i)
+[t]/vextenddouble/vextenddouble/vextenddoubleon levelt, taking both the propaga-
+tion error and the approximation error into account.
+Lemma 6.4. There is a Cf>0such that/vextenddouble/vextenddouble/vextenddouble∆(i)
+[t]/vextenddouble/vextenddouble/vextenddouble≤2Cf−i·2tfor alli∈N.
+Proof.Let/tildewidef(i)
+[t]:=f[t][[>t]/ρ(i)
+[>t]]. Observe that the coefficients of /tildewidef(i)
+[t]and thus its least fixed
+point/tildewideµ(i)
+[t]are monotonically increasing with i, because ρ(i)
+[>t]is monotonically increasing as well.
+Consider an arbitrary depth tand choose real numbers cmin>0 andµmin>0 and an integer i0
+suchthat,forall i≥i0,cminandµminarelowerboundsonthesmallestnonzerocoefficientof /tildewidef(i)
+[t]
+and the smallest coefficient of /tildewideµ(i)
+[t], respectively. Let µmaxbe the largest component of µf[t].
+31Let/tildewidek:=/ceilingleftig
+n·logµmax
+cmin·µmin·min{µmin,1}/ceilingrightig
+. Then it follows from Theorem 5.9 that performing /tildewidek+j
+Newtoniterations( j≥0)ondepth tyieldsjvalidbitsof/tildewideµ(i)
+[t]foranyi≥i0.Inparticular, /tildewidek+i·2t
+Newton iterations give i·2tvalid bits of/tildewideµ(i)
+[t]for anyi≥i0. So there exists a constant c1>0
+such that, for all i≥i0,/vextenddouble/vextenddouble/vextenddouble/tildewideµ(i)
+[t]−ρ(i)
+[t]/vextenddouble/vextenddouble/vextenddouble≤2c1−i·2t, (13)
+because DNM (see Figure 4) performs i·2titerations to compute ρ(i)
+SwhereSis an SCC of
+deptht. Choose c1large enough such that Equation (13) holds for all i≥0 and all depths t.
+Now we can prove the theorem by induction on t. In the base case ( t=h(f)) there is no
+propagation error, so the claim of the lemma follows from (13). Let t < h(f). Then
+/vextenddouble/vextenddouble/vextenddouble∆(i)
+[t]/vextenddouble/vextenddouble/vextenddouble=/vextenddouble/vextenddouble/vextenddoubleµf[t]−/tildewideµ(i)
+[t]+/tildewideµ(i)
+[t]−ρ(i)
+[t]/vextenddouble/vextenddouble/vextenddouble
+≤/vextenddouble/vextenddouble/vextenddoubleµf[t]−/tildewideµ(i)
+[t]/vextenddouble/vextenddouble/vextenddouble+/vextenddouble/vextenddouble/vextenddouble/tildewideµ(i)
+[t]−ρ(i)
+[t]/vextenddouble/vextenddouble/vextenddouble
+≤/vextenddouble/vextenddouble/vextenddoubleµf[t]−/tildewideµ(i)
+[t]/vextenddouble/vextenddouble/vextenddouble+2c1−i·2t(by (13))
+≤c2·/radicalbigg/vextenddouble/vextenddouble/vextenddouble∆(i)
+[>t]/vextenddouble/vextenddouble/vextenddouble+2c1−i·2t(Lemma 6.3)
+≤c2·/radicalbig
+2c3−i·2t+1+2c1−i·2t(induction hypothesis)
+≤2c4−i·2t
+for some constants c2,c3,c4>0. ⊓ ⊔
+Now Theorem 6.2 follows easily.
+Proof (of Theorem 6.2). From Lemma 6.4 we deduce that for each component j∈[t] there is a
+cjsuch that
+(µfj−ρ(i)
+j)/µfj≤2cj−i·2t≤2cj−i.
+Letkf≥cjfor all 1≤j≤n. Then
+(µfj−ρ(i+kf)
+j)/µfj≤2cj−(i+kf)≤2−i.
+⊓ ⊔
+Notice that, unfortunately, we cannot give a bound on kf, mainly because Lemma 6.3 does
+not provide a bound on Cf.
+6.2 Convergence Speed of Newton’s Method
+We useTheorem6.2toprovethe followingtheoremforthe regular(i.e . notdecomposed)Newton
+sequence ( ν(i))i∈N.
+Theorem 6.5. Letfbe a quadratic, clean and feasible SPP. There is a threshold kf∈Nsuch
+thatβ(kf+i·n·2n)≥β(kf+i·(h(f)+1)·2h(f))≥ifor alli∈N.
+In the rest of the section we prove this theorem by a sequence of le mmata. The following
+lemma states that a Newton step is not faster on an SCC, if the value s of the lower SCCs are
+fixed.
+32Lemma 6.6. Letfbe a clean and feasible SPP. Let 0≤x≤f(x)≤µfsuch that f′(x)∗
+exists. Let Sbe an SCC of fand letLdenote the set of components that are not in S, but on
+which a variable in Sdepends. Then (Nf(x))S≥ NfS[L/xL](xS).
+Proof.
+(Nf(x))S=/parenleftbig
+f′(x)∗(f(x)−x)/parenrightbig
+S
+=f′(x)∗
+SS(f(x)−x)S+f′(x)∗
+SL(f(x)−x)L
+≥f′(x)∗
+SS(f(x)−x)S
+=/parenleftbig
+(fS[L/xL])′(xS)/parenrightbig∗(fS[L/xL](xS)−xS)
+=NfS[L/xL](xS)
+⊓ ⊔
+Recall Lemma 4.9 which states that the Newton operator Nis monotone. This fact and
+Lemma 6.6 can be combined to the following lemma stating that i·(h(f)+1) iterations of the
+regular Newton’s method “dominate” a decomposed Newton’s metho d that performs iNewton
+steps in each SCC.
+Lemma 6.7. Let/tildewideν(i)denote the result of a decomposed Newton’s method which perf ormsi
+iterations of Newton’s method in each SCC. Let ν(i)denote the result of iiterations of the
+regular Newton’s method. Then ν(i·(h(f)+1))≥/tildewideν(i).
+Proof.Leth=h(f). Let [t] and [>t] again denote the set of components of depth tand> t,
+respectively. We show by induction on the depth t:
+ν(i·(h+1−t))
+[t]≥/tildewideν(i)
+[t]
+The induction base ( t=h) is clear, because for bottom SCCs the two methods are identical.
+Let now t < h. Then
+ν(i·(h+1−t))
+[t]=Ni
+f(ν(i·(h−t)))[t]
+≥ Ni
+f[t][[>t]/ν(i·(h−t))
+[>t]](ν(i·(h−t))
+[t]) (Lemma 6.6)
+≥ Ni
+f[t][[>t]//tildewideν(i)
+[>t]](ν(i·(h−t))
+[t]) (induction hypothesis)
+≥ Ni
+f[t][[>t]//tildewideν(i)
+[>t]](0[t]) (Lemma 4.9)
+=/tildewideν(i)
+[t](definition of /tildewideν(i))
+Now, the lemma itself follows by using Lemma 4.9 once more. ⊓ ⊔
+As a side note, observe that above proof of Lemma 6.7 implicitly benefi ts from the fact
+that SCCs of the same depth are independent. So, SCCs with the sa me depth are handled in
+parallel by the regular Newton’s method. Therefore, w(f), the width of f, is irrelevant here (cf.
+Proposition 6.1).
+Now we can prove Theorem 6.5.
+Proof (of Theorem 6.5). Letk2be thekfof Theorem 6.2, and let k1=k2·(h(f)+1)·2h(f).
+Then we have
+ν(k1+i·(h(f)+1)·2h(f))=ν((k2+i)·(h(f)+1)·2h(f))
+33≥/tildewideν((k2+i)·2h(f))(Lemma 6.7)
+≥ρ(k2+i),
+where the last step follows from the fact that DNM( f,k2+i) runs at most ( k2+i)·2h(f)
+iterations in every SCC. By Theorem 6.2, ρ(k2+i)and hence ν(k1+i·(h(f)+1)·2h(f))haveivalid
+bits ofµf. Therefore, Theorem 6.5 holds with kf=k1. ⊓ ⊔
+7 Upper Bounds on the Convergence
+In this section we show that the lower bounds on the convergence o rder of Newton’s method
+that we obtained in the previoussection areessentially tight, meanin g that an exponential (in n)
+number of iterations may be needed per bit.
+More precisely, we expose a family/parenleftig
+f(n)/parenrightig
+n≥1of SPPs with nvariables, such that more than
+k·2n−1iterations are needed for kvalid bits. Consider the following system.
+X=f(n)(X) =
+1
+2+1
+2X2
+1
+1
+4X2
+1+1
+2X1X2+1
+4X2
+2
+...
+1
+4X2
+n−1+1
+2Xn−1Xn+1
+4X2
+n
+(14)
+The only solution of (14) is µf(n)= (1,...,1)⊤. Notice that each component of f(n)is an SCC.
+We prove the following theorem.
+Theorem 7.1. The convergence order of Newton’s method applied to the SPP f(n)from(14)
+(withn≥2) satisfies
+β(k·2n−1)< kfor allk∈ {1,2,...}.
+In particular, β(2n−1) = 0.
+Proof.We write f:=f(n)for simplicity. Let
+∆(i):=µf−ν(i)= (1,...,1)⊤−ν(i).
+Notice that ( ν(i)
+1)i∈N= (0,1
+2,3
+4,7
+8,...) which is the same sequence as obtained by applying
+Newton’s method to the 1-dimensional system X1=1
+2+1
+2X2
+1. So we have ∆(i)
+1= 2−i, i.e., after
+iiterations we have exactly ivalid bits in the first component.
+We know from Theorem 4.1 that for all jwith 1≤j≤n−1 we have ν(i)
+j+1≤fj+1(ν(i)) =
+1
+4(ν(i)
+j)2+1
+2ν(i)
+jν(i)
+j+1+1
+4(ν(i)
+j+1)2andν(i)
+j+1≤1. It follows that ν(i)
+j+1is at most the least solution
+ofXj+1=1
+4(ν(i)
+j)2+1
+2ν(i)
+jXj+1+1
+4(Xj+1)2, and so ∆(i)
+j+1≥2/radicalig
+∆(i)
+j−∆(i)
+j>/radicalig
+∆(i)
+j.
+By induction it follows that ∆(i)
+j+1>(∆(i)
+1)2−j. In particular,
+∆(k·2n−1)
+n>/parenleftig
+∆(k·2n−1)
+1/parenrightig2−(n−1)
+= 2−k·2n−1·2−(n−1)= 2−k.
+Hence, after k·2n−1iterations we have fewer than kvalid bits. ⊓ ⊔
+Notice that the proof exploits that an error in the first component gets “amplified” along
+the DAG of SCCs. One can also show along those lines that computing µfis anill-conditioned
+problem: Consider the SPP g(n,ε)obtained from f(n)by replacing the first component by 1 −ε
+where 0 ≤ε <1. Ifε= 0 then ( µg(n,ε))n= 1, whereas if ε=1
+22n−1then (µg(n,ε))n<1
+2. In
+other words, to get 1 bit of precision of µgone needs exponentially in nmany bits in g. Note
+that this observation is independent from any particular method to compute or approximate
+the least fixed point.
+348 Geometrical Aspects of SPPs
+As shown in §4.4 we can assume that fconsists of quadratic polynomials. For quadratic
+polynomials the locus of zeros is also called a quadric surface , or more commonly quadric.
+Quadrics are one of the most fundamental class of hypersurface s. It is therefore natural to study
+the quadrics induced by a quadratic SPP f, and how the Newton sequence is connected to these
+surfaces.
+Let us write qforf−X. Every component qiofqis also a quadratic polynomial each
+defining a quadric denoted by
+Qi:={x∈Rn|qi(x) =fi(x)−xi= 0}.
+Findingµfthus corresponds to finding the least non-negative point of inters ection of these n
+quadrics Qi.
+Example 8.1. Consider the SPP fgiven by
+f(X,Y) =/parenleftbigg1
+2X2+1
+4Y2+1
+41
+4X+1
+4XY+1
+4Y2+1
+4/parenrightbigg
+leading to
+q1(X,Y) =1
+2X2+1
+4Y2+1
+4−Xandq2(X,Y) =1
+4X+1
+4XY+1
+4Y2+1
+4−Y.
+Using standard techniques from linear algebra one can show that q1defines an ellipse while q2
+describes a parabola (see Figure 5). ⊓ ⊔
+PSfrag replacements
+XYµfPSfrag replacements
+YYµf
+(a) (b)
+Fig.5.(a) The quadrics induced by the SPP from Example 8.1 with “ q1= 0” an ellipse, and “ q2= 0” a
+parabola. (b) Close-up view of the region important for dete rmining µf. The crosses show the Newton
+approximants of µf.
+Figure 5 shows the two quadrics induced by the SPP fdiscussed in the example above. In
+Figure 5 (a) one can recognize one of the two quadrics as an ellipse wh ile the other one is a
+35parabola. In this example the Newton approximants (depicted as cr osses) stay within the region
+enclosed by the coordinate axes and the two quadrics as shown in Fig ure 5 (b).
+In this section we want to show that the above picture in principle is th e same for all clean
+and feasible scSPPs. That is, we show that the Newton (and Kleene) approximants always stay
+in the region enclosed by the coordinate axes and the quadrics. We c haracterize this region and
+study some of the properties of the quadrics restricted to this re gion. This eventually leads to
+a generaliztion of Newton’s method (Theorem 8.13). We close the sec tion by showing that this
+new method converges at least as fast as Newton’s method. All miss ing proofs can be found in
+the appendix.
+Let us start with the properties of the quadrics Qi. We restrict our attention to the region
+[0,µf). For this we set
+Mi:=Qi∩[0,µf) ={x∈[0,µf)|qi(x) = 0}.
+We start by showing that for every x∈Mithe gradient q′
+i(x) inxatMidoes not vanish.
+Asq′
+i(x) is perpendicular to the tangent plane in xatMi, this means that the normal of the
+tangent plane is determined by q′
+i(x) (up to orientation). See Figure 6 for an example. This will
+later allow us to apply the implicit function theorem.
+Fig.6.The normals (scaled down) of the quadrics from Example 8.1.
+Lemma 8.2. For every quadric qiinduced by a clean and feasible scSPP fwe have
+q′
+i(x) = (∂X1qi(x),∂X2qi(x),...,∂ Xnqi(x))/\⌉}a⊔io\slash=0and∂Xiqi(x)<0∀x∈[0,µf).
+In the following, for i∈ {1,...,n}we write x−ifor the vector ( x1,...,x i−1,xi+1,...,x n)
+and define ( x−i,xi) to also denote the original vector x.
+We next show that there exists a complete parametrization of “the lower part” of Mi. With
+“lower part” we refer to the set
+Si:={x∈Mi| ∀y∈Mi: (x−i=y−i)⇒xi≤yi},
+i.e., the points x∈Misuch that there is no point ywith the same non- i-componentsbut smaller
+i-component. Taking a look at Figure 5, the surfaces S1andS2are those parts of M1, resp.M2,
+which delimit that part of R2
+≥0shown in Figure 5 (b).
+36Ifx∈Sithenxiis the least non-negative root of the (at most) quadratic polynomia l
+qi(Xi,x−i). As we will see, these roots can also be represented by the followin g functions:
+Definition 8.3. For a clean and feasible scSPP fwe define for all k∈Nthe polynomial h(k)
+i
+by
+h(0)
+i(X−i) :=fi[i/0](X−i), h(k+1)
+i(X−i) :=fi[i/h(k)
+i(X−i)](X−i)
+The function hi(X)is then defined pointwise by
+hi(x−i) := lim
+k→∞h(k)
+i(x−i)
+for allx−i∈[0,µf−i].
+We showin the appendix (see PropositionB.1) that the function hiis well-defined and exists.
+We therefore can parameterize the surface Siw.r.t. the remaining variables X−i, i.e.,hiis the
+“height” of the surface Siabove the “ground” Xi= 0.
+By the preceding proposition the map
+pi: [0,µf−i)→[0,µf] :x−i/maps⊔o→(x1,...,x i−1,hi(x−i),xi+1,...,x n)
+givesusapointwiseparametrizationof Si.Wewanttoshowthat piiscontinuouslydifferentiable.
+For this it suffices to show that hiis continuously differentiable which follows easily from the
+implicit function theorem (see e.g. [OR70]).
+Lemma 8.4. hiis continuously differentiable with
+∂Xjhi(x−i) =∂Xjfi(x)
+−∂Xiqi(x)=∂Xjqi(x)
+−∂Xiqi(x)forx∈Siandj/\⌉}a⊔io\slash=i.
+In particular, ∂Xjhiis monotonically increasing with x.
+Corollary 8.5. The map
+pi: [0,µf−i)→[0,µf] :x−i/maps⊔o→(x1,...,x i−1,hi(x−i),xi+1,...,x n)
+is continuously differentiable and a local parametrization of the manifold Si.
+Example 8.6. For the SPP fdefined in Example 8.1 we can simply solve q1(X,Y) forXleading
+to
+h1(Y) = 1−/radicalbigg
+1
+2(1−Y2).
+The important point is that by the previous result we know that this f unction has to be defined
+on [0,µf2], and differentiable on [0 ,µf2). Similarly, we get
+h2(X) = 2−1
+2X−1
+2/radicalbig
+X2−12X+12.
+Figure 5 (b) conveys the impression that the surfaces Siare convex w.r.t. the parameteri-
+zationspi. As we have seen, the functions hiare monotonically increasing. Thus, in the case
+of two dimensions the functions hieven have to be strictly monotonically increasing (as fis
+strongly-connected),so that the surfaces Siare indeed convex. (Recall that a surface Sis convex
+in a point x∈SifSis located completely on one side of the tangent plane at Sinx.) But in
+the case of more than two variables this no longer needs to hold.
+37(a) (b) (c)
+Fig.7.(a) The hyperbolic paraboloid defined by Z=1
+8X2+3
+4XY+1
+8Y2+1
+4forX,Y,Z∈[−10,10].
+(b) A visualization of an SPP consisting of three copies of th e quadric of (a) with µf= (1
+2,1
+2,1
+2) the
+upper apex. (c) One of the three quadrics of (b) over [ 0,µf]. Clearly, even limited to this range the
+surface is not convex.
+Example 8.7. The equation
+Z=1
+8X2+3
+4XY+1
+8Y2+1
+4
+is an admissible part of any SPP. It defines the hyperbolic paraboloid d epicted in Figure 7 which
+is clearly not convex.
+Still, as shown in Lemma 2.3 it holds for all 0≤x≤ythat
+x+f′(x)·y≤f(x+y).
+It now follows (see the following lemma) that the surfaces Sihave the property that for every
+x∈[0,µf) the “relevant” part of Sifor determining µf, i.e.Si∩[x,µf], is located on the same
+side of the tangent plane at Siinx(see Figure 8).
+Fig.8.The graphic shows the quadric defined by q1= 0 with the tangent and normal in xatS1.
+Every point yofS1abovexis located on the same side of the the tangent. More precisely , we have
+∇q1|x·(y−x)≤0.
+38Lemma 8.8. For allx∈Siwe have
+∀y∈Si∩[x,µf] :q′
+i(x)·(y−x)≤0.
+In particular
+∀y∈Si∩[x,µf] :yi≥xi+/summationdisplay
+j/\e}atio\slash=i∂Xjhi(x−i)·(yj−xj).
+Consider now the set
+R:=n/intersectiondisplay
+i=1{x∈[0,µf)|xi≤hi(x−i)},
+i.e., the region of [ 0,µf) delimited by the coordinate axes and the surfaces Si. Note that the
+gradient q′
+i(x) forx∈Sipoints from SiintoR(see Figure 6).
+Proposition 8.9. It holds
+x∈R⇔x∈[0,µf)∧q(x)≥0.
+From this last result it now easily follows that Ris indeed the region of [ 0,µf) where all
+Newton and Kleene steps are located in.
+Theorem 8.10. Letfbe a clean and feasible scSPP. All Newton and Kleene steps sta rting
+from0lie within R, i.e.
+ν(i),κ(i)∈R(∀i∈N).
+Proof.For an scSPP we have κ(i),ν(i)∈[0,µf) for alli. Further, κ(i)≤κ(i+1)=f(κ(i)) and
+ν(i)≤f(ν(i)) holds for all i, too. ⊓ ⊔
+In the rest of this section we will use the results regarding Rand the surfaces Sifor inter-
+preting Newton’s method geometrically and for obtaining a generaliza tion of Newton’s method.
+The preceding results suggest another way of determining µf(see Figure 9): Let xbe some
+point inside of R. We may move from xontoone ofthe surface Siby goingupward alongthe line
+x+t·eiwhich gives us the point pi(x−i) = (x−i,hi(x−i)). Asx∈R, we have x,pi(x−i)≤µf.
+Consider now the tangent plane
+Ti|x=/braceleftbig
+y∈Rn|q′
+i(pi(x−i))·/parenleftbig
+y−pi(x−i)/parenrightbig
+= 0/bracerightbig
+atSiinpi(x−i). Recall that by Lemma 8.8 we have
+∀y∈Si∩[pi(x−i),µf) :q′
+i(pi(x−i))·(y−pi(x−i))≤0,
+i.e., the part of Sirelevant for determining µfis located completely below (w.r.t. q′
+i(()pi(x−i)))
+this tangent plane. By continuity this also has to hold for y=µf. Hence, when taking the
+intersection of all the tangent planes T1toTnthis gives us again a point T(x) inside of R. That
+this point T(x) exists and is uniquely determined is shown in the following lemma.
+Lemma 8.11. Letfbe a clean and feasible scSPP. Let x(1),...,x(n)∈[0,µf). Then the
+matrix 
+q′
+1(x(1))
+...
+q′
+n(x(n))
+
+is regular, i.e., the vectors {q′
+i(x(i))|i= 1,...,n}are linearly independent.
+39Fig.9.Given a point xinside of Rthe intersection of the tangents at the quadrics in the point sp1(x2),
+resp.p2(x1) is also located inside of R, yielding a better approximation of µf.
+By this lemma the normals at the quadrics in the points pi(x−i) forx∈[0,µf) are linearly
+independent. Thus,thereexistsauniquepointofintersectionoft angentplanesatthequadricsin
+these points. Of course, in general the values hi(x−i) can be irrational. The following definition
+takes this in account by only requiring that underapproximations ηiofhi(x−i) are known.
+Definition 8.12. Letx∈R. Fori= 1,...,nfix some ηi∈[xi,hi(x−i)], and set η=
+(η1,...,η n). We then let Tη(x)denote the solution of
+q′
+i((x−i,ηi))(X−(x−i,ηi)) =−qi((x−i,ηi)) (i= 1,...,n).
+We drop the subscript and simply write Tin the case of ηi=hi(x−i)fori= 1,...,n.
+Note that the operator Txis the Newton operator N.
+Theorem 8.13. Letfbe a clean and feasible scSPP. Let x∈R. Fori= 1,...,nfix some
+ηi∈[xi,hi(x−i)], and set η= (η1,...,η n). We then have
+x≤ N(x)≤ Tη(x)≤ T(x)≤µf
+Further, the operator Tis monotone on R, i.e., for any y∈Rwithx≤yit holds that
+T(x)≤ T(y).
+By Theorem8.13,replacingthe Newton operator NbyTgivesavariantofNewton’s method
+which converges at least as fast.
+We do not know whether this variant is substantially faster. See Figu re 10 for a geometrical
+interpretation of both methods.
+9 Conclusions
+We have studied the convergenceorder and convergencerate of Newton’s method for fixed-point
+equations of systems of positive polynomials (SPP equations). Thes e equations appear naturally
+in the analysis of several stochastic computational models that ha ve been intensely studied in
+recent years, and they also play a central rˆ ole in the theory of st ochastic branching processes.
+40(a) (b)
+(c) (d)
+Fig.10. Geometrical interpretation of Newton’s method: (a) Given a pointx∈RNewton’s methodfirst
+considers the “enlarged” quadrics defined by qi(X) =qi(x) (drawn dashed and dotted) which contain
+the current approximation x. (b) Then the tangents in xat these enlarged quadrics are computed
+(drawn dotted), i.e., q′
+i(x)·(X−x) = 0. (c) Finally, these tangents are corrected by moving the m
+towards the actual quadrics, i.e. q′
+i(x)·(X−x) =−qi(x). The intersection of these corrected tangents
+gives the next Newton approximation. (d) A comparison betwe enN(x) andT(x):N(x), resp.T(x) is
+given by the intersection of the dotted, resp. dashed lines. Clearly, we have N(x)≤ T(x).
+41The restriction to positive coefficients leads to strong results. For arbitrary polynomial equa-
+tions Newton’s method may not converge or converge only locally, i.e., when started at a point
+sufficiently close to the solution. We have extended a result by Etess ami and Yannakakis [EY09],
+and shown that for SPP equations the method always converges st arting at 0. Moreover, we
+have proved that the method has at least linear convergence orde r, and have determined the
+asymptotic convergence rate. To the best of our knowledge, this is the first time that a lower
+bound on the convergence order is proved for a significant class of equations with a trivial
+membership test.4Finally, in the case of strongly connected SPPs we have also obtained upper
+bounds on the threshold, i.e., the number of iterations necessary t o reach the “steady state” in
+which valid bits are computed at the asymptotic rate. These results lead to practical tests for
+checking whether the least fixed point of a strongly connected SPP exceeds a given bound.
+It is worth mentioning that in a recent paper we study the behavior o f Newton’s method
+when arithmetic operations only have a fixed accuracy [EGK10]. We de velop an algorithm for a
+relevant class of SPPs that computes iterations of Newton’s metho d increasing the accuracy on
+demand. A simple test applied after each iteration decides if the roun d-off errors have become
+too large, in which case the accuracy is increased.
+There are still at least two important open questions. The first one is, can one provide a
+bound on the threshold valid for arbitrary SPPs, and not only for st rongly connected ones?
+Since SPPs cannot be solved exactly in general, we cannot first comp ute the exact solution
+for the bottom SCCs, insert it in the SCCs above them, and iterate. We can only compute an
+approximation, and we are not currently able to bound the propaga tion of the error. For the
+second question, say that Newton’s method is polynomial for a class of SPP equations if there
+is a polynomial p(x,y,z) such that for every k≥0 and for every system in the class with n
+equations and coefficients of size m, thep(n,m,k)-th Newton approximant ν(p(n,m,k))hask
+valid bits. We have proved in Theorem 5.12 that Newton’s method is poly nomial for strongly
+connected SPPs fsatisfying f(0)≻0; for this class one can take p(n,m,k) = 7mn+k. We
+have also exhibited in §7 a class for which computing the first bit of the least solution takes
+2niterations. The members of this class, however, are not strongly c onnected, and this is the
+fact we have exploited to construct them. So the following question remains open: Is Newton’s
+method polynomial for strongly connected SPPs?
+Acknowledgments. We thank Kousha Etessami for several illuminating discussions, and
+two anonymous referees for helpful suggestions.
+A Proof of Lemma 6.3
+TheproofofLemma6.3isbyasequenceoflemmata.TheproofofLem maA.1and,consequently,
+the proof of Lemma 6.3 are non-constructive in the sense that we c annot give a particular Cf.
+Therefore, we often use the equivalence of norms, disregard the constants that link them, and
+state the results in terms of an arbitrary norm.
+The following two Lemmata A.1 and A.2 provide a lower bound on /ba∇⌈blf(x)−x/ba∇⌈blfor an
+“almost-fixed-point” x.
+Lemma A.1. Letfbe a quadratic, clean and feasible SPP without linear terms, i.e.,f(X) =
+B(X,X)+cwhereBis a bilinear map, and cis a constant vector. Let f(X)be non-constant
+4Notice the contrast with the classical result stating thati f (Id−f′(µf))is non-singular, thenNewton’s
+method has exponential convergence order; here the members hip test is highly non-trivial, and, for
+what we know, as hard as computing µfitself.
+42in every component. Let R˙∪S={1,...,n}withS/\⌉}a⊔io\slash=∅. Let every component depend on every
+S-component and not on any R-component. Then there is a constant Cf>0such that
+/ba∇⌈blf(µf−δ)−(µf−δ)/ba∇⌈bl ≥Cf·/ba∇⌈blδ/ba∇⌈bl2
+for allδwith0≤δ≤µf.
+Proof.With the given component dependencies we can write f(X) as follows:
+f(X) =/parenleftbigg
+fR(X)
+fS(X)/parenrightbigg
+=/parenleftbigg
+BR(XS,XS)+cR
+BS(XS,XS)+cS/parenrightbigg
+A straightforward calculation shows
+e(δ) :=f(µf−δ)−(µf−δ) = (Id−f′(µf))δ+B(δ,δ).
+Furthermore, ∂XRfis constant zero in all entries, so
+eR(δ) =δR−∂XSfR(µf)·δS+BR(δS,δS) and
+eS(δ) =δS−∂XSfS(µf)·δS+BS(δS,δS).
+Notice that for every real number r >0 we have
+min
+0≤δ≤µf,/bardblδ/bardbl≥r/ba∇⌈ble(δ)/ba∇⌈bl
+/ba∇⌈blδ/ba∇⌈bl2>0,
+because otherwise µf−δ< µfwould be a fixed point of f. We have to show:
+inf
+0≤δ≤µf,/bardblδ/bardbl>0/ba∇⌈ble(δ)/ba∇⌈bl
+/ba∇⌈blδ/ba∇⌈bl2>0
+Assume,foracontradiction,thatthisinfimumequalszero. Thenth ereexistsasequence( δ(i))i∈N
+with0≤δ(i)≤µf,/vextenddouble/vextenddouble/vextenddoubleδ(i)/vextenddouble/vextenddouble/vextenddouble>0 such that lim i→∞/vextenddouble/vextenddouble/vextenddoubleδ(i)/vextenddouble/vextenddouble/vextenddouble= 0 and lim i→∞/ba∇⌈ble(δ(i))/ba∇⌈bl
+/ba∇⌈blδ(i)/ba∇⌈bl2= 0. Define
+r(i):=/vextenddouble/vextenddouble/vextenddoubleδ(i)/vextenddouble/vextenddouble/vextenddoubleandd(i):=δ(i)
+/ba∇⌈blδ(i)/ba∇⌈bl. Notice that d(i)∈ {d∈Rn
+≥0| /ba∇⌈bld/ba∇⌈bl= 1}=:DwhereDis
+compact. So some subsequence of ( d(i))i∈N, say w.l.o.g. the sequence ( d(i))i∈Nitself, converges
+to some vector d∗∈D. By our assumption we have
+/vextenddouble/vextenddouble/vextenddoublee(δ(i))/vextenddouble/vextenddouble/vextenddouble//vextenddouble/vextenddouble/vextenddoubleδ(i)/vextenddouble/vextenddouble/vextenddouble2
+=/vextenddouble/vextenddouble/vextenddouble/vextenddouble1
+r(i)(Id−f′(µf))d(i)+B(d(i),d(i))/vextenddouble/vextenddouble/vextenddouble/vextenddouble−→0. (15)
+AsB(d(i),d(i)) is bounded,1
+r(i)(Id−f′(µf))d(i)must be bounded, too. Since r(i)converges to
+0,/vextenddouble/vextenddouble/vextenddouble(Id−f′(µf))d(i)/vextenddouble/vextenddouble/vextenddoublemust converge to 0, so
+(Id−f′(µf))d∗=0.
+In particular,/parenleftbig
+(Id−f′(µf))d∗/parenrightbig
+R=d∗
+R−∂XSfR(µf)·d∗
+S=0. So we have d∗
+S>0, because
+d∗
+S=0would imply d∗
+R=0which would contradict d∗>0.
+In the remainder of the proof we focus on fS. Define the scSPP g(XS) :=fS(X). Notice
+thatµg=µfS. We can apply Lemma 5.3 to gandd∗
+Sand obtain d∗
+S≻0. AsfS(X) is non-
+constant we get BS(d∗
+S,d∗
+S)≻0. By (15),1
+r(i)(Id−g′(µg))d(i)
+Sconverges to −BS(d∗
+S,d∗
+S)≺0.
+43So there is a j∈Nsuch that (Id −g′(µg))d(j)
+S≺0. Let/tildewideδ:=rd(j)for some small enough r >0
+such that 0</tildewideδS≤µgand
+eS(/tildewideδ) = (Id−g′(µg))/tildewideδS+BS(/tildewideδS,/tildewideδS)
+=r(Id−g′(µg))d(j)
+S+r2BS(d(j)
+S,d(j)
+S)≺0.
+So we have g(µg−/tildewideδS)≺µg−/tildewideδS. However, µgis the least point xwithg(x)≤x. Thus we
+get the desired contradiction. ⊓ ⊔
+Lemma A.2. Letfbe a quadratic, clean and feasible scSPP. Then there is a cons tantCf>0
+such that
+/ba∇⌈blf(µf−δ)−(µf−δ)/ba∇⌈bl ≥Cf·/ba∇⌈blδ/ba∇⌈bl2
+for allδwith0≤δ≤µf.
+Proof.Writef(X) =B(X,X) +LX+cfor a bilinear map B, a matrix Land a constant
+vectorc. By Theorem 4.1.2. the matrix L∗= (Id−L)−1= (Id−f′(0))−1exists. Define the
+SPP/tildewidef(X) :=L∗B(X,X) +L∗c. A straightforward calculation shows that the sets of fixed
+points of fand/tildewidefcoincide and that
+f(µf−δ)−(µf−δ) = (Id−L)/parenleftig
+/tildewidef(µf−δ)−(µf−δ)/parenrightig
+.
+Further, if σn(Id−L) denotes the smallest singular value of Id −L, we have by basic facts about
+singular values (see [HJ91], Chapter 3) that
+/vextenddouble/vextenddouble/vextenddouble(Id−L)/parenleftig
+/tildewidef(µf−δ)−(µf−δ)/parenrightig/vextenddouble/vextenddouble/vextenddouble
+2≥σn(Id−L)/vextenddouble/vextenddouble/vextenddouble/tildewidef(µf−δ)−(µf−δ)/vextenddouble/vextenddouble/vextenddouble
+2.
+Note that σn(Id−L)>0 because Id −Lis invertible. So it suffices to show that
+/vextenddouble/vextenddouble/vextenddouble/tildewidef(µf−δ)−(µf−δ)/vextenddouble/vextenddouble/vextenddouble≥Cf·/ba∇⌈blδ/ba∇⌈bl2.
+Iff(X) is linear (i.e. B(X,X)≡0) then/tildewidef(X) is constant and we have/vextenddouble/vextenddouble/vextenddouble/tildewidef(µf−δ)−(µf−δ)/vextenddouble/vextenddouble/vextenddouble=/ba∇⌈blδ/ba∇⌈bl, so we are done in that case. Hence we can assume that some
+component of B(X,X) is not the zero polynomial. It remains to argue that /tildewidefsatisfies the
+preconditions of Lemma A.1. By definition, /tildewidefdoes not have linear terms. Define
+S:={i|1≤i≤n, Xiis contained in a component of B(X,X)}.
+Notice that Sis non-empty. Let i0,i1,...,im,im+1(m≥0) be any sequence such that, in f, for
+alljwith 0≤j < mthe component ijdepends directly on ij+1viaa linear term and imdepends
+directly on im+1via a quadratic term. Then i0depends directly on im+1via a quadratic term
+inLmB(X,X) and hence also in /tildewidef. So all components are non-constant and depend (directly
+or indirectly) on every S-component. Furthermore, no component depends on a compone nt that
+is not in S, because L∗B(X,X) contains only S-components. Thus, Lemma A.1 can be applied,
+and the statement follows. ⊓ ⊔
+The following lemma gives a bound on the propagation error for the ca se thatfhas a single
+top SCC.
+44Lemma A.3. Letfbe a quadratic, clean and feasible SPP. Let S⊆ {1,...,n}be the single
+top SCC of f. LetL:={1,...,n}\S. Then there is a constant Cf≥0such that
+/ba∇⌈blµfS−/tildewideµS/ba∇⌈bl ≤Cf·/radicalbig
+/ba∇⌈blµfL−xL/ba∇⌈bl
+for allxLwith0≤xL≤µfLwhere/tildewideµS:=µ(fS[XL/xL]).
+Proof.We write fS(X) =fS(XS,XL) in the following.
+IfSis a trivial SCC then µfS=fS(0,µfL) and/tildewideµS=fS(0,xL). In this case we have with
+Taylor’s theorem (cf. Lemma 2.3)
+/ba∇⌈blµfS−/tildewideµS/ba∇⌈bl=/ba∇⌈blfS(0,µfL)−fS(0,xL)/ba∇⌈bl
+≤ /ba∇⌈bl∂XfS(0,µfL)·(µfL−xL)/ba∇⌈bl
+≤ /ba∇⌈bl∂XfS(0,µfL)/ba∇⌈bl·/ba∇⌈blµfL−xL/ba∇⌈bl
+=/ba∇⌈bl∂XfS(0,µfL)/ba∇⌈bl·/radicalbig
+/ba∇⌈blµfL−xL/ba∇⌈bl·/radicalbig
+/ba∇⌈blµfL−xL/ba∇⌈bl
+≤ /ba∇⌈bl∂XfS(0,µfL)/ba∇⌈bl·/radicalbig
+/ba∇⌈blµfL/ba∇⌈bl·/radicalbig
+/ba∇⌈blµfL−xL/ba∇⌈bl
+and the statement follows by setting Cf:=/ba∇⌈bl∂XfS(0,µfL)/ba∇⌈bl·/radicalbig
+/ba∇⌈blµfL/ba∇⌈bl.
+Hence, in the following we can assume that Sis a non-trivial SCC. Set g(XS) :=
+fS(XS,µfL). Notice that gis an scSPP with µg=µfS. By applying Lemma A.2 to gand
+settingc:= 1//radicalbig
+Cg(theCgfrom Lemma A.2) we get
+/ba∇⌈blµfS−/tildewideµS/ba∇⌈bl ≤c·/radicalbig
+/ba∇⌈blg(µg−(µfS−/tildewideµS))−(µg−(µfS−/tildewideµS))/ba∇⌈bl
+=c·/radicalbig
+/ba∇⌈blfS(/tildewideµS,µfL)−/tildewideµS/ba∇⌈bl
+=c·/radicalbig
+/ba∇⌈blfS(/tildewideµS,µfL)−fS(/tildewideµS,xL)/ba∇⌈bl
+and with Taylor’s theorem (cf. Lemma 2.3)
+≤c·/radicalbig
+/ba∇⌈bl∂XLfS(/tildewideµS,µfL)(µfL−xL)/ba∇⌈bl
+≤c·/radicalbig
+/ba∇⌈bl∂XLfS(µfS,µfL)(µfL−xL)/ba∇⌈bl
+≤c·/radicalbig
+/ba∇⌈bl∂XLfS(µfS,µfL)/ba∇⌈bl·/radicalbig
+/ba∇⌈blµfL−xL/ba∇⌈bl.
+So the statement follows by setting Cf:=c·/radicalbig
+/ba∇⌈bl∂XLfS(µfS,µfL)/ba∇⌈bl. ⊓ ⊔
+Now we can extend Lemma A.3 to Lemma 6.3, restated here.
+Lemma 6.3. There is a constant Cf>0such that
+/vextenddouble/vextenddouble/vextenddoubleµf[t]−/tildewideµ[t]/vextenddouble/vextenddouble/vextenddouble≤Cf·/radicalbigg/vextenddouble/vextenddouble/vextenddoubleµf[>t]−ρ[>t]/vextenddouble/vextenddouble/vextenddouble
+holds for all ρ[>t]with0≤ρ[>t]≤µf[>t], where/tildewideµ[t]=µ/parenleftbig
+f[t][[>t]/ρ[>t]]/parenrightbig
+.
+Proof.Observe that µf[t],/tildewideµ[t],µf[>t]andρ[>t]do not depend on the components of depth < t.
+So we can assume w.l.o.g. that t= 0. Let SCC(0) ={S1,...,S k}.
+For anySifromSCC(0), letf(i)be obtained from fby removing all top SCCs except for Si.
+Lemma A.2 applied to f(i)guarantees a C(i)such that
+/vextenddouble/vextenddoubleµfSi−/tildewideµSi/vextenddouble/vextenddouble≤C(i)·/radicalbigg/vextenddouble/vextenddouble/vextenddoubleµf[>0]−ρ[>0]/vextenddouble/vextenddouble/vextenddouble
+45holds for all ρ[>0]with0≤ρ[>0]≤µf[>0]. Using the equivalence of norms let w.l.o.g. the norm
+/ba∇⌈bl·/ba∇⌈blbe the maximum-norm /ba∇⌈bl·/ba∇⌈bl∞. LetCf:= max 1≤i≤kC(i). Then we have
+/vextenddouble/vextenddouble/vextenddoubleµf[0]−/tildewideµ[0]/vextenddouble/vextenddouble/vextenddouble= max
+1≤i≤k/vextenddouble/vextenddoubleµfSi−/tildewideµSi/vextenddouble/vextenddouble≤Cf·/radicalbigg/vextenddouble/vextenddouble/vextenddoubleµf[>0]−ρ[>0]/vextenddouble/vextenddouble/vextenddouble
+for allρ[>0]with0≤ρ[>0]≤µf[>0]. ⊓ ⊔
+B Proofs of §8
+B.1 Proof of Lemma 8.2
+Lemma 8.2. For every quadric qiinduced by a clean and feasible scSPP fwe have
+q′
+i(x) = (∂X1qi(x),∂X2qi(x),...,∂ Xnqi(x))/\⌉}a⊔io\slash=0and∂Xiqi(x)<0∀x∈[0,µf).
+Proof.As shown by Etessami and Yannakakis in [EY09] under the above pre conditions it holds
+for allx∈[0,µf) that/parenleftbig
+Id−f′(x)/parenrightbig
+is invertible with
+/parenleftbig
+Id−f′(x)/parenrightbig−1=f′(x)∗.
+Thus, we have
+q′(x)−1=/parenleftbig
+f′(x)−Id/parenrightbig−1=−/parenleftbig
+f′(x)∗/parenrightbig
+,
+implying that q′
+i(x)/\⌉}a⊔io\slash=0for allx∈[0,µf) asq′(x) has to have full rank nin order for q′(x)−1
+to exist. Furthermore, it follows that all entries of q′(x)−1are non-positive as f′(x)∗is non-
+negative. Now, as qi(X) =fi(X)−Xiandfi(X) is a polynomial with non-negative coefficients,
+it holds that
+q′
+i(x)·ej=∂Xjqi(x) =∂Xjfi(x)≥0
+for allj/\⌉}a⊔io\slash=iandx≥0. With every entry of q′(x)−1non-positive, and
+q′
+i(x)·q′(x)−1=e⊤
+i,
+we conclude ∂Xiqi(x)<0. ⊓ ⊔
+B.2 Proof of Lemma 8.4
+We first summarize some properties of the functions hi:
+Proposition B.1. Letfbe a clean and feasible scSPP. Let x,y∈[0,µf]withx≤y.
+(a)0≤h(k)
+i(x−i)≤µfi.
+(b)h(k)
+i(x−i)≤h(k+1)
+i(x−i)for allk∈N.
+(c)h(k)
+i(x−i)≤h(k)
+i(y−i)for allk∈N.
+(d)hi(x−i)≤µfi, andhiis a map from [0,µf−i]to[0,µfi].
+Iffidepends on at least one other variable except Xi, we also have hi([0,µf−i))⊆[0,µfi).
+(e)hi(x−i)≤hi(y−i).
+(f)fi(x−i,hi(x−i)) =hi(x−i).
+(g) Forxi=fi(x)we have hi(x−i)≤xi.
+46(h)hi(µf−i) =µfi.
+Proof.Let0≤x≤y≤µf. Using the monotonicity of fioverRn
+≥0we proceed by induction on
+k.
+(a) Fork= 0 we have
+0≤h(0)
+i(x−i) =fi(0,x−i)≤fi(µf) =µfi.
+We then get
+0≤h(k+1)
+i(x−i) =fi(h(k)
+i(x−i),x−i)≤fi(µf) =µfi.
+(b) Fork= 0 we have
+h(0)
+i(x−i) =fi(0,x−i)≤fi(h(0)
+i(x−i),x−i) =h(1)
+i(x−i).
+Thus
+h(k+1)
+i(x−i) =fi(h(k)
+i(x−i),x−i)≤fi(h(k+1)
+i(x−i),x−i) =h(k+2)
+i(x−i)
+follows.
+(c) Asx≤y, we have for k= 0
+h(0)
+i(x−i) =fi(0,x−i)≤fi(0,y−i) =h(0)
+i(y−i).
+Hence, we get
+h(k+1)
+i(x−i) =fi(h(k)
+i(x−i),x−i)≤fi(h(k)
+i(y−i),y−i) =h(k+1)
+i(y−i).
+(d) As the sequence( h(k)
+i(x−i))k∈Nis monotonicallyincreasingand bounded fromaboveby µfi,
+the sequence converges. Thus, for every xthe value
+hi(x−i) = lim
+k→∞h(k)
+i(x−i)
+is well-defined, i.e., hiis a map from [ 0,µf−i] to [0,µfi].
+Iffidepends on at least one other variable except Xi, thenhiis a non-constant power series
+in this variable with non-negative coefficients. For x−i∈[0,µf−i) we thus always have
+hi(x−i)< hi(µf−i) =µfi
+asx−i≺µf−i.
+(e) This follows immediately from (b).
+(f) Asfiis continuous, we have
+fi(hi(x−i),x−i) =fi( lim
+k→∞h(k)
+i(x−i),x−i) = lim
+k→∞h(k+1)
+i(x−i) =hi(x−i),
+where the last equality holds because of (b).
+(g) Using induction similar to (a) replacing µfbyx, one gets h(k)
+i(x−i)≤xifor allk∈Nas
+fi(x−i) =xi. Thus,hi(x−i)≤xifollows similarly to (d).
+(h) By definition, we have µf= limk→∞fk(0). Fork= 0, we have
+(f0(0))i= 0≤fi(0,µf−i) =h(0)
+i(µf−i).
+We thus get by induction
+(f(k+1)(0))i=fi(fk(0))≤fi(h(k)
+i(µf−i),µf−i) =h(k+1)
+i(µf−i).
+Thus, we may conclude µfi≤hi(µf−i). Asµfi=fi(µf), we get by virtue of (g) that
+hi(µf−i)≤µfi, too.
+47⊓ ⊔
+With Proposition B.1 at hand, we now can show Lemma 8.4:
+Lemma 8.4. hiis continuously differentiable with
+∂Xjhi(x−i) =∂Xjfi(x)
+−∂Xiqi(x)=∂Xjqi(x)
+−∂Xiqi(x)forx∈Siandj/\⌉}a⊔io\slash=i.
+In particular, ∂Xjhiis monotonically increasing with x.
+Proof.By Lemma 8.2 the implicit function theorem is applicable for every x∈Si. We therefore
+findforevery x∈Sialocalparametrization hx:U/maps⊔o→Vwithhx(x−i) =xi.Thushx(x−i)isthe
+least non-negative solution of qi(Xi,x−i) = 0. By continuity of qiit is now easily shown that for
+ally−i∈Uit has to hold that hx(y−i) is also the least non-negative solution of qi(Xi,y−i) = 0
+(see below). By uniqueness we therefore have hx=hiand that hiis continuously differentiable
+for allx−i∈[0,µf−i).
+For every x−i∈[0,µf−i) we can solve the (at most) quadratic equation qi(Xi,x−i) = 0.
+We already know that hi(x−i) is the least non-negative solution of this equation. So, if there
+exists another solution, it has to be real, too.
+Assume first that this equation has two distinct solutions for some fi xedx−i∈[0,µf−i).
+Solvingqi(Xi,x−i) = 0 thus leads to an expression of the form
+−b(x−i)±/radicalbig
+b(x−i)2−4a·c(x−i)
+2a
+for the solutions where b,care (at most) quadratic polynomials in X−i,chaving non-negative
+coefficients, and ais a positive constant (leading coefficient of X2
+iinqi(X)). Asbandcare
+continuous,the discriminant b(·)2−4a·c(·) stayspositive forsome open ball around x−iincluded
+inside of U(it is positive in x−ias we assume that we have two distinct solutions). By making U
+smaller, we may assume that Uis this open ball. One of the two solutions must then be the least
+nonnegative solution. As hxis the least non-negative solution for x−i, andhxis continuous,
+this also has to hold for some open ball centered at x−i. W.l.o.g., Uis this ball. So, hxandhi
+coincide on U.
+We turn to the case that qi(Xi,x−i) = 0 has only a single solution, i.e. hi(x−i). Note
+thatqi(X) is linear in Xiif and only if qi(Xi,x−i) is linear in Xi. Obviously, if qilinear in
+Xi, thenhiandhxcoincide on U. Thus, consider the case that qi(X) is quadratic in Xi,
+butqi(Xi,x−i) has only a single solution. This means that x−iis a root of the discriminant,
+i.e.b(x−i)−4ac(x−i) = 0. As hi(y−i) is a solution of qi(Xi,y−i) = 0 for all y−i∈U, the
+discriminant is non-negativeon U. If it equal to zero on U, then we again havethat hiis equal to
+hxonU. Thereforeassumethat is positive in some point of U. As the discriminantis continuous,
+the solutions change continuously with x−i. But this implies that for some y−i∈Uthere are at
+least two yi,y∗
+i∈Vsuch that ( y−i,yi) and (y−i,y∗
+i) are both located on the quadric qi(X) = 0.
+But this contradicts the uniqueness of hxguaranteed by the implicit function theorem.
+Assume now that x∈Si. We then have
+qi(x) =qi(x−i,hi(x−i)) = 0,
+or equivalently
+fi(x−i,hi(x−i)) =hi(x−i).
+Calculating the gradient of both in xyields
+f′
+i(x)·p′
+i(x−i) =h′
+i(x−i).
+48For the Jacobian of piwe obtain
+p′
+i(x−i) =
+e⊤
+1
+...
+e⊤
+i−1
+h′
+i(x−i)
+e⊤
+i+1
+...
+e⊤
+n
+.
+This leads to
+∂Xjfi(x)+∂Xifi(x)·∂Xjhi(x−i) =∂Xjhi(x−i)
+which solved for ∂Xjhiyields
+∂Xjhi(x−i) =∂Xjfi(x)
+−∂Xiqi(x).
+As∂Xiqi(x)<0 and both ∂Xjfiand∂Xiqimonotonically increase with x, it follows that ∂Xjhi
+alsomonotonicallyincreaseswith x.Finally,for j/\⌉}a⊔io\slash=iwehavethat ∂Xjqi=∂Xjfiasqi=fi−Xi.
+⊓ ⊔
+B.3 Proof of Lemma 8.8
+Lemma 8.8. For allx∈Siwe have
+∀y∈Si∩[x,µf] :q′
+i(x)·(y−x)≤0.
+In particular
+∀y∈Si∩[x,µf] :yi≥xi+/summationdisplay
+j/\e}atio\slash=i∂Xjhi(x−i)·(yj−xj).
+Proof.Letx∈Si, i.e.fi(x) =xi. We want to show that
+q′
+i(x)·(y−x)≤0
+for ally∈Si∩[x,µf). Asfiis quadratic in X, we may write
+0 =qi(y)
+=−yi+fi(y)
+=−yi+fi(x)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+=xi+f′
+i(x)·(y−x)+(y−x)⊤·A·(y−x)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+≥0
+≥ −yi+xi+f′
+i(x)·(y−x)
+=f′
+i(x)·(y−x)−e⊤
+i·(y−x)
+=q′
+i(x)·(y−x)
+whereAis a symmetric square-matrix with non-negative components such t hat the quadric
+terms of fiare given by X⊤AX.
+The second claim is easily obtained by solving this inequality for yiand recalling that by
+Lemma 8.4 we have ∂Xjhi(x−i) =∂Xjqi(pi(x−i))
+−∂Xiqi(pi(x−i))and∂Xiqi(pi(x−i))<0. ⊓ ⊔
+49B.4 Proof of Proposition 8.9
+Proposition 8.9. It holds
+x∈R⇔x∈[0,µf)∧q(x)≥0.
+Proof.Letx∈Randi∈ {1,...,n}. Consider the function
+g(t) :=qi(pi(x−i)+tei).
+Asqiisaquadraticpolynomialin Xthereexistsasymmetricsquare-matrix Awithnon-negative
+entries, a vector b, and a constant csuch that
+qi(X) =X⊤AX+b⊤X+c.
+It then follows that
+qi(X+Y) =qi(X)+q′
+i(X)Y+Y⊤AY.
+Withqi(pi(x−i)) = 0 this implies
+g(t) =q′
+i(pi(x−i))tei+t2e⊤
+iAei/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+:=a≥0=t·(∂Xiqi(pi(x−i))+a·t).
+Aspi(x−i)≺µf(fis strongly connected and x∈[0,µf)), we know that ∂Xiqi(pi(x−i))<0.
+Thus,g(t) has at most two zeros, one at 0, the other for some t∗≥0.
+For the direction ( ⇒) we only have to show that xi≤hi(x−i) implies that qi(x)≥0. This
+now easily follows as xi≤hi(x−i) implies that there is a t′≤0 withpi(x−i)+tei=x. But for
+thist′≤0 we have qi(x) =g(t′)≥0.
+Consider therefore the other direction ( ⇐), that is x∈[0,µf) withq(x)≥0. Assume that
+x/\⌉}a⊔io\slash∈R, i.e., for at least one iwe have xi> hi(x−i). Asqi(x)≥0 there has to be a t′′>0 with
+pi(x−i)+t′′ei=xandg(t′′)≥0. This implies that a >0 has to hold as otherwise g(t) would
+be linear in tand negative for t >0. But then the second root t∗ofg(t) has to be positive. Set
+x∗=pi(x−i)+t∗eiwithqi(x∗) = 0, too.
+A calculation similar to the one from above leads to
+g(t+t∗) =qi(x∗+tei) =t·(∂Xiqi(x∗)+a·t).
+It follows that ∂Xiqi(x∗) has to be greater than zero for −t∗to be a root (as a >0). But we
+have shown that ∂Xiqi(x)<0 for allx∈[0,µf). ⊓ ⊔
+B.5 Proof of Lemma 8.11
+Lemma 8.11. Letfbe a clean and feasible scSPP. Let x(1),...,x(n)∈[0,µf). Then the
+matrix
+q′
+1(x(1))
+...
+q′
+n(x(n))
+
+is regular, i.e., the vectors {q′
+i(x(i))|i= 1,...,n}are linearly independent.
+50Proof.Definex∈[0,µf) by setting
+xi:= max{x(j)
+i|j= 1,...,n}.
+We then have x(i)≤xfor alli, andx≺µf. As mentioned above, we therefore have that q′(x)
+is regular with
+q′(x)−1=−/summationdisplay
+k∈Nf′(x)k.
+Asx(i)≤xit follows that
+f′
+1(x(1))
+...
+f′
+n(x(n))
+≤f′(x).
+Hence, we also have
+l/summationdisplay
+k=0
+f′
+1(x(1))
+...
+f′
+n(x(n))
+l
+≤l/summationdisplay
+k=0f′(x)
+implying that
+f′
+1(x(1))
+...
+f′
+n(x(n))
+∗
+and, thus,
+q′
+1(x(1))
+...
+q′
+n(x(n))
+−1
+exist.
+So, the vectors {q′
+1(x(1)),...,q′
+n(x(n))}have to be linearly independent. ⊓ ⊔
+B.6 Proof of Theorem 8.13
+Theorem 8.13. Letfbe a clean and feasible scSPP. Let x∈R. Fori= 1,...,nfix some
+ηi∈[xi,hi(x−i)], and set η= (η1,...,η n). We then have
+x≤ N(x)≤ Tη(x)≤ T(x)≤µf
+Further, the operator Tis monotone on R, i.e., for any y∈Rwithx≤yit holds that
+T(x)≤ T(y).
+Proof.Set
+πi:= (x−i,ηi) andh:= (h1(x−1),...,h n(x−n)).
+We first show that x≤ Tη(x):
+Tη(x) =/parenleftbig
+q′
+i(πi)/parenrightbig
+)−1
+i=1,...,n·/parenleftbig
+q′
+i(πi)·πi−qi(πi)/parenrightbig
+i=1,...,n
+=/parenleftbig
+f′
+i(πi)/parenrightbig∗
+i=1,...,n·/parenleftbig
+−q′
+i(πi)·πi+qi(πi)/parenrightbig
+i=1,...,n
+=/parenleftbig
+f′
+i(πi)/parenrightbig∗
+i=1,...,n·/parenleftbig
+−q′
+i(πi)·(x+(ηi−xi)·ei)+qi(πi)/parenrightbig
+i=1,...,n
+=/parenleftbig
+f′
+i(πi)/parenrightbig∗
+i=1,...,n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+≥0 in every comp.·/parenleftbig
+−q′
+i(πi)·x−∂Xiqi(πi)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+<0·(ηi−xi)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+≥0+qi(πi)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+≥0/parenrightbig
+i=1,...,n
+≥x.
+Tη(x) is by definition the (unique) solution of the equation system defined by
+q′
+i(πi)(X−πi) =−qi(πi) (i= 1,...,n).
+51AsTη(x)≥xwe can also consider this system with the origin of the coordinate sys tem moved
+intox, i.e.
+q′
+i(πi)(X+x−πi) =−qi(πi) (i= 1,...,n).
+We show that this system is equivalent to an SPP. For this, we solve th ese equations for Xi:
+q′
+i(πi)(X+x−πi) =−qi(πi)
+⇔q′
+i(πi)X=−qi(πi)+q′
+i(πi) (πi−x)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+=(ηi−xi)·ei
+⇔Xi=/summationtext
+j/\e}atio\slash=i∂Xjqi(πi)
+−∂Xiqi(πi)·Xj+qi(πi)
+−∂Xiqi(πi)+(ηi−xi).
+Again, we have ∂Xiqi(πi)<0≤∂Xjqi(πi) asπi∈R, andq′
+i(πi) monotonically increases with
+ηi. Hence, the abovelinear equation for Xiis indeed a polynomial with non-negative coefficients.
+Denote by fηthe SPP defined by these linear equations. We then have µfη=Tη(x)−xas
+the above equation system has Tη(x)−x≥0as its unique solution. Further, we know that the
+Kleene sequence/parenleftbig
+fk
+η(0)/parenrightbig
+k∈Nconverges to µfη. We show that all coefficients of fηincrease with
+η→h. This is straight-forward for
+∂Xjqi(πi)
+−∂Xiqi(πi)
+as∂Xiqi(πi)<0≤∂Xjqi(πi),andallthesetermsincreasewith ηi→hi(x−i).Considertherefore
+0≥qi(πi)
+−∂Xiqi(πi)+(ηi−xi) =qi(πi)−∂Xiqi(πi)(ηi−xi)
+−∂Xiqi(πi).
+Weshowthatthistermincreaseswith ηi.Setδi:=ηi−xi.Wecanfindanon-negative,symmetric
+square-matrix A, a vector b, and constant csuch that
+qi(X) =X⊤AX+b⊤X+candq′
+i(X) = 2X⊤A+b⊤.
+Asπi=x+δieiwe have
+qi(πi) =qi(x+δiei) =qi(x)+∂Xiqi(x)δi+δ2
+iAii,
+and
+∂Xiqi(πi)·δi=q′
+i(x+δiei)δiei=∂Xiqi(x)δi+2δ2
+iAii.
+This leads to
+qi(πi)−∂Xiqi(πi)δi
+−∂Xiqi(πi)=qi(x)−δ2
+iAii
+−∂Xiqi(x)−2δiAii.
+Taking the derivative w.r.t. δiyields:
+−2Aiiδi
+−∂Xiqi(x)+2Aiiδi−qi(x)−Aiiδ2
+i
+(−∂Xiqi(x)−2Aiiδi)2(−2Aii)
+=2Aii∂Xiqi(x)δi+4A2
+iiδ2
+i+2Aiiqi(x)−2A2
+iiδ2
+i
+(−∂Xiqi(x)−2Aiiδi)2
+= 2AiiAiiδ2
+i+∂Xiqi(x)δi+qi(x)
+(−∂Xiqi(x)−2Aiiδi)2
+= 2Aiiqi(πi)
+(−∂Xiqi(πi))2.
+Asqi(πi)≥0 andAii≥0, it follows that
+qi(πi)
+−∂Xiqi(πi)+(ηi−xi)
+52increases with ηi→hi(x−i). Thus, all coefficients of fηincrease with ηi→hi(x−i), and so for
+anyη′∈[η,h] it follows that
+fη(y)≤fη′(y) for ally≥0,
+and
+Tη(x)−x=µfη≤µfη′=Tη′(x)−x.
+AsN(X) =Tx(X) andT(X) =Th(X) we may therefore conclude that
+N(x)≤ Tη(x)≤ Tη′(x)≤ T(x).
+It remains to show that T(x)≤µf. This is equivalent to showing that µfh≤µf−x. For
+fh(X) we have by definition and Lemma 8.4
+/parenleftbig
+fh(X)/parenrightbig
+i=/summationdisplay
+j/\e}atio\slash=i∂Xjqi(pi(x−i))
+−∂Xiqi(pi(x−i))Xj+(hi(x−i)−xi) =/summationdisplay
+j/\e}atio\slash=i∂Xjhi(x−i)Xj+(hi(x−i)−xi).
+By virtue of Lemma 8.8 it follows that µfis above all the tangents, i.e.
+fh(µf−x)≤µf−x.
+By monotonicity of fhwe also have
+fh(0)≤fh(µf−x).
+A straight-forward induction therefore shows that
+fk
+h(0)≤µf−x(∀k∈N),
+and, thus,
+T(x)−x=µfh≤µf−x.
+We turn to the monotonicity of T. Lety∈Rwithx≤y. Assume that xandyare located on
+the surface Si, i.e.
+hi(x−i) =xiandhi(y−i) =yi.
+Thetangent Ti|xatSiinxisspannedbythe partialderivativesof piinx.Thepart Ti|x∩[x,µf]
+relevant for T(x) can therefore be parameterized by
+x+/summationdisplay
+j/\e}atio\slash=i∂Xjpi(x)·(uj−xj) withu−i∈[x−i,µf−i].
+Similarly for Ti|y.
+In particular, for u−i∈[y−i,µf−i] both points on the tangents defined by u−idiffer only
+in theith coordinate being (the remaining coordinates are simply u−i)
+ty=yi+/summationdisplay
+j/\e}atio\slash=i∂Xjhi(y)·(uj−yj), resp.tx=xi+/summationdisplay
+j/\e}atio\slash=i∂Xjhi(x)·(uj−xj).
+By Lemma 8.8 we have
+yi≥xi+/summationdisplay
+j/\e}atio\slash=i∂Xjhi(x)·(yj−xj).
+From Lemma 8.4 it follows that ∂Xjhi(y)≥∂Xjhi(x). Thusty≥tximmediately follows.
+Now forx,y∈Rwithx≤ywe can apply this result to the tangents at Siinpi(x−i), resp.
+pi(y−i), andT(x)≤ T(y) follows.
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diff --git a/1001.0341.txt b/1001.0341.txt
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@@ -0,0 +1,1086 @@
+arXiv:1001.0341v2  [q-bio.GN]  16 Oct 2010UCLA Statistics Preprints, Jan 2010
+ON WEIGHT MATRIX AND FREE ENERGY MODELS FOR
+SEQUENCE MOTIF DETECTION∗
+By Qing Zhou†
+University of California, Los Angeles
+The problem of motif detection can be formulated as the const ruc-
+tion of a discriminant function to separate sequences of a sp ecific
+pattern from background. In computational biology, motif d etection
+is used to predict DNA binding sites of a transcription facto r (TF),
+mostly based on the weight matrix (WM) model or the Gibbs free
+energy (FE) model. However, despite the wide applications, theoret-
+ical analysis of these two models and their predictions is st ill lacking.
+We derive asymptotic error rates of prediction procedures b ased on
+these models under different data generation assumptions. T his al-
+lows a theoretical comparison between the WM-based and the F E-
+based predictions in terms of asymptotic efficiency. Applica tions of
+the theoretical results are demonstrated with empirical st udies on
+ChIP-seq data and protein binding microarray data. We find th at,
+irrespective of underlying data generation mechanisms, th e FE ap-
+proach shows higher or comparable predictive power relativ e to the
+WM approach when the number of observed binding sites used fo r
+constructing a discriminant decision is not too small.
+Key words : asymptotic efficiency, discriminant analysis, protein-
+DNA interaction, predictive error, transcription factor b inding site.
+1. Introduction. Transcription factors (TFs), a class of proteins, regulate gene
+transcription through their physical interactions with pa rticular DNA sites. Such a
+DNA site is called a transcription factor binding site (TFBS ), which is usually a short
+piece of nucleotide sequence (e.g., ‘CATTGTC’). Typically , a TF can bind different
+sites and regulate a set of genes. A key observation is that si tes of the same TF share
+similarity in their sequence composition, which is charact erized by a motif. Since gene
+regulation has always been an important problem in biology, many computational
+methods have been developed to predict whether a given DNA se quence can be bound
+by a TF. Please see Elnitski et al. (2006), Ji and Wong (2006), and Vingron et al.
+(2009) for recent reviews on relevant methods.
+The prediction of TFBS’s considered in this article is formu lated as a classification
+problem. Denote by wthe width of the binding sites and code the four nucleotide
+bases, A, C, G and T, by a set of positive integers I={1,···,J}(J= 4). Suppose
+that we have observed a sample of labeled sequences of length w,Dn={(Yk,Xk)}n
+k=1,
+whereXk∈ IwandYk∈ {0,1}indicating whether Xkis bound by the TF ( Yk= 1)
+or not (Yk= 0). We call D+
+n={Xk:Yk= 1}observed binding sites (or motif sites)
+∗To appear in Journal of Computational Biology .
+†Department of Statistics, 8125 Mathematical Sciences Buil ding, University of California, Los An-
+geles, CA 90095 (email: zhou@stat.ucla.edu).
+12 Q. ZHOU
+andD−
+n={Xk:Yk= 0}background sites (or background sequences). Then, motif
+detection is to construct a discriminant function from Dnto predict the label of any
+new sequence x∈ Iw.
+Mostoftheexistingcomputational methodsformotifdetect ion canbeclassifiedinto
+two groups. The starting point of the first group is the sequen ce specificity of binding
+sites,whichisoftensummarizedbytheposition-specificwe ightmatrix(WM).Forearly
+developments ofWM, pleaseseeStormo(2000). UndertheWM mo del,each nucleotide
+(letter) in a binding site is assumed to be generated indepen dently from a multinomial
+distribution on {A, C, G, T }. This model has been widely used in search of TFBS’s
+(e.g., Hertz and Stormo, 1999; Kel et al., 2003; Rahmann et al ., 2003; Turatsinze
+et al., 2008), de novomotif finding (e.g., Stormo and Hartzell, 1989; Lawrence et a l.,
+1993; Bailey and Elkan, 1994; Roth et al., 1998; Liu et al., 20 02) and many other works
+reviewed in Vingron et al. (2009). Thesecond group aims at mo deling physical binding
+affinity between a TF and its binding sites via the concept of th e Gibbs free energy
+(FE) or binding energy (e.g., Berg and von Hippel, 1987; Stor mo and Fields, 1998;
+Gerland et al., 2002; Kinney et al., 2007). Assuming that eac h nucleotide in a DNA
+sequence of length w(w-mer) contributes additively to the interaction with the TF ,
+this approach often leads to a regression-type model for the conditional distribution of
+binding affinity given a piece of nucleotide sequence (e.g., D jordjevic et al. 2003; Foat
+et al. 2006). This group of methods have tight connections wi th predictive modeling
+approaches to gene regulation, reviewed in Bussemaker et al . (2007), which can be
+regardedasanaturalgeneralization tothefreeenergyfram ework(ZhouandLiu,2008).
+Although the standpoints are different, the two groups of appr oaches are in some sense
+closely related. They often give similar discriminant func tions for predicting TFBS’s,
+and there are many FE-based methods that use a weight matrix t o approximate Gibbs
+free energy (e.g., Granek and Clarke, 2005; Roider et al., 20 07).
+In spite of the fast methodological development on the WM and the FE models,
+there is still a lack of solid theoretical analysis to compar e model assumptions, pa-
+rameter estimations and response predictions of the two app roaches. Such theoretical
+analysis can provide insights into these methods by seeking answers to a series of ques-
+tions. For example, what are the common and distinct assumpt ions between the WM
+and the FE models, what is the relative performance between t he two approaches in
+predicting TFBS’s given a certain data generation mechanis m, and how to calculate
+their predictive error rates when the size of observed sampl eDnbecomes large? With-
+out answering these questions, one may find it difficult to unde rstand the nature of
+these methodsand cannot extract thefull information conta ined in extensive empirical
+comparisons between the two approaches.
+In this article, we compare model assumptions and parameter estimations of typical
+WM and FE approaches, derive asymptotic error rates of their predictions under dif-
+ferent data generation models, and perform comparative stu dies on large-scale binding
+data. The article is organized as follows. In Section 2 we rev iew the basic models of the
+two approaches. Asymptotic error rates of prediction proce dures based on these mod-
+els are derived and analyzed in Section 3. Computational app roaches are developed in
+Section 4 for practical applications of the theoretical res ults. Numerical analysis andON MOTIF DETECTION 3
+biological applications are presented in Sections 5 and 6, r espectively, with a compar-
+ison of the WM-based and the FE-based predictions on ChIP-se q data and protein
+binding microarray data. The paper concludes with discussi ons in Section 7. Some
+mathematical details are provided in Appendices. Although presented in the specific
+context of motif detection, the results in this article are g enerally applicable to the
+modeling and classification of categorical data.
+2. Models. Letcbe a scalar, u= (u1,···,uJ) be a (column) vector, v=
+(v1,···,vw)∈ Iw, andA= (aij)w×JandB= (bij)w×Jbe twow×Jmatrices.
+For notational ease, we define c±A:= (c±aij)w×J,A/B:= (aij/bij)w×Jprovided
+thatbij∝ne}ationslash= 0,vA:=/summationtextw
+i=1aivi,A(v) :=/producttextw
+i=1aiviandu(v) :=/producttextw
+i=1uvi. Furthermore,
+we definev[−k]:= (v1,···,vk−1,vk+1,···,vw) andA[−k]by removing the kth row
+fromA, fork= 1,···,w. Symbols ‘L→’ and ‘P→’ are used for convergence in law and
+in probability, respectively.
+Letθ0= (θ01,···,θ0J) bethecell probabilities (probability vector) of amultin omial
+distribution for i.i.d. background nucleotides, where/summationtextJ
+j=1θ0j= 1 andθ0j>0 for
+j= 1,···,J. Sinceθ0can be accurately estimated from a large number of genomic
+backgroundsequences,weassumethatitisgiveninthefollo winganalyses.Throughout
+the paper, we assume that the cell probabilities of any multi nomial distribution are
+bounded away from 0.
+2.1.The weight matrix model. LetX= (X1,···,Xw)∈ Iwbe a sequence of
+lengthw. In the weight matrix model (WMM), we assume that Xis generated from a
+mixturedistribution.Let Y∈ {0,1}labelthemixturecomponent. Withprobability q0,
+Y= 0 andXis generated from an i.i.d. background model (with paramete r)θ0, that
+is,P(X|Y= 0) =θ0(X). With probability q1= 1−q0,Y= 1 andXis generated
+from a weight matrix Θ= (θij)w×J= (θ1,···,θw)t, whereθi= (θi1,···,θiJ) is a
+probability vector for i= 1,···,wandXiis independent of other Xk(k∝ne}ationslash=i). To be
+specific,P(X|Y= 1) =Θ(X). From this model the log-odds ratio of YgivenXis
+logP(Y= 1|X)
+P(Y= 0|X)= logq1Θ(X)
+q0θ0(X)= log(q1/q0)+w/summationdisplay
+i=1log(θiXi/θ0Xi).(1)
+IntheWM-based prediction, q1istypically fixedbypriorexpectation ordeterminedby
+the relative cost of the two types of errors (false positive v s false negative). Effectively,
+we assume that q1is given. Let
+β0= log(q1/q0), βij= log(θij/θ0j), (2)
+for 1≤i≤w,1≤j≤Jandβ= (βij)w×J. We rewrite (1) as
+logP(Y= 1|X)
+P(Y= 0|X)=β0+w/summationdisplay
+i=1βiXi=β0+Xβ:=h(X), (3)
+which defines an additive discriminant function to predict YgivenX, i.e., to predict
+whether the sequence Xcan be bound by the TF. The label Ywill be predicted as4 Q. ZHOU
+1 ifh(X)≥0 and 0 otherwise. This prediction can be regarded as a naive B ayesian
+classifier.
+Givenobservedbindingsites D+
+n,weestimate Θbythemaximumlikelihood estima-
+tor (MLE) ˆΘm= (ˆθm
+1,···,ˆθm
+w)tand substitute it in equation (2) to obtain ˆβm. Here,
+the superscript ‘ m’ stands for estimators based on the WMM. Let dˆθm
+i=ˆθm
+i−θi,
+which is an infinitesimal in the order of 1 /√nasn→ ∞. The standard asymptotic
+theory (e.g., Ferguson 1996) implies that
+√ndˆθm
+iL→N(0,Σm
+i) restricted toJ/summationdisplay
+j=1dˆθm
+ij= 0,asn→ ∞, (4)
+and that√ndˆθm
+i,i= 1,···,w, are mutually independent. The ( j,k)th element of the
+covariance matrix Σm
+iis (δjkθij−θijθik)/q1whereδjkis the Kronecker delta symbol
+and 1≤j,k≤J. From equation (2) we have dˆβm
+ij=dˆθm
+ij/θij, which leads to the
+following limiting distribution,
+√ndˆβm
+ijL→N(0,(1−θij)/(θijq1)),forj= 1,···,J, asn→ ∞, (5)
+with√ndˆβm
+imutually independent for i= 1,···,w.
+2.2.The free energy model. LetF,X= (X1,···,Xw) andFXbe a TF, a DNA
+sequence, and the corresponding TF-DNA complex, respectiv ely. The process of the
+TF-DNA interaction can be described by the chemical reactio nF+X=FX. The
+concentrations of the three molecules at chemical equilibr ium, [F],[X] and [FX], are
+determined by the association constant Ka(X), that is,
+[FX]
+[F][X]=Ka(X) = exp/braceleftbigg
+−∆G(X)
+RT/bracerightbigg
+,
+where ∆G(X) is the Gibbs free energy (FE) for the interaction of FwithX,Ris the
+gas constant and Tthe temperature. We regard RT >0 as a constant. Suppose that
+the contribution of a single nucleotide Xito the FE is additive (von Hippel and Berg,
+1986; Benos et al., 2002) so that we may write −∆G(X)/(RT) =c+/summationtextw
+i=1biXi. Then
+we have
+log[FX]
+[X]= log[F]+c+w/summationdisplay
+i=1biXi:=b0+w/summationdisplay
+i=1biXi. (6)
+To avoid non-identifiability in estimation, we take Sref= (s1,···,sw) as a reference
+sequence to determine a baseline level of the FE, and define ˜βij=bij−bisifor alli,j
+and˜β0=b0+/summationtextw
+i=1bisi, such that ˜βisi≡0 fori= 1,···,wand
+b0+w/summationdisplay
+i=1biXi=˜β0+w/summationdisplay
+i=1˜βiXi (7)ON MOTIF DETECTION 5
+for everyX∈ Iw. LetYbe the indicator for whether Xis bound by the TF at
+chemical equilibrium. From the physical meaning of concent ration,
+P(Y= 1|X) =[FX]
+[X]+[FX]. (8)
+Combining equations (6), (7) and (8) leads to an additive dis criminant function for
+this free energy model (FEM),
+logP(Y= 1|X)
+P(Y= 0|X)=˜β0+w/summationdisplay
+i=1˜βiXi=˜β0+X˜β:=˜h(X). (9)
+Similarly as for the WMM, we assume that ˜β0is fixed by prior or a desired cost.
+Furthermore, it is conventional to assume that Xis sampled from an i.i.d. background
+modelθ0, i.e.,P(X) =θ0(X). The data generation process of the FEM has a clear
+biological meaning. Suppose that we have sampled nnucleotide sequences of length
+w,{Xk∈ Iw}n
+k=1, from the genomic background θ0. We mix these sequences with
+TF molecules in a container where the concentration of the TF is held as a constant.
+At chemical equilibrium we label the sequences Xkbound by the TF as Yk= 1
+and otherwise Yk= 0. The output of this experiment is the labeled sample Dn=
+{(Yk,Xk)}n
+k=1. Although there exist other models based on binding free ene rgy, we
+focus on this basic model in this paper, which makes a theoret ical analysis relatively
+clean while capturing main characteristics of FE-based app roaches.
+GivenDn, the MLE of ˜β, denoted by ˆβf=˜β+dˆβfwith the superscript ‘ f’ for
+FE-based estimators, can be calculated by the standard logi stic regression. Note that
+ˆβfmaximizes the conditional likelihood
+P(Y|X,˜β) =exp{(˜β0+X˜β)Y}
+1+exp( ˜β0+X˜β)(10)
+determined by equation (9). Similar to the results in Efron ( 1975), it is not difficult
+to demonstrate that ˆβfis consistent for ˜βwith asymptotic normality,
+√ndˆβfL→N(0,Σf),asn→ ∞, (11)
+wheredˆβfis regarded as a vector of ( J−1)wdimensions (recall that ˜βisi=ˆβf
+isi≡0
+fori= 1,···,w). The asymptotic covariance matrix
+Σf=/bracketleftbig
+Eθ0/braceleftbig
+p1(X)p0(X)CXCt
+X/bracerightbig/bracketrightbig−1, (12)
+wherepy(X) =P(Y=y|X) fory= 0,1,CXis a (J−1)w-dimensional column
+vector coding each Xias a factor of Jlevels, and Eθ0is taken with respect to (w.r.t.)
+the background model θ0that generates the sequence X.6 Q. ZHOU
+2.3.Comparison. Given (β0,β) in the WMM and the reference sequence Srefin
+the FEM, if we let
+˜β0=β0+w/summationdisplay
+i=1βisi,˜βij=βij−βisi, (13)
+fori= 1,···,w, j= 1,···,J, then the two models have the same conditional distri-
+bution [Y|X] (3, 9) for any X. To simplify notations, we shall denote the decision
+function (9) in the FEM by h(X) =˜β0+X˜βhereafter. Except for this condition
+distribution, other model assumptions are different. The WMM assumes that the nu-
+cleotides inXare generated independently given its label Y. But this is not true for
+the FEM, in which the conditional probability of XgivenYis
+P(X|Y,FEM)∝P(Y|X,FEM)P(X|FEM) =exp{(˜β0+X˜β)Y}
+1+exp( ˜β0+X˜β)θ0(X).(14)
+Since equation (14) cannot be written as a product of functio ns ofXi, this model
+implicitly allows dependenceamong X1,···,Xw. Consequently, theFEM may account
+for someobserved nucleotide dependenceswithin amotif suc has in Bulyk et al. (2002),
+Barash et al. (2003), Zhou and Liu (2004), and Zhao et al. (200 5) among others. On
+the other hand, under the FEM model the marginal distributio n ofXis simply the
+background nucleotide distribution, i.e., P(X|FEM) =θ0(X), but the marginal
+distribution of Xunder the WMM is a mixture,
+P(X|WMM) =q1Θ(X)+q0θ0(X). (15)
+The different model assumptions lead to different procedures fo r parameter estima-
+tion, in particular the coefficients β(˜β). As discussed in Sections 2.1 and 2.2, ˆβmand
+ˆβfare consistent under the WMM and under the FEM, respectively . Sinceˆβfmaxi-
+mizes the conditional likelihood P(Y|X,˜β) (10) which is identical between the two
+models, it is also consistent for βunderthe WMM upto the translation (13). However,
+ˆβfis expected to be less efficient than ˆβmin prediction if the WMM corresponds to
+the underlying data generation process, due to the ignoranc e of the information on
+Θcontained in the marginal likelihood P(X|Θ,WMM) (to be discussed in detail
+in Section 3.1). Conversely, if data are generated by the FEM ,ˆβmis biased and no
+longer consistent. We will analyze the bias and the resultin g incremental error rate in
+later sections.
+3. Theoretical results. For both WMM and FEM, the ideal decision function
+h(x) is obtained with the true parameters of the respective mode ls and the corre-
+sponding ideal error rate
+R∗=/summationdisplay
+x:h(x)≥0P(Y= 0,X=x)+/summationdisplay
+x:h(x)<0P(Y= 1,X=x). (16)
+Denote by
+R∗(x) = min
+y∈{0,1}P(Y=y|X=x)ON MOTIF DETECTION 7
+the ideal error rate for h(x) givenX=x. Consider a decision function ˆh(x) estimated
+fromDn. Given anyxfor whichh(x)ˆh(x)<0, the incremental error rate beyond
+R∗(x) is
+∆R(x) =|P(Y= 1|X=x)−P(Y= 0|X=x)|.
+Then the expectation of the total incremental error rate for ˆhis
+E[∆R(ˆh)] =E/bracketleftBigg/summationdisplay
+x∈Iw∆R(x)P(X=x)1/braceleftBig
+h(x)ˆh(x)<0/bracerightBig/bracketrightBigg
+=/summationdisplay
+x∈Iw∆R(x)P(X=x)P/braceleftBig
+h(x)ˆh(x)<0/bracerightBig
+, (17)
+where1(·) is the indicator function. Please note that ˆh, constructed from a sample
+of sizen, is a random function. Let ∆ ˆh(x) =ˆh(x)−h(x) be the deviation of ˆh(x)
+fromh(x). In what follows, we will derive two theorems on E[∆R(ˆh)] under different
+assumptions for ∆ ˆh(x). As we will see, the asymptotic error rates of the WM and the
+FE procedures under the data generation models discussed in this paper can all be
+calculated based on the two theorems.
+Suppose that, for every x,√n∆ˆh(x)L→N(0,V(ˆh,x)), whereV(ˆh,x) is the asymp-
+totic variance. As n→ ∞,
+E[∆R(ˆh)]→/summationdisplay
+x∈Iw∆R(x)P(X=x)P(∆ˆh(x)>|h(x)|)
+=/summationdisplay
+x∈Iw∆R(x)P(X=x) Φ/braceleftbigg
+−/radicalBig
+nh2(x)/V(ˆh,x)/bracerightbigg
+,(18)
+where Φ is the cdf of the standard normal distribution N(0,1). Let
+α(ˆh) = min
+h(x)/ne}ationslash=0h2(x)/V(ˆh,x), (19)
+andx∗be the corresponding minimum. Note that ∆ R(x) = 0 when h(x) = 0. Thus,
+asn→ ∞,E[∆R(ˆh)] is dominated by the term ∆ R(x∗)P(X=x∗) Φ/bracketleftBig
+−{nα(ˆh)}1/2/bracketrightBig
+,
+whereα(ˆh) determines the rate of convergence. Using the theory of lar ge deviations,
+we obtain:
+Theorem 1.If√n∆ˆh(x)L→N(0,V(ˆh,x))for everyx∈ Iwthen
+1
+nlogE[∆R(ˆh)]→ −α(ˆh)
+2,asn→ ∞.
+Letˆhaandˆhbbetwo estimated decisions constructedfromsamples ofsize naandnb,
+respectively. Suppose that both of them satisfy the conditi on in Theorem 1. We define
+theasymptotic relative efficiency (ARE) of ˆhawith respect to ˆhbby ARE( ˆha,ˆhb) =
+α(ˆha)/α(ˆhb), which is the limit ratio nb/narequired to achieve the same asymptotic
+performance.8 Q. ZHOU
+Ifˆh(x) is biased in the sense that√n{∆ˆh(x)−µ(ˆh,x)}L→N(0,V(ˆh,x)), where
+µ(ˆh,x) denotes the asymptotic bias of ˆh(x), then simple derivation from equation
+(17) gives that
+E[∆R(ˆh)]→/summationdisplay
+x∈Iw∆R(x)P(X=x) Φ
+−√nsign{h(x)}{h(x) +µ(ˆh,x)}/radicalBig
+V(ˆh,x)
+,
+asn→ ∞, where sign( y) is the sign of ywith sign(0) ≡0.
+Theorem 2.Suppose that√n{∆ˆh(x)−µ(ˆh,x)}L→N(0,V(ˆh,x))for everyx∈
+Iw. LetB(ˆh) ={x: sign{h(x)}{h(x) +µ(ˆh,x)}<0}. Then
+E[∆R(ˆh)]→/summationdisplay
+x∈B(ˆh)∆R(x)P(X=x),asn→ ∞. (20)
+We ignore the case {x:h(x)+µ(ˆh,x) = 0}which practically never happens. The
+setB(ˆh) is the collection of xfor which the estimated decision ˆhgives a different
+predicted label from the ideal decision hasn→ ∞. Note that E[∆R(ˆh)] does not
+vanish if B(ˆh) is nonempty. Thus, the incremental percentage over the ide al error rate,
+E[∆R(ˆh)]/R∗, is an appropriate measure of the predictive performance of ˆh.
+In the remainder of this section, we derive and compare the er ror rates of the WM
+and the FE procedures. From Sections 3.1 to 3.4, we assume tha t the constant term
+β0(˜β0) is fixed to its true value. The results are generalized to sit uations where the
+constant is mis-specified in Section 3.5. The computation of α(ˆh) (19) and E[∆R(ˆh)]
+(20) will be discussed in Section 4.
+3.1.Error rates under WMM. In this subsection we assume that the underlying
+data generation process is given by the WMM. Since both ˆβmandˆβfare consistent
+with asymptotic normality under the WMM, we may uniformly de note their deci-
+sion functions by ˆh(x) =β0+xˆβ=h(x) +xdˆβ, wheredˆβ=ˆβ−βand√ndˆβ
+follows a normal distribution N(0,Σ) asn→ ∞. This implies that√n∆ˆh(x) =√nxdˆβL→N(0,Vm(ˆβ,x)) withVm(ˆβ,x) being the asymptotic variance. The super-
+script ‘m’ indicates the WMM as the data generation model. Let E[∆Rm(ˆβ)] be the
+expected incremental error rate of ˆhindexed by ˆβ. Following Theorem 1,
+1
+nlogE[∆Rm(ˆβ)]→ −αm(ˆβ)
+2=−1
+2min
+h(x)/ne}ationslash=0h2(x)
+Vm(ˆβ,x), (21)
+asn→ ∞. Consequently, the ARE of the FE procedure w.r.t the WM proce dure,
+AREm(ˆβf,ˆβm), is determined by the ratio of αm(ˆβf) overαm(ˆβm).
+The decision function of the WM procedure is constructed wit hˆβm(Section 2.1).
+Note thatxdˆβ=/summationtext
+idˆβixiis a summation of w dˆβij’s, each from a different dˆβi. TheON MOTIF DETECTION 9
+limiting distribution of√ndˆβm
+ij(5) and the mutual independence among dˆβm
+iimply
+that the asymptotic variance of√nxdˆβmis
+1
+q1w/summationdisplay
+i=1(1−θixi)/θixi=1
+q1x{(1−Θ)/Θ},
+and consequently,
+αm(ˆβm) = min
+xβ/ne}ationslash=−β0q1(β0+xβ)2
+x{(1−Θ)/Θ}.
+Suppose that we have chosen ( s1,···,sw) as the reference sequence in the FE pro-
+cedure. Define ˜β0and˜βfrom the parameters ( β0,β) of the WMM by equation (13).
+Then the FE-based estimator ˆβfis consistent for ˜βwith asymptotic normality. Let
+dˆβf=ˆβf−˜β. Similar to equation (12), the asymptotic covariance matri x of√ndˆβf
+is/bracketleftbig
+E/braceleftbig
+p1(X)p0(X)CXCt
+X/bracerightbig/bracketrightbig−1,where the expectation is taken w.r.t. the marginal
+distribution of Xunder the WMM (15). Thus the covariance matrix can be written
+as
+Covm/parenleftBig√ndˆβf/parenrightBig
+=/bracketleftBigg
+q0Eθ0/braceleftBigg
+eh(X)
+eh(X)+1CXCt
+X/bracerightBigg/bracketrightBigg−1
+, (22)
+where the expectation Eθ0averages over X∈ Iwgenerated from the background
+modelθ0. Based on equation (22), one can calculate the variance of√nxdˆβffor every
+xand determine the convergence rate αm(ˆβf) of the expected incremental error rate
+E[∆Rm(ˆβf)] for the FE procedure.
+Because the estimation of ˆβfis only based on the conditional distribution [ Y|X]
+whileˆβmis estimated from the joint distribution of YandX, we expect ˆβfto be
+less efficient in prediction with αm(ˆβf)< αm(ˆβm). We will conduct a numerical
+study in Section 5 to evaluate AREm(ˆβf,ˆβm) on 200 transcription factors to confirm
+our conclusion. Here we demonstrate the lower efficiency of ˆβfby the loss of Fisher
+information in estimating an individual θijfrom the conditional likelihood only. For
+simplicity, suppose that Θ[−i]is given and collapse Xiinto two categories, Xi=jand
+Xi∝ne}ationslash=j. Because
+P(X,Y|Θ) =P(Y|X,Θ)P(X|Θ)
+undertheWMM, thelossofinformation equalstheFisherinfo rmationon θijcontained
+in the marginal likelihood P(X|Θ), denoted by I(θij|X). LetI(θij|X,Y) be the
+Fisher information on θijgivenXandYjointly. We define
+∆(θij|[Y|X]) =I(θij|X)/I(θij|X,Y) (23)
+as the fraction of the loss of information on θijin the conditional likelihood P(Y|
+X,Θ).
+Proposition 3.Let¯θij=q0θ0j+q1θij. If(Y,X)is drawn from the WMM then
+∆(θij|[Y|X])≥q1θij(1−θij)
+¯θij(1−¯θij):=B(q1,θij,θ0j).10 Q. ZHOU
+Aproofof thisproposition isgiven inAppendixA. Ifonechoo ses toincludean equal
+number of background sites ( Y= 0) and binding sites ( Y= 1) in logistic regression to
+estimate ˆβf, which effectively specifies q0=q1= 0.5 by design, then this lower bound
+may be substantial. For example, with a uniform background d istribution θ0j= 0.25
+forj= 1,···,4, the range of B(q1,θij,θ0j) is between 20% and 55% for most typical
+values ofθij(Table 1).
+Table 1
+Typical values of B(0.5,θij,0.25)
+θij 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
+B(0.5,θij,0.25)(%) 31 46 53 55 53 49 42 32 18
+3.2.WMM with Markov background. We generalize the background model to a
+Markov chain, which often represents a better fit to genomic b ackground in high
+organisms. We assume that given Y= 0,Xis generated by a first order Markov
+chain with a transition probability matrix ψ0= (ψ0(x,y))J×Jwherex,y∈ I. For any
+x= (x1,···,xw)∈ Iw,ψ0(x) :=/producttextw
+i=1ψ0(xi−1,xi), whereψ0(x0,x1) is interpreted as
+the probability of x1under the stationary distribution of the Markov chain. The i deal
+decision function under this model is
+h1(x) = logP(Y= 1|X=x)
+P(Y= 0|X=x)=β0+w/summationdisplay
+i=1logθixi−logψ0(xi−1,xi),(24)
+whereβ0= log(q1/q0) and the subscript ‘1’ [in h1(x) and ∆Rm
+1(26)] indicates a quan-
+tity whose definition involves a Markov background model. Si nceψ0can be accurately
+estimated with sufficient genomic background sequences, we a ssume that it is given.
+With the MLE ˆΘmfrom observed binding sites, the WM procedure constructs a d e-
+cision whose expected incremental error rate converges to z ero exponentially fast as
+n→ ∞, following Theorem 1.
+With a slight abuse of notations, let us denote by θ0the probability vector of the
+stationary distribution of the Markov chain, which is also t he marginal distribution
+of any nucleotide Xiin a background site. We still define β0andβby equation (2)
+withθ0being the stationary probabilities, and translate β0andβvia a reference
+sequence to ˜β0and˜β(13). Let (Y,X) be a sample from the WMM with Markov
+background. If the dependence among neighboring nucleotid es in a background site is
+ignored, the conditional likelihood P(Y|X,˜β), parameterized by ˜β, is then given by
+the same expression in equation (10). Because the FE-based e stimator ˆβfmaximizes
+this conditional likelihood, it is standard to show that ˆβfP→˜βand is asymptotically
+normal. Let ˆhf(x) =˜β0+xˆβfdenote the estimated decision function of the FE
+procedure. As n→ ∞,
+ˆhf(x)P→˜β0+x˜β=β0+w/summationdisplay
+i=1logθixi−logθ0xi. (25)ON MOTIF DETECTION 11
+Let ∆ˆhf(x) =ˆhf(x)−h1(x) be the deviation of ˆhf(x) from the ideal decision (24).
+Comparing equations (24) and (25) gives the asymptotic bias ,
+b(x) =w/summationdisplay
+i=1logψ0(xi−1,xi)−logθ0xi= logψ0(x)−logθ0(x).
+Duetotheasymptoticnormalityof ˆβf,wehave√n{∆ˆhf(x)−b(x)}L→N(0,Vm
+1(ˆβf,x)),
+whereVm
+1(ˆβf,x) is the corresponding asymptotic variance. Under this mode l,
+∆R(x)P(X=x) =|q1Θ(x)−q0ψ0(x)|=q0ψ0(x)/vextendsingle/vextendsingle/vextendsingleeh1(x)−1/vextendsingle/vextendsingle/vextendsingle.
+Following Theorem 2 with µ(ˆhf,x) =b(x), the expected incremental error rate
+E[∆Rm
+1(ˆβf)]→/summationdisplay
+Bm
+1(ˆβf)q0/vextendsingle/vextendsingle/vextendsingleeh1(x)−1/vextendsingle/vextendsingle/vextendsingleψ0(x),asn→ ∞, (26)
+whereBm
+1(ˆβf) ={x: sign{h1(x)}{h1(x) +b(x)}<0}. The incremental percentage
+over the ideal error rate, E[∆Rm
+1(ˆβf)]/(Rm
+1)∗, is appropriate for comparing the FE-
+based prediction with the WM-based prediction whose expect ed error rate converges
+to (Rm
+1)∗. A general expression for R∗is given in equation (16) which, undertheWMM
+with Markov background, is written as
+(Rm
+1)∗=q0Eψ0/braceleftBig
+1(h1(X)≥0)+eh1(X)1(h1(X)<0)/bracerightBig
+. (27)
+3.3.Error rates under FEM. We now analyze asymptotic error rates of the two
+procedures regarding the FEM as the underlying data generat ion mechanism.
+TheFE-based estimator ˆβfis consistent for ˜βunderthe FEM. Theasymptotic nor-
+mality of√ndˆβf(11, 12) implies that√nxdˆβfL→N(0,Vf(ˆβf,x)). LetE[∆Rf(ˆβf)]
+be the expected incremental error rate of the FE procedure un der the FEM. From
+Theorem 1, we have
+1
+nlogE[∆Rf(ˆβf)]→ −αf(ˆβf)
+2=−1
+2min
+h(x)/ne}ationslash=0h2(x)
+Vf(ˆβf,x),asn→ ∞.(28)
+Denote byθf
+i= (θf
+i1,···,θf
+iJ) the probability vector of the conditional distribution
+[Xi|Y= 1] under the FEM, i.e.,
+θf
+ij=P(Xi=j|Y= 1,FEM), (29)
+fori= 1,···,w, and call Θf= (θf
+ij)w×Jthe weight matrix. Recall that the WM-
+based estimator ˆβmis obtained by estimating θf
+iindividually from observed binding
+sitesD+
+nand then transforming the estimates via equation (2). Denot e the estimated
+weight matrix by ˆΘm. Since the data are generated by the FEM, ˆΘmP→Θfand√ndˆΘm=√n(ˆΘm−Θf) follows a multivariate normal distribution asymptotical ly,12 Q. ZHOU
+similar to (4), but dˆθm
+ianddˆθm
+kmay be correlated (1 ≤i∝ne}ationslash=k≤w). Given that the
+coefficients ˜βin the FEM are defined w.r.t. a reference sequence, we transfo rmˆΘmto
+ˆβm
+ij= log(ˆθm
+ij/θ0j)−log(ˆθm
+isi/θ0si),for alli,j, (30)
+wheresiis theith nucleotide of the reference sequence Sref. Let ∆ˆβm=ˆβm−˜βbe
+the deviation of ˆβm= (ˆβm
+ij)w×J. To obtain its asymptotic distribution, we determine
+the cell probability θf
+ij(29) from equation (14), that is,
+θf
+ij∝θ0je˜βij/summationdisplay
+x∈Iw−1e˜β0+x˜β[−i]
+1+e˜βije˜β0+x˜β[−i]θ0(x) =θ0je˜βijEθ0/braceleftbiggeUi
+1+e˜βijeUi/bracerightbigg
+,
+whereUi=˜β0+X˜β[−i]forX∈ Iw−1. In particular, θf
+isi∝θ0siEθ0/braceleftbig
+eUi/(1+eUi)/bracerightbig
+since˜βisi= 0. Fori= 1,···,wandj= 1,···,J, we define
+δij= logEθ0/braceleftbiggeUi
+1+e˜βijeUi/bracerightbigg
+−logEθ0/braceleftbiggeUi
+1+eUi/bracerightbigg
+(31)
+and rewrite θf
+ij=θ0je˜βij+δij/Zi, whereZi=/summationtext
+jθ0je˜βij+δijis the normalizing constant.
+Because ˆθm
+ijP→θf
+ijand˜βisi=δisi= 0, from equation (30) we have
+ˆβm
+ijP→log(θf
+ij/θ0j)−log(θf
+isi/θ0si) =˜βij+δij (32)
+for alliandjasn→ ∞. Thus,δ= (δij)w×Jis the asymptotic bias of ˆβm. From the
+asymptotic normality of√ndˆΘm, we see that√n(∆ˆβm−δ) follows a multivariate
+normal distribution with mean 0and finite covariance matrix as n→ ∞. Note that
+this multivariate normal distribution is defined on a ( J−1)w-dimensional space, since
+∆ˆβm
+isi=δisi≡0 fori= 1,···,w.
+Consider the WM-based decision function ˆhm(x) =˜β0+xˆβm=h(x)+x∆ˆβm. The
+above derivation shows that√n(x∆ˆβm−xδ)L→N(0,Vf(ˆβm,x)), where the variance
+is determined by the covariance matrix of ∆ ˆβm. Following Theorem 2, the expectation
+of the total incremental error rate of the WM procedure
+E[∆Rf(ˆβm)]→/summationdisplay
+Bf(ˆβm)tanh|h(x)/2|θ0(x),asn→ ∞, (33)
+whereBf(ˆβm) ={x: sign{h(x)}{h(x)+xδ}<0}. Similarly, the incremental percent-
+age over the ideal error rate E[∆Rf(ˆβm)]/(Rf)∗is used to compare the predictions of
+the WM and the FE procedures, given that E[∆Rf(ˆβf)]→0 (28). Under the FEM,
+the ideal error rate
+(Rf)∗=Eθ0{p0(X)1(h(X)≥0)+p1(X)1(h(X)<0)}. (34)
+Recall that py(X) =P(Y=y|X), i.e.,
+py(X) =eyh(X)
+eh(X)+1,fory= 0,1.ON MOTIF DETECTION 13
+3.4.FEM with Markov background. Next, we generalize the FEM to Markov back-
+ground and assume that any sequence X∈ Iwis generated marginally by a Markov
+chain. Consistent with Section 3.2, we denote by ψ0= (ψ0(x,y))J×Jthe transition
+probability matrix of the Markov chain.
+It is trivial to see that, with the Markov background model, t he ideal decision is
+stillh(x) =˜β0+x˜β. IfVf
+1(ˆβf,x) denotes the asymptotic variance of√nxdˆβfunder
+Markov background, then with Vf
+1in place ofVfequation (28) remains valid for the
+FE-based prediction. On the other hand, if we proceed with th e WM procedure, the
+expected incremental error rate
+E[∆Rf
+1(ˆβm)]→/summationdisplay
+Bf
+1(ˆβm)tanh|h(x)/2|ψ0(x),asn→ ∞, (35)
+whereBf
+1(ˆβm) ={x: sign{h(x)}{h(x)+δ(x)}<0}. Hereδ(x) denotestheasymptotic
+bias of the WM-based decision function for x. A detailed derivation of equation (35)
+and the bias δ(x) is provided in Appendix B. In analogy to the FEM with i.i.d. b ack-
+ground,E[∆Rf
+1(ˆβm)]/(Rf
+1)∗measures the increased error rate of the WM procedure
+relative to the FE procedure, where
+(Rf
+1)∗=Eψ0{p0(X)1(h(X)≥0)+p1(X)1(h(X)<0)}.
+3.5.Mis-specification of the constant term. In all the above derivations, we have
+assumed that the constant term β0(˜β0) is fixed to its true value. If this is not the case,
+then the deviation ∆ ˆβ0=ˆβ0−β0(˜β0) will be an extra bias term for an estimated
+decision in which the constant term is fixed to ˆβ0. More specifically, the set Bf(ˆβm)
+in equation (33) will be replaced by {x: sign{h(x)}{h(x) +xδ+ ∆ˆβ0}<0}, and
+similarly for Bm
+1(ˆβf) in equation (26) and Bf
+1(ˆβm) in equation (35).
+4. Computation. To apply the theoretical results, we need to solve the mini-
+mization (19) and the summation (20) involved in Theorems 1 a nd 2, respectively. If
+the width of a motif w≤12, brute-force enumeration of all w-mers is computationally
+feasible, which provides exact solutions for both the minim ization and the summation
+problems.
+For a motif of width w>12, we minimize (19) to find α(ˆh) by a two-step approach.
+We generate N= 5×106w-mers from the background model θ0and identify the
+minimum of (19) among them. Then we refine the obtained minimu m by simulated
+annealing for 5,000 iterations with temperature decreasin g linearly from one to zero.
+At each iteration, we randomly choose one nucleotide Xifrom thewpositions and
+propose to mutate Xito one of the other three nucleotide bases with equal probabi l-
+ity. The proposal is accepted according to a Metropolis-Has tings ratio with current
+temperature.
+Since the set B(ˆh), as in equations (26), (33) and (35), is usually small, it wi ll be
+very inefficient to approximate the summation by generating w-mers from background
+distributions. Thus, we develop an importance sampling app roach to approximate the
+summation (20) when w >12. Here, we use the calculation of E[∆Rf(ˆβm)] (33) to14 Q. ZHOU
+illustrate this approach. Note that one can bound xδin the definition of Bf(ˆβm)
+so thatxδ∈[M1,M2], whereM1=/summationtextw
+i=1minjδijandM2=/summationtextw
+i=1maxjδij. These
+bounds imply that if x∈ Bf(ˆβm) then
+h(x)∈(−M2,0)∪(0,−M1) :=H.
+We design a sequential proposal g(X) that is more likely to generate Xwithh(X)∈
+H. Suppose that we have generated X1,···,Xk−1(1≤k≤w) from this proposal. Let
+hk−1=˜β0+/summationtextk−1
+i=1˜βiXi, in particular h0=˜β0. We determine B(L)
+k+1=/summationtext
+i>kminj˜βij
+andB(U)
+k+1=/summationtext
+i>kmaxj˜βij, the bounds for/summationtext
+i>k˜βiXi. IfXk=jthen the range for
+h(X) is
+hk−1+˜βij+[B(L)
+k+1,B(U)
+k+1] := [Lkj,Ukj].
+The larger the overlap between this interval and H, the more likely that Xwill belong
+to the desired set Bf(ˆβm). Thus, we propose Xkwith probability
+gk(Xk=j|X1,···,Xk−1)∝ |[Lkj−ǫ,Ukj+ǫ]∩H|, (36)
+where| · |returns the length of an interval and ǫis a small positive number to allow
+the generation of Xk=jwhenLkj=Ukj∈ H. Proposing Xksequentially by (36)
+fork= 1,···,wgenerates an Xfromg(X). WithNproposed samples {X(t)}N
+t=1we
+estimate the summation (33) by
+1
+NN/summationdisplay
+t=1tanh/vextendsingle/vextendsingle/vextendsingleh(X(t))/2/vextendsingle/vextendsingle/vextendsingleθ0(X(t))1{X(t)∈ Bf(ˆβm)}
+g(X(t)).
+In this work, we propose N= 5×106samples for this importance samplingestimation.
+We verified that the estimations were very close to the exact s ummations. With differ-
+ent bounds for h1(x) andh(x), this approach is applied to other similar summations
+in (26) and (35).
+5. Numerical study. A numerical study was performed under the WMM to
+confirm and quantify the lower predictive efficiency of the FE- based estimator ˆβf
+compared to the WM-based estimator ˆβmdiscussed in Section 3.1. We randomly
+selected 200 TFs from the database TRANSFAC (Matys et al. 200 3). For each TF,
+experimentally verified binding sites were used to construc t a weight matrix with a
+small amount of pseudo counts. Then we randomly sampled 5,00 0 human upstream
+sequences, each of length 10 kilo bases, and calculated thei r nucleotide frequency ˆθ0=
+(0.263, 0.234, 0.237, 0.266). The 200 weight matrices displ ay large variability. The
+widthwranges from 6 to 21 and the information content,/summationtextw
+i=1{2+Eθi(log2θiXi)},
+ranges from 5.1 to 17.5 bits (Figure 1). These statistics sho w that our selection has
+covered the typical width and strength of DNA motifs.
+A constructed weight matrix was regarded as the parameter Θand the nucleotide
+frequency ˆθ0was usedforthei.i.d. backgroundintheWMM. Sincetheprior oddsratio
+(q1/q0) of a binding site over a background site is usually small, we chose three typicalON MOTIF DETECTION 15
+WidthCount
+5 10 15 200 5 10 20 30
+Information Content (bit)Count
+6 8 10 14 180 10 20 30 40
+Fig 1. Histograms of the width and the information content of 200 W Ms.
+Table 2
+Summary of AREm(ˆβf,ˆβm)
+λMin. Q1Median Q3Max.
+200 0.134 0.391 0.490 0.595 0.849
+500 0.183 0.475 0.555 0.638 0.849
+1000 0.217 0.508 0.560 0.675 0.918
+Q1,3: the first and the third quartiles.
+values for the inverse of the prior odds, λ=q0/q1= 200,500,1000, for numerical
+calculations. We evaluated the AREs of the FE-based predict ion w.r.t. the WM-based
+prediction, defined by AREm(ˆβf,ˆβm) =αm(ˆβf)/αm(ˆβm) in Section 3.1, for the 200
+WMs. As discussed in Section 4, our evaluation of AREs was exa ct for WMs of w≤12
+and was carried out with simulated annealing for w >12. In addition, Monte Carlo
+average was utilized, before simulated annealing, to appro ximate Covm(√ndˆβf) (22)
+by simulating 5 ×106w-mers from the i.i.d. background.
+The asymptotic relative efficiencies AREm(ˆβf,ˆβm) on the 200 TFs are summarized
+in Table 2 for the three inverse prior odds. It is seen that for all the WMs the FE-based
+prediction shows lower efficiency than the WM-based predicti on, and that the median
+AREs of ˆβftoˆβmare between 50% and 60% and the third quartiles ( Q3) between
+60% and 70%. Thus, for more than 75% of the TFs, the FE procedur e is less than
+70% as efficient as the WM procedure in terms of prediction. Thi s confirms the loss of
+efficiency of the FE-based prediction under the WMM, although both estimators are
+consistent. We note that the increase of ARE with higher λ(smallerq1) is consistent
+with the lower bound defined in Proposition 3.
+6. Applications. In this section, we apply the WM and the FE approaches to
+ChIP-seq data and protein binding microarray (PBM) data. We perform cross valida-
+tion (CV) with training data of different size, ranging from 20 to 500 binding sites,
+for two purposes. First, with the large scale of both types of data, we can compare16 Q. ZHOU
+empirical error rates in cross validation against theoreti cal error rates. This may allow
+us to verify some of the model assumptions and propose furthe r improvement on the
+models. Second, we are also interested in examining the prac tical performance of the
+two computational methods when the number of observed bindi ng sites varies in a
+wide range, which will provide useful guidance for future ap plications.
+6.1.ChIP-seq data. In the recent two years, the ChIP-seq technique (Johnson et
+al., 2007; Mikkelsen et al., 2007; Robertson et al., 2007) ha s become a powerful high-
+throughput method to detect TFBS’s in whole genome scale. A b inding peak in ChIP-
+seq data can usually narrow down the location of a TFBS to a nei ghborhood of 50 to
+200 bps (Johnson et al., 2007). ChIP-seq data that contain th ousands of binding sites
+for a number of TFs have been generated in a study on mouse embr yonic stem cells
+(Chen et al., 2008). We chose five TFs, Esrrb, Oct4, STAT3, Sox 2 and cMyc, in this
+study to compare the WM and the FE methods. The five TFs all have well-defined
+weight matrices in literature and each contains more than 2, 000 detected binding
+peaks in ChIP-seq, and their data quality was confirmed by mot if enrichment analysis
+in Chen et al. (2008). To identify the exact bindingsite of a C hIP-seq bindingpeak, we
+searched the 200-bp neighborhood of the peak, 100 bps on each side, to find the best
+match to the known weight matrix of the TF. Given the very smal l search space, the
+uncertainty in the exact location of the binding site should be minimal. If the motif
+width of a TF is w, background w-mers were extracted from genomic control regions
+that match the locations of the binding sites relative to nea rby genes. The ratio of the
+number of background sites over the number of binding sites w as set to 200 for every
+TF, that is, the inverse prior odds ratio λ=q0/q1= 200. A transition matrix was
+estimated from the extracted background sites for each TF, s ince the log Bayes factor
+of a Markov background model over an i.i.d model was >105.
+Based on the way we composed the data sets, the WMM with Markov background
+(Section 3.2) seems a more plausible data generation model. Clearly, a data set was a
+mixture of detected binding sites and random background sit es, and the background
+distribution was close to a Markov chain. If there is no withi n-motif dependence,
+binding sites can be regarded as being generated from a WM mod el, and consequently,
+the WM-based prediction is expected to have a smaller error r ate compared to the FE-
+based prediction. However, if there exists within-motif de pendencein bindingsites, the
+FEM, which is able to capture such dependence, may outperfor m the WM approach
+regardless of the mixture nature of the data sets. We compute d theoretical error rates
+of the two approaches under the WMM with Markov background. F or each TF, we
+estimatedaWMfromallthebindingsitesandatransitionmat rixfromthebackground
+sites. Regarding them as the model parameters, we calculate d the asymptotic error
+rateoftheWM-basedprediction,whichistheidealerrorrat e(27), andtheincremental
+rate of the FE-based prediction (26). Note that the bias due t o mis-specification of the
+constant term in the FE approach needs to be included for the c alculation of equation
+(26). These theoretical error rates are reported in Table 3 ( the column of n+=∞).
+To compare with theoretical results, we performed cross val idation to compute em-
+pirical error rates of the WM and the FE procedures on each dat a set. We randomlyON MOTIF DETECTION 17
+Table 3
+Predictive error rates (in the unit of 10−3) for ChIP-seq data
+TF n+20 50 100 200 500 ∞
+WM 2.66 2.49 2.43 2.40 2.36 2.30
+Esrrb FE 4.30 2.77 2.54 2.45 2.37 2.45
+(FE-WM)/WM (%) 61.7 11.2 4.5 2.1 0.4 6.5
+WM 3.68 3.54 3.50 3.47 3.44 2.98
+Oct4 FE 4.81 3.71 3.53 3.45 3.41 3.06
+(FE-WM)/WM (%) 30.7 4.8 0.9 −0.6−0.9 2.7
+WM 3.03 2.84 2.78 2.72 2.69 2.57
+STAT3 FE 4.69 3.09 2.84 2.75 2.70 2.74
+(FE-WM)/WM (%) 54.8 8.8 2.2 1.1 0.3 6.6
+WM 2.95 2.75 2.68 2.65 2.63 2.53
+Sox2 FE 3.44 2.89 2.73 2.68 2.66 2.59
+(FE-WM)/WM (%) 16.6 5.1 1.9 1.1 1.1 2.4
+WM 2.67 2.49 2.42 2.38 2.34 2.07
+cMyc FE 3.30 2.51 2.34 2.26 2.23 2.24
+(FE-WM)/WM (%) 23.6 0.8 −3.3−5.0−4.7 8.2
+Note: Reported are average error rates over 100 CVs.
+sampled (without replacement) n+binding sites and λ·n+background sites from
+a full data set to form a training set. Both approaches were ap plied to the training
+set to estimate their respective decision functions. For WM -based prediction, a WM
+and a transition matrix were estimated from the training dat a set to construct a de-
+cision function (24) with β0=−log(λ). For FE-based prediction, we applied logistic
+regression to the training set to obtain ˆhf(x) =ˆβ0+xˆβf. Then we predicted the
+class labels of the remaining unused sequences (test set) by each of the two decision
+functions and calculated empirical error rates (CV error ra tes). This procedure was
+repeated 100 times independently for each value of n+to obtain the average CV error
+rate. To examine performance with a varying sample size (the number of sequences in
+a training set), we chose n+from 20 to 500.
+The average CV error rates are reported in Table 3. The theore tical results give a
+reasonable approximation to the CV error rates for both appr oaches when the training
+sample size n+≥200. The asymptotic error rates of the WM approach are unifor mly
+lower than its CV error rates for all the TFs, while the FE appr oach achieves a smaller
+CV error rate with n+= 500 than its asymptotic rate for three TFs. Consequently, t he
+incremental percentage of the FE-based prediction for n+= 500 is less than the ex-
+pected level calculated from the theory. This comparison im plies that the WMM may
+not match the exact underlying data generation process, alt hough it is more plausible
+than the FEM given the mixture composition of the data sets. A s we discussed, poten-
+tial dependence within a motif may cause possible violation to the WMM. To verify
+our hypothesis, we conducted the χ2-test for every pair of motif positions ( XiandXk,
+1≤i < k≤w) given the binding sites in each data set. At the significance level of
+0.005, we identified 25, 19, 17, 8, and 12 pairwise correlatio ns for Esrrb, Oct4, STAT3,
+Sox2, and cMyc binding sites, respectively, which gives a fa lse discovery rate of <2%18 Q. ZHOU
+for all the TFs. By capturing such correlations the FEM is abl e to achieve compara-
+ble or even slightly better prediction than the WMM with a mod erate-size training
+sample (n+≥100, Table 3). Finally, it is important to note that even unde r the exact
+model assumptions of the WMM, the FE-based prediction only r esults in a marginal
+increment in error rate ( <10%) compared to the WM approach asymptotically (Ta-
+ble 3,n+=∞). Together with the superior or comparable CV performance w hen the
+training size is reasonably large, this result suggests the use of the FE approach, when
+we have a sufficient number of observed binding sites.
+6.2.PBM data. Protein binding microarrays (Mukherjee et al., 2004) provi de a
+high throughput means to interrogate protein binding speci ficity to DNA sequences.
+Quantitative measurement of the binding specificity of a pro tein to every short nu-
+cleotide sequence designed on a DNA microarray can be obtain ed simultaneously. The
+PBMdatainBergeret al.(2008) quantifiedDNAbindingofhome odomainproteinsvia
+the calculation of an enrichment score, with an expected fal se discovery rate (FDR),
+for each double-stranded nucleotide sequence of length eig ht (w= 8). The data set for
+each protein contains 32,896 8-mers, each with an enrichmen t score and an FDR. We
+identified as the consensus binding pattern for a protein the 8-mer with the highest
+enrichment score, and then labeled as binding sites those 8- mers whose FDR <0.005
+and which differ by no more than three nucleotides from the cons ensus after consid-
+ering both the forward and the reverse complement strands. T he remaining 8-mers
+were labeled as background sites and we randomly determined their strands (orien-
+tations) to avoid potential artifacts. In this study we incl uded five proteins, Hoxa11,
+Irx3, Lhx3, Nkx2.5, and Pou2f2, each from a different family, a nd called 134, 190, 267,
+145, and 213 binding sites, respectively.
+The FEM, developed by the biophysics of protein-DNA binding , is expected to
+be a better model that matches the design of PBM data than the W MM. Thus,
+theoretical analysis was conducted under the FEM for the five PBM data sets. We
+applied logistic regression to estimate ˜βand˜β0(9) with all the labeled 8-mers in a
+data set, where the 8-mer ‘AAAAAAAA’ was regarded as the refe rence sequence, i.e.,
+βi1≡0. We calculated the ideal error rate ( Rf)∗(34) of the FE-based prediction,
+with an i.i.d. uniform background (by design the background distribution is uniform).
+For the WM approach, we chose ˆβ0as the log-ratio of the number of binding sites
+over that of background sites, and calculated its asymptoti c error rate by equation
+(33), in which the bias in the constant term (∆ ˆβ0) was included. The theoretical error
+rates are reported in Table 4 ( n+=∞), where we find that the WM approach gives
+a significantly higher error rate, between 14% and 56%, than t he FE approach.
+The same CV procedure as in the previous section was performe d on the PBM data
+sets to compare the empirical predictive error rates of the t wo approaches, with n+
+varying between 20 and 100 (Table 4). There is a clear decreas ing trend in error rate
+for both approaches with the increase of the training sample sizen+, although for
+some data sets the difference between the CV error rate for n+= 100 and the asymp-
+totic rate is still quite obvious. Such discrepancy is proba bly due to the following two
+reasons. First, the parameters ( ˜β,˜β0) used for the calculation of asymptotic rates wereON MOTIF DETECTION 19
+Table 4
+Predictive error rates (in the unit of 10−3) for PBM data
+Protein n+20 50 100 ∞
+FE 3.57 2.62 2.43 1.63
+Hoxa11 WM 3.51 3.22 3.05 2.10
+(WM-FE)/FE (%) −1.7 22.9 25.5 28.8
+FE 5.91 4.71 4.54 3.26
+Irx3 WM 5.06 4.85 4.79 3.71
+(WM-FE)/FE (%) −14.4 3.0 5.5 13.8
+FE 7.51 4.64 4.20 3.18
+Lhx3 WM 6.90 6.54 6.47 4.97
+(WM-FE)/FE (%) −8.1 40.9 54.0 56.3
+FE 3.73 2.31 2.12 1.80
+Nkx2.5 WM 3.64 3.29 3.16 2.36
+(WM-FE)/FE (%) −2.4 42.4 49.1 31.1
+FE 6.25 4.91 4.56 3.31
+Pou2f2 WM 5.79 5.57 5.49 4.02
+(WM-FE)/FE (%) −7.3 13.4 20.4 21.5
+estimated from data sets which only contain 100 to 200 bindin g sites. This resulted in
+a high variance in the estimated parameters: The median rati o of the standard error
+over the absolute value of an estimated coefficient was betwee n 10% and 30% for the
+five data sets. Second, the training sample size, n+= 100, is still too small to achieve
+a comparable error rate as n+→ ∞. However, we have already seen substantially in-
+creased error rates of theWM-based predictions comparedto theFE-based predictions
+forn+= 100, which is very consistent with the theoretical results . This comparison
+confirms that unless the training sample size is really small , using the WM approach
+may degrade predictive performance dramatically if the dat a generation mechanism is
+close to the FEM.
+7. Discussion. Combining results on the ChIP-seq data and the PBM data, this
+study provides some general guidance for practical applica tions of the WM and the
+FE approaches, irrespective of underlying data generation . When the training sample
+size is small, the WM procedure seems to produce fewer errors than the FE procedure.
+But when we have observed enough binding sites, the advantag e of the FE procedure
+is clearly seen. On one hand, it gives a comparable or slightl y better prediction than
+the WM approach even if the WMM is more likely for the data (Tab le 3,n+≥100).
+On the other hand, when the data are generated in a way that mat ches the biophysical
+process of protein-DNA binding such as the PBM data, the redu ction in error rate
+of the FE approach can be substantial compared to the WM appro ach (Table 4,
+n+≥50). The relative performance between the two approaches re flects a typical
+variance-bias tradeoff. Estimation under the WMM is simple an d more robust, which
+typically has a smaller variance than the FEM. For a small sam ple size, predictive
+errors are mostly caused by variance in estimation and thus, WM-based predictions
+may outperform FE-based predictions. When the sample size i ncreases, estimation
+variance decreases for both approaches and the potential bi as in the WM approach20 Q. ZHOU
+becomes the main factor for predictive errors. Given that it s primary principle comes
+fromthebiophysicsofprotein-DNAinteractions, theFEMha sbecomemoreattractive,
+based on which many computational methods have been develop ed for predicting TF-
+DNA binding. In these methods a weight matrix is sometimes us ed as a first order
+approximation for computing free energy-based binding affin ity. This work suggests
+that this approximation must be applied with caution. The re sults on the PBM data
+have demonstrated that the WM procedure may give a predictio n with 50% or more
+errors compared to the FE-based decision for a reasonably la rge sample size (Table 4).
+Inrecent years, asubstantial amount of large-scale TF-DNA bindingdata have been
+generated for many important biological processes. As demo nstrated by the applica-
+tions to ChIP-seq data and PBM data, large-sample theory is a ble to provide valuable
+insights on statistical estimation and prediction for such large-scale data. The results
+in this article can be regarded as a first step towards a theore tical development on
+computational approaches for gene regulation analysis. In corporation of within-motif
+dependence in the WMM and interaction effects in the FEM is a dir ect next step
+of this work, for which the model selection component needs t o be considered in a
+theoretical analysis. Although desired, further generali zations to methods for de novo
+motif discovery, identification of cis-regulatory modules and predictive modeling of
+gene regulation will be more challenging future directions .
+Appendices.
+Appendix A: Proof of Proposition 3.
+Proof. Letθk(−j)= 1−θkjfork= 0,i. The second order partial derivative of the
+marginal log-likelihood l(Θ|X) = log{q0θ0(X)+q1Θ(X)}w.r.t.θijis
+∂2l(Θ|X)
+∂θ2
+ij=−q2
+1{Θ[−i](X[−i])}2
+{q0θ0(X)+q1Θ(X)}2,
+whereΘ(X) =Θ[−i](X[−i])·θiXiforXi=j,(−j) and similarly for θ0(X). Thus, the
+Fisher information on θijgivenXis
+I(θij|X) =−E/braceleftBigg
+∂2l(Θ|X)
+∂θ2
+ij/bracerightBigg
+=/summationdisplay
+xq2
+1{Θ[−i](x[−i])}2
+q0θ0(x)+q1Θ(x)
+=q1/summationdisplay
+xq1Θ(x)
+q0θ0(x)+q1Θ(x)·1
+θixi·Θ[−i](x[−i])
+=q1/summationdisplay
+x∈{j,(−j)}1
+θix·EΘ[−i]/braceleftBigg/parenleftbiggq0θ0xθ0(X[−i])
+q1θixΘ[−i](X[−i])+1/parenrightbigg−1/bracerightBigg
+.
+Because EΘ[−i]/braceleftbig
+θ0(X[−i])/Θ[−i](X[−i])/bracerightbig
+= 1, Jensen’s inequality implies that
+I(θij|X)≥/summationdisplay
+x∈{j,(−j)}q2
+1
+q0θ0x+q1θix=q2
+1
+¯θij(1−¯θij).ON MOTIF DETECTION 21
+The lower bound B(q1,θij,θ0j) is obtained by dividing the R.H.S. of this inequality
+by the Fisher information on θijgivenXandYjointly,
+I(θij|X,Y) =−E/braceleftBigg
+∂2l(Θ|X,Y)
+∂θ2
+ij/bracerightBigg
+=q1
+θij(1−θij),
+wherel(Θ|X,Y) = logP(X,Y|Θ) is the joint log-likelihood.
+Appendix B: Derivation of E[∆Rf
+1(ˆβm)](35).Given the estimated weight matrix
+ˆΘmbased on observed binding sites D+
+n, the constructed decision function of the WM
+approach
+ˆhm
+1(x) = log( q1/q0)+w/summationdisplay
+i=1[logˆθm
+ixi−logψ0(xi−1,xi)]
+P→˜β0+w/summationdisplay
+i=1/braceleftBigg
+logθf
+ixi
+θf
+isi−logψ0(xi−1,xi)
+ψ0(si−1,si)/bracerightBigg
+,asn→ ∞, (37)
+whereθf
+ij=P(Xi=j|Y= 1) under the FEM with Markov background, ( s1,···,sw)
+is the reference sequence, and
+˜β0= log(q1/q0)+w/summationdisplay
+i=1log{θf
+isi/ψ0(si−1,si)}.
+Letx[−i](s) = (x1,···,xi−1,s,xi+1,···,xw). Following a similar derivation in Sec-
+tion 3.3, we have θf
+ij∝exp(˜βij+ηij), where
+ηij= log
+
+/summationdisplay
+x[−i]eui
+1+e˜βijeuiψ0(x[−i](j))
+
+−log
+
+/summationdisplay
+x[−i]eui
+1+euiψ0(x[−i](si))
+
+
+withui=˜β0+x[−i]˜β[−i]. Since˜βisi=ηisi= 0, log(θf
+ij/θf
+isi) =˜βij+ηijfor alliandj.
+Thus, equation (37) becomes
+ˆhm
+1(x)P→˜β0+w/summationdisplay
+i=1/braceleftbigg
+˜βixi+ηixi−logψ0(xi−1,xi)
+ψ0(si−1,si)/bracerightbigg
+=h(x)+δ(x),
+whereδ(x) =/summationtextw
+i=1ηixi−log{ψ0(xi−1,xi)/ψ0(si−1,si)}. Let ∆ˆhm
+1(x) =ˆhm
+1(x)−h(x).
+The asymptotic normality of√ndˆΘmimplies that√n{∆ˆhm
+1(x)−δ(x)}follows a
+limiting normal distribution with mean 0 and a finite (possib ly zero) variance for
+everyx. Equation (35) then follows from Theorem 2.
+Acknowledgements. TheauthorthanksWingH.Wong,JunS.LiuandZhengqing
+Ouyang for helpful discussions. This work was supportedby N SF grant DMS-0805491.22 Q. ZHOU
+References.
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\ No newline at end of file
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--- /dev/null
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+QUANTUM FLUCTUATIONS CONTRIBUTION TO THE RANDOM WALK OF A 
+SINGLE MOLECULE AND NEW ESTIMATE  OF THE PLANCK CONSTANT 
+ 
+Jean Paul Mbelek1 
+1. Service d'Astrophysique, CEA-Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette, France  
+ 
+Correspondence to: Jean Paul Mbelek1 Correspondence and requests for materials should be 
+addressed to J.P.M. (Email : mbelek.jp@wanadoo.fr).  
+ 
+Abstract : It is shown, by considering the ca se of the harmonic oscillator, that quantum 
+fluctuations may be the most significant contri bution to the random walk of a single molecule. 
+From this point, the controversy on the existence of a “standard quantum limit” (SQL) is addressed and settled on the experimental ground. Comparisons to the experimental data yet 
+avalaible in the literature provide a new estimate of the reduced Planck constant yielding ħ = 
+(1.1 ± 0.2) × 10
+-34 J.s. 
+ 
+ I. Introduction 
+ 
+The uncertainty principle was first introduced in quantum mechanics by W. Heisenberg
+1 to 
+express quantitatively the necessity to give up classical notions like the orbits of electrons in 
+atoms. A more rigorous and general mathematical  derivation of the uncertainty relations was 
+next provided by E. H. Kennard2 then H. P. Robertson3, by defining properly the uncertainties 
+Δx and ΔPx as standard deviations. These authors show in particular that the commutation 
+relation [x , P x] = i ħ implies Δx ΔPx ≥ ½  ⎢<[x , P x]> ⎢or otherwise stated Δx ΔPx ≥ ħ/2, the 
+equality Δx ΔPx = ħ/2 being obtained for the harmonic oscillator. However, the definition of 
+the standard deviation may change from one experimental situation to another. So, carrying 
+the analysis on the uncertainty principle, D. Bohm4 gave an estimate of the time variation of 
+the fluctuation Δx of the position x of a particle (§ 5.4) which reads 
+ 
+Δx = ħ t/m Δx0 (1) 
+ 
+and also follows from the spread of the wave packet of initial width Δx0 in the limit t → ∞ 
+(see § 3.13). Now, by assuming | Δx - Δx0 | << Δx0/2, clearly relation (1) yields Δx2 ≅ (ħ/m)  t. 
+Further, the complete expression Δx = Δx0 [1 + (ħt/mΔx02)2]1/2 that holds at any time implies 
+Δx ≥ Δx0 and Δx2 ≥ (ħ/m)  t, therby suggesting a SQL. However, let us notice that by setting 
+ 
+Δ(dx/dt) = d Δx/dt  (2) 
+ 
+and considering the case of the harmonic oscillator, it is straightforeward to derive after 
+integration the equality 
+ 
+Δx2 = (ħ/m)  t (3) 
+ 
+On the other hand, still considering the case of the harmonic oscillator but assuming d Δx/dt ≤ 
+ΔPx/m, one would obtain instead 
+Δx2 ≤ (ħ/m)  t (4)  
+without necessarily being in conflict with th e uncertainty relations. Relation (3), where t 
+stands for the averaging time, has been put foreward (up to a factor 2 in ref. 6) by V. B. Braginsky et al.
+5,6,7 in their study of the possibility to evade the confrontation with the 
+uncertainty relations. Nevertheless, H. P. Yuen8 pointed out that the derivation of relation (3) 
+from the uncertainty principle is incorrect, and even needs not hold at all. Relation (4) seems 
+rather consistent with the claims by H. P. Yuen9, 10, M. Ozawa11, 12 and M.T. Jaekel and S. 
+Reynaud13. Moreover, we have found some 
+experimentalcasesintheliterature14,15,16,17,18whichplead forrelation (4). It turns out 
+that the controversy on the existence of a SQL9, 10, 11, 12, 13,19, which is of importance for the 
+interferometric detection of gravitational waves,  may be settled on the experimental ground. It 
+already seems that the SQL does not hold. 
+Asstatedabove,eitherrelation(3)or(4)may apply,depending on howd Δx/dt 
+compares to Δvx = ΔPx/m. In the following, we shall consider the case of the harmonic 
+oscillator where the avoidance of the SQL is realized. Thence, by comparing the theoretical results obtained in this study with the avalaible experimental data, an estimate of the reduced 
+Planck constant is derived. 
+  
+II. Quantum fluctuations contribution to the random walk 
+ 
+As one can see, relation (3) compares to the 1D Brownian motion relation Δx
+2 = 2D 0 t. Hence, 
+assuming that the condition (2) is realized, by taking into account both the classical and 
+quantum contribution to the random motion of  a particle, we predict a total diffusion 
+coefficient 
+ 
+D = D 0 + ħ/2m (5)  
+ 
+Let us emphasize that the quantum contribution ħ/2m to the diffusion coefficient is supported 
+by the Schrödinger equation - ( ħ2/2m) Δψ = iħ ∂ ψ/∂t of a free particle of mass m. Indeed, the 
+latter rewrites in imaginary time τ = i  t : (ħ/2m) Δψ = ∂ψ/∂τ, which looks like the diffusion 
+equation with a diffusion coefficient equal to ħ/2m. 
+ 
+ 
+III. Comparison with experimental data and estimate of the reduced Planck constant 
+ 
+Hereafter, we consider various experimental situations where a single molecule is both subject to harmonic oscillations and diffusion along a polymer chain. While the molecule under 
+consideration oscillates and undergoes a random walk, it sweeps out a volume V defined by a 
+lateral surface S and the displacement x. Since quantum fluctuations occur, those of the latter 
+geometrical quantities amount respectively to  ΔV = Δx Δy Δz, ΔS = Δy Δz and Δx. Now, one 
+may write accordingly   
+dΔV = ΔS × Δv
+x × dt  (6) 
+ 
+Hence, the 1D mean-square displacement (MSD) reads finaly 
+ 
+Δz2 = Δy2 = Δx2 = (ħ/3m)  t (7) 
+ which is less then the SQL by a factor 3. So, the total diffusion coefficient rewrites D = D 0 + ħ/6m (8) 
+ 
+Table 1 and Figure 1 below give respectively some  experimental data found in the literature 
+and the fit to these data. The results are found in good agreement with relations (7) and (8) by 
+neglecting the classical diffusion coefficient, D 0, with respect to the quantum mechanical 
+contribution ħ/6m. Also, we have found one case in the literature22 where the lower bound of 
+D is equal to ħ/6m, the upper bound of D 0 is then estimated to 5.22 μm2/s on account of 
+relation (8). Let us emphasize that diffusion coefficients that are less than ħ/2m, in 
+consistency with relation (4) by neglecting D 0, have been reported too in the literature14. All 
+other data, we have found in the literature20,21 are consistent with relations (7) and (8) by 
+neglecting instead ħ/6m with respect to D 0. 
+  
+ 
+ 
+ 
+Table 1 : 1D Diffusion coefficient, D, of the Brownian motion of a single molecule of mass 
+m. The data are taken from the references. The uncertainties* of the first and fourth raws have 
+been derived by us from the error bars of th e experimental curves of the authors (1 Da = 
+1,66054 × 10-27 kg ; 1 kDa = 103 Da). 
+ 
+Figure 1 : 1D diffusion coefficient D ( μm2/s) versus 1/6m (1022 kg-1). The linear fit to the data 
+(see table 1) yields a slope equal to (1.1 ± 0.2) × 10-34 J.s (reduced Chi-square χν2 = 1.19), 
+which is a fairly good estimate of ħ as compared to the 2006 CODATA value ħ = 
+(1.054 571 628 ± 0.000 000 053) × 10-34 J.s.   M (kDa)  [Ref.]  D (μm2/s)  [Ref.] 
+1050 15 0.0082 ± 0.0014* 15 
+113 16 0.11 ± 0.01 16 
+61 16 0.18 ± 0.02 16 
+61 17 0.16 ± 0.03* 17 
+50 18 0.23 ± 0.08 18 IV. Conclusion 
+ 
+We have shown that quantum fluctuations contribute to the random walk of a single molecule. In addition, referring to the experimental data, it turns out that the SQL is evaded in most 
+cases involving the random walk of a single molecule. As a consequence, we have derived a 
+fairly good estimate of the reduced Planck constant equal to ħ = (1.1 ± 0.2) × 10
+-34 J.s. 
+ 
+ References 
+ 
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+ 
+3. Robertson, H. P., Phys. Rev., The Uncertainty Principle, 34, 163, 1929. 
+ 
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\ No newline at end of file
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+arXiv:1001.0343v2  [quant-ph]  1 Jul 2011Entanglement and symmetry in permutation symmetric states
+Damian. J. H. Markham∗
+CNRS, LTCI, Telecom ParisTech, 37/39 rue Dareau, 75014 Pari s, France
+We investigate the relationship between multipartite enta nglement and symmetry, focusing on
+permutation symmetric states. We give a highly intuitive ge ometric interpretation to entanglement
+via the Majorana representation, where these states corres pond to points on a unit sphere. We use
+this to show how various entanglement properties are determ ined by the symmetry properties of the
+states. The geometric measure of entanglement is thus phras ed entirely as a geometric optimisation,
+and a condition for the equivalence of entanglement measure s written in terms of point symmetries.
+Finally we see that different symmetries of the states corres pond to different types of entanglement
+with respect to interconvertibility under stochastic loca l operations and classical communication
+(SLOCC).
+I. INTRODUCTION
+Entanglement and symmetry are two main concepts
+at the heart of quantum mechanics. For a while now
+there havebeen enticing hints ofa generalconnection be-
+tween the two. On an intuitive level, we may understand
+that changing global symmetry (or topological proper-
+ties) should be a global operation, so one that effects the
+entanglement of the system at hand. The relationship is
+of great interest in particular because of the relation be-
+tweensymmetryandphasetransitions. Therearebynow
+a vast array of examples where entanglement shows some
+relationship to symmetry breaking, for example in quan-
+tum phase transitions where phase transition coincides
+with change in entanglement properties [1–3]. Indeed it
+has been suggested that entanglement may be able to see
+phase transitions where conventional order parameters
+fail. However, a concrete relationship remains unclear,
+for example, it is known change in some symmetry prop-
+erties need not effect the entanglement, and vice versa.
+For a recent review of these issues see [4].
+At the same time symmetry properties of states have
+been used to simplify the study of their entanglement for
+example in its calculation [5, 6] and questions of sepa-
+rability [7]. A particular feature of multipartite entan-
+glement is that it is possible to have different ‘types’ of
+entanglement, where-by we mean different classes under
+SLOCC(StochasticLocalOperationsandClassicalCom-
+munication) [8]. This property has been largely over-
+looked in the in the study of phase transitions and the
+use of symmetries in entanglement theory. Two states
+are SLOCC inequivalent (belong to different classes) if
+they cannot be converted to one-another via local op-
+erations and classical communications, even probabilis-
+tically, which signifies them as potentially different re-
+sourcesin the contextofquantum informationprocessing
+(QIP). Alongside this comes a plethora of entanglement
+measures, with a variety of different operational interpre-
+tations, and which may be suited to quantifying one type
+∗Electronic address: markham@enst.frof entanglement more than another. The question nat-
+urally arises, can symmetry help us to explore this vast
+landscape, and can a relationship between symmetry and
+the types of entanglement be made.
+Inthisworkwefocusonpermutationsymmetricstates.
+These states are useful in a variety of QIP tasks, oc-
+cur naturally as ground states for example in some Hub-
+bard models, and certain of these states have been im-
+plemented experimentally recently [9, 10]. Various en-
+tanglement properties have also been studied of these
+states, such as the clarification of separability conditions
+[7], the calculation of the geometric measure of entan-
+glement [11, 12] and the identification of SLOCC classes
+[13]. In all these cases however, permutation symmetry
+is essentially used as a tool for simplification in calcula-
+tions. We would like to see if further symmetry proper-
+ties can be useful, if a deeper connection between sym-
+metry and entanglement properties can be found, and if
+there may be some insight into the role of entanglement
+in many body physics. To this extent we observe that
+symmetries with respect to local operations (rather than
+permutation) determine severalentanglement properties,
+with intriguing mirrors in spinor Bose-Einstein Conden-
+sates (BEC).
+In particular, by using the Majorana representation
+[14], we see how symmetry allows us to calculate the ge-
+ometric measure of entanglement [15] and identify the
+most entangled state [16]. Then, we show that the ex-
+istence of certain symmetry guarantees equivalence of
+three different entanglement measures - the geometric
+measure of entanglement, the logarithmic robustness of
+entanglement [17] and the relative entropy of entangle-
+ment [18]. Finally we will see how the different sym-
+metries reflect different types of entanglement, (in terms
+of SLOCC classes) indicating an intriguing relationship
+between symmetries and types of entanglement. We will
+closewith some remarkson occasionsthese same symme-
+tries coincide with different phases for spinor BEC, and
+how these states may be generated experimentally.2
+FIG. 1: (Colour online) The Majorana representation of the
+n-party GHZ state |GHZn/angbracketright:= (|0/angbracketright⊗n+|1/angbracketright⊗n)/√
+2), hasn
+MPsequallyspacedaroundtheequator, herefor n= 6insolid
+points. The hollow point at the north pole is point of the clos -
+est product state, maximizing/producttext
+i|/angbracketleftφ|ηi/angbracketright|2=/producttext
+i(cos(θi/2))2.
+II. ENTANGLEMENT IN THE MAJORANA
+REPRESENTATION
+We first present the Majorana representation [14].
+This way of seeing states has been used recently to sim-
+plify the classification of symmetric states into SLOCC
+classes [13, 19]. For nqubits, all permutation symmetric
+states can be written in the form [14]
+|ψ/an}bracketri}ht=eiα
+√
+K/summationdisplay
+PERM|η1/an}bracketri}ht|η2/an}bracketri}ht...|ηn/an}bracketri}ht, (1)
+wherethesumisoverallpermutationsand Kisanormal-
+isation constant. The Majorana representation consists
+ofnpoints corresponding to the nstates from this de-
+composition |ηi/an}bracketri}ht=cos(θi/2)|0/an}bracketri}ht+eiφisin(θi/2)|1/an}bracketri}htvia the
+standard Bloch sphere - i.e. a point on the unit sphere
+at position θi,φi. We call these the Majorana Points
+(MPs), and they define the state up to global phase eiα
+(see Fig. 1). For more details see the Appendix A.
+To see how entanglement can be visualised in the Ma-
+jorana representation, we first notice that the product of
+local unitaries on a symmetric state U⊗U⊗...U|ψ/an}bracketri}htis
+just a rotation of the Majorana sphere, since each point
+gets rotated by the same U. In fact it can be shown
+that if two symmetric states |ψ/an}bracketri}ht,|φ/an}bracketri}htare related by local
+unitariesU1⊗U2⊗...Un|ψ/an}bracketri}ht=|φ/an}bracketri}ht, there is always some
+Usuch that they can be connected by the same unitary
+U⊗U⊗...U|ψ/an}bracketri}ht=|φ/an}bracketri}ht. This fact is shown for the more
+general case of local invertible operations in [13, 19], and
+the same proof works for unitaries. Further, this shows
+that the symmetry of the state under local unitaries U⊗n
+is reflected by the symmetry of the MPs (see also the
+Appendix A). This will be a main tool throughout the
+paper.
+We can make the connection to entanglement more ex-
+plicit, using the geometric measure of entanglement [15],
+Eg(|ψ/an}bracketri}ht) = min
+|Φ/angbracketright∈PROD−log2(|/an}bracketle{tΦ|ψ/an}bracketri}ht|2), (2)
+where PROD is the set of product states. It has
+recently been shown that for permutation symmetric
+states, we can always take a symmetric product statep<m
+p<mm
+FIG. 2: (Colour online) States of n=m+ 2pqubits which
+are totally invariant for the Dihedral symmetry groups Dm,
+|Dm(n,p)/angbracketright= 1/√
+2(|S(n,p)/angbracketright+|S(n,n−p)/angbracketright). The state has
+pMPs at each pole and m=n−2pMPs distributed evenly
+around the equator. For all n,p, these states satisfy E=
+ERob=ER=EG.
+|Φ/an}bracketri}ht=|φ/an}bracketri}ht|φ/an}bracketri}ht...|φ/an}bracketri}htin this optimisation [11], for which the
+Majorana representation consists of npoints at the po-
+sition of |φ/an}bracketri}ht. For the general npartite symmetric state
+(1) we then have
+Eg(|ψ/an}bracketri}ht) =−log2(Λ(ψ)),
+Λ(ψ) = max
+|φ/angbracketright|/an}bracketle{tφ|⊗n|ψ/an}bracketri}ht|2
+=1
+Kn!2max
+|φ/angbracketright|/an}bracketle{tφ|η1/an}bracketri}ht|2|/an}bracketle{tφ|η2/an}bracketri}ht|2...|/an}bracketle{tφ|ηn/an}bracketri}ht|2.
+Hence, the optimisation problem of finding the closest
+product state has the geometric interpretation of max-
+imising the product of angles |/an}bracketle{tφ|ηi/an}bracketri}ht|2. The example of
+|GHZ6/an}bracketri}htis illustrated in Fig. 1.
+This geometric phrasing of the problem allows us to
+use geometric properties, for example symmetry of the
+MP distribution to calculate entanglement and to search
+for the most entangled states in this class. In a sense,
+we can say that the most entangled states will be those
+which spread the points out the most (though this does
+not necessarily coincide with other definitions of ‘spread’
+such as Tammes’ problem). This direction is pursued in
+detail in follow up work [20], and has been independantly
+studied in [27].
+III. EQUIVALENCE OF ENTANGLEMENT
+MEASURES
+We will now proceed to see how the Majorana rep-
+resentation can allow us to identify symmetries indicat-
+ing states for which several distance like entanglement
+measures coincide, and show that these states represent
+different SLOCC classes of entangled states.
+In ref. [12] the relationship between the geometric
+measure of entanglement and two other distance like en-
+tanglement measures, the relative entropy of entangle-
+mentER[18], the logarithmic robustness of entangle-
+ment [17] is studied. In particular, it was shown that if
+there exists a local unitary group for which the state in
+questionψ=|ψ/an}bracketri}ht/an}bracketle{tψ|is itself an invariant subspace of the
+group, then we have EG(|ψ/an}bracketri}ht) =ER(|ψ/an}bracketri}ht) =ERob(|ψ/an}bracketri}ht).3
+Equivalence of measures is desirable for several rea-
+sons. Primarily because the different measures have dif-
+ferent interpretations. For example, ERobsignifies the
+ability of the state to withstand noise [17] and the rela-
+tive entropy being an entropic quantity ERnaturally has
+several information theoretic interpretations [18]. Since
+EGis easier to calculate, it is easier to verify these oper-
+ational properties also. Further significance of the equiv-
+alence is discussed in [12] in particular it significance for
+local accessibility of information and the construction of
+optimal entanglement witnesses.
+Using the equivalence between symmetry of points
+and of states, we are able to phrase the problem soley in
+terms of the Majorana representation (see the Appendix
+B for more details)
+Lemma 1 :If a permutation symmetric state |ψ/an}bracketri}hthas
+MPs such that they are invariant under some subgroup
+X⊂SO(3), and that for any small change of the points
+this invariance disappears, it satisfies
+EG(|ψ/an}bracketri}ht) =ER(|ψ/an}bracketri}ht) =ERob(|ψ/an}bracketri}ht).
+We call such subgroups X⊂SO(3) the symmetry
+groups, and say such states are totally invariant. In-
+triguingly, this is exactly the condition for finding inert
+states in the context of spinor condensates [21], which
+will be discussed more in the concluding remarks.
+TheMajoranarepresentationthen allowsustoidentify
+symmetries to show equality of the entanglement mea-
+sures for many new sets of states. The complete set of
+all the possible subgroups of SO(3) are the continuous
+groups, orthogonal O(2) and special orthogonal SO(2),
+and discrete groups Cyclic Cm, Dihedral Dm, Tetrahe-
+dralT, Octehedral Oand Isocahedral Y. One can then
+systematicallygothroughallofthese groupstofindthese
+special states, as done in [21] in the context of inert
+states. For the subgroup of arbitrary rotations about
+a fixed axis SO(2), we see that states with MPs only at
+either pole of the rotation axis satisfy our condition. If
+the rotations are around the Z-axis, these are the states
+|S(n,k)/an}bracketri}ht:=1/radicalBig/parenleftbign
+k/parenrightbig(/summationdisplay
+PERM|00...0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+n−k11..1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+k/an}bracketri}ht),
+also known as Dicke states, and we can see here pictori-
+ally the proof of equivalence for these states reported in
+[12]. Note that for even nandk=n/2, these states also
+satisfy our condition for the group O(2) (arbitrary rota-
+tions around the Z-axis, and a flip on some axis in the
+X−Yplane). In such cases we associate the state with
+the smallest subgroup. The cyclic group Cnhas no truly
+invariantstates - since if all points aremovedtogether up
+and down the axis of rotation the symmetry is not lost.
+The dihedral group Dm(consisting of rotations through
+2π/mand a flip on the axis of rotation) has mtotally
+invariant states for each value m(see Fig. 2). T,Oand
+FIG. 3: (Colour online) Different symmetries for four qubits
+giving states such that E=ERob=ER=EG. The symmetry
+group and the entanglement are written below the sphere.
+Each state is in a different SLOCC class.
+Yonly have truly invariant states for certain n. For the
+tetrahedral group Ttruly invariant states are made up of
+tetrahedrons, their antipode tetrahedrons, and octagons
+with at most 2 MPs on any tetrahedron point and 3 MPs
+at any octahedron point, so that there are only truly in-
+variant states for n≤34. For the octahhedral group O
+truly invariant states have MPs at the points of the cube
+and the octahedron with at most 3 and 2 MPs at each re-
+spectively, sothattheyonlyexistfor n≤34. Alltrulyin-
+variant states of the Isocahedral group Yare made up of
+combinations of isocahedrons (with 12 vertices) and do-
+decahedrons (with 20 vertices) with at most 3 and 2 MPs
+ateachrespectively,hencetheyexistonlyfor n≤88. For
+four qubits there are four entangled states satisfying the
+condition, |T/an}bracketri}ht= 1/√
+3|S(4,0)/an}bracketri}ht+/radicalbig
+2/3|S(4,3)/an}bracketri}ht,|GHZ4/an}bracketri}ht,
+|S(4,2)/an}bracketri}htand|W4/an}bracketri}ht=|S(4,1)/an}bracketri}htas shown in Fig. 3.
+IV. SYMMETRY AND SLOCC
+ENTANGLEMENT CLASSES
+We now look at how these different symmetries also
+correspond to different SLOCC entanglement classes.
+First of all, it is shown in [13, 19] that if two states have
+different degeneracies of MPs (that is, the number of
+MPs which are on top of each other is different), they
+are SLOCC inequivalent. From this it is clear that:
+Lemma 2 :For any number of qubits greater than two
+the totally invariant states with respect to the groups
+O(2),SO(2)andDmare of different entanglement types.
+Thisistruesincetheyhavedifferentdegenerecies. This
+fact also means that in addition the totally invariant
+states for the dihedral group |Dm(n,p)/an}bracketri}htare SLOCC in-
+equivalent for all p>0 (see Fig. 2).
+For the remaining symmetry groups T,OandYthere
+are only a finite number of possible totally invariant
+states. We can then use a combination of the degen-
+eracy and other methods to attempt to show the same
+for these all subgroups. Consider the four qubit case in
+Fig. 3. From the above, we can see that |S(4,2)/an}bracketri}ht(with
+two sets of two degenerate MPs) is in a different class
+to|W4/an}bracketri}ht(with a three degenerate point), and they are4
+both in different classes to |T/an}bracketri}ht,|GHZ4/an}bracketri}ht(which have all
+four MPs separate). To see that the |T/an}bracketri}ht,|GHZ4/an}bracketri}htare
+different, we use the fact [8] that under SLOCC the min-
+imal number of terms rfor any expansion of the state
+in terms of only product states (the log of which is the
+Schmidt measure [22]) remains unchanged. It is straight-
+forwardtoseethattakingsomeminimumdecomposition,
+from definition (2) we have EG≤log2(r). We know
+thatr(|GHZ4/an}bracketri}ht) = 2 [8], and in [20] it is shown that
+EG(|T/an}bracketri}ht) = log2(3), hencer(|T/an}bracketri}ht)≥3> r(|GHZ4/an}bracketri}ht) and
+so they are in different SLOCC classes also. For larger
+none can in principle go through all cases individually
+(since there are only finitely many) and check using sim-
+ilar methods. Such an exhaustive search was beyond the
+scope of this manuscript, however, the same techniques
+as above can be used to show the SLOCC inequivalence
+forallthe totally invariant states up to seven qubits, and
+it seems plausible that indeed all different symmetries do
+imply different class of entanglement.
+V. DISCUSSIONS
+In this work we have presented a geometrical repre-
+sentation of the entanglement of permutation symmetric
+states in the form of the Majorana representation. This
+has allowed us to phrase the geometric measure of en-
+tanglement in a simple way, and further to look at how
+the further symmetry properties of states effect their en-
+tanglement properties, in particular showing equivalence
+of three differen distance like measures. This equiva-
+lence simplifies calculation, and allows for wider oper-
+ational understanding as the operational interpretations
+coincide. Finally we show that for these states the dif-
+ferent symmetries correspond to different types of entan-
+glement. This presents a very interesting relationship
+between symmetries and types of entanglement. Though
+we are not able to confirm this is a general connection,
+it is very interesting, and seems possible, and worth in-
+vesgtigating deeper.
+Intriguingly, in the context of spinor condensates sim-
+ilar symmetry arguments have been used to identify and
+characterise different phases of matter [23–25]. In this
+case the Majorana representation is used not to describe
+nsymmetric qubits, but rathera singlespin S=n/2sys-
+tem (through the well known isometry between the two)
+[14]. Because of this caution is required when talking
+about entanglement in this context, but it is not impossi-
+ble forthe twopicturesto coincide, for example, the total
+spin can be a result of combined spin half systems in ex-
+actly the permutation symmetric space we look at, which
+reallywouldbeentangledaswediscusshere. Inthissense
+we would see that phase transitions through symmetry
+are incidental with phase transitions through entangle-
+ment, raising the prospect of entanglement type as an
+indicator of different phases. Indeed in [23], a phase dia-
+gram is presented for a spin two spinor condensate where
+each phase is identified exactly with different symmetrytypes presented in Fig. 3. Where these connections are
+most explicit and general is in the case of inert states -
+often good candidates for ground states in spinor BEC
+- where the conditions of equivalence of EG,ERoband
+ERcoincide exactly in terms of the MPs [21], pointing
+to deeper possible connections.
+The states discussed here can also be experimentally
+prepared in a variety of ways and media. For example in
+opticsthesixparty |S(6,3)/an}bracketri}ht(Dicke)stateandseveralfive
+and four party states have recently been generated, and
+their entanglement properties verified [9, 10]. Further,
+recently a general scheme has been developed which can
+generate any symmetric state [26] (including all those
+here) which works for any Λ-system photon emitters,
+such as trapped ions or neutral atoms or quantum dots,
+so may be long lived, and which may be in reach of cur-
+rent experiment.
+Since completion of this work several related works
+have emerged [27–31].
+Acknowledgements We are very grateful to Shashank
+Virmani, Martin Aulbach, Mio Muraoand Vlatko Vedral
+for insightful comments and discussions.
+Appendix A: Majorana Representation
+The permutation symmetric subspace of nqubits is
+spanned by the Dicke states
+|S(n,k)/an}bracketri}ht:=1/radicalBig/parenleftbign
+k/parenrightbig(/summationdisplay
+PERM|00...0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+n−k11..1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+k/an}bracketri}ht),(A1)
+which can be understood as the symmetric states with k
+excitations. Thus any permutation symmetric state can
+be written as
+|ψ/an}bracketri}ht=/summationdisplay
+ak|S(n,k)/an}bracketri}ht. (A2)
+Alternatively all symmetric states can be written in the
+Majorana representation [14]
+|ψ/an}bracketri}ht=eiα
+√
+K/summationdisplay
+PERM|η1/an}bracketri}ht|η2/an}bracketri}ht...|ηn/an}bracketri}ht, (A3)
+where the sum is over all permutations and Kis a nor-
+malisation constant.
+To find the Majorana representation (A3) we consider
+the overlap with product state |φ/an}bracketri}ht⊗n,|φ/an}bracketri}ht= cos/parenleftbigθ
+2/parenrightbig
+|0/an}bracketri}ht+
+eiϕsin/parenleftbigθ
+2/parenrightbig
+|1/an}bracketri}ht. It is clear by comparison to equation (A3)
+that|φ/an}bracketri}htorthoganol to the MP |ηi/an}bracketri}htwill give zero over-
+lap. This is exactly how we find the MPs. For simplicity
+we take a multiple of the overlap, sometimes called the
+characteristic polynomial ,Majorana polynomial ,ampli-
+tude function orcoherent state decomposition
+f(ψ) := cos−n/parenleftbiggθ
+2/parenrightbigg
+/an}bracketle{tφ|⊗n|ψ/an}bracketri}ht=n/summationdisplay
+k=0/radicalBigg/parenleftbiggn
+k/parenrightbigg
+akαk,(A4)5
+which is a complex polynomial in α:=e−iϕtan/parenleftbigθ
+2/parenrightbig
+. By
+the fundamental theorem of algebrathis has unique zeros
+up to multiplication by some complex. Hence the zeros
+αj=e−iϕjtan/parenleftBig
+θj
+2/parenrightBig
+define the state |ψ/an}bracketri}htup to a global
+phase. The corresponding MPs are at position θ′
+j=θj+
+π,ϕ′
+j=ϕj+π.
+Note that we can understand the state |φ/an}bracketri}ht⊗nas a kind
+of generalized coherent state [32, 33], defined by the ac-
+tion of a group on some chosen fiducial state (so that
+certain properties apply such as overcompleteness). For
+our case the group is SU(2) as represented by U⊗n, with
+Ua rotation through θ,ϕand the fiducial state |0/an}bracketri}ht⊗n,
+that is
+|φ/an}bracketri}ht⊗n=U⊗n|0/an}bracketri}ht⊗n. (A5)
+When viewing the symmetric subspace as one spin S=
+n/2 system, these are equivalent to spin coherent states
+[34, 35]. In this sense the Majorana representation is a
+kind of condensed coherent state representation of states
+(since it is only concerned with the zeros of the coherent
+state decomposition (A4)).
+Appendix B: Equivalence of entanglement measures
+We now come to the proof of the equivalence between
+entanglement measures and the symmetry of the MP dis-
+tributions. The entanglement measures in question are
+the geometric measure of entanglement [15], the relative
+entropy of entanglement ER[18] and the logarithmic ro-
+bustness of entanglement [17] is studied. Equality be-
+tween the measures is guarenteed for a state |ψ/an}bracketri}ht, if it is
+possible to find a separable state of the form [12]
+ωsep= Λ(ψ)|ψ/an}bracketri}ht/an}bracketle{tψ|+(1−Λ(ψ))∆,(B1)
+where ∆ can be any density matrix and Λ( ψ) is the max-
+imum overlap with a product state as defined in (3).
+The state (B1) can be understood as the ‘closest’ sep-
+arable state with respect to the robustness of entangle-
+ment, which is deemed equal to the geometric measure
+of entanglement by its form [12].
+The trick used in [12] is to take techniques from group
+averaging to find such a state (see also [5]). For a group
+G, any particular representation U(g),g∈Gcan always
+be expanded as a product sum over irreducible represen-
+tations (irreps, which we enumerate by k), and the irreps
+give a decomposition of the total Hilbert space,
+U(g) =K/circleplusdisplay
+k=1I 1Ak⊗UBk(g) (B2)
+H=K/circleplusdisplay
+k=1HAk⊗HBk, (B3)
+whereUBk(g) is the representation of g∈Gfor irrepk
+acting on Hilbert space HBk. The role of HAkis just togive a compact form to express multiplicity - the multi-
+plicity of irrep kis given by dim(HAk) =Tr(I 1Ak). Note
+that the tensor product in the above is nothing to do
+with the separation of parties defining entanglement. By
+direct application of Shur’s lemma, averaging over the
+group gives [12]
+ω=/integraldisplay
+U(g)ρU(g)†dg
+=/summationdisplay
+k=11
+dim(HBk)TrBk{PAk⊗BkρPAk⊗Bk}⊗I 1Bk.
+(B4)
+If we nowaverageovera local unitary groupon a prod-
+uct state |Φ/an}bracketri}htwhich achieves the maximum overlap Λ( ψ),
+we will get a separable state, which is our candidate for
+(B1). If, further, the state ψ=|ψ/an}bracketri}ht/an}bracketle{tψ|isaninvariantsub-
+space associated to a one-dimensional irrep (say k= 1)
+with multiplicity one we have
+ωsep=/integraldisplay
+U(g)ΦU(g)†dg
+=Λ(ψ)|ψ/an}bracketri}ht/an}bracketle{tψ|
++/summationdisplay
+k=2TrBk{PAk⊗Bk|Φ/an}bracketri}ht/an}bracketle{tΦ|PAk⊗Bk}
+dim(HBk)⊗I 1Bk,
+(B5)
+which is indeed of form (B1), implying equality of the
+entanglement measures.
+In terms of states, ψ=|ψ/an}bracketri}ht/an}bracketle{tψ|corresponds to a one-
+dimensional irrep if it is invariant under group action. A
+one dimensional irrep is a phase which acts over a space
+of dimension equal to the multiplicity. Any state (one
+dimensional matrix) in this space is unchanged and so
+can itself be considered a 1D irrep. Since it is possible to
+continuously change states through this space, it means
+that a state which is a 1D irrep, therefore invariant, can
+be continuously changed to another state which is also
+a 1D irrep, and hence also invariant. If, on the other
+hand, a small shift breaks the invariance, the state has
+multiplicity only one, as we want.
+The groups we consider in this work are naturally
+enough subgroups of SU(2), as represented by the lo-
+cal unitaries U⊗n. Again, we see from the definition (1),
+such operations are simply rotations (in SO(3)) of the
+Majorana sphere itself. Since we are only interested in
+the state matrix ψ(where global phases do not matter),
+the invariance the MPs implies a state is a 1D irrep. If
+no small change in the positions of the points is also in-
+variant, this implies there is no multiplicity within the
+symmetric subspace. Although this is not immediately
+enough to show the group averaged state is of the form
+(B1), it can be proved as follows. The only remaining
+possibility for multiplicity is if it has part outside the
+symmetric subspace. In fact, a projection onto it (say for
+irrepk) must be of the form PAk⊗Bk=|ψ/an}bracketri}ht/an}bracketle{tψ|+|ψ⊥/an}bracketri}ht/an}bracketle{tψ⊥|
+where|ψ⊥/an}bracketri}hthas no components in the symmetric sub-
+space. This is true since its representation is U⊗nand so6
+any 1D irrep cannot stretch over the symmetric subspace
+and another subspace, but must be distinctly in one or
+the other. If we put this into (B4) (with again ρ= Φ),
+we indeed get the form (B5).
+Thus the condition for equality of measures stated in
+the main text is correct and complete. For example, forthe subgroup of arbitrary rotations about a fixed axis
+SO(2), we see that states with MPs only at either pole
+of the rotation axis satisfy our condition. If the rotations
+are around the Z-axis, these are the Dicke states, and we
+can see here pictorially the proof of equivalence for these
+states reported in [12].
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\ No newline at end of file
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+arXiv:1001.0344v1  [quant-ph]  3 Jan 2010Topological quantum order: stability under local
+perturbations
+Sergey Bravyi∗, Matthew Hastings†, and Spyridon Michalakis‡
+January 3, 2010
+Abstract
+We study zero-temperature stability of topological phases of matter under weak time-
+independent perturbations. Our results apply to quantum sp in Hamiltonians that can be written
+as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain
+topological order conditions. Given such a Hamiltonian H0we prove that there exists a con-
+stant threshold ǫ >0 such that for any perturbation Vrepresentable as a sum of short-range
+bounded-norm interactions the perturbed Hamiltonian H=H0+ǫVhas well-defined spectral
+bands originating from O(1) smallest eigenvalues of H0. These bands are separated from the
+rest of the spectrum and from each other by a constant gap. The band originating from the
+smallest eigenvalue of H0has exponentially small width (as a function of the lattice s ize).
+Our proof exploits a discrete version of Hamiltonian flow equ ations, the theory of relatively
+bounded operators, and the Lieb-Robinson bound.
+∗IBM Watson Research Center, Yorktown Heights NY 10594 (USA) ;sbravyi@us.ibm.com
+†Microsoft Research Station Q, CNSI Building, University of California, Santa Barbara, CA, 93106 (USA);
+mahastin@microsoft.com
+‡T-4 and CNLS, LANL - Los Alamos, NM, 87544 (USA); spiros@lanl.gov
+1Contents
+1 Introduction 3
+1.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
+1.2 Sketch of the stability proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
+2 Hamiltonians describing TQO 7
+2.1 Frustration-free commuting Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 7
+2.2 Formal definition of TQO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
+2.3 Verification of TQO conditions for stabilizer Hamiltoni ans . . . . . . . . . . . . . 10
+2.4 Unstable version of the toric code model . . . . . . . . . . . . . . . . . . . . . . . 11
+3 Relatively bounded perturbations 12
+3.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
+3.2 Stability of TQO under block-diagonal perturbations . . . . . . . . . . . . . . . . 13
+4 Hamiltonian flow equations 16
+4.1 Outline of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
+4.2 Local decompositions of Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . 18
+4.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
+5 Linearized block-diagonalization problem 23
+5.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
+5.2 Finding the transformation S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
+5.3 Local decomposition of the transformed Hamiltonian . . . . . . . . . . . . . . . . 28
+6 Lieb-Robinson bounds 29
+7 Adiabatic continuation of logical operators 34
+7.1 Dressed Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
+21 Introduction
+The traditional classification of different phases of matter due to Landau rests on symmetry
+breaking. Given a pair of gapped Hamiltonians H1,H2with some symmetry group G, the ground
+states ofH1andH2were considered to be in different phases if their symmetry br eaking patterns
+are different. The discovery of topologically ordered phase s, however, changes this paradigm.
+Models such as Kitaev’s toric code [1] have “topologically n on-trivial” ground states despite
+lacking any symmetry breaking. Such states cannot be change d into a “topologically trivial”
+state such as a product state by any unitary locality-preser ving operator [2].
+One possible approach to classifying topological phases is to call a pair of gapped Hamilto-
+niansH1,H2topologically equivalent iff it is possible to connect H1andH2by a continuous
+path in the space of local gapped Hamiltonians. Using the ide a of quasi-adiabatic continua-
+tion [3], one can describe the evolution of the ground state s ubspace along such a path by a
+unitary locality-preserving operator. In particular grou nd state degeneracy and the geometry
+of “logical operators” acting on the ground subspace is the s ame forH1andH2.
+Most of the Hamiltonians describing TQO models such as Kitae v’s quantum double model [1]
+or Levin-Wen string-net model [4] are not quite physical sin ce they involve interactions affecting
+more than two spins at a time. One may hope however that such mo dels emerge as low-energy
+effective Hamiltonians describing some simpler high-energ y theories [5, 6, 7]. For example, the
+toric code model with four-spin interactions can be “implem ented” as the fourth-order effective
+Hamiltonian describing low-energy limit of the honeycomb m odel [8] which involves only two-
+spin interactions. The higher-order corrections to the effe ctive Hamiltonian must be regarded
+as a perturbation. Thus in order to show that the honeycomb mo del is topologically equivalent
+to the toric code (in the Jz≫Jx,Jyphase) one has to prove that the spectral gap in the toric
+code Hamiltonian does not close in a presence of weak perturb ationsVthat can be represented
+as a sum of bounded-norm short-range (exponentially decayi ng) interactions.
+Even if one leaves aside the question of how multi-spin inter actions can be implemented
+in a lab, one has to worry about precision up to which an ideal m odel Hamiltonian can be
+approximated in a real life. If the presence or absence of the gap depends on tiny variations of
+the Hamitonian parameters that are beyond experimentalist ’s control, the distinction between
+gapped and gapless Hamiltonians is meaningless. The best we can hope for is to approximate
+individual interactions of the ideal model with some consta nt precision ǫindependent of the
+system size N. Accordingly, the ideal Hamiltonian can be approximated on ly up to an extensive
+errorO(ǫN). Proving stability of topological phases thus reduces to p roving that the spectral
+gap of the ideal TQO models does not close in the presence of su ch extensive perturbations.
+Currently, the tools for proving lower bounds on the spectra l gap are fairly limited. For
+example, one of the outstanding problems in mathematical ph ysics is to prove the existence of
+a spectral gap for the spin-1 Heisenberg chain, making rigor ous the arguments of Haldane [9].
+Some progress toward this was obtained by Yarotsky [10], who showed the stability of the gap
+near the AKLT point [11]. Yarotsky’s tools however are limit ed to perturbations of Hamiltonians
+which are topologically trivial. Thus, new methods are need ed to analyze topologically ordered
+phases. Some partial results were recently obtained by Treb st et al [12] and Klich [13] who
+proved gap stability for the toric code under a special type o f perturbations diagonal in the
+3z-basis as well as for anyon lattices on a sphere.
+In the present paper we succeed in proving gap stability unde r generic local perturbations.
+Our results are valid not just for the toric code, but more gen erally for any Hamiltonian which
+can be written as a sum of geometrically local commuting proj ectors on a D-dimensional lattice
+with certain topological order conditions that we define lat er. This includes models such as
+Kitaev’s quantum double model [1] and the Levin-Wen string- net model [4]. Furthermore, we
+prove stability of the spectral gaps separating sufficiently low-lying eigenvalues of the unper-
+turbed Hamiltonian. In the case of 2D models with anyonic exc itations it allows us to define
+string-like operators that create particle excitations fo r the perturbed Hamiltonian and prove
+stability of invariants describing the braiding statistic s of excitations. We explain how this may
+be used to adiabatically control a perturbed topological mo del to perform braiding operations
+to manipulate topologically protected quantum informatio n.
+1.1 Summary of results
+Consider a system composed of finite-dimensional quantum pa rticles (qudits) occupying sites
+of aD-dimensional lattice Λ of linear size L. The corresponding Hilbert space is a tensor
+product of the local Hilbert spaces, H=/circlemultiplytext
+u∈ΛHu, dim (Hu) =O(1). Suppose the unperturbed
+Hamiltonian H0can be written as a sum of geometrically local pairwise commu ting projectors,
+H0=/summationdisplay
+A⊆ΛQA,
+where the sum runs over all subsets of the lattice of diameter O(1) andQAis a projector acting
+non-trivially only on sites of A(one may have QA= 0 for some subsets A). The commutativity
+assumption implies that all projectors QAcan be diagonalized in the same basis. Accordingly,
+all eigenvalues of H0are non-negative integers. We assume that the smallest eige nvalue ofH0
+is zero, that is, ground states of H0are annihilated by every projector QA. Such states span
+the ground subspace P,
+P={|ψ/a\}⌊ra⌋ketri}ht ∈ H :QA|ψ/a\}⌊ra⌋ketri}ht= 0 for all A⊆Λ}.
+For any subset B⊆Λ we shall also define a local ground subspace as
+PB={|ψ/a\}⌊ra⌋ketri}ht ∈ H :QA|ψ/a\}⌊ra⌋ketri}ht= 0 for all A⊆B}.
+We shall use the notations PandPBboth for linear subspaces and for the corresponding
+projectors. Note that the projector PBacts non-trivially only on the subset B.
+We shall impose two extra conditions on H0and the ground subspace Pthat guarantee the
+gap stability. Let us first state these conditions informall y (see Section 2.2 for formal definitions):
+TQO-1: The ground subspace Pis a quantum code with a macroscopic distance1,
+TQO-2: Local ground subspaces are consistent with the global one
+1For our purposes it suffices that the distance grows as a positi ve power of the lattice size L
+4Condition TQO-1 is the traditional definition of TQO. It guar antees that a local operator cannot
+induce transitions between orthogonal ground states or dis tinguish a pair of orthogonal ground
+states from each other. Thus a local perturbation can lift th e ground state degeneracy only in
+then-th order of perturbation theory, where ncan be made arbitrarily large by increasing the
+lattice size, see [1]. Surprising, condition TQO-1 by itsel f is not sufficient for stability, see a
+simple counter-example in Section 2.4.
+Condition TQO-2 demands that a local ground subspace PBand the global ground subspace
+Pmust be consistent, namely, the projectors PBandPmust have the same reductions on any
+subsetA⊂Bwhich is ”sufficiently far” from the boundary of B. We need to impose TQO-2
+only for regions with trivial topology such a cube or a ball. T he consistency between the global
+and the local ground subspaces may be violated for regions wi th non-trivial topology. For
+example, if Bhas a hole, the local ground subspace PBmay include sectors with a non-trivial
+topological charge inside the hole as opposed to the global g round subspace. Condition TQO-2
+by itself is also not sufficient for stability, see a counter-e xample in Section 2.4.
+Let us emphasize that all our results apply also to the specia l case when H0has non-
+degenerate ground state. In this case TQO-1 is automaticall y satisfied since Pis a one-
+dimensional subspace and thus condition TQO-2 alone guaran tees the gap stability.
+We consider a perturbation Vthat can be written as a sum of bounded-norm interactions
+V=/summationdisplay
+r≥1/summationdisplay
+A∈S(r)Vr,A,
+whereS(r) is a set of cubes of linear size randVr,Ais an operator acting on sites of A. We
+assume that the magnitude of interactions decays exponenti ally for large r,
+max
+A∈S(r)/⌊ard⌊lVr,A/⌊ard⌊l ≤Je−µr,
+whereJ,µ> 0 are some constants independent of L. Our main result is the following.
+Theorem 1. SupposeH0obeys TQO-1,2. Then there exist constants J0,c1,c2>0depending
+only onµand the spatial dimension Dsuch that for all J≤J0the spectrum of H0+Vis
+contained (up to an overall energy shift) in the union of inte rvals/uniontext
+k≥0Ik, wherekruns over
+the spectrum of H0and
+Ik={λ∈R:k(1−c1J)−δ≤λ≤k(1 +c1J) +δ},
+and
+δ=poly(L) exp (−c2L3/8).
+In other words, the perturbation Vchanges positive eigenvalues of H0at most by a constant
+factor 1±c1J(neglecting the exponentially small correction δ) while the smallest eigenvalue
+k= 0 is transformed into a band I0of exponentially small width 2 δ, see Fig. 1. One can easily
+5check that for any fixed kthe bandIkis separated from all other bands Im,m/\e}atio\slash=k, by a gap
+at least 1/2 provided that J <Jk, where
+Jk=1
+c1(4k+ 2).
+Thus the bands originating from eigenvalues 0 ,1,...,k ofH0are separated from each other and
+from the rest of the spectrum by a gap at least 1 /2 provided that J <Jk.
+In the case when excitations of H0are anyons, one can infer all topological invariants such
+asS,R, andF-matrices by evaluating fusion and braiding diagrams with o nly a few particles
+(for example 4 particles suffice to compute all Fmatrices). Accordingly, any matrix element of,
+say,F-matrix, can be represented as an expectation value /a\}⌊ra⌋ketle{tψ0|Om...O2O1|ψ0/a\}⌊ra⌋ketri}htwhere|ψ0/a\}⌊ra⌋ketri}htis the
+ground state and Oiare operators creating pairs of excitations from the ground state, moving,
+fusing, and annihilating them. Our stability result for exc ited states with O(1) excitations allows
+us to construct quasi-adiabatic continuation of operators Oithus explicitly demonstrating that
+the perturbed Hamiltonian has the same S,R, andFmatrices as the ideal one, see Section 7.
+I0I1I2
+/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1Energy
+J012
+Figure 1: Energy bands Ikdescribing the spectrum of a perturbed Hamiltonian H0+V.
+1.2 Sketch of the stability proof
+Let us sketch the main steps of the proof of Theorem 1. We start from proving the theorem for
+a special class of perturbation Vsuch that all individual interactions Vr,Apreserve the ground
+subspaceP, that is, [P,Vr,A] = 0. We call such perturbations block-diagonal . In Section 3 we
+prove that block-diagonal perturbations are relatively bounded byH0, that is, /⌊ard⌊lVψ/⌊ard⌊l ≤b/⌊ard⌊lH0ψ/⌊ard⌊l
+for any state |ψ/a\}⌊ra⌋ketri}ht ∈ H and for some coefficient b=O(J). Here for simplicity we ignore some
+exponentially small corrections. A nice feature of relativ ely bounded perturbations is that the
+spectrum of a perturbed Hamiltonian H0+Vis contained in the union of intervals Ikwherek
+runs over the spectrum of H0andIk= (k(1−b),k(1 +b)), see Section 3.1. The proof of the
+relative boundness is rather elementary and uses certain de composition of the Hilbert space in
+terms of syndrome subspaces which is a standard tool in the theory of quantum error correc ting
+codes. In order to get a strong enough bound on the coefficient bwe use a novel technique of
+“coarse-graining” the syndrome subspaces, see Section 3.2 for details.
+6In the second part of the proof we reduce generic perturbatio nsVto block-diagonal pertur-
+bations. Specifically, we construct a unitary operator Usuch thatU(H0+V)U†≈H0+W,
+whereWis a block-diagonal perturbation. Since Udoes not change eigenvalues, we can use
+the techniques described above to analyze the spectrum of H0+V. The operator Uis con-
+structed using a discrete version of Hamiltonian flow equations developed by Glazek, Wilson,
+and Wegner [14]. Specifically, we define a hierarchy of Hamilt oniansH(n) =H0+V(n) +W(n)
+labeled by an integer level n≥0, such that W(n) is a block-diagonal perturbation while V(n)
+is a generic perturbation. We start at the level n= 0 with the perturbed H0+V, that is,
+V(0) =VandW(0) = 0. As we go to higher levels, the Hamiltonian H(n) becomes more close
+to a block-diagonal form. The transformation from H(n) toH(n+ 1) is described by a unitary
+operatorU(n) that block-diagonalizes H(n) up to errors of order V(n)2. These errors are dealt
+with at the next level of the hierarchy, see Section 4 for deta ils. We construct U(n) by solving a
+linearized block-diagonalization problem, see Section 5. The solution can be easily constructed
+in terms of the series while convergence of the series follow s from the fact that W(n) is relatively
+bounded by H0.
+We prove that the strength of V(n) decays doubly-exponentially as a function of n, while
+W(n) does not change essentially after the first few levels. We th en choose the desired unitary
+operatorUasU=U(nf)···U(1)U(0) where the highest level nf∼log (L) is chosen to make
+the norm of V(nf) exponentially small (as a function of L). The most technical part of the proof
+is to show that the unitary operators U(n) are locality preserving such that all Hamiltonians
+V(n) andW(n) remain sufficiently local. To this end we first prove that U(n) can be generated
+by a quasi-local Hamiltonian, see Section 5, and then employ the Lieb-Robinson bound, see
+Section 6.
+2 Hamiltonians describing TQO
+2.1 Frustration-free commuting Hamiltonians
+To simplify notations we shall restrict ourselves to the spa tial dimension D= 2. A generalization
+to an arbitrary Dis straightforward. Let Λ = ZL×ZLbe a two-dimensional square lattice of
+linear sizeLwith periodic boundary conditions. We assume that every sit eu∈Λ is occupied
+by a finite-dimensional quantum particle (qudit) such that t he Hilbert space describing Λ is a
+tensor product
+H=/circlemultiplydisplay
+u∈ΛHu,dimHu=O(1).
+LetS(r) be a set of all square blocks A⊆Λ of sizer×r, whereris a positive integer. Note that
+S(r) containsL2translations of some elementary square of size r×rfor allr <L ,S(L) = Λ,
+andS(r) =∅forr>L . We can always assume that the unperturbed Hamiltonian H0involves
+only 2×2 interactions (otherwise consider a coarse-grained latti ce):
+H0=/summationdisplay
+A∈S(2)GA. (2.1)
+7There will be three essential restrictions on the form of int eractionsGA. Firstly, we require that
+GAarepairwise commuting operators, that is,
+GAGB=GBGAfor allA,B∈ S(2).
+Thus all interactions GAcan be diagonalized in the same basis. Secondly, we require t hatH0
+is afrustration free Hamiltonian, that is, the ground state of H0minimizes energy of every
+individual term GA. Performing an overall energy shift we can always assume tha t allGAare
+positive-semidefinite operators,
+GA≥0.
+Then the condition of being frustration-free demands that g round states of H0are common zero
+eigenvectors of every term GA. Thus the ground subspace of H0is
+P={|ψ/a\}⌊ra⌋ketri}ht ∈ H :GA|ψ/a\}⌊ra⌋ketri}ht= 0 for all A∈ S(2)}. (2.2)
+Thirdly, we shall assume that every operator GAhas aconstant spectral gap , that is, the smallest
+positive eigenvalue of GAis bounded from below by a constant independent of the lattic e size
+L. We can always normalize the Hamiltonian H0such that the spectral gap of any GAis at
+least 1. This is equivalent to a condition
+G2
+A≥GA.
+LetPAbe the projector onto the zero subspace of GAandQA=I−PA. Note that all the
+projectorsPA,QAare pairwise commuting. For any square B∈ S(r),r≥2 define a projector
+onto the local ground subspace
+PB=/productdisplay
+A∈S(2)
+A⊆BPA (2.3)
+andQB=I−PB. Note that PBandQBhave support on B. We shall often use the same
+notation for a subspace and for the corresponding projector .
+2.2 Formal definition of TQO
+We shall need two extra property of H0and the ground subspace Pthat guarantee the gap
+stability and robustness of the ground state degeneracy. We shall assume that there exists
+a constant c > 0 such that the following conditions hold for some integer L∗≥cLfor all
+sufficiently large L:
+8TQO-1: LetA∈ S(r) be any square of size r≤L∗. LetOAbe any operator acting on A.
+Then
+POAP=cP
+for some complex number c.
+TQO-2: LetA∈ S(r) be any square of size r≤L∗and letB∈ S(r+ 2) be the square
+that contains Aand all nearest neighbors of A. Define reduced density matrices
+ρA= TrAc(P) andρ(B)
+A= TrAc(PB). Then the kernel of ρAcoincides with the
+kernel ofρ(B)
+A.
+Remark 1. Using the language of quantum error correcting codes one can define the minimum
+distance ofPas the smallest integer dsuch that erasure of any subset of dparticles can be
+corrected for any encoded state |ψ/a\}⌊ra⌋ketri}ht ∈P, see [15] for details. Note that TQO-1 holds for
+L∗=⌊√
+d⌋since the reduced state of any square A∈ S(L∗) does not depend on the encoded
+state. What is less trivial, TQO-1 holds also for L∗= Ω(d), see [15]. Thus L∗coincides with
+the distance of the code Pup to a constant coefficient (as far as condition TQO-1 is conce rned).
+Remark 2. Condition TQO-2 can be easily ‘proved’ if the excitations of H0are anyons
+(since the latter assumption lacks a rigorous formulation, the argument given below is not
+completely rigorous either). Indeed, in this case we can cho ose a complete basis of the excited
+subspaceQsuch that the basis vectors correspond to various configurat ions of anyons. For
+non-abelian theories one may have several basis vectors for a fixed configuration of anyons that
+describe different fusion channels, see [8]. Note that any st ate|ψ/a\}⌊ra⌋ketri}ht ∈PBis a superposition
+of configurations with no anyons inside B. SinceAis a topological trivial region, any such
+configuration can be prepared from the vacuum Pby some unitary operator UAcacting on
+complementary region Ac= Λ\A. Thus|ψ/a\}⌊ra⌋ketri}ht=UAc|ψ0/a\}⌊ra⌋ketri}htfor some ground state |ψ0/a\}⌊ra⌋ketri}ht ∈P. Since
+all ground states |ψ0/a\}⌊ra⌋ketri}hthave the same reduced matrix on A, it means that |ψ/a\}⌊ra⌋ketri}htandPhave the
+same reduced matrix on A. This implies TQO-2. The above arguments suggest that TQO-2
+holds for all 2D models of TQO that can be described by commuti ng frustration-free such as
+quantum double models [1] and Levin-Wen string-net models [ 4].
+Remark 3. As was already mentioned, the consistency between the globa l and the local
+ground subspaces may be violated for regions with non-trivi al topology. For example, if Ahas
+a hole, the local ground subspace PAmay include sectors with a non-trivial topological charge
+inside the hole as opposed to the global ground subspace.
+We shall need the following corollary of TQO-2.
+Corollary 2.1. LetA∈ S(r)be any square of size r≤L∗andOAbe any operator acting on
+Asuch thatOAP= 0. LetB∈ S(r+ 2)be the square that contains Aand all nearest neighbors
+ofA. ThenOAPB= 0.
+Proof. LetρA= TrAc(P). The assumption OAPO†
+A= 0 implies that OAρAO†
+A= 0, that is, OA
+annihilates any state in the range of ρA. From TQO-2, the range of Tr Ac(PB) coincides with
+the range of ρA, and thus Tr( OAPBO†
+A) = 0. It implies OAPB= 0.
+92.3 Verification of TQO conditions for stabilizer Hamiltonians
+Conditions TQO-1,2 can be easily checked for those models of TQO that can be described using
+the stabilizer formalism such as the toric code model [1] or t opological color codes [16]. For
+such models each site of the lattice Λ represents one or sever al qubits, while the ground state
+subspacePis a stabilizer code, i.e., the invariant subspace of some ab elian stabilizer group
+G ⊆ Pauli(Λ). Here Pauli(Λ) is a group generated by single-qubi t Pauli operators σx
+i,σy
+i,σz
+i.
+The stabilizer group must have a set of geometrically local g enerators, that is, G=/a\}⌊ra⌋ketle{tS1,...,S M/a\}⌊ra⌋ketri}ht
+where any generator Sa∈Pauli(Λ) acts non-trivially only on O(1) qubits located within distance
+O(1) from each other. Note that the generators need not to be in dependent. We choose the
+corresponding stabilizer Hamiltonian H0as
+H0=/summationdisplay
+a(I−Sa)/2
+such that states invariant under action of stabilizers have zero energy. The minimal distance of
+the code is the smallest integer dsuch that there exists a Pauli operator Othat commutes with
+all elements of Gbut does not belong to G. Such an operator Ocan be regarded as a logical
+Pauli operator acting on encoded states. It follows from res ults of [15] that condition TQO-1
+holds if we choose L∗= Ω(d).
+Assume that the set of qubits is coarse-grained into sites of the lattice Λ such that the
+support of any generator Sais contained in at least one 2 ×2 square. One can bring this
+Hamiltonian into the form Eq. (2.1) by distributing the gene rators over 2 ×2 squares in an
+arbitrary way. For any square B∈ S(r) one can define two subgroups of G: (i) a subgroup GB
+generated by generators Sawhose support is contained in B, and (ii) a subgroup G(B) that
+includes all stabilizers S∈ Gwhose support is contained in B. By definition, GB⊆ G(B), but
+in general GB/\e}atio\slash=G(B).
+Lemma 2.1. The stabilizer Hamiltonian H0satisfies condition TQO-2 iff for any square A∈
+S(r),r≤L∗, one has G(A)⊆ GB, whereB=b1(A).
+Thus TQO-2 demands that any element of the stabilizer group w hose support is contained
+in a square Acan be written as a product of generators whose support is con tained inAand a
+small neighborhood of A. We leave verification of this condition for the toric code mod el as an
+exercise for the reader.
+Proof. Indeed, the reduced density matrix ρAcomputed using the global ground subspace Pis
+proportional to the projector onto the codespace of the stab ilizer code G(A). The reduced density
+matrixρAcomputed using the local ground subspace PBis proportional onto the codespace of
+the stabilizer code GB(A), where GB(A) includes all elements of GBwhose support is contained
+inA. Thus TQO-2 holds iff G(A) =GB(A). This is equivalent to the condition of the lemma.
+102.4 Unstable version of the toric code model
+In this section we demonstrate that condition TQO-1 alone is not sufficient for stability. Let us
+start from the standard toric code model [1],
+Htc=−/summationdisplay
+pBp−/summationdisplay
+sAs,
+where qubits live on edges of a square 2D lattice, pandslabels plaquettes and sites of the
+lattice,Bpis a product of σzover the four boundary edges of p, andAsis a product of four σx
+over the four edges incident to s. We shall refer to BpandAsas plaquette and star operators.
+The ground subspace Pis defined by eigenvalue equations Bp= 1 for all pandAs= 1 for all
+s. It is well known that Pis a quantum code with the minimal distance d=L−1. HenceP
+obeys TQO-1,2 with L∗=L−1.
+Consider now a modified toric code model
+H′
+tc=−/summationdisplay
+(p,q)BpBq−/summationdisplay
+sAs−Bp∗
+where (p,q) labels pairs of adjacent plaquettes and p∗is some selected plaquette. We assume
+that the total number of plaquettes Npis even. Note that H′
+tcis a frustration-free commuting
+Hamiltonian. In addition, H′
+tcandHtchave the same ground subspace Pcorresponding to
+Bp= 1,As= 1 for all sandp. HenceH′
+tcobeys TQO-1. We claim that H′
+tcviolates TQO-2.
+Indeed, choose any square Alocated sufficiently far from the selected plaquette p∗. Then the
+local ground subspace PAhas equal contributions from sectors Bp= 1,As= 1 andBp=−1,
+As= 1 thus being inconsistent with the global ground state.
+Let us now argue that the spectral gap of H′
+tccloses in a presence of local perturbations with
+a strength of order 1 /Np. This instability has the same origin as the instability of t he classical
+2D Ising model under external magnetic field. Note that H′
+tchas spectral gap ∆ = 2 and the
+second smallest eigenvalue belongs to the sector As= 1,Bp=−1 for allsandp. Consider a
+perturbation describing an “external magnetic field”,
+V=h/summationdisplay
+pBp, h> 0.
+For sufficiently large h, say,h= 4/Np, the ground state of H′
+tc+Vmoves from the sector
+As= 1,Bp= 1 to the sector As= 1,Bp=−1. Hence the gap above the ground state closes for
+some intermediate value of h.
+Needless to say, condition TQO-2 alone is also not sufficient f or stability. The simplest
+counter-example is the 2D classical Ising model in which the gap is unstable under external
+magnetic field.
+113 Relatively bounded perturbations
+3.1 Definition and basic properties
+In this section we introduce necessary facts from the theory of relatively bounded perturbations.
+It mostly follows Chapter IV of [17] although our definitions and proofs are much simpler since
+we are interested only in finite-dimensional Hilbert spaces .
+LetH0andWbe any Hamiltonians acting on some Hilbert space H. We shall say that W
+isrelatively bounded byH0iff there exist 0 ≤b<1 such that
+/⌊ard⌊lWψ/⌊ard⌊l ≤b/⌊ard⌊lH0ψ/⌊ard⌊lfor all|ψ/a\}⌊ra⌋ketri}ht ∈ H. (3.1)
+The notion of a relatively bounded perturbation allows one t o define a “weak perturbation”
+and rigorously justify application of perturbative expans ions even when the norm of Wis much
+larger than the spectral gap of H0. We shall be mostly interested in the case when bis a constant
+independent of the lattice size L. Note that the condition Eq. (3.1) is equivalent to an operat or
+inequalityW2≤b2H2
+0.
+The following lemma asserts that a relatively bounded pertu rbation can change eigenvalues
+ofH0at most by a factor 1 ±b.
+Lemma 3.1. SupposeWis relatively bounded by H0. Then the spectrum of H0+Wis contained
+in the union of intervals [λ0(1−b),λ0(1 +b)]whereλ0runs over the spectrum of H0.
+Proof. Indeed, suppose ( H0+W)|ψ/a\}⌊ra⌋ketri}ht=λ|ψ/a\}⌊ra⌋ketri}ht, that is,
+(H0−λI)|ψ/a\}⌊ra⌋ketri}ht=−W|ψ/a\}⌊ra⌋ketri}ht. (3.2)
+The relative boundness then implies /⌊ard⌊l(H0−λI)ψ/⌊ard⌊l ≤b/⌊ard⌊lH0ψ/⌊ard⌊l, that is,
+/a\}⌊ra⌋ketle{tψ|(H0−λI)2|ψ/a\}⌊ra⌋ketri}ht ≤b2/a\}⌊ra⌋ketle{tψ|H2
+0|ψ/a\}⌊ra⌋ketri}ht. (3.3)
+LetH0=/summationtext
+λ0λ0Pλ0be the spectral decomposition of H0. Here the sum runs over the spectrum
+ofH0andPλ0is a projector onto the eigenspace with an eigenvalue λ0. Define a probability
+distribution p(λ0) =/a\}⌊ra⌋ketle{tψ|Pλ0|ψ/a\}⌊ra⌋ketri}ht. Substituting it into Eq. (3.3) one gets
+/summationdisplay
+λ0(λ0−λ)2p(λ0)≤/summationdisplay
+λ0b2λ2
+0p(λ0). (3.4)
+Therefore there exists at least one eigenvalue λ0such that
+(λ0−λ)2≤b2λ2
+0. (3.5)
+This is equivalent to λ0(1−b)≤λ≤λ0(1 +b).
+123.2 Stability of TQO under block-diagonal perturbations
+In this section we shall consider perturbations
+W=/summationdisplay
+A∈S(q)WA
+such that all local terms WAare block-diagonal,
+[WA,P] = 0 for all A∈ S(q).
+We shall assume that q≤L∗, so condition TQO-1 implies that the restriction of WAonto the
+P-subspace is a multiple of the identity. Since we are not inte rested in the overall shift in energy,
+we can assume that
+WAP= 0 for all A∈ S(q). (3.6)
+The interaction strength of Wwill be measured by a parameter
+w= max
+A∈S(q)/⌊ard⌊lWA/⌊ard⌊l. (3.7)
+Lemma 3.2. LetWbe a perturbation satisfying Eqs. (3.6,3.7). Then Wis relatively bounded
+byH0with a constant
+b=O(wq2).
+Proof. Let us start from introducing some notations. A syndromes:S(2)→ {0,1}is a function
+that assigns an eigenvalue sA∈ {0,1}to every projector QA,A∈ S(2). Given a syndrome s
+and a square A∈ S(2) we shall say that Ais adefect iffsA= 1. Thus one can consider sas a
+configuration of defects. For any syndrome sdefine a projector
+Rs=/productdisplay
+A∈S(2)[sAQA+ (1−sA)(I−QA)]
+projecting onto a subspace spanned by states with a syndrome s. Clearly the family of projectors
+Rsdefines an orthogonal decomposition of the Hilbert space, th at is,/summationtext
+sRs=I.
+Let us fix some partition of the lattice into contiguous q×qsquaresB1,...,B M∈ S(q) (if
+Lis not a multiple of q, the squares Bimay have size q±O(1)). We shall refer to a set of 2 ×2
+squares contained in a particular square Bjas abox. We shall need the following properties:
+1. Every square A∈ S(2) is contained in exactly one box Bi,
+2. Each box Bioverlaps with O(q2) squaresC∈ S(q),
+3. Each square C∈ S(q) overlaps with O(1) boxesBi.
+Given a syndrome sand a boxBiwe shall say that Biisoccupied ifBicontains at least one
+defect, that is, there is a 2 ×2 squareA⊂Bsuch thatsA= 1. Otherwise we shall say that the
+boxBiisempty .
+13Given a syndrome sletb(s)⊆[M] be the subset of occupied boxes. For any subset of boxes
+Y ⊆ [M] define a projector
+RY=/summationdisplay
+s:b(s)=YRs.
+It projects onto the subspace in which all boxes in Yare occupied and the remaining boxes
+are empty. Clearly, the family of projectors RYdefines an orthogonal decomposition, that is,/summationtext
+Y⊆[M]RY=I. We claim that any operator WAacting on a square A∈ S(q) and satisfying
+Eq. (3.6) has only a few off-diagonal blocks with respect to th is decomposition. Specifically,
+Corollary 2.1 implies that
+RYWARZ/\e}atio\slash= 0 (3.8)
+only ifAhas distance O(1) from some occupied box in YandAhas distance O(1) from some
+occupied box in Z,andthe configurations Y,Zdiffer only at those boxes that overlap with
+A. Clearly, for any fixed Y ⊆ [M] such that Yhaskoccupied boxes the number of pairs
+(A∈ S(q),Z ⊆ [M]) that could satisfy Eq. (3.8) is at most O(kq2). Thus for any state |ψ/a\}⌊ra⌋ketri}htwe
+get
+/a\}⌊ra⌋ketle{tψ|W2|ψ/a\}⌊ra⌋ketri}ht=/summationdisplay
+Y,Z,V⊆[M]/a\}⌊ra⌋ketle{tψ|RYWRZWRV|ψ/a\}⌊ra⌋ketri}ht
+≤/summationdisplay
+Y,Z,V⊆[M]/⌊ard⌊lRYWRZ/⌊ard⌊l·/⌊ard⌊lRZWRV/⌊ard⌊l·/⌊ard⌊lRYψ/⌊ard⌊l·/⌊ard⌊lRVψ/⌊ard⌊l
+≤/summationdisplay
+Y,Z,V⊆[M]/⌊ard⌊lRYWRZ/⌊ard⌊l·/⌊ard⌊lRZWRV/⌊ard⌊l·1
+2(/a\}⌊ra⌋ketle{tψ|RY|ψ/a\}⌊ra⌋ketri}ht+/a\}⌊ra⌋ketle{tψ|RV|ψ/a\}⌊ra⌋ketri}ht)
+=/summationdisplay
+Y,Z,V⊆[M]/⌊ard⌊lRYWRZ/⌊ard⌊l·/⌊ard⌊lRZWRV/⌊ard⌊l·/a\}⌊ra⌋ketle{tψ|RY|ψ/a\}⌊ra⌋ketri}ht
+≤/summationdisplay
+k≥0/summationdisplay
+Y:|Y|=kO(k2q4w2)/a\}⌊ra⌋ketle{tψ|RY|ψ/a\}⌊ra⌋ketri}ht=O(w2q4)/a\}⌊ra⌋ketle{tψ|G|ψ/a\}⌊ra⌋ketri}ht, (3.9)
+where
+G=/summationdisplay
+k≥0/summationdisplay
+Y:|Y|=kk2RY. (3.10)
+The inequality Eq. (3.9) follows from the fact that YandZdiffer at at most O(1) boxes and
+an obvious bound k(k+O(1)) =O(k2). Finally, note that G≤H2
+0since any configuration of
+defects with koccupied boxes must have at least kdefects and since creating a defect costs at
+least a unit of energy. We arrive at
+/a\}⌊ra⌋ketle{tψ|W2|ψ/a\}⌊ra⌋ketri}ht ≤b2/a\}⌊ra⌋ketle{tψ|H2
+0|ψ/a\}⌊ra⌋ketri}ht, b =O(wq2). (3.11)
+It completes the proof.
+We shall also need a local version of Lemma 3.2. For any region C⊆Λ define a local version
+of the Hamiltonian H0,
+H0(C) =/summationdisplay
+A∈S(2)
+A⊆CGA. (3.12)
+14LetPCandQCbe the local versions of the projectors PandQdefined in Eq. (2.3). The
+following is a straightforward corollary of Lemma 3.2.
+Corollary 3.1. LetC⊆Λbe any square of size smaller than L∗. LetW=/summationtext
+A∈S(q)WAbe a
+perturbation satisfying Eqs. (3.6,3.7). Suppose also that WA= 0unlessCcontains both Aand
+the nearest neighbors of A. ThenWis relatively bounded by H0(C)with a constant b=O(wq2).
+Proof. Combining Eq. (3.6) and TQO-2 we conclude that WAPC= 0 for all A. Thus we can
+apply all steps in the proof of Lemma 3.2 to the square Cconsidered as the entire lattice Λ.
+We shall need another technical lemma that provides a bound o n the norm of a commutator
+involving a block-diagonal Hamiltonian. Let us start from t he simplest scenario. Let Wbe any
+operator such that Wis relatively bounded by H0with a constant 0 ≤b <1. Then for any
+operatorSwe have
+/⌊ard⌊lQ[S,W ]P/⌊ard⌊l=/⌊ard⌊lQWSP/⌊ard⌊l=/⌊ard⌊lQWH−1
+0QH0SP/⌊ard⌊l
+=/⌊ard⌊lQWH−1
+0Q[H0,S]P/⌊ard⌊l ≤ /⌊ard⌊lWH−1
+0Q/⌊ard⌊l·/⌊ard⌊lQ[H0,S]P/⌊ard⌊l.
+Here the first equality follows from WP = 0 and the third equality uses H0P= 0. Let |ψ/a\}⌊ra⌋ketri}ht ∈Qbe
+a normalized state such that /⌊ard⌊lWH−1
+0Q/⌊ard⌊l=/⌊ard⌊lWH−1
+0ψ/⌊ard⌊l. Using the relative boundness assumption
+we get
+/⌊ard⌊lWH−1
+0Q/⌊ard⌊l=/⌊ard⌊lWH−1
+0ψ/⌊ard⌊l ≤b/⌊ard⌊lH0H−1
+0ψ/⌊ard⌊l=b.
+To conclude, we have proved that
+/⌊ard⌊lQ[S,W ]P/⌊ard⌊l ≤b/⌊ard⌊lQ[S,H0]P/⌊ard⌊l.
+Applying the same arguments as above with PandQreplaced by their local versions PCand
+QC, see Eq. (2.3), we arrive at the following lemma.
+Lemma 3.3. LetC⊆Λbe any region. Let Wbe any Hamiltonian such that Wis relatively
+bounded by H0(C)with a constant 0≤b<1. Then for any operator Sone has
+/⌊ard⌊lQC[S,W ]PC/⌊ard⌊l ≤b/⌊ard⌊lQC[S,H0(C)]PC/⌊ard⌊l. (3.13)
+Combining this lemma and Corollary 3.1 we get a simple upper b ound on /⌊ard⌊lQC[S,W ]PC/⌊ard⌊l.
+Corollary 3.2. LetC⊆Λbe any square of size smaller than L∗. LetW=/summationtext
+A∈S(q)WAbe a
+perturbation satisfying Eqs. (3.6,3.7). Suppose also that WA= 0unlessCcontains both Aand
+the nearest neighbors of A. Then for any operator Sone has
+/⌊ard⌊lQC[S,W ]PC/⌊ard⌊l ≤O(wq2)/⌊ard⌊lQC[S,H0(C)]PC/⌊ard⌊l. (3.14)
+154 Hamiltonian flow equations
+4.1 Outline of the method
+We are interested in the low-energy spectrum of a perturbed H amiltonian H0+V, whereVis
+a local perturbation specified by a list of local interaction s,
+V=/summationdisplay
+r≥1/summationdisplay
+A∈S(r)Vr,A. (4.1)
+HereVr,Ais some interaction supported on a square A∈ S(r). Our strategy will be to reduce the
+case of a generic perturbation to the special case of a block- diagonal perturbation which we can
+analyze using techniques of Section 3. To this end we shall de fine a hierarchy of Hamiltonians
+unitarily equivalent to H,
+H(n) =H0+n/summationdisplay
+k=1W(k) +V(n) +E(n) +λ(n)I, n = 0,1,2,..., (4.2)
+such thatH(0) =H0+VandH(n+1) can be obtained from H(n) by a unitary transformation,
+H(n+ 1) =U(n)H(n)U(n)†, U (n)U(n)†=I. (4.3)
+Accordingly, the spectrum of H(n) is the same for all n≥0. The purpose of the transformation
+U(n) is to make the Hamiltonian more close to the block-diagonal form. We shall refer to H(n)
+as alevel-nHamiltonian .
+Let us describe the purpose of various terms in Eq. (4.2). The Hamiltonian W(k) represents
+a block-diagonal contribution to the total Hamiltonian H(n) that has been created at the level
+k. The Hamiltonian V(n) represents the part of the total Hamiltonian H(n) that has to be
+block-diagonalized at the level n+ 1. The Hamiltonians V(n) andW(n) will be represented by
+a sum of local interactions supported in squares of size r≤L∗, and such that all interactions
+involved in W(n) are individually block-diagonal,
+V(n) =/summationdisplay
+1≤r≤L∗/summationdisplay
+A∈S(r)Vr,A(n),
+and
+W(n) =/summationdisplay
+1≤r≤L∗/summationdisplay
+A∈S(r)Wr,A(n),whereQWr,A(n)P= 0.
+All contributions from squares of size r>L∗will be collected into the third Hamiltonian E(n)
+which can be regarded as an error Hamiltonian. The norm of E(n) will be exponentially small in
+Lfor alln. The norms of W(n) andV(n) will decay roughly as doubly-exponential functions of
+n. Thus at level n∼logLthe total Hamiltonian H(n) will be block-diagonal up to corrections
+of order exp ( −poly(L)) resulting from V(n) andE(n). Finally,λ(n) is an overall energy shift
+that we shall often ignore.
+16We start at the level n= 0 with initial conditions
+Wr,A(0) = 0, Vr,A(0) =Vr,A forr≤L∗, E (0) =/summationdisplay
+r>L∗/summationdisplay
+A∈S(r)Vr,A.
+Accordingly, H(0) =H0+Vis the Hamiltonian we are interested in. Suppose we have alre ady
+defined the Hamiltonians W(0),...,W (n),V≡V(n),E≡E(n) for some level n. Let
+W=n/summationdisplay
+k=1W(k)
+be the overall block-diagonal part of H(n). Let
+H≡H(n) =H0+W+V+E.
+We shall define the operator U(n) in Eq. (4.3) as U(n) = exp (S), whereSis the solution of a
+linearized block-diagonalization problem
+Q([S,H0+W] +V)P= 0, S†=−S. (4.4)
+The meaning of this equation can be easily understood if one t reatsVas a perturbation and
+H0+Was an unperturbed Hamiltonian. Expanding the transformed H amiltonian eSHe−Sin
+powers ofSwe get
+eSHe−S=H0+W+ ([S,H0+W] +V) +O(S2) +O(SV) +O(E).
+Thus Eq. (4.4) says that the transformed Hamiltonian must be block-diagonal up to terms
+O(V2) andO(E). The solution of Eq. (4.4) is constructed in Section 5, see L emma 5.1. We
+start from defining raw versions of W(n+ 1) andV(n+ 1) which we shall denote ˜Wand ˜V
+respectively, namely
+˜W= [S,H0+W] +V, (4.5)
+and
+˜V=eS(H0+W+V)e−S−(H0+W+V+ [S,H0+W]). (4.6)
+A simple algebra shows that
+eSHe−S=H0+W+˜W+˜V+eSEe−S.
+Note also that ˜Wis block-diagonal due to Eq. (4.4). We shall construct a deco mposition of
+˜Winto a sum of local interactions that are individually block -diagonal using techniques of
+Section 5, see Corollary 5.1 of Lemma 5.1. It will yield
+˜W=/summationdisplay
+r≥1/summationdisplay
+A∈S(r)˜Wr,A,whereQ˜Wr,AP= 0.
+We shall construct a decomposition of ˜Vinto a sum of local interactions using Lemma 4.1 and
+Lemma 6.1 arriving at
+˜V=/summationdisplay
+r≥1/summationdisplay
+A∈S(r)˜Vr,A.
+17Next we use ˜Wand ˜Vto defineW(n+ 1) andV(n+ 1) by taking out all contributions from
+squares of size r>L∗and adding these contributions to the error Hamiltonian, th at is,
+W(n+ 1) =/summationdisplay
+1≤r≤L∗/summationdisplay
+A∈S(r)˜Wr,A,
+V(n+ 1) =/summationdisplay
+1≤r≤L∗/summationdisplay
+A∈S(r)˜Vr,A,
+and
+E(n+ 1) =eSEe−S+/summationdisplay
+r>L∗/summationdisplay
+A∈S(r)˜Wr,A+˜Vr,A.
+For the detailed analysis of these flow equations see Section 4.3
+4.2 Local decompositions of Hamiltonians
+In order to analyze convergence of the flow equations we shall need to set up some notations
+and terminology. Recall that S(r) is a set of all r×rsquares.
+Definition 4.1. LetVbe any operator acting on H. A local decomposition of Vis a list of
+operators {Vr,A}r,Awherer≥1andA∈ S(r)such thatVr,Ahas support on a square Aand
+V=/summationdisplay
+r≥1/summationdisplay
+A∈S(r)Vr,A (4.7)
+Note that a local decomposition of an operator is not unique s ince the squares involved in
+the decomposition overlap with each other. Nevertheless, w e shall often identify an operator
+and its local decomposition unless it may lead to confusions .
+Definition 4.2. A local decomposition of an operator Vis(J,µ,α )-decaying iff
+max
+r≥1max
+A∈S(r)/⌊ard⌊lVr,A/⌊ard⌊lrαeµr≤J. (4.8)
+Here we mean that J,µ,α are some constants independent of L. We shall often use a term
+(J,µ,α )-decaying operator meaning that this operator has a local d ecomposition which is
+(J,µ,α )-decaying. To simplify notations we shall often use an abbr eviation (J,µ)-decay for
+(J,µ,0)-decay.
+In the rest of this section we derive several auxiliary techn ical results that can be skipped
+at the first reading.
+We shall often need to construct a local decomposition for a c ommutator [ S,V] given the
+local decompositions of SandV.
+Lemma 4.1. SupposeSis(K,µ,α )-decaying and Vis(J,µ,β )-decaying for some α,β≥4.
+Then [S,V]has a local decomposition which is (cKJ,µ )-decaying for some constant c.
+18Proof. Consider local decompositions of SandV,
+S=/summationdisplay
+p≥1/summationdisplay
+A∈S(p)Sp,A,/⌊ard⌊lSp,A/⌊ard⌊l ≤Kp−αe−µp, (4.9)
+V=/summationdisplay
+q≥1/summationdisplay
+B∈S(q)Vq,B,/⌊ard⌊lVq,B/⌊ard⌊l ≤Jq−βe−µq. (4.10)
+IfA∈ S(p) andB∈ S(q) is a pair of non-overlapping squares, A∩B/\e}atio\slash=∅, then clearly A∪B
+can be covered by some square C∈ S(p+q). Thus we can choose a local decomposition of
+[S,V] as
+[S,V] =/summationdisplay
+r≥2/summationdisplay
+C∈S(r)Dr,C, (4.11)
+where
+Dr,C=/summationdisplay
+p+q=r/summationdisplay
+A∈S(p)
+A⊆C/summationdisplay
+B∈S(q)
+B⊆C[Sp,A,Vq,B]χ(A,B,C ) (4.12)
+whereχ(A,B,C ) = 0,1 is some function that ‘distributes’ the commutators [ Sp,A,Vq,B] over
+different terms Dr,C. A specific form of this function is not important for us. For fi xedp,qsuch
+thatp+q=rand a fixed C∈ S(r) we can bound the number of squares A,B⊆Cas
+#{A∈ S(p) :A⊆C}= (r−p)2=q2
+and
+#{B∈ S(q) :B⊆C}= (r−q)2=p2.
+It yields
+/⌊ard⌊lDr,C/⌊ard⌊l ≤2KJe−µr/summationdisplay
+p+q=rp2−αq2−β≤2KJe−µr/summationdisplay
+p,q≥1p−2q−2=cKJe−µr(4.13)
+for some constant cprovided that α,β≥4.
+We shall also need the following simple lemma that will allow us to amplify the degree α
+by any constant by “borrowing” some decay from the exponenti al function. It makes the decay
+rateµa bit smaller and the amplitude Ja bit larger.
+Lemma 4.2 (Degree Reset). SupposeVis(J,µ,β )-decaying. Let 0< ǫ < 1andα > 0be
+any constants. Then Vis also (J′,µ′,α+β)-decaying where
+J′=cJ1−ǫandµ′=µ−Jǫ
+α. (4.14)
+Herecis a constant depending on ǫandα.
+Proof. Indeed, let µ′=µ−δwhereδwill be chosen later. Then
+/⌊ard⌊lVr,A/⌊ard⌊l ≤Jr−βe−µr≤Jr−α−βe−µ′rmax
+q≥0qαe−δq≤cδ−αJr−α−βe−µ′r, (4.15)
+wherec= max x≥0xαe−x=O(1) is a constant. Choosing δ=Jǫ/αwe achieve the desired
+scaling.
+19Finally, we shall use the following trivial observation.
+Lemma 4.3. SupposeVis(J,µ,α )-decaying. Then Valso has a local decomposition which is
+(cJe2µ,µ,α )-decaying with an extra property that Vr,Aacts trivially on all sites that are adjacent
+to the boundary of A. Herec=O(1)is some constant.
+Proof. Indeed, replace each square A∈ S(r) in the decomposition of Vby a square A′∈
+S(r+ 2) that contains Aand all nearest neighbors of A. LetV′
+r+2,A′=Vr,A. ThenV=/summationtext
+r≥1/summationtext
+A∈S(r)V′
+r,Ais the desired decomposition.
+4.3 Proof of the main theorem
+In this section we prove Theorem 1.
+Proof. We shall derive simplified flow equations for a triple of param etersJ(n),Jd(n), andµ(n)
+such thatV(n) is (J(n),µ(n),β)-decaying and W(n) is (Jd(n),µ(n),α)-decaying for all n≥0.
+Hereα,βare sufficiently large constants that will be chosen later. Ou r manipulations with
+local decompositions will typically decrease the constant sαandβ, so after each step of the flow
+equations we shall need to reset these constants back to thei r original values using Lemma 4.2.
+SinceW(0) = 0 we can choose initial conditions
+J(0) =J, Jd(0) = 0,andµ(0) =µ. (4.16)
+Let us prove that for any constant 0 <ǫ< 1 there exist constants c1,c2,c3>0 such that for all
+k≥0 one has
+J(k+ 1)≤c1J(k)2(1−ǫ), (4.17)
+Jd(k+ 1)≤c2J(k)1−ǫ, (4.18)
+µ(k+ 1) =1
+2µ(k)−c3J(k+ 1)ǫ
+10, (4.19)
+/⌊ard⌊lE(k+ 1)/⌊ard⌊l ≤ /⌊ard⌊lE(k)/⌊ard⌊l+O(L3)J(k)e−c3Lµ(k)(4.20)
+provided that J(0) is below some constant threshold value. Note that althou ghǫcan be chosen
+arbitrarily close to 0, Eq. (4.19) does not permit one to choo seǫ= 0 since otherwise the decay
+rateµ(n) becomes negative after O(1) iterations.
+Supposed we have already proved Eqs. (4.17-4.20) for k= 0,1,...,n . Let us denote V≡
+V(n),E≡E(n),
+W≡n/summationdisplay
+k=0W(k)
+and
+Jd≡n/summationdisplay
+k=0Jd(k).
+20Sinceµ(k) is monotone decreasing for k= 0,...,n , we can safely assume that Wis (Jd,µ(n),α)-
+decaying. Also, since J(k) decreases doubly exponentially for k= 0,...,n we can assume (for
+k>0) that
+Jd=O(Jd(1)) =O(J1−ǫ). (4.21)
+LetSbe the solution of the linearized block-diagonalization pr oblem Eq. (4.4) constructed in
+Lemma 5.1. By construction Sis (K,µ(n),β)-decaying, where
+K=c1J(n)
+1−c2Jd=O(J(n)). (4.22)
+Here we assumed that Jis sufficiently small, so that c2Jd≤1/2. Note that the assumptions of
+Lemma 5.1 require β≥2 andα≥β+ 4.
+Recall that ˜Wand ˜Vare raw versions of W(n+ 1) andV(n+ 1) defined in Eq. (4.5) and
+Eq. (4.6). From Corollary 5.1, Section 5.3, we infer that the operator ˜Whas a local decompo-
+sition with block-diagonal terms which is ( cJ(n),µ(n))-decaying. Note that the assumptions of
+Corollary 5.1 require β≥2 andα≥β+ 4. The operator ˜Vcan be rewritten as
+˜V= [S,V] +ω(H), (4.23)
+whereH≡H0+W+Vandω(H)≡eSHe−S−H−[S,H]. We can assume that His (c,µ(n),β)-
+decaying where c=O(1) and we assumed that β≤α. Applying Lemma 4.1 from Section 4.2
+we get a local decomposition for [ S,V] which is ( c1J(n)2,µ(n))-decaying for some constant c1
+provided that β≥4. Applying Lemma 6.1 from Section 6 we get a local decomposit ion forω(H)
+which is (c2J(n)2,µ(n)/2,−1)-decaying for some constant c2provided that β≥6. Summarizing,
+˜Vis (cJ(n)2,µ(n)/2,−1)-decaying where cis a constant. It will be convenient to keep the decay
+rates of ˜Wand ˜Vthe same. Thus we shall assume that ˜Wis (cJ(n),µ(n)/2)-decaying (which
+is a weaker version of what we proved above). One can easily ch eck that a choice
+α= 10 and β= 6 (4.24)
+satisfies conditions of all lemmas used above.
+Recall that W(n+ 1) andV(n+ 1) are defined by taking the local decompositions of ˜Wand
+˜Vand removing all terms associated with squares of size large r thanL∗. Therefore W(n+ 1)
+andV(n+ 1) have the same decay parameters as ˜Wand ˜V, that is, we get
+J(n+ 1)≤c1J(n)2, Jd(n+ 1)≤c2J(n), µ(n+ 1) =1
+2µ(n) (4.25)
+for some constants c1,c2. Note that we have not reset α,βto their original values yet. The total
+number of squares of size r>L∗is at mostL3. Thus the contribution to the error Hamiltonian
+E(n+ 1) can be estimated as
+/⌊ard⌊lE(n+ 1)/⌊ard⌊l ≤ /⌊ard⌊lE(n)/⌊ard⌊l+O(L3)J(n)e−µ(n)L∗/2=/⌊ard⌊lE(n)/⌊ard⌊l+O(L3)J(n)e−c3µ(n)L(4.26)
+for some constant c3. Resetting the constants α,β using Lemma 4.2 we arrive at the desired
+flow equations Eqs. (4.17-4.20).
+21Solving Eq. (4.17) we get
+J(n)≤J/parenleftbiggJ
+J0/parenrightbiggθn
+, θ = 2(1−ǫ) (4.27)
+for some constant J0>0. Note that we are free to choose the constant ǫas small as possible.
+To simplify the formulas let us set θ= 2. Also note that for small enough J0we can assume
+thatµ(n) decays exponentially with exponent arbitrarily close to 1 /2. To simplify the formulas
+let us assume that
+J(n)≤J/parenleftbiggJ
+J0/parenrightbigg2n
+andµ(n) =µ2−n. (4.28)
+Simple algebra shows that choosing the number of steps n=clogLfor some constant cone can
+achieve the bounds
+/⌊ard⌊lV(n)/⌊ard⌊l,/⌊ard⌊lE(n)/⌊ard⌊l ≤poly(L) exp (−c√
+L). (4.29)
+Here the exponent −c√
+Lis determined by a tradeoff between the doubly exponential de cay
+ofJ(n) and the exponential decay of µ(n). Neglecting these exponentially small errors we can
+assume for simplicity that the Hamiltonian H(n) contains only block-diagonal contributions,
+that is,
+H(n) =H0+W, W =n/summationdisplay
+k=0W(k). (4.30)
+Here the local decomposition of Wis (Jd,0,α)-decaying with Jd=O(J1−ǫ),α= 10, and all
+terms in this decomposition are individually block-diagon al. In addition, by definition of W(k),
+this decomposition contains only squares of size r≤L∗. By performing an overall energy shift
+and using TQO-1 we can guarantee that every local term in the d ecomposition of W(k) has
+zero restriction on the Psubspace (note that it increases the strength Jdat most by a factor of
+two). It allows us to apply the machinery of Section 3. In part icular, Lemma 3.2 says that W
+is relatively bounded by H0with a constant
+b≤O(Jd)/summationdisplay
+r≥1r2−α=O(Jd). (4.31)
+Assuming that b<1, Lemma 3.1 implies that the spectrum of H0+Wis contained in the union
+of intervals Ik= (k(1−b),k(1 +b)), wherek= 0,1,2,.... The effect of V(n) andE(n) can
+now be taken into account using the standard perturbation th eory by considering H0+Was
+an unperturbed Hamiltonian and using Eq. (4.29).
+Finally, notice that since the dominant contribution to Wcomes from the first level k= 1,
+we do not really need to perform the degree reset to estimate b(since the degree reset only
+changes our description of an operator but does not change th e operator itself). Hence we can
+setJd=O(J). The theorem is proved.
+225 Linearized block-diagonalization problem
+5.1 Statement of the problem
+Consider a pair of perturbations
+V=/summationdisplay
+1≤r≤L∗/summationdisplay
+A∈S(r)Vr,A, (5.1)
+W=/summationdisplay
+1≤r≤L∗/summationdisplay
+A∈S(r)Wr,A, (5.2)
+such that all terms Wr,Aare block-diagonal,
+QWr,AP= 0 for all r,A. (5.3)
+A linearized block-diagonalization problem can be divided into two parts. The first part is
+to find an anti-hermitian operator Ssuch that
+Q([S,H0+W] +V)P= 0, S†=−S (5.4)
+and construct a local decomposition of S. The second part is to construct a local decomposition
+for the transformed Hamiltonian
+˜W= [S,H0+W] +V=/summationdisplay
+r≥1/summationdisplay
+A∈S(r)˜Wr,A (5.5)
+such that every term in the local decomposition is block-dia gonal, that is, Q˜Wr,AP= 0 for all
+r,A. We solve the two parts of the problem in Lemma 5.1 and its Coro llary 5.1 respectively.
+Throughout this section we assume that H0is a Hamiltonian defined in Eq. (2.1) that obeys
+conditions TQO-1 and TQO-2.
+5.2 Finding the transformation S
+In this section we prove the following
+Lemma 5.1. LetVandWbe perturbations defined in Eqs. (5.1,5.2,5.3). Suppose tha tVis
+(J,µ,β )-decaying and Wis(Jd,µ,α )-decaying such that β≥2andα≥β+ 4. Then there exist
+constantsc1,c2>0such that Eq. (5.4) has a solution Swhich is (K,µ,β )-decaying with
+K=c1J
+1−c2Jd, (5.6)
+Proof. Let us start from several simplifying assumptions. Without loss of generality we can
+assume that
+Wr,AP= 0 for all r,A. (5.7)
+23Indeed, condition TQO-1 guarantees that Wr,APis a multiple of P. Shifting Wr,Aby the
+corresponding multiple of the identity we can satisfy Eq. (5 .7). On the other hand, one can
+easily check that Eq. (5.4) in invariant under such a shift. N ote also that the shift can increase
+the norm /⌊ard⌊lWr,A/⌊ard⌊lat most by a factor of two. Indeed, if Wr,AP=cPthen|c| ≤ /⌊ard⌊lWr,A/⌊ard⌊land thus
+/⌊ard⌊lWr,A−cI/⌊ard⌊l ≤ /⌊ard⌊lWr,A/⌊ard⌊l+|c| ≤2/⌊ard⌊lWr,A/⌊ard⌊l. Thus we can assume that Wsatisfies Eq. (5.7) provided
+that we change Jdto 2Jdin the final answer. By the same reasons, we can assume that
+PVr,AP= 0 for all r,A (5.8)
+provided that we change Jto 2Jin the final answer. In addition, we can assume that Vr,A
+commutes with GB,B∈ S(2), whenever Bis not contained in A,
+[Vr,A,GB] = 0 for all B∈ S(2) such that B∩Ac/\e}atio\slash=∅. (5.9)
+Indeed, by adding an idle layer of sites to each square in the d ecomposition of Vas explained
+in Lemma 4.3 we guarantee Eq. (5.9). The price we pay for this s implification is that Jhas to
+be changed to cJe2µ=O(J) in the final answer, see Lemma 4.3. Note also that now the loca l
+decomposition of Vin Eq. (5.1) starts from squares of size r= 3,
+V=/summationdisplay
+3≤r≤L∗/summationdisplay
+A∈S(r)Vr,A. (5.10)
+We shall construct a solution Sas a seriesS=/summationtext∞
+i=1S(i), whereS(i)is anti-hermitian for
+alliand
+Q([S(1),H0] +V)P= 0, (5.11)
+Q([S(i+1),H0] + [S(i),W])P= 0,fori≥1. (5.12)
+For any region A⊆Λ define a restricted Hamiltonian
+H0(A) =/summationdisplay
+B∈S(2)
+B⊆AGB. (5.13)
+Define also a super-operator EAthat takes as argument an arbitrary operator Oand returns an
+operator
+EA(O) =QAH0(A)−1OPA−PAOH0(A)−1QA. (5.14)
+Note that the PAis the zero-subspace of H0(A), so thatQAH0(A)−1is well-defined.
+Proposition 5.1. LetOAbe any operator acting on Asuch thatPOAP= 0. Suppose OA
+commutes with GB,B∈ S(2), whenever Bis not contained in A. Then
+Q([EA(OA),H0] +OA)P= 0. (5.15)
+IfOAis hermitian then EA(OA)is anti-hermitian.
+24Proof. Indeed, since all terms in H0which are not contained in Acommute with EA(OA) while
+H0(A)PA= 0 we have
+[EA(OA),H0] =−QAOAPA−PAOAQA.
+It yields
+Q[EA(OA),H0]P=−QQAOAP=−Q(QA+PA)OAP=−QOAP,
+where the first equality follows from PAP=P,QAP= 0, and the second equality uses identity
+PAOAP=POAP= 0. Thus we have proved Eq. (5.15). The last statement of the p roposition
+is obvious.
+Using the assumptions Eqs. (5.8,5.9) and the proposition we can choose S(1)in Eq. (5.11)
+as
+S(1)=/summationdisplay
+r≥3/summationdisplay
+A∈S(r)S(1)
+r,A, S(1)
+r,A=EA(Vr,A). (5.16)
+Taking into account that
+/⌊ard⌊lEA(OA)/⌊ard⌊l ≤ /⌊ard⌊lOA/⌊ard⌊lfor anyOA (5.17)
+we conclude that Eq. (5.16) is a local decomposition of S(1)which is (K1,µ,β )-decaying where
+K1=J. (5.18)
+This decomposition has an extra property that S(1)
+r,Ais block-off-diagonal with respect to PA,
+QA, andS(1)
+r,Acommutes with GB,B∈ S(2), whenever Bis not contained in A.
+Let us now solve Eq. (5.12). We shall assume as our induction h ypothesis that S(i)possesses
+a local decomposition
+S(i)=/summationdisplay
+p≥3/summationdisplay
+A∈S(p)S(i)
+p,A(5.19)
+such that
+I1S(i)
+p,Ais block-off-diagonal with respect to PA,QA
+I2S(i)
+p,Acommutes with GB,B∈ S(2), whenever Bis not contained in A
+I3/⌊ard⌊l[H0(A),S(i)
+p,A]/⌊ard⌊l ≤Kip−βe−µp
+Here the coefficient Kiwill be determined inductively in terms of Ki−1. Note that combining
+(I1), (I3) with the fact that QAH0(A)≥Ione gets
+/⌊ard⌊lS(i)
+p,A/⌊ard⌊l=/⌊ard⌊lQAS(i)
+p,APA/⌊ard⌊l ≤ /⌊ard⌊lQAH0(A)S(i)
+p,APA/⌊ard⌊l=/⌊ard⌊l[H0(A),S(i)
+p,A]/⌊ard⌊l ≤Kip−βe−µp,
+that is,S(i)is (Ki,µ,β )-decaying.
+The base of induction is i= 1 which we have already proved. Our first step will be choosin g
+a local decomposition for the commutator [ S(i),W] in Eq. (5.12). We shall need the following
+geometrical fact.
+25Proposition 5.2. LetA∈ S(p),p≥2, andB∈ S(q),q < L, be any squares such that
+A∩B/\e}atio\slash=∅. Then there exists a square C∈ S(r),r= min (p+q,L), such that A∪B⊆Cand
+Ccontains all nearest neighbors of B.
+Proof. Ifr=Lthe statement is obvious, so assume r=p+q < L . Define a metric on the
+lattice using the l∞-norm, that is, if u= (ux,uy) andv= (vx,vy) is a pair of sites, then
+D(u,v) = max{|ux−vx|,|uy−vy|}.
+For any region M⊆Λ letD(M) be the diameter of M, i.e., the largest distance between
+a pair of sites u,v∈M. ClearlyD(A) =p−1 andD(B) =q−1. SinceA∩B/\e}atio\slash=∅we
+haveD(A∪B)≤D(A) +D(B) =p+q−2. Therefore, A∪Bcan be covered by a square
+C′∈ S(p+q−1). IfC′=Bone actually has C′∈ S(q). Now one can choose Cas an arbitrary
+square of size rthat contains C′and all nearest neighbors of C′. Suppose now that C′/\e}atio\slash=B.
+Then either C′contains all nearest neighbors of B, orC′shares an edge or a corner with B. In
+the latter case, we can extend the size of C′by one obtaining a square C∈ S(p+q) with the
+desired properties.
+The proposition implies that
+[S(i),W] =/summationdisplay
+r≥3/summationdisplay
+C∈S(r)D(i)
+r,C, (5.20)
+where
+D(i)
+r,C=/summationdisplay
+p+q=r/summationdisplay
+A∈S(p)
+A⊆C/summationdisplay
+B∈S(q)
+B⊆C[S(i)
+p,A,Wq,B]χ(A,B,C ) (5.21)
+andχ(A,B,C ) = 0,1 is some function that ‘distributes’ the commutators over d ifferent terms
+D(i)
+r,C. A particular choice of such distribution is not important f or us. Using Proposition 5.2 we
+can assume that χ(A,B,C ) = 0 unless A∪B⊆CandCcontains all nearest neighbors of B.
+By construction, D(i)
+r,Chas support on C, that is, Eqs. (5.20,5.21) define a local decomposition
+of the commutator [ S(i),W].
+Using Proposition 5.1 we can choose a solution S(i+1)of Eq. (5.12) as
+S(i+1)=/summationdisplay
+r≥3/summationdisplay
+C∈S(r)S(i+1)
+r,C, S(i+1)
+r,C=EC(D(i)
+r,C). (5.22)
+This is a local decomposition of S(i+1)that satisfies (I1) by definition of the map EC. Let us
+check that it satisfies (I2). Indeed, let GF,F∈ S(2), be such that Fis not contained in C.
+ThenFis not contained in A⊆Cand thusGFcommutes with all S(i)
+p,Ain Eq. (5.21). Since
+for all terms Wq,Bin Eq. (5.21) the square Ccontains both Band the nearest neighbors of B,
+we conclude that Fdoes not overlap with B, that is,GFcommutes with Wq,B. Finally,GF
+commutes with all Krauss operators involved in the map EC. ThusGFcommutes with S(i+1)
+r,C
+which proves (I2).
+26It remains to verify that the local decomposition Eq. (5.22) satisfies (I3). It will be convenient
+to introduce an auxiliary Hamiltonian
+Wq(A,C) =/summationdisplay
+B∈S(q)
+B⊆Cχ(A,B,C )Wq,B. (5.23)
+It allows us to rewrite Eq. (5.21) as
+D(i)
+r,C=/summationdisplay
+p+q=r/summationdisplay
+A∈S(p)
+A⊆C[S(i)
+p,A,Wq(A,C)]. (5.24)
+Applying Corollary 3.2, using Eq. (5.7), and taking into acc ount thatWis (Jd,µ,α )-decaying
+we get
+/⌊ard⌊l[H0(C),S(i+1)
+r,C]/⌊ard⌊l=/⌊ard⌊lQCD(i)
+r,CPC/⌊ard⌊l ≤/summationdisplay
+p+q=r/summationdisplay
+A∈S(p)
+A⊆C/⌊ard⌊lQC[S(i)
+p,A,Wq(A,C)]PC/⌊ard⌊l
+≤/summationdisplay
+p+q=r/summationdisplay
+A∈S(p)
+A⊆Cbq/⌊ard⌊lQC[H0(C),S(i)
+p,A]PC/⌊ard⌊l, (5.25)
+where
+bq≤cJdq2−αe−µq(5.26)
+for some constant c. From (I1) we infer that S(i)
+p,Ais block-off-diagonal with respect to PA,QA
+while (I2) implies [ H0(C),S(i)
+p,A] = [H0(A),S(i)
+p,A]. Taking into account that PC=PAPCwe get
+a bound
+/⌊ard⌊lQC[H0(C),S(i)
+p,A]PC/⌊ard⌊l ≤ /⌊ard⌊lQA[H0(A),S(i)
+p,A]PA/⌊ard⌊l=/⌊ard⌊l[H0(A),S(i)
+p,A]/⌊ard⌊l ≤Kip−βe−µp, (5.27)
+where the last inequality follows from (I3). Combining Eqs. (5.25,5.27) and noting that the
+number of squares A∈ S(p) such that A⊆Cis equal to ( r−p)2=q2we get
+/⌊ard⌊l[H0(C),S(i+1)
+r,C]/⌊ard⌊l ≤Kir−1/summationdisplay
+q=1q2bq(r−q)−βe−µ(r−q)
+≤cJdKie−µrr−1/summationdisplay
+q=1q4−α(r−q)−β. (5.28)
+It is convenient to split the sum over qinto two parts:
+/summationdisplay
+1≤q≤r/2q4−α(r−q)−β≤cr−β/summationdisplay
+q≥1q4−α=c′r−β(5.29)
+for some constants c,c′since we assumed α≥β+ 4≥6. As for the other part, we have
+/summationdisplay
+r/2≤q≤r−1q4−α(r−q)−β≤cr4−α/summationdisplay
+q≥1q−β=c′r4−α≤c′r−β(5.30)
+27for some constants c,c′since we assumed that β≥2 andα≥β+ 4. Therefore we arrive at
+/⌊ard⌊l[H0(C),S(i+1)
+r,C]/⌊ard⌊l ≤cJdKir−βe−µr(5.31)
+for some constant cwhich proves (I3) for
+Ki+1=cJdKi. (5.32)
+Thus all induction assumptions are proved for S(i+1). By obvious reasons S=/summationtext
+i≥1Siis
+(K,µ,β )-decaying with
+K≤/summationdisplay
+i≥1Ki=c1J
+1−c2Jd. (5.33)
+5.3 Local decomposition of the transformed Hamiltonian
+Lemma 5.1 has the following corollary.
+Corollary 5.1. LetV,W, andSbe as in Lemma 5.1. Suppose Vis(J,µ,β )-decaying and
+Wis(Jd,µ,α )-decaying for some β≥2andα≥β+ 4. Then a transformed Hamiltonian
+˜W= [S,H0+W] +Vhas a local decomposition
+˜W=/summationdisplay
+r≥1/summationdisplay
+A∈S(r)˜Wr,A (5.34)
+such thatQ˜Wr,AP= 0for allr,A. This decomposition is (˜Jd,µ)-decaying, where
+˜Jd≤cJ
+1−cJd(5.35)
+for some constant c.
+Proof. We shall use notations and techniques introduced in the proo f of Lemma 5.1. By defini-
+tion ofSwe have
+˜W=/summationdisplay
+i≥1W(i), (5.36)
+where
+W(1)= [S(1),H0] +V, andW(i)= [S(i),H0] + [S(i−1),W] fori≥2. (5.37)
+Let us choose the local decomposition of W(1)as
+W(1)=/summationdisplay
+r≥3/summationdisplay
+A∈S(r)W(1)
+r,A, (5.38)
+where
+W(1)
+r,A= [EA(Vr,A),H0] +Vr,A=PAVr,APA+QAVr,AQA. (5.39)
+28Here the last equality uses Proposition 5.1 (see the first equ ation in the proof of the proposition).
+Obviously, /⌊ard⌊lW(1)
+r,A/⌊ard⌊l ≤ /⌊ard⌊lVr,A/⌊ard⌊land thusW(1)is (J,µ,β )-decaying. As was noticed in the proof of
+Lemma 5.1 we can assume that Vr,Acommutes with all operators GB,B∈ S(2) for which Bis
+not contained in A. It means that
+PW(1)
+r,A=PVr,APA=PVr,AP=W(1)
+r,AP, (5.40)
+that is,W(1)
+r,Ais block-diagonal.
+Recall that we have a decomposition
+[S(i),W] =/summationdisplay
+r≥3/summationdisplay
+C∈S(r)D(i)
+r,C, (5.41)
+whereD(i)
+r,Ccommutes with all operators GB,B∈ S(2) for which Bis not contained in C, see
+Eqs. (5.20,5.21) in the proof of Lemma 5.1. It means that we ca n choose the local decomposition
+ofW(i)withi≥2 as
+W(i)=/summationdisplay
+r≥3/summationdisplay
+C∈S(r)W(i)
+r,C, (5.42)
+where
+W(i)
+r,C= [EC(D(i−1)
+r,C),H0] +D(i−1)
+r,C=PCD(i−1)
+r,CPC+QCD(i−1)
+r,CQC, (5.43)
+see Eq. (5.22). Obviously, /⌊ard⌊lW(i)
+r,C/⌊ard⌊l ≤ /⌊ard⌊lD(i−1)
+r,C/⌊ard⌊landW(i)
+r,Cis block-diagonal. Using the fact that
+S(i)is (Ki,µ,β )-decaying where Kiis defined by Eqs. (5.18,5.32), and using Lemma 4.1 we
+conclude that W(i)is (cJdKi−1,µ)-decaying. Using Eq. (5.36) we obtain a local decompositio n
+of˜Wwith individually block-diagonal local terms which is ( ˜Jd,µ)-decaying with
+˜Jd=J+/summationdisplay
+i≥2cJdKi−1≤J+/summationdisplay
+i≥1J(cJd)i=J
+1−cJd. (5.44)
+Finally we have to replace JandJdbyO(J) andO(Jd) to justify our assumptions Eqs. (5.7,5.8,5.9).
+6 Lieb-Robinson bounds
+LetSbe some anti-hermitian operator and Vbe an arbitrary operator. We shall use notations
+τ(V) =eSVe−S,
+ω(V) =τ(V)−V−[S,V]. (6.1)
+Suppose we are given some local decompositions of SandV. In order to get a closed system of
+flow equations we need to construct a local decomposition for ω(V), see Section 4. The main
+result of this section is the following lemma.
+29Lemma 6.1. SupposeSis(K,µ,α )-decaying for some α≥6. SupposeVis(J,µ,β )-decaying
+for someβ≥6. Thenω(V)has a local decomposition which is (cJK2,µ/2,−1)-decaying for
+some constant c.
+Thus the magnitude of interactions of range rinω(V) decays as cJK2re−µr/2. Our ar-
+guments will rely on the Lieb-Robinson bound, see [20]. More specifically, we shall exploit
+quasi-locality of dynamics in quantum spin systems with a fa st decay of interactions as pre-
+sented in [19]. The extra factor 1 /2 in the decay rate of ω(V) represents a simple geometrical
+fact that the diameter of the light-cone of any local region i ncreases with a rate 2 v, wherev
+is the Lieb-Robinson velocity. This extra factor 1 /2 is the price one has to pay for using the
+powerful machinery built on the Lieb-Robinson bound.
+We shall start from solving a somewhat simpler problem. Let Obe any operator with
+support on some square B∈ S(q). Consider a local decomposition of S,
+S=/summationdisplay
+p≥2/summationdisplay
+A∈S(p)Sp,A. (6.2)
+LetC∈ S(q+ 2j) be a square that contains Band all sites within distance jfromB(with
+respect to the l∞-distance), that is,
+C={u∈Λ :D(u,B)≤j}. (6.3)
+LetSCbe a localized version of Sobtained by taking out all interactions whose support is not
+contained in C,
+SC=/summationdisplay
+p≥2/summationdisplay
+A∈S(p)
+A⊆CSp,A. (6.4)
+Define also a localized version of ω(O), that is,
+ωC(O) =eSCOe−SC−O−[SC,O]. (6.5)
+By definition, ωC(O) has support on C. We shall need a bound on the difference /⌊ard⌊lω(O)−ωC(O)/⌊ard⌊l
+that is proportional to /⌊ard⌊lO/⌊ard⌊l·K2e−µj.
+Lemma 6.2 (Quasi-Local Dynamics). LetS,SCandObe the operators defined above.
+SupposeSis(K,µ,α )-decaying for some α≥6. Then there exist constants c0,c1>0such that
+/⌊ard⌊lωC(O)−ω(O)/⌊ard⌊l ≤c0(q+j)q4K2/⌊ard⌊lO/⌊ard⌊le−µj(6.6)
+wheneverK≤c1.
+Proof. Define
+τt(O) =eStOe−St, τt
+C(O) =eSCtOe−SCt. (6.7)
+For any 0 ≤t≤1 define an operator
+f(t) =τt
+C(O)−τt(O)−[SC−S,O]t. (6.8)
+30Note thatf(t) is an analytic function and
+f(0) = ˙f(0) = 0, f(1) =ωC(O)−ω(O). (6.9)
+Computing the derivatives over twe get
+˙f(t) = [SC,τt
+C(O)]−[S,τt(O)]−[SC−S,O], (6.10)
+¨f(t) = [SC,[SC,τt
+C(O)]]−[S,[S,τt(O)]]. (6.11)
+Let us extract from ¨f(t) a norm preserving term [ S,˙f(t)]. After simple algebra we get
+¨f(t) = [S,˙f(t)]−[S−SC,[SC,τt
+C(O)]]−[S,[S−SC,O]]. (6.12)
+It means that
+˙f(t) =τt/parenleftbigg/integraldisplayt
+0τ−s([[SC,τs
+C(O)],∆SC] + [[∆SC,O],S])ds/parenrightbigg
+, (6.13)
+where ∆SC≡S−SC. Taking into account the initial conditions Eq. (6.9) we get
+/⌊ard⌊lf(1)/⌊ard⌊l ≤/integraldisplay1
+0dt1/⌊ard⌊l˙f(t1)/⌊ard⌊l ≤1
+2/⌊ard⌊l[[∆SC,O],S]/⌊ard⌊l+/integraldisplay1
+0dt1/integraldisplayt1
+0dt2/⌊ard⌊l[τt2
+C([SC,O]),∆SC]/⌊ard⌊l.(6.14)
+Let us start from bounding the time-independent term /⌊ard⌊l[[∆SC,O],S]/⌊ard⌊l. Denote
+Γp2,p1= #{(E2,E1) :E2∈ S(p2), E1∈ S(p1), E2∩(E1∪B)/\e}atio\slash=∅, E1∩B/\e}atio\slash=∅}.
+One can easily check that
+Γp2,p1≤(q+p1+p2)2(q+p1)2≤2(q+p1)4+ 2(q+p1)2p2
+2.
+The commutator [∆ SC,O] has contributions only from squares of size ≥jin the decomposition
+ofS, which implies
+/⌊ard⌊l[[∆SC,O],S]/⌊ard⌊l ≤cK2/⌊ard⌊lO/⌊ard⌊l/summationdisplay
+p1≥j/summationdisplay
+p2≥1Γp2,p1p−α
+1p−α
+2e−µ(p1+p2).
+Here and below cstands for a constant factor. Since α≥6 the sum over p2is bounded by a
+constant and we arrive at
+/⌊ard⌊l[[∆SC,O],S]/⌊ard⌊l ≤cK2/⌊ard⌊lO/⌊ard⌊le−µj/summationdisplay
+p1≥j(q+p1)4p−α
+1=cq4K2/⌊ard⌊lO/⌊ard⌊le−µj. (6.15)
+Let us now bound /⌊ard⌊l[τt2
+C([SC,O]),∆SC]/⌊ard⌊l. Note that the commutator has contributions only
+from those terms in the decomposition of ∆ SCthat overlap with both Cand its complement
+Cc. Define
+Ωp1,p2(t) = max
+E1∈S(p1)
+E1∩B/negationslash=∅max
+E2∈S(p2)
+E2∩C/negationslash=∅
+E2∩Cc/negationslash=∅max
+O1,O2/⌊ard⌊l[τt
+C([O1,O]),O2]/⌊ard⌊l,
+31whereO1,O2are unit-norm operators acting on E1,E2respectively. Counting the number of
+squares contributing to the double commutator yields
+/⌊ard⌊l[τt
+C([SC,O]),∆SC]/⌊ard⌊l ≤c(q+j)K2/summationdisplay
+p1,p2≥1(q+p1)2p−α
+1p2−α
+2e−µ(p1+p2)Ωp1,p2(t).
+As we show below, the unitary evolution under SCcan be characterized by a finite Lieb-Robinson
+velocityvLR=O(K). Using the Lieb-Robinson bound from [19] that governs unit ary evolution
+under Hamiltonians with exponentially decaying interacti ons one gets
+Ωp1,p2(t)≤c/⌊ard⌊lO/⌊ard⌊l(p1+q)2p2
+2exp [vLRt−µθ(j−p1−p2)],
+whereθ(x) =xforx≥0 andθ(x) = 0 forx < 0. Note that θ(j−p1−p2) is a lower
+bound on the distance between supports of [ O1,O] andO2in the definition of Ω p1,p2. Clearly,
+p1+p2+θ(j−p1−p2)≥j. Since 0 ≤t≤1 we can assume that vLRt=O(Kt) =O(1). Hence
+/⌊ard⌊l[τt
+C([SC,O]),∆SC]/⌊ard⌊l ≤c(q+j)/⌊ard⌊lO/⌊ard⌊lK2e−µj/summationdisplay
+p1,p2≥1(q+p1)4p−α
+1p4−α
+2≤c(q+j)q4/⌊ard⌊lO/⌊ard⌊lK2e−µj
+(6.16)
+forα≥6. Combining Eqs. (6.15,6.16) and computing the integrals i n Eq. (6.14) we arrive at
+|f(1)| ≤c(q+j)q4/⌊ard⌊lO/⌊ard⌊lK2e−µj.
+It remains to check that we have fast enough decay of interact ions inSto use the Lieb-
+Robinson bound from Ref. [19]. Below, we use notations from R ef. [19]. Define a function
+Fµ(x) =exp (−µx)
+1 +x2. (6.17)
+As was shown in Ref. [19] the Lieb-Robinson velocity is bound ed by a multiple of:
+/⌊ard⌊lS/⌊ard⌊lµ:= sup
+u,v∈Λ/summationdisplay
+r≥1/summationdisplay
+A∈S(r)
+A∋u,v/⌊ard⌊lSr,A/⌊ard⌊l
+Fµ(D(u,v)), (6.18)
+with the multiplying factor being 2 C0(1), whereC0(1) is a numerical constant (depending only
+on the dimensionality of the system; in our case the dimensio n is 2.) For any pair of sites u,v∈A
+withA∈ S(r) one hasD(u,v)≤r(here and below we use the l∞-distance). Since Fµ(x) is
+monotone-decreasing we have Fµ(D(u,v))≥Fµ(r). Taking into account that the number of
+squaresA∈ S(r) such that A∋uis at mostr2we get
+/⌊ard⌊lS/⌊ard⌊lµ≤K/summationdisplay
+r≥1r2(1 +r2)r−α≤cK, c =/summationdisplay
+r≥1(r−2+r−4)≤4, (6.19)
+provided that α≥6. Thus, the Lieb-Robinson velocity is bounded by 8 C0(1)K.
+32Let us use Lemma 6.2 to construct a local decomposition for ω(O). This local decomposition
+will involve a sequence of squares
+B0=B⊂B1⊂B2⊂...⊂Λ (6.20)
+such thatBj∈ S(q+2j) is the square obtained from Bby adding a boundary region of thickness
+jon all sides of B, that is,
+Bj={u∈Λ :D(u,B)≤j}. (6.21)
+Note thatBj= Λ for large enough j. Then we can write ω(O) as
+ω(O) =DB0(O) +/summationdisplay
+j≥1DBj(O), DB0(O) =ωB(O), DBj(O) =ωBj(O)−ωBj−1(O).(6.22)
+Note thatDBj(O) acts only on Bj, so Eq. (6.22) defines a local decomposition of ω(O). Using
+Lemma 6.2 we infer that
+/⌊ard⌊lDBj(O)/⌊ard⌊l ≤ /⌊ard⌊lωBj(O)−ω(O)/⌊ard⌊l+/⌊ard⌊lωBj−1(O)−ω(O)/⌊ard⌊l ≤c(q+j)q4K2·/⌊ard⌊lO/⌊ard⌊le−µj(6.23)
+for some constant c. In addition we have a standard bound (see for instance Ref. [ 7])
+/⌊ard⌊lDB0(O)/⌊ard⌊l=/⌊ard⌊lωB(O)/⌊ard⌊l ≤1
+2/⌊ard⌊l[SB,[SB,O]]/⌊ard⌊l ≤2/⌊ard⌊lSB/⌊ard⌊l2/⌊ard⌊lO/⌊ard⌊l. (6.24)
+Taking into account that
+/⌊ard⌊lSB/⌊ard⌊l ≤q/summationdisplay
+p=2/summationdisplay
+A∈S(p)
+A⊆B/⌊ard⌊lSp,A/⌊ard⌊l ≤q/summationdisplay
+p=2(q−p)2Kp−αe−µp≤cq2K (6.25)
+for some constant c. Thus for all j≥0 we have
+/⌊ard⌊lDBj(O)/⌊ard⌊l ≤cq4(q+j)K2/⌊ard⌊lO/⌊ard⌊le−µj. (6.26)
+Having finished this warmup we can easily prove Lemma 6.1.
+Proof of Lemma 6.1. Consider a local decomposition of V,
+V=/summationdisplay
+q≥2/summationdisplay
+B∈S(q)Vq,B,/⌊ard⌊lVq,B/⌊ard⌊l ≤Jq−βe−µq. (6.27)
+We construct a local decomposition of ω(Vq,B) as in Eq. (6.22), where O≡Vq,B. We get
+ω(V) =/summationdisplay
+r≥2/summationdisplay
+A∈S(r)Ωr,A, (6.28)
+where
+Ωr,A=r/2−1/summationdisplay
+j=0/summationdisplay
+B∈S(r−2j)
+A=BjDBj(Vr−2j,B). (6.29)
+33Note that for fixed jthe sum over Bcontains a single square Bsuch thatA=Bj, see Eq. (6.21).
+Using Eq. (6.26) with q=r−2j, we arrive at:
+/⌊ard⌊lΩr,A/⌊ard⌊l ≤r/2−1/summationdisplay
+j=0/summationdisplay
+B∈S(r−2j)
+A=BjK2/⌊ard⌊lVr−2j,B/⌊ard⌊l(r−2j)4re−µj
+≤r/2−1/summationdisplay
+j=0K2J(r−2j)4−βre−µ(r−j)≤cK2rJe−µr
+2.
+for some constant cprovided that β≥6. We have shown that ω(V) is (cK2J,µ/ 2,−1)-decaying.
+7 Adiabatic continuation of logical operators
+7.1 Dressed Operators
+Since the gap remains open up to a certain strength of perturb ation, this means that the
+perturbed Hamiltonian H0+Vis adiabatically connected to the original Hamiltonian, H0. This
+adiabatic connection suggests that the perturbed Hamilton ian should have similar properties to
+the unperturbed Hamiltonian. For example, following [3], w e can define string operators which
+have nontrivial action in the ground state subspace of the pe rturbed Hamiltonian and which
+have the correct commutation relations and expectation val ues. In fact, Theorem 1 will allow
+us to do even more, to define local operators that create defec t excitations with well-defined
+energies.
+To construct these operators, we use quasi-adiabatic conti nuation. We define a continuous
+family of Hamiltonians,
+Hs=H0+sV, (7.1)
+so that assvaries from 0 to 1, Hscontinuously interpolates between H0and the perturbed
+Hamiltonian.
+We define a quasi-adiabatic continuation operator, Dsby
+Ds≡i/integraldisplay
+dtF(t) exp(iHst)/parenleftBig
+∂sHs/parenrightBig
+exp(−iHst), (7.2)
+where the function F(t) is defined to have the following properties. First, the Four ier transform
+ofF(t), which we denote ˜F(ω), obeys
+|ω| ≥1/2→ ˜F(ω) =−1/ω. (7.3)
+Second, ˜F(ω) is infinitely differentiable, so that F(t) decays faster than any negative power of
+time for large |t|. Third,F(t) =−F(−t), so that Dsis anti-Hermitian.
+We define a unitary operator Usby
+Us≡ S′exp/braceleftBig
+i/integraldisplays
+0ds′Ds/bracerightBig
+, (7.4)
+34where the notation S′denotes that the above equation (7.4) is an s′-ordered exponential.
+The motivation for defining the above unitary operator is con tained in the following lemmas:
+Lemma 7.1. LetHsbe a differentiable family of Hamiltonians. Let |Ψi(s)/a\}⌊ra⌋ketri}htdenote eigenstates
+ofHswith energies Ei. LetEmin(s)<Emax(s)be continuous functions of sand
+I(s) ={λ∈R:Emin(s)≤λ≤Emax(s)}.
+Define a projector P(s)onto an eigenspace of Hsby
+P(s) =/summationdisplay
+i:Ei∈I(s)|Ψi(s)/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{tΨi(s)|. (7.5)
+Assume that the space that P(s)projects onto is separated from the rest of the spectrum by a
+gap of at least 1/2for allswith 0≤s≤1. That is, any eigenvalue of Hseither belongs to I(s)
+or is separated from I(s)by a gap at least 1/2. Then, for all swith 0≤s≤1, we have
+P(s) =UsP(0)U†
+s. (7.6)
+Proof. By linear perturbation theory,
+∂sP(s) =/summationdisplay
+i∈I(s)/summationdisplay
+j/∈I(s)1
+Ei−Ej|Ψj(s)/a\}⌊ra⌋ketri}ht/parenleftBig
+/a\}⌊ra⌋ketle{tΨj(s)|∂sH(s)|Ψi(s)/a\}⌊ra⌋ketri}ht/parenrightBig
+/a\}⌊ra⌋ketle{tΨi(s)|+h.c (7.7)
+=−/summationdisplay
+i∈I(s)/summationdisplay
+j/∈I(s)|Ψj(s)/a\}⌊ra⌋ketri}ht/parenleftBig
+/a\}⌊ra⌋ketle{tΨj(s)|/integraldisplay
+dtF(t) exp(iHst)∂sH(s) exp(−iHst)|Ψi(s)/a\}⌊ra⌋ketri}ht/parenrightBig
+/a\}⌊ra⌋ketle{tΨi(s)|+h.c
+=i[Ds,Ps].
+The first equality in the above equation holds because
+/a\}⌊ra⌋ketle{tΨj(s)|/integraldisplay
+dtF(t) exp(iHst)∂sH(s) exp(−iHst)|Ψi(s)/a\}⌊ra⌋ketri}ht (7.8)
+=/a\}⌊ra⌋ketle{tΨj(s)|/integraldisplay
+dtF(t) exp[i(Ej−Ei)t]∂sH(s)|Ψi(s)/a\}⌊ra⌋ketri}ht
+=˜F(Ej−Ei)/a\}⌊ra⌋ketle{tΨj(s)|∂sH(s)|Ψi(s)/a\}⌊ra⌋ketri}ht,
+where we use Eq. (7.3) to show ˜F(Ej−Ei) =−1/(Ej−Ei) using the assumption on the gap
+in the spectrum that |Ej−Ei| ≥1/2.
+Since∂s(UsP(0)U†
+s) =i[Ds,UsP(0)U†
+s], andU0=I, Eq. (7.6) follows from Eq. (7.7).
+This purpose for introducing this quasi-adiabatic continu ation operator is to define certain
+“dressed” operators, following the idea introduced in [3]. LetO1,O2,...be some operators that
+create defects when acting on the ground state of H0. These operators may be defined to
+have certain commutation or anti-commutation requirement s. For example, if we have certain
+operatorsOE
+iwhich create electric defects on a given neighboring pair of sites, and operators
+OM
+iwhich create magnetic defects on a given neighboring pair of plaquettes in a toric code
+35state, then these operators all commute with each other, exc ept an electric and a magnetic
+operator anti-commute if the bond connecting the sites and t he bond connecting the plaquettes
+on the dual lattice intersect. Then we define “dressed” opera torsOi(s) by
+Oi(s) =UsOiU†
+s. (7.9)
+SinceUsis unitary, these operators Oi(s) obeys the same commutation or anti-commutation
+requirements:
+[Oi,Oj] = 0→[Oi(s),Oj(s)] = 0, (7.10)
+{Oi,Oj}= 0→ {Oi(s),Oj(s)}= 0.
+LetPn(s) project onto the eigenspace of H(s) with energy in the interval In, see Theorem 1.
+We assume that Jis chosen sufficiently small (see Section 1.1) so that Inis separated from the
+rest of the spectrum by a gap at least 1 /2. So, Lemma 7.1 implies that if a product of operators
+O1O2...Omacting any ground state of H0creates an eigenstate of H0with energy n, then the
+product of operators O1(s)O2(s)...Om(s) acting on any ground state state of Hscreates a state
+with energy in the interval In. To see this, note that a ground state Ψ 0(s) ofHscan be written
+asUsΨ0(0) for some ground state Ψ 0(0) ofH0. Then,
+Pn(s)O1(s)...Om(s)|Ψ0(s)/a\}⌊ra⌋ketri}ht=UsPn(0)O1(0)...Om(0)|Ψ0/a\}⌊ra⌋ketri}ht (7.11)
+=Us(0)O1(0)...Om(0)|Ψ0/a\}⌊ra⌋ketri}ht=O1(s)...Om(s)|Ψ0(s)/a\}⌊ra⌋ketri}ht.
+Further, if some product of operators O1O2...has a given expectation value in the ground
+state ofH0, then the corresponding dressed operators have the same exp ectation value in the
+ground state of Hs. For example, since a product of σxaround a contractable loop has expec-
+tation value unity in the ground state of the toric code, the s ame product of dressed operators
+will have the same expectation value in the ground state of Hs.
+The next important property we want to show is that the operat orsOi(s) are local. To
+do these, we need a Lieb-Robinson bound for quasi-adiabatic continuation. Before describing
+this, some comments. We have chosen to define the quasi-adiab atic continuation using “exact”
+expressions. That is, we have chosen a filter function F(t) such that its Fourier transform is
+exactly equal to 1 /ωoutside a certain interval. This is Osborne’s[21] modificat ion of the quasi-
+adiabatic continuation of [3]. As a result, our filter functi onF(t) decays faster than any negative
+power oft, but does notdecay exponentially in t(however, there do exist such filter functions
+with|F(t)|decaying exponentially in a polynomial of t[23]). This contrasts with the approach
+in [3], where an approximation was used that gave a filter func tion decaying exponentially in t2.
+Each approach has certain advantages and disadvantages. Th e approach in [3] has the advantage
+that one can often get exponentially good approximations, r ather than approximations which
+merely decay faster than any power. However, Osborne’s modi fication which we use here has
+a few advantages also. First, we get an exact result that Ψ 0(s) =UsΨ0(0) above, while the
+approach in [3] only gives approximate estimates. Second, i t is much easier to derive locality
+estimates. In particular, the Lieb-Robinson bound for quas i-adiabatic continuation in the next
+lemma, which is the key step to prove locality of the dressed o perators, is much easier than
+[3, 22]. The reason that this bound is so much simpler is the fo llowing: the Fourier transform
+36of the filter function here can be chosen to be some given bound ed function of ω. In [3, 22],
+we have a parameter-dependent family of filter functions. As this parameter is increased, the
+accuracy of the approximation improves, but also the upper b ound on the Fourier transform of
+the filter function gets worse, and hence the bound on the norm ofDworsens.
+The locality of the dressed operators depends on locality pr operties of H0,V. We require
+thatH0,Vare both sums of of local operators H0,Z,VZ, where both H0,Z,VZobey a bound
+that, for all sites u∈Λ,
+/summationdisplay
+Z∋u/⌊ard⌊lH0,Z/⌊ard⌊l|Z|exp(µdiam(Z)) =O(1), (7.12)
+/summationdisplay
+Z∋u/⌊ard⌊lVZ/⌊ard⌊l|Z|exp(µdiam(Z))≤J <∞,
+where|Z|denotes the cardinality of Z, for some positive constants µ,J.
+Given such locality property, for any finite dimensional sys tem we have a Lieb-Robinson
+bound forH0+sVthat, for any operator OAsupported on set Aand anyOBsupported on set
+B,
+/⌊ard⌊l[exp(iHst)OAexp(−iHst),OB]/⌊ard⌊l ≤exp[−µ(dist(A,B)−vLRt)]|A|/⌊ard⌊lOA/⌊ard⌊l/⌊ard⌊lOB/⌊ard⌊l, (7.13)
+wherevLRis some constant which depends on µ,J. This bound is shown in [20].
+Lemma 7.2. LetH0andVobey Eq. (7.12). Then, if OAis supported on set AandOBis
+supported on set B,
+/⌊ard⌊l[UsOAU†
+s,OB]/⌊ard⌊l ≤h(dist(A,B))|A|/⌊ard⌊lOA/⌊ard⌊l/⌊ard⌊lOB/⌊ard⌊l, (7.14)
+where|A|denotes the cardinality of A, for 0≤s≤1, for some function h(l)which decays faster
+than any negative power of l. Similarly,
+/⌊ard⌊l[U†
+sOAUs,OB]/⌊ard⌊l ≤h(dist(A,B))|A|/⌊ard⌊lOA/⌊ard⌊l/⌊ard⌊lOB/⌊ard⌊l, (7.15)
+Further, the operator Dsis a sum of operators Ds(Z), with
+/⌊ard⌊lDs(Z)/⌊ard⌊l ≤const.×/⌊ard⌊lVZ/⌊ard⌊l, (7.16)
+and with the property that each such operator Ds(Z)obeys, for any operator OB,
+/⌊ard⌊l[Ds(Z),OB]/⌊ard⌊l ≤h′(dist(A,B))|Z|/⌊ard⌊lVZ/⌊ard⌊l/⌊ard⌊lOB/⌊ard⌊l, (7.17)
+for some function h′(l)which decays faster than any negative power of l.
+Proof. We have
+Ds=/summationdisplay
+ZDs(Z), (7.18)
+where
+Ds(Z) =i/integraldisplay
+dtF(t) exp(iHst)VZexp(−iHst). (7.19)
+37Eq. (7.16) follows immediately from a triangle inequality b ecause/integraltext
+dt|F(t)|converges.
+LetOBbe an operator supported on set B. Then
+/⌊ard⌊l[Ds(Z),OB]/⌊ard⌊l ≤/integraldisplay
+dt|F(t)|/⌊ard⌊l[exp(iHst)VZexp(−iHst),OB]/⌊ard⌊l. (7.20)
+Fort≤dist(Z,B)/2vLR, the above expression is exponentially small in dist( Z,B) by the Lieb-
+Robinson bound, while for larger t, the expression is bounded by |F(t)|, and hence this com-
+mutator decays faster thany any negative power of dist( Z,B).
+This decomposition and locality bound (7.20) implies that t here is a Lieb-Robinson bound
+for evolution using Dsas a Hamiltonian. Using the Lieb-Robinson bound in [20] (the use of
+this bound does require some geometric properties of the lat tice, which hold for any finite
+dimensional lattice), we have that
+/⌊ard⌊l[UsOAU†
+s,OB]/⌊ard⌊l ≤exp(cs)h′(dist(A,B))|A|/⌊ard⌊lOA/⌊ard⌊l/⌊ard⌊lOB/⌊ard⌊l, (7.21)
+for some constant cwhich depends on F,J,µ and on the geometric properties of the lattice,
+and for some function h′(l) which decays faster than any negative power of l. Since we assume
+s≤1, Eq. (7.13) follows from Eq. (7.21).
+Eq. (7.13) implies that OA(s) =UsOAU†
+scan be approximated by an operator Ollocalized
+on the set of sites within distance lof setA. To see this, define Ol=/integraltext
+dUUOA(s)U†, where
+the integral ranges over unitary rotations, with Haar measu re, supported on sites with distance
+greater than lfrom setA. Then, following [2], the desired result follows.
+The string operators can be used to manipulate the ground sta tes. We will assume that the
+operatorsO1which create defects are, in fact, unitaries. For example, i f a certain product of
+electric operators, O1O2...creates a nontrivial loop around the torus and has nontrivia l action
+on the ground state of H0, then the product of dressed operators O1(s)O2(s)...has the same
+action on the ground state of Hs. However, one might wonder how to create these dressed
+operators; it would be preferable not to have to solve quasi- adiabatic evolution equations to
+calculate what the operators should be, and then to produce t hem by careful control of a time-
+dependent Hamiltonian. Fortunately, we can use the gaps in t he excited state spectrum to
+argue that it is possible to drag defects in the perturbed Ham iltonianHsin an identical way to
+what is done in H0. First, we apply some local operator in an attempt to create a defect pair
+on neighboring sites. If this operator is not exactly equal t o the desired dressed operator, then
+this attempt may fail, in the sense that the energy may not fal l into the desired range I2, see
+Theorem 1. We can detect this failure by measuring the energy , and cooling back to the ground
+state: that is, if there are extra defects created, we will dr ag them together, using the procedure
+described in the next paragraph, to annihilate them. Since t he dressed operator is local, then
+some local operator (for example, the undressed operator) i s expected to have non-vanishing
+matrix elements between the ground state and the desired sta te. Eventually we will succeed in
+creating the defect pair.
+We now try to move the defect pair by weak changes in the Hamilt onian. We add a pertur-
+bationUto the Hamiltonian with norm bounded by 1 /4. Such a weak perturbation will keep
+the gap open to the states with 3 defects (of course, there are no such three defect states in the
+38toric code, but in more general models there are). So, by addi ng this perturbation and changing
+it adiabatically, we remain in the subspace with a defect pai r.
+In a toric code Hamiltonian, every state in this subspace is a linear combination of states
+which are given by acting on a ground state Ψ 0(s) by a linear combination of products of strings
+of dressed operators. Consider a given state created by a sin gle string of dressed operators, with
+endpointsiandjwhich are far separated. Call this state Ψ( S), whereSis a string with
+endpoints at i,j. Note that using the property that any contractable loop of d ressed operators
+has expectation value unity in the ground state of Hs, we can show that we can deform these
+strings in any way that leaves the endpoints fixed while leavi ng the state unchanged. Let us
+now see how the expectation value of local operators can chan ge in the state Ψ( S). IfOis
+an operator which is far separated from i,j, we claim that the expectation value of Oin state
+Ψ(S) is close to its value in the ground state. To see these, note t hat that we can define a state
+Ψ(S′) = Ψ(S) whereS′is a string which stays far from O. So, without loss of generality, we
+can assume that string Sis far from O. Then, the using the locality properties of the dressed
+operators, they almost commute with O, and so
+/a\}⌊ra⌋ketle{tΨ(S)|O|Ψ(S)/a\}⌊ra⌋ketri}ht=/a\}⌊ra⌋ketle{tΨ0(s)|O†
+n...O†
+1OO1...On|Ψ0(s)/a\}⌊ra⌋ketri}ht (7.22)
+≈ /a\}⌊ra⌋ketle{t Ψ0(s)|O†
+n...O†
+1O1...OnO|Ψ0(s)/a\}⌊ra⌋ketri}ht
+=/a\}⌊ra⌋ketle{tΨ0(s)|O|Ψ0(s)/a\}⌊ra⌋ketri}ht.
+So, the perturbation Ucan effect the state, but only if Uacts close to one of the endpoints
+of the string. Further, if Uis local, then one may show that Uwill have small matrix elements
+except between Ψ( S) and Ψ(S′) for strings S′which differ from Sonly by small motion of one
+of the endpoints. Thus, if Uis chosen to have the property that it reduces the energy when
+an end of the string is close to a certain point, the operator Ucan indeed by used to drag the
+defects.
+So, we have established that it is possible to create a defect pair and drag it. The gap to
+the rest of the spectrum prevents additional defects from be ing created. If they are created, we
+can detect them by measuring the energy, and we can drag them t o destroy them and correct
+errors.
+We can drag one of the defects around the system. Since at ever y step we remain in a state
+which is a linear combination of states Ψ( S), and eventually we succeed in dragging a defect
+all the way around the system, if we are able to return to a grou nd state Ψ′
+0(s), then Ψ′
+0(s) is
+produced by a local operation Oon a state Ψ( S) whereSis a string with two nearby endpoints
+connected by a string that goes around the sample. Any such st ate Ψ(S) is a product of dressed
+operators acting on Ψ 0(s). So, it is equal to Ψ( S) =UsO1...OnΨ0(0). Suppose Ψ′
+0(s) is a ground
+state such that /a\}⌊ra⌋ketle{tΨ′
+0(s),OΨ(S)/a\}⌊ra⌋ketri}htis non-negligible. This expectation value is equal to
+/a\}⌊ra⌋ketle{tΨ′
+0(0)|/parenleftBig
+U†
+sOUs/parenrightBig
+O1...On|Ψ0(0)/a\}⌊ra⌋ketri}ht. (7.23)
+However, due to the locality properties of quasi-adiabatic continuation, the operator U†
+sOUsis
+approximately local. Hence, the ground state Ψ′
+0(0) of the unperturbed system is connected
+by a local operator to the state O1...On|Ψ0(0)/a\}⌊ra⌋ketri}ht. So, the state Ψ′
+0(0) must be close to the state
+39formed by acting on Ψ 0(0) with a noncontractible string that winds around the samp le. Thus, by
+dragging the defects in the perturbed system, we succeed in e ffecting the desired transformation
+on the ground state sector of the perturbed system.
+One thing that can go wrong in this procedure is that we might i nadvertently create the
+wrong type of defect. We might accidentally create a magneti c defect pair rather than an electric
+defect pair (one may show that the perturbation Uwill have small amplitude to change electric
+into magnetic defects if the defects are far separated and Uis local). However, we expect to be
+able to locally tell the difference between these types of def ects and thus determine what kind
+of defect has been created, and, if the wrong type has been cre ated, to destroy the defect pair
+by bringing them together.
+One would like to improve the arguments here to a full formal p roof that using classical
+control it is possible to correct local errors and perform co ntrolled operations in this perturbed
+toric code. We will leave a complete proof of this to the futur e, but we have established the
+existence of operators with the necessary properties and of the gaps in the spectrum.
+Acknowledgments
+We thank Barbara Terhal and David DiVincenzo for useful disc ussions. Part of this work was
+done while SB and SM were visiting the Erwin Schr¨ odinger Int ernational Institute for Mathe-
+matical Physics at Vienna. SB was partially supported by the DARPA QUEST program under
+contract number HR0011-09-C-0047. SM thanks the organizer s of the program on “Quantum
+Information Science” at the KITP at UC Santa Barbara, where p art of this work was completed.
+SM was supported by NSF Grant DMS-07-57581 and DOE Contract D E-AC52-06NA25396.
+References
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+41
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+arXiv:1001.0345v1  [physics.atom-ph]  3 Jan 2010Isotropic magnetometry with simultaneous excitation of or ientation and alignment
+CPT resonances
+A. Ben-Kish, M. V. Romalis
+Department of Physics, Princeton University, Princeton, N ew Jersey 08544, USA
+(Dated: October 25, 2018)
+Atomic magnetometers have very high absolute precision and sensitivity to magnetic fields but
+suffer from a fundamental problem: the vectorial or tensoria l interaction of light with atoms leads
+to ”dead zones”, certain orientations of magnetic field wher e the magnetometer loses its sensitivity.
+We demonstrate a simple polarization modulation scheme tha t simultaneously creates coherent
+population trapping (CPT) in orientation and alignment, th ereby eliminating dead zones. Using
+87Rb in a 10 Torr buffer gas cell we measure narrow, high-contras t CPT transparency peaks in all
+orientations and also show absence of systematic effects ass ociated with non-linear Zeeman splitting.
+PACS numbers: 42.50.Gy, 07.55.Ge, 32.10.Dk
+Atomic magnetometers measure the magnetic field by
+detecting the Zeeman spin precession frequency and can
+achieve sensitivity surpassing even the best supercon-
+ducting quantum interference devices (SQUID) [1]. They
+are particularly suitable for operation in Earth’s mag-
+netic field because measurements of the spin precession
+frequency can be performed with high fractional resolu-
+tion, are related to the magnetic field only through fun-
+damental constants, and are relatively insensitive to the
+orientation of the magnetometer. As a result, atomic
+magnetometers are widely used for the most demanding
+applications, such as measurements of magnetic fields in
+space [2–4], mineral exploration [5], searches for archeo-
+logical artifacts [6], and unexploded ordnance [7].
+Since their inception, all types of atomic magnetome-
+ters have suffered from a fundamental problem known
+as a ”dead zone” [8]: for certain orientation of the mag-
+netic field relative to the device the magnetometer sig-
+nal goes to zero. Dead zones are an inherent feature of
+the vector or tensor interactions used for optical pump-
+ing and detection of spin oscillations: for certain ori-
+entations the interaction term goes to zero. Previous
+solutions to this problem included using multiple mag-
+netometer cells or beams [9, 10], mechanical rotation of
+some components[11], and use of unpolarized light and
+spatially-varying microwave fields [12]. Particularly for
+space magnetometers it presents a serious challenge that
+requires increased complexity of the system [3, 4].
+Here we demonstrate a dead-zone-free Rb magnetome-
+ter utilizing a single interaction region. The magnetome-
+ter uses a single laser beam with polarization modula-
+tion at the Zeeman frequency to simultaneously excite
+and detect the precession of the dipole and quadrupole
+(alignment and orientation) moments of the atomic den-
+sity matrix. Because the vector nature of the two inter-
+actions is different, the transmission of the laser beam is
+sensitive to the magnetic field for all orientations.
+Excitation of magnetic resonances using light intensity
+modulation was first demonstrated by Bell and Bloom
+[13] and polarizationmodulation wasexploredin [14, 15].Magnetometerybased on precession ofalignment was ex-
+tensively explored in non-linear magneto-opticalrotation
+(NMOR) magnetometers [16] while transmission mon-
+itoring has been used in CPT magnetometers [17–19].
+However, in all these cases the magnetometer signals go
+to zero for certain orientations of the magnetic field. We
+showthatwithaparticularchoiceoflightmodulationpa-
+rametersandthedetectionmethodonecanrealizeaCPT
+magnetometer operating simultaneously on two different
+coherences without any active adjustments, resulting in
+a large response for all orientations of the magnetic field.
+In addition to avoiding dead zones our arrangement
+also largely eliminates ”heading errors”, another long-
+standing problem, particulary for alkali-metal magne-
+tometers. Heading errors are due to non-linear Zeeman
+splitting of the magnetic sublevels given by the Breit-
+Rabi formula. By using symmetric optical pumping with
+equal intensities of σ+andσ−light we eliminate spin po-
+larization along the magnetic field. The remaining third-
+order heading error [20] is suppressed by the square of
+the ratio of the Zeeman frequency to the hyperfine fre-
+quency. The error due to nuclear magnetic moment is
+also avoided by optical pumping on only one of the hy-
+perfine ground states.
+Considerlight polarizedlinearlyin the ˆ xdirectionwith
+intensity Ipropagating parallel to the ˆ zaxis, passing
+through a polarization modulator with an optic axis at
+45◦to ˆxand a retardation angle φ=φ0cosωt. The fol-
+lowing Stokes parameters of the light will be modulated
+Sx=Icos(φ0cosωt), (1)
+Sz=Isin(φ0cosωt). (2)
+The interaction of light with atoms can be written as an
+effective non-hermitian ground-state Hamiltonian [21]
+∝angbracketleftn|H|m∝angbracketright=/summationdisplay
+m′E∗·dnm′dm′m·E
+/planckover2pi1(ω0−ωF,F′+iΓ/2),(3)
+whereEanddare the electric field and electric dipole
+operator, ω0is the laser frequency, ωF,F′is the transition2
+frequency from the ground state Fto the excited state
+F′, and Γ is the excited state decay rate. The Hamilto-
+nian can be decomposed into a sum of scalar, vector and
+tensor components[21, 22]. Retaining only terms propor-
+tional to the modulated Stokes parameters we obtain
+Hmod=2
+cǫ/summationdisplay
+F,F′α(1)
+F,F′SzFz+α(2)
+F,F′Sx(F2
+x−F2
+y)
+/planckover2pi1(ω0−ωF,F′+iΓ/2),(4)
+whereα(1)
+F,F′,α(2)
+F,F′are the vector and tensor atomic po-
+larizability constants respectively, defined in [22]. Light
+absorption in the vapor and the de-population optical
+pumping rateare both proportionalto the non-hermition
+partoftheHamiltonian H−H†[21,23,24]. Inparticular,
+forweaklightintensitythe atomswilldeveloporientation
+∝angbracketleftFz∝angbracketright ∝ −Szand alignment/angbracketleftbig
+F2
+x−F2
+y/angbracketrightbig
+∝ −Sxdue tode-
+population optical pumping. The effects of repopulation
+optical pumping are relatively small in the presence of
+buffer gas due to fast excited state spin relaxation.
+The modulation of the Stokes parameters can be ex-
+panded in terms of Bessel functions:
+Sx=J0(φ0)+2∞/summationdisplay
+k=1(−1)kJ2k(φ0)cos2kωt(5)
+Sz= 2∞/summationdisplay
+k=0(−1)kJ2k+1(φ0)cos(2k+1)ωt(6)
+One can see that for small φ0we get modulation of Sz
+atωand modulation of Sxat 2ω. The Bell-Bloom (BB)
+magnetometer [13] (Fig. 1A) relies on the Larmor pre-
+cession of the atomic orientation ∝angbracketleftFz∝angbracketright.If the frequency
+of the modulation ωmatches the Larmor precession fre-
+quencyωL, then∝angbracketleftFz∝angbracketrightwill be synchronously excited due
+to depopulation pumping by the modulation of Sz.The
+average transmission of the light through the cell will
+be increased due to the term SzFz∝ −cos2(ωt). How-
+ever, when the magnetic field is parallel to the ˆ zaxis,
+∝angbracketleftFz∝angbracketrightdoes not precess and the Bell-Bloom magnetome-
+ter has a dead zone for Bparallel to the light direction.
+For this case, however, one can see that the alignment/angbracketleftbig
+F2
+x−F2
+y/angbracketrightbig
+will precess at 2 ωL(Fig. 1B). Hence it can be
+synchronouslyexcited by modulation in Sxwhenω=ωL
+and the transmission through the cell will increase due
+to theSx(F2
+x−F2
+y)∝ −cos2(2ωt) term. Thus, average
+transmission through the cell will exhibit a CPT trans-
+mission resonance at the same frequency ω=ωLfor all
+orientations of the magnetic field.
+The experimental apparatus and the atomic levels di-
+agrams are presented in Fig. 1. A 795 nm DFB (Dis-
+tributed Feedback) diode laser is used for excitation and
+detection of the D1 transition in87Rb. A linear polarizer
+and an Electro-Optic Modulator (EOM) with its princi-
+ple axes at 45◦degrees relative to the input polarization
+are used for generating the time dependent polarization
+modulation seen in Fig. 1. The modulated light is sentthrough a 2 ×2×5 cm3glass cell with isotopically en-
+riched87Rb metal and 10 Torr of N 2buffer gas. The
+walls of the cell are coated with OTS coating and allow
+about 800 bounces before spin relaxation [25]. The cell
+is heated to 78◦C and located inside 3 layers of µmetal
+shielding,achievingmagneticfieldscreeningof ∼4orders
+of magnitude. A set of coils insides the shields is used
+for applying the needed magnetic fields in any orienta-
+tion. Another set of coils was used for nullifying residual
+magnetic field gradients. The transmitted light intensity
+through the cell is measured with a photo-detector.
+The wavelength dependance of the vector and tensor
+interactions for87Rb vapor in the presence of buffer gas
+and Doppler broadeninghas been consideredin [24]. The
+imaginary part of His maximized for both vector and
+tensor components close to the F= 2→F′= 1 D1
+transition. The maximum of the scalar absorption also
+occursnearthe samefrequency, hencethe laserfrequency
+can be locked to the transmission minimum using laser
+frequency modulation.
+σ+/σ-σ+ σ- π
+mF   =   -2        -1        0        1        2 -σ+±σ-
+mF=   -2        -1        0        1        2 -
+DFB LaserPol EOM
+detectorShield
+RbcellcoilsCF=2
+F=1F’=2
+’=1’
+F’
+F=2
+F=1F=2
+F=1’
+’A B
+σ+
+σ-
+FIG. 1: Energy levels diagrams for Bell-Bloom magnetometer
+withσ+orσ−polarized light (A), CPT magnetometer with
+linearσ+±σ−light (B), and the experimental apparatus (C).
+The applied voltage on the EOM was set to oscil-
+late sinusoidally with a retardation amplitude of ap-
+proximately φ0∼2 and the modulation frequency was
+scanned around the Larmor frequency ωL. In order to
+check the response of the magnetometer to magnetic
+fields in any orientation, superposition of electrical cur-
+rents was injected to the three axes of the Helmholtz
+coils inside the shields. An example of the measured
+transparency peak is shown in the inset of Fig. 2. The
+applied magnetic field was 28.6 mG, which corresponds
+to a Larmor frequency of 20 kHz. The contrast in this
+case is 50% and the FWHM of the CPT transmissionres-
+onance is 350 Hz. The measured contrast detected from
+each trace, for equally spread 58 orientations, was used
+for constructing the 3D plot in Fig. 2. Cubic spline in-
+terpolation was used for obtaining clearer 3D view of the
+measured results. As shown in the 3D plot, there are no
+dead zones. One can also see that the measured contrast
+along the three major axes is different.
+The natural axes of our scheme are set by the light3
+200-20
+−60−40−200204060−20020[%] 
+ZY
+ X
+2030405060
+16 17 18 19 20 21 22 23 2401020304050
+Freq uency  (kHz)Transmission [%]  
+FIG. 2: Contrast measurements as a function of magnetic
+field direction at 58 orientations, represented in percent o f
+DC transmission. An example of the measured contrast in
+one orientation is shown in the inset.
+propagation direction, ˆkz, the orientation of the linear
+polarization, ˆ εxand the magnetic field. In the case of
+Bzfield parallel to the propagation vector, the linear
+polarization component of the light modulated at 2 ωL
+generates CPT between appropriate Zeeman sub-levels
+(∆m= 2) (Fig. 1B). Destructive interference of the
+transitionamplitudesinthisΛsystemwillinduceatrans-
+parency peak at resonance [17]. The BB signal is zero for
+this orientation. For a magnetic field oriented in a plane
+perpendicular to the propagation vector ˆkz(Bx,By), the
+circular polarization components ( σ+orσ−) modulated
+at the Larmor frequency ( ωL) will induce a BB trans-
+parency peak at the Zeeman resonance (∆ m= 1, Fig.
+1A). With the quantization axis in the transverse plane
+the circularlypolarizedlight canbe thought ofas “ π”po-
+larizationand “ σ+±σ−” polarizationgeneratinga trans-
+parencypeak resonanceat ωL. This orientationterm will
+give a symmetric contribution to the transparency along
+BxandByaxes. In addition, the CPT resonance at
+2ωL(alignment term) will also contribute to the signal
+in this plane. One can see from Eq. (4) that there are
+two dead zones in CPT resonance for magnetic field in
+±ˆx±ˆydirections, as expected for a rank-2 tensor. For
+B field at ±45◦to ˆx, the signal is entirely due to BB
+resonance. When both CPT and BB signals are present,
+their interaction is complicated by the fact that large
+spin orientation created by BB modulation also leads to
+a significant alignment. The sign of this contribution
+to the second term in Eq. (4) is opposite for magnetic
+field in ˆxand ˆydirections. As a result, CPT and BB
+resonances add constructively for Byand destructively
+forBx. By choosing an appropriate modulation depth
+and laser power one can adjust the relative strength of
+the two signals and make the ratio of maximum to mini-
+mum contrast to be approximately 3:1 while maximizing
+the overall contrast. Experimentally we find the mea-
+sured contrast ranges between 15-60% while the FWHMis 350-700Hz, without any dead zones, as seen in Fig. 2.
+In Fig. 3 detailed contrast measurements along the
+threemajorplanesareshown. Theuppersetcorresponds
+to the signal with both CPT and BB signals, with pa-
+rameters similar to data in Fig. 2. The lower set is a ref-
+erence measurement where the CPT contribution to the
+signal is eliminated by choosing a specific laser detuning
+to the midpoint between F′= 1,2 upper levels where
+the constructive interference of the CPT signal is exactly
+cancelled due to different phase contribution of the two
+upper levels. It isevident from the figurethat nosignalis
+measured along the ˆ zaxis in the BB scheme whereas this
+”dead zone” is totaly avoided in the new scheme. More-
+over, along one axis (ˆ y), the signal’s strength is much
+larger and narrower than common BB magnetometers’
+signal, demonstrating a better signal to noise ratio.
+The fast and balanced alternation between right and
+left circular polarization at the Larmor frequency also
+eliminates the problem with heading errors. Heading er-
+rorinducedbyanasymmetricilluminationoftheatomsis
+manifested at magnetic fields on the order of the Earth’s
+magnetic field ( ∼0.5Gauss) due to the non-linear terms
+intheBreit-Rabiformula. Inordertocheckthisissue, we
+used the same experimental setup at a higher magnetic
+field. The EOM was driven at 1.9-2.2 MHz, correspond-
+ing to about 3 Gauss for87Rb atoms. By working at a
+field whichis ∼6times the Earth’smagneticfield anydis-
+tortion in the transparency due to the quadratic nature
+of the Breit-Rabi splitting should be observed.
+The measured signal in Fig. 4 corresponds to a case
+where a 2.85 Gauss magnetic field is tilted toward the
+Z axis by 26.5◦from either the ˆ xorˆyaxis. This com-
+paressignalswith the largestpossible contrastdifference,
+corresponding to the sum and the difference of the BB
+and CPT contributions. Whereas at low fields a sin-
+gle narrow peak is observed (e.g. Fig. 2), in this large
+field regime the Zeeman resonance is split into multiple
+peaks, particularly pronounced in the X-Z plane. The
+crucial observation here is that the signals are centered
+andsymmetric . Therefore, changes between the strength
+of BB and CPT contributions associated with rotation of
+the magnetometer in an external field will not result in
+a shift of the central frequency.
+In Fig. 4 we also present a case where unbalanced
+σ+/σ−contribution in the modulated light field is ob-
+tained by inducing an additional bias to the EOM’s driv-
+ing voltage. In this case the asymmetric signal will in-
+duce a heading error. This error is avoided in the current
+scheme due to the symmetric contribution of the vari-
+ous polarization components. In practice, one would also
+have to minimize residual polarization bias from stress-
+induced birefrigence in various optical elements to obtain
+good suppression of heading errors.
+In summary, a fundamental problem of “dead zones”
+in atomic magnetometers caused by the vectorial or ten-
+sorial interaction with the light has been resolved by a4
+  0.2  0.4  0.6
+30
+21060
+24090
+270120
+300150
+330180 0XY
+  0.2  0.4  0.6
+30
+21060
+24090
+270120
+300150
+330180 0XZ
+  0.2  0.4  0.6
+30
+21060
+24090
+270120
+300150
+330180 0YZ
+  0.02  0.06  0.1
+30
+21060
+24090
+270120
+300150
+330180 0  0.02  0.06  0.1
+30
+21060
+24090
+270120
+300150
+330180 0  0.02  0.06  0.1
+30
+21060
+24090
+270120
+300150
+330180 0XZ YZ XY
+FIG. 3: Contrast measurements along the three major planes.
+Upper set corresponds to the new magnetometer. The lower
+set corresponds to a specific laser light detuning (see text) in
+which the CPT contribution to the signal is canceled and only
+“regular” Bell-Bloom contribution appears with a clear Dea d
+Zone along the ˆ zaxis. Note the factor of 6 difference in scales
+between the two sets and that the chosen laser detuning for
+canceling the CPT signal is not necessarily an optimized cas e
+for the BB contrast.
+-20-15-10-505101520-0.0200.020.040.060.080.10.12
+Frequency [kHz]Transmission [a.u.]
+ YZ plane (26.5˚)
+XZ plane (26.5˚)
+unbalanced σ+ / σ
+FIG. 4: The magnetometer’s response at 2.85 Gauss (2 MHz).
+The two symmetric traces correspond to a magnetic field
+tilted toward the ˆ zaxis by 26.5◦from the ˆ xor the ˆyaxis.
+The asymmetric signal (in the X-Z plane) corresponds to an
+unbalanced σ+/σ−contribution, obtained by inducing an ad-
+ditional bias to the EOM’s driving voltage.
+simple polarization modulation scheme. Such technique
+can be applied to both alkali-metal and metastable4He
+magnetometers. We expect that the sensitivity of such
+magnetometer can reach picotesla level since measured
+CPT contrast ratio and linewidth are similar or better
+than in previous CPT magnetometers [18, 19]. In the
+simplest implementation of the magnetometer, frequency
+modulation at a few hundred hertz of the voltage applied
+to the EOM can be used to lock to the maximum of theaverage transmission through the cell. It is also possible
+to further improve the performance of the magnetometer
+by measuring the phase of the second and fourth har-
+monics of the Larmor frequency in the transmission sig-
+nal, analyzingthe polarizationof the transmitted light or
+using a separate probe laser. The amplitude of the EOM
+excitation can also be continuously adjusted to maximize
+the signal for any given orientation of the magnetic field.
+The headingerrorsspecific to alkali-metalmagnetome-
+ters in geomagnetic field are also largely eliminated due
+to symmetric optical pumping. Thus we expect that this
+technique will enable high precision isotropic magnetom-
+etry in many Earth-boundand spaceapplications. More-
+over, controlled excitation of orientation and alignment
+by polarization modulation of light can be used for bet-
+ter controlofthe atomic density matrixin variousatomic
+physics experiments.
+We would like to thank Vishal Shah for assistance with
+the measurements and Tom Kornack and Scott Seltzer
+for fabricating the cell. This work was supported by an
+ONR MURI award.
+[1] D. Budker and M. Romalis, Nature Phys. 3, 227 (2007).
+[2] M. Dougherty et al., Space Science Rev. 114, 331 (2004).
+[3] S. Matousek, Acta Astronautica 61, 932 (2007).
+[4] E. Friis-Christensen, H. L¨ uhr, and G. Hulot, Earth Plan -
+ets Space 58, 351 (2006).
+[5] M. Nabighian et al., Geophysics 70, 33ND (2005).
+[6] A. David et al., Antiquity 78, 341 (2004).
+[7] H. Nelson, J. McDonald, and D. Wright, Tech. Rep.
+NRL/MR/6110–05-8874, NRL (2005).
+[8] A. L. Bloom, Appl. Opt. 1, 61 (1962).
+[9] B. Cheron et al., J. Phys. III France 7, 1735 (1997).
+[10] B. Ch´ eron, H. Gilles, and J. Hamel, Eur. Phys. J. AP
+13, 143 (2001).
+[11] C. Guttin, J. Leger, and F. Stoeckel, J. Phys. IV Collocq
+France55, C4 665 (1994).
+[12] E. B.Aleksandrov, A.K.Vershovskii, andA.S.Pazgalev ,
+Technical Physics 51, 919 (2006).
+[13] W.E.BellandA.L.Bloom, Phys.Rev. 107, 1559(1957).
+[14] H. Gilles, J. Cheron, and J. Hamel, Opt. Comm 61, 369
+(1991).
+[15] H. Klepel and D. Suter, Opt. Comm. 90, 46 (1992).
+[16] D. Budker et al., Rev. Mod. Phys. 74, 1153 (2002).
+[17] A. Nagel et al., Europhys. Lett. 44, 31 (1998).
+[18] M. Stahler et al., Europhys. Lett. 54, 323 (2001).
+[19] P. D. Schwindt et al., Appl. Phys. Lett. 85, 6409 (2004).
+[20] S. J. Seltzer, P. J. Meares, and M. V. Romalis, Phys.
+Rev. A75, 051407(R) (2007).
+[21] W. Happer, Rev. Mod. Phys. 44, 169 (1972).
+[22] J. M. Geremia, J. K. Stockton, and H. Mabuchi, Phys.
+Rev. A73, 042112 (2006).
+[23] W. Happer and B. S. Mathur, Phys. Rev. 163, 12 (1967).
+[24] B. S. Mathur, H. Y. Tang, and W. Happer, Phys. Rev.
+A2, 648 (1970).
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+114905 (2009).
\ No newline at end of file
diff --git a/1001.0346.txt b/1001.0346.txt
new file mode 100644
index 0000000000000000000000000000000000000000..b7fc6fd400aaa2e58ab482acb7a99fa70a6778a1
--- /dev/null
+++ b/1001.0346.txt
@@ -0,0 +1,904 @@
+arXiv:1001.0346v1  [cs.IT]  3 Jan 20101
+Protocol Design and Stability/Delay Analysis of
+Half-Duplex Buffered Cognitive Relay Systems
+Yan Chen, Vincent K. N. Lau, Shunqing Zhang, and Peiliang Qiu
+Abstract—In this paper, we quantify the benefits of employing
+relay station in large-coverage cognitive radio systems wh ich
+opportunistically access the licensed spectrum of some sma ll-
+coverage primary systems scattered inside. Through analyt ical
+study, we show that even a simple decode-and-forward (SDF)
+relay, which can hold only one packet, offers significant path-
+loss gain in terms of the spatial transmission opportunities
+and link reliability. However, such scheme fails to capture the
+spatial-temporal burstiness of the primary activities, th at is,
+when either the source-relay (SR) link or relay-destinatio n (RD)
+link is blocked by the primary activities, the cognitive spe ctrum
+access has to stop. To overcome this obstacle, we further pro pose
+buffered decode-and-forward (BDF) protocol. By exploitin g the
+infinitely long buffer at the relay, the blockage time on eith er SR
+or RD link is saved for cognitive spectrum access. The buffer gain
+is shown analytically to improve the stability region and av erage
+end-to-end delay performance of the cognitive relay system .
+I. INTRODUCTION
+The core idea behind the cognitive transmission is to let
+the secondary users (SUs) exploit the under-utilized spec-
+trum holes left by the primary communication systems [1]–
+[3], either in temporal, frequency or spatial domain, witho ut
+interferingthe regulartransmissionsofthe primaryusers (PU).
+One key issue of the cognitive radio (CR) system is on the
+efficiency of spectrum sharing with the PU system. Direct
+transmission, which demands large transmit power, ends up
+with small opportunity of access and hence low spectrum
+sharing efficiency. As such, CR combined with relay station
+(RS), referred to as cognitive relay system (CRS), appears as
+an attractive solution to boost the spectrum sharing efficie ncy.
+The majority of the existing works on CRS focused on the
+physical layer aspects of the problem [4]–[6]. For example, a
+distributed algorithm for channel access and power control
+was proposed for cognitive multi-hop relays in [6], and a
+channelselection policy for multi-hopcognitivemesh netw ork
+was considered by [5]. When delay-sensitive applications are
+considered, other performance measures such as the stability
+regionand theaverage end-to-end packet delay become criti-
+cal. In [7], the authors analyzed the delay of a cognitive rel ay
+assisted multi-access network, however, they did not consi der
+Manuscript received 1 Dec 2008; revised 7 May 2009 and 3 Sep 20 09;
+accepted 11 Nov 2009. The associate editor coordinating the review of this
+paper and approving it for publication was Prof. Ashutosh Sa bharwal.
+Yan Chen and Peiliang Qiu (e-mail: qiupl418,qiupl@zju.edu.cn )
+are with Institute of Information and Communication Engine ering, Zhe-
+jiang University, Hangzhou, China. Dr. Yan Chen is now worki ng in
+Huawei Technologies, Shanghai, China. V. K. N. Lau and S. Zha ng (e-
+mail:eeknlau,eezsq@ee.ust.hk )is with Department of Electronic and
+Computer Engineering, Hong Kong University of Science and T echnology
+(HKUST), Clear Water Bay, Kowloon, Hong Kong.
+This paper is funded by RGC 615407.
+Fig. 1. An example of large-coverage CRS opportunistically access the
+licensed spectrum of some randomly distributed small-cove rage PU systems.
+the impact of PU activities and dynamic spectrum-sharing.
+Moreover,intheexistingworks,thecoverageofthePUsyste m
+is assumed to be much larger than the SU’s, so the spatial
+burstiness of the primary traffic and its impact on the CRS
+have not been fully investigated.
+In this paper, we try to shed a light on the protocol design
+of CRS that addresses the above issues. We are interested
+in the scenario where the CRS coverage is much larger than
+the PU coverage and try to design compatible cognitive relay
+protocols that could effectively deal with the uncertainty of
+PU locations and the spatial burstiness of PU activities. In
+addition, we study the stability region and the average end- to-
+enddelayoftheCRS underdifferentrelayingprotocols,whi ch
+are critical for delay-sensitive applications. We shall sh ow
+that the introduction of a cognitive relay provides two leve ls
+of potential gains. In particular, a conventional decode-a nd-
+forward(DF)relayunderasimpleDF(SDF)protocolprovides
+increased spatial transmission opportunitiesand enhance d link
+reliability, which is referred to as path-loss gain . On top of
+it, by enabling buffering at the relay, the blockage time on
+either the source-relay or relay-destination link can be sa ved
+to further increase spectrum access opportunities and redu ce
+the end-to-end delay. This is referred to as buffer gain and
+is shown via the analysis of our proposed buffered DF (BDF)
+protocol.Wederivetheclosed-formexpressionsofthestab ility
+region and the average end-to-end delay for each protocol an d
+quantify the two types of gains based on the derived results.
+II. SYSTEMMODEL
+WeconsiderascenariowherethecoverageoftheSUsystem
+is much larger than that of the PU systems. One example of
+such CR network is the WRAN system covering a suburb
+college town or rural areas, whose cell radius ranges from 2-
+10km or even larger. While the PU systems inside are Part74
+devices (wireless microphone), whose transmission ranges are2
+Fig. 2. Illustrative diagram of the CRS under three protocol s.
+about100-200m. So the transmission between a pair of SU
+nodes may affect multiple PU systems simultaneously.
+A. Assumptions for SU Transmission
+We consider a CRS with a SU transmitter (SU-Tx), a SU
+receiver (SU-Rx), and a half-duplex cognitive RS (SU-RS),
+as shown in Fig. 2. The links between SU-Tx and SU-Rx,
+SU-Tx and SU-RS, SU-RS and SU-Rx are referred to as
+SD, SR and RD links, respectively. In order to reduce the
+interference region caused by SU transmission, we assume
+that antenna arrays are equipped at both SU-Tx and SU-RS
+for beamforming1with beamwidth θand transmit antenna
+gainGt. Since SU-RS works in half-duplex mode, it uses the
+antenna array to obtain receiving antenna gain Gr. At SU-
+Rx, however, only one omnidirectional antenna is available .
+Further assume the transmission time of the CRS is slotted.
+In any slot, using power Pijto transmit signal X(with unit
+signal energy) on link ij, the received signal at jis
+Yj=/braceleftigg
+a(Pij)hij/radicalbig
+PijLijGtX+Ij+Zj, i=S,R,j=D
+a(Pij)hij/radicalbig
+PijLijGtGrX+Ij+Zj, i=S,j=R
+respectively, where a(Pij)is an indicatior variable and is a
+functionof Pij. It indicates whetherthe transmission from ito
+jusing power Pijis blocked by any PU, a(Pij) = 0indicates
+the transmission is blocked and a(Pij) = 1otherwise. hij
+stands for the channel fading coefficient, which is assumed
+to be flat Rayleigh so that the power gain on the link ij, i.e.
+Hij=|hij|2, is exponentially distributed with parameter 1.
+Hijis assumed to be quasi-static within a slot but identically
+and independently distributed (i.i.d) between different s lots.
+Lij=κ0·D−α
+ijis the large-scale path-loss between iandj,
+whereDijis the distance between iandj,κ0andαare path-
+loss coefficient and exponent, respectively. Ijstands for the
+sum signals received from all neighboring active PUs at node
+j. Since the PU’s coverage is much smaller and we assume
+SU-Rx is not inside the coverage of any active PU, then we
+can treat Ijas white noise with power E[Ij] =σ2
+I,∀j.Zj
+1Beamforming increases the ave. Rx SNR at the SU-Rx but the ins tanta-
+neous Rx SNR still follows Rayleigh fading due to the local sc attering cluster.is the white Gaussian noise at receiver j, i.e.Zj∼ N(0,σ2
+j)
+and we assume σ2
+j=σ2,∀j. Further define the instantaneous
+received signal-to-interference-and-noise ratio (SINR) on link
+ijasγij=a(Pij)·C0·Pij
+Dα
+ij·Hij, j=DandγSR=
+a(PSR)·C0Gr·PSR
+Dα
+SR·HSR, respectively, where C0=κ0Gt
+σ2+σ2
+Iis a constant independent of the transmit power. Moreover,w e
+assumethemaximumtransmitpowerconstraintforSU-Txand
+SU-RS are both Pmaxrespectively.
+B. Assumptions for PU Distribution and Activities
+We assume the PUs are uniformly(randomly)distributedon
+the two-dimensional plane with density ρ. In any given slot,
+each PU can be either active (ON) (with probability π1) or
+inactive(OFF) (withprobability π0= 1−π1). By the indicator
+variablea(Pij), we have related the impact of PU activities to
+received SINRs on link ij. When transmitting with power Pij,
+the average interference-to-noiseratio (INR) received by a PU
+DSPdistance away is ¯γSP(Pij,DSP) =C0·Pij
+σ2
+PDα
+SP. When
+the INR is higher than threshold γth, the PU transmission
+would be interfered. Setting ¯γSP(Pij,D∗
+SP) =γth, we can
+find the radius of the maximum interference region as D∗
+SP=/parenleftig
+C0
+σ2
+Pγth·Pij/parenrightig1/α
+. Since the directional beam with beamwidth
+θrad can be approximated by a sector with angleθ
+2π, the
+area of the interference region can be approximated by the
+area of that sector with radius D∗
+SP, which is ASP(Pij) =
+πD∗
+SP2·θ
+2π. Accordingtothe uniformdistributionassumption,
+the average number of PUs in the interference region can thus
+be calculated as N(Pij) =ρ·ASP(Pij). The SU transmission
+on linkijis “blocked” if any of the N(Pij)PUs is active and
+a(Pij) = 0. RecallSkis the activity state of the k-th PU in
+the region and the probability that the SU transmission with
+powerPijwould not be blocked is
+Pr{a(Pij) = 1}=π0N(Pij)= exp/braceleftig
+−C1(ρ,π0)·P2/α
+ij/bracerightig
+whereC1(ρ,π0) =ρθ
+2/parenleftig
+C0
+σ2
+Pγth/parenrightig2/α
+ln1
+π0>0is a constant
+independent of the transmit power Pijbut directly related to
+the PU distribution density ρand activity intensity π0.
+C. Probability of Successful Transmission over a Link
+We assume the data of the SU system is encapsulated into
+small packets with Mbits each. For each time slot, at most
+one packet can be transmitted which requireschannel capaci ty
+larger than M. Using Shannon formula, the channel capacity
+between node iandjcan be expressed as Φij=Blog2(1+
+γij), whereBis the bandwidth of the channel and γijis
+the SINR at receiver j. Thus, the probability of successful
+transmission of a packet over link ij, denoted as psucc
+ij, is
+psucc
+ij(Pij) = Pr{Blog2(1+γij)> M}
+= Pr/braceleftigg
+Hij>(2M/B−1)Dα
+ij
+C0Pij/bracerightigg
+·Pr{a(Pij) = 1}
+(1)= exp/braceleftig
+−C1(ρ,π0)·P2/α
+ij−C2(M/B,D ij)·P−1
+ij/bracerightig
+,(1)3
+fori=S,R,j=D, whereC2(M/B,D ij) =(2M/B−1)
+C0·
+Dα
+ij>0is a constant independent of Pijbut directly related
+to the ratio of packet size over the channel bandwidth M/B
+and the distance between the two ends of the link Dij. Step
+(1)is from equation (1). For the case j=R,
+psucc
+SR(PSR) = exp/braceleftigg
+−C1(ρ,π0)
+P−2/α
+SR−C2(M
+B,DSR)
+GrPSR/bracerightigg
+.(2)
+Property 1: For a given set of dij,R,ρ, andπ0, there
+exists a unique transmit power Popt
+ij>0that maximize
+the probability of successful transmission over link ij, i.e.
+∃Popt
+ij, s.t. Popt
+ij= argmax Pijpsucc
+ij(Pij).. The optimal
+transmit power for SD, SR and RD links (in the interference
+limited case, i.e. Popt
+ij< Pmax) are (i=S,R)
+Popt
+iD=/parenleftigg
+αC2(M
+B,DiD)
+2C1(ρ,π0)/parenrightiggα
+α+2
+Popt
+SR=/parenleftigg
+αC2(M
+B,DSR)
+2GrC1(ρ,π0)/parenrightiggα
+α+2
+(3)
+respectively, and the maximized probabilities of successf ul
+transmission on SD, RD and SR links are ( i=S,R)
+psucc,opt
+iD= exp/braceleftbigg
+−C3Cα
+α+2
+1(ρ,π0)C2
+α+2
+2(M
+B,DiD)/bracerightbigg
+(4)
+psucc,opt
+SR= exp/braceleftigg
+−C3
+G2
+α+2rCα
+α+2
+1(ρ,π0)C2
+α+2
+2(M
+B,DSR)/bracerightigg
+(5)
+whereC3=/parenleftbig
+1+α
+2/parenrightbig/parenleftbigα
+2/parenrightbig−α
+α+2is a constant related to α.
+Remark 1: For fixed transmit power Pij, the impact of PU
+activity on the SU transmission is like a good/bad fading
+process, with states a(Pij) = 1(good) and a(Pij) = 0
+(bad).However,it isdifferentfromatraditionalchannelf ading
+process ( ρ= 0,π0= 1) when the transmit power can be
+adjusted. Specifically, to combat a traditional channel fad ing,
+increasing the transmit power always helps to increase psucc
+ij,
+while with random PU distribution and activities, psucc
+ij(Pij)
+is not a monotonicincreasing functionof Pij. As shown in the
+Property 1, there exist a unique Popt
+ijthat maximize psucc
+ij(Pt)
+for a specific link ij.
+III. PROTOCOL DESCRIPTIONS
+In each protocol, an infinitely long buffer is assumed at
+SU-Tx and the first-in-first-out rule is applied. Recall that
+the transmission time is slotted and the packet transmissio n
+starts at the beginning of a slot. In each slot, only one packe t
+can be transmitted. When transmitting, SU-Tx and SU-RS are
+supposed to use the optimal transmit power to maximize the
+probability of successful transmission over a link.
+Baseline (BL) Protocol: In the BL protocol, SU-Tx trans-
+mits a packet directlyto SU-Rx if the SD link is not blocked.
+The packet is removedfrom SU-Tx’squeuewhen an acknowl-
+edge (ACK) message is received from SU-Rx2.
+2A low rate control channel is assumed for control signal exch ange, e.g.
+ACK/NACK message from SU-Rx or SU-RS. We also assume all the n odes
+can synchronize their behaviors via this channel. In additi on, since the use of
+microphones is usually restricted within a building or a squ are, their activity
+states can be sensed by sensors placed in the buildings or on t he squares. The
+sensing results are assumed known at both SU-Tx and SU-RS.Simple Decode-and-Forward (SDF) Protocol: In the SDF
+protocol, SU-RS can successfully receive a packet from SU-
+Txis the SRlink isnot blocked.It decodesthe receivedpacke t
+and forwards to the SU-Rx as a conventional DF relay does.
+The SU-RS can not receive new packet from SU-Tx before
+the currently holding packet has been successfully forward ed
+to the SU-Rx. A packet can be removed from SU-Tx’s queue
+if an ACK is received from SU-Rx.
+Buffered Decode-and-Forward (BDF) Protocol: In BDF
+protocol, infinitely long buffer is also assumed at SU-RS. SU -
+Tx transmits a packet to SU-RS only when SU-RS is not
+transmitting on the RD link and the SR link is not blocked.
+The packet is removed from SU-Tx’s queue if an ACK is
+receivedfrom SU-RS. Higher transmission priorityis assum ed
+at SU-RS such that it transmits to SU-Rx whenever the RD
+link is not blocked. Moreover, we assume SU-RS has the
+channel state information (CSI) of the RD link so that the
+channel outage time of the RD link can be further saved for
+SR link transmission. The packet is removed from SU-RS’s
+relay queue when an ACK is received from SU-Rx.
+IV. ANALYSIS OF STABILITY /DELAYPERFORMANCE
+A. Queue Dynamics and Stability/Delay Definitions
+Queue Dynamics: We adopt a similar model used in
+[7], [8] to depict the buffer dynamics in a slotted system.
+LetQS(t),t=mτ,m= 0,1,...denote the queue length
+at the SU-Tx observed at the end of slot t. It evolves as
+QS(t) = (QS(t−1)−YS(t))++XS(t),∀t. In the equation,
+XS(t)representsthenumberofpacketarrivalsinslot t(cannot
+be transmitted in the same slot), which is assumed to be a
+Bernoulli process with mean E[XS(t)] =λ, i.e.XS(t)only
+takes value 0or1with probability 1−λandλ, respectively.
+YS(t)denotes the number of packets that depart from SU-Tx
+in slott. According to the protocols, YS(t)also takes value
+from{0,1}, depending on the states of PU activities, channel
+fading,and the interactionwith the queue dynamicsat SU-RS .
+SinceQS(t+1)only depends on QS(t)andXS(t),YS(t)is
+either0or1,{QS}is a discrete time Markov Chain and
+its state transitions only happen between neighboring stat es,
+i.e.{QS}is a discrete time birth-death process (DTBDP).
+Forn≥0, letλn
+Sbe the state transition probabilities from
+QS(t) =ntoQS(t+1) =n+1andµn
+Sthe state transition
+probabilities from QS(t) =ntoQS(t+ 1) = n−1for
+n≥1. Similarly, the evolution for the queue at SU-RS can be
+defined as QR(t) = (QR(t−1)−YR(t))++XR(t),∀t. Note
+the arrival process XR(t)depends on the departure process
+of the source queue (i.e. YS(t)) and the half-duplex constraint
+makes the interaction between QS(t)andQR(t)complicated.
+{QR}is also a DTBDP, whose state transition probabilities
+can be definedin a similar manneras λn
+Randµn
+R, respectively.
+Stability of a CRS: A queue Qiisstableif and only if
+limt→∞Pr{Qi(t) = 0}>0[8], fori=S,R. In the BL and
+SDF protocols, when the queue at SU-Tx is stable, the whole
+system is stable, but in the BDF protocol, system stability
+requires both the queue at SU-Tx and the queue at SU-RS
+to be stable simultaneously. Denote by λ∗the stability region
+of the CRS in terms of the maximum exogenous arrival rate,
+which has unit “packet/slot”.4
+End-to-End Delay of a CRS: Theend-to-end delay of
+a packet in the CRS is the time from a packet arrives at the
+queueofSU-TxtillthepacketreachesSU-Rx.Little’stheor em
+[9] enables the study from the angle of average queue length.
+Given the exogenous arrival rate λ, the average end-to-end
+delay for the three protocols can be defined as (unit: slots)
+WBL=1
+λlim
+T→∞/summationtextT
+t=1QS(t)
+T=E[QS]
+λ, (6)
+WSDF/BDF =1
+λlim
+T→∞/summationtextT
+t=1QS(t)
+T+1
+λlim
+T→∞/summationtextT
+t=1QR(t)
+T
+=E[QS]+E[QR]
+λ(7)
+whereEis taken w.r.t the steady distribution of queue length.
+B. Stability/Delay Analysis for the BL protocol
+IntheBLprotocol,only {QS}isinvolved.Thequeuelength
+increasesbyoneifanewpacketarrivesandnopackethasbeen
+successfully transmitted, and decrease by one if an existin g
+packet is successfully transmitted and no new packet has
+arrived,whichimplies λn=0
+S=λandλn≥1
+S=λ·(1−psucc,opt
+SD),
+andµn≥1
+S= (1−λ)·psucc,opt
+SD. We derivethestable distribution
+of{QS}by solving the detailed balance equation related to
+the DTBDP. Define qn
+SasPr/bracketleftbig
+QS=n/bracketrightbig
+, we have
+q0
+S=/parenleftigg
+1+∞/summationdisplay
+n=1n−1/productdisplay
+k=0λk
+S
+µk+1
+S/parenrightigg−1
+= 1−λ
+psucc,opt
+SD, (8)
+qn
+S=q0
+Sn−1/productdisplay
+k=0λk
+S
+µk+1
+S=q0
+S
+1−psucc,opt
+SD/parenleftigg
+λ(1−psucc,opt
+SD)
+(1−λ)·psucc,opt
+SD/parenrightiggn
+.(9)
+System stability requires q0
+S>0, which implies λ∗=
+psucc,opt
+SD. Furthermore, the end-to-end delay can be calculated
+as according to (6). Theorem 1 summarized the above results.
+Theorem 1: The stability regionand the average end-to-end
+delay under the BL protocol are
+λ∗
+BL=psucc,opt
+SD,WBL=/parenleftbig
+1−λ/parenrightbig
+//parenleftbig
+psucc,opt
+SD−λ/parenrightbig
+.(10)
+C. Stability/Delay Analysis for the SDF/BDF protocol
+In the following, we shall first analyze a general case of
+SU-RS having buffer length L.
+General Case: Similar as in the BL protocol, the decrease
+inQSimplies a packet has been successfully transmitted to
+SU-RS and no new packet arrives. However, under the SDF
+or BDF protocol, the departure process of QS(t)is related
+to the dynamics of QR(t). The departure of a packet implies
+that the following three events must be true simultaneously :
+1) The queue length at SU-RS is not L, denoted as E{QR/ne}ationslash=
+L}; 2) SU-RS is not transmitting on the RD link, denoted as
+E{RD link idle }; 3) The SR link is able to support the packettransmission, denoted as E{SR link successful }. Therefore3,
+µn≥1
+S= (1−λ)Pr/parenleftbig
+E{SR link successful }/parenrightbig
+×Pr/parenleftbig
+E{RD link idle }/vextendsingle/vextendsingleE{QR/ne}ationslash=L}/parenrightbig
+Pr/parenleftbig
+E{QR/ne}ationslash=L}/parenrightbig
+= (1−λ)/bracketleftbig
+q0
+R+(1−q0
+R−qL
+R)(1−psucc
+RD)/bracketrightbig
+psucc,opt
+SR
+λn≥1
+S=λ/braceleftig
+1−/bracketleftbig
+q0
+R+(1−q0
+R−qL
+R)(1−psucc
+RD)/bracketrightbig
+psucc,opt
+SR/bracerightig
+andλn=0
+S=λ. Using the same method as in obtaining (8),
+the steady state probability of QS= 0is
+q0
+S= 1−λ/bracketleftbig
+q0
+R+(1−q0
+R−qL
+R)(1−psucc
+RD)/bracketrightbig
+psucc,opt
+SR(11)
+which implies the stability region is
+λ∗=/bracketleftbig
+q0
+R+(1−q0
+R−qL
+R)(1−psucc
+RD)/bracketrightbig
+psucc,opt
+SR.(12)
+For{QR}, due to the half-duplex constraint, packet arrival
+and packet departure would not happen in the same slot. As
+the increase of QRhas lower priority than its decrease. So the
+probability that QRdeceases by one equals the probability
+of successful transmission on the RD link. Thus, the state
+transition probabilities of {QR}are
+λn=0
+R= (1−q0
+S)psucc,opt
+SR,
+λ1≤n≤L−1
+R = (1−q0
+S)psucc,opt
+SR(1−psucc,opt
+RD),
+µ1≤n≤L
+R=psucc,opt
+RD.
+So the steady state probability of QR= 0is
+q0
+R=
+1+/summationtextL
+k=1/parenleftig(1−q0
+S)psucc,opt
+SR(1−psucc,opt
+RD)
+psucc,opt
+RD/parenrightigk
+1−psucc,opt
+RD
+−1
+.(13)
+Thesteadystateprobabilityof QR=kcanbefurtherobtained
+via similar calculations used in (9) as
+qk
+R=q0
+R
+1−psucc,opt
+RD/parenleftbig
+(1−q0
+S)psucc,opt
+SR(1−psucc,opt
+RD)/slashbig
+psucc,opt
+RD/parenrightbigk
+SinceqL
+Ris function of q0
+R, solving the combined equations of
+(11) and (13), we can get the solutions for q0
+Sandq0
+R, which
+further lead to the results of the stability region λ∗and the
+average end-to-end delay4. We now apply the general results
+derived above to the SDF and BDF protocols, respectively.
+Special Case I - SDF Protocol: In this case, L= 1and
+{QR}reduces to a two-state Markov chain with qL
+R= 1−q0
+R.
+Therefore, (11) and (13) reduce to
+q0
+S= 1−λ/q0
+Rpsucc,opt
+SR,
+q0
+R=psucc,opt
+RD/slashbig/parenleftbig
+(1−q0
+S)psucc,opt
+SR+psucc,opt
+RD/parenrightbig
+.
+3Here we approximate Pr{QR(t) =m|QS(t) =n}asPr{QR(t) =
+m}, which is asymptotically tight for large Lcase (e.g. the BDF case) but
+is an upper bound on µn
+Sin the SDF case. As a result, the final expression
+of the stability region (end-to-end delay) of the SDF case is an upper bound
+(lower bound), which is verified by the simulation in Section V.
+4There is no closed-form expressions for q0
+Randq0
+Sin the general case,
+which involves finding the roots of an polynomial to a power of L. However,
+for the two special cases L= 1andL=∞, the polynomials reduce to
+quadratic forms and closed-form solutions are ready to get.5
+Solving the two equations, the steady state probabilities a re
+q0
+S= 1−λ//parenleftbig
+psucc,opt
+SR−psucc,opt
+SRλ/slashbig
+psucc,opt
+RD/parenrightbig
+,(14)
+q0
+R= 1−λ/psucc,opt
+RD, (15)
+respectively. The stability region can be obtained from the
+requirement q0
+S>0and the average end-to-end delay can be
+calculatedaccordingto (7). Theorem2summarizesthe resul ts.
+Theorem 2: The stability regionand the average end-to-end
+delay for the SDF protocol are
+λ∗
+SDF=psucc,opt
+RDpsucc,opt
+SR
+psucc,opt
+RD+psucc,opt
+SR(16)
+WSDF=1−λ
+psucc,opt
+SR−psucc,opt
+SR
+psucc,opt
+RDλ−λ+1
+psucc,opt
+RD.(17)
+Special Case II - BDF Protocol: In this case, L=∞and
+then for a stable system qL
+R= 0. Correspondingly, we have
+q0
+S= 1−λ/bracketleftbig
+1−(1−q0
+R)psucc,opt
+RD/bracketrightbig
+psucc,opt
+SR,
+q0
+R= 1−(1−q0
+S)psucc,opt
+SR
+psucc,opt
+RD/bracketleftbig
+1+(1−q0
+S)psucc,opt
+SR/bracketrightbig.
+Solving the two equations, we get
+q0
+S= 1−λ/slashbig/bracketleftbig
+(1−λ)psucc,opt
+SR/bracketrightbig
+, q0
+R= 1−λ/slashbig
+psucc,opt
+RD.(18)
+respectively. The stability region can be obtained from sat is-
+fying the requirements of both q0
+S>0andq0
+R>0, and the
+average end-to-end delay can be calculated according to (7) .
+Theorem 3 summarizes the above results.
+Theorem 3: The stability regionand the average end-to-end
+delay for the BDF protocol are
+λ∗
+BDF= min/braceleftigg
+psucc,opt
+SR
+1+psucc,opt
+SR,psucc,opt
+RD/bracerightigg
+,(19)
+WBDF=1−λ
+(1−λ)psucc,opt
+SR−λ+1−λ
+psucc,opt
+RD−λ.(20)
+V. ANALYSIS OF PATH-LOSSGAIN AND BUFFERGAIN
+A. Analysis of Path-loss Gain
+Path-loss gain is referred to as the gain that a CRS obtains
+from the reduction in path-loss over the BL system. We shall
+use the metric called “throughput per Watt” to illustrate th is
+gain, which is defined as the average throughput (in bits/slo t)
+of delivering a single packet from SU-Tx to SU-Rx divided
+by the transmit power needed. In this case, the SDF and BDF
+protocols behave alike, so we only compare the SDF protocol
+with the BL protocol.
+To transmit a single packet with Rbits on SD, SR,
+and RD links take 1/psucc,opt
+SD,1/psucc,opt
+SRand1/psucc,opt
+RD
+time slots, respectively, so the average throughput for the
+BL and SDF protocols are TBL=R·psucc,opt
+SDand
+TSDF=R
+1/psucc,opt
+SR+1/psucc,opt
+RD=R·psucc,opt
+SRpsucc,opt
+RD
+psucc,opt
+SR+psucc,opt
+RD, respec-
+tively. While the transmit power needed for the two proto-
+cols arePopt
+SDandPopt
+SR+Popt
+RD, respectively. Further define
+∆T=TSDF/TBL−1and∆P=/parenleftbig
+Popt
+SR+Popt
+RD/parenrightbig
+/Popt
+SD−1as the throughput gain and power gain of the SDF protocol
+over the BL protocol, then the path-loss gain in throughput
+per Watt is thus given by ∆TP=1+∆T
+1+∆P−1. Fig. 3 depicts
+the path-loss gain of a CRS over the BL system as a function
+of SU-RS’s location. Here we assume SU-Tx and SU-Rx are
+fixed at(0,0)and(2,0)km while SU-RS is at (DSR,0). It
+can be shown that setting Popt
+SR(DSR) =Popt
+RD(DSD−DSR)
+achieves the peak of the path-loss gain, which implies D∗
+SR=
+(1−C4)DSD,D∗
+RD=C4DSD, whereC4=G−1/α
+r
+1+G−1/α
+r.
+Property 2: The path-loss gain achieves its peak when SU-
+RS is located at (D∗
+SR,0), i.e.,
+∆T∗
+P=1+∆T
+1+∆P−1 =1
+2/parenleftbig
+psucc,opt
+SD/parenrightbigC42α
+α+2−1
+Gr−α
+α+2(1−C4)α2
+α+2+C4α2
+α+2−1.
+Given such location of SU-RS, the condition for having
+positive path-loss gain is given by
+psucc,opt
+SD</parenleftig
+2Gr−α
+α+2(1−C4)α2
+α+2+2C4α2
+α+2/parenrightig1
+C42α
+α+2−1.
+B. Analysis of Buffer Gain
+The buffer gain is referred to the gain that comes from
+enabling the buffering capability at SU-RS. We shall use
+the results derived in section IV to illustrate this gain. In
+particular, we shall show the relative increase in the stabi lity
+regionand the reductionin the averageend-to-enddelay und er
+the BDF protocol (infinite buffer) compared with the SDF
+protocol(singlepacketstorage).Thebuffergaininthesta bility
+region and in the average end-to-end delay are defined as
+∆λ∗=λ∗
+BDF/λ∗
+SDF−1and∆¯W= 1−¯WBDF/¯WSDF,
+respectively.
+Property 3: Given SU-RS located at (D∗
+SR,0), and let
+ζ∆=psucc,opt
+RD/vextendsingle/vextendsingle
+DRD=D∗
+RD=/parenleftbig
+psucc,opt
+SD/parenrightbigC42α
+α+2, the buffer gain
+in stability region and in average end-to-end delay are
+∆λ∗=1−ζ
+1+ζ>0,∆¯W=λ2
+1−λ(1−ζ)
+(ζ−λ)(ζ−λ
+1−λ)>0.(21)
+The value of ζin (21) depends on the system parameters
+such asαandGras well as PU distribution density ρand PU
+activity intensity π0. Fig. 4 depicts how the buffer gain given
+by equation (21) varies with PU activity intensity π0and PU
+distributiondensity.Fromthe figurewe see that thebufferg ain
+increases when the cognitive environmentis more unfavorab le
+(e.g. larger intensity of PU activity and higher density of P U
+distribution), which verifies that with the buffering capab ility
+enabled at the cognitive relay, a CRS can better adjust to
+the environment and more efficiently deal with the spatial
+burstiness of the PU activities.
+C. Numerical Discussions
+In this subsection, we shall discuss the end-to-end delay
+performanceunderthethreeprotocolsandverifyouranalyt ical
+results via Monte Carlo simulations. The packet size M=
+16kbits and the channel bandwidth is B= 16kHz. Other
+parameters are set as follows: κ0= 1,α= 4,σ2=σ2
+I= 1,
+Gt=Gr= 2,θ=π/3rad andγth= 1. Fig. 5 depicts the the6
+00.511.52−0.100.10.20.30.40.50.60.70.8
+Distance of SU−RSThroughput gain and power gain
+  
+throughput gain
+power gain
+00.511.520123456
+Distance of SU−RSPath−loss gain
+  
+Gr = 2
+Gr = 1
+Gr = 1Gr = 2
+Fig. 3. Path-loss gain of CRS over BL system with different di stances of
+SU-RS.DSD= 2km,ρ= 2per km2,π0= 0.8.
+0.50.55 0.60.65 0.70.75 0.80.85 0.900.10.20.30.40.50.60.70.80.91
+PU activity intensity π0Buffer Gain
+  
+Buffer gain in delay
+λ = 0.2
+Buffer gain in
+stable region
+Buffer gain in delay
+λ = 0.1ρ = 2ρ = 1.5
+ρ = 1.5ρ = 2ρ = 2
+ρ = 1.5
+Fig. 4. Buffer gain in both stability region and average end- to-end delay of
+CRS under the BDF protocol over that under the SDF protocol wi th different
+PU activity intensity π0.ρ= 2per km2,DSD= 2km, and SU-RS is fixed
+at(D∗
+SR,0) = ((1 −C4)DSD,0) = (1.086,0)km.
+performance of the average end-to-end delay with increasin g
+PU distribution density, while Fig. 6 shows the trend that th e
+average end-to-end delay varies with decreasing PU activit y
+intensity. We can see that the simulation results matches we ll
+withouranalyticalresults.Moreover,theBDF protocolalw ays
+achieveslargerstability regionand smaller delaythan the SDF
+protocol, as claimed in Property 3.
+VI. CONCLUSION
+We have shown two levels of gains provided by a CRS, i.e.
+path-lossgain andbuffergain .Thepath-lossgainsubstantially
+increases the spectrum access opportunities by reducing th e
+transmit power while the buffering capability at the relay
+further saves the blockage time of either the SR or RD
+link and reduces the end-to-end delay to a larger extent. We
+emphasize the importance of exploiting relay buffers to dea l
+withtheuncertaintyofPUactivities,whichisanintrinsic issue
+associated with large-coverage cognitive systems.1 1.5 2 2.5 3 3.5 4051015202530
+PU distribution density ρAverage end−to−end delay
+  
+BL simu.
+SDF simu.
+BDF simu.
+analysis results
+λ=0.2
+analysis results, λ=0.1
+Fig. 5. The average end-to-end delay under the three protoco ls with different
+PU distribution density ρper km2.DSD= 2km and SU-RS is fixed at
+(D∗
+SR,0) = ((1 −C4)DSD,0) = (1.086,0)km,π0= 0.8.
+0.50.55 0.60.65 0.70.75 0.80.85 0.9051015202530
+PU activity intensity π0Average end−to−end delay
+  
+BL simu.
+SDF simu.
+BDF simu.analysis results
+λ = 0.2
+analysis results, λ = 0.1
+Fig. 6. The average end-to-end delay under the three protoco ls with different
+PU activity intensity π0.DSD= 2km and SU-RS is fixed at (D∗
+SR,0) =
+((1−C4)DSD,0) = (1.086,0)km,ρ= 2per km2.
+REFERENCES
+[1] J. Mitola and G. Q. Maguire, “Cognitive radio: making sof tware radios
+more personal,” IEEE Personal Commun. Mag. , vol. 6, no. 4, pp. 13 –
+18, Aug. 1999.
+[2] S. Haykin, “Cognitive radio: brain-empowered wireless communica-
+tions,”IEEE J. Sel. Areas Commun. , vol. 23, no. 2, pp. 201 – 220,
+Feb. 2005.
+[3] Q. Zhao, L. Tong, A. Swami, and Y. Chen, “Decentralized co gnitive
+mac for opportunistic spectrum access in ad hoc networks: A p omdp
+framework,” IEEE J. Sel. Areas Commun. , vol. 25, no. 3, pp. 589 –
+600, Apr. 2007.
+[4] K. Lee and A. Yener, “Outage performance of cognitive wir eless relay
+networks,” in Proc. IEEE GLOBECOM ’06 , Nov. 2006.
+[5] M. Sharma, A. Sahoo, and K. D. Nayak, “Channel selection u nder
+interference temperature model in multi-hop cognitive mes h networks,”
+inProc. IEEE DySPAN ’07 , Apr. 2007, pp. 133 –136.
+[6] Y. T. Hou, Y. Shi, and H. D. Sherali, “Spectrum sharing for multi-hop
+networking with cognitive radios,” IEEE J. Sel. Areas Commun. , vol. 26,
+no. 1, pp. 146 – 155, Jan. 2008.
+[7] A. K. Sadek, K. J. R. Liu, and A. Ephremides, “Cognitive mu ltiple
+access viacooperation: Protocol design and performance an alysis,”IEEE
+Trans. Inf. Theory , vol. 53, no. 10, pp. 3677 – 3696, Oct. 2007.
+[8] R. R. Rao and A. Ephremides, “On the stability of interact ing queues
+in a multiple-access system,” IEEE Trans. Inf. Theory , vol. 34, pp. 918
+– 930, Sept 1988.7
+[9] L. Kleinrock, Queueing Systems Volume I: Theory . John Wiley & Sons,
+1975.
+[10] [Online]. Available: http://arxiv.org/abs/0809.05 33
\ No newline at end of file
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+arXiv:1001.0347v1  [math.RT]  3 Jan 2010Decompositionofsplittinginvariantsinsplitrealgroups
+TashoKaletha
+Abstract
+To a maximal torus in a quasi-split semi-simple simply-conn ected group over
+a local fieldofcharacteristic 0, Langlands and Shelstadcon struct in[LS1] acoho-
+mologicalinvariantcalledthesplittinginvariant,which isanimportantcomponent
+of their endoscopic transfer factors. We study this invaria nt in the case of a split
+realgroupandproveadecomposition theoremwhichexpresse s thisinvariantfora
+general torus as a product of the corresponding invariants f or simple tori. We also
+show how this reduction formula allows for the comparison of splitting invariants
+between different toriinthe given real group.
+Inapplicationsofharmonicanalysisandrepresentationth eoryofreductivegroupsover
+local fields to questionsin numbertheory,a central role is p layed by the theoryof en-
+doscopy. This theory associates to a given connected reduct ive group Gover a local
+fieldFa collection of connected reductive groupsover F, often denoted by H, which
+have smaller dimension (except when H=G), but are usually not subgroups of G.
+Thegeometricsideofthetheoryisthenconcernedwithtrans ferringfunctionson G(F)
+to functions on H(F) in such a way that suitable linear combinations of their orb ital
+integralsare comparable,while the spectral side is concer nedwith transferring“pack-
+ets” ofrepresentationson H(F)to “packets”ofrepresentationson G(F)in sucha way
+that suitablelinear combinationsoftheir charactersare c omparable. In bothcases, the
+comparison involves certain normalizing factors, called g eometric or spectral transfer
+factors.
+Overthe real numbers,the theoryof endoscopywas developed in a seriesof profound
+papersby Diana Shelstad, in whichshe definesgeometricand s pectral transferfactors
+andprovesthatthesefactorsindeedgiveacomparisonoforb italintegralsandcharacter
+formulasbetween GandH. A verysubtle and complicatedfeatureof the transfer fac-
+torswas the need to assign a ±-sign to each maximal torusin Gin a coherentmanner,
+andShelstadwasabletoprovethatthisispossible. Aunifor mandexplicitdefinitionof
+geometrictransferfactorsforall local fieldswas givenin [ LS1]. An explicitconstruc-
+tionofspectraltransferfactorsovertherealnumberswasg ivenin[S2],whileoverthe
+p-adic numbers their existence is still conjectural. The str ucture of transfer factors is
+quitecomplex–boththegeometricandtherealspectralones areaproductofmultiple
+termsofgroup-theoreticorGalois-cohomologicalnature. Therearenumerouschoices
+involved in the construction of each individual term, but th e product is independent
+of most choices. One term that is common for both the geometri c and the real spec-
+tral transfer factors is called ∆I. It is regarded as the most subtle and is the one that
+makesexplicitthechoiceofcoherentcollectionofsignsin Shelstad’searlierwork. At
+its heart is a Galois-cohomological object, called the spli tting invariant. The splitting
+invariant is an element of H1(F,T) associated to any maximal torus Tof a quasi-split
+semi-simple simply-connected group G, whose construction occupies the first half of
+[LS1,Sec.2]. Itisdependsonthechoiceofasplitting( T0,B0,{Xα}α∈∆)ofGaswell as
+a-data{aβ}β∈R(T,G).
+This paper addresses the following question: If one has two t ori in a given real group
+whichoriginatefromthe same endoscopicgroup,how canone c omparetheir splitting
+AMS subject classification: Primary 11F70, 22E47; Secondar y 11S37, 11F72, 17B22
+1invariants? While there will in general be no direct relatio n between H1(F,T1) and
+H1(F,T2)fortwotori T1andT2ofG,ifboththosetorioriginatefrom Hthenthereare
+certain natural quotients of their cohomology groups which are comparable, and it is
+theimageofthesplittinginvariantinthosequotientsthat isrelevanttotheconstruction
+of∆I. An example of a situation where this problem arises is the st abilization of the
+topologicaltraceformulaofGoresky-MacPherson.Oneisle dtoconsidercharactersof
+virtual representationswhich occur as sums indexedoverto ri inGthat originate from
+thesame endoscopicgroup H, andeachsummandcarriesa ∆I-factorassociated to the
+correspondingtorus.
+To describe the results of this paper, let Gbe a split simply-connected real group and
+(T0,B0,{Xα}α∈∆)beafixedsplitting. Forasubset A⊂R,consistingofstronglyorthog-
+onal roots, let SAdenote the element of the Weyl group of T0given by the product of
+the reflectionsassociated to the elementsof A(the order in which the productis taken
+isirrelevant). We showthatassociatedto Athereisacanonicalmaximaltorus TAofG
+and a set of isomorphismsof real tori TSA
+0→TA, whereTSA
+0is the twist by SAofT0.
+Anymaximaltorusin GisG(R)-conjugatetooneofthe TA,soitisenoughtostudythe
+toriTA. We giveanexpressioninpurelyroot-theoretictermsfora c ertain1-cocyclein
+Z1(R,TSA
+0). This cocycle has the propertythat its image in Z1(R,TA) under any of the
+isomorphisms TSA→TAabove is the same, and the class in H1(R,TA) of that image
+is the splitting invariant of TA(associated to a specific choice of a-data). Moreover,
+we prove a reduction theorem which shows that this cocycle is a product over α∈A
+of the cocycles associated to the canonical tori T{α}, thereby reducing the study of the
+splittinginvariantof TAto thoseof thevarious T{α}. Thisproductdecompositiontakes
+place inside the group Z1(R,TSA
+0) – that is, we show that the elements of Z1(R,Tsα
+0)
+associated to the various T{α}withα∈Aalso lie in Z1(R,TSA
+0) and that their product
+is the element associated to TA. Finally we show that if A′⊂Aand the tori TA′and
+TAoriginate from the same endoscopic group, then the endoscop ic characters on the
+cohomology groups H1(R,TA′) andH1(R,TA) factor through certain explicitly given
+quotientsofthesegroups,andthequotientof H1(R,TA′)iscanonicallyembeddedinto
+that ofH1(R,TA). This, together with the reduction theorem, allows for a di rect com-
+parisonofthevaluesthattheendoscopiccharactersassoci atetothesplittinginvariants
+forTA′andTA.
+The paper is organized as follows: Section 1 contains a few ba sic facts and serves
+mainly to fix notation for the rest of the paper. Section 2 cont ains proofs of general
+facts about subsets of strongly orthogonal roots in reduced root systems, which are
+needed as a preparation for the reduction theorem mentioned above. The study of the
+splittinginvariantstakesplaceinsection3,wherefirstth esplittinginvariantforthetori
+T{α}is computed, and after that the results of section 2 are used t o reduce the case of
+TAto that of T{α}. While the statement of the reduction theorem appears natur al and
+clear, the proof contains some subtle points. First, one has to choose the Borel B0in
+thesplittingof Gwithcareaccordingto thestronglyorthogonalset A. Asremarkedin
+section 3, this choice does not a ffect the splitting invariant, but it significantly a ffects
+its computation. Moreover,the root system G2exhibitsa singular behavior among all
+reducedrootsystemsasfaraspairsofstrongly-orthogonal rootsareconcerned. Section
+4 contains explicit computations of the splitting invarian ts of the tori T{α}for all split
+almost-simpleclassicalgroups. Insection5weconstructt heaforementionedquotients
+ofthecohomologygroupsandtheembeddingbetweenthem.
+21 Notationand preliminaries
+Throughout this paper Gwill stand for a split semi-simple simply-connected group
+overRand (B0,T0,{Xα}) will be a splitting of G. We write R=R(T0,G) for the set
+of roots of T0inG, setα >0 ifα∈R(T0,B0), denote by∆the set of simple roots in
+R(T0,B0)andbyΩtheWeyl-groupof R,whichisidentifiedwith N(T0)/T0. Moreover
+we putΓ=Gal(C/R) and denote by σboth the non-trivial element in that group, as
+well as its action on T0. The notation g∈Gwill be shorthand for g∈G(C), and
+Int(g)h=ghg−1.
+1.1sl2-triples
+For anyα∈R(T0,B0) we have the coroot α∨:Gm→T0and its differentialdα∨:
+Ga→Lie(T0). We put Hα:=dα∨(1)∈Lie(T0). GivenXα∈Lie(G)αnon-zero, there
+existsa unique X−α∈Lie(G)−αso that[Hα,Xα,X−α]is ansl2-triple. Themap
+/parenleftigg1 0
+0−1/parenrightigg
+/ma√s⊔o→Hα,/parenleftigg0 1
+0 0/parenrightigg
+/ma√s⊔o→Xα,/parenleftigg0 0
+1 0/parenrightigg
+/ma√s⊔o→X−α
+givesa homomorphism sl2→Lie(G) which integratesto a homomorphismSL 2→G
+andonehas
+sl2✲Lie(G)
+SL2exp
+❄
+✲Gexp
+❄
+The image of/parenleftig
+a b
+c d/parenrightig
+∈SL2under this homomorphism will be called/parenleftig
+a b
+d c/parenrightig
+Xα.
+Noticethat/parenleftig
+t0
+0t−1/parenrightig
+Xα=α∨(t).
+Fact 1.1. Letα,β∈R be s.t.α+β/nelementR andα−β/nelementR. For any non-zero elements
+Xα∈Lie(G)αand Xβ∈Lie(G)β, the homomorphisms ϕXα,ϕXβ: SL2→G given by X α
+and Xβcommute.
+Proof:Sinceforanyfield k, SL2(k) isgeneratedbyitstwo subgroups
+/braceleftigg/parenleftigg1u
+0 1/parenrightigg
+|u∈k/bracerightigg /braceleftigg/parenleftigg1 0
+u1/parenrightigg
+|u∈k/bracerightigg
+it is enough to show that, for any u,v∈C, each of exp( uXα) and exp( uX−α) com-
+mutes with each of exp( vXβ) and exp( vX−β). This follows from [Spr, 10.1.4] and our
+assumptionon α,β. /square
+1.2 Chevalleybases
+Forα∈∆letnα=exp(Xα)exp(−X−α)exp(Xα)=/parenleftig
+0 1
+−1 0/parenrightig
+Xα. Givenµ∈Ωwe havethe
+liftn(µ)∈N(T0)givenby
+n(µ)=nα1···nαq
+3wheresα1···sαq=µisanyreducedexpression(by[Spr,11.2.9]thisliftisinde pendent
+ofthechoiceofreducedexpression). Notice n(µ)∈N(T0)(R) sinceT0is split. Put
+Xµ|α:=Int(n(µ))·Xα
+ThenXµ|α∈Lie(G)µαisa non-zeroelement.
+Lemma1.2. Ifα,α′∈∆andµ,µ′∈Ωares.t.µα=µ′α′thenwehavein Lie(G)µαthe
+equality
+Xµ′|α′=/productdisplay
+β>0
+(µ′)−1β<0
+µ−1β>0(−1)/a\}bracke⊔le{⊔β∨,µα/a\}bracke⊔ri}h⊔·Xµ|α
+Proof:By [Spr, 11.2.11]the relation( µ′)−1·µα=α′implies
+Xα′=Int/bracketleftig
+n/parenleftig/parenleftbigµ′/parenrightbig−1·µ/parenrightig/bracketrightig
+Xα
+Theclaimnowfollowsfrom[LS1, 2.1.A]andthe followingcom putation
+Xµ′|α′=Int/parenleftbign/parenleftbigµ′/parenrightbig/parenrightbigXα′
+=Int/bracketleftig
+n/parenleftbigµ′/parenrightbign/parenleftig/parenleftbigµ′/parenrightbig−1µ/parenrightig/bracketrightig
+Xα
+=Int/bracketleftig
+t/parenleftig
+µ′,(µ′)−1µ/parenrightig
+·n(µ)/bracketrightig
+Xα
+=Int/bracketleftig
+t/parenleftig
+µ′,(µ′)−1µ/parenrightig/bracketrightig
+Xµ|α
+=(µα)/parenleftig
+t(µ′,(µ′)−1µ)/parenrightig
+·Xµ|α
+/square
+Remark: Weseethatwhilethe”absolutevalue”of Xµ|αonlydependsontheroot µ·α,
+its”sign”doesdependonboth µandα.
+Definition1.3. Forγ∈R,µ,µ′∈Ωput
+ǫ(µ′,γ,µ) :=/productdisplay
+β>0
+(µ′)−1β<0
+µ−1β>0(−1)/a\}bracke⊔le{⊔β∨,γ/a\}bracke⊔ri}h⊔
+Remark: Withthisdefinitionwecanreformulatethe abovelemmaasfol lows
+Corollary1.4. Ifγ∈R andµ,µ′∈Ωares.t.µ−1γ,(µ′)−1γ∈∆then
+Xµ′|(µ′)−1γ=ǫ(µ′,γ,µ)·Xµ|µ−1γ
+Remark: If for eachγ∈Rwe chooseµγ∈Ωso thatµ−1
+γγ∈∆, then{Xµγ|µ−1γγ}γ∈Ris a
+Chevalleysysteminthe senseof[SGAIII.3, expXXIII §6].
+1.3 Cayley-transforms
+Letα∈R(T0,B0)andchoose Xα∈Lie(G)α(R)−{0}. Put
+gα:=exp/parenleftbiggiπ
+4(Xα+X−α)/parenrightbigg
+4Thenσ(gα)=exp/parenleftig
+−iπ
+4(Xα+X−α)/parenrightig
+=g−1
+αandσ(gα)−1·gα=g2
+α=exp/parenleftigiπ
+2(Xα+X−α)/parenrightig
+.
+We have
+gα=√
+2
+2/parenleftigg1i
+i1/parenrightigg
+Xα,g2
+α=/parenleftigg0i
+i0/parenrightigg
+Xα,g4
+α=/parenleftigg−1 0
+0−1/parenrightigg
+Xα=α∨(−1)
+Fact 1.5. The images of T 0underInt(gα)andInt(g−1
+α)are the same. They are a torus
+T definedoverRandthetransportsof the Γ-actiononT toT 0viaInt(g−1
+α)andInt(gα)
+bothequal s α⋊σ.
+Proof:
+Int(gα)T0=Int(g−1
+α)Int(g2
+α)T0=Int(g−1
+α)sαT0=Int(g−1
+α)T0
+σ(Int(gα)T0)=Int(σ(gα))T0=Int(g−1
+α)T0
+Int(σ(gα)−1gα)=Int(g2
+α)=sα=Int(g−2
+α)=Int(σ(gα)g−1
+α)
+/square
+Note:Different choices of Xαwill lead to different (yet conjugate) tori T. However,
+since we have fixed a splitting there is up to a sign a canonical Xα. Changingthe sign
+ofXαchangesgαtog−1
+α,henceTdoesnotchange. Thusweconclude:
+Thechoiceofa splittinggivesforeach α∈R(T0,B0) thefollowingcanonicaldata:
+1. a pair{X,X′}⊆Lie(G)α(R)−{0}withX′=−X
+2. a torus TαonwhichΓactsviasα⋊σ,
+3. a pairϕ,ϕ′of isomorphisms Tsα
+0→Tαs.t.ϕ′=ϕ◦sα, given by the Cayley-
+transformswith respectto X,X′.
+Corollary1.6. Forα∈R(T0,B0)let Tαbe the canonicallygiven torus as above. For
+µ,µ′∈Ωs.t.µ−1α,(µ′)−1α∈∆letϕ,ϕ′:Tsα
+0→Tαbe the isomorphisms given by
+Int(gXµ|µ−1α)andInt(gXµ′|(µ′)−1α). Then
+ϕ′=ϕ ,ǫ (µ′,α,µ)=1
+ϕ◦sα,ǫ(µ′,α,µ)=−1
+Proof:Clear. /square
+Notation: From now on we will write gµ,αinstead of gXµ|µ−1α. This notation will only
+beemployedin thecasethat α∈R(T0,B0) andµ−1α∈∆.
+2 Strongly orthogonalsubsetsof rootsystems
+In thissection, a few technicalfacts aboutstronglyorthog onalsubsets of rootsystems
+areproved.
+Definition2.1.
+1.α,β∈Rare calledstrongly orthogonalif α+β/nelementR andα−β/nelementR.
+2. A⊂R is called a strongly orthogonal subset (SOS) if it consists of pairwise
+stronglyorthogonalroots.
+53. A⊂R is calleda maximalstrongly orthogonalsubset (MSOS)if it isa SOSand
+isnotproperlycontainedin aSOS.
+A classification of the Weyl group orbits of MSOS in irreducib le root systems was
+givenin [AK]. In some cases, there exists morethan one orbit . To handle these cases,
+wewill use thefollowingdefinitionandlemma.
+Definition 2.2. Let A1,A2be SOS in R. A 2will be called adaptedto A 1ifspan(A2)⊂
+span(A1)andforalldistinct α,β∈A2
+{a∈A1: (a,α)/nequal0}∩{a∈A1: (a,β)/nequal0}=∅
+where()is anyΩ-invariantscalarproductontherealvectorspacespannedb yR.
+Notethatany Aisadaptedto itself.
+Lemma 2.3. There exist representatives A 1,...,Akof the Weyl group orbits of MSOS
+s.t. A1hasmaximallengthand A 2,...,Akare adaptedto A 1.
+Proof:Thisfollowsfromtheexplicitclassificationin [AK]. /square
+Notation: IfAis a SOS then all reflections with respect to elements in Acommute.
+Theirproductwill bedenotedby SA.
+Definition 2.4. For a root system R, a choice of positive roots >and a subset A of R
+let#(R,>,A)bethefollowingstatement
+∀α1,α2∈A∀β>0
+α1/nequalα2∧sα1(β)<0=⇒sα2(β)>0
+andlet##(R,>,A)be thefollowingstatement
+∀A1,A2⊂A∀β>0
+A1∩A2=∅∧SA1(β)<0=⇒SA2(β)>0∧SA1SA2(β)<0
+Remark: We will soon show that these statements are equivalent. More over we will
+showthatforanySOS A⊂Rwecanchoose >so thatthetriple( R,>,A)verifiesthese
+statements. Forthisitismoreconvenienttoworkwith#. For theapplicationshowever,
+weneed##.
+Lemma 2.5. Let R be a reduced root system and A ⊂R a SOS. There exists a choice
+ofpositiveroots >s.t.#(R,>,A′)holdsforany A′adaptedto A.
+Proof:LetVdenote the real vector space spanned by R, and (,) be anΩ-invariant
+scalar product on V. The elements of Aare orthogonal wrt ( ,). Extend Ato an
+orthogonalbasis( a1,...,an)ofV. Define thefollowingnotionofpositivityon R
+α>0⇐⇒(α,ai0)>0 for i0=min{i: (α,ai)/nequal0}
+It is clear from the construction that with this notion #( R,>,A′) is satisfied for any A′
+adapted to A. We just need to check that >0 defines a choice of positive roots, which
+wewill nowdo.
+6It is clear that for each α∈Rprecisely one of α >0 or−α >0 is true. We will
+construct p∈Vs.t. forallα∈R
+α>0⇐⇒(α,p)>0
+Let
+m=min{|(α,ai)|:α∈R,1≤i≤n,(α,ai)/nequal0}
+M=max{|(α,ai)|:α∈R,1≤i≤n}
+Constructrecursivelyrealnumbers p1,...,pns.t.
+pn=1pi>M
+m/summationdisplay
+k>ipk
+andputp=/summationtextpiai. Ifα∈Riss.t.α>0 andi0is thesmallest is.t. (α,ai)/nequal0then
+(α,p)=n/summationdisplay
+i=i0pi(α,ai)>mpi0−M/summationdisplay
+k>i0pk>0
+Thus
+α>0=⇒(α,p)>0
+Theconverseimplicationfollowsformally:
+¬(α>0)⇔−α>0⇒(−α,p)>0⇒¬((α,p)>0)
+/square
+Remark: Thetruthvalueofthestatement#( R,>,A)andthenotionofbeingadaptedto
+Aare unchangedif onereplaceselementsof Abytheirnegatives. Thuswe canalways
+assumethattheelementsof Aarepositive.
+Remark: It is necessary to choose the set of positive roots based on Ain order for
+#(R,>,A) to be true. An example that #( R,>,A) may be false is provided by V=
+R3,R=D3withpositiveroots
+1
+−1
+0,1
+0
+−1,0
+1
+−1,1
+1
+0,1
+0
+1,0
+1
+1
+and
+A=1
+0
+−1,1
+0
+1,β=1
+−1
+0
+Fact 2.6. Let R=G2and>any choice of positive roots. All MSOS A of R lie in
+the same Weyl-orbit and moreover automatically satisfy #(R,>,A). Some of these A
+containsimpleroots.
+Proof:Thisisan immediateobservation. /square
+Proposition 2.7. Let A⊂R be a SOS and >be a choice of positive roots. Then the
+statements #(R,>,A)and##(R,>,A)areequivalent.
+7Proof:First, weshowthat# impliesthefollowingstatement,to bec alled# 1:
+∀α1,α2∈A∀β>0
+α1/nequalα2∧sα1(β)<0=⇒sα2(β)>0∧sα1sα2(β)<0
+Letα1,α2∈Aandβ >0 be s.t. sα1(β)<0. Putβ′=−sα1(β). Thenβ′>0 and
+sα1(β′)=−β<0. Then#impliesthat sα2sα1(β)=−sα2(β′)<0.
+Nextweshowthat# 1impliesthefollowingstatement,tobecalled# 2:
+∀α1∈A∀A2⊂A∀β>0
+α1/nelementA2∧sα1(β)<0=⇒SA2(β)>0∧sα1SA2(β)<0
+Wedothisbyinductionofthecardinalityof A2,thecaseof A2singletonbeingprecisely
+#1. Now letα1∈A,A2⊂A\{α1}, andβ >0 be s.t.sα1(β)<0. Chooseα2∈A2and
+putβ′:=sα2(β). Then by # 1we haveβ′>0 andsα1(β′)<0. Applying the inductive
+hypothesiswe obtain SA2(β)=SA2\{α2}(β′)>0 andsα1SA2(β)=sα1SA2\{α2}(β′)<0.
+Nowweshowthat # 2impliesthe followingstatement,tobecalled ♭:
+IfA2⊂Aandβ>0ares.t.SA2(β)<0 thenthereexists α2∈A2s.t.sα2(β)<0.
+To see this, let A3⊂A2be a subset of minimal size s.t. SA3(β)<0. Takeα3∈A3
+and putβ′=SA3\{α3}(β). By minimalityof A3we haveβ′>0, andmoreover sα3(β′)=
+SA3(β)<0. Then# 2impliesthat sα3(β)=sα3SA3\{α3}(β′)<0.
+Finallyweshowthat# 2impliesthestatement##. Take A1,A2⊂As.t.A1∩A2=∅and
+β >0 s.t.SA1(β)<0. By♭there exists α1∈A1s.t.sα1(β)<0. Sinceα1/nelementA2we get
+from# 2thatSA2(β)>0andSA1SA2(β)=sα1SA2SA1\{α1}(β)<0.
+Thisshowsthat #imples##. Theconverseimplicationistriv ial. /square
+Proposition2.8. ForanSOS A⊂R, andachoice >ofpositiveroots, let
+R+
+A={β∈R:β>0∧SAβ<0}
+Assume that >is chosenso that ##(R,>,A)is true. Then if A′,A′′⊂A are disjoint, so
+are R+
+A′and R+
+A′′, and R+
+A′∪A′′=R+
+A′∪R+
+A′′. Moreover, the action of S A′on R preserves
+R+
+A′′.
+Proof:Thisfollowsimmediately. /square
+Corollary2.9. If A isaSOSand >ischosenso that #(R,>,A)istruethen
+R+
+A=/coproductdisplay
+α∈AR+
+α
+Proof:Clear.
+Lemma2.10. LetRbearootsystem,V therealvectorspacespannedbyit, Q ⊂V the
+rootlattice,and (,)aWeyl-invariantscalarproductonV. If v ∈Q iss.t.
+|v|≤min{|α|:α∈R}
+where||is the Euclidiannormarising from (,)then v∈R andthe aboveinequalityis
+anequality.
+8Proof:Chooseapresentation
+v=/summationdisplay
+α∈Rnαα,nα∈Z≥0
+s.t./summationtext
+αnαis minimal. First we claim that if α,β∈Rcontribute to this sum, then
+(α,β)≥0. If that were not the case, then by [Bou, Ch.VI, §1,no.3,Thm.1] we have
+thatγ:=α+β∈R∪{0}and we can replace the contribution α+βin the sum by γ,
+contradictingitsminimality. Now,if γ∈Ris anyrootcontributingto thesum,we get
+|v|2=/summationdisplay
+α,β∈Rnαnβ(α,β)≥n2
+γ(γ,γ)≥(γ,γ)=|γ|2
+withequalitypreciselywhen v=γ. /square
+Lemma 2.11. Let R be a root system and α,β∈R two strongly orthogonal roots. If
+α∨+β∨∈2Q∨,thenα,βbelongto thesamecopyofG 2.
+Proof:LetVdenote the real vector space spanned by R. Choose a Weyl-invariant
+scalarproduct( ,)anduseit toidentify Vwithitsdualandregard R∨asarootsystem
+inV.
+Assume nowthat α∨+β∨∈2Q∨. Note thatα∨andβ∨are orthogonal(butmaynot be
+stronglyorthogonalelementsof R∨).
+First we show that then α,βbelong to the same irreducible piece of R. To that end,
+assumethat Rdecomposesas R=R1⊔R2andVdecomposesaccordinglyas V1⊕V2.
+Ifα∈R1andβ∈R2, thenα∨∈V1andβ∨∈V2. Then1
+2(α∨+β∨)∈Q∨implies
+1
+2α∨∈Q∨
+1,1
+2β∨∈Q∨
+2(projectorthogonallyonto V1respV2). Thishowevercontradicts
+theabovelemma,because1
+2α∨haslengthstrictlylessthentheshortestelementsin R∨
+1.
+Knowing that α,βlie in the same irreducible piece we can now assume wlog that R
+is irreducible. Normalize ( ,) so that the short roots in Rhave length 1. We have the
+followingcases
+•All elements of Rhave length 1. Then all elements of R∨have length 2. The
+length of1
+2(α∨+β∨) is√
+2, which by the above lemma is not a length of an
+elementin Q∨.
+•Rcontainselementsof lengths1 and√
+2. ThenR∨containselementsoflengths√
+2and2.
+–If bothα∨,β∨have length√
+2, then1
+2(α∨+β∨) has length 1, so is not in
+Q∨.
+–Ifα∨haslength√
+2andβ∨haslength2,then1
+2(α∨+β∨)haslength√
+6
+2,so
+againisnotin Q∨.
+–Ifbothα∨,β∨havelength2,then1
+2(α∨+β∨)haslength√
+2andthuscould
+potentially be in Q∨. If it is, then by the above lemma it is also in R∨,
+so1
+2(α∨+β∨)∨must be an element of R. One immediately computesthat
+[1
+2(α∨+β∨)]∨=α+β,butthelatterisnotanelementof Rbecauseα,βare
+stronglyorthogonal.
+•Rhas elements of lengths 1 and√
+3. Then RisG2andR∨is alsoG2. As
+one sees immediately, up to the action of its Weyl-group, G2has a unique pair
+of orthogonal roots, which are then automatically strongly orthogonal and half
+theirsumis alsoa root.
+/square
+93 Splitting invariants
+Recall that we have fixed a split semi-simple and simply-conn ected group GoverR
+and a splitting ( T0,B0,{Xα}) of it. Given a maximal torus T, an element h∈Gs.t.
+Int(h)T0=Tand a-data{aβ}forR(T,G), Langlands and Shelstad construct in [LS1,
+2.3]a certainelementof Z1(Γ,T),whoseimagein H1(Γ,T)theycallλ(T)–the”split-
+ting invariant” of T. They show that this image is independent of the choice of h. In
+this section we want to study this splitting invariant in suc h a way that enables us to
+see howit varieswhenthetorusvaries. It turnsoutthata cer taintypeofa-datais very
+wellsuited forthis. Thisa-dataisdeterminedbyaBorel B⊃Tasfollows:
+αβ=i,β>0∧σT(β)<0
+−i,β<0∧σT(β)>0
+1,β>0∧σT(β)>0
+−1,β<0∧σT(β)<0
+whereσTdenotes the Galois-action on X∗(T) andβ >0 meansβ∈R(T,B). We will
+call this a-data B-a-data. It should not be confused with Shelstad’s terminol ogy of
+baseda-data,whichis also givenbya Borel – forbaseda-data,the p ositiveimaginary
+roots are assigned iwhile all other positive roots are assigned 1; for B-a-data, any
+positive root whose Galois-conjugate is negative is assign edi. Therefore a splitting
+invariantcomputedusing Shelstad’sbased a-data will in ge neralbe differentfromone
+computedusing B-a-data. Theprecisedi fferenceisgivenby[LS1,2.3.2]. Itishowever
+moreimportanttonotethataccordingto[LS1,Lemma2.3.C]t hisdifferencedisappears
+once the splitting invarianthas been paired with an endosco piccharacter. Thus, as far
+applicationstotransferfactorsare concerned,baseda-da taandB-a-datagivethesame
+result.
+In view of the reductiontheoremwhich we will provein sectio n 3.2, it will be helpful
+to consider not just the cohomologyclass, but the actual coc ycle constructed in [LS1,
+2.3]. We will denote this cocycle by λ(T,B,h) to record its dependence on the B-a-
+data and the element h, while the splitting ( T0,B0,{Xα}) is not present in the notation
+becauseitisassumedfixed. Sinceweareworkingover R,wewillidentifya1-cocycle
+and its value at σ∈Gal(C/R), and hence we will view λ(T,B,h)as an element of T.
+Givenh, there is an obvious choice for B, namely Int( h)B0. We will write λ(T,h) for
+λ(T,Int(h)B0,h). Note that in this notation, Tis clearly redundant, because it equals
+Int(h)T0. However,we keep it so that the notation is close to that in [L S1]. We would
+like to alert the reader of one potential confusion – while th e cohomology class of
+λ(T,B,h)is independentof the choice of h, that ofλ(T,h) is not, because in the latter
+hinfluencesnotonlytheidentificationof T0withTbutalso thechoiceof B-a-datafor
+T.
+3.1 The splitting invariantfor Tα
+Recall from section 1.3 that for each α∈R(T0,B0) there is a canonicalmaximal torus
+Tαandapairofisomorphisms Tsα
+0→Tα. Tofixoneofthetwo,fix µ∈Ωs.t.µ−1α∈∆.
+ThenInt( gµ,α)is oneof thetwoisomorphisms Tsα
+0→Tα. Thegoalofthissectionisto
+computeλ(Tα,B,gµ,α) fora givenBorel B⊃Tα, and in particular λ(Tα,gµ,α). We will
+givea formulaforthelatterin purelyroot-theoreticterms .
+Lemma 3.1. With g:=gµ,αwe have
+λ(Tα,B,g)=Int(g)/parenleftig
+α∨(i·aα◦Int(g−1))·sα(σ(δ))δ−1/parenrightig
+10where
+δ=/productdisplay
+β>0
+µ−1β<0β∨(aβ◦Int(g−1))−1
+andσdenotescomplexconjugationonT 0.
+Proof:Putu=n(µ). We will first computethecocycle λ(Tα,B,gu). Thenotationwill
+beasin[LS1, 2.3]. Thepullbackofthe Γ-actionon TαtoT0viagudiffersfromσby
+ωTα(σ) :=Int/parenleftig
+(gu)−1σ(gu)/parenrightig
+=Int/parenleftig
+n(µ)−1g−1σ(g)n(µ)/parenrightig
+=µ−1sαµ
+=sµ−1α
+Usingthatµ−1αissimple,wecomputethethreefactorsofInt( gu)−1λ(Tα,B,gu):
+x(σ)=/productdisplay
+β>0
+ωTα(σ)β<0β∨(aβ◦Int(gu)−1)
+=(µ−1α)∨(aµ−1α◦Int(u−1g−1))
+=Int(u−1)(α∨(aα◦Int(g−1)))
+n(ωTα(σ))=/parenleftigg0 1
+−1 0/parenrightigg
+Xµ−1α=Int(u−1)/parenleftigg0 1
+−1 0/parenrightigg
+Xµ|µ−1α
+σ(gu)−1(gu)=Int(u−1)g2=Int(u−1)/parenleftigg0i
+i0/parenrightigg
+Xµ|µ−1α
+Thus
+λ(Tα,B,gu)=Int(gu)Int(u−1)/parenleftig
+α∨(aα◦Int(g−1))α∨(i)/parenrightig
+Fromthe proofsof[LS1, 2.3.A]and[LS1,2.3.B]oneseesthat
+λ(Tα,B,gu)=Int(g)/parenleftig
+δσTα(δ)−1/parenrightig
+·λ(Tα,B,g)
+whereσTαisthetransportoftheactionofcomplexconjugationon TαtoT0viag. This
+actionissα⋊σ. Notice that the term λ−1σT(λ) appearingin the proofof [LS1, 2.3.A]
+istrivialsinceforus u=n(µ)andhence λ=1. Theclaimnowfollows. /square
+Beforewe turnto the computationof sα(σ(δ))δ−1we will needto takea closerlookat
+thefollowingset.
+Definition3.2. Forα>0putR+
+α=/braceleftig
+β∈R|β>0∧sα(β)<0/bracerightig
+Lemma 3.3. Letα>0andµ∈Ωbes.t.µ−1α∈∆. Thenthesets
+/braceleftig
+β∈R|β>0∧sα(β)<0∧µ−1β<0/bracerightig
+and/braceleftig
+β∈R|β>0∧sα(β)<0∧µ−1β>0∧β/nequalα/bracerightig
+aredisjointandtheirunionisR+
+α−{α}. Themap
+β/ma√s⊔o→−sα(β)
+isaninvolutiononR+
+α−{α}whichinterchangestheabovetwo sets.
+11Proof:Everyβin the first set satisfies β/nequalαbecauseµ−1αis positive. Hence the first
+set lies in R+
+α−{α}and clearly the second also does. The fact that the two are dis joint
+andcover R+
+α−{α}isobvious. Nowtothebijection. Let βbeanelementinthefirstset,
+andconsider ˜β=−sα(β). We have
+β/nequalα⇒˜β/nequalα
+sα(β)<0⇒˜β>0
+β>0⇒sα˜β=−β<0
+µ−1β<0⇒µ−1˜β=µ−1sα(−β)=sµ−1α(−µ−1β)>0
+wherethe last inequalityholdsbecause µ−1αissimple and
+β>0⇒β/nequal−α⇒−µ−1β/nequalµ−1α
+/square
+Remark: Asimilar observationappearsin[LS2, §4.3].
+Lemma 3.4. We have
+sα(σ(δ))δ−1=/productdisplay
+β∈R+
+α
+µ−1β<0/bracketleftig
+β∨/parenleftbigaβ◦Int(g−1)/parenrightbigsαβ∨/parenleftbigasαβ◦Int(g−1)/parenrightbig−1/bracketrightig
+Proof:Accordingtotheproofofpart(a)of[LS1,2.3.B]thecontrib utionstoδsα(σ(δ))−1
+areasfollows:
+(1){β|β>0∧µ−1β<0∧sαβ<0}:β∨(aβ◦Int(g−1))−1
+(2){β|β<0∧µ−1β<0∧µ−1sαβ<0∧sαβ>0}:β∨(aβ◦Int(g−1))
+(3){β|β>0∧µ−1β<0∧sαβ>0∧µ−1sαβ>0}:β∨(aβ◦Int(g−1))−1
+(4){β|µ−1β>0∧sαβ>0∧µ−1sαβ<0}:β∨(aβ◦Int(g−1))
+We will use µ−1sα(β)=sµ−1α(µ−1β) and the fact that µ−1αis simple to show that the
+last two sets are empty. In set (3), the conditions µ−1β <0 andµ−1sαβ >0 imply
+µ−1β=−µ−1α, i.e.β=−α, which contradicts β >0. In set (4), the conditions
+µ−1β>0 andµ−1sαβ<0implyβ=α. Sinceα>0thiscontradicts sαβ>0.
+Nextweclaim(2) =sα((1)). We have
+µ−1β<0∧µ−1sαβ<0⇔µ−1β<0∧µ−1β/nequal−µ−1α
+fromwhichweget
+(2)={−β|β>0∧sαβ<0∧µ−1β>0∧β/nequalα}
+Now(2)=sα((1))followsfromLemma3.3.
+Fromthese considerationsit followsthat
+δsα(σ(δ))−1=/productdisplay
+β∈(1)β∨(aβ◦Int(g−1))−1/productdisplay
+β∈(2)β∨(aβ◦Int(g−1))
+=/productdisplay
+β∈(1)/bracketleftig
+β∨/parenleftbigaβ◦Int(g−1)/parenrightbig−1sαβ∨/parenleftbigasαβ◦Int(g−1)/parenrightbig/bracketrightig
+12/square
+Let us recall our notation: α∈Ris any positive root, µ∈Ωis s.t.µ−1α∈∆, and
+g=gµ,αistheCayley-transformcorrespondingto Xµ|µ−1α.
+FromLemmas3.1and3.4weimmediatelyget
+Corollary3.5.
+λ(Tα,B,g)=Int(g)α∨(iaα◦Int(g−1))·/productdisplay
+β∈R+
+α
+µ−1β<0/bracketleftig
+β∨/parenleftbigaβ◦Int(g−1)/parenrightbigsαβ∨/parenleftbigasαβ◦Int(g−1)/parenrightbig−1/bracketrightig
+Inthecase B=Int(g)B0thisformulabecomessimpler.
+Corollary3.6.
+λ(Tα,gµ,α)=Int(gµ,α)α∨(−1)·/productdisplay
+β∈R+
+α
+µ−1β<0(β∨·sαβ∨)(i)
+Definition3.7. Put
+ρ(µ,α) :=α∨(−1)·/productdisplay
+β∈R+
+α
+µ−1β<0(β∨·sαβ∨)(i)∈T0
+By Corollary 3.6 and the work of [LS1, 2.3] we know that ρ(µ,α)∈Z1(Γ,Tsα
+0) and
+Int(gµ,α)ρ(µ,α)=λ(Tα,gµ,α).
+Proposition3.8.
+1.ρ(µ,α)=/producttext
+β∈R+αβ∨(i)n(sα)g2
+µ,α.
+2. sαρ(µ,α)=ρ(µ,α),σ(ρ(µ,α))=ρ(µ,α)−1.
+3. The image of ρ(µ,α)under the two canonical isomorphisms Tsα
+0→Tαis the
+same
+4. Ifµ′∈ΩisanotherWeyl-elements.t. (µ′)−1α∈∆, then
+ρ(µ′,α)=α∨/parenleftig
+ǫ(µ′,α,µ)/parenrightig
+ρ(µ,α)
+Inparticular
+λ(Tα,gµ′,α)=λ(Tα,gµ,α)·Int(gµ,α)/bracketleftig
+α∨(ǫ(µ′,α,µ))/bracketrightig
+Proof:The first point follows from Corollary 3.6, because the right hand side is by
+construction Int( gµ,α)λ(Tα,gµ,α). The second point is evident from the structure of ρ.
+The third point is now clear because as remarked in section 1. 3 the two canonical
+isomorphismsdifferbyprecompositionwith sα.
+Forthelast point,
+ρ(µ′,α)=/productdisplay
+β∈R+αβ∨(i)n(sα)g2
+µ′,α
+13Ifǫ(µ′,α,µ)= +1 thengµ′,α=gµ,αand the statement is clear. Assume now that
+ǫ(µ′,α,µ)=−1. Thengµ′,α=g−1
+µ,α. We see
+ρ(µ′,α)=/productdisplay
+β∈R+αβ∨(i)n(sα)g2
+µ,αg−4
+µ,α
+Butg−4
+µ,α=α∨(−1),hencetheclaim. /square
+3.2 The splitting invariantfor TA
+Fact3.9. Let A beaSOSinR. Considerthe set ofautomorphismsofG given by
+Int(g)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg=/productdisplay
+α∈Agµα,α,µα∈Ω,µ−1
+αα∈∆
+TheimageofT 0underanyelementofthatsetisthesame. CallitT A. Thenanyelement
+ofthatset inducesanisomorphismofreal tori
+TSA
+0→TA
+Proof:LetInt(g1),Int(g2)beelementsoftheaboveset, with
+gi=/productdisplay
+α∈Agµiα,α
+and letA′⊂Abe the subset of those αs.t.gµ1α,α/nequalgµ2α,α. For those αwe have
+thengµ1α,α=g−1
+µ2α,α, hence Int( g1g−1
+2)|T0=/producttext
+α∈A′g2
+µ1α,α|T0=SA′which normalizes
+T0. This shows that the images of T0under these two automorphisms are the same.
+Moreover, the transport of the Γ-action on TAtoT0via Int(g−1
+1) differs fromσby
+Int(σ(g−1
+1)g1)|T0=Int(g2
+1)|T0=SA. /square
+Definition3.10. ForaSOS A⊂R,we will calltheset
+Int(g)|T0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg=/productdisplay
+α∈Agµα,α,µα∈Ω,µ−1
+αα∈∆
+the canonical set of isomorphisms TSA
+0→TA. More generally, if A′⊂A, we will call
+theset Int(g)|TA′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg=/productdisplay
+α∈A\A′gµα,α,µα∈Ω,µ−1
+αα∈∆
+thecanonicalset ofisomorphismsTSA\A′
+A′→TA.
+Fact3.11. AnymaximaltorusT ⊂G isG(R)-conjugatetooneoftheT A.
+Proof:Chooseg∈Gs.t. Int(g)T0=T. The transport of the Γ-action on TtoT0
+via Int(g−1) differs fromσby an element of Z1(Γ,Ω)=Hom(Γ,Ω) and this element
+sendscomplexconjugationto anelement of Ωof order2. By [Bou, Ch.VI.Ex §1(15)]
+there exists a SOS As.t. this element equals SA. If Int(gA) is one of the canonical
+isomorphisms TSA
+0→TA,thenInt( gAg−1) :T→TAisanisomorphismofrealtori. By
+[S1, Thm. 2.1]thereexists g′∈G(R)s.t. Int(g′)T=TA. /square
+If we conjugate AbyΩto anA′, then the tori TAandTA′are also conjugateby G(R).
+Thuswemayfixrepresentatives A1,...,AkfortheΩ-orbitsofMSOSin Randstudythe
+toriTAforAinside one of the Ai. We assume that the fixed splitting ( T0,B0,{Xα}) is
+compatiblewith thechoiceofrepresentativesinthefollow ingsense
+14•##(R,>,Ai)holdsforall Ai.
+•Ifα,β∈Ailie inthesame G2-factorthenoneofthemissimple
+This can always be arranged, as Lemmas 2.3, 2.5, Fact 2.6 and P roposition 2.7 show.
+Notice that this condition does not reduce generality – it is only a condition on B0,
+but all Borels containing T0are conjugate under NT0(R) and thus by [LS1, 2.3.1] the
+splittinginvariantsareindependentofthechoiceof B0.
+Lemma 3.12. If A′,A′′are disjointsubsetsof some A ithen
+n(SA′)n(SA′′)=n(SA′∪A′′)
+Inparticular,n (SA′)andn(SA′′)commute.
+Proof:Thisfollowsimmediatelyfrom[LS1, Lemma2.1.A],becauseb y##theset
+{β∈R:β>0∧SA′(β)<0∧SA′SA′′(β)>0}
+isempty. /square
+Proposition 3.13. Letα,γbe distinct elements of one of the A i, andµ∈Ωbe s.t.
+µ−1α∈∆. Thenρ(µ,α)isfixedby s γ.
+Proof:We first show
+Claim:
+sγρ(µ,α)=ρ(µ,α)α∨(ǫ(sγµ,α,µ))
+Proof:We have
+sγ(ρ(µ,α))=sγα∨(−1)/productdisplay
+β∈R+
+α
+µ−1β<0β∨(i)sαβ∨(i)
+Nowsγpreservesα∨, commuteswith sα, andbyProposition2.8also preservesthe set
+R+
+α,hencethe last expressionequals
+α∨(−1)/productdisplay
+β∈R+
+α
+µ−1β<0sγβ∨(i)sαsγβ∨(i)=α∨(−1)/productdisplay
+sγβ∈R+
+α
+µ−1sγβ<0β∨(i)sαβ∨(i)
+=α∨(−1)/productdisplay
+β∈R+
+α
+µ−1sγβ<0β∨(i)sαβ∨(i)
+=ρ(sγµ,α)
+=ρ(µ,α)·α∨/parenleftig
+ǫ(sγµ,α,µ)/parenrightig
+thelast equalitycomingfromProposition3.8. /square(claim)
+We wanttoshow α∨/parenleftig
+ǫ(sγµ,α,µ)/parenrightig
+=1. Chooseν∈Ωs.t.ν−1γ∈∆. Wewill deriveand
+comparetwo expressionsfor
+/productdisplay
+β∈R+
+{α,γ}β∨(i)n(sαsγ)g2
+µ,αg2
+ν,γ (1)
+ByCorollary2.9wehave
+/productdisplay
+β∈R+
+{α,γ}β∨(i)=/productdisplay
+β∈R+αβ∨(i)/productdisplay
+β∈R+γβ∨(i)
+15ByProposition2.8 sαis apermutationoftheset R+
+γ, hence
+n(sα)/productdisplay
+β∈R+γβ∨(i)n(sα)−1=/productdisplay
+β∈R+γβ∨(i)
+By Lemma 3.12, the elements n(sα) andn(sγ) ofN(T0) commute. Moreover, by Fact
+1.1theelements g2
+µ,αandg2
+ν,γcommute. Thuswe getontheonehand
+(1)=/productdisplay
+β∈R+γβ∨(i)n(sγ)/productdisplay
+β∈R+αβ∨(i)n(sα)g2
+µ,αg2
+ν,γ
+=ρ(ν,γ)Int(g−2
+ν,γ)/bracketleftig
+ρ(µ,α)/bracketrightig
+=ρ(ν,γ)ρ(µ,α)α∨(ǫ(sγµ,α,µ))
+wherethelastequalityfollowsfromaboveclaim. Analogous ly,weobtainontheother
+hand
+(1)=/productdisplay
+β∈R+αβ∨(i)n(sα)/productdisplay
+β∈R+γβ∨(i)n(sγ)g2
+ν,γg2
+µ,α
+=ρ(µ,α)Int(g−2
+µ,α)/bracketleftig
+ρ(ν,γ)/bracketrightig
+=ρ(µ,α)ρ(ν,γ)γ∨(ǫ(sαν,γ,ν))
+We concludethat
+α∨(ǫ(sγµ,α,µ))=γ∨(ǫ(sαν,γ,ν))
+We claim that both sides of this equality are trivial. Assume by way of contradiction
+thatthisisnotthecase. Thenwehave
+α∨(−1)=γ∨(−1)⇔1=(−1)(α∨−γ∨)=(−1)α∨+γ∨∈C×⊗X∗(T0)=C×⊗Q∨
+⇔α∨+γ∨∈2Q∨
+whereQ∨is the coroot-lattice of T0, which coincides with X∗(T0) sinceGis simply-
+connected. ByLemma2.11 α,γmustlieinthesame G2-factorof R. Inthiscase{α,γ}
+isaMSOSforthat G2-factorandbyourassumptionfromthebeginningofthissect ion
+oneofα,γmustbesimple. Say wlog αissimple. By Proposition3.8
+ρ(µ,α)=α∨(ǫ(µ,α,1))ρ(1,α)=α∨(−ǫ(µ,α,1))
+whichis clearlyfixedby sγ,thuswesee
+1=sγ(ρ(µ,α))ρ(µ,α)−1=α∨(ǫ(sγµ,α,µ))=γ∨(ǫ(sαν,γ,ν))
+/square
+Corollary 3.14. Let A be a subset of some A i,α∈A andµ∈Ωs.t.µ−1α∈∆. Then
+ρ(µ,α)∈Z1(Γ,TSA
+0)andits image in T Aunderany canonicalisomorphism TSA
+0→TA
+isthesame.
+Proof:By Propositions 3.8 and 3.13 ρ=ρ(µ,α) is fixed by sγfor anyγ∈A. The
+first statement now follows from ρSAσ(ρ)=ρσ(ρ)=1 showingρ∈Z1(Γ,TSA
+0). The
+secondholdsbecauseanytwocanonicalisomorphisms TSA
+0→TAdifferbyprecompo-
+sitionwith SA′forsome A′⊂A /square
+16Corollary 3.15. Let A be a subset of some A i. Choose, for each α∈A,µα∈Ωs.t.
+µ−1
+αα∈∆. Put
+ρ({µα}α∈A,A)=/productdisplay
+α∈Aρ(µα,α)
+Then
+1.ρ({µα}α∈A,A)isfixedby s γforallγ∈A(evenallγ∈Ai)
+2. The image of ρ({µα}α∈A,A)underany of the canonicalisomorphismsTSA
+0→TA
+isthe same.
+Proof:ClearbytheprecedingCorollary.
+Proposition 3.16. Let A be a subset of some A i. For each α∈A chooseµα∈Ω
+s.t.µ−1
+αα∈∆and put g A=/producttext
+α∈Agµα,α. Thenλ(TA,gA)is the common image of
+ρ({µα}α∈A,A)underthecanonicalisomorphismsTSA
+0→TA. Inparticular
+λ(TA,gA)=/productdisplay
+α∈AInt(gA−{α})λ(Tα,gµα,α)
+isadecompositionof λ(TA,gA)asaproductofelementsof Z1(Γ,TA).
+Proof:The factors of the cocycle Int( g−1
+A)λ(TA,gA)∈Z1(Γ,TSA
+0) associated to these
+choicesare asfollows:
+x(σT)=/productdisplay
+β∈R+
+Aβ∨(i)=/productdisplay
+α∈A/productdisplay
+β∈R+αβ∨(i)
+wherethe secondequalityisduetoCorollary2.9,
+n(ωT(σ))=n(SA)=/productdisplay
+α∈An(sα)
+wherethe secondequalityisduetoLemma3.12,and
+σ(gA)−1gA=/productdisplay
+α∈Aσ(gα)−1gα=/productdisplay
+α∈Ag2
+α
+Theirproduct,whichequalsInt( g−1
+A)λ(TA,gA),isthus
+x(σT)n(ωT(σ))σ(gA)−1gA=/productdisplay
+α∈A/productdisplay
+β∈R+αβ∨(i)/productdisplay
+α∈An(sα)/productdisplay
+α∈Ag2
+α
+Just asintheproofofProposition3.13we canrewritethispr oductas
+/productdisplay
+α∈A/productdisplay
+β∈R+αβ∨(i)n(sα)/productdisplay
+α∈Ag2
+α
+17Nowweinductonthe size of A, with|A|=1beingclear. Choose α1∈A. Then
+/productdisplay
+α∈A/productdisplay
+β∈R+αβ∨(i)n(sα)/productdisplay
+α∈Ag2
+α
+=/productdisplay
+α∈A\{α1}/productdisplay
+β∈R+αβ∨(i)n(sα)/productdisplay
+β∈R+α1β∨(i)n(sα1)g2
+α1/productdisplay
+α∈A\{α1}g2
+α
+=/productdisplay
+α∈A\{α1}/productdisplay
+β∈R+αβ∨(i)n(sα)ρ(µα1,α1)/productdisplay
+α∈A\{α1}g2
+α
+=/productdisplay
+α∈A\{α1}/productdisplay
+β∈R+αβ∨(i)n(sα)/productdisplay
+α∈A\{α1}g2
+α·k/productdisplay
+α∈A\{α1}sα(ρ(µα1,α1))
+=/productdisplay
+α∈A\{α1}ρ(µα,α)·ρ(µα1,α1)
+where the last equality follows from Proposition 3.13 and th e inductive hypothesis.
+Thisshowsthat Int( gA)ρ({µα}α∈A,A)=λ(TA,gA) andthe resultfollows. /square
+4 Explicit computations
+Inthissectionwe aregoingtouse theclassificationofMSOS g ivenin[AK]to explic-
+itly compute λ(TA,gA) for the split simply-connected semi-simple groups associ ated
+to the classical irreducible root systems. By Propositions 3.8 and 3.16 it is enough to
+compute the cocycles ρ(µ,α) for eachα∈Ai, and some µ∈Ωwithµ−1α∈∆, where
+A1,...,Akis a set of representativesfor the Ω-classes of MSOS. We will use the nota-
+tionfrom[AK],whichis also thenotationused inthe Plateso f[Bou, Ch.VI].Thereis
+only one cosmetic di fference – in [Bou] the standard basis of Rkis denoted by ( ǫi), in
+[AK]by(λi),andweareusing( ei). Thedualbasiswillbedenotedby( e∗
+i). Onechecks
+easilyineachcasethatthechoicesofpositiverootsgiveni nthePlatesof[Bou,Ch.VI]
+andthe MSOSgivenin [AK] satisfycondition#ofsection2.
+4.1 Case An
+Thereis only oneΩ-equivalenceclass of MSOS, and the representativegivenin [AK]
+is
+A1={e2i−1−e2i|1≤i≤[(n+1)/2]}
+AllelementsofthisMSOSaresimplerootsandforeachofthem wecanchoose µ=1.
+Thenforany α∈A1wehave
+ρ(1,α)=α∨(−1)
+4.2 Case Bn
+Ifn=2k+1thenthereis auniqueequivalenceclassofMSOS, represent edby
+A1={e2i−1±e2i|1≤i≤k}∪{en}
+18Ifn=2kthentherearetwo equivalenceclassesofMSOS, represented by
+A1={e2i−1±e2i|1≤i≤k−1}∪{en}
+A2={e2i−1±e2i|1≤i≤k}
+Ifα=e2i−1−e2iorα=enthenαis simple,wecanchoose µ=1andhave
+ρ(1,α)=α∨(−1)
+Ifα=e2i−1+e2ithen we take µ=se2iand haveµ−1α=e2i−1−e2i∈∆. To compute
+ρ(µ,α)we first observe
+{β∈R|β>0∧µ−1β<0∧sαβ<0}={β∈R|β>0∧µ−1β<0}
+={e2i}∪{e2i±ej|2i<j}
+hence/summationdisplay
+β∈R+
+α
+µ−1β<0(β∨+sαβ∨)
+=2e∗
+2i+sα(2e∗
+2i)+n/summationdisplay
+j=2i+1(e∗
+2i−e∗
+j+sα(e∗
+2i−e∗
+j)+e∗
+2i+e∗
+j+sα(e∗
+2i+e∗
+j))
+=−2(e∗
+2i−1−e∗
+2i)−2(e∗
+2i−1−e∗
+2i)(n−2i)
+=−2(n+1−2i)(e∗
+2i−1−e∗
+2i)
+=−2(n+1−2i)α∨
+We get
+ρ(µ,α)=α∨((−1)n)
+4.3 Case Cn
+The root system family Cnis the only family for which the number of equivalence
+classes of MSOS growswhen ngrows. Representativesfor the equivalence classes of
+MSOS aregivenby
+As={e2i−1−e2i|1≤i≤s}∪{2ei|2s+1≤i≤n} 0≤s≤[n/2]
+Ifα=e2i−1−e2ithenαissimpleand
+ρ(1,α)=α∨(−1)
+Ifα=2eiandwe take µ=sei−enwehaveµ−1α=2en∈∆. Againwefirst observe
+{β∈R|β>0∧µ−1β<0∧sαβ<0}={β∈R|β>0∧µ−1β<0}
+={ei−ej|i<j}
+hence/summationdisplay
+β∈R+
+α
+µ−1β<0(β∨+sαβ∨)=n/summationdisplay
+j=i+1(e∗
+i−e∗
+j)+(−e∗
+i−e∗
+j)=−2n/summationdisplay
+j=i+1e∗
+j
+We get
+ρ(µ,α)=n/productdisplay
+j=ie∗
+i(−1)
+194.4 Case Dn
+Thereisa uniqueequivalenceclassofMSOS representedby
+A1={e2i−1±e2i|1≤i≤[n/2]}
+Ifα=e2i−1−e2iorα=en−1+enthenαissimple and
+ρ(1,α)=α∨(−1)
+Ifα=e2i−1+e2iwith 2i/nequalnthen we take µ=se2i−1−en−1◦se2i−enand haveµ−1α=
+en−1−en∈∆. Then
+{β∈R|β>0∧µ−1β<0∧sαβ<0}={β∈R|β>0∧µ−1β<0}
+={e2i−1−ej|2i<j}∪{e2i−ej|2i<j}
+hence
+/summationdisplay
+β∈R+
+α
+µ−1β<0(β∨+sαβ∨)=−2n/summationdisplay
+j=2i+12e∗
+j∈4Q∨,2|n
+4Q∨+4e∗
+n,2|n+1
+Noticethat2 e∗
+n∈Q∨,whilee∗
+n/nelementQ∨. We get
+ρ(µ,α)=α∨(−1)·2e∗
+n((−1)n)
+5 Comparisonofsplitting invariants
+Now we would like to employ the results from the previoussect ionsto comparesplit-
+ting invariants of di fferent tori. More precisely, we’d like to do the following: Le t
+(H,s,η) be an endoscopic triple for G, andT1,T2be two maximal tori of Gthat orig-
+inate from H. We’d like to compare the results of pairing the endoscopic d atums
+against the splitting invariantsfor T1andT2. In order for this to make sense, we need
+to show that those invariants reside in a common space. We wil l show that the en-
+doscopic characters on H1(Γ,T1) andH1(Γ,T2) arising from sfactor through certain
+quotientsofthesegroupsandthatthosequotientscanberel ated.
+Foramaximaltorus TinGput
+X∗(T)−1={λ∈X∗(T)|σT(λ)=−λ}
+IX∗(T)={λ−σT(λ)|λ∈X∗(T)}
+whereσTistheactionof σonT. Recall theTate-Nakayamaisomorphism
+X∗(T)−1
+IX∗(T)=H−1
+T(Γ,X∗(T))→H1(Γ,T)
+given by taking cup-product with the canonical class in H2(Γ,C×). Via this isomor-
+phism,thecanonicalpairing /hatwideT×X∗(T)→C×inducesa pairing
+/hatwideT×H1(Γ,T)→C×
+The splitting invariant enters into the construction of the Langlands-Shelstad transfer
+factorsviathispairing.
+20Lemma 5.1. Let A′⊂A be SOS in R. Then each element in the canonical set of
+isomorphismsTSA\A′
+A′→TA(Definition3.10)inducesthesameembedding
+iA′,A:X∗(TA′)−1֒→X∗(TA)−1
+Proof:Foranelement ω∈Ωput
+X∗(T0)ω=−1={λ∈X∗(T0)|ω(λ)=−λ}
+ForanySOS B
+X∗(T0)SB=−1=spanQ(B)∩X∗(T0)
+and any canonical isomorphism TSB
+0→TBidentifies X∗(T0)SB=−1withX∗(TB)−1(this
+identificationwill ofcoursedependonthechosenisomorphi sm).
+Fix one canonical isomorphism TSA\A′
+A′→TA. It is the composition of canonical iso-
+morphisms
+TSA\A′
+A′ϕ−1
+−→TSA
+0ψ→TA
+andhenceinducesaninclusionasclaimed,because X∗(T0)SA′=−1⊂X∗(T0)SA=−1. More-
+over,anyothercanonicalisomorphism TSA\A′
+A′→TAisgivenby
+TSA\A′
+A′ϕ−1
+−→TSA
+0SA′′−→TSA
+0ψ→TA
+forA′′⊂A\A′andclearly SA′′actstriviallyon X∗(T0)SA′=−1. /square
+Theembedding iA′,Ainducesanembedding
+¯iA′,A:X∗(TA′)−1
+IX∗(TA′)+i−1
+A′,A(IX∗(TA))֒→X∗(TA)−1
+iA′,A(IX∗(TA′))+IX∗(TA)
+and via the Tate-Nakayama isomorphism these quotients corr espond to quotients of
+H1(Γ,TA′) andH1(Γ,TA) respectively. We will argue that if the tori TA′andTAorig-
+inate in an endoscopic group H, then the endoscopic character factors through these
+quotients. This providesa meansof comparingthe valuesof t he endoscopiccharacter
+onthecohomologyofbothtori.
+Lemma 5.2. Let(H,s,η)be an endoscopic triple for G and assume that T A′and TA
+(A′⊂A) originate from H, that is, there exist tori T 1,T2⊂H and admissible iso-
+morphisms T 1→TA′and T2→TA. Write s TA′∈/hatwideTA′and sTA∈/hatwideTAfor the images
+of s under the duals of these isomorphisms. Assume that there exists a canonical iso-
+morphism j :TSA\A′
+A′→TAs.t./hatwidej(sTA)=sTA′(this can always be arranged). Then
+the characters s TA′and sTAon H1(Γ,TA′)resp. H1(Γ,TA)factor through the above
+quotients,andthepull-backofthecharacter s TAvia¯iA′,Aequals s TA′.
+Proof:We identify H1(Γ,−)withH−1
+T(Γ,X∗(−))viatheTate-Nakayamaisomorphism.
+Becausethe element sTA∈X∗(TA)⊗C×isΓ-invariant,its actionon X∗(TA)annihilates
+the submodule IX∗(TA). Thus, the action of j∗(sTA)∈X∗(TA′)⊗C×onX∗(TA′) annihi-
+lates the submodule j−1
+∗(IX∗(TA)). But we have arranged things so that j∗(sTA)=sTA′
+andweseethattheactionof sTA′onX∗(TA′)viathestandardpairingannihilatesthesub-
+moduleIX∗(TA′)+j−1
+∗(IX∗(TA)). By the same argument, the action of sTAonX∗(TA)
+annihilates the submodule IX∗(TA)+j∗(IX∗(TA′)). Finally notice that by Lemma 5.1
+therestrictionof j∗toX∗(TA′)−1coincideswith iA′,A. /square
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+tkaletha@math.uchicago.edu
+UniversityofChicago
+5734S.UniversityAvenue
+Chicago,Illinois60637
+22
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+arXiv:1001.0348v2  [math.NT]  23 May 2011Publ. Math. Debrecen 79(2011), in press.
+ON HARMONIC NUMBERS AND LUCAS SEQUENCES
+Zhi-Wei Sun
+Department of Mathematics, Nanjing University
+Nanjing 210093, People’s Republic of China
+zwsun@nju.edu.cn
+http://math.nju.edu.cn/ ∼zwsun
+Abstract. Harmonic numbers Hk=/summationtext
+0<j/lessorequalslantk1/j(k= 0,1,2,...) arise
+naturallyin manyfieldsof mathematics. In thispaper weiniti atethe study
+of congruences involving both harmonic numbers and Lucas seque nces.
+One of our three theorems is as follows: Let u0= 0, u1= 1,andun+1=
+un−4un−1forn= 1,2,3,.... Then, for any prime p >5 we have
+p−1/summationdisplay
+k=0Hk
+2kuk+δ≡0 (modp),
+whereδ= 0 ifp≡1,2,4,8 (mod 15), and δ= 1 otherwise.
+1. Introduction
+Harmonic numbers are those rational numbers given by
+Hn=/summationdisplay
+0<k/lessorequalslantn1
+k(n∈N={0,1,2,...}).
+They play important roles in mathematics; see, e.g., [BPQ] a nd [BB].
+In 1862 J. Wolstenholme [W] (see also [HT]) discovered that f or any
+primep >3 we have
+Hp−1=p−1/summationdisplay
+k=11
+k≡0 (modp2).
+2010Mathematics Subject Classification .Primary 11A07, 11B39; Secondary 05A10,
+33B99.
+Keywords . Congruences, harmonic numbers, Lucas sequences.
+Supported by the National Natural Science Foundation (grant 10 871087) and the
+Overseas Cooperation Fund (grant 10928101) of China.
+12 ZHI-WEI SUN
+In a previous paper [Su] the author developed the arithmetic theory of
+harmonic numbers by proving the following fundamental cong ruences for
+primesp >3:
+p−1/summationdisplay
+k=1H2
+k≡2p−2 (modp2),p−1/summationdisplay
+k=1H3
+k≡6 (modp),
+and
+p−1/summationdisplay
+k=1H2
+k
+k2≡0 (modp) provided p >5.
+In this paper we initiate the investigation of congruences i nvolving both
+harmonic numbers and Lucas sequences.
+ForA,B∈Zthe Lucas sequences un=un(A,B) (n∈N) andvn=
+vn(A,B) (n∈N) are defined as follows:
+u0= 0, u1= 1,andun+1=Aun−Bun−1(n= 1,2,3,...);
+v0= 2, v1=A,andvn+1=Avn−Bvn−1(n= 1,2,3,...).
+The sequence {vn}n/greaterorequalslant0is called the companion sequence of {un}n/greaterorequalslant0. The
+characteristic equation x2−Ax+B= 0 of the sequences {un}n/greaterorequalslant0and
+{vn}n/greaterorequalslant0has two roots
+α=A+√
+∆
+2andβ=A−√
+∆
+2,
+where ∆ = A2−4B. It is well known that for any n∈Nwe have
+Aun+vn= 2un+1,(α−β)un=αn−βn,andvn=αn+βn.
+(See, e.g., [R, pp.41–44].) Note that those Fn=un(1,−1) andLn=
+vn(1,−1) are well-known Fibonacci numbers and Lucas numbers respe c-
+tively.
+Here is our first theorem.
+Theorem 1.1. Letp >3be a prime and let A,B∈Zwithp∤A. Then
+p−1/summationdisplay
+k=1vk(A,B)Hk
+kAk≡0 (modp) (1.1)
+and
+p−1/summationdisplay
+k=1uk(A,B)Hk
+kAk≡2
+pp−1/summationdisplay
+k=1uk(A,B)
+kAk(modp). (1.2)
+Sincevk(2,1) = 2 for all k∈N, Theorem 1.1 yields the following
+consequence.ON HARMONIC NUMBERS AND LUCAS SEQUENCES 3
+Corollary 1.1 ([Su]).For any prime p >3we have
+p−1/summationdisplay
+k=1Hk
+k2k≡0 (modp).
+Remark. In 1987 S. W. Coffman [C] proved that/summationtext∞
+k=1Hk/(k2k) =π2/12.
+Applying Theorem 1.1 with A= 1 and B=−1 we get the following
+corollary.
+Corollary 1.2. Letp >3be a prime. Then
+p−1/summationdisplay
+k=1HkLk
+k≡0 (modp)andp−1/summationdisplay
+k=1FkHk
+k≡2
+pp−1/summationdisplay
+k=1Fk
+k(modp).(1.3)
+Letωdenote the cubic root ( −1+√−3)/2. Forn∈Nwe have
+un(−1,1) =un(ω+ ¯ω,ω¯ω) =ωn−¯ωn
+√−3=/parenleftBign
+3/parenrightBig
+and
+un(1,1) = (−1)n−1un(−1,1) = (−1)n−1/parenleftBign
+3/parenrightBig
+,
+where (−) denotes the Jacobi symbol. By induction, for any k∈Nwe
+have
+u4k(2,2) = 0, u4k+1(2,2) = (−4)kandu4k+2(2,2) =u4k+3(2,2) = 2(−4)k.
+Now we state our second theorem.
+Theorem 1.2. Letp >3be a prime.
+(i)LetA,B∈Zwithp∤Band(A2−4B
+p) = 1. Then, for any n∈Nwe
+have
+p−1/summationdisplay
+k=0(1+B−k)uk(A,B)Hn
+k≡0 (modp) (1.4)
+and
+p−1/summationdisplay
+k=0(1−B−k)vk(A,B)Hn
+k≡0 (modp). (1.5)
+(ii)We have
+p−1/summationdisplay
+k=0(−1)k/parenleftbiggk
+3/parenrightbigg
+Hk≡0 (modp) (1.6)4 ZHI-WEI SUN
+and
+p−1/summationdisplay
+k=0/parenleftbiggk
+3/parenrightbigg
+Hk≡(p
+3)−1
+4qp(3) (mod p), (1.7)
+whereqp(3)refers to the Fermat quotient (3p−1−1)/p. Also,
+p−1/summationdisplay
+k=0(−1)k/parenleftbiggk
+3/parenrightbigg
+kHk≡1−(p
+3)
+2(modp) (1.8)
+and
+p−1/summationdisplay
+k=0(1+2−k)uk(2,2)Hk≡0 (modp). (1.9)
+SinceF2k=uk(3,1),Fk=uk(1,−1) andLk=vk(1,−1) for all k∈N,
+Theorem 1.2(i) implies the following result.
+Corollary 1.3. Letpbe a prime with p≡ ±1 (mod 5) . Then
+p−1/summationdisplay
+k=0F2kHn
+k≡p−1/summationdisplay
+k=0
+2|kFkHn
+k≡p−1/summationdisplay
+k=0
+2∤kLkHn
+k≡0 (modp) (1.10)
+for every n= 0,1,2,....
+Our third theorem seems curious and unexpected.
+Theorem1.3. Letp >5be a prime. If (p
+15) = 1, i.e.,p≡1,2,4,8(mod 15) ,
+then
+p−1/summationdisplay
+k=0uk(1,4)
+2kHk≡0 (modp). (1.11)
+If(p
+15) =−1, i.e.,p≡7,11,13,14 (mod 15) , then
+p−1/summationdisplay
+k=0uk+1(1,4)
+2kHk≡0 (modp). (1.12)
+In the next section we are going to prove Theorems 1.1 and 1.2. Section
+3 is devoted to the proof of Theorem 1.3.
+To conclude this section we pose three related conjectures.
+Recall that harmonic numbers of the second order are given by
+H(2)
+n=/summationdisplay
+0<k/lessorequalslantn1
+k2(n= 0,1,2,...).ON HARMONIC NUMBERS AND LUCAS SEQUENCES 5
+Conjecture 1.1. Letp >3be a prime. Then
+p−1/summationdisplay
+k=0(−2)k/parenleftbigg2k
+k/parenrightbigg
+H(2)
+k≡2
+3qp(2)2(modp)
+whereqp(2) = (2p−1−1)/p. Ifp >5, then we have
+p−1/summationdisplay
+k=0(−1)k/parenleftbigg2k
+k/parenrightbigg
+H(2)
+k≡5
+2/parenleftBigp
+5/parenrightBigF2
+p−(p
+5)
+p2(modp).
+Conjecture 1.2. Letpbe an odd prime. Then
+p−1/summationdisplay
+k=0uk(2,−1)
+(−8)k/parenleftbigg2k
+k/parenrightbigg2
+≡0 (modp)ifp≡5 (mod 8) ,
+p−1/summationdisplay
+k=0uk(2,−1)
+32k/parenleftbigg2k
+k/parenrightbigg2
+≡0 (modp)ifp≡7 (mod 8) ,
+p−1/summationdisplay
+k=0vk(2,−1)
+(−8)k/parenleftbigg2k
+k/parenrightbigg2
+≡0 (modp)ifp≡5,7 (mod 8) ,
+p−1/summationdisplay
+k=0vk(2,−1)
+32k/parenleftbigg2k
+k/parenrightbigg2
+≡0 (modp)ifp≡5 (mod 8) .
+Also,
+p−1/summationdisplay
+k=0uk(4,1)
+4k/parenleftbigg2k
+k/parenrightbigg2
+≡0 (modp)ifp≡2 (mod 3) ,
+p−1/summationdisplay
+k=0uk(4,1)
+64k/parenleftbigg2k
+k/parenrightbigg2
+≡0 (modp)ifp≡11 (mod 12) ,
+p−1/summationdisplay
+k=0vk(4,1)
+4k/parenleftbigg2k
+k/parenrightbigg2
+≡p−1/summationdisplay
+k=0vk(4,1)
+64k/parenleftbigg2k
+k/parenrightbigg2
+≡0 (modp)ifp≡5 (mod 12) .
+Conjecture 1.3. Letp >3be a prime. Then
+p−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbigg/parenleftbigg2k
+k/parenrightbigg
+((−1)k−(−3)−k)≡/parenleftBigp
+3/parenrightBig
+(3p−1−1) (mod p3).6 ZHI-WEI SUN
+Ifp≡ ±1 (mod 12) , then
+p−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbigg/parenleftbigg2k
+k/parenrightbigg
+(−1)kuk(4,1)≡(−1)(p−1)/2up−1(4,1) (mod p3).
+Ifp≡ ±1 (mod 8) , then
+p−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbigg/parenleftbigg2k
+k/parenrightbigguk(4,2)
+(−2)k≡(−1)(p−1)/2up−1(4,2) (mod p3).
+2. Proofs of Theorems 1.1 and 1.2
+Letpbe an odd prime. For k= 0,1,... ,p−1 we obviously have
+(−1)k/parenleftbiggp−1
+k/parenrightbigg
+=/productdisplay
+0<j/lessorequalslantk/parenleftbigg
+1−p
+j/parenrightbigg
+≡1−pHk(modp2).(2.1)
+This basic fact is useful in the study of congruences involvi ng harmonic
+numbers.
+Lemma 2.1. Letn/greaterorequalslantj >0be integers. Then
+n/summationdisplay
+k=j/parenleftbiggk−1
+j−1/parenrightbigg
+=/parenleftbiggn
+j/parenrightbigg
+.
+Proof. This is a known identity due to Shih-chieh Chu (cf. (5.26) of [GKP,
+p.169]). Bycomparing the coefficients of xj−1onboth sides ofthe identity
+n/summationdisplay
+k=1(1+x)k−1=(1+x)n−1
+(1+x)−1,
+we get a simple proof of the desired identity. /square
+Proof of Theorem 1.1 . Letαandβbe the two roots of the equationON HARMONIC NUMBERS AND LUCAS SEQUENCES 7
+x2−Ax+B= 0. In view of Lemma 2.1,
+p−1/summationdisplay
+j=1vj(A,B)
+jAj(−1)j/parenleftbiggp−1
+j/parenrightbigg
+=p−1/summationdisplay
+j=1vj(A,B)
+jAj(−1)jp−1/summationdisplay
+k=j/parenleftbiggk−1
+j−1/parenrightbigg
+=p−1/summationdisplay
+k=1k/summationdisplay
+j=1/parenleftbiggk−1
+j−1/parenrightbigg(−1)jvj(A,B)
+jAj
+=p−1/summationdisplay
+k=11
+kk/summationdisplay
+j=1/parenleftbiggk
+j/parenrightbigg/parenleftbigg/parenleftBig
+−α
+A/parenrightBigj
++/parenleftbigg
+−β
+A/parenrightbiggj/parenrightbigg
+=p−1/summationdisplay
+k=1(1−α/A)k+(1−β/A)k−2
+k=p−1/summationdisplay
+k=1βk+αk
+kAk−2p−1/summationdisplay
+k=11
+k
+≡p−1/summationdisplay
+k=1vk(A,B)
+kAk(modp2).
+Since
+(−1)k/parenleftbiggp−1
+k/parenrightbigg
+−1≡ −pHk(modp2) fork= 1,... ,p−1,
+(1.1) follows from the above.
+Similarly,
+(α−β)p−1/summationdisplay
+j=1uj(A,B)
+jAj(−1)j/parenleftbiggp−1
+j/parenrightbigg
+=p−1/summationdisplay
+j=1(α−β)uj(A,B)
+jAj(−1)jp−1/summationdisplay
+k=j/parenleftbiggk−1
+j−1/parenrightbigg
+=p−1/summationdisplay
+k=1k/summationdisplay
+j=1/parenleftbiggk−1
+j−1/parenrightbigg(−1)j(α−β)uj(A,B)
+jAj
+=p−1/summationdisplay
+k=11
+kk/summationdisplay
+j=1/parenleftbiggk
+j/parenrightbigg/parenleftbigg/parenleftBig
+−α
+A/parenrightBigj
+−/parenleftbigg
+−β
+A/parenrightbiggj/parenrightbigg
+=p−1/summationdisplay
+k=1(1−α/A)k−(1−β/A)k
+k=p−1/summationdisplay
+k=1βk−αk
+kAk
+=(β−α)p−1/summationdisplay
+k=1uk(A,B)
+kAk.8 ZHI-WEI SUN
+Thus, if ∆ = A2−4B/ne}ationslash= 0 then
+p−1/summationdisplay
+k=1uk(A,B)
+kAk(1−pHk)≡ −p−1/summationdisplay
+k=1uk(A,B)
+kAk(modp2)
+and hence (1.2) follows.
+Now suppose that ∆ = 0. By induction, uk=k(A/2)k−1fork=
+0,1,2,.... Thus
+p−1/summationdisplay
+j=1uj(A,B)
+jAj(−1)j/parenleftbiggp−1
+j/parenrightbigg
+=p−1/summationdisplay
+j=1(−1)j
+2j−1A/parenleftbiggp−1
+j/parenrightbigg
+=2
+Ap−1/summationdisplay
+j=1/parenleftbiggp−1
+j/parenrightbigg/parenleftbigg
+−1
+2/parenrightbiggj
+=2
+A/parenleftbigg/parenleftbigg
+1−1
+2/parenrightbiggp−1
+−1/parenrightbigg
+=2
+A·1−2p−1
+2p−1
+=−2
+Ap−2/summationdisplay
+j=02j
+2p−1=−2
+Ap−1/summationdisplay
+k=11
+2k=−p−1/summationdisplay
+k=1uk(A,B)
+kAk.
+This yields (1.2) with the help of (2.1).
+So far we have completed the proof of Theorem 1.1. /square
+Lemma 2.2. LetA,B∈Zand∆ =A2−4B. Suppose that pis an odd
+prime with p∤B∆. Then we have the congruence
+/parenleftBigg
+A±√
+∆
+2/parenrightBiggp−(∆
+p)
+≡B(1−(∆
+p))/2(modp) (2.2)
+in the ring of algebraic integers.
+Proof. Bothα= (A+√
+∆)/2 andβ= (A−√
+∆)/2 are roots of the
+equation x2−Ax+B= 0. Observe that
+2αp≡2pαp= (A+√
+∆)p≡Ap+(√
+∆)p≡A+/parenleftbigg∆
+p/parenrightbigg√
+∆ (mod p).
+Similarly,
+2βp≡A−/parenleftbigg∆
+p/parenrightbigg√
+∆ (mod p).
+Thus, if (∆
+p) = 1, then
+αp−1B=αpβ≡αβ=B(modp) andβp−1B=αβp≡αβ=B(modp),ON HARMONIC NUMBERS AND LUCAS SEQUENCES 9
+henceαp−1≡1≡βp−1(modp). When (∆
+p) =−1, by the above we have
+αp+1=ααp≡αβ=B(modp) andβp+1=βpβ≡αβ=B(modp).
+This concludes the proof. /square
+Proof of Theorem 1.2 . (i) The equation x2−Ax+B= 0 has two roots
+α=A+√
+∆
+2andβ=A−√
+∆
+2,
+where ∆ = A2−4B. Also,
+Hp−1−k=Hp−1−/summationdisplay
+0<j/lessorequalslantk1
+p−j≡Hk(modp) fork= 0,1,... ,p−1.
+As (∆
+p) = 1, with the help of Lemma 2.2, for any n∈Nwe have
+p−1/summationdisplay
+k=0vk(A,B)Hn
+k=p−1/summationdisplay
+k=0vp−1−k(A,B)Hn
+p−1−k
+≡p−1/summationdisplay
+k=0/parenleftbig
+αp−1−k+βp−1−k/parenrightbig
+Hn
+k
+≡p−1/summationdisplay
+k=0/parenleftbiggβk
+Bk+αk
+Bk/parenrightbigg
+Hn
+k=p−1/summationdisplay
+k=0B−kvk(A,B)Hn
+k(modp).
+This proves (1.5). Similarly,
+(α−β)p−1/summationdisplay
+k=0uk(A,B)Hn
+k=p−1/summationdisplay
+k=0(αp−1−k−βp−1−k)Hn
+p−1−k
+≡p−1/summationdisplay
+k=0/parenleftbiggβk
+Bk−αk
+Bk/parenrightbigg
+Hn
+k= (β−α)p−1/summationdisplay
+k=0B−kuk(A,B)Hn
+k(modp).
+As (α−β)2= ∆/ne}ationslash≡0 (modp), (1.4) follows.
+(ii) Ifp≡1 (mod 3) (i.e., (−3
+p) = 1), then by putting A=±1 and
+B= 1 in (1.4) we get (1.6) and (1.7). Now we prove (1.6) and (1.7) in the10 ZHI-WEI SUN
+casep≡2 (mod 3). Observe that
+p−1/summationdisplay
+k=0(−1)k/parenleftbiggk
+3/parenrightbigg/parenleftbigg/parenleftbiggp−1
+k/parenrightbigg
+(−1)k−1/parenrightbigg
+=p−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbigg/parenleftbiggk
+3/parenrightbigg
+−p−1/summationdisplay
+k=0(−1)k/parenleftbiggk
+3/parenrightbigg
+=p−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbiggωk−¯ωk
+√−3−p−1/summationdisplay
+k=0(−1)kωk−¯ωk
+√−3
+=1√−3/parenleftbig
+(1+ω)p−1−(1+ ¯ω)p−1/parenrightbig
+−1√−3/parenleftbigg1+ωp
+1+ω−1+ ¯ωp
+1+ ¯ω/parenrightbigg
+=1√−3/parenleftbigg
+(−ω2)p−1−(−ω)p−1−−ω2p
+−ω2+−ωp
+−ω/parenrightbigg
+= 0.
+Also,
+p−1/summationdisplay
+k=0/parenleftbiggk
+3/parenrightbigg/parenleftbigg/parenleftbiggp−1
+k/parenrightbigg
+(−1)k−1/parenrightbigg
+=p−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbigg
+(−1)kωk−¯ωk
+√−3−(p−2)/3/summationdisplay
+j=12/summationdisplay
+d=0/parenleftbigg3j−d
+3/parenrightbigg
+−/parenleftbiggp−1
+3/parenrightbigg
+=1√−3/parenleftbig
+(1−ω)p−1−(1−ω2)p−1/parenrightbig
+−/parenleftbiggp−1
+3/parenrightbigg
+=1√−3(1−ω)p−1/parenleftbig
+1−(−ω2)p−1/parenrightbig
+−1
+and hence
+p−1/summationdisplay
+k=0/parenleftbiggk
+3/parenrightbigg/parenleftbigg/parenleftbiggp−1
+k/parenrightbigg
+(−1)k−1/parenrightbigg
+=1−ω2
+√−3(1−ω)p−1−1 =−ω2
+√−3(1−ω)p−1
+=−ω2
+√−3(√
+−3ω2)p−1 =−(−3)(p−1)/2ω2+2p−1
+=−/parenleftbigg
+(−3)(p−1)/2−/parenleftbigg−3
+p/parenrightbigg/parenrightbigg
+≡−(−3
+p)
+2/parenleftBigg
+(−3)p−1−/parenleftbigg−3
+p/parenrightbigg2/parenrightBigg
+=3p−1−1
+2(modp2).ON HARMONIC NUMBERS AND LUCAS SEQUENCES 11
+Combining these with (2.1) we immediately obtain (1.6) and ( 1.7).
+Next we show (1.8). Observe that
+p−(p
+3)/summationdisplay
+k=0(−1)k/parenleftbiggk
+3/parenrightbigg
+k=(p−(p
+3))/6/summationdisplay
+q=15/summationdisplay
+r=0(−1)6q−r/parenleftbigg6q−r
+3/parenrightbigg
+(6q−r)
+=(p−(p
+3))/6/summationdisplay
+q=1((6q−1)+(6q−2)−(6q−4)−(6q−5)) =p−/parenleftBigp
+3/parenrightBig
+and hence
+p−1/summationdisplay
+k=0(−1)k/parenleftbiggk
+3/parenrightbigg
+k=1+(p
+3)
+2p−/parenleftBigp
+3/parenrightBig
+.
+Also,
+p−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbigg/parenleftbiggk
+3/parenrightbigg
+k= (p−1)p−1/summationdisplay
+k=1/parenleftbiggp−2
+k−1/parenrightbigg/parenleftbiggk
+3/parenrightbigg
+=(p−1)p−2/summationdisplay
+j=0/parenleftbiggp−2
+j/parenrightbiggωj+1−¯ωj+1
+√−3=p−1√−3/parenleftbig
+ω(1+ω)p−2−¯ω(1+ ¯ω)p−2/parenrightbig
+=p−1√−3/parenleftbig
+ω(−ω2)p−2−ω2(−ω)p−2/parenrightbig
+=p−1√−3/parenleftbig
+ωp−ω2p/parenrightbig
+= (p−1)/parenleftBigp
+3/parenrightBig
+.
+Thus
+p−1/summationdisplay
+k=0(−1)k/parenleftbiggk
+3/parenrightbigg
+k/parenleftbigg/parenleftbiggp−1
+k/parenrightbigg
+(−1)k−1/parenrightbigg
+=(p−1)/parenleftBigp
+3/parenrightBig
+−/parenleftbigg1+(p
+3)
+2p−/parenleftBigp
+3/parenrightBig/parenrightbigg
+=(p
+3)−1
+2p.
+This implies (1.8) due to (2.1).
+Finally we prove (1.9). If p≡1 (mod 4) (i.e., (−4
+p) = 1), then (1.4) in
+the case A=B= 2 yields (1.9). Below we assume that p≡3 (mod 4).
+Note that
+uk(2,2) =(1+i)k−(1−i)k
+2ifor allk∈N.12 ZHI-WEI SUN
+Thus
+2ip−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbigg
+(−1)k(2−k+1)uk(2,2)
+=p−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbigg
+((−2)−k+(−1)k)/parenleftbig
+(1+i)k−(1−i)k/parenrightbig
+=/parenleftbigg
+1−1+i
+2/parenrightbiggp−1
+−/parenleftbigg
+1−1−i
+2/parenrightbiggp−1
++(1−(1+i))p−1−(1−(1−i))p−1
+=/parenleftbigg1−i
+2/parenrightbiggp−1
+−/parenleftbigg1+i
+2/parenrightbiggp−1
++(−i)p−1−ip−1
+=/parenleftbigg−2i
+4/parenrightbigg(p−1)/2
+−/parenleftbigg2i
+4/parenrightbigg(p−1)/2
+= 2ii(p+1)/2
+2(p−1)/2
+and hence
+p−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbigg
+(−1)k(2−k+1)uk(2,2) =(−1)(p+1)/4
+2(p−1)/2.(2.3)
+Also,
+2ip−1/summationdisplay
+k=0(2−k+1)uk(2,2)
+=p−1/summationdisplay
+k=0(2−k+1)/parenleftbig
+(1+i)k−(1−i)k/parenrightbig
+=1−((1+i)/2)p
+1−(1+i)/2−1−((1−i)/2)p
+1−(1−i)/2+1−(1+i)p
+1−(1+i)−1−(1−i)p
+1−(1−i)
+=(1+i)−2/parenleftbigg1+i
+2/parenrightbiggp+1
+−/parenleftBigg
+(1−i)−2/parenleftbigg1−i
+2/parenrightbiggp+1/parenrightBigg
++i−i(1+i)(1+i)p−1+i−i(1−i)(1−i)p−1
+=2i−2/parenleftbigg2i
+4/parenrightbigg(p+1)/2
++2/parenleftbigg−2i
+4/parenrightbigg(p+1)/2
++2i+(1−i)(2i)(p−1)/2−(1+i)(−2i)(p−1)/2
+=4i+2(2i)(p−1)/2= 2i/parenleftBig
+2+i(p−3)/22(p−1)/2/parenrightBig
+and hence
+p−1/summationdisplay
+k=0(2−k+1)uk(2,2) = 2−(−1)(p+1)/42(p−1)/2. (2.4)ON HARMONIC NUMBERS AND LUCAS SEQUENCES 13
+Combining (2.1), (2.3) and (2.4) we obtain
+−pp−1/summationdisplay
+k=0(2−k+1)uk(2,2)Hk
+≡(−1)(p+1)/4
+2(p−1)/2−2+(−1)(p+1)/42(p−1)/2
+≡(−1)(p+1)/4
+2(p−1)/2/parenleftBig
+2(p−1)/2−(−1)(p+1)/4/parenrightBig2
+≡0 (modp2)
+since
+/parenleftbigg2
+p/parenrightbigg
+= (−1)(p2−1)/8= (−1)(p+1)/4×(p−1)/2= (−1)(p+1)/4.
+Therefore (1.9) holds.
+By the above, we have completed the proof of Theorem 1.2. /square
+3. Proof of Theorem 1.3
+Lemma 3.1. LetA,B∈Z. Letpbe an odd prime with (B
+p) = 1. Suppose
+thatb2≡B(modp)whereb∈Z. Then
+u(p−1)/2(A,B)≡/braceleftBigg0 (modp) if(A2−4B
+p) = 1,
+1
+b(A−2b
+p) (modp)if(A2−4B
+p) =−1;
+and
+u(p+1)/2(A,B)≡/braceleftBigg(A−2b
+p) (modp)if(A2−4B
+p) = 1,
+0 (modp) if(A2−4B
+p) =−1.
+Proof. The congruences are known results, see, e.g., [S]. /square
+Lemma 3.2. Letun=un(1,4)forn∈N. Then, for any prime p >5we
+have
+up−2p−1/parenleftBigp
+15/parenrightBig
+≡2(p
+15)−2up−(p
+15)(modp2). (3.1)
+Proof. The two roots
+α=1+√−15
+2andβ=1−√−15
+2
+of the equation x2−x+4 = 0 are algebraic integers. Clearly
+−15up= (α−β)2up= (α−β)(αp−βp)≡(α−β)p+1= (−15)(p+1)/2(modp)14 ZHI-WEI SUN
+and hence
+up≡(−15)(p−1)/2≡/parenleftbigg−15
+p/parenrightbigg
+=/parenleftBigp
+15/parenrightBig
+(modp).
+Also,
+vp=αp+βp≡(α+β)p= 1 (mod p),
+wherevnrefers to vn(1,4). (In fact, both up(A,B) andvp(A,B) modulo
+pare known for any A,B∈Z.) By induction, un+vn= 2un+1for any
+n∈N.
+Case1. (p
+15) = 1.In this case,
+4up−1=up−up+1=up−vp
+2≡1−1
+2= 0 (mod p)
+and
+vp−1= 2up−up−1≡2≡2p(modp).
+Since
+(vp−1−2p)(vp−1+2p) =(αp−1+βp−1)2−4(αβ)p−1
+=(αp−1−βp−1)2=−15u2
+p−1≡0 (modp2),
+we must have vp−1≡2p(modp2) and hence
+2up=up−1+vp−1≡up−1+2p(modp2).
+Case2. (p
+15) =−1. In this case,
+2up+1=up+vp≡ −1+1 = 0 (mod p)
+and
+vp+1= 2up+2−up+1=up+1−8up≡8≡2p+2(modp).
+As
+(vp+1−2p+2)(vp+1+2p+2) =(αp+1+βp+1)2−4(αβ)p+1
+=(αp+1−βp+1)2=−15u2
+p+1≡0 (modp2),
+we must have vp+1≡2p+2(modp2) and hence
+8up=2(up+1−up+2) = 2up+1−(up+1+vp+1)
+=up+1−vp+1≡up+1−2p+2(modp2).
+Combining the above, we immediately obtain the desired resu lt./squareON HARMONIC NUMBERS AND LUCAS SEQUENCES 15
+Proof of Theorem 1.3 . Setδ= (1−(p
+15))/2. Then
+p−1/summationdisplay
+k=0uk+δ
+2kHk≡p−1/summationdisplay
+k=0uk+δ
+2k·1−(−1)k/parenleftbigp−1
+k/parenrightbig
+p(modp).
+So it suffices to show
+p−1/summationdisplay
+k=0uk+δ2p−1−k≡p−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbigg
+uk+δ(−2)p−1−k(modp2),(3.2)
+which implies (1.11) and (1.12) in the cases δ= 0,1 respectively. Recall
+that
+uk+δ=αk+δ−βk+δ
+α−β,
+where
+α=1+√−15
+2andβ=1−√−15
+2
+are the two roots of the equation x2−x+4 = 0. Since
+p−1/summationdisplay
+k=0xkyp−1−k=xp−yp
+x−yandp−1/summationdisplay
+k=0/parenleftbiggp−1
+k/parenrightbigg
+xkyp−1−k= (x+y)p−1,
+(3.2) can be rewritten as follows:
+1
+α−β/parenleftbigg
+αδαp−2p
+α−2−βδβp−2p
+β−2/parenrightbigg
+≡αδ(α−2)p−1−βδ(β−2)p−1
+α−β(modp2).(3.3)
+Note that ( α−2)(β−2) = 4+ αβ−2(α+β) = 4+4 −2 = 6 and
+αδ(αp−2p)(β−2)−βδ(α−2)(βp−2p)
+=(2p−αp)(αδ+1+αδ)+(βδ+1+βδ)(βp−2p)
+=2p(αδ+1−βδ+1+αδ−βδ)−(αp+δ+1−βp+δ+1)−(αp+δ−ββ+δ)
+=(α−β)(2p(uδ+1+uδ)−(up+δ+1+up+δ))
+=(α−β)/parenleftbig
+2p+δ−2up+δ+4up+δ−1/parenrightbig
+.
+So the left-hand side of (3.3) coincides with
+2p+δ−2up+δ+4up+δ−1
+6=2p+δ−1−up+δ+2up+δ−1
+3.16 ZHI-WEI SUN
+Since (α−2)2=−3αand (β−2)2=−3β, we have
+αδ(α−2)p−1−βδ(β−2)p−1
+α−β
+=αδ(−3α)(p−1)/2−βδ(−3β)(p−1)/2
+α−β
+=(−3)(p−1)/2α(p−1)/2+δ−β(p−1)/2+δ
+α−β= (−3)(p−1)/2u(p−(p
+15))/2.
+Applying Lemma 3.1 with A= 1 and B= 4, we get that
+u(p−(p
+15))/2≡0 (modp) andu(p+(p
+15))/2≡/parenleftbigg−3
+p/parenrightbigg
+2((p
+15)−1)/2(modp).
+(3.4)
+For anyn∈Nwe have
+u2n=α2n−β2n
+α−β=αn−βn
+α−β(αn+βn) =unvn.
+If (p
+15) = 1, then by (3.4) we have u(p−1)/2≡0 (modp) and
+v(p−1)/2= 2u(p+1)/2−u(p−1)/2≡2/parenleftbigg−3
+p/parenrightbigg
+(modp),
+hence
+up−1=u(p−1)/2v(p−1)/2
+≡2/parenleftbigg−3
+p/parenrightbigg
+u(p−1)/2≡2(−3)(p−1)/2u(p−1)/2(modp2).
+If (p
+15) =−1, then by (3.4) we have u(p+1)/2≡0 (modp) and
+v(p+1)/2=2u(p+3)/2−u(p+1)/2=u(p+1)/2−8u(p−1)/2
+≡−8/parenleftbigg−3
+p/parenrightbigg
+2−1=−4/parenleftbigg−3
+p/parenrightbigg
+(modp),
+hence
+up+1=u(p+1)/2v(p+1)/2
+≡−4/parenleftbigg−3
+p/parenrightbigg
+u(p+1)/2≡ −4(−3)(p−1)/2u(p+1)/2(modp2).
+Thus the right-hand side of (3.3) is congruent to up−(p
+15)/(2(−2)δ) mod
+p2.ON HARMONIC NUMBERS AND LUCAS SEQUENCES 17
+By the above, (3.3) is equivalent to the following congruenc e
+2p+δ−1−up+δ+2up+δ−1
+3≡up−(p
+15)
+2(−2)δ(modp2). (3.5)
+If (p
+15) = 1, then δ= 0, and hence (3.5) reduces to the congruence
+2(2p−1−up+2up−1)≡3up−1(modp2)
+which is equivalent to (3.1) since (p
+15) = 1. When (p
+15) =−1, we have
+δ= 1 and hence (3.5) can be rewritten as
+−4(2p−up+1+2up)≡3up+1(modp2)
+which follows from (3.1) since (p
+15) =−1. This concludes the proof. /square
+Acknowledgments . The author is grateful to the two referees for their
+helpful comments.
+References
+[BPQ] A. T. Benjamin, G. O. Preston and J. J. Quinn, A Stirling encounter with
+harmonic numbers , Math. Mag. 75(2002), 95–103.
+[BB] D.Borweinand J. M.Borwein, On an intriguing integral and some series related
+toζ(4), Proc. Amer. Math. Soc. 123(1995), 1191–1198.
+[C] S. W. Coffman, Problem 1240 and Solution: An infinite series with harmonic
+numbers , Math. Mag. 60(1987), 118–119.
+[GKP] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics , 2nd ed.,
+Addison-Wesley, New York, 1994.
+[HT] C. Helou and G. Terjanian, On Wolstenholme’s theorem and its converse , J.
+Number Theory 128(2008), 475–499.
+[R] P. Ribenboim, The Book of Prime Number Records , Springer, New York, 1989.
+[S] Z. H. Sun, Values of Lucas sequences modulo primes , Rocky Mount. J. Math.
+33(2003), 1123–1145.
+[Su] Z. W. Sun, Arithmetic theory of harmonic numbers , Proc. Amer. Math. Soc.,
+in press.
+[W] J. Wolstenholme, On certain properties of prime numbers , Quart. J. Math. 5
+(1862), 35–39.
\ No newline at end of file
diff --git a/1001.0349.txt b/1001.0349.txt
new file mode 100644
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@@ -0,0 +1,443 @@
+arXiv:1001.0349v1  [astro-ph.SR]  3 Jan 2010Selection, Prioritization, and Characteristics of KeplerTarget
+Stars
+Natalie M. Batalha
+Department of Physics and Astronomy, San Jose State University , San Jose, CA 95192
+Natalie.Batalha@sjsu.edu
+William J. Borucki, David G. Koch, Stephen T. Bryson, and Michael.R. Ha as
+NASA Ames Research Center, Moffett Field, CA
+Timothy M. Brown
+Las Cumbres Observatory Global Observatory Telescope Networ k, Goleta, CA 93117
+Douglas A. Caldwell
+SETI Institute, Mountain View, CA 94043
+Jennifer R. Hall
+Orbital Sciences Corporation/NASA Ames Research Center, Moffe tt Field, CA 94035
+Ronald L. Gilliland
+STScI, Baltimore, MD 21218
+David W. Latham and Soren Meibom
+Harvard-Smithsonian, CfA, Cambridge, MA 02138
+and
+David G. Monet
+U.S. Naval Observatory, Flagstaff, AZ 86001
+Received ; accepted– 2 –
+ABSTRACT
+TheKepler Mission began its 3.5-year photometric monitoring campaign in
+May 2009 on a select group of approximately 150,000 stars. The sta rs were
+chosen from the ∼half million in the field of view that are brighter than 16th
+magnitude. The selection criteria are quantitative metrics designed to optimize
+the scientific yield of the mission with regards to the detection of Ear th-size
+planets in the habitable zone. This yields more than 90,000 G-type sta rs on or
+close to the Main Sequence, >20,000of which are brighter than 14thmagnitude.
+At the temperature extremes, the sample includes approximately 3 ,000 M-type
+dwarfs and a small sample of O and B-type MS stars ( <200). Small numbers
+of giants are included in the sample which contains ∼5,000 stars with surface
+gravities log( g)<3.5. We present a brief summary of the selection process and
+the stellar populations it yields in terms of surface gravity, effective temperature,
+and apparent magnitude. In addition to the primary, statistically-d erived target
+set, several ancillary target lists were manually generated to enha nce the science
+of the mission, examples being: known eclipsing binaries, open cluster members,
+and high proper-motion stars.
+Subject headings: stars: fundamental parameters — techniques: photometric —
+catalogs– 3 –
+1. Introduction
+Kepler’s primary objective is to detect Earth-size planets in the habitable zo ne of
+Main Sequence stars. The large field of view (FOV) targeting the rich star field in Cygnus
+(∼10 degrees off the galactic plane) was carefully chosen to optimize th e science yield
+with regards to that specific objective (Batalha et al. 2006; Jenkin s et al. 2005). Even so,
+approximately 40% of the stars brighter than R=16.5 that fall on silic on are expected
+to be giants and supergiants according to simulations (Robin et al. 19 93) of the stellar
+populations in that region of the sky (Batalha et al. 2006). During th e mission design
+phase, it was recognized that significant improvements in efficiency c ould be realized by
+pre-selecting the target stars. On the order of 105stars brighter than 16th magnitude in
+theKeplerpassband (denoted herein as Kp) were expected to have the right properties for
+detectability of terrestrial-size planets, implying that less than 10% of the pixels need to
+be downloaded from the spacecraft. Furthermore, pre-selectio n of the Main Sequence stars
+eliminates entire classes of astrophysical false-positives, reducin g the resources required by
+the ground-based follow-up teams.
+Beginning with the pre-launch stellar classification effort (Section 2) , we describe the
+methodology for cherry-picking 150,000 stars from the ∼0.5 million in the field of view
+brighter than 16th magnitude (Section 3). We summarize the criter ia used to prioritize
+the stellar targets (Section 4) and present a summary of the char acteristics of the sample
+(Section 5). The stellar sample described here is specific to the Quar ter 4 (mid-December
+2009 through mid-March 2010) targets. The populations differ sligh tly from those observed
+May-December 2009 (Batalha et al. 2009) due to revised estimates of the expected per
+cadence noise and a change in the definition of the crowding metric. T he target selection
+described herein utilizes a crowding metric computed using a set of pix els that define the
+optimal photometric aperture (Bryson et al. 2010) for each individ ual star as opposed to a– 4 –
+fixed 11×11 pixel aperture centered on each star. Lastly, we provide an ov erview of the
+ancillary target lists drawn from specialized sources (Section 6). Wh ile these comprise a
+small percentage of the sample, they contribute significantly to th e primary objectives of
+the mission.
+2. The Kepler Input Catalog
+A ground-based observing campaign was initiated in 2004 to determin e the apparent
+magnitude ( Kp), surface gravity, effective temperature, metallicity, and inters tellar
+extinction of stars in the Keplerfield from broad and intermediate band photometry.
+An estimate of the stellar radius and mass follows via isochrone interp olation. The
+calibrated photometry and resulting stellar parameters are archiv ed in the Kepler
+Input Catalog (KIC) which is publicly available at the Multi-Mission Archive at Space
+Telescope Science Institute1. We provide, here, a brief summary of the process
+leading to stellar classification. A full description of the algorithms is a vailable at
+http://www.cfa.harvard.edu/kepler/kic/kicindex.html .
+A custom photometer (KeplerCam) was built for the 48-inch telesco pe at Whipple
+Observatory on Mt. Hopkins utilizing a Fairchild CCD486 (Szentgyorg yi et al. 2005).
+The campaign required multiple visits to >1,600 pointings (Latham et al. 2005) covering
+theKeplerFOV and the open cluster, M67 (used for color calibrations and reliab ility
+tests). Multiple exposures were taken in the (Sloan-like) g,r,i,zfilters as well as a custom
+intermediate-band filter, D51, centered at 510 nm (Mg b line). A sma ll number of pointings
+included uandGred(a custom intermediate-band filter centered at 432 nm) observat ions.
+However, the long exposure times required for the requisite SNR we re prohibitively costly,
+1http://archive.mast.edu/kepler– 5 –
+and these filters were dropped during the earlier stages of the pro gram. Surface gravity
+dependence is largely provided by the D51 measurements. For the M -type stars, J−Kis a
+more sensitive diagnostic. The 2MASSJ,H,K magnitudes are federated with the calibrated
+g,r,i,z, and D51 magnitudes for the purpose of stellar classification. Corr ections for
+atmospheric extinction are based on nightly observations of ∼1500 secondary photometric
+standards within the Kepler field. These in turn are tied to >300 stars from SDSS DR1
+(Smith et al. 2002), selected for distribution on the sky and range in color.
+Theg,r,i,z, D51, and J,H,K magnitudes yield 7 independent color measurements t hat
+are calibrated by zero-point offsets derived from cluster observa tions and Sloan standard
+stars. These 7 observables are used together with the rmagnitude and the galactic latitude
+to derive 11 parameters, namely: effective temperature (T eff), surface gravity (log( g)),
+metallicity (log( Z)), mass, radius, luminosity, bolometric correction, distance, inte rstellar
+extinction ( ArandAV), and reddening ( EB−V). Functions relating these parameters
+(e.g. extinction laws (Cardelli, Clayton, & Mathis 1989), the radius de pendence on mass
+and surface gravity, the luminosity dependence on effective tempe rature and radius, and
+the apparent magnitude dependence on luminosity, the bolometric c orrection, interstellar
+extinction, and distance) reduce the number of unknowns from 11 to 5.
+Stellar classification requires choosing among model stellar atmosph eres that best fit the
+observables. The Castelli & Kurucz (2004) models provide flux as a f unction of wavelength
+for temperatures ranging from 3500 to 50000 K, surface gravitie s of 0 to 5.5 dex, and
+metallicities of −3.5 to +0.5. Fluxes are transformed to magnitudes by multiplying by an
+estimate of the CCD response and filter transmission and integratin g over wavelength. The
+model colors (formed taking relevant magnitude differences) are c alibrated to the cluster
+observations using a zero-point offset and a term linear in ( g−r). The evolutionary tracks
+of Girardi et al. (2000) are used to constrain stellar luminosity (and , ultimately, radius and– 6 –
+mass) given effective temperature and surface gravity (assuming solar metallicity).
+The photometric data are minimally sufficient for obtaining reliable surf ace gravity
+determinations. The parameter estimation is, therefore, carried out using additional
+astrophysical information to form Bayesian priors. A star’s physic al parameters (given as
+the vector /vector x) are those which maximize the posterior probability of obtaining the o bserved
+magnitudes and colors (given as the vector /vector q). The posterior probability, P(/vector x|/vector q), is given
+by:
+P(/vector x|/vector q) = [P0(x)P(/vector q|/vector x)]/P0(/vector q). (1)
+The denominator in this expression, the a priori probability of obser ving photometric
+indices given by /vector q, is a normalization that does not depend upon the parameters /vector xof
+interest; for purposes of maximizing the posterior probability, it ma y be ignored.
+Assuming independent, Gaussian-distributed errors, one can writ e
+P(/vector q|/vector x) =exp(−χ2), (2)
+whereχ2is the usual goodness-of-fit statistic. The prior probability, P0(/vector x), is parametrized
+as
+P0(/vector x) =PHR[Teff,log(g)]PZ[logZ]Pz(z), (3)
+wherePHRis the probability of finding a star in a given region of the HR diagram
+(parametrized in terms of effective temperature and surface gra vity),PZis the probability
+of having a metallicity Z, andPzis the probability of being situated at a height, z, above
+the galactic plane. PHRis constructed from the 9,590 Hipparcos stars having parallaxes
+known to better than 10%. The metallicity distribution is taken from t he compilation of
+Nordstr¨ om et al. (2004) for 14,000 bright, nearby stars. The st ellar density is assumed to
+decrease exponentially with a scale height of 300 pc. It allows for the cubic increase in
+volume with increasing distance.– 7 –
+The advantage of employing the Bayesian method is that implausible st ellar
+classifications are ruled out a priori. The disadvantage is that stars with rare properties
+are liekly to be misclassified. For our primary objective – that of distin guishing giants
+from dwarfs – the Bayesian approach can be expected to work well. Stellar parameters are
+derived for ∼80% of the stars brighter than 17th magnitude. The classification c an fail
+for a multitude of reasons, the most common of which is related to th e completeness of
+the photometric measurements. Classification is not attempted if t here are fewer than 2
+valid optical magnitudes or 3 NIR magnitudes in the database. This co mpleteness rate is
+independent of magnitude for stars brighter than r= 17.
+The photometry and derived parameters are archived in the KIC to gether with results
+from other photometric survey catalogs (e.g. 2MASS, USNO-B, Ty cho, and Hipparcos)
+in order to provide a full census of point-sources in the field down to SDSSg= 20
+corresponding to the completeness of the USNO-B catalog Monet e t al. (2003). The stellar
+parameters form the basis of target selection and prioritization.
+During the commissioning phase of Kepleroperations, 9.7 days of 30-minute cadence
+photometry was collected on a sample of 52,496 of the brightest sta rs (excluding those
+known to be highly crowded by neighboring stars). From this sample, one-thousand
+bright, unsaturated, known giants (well-studied astrometric grid stars) and MS dwarfs (as
+indicated by KIC classifications and, in some cases, confirmed by Hipp arcos parallaxes)
+were used to assess the reliability of the classifications in the Kepler Input Catalog . A study
+of 7 days of HST photometry (5 samples per 96-minute orbit) of bulg e stars suggests that
+high-precision light curves at sufficiently high cadence can be used as a luminosity class
+discriminator (Gilliland 2008). The RMS variability of the giant light curve s is typically
+at least one order of magnitude larger than that of dwarfs. Giants also present distinct
+frequency characteristics: quasi-periodic variations with severa l distinct frequencies over a– 8 –
+narrow range while dwarfs exhibit predominantly white noise over a 10 -day window. This
+photometric luminosity class discriminator was applied to the Keplerlight curves of the
+giant/dwarf control samples. The results were inspected manually to assess the reliability
+of the stellar classifications in the Kepler Input Catalog with regards to giant/dwarf
+separation. The KIC designations agree with the variability discrimina tor>90% of the
+time for unsaturated stars brighter than 13th magnitude.
+3. Target Selection
+We approach target selection by computing, for every KIC-classifi ed star in the field,
+the radius of the smallest planet detectable at the 7.1-sigma level (t hreshold at which we
+expect less than one statistical false-positive) over the mission life time. The total SNR is
+the ratio of the transit depth (which depends on the ratio of the pla net to star surface
+area) to the total noise, σtot, in the folded and binned light curve. The calculation requires
+knowledge of the stellar parameters as well as the instrument perf ormance. The per-cadence
+noise is modeled and scaled by 1 /√NtrwhereNtris the number of transits observed over
+the 3.5-year mission duration and by 1 //radicalbig
+NsamplewhereNsampleis the number of 30-minute
+samples per transit event to give σtot. The number of transits depends on the orbital period
+for a given stellar mass, M∗, and semi-major axis, a, while the number of samples per
+transit depends on the transit duration ( ttr) which, assuming a circular orbit, is given by:
+ttr=π
+4t0= 2R∗/radicalbigga
+GM∗(4)
+wheret0is the central transit duration. It is scaled by π/4 to give the mean transit duration
+for stars randomly distributed in inclination.
+The minimum detectable planet radius, Rp,min, is computed at three semi-major axes:
+1) the inner radius of the habitable zone (HZ), 2) half that distance , and 3) five stellar radii.– 9 –
+A planet is not considered detectable unless the total SNR exceeds 7.1-sigma and at least
+three transits occur over the 3.5-year mission lifetime. We use the s tandard solar-system
+inner HZ radius of Kasting, Whitmire, & Reynolds (1992) (0.95 AU) and scale it by the
+ratio of the stellar luminosity (computed from the TeffandR∗archived in the KIC) to that
+of the Sun.
+The completeness of the KIC with regards to point sources down to SDSSg= 20
+allows us to simulate the sky and estimate the crowdedness of every point source. Crowding
+can lead to flux contamination in the photometric aperture of the ta rget star. The excess
+flux dilutes the transits in the photometric light curve, thereby red ucing their detectability.
+A crowding metric, r, is defined to be the flux of the target star, F∗, in the photometric
+aperture expressed as a fraction of the total flux (star plus bac kground flux, Fbg, which
+includes contributions from zodiacal emission as well as background stars):
+r=F∗
+F∗+Fbg. (5)
+The photometric aperture is defined as the set of pixels that optimiz es the total SNR and is
+referred to as the Optimal Aperture . It is dependent on the local Pixel Response Function
+(PRF), measured on-orbit during the commissioning period (Bryson et al. 2010), as well as
+the distribution of stellar flux on the sky near the target. The latte r relies on information
+from the KIC. An optimal aperture and crowding metric is computed for every po tential
+target star.
+Figure 1 shows a small region (2 ×2 arcmin) of a simulated image (left), sized to
+highlight individual stars. Also shown (right) is the same region of an a ctual full-frame
+image taken on orbit by Kepler. Individual stars are well-reproduced by simulations as are
+the pixel-to-pixel brightness differences. Note that the same gra y-scale mapping and range
+is used for each image. The middle panel shows the same region with th e optimal aperture
+pixels blackened out. The diluted fractional transit depth, f, is related to the intrinsic– 10 –
+fractional depth, f0, via the crowding metric, r:
+f=r×f0. (6)
+Combining the above formulation, the minimum detectable planet radiu s,Rp,min, is
+given by:
+Rp,min=R∗/radicalbigg
+7.1σtot
+r, (7)
+whereR∗is the stellar radius. More than 200,000 stars in Kepler’s FOV have a
+minimum detectable planet radius smaller than two earth-radii (2 Re) at a semi-major axis
+corresponding to the inner HZ for that star as well as 3 or more tra nsits over the lifetime
+of the mission. However, more than 60% are fainter than 15th magn itude and are not,
+consequently, amenable to high-precision RV follow-up. Clearly, a st rict sorting on Rp,min
+does not yield the desired population. A more suitable prioritization sc heme is required.
+4. Prioritization
+Table??lists the prioritization criteria, ordered from high to low priority, and the
+resulting cumulative star counts. The highest priority targets are those for which a
+terrestrial-size planet is detectable in the HZ and for which 1-3 m/s r adial velocity precision
+can be readily achieved with existing instruments and methods ( Kp≤14). It is recognized
+that the very late-type stars will tend to be faint, especially in the o ptical, and that these
+populations will benefit considerably from the high-resolution infrar ed spectrometers and
+RV templates that are being developed to make use of the proportio nately higher photon
+fluxes these objects emit in the IR. Consequently, the next highes t priority targets are
+those for which Earth-size planets are detectable in the HZ even if t hey are as faint as
+15-16 magnitudes. The fainter K and M dwarfs will be captured here . The HZ constraint is
+relaxed next, bringing in stars that did not meet the 3-transit crite rion ata=HZas well– 11 –
+as those that benefit from the additional SNR built up by observing m ore transits at the
+shorter period orbits.
+The end product is a sample of 261 ,636 stars brighter than 16th magnitude that fall
+on silicon for which a planet of radius ≤2Reis detectable (at acceptable confidence levels)
+over the duration of the mission. We do not make an attempt to includ e predictions with
+regards to stellar variability in the noise models used for target selec tion (since so little is
+known about stellar variability across the HR diagram at this precision ). Such a metric can
+be included in the future as we learn more about the target stars an d will be important in
+the event we are forced to downselect. The final target list is draw n from this prioritized
+list of stars, trimmed down to meet the mission constraints: number of targets <170,000
+and number of pixels <5.44 million.
+5. Characteristics of the Target Stars
+Table 1 reports the star counts from amongst the 150,000 highest priority targets,
+binned by apparent magnitude and effective temperature and sepa rated by surface gravity
+(logg≥3.5 and log g <3.5). Some giants make it into the target sample when the
+combination of apparent magnitude and stellar radius allows for the d etection of planets
+as small as 2 Rein an orbit as close as 5 R∗. The sample is dominated by G-type stars on
+or near the Main Sequence and stars fainter than 14th magnitude. At the temperature
+extremes, we have ∼3,000 M-type Main Sequence stars and <200 O and B-type stars.
+OB-type stars are rare because of their short MS lifetimes and the fact that the field of view
+was chosen to avoid young stellar populations (OB associations and s tar forming regions).
+Table 2 reports star counts for the populations that did not satisf y the detectability
+criteria. These are the stars brighter than 16th magnitude which c an be included in– 12 –
+proposals for the Guest Observer Program2. The targets brighter than 14th magnitude are
+almost exclusively evolved stars. Main Sequence stars not being obs erved by the mission
+are predominantly G-type stars fainter than 15th magnitude.
+6. Ancillary Target Lists
+Table 1 represents the statistically derived PLANETARY target list. Several smaller,
+ancillary target lists have been constructed to ensure no high prior ity targets are missed. All
+known eclipsing binaries ( EB) in the field ( >600) are included on an EBtarget list drawn
+from ground-based surveys (Pigulski et al. 2009; Mjaseth et al. 2 007; Hartman et al. 2004)
+and from EB’s already flagged in the SIMBAD database. EB light curve s will be combed
+for transit events and subjected to eclipse timing using techniques developed by Kepler
+Participating Scientists . There are four open clusters in the Keplerfield of view: NGC 6866
+(∼0.5 Gyr), NGC 6811 ( ∼1 Gyr), NGC 6819 ( ∼2.5 Gyr), and NGC 6791 ( ∼10 Gyr). All
+known members are included in a CLUSTER target list. The clusters will be used to extend
+the age-rotation relation beyond the age of the Hyades (Meibom 20 09). This calibration will
+help to define the ages of stars with planets. The closest Main Seque nce stars are explicitly
+added to the target list by including a) 141 high proper-motion K and M dwarfs from
+the LSPM-North catalog (Lepine & Shara 2005), b) stars with Hippa rcos parallaxes that
+confirm their Main Sequence status ( ∼200 stars), and c) M-dwarfs from the Gliese catalog
+(∼10). Many of the targets captured in the above lists are also identifi ed by the statistical
+selection process based on detectability metrics. However, explicit inclusion ensures that
+these high priority targets will not be inadvertently dropped in futu re down-selects.
+Finally, as noted in Section 2, approximately 20% of the stars in the Kepler Input
+2http://keplergo.arc.nasa.gov– 13 –
+Catalogbrighter than Kp= 17 lack effective temperature, surface gravity, and radius
+estimates. Consequently, we began science operations observing ∼10,000 unclassified
+stars brighter than 14th magnitude. The number has been reduce d by applying the
+photometric luminosity class discriminator described in Section 2 to th e data collected
+during commissioning and the first 33 days of science operations. 1,9 96 of the ∼10,000
+stars have been explicitly included as a result of the dwarf/giant disc riminator while 2,272
+of the∼10,000 stars have been explicitly excluded. The remaining ∼6,000 remain on the
+unclassified target list pending additional data. The unclassified sta rs that were determined
+to be giants comprise part of the Quarter 2 Dropped Target List. T he light curves of
+dropped targets are made available to the public 60 days after relea se (Haas et al. 2010).
+7. Summary
+Kepleris observing more than 150,000 stars during the first year of mission operations,
+more than 90,000 of which are G-type stars on or near the Main Sequ ence. 10,575 are
+G-type stars brighter than 13th magnitude. 28,519 are brighter t han 14th magnitude.
+Over 90% of the target stars are selected based on detectability m etrics that suggest a
+terrestrial-size planet ( R <2Re) is detectable in 3.5 years. The detectability metrics are
+derived from stellar properties in the Kepler Input Catalog . Stellar classifications (surface
+gravity, effective temperature, and the inferred stellar radius) a re complete at the 80%
+level, independent of magnitude. A photometric luminosity class discr iminator is applied to
+1,000 Main Sequence and 1,000 Giant light curves (as indicated by KIC c lassifications and
+confirmed, in some cases, by Hipparcos parallaxes) from commission ing data. We report
+agreement between the KIC and photometric discriminators >90% of the time for stars
+brighter than 13th magnitude. Priority on the target list is given to s tars that can be
+followed up with high-precision radial velocity measurements from cu rrent ground-based– 14 –
+facilities. High priority is also assigned to the fainter (14 ≤Kp <16) K and M-type dwarfs
+that will benefit from future IR spectrometers. Ancillary target lis ts are generated for
+special purposes (e.g. to detect planets around EBs, to extend t he age-rotation relation for
+Main Sequence stars, and to ensure that the closest Main Sequenc e stars are included).
+Finally the characteristics of the stars that are not being observe d are provided for
+prospective Guest Observers.
+Funding for this Discovery mission is provided by NASA’s Science Mission Directorate.– 15 –
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+Jenkins, J., Koch, D. G. 2006, BAAS, 38, 1188
+Batalha, N. M., Borucki, W. J., Koch, D. G., Brown, T. M., Caldwell, D. A., Latham, D. W.
+2009, Planetary Systems as Potential Sites for Life, proceedings of the International
+Astronomical Union, IAU Symposium, in press
+Brown, T. M., Everett, M., Latham, D. W., Monet, D. G. 2005, BAAS, 37, 1340
+Bryson, S. T., Borucki, W. J., Koch, D. G., Caldwell, D. A., Gilliland, R. L., J enkins, J.
+M., Dotson, J. L., Haas, M. R., Allen, C., Klaus, T. C., Tenenbaum, P. 20 10, ApJ,
+submitted
+Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245
+Castelli, F. & Kurucz, R. L. 2004, arXiv:astro-ph/0405087
+Demarque, P., Woo J., Kim, Y., Yi, S. 2004, ApJS, 155, 667
+Gilliland, R. L. 2008, AJ, 136, 566
+Girardi, L., Bressan, A., Bertelli, G., Chiosi, C. 2000, A&AS, 141, 371
+Haas, M. R., Batalha, N. M., Bryson, S. T., Caldwell, D. A., Dotson, J. L ., Hall, J.,
+Jenkins, J. M., Klaus, T. C., Koch, D. G., Kolodziejczak, J. J., Middour , C., Smith,
+M., Sobeck, C., Stober, F., Thompson, R., Van Cleve, J. E. 2010, ApJ , submitted
+Hartman, J. D., Bakos, G., Stanek, K. Z., Noyes, R. W. 2004, AJ, 12 8, 1761
+Jenkins, J. M., Chandrasekharan, H., Caldwell, D. A., Batalha, N. M., S ilva, N., Koch, D.
+G., Gautier, T. N. 2005, BAAS, 37, 684– 16 –
+Kasting, J. F., Whitmire, D. P., Reynolds, R. T. 1993, Icarus, 101, 1 08
+Koch, D. G. et al. 2010, ApJ, submitted
+Latham, D. W., Brown, T. M., Monet, D. G., Everett, M., Esquerdo, G . A., Hergenrother,
+C. W. 2005, BAAS, 37, 1340
+Lepine, S., Shara, M. M. 2005, AJ, 129, 1483
+Meibom, S. 2009, The Ages of Stars, IAU Symposium, 258, 357
+Mjaseth, K., Batalha, N., Borucki, W., Caldwell, D., Latham, D., Martin, K. R., Rabbette,
+M., Witteborn, F. 2007, BAAS, 38, 98
+Monet, D. G., Levine, S. E., Canzian, B., Ables, H. D., Bird, A. R., Dahn, C. C., Gietter,
+H. H., Harris, H. C., Henden, A. A., Leggett, S. K., Levison, H. F., Lug inbuhl, C.
+B., Martini, J., Monet, A. K. B., Munn, J. A., Pier, J. R., Rhodes, A. R., R iepe, B.,
+Sell, S., Stone, R. C., Vrba, F. J., Walker, R. L., Westerhoult, G. 2003 , AJ, 125, 984
+Nordstr¨ om, B., Mayor, M., Andersen, J., Holmberg, J., Pont, F., Jo rgensen, B. R., Olsen,
+E. H., Udry, S., Mowlavi, N. 2004,˚ a, 418, 989
+Pigulski, A., Pojmanski, G., Pilecki, B., Szczygiel, D. M. 2009, Acta Astr onomia, 59, 33
+Robin, A. C., Reyle, C., Derriere, S., Picaud, S. 1993,˚ a, 409, 523
+Smith, J. A., Tucker, D. L., Kent, S., Richmond, M. W., Fukugita, M., Ic hikawa, T.,
+Ichikawa, S., Jorgensen, A. M., Uomoto, A., Gunn, J. E., Hamabe, M., Watanabe,
+M., Tolea, A., Henden, A., Annis, J., Pier, J. R., McKay, T. A., Brinkmann , J.,
+Chen, B., Holtzman, J., Shimasaku, K., & York, D. G. 2002, AJ, 123, 2 121
+Stoughton, C. & Jester, S. 2004, AGN Physics with the Sloan Digital Sky Survey, eds. G.
+T. Richards & P. B. Hall, ASP Conference Series, 311, 475– 17 –
+Szentgyorgyi, A. H., Geary, J. G., Latham, D. W., Groner, L. T., Ama to, S., Bennett, K.,
+Falco, E. E., Peters, W. A., Ordway, M. P., Fata, R. G. 2005, BAAS, 3 7, 1339
+This manuscript was prepared with the AAS L ATEX macros v5.2.– 18 –
+Simulated
+705 715 725485495505Optimal Apertures
+705 715 725Observed
+705 715 725
+Fig. 1.— A comparison of the thumbnails of 1) a simulated image (left) an d 2) a full-frame
+image taken during flight (right) shows that the sky is well-reproduc ed by models at even the
+pixel level. Simulated images use the Kepler Input Catalog as input as well as models of the
+instrument characteristics produced during the commissioning per iod (e.g. Pixel Response
+Function, Focal Planet Geometry, etc). The middle panel shows th e mask that defines the
+optimal aperture pixels for the target stars in this region of the sk y.– 19 –
+Table 1: Star counts as a function of effective temperature and ap parent magnitude ( Kp)
+for the 150,000 highest priority target stars.
+MAG 10500 9500 8500 7500 6500 5500 4500 3500 TOTAL
+logg≥3.5
+6.50 1 0 1 2 0 1 0 0 5
+7.50 1 8 9 6 8 6 0 0 38
+8.50 8 20 25 24 49 15 7 8 156
+9.50 9 31 81 66 116 88 11 4 406
+10.50 27 37 100 209 405 359 40 9 1186
+11.50 24 58 171 398 1499 1356 158 37 3701
+12.50 30 44 231 676 4146 4760 626 62 10575
+13.50 34 53 170 747 9279 15866 2213 157 28519
+14.50 3 0 0 0 4855 29352 4227 554 38991
+15.50 7 4 0 0 4449 42627 12093 1961 61141
+TOTAL 144 255 788 2128 24806 94430 19375 2792 144718
+logg <3.5
+6.50 0 0 0 0 0 0 0 0 0
+7.50 0 0 1 1 2 2 7 0 13
+8.50 0 0 1 2 2 9 80 2 96
+9.50 0 0 2 15 2 27 220 1 267
+10.50 1 0 5 21 7 99 452 2 587
+11.50 0 0 1 25 11 186 674 2 899
+12.50 0 0 1 12 14 347 1114 0 1488
+13.50 0 0 0 6 5 518 1403 0 1932
+14.50 0 0 0 0 0 0 0 0 0
+15.50 0 0 0 0 0 0 0 0 0
+TOTAL 1 0 11 82 43 1188 3950 7 5282– 20 –
+Table 2: Star counts as a function of effective temperature and ap parent magnitude ( Kp) for
+the remaining KIC stars with stellar classifications that are brighter than 16th magnitude.
+MAG 10500 9500 8500 7500 6500 5500 4500 3500 TOTAL
+logg≥3.5
+6.50 0 0 0 0 0 0 0 0 0
+7.50 0 0 0 0 0 0 0 0 0
+8.50 0 0 0 0 0 0 0 0 0
+9.50 0 0 0 0 0 0 0 0 0
+10.50 0 0 0 0 0 0 0 0 0
+11.50 0 0 0 0 0 0 0 0 0
+12.50 0 0 0 0 0 0 0 0 0
+13.50 0 0 0 0 0 0 0 0 0
+14.50 50 74 212 707 11340 14083 2998 1 29465
+15.50 66 87 277 1038 25476 55506 7737 107 90294
+TOTAL 116 161 489 1745 36816 69589 10735 108 119759
+logg <3.5
+6.50 0 0 0 0 0 0 1 0 1
+7.50 0 0 0 0 0 0 4 2 6
+8.50 0 0 0 0 0 1 81 52 134
+9.50 0 0 0 0 0 8 295 125 428
+10.50 0 0 0 0 0 37 989 273 1299
+11.50 0 0 0 0 1 98 2691 404 3194
+12.50 0 0 0 1 1 209 5673 505 6389
+13.50 0 0 1 1 2 375 10254 424 11057
+14.50 0 0 0 2 13 1251 15328 311 16905
+15.50 0 0 0 5 11 1672 15657 252 17597
+TOTAL 0 0 1 9 28 3651 50973 2348 57010– 21 –
+Table 3: Prioritization: cumulative star counts
+Criteria Counts
+Ntr≥3;Rp,min≤1Re;a=HZ;Kp <13 3742
+Ntr≥3;Rp,min≤2Re;a=HZ;Kp <13 4169
+Ntr≥3;Rp,min≤1Re;a=HZ;Kp <14 9323
+Ntr≥3;Rp,min≤2Re;a=HZ;Kp <14 17375
+Ntr≥3;Rp,min≤1Re;a=HZ;Kp <15 22714
+Ntr≥3;Rp,min≤1Re;a=HZ;Kp <16 25610
+Ntr≥3;Rp,min≤1Re;a=1
+2HZ;Kp <14 30802
+Ntr≥3;Rp,min≤2Re;a=1
+2HZ;Kp <14 46366
+Ntr≥3;Rp,min≤2Re;a= 5R∗;Kp <14 58103
+Ntr≥3;Rp,min≤2Re;a=HZ;Kp <15 91755
+Ntr≥3;Rp,min≤2Re;a=HZ;Kp <16 154008
+Ntr≥3;Rp,min≤2Re;a= 5R∗;Kp <15 183563
+Ntr≥3;Rp,min≤2Re;a= 5R∗;Kp <16 261636
\ No newline at end of file
diff --git a/1001.0350.txt b/1001.0350.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2fa732ea1d1b53874a766f149c28ce6185773c52
--- /dev/null
+++ b/1001.0350.txt
@@ -0,0 +1,94 @@
+arXiv:1001.0350v2  [physics.optics]  5 Jan 2010Note on the positiveness of magnetic energy stored in the Wir e-SRR metamaterial
+Pi-Gang Luan
+Wave Engineering Laboratory, Department of Optics and Phot onics,
+National Central University, Jhungli 320, Taiwan
+Recently, a new energy density formula for electromagnetic waves in the metamaterial consisting
+of arrays of conducting wires and split-ring-resonators (S RR) has been derived [Phys. Rev. E 80,
+046601 (2009)]. According to that formula, the positivenes s is obvious for the electric part of the
+energy density but not clear for the magnetic part. In this pa per, I show that the magnetic energy
+density is also positively definite.
+PACS numbers: 41.20.Jb, 03.50.De, 78.20.Ci
+In a recent paper, Luan derived a new expression for
+energydensityofelectromagneticwavesinthemetamate-
+rial medium consisting of wires and split-ring-resonators
+(SRR) [1]. The motivation of that paper was to resolve
+the apparent contradictions between the results given by
+the equivalent circuit (EC) and electrodynamic (ED) ap-
+proaches [2, 3]. Luan pointed out a calculation error
+made in [2], which can be corrected so that EC approach
+can really give the correct formula. On the other hand,
+usually the ED approach [3] does not give unique result,
+and the analysis in [1] revealed that the expression of the
+energy density is in fact determined by the power loss we
+choose. For a lossy medium, Luan suggested to find the
+correct expression of power loss first, then the correct
+energy density formula will emerge automatically. The
+equivalence between EC and ED approaches was then
+confirmed by reproducing the EC energy formula after
+time averaging the ED formula.
+The positiveness of the electric energy density can be
+easily observed from the following expressions
+We=ǫ0E2
+2+1
+2ω2pǫ0/parenleftbigg∂P
+∂t/parenrightbigg2
+, (1)
+/angbracketleftWe/angbracketright=ǫ0|E|2
+4/parenleftBigg
+1+ω2
+p
+ω2+ν2/parenrightBigg
+. (2)
+Although the positiveness of magnetic energy density in
+the time-domain formula is obvious (note that 0 < F <1
+is assumed)
+Wb=µ0(1−F)
+2H2+µ0
+2F(M+FH)2(3)
++µ0
+2ω2
+0F/parenleftbigg∂M
+∂t+F∂H
+∂t+γM/parenrightbigg2
+,(4)this property is not clear in the time averaged formula
+/angbracketleftWb/angbracketright=µ0|H|2
+4/bracketleftBigg
+1+Fω2/parenleftbig
+3ω2
+0−ω2/parenrightbig
+(ω2
+0−ω2)2+ω2γ2/bracketrightBigg
+.(5)
+Whenω >√
+3ω0, the above formula gives a /angbracketleftWb/angbracketright
+smaller than the vacuum value µ0|H|2/4. Will this ex-
+pression always give a positive /angbracketleftWb/angbracketright? If we add the two
+terms in the square bracket directly, we get
+4/angbracketleftWb/angbracketright
+µ0|H|2=(1−F)ω4+(3F−2)ω2
+0ω2+ω4
+0+ω2γ2
+(ω2
+0−ω2)2+ω2γ2,
+(6)
+which is still hard to judge its sign. However, if we con-
+sider the identity
+(ω2
+0−ω2)2+ω2γ2+Fω2(3ω2
+0−ω2)
+= (1−F)(ω2−ω2
+0)2+Fω2
+0(ω2+ω2
+0)+ω2γ2(7)
+the positiveness of the magnetic energy becomes very ob-
+vious (remember that both Fand 1−Fare positive).
+According to this analysis, both the electric and mag-
+netic energy densities in the energy density formula of
+[1] are positively definite.
+ACKNOWLEDGEMENT
+The author gratefully acknowledge financial support
+from National Science Council (Grant No. NSC 98-2112-
+M-008-014-MY3) of Republic of China, Taiwan.
+[1] Pi-Gang Luan, Phys. Rev. E 80, 046601 (2009).
+[2] S.A. Tretyakov, Phys. Lett. A, 343, 231 (2005)
+[3] A. D. Boardman, K. Marinov, Phys. Rev. B 73, 165110(2006)
\ No newline at end of file
diff --git a/1001.0351.txt b/1001.0351.txt
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diff --git a/1001.0352.txt b/1001.0352.txt
new file mode 100644
index 0000000000000000000000000000000000000000..45ac2a27ef1234ff04a1235286a4bb639f1ae395
--- /dev/null
+++ b/1001.0352.txt
@@ -0,0 +1,346 @@
+arXiv:1001.0352v1  [astro-ph.EP]  3 Jan 2010Version V4.1 2-Jan-2010
+TheKeplerFollow-up Observation Program
+Thomas N. Gautier III1, Natalie M. Batalha2, William J. Borucki3, William D. Cochran4,
+Edward W. Dunham5, Steve B. Howell6, David G. Koch3, David W. Latham7,
+Geoff W. Marcy8, Lars A. Buchhave7,9, David R. Ciardi10, Michael Endl4, Gabor F˝ ur´ esz7,
+Howard Isaacson8, Phillip MacQueen4, Georgi Mandushev5, Lucianne Walkowicz8
+ABSTRACT
+TheKepler Mission was launched on March 6, 2009 to perform a photometric
+survey of more than 100,000 dwarf stars to search for terrestr ial-size planets with
+the transit technique. Follow-up observations of planetary candid ates identified
+by detection of transit-like events are needed both for identificat ion of astrophys-
+ical phenomena that mimic planetary transits and for characteriza tion of the
+true planets and planetary systems found by Kepler. We have developed tech-
+niques and protocols for detection of false planetary transits and are currently
+conducting observations on 177 Keplertargets that have been selected for follow-
+up. A preliminary estimate indicates that between 24% and 62% of plan etary
+candidates selected for follow-up will turn out to be true planets.
+Subject headings: techniques: radial velocities, techniques: spectroscopic
+1Jet Propulsion Laboratory/CaliforniaInstitute of Technology, 48 00 Oak Grove Dr, Pasadena, California
+91109; Thomas.N.Gautier@jpl.nasa.gov
+2Department of Physics and Astronomy, San Jose State University , San Jose, CA 95192
+3NASA Ames Research Center, Moffett Field, CA 94035
+4McDonald Observatory, The University of Texas at Austin, Austin, TX 78712
+5Lowell Observatory, Flagstaff, AZ 86001
+6National Optical Astronomy Observatory, Tucson, AZ 85719
+7Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02 138
+8University of California, Berkeley, Berkeley, CA 94720
+9Niels Bohr Institute, Copenhagen University, DK-2100 Copenhage n, Denmark
+10NASA Exoplanet Science Institute, California Institute of Technolo gy, Pasadena, CA 91125– 2 –
+1. Introduction
+Kepler, a NASA Discovery Mission designed to determine the frequency of t errestrial-
+size planets in and near the habitable zone of solar-like stars, was lau nched on March 6,
+2009 and has been returning scientific data since early May 2009. Th e specific goals of the
+mission are:
+•Determine the frequency of Earth-size and larger planets in or nea r the habitable zone
+of a wide variety of stars.
+•Determine the distributions of planet sizes and orbital semi-major a xes
+•Estimate the frequency and orbital distribution of planets in multiple -star systems.
+•Determine the distributions of semi-major axis, albedo, size, mass, and density of
+short-period planets giant planets.
+•Identify additional members of each photometrically discovered pla netary system using
+complementary techniques.
+•Determine the association of stellar properties with planetary char acteristics.
+Keplerwill survey more than 100,000 late-type dwarf stars in the solar neig hborhood
+with visual magnitudes between 9 and 16 for a period of 3.5 years look ing for transits of
+planets around those stars. Details of the Kepler Mission , the photometer and its operating
+modes are given in Borucki, et al. (2010a) and Koch, et al. (2010a).
+Whilethe KeplerphotometeriscapableofdetectingtransitsofEarth-sizeplanets around
+its target stars the photometric detections must be supplemente d by follow-up observations
+with other facilities to identify transit-like signals from non-planetar y sources and to ac-
+complish goals of the mission beyond just detection of the planets. T heKeplerFollow-up
+Observation Program (FOP) is designed to provide these supplemen tal observations.
+2. Purpose of KeplerFollow-up Program
+The detection of a transit-like signal in a Keplertarget star is not sufficient evidence to
+confirm the presence of a planet orbiting the target star, nor wou ld it be for any transiting
+planet survey. Brown (2003) distinguishes 12 combinations of giant planets and stars in
+eclipsing and transiting systems that can produce light curves mimick ing a planet transiting
+a solitary primary star. Six of the combinations do not involve planets at all and four others– 3 –
+distort the transit light curve so that the size of the planet is indete rminate. In the case of
+Keplerwhere it is practical to detect transits of planets down to the size o f Earth and below
+additional opportunities for confusion exist.
+The principal causes of false positive or ambiguous planet size detec tions for Keplerare:
+1. Background eclipsing binaries where a distant eclipsing star syste m’s light is confused
+with and diluted by the light of the Keplertarget star. The deep eclipse of the binary
+will appear as a shallow transit characteristic of a planet around the target star.
+2. Eclipsing binaries in multiple-star systems where the multiple system is theKepler
+target but the deep eclipse of the binary is diluted by the light of othe r stars in the
+system so as to appear as shallow as a planetary transit.
+3. Grazing eclipses of binary stars that are Keplertarget stars
+4. Transits in a binary system consisting of a giant star and a main seq uence star
+5. Transits of background main sequence stars by giant planets th at are diluted by the
+lightofthe KeplertargetstarsoastoappearasshallowastransitsofEarth-sizepla nets
+6. Transits of giant stars by giant planets that appear as shallow as Earth-size planets
+transiting main sequence stars.
+In general light curve data alone cannot distinguish between these six configurations
+and the principle objective of the Keplersurvey, Earth-size planets transiting main sequence
+stars.Keplerdata is collected in a way that allows determination of the photo-cent er of
+light around the target star which can often identify causes 1 and 5 above (see sections 3.1
+and 4.1 below). Careful spectroscopic and photometric follow-up o bservations of candidate
+transiting planets are needed to fully determine the source of a Keplertransit signal.
+TheKeplerfollow-up observation program therefore has two purposes. Firs t, the follow-
+upprogrammust separatefalsepositive transit indications fromtr ueplanets toa highdegree
+of reliability. Second, the follow-up program will characterize a repr esentative sample of
+planets and planetary systems by determining the mass and orbit of the transiting planet
+and any other detectable, non-transiting planets.– 4 –
+3. False Positive Elimination
+3.1. Elimination with Keplerdata
+False positive elimination for Keplertransit detections begins in the KeplerPipeline
+(Jenkins, et al. 2010a) using only Keplerdata. After transit-like signals are detected and
+matched againsteach other tofindperiodicseries oftransits, the series, nowcalledThreshold
+Crossing Events (TCEs), are checked for adequate signal to nois e ratio (SNR), regularity
+of period, transit depth, transit shape and consistency with Keple r’s (the astronomer) laws
+of planetary motion. TCEs that pass these tests are subject to a preliminary photocenter
+motionanalysiswherein themotionofthecentroidofthelight distribu tioninthephotometer
+aperture, in and out of transit, is measured. Large centroid motio ns likely indicate that a
+nearby object, not the Keplertarget, is the source of the transit signal. In these cases
+examination of the light curves from individual pixels in the photomete r aperture usually
+identifies the background object. These steps will attack false po sitive causes 1 and 5 in
+section 2.
+The full light curve of the target star, in and out of transit, is exam ined by a human
+to look for characteristic signs of an eclipsing binary star or other b ehavior unexpected of
+a transiting planet. A power spectrum of the light curve may be exam ined to determine if
+the target star is a giant or a dwarf as giants will display noise levels ty pically an order of
+magnitude higher than dwarfs (see, for example, the analysis of KO I-145 in Gilliland, et al.
+2010). (The Keplerplanetary target set is intended to exclude giants except in specific
+cases; see Batalha, et al. (2010b). However, some giants will inevit ably have leaked in.)
+Examination of the light curve can also help detect grazing transits o f stellar companions
+or giant planets which can be mistaken for transits of small planets. These steps will attack
+causes 2, 3 and 4 in section 2.
+Allofthesepre-FOPstepsaredescribedmorefullyinthecompanion paperbyBatalha, et al.
+(2010a). At the time of this writing these steps have, for the most part, been carried out
+manually. With the delivery of the next version of the KeplerPipeline many of the steps
+will be automated.
+3.2. Elimination with Follow-up Observations
+Planetary candidates that survive elimination on the basis of the Keplerdata are sent
+for follow-up observation (see section 4 below) where moderate pr ecision (∼300 m/s), mod-
+erate SNR radial velocity (RV) measurements are used to detect a nd eliminate stellar mass– 5 –
+companions as the cause of the transit signal. Spectra for these m easurements are usually
+taken at high resolution, as for high precision radial velocity work, b ut without enhance-
+ments such as exposure through an iodine cell that enable high RV pr ecision. Imaging under
+good seeing conditions, active optic imaging and speckle imaging are us ed in combination
+with the previously obtained centroid motion analysis to determine if t heKeplertarget is
+the true source of the transit signal. Moderate precision, high SNR spectra may be used for
+line bisector analysis to detect multiple-star systems (see Torres, et al. 2004, and references
+therein). The moderate precision spectra are also used for classifi cation of the target stars.
+These methods are generally the same as those discussed by Alonso , et al. (2004).
+These follow-upobservations will detect falsepositives due toeclips ing binary starsboth
+in the background of the Keplertarget and as the target star itself, either as a solitary binary
+or in a multiple-star system where a third light dilutes the transit. Plan etary candidates
+surviving all of these observations and tests should have a high pro bability of being a true
+planet.
+The current gold standard for extrasolar transiting planet confir mation is radial velocity
+determination of an orbit for the planet that is consistent with the t iming and duration of
+the observed transits. Measurement of the planet’s mass is also co nfirmatory. This sort of
+determination will be possible only for a fraction of the planets anticip ated to be detected by
+Kepler, given the limitations of the sensitivity of the RV method and the reso urces available
+toKepler. Giant planets around late type main sequence stars with orbital pe riods up to
+a few years and Earth mass planets in short period orbits around low mass stars will pro-
+duce enough reflex motion in their parent stars to be measurable wit h current spectroscopic
+techniques. Some examples of this kind of confirmation for Keplerplanet detections can be
+found in papers in this volume (Borucki, et al. 2010b; Koch, et al. 201 0b; Dunham, et al.
+2010; Latham, et al. 2010; Jenkins, et al. 2010b). The low mass plan ets in long period or-
+bits expected to be detected by Keplerwill not produce enough reflex motion to be seen
+by current instrumentation. Additionally, the bulk of Keplerplanetary survey targets are
+fainter thanabout 13.5visual magnitudemaking highprecision radial velocity measurements
+prohibitively expensive or impossible. Confidence in most low mass plane ts found by Kepler
+will have to be based on the quality of the photometric data and elimina tion of false positive
+possibilities.
+4. Operation of Follow-up Program
+As the results of the analyses of the Keplerdata described in section 3.1 become avail-
+able, the KeplerTCEReview Team(TCERT) convenes toreview theresultsandselect TCEs– 6 –
+that are likely to be true transiting planets. The selected TCEs are a ssignedKeplerObject
+of Interest numbers (KOIs) and the TCERT requests the FOP Obs erver Group (FOG) to
+begin false positive elimination on the KOIs. These requests include a s tatement of priority
+and a discussion of the analytical results that led to the selection of the TCEs and KOIs.
+When requests are received from the TCERT the FOG reviews the pr iorities and anal-
+yses and begins observations of the KOIs following a protocol for f alse positive elimination.
+As data and results of the follow-up observations accumulate, the FOG discusses the status
+of each KOI under active observation to monitor the course of the observations. When work
+is completed and the FOG has decided the disposition of the KOI the re sults are passed back
+to the TCERT and the KeplerScience Analysis System (KSAS, described below).
+The TCERT then reviews the results of false positive elimination and ma y ask the FOG
+to perform more observations on selected KOIs to further chara cterize a planet or planetary
+system.
+4.1. Follow-up Protocol
+The protocol for elimination of false positives consists of a series of observations with
+1-3 meter class telescopes with good spectroscopic or high spatial resolution capability.
+1. Begin with a reconnaissance spectrum taken at moderate precis ion and with moderate
+signal to noise ratio to provide radial velocity precision of ∼300 m/s. This spectrum
+will verify andimprove stellar parameters fromthe Kepler Input Catalog anddetermine
+thestar’srotationvelocity. Fastrotatingstars( vsini≥15km/s)andhotstars, earlyF
+andearlier, that show nearly featureless spectra are moredifficult for highprecision RV
+measurements and may be given low priority for characterization st udies. Double-lined
+spectroscopic binaries that escaped pre-FOP detection due to gr azing eclipses shallow
+enough to be mistaken for planetary transits may be detected at t his point. This
+reconnaissance spectrum is also the first point in the RV time series n eeded to detect
+stellar mass companions. This step attacks false positive causes 3 a nd 6 described in
+section 2.
+2. Obtain images of the field around the KOI using active optics, spec kle imaging and
+conventional imaging to search for background objects that migh t be the source of the
+transit signal. Out-of-transit images provide the light distribution in theKeplerpho-
+tometric aperture which allows detailed interpretation of the centr oid motion analysis
+described in section 3.1. This analysis can often distinguish if the KOI is the source– 7 –
+of the signal. Latham, et al. (2010), Dunham, et al. (2010), Koch, et al. (2010b) and
+Jenkins, et al. (2010b) give examples of this analysis.
+In-transit images, though harder to obtain due to observation sc heduling difficulties,
+will occasionally be useful. For large transit signals, ∼1%, in-transit photometry
+can provide a definitive indication of which object in the aperture is pr oducing the
+transit signal by observing which object dims during transit. For sm all signals, ∼
+0.01% as expected for Earth-size planets, where centroid motion an alysis may be too
+insensitive for definite determination of the signal source, in-tran sit photometry can
+provide falsification of background objects as the signal source b y showing that any
+observed background object variations are too small to provide t he transit signal (after
+Deeg, et al. 2010).
+This step attacks causes 1 and 5 in section 2.
+3. If the KOI is still of interest, continue moderate precision RV mea surements. Stellar
+mass companions responsible for the transit signal will reveal them selves by large
+velocity variations consistent with an orbit commensurate with the t ransit light curve.
+This step attacks causes 3 and 4 in section 2.
+4. Obtain a short time series of high SNR, moderate resolution spect ra for line bisector
+analysis to detect KOIs which are multiple-stars where the transit s ignal is diluted by
+non-transited components of the system. This step attacks cau se 2 in section 2.
+This protocol also detects dilution of transits of true giant planets and can allow for
+correction of the dilution for proper measurement of planetary dia meter.
+KOIs determined to be background eclipsing binary stars will genera lly be eliminated
+from further follow-up observation. KOIs in which the target star is found to be an eclipsing
+binary may be removed from further observations if RV techniques cannot yield a planet
+confirmation. However, eclipsing binary systems are inherently inte resting for a transiting
+planet search since the planetary plane is likely to be close to the binar y plane. The TCERT
+may ask members of the KeplerScience Team to use other methods, such as transit timing
+variation, to search for planets around these stars.
+4.2. Further characterization
+AftertheFOPhasreportedtotheTCERTthataKOIislikelytobeapla nettheTCERT
+may request, as mentioned above, that further characterizatio n of the KOI be done. These
+requests are expected to be mainly for determination of planetary orbits and masses or for– 8 –
+precise measurements of stellar and planetary parameters, includ ing planetary atmosphere
+temperature and composition. RV searches for additional, non-tr ansiting planets may also
+be made. Methods employed include:
+•High precision RV measurements (1-10 m/s) for orbit and mass dete rmination and for
+searches for non-transiting planets. Measurements of the Ross iter-McLaughlin effect
+(Rossiter 1924; McLaughlin 1924; Kopal 1990) may be made, espec ially for stars with
+higher rotation rates for which RV is more difficult and RM is more sensit ive. We have
+currently obtained RM data for one planet, Kepler-8b (Jenkins, et al. 2010b).
+•Observation of secondary eclipses with the Spitzer Space Telescop e to measure tem-
+peratures in the planetary atmosphere and to determine atmosph eric composition.
+4.3. Planet Confirmation
+TheKeplerteam is currently working to develop and validate methods other tha n
+spectroscopic orbitdeterminationforconfirmationoftransitingp lanets. Orbitdetermination
+is excessively expensive or impossible for the majority of planets exp ected to be detected
+byKepler. Therefore, analysis of Keplerphotometry combined with follow-up observations
+using moderate precision spectroscopy and high spatial resolution imaging, as described
+in section 4.1, or other methods that may be developed must be used alone to eliminate
+false positives, leaving a statistically reliable, but not perfect, set o f planet detections. The
+standard method of spectroscopic orbit determination and Rossit er-McLaughlin detection
+are being used on the brighter Keplertargets harboring giant planets and low mass planets
+in short period orbits around low mass stars to determine the reliabilit y of planets surviving
+the protocol of section 4.1.
+We expect that Kepler’s extraordinary photometric precision, which allows detection of
+the odd-even effect in binary star eclipses, transit shapes that ar e inconsistent with plan-
+ets and secondary occultations of actual transiting planets, can be combined with centroid
+motion analysis and follow-up spectroscopy to achieve the required degree of false positive
+rejection.
+4.4. Resources
+The follow-upprogramhas ateam of16 observers and5 other astr onomers who dedicate
+atleastpartoftheirtimeto Keplerfollow-upobservations. Theinstrument facilitiesavailable– 9 –
+are shown in Table 1. In the 2009 observing season from June throu gh November 13 nights
+of Keck time and more than 100 nights on other telescopes were use d forKeplerfollow-up
+observations.
+5. Current Results
+At the end of the 2009 observing season 177 Kepler targets have b een identified as KOIs
+from commissioning (Q0) and quarter 1 (Q1) data sets and sent to t he FOP for follow-up
+observation (see Batalha, et al. 2010a,b, for a description of thes e data). These 177 KOIs
+were selected from the Kepler planetary targets brighter than 15thKepler magnitude and
+have been classified as shown in Table 2. This table is intended only to pr esent the current
+status of Kepler follow-up observation and is not suitable for drawin g statistical inferences.
+Obviously interesting numbers like completeness estimates for plane t detection cannot yet
+be sensibly derived. Final versions, optimized for actual flight data , of the algorithms for
+transit detection, background eclipsing binary elimination and other procedures described
+in section 3.1 were not in place when these KOIs were selected. Indee d some steps of
+the selection were performed by human examination of the light curv es and the number
+of background eclipsing binaries reflects early, inefficient versions o f pre-FOP elimination.
+When improved versions of the algorithms are applied to the Q0 and Q1 data new KOIs may
+appear which will be followed up in future observing seasons. Sensible preliminary estimates
+of survey completeness and other statistics should begin to be ava ilable after the end of the
+2010 observing season when our KOI selection methods are better understood and more
+KOIs currently under reconnaissance are resolved.
+A preliminary estimate of the fraction of planets expected in the KOI s sent for follow-up
+observation is possible. A well studied set of 70 transit detections f rom planetary targets
+brighter than 14thmagnitude have been subject to our current best versions of det ection
+validation and background eclipsing binary elimination as described in se ction 3.1. Table
+3 presents the state of follow-up observations for these 70 targ ets. Seventy percent of the
+initially interesting transit detections were found to be false positive in the pre-FOP vetting
+process. Of the 21 KOIs left for FOP follow-up, at least 24% are goo d planets. All 8 of the
+KOIs not yet rejected nor confirmed as planets could prove to be g ood planets, providing a
+first estimate of between 24 and 60% good planets in KOIs sent for f ollow-up observation.
+Small number statistics and incomplete analysis of the 8 KOIs curren tly prevent a better
+estimate.
+We gratefully acknowledge the outstanding work of the enormous n umber of people on– 10 –
+theKeplerteam who have contributed to the success of the mission, and witho ut whom the
+Follow-up Program would have nothing to follow.
+Keplerwas competitively selected as the tenth Discovery mission. Funding f or this
+mission is provided by NASA’s Science Mission Directorate. The NASA Ex oplanet Science
+Institute developed the FOP coordination data server with the cap able services of Megan
+Crane as developer and David Imel as manager.
+Facilities: The Kepler Mission
+REFERENCES
+Alonso, R., Deeg, H. J., Brown, T. M. & Belmonte, J. A., 2004, Astron . Nachr., 325, 594
+Batalha, N., etal., 2010a, ApJ, in prep.
+Batalha, N., etal., 2010b, ApJ, in prep.
+Borucki, W., et al., 2010a, Science, in prep.
+Borucki, etal., 2010b, ApJ, in prep.
+Brown, T. M., 2003, ApJ, 593, L125
+Caldwell, D., etal., 2010, ApJ, in prep.
+Deeg, H. J., etal., 2009, A&A, 506, 343
+Dunham, etal., 2010, ApJ, in prep.
+Gilliland, R., etal., 2010, PASP, in prep.
+Jenkins, J., etal., 2010a, ApJ, in prep.
+Jenkins, J., etal., 2010b, ApJ, in prep.
+Koch, D., etal., 2010a, PASP, in prep.
+Koch, etal., 2010b, ApJ, in prep.
+Kopal, Z. 1990, Mathematical Theory of Stellar Eclipses (Dordrech t: Kluwer)
+Latham, etal., 2010, ApJ, in prep.
+McLaughlin, D. B. 1924, ApJ, 60, 22– 11 –
+Rossiter, R. A. 1924, ApJ, 69, 15
+Torres, G., etal., 2004, ApJ, 609, 1071
+This preprint was prepared with the AAS L ATEX macros v5.2.– 12 –
+Table 1. Follow-up Resources
+Telescope Instrument Institution Location
+Tillinghast reflector TRES Spectrograph CfA Arizona
+Shane telescope Hamilton Spectrograph Lick observatory Ca lifornia
+Nordic Optical Telescope FIES Spectrograph Canary Islands
+Keck Telescope HIRES Spectrograph NASA Hawaii
+Palomar 5m Hale Telescope PHARO/NGS AO camera Caltech/JPL C alifornia
+WIYN Telescope Speckle Camera KPNO Arizona
+KPNO 2.1m Telescope Gcam Spectrograph KPNO Arizona
+KPNO 4m Telescope RCSpec KPNO Arizona
+Hobby Eberly Telescope High Resolution Spectrograph Texas
+Harlan J. Smith Telescope Tull Coude Spectrograph McDonald Observatory Texas
+MMT ARIES AO system SAO/Univ. Arizona Arizona
+1.1-m Hall NASA42 CCD Camera Lowell Observatory Arizona
+1.8-m Perkins PRISM CCD Camera Lowell Observatory Arizona
+William Herschel Telescope HARPS NEF spectrometer (when co mpleted) Canary Islands
+Spitzer Space Telescope IRAC camera (warm) NASA Earth trail ing orbit
+Hubble Space Telescope NASA low Earth orbit
+Table 2. Current Status of Observations
+Type Number
+Total KOIs 177 From targets mkepler≤15 in quarters 0 and 1
+Planet 5 Good rv orbit matches light curve.
+Possible planet 52 Radial velocity variation is small enoug h for a planetary mass companion.
+Recon 65 Still under reconnaissance. No type assigned.
+Double lined spectrum 5
+Stellar companion 8 RV variations indicate a stellar mass co mpanion.
+Triple system 1 Transit source is in a triple (or greater) sys tem.
+Background eclipsing binary 11
+Fast rotator 13 Star is rotating too fast for very precise vel ocities.
+Withdrawn 14 Withdrawn by TCERT after re-examination of lig ht curve
+Unsuitable 3 Featureless spectrum unsuitable for RV work or no star apparent at target location
+Table 3. False Positive Rejection Statistics
+Type Number Fraction
+Number of targets in well studied sample (see text) 70
+Rejected by photometric appearance or centroid motion 49 70 %
+KOIs left after pre-follow-up vetting 21
+Planet 5 24%
+Not yet rejected nor confirmed as planets 8 38%
+Rejected by FOP observation 7 33%
+Dropped due to confusion with nearby stars 1 5%
\ No newline at end of file
diff --git a/1001.0353.txt b/1001.0353.txt
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+arXiv:1001.0353v1  [astro-ph.SR]  3 Jan 2010Pulsars: Theory, Categories and Applications,
+Editor: Frank Columbus et al.,pp. 1-45ISBN0000000000
+c/ci∇cleco√y∇t2010 NovaScience Publishers, Inc.
+Chapter 1
+NUCLEAR LIMITS ON PROPERTIES OF PULSARS AND
+GRAVITATIONAL WAVES
+PlamenG. Krastev∗
+DepartmentofPhysicsand Astronomy,TexasA&MUniversity- Commerce,
+P.O. Box 3011,Commerce, TX75429,U.S.A.
+Department ofPhysics,San DiegoState University,
+5500 CampanileDrive,San DiegoCA 92182-1233,U.S.A.
+Bao-An Li†
+DepartmentofPhysicsand Astronomy,TexasA&MUniversity- Commerce,
+P.O. Box 3011,Commerce, TX75429,U.S.A.
+November13,2018
+PACS:04.30.-w, 97.60.Gb, 97.60.Jd, 21.65.Mn
+Keywords: pulsars, dense matter, rapid rotation, gravitational wave s
+∗E-mail address: pkrastev@sciences.sdsu.edu
+†E-mail address: Bao-An Li@tamu-commerce.edu2 Plamen G.Krastev and Bao-An Li
+Abstract
+Pulsars are among the most mysterious astrophysical object s in the Universe and
+are believed to be rotating neutron stars formed in supernov a explosions. They are
+unique testing grounds of dense matter theories and gravita tional physics and also
+providelinksamongnuclearphysics,particlephysicsandG eneralRelativity. Neutron
+stars may exhibit some of the most extreme and exotic charact eristics that could not
+be found elsewhere in the Universe. Their properties are lar gely determined by the
+equationof state (EOS) of neutron-richmatter, which is the chief ingredientin calcu-
+latingneutronstarstructureandpropertiesofrelatedphe nomena,suchasgravitational
+wave emission from deformed pulsars. Presently, the EOS of n eutron-rich matter is
+still veryuncertainmainlyduetothepoorlyknowndensityd ependenceofthenuclear
+symmetry energy especially at supra-saturation densities . Nevertheless, significant
+progresshasbeenmaderecentlyinconstrainingthedensity dependenceofthenuclear
+symmetryenergymostlyatsub-saturationdensitiesusingt errestrialnuclearreactions.
+While there are still some uncertainties especially at supr a-saturation densities, these
+constraints could provide useful information on the limits of the global properties of
+pulsars and the gravitational waves to be expected from them . Here we review our
+recentworkonconstrainingpropertiesofpulsarsandgravi tationalradiationwithdata
+fromterrestrialnuclearlaboratories.
+Contents
+1 Introduction 3
+2 The equation of state of neutron-rich nuclear matter parti ally constrained by
+recent datafrom terrestrial heavy-ion reactions 5
+3 Equationsdetermining thestructure of neutron stars 11
+3.1 Static stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
+3.2 Rotating stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
+4 Constraining global properties of pulsars 15
+4.1 Keplerian (and static) sequences . . . . . . . . . . . . . . . . . . . . . . . 15
+4.2 Rotation at various frequencies . . . . . . . . . . . . . . . . . . . . . . . . 18
+5 Neutron star momentof inertia 20
+5.1 Slow rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
+5.2 Rapid rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
+5.3 Fractional moment of inertia of the neutron star crust . . . . . . . . . . . . 26
+6 Rotation and proton fraction 29
+7 Constraining gravitational waves from elliptically defo rmed pulsars 31
+7.1 Formalism relating theEOSof neutron-rich matter tothe strength of gravi-
+tational waves from slowly rotating neutron stars . . . . . . . . . . . . . . 32
+7.2 Gravitational waves from slowlyrotating neutron stars . . . . . . . . . . . 33
+7.3 Gravitational waves from rapidlyrotating neutron stars . . . . . . . . . . . 37Nuclear limits on properties of pulsars and gravitational w aves 3
+8 Summaryandoutlook 38
+1 Introduction
+Pulsars exhibit a large array of extreme characteristics. T hey are generally accepted to
+be rotating neutron stars – the smallest and densest stars kn own to exist. Matter in their
+cores is compressed to huge densities ranging from the densi ty of normal nuclear matter,
+ρ0≈0.16fm−1, to an order of magnitude higher [1]. The number of baryons fo rming a
+neutron star is in the order of A≈1057. Understanding properties of matter under such
+extreme conditions of density (and pressure) is still far fr om complete and represents one
+of the most important but also challenging problems in moder n physics.
+Because of their strong gravitational binding neutron star s can rotate very fast [2]. The
+first millisecond pulsar PSR1937+214, spinning at ν= 642Hz[3], was discovered in
+1982, and during the next decade or so almost every year a new o ne was reported. In
+the recent years the situation changed considerably with th e discovery of an anomalously
+large population of millisecond pulsars in globular cluste rs [4], where the density of stars
+is roughly 1000 times that in the field of the galaxy and which a re therefore very favorable
+sites for formation of rapidly rotating neutron stars which have been spun up by the means
+of mass accretion from a binary companion. Presently, the nu mber of the observed pulsars
+isclose to2000, and thedetection rate is rather high.
+In 2006 Hessels et al. [5] reported the discovery of a very rap id pulsar J1748-2446ad,
+rotating at ν= 716Hzand thus breaking the previous record (of 642Hz). Rapid rotation
+could affect significantly the neutron star structure and pr operties, especially if the rota-
+tional frequency is near the Kepler frequency for the star. ( The Kepler, or mass-shedding
+frequency, is the highest possible frequency an object boun d by gravity can have before it
+startstoloosemassattheequator.) Pulsarswithmassesabo ve1M⊙entertherapid-rotation
+regime, if their rotational frequencies are higher than 100 0Hz [2]. Although, presently no
+pulsar is confirmed to rotate above the 1000Hz limit, theory d oes not exclude the possi-
+bility for existence of such rapidly rotating neutron stars and therefore it is important to
+predict their properties. (Here we should mention the recen t discovery of X-ray burst os-
+cillations from the X-ray transient XTEJ1739285 [6], which could suggest that it contains
+an extremely rapidly rotating neutron star spinning at 1122 Hz. However, this observation
+has not been confirmed yet.) Neutron stars are natural astrop hysical laboratories of dense
+matter [4]. Rotating neutron stars appear as much better pro bes for the structure of dense
+baryonic matter than static ones, primarily because the par ticle compositions in rotating
+neutron stars are not frozen in, as it is the case for non-rota ting neutron stars, but are vary-
+ing with time [7]. The associated density changes could be as large as 60% [4] in neutron
+stars in binary stellar systems (e.g., low-mass X-ray binar ies), which are being spun up to
+higher rotational frequencies, or isolated rotating neutr on stars (e.g. isolated millisecond
+pulsars) which are spinning down to lower frequencies becau se of the gravitational radia-
+tion, electromagnetic dipole radiation, and awind of elect ron-positron pairs [7].
+Since neutron stars are objects of extremely condensed matt er, the geometry of space-
+timeisconsiderably alteredfromthatofaflatspace. Theref ore, theconstruction ofrealistic
+models of neutron stars has to be done in the framework of Gene ral Relativity [1, 4]. De-
+tailed knowledge of the equation of state (EOS) of stellar ma tter over a very wide range of4 Plamen G.Krastev and Bao-An Li
+densitiesisrequiredforsolvingtheneutron-star structu reequations. Atpresenttimethebe-
+haviorofmatterunderextremedensitiessuchasthosefound intheinteriorsofneutronstars
+is still highly uncertain and relies upon, often, rather con troversial theoretical predictions.
+On the other hand, fortunately, heavy-ion reactions provid e unique means to constrain the
+EOS of dense nuclear matter in terrestrial laboratories [8] . One of the major uncertainties
+oftheEOSisthedensitydependence ofthenuclear symmetrye nergy [9,10,11], Esym(ρ),
+which is the difference between the nucleon specific energie s in pure neutron matter and
+symmetric nuclear matter. Due to its importance for the neut ron star structure and many
+other ramifications in astrophysics and cosmology, determi ning the density dependence
+of the nuclear symmetry energy has been a major thrust of rese arch for the intermediate
+energy heavy-ion community. Although extracting informat ion about the density depen-
+dence of the nuclear symmetry energy from heavy-ion reactio ns is not an easy task due
+to the complicated roles of the isospin degree of freedom in t he reaction dynamics, sev-
+eral promising probes of the symmetry energy have been sugge sted and tested in recent
+years[12, 13, 14, 15, 16] (see also Refs. [8, 17, 18] for revie ws). In fact, some signifi-
+cant progress has been made in determining the Esym(ρ)at subsaturation densities using:
+(1) isospin diffusion [19] and isoscaling [20] in heavy-ion reactions at intermediate ener-
+gies [21, 22, 23, 24, 25], (2) sizes of neutron skins in heavy n uclei [10, 26, 27, 28], (3) the
+Pygmy dipole resonance [29] and (4) masses of nuclei [30]. At supranormal densities, a
+number of potential probes of the symmetry energy have been p roposed [16, 31] although
+the available data has been very limited so far. Interesting ly, circumstantial evidence for
+a super-soft Esym(ρ)at supra-saturation densities was reported recently [32] b ased on the
+analysis of the FOPI/GSI data on pion production [33]. It is a lso very encouraging to no-
+tice that several dedicated experiments are being planned a t several laboratories tostudy in
+more detail the symmetry energy at supra-saturation densit ies using high energy heavy-ion
+reactions induced by both stable and radioactive beams.
+While global properties of spherically symmetric static ne utron stars have been studied
+extensively and comprehensive literature exists, e.g. Ref s. [9, 10, 11, 34, 35, 36, 37, 38],
+properties of(rapidly) rotatingneutronstarshavebeenin vestigated tolesserextent. Models
+of (rapidly) rotating neutron stars have been constructed b y several research groups with
+various degree of approximation [4, 39, 40, 41, 42, 43, 44, 45 , 46, 47, 48, 49, 50] (see
+Ref. [51] for a review). (Rapidly) rotating, elliptically d eformed neutron stars could be
+one of themajor candidates for emitting gravitational wave s(GWs) –tiny ripples inspace-
+time predicted by the theory of General Relativity [52]. Alt hough GWs have not been
+detected directly yet, indirect evidence do exist [53]. GWs are characterized by a small
+dimensionless strain amplitude, h0, which, in addition to the pulsar’s distance to detector,
+depends on the neutron star structure determined by the unde rlying EOS of stellar matter.
+Thus the EOS is instrumental for predicting the strength of g ravitational radiation emitted
+from neutron stars.
+The partially constrained EOS by the available terrestrial nuclear laboratory data has
+been used in studying several properties of neutron stars, i ncluding the mass-radius corre-
+lation [54], the surface temperature of neutron stars in con nection with the changing rate
+of the gravitational constant G[55], the core-crust transition density [56], the frequenc y
+and damping time of the w-mode of gravitational waves [57] an d the gravitational binding
+energy of neutron stars [58]. In this chapter we concentrate on reviewing several selectedNuclear limits on properties of pulsars and gravitational w aves 5
+properties of fast pulsars and the gravitational waves expe cted from them mostly based on
+our recent workreported inRefs. [59,60, 61].
+2 Theequationofstateofneutron-richnuclearmatterparti ally
+constrained by recent data from terrestrial heavy-ion reac -
+tions
+In this section, we first outline the theoretical tools one us es to extract information about
+the EOS of neutron-rich nuclear matter from heavy-ion colli sions. We put the special em-
+phasis on exploring the density-dependence of the symmetry energy as the study on the
+EOS of symmetric nuclear matter with heavy-ion reactions is better known to the astro-
+physical community and it has been extensively reviewed, se e e.g., Refs. [8, 10, 62] for
+recent reviews. We will then summarize the latest constrain ts on the density dependence
+of the symmetry energy at sub-saturation densities extract ed from heavy-ion reactions at
+intermediate energies. Finally, weaddress thequestion of what kind of isospin-asymmetry,
+especially for dense matter, can bereached inheavy-ion rea ctions.
+Heavy-ion reactions provide unique means to create dense nu clear matter in terrestrial
+laboratories similar to those found in the core of neutron st ars. While unlike the matter
+in neutron stars the dense matter created in heavy-ion react ions is hot, accurate informa-
+tion about the EOS of cold matter can be extracted reliably by modeling carefully the
+kinetic part of the EOS during heavy-ion reactions. Dependi ng on the beam energy, im-
+pact parameter and the reaction system, various hadrons and /or partons may be created
+during the reaction. To extract information about the EOS of dense matter from heavy-
+ion reactions requires careful modeling of the reaction dyn amics and selection of sensitive
+observables. Among the available tools, computer simulati ons based on the Boltzmann-
+Uehling-Uhlenbeck ( BUU) transport theory have been very useful, see, e.g., Refs. [8 , 63]
+forreviews. Theevolution ofthephasespacedistribution f unctionfi(/vector r,/vector p,t)ofnucleon iis
+governed byboththemeanfieldpotential Uandthecollision integral IcollisionviatheBUU
+equation
+∂fi
+∂t+/vector∇pU·/vector∇rfi−/vector∇rU·/vector∇pfi=Icollision. (1)
+Normally, effects of the collision integral Icollisionvia both elastic and inelastic channels
+including particle productions, such aspions, aremodeled viaMonte Carlosampling using
+either free-space experimental data or calculated in-medi um cross sections for the elemen-
+taryhadron-hadron scattering[63]. Thecollisionintegra liscriticalformodelingthekinetic
+part oftheEOS.Information about theEOSof coldnuclear mat terisobtained fromtheun-
+derlying mean-field potential U which is an input to the trans port model. By comparing
+experimental data on some carefully selected observables w ith transport model predictions
+using different mean-field potentials corresponding to var ious EOSs, one can then con-
+strain the corresponding EOS. The specific constrains on the density dependence of the
+nuclear symmetry energy that we are using in this work were ob tained by analyzing the
+isospin diffusion data[19]withintheIBUU04versionof ani sospin andmomentum depen-
+dent transport model [64]. In this model, an isospin and mome ntum-dependent interaction
+(MDI) [65] is used. With this interaction, the potential ene rgy density V(ρ,T,δ)at total6 Plamen G.Krastev and Bao-An Li
+densityρ, temperature Tand isospin asymmetry δis
+V(ρ,T,δ) =Auρnρp
+ρ0+Al
+2ρ0(ρ2
+n+ρ2
+p)+B
+σ+1ρσ+1
+ρσ
+0(1−xδ2)
++/summationdisplay
+τ,τ′Cτ,τ′
+ρ0/integraldisplay /integraldisplay
+d3pd3p′fτ(/vector r,/vector p)fτ′(/vector r,/vector p′)
+1+(/vector p−/vector p′)2/Λ2(2)
+In the mean field approximation, Eq. (2) leads to the followin g single particle potential for
+anucleon withmomentum /vector pand isospin τ
+Uτ(ρ,T,δ,/vector p,x ) =Au(x)ρ−τ
+ρ0+Al(x)ρτ
+ρ0+B/parenleftbiggρ
+ρ0/parenrightbiggσ
+(1−xδ2)
+−8τxB
+σ+1ρσ−1
+ρσ
+0δρ−τ+/summationdisplay
+t=τ,−τ2Cτ,t
+ρ0/integraldisplay
+d3/vector p′ft(/vector r,/vector p′)
+1+(/vector p−/vector p′)2/Λ2,(3)
+whereτ= 1/2(−1/2) for neutrons (protons), x,Au(x),Aℓ(x),B,Cτ,τ,Cτ,−τ,σ, and
+Λare all parameters given in Ref. [65]. The last two terms in Eq . (3) contain the mo-
+mentum dependence of the single-particle potential, inclu ding that of the symmetry po-
+tential if one allows for different interaction strength pa rameters Cτ,−τandCτ,τfor a
+nucleon of isospin τinteracting, respectively, with unlike and like nucleons i n the back-
+ground fields. It is worth mentioning that the nucleon isosca lar potential estimated from
+Uisoscalar≈(Un+Up)/2agrees withtheprediction of variational many-body calcul ations
+forsymmetricnuclear matter[66]inabroad densityandmome ntumrange[64]. Moreover,
+the EOS of symmetric nuclear matter for this interaction is c onsistent with that extracted
+from the available data on collective flow and particle produ ction in relativistic heavy-ion
+collisions up to five times the normal nuclear matter [8]. On t he other hand, the corre-
+sponding isovector (symmetry) potential canbeestimated f romUsym≈(Un−Up)/2δ. At
+normal nuclear matter density, the MDIsymmetry potential a grees verywell withtheLane
+potential extracted from nucleon-nucleus and (n,p) charge exchange reactions available for
+nucleon kinetic energies up to about 100MeV [64]. At abnormal densities and higher
+nucleon energies, however, there is no experimental constr ain on the symmetry potential
+available at present.
+The different xvalues in the MDI interaction are introduced to vary the dens ity de-
+pendence of the nuclear symmetry energy while keeping other properties of the nuclear
+equation of state fixed. Specifically, choosing the incompre ssibilityK0of cold symmet-
+ric nuclear matter at saturation density ρ0to be211MeV leads to the dependence of the
+parameters AuandAlon thexparameter according to
+Au(x) =−95.98−x2B
+σ+1, Al(x) =−120.57+x2B
+σ+1, (4)
+withB= 106.35 MeV.
+With the potential contribution from Eq. (2) and the well-kn own contribution from nu-
+cleon kinetic energies in the Fermi gas model, the EOS and the symmetry energy at zero
+temperature can be easily obtained. As shown in Fig. (1), var ying the parameter xleads to
+a broad range of the density dependence of the nuclear symmet ry energy, similar to thoseNuclear limits on properties of pulsars and gravitational w aves 7
+0.0 0.5 1.0 1.5 2.0
+ρ/ρ0020406080Esym(ρ) (MeV)MDI interaction
+x=1    0   -1   -2
+Figure 1: The density dependence of the nuclear symmetry ene rgy for different values of
+the parameter xinthe MDIinteraction. Taken from [72].
+predicted by various microscopic and/or phenomenological many-body theories. Asprevi-
+ously demonstrated in [24] and [54], only equations of state withxbetween -1 and 0 have
+symmetryenergies inthesub-saturation densityregioncon sistent withtheisospindiffusion
+data and the available measurements of the skin thickness of208Pbusing hadronic probes.
+Morerecentlyhowever,itwasshownthatatsupra-saturatio n densitiesthesymmetryenergy
+could be much softer than the extrapolation of the symmetry e nergy with x= 0[32]. The
+possibility forasuper-soft symmetryenergyatsupra-satu ration densitiescouldhaveimpor-
+tant consequences for the neutron star stability and the nat ure of gravity at sub-millimeter
+distances [67]. Nevertheless, as the first and a conservativ e step in our studies of the prop-
+erties of (rotating) neutron stars [59, 60] and the gravitat ional waves expected from them
+[61], we have used x=−1andx= 0to compute the range of possible (rotating) neutron
+star configurations. The Esym(ρ)withx= 0is also consistent with the RMF prediction
+using the FSUGold interaction [68]. In this review, we thus c onsider only the two limiting
+cases with x= 0andx=−1as boundaries of the symmetry energy consistent with the
+available terrestrial nuclear laboratory data at sub-satu ration densities.
+Tofacilitate comparisons withothermodelsintheliteratu re, itisuseful toparameterize
+theEsym(ρ)fromtheMDIinteractionandlistitscharacteristics. With inphenomenological
+models itiscustomary toseparate thesymmetry energy intot hekinetic andpotential parts,
+Table 1: The parameters F(MeV),G,Ksym(MeV),L(MeV), and Kasy(MeV) for different
+valuesofx. Takenfrom[22].
+x F G K sym L K asy
+1 107.232 1.246−270.4 16.4-368.8
+0 129.981 1.059−88.6 62.1-461.2
+−1 3.673 1.569 94 .1 107.4-550.3
+−2−38.395 1.416 276 .3 153.0-641.78 Plamen G.Krastev and Bao-An Li
+Figure2: Centralbaryondensity(upperpanel)andisospina symmetry(lowerpanel)ofhigh
+density region in the reaction of132Sn+124Snat a beam energy of 400 MeV/nucleon and
+an impact parameter of 1 fm. Taken from Ref. [72].
+see, e.g. [69],
+Esym(ρ) = (22/3−1)3
+5E0
+F(ρ/ρ0)2/3+Epot
+sym(ρ). (5)
+With the MDI interaction, the potential part of the nuclear s ymmetry energy can be well
+parameterized by
+Epot
+sym(ρ) =F(x)ρ/ρ0+(18.6−F(x))(ρ/ρ0)G(x), (6)
+withF(x)andG(x)given in Table 1 for x= 1,0,−1and−2. The MDI parameteriza-
+tions for the Epot
+sym(ρ)are similar to but significantly different from those used in Ref. [69].
+Also shown in Table 1 are other characteristics of the symmet ry energy, including its slope
+parameter Land curvature parameter Ksymatρ0, as well as the isospin-dependent part
+Kasyof the isobaric incompressibility of asymmetric nuclear ma tter [22]. The symmetry
+energy in the subsaturation density region with x=0 and -1 ca n be roughly approximately
+byEsym(ρ)≈31.6(ρ/ρ0)0.69andEsym(ρ)≈31.6(ρ/ρ0)1.05,respectively.
+What is the maximum isospin asymmetry reached, especially i n the supra-normal den-
+sity regions, in typical heavy-ion reactions? How does it de pend on the symmetry energy?
+Do both the density and isospin asymmetry reached have to be h igh simultaneously in or-
+der to probe the symmetry energy at supra-normal densities w ith heavy-ion reactions? The
+answerstothesequestions areimportantforustobetterund erstand theadvantages andlim-
+itations of using heavy-ion reactions to probe the EOS of neu tron-rich nuclear matter and
+properly evaluate their impacts onastrophysics.
+ToanswerthesequestionswefirstshowinFig.2thecentralba ryondensity(upperwin-
+dow) and the average (n/p)ρ≥ρ0ratio (lower window) of all regions with baryon densities
+higher than ρ0inthereaction of132Sn+124Snatabeam energy of400MeV/nucleon and
+an impact parameter of 1 fm. It is seen that the maximum baryon density is about 2 times
+the normal nuclear matter density. Moreover, the compressi on is rather insensitive to the
+symmetry energy because the latter is relatively small comp ared to the EOS of symmetricNuclear limits on properties of pulsars and gravitational w aves 9
+nuclear matteraround thisdensity. Thehighdensity phasel astsforabout 15fm/cfrom5to
+20 fm/c for this reaction. It isinteresting to see in the lowe r window that the isospin asym-
+metryofthehighdensityregionisquitesensitivetotheden sitydependenceofthesymmetry
+energy used in the calculation. The soft (e.g., x= 1) symmetry energy leads to a signifi-
+cantly higher value of (n/p)ρ≥ρ0than the stiff one (e.g., x=−2). This is consistent with
+the well-known isospin fractionation phenomenon in asymme tric nuclear matter [70, 71].
+Because of the Esym(ρ)δ2term in the EOS of asymmetric nuclear matter, it is energeti-
+callymorefavorable tohaveahigher isospin asymmetry δinthehighdensity region witha
+softer symmetry energy functional Esym(ρ). In the supra-normal density region, as shown
+inFig.1,thesymmetryenergychanges frombeingsofttostif fwhentheparameter xvaries
+from 1 to-2. Thusthe value of (n/p)ρ≥ρ0becomes lower asthe parameter xchanges from
+1 to -2. It is worth mentioning that the initial value of the qu antity(n/p)ρ≥ρ0is about 1.4
+which is less than the average n/p ratio of 1.56 of the reactio n system. This is because
+of the neutron-skins of the colliding nuclei, especially th at of the projectile132Sn. In the
+neutron-rich nuclei,then/pratioonthelow-densitysurfa ceismuchhigherthanthatintheir
+interior. Alsobecause ofthe Esym(ρ)δ2termintheEOS,theisospin-asymmetry inthelow
+density region ismuchlower thanthesupra-normal density r egion aslong asthesymmetry
+increases with density. In fact, as shown in Fig. 2 of Ref. [54 ], the isospin-asymmetry of
+thelowdensity region canbecome muchhigher thantheisospi n asymmetry of thereaction
+system.
+It is clearly seen that the dense region can become either neu tron-richer or neutron-
+poorerwithrespecttotheinitialstatedependingonthesym metryenergyfunctional Esym(ρ)
+used. As long as the symmetry energy increases with the incre asing density, the isospin
+asymmetry of the supra-normal density region is always lowe r than the isospin asymmetry
+of the reaction system. Thus, even with radioactive beams, t he supra-normal density re-
+gioncannot bebothdense andneutron-rich simultaneously, unlike thesituation inthecore
+of neutron stars, unless the symmetry energy starts decreas ing at high densities. The high
+density behavior of the symmetry energy is probably among th e most uncertain properties
+of dense matter as stressed by [73, 74]. Indeed, some predict ions show that the symmetry
+energy candecrease withincreasing density above certain d ensity andmayevenfinally be-
+comesnegative. Thisextremebehaviorwasfirstpredictedby somemicroscopicmany-body
+theories, see e.g., Refs. [11, 75, 76]. It has also been shown that the symmetry energy can
+becomenegativeat various highdensities withintheHartre e-Fock approach using theorig-
+inalGognyforce[77],thedensity-dependent M3Yinteracti on [78,79]andabout2/3ofthe
+87 Skyrme interactions that have been widely used in the lite rature [80]. The mechanism
+leading toandphysical meaning ofanegativesymmetryenerg y arethetopics ofsomevery
+recent studies [81]. It was found that the spin-isopsin depe ndence of the three-body force
+and the in-medium properties of the short-range tensor forc e due to the ρ-meson exchange
+are the main sources of the very uncertain high-density beha vior of the nuclear symmetry
+energy.
+The isospin effects in heavy-ion reactions are determined m ainly by the Esym(ρ)δ2
+term in the EOS. One expects a larger effect if the isospin-as ymmetry is higher. Thus,
+ideally, one would like to have situations where both the den sity and isospin asymmetry
+are sufficiently high simultaneously as in the cores of neutr on stars in order to observe the
+strongest effects due to the symmetry energy at supra-norma l densities. However, since it10 Plamen G.Krastev and Bao-An Li
+      110100P (MeV fm-3)
+APR
+MDI
+DBHF+Bonn B
+ExperimentSymmetric matter
+(a)
+012345
+ρ/ρ0110100P (MeV fm-3)Neutron matter
+APR
+MDI (x=0)
+MDI (x=-1)
+DBHF+Bonn B
+Exp.+MDI(b)
+Figure 3: (Color online) Pressure as a function of density fo r symmetric (upper panel) and
+pure neutron (lower panel) matter. The green area in the uppe r panel is the experimental
+constraint on symmetric matter extracted by Danielewicz, L acey and Lynch [8] from ana-
+lyzing the collective flow in relativistic heavy-ion collis ions. The corresponding constraint
+on thepressure of pure neutron matter, obtained by combinin g the flowdata and an extrap-
+olation of the symmetry energy functionals constrained bel ow1.2ρ0(ρ0= 0.16fm−3) by
+the isospin diffusion data, is the shaded black area in the lo wer panel. Results are taken
+partially from Ref. [8].
+istheproduct ofthesymmetryenergy andtheisospin-asymme try thatmatters, onecanstill
+probe the symmetry energy at high densities where the isospi n asymmetry is generally low
+with symmetry energy functionals that increase with densit y. Therefore, even if the high
+density region may not be as neutron-rich as in neutron stars , heavy-ion collisions can still
+be used to probe the symmetry energy at high densities useful for studying properties of
+neutron stars.
+For many astrophysical studies (as those in this chapter), i t is more convenient to ex-
+press the EOS in terms of the pressure as a function of density and isospin asymmetry.
+In Fig. 3 we show pressure as a function of density for two extr eme cases: symmetric
+(upper panel) and pure neutron matter (lower panel). The gre en area in the density range
+ofρ0= [2.0−4.6]is the experimental constraint on the pressure P0of symmetric nu-
+clear matter extracted by Danielewicz, Lacey and Lynch from analyzing the collective
+flow data from relativistic heavy-ion collisions [8]. The pr essure of pure neutron matter
+PPNM=P0+ρ2dEsym(ρ)/dρdepends on the density behavior of the nuclear symmetryNuclear limits on properties of pulsars and gravitational w aves 11
+Table 2: Saturation properties of the nuclear EOSs (for symm etric nuclear matter) shown
+inFig. 3.
+EOS ρ0Es κ m∗(ρ0)Esym(ρ0)
+(fm−3) (MeV) (MeV) (MeV/c2) (MeV)
+MDI 0.160 -16.08 211.00 629.08 31.62
+APR 0.160 -16.00 266.00 657.25 32.60
+DBHF+BonnB 0.185 -16.14 259.04 610.30 33.71
+Thefirstcolumnidentifiestheequationofstate. Theremaini ngcolumnsexhibitthefollowingquan-
+titiesatthenuclearsaturationdensity: saturation(bary on)density;energy-per-particle;compression
+modulus;nucleoneffectivemass; symmetryenergy.
+energyEsym(ρ). Since the experimental constraints on the symmetry energy from terres-
+trial laboratory experiments areavailable mostlyfor dens ities less thanabout 1.2ρ0asindi-
+cated by the green and red squares in the lower panel, which is in contrast to the constraint
+on the symmetric EOS that is only available at much higher den sities, the most reliable
+estimateoftheEOSofneutron-rich mattercanthusbeobtain edbyextrapolating theunder-
+lyingmodel EOSfor symmetricmatterandthesymmetryenergy intheir respective density
+ranges to all densities. Shown by the shaded black area in the lower panel is the resulting
+best estimate of the pressure of high density pure neutron ma tter based on the predictions
+from the MDI interaction with x= 0andx=−1as the lower and upper bounds on the
+symmetry energy and the flow-constrained symmetric EOS.As o ne expects and consistent
+withtheestimateinRef.[8],theestimatederrorbarsofthe highdensitypureneutronmatter
+EOSaremuchwiderthantheuncertainty rangeofthesymmetri cEOS.Forthefourinterac-
+tions indicated in the figure, their predicted EOSscannot be distinguished by the estimated
+constraint on the high density pure neutron matter. In addit ion to the MDI EOS, in Fig. 3
+we show results by Akmal et al. [82] with the A18 +δυ+UIX∗interaction (APR) and
+recent Dirac-Brueckner-Hartree-Fock (DBHF) calculation s [83] with Bonn B One-Boson-
+Exchange (OBE)potential (DBHF+Bonn B) [85]. (Older calcul ations of the DBHF+Bonn
+B EOS can be found in Refs. [84, 11].) The saturation properti es of the nuclear equations
+of state applied in this workare summarized in Table2.
+3 Equations determining the structure ofneutron stars
+Forcompleteness, inthefollowingwebrieflyrecall theequa tions determining thestructure
+of both static and (rapidly) rotating neutron stars based on the well-established formalism
+in the astrophysical literature. As already mentioned in th e introduction, neutron stars are
+objectsofextremelycompressed matterandthereforeprope r understanding oftheirproper-
+ties requires application of both General Relativity and th etheories of dense matter. In this
+respect, neutron stars provide a direct link between two of t he frontiers of modern physics
+- General Relativity and strong interactions in dense matte r [4]. The connection between12 Plamen G.Krastev and Bao-An Li
+both branches of physics isprovided byEinstein’s fieldequa tions
+Gµν=Rµν−1
+2gµνR= 8πTµν(ǫ,P(ǫ)), (7)
+(µ,ν= 0,1,2,3)whichcoupletheEinsteincurvaturetensor, Gµν,totheenergy-momentum
+tensor,
+Tµν= (ǫ+P)uµuν+Pgµν, (8)
+of stellar matter. In the above equations Pandǫdenote pressure and mass energy den-
+sity, while Rµν,gµν, andRdenote the Ricci tensor, the metric tensor, and the Ricci sca lar
+curvature respectively(see e.g., [1]). In Eq. (8) uµis the unit time-like four-velocity satis-
+fyinguµuµ=−1. The tensor Tµνcontains the EOS of stellar matter in the form P(ǫ).
+In general, Einstein’s field equations and those of the nucle ar many-body problem were to
+be solved simultaneously since the baryons and quarks follo w the geodesics of the curved
+space-time whose geometry, determined by the Einstein’s fie ld equations, is coupled to
+the total mass energy density of matter [4]. In the case of neu tron stars, as for all astro-
+physical situations for which the long-range gravitationa l force can be separated from the
+short-range strong force, thedeviation fromflat space-tim e at thelength-scale of thestrong
+interactions ( ∼1fm) is practically zero up to the highest densities achieved in the neu-
+tron star interiors. (This is not to be confused with the glob al length-scale of neutron stars
+(∼10km) for which M/R∼0.3depending on the star’s mass (in units c=G= 1so
+thatM⊙≈1.475km).) In other words, gravity curves space-time only on a macro scopic
+scale but to a very good approximation leaves it flat on a micro scopic scale. To achieve an
+appreciable curvature on a microscopic level at which the st rong interactions dominate the
+particle dynamics mass densities greater than ∼1040g cm−3would be necessary [4, 86].
+Under this circumstances the problem of constructing model s of neutron stars separates
+into two distinct tasks. First, the short-range effects of t he nuclear forces are described by
+the principles of many-body nuclear physics in a local inert ial frame (co-moving proper
+reference frame) in which space-time is flat. Second, the cou pling between the long-range
+gravitational force and matter is accounted for by solving t he general relativistic equations
+for the gravitational field described by the curvature of spa ce-time, leading to the global
+structure of stellar configurations.
+3.1 Static stars
+In the case of spherically symmetric static (non-rotating) stars the metric has the famous
+Schwarzschild form:
+ds2=−e2φ(r)dt2+e2Λ(r)dr2+r2(dθ2+sin2θdφ2), (9)
+(c=G= 1)where the metric functions φ(r)andΛ(r)are given by:
+e2Λ(r)= (1−γ(r))−1, (10)
+e2φ(r)=e−2Λ(r)= (1−γ(r))r > Rstar, (11)Nuclear limits on properties of pulsars and gravitational w aves 13
+with
+γ(r) =
+
+2m(r)
+rifr < Rstar
+2M
+rifr > Rstar(12)
+For a static star Einstein’s field equations (Eq. (7)) reduce then to the familiar Tolman-
+Oppenheimer-Volkoff equation (TOV) [87,88]:
+dP(r)
+dr=−ǫ(r)m(r)
+r2/bracketleftbigg
+1+P(r)
+ǫ(r)/bracketrightbigg/bracketleftBigg
+1+4πr3p(r)
+m(r)/bracketrightBigg/bracketleftbigg
+1−2m(r)
+r/bracketrightbigg−1
+(13)
+where the gravitational mass within asphere of radius ris determined by
+dm(r)
+dr= 4πǫ(r)r2(14)
+Themetric function φ(r)isdetermined through thefollowing differential equation :
+dφ(r)
+dr=−1
+ǫ(r)+P(r)dP(r)
+dr, (15)
+withthe boundary condition at r=R
+φ(r=R) =1
+2ln/parenleftbigg
+1−2M
+R/parenrightbigg
+(16)
+To proceed to the solution of these equations, it is necessar y to provide the EOS of
+stellar matter in the form P(ǫ). Starting from some central energy density ǫc=ǫ(0)at the
+center of the star (r= 0), and with the initial condition m(0) = 0, the above equations
+can be integrated outward until the pressure vanishes, sign ifying that the stellar edge is
+reached. Some care should be taken at r= 0since, as seen above, the TOV equation is
+singular there. The point r=Rwhere the pressure vanishes defines the radius of the star
+andM=m(R) = 4π/integraltextR
+0ǫ(r′)r′2dr′its gravitational mass.
+ForagivenEOS,there isaunique relationship between thest ellar massandthe central
+densityǫc. Thus, for aparticular EOS,thereisaunique sequence of sta rs parameterized by
+the central density (or equivalently thecentral pressure P(0)).
+3.2 Rotating stars
+Equationsofstellar structure of(rapidly) rotatingneutr on starsareconsiderably morecom-
+plex than those of spherically symmetric stars [4]. These co mplications arise due to the
+rotational deformations inrotatingstars(i.e.,flattenin gatthepolesandbulging attheequa-
+tor), which lead to a dependence of the star’s metric on the po lar coordinate θ. In addition,
+rotation stabilizes the star against gravitational collap se and therefore rotating neuron stars
+are more massive than static ones. A larger mass, however, ca uses greater curvature of
+space-time. Thisrendersthemetricfunctionsfrequency-d ependent. Finally,thegeneralrel-
+ativistic effect of dragging the local inertial frames impl ies the occurrence of an additional
+non-diagonal term, gtφ,inthemetrictensor gµν. Thistermimposes aself-consistency con-
+dition on the stellar structure equations, since the degree at which the local inertial frames14 Plamen G.Krastev and Bao-An Li
+are dragged along by the star, is determined by the initially unknown stellar properties like
+mass and rotational frequency [4].
+The structure equations of rapidly rotating neutron stars h ave been computed with the
+RNS1codedevelopedandmadeavailabletothepublicbyNikolaosS tergioulas [46]. Here
+we outline briefly the equations solved by the RNScode. The coordinates of the station-
+ary, axial symmetric space-time used to model a (rapidly) ro tating neutron star are defined
+through ageneralization of Bardeen’s metric [51]:
+ds2=−eγ+ρdt2+e2α(dr2+r2dθ2)
++eγ−ρr2sin2θ(dφ−ωdt2), (17)
+where the metric potentials γ,ρ,α, and the angular velocity of the stellar fluid relative to
+thelocal inertial frame, ω,arefunctions ofthequasi-isotropic radial coordinates, r,andthe
+polar angle θonly. The matter inside a rigidly rotating star is approxima ted as a perfect
+fluid [51], whose energy momentum tensor is given by Eq. (8). T he proper velocity of
+matter,υ, relative to the local Zero Angular Momentum Observer (ZAMO )[89] is defined
+as
+υ=rsin(θ)(Ω−ω)e−ρ(r)(18)
+withΩ =u3/u0the angular velocity of afluid element. Thefour-velocity is given by
+uµ=e−(γ+ρ)/2
+/radicalbig
+(1−υ2)(1,0,0,Ω) (19)
+Intheaboveequation thefunction (γ+ρ)/2represents therelativistic generalization ofthe
+Newtoniangravitational potential, while exp[(γ+ρ)/2]isatimedilationfactorbetweenan
+observer moving with angular velocity ωand one at infinity. Substitution of Eq. (19) into
+Einstein’s fields equations projected onto the ZAMO referen ce frame gives three elliptic
+partial differential equations for the metric potentials γ,ρ, andω, and two linear ordinary
+differential equations for the metric potential α. Technically, the elliptic differential equa-
+tions for the metric functions are converted into integral e quations which are then solved
+iteratively applying Green’s function approach [44,51].
+Fromtherelativistic equations ofmotion, theequations of hydrostatic equilibrium for a
+barotropic fluid maybe obtained as [51, 89]:
+h(P)−hp=1
+2[ωp+ρp−γ−ρ−ln(1−υ2)+A2(ω−Ωc)2], (20)
+withh(P)the specific enthalpy, Ppthe re-scaled pressure, hpthe specific enthalpy at the
+pole,γpandρpthevalues ofthemetricpotentials at thepole, Ωc=reΩ,andAarotational
+constant [51]. The subscripts p,e, andclabel the corresponding quantities at the pole,
+equator and center respectively. The RNScode solves iteratively the integral equations
+forρ,γandω, and the ordinary differential equation for the metric func tionαcoupled
+withEq.(20) and theequations for hydrostatic equilibrium at the stellar center andequator
+(givenh(Pc)andh(Pe) = 0) to obtain ρ,γ,α,ω, the equatorial coordinate radius re,
+angular velocity Ω,energy density ǫ,and pressure Pthroughout the star.
+1Thanks to Nikolaos Stergioulas the RNScode is available as a public domain program at
+http://www.gravity.phys.uwm.edu/rns/Nuclear limits on properties of pulsars and gravitational w aves 15
+     101214161820Req (km)
+01234
+εc (× 1015 g cm-3)0.00.51.01.52.02.5M/MO •
+APR, ν = 0Hz
+APR, ν = νKDBHF+Bonn B, ν = 0Hz
+DBHF+Bonn B, ν = νKMDI (x=0), ν = 0Hz
+MDI (x=0), ν = νKMDI (x=-1), ν = 0Hz
+MDI (x=-1), ν = νK
+Figure4: (Color online) Neutron star massesandradii. Neut ronstar equatorial radii (upper
+panel) and total gravitational mass (lower panel) versus ce ntral energy density ǫc. Both
+static (solid lines) and Keplerian (broken lines) models ar eshown. Taken from Ref. [59].
+4 Constraining globalproperties ofpulsars
+Weconstructone-parameter2-Dstationaryconfigurationso f(rapidly)rotatingneutronstars
+employing several nucleonic EOSsandthe RNScode. Weassume asimple model of stel-
+lar matter of nucleons and light leptons (electrons and muon s) in beta-equilibrium. Below
+thebaryon density of approximately 0.07fm−3theequations of state applied here aresup-
+plemented by a crustal EOS, which is more suitable for the low density regime. Namely,
+we apply the EOS by Pethick et al. [90] for the inner crust and t he one by Haensel and
+Pichon [91] for the outer crust. In the core we assume a contin uous functional for the
+EOSs employed in this work. (See Ref. [11] for a detailed desc ription of the extrapolation
+procedure for the DBHF+BonnB EOS.)
+4.1 Keplerian(andstatic)sequences
+In what follows we examine the effect of ultra-fast rotation at the Kepler frequency on the
+neutron star gravitational masses and radii. The equilibri um configurations for both static
+and (rapidly) rotating neutron stars are parameterized in t erms of the central mass energy
+density,ǫc=ǫ(0)(or equivalently central pressure, Pc=P(0)). This functional depen-
+dence is shown in Fig. 4, where we display the stellar equatorial radius (upper frame) and
+total gravitational mass(lowerframe)versuscentral ener gy density fortheEOSsappliedin16 Plamen G.Krastev and Bao-An Li
+810121416182022
+Req (km)0.00.51.01.52.02.53.0M/MO •DBHF+Bonn B
+APR
+x = -1
+x = 0DBHF+Bonn B
+APR
+x = -1
+x = 0EXO 0748-676causality
+Figure 5: (Color online) Mass-radius relation. Both static (solid lines) and Keplerian (bro-
+kenlines) sequences areshown. The 1−σerror barcorresponds tothemeasurement ofthe
+mass and radius of EXO0748-676 [95]. Taken from Ref. [59].
+0 1 2 3 4
+εc (× 1015 g cm-3)6810121416Rpol (km)
+Figure6: (Coloronline) Neutron-star polarradiusversusc entralenergydensity. Bothstatic
+(solid lines) and Keplerian (broken lines) sequences are sh own. The labeling of the curves
+isthe same asin Fig.4. Takenfrom Ref. [59].
+this work. Predictions for both static and maximally rotati ng models are shown. Themain
+feature of the mass-density plot is that there exists a maxim um value of the gravitational
+mass of a neutron star that a given EOS can support [4]. This ho lds for both static and
+(rapidly) rotating stars. The sequences shown in Fig. 4 term inate at the ”maximum mass”
+point. Comparingtheresultsforstaticandrotatingstars, itisseenclearlythattherapidrota-
+tionincreasesnoticeably themassthatcanbesupported aga instcollapsewhileloweringthe
+central density of themaximum-mass configuration. Thisisw hat one should expect, since,
+as already mentioned, rotation stabilizes the star against the gravitational pull providing anNuclear limits on properties of pulsars and gravitational w aves 17
+Table 3: Maximum-mass static (non-rotating) models.
+EOS Mmax(M⊙)R(km)ǫc(×1015g cm−3)
+MDI(x=0) 1.91 9.89 3.02
+APR 2.19 9.98 2.73
+MDI(x=-1) 1.97 10.85 2.57
+DBHF+BonnB 2.26 10.91 2.37
+The first column identifies the equation of state. The remaini ng columns exhibit the following
+quantitiesforthestaticmodelswithmaximumgravitationa lmass: gravitationalmass;radius;central
+massenergydensity.
+Table 4: Maximum-mass rapidly rotating models at the Kepler frequency ν=νk.
+EOS Mmax(M⊙)Increase (%) ǫc(×1015g cm−3)νk(Hz)νFIP
+k(Hz)
+MDI(x=0) 2.25 15 2.59 1742 1610
+APR 2.61 17 2.53 1963 1699
+MDI(x=-1) 2.30 14 2.21 1512 1423
+DBHF+BonnB 2.71 17 1.94 1644 1510
+The first column identifies the equation of state. The remaini ng columns exhibit the following
+quantitiesforthemaximallyrotatingmodelswithmaximumg ravitationalmass: gravitationalmass;
+its percentage increase over the maximum gravitational mas s of static models; central mass energy
+density; maximum rotational frequency; Kepler frequency a s computed via the empirical relation
+givenbyEq. (21)[93].
+extra (centrifugal) repulsion. The rotational effect on th e mass-radius relation is illustrated
+in Fig. 5 where the gravitational mass is given as a function o f the circumferential radius.
+For rapid rotation at the Kepler frequency, a mass increase u p to∼17%(Table 4) is ob-
+tained, depending ontheEOS.Theequatorial radius increas es byseveral kilometers, while
+the polar radius decreases by several kilometers (see Fig. 6 ) leading to an overall oblate
+shape of the rotating star. Table 3 summarizes the propertie s (masses, radii and central en-
+ergy densities) of themaximum-mass non-rotating neutron s tar configurations. Ourstudies
+onthe effect of rapid rotation ontheupper masslimits for th efour EOSsconsidered inthis
+chapter are presented in Table 4. In each case the upper mass l imit is obtained for a model
+at the mass-shedding limit where ν=νk, with central density ∼15%below that of the
+static model with the largest mass. These findings are consis tent with those in Refs. [92]
+and [46]. Table 4 also provides an estimate of the upper limit ing rotation rate of a neutron
+star. In general, softer EOSs permit larger rotational freq uencies since the resulting stellar
+models are more centrally condensed, see e.g., Ref. [92]. In the last column of Table 4 we
+show the Kepler frequencies computed via the empirical rela tion
+Ωk
+104s−1= 0.72/parenleftbiggMs
+M⊙/parenrightbigg1/2/parenleftbiggRs
+10km/parenrightbigg−3/2
+(21)18 Plamen G.Krastev and Bao-An Li
+0500100015002000
+νK (Hz)0.00.51.01.52.02.5M/MO •ν = 716 HzJ1748-2446adAPRDBHF+Bonn B
+x = -1x = 0
+Figure 7: (Color online) Mass versus Keplerian (mass-shedd ing) frequency νk. Adapted
+from Ref. [59].
+proposed inRef. [93]. Theuncertainty ofEq.(21)is ∼10%(seeRef.[94]foranimproved
+version of the empirical formula). At the time of constructi ng the above relation Friedman
+et al. [93] did not consider a then unknown class of minimum pe riod EOSs [96] which
+explains why the numbers in the last column of Table 4 exhibit larger deviation from the
+exact numerical solutions (in column five). This is particul arly pronounced for the APR
+EOSforwhichtheapproximatedKeplerfrequencydeviates ∼14%fromtheexactsolution.
+(Note that the APREOShas thelowest period among the EOSscon sidered here.)
+Fig.7displaystheneutronstargravitational massasafunc tionoftheKeplerfrequency,
+νk. Thevertical dotedlinecorresponds tothefrequency ofthe fastest neutronstarpresently
+known(J1748-2446ad [5]). Thefiguresuggests thatthehighe r therotational frequency, the
+larger the mass for the possible stable neutron star configur ations. We discuss this further
+inthe next subsection.
+4.2 Rotationatvariousfrequencies
+In this subsection we study neutron stars rotating at variou s (fixed) frequencies. Stability
+withrespecttothemass-sheddingfromequatorimpliesthat atagivengravitationalmassthe
+equatorial radius Reqshouldbesmallerthan Rmax
+eqcorresponding totheKeplerianlimit[2].
+Thevalueof Rmax
+eqresultsfromthecondition that thefrequency ofatest parti cle atcircular
+equatorial orbit of radius Rmax
+eqjust above the equator of the actual rotating star is equal to
+the rotational frequency of the star. As reported by Bejger e t al. [2] the relation between
+MandReqat the “mass-shedding point” is very well approximated by th e expression for
+theorbital frequency for atest particle orbiting at r=ReqintheSchwarzschild space-time
+created by a spherical mass. The formula satisfying νSchw.
+orb=ν, represented by the dotted
+line in Figs. 8, 9and 10is given by
+1
+2π/parenleftBigg
+GM
+R3eq/parenrightBigg
+=ν, (22)Nuclear limits on properties of pulsars and gravitational w aves 19
+8101214161820
+Req (km)0.51.01.52.02.5M/MO •ν = 642 Hz
+APR
+x = 0
+x = -1
+DBHF+Bonn B
+Figure 8: (Color online) Gravitational mass versus circumf erential radius for neutron stars
+rotating at ν= 642Hz. Seetext for details.
+8101214161820
+Req (km)0.51.01.52.02.5M/MO •ν = 716 Hz
+APR
+x = 0
+x = -1
+DBHF+Bonn B
+Figure 9: (Color online) Gravitational mass versus circumf erential radius for neutron stars
+rotating at ν= 716Hz.
+whereν= 642Hzin Fig. 8, ν= 716Hzin Fig. 9, and ν= 1000Hzin Fig. 10 respec-
+tively. (Rotational frequencies 642Hzand716Hzare the spinning rates of the two fastest
+pulsars PSR1937+214[3]andJ1748-2446ad [5].) Thisformul afortheSchwarzschild met-
+ric coincides with the one obtained in Newtonian gravity for a point mass M[2]. Eq. (22)
+implies
+Rmax=χ/parenleftbiggM
+1.4M⊙/parenrightbigg1/3
+km, (23)
+withχ= 22.52for rotational frequency ν= 642Hz(Fig. 8),χ= 20.94forν= 716Hz
+(Fig. 9), and χ= 16.76forν= 1000Hz(Fig. 10).
+InFigs.8-10weobservethattherangeoftheallowedmassess upportedbyagivenEOS
+for rapidly rotating neutron stars becomes narrower than th e one of static configurations.20 Plamen G.Krastev and Bao-An Li
+8101214161820
+Req (km)1.21.41.61.82.02.22.4M/MO •ν = 1000 Hz
+DBHF+Bonn B APR
+x = -1
+x = 0
+Figure 10: (Color online) Gravitational mass versus circum ferential radius for neutron star
+models rotating at ν= 1000Hz.
+This effect becomes stronger with increasing frequency and depends upon the EOS. For
+instance, for models rotating at 1000Hz(Fig. 10) for the x=−1EOS the allowed mass
+rangeis∼0.2M⊙. Asalreadydiscussed intheliterature [2],thisobservati on couldexplain
+whysuch rapidly rotating neutron starsaresorare–their al lowed massesfall within avery
+narrow range.
+5 Neutron starmomentof inertia
+In this section we turn our attention to the moment of inertia of neutron stars. Such stud-
+ies are important and timely as they are related to astrophys ical observations in the near
+future. In particular, the moment of inertia of pulsar A in th e extremely relativistic neu-
+tron star binary PSR J0737-3039 [97] may be determined in a fe w years through detailed
+measurements of the periastron advance [98].
+Employing the EOSs described briefly in section 2, we compute the neutron star mo-
+ment of inertia with the RNScode. It solves the hydrostatic and Einstein’s field equatio ns
+for massdistributions rotating rigidly under the assumpti on of stationary andaxial symme-
+try about the rotational axis, and reflectional symmetry abo ut the equatorial plane. RNS
+calculates the angular momentum Jas [51]
+J=/integraldisplay
+Tµνξν
+(φ)dV, (24)
+whereTµνisthe energy-momentum tensor of stellar matter
+Tµν= (ǫ+P)uµuν+Pgµν, (25)
+ξν
+(φ)is the Killing vector in azimuthal direction reflecting axia l symmetry, and dV=√−gd3xis a proper 3-volume element ( g≡det(gαβ)is the determinant of the 3-metric).
+In Eq. (25) Pis the pressure, ǫis the mass-energy density, and uµis the unit time-likeNuclear limits on properties of pulsars and gravitational w aves 21
+four-velocity satisfying uµuµ=−1. For axial-symmetric stars it takes the form uµ=
+ut(1,0,0,Ω), whereΩis the star’s angular velocity. Under this condition Eq. (24 ) reduces
+to
+J=/integraldisplay
+(ǫ+P)ut(gφφuφ+gφtut)√−gd3x (26)
+It should be noted that the moment of inertia cannot be calcul ated directly as an integral
+quantity over the source of gravitational field [51]. In addi tion, there exists no unique
+generalization of the Newtonian definition of the moment of i nertia in General Relativity
+and therefore I=J/Ωisa natural choice for calculating this important quantity .
+For rotational frequencies much lower than the Kepler frequ ency (the highest possible
+rotational rate supported by a given EOS), i.e. ν/νk<<1(ν= Ω/(2π)), the deviations
+fromspherical symmetryareverysmall,sothatthemomentof inertiacanbeapproximated
+fromspherical stellar models. Inwhatfollowswereviewbri eflythisslow-rotation approxi-
+mation,seee.g. Ref. [39]. Intheslow-rotational limitthe metriccanbewritteninspherical
+coordinates as(in geometrized units G=c= 1)
+ds2=−e2φ(r)dt2+/parenleftbigg
+1−2m(r)
+r/parenrightbigg−1
+dr2−2ωr2sin2θdtdφ+r2(dθ2+sin2θdφ2)(27)
+In the above equation m(r)is the total gravitational mass within radius rsatisfying the
+usual equation
+dm(r)
+dr= 4πǫ(r)r2(28)
+andω(r)≡(dφ/dt)ZAMOistheLense-Thirringangularvelocityofazero-angular-m omentum
+observer (ZAMO).Uptofirst order in ωall metric functions remain spherically symmetric
+and depend only on r[99]. Inthe stellar interior the Einstein’s fieldequations reduce to
+dφ(r)
+dr=m(r)/bracketleftBigg
+1+4πr3P(r)
+m(r)/bracketrightBigg/bracketleftbigg
+1−2m(r)
+r/bracketrightbigg−1
+(r < Rstar) (29)
+and
+1
+r3d
+dr/parenleftbigg
+r4j(r)d¯ω(r)
+dr/parenrightbigg
++4dj(r)
+dr¯ω(r) = 0 (r < Rstar), (30)
+with¯ω≡Ω−ωthe dragging angular velocity (the angular velocity of thes tar relative to a
+local inertial framerotating at ω) and
+j≡/parenleftbigg
+1−2m(r)
+r/parenrightbigg1/2
+e−φ(r)(31)
+Outside thestar the metric functions become
+e2φ=/parenleftbigg
+1−2M
+r/parenrightbigg
+(r > Rstar) (32)
+and
+ω=2J
+r3(r > Rstar), (33)
+whereM=m(r=R) = 4π/integraltextR
+0ǫ(r′)r′2dr′is the total gravitational mass and Ris the
+stellar radius defined as the radius at which the pressure dro ps to zero ( P(r=R) = 0). At22 Plamen G.Krastev and Bao-An Li
+thestar’ssurfacetheinteriorandexteriorsolutionsarem atchedbysatisfyingtheappropriate
+boundary conditions
+¯ω(R) = Ω−R
+3/parenleftbiggd¯ω
+dr/parenrightbigg
+r=R(34)
+and
+φ(r) =1
+2ln/parenleftbigg
+1−2M
+R/parenrightbigg
+(35)
+Themomentofinertia I=J/ΩthencanbecomputedfromEq.(26). With Ω =uφ/utand
+retaining only first order terms in ωandΩ,the moment of inertia reads [99,35]
+I≈8π
+3/integraldisplayR
+0(ǫ+P)e−φ(r)/bracketleftbigg
+1−2m(r)
+r/bracketrightbigg−1¯ω
+Ωr4dr (36)
+This slow-rotation approximation for the neutron-star mom ent of inertia neglects devia-
+tions from spherical symmetry and is independent of the angu lar velocity Ω[99]. For
+neutron stars with masses greater than 1M⊙Lattimer and Schutz [100] found that, for
+slow-rotations, the moments of inertia computed through th e above formalism (Eq. (36))
+can beapproximated very well by the following empirical rel ation:
+I≈(0.237±0.008)MR2/bracketleftBigg
+1+4.2Mkm
+M⊙R+90/parenleftbiggMkm
+M⊙R/parenrightbigg4/bracketrightBigg
+(37)
+The above equation is shown [100] to hold for a wide class of EO Ss except for ones with
+appreciable degree of softening, usually indicated by achi eving a maximum mass of ∼
+1.6M⊙or less. Since none of the EOSs employed in this paper exhibit such pronounced
+softening, Eq. (37) is a good approximation for the momenta o f inertia of slowlyrotating
+stars.
+5.1 Slow rotation
+If the rotational frequency is much smaller than the Kepler f requency, the deviations from
+spherical symmetry are negligible and the moment of inertia can be calculated applying
+theslow-rotation approximation discussed brieflyabove. F orthis caseLattimer and Schutz
+[100] showed that the moment of inertia can be very well appro ximated by Eq. (37). In
+Fig. 11 we display the moment of inertia as a function of stell ar mass for slowly rotat-
+ing neutron stars as computed with the empirical relation Eq . (37). As shown in Fig. 5,
+above∼1.0M⊙the neutron star radius remains approximately constant bef ore reaching
+the maximum mass supported by a given EOS. The moment of inert ia (I∼MR2) thus
+increases almost linearly withstellar mass for all models. Right before the maximum mass
+is achieved, the neutron star radius starts to decrease (Fig . 5), which causes the sharp drop
+in the moment of inertia observed in Fig. 11. Since Iis proportional to the mass and the
+square of theradius, it ismoresensitive tothedensity depe ndence ofthenuclear symmetry
+energy, which determines the neutron star radius. Here we re call that the x=−1EOShas
+much stiffer symmetry energy (with respect to the one of the x= 0EOS), which results
+in neutron star models with larger radii and, in turn, moment a of inertia. For instance, for
+a “canonical” neutron star ( M= 1.4M⊙), the difference in the moment of inertia is moreNuclear limits on properties of pulsars and gravitational w aves 23
+1.01.21.41.61.82.02.2
+M (MO • )0.51.01.52.02.5I (× 1045 g cm2)
+APR
+x = 0
+x = -1
+DBHF+Bonn B
+Figure 11: (Color online) Total moment of inertia of neutron stars estimated with Eq. (37).
+Taken from Ref. [60].
+1.01.21.41.61.82.02.2
+M (MO • )2030405060I / M3/2 (km2/MO •1/2 )
+APR
+x=0
+x=-1
+DBHF+Bonn B
+Figure12: (Coloronline) Themoment ofinertia scaledby M3/2asafunction ofthestellar
+massM. Theshaded band illustrates a10% error of hypothetical I/M3/2measurement of
+50km2M−1/2
+⊙. The error bar shows the specific case in which the mass is 1.34M⊙(after
+Lattimer and Schutz [100]). Taken from Ref. [60].
+than30%with thex= 0and thex=−1EOSs. In Fig. 12 we take another view of the
+moment of inertia where Iisscaled by M3/2asafunction of the stellar mass(after [100]).
+The discovery of the extremely relativistic binary pulsar P SRJ0737-3039A,B provides
+an unprecedented opportunity to test General Relativity an d physics of pulsars [97]. Lat-
+timer and Schutz [100] estimated that the moment of inertia o f the A component of the
+system should be measurable with an accuracy of about 10%. Gi ven that the masses of
+both stars are already accurately determined by observatio ns, a measurement of the mo-24 Plamen G.Krastev and Bao-An Li
+Table 5: Numerical results for PSRJ0737-3039A ( MA= 1.338M⊙,νA= 44.05Hz).
+EOS ǫc(×1014g cm−3)Req(km)I(×1045g cm2)ILS(×1045g cm2)
+MDI(x=-1) 7.04 13.75 (13.64) 1.63 1.67
+DBHF+BonnB 7.34 12.56 (12.47) 1.57 1.43
+MDI(x=0) 9.85 12.00 (11.90) 1.30 1.34
+APR 9.58 11.60 (11.52) 1.25 1.26
+The first column identifies the equation of state. The remaini ng columns exhibit the following
+quantities: central mass-energy density, equatorial radi us (the numbers in the parenthesis are the
+radiiofthesphericalmodels;thedeviationsfromspherici tyduetorotationare ∼1%),totalmoment
+ofinertia,totalmomentofinertia ILSascomputedwithEq.(37).
+ment of inertia of even one neutron star could have enormous i mportance for the neutron
+star physics [100]. (The significance of such a measurement i s illustrated in Fig. 12. As
+pointed by[100],itisclear thatveryfewEOSswouldsurvive theseconstraints.) Thus,the-
+oretical predictions ofthemomentofinertiaareverytimel y. Calculations ofthemomentof
+inertia of pulsar A ( MA= 1.338M⊙,νA= 44.05Hz) have been reported by Morrison et
+al.[99]andBejgeretal.[98]. InTable5weshowthemomentof inertia(andother selected
+quantities) of PSR J0737-3039A computed with the RNScode using the EOSsemployed
+inthis study. Ourresults withthe APREOSareinvery good agr eement withthose by [99]
+(IAPR= 1.24×1045g cm2) and [98] ( IAPR= 1.23×1045g cm2). In the last column
+of Table 5 we also include results computed with the empirica l relation (Eq. (37)). From a
+comparison withtheresults fromtheexact numerical calcul ation weconclude that Eq.(37)
+is an excellent approximation for the moment of inertia of sl owly-rotating neutron stars.
+(Theaverageuncertainty ofEq.(37)is ∼2%,exceptfortheDBHF+BonnBEOSforwhich
+it is∼8%.) Our results (with the MDIEOS)allowed us toconstrain the m oment of inertia
+of pulsar A tobe inthe range I= (1.30−1.63)×1045(g cm2).
+5.2 Rapidrotation
+Inthis subsection weturn our attention tothe moment of iner tia of rapidly rotating neutron
+stars. In Fig. 13 we show the moment of inertia as a function of stellar mass for neutron
+starmodelsspinningatthemass-shedding (Kepler)frequen cy. Thenumerical calculationis
+performed with the RNScode. Weobserve that the moments of inertia of rapidly rotat ing
+neutronstarsaresignificantly largerthanthoseofslowlyr otatingmodels(forafixedmass).
+Thisis easily understood in termsof the increased (equator ial) radius (Fig. 5).
+We also compute the moments of inertia of pulsars spinning at 642Hz, 715Hz, and
+1000Hz (642Hz [3] and 716Hz [6] are the rotational frequenci es of the fastest pulsars as
+of today). The numerical results are presented in Fig. 14. As demonstrated by Bejger et
+al. [2] and most recently by Krastev et al. [59], the range of t he allowed masses supported
+by a given EOS for rapidly rotating neutron stars becomes nar rower than the one for static
+configurations. The effect becomes stronger with increasin g frequency and depends upon
+the EOS. This is also illustrated in Fig. 14, particularly in panel (c). Additionally, theNuclear limits on properties of pulsars and gravitational w aves 25
+1.0 1.5 2.0 2.5
+M(MO • )1234I (×1045g cm2)
+APR
+x = 0
+x = -1
+DBHF+Bonn Bν=νk
+Figure 13: (Color online) Total moment of inertia for Kepler ian models. The neutron star
+sequences are computed with the RNScode. Taken from Ref. [60].
+       1·10452·10453·10454·1045I (g cm2)ν = 642HzAPR
+x = 0
+x = -1
+DBHF+Bonn B
+(a)
+       1·10452·10453·10454·1045I (g cm2)ν = 716Hz
+(b)
+1.21.41.61.82.02.22.4
+M/MO •1·10452·10453·10454·1045I (g cm2)ν = 1000Hz
+(c)
+Figure 14: (Color online) Total moment of inertia as a functi on of stellar mass for models
+rotating at 642Hz (a), 716Hz (b), and 1000Hz (c). Data ispart ially taken from Ref. [60].26 Plamen G.Krastev and Bao-An Li
+020040060080010001200
+ν(Hz)1.01.21.41.61.82.02.22.4I (× 1045 g cm2)APR
+x = 0
+x = -1
+DBHF+BonnBM=1.4MO •
+Figure 15: (Color online) Total moment of inertia as a functi on of rotational frequency for
+stellar models with mass M= 1.4M⊙. Taken from Ref. [60].
+moment of inertia shows increase with rotational frequency at a rate dependent upon the
+details ofthe EOS.Thisisbest seeninFig. 15wherewedispla y themoment of inertia asa
+function of the rotational frequency for stellar models wit hafixed mass( M= 1.4M⊙).
+The neutron star sequences shown in Fig. 14 are terminated at the mass-shedding fre-
+quency. At the lowest frequencies the moment of inertia rema ins roughly constant for all
+EOSs(which justifies the application of the slow-rotation a pproximation and Eq.(37)). As
+the stellar models approach the Kepler frequency, the momen t of inertia exhibits a sharp
+rise. This is attributed to the large increase of the circumf erential radius as the star ap-
+proaches the “mass-shedding point”. As pointed by Friedman et al. [92], properties of
+rapidly rotating neutron stars display greater deviations from those of spherically symmet-
+ric (static) stars for models computed with stiffer EOSs. Th is is because such models are
+less centrally condensed and gravitationally bound. This a lso explains why the momenta
+of inertia of rapidly rotating neuron star configurations fr om thex=−1EOS show the
+greatest deviation from those of static models.
+5.3 Fractional momentofinertia ofthe neutron starcrust
+Asit wasdiscussed extensively by Lattimer and Prakash [35] (and others), the neutron star
+crust thickness might be measurable from observations of pu lsar glitches, the occasional
+disrupts of the otherwise extremely regular pulsation from magnetized, rotating neutron
+stars. Thecanonical model of Linket al. [101]suggests that glitches are due totheangular
+momentum transfer from superfluid neutrons tonormal matter inthe neutron star crust, the
+regionofthestarcontainingnucleiandnucleonsthathaved rippedoutofnuclei. Thisregion
+is bound by the neutron drip density at which nuclei merge int o uniform nucleonic matter.
+Linketal. [101]concludedfromtheobservations oftheVela pulsarthatatleast 1.4%ofthe
+total moment of inertia resides in the crust of the Vela pulsa r. For slowly rotating neutron
+stars, applying several realistic hadronic EOSs that permi t maximum masses of at least
+∼1.6M⊙Lattimer and Prakash [35] found that the fractional moment o f inertia, ∆I/I,Nuclear limits on properties of pulsars and gravitational w aves 27
+Table 6: Transition densities and pressures for theEOSsuse d inthis paper.
+EOS MDI(x=0) MDI(x=-1) APR DBHF+BonnB
+ρt(fm−3) 0.091 (0.095) 0.093 (0.092) 0.087 0.100
+Pt(MeV fm−3)0.645 0.982 0.513 0.393
+The first row identifies the equation of state. The remaining r ows exhibit the following quantities:
+transition density, transition pressure. The numbers in th e parenthesis are the transition densities
+calculatedbyKubis[102].
+can beexpressed approximately as
+∆I
+I≃28πPtR3
+3Mc2(1−1.67β−0.6β2)
+β/bracketleftBigg
+1+2Pt(1+5β−14β2)
+ρtmbc2β2/bracketrightBigg−1
+(38)
+In the above equation ∆Iis the moment of inertia of the neutron star crust, Iis the total
+moment of inertia, β=GM/Rc2isthecompactness parameter, mbistheaverage nucleon
+mass,ρtisthetransitiondensityatthecrust-coreboundary, and Ptisthetransitionpressure.
+The determination of the transition density itself is a very complicated problem. Different
+approaches often give quite different results. Similar to d etermining the critical density for
+thespinodaldecompositionfortheliquid-gasphasetransi tioninnuclearmatter,foruniform
+npe-matter, Lattimer and Prakash [35] and more recently Kubis [ 102] have evaluated the
+crust transition density byinvestigating whenthe incompr essibility of npe-matter becomes
+negative, i.e
+Kµ=ρ2d2E0
+dρ2+2ρdE0
+dρ+δ2/bracketleftBigg
+ρ2d2Esym
+dρ2+2ρdEsym
+dρ−2E−1
+sym/parenleftbigg
+ρdEsym
+dρ/parenrightbigg2/bracketrightBigg
+<0(39)
+0.050.100.150.20
+ρ(fm-3)0204060Kµ(MeV)
+x = 0
+x = -1
+Figure 16: (Color online) The incompressibility, Kµ, as a function of baryon density ρ.
+Taken from Ref. [60].28 Plamen G.Krastev and Bao-An Li
+0.00.51.01.52.0
+M/MO •0.000.020.040.060.080.100.120.14∆ I / I
+∆ I / I = 0.014APR
+DBHF+Bonn B
+MDI (x = 0)
+MDI (x = -1)
+9101112131415
+R (km)        
+Figure 17: (Color online) The fractional moment of inertia o f the neutron star crust as
+a function of the neutron star mass (left panel) and radius (r ight panel) estimated with
+Eq. (39). The constraint from the glitches of the Vela pulsar is also shown. Taken from
+Ref. [60].
+(seeFig.16)where E0(ρ)istheEOSofsymmetricnuclearmatter, Esymisthenuclearsym-
+metryenergy,and δ= (ρn−ρp)/(ρn+ρp)istheasymmetryparameter. Usingthisapproach
+andtheMDIinteraction, Kubis[102]foundthetransitionde nsityof0.119,0.092,0.095and
+0.160fm−3for thexparameter of 1,0,−1and−2, respectively. Similarly, we have cal-
+culated the transition densities and pressures for the EOSs employed in this work. Our
+results are summarized in Table 6. We find good agreement betw een our results and those
+by Kubis [102] with the MDI interaction. It is interesting to notice that the transition den-
+sities predicted by all EOSs are in the same density range exp lored by heavy-ion reactions
+at intermediate energies. The MDI interaction with x= 0andx=−1constrained by
+the available data on isospin diffusion in heavy-ion reacti on at intermediate energies thus
+limits the transition density rather tightly in the range of ρt= [0.091−0.093](fm−3).
+It is worth noticing, however, that the transition density i s estimated here by using the
+parabolic approximation of the EOS for isospin asymmetric n uclear matter. Relaxing this
+approximation, usingboth dynamical andthermodynamical a pproaches, thetransition den-
+sity and pressure have been studied in great detail in Ref. [5 6]. A somewhat lower values
+ofρt= [0.040−0.065](fm−3)wereobtained. Nevertheless, tobeconsistent withthe rest
+of the study in this review, we stick to the parabolic approxi mation of the EOS in studying
+the∆I/I.
+The fractional momenta of inertia ∆I/Iof the neutron star crusts are shown in Fig. 17
+as computed through Eq. (38) with the parameters listed in Ta ble 6. It is seen that the
+condition ∆I/I >0.014extracted from studying the glitches of the Vela pulsar does put a
+strict lower limit on the radius for agiven EOS.It also limit s the maximum mass to be less
+than about 2M⊙for all of the EOSsconsidered. Similar tothe total momenta o f inertia the
+ratio∆I/Ichanges more sensitively withthe radius asthe EOSis varied .Nuclear limits on properties of pulsars and gravitational w aves 29
+      234567ρc / ρ0716 Hzx = 0, 1.9MO •
+x = -1,  1.1MO •
+20040060080010001200
+ν (Hz)0.1150.1250.1350.1450.1550.165Ypc 716 Hz
+Direct URCA
+x = 0, 1.9MO •x =-1,  1.1MO •
+Figure 18: (Color online) Density (upper panel) and proton f raction (lower panel) versus
+rotational frequency for fixedneutron star mass. Taken from Ref. [59].
+6 Rotationand protonfraction
+Finally, westudy the effect of (fast) rotation on the proton fraction inthe neutron star core.
+In Fig. 18 we show the central baryon density (upper frame) an d central proton fraction
+(lower frame) as a function of the rotational frequency for fi xed-mass models. Predictions
+from both x= 0andx=−1EOSs are shown. We observe that central density decreases
+with increasing frequency. This reduction is more pronounc ed in heavier neutron stars.
+Most importantly, we also observe decrease in the proton fra ctionYc
+pin the star’s core.
+We recall that large proton fraction (above ∼0.14fornpeµ-stars) leads to fast cooling
+of neutron stars through direct Urca reactions. Our results demonstrate that depending on
+the stellar mass and rotational frequency, the central prot on fraction could, in principle,
+drop below the threshold for the direct nucleonic Urca channel and thus making the fast
+coolinginrotatingneutronstarsimpossible. Themassesof themodelsshowninFig.18are
+chosen so that the proton fraction in stellar core is just abo ve the direct Urca limit for the
+staticconfigurations, see Fig. 19 upper and lower frames. The stell ar sequences in Fig. 18
+are terminated at the Kepler (mass-shedding) frequency. In both cases the central proton
+fraction drops below the direct Urca limit at frequencies lo wer than that of PSR J1748-
+244ad [5]. This implies that the fast cooling can be effectiv ely blocked in millisecond
+pulsarsdepending ontheexact massandspinrate. Itmightal soexplainwhyheavyneutron
+stars (could) exhibit slow instead of fast cooling. For inst ance, with the x= 0EOS (with
+softer symmetry energy) for a neutron star of mass approxima tely1.9M⊙, the Direct Urca
+channelclosesat ν≈470Hz. Ontheotherhand,withthex=-1EOS(withstiffersymmetry
+energy) the direct Urca channel can close only for low mass ne utron stars, in fact only for30 Plamen G.Krastev and Bao-An Li
+      0.00.51.01.52.0M/MO • MDI (x=0)
+MDI (x=-1)
+      050100150200esym(MeV)
+0246810
+ρ/ρ00.000.050.100.150.200.250.30Yp Yp = 0.14 Direct Urca
+Figure 19: (Color online) Neutron star mass(upper panel), n uclear symmetry energy (mid-
+dle panel) and proton fraction (lower panel) versus baryon d ensity. Predictions from the
+MDI(x=-1,0) are shown. Taken from Ref. [59].
+masses well below the canonical mass of 1.4M⊙. Thisis due tothe much stiffer symmetry
+energy(seeFig.19middleframe)becauseofwhichthedirect Urcathreshold(Fig.19lower
+frame) isreached at much lower densities and stellar masses (Fig. 19 upper frame).
+Beforeclosingthediscussion inthissectionafewcomments areinorder. Inthepresent
+study we do not consider ”exotic” states of matter in neutron stars. On the other hand,
+due to the rapid rise of the baryon chemical potentials sever al other species of particles,
+such as strange hyperons Λ0andΣ−, are expected to appear once their mass thresholds
+are reached [103]. The appearance of hyperonic degrees of fr eedom lowers the energy-
+per-particle inthestellar medium andcauses morecentrall y condensed configurations with
+lowermassesandradii. Additionally, hyperons helpthecon dition for nucleonic directUrca
+process to be satisfied at lower densities due to the increase d proton fraction [104], and
+depending on their exact concentrations could potentially contribute to the fast cooling of
+the star through hyperonic direct Urca processes [105]. These considerations would al ter
+the balance between the curves in Fig. 19 and ultimately the r esults displaced in Fig. 18
+in favor of the direct Urca process, i.e. smaller masses and h igher frequencies would be
+necessary toclose thefast cooling channel. Thisisduetoth efact that theoverall impact of
+(rapid)rotationontheneutronstarstructureissmallerfo rmorecentrallycondensedmodels
+resulting from “softer” EOSs [92]. Therefore, for such mode ls there is smaller deviation
+from properties and structure of static configurations. In a ddition, at even higher densities
+matter is expected to undergo a transition to quark-gluon pl asma [4, 103], which favors a
+fast cooling through enhanced nucleonic direct Urca and qua rk direct Urca processes, seeNuclear limits on properties of pulsars and gravitational w aves 31
+e.g., Ref. [105].
+7 Constraining gravitational waves from elliptically defo rmed
+pulsars
+Gravitational waves are tiny disturbances in space-time an d are a fundamental, although
+not yet directly confirmed, prediction of General Relativit y. They can be triggered in cat-
+aclysmic events involving (compact) stars and/or black hol es. They could even have been
+produced during the very early Universe, well before any sta rs had been formed, merely
+as a consequence of the dynamics and expansion of the Univers e. Because gravity interact
+extremely weakly with matter, gravitational waves would ca rry a genuine picture of their
+sourcesandthusprovideundisturbedinformationthatnoot hermessengercandeliver[106].
+Gravitational wave astrophysics would open an entirely new non-electromagnetic window
+making it possible toprobe physics that ishidden or dark toc urrent electromagnetic obser-
+vations [107].
+(Rapidly)rotatingneutronstarscouldbeoneofthemajorca ndidatesforsourcesofcon-
+tinuousgravitational wavesinthefrequencybandwidthoft heLIGO[108]andVIRGO(e.g.
+Ref.[109])laserinterferometric detectors. Itiswellkno wnthatarotatingobjectself-bound
+by gravity and which is perfectly symmetric about the axis of rotation does not emit grav-
+itational waves. In order to generate gravitational radiat ion over extended period of time,
+a rotating neutron star must have some kind of long-living ax ial asymmetry [110]. Several
+mechanismsleading tosuchanasymmetryhavebeenstudied in theliterature: (1)Sincethe
+neutron star crust is solid, its shape might not be necessari ly symmetric, as it would be for
+afluid, withasymmetries supported by anisotropic stress bu ilt up during the crystallization
+periodofthecrust[111]. (2)Additionally, duetoitsviole ntformation(supernova) ordueto
+its environment (accretion disc), the rotational axis may n ot coincide with a principal axis
+of the moment of inertia of the neutron star which make the sta r precess [112]. Even if the
+star remainsperfectly symmetricabout therotational axis , sinceitprecesses, itemitsgravi-
+tational waves [112,113]. (3) Also, the extreme magnetic fie lds presented inaneutron star
+causemagneticpressure(Lorenzforcesexertedonconducti ng matter)whichcandistortthe
+star if the magnetic axis is not aligned with the axis of rotat ion [114], which is widely sup-
+posed to occur in order to explain the pulsar phenomenon. Sev eral other mechanisms exist
+thatcanproducegravitational wavesfromneutronstars. Fo rinstance,accretionofmatteron
+aneutronstarcandriveitintoanon-axisymmetricconfigura tionandpowersteadyradiation
+with a considerable amplitude [115]. This mechanism applie s to a certain class of neutron
+stars, including accreting stars in binary systems that hav e been spun up to the first insta-
+bility point of the so-called Chadrasekhar-Friedman-Schu tz (CFS) instability [116]. Also,
+Andersson [117] suggested a similar instability in r-modes of (rapidly) rotating relativistic
+stars. Ithasbeenshownthat theeffectiveness oftheseinst abilities depends ontheviscosity
+of stellar matter which in turn isdetermined by thestar’s te mperature.
+Gravitational waves are characterized by tiny dimensionle ss strain amplitude, which
+depends on the degree to which the neutron star is deformed fr om axial symmetry which,
+inturn, depends upon theequation of state(EOS)of neutron- rich stellar matter. Asalready
+mentioned, presently theEOSof matter under extreme condit ions (densities, pressures and32 Plamen G.Krastev and Bao-An Li
+isospin asymmetries) is still rather uncertain, mainly due to the poorly known density de-
+pendence of the nuclear symmetry energy, Esym(ρ), e.g. [9]. Applying several nucleonic
+EOSs (discussed in section 2), we calculate the gravitation al wave strain amplitude for se-
+lected neutron star configurations. Particular attention i s paid to predictions with the EOS
+partially constrained by the nuclear laboratory data. Thes e results set an upper limit on the
+strain amplitude of gravitational radiation expected from rotating neutron stars.
+The pulsar population is such that most have spin frequencie s that fall below the sensi-
+tivitybandofcurrentdetectors. Inthefuture,thelow-fre quencysensitivityofVIRGO[118]
+and Advanced LIGO [119] should allow studies of a significant ly larger sample of pulsars.
+Moreover, LISA (the Laser Interferometric Space Antenna) i s currently being jointly de-
+signed by NASA in the United States and ESA (the European Spac e Agency), and will be
+launched into orbit in the near future providing an unpreced ented instrument for gravita-
+tional wavessearch anddetection [107]. Thediscussion int hissection is(mainly) basedon
+Ref. [61].
+7.1 Formalism relating the EOS of neutron-rich matter to the strength of
+gravitationalwavesfrom slowlyrotating neutron stars
+In the following we review briefly the formalism used to calcu late the gravitational wave
+strain amplitude from slowly rotating neutron stars. A spin ning neutron star is expected
+to emit GWs if it is not perfectly symmetric about its rotatio nal axis. As already men-
+tioned, non-axial asymmetries can be achieved through seve ral mechanisms such as elas-
+tic deformations of the solid crust or core or distortion of t he whole star by extremely
+strong misaligned magnetic fields. Suchprocesses generall y result inatriaxial neutron star
+configuration [108] which, in the quadrupole approximation and with rotation and angu-
+lar momentum axes aligned, would cause gravitational waves attwicethe star’s rotational
+frequency [108]. These waves have characteristic strain am plitude at the Earth’s vicinity
+(assuming an optimal orientation of therotation axis with r espect tothe observer) of [120]
+h0=16π2G
+c4ǫIzzν2
+r, (40)
+whereνis the neutron star rotational frequency, Izzits principal moment of inertia, ǫ=
+(Ixx−Iyy)/Izzitsequatorial ellipticity, and ritsdistancetoEarth. Theellipticity isrelated
+tothe neutron star maximum quadrupole moment (with m= 2) via [121]
+ǫ=/radicalbigg8π
+15Φ22
+Izz, (41)
+where for slowlyrotating (and static) neutron stars Φ22can be written as[121]
+Φ22,max= 2.4×1038g cm2/parenleftbiggσ
+10−2/parenrightbigg/parenleftbiggR
+10km/parenrightbigg6.26/parenleftbigg1.4M⊙
+M/parenrightbigg1.2
+(42)
+In the above expression σis the breaking strain of the neutron star crust which is rath er
+uncertain at present time. Earlier studies have estimated σto be in the range σ= [10−5−
+10−2][120]. More recently, using molecular dynamics simulation s, it was estimated to be
+about0.1whichisconsiderablylargerthanthepreviousfindings[122 ]. Inourwork[61],weNuclear limits on properties of pulsars and gravitational w aves 33
+have used σ= 10−2, which, compared to the latest estimate [122], is rather con servative.
+Nevertheless, ourresultsaresimplyamplifiedbyafactorof 100ifweapplytheestimateby
+Horowitzetal.[122]inourcalculations. FromEqs.(40)and (41)itisclearthat h0doesnot
+depend onthemoment ofinertia Izz,andthat thetotal dependence upon theEOSiscarried
+by thequadrupole moment Φ22. Thus Eq.(40) can be rewritten as
+h0=χΦ22ν2
+r, (43)
+withχ=/radicalbig
+2045π5/15G/c4. Eq. (43) establishes the link between the gravitational
+wave strain amplitude and the EOS of neutron-rich matter–th e EOS is the main ingredi-
+ent for determining the neutron star properties, including its quadrupole moment. In a
+recent work [60] we have calculated the neutron star moment o f inertia of both static and
+(rapidly) rotating neutronstars. Forslowlyrotatingneut ronstarsLattimerandSchutz[100]
+derived an empirical relation (Eq. (37)) for Izzwhich is shown to hold for a wide class of
+equations of state which do not exhibit considerable soften ing and for neutron star models
+with masses above 1M⊙[100]. Using Eq. (37) to calculate the neutron star moment of
+inertia and Eq. (42) the corresponding quadrupole moment, t he ellipticity ǫcan be readily
+computed (via Eq. (41)). Since the global properties of spin ning neutron stars (in partic-
+ular the moment of inertia) remain approximately constant f or rotating configurations at
+frequencies upto ∼300Hz[60],the aboveformalism can bereadily employed toestimat e
+the gravitational wave strain amplitude, provided one know s the exact rotational frequency
+and distance to Earth, and that the frequency is relatively l ow (below ∼300Hz). These
+estimates are then to be compared with the current upper limi ts for the sensitivity of the
+laser interferometric observatories (e.g. LIGO).
+7.2 Gravitationalwavesfrom slowlyrotating neutron stars
+We calculate the gravitational wave strain amplitude h0for several selected pulsars with
+rotational frequencies ∼200Hzand relatively close to Earth, employing several nucleonic
+equationsofstate. Weassumeasimplemodelofstellarmatte rofnucleonsandlightleptons
+(electronsandmuons)inbeta-equilibrium. Theequations o fstatethatweemployherehave
+been discussed insection 2and shown in Fig. 3.
+Fig. 20 displays the neutron star quadrupole moment (left pa nel) and ellipticity (right
+panel). The quadrupole moment is calculated through Eq. (42 ) and the ellipticity through
+Eq. (41). Note that Eq. (42) is valid only for slowlyrotating neutron star models. We
+notice that Φ22decreases withincreasing stellar massforall EOSsconside red inthisstudy.
+The rate of this decrease depends upon the EOS and is largest f or thex=−1EOS. This
+behavior is easily understood in terms of the increased cent ral density with stellar mass
+– more massive stars are more compact and, since the quadrupo le moment is a measure
+of the star’s deformation (see Eq. (41)), they are also less d eformed with respect to less
+centrally condensed models. Moreover,itiswellknownthat themassismainlydetermined
+by the symmetric part of the EOS while the radius of a neutron s tar is strongly affected
+by the density slope of the symmetry energy. More quantitati vely, an EOS with a stiffer
+symmetry energy, such as the x=−1EOS, results in less compact stellar models, and
+hence more deformed pulsars. Here we recall specifically tha t thex=−1EOS yields34 Plamen G.Krastev and Bao-An Li
+1.01.21.41.61.82.02.2
+M (MO • )0.00.51.01.52.02.53.0Φ22 (× 1039 g cm2)
+1.01.21.41.61.82.02.2
+M/MO •0.00.51.01.52.02.53.03.5ε (× 10-6)APR
+x = 0
+x = -1
+DBHF+Bonn B
+Figure20: (Coloronline) Neutronstarquadrupole moment(l eftpanel) andellipticity (right
+panel). Taken from Ref. [61].
+neutron star configurations withlarger radii thanthose ofm odels form therest of theEOSs
+considered inthisstudy(e.g. seeFig.5). Theseresultsare consistent withpreviousfindings
+which suggest that more compact neutron star models are less altered by rotation, e.g. see
+Ref. [92]. Consequently, it is reasonable also to expect suc h configurations to be more
+“resistant” toanykindofdeformation. Theneutron starell ipticity,ǫ,isshownasafunction
+of the neutron star mass (Fig. 20, right panel). Since ǫis proportional to the quadrupole
+momentΦ22(scaledbythemomentofinertia Izz),itdecreaseswithincreasingstellarmass.
+The results shown in the right panel of Fig. 20 are consistent with the maximum ellipticity
+ǫmax≈2.4×10−6corresponding to the largest crust “mountain” one could exp ect on
+a neutron star [120, 123]. (The estimate of ǫmaxin Refs. [120, 123] has been obtained
+assuming breaking strain of the crust σ= 10−2, as we have assumed in the present paper
+incalculating the neutron star quadrupole moment.)
+InFig.21wedisplaytheGWstrainamplitude, h0,asafunctionofstellar mass. Predic-
+tions are shown for three selected millisecond pulsars, whi ch are relatively close to Earth
+(r <0.4kpc), and have rotational frequencies below 300Hzso that the corresponding mo-
+mentaofinertiaandquadrupolemomentscanbecomputedappr oximatelyviaEqs.(37)and
+(42) respectively. The properties of these pulsars (of inte rest to this study) are summarized
+in Table 7. The error bars in Fig. 21 between the x= 0andx=−1EOSs provide a
+constraint on the maximal strain amplitude of the gravitational waves emitted by the m il-
+lisecond pulsars considered here. The specific case shown in the figure is for neutron star
+models of 1.4M⊙. Depending on the exact rotational frequency, distance to d etector, and
+detailsoftheEOS,the maximalh0isintherange ∼[0.4−1.5]×10−24. Theseestimatesdo
+not take into account the uncertainties in the distance meas urements. They also should be
+regarded as upper limits since the quadrupole moment (Eq. (4 2)) has been calculated with
+σ= 10−2(whereσcan go as low as 10−5). Here we recall that the mass of PSR J0437-
+4715(Fig.21rightpanel)is 1.3±0.2M⊙[125]. (Anothermassconstraint, 1.58±0.18M⊙,
+wasgiven previously byvanStraten et al. [126].) Theresult s shown inFig.21 suggest that
+the GW strain amplitude depends on the EOS of stellar matter, where this dependence is
+stronger for lighter neutron star models. In addition, it is also greater for stellar configura-Nuclear limits on properties of pulsars and gravitational w aves 35
+1.01.21.41.61.82.02.2
+M/MO •0.00.51.01.52.02.53.0h0 (× 10-24)APR
+x = 0
+x = -1
+DBHF+Bonn Bν = 202.79Hz, r = 0.25kpc
+PSR J2124-3358
+1.01.21.41.61.82.02.2
+M/MO •       
+ν = 193.72Hz, r = 0.35kpc
+PSR J1024-0719
+1.01.21.41.61.82.02.2
+M/MO •       
+ν = 173.91Hz, r = 0.18kpc
+PSR J0437-4715
+Figure 21: (Color online) Gravitational-wave strain ampli tude as a function of the neutron
+starmass. Theerrorbarsbetweenthe x= 0andx=−1EOSsprovidealimitonthestrain
+amplitude of the gravitational waves to be expected from the se neutron stars, and show a
+specific case for stellar models of 1.4M⊙. Taken form Ref.[61].
+Table 7: Properties of thepulsars considered in this study.
+Pulsar ν(Hz)M(M⊙)r(kpc)Reference
+PSRJ2124-3358 202.79 - 0.25 [124]
+PSRJ1024-0719 193.72 - 0.35 [124]
+PSRJ0437-4715 173.91 1.3±0.20.18 [125, 126]
+The first column identifies the pulsar. The remaining columns exhibit the following quantities:
+rotational frequency; mass (if known); distance to Earth; c orresponding references. Notice that
+only the mass of PSR J0437-4715 is known from orbital dynamic s [126, 125] (as the pulsar has a
+low-masswhite dwarfcompanion). The massesof PSRs J2124-3 358andJ1024-0719are presently
+unknownastheyarebothisolatedneuronstars[124].
+tions computed withstiffer EOS.Asexplained, such models a re less compact and thus less
+gravitationally bound. As a result, they could be more easil y deformed by rotation or/and
+other deformation driving mechanisms and phenomena, and th erefore are expected to emit
+stronger gravitational radiation (Eq. (40)).
+In Fig. 22 we take another view of the results shown in Fig. 21. We display the maxi-
+mal GW strain amplitude as a function of the GW frequency and c ompare our predictions
+with the best current detection limit of LIGO. The specific ca se shown is for neutron star
+models with mass 1.4M⊙computed with the x= 0andx=−1EOSs. Since these EOSs
+areconstrained bytheavailable nuclear laboratory data th eyprovide alimit onthepossible
+neutron star configurations and thus gravitational emissio n from them. The results shown
+in Fig. 22 would suggest that presently the gravitational ra diation from the three selected
+pulsars should be within the detection capabilities of LIGO . The fact that such a detection36 Plamen G.Krastev and Bao-An Li
+100 1000
+Frequency (Hz)10-2610-2510-2410-2310-2210-21Gravitational wave strain amplitude h0 M=1.4MO •
+PSR J2124-3358 (x=0)
+PSR J2124-3358 (x=-1)
+PSR J1024-0719 (x=0)
+PSR J1024-0719 (x=-1)
+PSR J0437-4715 (x=0)
+PSR J0437-4715 (x=-1)
+Figure 22: (Color online) Gravitational wave strain amplit ude as a function of the gravita-
+tional wave frequency. The characters denote the strain amp litude of the GWs expected to
+beemitted from spinning neutron stars ( ν <300Hz) withmass 1.4M⊙. Solid linedenotes
+the current upper limit of the LIGOsensitivity. Adapted fro m Ref. [108].
+has not been made yet deserves a few comments at this point. Fi rst, as we mentioned, in
+thepresent calculation weassumebreaking strain oftheneu tron star crust σ= 10−2which
+might be too optimistic. As pointed by Haskell et al. [120], w e are still away from testing
+more conservative models and if the true value of σlies in the low end of its range, we
+wouldbestillfarawayfromadirectdetectionofagravitati onal wavesignal. Second,while
+we have assumed a specific neutron star mass of 1.4M⊙, Fig. 21 tells us that h0decreases
+with increasing stellar mass, i.e. heavier neutron stars wi ll emit weaker GWs. Here we re-
+callthatfromtheselectedpulsars onlythemassofPSRJ0437 -4715 isknown(withinsome
+accuracy [125, 126]). The masses of PSR J2124-3358 and PSR J1 024-0719 are unknown.
+Third, inthepresent study weassume averysimplemodel ofst ellar matter consisting only
+beta equilibrated nucleons and light leptons (electrons an d muons). On the other hand,
+in the core of neutron stars conditions are such that other mo re exotic species of particles
+could readily abound. Such novel phases of matter would soft en considerably the EOS of
+stellar medium [103] leading to ultimately more compact and gravitationally tightly bound
+objects which could withstand larger deformation forces (a nd torques). Lastly, the exis-
+tence of quark stars, truly exotic self-bound compact objec ts, is not excluded from further
+considerations and studies. Such stars would be able to resi st huge forces (such as those
+resulting from extremely rapid rotation beyond the Kepler, or mass-shedding, frequency)
+and as aresult retain their axial symmetric shapes effectiv ely dumping the gravitational ra-
+diation (e.g. Ref. [4]). At the end, we recall that Eq. (40) im plies that the best possible
+candidates for gravitational radiation (from spinning rel ativistic stars) are rapidlyrotating
+pulsars relatively close to Earth ( h0∼Φ22ν2/r). Increasing rotational frequency (and/or
+decreasing distance to detector, r) would alter the results shown in Figs. 21 and 22in favor
+ofadetectable signal bythecurrent observational facilit ies (e.g. LIGO).Ontheother hand,
+for more realistic and quantitative calculations, the neut ron star quadrupole moment mustNuclear limits on properties of pulsars and gravitational w aves 37
+Table8: Propertiesofthe rapidlyrotating pulsars considered inthisstudy. Thefirstcolumn
+identifies the pulsar. The remaining columns exhibit the fol lowing quantities: rotational
+frequency; first derivative of the rotational frequency; di stance to Earth; corresponding
+reference.
+Pulsar ν(Hz) ˙ν(Hz s−1)r(kpc)Reference
+PSRB1937+21 641.93 −4.33×10−143.60 [3,130]
+PSRJ1748-2446ad 716.35 – 8.70 [5,131]
+be calculated numerically exactly by solving the Einstein fi eld equations for rapidly rotat-
+ing neutron stars. (Such calculations have been reported, f or instance, by Laarakkers and
+Piosson [127].)
+7.3 Gravitationalwavesfrom rapidlyrotating neutron stars
+For rapidly rotating neutron stars the estimate of the quadr upole deformation in Eq. (42) is
+no longer valid. Moreover, the moment of inertia has to be cal culated differently as well.
+Often,oneusesthespin-downratetoestimatethestrainamp litudeforfastpulsars. Provided
+that the spin-down rate, ˙ν,for agiven pulsar isknown, h0could be estimated from [128]
+hsd
+0=5
+2/parenleftbiggGIzz|˙ν|
+c3r2ν/parenrightbigg1/2
+. (44)
+Here we should mention that in such calculations one assumes that theonlymechanism
+contributing to the pulsar’s observed spin-down is gravita tional radiation. However, other
+mechanismscouldalsoaccountforthestar’sobserveddecre aseinrotationalfrequencysuch
+asmagnetic dipole radiation, andparticle acceleration in themagnetosphere [129]. Despite
+theseuncertainties, calculations ofgravitational waves train amplitude through Eq.(44)are
+still important because, inaddition toproviding arather c onservative upper limit ontheex-
+pected gravitational radiation, theyalsoservetoestimat e another veryuncertain but impor-
+tantquantity–theellipticity ǫ. SofarthefastestpulsarsreportedarethePSRB1937+21[3]
+and PSRJ1748-2446ad [5]. Their known properties are summar ized inTable 8.
+TheRNScode [46] was used to calculate the principal moment of inert ia (and other
+properties) of rapidly rotating neutron stars. Theneutron star moment of inertia for the two
+fastest pulsars isshowninFig14. Panel(a)showsthemoment ofinertia ofPSRB1937+21
+and panel (b) displays the moment of inertia of PSR J1748-244 6ad. As already observed
+previously, the moment of inertia increases with rotationa l frequency, while the range of
+possible neutron star configurations decreases (see, for in stance, Refs. [59, 60]).
+A particularly interesting example is the neutron star rota ting at 642 Hz [3]. Since its
+first observation in 1982, this pulsar has been studied exten sively and an observed spin-
+down rate has been measured (see Table 1). Using the spin-dow n rate, an upper limit on
+the gravitational wave strain amplitude was obtained. The s pin-down rate corresponds to a
+loss in kinetic energy at arate of ˙E=4π2Izzν|˙ν| ∼[0.6 –3.1] ×1036erg/s, depending on
+theEOS.Assumingthat theenergy lossiscompletely duetoth egravitational radiation, the
+gravitational wavestrainamplitudecanbecalculated thro ughEq.(44). Similarcalculations
+for this pulsar and others with an observed spin-down rate ha ve been performed in the38 Plamen G.Krastev and Bao-An Li
+1.01.21.41.61.82.02.22.4
+M/MO •1.5·10-272.0·10-272.5·10-273.0·10-273.5·10-27h0sd
+APR
+x = 0
+x = -1
+DBHF+Bonn Bν = 641.93Hz, r = 3.6kpc
+PSR B1937+21
+1.01.21.41.61.82.02.22.4
+M/MO •2.5·10-93.0·10-93.5·10-94.0·10-9εsdν = 641.93Hz, r = 3.6kpc
+PSR B1937+21
+Figure23: (Coloronline)Gravitational wavestrainamplit udehsd
+0(leftpanel)andellipticity
+(right panel), deduced from the spin-down rate of PSRB1937+ 21. Seetext for details.
+past [108, 128]. These calculations have provided estimate s for the gravitational wave
+strainamplitudeofselectedpulsarsforwhichthespin-dow nratesareknown,andalsoupper
+bounds for their ellipticities using thequadrupole model [ 108]. However, such calculations
+simplyusedthe“fiducial”valueof 1045g cm2forthemomentofinertia Izzinallestimates.
+On the other hand, the neutron star moment of inertia is sensi tive to the details of the
+EOS of stellar matter, and especially to the density depende nce of the nuclear symmetry
+energy[60]. Moreover, Izzincreaseswithincreasingrotational frequency(see,e.g. Fig.15)
+and the differences with the static values of the moment of in ertia could be significant,
+particularly for rapidly rotating neutron stars.
+The gravitational wave strain amplitude of PSR B1937+21 is s hown in Fig. 23 (left
+panel). Because the MDI EOS is constrained by available nucl ear laboratory data, our re-
+sultswiththe x= 0andx=−1EOSsallowedustoplacearatherconservative upperlimit
+on the gravitational waves to be expected from this pulsar, p rovided the onlymechanism
+accounting for its spin-down rate is gravitational radiati on. Under these circumstances, the
+upper limit of the strain amplitude, hsd
+0, for neutron star models of 1.4M⊙is in the range
+hsd
+0= [2.24−2.61]×10−27. Similarly,wehaveconstrainedtheupperlimitoftheellip ticity
+of PSRB1937+21 to bein the range ǫsd= [2.68−3.12]×10−9(Fig. 23, right panel).
+8 Summary and outlook
+In this chapter we have summarized our recent studies on prop erties of (rapidly) rotating
+neutron stars and the gravitational waves expected from def ormed pulsars using several
+nuclear EOSs constrained partially by terrestrial laborat ory experiments. In particular, the
+latest heavy-ion reaction experiments have constrained pa rtially the density dependence of
+the nuclear symmetry energy and thus the EOS of neutron-rich nuclear matter. These lim-
+its, while being incomplete and still suffer from some remai ning uncertainties, can already
+provide someuseful information about thepossible stable c onfigurations of (rapidly) rotat-
+ing neutron stars and the gravitational radiation expected from them. Specifically, we have
+studied properties of (rapidly) rotating neutron stars emp loying four nucleonic EOSs and
+find that the rapid rotation affects the neutron star structu re significantly. It increases theNuclear limits on properties of pulsars and gravitational w aves 39
+maximum possible mass up to ∼17%and increases/decreases the equatorial/polar radius
+by several kilometers. Theneutron star moment of inertia ha s been studied for both slowly
+andrapidlyrotatingmodelswithinawellestablished forma lism. Wefoundthatthemoment
+ofinertiaofPSRJ0737-3039A islimitedintherangeof I= (1.30−1.63)×1045(g cm2).
+The fractional momenta of inertia ∆I/Iof the neutron star crust are also constrained. It is
+also found that the moment of inertia increases with rotatio nal frequency at a rate strongly
+depending upon the EOS used. Additionally, rotation reduce s central density and proton
+fraction in the neutron star core, and depending on the exact stellar mass and rotational
+frequency could effectively close the fast cooling channel in millisecond pulsars. This cir-
+cumstancemayhaveimportant consequences forboththeinte rpretation ofcoolingdataand
+the thermal evolution modelling.
+We have reported predictions on the upper limit of the strain amplitude of the gravi-
+tational waves to be expected from elliptically deformed pu lsars at frequencies <300Hz.
+Our results are intended to provide guidance to the ground-b ased gravitational wave ob-
+servatories. By applying an EOS with symmetry energy constr ained by recent nuclear
+laboratory data, we obtained an upper limit on the gravitati onal-wave signal to be ex-
+pected from several pulsars. Depending on details of the EOS , for several millisecond
+pulsars0.18kpcto0.35Kpcfrom Earth, the maximal h0is found to be in the range of
+∼[0.4−1.5]×10−24. Finally, from the spin-down rate of PSR B1937+21 we have de-
+duced the upper limit of the strain amplitude, hsd
+0, for neutron star models of 1.4M⊙to be
+in the range hsd
+0= [2.24−2.61]×10−27. We have also constrained the upper limit of the
+ellipticity of PSR B1937+21 to be in the range ǫsd= [2.68−3.12]×10−9. These pre-
+dictions set the first direct nuclear constraints on the grav itational waves from elliptically
+deformed pulsars.
+Looking forward, new experiments in terrestrial nuclear la boratories are expected to
+improveourunderstandingabouttheEOSofdenseneutron-ri chnuclearmatterdramatically
+in the next few years. The EOS of neutron-rich matter will be b etter constrained in a wide
+density range. Most of the studies presented in this review w ill then be further refined.
+Conversely, rapid progress in astrophysical observations of neutron stars and gravitational
+waves surely will also help us better understand the EOS of ne utron-rich matter, and thus
+stimulate the progress in nuclear physics. Eventually, the ultimate goal of understanding
+thoroughly all mysteries of pulsars can only be realized by u tilizing the progress made in
+both nuclear physics and astrophysics.
+Acknowledgements
+WethankAaronWorleyforhiscollaboration onsomeofthewor kreportedhere. Thiswork
+issupportedinpartbytheUSNationalScienceFoundationun derGrantsNo. PHY0757839,
+the Research Corporation under Award No. 7123 and the Texas C oordinating Board of
+Higher Education Grant No.003565-0004-2007.40 Plamen G.Krastev and Bao-An Li
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\ No newline at end of file
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+arXiv:1001.0354v3  [math.GT]  31 Jan 2010TopologicalQuantumInformation,KhovanovHomologyand
+theJonesPolynomial
+LouisH. Kauffman
+DepartmentofMathematics, Statistics
+andComputerScience (m/c 249)
+851South MorganStreet
+University ofIllinois atChicago
+Chicago,Illinois 60607-7045
+<kauffman@uic.edu >
+Abstract
+This paperisdedicatedto AnatolyLibgoberon his60-thbirt hday.
+1 Introduction
+In this paper we give a quantum statistical interpretation f or the bracket polynomial state sum
+/an}bracketle{tK/an}bracketri}htand for theJones polynomial VK(t).We use this quantummechanical interpretationto give
+a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its
+conceptual simplicity and it applies to all values of the pol ynomial variable that lie on the unit
+circle in the complex plane. Letting C(K)denote the Hilbert space for this model, there is a
+natural unitarytransformation
+U:C(K)−→ C(K)
+such that
+/an}bracketle{tK/an}bracketri}ht=<ψ|U|ψ>
+where|ψ/an}bracketri}htis a sum over basis states for C(K).The quantum algorithm comes directly from this
+formula via the Hadamard Test. We then show that the framewor k for our quantum model for
+the bracket polynomial is a natural setting for Khovanov hom ology. The Hilbert space C(K)of
+our model has basis in one-to-one correspondence with the en hanced states of the bracket state
+summmationandisisomorphicwiththechaincomplexforKhov anovhomologywithcoefficients
+inthecomplexnumbers. WeshowthatfortheKhovanovboundar yoperator∂:C(K)−→ C(K),
+we have the relationship ∂U+U∂= 0.Consequently, the operator Uacts on the Khovanov
+homology, and we therefore obtain a direct relationship bet ween Khovanov homology and thisquantum algorithm for the Jones polynomial. The quantum alg orithm given here is inefficient,
+and so it remains an open problem to determine better quantum algorithmsthat involveboth the
+Jonespolynomialand theKhovanovhomology.
+Thepaper isorganized as follows. Section 2 reviewsthestru ctureofthebracket polynomial,
+its state summation, the use of enhanced states and the relat ionship with the Jones polynomial.
+Section 3 describes the quantum statistical model for the br acket polynomial and refers to Sec-
+tion 7 (the Appendix) for the details of the Hadamard Test and the structure of the quantum
+algorithm. Section 4 describes the relationship of the quan tum model with Khovanovhomology
+axiomatically, without using the specific details of the Kho vanov chain complex that give these
+axioms. In Section 5 weconstruct theKhovanovchain complex in detail and showhowsomeof
+its properties follow uniquely from the axioms of the previo us section. In Section 6 we discuss
+how the framework of this paper can be generalized to other si tuations where the Hilbert space
+of a quantum information system is also a chain complex. We gi ve an example using DeRahm
+cohomology. In Section 7 (theAppendix)wedetailtheHadama rd test.
+2 BracketPolynomialandJonesPolynomial
+The bracket polynomial [11] model for the Jones polynomial [ 8, 9, 10, 29] is usually described
+by theexpansion
+/an}bracketle{t/an}bracketri}ht=A/an}bracketle{t/an}bracketri}ht+A−1/an}bracketle{t/an}bracketri}ht
+Here the small diagrams indicate parts of otherwise identic al larger knot or link diagrams. The
+two types of smoothing(local diagram with no crossing)in th is formulaare said to be of type A
+(Aabove)and type B(A−1above).
+/an}bracketle{t/circleco√yrt/an}bracketri}ht=−A2−A−2
+/an}bracketle{tK/circleco√yrt/an}bracketri}ht= (−A2−A−2)/an}bracketle{tK/an}bracketri}ht
+/an}bracketle{t/an}bracketri}ht= (−A3)/an}bracketle{t/an}bracketri}ht
+/an}bracketle{t/an}bracketri}ht= (−A−3)/an}bracketle{t/an}bracketri}ht
+One uses these equations to normalize the invariant and make a model of the Jones polynomial.
+In thenormalizedversionwedefine
+fK(A) = (−A3)−wr(K)/an}bracketle{tK/an}bracketri}ht//an}bracketle{t/circleco√yrt/an}bracketri}ht
+where the writhe wr(K)is the sum of the oriented crossing signs for a choice of orien tation of
+the linkK.Since we shall not use oriented links in this paper, we refer t he reader to [11] for
+the details about the writhe. One then has that fK(A)is invariant under the Reidemeister moves
+(againsee [11]) andtheoriginalJonespolynonmial VK(t)isgivenbytheformula
+VK(t) =fK(t−1/4).
+2TheJonespolynomialhasbeenofgreatinterestsinceitsdis coveryin1983duetoitsrelationships
+with statistical mechanics, due to its ability to often dete ct the difference between a knot and its
+mirror image and due to the many open problems and relationsh ips of this invariant with other
+aspects oflowdimensionaltopology.
+The State Summation. In order to obtain a closed formula for the bracket, we now des cribe it
+as a state summation. Let Kbe any unoriented link diagram. Define a state,S, ofKto be the
+collection of planar loops resulting from a choice of smooth ing for each crossing of K.There
+are two choices ( AandB) for smoothing a given crossing, and thus there are 2c(K)states of a
+diagram with c(K)crossings. In a state we label each smoothing with AorA−1according to
+the convention indicated by the expansion formula for the br acket. These labels are the vertex
+weightsof the state. There are two evaluations related to a state. Th e first is the product of the
+vertexweights,denoted /an}bracketle{tK|S/an}bracketri}ht.Thesecondis thenumberofloopsin thestate S, denoted ||S||.
+Define the statesummation ,/an}bracketle{tK/an}bracketri}ht, bytheformula
+/an}bracketle{tK/an}bracketri}ht=/summationdisplay
+S<K|S >δ||S||
+whereδ=−A2−A−2.This is the state expansion of the bracket. It is possible to r ewrite this
+expansioninotherways. Forourpurposesinthispaperitism oreconvenienttothinkoftheloop
+evaluation as a sum of twoloop evaluations, one giving −A2and one giving −A−2.This can
+be accomplished by letting each state curve carry an extra la bel of+1or−1.We describe these
+enhanced states below.
+Changing Variables. Lettingc(K)denote the number of crossings in the diagram K,if we
+replace/an}bracketle{tK/an}bracketri}htbyA−c(K)/an}bracketle{tK/an}bracketri}ht,and then replace A2by−q−1,the bracket is then rewritten in the
+followingform:
+/an}bracketle{t/an}bracketri}ht=/an}bracketle{t/an}bracketri}ht−q/an}bracketle{t/an}bracketri}ht
+with/an}bracketle{t/circleco√yrt/an}bracketri}ht= (q+q−1). It is useful to use this form of the bracket state sum for the s ake of the
+grading in the Khovanov homology (to be described below). We shall continue to refer to the
+smoothingslabeled q(orA−1in theoriginalbracket formulation)as B-smoothings .
+Using Enhanced States. We now use the convention of enhanced states where an enhanced
+statehasa labelof 1or−1on each ofitscomponentloops. Wethen regardthevalueofthe loop
+q+q−1asthesumofthevalueofacirclelabeledwitha 1(thevalueis q)addedtothevalueofa
+circle labeled with an −1(the value is q−1).We could have chosen the less neutral labels of +1
+andXso that
+q+1⇐⇒+1⇐⇒1
+and
+q−1⇐⇒ −1⇐⇒X,
+since an algebra involving 1andXnaturally appears later in relation to Khovanovhomology. I t
+does noharm totakethisformoflabelingfrom thebeginning.
+3Consider the form of the expansion of this version of the brac ket polynonmial in enhanced
+states. Wehavetheformulaas asumoverenhanced states s:
+/an}bracketle{tK/an}bracketri}ht=/summationdisplay
+s(−1)i(s)qj(s)
+wherei(s)is the number of B-type smoothings in sandj(s) =i(s)+λ(s), withλ(s)equal to
+thenumberofloopslabeled 1minusthenumberofloopslabeled −1intheenhanced state s.
+One advantage of the expression of the bracket polynomial vi a enhanced states is that it is
+nowasumofmonomials. Weshallmakeuseofthisproperty thro ughouttherest ofthepaper.
+3 QuantumStatisticsandtheJonesPolynomial
+We now use the enhanced state summation for the bracket polyn omial with variable qto give a
+quantum formulation of the state sum. Letqbe on the unit circle in the complex plane. (This is
+equivalentto lettingtheoriginalbracket variable Abeon theunit circleand equivalenttoletting
+the Jones polynmial variable tbe on the unit circle.) Let C(K)denote the complex vector space
+withorthonormalbasis {|s/an}bracketri}ht}wheresrunsovertheenhancedstatesofthediagram K.Thevector
+spaceC(K)is the (finite dimensional) Hilbert space for our quantum for mulation of the Jones
+polynomial. While it is customary for a Hilbert space to be wr itten with the letter H,we do not
+follow that convention here, due to the fact that we shall soo n regardC(K)as a chain complex
+and takeitshomology. Onecan hardly avoidusing Hforhomology.
+Withqontheunitcircle, wedefine aunitarytransformation
+U:C(K)−→ C(K)
+by theformula
+U|s/an}bracketri}ht= (−1)i(s)qj(s)|s/an}bracketri}ht
+foreach enhanced state s.Herei(s)andj(s)are asdefined intheprevioussectionofthispaper.
+Let
+|ψ/an}bracketri}ht=/summationdisplay
+s|s/an}bracketri}ht.
+Thestatevector |ψ/an}bracketri}htisthesumoverthebasisstates ofourHilbertspace C(K).Forconvenience,
+wedo notnormalize |ψ/an}bracketri}htto lengthoneintheHilbertspace C(K).Wethen havethe
+Lemma. Theevaluationofthebracket polynomialisgivenby thefoll owingformula
+/an}bracketle{tK/an}bracketri}ht=/an}bracketle{tψ|U|ψ/an}bracketri}ht.
+Proof.
+/an}bracketle{tψ|U|ψ/an}bracketri}ht=/summationdisplay
+s′/summationdisplay
+s/an}bracketle{ts′|(−1)i(s)qj(s)|s/an}bracketri}ht=/summationdisplay
+s′/summationdisplay
+s(−1)i(s)qj(s)/an}bracketle{ts′|s/an}bracketri}ht
+4=/summationdisplay
+s(−1)i(s)qj(s)=/an}bracketle{tK/an}bracketri}ht,
+since
+/an}bracketle{ts′|s/an}bracketri}ht=δ(s,s′)
+whereδ(s,s′)istheKroneckerdelta, equalto 1whens=s′andequal to 0otherwise. //
+Thus the bracket polyomial evaluation is a quantum amplitud e for the measurement of the
+stateU|ψ/an}bracketri}htin the/an}bracketle{tψ|direction. Since /an}bracketle{tψ|U|ψ/an}bracketri}htcan be regarded as a diagonal element of the
+transformation Uwith respect to a basis containing |ψ/an}bracketri}ht,this formula can be taken as the foun-
+dation for a quantum algorithm that computes the bracket of Kvia the Hadamard test. See the
+Appendix to this paper, for a discussion of the Hadamard test and the corresponding quantum
+algorithm.
+Afewwordsaboutquantumalgorithmsareappropropiateatth ispoint. Ingeneralonebegins
+with an intial state |ψ/an}bracketri}htand a unitary transformation U.Quantum processes are modeled by uni-
+tary transformations, and it is in principle possibleto cre ate a physical process corresponding to
+any given unitary transformation. In practice, one is limit ed by the dimensions of the spaces in-
+volvedandby thefact thatphysicalquantumstatesare veryd elicateand subjecttodecoherence.
+Nevertheless, onedesigns quantum algorithmsat first by find ing unitary operators that represent
+the informaton in the problem one wishes to calculate. Then t he quantum part of the quantum
+computationisthephysicalprocessthatproducesthestate U|ψ/an}bracketri}htfromtheinitialstate |ψ/an}bracketri}ht.Atthis
+pointU|ψ/an}bracketri}htis available for measurement. Measurement means interacti on with the enviroment,
+andwillhappeninanycase,butoneintendsacontrolledcirc umstanceinwhichthemeasurement
+can occur. If {|e1/an}bracketri}ht,···|en/an}bracketri}ht}is a basis for the Hilbert space (here denoted by C(K)) then we
+wouldhave
+U|ψ/an}bracketri}ht=/summationdisplay
+kzk|ek/an}bracketri}ht,
+alinearcombinationofthebasiselements. Onmeasuringwit hrespecttothisbasis,each |ek/an}bracketri}htcor-
+responds to an observable outcome and this outcome will occur with frequency |zk|2/(/summationtext
+i|zi|2).
+The form of quantum computation consists in repeatedly prep aring and running the process to
+find the frequency with which certain key observations occur . In the case of our algorithm we
+are interested in the frequency of measuring |ψ/an}bracketri}htitself and this corresponds to the inner product
+/an}bracketle{tψ|U|ψ/an}bracketri}ht.However,adirectmeasurementfrom U|ψ/an}bracketri}htwillyieldonlyapproximationstotheabso-
+lutesquareof /an}bracketle{tψ|U|ψ/an}bracketri}ht.ForthisreasonweusethemorerefinedHadamardtestasdescri bedinthe
+Appendix. There is much more to be said about quantum algorit hmsin general and about quan-
+tumalgorithmsfortheJonespolynomialinparticular. Ther eadercanexamine[1,14,15,16,26]
+formoreinformation.
+It is useful to formalize the bracket evaluation as a quantum amplitude. This is a direct way
+to give a physical interpretation of the bracket state sum an d the Jones polynomial. Just how
+this process can be implementedphysically depends upon the interpretation of the Hilbert space
+C(K).It is common practice in theorizing about quantum computing and quantum information
+todefineaHilbertspaceintermsofsomemathematicallyconv enientbasis(suchastheenhanced
+5states of the knot or link diagram K) and leave open the possibility of a realization of the space
+andthequantumevolutionoperatorsthathavebeendefinedup onit. Inprincipleanyfinitedimen-
+sionalunitaryoperatorcanberealizedbysomephysicalsys tem. Inpractice,thisistheproblemof
+constructingquantumcomputersand quantuminformationde vices. It is not so easy to construct
+whatcanbedoneinprinciple,andthequantumstatesthatare producedmaybealltooshort-lived
+to produce reliable computation. Nevertheless, one has the freedom to create spaces and opera-
+tors on the mathematical level and to conceptualize these in a quantum mechanical framework.
+Theresultingstructuresmayberealizedinnatureandinpre sentorfuturetechnology. Inthecase
+of our Hilbert space associated with the bracket state sum an d its corresponding unitary trans-
+formationU,there is rich extra structure related to Khovanov homology t hat we discuss in the
+nextsection. Onehopesthatina(future)realizationofthe sespacesandoperators,theKhovanov
+homologywillplaya keyroleinquantuminformationrelated totheknotorlink K.
+There are a number of conclusions that we can draw from the for mula/an}bracketle{tK/an}bracketri}ht=/an}bracketle{tψ|U|ψ/an}bracketri}ht.
+First of all, this formulation constitutes a quantum algori thm for the computation of the bracket
+polynomial (and hence the Jones polynomial) at any speciali zation where the variable is on the
+unit circle. We have defined a unitary transformation Uand then shown that the bracket is an
+evaluation in the form /an}bracketle{tψ|U|ψ/an}bracketri}ht.This evaluation can be computed via the Hadamard test [24]
+and this gives the desired quantum algorithm. Once the unita ry transformation is given as a
+physical construction, the algorithm will be as efficient as any application of the Hadamard test.
+The present algorithm requires an exponentially increasin g complexity of construction for the
+associated unitary transformation, since the dimension of the Hilbert space is equal to the 2e(K)
+wheree(K)is the number of enhanced states of the diagram K. (Note that e(K) =/summationtext
+S2||S||
+whereSruns over the 2c(K)standard bracket states, c(K)is the number of crossings in the
+diagram and ||S||is the number of loops in the state S.) Nevertheless, it is significant that the
+Jones polynomial can be formulated in such a direct way in ter ms of a quantum algorithm. By
+the same token, we can take the basic result of Khovanovhomol ogy that says that the bracket is
+agradedEulercharacteristicoftheKhovanovhomologyaste llingusthatweare takingastepin
+the direction of a quantum algorithm for the Khovanov homolo gy itself. This will be discussed
+below.
+4 Khovanov Homology and a Quantum Model for the Jones
+Polynomial
+In this sectionwe outlinehowtheKhovanovhomologyis relat ed with ourquantummodel. This
+can be done essentially axiomatically, without giving the d etails of the Khovanov construction.
+Wegivethesedetailsin thenextsection. Theoutlineisas fo llows:
+1. There is a boundary operator ∂defined on the Hilbert space of enhanced states of a link
+diagramK
+∂:C(K)−→ C(K)
+6such that∂∂= 0and so that if Ci,j=Ci,j(K)denotes the subspace of C(K)spanned by
+enhanced states |s/an}bracketri}htwithi=i(s)andj=j(s),then
+∂:Cij−→ Ci+1,j.
+Thatis, wehavetheformulas
+i(∂|s/an}bracketri}ht) =i(|s/an}bracketri}ht)+1
+and
+j(∂|s/an}bracketri}ht) =j(|s/an}bracketri}ht)
+for each enhanced state s.In thenext section, we shall explainhow the boundary operat or
+isconstructed.
+2.Lemma. By defining U:C(K)−→ C(K)as intheprevioussection,via
+U|s/an}bracketri}ht= (−1)i(s)qj(s)|s/an}bracketri}ht,
+wehavethefollowingbasicrelationshipbetween Uand theboundaryoperator ∂:
+U∂+∂U= 0.
+Proof.This follows at once from the definition of Uand the fact that ∂preservesjand
+increasesitoi+1.//
+3. From this Lemma we conclude that the operator Uacts on the homology of C(K).We
+can regardH(C(K)) =Ker(∂)/Image(∂)as a new Hilbert space on which the unitary
+operatorUacts. In this way, the Khovanov homology and its relationshi p with the Jones
+polynomialhasa naturalquantumcontext.
+4. Forafixedvalueof j,
+C•,j=⊕iCi,j
+is a subcomplex of C(K)with the boundary operator ∂.Consequently, we can speak of
+the homology H(C•,j).Note that the dimension of Cijis equal to the number of enhanced
+states|s/an}bracketri}htwithi=i(s)andj=j(s).Consequently,wehave
+/an}bracketle{tK/an}bracketri}ht=/summationdisplay
+sqj(s)(−1)i(s)=/summationdisplay
+jqj/summationdisplay
+i(−1)idim(Cij)
+=/summationdisplay
+jqjχ(C•,j) =/summationdisplay
+jqjχ(H(C•,j)).
+Hereweusethedefinitionofthe Eulercharacteristicofa chaincomplex
+χ(C•,j) =/summationdisplay
+i(−1)idim(Cij)
+7and thefact that the Euler characteristic of thecomplex is e qual to the Euler characteristic
+of its homology. The quantum amplitude associated with the o peratorUis given in terms
+oftheEulercharacteristics oftheKhovanovhomologyofthe linkK.
+/an}bracketle{tK/an}bracketri}ht=/an}bracketle{tψ|U|ψ/an}bracketri}ht=/summationdisplay
+jqjχ(H(C•,j(K))).
+Our reformulation of the bracket polynomial in terms of the u nitary operator Uleads to a
+new viewpoint on the Khovanov homology as a representation s pace for the action of U.The
+bracket polynomial is then a quantum amplitude that express ing the Euler characteristics of the
+homology associated with this action. The decomposition of the chain complex into the parts
+Ci,j(K)corresponds to the eigenspace decomposition of the operato rU.The reader will note
+that in this case the operator Uis already diagonal in the basis of enhanced states for the ch ain
+complexC(K).Weregardthisreformulationasaguidetofurtherquestions abouttherelationship
+oftheKhovanovhomologywithquantuminformationassociat edwiththelink K.
+Asweshallseeinthenextsection,theinternalcombinatori alstructureofthesetofenhanced
+states for the bracket summation leads to the Khovanov homol ogy theory, whose graded Euler
+characteristic yields the bracket state sum. Thus we have a q uantum statistical interpretation of
+the Euler characteristics of the Khovanov homology theory, and a conceptual puzzle about the
+nature of this relationship with the Hilbert space of that qu antum theory. It is that relationship
+that is thesubjectof thispaper. Theunusual pointabout the Hilbertspace is theeach ofits basis
+elementshasaspecificcombinatorialstructurethatisrela tedtothetopologyoftheknot K.Thus
+this Hilbert space is, from the pointof view of itsbasis elem ents, a form of“taking apart” ofthe
+topologicalstructureoftheknotthat weareinterested ins tudying.
+Homological structure of the unitary transformation. We now prove a general result about
+the structure of a chain complex that is also a finite dimensio nal Hilbert space. Let Cbe a chain
+complexoverthecomplexnumberswithboundaryoperator
+∂:Ci−→ Ci+1,
+withCdenotingthedirect sumofall the Ci,i= 0,1,2,···n(for somen). Let
+U:C −→ C
+beaunitaryoperatorthatsatisfiestheequation U∂+∂U= 0.Wedonotassumeasecondgrading
+jasoccursintheKhovanovhomology. However,since Uisunitary,itfollows[22]thatthereisa
+basisforCinwhichUisdiagonal. Let B={|s/an}bracketri}ht}denotethisbasis. Let λsdenotetheeigenvalue
+ofUcorrespondingto |s/an}bracketri}htsothatU|s/an}bracketri}ht=λs|s/an}bracketri}ht.Letαs,s′bethematrixelementfor ∂sothat
+∂|s/an}bracketri}ht=/summationdisplay
+s′αs,s′|s′/an}bracketri}ht
+wheres′runs overaset ofbasis elementssothat i(s′) =i(s)+1.
+8Lemma. With the above conventions, we have that for |s′/an}bracketri}hta basis element such that αs,s′/ne}ationslash= 0
+thenλs′=−λs.
+Proof.Notethat
+U∂|s/an}bracketri}ht=U(/summationdisplay
+s′αs,s′|s′/an}bracketri}ht) =/summationdisplay
+s′αs,s′λs′|s′/an}bracketri}ht
+while
+∂U|s/an}bracketri}ht=∂λs|s/an}bracketri}ht=/summationdisplay
+s′αs,s′λs|s′/an}bracketri}ht.
+SinceU∂+∂U= 0,theconclusionoftheLemmafollowsfromtheindependenceof theelements
+inthebasisfortheHilbertspace. //
+In this way we see that eigenvalues will propagate forward fr omC0with alternating signs ac-
+cording to the appearance of successive basis elements in th e boundary formulas for the chain
+complex. Variousstatesofaffairsarepossibleingeneral, withneweigenvaluuesstartingatsome
+Ckfork >0.The simplest state of affairs would be if all the possible eig envalues (up to multi-
+plicationby −1)forUoccurred in C0sothat
+C0=⊕λC0
+λ
+whereλruns over all the distinct eigenvalues of Urestricted to C0,andC0
+λis spanned by all
+|s/an}bracketri}htinC0withU|s/an}bracketri}ht=λ|s/an}bracketri}ht.Let us take the further assumption that for each λas above, the
+subcomplexes
+C•
+λ:C0
+λ−→ C1
+−λ−→ C2
++λ−→ ···Cn
+(−1)nλ
+haveC=⊕λC•
+λas their direct sum. With this assumption about the chain com plex, define
+|ψ/an}bracketri}ht=/summationtext
+s|s/an}bracketri}htas before,with |s/an}bracketri}htrunningoverthewholebasisfor C.Thenitfollowsjustasinthe
+beginningofthissectionthat
+/an}bracketle{tψ|U|ψ/an}bracketri}ht=/summationdisplay
+λλχ(H(C•
+λ)).
+Hereχdenotes the Euler characteristic of the homology. The point is, that this formula for
+/an}bracketle{tψ|U|ψ/an}bracketri}httakes exactly the form we had for thespecial case of Khovanov homology(with eigen-
+values(−1)iqj),butheretheformulaoccursjustintermsoftheeigenspace decompositionofthe
+unitarytransformation Uinrelationtothechain complex. Clearly thereismorework t obedone
+hereand wewillreturn toit inasubsequentpaper.
+Remarkon thedensity matrix. Giventhestate |ψ/an}bracketri}ht,wecan define the densitymatrix
+ρ=|ψ/an}bracketri}ht/an}bracketle{tψ|.
+Withthisdefinitionitis immediatethat
+Tr(Uρ) =/an}bracketle{tψ|U|ψ/an}bracketri}ht
+whereTr(M)denotes the trace of a matrix M.Thus we can restate the form of our result about
+Eulercharacteristicsas
+Tr(Uρ) =/summationdisplay
+λλχ(H(C•
+λ)).
+9InsearchingforaninterpretationoftheKhovanovcomplexi nthisquantumcontextitisusefulto
+usethisreformulation. Forthebracket wehave
+/an}bracketle{tK/an}bracketri}ht=<ψ|U|ψ>=Tr(Uρ).
+Remark on physics. We are taking the wholeHilbert space as the statespace of a ph ysical sys-
+tem. Ifthesystemwereaclassicalsystemthen theenergetic states oftheclassicalsystemwould
+correspond tothebasiswehavechosen fortheHilbertspace. Thisisin linewiththeconceptsof
+quantum mechanics, since the classical states are then in co rrespondence with a basis for mea-
+surement. In quantum context we think of the elements in the H ilbert space as corresponding
+to superpositions of possible measurements. All this is the n transposed to topological configu-
+rations for the knot or link. But the new ingredient is the one from topology that will make the
+statesoftheknotintoachaincomplexandleadtohomologica lcomputations. Sinceweareusing
+enhanced states, each state is a generator for the chain comp lex. Thus we can regard C(K)itself
+as the chain complex (over the complex numbers) and add in ext ra grading structure to boot.
+There is also the fine structure of the underlying state circl es, but this is not seen by the Hilbert
+space itself and the chain complex structure can certainly b e written just in terms of the basis of
+the Hilbert space. We want to know what is the relationship be tween the unitary transformation
+and thehomological structure. This relation is givenby the way the grading works in relation to
+(−1)iqjfor each state: the constancy of the quantum grading junder the action of the differen-
+tial. It isthisthatmakes /an}bracketle{tψ|U|ψ/an}bracketri}htagraded Eulercharacteristicforthehomology. In thisway t he
+unitary transformation is linked with the structure of stat e transitions that govern the homology.
+At theend ofthissectionweindicateamoregeneral approach to thispattern.
+The key conceptual issue is the construction of a Hilbert spa ce whose basis is the set of
+observablestatesofaphysicalsystem. Wecanalwaysdothis forastatisticalmechanicalsystem.
+Letsdenote such states and E(s)denote the energy of a state s(observable and real). Define a
+unitarytransformation U(t)(wheretdenotestime)by
+U(t)|s/an}bracketri}ht=e(it/¯h)E(s)|s/an}bracketri}ht.
+Define theamplitude
+A(t) = Σse(it/¯h)E(s).
+Notethatwehavetheformula
+A(t) =/an}bracketle{tψ|U(t)|ψ/an}bracketri}ht,
+where|ψ/an}bracketri}ht= Σs|s/an}bracketri}htjust as before. Now U(t)can be regarded as the quantum evolution of the
+stateofthesystemandwehavewrittenthisamplitudeinanal ogywithapartitionfunctionforthe
+statisticalmechanics model.
+Onemoreconceptualproblem: Weconsideran amplitudeofthe form
+/an}bracketle{tψ|U|ψ/an}bracketri}ht=/summationdisplay
+s(−1)i(s)qj(s)
+10whereq=eiθisaunitcomplexnumber. Thequestionis, howdoesthisfit the pattern
+A(t) =/summationdisplay
+se(it/¯h)E(s)?
+So wewant
+(t/¯h)E(s) =πi(s)+θj(s)
+whichgives
+E(s) = (¯h/t)(πi(s)+θj(s)).
+Evolution from time t= 0is out of the question. Probably the best interpretation is t o think of
+theevolutionstartingat t= 1.
+In this point of view, the Hilbert space for expressing the br acket polynomial as a quantum
+statistical amplitudeis quitenaturally the chain complex for Khovanovhomology with complex
+coefficients,andtheunitarytransformationthatisthestr uctureofthebracketpolynomialactson
+the homologyof this chain complex. This means that the homol ogy classes contain information
+preservedbythequantumprocessthatunderliesthebracket polynomial. Wewouldliketoexploit
+thisdirectrelationshipbetweenthequantummodelandtheK hovanovhomologytoobtaindeeper
+information about the relationship of topology and quantum information theory, and we would
+liketo usethisrelationshiptoprobetheproperties ofthes etopologicalinvariants.
+5 BackgroundonKhovanovHomology
+In this section, we describe Khovanov homology along the lin es of [19, 2], and we tell the story
+sothatthegradingsandthestructureofthedifferentialem ergeinanaturalway. Thisapproachto
+motivatingtheKhovanovhomologyuseselementsofKhovanov ’soriginalapproach,Viro’suseof
+enhancedstatesforthebracketpolynomial[28],andBar-Na tan’semphasisontanglecobordisms
+[3]. Weusesimilarconsiderationsinourpaper[17].
+Two key motivating ideas are involved in finding the Khovanov invariant. First of all, one
+would like to categorify a link polynomial such as /an}bracketle{tK/an}bracketri}ht.There are many meanings to the term
+categorify, but here the quest is to find a way to express the li nk polynomial as a graded Euler
+characteristic /an}bracketle{tK/an}bracketri}ht=χq/an}bracketle{tH(K)/an}bracketri}htforsomehomologytheory associatedwith /an}bracketle{tK/an}bracketri}ht.
+We will use the bracket polynomial and its enhanced states as described in the previous sec-
+tions of this paper. To see how the Khovanovgrading arises, c onsider the form of the expansion
+of this version of the bracket polynomialin enhanced states . We havethe formula as a sum over
+enhanced states s:
+/an}bracketle{tK/an}bracketri}ht=/summationdisplay
+s(−1)i(s)qj(s)
+11wherei(s)is the number of B-type smoothings in s,λ(s)is the number of loops in slabeled
+1minus the number of loops labeled X,andj(s) =i(s) +λ(s). This can be rewritten in the
+followingform:
+/an}bracketle{tK/an}bracketri}ht=/summationdisplay
+i,j(−1)iqjdim(Cij)
+wherewedefine Cijtobethelinearspan(overthecomplexnumbersforthepurpos eofthispaper,
+but over the integers or the integers modulo two for other con texts) of the set of enhanced states
+withi(s) =iandj(s) =j.Then thenumberofsuch statesisthedimension dim(Cij).
+Wewouldliketohavea bigradedcomplexcomposedofthe Cijwithadifferential
+∂:Cij−→ Ci+1j.
+Thedifferentialshouldincreasethe homologicalgrading iby1andpreservethe quantumgrading
+j.Thenwecould write
+/an}bracketle{tK/an}bracketri}ht=/summationdisplay
+jqj/summationdisplay
+i(−1)idim(Cij) =/summationdisplay
+jqjχ(C•j),
+whereχ(C•j)istheEulercharacteristicofthesubcomplex C•jforafixed valueof j.
+This formula would constitute a categorification of the brac ket polynomial. Below, we shall see
+howthe original Khovanov differential ∂is uniquely determined by the restriction that j(∂s) =
+j(s)for each enhanced state s.Sincejis preserved by the differential, these subcomplexes C•j
+havetheirownEulercharacteristicsand homology. Wehave
+χ(H(C•j)) =χ(C•j)
+whereH(C•j)denotes thehomologyofthecomplex C•j. Wecan write
+/an}bracketle{tK/an}bracketri}ht=/summationdisplay
+jqjχ(H(C•j)).
+Thelastformulaexpressesthebracketpolynomialasa gradedEulercharacteristic ofahomology
+theory associatedwith theenhanced statesof thebracket st atesummation. Thisis thecategorifi-
+cation of the bracket polynomial. Khovanov proves that this homology theory is an invariant of
+knots and links (via the Reidemeister moves of Figure 1), cre ating a new and stronger invariant
+thantheoriginalJonespolynomial.
+Wewillconstructthedifferentialinthiscomplexfirstform od-2coefficients. Thedifferential
+is based on regarding two states as adjacentif one differs from the other by a single smoothing
+at some site. Thus if (s,τ)denotes a pair consisting in an enhanced state sand siteτof that
+state withτof typeA, then we consider all enhanced states s′obtained from sby smoothing at
+τand relabeling only thoseloops that are affected by the resm oothing. Call this set of enhanced
+12I
+II
+III
+Figure1: ReidemeisterMoves
+statesS′[s,τ].Then we shall define the partial differential ∂τ(s)as a sum over certain elements
+inS′[s,τ],and thedifferentialby theformula
+∂(s) =/summationdisplay
+τ∂τ(s)
+withthesumoveralltype Asitesτins.Itthenremainstoseewhatarethepossibilitiesfor ∂τ(s)
+sothatj(s)ispreserved.
+Notethatif s′∈S′[s,τ], theni(s′) =i(s)+1.Thus
+j(s′) =i(s′)+λ(s′) = 1+i(s)+λ(s′).
+From thisweconcludethat j(s) =j(s′)ifand onlyif λ(s′) =λ(s)−1.Recall that
+λ(s) = [s: +]−[s:−]
+where[s: +]is the number of loops in slabeled+1,[s:−]is the number of loops labeled −1
+(sameas labeledwith X)andj(s) =i(s)+λ(s).
+Proposition. Thepartialdifferentials ∂τ(s)areuniquelydeterminedbytheconditionthat j(s′) =
+j(s)foralls′involvedintheactionofthepartialdifferentialontheenh ancedstates.Thisunique
+form ofthepartial differentialcan bedescribed by thefoll owingstructures ofmultiplicationand
+comultiplication on the algebra A = k[X]/(X2)wherek=Z/2Zfor mod-2 coefficients, or
+k=Zforintegralcoefficients.
+1. Theelement 1isamultiplicativeunitand X2= 0.
+2.∆(1) = 1 ⊗X+X⊗1and∆(X) =X⊗X.
+These rules describe the local relabeling process for loops in a state. Multiplicationcorresponds
+to the case where two loops merge to a single loop, while comul tiplication corresponds to the
+casewhere oneloopbifurcates intotwoloops.
+13Proof.Using the above description of the differential, suppose th at there are two loops at τthat
+merge in the smoothing. If both loops are labeled 1insthen the local contribution to λ(s)is2.
+Lets′denote a smoothing in S[s,τ].In order for the local λcontribution to become 1, we see
+that the merged loop must be labeled 1. Similarly if the two loops are labeled 1andX,then the
+merged loop must be labeled Xso that the local contribution for λgoes from 0to−1.Finally,
+if the two loops are labeled XandX,then there is no label available for a single loop that will
+give−3,sowedefine ∂tobezero inthiscase. Wecan summarizetheresultbysayingt hatthere
+is a multiplicativestructure msuch thatm(1,1) = 1,m(1,X) =m(X,1) =x,m(X,X) = 0,
+and this multiplication describes the structure of the part ial differential when two loops merge.
+Since this is the multiplicativestructure of the algebra A=k[X]/(X2),we take this algebra as
+summarizingthedifferential.
+Nowconsiderthecasewhere shasasingleloopatthesite τ.Smoothingproducestwoloops.
+Ifthesingleloopislabeled X,thenwemustlabeleachofthetwoloopsby Xinordertomake λ
+decreaseby 1. Ifthesingleloopislabeled 1,thenwecanlabelthetwoloopsby Xand1ineither
+order. In this second case we take the partial differential o fsto be the sum of these two labeled
+states. Thisstructurecan bedescribedbytakingacoproduc tstructurewith ∆(X) =X⊗Xand
+∆(1) = 1 ⊗X+X⊗1.Wenowhavethealgebra A=k[X]/(X2)withproduct m:A⊗A −→ A
+and coproduct ∆ :A −→ A ⊗ A ,describing the differential completely. This completes th e
+proof. //
+Partial differentials are defined on each enhanced state sand a siteτof typeAin that state.
+We consider states obtained from the given state by smoothin g the given site τ. The result of
+smoothingτis to produce a new state s′with one more siteof type Bthans.Formings′froms
+we either amalgamate two loops to a single loop at τ, or we divide a loop at τinto two distinct
+loops. In the case ofamalgamation,thenew state sacquires the label on theamalgamated circle
+that is the product of the labels on the two circles that are it s ancestors in s. This case of the
+partial differential is described by the multiplication in the algebra. If one circle becomes two
+circles, then we apply the coproduct. Thus if the circle is la beledX, then the resultant two
+circles are each labeled Xcorresponding to ∆(X) =X⊗X. If the orginal circle is labeled
+1then we take the partial boundary to be a sum of two enhanced st ates with labels 1andX
+in one case, and labels Xand1in the other case, on the respective circles. This correspon ds to
+∆(1) = 1 ⊗X+X⊗1.Modulotwo,theboundaryofanenhancedstateisthesum,over allsitesof
+typeAinthestate,ofthepartialboundariesatthesesites. Itisn othardtoverifydirectlythatthe
+squareoftheboundarymappingiszero(thisistheidentityo fmixedpartials!) andthatitbehaves
+as advertised, keeping j(s)constant. There is more to say about the nature of this constr uction
+withrespecttoFrobeniusalgebrasandtanglecobordisms. I nFigures2,3and4weillustratehow
+the partial boundaries can be conceptualized in terms of sur face cobordisms. The equality of
+mixed partials corresponds to topological equivalence of t he corresponding surface cobordisms,
+and to the relationships between Frobenius algebras and the surface cobordism category. In
+particular,inFigure4weshowhowinakeycaseoftwosites(l abeled1and2inthatFigure)the
+twoorders ofpartial boundaryare
+∂2∂1= (1⊗m)◦(∆⊗1)
+14AA-1
+A
+A-1
+A-1
+AΔ
+m
+Figure2: SaddlePoints andState Smoothings
+and
+∂1∂2= ∆◦m.
+In theFrobeniusalgebra A=k[X]/(X2)wehavetheidentity
+(1⊗m)◦(∆⊗1) = ∆◦m.
+Thus the Frobenius algebra implies the identity of the mixed partials. Furthermore, in Figure
+3 we see that this identity corresponds to the topological eq uivalence of cobordisms under an
+exchange of saddle points. There is more to say about all of th is, but we will stop here. The
+proof of invariance of Khovanov homology with respect to the Reidemeister moves (respecting
+gradingchanges)willnotbegivenhere. See [19, 2, 3]. It isr emarkablethatthisversionofKho-
+vanov homology is uniquely specified by natural ideas about a djacency of states in the bracket
+polynomial.
+Remark on Integral Differentials . Choose an ordering for the crossings in the link diagram K
+and denote them by 1,2,···n.Letsbe any enhanced state of Kand let∂i(s)denote the chain
+obtainedfrom sbyapplyingapartial boundaryat the i-th siteofs.If thei-th siteisasmoothing
+of typeA−1, then∂i(s) = 0.If thei-th site is a smoothing of type A, then∂i(s)is given by the
+rulesdiscussedabove(withthesamesigns). Thecompatibil ityconditionsthatwehavediscussed
+showthatpartialscommuteinthesensethat ∂i(∂j(s)) =∂j(∂i(s))foralliandj.Onethendefines
+signedboundaryformulasintheusualwayofalgebraictopol ogy. Onewaytothinkofthisregards
+the complex as the analogue of a complex in de Rahm cohomology . Let{dx1,dx2,···,dxn}be
+a formal basis for a Grassmann algebra so that dxi∧dxj=−dxj∧dxiStarting with enhanced
+statessinC0(K)(that is, state with all A-type smoothings) Define formally, di(s) =∂i(s)dxi
+and regarddi(s)as identical with ∂i(s)as we have previously regarded it in C1(K).In general,
+15Δm
+F G1m1Δ
+Δm
+Figure3: Surface Cobordisms
+Δm
+1Δ
+1m12
+12=1m 1Δ ( ())
+12=Δm)) ((
+12 12=1'22'1
+1' 2'
+Figure4: Local Boundaries Commute
+16given an enhanced state sinCk(K)withB-smoothings at locations i1< i2<···< ik,we
+represent thischain as sdxi1∧···∧dxikand define
+∂(sdxi1∧···∧dxik) =n/summationdisplay
+j=1∂j(s)dxj∧dxi1∧···∧dxik,
+just as in a de Rahm complex. TheGrassmann algebra automatic ally computesthe correct signs
+in the chain complex, and this boundary formula gives the ori ginal boundary formula when we
+take coefficients modulo two. Note, that in this formalism, p artial differentials ∂iof enhanced
+states with a B-smoothing at the site iare zero due to the fact that dxi∧dxi= 0in the Grass-
+mann algebra. There is more to discuss about the use of Grassm ann algebra in this context. For
+example,thisapproach clarifies parts oftheconstructioni n[18].
+It of interest to examine this analogy between the Khovanov ( co)homology and de Rahm
+cohomology. In that analogy the enhanced states correspond to the differentiable functions on a
+manifold. TheKhovanovcomplex Ck(K)isgeneratedbyelementsoftheform sdxi1∧···∧dxik
+wheretheenhanced state shasB-smoothingsat exactlythesites i1,···,ik.If wewere tofollow
+theanalogywithdeRahmcohomologyliterally,wewoulddefin eanewcomplex DR(K)where
+DRk(K)is generated by elements sdxi1∧···∧dxikwheresisanyenhanced state of the link
+K.Thepartialboundariesaredefinedinthesamewayasbeforean dtheglobalboundaryformula
+is just as we have written it above. This gives a newchain complex associated with the link K.
+Whetheritshomologycontainsnewtopologicalinformation aboutthelink Kwillbethesubject
+ofasubsequentpaper.
+A further remark on de Rham cohomology. There is another deep relation with the de Rham
+complex: In [21] it was observed that Khovanov homology is re lated to Hochschild homology
+andHochschildhomologyisthoughttobeanalgebraicversio nofdeRhamchaincomplex(cyclic
+cohomologycorrespondsto deRham cohomology),compare[23 ].
+6 OtherHomologicalStates
+The formalism that we have pursued in this paper to relate Kho vanov homology and the Jones
+polynomialwithquantumstatisticscanbegeneralizedtoap plytoothersituations. Itispossibleto
+associatea finite dimensionalHilbert space that is also a ch ain complex(or cochain complex)to
+topologicalstructures otherthan knotsand links. To formu latethis,let Xdenotethe topological
+structure and C(X)denote the associated linear space endowed with a boundary o perator∂:
+C(X)−→ C(X),with∂∂= 0.We want to consider situations where there is a unitary opera tor
+U:C(X)−→ C(X)such thatU∂+∂U= 0so thatUinduces a unitary action on H(X),the
+homologyof C(X)withrespect to ∂.
+17For example, let Mbe a differentiable manifold and C(M)denote the DeRham complex of
+Moverthecomplexnumbers. Thenforadifferentialformofthe typef(x)ωinlocalcoordinates
+x1,···,xnandωa wedgeproduct ofasubsetof dx1···dxn,wehave
+d(fω) =n/summationdisplay
+i=1(∂f/∂x i)dxi∧ω.
+Heredis the differential for the DeRham complex. Then C(M)has basis the set of |f(x)ω/an}bracketri}ht
+whereω=dxi1∧ ··· ∧dxikwithi1<···< ik.We could achieve Ud+dU= 0ifUis a
+verysimpleunitaryoperator(e.g. multiplicationbyphase sthatdonotdependonthecoordinates
+xi) but in general it will be an interesting problem to determin eall unitary operators Uwith this
+property.
+Even in the case of Khovanov homology, we could keep the homol ogy theory fixed and ask
+for other unitary operators Uthat satisfyU∂+∂U= 0.Knowing other examples of such oper-
+ators would shed light on the nature of Khovanov homology fro m the point of view of quantum
+statistics.
+7 Appendix- TheHadamardTest
+In order to make a quantum computation of the trace of a unitar y matrixU, one can use the
+Hadamardtest toobtainthediagonalmatrixelements /an}bracketle{tψ|U|ψ/an}bracketri}htofU.Thetraceisthenthesumof
+thesematrix elements as |ψ/an}bracketri}htruns overan orthonormal basisfor thevectorspace. In theap plica-
+tiontothealgorithmdescribedherefortheJonespolynomia litisonlynecessarytocomputeone
+numberoftheform /an}bracketle{tψ|U|ψ/an}bracketri}ht.TheHadamard testproceeds as follows.
+Wefirst obtain1
+2+1
+2Re/an}bracketle{tψ|U|ψ/an}bracketri}ht
+as an expectationbyapplyingtheHadamard gate H
+H|0/an}bracketri}ht=1√
+2(|0/an}bracketri}ht+|1/an}bracketri}ht)
+H|1/an}bracketri}ht=1√
+2(|0/an}bracketri}ht−|1/an}bracketri}ht)
+tothefirst qubitof
+CU◦(H⊗1)|0/an}bracketri}ht|ψ/an}bracketri}ht=1√
+2(|0/an}bracketri}ht⊗|ψ/an}bracketri}ht+|1/an}bracketri}ht⊗U|ψ/an}bracketri}ht.
+HereCUdenotes controlled U,acting asUwhen the control bit is |1/an}bracketri}htand the identity mapping
+whenthecontrolbitis |0/an}bracketri}ht.Wemeasuretheexpectationforthefirstqubit |0/an}bracketri}htoftheresultingstate
+(H⊗1)◦CU◦(H⊗1)|0/an}bracketri}ht|ψ/an}bracketri}ht=
+18|ψ>UH H |0>
+Figure5: HadamardTest
+1
+2(H|0/an}bracketri}ht⊗|ψ/an}bracketri}ht+H|1/an}bracketri}ht⊗U|ψ/an}bracketri}ht) =1
+2((|0/an}bracketri}ht+|1/an}bracketri}ht)⊗|ψ/an}bracketri}ht+(|0/an}bracketri}ht−|1/an}bracketri}ht)⊗U|ψ/an}bracketri}ht)
+=1
+2(|0/an}bracketri}ht⊗(|ψ/an}bracketri}ht+U|ψ/an}bracketri}ht)+|1/an}bracketri}ht⊗(|ψ/an}bracketri}ht−U|ψ/an}bracketri}ht)).
+Thisexpectationis
+1
+4(/an}bracketle{tψ|+/an}bracketle{tψ|U†)(|ψ/an}bracketri}ht+U|ψ/an}bracketri}ht) =1
+2+1
+2Re/an}bracketle{tψ|U|ψ/an}bracketri}ht.
+In Figure 5, we illustrate this computation with a diagram th at indicates the structure of the
+testwithparallellinescorrespondingtotensorproductso fthesinglequbitspace(withthreelines
+chosenforillustrationasthesizeof U). Theextratensorfactorisindicatedonthetoplinewithth e
+Hadamard matrix Hindicated by abox and thecontrol of Uindicated by a circle with a vertical
+control line extendingdown to the U-box. The half -circle on the top line on theright stands for
+the measurement of that line that is used for the computation . Thus Figure 5 represents a circuit
+diagram for the quantum computation of1
+2+1
+2Re/an}bracketle{tψ|U|ψ/an}bracketri}htand hence the quantum computation
+of1
+2+1
+2Re/an}bracketle{tK/an}bracketri}htwhenUis taken to be the unitary tranformation corresponding to th e bracket
+polynomial,as discussedin previoussectionsofthispaper .
+Theimaginarypart isobtainedbyapplyingthesameprocedur e to
+1√
+2(|0/an}bracketri}ht⊗|ψ/an}bracketri}ht−i|1/an}bracketri}ht⊗U|ψ/an}bracketri}ht
+This is the method used in [1, 14, 15], and the reader may wish t o contemplate its efficiency in
+thecontextofthissimplemodel. NotethattheHadamardtest enables thisquantumcomputation
+to estimate the trace of any unitary matrix Uby repeated trials that estimate individual matrix
+entries/an}bracketle{tψ|U|ψ/an}bracketri}ht.
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+21
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+arXiv:1001.0355v2  [math.PR]  11 Jul 2010ENTROPY OF RANDOM WALK RANGE ON UNIFORMLY
+TRANSIENT AND ON UNIFORMLY RECURRENT GRAPHS
+DAVID WINDISCH
+The Weizmann Institute of Science
+Faculty of Mathematics and Computer Science
+POB 26
+Rehovot 76100
+Israel
+david.windisch@weizmann.ac.il
+Abstract. We study the entropy of the distribution of the set Rnof vertices
+visited by a simple random walk on a graph with bounded degree s in its first
+nsteps. It is shown that this quantity grows linearly in the ex pected size
+ofRnif the graph is uniformly transient, and sublinearly in the e xpected
+size if the graph is uniformly recurrent with subexponentia l volume growth.
+This in particular answers a question asked by Benjamini, Ko zma, Yadin and
+Yehudayoff [1].
+1.Introduction
+Let (Xn)n≥0be a simple random walk on a graph G= (V,E) starting at some
+vertexo∈V. The entropy of the distribution of Xnfor large nhas been studied
+on Cayley graphs and is known to be related to other objects of inte rest, such as
+the rate of escape and the existence of non-trivial bounded harm onic functions, see
+[5], [6], [10]. This work is devoted to the entropy of a similar observable, to that of
+the range of the random walk. Let Rn={X0,X1,...,X n}be the set of vertices
+visited by the random walk in its first nsteps. The entropy of Rnis defined as
+Ho(Rn) =Eo/bracketleftBigg
+log/parenleftbigg1
+po(Rn)/parenrightbigg/bracketrightBigg
+, (1.1)
+where the random walk starts at o∈V,Po-a.s.,po(A) =Po[Rn=A] for any
+finite connected set A⊆Vcontaining o, and 0log(1 /0) is defined to be 0 as
+usual. According to Shannon’s noiseless coding theorem [9], Ho(Rn)/log2 can
+be interpreted as the approximate number of 0-1-bits required pe r realization in
+order to encode a large number of independent realizations of Rnwith negligible
+probability of error. Up to an additive constant, Ho(Rn)/log2 can also be viewed
+as the expected number of bits necessary and sufficient to encode Rn(see [3],
+Theorem 5.4.1), and as the expected number of fair coin tosses req uired for the
+simulation of Rn(see [3], Theorem 5.12.3).
+The entropy of Rnfor random walk on Zd,d≥1, is investigated in a recent work
+by Benjamini, Kozma, Yadin and Yehudayoff [1]. There, the authors s how that the
+1largenbehavior of Ho(Rn) is of order nford≥3, of order n/log2nford= 2 and
+of order log nford= 1. Comparing this behavior with that of the expected size
+/a\}bracketle{tRn/a\}bracketri}htoofRn,
+/a\}bracketle{tRn/a\}bracketri}hto=Eo[|Rn|], (1.2)
+known to be of order nford≥3, of order n/lognford= 2 and of order√nfor
+d= 1 (cf. [8], p. 221) , we observethat on Zd,Ho(Rn) growslinearly in /a\}bracketle{tRn/a\}bracketri}htoin the
+transient case and only sublinearly in the recurrent case. On gener al graphs with
+bounded degrees, Ho(Rn) can grow at most linearly in /a\}bracketle{tRn/a\}bracketri}hto(see Proposition 2.4),
+but the assumptions on recurrence and transience alone do not allo w us to conclu-
+sively compare Ho(Rn) with/a\}bracketle{tRn/a\}bracketri}hto. Under slightly stronger assumptions, however,
+we can generalize the observation just made for Zdto large classes of graphs.
+The graphswe considerin this workarecharacterizedbystrength enedtransience
+and recurrence conditions. Recall that a graph is transient if the e scape probability
+Po[o /∈ {X1,X2,...}] is strictly positive for any starting vertex o∈V. We call a
+graphGuniformly transient if such an estimate holds uniformly in o, that is, if
+inf
+o∈VPo[o /∈ {X1,X2,...}]>0. (1.3)
+Theorem 1.1 shows that Ho(Rn) grows linearly in /a\}bracketle{tRn/a\}bracketri}htoon uniformly transient
+graphswithboundeddegrees. Suchastatementdoesnotholdund ertheassumption
+of transience alone, see Remark 3.5 for a counterexample.
+Theorem 1.1. LetGbe a uniformly transient graph with bounded degrees. Then
+liminf
+n→∞inf
+o∈VHo(Rn)
+/a\}bracketle{tRn/a\}bracketri}hto>0. (1.4)
+On the other hand, we call a graph Guniformly recurrent if
+sup
+o∈VPo[o /∈ {X1,...,X n}]−→0,asn→ ∞. (1.5)
+In addition, let |∂eB(x,r)|be the number of vertices at distance r+1 from x∈V
+with respect to the usual graph distance. If the degrees of the g raph are bounded
+byd≥2, then|∂eB(x,r)| ≤dr+1. The following condition requires slightly more:
+for anyǫ >0, sup
+x∈V|∂eB(x,r)| ≤eǫr,for infinitely many r∈N. (1.6)
+Under these conditions, we prove that Ho(Rn) grows only sublinearly in /a\}bracketle{tRn/a\}bracketri}hto. We
+note that recurrence alone does not imply such a statement, see R emark 4.1 and
+also Remark 5.4.
+Theorem 1.2. LetGbe any uniformly recurrent graph with bounded degrees sat-
+isfying(1.6). Then
+sup
+o∈VHo(Rn)
+/a\}bracketle{tRn/a\}bracketri}hto−→0,asn→ ∞. (1.7)
+The above results apply in particular to all vertex-transitive graph s. Recall that
+a graphGisvertex-transitive , if for every pair of vertices ( x,x′), there is a bijection
+φfromVtoVsuch that φ(x) =x′andd(y,y′) =d(φ(y),φ(y′)) for all y,y′∈V,
+whered(.,.) denotesthe usual graphdistance. Forvertex-transitivegrap hs,Ho(Rn)
+and/a\}bracketle{tRn/a\}bracketri}htodo not depend on the starting vertex oof the random walk, so we omit o
+from the notation. We can deduce the following dichotomy from the r esults above:
+2Theorem 1.3. LetGbe a vertex-transitive graph.
+IfGis transient, then liminf
+n→∞H(Rn)
+/a\}bracketle{tRn/a\}bracketri}ht>0, (1.8)
+ifGis recurrent, thenH(Rn)
+/a\}bracketle{tRn/a\}bracketri}ht−→0,asn→ ∞. (1.9)
+For uniformly transient graphs, /a\}bracketle{tRn/a\}bracketri}htgrows linearly in n(see Proposition 5.2).
+Hence, (1.8) in particular shows that H(Rn) grows linearly in non all vertex-
+transitive and transient graphs, thus answering a question asked in [1], Section 3.3.
+We now comment on the proofs, starting with the one of Theorem 1.1 , given
+in Section 3. A simple observation made in [1] shows that Ho(Rn) is bounded
+from below by a constant times the expected size of the interior bou ndary∂iRn
+ofRn. It thus suffices to prove a lower bound of order /a\}bracketle{tRn/a\}bracketri}htoonEo[|∂iRn|], or,
+equivalently, to prove that the fraction of the visited vertices belo nging to ∂iRnis
+non-degenerate. Note that a vertex belongs to ∂iRnif it is visited by the random
+walk, but at least one of its neighbors is not. Not all visited vertices a re equally
+likely to belong to ∂iRn. Indeed, a vertex in the middle of a long path of vertices
+of degree 2 is very unlikely to end up in the boundary of Rn, because the random
+walk typically returns many times and visits both of its neighbors befo re escaping
+from it. In order to avoid such situations, we observe that on unifo rmly transient
+graphs, vertices of degree at least 3 exist within a bounded distanc e of every vertex
+(cf. Lemma 3.3), and use uniform transience to prove that whenev er the random
+walk visits such a vertex x, there is a non-degenerateprobability that the walk then
+escapes from the ball B(x,1) of radius 1 around it and leaves xin∂iRn. This yields
+the lowerbound on Eo[|∂iRn|] requiredtoproveTheorem1.1. In orderto showthat
+transience alone is not a sufficient assumption for the conclusion of T heorem 1.1,
+we consider a binary tree, where every edge at depth lis replaced by a path of
+lengthl+3, and show that this graph satisfies (1.7), see Remark 3.5 and Figu re 1.
+In the proof of Theorem 1.2 in Section 4, we cover every possible rea lization of
+Rnwith one of at most supx∈V|∂eB(x,r)|(const.)|Rn|/rdifferent collections of balls
+of radius r≥1. Uniform recurrence implies that most of these balls are typically
+completely covered by Rn. We then consider the conditional entropy of Rn, given
+the number Kof such balls intersected by Rnand the number Lof balls that are
+not completely filled by Rn(the definition of conditional entropy is recalled in (2.4)
+below). The conditional entropy can then be bounded by the expec ted logarithm
+of the number of possible configurations Rncan belong to, given ( K,L) (cf. (2.7)).
+By assumption (1.6), the expected logarithm of the number of poss ible collections
+of covering balls can be made smaller than ǫ/a\}bracketle{tRn/a\}bracketri}htofor any fixed ǫ >0 by choosing r
+appropriately,while the assumptionof uniform recurrenceyields a s imilarbound on
+the expected logarithm ofthe number of possible choices of the exa ct configurations
+ofRnin the few unfilled balls. This argument proves the required estimate o n the
+conditional entropy of Rngiven (K,L). Since the entropy of ( K,L) itself is only of
+orderatmostlog n, thisissufficientforTheorem1.2. Theorem1.2cannotbeproved
+for every recurrent graph. As a counterexample, we consider a s equence of finite
+binary trees with rapidly increasing depths, connected together b y a half-infinite
+path, see Remark 4.1 and Figure 2.
+3The article is organized as follows: In Section 2, we introduce notatio n and
+preliminary results on entropy, derive a general lower bound on /a\}bracketle{tRn/a\}bracketri}htoand show
+thatHo(Rn) grows at most linearly in /a\}bracketle{tRn/a\}bracketri}hto. Section 3 contains the proof of
+Theorem 1.1 and Section 4 proves Theorem 1.2. Finally, the dichotomy for vertex-
+transitive graphs in Theorem 1.3 is deduced in Section 5.
+Acknowledgement. The author is grateful to Itai Benjamini for helpful con-
+versations and to an anonymous referee for helpful remarks.
+2.Notation and preliminaries
+In this section, we introduce the notation and prove some preliminar y results.
+These include a lower bound of order√non the expected size of Rnon a general
+infinite graph with bounded degrees, obtained in Proposition 2.3 by ad apting an
+argument in [7], as well as the observation that Ho(Rn) grows at most as fast as
+the expected size /a\}bracketle{tRn/a\}bracketri}htoon any infinite connected graph with bounded degrees.
+Throughout this article, we let G= (V,E) be a graph. Vdenotes the set
+of vertices and Ethe set of edges consisting of unordered pairs of vertices in V.
+Whenever {x,y} ∈E, we write x∼yand call the vertices xandyneighbors. The
+number of neighbors of a vertex xis always assumed to be finite and referred to as
+the degree of x, denoted deg( x). A path of length lis a sequence ( x0,x1,...,x l) of
+vertices in Vsuch that xi∼xi+1for 0≤i≤l−1. The distance d(x,y) between
+any two vertices xandyis defined as the length of the shortest path starting at
+x0=xand ending at xd(x,y)=y, and defined to be ∞if no such path exists. G
+is said to be connected if d(x,y)<∞for allx,y∈V. For any x∈V,r≥0, we
+define the ball B(x,r) ={y∈V:d(x,y)≤r}. The graph Ghas bounded degrees
+if supx∈Vdeg(x)≤dfor some d≥1. For any set Aof vertices, we define the
+interior and exterior boundaries of Aby∂iA={x∈A:x∼yfor some y∈V\A}
+and∂eA=∂i(V\A). The subgraph G(A) = (A,E(A)) induced by A⊆Vconsists
+of the vertices in Aand the set of edges E(A) ={{x,y} ∈E:{x,y} ⊆A}. We say
+thatAis connected if G(A) is connected in the above sense. The cardinality of any
+setAis denoted by |A|, the largest integer less than a∈Rby [a] and the minimum
+of numbers a,b∈Rbya∧b. Throughout the text, corc′are used to denote
+strictly positive constants with values changing from place to place. Dependence
+of constants on additional parameters appears in the notation. F or example, cd
+denotes a positive constant possibly depending on d.
+For anyx∈V, we denote by Pxthe law on VN(equipped with the canonical σ-
+algebra generated by the coordinate projections) of the Markov chain on Vstarting
+atxwith transition probability
+p(x,y) =/braceleftbigg
+1/deg(x),ifx∼y,
+0, otherwise,
+and write ( Xn)n≥0for the canonical coordinate process, referred to as the simple
+random walk on G. The corresponding expectation is denoted by Ex, the canonical
+shift-operatorson VNby(θn)n≥0. Notethatdeg( x)p(x,y) = deg(y)p(y,x), meaning
+that the measure π:A/ma√sto→/summationtext
+x∈Adeg(x) onVis a reversible measure for the simple
+random walk. The first entrance and hitting times of a set Aof vertices are defined
+as
+τA= inf{n≥0 :Xn∈A}, τ+
+A= inf{n≥1 :Xn∈A}, (2.1)
+4where we write τxrather than τ{x}ifAconsists of a single element x. The capacity
+of a finite nonempty subset AofVis defined as
+cap(A) =/summationdisplay
+x∈APx[τ+
+A=∞]deg(x). (2.2)
+We will generally refer to the starting vertex of the random walk as o. It will be
+convenient to define the collection Coas
+Co={A⊆V:Ais finite, connected, and o∈A}. (2.3)
+Note that Rn∈ Co,Po-a.s. The entropy of the range Rnof the random walk has
+been defined in (1.1). More generally, for any random variable Xtaking values in
+a countable set X, we define the entropy of Xby
+Ho(X) =Eo/bracketleftbigg
+log/parenleftbigg1
+po(X)/parenrightbigg/bracketrightbigg
+,wherepo(x) =Po[X=x],forx∈ X.
+Givenanotherrandomvariable Ytakingvaluesinacountableset Y, theconditional
+entropy of XgivenYis defined by
+Ho(X|Y) =Ho/parenleftbig
+(X,Y)/parenrightbig
+−Ho(Y). (2.4)
+An application of Jensen’s inequality shows that
+Ho(X)≤log|X|, (2.5)
+while the following estimate is elementary:
+Ho(X)≤Ho(X|Y)+Ho(Y). (2.6)
+Moreover, we have the following useful lemma:
+Lemma 2.1. For random variables XandYwith values in countable sets Xand
+Y,
+Ho(X|Y)≤Eo[log|XY|],where (2.7)
+Xy={x∈ X:Po[(X,Y) = (x,y)]>0},fory∈ Y.
+Proof.Using Jensen’s inequality, we have
+Ho(X|Y) =/summationdisplay
+(x,y)∈X×Y:
+Po[(X,Y)=(x,y)]>0Po/bracketleftbig
+(X,Y) = (x,y)/bracketrightbig
+log/parenleftbiggPo[Y=y]
+Po/bracketleftbig
+(X,Y) = (x,y)/bracketrightbig/parenrightbigg
+=/summationdisplay
+y∈YPo[Y=y]/summationdisplay
+x∈XyPo[X=x|Y=y]log/parenleftbigg1
+Po[X=x|Y=y]/parenrightbigg
+≤Eo[log|XY|]. /square
+The following simple lemma, combined with a covering argument, will be ins tru-
+mental in proving the bound on Ho(Rn) in Theorem 1.2.
+Lemma 2.2. LetG= (V,E)be any graph with bounded degrees, o∈V, and let
+A∈ Co(cf.(2.3)). Then there exists a nearest-neighbor path in G(A)starting at o,
+covering A, and of length at most 2|A|.
+5Proof.The standard depth-first search algorithm (see [2]) yields a spannin g tree
+TA= (A,ET) ofG(A) (i.e. a tree with vertices Aand edges ET⊆E(A)), as well as
+a nearest-neighbor path in TAcovering Aand traversing every edge in ETat most
+once in every direction. Since the length of such a path is bounded fr om above by
+2|ET|= 2(|A|−1)<2|A|, the statement follows. /square
+We now prove a general lower bound on the expected range of rand om walk on
+an infinite connected graph with bounded degrees. This lemma will allow us to
+disregard small errors when relating Ho(Rn) to/a\}bracketle{tRn/a\}bracketri}hto.
+Proposition 2.3. LetG= (V,E)be any infinite connected graph with bounded
+degrees. Then
+liminf
+n→∞inf
+o∈V/a\}bracketle{tRn/a\}bracketri}hto√n>0. (2.8)
+Proof.This proof is an adaptation of an argument appearing in [7] in the cont ext
+of random walk on Cayley graphs. Let ( Nk)k≥1be the successive times when the
+random walk visits a new vertex, that is, N1= 0, and for k≥1,Nk= inf{n >
+Nk−1:Xn/∈ {X0,...XNk−1}}. Then we have for k≥1 ando∈V, by the
+Chebychev inequality,
+Po[|Rn| ≤k] =Po/bracketleftBig/vextendsingle/vextendsingle/braceleftbig
+0≤l≤n:Xl∈ {XN1,...,X Nk}/bracerightbig/vextendsingle/vextendsingle=n+1/bracketrightBig
+≤1
+n+1/summationdisplay
+1≤i≤kEo/bracketleftbigg/summationdisplay
+0≤l≤n1{Xl=XNi}/bracketrightbigg
+.
+Noting that Xl/\e}atio\slash=XNiforl < Niand applying the strong Markov property at time
+Ni, we deduce that
+Po[|Rn| ≤k]≤k
+n/summationdisplay
+0≤l≤nsup
+x∈VPx[Xl=x].
+By a general heat-kernel upper bound, we have supx,y∈VPx[Xl=y]≤cd/√
+l+1,
+for some constant cd>0 depending on the uniform bound don degrees (see, for
+example, [11], Corollary 14.6, p. 149). Hence, we find
+Po[|Rn| ≤k]≤cdk√n,fork≥1.
+This last inequality, applied with k= [√n/(2cd)], yields
+Eo[|Rn|]≥√n
+2cdPo[|Rn|>[√n/(2cd)]]≥√n
+4cd. /square
+Finally, we prove that on any infinite connected graph with bounded d egrees,
+Ho(Rn) does not grow faster than /a\}bracketle{tRn/a\}bracketri}hto.
+Proposition 2.4. LetG= (V,E)be any infinite connected graph with degrees
+bounded by d. Then
+limsup
+n→∞sup
+o∈VHo(Rn)
+/a\}bracketle{tRn/a\}bracketri}hto≤2logd.
+6Proof.Lemma 2.1 and Lemma 2.2 together imply that
+Ho(Rn||Rn|)≤ /a\}bracketle{tRn/a\}bracketri}hto2logd,
+whereas it follows from (2.5) that
+Ho(|Rn|)≤log(n+1).
+Hence by (2.6), Ho(Rn)≤ /a\}bracketle{tRn/a\}bracketri}hto2logd+log(n+1). Proposition 2.3 completes the
+proof. /square
+3.The transient case
+In this section, we prove Theorem 1.1 asserting that the entropy o fRngrows at
+least linearly in its expected size /a\}bracketle{tRn/a\}bracketri}htoon any uniformly transient graph.
+Lemma 3.1 and Corollary 3.2 show that Ho(Rn) grows at least linearly in the
+expected size /a\}bracketle{t∂iRn/a\}bracketri}htoof∂iRn. These two statements and their proofs are straight-
+forward adaptations of Lemma 3 and Corollary 4 in [1].
+Lemma 3.1. For any graph Gwith degrees bounded by d≥2,
+Po[Rn=A]≤(1−d−1)|∂iA|−1,for allo∈A⊆V. (3.1)
+Proof.FixA⊆Vand define the successive entrance times to ∂iAbyT1=τ∂iA
+and fork≥2,Tk= inf{n > Tk−1:Xn∈∂iA}. Then{Rn=A} ⊆ {T|∂iA|≤n} ⊆
+{T|∂iA|−1< n}. Note that on the event {Tk<∞}, we have PXTk[XTk+1∈A]≤
+1−(1/d). An inductive application of the strong Markov property at the tim es
+T|∂iA|−1,T|∂iA|−2,...,T 1therefore yields
+Po[Rn=A]≤Po/bracketleftbigg/intersectiondisplay
+1≤k≤|∂iA|−1{Tk< n,X Tk+1∈A}/bracketrightbigg
+≤/parenleftBig
+1−1
+d/parenrightBig|∂iA|−1
+. /square
+Corollary 3.2. For any graph Gwith degrees bounded by d≥2,
+Ho(Rn)≥(/a\}bracketle{t∂iRn/a\}bracketri}hto−1)log/parenleftbig
+(1−d−1)−1/parenrightbig
+, (3.2)
+where/a\}bracketle{t∂iRn/a\}bracketri}hto=Eo[|∂iRn|].
+Proof.By Lemma 3.1,
+Ho(Rn)≥log/parenleftbig
+(1−d−1)−1/parenrightbig/summationdisplay
+A∈CoP[Rn=A]/parenleftbig
+|∂iA|−1/parenrightbig
+. /square
+By the last corollary, it suffices to control /a\}bracketle{t∂iRn/a\}bracketri}htoin order to prove a lower
+bound on Ho(Rn). We will eventually prove that many vertices of degree at least
+3 typically belong to ∂iRn. To this end, we prove in the following lemma that such
+vertices exist within constant distance of any vertex in uniformly tr ansient graphs.
+For notational convenience, we introduce the parameter
+α= inf
+x∈VPx[τ+
+x=∞], (3.3)
+which is strictly positive if Gis uniformly transient (cf. (1.3)).
+Lemma 3.3. LetGbe a uniformly transient graph with αas in(3.3). Then for
+anyx∈V, there is a vertex ywithdeg(y)≥3andd(x,y)≤1/α.
+7Proof.Letx∈V, and let rbe the distance from xto the vertex of degree at least
+3 closest to x. Then
+α≤Px[τ+
+x=∞]≤Px[τ∂iB(x,r)< τ+
+x] (3.4)
+=/summationdisplay
+y:y∼x1
+deg(x)Py[τ∂iB(x,r)< τx].
+Consider any neighbor yofxand letIybe the set of vertices in B(x,r)\∂iB(x,r)
+that are connected to yin (B(x,r)\∂iB(x,r))\ {x}. ThenIyis connected and
+consists of vertices of degree at most 2. If no vertex in Iyis a neighbor of a vertex
+in∂iB(x,r), then the escape probability on the right-hand side of (3.4) equals 0.
+Otherwise, the escape probability is equal to the probability that a s imple random
+walk on Zstarted at 1 reaches rbefore 0, hence to 1 /r(see, for example, [4],
+Chapter 3, Example 1.5, p. 179). It follows that α≤1/r. /square
+We will also require the following simple consequence of monotonicity of the
+capacity (cf. (2.2)):
+Lemma 3.4. LetGbe a uniformly transient graph. Then
+inf
+A⊆V,A/ne}ationslash=∅cap(A)≥α. (3.5)
+Proof.By uniform transience, cap( {x})≥αdeg(x)≥αfor allx∈V(see (1.3)).
+The statement (3.5) thus follows immediately from monotonicity of ca p:
+for finite, non-empty sets A⊆B⊆V,cap(A)≤cap(B). (3.6)
+Here is a direct proof of this well-known property: by summing over a ll possible
+timesnand locations yof the last visit of the random walk to the set Band using
+the simple Markov property at time n, we find that, for A,Bas above,
+cap(A) =/summationdisplay
+x∈∂iA∞/summationdisplay
+n=0/summationdisplay
+y∈∂iBPx[τ+
+A> n,X n=y,τ+
+B◦θn=∞]deg(x) (3.7)
+=/summationdisplay
+x∈∂iA∞/summationdisplay
+n=0/summationdisplay
+y∈∂iBPx[τ+
+A> n,X n=y]Py[τ+
+B=∞]deg(x).
+Reversibility of the simple random walk implies that
+deg(x)Px[τ+
+A> n,X n=y] = deg(y)Py[τA=n,Xn=x],
+hence by (3.7) that
+cap(A) =/summationdisplay
+y∈∂iBdeg(y)Py[τA<∞]Py[τ+
+B=∞]≤cap(B),
+proving (3.6). /square
+We now come to the main objective of this section.
+Proof of Theorem 1.1. We will prove that
+liminf
+n→∞inf
+o∈V/a\}bracketle{t∂iRn/a\}bracketri}hto
+/a\}bracketle{tRn/a\}bracketri}hto>0. (3.8)
+The desired statement then follows by Corollary 3.2 and the fact tha t
+inf
+o∈V/a\}bracketle{tRn/a\}bracketri}hto→ ∞,
+8proved in Proposition 2.3 (indeed, all connected components of a tr ansient graph
+are infinite). Denote the set of vertices with degree at least 3 by V≥3and the
+uniform bound on the degrees by d. Observe that any vertex xinV≥3belongs to
+∂iRnif it is visited by the random walk, but at least one of its neighbors is not .
+Hence, forany x∈V≥3, the probabilityof {x∈∂iRn}isbounded from belowbythe
+probability that the random walk first reaches B(x,1) at some vertex y∼x, then
+follows the path ( y,x,z), where zis the neighbor of xmaximizing Pz[τ+
+B(x,1)=∞],
+and does not return to B(x,1) until time n. By the strong Markovpropertyapplied
+at timeτB(x,1)+2 and at time τB(x,1), we hence obtain that for any o∈V,
+Po[x∈∂iRn]≥ (3.9)
+Po[τB(x,1)≤n−2,(X1,X2)◦θτB(x,1)= (x,z),τ+
+B(x,1)◦θτB(x,1)+2=∞]
+≥Po[τB(x,1)≤n−2]1
+d2α
+d2,for anyx∈V≥3,
+where we have applied Lemma 3.4 with A=B(x,1) in order to bound the escape
+probability Pz[τ+
+B(x,1)=∞] from below by α/d2. For any r≥1, the random walk
+reachesB(x,1) if it enters B(x,r) and then follows the shortest path to B(x,1),
+hence,
+Po[τB(x,1)≤n−2]≥Po[τB(x,r)≤n−1−r]d−(r−1).
+Using this estimate in (3.9), we obtain
+/a\}bracketle{t∂iRn/a\}bracketri}hto≥/summationdisplay
+x∈V≥3Po[x∈∂iRn] (3.10)
+≥/summationdisplay
+x∈V≥3Po[τB(x,r)≤n−1−r]α
+d3+r
+We now fix r= [1/α] + 1 and note that, by Lemma 3.3, ∪x∈V≥3B(x,r) =V. In
+particular, the random walk is in one of the balls B(x,r),x∈V≥3, at every step,
+and therefore visits at least [ |Rk|/supx|B(x,r)|] of them in its first k≥0 steps.
+Hence, we can deduce from (3.10) that
+/a\}bracketle{t∂iRn/a\}bracketri}hto≥/parenleftbiggEo[|Rn−1−r|]
+supx∈V|B(x,r)|−1/parenrightbiggα
+d3+r
+≥/parenleftbigg/a\}bracketle{tRn/a\}bracketri}hto−1−r
+d1+r−1/parenrightbiggα
+d3+r,for anyo∈V.
+Using again that inf o∈V/a\}bracketle{tRn/a\}bracketri}hto→ ∞by Proposition 2.3, we obtain (3.8). /square
+Remark3.5. Theconclusion(1.4)ofTheorem1.1doesnotholdforeverytransien t
+graph with bounded degrees. For a counterexample, consider a bin ary tree Tb=
+(Vb,Eb) and denote by Sl={x∈Vb:dTb(o,x) =l}the set of vertices at distance l
+from the root. We now split every edge connecting a vertex in Sl−1to a vertex in
+Slinto a path of length l+2, forl≥1, and thereby obtain the stretched tree G=
+(V,E), illustrated in Figure 1. Let now ( σn)n≥1be the successive displacements of
+the random walk in Vb, that is, σ1=τ+
+Vband fori≥2,σi= inf{n > σi−1:Xn∈
+τVb}. Then elementary computations using the simple Markov property a t time 1
+9PSfrag replacementsS1
+S2o
+Figure 1. The stretched binary tree constructed in Remark 3.5. The
+filled vertices are the ones present in the original binary tr ee
+Tb, the stretched tree Gis obtained by adding the unfilled
+ones.
+(cf. [4], Chapter 4, Example 7.1, p. 271-272) show that for any ver texxinSl,
+Px[dTb(Xσ1,o)> l] =2
+2+(l+3)/(l+2)≥6
+10>1
+2,forl≥1. (3.11)
+Hence, the process ( Xσk)k≥1is transient. In particular, the stretched graph G
+remains transient. Denote by R′
+n=Rn∩Vbthe intersection of Rnwith the vertices
+in the original tree. Then from (2.6), we obtain that
+Ho(Rn)≤Ho(Rn|R′
+n)+Ho(R′
+n). (3.12)
+For any fixed realization of R′
+nof diameter mwith respect to the original tree Tb,
+the setRnis determined up to the precise location of at most 2 |R′
+n|boundary
+vertices on paths of length at most m+3. Hence, for any such realization, there
+are at most ( m+ 3)2|R′
+n|≤(c|Rn|)2|R′
+n|choices for Rn. It therefore follows from
+Lemma 2.1 that
+Ho(Rn|R′
+n)≤cEo[|R′
+n|]logn.
+The amount of time the process Xσ.spends in any given set Skis by (3.11) and an
+elementary estimate on biased random walk stochastically dominated by a geomet-
+rically distributed random variable with success probability 1 /5 (cf. [4], Chapter 4,
+Example 7.1 (b), p. 271-272). It follows that
+Eo[|R′
+n|]≤1+5Eo/bracketleftbigg/summationdisplay
+1≤k≤n1{τSk≤n}/bracketrightbigg
+(3.13)
+≤cEo/bracketleftBig
+max
+0≤k≤nd(o,Xk)1/2/bracketrightBig
+≤cEo[/radicalbig
+|Rn|].
+Applying Jensen’s inequality on the right-hand side, we deduce that
+Ho(Rn|R′
+n)≤c/a\}bracketle{tRn/a\}bracketri}ht1/2
+ologn.
+10Regarding the other term on the right-hand side of (3.12), we have by the same
+argument as in the proof of Proposition 2.4,
+Ho(R′
+n)≤cEo[|R′
+n|]≤c/a\}bracketle{tRn/a\}bracketri}ht1/2
+o,by (3.13) .
+Upon inserting the last two estimates into (3.12), it follows from Prop osition 2.3
+that supo∈VHo(Rn)//a\}bracketle{tRn/a\}bracketri}htotends to 0 as n→ ∞, although Gis transient.
+4.The recurrent case
+In this section we prove Theorem 1.2, showing that supo∈VHo(Rn)//a\}bracketle{tRn/a\}bracketri}htotends
+to zero as n→ ∞on uniformly recurrent graphs satisfying the growth condition
+(1.6).
+Recall that any realization of Rnbelongs to Co,P-a.s. (cf. (2.3)). Our argument
+is based on Lemma 2.2 showing that any set A∈ Cocan be explored by a path not
+longer than 2 |A|. We will use this fact in the proof of Theorem 1.2 in order to show
+thatanyrealizationof Rncanbecoveredbyoneofatmostsupx∈V|∂eB(x,r)|d|Rn|/r
+different collections of balls of radius r. The second key observation will be that
+due to uniform recurrence, most of these balls of radius rvisited are typically
+completely covered by the random walk. This will show that the distrib ution ofRn
+is sufficiently concentrated to admit a bound on Ho(Rn) sublinear in /a\}bracketle{tRn/a\}bracketri}hto.
+Proof of Theorem 1.2. We denotethe uniformbound ondegreesby d. Notethat we
+canassumewithoutlossofgeneralitythat Gisconnected, forwemayotherwiseonly
+considerthe componentof Gcontainingthe startingvertex oofthe randomwalk. If
+Gis finite, then (1.7) immediately follows from the fact that lim nPo[Rn=V] = 1.
+Hence, we can from now on assume that Gis connected and infinite. In particular,
+Proposition 2.3 is available for application.
+Consider any o∈Vandr≥1. For any A∈ Co(cf. (2.3)), we define the finite
+sequence of vertices
+Fr(A) = (x1,...,x k) (4.1)
+such that the balls with radius rcentered at x1,...,x kcoverA, trying to keep k
+as small as possible. We define Fk(A) inductively as follows: by Lemma 2.2, there
+is a nearest-neighbor path pA= (p(0),...,p(l)) inG(A) starting at o=pA(0) and
+visiting all vertices in Ainl≤2|A|steps. Set x1=o,t1= 0, and for i≥2 such
+thatti−1<∞, definetias
+ti=/braceleftbigginf{t > ti−1:pA(t)/∈B(xi−1,r)},if∪1≤j≤i−1B(xj,r)/notsuperseteqlA,
+∞, otherwise,
+and, provided ti<∞, definexi=pA(ti). Since pAis of length at most 2 |A|and
+covers all of A, this yields a finite sequence as in (4.1), where kis the largest index
+such that tk<∞and satisfies
+k≤1+[2|A|/r]. (4.2)
+Finally, the construction implies that
+∪k
+i=1B(xi,r)⊇A. (4.3)
+We now partition the collection Coaccording to the cardinality of Fr(A) and the
+number of vertices xinFr(A) with the property that B(x,r) is not completely
+11filled byA: we have Co=∪∞
+k=1∪k
+l=0C(k,l), where the disjoint collections C(k,l) are
+defined as
+C(k,l) ={A∈ Co:|Fr(A)|=k,|{x∈Fr(A) :A/notsuperseteqlB(x,r)}|=l}. (4.4)
+We introduce the random variables KandL, defined as
+K=|Fr(Rn)|, L=|{x∈Fr(Rn) :Rn/notsuperseteqlB(x,r)}|. (4.5)
+Observe that since |Rn| ≤n+1, (4.2) implies that
+0≤L≤K≤[2(n+1)/r]+1, Po-a.s.
+Using the elementary estimate (2.6) together with (2.5) and Lemma 2 .1, we obtain
+the following bound on the entropy of Rn:
+Ho(Rn)≤Ho/parenleftbig
+(K,L)/parenrightbig
++Ho/parenleftbig
+Rn|(K,L)/parenrightbig
+(4.6)
+≤cdlogn+Eo/bracketleftbig
+log|C(K,L)|/bracketrightbig
+.
+In order to bound the cardinality of C(k,l) fork≥1,l≥0, we denote the maximal
+size of the ball and of the sphere of radius r≥1 by
+br= sup
+x∈V|B(x,r)| ≤d1+randsr= sup
+x∈V|∂eB(x,r)|. (4.7)
+We now bound the number of choices we have when selecting a set Afrom the
+collection C(k,l). For a set Ato belong to C(k,l),Fr(A) must consist of kvertices.
+The initial vertex x1must beoand each successive vertex must be at distance r+1
+from the previous one. Hence, Fr(A) can take at most sk
+rdifferent values. For every
+choice of Fr(A) = (x1,...,x k), there are at most/parenleftbigk
+l/parenrightbig
+≤2kdifferent choices of size- l
+subsets of vertices xiwith the property that B(xi,r) is not completely filled by A.
+After selecting Fr(A) and such a subset, Ais by (4.3) determined up to the at most
+(2br)lpossible configurations in the balls with centers xithat are not subsets of A.
+Hence, we have
+log|C(k,l)| ≤k(logsr+log2)+ brllog2. (4.8)
+Inserting this estimate into the above bound (4.6) on Ho(Rn), we obtain
+Ho(Rn)≤Eo[K](logsr+log2)+ Eo[L]brlog2+cdlogn. (4.9)
+By (4.2), Fr(Rn) is of cardinality at most (2 |Rn|/r)+1, hence
+Eo[K]≤2
+r/a\}bracketle{tRn/a\}bracketri}hto+1. (4.10)
+We now write P(r,n) = supx∈VPx[R[n1/4]/notsuperseteqlB(x,r)] and bound the other expec-
+tation on the right-hand side of (4.9). Since Fr(Rn)⊆Rn,
+Eo[L]≤/summationdisplay
+x∈VPo[τx≤n,Rn/notsuperseteqlB(x,r)] (4.11)
+=/summationdisplay
+x∈Vn/summationdisplay
+m=0Po[τx=m]Px[Rn−m/notsuperseteqlB(x,r)] (Markov prop.)
+≤/summationdisplay
+x∈V/parenleftBiggn−[n1/4]/summationdisplay
+m=0Po[τx=m]P(r,n)+n/summationdisplay
+m=n−[n1/4]+1Po[τx=m]/parenrightBigg
+≤ /a\}bracketle{tRn/a\}bracketri}htoP(r,n)+n1/4.
+12PSfrag replacementsa1
+a2o1 o2 o3
+Figure 2. The graph constructed in Remark 4.1.
+In order to bound P(r,n), fix any x∈V. Denote the successive return times to x
+by (Tk)k≥1, that is, T1=τ+
+x, and for k≥2,Tk= inf{n > T k−1:Xn=x}.Since
+anyy∈B(x,r) is within distance rofx, we have Px[τy≤τ+
+x]≥d−rfor any such
+y. With the strong Markov property applied at the times Tk, we infer that for any
+m≥1,
+P(r,n)≤sup
+x∈V/summationdisplay
+y∈B(x,r)Px[τy> n1/4] (4.12)
+≤sup
+x∈V/summationdisplay
+y∈B(x,r)/parenleftBig
+Px[τy> Tm]+Px[Tm> n1/4]/parenrightBig
+≤sup
+x∈V|B(x,r)|/parenleftBig
+(1−d−r)m+mPx[τ+
+x≥n1/4/m]/parenrightBig
+.
+Inserting the above estimates into (4.9), we deduce that
+Ho(Rn)
+/a\}bracketle{tRn/a\}bracketri}hto≤c2logsr
+r+cb2
+r(1−d−r)m+cb2
+rmsup
+x∈VPx/bracketleftBig
+τ+
+x>n1/4
+m/bracketrightBig
++cdbrn1/4
+/a\}bracketle{tRn/a\}bracketri}hto,
+for anyr,m,n≥1 ando∈V. By assumption (1.5) and Proposition 2.3, applied
+to the last two terms, the supremum over o∈Vof right-hand side converges as n
+tends to infinity to ( c(logsr)/r)+cb2
+r(1−cd,r)m. Letting mtend to infinity, then
+using (1.6), we obtain (1.7). /square
+Remark 4.1. The conclusion (1.7) of Theorem 1.2 does not hold for every re-
+current graph with bounded degrees, not even without the supre mum over o∈V.
+As a counterexample, consider an infinite sequence of binary trees of finite depths
+a1< a2<···, with roots o=o1,o2,.... Regardless of the choice of a1,a2,..., after
+addition of edges
+{o1,o2},{o2,o3},...
+one obtains a connected and recurrent graph, illustrated in Figure 2. The sequence
+of depths can be chosen recursively such that
+Po[τon≥√an−1]≤1/2 (4.13)
+13(observe that the distribution of τondepends on a1,...,a n−1, but not on an).
+Denote the set of vertices in the n-th tree at distance lfromonbySl,l≥1.
+Observe that by an elementary estimate on one-dimensional biased random walk,
+whenever the random walk reaches a previously unvisited level Slat some vertex
+x∈Sl, the probability that the random walk never returns to Sluntil time an
+and thereby leaves xin∂iRanis at least 1 /3 (see, for example, [4], Chapter 4,
+Example 7.1, p. 271-272). Hence,
+Eo[|∂iRan|]≥1
+3an/summationdisplay
+l=0Po[τSl≤an]
+≥can/summationdisplay
+l=0Po[τS1≤√an]PS1[τSl≤an−√an] (Markov prop.)
+≥can/summationdisplay
+l=0PS1[τSl≤an−√an] (by (4.13))
+≥cES1[d(on,X[an−√an]∧τon)].
+Since/parenleftbig
+d(on,Xk∧τon)−(k∧τon)/3/parenrightbig
+k≥0is a martingale (see again [4], p. 272), the
+right-hand side is bounded from below by
+cES1[(an−√an)∧τon]≥c(an−√an)PS1[τon> an]≥c′an≥c′/a\}bracketle{tRan/a\}bracketri}ht,
+using again an elementary estimate on one-dimensional biased rando m walk for the
+second inequality. Hence, Corollary 3.2 shows that Ho(Rn)//a\}bracketle{tRn/a\}bracketri}htodoes not tend to
+0 for the recurrent graph Gdefined above.
+5.Vertex-transitive graphs
+This final section contains the proof of Theorem 1.3 on the dichotom y for vertex-
+transitive graphs. Note that, due to vertex-transitivity, Ho(Rn), and/a\}bracketle{tRn/a\}bracketri}htodo not
+depend on o, soocan be omitted from the notation. We first deduce the estimate
+in the recurrent case.
+Corollary 5.1. LetGbe any vertex-transitive and recurrent graph. Then
+H(Rn)
+/a\}bracketle{tRn/a\}bracketri}ht−→0,asn→ ∞. (5.1)
+Proof.By Theorem 1.2, we only need to check conditions (1.5) and (1.6). Due to
+vertex-transitivity, the suprema become superfluous in both con ditions. The asser-
+tion (1.5) is thus a consequence of recurrence and monotone conv ergence. Simple
+random walk on a vertex-transitive graph is quasi-homogeneous, s ee [11], Theo-
+rem 4.18, p. 47. As a consequence, if liminf r|B(o,r)|/r3>0, thenGsatisfies
+a three-dimensional isoperimetric inequality, which implies in particular that the
+random walk is transient (we refer to [11], Corollary 4.16, p. 47, and T heorem 5.2,
+p. 49, for the proofs of these claims). Hence, we must have liminf r|B(o,r)|/r3= 0,
+from which (1.6) immediately follows. /square
+Theorem 1.3 follows:
+Proof of Theorem 1.3. Sincetheinfimumincondition(1.3)issuperfluousforvertex-
+transitivegraphs,everyvertex-transitiveandtransientgraph isuniformlytransient.
+14Theorem 1.1 hence shows that
+liminf
+n→∞H(Rn)//a\}bracketle{tRn/a\}bracketri}ht>0
+in the transient case. Corollary 5.1 shows that H(Rn)//a\}bracketle{tRn/a\}bracketri}httends to 0 for vertex-
+transitive recurrent graphs. Since every vertex-transitive gra ph is either transient
+or recurrent, this proves Theorem 1.3. /square
+In order to answer the question from [1] mentioned after Theorem 1.3, it remains
+to prove that /a\}bracketle{tRn/a\}bracketri}htgrows linearly in nfor vertex-transitive transient graphs. This
+is surely well-known, but we could not find a proof in the literature, so we give a
+proof here. We denote the Green function of the simple random walk (evaluated
+on the diagonal) by
+g(x) =Ex/bracketleftBigg∞/summationdisplay
+k=01{Xk=x}/bracketrightBigg
+,forx∈V,
+and the Green function of the random walk killed after nsteps by
+gn(x) =Ex/bracketleftBiggn/summationdisplay
+k=01{Xk=x}/bracketrightBigg
+,forx∈V.
+Proposition 5.2. LetGbe any transient graph and let
+e(x) =Px[τ+
+x=∞]∈(0,1)
+be the escape probability from vertex x∈V. Then
+inf
+o∈Ve(o)≤liminf
+n→∞inf
+o∈V/a\}bracketle{tRn/a\}bracketri}hto
+n. (5.2)
+Moreover, if Gis vertex-transitive and transient, then
+/a\}bracketle{tRn/a\}bracketri}ht
+n−→e(o)>0,asn→ ∞. (5.3)
+Proof.LetGbe any transient graph and o∈V. By the Markov property applied
+at time 1, g(x) = 1+(1 −e(x))g(x),hence
+g(x) = 1/e(x),for anyx∈V. (5.4)
+Forn≥1 andx∈V, we denote by Nn
+xthe total number of visits to xin the first
+nsteps,
+Nn
+x=/summationdisplay
+0≤k≤n1{Xk=x}.
+Summation over all vertices yields the total number of steps:
+n+1 =/summationdisplay
+x∈VNn
+x. (5.5)
+15Taking expectations in this equation and applying the strong Markov property at
+timeτx, we obtain that
+n+1 =/summationdisplay
+x∈VEo[Nn
+x] (5.6)
+≤/summationdisplay
+x∈VPo[τx≤n]g(x)
+≤ /a\}bracketle{tRn/a\}bracketri}htosup
+x∈Vg(x).
+The estimate (5.2) is trivial if inf o∈Ve(o) = 0 and follows from (5.4) and (5.6)
+otherwise.
+Let now Gbe a vertex-transitive transient graph. In particular, (5.2) holds
+without the infimum on either side. Replacing nby [λn] withλ >1 in (5.5), we
+have for n≥cλ,
+[λn]+1≥/summationdisplay
+x∈VEo[N[λn]
+x1{τx≤n}]
+≥/summationdisplay
+x∈VPo[τx≤n]g[λn]−n(o)
+=/a\}bracketle{tRn/a\}bracketri}htog[λn]−n(o).
+Lettingntend to infinity and using (5.4), it follows that for any λ >1,
+limsup
+n→∞/a\}bracketle{tRn/a\}bracketri}hto
+n≤λe(o).
+We now let λtend to 1 and together with (5.2) obtain (5.3). /square
+Remark 5.3. Theorem 1.1 and (5.2) prove that Ho(Rn) grows linearly in nfor all
+uniformly transient graphs with bounded degrees, while Theorem 1.2 in particular
+proves that Ho(Rn) grows sublinearly in non uniformly recurrent graphs satisfying
+the growth condition (1.6).
+Remark 5.4. The example presented in Remark 4.1 does not satisfy the volume
+growthassumption (1.6) and hence does not show that (1.6) is actu ally necessaryin
+Theorem 1.2. We do not know of an example showing necessity of (1.6) , or indeed
+if there even exists a graph satisfying (1.5) but not (1.6).
+Remark5.5. Giventheresultsofthepresentwork,itisnaturaltowonderwhet her
+onecanobtainmorepreciseestimateson Ho(Rn)onvertex-transitivegraphsoreven
+proveananalogueofShannon’stheoremonthealmost-surebehav ioroflog( po(Rn)).
+References
+[1] I. Benjamini, G. Kozma, A. Yadin, A. Yehudayoff. Entropy o f random walk range.
+Ann. Inst. H. Poincar´ e Probab. Statist. , to appear.
+[2] T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein. Introduction to algorithms . (second
+edition) MIT Press, Cambridge, MA, 2001.
+[3] T.M. Cover, J.A. Thomas. Elements of information theory. John Wiley & Sons, Inc., New
+York, 1991.
+[4] R. Durrett. Probability: Theory and Examples. (third edition) Brooks/Cole, Belmont, 2005.
+[5] A. Erschler. On drift and entropy growth for random walks on groups. Ann. Probab. 31/3,
+pp. 1193-1204, 2003.
+[6] V.A. Ka˘ ımanovich, A.M. Vershik. Random walks on discre te groups: boundary and entropy.
+Ann. Probab. , 11/3, pp. 457-490, 1983.
+16[7] A. Naor, Y. Peres. Embeddings of discrete groups and the s peed of random walks. Interna-
+tional Mathematics Research Notices 2008, Art. ID rnn 076.
+[8] P. R´ ev´ esz. Random walk in random and non-random environments . Second edition. World
+Scientific Publishing Co. Pte. Ltd., Singapore, 2005.
+[9] C.E.Shannon. A mathematical theory of communication. The Bell System Technical Journal ,
+Vol. 27, pp. 379-423, 623-656, 1948.
+[10] N.T. Varopoulos. Long range estimates for Markov chain s.Bull. Sci. Math. (2) 109, pp. 225-
+252, 1985.
+[11] W. Woess. Random walks on infinite graphs and groups. Cambridge University Press, Cam-
+bridge, 2000.
+17
\ No newline at end of file
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+++ b/1001.0356.txt
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+arXiv:1001.0356v1  [astro-ph.HE]  3 Jan 2010Suzaku Monitoring of the Iron K Emission Line in the Type 1
+AGN NGC 5548
+Yuan Liu1,2, Martin Elvis1, Ian M. McHardy3, Dirk Grupe4, Belinda J. Wilkes1, James
+Reeves5, Nancy Brickhouse1, Yair Krongold6, Smita Mathur7, Takeo Minezaki8, Fabrizio
+Nicastro1, Yuzuru Yoshii8, Shuang Nan Zhang2,9,10
+ABSTRACT
+We present 7 sequential weekly observations of NGC 5548 conduct ed in 2007
+with the SuzakuX-ray Imaging Spectrometer (XIS) in the 0.2-12 keV band and
+Hard X-ray Detector (HXD) in 10-600 keV band. The iron K αline is well
+detectedinallsevenobservationsandK βlineisalsodetectedinfourobservations.
+In this paper, we investigate the origin of the Fe K lines using both the width of
+the line and the reverberation mapping method.
+With the co-added XIS and HXD spectra, we identify Fe K αand Kβline at
+6.396+0.009
+−0.007keV and 7.08+0.05
+−0.05keV, respectively. The width of line obtained from
+the co-added spectra is 38+16
+−18eV (FWHM = 4200+1800
+−2000km/s) which corresponds
+1Harvard-Smithsonian Center for Astrophysics, 60 Garden Stree t, Cambridge, MA 02138, USA;
+yliu@cfa.harvard.edu, elvis@cfa.harvard.edu
+2Physics Department and Center for Astrophysics, Tsinghua Unive rsity, Beijing 100084, China; yuan-
+liu@mails.tsinghua.edu.cn
+3School of Physics and Astronomy, University of Southampton, So uthampton SO17 1BJ, UK
+4Department of Astronomy and Astrophysics, Pennsylvania State University, 525 Davey Lab, University
+Park, PA 16 802, USA
+5AstrophysicsGroup, SchoolofPhysicaland GeographicalScienc es, KeeleUniversity, Keele, Staffordshire
+ST5 5BG, UK
+6Instituto de Astronomia, Universidad Nacional Autonoma de Mexico , Apartado Postal 70-264, 04510
+Mexico DF, Mexico
+7Department of Astronomy, The Ohio State University, 140 West 18 th Avenue, Columbus, OH 43 210,
+USA
+8Institute of Astronomy, School of Science, University of Tokyo, Mitaka, Tokyo 181-0015, Japan
+9KeyLaboratoryofParticleAstrophysics,Institute ofHighEnerg yPhysics,ChineseAcademyofSciences,
+P.O.Box 918-3, Beijing 100049, China
+10Physics Department, University of Alabama in Huntsville, Huntsville, A L 35899, USA– 2 –
+to a radius of 20+50
+−10light days, for the virial production of 1 .220×107M⊙in
+NCG 5548.
+To quantitatively investigate the origin of the narrow Fe line by the re ver-
+beration mapping method, we compare the observed light curves of Fe Kαline
+with the predicted ones, which are obtained by convolving the contin uum light
+curve with the transfer functions in a thin shell and an inclined disk. T he best-
+fit result is given by the disk case with i= 30◦which is better than a fit to a
+constant flux of the Fe K line at the 92.7% level (F-test). However, the results
+with other geometries are also acceptable (P >50%). We find that the emitting
+radius obtained from the light curve is 25-37 light days, which is consis tent with
+the radius derived from the Fe K line width. Combining the results of th e line
+width and variation, the most likely site for the origin of the narrow iro n lines
+is 20-40 light days away from the central engine, though other pos sibilities are
+not completely ruled out. This radius is larger than the H βemitting parts of
+the broad line region at 6-10 light days (obtained by the simultaneous optical
+observation), and smaller than the inner radius of the hot dust in NG C 5548 (at
+about 50 light days).
+Subject headings: galaxies : active - galaxies : individual (NGC 5548) - galaxies
+: Seyfert - quasars: emission lines - X-rays: galaxies
+1. Introduction
+The neutral, photoionized iron K emission line commonly found in the X-r ay spectra of
+active galactic nuclei (AGNs) can be decomposed into a narrow core around 6.4 keV and a
+broad redshifted component, though sometimes only one of them is present (Fabian et al.
+2000; Yaqoob et al. 2001; Nandra et al. 1997, 2007). The broad, r elativistic component
+could be modelled as being emitted from the vicinity of the central blac k hole and used
+to determine the parameters of the black hole (e.g. Brenneman & Re ynolds 2006). An
+alternative explanation of this feature is the complex absorption or igin (Miller, Turner &
+Reeves 2008). However, the origin of the narrow component is still unclear, but could be
+criticaltounderstandingthecentralstructureofAGNs. Thena rrowwidth(FWHM ∼several
+thousand km/s or less, e.g. Yaqoob & Padmanabhan 2004) suggest s an origin well outside
+the continuum producing region. One of the proposed sites is the pu tative ‘obscuring torus’
+(e.g. Nandra 2006) at ∼0.1-1 pc. The evaporation radius of the dust can be estimated by
+r= 1.3L1/2uv,46T−2.8
+1500pc (Barvainis 1987), where Luv,46is the ultraviolet luminosity in units of
+1046ergs/s, and T1500isthe evaporationtemperature inunits of 1500K. ForNGC5548, sin ce– 3 –
+Luv,46∼0.01 and assuming T1500= 1, the evaporation radius is about 0.1 pc. However, the
+broad emission lines have similar line widths, so the more compact broad line region (BLR)
+is another possible location which cannot be ruled out simply. A BLR orig in is supported by
+the quasi-simultaneous optical spectroscopic observation with Chandra observation of NGC
+7213 (Bianchi et al. 2008), which shows consistent Fe K and H βline widths, and by the
+rapidNHchanges seen in several AGNs (Elvis et al. 2004; Puccetti et al. 200 7; Risaliti et
+al. 2002), which require a BLR-like radius for the NH/greaterorsimilar1023cm−2absorbers. Seen from
+another angle, these absorbers must re-emit in Fe K.
+The lag between the variation of the flux of the continuum and the line can be used
+to measure the location of an emission line region (Blandford & McKee 1 982) and this
+‘reverberation mapping’ methodologyhas been applied tothe optica l andUVbroademission
+lines(BELs)withgreatsuccess (Petersonetal. 2004). However, untilnowthe10%orgreater
+error on the flux of the 6.4 keV Fe K line (compared with the usual 1%- 5% error on the
+flux of BELs), and the low sampling frequency of the X-ray observa tions, have made it
+hard to apply this method to determine lags for Fe K lines, especially on relatively short
+timescale expected ( ∼10 days for a BLR origin). For example, Chiang et al. (2000) found
+the flux of the iron K line in NGC 5548 was consistent with being constan t using the four
+simultaneous observations of ASCAandRXTEwithin 25 days and another observation
+after about half a year. However, some response of the Fe K line to the continuum changes
+has been found. Markowitz et al. (2003) tried this method, analyzin g the long-term RXTE
+spectra for seven sources. Althoughthey foundnoevidence for correlatedvariability between
+the line and continuum, comparable systematic long-term ( ∼3-4 years) decreases in the line
+and continuum were present in NGC 5548.
+We report here on a series of seven sequential X-ray observation s of NGC 5548 by
+Suzaku, spaced roughly weekly. This observing campaign was designed, amo ng other goals,
+to constrain the flux of the iron line to about 10% in each observation . This allows us, for
+the first time, to apply the reverberation mapping technique to Fe K to try to distinguish
+different geometries of the Fe K emitting region. NGC 5548 is the sour ce best studied by the
+optical reverberation mapping technique (e.g. Peterson & Wandel 1999) and so has the best
+determined radial structure. Therefore, it is easy to determine t he relative location of the
+Fe K line emitting region. In previous observations, only the Fe K αline was detected. As
+we will present in this paper, the K βline is also well detected in the Suzakuobservations.
+Although very weak in some observations, K βis useful to constrain the ionization state of
+iron.
+The iron line in NGG 5548 has been observed by several X-ray satellite s:ASCAspectra
+suggested a relativistic broad iron K αline in NGC 5548 with σ= 340+190
+−120eV (68% error– 4 –
+for 4 interesting parameters, Nandra et al. 1997) and σ∼400 eV (Chiang et al. 2000).
+However, less than two years later, only the narrow K αline was detected in a Chandra
+observation with much higher energy resolution (38 eV vs 160 eV) bu t lower S/N (Yaqoob et
+al. 2001). An XMM-Newton EPIC CCD observation confirmed the Chandra result (Pounds
+et al. 2003).
+In§2, we describe the observations and the procedure of the data re duction. In this
+paper we intend to investigate the origin of the Fe K line using the reve rberation mapping
+method andthewidth of the line. Therefore, in §3, we first fit the spectra ofeach observation
+to determine the flux of the continuum and the iron line. We then fit th e co-added spectra
+in§3.2 to determine the mean parameters, especially the width of the iro n line. In §4.2 and
+§4.3, we calculate the transfer functions in different geometries and investigate the possible
+emitting region of iron line. In §5.1, we discuss the possible origin of the iron line. In §5.3,
+we briefly discuss the implications of the intensity and the equivalent w idth of iron line.
+In§6 we give our conclusions. Throughout this paper we adopt the reds hift of NGC 5548
+obtained from 21 cm H I measurements, i.e. z=0.017175 (de Vaucoule urs et al. 1991). The
+errors quoted in this paper correspond to 90% confidence level (∆ χ2= 2.706, Avni 1976) if
+not otherwise specified.
+2. Observations and data reduction
+2.1. Observations
+During 2007 June to 2007 August, NGC 5548 was observed by the CC D X-ray Imaging
+Spectrometers (XIS 0, 1, and 3) in 0.2-12 keV band (Koyama et al. 2 007) and by the Hard
+X-ray Detector (HXD) in 10-600 keV band (Takahashi et al. 2007) onSuzaku(Mitsuda et
+al. 2007) seven times for 28.9 ks-38.7 ks each. We denote these as o bservations 1-7. The
+details of the observations are summarized in Table 1.
+2.2. Data reduction
+Following the standard procedures outlined in the “Suzaku Data Red uction (ABC)
+Guide (version 2)1”, we used the updated Charge Transfer Inefficiency (CTI) calibra tion
+(Suzaku XIS CALDB 20081110) and screened the events using the xispi(Ftools 6.5) and
+1http://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/abc/– 5 –
+xselect scripts provided by Suzakuteam (we adopted the standard criterion in xis event.sel
+and xismkf.sel)2, respectively.
+X-ray spectra were extracted using xselect3from all the XISs with a circular ex-
+traction region of radius 260 arcsec centered on NGC 5548 ( α=14h 17m 59.5s, δ=+25d
+08m 12.4s, J2000). Background spectra were obtained from a larg er annulus around the
+source (but avoiding the calibration sources on the corners of the chips). Response matrices
+(rmf) and effective area ( arf) files were generated with the xisrmfgen4andxissimarfgen5
+(estepfile=dense andnumphoton=300000 ), respectively. We then combined the spectra,
+background, rmf, andarffiles from the two front-side illuminated CCDs, XIS 0 and 3, for
+each observation with addascaspec6(we denote the combined spectra by ‘XIS03’). The
+spectra of XIS 1 are considered separately, as this detector use s a back-side illuminated
+CCDs.
+NGC 5548 was detected by the silicon diode PIN instrument of the HXD , but below the
+sensitivity limit of the GSO crystal scintillator instrument. We downloa ded the tuned non-
+X-ray background (NXB, version 2.0) files from the SuzakuGuest Observer Facility (GOF)7
+and then merged the good-time interval (GTI) of the NXB with that of the screened event
+files to produce a common GTI using mgtime. Then the spectra of the source and the NXB
+were extracted by xselect using the common GTI. And the dead time of the source spectra
+was corrected by hxddtcor . Since the event rate in the PIN background event file is 10
+times higher than the real background to suppress the Poisson er rors, the exposure time of
+the spectra of NXB was increased by a factor of 10. The cosmic X-r ay background (CXB)
+was not taken into account in the NXB file, therefore we also added a CXB component in
+the spectral fitting using the model given by the ABC guide (Section 7.3.3). The response
+fileaehxdpinxinome3 20080129.rsp was used for observation 1-5, while the response file
+aehxdpinxinome4 20080129.rsp was used for observation 6 and 7 due to the changes in
+instrumental settings of Suzakuduring different observation epochs. The PIN spectra were
+multiplied by a constant cross-normalization factor of 1.16 to accou nt for the differences in
+2http://suzaku.gsfc.nasa.gov/docs/suzaku/analysis/xisrepro.x co,http://suzaku.gsfc.nasa.gov/docs/suzaku/analysis/xis event.sel,
+http://suzaku.gsfc.nasa.gov/docs/suzaku/analysis/xis mkf.sel
+3http://heasarc.nasa.gov/docs/software/lheasoft/ftools/xs elect/index.html
+4http://heasarc.nasa.gov/docs/suzaku/analysis/xisrmfgen.htm l
+5http://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/xissimarf gen/
+6http://heasarc.gsfc.nasa.gov/lheasoft/ftools/fhelp/addasca spec.txt
+7ftp://legacy.gsfc.nasa.gov/suzaku/data/background/pinnxb ver2.0tuned/– 6 –
+calibration between XIS and PIN8.
+3. Spectral fitting
+To perform the reverberation mapping calculation, we should first d etermine the flux of
+the continuum and the Fe K line. In order to avoid the influence of the complex absorption
+below 3 keV (Detmers et al. 2008; Steenbrugge et al. 2003), we only analyze the XIS spectra
+in 3-10 keV band. The warm absorber seen in the <3 keV spectra is presented by Krongold
+et al. (2009). In this paper, we will focus on the property and origin of the iron emission
+line. A global fit to the entire Suzakuspectral band for the entire campaign will be presented
+in a subsequent paper.
+For each observation, we fitted the spectra of XIS03 simultaneou sly with the spectra of
+XIS 1 in XSPEC (version 12.4). Although both of the energy resolutio n and the effective
+area of XIS 1 are lower than XIS 0, 3 in 4-10 keV band (the backgrou nd level of XIS 1 is
+also higher than XIS 0, 3 in this band), it is still useful to reduce the e rror of the intensity of
+the Fe K line. We fixed the Galactic absorption column at 1.63 ×1020cm−2(Murphy et al.
+1996), and fitted the spectra with a single power law. The continua w ere well described by
+a single power law except for the region of the iron K αand Kβlines around 6.4 keV and 7.0
+keV, respectively (see Figure 1). Therefore, we added two Gauss ian lines to describe these
+features. It is important to construct a self-consistent model t o describe all the components
+of the continuum and theoretically predict the strength of the Fe K line, as in Murphy &
+Yaqoob (2009). However, this is not the purpose of this paper. Ou r purpose is only to
+simply model the Fe K line using a Gaussian line and determine the flux and width of it
+and then to perform the reverberation calculation. A K αline is required by all the seven
+spectra ( >6σ), while K βline is only required by four observations ( >2σfor 1, 2, 4, and
+7). The 90% upper limit of the flux of the K βline is determined in observation 3, 5, and
+6. Since the K βline is weak, we fixed the width of K βto be the same as that of K α. Fe
+Kβline was also detected in other sources, e.g. NGC 2992 (Yaqoob et al. 2007) and Mrk
+3 (Awaki et al. 2008). The detailed result of the fitting is given in Table 3, though we are
+mainly concerned about the flux of the continuum and the intensity o f the Fe K line. Due to
+the weak reflection component in NCG 5548 (e.g. Pounds et al. 2003) , if we simultaneously
+fit the continua in the XIS and PIN spectra of each observation usin g thepexravmodel
+instead of the power law model, the intensity of the Fe K line will system atically decrease by
+10% for each observation (the strength of the reflection compon entR∼0.2−1.5). However,
+8http://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/watchou t.html– 7 –
+this change will not influence the reverberation mapping result in §4.3, since it degenerates
+with the normalization of the transfer function (see the detail of t he transfer function in
+§4.2). The PIN data is also not helpful to reduce the error of the flux of the Fe K line
+due to the additional uncertainty introduced by the reflection com ponent and large error in
+the PIN spectra. The detailed discussion about the variation of the X-ray continuum and
+simultaneous UV/optical data will be presented in another paper.
+3.1. Continuum light curve
+Having fitted the spectra, we calculated the observed flux of the c ontinuum in the 3-10
+keV band. The result light curve is shown in Figure 3a. To better samp le the continuum
+light curve, we also utilized one observation of the continuum from th e simultaneous Swift
+campaign. Since the observation times and the flux of other Swiftdata are quite similar to
+that of the Suzakudata, we will not include them in this paper. The details of the Swift
+campaign will be discussed in Grupe et al. (2009). 28 observations of the continuum with
+PCA on RXTEbefore the Suzakucampaign are also used, since we are looking for the lag
+between the variation of continuum and line. However, due to the sh ort exposure time, we
+cannot determine the flux of the line in any of these additional obser vations. The details of
+theRXTEandSwiftobservations are summarized in Table 2.
+3.2. Narrow iron lines
+TodeterminethemeanparametersoftheFeK αandKβlinesmoreaccurately, especially
+the width of the line, we added the spectra from the seven observa tions together using
+addascaspec (the XIS03, XIS1, and PIN spectra were added separately and th en fitted
+simultaneously). The net counts of the source in 3-10 keV band in th e total XIS03 and XIS1
+spectra are 160349 and 75332, respectively. The net counts of t he source in 12-35 keV band
+in the total PIN spectra is 22243. We found that the XIS1 spectra could not provide any
+useful constraint on the width of the Fe K line, since the σof the Gaussian line is pegged
+at 0 eV (the 90% upper limit is 30 eV). Therefore, we will only utilize the c o-added XIS 03
+and PIN spectra to determine the width of the Fe K line.
+If we only fitted the co-add XIS03 spectra using the model in §3, i.e. a power law and
+two Gaussian lines, the width is σ= 50+14
+−15eV. However, since the weak reflection component
+could influence the width of the Fe K line, we then simultaneously fitted the co-added XIS 03
+and PIN spectra using the pexravmodel and two Gaussian lines. The derived parameters– 8 –
+are given in Table 4, where the strength of the reflection componen t (R∼0.8) could explain
+the Fe K line. As shown in Figure 1, only a narrow iron line is clearly presen t in the spectra,
+with a width, σ=38+16
+−18eV. This value is consistent with previous results: i.e. σ=41+32
+−24eV
+obtained by HEG+MEG on Chandra (Yaqoob et al. 2001) and σ=40+40
+−40eV (MOS) and
+64+24
+−24eV (PN) obtained by XMM-Newton (Pounds et al. 2003).
+Since the peak energies of the line and the parameters of the contin ua are somewhat
+different for each observation, we simultaneously fitted the spect ra of all observations in
+XSPEC to test whether the line was broadened artificially in the co-ad ded process. We
+required the width of line to be the same for all observations and kep t other parameters free.
+The obtained width is only slightly smaller than that from the co-add sp ectra by about 2
+eV, which implies the co-added method has not significantly broadene d the width.
+The width obtained by the co-add spectra is inconsistent with zero a t 2.2σand corre-
+sponds to FWHM= 4200 km/s and a radius of 5.2 ×1016cm or 5.2 ×103rg(rg=GM/c2), for
+the 6.71×107M⊙black hole in NGC 5548 (Peterson et al. 2004). In §5.1, we will estimate
+the location of the emitting material of the line and discuss the origin o f the line.
+To test the presence of the broad, relativistic Fe K αline found by ASCA, we tried the
+diskline model in XSPEC to fit the K αline. We fixed the inner and outer disk radii at 6 rg
+and 1000 rg(rg=GM/c2), respectively. The index of the power law emissivity was frozen
+at -2.5 (the mean value obtained in Nandra et al. 1997). If we thaw th e index, it will be
+pegged at the positive upper limit in XSPEC, i.e., the flux of the line is domin ated by the
+outer disk and therefore it is not a disk line at all. It was found that th ediskline model
+cannot describe the K αline alone, since χ2was higher by 128 than for a narrow line with
+the same number of parameters. The same conclusion was also obta ined by Yaqoob et al.
+(2001).
+Next, we investigated the result if we fit the K αline with a disk line and a Gaussian
+line. Besides the constraint on the diskline model mentioned above, we also fixed the value
+of the inclination angle at 31 degrees, which is the best-constrained value from the ASCA
+data (Yaqoob et al. 2001). We found the central value of the inten sity of the disk line
+is pegged at 0 and the 90% upper limit is 4 ×10−6photons/cm2/s. Therefore, any broad
+component must be >5 times weaker than the narrow component and we will not include
+this component in the following discussion.
+We show the confidence regions for the peak energies of K αand Kβin Figure 2a. The
+expected values of different iron ionization states (Palmeri et al. 20 03; Mendoza et al. 2004;
+Yaqoobet al. 2007)arealso shown. Since thepeak energymay beinfl uenced bytheresiduals
+in the energy scale calibration, we extracted the spectra of the55Fe calibration sources on– 9 –
+the corners of the CCDs and added them together (see the inset in Figure 1b). Using a
+Gaussian line to fit the Mn K αline, we found the peak energies are 5.892+0.001
+−0.001keV and
+5.893+0.002
+−0.001keV for XIS03 and XIS1, respectively. The expected value of the M n Kαline is
+5.895 keV. This result is well within the accuracy of the absolute XIS e nergy scale given by
+theSuzakuTechnical Description9as 0.2%10. The fit widths of Mn K αline are 12+3
+−5eV
+and 16+5
+−8eV for XIS03 and XIS1, respectively. Therefore, the peak energ y obtained by the
+spectral fitting is reliable and we could conclude that the iron emission line in NGC 5548 is
+most likely to be dominated by relatively low ionization states, <Fe XIII (99% confidence).
+Actually, the observed Fe K line could be a blend of several ionization s tates.
+The confidence regions of the intensities of the Fe K αand Kβlines are shown in Figure
+2b. The expected ratio of I(K β) to I(Kα) varies from 0.12 to 0.20 for different ionization
+states (dashed lines in Figure 2b, Palmeri et al. 2003; Mendoza et al. 2004;). Our result is
+fully consistent with the expected value. However, due to the weak ness of K β, the error is
+still too large to constrain the ionization state precisely.
+4. Time variability analysis
+4.1. Fe K αline light curves
+The light curve of Fe K αline is shown in Figure 3b each being determined to ∼10%.
+As Kβis weak in some observations, we will only discuss the result for K αbelow. As shown
+in Figure 3a, the flux of the continuum changed strongly during the s even observations. The
+highest value (2.65 ×10−11ergs/cm2/s, for observation 5) is about 4 times that of the lowest
+(6.89×10−12ergs/cm2/s, for observation 1). However, perhaps due to the relatively lar ge
+error, the flux of the line is consistent with being constant, χ2
+min=3.05 (P=80%) and the
+value of the corresponding line constant flux is 2.24 ×10−5photons/cm2/s.
+A constant Fe K αflux is the simplest solution to the Fe K αlight curve, and it requires
+the emitting region should be far from the central engine to smooth out the variation in
+short time scale. According to the result in §4.3 and Figure 6b, it should be larger than 100
+light days. However, as shown in Figure 1 and 2 in Markowitz et al. (200 3), the presence
+of the variation of the Fe K flux in time scale shorter than 100 days ind icates the emitting
+9http://www.astro.isas.jaxa.jp/suzaku/doc/suzaku td/
+10We found that using the calibration taken for the same time interval was important. A preliminary
+analysis using the calibration available when the observations were ma de showed a 20 eV offset, which was
+puzzling. The amplitude of the Mn K αapparent energy variation was also about 20 eV– 10 –
+regionmust besmaller than100lightdays. The emitting regionrequire d bytheconstant flux
+fitting is also inconsistent with that required by the line width (see §5.1). Therefore, we have
+investigated whether there is a better and more self-consistent s olution than the constant
+flux fitting. To quantitatively access the emitting location of the Fe K αline, we should
+perform the reverberation calculation. The cross-correlation fu nction (CCF) is usually used
+in the optical reverberation mapping to determine the lag between t he continuum and the
+emission line. However, due to the very few data points of the light cu rve of the Fe K line
+and the large error of them, we did not find any significant peak in CCF . Therefore, we will
+approach this problem in a different way. To do so we first calculate th e transfer function
+for Fe Kαin§4.2.
+4.2. Transfer function
+As in the case of the optical broad emission lines, we calculate a trans fer function for
+Fe Kα, i.e. the response of the flux of the line to a δfunction change in the continuum. We
+consider two cases: (1) a spherical region and (2) an inclined disk.
+(1) We assume the emitting material is spherically distributed around the center with
+an inner ( rmin) and outer ( rmax) radius. The transfer function in this model is then simply a
+constant between 0 and2 rmin/c, and then decays to 0 at 2 rmax/c(see Figure4a andPeterson
+1993).
+(2) For a thin disk, the general form of the transfer function is tw o-peaked (Welsh &
+Horne 1991). Figure 4b shows the result for different values of the inclination angle i(the
+angle between the normal of the disk and the line of sight).
+The shape of the transfer function depends on the form of the “r esponsivity” ε(r) (see
+Figure 4a), which combines the effects of the distributions of the nu mber density of clouds
+and the emissivity per cloud. We assumed that the responsivity is a po wer lawε(r)∝r−α
+and the normalization is adjusted to fit the observed light curve. Th e power law form is
+simple and somewhat arbitrary. However, since we only consider the thin shell case (i.e.
+∆r≪r, or equivalent to the locally optimized clouds model), the result is not s ensitive to
+the detailed form of the responsivity nor the index of the power law ( see Figure 6a). α= 3
+is adopted in the calculation in §4.3.
+We could then obtain the predicted light curve of the line by convolving the transfer
+function with the light curve of the continuum flux. In the thin spher ical shell and disk
+cases discussed in §4.3, it can be proved that the the lag time, τ, obtained by the CCF just
+corresponds to the radius of the emitting region, i.e. τ=r/c(Koratkar & Gaskell 1991).– 11 –
+4.3. Comparison with the observed light curve
+Since the observation data of the continuum are still too few to pro duce a complete light
+curve, we performedlinear interpolationbetween datapoints to co nvolve the light curve with
+the transfer function.
+We used the light curve of the continuum in 3-10 keV band in order to h ave precision
+measurements. Similar bands (e.g. 2-10 keV) have been adopted in t he previous attempts to
+determine the lag between the variation of the continuum and the Fe K line (e.g. Markowitz
+et al. 2003, 2009), although only photons from above the ionization threshold can actually
+lead the emission of an Fe K photon11. As shown in Figure 5, the flux in 3-6 keV band is
+tightly and nearly proportionally correlated with the flux in 8-10 keV b and. Any possible
+effects due to rapidly changing NH(Risaliti et al. 2005) must therefore be small. As a
+result, the results of the following calculations will not be sensitive to the adopted energy
+band of the continuum. The change of the continuum slope is also rela ted to the total
+amount of the ionization flux. However, this is a minor factor compar ed with the change
+of the normalization of the continuum. Specially, this effect should be even small for NGC
+5548, since e.g. Sobolewska and Papadakis (2009) showed that the continuum slope is nearly
+independent on the flux. We find the same lack of dependence (Table 3).
+Wefixedthewidthoftheemittingregionat0.1lightdays, substantially smaller thanthe
+likely radius of the Fe K emitting region, and varied rmin. After convolving the light curve of
+the continuum with the transfer functions in different geometries a ndrmin, we compared the
+predicted light curve of the line with the observed one to find the minim um value of χ2(since
+the error on the flux is asymmetric, we conservatively adopt the lar ger one to calculate χ2).
+Figure 6b shows χ2vsrmin. All the curves show pronounced minima (somewhat surprisingly,
+given the weak structure in the Fe K αlight curve) and finally decrease towards the result of
+the constant fitting (the horizontal dashed in Figure 6b) with incre asingrmin, as the result of
+the significant smoothing effect with large rmin. The values of minimum χ2in the spherical
+thin shell case and disk case with i= 15◦andi= 30◦are well below that of the constant
+fitting, and we will only discuss these three cases below. With the sma llest value of χ2, the
+best-fitting light curve in the disk with i= 30◦is improved at 92.7% level compared with
+the constant fitting. We show the predicted light curves correspo nding to the minima of χ2
+in these three cases in Figure 7. Due to the few data points, we cann ot tightly constrain the
+inclination angle. Except for the the disk case with i= 15◦, the positions of the minimum
+χ2in the spherical case (27+22
+−7light days) and the disk case with i= 30◦(37+7
+−5light days)
+11A similar approximation is used in reverberation mapping of AGNs (e.g. P eterson et al. 2002; Bentz et
+al. 2007)– 12 –
+are similar, and they are smaller than the inner radius of the dust (47 -53 light days, see the
+discussion in §5.1). The value of the best-fitting rminin the disk case with i= 15◦is 61+6
+−7
+light days, which is beyond the inner radius of the dust of NGC 5548 (s uch optically thick
+region will significantly absorb the photons of Fe K lines) and only marg inally consistent
+with the small tail of the 90% confidence interval of the emitting reg ion inferred from the
+widths of FeK αline (see the error bar inFigure 9 and it should benoted that the erro r bar is
+quite asymmetric). In addition, i= 15◦is also well smaller than the best-fitting inclination
+angle of the disk line in the ASCA(i= 31.5◦±6.5◦, Yaqoob et al. 2001), though it is not
+very reliable, due to the absence of the broad component of the Fe K line in the following
+Chandra,XMM-Newton , and our Suzakuobservations. Therefore, the disk case with i= 15◦
+is quite unlikely, and we will only focus on the spherical case and the dis k case with i= 30◦
+when discussing the origin of the narrow Fe K αline in detail in §5.1.
+5. Discussion
+5.1. Location of the Fe K emitting region
+Assuming the geometry and dynamics of the emitting region of the Fe K line are the
+same as that of the H βline, and using the virial relation for the H βline,σ2r/G= 1.220×107
+M⊙(Peterson et al. 2004), the radius of the Fe K emitting region can be derived using the Fe
+Kαwidth,σ, obtained in §3.2. The derived radius using the width obtained by the co-added
+XIS03 and PIN spectra is 20+50
+−10light days12.
+In Figure 8, we plot the line width size against rminfrom the reverberation analysis
+(Figure 6b) for both the spherical thin shell case and the disk case withi= 30◦with 90%
+confidence intervals for both quantities. The disk case with i= 30◦and the spherical case
+are consistent with the width of the co-added Fe K αline13. Combining the above results,
+we conclude that the origin of the narrow iron line in NGC 5548 is likely to b e∼20−40
+light days away from the continuum source, for the geometries con sidered. However, the
+12Since the virial production is an observed quantity and the radius de rived from width will be compared
+with that also obtained from the reverberation mapping method, no additional geometry factor is required.
+13The calibration residual in the width of Mn K αline derived in §3.2 (12+3
+−5eV and 16+5
+−8eV for XIS03
+and XIS1, respectively) is partly due to the systematic error on th e calibration of the non-Gaussian response
+function of XIS (Koyama et al. 2007). The energy resolution at the center of the CCD chip is also slightly
+better than that at the corner, but this difference is smaller than t he systematic error on the calibration.
+Therefore, the true width of the Fe K line could be smaller than the ob served one by a few eVs due to the
+above factors. However, these effects could not be accurately c orrected simply.– 13 –
+other possible origins are not completely ruled out due to the incomple teness of the light
+curves, our model-dependent method, and the sizeable error on t he width of the Fe K line.
+Since NCG 5548 is one of the best-studied AGNs with reverberation m apping, the
+locations of the BELs are well-determined, especially for the H βline. The lag between the
+flux of H βline and the continuum varies from 6.5 light days to 26.5 light day depend ing
+on the luminosity of the continuum (Bentz et al. 2007; Cackett & Hor ne 2006). Since
+the correlation between the lag time and the continuum flux is more sig nificant than the
+correlation between the lag time and the width of H βline (Bentz et al. 2007), and the broad
+component of the H βline is weak during the Suzakucampaign, we will utilize the continuum
+flux to predict the radius of the H βline region.
+From the simultaneous optical spectra of NGC 5548 from FLWO FAST spectrograph
+(June 19-23, 2007), we measured the monochromatic flux at 5100 ˚A and used this flux to
+estimate the radius of the H βemitting region at the time of the Suzakucampaign. After
+calibration with a standard star, the optical spectra were put ona n absolute calibration scale
+assuming a constant flux of F([OIII] 5007)standard= 5.58×10−13ergs/s/cm2(Peterson et
+al. 1991) and found the monochromatic flux at 5100 ˚A,F(5100˚A)observed= 5.13×10−15
+ergs/s/cm2/˚A. For the aperture size of 5 arcsec ×7.5 arcsec, the contribution of the host
+galaxy at 5100 ˚A is 4.47 ×10−15ergs/s/cm2/˚A (Bentz et al. 2006). To account for the
+effect of the aperture and the difference between telescopes, th e flux measured by FLWO
+should be converted by the coefficients given in Peterson et al. (200 2), i.e.F(5100˚A)true=
+ϕF(5100˚A)observed−G. We averaged the results using the different coefficients (i.e. ϕand
+G) determined in years 8 and 9-13 of the campaign (see details in Peter son et al. 2002), and
+then excluded the flux of the host galaxy (4.47 ×10−15ergs/s/cm2/˚A, Bentz et al. [2006]) .
+The final 5100 ˚A flux of the AGN in NGC 5548 is (1 .7±0.7)×10−15ergs/s/cm2/˚A during
+theSuzakucampaign. With the relation between the emitting region of H βline andF(5100
+˚A) (Figure 5 in Bentz et al. [2007]), we found r(H β)= 8±2 light days during the Suzaku
+observations.
+From the result of dust reverberation of NGC 5548, we also know th e inner radius of
+the hot dust in this source is 47-53 light days (from the lag of K-band relative to V-band,
+Suganuma et al. 2006).
+We plot the radial locations of Fe K α, Hβand the inner dust in Figure 9 (note that this
+is a linear plot). It is likely that the location of the Fe K emitting region is c loser in than the
+inner radius of the dust, but slightly farther out than H β, i.e. the origin of the narrow iron
+line lies in the outer part of the BLR [(0 .5−1.0)×104rg]. This result is consistent with the
+regionemitting the optical FeIIemission lines, which is less thansever al light weeks fromthe
+continuum emitter (Vestergaard & Peterson 2005). This region is a lso consistent with the– 14 –
+intermediate line region proposed by Zhu et al. (2009) and the radius of the low ionization,
+low velocity component of the warm absorber in NGC 5548 (r <3 pc, Krongold et al. 2009).
+Despite the uncertainty of the column density of the Fe K emitting re gion, it is surely much
+higher (see §5.3) than that of the warm absorber found in NGC 5548 (1021∼1022cm−2,
+Krongold et al. 2009). Therefore, It clearly does not lie along our line of sight.
+5.2. Physical condition
+We can use the location derived above to estimate the density of the emitting region
+from the ionization parameter and the distance. Kallman et al. (2004 ) investigated the
+dependence of the line profile of K αand Kβon the ionization parameter. The peak energies
+of Kαand Kβin NGC 5548 (see Figure 2a) prefer a relatively low ionization paramete r,
+i.e. logξ=−1±1, where ξ=L/nR2(erg s−1cm),Lis the Fe K ionizing luminosity of
+the continuum source (erg s−1),nis the density of the emitting region (cm−3), andRis
+the distance to the continuum source (cm). With log L= 43.7±0.3, logξ=−1±1, and
+R= 30±10 light days, we obtain log n= 10.9±1.1, which is comparable with the density
+of the gas in the BLR of NGC 5548 (Ferland et al. 1992; Goad & Koratk ar 1998; Kaspi &
+Netzer 1999).
+5.3. Theoretical intensity of the Fe line
+The intensity and the equivalent width of the iron line can be estimated theoretically
+(Krolik& Kallman1987; Yaqoobet al. 2001). Wefoundthecolumn dens ityNH>1023cm−2
+is required to produce the observed intensity and the equivalent wid th of the Fe K αline in
+ourSuzakuobservations. However, as pointed out by Miller et al. (2009) and Ya qoob et al.
+(2009), due to the self-absorption effect and the Compton scatt ering, for NH>1023cm−2,
+the relation between the intensity of the Fe K line and NHor the abundance of iron is quite
+non-linear, and the intensity of the Fe K line also significantly depends on the geometry of
+the emitting regionand theobserving angle. Since thedetailed invest igation of thegeometry,
+NH, and the abundance of the emitting region of Fe K lines is much beyond the scope of
+this paper, we will not further discuss the constraints obtained fr om the intensity and the
+equivalent width of the Fe K lines.– 15 –
+6. Conclusions
+We analyzed the iron K αand Kβlines in spectra of NGC 5548 obtained by SuzakuXIS
+and summarize our results as follows.
+1. The iron K αline was well detected ( >6σ) in all seven observations and the K βline
+was also detected ( >2σ) in four observations (1, 2, 4, and 7).
+2. Only a narrow iron line was found in the spectra. The line width obtain ed by the
+addedspectrais38+16
+−18eV,which isconsistent withtheresultsof Chandra andXMM-Newton .
+Assuming the same virial relation as that of the H βline, this width corresponds to a radius
+of 20+50
+−10light days. Any relativistically broadened disk line must be a factor of 5 weaker
+than the narrow component in flux at 90% confidence level.
+3. We compared the observed peak energies and intensity ratios of Kαand Kβlines
+with the expected value and found they are consistent with the low io nization states of iron,
+i.e. lower than Fe XIII, at the 99% confidence level.
+4. The Fe K αline is consistent with being constant over the 50 days of the Suzaku
+campaign, although the 3-10 keV continuum varies by a factor of 4. It is shown that a
+location at >100 light days is consistent with the data (see the discussion in §4.3 and Figure
+6b), but this is not a unique result.
+5. To further access the location of the iron lines using the light curv e, we calculated the
+transfer functions in spherical and disk geometries, and compare d the predicted light curves
+with the observed one. The value of χ2is smallest in the disk case with i= 30◦, which
+is better than the constant fitting at the 92.7% level. The spherical thin shell case is also
+acceptable (P=81%). The inferred emitting radii are 27+22
+−7light days in the spherical case
+and 37+7
+−5light days in the disk case with i= 30◦, which are consistent with that obtained
+from the width of iron lines.
+6. Combining the Suzakuconstraints, the most likely origin of the narrow iron lines is
+about 20-40 light days away from the central engine, i.e. the outer part of BLR (5.2 ×103-
+1.0×104rg). However, we could not completely rule out other possible origins.
+The approaches used in this paper offer a valuable tool for determin ing the size and
+structure of the inner regions of AGNs, although we stress again t his method is model
+dependent. The constraint on the emitting region of the narrow Fe K line obtained by the
+width will be greatly improved by upcoming calorimeters, which have >10 time better the
+energy resolution, of only a few eV. If future X-ray satellites (e.g. Astro-H, IXO, and Gen-X)
+with larger effective area could reduce the error of the flux of the F e K emission line by even
+a factor of 2, then it will be possible to distinguish different geometrie s from the constant– 16 –
+flux fitting. Higher sampling frequency campaign, preferably over a longer baseline, is also
+desirable to obtain a cross-correlation function with a quality compa rable to or better than
+current optical observations.
+Y. L. Thanks Junfeng Wang and Joanna Kuraszkiewicz for the helpf ul discussions.
+S.N.Z. acknowledges partial funding support by the Yangtze Endow ment from the Ministry
+ofEducationatTsinghuaUniversity, DirectionalResearchProjec toftheChineseAcademyof
+Sciences underprojectno. KJCX2-YW-T03andbytheNationalNa turalScienceFoundation
+of China under grant nos. 10821061, 10733010, 10725313, and b y 973 Program of China
+under grant 2009CB824800. This work has been funded by CXC gra nt GO7-8136A and
+Suzaku Grant NNX08AB81G.– 17 –
+3 5 10 20 3010−310−210−1counts/s/keV
+6 6.5 7 7.5−0.0100.010.020.03
+E (keV)counts/s/keV5.85.960.020.040.060.080.1
+E (keV)counts/s/keV
+3 5 10 20 300.811.2
+E (keV)ratioa b 
+Fig. 1.— The best fit model (solid lines) and the added spectra of XIS0 3 (plus). The inset
+(a) is the residual of the spectra of XIS03 to the best fit of a single power law. The inset (b)
+is the added spectra of Mn K αline of the calibration sources on XIS03.– 18 –
+6.385 6.39 6.395 6.4 6.405 6.41 6.41577.027.047.067.087.17.127.147.167.18
+EKα (keV)EKβ (keV)a 
+Fe I Fe XIV 
+Fe XIII
+1.8 1.9 2 2.1 2.2 2.3 2.411.522.533.544.555.5IKβ (10−6 photons/cm2/s)
+IKα (10−5 photons/cm2/s)b 
+Fig. 2.— The confidence region of the peak energies (a) and intensitie s (b) of K αand Kβ
+obtained from the fitting of the co-added XIS03 and PIN spectra. The contours from inner
+to outer correspond to ∆ χ2=1.00, 2.71, and 6.63 (68%, 90%, 99%), respectively. The crosses
+in (a) are the predicted values of the peak energies in different ioniza tion states of iron. The
+lower and upper dashed lines in (b) correspond to the I(K β)/I(Kα) ratios of 0.12 and 0.20,
+respectively.– 19 –
+−200 −150 −100 −50 0 50051015202530354045Flux (10−12 ergs/cm2/s)
+Time (da y)   Suzaku
++  Swift      
+×  RXTE  a 
+−5 051015202530354045501.61.822.22.42.62.83
+Time (day)Flux (10−5 photons/cm2/s)χ2=3.05 b 
+Fig. 3.— (a) The light curve of the flux of the continuum in 3-10 keV ban d. The small error
+bar of the flux of the Suzakuobservation is omitted. (b) The light curve of the flux of K α
+line. The solid line in (b) is the result of the constant fitting (see the te xt in§4.1).– 20 –
+0 5 10 15 20 25 30 35 4000.20.40.60.81
+τ (day)Responsea 
+0 5 10 15 20 25 3000.20.40.60.81
+τ (day)Respnseb 
+Fig. 4.—The transfer functions forthe spherical (a) anddisk case s (b). We choose rmin= 10
+light days and rmax= 20light days forall curves, which arenormalized to 1at themaximum .
+(a)α= 2 (blue), α= 3 (red), and α= 4 (green). (b) α= 3 for all curves. i= 30◦(blue),
+i= 45◦(red), and i= 60◦(green).– 21 –
+2 2.5 3 3.5 4 4.5 5 5.5468101214
+Flux 8−10 keV (10−12 ergs/cm2/s)Flux 3−6 keV (10−12 ergs/cm2/s)
+2 2.5 3 3.5 4 4.5 5 5.50.30.350.40.450.50.55Flux (8−10 keV)/Flux (3−6 keV)
+Fig. 5.— The top panel shows the correlation between the flux in 3-6 k eV band and that in
+8-10 keV band of all seven observations. The solid line is the best-fit ting straight line across
+the origin. The error bars of the fluxes are omitted, since they are smaller than the symbols.
+The bottom panel shows the ratios between the flux in 8-10 keV ban d and that in 3-6 keV
+band.– 22 –
+0 10 20 30 40 50 60 70 80 9002468101214161820
+rmin (light day)χ2
+a 
+0 50 100 150051015202530354045
+rmin (light day)χ2
+b – 23 –
+10 20 30 40 50 60 7001234567
+rmin (light day)χ2
+c 
+Fig. 6.— (a) Comparison with different values of αin the spherical case. α= 2 (blue),
+α= 3 (red), and α= 4 (green). Since the original curves are nearly identical, the curv es
+ofα= 2 and α= 4 are vertically shifted by -0.5 and 0.5, respectively. (b) The curve s of
+χ2with different rminfor spherical and disk cases. Spherical (black), disk i= 15◦(blue),
+i= 30◦(red),i= 45◦(green), and i= 60◦(yellow). The horizontal dashed line is the χ2of
+the constant fitting (see the text). (c) Enlarged plot of (b). The error bar at the bottom is
+the 90% confidence interval inferred from the line width obtained by the simultaneous fitting
+of all spectra (see Figure 8).– 24 –
+1.61.82  Flux (10−5 photons/cm2/s)
+22.53
+0 10 20 30 40 5022.53
+Time (day)a 
+b 
+crmin=27 light days 
+        χ2=2.25 
+i=15° rmin=61 light days 
+        χ2=2.12 
+i= 30° rmin=37 light days 
+           χ2=0.821.61.82  Flux (10−5 photons/cm2/s)
+22.53
+0 10 20 30 40 5022.53
+Time (day)a 
+b 
+crmin=27 light days 
+        χ2=2.25 
+i=15° rmin=61 light days 
+        χ2=2.12 
+i= 30° rmin=37 light days 
+           χ2=0.82
+Fig. 7.— The comparison between the observed and predicted light cu rve of the flux of the
+line with the minimum of χ2in different cases, i.e. (a) spherical case, (b) disk case with
+i= 15◦, (c) disk case with i= 30◦.– 25 –
+Fig. 8.—The relationship between σof thelineandtheradius oftheemitting regioninferred
+from the black hole mass. The vertical dashed and solid lines corresp ond the central value
+and the 90% confidence intervals of the line width obtained by the co- added XIS 03 and PIN
+spectra, respectively. The horizontal lines correspond to the 90 % confidence intervals of the
+emitting region inferred from the light curve in the spherical case (s olid) and the disk case
+withi= 30◦(dashed, see Figure 6c).– 26 –
+0 10 20 30 40 50 60 700 2.6 5.2 7.9 10.5 13.1 15.7 18.3
+Spherical 
+Disk i=30 ° Disk i=15 ° 
+Co−added
+Hβ Inner radius of dust
+r (light day)Fe K reverberation 
+Fe K width 
+Other reverberations r (×103 rg)
+Fig. 9.— The summary of the location of the Fe K αline emitting region. We show intervals
+derived from the variation of the flux and the width of the Fe K line, wit h the locations
+of Hβand the inner radius of the dust for comparison. The solid and dashe d error bars
+correspond to 68% and 90% confidence intervals, respectively.– 27 –
+Table 1: Suzakuobservation log.
+Sequence number Obs. ID Start Date & Time Exposure time (ks) 3-1 0 keV Count Rate
+(10−3photons/cm2/s)
+1 702042010 2007-06-18 UT 22:28:15 31.1 0.772
+2 702042020 2007-06-24 UT 21:53:31 35.9 1.29
+3 702042040 2007-07-08 UT 10:02:55 30.7 2.35
+4 702042050 2007-07-15 UT 13:57:39 30.0 1.62
+5 702042060 2007-07-22 UT 10:40:25 28.9 3.05
+6 702042070 2007-07-29 UT 04:20:44 31.8 2.04
+7 702042080 2007-08-05 UT 00:37:46 38.8 1.12– 28 –
+Table 2: RXTEandSwiftobservation logs.
+Obs. ID Start Date & Time Exposure time (s) 3-10 keV Flux
+(10−12ergs/cm2/s)
+RXTE
+92113-07-40-00 2006-12-05 UT 03:35:48 919 12.0 ±0.9
+92113-07-41-00 2006-12-11 UT 12:59:06 885 18.1 ±1.0
+92113-07-42-00 2006-12-21 UT 16:34:27 961 12.2 ±0.9
+92113-07-43-00 2006-12-23 UT 11:08:04 830 15.8 ±1.0
+92113-07-44-00 2007-01-02 UT 13:09:22 914 24.0 ±1.0
+92113-07-45-00 2007-01-10 UT 10:09:27 1259 43.6 ±1.0
+92113-07-46-00 2007-01-14 UT 21:15:16 926 29.3 ±1.1
+92113-07-47-00 2007-01-21 UT 16:35:22 913 28.3 ±1.0
+92113-07-48-00 2007-01-29 UT 17:19:33 948 21.5 ±0.9
+92113-07-49-00 2007-02-05 UT 20:22:13 1171 20.4 ±0.9
+92113-07-50-00 2007-02-13 UT 12:12:30 895 36.0 ±1.1
+92113-07-51-00 2007-02-19 UT 08:05:31 896 41.9 ±1.1
+92113-07-52-00 2007-02-26 UT 07:25:07 1002 30.2 ±1.1
+92113-07-53-00 2007-03-04 UT 07:48:42 1354 39.6 ±0.9
+92113-07-54-00 2007-03-13 UT 11:30:51 614 23.2 ±1.2
+92113-07-55-00 2007-03-18 UT 18:56:43 572 26.4 ±1.3
+92113-07-56-00 2007-03-26 UT 13:58:42 994 19.1 ±0.9
+92113-07-57-00 2007-04-01 UT 16:10:48 982 15.7 ±1.0
+92113-07-58-00 2007-04-09 UT 04:12:03 893 19.0 ±1.0
+92113-07-59-00 2007-04-16 UT 03:23:00 955 14.2 ±1.1
+92113-07-60-00 2007-04-23 UT 06:05:57 961 16.3 ±0.9
+92113-07-61-00 2007-04-29 UT 12:54:16 1041 10.7 ±0.8
+92113-07-62-00 2007-05-07 UT 17:14:43 928 9.4 ±0.9
+92113-07-63-00 2007-05-14 UT 11:42:04 950 15.5 ±0.9
+92113-07-64-00 2007-05-21 UT 11:48:48 922 13.3 ±0.9
+92113-07-65-00 2007-05-28 UT 03:30:13 931 15.9 ±1.0
+92113-07-66-00 2007-06-04 UT 01:59:35 949 22.7 ±1.0
+92113-07-67-00 2007-06-11 UT 07:33:05 888 18.3 ±1.1
+Swift
+00030022061 2007-07-01 UT20:25:01 2503.18 10.5– 29 –Table 3: The derived parameters from the spectral fits of the XIS data of each observation (with error corresponding
+to 90% confidence level except flux and intensity).
+Sequence Continuum Kα Kβ
+number ΓaFluxbEcσdIeEWfEcIgEWfχ2/dof ∆ χ21h∆χ22i
+11.41+0.07
+−0.066.89+0.24
+−0.346.405+0.016
+−0.01744+32−442.02+0.23
+−0.24215+83−77
+7.00+0.08
+−0.085.6+1.7
+−1.868+72−6878.9/114 10.2 122.4
+21.55+0.05
+−0.0511.28+0.23
+−0.306.384+0.017
+−0.01646+29−462.40+0.25
+−0.29155+58−56
+7.08+0.06
+−0.065.5+2.0
+−2.042+51−42197.7/177 7.4 122.8
+31.68+0.03
+−0.0320.25+0.13
+−0.166.374+0.027
+−0.02562+31−372.16+0.30
+−0.3177+39−38
+7.04j<5.5<40355.1/338 65.7
+41.53+0.04
+−0.0414.24+0.24
+−0.276.380+0.019
+−0.01834+31−342.21+0.28
+−0.27112+48−48
+7.05+0.04
+−0.047.5+2.2
+−2.144+44−44227.3/214 11.7 98.9
+51.61+0.03
+−0.0326.46+0.14
+−0.166.401+0.029
+−0.03162+49−622.25+0.43
+−0.3661+34−34
+7.04j<5.3<31427.9/406 54.6
+61.57+0.03
+−0.0317.80+0.18
+−0.226.393+0.015
+−0.01534+29−342.62+0.29
+−0.26106+39−39
+7.04j<7.0<50303.2/308 124.1
+71.53+0.04
+−0.049.85+0.20
+−0.246.416+0.012
+−0.01223+28−232.17+0.19
+−0.21162+52−53
+6.97+0.09
+−0.084.0+1.5
+−1.634+43−34205.0/216 6.9 166.1
+aPhoton index
+bObserved flux in 3-10 keV band (10−12ergs/cm2/s) with 68% error
+cRest energy of line (keV)
+dIntrinsic width of line (eV)
+eObserved intensity of line (10−5photons/cm2/s) with 68% error
+fEquivalent width of line (eV). 90% upper limit is shown if the K βline is very weak.
+gObserved intensity of line (10−6photons/cm2/s) with 68% error. 90% upper limit is shown if the K βline is
+very weak.
+hThe increase of χ2after removing K βline from the best-fit model
+iThe increase of χ2after removing K αand Kβlines from the best-fit model
+jThe rest energy of line is fixed at the value obtained by the co-add sp ectra, since the K βline is very weak
+in this observation– 30 –Table 4: The derived parameters from the spectral fits of the co- added XIS03 and PIN data (with error corresponding
+to 90% confidence level except flux and intensity).
+Sequence ContinuumaKα Kβ
+number ΓbEcut-offcRdFluxbEbσbIbEWbEbIbEWbχ2/dofb∆χ21b∆χ22b
+11.59+0.03
+−0.0375+85−150.79+0.35
+−0.3215.1+0.1
+−0.36.396+0.009
+−0.00738+16−182.02+0.13
+−0.1195+21−19
+7.08+0.05
+−0.053.4+0.9
+−0.919+17−17423.8/396 14.7 405.5
+aThe cosine of the disc inclination was fixed at 0.87
+bThe meaning is the same as that in Table 3
+cEnergy cut-off (keV)
+dReflection parameter of cold matter; R= Ω/2π– 31 –
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\ No newline at end of file
diff --git a/1001.0357.txt b/1001.0357.txt
new file mode 100644
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+arXiv:1001.0357v1  [cs.IT]  3 Jan 2010Orthogonal vs Non-Orthogonal Multiple Access
+with Finite Input Alphabet and Finite Bandwidth
+J. Harshan
+Dept. of ECE, Indian Institute of Science
+Bangalore 560012, India
+Email:harshan@ece.iisc.ernet.inB. Sundar Rajan
+Dept. of ECE, Indian Institute of Science
+Bangalore 560012, India
+Email:bsrajan@ece.iisc.ernet.in
+Abstract—For a two-user Gaussian multiple access channel
+(GMAC), frequency division multiple access (FDMA), a well
+known orthogonal-multiple-access (O-MA) scheme has been
+preferred to non-orthogonal-multiple-access (NO-MA) sch emes
+since FDMA can achieve the sum-capacity of the channel with
+only single-user decoding complexity [ Chapter 14, Elements of
+Information Theory by Cover and Thomas ]. However, with finite
+alphabets, in this paper, we show that NO-MA is better than O-
+MA for a two-user GMAC. We plot the constellation constraine d
+(CC) capacity regions of a two-user GMAC with FDMA and time
+division multiple access (TDMA) and compare them with the
+CC capacity regions with trellis coded multiple access (TCM A),
+a recently introduced NO-MA scheme. Unlike the Gaussian
+alphabets case, it is shown that the CC capacity region with
+FDMA is strictly contained inside the CC capacity region wit h
+TCMA. In particular, for a given bandwidth, the gap between
+the CC capacity regions with TCMA and FDMA is shown to
+increase with the increase in the average power constraint. Also,
+for a given power constraint, the gap between the CC capacity
+regions with TCMA and FDMA is shown to decrease with the
+increase in the bandwidth. Hence, for finite alphabets, a NO-
+MA scheme such as TCMA is better than the well known O-
+MACschemes, FDMAandTDMAwhichmakes NO-MAschemes
+worth pursuing in practice for a two-user GMAC.
+I. INTRODUCTION
+Works on coding for Gaussian multiple access channel
+(GMAC) with finite input alphabets have been reported in
+[1]-[4]. Recently, constellation constrained (CC) capaci ty re-
+gions [5] of a two-user GMAC (shown in Fig. 1) have been
+computed in [6] wherein the angles of rotation between the
+alphabets of the users which enlarge the CC capacity regions
+have also been computed. For this channel model, code pairs
+based on trellis coded modulation (TCM) [7] are proposed
+in [8]-[10] such that any rate pair within the CC capacity
+region can be approached. Such a multiple access scheme
+whichemployscapacityapproachingtrelliscodesisreferr edas
+trellis coded multiple access (TCMA). In this paper, multip le
+access schemes are referred to as non-orthogonal multiple
+access (NO-MA) schemes whenever the two users transmit si-
+multaneously during the same time and in the same frequency
+band.TCMAisanexampleforaNO-MAscheme.Henceforth,
+throughout the paper, CC capacity regions obtained in [6] are
+referred to as CC capacity regions with TCMA since TCMA
+is shown to achieve any rate pair on the CC capacity region
+[10].
+For a two-userGMAC with Gaussian distributed continuousz
+yUser 1
+User 2AWGNDestination
+x2x1
+x1+x2
+Fig. 1. Two-user Gaussian MAC model
+input alphabets, it is well known that successive interfere nce
+cancellation decoder can achieve any point on the capac-
+ity region, provided the codebooks contain infinite length
+codewords [11], [12]. It is also known that time division
+multiple access (TDMA) and frequency division multiple
+access (FDMA), two of the widely knownorthogonalmultiple
+access (O-MA) techniques do not achieve all the points on the
+capacity region. In particular, if FDMA is used such that the
+bandwidthallocated to each user is proportionalto its tran smit
+power, then one of the points on the capacity region can be
+achieved. Note that such an achievable point lies on the sum-
+capacity line segment of the capacity region [11] (note that
+apart from the point on the sum capacity line, FDMA also
+intersects the capacity region at the axes points). The set o f
+rate pairs that can be achieved using TDMA and FDMA are
+provided in Fig. 2 along with the capacity region wherein the
+total bandwidth (for both the users) is WHertz, the power
+constraint for each user is ρWatts and the variance of the
+AWGN is σ2.
+In this paper, we compute the CC capacity regions of a
+two-user GMAC with the well known O-MA schemes such
+as TDMA and FDMA. On the similar lines of the study
+for Gaussian input alphabets (as shown in Fig. 2), we study
+the differences among the CC capacity regions of a two-user
+GMAC with TDMA, FDMA and TCMA. Throughout the pa-
+per,thetermsalphabetandsignal set are usedinterchangea bly.
+The main contributions of this paper may be summarized as
+below:
+•We compute the CC capacity regions of a two-user
+GMACwhenO-MA schemessuchas TDMA andFDMATDMAFDMAGaussian capacity region
+Point of intersectionC(W, ρ, σ2)C(W, ρ, σ2) =W /D0/D3/CV(1 +ρ
+σ2W)
+R2
+C(W, ρ, σ2)R1
+Fig. 2. Achievable rate pairs (in bits per second) for TDMA an d FDMA for a total bandwidth of WHertz
+are employed for finite bandwidth.
+•Since FDMA with Gaussian alphabets achieve one of the
+sum-capacity points with single-user decoding complex-
+ity, the authors of [11] have stated that ”the improvement
+in the capacity due to multiple access ideas such as the
+one achieved by the successive interference decoder (a
+NO-MA scheme) may not be sufficient to warrant the
+increased complexity” (see page 407, Section 14.3.6 of
+Chapter 14 in [11]). In this paper, we point out that the
+above comment in [11] is not valid for a GMAC with
+finite alphabetswhichisthe case inpracticalscenarios.In
+particular,we show that NO-MAschemessuch asTCMA
+canprovidesubstantialimprovementinthecapacitywhen
+compared to O-MA schemes such as FDMA and TDMA
+for finite alphabets which in-turn makes TCMA worth
+pursuing in practice for a two-user GMAC. We highlight
+that this result is not apparent unless CC capacity regions
+with FDMA and TCMA are plotted (Section IV).
+•We also show that the gap between the CC capacity
+regions with TCMA and FDMA is a function of the
+bandwidth, WHertz and the average power constraint,
+ρWatts. It is shown that, (i) for a fixed W, the gap
+betweentheCC capacityregionswith FDMA andTCMA
+increases with the increase in ρ(see Fig. 4, Fig. 5 and
+Fig. 6 for a fixed Wand varying ρ), and (ii) for a fixed
+ρ, the gap between the CC capacity regions with FDMA
+and TCMA decreases with the increase in W(see Fig. 4
+and Fig. 7 for a fixed ρand varying W) (Section IV-A).
+•On the similar lines of the results for Gaussian alphabets,
+the CC capacity region with TDMA is shown to be well
+contained inside the CC capacity region with TCMA
+(Section IV-B).
+The result presented in this paper is another example to
+illustrate the differences in the system behaviour in terms of
+capacity when the input alphabets are constrained to have
+finite cardinality. An earlier example can be found in [6],whereinarelativeangleofrotationbetweentheinputalpha bets
+is shown to enlarge the CC capacity region with TCMA1.
+Note that such a capacity advantage is not applicable for
+Gaussian alphabets. For example,see Fig. 3, which depictst he
+advantage of rotations on the CC capacity region with TCMA
+for QPSK signal sets. For more details on the advantages of
+rotation on the CC capacity region, we refer the readers to [9 ].
+Notations: Cardinalityofaset Sisdenotedby |S|. Absolute
+value of a complex number xis denoted by |x|andE[x]
+denotes the expectation of the random variable x. A circularly
+symmetric complex Gaussian random vector, xwith mean µ
+and covariance matrix Γis denoted by x∼ CN(µ,Γ).
+The remaining content of the paper is organized as follows:
+In Section II, the two-user GMAC model considered in this
+paper is presented wherein we recall the CC capacity regions
+with TCMA. In Section III, we revisit the achievablerate pai rs
+for FDMA and TDMA with Gaussian alphabets. In Section
+IV, we compute the CC capacity regions with FDMA and
+TDMA and show the optimality of TCMA over FDMA and
+TDMA. Section V constitutes conclusion and some directions
+for possible future work.
+II. CHANNEL MODEL AND CCCAPACITY REGIONS OF
+GMAC
+The model of a two-user Gaussian MAC as shown in Fig.
+1 consists of two users that need to convey information to a
+single destination. It is assumed that User-1 and User-2 com -
+municate to the destination at the same time and in the same
+frequency band of WHertz. Symbol level synchronization is
+assumed at the destination. The two users are equipped with
+alphabets S1andS2of sizeN1andN2respectively such
+that forxi∈ Si, we have E[|xi|2] = 1. In other words, the
+signal sets S1andS2have unit average energy. When User-1
+1Note that the CC capacity region for a two-user GMAC [6] is ref erred as
+the CC capacity region with TCMA since TCMA is shown to achiev e any
+rate pair on the CC capacity region [9]I(/radicalbigg
+ρ
+Wx2:y) =log2(N2)−1
+N1N2N1−1/summationdisplay
+k1=0N2−1/summationdisplay
+k2=0E/bracketleftBigg
+log2/bracketleftBigg/summationtextN1−1
+i1=0/summationtextN2−1
+i2=0exp/parenleftbig
+−|/radicalbigρ
+W(x1(k1)+x2(k2)−x1(i1)−x2(i2))+z|2/σ2/parenrightbig
+/summationtextN1−1
+i1=0exp/parenleftbig
+−|/radicalbigρ
+Wx1(k1)−/radicalbigρ
+Wx1(i1)+z|2/σ2/parenrightbig/bracketrightBigg/bracketrightBigg
+.
+(1)
+I(/radicalbiggρ
+Wx1:y|x2) =log2(N1)−1
+N1N1−1/summationdisplay
+k1=0E/bracketleftBigg
+log2/bracketleftBigg/summationtextN1−1
+i1=0exp/parenleftbig
+−|/radicalbigρ
+Wx1(k1)−/radicalbigρ
+Wx1(i1)+z|2/σ2/parenrightbig
+exp(−|z|2/σ2)/bracketrightBigg/bracketrightBigg
+.(2)
+0.40.60.811.21.41.61.822.20.40.60.811.21.41.61.822.22.4
+R1R2
+snr = 0 dBsnr = 6 dBCC capacity
+region with rotation
+CC capacity
+region without rotation
+2 dB4 dBUnconstrained
+capacity region
+at 6 dB
+Fig. 3. CC capacity regions with TCMA of QPSK signal sets with and
+without rotation for two-user GMAC where snr =ρ
+Wandσ2= 1.
+and User-2 transmit symbols x1andx2simultaneously, the
+destination receives a symbol ygiven by,
+y=/radicalbiggρ
+Wx1+/radicalbiggρ
+Wx2+zwherez∼ CN/parenleftbig
+0,σ2/parenrightbig
+(3)
+whereρis the average power constraint for the two users.
+Throughout the paper, we assume equal average power con-
+straint for the two users. Extension of the results presente d in
+this paper to the unequal power case is straight forward. We
+assume that every channel use consumes Tseconds for each
+user (where we assume1
+T=WHertz) and henceρ
+Wbecomes
+the average energy constraint for the two users.
+In [6], CC capacity regions for two-user Gaussian multiple
+access channels have been computed for some well known
+alphabets such as M-PSK,M-QAM etc. Applying the results
+of [6] to the channel model in (3), the set of CC capacity
+values (in bits per channel use) that define the boundary of
+the CC capacity region are as given below,
+R1≤I(/radicalbiggρ
+Wx1:y|x2),
+R2≤I(/radicalbiggρ
+Wx2:y|x1)and
+R1+R2≤I(/radicalbiggρ
+Wx1+/radicalbiggρ
+Wx2:y),(4)where the expressions for I(/radicalbigρ
+Wx2:y),I(/radicalbigρ
+Wx1:y|x2)
+are shown in (1) and (2) at the top of this page. The term
+I(/radicalbigρ
+Wx1+/radicalbigρ
+Wx2:y)can be calculated as I(/radicalbigρ
+Wx2:y)
++I(/radicalbigρ
+Wx1:y|x2). Examples of CC capacity regions are
+shown in Fig. 3 for QPSK alphabets. Since every channel
+use consumes Tseconds, the CC capacity pairs (in bits per
+seconds) that define the CC capacity region are given by
+R1≤WI(/radicalbiggρ
+Wx1:y|x2),
+R2≤WI(/radicalbiggρ
+Wx2:y|x1)and
+R1+R2≤WI(/radicalbiggρ
+Wx1+/radicalbiggρ
+Wx2:y).(5)
+In the following section, we will revisit the capacity regio n
+of a two-user Gaussian MAC (referred as the unconstrained
+capacity region of two-user GMAC) and recall the set of
+achievable rate pairs when FDMA and TDMA are employed.
+III. UNCONSTRAINED CAPACITY REGIONS OF TWO -USER
+GMAC
+For computing the CC capacity regions, the input alphabets
+are assumed to take values with uniform distribution [6].
+However, if the input alphabets are continuous and distribu ted
+asx1,x2∼ CN(0,1), thenthereceivedsymbol, yis Gaussian
+distributed as CN/parenleftbig
+0,2ρ
+W+σ2/parenrightbig
+. Considering the bandwidth of
+W Hertz, the unconstrained capacity region in bits per secon d
+is given by
+R1≤Wlog2(1+ρ
+σ2W) =C(W,ρ,σ2),
+R2≤Wlog2(1+ρ
+σ2W) =C(W,ρ,σ2)and
+R1+R2≤Wlog2(1+2ρ
+σ2W). (6)
+From the results of [11], [12], it is well known that any
+rate pair within the capacity region can be achieved by us-
+ing successive interference cancellation decoder provide d the
+codebookshaveinfinitelengthcodewords.Whenthe twousers
+employ FDMA, let W1=αWandW2= (1−α)Wbe the
+bandwidth occupied by User-1 and User-2 respectively where
+0< α <1. For such a scheme, the maximum achievable rate00.20.40.60.811.21.41.61.800.20.40.60.811.21.41.61.8
+R1R2CC capacity
+region
+with TDMACC capacity
+region with
+FDMAUnconstrained
+capacity region
+CC capacity
+region with
+TCMA
+Fig. 4. CC Capacity regions (in bits/sec) with FDMA, TDMA and TCMA
+for two-user GMAC with QPSK signal sets when ρ= 2 dB,σ2= 1, and W
+= 2 Hertz.
+pairs (in bits per second) for the two users are given by
+R1≤W1log2(1+ρ
+σ2W1)andR2≤W2log2(1+ρ
+σ2W2).
+(7)
+The maximum achievable sum rate when α= 0.5is given by,
+R1+R2≤Wlog2(1+2ρ
+σ2W) (8)
+which is equal to the sum capacity of the two-user Gaussian
+MAC given in (6). Therefore, for Gaussian alphabets, FDMA
+can achieve one of the points on the maximum sum rate line
+of the unconstrained capacity region. Proceeding similarl y for
+the case of TDMA, one obtains the set of achievable rate pairs
+as shown in Fig. 2.
+On the similar lines of the discussion in this section, in the
+followingsection, we will obtain the CC capacity regionswi th
+FDMA and TDMA.
+IV. CC C APACITY REGIONS WITH FDMA ANDTDMA
+In this section, we first compute the CC capacity regions
+with FDMA. Let W1=αWandW2= (1−α)Wbe the
+disjoint band of frequencies occupied by User-1 and User-2
+respectively where 0< α <1. Hence, for each i= 1,2, User-
+iviews a Single-Input Single-Output (SISO) AWGN channel
+to the destination with the input alphabet Si, bandwidth Wi
+and energy constraint,ρ
+Wi. The CC capacity values (in bits
+per second) for the two users are given by,
+R1≤W1I/parenleftbigg/radicalbiggρ
+W1x1:y|x2/parenrightbigg
+and
+R2≤W2I/parenleftbigg/radicalbiggρ
+W2x2:y|x1/parenrightbigg
+.
+Note that when α= 0.5, the CC sum capacity with FDMA is
+given by,0 0.5 1 1.5 2 2.5 300.511.522.53
+R1R2
+CC capacity region
+with TDMACC capacity region
+with FDMACC capacity
+region
+with TCMAUnconstrained
+ capacity region
+Fig. 5. CC Capacity regions (in bits/sec) with FDMA, TDMA and TCMA
+for two-user GMAC with QPSK signal sets when ρ= 5 dB,σ2= 1, and W
+= 2 Hertz.
+00.511.522.533.544.500.511.522.533.544.5
+R1R2
+CC
+capacity
+region
+with
+TDMA
+CC capacity
+region with
+FDMAUnconstrained capacity
+region
+CC capacity region
+with  TCMA
+Fig. 6. CC Capacity regions (in bits/sec) with FDMA, TDMA and TCMA
+for two-user GMAC with QPSK signal sets when ρ= 8 dB,σ2= 1, and W
+= 2 Hertz.
+R1+R2≤W
+2I/parenleftBigg/radicalbigg
+2ρ
+Wx1:y|x2/parenrightBigg
++W
+2I/parenleftBigg/radicalbigg
+2ρ
+Wx2:y|x1/parenrightBigg
+.
+Ifweassumethetwouserstoemployidenticalsignalsets,th en
+the CC sum capacity (denoted by R1+R2|FDMA ,α=0.5)is
+given by
+R1+R2|FDMA ,α=0.5≤WI/parenleftBigg/radicalbigg
+2ρ
+Wσ2x1:y|x2/parenrightBigg
+(9)
+wherein without loss of generality, we have used the variabl e
+x1for both the users.0.8 1 1.2 1.4 1.6 1.8 20.811.21.41.61.82
+R1R2
+CC capacity region
+with TDMACC capacity region
+with FDMACC capacity region
+with TCMAUnconstrained
+capacity region
+Fig. 7. CC Capacity regions (in bits/sec) with FDMA, TDMA and TCMA
+for two-user GMAC with QPSK signal sets when ρ= 2 dB,σ2= 1, and W
+= 5 Hertz.
+0.5 1 1.5 2 2.5 3 3.5 40.511.522.533.54
+R1R2
+CC capacity region
+with TDMACC capacity region
+with FDMACC capacity region
+with TCMAUnconstained
+capacity region
+Fig. 8. CC Capacity regions (in bits/sec) with FDMA, TDMA and TCMA
+for two-user GMAC with QPSK signal sets when ρ= 5 dB,σ2= 1, and W
+= 5 Hertz.
+A. Comparing the CC capacity regions of TCMA and FDMA
+The CC sum capacity with TCMA (denoted by R1+
+R2|TCMA) given in (5) is
+R1+R2|TCMA≤WI/parenleftbigg/radicalbiggρ
+Wx1+/radicalbiggρ
+Wx2:y/parenrightbigg
+.(10)
+Comparing (9) and (10), it is not straightforward to comment
+whether, the CC sum capacity offered by FDMA is equal to
+or away from the CC sum capacity with TCMA. Therefore, in
+Fig. 4, Fig. 5 and Fig. 6, we plot the CC capacity regions with
+TCMA and FDMA for QPSK (quadrature phase shift keying)
+signalsets whenbandwidth, W= 2Hertz.Similarly,inFig. 7,
+Fig.8andFig.9,CC capacityregionswithTCMA andFDMA
+are presented for QPSK signal sets when bandwidth, W= 50 1 2 3 4 5 60123456
+R1R2
+CC capacity
+region
+with TDMACC capacity
+region
+with TCMA
+CC capacity
+region
+with FDMAUnconstrained
+ capacity
+region
+Fig. 9. CC Capacity regions (in bits/sec) with FDMA, TDMA and TCMA
+for two-user GMAC with QPSK signal sets when ρ= 8 dB,σ2= 1, and W
+= 5 Hertz.
+−2 0 2 4 6 8 10 1211.522.533.54
+SNR in dBsum rate in bps for unit bandwidthFDMA
+TCMA
+Fig. 10. CC sum capacity in bits per second per Hertz where SNR =γ.
+Hertz.Theplotsshowthat the CC capacityregionwith FDMA
+is strictly enclosedwithin the CC capacityregionwith TCMA .
+Note that, for a given value of W, the difference between
+the regions with FDMA and TCMA becomes significant with
+the increased value of ρ(see Fig. 4, Fig. 5 and Fig. 6). In
+particular, the plots show the following inequality,
+R1+R2|FDMA ,α=0.5≤R1+R2|TCMA.
+Note that the difference between R1+R2|FDMA ,α=0.5and
+R1+R2|TCMAdepends on Wfor a given value of ρandσ2.
+In order to make the above difference component independent
+of the bandwidth W, we calculate the percentage increase in
+the CC sum capacity from R1+R2|FDMA ,α=0.5toR1+TABLE I
+THE VALUE OF µIN PERCENTAGE FOR DIFFERENT VALUES OF γ
+γin dB -2024681012
+µin%5.469.3117.7932.4850.5568.3382.7092.78
+R2|TCMA(denoted as µ) given by,
+µ=WI(/radicalbigρ
+Wx1+/radicalbigρ
+Wx2:y)−WI(/radicalBig
+2ρ
+Wσ2x1:y|x2)
+WI(/radicalBig
+2ρ
+Wx1:y|x2)
+=I(/radicalbigρ
+Wx1+/radicalbigρ
+Wx2:y)−I(/radicalBig
+2ρ
+Wx1:y|x2)
+I(/radicalBig
+2ρ
+Wx1:y|x2).
+Replacing γ=ρ
+W, the right hand-side of the above inequality
+can be written as
+µ=I(√γx1+√γx2:y)−I(√2γx1:y|x2)
+I(√2γx1:y|x2)
+whereinWhas been absorbed in the term γ. In the Table I,
+we provide the values of µfor different values of γwhen
+σ2= 1and the input alphabets are QPSK signal sets. The
+values of I(√γx1+√γx2:y)andI(√2γx1:y)have also
+been plotted as a function of γin Fig. 10. The values of µ
+can also be plotted against γfor larger signal sets as well.
+From Table I, it is clear that µincreases as the value of
+γincrease. An intuitive reasoning for such a behaviour is as
+follows: Theterm I(√γx1+√γx2:y)is the CC capacity ofa
+16 point constellation (sum alphabet of two QPSK signal sets )
+with average energy of 2γwhereasI(√2γx1:y)is the CC
+capacity of a 4 point constellation (QPSK signal set) with th e
+sameaverageenergyof 2γ.Notethat,asymptotically(forlarge
+values of γ),I(√2γx1:y)andI(√γx1+√γx2:y)saturate
+to 2 bits and 4 bits respectively. Therefore, as γincreases,
+the term I(√2γx1:y)increases at a slower rate to saturate
+to 2 bits however, I(√γx1+√γx2:y)increases at a faster
+rate as its saturation is only at 4 bits. A similar reasoning a lso
+holdsforalphabetswitharbitrarysize.However,the diffe rence
+in the sum capacity may differ depending on the size of the
+alphabets.
+B. CC capacity region with TDMA
+In subsection, we obtain the CC capacity pairs when the
+two users employ TDMA. If User-1 uses the channel for α
+seconds and User-2 uses the channel for (1−α)seconds for
+some0< α <1, then the CC capacity values (in bits per
+second) for the two users are given by
+R1≤αWI/parenleftbigg/radicalbiggρ
+Wx1:y|x2/parenrightbigg
+and
+R2≤(1−α)WI/parenleftbigg/radicalbiggρ
+Wx2:y|x1/parenrightbigg
+.Assuming identical alphabets for the two users, the CC sum
+capacity with TDMA is given by,
+R1+R2≤WI/parenleftbigg/radicalbiggρ
+Wx1:y|x2/parenrightbigg
+.
+The set of CC capacity pairs when the two users employ
+TDMA are shown in Fig.4, Fig.5 and Fig. 6 which shows
+that TCMA is better than TDMA for finite alphabets as well.
+We highlightthat,alongwiththesubstantialimprovementi n
+the capacity, low complexitytrellis codes proposedfor TCM A
+in [8] makes TCMA worth pursuing in practice for a two-user
+GMAC.
+V. DISCUSSION
+In this paper,we computethe CC capacity regionsof a two-
+user GMAC when multiple access schemes such as TDMA
+and FDMA are employed. Unlike the Gaussian alphabets
+case, it is shown that the CC capacity region with FDMA is
+strictly contained inside the CC capacity region with TCMA,
+essentially showing that TCMA is better than FDMA for
+finite alphabets. After [6], the result presented in this pap er
+is another example to illustrate the differences in the capa city
+behaviour when the input alphabets are constrained to have
+finite cardinality. Some possible directions for further wo rk
+are as follows:
+•In this paper, we have assumed equal average power
+constraint for both the users. The results presented in this
+paper can be extended to alphabets with unequal power
+constraints as well
+•The presented result can also be extended to multiple
+access channels with more than two users.
+ACKNOWLEDGMENT
+Thisworkwas partlysupportedby theDRDO-IISc Program
+on Advanced Research in Mathematical Engineering to B.S.
+Rajan.
+REFERENCES
+[1] F. N. Brannstrom, T. M. Aulin and L. K. Rasmussen, ”Conste llation-
+Constrained capacity for Trellis code Multiple Access Syst ems” in the
+proceedings of IEEE GLOBECOM 2001 , vol. 2, San Antonio, Texas,
+Nov, 2001, pp. 11-15.
+[2] T. Aulin and R. Espineira, ”Trellis coded multiple acces s (TCMA)” in the
+proceedings of ICC ’99, Vancouver, BC, Canada, June 1999, pp. 1177-
+1181.
+[3] Fredrik N Brannstrom, Tor M. Aulin and Lars K. Rasmussen ” Iterative
+Multi-User Detection of Trellis Code Multiple Access using a posteriori
+probabilities” in the proceedings of IEEE ICC 2001 , Finland, June 2001,
+pp. 11-15.
+[4] Wei Zhang, C. D’Amours and A. Yongacoglu, ”Trellis Coded Modulation
+Design formMulti-User Systems onAWGNChannels” intheproc eedings
+ofIEEE Vehicular Technology Conference 2004 , pp. 1722 - 1726.[5] Ezio Biglieri, ”Coding for wireless channels”, Springer-Verlag New York,
+Inc, 2005.
+[6] J. Harshan and B. Sundar Rajan, ”Finite Signal-set Capac ity of Two-
+user Gaussian Multiple Access Channel” in the proceedings o fIEEE
+International Symposium on Information Theory, (ISIT 2008 ), Toronto,
+Canada, July 06-11, 2008. pp. 1203 - 1207.
+[7] G. Ungerboeck, ”Channel coding with multilevel/phase s ignals,”IEEE
+Trans. Inform. Theory , vol. 28, no. 01, 1982, pp. 55-67.
+[8] J. Harshan and B. Sundar Rajan, ”Coding for Two-User Gaus sian MAC
+with PSK and PAMSignal Sets” in the proceedings of IEEEInternational
+Symposium on Information Theory, (ISIT 2009) , Seoul, South Korea, June
+28- July 03, 2009, pp. 1859-1863.
+[9] J. Harshan and B. Sundar Rajan, ”Coding for two-user SISO and MIMO
+multiple access channels”, available online at arXiv:0901 .0168v3 [cs.IT],
+January 2009.
+[10] J. Harshan and B. Sundar Rajan, ”Trellis coded modulati on for two-
+user unequal-rate Gaussian MAC”, available online at arXiv :0908.1163v1
+[cs.IT], August 2009.
+[11] Thomas M Cover and J. A. Thomas, ”Elements of informatio n theory”,
+second edition - Wiley Series in Telecommunications and Sig nal Process-
+ing
+[12] Rimoldi, B. and Urbanke, R, ”A rate-splitting approach to the Gaussian
+multiple-access channel”, IEEE Transactions on Information Theory , vol.
+42, No. 02, pp. 364-375, 1996.
\ No newline at end of file
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+arXiv:1001.0358v1  [math.AT]  3 Jan 2010FINITENESS OF RANK INVARIANTS OF MULTIDIMENSIONAL
+PERSISTENT HOMOLOGY GROUPS
+FRANCESCA CAGLIARI AND CLAUDIA LANDI
+Abstract. Rank invariants are a parametrized version of Betti numbers of a
+space multi-filtered by a continuous vector-valued functio n. In this note we
+give a sufficient condition for their finiteness. This conditi on is sharp for spaces
+embeddable in Rn.
+Introduction
+Persistent Topology is a theory for studying objects related to co mputer vision
+and computer graphics, which involves analyzing the qualitative and q uantitative
+behavior of real-valued functions defined over topological spaces . More precisely,
+it studies the sequence of nested lower level sets of the considere d functions and
+encodes the scale at which a topological feature (e.g., a connected component, a
+tunnel, a void) is created, and when it is annihilated along this filtration . In this
+framework,multidimensionalpersistenthomologygroupscapture thehomologyofa
+multi-parameterincreasingfamily of spaces. For applicationpurpos es, these groups
+are further encoded by simply considering their rank, yielding to a pa rametrized
+version of Betti numbers, called rank invariants [3].
+The aim of this note is to prove the following theorem, providing a suffic ient
+condition in order that multidimensional persistent homology groups are finitely
+generated (or, equivalently, rank invariants are finite):
+Theorem 1. IfXis a compact, locally contractible subspace of Rn, and/vectorf=
+(f1,f2,...,f k) :X→Rkis a continuous function, then, for any q∈Z, the rank
+invariant ρ(X,/vectorf),qtakes only finite values: for any /vector u= (u1,u2,...,u k)and/vector v=
+(v1,v2,...,v k)inRk, withuj< vj(j= 1,2,...,k),
+ρ(X,/vectorf),q(/vector u,/vector v)def= rankim Hq(X/vectorf≤/vector u֒→X/vectorf≤/vector v)<+∞,
+the map being the inclusion, and X/vectorf≤/vector uandX/vectorf≤/vector vdenoting the lower level sets
+{x∈X:fj(x)≤uj, j= 1,2,...,k}and{x∈X:fj(x)≤vj, j= 1,2,...,k},
+respectively.
+The finiteness condition for rank invariants has proved to be crucia l for the
+stabilityofpersistence diagrams[6], andthe stability ofmultidimension al persistent
+homology groups [2, 4]. In each of these papers, finiteness of pers istent homology
+rankswasgenerallyguaranteedbyrequiringthe topologicalspace to be triangulable
+or imposing a tameness condition on the filtering function. In [5] it is ar gued that
+2000Mathematics Subject Classification. Primary: 55N99; Secondary: 68T10, 54C15.
+Key words and phrases. Persistent topology, Shape analysis, Betti numbers, Eucli dean neigh-
+borhood retract.
+Research partially carried out within the activities of ARC ES (Universit` a di Bologna).
+12 FRANCESCA CAGLIARI AND CLAUDIA LANDI
+this functional setting is not large enough to address the problems encountered in
+practical applications, and stability of persistence diagrams is revis ited. The basic
+assumption to prove stability in [5] is the finiteness of persistent hom ology ranks,
+but the question of how achieving this is left unanswered. Neverthe less, it is known
+that the ranks of 0th persistent homology groups (i.e. size functio ns) are finite
+provided that the space is only compact and locally connected [2]. Th eorem 1
+settles this issue also for multidimensional persistent homology grou ps of spaces
+embeddable in the Euclidean space.
+1.Proof of Theorem 1
+We begin recalling the definition of ENR and the criteria for a space to b e an
+ENR, following [1].
+Definizione 2. A topological subspace YofRnis called a Euclidean neighborhood
+retract(ENR) if there is a neighborhood in Rnof which Yis a retract.
+Theorem 3 (cf. [1]).IfY⊆Rnis locally compact and locally contractible then Y
+is an ENR.
+In order to prove Theorem 1 we anticipate a lemma.
+Lemma 4. LetXbe a compact, locally contractible subspace of Rn. LetP,Qbe
+subspaces of XwithclX(Q)⊆intX(P). Then, for any q∈Z, it holds that
+rank imHq(Q ֒→P)<+∞,
+the map being the inclusion.
+Proof.We take L1= clX(Q),L2= intX(P). It is sufficient to show that, for
+anyq∈Z, the rank of im Hq(ι:L1֒→L2) is finite, ιbeing the inclusion map.
+To this aim we observe that L1is closed in X, and hence compact. Moreover, L2,
+being open in X⊆Rn, islocally compactand locallycontractible, and therefore, by
+Theorem3, L2is an ENR. Thus, by Definition 2, there exists an open neighborhood
+L3ofL2inRnand a retraction r:L3→L2. SinceL3is open in Rn, for any x∈L3
+there is an open n-dimensional cube Q(x) centered at x, whose closure is contained
+inL3. Let us consider the open cover of L1given by Q={Q(x)∩L1}x∈L3. By
+compactness, L1admits a finite subcover: L1=/uniontextr
+s=1Q(xs)∩L1. Let us set
+K=/uniontextr
+s=1clRn(Q(xs)). Clearly L1is contained in K, andKis contained in L3.
+SinceKis a finite union of compact polyhedra, it is a polyhedron itself (cf. [7]) ,
+and its homology groups are finitely generated.
+Let us now consider the inclusion j:L1֒→K, and the restriction of the retrac-
+tionrtoK,r|K:K→L2. Sinceris a retraction onto L2⊇L1, it holds that, for
+everyx∈L1,r|K(x) =x. Henceι=r|K◦j:L1֒→K→L2. Therefore
+rankimHq(ι) = rankim/parenleftbig
+Hq(r|K)◦Hq(j)/parenrightbig
+≤rankimHq(j)≤rankHq(K)<+∞.
+/square
+Proof of Theorem 1. ItissufficienttoapplyLemma4,forevery /vector u= (u1,u2,...,u k),
+/vector v= (v1,v2,...,v k) inRk, withuj< vj(j= 1,2,...,k), setting Q=X/vectorf≤/vector uand
+P=X/vectorf≤/vector v. /squareFINITENESS OF RANK INVARIANTS OF MULTIDIMENSIONAL PERSIST ENT HOMOLOGY GROUPS 3
+We conclude this note observing that, for a space Xembeddable in Rn, the
+conditions of compactness and local contractibility cannot be weak ened in order to
+guarantee finiteness for rank invariants. Indeed, taking the clos ed topologist’s sine
+curve
+X={(x,y)∈R2:x∈(0,1], y= sin(1/x)}∪{(x,y)∈R2:x= 0, y∈[−1,1]},
+we have an example of a compact but not locally contractible subspac e ofR2whose
+0-th rank invariant, when Xis filtered using the height function f(x,y) =y, is
+unbounded. Analogously, taking X={(x,y)∈R2:x∈(0,1], y= sin(1/x)}, we
+have an example of a locally contractible but non-compact subspace ofR2whose
+0-th rank invariant, when Xis filtered using the height function, is unbounded.
+References
+1. G. E. Bredon, Topology and geometry , Springer-Verlag, 1993.
+2. F. Cagliari, B. Di Fabio, and M. Ferri, One-dimensional reduction of multidimensional persis-
+tent homology , to appear in Proc. Am. Math. Soc.
+3. G. Carlsson and A. Zomorodian, The theory of multidimensional persistence , Discrete & Com-
+putational Geometry 42(2009), no. 1, 71–93.
+4. A. Cerri, B. Di Fabio, M. Ferri, P. Frosini, and C. Landi, Multidimensional per-
+sistent homology is stable , Technical report, University of Bologna, Luglio 2009,
+http://amsacta.cib.unibo.it/2603/.
+5. F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S . Y. Oudot, Proximity of persis-
+tence modules and their diagrams , SCG ’09: Proceedings of the 25th annual symposium on
+Computational geometry (New York, NY, USA), ACM, 2009, pp. 2 37–246.
+6. D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, Stability of persistence diagrams , Discrete
+Comput. Geom. 37(2007), no. 1, 103–120.
+7. C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology , Springer-Verlag,
+New York, 1972.
+Dipartimento di Matematica, Universit `a di Bologna, P.zza di Porta S. Donato 5,
+I-40126Bologna, Italia
+E-mail address :cagliari@dm.unibo.it
+Dipartimento di Scienze e Metodi dell’Ingegneria, Universi t`a di Modena e Reggio
+Emilia, Via Amendola 2, Pad. Morselli, I-42100 Reggio Emilia, Italia
+E-mail address :clandi@unimore.it
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+arXiv:1001.0359v2  [gr-qc]  1 Apr 2010Circular geodesics in Extremal Reissner Nordstrom
+Spacetimes
+Parthapratim Pradhan∗
+Department of Physics
+Vivekananda Satabarshiki Mahavidyalaya
+Manikpara, Paschim Medinipur 721513, India.
+and
+Parthasarathi Majumdar†
+Saha Institute of Nuclear Physics
+Kolkata 700 064, India.
+October 29, 2018
+Abstract
+Circular null geodesic orbits, in extremal Reissner-Nords trom spacetimes, are
+examined with regard to their stability, and compared with s imilar orbits in the
+near-extremal situation. Extremization of the effective pot ential for null circular
+orbits shows the existence of a stable circular geodesic in t he extremal spacetime,
+precisely onthe event horizon, which coincides with its null geodesic ge nerator.
+Such an orbit also emerges as a global minimum of the effective p otential for circular
+timelikeorbits. This type of geodesic is of course absent in the corre sponding near-
+extremal spacetime, as we show here, testifying to difference s between the extremal
+limit of a generic RN spacetime and the exactly extremal geom etry.
+1 Introduction
+A number of features of near-extremal black hole spacetimes indic ate the absence of a
+smooth limit to extremality [1]. One aspect related to black hole thermo dynamics is
+the definition of the so-called Entropy Function [2] used widely nowad ays to match the
+∗pppradhan5@rediffmail.com
+†parthasarathi.majumdar@saha.ac.in
+1‘macroscopic’ entropy of a class of extremal black holes emerging in the supergravity limit
+of string theories, to the ‘microscopic’ entropy obtained from cou nting of string states [3].
+While the state counting is strictly restricted to BPS states, the us e of the Entropy
+Function is stymied by the fact that its existence depends on that o f abifurcation two-
+sphere(in four spacetime dimensions). This bifurcation sphere only exists a way from
+extremality, which forces one to begin with a near-extremal situation, and then proceed
+eventually to the extremal limit.
+However, as has been suspected earlier [4] and succinctly pointed out recently [5], the
+existence of this limit cannot be taken for granted. In other words the extremal limit of a
+near-extremal spacetime is not necessarily the extremal sp acetime. One manifestation of
+this concerns the near horizon limit: Consider for instance the near -horizon geometry of
+a non-extremal Reissner Nordstrom (RN) spacetime
+ds2=−/bracketleftbigg(r−r+)(r−r−)
+r2/bracketrightbigg
+dt2+/bracketleftbigg(r−r+)(r−r−)
+r2/bracketrightbigg−1
+dr2+r2dΩ2. (1)
+Near extremality, defining δ≡(r+−r−)/r0<<1, ǫ≡(r−r0)/r0<<1 wherer0is the
+radius of the horizon in the extremal case, this metric reduces, clo se to the event horizon,
+to
+ds2=−ǫ(ǫ+δ)
+(1+ǫ)2dt2+(1+ǫ)2
+ǫ(ǫ+δ)dr2+r2
+0(1+ǫ)2dΩ2(2)
+which leads to two distinct outcomes depending on the order in which t he limits δ→0
+andǫ→0 are taken. If the extremal limit δ→0 is taken first and then the near-horizon
+limit is taken, the localgeometry is that of an AdS2×S2
+ds2≃ −ǫ2dt2+r2
+0
+ǫ2dǫ2+r2
+0dΩ2. (3)
+On the other hand, if the near horizon limit is taken before the extre mal limit, one gets
+ds2≃ −ǫδdt2+r2
+0
+ǫδdǫ2+r2
+0dΩ2, (4)
+which certainly does not correspond to an AdS2×S2; in fact the extremal limit is now
+singular.
+Indeed, it is known [1] that extremal spacetimes do not have any tr apped surface
+inside the event horizon (itself usually a marginal outer trapped sur face). This makes use
+of the fact, first pointed out in [6], that the proper distance betwe en the event and Cauchy
+horizons in the extremal geometry (in the RN case, for instance) is actually infinite, even
+though the coordinate distance vanishes. This is certainly not the s ituation in the non-
+extremal situation where the proper distance between the inner a nd outer horizons is
+finite, as is the coordinate distance. The continued use of the extr emal limit of a generic
+spacetime asthe extremal geometry may not thus be above suspicion.
+2One aspect that has not been studied in detail is the behaviour of ge odesics, in the
+exterior of such spacetimes, both in the near-extremal and extr emal cases. The interest
+here is in what happens to the geodesics near the horizon. There is a class of timelike
+geodesics in spherically symmetric spacetimes like Schwarzschild and R N, which encircle
+the event horizon in a stable orbit (the so-called ISCOs). There is ho wever no stable null
+geodesic orbit for any generic spacetime. Since our interest is in the behaviour of circular
+null orbits near the event horizon of an extremal RN black hole, the use of the Kruskal-
+Szekeres extension of such spacetimes becomes crucial. Already a t this level, there seems
+to be a discontinuity in the extremal limit of the Kruskal-Szekeres ex tension of a generic
+RN spacetime, vis-a-vis the extension of the extremal RN spacetim e. It further ensues
+that circular null geodesics near the event horizon of an extremal RN spacetime exhibit
+a behaviour quite different from that in near extremal situations. I n what follows, we
+consider the behaviour of such orbits in some detail for near-extr emal and extremal RN
+spacetimes.
+A further motivation for the work comes from Hawking radiation, wh ich is known
+to be absent for the extremal spacetime, as the surface gravity on the event horizon
+which measures the equilibrium temperature for the thermal distrib ution of the radiation
+vanishes in this case. Once again, this thermal state cannot be ach ieved in a continuous
+manner from a radiant black hole (however weakly) without violating e nergy conditions
+[7]. The behaviour of circular orbits is relevant to this in order to asce rtain what really
+happens at extremality. This is animportant issue for rotating black holes for which black
+hole radiance also includes superradiance in addition to Hawking radiation. While we do
+not consider rotating black holes in this paper, these issues serve a s motivation for the
+present work.
+The plan of the paper is as follows: in section 2 we exhibit the Kruskal-S zekeres
+extension for non-extremal RN spacetime. We show explicitly how th e extremal limit of
+this extension is singular, implying that extremal RN geometry has to be thus extended
+directly (e.g., ´ a la Carter [9]), instead of by a limiting procedure on the extension of
+the non-extremal geometry. In section 3 we calculate the radial lo cation of a possible
+circular orbit for extremal RN spacetime, and compare this with the radial location of
+circular orbits in near-extremal RN black holes. We point out that there is a stable
+circular null orbit on the event horizon in the extremal spacetime, which disappears in
+the near extremal geometry. This is a key result of our analysis. Th e concluding section
+(4) includes a discussion of our results in the light of trapped surfac es and also presents
+our future outlook.
+2 Kruskal-Szekeres extension of RN spacetime
+The Kruskal-extended RN spacetime was first worked out by Carte r [9] who gave the
+extended geometry only for the extremal case. Here we consider first the extension of the
+3non-extremal or generic RN spacetime. While the procedure is now stan dard textbook
+material, we have not been able to locate in the literature an adequat e discussion of the
+pathologies in this extension, that manifest in the extremal limit r+→r−. The singular
+behaviour discerned here bears an unmistakable stamp of the subt leties of the extremal
+limit. In other words, we show that the extremal limit of the Kruskal extension is notthe
+same as the Kruskal-extended extremal RN spacetime found in [9].
+2.1 Non-extremal case
+We begin by defining ‘tortoise’ coordinates, and then use that to de rive the Kruskal
+extension. The tortoise coordinate r∗is given by
+dr∗=r2dr
+∆=r2dr
+(r−r−)(r−r+). (5)
+Integrating this equation, we obtain
+r∗=r+r2
++
+(r+−r−)ln|r−r+|−r2
+−
+(r+−r−)ln|r−r−|+c (6)
+where as usual r±=M±/radicalbig
+M2−Q2andcis an integration constant. The outer horizon
+r+is an event horizon and the inner horizon r−is a Cauchy horizon. Now, near the event
+horizonr=r+, the tortoise coordinate is given by
+r∗≈r2
++
+(r+−r−)ln|r−r+| (7)
+Herer∗has logarithmic dependence on r−r+and is singular at r=r+. Introducing the
+radial null coordinates uandv, given by u=t−r∗,v=t+r∗, we observe that the surface
+r=r+appears at v−u=−∞. This implies that we need another transformation, which
+is the Kruskal-Szekeres extension. The null coordinates
+du=dt−dr∗=dt−r2dr
+∆(8)
+dv=dt+dr∗=dt+r2dr
+∆, (9)
+sincedr∗=r2dr
+∆, whereu=const.are outgoing radial null geodesics and v=const.
+ingoing radial null geodesics respectively. The RN metric now assume s the form
+ds2=−∆
+r2(dt2−dr∗2)+r2dΩ2
+=−∆
+r2dudv+r2dΩ2(10)
+4wheredudv=dt2−dr∗2.
+The Kruskal-Szekeres frame is now defined by the transformation s
+U+=−exp(−κ+u) (11)
+V+= exp(κ+v) (12)
+whereκ±=r±−r∓
+2r2
+±is the surface gravity of the null hypersurfaces.
+Therefore the metric, near r=r+, becomes
+ds2=−r+r−
+κ2
++exp(−2κ+r)
+r2(r−
+r−r−)κ+/κ−−1dU+dV++r2
++dΩ2(13)
+where
+U+V+=−exp(2κ+r)(r−r+
+r+)(r−r−
+r−)κ+
+κ− (14)
+This implies that, when in terms of the Kruskal coordinates ( U+,V+), the metric is well
+behaved at the event horizon ( r=r+) but is singular at the inner (Cauchy) horizon
+(r=r−). From this we can conclude that these coordinates ( U+,V+) are valid for the
+region (r−< r < r +) which is different from the original patch covering the region r > r+.
+These coordinates do not cover r≤r−because of the singularity at r=r−, so another
+new coordinate patch is required to cover this region. In this region gtt>0 andgrr<0
+such that tis spacelike and ris timelike. Note that in the above metric if we take the
+extremal limit r−→r+then the metric diverges, proving that the extremal limit is not
+continuous insofar as this extension is concerned. This follows from the fact that the
+coordinate chart U+,V+considered here does not extend to the Cauchy horizon.
+Now consider what happens for the Cauchy horizon r=r−. Near the Cauchy horizon
+r=r−, the tortoise coordinates is given by
+r∗≈r2
+−
+(r+−r−)ln|r−r−| (15)
+Herer∗has a logarithmic singularity at r=r−. The radial null coordinates uandvare
+u=t−r∗,v=t+r∗, then the surface r=r−appears at v−u= +∞. Therefore the
+Kruskal-Szekeres transformations are
+U−=−exp(−κ−u) (16)
+V−= exp(κ−v) (17)
+whereκ−has been previously defined. Therefore near r=r−, the metric becomes
+ds2=−r+r−
+κ2
+−exp(−2κ−r)
+r2(r+
+r−r+)κ−/κ+−1dU−dV−+r2
+−dΩ2(18)
+5where
+U−V−=−exp(2κ−r)(r−r−
+r−)(r−r+
+r+)κ−
+κ+ (19)
+This implies that, when expressed in terms of Kruskal coordinates ( U−,V−), the metric
+is regular at the Cauchy (inner) horizon ( r=r−) but singular at the event horizon
+(r=r+). From this we can conclude that these coordinates ( U−,V−) are valid for the
+region (0 < r < r −). Once again, the extended metric is singular in the extremal limit,
+thwarting any attempt to derive the Kruskal extension of the ext remal spacetime by a
+straightforward limiting procedure on the non-extremal extende d geometry.
+2.2 Extremal case
+Now we want to see what happens for the extremal case. The tort oise coordinate is given
+by
+r∗=/integraldisplayr2dr
+(r−M)2=r+2M[ln|r−M|−M
+2(r−M)]. (20)
+Near the horizon this becomes
+r∗≈M2
+(r−M)(21)
+Herer∗has no logarithmic dependence, but an extra pole : M2/(r−M); it is singular
+atr=M. Introducing the double null coordinates uandvasu=t−r∗,v=t+r∗, the
+surfacer=r+appears at v−u=∞, hence these coordinates are inappropriate there.
+We need another transformation as in the previous generic cases.
+The metric in terms of double null coordinates uandvis given by
+ds2=−(1−M/r)2dudv+r2(u,v)dΩ2(22)
+Now from eqn. (20)
+exp(αr∗) = exp(αr)(r−M)2exp(−M/(r−M)), α= 1/M (23)
+The maximally extended spacetime can be obtained by substituting
+tanU=−exp(−αu) =−exp(−αt)[exp(αr)(r−M)2exp(−M/(r−M))]
+tanV= +exp(+ αv) = exp(αt)[exp(αr)(r−M)2exp(−M/(r−M))] (24)
+Therefore the complete extremal RN metric in Kruskal-Szekeres c oordinate system is
+given by (after substituting the value of α= 1/M)
+ds2=−4M2(1−M
+r)2csc2Ucsc2VdUdV+r2dΩ2
+tanUtanV=−exp(2r/M)(r−M)4exp(−2M/(r−M)) (25)
+This is the result of Carter [9].
+63 Circular Orbits in RN spacetime
+3.1 Extremal spacetime
+As is well-known, The RN spacetime has a timelike Killing vector ξ≡∂twhose projection
+along the 4-velocity u(u2=−1 for timelike and u2= 0 for null) of geodesics: ξ·u=−E,
+is conserved along such geodesics. Recall that the timelike Killing vect or becomes null on
+the event horizon. There is also the ‘angular momentum’ L≡ζ·u(whereζ≡∂φ) which
+is similarly conserved. It is straightforward to show that (see, e.g., [8]) for circular null
+orbits (r(U,V) =R),Eobeys the equation
+E2=Ueff(R) =/parenleftbiggL2
+R2/parenrightbigg /parenleftbigg
+1−M
+R/parenrightbigg2
+, (26)
+whereVeff(R) is called the effective potential. For timelike orbits, the correspond ing
+effective potential is given by
+E2=Veff(R) =/parenleftbigg
+1+L2
+R2/parenrightbigg /parenleftbigg
+1−M
+R/parenrightbigg2
+. (27)
+If this effective potential has an absolute minimum, this is identified wit h the radial
+location of a stablecircular orbit. Alternatively, other extrema are identified as radial
+locations of unstable circular orbits.
+It is convenient to define dimensionless quantities r≡R/M , ℓ ≡L/Mandq≡Q/M.
+Thus, the extremal case corresponds to q= 1.
+3.1.1 Null orbits
+In this notation, the effective potential becomes
+Ueff(r) =ℓ2
+r2/parenleftbigg
+1−1
+r/parenrightbigg2
+. (28)
+SettingU′
+eff(r) = 0 one obtains
+2ℓ2
+r3/parenleftbigg
+1−1
+r/parenrightbigg/parenleftbigg2
+r−1/parenrightbigg
+= 0. (29)
+which has the solutions as circular orbits at r= 1,2.
+It is easy to check that U′′
+eff(1)>0 which corresponds to a global minimum of Ueff.
+This can be taken to correspond to a stable circular orbit for a phot on withE= 0. On
+the other hand, U′′
+eff(2)<0, implying that there is no other stable circular photon orbit.
+Thus, there is a null stable circular orbit onthe event horizon of an extremal RN
+spacetime, a feature which appears to be novel. We have not seen a ny discussion of this
+7type of orbit in extremal RN spacetime anywhere in the literature. T his circular geodesic
+can be seen to coincide with the null geodesic generator of the even t horizon. The real
+novelty here is that it appears as a global minimum of the effective pot ential. While a null
+geodesic generator of the event horizon is not thought of as a sta ble circular orbit, since
+there is no such orbit in generic black hole spacetimes (derived as an a bsolute minimum
+of the effective potential), we would like to point out that the extrem al RN is special in
+this sense. However, we expect similar orbits to exist in other extre mal spacetimes as
+well. Before considering whether such an orbit exists in the nearextremal spacetime, we
+record an interesting finding for timelike orbits.
+3.1.2 Timelike orbits
+The effective potential in this case is given by
+Veff(r) =/parenleftbigg
+1+ℓ2
+r2/parenrightbigg /parenleftbigg
+1−1
+r/parenrightbigg2
+. (30)
+SettingV′
+eff(r) = 0 one obtains
+r2(r−1)−ℓ2(r−2)(r−1) = 0, (31)
+2ℓ2
+r3/parenleftbigg
+1−1
+r/parenrightbigg/parenleftbigg2
+r−1/parenrightbigg
+= 0. (32)
+The solutions are given by
+r0= 1
+r±=ℓ2
+2/bracketleftBigg
+1±/parenleftbigg
+1−8
+ℓ2/parenrightbigg1/2/bracketrightBigg
+(33)
+It is easy to check that the circular orbit with r=r0= 1indeed corresponds to the
+likely position of a stable circular orbit , being a stable global minimum of Veffgiven in
+eq. (30), since V′′
+eff(1)>0. This orbit is located exactly as the null orbit found above
+at the event horizon, and must be the same orbit, even though it sh ows up as a global
+minimum of the potential for timelike orbits. The likely reason is that th e Killing vector
+field which is timelike everywhere in the exterior RN spacetime, turns n ull on the event
+horizon.
+Of the two other orbits at r=r±, clearly one must choose ℓ2≥8 for them to be
+real; we choose ℓ2= 9 for simplicity and get r+= 6,r−= 3. One obtains V′′
+eff(6)>0,
+showing that the orbit with radius r+is a stable localminimum, possibly the ISCO, while
+V′′
+eff(3)<0, so that the orbit with radius r−is unstable.
+The issue is now: what happens to the orbit on the event horizon in th e near extremal
+case, when we move infinitesimally away from extremality in the black ho le parameter
+space. To this we turn in the next subsection.
+83.2 Near extremal RN spacetime
+3.2.1 Null Orbits
+The effective potential for null circular geodesics in generic non-ex tremal RN spacetime
+is determined exactly as in the last subsection. In terms of the dimen sionless quantities
+introduced in that section, this effective potential can be express ed as
+Ueff(r) =ℓ2
+r2∆(r) (34)
+where,
+∆(r)≡1−2
+r+q2
+r2. (35)
+We solve the extremization equation U′
+eff(r) = 0 perturbatively around the extremal
+solutions satisfying eq. (29), corresponding to the near extremal geometry. To this end,
+we define χ≡1−q >0, ρ≡r−1, where, recall that for the extremal situation q= 1,
+the stable circular orbit is at r=r0= 1. In our chosen units, the event horizon is located
+atr+= 1+(2χ)1/2+O(χ3/2) while the Cauchy horizon is at r−= 1−(2χ)1/2+O(χ3/2).
+The extremization equation equation is to be solved perturbatively a round the extremal
+solution to yield ρtolinearorder in the perturbation χ. Likewise, terms of O(ρ2) or
+higher are ignored. In terms of the linear perturbation χ, the effective potential for null
+circular orbits is given by
+Ueff(r) =ℓ2
+r2/bracketleftBigg/parenleftbigg
+1−1
+r/parenrightbigg2
+−2χ
+r2/bracketrightBigg
+. (36)
+The extremization equation U′
+eff(r) = 0 yields the quadratic equation
+r2−3r+2(2−χ) = 0. (37)
+leading to circular orbits with radii r= 1−4χ ,2+4χ. It is easy to show that U′′
+eff(r)<0
+for both these radii, demonstrating that the near extremal spac etime does not admit any
+stable null circular orbit at all. This is merely a rephrasing of what is well- known in the
+literature [8] : the generic RN spacetime does not admit a stable null c ircular orbit, just
+like the Schwarzschild spacetime. The excercise however underlines the key point we wish
+to make in this paper : the stable circular null geodesic on the event horizon prese nt in
+the extremal case as the global minimum of the effective poten tial, is absent in the near
+extremal case . This adds to the list of disparities between the extremal limit of the g eneric
+RN spacetime and the actual extremal spacetime.
+For completeness, we include a treatment for timelike orbits in the ne ar extremal
+geometry.
+93.2.2 Timelike orbits
+In terms of the dimensionless quantities introduced in that section, this effective potential
+can be expressed as
+Veff(r) =/parenleftbigg
+1+ℓ2
+r2/parenrightbigg
+∆(r) (38)
+where, ∆( r) is given above in (35). The extremization condition V′
+eff(r) = 0 leads to the
+cubic equation
+r3−(q2+ℓ2)r2+3ℓ2r−2q2ℓ2= 0. (39)
+It is sufficient for us to solve eq. (39) in the near extremal approximation, to com-
+pare the results with those in the extremal situation. One gets, to linear order in the
+perturbation
+ρ=−2χ/parenleftbigg1+2ℓ2
+1+ℓ2/parenrightbigg
++O(χ2), (40)
+leading to a circular orbit with radius
+rnex
+0≃1−2χ/parenleftbigg1+2ℓ2
+ℓ2+1/parenrightbigg
+< r+! (41)
+Thus, the stable circular orbit at r0= 1 found for the extremal spacetime has no analogue
+in the near-extremal situation.
+To find a stable circular orbit in this case, we proceed as before. Per turbing around
+the unstable orbit at r−= (1/2)ℓ2[1−(1−8/ℓ2)1/2] we obtain the perturbation
+ρ−=−2χ/bracketleftbiggℓ2r−
+(ℓ2−2)r−−3ℓ2/bracketrightbigg
+. (42)
+It is obvious that with ℓ2>8 one gets ρ−<0, this orbit does indeed correspond to an
+unstable circular orbit in the near extremal spacetime. In fact, fo rℓ2= 9, ρ−= 9χ. On
+the other hand, perturbation around the stable circular orbit at r+= (1/2)ℓ2[1 + (1−
+8/ℓ2)1/2] in the extremal case, leads to a perturbed orbit with
+ρ+=−2χ/bracketleftbiggℓ2r+
+(ℓ2−2)r+−3ℓ2/bracketrightbigg
+. (43)
+It is not difficult to see that for ℓ2>8 we have ρ+>0. In fact, for ℓ2= 9,ρ+=−(36/5)χ.
+This does indeed correspond to a stable circular orbit in the near ext remal spacetime,
+indeed it is the ISCO in this case.
+104 Discussion
+The class of stable circular orbits on the event horizon of the extre mal RN spacetime,
+discerned above, coincides with the null geodesic generator of the horizon. However,
+what is interesting is that this happens only in the extremal spacetim e. For the near
+extremal geometry, the effective potential has no global minimum corresponding to such
+a geodesic. Indeed, neither for the Schwarzschild nor for any oth er generic spherically
+symmetric spacetime is the null geodesic generator on the event ho rizon represented by a
+stablecircular orbit onthe horizon, characterized by an absolute minimum of the effective
+potential for circular orbits. The fact that such a class of orbits is absent even in the near
+extremal geometry is another illustration of the subtlety associat ed with the extremal
+limit. This can be seen to have arisen from the absence of outer trappe d surfaces within
+the horizon in the extremal geometry in contrast to a more generic situation.
+4.1 Absence of trapped surfaces in extremal RN spacetime
+Inageneral spacetime ( M,gµν)withthemetric gµνhavingsignature( −+++), onedefines
+two future directed null vectors lµandnµwhose expansion scalars are given by
+θ(l)=qµν∇µlνθ(n)=qµν∇µnν. (44)
+whereqµν=gµν+lµnν+nµlνisthemetric induced by gµνonthetwo dimensional spacelike
+surface formed by spatial foliation of the null hypersurface gene rated by lµandnµ.
+Then (i) a two dimensional spacelike surface S is said to be a trappedsurface if both
+θ(l)<0 andθ(n)<0; (ii) S is to be marginally trapped surface if one of two null expansions
+vanish i.e. θ(l)= 0 orθ(n)= 0. The null vectors for non-extremal RN black hole are given
+by
+lµ=1
+∆(r2,−∆,0,0)nµ=1
+2r2(r2,∆,0,0) (45)
+lµ=1
+∆(−∆,−r2,0,0)nµ=1
+2r2(−∆,r2,0,0) (46)
+where ∆ = ( r−r+)(r−r−) andr±=M±/radicalbig
+M2−Q2. The null vectors satisfy the
+following conditions :
+lµnµ=−1lµlµ= 0nµnµ= 0 (47)
+Using (44), one obtains
+θ(l)=−2
+rθ(n)=(r−r+)(r−r−)
+r3(48)
+11In the region ( r−< r < r +),θ(l)<0 andθ(n)<0. This implies that trapped surfaces
+exist for non extreme RN black hole in this region. In contrast, for t he extreme RN black
+hole
+θ(l)=−2
+rθ(n)=(r−M)2
+r3(49)
+Here inside or outside extremal horizon r < Morr > M,θ(l)<0 andθ(n)>0. This
+implies that there are no trapped surfaces for extremal RN black h ole beyond the event
+horizon .
+4.2 Outlook
+Ouranalysisreinforcesearlierassertionsintheliterature[4],[5]tha ttheextremallimitofa
+generic charged black hole is not necessarily the extremal black hole spacetime. However,
+recent work [10] has conclusively shown that extremal black holes c an be modeled by
+isolated horizons on par with generic non-extremal black holes. One expects this to le ad
+to a well-defined microcanonical entropy obtained for extremal ma croscopic black holes as
+aninfinite series in horizonarea, with theleading Bekenstein-Hawking areaterm receiving
+precise subleading logarithmic and power law corrections [11], for sp herically symmetric
+horizons, just like more generic black holes. Notethat this isgenuine gravitational entropy
+of extremal black holes, and has little to do with entanglement or suc h non-gravitational
+phenomena. This understanding of extremal black hole entropy, b ased on Loop Quantum
+Gravity, ought to find extensive applications in superstring theore tic black holes in four
+dimensional spacetime [12]. The object now is to discern whether this approach works for
+rotating black holes with a similar degree of precision. Apart from que stions pertaining
+to geodesics and circular orbits in extremal Kerr and Kerr-Newman spacetimes, there is
+the issue of stability of such spacetimes with respect to superradiance . Isolated horizons
+are not expected to be very useful in this respect, since the prop erties of the ergosphere
+play a crucial role for superradiance. The aim in future then ought t o be a study of
+perturbations of extremal Kerr black holes with regard to superr adiance.
+Acknowledgment : We thank R. Basu, A. Chatterjee and A. Ghosh for helpful discus-
+sions.
+References
+[1] I. Racz, Class. Quant. Grav. 17(2000) 4353; arXiv:gr-qc/0009049.
+[2] R. M. Wald, Phys. Rev. D 48(1993) 3427; arXiv:gr-qc/9307028. See also, V. Iyer
+and R. M. Wald, Phys. Rev. D 50(1994) 846; arXiv:gr-qc/9403028.
+[3] A. Sen, Int. J. Mod. Phys. A24(2009) 4225 and references therein;
+arXiv:0809.3304v2.
+12[4] S. Das, A. Dasgupta and P. Ramadevi, Mod. Phys. Lett. A12(1997) 3067;
+arXiv:9608162.
+[5] S. M. Carroll, M. C. Johnson, L. Randall, JHEP0911(2009) 109; arXiv:0901.0931.
+[6] J. D. Bekenstein, Phys. Rev. D 7(1973) 2333.
+[7] W. Israel, Phys. Rev. Lett. 57(1986) 397.
+[8] S. Chandrasekhar, The Mathematical Theory of Black Holes , Clarendon Press, Ox-
+ford (1983).
+[9] B. Carter, Phys. Lett. 21(1966) 423.
+[10] A. Chatterjee and A. Ghosh, Phys. Rev. D80(2009) 064036.
+[11] R. K. Kaul and P. Majumdar, Phys. Rev. Lett. 84(2000) 5255; arXiv:gr-qc/0002040.
+[12] A. Chatterjee, Entropy of Black Holes in N= 2Supergravity , arXiv:0907.1556 [gr-
+qc].
+13
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+arXiv:1001.0360v1  [math.GT]  3 Jan 2010An equivalence between the set of graph-knots
+and the set of homotopy classes of looped
+graphs
+D.P. Ilyutko∗
+Abstract
+In the present paper we construct a one-to-one corresponden ce
+between the set of graph-knots and the set of homotopy classe s of
+looped graphs. Moreover, the graph-knot and the homotopy cl ass
+constructed from a given knot are related with this correspo ndence.
+This correspondence is given by a simple formula.
+1 Introduction
+The discovery of virtual knot theory by Kauffman [Kau] in mid 19 90s
+was an important step in generalising combinatorial and top ological
+knot theoretical techniques into a larger domain (knots in t hickened
+surfaces), which is an important step towards generalisati on of these
+techniques for knots in arbitrary manifolds.
+It turned out that some invariants (Kauffman bracket polynomi al)
+can be generalised for virtual knots immediately [Kau], and some
+other theories (Khovanov homology theory) need a complete r evision
+of the original construction for a generalisation for the ca se of vir-
+tual knots [Ma1]. Virtual knots also sharpened several prob lems and
+elicited some phenomena which do not appear in the classical knot
+theory [FKM], e.g., the existence of a virtual knot with non- trivial
+∗Partially supported by grants of RF President NSh – 660.2008.1, RFB R 07–01–00648,
+RNP 2.1.1.3704, the Federal Agency for Education NK-421P/108.
+12
+12
+3
+11
+1 11
+22
+222
+3
+3
+333
+Figure 1: A Gauss Circuit; A Rotating Circuit
+Jones polynomial and trivial fundamental group emphasises the diffi-
+culty of extracting the Jones polynomial information out of the knot
+group.
+In the present paper, we consider two new theories: graph-links in-
+troduced in [IM1, IM2] and homotopy classes of looped interlacement
+graphsintroduced by L. Traldi and L. Zulli [TZ]. The two theories ar e
+closely connected to both classical and virtual knot theori es and, in
+somesense, are generalisations of virtual knots. Likewise virtual knots
+appear out of non-realisable Gauss code and thus generalise classical
+knots (which have realisable Gauss codes), graphs-links an d looped
+graphs come out of intersection graphs : we may consider graphs which
+realise chord diagrams, and, in turn, virtual knots, and pas s to arbi-
+trarygraphswhichcorrespondtosomemysteriousobjectsge neralising
+knots and virtual knots.
+There is a way of coding all virtual knots by Gauss diagrams
+and there is another way of coding all virtual links by rotating cir-
+cuits(see, e.g., [Ma2]), see Fig. 1. Looped graphs are a generalis a-
+tion of intersection graphs of chord diagrams constructed b y using
+the first way of coding and graph-links come out from consider ing a
+rotating circuit. In low-dimensional topology both approa ches, the
+Gauss circuit approach and the rotating circuit approach, a re very
+widely used. The Gauss circuit approach is applied in knot th eory,
+namely in the construction of finite-type invariants, Vassi liev invari-
+ants [BN, GPV, CDL], and in the planarity problem of immersed
+curves, [CE1, CE2, RR]. However, for detecting planarity of a framed
+4-graph it is more convenient to use the rotating circuit app roach,3
+Figure 2: Non-Realisable Bouchet Graphs
+see [Ma3, Ma4]. The criterion of the planarity of an immersed curve,
+which is the framed 4-graph, is formulated very easy: an imme rsed
+curve is planar if and only if the chord diagram obtained from a ro-
+tating circuit is a d-diagram , i.e. the set of all the chords can be split
+into two sets and the chords from one set do not intersect each other,
+see [Ma5]. If we want to generalise the planarity problem to t he prob-
+lem of finding the minimum genus of a closed surface which a giv en
+curve can be immersed in, the rotating circuit approach is al so more
+useful. There are criteria giving us the answer to the questi on what
+is the minimum genus for a given curve, see [Ma3].
+In spite of the fact that graph-links and the homotopy classe s of
+looped graphs are abstract objects the Kauffman bracket polyn omial
+and the Jones polynomial were constructed for them, see [IM1 , IM2,
+TZ]. But the first question which has arisen after the constru ctions of
+these two theories was whether or not in each graph-link (eac h homo-
+topy class of looped graphs) there exists a ‘realisable’ rep resentative,
+i.e. a graph which is the intersection graph of a chord diagra m. Some
+graphs can not be represented by chord diagrams at all [Bou], see
+Fig. 2. The answer to this question is negative and this probl em was
+first solved by V.O. Manturov in [Ma6, Ma7] for homotopy class es of
+looped graphs. The second question is whether there is an equ ivalence
+between the two theories. In the present paper we give an expl icit for-
+mula which gives us an equivalence between the set of graph-k nots
+(graph-links with one component) and the set of homotopy cla sses of
+looped graphs. Moreover, under this equivalence ‘realisab le’ objects
+correspond to ‘realisable’ ones. Therefore, by using this e quivalence
+we can give the negative answer to the question of existing a r ealisable
+representative for graph-knots.4
+Acknowledgments
+The author is grateful to V.O. Manturov, A.T. Fomenko, D.M. A fa-
+nasiev, I.M. Nikonov for their interest to this work and to L. Traldi
+for his discussion of [TZ] and useful comments.
+2 Basic Definitions and Constructions
+2.1 Chord diagrams and Framed 4-Graphs
+Throughout the paper all graphs are finite. Let Gbe a graph with the
+set of vertices V(G) and the set of edges E(G). We think of an edge
+as an equivalence class of the two half-edges. We say that a ve rtex
+v∈V(G) hasdegreekifvis incident to khalf-edges. A graph whose
+vertices have the same degree kis calledk-valentor ak-graph. The
+free loop, i.e. the graph without vertices, is also consider ed ask-graph
+for anyk.
+Definition 2.1. A 4-graph is framedif for every vertex the four
+emanating half-edges are split into two pairs of (formally) opposite
+edges. The edges from one pair are called opposite to each other .
+Avirtual diagram is a framed 4-graph embedded into R2where
+eachcrossingiseitherendowedwithaclassical crossingst ructure(with
+a choice for underpassand overpass specified) or just said to bevirtual
+and marked by a circle. A virtual link is an equivalence class of virtual
+diagrams modulo generalised Reidemeister moves. The latte r consist
+of usual Reidemeister moves referring to classical crossin gs and the
+detour move that replaces one arc containing only virtual intersection s
+and self-intersections by another arc of such sort in any oth er place
+of the plane, see Fig. 3. A projection of a virtual diagram is a framed
+4-graph obtained from the diagram by considering classical crossings
+as vertices and virtual crossings are just intersection poi nts of images
+of different edges. A virtual diagram is connected if its projection is
+connected. Withoutlossofgenerality, all virtual diagrams are assumed
+to be connected and contain at least one classical crossing [IM1, IM2].
+Definition 2.2. Achord diagram is a cubic graph consisting of a se-
+lected cycle (the circle) and several non-oriented edges ( chords) con-
+necting points on the circle in such a way that every point on t he5
+AB
+AB
+Figure 3: The detour move
+Figure 4: Virtualisation
+circle is incident to at most one chord. A chord diagram is labeled
+if every chord is endowed with a label ( a,α), wherea∈ {0,1}is the
+framingof the chord, and α∈ {±}is the sign of the chord. If no labels
+are indicated, we assume the chord diagram has all chords wit h label
+(0,+). Two chords of a chord diagram are called linkedif the ends of
+one chord lie in different connected components of the circle w ith the
+end-points of the second chord removed.
+Definition 2.3. By avirtualisation of a classical crossing of a virtual
+diagram we mean a local transformation shown in Fig. 4.
+Having a labeled chord diagram D, onecan construct a virtual link
+diagramK(D) (up to virtualisation) as follows. Let us immerse this
+diagram in R2by taking an embedding of the circle and placing some
+chords inside the circle and the other ones outside the circl e. After
+that we remove neighbourhoods of each of the chord ends and re place
+them by a pair of lines (connecting four points on the circle w hich are
+obtained after removing two neighbourhoods) with a classic al crossing
+if the chord is framed by 0 and a couple of lines with a classica l
+crossing and a virtual crossing if the chord is framed by 1 in t he6
+1
+234
+567
+891012
+1311
+1413
+245
+6
+7
+8
+9
+10
+11 121314
+:2-8,3-14,4-6,7-10,
+11-13:1-12,5-9
+Figure 5: Rotating Circuit Shown by a Thick Line; Chord Diagram
+following way. The choice for underpass and overpass is spec ified as
+follows. A crossing can be smoothed in two ways: AandBas in the
+Kauffman bracket polynomial; we require that the initial piec e of the
+circle corresponds to the A-smoothing if the chord is positive and to
+theB-smoothing if it is negative: A:→,B:→.
+Conversely, having a connected virtual diagram K, one can get a
+labeled chord diagram DC(K), see Fig. 5. Indeed, one takes a circuit
+CofKwhich is a map from S1to the projection of K. This map
+is bijective outside classical and virtual crossings, has e xactly two
+preimages at each classical and virtual crossing, goes tran sversally at
+each virtual crossing and turns from an half-edge to an adjac ent (non-
+opposite) half-edge at each classical crossing. Connectin g the two
+preimages of a classical crossing by a chord we get a chord dia gram,
+where the sign of the chord is + if the circuit locally agrees w ith
+theA-smoothing, and −if it agrees with the B-smoothing, and the
+framing of a chord is 0 (resp., 1) if two opposite half-edges h ave the
+opposite (resp., the same) orientation. It can be easily che cked that
+this operation is indeed inverse to the operation of constru cting a
+virtual link out of a chord diagram: if we take a chord diagram D,
+and construct a virtual diagram K(D) out of it, then for some circuit
+Cthe chord diagram DC(K(D)) will coincide with D. The rule for
+setting classical crossings here agrees with the rule descr ibed above.
+This proves the following
+Theorem 2.1 ([Ma5]).For any connected virtual diagram Lthere is
+a certain labeled chord diagram Dsuch thatL=K(D).7
+The Reidemeister moves on virtual diagrams generate the Rei de-
+meister moves on labeled chord diagrams [IM1, IM2].
+2.2 Reidemeister Moves for Looped Interlace-
+ment Graphs and Graph-links
+Now we are describing moves on graphs obtained from virtual d ia-
+grams by using rotating circuit [IM1, IM2] and the Gauss circ uit [TZ].
+These moves in both cases will correspond to the “real” Reide meis-
+ter moves on diagrams. Then we shall extend these moves to all
+graphs (not only to realisable ones). As a result we get new ob jects,
+agraph-link and ahomotopy class of looped interlacement graphs , in
+a way similar to the generalisation of classical knots to vir tual knots:
+the passage from realisable Gauss diagrams (classical knot s) to arbi-
+trary chord diagrams leads to the concept of a virtual knot, a nd the
+passage from realisable (by means of chord diagrams) graphs to ar-
+bitrary graphs leads to the concept of two new objects: graph -links
+and homotopy classes of looped interlacement graphs (here ‘ looped’
+corresponds to the writher number, if the writher number is - 1 then
+the corresponding vertex has a loop). To construct the first o bject we
+shall use simple labeled graphs, and for the second one we sha ll use
+(unlabeled) graphs without multiple edges, but loops are al lowed.
+Definition 2.4. A graph is labeledif every vertex vof it is endowed
+with a pair ( a,α), wherea∈ {0,1}is the framing of v, andα∈ {±}
+is the sign of v. LetDbe a labeled chord diagram D. Thelabeled
+intersection graph , cf. [CDL], G(D) ofDis the labeled graph: 1)
+whose vertices are in one-to-one correspondence with chord s ofD, 2)
+the label of each vertex corresponding to a chord coincides w ith that
+of the chord, and 3) two vertices are connected by an edge if an d only
+if the corresponding chords are linked.
+Definition 2.5. A simple (labeled) graph His called realisable if
+there is a (labeled) chord diagram Dsuch thatH=G(D).
+The following lemma is evident.
+Lemma 2.1. A simple (labeled)graph is realisable if and only if each
+its connected component is realisable.8
+u v
+LC(G;u)
+G
+P(G;u,v)u v
+u v
+Figure 6: Local Complementation and Pivot
+Definition 2.6. LetGbe a graph and let v∈V(G). The set of all
+vertices adjacent to vis called the neighbourhood of a vertex vand
+denoted by N(v) orNG(v).
+Let us define two operations on simple unlabeled graphs.
+Definition 2.7. (Local Complementation) Let Gbe a graph. The
+local complementation ofGatv∈V(G) is the operation which toggles
+adjacencies between a,b∈N(v),a/\e}atio\slash=b, and doesn’t change the rest of
+G. Denote the graph obtained from Gby the local complementation
+at a vertex vby LC(G;v).
+Definition 2.8. (Pivot) Let Gbeagraphwithdistinctvertices uand
+v. Thepivoting operation of a graph Gatuandvis the operation
+which toggles adjacencies between x, ysuch thatx, y /∈ {u,v},x∈
+N(u), y∈N(v) and either x /∈N(v) ory /∈N(u), and doesn’t change
+the rest of G. Denote the graph obtained from Gby the pivoting
+operation at vertices uandvby piv(G;u,v).
+Example. In Fig. 6 the graphs G, LC(G;u) and piv(G;u,v) are de-
+picted.
+The following lemma can be easily checked.
+Lemma 2.2. Ifuandvare adjacent then there is an isomorphism
+piv(G;u,v)∼=LC(LC(LC( G;u);v);u).9
+Let us define graph-moves by considering intersection graph s of
+chord diagrams constructed by using a rotating circuit , and these
+moves correspond to the Reidemeister moves on virtual diagr ams. As
+a result we obtain a new object — an equivalence class of label ed
+graphs under formal moves. These moves were defined in [IM1, I M2].
+Definition 2.9. Ωg1. The first Reidemeister graph-move is an addi-
+tion/removal of an isolated vertex labeled (0 ,α),α∈ {±}.
+Ωg2. The second Reidemeister graph-move is an addition/remov al
+of two non-adjacent (resp., adjacent) vertices having (0 ,±α) (resp.,
+(1,±α)) and the same adjacencies with other vertices.
+Ωg3. The third Reidemeister graph-move is defined as follows.
+Letu, v, wbe three vertices of Gall having label (0 ,−) so thatuis
+adjacent only to vandwinG. Then we only change the adjacency
+ofuwith the vertices t∈N(v)\N(w)/uniontextN(w)\N(v) (for other pairs
+of vertices we do not change their adjacency). In addition, w e switch
+the signs of vandwto +. The inverse operation is also called the
+third Reidemeister graph-move.
+Ωg4. The fourth graph-move for Gis defined as follows. We take
+two adjacent vertices uandvlabeled (0,α) and (0,β) respectively.
+ReplaceGwith piv(G;u,v) and change signs of uandvso that the
+sign ofubecomes −βand the sign of vbecomes −α.
+Ωg4′. In this fourth graph-move we take a vertex vwith the label
+(1,α). Replace Gwith LC(G;v) and change the sign of vand the
+framing for each u∈N(v).
+The comparison of the graph-moves with the Reidemeister mov es
+yields the following theorem.
+Theorem 2.2. LetK1andK2be two connected virtual diagrams, and
+letG1andG2be two labeled intersection graphs obtained from K1and
+K2, respectively. If K1andK2are equivalent in the class of connected
+diagrams then G1andG2are obtained from one another by a sequence
+ofΩg1−Ωg4′graph-moves.
+Definition 2.10. Agraph-link is an equivalence class of simple la-
+beled graphs modulo Ω g1−Ωg4′graph-moves.
+LetDG(K) be the Gauss diagram of a virtual diagram K. Let us
+constructthegraph obtained from theintersection graph of DG(K) by
+adding loops to vertices corresponding to chords with negat ive writhe10
+number [TZ]. We refer to this graph as the looped interlacement graph
+or thelooped graph . Let us construct the moves on graphs. These
+moves are similar to the moves for graph-links and also corre spond to
+the Reidemeister moves on virtual diagrams.
+Definition 2.11. Ω1. The first Reidemeister move for looped in-
+terlacement graphs is an addition/removal of an isolated lo oped or
+unlooped vertex.
+Ω2. ThesecondReidemeistermoveforloopedinterlacement g raphs
+is an addition/removal of two vertices having the same adjac encies
+with other vertices and, moreover, one of which is looped and the
+other one is unlooped.
+Ω3. The third Reidemeister move for looped interlacement gr aphs
+isdefinedasfollows. Let u, v, wbethreeverticessuchthat vislooped,
+wis unlooped, vandware adjacent, uis adjacent to neither vnorw,
+and every vertex x /∈ {u, v, w}is adjacent to either 0 or precisely two
+ofu, v, w. Then we only remove all three edges uv,uwandvw. The
+inverse operation is also called the third Reidemeister mov e.
+Remark 2.1. The two third Reidemeister moves do not exhaust all
+thepossibilities forrepresentingthethirdReidemeister move onGauss
+diagrams, see [TZ]. It can be shown that all the other version s of the
+third Reidemeister move, which involve toggling the non-lo op edges in
+one of the six pictured configurations in Figure 7 (every vert ex outside
+the picture must have either 0 or precisely 2 neighbours amon g the
+three vertices that are pictured), are combinations of the s econd and
+third Reidemeister moves, see [ ¨Ost] for details.
+Definition 2.12. We call an equivalence class of graphs (without
+multiple edges, but loops are allowed) modulo the three move s listed
+in Definition 2.11 a homotopy class of looped interlacement graphs.
+Remark 2.2. Looped interlacement graphs encode only knot dia-
+grams but graph-links can encode virtual diagrams with any n umber
+of components. The approach using a rotating circuit has an a dvan-
+tage in this sense. In [Tr] L. Traldi introduced the notion of a marked
+graph by considering any Euler tour (we have vertices which w e go
+transversally and in which we rotate). In the paper we don’t c onsider
+“mixes circuits”.11
+Figure 7: The Possible Configurations of the 3-rd Reidemeister move12
+3 Graph-Links and Homotopy Classes
+of Looped Graphs: Their Equivalence
+The main goal of this section is to construct an equivalence b etween
+the set of graph-knots, see ahead, and the set of homotopy cla sses
+of looped graphs such that the graph-knot and the homotopy cl ass
+of looped graphs constructed from a given knot are related by the
+equivalence.
+Before constructing an equivalence we need some definitions and
+statements.
+Definition 3.1. LetGbe a labeled graph on vertices from the
+enumerated set V(G) ={v1,...,vn}. Define the adjacency matrix
+A(G) = (aij) ofGoverZ2as follows:aiiis equal to the framing of vi,
+aij= 1,i/\e}atio\slash=j, if and only if viis adjacent to vjandaij= 0 otherwise.
+Remark 3.1. Throughout the paper all the matrices are over Z2.
+Therefore, corank’s and det’s are calculated over Z2.
+IfGandG′represent the same graph-link then corank Z2(A(G) +
+E) = corank Z2(A(G′)+E), whereEis identity matrix, see [IM1, IM2].
+Definition 3.2. Let us define the number of components in a graph-
+linkFas corank Z2(A(G)+E)+1, here Gis a representative of F. A
+graph-link Fwith corank Z2(A(G) +E) = 0 for any representative G
+ofFis called a graph-knot .
+Let corank Z2(A(G)+E) = 0,Bi(G) =A(G)+E+Eii(all elements
+ofEiiexcept for the one in the i-th column and i-th row which is one
+are 0) for each vertex vi∈V(G).
+Definition 3.3. Thewrithe number wiofG(with corank Z2(A(G)+
+E) = 0) atviiswi= (−1)corank Z2Bi(G)signviand thewrithe number
+ofGis
+w(G) =n/summationdisplay
+i=1wi.
+IfGis a realisable graph by a chord diagram and, therefore, by a
+virtual diagram then wiis the “real” writhe number of the crossing
+corresponding to vi.
+By/hatwideCi,j,...,kdenotethematrix obtained froma matrix Cby deleting
+thei,j,...,k -th rows and the i,j,...,k -th columns. We shall write
+/hatwideBi,j,...,k(G) instead of /hatwiderB(G)i,j,...,k.13
+Lemma 3.1. wi(G) = (−1)corank Z2/hatwideBi(G)+1signvi.
+Proof.Without loss ofgenerality weprovethestatement of thelemm a
+forw1. Let
+A(G)+E=/parenleftbiggab⊤
+bC/parenrightbigg
+and
+corank Z2(A(G)+E) = 0⇐⇒det(A(G)+E) = 1,
+where bold characters indicate column vectors. We have
+detB1(G) = det/parenleftbigga+1b⊤
+bC/parenrightbigg
+= det/parenleftbiggab⊤
+bC/parenrightbigg
++det/parenleftbigg10⊤
+bC/parenrightbigg
+and
+/hatwideB1(G) =C,det(B1(G)) = detA(G)+detC= 1+det/hatwideB1(G).
+The last equality gives us the statement of the lemma.
+Lemma 3.2. LetGbe labeled graph with detB(G) = 1andB(G)−1=
+(bij). Thenbii=1−wi(G)signvi
+2.
+Proof.We have
+wi(G) = (−1)corank Z2/hatwideBi(G)+1signvi⇐⇒wi(G)signvi= (−1)corank Z2/hatwideBi(G)+1
+⇐⇒corank Z2/hatwideBi(G) =wi(G)signvi+1
+2⇐⇒bii= det/hatwideBi(G)
+= 1−corank Z2/hatwideBi(G) =1−wi(G)signvi
+2.
+Definition 3.4. Define the adjacency matrix A(L) = (aij) overZ2
+for a looped graph Lwith enumerated set of vertices as: aii= 1 if and
+only if the vertex numbered iis looped and aii= 0 otherwise; aij= 1,
+i/\e}atio\slash=j, if and only if the vertex with the number iis adjacent to the
+vertex with the number jandaij= 0 otherwise.
+Definition 3.5. We say that an n×nmatrixA= (aij)coincides
+with ann×nmatrixB= (bkl)up to diagonal elements ifaij=bij,
+i/\e}atio\slash=j.14
+Lemma 3.3 ([Ily]).LetAbe a symmetric matrix over Z2. Then there
+exists a symmetric matrix /tildewideAwithdet/tildewideA= 1equal toAup to diagonal
+elements.
+The main theorem of the paper is the following one.
+Theorem 3.1. There is a one-to-one correspondence between the set
+of all graph-knots and the set of all homotopy classes of loop ed graphs.
+Moreover, if Kis a knot, and Fis a graph-knot constructed from K
+andLis the homotopy class of looped graphs constructed from Kthen
+FandLare related by this correspondence.
+Toprovethistheorem weconstructamapfromtheset ofall gra ph-
+knots to the set of all homotopy classes of looped graphs. We s hall
+show that this map has the inverse map.
+Letusconstructthemap χfromthesetofallgraph-knotstotheset
+ofallhomotopyclasses ofloopedgraphs. Let Fbeagraph-knotandlet
+Gbe its representative. Let us consider the simple graph Hhaving
+the adjacency matrix coinciding with ( A(G) +E)−1up to diagonal
+elements and construct the graph L(G) fromHby just adding loops
+to any vertex of Hcorresponding to a vertex of Gwith the negative
+writhe number. By definition, put χ(F) =L, hereLis the homotopy
+class ofL(G).
+Theorem 3.2. The mapχfrom the set of all graph-knots to the set
+of all homotopy classes of looped graphs defined by χ(F) =L, here
+Lis the homotopy class of L(G)andGis a representative of F, is
+well-defined.
+Proof.LetG1, G2betworepresentatives of F,andletB(Gi) =A(Gi)+
+E,B(Gi)−1= (bkl
+i). We have to show that the homotopy classes of
+L(G1) andL(G2) are the same, i.e. L1=L(G1) andL2=L(G2) are
+related to each other by the Reidemeister moves on looped gra phs.
+Let us consider four cases.
+1) We know that if G1andG2are obtained from each other by
+Ωg4 and/or Ω g4′(these moves correspond to changing of a rotating
+circuit in the case of realisable graphs) then the writhe num bers of
+correspondingvertices of G1andG2arethesame, see [IM1, IM2], and
+the graphs L1andL2are isomorphic up to loops, see [Ily]. Therefore,
+L1andL2are isomorphic.15
+2) LetG1andG2be obtained from each other by Ω g1 (we remove
+the vertex with the number 1). We have
+B(G1) =/parenleftbigg10⊤
+0A(G2)+E/parenrightbigg
+, B(G2) =A(G2)+E,
+B(G1)−1=/parenleftbigg1 0⊤
+0(A(G2)+E)−1/parenrightbigg
+, B(G2)−1= (A(G2)+E)−1,
+where0indicates column vector with all entries 0. Therefore, L2is
+obtained from L1by Ω1.
+3) LetG1andG2be obtained from each other by Ω g2 (we remove
+the vertex with the numbers 1 and 2). We have two cases. The firs t
+case
+B(G1) =
+1 0 a⊤
+0 1 a⊤
+a aA(G2)+E
+, B(G2) =A(G2)+E,
+and the second case
+B(G1) =
+0 1 a⊤
+1 0 a⊤
+a aA(G2)+E
+, B(G2) =A(G2)+E.
+Let us consider only the first case. We know that w1(G1) =−w2(G1)
+inG1[IM1, IM2]. Moreover, as
+det
+1 0/tildewidea⊤
+0 1/tildewidea⊤
+/tildewidea/tildewideaC
+= det
+1 0/tildewidea⊤
+1 10⊤
+/tildewidea/tildewideaC
+= det
+1 0/tildewidea⊤
+1 10⊤
+0 0C
+= detC,
+then
+B(G1)−1=
+b c d⊤
+c b d⊤
+d d(A(G2)+E)−1
+, B(G2)−1= (A(G2)+E)−1.
+This means that L1andL2are obtained from each other by Ω2.
+4) Now assume that G1andG2are obtained from each other by
+Ωg3. The corresponding vertices of G1andG2under Ω g3 have the
+samenumbers(in[IM1]wehaveanotherenumeration). Weshal lprove
+thatL1andL2are obtained from each other by a sequence of Ω2, Ω3
+moves.16
+We have
+B(G1) =
+1 1 1 0⊤
+1 1 0 a⊤
+1 0 1 b⊤
+0 a bC
+,
+1 = detB(G1) = det
+1 1 1 0⊤
+1 1 0 a⊤
+1 0 1 b⊤
+0 a bC
+= det
+0 1a⊤
+1 0b⊤
+a bC
+,
+B(G2) =
+1 0 0 ( a+b)⊤
+0 1 0 b⊤
+0 0 1 a⊤
+a+b b a C
+,detB(G2) = 1
+Let us show that we have a structure either for v1, v2, v3∈V(L1)
+or/tildewidev1,/tildewidev2,/tildewidev3∈V(L2) as in Figure 7.
+We have ([IM1])
+wi(G1) =wi(G2), i= 1,2,3,
+det/hatwideB1(G1) = det
+1 0a⊤
+0 1b⊤
+a bC
+
+= det
+1 0a⊤
+1 1b⊤
+0 bC
++det
+0 0a⊤
+1 1b⊤
+a bC
+
+= det
+1 0a⊤
+1 1b⊤
+0 bC
++det
+0 1a⊤
+1 1b⊤
+a 0C
++det
+0 1a⊤
+1 0b⊤
+a bC
+,
+det/hatwideB2(G1) = det
+1 10⊤
+1 1b⊤
+0 bC
+= det/parenleftbigg0b⊤
+bC/parenrightbigg
+,
+det/hatwideB3(G1) = det
+1 10⊤
+1 1a⊤
+0 aC
+= det/parenleftbigg0a⊤
+aC/parenrightbigg
+,
+b12
+1= det
+1 0a⊤
+1 1b⊤
+0 bC
+= det/parenleftbigg1a⊤+b⊤
+bC/parenrightbigg17
+= det/parenleftbigg1a⊤
+bC/parenrightbigg
++det/parenleftbigg0b⊤
+bC/parenrightbigg
+,
+b13
+1= det
+1 0a⊤
+1 1b⊤
+0 aC
+= det/parenleftbigg1a⊤+b⊤
+aC/parenrightbigg
+= det/parenleftbigg1b⊤
+aC/parenrightbigg
++det/parenleftbigg0a⊤
+aC/parenrightbigg
+,
+b23
+1= det
+1 10⊤
+1 0b⊤
+0 aC
+= det/parenleftbigg1b⊤
+aC/parenrightbigg
+,
+b12
+2= det
+0 0 b⊤
+0 1 a⊤
+a+b aC
+= det
+0 0b⊤
+1 1a⊤
+b aC
+
+= det
+0 1b⊤
+1 0a⊤
+b aC
++det
+0 1b⊤
+1 1a⊤
+b 0C
+= 1+b12
+1,
+b13
+2= det
+0 1 b⊤
+0 0 a⊤
+a+b bC
+= det
+1 1b⊤
+0 0a⊤
+a bC
+
+= det
+1 0b⊤
+0 1a⊤
+a bC
++det
+1 1b⊤
+0 1a⊤
+a 0C
+= 1+b13
+1,
+b23
+2= det
+1 0 ( a+b)⊤
+0 0 a⊤
+a+b bC
+= det
+1 0b⊤
+0 0a⊤
+a bC
+
+= det
+1 0b⊤
+0 1a⊤
+a bC
++det
+1 0b⊤
+0 1a⊤
+a 0C
+= 1+b23
+2.
+b12
+1=b23
+1+det/hatwideB2(G1), b13
+1=b23
+1+det/hatwideB3(G1),
+b12
+1+b13
+1= det/hatwideB1(G1)+1,
+b12
+2=b12
+1+1, b13
+2=b13
+1+1, b23
+2=b23
+1+1.
+It is not difficult to show that the last equalities guarantee u s that
+eitherv1, v2, v3∈V(L1) or/tildewidev1,/tildewidev2,/tildewidev3∈V(L2) have a structure as in18
+Figure 7. The structure of the other triple is obtained from t he first
+triple by toggling the non-loop edges.
+Denote by/hatwidefithe column vector obtained from fby deleting the
+i-th element and denote by /hatwideCj(resp.,/hatwideCi
+j) the matrix obtained from C
+by deleting the j-th column (resp., the i-th row and the j-th column).
+Other equalities for i,j >3 are as follows:
+bij
+1= det
+1 1 1 0⊤
+1 1 0 ( /hatwideaj−3)⊤
+1 0 1 ( /hatwidebj−3)⊤
+0/hatwideai−3/hatwidebi−3/hatwideCi−3
+j−3
+
+= det
+1 0 0/parenleftbig/hatwider(a+b)j−3/parenrightbig⊤
+0 1 0 ( /hatwidebj−3)⊤
+0 0 1 ( /hatwideaj−3)⊤
+/hatwider(a+b)i−3/hatwidebi−3/hatwideai−3/hatwideCi−3
+j−3
+=bij
+2,
+b1j
+2= det
+0 1 0 ( /hatwidebj−3)⊤
+0 0 1 ( /hatwideaj−3)⊤
+(a+b)b a/hatwideCj−3
+
+= det
+1 1 0 (/hatwidebj−3)⊤
+1 0 1 (/hatwideaj−3)⊤
+0 b a/hatwideCj−3
+=b1j
+1,
+b2j
+2= det
+1 0 0/parenleftbig/hatwider(a+b)j−3/parenrightbig⊤
+0 0 1 ( /hatwideaj−3)⊤
+a+b b a /hatwideCj−3
+
+= det
+1 0 0/parenleftbig/hatwider(a+b)j−3/parenrightbig⊤
+1 0 1 ( /hatwideaj−3)⊤
+0 b a /hatwideCj−3
+= det
+1 0 0 (/hatwidebj−3)⊤
+1 0 1 0⊤
+0 b a/hatwideCj−3
+
+= det
+1 1 0 (/hatwidebj−3)⊤
+1 1 1 0⊤
+0 b a/hatwideCj−3
+=b2j
+1,
+analogously b3j
+1=b3j
+2.
+We have to verify that every vertex x /∈ {v1, v2, v3}is adjacent to
+either noneof v1, v2, v3or precisely two of them in L1andanalogously
+forL2. This statement is equivalent to both b1j
+1+b2j
+1+b3j
+1= 0 and19
+b1j
+2+b2j
+2+b3j
+2= 0. By using the above equalities it is enough to verify
+only the first equality.
+We have
+b2j
+1= det
+1 1 1 0⊤
+1 0 1 (/hatwidebj−3)⊤
+0 a b/hatwideCj−3
+= det/parenleftigg
+1 0 (/hatwidebj−3)⊤
+a b/hatwideCj−3/parenrightigg
+,
+b3j
+1= det
+1 1 1 0⊤
+1 1 0 (/hatwideaj−3)⊤
+0 a b/hatwideCj−3
+= det/parenleftbigg0 1 (/hatwideaj−3)⊤
+a b/hatwideCj−3/parenrightbigg
+,
+b1j
+1= det
+1 1 0 (/hatwideaj−3)⊤
+1 0 1 (/hatwidebj−3)⊤
+0 a b/hatwideCj−3
+
+= det/parenleftigg
+1 1/parenleftbig/hatwider(a+b)j−3/parenrightbig⊤
+a b/hatwideCj−3/parenrightigg
+=b2j
+1+b3j
+1.
+We have proven that our triples of vertices are related with e ach
+other as in Remark 2.2, therefore, L1andL2are obtained from each
+other by a sequence of Ω2 and Ω3 moves.
+Let us define the map ψfrom the set of all homotopy classes of
+looped graphs to the set of all graph-knots. Let Lbe the homotopy
+class ofL. By using Lemma 3.3 we can construct a symmetric matrix
+/tildewideA(L) = (aij) overZ2coinciding with the adjacency matrix of Lup to
+diagonal elements and det /tildewideA(L) = 1. LetG(L) be the labeled simple
+graphhavingthematrix /tildewideA(L)−1+Easitsadjacencymatrix(therefore,
+the first component of the label is equal to the corresponding diagonal
+element of/tildewideA(L)−1+E), the second component of the label of the
+vertex with the number iiswi(1−2aii), herewi= 1 if the vertex of L
+with the number idoes not have a loop, and wi=−1 otherwise. Set
+ψ(L) =F, hereG(L) is a representative of F.
+Theorem 3.3. The mapψfrom the set of all homotopy classes of
+looped graphs to the set of all graph-knots defined above is we ll-defined.
+Proof.LetL1, L2be two representatives of L. We have to show that
+the graph-knots having representatives G1=G(L1) andG2=G(L2)
+respectively are the same. We shall show that a graph-link do es not
+depend on the choice of diagonal elements, and G1andG2are related
+to each other by Reidemeister graph-moves.20
+1) The independence (up to the second component of the label o f
+a vertex) on the choice of diagonal elements follows from the results
+of [Ily]. Namely, graph-knots obtained from matrices havin g different
+diagonal elements are related by Ω g4 and/or Ω g4′graph-moves (up to
+the second component of the label). We have defined the sign of a ver-
+tex in such a way that looped vertices correspond to vertices with the
+negative writhe number and unlooped vertices correspond to vertices
+with the positive writhe number (Lemma 3.2). The writhe numb er
+doesn’t change under Ω g4, Ωg4′graph-moves, and both the writhe
+number and the framing allow one to determine the sign of a ver -
+tex. Therefore, graph-knots obtained from matrices having different
+diagonal elements are related by Ω g4 and/or Ω g4′graph-moves.
+Now we pass to moves on looped graphs.
+2) LetL1andL2be obtained from each other by Ω1 (we remove
+the vertex with the number 1). We have
+A(L1) =/parenleftbigga0⊤
+0A(L2)/parenrightbigg
+, a∈ {0,1}.
+Assume det/tildewideA(L2) = 1, where/tildewideA(L2) coincides with A(L2) up to diag-
+onal elements, and
+/tildewideA(L1) =/parenleftbigg10⊤
+0/tildewideA(L2)/parenrightbigg
+.
+Then
+/tildewideA(L1)−1=/parenleftbigg10⊤
+0/tildewideA(L2)−1/parenrightbigg
+,
+therefore,G1andG2are related by a sequence of Ω g1, Ωg4, Ωg4′
+graph-moves.
+3) LetL1andL2be obtained from each other by Ω2 (we remove
+the vertex with the numbers 1 and 2). We have
+A(L1) =
+0ba⊤
+b1a⊤
+a aA(L2)
+, b∈ {0,1}.
+Assume det/tildewideA(L2) = 1, where/tildewideA(L2) coincides with A(L2) up to diag-
+onal elements, and
+/tildewideA(L1) =
+1+b b a⊤
+b1+ba⊤
+a a/tildewideA(L2)
+,det/tildewideA(L1) = 1.21
+As
+det
+1+b b/tildewidea⊤
+b1+b/tildewidea⊤
+/tildewidea/tildewideaC
+= det
+1+b b/tildewidea⊤
+1 10⊤
+/tildewidea/tildewideaC
+
+= det
+1+b b/tildewidea⊤
+1 10⊤
+0 0C
+= detC,
+det/parenleftbigg1+ba⊤
+a/tildewideA(L2)/parenrightbigg
+= det/parenleftbiggba⊤
+a/tildewideA(L2)/parenrightbigg
++det/tildewideA(L2),
+then
+/tildewideA(L1)−1=
+f f+1d⊤
+f+1fd⊤
+d d/tildewideA(L2)−1
+.
+From the structure of the matrix /tildewideA(L1)−1it follows that the two
+vertices (which don’t belong to G2) have the same framing and the
+necessary adjacencies, and from the structureof the matrix A(L1) and
+the coincidence of vertices’ framings it follows that the tw o vertices
+have the different signs. This means that G1andG2are obtained
+from each other by a sequence of Ω g2, Ωg4, Ωg4′graph-moves.
+4) Now assume that L1andL2are obtained from each other by
+Ω3. Let us enumerate all the vertices of V(Li) ={vi
+1,...,vi
+n}in such
+a way that corresponding vertices of L1andL2under Ω3 move have
+the same number, and without loss of generality we assume tha tv1
+1
+andv1
+3are looped, v1
+2is unlooped, v1
+1is adjacent to v1
+2;v1
+3is adjacent
+to neitherv1
+1norv1
+2inL1. The case when v1
+3is unlooped is obtained
+from the first case by applying second Reidemeister and one of the
+third Reidemeister moves.
+We have
+A(L1) =
+1 1 0 a⊤
+1 0 0 b⊤
+0 0 1 c⊤
+a b cD
+, A(L2) =
+1 0 1 a⊤
+0 0 1 b⊤
+1 1 1 c⊤
+a b cD
+,
+a+b+c=0.
+Without loss of generality (if necessary we apply second Rei de-22
+meister moves) we may assume that c/\e}atio\slash=0and (Lemma 3.3)
+/tildewideA(L1) =
+0 1 0 a⊤
+1 1 0 b⊤
+0 0 0 c⊤
+a b c/tildewideD
+
+and det/tildewideA(L1) = 1.
+As
+det
+0 0 1 a⊤
+0 0 1 b⊤
+1 1 1 c⊤
+a b c/tildewideD
+= det
+0 0 1 a⊤
+0 0 0 c⊤
+1 0 1 c⊤
+a c c/tildewideD
+
+= det
+0 0 1 a⊤
+0 0 0 c⊤
+1 0 1 b⊤
+a c b/tildewideD
+= det/tildewideA(L1) = 1, (1)
+we have
+/tildewideA(L2) =
+0 0 1 a⊤
+0 0 1 b⊤
+1 1 1 c⊤
+a b c/tildewideD
+.
+Let/tildewideA(L1)−1= (/tildewideaij
+1),/tildewideA(L2)−1= (/tildewideaij
+2), and letGi=G(Li),i= 1,2, be
+the two graph-knots having the adjacency matrices /tildewideA(Li)−1+E. Let
+us show that G1andG2are obtained from each other by a sequence
+of Ωg2, Ωg3, Ωg4, Ωg4′graph-moves.
+Performing the same elementary manipulations as in (1) we ha ve
+/tildewideaij
+1=/tildewideaij
+2fori,j/greaterorequalslant3. Further, we get
+1 = det/tildewideA(L1) = det
+0 1 0 a⊤
+1 1 0 b⊤
+0 0 0 c⊤
+a b c/tildewideD
+= det
+0 1 0 a⊤
+1 1 0 b⊤
+1 0 0 0⊤
+a b c/tildewideD
+
+= det
+1 0a⊤
+1 0b⊤
+b c/tildewideD
+= det
+1 0b⊤
+0 0c⊤
+b c/tildewideD
+=/tildewidea11
+1,23
+/tildewidea12
+1= det
+1 0b⊤
+0 0c⊤
+a c/tildewideD
+= det
+1 0b⊤
+0 0c⊤
+b c/tildewideD
+= 1,
+/tildewidea13
+1= det
+1 1b⊤
+0 0c⊤
+a b/tildewideD
+= det
+0 1b⊤
+0 0c⊤
+c b/tildewideD
+= 1,
+/tildewidea1j
+1= det
+1 1 0 (/hatwidebj−3)⊤
+0 0 0 (/hatwidecj−3)⊤
+a b c/hatwide/tildewideDj−3
+= 0, j/greaterorequalslant3 (a+b+c=0),
+/tildewidea2j
+1= det
+0 1 0 (/hatwideaj−3)⊤
+0 0 0 (/hatwidecj−3)⊤
+a b c/hatwide/tildewideDj−3
+= det
+0 1 0 (/hatwideaj−3)⊤
+0 0 0 (/hatwidecj−3)⊤
+a 0 c/hatwide/tildewideDj−3
+,
+/tildewidea3j
+1= det
+0 1 0 (/hatwideaj−3)⊤
+1 1 0 (/hatwidebj−3)⊤
+a b c/hatwide/tildewideDj−3
+= det
+0 1 0 (/hatwideaj−3)⊤
+1 0 0 (/hatwidebj−3)⊤
+a 0 c/hatwide/tildewideDj−3
+.
+If either/tildewidea22
+1= 0 or/tildewidea33
+1= 0 we can apply the same second Reide-
+meister graph-moves to G1andG2and after that applying the Ω′
+g4
+graph-move we get that the corresponding vertices have the f raming
+0. Analogously, if /tildewidea32
+1= 1 we can apply the same second Reidemeister
+graph-moves to G1andG2and after that applying the Ω g4 graph-
+move we get that v1
+2andv1
+3are not adjacent to each other. Therefore,
+without loss of generality we may assume that /tildewidea22
+1=/tildewidea33
+1= 1,/tildewidea32
+1= 0.
+By using the last equalities we get
+/tildewidea22
+1= det
+0 0a⊤
+0 0c⊤
+a c/tildewideD
+= det
+0 0b⊤
+0 0c⊤
+b c/tildewideD
+
+= 1+det/parenleftbigg0c⊤
+c/tildewideD/parenrightbigg
+= 1,
+/tildewidea33
+1= det
+0 1a⊤
+1 1b⊤
+a b/tildewideD
+= det
+1 0c⊤
+0 1b⊤
+c b/tildewideD
+
+= 1+det/parenleftbigg1b⊤
+b/tildewideD/parenrightbigg
+= 1,24
+/tildewidea23
+1= det
+0 1a⊤
+0 0c⊤
+a b/tildewideD
+= det
+1 1b⊤
+0 0c⊤
+c b/tildewideD
+
+= 1+det/parenleftbigg0c⊤
+b/tildewideD/parenrightbigg
+= 0.
+Let us find the remaining elements of /tildewideA(L2)−1. We have
+/tildewidea11
+2= det
+0 1b⊤
+1 1c⊤
+b c/tildewideD
+= det
+0 1b⊤
+1 0a⊤
+b c/tildewideD
+= det
+1 1b⊤
+1 0a⊤
+b c/tildewideD
+
++det/parenleftbigg0a⊤
+c/tildewideD/parenrightbigg
+= det
+1 1b⊤
+0 1c⊤
+b c/tildewideD
++det/parenleftbigg0b⊤
+c/tildewideD/parenrightbigg
++det/parenleftbigg0c⊤
+c/tildewideD/parenrightbigg
+= det
+1 1b⊤
+0 0c⊤
+b c/tildewideD
++det/parenleftbigg1b⊤
+b/tildewideD/parenrightbigg
++1 = 1,
+/tildewidea22
+2= det
+0 1a⊤
+1 1c⊤
+a c/tildewideD
+= det
+1 1b⊤
+1 1c⊤
+b c/tildewideD
+= 1,
+/tildewidea33
+2= det
+0 0a⊤
+0 0b⊤
+a b/tildewideD
+= det
+0 0b⊤
+0 0c⊤
+b c/tildewideD
+= 1,
+/tildewidea12
+2= det
+0 1b⊤
+1 1c⊤
+a c/tildewideD
+= det
+1 1b⊤
+0 1c⊤
+b c/tildewideD
+
+= det
+1 1b⊤
+0 0c⊤
+b c/tildewideD
++det/parenleftbigg1b⊤
+b/tildewideD/parenrightbigg
+= 0,
+/tildewidea13
+2= det
+0 0b⊤
+1 1c⊤
+a b/tildewideD
+= det
+0 0b⊤
+0 1c⊤
+c b/tildewideD
+
+= det
+0 0b⊤
+0 0c⊤
+c b/tildewideD
++det/parenleftbigg0b⊤
+c/tildewideD/parenrightbigg
+= 0,25
+/tildewidea23
+2= det
+0 0a⊤
+1 1c⊤
+a b/tildewideD
+= det
+0 1b⊤
+0 1c⊤
+c b/tildewideD
+= 0,
+/tildewidea1j
+2= det
+0 0 1 (/hatwidebj−3)⊤
+1 1 1 (/hatwidecj−3)⊤
+a b c/hatwide/tildewideDj−3
+= det
+0 0 1 (/hatwidebj−3)⊤
+1 1 0 (/hatwideaj−3)⊤
+a b c/hatwide/tildewideDj−3
+
+= det
+1 0 1 (/hatwidebj−3)⊤
+0 1 0 (/hatwideaj−3)⊤
+0 b c/hatwide/tildewideDj−3
+= det/parenleftigg
+1 0 (/hatwideaj−3)⊤
+b c/hatwide/tildewideDj−3/parenrightigg
+=/tildewidea2j
+1+/tildewidea3j
+1,
+/tildewidea2j
+2= det
+0 0 1 (/hatwideaj−3)⊤
+1 1 1 (/hatwidecj−3)⊤
+a b c/hatwide/tildewideDj−3
+= det
+0 1 1 (/hatwideaj−3)⊤
+1 0 0 (/hatwidebj−3)⊤
+a 0 c/hatwide/tildewideDj−3
+=/tildewidea3j
+1,
+/tildewidea3j
+2= det
+0 0 1 (/hatwideaj−3)⊤
+0 0 1 (/hatwidebj−3)⊤
+a b c/hatwide/tildewideDj−3
+= det
+0 0 1 (/hatwideaj−3)⊤
+0 0 0 (/hatwidecj−3)⊤
+a b 0/hatwide/tildewideDj−3
+=/tildewidea2j
+1.
+We see that
+/tildewideA(L1)−1+E=
+0 1 1 0⊤
+1 0 0 f⊤
+1 0 0 g⊤
+0 f gH
+,
+/tildewideA(L2)−1+E=
+0 0 0 f⊤+g⊤
+0 0 0 g⊤
+0 0 0 f⊤
+f+g g f H
+.
+It is easy to see that the corresponding vertices have the str ucture as
+in Definition 2.9. Therefore, G2is obtained from G1by a sequence of
+Ωg2, Ωg3, Ωg4, Ωg4′graph-moves.
+From Theorems 3.2, 3.3 and from the definitions of the map χand
+ψit follows that these maps are mutually inverse. Therefore, we have
+proven Theorem 3.1.
+We conclude the paper with the example of non-realisable gra ph-
+knot.
+Definition 3.6. A graph-link (a homotopy class of looped graphs) is
+callednon-realisable if it has no realisable representative.26
+Figure 8: The First Bouchet Graph Gives a Non-Realisable Graph-Kno t
+Corollary 3.1. A graph-link Fis non-realisable if and only if χ(F)is
+non-realisable.
+LetGbe the labeled graph depicted in Fig. 8 with each vertex
+having the framing 1 and signs are chosen arbitrary.
+Corollary 3.2. The graph-knot Fgenerated by Gis non-realisable.
+Proof.LetL=χ(F). It is not difficult to see that the looped graph
+isomorphic to G(we have no loop) is a representative of L. Therefore,
+Lis non-realisable, see [Ma6, Ma7], and, in turn, Fis non-realisable
+graph-knot.
+References
+[BN] D. Bar–Natan, On the Vassiliev Knot Invariants, Topology 34
+(1995), pp. 423–472.
+[Bou] A.Bouchet, Circlegraphobstructions, J. Combinatorial Theory
+B60(1994), pp. 107–144.
+[CE1] G. Cairns and D. Elton, The planarity problem for signe d
+Gauss words, Journal of Knot Theory and Its Ramifications 2
+(1993), pp. 359–367.
+[CE2] G. Cairns and D. Elton, The planarity problem. II, Journal of
+Knot Theory and Its Ramifications 5(1996), pp. 137–144.
+[CDL] S.V. Chmutov, S.V. Duzhin, S.K. Lando, Vassiliev Knot In-
+variants (1994). I, II, III Adv. Sov. Math. 21, pp. 117–147.
+[FKM] R.A. Fenn, L.H. Kauffman, V.O. Manturov, Virtual knots
+— unsolved problems, Fundamenta Mathematicae 188(2006)
+293–323.27
+[GPV] M. Goussarov, M. Polyak, and O. Viro, Finite type invar iants
+of classcial and virtual knots, Topology 39(2000) 1045–1068.
+[Ily] D.P. Ilyutko, Framed 4-Valent Graphs: Euler Tours, Ga uss Cir-
+cuits and Rotating Circuits, arXiv:math.CO/0911.5504.
+[IM1] D.P. Ilyutko, V.O. Manturov, Introduction to graph-l ink the-
+ory,Journal of Knot Theory and Its Ramifications 18:6 (2009),
+pp. 791–823.
+[IM2] D.P.Ilyutko, V.O.Manturov, Graph-links, Doklady Mathemat-
+ics80:2 (2009), pp. 739–742 (Original Russian Text in Doklady
+Akademii Nauk 428:5 (2009), pp. 591–594).
+[Kau] L.H. Kauffman, Virtual knot theory, Eur. J. Combinatorics
+20:7 (1999), pp. 662–690.
+[Ma1] V.O.Manturov, Khovanovhomologyofvirtualknotswit harbi-
+trary coefficients, Izvestiya: Mathematics 71:5 (2007), pp. 967–
+999 (Original Russian Text in Izvestiya RAN: Ser. Mat. 71:5
+(2007), pp. 111–148).
+[Ma2] V.O. Manturov, Embeddings of four-valent framed grap hs into
+2-surfaces, arxiv.math: GT/0804.4245.
+[Ma3] V.O. Manturov, Embeddings of 4-Valent Framed Graphs i nto
+2-Surfaces, Doklady Mathematics 79:1 (2009), pp. 56–58 (Orig-
+inal Russian Text in Doklady Akademii Nauk 424:3 (2009), pp.
+308–310).
+[Ma4] V.O. Manturov. A proof of Vassiliev’s conjecture on th e pla-
+narity of singular links, Izvestiya: Mathematics 69:5 (2005), pp.
+1025–1033 (Original Russian Text in Izvestiya RAN: Ser. Mat.
+69:5 (2005), pp. 169–178).
+[Ma5] V.O. Manturov, Teoriya Uzlov (Knot Theory), (Moscow-
+Izhevsk, RCD), 2005 (512 pp.).
+[Ma6] V.O. Manturov, On Free Knots, arXiv:Math.GT/0901.22 14.
+[Ma7] V.O. Manturov, On Free Knots and Links,
+arXiv:Math.GT/0902.0127.
+[¨Ost] Olof-Petter ¨Ostlund, Invariants of knot diagrams and relations
+among Reidemeister moves, arXiv:math.GT/0005108.
+[RR] R.C. Read, P. Rosenstiehl, Onthe Gauss crossing proble m,Col-
+loq. Math. Soc. Janos Bolyai , North-Holland, Amsterdam and
+New-York, (1976), pp. 843–876.28
+[TZ] L. Traldi, L. Zulli, A bracket polynomial for graphs,
+arXiv:math.GT/0808.3392.
+[Tr] L. Traldi, A bracket polynomial for graphs. II. Links, E uler cir-
+cuits and marked graphs, arXiv:math.GT, math.CO/0901.145 1.
\ No newline at end of file
diff --git a/1001.0361.txt b/1001.0361.txt
new file mode 100644
index 0000000000000000000000000000000000000000..cf591522f85e51a34ef6ea0dde8c80b67a0e3f6b
--- /dev/null
+++ b/1001.0361.txt
@@ -0,0 +1,968 @@
+1 
+ Self -Selected or Mandated, Open Access Increases Citation 
+Impact  for Higher Quality Research  
+ 
+Yassine Gargouri1, Chawki Hajjem1, Vincent Larivière2, Yves Gingras3, Les Carr5, 
+Tim Brody5 & Stevan Harnad,4,5 
+1Institut des sciences cognitives, U . du Québec à Montréal  
+2Observatoire des Sciences  et des Technologies , U. du Québec à Montréal  
+3Canada Research Chair in the History and Sociology of Science, U . du Québec à 
+Montréal  
+4Canada Research Chair in Cognitive Sciences, U . du Québec à Montréal  
+5School of Electronics and Computer Science, U . of Southampton  
+Corresponding Author : S. Harnad, Institut des sciences cognitives, Université du Québec 
+à Montréal , Montr éal, Québec CANADA H3C 3P8  harnad@uqam.ca  
+ 
+ 2 
+ Abstract :  
+Background: Articles whose authors make them Open Access (OA) by self -archiving 
+them online are cited significantly more than articles accessible only to subscribers. Some 
+have suggested that this “OA Advantage” may not be causal but just a self -selection bias, 
+because  authors preferentially make  higher -quality articles OA. To test this we  compared 
+self-selective self -archiving with mandatory self -archiving for a sample of 27,197 articles 
+published 2002 -2006 in 1,984 journals.  
+Methdology/Principal F indings: The OA Advantage proved just as high for both. 
+Logis tic regression showed that the  advantage is independent of other correlates of 
+citations (article age; journal impact factor; number of co -authors, references or pages; 
+field; article type; or co untry)  and greatest for the most highly cited articles . The OA 
+Advantage is real, independent and causal, but skewed. Its size is indeed correlated with 
+quality, just as citations themselves are (the top 20% of articles receive about 80% of all 
+citations).  
+Conclusions /Significance : The OA advantage  is greater for the more citeable articles, 
+not because of a quality bias  from authors self -selecting what to make OA, but because of 
+a quality advantage,  from users self-selecting  what to use and cite, freed by OA from the 
+constraints  of selective accessibility  to subscribers  only. It is hoped that these findings 
+will help motivate the adoption of OA mandates be universities, research institutions and 
+research funders.  
+ 
+One-sentence Summary:  We demonstrate that the greater citation impact of open access research 
+is causal rather than an artifact of author bias (i.e., authors self -selectively making higher quality 
+research open access) by showing that the citation increase is just as great when th e open access is 
+mandatory; the open access impact advantage is independent of other correlates of citation impact, 
+and greater for higher quality research.  
+Introduction  
+The 25,000  peer-reviewed journals and refereed conference proceedings that exist today 
+publish about 2.5 million articles per year, across all disciplines, languages and nations. 
+No university  or research  institution anywhere, not even the richest, can afford to 
+subscribe to all or most of the journals that its researchers may need to use (Odlyzko 
+2006). As a consequence, all articles are currently losing some portion of their potential 
+research impact (us age and citations), because they are not accessible online to all their 
+potential users (Hitchock 2009).  
+This is supported by recent evidence, independently confirmed by many studies , that 
+articles whose authors have supplemented subscription -based access to the publisher’s 
+version by self -archiving their own final draft to make it accessible free for all on the web 3 
+ (“Open Access”, OA) average twice as many citations as articles in the same jour nal and 
+year that have not been made OA. This “OA Impact Advantage” has been found in all 
+fields analyzed so far -- physical, technological, biological and social sciences, and 
+humanities (Lawrence 2001; Brody & Harnad 2004; Hajjem et al. 2005; Moed 2005 b; 
+Eysenbach 2006; Giles et al. 1998; Kurtz & Brody, 2006; Norris et al. 2008; Evans 2008; 
+Evans & Reimer 2009).  
+Hence OA is not just about public access rights or the general dissemination of 
+knowledge: It is about increasing the impact and thereby the pr ogress of research itself. A 
+work’s research impact is an indication of how much it contributes to further research by 
+other scientists and scholars, how much it is used, applied and built upon  (Brin & Page 
+1998; Garfield 1955, 1976, 1988; Page et al. 1999 ). That is also why impact is valued, 
+measured and rewarded in researcher performance assessement as well as in research 
+funding (Harnad 200 9).  
+Self-archiving mandates  
+Only about 15% of the 2.5 million articles published annually worldwide are being self -
+archived by their authors today (Bjork et al 200 8; Hajjem and al., 2005). Creating an 
+Institutional Repository (IR)  and encouraging faculty to self -archive their articles therein 
+is a good first step, but that is not sufficient to raise the self -archiving rate appreciably 
+above its current spontaneous self -selective baseline of 15% (Sale, 2006). Nor are mere 
+requests or recommendations by researchers’ institutions or funders, encouraging them to 
+self-archive, enough to  raise this 15% figure appreciably, even when coupled with offers 
+of help, rewards, incentives and offers to do the deposit on the author’s behalf. In two 
+international, multidisciplinary surveys, 95% of researchers reported that they would self -
+archive if  (but only if) required  to do so by their institutions or funders. (Eighty -one 
+percent reported that, if it was required, they would deposit willingly ; 14% said they 
+would deposit reluctantly, and only 5% would not comply with the deposit requirement; 
+Swan 2006 .) Subsequent studies on actual mandate compliance have gone on to confirm 
+that researchers do indeed do as they reported they would, with mandat ed IRs generating 
+deposit rates several times greater than the 15% self -selective baseline and well on the 
+road toward 100% within about two years of adoption (Sale, 2006).  
+Universities' own IRs are the natural locus for the direct deposit of their own research 
+output: Universities (and research institutions) are the universal providers of all research 
+output, in all scientific and scholarly disciplines; they accordingly have a direct interest in 
+hosting, archiving, monitoring, measuring, managing, evalu ating, and showcasing their 
+own research output in their own IRs, as well as in maximizing its uptake, usage, and 
+impact  (Holmes & Oppenheim 2001; Oppenheim 1996; . OA self -archiving mandates 4 
+ hence add visibility and value at both the individual and instit utional level (Swan & Carr 
+2008) . 
+In 2002 , The University of Southampton’s School of Electronics & Computer Science 
+(ECS ) became the first in the world to adopt an official self -archiving mandate. Since 
+then, a growing number of departments, faculties and institutions worldwide (including 
+Harvard, Stanford, and MIT) as well as research fun ders (including all seven UK 
+Research Funding Councils, the US National Institutes of Health , and the European 
+Research Council) have likewise adopted OA self -archiving mandates. Over 100  
+mandates had already been adopted and registered and charted in ROARMAP1 as of 
+autumn 2009.  
+In 2008, mindful of the benefits of mandating OA, the council of the European 
+Universities Association (EUA )2 unanimously recommended that all European 
+Universities should create IRs and mandate that all their research output should be  
+deposited in them immediately upon publication (to be made OA as soon as possible 
+thereafter). The EUA further recommended that these self -archiving mandates be 
+extended to all research results arising from EU research project funding. A similar 
+recommend ation was made by EURAB  (European Research Advisory Board). In the US, 
+the FRPAA  has proposed similar mandates for all research funded by the major US 
+research funding agencies.  
+Some studie s, however, have suggested that the “OA Advantage” might just be a self -
+selection bias rather than a causal factor, with authors selectively tending to make higher -
+quality (hence more citeable) articles OA (Craig et al. 2007; Davis & Fromerth 2007; 
+Henneken et al 2006; Moed 200 6). The present study was car ried out to test this 
+hypothesis by comparing self -selected OA with  mandated OA on the basis of  the research 
+article output of the four institutions with the longest -standing OA mandates: ( i) 
+Southampton University (School of Electronics & Computer Science ) in the UK (since 
+2002); ( ii) CERN (European Organization for Nuclear Research) in Switzerland (since 
+November, 2003); ( iii) Queensland University of Technology in Australia (since 
+February 2004); ( iv) Minho University in Portugal (since December, 2004).  
+Method  
+The objective was to compare citation counts  -- always within the same journal/year  -- 
+for OA (O) and non -OA (Ø) articles, comparing the O/Ø citation ratios for OA that was 
+self-selected (S) vs. mandated (M). (The critical comparisons were hence SO /Ø vs. 
+MO/Ø.) The sample covered articles published between 2002 and 2006.3 The metadata 
+                                                           
+1 ROARMAP (Registry of Open Access Repository Material Archiving Policies)  
+http://www.eprints.org/openaccess/policysignup/  
+2 EUA consists of  more than 800  universities, in 46 countries (in January  2009 ) 
+3 About two years need to elapse for the citations from the most recent year to stabilize.  5 
+ for the articles were collected from the four institutional repositories, as well as from the 
+Thomson -Reuters citation database.4  
+The effect of OA on citation impact cannot be reliably tested by comparing OA and non -
+OA journals because no two journals have identical subject matter, track -records and 
+quality -standards (nor are there as yet enough established OA journals in most fields ).  
+The comparison must hence be between OA and non -OA articles published within the 
+same (non -OA) journals (Harnad and Brody, 2004). For each  mandated article, M i, 
+deposited in our four mandated IRs we accordingly collected, as our pool of 
+nonmandated con trols for comparison, all articles N j published in the same journal, 
+volume and year. Our sample of deposited articles from 2002 to 2006 was distributed 
+across 1,984 non -OA journals in the Thomson -Reuters database ( Table 1 ).5  
+ 
+  Journal  Count  
+2002  331 
+2003  367 
+2004  415 
+2005  445 
+2006  426 
+TOTAL  1984  
+Table 1: Journal  counts per year  
+To reduce our nonmandated comparison sample to a reasonable processing size, we 
+restricted the number of journal/year -matched controls to the 10 Øj articles that were 
+semantically closest to their corresponding target M i (as computed on the basis of shared 
+words in titles, omitting stop words). This tightening of content similarity also made the 
+control articles even more comparable to their targets than using the full spectrum of 
+same -journal content. The total size of the article sample (6215 mandated targets plus 
+their 20982 corresponding controls6) from 2002 to 2006 was 27197 .  
+                                                           
+4 Citation counts were extracted from the Thomson -Reuters database November, 2008.  
+5 Based on the Directory of Open Access Journals (DOAJ), 2% of journals indexed by Thomson -Reuters in 
+2006 were OA journals. Articles from these journals were removed from our pool because for them O/Ø 
+comparisons were not possible.  
+6 When more than one M article was published in the same journal/volume/ year (which represents 66% 
+of M ar ticles), only 10 articles were selected as controls, using keyword matching for one of these M 
+articles.    6 
+ The full -text OA status of the articles in our sample was verified usi ng an automated 
+webwide search -robot (Hajjem and al. 2005 ) as well as  an automated Google Scholar 
+search.  Figure 1  shows each of our four mandated institutions’ verified annual OA article 
+deposits as a percentage of the institution’s total published article output for each year 
+based (only) on those articles published in the journals indexed by the Thomson -Reuters 
+citation database; the resulting estimate of the overall OA mandate compliance rate is 
+about 60%. Note also the robot data’s confirmation of the ~15% baseline for 
+spontaneous, self -selected (i.e., non -mandated) OA self -archiving among the control 
+articles in the same journal/years.  
+ 
+Figure 1:  OA Self -Archiving Levels for Mandate Compliance and Self -Selected 
+Controls: As estimated from their Thomson -Reuters -indexed portion, about 60% 
+of each mandated institution’s total yearly article output was deposited and hence 
+made OA as mandated. The corresponding percentage OA among the control 
+articles published in the same journal/year (but originating from other, 
+presumably nonmandated  institutions) was close to the previously reported global 
+spontaneous baseline rate of about 15% for self -selected ( nonmandated ) self -
+archiving (Bjork et al 2008).  
+This mandated deposit rate of 60% is substantially higher than  the self -selected deposit 
+rate of 15%. Although with anything short of 100% compliance it is always logically 
+possible to hold onto the hypothesis that the OA citation advantage could be solely a self -
+selection bias -- arguing that, with a mandate, the fo rmer bias in favor of self -selectively 
+self-archiving one’s more citeable articles takes the form of a selective bias against 
+compliance with the mandate for one’s less citeable articles), but a reasonable 
+expectation would be at least a diminution in the size of the OA impact advantage with a 
+mandated deposit rate four times as high as the spontaneous rate, if it is indeed true that 
+the OA advantage is solely or largely due to self -selection bias.                
+To test whether mandated OA diminishes the OA  citation advantage, 4 kinds of articles 
+need to be compared:  
+- O M : OA, Mandated,  
+7 
+ - Ø M : Non -OA, Mandated,  
+- O S : OA, Self -Selected  
+- Ø S : Non -OA, Self -Selected  
+The analysis uses the citation counts within each journal/year. Because the date on which 
+the mandate was first adopted varies (from 2002 to 2004) for the four institutions, we 
+analyzed the data for the four institutions separately as well as their joint averages. The 
+separate analyses show the time -course of mandate compliance more clearly; th e global 
+analysis combines data, enlarges the sample size and smoothes out incidental effects of 
+institutional and timing differences.  
+We compare the following ratios: O/Ø, OM/OS, OS/ØS, OM/ØM, OM/Ø, OS/Ø and 
+OM/OS  using their mean log citation ratios. For example, to compare mandated OA with 
+self-selected OA, we compute the log of the ratio OM j/OS j for each journal j and then,  we 
+compute the arithmetic mean of all the log ratios for all journals. There is an  advantage in 
+favor of OM when the log  ratio is g reater than 0, and in favor of OS otherwise.  
+
+n
+j jj
+OSOM
+nOS OM
+1log1/
+ 
+The logarithm is used to normalize the data and reduce any effect coming from articles 
+having a relatively high citation count, compared to the whole sample. The comparison is 
+within -journal, to minimize between -journal differences in citation average (“journal 
+impact factor”) , and it is keyword -matched to mi nimize differences still further.     
+Results  
+Overall, OA articles are cited significantly more than non -OA articles, confi rming the 
+repeatedly observed OA Advantage (O/Ø). There is also no evidence at all that mandated 
+OA has a smaller citation advantage than self -selected OA, but rather the contrary. 
+Figure 2  shows the results for the four institutions together. Appendix 1  shows each 
+institution separately.  The pattern for the individual institutional data is largely the same 
+as for the average across the four institutions.  8 
+  
+Figure 2:  Comparing the OA Impact Advantage for Self -Selected vs Mandatory 
+OA. Averages across the sample of four institutions confirm the significantly 
+higher citation rates for OA vs. matched control Non -OA articles ( O/Ø) published 
+in the same journal and year. OA articles are more highly cited irrespective of 
+whether the OA is Self -Selected  (S) or M andated (M). The O/Ø Advantage is 
+present for mandated OA (OM/ØS), and of about the same magnitude, whether S 
+vs M are compared on the basis of the entire control sample (OS/Ø vs OM/Ø) or 
+just S alone vs M alone (OS/ØS vs OM/ØM). Far from the OA Advantage 
+diminishing when it is mandatory (compliance rate, 60%) rather than self -
+selective (15%), it is, according to this sample, if anything, slightly greater 
+(OM/OS) (although this, being the smallest effect, might be due to chance or 
+sampling error).  
+Table 2:  Paired Samples Test  
+ 
+  
+ 00.10.20.30.40.50.60.7
+O/Ø O M/Ø S O S/Ø O M/Ø O S/Ø S O M/Ø M O M/O S2002
+2003
+2004
+2005
+20064 institutions
+9 
+ For all OA vs Non -OA (O/Ø) comparisons, regardless of whether the OA was Self -
+Selected (S) or Mandated (M), the mean log citation differences are significantly greater 
+than 0 (based on correlated -sample t -tests for within -journal differences; Table 2 ). As the 
+last of the four institutional mandates was adopted in 2004, the test was based on a 
+sample of M and S articles published between 2004 and 20067. There is no detectable 
+reduction in the size of the OA Advantage for Mandated OA  (60%) compared to Self -
+Selected OA (15%). (The Mandated OA Advantage is, if anything, greater, and also 
+grows with time.) It would require a very complicated argument indeed (“self -selective 
+noncompliance for less citeable articles”) to resurrect the hypo thesis that the OA 
+Advantage is only or mostly a self -selection bias in the face of these findings. (Such an 
+argument does remain a logical possibility until there is 100% mandate compliance, but 
+an increasingly implausible one.)  
+Logistic regression  
+The nu mber of citations an article receives can be correlated with and hence influenced 
+by a variety of variables. Those variables, in turn, could create another kind of bias. For 
+example, older articles tend to have more citations than younger articles simply b ecause 
+there has been more time to cite them. If OA articles tended to be older than non -OA 
+articles, then article age, rather than OA, could be the cause of the OA Advantage. A way 
+to test whether correlates of citation other than OA are responsible for t he OA Advantage 
+is to perform a logistic regression analysis to see whether OA alone is still significantly 
+correlated with higher citations when the correlation with other variables has been 
+partialled out. We have accordingly analyzed the following set o f variables potentially 
+influencing citations. Variables 1 -8 are known to be correlated with citation counts. 
+                                                           
+7 The greater OA Advantage for 2006 might be due to a variety of factors that will be analyzed in future 
+more detailed studies over a longer time bas e (and taking deposit date into account, alongside publication 
+date and the date at which citations are counted):  
+ (a) These results were analyzed in 2009, and 2006 was the most recent full year analyzed. If the 
+results for a year are analyzed before at le ast 1.5 years have elapsed, citations are still incomplete, the 
+data are unstable, and the OA Advantage may not yet be detectable.  
+ (b) Some of the compliance rates for 2002 -2006 may have been retroactive, with the older 
+articles deposited several years a fter they were published. That would mean that older articles were not 
+receiving their full OA Advantage (and the finding of an Early Access Advantage by Kurtz et al 2005 and  
+Kurtz & Henneken 2007 suggests that later deposits may never gain all the citati ons they would have 
+received if deposited earlier). The mandates were adopted between 2003 and 2005. So perhaps only 2006 
+was receiving its full OA Advantage.  
+ (c) Citations grow with article age but our multiple regression analyses also reveal an OA*Age 
+interaction, with the OA Advantage growing faster than article age. Hence the OA Advantage becomes 
+bigger for older articles, when measured independently, with other variables that increase citations (such 
+as age, journal impact factor, number of co -authors ) partialled out.  
+ (d) In contrast to (c), however, it is also possible that as global OA is growing, global OA use is 
+rising, which would mean that the OA Advantage itself is growing; this too could help explain the higher 
+Advantage for the most recent year in our current sample (2006).  10 
+ Variable 9 is OA itself; and variable 10 is a measure of the degree to which the relation 
+between OA and Age is non -additive. Variable 11 is wheth er or not the OA is mandated. 
+Variables 12 -15 are just the four mandating institutions that are our reference points in 
+this study.  
+1. Age : How old is the article (articles published from 2002 to 2006)?  
+2. JIF : What is the Thomson -Reuters "Impact Factor" (average citations per article 
+in 2-year window) of the journal in which the article was published (from 0 to 
+30)? 
+3. Auth_N : How many co -authors does the article have?  
+4. Ref_N  : How many references does the article cite?  
+5. Page_N : How many pages in the article ? 
+6. Sci : Is the article classified by Thomson -Reuters as Science (1) or Social Science 
+(0) 
+7. Review  : Is the article classified by Thomson -Reuters as a "review" article (1) or 
+not (0)?  
+8. USA : What is the country of the first author (USA 1, other 0)?  
+9. OA: Is the  article Open Access (1) or Not (0)?  
+10. Age*OA : The interaction between Age and OA  
+11. M: Does the author's institution Mandate Open Access (1) or Not (0)?  
+12. CERN  : Is the first author from CERN (1/0)?  
+13. South : Is the first author from Southampton  (1/0)?  
+14. Minho : Is the first author from Minho (1/0)?  
+15. Queens : Is the first author from Queensland University of Technology (1/0)?  
+ 
+ 
+Figure 3 : Citation count distribution (minus self -citations ). 23% of our sample 
+of 27197 articles had zero citations; 51% had 1 -5 citations; 12% had 6 -10 
+citations; 8% had 11 -20 citations; 6% had  20+ citations.   
+11 
+  
+Model 
+N. Dependent 
+V. Age JIF Ref_N  Auth_N  Page_N  OA M USA  Review  Sci CERN  South  Minho  Queens  Age*OA  
+    M_1  Cit_a_0&1 -5 1.494  2.229  1.020  1.007  0.993  0.957      0.627  1.249  0.789      1.476  1.209  
+M_2  Cit_a_1 -5&5-
+10 1.490  1.514  1.016  1.002  0.986  1.323  1.889  1.415  0.777  1.475            
+M_3  Cit_a_1 -
+5&10 -20 1.786  1.776  1.020  1.002  0.992  1.392  1.716  1.406  0.992  1.887       
+M_4  Cit_a_1 -
+5&20+  2.439  2.114  1.019  0.999    8.953    1.860  1.914  3.050  2.306        0.968  
+Table 3:  The Exp(ß) values for logistic regressions  
+ 
+Bold:  p<0.01  
+Italic: 0.01≤p<0.05  
+ 
+All self -citations were subtracted from the citation counts. (About 32% of the articles in 
+our sample have at least 1 self -citation, with an average of about 2 self -citations per 
+article.) As is well-known and evident from Figure 3 , citation coun ts are not  normally 
+distributed and instead follow a power -law or stretched -exponential  function ( Lariviere et 
+al. 2009; Wallace et al. 2009) . We accordingly used binary stepwise logistic regression 
+analysis, with a dichotomous dependent variable, selecting for each  test the model that 
+maximizes the chi -square likelihood ratio. To make the interpretation of the coefficients 
+easier, we exponentiated the ß coefficients ( Exp(ß) ) and interpreted them as odds -ratios. 
+For example, we can say for the first model that for a one unit increase in OA, the odds of 
+receiving 1 -5 citations (versus zero citations) increased by a factor of 0.957. Table 3  and 
+Figure 4  show  Exp(ß) values for each model having " Cit_a_x -y&y-z" as dependent 
+variables   ((x,y,z) {1, 2, 3, ..., 20}), where Cit_a_x -y&y-z = 1 if the citation count 
+(minus self -citations) is between y and z and 0 if between x and y. Models are referred to 
+as "M_r ". (The Exp(ß) values of variables turned out to have the same polarity and to be 
+quite si milar, with and without self -citations).  
+ 12 
+  
+Figure 4:  The Exp(ß) values for logistic regressions. In most of the four citation 
+range comparisons (zero/low, low/med1, low/med2, low/high) citation counts are 
+positively correlated with Age, Journal Impact Factor, Number of Authors, 
+Number of References, Number of Pages, Science, Review, USA Author, OA, 
+and Mandatedness. There is also an OA*Age interaction in the top and bottom 
+range. (Citations grow with time; for age -matched articles, the OA Advantage 
+grows even faster with time; Figure 5 ). OA is a significant independent 
+contributor in every citation ran ge, but especially at the high end.  
+Figure 4  shows that citations are, as already known, positively correlated with the first 
+eight variables listed earlier (age, journal impact factor, authors, references, pages, 
+science, review, USA) as well as with OA. Articles that are made OA have significantly 
+higher citation counts. This significant OA advantage, which this analysis now shows to 
+be independent of the other variables, is present in every citation range but highest in the 
+highest citation range (1 -5 citations vs 20+ citations): In other words, the OA advantage 
+is greatest for highly cited articles.  
+In our sample, articles by authors at institutions that have OA Mandates have higher 
+citation counts; this effect is present only in the medium -high citation  ranges (and is of 
+course also influenced by the level of author compliance with the institutional Mandate, 
+discussed further below). CERN articles have higher citation counts in the lowest and 
+especially the highest citation range. However, when all CERN articles are excluded from 
+our sample, there is no significant change in the other variables.  
+There is a significant interaction between Age and OA (Age*OA) for the lowest citation 
+range (0 vs. 1 -5 citations) as well for highest citation range (1 -5 citatio ns vs. 20 citations 
+and more). Both the linear main effect of age and OA, and this nonlinear interaction are 
+statistically significant. Figure 5  shows the citation mean ( Cit_a_1 -5&20 +) for OA and 
+NOA articles corresponding to each Age value. This figure confirms the OA advantage 
+and shows the interaction with age: The OA Advantage becomes even bigger for older 012345678910
+Cit_a_0&1 -5 Cit_a_1 -5&5 -10 Cit_a_1 -5&10 -20 Cit_a_1 -5&20+Age
+JIF
+Auth_N
+Ref_N
+Page_N
+Sci
+Review
+USA
+OA
+Age*OA
+M
+CERN
+South
+Minho
+Queens13 
+ articles.   
+ 
+Figure 5 : Interaction Between Article Age and OA. The size of the OA 
+Advantage increases as articles get older, over and above the sum of the 
+independent positive effects of age alone and of OA alone on citations.  
+Logistic regression by Impact Factor interval:  
+In order to compare articles belonging to com parable journals and to see the profile for 
+journals in increasing impact ranges  (see distribution, Figure 6 ), we divided our sample 
+into 4 quartiles by Journal Impact Factor (JIF), each range covering 25% of the articles:  
+        JIF_1 :           0 ≤ JIF < 0.633  
+        JIF_2 :    0.633 ≤ JIF < 1.053 Tav 
+        JIF_3 :    1.035 ≤ JIF < 1.782  
+        JIF_4 :    1.782 ≤ JIF < 29.957  
+Only the top quartile contains journals with JIFs from 1.782 to 29.957.  As we are also 
+interested in the variability within this quartile, we further subdivided this top quartile 
+into two octiles, each covering 12.5% of the articles. Subdividing more minutely would 
+make the sample sizes too small to detect effects of interest. This yielded a total of five 
+ranges for the JIF variable : 
+        JIF_1 :          0 ≤ JIF < 0.633  
+        JIF_2 :    0.633 ≤ JIF < 1.053  
+        JIF_3 :    1.053 ≤ JIF < 1.782  
+        JIF_4 :    1.782 ≤ JIF < 2.468  
+        JIF_5 :    2.468 ≤ JIF ≤ 29.957  
+14 
+  
+Figure 6 :  Distribution of journals by Journal Impact Factor (JIF).  
+The same regression is done separately for each JIF range by controlling all the variables 
+(except JIF). Figures 7 -11 (and Appendix Tables 4 -8) summarize the values of Exp(ß) 
+corresponding to the controlled variables for each JIF range8.  
+When articles are published in a low JIF journal, citation counts for their individual 
+articles are positively correlated with Age, References, Authors, OA and M. The OA 
+advantage is greater in the higher citation ranges. For the lowest range of individual  
+article citations, the Age*OA interaction is significant, but OA itself is not.  
+ 
+                                                           
+8 As noted earlier, our Exp(ß) values for these variables have the same polarity and pattern whether or not 
+we exclude self -citations from the citation count . 
+JIF30,00 25,00 20,00 15,00 10,00 5,00 0,00Frequency8 000
+6 000
+4 000
+2 000
+0Mean =1,45
+Std. Dev. =1,849
+N =27 1910.00.51.01.52.02.53.03.54.04.55.0
+Cit_a_0&1 -5 Cit_a_1 -5&5 -10 Cit_a_1 -5&10 -20 Cit_a_1 -5&20+Age
+Auth_N
+Ref_N
+Page_N
+Sci
+Review
+USA
+OA
+Age*OA
+M
+CERN
+South
+Minho
+Queens15 
+ Figure 7 :  Exp(ß) values for logistic regressions (Lowest JIF Range: 0.0 -.0.633)  
+For articles in journals with JIFs between 0.633 and 1.053, the pattern is quite similar, 
+except the Age*OA interaction is absent and OA itself (alongside Age, as separate 
+variables) is significant.  
+ 
+Figure 8:   Exp(ß) values for logistic regressions (JIF range 0.633 -1.053)  
+For articles in journals with JIFs between 1.053 and 1.782, the pattern is again quite 
+similar. The USA and Review variables now also correlate with citation increase. In this 
+JIF range, three of the institutions (QUT, Southampton and CERN) have a small citation 
+advantage in some of the comparisons. Removing the articles from any one of these 
+institutions, however, does not change the pattern for the other variables.   
+ 
+ Figure 9:  Exp(ß) values for logistic regressions (JIF range 1.053 -1.782)  0.00.51.01.52.02.53.03.54.04.55.0
+Cit_a_0&1 -5 Cit_a_1 -5&5 -10 Cit_a_1 -5&10 -20 Cit_a_1 -5&20+Age
+Auth_N
+Ref_N
+Page_N
+Sci
+Review
+USA
+OA
+Age*OA
+M
+CERN
+South
+Minho
+Queens
+0.00.51.01.52.02.53.03.54.04.55.0
+Cit_a_0&1 -5 Cit_a_1 -5&5 -10 Cit_a_1 -5&10 -20 Cit_a_1 -5&20+Age
+Auth_N
+Ref_N
+Page_N
+Sci
+Review
+USA
+OA
+Age*OA
+M
+CERN
+South
+Minho
+Queens16 
+ For journals with JIFs between 1.782 and 2.468, longer articles (more pages) have more 
+citations. Here the OA advantage is significant only  in the highe st citation count ranges. 
+The number of authors is also less correlated with increased citations as the citation range 
+gets higher. CERN has a citation advantage in this JIF range. Howev er, removing CERN 
+articles does not alter the pattern for the other variables.   
+ 
+ 
+ Figure 10:  Exp(ß) values for logistic regressions (JIF range 1.782 -2.468)  
+ 
+For journals with JIFs between 2.468 and 29.957. The OA advantage is again significant 
+for the highest citation ranges. (The increased citations for USA and Review articles also 
+increase in significance .) 
+ 
+Figure 11:  The Exp(ß) values for logistic regressions (JIF 2.468 -29.957)  0.00.51.01.52.02.53.03.54.04.55.0
+Cit_a_0&1 -5 Cit_a_1 -5&5 -10 Cit_a_1 -5&10 -20 Cit_a_1 -5&20+Age
+Auth_N
+Ref_N
+Page_N
+Sci
+Review
+USA
+OA
+Age*OA
+M
+CERN
+South
+Minho
+Queens
+0.00.51.01.52.02.53.03.54.04.55.0
+Cit_a_0&1 -5 Cit_a_1 -5&5 -10 Cit_a_1 -5&10 -20 Cit_a_1 -5&20+Age
+Auth_N
+Ref_N
+Page_N
+Sci
+Review
+USA
+OA
+Age*OA
+M
+CERN
+South
+Minho
+Queens17 
+ Overall, OA is correlated with a significant citation advantage for all  journal JIF intervals 
+as well as for the sample as a whole. This advantage is greatest for the highest citation 
+ranges. When regressions are done separately for the different JIF ranges, the Age*OA 
+interaction disappears, but OA and Age (as separate varia bles) remain significant. (There 
+is no significant effect of a specific institution compared to the rest of the institutions, 
+hence there is no need to exclude any specific institution from our sample.)  
+Discussion  
+ 
+This study confirms that the OA advantage  is a statistically significant, independent 
+positive increase in citations, even when we control the independent contributions of 
+many other salient variables (article age, journal impact factor, number of authors, 
+number of pages, number of references ci ted, Review, Science, USA author). All these 
+other variables are of course correlated with citation counts, so the fact that OA continues 
+to correlate with an independent positive increase in citation counts even when all these 
+other correlates are partial led out is quite a strong outcome.  It means that the OA 
+Advantage is not just a bias arising from either a chance or systematic imbalance in the 
+other correlates of citations.  
+Moreover, the OA advantage is just as great when the OA is mandated (with mandat e 
+compliance rate  ~60%) as when it is self -selective  (self-selection rate ~15%) . That makes 
+it highly unlikely that the OA advantage is either entirely or mostly the result of an author 
+bias toward selectively self -archiving higher quality – hence higher citeability – 
+articles.   Nor are the main effects the result of institutional citation advantages, as the 
+institutions were among the independent predictor variables partialled out in the logistic 
+regression; the outcome pattern and significance is also un altered by removing CERN, 
+the only one of the four institutions that might conceivably have biased the outcome 
+because its papers were all in one field and tended to be of higher quality, hence higher 
+citeability  overall.  
+Since, with the exception of our o ne unidisciplinary  institute,  CERN  (high energy 
+physics), the pluridisciplinary articles from the three other mandated institutional 
+repositories are mostly not in fields that habitually self -archive their unrefereed preprints 
+well before publication (as many in high energy physics do), nor in fields that already 
+have effective OA for their published postprints (as astronomy does: Henneken et al 
+2006, 2008; Kurtz & Brody 2006 ), it is also unlikely that the OA advantage is either 
+entirely or mostly just an early (prepublication) access advantage (Kurtz et al 200 5; Kurtz 
+& Henneken 2007). This will eventually be testable once there are enough reliable 
+deposit -date data, relative to publication -date, for a large enough body of self -archived 
+OA articles. In any  case, an early -access ad vantage in a preprint self -archiving  field 
+translates into a generic postpublication OA advantage in fields in which  authors do not 18 
+ self-archive prepublication preprints and published postprints are otherwise accessible 
+only to subscribers. The OA mandates all apply only to refereed postprints, not to pre -
+refereeing preprints.  
+This study confirms that the OA advantage is substantially greater for articles that have 
+successfully met the quality standards of higher -impact journals and it is also greater in 
+the higher -citation ranges for individual papers within each journal -impact level.  The 
+Seglen (1992) “skewness” effect, whereby the top 10 -20% of articles receive about 80-
+90% of all citations, is confirmed in our own sample of 70 8,219 articles extracted from 
+Thomson -Reuters from  1998 to 2007: In our sample, 20% of articles  receive d 80% of all 
+citations (29%  received 90%). In addition, 10% of journals  receive 90% of all citations  
+The implication is that OA will not make an unuseab le (hence unciteable ) paper more 
+used and cited (although the proportion of uncited papers has been diminishing with time; 
+Wallace et al. 2009) . But wherever there are subscription -based constraints on 
+access ibility , providing OA will increase the usage and citation of the more useable and 
+citeable papers, probably in proportion to their importance and quality, hence citeability. 
+The most likely cause of the OA citation advantage is accordingly not author self -
+select ion toward making more citeable articles OA, but user self -selection  toward using 
+and citing the more citeable articles – once OA self -archiving has made them accessible 
+to all users, rather than just to those whose institutions could afford subscription a ccess.  
+In other words, the OA advantage is a quality advantage , rather than a quality bias : it is 
+not that the higher quality articles – the ones that are more likely to be selectively cited 
+anyway -- are more likely to be made OA self-selectively by their  authors, but that the 
+higher quality articles that are more likely to be selectively cited  are made more 
+accessible, hence more citeable, by being made OA.  
+Our results also suggest that mandated OA might have some further independent citation 
+advantage of  its own, over self -selected OA  -- but until replicated, it is more likely that 
+this small, previously unreported effect was an effect of chance or sampling error. If there 
+does indeed prove to be a n ind ependent  “mandate advantage” over and above OA itself , a 
+possible interpretation would be the reverse of the self -selection hypothesis: There may 
+be a higher proportion of higher -quality work among the 85% that are not  made  OA on a 
+self-selective basis today than among the 15% that  are, so the mandates serve  to help 
+bring this “cream of science” to the top.  
+It also needs to be noted that some of the factors contributing to the OA advantage are 
+permanent, whereas others  will shrink as OA rises from its current 15% level and will 
+disappear at 100%  OA. All competitive advantage  of OA over non -OA (because OA is 
+more accessible) will of course vanish at 100% OA  (as will the possibility of concurrent 
+measurement of the OA Advantage) . Any self-selective bias  (whether positive or 19 
+ negative)  too will disappear at 1 00% OA . What will remain will be the quality advantage  
+itself (the tendency of researchers to selectively use and cite the best research , if they can 
+access it ), now maximized by making everything accessible to every user online.  
+There will continue to be the early -access advantage in fast turnaround fields: I t is not 
+that making findings accessible earlier merely gets  them th eir citation “quota” earlier; it 
+significantly increases that quota, probably by both accelerating and br oadening their 
+uptake in further research (Kutz et al. 2005).  And even after the competitive advantage is 
+gone because all articles are OA, the download advantage  will continu e to be enjoyed by 
+all articles (Bollen et al 2009; Davis et al. 2008), while the  quality advantage will see to it 
+that for the best work, increased downloads are translated into uptake, usage and eventual 
+increased citations, of which earlier download increases have been found to be predictive 
+(Brody et al. 2006).  
+ 
+Summary and Conclus ion 
+  
+The assumption that increasing access to research will increase its usage and impact is the 
+main rationale for the worldwide OA movement. Many prior studies have by now shown 
+across all fields that those journal articles whose authors have made them Open A ccess 
+by self -archiving them online, freely accessible to all potential users, are cited 
+significantly more than articles that are accessible only to subscribers. There  is prior 
+evidence  for a self -selection bias in a few special fields (such as astronomy and some 
+areas of physics) where most articles are made OA in unrefereed preprint form long 
+before they are refereed and published, and where the published version is effectively 
+accessible to all potential users as soon as it is published . Authors may ind eed be more 
+reluctant to make articles about which they have doubts OA before they are refereed 
+(Kurtz et al. 2005; Moed 2006). We have now shown that for most other fields (i) the OA 
+Advantage remains just as high for mandatory self -archiving as for self -selected self -
+archiving and that (ii) this is not an artifact of biases in other correlates of citation counts.  
+Both the self -archiving and the mandates apply to refereed postprints, upon acceptance 
+for publication, not to unrefereed preprints.  
+Hence the O A Advantage is real, independent and causal. It is indeed true that the size of 
+the advantage is correlated with quality, just as citations themselves are (the top 20% of 
+articles receiving about 80% of all citations); but what that means is that the OA 
+advantage is higher for the more citeable articles, not because of a quality bias from 
+author self -selection but because of a quality advantage that OA enhances  by maximizing 
+accessibility, and thereby also citeability. On a playing field leveled by OA, users can 20 
+ selectively access, use and cite those articles that they judge to be of the highest relevance 
+and quality, no longer constrained by their accessibility .  
+Overall, only about 15% of articles are being spontaneously self -archived today, self -
+selectively. To reach 100% OA globally, researchers' institutions and funders need to 
+mandate self -archiving, as they are now increasingly beginning to do . We hope that this 
+demonstration that the OA Impact Advantage is real and causal will give further 
+incentive and impetus for the adoption of OA mandates worldwide  in order to  allow 
+research to achieve its f ull impact potential , no longer constrained by limits on 
+accessibility  (Brody et al. 2007; Bernius & Hanauske 2009 ; Carr & Harnad 2009; Dror & 
+Harnad 2009 ). 
+To measure that maximized research impact, we and others are already developing new 
+OA metrics for monitoring, analyzing, evaluating, crediting and rewarding research 
+productivity and progress ( Adler & Harzing 2009; Bollen et al 2009; Brody 2003; Brody 
+et al 2006; Cronin 1984; Cronin & Meho 2006;  De Bellis 2009; De Robbio 2009; 
+Diamond 1986; Harzing 20 08; Harnad 2009 ; Jacso 2006;  Moed 2005a ). Hence there is 
+no need to have any penalties or sanctions for non -compliance with OA self -archiving 
+mandates. As the experience of Southampton ECS, Minho, QUT and CERN has already 
+demonstrated, OA mandates, togeth er with OA’s own rewards (enhanced research access, 
+usage and impact), will be enough to establish the causal connection between providing 
+access and reaping its impact, through the research community’s existing system for 
+evaluating and rewarding research  productivity. In the online era, researchers’ own 
+“mandate ” will no longer just be “publish -or-perish” but “self -archive to flourish.”  
+ 
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+Hajjem, C., Harnad, S. and Gingras, Y. (2005 ) Ten-Year Cross -Disciplinary Comparison 
+of the Growth of Open Access and How it Increases Research Citation Impact . IEEE 
+Data Engineering Bulletin 28(4) 39 -47. 
+ 
+Harnad, S. (2009) Open Access Scientometrics and the UK Research Assessment 
+Exercise . Scientometrics  79 (1)  
+Harnad, S. & Brody, T. (2004) Comparing the Impact of Open Access (OA) vs. Non -OA 
+Articles in the Same Journals, D-Lib Magazine 10 (6) 
+http://www.dlib.org/dlib/june04/harnad/06harnad.html  
+ 
+Harzing, A.W.K.; Wal, R. van der (2008) Google Scholar as a new source for citation 
+analysis?  Ethics in Science and Environmental Politics , vol. 8, no. 1, pp. 62 -71.  
+ 23 
+ Henneken, E. A., Kurtz, M. J., Accomazzi, A., Grant, C. S., Thomson , D., Bohlen, E. and 
+Murray, S. S. (2008)  Use of Astronomical Literature - A Report on Usage Pa tterns 
+Journal of Informetrics  3(1) 1 -90 http://arxiv.org/abs/0808.0103  
+ 
+Henneken, E. A., Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C., Thomson , D., and 
+Murray, S. S. (2006) Effect of E -printing on Citation Rates in Astronomy and Physics . 
+Journal of Electronic Publishing   9(2)  
+ 
+Hitchcock, S. (2009) The effect of open access and downloads ('hits') o n citation impact: 
+a bibliography of studies   
+ 
+Holmes A & Oppenheim C (2001 ) Use of citation analysis to predict the outcome of the 
+2001 Research Assessment Exercise for Unit of Assessment (UoA)  Library and 
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+Jacso, P (2006) Testing the Calculation of a Realistic h -index in Google Scholar, Scopus, 
+and Web of Science for F. W. Lancaster . Library Trends  56(4) 784 -815.  
+ 
+Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C. S., Demleitner, M., Murray, S. S. 
+(2005)  The Effect of Use and Access on Citations . Information Processing and 
+Managem ent 41 (6) 1395 -1402   
+ 
+Kurtz, M. & Brody, T. (2006) The impact loss to authors and research, in Jacobs,  
+Neil, Eds. Open Access: Key Strategic, Technical and Economic Aspects . Chandos 
+Publishing (Oxford) Limited.  
+ 
+Kurtz, M. J. and Henneken, E. A. (2007)  Open Access does not increase citations for 
+research articles from The Astrophysical Journal   
+ 
+Lariviere, V; Y Gingras & E Archambault (2009) The decline in the concentration of 
+citations, 1900 -2007 . JASIST 60(4) 858 -862.   
+ 
+Lawrence, S. (2001) Free online availability substantially increases a paper's impact  
+Nature  411:521  
+ 
+Moed,  H. F. (2005 a) Citation Analysis in Research Evaluation . NY Springer.  
+ 
+Moed, H. F. (2005 b) Statistical Relationships Between Downloads and Citations at the 
+Level of Individual Documents Within a Single Journal. Journal of the American Society 
+for Information Science and Technology  56(10) 1088 -1097  
+ 
+Moed, H. F. (2006) The effect of 'Open Access' upon citation impact: An analysis of 
+ArXiv's Condensed Matter Section  Journal of the American Society for Information 
+Science and Technology  58(13) 2145 -2156  
+ 24 
+ Norris, M., Oppenheim, C., & Rowland, F. (200 8) The citation advantage of open -access 
+articles   
+Journal of the American Society for Information Science and Technology   59(12) 1963 -
+1972  
+ 
+Odlyzko, A. (2006) The economic costs of toll access, in Jacobs, Nei l, Eds. Open Access: 
+Key Strategic, Technical and Economic Aspects . Chandos Publishing (Oxford) Limited.  
+ 
+Oppenheim, Charles (1996) Do citations count? Citation indexing and the research 
+assessment exercise , Serials , 9:155 -61  
+ 
+Page, L., Brin, S., Motwani, R., Winograd, T. (1999) The PageRank Citation Ranking: 
+Bringing Order to the Web .   
+ 
+Sale, A. (2006) The acquisition of open access research articles . First Monday , 11(9), 
+October 2006.  
+ 
+Seglen, PO (1992) The skewness of science. Journal of the American Society for 
+Information Science  43:628 -38. 
+ 
+Swan, A. (2006) The culture of Open Access: researchers’ views and responses . In : 
+Jacobs, N., Eds. Open Access: Key Strategic, Technical and Economic Aspects . Oxford : 
+Chandos/ 52 -59, 2006.  
+ 
+Swan, A. and Carr, L (2008). Institutions, their repositories and the Web . Serials Review  
+34 (1) 2008.   
+ 
+Wallace, ML, V Larivière,  Gingras, (2009) Modeling a Century of Citation 
+Distributions . Journal of Informetrics  3(4): 296 -303 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 25 
+  
+Appendix 1  
+ 
+Figure 12:  OA Impact Advantage for Self -Selected vs Mandatory OA  for Southampton 
+ECS  
+ 
+ 
+ 
+ -0.200.20.40.60.81
+O/Ø O M/Ø S O S/Ø O M/Ø O S/Ø S O M/Ø M O M/O S2002
+2003
+2004
+2005
+2006Southampton26 
+  
+Figure 13:  OA Impact Advantage for Self-Selected vs Mandatory OA  for 
+Queensland UT  
+ 
+ 
+ 
+ -0.200.20.40.60.81
+O/Ø O M/Ø S O S/Ø O M/Ø O S/Ø S O M/Ø M O M/O S2002
+2003
+2004
+2005
+2006Queensland27 
+  
+Figure 14:  OA Impact Advantage for Self -Selected vs Mandatory OA  for Minho  
+  -0.200.20.40.60.81
+O/Ø O M/Ø S O S/Ø O M/Ø O S/Ø S O M/Ø M O M/O S2002
+2003
+2004
+2005
+2006Minho28 
+  
+ 
+Figure 15:  OA Impact Advantage for Self -Selected vs Mandatory OA  for CERN  
+ 
+  00.10.20.30.40.50.60.70.80.91
+O/Ø O M/Ø S O S/Ø O M/Ø O S/Ø S O M/Ø M O M/O S2002
+2003
+2004
+2005
+2006CERN29 
+  
+  
+Appendix 2: Multiple regression by JIF – Beta values  
+A. JIF1 (JIF < 0,633)  
+Model N.  Dependent Var.  Age Ref_N  Auth_N  Page_N  OA M USA  Review  Sci CERN  South  Minho  Queens  Age*OA  
+M_1  Cit_a_0&1 -5 1,537  1,017  1,079          0,701   1,093  
+M_2  Cit_a_1 -5&5-10 1,847  1,013  1,066    1,881         1,059  
+M_3  Cit_a_1 -5&10 -20 2,071  1,026  1,054  0,962  1,533  1,902          
+M_4  Cit_a_1 -5&20+  2,689  1,020  1,087   2,406    4,760  3,214       
+Table 4:  The Exp(ß) values for logistic regressions for GIF1  
+ 
+B. JIF2 (0,633 <= JIF < 1,053)  
+ 
+Model N.  Dependent Var.  Age Ref_N  Auth_N  Page_N  OA M USA  Review  Sci CERN  South  Minho  Queens  Age*OA  
+M_1  Cit_a_0&1 -5 1,407  1,016  1,028    1,265   0,605   0,511      
+M_2  Cit_a_1 -5&5-10 1,548  1,012    1,346  1,963          
+M_3  Cit_a_1 -5&10 -
+20 1,869  1,018  1,007   1,337  1,722          
+M_4  Cit_a_1 -5&20+  2,117  1,011    2,322    3,106        
+Table 5:  The Exp(ß) values for logistic regressions for GIF2  
+ 
+C. JIF3 (1,053 <= JIF < 1,743)  
+ 
+Model N.  Dependent Var.  Age Ref_N  Auth_N  Page_N  OA M USA  Review  Sci CERN  South  Minho  Queens  Age*OA  
+M_1  Cit_a_0&1 -5 1,581  1,012  1,032   1,236      0,401    1,856   
+M_2  Cit_a_1 -5&5-10 1,540  1,007  1,033    1,428  1,330         
+M_3  Cit_a_1 -5&10 -20 1,879  1,013  1,026   1,263         1,382   
+M_4  Cit_a_1 -5&20+  2,305  1,009  1,041  1,026  1,449  1,492  1,791  1,939    3,734     
+Table 6:  The Exp(ß) values for logistic regressions for GIF3  
+ 
+ 30 
+  
+ 
+ 
+ 
+D. JIF4 (1,743 <= JIF < 2,468)  
+ 
+Model N.  Dependent Var.  Age Ref_N  Auth_N  Page_N  OA M USA  Review  Sci CERN  South  Minho  Queens  Age*OA  
+M_1  Cit_a_0&1 -5 1,690  1,020        2,090  0,554  0,233     
+M_2  Cit_a_1 -5&5-10 1,427  1,010     1,645     1,657      
+M_3  Cit_a_1 -5&10 -20 1,800  1,019     1,729     1,615      
+M_4  Cit_a_1 -5&20+  2,540  1,024  0,994  1,028  1,747   1,822    3,974      
+Table 7:  The Exp(ß) values for logistic regressions for GIF4  
+ 
+ 
+E. JIF5 (2.468 <= JIF < 29,957)  
+ 
+ 
+Model N.  Dependent Var.  Age Ref_N  Auth_N  Page_N  OA M USA  Review  Sci CERN  South  Minho  Queens  Age*OA  
+M_1  Cit_a_0&1 -5 1,484  1,016       0,182   0,446      
+M_2  Cit_a_1 -5&5-10 1,312  1,010   0,976   1,468  1,391  0,586        
+M_3  Cit_a_1 -5&10 -20 1,590  1,007  0,998    1,360       1,751   
+M_4  Cit_a_1 -5&20+  2,259  1,009  0,995   1,722   1,635  1,650  2,007       
+Table 8:  The Exp(ß) values for logistic regressions for JIF5  
+ 
+ 
\ No newline at end of file
diff --git a/1001.0362.txt b/1001.0362.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9d748998663d53a7720afffa893abf99109fbd2d
--- /dev/null
+++ b/1001.0362.txt
@@ -0,0 +1,598 @@
+arXiv:1001.0362v1  [nucl-th]  3 Jan 2010Photodisintegration of3Hin athree dimensional faddeev approach
+S. Bayegan1,a, M.A. Shalchi1,andM.R. Hadizadeh2
+1DepartmentofPhysics,UniversityofTehran,P.O.Box14395 -547,Tehran,Iran
+2Instituto de F´ ısica Te´ orica, Universidade Estadual Paul ista, Rua Dr. Bento Teobaldo Ferraz 271, Bl. II, Barra Funda,
+01140-070,S˜ aoPaulo,Brazil
+Abstract. An interaction of a photon with3His invstigated based on a three dimensional Faddeev approac h.
+Inthis approach the three-nucleon Faddeevequations witht wo-nucleon interactions areformulated withconsid-
+eration of the magnitude of the vector Jacobi momenta and the angle between them with the inclusion of the
+spin-isospin quantum numbers, without employing a partial wave decomposition. In this formulation the two
+body t-matricesand tritonwave function are calculated int he three dimensional approach using AV18potential.
+Inthe firststepwe use the standard single nucleon current in thisarticle.
+1 Introduction
+Since the early days of the study of the nuclear physicsso
+manyeffortshavebeenperformedon3Nsystemsconsider-
+ingreal or virtualphotoninteractions[1]-[2].Also sever al
+studiesonthebehaviorof3Nboundstatesinrealorvirtual
+photon absorbtion have been reported[3]-[4]. Before six-
+ties variational approach was used for these calculations
+and works using this approach are still continuing. After
+introducing Faddeev formulation for three body systems
+[5]-[6], new efforts using this scheme were started. As an
+example one can point out the early calculations of elec-
+trodisintegration [7] and photodisintegration[8] with3He
+and3H.Animprovementinthephotodisintegrationcalcu-
+lation of the bound and 3N continuum with the same 3N
+hamiltonianhavebeenperformed[9].Therearealso other
+approaches to calculate electromagnetic interactions wit h
+light nuclei such as Green-function-Monte-Carlo method
+[10],hypersphericalharmonicexpansions[11],andLorent z
+integraltransform(LIT)method[12].Thereis a verygood
+review of Faddeev calculations on the interaction of real
+or virtual photon with3He[13]. In this work like previ-
+ous calculations the partial wave decomposition has been
+used. In PW approach one should sum all PW to max-
+imum angular momentum where the calculation is con-
+verged. The problem is that in higher energies this max-
+imum angular momentum increases and we should solve
+morecomplicatedequations.Toavoidthiscomplexityone
+should use vector momentum as basis states [14]. To this
+aim in the past decade the main steps have been taken
+byOhio-Bochumcollaboration(Elster,Gl¨ ockle et al.) and
+Bayeganet al. to implementthe 3D approachin few-body
+bound and scattering calculations (see for examples Refs.
+[15]-[22]).Itshouldbeclearthatthebuildingblockstoth e
+few-bodycalculationswithoutangularmomentumdecom-
+ae-mail:bayegan@khayam.ut.ac.irposition are two-body o ff-shell t-matrices, which depend
+on the magnitudesof the initial and final Jacobi momenta
+and the angle between them. Fachruddin et al. have cal-
+culated the NN bound and scattering states in a 3D repre-
+sentation using the Bonn-B and the AV18 potentials [15]-
+[16]. Recentlythere hasbeene ffortsto dothe same calcu-
+lation using chiral potential [23]. our aim in this work is
+to formulate photodisintegration of3Hin a three dimen-
+sional Faddeev approach. In the first step we ignore three
+bodyforcesandwejustusethesinglenucleoncurrent.We
+willuseAV18potentialandtritonwavefunctionwhichhas
+beencalculatedin ourpreviouswork[20].
+This manuscript has been organized as follow: in sec-
+tion 2 we explain our basic states and we evaluate all of
+the matrix elements in these basis. In section 3 we intro-
+duceoursingularityproblemandits solution.We finish in
+section4 witha summaryandoutlook.
+2 Integralequationofnuclearmatrix
+elementswithoutpartialwave
+decomposition
+Tocalculatethephotodisintegrationobservablewefirstne ed
+tocalculatenuclearmatrixelementsintheFaddeevscheme.
+Formoredetailssee Ref.[13].
+N=1
+2/an}bracketle{tφ0|(1+tG0)P|U/an}bracketri}ht (1)
+|U/an}bracketri}ht=(1+P)J|ψ/an}bracketri}ht+tG0P|U/an}bracketri}ht (2)
+Inaboveequations tisNNt-operatorwhichobeysLipmann-
+Schwinger equations, G0is free propagator, Pis permuta-
+tion operator,|ψ/an}bracketri}htis three body bound state and |U/an}bracketri}htis an
+axillarystate.Threebodyforceshavebeenignored. |φ0/an}bracketri}htisEPJWeb ofConferences
+a subsection of the fully antisymmetric free state, |Φ0/an}bracketri}ht, in
+whichnucleons2and3 areinsubsystem.
+|Φ0/an}bracketri}ht=(1+P)|φ0/an}bracketri}ht (3)
+|φ0/an}bracketri}htisalsoourbasicstatetosolvetheintegralequation
+(2) and is antisymmetric under permutation of nucleons 2
+and3.
+|φ0/an}bracketri}ht≡|pqm1m2m3ν1ν2ν3/an}bracketri}hta(4)
+In equation (4) pandqare jacobi momenta and m’s
+andν’s are the spin and isospin of the individualnucleons
+respectively.
+Orthonormalityandcompletenessrelationsoftheseba-
+sic statescanbeconsideredasbellow:
+a/an}bracketle{tpqm1m2m3ν1ν2ν3|p′q′m′
+1m′
+2m′
+3ν′
+1ν′
+2ν′
+3/an}bracketri}hta
+=1
+2{δ(p−p′)δm2m′
+2δm3m′
+3δν2ν′
+2δν3ν′
+3
+−δ(p+p′)δm2m′
+3δm3m′
+2δν2ν′
+3δν3ν′
+2}δ(q−q′)δm1m′
+1δν1ν′
+1(5)
+/summationdisplay
+m1m2m3
+ν1ν2ν3/integraldisplay
+d3pd3q|pqm1m2m3ν1ν2ν3/an}bracketri}hta
+×a/an}bracketle{tpqm1m2m3ν1ν2ν3|≡1 (6)
+Considering these properties we can rewrite the integral
+equations(1)and(2)inourbasicstats.
+N=1
+2a/an}bracketle{tpqm1m2m3ν1ν2ν3|(1+tG0)P|U/an}bracketri}ht
+=1
+2a/an}bracketle{tpqm1m2m3ν1ν2ν3|P|U/an}bracketri}ht
++1
+2a/an}bracketle{tpqm1m2m3ν1ν2ν3|tG0P|U/an}bracketri}ht (7)
+a/an}bracketle{tpqm1m2m3ν1ν2ν3|U/an}bracketri}ht
+=a/an}bracketle{tpqm1m2m3ν1ν2ν3|(1+P)J|ψ/an}bracketri}ht
++a/an}bracketle{tpqm1m2m3ν1ν2ν3|tG0P|U/an}bracketri}ht (8)
+The effect of permutation operator on our basic states
+canbeconsideredasfollow:
+/an}bracketle{tpqm1m2m3ν1ν2ν3|P|p′q′m′
+1m′
+2m′
+3ν′
+1ν′
+2ν′
+3/an}bracketri}ht
+=δ(p+1
+2p′+3
+4q′)δ(q−p′+1
+2q′)
+×δm1m′
+2δm2m′
+3δm3m′
+1δν1ν′
+2δν2ν′
+3δν3ν′
+1
++δ(p+1
+2p′−3
+4q′)δ(q+p′+1
+2q′)
+×δm1m′
+3δm2m′
+1δm3m′
+2δν1ν′
+3δν2ν′
+1δν3ν′
+2
+=δ(p+π2)δ(p′−π1)
+×δm1m′
+2δm2m′
+3δm3m′
+1δν1ν′
+2δν2ν′
+3δν3ν′
+1
++δ(p−π2)δ(p′+π1)
+×δm1m′
+3δm2m′
+1δm3m′
+2δν1ν′
+3δν2ν′
+1δν3ν′
+2(9)Where:
+π1=q+1
+2q′π2=1
+2q+q′(10)
+Now with respect to aboverelation and symmetry consid-
+erationswe canevaluateequations(7)and(8)asfollow:
+N=1
+2{/an}bracketle{t−1
+2p−3
+4q,p−1
+2qm2m3m1ν2ν3ν1|U/an}bracketri}ht
+/an}bracketle{t−1
+2p+3
+4q,−p−1
+2qm3m1m2ν3ν1ν2|U/an}bracketri}ht}
++/summationdisplay
+m′
+2m′
+3ν′
+2ν′
+3/integraldisplay
+d3qa/an}bracketle{tpm2m3ν2ν3|t|1
+2q+q′,m′
+2m′
+3ν′
+2ν′
+3/an}bracketri}hta
+1
+E−q2+q′2+q·q′
+m/an}bracketle{t−1
+2q′−q,q′m′
+2m′
+3m1ν′
+2ν′
+3ν1|U/an}bracketri}ht(11)
+/an}bracketle{tpq,m1m2m3ν1ν2ν3|U/an}bracketri}ht
+=/an}bracketle{tpq,m1m2m3ν1ν2ν3|(1+P)J|ψ/an}bracketri}ht
++/summationdisplay
+m′
+2m′
+3ν′
+2ν′
+3/integraldisplay
+d3qa/an}bracketle{tpm2m3ν2ν3|t|1
+2q+q′,m′
+2m′
+3ν′
+2ν′
+3/an}bracketri}hta
+1
+E−q2+q′2+q·q′
+m/an}bracketle{t−1
+2q′−q,q′m′
+2m′
+3m1ν′
+2ν′
+3ν1|U/an}bracketri}ht(12)
+Thefirstermintheequation(12)canbeevaluatedas:
+a/an}bracketle{tpq,m1m2m3ν1ν2ν3|(1+P)J|ψ/an}bracketri}ht
+=/summationdisplay
+m′,ν′/integraldisplay
+d3p′d3q′
+a/an}bracketle{tpq,m1m2m3ν1ν2ν3|(1+P)J|p′q′,m′
+1m′
+2m′
+3ν′
+1ν′
+2ν′
+3/an}bracketri}hta
+×a/an}bracketle{tp′q′,m′
+1m′
+2m′
+3ν′
+1ν′
+2ν′
+3|ψ/an}bracketri}ht (13)
+Nowweconcentrateontheelementsoftheseequationsi.e.
+current,two-bodyt-matrix and triton wave function,more
+precisely.
+2.1 current
+Consideringthesymmetrypropertieswehave:
+a/an}bracketle{tpq,m1m2m3ν1ν2ν3|(1+P)JSN|ψ/an}bracketri}ht
+=3a/an}bracketle{tpq,m1m2m3ν1ν2ν3|(1+P)JSN(1)|ψ/an}bracketri}ht
+(14)
+Matrixelementsofsinglenucleoncurrentcanbeevaluated
+asfollow:
+a/an}bracketle{tpq,m1m2m3ν1ν2ν3|J(1)|p′q′,m′
+1m′
+2m′
+3ν′
+1ν′
+2ν′
+3/an}bracketri}ht
+=δ(q′−q+2
+3Q)
+×1
+2[δ(p−p′)δm2m′
+2δm3m′
+3δν2ν′
+2δν3ν′
+3
+−δ(p+p′)δm2m′
+3δm3m′
+2δν2ν′
+3δν3ν′
+2]×Jm1m′
+1ν1ν′
+1(Q,q)
+(15)19thInternationalIUPAP ConferenceonFew-BodyProblemsinPhy sics
+Inaboveequation Qisthemomentumofphoton.We need
+to rewrite the single nucleon current operator in a form
+which is suitable for our basic states. The current opera-
+torwhichwewill use is:
+J=GE(Q)k1+k′
+1
+2mN+i
+2mNGM(Q)σ×(k1−k′
+1) (16)
+Which is summation of convection current and spin cur-
+rent.GE(Q)andGM(Q)areelectricandmagneticformfac-
+torsrespectively.Fortheconvectionpartwehave:
+k1+k′
+1=2q+Q+2
+3K (17)
+As we will show we have to choose coordinate system in
+which the zaxis is along the Qvector and we also need
+tensor component of current so the second and the third
+terms of the right hand side of equation(17) will vanish.
+Thusforthe convectioncurrentwehave:
+Jconvec
+±1=GE(Q)q±1
+mN(18)
+Andthetensorcomponentofspinpartcanalsobeeval-
+uatedas:
+JSpin
+±1=−√
+2Q
+2mNGM(Q)S± (19)
+2.2 two-body t-matrix
+Two body t-matrices can be related to the one which cal-
+culatedinhelicitybasis:
+a/an}bracketle{tpm1m2nν1ν2|t|p′m1m2ν′
+1ν′
+2/an}bracketri}hta=1
+4δ(ν1+ν2),(ν′
+1+ν′
+2)
+ei(Λ0φp−Λ′
+0φ′
+p)/summationdisplay
+πst(1−ηπ)C(1
+21
+2t,ν1ν2)C(1
+21
+2t,ν′
+1ν′
+2)
+C(1
+21
+2S,m1m2Λ0)C(1
+21
+2S,m′
+1m′
+2Λ′
+0)
+/summationdisplay
+ΛΛ′dS
+Λ0Λ(θp)dS
+Λ′
+0Λ′(θ′
+p)/summationtexteiN(φp−φ′
+p)dS
+NΛ(θp)dS
+NΛ′(θ′
+p)
+dS
+Λ′Λ(θpp′)
+tπSt
+ΛΛ′(p,p′,cosθpp′,z) (20)
+In the above relation z=E−3q2
+4mis the energy of subsys-
+tem. As we know two-body function has a singularity in
+the energyof deuteron, z=Ed. To removethis singularity
+we shouldconsidert-operatorasfollow:
+a/an}bracketle{tpm1m2nν1ν2|t|p′m1m2ν′
+1ν′
+2/an}bracketri}hta
+=a/an}bracketle{tpm1m2nν1ν2|ˆt|p′m1m2ν′
+1ν′
+2/an}bracketri}hta
+z−Ed(21)
+Twobodyt-matrixinhelicitybasishasbeencalculated
+before[15].tπSt
+Λ′Λ/parenleftbigp′,p,cosθ/parenrightbig=VπSt
+Λ′Λ/parenleftbigp′,p,θ/parenrightbig
++1
+2/integraldisplay
+dp′′p′′21/integraldisplay
+−1d/parenleftbigcosθ′′/parenrightbigvπSt,Λ
+Λ′1/parenleftbigp′,p′′,θ′,θ′′/parenrightbigG0/parenleftbigp′′/parenrightbig
+tπSt
+1Λ/parenleftbigp′′,p,θ′′/parenrightbig
++1
+2/integraldisplay
+dp′′p′′21/integraldisplay
+−1d/parenleftbigcosθ′′/parenrightbigvπSt,Λ
+Λ′0/parenleftbigp′,p′′,θ′,θ′′/parenrightbigG0/parenleftbigp′′/parenrightbig
+tπSt
+0Λ/parenleftbigp′′,p,θ′′/parenrightbig(22)
+Where
+vπSt,Λ
+Λ′Λ′′(p′,p′′,θ′,θ′′)=2π/integraldisplay
+0dφ′′e−iΛ(φ′−φ′′)VπSt
+Λ′Λ′′/parenleftbigp′,p′′/parenrightbig
+(23)
+2.3 tritonwave function
+For evaluating the Triton wave function we need to make
+a relationbetweenthiswavefunctioninourbasicstatesto
+theonewhichhasbeencalculatedinthefollowingbasis[20] :
+/an}bracketle{tpq(s1
+2)SmS(t1
+2)TmT|ψ/an}bracketri}ht=/an}bracketle{tpq,Xpq,α|ψ/an}bracketri}ht(24)
+Wecanrelatethesestatestoourfreespinandisospinstates
+with Clebsch-Gordancoe fficients.
+gγα=/an}bracketle{tγ|α/an}bracketri}ht (25)
+Where
+|α/an}bracketri}ht=|(s1
+2)SmS,(t1
+2)TmT/an}bracketri}ht
+|γ/an}bracketri}ht=|m1m2m3ν1ν2ν3/an}bracketri}ht (26)
+It is very important to mention that the spin of the nucle-
+ons is quantized in the direction of the zaxis which in the
+calculationofwavefunctionithasbeenchosentobeinthe
+directionof q.Butwehavetoconsiderthe zaxisalongthe
+direction of incident photon Q. So we should first rotate
+the spin of the nucleons in our basis to be settled in the
+direction of qaxis. Then we should use Clebsch-Gordan
+coefficients to obtain the wave function in the calculated
+basismentionedin theequation(24):
+/an}bracketle{tpqm1m2m3ν1ν2ν3|ψ/an}bracketri}ht
+=/summationdisplay
+m′
+1m′
+2m′
+3/summationdisplay
+αDm1m′
+1(θq,φq)Dm2m′
+2(θq,φq)Dm3m′
+3(θq,φq)gγα
+×/an}bracketle{tpq,Xpq,α|ψ/an}bracketri}ht (27)EPJWeb ofConferences
+3 Singularity problem
+Inordertoconsiderthesingularityproblemwecanrewrit-
+ten the equation (11) an (12) in a unified form ignoring
+isospindependentwhichissimilarto spindependent.
+Um1m2m3(p,q,Q)=U′
+m1m2m3(p,q,Q)
++/summationdisplay
+m′
+2m′
+3/integraldisplay
+d3q′′Um′
+2m′
+3m1(π2,q′′,Q)
+E−q2+q′′2−q·q′′
+mˆta
+m2m3m′
+3m′
+3(p,π1,z)
+E+iǫ−Ed−3q2
+4m
+(28)
+Tosolvethisintegralequationweshouldevaluatesingular -
+ity in the denominator of the propagator which is a func-
+tion ofq′′and angle between q′′andq. So instead of sin-
+gularpointwehavearegionofsingularityin q−q′′plane.
+There is a solution to this moving singularity in Ref.[24].
+For using this method we have to put zaxis along the q.
+But because of simplification in current operatorand final
+cross section we should choose the zaxis in the direction
+ofthemomentumofthephoton, Q.Soinordertoevaluate
+the singularitywe shoulduse another methodwhich is in-
+troducedin Ref.[25]. Therefore one should separate angle
+partofdeltafunctionsasfollow:
+δ(p′+π1)δ(p′′−π1)=δ(p′−π1)
+p′2δ(p′′−π1)
+p′′2
+δ(ˆp′+ˆπ1)δ(ˆp′′−ˆπ1)
+δ(p′−π1)δ(p′′+π1)=δ(p′−π1)
+p′2δ(p′′−π1)
+p′′2
+δ(ˆp′−ˆπ1)δ(ˆp′′+ˆπ1) (29)
+Andthenthe integralequationcanbe rewriteasfollow:
+Um1m2m3(p,q,Q)=U′
+m1m2m3(p,q,Q)
++/summationdisplay
+m′
+2m′
+3/integraldisplay
+d3q′′dp′dp′′
+Um′
+2m′
+3m1(π2,q′′,Q)
+E−1
+m(p′′2+3
+4q′′2)ˆta
+m2m3m′
+3m′
+3(p,π1,z)
+E+iǫ−Ed−3q2
+4m
+(30)
+After some simplification the integralequationtransforms
+tothisequation:
+Um1m2m3(p,q,Q)=U′
+m1m2m3(p,q,Q)
++2
+q/summationdisplay
+m′
+2m′
+3∞/integraldisplay
+0dp′p′ 1
+E+iǫ−1
+m(p′2+3
+4q2)
+q/2+p′/integraldisplay
+|q/2−p′|dq′′q′′¯G(q,q′′,p′)/integraldisplay
+dˆq′′δ(x′′−x0)
+Um′
+2m′
+3m1(p′′ˆπ2,q′′,Q)ˆta
+m2m3m′
+3m′
+3(p,p′ˆπ1,z)
+−2
+q/summationdisplay
+m′
+2m′
+3∞/integraldisplay
+0dq′′q′′ 1
+E+iǫ−Ed−3q2
+4mq/2+q′′/integraldisplay
+|q/2−q′′|dp′p′¯G(q,q′′,p′)/integraldisplay
+dˆq′′δ(x′′−x0)
+Um′
+2m′
+3m1(p′′ˆπ2,q′′,Q)ˆta
+m2m3m′
+3m′
+3(p,p′ˆπ1,z) (31)
+Intheaboveequation ¯Gwhichisalwayspositiveisdefined
+as:
+¯G(q,q′′,p′)=1
+−Ed−3q′′2
+4m+1
+m(p′2+3
+4q2)(32)
+x′′=cosθ′′indicatesthe angle between qandq′′and
+x0isintroducedasfollow:
+x0=1
+qq′′(p′2−1
+4q2−q′′2)=1
+qq′′(p′′2−1
+4q′′2−q2)
+(33)
+4 Summaryandoutlook
+InthispaperwehaveformulatedtheFaddeevintegralequa-
+tions for calculating the photodisintegrationobservable of
+triton in a three dimensional approach.To this aim we in-
+troduced our basic states which contains jacobi momenta
+in vector forms as well as individual spin and isospin of
+each nucleon. So we have avoided to decompose angle
+states in terms of angular momentum states (partial wave
+approach)whichistraditionallyusedtosolvethesekindof
+equations.Thefinalintegralequationsarelesscomplicate d
+than the PW ones and are unique in number of the equa-
+tions in all energies. We have also explained about over-
+comingofthemovingsingularityin ourwork.
+The calculation of this observable using the AV18 po-
+tential isunderwayandtheresultswill be publishedsoon.
+Adding two and three body currents as well as three
+bodyforcesinourcalculationsareotherfuturemajorworks .
+The same calculation for radiative capture is also under
+consideration.
+Acknowledgments
+Thisworkwassupportedbycenterofexcellenceonstruc-
+tureofmatter,DepartmentofPhysics,UniversityofTehran .
+References
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\ No newline at end of file
diff --git a/1001.0363.txt b/1001.0363.txt
new file mode 100644
index 0000000000000000000000000000000000000000..417fcc7b6b9bcda2398eddb9e7ec840f2e469770
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+++ b/1001.0363.txt
@@ -0,0 +1,693 @@
+arXiv:1001.0363v1  [physics.gen-ph]  3 Jan 2010Generalized Gravi - Electromagnetism∗
+P. S. Bisht, Gaurav Karnatak and O. P. S. Negi
+October 29, 2018
+Department of Physics,
+Kumaun University,
+S. S. J. Campus,
+Almora-263601(Uttarakhand) India
+Email- ps_bisht123@rediffmail.com,
+gauravkarnatak2009@yahoo.in,
+ops_negi@yahoo.com
+Abstract
+A self consistant and manifestly covariant theory for the dy namics of four charges (masses)
+(namely electric, magnetic, gravitational, Heavisidian) has been developed in simple, compact
+and consistent manner. Starting with an invariant Lagrangi an density and its quaternionic
+representation, we have obtained the consistent field equat ion for the dynamics of four charges.
+It has been shown that the present reformulation reproduces the dynamics of individual charges
+(masses) in the absence of other charge (masses) as well as th e generalized theory of dyons
+(gravito - dyons) in the absence gravito - dyons (dyons).
+key words : dyons, gravito - dyons, quaternion
+PACS NO : 14.80Hv
+1 Introduction
+Magnetic monopoles [1] were advocated to symmetrize Maxwel l’s equations in a manifest way
+that the mere existence of an isolated magnetic charge impli es the quantization of electric charge.
+The fresh interests in this subject are enhanced with the ide a of t’ Hooft and Polyakov [2] that the
+classical solutions having the properties of magnetic mono poles may be found in Yang - Mills gauge
+theories. The Dirac monopoles were elementary particles bu t the t’ Hooft - Polyakov [2] monopoles
+are complicated extended object having a definite mass and fin ite size inside of which massive fields
+play an important role in providing a smooth structure and ou tside it they vanish rapidly leaving the
+field configuration identical to abelian Dirac monopole. Jul ia and Zee [3] extended the theory of non-
+Abelian monopoles of t’ Hooft - Polyakov [2] to the theory of n on Abelian dyons (particles carrying
+∗Oral presentation by Gaurav Karnatak in, ‘Conference on Gra vitation Theories and Astronomy
+(CGTA- 2009)’ on the eve of International year of Astronomy a t Nilkanthrao Shinde Science & Arts
+College, Bhadrawati, District-Chandrapur, Maharstra Sta te, December 28-30, 2009
+1simultaneously electric and magnetic charges). The quantu m mechanical excitation of fundamental
+monopoles include dyons which are automatically arisen fro m the semi-classical quantization of global
+charge rotation degree of freedom of monopoles. Despite of t he potential importance of monopoles
+[1, 2] and dyons [3], the formalism necessary to describe the m has been clumsy and not manifestly
+covariant. So, a self consistent and manifestly covariant t heory of generalized electromagnetic fields
+of dyons (particle carrying electric and magnetic charges) and those for generalized fields of gravi-
+todyons, has been constructed [4, 5] in terms of two four - pot entials to avoid the use of controversial
+string variables. On the other hand,the quaternionic formu lation [6, 7] of electrodynamics has a long
+history [8, 9, 10] , stretching back to Maxwell himself [8] wh o used real (Hamilton) quaternion in his
+original manuscript ’on the application of quaternion to el ectromagnetism’ and in his celebrated book
+“Treatise on Electricity and Magnetism”. Quaternion analy sis has since been rediscovered at regular
+intervals and accordingly the Maxwell’s Equations of elect romagnetism were rewritten as one quater-
+nion equations [11, 12]. Negi and coworkers [13], have also s tudied the quaternionic formulation for
+generalized electromagnetic fields of dyons (particles car rying simultaneous existence of electric and
+magnetic charges) in unique, simpler and compact notations . Kravchenko and co-authors [14, 15],
+discussed the Maxwell’s equations in homogeneous media, ch iral media and inhomogeneous media.
+Accordingly they have developed the quaternionic reformul ation of the time-dependent Maxwell’s
+equations along with the classical solution of a moving sour ce i.e. electron. In the series of papers
+Negi and coworkers [16, 17, 18, 19, 20] have derived the gener alized Dirac - Maxwell (GDM) equations
+in presence of electric and magnetic sources in an isotropic (homogeneous) medium. They have also
+analyzed the other quantum equations of dyons in consistent and manifest covariant way [16]. This
+theory has been shown to remain invariant under the duality t ransformations in isotropic homoge-
+neous medium. Quaternion analysis of time dependent Maxwel l’s equations has been developed by
+them [17] in presence of electric and magnetic charges and th e solutions for the classical problem of
+moving charge (electric and magnetic) are obtained consist ently. The time dependent generalized
+Dirac - Maxwell’s (GDM) equations of dyons are also discusse d [18] in chiral and inhomogeneous
+media and the solutions for the classical problem are obtain ed. The quaternion reformulation of gen-
+eralized electromagnetic fields of dyons in chiral and inhom ogeneous media has also been analyzed
+[19]. The monochromatic fields of generalized electromagne tic fields of dyons have also discussed
+[20] in slowly changing media in a consistent manner. The qua ternion analysis has also been used
+[21] to combine the complex desciption of dyons and gravito - dyons and accordingly the unified
+quaternionic angular momentum for generalized fields of dyo ns and gravito - dyons along with their
+commutation relations has been analyzed in unique and consi stent manner. In this paper, we have
+made an attempt to developed a self consistant and manifestl y covariant theory for the dynamics
+of four charges (masses) (namely electric, magnetic, gravi tational, Heavisidian) has been developed
+in simple, compact and consistent manner. Starting with an i nvariant Lagrangian density and its
+quaternionic representation, we have obtained the consist ent field equation and equation of motion
+for the dynamics of four charges. It has been shown that the pr esent reformulation reproduces
+the dynamics of individual charges (masses) in the absence o f other charge (masses) as well as the
+generalized theory of dyons (gravito - dyons) in the absence gravito - dyons (dyons) and vice versa..
+2 Quaternionic Unified Charge of Dyons and Gravito-dyons
+Let us describe the property of quaternion algebra with the u se of natural units (c=/planckover2pi1=G= 1)in
+order to reformulate the unified theory of generalized elect romagnetic fields (associated with dyons)
+2and generalized gravito - Heavisidian fields (associated wi th gravito - dyons) of linear gravity. So we
+define the unified charge [21, 22] as
+Q= (e, g, m, h ) =e−ig−jm−kh, (1)
+wheree, g, m andhare respectively described as the electric, magnetic, grav itational and Heavisidian
+charges (masses).In equation (1), i, jandkare the quaternion units satisfy the following properties
+ij=−ji=k(say) (2)
+and
+i(ij) = (ii)j=−j,
+(ij)j=i(jj) =−i,
+ik=−ki=−j,
+kj=−jk=−i,
+i2=j2=k2=−1. (3)
+Complex structure (e, g)represents the generalized charge of dyons of electromagne tic fields while
+(m, h)denotes the generalized charge of gravito - dyons. The norm o f unified quaternion charge (1)
+is
+N(Q) =QQ= (e2+g2+m2+h2), (4)
+where
+Q= (e,−g,−m,−h) =e+ig+jm+kh. (5)
+Unified quaternion valued four-potential may then be defined as
+{Vµ}={Aµ}−i{Bµ}−j{Cµ}−k{Dµ}, (6)
+where{Aµ}is the four - potential associated with the dynamics of elect ric charge, {Bµ}is used for
+magnetic charge, {Cµ}describes the gravitational charge (mass) while {Dµ}has been associated
+with the gravi - magnetic (Heavisidian) charge (mass). Then the varius potentials of equation (6)
+are written in the following quaternionic forms,
+3A=A0−iA1−jA2−kA3,
+B=B0−iB1−jB2−kB3,
+C=C0−iC1−jC2−kC3,
+D=D0−iD1−jD2−kD3. (7)
+Similarly one may define [22] the quaternion valued unified fie ld tensor as
+ℑµν=Fµν−iMµν−jfµν−kNµν, (8)
+where
+Fµν=Aµ,ν−Aν,µ−iεµνρσBρσ,
+Mµν=Bµ,ν−Bν,µ−iεµνρσAρσ,
+fµν=Cµ,ν−Cν,µ−iεµνρσDρσ,
+Nµν=Dµ,ν−Dν,µ−iεµνρσCρσ(9)
+are the field tensors respectively associated with the dynam ics of the electric, magnetic, gravitational
+and Heavisidian charges (masses). These field tensors satis fy the following Maxwellian field equations
+Fµν,ν=j(e)
+µ,
+Mµν,ν=j(m)
+µ,
+fµν,ν=j(G)
+µ,
+Nµν,ν=j(H)
+µ, (10)
+wherej(e)
+µis the four - current associated with electric charge, j(m)
+µis the four - current associated
+with magnetic charge, j(G)
+µis the four - current for gravitational charge(mass) while j(H)
+µis the four
+- current for Heavisidian charge (mass). In equations (8 - 10 ), the field tensors FµνandMµνare
+associated with dyons, while fµνandNµνare associated with gravito - dyons. As such the quaternion
+valued current may then be defined as
+Jµ=j(e)
+µ−ij(m)
+µ−jj(G)
+µ−kj(H)
+µ. (11)
+3 Lagrangian Formulation and field equations
+The Lagrangian density for the unified charged particle cont aining the electric, magnetic, gravita-
+tional and Heavisidian charges and the rest mass of the unifie d particle M, may be written in the
+following form
+4L=−M−1
+4/bracketleftbig
+α/braceleftbig
+(Aµ,ν−Aν,µ)2−(Bµ,ν−Bν,µ)2−(Cµ,ν−Cν,µ)2−(Dµ,ν−Dν,µ)2/bracerightbig/bracketrightbig
+−2β[(Aµ,ν−Aν,µ)(Bµ,ν−Bν,µ)+(Cµ,ν−Cν,µ)(Dµ,ν−Dν,µ)]
+−2γ[(Aµ,ν−Aν,µ)(Cµ,ν−Cν,µ)+(Bµ,ν−Bν,µ)(Dµ,ν−Dν,µ)]
+−2∆[(Aµ,ν−Aν,µ)(Dµ,ν−Dν,µ)+(Bµ,ν−Bν,µ)(Cµ,ν−Cν,µ)]
++(αAµ−βBµ+γCµ−∆Dµ)j(e)
+µ−(αBµ−βAµ+γDµ−∆Cµ)j(m)
+µ
++(αCµ−βDµ+γAµ−∆Bµ)j(G)
+µ−(αDµ−βCµ+γBµ−∆Aµ)j(H)
+µ
+=LP+LF+LI, (12)
+whereLPis the free particle Lagrangian, LFis the field Lagrangian and LIis the interaction
+Lagrangian and α, β, γ,∆are real positive arbitrary uni modular parameters satisfy ing the following
+conditions
+α−iβ−jγ−k∆ =e−θˆn= cosθ−ˆnsinθ, (13)
+and
+α+iβ+jγ+k∆ =eθˆn= cosθ+ ˆnsinθ. (14)
+From equations (13) and (14), we get
+α2+β2+γ2+∆2= 1. (15)
+As such, we may write the constancy condition [13] as,
+tanθ=g
+e=h
+m=Bµ
+Aµ=Dµ
+Cµ=j(m)
+µ
+j(e)
+µ=j(H)
+µ
+j(G)
+µ. (16)
+The action integral of the above unified system may be written as,
+S=ˆ
+Ldt=SP+SF+SI, (17)
+where the action part SPdepends upon the properties of the particle, SFdepends on the properties
+of the field and SIdepends on the parameters of the particle and field both.
+In deriving the equation of motion of the particle, we vary th e trajectory of the particle without
+changing the field parameters and as such the action SFdoes not affect the motion. On the other
+hand,in order to find the field equations we take the variation with respect to field parameters
+assuming the trajectory of the particle fixed. For deriving t he equation of motion, we write the
+concerned part of action in the following form,
+5S=SP+SI=−t2ˆ
+t1Mdt
++t2ˆ
+t1(αAµ−βBµ+γCµ−∆Dµ)j(e)
+0dxµ
+dtdt
+−t2ˆ
+t1(αBµ−βAµ+γDµ−∆Cµ)j(m)
+0dxµ
+dtdt
++t2ˆ
+t1(αCµ−βDµ+γAµ−∆Bµ)j(G)
+0dxµ
+dtdt
+−t2ˆ
+t1(αDµ−βCµ+γBµ−∆Aµ)j(H)
+0dxµ
+dtdt, (18)
+wherej(e)
+µ,j(m)
+µ,j(G)
+µandj(H)
+µare expressed in terms of j(e)
+0,j(m)
+0,j(G)
+0andj(H)
+0anddxµ
+dt. Equation
+(18) may also be written as,
+S=−Mˆb
+adS
++ˆb
+a[(αAµ−βBµ+γCµ−∆Dµ)j(e)
+0dxµ
+−ˆb
+a(αBµ−βAµ+γDµ−∆Cµ)j(m)
+0dxµ
++ˆb
+a(αCµ−βDµ+γAµ−∆Bµ)j(G)
+0dxµ
+−ˆb
+a(αDµ−βCµ+γBµ−∆Aµ)j(H)
+0dxµ, (19)
+where the first term is an integral along the world line of the p article between two events aandbi.e
+the presence of the particle at its initial and final position s at time t1andt2. The variation of the
+actionSmay be written as,
+δS=δ{ˆb
+a−MdS
++ˆb
+a[(αAµ−βBµ+γCµ−∆Dµ)j(e)
+0dxµ
+−(αBµ−βAµ+γDµ−∆Cµ)j(m)
+0dxµ
++(αCµ−βDµ+γAµ−∆Bµ)j(G)
+0dxµ
+−(αDµ−βCµ+γBµ−∆Aµ)j(H)
+0dxµ]}= 0. (20)
+Taking the variation of the terms one by one, we get
+6I=δˆb
+a−MdS=ˆb
+aMuµdδxµ, (21)
+whereuµ=dxµ
+dS(four - velocity). Integrationg equation (21) by parts, we g et
+I=|Muµδxµ|b
+a−ˆb
+aMduµδxµ, (22)
+where first term is zero, since the integral is varied between fixed limits,
+(δxµ)b= (δxµ)a= 0. (23)
+Thus
+I=−´b
+aMduµδxµ=−ˆb
+aMduµ
+dSdSδxµ. (24)
+Now. it is convenient to take the following variations in ord er to solve the equations of motion of
+varius charges,
+δˆb
+aαAµj(e)
+0dxµ=ˆb
+aαj(e)
+0(Aν,µ−Aµ,ν)uνδxµdS=ˆb
+aαj(e)
+0FµνuνδxµdS;
+δˆb
+aβBµj(e)
+0dxµ=ˆb
+aβj(e)
+0(Bν,µ−Bµ,ν)uνδxµdS=ˆb
+aβj(e)
+0MµνuνδxµdS;
+δˆb
+aγCµj(e)
+0dxµ=ˆb
+aγj(e)
+0(Cν,µ−Cµ,ν)uνδxµdS=ˆb
+aγj(e)
+0fµνuνδxµdS;
+δˆb
+a∆Dµj(e)
+0dxµ=ˆb
+a∆j(e)
+0(Dν,µ−Dµ,ν)uνδxµdS=ˆb
+a∆j(e)
+0NµνuνδxµdS; (25)
+δˆb
+aαBµj(m)
+0dxµ=ˆb
+aαj(m)
+0(Bν,µ−Bµ,ν)uνδxµdS=ˆb
+aαj(m)
+0MµνuνδxµdS;
+δˆb
+aβAµj(m)
+0dxµ=ˆb
+aβj(m)
+0(Aν,µ−Aµ,ν)uνδxµdS=ˆb
+aβj(m)
+0FµνuνδxµdS;
+δˆb
+aγDµj(m)
+0dxµ=ˆb
+aγj(m)
+0(Dν,µ−Dµ,ν)uνδxµdS=ˆb
+aγj(m)
+0NµνuνδxµdS;
+δˆb
+a∆Cµj(m)
+0dxµ=ˆb
+a∆j(m)
+0(Cν,µ−Cµ,ν)uνδxµdS=ˆb
+a∆j(m)
+0fµνuνδxµdS; (26)
+7δˆb
+aαCµj(G)
+0dxµ=ˆb
+aαj(G)
+0(Cν,µ−Cµ,ν)uνδxµdS=ˆb
+aαj(G)
+0fµνuνδxµdS;
+δˆb
+aβDµj(G)
+0dxµ=ˆb
+aβj(G)
+0(Dν,µ−Dµ,ν)uνδxµdS=ˆb
+aβj(G)
+0NµνuνδxµdS;
+δˆb
+aγAµj(G)
+0dxµ=ˆb
+aγj(G)
+0(Aν,µ−Aµ,ν)uνδxµdS=ˆb
+aγj(G)
+0FµνuνδxµdS;
+δˆb
+a∆Bµj(G)
+0dxµ=ˆb
+a∆j(G)
+0(Bν,µ−Bµ,ν)uνδxµdS=ˆb
+a∆j(G)
+0MµνuνδxµdS; (27)
+and
+δˆb
+aαDµj(H)
+0dxµ=ˆb
+aαj(H)
+0(Dν,µ−Dµ,ν)uνδxµdS=ˆb
+aαj(H)
+0NµνuνδxµdS;
+δˆb
+aβCµj(H)
+0dxµ=ˆb
+aβj(H)
+0(Cν,µ−Cµ,ν)uνδxµdS=ˆb
+aβj(H)
+0fµνuνδxµdS;
+δˆb
+aγAµj(H)
+0dxµ=ˆb
+aγj(H)
+0(Aν,µ−Aµ,ν)uνδxµdS=ˆb
+aγj(H)
+0FµνuνδxµdS;
+δˆb
+a∆Bµj(H)
+0dxµ=ˆb
+a∆j(H)
+0(Bν,µ−Bµ,ν)uνδxµdS=ˆb
+a∆j(H)
+0MµνuνδxµdS. (28)
+Substituting equations (24-28) in equation (20), we get the following equations of motion for the
+dynamics of fields of a particle containing four charges as,
+Md2xµ
+dt2=eFµνuν;
+Md2xµ
+dt2=gMµνuν;
+Md2xµ
+dt2=mfµνuν;
+Md2xµ
+dt2=hNµνuν. (29)
+HereMis the effective mass given by
+M=m−κ−1
+2h (30)
+whereκ= +1 for elcrtomagnetic fields and κ=−1for gravito-Heavisidian fields. The field equations
+(29) may also be obtained from the action (17) by taking the tr ajectory of the unified particle fixed
+and considering the variation of the field parameters (i.e. p otentials) only. So, the parameters
+associated unified charged particle such as the free particl e action and four - current density Jµare
+treated to be constant i.e. δSP= 0.From equation (12), we may write the field and interaction par ts
+of the Lagrangian required for the variation as,
+8LF+LI=−1
+4[α(FµνFµν−MµνMµν−fµνfµν−NµνNµν)]
+−2β[FµνMµν+fµνNµν]−2γ[Fµνfµν+MµνNµν]−2∆[FµνNµν+fµνMµν]
++(αAµ−βBµ+γCµ−∆Dµ)j(e)
+µ−(αBµ−βAµ+γDµ−∆Cµ)j(m)
+µ
++(αCµ−βDµ+γAµ−∆Bµ)j(G)
+µ−(αDµ−βCµ+γBµ−∆Aµ)j(H)
+µ. (31)
+Now considering the term wise variation in equation (31), we get
+δS1=ˆ−α
+4δF2
+µνdΩ =ˆ−α
+2FµνδFµνdΩ;
+δS2=ˆ−α
+4δM2
+µνdΩ =ˆ−α
+2MµνδMµνdΩ;
+δS3=ˆ−α
+4δf2
+µνdΩ =ˆ−α
+2fµνδfµνdΩ;
+δS4=ˆ−α
+4δN2
+µνdΩ =ˆ−α
+2NµνδNµνdΩ; (32)
+δS5=ˆβ
+2δ(FµνMµν+fµνNµν)dΩ
+=ˆβ
+2{[FµνδMµν+MµνδFµν]+[fµνδNµν+Nµνδfµν]}dΩ; (33)
+δS6=ˆγ
+2δ(Fµνfµν+MµνNµν)dΩ
+=ˆγ
+2{[Fµνδfµν+fµνδFµν]+[MµνδNµν+NµνδMµν]}dΩ; (34)
+δS7=ˆ∆
+2δ(FµνNµν+Mµνfµν)dΩ
+=ˆ∆
+2{[FµνδNµν+NµνδFµν]+[Mµνδfµν+fµνδMµν]}dΩ; (35)
+δS8=ˆ
+αj(e)
+µδAµ−ˆ
+βj(e)
+µδBµ+ˆ
+γj(e)
+µδCµ−ˆ
+∆j(e)
+µδDµ
+−ˆ
+αj(m)
+µδBµ+ˆ
+βj(m)
+µδAµ−ˆ
+γj(m)
+µδDµ+ˆ
+△j(m)
+µδCµ
++ˆ
+αj(G)
+µδCµ−ˆ
+βj(G)
+µδDµ+ˆ
+γj(G)
+µδAµ−ˆ
+∆j(G)
+µδDµ
+−ˆ
+αj(H)
+µδDµ+ˆ
+βj(H)
+µδCµ−ˆ
+γj(H)
+µδBµ+ˆ
+△j(H)
+µδAµ. (36)
+δS1may be calculated after integrating it by parts and applying Gauss theorem as
+9δS1=−αˆ∂Fµν
+∂xµδAµdΩ. (37)
+Similarly, other variations may be calculated. taking into account the equation
+δS=δS1+δS2+δS3+δS4+δS5+δS6+δS7+δS8= 0 (38)
+from which we get
+ˆ/parenleftBig
+j(e)
+µδAµ+j(m)
+µδBµ+j(G)
+µδCµ+j(H)
+µδDµ/parenrightBig
+αdΩ
+−ˆ/parenleftbigg∂Fµν
+∂xµδAµ−∂Mµν
+∂xµδBµ−∂fµν
+∂xµδCµ−∂Nµν
+∂xµδDµ/parenrightbigg
+αdΩ
+−ˆ/parenleftBig
+j(e)
+µδBµ+j(m)
+µδAµ+j(G)
+µδDµ+j(H)
+µδCµ/parenrightBig
+αdΩ
+−ˆ/parenleftbigg∂Fµν
+∂xµδAµ−∂Mµν
+∂xµδBµ−∂fµν
+∂xµδCµ−∂Nµν
+∂xµδDµ/parenrightbigg
+αdΩ|
+−ˆ
+(j(e)
+µ−ij(m)
+µ−jj(G)
+µ−kj(H)
+µ)δ(Aµ+iBµ+jCµ+kDµ)βdΩ
+−ˆ
+(Fµν,ν−iMµν,ν−jfµν,ν−kNµν,ν)δ(Aµ+iBµ+jCµ+Dµ)βdΩ
+−ˆ
+(j(e)
+µ−ij(m)
+µ−jj(G)
+µ−kj(H)
+µ)δ(Bµ+iAµ+jDµ+kCµ)βdΩ
+−(Fµν,ν−iMµν,ν−jfµν,ν−kNµν,ν)δ(Bµ+iAµ+jDµ+kCµ)βdΩ
+=0 (39)
+which yields the Maxwellian field equations given by equatio n (10) for the dynamics of electric,
+magnetic, gravitational and Heavisidian charges (masses) . These equations thus provide the following
+form of unified field equations of a particle simultaneously c ontains these four charges (masses)
+ℑµν,ν=Jµ (40)
+whereℑµνis the unified field tensor defind by equation (8) and Jµis the unified current density given
+by equation (11).
+4 Equation of motion of a unified charged in quaternionic form
+From equation (26), we may write the unified equation of motio n for a particle simultaneously
+contains four charges (masses) in terms of quaternion as
+Md2xµ
+dt2=/braceleftbig
+Re/parenleftbig¯Qℑµν/parenrightbig/bracerightbig
+uν(41)
+where¨xµ=d2xµ
+dt2is particle acceleration, uνis the four - velocity, and Qis the quaternion conjugate
+of the unified charge given by equation (5). The right hand sid e of the equation (41) may also be
+10written as
+/braceleftbig
+Re/parenleftbig¯Qℑµν/parenrightbig/bracerightbig
+uν= (eFµν+gMµν+mfµν+hNµν)uν(42)
+which can be reduced as
+Re[Q,ℑµν]uν=1
+2/parenleftbig
+Q¯ℑµν+¯Qℑµν/parenrightbig
+uν
+=(eFµν+gMµν+mfµν+hNµν)uν
+=/braceleftBig
+e/bracketleftBig− →E+− →v×− →H/bracketrightBig
++g/bracketleftBig− →H−− →v×− →E/bracketrightBig/bracerightBig
++/braceleftBig
+m/bracketleftBig− →G−− →v×− →H/bracketrightBig
++h/bracketleftBig− →H+− →v×− →G/bracketrightBig/bracerightBig
+(43)
+where− →Eis the electric field,− →His the magnetic field,− →Gis the gravitational field and− →His Heavisidian
+field.
+5 Euler’s Lagrangian equation of motion of unified charged
+The Lagrangian density for unified fields of a paricle contain ing simultaneously the four charges
+namely electric, magnetic, gravitational and Heavisidian may also be expressed as
+L=−1
+4FµνFµν−1
+4MµνMµν−1
+4fµνfµν−1
+4NµνNµν+Aµj(e)
+µ+Bµj(m)
+µ+Cµj(G)
+µ+Dµj(H)
+µ.(44)
+This Lagrangian density may also be written in terms of Grass mann product as
+L= =1
+8/bracketleftbig
+ℑµν,ℑµν/bracketrightbig
++/bracketleftbig
+Vµ,Jµ/bracketrightbig
+(45)
+where/bracketleftbig
+Vµ,Jµ/bracketrightbig
+=1
+2/bracketleftbig
+VµJµ+VµJµ/bracketrightbig
+, withVµ(the quaternion conjugate of the unified four - po-
+tential), Jµ(the quaternion conjugate of unified four - current density) andℑµν(the quaternion
+conjugate of unified field tensor) are defined as follows,
+Vµ=Aµ+iBµ+jCµ+kDµ;
+Jµ=j(e)
+µ+ij(m)
+µ+jj(G)
+µ+kj(H)
+µ;
+ℑµν=Fµν+iMµν+jfµν+kNµν. (46)
+Then
+/bracketleftbig
+Vµ,Jµ/bracketrightbig
+=/bracketleftBig
+Aµj(e)
+µ+Bµj(m)
+µ+Cµj(G)
+µ+Dµj(H)
+µ/bracketrightBig
+. (47)
+Similarly, we have
+111
+8/bracketleftbig
+ℑµν,ℑµν/bracketrightbig
+=1
+4[FµνFµν+MµνMµν+fµνfµν+NµνNµν]. (48)
+From the Lagrangian density (44), we get the Euler’s Lagrang ian equations in the following forms;
+∂L
+∂Aµ−∂ν∂L
+∂(∂νAµ)= 0;
+∂L
+∂Bµ−∂ν∂L
+∂(∂νBµ)= 0;
+∂L
+∂Cµ−∂ν∂L
+∂(∂νCµ)= 0;
+∂L
+∂Dµ−∂ν∂L
+∂(∂νDµ)= 0. (49)
+This equation provides the field equations associated respe ctively with the dynamics of electric,
+magnetic, gravitational and Heavisidian charges (masses) given by equation (10) after taking care
+the usual method of variations with respect to potential. Th ese equation may immediately be
+genralized to equation (40) as unified field equations of a par ticle simultaneously containing four
+charges namely electric, magnetic, gravitational and Heav isidian charges (masses).
+6 Conclusion
+A unified Lagrangian density of generalized electromagneti c fields and Heavisidian fields have been
+developed with the fact that these fields essentially posses s the structural symmetry. The action
+integral (17) of the unified quaternionic charge (i.e. the co mplex structure of dyons and gravito -
+dyons) in the field of others constructed by choosing the suit able Lagrangian density (12), does not
+make the use of string variables. Rather, it has been written in terms of four - potentials associated
+with four charges namely electric, magnetic, gravitationa l and Heavisidian charges. Following the
+usual method of variation, it is shown that this action leads to equation of motion (29) and unified
+field equations (40). The beauty of the Lagrangian (12) lies i n the fact that individual components of
+four - currents and four - potentials have been embodied in th e corresponding generalized quantities
+through the unknown parameters α, β, γ and∆which satisfy the constraints described by equations
+(13, 14) and (15). It has already been shown that the Lagrangi an density (12), equation of motion
+(29) and the unified field equations (40) are invariant under t he rotation in charge space or its
+combination with space and time reflections and also under th e reflection in charge space combined
+with time reversal or space reflection. It means that the syst em possess strong symmetry under
+rotation in charge space.
+References
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+14
\ No newline at end of file
diff --git a/1001.0364.txt b/1001.0364.txt
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@@ -0,0 +1,234 @@
+arXiv:1001.0364v1  [cond-mat.mes-hall]  3 Jan 2010On the Temperature Dependence of the Lifetime
+of Thermally Isolated Metastable Clusters
+L. A. Openov∗, D. A. Lobanov, and A. I. Podlivaev
+Moscow Engineering Physics Institute (State University), 115409 Moscow, Russia
+∗E-mail: LAOpenov@mephi.ru
+ABSTRACT
+The temperature dependence of the lifetime of the thermally isolate d metastable N 8
+cubane up to its decay into N 2molecules has been calculated by the molecular dynamics
+method. It has been demonstrated that this dependence significa ntly deviates from the
+Arrhenius law. The applicability of the finite heat bath theory to the d escription of
+thermally isolated atomic clusters has been proved using statistical analysis of the results
+obtained.
+PACS: 36.40.-c, 36.40.Qv, 71.15.Pd
+1Simulations of the time evolution of atomic clusters at high temperatu res are of interest
+fromseveral pointsofview. Ananalysisofdistortionsofthecluste rstructureandproducts
+of its decay can provide information on the mechanism of the final st age of the cluster
+formation and possible components for its synthesis, respectively . For example, when the
+C60fullerene is heated, defects are formed on its “surface” because of the Stone-Wales
+transformations, while the formation of the fullerene from an initial imperfect C 60cluster
+occurs due to the same transformations going in the reverse orde r [1, 2]. In turn, the
+decomposition of the hydrocarbon cubane C 8H8into benzene and acetylene molecules
+suggests the use of these compounds for the synthesis of the cu bane (applying catalysts
+to speed up the reverse reaction) [3].
+The main parameter determining the thermal stability of a metastab le cluster is the
+activation energy of its decomposition (or the transformation into another isomer) Ea.
+If the height of the minimum energy barrier preventing the decompo sition is U, the
+inequality Ea> Uholds true. If there is only one decomposition channel, the equality
+Ea=Uis satisfied. The magnitude of Eais determined [4] in experiments by measuring
+the lifetime τof the cluster at different temperatures Tand making use of the Arrhenius
+law
+τ−1(T) =Aexp(−Ea/kBT), (1)
+whereAis the frequency factor and kBis the Boltzmann constant. The theoretical depen-
+dence of τonT, in principle, can be determined by numerical simulations of the cluste r
+lifetime using the molecular dynamics method (the so-called “compute r experiment”).
+However, it should be noted that Eq. (1) is valid only when the cluster is in thermal
+equilibrium with the environment (it is thermalized). In the experiment , the thermal
+2contact is ensured by placing clusters in a “buffer gas” (usually, heliu m is used) heated to
+a required temperature [4]. In the theory, the canonical NVTensemble is used to describe
+thermalized clusters. However, sometimes cluster disassociates s o quickly that it does not
+have enough time to come into thermal equilibrium with its environment . It happens,
+for example, during photofragmentation, when the clusters excit ed by a laser pulse do
+not collide with each other for a time t <1µs required for their decomposition. In this
+case, the dynamics of the cluster is simulated using the microcanonic alNVEensemble [5]
+and the total energy of the cluster E=Ekin+Epot(the sum of the kinetic and potential
+energies) remainsconstant duringitsevolution. Here, theroleoft hetemperatureisplayed
+by the so called “microcanonical” (or “dynamic”) temperature Tm, which is a measure of
+the kinetic energy of relative motion of atoms in the cluster, which is a t rest and does not
+rotate as a whole (i.e., it is basically the measure of the excitation ener gy of the cluster).
+It iscalculated using theformula [6, 7] /angbracketleftEkin/angbracketright=1
+2kBTm(3n−6), where /angbracketleftEkin/angbracketrightis thekinetic
+energy of the cluster averaged over the microcanonical ensemble (or over the time) and n
+is the number of atoms in it.
+The question arises: is it possible to use Eq. (1) to determine the dec omposition
+activation energy of a thermally isolated cluster by simply substitutin gTbyTmin it?
+According to the so-called finite heat bath theory [8, 9], in which the c luster itself plays
+the role of the reservoir for the degree of freedom along the reac tion coordinate that leads
+to the decomposition, this is possible. But there is a reservation: th e equation should
+include the quantity T∗=Tm−Ea/2Cinstead of Tm, whereC= (3n−6)kBis the specific
+heat of the cluster.
+Inorder toexperimentally test this assertion, itisnecessary, firs t, to ensurethat thereis
+3onlyonechannel ofdecomposition (otherwise, thedependences o fτonTand, correspond-
+ingly,τonTmbecome more complex) and, second, to know Eafor the decomposition in
+the channel. However, usually the cluster has several decomposit ion channels (sometimes
+they are completely different [10]), and even if the decomposition goe s mainly over one of
+them, the corresponding activation energy is not known a priori. Sin ce, for medium-sized
+clusters and temperatures, at which the cluster decomposition tim e ist <1µs, the correc-
+tionEa/2CTmis usually very small ( ≈0.07 for C 60fullerene [2]), the use of T∗instead of
+Tmin Eq. (1) does not lead to any substantial change in the function τ(Tm), even though
+it leads to somewhat better agreement of numerical simulations with experiment [2].
+The purpose of the present paper is to directly verify the finite hea t bath theory via
+numerical simulations of dynamics of a metastable cluster for which t here is only one
+decompositionchannel andtheheightofthecorrespondingpoten tialbarrierisknownfrom
+independent calculations. We choose the N 8cubane (Fig. 1), which has been predicted
+theoretically [11-13] but, as far as we know, has not been observe d in experiment yet, to
+play the role of such a cluster. Our choice is stipulated mainly by a small number of atoms
+in the cubane (and, correspondingly, a comparatively large magnitu de of the correction to
+microcanonical temperature), and also by the fact that the prod ucts of its decomposition
+(four N 2molecules) are known, which simplifies the search for barriers that p revent the
+decomposition (the more so because the initial cluster has a high sym metry). Another
+reason for the choice of so exotic cluster for testing of the statis tical laws is irreversible
+character of decomposition of the cubane after going over the ac tivation barrier nearest to
+the local energy minimum. The majority of clusters do not possess t his feature and come
+from the initial configuration to adjacent metastable configuratio ns and back many times
+4during their evolution, because of which it is quite difficult to determine the lifetime of
+the cluster [2, 4].
+We calculated the interatomic distance a(the length of the N-N bond) and the disasso-
+ciation energy of the cubane Ediss=E(N8)−4E(N2) using the density functional theory
+with the 6-31G∗basis set and the exchange-correlation functional B3LYP. Our re sults
+(a= 1.52˚A,Ediss= 2.27 eV/atom) agree with the data of other authors [11-13] obtaine d
+using different ab initio methods ( a= 1.46−1.54˚A,Ediss= 1.4−3.1 eV/atom). We
+also found the reaction path for the cubane decomposition along th e reaction coordinate,
+which connects the initial (N 8cubane) and final (four N 2molecules) configurations in the
+space of atomic coordinates. It turned out that the path contain s several barriers that
+correspond to sequential transitions of the N 8cluster to different intermediate configura-
+tions (Fig. 2). Each barrier has the corresponding saddle point at t he potential energy
+hypersurface. Maximum energy corresponds to the saddle point o f the first (the nearest
+to the cubane) barrier. Therefore, this barrier determines the s tability of the cubane. Its
+height is U= 0.88 eV. We obtained similar results using the Hartree-Fock method an d
+the Moller-Plesset second-order perturbation theory (in particu lar,U= 0.80 eV).
+Inprinciple, at this stage, we can proceed to the determination of t he dependence of the
+lifetime of the cubane τon the microcanonical temperature Tmby setting different initial
+excitation energies for the cluster and employing the molecular dyna mics method to an-
+alyze its evolution. However, the ab initio calculations require very lar ge amounts of the
+computer time, because of which the “life” of the cluster can be obs erved only over a very
+short period of time (1 - 10 ps), which is insufficient to collect necessa ry statistics. There-
+fore, we chose another way: we found the tight binding potential f or nitrogen systems. It
+5is known that tight binding models produce a reasonable compromise b etween rigorous ab
+initio approaches and oversimplified classic potentials of the interact ion between atoms.
+Often, they incur only slight loss in accuracy as compared to the for mer approaches (be-
+cause they take into account explicitly the quantum-mechanical “b and” contribution of
+the electron subsystem to the total energy) [1, 14] and simultane ously make it possible to
+perform the “computer experiment” for macroscopic (on atomic s cales) times ∼1µs [3,
+15].
+We start from the nonorthogonal tight binding potential [16], which we successfully
+used before to simulate dynamics of the hydrocarbon cubane C 8H8[3, 15]. We choose
+parameters of the potential for the case of nitrogen systems to achieve the best possible
+correspondence of interatomic distances, binding energies, and m inimum oscillation fre-
+quencies to experimental values (for N 2molecules) or to results of ab initio calculations
+(for the N 8cubane and some other metastable nitrogen clusters). In particu lar, for the
+case of the cubane, our model leads to a= 1.54˚A, andEdiss= 2.22 eV/atom. The anal-
+ysis shows that the dependence of the potential energy of the N 8system on the reaction
+coordinate along the decomposition path of the cubane has the sam e form as in Fig. 2;
+i.e., in agreement with the ab initio calculations, the stability of the cuba ne is determined
+by the height of the nearest barrier. It turns out to be 0.384 eV wit hin the tight binding
+model, which is approximately two times smaller than the result of our d ensity functional
+theory calculations. It is not surprising because it is known that the ab initio methods,
+in general, and the density functional theory, in particular, tend t o overstimate heights
+of potential barriers [3]. However, at present, we are interested not in which of the the-
+oretical approaches provides more precise calculations, but in the fact that, now, using
+6the tight binding potential and the value of Udetermined using the same potential, we
+can study the dynamics of the cubane to its decomposition over a ve ry wide range of
+durations 10−13−10−6s and then compare the temperature dependences of the lifetime
+of the cubane both with the usual Arrhenius law and with predictions of the finite heat
+bath theory [8, 9]. The criterion of the validity of one or another app roach is coincidence
+of the activation energy found from the “computer experiment” w ith the barrier height
+U.
+We simulate the dynamics of the thermally isolated N 8cubane in the following way.
+At the initial instant of time, we set each atoms in motion in such a way t hat the total
+momentum and the angular momentum of the cluster as a whole are eq ual to zero (a
+special case of the NVEPJ ensemble [15]). Then, forces that act on atoms are calculated.
+The classical Newton equations aresolved numerically using theveloc ity Verlet algorithm.
+The time step used is t0= 2.72·10−16s.
+We study the evolution of the cubane for 74 different sets of initial v elocities and dis-
+placements of atoms that correspond to the initial microcanonical temperatures Tm=
+350−1000 K. According to the ergodic hypothesis, Tmis equal to the “dynamic” tem-
+perature [15], which is determined by averaging the kinetic energy of the cluster over
+1000-10000 steps of the molecular dynamics simulation [6, 7]. We foun d that the cubane
+decomposes into either four N 2molecules or one N 2molecule and two quasi-linear N 3
+radicals, each of which, in turn, decomposes into the N 2molecule and the nitrogen atom
+(according to our calculations, E(N 3)≈E(N2) + E(N), which is in agreement with results
+of other authors [17]). The decomposition goes on very quickly (ove r 0.1 - 1.0 ps). Since
+the total energy of the thermally isolated cluster is Epot+Ekin= const, a drastic decrease
+7in the potential energy with the decomposition leads to the corresp onding increase in the
+kinetic energy, i.e., to the increase in Tm(by several thousands of degrees). The detailed
+analysis oftheatomicconfigurationattheinitial stageofthedecom position process shows
+that it corresponds (with very rare exceptions) to the configura tion of the saddle point
+in the potential energy of the cluster with the energy barrier U= 0.384 eV. When this
+barrier is overcome, the decomposition becomes irreversible, even though, then, it can go
+via different paths (Fig. 2 shows the scheme of the cubane decay int o four N 2molecules,
+but the final products can also be three N 2molecules and two nitrogen atoms).
+Figure 3 shows the calculated values of ln( τ) for different microcanonical temperatures
+Tm. It can be seen that, over the entire range of Tmunder study, the dependence ln( τ) on
+1/Tmcan be fit well by a straight line, which is in agreement with the Arrheniu s formula
+following for the microcanonical ensemble from Eq. (1) at T=Tm. The temperature
+dependence of the frequency factor A, if any, is so weak that it cannot be revealed using
+our data. Therefore, at first glance, the Arrhenius formula desc ribes well the results of
+the simulations of the decomposition of the thermally isolated cubane . However, the
+activation energy determined from the slope of the approximating s traight line in Fig. 3
+turns out to be Ea= 0.626±0.022 eV, which is much larger than the height U= 0.384 eV
+of the barrier that is overcome during the decomposition. Therefo re, the inference can be
+made that the usual Arrhenius law is inapplicable to the description of the decomposition
+of the thermally isolated cluster.
+Let us see what happens with corrections for finite size of heat bat h [8, 9]. By as-
+suming that Tin Eq. (1) is equal to T∗=Tm−Ea/2Cand considering the same set
+of ln(τ) values as a function of 1/ T∗(Fig. 4) instead of the function of 1/ Tm, we see
+8that ln(τ) as a function of 1/ T∗also fits straight line very well, but, in this case, the
+activation energy (which was determined by successive iterations) isEa= 0.367±0.013
+eV in excellent agreement with U= 0.384 eV. Notice that the parameter Ea/2CTmthat
+describes magnitude of the correction to Tmis 0.10-0.35 for the range of Tmunder study;
+i.e., it is substantially larger than that, for example, in simulations of th e C60fullerene [2].
+Ultimately, this is the fact that allows us to find the clear distinction (m uch larger than
+statistical uncertainty) between the functional dependence of τonTmwith and without
+corrections for the finite size of the heat bath.
+The frequency factors determined by crossings of the straight lin es in Figs. 3 and 4
+with the vertical axis are A= 8.5·1015c−1and 4.1·1014c−1, respectively. It is worth
+noting that one and the same set of data can be equally well fit using a bsolutely different
+pairs of fitting parameters ( Ea,A). The sensitivity of the parameters to the form of
+the functional dependence τ(Tm) indicates once again that it is necessary to choose the
+correct one to analyze the data on the lifetime of metastable cluste rs. In the present
+paper, we directly prove that the finite heat bath theory is applicab le to the quantitative
+descriptionofthermallyisolatedclustersandthatitcanbeusedinst udiesoffastprocesses
+of decomposition of clusters and their interaction with each other.
+References
+[1] A. I. Podlivaev and L. A. Openov, Pisma Zh. Eksp. Teor. Fiz. 81, 656 (2005) [JETP
+Lett.81, 533 (2005)].
+[2] L. A. Openov and A. I. Podlivaev, Pisma Zh. Eksp. Teor. Fiz. 84, 73 (2006) [JETP
+Lett.84, 68 (2006)].
+9[3] M. M. Maslov, D. A. Lobanov, A. I. Podlivaev, and L. A. Openov, F iz. Tverd. Tela
+(St. Petersburg) 51, 609 (2009) [Phys. Solid State 51, 645 (2009)].
+[4] C.Lifshitz, Int. J. Mass. Spectrom. 198, 1 (2000).
+[5] E. M. Pearson, T. Halicioglu, and W. A. Tiller, Phys. Rev. A 32, 3030 (1985).
+[6] C. Xu and G. E. Scuseria, Phys. Rev. Lett. 72, 669 (1994).
+[7] J. Jellinek and A. Goldberg, J. Chem. Phys. 113, 2570 (2000).
+[8] C. E. Klots, Z. Phys. D 20, 105 (1991).
+[9] J. V. Andersen, E. Bonderup, and K. Hansen, J. Chem. Phys. 114, 6518 (2001).
+[10] L. A. Openov and A. I. Podlivaev, Pisma Zh. Eksp. Teor. Fiz. 84, 217 (2006) [JETP
+Lett. 84 (4), 190 (2006)].
+[11] R. Engelke and J. R. Stine, J. Phys. Chem. 94, 5689 (1990).
+[12] W. J. Lauderdale, J. F. Stanton, and R. J. Bartlett, J. Phys. Chem.96, 1173 (1992).
+[13] M. L. Leininger, C. D. Sherrill, and H. F. Schaefer, J. Phys. Che m.99, 2324 (1995).
+[14] I. V. Davydov, A. I. Podlivaev, and L. A. Openov, Fiz. Tverd. T ela (St. Petersburg)
+47, 751 (2005) [Phys. Solid State 47, 778 (2005)].
+[15] L. A. Openov and A. I. Podlivaev, Fiz. Tverd. Tela (St. Petersb urg)50, 1146 (2008)
+[Phys. Solid State 50, 1195 (2008)].
+[16] M. M. Maslov, A. I. Podlivaev, and L. A. Openov, Phys. Lett. A 373, 1653 (2009).
+[17] P. Zang, K. Morokuma, and A. M. Wodtke, J. Chem. Phys. 122, 014106 (2005).
+10Fig. 1. N 8cubane.
+11Reaction coordinateEpot1
+2U
+Fig. 2. Schematic dependence of the potential energy Epotof the N 8cubane (1) on the
+reaction coordinate along the path cubane →four N 2molecules (2).
+120.001 0.002 0.003 0.004 
+1/Tm-30 -25 -20 -15 ln(τ)
+ 
+Fig. 3. Logarithm of the lifetime τ(in seconds) of the N 8cubane as a function of
+the inverse microcanonical temperature 1/ Tm(in K−1. Points indicate the results of
+the calculation. The solid line represents the linear approximation by t he least-squares
+13method.
+140.001 0.002 0.003 0.004 0.005 
+1/T*-30 -25 -20 -15 ln(τ)
+ 
+Fig. 4. The same as in Fig. 3 but in the other coordinates. The quantit y inverse to
+T∗=Tm−Ea/2Cis plotted along the abscissa axis (see text).
+15
\ No newline at end of file
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+arXiv:1001.0365v1  [hep-ph]  3 Jan 2010J/ψandΥat high temperature
+I.M.Narodetskii, Yu.A.Simonov, and A.I.Veselov
+Institute of Theoretical and Experimental Physics, Moscow 117218, Russia
+Abstract
+We use the screened Coulomb potential with r-dependent coupling constant and the non–
+perturbative quark–antiquark potential derived within the Field Co rrelator Method (FCM)
+to calculate J/ψand Υ binding energies and melting temperatures in the deconfined ph ase
+of quark-gluon plasma.
+keywords: Quark-gluon plasma, non–perturbative potential, heavy mesons
+PACS:12.38Lg, 14.20Lq, 25.75Mq
+1 INTRODUCTION
+Since1986,thegold-platedsignatureofdeconfinement wasthoug httobeJ/ψsuppression[1].
+If Debye screening of the Coulomb potential above Tcis strong enough then J/ψproduction
+in A+A collisions will be suppressed. Indeed, applying the Bargmann co ndition [2] for the
+screened Coulomb potential
+VC(r) =−4
+3·αs(r)
+r·e−mdr, (1)
+wheremdis the Debye mass, we obtain the simple estimate for the number of, s ay,ccS–wave
+bound states
+n≤µc∞/integraldisplay
+0|VC(r)|rdr=4αs
+3·µc
+md, (2)
+whereµcis the constituent mass of the c-quark and for the moment we neglect the r-
+dependence of αs. Takingµc= 1.4 GeV and αs= 0.39 we conclude that if md≥0.7
+GeV, there is no J/ψbound state. Parenthetically, we note that no light or strange mes ons
+(µ∼300−500 MeV) survive. But this is not the full story.There is a significant change of views on physical properties and und erlying dynamics of
+quark–gluon plasma (QGP), produced at RHIC, see e.g.[3] and references there in. Instead
+of behaving like a gas of free quasiparticles – quarks and gluons, the matter created in
+RHIC interacts much more strongly than originally expected. Also, t he interaction deduced
+from lattice studies is strong enough to support QQbound states. It is more appropriate
+to describe the non-perturbative (NP) properties of the QCD pha se close to Tcin terms
+remnants of the non–perturbative part of the QCD force rather than a strongly coupled
+Coulomb force.
+In the QCD vacuum, the NP quark-antiquark potential is V=σr. AtT≥Tc,σ= 0, but
+that does not mean that the NP potential disappears. In a recent paper [4] we calculated
+binding energies for the lowest QQandQQQeigenstates ( Q=c,b) aboveTcusing the NP
+QQpotential derived in the Field Correlator Method (FCM) [5] and the sc reened Coulomb
+potential with the strong coupling constant αs= 0.35 in Eq. (1). In this talk we extend our
+analysis to the case of the running αs(r).
+2 The Field Correlator Method as applied to finite T
+The NPQQpotential can be studied through the modification of the correlato r functions,
+which define the quadratic field correlators of the nonperturbativ e vaccuum fields
+<trFµν(x)Φ(x,0)Fλσ(0)>=Aµν;λσD(x)+Bµν;λσD1(x), (3)
+whereAµν;λσandBµν;λσare the two covariant tensors constructed from gµνandxµxν, Φ(x,0)
+is the Schwinger parallel transporter, xEuclidian.
+AtT≥Tc, one should distinguish the color electric correlators DE(x), DE
+1(x) and color
+magnetic correlators DH(x), DH
+1(x). AboveTc, the color electric correlator DE(x) that
+defines the string tension at T= 0 becomes zero [6] and, correspondingly, σE= 0. The color
+magnetic correlators DH(x) andDH
+1(x) do not produce static quark–antiquark potentials,
+they only define the spatial string tension σs=σHand the Debye mass md∝√σsthat
+grows with T.
+The main source of the NP static QQpotential at T≥Tcoriginates from the color–
+electric correlator function DE
+1(x)
+Vnp(r,T) =1/T/integraldisplay
+0dν(1−νT)r/integraldisplay
+0λdλDE
+1(x). (4)
+In the confinement region the function DE
+1(x) was calculated in [7]
+DE
+1(x) =Bexp(−M0x)
+x, (5)
+whereB= 6αf
+sσfM0,αf
+sbeing the freezing value of the strong coupling constant to be
+specified later, σfis the sting tension at T= 0, and the parameter M0has the meaning of
+the gluelump mass. In what follows we take σf= 0.18 GeV2andM0= 1 GeV. Above Tcthe
+analytical form of DE
+1should stay unchanged at least up to T∼2Tc. The only change is
+2B→B(T) =ξ(T)B, where the factor ξ(T) =/parenleftBig
+1−0.36M0
+BT−Tc
+Tc/parenrightBig
+is determined by lattice
+data [9]. Integrating over λone obtains
+Vnp(r,T) =B(T)
+M01/T/integraldisplay
+0(1−νT)/parenleftBig
+e−νM0−e−√
+ν2+r2M0/parenrightBig
+dν=V(∞,T)−V(r,T) (6)
+Note thatV(∞,Tc)≈0.5 GeV that agrees with lattice estimate for the free quark-antiqua rk
+energy.
+In the framework of the FCM, the masses of heavy quarkonia are d efined as
+MQ¯Q=m2
+Q
+µQ+µQ+E0(mQ,µQ), (7)
+E0(mQ,µQ) is an eigenvalue of the Hamiltonian H=H0+Vnp+VC,mQare the bare
+quark masses, µQare the auxiliary fields that are introduced to simplify the treatment of
+relativistic kinematics. The auxiliary fields are treated as c–number v ariational parameters
+to be found from the extremum condition imposed on MQ¯Qin Eq. (7). Such an approach
+allows for a very transparent interpretation of axiliary fields as the constituent masses that
+appear due to the interaction. Once mQis fixed, the quarkonia spectrum is described. The
+dissociation points are defined as those temperature values for wh ich the energy gap between
+V(∞,T) andE0disappears.
+3 Coulomb potential
+We use the perturbative screened Coulomb potential (1) with the r-dependent QCD cou-
+pling constant αs(r,T). Note that in the entire regime of distances for which at T= 0 the
+heavy quark potential can be described well by QCD perturbation t heoryαs(r,T) remains
+unaffected by temperature effects at least up to T/lessorequalslant3Tcand agrees with the zero tempera-
+ture running coupling αs(r,0) =αs(r). For our purposes, we find it convenient to define the
+r–dependent coupling constant in terms of the q2dependent constant αB(q2) calculated in
+the background perturbation theory (BPTh) [8]
+αs(r) =2
+π∞/integraldisplay
+0dqsinqr
+qαB(q2). (8)
+The formula for αB(q2) is obtained by solving the two-loop renormalization group equation
+for the running coupling constant in QCD:
+αB(q2) =4π
+β0t/parenleftbigg
+1−β1
+β02lnt
+t/parenrightbigg
+, t= lnq2+m2
+B
+Λ2
+V, (9)
+whereβiare the coefficients of the QCD β-function.
+In Eq. (9) the parameter mB∼1 Gev has the meaning of the mass of the lowest
+hybrid excitation. The result can be viewed as arising from the intera ction of a gluon with
+3Table 1:J/ψabove the deconfinement region. V(∞,T) is the continuum threshold (a
+constant shift in the potential). Units are GeV or GeV−1.
+T/TcV(∞)µbE0−V(∞)r0MJ/ψ
+1 0.445 1.443 -0.011 8.23 3.235
+1.2 0.368 1.423 -0.003 10.07 3.171
+background vacuum fields. We employ the values Λ V= 0.36GeV, mB= 0.95GeV, which
+lie within the range determined in Ref. [10]. The result is consistent with the freezing of
+αB(r) with a magnitude 0.563 (see Table 4 of Ref. [11]. The zero temperat ure potential
+with the above choice of the parameters gives a fairly good descript ion of the quarkonium
+spectrum [10]. At finite temperature we utilize the information on mdin Eq. (1) from Ref.
+[12]. For pure-gauge SU(3) theory ( Tc= 275 MeV) mdvaries between 0.8 GeV and 1.4 GeV,
+whenTvaries between Tcand 2Tc.
+4 Results
+The solutions for the binding energy for the 1 S J/ψand Υ states are shown in Tables 1,
+2. In these Tables we present the constituent quark masses µQforccandbb, the differences
+εQ=E0−VQQ(∞), the mean squared radii r0=/radicalbig
+r2, and the masses of the QQmesons.
+We employ mc= 1.4 GeV and mb= 4.8 GeV. Note that, as in the confinement region, the
+constituent masses µQonly slightly exceed bare quark masses mQthat reflect smallness of
+the kinetic energies of heavy quarks.
+AtT=Tcwe obtain the weakly bound ccstate that disappears at T∼1.3Tc. The
+charmonium masses lie in the interval 3.2 - 3.3 GeV, that agrees with th e results of Ref. [9].
+Note that immediately above Tcthe mass of the ccstate is about 0.2 GeV higher than that of
+J/ψ. As expected, the Υ state remains intact up to the larger tempera tures,T∼2.3Tc, see
+Table 2. The masses of the L = 0 bottomonium lie in the interval 9.7–9.8 G eV, about 0.2–0.3
+GeV higher than 9.460 GeV, the mass of Υ(1 S) atT= 0. AtT=Tcthebbseparationr0is
+0.25 fm that compatible with r0= 0.28 fm atT= 0. The 1S bottomoniium undergo very
+little modification till T∼2Tc. The results agreewith those found previously for a constant
+αs= 0.35 [4]. We also mention that the melting temperatures for Ω cand Ωbcalculated in
+[4] practically coincide with those for J/ψand Υ.
+Our results for 1 S(J/ψ) are qualitatively agree with those of Refs. [13], [14] based on
+phenomenological QQpotentials identified with the free quark-antiquark energy measur ed
+on the lattice while our melting temperature for 1 S(Υ) is much smaller than T∼(4−6)Tc
+found in Ref. [14].
+This work was supported in part by RFBR Grants # 08-02-00657, # 0 8-02-00677, # 09-
+02-00629 and by the grant for scientific schools # NSh.4961.2008.2.
+4Table 2: 1S bbstate above the deconfinement region.
+T/TcV(∞,T)µbE0(T)−V(∞,T)r0Mbb
+1 0.445 4.948 -0.255 1.39 9.796
+1.3 0.332 4.922 -0.158 1.69 9.777
+1.6 0.237 4.894 -0.084 2.23 9.755
+2.0 0.134 4.854 -0.022 4.23 9.712
+2.2 0.090 4.831 -0.006 6.77 9.684
+2.3 0.070 4.821 -0.002 8.32 9.668
+References
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+[2] V. Bargmann, Proc. Nat. Acad. Sci. (USA), 38961 (1952)
+[3] M. J. Tannenbaum, Rep. Prog. Phys. 69, 2005 (2006)
+[4] I. M. Narodetskiy, Yu. A. Simonov, A. I. Veselov, JETP Lett. 90, 232 (2009) [Pisma
+Zh. Eksp. Teor. Fiz. 90, 254 (2009)];
+[5] for a recent review see A. V. Nefediev, Yu. A. Simonov, M. A. Tru sov, Int. J. Mod.
+Phys.E18, 549 (2009) and references therein
+[6] Yu. A. Simonov, JETP Lett. 54, 249 (1991); Phys. At. Nucl. 58, 309 (1995)
+[7] Yu. A. Simonov, Phys. Lett. B 619, 293 (2005)
+[8] Yu. A. Simonov, Phys. Atom. Nucl. 58, 107 (1995), [Yad. Fiz. 58, 113 (1995)].
+[9] A. DiGiacomo, E. Meggiolaro, Yu. A. Simonov, and A. I. Veselov, Ph ys. Atom. Nucl.
+70, 908 (2007)
+[10] A. M. Badalian and D. S. Kuzmenko, Phys. Rev. D 65, 016004 (2002).
+[11] R. Ya. Kezerashvili, I. M. Narodetskiy, A. I. Veselov, Phys. Rev. D 79, 034003 (2009)
+[12] N. O. Agasian, Phys. Lett. B 562, 257 (2003), [arXiv:hep-ph/0303127].
+[13] D. Blaschke, O. Kaczmarek, E. Laermann, V. Yudichev, Eur. P hys. J. C 43, 81 (2005)
+[14] W. M. Alberico, A. Beraudo, A. De Pace, and A. Molinari, arXiv: he p-ph/0507084
+5
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+arXiv:1001.0366v1  [math.NA]  3 Jan 2010On a new notion of the solution to an ill-posed
+problem∗†
+A.G. Ramm
+Mathematics Department, Kansas State University,
+Manhattan, KS 66506-2602, USA
+ramm@math.ksu.edu
+Abstract
+A new understanding of the notion of the stable solution to il l-posed problems
+is proposed. The new notion is more realistic than the old one and better fits
+the practical computational needs. A method for constructi ng stable solutions in
+the new sense is proposed and justified. The basic point is: in the traditional
+definition of the stable solution to an ill-posed problem Au=f, whereAis a linear
+or nonlinear operator in a Hilbert space H, it is assumed that the noisy data {fδ,δ}
+are given, ||f−fδ|| ≤δ, and a stable solution uδ:=Rδfδis defined by the relation
+limδ→0||Rδfδ−y||= 0, where ysolves the equation Au=f, i.e.,Ay=f. In this
+definition yandfare unknown. Any f∈B(fδ,δ) can be the exact data, where
+B(fδ,δ) :={f:||f−fδ|| ≤δ}.
+The new notion of the stable solution excludes the unknown yandffrom the
+definition of the solution.
+1 Introduction
+Let
+Au=f, (1.1)
+whereA:H→His a linear closed operator, densely defined in a Hilbert space H.
+Problem (1.1) is called ill-posed if Ais not a homeomorphism of HontoH, that is, either
+equation (1.1) does not have a solution, or the solution is non-unique , or the solution
+does not depend on fcontinuously. Let us assume that (1.1) has a solution, possibly
+non-unique. Let N(A) be the null space of A, andybe the unique normal solution to
+∗key words: ill-posed problems, regularizer, stable solution of ill-pose d problems
+†AMS2010 subject classification: 47A52, 65F22, 65J20(1.1), i.e., y⊥N(A). Given noisy data fδ,/bardblfδ−f/bardbl ≤δ, one wants to construct a stable
+approximation uδ:=Rδfδof the solution y,/bardbluδ−y/bardbl →0 asδ→0.
+Traditionally(see, e.g., [2]) onecallsafamilyofoperators Rharegularizer forproblem
+(1.1) (with not necessarily linear operator A) if
+a)RhA(u)→uash→0 for any u∈D(A),
+b)Rhfδis defined for any fδ∈Hand there exists h=h(δ)→0 asδ→0 such that
+/bardblRh(δ)fδ−y/bardbl →0 asδ→0. (∗)
+Inthisdefinition yisfixedand( ∗)mustholdforany fδ∈B(f,δ) :={fδ:/bardblfδ−f/bardbl ≤δ}.
+In practice one does not know the solution yand the exact data f. The only available
+information is a family fδand some a priori information about for about the solution
+y. This a priori information often consists of the knowledge that y∈ K, whereKis a
+compactum in H. Thus
+y∈Sδ:={v:/bardblA(v)−fδ/bardbl ≤δ, v∈ K}.
+Weassumethattheoperator Aisknownexactly, andwealwaysassume that fδ∈B(f,δ),
+wheref=A(y).
+Definition: We call a family of operators R(δ)a regularizer if
+sup
+v∈Sδ/bardblR(δ)fδ−v/bardbl ≤η(δ)→0 asδ→0. (1.2)
+There is a crucial difference between our new Definition (1.2) and the standard definition
+(∗):
+In(∗)uis fixed, while in (1.2)vis an arbitrary element of Sδand the supremum of
+the norm in (1.2)over all such vmust tend to zero as δ→0.
+The new definition is more realistic and better fits computational nee ds because not
+only the solution yto (1.1) satisfies the inequality /bardblAy−fδ/bardbl ≤δ, but any v∈Sδsatisfies
+this inequality /bardblAv−fδ/bardbl ≤δ,v∈ K. The data fδmay correspond to any f=Av, where
+v∈Sδ, and not only to f=Ay, whereyis a solution of equation (1.1). Therefore it is
+more natural to use definition (1.2) than ( ∗).
+Our goal is to illustrate the practical difference between these two definitions, and to
+construct regularizer inthesense (1.2)forproblem(1.1)withanar bitrary, notnecessarily
+bounded, linear operator A, which is closed and densely defined in H. This is done in
+Section 2. In Section 1 this is done for a class of equations (1.1) with n onlinear operators
+A:X→Y, whereXandYare Banach spaces. In this case we assume that
+A1)A:X→Yis a closed, nonlinear, injective map ,f∈ R(A),R(A) it is the range of
+A,
+and
+A2)φ:D(φ)→[0,∞),φ(u)>0ifu/ne}ationslash= 0,D(φ)⊆D(A),the setsK=Kc:={v:φ(v)≤
+c}are compactin Xfor every c=const >0,and ifvn→v,thenφ(v)≤liminf n→∞φ(vn).
+The last inequality holds if φislower semicontinuous . In Hilbert spaces and in
+reflexive Banach spaces norms are lower semicontinuous.
+2Let us give some examples of equations for which assumptions A1) an d A2) are
+satisfied.
+Example 1. Ais a linear injective compact operator, f∈ R(A),φ(v) is a norm on
+X1⊂X, whereX1is densely imbedded in X, the embedding i:X1→Xis compact,
+andφ(v) is lower semicontinuous.
+Example 2. Ais a nonlinear injective continuous operator f∈ R(A),A−1is not
+continuous, φis as in Example 1.
+Example 3. Ais linear, injective, densely defined, closed operator, f∈ R(A),A−1is
+unbounded, φis as in Example 1, X1⊆D(A).
+Let us demonstrate by Example A that a regularizer in the sense (∗)may be not a
+regularizer in the sense (1.2).
+In Example B a theoretical construction of a regularizer in the sens e (1.2) is given for
+some equations (1.1) with nonlinear operators.
+In Section 2 a novel theoretical construction of a regularizer in th e sense (1.2) is given
+for a very wide class of equations (1.1) with linear operators A.
+Example A :Stable numerical differentiation.
+In this Example the results from [3] - [11] are used. This Example is bor rowed from
+[10].
+Consider stable numerical differentiation of noisy data. The problem is:
+Au:=/integraldisplayx
+0u(s)ds=f(x), f(0) = 0,0≤x≤1. (1.3)
+The data are: fδand a constant Ma, which defines a compact K, where/bardblfδ−f/bardbl ≤δ, the
+norm isL∞(0,1) norm, and Kconsists of the L∞functions which satisfy the inequality
+/bardblu/bardbla≤Ma,a≥0. The norm
+/bardblu/bardbla:= sup
+x,y∈[0,1]
+x/negationslash=y|u(x)−u(y)|
+|x−y|a+ sup
+0≤x≤1|u(x)|if 0≤a≤1,
+/bardblu/bardbla:= sup
+0≤x≤1(|u(x)|+|u′(x)|)+ sup
+x,y∈[0,1]
+x/negationslash=y|u′(x)−u′(y)|
+|x−y|a−1,1< a≤2.
+Ifa >1, then we define
+R(δ)fδ:=
+
+fδ(x+h(δ))−fδ(x−h(δ))
+2h(δ), h(δ)≤x≤1−h(δ),
+fδ(x+h(δ))−fδ(x)
+h(δ),0≤x < h(δ),
+fδ(x)−fδ(x−h(δ))
+h(δ),1−h(δ)< x≤1,(1.4)
+where
+h(δ) =caδ1
+a, (1.5)
+andcais a constant given explicitly (cf [4]).
+3We prove that (1.4) is a regularizer for (1.3) in the sense (1.2), and K:={v:/bardblv/bardbla≤
+Ma, a >1}. In this example we do not use lower semicontinuity of the norm φ(v) and
+do not define φ.
+LetSδ,a:={v:/bardblAv−fδ/bardbl ≤δ,/bardblv/bardbla≤Ma}. To prove that (1.4)-(1.5) is a regularizer
+in the sense (1.2) we use the estimate
+sup
+v∈Sδ,a/bardblR(δ)fδ−v/bardbl ≤sup
+v∈Sδ,a{/bardblR(δ)(fδ−Av)/bardbl+/bardblR(δ)Av−v/bardbl} ≤δ
+h(δ)+Maha−1(δ)≤
+≤caδ1−1
+a:=η(δ)→0 asδ→0.
+(1.6)
+Thus we have proved that (1.4)-(1.5) is a regularizer in the sense (1 .2).
+Ifa= 1, and M1<∞, then one can prove the following result:
+Claim:There is no regularizer for problem (1.3)in the sense (1.2)even if the regu-
+larizer is sought in the set of all operators, including nonl inear ones .
+More precisely, it is proved in [5], p.345, (see also [8], pp 197-235, wher e the stable
+numerical differentiation problem is discussed in detail) that
+inf
+R(δ)sup
+v∈Sδ,1/bardblR(δ)fδ−v/bardbl ≥c >0,
+wherec >0 is a constant independent of δand the infimum is taken over all operators
+R(δ)acting from L∞(0,1)intoL∞(0,1), including nonlinear ones .
+On the other hand, if a= 1 and M1<∞, then a regularizer in the sense ( ∗) does
+exist, but the rate of convergence in (*) may be as slow as one wishe s, ifu(x) is chosen
+suitably (see [4], [8]).
+Example B: Construction of a regularizer in the sense (1.2)for some nonlinear
+equations.
+Assuming A1) and A2), let us construct a regularizer for (1.1) in the sense (1.2). We
+use the ideas from [10] and [11].
+DefineFδ(v) :=/bardblAv−fδ/bardbl+δφ(v) and consider the minimization problem of finding
+the infimum m(δ) of the functional Fδ(v) on a set Sδ:
+m(δ) := inf
+v∈SδFδ(v), S δ:={v:/bardblAv−fδ/bardbl ≤δ, φ(v)≤c}. (1.7)
+Here
+K=Kc:={v:φ(v)≤c}.
+The constant c >0 can be chosen arbitrary large and fixed at the beginning of the
+argument, and then one can choose a smaller constant c1, specified below. Since Fδ(u) =
+δ+δφ(u) :=c1δ,c1:= 1+φ(u), where usolves (1.1), one concludes that
+m(δ)≤c1δ. (1.8)
+Letvjbe a minimizing sequence and Fδ(vj)≤2m(δ). Thenφ(vj)≤2c1. By assumption
+A2), asj→ ∞, one has:
+vj→vδ, φ(vδ)≤2c1. (1.9)
+4Takeδ=δm→0 and denote vδm:=wm. Then (1.9) and Assumption A2) imply the
+existence of a subsequence, denoted again wm, such that:
+wm→w, A (wm)→A(w),/bardblA(w)−g/bardbl= 0. (1.10)
+ThusA(w) =gand, since Ais injective by Assumption A1), it follows that w=u, where
+uis the unique solution to (1.1).
+Define now R(δ)fδby the formula R(δ)fδ:=vδ, wherevδis defined in (1.9).
+Theorem 1.1. R(δ)is a regularizer for problem (1.1)in the sense (1.2).
+Proof.Assume the contrary:
+sup
+v∈Sδ/bardblR(δ)fδ−v/bardbl= sup
+v∈Sδ/bardblvδ−v/bardbl ≥γ >0, (1.11)
+whereγ >0 is a constant independent of δ. Sinceφ(vδ)≤2c1by (1.9), and φ(v)≤c, one
+can choose convergent in Xsequences wm:=vδm→˜w,δm→0, andvm→˜v, such that
+/bardblwm−vm/bardbl ≥γ
+2,/bardbl˜w−˜v/bardbl ≥γ
+2, andA(˜w) =g,A(˜v) =g. By the injectivity of Ait follows
+that ˜w= ˜v=u. This contradicts the inequality /bardbl˜w−˜v/bardbl ≥γ
+2>0. This contradiction
+proves the theorem.
+The conclusions A(˜w) =gandA(˜v) =g, that we have used above, follow from the
+inequalities /bardblA(vδ)−fδ/bardbl ≤δand/bardblA(v)−fδ/bardbl ≤δafter passing to the limit δ→0, using
+assumption A2). ✷
+2 Construction of a regularizer in the sense (1.2)for
+linear equations
+IfAis a linear closed densely defined in Hoperator, then T=A∗Ais a densely defined
+selfadjoint operator. Let Ta:=T+aI, wherea=const > 0. The operator T−1
+aA∗is
+densely defined and closable. Its closure is a bounded operator, de fined on all of H, and
+||T−1
+aA∗|| ≤1
+2√a.See [12]-[15] for details and other results. Let Esbe the resolution of the
+identity of the selfadjoint operator T,dρ:=d(Esy,y), andK:={u:/integraltext∞
+0s−2pdρ≤k2
+p},
+wherep∈(0,1) andkp>0 are constants.
+Our basic result is:
+Theorem 2.1. The operator Rδ=T−1
+a(δ)A∗is a regularizer for problem (1.1)in the
+sense(1.2)iflimδ→0δ
+a(δ)1/2= 0andlimδ→0a(δ) = 0. Moreover, if a(δ) =bpδ2
+2p+1,then
+sup
+y∈K,||Ay−fδ||≤δ/bardblR(δ)fδ−y/bardbl ≤Cpδ2p
+2p+1, (2.1)
+where
+Cp=1
+2/radicalbig
+bp+cpkpbp
+p, c p=pp(1−p)1−p, b p:= (4pcpkp)−2
+2p+1.
+5The above choice of a(δ)is optimal in the sense that the right-hand side of (2.2)(see
+below) is minimal for this choice of a(δ).
+Proof.
+Let
+ǫ:= sup
+y∈K,||Ay−fδ||≤δ||T−1
+aA∗fδ−y||:= sup||T−1
+aA∗fδ−y||.
+Then, with Ay=f, one has
+ǫ≤sup||T−1
+aA∗(fδ−f)||+sup||T−1
+aA∗Ay−y||:=J1+J2,
+where
+J1≤δ
+2√a,
+and
+J2
+2≤sup{a2||T−1
+ay||2} ≤sup/integraldisplay∞
+0a2
+(s+a)2d(Esy,y).
+Thus,
+J2
+2≤/parenleftbig
+max
+s≥0asp
+a+s/parenrightbig2k2
+p=c2
+pk2
+pa2p,
+because max s≥0asp
+a+sis attained at s=pa
+1−pand is equal to cpap, where
+cp:=pp(1−p)1−p, k2
+p:= sup
+y∈K/integraldisplay∞
+0s−2pd(Esy,y).
+Consequently,
+J2≤cpkpap,
+and
+ǫ≤δ
+2√a+cpkpap. (2.2)
+Minimizing the right-hand side of (2.2) with respect to a >0, one obtains inequality
+(2.1).
+The minimizer of the right-hand side of (2.2) is
+a=a(δ) =bpδ2
+2p+1, b p:= (4pcpkp)−2
+2p+1,
+and the minimum of the right-hand side of (2.2) is Cpδ2p
+2p+1, where
+Cp:=1
+2/radicalbig
+bp+cpkpbp
+p. (2.3)
+Theorem 2.1 is proved. ✷
+6References
+[1] Dunford, N., Schwartz, J., Linear Operators , Interscience, New York, 1958.
+[2] V. Morozov, Methods of solving incorrectly posed problems , Springer Verlag, New
+York, 1984.
+[3] Ramm, A.G., On numerical differentiation, Mathematics, Izvestija vuzov, 11,
+(1968), 131-135. (In Russian).
+[4] Ramm, A.G., Stable solutions of some ill-posed problems, Math. Meth . in the
+appl. Sci. 3, (1981), 336-363.
+[5] Ramm, A.G., Scattering by obstacles , D.Reidel, Dordrecht, 1986, pp.1-442.
+[6] Ramm, A.G., Random fields estimation theory , Longman Scientific and Wi-
+ley, New York, 1990.
+[7] Ramm,A.G., Inequalitiesforthederivatives, Math.Ineq.andApp l., 3,N1, (2000),
+129-132.
+[8] Ramm, A.G., Dynamical systems method for solving operator equations ,
+Elsevier, Amsterdam, 2007.
+[9] Ramm, A.G., Dynamical systems method for solving linear ill-posed pr oblems,
+Ann. Polon. Math., 95, N3, (2009), 253-272.
+[10] Ramm, A.G., On a new notion of regularizer, J.Phys A, 36 (2003), 2 191-2195.
+[11] Ramm, A.G., Regularization of ill-posed problems with unbounded op erators, J.
+Math. Anal. Appl., 271, (2002), 447-450.
+[12] Ramm, A.G., Ill-posedproblemswithunbounded operators, J. Ma th.Anal. Appl.,
+325, (2007), 490-495.
+[13] Ramm, A.G., Dynamical systems method (DSM) for selfadjoint op erators, J.
+Math. Anal. Appl., 328, (2007), 1290-1296.
+[14] Ramm, A.G., Iterative solution of linear equations with unbounded operators, J.
+Math. Anal. Appl., 330, N2, (2007), 1338-1346.
+[15] Ramm, A.G., On unbounded operators and applications, Appl. Mat h. Lett., 21,
+(2008), 377-382.
+7
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+arXiv:1001.0367v1  [math.CV]  3 Jan 2010A uniqueness theorem for entire functions
+A G Ramm
+Department of Mathematics
+Kansas State University, Manhattan, KS 66506-2602, USA
+ramm@math.ksu.edu
+Abstract
+LetG(k) =/integraltext1
+0g(x)ekxdx,g∈L1(0,1). The main result of this
+paper is the following theorem.
+Theorem .Iflimsupk→+∞|G(k)|<∞, theng= 0. There exists
+g/negationslash≡0,g∈L1(0,1), such that G(kj) = 0,kj< kj+1,limj→∞kj=
+∞,limk→∞|G(k)|does not exist, limsupk→+∞|G(k)|=∞. Thisg
+oscillates infinitely often in any interval [1−δ,1], however small δ >0
+is.
+MSC: 30D15, 42A38, 42A63
+Key words: Fourier integrals; entire functions; uniqueness theorem
+1 Introduction
+Letg∈L1(0,1) andG(k) =/integraltext1
+0g(x)ekxdx. The function Gis an entire
+function of the complex variable k. It satisfies for all k∈Cthe estimate
+|G(k)| ≤c0e|k|, c 0:=/integraldisplay1
+0|g(x)|dx, (1)
+it is bounded for ℜk≤0:
+|G(k)| ≤c0,ℜk≤0, (2)
+and it is bounded on the imaginary axis:
+sup
+τ∈R|G(iτ)| ≤c0, (3)
+whereRdenotes the real axis.
+1It is not difficult to prove that if gkeeps sign in any interval [1 −δ,1],
+and/integraltext1
+1−δ|g(x)|dx >0 then lim k→∞|G(k)|=∞. By∞we mean + ∞in this
+paper.
+However, there exists a g/negationslash≡0,g∈L1(0,1), such that lim k→∞|G(k)|
+does not exist, but limsupk→∞|G(k)|=∞. Thisgchanges sign (oscillates)
+infinitely often in any interval [1 −δ,1], however small δ >0 is, and the
+corresponding G(k) changes sign infinitely often in any neighborhood of
++∞, i.e., it has infinitely many isolated positive zeros kj, which tend to ∞.
+Our main result is:
+Theorem 1.1. Ifc1:= limsupk→∞|G(k)|<∞, theng= 0. There exists
+g/negationslash≡0,g∈L1(0,1), such that limsupk→∞|G(k)|=∞,limk→∞|G(k)|does
+not exist, G(kj) = 0, wherekj< kj+1,limj→∞kj=∞, andgoscillates
+infinitely often in any interval [1−δ,1], however small δ >0is.
+In Section 2 proofs are given. The main tools in the proofs are a
+Phragmen-Lindel¨ of (PL) theorem (see, e.g.,[1], [3],) and a construction, de-
+veloped in [2] for a study of resonances in quantum scatterin g on a half
+line.
+For convenience of the reader we formulate the Phragmen-Lin del¨ of the-
+orem, used in Section 2. A proof of this theorem can be found, e .g., in
+[1].
+By∂Qthe boundary of the set Qis denoted, M >0 is a constant, and
+0< α <2π.
+Theorem (PL) .LetG(k)be holomorphic in an angle Qof openingπ
+α
+and continuous up to its boundary. If supk∈∂Q|G(k)| ≤M, and the order ρ
+ofG(k)is less than α, i.e.,
+ρ:= limsup
+r→∞lnlnmax |k|=r|G(k)|
+lnr< α,
+thensupk∈Q|G(k)| ≤M.
+In Section 2 this Theorem will be used for α=π
+2andρ= 1< α.
+2 Proofs
+The proof of Theorem 1.1 is based on two lemmas, and the conclu sion of
+Theorem 1.1 follows immediately from these lemmas.
+Lemma 2.1. If
+limsup
+k→∞|G(k)|<∞, (4)
+2then
+g= 0. (5)
+Proof.The entire function Gin the first quadrant Q1of the complex plane
+k, that is, in the region 0 ≤argk≤π
+2,|k| ∈[0,∞), satisfies estimate (1)
+and is bounded on the boundary of Q1:
+sup
+k∈∂Q1|G(k)| ≤max(c0,c1) :=c2. (6)
+By a Phragmen-Lindel¨ of theorem (see, e.g., [3], p.276) one concludes that
+sup
+k∈Q1|G(k)| ≤c2. (7)
+A similar argument shows that
+sup
+k∈Qm|G(k)| ≤c2, m= 1,2,3,4, (8)
+whereQmarequadrants of the complex k−plane. Consequently, |G(k)| ≤c2
+for allk∈C, and, by the Liouville theorem, G(k) =const. Thisconstis
+equal tozero, because, bytheRiemann-Lebesguelemma, lim τ→∞G(iτ) = 0.
+IfG= 0, then g= 0 by the injectivity of the Fourier transform.
+Lemma 2.1 is proved.
+Lemma 2.2. There exists g/negationslash≡0,g∈C∞(0,1), such that
+limsup
+k→∞|G(k)|=∞,
+limk→∞|G(k)|does not exist, G(kj) = 0, wherekj< kj+1,limj→∞kj=∞,
+andgoscillates infinitely often in any interval [1−δ,1], however small δ >0
+is.
+Proof.Let ∆j= [xj,xj+1], 0< xj< xj+1<1, limj→∞xj= 1,j= 2,3,....,
+fj(x)≥0,fj∈C∞
+0(∆j),/integraltext
+∆jfj(x)dx= 1, and ǫj>0,ǫj+1< ǫj, be a
+sequence of numbers such that f(x)∈C∞(0,1), where
+f(x) :=∞/summationdisplay
+j=2ǫjfj(x). (9)
+Define
+g(x) :=∞/summationdisplay
+j=2(−1)jǫjfj(x), g/negationslash≡0. (10)
+3The function goscillates infinitely often in any interval [1 −δ,1], however
+smallδ >0 is.
+Let
+G(k) :=∞/summationdisplay
+j=2(−1)jǫjGj(k), G j(k) :=/integraldisplay
+∆jfj(x)ekxdx. (11)
+Note that Gj(k)>0 for all j= 2,3,...and allk >0, and
+lim
+k→∞Gm(k)
+Gj(k)=∞, m > j, (12)
+because intervals ∆ jand ∆ mdo not intersect if m > j.
+Fix an arbitrary small number ω >0.
+Let us construct a sequence of numbers
+bj>0, b j+1> bj,lim
+j→∞bj=∞,
+such that
+G(b2m)≥ω, G (b2m+1)≤ −ω, m = 1,2,.... (13)
+This implies the existence of kj∈(b2m,b2m+1), such that
+G(kj) = 0, k j< kj+1,lim
+j→∞kj=∞.
+Moreover,
+limsup
+k→∞|G(k)|=∞, (14)
+because otherwise G= 0 by Lemma 1, so g= 0contrary to theconstruction,
+see (10).
+Let us choose b2>0 arbitrary. Then G2(b2)>0. Making ǫ3>0 smaller,
+if necessary, one gets
+ǫ2G2(b2)−ǫ3G3(b2)≥ω. (15)
+Choosing b3> b2sufficiently large, one gets
+ǫ2G2(b3)−ǫ3G3(b3)≤ −ω. (16)
+This is possible because of (12).
+4Suppose that b2,....b2m+1are constructed, so that
+2m/summationdisplay
+j=2(−1)jǫjGj(b2p)≥ω, p = 1,2,..m, (17)
+and
+2m+1/summationdisplay
+j=2(−1)jǫjGj(b2p+1)≤ −ω, p = 1,....,m. (18)
+Letm→ ∞. Then
+lim
+m→∞2m/summationdisplay
+j=2(−1)jǫjGj(k) = lim
+m→∞2m+1/summationdisplay
+j=2(−1)jǫjGj(k) =G(k),(19)
+whereG(k) is defined in (11), and the convergence in (19) is uniform on
+compact subsets of [0 ,∞). Therefore inequalities (13) hold. Lemma 2.2 is
+proved.
+Theorem 1 follows from Lemmas 1 and 2. ✷
+References
+[1] G.Polya, G.Szeg¨ o, Problems and theorems in analysis , Springer-Verlag,
+Berlin, 1983.
+[2] A.G.Ramm, Multidimensional inverse scattering problems, Long-
+man/Wiley, New York, 1992.
+[3] W.Rudin, Real and complex analysis , McGraw Hill, New York, 1974.
+5
\ No newline at end of file
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+arXiv:1001.0368v1  [math.DS]  3 Jan 2010Dynamical Systems Method (DSM) for solving
+nonlinear operator equations in Banach spaces
+A G Ramm
+Department of Mathematics
+Kansas State University, Manhattan, KS 66506-2602, USA
+ramm@math.ksu.edu
+Abstract
+LetF(u) =hbe an operator equation in a Banach space X,/bardblF′(u)−
+F′(v)/bardbl ≤ω(/bardblu−v/bardbl), where ω∈C([0,∞)),ω(0) = 0, ω(r)>0 ifr >0,
+ω(r) is strictly growing on [0 ,∞). Denote A(u) :=F′(u), where F′(u)
+is the Fr´ echet derivative of F, andAa:=A+aI.Assume that (*)
+/bardblA−1
+a(u)/bardbl ≤c1
+|a|b,|a|>0,b >0,a∈L. Hereamay be a complex
+number, and Lis a smooth path on the complex a-plane, joining the ori-
+gin and some point on the complex a−plane, 0 <|a|< ǫ0, whereǫ0>0
+is a small fixed number, such that for any a∈Lestimate (*) holds. It is
+proved that the DSM (Dynamical Systems Method)
+˙u(t) =−A−1
+a(t)(u(t))[F(u(t))+a(t)u(t)−f], u(0) =u0,˙u=du
+dt,
+converges to yast→+∞, wherea(t)∈L, F(y) =f,r(t) :=|a(t)|, and
+r(t) =c4(t+c2)−c3, where cj>0 are some suitably chosen constants,
+j= 2,3,4.Existence of a solution yto the equation F(u) =fis assumed.
+It is also assumed that the equation F(wa) +awa−f= 0 is uniquely
+solvable for any f∈X,a∈L, and lim |a|→0,a∈L/bardblwa−y/bardbl= 0.
+MSC 2000, 47J05, 47J06, 47J35
+Key words: Nonlinear operator equations; DSM (Dynamical Systems Method);
+Banach spaces
+1 Introduction
+Consider an operator equation
+F(u) =f, (1)
+whereFis an operator in a Banach space X. ByX∗denote the dual space of
+bounded linear functionals on X.
+Assume that Fis continuously Fr´ echet differentiable, F′(u) :=A(u), and
+/bardblA(u)−A(v)/bardbl ≤ω(/bardblu−v/bardbl), ω(r) =c0rκ, κ∈(0,1], (2)
+1c0>0 is a constant. The function ω(r), in general, is a continuous strictly
+growing function, ω(0) = 0.
+Assume that
+/bardblA−1
+a(u)/bardbl ≤c1
+|a|b;|a|>0, Aa:=A+aI, c1=const>0, b>0.(3)
+Hereamay be a complex number, |a|>0, and there exists a smooth path Lon
+the complex plane C, such that for any a∈L,|a|<ǫ0, whereǫ0>0 is a small
+fixed number independent of u, estimate (3) holds, and Ljoins the origin and
+some point a0, 0<|a0|< ǫ0. Assumption (3) holds if there is a smooth path
+Lon a complex a-plane, consisting of regular points of the operator A(u), such
+that the norm of the resolvent A−1
+a(u) grows, as a→0, not faster than a power
+|a|−b. Thus, assumption (3) is a weak assumption. For example, assumpt ion (3)
+is satisfied for the class of linear operators A, satisfying the spectral assumption,
+introduced in [10], Chapter 8. This spectral assumption says, that the set
+{a:|arga−π| ≤φ0,0<|a|<ǫ0}consists of the regular points of the operator
+A. This assumption implies the estimate ||A−1
+a|| ≤c1
+a,0< a < ǫ 0,similar to
+estimate (3).
+Assume additionally that the equation
+F(wa)+awa−f= 0, a∈L, (4)
+is uniquely solvable for any f∈X, and
+lim
+a→0,a∈L/bardblwa−y/bardbl= 0, F(y) =f. (5)
+All the above assumptions are standing and are not repeated i n the formulation
+of Theorem 2.1, which is our main result.
+Theseassumptionsaresatisfied, e.g., if Fisamonotoneoperatorin aHilbert
+spaceHandLis a segment [0 ,ǫ0], in which case c1= 1 andb= 1 (see [10]).
+Every equation (1) with a linear, closed, densely defined in a Hilbert sp ace
+HoperatorF=Acan be reduced to an equation with a monotone operator
+A∗A, whereA∗is the adjoint to A. The operator T:=A∗Ais selfadjoint and
+densely defined in H. Iff∈D(A∗), whereD(A∗) is the domain of A∗, then
+the equation Au=fis equivalent to Tu=A∗f, provided that Au=fhas a
+solution, i.e., f∈R(A), whereR(A) is the range of A. Recall that D(A∗) is
+dense inHifAis closed and densely defined in H. Iff∈R(A) butf/ne}ationslash∈D(A∗),
+then equation Tu=A∗fstill makes sense and its normal solution y, i.e., the
+solution with minimal norm, can be defined as
+y= lim
+a→0T−1
+aA∗f. (6)
+One proves that Ay=f, andy⊥N(A),whereN(A) is the null-space of A.
+These results are proved in [11], [13], [14].
+Our aim is to prove convergence of the DSM (Dynamical Systems Met hod)
+for solving equation (1), which is of the form:
+˙u=−A−1
+a(t)[F(u(t))+a(t)u(t)−f], u(0) =u0, (7)
+2whereu0∈Xis an initial element, a(t)∈C1[0,∞),a(t)∈L. Our main result
+is formulated in Theorem 2.1, in Section 2.
+The DSM for solving operator equations has been developed in the mo no-
+graph [10] and in a series of papers [11]-[27]. It was used as an efficient compu-
+tational tool in [5]-[9]. One of the earliest papers on the continuous a nalog of
+Newton’s method for solving well-posed nonlinear operator equation s was [3].
+The novel points in the current paper include the larger class of the oper-
+ator equations than earlier considered, and the weakened assump tions on the
+smoothness of the nonlinear operator F. While in [10] it was often assumed
+thatF′′(u) is locally bounded, in the current paper a weaker assumption (2) is
+used.
+Our proof of Theorem 2.1 uses the following result from [4].
+Lemma 1. Assume that g(t)≥0is continuously differentiable on any interval
+[0,T)on which it is defined and satisfies the following inequality:
+˙g(t)≤ −γ(t)g(t)+α(t)gp(t)+β(t), t∈[0,T), (8)
+wherep>1is a constant, α(t)>0,γ(t)andβ(t)are three continuous on [0,∞)
+functions. Suppose that there exists a µ(t)>0,µ(t)∈C1[0,∞), such that
+α(t)µ−p(t)+β(t)≤µ−1(t)[γ(t)−˙µ(t)µ−1(t)], t≥0, (9)
+and
+µ(0)g(0)<1. (10)
+ThenT=∞, i.e.,gexists on [0,∞), and
+0≤g(t)≤µ−1(t), t≥0. (11)
+This lemma generalizes a similar result for p= 2 proved in [10].
+In Section 2 a method is given for a proof of the following conclusions:
+there exists a unique solution u(t) to problem (7) for all t≥0, there exists
+u(∞) := lim t→∞u(t), andF(u(∞)) =f:
+∃!u(t)∀t≥0;∃u(∞);F(u(∞)) =f. (12)
+The assumptions on u0anda(t) under which (12) holds for the solution
+to (7) are formulated in Theorem 2.1 in Section 2. Theorem Theorem 2 .1 in
+Section 2 is our main result. Roughly speaking, this result says that c onclusions
+(12) hold for the solution to problem (7), provided that a(t) is suitably chosen.
+2 Proofs
+Let|a(t)|:=r(t)>0. Ifa(t) =a1(t) +a2(t), wherea1(t) = Rea(t), a2(t) =
+Ima(t), then
+|˙r(t)| ≤ |˙a(t)|. (13)
+3Indeed,
+|˙r(t)|=|a1˙a1+a2˙a2|
+r(t)≤r(t)|˙a(t)|
+r(t), (14)
+and (14) implies (13).
+Leth∈X∗be arbitrary with /bardblh/bardbl= 1, and
+g(t) := (z(t),h);z(t) :=u(t)−wa(t), (15)
+whereu(t) solves (7) and w0(t) solves (4) with a=a(t). By the assumption,
+wa(t) exists for every t≥0. The local existence of u(t), the solution to (7), is
+the conclusion of Lemma 2.
+Lemma 2. If(4)-(5)hold, thenu(t), the solution to (7), exists locally.
+Proof.Differentiate equation (5) with respect to t. The result is
+Aa(t)(wa(t)) ˙wa(t) =−˙a(t)wa(t), (16)
+or
+˙wa(t) =−˙a(t)A−1
+a(t)(wa(t))wa(t). (17)
+Denote
+ψ(t) :=F(u(t))+a(t)u(t)−f. (18)
+For anyψ∈Hequation (18) is uniquely solvable for u(t) by the inversefunction
+theorem, because, by our assumption (3), the Fr´ echet derivat iveF′(u(t)) +
+a(t)Iis boundedly invertible, and (2) implies that the solution u(t) to (18) is
+continuously differentiable with respect to tifψ(t) is. One may solve (18) for
+uand writeu=G(ψ), where the map Gis continuously Fr´ echet differentiable
+becauseFis.
+Differentiate (18) and get
+˙ψ(t) =Aa(t)(u(t))˙u(t)+ ˙a(t)u. (19)
+If one wants the solution to (18) to be a solution to (7), then one ha s to require
+that
+Aa(t)(u(t))˙u=−ψ(t). (20)
+If (20) holds, then (19) can be written as
+˙ψ(t) =−ψ+ ˙a(t)G(ψ), G(ψ) :=u(t), (21)
+whereG(ψ) is continuously Fr´ echet differentiable. Thus, equation (21) is equ iv-
+alent to (7) at all t≥0 if
+ψ(0) =F(u0)+a(0)u0−f. (22)
+Indeed, ifusolves (7) then ψ, defined in (18), solves (21)-(22). Conversely,
+ifψsolves (21)-(22), then u(t), defined as the unique solution to (18), solves
+(7). Since the right-hand side of (21) is Fr´ echet differentiable, it s atisfies a local
+Lipschitz condition. Thus, problem (21)-(22) is locally, solvable. The refore,
+problem (7) is locally solvable.
+Lemma 2 is proved.
+4To prove that the solution u(t) to (7) exists globally, it is sufficient to prove
+the following estimate
+sup
+t≥0/bardblu(t)/bardbl<∞. (23)
+Lemma 3. Estimate (23)holds.
+Proof.Denote
+z(t) :=u(t)−w(t), (24)
+whereu(t) solves (7) and w(t) solves (4) with a=a(t). If one proves that
+lim
+t→∞/bardblz(t)/bardbl= 0, (25)
+then (23) follows from (25) and (5):
+sup
+t≥0/bardblu(t)/bardbl ≤sup
+t≥0/bardblz(t)/bardbl+sup
+t≥0/bardblw(t)/bardbl<∞. (26)
+To prove (25) we use Lemma 1.
+Let
+g(t) :=/bardblz(t)/bardbl. (27)
+Rewrite (7) as
+˙z=−˙w−A−1
+a(t)(u(t))[F(u(t))−F(w(t))+a(t)z(t)]. (28)
+Note that:
+sup
+h∈X∗,/bardblh/bardbl=1( ˙w(t),h) =/bardbl˙w(t)/bardbl. (29)
+Lemma 4. If the norm /bardblw(t)/bardblinXis differentiable, then
+/bardblw(t)/bardbl.≤ /bardbl˙w(t)/bardbl. (30)
+Proof.The triangle inequality implies:
+/bardblw(t+s)/bardbl−/bardblw(t)/bardbl
+s≤/bardblw(t+s)−w(t)/bardbl
+s, s>0. (31)
+Passing to the limit sց0, one gets (30).
+Lemma 4 is proved.
+The norm is differentiable if Xis strictly convex (see, e.g, [1]). A Banach
+spaceXis called strictly convex if ||u+v||<2 for anyu/ne}ationslash=v∈Xsuch that
+||u||=||v||= 1. A Banach space Xis called uniformly convex if for any ǫ>0
+there is aδ >0 such that for all u,v∈B(0,1) with||u−v||=ǫone has
+||u+v|| ≤2(1−δ). HereB(0,1) is the closed ball in X, centered at the origin
+and of radius one.
+5Various necessary and sufficient conditions for the Fr´ echet differ entiability
+of the norm in Banach spaces are known in the literature (see [2]), s tarting with
+Shmulian’s paper of 1940, see [28].
+Hilbertspaces, Lp(D)andℓp-spaces,p∈(1,∞),andSobolevspaces Wℓ,p(D),
+p∈(1,∞),D∈Rnis a bounded domain, have Fr´ echet differentiable norms.
+Thesespacesareuniformlyconvexandtheyhavethe E−property,i.e., if un⇀u
+and||un|| → ||u||asn→ ∞, then lim n→∞||un−u||= 0.
+From (17), (29) and (3) one gets
+/bardbl˙w/bardbl ≤c1|˙a(t)|r−b(t)/bardblw(t)/bardbl, r(t) =|a(t)|, (32)
+wherew(t) =wa(t). Since lim t→∞|a(t)|= 0, (32) and (5) imply
+/bardbl˙w/bardbl ≤c2|˙a(t)|r−b(t), c2=const>0, (33)
+because (5) implies the following estimate:
+c1/bardblw(t)/bardbl ≤c2, t≥0. (34)
+Inequality (13) implies that inequality (33) holds if
+/bardbl˙w/bardbl ≤c2|˙r(t)|r−b(t), t≥0. (35)
+Note that
+F(u)−F(w) =/integraldisplay1
+0F′(w+sz)dsz=A(u)z+/integraldisplay1
+0[A(w+sz)−A(u)]dsz.(36)
+Applyhto (28), take suph∈X∗,/bardblh/bardbl=1, and use Lemma 4, relation (36), estimate
+(3), and inequality (2), to get:
+˙g(t)≤ /bardbl˙z(t)/bardbl ≤c2|˙r(t)|r−b(t)+c3r−b(t)gp−g, (37)
+whereg(t) is defined in (27),
+p= 1+κ, c3:=c0c1. (38)
+Inequality (37) is of the form (8) with
+γ(t) = 1, α(t) =c2r−b(t), β(t) =c2|˙r(t)|r−b(t). (39)
+Choose
+µ(t) =λr−k(t), λ=const>0, k=const>0. (40)
+Then
+˙µµ−1=−k˙rr−1. (41)
+Let us assume that
+r(t)ց0,˙r<0,|˙r| ց0. (42)
+6Condition (10) implies
+g(0)λ
+rk(0)<1, (43)
+and inequality (9) holds if
+c3r−b(t)rkp
+λp+c2|˙r(t)|r−b(t)≤rk(t)
+λ(1−k|˙r(t)|r−1(t)), t≥0.(44)
+Inequality (44) can be written as
+c3rk(p−1)−b(t)
+λp−1+c2λ|˙r(t)|
+rk+b(t)+k|˙r(t)|
+r(t)≤1. (45)
+Let us choose kso that
+k(p−1)−b= 1,
+that is,
+k=b+1
+p−1. (46)
+Chooseλas follows:
+λ=rk(0)
+2g(0). (47)
+Then inequality (43) holds, and inequality (45) can be written as:
+c3r(t)[2g(0)]p−1
+[rk(0)]p−1+c2rk(0)
+2g(0)|˙r(t)|
+rk+b(t)+k|˙r(t)|
+r(t)≤1, t≥0.(48)
+Note that (46) implies:
+k+b=kp−1. (49)
+Chooser(t) so that relations (42) hold and
+k|˙r(t)|
+r(t)≤1
+2, t≥0. (50)
+Then inequality (48) holds if
+c2[2g(0)]p−1
+rb+1(0)+c2rk(0)
+2g(0)|˙r(t)|
+rkp−1≤1
+2, t≥0. (51)
+Denote
+c2rk(0)
+2g(0)=c2λ:=c4. (52)
+Let
+c4|˙r(t)|
+rkp−1=1
+4, t≥0, (53)
+7andkp>2. Then equation (53) implies
+r(t) =/bracketleftbigg
+t+4c4
+kp−21
+rkp−2/bracketrightbigg−1
+kp−2/parenleftbiggkp−2
+4c4/parenrightbigg−1
+kp−2
+. (54)
+Thisr(t) satisfies conditions (42), and equation (53) implies:
+k|˙r(t)|
+r(t)=krkp−2(t)
+4c4, t≥0. (55)
+Recall that r(t) decays monotonically. Therefore, inequality (50) holds if
+krkp−2(0)
+4c4≤1
+2. (56)
+Inequality (56) holds if
+k
+2rkp−2−k(0)
+c22g(0) =kg(0)
+c2rk(p−1)−2(0)≤1. (57)
+Note that (46) implies:
+k(p−1)−2 =b−1. (58)
+Condition (57) holds if g(0) is sufficiently small or rb−1(0) is sufficiently large:
+g(0)≤c2
+krb−1(0). (59)
+Ifb >1, then condition (59) holds for any fixed g(0) ifr(0) is sufficiently
+large. Ifb= 1, then (59) holds if g(0)≤c2
+k. Ifb∈(0,1) then (59) holds either
+ifg(0) is sufficiently small or r(0) is sufficiently small.
+Consequently, if (54) and (59) hold, then (53) holds. Therefore, (51) holds
+if
+c3[2g(0)]p−1
+rb+1(0)≤1
+4. (60)
+It follows from (59) that (60) holds if
+c32p−1/parenleftBigc2
+k/parenrightBigp−11
+rp+2b−bp(0)≤1
+4. (61)
+Ifb∈(0,1] then
+p−pb+2b>0. (62)
+Thus, (61) always holds if r(0) is sufficiently large, specifically, if
+r(0)≥[4c3/parenleftbig
+2c2k−1/parenrightbigp−1]1
+p(1−b)+2b. (63)
+We have proved the following theorem.
+8Theorem 2.1. Ifr(t) =|a(t)|is defined in (54), and if (59)and(63)hold,
+then
+/bardblz(t)/bardbl<rk(t)λ−1,lim
+t→∞/bardblz(t)/bardbl= 0. (64)
+Thus, problem (7)has a unique global solution u(t)and
+lim
+t→∞/bardblu(t)−y/bardbl= 0, (65)
+where
+F(y) =f. (66)
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+11
\ No newline at end of file
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+arXiv:1001.0369v3  [hep-ph]  3 Feb 2010EPJ manuscript No.
+(will be inserted by the editor)
+FZJ–IKP(TH)–2009–39
+Interplay of quark and meson degrees of freedom in a
+near-threshold resonance
+V. Baru1,2, C. Hanhart2, Yu. S. Kalashnikova1, A. E. Kudryavtsev1, A. V. Nefediev1
+1Institute for Theoretical and Experimental Physics, 11721 8, B. Cheremushkinskaya 25, Moscow, Russia
+2Forschungszentrum J¨ ulich, Institut f¨ ur Kernphysik (The orie) and J¨ ulich Center for Hadron Physics, D-52425 J¨ ulic h, Germany
+Abstract. We investigate the interplay of quark and meson degrees of fr eedom in a physical state repre-
+senting a near-threshold resonance for the case of a single c ontinuum channel. We demonstrate that such a
+near-threshold resonance may possess quite peculiar prope rties if both quark and meson dynamics generate
+weakly coupled near-threshold poles in the S-matrix. In par ticular, the scattering t-matrix may possess
+zeros in this case. We also discuss possible implications fo r production reactions as well as studies within
+lattice QCD.
+PACS.12.38.Lg – 11.55.Bq – 12.39.Mk
+1 Introduction
+With recent developments of B-factories, a wealth of new
+charmonia states was reported, with properties of some of
+those being incompatible with simple quark model predic-
+tions. Exotic explanations, such as hybrids or tetraquarks,
+are suggested to describe the new states. On the other
+hand, these new enigmatic states are above open charm
+threshold, so that the spectrum of the “quenched” quark
+model is to be significantly distorted by charmed-meson
+pairs (for reviews, see e.g. Refs. [1,2,3,4,5]).
+Moreover, some of the new states reside in the vicin-
+ity ofS-wave thresholds. A most prominent example here
+is the famous X(3872) state, first observed in B-meson
+decays [6]. The X(3872) is extremely close to the D¯D∗
+threshold, with D¯D∗being in the S-wave, if the quan-
+tum numbers of the Xare 1++, as suggested by the data.
+There are also the Y(4260) and Y(4325) vector states
+[7,8], with relevant S-wave thresholds being the D¯D1at
+4.285 GeV and D∗¯D0at about 4.360 GeV (D1andD0are
+correspondingly 1+and 0+D-mesons). The purely exotic
+charged state Z+(4430) [9] is close to the D∗¯D1thresh-
+old and, for the charged Z+
+1(4050) and Z+
+2(4250) [10], the
+relevant thresholds are D∗¯D∗andD1¯D, respectively1.
+Thresholdproximityimpliesthat,independentlyofthe
+binding mechanism, there should be a significant compo-
+nent of a hadronic molecule in the wave function of the
+state. So the question arises of how to distinguish between
+a genuine “elementary” particle ( q¯q, hybrid, or compact
+1These charged states are to be considered with caution,
+as seen only in one experiment (Belle). Besides, as quantum
+numbers of the Z-particles are not known, the thresholds are
+not necessarily the S-wave ones.tetraquark) and a composite state (hadronic molecule),
+and how to estimate the admixture of the latter. It was
+suggested in Ref. [11,12,13] that it is possible, in the case
+ofa near-thresholdbound state, to answerthis question in
+a model-independent way: the state is mostly elementary
+if the effective radius is large and negative. The approach
+was generalised in Ref. [14,15] to the case of presence
+of inelastic channels, as well as to the case of an above-
+threshold resonance. Related to this is the pole counting
+approach[16], in which the structureofthe near-threshold
+singularities of the scattering amplitude is studied. It ap-
+pears that the state is mostly elementary if there are two
+nearby poles in the scattering amplitude, while a compos-
+ite particle corresponds to a single near-threshold pole.
+The approaches of Ref. [11,12,13,14,15,16] are based
+on the effective-range expansion of the scattering ampli-
+tude and, as such, are expected to be valid for the mo-
+menta involvedmuch smaller than the inverserangeofthe
+force. However,as noticed in Ref. [11,12,13], the effective-
+range formulae are not the most general ones even in the
+small-momentalimit. The scatteringamplitude, as afunc-
+tion of energy, can have a zero and, if this zero is situated
+in the near-thresholdregion, the effective-range expansion
+would fail. In the present paper we develop a formalism
+which allows one to pinpoint the source of this failure, to
+studythehadronicobservablesinthepresenceofthiszero,
+and to identify the physical situation in which it occurs.
+2 General formalism
+We consider a physical state which is a mixture of a bare
+state (q¯qor compact tetraquark) and a dynamical (mole-2 V. Baru et al.: Interplay of quark and meson degrees of freedom in a near-th reshold resonance
+cule) component and represent its wave function as:
+|Ψ/an}bracketri}ht=/parenleftbiggc|ψ0/an}bracketri}ht
+χ|M1M2/an}bracketri}ht/parenrightbigg
+, (1)
+where|ψ0/an}bracketri}htis the bare elementary state with the probabil-
+ity amplitude c, whileχ(p) describes the relative motion
+in the system of two mesons ( M1M2), with the masses m1
+andm2, respectively, and with the relative momentum p.
+The wave function |Ψ/an}bracketri}htobeys a Schr¨ odinger-like equation:
+H|Ψ/an}bracketri}ht=E|Ψ/an}bracketri}ht, (2)
+with the Hamiltonian
+H=/parenleftbigg
+H0Vqh
+VhqHh/parenrightbigg
+, (3)
+where
+H0|ψ0/an}bracketri}ht=E0|ψ0/an}bracketri}ht, (4)
+m1+m2+E0is the bare state mass, and
+Hh(p,p′) =p2
+2µδ(p−p′)+V(p,p′),(5)
+whereµis the reduced mass.
+The termVqhis responsiblefor the dressingofthe bare
+state, which is given by the transition form factor f(p):
+/an}bracketle{tψ0|Vqh|M1M2/an}bracketri}ht=f(p). (6)
+The Schr¨ odinger-like equation (2) is equivalent to the
+system of equations for the candχ, which reads:
+
+
+c(E)E0+/integraldisplay
+f(p)χ(p)d3p=c(E)E,
+p2
+2µχ(p)+c(E)f(p)+/integraldisplay
+V(p,k)χ(k)d3k=Eχ(p).
+(7)
+On substituting c(E) from the first equation to the
+second one, we arrive at the Schr¨ odinger equation in the
+mesonic channel, with an effective potential
+Veff(p,p′,E) =V(p,p′)+f(p)f(p′)
+E−E0.(8)
+The off-shell mesonic t-matrixt(p,p′,E) is a solution of
+the Lippmann–Schwinger equation2,
+t(p,p′,E) =Veff(p,p′,E)−/integraldisplay
+d3qVeff(p,q,E)t(q,p′,E)
+q2/(2µ)−E−i0.
+(9)
+The solution of this equation can be written as
+t(p,p′,E) =tV(p,p′,E)+φ(p,E)¯φ(p′,E)
+E−E0+G(E),(10)
+2Normalisation of the t-matrix is such that the M1M2scat-
+tering amplitude is given by f(k,k,E) =−4π2µt(k,k,E),
+withE=k2/(2µ).wheretV(p,p′,E) is thet-matrix for the potential prob-
+lem,
+tV(p,p′,E) =V(p,p′)−/integraldisplay
+d3qV(p,q)tV(q,p′,E)
+q2/(2µ)−E−i0,(11)
+while the dressed vertex functions are
+φ(p,E) =f(p)−/integraldisplay
+d3qtV(p,q,E)f(q)
+q2/(2µ)−E−i0,(12)
+¯φ(p,E) =f(p)−/integraldisplay
+d3qtV(q,p,E)f(q)
+q2/(2µ)−E−i0,(13)
+and
+G(E) =/integraldisplay
+d3qf(q)φ(q,E)
+q2/(2µ)−E−i0. (14)
+The system of equations (7) can possess bound states
+(generally, more than one). For a bound state iwith the
+binding energy ǫ(i)
+B, the solution of the system (7) may be
+written as
+c(i)
+B= cosθi, χ(i)
+B(p) =ψi(p)sinθi,(15)
+whereψi(p) is normalised to unity. Then the wave func-
+tion of this bound state,
+|Ψ/an}bracketri}ht(i)
+B=/parenleftbiggcosθi|ψ0/an}bracketri}ht
+sinθiψi(p)|M1M2/an}bracketri}ht/parenrightbigg
+, (16)
+is also normalised,(i)
+B/an}bracketle{tΨ|Ψ/an}bracketri}ht(i)
+B= 1.
+The quantity
+Zi=|/an}bracketle{tψ0|Ψ/an}bracketri}ht(i)
+B|2= cos2θi, (17)
+introduced in Ref. [11,12,13], gives the probability to find
+a bare state in the wave function of the bound state i.
+The solution of the system (7) for the continuum is
+ck(E) =1
+E−E0/integraldisplay
+d3pχk(p)f(p) =¯φ(k,E)
+E−E0+G(E),
+(18)
+χk(p) =δ(p−k)−t(p,k,E)
+p2/(2µ)−E−i0, E=k2
+2µ,(19)
+so that the continuum counterpart of the quantity (17),
+the spectraldensity w(E), givesthe probabilityto find the
+bare state in the continuum wave function [17]. It can be
+found as:
+w(E) = 4πµk|ck(E)|2Θ(E), (20)
+withck(E) given by Eq. (18). As shown in Ref. [17], the
+normalisation condition for the distribution w(E) reads:
+/integraldisplay∞
+0w(E)dE= 1−/summationdisplay
+iZi, (21)
+where the sum goes over all bound states present in the
+system, while the corresponding Z-factors are given by
+Eq. (17).
+We are interested in the energy range close to the
+threshold. To perform a low-energy reduction, let us as-
+sume that the scattering length approximation is valid forV. Baru et al.: Interplay of quark and meson degrees of freedom in a near-th reshold resonance 3
+the potential problem, so that the potential t-matrixtV
+takes the form:
+tV(q,p,E) =−1
+4π2µ(−a−1
+V−ik)+..., (22)
+whereaVisthe scatteringlengthforthe potential V(p,p′)
+and ellipsis stands for the range of forces corrections. We
+usethesignconvention,whereanegativescatteringlength
+corresponds to an attractive potential, however, too weak
+toproduceaboundstatewhileapositivescatteringlength
+reverses to either a repulsive potential (which cannot pro-
+duce a singularity of the S-matrix, thus we automatically
+have 1/aV∼β,βis the range of forces) or a bound state,
+which is the case of interest here. Note, the above expres-
+sion is useful only for aVβ≫1 and then applicable only
+for momenta of order 1 /aV. In this case tVhas a near-
+threshold pole. However, it should be stressed that this
+singularity is observable only in the weak coupling limit
+to the quark states located nearby, for otherwise all singu-
+larities undergo a significant mixing, the origin of which
+is the appearance of tVin the dressed vertex functions of
+Eqs. (12) and (13). For a detailed discussion of this effect
+see Ref. [18,19].
+It is convenient to define the loop functions:
+g(E) =/integraldisplay
+d3qf2(q)
+q2/(2µ)−E−i0=f2
+0(R+4π2µik),
+(23)
+g′(E) =/integraldisplay
+d3qf(q)
+q2/(2µ)−E−i0=f0(R′+4π2µik),
+where the constants RandR′are expected to be of order
+µβand corrections O(k2/β2) where dropped; in addition
+we introduced f0≡f(0). Defining
+RV=4π2µ
+aV, (24)
+one arrives at the following form for the on-shell t-matrix
+(10):
+t(E) =E−EC
+[E−E0]RV+f2
+0[RRV−R′2]+4π2µik[E−EC],
+(25)
+EC=E0−f2
+0(R+RV−2R′).
+Expression (25) for the scattering amplitude has a zero at
+E=EC, and the effective-range expansion for the am-
+plitude (25) is not valid, if ECis in the near-threshold
+region.
+3 Flatt´ e formulae
+In this chapter we develop a generalisation of the well-
+known Flatt´ e parameterisation [20] for the near-threshold
+scattering amplitude, and express the t-matrix of Eq. (25)
+in terms of Flatt´ e-type parameters.
+Let us define the Flatt´ e energy Efsuch that the real
+part of the denominator in Eq. (25) has a zero when E=
+Ef,
+[Ef−E0]RV+f2
+0[RRV−R′2] = 0,(26)and get rid of E0this way. Then one can express ECin
+terms ofEf:
+EC=Ef−f2
+0
+RV(R′−RV)2, (27)
+and, using the identity
+E−Ef
+E−EC=E−Ef
+Ef−EC−(E−Ef)2
+(E−EC)(Ef−EC),(28)
+rewrite the t-matrix in the Flatt´ e-type form as:
+t(E) =gf
+8π2µDF(E). (29)
+Here
+gf
+2= 4π2µEf−EC
+RV=4π2µf2
+0
+R2
+V(R−RV)2(30)
+and
+DF(E) =E−Ef−(E−Ef)2
+E−EC+i
+2gfk.(31)
+If|EC| ≫ |Ef|, we are back to the standard Flatt´ e ap-
+proximation for the near-threshold scattering amplitude
+[20],
+tF(E) =1
+8π2µgf
+E−Ef+i
+2gfk, (32)
+which depends on two parameters, gfandEf, in contrast
+to the expression (29), which depends on three parame-
+ters:gf,Ef, andEC. One can see that, for |EC| ∼ |Ef|,
+the standard Flatt´ e formula is severely distorted by the
+presence of the t-matrix zero, with a drastic effect on the
+behaviour of the elastic scattering cross section.
+The Flatt´ e-type formulae given above define the t-
+matrix in terms of the parameters Ef,gf, andEC. Fol-
+lowing S. Weinberg [11,12,13], it is instructive to consider
+the caseofa singlebound state present,and to expressthe
+t-matrix in terms of the binding energy ǫB, theZ-factor
+(which is the probability to find a bare state in the physi-
+cal bound state wave function), and the scattering length
+aV.
+We start from the expression for the inverse t-matrix
+(25),
+t−1(E) =E−Ef
+E−ECRV+4π2µik. (33)
+In the vicinity of the bound-state pole one has
+t−1(E)≃ǫB+E
+g2
+eff, (34)
+with (see Ref. [11,12,13])
+g2
+eff=√2µεB
+4π2µ2(1−Z). (35)
+This defines
+Z
+1−Z=2ǫB
+ǫB+EC/parenleftbigg
+1−γV√2µεB/parenrightbigg
+,(36)4 V. Baru et al.: Interplay of quark and meson degrees of freedom in a near-th reshold resonance
+withγV= 1/aVbeing the inverse scattering length for
+the potential problem in the absence of the quark state.
+Thus the following formulae can be obtained [11,12,13]:
+kcotδ=−/radicalbig
+2µεB+√2µεB(E+ǫB)(ǫB+EC)
+2ǫB(E−EC)Z
+1−Z,
+(37)
+and
+EC=−ǫB/parenleftbigg
+1−2(1−Z)
+Z+2(1−Z)
+ZγV√2µεB/parenrightbigg
+.(38)
+Now, for |EC| ≫ǫB, the effective-range expansion is
+recovered from Eq. (37), with the scattering length aand
+the effective radius regiven by:
+a=2(1−Z)
+(2−Z)1√2µεB, re=−Z
+(1−Z)1√2µεB,(39)
+which allows one to define the quantity Zin a model-
+independent way, if the effective-range parameters aand
+reare available.
+On the contrary, for |EC|/lessorsimilarǫBthe effective-range
+expansion does not converge anymore for energies larger
+than|EC|. Correspondingly, formulae from Eq. (39) do
+not hold anymore. In the next section we discuss the cir-
+cumstances, when the described unusual behaviour may
+occur.
+4 Pole structure in the presence of the
+t-matrix zero
+Due to a very simple relation (30) between ECandγV,
+it is instructive to study the amplitude singularities in
+terms of the variables Ef,gf, andγV. The expression for
+the scattering t-matrix in this case takes the form:
+t(E) =1
+4π2µE−Ef+1
+2gfγV
+(E−Ef)(γV+ik)+i
+2gfγVk.(40)
+As seen from expression (40), there generally could be up
+to three near-threshold poles, and the equation defining
+the pole positions in the k-plane reads:
+(k2−2µEf)(γV+ik)+iµgfγVk= 0.(41)
+Denote the solutions of Eq. (41) as k1,k2, andk3. As
+follows from the explicit form of Eq. (41), one of these so-
+lutions is always imaginary, while the other two are either
+both imaginary or placed at the lower half–plane symmet-
+rically with respect to the imaginary axis, as required by
+the general properties of the S-matrix. Thus, the set of
+parameters {Ef,gf,γV}used in Eq. (40) is fully defined
+by the poles of the scattering t-matrix{k1,k2,k3}:
+Ef=−1
+2µk1k2k3
+k1+k2+k3, (42)
+gf=−(k1+k2)(k1+k3)(k2+k3)
+iµ(k1+k2+k3)2,(43)
+γV=−i(k1+k2+k3). (44)The position ECof thet-matrix zero can be also ex-
+pressed in terms of {k1,k2,k3}:
+EC=Ef−1
+2gfγV=−1
+2µ(k1k2+k1k3+k2k3).(45)
+Notice that, for the given EfandγV, the range of values
+forECis restricted by the condition gf≥0 (see Eq. (30)),
+which follows from causality.
+It is illustrating to discuss the behaviour of the three
+poles when the effective coupling gfis varied from zero to
+some large value. To study the properties of the system
+we introduce the scale ∆≪βthat defines the region of
+applicability of the equations3. If the coupling of the bare
+state to the mesonic channel is switched off, gf= 0, the
+solutions of Eq. (41) are
+k(0)
+1,2=±/radicalbig
+2µEf, k(0)
+3=iγV. (46)
+The first pair of poles corresponds to the bare quark state
+(uncoupled from the mesonic channel) with the energy
+Ef=E0, and the third one is due to the direct interac-
+tionV(p,p′) in the mesonic channel, as already described
+above. Note, however, that the numerator of the t-matrix
+(see Eq. (40)) has a zero exactly at the momenta k(0)
+1,2
+which eliminates the contribution of the corresponding
+poles. Thus, in the limit of an extremely small coupling,
+gf≪ |Ef|/|γV|(see Eq. (45)), there is only one pole k3in
+thescattering t-matrixthat correspondstoaboundorvir-
+tual state in the mesonic channel, depending on the sign
+ofγV. As the interaction gfis switched on and increases,
+gf∼ |Ef|/|γV|, the poles start to move and to mix, and
+thet-matrix may have a zero and three poles in different
+places of the k-plane. Note also that in this case there is
+no straightforward interpretation of γVpossible anymore
+duetothemixingofpoles.Clearly,thezeroofthe t-matrix
+can have an impact on the observables only if it is in the
+regionofapplicability definedas ∆. From Eq.(45) onecan
+see then that EC∼∆only if all three poles are located
+very close to the threshold. In other words, if either the
+potential pole or the quark poles are outside the region of
+applicability of our formalism, also the t-matrix zero lies
+outside that region. In the latter case the effective-range
+approximation is adequate for the problem under consid-
+eration, so that one can set EC≫E,∆in all equations,
+thus recovering the standard Flatt´ e formulae.
+Let us now assume that all three bare poles appear
+in the vicinity of the threshold ( |Ef| ≃γ2
+V/(2µ)/lessorsimilar∆)
+but the coupling is strong ( gf≫ |Ef|/|γV|). Clearly, in
+this caseEC≫∆and we again recover the standard
+lineshapes together with the standard effective range ex-
+pansion; only a single pole remains in the near-threshold
+regime — the expressions of Eq. (39) hold again and we
+find, as expected, that this single pole corresponds to a
+state of predominantly dynamical (=molecular) origin.
+To summarise, the most peculiar situation takes place
+if:
+3Note, however, that this applicability region might be fur-
+therrestricted byadditional scales presentinaparticula r prob-
+lem under consideration.V. Baru et al.: Interplay of quark and meson degrees of freedom in a near-th reshold resonance 5
+1. the bare quark and potential poles appear acciden-
+tallyclose to each other and to the threshold ( |Ef| ≃
+γ2
+V/(2µ)/lessorsimilar∆);
+2. the quark state is relatively weakly coupled to the
+hadronic channel ( gf∼ |Ef|/|γV|).
+Thus, the appearanceofazeroin the scatteringamplitude
+very close to threshold, signaled also by an early break-
+down of the effective-range expansion, is a clear signal of
+the presence of both potential as well as quark poles.
+To detail these statements and to mimic the peculiar
+situation described above we choose |γV| ∼/radicalbig
+2µ|Ef|for
+gf= 0. We choose for the reduced mass µ= 1 GeV as
+is appropriate for the X(3872) case, and define the near-
+threshold region by ∆∼1 MeV — the energy range rele-
+vant for the X(3872) particle (in case of the X(3872) the
+range of validity of the approach is not set by the range of
+forces, but by the closet threshold, which is only 8 MeV
+away). We then study the system for different values of
+gffor six representative scenarios (recall: γV<0 refers to
+an attractive potential, which does not support a bound
+state, while γV>0 refers to a bound state present from
+the pure potential scattering). For each case we fix the
+parameters EfandγV, and change gffrom very small
+values to the ones for which the t-matrix zero leaves the
+near-threshold region.
+–Case (i):Ef>0,γV>0 (Ef= 1 MeV, γV=
+45 MeV). See Fig. 1, 2.
+–Case (ii):Ef>0,γV<0 (Ef= 1 MeV, γV=
+−45 MeV). See Fig. 3.
+–Case (iii-a): Ef<0,γV>0,/radicalbig
+2µ|Ef|> γV(Ef=
+−1 MeV,γV= 40 MeV). See Fig. 4.
+–Case (iii-b): Ef<0,γV>0,/radicalbig
+2µ|Ef|< γV(Ef=
+−1 MeV,γV= 50 MeV). See Fig. 5.
+–Case (iv-a): Ef<0,γV<0,/radicalbig
+2µ|Ef|>|γV|(Ef=
+−1 MeV,γV=−20 MeV). See Fig. 6.
+–Case (iv-b): Ef<0,γV<0,/radicalbig
+2µ|Ef|<|γV|(Ef=
+−1 MeV,γV=−55 MeV). See Fig. 7.
+For cases (i) and (ii), the hadronic cross section van-
+ishes atE=ECaccompanied by a very peculiar energy
+dependence. For small couplings, the spectral density as
+well as the elastic cross section peaks at around Efand
+the bound state, present in case (i), is a purely mesonic
+one. Then, with the increase of the coupling, the afore-
+mentioned bound state leaves the near-threshold region
+and acquires a large admixture of the quark component.
+In this strong-coupling regime the dynamics is defined by
+the presence of a single pole which corresponds to the vir-
+tual state. Due to the normalisation condition (21), the
+spectral density decreases with the increase of the cou-
+pling.
+It is instructive to discuss, for example, Case (i) in
+some more detail. In particular, in Fig. 2, the movement
+of the poles in the s-plane is depicted on both first (left
+plot) and second (right plot) Riemann sheets. The arrows
+indicate how the poles start to move when gfis increased.
+Clearly this plot is equivalent to the pole movement in the
+k-planedepicted intheupperleft panelofFig.1.Asusual,
+foranarrowresonance(small gf)onlythepoleinthelowerhalf plane on the second sheet (see Fig. 2) influences the
+physics. Here this state is nearly a pure quark state. As gf
+gets larger the width of the state grows (and the mesonic
+admixture increases). At latest, when the real part of the
+pole position reachesthe threshold (indicated in the figure
+as the perpendicular, dashed line), both poles are equally
+important. Notice that, in this regime, the interpretation
+of the imaginary part of the pole position as half of the
+width of the state is lost. Similar pole movements in the
+subthreshold regime were reported previously in Refs. [21,
+22].
+In Cases (iii-a,b) one has EC<0, so that the t-matrix
+zero does not manifest itself in the hadronic cross sec-
+tion defined for positive energies only. In the meantime,
+with|EC| ∼∆, there are two bound states in the near-
+threshold region. With the increase of the coupling, only
+one bound state survives in the near-threshold region,
+withZ→0. The spectral density is small for all values of
+the coupling, as two bound states saturate the normalisa-
+tion condition (21).
+Finally, in cases (iv-a) and (iv-b) and for small cou-
+plings, the bound state is almost purely of quark nature,
+andEC<0. With the increase of the coupling, ECbe-
+comes positive and shows up in the hadronic cross sec-
+tion, while the bound state acquires a significant mesonic
+admixture. Further increase of the coupling forces the t-
+matrixzeroto leavethe near-thresholdregion,and the dy-
+namics is defined by a single bound state, which is purely
+molecular, with Z→0.
+So, for all cases considered, there is only one near-
+threshold pole in the strong-coupling regime which cor-
+responds to the scattering-length approximation for the
+mesonict-matrix,independentlyoftheunderlyingdynam-
+ics.
+5 Production reactions
+Unfortunatelythereisnoexperimentalpossibilitytostudy
+elastic scattering of, say, charmed mesons, and our knowl-
+edge of the resonance properties comes from production
+experiments. In general there are two production mech-
+anisms possible, namely a production of the resonance
+of interest via its hadronic component or via its quark
+component. If the production source can be considered
+as point-like, then the production amplitude of the me-
+son pair (M1M2) through the resonance from the former
+mechanism can be written as
+Mh(E) =Fh/parenleftbigg
+1−/integraldisplay
+d3pt(p,k,E)
+p2/(2µ)−E−i0/parenrightbigg
+|k2/(2µ)=E,
+(47)
+whereFhis the initial state production amplitude from
+the point-like source, unity stands for the Born term, and
+the second term defines the final-state interaction — see
+the left panel of Fig. 8. For a point-like source and small
+energies, the amplitude (47) is
+Mh(E) =Fh(1−L(E)t(E)), (48)6 V. Baru et al.: Interplay of quark and meson degrees of freedom in a near-th reshold resonance
+Lbeing the loop function describing propagation of the
+intermediate mesonic state,
+L(E) = 4π2µ(l0+ik), (49)
+withl0∼β, whereβis the range of the force. We may
+therefore write
+Mh(E) =Fh(E−Ef)(Ef−EC)−1
+2l0gf(E−EC)
+(E−Ef)(Ef−EC)+i
+2kgf(E−EC).
+(50)
+Thus, in Mh(E) the zero in the production amplitude is
+shifted with respect to the zero of the t-matrix. However,
+for the energies Ef≃EC≪βthis shift is small. The sec-
+ond term in the numerator of Eq. (50) dominates, and the
+production rate through the hadronic component alone is
+dBrh(M1M2)
+dE= const×k|t(E)|2Θ(E)
+= const×/parenleftbigggf
+8π2µ/parenrightbigg2k
+|DF|2,(51)
+where the denominator DFis given by Eq. (31). Thus
+one might expect that all said above about the phys-
+ical content of a possible zero in t-matrix in the near-
+threshold regime translates one-to-one also to production
+reactions. However, this is not correct for there is, in ad-
+dition to the hadronic production, also the production via
+thequarkcomponentpossible.Thispiececanbeexpressed
+via the spectral density. Indeed, as seen from the right
+panel Fig. 8, the corresponding production amplitude is
+given by:
+Mq=−FqGq0(E)tqh(k,E),k2
+2µ=E,(52)
+whereFqis the production amplitude for the bare quark
+state by the point-like source, Gq0(E) = 1/(E0−E) is
+the quark state free Green’s function, and tqh(k,E) is the
+t-matrix element responsible for the quark–meson transi-
+tion. From the corresponding Lippmann–Schwinger equa-
+tion,
+tqh(k,E) =f(k)−/integraldisplay
+d3pf(p)t(p,k,E)
+p2/(2µ)−E−i0,(53)
+and with the help of Eqs. (18), (19), the latter can be
+found in the form:
+tqh(k,E) =/integraldisplay
+d3pχk(p)f(p) = (E−E0)ck(E).(54)
+Thus, the near-threshold production rate via the quark
+component of the wave function is simply
+dBrq(M1M2)
+dE= const×w(E), (55)
+wherew(E) is defined in Eq. (20). Then, using definition
+(20) and the explicit form of the coefficient ck(E), which
+follows from Eq. (18),
+ck(E) =/radicalbigggf
+8π2µEf−EC
+E−EC1
+DF, (56)one finds:
+w(E) =1
+2πkgf
+|DF|2(Ef−EC)2
+(E−EC)2. (57)
+Obviously, this implies that dBrq/dEdoes not employ a
+zero.
+In reality one expects both mentioned mechanisms to
+contribute to the production of the physical resonance,
+with a relative importance depending on the production
+mechanism. Therefore, we may write for the full produc-
+tion rate:
+dBr(M1M2)
+dE= const×k|E−EC+r(Ef−EC)|2
+|DF|2(E−EC)2,
+(58)
+whereris a real number containing, among other contri-
+butions, the ratio of the production rates via the quark
+state and the hadronic state. A priori no estimate of r
+is possible thus, even if there were a system that shows a
+zero in the scattering amplitude near threshold and allow-
+ing for all the conclusions ofthe previoussection, this zero
+might well be shielded in the production reaction. On the
+other hand, if Ec≫∆it requires a delicate fine tuning
+of the parameter rto produce a zero in the amplitude of
+Eq. (58). Thus, if there is a zero observed in a production
+amplitude, it at least suggests the presence of both quark
+states as well as a hadronic molecule.
+Note that the zero in the lineshape of the X(3872)
+predicted to occur in a B-decay amplitude reported in
+Ref. [23,24] is of a different kind, for it originates as a
+coupled-channel effect of two nearby continuum channels.
+In this work, on the other hand, the effects in the presence
+of only a single continuum channel were discussed.
+6 Implications for lattice QCD
+From the previous discussions it should be clear that the
+presenceof a zeroin the scatteringamplitude contains im-
+portant information about the system studied. However,
+this information is no longer visible in the production am-
+plitude, which is the quantity accessible experimentally.
+Fortunately scattering observables, in particular phase
+shifts,areaccessibleinlatticeQCDfromastudyofvolume
+dependencies of the energy spectra [25]. In simulations of
+full QCD both quark states as well as molecular states are
+present and will influence all correlators, as long as some
+overlapexists with the interpolating fields used. Thus, the
+results of those simulations need to be interpreted in the
+same way as experimental results call for an interpreta-
+tion, if one wants to gain some insight into the dynami-
+cal mechanisms that lead to the structure formation. One
+method pushed recently is to work with a large basis of
+interpolatingfields and to use the resulting eigenvectorsof
+the correlatormatrices to interpret the findings [26,27]. In
+Ref. [22] the scatteringlength is shownto be an important
+quantity also to extract information about the nature of
+states from lattice studies. The discussion above provides
+one with an additional approach: in order to understandV. Baru et al.: Interplay of quark and meson degrees of freedom in a near-th reshold resonance 7
+the interplay of quark and meson degrees of freedom one
+may investigate the near-threshold phase shifts. If a zero
+inthet-matrix,correspondingtoazerointhephaseshifts,
+occurs, it immediately signals the presence of both quark
+poles as well as potential scattering poles with a relatively
+weak coupling as described in Sec. 4.
+To illustrate this point we discuss briefly the phase
+shift (its cotangent) for Case (i) introduced above,
+kcotδ=−γV(E−Ef)
+E−Ef+1
+2gfγV. (59)
+This expression employs a pole at the position of the t-
+matrix zero, as shown in Fig. 9. We expect that such a
+structure could be extracted from studies within lattice
+QCD.
+7 Conclusions
+In this paper we have presented a dynamical scheme in
+which a near-threshold zero in the mesonic t-matrix ap-
+pears as a consequence of the interplay between quark
+statesand anonperturbativelyinteractinghadron–hadron
+continuum. The appearance of such a zero invalidates the
+effective-range expansion and corresponds to three near-
+threshold poles of the t-matrix. The effect on hadronic
+observables in elastic scattering is proved to be drastic.
+However, a near-threshold t-matrix zero could exist
+only if several requirements are met. First, one needs the
+direct interaction in the mesonic channel to be strong
+enough to support a bound or virtual state. Second, a
+nearby bare quark state should exist, with a weak cou-
+pling to the mesonic channel. While, in principle, such a
+situation cannot be excluded, it is highly accidental.
+Without such special arrangements the effective-range
+formulae are valid, and the conclusions of Refs. [11,12,13,
+14,15,16] hold true: a large and negative effective range
+corresponds to a compact quark state, while a small ef-
+fective range means that the state is composite. In the
+former case there are two near-threshold poles in the t-
+matrix, while in the latter case there is only one pole.
+In the two-pole case one can definitely state that the
+resonance is generated by an s-channel diagram. The cou-
+pling of this quark state to the mesonic continuum is not
+large. In the one-pole situation no model-independent in-
+sight into the underlying dynamics is possible and one can
+only indicate that the resonance is generated dynamically
+but, in the language of meson exchange, the binding po-
+tential could emerge from either s-channel of t-channel
+exchanges — or a mixture of both. It seems not pos-
+sible to decide between these scenarios model indepen-
+dently. The three pole scenario, on the other hand, calls
+for the presence of both s-channel interactions as well
+as potential scattering — traditionally identified with t-
+channel exchanges. This situation can be identified easily
+by the break-down of the effective-range expansion. Al-
+though scattering experiments are not possible for most
+unstable particles, an alternative access to scattering ob-
+servables could be provided by studies of lattice QCD.We alsodiscussedtheimplicationsforproductionreac-
+tions. Unfortunately it turns out that for the lineshapes of
+production reactions no statement is possible model inde-
+pendently based solely on information from the scattering
+amplitudes.
+The work was supported in parts by funds provided from the
+Helmholtz Association (grants VH-NG-222, VH-VI-231), by
+the DFG (grants SFB/TR 16 and 436 RUS 113/991/0-1), by
+theEUHadronPhysics2project, bytheRFFI(grantsRFFI-09-
+02-91342-NNIOa and RFFI-09-02-00629a), and by the Presi-
+dential programme for support of the leading scientific scho ols
+(grants NSh-4961.2008.2 and NSh-4568.2008.2). The work of
+V.B., Yu.K., A.K., and A.N. was supported by the State Cor-
+poration of Russian Federation “Rosatom”. A. N. would like
+to acknowledge the support of the non-profit “Dynasty” foun-
+dation and ICFPM and of the grant PTDC/FIS/70843/2006-
+Fisica.
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+Meißner, Phys. Lett. B 681(2009) 26.
+19. M. D¨ oring, C. Hanhart, F. Huang, S. Krewald and U. G.
+Meißner, Nucl. Phys. A 829(2009) 170.
+20. S. Flatt´ e, Phys. Lett. B 63(1976) 224.
+21. C. Hanhart, J. R. Pelaez, and G. Rios, Phys. Rev. Lett.
+100(2008) 152001.
+22. F. K. Guo, C. Hanhart, and U. G. Meißner, Eur. Phys. J.
+A40(2009) 171.
+23. E. Braaten and M. Lu, Phys. Rev. D 77(2008) 014029.
+24. M. B. Voloshin, Phys. Rev. D 76(2007) 014007.
+25. M. L¨ uscher, Commun. Math. Phys. 105(1986) 153.
+26. J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G.
+Richards, and C. E. Thomas, Phys. Rev. Lett. 103(2009)
+262001.
+27. J. J. Dudek, R. G. Edwards, M. Mathur, and D. G.
+Richards, Phys. Rev. D 77(2008) 034501.8 V. Baru et al.: Interplay of quark and meson degrees of freedom in a near-th reshold resonance
+Fig. 1.Case (i). Upper panel: the pole structure in the k-plane (left plot) and the Z-factor — see Eq. (36) (right plot) versus
+the coupling constant gf. Lower panel: the elastic scattering cross section (left pl ot) and the spectral density (right plot) versus
+the energy for gf= 0.01 (solid line), gf= 0.03 (dashed line), and gf= 0.1 (dotted line).
+Fig. 2. The poles motion in the s-plane for Case (i) at the first Riemann sheet (left plot) and o n the second Riemann sheet
+(right plot). The unitarity cut is depicted in grey.V. Baru et al.: Interplay of quark and meson degrees of freedom in a near-th reshold resonance 9
+Fig. 3.Case (ii). Upper panel: the pole structure in the k-plane. Lower panel: the elastic scattering cross section ( left plot) and
+the spectral density (right plot) versus the energy for gf= 0.01 (solid line), gf= 0.03 (dashed line), and gf= 0.1 (dotted line).
+Fig. 4. Case (iii-a). Upper panel: the pole structure in the k-plane (left plot) and the Z-factors — see Eq. (36) (right plot)
+versus the coupling constant gf. Lower panel: the elastic scattering cross section (left pl ot) and the spectral density (right plot)
+versus the energy for gf= 0.001 (solid line), gf= 0.02 (dashed line), and gf= 0.3 (dotted line).10 V. Baru et al.: Interplay of quark and meson degrees of freedom in a near-th reshold resonance
+Fig. 5.Case (iii-b). Upper panel: the pole structure in the k-plane (left plot) and the Z-factors (right plot) versus the coupling
+constant gf. Lower panel: the elastic scattering cross section (left pl ot) and the spectral density (right plot) versus the energy
+forgf= 0.001 (solid line), gf= 0.02 (dashed line), and gf= 0.3 (dotted line).
+Fig. 6. Case (iv-a). Upper panel: the pole structure in the k-plane (left plot) and the Z-factor — see Eq. (36) (right plot)
+versus the coupling constant gf. Lower panel: the elastic scattering cross section (left pl ot) and the spectral density (right plot)
+versus the energy for gf= 0.01 (solid line), gf= 0.13 (dashed line), gf= 0.3 (dotted line), and g= 0.5 (dash-dotted line).V. Baru et al.: Interplay of quark and meson degrees of freedom in a near-th reshold resonance 11
+Fig. 7. Case (iv-b). Upper panel: the pole structure in the k-plane (left plot) and the Z-factor — see Eq. (36) (right plot)
+versus the coupling constant gf. Lower panel: the elastic scattering cross section (left pl ot) and the spectral density (right plot)
+versus the energy for gf= 0.01 (solid line), gf= 0.07 (dashed line), gf= 0.15 (dotted line), and g= 0.5 (dash-dotted line).
+Fig. 8. Diagrams for the two meson production via the hadronic compo nent (left plot) and via the quark component (right
+plot).
+Fig. 9.The phase shift for Case (i) for gf= 0.01 (solid line), gf= 0.02 (dashed line), and gf= 0.1 (dotted line).
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+arXiv:1001.0371v3  [hep-th]  14 May 2010Acoustic black holesfor relativistic fluids
+Xian-HuiGe1, andSang-Jin Sin2
+1DepartmentofPhysics,ShanghaiUniversity,Shanghai 2004 44,China
+2DepartmentofPhysics,HanyangUniversity,Seoul133-791, Korea
+E-mail: gexh@shu.edu.cn, sjsin@hanyang.ac.kr
+Abstract
+WederiveanewacousticblackholemetricfromtheAbelianHi ggsmodel. Inthenon-relativistic
+limit, while the Abelian Higgs model becomes the Ginzburg-L andau model, the metric reduces to
+an ordinary Unruh type. We investigate the possibility of us ing (type I and II) superconductors as
+the acoustic black holes. We propose to realize experimenta l acoustic black holes by using spiral
+vortices solutions from theNavier-stokes equation in the n on-relativistic classical fluids.
+1Contents
+1 Introduction 2
+2 Acousticblackholes from AbelianHiggsmodel 3
+2.1 Dispersionrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
+3 Non-relativisticlimit: Acousticblackholes insupercon ductors 8
+3.1 Spiral-Vortexgeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
+4 Conclusion 14
+1 Introduction
+Black hole is characterized by the event horizon, a spherica l boundary out of which even the light
+cannot escape. In 1974, Hawking announced that black holes a re not black at all, and emit thermal
+radiationatatemperatureproportionaltothehorizonsurf acegravity. Althoughseveraldecadespassed,
+theexperimentalverificationofHawkingradiationis still elusive.
+Compared with the difficulties on the astrophysical side, an alog models of general relativity is
+more appealing since these models are shedding light on poss ible experimental verifications of the
+evaporation of black holes. In the remarkable paper of Unruh [1], the idea of using hydrodynamical
+flows as analog systems to mimic a few properties of black hole physics was proposed. In this model,
+soundwavesratherthanlightwaves,cannotescapefromtheh orizonandthereforeitisnamed“acoustic
+(sonic) black hole”. A moving fluid with speed exceeding the l ocal sound velocity through a spherical
+surface could in principle form an acoustic black hole. The e vent horizon is located on the boundary
+between subsonicand supersonicflowregions.
+In general, any fluid flows could be the candidate of acoustic b lack holes. Many classical and
+quantum systems have been investigated as black hole analog ues, including gravity wave [3], water
+[4], slow light [5–7], an optical fiber [8], and an electromag netic waveguide [9]. But considered the
+detection of “Hawking radiation” of acoustic black holes, q uantum fluid systems, rather than classical
+fluid systems are specifically preferred in that Hawking radi ation is actually a quantum phenomenon.
+Up to now, superfluid helium II [10], atomic Bose-Einstein co ndensates [11], one-dimensional Fermi-
+degenerate noninteracting gas [12] have been proposed to cr eated an acoustic black hole geometry in
+thelaboratory. Thepossibleapplicationofacousticblack holestoquantuminformationtheoryhasbeen
+suggestedin [13].
+2Thefirst experimentalrealizationofacousticblack holere portedwasconductedinaBose-Einstein
+condensate [14]. So far it has been assumed that the system is non-relativistic, because all the atomic
+system is likely in such regime. However, some strongly inte racting system shows some evidences
+for exotic behavior that the matter is based on the heavy atom s but the basic excitations has massless
+character [15,16]. In fact thelinearandmasslessdispersi onrelationis canonicalforfermionicsystem.
+Thereforeitisinterestingtoaskwhathappentotheacousti cblackholefortherelativisticfluidsystem.
+Inthispaper,weexplorearelativisticversionofacoustic blackholesfromtheAbelianHiggsmodel.
+Another motivation comes from the feasibility of detecting Hawking temperature associated with the
+acoustic black hole. Since the Hawking temperature depends on the gradients of flow speed at the
+horizon, detecting thermal phonons radiating from the hori zons is very difficult. In fact the Hawking
+temperature calculated from models in Bose-Einstein conde nsates so far is very low ( ∼nano Kelvin).
+In constrast, the acoustic black holes created in the color s uperconducting phase in the dense quark
+matter and the pion superfluid phase at finite isospin density may provide us examples of relativistic
+superfluid and higher temperature acoustic black hole. The l ine elements of the black hole metric we
+obtained are found to be different from the non-relativisti c version, which has been widely used for
+almost30 years.
+2 AcousticblackholesfromAbelianHiggsmodel
+TheAbelian Higgsmodelis theMexican-hatmodelcoupled toe lectromagnetismwiththeaction
+S=/integraldisplay
+d4x(−1
+4FµνFµν+|(∂µ−ieAµ)φ|2+m2|φ|2−b|φ|4), (2.1)
+whereFµν=∂νAµ−∂µAν. The Planck units /planckover2pi1=c=kB= 1are used here. We will turn on the
+Planck constant /planckover2pi1in section3. Thecorresponding equationsofmotioncan bede duced from theaction
+with one governing on the electromagnetic field and the other on the scalar field. We write down only
+thesecond equationofmotionforthepurposeofderivingthe acousticblack holegeometry
+/squareφ+2ieAµ∇µφ−e2AµAµφ+m2φ−b|φ|2φ= 0, (2.2)
+in the∂µAµ= 0gauge.
+Multiplying(2.2)with φ∗, weobtain
+φ∗/parenleftbig
+−∂t∂tφ+∇i∇iφ+2ieAµ∇µφ−e2AµAµφ/parenrightbig
++m2φφ∗−b|φ|4= 0, (2.3)
+where the index µ=t,x1...x3andi=x1...x3. We can obtain another equation by multiplying the
+conjugationof(2.2)with φ
+φ/parenleftbig
+−∂2
+tφ∗+∇i∇iφ∗+2ieAµ∇µφ∗−e2AµAµφ∗/parenrightbig
++m2φφ∗−b|φ|4= 0. (2.4)
+3Thecombinationof(2.3)and (2.4) yieldstwo equations
+−∂t/parenleftbig
+φ∗∂tφ−φ∂tφ∗/parenrightbig
++∇i/parenleftbig
+φ∗∇iφ−φ∇iφ∗/parenrightbig
++2ieAµ(φ∗∇µφ+φ∇µφ∗) = 0.(2.5)
+and
+−∂t/parenleftbig
+φ∗∂tφ+φ∂tφ∗/parenrightbig
++2|∂tφ|2+∇i/parenleftbig
+φ∗∇iφ+φ∇iφ∗/parenrightbig
+−2|∂iφ|2+2m2|φ|2
++2ieAµ(φ∗∇µφ−φ∇µφ∗)−2b|φ|4−2e2AµAµ|φ|2= 0. (2.6)
+With theassumption φ=/radicalbig
+ρ(/vector x,t)eiθ(/vector x,t), theaboveequationsreduce to
+−∂t/bracketleftBig
+ρ(˙θ−eAt)/bracketrightBig
++∇i/bracketleftbig
+ρ/parenleftbig
+∇iθ+eAi/parenrightbig/bracketrightbig
+= 0, (2.7)
+/square√ρ√ρ+(˙θ−eAt)2−(∇iθ+eAi)2+m2−bρ= 0, (2.8)
+where(2.7)isthecontinuityequationand(2.8)lookslikea nequationdescribinghydrodynamicalfluid
+withanextraquantumcorrectionterm/square√ρ√ρ,whichisthesocalledquantumpotential. (2.7)and(2.8)ar e
+actually relativisticand thus are Lorentz invariant. This is a crucial difference from previous literature
+whereonlytheGalileaninvariantfluidequationswereconsi dered. Hydrodynamicsisalongwavelength
+effectivetheoryforphysicalfluids. Quantumgravity,orev enveryexoticfieldtheories,whenheatedup
+to finitetemperature,alsobehavehydrodynamicallyatlong -wavelengthand long-timescales. Nowwe
+consider perturbationsaround thebackground (ρ0,θ0):ρ=ρ0+ρ1andθ=θ0+θ1, and rewrite(2.7)
+and (2.8)as
+−∂t/bracketleftBig
+ρ0˙θ1+(˙θ0−eAt)ρ1/bracketrightBig
++∇·[ρ0∇θ1+ρ1(∇θ0+eAi)] = 0, (2.9)
+2(˙θ0−eAt)˙θ1−2(∇θ0+eAi)·∇θ1−bρ1+D2ρ1= 0, (2.10)
+where wehavedefined
+D2ρ1=−1
+2ρ−3
+2
+0(/square√ρ0)ρ1+1
+2ρ−1
+2
+0/square(ρ−1
+2
+0ρ1).
+Inthiswork,wedonotconsiderfluctuationsoftheelectroma gneticfieldAµ. Itisabackgroundfieldthat
+canaffecttheacousticblackholegeometry. Theequations( 2.9)and(2.10)correspondstoarelativistic
+fluids equation if bρ1term andD2ρ1(the quantum potential) term are absent. Actually, the D2term
+containsthesecondderivativeofslowlyvarying ρ,whichcanbenegligibleinthehydrodynamicregion
+wherek,ωare small. We will turn on the quantum potential term only in n ext subsection. In fact it is
+well-knownthatin theMadelungrepresentationoftheconde nsation,
+φ=/radicalbig
+ρ(/vector x,t)eiθ(/vector x,t),
+4theSchrodingerequationcan berewrittenas acontinuityeq uationplusan Eulerequationforanirrota-
+tional and inviscid fluid when the quantum potential can be ne glected. The Abelian Higgs model here
+showsthatthesamepropertyholdsforrelativisticfluids.
+Forthesimplicityofcalculation,letus introducevariabl esω0and/vector v0withthedefinition
+ω0=−˙θ0+eAtand/vector v0=∇θ0+e/vectorA.
+We thenobtain
+−∂t/bracketleftbigg
+ρ0˙θ1+2
+b/parenleftBig
+ω2
+0˙θ1+ω0/vector v0·∇θ1/parenrightBig/bracketrightbigg
++∇·/bracketleftbigg
+ρ0∇θ1+2
+b/parenleftBig
+−/vector v0ω0˙θ1−/vector v0·∇θ1/vector v0/parenrightBig/bracketrightbigg
+= 0. (2.11)
+Comparing theaboveequationwiththemasslessKlein-Gordo nequation
+1√−g∂ν(√−ggµν∂µθ1) = 0, (2.12)
+onemay find
+√−ggµν≡
+−b
+2ρ0−ω2
+0...−vj
+0ω0
+······ · ············
+−vi
+0ω0...(b
+2ρ0δij−vi
+0vj
+0)
+. (2.13)
+Defining thelocalspeed ofsound
+c2
+s=b
+2ω2
+0ρ0, (2.14)
+themetricforacousticblack holeyieldstheform
+gµν≡bρ0
+2cs/radicalbig
+1+c2
+s−v2
+−(c2
+s−v2)... −vj
+······ · ············
+−vi...(1+c2
+s−v2)δij+vivj
+, (2.15)
+where we madereplacement vi
+0→viω0and notation v2=/summationtext
+ivivi. We nowhavea relativisticversion
+of acousticblack holewhich is ageneralization of [1]. It is worth notingthat thesound velocity csis a
+functionoftheelectromagneticfield At, whichis treated asa slowlyvaryingquantity.
+In the non-relativistic limit, cs≪1,/vector v≪1, and keeping only the leading order term of each line
+element,wecanrecoverthemetricfromnon-relativisticth eoryfirstobtainedbyUnruh[1],butwithan
+overallfactordifference
+gµν≡/parenleftbiggρ0
+cs/parenrightbigg
+−(c2
+s−v2)...−vj
+······ · ············
+−vi...δij
+. (2.16)
+5Equivalently,wecan rewrite(2.15)as
+ds2=b
+2ρ0
+cs1/radicalbig
+1+c2
+s−v2/bracketleftbig
+−(c2
+s−v2)dt2−2/vector v·d/vector xdt+(/vector v·d/vector x)2+(1+c2
+s−v2)d/vector x2/bracketrightbig
+=b
+2ρ0
+cs1/radicalbig
+1+c2
+s−v2/bracketleftbigg
+−(c2
+s−v2)dτ2+(1+c2
+s−v2)/parenleftbiggvivj
+c2s−v2+δij/parenrightbigg
+dxidxj/bracketrightbigg
+,(2.17)
+where we have defined a new time coordinate by dτ=dt+/vector v·d/vector x
+c2s−v2. In this coordinate the acoustic
+geometry isactuallystationary.
+In thecasevz= 0,butvr∝ne}ationslash= 0andvφ∝ne}ationslash= 0,we can simplifythemetricinaKerr-likeform
+ds2=b
+2ρ0
+cs/bracketleftbigg
+−N2(vr,vφ)dτ2+1
+N2(vr,vφ)dr2+/radicalbig
+1+c2s−v2dz2
++N2(vr,vφ)r2dϕ2+(vφdτ−rdϕ)2
+/radicalbig
+1+c2s−v2/bracketrightBigg
+, (2.18)
+where
+N2(vr,vφ) =c2
+s−v2
+r/radicalbig
+1+c2
+s−v2, (2.19)
+and thecoordinatetransformationshavebeen used
+dt=dτ+vr
+vr2−c2
+sdr, dφ =dϕ+vrvφ
+vr2−c2
+sdr
+r. (2.20)
+Whenvz∝ne}ationslash= 0,themetricbecomesnon-sphericallysymmetricandnon-axi symmetric. Innon-relativistic
+limit,thegeometry(2.18)reduces to a“drainingbathtub”m etricfirst appeared in[2].
+At the region far from the acoustic horizon, the normal modes of theθ1field are purely outgoing.
+The operator θ1can be expanded in terms of modes in the region outside the hor izon. An observer
+traveling with the fluid as it flows through the acoustic horiz on will see that the field θ1is essentially
+the vacuum state in analogy with the behavior of the normal mo des of a scalar field in Schwarzschild
+coordinates by a freely falling observer [1]. The correspon ding Hawking temperature at the event
+horizon reads
+TH=/planckover2pi1
+2πkB∂(cs−vi)
+∂xi|horizon. (2.21)
+The relativistic version of Hawking temperature from Abeli an Higgs model is identical with the non-
+relativisticone.
+Since there is no real gravity, the Newton constant GNis not involved in the formulation of the
+acoustic“surfacegravity”andHawkingtemperature. Onema ycomparetheacousticHawkingtemper-
+ature withUnruh temperaturein Rindlercoordinates. Theac oustictemperatureand theUnruh temper-
+ature yieldalmostthesameform: Theacoustictemperaturei sproportionalto thegradients ofthefluid
+6kinetic energy (i.e. ∝∂i(c2
+s−v2
+i)|horizon) at the horizon, while the Unruh temperature ( T=aR
+2π), ob-
+servedbyanaccelerating observerisproportionaltotheac celerationaR(=∂v
+∂t). AccordingtoEinstein
+equivalence principle, no experiment can distinguish the a cceleration due to gravity from the inertial
+acceleration due to a change of velocity. Thus, the Unruh eff ects in Rindler coordinates and Hawking
+effects in astrophysicalblack hole spacetimeare equivale nt. It is surprising that the acoustic geometry
+provides us anotherway to realize the equivalenceprincipl eby therelation (Acceleration) = (Intensity
+ofthegravitationalfield)=(Gradientsofthekineticenerg y). Onepointthatshouldbemadeclearisthat
+acoustic analog models of black holes could be used to mimic t he kinematical but not the dynamical
+aspectsofgravitationalsystems. Forexample,itisimposs ibletoreproduceblackholethermodynamics
+usingacousticanaloguemodelsofblack holes[18].
+2.1 Dispersion relation
+We have shown that by studying the linearized equations of mo tion of the complex scalar fields of
+Abelian Higgs model, the perturbations obey a Klein-Gordon equation in the curved space-time. Here
+we stressthat sinceweworked inthelong-wavelengthand slo wlyvaryinglimit,thequantumpotential
+term (theD2term)isneglible.
+In the high-momentum or eikonal regime, the contribution of the quantum potential would be im-
+portant. In [17], the authors explored the dispersion relat ion with the contribution of the quantum
+potential. In our case, both the quantum potential and the re lativistic effects will contribute to the dis-
+persion relation. IN this regime, the phase fluctuation θ1and the density fluctuation ρ1can be treated
+as a functionwith aslowlyvaryingamplitudeand rapidlyvar yingphase:
+θ1∼ρ1∼Re{exp(−iΩt +i/vectork·/vectorx)} (2.22)
+Ω =∂tθ1
+θ1=∂tρ1
+ρ1, ki=−∇iθ1
+θ1=−∇iρ1
+ρ1(2.23)
+Theoperator D2can beapproximatelywrittenas
+D2ρ1≡ −1
+2ρ−3
+2
+0(/square√ρ0)ρ1+1
+2ρ−1
+2
+0/square(ρ−1
+2
+0ρ1)
+=ρ1(∂µρ0)2
+2ρ3
+0−∂µρ0∂µρ1
+2ρ2
+0−ρ1/squareρ0
+2ρ2
+0+/squareρ1
+2ρ0
+≈/squareρ1
+2ρ0
+=−1
+2ρ−1
+0(k2−Ω2)ρ1, (2.24)
+7whereµ=t,x1...x3. We havesimilarresult for D2θ1. Therefore, we can treat D2=−1
+2ρ−1
+0(k2−Ω2)
+in theeikonalregime. TheKlein-Gordonequation(2.9)and ( 2.10) become
+D2−b
+2ρ0(kikjδij−Ω2)−(ω0Ω+vi
+0kj)2= 0. (2.25)
+With theexpressionof D2above,Eq.(2.25)becomes
+bρ0
+2/parenleftbig
+k2−Ω2/parenrightbig
+−/parenleftbig
+ω0Ω+vi
+0ki/parenrightbig2+1
+4/parenleftbig
+k2−Ω2/parenrightbig2= 0. (2.26)
+In the eikonal limit, the last term in the above equation will become dominant and we can neglect the
+othertwoterms, whichgives
+Ω =k. (2.27)
+On theotherhand, inthehydrodynamicsregime, wecan neglec t theD2term again intheleft hand
+of Eq.(2.25). To be definite, we choose ki=kδi1, and define vi=vi
+0/ω0. Then we easily obtain the
+dispersionrelation
+Ω =−v1+cs/radicalbig
+1+c2s−(v1)2
+1+c2
+sk. (2.28)
+The dispersion relation is quiet different from the result o f [2]. For large wavelength and in the non-
+relativisticlimit,itreads
+Ω = (cs−v1)(1+(cs+v1)/2)k. (2.29)
+Thesoundvelocityinhydrodynamiclimithasacorrectionto thefamiliarvalue cs. Themaincorrection
+is simply due to the fluid motion and the next term is higher ord er. Sincev1, the generalized physical
+velocity in the presence of the electro-magnetic fields, dep ends on background electric and magnetic
+fields, as onecan seefrom
+v1= (∂xθ0+eAx)/(−˙θ+eAt), (2.30)
+so isthesoundvelocity.
+We stress that we assume that the back ground fields change ver y little within a wave length. If c2
+s
+andv2canbeneglected,comparedwiththespeedoflight,thenther elationreturntoalineardispersion
+relationΩ≈cs|k|.
+3 Non-relativisticlimit: Acousticblackholesin supercon ductors
+In thenon-relativisticlimit,theAbelian Higgsmodelredu ces totheGinzburg-Landautheory
+i/planckover2pi1∂
+∂tψ(/vector r,t) =−/planckover2pi12
+2m∗/vectorD2ψ(/vector r,t)+a(T)ψ(/vector r,t)+b(T)|ψ(/vector r,t)|2ψ(/vector r,t), (3.1)
+8where/vectorD=∇+2ie
+/planckover2pi1/vectorA,m∗is the mass of a cooper pair, a(T)andb(T)are two parameters that depend
+on temperature. We have turned on the Planck constant /planckover2pi1. The Ginzburg-Landau coherence length is
+defined by
+ξ(T) =/parenleftbigg
+/planckover2pi12
+2m∗|a(T)|/parenrightbigg1/2
+. (3.2)
+Assumingψ(/vector r,t) =/radicalbig
+ρ(/vector r,t)eiθ(/vector r,t),wehave
+∂ρ
+∂t+/vector∇·(ρ/vector v) = 0, (3.3)
+/planckover2pi1∂θ
+∂t=/planckover2pi12
+2m∗∇2√ρ√ρ−m∗
+2/vector v2−a(T)−b(T)ρ. (3.4)
+wherewehaveusedthegauge ∇·/vectorA= 0,andthevelocityisdenotedby /vector v=/planckover2pi1∇θ
+m∗+2e
+m∗/vectorA. Thefirstterm
+intherighthandof(3.4)correspondstothequantumpotenti al. Theabovetwoequationsarecompletely
+equivalent to the hydrodynamical equations for irrotation al and inviscid fluid apart from the quantum
+potential. In thelong-wavelengthapproximation,thecont ributioncomingfrom thelinearization ofthe
+quantumpotentialcan beneglected.
+LinearizingEqs.(3.3)and (3.4)aroundthebackground( ρ0,θ0), weget
+∂ρ1
+∂t+∇·(ρ1/vector v0+ρ0∇θ1) = 0, (3.5)
+∂θ1
+∂t+/vector v0·∇θ1+b(T)
+m∗ρ1= 0. (3.6)
+Eliminating ρ1,theaboveequationslead to
+∂
+∂t/parenleftbigg
+−m∗
+b(T)∂θ1
+∂t−m∗
+b(T)/vector v0·∇θ1/parenrightbigg
++∇·/parenleftbigg
+ρ0∇θ1−m∗
+b(T)/vector v0∂θ1
+∂t−m∗/vector v0
+b(T)/vector v0·∇θ1/parenrightbigg
+= 0,(3.7)
+where/vector v0=/planckover2pi1∇θ0
+m∗+2e
+m∗/vectorA. Onecanextractametricfrom(3.7)becauseithasthesamefo rmasamassless
+Klein-Gordornequationincurvedspace-time. Thenon-rela tivisticversionofacousticmetricfrom(3.7)
+is read as
+ds2=ρ0
+m∗cs/braceleftBig
+−(c2
+s−v2
+0)dt2−2/vector v0·d/vector rdt+d/vector r·d/vector r/bracerightBig
+, (3.8)
+wherecs=/radicalBig
+b(T)
+m∗ρ0is denoted as the “sound velocity”. Actually, one can write t he sound velocity in
+terms ofthecoherent length
+cs=/planckover2pi1√
+2m∗ξ(T). (3.9)
+9One should not identify the “sound velocity” defined here wit h the sound velocity produced by the
+crystal lattices. Here csis a function of the density ρ0of superconducting electrons and temperature-
+dependent parameter b(T).
+Wewouldliketodosomecalculationson thenecessary condit ionsfortheformationofan acoustic
+eventhorizoninGinzburg-Landautheory. Wewillshowthati twouldbeatoughtasktorealizeacoustic
+black holes by using typeI and type II superconductors. The d efinition of acoustic black holerequires
+v0
+cs>1in someregion. From theLondonequation
+∇2/vectorB−1
+λ(T)/vectorB= 0, (3.10)
+and theMaxwellequation
+∇×/vectorB=µ0/vectorjs, (3.11)
+whereλ(T) = (me
+µ0nse2)1/2and/vectorjsdenote the penetration depth and the supercurrent, respect ively. We
+have a finite magnetic field , say /vectorB= (0,By(x),0), outside a superconducting region. After simple
+computation,wefind thesupercurrent yields[19]
+jsz=H(T)
+λ(T). (3.12)
+Thecriticalcurrentdensityaboveisonlyvalidforcleanty peIsuperconductors. Ontheotherhand,the
+current densityhas theform
+jsz=ensv0s, (3.13)
+wherens= 2|ψ2|= 2ρ0denotes thesuperfluid density,and v0sis thevelocityof theCooperpairs. We
+then have
+v0s
+cs=√
+2κH(T)
+Hc2=H(T)
+Hc, (3.14)
+whereκ=λ(T)
+ξ(T), andHc2=h
+4πeµ0ξ2(T)=√
+2κHc. In thestandard textbook, κdenotes theG-L param-
+eter, which describes the difference between type I and type II superconductors: κ <1√
+2is for type I,
+andκ >1√
+2is for type II. Hc2denotes the critical magnetic field, and Hcthe thermodynamic critical
+field. From (3.14), we can see that for type I superconductors , whenv0s=cs, the superconducting
+phase is broken and return to Normal states. Now we are going t o solve (3.10) in the high κlimit,
+λ≫ξ, which is valid for the cooper-oxide superconductors. We wi ll show that even near the normal
+coreofthevortex,itishardforthespeedofsuperconductin gelectronstoexceedthe“soundvelocity”–
+cs. Forsimplicity,thevortexisassumedtobeinfinitelylonga ndaxiallysymmetricsothatthereareno
+zorangulardependenceofitsfield distribution. Wethen writ eEq.(3.10)incylindricalcoordinates
+1
+r∂r(r∂rB)−1
+λ2B=0. (3.15)
+10This solutionisgivenby [19]
+B(r) =φ0
+2πλ2K0(r/λ), (3.16)
+whereK0(r/λ)is the zeroth-order modified Bessel function. The current de nsity is derived by the
+Maxwell equation(3.11),
+/vectorjs(r) =φ0
+2πµ0λ3K1(r/λ)/vectoreφ (3.17)
+We are interested in the asymptotic behaviors at small radia l distancesr≪λ, whereK1(r/λ)≈λ
+r.
+Thespeed ofCooperpairs isthen givenby
+v0s=js(r)
+nse(3.18)
+We thenfind
+v0s
+cs=√
+2ξ
+r. (3.19)
+The formation of acoustic black holes requires v0s/cs>1, i.e.r <√
+2ξ1. But at the core region of
+the vortexr < ξ, the order parameter and the superconducting current would decay asr→0, since it
+correspondstoanormalstate. (3.13)tellsusthatforfixede lectronvelocity v0s,thecurrent jszdecreases
+asns(= 2ρ0)decreases. Since the“sound velocity” csdepends on thedensity ρ0, weexpect that in the
+regionξ <r<√
+2ξ,electronvelocitycan exceedthesoundvelocity. However, itwouldbeinteresting
+ifonecan verify thispointexperimentally.
+In order to trap phonons, one need input a sink so that electro ns can propagate along φ- andr-
+direction simultaneously. For this purpose, we need turn to spiral flux vortices in the type II supercon-
+ductors. The spiral flux vortices [20] in current-carrying t ype-II superconducting cylinders subjected
+to longitudinal magnetic fields can be used to mimic acoustic black holes. Consider a type-II super-
+conducting cylinder of radius acontaining a spiral vortex, in the presence of a longitudina l applied
+magneticfield Hαand transport current I, whereHαandIare positiveinthe zdirection. TheLorentz
+force imposedon the vortexwere found to beradially inward. Thus thespiral vortexhas a tendency to
+move to the center of the cylinder. One can determine the term inal speedvr
+0sof the contracting vortex
+bybalancingtheLorentzforceandtheviscousdragforce ηvr
+0s,whereηisaphenomenologicalviscous
+drag coefficient perunitlengthofvortex[20].
+Actually, the spiral vortex in the presence of a sufficiently large current density parallel to its axis,
+is unstable against a helical perturbation [20]. But this do es not prevent the formation of acoustic
+black hole geometry. In the next section, we investigate the spiral vortex geometry from classical
+hydrodynamicsin thenon-relativisticframeand givetheex act metricforaspiral-vortexgeometry.
+1Theorderparameteralso reachesitsmaximumat r=√
+2ξ[19].
+113.1 Spiral-Vortex geometry
+In[2],theauthorconstructeda“drainingbathtub”blackho lemetricbya (2+1)dimensionalNewtonian
+flow with a sink at the origin. An stable vortex solution to the Navier-Stokes equation requires each
+component ofthefluid velocityto benon-vanishing. Such an e xact solutionto Navier-Stokes equation
+was first found by Sullivan [21] and later observed in experim ents [22]. Now, we utilize the Sullivan
+vortex solution of Navier-Stokes equation to model a 4-dime nsional acoustic black hole metric, which
+could berealized inexperiments. Thenon-relativisticNav ier-Stokes equationisgivenby
+∂t/vector v+/vector v·∇/vector v=−/vector∇p/ρ+ν∇2/vector v+/vectorf/ρ, (3.20)
+where/vector vis the fluid velocity, pthe fluid pressure, ρthe mass density, νthe shear viscosity and /vectorfan
+externallyspecifiedforcingfunction. Thecontinuityequa tiongenerallyaccompaniestheNavier-Stokes
+equation
+∂tρ+∇·(ρ/vector v) = 0. (3.21)
+By defining /vector v=∇ψ, theNavier-Stokesequationcan berewrittenas
+∂tψ+1
+2(∇ψ)2+/integraldisplayp
+0dp′
+ρ−ν∇2ψ= 0, (3.22)
+where we have chosen vanishing external force /vectorf= 0. Now linearize these equations around a fixed
+background (ρ0,p0,ψ0)and setρ=ρ0+ρ1,p=p0+p1, andψ=ψ0+ψ1. After linearizing the
+equationofmotionandthecontinuityequation,theequatio nsforρ1andψ1yield
+∂ρ1
+∂t+∇·(ρ1/vector v0+ρ0∇ψ1) = 0, (3.23)
+∂ψ1
+∂t+/vector v0·∇ψ1+p1
+ρ0−ν∇2ψ1= 0. (3.24)
+One can further assume the fluid is barotropic, which implies thatρis a function of ponly2. The
+barotropicassumptiongives
+ρ1=∂ρ
+∂pp1, (3.25)
+After rearranging (3.24), wehave
+ρ1=−∂ρ
+∂pρ0/parenleftbig
+∂tψ1+/vector v0·∇ψ1−ν∇2ψ1/parenrightbig
+, (3.26)
+2Thisassumptionwasfirst usedimplicitlyin[1] (seealso [2] ).
+12Substitutingtheabovelinearizedequationofmotionback i nto(3.23), weobtain
+−∂t/bracketleftBig∂ρ
+∂pρ0/parenleftbig
+∂tψ1+/vector v0·∇ψ1−ν∇2ψ1/parenrightbig/bracketrightBig
++∇·/bracketleftBig
+ρ0∇ψ1−∂ρ
+∂pρ0/vector v0/parenleftbig
+∂tψ1+/vector v0·∇ψ1−ν∇2ψ1/parenrightbig/bracketrightBig
+= 0. (3.27)
+We can rewritetheaboveequationinaKlein-Gordonlikeform
+∂µ(√−ggµν∂νψ1) =−∂ρ
+∂pρ0ν(∂t+/vector v0·∇)∇2ψ1, (3.28)
+where theinversemetricisgivenby
+gµν≡/parenleftbigg1
+ρ0cs/parenrightbigg
+−1...−vj
+0
+······ · ············
+−vi
+0...−(c2
+sδij−vi
+0vj
+0)
+, (3.29)
+and thelocalspeed ofsoundisdefined by c−2
+s=∂ρ
+∂p. Theacousticmetric gµνis givenin (2.16).
+What is different from before is that we have third order deri vative term of ψ1in the right hand
+of (3.28). The similar situation was discussed in [2] and Lor entz symmetry breaking was observed.
+Clearly, by writing ψ1asψ1∼exp(−iωt + i/vectork·/vectorx), one see that in the eikonal approximation, the
+right hand of(3.28) willbe dominant,whilein the hydrodyna miclimit, theright hand of (3.28) can be
+ignored. In thefollowing,wewillwork inlong-wavelengtha pproximationand neglectthoseterms.
+In thecylindricalcoordinate (r,φ,z), theexactsolutionto Navier-Stokes equationyields[21]
+vr
+0=−αr+2βν
+r/parenleftBig
+1−e−αr2
+2ν/parenrightBig
+, (3.30)
+vφ
+0=Γ0
+2πrH(x)
+H(∞), (3.31)
+vz
+0= 2αz/bracketleftBig
+1−βe−αr2
+2ν/bracketrightBig
+(3.32)
+where
+H(x) =H(ar2
+2ν) =/integraldisplayx
+0exp/parenleftbigg
+−s+β/integraldisplays
+01−e−τ
+τdτ/parenrightbigg
+ds, (3.33)
+Γ0isaconstantthatdependsontheboundarycondition, νdenotestheshearviscosity, α>0andβ >1
+are free parameters thatcan befixed inexperiments. By using thecoordinatetransformation
+dt=dτ+vr
+0
+vr
+02+vz
+02−c2sdr+vz
+0
+vr
+02+vz
+02−c2sdz, (3.34)
+dφ=dϕ+vr
+0vφ
+0
+vr
+02+vz
+02−c2
+sdr
+r+vz
+0vφ
+0
+vr
+02+vz
+02−c2
+sdz
+r. (3.35)
+13We obtainanon-sphericallysymmetricmetricby rewriting( 2.16)as
+ds2=ρ0
+cs/braceleftBig
+−(c2
+s−v2
+0)dτ2+/parenleftbigg
+1−vr
+02
+vr
+02+vz
+02−c2
+s/parenrightbigg
+dr2+/parenleftbigg
+1−vz
+02
+vr
+02+vz
+02−c2
+s/parenrightbigg
+dz2
+−2vr
+0vz
+0
+vr
+02+vz
+02−c2
+sdrdz−2vφ
+0rdϕdτ+r2dϕ2/bracerightBig
+. (3.36)
+It is worthnotingthat when vz= 0, wecan recoverthe“ drainingbathtub”metricobtainedin [2 ], that
+is tosay
+ds2=ρ0
+cs/braceleftBig
+−(c2
+s−v2
+0)dτ2+c2
+s
+c2s−vr
+02dr2+dz2−2vφ
+0rdϕdτ+r2dφ2/bracerightBig
+.(3.37)
+The form of the metric obtained in (3.36) is non-spherically symmetric and even non axi-symmetric.
+There exists an event horizon for the metric. The event horiz on can be fixed by using the Null surface
+equation[24]
+gµν∂f
+∂xµ∂f
+∂xν= 0, (3.38)
+wheref=f(τ,r,ϕ,z) = 0denotesthehypersurface. Sincetheradiusoftheeventhori zonwouldvary
+as a functionof τandz, wefind thattheeventhorizonisdetermined by
+˙rH=/bracketleftbig
+vr
+02(rH)+vz
+02(rH)−c2
+s/bracketrightbig/bracketleftBig
+vr
+02(rH)−r′
+Hvr
+0(rH)vz
+0(rH)+r′2
+Hvz
+02(rH)−c2
+s(1+r′2
+H)/bracketrightBig
+c−2
+s,(3.39)
+where thedot andtheprimedenotethederivativewithrespec t toτandz, respectively.
+4 Conclusion
+In summary, a new acoustic black hole metric has been derived from the Abelian Higgs model. In
+the non-relativistic limit, the metric can reduce to an acou stic black hole geometry given by Unruh.
+Since the Abelian Higgs model describes high energy physics , our results indicate that acoustic black
+holes might be created in high energy physical process, such as quark matters and neutron stars. The
+dispersion relation was given and in the non-relativistic l imit, we can recover the linear dispersion
+relation. Wehaveinterpretedtherequirementsofcreating anacousticblackholebyusingthelanguage
+of superconductor physics. Our results have demonstrated t hat although it would be very difficult
+to mimic the acoustic black hole in type I superconductors, p eople would be able to to discover the
+acoustic black holes near the magnetic vortex in type II supe rconductors . We developed an acoustic
+black holewithspiralvortexgeometryfrom theNavier-Stok es equationin classicalNewtonianfluids.
+One point to be figured out is that we have used the vortices to m imic the black hole. Actually,
+when a superfluid flow exceeds the Landau critical velocity, v ortices can be created and excitations
+14with negative energy appear, which means that black hole con figurations are energetically unstable
+since the speed of sound is also the the Landau critical veloc ity. However, this does not mean that
+the supercritical flow of the superfluid vacuum is not possibl e. In superfluid3Hethe Landau velocity
+for vortex nucleation can be exceeded by several orders of ma gnitude, without vortices creation. In
+[23], the event horizon was proposed to be realized in superfl uid3Heby arranging the superfluid flow
+velocity to be constant but the “sound velocity” to change in space and in some region becomes less
+than the flow velocity of the liquid. In our above discussions , we have shown that in the ideal type
+II superconductors, the “sound velocity” (3.9) indeed depe nds on the superfluid density ρ0, but the
+superconducting (superfluid) flow velocity can be chosen to b e constant by varying the current and
+the superfluid density to be ρ0simultaneously in (3.13). Therefore, if the “sound velocit y” changes in
+space, superconducting(superfluid)flowvelocitywouldexc eed the“soundvelocity”insomeregions.
+Acknowledgments
+XHG thanks Hanyang university and CQUeST for warm hospitali ty. XHG is indebted to Yunping
+Wang for helpful discussions. This work was supported in par t by the WCU project of Korean Min-
+istry of Education, Science and Technology (R33-2008-000- 10087- 0). The work of XHG was partly
+supported by NSFC, China (No. 10947116), Shanghai Rising-S tar Program(No. 10QA1402300) and
+Shanghai Leading Academic DisciplineProject (No. S30105) . The work of SJS was supported in part
+by KOSEF Grant R01-2007- 000-10214-0 and by the National Res earch Foundation of Korea(NRF)
+grant funded bytheKoreagovernment(MEST)(No.2005-00494 09).
+References
+[1] W. G. Unruh,Phys.Rev.Lett, 46,1351(1981).
+[2] M. Visser,Class. Quant.Grav. 151767(1998)[arXiv:gr-qc/9712010].
+[3] R. Sch ¨ utzholdand W. G.Unruh, Phys.Rev.D 66,044019(2002).
+[4] G. Rousseaux,C. Mathis,P. Ma ¨ ıssa,T.G. Philbin,and U. Leonhardt,NewJournalofPhysics 10,
+053015(2008).
+[5] U. Leonhardtand P. Piwnicki,Phys.Rev. Lett.84, 822(20 00).
+[6] U. Leonhardt,Nature415,406 (2002).
+15[7] W. G. Unruhand R. Sch ¨ utzhold,Phys.Rev.D 68,024008(2003).
+[8] T.G.Philbin,C.Kuklewicz,S.Robertson,S.Hill,F.K ¨ onig,andU.Leonhardt,Science 319,1367
+(2008).
+[9] R. Sch ¨ utzholdand W. G.Unruh, Phys.Rev.Lett.95, 031301(2005).
+[10] M.Novello,M.VisserandG. Volovik(Eds.), ArtificialBlackHoles ,World Scientific, Singapore,
+(2002).
+[11] L. J.Garay, J. R. Anglin,J.I. Cirac, and P. Zoller,Phys .Rev.Lett. 85,4643(2000).
+[12] S. Giovanazzi,Phys.Rev.Lett. 94061302(2005).
+[13] X. H. Geand Y.G. Shen, Phys.Lett. B623,141 (2005)[quant-ph/0507166].
+[14] O. Lahav,A. Itah, A.Blumkin,C. Gordon andJ. Steinhaue r, [arXiv0906.1337]
+[15] S. J. Sin and I. Zahed,JHEP 0912,015 (2009)[arXiv:0907.1434[hep-th]].
+[16] S. S. Gubserand F. D. Rocha, arXiv:0911.2898[hep-th].
+[17] C. Barcelo, S. Liberati andM. Visser,LivingRev. Rel. 812(2005)[gr-qc/0505065].
+[18] M. Visser,Phys.Rev.Lett., 80, 3436(1998). [arXiv:gr -qc/9712016].
+[19] C. P. Poole, H. A. Farach and R. J. Creswick, Superconductivity , Academic Press, Netherlands,
+page52-167,(2007).
+[20] J. R. Clem, Phys.Rev.Lett. 381425(1977)
+[21] R. D. Sullivan,J. Aero/SpaceSci. 26,767(1959).
+[22] F. W. LeslieandJ. T.Snow,AIAA J. 111272(1980).
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+[24] S.W.HawkingandG.F.R.Ellis, Thelargescalestructureofspace-time ,Cambridge: Cambridge
+UniversityPress, (1973)
+16
\ No newline at end of file
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+arXiv:1001.0372v2  [cond-mat.other]  18 Sep 2014A generalised Landau-Lifshitz equation for isotropic
+SU(3) magnet
+J Bernatska1,2and P Holod1,2
+1National University of ‘Kiev-Mohyla Academy’, 2 Skovorody Str., 040 70 Kiev, UA
+2Bogolyubov Institute for Theoretical Physics, 14b Metrologichna Str., 03680 Kiev,
+UA
+E-mail:BernatskaJM@ukma.kiev.ua, Holod@ukma.kiev.ua
+Abstract. In the paper we obtain equations for large-scale fluctuations of th e mean
+field (the field of magnetization and quadrupole moments) in a magnet ic system
+realized by a square (cubic) lattice of atoms with spin s/greaterorequalslant1 at each site. We use the
+generalized Heisenberg Hamiltonian with biquadratic exchange as a qu antum model.
+A quantum thermodynamical averaging gives classical effective mod els, which are
+interpreted as Hamiltonian systems on coadjoint orbits of Lie group SU(3).
+AMS classification scheme numbers: 37K65, 82D40
+Submitted to: J. Phys. A: Math. Gen.A generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 2
+1. Introduction
+Being a multiparticle quantum system, a magnet can be considered on different
+levels of hierarchy: a quantum (microscopic) level and a classical (m acroscopic) one.
+The quantum level is described by means of quantum electrodynamic s, or by simpler
+models like the Hubbard model or the Heisenberg one. The most comm on model for
+the classical level is the mean field model. Dynamics of a mean field is des cribed by the
+equations of Landau-Lifshitz type.
+Each model is suitable to describe certain phenomena. For example, the problems
+of formation of large-scale structures (domain walls, topological s olitons, nonlinear
+magnetization waves and so on) are naturally investigated from a cla ssical point of
+view. More tenuous problems, like renormalization of the order para meter according to
+a temperature or an effective interaction constant, require a qua ntum point of view [1].
+Here we start from the quantum level described by the Heisenberg model.
+In addition to the usual Heisenberg bilinear interaction −J(ˆSn,ˆSm), we consider the
+biquadratic one −K(ˆSn,ˆSm)2. By many theoretical and experimental researches it was
+shown that the biquadratic interactions have significant effects on magnetic properties.
+For example, a new ordered state (a nematic state, with zero magn etization) occurs as
+a separate phase transition [2]. Note, that the biquadratic interac tion can be taken into
+account only if a magnetic system has the spin s/greaterorequalslant1.
+In this paper we propose a classical generalization of the isotropic L andau-Lifshitz
+equation corresponding to the Heisenberg model with biquadratic e xchange interaction.
+A transition from the quantum level to the classical one is performe d by the mean
+field approximation. The classical model can be interpreted as a Ham iltonian system
+on a coadjoint orbit of the unitary group SU(3). Therefore, we ac quire an additional
+mathematical apparatus, which gives a significant advantage.
+Themeanfieldapproximationgivesaqualitativeanalysisoforderedst ates[3,4], but
+has no answer about their stability. Moreover, in this approximation the temperature
+dependencies of order parameters considerably differ from the ob served dependencies.
+Thatprovesanecessity totakeintoaccount fluctuationsofthem eanfield. Theproposed
+effective classical models describe large-scale (or slow) fluctuation s of mean field.
+One can come to slow fluctuations by an averaging over high frequen cies [1]. However,
+remaining in the context of theory of magnetism, we choose the mod els associated with
+the equations of Landau-Lifshitz type.
+The paper is organized as follows. Section 2 is devoted to the quantu m model
+based on the spin Hamiltonian with biquadratic exchange interactions . We consider the
+SU(3)-invariant case. In section 3 we construct two effective mod els that describe large-
+scale fluctuations of mean field (the field of magnetization and quadr upole moments).
+WeobtainoneofthembyanaveragingofthequantumHamiltonianove rcoherentstates.
+The other effective model is a result of an averaging over mixed stat es. These classical
+models appear to be Hamiltonian systems on coadjoint orbits of the g roup SU(3), that
+follows from SU(3)-invariance of the original quantum model. Each c oadjoint orbit isA generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 3
+determined by constrains, which are observed quantities becoming rigid after averaging.
+In section 4 we summarize results and give some ideas how to extend t he proposed
+scheme to magnetic systems with higher spins.
+2. Quantum model of magnetic system
+2.1. Description of the model
+The magnetic system in question is realized by a homogeneous lattice o f atoms with
+the spins/greaterorequalslant1 at each site. The lattice can be one-, two-, or three-dimensional, and
+has the distance lbetween the nearest-neighbor sites. We assign three spin operators
+(ˆS1
+n,ˆS2
+n,ˆS3
+n) to each site n; they obey the standard commutation relations:
+[ˆSα
+n,ˆSβ
+m] = iεαβγˆSγ
+nδnm,
+whereα,β,γrun over the set {1,2,3}, andδnmdenotes the Kronecker symbol.
+We use the localized spin model for the magnetic system. In many cas es this
+model adequately describes a magnetic system by the Heisenberg H amiltonian, which
+includes only the bilinear exchange interaction. Nevertheless, ther e are a lot of magnets
+that require taking into account higher powers of exchange intera ction. Our model is
+applicable to magnets with the spin s/greaterorequalslant1.
+In the present paper we consider the Hamiltonian with biquadratic ex change and
+call itbilinear-biquadratic :
+ˆH=−/summationdisplay
+n,δ{J(ˆSn,ˆSn+δ)+K(ˆSn,ˆSn+δ)2}, (1)
+whereˆSn= (ˆS1
+n,ˆS2
+n,ˆS3
+n) is a vector of spin operators at site n, andδruns over the
+nearest-neighbor sites. This Hamiltonian was discussed, for examp le, in [2–5]. The
+constantsJandKserve as exchange integrals. We suppose that JandKare positive.
+It means that we consider a ferromagnetic interaction in preferen ce.
+The operators {ˆSα
+n}(herenis fixed) aredefined over the (2 s+1)-dimensional space
+of irreducible representation of the group SU(2). They generate an associative matrix
+algebra over this space. The complete matrix algebra can be repres ented as a direct
+sum of irreducible sets of tensor operators with respect to the ac tion ad ˆSα. In the case
+ofs=1, we have: Mat 3×3≃[9]=[1]+[3]+[5]. Evidently, the operators {ˆSα
+n}form a
+basisinthe3-dimensional irreducibleset. Onecanconstruct abasis inthe5-dimensional
+irreducible set from the tensor operators of weight 2. These are t hequadrupole operators
+{ˆQ12
+n,ˆQ13
+n,ˆQ23
+n,ˆQ[2,2]
+n,ˆQ[2,0]
+n}defined by the formulas:
+ˆQαβ
+n=ˆSα
+nˆSβ
+n+ˆSβ
+nˆSα
+n, α/ne}ationslash=β,
+ˆQ[2,2]
+n= (ˆS1
+n)2−(ˆS2
+n)2,ˆQ[2,0]
+n=√
+3/parenleftig
+(ˆS3
+n)2−2
+3/parenrightig
+.
+The spin and quadrupole operators are normalized by the following re lation:
+Tr(ˆP)2=1
+3s(s+1)(2s+1).A generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 4
+Ass=1 we have Tr( ˆP)2=2 . The chosen normalization is matched to the relation
+(ˆS1
+n)2+(ˆS2
+n)2+(ˆS3
+n)2=s(s+1).
+Now, fix the canonical basis {|+1/an}bracketri}ht,|−1/an}bracketri}ht,|0/an}bracketri}ht}in the space of representation. Then
+one obtains the following matrix representation for the spin and qua drupole operators:
+ˆS1
+n=1√
+2
+0 0 1
+0 0 1
+1 1 0
+,ˆS2
+n=1√
+2
+0 0−i
+0 0 i
+i−i 0
+,
+ˆS3
+n=
+1 0 0
+0−1 0
+0 0 0
+,ˆQ[2,0]
+n=1√
+3
+1 0 0
+0 1 0
+0 0−2
+,
+ˆQ12
+n=
+0−i 0
+i 0 0
+0 0 0
+,ˆQ13
+n=1√
+2
+0 0 1
+0 0−1
+1−1 0
+,
+ˆQ23
+n=1√
+2
+0 0−i
+0 0−i
+i i 0
+,ˆQ[2,2]
+n=
+0 1 0
+1 0 0
+0 0 0
+.
+We denote all spin and quadrupole operators: {ˆS1
+n,ˆS2
+n,ˆS3
+n,ˆQ12
+n,ˆQ13
+n,ˆQ23
+n,ˆQ[2,2]
+n,ˆQ[2,0]
+n}
+by{ˆPa
+n}8
+a=1. The operators {ˆPa
+n}obey the following commutation relations:
+[ˆPa
+n,ˆPb
+m] = iCabcˆPc
+nδnm,
+whereCabcare structure constants; nonzero components are
+C123=C145=C167=C264=C257=C356= 1,
+C168=C528=√
+3, C437= 2.
+The Hamiltonian (1) becomes bilinear in the terms of {ˆPa
+n}:
+ˆH=−(J−1
+2K)/summationdisplay
+n,δ/summationdisplay
+αˆSα
+nˆSα
+n+δ−1
+2K/summationdisplay
+n,δ/summationdisplay
+aˆQa
+nˆQa
+n+δ−4
+3KN, (2)
+whereNdenotes the total number of sites. Obviously, the Hamiltonian is SU( 2)-
+invariant, and one can transform the operators {ˆSα
+n}and{ˆQa
+n}by the formulas of
+adjoint representation
+ˆUˆSα
+nˆU−1=/summationdisplay
+βˆDαβ(ˆU)ˆSβ
+n,ˆDαβ∈SO(3),
+ˆUˆQa
+nˆU−1=/summationdisplay
+bˆDab(ˆU)ˆQb
+n,ˆDab∈SO(5),
+whereˆDαβ(ˆU) andˆDab(ˆU) are matrices of the real irreducible 3- and 5-dimensional
+representations of the group SU(2) respectively, and ˆU= exp{/summationtext
+αϕαˆSα
+n}, where{ϕα}
+are group parameters. As K=Jthe SU(2)-symmetry is extended to the SU(3)-one,
+and the Hamiltonian (2) gets the form
+ˆH=−1
+2J/summationdisplay
+n,δ/summationdisplay
+aˆPa
+nˆPa
+n+δ−4
+3JN. (3)A generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 5
+2.2. Mean field approach and ordered states
+Insteadofinteractions between thespinandquadrupoleoperato rs{ˆPa
+n}accordingtothe
+Hamiltonian (2), we consider effective interactions of the operator s{ˆPa
+n}with a classical
+mean field. We suppose that components of the mean field at site nare proportional to
+averages (quasiaverages) of the quantum operators {ˆPa
+n}.
+In the mean field approximation the Hamiltonian (2) has the form
+ˆHMF=−(J−1
+2K)z/summationdisplay
+n/summationdisplay
+αˆSα
+n/an}bracketle{tˆSα
+n/an}bracketri}ht−1
+2Kz/summationdisplay
+n/summationdisplay
+aˆQa
+n/an}bracketle{tˆQa
+n/an}bracketri}ht−4
+3KNz,(4)
+wherezis a number of the nearest-neighbor sites. We have to give a warning about
+averages of {ˆPα
+n}. If one calculates the averages by means of the density matrix
+ˆρ(T)= exp{−H
+kT}, one obtains zeros. This follows from the SU(2)-symmetry of the
+Hamiltonian (2). Nonzero values of the averages appear if the symm etry is broken.
+Symmetry breaking can be stimulated by an external magnetic field t hat vanishes after
+specifying an order in the magnetic system. Such averages are calle dquasiaverages [6].
+Suppose that the magnetic system in question has nonzero quasiav erages{/an}bracketle{tˆPα
+n/an}bracketri}ht}.
+They form a classical 8-component vector field {µa(xn)}8
+a=1, which we call a mean field .
+Suppose that the mean field is constant over the whole magnetic sys tem. This happens
+in the case of thermodynamic equilibrium and an infinite lattice. Then un der an action
+of the group SU(2) the Hamiltonian (4) can be reduced to a diagonal form, namely:
+ˆHMF=−(J−1
+2K)z/summationdisplay
+nˆS3
+n/an}bracketle{tˆS3
+n/an}bracketri}ht−1
+2Kz/summationdisplay
+nˆQ[2,0]
+n/an}bracketle{tˆQ[2,0]
+n/an}bracketri}ht−4
+3KNz=
+=−z/summationdisplay
+n/braceleftig
+(J−1
+2K)ˆS3
+nµ3+1
+2KˆQ[2,0]
+nµ8+4
+3K/bracerightig
+,
+where the components µ3=/an}bracketle{tˆS3/an}bracketri}htandµ8=/an}bracketle{tˆQ[2,0]/an}bracketri}htdo not depend on the spatial point xn.
+These components are suitable to be order parameters . Evidently, µ3describes a
+normalized magnetization (a ratio of z-projection of magnetic moment to a saturation
+magnetization), µ8is similarly connected to a quadrupole moment.
+Now we briefly show that the proposed quantum model admits order ed states. In
+the mean field approximation a partition function is calculated by the f ormula
+Z(µ3,µ8,T) = Tre−hMF
+kT,
+wherehMFdenotes the one-site Hamiltonian
+hMF=−(J−1
+2K)µ3ˆS3−1
+2Kµ8ˆQ[2,0]−4
+3K.
+The mentioned mean field exists if self-consistent relations are held, in other words, if
+the system
+µ3=/an}bracketle{tˆS3/an}bracketri}htMF=TrˆS3e−hMF
+kT
+Tre−hMF
+kT,
+µ8=/an}bracketle{tˆQ[2,0]/an}bracketri}htMF=TrˆQ[2,0]e−hMF
+kT
+Tre−hMF
+kT.A generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 6
+hasasolution. Aftercalculationofthemeanfieldaveragesoneobta instheself-consistent
+relations in the form
+µ3=2sinh(J−K
+2)µ3
+kT
+exp/braceleftig
+−√
+3Kµ8
+2kT/bracerightig
++2cosh(J−K
+2)µ3
+kT,
+µ8=2√
+3cosh(J−K
+2)µ3
+kT−exp/braceleftig
+−√
+3Kµ8
+2kT/bracerightig
+exp/braceleftig
+−√
+3Kµ8
+2kT/bracerightig
++2sinh(J−K
+2)µ3
+kT.
+Solutions of the system correspond to ordered states of the mag netic system in question.
+An evident solution is the paramagnetic state ( µ3=0, µ8=0). All other solutions
+depend on a temperature T, and the exchange integrals JandK. Note, that we
+consider the ferromagnetic interaction in preference: J>0. Nontrivial solutions appear
+at temperatures low than the critical one Tcrit=2
+3k(J−1
+2K). AsK<0 there exists
+a ferromagnetic state with the values ( µ3=1, µ8=1√
+3) at zero temperature, and a
+nematic state with the values ( µ3=0, µ8=1√
+3) at zero temperature. As K>0 there
+exist fournontrivialsolutions: twoferromagneticstateswiththe values(µ3=1, µ8=1√
+3)
+and (µ3=2
+3, µ8=−1
+2√
+3) at zero temperature, and two nematic states with the values
+(µ3=0, µ8=−2√
+3) and (µ3=0, µ8=1√
+3) at zero temperature. The same states are
+declared in [3,4]. The states ( µ3=1, µ8=1√
+3) and (µ3=0, µ8=−2√
+3) are stable. The
+problem of transient processes in the mean field approach is discuss ed, for example,
+in [4]. The analysis of solutions of the self-consistent relations prove s that ordered
+states in the proposed model exist.
+In the sequel we deal with the case J=K, which corresponds to the boundary
+between the ferromagnetic and the nematic regions (see the phas e diagram of the
+bilinear-biquadratic s=1 model in [5]). In this case, the Hamiltonian (2) and its mean
+field approximation are SU(3)-invariant. The latter gets the form
+ˆHMF=−1
+2Jz/summationdisplay
+n/summationdisplay
+aˆPa
+n/an}bracketle{tˆPa
+n/an}bracketri}ht−4
+3JNz=−1
+2Jz/summationdisplay
+n/summationdisplay
+aˆPa
+nµa−4
+3JNz.(5)
+2.3. Motion equations for large-scale fluctuations of mean fi eld
+Return to the quantum SU(3)-invariant spin model with the Hamilton ian (3). The
+Heisenberg equation for an evolution of ˆPa
+nhas the form
+i/planckover2pi1dˆPa
+n
+dt= [ˆPa
+n,ˆH]. (6)
+Wesupposethatthemagneticsystem isordered, thenwetakeana verage ofequation(6)
+over the Heisenberg (time independent) coherent states
+|ψ(n)/an}bracketri}ht=1√
+N/parenleftig
+c1(n)|1/an}bracketri}ht+c−1(n)|−1/an}bracketri}ht+c0(n)|0/an}bracketri}ht/parenrightig
+,
+|c1|2+|c−1|2+|c0|2= 1.A generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 7
+Alternatively, one can take an average by means of the density mat rix. In the both cases
+we neglect correlations between fluctuations of the quantum fields {ˆPa
+n}8
+a=1at distinct
+sites, that is
+/an}bracketle{tˆPa
+nˆPb
+m/an}bracketri}ht ≈ /an}bracketle{tˆPa
+n/an}bracketri}ht/an}bracketle{tˆPb
+m/an}bracketri}ht=µa(xn)µb(xm). (7)
+An averaging of equation (6) results in the following equation for µa(xn):
+/planckover2pi1∂µa(xn)
+∂t= 2Jl2Cabcµb(xn)/parenleftig
+µc,xx(xn)+µc,yy(xn)/parenrightig
+, (8)
+which is a Hamiltonian one with respect to the Lie-Poisson bracket.
+In order to investigate large-scale fluctuations of the mean field {µa(xn)}8
+a=1, we
+consider a continuum space instead of the discrete lattice. It can b e achieved by
+the well-known limiting process. In the case of SU(2)-magnetic syst em (only bilinear
+intereations are taken into account), this limiting process underlies the macroscopic
+phenomenological theory of magnetism [7]. The limiting process replac es quantum
+operators by densities of their averages, which serve as dynamica l variables. In our
+case, we deal with the densities Maof averages of the spin and quadrupole moments:
+Ma(x) =/summationdisplay
+nµa(xn)δ(x,xn), δ(x,xn) =/braceleftigg
+1
+V0xn∈U(x)
+0xn/ne}ationslash∈U(x),
+whereV0denotes a physically infinitesimal region of the lattice, and U(x) is the
+infinitesimal neighborhood of x. The Lie-Poisson bracket for {Ma(x)}is defined by
+{Ma(x),Mb(y)}=CabcMc(x)δ(x−y),
+whereδ(x) is the Dirac function. Since dimensionless quantities are more suitab le,
+we introduce µa(x)=V0Ma(x) instead of Ma(x). Then equation (8) gets the form
+/planckover2pi1∂µa(x)
+∂t={Heff, µa(x)}=V0Cabcµb(x)δHeff
+δµc, (9)
+Heff=J
+ld−2/integraldisplay/summationdisplay
+a/angbracketleftig∂µa
+∂x,∂µa
+∂x/angbracketrightig
+ddx,
+wherelis the lattice distance, and dis the lattice dimension. Note, that in the
+2-dimensional case we obtain a scale-invariant Hamiltonian.
+Evidently, (9) is a generalization of the well-known Landau-Lifshitz e quation to the
+case of 8-component vector field {µa}. In the same way one can obtain the standard
+Landau-Lifshitz equation, if considers a spin system with s=1
+2over the 2-dimensional
+space of representation of SU(2).
+We rewrite (9) in the matrix form
+/planckover2pi1∂ˆµ
+∂t=2JV0
+ld−2[ˆµ,∆ˆµ],ˆµ=/summationdisplay
+aµaˆPa. (10)
+Here ˆµis a Hermitian 3 ×3 matrix, [ ·,·] denotes the matrix commutator, ∆ is the Laplas
+operator. Being SU(3)-invariant equation (10) as well as (9) pres erves the quantities
+h0=1
+2Tr ˆµ2andf0=1
+2Tr ˆµ3, which we call invariants. They serve as constrains for the
+Hamiltonian system and define the manifold where the vector field {µa}lives. At the
+same time, this manifold is an orbit of coadjoint representation of th e group SU(3).A generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 8
+3. Classical Hamiltonian systems on coadjoint orbits of SU( 3)
+In the 1-dimensional case the Hamiltonian system (9) appears to be integrable, what is
+shown below by means of the orbital approach.
+3.1. Phase space for SU(3)-symmetric generalization of Lan dau-Lifshitz equation
+In this section we briefly construct the orbital interpretation of a finite-zone phase space
+for the SU(3)-symmetric generalization of the Landau-Lifshitz eq uation.
+Consider analgebra ofpolynomials in λwith coefficients fromtheLie algebra su(3).
+Denote by /tildewideg+the algebra su(3)⊗P(λ), where P(λ) is a ring of polynomials in λwith
+the standard multiplication. Let A, B∈/tildewideg+have the form:
+A(λ) =N+1/summationdisplay
+n=0ˆAnλn, B(λ) =N+1/summationdisplay
+k=0ˆBkλk,ˆAn,ˆBk∈su(3).
+Then
+[A, B] =/summationdisplay
+n,k[ˆAn,ˆBk]λn+k∈/tildewideg+. (11)
+The operation (11) turns /tildewideg+into a graded Lie algebra.
+LetˆPa,n=λnˆPa, wherearuns from 1 to 8. The set {ˆPa,n}serves as a basis in /tildewideg+.
+Recall that [ ˆPa,ˆPb] = iCabcˆPc; the nonzero components Cabchave the following values:
+C123=C145=C167=C264=C257=C356= 1,
+C168=C528=√
+3, C437= 2.
+Introduce a bilinear ad-invariant form on /tildewideg+by
+/an}bracketle{tA,B/an}bracketri}ht=1
+2resλ−N−2TrA(λ)B(λ). (12)
+The basis {ˆPa,n}is orthonormal with respect to the bilinear form. Let M=/tildewideg∗
++be a
+dual space to the algebra /tildewideg+with respect to (12). Orthonormality of {ˆPa,n}implies
+that{ˆPa,n}also form a basis in M. Consider the following elements of M:
+ˆµ(λ) =N/summationdisplay
+n=08/summationdisplay
+a=1µn
+aλnˆPa+(µN+1
+3ˆP3+µN+1
+8ˆP8)λN+1.
+The functions ˆ µ(λ) form a closed ad-invariant subset of M, we denote it by MN+1. One
+can compute the coordinate µn
+aof ˆµ(λ) by the formula
+µn
+a=/an}bracketle{tˆµ(λ),ˆPa,−n+N+1/an}bracketri}ht.
+Define a Lie-Poisson bracket in C(MN+1) as
+{f1,f2}=/summationdisplay
+m,n8/summationdisplay
+a,bWmn
+ab∂f1
+∂µm
+a∂f2
+∂µn
+b(13)A generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 9
+with the Poisson tensor field
+Wmn
+ab=/an}bracketle{tˆµ(λ),[ˆPa,−m+N+1,ˆPb,−n+N+1]/an}bracketri}ht.
+Introduce also two ad-invariant functions I2(λ) andI3(λ) by the formulas
+I2(λ) =1
+2Trˆµ2(λ) =/summationdisplay
+aµ2
+a(λ),
+I3(λ) =1
+2Trˆµ3(λ) =/radicalig
+5
+3dabcµa(λ)µb(λ)µc(λ),
+wheredabc=√
+3
+4√
+5Tr(ˆPaˆPbˆPc+ˆPbˆPaˆPc), andµa(λ) denotes the polynomial
+µa(λ) =µ0
+a+µ1
+aλ+µ2
+aλ2+···+µN+1
+aλN+1.
+The invariant functions are also polynomials in λ:
+I2(λ) =h0+h1λ+···+h2N+2λ2N+2,
+I3(λ) =f0+f1λ+···+f3N+3λ3N+3.
+It is easy to prove that the coefficients {h0,...,hN+1,f0,...,fN+1}are annihilators
+with respect to the bracket (13). We fix these coefficients and obt ain the system of
+algebraic equations
+hn= const, fn= const, n= 0, ..., N+1, (14)
+whichdetermines anembedding ofanorbit ON+1ofdimension6( N+1)into MN+1. The
+coefficients {hN+2,..., h 2N+2,fN+2,..., f 3N+3}are pairwise commutative integrals
+of motion. We call them Hamiltonians. In the 1-dimensional case the n umber of
+Hamiltonians is sufficient for integrability of the Hamiltonian system on a n orbit.
+Here we are interested in two Hamiltonians: hN+2,hN+3, and the corresponding
+Hamiltonian flows. The Hamiltonian hN+2gives rise to the stationary flow
+∂µn
+a
+∂x={µn
+a,hN+2}= 2Cabcµ0
+bµn+1
+c, a= 1, ...,8. (15)
+The Hamiltonian hN+3gives rise to the evolutionary flow
+∂µn
+a
+∂t={µn
+a,hN+3}= 2Cabc(µ0
+bµn+2
+c+µ1
+bµn+1
+c), a= 1, ...,8.(16)
+Equations (15) and (16) are compatible, for the corresponding Ha miltonians
+commute: {hN+2, hN+3}=0. Thus, (16) describes an evolution on the trajectories of
+(15), that is the dynamical variables {µn
+a}in (16) depend on x. From (15) and (16) we
+have:
+∂µ0
+a
+∂t= 2Cabcµ0
+bµ2
+c=∂µ1
+a
+∂x. (17)
+The variables {µ1
+a}canbe expressed in theterms of {µ0
+a}and{∂
+∂xµ0
+a}, then (17) becomes
+a closed system of partial equations for {µ0
+a}. In order to compute the variables {µ1
+a}
+one has to solve the following degenerate system of equations of th e stationary flow:
+∂µ0
+a
+∂x= 2Cabcµ0
+bµ1
+c, a= 1, ...,8. (18)
+It becomes possible if one restricts the system to the orbit ON+1⊂MN+1.A generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 10
+3.2. Classification of orbits of SU(3)
+It is evident, that the orbit ON+1defined by (14) is a vector bundle over a coadjoint
+orbit of the group SU(3). That is why we need to classify orbits of SU (3).
+The group SU(3) is simple [8], hence its algebra g≃su(3) coincides with the dual
+spaceg∗. Consequently, the coordinates {µa}ing∗can be regarded as coordinates in
+su(3) as well as in su∗(3). A generic element ˆ µ∈su∗(3) has the form
+ˆµ=
+µ3+1√
+3µ8 µ7−iµ41√
+2(µ1−iµ6+µ5−iµ2)
+µ7+iµ4 −µ3+1√
+3µ81√
+2(µ1−iµ6−µ5+iµ2)
+1√
+2(µ1+iµ6+µ5+iµ2)1√
+2(µ1+iµ6−µ5−iµ2) −2√
+3µ8
+.(19)
+Lethbe the maximal commutative subalgebra (also called the Cartan suba lgebra) of g.
+The dual space h∗to the Cartan subalgebra hcoincides with h.
+By definition the set Oˆµin={g−1ˆµing,∀g∈SU(3)}is thecoadjoint orbit of SU(3)
+through an initial point ˆµin∈su∗(3). All elements g′∈SU(3) such that g′−1ˆµing′= ˆµ0
+form the stationary subgroup S ˆµinat ˆµin. The orbit Oˆµinis a homogeneous space,
+which is diffeomorphic to the coset space SU(3) /Sˆµin. There exist two types of orbits of
+SU(3): the genericOgen=SU(3)
+U(1)×U(1)of dimension 6, and the degenerate Odeg=SU(3)
+SU(2)×U(1)
+of dimension 4.
+It is proven by R. Bott that each orbit of coadjoint action of a semisimple group G
+intersects h∗precisely in an orbit of the Weyl group W(G) .
+ThefullWeylgroupofSU(3)consistsofsixelements {e, σ1, σ2, σ1σ2, σ2σ1, σ1σ2σ1},
+whereσ1,σ2are reflections across the hyperplanes orthogonal to the simple r oots
+α1,α2(see figure 1). The open domain C={ˆµ∈h∗,/an}bracketle{tˆµ,α/an}bracketri}ht>0,∀α∈∆+}is called a
+positive Weyl chamber. Here ∆+denotes the set of positive roots. We call the set
+Γα={ˆµ∈h∗,/an}bracketle{tˆµ,α/an}bracketri}ht=0}a wall of the Weyl chamber. An orbit of the Weyl group W(G)
+is obtained by the action of W(G) on a point of C.
+12
+/c97/c97
+1/c1152/c115Weyl□chamber
+Figure 1. Root diagram of SU(3).
+Each orbit of the Weyl group W(G), and consequently, each coa djoint orbit of G
+intersects the positive Weyl chamber in an only point. That is why we can classify
+coadjoint orbits of G by points of the positive Weyl chamber.
+In the case of group SU(3), there exist two types of orbits of the Weyl group.
+A generic orbit contains six elements and passes through the interio r of the positiveA generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 11
+Weyl chamber. A degenerate orbit contains three elements and pa sses through a wall
+of the positive Weyl chamber. According to this, we call an orbit of S U(3) ageneric
+one if ˆµinlies in the interior of the positive Weyl chamber, and a degenerate one if ˆµin
+belongs to a wall of the positive Weyl chamber.
+In our case, ˆ µinhas the following diagonal form
+ˆµin= diag(m+1√
+3q,−m+1√
+3q,−2√
+3q),
+wheremandqdenote initial values of the variables µ3andµ8respectively, or boundary
+values (at zero temperature) of the corresponding components of the mean field.
+Asm>0,q>0 the coadjoint action of SU(3) gives a generic orbit. If m=0 orm=√
+3q,
+we obtain a degenerate orbit. In the sequel we consider degenera te orbits with m=0.
+3.3. Hamiltonian equations on orbits of SU(3)
+Return to the system of equations (18), which is degenerate in MN+1. However, it
+can be solved if one restricts the system to the orbit ON+1⊂MN+1. Each orbit is
+determined by the following equation [9]:
+χmin(ˆµ) = 0, (20)
+whereχmin(ˆµ) is the minimal characteristic polynomial in ˆ µ∈ON+1. Equation (20)
+serves as a constrain for the system (18), which has the form
+∂ˆµ0
+∂x= Adˆµ0ˆµ1. (21)
+Now we solve (21) on orbits of the group SU(3).
+A degenerate orbit is determined by the equation
+ˆµ2+/radicalig
+h0
+3ˆµ−2h0
+3= 0,
+whereh0=q2=const. Using this constrain, one obtains the following solution of (2 1):
+µ1
+a=1
+6h0Cabcµ0
+bµ0
+c,x+h1
+2h0µ0
+a, whereh1
+2h0µ0
+ais an element of KerAd ˆµ0. The motion
+equation (17) on the degenerate orbit has the form
+∂µa
+∂t=8A
+3h0Cabcµbµc,xx+8Ah1
+h0µa,x, (22)
+where we write µainstead ofµ0
+aand scale the flow parameter tby 16A. The dimensional
+constant Aprovides a correspondence between (22) as h1=0 and (10) as d=1. That
+is (22) describes large-scale fluctuations of the mean field {µa}.
+A generic orbit is determined by the characteristic equation
+ˆµ3−h0ˆµ−2
+3f0= 0,
+whereh0=m2+q2, andf0=1√
+3(3m2q−q3). On this orbit we obtain the following
+solution of (21):
+µ1
+a=1
+8(h3
+0−3f2
+0)/parenleftbig
+h2
+0Cabcµ0
+bµ0
+c,x−2√
+3f0Cabcη0
+bµ0
+c,x+h0Cabcη0
+bη0
+c,x/parenrightbig
++
++2f0f1−3h2
+0h1
+6(f2
+0−h3
+0)µ0
+a+3f0h1−2h0f1
+6√
+3(f2
+0−h3
+0)η0
+a,A generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 12
+whereη0
+a=√
+5dabcµ0
+bµ0
+c. The motion equation (17) on the generic orbit has the form
+∂µa
+∂t=2A
+h3
+0−3f2
+0/parenleftig
+h2
+0Cabcµbµc,xx−√
+3f0Cabcµbηc,xx−
+−√
+3f0Cabcηbµc,xx+h0Cabcηbηc,xx/parenrightig
++ (23)
++8A
+32f0f1−3h2
+0h1
+f2
+0−h3
+0µa,x+8A
+3√
+33f0h1−2h0f1
+f2
+0−h3
+0ηa,x.
+Ash1=0,f1=0 equation (23) also describes large-scale fluctuations of the mea n field.
+One can obtain (23) from (6) by averaging with a more complicate cor relation rule.
+Equations (22) and (23) imply the following Hamiltonians respectively:
+Hdeg=4A//planckover2pi1
+3h0/integraldisplay8/summationdisplay
+a=1(µa,x)2dx,
+Hgen=A//planckover2pi1
+h3
+0−3f2
+0/integraldisplay8/summationdisplay
+a=1/parenleftig
+h2
+0(µa,x)2+h0(ηa,x)2−2√
+3f0µa,xηa,x/parenrightig
+dx.
+In addition to the 1-dimensional case one can consider the corresp onding 2- or
+3-dimensional Hamiltonian systems with the effective Hamiltonians
+Heff=J/integraldisplay
+H(µ)ddx, (24)
+whereµdenotes{µa}8
+a=1. The exchange integral J=A//planckover2pi1gives the Hamiltonian the
+required physical dimension. By Hwe denote the Hamiltonian density
+Hdeg=4
+3h0d/summationdisplay
+k=18/summationdisplay
+a=1(µa,xk)2,or
+Hgen=1
+h3
+0−3f2
+0d/summationdisplay
+k=18/summationdisplay
+a=1/parenleftig
+h2
+0(µa,xk)2+h0(ηa,xk)2−2√
+3f0µa,xkηa,xk/parenrightig
+.
+One can use these effective Hamiltonians for describing the magnetic system considered
+in section 2. Note, that Hdegis the same as the Hamiltonian of (9).
+The proposed Hamiltonians describe large-scale (slow) fluctuations of the mean
+fieldµ. After averaging over high frequencies some observed quantities become rigid
+(or invariant); these quantities are h0=δabµaµb, andf0=/radicalbig
+5/3dabcµaµbµc. They serve
+as constrains for the Hamiltonian systems, and are equivalent to (2 0). The constrains
+determine the orbit where the system has to be considered.
+In the case of SU(3)-invariant model we deal with the magnet whos e ferromagnetic
+and nematic states are equiprobable. A generic orbit corresponds to a state with the
+ferromagnetic order at zero temperature, because of nonzero magnetization ( m/ne}ationslash=0).
+A degenerate orbit ( m=0) corresponds to a state with the nematic order at zero
+temperature. So equations (22) and (23) describe fluctuations o f the mean field µ
+near a nematic and a ferromagnetic ordered states respectively.A generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 13
+3.4. SU(3)-invariance of effective Hamiltonians
+AsmentionedinSection2,thequantumHamiltonian(2)andthemeanfi eldHamiltonian
+(4) are SU(3)-invariant as K=J. Here we show that the proposed classical effective
+Hamiltonians (24) are also SU(3)-invariant.
+Recall, thatthe meanfield {µa}belongstothe real8-dimensional space ofcoadjoint
+representation of SU(3). Hence, an action of SU(3) transforms {µa}by the formula
+µa=ˆDabµb,ˆDab∈SO(8),
+ˆDabis a matrix of the real irreducible 8-dimensional representation of t he group SU(3).
+Note, thatthetensor dabcsatisfies therelation dabcdqbc=δaq. Thecomponents {dabc}
+serveasClebsch-Gordoncoefficientsforadecompositionoftenso rsquareofthecoadjoint
+representation into irreducible components. In this connection, w e have the following
+relation, well-known in theory of representations, ˆDbb′ˆDcc′=dqbcdq′b′c′ˆDqq′. Then as a
+result of the action of SU(3) on {ηa}we get
+ηa=√
+5dabcˆDbb′µb′ˆDcc′µc′=ˆDqq′ηq′.
+The action of SU(3) on the vector fields {µa,x}and{ηa,x}is the same. Therefore, the
+densitiesHgenandHdegare SU(3)-invariant.
+Densities of the effective Hamiltonians can be expressed as
+H=/summationdisplay
+jk/summationdisplay
+abgab(µ)∂µa
+∂xj∂µb
+∂xkGjk(x), (25)
+wheregab(µ) serves as a metrics invariant under an action of the group that
+transforms µ, andGjk(x) is a metrics in the x-space. For the proposed effective
+Hamiltonians the x-space is Euclidean: Gjk(x) =δjk. The metrics in µ-space is trivial:
+gab(µ)=4
+3h0δabin the case of a degenerate orbit, and has a more complicate form:
+gab(µ) =1
+h3
+0−3f2
+0/parenleftig
+h2
+0δab+20h0dcpadcqbµpµq−4√
+15f0dabcµc/parenrightig
+in the case of a generic orbit.
+The density (25) can be interpreted as a Lagrangian density of rela tivisticσ-
+model; in this case Gjkis the metrics of the Minkowski space. After quantization one
+obtains a Hamiltonian system that describes slow fluctuations. Quick fluctuations can
+be taken into account by means of a renormalization group [1]. It ma kes the coefficients
+1
+h3
+0−3f2
+0and4
+3h0dependent on parameters of the renormalization group, for exam ple on
+a temperature.
+3.5. Parametrization of orbits
+Remarkably, that the effective models are entirely defined by geome try of orbits. We
+will prove this statement, if perform a parametrization of orbits an dexpress the effective
+Hamiltonians in terms of these parameters.A generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 14
+A generalized stereographic projection gives a suitable way of para metrization for
+coadjoint orbits of a semisimple Lie group [10]. In the case of group S U(3) we have
+µa=−m−√
+3q
+2ζa+mξa, η a=√
+3(m2−q2)−2mq
+2ζa+2mqξa,
+where
+ζ1=−1√
+2w2+w3+¯w2+¯w3
+1+|w2|2+|w3|2ξ1=−1√
+2(1−w1)(¯w3−¯w1¯w2)+(1−¯w1)(w3−w1w2)
+1+|w1|2+|w3−w1w2|2,
+ζ2=−i√
+2w2−w3−¯w2+¯w3
+1+|w2|2+|w3|2ξ2=−i√
+2(1+w1)(¯w3−¯w1¯w2)−(1+¯w1)(w3−w1w2)
+1+|w1|2+|w3−w1w2|2,
+ζ3=|w2|2−|w3|2
+1+|w2|2+|w3|2ξ3=1−|w1|2
+1+|w1|2+|w3−w1w2|2,
+ζ4=i¯w2w3−w2¯w3
+1+|w2|2+|w3|2ξ4=iw1−¯w1
+1+|w1|2+|w3−w1w2|2,
+ζ5=1√
+2w2−w3+¯w2−¯w3
+1+|w2|2+|w3|2ξ5=−1√
+2(1+w1)(¯w3−¯w1¯w2)+(1+¯w1)(w3−w1w2)
+1+|w1|2+|w3−w1w2|2,
+ζ6=i√
+2w2+w3−¯w2−¯w3
+1+|w2|2+|w3|2ξ6=i√
+2(1−¯w1)(w3−w1w2)−(1−w1)(¯w3−¯w1¯w2)
+1+|w1|2+|w3−w1w2|2,
+ζ7=−¯w2w3+w2¯w3
+1+|w2|2+|w3|2ξ7=−w1+¯w1
+1+|w1|2+|w3−w1w2|2,
+ζ8=1√
+32−|w2|2−|w3|2
+1+|w2|2+|w3|2ξ8=1√
+31+|w1|2−2|w3−w1w2|2
+1+|w1|2+|w3−w1w2|2.
+Herew1,w2,w3are complex parameters on a generic orbit, mandqare initial values
+ofµ3andµ8respectively. The initial values fix an orbit. For a degenerate orbit o ne has
+to assignm= 0 andw1= 0.
+After this parameterization the effective Hamiltonians get the form
+Heff=/integraldisplayd/summationdisplay
+k=1/summationdisplay
+α,βgαβ(w)∂wα
+∂xk∂wβ
+∂xkddx,
+gdeg
+αβ=/summationdisplay
+a∂ζa
+∂wα∂ζa
+∂wβ,
+ggen
+αβ=/summationdisplay
+a/parenleftbigg∂ζa
+∂wα∂ζa
+∂wβ−∂ζa
+∂wα∂ξa
+∂wβ+∂ξa
+∂wα∂ξa
+∂wβ/parenrightbigg
+.
+The tensors ggenandgdegserve as metrics on orbits in terms of the complex parameters
+w={w1,¯w1, w2,¯w2, w3,¯w3}for a generic orbit, and w={w2,¯w2, w3,¯w3}for a
+degenerate orbit. Note, that the metrics do not depend on the init ial valuesmandq,
+fixing an orbit. All generic orbits have the same metrics, as well as de generate orbits.
+4. Results and discussion
+Our main result is the following. For a magnetic system with the spin s/greaterorequalslant1 we propose
+two effective classical models that describe fluctuations of the mea n field by the Landau-
+Lifshitz like equations. We consider the 8-component mean field µ={µa}8
+a=1, taking
+into account not only magnetization but also quadrupole moments.
+The effective models deal with large-scale (slow) fluctuations of the mean field.
+Small-scale (quick) fluctuations are cut off by quasiaveraging. In th is process some
+observedquantitiesbecomerigidandserveasconstrainsdetermin ing themanifoldwhere
+the mean field lives. This manifold appears to be a coadjoint orbit of th e group SU(3).
+Also we propose a complex parametrization for the manifold and redu ce the mean
+field and the Hamiltonian density to complex parameters. Remarkably , that in termsA generalised Landau-Lifshitz equation for isotropic SU(3 ) magnet 15
+of the complex parameters the density becames independent on bo undary values of µ.
+Moreover, the Hamiltonian density serve as a Riemannian metrics on t he manifold.
+In the case of SU(3)-invariant model we deal with the magnet whos e ferromagnetic
+and nematic states at zero temperature are equiprobable. That is why we propose two
+effectiveHamiltonians: Hgenforstateswiththeferromagneticorderatzerotemperature,
+andHdegfor states with the nematic order (when magnetization is zero) at z ero
+temperature. Alsoweproduceequations(22)and(23)describing large-scalefluctuations
+of the mean field µnear a nematic and a ferromagnetic ordered states respectively.
+The proposed classical models can be used to construct topologica l excitations [11],
+which are stationary solutions of the Landau-Lifshitz like equations . These excitations
+realize destruction of a long-range order in 2-dimensional spin syst ems at nonzero
+temperatures, according to the Mermin-Wagner theorem.
+Theconsideredscheme iseasilyextendedtothecasewithhigherpow ersofexchange
+interaction. For an arbitrary spin sthe spin operators {ˆSα
+n}are defined over the
+(2s+1)-dimensional space of representation of the group SU(2). Th e complete matrix
+algebra generated by the spin operators is Mat (2s+1)×(2s+1). Then one can consider a
+spin Hamiltonian with powers of exchange interaction up to 2 s. Such Hamiltonian
+admits a bilinear form, if one takes into account multipole moments. In the mean field
+approximation this quantum model corresponds to a Hamiltonian sys tem on a coadjoint
+orbit of the group SU(2 s+1). Each orbit has a Hamiltonian system, which serves as an
+effective classical model.
+Acknowledgements
+The research presented in the paper was conducted with the finan cial support of the
+Julian Wynnyckyj professorship in natural sciences at Kyiv-Mohyla Academy, and the
+grant of the International Charitable Fund for Renaissance of Ky iv-Mohyla Academy.
+References
+[1] TsvelikA Quantum Field Theory in Condensed Matter Physics (Cambridge: CambridgeUniversity
+Press) p 348
+[2] Blume M and Hsien Y Y 1969 J. Appl. Phys. 401249
+[3] Matveev V M 1973 Zh. Eksp. Teor. Fiz. 651626–36
+[4] Nauciel-Bloch M, Sarma G and Castets A 1972 Phys. Rev. B 54603
+[5] Buchta K, Fath G, Legeza ¨O and Solyom J 2005 Phys. Rev. B 72054433
+[6] Bogolyubov N N and Bogolyubov N N (Jr.) 1984 Introduction to Quantum Statistical Mechanics
+[in Russian] (Moscow: Nauka) p 384
+[7] Herring C and Kittel C 1951 Phys. Rev. 81869
+[8] Helgason S 1978 Differential Geometry, Lie Groups and Symmetric Spaces (New York: Academic
+Press) p 644
+[9] Boyarskii A and Skrypnik T 1996 Russ. Math. Surv. 51541-2.
+[10] Bernatska J and Holod P 2008 Proceedings of the IXth International Conference on Geomet ry,
+Integrability and Quantization (Sofia) 146–66
+[11] Bernatska J and Holod P Theor. and Math. Phys. to be published
\ No newline at end of file
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+++ b/1001.0373.txt
@@ -0,0 +1,1315 @@
+arXiv:1001.0373v1  [cond-mat.str-el]  3 Jan 2010J.M. BERNATSKA, P.I. HOLOD
+ORDERED STATES AND NONLINEAR LARGE-SCALE
+EXCITATIONS IN A PLANE MAGNET WITH SPIN s=1
+J.M. BERNATSKA, P.I. HOLOD
+UDC 538.221
+c/circlecopyrt2008National University “Kyiv-Mohyla Academy”
+(2, Skovoroda Str., Kyiv 04070, Ukraine),
+1M.M. Bogolyubov Institute for Theoretical Physics, Nat. Ac ad. of Sci. of Ukraine
+(14b, Metrolohichna Str., Kyiv 03143, Ukraine)
+We study ordered states and topological excitations in a qua si-
+two-dimensional magnet, modeled by a square lattice with sp ins
+s=1at all sites, and the Hamiltonian with biquadratic exchange
+interaction between nearest neighbor sites. We propose two
+effective Hamiltonians for description of large-scale exci tations
+in the two-dimensional case. They describe excitations of t he
+mean field in a nematic phase and a mixed ferromagnetic-nemat ic
+phase. It is shown that the effective Hamiltonians are minimi zed
+on configurations with fixed topological charge. These topol ogical
+excitations can arise at low temperatures and cause a destru ction
+of a long-range order in the two-dimensional system.
+1. Introduction
+Quasi-two-dimensional magnets have various technolo-
+gical applications. They serve as magnetic films used
+for recording of information, thin ferromagnetic layers
+in Josephson semiconductor junctions, layered resistive
+systems, etc.
+Here, we will not deal with applied aspects of
+theory of magnetism. However, we note that a study
+of two-dimensional systems has also a significant value.
+Studying ordered states, their stability, and excitation
+spectra, we obtain model scenarios of a self-organization
+of a substance with decrease in a temperature or under
+an action of external fields. To support the above-
+presented assertion, it is worth to recall an important
+role played by the Onsager’s results [1] on the two-
+dimensional Ising model, or the Kosterlitz–Thouless
+theory of topological phase transitions [2,3]. This is also
+related to the study of the two-dimensional O(3)-sigma
+model or a planar Heisenberg magnet [4]. As a natural
+continuation of this trend, we mention a significant
+number of scientific papers devoted to investigations of
+two-dimensional continuous or lattice systems with high
+spins at sites.
+In the present paper we consider a generalized
+Heisenberg magnet taking into account bilinear and
+biquadratic interactions at nearest-neighbor sites of asquare or cubic lattice. This Hamiltonian for magnets
+of spins=1was proposed and studied long ago [5,
+6] without any restriction on dimensionality. At the
+beginning of 1970s, an existence of ordered phases
+different from the ferromagnetic or antiferromagnetic
+ones was established by the mean-field methods. In
+particular, if the constant of biquadratic interaction is
+larger than that of bilinear interaction, then a pure
+quadrupole ordering or a spin nematic state can be
+realized in the system [7,8].
+It is known that a two-dimensional system with
+a continuous group of symmetries has no long-range
+order atT>0(the Mermin–Wagner theorem). In many
+cases, an instability of ordered phases in two-dimensional
+systems implies an existence of nonlinear topological
+excitations caused by small fluctuations of temperature.
+The role of such excitations in destruction of a long-range
+order is proven within the model of plane rotators [2,3]
+and for the two-dimensional Heisenberg ferromagnet [4].
+The main result of our work is a proof of an existence
+of topological excitations in the model with biquadratic
+interaction between nearest spins s=1at sites of a
+square lattice. We will consider the boundary case of
+a nematic phase, where the constants of bilinear and
+biquadratic interactions are identical. It is known that,
+in this case, energies of both possible phases (nematic
+and ferromagnetic-quadrupole ones) are equal. In order
+to study excitations of nematic phase, we assume that
+K−J=ε, andεis a small positive value. It is obvious
+that topological excitations exist at these parameters,
+and differ slightly from those in the case ε=0. Ifε>0,
+the manifold of degeneration of a ground state of the
+system is deformed, but the topology is not sensitive
+to smooth deformations. Therefore, our conclusion of
+existence of topological excitations at J=Kremains
+valid also in the case K>J> 0.
+The present paper contains two parts. The first part
+is a survey. In the mean field approximation, we reveal
+existence conditions for ordered phases and solve an
+1208 ISSN 2071-0186. Ukr. J. Phys. 2008. V. 53, N 12ORDERED STATES AND NONLINEAR EXCITATIONS
+equation of self-consistent relations for order parameter s.
+Comparing with results of other researchers on this
+topic, we obtain conditions of occurrence of a nematic
+state. In the second part, averaging equations of motion
+over coherent states, and passing from a plane square
+lattice to a continuous plane, we obtain formulas for free
+energy of an inhomogeneous distribution of the mean
+field. Topological excitations are described in terms of
+the inhomogeneous distribution. Depending on a choice
+of an equilibrium state and a way of averaging, we
+get two formulas for the free energy: the first formula
+corresponds to excitations of a pure nematic state, and
+the second one is related to excitations in the state
+with nonzero magnetization and quadrupole moment.
+In the case of SU(3) -symmetry, we determine self-dual
+solutions of the problem of minimization. The obtained
+topological excitations give the absolute minimum for
+the free energy, and its value is proportional to a
+topological charge.
+2. The Quantum Model of a Planar Magnet
+Let us consider a plane square lattice containing atoms
+of spinsat each site. Each atom is assigned by
+three spin operators {ˆS1
+n,ˆS2
+n,ˆS3
+n}obeying the standard
+commutation relations
+[ˆSα
+n,ˆSβ
+n] =iεαβγˆSγ
+nδnm,
+where the indices α,β, andγrun the values {1,2,3}
+for each site n, andδnmis the Kronecker symbol.
+Usually, such system is described by the Heisenberg
+Hamiltonian. As s/greaterorequalslant1we can include higher orders of the
+exchange interaction in the Hamiltonian. In particular,
+magnets with the biquadratic interaction were studied
+in the 1970s [9, 10]. The latter Hamiltonian will be
+considered in what follows. Let
+ˆH=−/summationdisplay
+n,δ{J(ˆSn,ˆSn+δ)+K(ˆSn,ˆSn+δ)2},(1)
+whereˆSndenotes the vector (ˆS1
+n,ˆS2
+n,ˆS3
+n)of the spin
+operators at site n, andδruns over the nearest-neighbour
+sites. We assume that the exchange integrals JandK
+are positive, i.e. we consider mainly the ferromagnetic
+interaction.
+The operators {ˆSα}are defined over the (2s+1)-
+dimensional space of an irreducible representation of
+the group SU(2) . The spin operators generate the
+complete matrix algebra over this space (the Burnside
+theorem). With respect to the adjoint action adˆSα, the
+complete matrix algebra is divided into a direct sum ofirreducible collections of tensor operators. For example,
+let us consider the case of s=1. Then for the complete
+matrix algebra over the representation space, we have
+dimMat 3×3≃[9]=[1]+[3]+[5] . Obviously, bases in the
+three- and five-dimensional irreducible collections are
+formed, respectively, by the operators ˆSα, and by the
+tensor operators of weight 2. The latter are the operators
+of quadrupole moment chosen in the form
+ˆQαβ
+n=ˆSα
+nˆSβ
+n+ˆSβ
+nˆSα
+n, α/ne}ationslash=β,
+ˆQ[2,2]
+n= (ˆS1
+n)2−(ˆS2
+n)2,
+ˆQ[2,0]
+n=√
+3/parenleftbig
+(ˆS3
+n)2−2
+3/parenrightbig
+.
+A normalization of the operators ˆSαis defined by the
+relation
+(ˆS1)2+(ˆS2)2+(ˆS3)2=s(s+1)I3,
+which yields Tr(ˆSα)2=1
+3s(s+1)(2s+1). Fors=1we
+haveTr(ˆSα)2=2. We extend such normalization for all
+other operators.
+Now we fix the canonical basis {|+1/an}bracketri}ht,|−1/an}bracketri}ht,|0/an}bracketri}ht}in
+the representation space. Then a matrix representation
+of the operators of spin and quadrupole moment is as
+follows:
+ˆS1
+n=1√
+2
+0 0 1
+0 0 1
+1 1 0
+,ˆS2
+n=1√
+2
+0 0−i
+0 0i
+i−i0
+,
+ˆS3
+n=
+1 0 0
+0−1 0
+0 0 0
+,ˆQ[2,0]
+n=1√
+3
+1 0 0
+0 1 0
+0 0−2
+,
+ˆQ12
+n=
+0−i0
+i0 0
+0 0 0
+,ˆQ13
+n=1√
+2
+0 0 1
+0 0 −1
+1−1 0
+,
+ˆQ23
+n=1√
+3
+0 0−i
+0 0−i
+i i0
+,ˆQ[2,2]
+n=
+0 1 0
+1 0 0
+0 0 0
+.
+It is easy to see that the proposed matrices are connected
+with the Gell-Mann matrices ˆΛa,a=1,2,...,8, which
+also form a basis in isu(3). The connection is given by
+the linear transformations
+ˆS1
+n=1√
+2(ˆΛ4+ˆΛ6),ˆS2
+n=1√
+2(ˆΛ5−ˆΛ7),ˆS3
+n=ˆΛ3,
+ˆQ12
+n=ˆΛ2,ˆQ[2,0]
+n=ˆΛ8,ˆQ[2,2]
+n=ˆΛ1,
+ˆQ13
+n=1√
+2(ˆΛ5+ˆΛ7),ˆQ23
+n=1√
+2(ˆΛ4−ˆΛ6).
+ISSN 2071-0186. Ukr. J. Phys. 2008. V. 53, N 12 1209J.M. BERNATSKA, P.I. HOLOD
+By{ˆPa}we denote the collection of operators {ˆS1
+n,
+ˆS2
+n,ˆS3
+n,ˆQ12
+n,ˆQ13
+n,ˆQ23
+n,ˆQ[2,2]
+n,ˆQ[2,0]
+n}. It is easy to prove
+that the commutation relations
+[ˆPa
+n,ˆPb
+m] =iCabcˆPc
+nδnm,
+hold true. Here, the tensor of structure constants Cabc
+is antisymmetric under a permutation of any pair of
+indices, and its nonzero components are
+C123=C145=C167=C264=C257=C356= 1,
+C168=C528=√
+3, C437= 2,
+In terms of the operators of spin and quadrupole
+moment, Hamiltonian (1) gets the form
+ˆH=−/parenleftbig
+J−1
+2K/parenrightbig/summationdisplay
+n,δ/summationdisplay
+αˆSα
+n,ˆSα
+n+δ−
+−1
+2K/summationdisplay
+n,δ/summationdisplay
+aˆQa
+nˆQa
+n+δ−4
+3KN, (2)
+whereNdenotes the total number of sites of the lattice.
+Obviously Hamiltonian remains SU(2) -invariant; hence,
+the operators ˆSα
+nandˆQa
+nare transformed by formulas
+of the adjoint representation
+ˆUˆSα
+nˆU−1=/summationdisplay
+βˆDαβ(ˆU)ˆSβ
+n,
+ˆUˆQa
+nˆU−1=/summationdisplay
+bˆDab(ˆU)ˆQb
+n,∀ˆU∈SU(2),
+whereˆDαβ(ˆU)andˆDab(ˆU)are matrices of the real
+irreducible representations of SU(2) with dimensions 3
+and 5, respectively. If K=J, then the SU(2) -symmetry
+can be extended to the group SU(3) . In this case, the
+Hamiltonian (2) gets the form
+ˆH=−1
+2J/summationdisplay
+n,δ/summationdisplay
+aˆPa
+nˆPa
+n+δ−4
+3JN. (3)
+To study possible ordered phases of such spin system,
+we use the mean-field approximation.
+3. Mean-Field Approximation. Ordered States
+Now we replace the interaction between spin and
+quadrupole operators that is described by Hamiltonian
+(2) with an effective interaction between the operators
+ˆPa
+nand the classical mean field. Components of the
+mean field are considered proportional to averages
+(quasiaverages) of the quantum operators {ˆPa
+n}. TheHamiltonian in the mean field approximation has the
+form
+ˆHMF=−(J−1
+2K)/summationdisplay
+n,δ/summationdisplay
+αˆSα
+n/an}bracketle{tˆSα
+n+δ/an}bracketri}ht−
+−1
+2K/summationdisplay
+n,δ/summationdisplay
+aˆQa
+n/an}bracketle{tˆQa
+n+δ/an}bracketri}ht−4
+3KN. (4)
+It is worth to give a warning that a direct calculation
+of the averages {/an}bracketle{tˆSα
+n/an}bracketri}ht}and{/an}bracketle{tˆQa
+n/an}bracketri}ht}, for example by
+means of the density matrix ˆρ(T)= exp{−H
+kT}, results
+in the zero values. This follows from the SU(2) -
+symmetry of Hamiltonian (2). Nonzero values of the
+averages appear if the symmetry is broken. Symmetry
+breaking can be stimulated by an external field, which
+vanishes after specifying an order in the magnetic
+system. The quantities calculated in this way are called
+“quasiaverages” [11].
+Hence, we assume that in our system the nonzero
+quasiaverages {/an}bracketle{tˆSα
+n/an}bracketri}ht}and{/an}bracketle{tˆQa
+n/an}bracketri}ht}exist, and form a
+classical 8-component vector field µa(xn),a=1,2,...,8.
+In order to obtain nonzero values of {/an}bracketle{tˆQa
+n/an}bracketri}ht}, the external
+field must have nonzero gradient. If the mean field is
+homogeneous, an action of the group SU(2) transforms
+Hamiltonian (4) to the form
+ˆHMF=−(J−1
+2K)/summationdisplay
+nˆS3
+n/an}bracketle{tˆS3/an}bracketri}ht−
+−1
+2K/summationdisplay
+nˆQ[2,0]
+n/an}bracketle{tˆQ[2,0]/an}bracketri}ht−4
+3KN=−4
+3KN−
+−/summationdisplay
+n/braceleftig
+(J−1
+2K)ˆS3
+nµ3+1
+2KˆQ[2,0]
+nµ8/bracerightig
+.
+In the case of thermodynamic equilibrium and an
+infinite lattice, the fields µ3=/an}bracketle{tˆS3/an}bracketri}htandµ8=/an}bracketle{tˆQ[2,0]/an}bracketri}htare
+constant, i.e. have the same values at all points xn
+(a homogeneous mean field). These quantities serve as
+order parameters . Obviously, µ3describes a normalized
+magnetization (a ratio of z-projection of magnetic
+moment to a saturation magnetization), and µ8is
+analogously related to a quadrupole moment.
+In the mean field approximation, it is easy to
+calculate a partition function for the homogeneous case
+NZ(µ3,µ8,T) = Tre−HMF
+kT= Tre−NhMF
+kT,
+wherehMFdenotes a one-site Hamiltonian
+hMF=−(J−1
+2K)µ3ˆS3−1
+2Kµ8ˆQ[2,0]−4
+3K.(5)
+The introduced mean field makes sense if self-
+consistent relations are held:
+µ3=/an}bracketle{tˆS3/an}bracketri}htMF=TrˆS3e−NhMF
+kT
+Tre−NhMF
+kT,
+1210 ISSN 2071-0186. Ukr. J. Phys. 2008. V. 53, N 12ORDERED STATES AND NONLINEAR EXCITATIONS
+µ8=/an}bracketle{tˆQ[2,0]/an}bracketri}htMF=TrˆQ[2,0]e−NhMF
+kT
+Tre−NhMF
+kT.
+these relations serve as an analog of the Weiss
+equation from theory of ferromagnetism. The averages
+of operators can be presented via the partition function:
+µ3=kT
+(J−K
+2)∂
+∂µ3lnZ(µ3,µ8,T),
+µ8=2kT
+K∂
+∂µ8lnZ(µ3,µ8,T).
+For the system described by one-site Hamiltonian (5),
+the self-consistent relations get the form
+µ3=2sh(J−K
+2)µ3
+kT
+exp/braceleftig
+−√
+3Kµ8
+2kT/bracerightig
++2ch(J−K
+2)µ3
+kT,
+µ8=2√
+3ch(J−K
+2)µ3
+kT−exp/braceleftig
+−√
+3Kµ8
+2kT/bracerightig
+exp/braceleftig
+−√
+3Kµ8
+2kT/bracerightig
++2ch(J−K
+2)µ3
+kT. (6)
+Note, that the true averages are always less than their
+expectation values calculated from the self-consistent
+relations. Therefore, solutions of (6) have a qualitative
+sense only.
+Here we analyze Eqs. (6) and make comparison
+with results described in the literature. The obvious
+solution corresponds to the paramagnetic state with
+µ3=0andµ8=0; this state is realized at temperatures
+kT>J−K/2. At the same time, this inequality shows
+that the model, oriented to ferromagnetic materials,
+gives a meaningful result only in the region J−K/2<0,
+which contains areas with the ferromagnetic and
+quadrupole orderings, according to the well-known phase
+diagram (Fig. 1) for the bilinear-biquadratic s=11-
+dimensional spin model [12].
+/c45K
+/c45J HaldaneFerro-
+magneticTrimerized
+Spin nematicDimerized/c45/c112/4/c112/4/c112/2
+/c48/c44/c54/c55/c60 /c47
+/c60/c49/c44/c50/c50J KFig. 1. Phase diagram of a one-dimensional system of spins s=1
+In the case of K<0, the self-consistent relations
+have a unique nontrivial solution, corresponding to the
+ferromagnetic ordering, because this solution tends to
+the boundary values µ3=1andµ8=1√
+3as temperature
+decreases to zero. The critical temperature of transition
+from a ferromagnetic state into a paramagnetic one
+is determined in terms of the constants JandKas
+Tc=2
+3k(J−K/2).
+In the case of K>0(the light-grey region in
+Fig. 1), Eqs. (6) have more than one nontrivial solution:
+two solutions corresponding to ferromagnetic states
+with the boundary values µ(1)
+3=1andµ(2)
+3=2/3(and
+the corresponding values of µ8), and two solutions
+describing nematic states ( µ3=0) with the boundary
+valuesµ(1)
+8=−2√
+3andµ(2)
+8=1√
+3. Existence of four ordered
+states in ferromagnets is also reported in [7]: they
+are a ferromagnetic state with µ(1)
+3=s, a quadrupole
+(or nematic) state with µ3=0,µ(1)
+8=−s(s+ 1)/√
+3, a
+partially ordered quadrupole state with µ3=0,µ(1)
+8>0,
+and a partially ordered ferromagnetic state with µ(1)
+3<s.
+Partially ordered states are unstable [7].
+Analyzing the temperature evolution of solutions of
+(6) asK>0,J>0, we reveal two critical temperatures,
+which are solutions of the equation
+2/parenleftigJ−K/2
+kT−1/parenrightig
+= exp/braceleftigK
+kT/parenleftig
+1−3kT
+2(J−K/2)/parenrightig/bracerightig
+.
+An obvious solution is Tc1=2
+3k(J−K/2). The other
+solutionTc2is calculated numerically. In the region
+J>K , i.e. for ferromagnetic materials, the temperature
+Tc2is less than Tc1, whereas the reversed situation
+takes place for nematics in the region J<K . At smaller
+critical temperature the solution µ(2)
+3disappears. Then
+only the solution µ(1)
+3exists in a certain interval of
+temperatures. A comparison with results of the paper [8]
+shows that at a higher critical temperature we have a
+second order phase transition from a ferromagnetic state
+into a paramagnetic one.
+A somewhat different behavior is observed
+in materials with J≈K, or more precisely
+0.67<J/K< 1.22, i.e. on the boundary between the
+ferromagnetic and nematic regions (the dark-grey
+region in Fig. 1). Disappearing at a lower critical
+temperature, the solution µ(2)
+3arises again at a higher
+critical temperature, and grows continuously from zero
+towardµ(1)
+3. When two solutions coincide at a certain
+ISSN 2071-0186. Ukr. J. Phys. 2008. V. 53, N 12 1211J.M. BERNATSKA, P.I. HOLOD
+temperature T0, they disappear by jump. We may
+conclude that the phase transition from a ferromagnetic
+state to a paramagnetic one is a transition of the first
+order. This well agrees with results of the paper [8],
+where the change of a second order phase transition
+into a first order one in the region 2J/3<K<J was
+considered (for ferromagnetic materials).
+Further we consider the case J=K, corresponding to
+the boundary between the ferromagnetic and quadrupole
+regions. Moreover, as J=Kthe quantum Hamiltonian
+(2) and the Hamiltonian in the mean-field approximation
+areSU(3) -invariant. The latter has the form
+ˆHMF=−1
+2J/summationdisplay
+n/summationdisplay
+aˆPa
+n/an}bracketle{tˆPa/an}bracketri}ht−4
+3JN=
+=−1
+2J/summationdisplay
+n/summationdisplay
+aˆPa
+nµa−4
+3JN. (7)
+4. Equations of Motion for Large-Scale
+Fluctuations of the Mean Field
+Now we return to the quantum SU(3) -invariant spin
+model with Hamiltonian (3). The Heisenberg evolution
+equations for the operators ˆPa
+nhave the form
+i/planckover2pi1dˆPa
+n
+dt= [ˆPa
+n,ˆH]. (8)
+We assuming that the system is ordered state, and
+take an average of the both sides of Eq. (8) over
+Heisenberg (time independent) coherent states
+|ψ(n)/an}bracketri}ht=1√
+N/parenleftbig
+c1(n)|1/an}bracketri}ht+c−1(n)|−1/an}bracketri}ht+c0(n)|0/an}bracketri}ht/parenrightbig
+,
+|c1|2+|c−1|2+|c0|2= 1.
+On the other hand, an averaging can be performed
+by means of the density matrix (thermodynamical
+averaging) as T>0[13]. In both cases, we suppose
+/an}bracketle{tˆPa
+nˆPb
+m/an}bracketri}ht ≈ /an}bracketle{tˆPa
+n/an}bracketri}ht/an}bracketle{tˆPb
+m/an}bracketri}ht=µa(n)µb(m), (9)
+i.e. we neglect correlations between fluctuations of the
+quantum fields ˆPa
+n. Then we obtain the following system
+ofHamiltonian equations for the averages µa(n):
+/planckover2pi1dµa(n)
+dt=Cabcµb(n)∂/an}bracketle{tH/an}bracketri}ht
+∂µc(n)={µa(n),/an}bracketle{tH/an}bracketri}ht}.(10)
+Taking (9) into account we have /an}bracketle{tH/an}bracketri}ht=/an}bracketle{tHMF/an}bracketri}ht.
+In order to investigate large-scale fluctuations of the
+fieldµa(n)we take a two-dimensional continuum insteadof the discrete lattice. Such transition is well known
+for anSU(2) -magnet [14] and underlies the macroscopic
+phenomenological theory of magnetism [15]. In the
+continuous theory dynamical variables are densities of
+averaged spin and quadrupole moments:
+Ma(x) = lim
+S→01
+S/summationdisplay
+n∈Sµa(n)δx,xn=/summationdisplay
+n∈Sµa(n)δ(x−xn).
+Here,Sis a ‘physically’ infinitesimal region of the lattice,
+andδ(x−xn)is the Dirac delta-function, which has the
+dimension of reciprocal area. A Poisson bracket on the
+space of {Ma(x)}is calculated by the formula
+{Ma(x),Mb(y)}=CabcMc(x)δ(x−y).
+In what follows, we deal with the dimensionless field
+µa(x)=l2Ma(x), wherelis a distance between the
+nearest neighbors of the square lattice.
+Considering (j±1,k),(j,k±1)as the nearest
+neighbors of the site n=(j,k), in Eqs. (10) we perform a
+transition from the discrete variable xnto a continuous
+onex. Then the field {µa(x)}satisfies the equations
+/planckover2pi1∂µa(x)
+∂t={µa(x),Heff}=−CabcµbδHeff
+δµc,(11)
+where
+Heff=J/integraldisplay/summationdisplay
+a/parenleftig∂µa
+∂x/parenrightig2
+d2x.
+A suitable representation of the system of Hamilton
+equations (11) is the matrix equation
+∂ˆµ
+∂t=2Jl2
+/planckover2pi1[ˆµ,∆ˆµ], (12)
+where
+ˆµ=−i
+2/summationdisplay
+aµa(x)ˆPa.
+Obviously, ˆµis a Hermitian 3×3matrix, the bracket [·,·]
+denotes a matrix commutator, and ∆=∂
+∂x2+∂
+∂y2is the
+Laplace operator over a two-dimensional space. Equation
+(11) generalizes the Landau–Lifshits equation [16] for
+isotropic ferromagnets to the case of 8-component mean
+fieldµa(x). The Landau–Lifshits equation is well-known
+in the macroscopic theory of magnetism, and suitable for
+describing large-scale excitations in planar magnets and
+exploring the ferromagnetic resonance.
+1212 ISSN 2071-0186. Ukr. J. Phys. 2008. V. 53, N 12ORDERED STATES AND NONLINEAR EXCITATIONS
+5. Free Energy and Topological Charge
+As mentioned above, a Hamiltonian coincides with a free
+energy at constant temperature and volume. Therefore,
+we use the notion of free energy in what follows. As
+shown in Appendix (Section 6) by means of an algebraic
+approach, Eq. (12) coincides with the two-dimensional
+generalization of Eq. (25) on a degenerate orbit. This
+equation is a Hamiltonian one, though it is nonintegrable
+in the two-dimensional case; and its Hamiltonian can be
+used as an effective free energy for the spin system in
+question, namely:
+Feff
+1=2
+3h0/integraldisplay/integraldisplay/parenleftbig
+µ2
+a,x+µ2
+a,y/parenrightbig
+dxdy.
+Obviously, Feff
+1is a part of the total free energy of a
+magnet, and arises from an inhomogeneous distribution
+of the average values {µa(x)}.
+The algebraic approach yields one more equation of
+motion, associated with a generic orbit. Evidently, this
+equation can be obtained from the quantum-mechanicalapproach, by performing a relevant averaging that takes
+correlations into account. Therefore, we consider another
+effective free energy
+Feff
+2=1
+2(3f2
+0−h3
+0)/integraldisplay/integraldisplay/parenleftbig
+h2
+0(µ2
+a,x+µ2
+a,y)−
+−6f0(µa,xTa,x+µa,yTa,y)+3h0(T2
+a,x+T2
+a,y)/parenrightbig
+dxdy.
+Equations of extremals for the functionals of free energy
+are the two-dimensional generalization of Eqs. (25) and
+(26).
+If equations of constrains determining orbits are
+solved or, in other words, orbits are parameterized, then
+the formulas for the free energy can be simplified. Orbits
+of co-adjoint representation of semisimple compact Lie
+groups are compact K¨ ahlerian manifolds. Therefore,
+it is suitable to use a complex parameterization. In
+order to parameterize a generic orbit, it requires three
+complex variables w1,w2, dw3. Explicit formulas for the
+parameterization of a generic orbit are the following:
+µ1=m−√
+3q
+2√
+2·w2+w3+ ¯w2+ ¯w3
+1+|w2|2+|w3|2−m√
+2(1−w1)(¯w3−¯w1¯w2)+(1−¯w1)(w3−w1w2)
+1+|w1|2+|w3−w1w2|2,
+µ2=m−√
+3q
+2i√
+2·w3−w2−¯w3+ ¯w2
+1+|w2|2+|w3|2−im√
+2(1+w1)(¯w3−¯w1¯w2)−(1+ ¯w1)(w3−w1w2)
+1+|w1|2+|w3−w1w2|2,
+µ3=−m−√
+3q
+2·|w2|2−|w3|2
+1+|w2|2+|w3|2+m(1−|w1|2)
+1+|w1|2+|w3−w1w2|2,
+µ4=m−√
+3q
+2i·¯w2w3−w2¯w3
+1+|w2|2+|w3|2+im(w1−¯w1)
+1+|w1|2+|w3−w1w2|2,
+µ5=m−√
+3q
+2√
+2·w3−w2+ ¯w3−¯w2
+1+|w2|2+|w3|2−m√
+2(1+w1)(¯w3−¯w1¯w2)+(1+ ¯w1)(w3−w1w2)
+1+|w1|2+|w3−w1w2|2,
+µ6=m−√
+3q
+2i√
+2·w2+w3−¯w2−¯w3
+1+|w2|2+|w3|2+im√
+2(1−¯w1)(w3−w1w2)−(1−w1)(¯w3−¯w1¯w2)
+1+|w1|2+|w3−w1w2|2,
+µ7=m−√
+3q
+2·¯w2w3+w2¯w3
+1+|w2|2+|w3|2−m(w1+ ¯w1)
+1+|w1|2+|w3−w1w2|2,
+µ8=−m−√
+3q
+2√
+3·2−|w2|2−|w3|2
+1+|w2|2+|w3|2+m√
+3·1+|w1|2−2|w3−w1w2|2
+1+|w1|2+|w3−w1w2|2. (13)
+Here,mandqdenote boundary values of the quantities
+µ3andµ8, respectively. For a degenerate orbit, it is
+sufficient to have two complex variables, for instance w2
+andw3; in this case, we assign w1=0.
+After the restriction onto an orbit by formulas (13),the expressions for free energy get the form
+Feff=/integraldisplay/summationdisplay
+α,βgα¯β/parenleftbigg∂wα
+∂z∂¯wβ
+∂¯z+∂wα
+∂¯z∂¯wβ
+∂z/parenrightbigg
+dzd¯z,(14)
+wheregα¯βdenote components of a metrics on the orbit,
+ISSN 2071-0186. Ukr. J. Phys. 2008. V. 53, N 12 1213J.M. BERNATSKA, P.I. HOLOD
+and real coordinates x,yon a plane are changed into
+complex ones z,¯z.
+While coadjoint orbits are K¨ ahlerial manifolds they
+possess K¨ ahler potentials, which generate related metric s
+tensorgand 2-form h; their components are calculated
+by the formulas
+gα¯β=∂2Φ
+∂wα∂¯wβ, hα¯β=i∂2Φ
+∂wα∂¯wβ.
+A 2-form gives rise to a topological charge
+Q=1
+4π/integraldisplay/summationdisplay
+αβhα¯β/parenleftbigg∂wα
+∂z∂¯wβ
+∂¯z−∂wα
+∂¯z∂¯wβ
+∂z/parenrightbigg
+dz∧d¯z,
+which means a degree of mapping of a plane into an
+orbit, realized by the function w1,w2,w3.
+On a degenerate orbit of SU(3) the function
+Φ2= ln(1+ |w2|2+|w3|2)serves as a K¨ ahlerian
+potential, and the metrics tensor from (14) is a K¨ ahlerian
+one. Then the formulas for topological charge and free
+energy differ only in a sign (‘ +’ for a free energy, and
+‘−’ for a topological charge), hence,
+F[ξ]/greaterorequalslant4π|Q|.
+The equality holds if the second term in the brackets
+vanishes, that happens if the functions {wα}are
+holomorphic or antiholomorphic.
+Consequently, holomorphic functions form a class
+of solutions with quadrupole ordering ( m=0) that
+correspond to the minimum of free energy. The same
+takes place for antiholomorphic functions.
+Now we consider the case of a generic orbit.
+The cohomology class of rank 2 for the orbit
+is two-dimensional, then there exist two basis 2-
+forms, generated by the K¨ ahlerian potentials Φ2,
+andΦ1= ln(1+ |w1|2+|w3−w1w2|2). As a unique
+K¨ ahlerian potential we take the one corresponding to
+the Kirillov-Costant-Suriau form
+Φ =mΦ1−m−√
+3q
+2Φ2. (15)
+Generally speaking, the metrics tensor of free energy (14)
+is not K¨ ahlerian. However, we can construct a K¨ ahlerian
+metrics of the form
+Feff
+2=1
+2(3f2
+0−h3
+0)/integraldisplay/integraldisplay/parenleftig
+C1(µ2
+a,x+µ2
+a,y)−
++C2(µa,xTa,x+µa,yTa,y)+C3(T2
+a,x+T2
+a,y)/parenrightig
+dxdy,
+by choosing appropriate coefficients C1,C2,C3.Then all conclusions relative to a degenerate orbit
+can be extended to a generic one. That is, the class of
+solutions with ferromagnetic ordering that correspond to
+the minimum of free energy are realized by holomorphic
+(or antiholomorphic) functions.
+5.1. Large-scale topological excitations
+Excitations of a state with quadrupole ordering are
+described by spatially inhomogeneous distributions
+of the 8-component vector field µa(x), living on a
+degenerate orbit
+O(µ3= 0,µ8)≃SU(3)
+SU(2)×U(1).
+Mappings of topological charge 2 can be modeled by the
+holomorphic functions
+w2(z) =a1
+z−z1, w 3(z) =a2
+z−z2,
+wherea1,a2,z1, andz2are fixed complex numbers.
+The components µ3andµ8of the mean field,
+whose boundary values are called order parameters, are
+presented in Figs. 2 and 3 (we show a cut along the
+straight line joining poles of the functions w2(z)and
+w3(z), the origin of coordinates being at a middle of
+the interval z1z2). In Figs. 2 and 3, qis a value of the
+component µ8at an initial point of a degenerate orbit
+(µ3=0).
+q3
+2/c214
+/c45z1
+z2 z/c1093
+3
+2/c214q
+Fig. 2. Contour of µ3(z,¯z),µ3(∞)→0
+2/c45z1 z2 z8/c109
+q
+q
+1214 ISSN 2071-0186. Ukr. J. Phys. 2008. V. 53, N 12ORDERED STATES AND NONLINEAR EXCITATIONS
+Fig. 3. Contour of µ8(z,¯z),µ8(∞)→q
+These excitations are analogous to Belavin–Polyakov
+solitons, well-known for planar Heisenberg ferromagnets
+(the quantities a1anda2define widths of solitons,
+and the quantities z1,z2give their positions). It is
+easy to see that an energy of a configuration does
+not depend on a width of soliton, that proves scale
+invariance of the energy in the two-dimensional case.
+Hence, topological perturbations can have arbitrary
+large size. Such instability (an unrestricted increase of
+soliton width without pumping of energy) can cause a
+destruction of the nematic order in the considered model.
+6. Appendix. Integrability of SU(3) -Symmetric
+Equations of the Landau–Lifshits Type in a
+One-Dimensional Space
+It is known that system of equations (12) in the one-
+dimensional case is an integrable Hamiltonian system on
+a degenerate orbit of the group SU(3) [17]. We generalize
+Eq. (12) to the case of a generic orbit. The algebraic-
+geometric nature of equations like (12) is revealed in
+the frame of so called orbit approach to nonlinear
+Hamiltonian systems. Below, we construct integrable
+Hamiltonian equations on orbits of the group SU(3) .
+Consider polynomials in a complex variable λ,
+whose coefficients are anti-Hermitian matrices of the
+algebra su(3). We denote the set of polynomials by
+/tildewideg+≃su(3)⊗P(λ), whereP(λ)is a ring of polynomials
+with the standard multiplication operation. If A,B∈/tildewideg+
+have the form A(λ)=/summationtext
+nˆAnλn,B(λ)=/summationtext
+kˆBkλk,then
+[A, B] =/summationdisplay
+n,k[ˆAn,ˆBk]λn+k∈/tildewideg+. (16)
+Operation (16) shows a structure of graded Lie algebra in
+/tildewideg+. LetXn
+a=λnˆXabe a basis in /tildewideg+, whereˆXa=−i
+2ˆΛa,
+a=1,2,...,8,{ˆΛa}8
+a=1denote the Gell-Mann matrices.
+In/tildewideg+we introduce a bilinear ad-invariant form
+/an}bracketle{tA,B/an}bracketri}ht=−2resλ−N−2TrA(λ)B(λ), (17)
+whereN+1is the maximum degree of matrix
+polynomials AandB. LetM=(/tildewideg+)∗be a space dual to
+/tildewideg+with respect to form (17). A collection of the linear
+forms
+ξ(λ) =N/summationdisplay
+n=08/summationdisplay
+a=1ξn
+aλnˆXa+/parenleftbig
+ξ3ˆX3+ξ8ˆX8/parenrightbig
+λN+1creates a closed ad-invariant subspace MNinM. The
+coordinates ξn
+aofξ(λ)∈MNare calculated by the
+formula
+ξn
+a=/an}bracketle{tξ(λ),X−n+N+1
+a/an}bracketri}ht.
+In the linear space MNwith coordinates ξn
+a,
+n=0,1,...,N , we define the Lie–Poisson bracket
+{f1,f2}=/summationdisplay
+m,n8/summationdisplay
+a,bWmn
+ab∂f1
+∂ξma∂f2
+∂ξn
+b(18)
+with the Poisson tensor field
+Wmn
+ab=/an}bracketle{tξ(λ),[X−m+N+1
+a,X−n+N+1
+b]/an}bracketri}ht.
+We also define two ad-invariant functions I2(λ)andI3(λ)
+by the formulas
+I2(λ) =−2Trξ2(λ) =/summationdisplay
+aξ2
+a(λ),
+I3(λ) =−4iTrξ3(λ) =dabcξa(λ)ξb(λ)ξc(λ),
+Here,dabc=−2iTr(XaXbXc+XbXaXc), andξa(λ)is a
+polynomial
+ξa(λ) =ξ0
+a+ξ1
+aλ+ξ2
+aλ2+···+ξN+1
+aλN+1.
+The invariant functions are also polynomials in the
+complex parameter λ:
+I2(λ) =h0+h1λ+···+h2N+2λ2N+2,
+I3(λ) =f0+f1λ+···+f3N+3λ3N+3.
+It is easy to prove that the coefficients h0,...,hN+1,
+f0,... fN+1are annihilators of bracket (18). Fixing
+them we obtain the system of algebraic equations
+hn= const, fn= const, n= 0,...,N+1,(19)
+which give an embedding of the orbit ON+1of dimension
+6(N+ 1) into the linear space MN+1. The rest of
+coefficients {hN+2,..., h2N+2,fN+2,..., f3N+3}form
+a pairwise commutative collection of integrals of motion,
+which is necessary to integrate the Hamiltonian system.
+We are interested in the functions hN+2andhN+3and
+the corresponding Hamiltonian equations. In particular,
+the Hamiltonian hN+2gives rise to so-called stationary
+equations. In terms of the coordinates ξn
+a, they are
+∂ξn
+a
+∂x= 2fabcξ0
+cξn+1
+b, (20)
+where{fabc}are antisymmetric structure constants of
+the algebra su(3)in the basis of Gell-Mann matrices:
+[Xa,Xb] =fabcXc,
+ISSN 2071-0186. Ukr. J. Phys. 2008. V. 53, N 12 1215J.M. BERNATSKA, P.I. HOLOD
+f123= 1, f458=f786=√
+3
+2,
+f147=f165=f246=f257=f345=f376=1
+2,
+Byxwe denote a “time” with respect to the Hamiltonian
+hN+2, which corresponds to evolution equations
+∂ξn
+a
+∂t= 2fabc(ξ0
+cξn+2
+b+ξ1
+cξn+1
+b). (21)
+Equations (20) and (21) are consistent,
+because the corresponding Hamiltonians commute:
+{hN+2, hN+3}=0. Hence, we can determine evolution
+(21) on trajectories of system (20), i.e. we suppose the
+dynamical variables ξn
+ain Eq. (21) to be dependent on
+x. Combining (20) and (21), we have
+∂ξ0
+a
+∂t= 2fabcξ0
+cξ2
+b=∂ξ1
+a
+∂x. (22)
+The variables {ξ1
+a}can be expressed in terms of {ξ0
+a}
+and{∂
+∂xξ0
+a}, then (22) becomes a closed system of partial
+differential equations for {ξ0
+a}. For{ξ1
+a}it is necessary
+to solve the degenerate system of equations
+∂ξ0
+a
+∂x= 2fabcξ0
+cξ1
+b, (23)
+that becomes possible after restriction to an orbit
+ON+1⊂ MN+1.
+6.1. Classification of orbits
+It follows from Eqs. (19) that the orbit ON+1inMN+1
+has the structure of a vector bundle over a co-adjoint
+orbit of the group SU(3) . Hence, a classification of the
+orbitsON+1is reduced to that of orbits of SU(3) .
+Since the group SU(3) is simple, we have g∗=g.
+Therefore, we consider {ξ0
+a}also as coordinates in the
+algebrag≃su(3). Then a generic element ˆξ∈su(3)is
+represented by the matrix
+/hatwideξ=−i
+2
+ξ0
+3+1√
+3ξ0
+8ξ0
+1−iξ0
+2ξ0
+4−iξ0
+5
+ξ0
+1+iξ0
+2−ξ3+1√
+3ξ8ξ0
+6−iξ0
+7
+ξ0
+4+iξ0
+5ξ0
+6+iξ0
+7−2√
+3ξ8
+.(24)
+Letˆξ(0)be a fixed element of su(3). By definition,
+the set Oˆξ(0)={gˆξ(0)g−1,∀g∈SU(3)}is an orbit of
+SU(3) through the point ˆξ(0). Elements g′of the group
+SU(3) such thatg′ˆξ(0)g′−1=ˆξ(0)form the stationary
+subgroup at the point ˆξ(0). An orbit Oˆξ(0), being a
+homogeneous space, is a coset space SU(3)/G′≃Oˆξ(0),
+whereG′denotes the stationary subgroup.A maximal commutative subalgebra of a semisimple
+algebrag, which is called Cartan, can be diagonalized.
+The Cartan subalgebra of su(3)is formed from diagonal
+matrices depending on two parameters ξ0
+3andξ0
+8. It is
+well known that a proper transformation puts any
+elementˆξ∈ginto the Cartan subalgebra of g. This yields
+thateach orbit intersects the Cartan subalgebra at least
+once. In fact, there is more than one intersection point
+number, more precisely as many as an order of the Weyl
+groupW(G) . We discuss this in what follows.
+Nontrivial similarity transformations ˆξ→gˆξg−1that
+preserve a subalgebra hform a discrete subgroup
+W(G)⊂G, which is called a Weyl group [18]. An
+action of the group W(G) on the subalgebra h=h∗
+is generated by reflections in planes orthogonal to
+simple roots. The Weyl group of SU(3) is generated
+by two reflections σ1andσ2in the planes shown in
+Fig. 4 by dotted lines. The full Weyl group consists of
+six elements {e,σ1, σ2, σ1σ2,σ2σ1,σ1σ2σ1≃σ2σ1σ2},
+and is isomorphic to the group of permutations S3,
+ordS3=3!.
+12
+/c97/c97
+1/c1152/c115Weyl□chamber
+Fig. 4. Root diagram of the group SU(3)
+Since a Weyl group acts over a Cartan subalgebra,
+every point σˆξ(0)σ−1,σ∈W(G) , belongs to an orbit
+through ˆξ(0)∈h. The group W(G) acts efficiently
+(change an element ˆξinto another, if the element does
+not belong to reflection planes). An open domain in a
+Cartan subalgebra where a Weyl group acts efficiently, is
+called a Weyl chamber (see Fig. 4). Elements of different
+Weyl chambers are adjoint by elements σ∈W(G) , this
+implies that each orbit through a point ˆξ(0)of Weyl
+chamber intersects a Cartan subalgebra as many times
+as an order of W(G) . Ifˆξ(0)is an interior point point
+of a Weyl chamber, we call the orbit a generic one, and
+call the points σˆξ(0)σ−1,∀σ∈W(G) poles of an orbit .
+For a generic orbit a stationary subgroup G′
+coincides with a maximum torus Trfor a group G
+of rankr. In the case of group SU(3) , we have
+T2=U(1)×U(1). Hence, generic orbits are coset spaces
+Ogen≃SU(3)/U(1)×U(1).
+1216 ISSN 2071-0186. Ukr. J. Phys. 2008. V. 53, N 12ORDERED STATES AND NONLINEAR EXCITATIONS
+If an initial point ˆξ(0)belongs to a wall of a Weyl
+chamber (in the case of SU(3) , belongs to one of
+the reflection lines), then we deal with a degenerate
+orbit. In this case, a stationary subgroup G′contains
+a semisimple subgroup generated by roots orthogonal
+to an initial point ˆξ(0). Consider the group SU(3) .
+Ifˆξ(0)lies the vertical reflection line, α1and−α1
+are orthogonal to this element. The corresponding
+sl(2)-triple{Xα1, X−α1, Hα1=[Xα1, X−α1]}generates
+a subgroup SU(2)⊂SU(3) . Obviously, the element
+ˆξ(0)=−i
+2√
+3ξ0
+8diag(1,1,−2)is invariant under a
+transformation g′ˆξ(0)g′−1, whereg′is the unitary matrix
+g′=
+α β 0
+−β∗α∗0
+0 0 1
+
+eiϕ/20 0
+0eiϕ/20
+0 0 e−iϕ
+.
+Hence,g′∈SU(2)×U(1), and a degenerate orbit is a
+coset space Odeg≃SU(3)/SU(2)×U(1).
+6.2. Equations on orbits and their Hamiltonians
+Return to construction of integrable systems on orbits
+of loop groups, and consider Eq. (22). In order to solve
+this equation we have to restrict the degenerate system
+(23) into an orbit and solve.
+Ifξ0
+a∈Odeg, then the matrix 2fabcξ0
+chas rank 4, and
+its inversion gives a solution
+ξ1
+a=2
+3h0fabcξ0
+bξ0
+c,x+h1
+2h0ξ0
+a,
+where the constants h0,h1define an orbit by Eqs. (19).
+Ifξ0
+a∈Ogen, then the matrix 2fabcξ0
+chas rank 6 and
+is invertable on a generic orbit. Then we have
+ξ1
+a=1
+2(h3
+0−3f2
+0)/parenleftig
+h2
+0fabcξ0
+bξ0
+c,x+
++3h0fabcη0
+bη0
+c,x−6f0fabcξ0
+bη0
+c,x/parenrightig
++
++2f0f1−3h2
+0h1
+6(f2
+0−h3
+0)ξ0
+a+3f0h1−2h0f1
+6(f2
+0−h3
+0)η0
+a,
+whereη0
+a=dabcξ0
+bξ0
+c, and the constants h0,h1,f0,f1
+come from Eqs. (19).
+Substituting the obtained expressions in the right-
+hand side of (22), we get two equations for the functions
+ξa(x,t)≡ξ0
+a:
+∂ξa
+∂t=2
+3h0fabcξbξc,xx+h1
+h0ξa,x, ξa∈ Odeg,(25)
+∂ξa
+∂t=1
+2(h3
+0−3f2
+0)/parenleftig
+h2
+0fabcξbξc,xx−3f0fabcξbηc,xx++3h0fabcηbηc,xx−3f0fabcηbξc,xx/parenrightig
++
++2f0f1−3h2
+0h1
+6(f2
+0−h3
+0)ξa,x+3f0h1−2h0f1
+6(f2
+0−h3
+0)ηa,x, ξa∈ Ogen.(26)
+Leth1=0in Eq. (25), and replace the variables ξaby
+µa. Then its generalization to the two-dimensional case
+gets the form
+∂µa
+∂t=1
+6h0Cabcµb∆µc,
+where∆=∂2
+∂x2+∂2
+∂y2. Obviously, this equation has the
+Hamiltonian
+Heff=1
+12h0/integraldisplay/integraldisplay/parenleftbig
+µ2
+a,x+µ2
+a,y/parenrightbig
+dxdy.
+It is easy to see that (25) coincides with a one-
+dimensional analog of (12). In other words, Eq. (12) can
+be considered as a two-dimensional generalization of the
+integrable equation (25).
+In the same way we treat with Eq. (26), namely,
+replaceξabyµaand assign f1=h1=0. Generalized to
+two dimensions, the obtained equations get the form
+∂µa
+∂t=1
+8(h3
+0−3f2
+0)/parenleftig
+h2
+0Cabcµb∆µc−3f0Cabc/tildewideηb∆µc+
++3h0Cabc/tildewideηb∆/tildewideηc−3f0Cabcµb∆/tildewideηc/parenrightig
+. (27)
+Here,/tildewideηaare quadratic forms in µa:/tildewideηa=/tildewidedabcµbµc,
+where/tildewidedabc=1
+4Tr(PaPbPc+PbPaPc). Obviously, Eq.
+(27) is Hamiltonian, and give the following effective
+Hamiltonian
+Heff=1
+16(h3
+0−3f2
+0)/integraldisplay/integraldisplay/parenleftbig
+h2
+0µ2
+a,x+h2
+0µ2
+a,y−
+−6f0µa,x/tildewideηa,x−6f0µa,y/tildewideηa,y+3h0/tildewideη2
+a,x+3h0/tildewideη2
+a,y/parenrightbig
+dxdy.
+7. Conclusions
+In the present paper, we have constructed nonlinear
+stationary excitations appearing in the nematic phase
+of a planar magnet of spin s=1, modeled by a square
+lattice with a biquadratic interaction between nearest-
+neighbor sites. These excitations are characterized by
+an integer topological charge, and reveal themselves
+as regions with nonzero magnetization and mean
+quadrupole moment. Topological excitations in a two-
+dimensional system (without taking into account an
+anisotropy and a demagnetizing field) can increase
+unrestrictedly without pumping of energy. This destroys
+ISSN 2071-0186. Ukr. J. Phys. 2008. V. 53, N 12 1217J.M. BERNATSKA, P.I. HOLOD
+a nematic state in the system, according to the Mermin–
+Wagner theorem on absence of a long-range order in one-
+and two-dimensional systems.
+This work is partly supported by grant DFFD
+UkrF16/457-2007 and the grant of the International
+Charitable Fund for Renaissance of Kyiv-Mohyla
+Academy.
+1. L. Onsager, Phys. Rev. 65, 117 (1944).
+2. J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6, 1181 (1973).
+3. V.L. Berezinskii, Zh. Eksp. Teor. Fiz. 59, 907 (1971); Ibid. 61,
+1144 (1971).
+4. A.M. Tsvelik, Quantum Field Theory in Condensed Matter
+Physics (Cambridge Univ. Press, Cambridge, 1995).
+5. M. Blume and Y.Y. Hsien, J. Appl. Phys. 40, 1249 (1969).
+6. H.H. Chen and P.M. Levy, Phys. Rev. B 7, 4267 (1974).
+7. V.M. Matveev, Zh. Eksp. Teor. Fiz. 65, 1626 (1973).
+8. M. Nauciel-Bloch, G. Sarma, and A. Castets, Phys. Rev. B 5,
+4603 (1972).
+9. E.A. Harris and J. Owen, Phys. Rev. Lett. 11, 9 (1963).
+10. D.S. Rodbell, I.S. Jacobs, and J. Owen, Phys. Rev. Lett. 11,
+10 (1963).
+11. N.N. Bogolyubov and N.N. Bogolyubov (jr.), Introduction in
+Quantum Statistical Mechanics (Nauka, Moscow, 1984) (in
+Russian).
+12. K. Buchta, G. Fath, and O. Hegezand, Phys. Rev. B 72,
+054433 (2005).
+13. V.M. Loktev and V.S. Ostrovsky, Fiz. Nizk. Temp. 20, 983
+(1994).
+14. C. Herring and C. Kittel, Phys. Rev. 81, 869 (1951).15. V.G. Bar’yakhtar, V.N. Krivoruchko, and D.A. Yablonsky ,
+Green Functions in Magnetism Theory (Naukova Dumka,
+Kiev, 1984) (in Russian).
+16. L.D. Landau and E.M. Lifshits, Electrodynamics of
+Continuous Media (New York: Pergamon, 1984).
+17. P.I. Holod and O. Kisilevych, Integrable Dynamics of SU(3)
+Magnets , Preprint ITF-93-30U (Inst. Theor. Phys., Kiev,
+1993) (in Ukrainian).
+18. S. Helgason, Differential Geometry and Symmetric Spaces
+(Academic Press, New York, 1962).
+Received 02.04.08.
+Translated from Ukrainian by V.V. Kukhtin
+ВПОРЯДКОВАНI СТАНИ ТА НЕЛIНIЙНI
+ВЕЛИКОМАСШТАБНI ЗБУДЖЕННЯ
+У ПЛОСКОМУ МАГНЕТИКУ
+ЗI СПIНОМ s=1
+Ю.М. Бернацька, П.I. Голод
+Р е з ю м е
+Дослiджено впорядкованi стани та топологiчнi збудження
+у квазiдвовимiрному магнетику, змодельованому квадратно ю
+ґраткою зi спiнами s=1у вузлах та гамiльтонiаном з бiквад-
+ратною обмiнною взаємодiєю найближчих сусiдiв. Запропоно -
+вано два ефективних гамiльтонiани для опису великомасштаб -
+них збуджень у строго двовимiрному випадку. Один з них опи-
+сує збудження середнього поля в нематичнiй фазi, iнший /emdash.cyr у
+змiшанiй феромагнiтно-нематичнiй фазi. Показано, що ефек -
+тивнi гамiльтонiани мiнiмiзуються на конфiгурацiях, якi м ають
+фiксований топологiчний заряд. Цi топологiчнi збудження м о-
+жуть виникати при незначних температурах i бути причиною
+руйнування дальнього порядку у строго двовимiрнiй системi .
+1218 ISSN 2071-0186. Ukr. J. Phys. 2008. V. 53, N 12
\ No newline at end of file
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+arXiv:1001.0374v1  [hep-ex]  3 Jan 2010IHEP 2008-27
+OEF
+Observation of the destructive interference in
+the radiative kaon decay K−→µ−¯νγ
+O.G. Tchikilev, S.A. Akimenko, G.I. Britvich, A.P. Filin,
+A.S. Konstantinov, I.Y. Korolkov, V.M. Leontiev, V.F. Obra ztsov,
+V.A. Polyakov, V.I. Romanovsky, V.K. Semenov, V.A. Uvarov,
+O.P.Yushchenko
+Institute for High Energy Physics, Protvino, Russia
+V.N. Bolotov, V.A. Duk, A.I. Makarov, A.A. Khudiakov, V.P. N ovikov,
+A.Yu. Polyarush
+Institute for Nuclear Research, Moscow, RussiaAbstract
+Using data collected with the “ISTRA+” spectrometer at U70 p roton
+synchrotron of IHEP, we report the first measurement of the de structive
+interference in the radiative kaon decay K−→µ−νγ. We find the difference of
+the vector and axial form factors FV−FA= 0.126±0.027(stat)±0.043(syst).
+The measured value is two standard deviations above the O(p4) ChPT prediction
+equal to 0.055. Inclusion of exotic tensor interaction give sFV−FA= 0.144±
+0.044(stat)±0.035(syst)andFT=−0.0079±0.0113(stat)±0.0073(syst), i.e.
+−0.03< FT<0.01at90%CL, consistent both with zero and with recent
+theoretical prediction equal to |FT|= 0.022.
+1 Introduction
+The decay K−→µ−νγproceeds via two distinct mechanisms: the internal
+Bremsstrahlung (IB) with the photon emitted by the kaon or th e muon, and
+the structure-dependent(SD) decay involving emission of t he photon from
+intermediate states. SD is sensitive to the electroweak str ucture of the kaon
+and allows for good test of theories describing hadron inter actions and decays,
+like Chiral Perturbation Theory (ChPT)[1, 2]. This decay is also a good place
+for searches for possible tensor interactions, see theoret ical [3, 4, 5, 6, 7] and
+experimental [8, 9, 10, 11] papers, devoted mainly to the π→eνγdecay.
+The differential probability of the decay can be written in te rms ofx=2Eγ
+MK
+andy=2Eµ
+MK(whereMKis the kaon mass and EγandEµare the photon
+and muon energies in the kaon rest frame). In the most general case, which
+includes hypothetical tensor term it reads:
+dΓ
+dxdy=AIBfIB(x,y)+ASD[(FV+FA)2fSD+(x,y)+(FV−FA)2fSD−(x,y)]
+−AINT[(FV+FA)fINT+(x,y)+(FV−FA)fINT−(x,y)]+ATF2
+TfT(x,y)
+−AIBTFTfIBT(x,y)−ASDTFT(FV−FA)fSDT(x,y).
+fIB(x,y) = [1−y+r
+x2(x+y−1−r)][x2+2(1−x)(1−r)−2xr(1−r)
+x+y−1−r],(1)
+fSD+(x,y) = [x+y−1−r][(x+y−1)(1−x)−r], (2)
+1fSD−(x,y) = [1−y+r][(1−x)(1−y)+r], (3)
+fINT+(x,y) = [1−y+r
+x(x+y−1−r)][(1−x)(1−x−y)+r], (4)
+fINT−(x,y) = [1−y+r
+x(x+y−1−r)][x2−(1−x)(1−x−y)−r],(5)
+fT(x,y) = (x+y−1−r)(1+r−y) (6)
+fIBT= (1+r−x+y−1−r
+x−rx
+x+y−1−r) (7)
+fSDT=x(1+r−y) (8)
+wherer= (Mµ
+MK)2withMµbeing the muon mass and
+ASD= ΓKµ2α
+8π1
+r(1−r)2[MK
+FK]2, (9)
+AIB= 4r(FK
+MK)2ASD,AINT= 4r(FK
+MK)ASD,AT= 4ASD,ASDT= 4√rASD
+andAIBT= 8√r(FK
+MK)ASD.
+In these formulas FVandFAare the vector and axial form factors, FT
+is the tensor form factor. We use the prescription [3, 16] for the tensor
+interaction. αis the fine structure constant, F Kis the charged kaon decay
+constant ( 155.5±0.2±0.8±0.2MeV [13], and ΓKµ2is the width of the
+Kµ2decay. As in the paper[19] a minus sign precedes the interfer ence terms,
+thus changing the sign of the form factors. Without this chan ge in sign the
+formulas coincide with the ones given in [3, 16]. SD+and SD−refer to the
+terms with different photon polarizations and do not mutuall y interfere. Their
+interference with IB leads to the terms labeled INT+and INT−. The term
+SDT refers to SD−T interference, the interference with SD+is possible only
+in rather exotic models [4, 5, 6]. The interference between t he tensor and
+inner bremsstrahlung amplitudes is labeled as IBT.
+The x vs y plots for different terms are illustrated in Figs.1 a nd 2. The
+quadrangle area in these figures is restricted by the conditi onxb< x <
+xb+ 0.1for theyinterval 0.49/emdash.cyr1.0 , where xb= 1.0−0.5(y+√y2−4r)
+is the border of the Dalitz plot. This area, characterized by considerable
+contribution from INT−,SD−, IBT and SDT terms, and favourable background
+conditions is used in our analysis.
+Generally form factors can depend on q2= (pK−pγ)2=M2
+K(1−x). In
+our analysis we assume either constant form factors or the ph enomenological
+2dependence for the vector and axial form factors as in [19]: FV(q2) =FV(0)/(1−
+q2/M2
+V)andFA(q2) =FA(0)/(1−q2/M2
+A)withMV= 0.870GeV and MA=
+1.270GeV. The theoretical situation with q2dependence of the form factors
+is controversial. For O(p4)ChPT calculations [1] there is no q2dependence,
+O(p6)ChPT calculations [17, 18] predict nearly linear dependenc e, similar
+to the phenomenological one. At the same time calculations [ 17, 18] in the
+framework of the light-front quark model(LFQM) give quite d ifferent dependence
+with form factors going to zero with q2increase.
+We assume also the q2independence of the tensor form factor FT.
+The absolute value of the sum of the vector and axial form fact ors is known
+with high precision: |FV+FA|= 0.155±0.008[19], whereas the difference
+FV−FAis still poorly known. The latest measurent [19] gives for FV−FA
+only the 90% confidence level: −0.04< FV−FA<0.24, whereas the O(p4)
+ChPT prediction is equal to 0.055 [1, 2].
+The situation with the tensor contribution is controversia l. The measurements
+have been done mainly for pion decays[8, 9, 10, 11], The obser vation of
+the ISTRA experiment[8] was confirmed in the preliminary res ult by the
+PIBETA Collaboration[9, 10] having statistically signific ant deviation from
+the standard model without tensor term. Latest PIBETA data[ 11] have however
+eliminated the evidence for the tensor term and give the limi t:−0.00052<
+FT<0.0004at the 90% CL. The searches for tensor terms are done also in
+the studies of the angular distributions in the Gamow-Telle rβdecays of the
+spin polarized atoms, see for example [15] and the reference s therein. These
+studies are still less restrictive than πe2γdata.
+It is of interest to note that in some theoretical models [16] FTfor kaon
+decay is larger by a factor of 1/tan(θC)∼4.4as compared with pion decay.
+HereθCis the Cabibbo angle.
+2 Experimental setup and event selection
+The experiment is performed at the IHEP 70 GeV proton synchro tron U70.
+The ISTRA+ spectrometer has been described in detail in pape rs onKe3
+[20, 21], Kµ3[22, 23] and π−π◦π◦decays [24]. Here we recall briefly the
+characteristics relevant to our analysis. The ISTRA+ setup is located in
+the negative unseparated secondary beam line 4A of the U70. T he beam
+momentum is ∼25GeV/c with ∆p/p∼1.5%. The admixture of K−in the
+beam is∼3%, the beam intensity is ∼3·106per 1.9 sec U70 spill.
+3Figure 1: Dalitz plots for IB, SD+, SD−and INT−contributions. The vertical
+scale is logarithmic. The quadrangle area shows the region s tudied.
+4Figure 2: The same as in Fig.1 for INT+, T, IBT and SD−T terms.
+5During the physics run in November-December 2001 350 millio n trigger
+events were collected with high beam intensity. This inform ation is complemented
+by 124 M Monte Carlo (MC) events generated using Geant3 [25] f or the
+dominant K−decay modes, 100 M of them are the mixture of the dominant
+decay modes with the branchings exceeding 1 % and 24 M MC event s are
+the radiative Kµ2decays.
+Some information on the data processing and reconstruction procedures
+is given in [20, 21, 22, 23, 24], here we briefly mention the det ails relevant
+for present analysis.
+The muon identification (see [22, 23]) is based on the informa tion from
+the electromagnetic calorimeter SP 1and hadron calorimeter HC. The energy
+deposition in the SP 1is required to be compatible with the MIP signal in
+order to suppress charged pions and electrons. The sum of the signals in the
+HC cells associated with charged track is required to be comp atible with the
+MIP signal. The muon selection is further enhanced by the req uirement that
+the ratio r3of the HC energy in last three layers to the total HC energy
+exceeds 5 %. The used cut values are the same as in [23].
+Events with one reconstructed charged track and one reconst ructed shower
+in the calorimeter SP 1are selected.
+A set of cuts is developed to suppress various backgrounds an d/or to do
+data cleaning:
+0) We select events with good charged track having two recons tucted
+(x−zandy−z) projections and the number of hits in the matrix hodoscope
+MH below 3.
+1) Events with the reconstructed vertex inside the interval 400< z <
+1600cm are selected.
+2) The measured missing energy Emis=Ebeam−Eµ−Eγis required to
+be above zero.
+3) The events with missing momentum pointing to the SP 1working
+aperture are selected in order to suppress π−π◦background ( r >10cm,
+hereris the distance between the impact point of the missing momen tum
+and the SP 1center in the SP 1transverse plane).
+4) We require also the absence of the signal above the thresho ld in the
+calorimeter SP 2and the guard veto system GS.
+5) In order to suppress the remaining K π2contribution at yaround 0.8 we
+use the cut cos(φ(µγ))>−0.95whereφ(µγ)is the angle between transverse
+momenta of the muon and the photon.
+We look for the signal in the distributions over the effective mass M(µ−γν),
+6whereνfour-momentum is calculated using the measured missing mom entum
+and assuming m ν= 0. Effective mass spectra for the quadrangular region in
+Figs. 1 and 2 are shown in Figs. 3,4 and 5 for y-interval 0.49/emdash.cyr1.00 with the
+stepδy= 0.03.
+The effective mass spectra have been parametrized by the sum o f the
+signal and of the background. The signal form have been found from the
+signal Monte Carlo events parametrized by the sum of two Gaus sians. The
+background have been found using the histogram smoothing of the MC
+background mass spectra by the HQUAD routine from the HBOOK p ackage [26].
+This background does not ideally describe the real data, esp ecially at low
+effective masses. This discrepance has been taken into accou nt by addition
+of the term
+(P(4)+P(5)∗M)∗exp(−P(6)∗M) (10)
+to the MC background. It should be noted that the contributio n of this term
+in the signal regionis rather small.
+First parameter of the fit gives the number of events in the kao n peak,
+second – the position of the peak, third – normalization of th e MC background.
+At small ythe signal is rather small, at large y, especially in the IB
+region, it dominates over the background. The peak at the effe ctive mass
+0.43–0.45 GeV, seen in the histogram o) is the reflection of th e remaining
+Kπ2background.
+The resulting event distribution in the interval 0.49< y <1.00have been
+parametrized by function constructed using Phase Space sig nal MC events,
+weighted with corresponding terms (1)-(8) calculated usin g “true” MC xand
+yvalues in the corresponding δyintervals. The results of the fit without
+tensor component are shown in Fig.6. Here the first parameter isFV+FA
+fixed at the value 0.155 taken from [19], the second parameter isFV−FA,
+the third parameter is the tensor form factor and the fourth p arameter is
+the normalization factor. In fact, the fit results are insens itive to the value
+FV+FAsince the SD+and INT+contributions are negligible, see Fig.6.
+The fit is not perfect, but satisfactory, χ2/NDF= 34.11/(17−2)with
+around three sigma deviation from the expected χ2andFV−FA= 0.126±
+0.027.
+Several sources of the systematic uncertainty have been stu died. The
+uncertainty given by poor knowledge of the background shape was obtained
+in two ways. First, by rescaling of the errors in the effective mass distributions
+in order to have χ2equal to one in each bin and refitting then. Second, by
+7Figure 3: Effective mass m (µ−νγ) spectra for the y-interval 0.49/emdash.cyr0.67 with
+the step δy= 0.03.
+8Figure 4: Effective mass m (µ−νγ)spectra for the y-interval 0.67/emdash.cyr0.85 with
+the step δy= 0.03.
+9Figure 5: Effective mass m (µ−νγ)spectra for the y-interval 0.85-1.00 with
+the step δy= 0.03.
+10Figure 6: Results of fit of the event distribution with FT= 0.
+11using polynomial parametrization instead of the form (10). First method
+leads to the deviation in the FV−FAequal to 0.0104, second /emdash.cyr to the
+0.0086, the maximum of these two values was used as the contri bution in the
+systematic uncertainty. The possible contribution from th ezvertex position
+was obtained using increased zinterval with maximum zequal to 1850 cm.
+This uncertainty is equal to 0.0329. The variation in the muo n selection
+criteria gives the value equal to 0.0167. And the fit in the diff erentyintervals
+gives the deviation equal to 0.0191. Adding the individual e rrors in quadrature
+we find a total systematic error of 0.0428.
+We have tried also the fit with non-zero tensor form factor. Th eχ2/NDF=
+31.97/(17−3),FV−FA= 0.144±0.044±0.035andFT=−0.0079±0.011±
+0.007.
+The fit with constant vector and axial form factors gives: χ2/NDF=
+33.4/(17−2),FV−FA= 0.146±0.028±0.046withFT= 0andχ2/NDF=
+31.29/(17−3),FV−FA= 0.136±0.060±0.045andFT= 0.006±0.035±0.007.
+3 Conclusions
+Our conclusions are as follows: The study of the radiative ka on decay K−
+µ2γin
+a new region where SD−and INT−terms have maximum gives the value FV−
+FA= 0.126±0.027(stat)±0.043(syst). This value is 1.4 standard deviations
+aboveO(p4)ChPT prediction, equal to 0.055, and probably indicates the
+need for higher order calculations or for more elaborate ana lysis of the q2
+dependence of the form factors.
+The inclusion of the tensor component gives: FV−FA= 0.144±0.044±
+0.035. andFT=−0.0079±0.011±0.007, i.e.−0.03< FT<0.01at 90%CL.
+The work is supported in part by the RFBR grant N07-02-00957( IHEP
+group).
+12References
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+14
\ No newline at end of file
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+arXiv:1001.0375v1  [cond-mat.mes-hall]  3 Jan 2010Topological excitations in 2D spin system with
+high spin s/greaterorequalslant1
+Julia Bernatska, Petro Holod
+Abstract
+We construct a class of topological excitations of a mean fiel d in a two-
+dimensional spin systemrepresented byaquantumHeisenber gmodel with
+high powers of exchange interaction. The quantum model is as sociated
+with a classical one (the continuous classical analogue) th at is based on
+a Landau-Lifshitz like equation, and describes large-scal e fluctuations of
+the mean field. On the other hand, the classical model is a Hami ltonian
+system on a coadjoint orbit of the unitary group SU(2 s+1) in the case
+of spins. We have found a class of mean field configurations that can be
+interpreted as topological excitations, because they have fixed topological
+charges. Such excitations change their shapes and grow pres erving an
+energy.
+1 Introduction
+According to Mermin and Wagner [1] there is no ferromagnetic or an tiferromag-
+netic order in the one- and two-dimensional isotropic Heisenberg mo dels with
+interactions of finite range at nonzero temperature. This statem ent is proven
+duetoBogoliubov’sinequalityinthe generalcase. Hereweconstruc texcitations
+thatcauseadestructionofalong-rangenematicormixedferroma gnetic-nematic
+order. This is an extension of the results of Belavin and Polyakov [2].
+We model a planar magnet by a square atomic lattice with the same spin s
+at each site. We describe this two-dimensional spin system by a gene ralized
+Heisenberg Hamiltonian, taking into account high powers of the exch ange inter-
+action (ˆSn,ˆSn+δ), where ˆSnis a vector of spin operators at site n. By a mean
+field approximation we obtain a classical long-range equation from th e quantum
+Heisenberg one.
+An equation for a mean field (the field of magnetization and multipole mo -
+ments) is a Hamiltonian equation on a coadjoint orbit of Lie group. At t he
+same time, this is a generalization of the well-known Landau-Lifshitz e quation
+for a magnetization field. In this context we obtain effective Hamilton ians for
+the magnetic system in question. Using K¨ ahlerian structure of coa djoint orbits,
+we construct effective Hamiltonians such that their minimums are pro portional
+to topological charges of excitations. In addition, we produce the se mean field
+configurations that give minimums to the Hamiltonians.
+12 Quantum and classical models
+As mentioned above, we represent the spin system by a planar atom ic lattice
+with the same spin sat all sites. We assign three spin operators ( ˆS1
+n,ˆS2
+n,ˆS3
+n) =
+ˆSnto each atom n; the operators obey the standard commutation relations
+[ˆSa
+n,ˆSb
+m] =iεabcˆSc
+nδnm,
+wherea, b, crun over the values {1,2,3}, andδnmis the Kronecker symbol.
+2.1 Generalized Heisenberg Hamiltonians
+We are interested in so called high spins s/greaterorequalslant1. In this case, we can describe the
+system by the following bilinear-biquadratic Hamiltonian
+ˆH2=−/summationdisplay
+n,δ/parenleftBig
+J(ˆSn,ˆSn+δ)+K(ˆSn,ˆSn+δ)2/parenrightBig
+.
+Hereδruns over the nearest-neighbour sites, nruns over all sites of the lattice,
+constants JandKdenote exchange integrals. In the case of spin s/greaterorequalslant3/2, the
+above Hamiltonian can include also the bicubic exchange, namely
+ˆH3=−/summationdisplay
+n,δ/parenleftBig
+J(ˆSn,ˆSn+δ)+K(ˆSn,ˆSn+δ)2+L(ˆSn,ˆSn+δ)3/parenrightBig
+,
+whereLdenotes the corresponding exchange integral.
+One can easily write a generalized Heisenberg Hamiltonian for the syst em of
+anarbitraryspin sorgreaterthan s. ThisHamiltoniancontainsallpowersofthe
+exchange interaction up to 2 s. It can be reduced to a bilinear form if one takes
+the 2s+1-dimensional space of irreducible representation of the group S U(2).
+The spin operators {ˆSa
+n}over this space generate a complete associative matrix
+algebra, which has a sufficient number of operators to reduce the c orresponding
+Hamiltonian to a bilinear form.
+For example, in the case of spin s=1 the appropriate space of representation
+is 3-dimensional, and we choose a canonical basis in the form: {|+1/an}bracketri}ht,|−1/an}bracketri}ht,|0/an}bracketri}ht}.
+The spin operators {ˆSa
+n}generate the algebra Mat 3×3. In order to form a basis
+in the algebra we take the tensor operators of weight 2
+ˆQab
+n=ˆSa
+nˆSb
+n+ˆSb
+nˆSa
+n, a/ne}ationslash=b,
+ˆQ22
+n= (ˆS1
+n)2−(ˆS2
+n)2,ˆQ20
+n=√
+3/parenleftbig
+(ˆS3
+n)2−2
+3/parenrightbig (1)
+inadditiontothespinoperators. Theintroducedoperatorsareca lledquadrupole
+operators .
+In the case of spin s=3/2 the appropriate space of representation is 4-
+dimensional, and {|+3
+2/an}bracketri}ht,|+1
+2/an}bracketri}ht,|−1
+2/an}bracketri}ht,|−3
+2/an}bracketri}ht}is a canonicalbasis. We complete the
+associative matrix algebra Mat 4×4of{ˆSa
+n}by the tensor operators of weights 2
+2and 3, defining them by the following formulas:
+ˆQab
+n=√
+5
+2√
+3/parenleftBig
+ˆSa
+nˆSb
+n+ˆSb
+nˆSa
+n/parenrightBig
+, a/ne}ationslash=b
+ˆQ22
+n=√
+5
+2√
+3/parenleftBig
+(ˆS1
+n)2−(ˆS2
+n)2/parenrightBig
+,ˆQ20
+n=√
+5
+2/parenleftbig
+(ˆS3
+n)2−5
+4/parenrightbig
+,(2)
+ˆTa3
+n= (ˆQa2
+nˆS3
+n+ˆS3
+nˆQa2
+n), a,b∈ {1,2}, a/ne}ationslash=b,
+ˆTab
+n=1√
+6/parenleftBig
+(ˆSa
+n)2ˆSb
+n+ˆSb
+n(ˆSa
+n)2+ˆSa
+nˆSb
+nˆSa
+n−(ˆSb
+n)3/parenrightBig
+,
+ˆT3a
+n=1√
+10/parenleftBig
+ˆQa3
+nˆS3
+n+ˆS3
+nˆQa3
+n+√
+3(ˆQ20
+nˆSa
+n+ˆSa
+nˆQ20
+n)/parenrightBig
+,
+ˆT30
+n=1
+12/parenleftbig
+41ˆS3
+n−20(ˆS3
+n)3/parenrightbig
+.(3)
+We call the tensor operators of weight 3 sextupole operators . In what follows we
+denote all tensor operators over the chosen space of represen tation by {ˆPa
+n}.
+In terms of representation operators a generalized Heisenberg H amiltonian
+gets a bilinear form. For the Hamiltonians considered above we have:
+ˆH2=−(J−1
+2K)/summationdisplay
+n,δ/summationdisplay
+bˆSb
+nˆSb
+n+δ−1
+2K/summationdisplay
+n,δ/summationdisplay
+αˆQα
+nˆQα
+n+δ−4
+3KN;
+ˆH3=−(J−1
+2K+587
+80L)/summationdisplay
+n,δ/summationdisplay
+bˆSb
+nˆSb
+n+δ−75
+32(4K−L)N−
+−6
+5(K−2L)/summationdisplay
+n,δ/summationdisplay
+αˆQα
+nˆQα
+n+δ−9
+10L/summationdisplay
+n,δ/summationdisplay
+βˆTβ
+nˆTβ
+n+δ,
+whereNis the overall number of sites in the lattice. Obviously, the obtained
+bilinear Hamiltonians are SU(2)-invariant, because they are constr ucted from
+representation operators of the group SU(2).
+2.2 Mean field approximation
+Here a mean field is a field of expectation values for the operators {ˆPa
+n}cal-
+culated after spontaneous breaking of symmetry. The breaking o f symmetry is
+performed by switching on an external magnetic field, which specifie s an order
+in the system; then the external field vanishes. Such kind of avera ges is also
+calledquasiaverages [3].
+We denote a mean field averaging by /an}bracketle{t·/an}bracketri}ht, and components of the mean field
+by{µa(xn)=/an}bracketle{tˆPa
+n/an}bracketri}ht}. A mean field approximation of the bilinear-biquadratic
+Hamiltonian has the form
+ˆH2
+MF=−(J−1
+2K)z/summationdisplay
+n3/summationdisplay
+a=1ˆPa
+nµa(xn)−1
+2Kz/summationdisplay
+n8/summationdisplay
+a=4ˆPa
+nµa(xn)−4
+3KzN,
+wherezis a number of the nearest-neighbour sites.
+Evidently, a mean field Hamiltonian remains SU(2)-invariant. Then by a n
+action of the group SU(2) it can be reduced to a diagonal form. Of c ourse,
+3this reduction is possible only in the case of thermodynamical equilibriu m and
+an infinite lattice, when the mean field becomes constant and the dep endance
+on sitencan be omitted. Moreover, almost all components of the mean field
+vanish, except the components corresponding to diagonal opera tors of{ˆPa
+n}.
+For the bilinear-biquadratic Hamiltonian a diagonal form is the following :
+ˆH2
+MF=−zN/parenleftBig
+(J−1
+2K)ˆS3µ3+1
+2KˆQ20µ8+4
+3K/parenrightBig
+.
+The remaining components are suitable to serve as order parameters . The com-
+ponentµ3describes a normalized magnetization (a ratio of the z-projection of
+magnetic moment to a saturation magnetization). The components µ8andµ15
+are normalized projections of quadrupole andsextupole moments , respectively.
+Possible values of order parameters can be obtained from the self-consistent
+equations
+µa=/an}bracketle{tˆPa/an}bracketri}htMF=TrˆPae−ˆhMF
+kT
+Tre−ˆhMF
+kT
+for all diagonal operators ˆPa. Here we use the density matrix with the one-site
+Hamiltonian ˆhMF=ˆHMF/N. Note, that for the standard Heisenberg Hamilto-
+nian, when only the spin operators are considered, a self-consiste nt equation
+turns into the well-known Weiss equation.
+In the case of bilinear-biquadratic Hamiltonian, an analysis of self-co nsistent
+equations gives the following. We adduce probable values of order pa rameters
+in the limit T→0 asJ, K>0: 1)|µ3|=1, µ8=2J−K√
+3K; 2)|µ3|=1
+2, µ8=J−K/2√
+3K;
+3)µ3=0,|µ8|=2√
+3; 4)µ3=0,|µ8|=1√
+3. Two of the solutions correspond
+to states with the ferromagnetic order, because they have a non zero magnetiza-
+tionµ3. The other two solutions correspond to states with the quadrupo le order
+(so called nematic states), which are states with zero magnetizatio n. Solutions
+2) and 4) are unstable and called partly ordered. If Kbecomes negative, only
+solution 1) remains. These results accord with the results of [4, 5] a nd with the
+phase diagram of ordered states in a one-dimensional spin system f rom [6].
+In what follows we consider an SU(3)-invariant system with the bilinea r-bi-
+quadratic Hamiltonian. The SU(3)-invariance is reached by assigning J=K;
+this is the boundary line between the ferromagnetic and the nematic regions
+(see [6]). It means the system can appear in the both states. In th e case of
+Hamiltonian ˆH3,themaximalSU(4)-invarianceisreachedas J=−81
+44K=−81
+16L,
+that is located within the ferromagnetic region.
+Now we apply the mean field averagingto the quantum Heisenberg equ ation
+i/planckover2pi1dˆPa
+n
+dt= [ˆPa
+n,ˆH]. (4)
+The averaging is performed with the assumption of zero correlation s between
+fluctuations of {ˆPa
+n}at distinct sites: /an}bracketle{tˆPa
+nˆPb
+m/an}bracketri}ht=/an}bracketle{tˆPa
+n/an}bracketri}ht/an}bracketle{tˆPb
+m/an}bracketri}ht. Then we take a
+large-scale limit and obtain the Landau-Lifshitz like equation
+/planckover2pi1∂µa
+∂t= 2Jl2Cabcµb(µc,xx+µc,yy), (5)
+4whereCabcare structure constants of the Lie algebra of {ˆPa
+n}with the commu-
+tation relations [ ˆPa
+n,ˆPb
+m] =iCabcˆPc
+nδnm. Equation (5) is an equation of motion
+for the mean field {µa(x)}over the plain {x= (x,y)|x, y∈R}that replaces
+the lattice.
+InthecaseofstandardHeisenbergHamiltonian(5)coincideswithth eLandau-
+Lifshitz equation for an isotropic magnet. Therefore, in the gener al case we call
+(5) ageneralized Landau-Lifshitz equation for the vector field {µa}. The vector
+field has 8 components if one exploits the bilinear-biquadratic Hamilton ian, and
+15 components for the Hamiltonian with the bicubic exchange.
+2.3 Effective Hamiltonians on coadjoint orbits
+The generalized Landau-Lifshitz equation (5) can be interpreted a s aHamilto-
+nian equation on a coadjoint orbit of Lie group . In the case of spin s=1 we deal
+with the group SU(3), in the case of an arbitrary spin sthe group is SU(2 s+1).
+Note, that the matrices {ˆPa}serve as a basis in the corresponding Lie algebra
+su(2s+1), and components of the mean field {µa}serve as coordinates in the
+dual space to su(2s+1).
+We start with brief description of the groups SU(3) and SU(4). For more
+material see, in particular, [7, 8].
+The group SU(3) has two types of orbits: the genericSU(3)
+U(1)×U(1)of dimen-
+sion 6, and the degenerateSU(3)
+SU(2)×U(1)of dimension 4. Each orbit of SU(3) is
+defined by two numbers mandq, which are values of the coordinates µ3and
+µ8at an initial point (a point in the positive Weyl chamber). Simultaneous ly,
+these numbers are limiting values of the mean field components µ3andµ8at
+zero temperature. For a degenerate orbit one has to assign m=0, orm=√
+3q.
+Evidently the degenerate orbits with m=0 are domains of mean field configura-
+tions that realize nematic states. Ferromagnetic states are realiz ed on all other
+orbits of SU(3).
+The group SU(4) has four types of orbits: the genericSU(4)
+U(1)×U(1)×U(1)of
+dimension 12, the degenerateSU(4)
+SU(2)×U(1)×U(1)of dimension 10, the degenerate
+SU(4)
+S(U(2)×U(2))of dimension 8, and the maximal degenerateSU(4)
+SU(3)×U(1)of dimen-
+sion 6. Each orbit is defined by numbers m,q,p, which are limiting values of the
+mean field components µ3,µ8,µ15. Almost all orbits are domains of ferromag-
+netic mean field configurations. Nematic states are realized on the d egenerate
+orbits of dimension 8 as m=p=0 andqis arbitrary. So it is probable to reveal
+a nematic state even in the ferromagnetic region of the phase diagr am [6].
+As shownabove, limiting values of diagonal components of a mean field serve
+as order parameters. Simultaneously, they define a coadjoint orbit where the
+corresponding mean filed configuration lives.
+Each orbit possesses a Hamiltonian system with an equation of motion and
+a group-invariant Hamiltonian. For a degenerate orbit of SU(3) the equation is
+∂µa
+∂t=4A
+3(m2+q2)Cabcµb(µc,xx+µc,yy), (6)
+5whereAdenotes a dimensional constant. The values m,qof order parameters
+(a magnetization and a projection of quadrupole moment) define an orbit via
+the following equations, which we call constraints:
+δabµaµb=m2+q2
+dabcµbµc=±/radicalBig
+m2+q2
+5µa.
+Heredabc=√
+3
+4√
+5Tr(ˆPaˆPbˆPc+ˆPbˆPaˆPc) is a symmetric tensor. The corresponding
+SU(3)-invariant Hamiltonian is the following
+Hdeg
+eff=2J
+3(m2+q2)/integraldisplay8/summationdisplay
+a=1/parenleftBig
+(µa,x)2+(µa,y)2/parenrightBig
+dxdy, (7)
+where the dimensional constant Jhas a meaning of exchange integral.
+Obviously, equations (6) and (5) coincide. It is easy to show, that ( 5) coin-
+cides with the equation of motion on a maximal degenerate orbit. Rec all, that
+(5) is obtained when correlations between fluctuations of {ˆPa
+n}are neglected.
+Presumably, equations of motion on other orbits are derived from ( 4) via the
+mean field averaging with more complicate correlation rules.
+A generic orbit of SU(3) is determined by the following equations:
+δabµaµb=m2+q2
+dabcµaµbµc=1√
+5q(3m2−q2).
+The SU(3)-invariant Hamiltonian on this orbit has the form
+Hgen
+eff=J
+2m2(m2−3q2)28/summationdisplay
+a=1/parenleftBig
+(m2+q2)2(µa,x)2+(m2+q2)(ηa,x)2−
+−2√
+3q(3m2−q2)µa,xηa,x/parenrightBig
+,(8)
+whereηais a quadraticform in {µa}:ηa=√
+5dabcµbµc. For more details see [9].
+TheHamiltoniansystemsoncoadjointorbitsofSU(3)serveas classical effec-
+tive models for the spin system of s/greaterorequalslant1 with biquadratic exchange. Evidently,
+these models describe large-scale (or slow) fluctuations of the mea n field. In
+this paper we suppose that the order parameters m,qare fixed numbers. But
+generally speaking, they depend on a temperature Tand the interaction con-
+stantJ. Taking into account these dependencies, one can consider small-s cale
+(or quick) fluctuations of the mean field.
+2.4 Geometrical properties of effective Hamitonians
+Each coadjoint orbit of a semisimple Lie group is a homogeneous space that ad-
+mits a K¨ ahlerian structure. Thus one can introduce a complex para meterization
+of an orbit. For this purpose we use a generalized stereographic projection (for
+6more details see [8]). In the case of group SU(3), the projection is r epresented
+by the following formulas:
+µa=−m−√
+3q
+2ζa+mξa, ηa=√
+3(m2−q2)−2mq
+2ζa+2mqξa,
+ζ1=−w2+ ¯w2+w3+ ¯w3√
+2(1 +|w2|2+|w3|2)ξ1=−(1−¯w1)(w3−w1w2) + (1−w1)(¯w3−¯w1¯w2)√
+2(1 +|w1|2+|w3−w1w2|2))
+ζ2=iw3−¯w3−w2+ ¯w2√
+2(1 +|w2|2+|w3|2)ξ2=i(1 + ¯w1)(w3−w1w2)−(1 +w1)(¯w3−¯w1¯w2)√
+2(1 +|w1|2+|w3−w1w2|2)
+ζ3=|w2|2− |w3|2
+1 +|w2|2+|w3|2ξ3=1− |w1|2
+1 +|w1|2+|w3−w1w2|2
+ζ4=i¯w2w3−w2¯w3
+1 +|w2|2+|w3|2ξ4=iw1−¯w1
+1 +|w1|2+|w3−w1w2|2(9)
+ζ5=w2+ ¯w2−w3−¯w3√
+2(1 +|w2|2+|w3|2)ξ5=−(1 + ¯w1)(w3−w1w2) + (1 + w1)(¯w3−¯w1¯w2)√
+2(1 +|w1|2+|w3−w1w2|2)
+ζ6=iw2−¯w2+w3−¯w3√
+2(1 +|w2|2+|w3|2)ξ6=i(1−¯w1)(w3−w1w2)−(1−w1)(¯w3−¯w1¯w2)√
+2(1 +|w1|2+|w3−w1w2|2)
+ζ7=−¯w2w3+w2¯w3
+1 +|w2|2+|w3|2ξ7=−w1+ ¯w1
+1 +|w1|2+|w3−w1w2|2
+ζ8=2− |w2|2− |w3|2
+√
+3(1 +|w2|2+|w3|2)ξ8=1 +|w1|2−2|w3−w1w2|2
+√
+3(1 +|w1|2+|w3−w1w2|2).
+The coordinates {w1, w2, w3}(Bruhat coordinates according to [7]) parameter-
+ize a generic orbit of SU(3). In the case of a degenerate orbit, one has to assign
+m=0 andw1=0, orm=√
+3qandw2=0.
+In terms of {wα}the effective Hamiltonians (7) and (8) have the form
+Heff=J/integraldisplay/summationdisplay
+α,βgα¯β/parenleftBig∂wα
+∂z∂¯wβ
+∂¯z+∂wα
+∂¯z∂¯wβ
+∂z/parenrightBig
+dzd¯z, α, β = 1,2,3,(10)
+wherez=x+iyis a complex coordinate on the plane obtained from the atomic
+lattice after a large-scale limiting process (see Section 2.2). The ten sorgis non-
+degenerate and positively defined, thus it can serve as a metrics on an orbit.
+Its components {gα¯β}come from (7) for a degenerate orbit, and from (8) for a
+generic one. In terms of the auxiliary vector fields {ζa}and{ξa}we have
+gdeg1
+α¯β=1
+2/summationdisplay
+a∂ζa
+∂wα∂ζa
+∂¯wβ, gdeg2
+α¯β=1
+2/summationdisplay
+a∂ξa
+∂wα∂ξa
+∂¯wβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+w2=0,
+ggen
+α¯β=1
+2/summationdisplay
+a/parenleftbigg∂ζa
+∂wα∂ζa
+∂¯wβ−∂ζa
+∂wα∂ξa
+∂¯wβ+∂ξa
+∂wα∂ξa
+∂¯wβ/parenrightbigg
+.
+Note, that in terms of {wα}the tensor gdoes not depend on a particular orbit.
+BeingaK¨ ahlerianmanifoldanorbitofSU(3)possessesaK¨ ahlerianp otential.
+For this purpose we use a potential Φ of the Kirillov-Kostant-Suoria u form:
+Φ =mΦ1−m−√
+3q
+2Φ2,
+Φ1= ln(1+ |w1|2+|w3−w1w2|2),Φ2= ln(1+ |w2|2+|w3|2),
+7A topological structure of the orbit is characterized by the secon d cohomology
+groupH2of dimension 2. That is why there exist two basis2-forms, forexamp le
+generated by the potentials Φ 1, Φ2. Each of them defines a topological charge
+Qk=1
+4π/integraldisplay/summationdisplay
+α,βi∂2Φk
+∂wα∂¯wβ/parenleftBig∂wα
+∂z∂¯wβ
+∂¯z−∂wα
+∂¯z∂¯wβ
+∂z/parenrightBig
+dz∧d¯z, k= 1,2.
+On a degenerate orbit only one potential is governing, and only one t opolo-
+gical charge exists. Then the expressions for QkandHdegk
+effdiffer only in a sign.
+Evidently,
+Hdegk
+eff/greaterorequalslant4πJ|Qk|. (11)
+Hence,on a degenerate orbit a minimum of Heffis realized if the equality holds,
+that takes place if {wα}are holomorphic or antiholomorphic functions . Here we
+use an idea of Belavin and Polyakov [2].
+For a generic orbit we define a topological charge by Q=Q1+Q2. In order
+to extend inequality (11) to generic orbits with the topological char geQ, we
+construct an effective Hamiltonian by the following formula:
+Hgen
+eff=J
+2m2(m2−3q2)28/summationdisplay
+a=1/parenleftBig
+C1(µa,x)2+C2(ηa,x)2+C3µa,xηa,x/parenrightBig
+,(12)
+C1=m4+q4−4√
+3mq(q2−m2)+14m2q2,
+C2=5
+3m2+q2−2√
+3mq,
+C3=2√
+3m3+2q3−26
+3m2q+2√
+3mq2.
+In terms of {wα}it is reduced to the form (10) with the metrics
+ggen
+α¯β=1
+2/summationdisplay
+a/parenleftbigg∂ζa
+∂wα∂ζa
+∂¯wβ+∂ξa
+∂wα∂ξa
+∂¯wβ/parenrightbigg
+.
+Then we get
+Hgen
+eff/greaterorequalslant4πJ|Q|,
+anda minimum of Hgen
+effis realized if the equality holds, that takes place if {wα}
+are holomorphic or antiholomorphic functions .
+3 Large-scale topological excitations
+Now we construct a particular class of topological excitations that give min-
+imums to the effective Hamiltonians Hdeg1
+eff, andHgen
+effdefined by (12). We
+describe these excitations by holomorphic functions {wα(z)}. Each set {w1(z),
+w2(z),w3(z)}represents a mean field configuration of the system in question.
+First, we consider a degenerate orbit of SU(3) with m=0,q=−2√
+3, where
+a nematic state is realized. In order to satisfy the limiting conditions: µ3→m,
+µ8→q, the functions {wα(z)}have to vanish as z→∞. Let
+w1(z) = 0, w2(z)=a2
+z−z2, w3(z)=a3
+z−z3, a 2, z2, a3, z3∈C,(13)
+8be a large-scale excitation in the magnet in question. The correspon ding mean
+field configuration is obtained by substitution of (13) into (9). A beh avior of
+the components µ3andµ8is represented on the Fig.1. As z→∞the values of
+µ3,µ8tend tom,q.
+1
+z2z3/c1093
+/c451
+32
+/c214/c4531
+/c214
+z/c1098
+z3 z2za2
+a3a21 a3/c40 /c41w□, w3 2 /c40 /c41w□, w3 2
+Fig. 1. Profiles of the mean field components µ3,µ8in the case of
+configuration (13).
+Suppose z2=z3. By shifting of the coordinate zone can easily reduce (13)
+to the configuration
+w1(z) = 0, w2(z)=a2
+z, w3(z)=a3
+z, a 2, a3∈C,
+and calculate its topological charge:
+Q=2i
+4π/integraldisplay/integraldisplay
+C(a2
+2+a2
+3)
+(|z|2+a2
+2+a2
+3)2dz∧d¯z= 1. (14)
+In the case of z2/ne}ationslash=z3the topological charge equals 2.
+It can be interpreted as follows. Each pole of a mean field configurat ion rep-
+resents a kind of Belavin-Plyakov soliton. Each soliton gives a unit top ological
+charge. Thus, two distinct solitons havethe topologicalcharge2. No continuous
+deformation take a configuration of topological charge 2 to a confi guration of
+topological charge 1. If we allow noncontinuous deformation, then two solitons
+can meet at any point and join into one, at the same time an energy is r eleased.
+From (11), it follows that the released energy equals 4 πJper one pole.
+Note, that the energy of configuration (13) does not depend on p arameters
+of solitons: a2,z2,a3,z3. It means that the excitation can grow (when |a2|and
+|a3|grow) preserving an energy. Such growth immediately leads to dest ruction
+of an order in the system.
+One can construct a configuration with more than two solitons:
+w1(z) = 0, w2(z) =a2/n/productdisplay
+k=1(z−z2k), w3(z) =a3/m/productdisplay
+k=1(z−z3k),(15)
+herea2,a3,{z2k}n
+k=1, and{z3k}m
+k=1are fixed complex numbers. If all values
+{z2k}n
+k=1and{z3k}m
+k=1are distinct, a topological charge equals n+m. When a
+pole of the function w2(z) coincides with a pole of w3(z), the topological charge
+decreasesby 1. But a coincidence of two poles ofthe same function (for example
+9w2(z)) does not lead to a decrease of the topological charge. It is easy to see,
+that the minimal energy of configuration (15) equals 4 πJ ·min(n,m).
+Now we consider a generic orbit, where a ferromagnetic state is rea lized.
+Suppose m=1 and q=−2√
+3. In this case, we describe a mean field in the
+magnet by the effective Hamiltonian Hgen
+eff, defined by (12). Let
+w1(z) =a1
+z−z1, w2(z)=a2
+z−z2, w3(z)=a3
+z−z3, ak,zk∈C, k= 1,2,3.(16)
+be a large-scaleexcitation ofthe mean field. A calculation oftopologic alcharges
+gives: 1) Q1=3,Q2=2, ifa1/ne}ationslash=a2/ne}ationslash=a3ora1=a2/ne}ationslash=a3, 2)Q1=2,Q2=2, if
+a1=a3/ne}ationslash=a2, 3)Q1=2,Q2=1, ifa1=a2=a3ora1/ne}ationslash=a2=a3.
+4 Conclusion and discussion
+Each generalized Heisenberg Hamiltonian with high powers of the exch ange in-
+teraction can be reduced to a bilinear form. By a mean field averaging we
+obtain a classical system from the original quantum one. An averag ing of the
+Heisenberg equation gives a Landau-Lifshitz like equation for a mean field. Us-
+ing Lie group apparatus, we construct effective Hamiltonians for th e classical
+system with SU(3) symmetry. One of them Hdeg
+effis an SU(3)-analogue of the
+Hamiltonian commonly used in theory of magnetism. In addition, we pro pose
+another one Hgen
+eff, which is biquadratic in the mean field. Further, we construct
+examples of topological excitations that give minimums to the Hamilton ians.
+Such excitations can change their shapes and grow preserving an e nergy. This
+is a probable scenario for the destruction of an ordered state in a 2 D magnet at
+nonzero temperature, that agrees with the Mermin-Wagner theo rem.
+5 Acknowledgments
+This work is partly supported by the grant of the International Ch aritable Fund
+for Renaissance of Kyiv-Mohyla Academy.
+References
+[1] Mermin N. D., Wagner H. Phys. Rev. Lett 171133 (1966).
+[2] Belavin A. A., Polyakov A. M. JETP Letters 22245 (1975).
+[3] Bogolyubov N N and Bogolyubov N N (Jr.) Introduction to Quantum Statistical Me-
+chanics [in Russian] (Moscow: Nauka) 1984 p 384
+[4] Matveev V M Zh. Eksp. Teor. Fiz. 651626 (1973)
+[5] Nauciel-Bloch M, Sarma G and Castets A Phys. Rev. B 54603 (1972)
+[6] Buchta K., Fath G., Legeza ¨O., Solyom J. Phys. Rev. B 72054433 (2005)
+[7] Picken R. F. J. Math. Phys. 31616 (1990).
+[8] Bernatska J., Holod P. Proceedings of the IXth International Conference on Geomet ry,
+Integrability and Quantization (Sofia) 2008, p. 146–166; arXiv: math.RT/0801.2913
+[9] Bernatska J., Holod P. J. Phys. A 42(to be published in 2009)
+10
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+arXiv:1001.0376v1  [nlin.SI]  3 Jan 2010Journal of Nonlinear Mathematical Physics Volume 14, Numbe r 3 (2007), 353–374 Article
+On Separation of Variables for Integrable
+Equations of Soliton Type
+Julia Bernatska†and Petro Holod‡and
+†The National University of ’Kiev-Mohyla Academy’, Ukraine
+E-mail: jnb@ukma.kiev.ua
+‡The National University of ’Kiev-Mohyla Academy’, Ukraine
+Bogolyubov Institute for Theoretical Physics, Ukraine
+E-mail: holod@ukma.kiev.ua
+Received August 11, 2006; Accepted October 6, 2006
+Abstract
+We propose a general scheme for separation of variables in the inte grable Hamilto-
+nian systems on orbits of the loop algebra sl(2,C)×P(λ,λ−1). In particular, we illus-
+trate the scheme by application to modified Korteweg—de Vries (MKd V), sin(sinh)-
+Gordon, nonlinear Schr¨ odinger, and Heisenberg magnetic equatio ns.
+Introduction
+Let us make a brief review of the problem.
+After the fundamental paper [23], B. Dubrovin (see [8]) prop osed a separation of vari-
+ables for finite gap KdV system. B. Dubrovin shows that the pol es of an appropriately
+normalized Baker—Akhiezer function for the auxiliary line ar spectral problem are the
+separation variables. The new variables evolve on a hyperel liptic Riemannian surface R
+of genusg. The genus coincides with the number of degrees of freedom of the finite gap
+phase space.
+In the papers [14,18,20,25] a separation of variables is rea lized for sin-Gordon equation,
+nonlinear Schr¨ odinger equation, and the classic Thirring model. The case of sin-Gordon
+equation appears to be completely similar to the KdV system. However, the cases of
+nonlinear Schr¨ odinger equation and Thirring model have a d istinction: the number of
+degrees of freedom is greater by one than the genus of the corr esponding spectral curve .
+Here the papers [14,18,25] suggest the separation of variab les on a reduced phase space.
+Later, the complex Liouville torus of nonlinear Schr¨ oding er equation was proven to be the
+generalized Jacobian of a singular Riemannian surface (see [24]).
+The ideas of the early papers on the integration of finite gap s ystems were generalized
+by E. Sklyanin [27,28] and partly extended to the quantum int egrable models [29,30].
+At the beginning of the 90s a new technique of separation of va riables appeared that
+effectively uses bi-hamiltonian, or multi-hamiltonian, pro perties of integrable systems,
+Copyright c/ci∇cleco√y∇t2007 by J Bernatska and P Holod354 J Bernatska and P Holod
+see [3,5–7] and [10–12,22]. The main result in this directio n is the diagonalization of
+recursive Nijenhuis operator. In the papers [11,22] the met hod is applied to KdV and
+Boussinesq hierarchies, and classical finite-dimensional systems.
+The papers [9,31,32] investigate the connection between th e problem of separation of
+variables and the parametrization of compact tori by symmet ric products of Riemannian
+surfaces. According to [31,32], if a change of variables reduces Liouville 1-form to a sum
+of meromorphic differentials on the corresponding Riemanni an surface, then we say that
+the new variables are the separation variables.
+We proposeamethod of separation of variables for integrabl e Hamiltonian systems that
+is connected with the orbit structure of affine Lie algebras. T he fact that finite gap phase
+space of an integrable soliton hierarchy has an orbital stru cturewas established in [15,16].1
+The Hamiltonian systems in question obey the equations of La x type, and hence the
+separation variablesarepointsonthecorresponding spectral curve . Notethat suchsystems
+are multi-Hamiltonian, which connects our results with the results of [10–12,22].
+In Sections 1 and 3 we reproduce the key results from [15,16] a bout finite gap phase
+spaces for integrable equations as orbits of loop algebra. W e illustrate our scheme by
+the examples of modified Korteweg-de Vries (MKdV) system, si n(sinh)-Gordon equation,
+nonlinear Schr¨ odinger equation, and Heisenberg magnetic chain.
+This paper is organized as follows. Sections 1 and 2 are devot ed to MKdV system and
+sin(sinh)-Gordon equation. In Section 1 we construct adjoi nt Poisson spaces and define
+the orbits regarded as phase spaces for MKdV system and sin(s inh)-Gordon equation.
+The construction is discussed in more detail in [4]. In Secti on 2 we describe the scheme
+for separation of variables and illustrate it by applicatio n to MKdV system and sin(sinh)-
+Gordon equation. We show that the separation of variables is achieved on both orbits
+simultaneously. In Sections 3 we construct adjoint Poisson spaces and define the orbits
+regarded as phase spaces for nonlinear Schr¨ odinger equati on and Heisenberg magnetic
+chain. In Sections 4 and 5 we similarly consider separation o f variables for nonlinear
+Schr¨ odinger equation and Heisenberg magnetic chain, acco rdingly.
+Acknowledgements. Theauthors aregrateful to participants ofthescientific se minar
+‘IntegrableHamiltonianSystemsandSolitons’T.Skrypnyk ,D.Leikin, N.Yorgov foruseful
+remarks and discussions.
+1 Phase spaces for MKdV system and sin-Gordon equation
+as orbits in sl(2,C)⊗P(λ,λ−1)
+First, let us recall some constructions from [15,16]. Take t he algebra sl(2,C) with the
+basis
+H=/parenleftbigg1
+20
+0−1
+2/parenrightbigg
+, X=/parenleftbigg0 1
+0 0/parenrightbigg
+, Y=/parenleftbigg0 0
+1 0/parenrightbigg
+.
+1The results of [15,16] are partially covered by [13]. Howeve r the authors of [13] took no notice of
+the remarkable dualitybetween pairs of soliton equations: MKdV and sin-Gordon equ ations, KdV and
+Liouville equations, nonlinear Schr¨ odinger and Heisenbe rg magnetic equations, etc. The duality is evident
+if one uses the orbital approach. The pairs of dual equations have common Liouville torus and separation
+variables. Sometimes, there exists a gauge equivalence bet ween the equations of a pair, and the equivalence
+extends to the total infinite phase space [26].On Separation of Variables for Integrable Equations of Soli ton Type 355
+Suppose P(λ,λ−1) is the algebra of Laurent polynomials in λ. Denote by /tildewidegthe algebra
+sl(2,C)⊗P(λ,λ−1). Then
+H2m=λmH, X2m+1=λmX, Y2m+1=λm+1Y (1.1)
+is abasisin/tildewideg.
+Consider the operator
+d= 2λd
+dλ+adH;
+we call it the operator of principal grading . It is easy to prove that the basis elements (1.1)
+are the eigenvectors of d. We call the eigenvalues of dthedegrees. The superscripts in the
+lefthand sides of (1.1) indicate the corresponding princip al degrees of the basis elements.
+Bygl,l∈Z, denote an eigenspace of principal degree l. It is evident that
+g2m= spanC{H2m},g2m+1= spanC{X2m+1,Y2m+1}.
+Decompose /tildewideginto two subalgebras
+/tildewideg+=/summationdisplay
+l/greaterorequalslant0gl,/tildewideg−=/summationdisplay
+l<0gl,/tildewideg=/tildewideg++/tildewideg−.
+Further, consider the ad -invariant bilinear forms
+/an}b∇acketle{tA(λ),B(λ)/an}b∇acket∇i}htk= resλ−k−1TrA(λ)B(λ), A(λ), B(λ)∈/tildewideg, k∈Z. (1.2)
+We use the forms to define the spaces dual to /tildewideg+and/tildewideg−.
+Example 1. Letk=−1. We have
+(/tildewideg−)∗=/tildewideg++g−1,(/tildewideg+)∗=/summationdisplay
+l/lessorequalslant−2gl, (1.3)
+where (/tildewideg−)∗and (/tildewideg+)∗contain only the nonzero functionals on /tildewideg±.
+Example 2. Letk=N/greaterorequalslant0. Then
+(/tildewideg−)∗=/summationdisplay
+l/greaterorequalslant2N+1gl,(/tildewideg+)∗=/summationdisplay
+l/lessorequalslant2Ngl.
+FixN/greaterorequalslant0. Consider MN+1⊂/tildewideg, where an element /hatwideµ(λ)∈MN+1has the form
+/hatwideµ(λ) =/parenleftbiggα(λ)β(λ)
+γ(λ)−α(λ)/parenrightbigg
+with
+α(λ) =N/summationdisplay
+m=0λmα2m, β(λ) =N+1/summationdisplay
+m=0λm−1β2m−1, γ(λ) =N+1/summationdisplay
+m=0λmγ2m−1.
+We callMN+1theN-gap sector of /tildewideg, or shortly the finite gap sector .356 J Bernatska and P Holod
+Because the factor-algebra /tildewideg−//summationtext
+l/lessorequalslant−2N−4glacts effectively on MN+1, the coadjoint
+action of /tildewideg−with respect to the form /an}b∇acketle{t,/an}b∇acket∇i}ht−1is well defined on MN+1. The same is true
+for the coadjoint action of /tildewideg+with respect to the form /an}b∇acketle{t,/an}b∇acket∇i}htN, indeed, the factor-algebra
+/tildewideg+//summationtext
+l/greaterorequalslant2N+2glacts effectively on MN+1.
+LetC(MN+1) be the space of smooth functions on MN+1. For allf1,f2∈C(MN+1)
+define the first Lie-Poisson bracket by the formula
+{f1,f2}1=N/summationdisplay
+m,n=03/summationdisplay
+a,b=1Pmn
+ab(−1)∂f1
+∂µam∂f2
+∂µbn, (1.4)
+where
+Pmn
+ab(−1) =/an}b∇acketle{t/hatwideµ(λ),[Z−m−1
+a,Z−n−1
+b]/an}b∇acket∇i}ht−1,
+Zm
+1=Hm, Zm
+2=Ym, Zm
+3=Xm,
+µ1
+m=αm, µ2
+m=βm, µ3
+m=γm.
+With the same notation, define the second Lie-Poisson bracket by the formula
+{f1,f2}2=N/summationdisplay
+m,n=03/summationdisplay
+a,b=1Pmn
+ab(N)∂f1
+∂µam∂f2
+∂µbn, (1.5)
+where
+Pmn
+ab(N) =/an}b∇acketle{t/hatwideµ(λ),[Z−m+N
+a,Z−n+N
+b]/an}b∇acket∇i}htN.
+One can see that the functions β2N+1andγ2N+1annihilate the bracket (1.4)
+{β2N+1,f}1=0,{γ2N+1,f}1=0 for all f∈C(MN+1).
+Thus, we can assume without loss of generality that
+β2N+1=γ2N+1=const (1.6)
+and restrict the bracket (1.4) to the subspace MN+1
+con⊂MN+1with the constraints (1.6),
+clearly, dim MN+1
+con= 3(N+1). The first Lie-Poisson bracket is nondegenerate on MN+1
+con.
+We use the set γ2m−1,β2m−1,α2m,m= 0,1, ..., N, ascoordinate functions inMN+1
+con.
+We call the fixed coordinates β2N+1,γ2N+1theexternal parameters .
+We see that, on one hand, MN+1
+con⊂(/tildewideg−)∗with respect to /an}b∇acketle{t,/an}b∇acket∇i}ht−1, see Example 1, and,
+on the other hand, MN+1
+con⊂(/tildewideg+)∗with respect to /an}b∇acketle{t,/an}b∇acket∇i}htN, see Example 2.
+In addition to the brackets (1.4) and (1.5), one can define Nintermediate brackets with
+the Poisson tensors
+Pmn
+ab(k) =/an}b∇acketle{t/hatwideµ(λ),[Z−m+k
+a,Z−n+k
+b]/an}b∇acket∇i}htk, k= 0,...,N−1. (1.7)
+Now, consider the ad∗-invariant function
+I(λ) =−det/tildewideµ(λ) =h−1λ−1+h0+···+h2N+1λ2N+1.On Separation of Variables for Integrable Equations of Soli ton Type 357
+Then we have
+hν=/summationdisplay
+m+n=ν(α2mα2n+γ2m−1β2n−1), ν=−1,0, ...,2N+1. (1.8)
+The Kostant-Adler scheme [1] implies the following asserti ons.
+Proposition 1. All functions hν,ν=−1,0, ...,2N+ 1determined by (1.8)mutually
+commute with respect to the brackets (1.4),(1.5), and the intermediate brackets with the
+Poisson tensors (1.7).
+Proposition 2. The functions hν,ν=N,...,2N, are functionally independent and
+annihilate the bracket (1.4).
+Consider the algebraic variety ON
+1⊂MN+1
+condefined by the set of equations hν=cν,
+ν=N,...,2N, wherecνare arbitrary fixed complex numbers. ON
+1is an orbit of coadjoint
+action of the subalgebra /tildewideg−anddimON
+1= 2(N+1).
+Proposition 3. The functions hν,ν=−1,..., N−1, are functionally independent and
+annihilate the bracket (1.5).
+Consider the algebraic variety ON
+2⊂MN+1
+condefined by the set of equations hν=cν,
+ν=−1,...,N−1, wherecνare arbitrary fixed complex numbers. ON
+2is an orbit of
+coadjoint action of the subalgebra /tildewideg+anddimON
+2= 2(N+1).
+It is obvious that the orbits ON
+1andON
+2are thesymplectic leaves with respect to the
+first and the second Lie-Poisson brackets, accordingly.
+Further, the functions h−1,h0, ...,hN−1, regarded as Hamiltonians with respect to
+the first Lie-Poisson bracket, generate non-trivial flows on MN+1
+con
+∂µa
+m
+∂τν={µa
+m,hν}1, ν=−1,0, ..., N−1. (1.9)
+The equations (1.9) can be written with the help of the second Lie-Poisson bracket and
+the functions hN, ...,h2Nregarded as Hamiltonians. Namely (see [16]), one has
+{µa
+m,hν}1=−{µa
+m,hν+N+1}2.
+Proposition 4. The system (1.9)reduced to the orbit ON
+1is equivalent to the finite gap
+complex MKdV hierarchy.
+The system (1.9)reduced to the orbit ON
+2is equivalent to finite gap sin(sinh)-Gordon
+equation.
+Below we give the outline of the proof which may be found in ful l detail in [4].
+First, rewrite (1.9) in matrix form
+∂/hatwideµ(λ)
+∂τν= [∇2hν+N+1,/hatwideµ(λ)] = [/hatwideµ(λ),∇1hν], (1.10)
+where
+∇1h=N/summationdisplay
+m=0/parenleftbigg∂h
+∂α2mH−2m−2+∂h
+∂β2m−1Y−2m−1+∂h
+∂γ2m−1X−2m−1/parenrightbigg
+,
+∇2h=N/summationdisplay
+m=0/parenleftbigg∂h
+∂α2mH−2m+2N+∂h
+∂β2m−1Y−2m+2N+1+∂h
+∂γ2m−1X−2m+2N+1/parenrightbigg
+.358 J Bernatska and P Holod
+Thehamiltonianflowsalong τνandτν′commute, whichimpliesthecompatibility condition
+in the form of zero curvature equations . In particular, assigning τN−1=x,τN−2=t, we
+obtain
+∂∇2h2N
+∂t−∂∇2h2N−1
+∂x+[∇2h2N,∇2h2N−1] = 0,
+where
+∇2h2N=/parenleftbiggα2Nβ2N+1
+λγ2N+1−α2N/parenrightbigg
+,
+∇2h2N−1=/parenleftbiggα2N−2+λα2Nβ2N−1+λβ2N+1
+λγ2N−1+λ2γ2N+1−(α2N−2+λα2N)/parenrightbigg
+.
+Recall that β2N+1andγ2N+1are the fixed external parameters.
+The reduction of (1.9) onto the orbit ON
+1gives the equation
+∂α2N
+∂t=∂α2N−2
+∂x, (1.11)
+equivalent to MKdV equation with respect to the function α2N(x,t) =u(x,t). Indeed,
+reducing the equation (1.9) as ν=N−1 to the orbit ON
+1we obtain
+β2N−1=1
+2β2N+1/parenleftbigg
+c2N−∂α2N
+∂x−α2
+2N/parenrightbigg
+, (1.12a)
+γ2N−1=1
+2β2N+1/parenleftbigg
+c2N+∂α2N
+∂x−α2
+2N/parenrightbigg
+, (1.12b)
+α2N−2=1
+4β2N+1/parenleftbigg∂2α2N
+∂x2−2α3
+2N+2c2Nα2N/parenrightbigg
+. (1.12c)
+It is readily seen that combining (1.11) and (1.12c) we get th e complex MKdV equation.
+Two real subalgebras su(2) andsu(1,1)∼=sl(2,R) ofsl(2,C) give rise to two real MKdV
+equations (the so-called ±MKdV).
+In the same time, the reduction of (1.9) onto the orbit ON
+2leads to sin(sinh)-Gordon
+equation. Let ON
+2∪g−1be thebasefor the orbit ON
+2. The 1-parameter subgroup G0=
+expg0parametrizes the base in a natural way
+γ−1=/radicalbig
+h−1eu, β −1=/radicalbig
+h−1e−u.
+Then the equations (1.9) imply
+α2N=1
+2∂
+∂xu, (1.13)
+wherex≡τN−1as above; the corresponding flow is called stationary . The Hamiltonian
+hNgives rise to an evolutionary flow. In the case of the subalgebra sl(2,R) we have
+∂α2N
+∂t= 2β2N+1/radicalbig
+h−1sinhu. (1.14)
+Combining (1.13) and (1.14) we obtain sinh-Gordon equation .
+In the case of the subalgebra su(2) we have to assign α2m=ia2m,a2m∈R, and
+γ2m−1=−β∗
+2m−1, thereforeβ2N+1=γ2N+1=ib,γ−1=−ireiu,β−1=−ire−iu. Then we
+come to sin-Gordon equation
+∂2u
+∂t∂x= 4rbsinu.On Separation of Variables for Integrable Equations of Soli ton Type 359
+2 Separation of variables for MKdV and sin(sinh)-Gordon
+equation
+Definition 1. Suppose we have the variables ( λk,wk),k= 1,...,N+1, such that
+(i) they are quasi-canonically conjugate, that is
+{λk,wl}1=f(λk)δkl,{λk,λl}1={wk,wl}1= 0,
+wheref(λ) is an arbitrary smooth function;
+(ii) they reduce Liouville 1-form2to a sum of meromorphic differentials on the corre-
+sponding Riemannian surface.
+We call (λk,wk),k= 1,...,N+1,separation variables .
+Consider the orbit ON
+1, dimON
+1= 2(N+ 1). One can parameterize the orbit using
+any subset of 2( N+1) variables from {α2m, β2m−1,γ2m−1},m= 0,1,...,N. The most
+natural way to obtain the parameterization is to eliminate o ne of the subsets {β2m−1}or
+{γ2m−1}. The reason for this is the nilpotency of the basis elements t hat correspond to
+the subsets.
+Note that the correspondence between the elimination varia bles, which we chose to
+parameterize the orbit, and the nilpotent elements of the ba sis of the algebra is a crucial
+feature of our scheme and applies to all examples.
+We chose to parameterize the orbit ON
+1by the variables {γ2m−1,α2m},m= 0,1,...,N,
+that is we eliminate the set {β2m−1}. From the orbit equations we find
+β2m−1=N+1/summationdisplay
+j=0(Γ+)−1
+mj(cN+j−AN+j), m= 0, ... N+1, c2N+1=β2N+1γ2N+1,(2.1)
+where
+Γ+=
+γ2N+1γ2N−1... γ 1γ−1
+0γ2N+1... γ 3γ1
+...............
+0 0 ... γ 2N+1γ2N−1
+0 0 ...0γ2N+1
+andAν=/summationdisplay
+m+n=ν,
+0/lessorequalslantm,n/lessorequalslantNα2mα2n.
+Now, using the parameterization (2.1), we find expressions f or the Hamiltonians h−1,h0,
+...,hN−1
+hn−1=N+1/summationdisplay
+m,j=0Γ−
+nm(Γ+)−1
+mj(cN+j−AN+j)+An−1, n= 0, ...N, (2.2)
+where
+Γ−=
+γ−10...0 0
+γ1γ−1...0 0
+...............
+γ2N−1γ2N−3... γ −10
+.
+2We call Ω Liouville 1-form if dΩ =ω, whereωis a symplectic 2-form.360 J Bernatska and P Holod
+Note that the expressions (2.2) are linear in cν,ν=N,...,2N+1.
+Clearly, one can obtain an analogous parametrization of the orbitON
+2by the set
+{α2m,γ2m−1},m= 0,...,N.
+To proceed we need to define the characteristic polynomial
+Q(κ,λ) = det/parenleftbig
+µ(λ)−κ·I/parenrightbig
+,
+whereIdenotes 2 ×2 identity matrix. By the substitution κ=wλ−1the equation
+Q(κ,λ) = 0 becomes transformed into the standard equation of a hype relliptic curve of
+genusN+1
+P(w,λ) =λ2Q(wλ−1,λ) =w2−λ(h−1+h0λ+···+h2N+1λ2N+2) = 0.(2.3)
+Recall that on the orbit ON
+1we havehν=cν,ν=N, ..., 2N. Denote by ( wk, λk) a
+root ofP(w,λ) on the orbit, that is
+w2
+k=λk(h−1+h0λk+···hN−1λN
+k+cNλN+1
+k+cN+1λN+2
+k+···+c2N+1λ2N+2
+k).(2.4)
+We proceed toshow that theset {(wk, λk)},k= 1,...,N+1, definesanother parametriza-
+tion of the orbit ON
+1. We have to find the explicit relation between the sets {(w1,λ1),...,
+(wN+1,λN+1)}and{α0,α2,...,α 2N,γ−1,γ1,...,γ2N−1}.
+Solving (2.4) for the Hamiltonians h−1,h0,...,h N−1one gets
+h−1=1
+W[W1/parenleftBig
+w2
+λ/parenrightBig
+−cNW1(λN+1)−···−c2N+1W1(λ2N+2)]
+h0=1
+W[W2/parenleftBig
+w2
+λ/parenrightBig
+−cNW2(λN+1)−···−c2N+1W2(λ2N+2)]
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+hN−1=1
+W[WN+1/parenleftBig
+w2
+λ/parenrightBig
+−cNWN+1(λN+1)−···−c2N+1WN+1(λ2N+2)].(2.5)
+whereW=/producttext(λi−λj) is Vandermonde determinant of λ1,λ2, ...,λN+1. ByWi(f(λ,w))
+we denote the determinant of Vandermonde matrix with the i-th column replaced by/parenleftbig
+f(λ1,w1),...,f(λN+1,wN+1)/parenrightbigt.
+On the orbit the formulas (2.2) and (2.5) define the same set of functions. We see that
+both (2.2) and (2.5) are linear in cν,ν=N,...,2N+1. As{cν}is the set of independent
+parameters one can equate the corresponding terms. Namely, we get
+γ2m−1
+γ2N+1=−Wm+1(λN+1)
+W, m= 0, ...N.
+This implies that the set {λk}is, in fact, the set of roots of the polynomial γ(λ)
+γ(λk) = 0,
+while the variables {wk}satisfy the equalities
+w2
+k=λ2
+k/parenleftbig
+α2(λk)−γ(λk)β(λk)/parenrightbig
+=λ2
+kα2(λk), k= 1,...,N+1.On Separation of Variables for Integrable Equations of Soli ton Type 361
+Theorem 1. Suppose the orbit ON
+1has the coordinates (α2m, γ2m−1),m= 0,1,...,N,
+as above. Then the new coordinates (λk,wk),k= 1,...,N+1, defined by the formulas
+γ(λk) = 0, w k=ελkα(λk),whereε2= 1, (2.6)
+have the following properties:
+(1)a pair(wk,λk)is a root of the characteristic polynomial (2.3).
+(2)a pair(λk,wk)is quasi-canonically conjugate with respect to the first Lie- Poisson
+bracket(1.4):
+{λk,λl}1= 0,{λk,wl}1=ελkδkl,{wk,wl}1= 0; (2.7)
+(3)the corresponding Liouville 1-form is
+Ω−1=/summationdisplay
+kελ−1
+kwkdλk.
+Proof.(1) The assertion is a direct consequence of (2.3) and (2.6).
+(2) It is evident that
+{λk,λl}1= 0.
+Indeed, since λk,k= 1, ...,N+ 1, depend only on γ2m−1,m= 0, ...,N, andγ2m−1
+mutually commute, λkalso mutually commute.
+Let us calculate the bracket of λkandwl
+{λk,wl}1=/summationdisplay
+m,n/parenleftbigg∂λk
+∂γ2m−1∂wl
+∂α2n−∂λk
+∂α2n∂wl
+∂γ2m−1/parenrightbigg
+{γ2m−1,α2n}1.
+From (2.6) we have
+∂λk
+∂α2n= 0,∂λk
+∂γ2m−1=−λm
+k
+γ′(λk),∂wl
+∂α2n=ελn+1
+l. (2.8)
+Further {γ2m−1,α2n}1=−γ2(m+n)+1whenm+n < Nand{γ2m−1,α2n}1= 0 when
+m+n/greaterorequalslantN. Thus, we obtain
+{λk,wl}1=/summationtext
+m+n<Nελm
+kλn+1
+lγ2(m+n)+1
+γ′(λk)=ελl
+γ′(λk)γ(λl)−γ(λk)
+λl−λk.
+Ask/ne}ationslash=lit is evident that {λk,wl}1= 0 whileγ(λl) =γ(λk) = 0. Ask=lwe get
+{λk,wk}1= lim
+λl→λkελl
+γ′(λk)γ(λl)−γ(λk)
+λl−λk=ελk.
+Thus,
+{λk,wl}1=ελkδkl.362 J Bernatska and P Holod
+Let us calculate the bracket of wkandwl
+{wk,wl}1=/summationdisplay
+m,n/parenleftbigg∂wk
+∂γ2m−1∂wl
+∂α2n−∂wk
+∂α2n∂wl
+∂γ2m−1/parenrightbigg
+{γ2m−1,α2n}1.
+From (2.6) follows that
+∂wk
+∂γ2m−1=ε/bracketleftbig
+α(λk)+λkα′(λk)/bracketrightbig∂λk
+∂γ2m−1,
+then, using (2.8), we obtain
+{wk,wl}1=/parenleftbiggλl[α(λk)+λkα′(λk)]
+γ′(λk)−λk[α(λl)+λlα′(λl)]
+γ′(λl)/parenrightbiggγ(λl)−γ(λk)
+λl−λk,
+hence
+{wk,wl}1= 0.
+(3) From (2.7) it follows that Liouville 1-form on the orbit ON
+1is
+Ω−1=/summationdisplay
+kελ−1
+kwkdλk.
+The reduction to Liouville torus is done by fixing the values o f Hamiltonians h−1,
+h0, ...,hN−1. On the torus wkis the algebraic function of λkdue to(2.3). After the
+reduction the form Ω −1becomes a sum of meromorphic differentials on the Riemann
+surfaceP(w,λ) = 0. /squaresolid
+The next theorem is proven similarly.
+Theorem 2. Suppose the orbit ON
+2has the coordinates (α2m, γ2m−1),m= 0,1,...,N.
+Then the new coordinates (λk,wk),k= 1,...,N+1, defined by the formulas
+γ(λk) = 0, w k=ελkα(λk),whereε2= 1,
+have the following properties:
+(1)a pair(wk,λk)is a root of the characteristic polynomial (2.3);
+(2)a pair(λk,wk)is quasi-canonically conjugate with respect to the second Li e-Poisson
+bracket(1.5):
+{λk,λl}2= 0,{λk,wl}2=−ελN+2
+kδkl,{wk,wl}2= 0; (2.9)
+(3)the corresponding Liouville 1-form is
+ΩN=−/summationdisplay
+kελ−(N+2)
+kwkdλk.On Separation of Variables for Integrable Equations of Soli ton Type 363
+Let us summarize our scheme of obtaining the separation vari ables. First, we param-
+eterize the orbit by eliminating a subset of group coordinat es corresponding to nilpotent
+basis elements. Next, we restrict the curve P(w,λ)=0 onto the orbit, where P(w,λ) is
+the characteristic polynomial
+P(w,λ) = det/parenleftbig
+µ(λ)−w·I/parenrightbig
+,
+Iis identity matrix. We use the set {λk,wk},k= 1,...,N+1, where P(wk,λk) = 0, to
+define another parametrization of the orbit. Then, we equate expressions for Hamiltonians
+in the coordinates of two parameterizations of the orbit in o rder to obtain the link between
+thetwo sets of orbit coordinates. Finally, theset {λk,wk}is the set of separation variables.
+Further, in Sections 4 and 5 we apply the scheme to nonlinear S chr¨ odinger equation
+and Heisenberg magnetic chain.
+3 Phase spaces for nonlinear Schr¨ odinger equation and
+Heisenberg magnetic chain as orbits in sl(2,C)⊗P(z,z−1)
+Here we use the construction from Section 1 with homogeneous grading . That is,
+Xl=zlX, Yl=zlY, Hl=zlH (3.1)
+be thebasisin/tildewideg≃sl(2,C)/circlemultiplytextP(z,z−1).
+Note the well-known fact that the Lie algebra from Sections 1 –2 can be realized as the
+subalgebra of sl(2,C)⊗P(z,z−1) invariant with respect to an automorphism of order 2,
+see [17], [19], [21].
+Bygl,l∈Z, denote an eigenspace of homogeneous degree l. It is evident that
+gl= spanC{Xl, Yl, Hl}.
+Decompose /tildewideginto two subalgebras
+/tildewideg+=/summationdisplay
+l/greaterorequalslant0gl,/tildewideg−=/summationdisplay
+l<0gl,/tildewideg=/tildewideg++/tildewideg−.
+Use the same ad -invariant bilinear forms (1.2) to define the spaces dual to /tildewideg+and/tildewideg−.
+FixN/greaterorequalslant0. Consider MN+1⊂/tildewideg, where an element /hatwideµ(z)∈MN+1has the form
+/hatwideµ(z) =/parenleftbiggα(z)β(z)
+γ(z)−α(z)/parenrightbigg
+with
+α(λ) =N+1/summationdisplay
+m=0zmαm, β(z) =N+1/summationdisplay
+m=0zmβm, γ(z) =N+1/summationdisplay
+m=0zmγm.
+As above, we call MN+1theN-gap sector of /tildewideg, or shortly the finite gap sector .364 J Bernatska and P Holod
+For allf1,f2∈C(MN+1) define two Lie-Poisson brackets
+{f1,f2}1=N+1/summationdisplay
+m,n=03/summationdisplay
+a,b=1Pmn
+ab(−1)∂f1
+∂µam∂f2
+∂µbn(3.2)
+and
+{f1,f2}2=N+1/summationdisplay
+m,n=03/summationdisplay
+a,b=1Pmn
+ab(N+1)∂f1
+∂µam∂f2
+∂µbn, (3.3)
+where
+Pmn
+ab(−1) =/an}b∇acketle{t/hatwideµ(z),[Z−m−1
+a,Z−n−1
+b]/an}b∇acket∇i}ht−1,
+Pmn
+ab(N+1) =/an}b∇acketle{t/hatwideµ(z),[Z−m+N+1
+a,Z−n+N+1
+b]/an}b∇acket∇i}htN+1,
+Zm
+1=Hm, Zm
+2=Ym, Zm
+3=Xm,
+µ1
+m=αm, µ2
+m=βm, µ3
+m=γm.
+One can see that MN+1⊂(/tildewideg−)∗with respect to /an}b∇acketle{t,/an}b∇acket∇i}ht−1and, in the same time,
+MN+1⊂(/tildewideg+)∗with respect to /an}b∇acketle{t,/an}b∇acket∇i}htN+1.
+Next, introduce the ad∗-invariant function
+I(z) =−det/tildewideµ(z) =h0+h1z+···+h2N+2z2N+2,
+where
+hν=/summationdisplay
+m+n=ν(αmαn+γmβn), ν= 0,1,...,2N+2. (3.4)
+One can easily prove that the functions αN+1,βN+1,γN+1annihilate the bracket (3.2).
+In order to obtain nonlinear Schr¨ odinger equation we have t o assign
+βN+1=γN+1= 0, α N+1=const/ne}ationslash= 0. (3.5)
+After the restriction of the bracket (3.2) to the subspace MN+1
+con⊂MN+1with the con-
+strains (3.5) we get dim MN+1
+con= 3(N+1). The bracket (3.2) is nondegenerate on MN+1
+con.
+We use the set γm,βm,αm,m= 0,1,...Nascoordinate functions inMN+1
+con. We call the
+fixed coordinate αN+1theexternal parameter .
+Ontheother hand, thefunctions αN+1,βN+1,γN+1commute withall Hamiltonians hν,
+ν=N+2,N+3,...,2N+2, with respect to the bracket (3.3) and give rise to nontriv ial
+flows onMN+1, therefore are Hamiltonians. That is, the bracket (3.3) is c onsidered on
+MN+1, dimMN+1= 3(N+ 2), and the set γm,βm,αm,m= 0,1,...N+ 1 serve as
+coordinate functions inMN+1.
+The following assertions are immediately derived from the K ostant-Adler scheme [1].
+Proposition 5. All functions hν,ν= 0,1,...,2N+ 2determined by (3.4)mutually
+commute with respect to the brackets (3.2)and(3.3).On Separation of Variables for Integrable Equations of Soli ton Type 365
+Proposition 6. The functions hν,ν=N+ 1,...,2N+ 1are functionally independent
+and annihilate the bracket (3.2).
+Consider the algebraic variety ON
+1⊂MN+1
+condefined by the set of equation hν=cν,
+ν=N+1,...,2N+1, wherecνare arbitrary fixed complex numbers. ON
+1is an orbit of
+coadjoint action of the subalgebra /tildewideg−anddimON
+1= 2(N+1).
+Proposition 7. The functions hν,ν= 0,...,N+ 1are functionally independent and
+annihilate the bracket (3.3).
+Consider the algebraic variety ON+1
+2⊂MN+1defined by the set of equation hν=cν,
+ν= 0,...,N+ 1, wherecνare arbitrary fixed complex numbers. ON+1
+2is an orbit of
+coadjoint action of the subalgebra /tildewideg+anddimON+1
+2= 2(N+2).
+Further, the functions h0,h1,...,h Nregarded as Hamiltonians with respect to the
+bracket (3.2) give rise to nontrivial flows on MN+1
+con
+∂µa
+m
+∂τν={µa
+m,hν}1, ν= 0,1,...N. (3.6)
+The functions hN+2,...,h 2N+2regarded as Hamiltonians with respect to the bracket (3.3)
+give rise to nontrivial flows on MN+1
+∂µa
+m
+∂τν=−{µa
+m,hν+N+2}2, ν= 0,1,...N. (3.7)
+Proposition 8. The system (3.6)reduced to the orbit ON
+1is equivalent to finite gap
+nonlinear Schr¨ odinger equation.
+The system (3.7)reduced to the orbit ON+1
+2is equivalent to finite gap Heisenberg mag-
+netic chain.
+Here we give the outline of the proof.
+Consider the orbit ON
+1. Rewrite (3.6) in matrix form. In particular, assigning τN=x,
+τN−1=t, we obtain
+∂/hatwideµ(z)
+∂x= [/hatwideµ(z),∇1hN] = [∇2h2N+1,/hatwideµ(z)], (3.8a)
+∂/hatwideµ(z)
+∂τ= [/hatwideµ(z),∇1hN−1] = [∇2h2N,/hatwideµ(z)], (3.8b)
+where
+∇2h2N+1=/parenleftbiggzα2N+1+αNβN
+γN −(zα2N+1+αN)/parenrightbigg
+,
+∇2h2N=/parenleftbiggz2α2N+1+zαN+αN−1zβN+βN−1
+zγN+γN−1 −(z2αN+1+zαN+αN−1)/parenrightbigg
+.
+We use the real subalgebra su(2) ofsl(2,C), that is assign αm=iam,γm=∓β∗
+m, then
+the compatibility condition for (3.8) gives
+2iaN+1∂βN
+∂t=−∂2βN
+∂x2−2βN|βN|2−2βNh2N,366 J Bernatska and P Holod
+whichcoincideswithnonlinearSchr¨ odingerequationwith respecttothefunction βN(x,t) =
+ψ(x,t) asaN+1=1
+2,hN=aN= 0,
+i∂ψ
+∂t=−∂2ψ
+∂x2+2εψ|ψ|2, ε2= 1.
+Consider the orbit ON+1
+2. Note, that dim ON+1
+2= 2(N+2) and the set hN+2,hN+3,
+...,h2N+2is insufficient to provide Liouville integrability. Since th e functions αN+1,
+βN+1, andγN+1are in involution with the set hN+2,hN+3, ...,h2N+2with respect to the
+bracket (3.3) one can take any of them as an extra Hamiltonian . Here we chose αN+1.
+Rewrite (3.7) in matrix form. In particular, by assigning τN+2=xandτN+3=twe
+obtain
+∂/hatwideµ(z)
+∂x= [∇2hN+2,/hatwideµ(z)] = [/hatwideµ(z),∇1h0], (3.9a)
+∂/hatwideµ(z)
+∂t= [∇2hN+3,/hatwideµ(z)] = [/hatwideµ(z),∇1h1], (3.9b)
+where
+∇1h0=z−1/parenleftbiggα0β0
+γ0−α0/parenrightbigg
+,
+∇1h1=z−1/parenleftbiggα1β1
+γ1−α1/parenrightbigg
++z−2/parenleftbiggα0β0
+γ1−α1/parenrightbigg
+.
+Let us replace the coordinates αm,βm,γm,m= 0,1,...,N+ 1, according to the
+formulas
+αm=iµ3
+m, β m=µ1
+m−iµ2
+m, γ m=−µ1
+m−iµ2
+m.
+Then
+{µi
+m,µj
+n}2=εijkµk
+m+n−N−1.
+Introduce the vector notation µm= (µ1
+m,µ2
+m,µ3
+m)t,m= 0,1,...,N+1. Now the orbit
+ON+1
+2is determined by the equations
+(µ0,µ0) =−c0,
+2(µ0,µ1) =−c1,
+. . . . . . . . . . . . . . . ./summationtext
+m+n=N+1(µm,µn) =−cN+1,
+where (·,·) denotes the dot product. The equations (3.9) are written in the form
+∂µm
+∂x= 2[µ0,µm+1], (3.10a)
+∂µm
+∂t= 2[µ1,µm+1]+2[µ0,µm+2], (3.10b)
+where [·,·] denotes the cross product.On Separation of Variables for Integrable Equations of Soli ton Type 367
+By reduction of (3.10) to the orbit ON+1
+2we obtain
+µ1=1
+2c0/bracketleftbigg
+µ0,∂µ0
+∂x/bracketrightbigg
++c1
+2c0µ0.
+Taking into account the compatibility condition
+∂µ0
+∂t=∂µ1
+∂x,
+we get
+∂µ0
+∂t=1
+2c0/bracketleftbigg
+µ0,∂2µ0
+∂x2/bracketrightbigg
++c1
+2c0∂µ0
+∂x. (3.11)
+Whenc1= 0, the equation (3.11) becomes the well-known classic Heisenberg magnetic
+equation, also called isotropic Landau-Livshits equation .
+4 Separation of variables for nonlinear Schr¨ odinger equa-
+tion
+Consider the orbit ON
+1, dimON
+1= 2(N+1). The most natural way to parameterize the
+orbit, which we already noted in Section 2, is to eliminate th e subset {βm}(or{γm}),
+m= 0,1,...,N+ 1. Then, roots of γ(λ) (orβ(λ)) give a half of separation variables.
+This way is applied in [18]. However, in the case of nonlinear Schr¨ odinger equation the
+finite gap phase space is determined by the constrains (3.5). That is, the polynomials β(λ)
+andγ(λ) have the order N, therefore the set of roots is insufficient to parameterize th e
+orbitON
+1.
+In order to solve this problem we change coordinates. We use c oordinates as in [24].
+Let
+T=/parenleftbigg0−1
+2
+−1
+20/parenrightbigg
+, R=/parenleftbigg1
+2−1
+21
+2−1
+2/parenrightbigg
+, S=/parenleftbigg1
+21
+2
+−1
+2−1
+2/parenrightbigg
+, (4.1)
+be the basis in sl(2,C). It is easily shown that [ T,S] =S, [T,R] =−R, [S,R] = 2T. Then
+Tm=zmT, Rm=zmR, Sm=zmS.
+is abasisin/tildewideg≃sl(2,C)⊗P(z,z−1). An element /hatwideµ(z)∈MN+1has the form
+/hatwideµ(z) =/parenleftbigg1
+2[r(z)+s(z)]1
+2[r(z)−s(z)−2t(z)]
+1
+2[s(z)−r(z)−2t(z)]−1
+2[r(z)+s(z)]/parenrightbigg
+,
+where
+t(z) =N+1/summationdisplay
+m=0zmtm, r(z) =N+1/summationdisplay
+m=0zmrm, s(z) =N+1/summationdisplay
+m=0zmsm.368 J Bernatska and P Holod
+Note thattm,rm,sm,m= 0,1,...,N+1 are defined by the formulas:
+tm=/an}b∇acketle{t/hatwideµ(z),T−m−1/an}b∇acket∇i}ht−1=/an}b∇acketle{t/hatwideµ(z),T−m+N+1/an}b∇acket∇i}htN+1,
+rm=/an}b∇acketle{t/hatwideµ(z),R−m−1/an}b∇acket∇i}ht−1=/an}b∇acketle{t/hatwideµ(z),R−m+N+1/an}b∇acket∇i}htN+1, (4.2)
+sm=/an}b∇acketle{t/hatwideµ(z),S−m−1/an}b∇acket∇i}ht−1=/an}b∇acketle{t/hatwideµ(z),S−m+N+1/an}b∇acket∇i}htN+1, m= 0,1,...,N+1.
+With the new coordinates MN+1
+conis defined by constrains
+tN+1= 0, r N+1=sN+1=√c2N+2/ne}ationslash= 0.
+One can see that the subsets {rm}and{sm},m= 0,1,...,N+ 1 have nilpotent
+corresponding basis elements. For the reason we chose one of these subsets, namely {rm}
+here, to parameterize the orbit ON
+1. From the orbit equations we find
+rm=N+1/summationdisplay
+j=0(S+)−1
+mj(cj+N+1−Bj+N+1), m= 0,...,N+1, c2N+2=rN+1sN+1,(4.3)
+where
+S+=
+sN+1sN... s 1s0
+0sN+1... s 2s1
+...............
+0 0 ... sN+1sN
+0 0 ...0sN+1
+andBν=/summationdisplay
+m+n=ν,
+0/lessorequalslantm,n/lessorequalslantNtmtn.
+Usingtheparameterization(4.3), wefindtheexpressionsfo rtheHamiltonians h0,h1,...,h N
+hn=N+1/summationdisplay
+mj=0S−
+nm(S+)−1
+mj(cj+N+1−Bj+N+1)+Bn, n= 0, ...N, (4.4)
+where
+S−=
+s00...0 0
+s1s0...0 0
+...............
+sNsN−1... s00
+.
+To proceed we define the characteristic polynomial
+P(w,z) = det/parenleftbig
+µ(z)−w·I/parenrightbig
+. (4.5)
+The equation P(w,z) = 0 has a form of the standard equation of a hyperelliptic cur ve of
+genusN+1
+P(w,z) =w2−(h0+h1z+···+h2N+2z2N+2) = 0. (4.6)
+On the orbit ON
+1we havehν=cν,ν=N+1,...,2N+1. Denote by ( wk,zk) a root
+ofP(w,z) on the orbit, that is
+w2
+k=h0+h1zk+···hNzN
+k+cN+1zN+1
+k+···c2N+2z2N+2
+k. (4.7)On Separation of Variables for Integrable Equations of Soli ton Type 369
+We proceed to show that the set {(wk,zk)},k= 0,1,...,N+ 1 defines another pa-
+rameterization of the orbit ON
+1. We have to find the explicit relation between the sets
+{(w1,z1),...,(wN+1,zN+1)}and{t0,t1,...,tN,s0,s1,...,sN}.
+Solving (4.7) for Hamiltonians h0,h1,...,h None gets
+h0=1
+W[W1/parenleftbig
+w2/parenrightbig
+−cN+1W1(zN+1)−···−c2N+2W1(z2N+2)]
+h1=1
+W[W2/parenleftbig
+w2/parenrightbig
+−cN+1W2(zN+1)−···−c2N+2W2(z2N+2)]
+................................................... ...................
+hN=1
+W[WN+1/parenleftbig
+w2/parenrightbig
+−cN+1WN+1(zN+1)−···−c2N+2WN+1(z2N+2)],(4.8)
+whereWandWi(f(z,w)) denote the same as in (2.5).
+On the orbit ON
+1the formulas (4.4) and (4.8) define the same set of functions. We see
+that both (4.4) and (4.8) are linear in cν,ν=N+ 1,...,2N+ 2. As {cν}is the set of
+independent parameters one can equate the corresponding terms. Namely, we obtain
+sm
+sN+1=Wm+1(zN+1)
+W, m= 0,1,...,N.
+This implies that the set {zk}is the set of roots of the polynomial s(z)
+s(zk) = 0,
+while the variables {wk}satisfy the equalities
+w2
+k=t2(zk)+s(zk)r(zk) =t2(zk), k= 1,...,N+1.
+Theorem 3. Suppose the orbit ON
+1has the coordinates (tm,sm),m= 0,1,...,N, as
+above. Then the new coordinates (zk,wk),k= 1,...,N+1, defined by the formulas
+s(zk) = 0, w k=εt(zk),whereε2= 1, (4.9)
+have the following properties:
+(1)a pair(wk,zk)is a root of the characteristic polynomial (4.6).
+(2)a pair(zk,wk)is canonically conjugate with respect to the Lie-Poisson bra cket(3.2):
+{zk,zl}1= 0,{zk,wl}1=εδkl,{wk,wl}1= 0; (4.10)
+(3)the corresponding Liouville 1-form is
+Ω−1=/summationdisplay
+kεwkdzk.
+Proof.(1) The assertion is a direct consequence of (4.6) and (4.9).
+(2) It is evident that
+{zk,zl}1= 0,370 J Bernatska and P Holod
+sincezk,k= 1,...,N+1 depend only on sm,m= 0,1,...,Nandsmmutually commute.
+Let us calculate the brackets {zk,wl}1and{wk,wl}1. From (4.9) we have
+∂zk
+∂tn= 0,∂zk
+∂sm=−zm
+k
+s′(zk),∂wl
+∂tn=εzn
+l,∂wl
+∂sm=εt′(zl)∂zl
+∂sm.
+Further{sm,tn}1=−sm+nwhenm+n/lessorequalslantNand{sm,tn}1= 0 whenm+n>N. Thus,
+we obtain
+{zk,wl}1=ε
+s′(zk)s(zk)−s(zl)
+zk−zl,{wk,wl}1=/parenleftbiggt′(zk)
+s′(zk)−t′(zl)
+s′(zl)/parenrightbiggs(zk)−s(zl)
+zk−zl.
+Thus,
+{zk,wl}1=εδkl,{wk,wl}1= 0.
+(3) From (4.10) it follows that Liouville 1-form on the orbit ON
+1is
+Ω−1=/summationdisplay
+kεwkdzk.
+Thereduction to Liouville torus is done by fixingthe values o f Hamiltonians h0,h1, ...,
+hN. On the torus wkis the algebraic function of zkdue to (4.6). After the reduction the
+form Ω −1becomes a sum of meromorphic differentials on the Riemann surf aceP(w,z)=0.
+/squaresolid
+Given a set of pairs ( zk,wk),k= 1,...,N+ 1, one can find the set ( tm,sm),m=
+0,...,N, such that the equations (4.9) are satisfied. Thus, we can defi nea homomorphism
+C2N+2→ ON
+1 (4.11)
+that takes the set of pairs ( zk,wk),k= 1,...,N+1, to a point of the orbit ON
+1. Since
+the orbit ON
+1is topologically trivial, this implies that the map (4.11) i s global.
+After the reduction (4.11) turns into the map that takes the ( N+1)th symmetric power
+of Riemann surface to Loiuville torus:
+Sym{R×R×···×R} /ma√sto→ TN+1.
+5 Separation of variables for Heisenberg magnetic chain
+Consider the orbit ON+2
+2, dimON+2
+2= 2(N+ 2). We chose to parameterize the orbit
+ON+2
+2by the variables {γm,αm},m= 0,1,...,N+1, that is we eliminate the set {βm},
+which corresponding basis elements are nilpotent.
+From the orbit equation we find
+βm=N+1/summationdisplay
+j=0(/tildewideΓ−)−1
+mj(cj−/tildewideAj), m= 0, ... N+1, (5.1)On Separation of Variables for Integrable Equations of Soli ton Type 371
+where
+/tildewideΓ−=
+γ00...0 0
+γ1γ0...0 0
+...............
+γNγN−1... γ 00
+γN+1γN... γ 1γ0
+and /tildewideAν=/summationdisplay
+m+n=ν,
+0/lessorequalslantm,n/lessorequalslantN+1αmαn.
+Now, using the parameterization (5.1), we find expressions f or the Hamiltonians hN+2,
+hN+3, ...,h2N+2
+hn+N+1=N+1/summationdisplay
+m,j=0/tildewideΓ+
+nm(/tildewideΓ−)−1
+mj(cj−/tildewideAj)+/tildewideAn+N+1, n= 1, ...N+1, (5.2)
+where
+/tildewideΓ+=
+0γN+1... γ 2γ1
+0 0... γ 3γ2
+...............
+0 0...0γN+1
+.
+Note that the expressions (5.2) are linear in cν,ν=N,...,2N+1.
+To proceed we use the same characteristic polynomial (4.5)
+On the orbit ON+1
+2we havehν,ν= 0,1,...,N+1. Denote by ( wk,zk) a root ofP(w,z)
+on the orbit, that is
+w2
+k=c0+c1zk+···cN+1zN+1
+k+hN+2zN+2
+k+···h2N+2z2N+2
+k. (5.3)
+The set {(wk,zk)},k= 0,1,...,N+ 1 is insufficient to parameterize ON+1
+2. Let us fix
+αN+1andγN+1regarded as Hamiltonians and consider below the reduced orb itON+1
+2red. If
+we find an explicit relation between the sets {(w1,z1),...,(wN+1,zN+1)}and{α0,α1, ...,
+αN,γ0,γ1, ...,γN}we show that the set {(wk,zk)},k= 0,1,...,N+1 defines another
+parameterization of the orbit ON+1
+2red.
+Theorem 4. Suppose the orbit ON+1
+2redhas the coordinates (αm,γm),m= 0,1,...,N, as
+above. Then the new coordinates (zk,wk),k= 1,...,N+1, defined by the formulas
+γ(zk) = 0, w k=εα(zk),whereε2= 1, (5.4)
+have the following properties:
+(1)a pair(wk,zk)is a root of the characteristic polynomial (4.6).
+(2)a pair(zk,wk)is quasi-canonically conjugate with respect to the Lie-Pois son bracket
+(3.3):
+{zk,zl}2= 0,{zk,wl}2=−εzN+2
+kδkl,{wk,wl}2= 0; (5.5)372 J Bernatska and P Holod
+(3)the corresponding Liouville 1-form is
+ΩN+1=−/summationdisplay
+kεz−(N+2)
+kwkdzk.
+Proof.(1) The assertion is a direct consequence of (4.6) and (5.4).
+(2) It is evident that
+{zk,zl}2= 0,
+sincezk,k= 1,...,N+1 depend only on γm,m= 0,1,...,Nandγmmutually commute.
+Let us calculate the brackets {zk,wl}2and{wk,wl}2. From (5.4) we have
+∂zk
+∂αn= 0,∂zk
+∂γm=−zm
+k
+γ′(zk),∂wl
+∂αn=εzn
+l,∂wl
+∂γm=εα′(zl)∂zl
+∂γm.
+Further{γm,αn}2=−γm+n−N−1whenm+n/lessorequalslantN+1 and{γm,αn}2= 0 whenm+n>
+N+1. Thus, we obtain
+{zk,wl}2=1
+γ′(zk)zN+2
+kγ(zl)−zN+2
+lγ(zk)
+zk−zl,
+{wk,wl}2=/parenleftbigg1
+γ′(zk)−1
+γ′(zl)/parenrightbiggzN+2
+kγ(zl)−zN+2
+lγ(zk)
+zk−zl,
+whence
+{zk,wl}2=−zN+2
+kδkl,{wk,wl}2= 0.
+(3) From (5.4) it follows that Liouville 1-form on the orbit ON+1
+2redis
+ΩN+1=−/summationdisplay
+kεz−(N+2)
+kwkdzk.
+Thereduction to Liouville torus is done by fixingthe values o f Hamiltonians h0,h1, ...,
+hN. On the torus wkis the algebraic function of zkdue to (4.6). After the reduction the
+form Ω −1becomes a sum of meromorphic differentials on the Riemann surf aceP(w,z)=0.
+/squaresolid
+Remark 1. WhenγN+1= 0,αN+1/ne}ationslash= 0 one can replace αm,βm,γmbytm,sm,rm,m=
+0,1,...,N, according to (4.2). In this case the new coordinates ( zk,wk),k= 1,...,N+1,
+defined by the formulas
+s(zk) = 0, w k=εt(zk),whereε2= 1. (5.6)
+and have the same properties as in Theorem 4.
+The results of Theorem 3 and 4 can be summarized as follows. Li ouville tori for the
+nonlinear Schr¨ odinger equation and Heisenberg magnetic c hain have the same number of
+parameterizing variables zkand each variable belongs to the hyperelliptic curve (4.6) o f
+genusg=N+1. In other words, the common Liouville torus is the Jacobi v ariety of the
+curve (4.6).On Separation of Variables for Integrable Equations of Soli ton Type 373
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+arXiv:1001.0377v2  [math.AP]  19 Jan 2010GENERALIZED ELLIPTIC FUNCTIONS AND THEIR
+APPLICATION TO A NONLINEAR EIGENVALUE
+PROBLEM WITH p-LAPLACIAN
+SHINGO TAKEUCHI
+Dedicated to Professor Yoshio Yamada on occasion of his 60thbirthday
+Abstract. The Jacobian elliptic functions are generalized and
+applied to a nonlinear eigenvalue problem with p-Laplacian. The
+eigenvalue and the corresponding eigenfunction are represented
+in terms of common parameters, and a complete description of
+the spectra and a closed form representation of the correspond ing
+eigenfunctionsareobtained. Asaby-productoftherepresenta tion,
+it turns out that a kind of solution is also a solution of another
+eigenvalue problem with p/2-Laplacian.
+1.Introduction
+In this paper we generalize the Jacobian elliptic functions and apply
+them to a nonlinear eigenvalue problem
+(PEpq)/braceleftBigg
+(φp(u′))′+λφq(u)(1−|u|q) = 0, t∈(0,T),
+u(0) =u(T) = 0,
+whereT, λ>0, p, q > 1 andφm(s) =|s|m−2s(s/\e}atio\slash= 0),= 0 (s= 0).
+Problem (PE pq) appears frequently in various articles as station-
+ary problems. In particular, the equation for p=q= 2 is called,
+e.g., the Allen-Cahn equation, the Chafee-Infante equation [3], and a
+bistable reaction-diffusion equation with logistic effect. The equation
+forp= 2< qis said to be a bistable reaction-diffusion equation with
+Allee effect. In case p=nandq= 2 with an n-dimensional domain,
+an equation of this type is known as the Euler-Lagrange equation of
+functional related to models introduced by Ginzburg and Landau fo r
+the study of phase transitions (cf. Problem 17 in [2]).
+As to (PE pq) for general p>1, we have to mention the work [8] by
+Guedda and V´ eron [8]. They showed that if p=q>1 then there exists
+a positive increasing sequence {λn}such that a pair of solutions ±unof
+1991Mathematics Subject Classification. Primary 34L40, Secondary 33E05.
+This work was supported by KAKENHI (No. 20740094).
+12 SHINGO TAKEUCHI
+(PEpq)with(n−1)-zeroszj=jT/n(j= 1,2,...,n−1) bifurcates from
+the trivial solution at λ=λnand|un| →1 uniformly on any compact
+set of (0,T)\{z1,z2,...,z n−1}asλ→ ∞. Moreover, they proved that
+ifp=q>2 then for each n∈Nthere exists Λ n>λnsuch thatλ>Λn
+implies|un|= 1 onflat cores [zj−1+T
+2n(Λn
+λ)1/p,zj−T
+2n(Λn
+λ)1/p] (j=
+1,2,...,n) ofun, wherez0= 0 andzn=T. This is a great contrast
+to case 1< p=q≤2, where |un|<1 in [0,T]. Since the equation in
+(PEpq) is autonomous, if un(n≥2) has flat cores, then there exists
+uncountable solution with ( n−1)-zeros near un, which is produced
+by expanding and contracting the flat cores with preserving its tot al
+lengthT(1−(Λn
+λ)1/p). In this sense, the n-th branch ( λ,un) bifurcating
+from (λn,0) causes the second bifurcation at (Λ n,uΛn) for eachn≥2.
+The phenomena of flat corein [8] above was generalized to case p>2
+andq >1 by the author and Yamada [11]. They also studied change
+in bifurcation depending on the relation between pandq(as far as
+the first bifurcation is concerned, their proof can be applied to cas e
+1<p≤2), and showed that for each n∈N, ifp>qthen there exists
+a pair of solutions ±unof (PE pq) with (n−1)-zeros for λ>0; ifp=q
+then there exists λn>0 such that (PE pq) has no solution with ( n−1)-
+zeros forλ≤λnand (PE pq) has a pair of solutions ±unforλ > λ n
+(the same result as [8]); if p < qthen there exists λ∗
+n>0 such that
+(PEpq) has no solution with ( n−1)-zeros for λ<λ∗
+nand (PE pq) has a
+pair of solutions ±unforλ=λ∗
+nand (PE pq) has two pairs of solutions
+±un,±vnsatisfying |un(t)|>|vn(t)|witht/\e}atio\slash=zj(j= 0,1,...,n)
+forλ > λ∗
+n. In this sense, the point ( λ∗
+n,uλ∗n) causes the spontaneous
+bifurcation . In any case, each solution unhas flat cores for sufficiently
+largeλ.
+The purpose of this paper is to obtain a complete description of the
+spectra and a closed form representation of the corresponding e igen-
+functions of (PE pq), while the studies [8] and [11] above are done in
+the way of phase-plane analysis and no exact solution is given there.
+For the description and representation, we first recall that the J aco-
+bian elliptic function sn( t,k) with modulus k∈[0,1) satisfies
+u′′+u(1+k2−2k2u2) = 0. (1.1)
+(e.g., Example 4 of p.516 in the book [13] of Whittaker and Watson).
+Eq.(1.1) reminds that the solution of nonlinear eigenvalue problem
+(PEpq) withp=q= 2 can be represented explicitly by using sn( t,k).
+Indeed, for any given k∈(0,1), the set of eigenvalues of (PE 22) is givenGENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 3
+by
+λn(k) = (1+k2)/parenleftbigg2nK(k)
+T/parenrightbigg2
+(1.2)
+for eachn∈N, with corresponding eigenfunctions ±un,k, where
+un,k(t) =/radicalbigg
+2k2
+1+k2sn/parenleftbigg2nK(k)
+Tt,k/parenrightbigg
+(1.3)
+andK(k) is the complete elliptic integral of the first kind
+K(k) =/integraldisplay1
+0ds/radicalbig
+(1−s2)(1−k2s2)
+(cf. Section 2 in [1]). Conversely, all nontrivial solutions are given by
+Eqs.(1.2) and (1.3), and in particular, it follows from Eq.(1.3) that a ll
+solutions satisfy |u|<1.
+In our study on (PE pq), after the fashion of Jacobi’s sn( t,k), we
+introduce a new transcendental function sn pq(t,k) with modulus k∈
+[0,1). This satisfies
+(φp(u′))′+q
+p∗φq(u)(1+kq−2kq|u|q) = 0, (1.4)
+wherep∗:=p/(p−1). Using sn pq(t,k), we can obtain a complete de-
+scription of the set of eigenvalues and the corresponding eigenfun ctions
+of (PE pq) as Eqs.(1.2) and (1.3) with
+Kpq(k) =/integraldisplay1
+0ds
+p/radicalbig
+(1−sq)(1−kqsq).
+It is important that Kpq(k) converges to Kp
+2,q(0) ask→1−0 if and
+only ifp>2. Indeed,
+lim
+k→1−0Kpq(k) =/integraldisplay1
+0ds
+(1−sq)2
+p=Kp
+2,q(0).
+Similarly, sn pq(t,k) converges to sn p
+2,q(t,0) ask→1−0. These conver-
+gent properties yield the existence of special solutions, not neces sarily
+|u|<1, and we can really construct the solutions of (PE pq) with flat
+cores. Moreover, sn p
+2,q(t,0) satisfies Eq.(1.4) with k= 0 andpreplaced
+byp/2 as well as Eq.(1.4) with k= 1. Thus, we obtain the following
+(curious) property: a kind of solution of (PE pq) is also a solution of the
+nonlinear eigenvalue problem with p/2-Laplacian
+/braceleftBigg
+(φp
+2(u′))′+λφq(u) = 0, t∈(0,T),
+u(0) =u(T) = 0.4 SHINGO TAKEUCHI
+Thispaperisorganizedasfollows. InSection2,weintroduceagener -
+alized trigonometric function sin pq(t) given by Dr´ abek and Man´ asevich
+[7] and define a new transcendental function sn pq(t,k), which is a gen-
+eralization of the Jacobian elliptic function sn( t,k) and an extension
+of sinpq(t) as sn pq(t,0) = sin pq(t). In Section 3, we apply them to non-
+linear eigenvalue problems, particularly to the problem considered in
+[8] and [11], and obtain complete descriptions of the set of eigenvalue s
+and the corresponding eigenfunctions.
+2.Transcendental Functions
+2.1.Generalized trigonometric functions. Generalizedtrigonomet-
+ric functions were introduced by Dr´ abek and Man´ asevich [7] (see also
+[6]). Forσ∈[0,1], we define (in a slightly different way from [7])
+arcsin pq(σ) :=/integraldisplayσ
+0ds
+(1−sq)1
+p, (2.1)
+wherep>1, q>0. Lettings=z1/q, we have
+arcsin pq(σ) =1
+q/integraldisplayσq
+0z1
+q−1(1−z)−1
+pdz=1
+q˜B/parenleftbigg1
+q,1
+p∗,σq/parenrightbigg
+,
+where˜B(s,t,u) denotes the incomplete beta function
+˜B(s,t,u) =/integraldisplayu
+0zs−1(1−z)t−1dz.
+We define the constant πpqas
+πpq:= 2arcsin pq(1) =2
+qB/parenleftbigg1
+q,1
+p∗/parenrightbigg
+,
+whereB(s,t) denotes the beta function
+B(s,t) =˜B(s,t,1) =/integraldisplay1
+0zs−1(1−z)t−1dz.
+We have that arcsin pq: [0,1]→[0,πpq/2], and is strictly increasing.
+Let us denote its inverse by sin pq. Then, sin pq: [0,πpq/2]→[0,1] and
+is strictly increasing. We extend sin pqto allR(and still denote this
+extension by sin pq) in the following form: for t∈[πpq/2,πpq], we set
+sinpq(t) := sin pq(πpq−t), then for t∈[−πpq,0], we define sin pq(t) :=
+−sinpq(−t), and finally we extend sin pqto allRas a 2πpqperiodic
+function.
+When 0< p≤1, we also define arcsin pqas Eq.(2.1) for σ∈[0,1).
+We have that arcsin pq: [0,1)→[0,∞), and is strictly increasing. Let
+us denote its inverse by sin pq. Then, sin pq: [0,∞)→[0,1) and isGENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 5
+strictly increasing. We extend sin pqto allRas sinpq(t) :=−sinpq(−t)
+fort∈(−∞,0] and still denote this extension by sin pq.
+Remark2.1. We immediately find that sin 22(t) = sin(t) andπ22=π
+from the properties of the beta function. Moreover, sin pp(t) = sin p(t)
+andπpp=πp=2π
+psinπ
+p, where sin pandπpare the generalized sine
+function and its half-period, respectively, appearing in [4], [5] and [6].
+We define for t∈[0,πpq/2] (in case 0 <p≤1, fort∈[0,∞))
+cospq(t) := (1−sinq
+pq(t))1
+p,
+then we obtain
+cosp
+pq(t)+sinq
+pq(t) = 1,
+d
+dtsinpq(t) = cos pq(t).
+Proposition 2.1. Forp, q >1,sinpqsatisfies for all R
+(2.2) ( φp(u′))′+q
+p∗φq(u) = 0.
+Proof.Fort∈(0,πpq/2) we have
+(φp(u′))′= (φp(cospq(t)))′
+= ((1−sinq
+pq(t))1
+p∗)′
+=1
+p∗(1−sinq
+pq(t))−1
+p·(−qsinq−1
+pq(t))·cospq(t)
+=−q
+p∗φq(u).
+By symmetry of sin pq, Eq.(2.2) holds true for t/\e}atio\slash=tn:=nπpq/2, n∈Z.
+Since lim t→tn(φp(u′))′exists,φp(u′) is differentiable also at t=tnand
+satisfies Eq.(2.2) for all Rin the classical sense. /square
+2.2.Generalized Jacobian elliptic functions. We shall introduce
+new transcendental functions, which generalize the Jacobian ellipt ic
+functions. For σ∈[0,1] andk∈[0,1), we define
+arcsnpq(σ) = arcsn pq(σ,k) :=/integraldisplayσ
+0ds
+p/radicalbig
+(1−sq)(1−kqsq), (2.3)
+wherep>1, q>0. We define the constant Kpq(k) as
+Kpq=Kpq(k) := arcsn pq(1,k) =/integraldisplay1
+0ds
+p/radicalbig
+(1−sq)(1−kqsq)6 SHINGO TAKEUCHI
+We have that arcsn pq: [0,1]→[0,Kpq], and is strictly increasing.
+Let us denote its inverse by sn pq(·) = sn pq(·,k). Then, sn pq: [0,Kpq]→
+[0,1] and is strictly increasing. We extend sn pqto allR(and still
+denotethis extension by sn pq) inthefollowing form: for t∈[Kpq,2Kpq],
+we set sn pq(t) := sn pq(2Kpq−t), then for t∈[−2Kpq,0], we define
+snpq(t) :=−snpq(−t), and finally we extend sn pqto allRas a 4Kpq
+periodic function.
+When 0< p≤1, we also define arcsn pqas Eq.(2.3) for σ∈[0,1).
+We have that arcsn pq: [0,1)→[0,∞), and is strictly increasing. Let
+us denotes its inverse by sn pq(·) = sn pq(·,k). Then, sn pq: [0,∞)→
+[0,1) and is strictly increasing. We extend sn pqto allRas snpq(t) :=
+−snpq(−t) fort∈(−∞,0] and still denote this extension by sn pq.
+The following proposition is crucial to our study.
+Proposition 2.2. Forp, q>0,Kpqis continuous and strictly increas-
+ing in[0,1),2Kpq(0) =πpqandsnpq(t,0) = sin pq(t). Moreover,
+lim
+k→1−02Kpq(k) =/braceleftBigg
+πp
+2,qifp>2,
+∞if0<p≤2,
+lim
+k→1−0snpq(t,k) = sin p
+2,q(t).
+Proof.The first half is trivial from the definitions of Kpqand sn pq. If
+p>2, then the monotone convergence theorem of Beppo Levi gives
+lim
+k→1−02Kpq(k) = 2/integraldisplay1
+0ds
+(1−sq)2
+p= 2arcsin p
+2,q(1) =πp
+2,q.
+If 0< p≤2, then 2Kpq(k) diverges to ∞ask→1−0 by Fatou’s
+lemma.
+The last property is proved as follows. By the symmetry of sn pq(·,k),
+we may assume t >0. Suppose p >2 and that there exist t0, ε >0
+and{kj}such thatkj→1 asj→ ∞and
+|σkj−sinp
+2,q(t0)| ≥ε, (2.4)
+whereσkj= snpq(t0,kj). Letn∈Zbe the number satisfying t0∈
+In:= [nπp
+2,q/2,(n+1)πp
+2,q/2) andj∈Na large number satisfying t0∈
+In(kj) := [nKpq(kj),(n+1)Kpq(kj)). We write sn(n)
+pq(·,kj) as sn pq(·,kj)
+onIn(kj) and sin(n)
+pq(·) as sin pq(·) onIn. Now, since σkjis bounded, we
+can choose a subsequence {kj′}of{kj}such thatσkj′→σfor some
+σ∈[−1,1] asj′→ ∞. Thus, asj′→ ∞
+t0=nKpq(kj′)+arcsn pq(σkj′)→nπp
+2,q
+2+arcsin p
+2,q(σ),GENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 7
+and henceσ= sin(n)
+p
+2,q(t0), which contradicts (2.4). The proof to case
+0<p≤2 is similar and we omit it. /square
+Remark2.2. In case p>2, 2Kpq(k) and sn pq(·,k) converge to the finite
+valueπp
+2,qandto thefinite-periodicfunction sin p
+2,qask→1−0, respec-
+tively. This is quite different from case p= 2, where 2 K2q(k) diverges
+to∞and sn 22(t,k) converges to the monotone increasing function
+sin12(t) = tanh(t) ask→1−0.
+We define for t∈[0,Kpq] (in case 0 <p≤1, fort∈[0,∞))
+cnpq(t) := (1−snq
+pq(t))1
+p,
+dnpq(t) := (1−kqsnq
+pq(t))1
+p,
+then we obtain
+cnp
+pq(t)+snq
+pq(t) = 1,
+d
+dtsnpq(t) = cn pq(t)dnpq(t).
+Proposition 2.3. Forp, q >1,snpqsatisfies for all R
+(2.5) ( φp(u′))′+q
+p∗φq(u)(1+kq−2kq|u|q) = 0,
+which includes Eq. (2.2)as casek= 0.
+Proof.Fort∈(0,Kpq(k)) we have
+(φp(u′))′= (φp(cnpq(t)dnpq(t)))′
+= (((1−sinq
+pq(t))(1−kqsinq
+pq(t)))1
+p∗)′
+=1
+p∗((1−sinq
+pq(t))(1−kqsinq
+pq(t)))−1
+p
+×(−qsinq−1
+pq(t)·(1+kq−2kqsinq
+pq(t)))·cnpq(t)dnpq(t)
+=−q
+p∗φq(u)(1+kq−2kquq).
+By symmetry of sn pq, Eq.(2.5) holds true for t/\e}atio\slash=tn:=nKpq(k), n∈Z.
+Since lim t→tn(φp(u′))′exists,φp(u′) is differentiable also at t=tnand
+satisfies Eq.(2.5) for all Rin the classical sense. /square
+Remark2.3. Letting s= sinpq(t) in Eq.(2.3), we have
+arcsnpq(σ,k) =/integraldisplayarcsinpq(σ)
+0dt
+p/radicalBig
+1−kqsinq
+pq(t).8 SHINGO TAKEUCHI
+We define the amplitude function am pq(·,k) : [0,Kpq(k)]→[0,πpq/2]
+by
+t=/integraldisplayampq(t,k)
+0dθ
+p/radicalBig
+1−kqsinq
+pq(θ),
+thus sn pqis represented by sin pqas
+snpq(t,k) = sin pq(ampq(t,k)).
+3.Applications
+3.1.The(p,q)-eigenvalue problem. LetT, λ>0 andp, q>1. We
+consider the nonlinear eigenvalue problem
+(Epq)/braceleftBigg
+(φp(u′))′+λφq(u) = 0, t∈(0,T),
+u(0) =u(T) = 0.
+Problem (E pq) has been studied by many authors. In particular, in pa-
+per [10] of ˆOtani, the existence of infinitely many multi-node solutions
+was proved by using subdifferential operators method and phase- plane
+analysis combined with symmetry properties of the solutions. After
+that, Dr´ abek and Man´ asevich [7] provided explicit forms of the wh ole
+spectrum and the corresponding eigenfunctions for (E pq) (see also [6]).
+We follow [7] to understand completely the set of all solutions of (E pq).
+It will be convenient to find first the solution to the initial value
+problem
+(3.1)/braceleftBigg
+(φp(u′))′+λφq(u) = 0,
+u(0) = 0, u′(0) =α,
+where without loss of generality we may assume α>0.
+Letube a solution to Eq.(3.1) and let t(α) be the first zero point of
+u′(t). On interval (0 ,t(α)),usatisfiesu(t)>0 andu′(t)>0, and thus
+u′(t)p
+p∗+λu(t)q
+q=λRq
+q=αp
+p∗,
+whereR=u(t(α))>0. Solving for u′and integrating, we find
+/parenleftbiggq
+λp∗/parenrightbigg1
+p/integraldisplayt
+0u′(s)
+(Rq−u(s)q)1
+pds=t,
+which after a change of variable can be written as
+t=/parenleftbiggq
+λp∗/parenrightbigg1
+p1
+Rq
+p−1/integraldisplayu(t)
+R
+0ds
+(1−sq)1
+p=/parenleftbiggq
+λp∗/parenrightbigg1
+p1
+Rq
+p−1arcsin pq/parenleftbiggu(t)
+R/parenrightbigg
+.GENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 9
+Thus we obtain the solution to Eq.(3.1) can be written as
+u(t) =Rsinpq/parenleftBigg/parenleftbiggλp∗
+q/parenrightbigg1
+p
+Rq
+p−1t/parenrightBigg
+, (3.2)
+whereR= (q
+λp∗)1
+qαp
+q.
+Theorem 3.1. All nontrivial solutions of (Epq)are given as follows.
+For any given R>0, the set of eigenvalues of (Epq)is given by
+λn(R) =q
+p∗/parenleftBignπpq
+T/parenrightBigp
+Rp−q(3.3)
+for eachn∈N, with corresponding eigenfunctions ±un,R, where
+un,R(t) =Rsinpq/parenleftBignπpq
+Tt/parenrightBig
+. (3.4)
+Proof.For givenR >0, by imposing that uin Eq.(3.2) satisfies the
+boundaryconditions in (E pq), we obtain that λisaneigenvalue of (E pq)
+if and only if
+/parenleftbiggλp∗
+q/parenrightbigg1
+p
+Rq
+p−1T=nπpq, n∈N,
+and hence Eq.(3.3) follows. Expression (3.4) for the eigenfunction s
+follows directly from Eq.(3.2). /square
+3.2.A perturbed (p,q)-eigenvalue problem. LetT, λ > 0 and
+p, q>1. We consider the nonlinear eigenvalue problem
+(PEpq)/braceleftBigg
+(φp(u′))′+λφq(u)(1−|u|q) = 0, t∈(0,T),
+u(0) =u(T) = 0.
+Problem(PE pq)hasbeenstudiedbyBergerandFraenkel[1]andChafee
+and Infante [3] ( p=q= 2), Wang and Kazarinoff [12] and Korman, Li
+and Ouyang [9] ( p= 2< q), Guedda and V´ eron [8] ( p=q >1), and
+Takeuchi and Yamada [11] ( p>2, q >1). However, there is no study
+providing explicit forms of the whole spectrum and the correspondin g
+eigenfunctions for (PE pq).
+As we have done for (E pq), it will be convenient to find first the
+solution to the initial value problem
+(3.5)/braceleftBigg
+(φp(u′))′+λφq(u)(1−|u|q) = 0,
+u(0) = 0, u′(0) =α,
+where without loss of generality we may assume α>0.10 SHINGO TAKEUCHI
+Letube a solution to Eq.(3.5) and let t(α) be the first zero point of
+u′(t). On interval (0 ,t(α)),usatisfiesu(t)>0 andu′(t)>0, and thus
+u′(t)p
+p∗+λF(u)
+q=λF(R)
+q=αp
+p∗,
+whereF(s) =sq−1
+2s2qandR=u(t(α)). Since we are interested
+in functions satisfying the boundary condition of (PE pq), it suffices to
+assume 0< R≤1, which means |u| ≤1. Moreover, we restrict to
+0<R<1 and concentrate solutions satisfying |u|<1 for a while.
+Solving for u′and integrating, we find
+/parenleftbiggq
+λp∗/parenrightbigg1
+p/integraldisplayt
+0u′(s)
+p/radicalbig
+F(R)−F(u(s))ds=t,
+which after a change of variable can be written as
+t=/parenleftbiggq
+λp∗/parenrightbigg1
+p/integraldisplayu(t)
+R
+0R
+p/radicalbig
+F(R)−F(Rs)ds. (3.6)
+It is easy to verify that
+F(R)−F(Rs) =F(R)(1−sq)/parenleftbigg
+1−Rq
+2−Rqsq/parenrightbigg
+,
+and hence
+t=/parenleftbiggq
+λp∗/parenrightbigg1
+pR
+F(R)1
+p/integraldisplayu(t)
+R
+0ds
+p/radicalbig
+(1−sq)(1−kqsq)/parenleftbigg
+kq:=Rq
+2−Rq/parenrightbigg
+=/parenleftbiggq
+λp∗/parenrightbigg1
+pR
+F(R)1
+parcsnpq/parenleftbiggu(t)
+R,k/parenrightbigg
+.
+Then we obtain that the solution to Eq.(3.5) can be written as
+u(t) =Rsnpq/parenleftBigg/parenleftbiggλp∗
+q/parenrightbigg1
+pF(R)1
+p
+Rt,k/parenrightBigg
+, (3.7)
+where
+k=/parenleftbiggRq
+2−Rq/parenrightbigg1
+q
+, (3.8)
+R=/parenleftbiggq
+λp∗/parenrightbigg1
+q
+αp
+q/parenleftbigg1
+2+1
+2/radicalbigg
+1−2q
+λp∗αp/parenrightbigg−1
+q
+.
+We first observe the structure of the set of all nontrivial solution s of
+(PEpq) satisfying |u|<1.GENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 11
+Theorem 3.2 (|u|<1).All nontrivial solutions for p∈(1,2]and all
+nontrivial solutions with |u|<1forp >2are given as follows. For
+any givenk∈(0,1), the set of eigenvalues of (PEpq)is given by
+λn(k) =q
+p∗(1+kq)/parenleftbigg2kq
+1+kq/parenrightbiggp
+q−1/parenleftbigg2nKpq(k)
+T/parenrightbiggp
+(3.9)
+for eachn∈N, with corresponding eigenfunctions ±un,k, where
+un,k(t) =/parenleftbigg2kq
+1+kq/parenrightbigg1
+q
+snpq/parenleftbigg2nKpq(k)
+Tt,k/parenrightbigg
+. (3.10)
+Proof.Fork∈(0,1) given, we impose that Function (3.7) with R∈
+(0,1)decidedfromEq.(3.8)satisfiestheboundaryconditionsin(PE pq).
+Then, we obtain that λis an eigenvalue of (PE pq) if and only if
+/parenleftbiggλp∗
+q/parenrightbigg1
+pF(R)1
+p
+RT= 2nKpq(k), n∈N.
+From Eq.(3.8) again we have
+F(R)1
+p
+R=/parenleftbigg2kq
+1+kq/parenrightbigg1
+p−1
+q
+(1+kq)−1
+p,
+and hence we obtain Eq.(3.9). Expression (3.10) for the eigenfunc tions
+follows then directly from Eq.(3.7).
+It remains to show that no other nontrivial solution of (PE pq) is
+obtained when 1 < p≤2. Assume the contrary. Then there exist
+t∗>0 and a nontrivial solution uof (PE pq) withR=u(t∗) = 1.
+However, the right-hand side of Eq.(3.6) with t=t∗diverges because
+p/radicalbig
+F(1)−F(s) =O((1−sq)2
+p) ass→1−0. Thus,t∗=∞, which is a
+contradiction. /square
+Next we find solutions of (PE pq) with|u| ≤1, except the solu-
+tions given by Theorem 3.2. From Proposition 2.2, one of solutions
+of Eq.(3.5) is obtained by k→1−0 in Eq.(3.7) with Eq.(3.8), namely
+u(t) = sin p
+2,q/parenleftBigg/parenleftbiggλp∗
+2q/parenrightbigg1
+p
+t/parenrightBigg
+.
+Now we assume p>2 and take a number t∗as (λp∗
+2q)1
+pt∗=πp
+2,q/2, then
+uattains 1 at t=t∗(note that it is impossible to obtainsuch a solution
+when 1< p≤2). Using this u, we can make the other solutions of
+Eq.(3.5) as follows. In the phase-plane, the orbit ( u(t),u′(t)) arrives at
+the equilibrium point (1 ,0) att=t∗and can stay there for any finite
+timeτbefore it begins to leave there. Then, the interval [ t∗,t∗+τ] is a12 SHINGO TAKEUCHI
+flat core of the solution. Similarly, there is the other equilibrium point
+(−1,0), where the orbit can stay, and the solution has another flat cor e
+of any finite length. Thus we have solutions of Eq.(3.5) attaining ±1
+with any number of flat cores.
+Theorem 3.3 (|u| ≤1).Letp >2, then all nontrivial solutions are
+given as follows, in addition to Theorem 3.2. For any given τ∈[0,T),
+the set of eigenvalues of (PEpq)is given by
+Λn(τ) =2q
+p∗/parenleftbiggnπp
+2,q
+T−τ/parenrightbiggp
+for eachn∈N, with corresponding eigenfunctions ±un,{τi}, where
+un,{τi}is any function given as follows: for any {τi}n
+i=1withτi≥0
+and/summationtextn
+i=1τi=τ
+un,{τi}(t) =
+
+(−1)j−1sinp
+2,q/parenleftBignπp
+2,q
+T−τ(t−Tj−1)/parenrightBig
+ifTj−1≤t≤Tj−1+T−τ
+2n,
+(−1)j−1ifTj−1+T−τ
+2n≤t≤Tj−T−τ
+2n,
+(−1)j−1sinp
+2,q/parenleftBignπp
+2,q
+T−τ(Tj−t)/parenrightBig
+ifTj−T−τ
+2n≤t≤Tj,
+j= 1,2,...,n,(3.11)
+whereT0= 0andTj=(T−τ)j
+n+/summationtextj
+i=1τiforj= 1,2,...,n.
+Proof.For eachn∈N, it suffices to construct solutions with ( n−1)-
+zeros. Letτ∈[0,T). They are all generated by the eigenvalue and the
+corresponding eigenfunction of (PE pq) withTreplaced by T−τ
+Λn(τ) =2q
+p∗/parenleftbiggnπp
+2,q
+T−τ/parenrightbiggp
+,
+un,τ(t) = sin p
+2,q/parenleftbiggnπp
+2,q
+T−τt/parenrightbigg
+,
+which are obtained from Eqs.(3.9) and (3.10) with k→1−0, re-
+spectively. In the phase-plane, the orbit ( un,τ(t),u′
+n,τ(t)) goes through
+the equilibrium points ( ±1,0) inn-times without staying there as t
+increases from 0 to T−τ. Therefore, if the orbit stays the i-th equilib-
+rium point for time τi, whereτ1+τ2+···+τn=τ, then we can obtain
+Solution (3.11) with n-flat cores in [0 ,T]. /square
+In Theorems 3.2 and 3.3, we give parameters kandτto obtain the
+eigenvalue and the corresponding eigenfunction of (PE pq). Conversely,
+giving any λ>0, we can observe the set Sλof all solutions of (PE pq)
+by considering the inverses of λnand Λ n.GENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 13
+Theorem 3.4. Letp>1andq>1.
+Casep>q. For anyλ>0there exists a strictly decreasing positive
+sequence {kj}∞
+j=1such thatkj→0asj→ ∞and
+Sλ={0}∪∞/uniondisplay
+j=1{±uj,kj}.
+Casep=q. If
+0<λ≤q
+p∗/parenleftBigπpq
+T/parenrightBigp
+,
+thenSλ={0}. If
+q
+p∗/parenleftBignπpq
+T/parenrightBigp
+<λ≤q
+p∗/parenleftbigg(n+1)πpq
+T/parenrightbiggp
+, n∈N,
+then there exists a strictly decreasing positive sequence {kj}n
+j=1such
+that
+Sλ={0}∪n/uniondisplay
+j=1{±uj,kj}.
+Casep < q. There exists λ1>0such that if 0< λ < λ 1, then
+Sλ={0}. Ifnpλ1≤λ<(n+1)pλ1, n∈N, then there exist a strictly
+decreasing positive sequence {kj}n
+j=1and a strictly increasing positive
+sequence {ℓj}n
+j=1such thatkj>ℓj, j= 1,2,...,n−1and
+Sλ={0}∪n/uniondisplay
+j=1{±uj,kj}∪n/uniondisplay
+j=1{±uj,ℓj},
+whereun,kn=un,ℓnwithkn=ℓnforλ=npλ1and|un,kn|>|un,ℓn|(t/\e}atio\slash=
+jT/n, j= 1,2,...,n−1)withkn>ℓnotherwise.
+In any case, each kj, ℓjis calculated by Eq. (3.9)forλj, and the
+corresponding solution is given in Form (3.10).
+When1<p≤2, we havekj<1. Whenp>2, in addition, if
+λ≥2q
+p∗/parenleftbiggmπp
+2,q
+T/parenrightbiggp
+, m∈N,
+then for each j= 1,2,...,m, the set {±uj,kj}above is replaced by
+∪{τi}{±uj,{τi}}, where∪{τi}is the union for all {τi}j
+i=1satisfyingτi≥0
+and
+j/summationdisplay
+i=1τi=T−jπp
+2,q/parenleftbigg2q
+λp∗/parenrightbigg1
+p
+.
+The nontrivial solution uj,{τi}is given in Form (3.11).14 SHINGO TAKEUCHI
+Proof.First we assume 1 <p≤2. In this case, we have already known
+that all nontrivial solutions of (PE pq) are obtained by Theorem 3.2.
+Now we fix λ>0. We obtain that λis thej-th eigenvalue of (PE pq)
+if and only if from Eq.(3.9) there exists k∈(0,1) such that
+T
+2j/parenleftbiggλp∗
+q/parenrightbigg1
+p
+= (1+kq)1
+p/parenleftbigg2kq
+1+kq/parenrightbigg1
+q−1
+p
+Kpq(k) =: Φ(k). (3.12)
+Casep > q. Φ(k) is strictly increasing in (0 ,1) and it follows from
+Proposition 2.2 that Φ(0) = 0 and lim k→1−0Φ(k) =∞. Thus, there
+exists a unique k=kj(λ) satisfying Eq.(3.12). For jandkj, a unique
+solutionuj,kjof (PE pq) is obtained by Eq.(3.10).
+Casep=q. Φ(k) is strictly increasing in (0 ,1) and it follows from
+Proposition 2.2 that Φ(0) = πpq/2 and lim k→1−0Φ(k) =∞. Thus, if
+T
+2j/parenleftbiggλp∗
+q/parenrightbigg1
+p
+>πpq
+2,
+namely,
+λ>q
+p∗/parenleftbiggjπpq
+T/parenrightbiggp
+,
+then there exists a unique k=kj(λ) satisfying Eq.(3.12). For jand
+kj, a unique solution uj,kjof (PE pq) is obtained by Eq.(3.10).
+Casep < q. It is clear that lim k→+0Φ(k) = lim k→1−0Φ(k) =∞.
+Changing variable r=kq
+1+kq, we can write Φ as
+Ψ(r) =/integraldisplay1
+0(1+sq)1
+p−1
+q
+(1−sq)1
+pψ((1+sq)r)ds, r∈(0,1/2),
+whereψ(t) =t1
+q−1
+p(1−t)−1
+p. It is easy to see that ψis convex in (0 ,1)
+becauseψ(t)>0 and
+(logψ(t))′′=/parenleftbigg1
+p−1
+q/parenrightbigg1
+t2+1
+p1
+(1−t)2>0.
+Then, Ψ is twice-differentiable in (0 ,1/2) and
+Ψ′′(r) =/integraldisplay1
+0(1+sq)1
+p−1
+q+2
+(1−sq)1
+pψ′′((1+sq)r)ds>0.
+Thus, Ψ is convex and there exists k∗∈(0,1) such that Φ( k∗) is the
+only one critical value, and hence the minimum of Φ in (0 ,1).GENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 15
+If
+T
+2j/parenleftbiggλp∗
+q/parenrightbigg1
+p
+= Φ(k∗),
+namely,
+λ=jpλ1:=q
+p∗/parenleftbigg2jΦ(k∗)
+T/parenrightbiggp
+,
+thenk∗satisfies Eq.(3.12). For jandk∗, a unique solution uj,k∗of
+(PEpq) is obtained by Eq.(3.10). Moreover, if
+T
+2j/parenleftbiggλp∗
+q/parenrightbigg1
+p
+>Φ(k∗),
+namely,λ>jpλ1, then there exist k=kj(λ) andℓj(λ) such that
+kj(λ) = Φ−1/parenleftBigg
+T
+2j/parenleftbiggλp∗
+q/parenrightbigg1
+p/parenrightBigg
+∈(k∗,1),
+ℓj(λ) = Φ−1/parenleftBigg
+T
+2j/parenleftbiggλp∗
+q/parenrightbigg1
+p/parenrightBigg
+∈(0,k∗).
+Forj,kjandℓj, solutions uj,kjanduj,ℓjof (PE pq) are obtained by
+Eq.(3.10).
+Next, we assume p >2. In any case, a similar proof as above with
+limk→1−0Φ(k) = 21
+p−1πp
+2,qinstead of lim k→1−0Φ(k) =∞gives that it
+is impossible to find km∈(0,1) above satisfying Eq.(3.12), provided
+λ≥2q
+p∗/parenleftbiggmπp
+2,q
+T/parenrightbiggp
+, m∈N.
+Then, however, for each j= 1,2,...,m, we can take τ∈[0,T) such
+that
+λ=2q
+p∗/parenleftbiggjπp
+2,q
+T−τ/parenrightbiggp
+,
+and Theorem 3.3 yields the solutions uj,{τi}, where{τi}j
+i=1is any se-
+quence satisfying that τi≥0,/summationtextj
+i=1τi=τ. /square
+It follows directly from Representation (3.11) of Theorem 3.3 that a
+kind of solution of (PE pq) withp-Laplacian is also an eigenfunction of
+(Ep
+2,q) withp/2-Laplacian.16 SHINGO TAKEUCHI
+Corollary 3.1. Letp >2. For each n∈N, any solution un,{τi}of
+(PEpq)in Theorem 3.3 satisfies
+(φp
+2(u′))′+(p−2)q
+p/parenleftbiggnπp
+2,q
+T−τ/parenrightbiggp
+2
+φq(u) = 0
+in the intervals where |un,{τi}|<1, whereτ=/summationtextn
+i=1τi. In particu-
+lar, for each n∈N, the solution un,{0}of(PEpq)withτ= 0is an
+eigenfunction of (Ep
+2,q), that is,
+
+
+(φp
+2(u′))′+(p−2)q
+p/parenleftBignπp
+2,q
+T/parenrightBigp
+2φq(u) = 0, t∈(0,T),
+u(0) =u(T) = 0.
+Moreover,un,{0}is characterized by un,RwithR= 1in Eq.(3.4)with
+preplaced by p/2.
+References
+[1] M.S.Berger and L.E.Fraenkel, On the asymptotic solution of a non linear
+Dirichlet problem, J. Math. Mech. 19(1969/1970), 553–585.
+[2] F.Bethuel, H.BrezisandF.Helein, Ginzburg-Landau vortices , ProgressinNon-
+linear Differential Equations and their Applications, 13. Birkhauser B oston,
+Inc., Boston, MA, 1994.
+[3] N.Chafee and E.F.Infante, A bifurcation problem for a nonlinear partial dif-
+ferential equation of parabolic type, Applicable Anal. 4(1974/75), 17–37.
+[4] O.Doˇ sl´ y, Half-linear differential equations, Handbook of differential equations ,
+161–357, Elsevier/North-Holland, Amsterdam, 2004.
+[5] O.Doˇ sl´ yand P. ˇReh´ ak,Half-linear differential equations , North-Holland Math-
+ematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005.
+[6] P.Dr´ abek, P.Krejˇ c´ ı and P.Tak´ aˇ c, Nonlinear differential equations , Papers
+from the Seminar on Differential Equations held in Chvalatice, June 29 –July3,
+1998. Chapman & Hall/CRC Research Notes in Mathematics, 404. Cha pman
+& Hall/CRC, Boca Raton, FL, 1999.
+[7] P.Dr´ abek and R.Man´ asevich, On the closed solution to some no nhomoge-
+neous eigenvalue problems with p-Laplacian, Differential Integral Equations
+12(1999), 773–788.
+[8] M.Guedda and L.V´ eron, Bifurcation phenomena associated to thep-Laplace
+operator, Trans. Amer. Math. Soc. 310(1988), 419–431.
+[9] P.Korman, Y.Li and T.Ouyang, Exact multiplicity results for bou ndary value
+problems with nonlinearities generalising cubic, Proc. Roy. Soc. Edinburgh
+Sect. A126(1996), 599–616.
+[10] M.ˆOtani, On certain second order ordinary differential equations ass ociated
+with Sobolev-Poincar´ e-typeinequalities, Nonlinear Anal. 8(1984), 1255–1270.
+[11] S.Takeuchi and Y.Yamada, Asymptotic properties of a react ion-diffusion
+equation with degenerate p-Laplacian, Nonlinear Anal. 42(2000), 41–61.
+[12] S-H.Wang and N.D.Kazarinoff, Bifurcation and stability of positiv e solutions
+of a two-point boundary value problem, J. Austral. Math. Soc. Ser. A 52
+(1992), 334–342.GENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION 17
+[13] E.T.Whittaker and G.N.Watson, A course of modern analysis , An introduc-
+tion to the general theory of infinite processes and of analytic fun ctions; with
+an account of the principal transcendental functions. Reprint o f the fourth
+(1927) edition. Cambridge Mathematical Library.CambridgeUniver sity Press,
+Cambridge, 1996.
+Department ofGeneral Education, Kogakuin University, 266 5-1 Nakano,
+Hachioji, Tokyo 192-0015, JAPAN
+E-mail address :shingo@cc.kogakuin.ac.jp
\ No newline at end of file
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+arXiv:1001.0378v1  [math.AP]  3 Jan 2010L∞estimates and integrability by compensation
+in Besov-Morrey spaces and applications
+Laura Gioia Andrea Keller∗
+October 30, 2018
+Abstract
+L∞estimates in the integrability by compensation result of H. Wente ([26])
+fail in dimension larger than two when Sobolev spaces are rep laced by the ad-
+hoc Morrey spaces (in dimension n≥3).
+However, in this paper we prove that L∞estimates hold in arbitrary dimension
+whenMorrey spaces are replaced bytheir Littlewood Paley co unterparts: Besov-
+Morrey spaces.
+As an application we prove the existence of conservation law s to solution of
+elliptic systems of the form
+−∆u= Ω·∇u
+where Ω is antisymmetric and both ∇uand Ω belong to these Besov-Morrey
+spaces for which the system is critical.
+1 Introduction
+In this section we will give the precise statement of our results and a dd some
+remarks.
+Forthesakeofsimplicity, inwhatfollowswewillusetheabbreviation axfor∂
+∂xa.
+Our work was motivated by Rivi` ere’s [14] article about Schr¨ odinge r systems
+with antisymmetric potentials, i.e. systems of the form
+−∆u= Ω·∇u (1)
+withu∈W1,2(ω,Rm) and Ω∈L2(ω,so(m)⊗Λ1Rn),ω⊂Rn.
+The differential equation (1) has to be understood in the following se nse:
+For all indices i∈ {1,...,m}we have−∆ui=/summationtextm
+j=1Ωi
+j·∇ujandL2(ω,so(m)⊗
+Λ1Rn) means that ∀i,j∈ {1,...,m},Ωi
+j∈L2(ω,Λ1Rn) and Ωi
+j=−Ωj
+i.
+In particular, it was the result that in dimension n= 2 solutions to (1) are
+continuous which attracted our interest.
+The interest for such systems originatesin the fact that they ”en code” all Euler-
+Lagrange equations for conformally invariant quadratic Lagrangia ns in dimen-
+sion 2 (see [14] and also [9]).
+∗Department of Mathematics, ETH Z¨ urich, 8092 Z¨ urich, Swit zerland
+1In what follows we will take ω=Bn
+1(0), the n-dimensional unit ball.
+In the above cited work, there were three crucial ideas:
+•Antisymmetry of Ω
+If we drop the assumption that Ω is symmetric, there may occur solu tions
+which are not continuous as the following example shows:
+Letn= 2,ui= 2loglog1
+rfori= 1,2 and let
+Ω =/parenleftbigg
+∇u10
+0∇u2/parenrightbigg
+Obviously, usatisfies equation (1) with the given Ω but is not continuous.
+•Construction of conservation laws
+In fact, once there exist A∈L∞(Bn
+1(0),Mm(R))∩W1,2(Bn
+1(0),Mm(R)) such
+that
+d∗(dA−AΩ) = 0. (2)
+for given Ω ∈L2(Bn
+1(0),so(m)⊗Λ1Rn), then any solution uof (1) satisfies
+the following conservation law
+d(∗Adu+(−1)n−1(∗B)∧du) = 0 (3)
+whereBsatisfies−d∗B=dA−AΩ.
+The existence of such an A(andB) is proved by Rivi` ere in [14] and relies
+on anon linear Hodge decomposition which can also be interpreted as
+achange of gauge . (see in our case theorem 1.5)
+•Understanding the linear problem
+The proof of the above mentioned regularity result uses the result below for
+the linear problem:
+Theorem 1.1 ([26],[7], [24])
+Leta,bsatisfy∇a,∇b∈L2and letϕbe the unique solution to
+/braceleftbigg
+−∆ϕ=∇a·∇⊥b=∗(da∧db) =axby−aybxinBn
+1(0)
+ϕ= 0on∂Bn
+1(0).(4)
+Thenϕis continuous and it holds that
+||ϕ||∞+||∇ϕ||2+||∇2ϕ||1≤C||∇a||2||∇b||2. (5)
+Note that the L∞estimate in (5) is the key point for the existence of A,B
+satisfying (2).
+A moredetailed explanation ofthese key points and their interplayca n be found
+in Rivi` ere’s overview [15].
+Our strategy to extend the cited regularity result to domains of ar bitrary di-
+mension is to find first of all a good generalisation of Wente’s estimate . Here,
+the first question is to detect a suitable substitute for L2since obviously for
+n≥3 from the fact that a,b∈W1,2we can not conclude that ϕis continuous.
+2So we have to reduce our interest to a smaller space than L2. A first idea is to
+look at the Morrey space Mn
+2, i.e. at the spaces of all functions f∈L2
+loc(Rn)
+such that
+||f|Mn
+2||= sup
+x0∈Rnsup
+R>0R1−n/2||f|L2(B(x0,R))||<∞.
+The choice of this space was motivated by the following observation ( for details
+see [16]):
+For stationary harmonic maps uwe have the following monotonicity estimate
+r2−n/integraldisplay
+Bnr(x0)|∇u|2≤R2−n/integraldisplay
+Bn
+R(x0)|∇u|2
+for allr≤R. From this, it is rather natural to look at the Morrey space Mn
+2.
+Unfortunately, this first try is not successful as the following cou nterexample in
+dimensionn= 3 shows:
+Leta=x1
+|x|andb=x2
+|x|. As required ∇a,∇b∈ M3
+2(B3
+1(0)). The results in
+([7]) imply that the unique solution ϕof (4) satisfies ∇2ϕ∈ M3
+2
+1butϕis not
+bounded!
+Therefore, in [17] the attempt to construct conservation laws fo r (1) in the
+framework of Morrey spaces fails.
+Another drawback is that C∞is not dense in Mn
+2. This point is particularly
+important if one has in mind the proof via paraproducts of Wente’s L∞bound
+for the solution ϕ.
+Inthispaperweshallstudy L∞estimatesbyreplacingtheMorreyspaces Mn
+2by
+their ”nearest” Littlewood Paley counterpart, the Besov-Morre y spacesB0
+Mn
+2,2,
+i.e. the spaces of f∈ S′such that
+/parenleftBig∞/summationdisplay
+j=0||F−1ϕjFf|Mn
+2(Rn)||2/parenrightBig1
+2<∞
+whereϕ={ϕj}∞
+j=0is a suitable partition of unity.
+It turns out that we have a suitable density result at hand, see lemm a 2.14.
+These spaces were introduced by Kozono and Yamazaki in [10] and applied to
+the study of the Cauchy problem for the Navier-Stokes equation a nd semilinear
+heat equation (see also [12]).
+Note, that we have the following natural embeddings :B0
+Mn
+2,2⊂ Mn
+2(see
+lemma 2.10) and on compact subsets B0
+Mn
+2,2is a natural subset of L2(see
+lemma 2.13).
+The success to which these Besov-Morrey spaces give rise relies cr ucially on the
+fact that we first integrate and then sum !
+In the spirit of the scales of Triebel-Lizorkin and Besov spaces (defi nition are
+restated in the next section) where we have for 0 <q≤ ∞and 0<p<∞
+Bs
+p,min{p,q}⊂Fs
+p,q⊂Bs
+max{p,q}
+3and due to the fact that for 1 <q≤p<∞
+||f||Mp
+q≃/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBig∞/summationdisplay
+j=0|F−1ϕjFf|2/parenrightBig1
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+Mp
+q
+it is obvious to exchange the order of summability and integrability in or der to
+find a smaller space starting from a given one.
+A more detailed exposition of the framework of Besov-Morrey spac es is given in
+the next section.
+We have
+Theorem 1.2 i) Assume that a,b∈B0
+Mn
+2,2, and assume further that
+ax,ay,bx,by∈B0
+Mn
+2,2wherex,y=zi,zjwithi,j,∈ {1,...,n}.
+Then any solution of
+−∆u=axby−aybx
+is continuous and bounded.
+ii) Assume that ax,ay,bxandbyare distributions whose support is contained in
+Bn
+1(0)and belong to B0
+Mn
+2,2,n≥3.
+Moreover, let ube a solution ( in the sense of distributions ) of
+−∆u=axby−bxay.
+Then it holds
+∇u∈B0
+Mn
+2,1.
+iii) Assume that ax,ay,bxandbyare distributions whose support in Bn
+1(0)and
+belong toB0
+Mn
+2,2.
+Moreover, let ube a solution ( in the sense of distributions ) of
+−∆u=axby−bxay.
+Then it holds
+∇2u∈B−1
+Mn
+2,1⊂B−2
+∞,1.
+Remark 1.3
+•If we reduce our interest to dimension n= 2, our assumption in the theorem
+below coincide with the original ones in Wente’s framework due to the f act
+thatM2
+2=L2andB0
+2,2=L2=F0
+2,2.
+•Obviously we have the a-priori bound
+||u||∞≤C/parenleftbig
+||a|B0
+Mn
+2,2||+||∇a|B0
+Mn
+2,2||/parenrightbig/parenleftbig
+||b|B0
+Mn
+2,2||+||∇b|B0
+Mn
+2,2||/parenrightbig
+.
+•Now, if we use a homogeneouspartitionof unity instead ofan inhomog eneous
+asbefore, ourresultholdsifwereplacethespaces B0
+Mn
+2,2bythe spaces N0
+n,2,2.
+For further information about these homogeneous function spac es we refer to
+Mazzucato’s article [12].
+4•Note that the estimate ∇u∈B0
+Mn
+2,1implies that uis bounded and continu-
+ous.
+As an application of what we did so far, we would like to present an adap tation
+ofRivi` ere’sconstructionofconservationlawsviagaugetransfo rmation(see [14])
+to our setting, more precisely we are able to prove the following asse rtion:
+Theorem 1.4 Letn≥3. There exist constants ε(m)>0andC(m)>0such
+that for every Ω∈B0
+Mn
+2,2(Bn
+1(0),so(m)⊗Λ1Rn)which satisfies
+||Ω|B0
+Mn
+2,2|| ≤ε(m)
+there existA∈L∞(Bn
+1(0),Glm(R))∩B1
+Mn
+2,2andB∈B1
+Mn
+2,2(Bn
+1(0),Mm(R)⊗
+Λ2Rn)such that
+i)
+dΩ:=dA−AΩ =−d∗B=−∗d∗B
+ii)
+||∇A|B0
+Mn
+2,2||+||∇A−1|B0
+Mn
+2,2||+/integraldisplay
+Bn
+1(0)||dist(A,SO(m))||2
+∞≤C(M)||Ω|B0
+Mn
+2,2||
+iii)
+||∇B|B0
+Mn
+2,2|| ≤C(m)||Ω|B0
+Mn
+2,2||.
+This finally leads to the following regularity result:
+Corollary 1.5 Let the dimension nsatisfyn≥3. Letε(m),Ω,AandBbe as
+in theorem 1.4. Then any solution uof
+−∆u= Ω·∇u
+satisfies the conservation law
+d(∗Adu+(−1)n−1(∗B)∧du) = 0.
+Moreover, any distributional solution of ∆u=−Ω·∇uwhich satisfies in addition
+∇u∈B0
+Mn
+2,2
+is continuous.
+Remark 1.6 Note that the continuity assertion of the above corollary is al-
+ready contained in [16], but our result differs from [16] (see also [18] f or a
+modification of the proof of Rivi` ere and Struwe) in so far, as on on e hand we do
+not impose any smallness of the norm of the gradient of a solution and really
+constructAandB(see theorem 1.4) and not only construct Ω and ξsuch that
+P−1dP+P−1ΩP=∗dξ, but on the other hand work in a slightly smaller space.
+The present article is organised as follows: After recalling some basic definitions
+and preliminary facts in section 2 we give in the third section the proof s of the
+statements claimed before.
+Acknowledgement The author would like to thank Professor T. Rivi` ere for
+having pointed out the present problem to her and for his support.
+52 Definitions and preliminary results
+We recall the important definitions and state basic results we will use .
+2.1 Besov and Triebel-Lizorkin spaces
+2.1.1 Non-homogeneous Besov and Triebel-Lizorkin spaces
+In order to define them we have to introduce some additional notion s:
+Definition 2.1 (Φ (Rn))LetΦ(Rn)be the collection of all systems
+ϕ={ϕj(x)}∞
+j=0⊂ S(Rn)such that
+/braceleftbiggsuppϕ 0⊂ {x| |x| ≤2}
+suppϕ j⊂/braceleftbig
+x|2j−1≤ |x| ≤2j+1/bracerightbig
+ifj= 1,2,3,...,
+for every multi-index αthere exists a positive number Cαsuch that
+2j|α||Dαϕj(x)| ≤Cαfor allj= 1,2,3,...and allx∈Rn
+and∞/summationdisplay
+j=0ϕj(x) = 1∀x∈Rn
+Remark 2.2
+•Note that in the above expression/summationtext∞
+j=0ϕj(x) = 1 the sum is locally finite!
+•Example of a system ϕwhich belongs to Φ( Rn):
+We start with an arbitrary C∞
+0(Rn) functionψwhich has the following prop-
+erties:ψ(x) = 1 for |x| ≤1 andψ(x) = 0 for |x| ≥3
+2. We setϕ0(x) =ψ(x),
+ϕ1(x) =ψ(x
+2)−ψ(x), andϕj(x) =ϕ1(2−j+1x),j≥2. Then it is easy to
+check that this family ϕsatisfies the requirements of our definition.
+Moreover, we have/summationtextn
+j=0ϕj(x) =ψ(2−nx),n≥0.
+By the way, other examples of ϕ∈Φ, apart from this one, can be found in
+[17], [25] or [6]. )
+Now, we can state the definitions of the above mentioned Besov and Triebel-
+Lizorkin spaces.
+Definition 2.3 (Besov spaces and Triebel-Lizorkin spaces) Let−∞<s<
+∞, let0<q≤ ∞and letϕ∈Φ(Rn).
+i) If0<p≤ ∞then the (non-homogeneous) Besov spaces Bs
+p,q(Rn)consist of
+allf∈ S′such that the following inequality holds
+||f|Bs
+p,q(Rn)||ϕ=||2jsF−1ϕjFf|lq(Lp(Rn))||<∞
+ii) If0<p<∞then the (non-homogeneous) Triebel-Lizorkin spaces Fs
+p,q(Rn)
+consist of all f∈ S′such that the following inequality holds
+||f|Fs
+p,q(Rn)||ϕ=||2jsF−1ϕjFf|Lp(Rn,lq)||<∞
+6iii) Ifp=∞then the spaces Fs
+∞,q(Rn)consist of all f∈ S′such that
+∃{fk(x)}∞
+k=0⊂L∞(Rn)such that the following holds
+f=∞/summationdisplay
+k=0F−1ϕkFfkinS′(Rn)
+and
+||2skfk|L∞(Rn,lq)||<∞.
+Moreover we set
+||f|Fs
+∞,q(Rn)||ϕ= inf||2skfk|L∞(Rn,lq)||
+where the infimum is taken over all admissible representatio ns off.
+HereFdenotes the Fourier transform and
+||fk|lq(Lp(Rn))||=/parenleftBigg∞/summationdisplay
+k=0/parenleftbig/integraldisplay
+|fk(x)|pdx/parenrightbigq
+p/parenrightBigg1
+q
+and
+||fk|Lp(Rn,lq)||=/parenleftBigg/integraldisplay/parenleftbig∞/summationdisplay
+k=0|fk(x)|q/parenrightbigp
+qdx/parenrightBigg1
+p
+.
+Recall that the spaces Bs
+p,qandFs
+p,qareindependent ofthe choiceof ϕ(see [25]).
+Most of the important fact (embeddings, relation with other funct ion spaces,
+multiplier assertions and so on) about these spaces can be found in [ 17] and [25].
+In what follows we will give precise indications where a result we use is pr oved.
+2.1.2 Besov-Morrey spaces
+In stead of combining Lp-norms ans lq-norm one can also combine Morrey-
+(respectively Morrey-Campanato-) norms with lq-norms. This idea was first
+introduced and applied by Kozono and Yamazaki in [10].
+In order to make the whole notation clear and to avoid misunderstan ding we
+will recall some definitions.
+We start with the definition of Morrey spaces
+Definition 2.4 (Morrey spaces) Let1≤q≤p<∞.
+i) The Morrey spaces Mp
+q(Rn)consist of all f∈Lq
+loc(Rn)such that
+||f|Mp
+q||= sup
+x0∈Rnsup
+R>0Rn/p−n/q||f|Lq(B(x0,R))||<∞
+ii) The local Morrey spaces Mp
+q(Rn)consist of all f∈Lq
+loc(Rn)such that
+||f|Mp
+q||= sup
+x0∈Rnsup
+0<R≤1Rn/p−n/q||f|Lq(B(x0,R))||<∞
+whereB(x0,R)denotes the closed ball in Rnwith center x0and radius R.
+7Note that it is easy to see that the spaces Mp
+qandMp
+qcoincide on compactly
+supported functions.
+Apart from these spaces of regular distributions, i.e. function belo nging toL1
+loc,
+in the case q= 1 we are even allowed to look at measures in stead of functions.
+More precisely we have the following measure spaces of Morrey type . They will
+become useful later on in a rather technical context.
+Definition 2.5 (Measure spaces of Morrey type) Let1≤p<∞.
+i) The measure spaces of Morrey type Mp(Rn) =Mpconsist of all Radon
+measuresµsuch that
+||µ|Mp||= sup
+x0∈Rnsup
+R>0Rn/p−n|µ|(B(x0,R))<∞.
+ii) The local measure spaces of Morrey type Mq(Rn) =Mpconsist of all Radon
+measuresµsuch that
+|µ|Mp||= sup
+x0∈Rnsup
+0<R≤1Rn/p−n|µ|(B(x0,R))<∞
+where as above B(x0,R)denotes the closed ball in Rnwith center x0and
+radiusR.
+Remember that all the spaces we have seen so far, i. e. Mp
+q,Mp
+q,MpandMp
+are Banach spaces with the norms indicated before. Moreover, Mp
+1andMp
+1can
+be considered as closed subspaces of MpandMprespectively, consisting of all
+those measures which are absolutely continuous with respect to th e Lebesgue
+measure.
+For details, see e.g. [10].
+Once we have the above definition of Morrey spaces (of regular dist ributions),
+we now define the Besov-Morrey spaces in the same way as we const ructed the
+Besov spaces, of course with the necessary changes.
+Definition 2.6 (Besov-Morrey spaces) Let1≤q≤p <∞,1≤r≤ ∞
+ands∈R.
+i) Letϕ∈˙Φ(Rn). The homogeneous Besov-Morrey spaces Ns
+p,q,rconsist of all
+f∈ Z′such that
+||f|Ns
+p,q,r(Rn)||ϕ=/parenleftBig∞/summationdisplay
+j=−∞2jsr||F−1ϕjFf|Mp
+q(Rn)||r/parenrightBig1
+r<∞.
+ii) Letϕ∈Φ(Rn). The inhomogeneous Besov-Morrey spaces Ns
+p,q,rconsist of
+allf∈ S′such that
+||f|Ns
+p,q,r(Rn)||ϕ=/parenleftBig∞/summationdisplay
+j=02jsr||F−1ϕjFf|Mp
+q(Rn)||r/parenrightBig1
+r<∞.
+Note that since Lp(Rn) =Mp
+p(Rn) the framework of the Ns
+p,q,r(Rn) can be seen
+as a generalisation of the framework of the homogeneous Besov sp aces.
+In our further work we will crucially use still another variant of spac es which
+are defined via Paley-Littlewood decomposition. We will use the decom position
+into frequencies of positive power but measure the single contribut ions in a
+homogeneous Morrey norm:
+8Definition 2.7 (The spaces Bs
+Mp
+q,r)i) Let1≤q≤p <∞,1≤r≤ ∞and
+s∈R. Letϕ∈Φ(Rn). The spaces Bs
+Mp
+q,rconsist of all f∈ S′such that
+||f|Bs
+Mp
+q,r(Rn)||ϕ=/parenleftBig∞/summationdisplay
+j=02jsr||F−1ϕjFf|Mp
+q(Rn)||r/parenrightBig1
+r<∞.
+ii) The spaces Bs
+Mp
+q,r(Ω)whereΩis a bounded domain in Rnconsist of all
+f∈Bs
+Mp
+q,rwhich in addition have compact support contained in Ω.
+Remark 2.8
+i) Again, as in the case of Besov and Triebel-Lizorkin spaces, all the s paces
+defined above do not depend on the choice of ϕ.
+ii) Previously we mentioned that our interest in these latter spaces w as moti-
+vated by the work of Rivi` ere and Struwe (see [17]) let us say a few words
+about this. In [17] the authors used the homogeneous Morrey spa ceL2,n−2
+1
+with norm
+||f||2
+L2,n−2
+1= sup
+x0∈Rnsup
+r>0/parenleftBig1
+rn−2/integraldisplay
+B−r(x0)|∇u|2/parenrightBig
+.
+Note thatu∈L2,n−2
+1is equivalent to the fact that for all radii r>0 and all
+x0∈Rnwe have the inequality
+||∇u||L2(Br(x0))≤Cr(n−2)/p=Crn
+2−2
+2
+but this latter estimateis againequivalent to the fact that ∇u∈ Mn
+2. Finally
+we remember that Mn
+2=N0
+n,2,2(see for instance [12]) and note that ∇u∈
+N0
+n,2,2is equivalent to u∈ N1
+n,2,2since for all s- even for the negative ones -
+we have the equivalence 2s||us||Mn
+2≃ ||(∇u)s||Mn
+2because we always avoid
+the origin in the Fourier space and also near the origin work with annuli with
+radiir≃2s.
+Before we continue, let us state a few facts concerning the space sBs
+Mp
+q,rwhich
+are interesting and important.
+Lemma 2.9 i) The spaces Bs
+Mp
+q,rare complete for all possible choices of in-
+dices.
+ii) a) Let s>0,1≤q≤p<∞,1≤r≤ ∞andλ>0. Then
+||f(λ·)|Bs
+Mp
+q,r|| ≤Cλ−n
+psup{1,λ}s||f|Bs
+Mp
+q,r||.
+b) Lets= 0,1≤q≤p<∞,1≤r≤ ∞andλ>0. Then
+||f(λ·)|Bs
+Mp
+q,r|| ≤Cλ−n
+p(1+|logλ|)α||f|Bs
+Mp
+q,r||
+where
+α=1
+rifλ>1andα= 1−1
+r=1
+r′if0<λ<1.
+9The first assertion is obtained by the same proof as the correspon ding claim for
+the spacesNs
+p,q,rin [10].
+The second fact is a variation of a well known proof given in [5].
+Furthermore we have the following embedding result which relates th e spaces
+B0
+Mp
+q,rto the Morrey spaces with the same indices respectively, similar for t he
+spacesN0
+p,q,r.
+Lemma 2.10 Let1<q≤2,1<q≤p<∞andr≤q. Then
+B0
+Mp
+q,r⊂ Mp
+q
+and
+N0
+pq,r⊂Mp
+q.
+From this result we immediately deduce the following corollary.
+Corollary 2.11 Let1< q≤2,1< q≤p <∞andr≤qand assume that
+f∈B0
+Mp
+q,rhas compact support. Then f∈Lq.
+This holds because of the preceding lemma and the fact that for a bo unded
+domain Ω we have the embedding Mp
+q(Ω)⊂Lq(Ω).
+Similar to the result that W1,p=F1
+p,2, 1<p<∞we have the following lemma.
+Lemma 2.12 Assume that fis a compactly supported distribution. Then, if
+1<q≤2,1<q≤p<∞andr≤q, the following two norms are equivalent
+||f|B0
+Mp
+q,r||+||∇f|B0
+Mp
+q,r||
+||f|B1
+Mp
+q,r||.
+Moreover, also the fact that for a compactly supported distribut ion the ho-
+mogeneous and the inhomogeneous Sobolev norms are equivalent, w e have the
+following result.
+Lemma 2.13 Let1<q≤2,1<q≤p<∞,2≤p,r≤qandn≥3. Assume
+that the distribution fhas the following properties: fhas compact support and
+∇f∈B0
+Mp
+q,r. Then
+f∈B1
+Mp
+q,r.
+As a by-product of our studies we have the following density result.
+Lemma 2.14 Let1≤q≤p<∞,1≤r≤ ∞ands∈R. ThenOMis dense
+inNs
+p,q,rrespectively in Ns
+p,q,randBs
+Mp
+q,rwhereOMdenotes the space of all
+C∞-functions such that ∀β∈Nnthere exist constants Cβ>0andmβ∈Nsuch
+that
+|∂βf(x)| ≤Cβ(1+|x|)mβ∀x∈Rn.
+Moreover, if f∈Ns
+p,q,rorf∈Bs
+Mq
+pr,withs≥0,1≤q≤2and1≦≤p≤ ∞
+has compact support, it can be approximated by elements in C∞
+0.
+10Last, but not least we would like to mention a stability result which we will
+apply later on.
+Lemma 2.15 Letg∈B0
+Mn
+2,2andf∈B1
+Mn
+2,2,∩L∞. Then
+||gf|B0
+Mn
+2,2|| ≤C||g|B0
+Mn
+2,2||(||f|B1
+Mn
+2,2||+||f||∞),
+i.e.B0
+Mn
+2,2is stable under multiplication with a function in B1
+Mn
+2,2∩L∞.
+The proofs of lemma 2.10, 2.12, 2.13, 2.14 and 2.15are given in the next section.
+For further information about the Besov-Morrey spaces, see [10 ], [11] and [12].
+2.2 Spaces involving Choquet integrals
+In what follows, we will use a certain description of the pre-dual spa ce ofM1.
+Before we can state this assertion we have to introduce some func tion spaces
+involving the so-called Choquet integral. A general reference for t his section is
+[1] and the references given therein.
+We start with the notion of Hausdorff capacity:
+Definition 2.16 (Hausdorff capacity) LetEbe a subset of Rnand let
+{Bj}, j= 1,2,...be a cover of E, i.e.{Bj}is a countable collection of open
+ballsBjwith radius rjsuch thatE⊂ ∪jBj. Then we define the Hausdorff
+capacity of Eof dimension d,0<d≤nto be the following quantity
+Hd
+∞(E) = inf/summationdisplay
+jrd
+j
+where the infimum is taken over all possible covers of E.
+Remark 2.17 The name capacity may lead to confusion. Here we use this
+expression in the sense of N. Meyers. See [13], page 257.
+Oncewehavethis capacity,wecanpasstothe Choquetintegralof φ∈C0(Rn)+:
+Definition 2.18 (Choquet integral and L1(Hd
+∞))Letφ∈C0(Rn)+. Then
+theChoquet integral ofφwith respect to the Hausdorff capacity Hd
+∞is defined
+to be the following Riemann integral:
+/integraldisplay
+φdHd
+∞≡/integraldisplay∞
+0Hd
+∞[φ>λ]dλ.
+The space L1(Hd
+∞)is now the completion of C0(Rn)under the functional/integraltext
+|φ|dHd
+∞.
+Two important facts about L1(Hd
+∞) are summarised below, again for instance
+see [1] and also the references given there.
+Remark 2.19
+•L1(Hd
+∞) can also be characterisedto be the space of all Hd
+∞-quasi continuous
+functionsφwhich satisfy/integraltext
+|φ|dHd
+∞<∞, i.e. for all ε >0 there exists an
+open setGsuch thatHd
+∞[G]<εand thatφrestricted to the complement of
+Gis continuous there.
+11•One can show that L1(Hd
+∞) is a quasi-Banach space with respect to the
+quasi-norm/integraltext
+|φ|dHd
+∞.
+Now, we can state the duality result we mentioned earlier. A proof of this
+assertion is given in [1], but take care of the notation which differs fro m our
+notation!
+Proposition 2.20 We have (L1(Hd
+∞))∗=Mn
+n−dand in particular the esti-
+mate/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+udµ/vextendsingle/vextendsingle/vextendsingle≤ ||u||L1(Hd∞)||µ||Mn
+n−d
+holds and
+||µ||(L1(Hd∞))∗= sup
+||u||L1(Hd∞)≤1/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+udµ/vextendsingle/vextendsingle/vextendsingle≃ ||µ||Mn
+n−d.
+Note that in order to show that a certain function belongs to Mn
+n−d, it is
+enough to show that it defines a linear functional on L1(Hd
+∞), i. e. that
+sup||u||L1(Hd∞)≤1/vextendsingle/vextendsingle/vextendsingle/integraltext
+udµ/vextendsingle/vextendsingle/vextendsingle<∞. This does not require that L1(Hd
+∞) is a Banach
+space and is quite different from the case when you use the dual cha racterisation
+of a norm in order to show that a certain distribution belongs to a cer tain space.
+Remark 2.21 The above proposition is just a special case of a more general
+result which involves also spaces Lp(Hd
+∞), see for instance [2].
+Before ending this section we will state some useful remarks for lat er applica-
+tions.
+Remark 2.22
+•Observe that Mp⊂ S′(in particular for p=n
+n−d). In order to verify this,
+note that Mp⊂N0
+p,1,∞⊂ S′: Letµ∈ Mpand let as usual ϕ∈Φ(Rn) then
+we have
+||µ|N0
+p,1,∞||= sup
+k∈N||ˇϕk∗µ|Mp
+1||
+= sup
+k∈N||ˇϕk∗µ|Mp||
+note that ˇϕk∗µ∈C∞⊂L1
+locsinceµ∈ D′
+and ˇϕk∗µcan be seen as a measure
+≤sup
+k∈N||ˇϕk||1||µ|Mp||
+because of [10], lemma 1.8
+≤C||µ|Mp||
+<∞
+according to our hypothesis.
+Once we have this, we apply the continuous embedding of N0
+p,1,∞intoS′(see
+e.g. [12]) and conclude that actually Mp⊂ S′.
+Note also that S ⊂L1(Hd
+∞)
+12•Using the duality asserted above, we can show that L1(Hd
+∞)⊂ S′: We start
+withf∈C∞
+0(Rn). Sincef∈L∞it is easy to check that f∈Mp
+q,1≤q≤
+p <∞,with||f|Mp
+q||=||f||∞. Moreover, feven belongs to Mp
+q. In order
+to establish this, it remains to show that there is a constant C, independent
+onf, such that ∀x∈Rnand for 1 ≤r
+||f||L1(Br(x))≤Crn
+q−n
+p.
+In fact, it holds ∀x∈Rnand∀r≥1
+||f||L1(Br(x))≤ ||f||1
+≤ ||f||1rn
+q−n
+p
+since due to the choice of pandqwe have
+n
+q−n
+p≥0.
+If we put together all these information we find
+||f|Mp
+q|| ≤ ||f||∞+||f||1.
+Now, recall that the duality between L1(Hd
+∞) andMn
+n−dis given by
+<µ,u>(L1(Hd
+∞))∗=Mn
+n−d,L1(Hd
+∞)=/integraldisplay
+udµ
+whereu∈L1(Hd
+∞) andµ∈ Mn
+n−d.
+In a next step we define the action of u∈L1(H∞) onf∈C∞
+0as follows
+<u,f > D′,C∞
+0:=<f,u>Mn
+n−d,L1(Hd∞).
+Last, but not least, we observe that for ϕ∈ Swe have
+||ϕ||∞+||ϕ||1≤C(n)||ϕ||S.
+This finally leads to the conclusion that in fact, L1(Hd
+∞)⊂ S′.
+This last remark enables us to use the above introduced L1(Hd
+∞)-quasi norm
+to construct - in analogy to the case of Besov- or Besov-Morrey- spaces - a new
+space of functions.
+Definition 2.23 (Besov-Choquet spaces) Letϕ∈Φ(Rn).
+We say that f∈ S′belongs to B0
+L1(Hd∞),∞if∃{fk(x)}∞
+k=0⊂L1(Hd
+∞)such that
+the following holds
+f=∞/summationdisplay
+k=0F−1ϕkFfkinS′(Rn)
+and
+sup
+k||fk|L1(Hd
+∞)||<∞.
+Moreover we set
+||f|B0
+L1(Hd∞),∞||= infsup
+k||fk|L1(Hd
+∞)||
+where the infimum is taken over all admissible representatio ns off.
+Moreover, we denote by b0
+L1(Hd∞),∞the closure of Sunder the construction
+explained above.
+13Remark 2.24 In complete analogy to the construction of the Besov spaces
+(respectively the Besov-Morrey-spaces) one could also constru ct new spaces if
+we replacethe Lebesgue Lp-norms(respectivelythe Morrey-norms)by Lp(Hd
+∞)-
+quasi-norms.
+3 Proofs
+3.1 Some preliminary remarks
+In what follows we set
+fj(x) =F−1(ϕjFf)(x)
+whereϕ={ϕj(x)}∞
+j=0∈Φ(Rn).
+Recall that oncewe can controlthe paraproducts π1(f,g) =/summationtext∞
+k=2/summationtextk−2
+l=0flgk,
+π2(f,g) =/summationtext∞
+k=0/summationtextk+1
+l=k−1flgkandπ3(f,g) =/summationtext∞
+l=2/summationtextl−2
+k=0flgk(fi= 0 ifi≤ −1
+and similarly for g) we are also able to control the product fg(see e.g. [17]).
+Since in the sequel we want to take into account cancellation phenom ena, we
+will analysis
+π1(ax,by), π1(ay,bx),π3(ax,by),π3(ay,bx) and∞/summationdisplay
+s=0s+1/summationdisplay
+t=s−1at
+xbs
+y−at
+ybs
+x.(6)
+Last but not least, remember that
+suppF/parenleftBigl−2/summationdisplay
+i=0ai
+xbl
+y/parenrightBig
+⊂/braceleftbig
+ξ|2l−3≤ |ξ| ≤2l+3/bracerightbig
+forl≥2.
+and
+suppF/parenleftBigl+1/summationdisplay
+i=l−1ai
+xbl
+y/parenrightBig
+⊂/braceleftbig
+ξ| |ξ| ≤5·2l/bracerightbig
+forl≥0.
+3.2 Proof of theorem 1.2 i)
+The proof of this assertion is split into severalparts: In a first ste p we show that
+π1(ax,by), π3(ax,by), π3(ay,bx)andπ1(ay,bx)∈B−1
+∞,1and/summationtext∞
+s=0/summationtexts+1
+t=s−1at
+xbs
+y−
+at
+ybs
+x∈B−2
+∞,1. Once we have this we show in a second step that under this hy-
+pothesis the solution uof
+−∆u=fwheref∈B−2
+∞,1
+is continuous.
+Claim:π1(ax,by)∈B−2
+∞,1
+Our hypotheses together with [10], theorem 2.5, ensures us that ax, by∈B−1
+∞,2.
+Next, due to [17], proposition 1, chapter 2.3.2, it is enough to prove t hat
+||2−2jcj|l1(L∞)||<∞
+14where as before cj:=/summationtextk−2
+t=0at
+xbj
+y.
+We actually have
+||2−2jcj|l1(L∞)||=∞/summationdisplay
+j=02−2j||j−2/summationdisplay
+t=0at
+xbj
+y||∞
+≤∞/summationdisplay
+j=02−2j||j−2/summationdisplay
+t=0at
+x||∞||bj
+y||∞
+=∞/summationdisplay
+j=02−j||j−2/summationdisplay
+t=0at
+x||∞2−j||bj
+y||∞
+≤/parenleftBig∞/summationdisplay
+j=02−2j||j−2/summationdisplay
+t=0at
+x||2
+∞/parenrightBig1
+2/parenleftBig∞/summationdisplay
+j=02−2j||bj
+y||2
+∞/parenrightBig1
+2
+due to H¨ older’s inequality
+=||2−j|j−2/summationdisplay
+t=0at
+x| |l2(L∞)||||by|B−1
+∞||
+≤C||2−j|j/summationdisplay
+t=0at
+x| |l2(L∞)||||by|B−1
+∞||
+≤C||ax|B−1
+∞,2||||by|B−1
+∞||because of [17], first lemma in chapter 4.4.2
+<∞thanks to our hypothesis.
+This shows that in fact π1(ax,by)∈B−2
+∞,1. Similarly one proves that also
+π1(ay,bx), π3(ax,by) andπ1(ay,bx) belong to the same space.
+In remains to analyse the contribution where the frequencies are c omparable.
+This is our next goal.
+Analysis of/summationtext∞
+s=0/summationtexts+1
+t=s−1at
+xbs
+y−at
+ybs
+x
+In stead of first applying the embedding result of Kozono/Yamazak i which em-
+beds Morrey-Besovspaces into Besov spaces and then analysing a certain quan-
+tity, we invert the order of these steps in order to estimate/summationtext∞
+s=0/summationtexts+1
+t=s−1at
+xbs
+y−
+at
+ybs
+x.
+We will use the following result concerning predual spaces of Morrey spaces .
+Proposition 3.1 The dual space of b0
+L1(Hn−2
+∞),∞is the space B0
+Mn
+2
+1,1.
+Remark 3.2 The above result has the same flavour as (see for instance [17])
+(b0
+∞,∞)∗=B0
+1,1
+Proof of proposition 3.1:
+We have to show the two inclusion relations.
+We start with ( b0
+L1(Hn−2
+∞),∞)∗⊃B0
+Mn
+2
+1,1:
+Assume that f∈B0
+Mn
+2
+1,1⊂N0n
+2,1,1⊂ S′and assume that ψ∈b0
+L1(Hn−2
+∞),∞. By
+15density we may assume that ψ∈ S. We have to show that f∈(b0
+L1(Hn−2
+∞),∞)∗.
+To this end let/summationtext∞
+k=0ˇϕk∗ψkbe a representation of ψwith
+sup
+k||ψk||L1(Hn−2
+∞)≤2||ψ|b0
+L1(Hn−2
+∞),∞||.
+Note that in our case - as a tempered distribution - facts onψand we estimate
+|f(ψ)|=|f(/summationdisplay
+k≥0ˇϕk∗ψk)|
+=/vextendsingle/vextendsingle/vextendsinglef/parenleftBig∞/summationdisplay
+k=oF−1(ϕkFψk)/parenrightBig/vextendsingle/vextendsingle/vextendsingle
+=/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+k=0f/parenleftBig
+F−1(ϕkFψk)/parenrightBig/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+k=0/integraldisplay
+fF−1(ϕkFψk/parenrightBig/vextendsingle/vextendsingle/vextendsingle
+=/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+k=0ψkF(ϕkF−1f)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+k=0/integraldisplay
+ψkdf/vextendsingle/vextendsingle/vextendsingle
+wheredf=F(ϕkF−1f)dλwithλthe Lebesgue measure
+≤∞/summationdisplay
+k=0|ψkF(ϕkF−1f)|
+≤sup
+k≥0||ψk||L1(Hn−2
+∞)∞/summationdisplay
+k=o||F(ϕkF−1f)||Mn
+2recall proposition 2.20
+= sup
+k≥0||ψk||L1(Hn−2
+∞)∞/summationdisplay
+k=o||F(ϕkF−1f)||
+Mn
+2
+1
+cf. also remark 2.22
+≤Csup
+k≥0||ψk||L1(Hn−2
+∞)∞/summationdisplay
+k=o||F−1(ϕkFf)||
+Mn
+2
+1
+≤C||ψ|b0
+L1(Hn−2
+∞),∞|| ||f|B0
+Mn
+2
+1,1||
+<∞
+thanks to our assumptions.
+Now we show the other inclusion, ( b0
+L1(Hn−2
+∞),∞)∗⊂B0
+Mn
+2
+1,1:
+We start with f∈(b0
+L1(Hn−2
+∞),∞)∗and we have to show that fbelongs also to
+B0
+Mn
+2
+1,1: First of all, note that fgives also rise to elements of ( L1(Hn−2
+∞))∗as
+follows: Each ψ∈b0
+L1(Hn−2
+∞),∞can be seen as a sequence {ψk}∞
+k=0⊂L1(Hn−2
+∞),
+and of course ˇ ϕk∗ψk∈b0
+L1(Hn−2
+∞),∞∀k∈N. Moreover, for each k∈Nwe have
+16- again by density of S-
+f(δkj(ˇϕj∗ψj)) =<f,δkjψ>(b0
+L1(Hn−2∞),∞)∗,b0
+L1(Hn−2∞),∞
+=<f,ˇϕk∗ψk>(b0
+L1(Hn−2∞),∞)∗,b0
+L1(Hn−2∞),∞
+=<f,ˇϕk∗ψk>S′,S
+=<f,F−1(ϕkFψk)>S′,S
+=<F(ϕkF−1f),ψk>S′,S
+=<F(ϕkF−1f),ψk>Mn
+2,L1(Hn−2
+∞).
+Next we will construct a special element of b0
+L1(Hn−2
+∞),∞:
+Let 0<εsmall.
+We choose ψksuch that
+•ψk∈ S: Remember that we have density!
+• ||ψk||L1(Hn−2
+∞)≤1
+•0< <F(ϕkF−1f),ψk>Mn
+2,L1(Hn−2
+∞)
+•
+<F(ϕkF−1f),ψk>Mn
+2,L1(Hn−2
+∞)≥ ||F(ϕkF−1f)||Mn
+2−ε2−k
+=||F(ϕkF−1f)||(L1(Hn−2
+∞))∗−ε2−k
+= sup
+u∈L1(Hn−2
+∞)
+||u||L1(Hn−2∞)≤1|<F(ϕkF−1f),u>|−ε2−k.
+Note that like that ψ=/summationtext∞
+k=0ˇϕk∗ψk∈b0
+L1(Hn−2
+∞),∞with
+||ψ|b0
+L1(Hn−2
+∞),∞|| ≤1.
+If we put now all this together we find - recall that facts linearly! -
+∞/summationdisplay
+k=0||fk||
+Mn
+2
+1=∞/summationdisplay
+k=0||F−1(ϕkFf)||
+Mn
+2
+1
+=C∞/summationdisplay
+k=0||F(ϕkF−1f)||
+Mn
+2
+1
+≤2ε+f(ψ)
+ψas constructed above
+≤2ε+||f|(b0
+L1(Hn−2
+∞),∞)∗|| ||ψ|b0
+L1(Hn−2
+∞),∞||
+≤2ε+||f|(b0
+L1(Hn−2
+∞),∞)∗||.
+Since this holds for all 0 <εwe letεtend to zero and get the desired inclusion.
+All together we established the duality result we claimed above.
+✷
+17What concerns the next lemma, recall that Sis dense in b0
+L1(Hn−2
+∞),∞:
+Lemma 3.3 Letφ∈Φ(Rn)and assume that ψ∈ S∩L1(Hn−2
+∞)with represen-
+tation{ψk}∞
+k=0, i.e./summationtext∞
+k=0ˇϕk∗ψk=ψ, such that
+sup
+k||ψk||L1(Hn−2
+∞)≤2||ψ|b0
+L1(Hn−2
+∞),∞||.
+Then
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂
+∂xˇϕk∗ψk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+L1(Hn−2
+∞)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂
+∂x( ˇϕk∗ψk)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+L1(Hn−2
+∞)
+≤C2s||ψk||L1(Hn−2
+∞)≤C2s||ψ|b0
+L1(Hn−2
+∞),∞||.
+Proof:
+For the proof of this lemma, we need the fact that if f(x)≥0 is lower semi-
+continuous on Rnthen
+||f||L1(Hd
+∞)=/integraldisplay
+f dHd
+∞∼sup/braceleftbigg/integraldisplay
+f dµ|µ∈ Mn
+n−d
++and||µ||Mn
+n−d≤1/bracerightbigg
+.
+see Adams [1].
+It holds
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂
+∂xˇϕk∗ψk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+L1(Hn−2
+∞)≤ || |∂
+∂xˇϕk|∗|ψk| ||L1(Hn−2
+∞)
+≤Csup
+µ∈Mn
+2
++
+||µ||
+Mn
+2≤1/braceleftbigg/integraldisplay
+|∂
+∂xˇϕk|∗|ψk|dµ/bracerightbigg
+=Csup
+µ∈Mn
+2
++
+||µ||
+Mn
+2≤1/braceleftbigg/integraldisplay /integraldisplay
+|∂
+∂xˇϕk|(x−y)|ψk|(y)dλ(y)dµ(x)/bracerightbigg
+≤Csup
+µ∈Mn
+2
++
+||µ||
+Mn
+2≤1/braceleftbigg/integraldisplay
+|ψk|(y)/integraldisplay
+|∂
+∂xˇϕk|(x−y)dµ(x)dλ(y)/bracerightbigg
+by Tonelli’s theorem
+=Csup
+µ∈Mn
+2
++
+||µ||
+Mn
+2≤1/braceleftbigg/integraldisplay
+|ψk|(y)/integraldisplay
+|∂
+∂xˇϕk|(y−x)dµ(x)dλ(y)/bracerightbigg
+note thatϕkcan be chose radial
+which implies that ˇ ϕkand∂
+∂xˇϕkare radial
+see e.g. [22]
+=Csup
+µ∈Mn
+2
++
+||µ||
+Mn
+2≤1/braceleftbigg/integraldisplay
+|ψk|(y)∂
+∂xˇϕk|(y−x)∗µ(y)dλ(y)/bracerightbigg
+18and we continue
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂
+∂xˇϕk∗ψk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+L1(Hn−2
+∞)≤Csup
+µ∈Mn
+2
++
+||µ||
+Mn
+2≤1/braceleftbigg/integraldisplay
+|ψk|(y)dν(y)/bracerightbigg
+whereν:=∂
+∂xˇϕkλ∗µ
+≤Csup
+µ∈Mn
+2
++
+||µ||
+Mn
+2≤1/braceleftbigg
+||ψk||L1(Hn−2
+∞)||∂
+∂xˇϕkλ∗µ||Mn
+2/bracerightbigg
+≤Csup
+µ∈Mn
+2
++
+||µ||
+Mn
+2≤1/braceleftbigg
+||ψk||L1(Hn−2
+∞)||∂
+∂xˇϕk||L1||µ||Mn
+2/bracerightbigg
+by [10], lemma 1.8
+≤C||ψk||L1(Hn−2
+∞)||∂
+∂xˇϕk||L1
+≤C2k||ψk||L1(Hn−2
+∞)
+≤C2k||ψ|b0
+L1(Hn−2
+∞),∞||
+what we had to prove.
+✷
+The next lemma is a technical one:
+Lemma 3.4 Letaandbbelong toC∞
+0(Rn),t=s+jwherej∈ {−1,0,1}and
+ψwith representation {ψk}∞
+k=0, i.e./summationtext∞
+k=0ˇϕk∗ψk=ψ, such that
+sup
+k||ψk||L1(Hn−2
+∞)≤2||ψ|b0
+L1(Hn−2
+∞),∞|| ≤2
+Then
+/integraldisplay
+Rn∂
+∂x(atbs
+y)ψ−∂
+∂y(atbs
+x)ψ
+=/integraldisplay
+Rn∂
+∂x/parenleftBig
+atbs
+y/parenrightBig/parenleftBigs+3/summationdisplay
+k=0F−1(ϕkFψk)/parenrightBig
+−∂
+∂y/parenleftBig
+atbs
+x/parenrightBig/parenleftBigs+3/summationdisplay
+k=0F−1(ϕkFψk)/parenrightBig
+.
+Proof:
+First of all, note that h∈ S′andatbs
+yandatbs
+xbelong to Sindependently of
+the choices of sandt.
+We now calculate
+/integraldisplay
+Rn∂
+∂x(atbs
+y)ψ−∂
+∂y(atbs
+x)ψ=/integraldisplay
+Rn∂
+∂x(atbs
+y)ψ−/integraldisplay
+Rn∂
+∂y(atbs
+x)ψ
+=/integraldisplay
+Rn∂
+∂x(atbs
+y)∞/summationdisplay
+k=0F−1(ϕkFψk)
+−/integraldisplay
+Rn∂
+∂y(atbs
+x)∞/summationdisplay
+k=0F−1(ϕkFψk)
+19so
+/integraldisplay
+Rn∂
+∂x(atbs
+y)ψ−∂
+∂y(atbs
+x)ψ=/integraldisplay
+Rn∂
+∂x(atbs
+y)/bracketleftBigs+3/summationdisplay
+k=0F−1(ϕkFψk)+∞/summationdisplay
+k=s+4F−1(ϕkFψk)/bracketrightBig
+−/integraldisplay
+Rn∂
+∂y(atbs
+x)/bracketleftBigs+4/summationdisplay
+k=0F−1(ϕkFψk)+∞/summationdisplay
+k=s+4F−1(ϕkFψk)/bracketrightBig
+.
+These calculations show that we have to prove that
+/integraldisplay
+Rn∂
+∂x(atbs
+y)∞/summationdisplay
+k=s+4F−1(ϕkFψk) = 0
+and/integraldisplay
+Rn∂
+∂y(atbs
+x)∞/summationdisplay
+k=s+4F−1(ϕkFψk) = 0.
+In what follows, we will only discuss the first integral because the se cond one
+can be analysed in exactly the same way.
+So from now on we look at
+/integraldisplay
+Rn∂
+∂x(atbs
+y)∞/summationdisplay
+k=s+4F−1(ϕkFψk).
+Here we have
+/integraldisplay
+Rn∂
+∂x(atbs
+y)∞/summationdisplay
+k=s+4F−1(ϕkFψk) =/integraldisplay
+Rn∂
+∂x(atbs
+y)F−1/parenleftBig∞/summationdisplay
+k=s+4ϕkFψk/parenrightBig
+sine the sum is locally finite
+=/integraldisplay
+Rn∂
+∂x(atbs
+y)FF−1F−1/parenleftBig∞/summationdisplay
+k=s+4ϕkFψk/parenrightBig
+= (2π)n/integraldisplay
+RnF/parenleftBig∂
+∂x(atbs
+y)/parenrightBig∞/summationdisplay
+k=s+4ϕk(− ·)Fψk(− ·)
+because∂
+∂x(atbs
+y)∈ Sand∞/summationdisplay
+k=s+4ϕkFψk)∈ S′
+= 0.
+In the last step of the above calculations we used the fact that
+suppF(∂
+∂x(atbs
+y))⊂ {ξ| |ξ| ≤5·2s}
+and
+supp∞/summationdisplay
+k=s+4ϕk(− ·)⊂/braceleftbig
+ξ|2s+3≤ |ξ|/bracerightbig
+imply that
+suppF/parenleftBig∂
+∂x(atbs
+y)/parenrightBig
+∩supp∞/summationdisplay
+k=s+4ϕk=∅.
+This completes the proof.
+✷
+20Now, we can start with the estimate of/summationtext∞
+s=0/summationtexts+1
+t=s−1at
+xbs
+y−at
+ybs
+x. Our goal is
+to show that/summationtext∞
+s=0/summationtexts+1
+t=s−1at
+xbs
+y−at
+ybs
+xbelongs toB0
+Mn
+2
+1,1. Making use of the
+above duality result, see proposition 3.1, we will first show that
+s+1/summationdisplay
+t=s−1at
+xbs
+y−at
+ybs
+x∈B0
+Mn
+2
+1,1∀s∈N
+then we establish
+∞/summationdisplay
+s=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingles+1/summationdisplay
+t=s−1at
+xbs
+y−at
+ybs
+x/vextendsingle/vextendsingle/vextendsingleB0
+Mn
+2
+1,1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<∞.
+This ensures that
+∞/summationdisplay
+s=0s+1/summationdisplay
+t=s−1at
+xbs
+y−at
+ybs
+x∈B0
+Mn
+2
+1,1⊂N0n
+2,1,1.
+First of all, let us fix t=s+jwherej∈ {−1,0,1}.
+In order to show that at
+xbs
+y−at
+ybs
+x∈B0
+Mn
+2
+1,1it suffices to show that for all
+ψ∈b0
+L1(Hn−2
+∞),∞with||ψ|b0
+L1(Hn−2
+∞),∞|| ≤1 the following inequality holds
+/integraldisplay
+Rnψd(at
+xbs
+y−at
+ybs
+x) =/integraldisplay
+Rnψ(at
+xbs
+y−at
+ybs
+x)dλ<∞
+where as before λdenotes the Lebesgue measure.
+Moreover, in the subsequent calculations we assume that for ψwe have a rep-
+resentation {ψk}∞
+k=0, i.e./summationtext∞
+k=0ˇϕk∗ψk=ψ, such that
+sup
+k||ψk||L1(Hn−2
+∞)≤2||ψ|b0
+L1(Hn−2
+∞),∞|| ≤2
+and again, recall that we have density of Sinb0
+L1(Hn−2
+∞),∞.
+In this case we have
+/integraldisplay
+Rnψ(at
+xbs
+y−at
+ybs
+x) =/integraldisplay
+Rnψ∂
+∂x/parenleftbig
+atbs
+y/parenrightbig
+−ψ∂
+∂y/parenleftbig
+atbs
+x/parenrightbig
+=/integraldisplay
+Rn/bracketleftBig∂
+∂x/parenleftbig
+atbs
+y/parenrightbig/parenleftBigs+3/summationdisplay
+k=0F−1(ϕkFψk)/parenrightBig
+−∂
+∂y/parenleftbig
+atbs
+x/parenrightbig/parenleftBigs+3/summationdisplay
+k=0F−1(ϕkFψk)/parenrightBig/bracketrightBig
+because of the same reason as in lemma 3.4
+=/integraldisplay
+Rn/bracketleftBig
+−atbs
+y∂
+∂x/parenleftBigs+3/summationdisplay
+k=0F−1(ϕkFψk)/parenrightBig
++atbs
+x∂
+∂y/parenleftBigs+3/summationdisplay
+k=0F−1(ϕkFψk)/parenrightBig/bracketrightBig
+by a simple integration by parts
+21and further
+/integraldisplay
+Rnψ(at
+xbs
+y−at
+ybs
+x)≤/integraldisplay
+Rn/bracketleftBig
+−atbs
+y/parenleftBigs+3/summationdisplay
+k=0∂
+∂xˇϕk∗ψk/parenrightBig
++atbs
+x/parenleftBigs+3/summationdisplay
+k=0∂
+∂yˇϕk∗ψk/parenrightBig/bracketrightBig
+≤s+3/summationdisplay
+k=0/integraldisplay
+Rn/bracketleftBig
+−atbs
+y∂
+∂xˇϕk∗ψk
++atbs
+x∂
+∂yˇϕk∗ψk/bracketrightBig
+≤s+3/summationdisplay
+k=0/parenleftBig
+||atbs
+y|Mn
+2|| ||∂
+∂xˇϕk∗ψk|L1(Hn−2
+∞)||
++||atbs
+x|Mn
+2|| ||∂
+∂yˇϕk∗ψk|L1(Hn−2
+∞)||/parenrightBig
+by proposition 2.20
+≤s+3/summationdisplay
+k=0/parenleftBig
+||atbs
+y|Mn
+2
+1|| ||∂
+∂xˇϕk∗ψk|L1(Hn−2
+∞)||
++||atbs
+x|Mn
+2
+1|| ||∂
+∂yˇϕk∗ψk|L1(Hn−2
+∞)||/parenrightBig
+≤s+3/summationdisplay
+k=0/parenleftBig
+||at|Mn
+2|| ||bs
+y|Mn
+2|| ||∂
+∂xˇϕk∗ψk|L1(Hn−2
+∞)||
++||at|Mn
+2|| ||bs
+x|Mn
+2|| ||∂
+∂yˇϕk∗ψk|L1(Hn−2
+∞)||/parenrightBig
+because of H¨ older’s inequality with Morrey norms
+see also remark below
+≤s+3/summationdisplay
+k=0/parenleftBig
+||at|Mn
+2|| ||bs
+y|Mn
+2||2k||ψ|b0
+L1(Hn−2
+∞),∞||
++||at|Mn
+2|| ||bs
+x|Mn
+2||2k||ψ|b0
+L1(Hn−2
+∞),∞||/parenrightBig
+according to lemma 3.3
+≤C2s||at|Mn
+2|| ||bs
+y|Mn
+2||+C2s||at|Mn
+2|| ||bs
+x|Mn
+2||
+<∞
+due to our assumptions.
+Thus we have seen that for all s∈N
+at
+xbs
+y−at
+ybs
+x∈(b0
+L1(Hn−2
+∞),∞)∗=B0
+Mn
+2
+1,1⊂N0n
+2,1,1.
+Next, we study
+∞/summationdisplay
+s=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingles+1/summationdisplay
+t=s−1at
+xbs
+y−at
+ybs
+x/vextendsingle/vextendsingle/vextendsingleB0
+Mn
+2
+1,1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
+22What concerns this latter quantity, we will assume for the sake of s implicity
+thatt=s. Then we can estimate
+∞/summationdisplay
+s=0||as
+xbs
+y−as
+ybs
+x|B0
+Mn
+2
+1,1||=||a0
+xb0
+y−a0
+yb0
+x|B0
+Mn
+2
+1,1||+∞/summationdisplay
+s=1||as
+xbs
+y−as
+ybs
+x|B0
+Mn
+2
+1,1||
+≤C||a0|Mn
+2|| ||b0
+y|Mn
+2||+C||a0|Mn
+2|| ||b0
+x|Mn
+2||
++C∞/summationdisplay
+s=12s||as|Mn
+2|| ||bs
+y|Mn
+2||
++C∞/summationdisplay
+s=12s||as|Mn
+2|| ||bs
+x|Mn
+2||
+≤C||a0|Mn
+2|| ||b0
+y|Mn
+2||+C||a0|Mn
+2|| ||b0
+x|Mn
+2||
++C∞/summationdisplay
+s=1||as
+x|Mn
+2|| ||bs
+y|Mn
+2||
++C∞/summationdisplay
+s=1||as
+y|Mn
+2|| ||bs
+x|Mn
+2||
+similar to 2ms||g||p≃ ||∇mg||p(under appropriate assumptions)
+cf. also theorem 2.9 in [10]
+≤C||a0|Mn
+2|| ||b0
+y|Mn
+2||+C||a0|Mn
+2|| ||b0
+x|Mn
+2||
++C/parenleftBig∞/summationdisplay
+s=1||as
+x|Mn
+2||2/parenrightBig1
+2/parenleftBig∞/summationdisplay
+s=1||bs
+y|Mn
+2||2/parenrightBig1
+2
++C/parenleftBig∞/summationdisplay
+s=1||as
+y|Mn
+2||2/parenrightBig1
+2/parenleftBig∞/summationdisplay
+s=1||bs
+x|Mn
+2||2/parenrightBig1
+2
+by H¨ older’s inequality
+<∞
+thanks to our hypothesis.
+All together we have seen that
+∞/summationdisplay
+s=0as
+xbs
+y−as
+ybs
+x∈B0
+Mn
+2
+1,1⊂N0n
+2,1,1.
+Now, since the above estimate is independent of the choice of jwe immediately
+conclude that
+∞/summationdisplay
+s=0s+1/summationdisplay
+t=s−1at
+xbs
+y−at
+ybs
+x∈N0n
+2,1,1
+Now, as we know that/summationtext∞
+s=0/summationtexts+1
+t=s−1at
+xbs
+y−at
+ybs
+x∈B0
+Mn
+2
+1,1⊂N0n
+2,1,1we apply
+the embedding result of Kozono/Yamazaki, theorem 2.5 in [10], and fin d that
+∞/summationdisplay
+s=0s+1/summationdisplay
+t=s−1at
+xbs
+y−at
+ybs
+x∈B−2
+∞,1.
+23Remark 3.5 Assume that f,g∈ Mn
+2. Then we have for all 0 <rand for all
+x∈Rn
+||fg||L1(Br(x))≤ ||f||L2(Br(x))||g||L2(Br(x))
+≤C1rn
+2−1C2rn
+2−1
+=Crn−2.
+According to the definition, this shows that fg∈ Mn
+2
+1.
+Regularity
+We rewrite our equation ∆ u=fas ∆u=f0+/summationtext
+k≥1fk.
+And the solution ucan be written as
+u= ∆−1f0+∆−1(/summationdisplay
+k≥1fk)
+=:u1+u2.
+Our strategy is to show that u1as well asu2is continuous and bounded.
+What concerns u1, observethat due to the Paley-Wienertheorem f0is analytic,
+so in particular continuous. This implies immediately - by classical result s (see
+e.g. [8]) - that u1is continuous.
+On one hand we have that f0∈Bsn
+2,2for alls∈R(since∇a,∇b∈B0
+Mn
+2,2⊂
+Mn
+2⊂Ln) on the other hand we know that f0∈Bs
+∞,1for alls∈Rbecause
+f∈B−2
+∞,1. From that we can deduce by standard elliptic estimates (see also
+[17]) and the embedding result of Sickel and Triebel [21] that u1is not only
+continuous but also bounded!
+Next, we will show that u2is bounded and continuous. In order to reach this
+goal, we show that u2∈B0
+∞,1: We find the following estimates
+||u2|B0
+∞,1||=∞/summationdisplay
+s=0||us
+2||∞
+=∞/summationdisplay
+s=02−2s22s||us
+2||∞
+=C∞/summationdisplay
+s=02−2s||(∆u2)s||∞
+This last passage holds thanks to the fact that
+2ms||g||p≃ ||∇mg||p
+if the Fourier transform of gis supported on an annulus with radii comparable
+to 2s(see [23] for instance).
+Fors= 0 we observe
+F(−∆u2) =F(/summationdisplay
+k≥1fk)
+which implies
+supp(F(u2))⊂(B1(0))c
+24because of the fact that
+supp(F(/summationdisplay
+k≥1fk))⊂(B1(0))c.
+So in this case too, we can apply the above mentioned fact in order to conclude
+that also for s= 0 we have
+||u0
+2||∞≤C||(∆u2)0||∞.
+Back to our estimate, we continue
+||u2|B0
+∞,1|| ≤C∞/summationdisplay
+s=02−2s||(∆u2)s||∞
+=C∞/summationdisplay
+s=02−2s||(/summationdisplay
+k≥1fk)s||∞
+=C∞/summationdisplay
+s=02−2s||F−1(s+1/summationdisplay
+k=s−1ϕsϕkˆf)||∞
+≤∞/summationdisplay
+s=02−2s||fs||∞
+thanks to a Fourier multiplier result
+for further details we refer to [25]
+=C||f|B−2
+∞,1||
+<∞according to our assumptions.
+This shows that u2belongs toB0
+∞,1(Rn).
+Alternatively one could make use of the lifting property, see [17], cha pter 2.6,
+to show that u2∈C. (Recall that Cdenotes the space of all uniformly con-
+tinuous functions on Rn.) The last ingredient is the embedding result due to
+Sickel/Triebel (see [21]).
+This leads immediately to the assertion we claimed because uas a sum of two
+bounded continuous functions is again continuous and bounded.
+✷
+3.3 Proof of theorem 1.2 ii)
+In a first step we show that axby−aybx∈B−1
+Mn
+2,1:
+From the proof of theorem 1.2 we know that
+∞/summationdisplay
+k=0k+1/summationdisplay
+s=k−1ak
+xbs
+y−ak
+ybs
+x∈B0
+Mn
+2
+1,1⊂B−1
+Mn
+2,1.
+25Next, we observe that
+||π3(ax,bx)|B−1
+Mn
+2,1|| ≤C∞/summationdisplay
+s=02−s/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingles−2/summationdisplay
+k=0as
+xbk
+y/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+Mn
+2
+by a simple modification of lemma 3.16 in [12]
+≤C∞/summationdisplay
+s=02−s||as
+x||Mn
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingles−2/summationdisplay
+k=0bk
+y/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+∞
+≤C/parenleftBig∞/summationdisplay
+s=0||as
+x||2
+Mn
+2/parenrightBig1
+2/parenleftBig∞/summationdisplay
+s=02−2s/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingles−2/summationdisplay
+k=0bk
+y/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+∞/parenrightBig1
+2
+≤C||ax|B0
+Mn
+2,2||/parenleftBig∞/summationdisplay
+s=02−2s/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingles/summationdisplay
+k=0bk
+y/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+∞/parenrightBig1
+2
+≤C||ax|B0
+Mn
+2,2|| ||by|B−1
+Mn
+2,2||
+according to lemma 4.4.2 of [17]
+≤ ||ax|B0
+Mn
+2,2|| ||by|B0
+Mn
+2,2||.
+Now, since
+∂xiu=F−1/parenleftBig
+iξi
+|ξ|2F(∆u)/parenrightBig
+we note first, that due to the facts that ∆ u∈F0
+1,2⊂L1andr−1∈Ln
+n−1for
+n≥3,
+(∇u)0∈Ln⊂ Mn
+2
+which implies that ( ∇u)0∈B0
+Mn
+2,2.
+Second, for s≥1 we have
+||(∇u)s||Mn
+2≤C2−s||(∆u)s||Mn
+2
+which leads to the conclusion - remember the first step! - that/summationtext
+s≥1(∇u)s∈
+B0
+Mn
+2,1.
+Alternatively one could observe that
+/vextendsingle/vextendsingle/vextendsingle∂|α|/parenleftBigξi
+|ξ|2/parenrightBig/vextendsingle/vextendsingle/vextendsingle≤C|ξ|−1−|α|
+information,whichtogetherwiththeorem2.9in[10]leadstothesam econclusion
+as above, namely that
+∇u∈B0
+Mn
+2,1.
+These estimates complete the proof.
+✷
+3.4 Proof of theorem 1.2 iii)
+This proof is very similar to the one of theorem 1.2 ii).
+In stead of the observation/vextendsingle/vextendsingle/vextendsingle∂|α|/parenleftBig
+ξi
+|ξ|2/parenrightBig/vextendsingle/vextendsingle/vextendsingle≤C|ξ|−1−|α|here we use theorem 2.9 of
+[10] together with the fact that
+/vextendsingle/vextendsingle/vextendsingle∂|α|/parenleftBigξiξj
+|ξ|2/parenrightBig/vextendsingle/vextendsingle/vextendsingle≤C|ξ|−|α|.
+✷
+263.5 Proof of theorem 1.4
+Lemma 3.6 There exist constants ε(m)>0andC(m)>0such that for every
+Ω∈B0
+Mn
+2,2(Bn
+1(0),so(m)⊗Λ1Rn)which satisfies
+||Ω|B0
+Mn
+2,2|| ≤ε(m)
+there existξ∈B1
+Mn
+2,2(Bn
+1(0),so(m)⊗Λn−2Rn)andP∈B1
+Mn
+2,2(Bn
+1(0),SO(m))
+such that
+i)
+∗dξ=P−1dP+P−1ΩPinBn
+1(0)
+ii)
+ξ= 0on∂Bn
+1(0)
+iii)
+||ξ|B1
+Mn
+2,2||+||P|B1
+Mn
+2,2|| ≤C(m)||Ω|B0
+Mn
+2,2||.
+The proof of this lemma is a straightforward adaptation of the corr esponding
+assertion in [16].
+Now, letε(m),Pandξbe as in lemma 3.6. Note that since P∈SO(m) we
+have alsoP−1∈B1
+Mn
+2,2. Our goal is to find AandBsuch that
+dA−AΩ =−d∗B. (7)
+If we set ˜A:=APthen, according to equation (7) it has to satisfy
+d˜A+(d∗B)P=˜A+dξ.
+As a intermediate step we will first study the following problem
+
+
+∆ˆA =dˆA·∗dξ−d∗B·∇PinBn
+1(0)
+d(d∗B) =dˆA∧dP−1−d∗(ˆAdξP−1)−d∗(dξP−1)
+∂ˆA
+∂ν,= 0 andB= 0 on∂Bn
+1(0)/integraltext
+Bn
+1(0)ˆA=idm.
+For this system we have the a-priori-estimates(recall theorem 1.2 with its proof,
+lemma 2.15 and the fact that we are working on a bounded domain)
+||ˆA|B1
+Mn
+2,2||+||ˆA||∞≤C||ξ|B1
+Mn
+2,2|| ||ˆA|B1
+Mn
+2,2||
++C||P|B1
+Mn
+2,2|| ||B|B1
+Mn
+2,2||
+and
+||B|B1
+Mn
+2,2|| ≤C||P−1|B1
+Mn
+2,2|| ||ˆA|B1
+Mn
+2,2||+C||ξ|B1
+Mn
+2,2|| ||ˆA||∞
++C||ξ|B1Mn
+2,2||.
+Since the used norms of ξandP- as well as of P−1- can be bounded in terms
+ofC||Ω|B0
+Mn
+2,2||the above estimates together with standard fixpoint theory
+27guarantee the existence of ˆAandBsuch that they solve the above system and
+in addition satisfy
+||ˆA|B1
+Mn
+2,2||+||ˆA||∞+||B|B1
+Mm
+2,2|| ≤C||Ω|B0
+Mm
+2,2||. (8)
+Next, similar to the proof of corollary 1.5 we decompose for some D
+dˆA−ˆA∗dξ+d∗BP=d∗D.
+Then we set ˜A:=ˆA+idm, which satisfies for some n−2-formF
+d˜A−˜A∗dξ+d∗BP=d∗D−∗dξ=:∗dF.
+It is not difficult to show that ∗d(∗dFP−1) = 0 together with F= 0 on∂Bn
+1(0)
+imply that F≡0 (see also a similar assertion in [14] and remember that on
+bounded domains B0
+Mn
+2,2⊂L2).
+From this we conclude that in fact ˜Asatisfies the desired equation. If wet finally
+setA:=˜AP−1and letBas given in the above system we get that in fact these
+AandBsolve the required relation (7).
+So far, we have proved parts ii) and iii) of theorem 1.4 (recall also est imate (8)).
+Moreover, the invertibility of Afollows immediately from its construction, like-
+wise the estimates for ∇Aand∇A−1.
+Last but not least, the relation A=ˆAP−1+idmP−1implies that
+||dist(A,SO(m))||∞≤C||ˆA||∞≤C||Ω|B0
+Mn
+2,2||.
+This completes the proof of theorem 1.4.
+✷
+3.6 Proof of corollary 1.5
+The first part of the corollary is a straightforward calculation.
+LetAandBbe as in theorem 1.4.
+Then we have /braceleftbigg∗d∗(Adu) =−d∗B·∇u
+d(Adu) =dA∧du.
+These equations together with a classical Hodge decomposition for Adu
+Adu=d∗E+dDwithE,D∈W1,2
+lead to the following equations
+/braceleftbigg∆D=−d∗B·∇u
+∆E=dA∧du.
+Since the right hand sides are made of Jacobians we conclude that D,E∈B0
+∞,1.
+Next, we observe that
+du=A−1(d∗E+dD)∈B0
+Mn
+2,1⊂B−1
+∞,1.
+This holds because A−1∈B1
+Mn
+2,2∩L∞(see also theorem 1.4) and dD,d∗E∈
+B0
+Mn
+2,1(see also theorem 1.2 ii)).The proof of the above fact is the same as t he
+28proof of the assertion of lemma 2.15. In a last step we note that (re call the
+reasons why theorem 1.2 hold) thanks to the information we have so far
+u∈B0
+∞,1⊂C
+which completes the proof.
+✷
+3.7 Proof of lemma 2.10
+We start with the following observation.
+Letx0∈Rnandr>0 and recall that 1 <q≤2 andr≤q. Then forf∈B0
+Mp
+q,r
+we have
+/parenleftBig/integraldisplay
+Br(x0)/parenleftbig∞/summationdisplay
+s=0|fs|2/parenrightbigq
+2/parenrightBig1
+q≤/parenleftBig/integraldisplay
+Br(x0)∞/summationdisplay
+s=0|fs|q/parenrightBig1
+q
+≤/parenleftBig∞/summationdisplay
+s=0/integraldisplay
+Br(x0)|fs|q/parenrightBig1
+q
+≤/parenleftBig∞/summationdisplay
+s=0||fs||q
+Lq(Br(x0))/parenrightBig1
+q
+≤/parenleftBig∞/summationdisplay
+s=0||fs||q
+Mp
+q(rn
+q−n
+p)q/parenrightBig1
+q
+≤/parenleftBig
+(rn
+q−n
+p)q∞/summationdisplay
+s=0||fs||q
+Mp
+q/parenrightBig1
+q
+=rn
+q−n
+p/parenleftBig∞/summationdisplay
+s=0||fs||q
+Mp
+q/parenrightBig1
+q
+=rn
+q−n
+p||f|B0
+Mp
+q,q||
+≤Crn
+q−n
+p||f|B0
+Mp
+q,r||.
+From the last inequality we have that for all r>0 and for all x0∈Rn
+rn
+p−n
+q/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbig∞/summationdisplay
+s=0|fs|2/parenrightbigq
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+Lq(Br(x0))≤C||f|B0
+Mp
+q,r||.
+This last estimate together [11], proposition 4.1, implies that f∈ Mp
+q.
+The assertion in the case f∈N0
+p,q,ris the same.
+✷
+3.8 Proof of lemma 2.12
+i) In a first step we will show that if f∈B1
+Mp
+q,rthere exist a constant C-
+independent of f- such that
+||f|B0
+Mp
+q,r||+||∇f|B0
+Mp
+q,r|| ≤C||f|B1
+Mp
+q,r||.
+29Obviously, we have that
+||f|B0
+Mp
+q,r|| ≤ ||f|B1
+Mp
+q,r||.
+Moreover, we observe that
+||∇f|B0
+Mp
+q,r||=/parenleftBig∞/summationdisplay
+j=0||(∇f)j||r
+Mq
+p/parenrightBig1
+r
+≤/parenleftBig∞/summationdisplay
+j=1||(∇f)j||r
+Mq
+p/parenrightBig1
+r+||(∇f)0||Mp
+q
+≤C/parenleftBig∞/summationdisplay
+j=12jr||fj||r
+Mq
+p/parenrightBig1
+r+C||f||Mp
+q
+where for the first addend we used an estimate similar to (3.2)
+with the necessary adaptations to our situation
+see also [10]
+and for the second addend we used [10], lemma 1.8
+and the observation F−1(ξϕ0ˆf) =F−1(ξϕ0)∗f.
+≤C||f|B1
+Mp
+q,r||+C||f|B0
+Mp
+q,r||
+because of lemma 2.10
+≤C||f|B1
+Mp
+q,r||+C||f|B1
+Mp
+q,r||
+≤ ||f|B1
+Mp
+q,r||
+as desired.
+ii) Now, we assume that fsatisfies
+||f|B0
+Mp
+q,r||+||∇f|B0
+Mp
+q,r||<∞.
+We have to show that this last quantity controls
+||f|B1
+Mp
+q,r||.
+In fact, we calculate
+||f|B1
+Mp
+q,r||=/parenleftBig∞/summationdisplay
+j=02jr||fj||r
+Mq
+p/parenrightBig1
+r
+≤C||f0||Mp
+q+C/parenleftBig∞/summationdisplay
+j=12jr||fj||r
+Mq
+p/parenrightBig1
+r
+≤C||f0|B0
+Mp
+q,r+C||∇f|B0
+Mp
+q,r||
+again by an adaption of estimate (3.2)
+≤C(||f0|B0
+Mp
+q,r+||∇f|B0
+Mp
+q,r||).
+✷
+303.9 Proof of lemma 2.13
+According to lemma 2.12 it is enough to show that f∈B0
+Mp
+q,r. First of all, we
+observe that
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+j=1fj/vextendsingle/vextendsingle/vextendsingleB0
+Mp
+q,r/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+j=1fj/vextendsingle/vextendsingle/vextendsingleB1
+Mp
+q,r/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤C/parenleftBig∞/summationdisplay
+j=02jr||fj||r
+Mp
+q/parenrightBig1
+r
+≤C/parenleftBig∞/summationdisplay
+j=0||(∇f)j||r
+Mp
+q/parenrightBig1
+r
+≤ ||∇f|B0
+Mp
+q,r||.
+Now, it remains to estimate ||f0||Mp
+q:
+It holds
+f0=F−1/parenleftbign/summationdisplay
+i=1ξi
+|ξ|2ξiˆfϕ0/parenrightbig
+.
+Next, due to lemma 2.10and its corollarywe knowthat f∈Lqand in particular
+- sincefhas compact support f∈L1soξiˆf∈L∞for alli. Moreover, thanks
+to our assumptions
+ϕ01
+|ξ|∈Lp
+p−1wherep
+p−1∈[1,2].
+So, for all possible i
+ϕ0ξi
+|ξ|2ξiˆf∈Lp
+p−1.
+From this we conclude that
+f0∈Lp⊂ Mp
+q,
+and finally
+||f0|B0
+Mp
+q,r|| ≤ ||f0||Mp
+q+||f1||Mp
+q
+≤ ||f0||Lp+C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+j=1fj|B0
+Mp
+q,r/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤C||∇f|B0
+Mp
+q,r||+C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+j=1fj|B0
+Mp
+q,r/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤C||∇f|B0
+Mp
+q,r||.
+✷
+3.10 Proof of lemma 2.14
+Density ofOMinNs
+p,q,rrespectively in Bs
+Mp
+q,r
+The idea is to approximate f∈Ns
+p,q,rbyfn:=/summationtextn
+k=0fk.
+31From the definition of the spaces Ns
+p,q,rwe immediately deduce that there exists
+N∈Nsuch that/parenleftBig∞/summationdisplay
+j=N+12sjr||fj||r
+Mp
+q/parenrightBig1
+r<ε.
+What concerns the first contributions, i.e. f0-fN, we know that
+N/summationdisplay
+j=0fj=:fN∈OM.
+So,
+||f−fN|Ns
+p,q,r|| ≤C/parenleftBig∞/summationdisplay
+j=N+12sjr||fj||r
+Mp
+q/parenrightBig1
+r<Cε
+whereCdoes not depend on f. This shows that fNapproximates fin the
+desired way.
+The proof in the case Bs
+Mp
+q,ris the same - with the necessary modifications of
+course.
+Density ofOMinNs
+p,q,r
+The idea is the same as above.
+Observe that the definition implies that there exist integers nandmsuch that
+/parenleftBig/summationdisplay
+j/∈{−n,...,0...m}2sjr||fj||r
+Msp,q/parenrightBig1/r
+≤ε
+2.
+And as before, this gives us the result that OMis dense in Ns
+p,q,r.
+Another idea to prove the density of C∞inNs
+p,q,rarises from the usual mollifi-
+cation:
+We have to show that for any given εand any given function f∈Ns
+p,q,rthere
+exists a function g∈C∞such that
+||f−g|Ns
+p,q,r|| ≤ε.
+As indicated above, our candidate for gwill be a function of the form
+g=ϕδ∗f
+whereϕδis a mollifying sequence ( and δwill be specified later on).
+First of all, observe that due to Tonelli-Fubini we have ϕδ∗fj= (ϕδ∗f)j.
+Now, as above we observe that the fact that fbelongs to Ns
+p,q,rimplies that
+there exists N0∈Nsuch that
+/parenleftBig∞/summationdisplay
+N0+12jsr||fj|Mp
+q||r/parenrightBig1
+r≤˜ε
+which together with [10], lemma 1.8, immediately leads to the observatio n that
+/parenleftBig∞/summationdisplay
+N0+12jsr||(f−f∗ϕδ)j|Mp
+q||r/parenrightBig1
+r≤ε
+2.
+32For the remaining contributions we first of all observe that
+|fj−fj∗ϕδ| ≤ ||∇fj||∞δ≤C||f|Ns
+p,q,r||2jδ.
+In order to see this, note that fj∈Ns
+p,q,1which together with two results from
+[10] similar to the estimate (3.2) and the embedding of Besov-Morrey into Besov
+spaces (see also [10]) implies that
+||∇fj||∞≤C||f|Ns
+p,q,r||2j.
+In the case j= 0 observe that
+(∂xif)0=F−1(iξiˆfφ0)
+=F−1(iξiˆfφ0(φ0+φ1))
+=f0∗F−1(iξi(φ0+φ1))
+which implies that
+||∂xif0|Mp
+q|| ≤C||f0|Mp
+q||.
+Apart from this observation, the argument is the same as the usua l one known
+in the framework of Lebesgue spaces.
+Now, we can calculate for any radius R∈(0,1] and for any point x0∈Rn
+Rn
+p−n
+q||fj−fj∗ϕδ||Lq(BR(x0))=Rn
+p−n
+q/parenleftBig/integraldisplay
+BR(x0)|fj−fj∗ϕδ|q/parenrightBig1
+q
+≤CRn
+p−n
+q/parenleftBig
+||∇fj||q
+∞δqRn/parenrightBig1
+q
+≤CRn
+p−n
+q/parenleftBig
+||f|Ns
+p,q,r||q2jqδqRn/parenrightBig1
+q
+=CRn
+p||f|Ns
+p,q,r||δ2j
+≤C||f|Ns
+p,q,r||δ2j
+from which we conclude that
+/parenleftBigN0/summationdisplay
+j=02jsr||fj−fj∗ϕδ|Mp
+q||r/parenrightBig1
+r≤N0/summationdisplay
+j=0||f|Ns
+p,q,r||δ2N0+N0sr
+≤(N0+1)||f|Ns
+p,q,r||δ2N0+N0sr
+≤ε
+2
+if we choose δsufficiently small.
+This shows that f∈Ns
+p,q,rcan be approximated by compactly supported
+smooth function - the convolution f∗ϕδ∗fhas compact support.
+Now, we assume that f∈Bs
+Mp
+q,rwheres≥0, 1<q≤2 and 1≤q≤p≤ ∞has
+compact support. First of all, we observe that according to lemma 2 .10f∈ Mp
+q
+and since it has compact support, f∈Lq. From this we deduce that whenever
+O≤j≤N0,fj∈Bs
+q,mfor alls∈Rand arbitrary mand in particular, fj∈Lp.
+So for each jthere exists a δjsuch that
+||fj−fj∗ϕδj||m
+q≤/parenleftBigε
+2(N0+1)/parenrightBigm
+.
+33If we now choose δsmall enough, then
+/parenleftBigN0/summationdisplay
+j=02jsr||fj−fj∗ϕδ|Mp
+q||r/parenrightBig1
+r=/parenleftBigN0/summationdisplay
+j=02jsr||(f−f∗)jϕδ|Mp
+q||r/parenrightBig1
+r≤ε
+2.
+The other frequencies are estimated as above.
+Finally we observe that f∗ϕδis not only smooth but also compactly supported
+since it is a convolution of a compactly supported function with a comp actly
+supported distribution.
+✷
+Remark 3.7 A close look at the proof we just gave, shows that in fact
+∩m≥0Cm
+is dense in the above spaces.
+3.11 Proof of lemma 2.15
+We split the product fginto the three paraproducts π1(f,g),π2(f,g) and
+π3(f,g) and analyse each of them independently.
+i) We start with π1(f,g) =/summationtext∞
+k=2/summationtextk−2
+l=0flgk. It is easy to see that a simple
+adaptation of lemma 3.15 of [12] to our variant of Besov-Morrey, imp lies that
+it suffices to show that
+/parenleftBig∞/summationdisplay
+k=2||gkk−2/summationdisplay
+l=0fl||2
+Mn
+2/parenrightBig1
+2≤C||g|B0
+Mn
+2,2||(||f|B1
+Mn
+2,2||+||f||∞).
+In fact, we calculate
+/parenleftBig∞/summationdisplay
+k=2||gkk−2/summationdisplay
+l=0fl||2
+Mn
+2/parenrightBig1
+2≤/parenleftBig∞/summationdisplay
+k=2||gk(sup
+s|s/summationdisplay
+l=0fl|)||2
+Mn
+2/parenrightBig1
+2
+≤/parenleftBig∞/summationdisplay
+k=2||gk||2
+Mn
+2||sup
+s|s/summationdisplay
+l=0fl|||2
+∞/parenrightBig1
+2
+≤ ||sup
+s|s/summationdisplay
+l=0fl|||∞(∞/summationdisplay
+k=2||gk||2
+Mn
+2/parenrightBig1
+2
+≤ ||sup
+s|s/summationdisplay
+l=0fl|||∞||g|B0
+Mn
+2,2||
+≤ ||f||∞||g|B0
+Mn
+2,2||
+because of lemma 4.4.2 of [17]
+<∞.
+ii) Next, we study π2(f,g) =/summationtext∞
+k=0/summationtextk+1
+l=k−1flgk. For our further calculations
+we fixl=k. We will see that what follows will not depend on this choice, so
+||π2(f,g)|B0
+Mn
+2,2|| ≤Csup
+s∈{−1,0,1}||∞/summationdisplay
+k=0fk+sgk|B0
+Mn
+2,2||.
+34In fact, we will show a bit more, namely π2(f,g)∈B1
+Mn
+2
+1,1. Again a sim-
+ple adaptation of lemma 3.16 of [12] shows that we only have to estimat e/summationtext∞
+k=02k||fkgk||
+Mn
+2
+1. In fact, we have
+∞/summationdisplay
+k=02k||fkgk||
+Mn
+2
+1≤∞/summationdisplay
+k=02k||fk||Mn
+2||gk||Mn
+2
+≤/parenleftBig∞/summationdisplay
+k=022k||gk||2
+Mn
+2/parenrightBig1
+2/parenleftBig∞/summationdisplay
+k=0||fk||2
+Mn
+2/parenrightBig1
+2
+≤ ||g|B1
+Mn
+2,2|| ||f|B0
+Mn
+2,2||
+<∞.
+Once we have this, it implies together with the embedding of Besov-Mo rrey
+spaces into Besov spaces (see [10]) - adapted to our variant of Bes ov-Morrey
+spaces - and the fact that l1⊂l2immediately that/summationtext∞
+k=0fkgk∈B0
+Mn
+2,2.
+And finally we get that π2(f,g)∈B0
+Mn
+2,2.
+iii) The remaining addend is π3(f,g). Again, as in i) it is enough to show that
+we can estimate/parenleftBig/summationtext∞
+l=2||fl/summationtextl−2
+k=0gk||2
+Mn
+2/parenrightBig1
+2in the desired manner. In fact
+we observe that the following inequalities hold:
+/parenleftBig∞/summationdisplay
+l=2||fll−2/summationdisplay
+k=0gk||2
+Mn
+2/parenrightBig1
+2≤∞/summationdisplay
+l=2||fll−2/summationdisplay
+k=0gk||Mn
+2
+≤∞/summationdisplay
+l=2||fl||Mn
+2||l−2/summationdisplay
+k=0gk||∞
+=∞/summationdisplay
+l=22l||fl||Mn
+22−l||l−2/summationdisplay
+k=0gk||∞
+≤/parenleftBig∞/summationdisplay
+l=022l||fl||2
+Mn
+2/parenrightBig1
+2/parenleftBig∞/summationdisplay
+l=02−2l||l−2/summationdisplay
+k=0gk||2
+∞/parenrightBig1
+2
+≤C/parenleftBig∞/summationdisplay
+l=022l||fl||2
+Mn
+2/parenrightBig1
+2/parenleftBig∞/summationdisplay
+l=02−2l||l/summationdisplay
+k=0gk||2
+∞/parenrightBig1
+2
+≤C||f|B1
+Mn
+2,2||/parenleftBig∞/summationdisplay
+l=02−2l||l/summationdisplay
+k=0gk||2
+∞/parenrightBig1
+2
+≤C||f|B1
+Mn
+2,2|| ||g|B−1
+∞,2||
+according to lemma 4.4.2 of [17]
+≤C||f|B1
+Mn
+2,2|| ||g|N0
+n,2,2||
+due to the embedding result for Besov-Morrey spaces ([10])
+≤C||f|B1
+Mn
+2,2|| ||g|B0
+Mn
+2,2||
+<∞.
+If we put together all our results from i) to iii) we see that we have th e estimate
+||gf|B0
+Mn
+2,2|| ≤C||g|B0
+Mn
+2,2||(||f|B1
+Mn
+2,2||+||f||∞)
+35as claimed.
+✷
+References
+[1] David R. Adams, A note on Choquet integrals with respect to Haus dorff
+capacities,Functionspacesandapplications(Lund, 1986),115-1 24,Lecture
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+arXiv:1001.0379v1  [math.DG]  3 Jan 2010GRADED NILPOTENT LIE ALGEBRAS OF INFINITE
+TYPE
+BORIS DOUBROV, OLGA RADKO
+Abstract. The paper gives the complete characterization of all
+gradednilpotentLiealgebraswithinfinite-dimensionalTanakapro-
+longation as extensions of graded nilpotent Lie algebras of lower
+dimension by means of a commutative ideal. We introduce a no-
+tion of weak characteristics of a vector distribution and prove tha t
+if a bracket-generating distribution of constant type does not ha ve
+non-zero complex weak characteristics, then its symmetry algebr a
+is necessarily finite-dimensional. The paper also contains a num-
+ber of illustrative algebraic and geometric examples including the
+proof that any metabelian Lie algebra with a 2-dimensional center
+always has an infinite-dimensional Tanaka prolongation.
+1.Introduction
+This paper is devoted to the study of non-holonomic vector distribu -
+tionswithinfinite-dimensional symmetry algebras. The simplest exam -
+ples of such distributions are contact distributions on odd-dimensio nal
+manifoldsand,moregenerally, contactsystemsonjetspaces Jk(Rn,Rm).
+Thesymmetry algebrasofsuchdistributionsaregivenbyLie–Backlu nd
+theorem and are isomorphic to either the Lie algebra of all vector fie lds
+onJ0(Rn,Rm) =Rn+mform≥2 or to the Lie algebra of all contact
+vector fields on J1(Rn,R) form= 1. In both cases the symmetry
+algebras are infinite-dimensional.
+We are interested in only so-called bracket-generating distributions ,
+i.e., we always assume that repetitive brackets of vector fields lying in
+Dgenerate the whole tangent bundle TM. If this is not the case, then
+Dlies in a certain proper completely integrable vector distribution D′,
+and the geometry of Dcan be essentially reduced to the restrictions of
+Dto the fibers of D′.
+We shall also say that the distribution Disdegenerate , if it possesses
+non-zero Cauchy characterisitics , i.e., vector fields X∈Dsuch that
+2000Mathematics Subject Classification. 17B70, 53C30, 58A17.
+Key words and phrases. Graded nilpotent Lie algebras, Tanaka prolongation,
+metabelian Lie algebras, Lie algebra cohomology.
+12 BORIS DOUBROV, OLGA RADKO
+[X,D]⊂D. It is also clear that any degenerate distribution, even if
+it is bracket-generating, has an infinite-dimensional symmetry alge bra.
+For the proof and other basic properties of vector distributions w e refer
+to [2, Chapter 2].
+While it seems to be very difficult to provide the complete local de-
+scription of all vector distributions with infinite-dimensional symmet ry
+algebra, it appears that it is possible to give the complete description
+of the their symbol algebras.
+Namely, let Dbeabracket-generatingvectordistributiononasmooth
+manifold M. Taking repetitive brackets of vector fields lying in D, we
+can define a weak derived flag ofD:
+0⊂D⊂D2⊂ ··· ⊂Dµ=TM,
+D0= 0, D1=D, Di+1= [D,Di], i≥2.
+At each point p∈Mwe can defined the associated graded vector
+space
+(1) m(p) =/summationdisplay
+i<0m−i(p),m−i(p) =Di
+p/Di−1
+p.
+Since [Di,Dj]⊂Di+jfor alli,j≥0,m(p) is naturally equipped with
+a structure of a graded Lie algebra. Namely, if x∈m−i(p) andy∈
+m−j(p) are two homogeneous elements in m(p), andX∈Di,Y∈Dj
+are two vector fields such that Xp+Di−1
+p=xandYp+Dj−1
+p=y,
+then the value of [ X,Y]+Di+j−1
+pdepends only on xandy, and, thus,
+defines a graded Lie algebra structure on m(p). It is also clear from the
+definition, that m(p) is a nilpotent Lie algebra generated by m−1(p).
+This Lie algebra is called a symbol of the distribution Dat a point
+p∈M, and it plays essential role in study of Dand any geometric
+structures subordinate to D. For example, if Dis a contact struc-
+ture on a smooth manifold of dimension 2 n+1, i.e., a non-degenerate
+codimension 1 distribution on M, then its symbol is isomorphic to the
+(2n+1)-dimensional Heisenberg Lie algebra at any point p∈M.
+The family of graded Lie algebras m(p) is a basic invariant of any
+bracket-generatingdistribution, whichincludesnotonlythedimens ions
+of the weak derived series of D, but also a non-trivial algebraic infor-
+mation. In many cases the structure of these algebras has very im por-
+tant geometric consequences and allows to associate various geom etric
+structures with M. One of the most famous examples is E. Cartan
+paper [3], where he associates a G2-geometry with any non-degenerate
+2-dimensional vector distribution on a 5-dimensional manifold.
+We say that the distribution Dhas constant symbol mor is of type
+mif its symbols m(p) are isomorphic to mfor all points p∈M. ForGRADED NILPOTENT LIE ALGEBRAS OF INFINITE TYPE 3
+example, the contact distribution and all contact systems on jet s paces
+are of constant type.
+We shall use the term graded nilpotent Lie algebra or simply GNLA
+for any negatively graded Lie algebra m=/summationtextµ
+i=1m−igenerated by m−1.
+We shall call such Lie algebra non-degenerate , ifm−1does not include
+any non-zero central elements. This has a clear geometric meaning . It
+is easy to see that the symbol of a bracket-generating distributio nDis
+non-degenerate if andonlyif Dhas no non-zero Cauchy characteristics.
+The largest µsuch that m−µ/ne}ationslash= 0 is called the depth ofm.
+It appears that we can derive the exact bound for the dimension of
+the symmetry algebra of Dpurely in terms of its symbol m. Namely,
+Tanaka [15] has shown in his pioneer works on the geometry of filtere d
+manifolds, that there is a well-defined graded Lie algebra g(m) called
+Tanaka (or universal) prolongation of m. It is characterized by the
+following conditions:
+(1)gi(m) =mifor alli <0;
+(2) if [X,m] = 0 for certain X∈gi(m), thenX= 0;
+(3)g(m) is the largest graded Lie algebra satisfying the above two
+conditions.
+The Lie algebra g(m) has also a clear geometric meaning. Namely, let
+Mbe a connected simply connected Lie group with the Lie algebra
+m. For example, we can identify Mwithmand define the Lie group
+multiplicationin MbymeansofCampbell–Haussdorfseries. Definethe
+distribution DonMby assuming that it isleft invariant andthat De=
+m−1. Theng(m) canbenaturallyidentified with thegradedLiealgebra
+associated with the filtered Lie algebra of all germs of infinitesimal
+symmetries of Dat the identity. In particular, if g(m) is finite or
+infinite-dimensional, so is the symmetry algebra of the correspondin g
+filtration. Such distributions Dare called standard distributions of
+typem.
+One of the main result of Tanaka paper [15] can be reformulated as
+follows.
+Tanaka theorem ([15]).Ifg(m)is finite-dimensional, then with each
+distribution D⊂TMof typemwe can associate a canonical coframe
+on a certain bundle PoverMof dimension dimg(m). In particular,
+the symmetry algebra of the distribution Dis finite-dimensional, and
+its dimension is bounded by dimg(m).
+Thus, ifDis a holonomic distribution with infinite-dimensional sym-
+metry algebra, then the Tanaka prolongation of its symbol g(m) should
+also beinfinite-dimensional. The properties ofTanaka prolongationa re
+also studied in [18, 17, 10, 16, 11].4 BORIS DOUBROV, OLGA RADKO
+The main result of this paper is Theorem 1, which gives the the com-
+plete characterization of all GNLA with infinite-dimensional Tanaka
+prolongation as extensions of graded nilpotent Lie algebras of lower di-
+mension by means of a commutative ideal. Along with this description
+we also introduce a notion of weak characteristics of a vector distri-
+bution and prove that if a bracket-generating distribution of cons tant
+type does not have non-zero complex weak characteristics, then its
+symmetry algebra is necessarily finite-dimensional.
+This article is closely related to a series of papers devoted to the ge-
+ometryof2and3-dimensionaldistributions[1,4,5]. Thesepaperss how
+that under very mild non-degeneracy conditions the symmetry alge bra
+of a non-holonomic vector distribution becomes finite-dimensional. A s
+we show in Theorem 2, the set of all GNLA, whose Tanaka prolonga-
+tion is infinite-dimensional, forms a closed subvariety in the variety of
+all GNLA mwith fixed dimensions of subspaces m−i,i >0.
+Finally, in the last section of the paper we present a number of
+illustrative algebraic and geometric examples including the proof that
+any metabelian Lie algebra with a 2-dimensional center always has an
+infinite-dimensional Tanaka prolongation.
+We would like to express our gratitude to Ian Anderson, Tohru Mo-
+rimoto, Ben Warhurst, Igor Zelenko for stimulating discussions on t he
+topic of this paper.
+2.Tanaka and Spencer criteria
+Below we assume that all Lie algebras we consider are defined over
+an arbitrary field kof characteristic 0. We try to be as generic as
+possible in our algebraic considerations and we explicitly state, when
+we require the base field kto be algebraically closed.
+Letmbe an arbitrary GNLA and let g(m) be its Tanaka prolon-
+gation. We say that misof finite (infinite) type , ifg(m) is finite-
+dimensional (resp., infinite-dimensional). Note that this notion is sta -
+ble with respect to the base field extensions, since each subspace gi(m),
+i≥0 of the Tanaka prolongation can be computed by solving a system
+of linear equations. Namely, assuming that all subspaces gi(m),i < k
+are already computed, the space gk(m) can be defined as follows:
+gk(m) ={φ:m→/summationdisplay
+i<kgi(m)|φ(m−i)⊂gk−i(m),
+φ([x,y]) = [φ(x),y]+[x,φ(y)],∀x,y∈m}.
+Tanaka criterium reduces the question whether Tanaka prolongat ion
+g(m) ofmis finite-dimensional or not to the same question for theGRADED NILPOTENT LIE ALGEBRAS OF INFINITE TYPE 5
+standard prolongation (as described, for example, in [14]) of a cert ain
+linear Lie algebra. Namely, let Der 0(m) be theLie algebra of all degree-
+preserving derivations of m. Define the subalgebra h0⊂Der0(m) as
+follows:
+h0={d∈Der0(m)|d(x) = 0 for all x∈m−i,i≥2}.
+We can naturally identify h0with a subspace in End( m−1). Then
+Tanaka criteruim can be formulated as follows:
+Tanaka criterium ([15]).Tanaka prolongation g(m)ofmis finite-
+dimensional if and only if so is the standard prolongation of h0⊂
+End(m−1).
+In its turn, Spencer criterium provides a computationally efficient
+method of detecting whether the standard prolongation of a linear sub-
+spaceA⊂End(V) is finite-dimensional or not.
+Spencer criterium ([13, 7]).The standard prolongation of a subspace
+A⊂End(V)is finite-dimensional if and only if A¯k⊂End(V¯k)does
+not contain endomorphisms of rank 1.
+Here by ¯kwe mean the algebraic closure of our base field k, and
+byA¯kthe subspace in End( V¯k) obtained from Aby field extension.
+A detailed and self-contained proof of Spencer criterium can be fou nd
+in [12].
+We emphasize that this criterium if computationally efficient, as the
+set of all rank 1 endomorphisms in Ais described by a finite set of
+quadratic polynomials, and, for example, Gr¨ obner basis technique pro-
+vides an algorithm to determine whether this set of polynomials has a
+non-zero common root.
+In the next section we use both these criteria to prove that all fun -
+damental Lie algebras mof infinite type over an algebraically closed
+field possess a very special algebraic structure.
+3.Graded nilpotent Lie algebras of infinite type
+Letgbe an arbitrary finite-dimensional Lie algebra, and let Vbe
+an arbitrary g-module. We recall that a Lie algebra ¯gis called an
+extension of gby means of V, if¯gcan be included into the following
+exact sequence:
+0→V→¯g→g→0.
+In other words, Vis embedded into ¯gas a commutative ideal, the quo-
+tient¯g/Vis identified with gand the natural action of g=¯g/VonV
+coincideswiththepredefined g-modulestructureon V. Two extensions6 BORIS DOUBROV, OLGA RADKO
+¯g1and¯g2are called equivalent, if there exists an isomorphism ¯g1→¯g2
+identical on Vandg≡¯g1/V≡¯g2/V.
+It is well-known that equivalence classes of such extensions are de-
+scribed by the cohomology space H2(g,V). Namely, if [ α] is an element
+ofH2(g,V), where α∈Z2(g,V), then¯gcan be identified as a vector
+space with g×Vwith the Lie bracket given by:
+(2) [( x1,v1),(x2,v2)] = ([x1,x2],x1.v2−x2.v1+α(x1,x2)),
+x1,x2∈g,v1,v2∈V.
+Assume now that both the Lie algebra gand the g-module Vare
+graded, that is g=/summationtextgi,V=/summationtextVjandgi.Vj⊂Vi+j. Then the
+cohomology space H2(g,V) is naturally turned into the graded vector
+space as well:
+H2(g,V) =/summationdisplay
+H2
+i(g,V).
+It is easy to see that the Lie algebra ¯gdefined by (2) is also graded if
+and only if [ α]∈H2
+0(g,V). Thus, we see that the graded extensions
+¯gof the graded Lie algebra gby means of the graded g-module Vare
+in one to one correspondence with the elements of the vector spac e
+H2
+0(g,V).
+We can introduce an additional group action on H2
+0(g,V) as follows.
+Define Aut 0(g,V) as a subgroup in Aut 0(g)×GL0(V) of the form:
+Aut0(g,V) ={(f,g)∈Aut0(g)×GL0(V)|
+g(x.v) =f(x).g(v),∀x∈g,v∈V}.
+This group naturally acts on H2
+0(g,V):
+(f,g).[α] = [g◦α◦f−1].
+It is easy to see that elements of H2
+0(g,V) lying in the same orbit of
+this action correspond to the isomorphic Lie algebras.
+In this paper we shall consider special extensions of GNLA . Namely,
+letnbe an arbitrary (possibly degenerate) GNLA. Fix any subspace
+W⊂n−1and consider n−1/Was a graded commutative Lie alge-
+bra concentrated in degree −1. Take any graded n−1/W-module V=/summationtextµ
+i=1V−igenerated by V−1(as a module). Assuming that W.V= 0
+andn−i.V= 0 for all i≥2 we can consider Vas ann-module. We call
+any extension of the GNLA nby means of such n-module Va special
+GNLAextension. Let [ α] be the corresponding element of H2
+0(n,V). It
+is easy to see that such extension is non-degenerate if and only if th e
+following two conditions hold:
+(1) the condition n.v= 0 implies v= 0 forv∈V−1;
+(2) the cocycle αis non-degenerate on Z(n)∩n−1.GRADED NILPOTENT LIE ALGEBRAS OF INFINITE TYPE 7
+We note that unlike the extensions of GNLA considered in the pa-
+pers [10, 1], the special extensions defined above are not central exten-
+sions of Lie algebras, as the ideal Vabove does not in general belong
+to the center, and the action of nonVis non-trivial.
+The main result of the paper is:
+Theorem 1. Letmbe a non-degenerate GNLA oven an algebraically
+closed field. Then the following conditions are equivalent:
+(1)there exists such d∈Der0(m)thatd(m−i) = 0fori≥2and
+rankd= 1;
+(2)there exists such y∈m−1thatrankady= 1; in particular, in
+this case [y,m−i] = 0for alli≥2;
+(3)mcan be represented as a special extension of a certain (possi bly
+degenerate) GNLA nwithdimm−1= dimn−1+1anddimV−i=
+1for alli=−1,...,−µandµ≥2;
+(4)mis of infinite type.
+Proof.(1)⇒(2). Let hbe a subalgebra in Der 0(m) defined by con-
+ditiond(m−i) = 0 for any d∈h. Letd∈hand rank d= 1. Define
+W⊂masW= kerd. Choose non-zero x,y∈m−1such that d(x) =y.
+Then we have m−1=/an}bracketle{tx/an}bracketri}ht⊕W, andW⊃[m,m]. Next,
+[y,W] = [d(x),W] =−[x,d(W)] = 0.
+Sincemis non-degenerate, this implies that rankad y= 1 and [ x,y]/ne}ationslash=
+0. We also have
+[x,d(y)] =−[d(x),y] =−[y,y] = 0.
+On the other hand,
+[d(y),W] =−[y,d(W)] = 0.
+Hence, [d(y),m] = 0, and from non-degeneracy of mit follows that
+d(y) = 0. In particular, dis nilpotent.
+(2)⇒(1). Let y∈m−1such that rankad y= 1. Let W= kerady.
+Choose such x∈m−1that [x,y]/ne}ationslash= 0. Such xalways exists by non-
+degeneracy of m. It is clear that m=/an}bracketle{tx/an}bracketri}ht⊕Wandy∈W. It is also
+clear that W⊃[m,m]. Define d∈hasd(x) =yandd(W) = 0. It is
+easy to see that dis indeed a differentiation of m.
+(2)⇒(3). Again, let x,ybeelements in m−1suchthatrankad y= 1,
+[x,y]/ne}ationslash= 0,m=/an}bracketle{tx/an}bracketri}htW,W⊃[m,m] and [y,W] = 0. Set y1=yand
+defineyi+1= [x,yi] for all i≥2. Note that y2/ne}ationslash= 0. Let µbe the
+smallest integer such that yµ/ne}ationslash= 0 and yµ+1= 0. By induction we get:
+[yi+1,W] = [[x,yi],W] = [x,[W,yi]]+[yi,[x,W]] = 0.8 BORIS DOUBROV, OLGA RADKO
+LetV=/an}bracketle{ty1,...yµ. It is clear that V⊂W, [V,W] = 0] and [ x,V]⊂V.
+Hence,Vis a commutative ideal in m. Putn=m/V. It is easy to see
+thatVis generated by V−1as ann-module and the action of [ n,n] on
+Vis trivial. Finally, it is evident that nis also generated by n−1as a
+graded Lie algebra. Thus, mis represented as a special extension:
+(3) 0 →V→m→n→0.
+(3)⇒(2). Let mbe a special extension (3) of nwith dim V−i= 1,
+i= 1,···,µ. Takeyas a non-zero element in V−1. Since [ n,n] acts
+trivially on V, we see that [ y,m]⊂V−2. SinceVis generated by V−1
+as ann-module, we see that ad y/ne}ationslash= 0 and rankad y= 1.
+(4)⇔(1). Thisisexactlythecombined TanakaandSpencer criteria.
+/square
+Remark 1.Only the last implication uses the fact that the base field
+is algebraically closed. In case of arbitrary field of characteristic 0 t he
+items (1), (2) and (3) are still equivalent and imply that mhas infinite
+type. a simple counterexample for the implication (4) ⇒(1) overRis
+given by the 3-dimensional complex Heisenberg Lie algebra (with an
+obvious grading of depth 2) viewed as a 6-dimensional real Lie algebr a.
+Remark2.Condition (2) of the Theorem appears already in [11] as a
+sufficient condition for mto be of infinite type.
+Item (3) of Theorem 1 gives an inductive algorithm for constructing
+all GNLA of infinite type. Namely, to construct all GNLA mof infinite
+type and dim m−1=n+1 one needs to:
+•take an arbitrary GNLA n(of finite or infinite type) such that
+dimn−1=n;
+•take an arbitrary subspace W⊂n−1of codimension 1;
+•fix any integer s≥2 and construct a graded n−1/W-module
+V=/summationtexts
+i=1V−i, which is uniquely defined by the conditions
+dimV−i= 1,i= 1,...,sandV−i−1= (n−1/W).V−i;
+•compute cohomology space H2
+0(n,V), where Vis treated as an
+n-module with the trivial action of W⊕[n,n];
+•take any ω∈H2
+0(n,V) and define mas an extension of nby
+means of Vcorresponding to ω.
+Incoordinatesthisprocedurecanbereformulatedasfollows. Fixa ny
+convector α∈n∗
+−1and an element X∈n−1such that α(X) = 1. Ex-
+tendXto the basis {X,Z1,...,Z r}ofnconsisting of homogeneous ele-
+ments. Define masaLiealgebrawithabasis {X,Y1,...,Y s,Z1,...,Z r},GRADED NILPOTENT LIE ALGEBRAS OF INFINITE TYPE 9
+s≥2, where
+[X,Yi] =Yi+1, i= 1,...,s−1;
+[Zj,Yi] = 0, i= 1,...,s, j = 1,...,r;
+[Yi,Yj] = 0, i,j= 1,...,s;
+[X,Zi]m= [X,Zi]n+s/summationdisplay
+j=1aj
+iYj;
+[Zi,Zj]m= [Zi,Zj]n+s/summationdisplay
+k=1bk
+ijYk,
+where the constants aj
+iandbk
+ijdefine a cocycle ∧2n→Vof degree 0
+(i.e.,mis a graded Lie algebra) and are viewed modulo the changes
+of the basis X/mapsto→X+/summationtexts
+i=1ciYs,Zi/mapsto→Zi+/summationtexts
+j=1dj
+iYj. See the next
+section for the concrete examples.
+Theorem 1 provides also an easy way to prove that the set of all
+graded Lie algebras of infinite type forms forms a closed algebraic su b-
+variety in the variety of all graded nilpotent Lie algebras. Namely, fix
+a sequence of integers k1,...,k µand denote by M(k1,...,k µ) a vari-
+ety of all graded nilpotent Lie algebras msuch that dim m−i=kifor
+i= 1,...,µandm−i= 0 fori > µ. It is easy to see that it is a closed
+algebraic variety in the vector space of all skew-symmetric graded al-
+gebras (without any conditions on multiplication).
+The set of all fundamental graded Lie algebras in M(ki) is distin-
+guished by conditions [ m−1,m−i] =m−i−1for alli= 1,...,µ−1. It
+is easy to see that these conditions define an open subset (in Zariss ki
+topology) in M(ki), which we shall denote by MF(ki).
+DefineMF∞(ki) as the set of all fundamental Lie algebras of infinite
+type.
+Theorem 2. MF∞(ki)is a closed subvariety in MF(ki).
+Proof.The existence of an element y∈m−1such that rankad y= 1
+(and, hence, [ y,m−i] = 0 for i≥2 can be reformulated as an existence
+ofa non-zero solution ofa certain system of quadraticequations ( 2 by 2
+minors of the matrix of ad y), whose coefficients arestructure constants
+of the Lie algebra m. It is well-known [8, Theorem 3.12] that that the
+conditions when such non-zero solution exists are given in terms of t he
+algebraic equations on the coefficients of this system. /square
+Theorem 1 also has a number of immediate geometric applications.
+Namely, let Dbe a bracket generating distribution with a constat sym-
+bolm. Asusual, we assume that DhasnoCauchy characteristics. This10 BORIS DOUBROV, OLGA RADKO
+implies that mis non-degenerate, i.e., does not have non-zero central
+elements lying in m−1.
+We say that a vector field Y∈Disa weak characteristic of D, if
+the following conditions hold:
+(1)Y∈char(Di) for alli≥2, or, in other words, [ Y,(Di)]⊂(Di)
+for alli≥2;
+(2) thesubbundle D′⊂Dspannedbyall X∈Dsuchthat[ X,Y]⊂
+Dhas codimension 1 in D.
+Similarly, we can define complex weak characteristics of a vector distri-
+butionDas sections of the comlpexified subbundle DCof the comlpex-
+ified tangent bundle TCMsatisfying the conditions (1) and (2) above,
+where we replace powers Diby their complexified versions.
+Theorem 3. LetDbe a bracket generating distribution with a con-
+stant symbol m. Assume that Dhas no Cauchy characteristics and
+thatdimsym( D) =∞. Then Dcontains non-trivial complex week
+characteristics.
+Proof.This immediately follows from item (2) of Theorem 1 and the
+Tanaka theorem [15] that bounds the dimension of sym( D) by the
+dimension of the Tanaka prolongation of its symbol m. /square
+Corollary 1. IfD∩(∩i≥2char(Di)) = 0, then the symmetry algebra
+ofDis finite-dimensional.
+Proof.Assume that sym( D) is infinite-dimensional. Then it possesses
+complex week characteristic Y. Clearly both YandcY,c∈C, are
+also complex week characteristics. Therefore, the complex subbu ndle
+spanned by YandYis stable with respect to the complex conjugation.
+Hence, it also contains a real one-dimensional subbundle spanned b y a
+certain non-zero vector field X. While Xitself might no longer be a
+week characteristic, it still lies in D∩(∩i≥2char(Di)). /square
+Remark 3.We formulate Theorem 2 and its corollary only for the
+vector distribution with constant symbol, since the Tanaka theore m
+on the bound of dimsym( D) has been proved up to now only in this
+case. The Tanaka proof does not immediately generalize to the case of
+non-constant symbol, and this still remains an open problem accord ing
+to the authors knowledge.
+4.Examples
+4.1.Symbols of 2-dimensional distributions of infinite type.
+As a first application of Theorem 1 we describe the symbols of 2-
+dimensional bracket generating distributions of infinite type.GRADED NILPOTENT LIE ALGEBRAS OF INFINITE TYPE 11
+Using Theorem 1, it is easy to prove that in each dimension n≥2
+there exists a unique (up to isomorphism) GNLA mwith dim m−1= 2.
+It turns out to coincide with the symbol of the Goursat distribution ,
+which can coincides also with the canonical contact system on the je t
+spaceJn−2(R,R).
+Indeed, by Theorem 1 any such GNLA must be a special extension
+of a certain GNLA nwith dim n−1= 1. But then n=Ris a one-
+dimensional commutative Lie algebra, and all its special extensions a re
+trivial. In each dimension there is exactly one such extension. It can
+be described as a Lie algebra mwith a basis /an}bracketle{tX,Z1,...,Z n−1/an}bracketri}htand
+the only non-zero Lie brackets [ X,Zi] =Zi+1,i= 1,...,n−2. Here
+degX=−1 and deg Zi=−i,i= 1,...,n−1. It is easy to see that the
+contact system on Jn−2(R,R) is equivalent to the standarddistribution
+of typem.
+4.2.Symbols of 3-dimensional distributions of infinite type. In
+case of dim m−1= 3 we already get a lot of various examples, including
+the examples of non-trivial special extension of GNLA. As above, a ny
+GNLAmof infinite type with dim m−1= 3 is a special extension of a
+certain GNLA nwith dim n−1= 2. Note that nitself does not need to
+be of infinite type. As the classification of all such GNLA unknown at
+the moment, it makes also impossible to classify all infinite type GNLA
+with dim m−1= 3.
+As an example, let us describe all special extensions in case, when
+nis a three-dimensional Heisenberg algebra. Then all trivial special
+extensions of ncan be described as:
+m=/an}bracketle{tX,Z1,Z2,Y1,...,Y k/an}bracketri}ht,
+where deg X=−1, degYi=−i,i= 1,...,k, degZj=−j,j= 1,2.
+All non-zero Lie brackets are [ X,Z1] =Z2and [X,Yi] =Yi+1,i=
+1,...,k−1. The Lie algebra mis a semidirect product of its subal-
+gebran=/an}bracketle{tX,Z1,Z2/an}bracketri}htand the commutative ideal V=/an}bracketle{tY1,...,Y k/an}bracketri}ht.
+Geometrically the standard distribution of type mcan be described as
+a canonical contact system on the mixed jet bundle J1,k−1(R,R2).
+Any non-trivial special extension of ncan be obtained by adding
+non-trivial Lie brackets of the form:
+[X,Z1] =Z2+aY2;
+[X,Z2] =bY3;
+[Z1,Z2] =cY3.
+By changing Z1toZ1−aY1andZ2toZ2−bY2we can always obtain
+a=b= 0. Ifa/ne}ationslash= 0, then the Jacobi identity is satisfied only if k= 3.12 BORIS DOUBROV, OLGA RADKO
+In this case we can always scale ato 1. Thus, we see that up to the
+isomorphism there exists exactly one non-trivial special extension of
+the three-dimensional Heisenberg Lie algebra. It is given by:
+m=/an}bracketle{tX,Z1,Z2,Y1,Y2,Y3/an}bracketri}ht,
+where non-zero Lie brackets are given by:
+[X,Y1] =Y2,[X,Y2] =Y3;
+[X,Z1] =Z2;
+[Z1,Z2] =Y3.
+Direct computation shows that an arbitrary symmetry of the stan dard
+distribution of type mdepends on 4 functions of 1 variable.
+4.3.Metabelian Lie algebras. Letm=m−1⊕m−2be a GNLA of
+depth 2. The structure of this Lie algebra is completely determined b y
+the skew-symmetric map: S:∧2m−1→m−2defined as a restriction of
+the Lie bracket to m−1. Below we assume that mis non-degenerate.
+Consider the dual map S∗:m∗
+−2→ ∧2m∗
+−1. Sincemis generated
+bym−1, the map S∗is injective. Thus, up to the isomorphism the
+structure of mis determined by the subspace Im S∗⊂ ∧2m∗
+−1. This
+subspace is considered up to the natural action of the group GL(m−1)
+on∧2m∗
+−1.
+To simplify the notation, let V=m−1,n= dimV, and let P=
+ImS∗⊂ ∧2V∗,m= dimP. If we fix a basis in V, then∧2V∗can
+be identified with a space of all skew-symmetric nbynmatrices, Pis
+spanned by mmatrices B1,...,B mand the action of GL(V) on∧2V∗
+is given by:
+X.B=XtBX, X ∈GL(V),B∈ ∧2V∗.
+Letybe an arbitrary element in m−1=V. It is easy to see that
+the rank of ad yis the maximal number of linearly independent skew-
+symmetric matrices BinP, such that iyB/ne}ationslash= 0. In more detail, let Py
+be a subspace in Pdefined as:
+Py={B∈P|iyB= 0}.
+Then rank of ad yis equal to to codimension of PyinP.
+Theorem 4. Letm=m−1⊕m−2be a 2-step complex graded nilpotent
+Lie algebra with dimm−2= 2. Then Tanaka prolongation of mis
+infinite-dimensional.
+Proof.Since the dimension of the Tanaka prolongation is preserved
+under extension of the base field, we can assume that our base field is
+algebraically closed. Then according to Theorem 1 mis of infinite typeGRADED NILPOTENT LIE ALGEBRAS OF INFINITE TYPE 13
+if and only if there exists a non-zero element y∈m−1=Vsuch that Py
+is one-dimensional. Let B1,B2be the basis of P. Then the condition
+dimPy= 1 is equivalent to the existence of such λ1,λ2that:
+(λ1B1+λ2B2)y= 0.
+It is clear that this is equivalent to det( λ1B1+λ2B2) = 0, which always
+has non-trivial solutions in case of algebraically closed field. /square
+Note that pencils of skew-symmetric matrices over an algebraically
+closed field can be effectively classified using Kronecker results on pe n-
+cils of matrices. This has been done in the work of M. Gauger [6].
+Namely, any pair ( A,B) of skew-symmetric matrices can be written as
+one matrix P=µA+λB, whose entries are linear forms in λandµ.
+Such matrices are classified by the following data:
+•minimal indices 0 ≤m1≤m2≤mp,p≥0 (in particular, the
+set of minimal indices can be empty);
+•elementary divisors ( µ+a1λ)e1, ..., (µ+aqλ)er, (λ)f1, ..., (λ)fs
+(each of these divisors appears twice).
+The canonical form of the pencil Pfor this data is:
+(4)P=
+Mm1...
+Mmp
+Ee1(a1)
+...
+Eer(ar)
+Ff1...
+Ffs
+
+where
+Mm=/parenleftbigg
+0Mm
+−Mt
+m0/parenrightbigg
+,(2m+1)×(2m+1),M0= (0);
+En(a) =/parenleftbigg
+0En(a)
+−En(a)t0/parenrightbigg
+,(2n)×(2n);
+Fn=/parenleftbigg
+0Fn
+−Ft
+n0/parenrightbigg
+,(2n)×(2n);14 BORIS DOUBROV, OLGA RADKO
+and
+Mm=
+λ
+λ µ
+·µ
+· ·
+· ·
+λ·
+λ µ
+µ
+,(m+1)×m;
+En(a) =
+µ+aλ
+·λ
+·
+· ·
+·
+µ+aλ·
+µ+aλ λ
+, n×n;
+Fn=
+λ
+·µ
+·
+· ·
+·
+λ·
+λ µ
+, n×n;
+Inaddition,theelementarydivisorsareconsidereduptonon-dege nerate
+linear transformations of ( λ,µ).
+Note also, that the pencil Pabove corresponds to a non-degenerate
+2-step nilpotent Lie algebra if and only if there are no minimal indices
+0 among the set of minimal indices m1,...,m p. As we assume that
+these indices are ordered, this is equivalent to the condition m1>0.
+Define the following subgroups in GL(V) andthe corresponding sub-
+algebras in gl(V):
+Aut(P) ={X∈GL(V)|XtBX⊂Pfor anyB∈P},
+H0={X∈GL(V)|XtBX=Bfor anyB∈P},
+Der(P) ={X∈gl(V)|XtB+BX⊂Pfor anyB∈P},
+h0={X∈gl(V)|XtB+BX= 0 for any B∈P}.
+It is clear that:GRADED NILPOTENT LIE ALGEBRAS OF INFINITE TYPE 15
+•Der(P) andh0are subalgebras in gl(V) corresponding to the
+subgroups Aut( P) andH0respectively;
+•H0is a subgroup in Aut( P),h0is a subalgebra in Der( P);
+•Aut(P) is naturally identified with the the group Aut 0(m) of
+all grading-preserving automorphisms of the Lie algebra mand
+Der(P) is identified with the Lie algebra Der 0(m) of all grading-
+preserving derivations of m;
+•h0can be identified with all elements in Der 0(m) that act triv-
+ially onm−2.
+Let us show explicitly that h0always contains a rank 1 element.
+This will give another proof that the standard prolongation of h0is
+infinite-dimensional.
+LetQbe any of blocks that appear on the diagonal in the canonical
+form (4) of the pencil P. That is Qis one of the following matrices:
+Mmform≥0,Ee(a) fore≥1, orFf,f≥1. We shall call Q
+an elementary subpencil of the pencil Pand denote by h0(Q) the Lie
+algebra
+h0(Q) ={X∈gl(k,C)|XtB+BX= 0 for all B∈Q}.
+It is easy to see that h0(Q) can be embedded as a subalgebra into h0.
+So, to prove the theorem it is sufficient to show that h0(Q) contains
+rank 1 element for each elementary subpencil. Moreover, since elem en-
+tary divisors are considered up to linear transformations of ( λ,µ), we
+can restrict ourselves only to elementary pencils Mm,m≥1 andFr,
+r≥1.
+Inboththesecasesthesubalgebra h0(Q)iseasilycomputed. Namely,
+denote by Sk,l(C) the set of k×lmatrices ( aij) such that aij=ai+1,j+1
+for any 1 ≤i < k, 1≤j < l. Denote also by Tk(C) the set of all
+k×kupper triangular matrices that also lie in Sk,k(C). Note that
+bothSk,l(C) andTk(C) contain elements of rank 1 (for example, in the
+upper right corner).
+We have:
+h0(Mm) =/braceleftbigg/parenleftbigg
+xEm+1Y
+0−xEm/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglex∈C,Y∈Sm+1,m(C)/bracerightbigg
+;
+h0(Fr) =/braceleftbigg/parenleftbigg
+Y11Y12
+Y21−Y11/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleY11,Y12,Y21∈Tr(C)/bracerightbigg
+;
+As we see, in both cases h0(Q) contains a nilpotent element of rank 1.
+4.4.Remarks. 1. It follows form the proof of Theorem 4 that h0
+contains the block diagonal subalgebra, where each block consists of
+all elements in h0(Q) for the primitive subpencils Q. However, h0can16 BORIS DOUBROV, OLGA RADKO
+be larger than this direct sum. For example, this happens in the case
+when there no minimal indices and the set of ( λ) and (µ) where each of
+these divisors appears with multiplicity 2 pand 2qrespectively. In this
+casemis isomorphic to the direct sum of two Heisenberg lie algebras
+of dimension 2 p+ 1 and 2 q+ 1. The subalgebra h0is isomorphic to
+sp(2p,C)⊕sp(2q,C), whileh0(Q) for each primitive Qis justsll(2,C).
+2. It is well-known that there exist 2-step nilpotent Lie algebras with
+3-dimensional center, whose Tanaka prolongation is finite-dimensio nal.
+The simplest example is a free 2-step nilpotent Lie algebra with 3-
+dimensional set of generators and the 3-dimensional center:
+m−1=/an}bracketle{tX1,X2,X3/an}bracketri}ht,
+m−2=/an}bracketle{tX12,X13,X23/an}bracketri}ht,
+where [Xi,Xj] =Xijfor 1≤i < j≤3. Its Tanaka prolongation is 21-
+dimensional and is isomorphic to so(7,C). In this case the subalgebra
+h0is trivial.
+We can also generalize this example for arbitrary number of genera-
+tors. Namely, let:
+m−1=/an}bracketle{tX1,X2,...,X k/an}bracketri}ht,
+m−2=/an}bracketle{tY1,Y2,Y3/an}bracketri}ht,
+where all non-zero Lie brackets have the form:
+[X1,Xk] = [X2,Xk−1] =···=Y1;
+[X1,Xk−1] = [X2,Xk−2] =···=Y2;
+[X2,Xk] = [X3,Xk−1] =···=Y3.
+Direct computation shows that Tanaka prolongation g(m) ofmhas the
+form:
+•fork= 3,g(m)≡so(7,C);
+•fork= 4,g(m)≡sp(6,C);
+•fork= 5,g(m) =m⊕g0, whereg0≡gl(2,C) andg0-module
+g−1is irreducible;
+•fork≥6 and even, g(m) =m⊕g0, whereg0≡gl(2,C)×C
+andg0-module g−1splits into the sum of k/2gl(2,C)-modules
+/an}bracketle{tXi,Xk/2+i/an}bracketri}ht,i= 1,...,k/2−1;
+•fork≥7 and odd, g(m) =m⊕g0, whereg0=C2and the
+action of g0ong−1diagonalizes in the basis {X1,X2,...,X n}.
+3. The aboveexample of2-stepnilpotent Liealgebrawith 3-dimensi-
+onal center shows that the subalgebra h0is trivial for generic 2-step
+nilpotent Lie algebras with dim m−2≥3. In particular, a generic
+GNLA of depth 2 with at least 3-dimensional center is of finite type.GRADED NILPOTENT LIE ALGEBRAS OF INFINITE TYPE 17
+However, it is still an open problem to classify all 2-step nilpotent Lie
+algebras with infinite-dimensional Tanaka prolongation.
+4. Even though the theorem above is formulated, and proved over C,
+itisinfacttrueover anyfieldofcharacteristic0, and, inparticular, over
+R. Indeed, the results of Kronecker and, thus, Gauger hold over a ny
+algebraically closed filed of characteristic 0. And it is easy to see that
+the dimension of Tanaka prolongation does not change if we extend t he
+field of scalars.
+5. LetDbe a vector distribution of codimension 2 on a smooth
+manifold M, such that D2=TM. Then Theorem 4 implies that the
+symbolm(x) ofDis of infinite type for each x∈M. However, this
+does not imply that such distributions always have infinite-dimensiona l
+symmetry algebras. For example, such distributions with the symbo l
+corresponding to the pencil Mmare treated in [9]. It is proved that if
+a subbundle D′generated by all weak characteristics of Dis bracket
+generating, then the symmetry algebra of Dis finite-dimensional. Note
+that in case of the standard distribution Dof typem, wheremcorre-
+sponds to the pencil Mm, the subdistribution D′is involutive.
+References
+[1] I. Anderson, B. Kruglikov, Rank 2 distributions of Monge equations: symme-
+tries, equivalences, extensions , arXiv:0910.546v1 [math.DG].
+[2] R. Bryant, S.S. Chern, R.B. Gardner, H.L. Goltschmidt, P.A. Griffit hs,Exterior
+differential systems , Springer–Verlag, New York, 1991.
+[3]´E. Cartan, Les syst` emes de Pfaff ` a cinq variables et les ´ equations aux d´ eriv´ ees
+partielles du second ordre , Ann.´Ecole Norm. Sup., 27(1910), pp. 109–122.
+[4] B.Doubrov,I.Zelenko, On local geometry of nonholonomic rank 2 Distributions ,
+Journal of London Math. Society, 802009, pp. 545–566.
+[5] B. Doubrov, I. Zelenko, On local geometry of rank 3 distributions with 6-
+dimensional square , arXiv:math/0703662v1 [math.DG].
+[6] M. Gauger, On the classificaation of metabelian Lie algebras , Transaction AMS,
+179(1973), pp. 293–329.
+[7] V. Guillemin, D. Quillen, S. Sternberg, The classification of the irreducible com-
+plex algebras of infinite type , J. Anal. Math., 18(1967), pp. 107–112.
+[8] J. Harris, Algebraic geometry , Graduate Texts in Mathematics, Volume 133,
+Springer–Verlag, 1989.
+[9] W. Kry´ nski, I. Zelenko, Canonical frames for distributions of odd rank and
+corank 2 with maximal Kronecker index (in preparation).
+[10] O. Kuzmich, Graded nilpotent Lie algebras of low dimension , Lobachevskij
+J. Math., 3(1999), pp. 147–184.
+[11] A. Ottazzi, A sufficient condition for nonrigidity of Carnot groups , Math. Z.,
+259(2008), pp. 617–629.
+[12] A. Ottazzi, B. Warhurst, Rigidiry of Carnot groups (in preparation).
+[13] D.C. Spencer, Overdetermined systems of linear partial differential equa tions,
+Bull. AMS, 75(1969), pp. 179–239.18 BORIS DOUBROV, OLGA RADKO
+[14] S. Sternberg, Lectures on differential geometry , Prentice Hall, N.J., 1964.
+[15] N. Tanaka, On differential systems, graded Lie algebras and pseudo-gro ups, J.
+Math. Kyoto. Univ., 10(1970), pp. 1–82.
+[16] B. Warhurst, Tanaka prolongation of free Lie algebras , Geom. Dedicata, 130
+(2007), pp. 59–69.
+[17] T. Yatsui, On pseudo-product graded Lie algebras , Hokkaido Math. J., 17
+(1988), pp. 333–343.
+[18] K. Yamaguchi, Differential systems associated with simple graded Lie alge bras,
+Adv. Studies in Pure Math., 22(1993), pp. 413–494.
+Belarussian State University, Nezavisimosti av. 4, 220030 , Minsk,
+Belarus
+E-mail address :doubrov@islc.org
\ No newline at end of file
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+arXiv:1001.0380v3  [math.AP]  5 Oct 2010Blowup for the Euler and Euler-Poisson
+Equations with Repulsive Forces
+Manwai Yuen∗
+Department of Applied Mathematics,
+The Hong Kong Polytechnic University,
+Hung Hom, Kowloon, Hong Kong
+Revised 05-Oct-2010
+Abstract
+In this paper, we study the blowup of the N-dim Euler or Euler-Poisson equations with
+repulsive forces, in radial symmetry. We provide a novel int egration method to show that
+the non-trivial classical solutions ( ρ,V), with compact support in [0 ,R], where R >0 is a
+positive constant and in the sense which ρ(t,r) = 0 and V(t,r) = 0 for r≥R, under the
+initial condition
+H0=/integraldisplayR
+0rV0dr >0, (1)
+blow up on or before the finite time T=R3/(2H0) for pressureless fluids or γ >1.
+The main contribution of this article provides the blowup re sults of the Euler ( δ= 0)
+or Euler-Poisson ( δ= 1) equations with repulsive forces, and with pressure ( γ >1), as the
+previous blowup papers ([1] [2], [3] and [4]) cannot handle t he systems with the pressure term,
+forC1solutions.
+Key Words: Euler Equations, Euler-Poisson Equations, Inte gration Method, Blowup, Re-
+pulsive Forces, With Pressure, C1Solutions, No-Slip Condition
+∗E-mail address: nevetsyuen@hotmail.com
+12 M.W.Yuen
+1 Introduction
+The isentropic Euler ( δ= 0) or Euler-Poisson ( δ=±1) equations can be written in the following
+form: 
+
+ρt+∇·(ρu) =0,
+ρ[ut+(u·∇)u]+∇P=ρ∇Φ,
+∆Φ(t,x) =δα(N)ρ,(2)
+whereα(N) is a constant related to the unit ball in RN:α(1) = 1, α(2) = 2πandα(3) = 4π.
+And as usual, ρ=ρ(t,x)≥0 andu=u(t,x)∈RNare the density and the velocity respectively.
+P=P(ρ) is the pressure function. The γ-law can be applied on the pressure term P(ρ), i.e.
+P(ρ)=Kργ, (3)
+which is a common hypothesis. If the parameter is set as K >0, we call the system with pressure ;
+ifK= 0, we call it pressureless . The constant γ=cP/cv≥1, where cP,cvare the specific heats
+per unit mass under constant pressure and constant volume resp ectively, is the ratio of the specific
+heats, that is, the adiabatic exponent in the equation (3). In part icular, the fluid is called isother-
+mal ifγ= 1. IfK >0, we call the system with pressure; if K= 0, we call it pressureless.
+In the above systems, the self-gravitational potential field Φ = Φ( t,x) is determined by the density
+ρitself, through the Poisson equation (2) 3.
+Whenδ=−1, the system can model fluids that are self-gravitating , such as g aseous stars. In
+addition, the evolution of the simple cosmology can be modelled by the d ust distribution with-
+out pressure term. This describes the stellar systems of collisionles s and gravitational n-body
+systems [5]. And the pressureless Euler-Poissonequations can be d erived from the Vlasov-Poisson-
+Boltzmann model with the zero mean free path [6]. For N= 3 and δ=−1, the equations (2) are
+the classical (non-relativistic) descriptions of a galaxy in astrophy sics. See [7] and [8], for details
+about the systems.
+Whenδ= 1, the system is the compressible Euler-Poisson equations with rep ulsive forces. The
+equation (2) 3is the Poisson equation through which the potential with repulsive fo rces is deter-
+mined by the density distribution of the electrons. In this case, the system can be viewed as aBlowup for Euler and EP Equations 3
+semiconductor model. See [9] and [10] for detailed analysis of the sys tem.
+On the other hand, the Poisson equation (2) 3can be solved as
+Φ(t,x) =δ/integraldisplay
+RNG(x−y)ρ(t,y)dy, (4)
+whereGis Green’s function for the Poisson equation in the N-dimensional spaces defined by
+G(x).=
+
+|x|, N= 1;
+log|x|, N= 2;
+−1
+|x|N−2, N≥3.(5)
+Usually, the Euler-Poisson equations can be rewritten in the scalar f orm:
+
+
+∂ρ
+∂t+N
+Σ
+k=1uk∂ρ
+∂xk+ρN
+Σ
+k=1∂uk
+∂xk=0,
+ρ/parenleftbigg
+∂ui
+∂t+N
+Σ
+k=1uk∂ui
+∂xk/parenrightbigg
++∂P
+∂xi=ρ∂Φ
+∂xi, fori= 1,2,...N.(6)
+For the construction of the analytical solutions for the systems, interested readers should refer
+to [11], [12], [13], [14] and [15]. The results for local existence theories can be found in [16], [17]
+and [18]. The analysis of stabilities for the systems may be referred t o [19], [20], [21], [1], [2], [3],
+[22], [13], [23], [24], [4] and [25].
+We seek the radial symmetry solutions
+ρ(t,/vector x) =ρ(t,r) and/vector u=/vector x
+rV(t,r) =:/vector x
+rV, (7)
+with the radius r=/parenleftBig/summationtextN
+i=1x2
+i/parenrightBig1/2
+.
+For the solutions in spherical symmetry, the Poisson equation (2) 3is transformed to
+rN−1Φrr(t,x)+(N−1)rN−2Φr=α(N)δρrN−1, (8)
+Φr=α(N)δ
+rN−1/integraldisplayr
+0ρ(t,s)sN−1ds. (9)
+By standard computation, the Euler-Poisson equations in radial sy mmetry can be written in the
+following form:
+
+ρt+Vρr+ρVr+N−1
+rρV= 0,
+ρ(Vt+VVr)+Pr(ρ) =ρΦr(ρ).(10)4 M.W.Yuen
+Historically, Makino, Ukai and Kawashima initially defined the tame solut ions [1] for outside
+the compact of the solutions
+Vt+VVr= 0. (11)
+Following this, Makino and Perthame considered the tame solutions fo r the system with gravita-
+tional forces [2]. After that Perthame discovered the blowup resu lts for 3-dimensional pressureless
+system with repulsive forces [3] ( δ= 1). In short, all the results above rely on the solutions with
+radial symmetry:
+Vt+VVr=α(N)δ
+rN−1/integraldisplayr
+0ρ(t,s)sN−1ds. (12)
+And theEmdenordinarydifferentialequationswerededucedonthe boundarypointofthesolutions
+with compact support:
+D2R
+Dt2=δM
+RN−1, R(0,R0) =R0≥0,˙R(0,R0) = 0, (13)
+wheredR
+dt:=VandMis the mass of the solutions, along the characteristic curve. They s howed
+the blowup results for the C1solutions of the system (10).
+Recently, ChaeandTadmor[4]showedthefinitetime blowup, forthe pressurelessEuler-Poisson
+equations with attractive forces ( δ=−1), under the initial condition,
+S:={a∈RN/vextendsingle/vextendsingleρ0(a)>0,Ω0(a) = 0,∇·u(0,x(0)<0} /ne}ationslash=φ, (14)
+whereΩistherescaledvorticitymatrix(Ω0ij) =1
+2(∂iuj
+0−∂jui
+0)withthenotation u= (u1,u2,....,uN)
+in their paper and some point x0.
+They use the analysis of spectral dynamics to show the Racatti diff erential inequality,
+Ddivu
+Dt≤ −1
+N(divu)2. (15)
+The solution for the inequality (15) blows up on or before T=−N/(∇·u(0,x0(0)).
+However, their method cannot be applied to the system with repulsiv e forces to obtain the
+similar blowup result.
+On the other hand, in [24], we have the blowup results if the solutions w ith compact support
+under the condition,
+2/integraldisplay
+Ω(t)(ρ|u|2+2P)dx < M2−ǫ, (16)Blowup for Euler and EP Equations 5
+whereMis the mass of the solution.
+In this article, the alternative approach is adopted to show that th ere is no global existence of
+C1solutions for the system, (6) ( δ= 0 orδ= 1),with compact support without the condition
+(14). We notice that the conditions in our result are different from t he works of Engerlberg et. al
+[26].
+Theorem 1 Consider the N-dimensional Euler (δ= 0)or Euler-Poisson equations with repulsive
+forces(δ= 1)(2). The non-trivial classical solutions (ρ,V), in radial symmetry, with compact
+support in [0,R], whereR >0is a positive constant (which ρ(t,r) = 0andV(t,r) = 0forr≥R)
+and the initial velocity such that:
+H0=/integraldisplayR
+0rV0dr >0, (17)
+blow up on or before the finite time T=R3/(2H0),for pressureless fluids (K= 0)orγ >1.
+The solutions ( ρ,u) may lose their regularity, for example the velocity function V∈C0only or
+the shock waves appear on or before the finite time T.
+2 Integration Method
+In this section, we present the proof of Theorem 1. The technique of the proof was selected simply
+to deduce the partial differential equations to the Racatti equat ion, to show the blowup result.
+However, we note our integration method is novel to the studies of blowup for this kind of the
+systems.
+Proof.In general, we show that the ρ(t,x(t;x)) preserves its positive nature as the mass equation
+(6)1can be converted to be
+Dρ
+Dt+ρ∇·u= 0, (18)
+with the material derivative,
+D
+Dt=∂
+∂t+(u·∇). (19)
+We integrate the equation (18):
+ρ(t,x) =ρ0(x0(0,x0))exp/parenleftbigg
+−/integraldisplayt
+0∇·u(t,x(t;0,x0))dt/parenrightbigg
+≥0, (20)6 M.W.Yuen
+forρ0(x0(0,x0))≥0,along the characteristic curve.
+We use the momentum equation (10) 2with the non-trivialsolutions in radialsymmetry, ρ0/ne}ationslash= 0,
+to have:
+Vt+VVr+Kγργ−2ρr= Φr, (21)
+Vt+∂
+∂r(1
+2V2)+Kγργ−2ρr= Φr, (22)
+rVt+r∂
+∂r(1
+2V2)+Kγrργ−2ρr=rΦr, (23)
+with multiplying ron the both sides.
+We take integration with respect to r,to the above equation, for γ >1 orK≥0:
+/integraldisplayR
+0rVtdr+/integraldisplayR
+0rd
+dr(1
+2V2)+/integraldisplayR
+0Kγrργ−2ρrdr=/integraldisplayR
+0rΦrdr, (24)
+/integraldisplayR
+0rVtdr+/integraldisplayR
+0rd
+dr(1
+2V2)+/integraldisplayR
+0Kγr
+γ−1dργ−1=/integraldisplayR
+0/bracketleftbiggα(N)δr
+rN−1/integraldisplayr
+0ρ(t,s)sN−1ds/bracketrightbigg
+dr,(25)
+/integraldisplayR
+0rVtdr+/integraldisplayR
+0rd
+dr(1
+2V2)+/integraldisplayR
+0Kγr
+γ−1dργ−1≥0, (26)
+forδ≥0.
+It follows with integration by part:
+/integraldisplayR
+0rVtdr−1
+2/integraldisplayR
+0V2dr+1
+2/bracketleftbig
+RV(t,R)2−0·V(t,0)2/bracketrightbig
+−/integraldisplayR
+0Kγ
+γ−1ργ−1dr+Kγ
+γ−1/bracketleftbig
+Rργ−1(t,R)−0·ργ−1(t,0)/bracketrightbig
+≥0.
+(27)
+Theaboveinequalitywiththeboundarycompactconditionof V(t,R) = 0and ρ(t,R) = 0, becomes
+/integraldisplayR
+0rVtdr−1
+2/integraldisplayR
+0V2dr−/integraldisplayR
+0Kγ
+γ−1ργ−1dr= 0. (28)
+Asrandtare independent variables and VisC1in the domain [0 ,R] in the assumption of the
+theorem, we may change the differentiation and the integration as t he following:
+d
+dt/integraldisplayR
+0rVdr−1
+2/integraldisplayR
+0V2dr−/integraldisplayR
+0Kγ
+γ−1ργ−1dr≥0, (29)
+d
+dt1
+2/integraldisplayR
+0Vdr2−1
+2/integraldisplayR
+01
+2rV2dr2≥/integraldisplayR
+0Kγ
+γ−1ργ−1dr≥0, (30)
+forγ >1 orK= 0.
+For the non-trivial initial condition ρ0≥0, we have the following differential inequality:
+d
+dt1
+2/integraldisplayR
+0Vdr2−1
+2/integraldisplayR
+01
+2rV2dr2≥0, (31)Blowup for Euler and EP Equations 7
+d
+dt/integraldisplayR
+0Vdr2≥/integraldisplayR
+01
+2rV2dr2≥1
+2R/integraldisplayR
+0V2dr2, (32)
+d
+dt/integraldisplayR
+0Vdr2≥1
+2R/integraldisplayR
+0V2dr2. (33)
+By denoting
+H:=H(t) =/integraldisplayR
+0rVdr=1
+2/integraldisplayR
+0Vdr2, (34)
+and with the Cauchy-Schwarz inequality,
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayR
+0V·1dr2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftBigg/integraldisplayR
+0V2dr2/parenrightBigg1/2/parenleftBigg/integraldisplayR
+01dr2/parenrightBigg1/2
+, (35)
+/vextendsingle/vextendsingle/vextendsingle/integraltextR
+0Vdr2/vextendsingle/vextendsingle/vextendsingle
+R≤/parenleftBigg/integraldisplayR
+0V2dr2/parenrightBigg1/2
+, (36)
+4H2
+R2≤/integraldisplayR
+0V2dr2, (37)
+2H2
+R3≤1
+2R/integraldisplayR
+0V2dr2, (38)
+the inequality (32) becomes
+d
+dtH≥1
+2R/integraldisplayR
+0V2dr2≥2H2
+R3, (39)
+d
+dtH≥2H2
+R3. (40)
+With the initial condition: H0=/integraltextR
+0rV0dr >0,we can obtain
+H≥−R3H0
+2H0t−R3. (41)
+Therefore, the solutions blow up on or before the finite time T=R3/(2H0).
+This completes the proof.
+Remark 2 For controlled experiments in engineering, fluids are kept i n a fixed ball solid container
+with a radial R. Therefore, it requires the compact support condition for t≥0,
+ρ(t,r) = 0andV(t,r) = 0, (42)
+withr≥R. This corresponding condition is called no-slip condition (solid boundary condition)
+[27] and [28].
+On the other hand, in computing simulations, the systems are usually coupled with the similar8 M.W.Yuen
+boundary conditions for real applications. Therefore, the condition for compact support (no-slip
+condition) is reasonable in modelings. But for free boundar y problems, fluids may not be bounded
+by a fixed volume for all time. Therefore, further research is needed to study the corresponding
+result in future works.
+Remark 3 It is still an open question whether or not there exists time- localC1-solution with
+compact support for any initial condition with compact supp ort. On the other hand, if the global
+solutions with compact support whose radii expand unbounde dly as time tends to infinity, the dis-
+cussion of this paper can offer no information about this case .
+Remark 4 This article has shed new light on situations with the pressu re term. In particular, it
+provides the blowup results of the Euler (δ= 0)or Euler-Poisson (δ= 1)equations with repulsive
+forces, and with pressure (γ >1). This is the main contribution of the article, as the previou s
+blowup papers ([1] [2], [3] and [4]) cannot handle the system s with the pressure term, for C1
+solutions. A further refinement for the non-radial symmetry is expected in future studies.
+3 Acknowledgement
+The author would like to thank the comments of reviewers to improve the article.
+References
+[1] T. Makino, S. Ukai and S. Kawashima, On Compactly Supported Solutions of the Compressible
+Euler Equation , Recent Topics in Nonlinear PDE, III (Tokyo, 1986), 173–183, Nor th-Holland
+Math. Stud., 148, North-Holland, Amsterdam, 1987.
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+poisson Pour levolution d’etoiles gazeuses,(French) [On R adially Symmetric Solutions of the
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+165–170.Blowup for Euler and EP Equations 9
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+Forces, Japan J. Appl. Math. 7(1990), 363–367.
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+RN, Commun. Math. Sci. 6(2008), 785–789.
+[5] H. H. Fliche and R. Triay, Euler-Poisson-Newton Approach in Cosmology , Cosmology and
+Gravitation, 346–360, AIP Conf. Proc., 910, Amer. Inst. Phys., Melville, NY, 2007.
+[6] R. T. Glassey, The Cauchy Problem in Kinetic Theory , Society for Industrial and Applied
+Mathematics (SIAM), Philadelphia, PA, 1996.
+[7] J. Binney and S. Tremaine, Galactic Dynamics , Princeton Univ. Press, 1994.
+[8] S. Chandrasekhar, An Introduction to the Study of Stellar Structure , Univ. of Chicago Press,
+1939.
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+[10] P.L. Lions, Mathematical Topics in Fluid Mechanics . Vols.1,2, 1998, Oxford: Clarendon
+Press, 1998.
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+991-997.
+[12] T. Makino, Blowing up Solutions of the Euler-Poission Equation for the Evolution of the
+Gaseous Stars , Transport Theory and Statistical Physics 21(1992), 615–624.
+[13] Y.B. Deng, J.L. Xiang and T. Yang, Blowup Phenomena of Solutions to Euler-Poisson Equa-
+tions, J. Math. Anal. Appl. 286(2003), 295–306.
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+Appl. Anal .4(2005), 757–762.
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+tions of Gaseous Stars , J. Math. Anal. Appl. 341 (2008),445–456.10 M.W.Yuen
+[16] T. Makino, On a Local Existence Theorem for the Evolution Equation of Ga seous Stars ,
+Patterns and waves, 459–479, Stud. Math. Appl., 18, North-Holland, Amsterdam, 1986.
+[17] M. Bezard, Existence locale de solutions pour les equations d’Euler-P oisson (Local Existence
+of Solutions for Euler-Poisson Equations) , Japan J. Indust. Appl. Math. 10(1993), 431–450
+(in French).
+[18] P.Gamblin, Solution reguliere a temps petit pour l’equation d’Euler–P oisson (Small-time Regu-
+lar Solution for the Euler-Poisson Equation) , Comm. PartialDifferential Equations 18(1993),
+731–745 (in French).
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+Math. Phys. 101(1985), 475–485.
+[20] S. Alinhac, Blowup for Nonlinear Hyperbolic Equations . Progress in Nonlinear Differential
+Equations and their Applications, 17. Birkh¨ aser Boston, Inc., Bos ton, MA, 1995.
+[21] S. Engelberg, Formation of Singularities in the Euler and Euler-Poisson E quations, Phys. D,
+98, 67–74, 1996.
+[22] Y.B. Deng, T.P. Liu, T. Yangand Z.A. Yao, Solutions of Euler-Poisson Equations for Gaseous
+Stars, Arch. Ration. Mech. Anal. 164(2002) 261–285.
+[23] J. Jang, Nonlinear Instability in Gravitational Euler-Poisson Sys tems for γ= 6/5, Arch.
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+Anal. Appl. 344(2008), 145–156.
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+Commun. Math. Sci. 7(2009), 627–634.
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+ana Univ. Math. J. 50(2001), 109–157.Blowup for Euler and EP Equations 11
+[27] M. A. Day, The No-slip Condition of Fluid Dynamics , Erkenntnis 33(1990), 285–296.
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\ No newline at end of file
diff --git a/1001.0381.txt b/1001.0381.txt
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+arXiv:1001.0381v5  [gr-qc]  21 Jun 2011Some remarks on exact wormhole solutions
+Peter K. F. Kuhfittig
+Department of Mathematics
+Milwaukee School of Engineering
+Milwaukee, Wisconsin 53202-3109 USA
+Exact wormhole solutions, while eagerly saught after, ofte n have the appearance of being overly
+specialized or highly artificial. A case for the possible exi stence of traversable wormholes would be
+more compelling if an abundance of solutions could be found. It is shown in this note that for many
+of the wormhole geometries in the literature, the exact solu tions obtained imply the existence of
+large sets of additional solutions.
+PAC numbers: 04.20.Jb, 04.20.Gz
+Key words: wormholes, exact solutions
+I. INTRODUCTION
+One of the fundamental problems in the general theory
+of relativity is finding exact solutions of the Einstein field
+equations [1]. An example is the Schwarzschild solution,
+which proved to be a major milestone. Similarly, find-
+ing exact solutions of wormhole geometries has occupied
+many researchers. The difference is that in this setting
+exact solutions are hardly a virtue if they have the ap-
+pearance of being artificial or, at best, overly special-
+ized. It is shown in this note that exact solutions may be
+used to show the existence of entire families of solutions,
+which in turn helps strengthen the case for the possible
+existence of traversable wormholes.
+II. WORMHOLE SOLUTIONS
+A wormhole may be defined as a handle or tunnel in the
+spacetime topology linking different universes or widely
+separated regions of our own Universe [9]. A general line
+element is the following:
+ds2=−e2Φ(r)dt2+e2Λ(r)dr2+r2(dθ2+sin2θdφ2),(1)
+where Φ( r)→0 and Λ( r)→0 asr→ ∞, usually re-
+ferred to as asymptotic flatness . We also assume that
+Φ and Λ have continuous derivatives in their respective
+domains. The function Φ = Φ( r) is called the redshift
+function, which must be everywhere finite to prevent
+an event horizon. The function Λ = Λ( r) is related
+to theshape function b(r) =r(1−e−2Λ(r)), that is,
+e2Λ(r)= 1/[1−b(r)/r]. The minimum radius r=r0
+is thethroatof the wormhole, where b(r0) =r0. Also
+needed is b′(r0)<1, usually referred to as the flare-out
+condition . As a consequence, Λ( r) has a vertical asymp-
+tote atr=r0: limr→r0+Λ(r) = +∞. For the redshift
+function we also have the additional requirement that
+Φ′(r0) be finite. Finally, to hold a wormhole open, the
+null energy condition must be violated [9].
+ThecomponentsoftheEinsteintensorinthe orthonor-mal frame are listed next [4].
+Gˆtˆt=2
+re−2Λ(r)Λ′(r)+1
+r2/parenleftBig
+1−e−2Λ(r)/parenrightBig
+,(2)
+Gˆrˆr=2
+re−2Λ(r)Φ′(r)−1
+r2/parenleftBig
+1−e−2Λ(r)/parenrightBig
+,(3)
+Gˆθˆθ=Gˆφˆφ
+=e−2Λ(r)/parenleftbigg
+Φ′′(r)−Φ′(r)Λ′(r)+[Φ′(r)]2
++1
+rΦ′(r)−1
+rΛ′(r)/parenrightbigg
+.(4)
+The case that will be examined in some detail is a
+wormhole supported by phantom energy, whose equation
+of state (EoS) is p=−Kρ,K >1. For this case, ρ+p=
+ρ(1−K)<0, in violation of the null energy condition.
+From the Einstein field equations Gˆαˆβ= 8πTˆαˆβand
+Eqs. (2) and (3), we now get
+1
+8π/bracketleftbigg2
+re−2Λ(r)Φ′(r)−1
+r2/parenleftBig
+1−e−2Λ(r)/parenrightBig/bracketrightbigg
+=−K1
+8π/bracketleftbigg2
+re−2Λ(r)Λ′(r)+1
+r2/parenleftBig
+1−e−2Λ(r))/parenrightBig/bracketrightbigg
+.
+Solving for Λ′(r), we have
+Λ′(r) =1
+K/bracketleftbigg
+−Φ′(r)−1
+2r/parenleftBig
+e2Λ(r)−1/parenrightBig
+(K−1)/bracketrightbigg
+.(5)
+Observe that lim r→r0+Λ′(r) =−∞. Otherwise Λ′(r) is
+continuous in the interval ( r0,∞).
+The continuity assumptions play a critical role: thus
+fora > r 0, the integral/integraltextb
+aeΛ(r)drexists since Λ( r) is
+continuous. Now from the line element (1), the proper
+radial distance ℓ(r) from the throat r=r0to any point
+away from the throat is given by
+ℓ(r) =/integraldisplayr
+r0eΛ(r′)dr′. (6)2
+For an “arbitrary” Λ( r), this integral is not likely to be
+finite, given the asymptotic behavior of Λ( r). For cer-
+tain choices of Φ( r), however, ℓ(r) does exist, thereby
+producing a traversable wormhole. Examples are the ex-
+act solutions for Λ( r) obtained by choosing the following
+redshift functions: Φ( r)≡constant [6] and Φ( r) = lnk
+r
+[3, 10].
+Next, let us suppose that Λ 1(r) is one of the above
+exact solutions [corresponding to some Φ( r)], so that
+ℓ1(r) =/integraldisplayr
+r0eΛ1(r′)dr′
+is finite. Given that Φ′(r) is finite by assumption, we
+deduce from Eq. (5) that
+q1(r) = Λ′
+1(r)+1
+K1
+2r/parenleftBig
+e2Λ1(r)−1/parenrightBig
+(K−1) (7)
+is also finite. To generate a new solution, we let ǫ=ǫ(r)
+be a function with a continuous derivative satisfying the
+conditions ǫ(r0) =ǫ′(r0) = 0 and ǫ(r)→0 asr→ ∞.
+Define Λ 2(r) = Λ1(r)+ǫ(r). Then by Eq. (7)
+q2(r) = Λ′
+2(r)+1
+K1
+2r/parenleftBig
+e2Λ2(r)−1/parenrightBig
+(K−1) (8)
+is finite. Now consider the equation
+Λ′
+2(r) =
+1
+K/bracketleftbigg
+−[Φ′(r)+η′(r)]−1
+2r/parenleftBig
+e2Λ2(r)−1/parenrightBig
+(K−1)/bracketrightbigg
+(9)
+for some finite differentiable function η=η(r). Writing
+Eq. (9) in the form
+Λ′
+1(r)+ǫ′(r) =
+1
+K/bracketleftbigg
+−[Φ′(r)+η′(r)]−1
+2r/parenleftBig
+e2Λ1(r)+2ǫ(r)−1/parenrightBig
+(K−1)/bracketrightbigg
+,
+(10)
+we see that, by comparison to Eq. (5), η′(r) must sat-
+isfy the conditions η′(r0) = 0 and η(r)→0 asr→ ∞,
+thereby retaining the asymptotic flatness. The function
+η′(r) is defined only implicitly in Eq. (10), but since
+we are strictly interested in the existence of new solu-
+tions, it is sufficient to note that the formal relationship
+η′(r) =−Φ′(r)−Kq2(r) implies that η′(r) exists and is
+sectionally continuous, so that η=η(r) exists, as well.
+We conclude that Λ 2(r) is a solution to Eq. (5) cor-
+responding to some finite redshift function Φ( r) +η(r).
+Moreover,
+ℓ2(r) =/integraldisplayr
+r0eΛ2(r′)dr′(11)
+is finite since ǫ(r0) = 0. The result is the existence of an
+entire family of solutions, one for each ǫ=ǫ(r).
+Similar arguments can be used to obtain new classes
+of solutions from exact solutions of wormholes supportedbyChaplyginandgeneralizedChaplygingas[7], modified
+Chaplygin gas [2], and even van der Waals quintessence
+[8]. For the case of a generalized Chaplygin gas, the EoS
+isp=−A/ρα, 0< α≤1, and where Ais a constant [7].
+After substituting and solving for Λ′(r), we get [4]
+Λ′(r) =
+1
+2(8π)1+1/αr(r2)1/αA1/αe2Λ(r)
+/bracketleftbig
+1−e−2Λ(r)−2re−2Λ(r)Φ′(r)/bracketrightbig1/α−1
+2r/parenleftBig
+e2Λ(r)−1/parenrightBig
+.
+(12)
+According to Ref. [7], there is an exact solution for Φ′≡
+0, making ℓ(r) finite. In analogous manner, we now let
+Λ1(r) be an exact solution of Eq. (12). Then Λ 2(r)+ǫ(r)
+is a new solution, corresponding to some Φ( r) +η(r).
+(Assume that ǫ(r) andη(r) satisfy the same conditions
+as before.) The resulting proper distance ℓ2(r) is again
+finite.
+Anexamplethatillustratestheproblemofgeneralizing
+an exact solution even better comes from the modified
+Chaplygin gas model, whose EoS is p=Aρ−B/ρα,
+0< α≤1, and where AandBare constants. This time
+we begin with Λ( r) and determine Φ( r).
+According to Ref. [2], one may choose b(r) =r0+
+d(r−r0), where dmust be less than unity to meet the
+flare-out condition. The shape function b(r) determines
+Λ(r), as well as Φ( r) [2]:
+Φ(r) =φ0+1+Ad
+2(1−d)ln|r−r0|−1
+2lnr
+−32π2B
+d(1−d)r4
+0ln|r−r0|
+−32π2B
+d(1−d)/bracketleftbigg1
+4(r−r0)4+4r3
+0(r−r0)
++3r2
+0(r−r0)2+4
+3r0(r−r0)3/bracketrightbigg
+.(13)
+This Φ(r) is not finite unless Ahas the (corrected) value
+A=1
+d2(64π2Br4
+0−d). (14)
+Remark: SinceAis part of the EoS, it would be more
+realistic to state the condition as follows: determine d
+(0< d <1), if it exists, so that Eq. (14) is satisfied.
+For such a choice of d, we find that
+Φ(r) =φ0−1
+2lnr−32π2B
+d(1−d)/bracketleftbigg1
+4(r−r0)4
++4r3
+0(r−r0)+3r2
+0(r−r0)2+4
+3r0(r−r0)3/bracketrightbigg
+,(15)
+which has a well defined and continuous derivative. If we
+now substitute Eqs. (2) and (3) in the EoS and solve for3
+Φ′(r), we get
+Φ′(r) =1
+2r/parenleftBig
+e2Λ(r)−1/parenrightBig
++A/bracketleftbigg
+Λ′(r)+1
+2r/parenleftBig
+e2Λ(r)−1/parenrightBig/bracketrightbigg
+−(8π)1+αB(r
+2)e2Λ(r)
+/bracketleftbig2
+re−2Λ(r)Λ′(r)+1
+r2/parenleftbig
+1−e−2Λ(r)/parenrightbig/bracketrightbigα.(16)
+Since the left side is well behaved (due to the choice of
+d), the right side is also well behaved in the sense of
+being finite and sectionally continuous. As before, we
+can replace Λ( r) by Λ(r) +ǫ(r), where, once again, ǫ=
+ǫ(r) has a continuous derivative with ǫ(r0) =ǫ′(r0) = 0
+andǫ(r)→0 asr→ ∞. Now denote the left side by
+Φ′
+1(r). Since Φ′
+1(r) is also sectionally continuous, Φ 1(r)
+will exist, again showing the existence of an entire family
+of solutions.III. CONCLUSION
+Wehaveshowninthisnotethatexactsolutionsofcertain
+wormholegeometries,suchastraversablewormholessup-
+ported by phantom energy, Chaplygin and generalized
+Chaplygin gas, or modified Chaplygin gas, imply the ex-
+istence of entire families of additional solutions. Having
+an abundance of solutions is highly desirable since ex-
+act solutions often have the appearance of being overly
+specialized or even artificial.
+From a physical standpoint, the proliferation of solu-
+tions, given the various equations of state, would greatly
+increasetheprobabilityofwormholes’havingformednat-
+urally. In particular, it is argued in Ref. [5] that the EoS
+had very likely crossed the “phantom divide” some time
+in the past, so that wormholes could have formed natu-
+rally. Inapproachingthepresentdark-energyphase, such
+wormholes would have formed event horizons, thereby
+becoming black holes or (possibly) quasi-black holes.
+This outcome provides at least a possibleexplanation for
+the abundance of black holes and the complete lack of
+wormholes.
+[1] J. Bicak, Einstein equations: exact solutions, arXiv:
+gr-qc/0604102.
+[2] S. Chakraborty and T. Bandyopadhyay, Modified Chap-
+lygin traversable wormholes, International Journal of
+Modern Physics D, 18 (2007), 463-476.
+[3] P.K.F. Kuhfittig, Seeking exactly solvable models of
+traversable wormholes supported by phantom energy,
+Classical and Quantum Gravity, 23 (2006), 5853-5860.
+[4] P.K.F. Kuhfittig, Theoretical construction of wormhole s
+supported by Chaplygin gas, arXiv: 0802.3656.
+[5] P.K.F. Kuhfittig, Could some black holes have evolved
+from wormholes? Scholarly Research Exchange, 2008
+(2008), Article ID 296158, 5 pages.
+[6] F.S.N. Lobo, Phantom energy traversable wormholes,Physical ReviewD,71(2005), ArticleID084011, 8pages.
+[7] F.S.N. Lobo, Chaplygin traversable wormholes, Physica l
+Review D, 73 (2006), Article ID 064028, 9 pages.
+[8] F.S.N. Lobo, Van der Waals quintessence stars, Physical
+Review D, 75 (2007), Article ID 024023, 7 pages.
+[9] M.S. Morris and K.S. Thorne, Wormholes in spacetime
+and their use for interstellar travel: A tool for teach-
+ing general relativity, American Journal of Physics, 56
+(1988), 395-412.
+[10] O.B. Zaslavskii, Exactly solvable models of wormholes
+supported by phantom energy, Physical Review D, 72
+(2005), Article ID 061303, 3 pages.
\ No newline at end of file
diff --git a/1001.0382.txt b/1001.0382.txt
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+++ b/1001.0382.txt
@@ -0,0 +1,326 @@
+arXiv:1001.0382v3  [astro-ph.HE]  11 Sep 2011The extension of radiative viscosity to superfluid matter∗
+PI Chun-Mei1, YANG Shu-Hua2∗∗, and ZHENG Xiao-Ping2
+1Department of Physics and Electronics, Hubei University of Education, Wuhan 430205, P.R.China
+2Institute of Astrophysics, Huazhong Normal University, Wu han 430079, P.R.China,
+(Dated: Received: , Accepted:)
+The radiative viscosity of superfluid npematter is studied, and it is found that to the lowest or-
+der ofδµ/Tthe ratio of radiative viscosity to bulk viscosity is the sam e as that of the normal matter.
+PACS: 97.60.Jd, 21.65.-f, 95.30.Cq
+As one of the most important transport coefficients,
+bulk viscosities of simple npematter, of hyperon matter
+and even of quark matter, both in normal and superfluid
+states, have been extensively studied [1–18], for more ref-
+erences see [19].
+In fact, the mechanical energy of density perturbations
+isnotonlydissipatedtoheatviabulkviscosity,butalsois
+radiated awayvia neutrinos, this was first pointed out by
+Finiz and Wolf in 1968[1]. However, the damping mech-
+anism through neutrinos is ignored for several decades,
+until recently, Sa’d and Schaffner-Bielich [20] named this
+mechanism the radiative viscosity, and found it is 1.5
+times larger than the bulk viscosity to all Urca processes
+in the lowest order of δµ/T, both in nuclear matter and
+quark matter. Yang et al. [21] studied non-linear ef-
+fect of radiative viscosity of npematter in neutron stars
+for both direct Urca process and modified Urca process,
+and found that non-linear effect will decrease the ratio of
+radiative viscosity to bulk viscosity from 1 .5 to 0.5 (for
+direct Urca process) and 0.375 (for modified Urca pro-
+cess); which means that for small oscillations of neutron
+star the large fraction of oscillation energy is emitted as
+neutrinos, but for large enough ones bulk viscous dissi-
+pation dominates.
+It’s well known that below certain critical tempera-
+ture nucleons in neutron star matter will be in superfluid
+states. The bulk viscosity of superfluid nucleon matter
+have been studied [4, 5, 17], and it turns out that the
+superfluidity may strongly reduce bulk viscosity. This
+paper aims to study the radiative viscosity of superfluid
+nucleon matter, in other words, we will give the relation-
+ship between radiative viscosity and bulk viscosity in the
+superfluid case.
+Both the bulk viscosity and the radiative viscosity of
+npematterarerelatedtodirectUrcaprocess( n→p+e+
+νe,p+e→n+νe) and modified Urca process ( n+N→
+p+N+e+νe,p+N+e→n+N+νe) in different
+conditions. As shown by [22], the direct Urca process
+is allowed by the momentum conservation when pFn<
+pFp+pFe, and for pure npematter where pFp=pFe, it
+∗This research is supported by NFSC under Grants No. 11073008 .
+∗∗Email: ysh@phy.ccnu.edu.cn
+Telephone number: 13545359902, 13545119781corresponds to np/n >1/9. This happens if the density
+is several times larger than the standard nuclear matter
+densityρ0= 2.8×1014gcm−3. In the following, we focus
+on the direct Urca process.
+Let us first recall the formulae of radiative viscosity of
+simple non-superfluid npematter [7, 20]. Considering a
+periodic perturbation to the baryon number density
+nb(t) =nb0+Re(δnbeiωt). (1)
+It will lead to a deviation from β-equilibrium character-
+ized by
+δµ=µp+µe−µn=δµp+δµe−δµn,(2)
+whereµn,µpandµeare the chemical potentials of the
+neutrons, protons and electrons. δµcan be expressed in
+terms of the variations of two independent variables δnb
+andδXp,
+δµ=Cδnb
+nb+BδXp, (3)
+whereXp=np/nbis the proton fraction and the coeffi-
+cient functions CandBare given by
+C=np∂µp
+∂np+ne∂µe
+∂ne−nn∂µn
+∂nn, (4)
+B=nb/parenleftbigg∂µp
+∂np+∂µe
+∂ne+∂µn
+∂nn/parenrightbigg
+. (5)
+To the leading order, the net reaction rate and the incre-
+ments of neutrino emissivity due to off-equilibrium could
+be written as
+Γν−Γν=−λ0ξ2
+δµ, (6)
+˙Eloss=S0ξ2, (7)
+where[21, 23]
+λ0=714
+457ǫ(T,0), (8)
+S0=1071
+457ǫ(T,0), (9)2
+ξ=δµ/T, andǫ(T,0) is the neutrino emissivity in equi-
+librium
+ǫ(T,0) = 3.3×10−14/parenleftbiggxpρ
+ρ0/parenrightbigg1/3
+T6MeV5.(10)
+For a periodic process, the expansion and contraction
+of the system will induce not only the dissipation of os-
+cillation energy to heat, but also the loss of oscillation
+energy through an increasing of the neutrino emissivity.
+Bulkviscouscoefficient ζandradiativeviscouscoefficient
+Rcan be defined for the description of these dissipation
+mechanisms, respectively [20]
+/angb∇acketleft˙Ediss/angb∇acket∇ight=−ζ
+τ/integraldisplayτ
+0dt(∇·/vector v)2, (11)
+/angb∇acketleft˙Eloss/angb∇acket∇ight=R
+τ/integraldisplayτ
+0dt(∇·/vector v)2, (12)
+where/vector vis the hydrodynamic velocity associated with
+the density oscillations, and τ= 2π/ωis the oscillation
+period. Using the continuity equation, one obtains
+ζ=−2/angb∇acketleft˙Ediss/angb∇acket∇ight/parenleftBigυ0
+∆υ/parenrightBig2/parenleftBigτ
+2π/parenrightBig2
+, (13)
+R= 2/angb∇acketleft˙Eloss/angb∇acket∇ight/parenleftBigυ0
+∆υ/parenrightBig2/parenleftBigτ
+2π/parenrightBig2
+, (14)
+where the energy dissipation is
+/angb∇acketleft˙Ediss/angb∇acket∇ight=−/integraldisplayτ
+0(Γν−Γν)δµdt, (15)
+and the neutrino emissivity caused by the oscillation is
+/angb∇acketleft˙Eloss/angb∇acket∇ight=/integraldisplayτ
+0˙Elossdt. (16)
+Here, we only present the results, for detailed calcula-
+tions see [20]
+ζ=λC2
+ω2+(λB/nb)2, (17)
+R=/parenleftbiggS0
+λ0/parenrightbigg
+ζ. (18)
+From Eqs. (8)and(9), one gets
+R=3
+2ζ. (19)
+This relation only holds for small oscillations. If the per-
+turbation amplitude is largeenough, the non-lineareffect
+must be taken into account and this simple relation is no
+longer correct [21].Now let us consider the effect of nucleon superfluidity
+on radiative viscosity. Assuming
+Γν−Γν=−λξ2
+δµ, (20)
+˙Eloss=Sξ2. (21)
+Apparently,
+R=/parenleftbiggS
+λ/parenrightbigg
+ζ. (22)
+In the following, we will show how to calculate S/λin
+the lowest order of ξ.
+In non-beta equilibrium, the net reaction rate of direct
+Urca process with nucleon superfluidity is [24]
+Γν−Γν=4π
+(2π)8T5
+3/productdisplay
+j=1/integraldisplay
+dΩj
+δ(Pf−Pi)|Mfi|23/productdisplay
+j=1pFjm∗
+j
++∞/integraldisplay
+0dxνx2
+ν[J(xν−ξ,vj)−J(xν+ξ,vj)],(23)
+and the total neutrino emissivity is[25, 26]
+ǫ(ξ,vj) =4π
+(2π)8T6
+3/productdisplay
+j=1/integraldisplay
+dΩj
+δ(Pf−Pi)|Mfi|23/productdisplay
+j=1pFjm∗
+j
++∞/integraldisplay
+0dxνx3
+ν[J(xν−ξ,vj)+J(xν+ξ,vj)],(24)
+where
+J(x,vj) =+∞/integraldisplay
+−∞dx1dx2dx3f(z1)f(z2)f(x3)
+×δ(z1+z2+x3−x),(25)
+and the subscript j= 1,2,3 corresponds to n,p,ere-
+spectively. vj=∆j
+Tis the gap amplitude and zj=εj−µj
+T
+(wherej= 1,2),x3=εe−µe
+T.xν=εν
+Tis the dimen-
+sionless energy of the neutrino, and f(x) = (1+ ex)−1is
+the Fermi-Dirac functions of nucleons and electrons, pFj
+is the Fermi momentum and m∗
+jis the effective particle
+mass.dΩjis the solid angle element in the direction of
+the particle momentum, and |Mfi|2is the squared reac-
+tion amplitude.
+Here, we want to stress that z1andz2in the above
+two formula carry all the information about nucleon su-
+perfluidity. Whether for neutron superfluidity or proton
+superfluidity, near the Fermi surface we have
+ε−µ= sign(η)/radicalbig
+δ2+η2, (26)3
+whereη=vF(p−pF),vFandpFare the Fermi velocity
+and Fermi momentum, respectively; and δ2= ∆2F(ϑ),
+∆ is the energy gap and F(ϑ) describes the dependence
+of the gap on the angle ϑbetween the quantization axis
+and the particle momentum. For different types of nu-
+cleon superfluidity, the expression of F(ϑ) is completely
+different, which can be seen in [24–26].
+In the case of ξ≪1, to the lowest order we have
+J(xν−ξ,vj)−J(xν+ξ,vj) =−2ξ∂J(xν,vj)/∂xν,(27)
+J(xν−ξ,vj)+J(xν+ξ,vj)−2J(xν,vj) =ξ2∂2J(xν,vj)/∂x2
+ν,
+(28)
+then the net reaction rate is
+∆Γ = Γ ν(ξ,vj)−Γν(ξ,vj)
+=4π
+(2π)8T5
+3/productdisplay
+j=1/integraldisplay
+dΩj
+δ(Pf−Pi)|Mfi|23/productdisplay
+j=1pFjm∗
+j
+×(−4ξ)+∞/integraldisplay
+0dxνxνJ(xν,vj), (29)
+and the neutrino emissivity due to the departure from
+β-equilibrium is
+˙Eloss=ǫ(ξ,vj)−ǫ(0,vj)
+=4π
+(2π)8T6
+3/productdisplay
+j=1/integraldisplay
+dΩj
+δ(Pf−Pi)|Mfi|23/productdisplay
+j=1pFjm∗
+j
+×6ξ2+∞/integraldisplay
+0dxνxνJ(xν,vj). (30)
+Note that, the leading order of ∆Γ is the first order of ξ,
+while for ˙Elossit is the second order of ξ.
+Unlike normal matter, Eqs.(29) and (30) haven’t exact
+analytical solutions. However, one can find that ∆Γ δµis
+in proportion to ˙Eloss, thus we can easily obtain
+S/λ=3
+2, (31)which is the same as S0/λ0. This means that the nucleon
+superfluidity doesn’t change the value of R/ζ.
+In summary, we have studied the radiative viscous co-
+efficient of superfluid npematter, and find that for di-
+rect Urca process, the ratio of radiative viscosity to bulk
+viscosity in the lowest order of ξyielded by Sa’d and
+Schaffner-Bielich [20] for normal nucleons could be ex-
+tended to the superfluid case. In fact, it can be seen
+from our calculations that this is correct for modified
+Urca process, too. Thus, to the lowest order of ξ, the
+relation factor of3
+2between the radiative viscosity and
+the bulk viscosity of npematter is a generic one.
+The following are the extensive discussions to our re-
+sult:
+First, Although we have S/λ=S0/λ0in the low-
+est order, the viscosities in the superfluid case are dif-
+ferent from these in the non-superfluid case. The bulk
+and radiative viscous coefficients could be easily ex-
+pressed as ζ=ζ0RΓandR=R0Rǫ, where ζ0and
+R0are the viscosities in the non-superfluid case, RΓ=
+∆Γ(ξ,vj)/∆Γ(ξ) andRǫ=˙Eloss(ξ,vj)/˙Eloss(ξ) are the
+reduction factors caused by superfluidity. Of course, our
+result shows that to the lowest order of ξ,Rǫequals to
+RΓ(note that RΓhas been calculated numerically by
+Haensel et al. [4, 5]).
+Second, our result satisfies the general relationship
+∂˙Eloss/∂δµ= 3(Γν−Γν), whichhasbeenfound byFlores-
+Tuli´ an and Reisenegger [27]. They found that this rela-
+tionship holds both in the case of normal nucleons and
+in the case of superfluid nucleons, and both in linear case
+and for precise solutions.
+Finally, our result based on the fact that to the low-
+est order of ξ, the existence of superfluidity don’t change
+the relationship between the neutrino emissivity due to
+the departure from β-equilibrium and energy dissipation
+in the form of heat. Nevertheless, it is not correct for
+higher order calculations. As shown in [21, 23], for nor-
+mal matter ∆Γ and ˙Elosscan be solved analytically as
+polynomials of ξ, while for superfluid matter they must
+be solved numerically. As a result, we expect that in
+non-linear regime, the ratio of radiative viscosity to bulk
+viscosity of superfluid matter no longer equals to that
+of normal matter. However, the calculation in the non-
+linear case is far more complicated and we will consider
+it in our further study.
+[1] Finzi A and Wolf R A 1968 Astrophys. J. 153835
+[2] Sawyer R F 1989 Phys. Rev. D 393804
+[3] Haensel P and Schaeffer R 1992 Phys. Rev. D 454708
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+Pramana - J. Phys. 49443
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+[14] Pan N N, Zheng X P, and Li J R 2006 Mon. Not. Roy.4
+Astron. Soc. 371135
+[15] Sa’d B A, Shovkovy I A and Rischke D H 2007 Phys.
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+[16] Sa’d B A, Shovkovy I A and Rischke D H 2007 Phys.
+Rev. D75125004
+[17] Gusakov M E 2007 Phys. Rev. D 76083001
+[18] Gusakov M E and Kantor E M 2008 Phys. Rev. D 78
+083006
+[19] Dong H, Su N and Wang Q 2007 J. Phys. G 34S643
+[20] Sa’d B A and Schaffner-Bielich J arXiv:0908.4190
+[21] Yang S H, Zheng X P and Pi C M 2010 Phys. Lett. B683255
+[22] Lattimer J M, Pethick C J, Prakash M and Haensel P
+1991 Phys. Rev. Lett. 662701
+[23] Reisenegger A 1995 Astrophys. J. 442749
+[24] Villain L and Haensel P 2005 Astron. Astrophys. 444539
+[25] Yakovlev D G, Kaminker A D, Gnedin O Y and Haensel
+P 2001 Phys. Rept. 3541
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+045802
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+Astron. Soc. 372276
\ No newline at end of file
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+arXiv:1001.0383v2  [cs.CC]  3 Feb 2010Symposium on TheoreticalAspects of Computer Science 2010( Nancy, France), pp. 227-238
+www.stacs-conf.org
+RESTRICTED SPACE ALGORITHMS FOR ISOMORPHISM ON
+BOUNDED TREEWIDTH GRAPHS
+BIRESWAR DAS1AND JACOBO TOR ´AN2AND FABIAN WAGNER3
+1Institute of Mathematical Sciences, Chennai, India
+E-mail address :bireswar@imsc.res.in
+2Institut f¨ ur Theoretische Informatik, Universit¨ at Ulm, 89069 Ulm, Germany
+E-mail address :jacobo.toran@uni-ulm.de
+3Institut f¨ ur Theoretische Informatik, Universit¨ at Ulm, 89069 Ulm, Germany
+E-mail address :fabian.wagner@uni-ulm.de
+Abstract. The Graph Isomorphism problem restricted to graphs of bound ed treewidth
+or bounded tree distance width are known to be solvable in pol ynomial time [2],[19]. We
+give restricted space algorithms for these problems provin g the following results:
+•Isomorphism for bounded tree distance width graphs is in L an d thus complete for
+the class. We also show that for this kind of graphs a canon can be computed within
+logspace.
+•For bounded treewidth graphs, when both input graphs are giv en together with a
+tree decomposition, the problem of whether there is an isomo rphism which respects
+the decompositions (i.e. considering only isomorphisms ma pping bags in one decom-
+position blockwise onto bags in the other decomposition) is in L.
+•For bounded treewidth graphs, when one of the input graphs is given with a tree
+decomposition the isomorphism problem is in LogCFL.
+•As a corollary the isomorphism problem for bounded treewidt h graphs is in LogCFL.
+This improves the known TC1upper bound for the problem given by Grohe and
+Verbitsky [8].
+1. Introduction
+The Graph Isomorphism problem consists in deciding whether two given graphs are
+isomorphic, or in other words, whether there exists a biject ion between the vertices of both
+graphs preserving the edge relation. Graph Isomorphism is a well studied problem in NP
+becauseofitsmanyapplications andalsobecauseitisoneof thefewnaturalproblemsinthis
+class not known to be solvable in polynomial time nor known to be NP-complete. Although
+for the case of general graphs no efficient algorithm for the pr oblem is known, the situation
+is much better when certain parameters in the input graphs ar e bounded by a constant. For
+1998 ACM Subject Classification: Complexity Theory, Graph Algorithms.
+Key words and phrases: Complexity, Algorithms, Graph Isomorphism Problem, Treew idth, LogCFL.
+Supported by DFG grants TO 200/2-2.
+c/circlecopyrtB. Das,J.Tor ´an,andF.Wagner
+CC/circlecopyrtCreativeCommonsAttribution-NoDerivsLicense228 B. DAS, J. TOR ´AN, AND F. WAGNER
+example the isomorphism problem for graphs of bounded degre e [13], bounded genus [15],
+bounded color classes [14], or bounded treewidth [2] is know n to be in P. Recently some of
+these upper bounds have been improved with the development o f space efficient techniques,
+most notably Reingold’s deterministic logspace algorithm for connectivity in undirected
+graphs [16]. In some cases logspace algorithms have been obt ained. For example graph
+isomorphism for trees [12], planar graphs [5] or k-trees [10]. In other cases the problem has
+been classified in some other small complexity classes below P. The isomorphism problem
+for graphs of bounded treewidth is known to be in TC1[8] and the problem restricted to
+graphs of bounded color classes is known to be in the #L hierar chy [1].
+In this paper we address the question of whether the isomorph ism problem restricted
+to graphs of bounded treewidth and bounded tree distance wid th can be solved in logspace.
+Intuitively speaking, the treewidth of a graph measures how much it differs from a tree.
+Thisconcepthasbeenusedverysuccessfullyinalgorithmic sandfixed-parametertractability
+(see e.g. [3, 4]). For many complex problems, efficient algori thms have been found for the
+cases when the input structures have bounded treewidth. As m entioned above Bodlaender
+showed in [2] that Graph Isomorphism can be solved in polynom ial time when restricted
+to graphs of bounded treewidth. More recently Grohe and Verb itsky [8] improved this
+upper bound to TC1. In this paper we improve this result showing that the isomor phism
+problem for bounded treewidth graphs lies in LogCFL, the cla ss of problems logarithmic
+space reducible to a context free language. LogCFL can be alt ernatively characterized as
+the class of problems computable by a uniform family of polyn omial size and logarithmic
+depth circuits with bounded AND and unbounded OR gates, and i s therefore a subclass of
+TC1. LogCFL is also the best known upper bound for computing a tre e decomposition of
+bounded treewidth graphs [18, 7], which is one bottleneck in our isomorphism algorithm.
+We prove that if tree decompositions of both graphs are given as part of the input, the
+question of whether there is an isomorphism respecting the v ertex partition defined by the
+decompositions can be solved in logarithmic space. Our proo f techniques are based on
+methods from recent isomorphism results [5, 6] and are very d ifferent from those in [8].
+The notion of tree distance width, a stronger version of the t reewidth concept, was
+introduced in [19]. There it is shown that for graphs with bou nded tree distance width the
+isomorphism problem is fixed parameter tractable, somethin g that is not known to hold for
+the more general class of bounded treewidth graphs. We prove that for graphs of bounded
+tree distance width it is possible to obtain a tree distance d ecomposition within logspace.
+Using this result we show that graph isomorphism for bounded tree distance width graphs
+can also be solved in logarithmic space. Since it is known tha t the question is also hard
+for the class L under AC0reductions [9], this exactly characterizes the complexity of the
+problem. We show that in fact a canon for graphs of bounded tre e distance width, i.e.
+a fixed representative of the isomorphism equivalence class , can be computed in logspace.
+Due to space reasons, some proofs are omitted and will be prov ided in the full version of
+the paper.
+2. Preliminaries
+We introduce the complexity classes used in this paper. L is t he class of decision prob-
+lems computable by deterministic logarithmic space Turing machines. LogCFL consists of
+all decision problems that can be Turing reduced in logarith mic space to a context free lan-
+guage. There are several alternative more intuitive charac terizations of LogCFL. ProblemsRESTRICTED SPACE ALGORITHMS FOR ISOMORPHISM ON BOUNDED TRE EWIDTH GRAPHS 229
+in this class can be computed by uniform families of polynomi al size and logarithmic depth
+circuits over bounded fan-in AND gates and unbounded fan-in OR gates. We will also
+use the characterization of LogCFL as the class of decisiona l problems computable by non-
+deterministic auxiliary pushdown machines (NAuxPDA). The se are Turing machines with
+a logarithmic space work tape, an additional pushdown and a p olynomial time bound [17].
+The class TC1contains the problems computable by uniform families of pol ynomial size
+and logarithmic depth threshold circuits. The known relati onships among these classes are:
+L⊆LogCFL ⊆TC1.
+In this paper we consider undirected simple graphs with no se lf loops. For a graph
+G= (V,E) and two vertices u,v∈V,dG(u,v) denotes the distance between uandvin
+G(number of edges in the shortest path between uandvinG). For a set S⊆V, and
+a vertex u∈V,dG(S,u) denotes min v∈SdG(v,u). Γ(S) denotes the set of neighbors of
+SinG. In a connected graph G, a separating set is a set of vertices such that deleting
+the vertices in S(and the edges connected to them) produces more than one conn ected
+component. For G= (V,E) and two disjoint subsets U,WofVwe use the following
+notion for an induced bipartite subgraph BG[U,W] ofGon vertex set U∪Wwith edge set
+{{u,w} ∈E|u∈U,w∈W}. LetG[U] be theinduced subgraph ofGon vertex set V\U.
+Atree decomposition of a graph G= (V,E) is a pair ( {Xi|i∈I},T= (I,F)), where
+{Xi|i∈I}is a collection of subsets of Vcalled bags, and Tis a tree with node set Iand
+edge set F, satisfying the following properties:
+i)/uniontext
+i∈IXi=V
+ii) for each {u,v} ∈E, there is an i∈Iwithu,v∈Xiand
+iii) for each v∈V, the set of nodes {i|v∈Xi}forms a subtree of T.
+Thewidthof a tree decomposition of G, is defined as max {|Xi| |i∈I} −1. The
+treewidth ofGis the minimum width over all tree decompositions of G.
+Atree distance decomposition of a graph G= (V,E) is a triple ( {Xi|i∈I},T=
+(I,F),r), where {Xi|i∈I}is a collection of subsets of Vcalledbags,Xr=Sa set of
+vertices and Tis a tree with node set I, edge set Fand root r, satisfying:
+i)/uniontext
+i∈IXi=Vand for all i/\e}atio\slash=j,Xi∩Xj=∅
+ii) for each v∈V, ifv∈XithendG(Xr,v) =dT(r,i) and
+iii) for each {u,v} ∈E(G), there are i,j∈Iwithu∈Xi,v∈Xjandi=jor{i,j} ∈F
+(for every edge in Gits two endpoints belong to the same or to adjacent bags in T).
+LetD= ({Xi|i∈I},T= (I,F),r) be a tree distance decomposition of G.Xris the
+root bag ofD. ThewidthofDis the maximum number of elements of a bag Xi. Thetree
+distance width ofGis the minimum width over all tree distance decompositions o fG.
+The tree distance decomposition Dis calledminimal if for each i∈I, the set of vertices
+in the bags with labels in the subtree rooted at iinTinduce a connected subgraph in G.
+In [19] it is shown that for every root set S⊆Vthere is a unique minimal tree distance
+decomposition of Gwith root set S. The width of such a decomposition is minimal among
+the tree distance decompositions of Gwith root set S.
+An isomorphism from GontoHrespects their tree (distance) decompositions D,D′if
+vertices in a bag of DinGare mapped blockwise onto vertices in a bag of D′inH. Not
+every isomorphism has this property.
+Sym(V) is thesymmetric group on a set V.230 B. DAS, J. TOR ´AN, AND F. WAGNER
+3. Graphs of bounded tree distance width
+3.1. Tree distance decomposition in L
+We describe an algorithm that on input a graph Gand a subset S⊂Vproduces the
+minimal tree distance decomposition D= ({Xi|i∈I},T= (I,F),r) ofGwith root set
+Xr=S. The algorithm works within space c·klognfor some constant c, wherekis the
+width of the minimal tree distance decomposition of Gwith root set S. The output of
+the algorithm is a sequence of strings of the form ( bag label, bag depth, vi1,vi2,...,vil),
+indicating the number of the bag, the distance of its element s toSand the list of the
+elements in the bag.
+The algorithm basically performs a depth first traversal of t he treeTin the decompo-
+sition while constructing it. Starting at Sthe algorithm uses three functions for traversing
+T. These functions perform queries to a logspace subroutine c omputing reachability [16].
+Parent(Xi): On input the elements of a bag Xithe function returns the elements
+of the parent bag in T. These are the vertices v∈Vwith the following two properties:
+v∈Γ(Xi)\Xiandvis reachable from SinG\Xi. For a vertex vthese two properties can
+be tested in space O(logn) by an algorithm with input G,SandXi. In order to find all
+the vertices in the parent set, the algorithm searches throu gh all the vertices in V.
+First Child (Xi): This function returns the elements of the first child of iinT. This is
+the child with the vertex vj∈Vwith the smallest index j.vjsatisfies that vj∈Γ(Xi)\Xi
+and that vjis not reachable from SinG\Xi. It can be found cycling in order through
+the vertices of Guntil the first one satisfying the properties is found. The ot her elements
+w∈Ximust satisfy the same two properties as vjand additionally, they must be in the
+same connected component in G\Xiwherevjis contained. In case Xidoes not have any
+children, the function outputs some special symbol.
+Next Sibling (Xi): This function first computes Xp:=Parent( Xi) and then searches
+for the child of pinTnext toXi. Letvibe the vertex with the smallest label in Xi. This
+is done similarly as the computation of First Child. The next sibling is the bag containing
+the unique vertex vjwith the following properties: vjis the vertex with the smallest label
+in this bag, label(vj)> label(vi) and there is no other bag which has a vertex with a label
+> viand< vj. The vertex vjis not reachable from SinG\Xp. The other elements in
+the bag are the vertices satisfying these properties and whi ch are in the same connected
+component of G\Xpwherevjis contained.
+With these three functions the algorithm performs a depth-fi rst traversal of T. It only
+needs to remember theinitial bag X0=Swhich is part of the input, and the elements of the
+current bag. On a bag Xiit searches for its first child. If it does not exist then it sea rches
+for the next sibling. When there are no further siblings the n ext move goes up in the tree
+T. The algorithm finishes when it returns to S. It also keeps two counters in order to be
+able to output the number and depth of the bags. The three ment ioned functions only need
+to keep at most two bags ( Xiand its father) in memory, and work in logarithmic space.
+On input a graph Gwithnvertices, and a root set S, the space used by the algorithm is
+therfore bounded by c·klogn, for a constant c, andkbeing the minimum width of a tree
+distance decomposition of Gwith root set S. When considering how the three functions areRESTRICTED SPACE ALGORITHMS FOR ISOMORPHISM ON BOUNDED TRE EWIDTH GRAPHS 231
+defined it is clear that the algorithm constructs a tree dista nce decomposition with root set
+S. Also they make sure that for each ithe subgraph induced by the vertices of the bags in
+the subtree rooted at iis connected thus producing a minimal decomposition. As obs erved
+in [19], this is the unique minimal tree distance decomposit ion ofGwith root set S.
+3.2. Isomorphism Algorithm for Bounded Tree Distance Width Graphs
+For our isomorphism algorithm we use a tree called the augmented tree which is based
+on the underlying tree of a minimal tree distance decomposit ion. This augmented tree,
+apart from the bags, contains information about the separat ing sets which separate bags.
+Definition 3.1. LetGbe a bounded tree distance width graph with a minimal tree dis -
+tance decomposition D= ({Xi|i∈I},T= (I,F),r). Theaugmented tree T(G,D)=
+(I(G,D),F(G,D),r) corresponding to GandDis a tree defined as follows:
+•The set of nodes of T(G,D)isI(G,D)which contains two kinds of nodes, namely
+I(G,D)=I∪J. Those in Iform the set of bag nodes inD, and those in Jthe
+separating set nodes . For each bag node a∈Iand each child bofainTwe consider
+the setXa∩Γ(Xb), i.e. the minimum separating set inXawhich separates Xbfrom
+the root bag XrinG. LetMsa
+1,...,M sa
+l(a)be the set of all minimum separating sets
+inXa, free of duplicates. There are nodes for these sets sa
+1,...,sa
+l(a), the separating
+set nodes. We define J=/uniontext
+a∈I{sa
+1,...,sa
+l(a)}. The node r∈Iis the root in T(G,D).
+•InF(G,D)there are edges between bag nodes a∈Iand the separating set nodes
+sa
+1,...,sa
+l(a)∈J(edges between bagnodesandtheirchildren intheaugmented tree).
+There are also edges between nodes b∈Iandsa
+jifMsa
+jis the minimum separating
+set inXawhich separates XbfromXr(edges between bag nodes and their parents).
+To simplify notation, we later say for example that s1,...,slare the children of a bag
+nodeaif the context is clear. The odd levels of the augmented tree T′correspond to bag
+nodes and the even levels correspond to separating set nodes .
+Observe that for each node in the augmented tree, we associat e a bag to a bag node and
+a minimum separating set to a separating set node. Hence, eve ry vertex vin the original
+graph occurs in at least one associated component and it migh t occur in more than one,
+e.g. ifvis contained in a bag and in a minimum separating set.
+LetT(G,D)be an augmented tree of some minimal tree distance decomposi tionDof a
+graphG. Letabe a node of T(G,D). The subtree of T(G,D)rooted at ais denoted by Ta.
+Note that T(G,D)=TrwhereXris the bag corresponding to the root of the tree distance
+decomposition D. We define graph(Ta) as the subgraph of Ginduced by all the vertices
+associated to at least one of the nodes of Ta. ThesizeofTa, denoted |Ta|is the number of
+vertices which occur in at least one component which is assoc iated to a node in Ta. Note,
+|Ta|is polynomially related to |graph(Ta)|, i.e. the number of vertices in the corresponding
+subgraph of G.
+When given a tree distance decomposition the augmented tree can be computed in
+logspace. Using the result in Section 3.1 we immediately get :
+Lemma 3.2. LetGbe a graph of bounded tree distance width. The augmented tree f orG
+can be computed in logspace.232 B. DAS, J. TOR ´AN, AND F. WAGNER
+... ......
+... ......Sr Tr′
+a1,1 a2,1 a′
+1,1a′
+1,k1a′
+2,1
+Sa1,1 Sa2,1Sa2,k2Sal,klTa′
+1,1Ta′
+1,k1Ta′
+2,1Ta′
+2,k2Ta′
+l,kla1,k1 a2,k2al,kl
+Sa1,k1a′
+2,k2a′
+l,klr r′
+s1 s2 sl t1 t2 tl
+Figure 1: The augmented trees SrandTr′rooted at bag nodes randr′. Node rhas
+separating set nodes s1,...,slas children. The children of s1are again bag nodes
+a1,1,...,a 1,k1.Sai,jis the subtree rooted at ai,j. Bag nodes and separating set
+nodes alternate in the tree.
+Isomorphism Order of Augmented Trees. We describe an isomorphism order proce-
+dure for comparing two augmented trees S(G,D)andT(H,D′)corresponding to the graphs G
+andHand their tree distance decompositions DandD′, respectively. This isomorphism
+order algorithm is an extension of the one for trees given by L indell [12] and it is different
+from that for planar graphs given by Datta et.al. [5]. The tre esS(G,D)andT(H,D′)are
+rooted at bag nodes randr′. The rooted trees are denoted then SrandTr′as shown in
+Figure 1.
+We will show that two graphs of bounded tree distance width ar e isomorphic if and
+only if for some root nodes randr′the augmented trees corresponding to the minimal tree
+distance decompositions have the same isomorphism order.
+The isomorphism order depends on the order of the vertices in the bags randr′.
+LetXrandX′
+r′be the corresponding bags in DandD′. We define the sets of mappings
+Θ(r,r′)=Sym(Xr)×Sym(X′
+r′). Let (σ,σ′) be such a mapping, then the tuples ( G[Xr],σ)
+and (G[X′
+r′],σ′) describe a fixed ordering on the vertices of the induced subg raphs. If ris
+not the top-level root of the augmented tree then Θ (r,r′)may become restricted to a subset,
+when going into recursion. The isomorphism order is defined t o beSr<TTr′if there exist
+mappings ( σ,σ′)∈Θ(r,r′)such that one of the following holds:
+1) (G[Xr],σ)<(H[X′
+r′],σ′) via lexicographical comparison of both ordered subgraphs
+2) (G[Xr],σ) = (H[X′
+r′],σ′) but|Sr|<|Tr′|
+3) (G[Xr],σ) = (H[X′
+r′],σ′) and|Sr|=|Tr′|but #r <#r′where # rand #r′is the
+number of children of randr′
+4) (G[Xr],σ) = (H[X′
+r′],σ′) and|Sr|=|Tr′|and #r= #r′=lbut (Ss1,...,S sl)<T
+(Tt1,...,T tl) where we assume that Ss1≤T··· ≤TSslandTt1≤T··· ≤TTtlare
+ordered subtrees of SrandTr′, respectively. To compute the order between the
+subtrees Ssi≤TTtjwe consider
+i:the lexicographical order of the minimal separating sets ( siandtj) inXrand
+X′
+r′according to σandσ′, as the primary criterion (observe that the separating
+sets are subsets of Xr(resp.Xr′) and are therefore ordered by σandσ′) and
+ii:pairwise the children ai,i′ofsianda′
+j,j′oftj(for all i′andj′via
+cross-comparisons) such that the induced bipartite graphs BG[si,ai,i′] andRESTRICTED SPACE ALGORITHMS FOR ISOMORPHISM ON BOUNDED TRE EWIDTH GRAPHS 233
+BH[tj,a′
+j,j′] can be matched according to σandσ′(i.e.σσ′−1is an isomor-
+phism) and
+iii:recursively the subtrees rooted at the children of siandtj. Note, that these
+children are again bag nodes. For the cross camparison of bag nodesai,i′and
+a′
+j,j′we restrict the set Θ (ai,i′,a′
+j,j′)to a subset of Sym(Xai,i′)×Sym(X′
+a′
+j,j′).
+Namely, Θ (ai,i′,a′
+j,j′)contains the pair ( φ,φ′)∈Sym(Xai,i′)×Sym(X′
+a′
+j,j′) if
+φφ′−1extends thepartial isomorphism σσ′−1fromchild ai,i′ontoa′
+j,j′blockwise
+and which induces an isomorphism from BG[si,ai,i′] ontoBH[tj,a′
+j,j′].
+We say that two augmented trees SrandTr′areequal according to the isomorphism
+order, denoted Sr=TTr′, if neither Sr<TTr′norTr′<TSrholds.
+Isomorphism of two subtrees rooted at bag nodes randr′.We have constant size
+components associated to the bag nodes. A logspace machine c an easily run through all the
+mappings of XrandX′
+r′and record the mappings which gives the minimum isomorphism
+order. This can be done with cross-comparison of trees ( Sr,σ) and (Tr′,σ′) with all possible
+mappings σ,σ′. Later we will see, that in recursion not all possible mappin gs forσandσ′
+are considered. Observe that |Sym(Xr)| ∈O(1).
+The comparison of ( Sr,σ) and (Tr′,σ′) itself can be done simply by renaming the
+vertices of XrandX′
+raccording to the mappings σandσ′and then comparing the ordered
+sequence of edges lexicographically. When equality is foun d then we recursively compute
+the isomorphism order of the subtrees rooted at the children ofrandr′.
+Isomorphism of two subtrees rooted at separating set nodes siandtj.Datta
+et.al. [5] decompose biconnected planar graphs into tricon nected components and obtain a
+tree on these components and separating pairs, i.e. separat ing sets of size two. We have
+separating sets of arbitrary constant size.
+Sincesiandtjcorrespond to subgraphs of XrandX′
+r′, we have an order for them
+given by the fixed mappings σandσ′. Therefore, we can order the children s1,...,sland
+t1,...,tlaccording to their occurrence in XrandX′
+r′(e.g. assume si= (1,2,3,7) according
+to the mapping σand also sj= (1,2,4,7), then we get ( si,σ)<T(sj,σ)). Hence, when
+comparing siwithtjwe have to check whether both come on thesame position in that order
+ofs1,...,slandt1,...,tl. If so, then we go to the next level in the tree, to the children of
+siandtj.
+Now we have a cross comparison among the children of siand the children of tj. In
+Steps 4i, 4iiand 4iiiwe partition the children ai,1,...,a i,liofsianda′
+j,1,...,a′
+j,ljoftj,
+respectively, into isomorphism classes, step by step.
+The membership of a child to a class according to Step 4 iand 4iican be recomputed.
+It suffices to keep counters on the work-tape to notice the curr ent class and traversing the
+siblings from left to right. After these two steps, ai,i′anda′
+j,j′are in the same class if and
+only if vertices of siandtjappear lexicographically at the same positions in σandσ′and
+the bipartite graphs B[si,ai,i′] andB[tj,a′
+j,j′] are isomorphic where siis mapped onto tj
+blockwise corresponding to σσ′−1in an isomorphism. In Step 4 iiiwe go into recursion and
+compare members of one class which are rooted at subtrees of t he same size. When going
+into recursion at ai,i′anda′
+j,j′we consider only those mappings from ( φ,φ′)∈Θ(ai,i′,a′
+j,j′)
+which induce an isomorphism φφ′−1fromB[si,ai,i′] ontoB[tj,a′
+j,j′].234 B. DAS, J. TOR ´AN, AND F. WAGNER
+Correctness of the isomorphism order. Both, the bag nodes and the separating set
+nodes correspond to subgraphs which are basically separati ng sets. A bag separates all its
+subtrees from the root and the separating set nodes refine the bag to separating sets of
+minimum size. Hence, a partial isomorphism is constructed a nd extended from each node
+to its child nodes, traversing the augmented tree (the whole graph, accordingly) in depth
+first manner. In the recursion, the isomorphism between the r oots of the current subtrees,
+saySrandTr′, is partially fixed by the partial isomorphism between their parents. With an
+exhaustive search we check every possible remaining isomor phism from XrontoX′
+r′and go
+into recursion again partially fixing the isomorphism for th e subtrees rooted at children of
+randr′. By an inductive argument, the partial isomorphism describ ed for the augmented
+tree can be followed simultaneously in the original graph an d we get:
+Theorem 3.3. The graphs GandHof bounded tree distance width are isomorphic if and
+only if there is a choice of a root bag randr′producing augmented trees SrandTr′such
+thatSr=TTr′. The isomorphism order between two augmented trees of GandHcan be
+computed in logspace.
+The proof is based on a careful space analysis at each computa tional step building on
+concepts of the isomorphism order algorithm of Lindell [12] . The isomorphism order is the
+basis for a canonization procedure. This is shown in a full ve rsion of this paper.
+Theorem 3.4. A graph of bounded tree distance width can be canonized in log space.
+4. Graphs of bounded treewidth
+In this section we consider several isomorphism problems fo r graphs of bounded
+treewidth. We are interested in isomorphisms respecting the decompositions (i.e. vertices
+are mapped blockwise from a bag to another bag). We show first t hat if the tree decomposi-
+tion of both input graphs is part of the input then the isomorp hism problem can be decided
+in L. We also show that if a tree decomposition of only one of th e two given graphs is part
+of the input, then the isomorphism problem is in LogCFL. It fo llows that the isomorphism
+problem for graphs of bounded treewidth is also in LogCFL.
+Assume the decompositions of both input graphs are given. Le t (G,D),(H,D′) be two
+bounded treewidth graphs together with tree decomposition sDandD′, respectively. We
+look for an isomorphism between GandHsatisfying the condition that the images of the
+vertices in one bag in Dbelong to the same bag in D′.
+We prove that this problem is in L. For this we show that given t ree decompositions
+together with designated bags as roots for GandHthe question of whether there is an
+isomorphism between the graphs mapping root to root and resp ecting the decompositions
+(i.e. mapping bags in Gblockwise onto bags in H) can be reduced to the isomorphism
+problem for graphs of bounded tree distance decomposition. We argued in the previous
+section that this problem belongs to L.
+Theorem 4.1. The isomorphism problem for bounded treewidth graphs with gi ven tree
+decompositions reduces to isomorphism for bounded tree dis tance width graphs under AC0
+many-one reductions.
+Since bounded tree distance width GI is in L, this almost prov es the desired result. To
+obtain it, we have to find roots for the tree decompositions. W e fix an arbitrary bag in the
+one graph and try all bags from the decomposition of the other graph as roots. We get:RESTRICTED SPACE ALGORITHMS FOR ISOMORPHISM ON BOUNDED TRE EWIDTH GRAPHS 235
+Corollary 4.2. For every k≥1there is a logarithmic space algorithm that, on input a pair
+of graphs together with a tree decompositions of width kfor each of them, decides whether
+there is an isomorphism between the graphs, respecting the d ecompositions.
+4.1. ALogCFL algorithm for isomorphism
+We consider now the more difficult situation in which only one o f the input graphs is
+given together with a tree decomposition.
+Theorem 4.3. Isomorphism testing for two graphs of bounded treewidth, whe n a tree de-
+composition for one of them is given, can be done in LogCFL.
+Proof.We describe an algorithm which runs on a non-deterministic a uxiliary pushdown
+automaton (NAuxPDA). Besides a read-only input tape and a fin ite control, this machine
+has access to a stack of polynomial size and a O(logn) space bounded work-tape. On the
+input tape we have two graphs G,Hof treewidth kand a tree decomposition D= ({Xi|
+i∈I},T= (I,F),r) forG. Forj∈Iwe define Gjto be the subgraph of Ginduced on
+the vertex set {v|v∈Xi,i∈Iandi=joria descendant of jinT}. That is, Gj
+contains the vertices which are separated by the bag XjfromXrand those in Xj. We
+defineDj= ({Xi,|,i∈Ij},Tj= (Ij,Fj),j) as the tree decomposition of Gjcorresponding
+toTj, the subtree of Trooted at j. We also consider a way to order the children of a node
+in the tree decomposition:
+Definition 4.4. Let 1,...,lbe the children of rin the tree T. We define the lexicographical
+subtree order , as the order among the subtrees ( G1,D1),...,(Gl,Dl) which is given by:
+(Gi,Di)<(Gj,Dj) iff there is a vertex w∈V(Gi)\Xrwhich has a smaller label than every
+vertex in V(Gj)\Xr.
+The algorithm non-deterministically guesses two main stru ctures. First, we guess a tree
+decomposition of width kforH. This is done in a similar way as in the LogCFL algorithm
+from Wanke [18] for testing that a graph has bounded treewidt h. Second, we guess an
+isomorphism φfromGtoHby extending partial mappings from bag to bag.
+Very simplified, Wanke’s algorithm on input a graph Hstarts guessing a root bag and
+it guesses then non-deterministically further bags in the d ecomposition using the pushdown
+to test that these bags fulfill the properties of a tree decomp osition and that every edge in
+Gis included in some bag. Our algorithm simulates Wanke’s alg orithm as a subroutine. In
+the description of the new algorithm we concentrate on the is omorphism testing part and
+hide the details of how to choose the bags. For simplicity the sentence “guess a bag XjinH
+according to Wanke’s algorithm” means that we simulate the g uessing steps from Wanke,
+checking at the same time that the constructed structure is i n fact a tree decomposition.
+Note, if the bags were not chosen appropriately, then the alg orithm would halt and reject.
+We start guessing a root bag X′
+r′of size≤k+ 1 for a decomposition of H. With
+X′
+r′as root bag we guess the tree decomposition D′ofHwhich corresponds to Dand
+its root r. We also construct a mapping φdescribing a partial isomorphism from the
+vertices of Gonto the vertices of H. At the beginning, φis the empty mapping and
+we guess an extension of φfromXrontoX′
+r′. The algorithm starts with a=r(and
+a′=r′). Then we describe isomorphism classes for 1 ,...,l, the children of a. First, the
+children of acan be distinguished because X1,...,X lmay intersect with Xadifferently.
+Second, we further partition the children within one class a ccording to the number of236 B. DAS, J. TOR ´AN, AND F. WAGNER
+isomorphic siblings in that class. This can be done in logspa ce with cross comparisons of
+pairs among ( G1,D1),...,(Gl,Dl), see Corollary 4.2. It suffices to order the isomorphism
+classes according to the lexicographical subtree order of t he members in the classes. We
+compare then thechildren of awith guessed children of a′keeping the following information:
+For each isomorphism class we check whether there is the same number of isomorphic
+subtrees of a′inHand whether those intersect with X′
+a′, accordingly. For this we use the
+lexicographical subtree order to go through the isomorphic siblings from left to right, just
+keeping a pointer to the current child on the work tape. For tw o such children, say s1ofa
+andt1ofa′, we check then recursively whether ( G1,D1) is isomorphic to the corresponding
+subgraph of t1inH, by an extension of φ.
+When we go into recursion, we push on the stack O(logn) bits for a description of Xa
+andX′
+a′as well as a description of the partial mapping φfromXaontoX′
+a′.
+In general, we do not keep all the information of φon the stack. We only have the
+partial isomorphism φ:{v|v∈Xr∪···∪Xa} → {v|v∈X′
+r′∪···∪X′
+a′}, wherer,...,a
+(r′,...,a′, respectively) is a simple path in Tfrom the root to the node at the current level
+of recursion. After we ran through all children of some node w e go one level up in recursion
+and recompute all the other information which is given impli citly by the subtrees from
+which we returned. Suppose now, we returned to the bag Xa, we have to do the following:
+•Pop from the stack the partial isomorphism φof the bags XaontoX′
+a′
+•Compute the lexicographical next isomorphic sibling. For t his we consider the par-
+tition into isomorphism classes according to φand the lexicographical subtree order
+of Definition 4.4. Recall, isomorphism testing of two subtre es ofXacan be done in
+logspace.
+•If there is no such sibling then we compute the lexicographic al first child of Xa
+inside the same isomorphism class. From this child of Xawe compute the sibling
+which is not in the same isomorphism class and which comes nex t to the right in
+the lexicographical subtree order.
+•If there is neither a further sibling in the same isomorphism class nor a non-
+isomorphic sibling of higher lexicographical order then we ran through all children
+ofXaand we are ready to further return one level up in recursion.
+Also for X′
+a′we guess all children in an isomorphism class from left to rig ht in lexico-
+graphical subtree order. If there is no further level to go up in recursion then the stack is
+empty and we halt in an accepting state. Algorithm 1 summariz es the above considerations.
+In Line 1, we guess an extension of φto include a mapping from XaontoX′
+a′. We know
+the partial isomorphism of their parent bags since this info rmation can be found on the top
+of the stack. In Line 3, we have e.g. the partition E1={T1,...,T l1},E2={Tl1+1,...,T l2}
+and so on. It can be obtained in logspace by testing isomorphi sm of the tree structures
+(G1,D1),...,(Gl,Dl). Two subtrees rooted at XiandXjare in the same isomorphism
+class iff there is an automorphism in Gwhich maps XiontoXjand fixes their parent Xa
+setwise. In Lines 6 to 9, we guess X′
+i′inHwhich corresponds to Xi, we test recursively
+whether the corresponding subgraphs GiandHi′are isomorphic with an extension of φ.
+In Line 7, we check whether X′
+i′fulfills the properties of a correct tree-decomposition as i n
+Wanke’s algorithm (i.e. X′
+i′must be a separating set which separates its split component s
+from the vertices in X′
+a′\X′
+i′).
+To see that the algorithm correctly computes an isomorphism , we make the following
+observation. A bag Xais a separating set which definestheconnected subgraphs G1,...,G l.RESTRICTED SPACE ALGORITHMS FOR ISOMORPHISM ON BOUNDED TRE EWIDTH GRAPHS 237
+Algorithm 1 Treewidth Isomorphism with one tree decomposition
+Input:GraphsG,H, tree decomposition DforG, bagsXainGandX′
+a′inH.
+Top of Stack: Partial isomorphism φmapping the vertices in the parent bag of Xaonto
+the vertices in the parent bag of X′
+a′.
+Output: Accept, if Gis isomorphic to Hby an extension of φ.
+1:Guess an extension of φto a partial isomorphism from XaontoX′
+a′
+2:ifφcannot be extended to a partial isomorphism which maps XaontoX′
+a′thenreject
+3:Let 1,...,lbe the children of ainT. Partition the subtrees of Trooted at 1 ,...,linto
+pisomorphism classes E1,...,E p
+4:foreach class Ejfromj= 1 top
+5:foreach subtree Ti∈Ej(in lexicographical subtree order)
+6:guess a bag X′
+i′inH(in increasing lexicographical subtree order). Let Hi′be the
+subgraph of Hinduced by the vertices in X′
+i′and by those which are separated
+fromX′
+r′inH\X′
+i′
+7:ifX′
+i′is not a correct child bag of X′
+a′(see Wanke’s algorithm) thenreject.
+8:Invoke this algorithm with input ( Gi,Hi′,Di,Xi,X′
+i′) recursively and push Xa,X′
+a′
+and the partial isomorphism φon the stack
+9:After recursion pop these informations from the stack
+10:ifthe stack is not empty thengo one level up in recursion
+11:accept and halt
+These subgraphs do not contain the root XrandV(Gi)∩V(Gj)⊆Xasince we have a tree
+decomposition D(V(Gi) are the vertices of Gi). We guess and keep from the partial
+isomorphism φexactly those parts which correspond to the path from the roo tsXrand
+X′
+r′to the current bags XaandX′
+a′. Once we verified a partial isomorphism from one child
+component (e.g. Gi) ofXaonto a child component (e.g. Hi′) ofX′
+a′, for the other child
+components it suffices to know the partial mapping of φfromXaontoX′
+a′.
+Observe that for each vinGin a computation path from the algorithm there can only
+be a value for φ(v). Clearly, if GandHare isomorphic then the algorithm can guess
+the decomposition of Hwhich fits to D, and the extensions of φcorrectly. In this case
+the NAuxPDA has some accepting computation. On the other han d, if the input graphs
+are non-isomorphic then in every non-deterministic comput ation either the guessed tree
+decomposition of Hdoes not fulfill the conditions of a tree decomposition (and w ould be
+detected) or the partial isomorphism φcannot be extended at some point.
+Wanke’s algorithm decides in LogCFL whether the treewidth o f a graph is at most k
+by guessing all possible tree decompositions. Using a resul t from [7] it follows that there is
+also a (functional) LogCFL algorithm that on input a bounded treewidth graph computes
+a particular tree decomposition for it. Since LogCFL is clos ed under composition, from this
+result and Theorem 4.3 we get:
+Corollary 4.5. The isomorphism problem for bounded treewidth graphs is in LogCFL.
+Conclusions and open problems. We have shown that the isomorphism problem for
+graphs of bounded treewidth is in the class LogCFL and that is omorphism testing and
+canonization of bounded tree distance width graphs is compl ete for L. By using standard238 B. DAS, J. TOR ´AN, AND F. WAGNER
+techniques in the area it can beshown that the same upperboun dsapply for other problems
+related to isomorphism on these graph classes. For example t he automorphism problem or
+the functional versions of automorphism and isomorphism ca n be done within the same
+complexity classes. The main question remaining is whether the LogCFL upper bound
+for isomorphism of bounded treewidth graphs can be improved . On the one hand, no
+LogCFL-hardness result for the isomorphism problem is know n, so maybe the result can be
+improved. We believe that proving a logspace upper bound for the isomorphism problem of
+boundedtreewidth graphswouldrequireto computetree deco mpositions withinlogarithmic
+space, which is a long standing open question. Another inter esting open question is whether
+bounded treewidth graphs can be canonized in LogCFL.
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+arXiv:1001.0384v1  [math.GT]  3 Jan 2010Graph-Links
+Denis Petrovich Ilyutko∗, Vassily Olegovich Manturov†
+November 13, 2018
+Abstract
+The present paper is a review of the current state of Graph-Link
+Theory(graph-links are also closely related to homotopy classes of
+looped interlacement graphs ): theory suggested in [5, 6], see also [23],
+dealing with a generalisation of knots obtained by translat ing the Rei-
+demeister moves for links into the language of intersection graphs of
+chord diagrams. In this paper we show how some methods of clas sical
+andvirtual knottheory can betranslated intothelanguage o f abstract
+graphs, and some theorems can be reproved and generalised to this
+graphical setting. We construct various invariants, prove certain min-
+imality theorems and construct functorial mappings for gra ph-knots
+and graph-links. In this paper, we first show non-equivalenc e of some
+graph-links to virtual links.
+1 Introduction
+It is well known that classical and virtual knots can be repre sented
+by Gauss diagrams, and the whole information about the knot a nd
+its invariants can be read out of any Gauss diagram encoding i t, see
+Fig. 1. Whenever a Gauss diagram does not describe any embedded
+curveinR2(just because the corresponding Gauss code is not planar)
+as in Fig. 2, one gets a virtual knot; the generic immersion po int is
+encircled.
+∗Partially supported by grants of RF President NSh – 660.2008.1, RFB R 07–01–00648,
+RNP 2.1.1.3704, the Federal Agency for Education NK-421P/108.
+†Partially supported by grants of RFBR 07–01–00648
+112
+3
++
+33
+22
+11 +
++
+Figure 1: The Right Trefoil and Its Gauss Diagram
+1
+2++2
+211
+Figure 2: The Virtual Trefoil and Its Gauss Diagram
+It turns out that some information about the knot can be ob-
+tained from a more combinatorial data: intersection graphs of Gauss
+diagrams. The intersection (adjacency) graph is a simple graph , i.e.
+a graph without loops and multiple edges, whose vertices are in one-
+to-one correspondence with chords of the Gauss diagram of th e knot
+(the latter are, in turn, in one-to-one correspondence with crossings of
+the knot). Two vertices of the intersection graph are adjace nt when-
+ever the corresponding arrows of the Gauss diagram are linked, i.e.
+the corresponding chords “intersect” each other in the pict ure, see
+Fig. 3. Vertices of the intersection graph are endowed with t he local
+writhe number of the crossing. However, sometimes the Gauss dia-
+gram can be obtained from the intersection graph in a non-uni que
+way, see Fig. 4, and some graphs can not be represented by chor d
+diagrams at all [1], see Fig. 5.
+When passing to the intersection graph, we remember the writ he
+number information, but forget about the right-left information en-
+coded by the arrows. Principally, it is possible to describe analogous
+objects when all information is saved in the intersection gr aph; how-
+ever, already the writhe number information is sufficient to r ecover a
+lot of data, as we shall see. Even more, if we forget about the w rithe
+2+-
+++
+++
++++++
+-
+Figure 3: A Gauss Diagram and its Labeled Intersection Graph
+Figure 4: A Graph not Uniquely Represented by Chord Diagrams
+Figure 5: Non-Realisable Bouchet Graphs
+31 3
+Figure 6: Resmoothing along Two Chords Yields One or Three Circles
+number information and only have the structure of opposite e dges we
+shall get non-trivial objects (modulo Reidemeister’s move s).
+Probably, the simplest evidence that one can get some inform ation
+out of the intersection graph is the number of circles one get s in a cer-
+tain state after a smoothing, see Fig. 6. The simplest exampl e shows
+that the number of circles obtained after resmoothings in tw o cross-
+ings gives three circles if the corresponding chords are unl inked or one
+circle if they aren’t. The general theorem (Soboleva’s theo rem, see
+ahead) allows one to count the number of circles in Kauffman’s s tates
+outoftheintersection graph. Inparticular, thismeanstha t graphsnot
+necessarily corresponding to any knot admit a way of general ising the
+Kauffman bracket, which coincides with the usual Kauffman brack et
+when the graph is realisable by a knot. This was the initial po int
+of investigation for L. Traldi and L. Zulli [23] (looped inte rlacement
+graphs): they constructed a self-contained theory of “non- realisable
+knots” possessing lots of interesting knot theoretic prope rties. These
+objects are equivalent classes of (decorated) graphs modul o “Reide-
+meister moves” (translated into the language of intersecti on graphs).
+A significant disadvantage of this approach was that it had ap plica-
+tions only to knots, not links : in order to encode a link, one has to
+use a more complicated object rather than just a Gauss diagra m, a
+Gauss diagram on many circles . This approach was further developed
+in Traldi’s works [21, 22], and it allowed to encode not only k nots but
+also links with any number of components by decorated graphs . The
+important question arises here: whether or not every graph i s Reide-
+meister equivalent to the looped interlacement graph of a vi rtual knot
+diagram.
+We suggested another way of looking at knots and links and gen -
+eralising them: whence a Gauss diagram corresponds to a tran sverse
+passage along a knot, one may consider a rotating circuit which never
+goes straight and always turns right or left at a classical cr ossing.
+One can also encode the type of smoothing (Kauffman’s A-smoothing
+41
+234
+567
+891012
+1311
+1413
+245
+6
+7
+8
+9
+10
+11 121314
+:2-8,3-14,4-6,7-10,
+11-13:1-12,5-9
+Figure 7: Rotating Circuit Shown by a Thick Line; Chord Diagram
+12
+3
+11
+1 11
+22
+222
+3
+3
+333
+Figure 8: A Gauss Circuit; A Rotating Circuit
+or Kauffman’s B-smoothing) corresponding to the crossing where the
+circuit turns right or left and never goes straight, see Fig. 7. We note
+that chords of the diagrams are naturally split into two sets : those
+corresponding to crossings where two opposite directions c orrespond
+toemanating edges with respect to the circuit and the other two cor-
+respond to incoming edges, and those where we have two consecutive
+(opposite) edges one of which is incoming and the other one is ema-
+nating.
+Certainly, there is an intuitive way of transforming Gauss d ia-
+grams into rotating circuits and vice versa (in the case of kn ots), see
+Fig. 7, 8. The second important question arises here whether there
+is an equivalence between the set of homotopy classes of loop ed in-
+terlacement graphs introduced by L. Traldi and L. Zulli [23] and the
+5set of graph-knots introduced by ourselves [5, 6]. The equiv alence of
+these two sets was proved in [7, 8] and, moreover, the homotop y class
+of looped interlacement graphs and the graph-knot both cons tructed
+from a given virtual knot diagram are related by this equival ence.
+This construction will be described in detail in Section 2. M oreover,
+one can consider mixed circuits (which was initiated by Tral di [21]).
+In this paper we do not consider mixed circuits. The way of tra nslat-
+ing “rotating circuits” into “transverse circuits” can be r ewritten in
+a combinatorial way and translated into the language of inte rsection
+graphs. Insomesenseitisobviousthat whenever an intersection graph
+is realisable in the sense of Gauss diagrams, the correspond ing “rotat-
+ing” intersection graph is realisable in the sense of rotati ng diagrams ,
+just because if one of these two graphs is realisable, the cor responding
+4-valent graph can be just drawn on the plane (with virtual cr oss-
+ings) and the algorithm becomes a “real redrawing algorithm ”. This
+statement follows immediately from our equivalence. This a llows us
+to switch from Gauss diagrams to rotating diagrams and vice v ersa,
+whenever we are proving some non-realisability theorems. W e shall
+answer the question of whether or not every graph is Reidemei ster
+equivalent to the looped interlacement graph of a virtual kn ot dia-
+gram, and by using the equivalence we immediately get the ans wer
+to this question concerning graph-knots. We shall show that it is not
+true for looped interlacement graphs and by using the equiva lence it
+is not true for graph-links. The examples of “non-realisabl e” looped
+interlacement graphs appeared firstly in the papers [9, 10]. So, the
+theory of graph-links is interesting for various reasons:
+a) Insomecases it exhibits purelycombinatorial ways of ext racting
+invariants for knots.
+b) In some cases it produces heuristic approaches to new “kno t
+theories”.
+c) It highlights some “graphical” effects which are hardly vis ible
+in usual or virtual knot theory.
+We conclude the introduction part by a couple of examples of f ree
+knots which are not equivalent to any realisable knot, see Fi g. 9. We
+call a graph-link non-realisable if it has no realisable representative.
+Herewedonotindicateanycrossingdecoration because any graph-
+link with this underlying graph is non-realisable . More precisely, this
+example gives us a non-realisable free graph-link . Having a virtual
+link, we may forget about over/under and right/left informa tion and
+take care only about the 4-valent underlying graph with the s tructure
+6Figure 9: The First Bouchet Graph Gives a Non-Realisable Graph-Kno t
+of opposite edges. This leads to the notion of free knots andfree links
+considered in [9, 10]. It turns out that some information abo ut virtual
+knots can be caught just from the 4-valent graph, which prove s non-
+triviality of many free knots and free links. The same trick w orks for
+graph-links: however, here instead of the underlyingfour- valent graph
+we consider the abstract graph which plays the role of the int ersection
+graph of the non-existing chord diagram .
+Certainly, the graph shown in Fig. 10 (left upper) is itself n on-
+realisable (in the sense of looped interlacement graphs and Gauss dia-
+grams), but what if we decorate its crossings in some way and t hen try
+to apply Reidemeister moves hoping to make it realisable. Fo r some
+graph-links it is possible, see, e.g. Fig. 10.
+Thegraph-linkwith“Gauss diagram”showninFig. 10(leftup per)
+isrealisable. Indeed, thesecond Reidemeister move transl ated into the
+language of Gauss diagram is an addition/removal of two “par allel”
+chords. In the language of intersection graphs, chords corr espond to
+vertices, and “parallel” chords correspond to vertices hav ing the same
+set of adjacent vertices. So, the vertices AandA′in Fig. 10 are
+adjacent, and the removal of these two vertices makes our dia gram
+realisable.
+The problem of finding free links having no representative re al-
+isable by a chord diagram is a problem similar to the problem o f
+constructing virtual knots not equivalent to any classical link. Sur-
+prisingly, the solution to the problem in the case of free lin ks can be
+achieved by using parity considerations (Theorem 3.4): all the cross-
+ings shown in Fig. 9 are odd(each of them is adjacent to an odd
+number of other crossings) and there is no immediate way to co ntract
+any two them by using a second Reidemeister moves. This is ind eed
+sufficient for a graph-link to be non-realisable in a very stro ng sense:
+any diagram of this link has a spurof the initial diagram (we disre-
+7AA’
+Figure 10: A Non-Realisable Graph Representing a Trivial Graph-Link
+gard thewrithenumberinformation), which is non-realisab le, and, the
+graph-link is, in turn, itself non-realisable. Also we have an example
+of non-realisable graph all the vertices of it are even, see F ig. 11.
+In fact, parity arguments allow to prove very strong theorem s in
+the realm of virtual knots, and we strongly recommend the rea der to
+read the paper [11] in the present volume.
+The present paper is organized as follows.
+We first give definitions of graph-links from two points of vie w:
+rotating circuits and Gauss diagrams, and describe their in teractions.
+In the third section we introduce parity and prove non-trivi ality
+results. In particular, parity arguments allow one to const ruct graph-
+valued invariants of graph-links.
+The fourth section is devoted to the orientability aspects of 2-
+surfaces corresponding to virtual knots (atoms) and functo rial map-
+pings for virtual knots and free links defined by using pariti es.
+Insection 5, webrieflydescribetheway of extendingtheKauffm an
+bracket andsomeotherinvariants. Wealsoformulatesomemi nimality
+results for graph-links and conclude the paper by a list of un solved
+problems.
+8x
+Figure 11: An Even Non-Realisable Graph Representing a Non-Realisa ble
+Graph-Knot
+Acknowledgments
+TheauthorsaregratefultoL.H.Kauffman,V.A.Vassiliev, and A.T.Fomenko
+for their interest to this work.
+2 Graph-Links and Looped Interlace-
+ment Graphs
+2.1 Chord diagrams and Framed 4-Graphs
+Throughout the paper all graphs are finite. Let Gbe a graph with the
+set of vertices V(G) and the set of edges E(G). We think of an edge
+as an equivalence class of the two half-edges. We say that a ve rtex
+v∈V(G) hasdegreekifvis incident to khalf-edges. A graph whose
+vertices have the same degree kis called k-valentor ak-graph. The
+free loop, i.e. the graph without vertices, is also consider ed ask-graph
+for anyk.
+Definition 2.1. A 4-graph is framedif for every vertex the four
+emanating half-edges are split into two pairs of (formally) opposite
+edges. The edges from one pair are called opposite to each other .
+Avirtual diagram is a framed 4-graph embedded into R2where
+eachcrossingiseitherendowedwithaclassical crossingst ructure(with
+a choice for underpassand overpass specified) or just said to bevirtual
+9AB
+AB
+Figure 12: The detour move
+and marked by a circle. A virtual link is an equivalence class of virtual
+diagrams modulo generalised Reidemeister moves. The latte r consist
+of usual Reidemeister moves referring to classical crossin gs and the
+detour move that replaces one arc containing only virtual intersection s
+and self-intersections by another arc of such sort in any oth er place of
+the plane, see Fig. 12. A projection of a virtual diagram is a framed
+4-graph obtained from the diagram by considering classical crossings
+as vertices and virtual crossings are just intersection poi nts of images
+of different edges. A virtual diagram is connected if its projection is
+connected. Withoutlossofgenerality, all virtual diagrams are assumed
+to be connected and contain at least one classical crossing [5, 6].
+Definition 2.2. Achord diagram is a cubic graph consisting of a se-
+lected cycle (the circle) and several non-oriented edges ( chords) con-
+necting points on the circle in such a way that every point on t he
+circle is incident to at most one chord. A chord diagram is labeled
+if every chord is endowed with a label ( a,α), where a∈ {0,1}is the
+framingof the chord, and α∈ {±}is the sign of the chord. If no labels
+are indicated, we assume the chord diagram has all chords wit h label
+(0,+). Two chords of a chord diagram are called linkedif the ends of
+one chord lie in different connected components of the circle w ith the
+end-points of the second chord removed.
+Definition 2.3. By avirtualisation of a classical crossing of a virtual
+diagram we mean a local transformation shown in Fig. 13.
+Having a labeled chord diagram D, one can construct a virtual
+link diagram K(D) (up to virtualisation) as follows. Let us immerse
+this diagram in R2by taking an embedding of the circle and placing
+some chords inside the circle and the other ones outside the c ircle.
+After that we remove neighbourhoods of each of the chord ends and
+10Figure 13: Virtualisation
+replace them by a pair of lines (connecting four points on the circle
+which are obtained after removing neighbourhoods) with a cl assical
+crossingifthechordisframedby0andacoupleoflineswitha classical
+crossing and a virtual crossing if the chord is framed by 1 in t he
+following way. The choice for underpass and overpass is spec ified as
+follows. A crossing can be smoothed in two ways: AandBas in the
+Kauffman bracket polynomial; we require that the initial piec e of the
+circle corresponds to the A-smoothing if the chord is positive and to
+theB-smoothing if it is negative: A:→,B:→.
+Conversely, having a connected virtual diagram K, one can get a
+labeled chord diagram DC(K), see Fig. 7. Indeed, one takes a circuit
+CofKwhich is a map from S1to the projection of K. This map
+is bijective outside classical and virtual crossings, has e xactly two
+preimages at each classical and virtual crossing, goes tran sversally at
+each virtual crossing and turns from an half-edge to an adjac ent (non-
+opposite) half-edge at each classical crossing. Connectin g the two
+preimages of a classical crossing by a chord we get a chord dia gram,
+where the sign of the chord is + if the circuit locally agrees w ith
+theA-smoothing, and −if it agrees with the B-smoothing, and the
+framing of a chord is 0 (resp., 1) if two opposite half-edges h ave the
+opposite (resp., the same) orientation. It can be easily che cked that
+this operation is indeed inverse to the operation of constru cting a
+virtual link out of a chord diagram: if we take a chord diagram D,
+and construct a virtual diagram K(D) out of it, then for some circuit
+Cthe chord diagram DC(K(D)) will coincide with D. The rule for
+setting classical crossings here agrees with the rule descr ibed above.
+This proves the following
+Theorem 2.1. [12] For any connected virtual diagram Lthere is a
+certain labeled chord diagram Dsuch that L=K(D).
+11The Reidemeister moves on virtual diagrams generate the Rei de-
+meister moves on labeled chord diagrams [5, 6].
+2.2 Reidemeister Moves for Looped Interlace-
+ment Graphs and Graph-links
+Now we are describing moves on graphs obtained from virtual d ia-
+grams by usingrotating circuit [5, 6] and the Gauss circuit [ 23]. These
+moves in both cases will correspond to the “real” Reidemeist er moves
+on diagrams. Then we shall extend these moves to all graphs (n ot
+only to realisable ones). As a result we get new objects, a graph-link
+and ahomotopy class of looped interlacement graphs , in a way similar
+to the generalisation of classical knots to virtual knots: t he passage
+from realisable Gauss diagrams (classical knots) to arbitr ary chord di-
+agrams leads to the concept of a virtual knot, and the passage from
+realisable (by means of chord diagrams) graphs to arbitrary graphs
+leads to the concept of two new objects, graph-links andhomotopy
+classes of looped interlacement graphs (here ‘looped’ corresponds to
+the writher number, if the writher number is -1 then the corre spond-
+ingvertex has aloop). Toconstructthefirstobjectweshall u sesimple
+labeled graphs, and for the second one we shall use (unlabele d) graphs
+without multiple edges, but loops are allowed.
+Definition 2.4. A graph is labeledif every vertex vof it is endowed
+with a pair ( a,α), where a∈ {0,1}is the framing of v, andα∈ {±}
+is the sign of v. LetDbe a labeled chord diagram D. Thelabeled
+intersection graph , cf. [2], G(D) ofDis the labeled graph: 1) whose
+vertices are in one-to-one correspondence with chords of D, 2) the
+label of each vertex corresponding to a chord coincides with that of
+the chord, and 3) two vertices are connected by an edge if and o nly if
+the corresponding chords are linked.
+Definition 2.5. A simple graph His called realisable if there is a
+chord diagram Dsuch that H=G(D).
+The following lemma is evident.
+Lemma 2.1. A simple graph is realisable if and only if each its con-
+nected component is realisable.
+12u v
+LC(G;u)
+G
+P(G;u,v)u v
+u v
+Figure 14: Local Complementation and Pivot
+Definition 2.6. LetGbe a graph and let v∈V(G). The set of all
+vertices adjacent to vis called the neighbourhood of a vertex vand
+denoted by N(v) orNG(v).
+Let us define two operations on simple unlabeled graphs.
+Definition 2.7. (Local Complementation) Let Gbe a graph. The
+local complementation ofGatv∈V(G) is the operation which toggles
+adjacencies between a,b∈N(v),a/\e}atio\slash=b, and doesn’t change the rest of
+G. Denote the graph obtained from Gby the local complementation
+at a vertex vby LC(G;v).
+Definition 2.8. (Pivot) Let Gbeagraphwithdistinctvertices uand
+v. Thepivoting operation of a graph Gatuandvis the operation
+which toggles adjacencies between x, ysuch that x, y /∈ {u,v},x∈
+N(u), y∈N(v) and either x /∈N(v) ory /∈N(u), and doesn’t change
+the rest of G. Denote the graph obtained from Gby the pivoting
+operation at vertices uandvby piv(G;u,v).
+Example. In Fig. 14 the graphs G, LC(G;u) and piv( G;u,v) are
+depicted.
+The following lemma can be easily checked.
+Lemma 2.2. Ifuandvare adjacent then there is an isomorphism
+piv(G;u,v)∼=LC(LC(LC( G;u);v);u).
+13Let us define graph-moves by considering intersection graph s of
+chord diagrams constructed by using a rotating circuit , and these
+moves correspond to the Reidemeister moves on virtual diagr ams. As
+a result we obtain a new object — an equivalence class of label ed
+graphs under formal moves. These moves were defined in [5, 6].
+Definition 2.9. Ωg1. The first Reidemeister graph-move is an addi-
+tion/removal of an isolated vertex labeled (0 ,α),α∈ {±}.
+Ωg2. The second Reidemeister graph-move is an addition/remov al
+of two non-adjacent (resp., adjacent) vertices having (0 ,±α) (resp.,
+(1,±α)) and the same adjacencies with other vertices.
+Ωg3. The third Reidemeister graph-move is defined as follows.
+Letu, v, wbe three vertices of Gall having label (0 ,−) so that uis
+adjacent only to vandwinG. Then we only change the adjacency
+ofuwith the vertices t∈N(v)\N(w)/uniontextN(w)\N(v) (for other pairs
+of vertices we do not change their adjacency). In addition, w e switch
+the signs of vandwto +. The inverse operation is also called the
+third Reidemeister graph-move.
+Ωg4. The fourth graph-move for Gis defined as follows. We take
+two adjacent vertices uandvlabeled (0 ,α) and (0 ,β) respectively.
+Replace Gwith piv( G;u,v) and change signs of uandvso that the
+sign ofubecomes −βand the sign of vbecomes −α.
+Ωg4′. In this fourth graph-move we take a vertex vwith the label
+(1,α). Replace Gwith LC( G;v) and change the sign of vand the
+framing for each u∈N(v).
+The comparison of the graph-moves with the Reidemeister mov es
+yields the following theorem.
+Theorem 2.2. LetK1andK2be two connected virtual diagrams, and
+letG1andG2be two labeled intersection graphs obtained from K1and
+K2, respectively. If K1andK2are equivalent in the class of connected
+diagrams then G1andG2are obtained from one another by a sequence
+ofΩg1−Ωg4′graph-moves.
+Definition 2.10. Agraph-link is an equivalence class of simple la-
+beled graphs modulo Ω g1−Ωg4′graph-moves.
+Remark 2.1. For agraph-linkhavingrepresentatives withorientable
+atoms, see ahead, there are two formally different equivalenc e rela-
+tions. The first relation is described in [5] (which includes only dia-
+grams with orientable atoms) and the last one defines graph-l inks. We
+14do not know whether these equivalence relations coincide fo r graph-
+links (they do for graph-knots, see ahead). Nevertheless, w e use the
+same term ‘graph-link’ for the object introduced in this pap er.
+Remark 2.2. Let us consider a simple realisable labeled graph. Let
+us represent this graph as an intersection graph of a chord di agram
+D. Constructing a virtual diagram K(D) we have to restore the struc-
+ture of opposite edges at each vertex. The first components (w hich
+are called the framings of vertices) of labels of vertices ar e responsi-
+ble for the structure of opposite edges. Therefore, we shall consider
+framed graphs of two types. The first type is framed 4-graphs d e-
+fined in Definition 1. The second one is free framed graphs which are
+equivalence classes of simple labeled graphs with labels ha ving only
+framings modulo Ω g4 and Ω g4′graph-moves up to signs of labels (we
+disregard the sign of each vertex).
+Definition 2.11. Afree graph-link is an equivalence class of free
+framed graphs modulo Ω g1−Ωg3 graph-moves up to signs of labels.
+LetDG(K) be the Gauss diagram of a virtual diagram K. Let us
+constructthegraph obtained from theintersection graph of DG(K) by
+adding loops to vertices corresponding to chords with negat ive writhe
+number [23]. We refer to this graph as the looped interlacement graph
+or thelooped graph . Let us construct the moves on graphs. These
+moves are similar to the moves for graph-links and also corre spond to
+the Reidemeister moves on virtual diagrams.
+Definition 2.12. The first Reidemeister move for looped interlace-
+ment graphs is an addition/removal of an isolated looped or u nlooped
+vertex.
+The second Reidemeister move for looped interlacement grap hs is
+an addition/removal of two vertices having the same adjacen cies with
+other vertices and, moreover, one of which is looped and the o ther one
+is unlooped.
+The third Reidemeister move for looped interlacement graph s is
+defined as follows. Let u, v, wbe three vertices such that vis looped,
+wis unlooped, vandware adjacent, uis adjacent to neither vnorw,
+and every vertex x /∈ {u, v, w}is adjacent to either 0 or precisely two
+ofu, v, w. Then we only remove all three edges uv,uwandvw. The
+inverse operation is also called the third Reidemeister mov e.
+15The two third Reidemeister moves do not exhaust all the possi -
+bilities for representing the third Reidemeister move on Ga uss dia-
+grams [23]. It can be shown that all the other versions of the t hird
+Reidemeister move are combinations of the second and third R eide-
+meister moves, see [15] for details.
+Definition 2.13. We call an equivalence class of graphs (without
+multiple edges, but loops are allowed) modulo the three move s listed
+in Definition 12 a homotopy class of looped interlacement graphs. A
+free homotopy class is an equivalence class of simple graphs modulo
+the Reidemeister moves for looped interlacement graphs up t o loops,
+i.e. we forget about loops.
+Remark 2.3. Looped interlacement graphs encode only knot dia-
+grams but graph-links can encode virtual diagrams with any n umber
+of components. The approach using a rotating circuit has an a dvan-
+tage in this sense. In [21] L. Traldi introduced the notion of a marked
+graph by considering any Euler tour (we have vertices which w e go
+transversally and in which we rotate).
+2.3 Looped Interlacement Graphs and Graph-
+Links
+LetGbe a labeled graph on vertices from the enumerated set V(G) =
+{v1,...,vn}, and let A(G) be the adjacency matrix ofGoverZ2de-
+fined as follows: aiiis equal to the framing of vi,aij= 1,i/\e}atio\slash=j, if and
+only ifviis adjacent to vjandaij= 0 otherwise. If GandG′represent
+the same graph-link then corank Z2(A(G)+E) = corank Z2(A(G′)+E),
+whereEis identity matrix.
+Definition 2.14. Let usdefinethe number of components in a graph-
+linkFas corank Z2(A(G)+E)+1, here Gis a representative of F. A
+graph-link Fwith corank Z2(A(G) +E) = 0 for any representative G
+ofFis called a graph-knot .
+Let corank Z2(A(G)+E) = 0,Bi(G) =A(G)+E+Eii(all elements
+ofEiiexcept for the one in the i-th column and i-th row which is one
+are 0) for each vertex vi∈V(G).
+Definition 2.15. Let us define the writhe number wiofG(with
+corank Z2(A(G) +E) = 0) at viaswi= (−1)corank Z2Bi(G)signviand
+16thewrithe number ofGas
+w(G) =n/summationdisplay
+i=1wi.
+IfGis a realisable graph by a chord diagram and, therefore, by a
+virtual diagram then wiis the “real” writhe number of the crossing
+corresponding to vi.
+Definition 2.16. We say that an n×nmatrixA= (aij)coincides
+with ann×nmatrixB= (bkl)up to diagonal elements ifaij=bij,
+i/\e}atio\slash=j.
+Lemma 2.3 ([7]).LetAbe a symmetric matrix over Z2. Then there
+exists a symmetric matrix /tildewideAwithdet/tildewideA= 1equal toAup to diagonal
+elements.
+LetFbe a graph-knot and let Gbe its representative. Let us
+consider the simple graph Hhaving the adjacency matrix coinciding
+with (A(G) +E)−1up to diagonal elements and construct the graph
+L(G) fromHby just addingloops to any vertex of Hcorrespondingto
+a vertex of Gwith the negative writhe number. Let us define the map
+χfrom the set of graph-knots to the set of homotopy classes of l ooped
+interlacement graphs defined by χ(F) =L, hereLis the homotopy
+class ofL(G). It turns out that the map χis well-defined [8]. The
+inverse map is defined as follows [8]. Let Lbe the homotopy class
+ofL. By using Lemma 2.3 we can construct a symmetric matrix
+A= (aij) overZ2coinciding with the adjacency matrix of Lup to
+diagonal elements and det A= 1. Let G(L) be the labeled simple
+graph having the matrix A−1+Eas its adjacency matrix (therefore,
+the first component of the vertex label is equal to the corresp onding
+diagonal element of A−1+E), thesecond component of thelabel of the
+vertex with the number iiswi(1−2aii), herewi= 1 if the vertex of L
+with the number idoesn’t have a loop, and wi=−1 otherwise [7, 8].
+We have χ−1(L) =F, hereG(L) is a representative of F. Therefore,
+we get
+Theorem 2.3 ([7, 8]).There is a one-to-one correspondence between
+the set of graph-knots and the set of homotopy classes of loop ed inter-
+lacement graphs. Moreover, the graph-knot and the homotopy class of
+looped interlacement graphs both constructed from a given v irtual knot
+diagram are related by this map.
+17Figure 15: A surgery of the circuit along a chord
+Figure 16: The manifold m(D)
+2.4 Soboleva’s Theorem and its Corollaries
+Assume we are given a framed chord diagram, i.e. each chord of it
+is endowed only with framing. Define the surgery over the set of
+chords as follows . For every chord having the framing 0 (resp., 1),
+we draw a parallel (resp., intersecting) chord near it and re move the
+arc of the circle between adjacent ends of the chords as in Fig . 15.
+By a small perturbation, the picture in R2is transformed into a one-
+manifold in R3. This manifold m(D) is the result of surgery, see
+Fig. 16. Surprisingly, the number of the connected componen ts of
+m(D) can be determined from the intersection graph.
+Theorem 2.4 ([16, 20]) .LetDbe a labeled chord diagram, and let
+Gbe its labeled intersection graph. Then the number of connect ed
+components of m(D)equalscorank Z2A(G) + 1, where A(G)is the
+adjacency matrix of G.
+Remark 2.4. Theorem 2.4 allows us to define ‘the number of circles’
+after a “surgery” for a graph even when the given graph is not a n
+intersection graph of any chord diagram.
+We refer to this Theorem as Soboleva’s theorem although a par tial
+case of this Theorem, when all the chords have the framing 0, fi rstly
+appears in [3].
+Throughout the rest of the paper, we shall use the following main
+principle :
+18Assume there is an equality concerning numbers of circles in some
+states of chord diagrams, which can be formulated in terms of the
+intersection graph. Then the corresponding equality usuall y holds even
+for non-realisable graphs.
+The reason for this principle to hold is the following: every time
+we have a picture where some two numbers of circles are equal t o each
+other (or differ by a constant), this can be expressed in terms o f the
+corresponding adjacency matrices, and the proof does not ge nerally
+depend on the behaviour of the matrix outside of the crossing s in
+question. Thismeans that theequality holdstrueforgeneri c matrices,
+thus, it works for general intersection graphs.
+This principle has lots of consequences. We shall demonstra te it
+for three examples.
+The first example comes from Definition 2.14. It shows that the
+number of components defined for non-realisable graphs by th e same
+formula as for realisable ones doesn’t change under the Reid emeister
+moves.
+The second example comes from Definition 2.15. We have defined
+the writhe number of a vertex for an arbitrary labeled graph b y using
+the definition of the writhe number for virtual diagrams and w riting
+this definition in terms of matrices. Then we definethe writhe number
+for a graph. Since the proof exists in the realisable case, it can be
+rewritten in terms in matrices, thus, we get that the writhe n umber
+of a graph doesn’t change under the second and third Reidemei ster
+graph-moves and changes by ±1 under the first Reidemeister graph-
+moves.
+The third example is as follows. Assume we have a framed 4-gra ph
+Kwith a vertex vand we would like to know whether this vertex
+belongs to one component or it belongs to different components of
+the corresponding graph-link. Then we may take the two smoot hings
+Ka,Kbof theKatvand see how many components we get. If v
+belongs to two branches of the same component of Kthen the number
+ofcomponentsofoneof Ka,Kbisequaltothatof K,andthenumberof
+componentsoftheotheroneisequaltothatof Kplusone. If vbelongs
+to two different components of K, then the number of components of
+each ofKa,Kbis that of Kminus one. Now, turning to graph-links,
+by taking appropriate matrix ranks, we may see whether each v ertex
+belongstoonecomponent ortotwodifferent componentsof theg raph-
+link. This method is used for proving the invariance of the Ka uffman
+bracket polynomial.
+19a
+bc d
+aa
+bb
+cc
+dd
+Figure 17: Two smoothings at a vertex for a framed graph
+2.5 Smoothing Operations and Turaev’s ∆
+By asmoothing of a framed 4-graph at a vertex vwe mean any of the
+two framed 4-graphs obtained by removing vand repasting the edges
+asa−b,c−dor asa−d,b−c, see Fig. 17. Generally, a smoothing
+of a framed 4-graph in a collection of vertices is the framed 4 -graph
+obtained by a sequence of smoothings.
+Now, we can mimic this definition for the case of free graph-li nks.
+LetGbe a free framed graph, i.e. an equivalence class of labeled
+simple graphs, and let v∈V(G) (the set of vertices is the same for
+any representative of G). Let us consider two cases. In the first case
+there exists a representative HofGfor which vhas either framing 1
+or the degree more than 0. It is not difficult to see that vhas the same
+propertyforeachrepresentativeof G, andtherearetworepresentatives
+H1andH2ofGwhich distinguish from each other by Ω g4 or Ω g4′at
+v. By asmoothing of a free framed graph Gat a vertex vwe mean
+any of the two free framed graphs having the representatives H1\{v}
+andH2\{v}, respectively. In the second case vhas framing 0 and is
+isolated for each representative of G. LetHbe a representative of G.
+Let us construct the new graph H′obtained from Hby adding a new
+vertexuwith framing 0 to Hwhich is adjacent only to v, see Fig. 18
+for the case of realisable graphs (the dashed line is a rotati ng circuit).
+By asmoothing of a free framed graph Gat a vertex vwe mean
+any of the two free framed graphs having the representatives H\{v}
+andH′, respectively. It is not difficult to see that the numbers of th e
+components of two different smoothings at any vertex distingu ishfrom
+20Figure 18: One of the two smoothings at an isolated vertex
+each other by 1. Generally, a smoothing of a free framed graph in a
+collection of vertices is the free framed graph obtained by a sequence
+of smoothings.
+We call a free framed graph Grealisable if any of its representative
+is realisable by a chord diagram. It is not difficult to show tha tGis
+realisable if and only if there exists a realisable represen tative of it,
+we just redraw the picture.
+Statement 2.1. Assume G,G′are free framed graphs and Gcan be
+obtained from G′as a result of smoothings at some vertices. Then if
+G′is realisable by a chord diagram, then so is G.
+The proof is obvious: if G′is realisable and Gis obtained from
+G′by applying a fourth graph-move and/or deleting a vertex, th en
+one can draw the corresponding framed 4-graph, and take the c orre-
+sponding resmoothing which will yield the chord diagram for G. In
+the other case, if G′is realisable then the realisability of Gfollows from
+Lemma 2.1.
+Leti >1 be a natural number. Define the set Z2Gito be the
+Z2-linear space generated by the set of free framed graphs Gwith
+corank Z2(A(G)+E) =i−1 modulo the following relations:
+1) the second Reidemeister graph-moves;
+2)G= 0, ifGhas two vertices with framing 0 which are adjacent
+only to each other.
+Fori= 1, we define Z2G1analogously with respect to equivalence
+1) and not 2).
+Let us define the map ∆: Z2G1→Z2G1, cf. [11]. Given a free
+framed graph Gwith corank Z2(A(G)+E) = 0. We shall construct an
+element ∆( G) fromZ2G2as follows. For each vertex vofG, there are
+21two ways of smoothing it. One way gives a graph from Z2G1, and the
+other smoothing gives a free framed graph GvfromZ2G2. We take Gv
+and set
+∆(G) =/summationdisplay
+vGv∈Z2G2.
+Theorem 2.5. ∆is a well defined mapping from Z2G1toZ2G2.
+By using the main principle from subsection 2.4 we can define
+whether a vertex belongs to one “component” or different compo nents
+of a free framed graph. Namely, we call a vertex viof a free framed
+graphGoriented if corank Z2(A(G)+E)/lessorequalslantcorank Z2(Bi(G)). It is not
+difficult to show that corank Z2Bi(G)/\e}atio\slash= corank Z2B(G \ {vi}) ifviis
+oriented.
+By usingthe notion of an oriented vertex we can definethe map ∆i
+(iteration) by considering smoothings at oriented vertice s and taking
+that smoothing which has more components than other in each s tep.
+Corollary 2.1. ∆iis a well defined mapping from Z2G1toZ2Gi+1.
+3 Parity, Minimalityand Non-Triviality
+Examples
+In the present section we consider the parity for free graph- knots and
+free graph-links in spirit of [9, 10]. As we have constructed the one-
+to-one correspondence between the set of graph-knots and th e set of
+homotopy classes of looped interlacement graphs it is suffici ent to
+construct a parity for free homotopy classes of looped inter lacement
+graphs and for graph-links with more than one components.
+Consider the category of free homotopy classes of looped int erlace-
+ment graphs.
+Definition 3.1. For every looped interlacement graph Lwe call a
+vertexvevenifvis adjacent to an even number of vertices distinct
+fromv; otherwise we call vodd.
+Theorem 3.1. The parity defined on homotopy classes of looped in-
+terlacement graphs (resp., free homotopy classes )by using even and
+odd vertices satisfies the parity axioms from [11].
+22The proof evidently follows from the consideration of the Re ide-
+meister moves for looped interlacement graphs.
+By using this parity we can define the map ∆i
+oddwhere the sum is
+taken over all odd oriented vertices or ∆i
+evenwhere the sum is taken
+over all even oriented vertices. We have to define the notion o feven
+andoddvertex for free framed graphs with many components.
+We call a vertex vofGwith one component even(resp.,odd) if the
+vertex corresponding to vof the looped interlacement graph χ(G) is
+even(resp., odd). Letusconsiderthefreeframedgraph Gv1,...,vk−1with
+kcomponents which is obtained from Gby smoothing Gconsequently
+atv1,...,vk−1;viis a oriented vertex in Gv1,...,vi−1. An oriented vertex
+uofGv1,...,vk−1iseven with respect to the smoothing at v1,...,vk−1
+(resp.,odd with respect to the smoothing at v1,...,vk−1) if the number
+of oriented vertices in Gv1,...,vk−1which are incident to uinχ(G) is even
+(resp., odd).
+Remark 3.1. We have defined even vertices only for those free
+framed graphs with many components which originate from giv en free
+framedgraphs with 1 component. It is sufficient to definetheit eration
+∆i
+oddand ∆i
+even.
+Statement 3.1. ∆i
+oddis a well defined mapping from Z2G1toZ2Gi+1.
+Let us consider another parity for graph-links with two comp o-
+nents.
+Definition 3.2. For a graph Grepresenting a two-component graph-
+link we call a vertex evenif it isoriented andoddotherwise.
+Theorem 3.2. The parity defined above satisfies the parity axioms
+given in [11].
+The proof again follows from the definition of parity.
+Remark 3.2. The parity used in Theorem 3.1 is easier to define via
+Gauss diagram approach : assuming a graph is the “intersection graph
+of a non-existing Gauss diagram”, we take those chords which are
+linked with even number of chords to be even, and the remainin g ones
+to be odd.
+In the language of the rotating circuit approach, this is mor e diffi-
+cult: justtakingan arbitraryrotatingcircuit andcountin gthenumber
+of adjacent chords can lead to different results: the same vert ex may
+have an even or odd degree for different rotating circuits.
+23Now, having these parities in hand, we can define the brackets [·]
+for graph-knots and {·}for graph-links analogously to the case of [11].
+For a free graph-knot G(resp., free two-component graph-link H),
+consider the following sums
+[G] =/summationdisplay
+s even.,1compGs,
+and
+{H}=/summationdisplay
+s even. non −trivialHs,
+where the sums are taken over all smoothings at all evenvertices
+(with respect to the definitions above), and only those summa nds
+are taken into account where corank Z2(A(Gs)+E) = 0 (resp., Hsis
+not equivalent to any simple graph having two vertices with f raming 0
+which are adjacent only to each other). Thus, if Ghaskeven vertices,
+then [G] will contain at most 2ksummands, and if all vertices of G
+are odd, then we shall have exactly one summand, the graph Gitself.
+The same is true for Hand{H}.
+Now, we are ready to formulate the main theorems of this secti on:
+Theorem 3.3. IfGandG′represent the same free graph-knot then
+the following equality holds: [G] = [G′].
+Analogously, if HandH′represent the same free graph-link with
+two components then {H}={H′}.
+The proof of 3.3 verbally reproduces the proof of the main the -
+orem from [11], according to the main principle or, maybe, a slight
+modification of it.
+Definition 3.3. We call a labeled graph G(resp., a looped graph L)
+minimal if there is no representative of the graph-link correspondi ng
+toG(resp., the homotopy class of L) having strictly smaller number
+of vertices than G(resp.,L) has.
+Theorem 3.4. LetG(resp.,H)be a simple labeled graph representing
+a free graph-knot (a two-component graph-link )with all odd vertices in
+the sense of Theorem 3.1 (resp., Theorem 3.2 ), such that no decreasing
+second Reidemeister move is applicable to G(resp.,H). Then there
+is no simple graph equivalent to G(resp.,H)with strictly smaller
+number of vertices.
+24As a consequence of this theorem we may deduce the following
+Corollary 3.1. The free graph-knot Gshown in Fig. 9 is minimal;
+in particular, it is non-trivial and has no realisable repre sentatives.
+Moreover, Ghas no representative realisable as an intersection
+graph of a chord diagram.
+The first claim of the corollary is trivial: we just check the c ondi-
+tions of the theorem and see that Gis minimal; to see that the second
+claim indeed holds, we shall need to go through the proof of Th eo-
+rem 3.3, where we see that any representative G′of the graph-knot G
+hasGas asmoothing , that is, Glies inside each representative G′of
+the same graph-link, and if Gis not realisable, then so is G′according
+to Statement 2.1. Analogously, one sees that the free two-co mponent
+graph-link Hwith therepresentative Hshownin Fig. 19(left part) has
+no realisable representative because {H}=H. Note that Hin Fig. 19
+is equivalent to the Bouchet graph shown in the right part of t he pic-
+ture by Ω 4-graph move; so they represent the same two-component
+free graph-link.
+Let us consider one more example.
+Statement 3.2. The looped graph Krepresented by the “Gauss dia-
+gram” shown in Fig. 11 is minimal and non-realisable.
+Theproofconsistsofthefollowingstepsverysimilartothe example
+from [11].
+First, note that ∆( K) consists of 7 summands L+/summationtext
+iLi, where
+only one summand (corresponding to the vertex x) is a 2-component
+free graph-link with all oddvertices; for each of the remaining sum-
+mandsLi, there is at least one even vertex. This is, indeed, very
+easy: the graph Khas only one vertex xwhich is adjacent to all
+the remaining vertices, then the argument just repeats the a rgument
+from [11].
+Now, the 2-component free graph-link Lhas a rotating circuit di-
+agram shown in Fig. 19; all framings of the vertices are 0. To s ee
+it, one should consecutively perform the following operati ons forK:
+first, we “smooth” its crossing x; it can be done only at the expense of
+changing our circuit to the rotating one at some vertex differe nt from
+x(a Gauss circuit can not represent link); after that, we have to make
+the circuit rotating at all other vertices. All these steps a re shown in
+Fig. 20.
+25=
+Figure 19: The Two-Component Free Link
+Now, consider the bracket {∆(K)}=L+/summationtext
+i{Li}. Note that all
+summands {Li}have representatives with strictly less than 6 vertices
+since each of Lihas at least one even vertex; on the other hand, Lhas
+no representative with less than 6 crossings; so, this eleme ntLis not
+canceled in the sum. Since it is not realisable, the free fram ed knotK
+is not realisable either.
+4 Atoms and Orientability
+In this section we introduce a object: atomwhich is a 2-manifold
+with additional structure. This object helps us define new eq uivalence
+relation of graphs, see [5], and is very useful for minimalit y theorems,
+see ahead.
+Definition 4.1. Anatom[4] is a pair ( M,Γ) consisting of a closed
+2-manifold Mand a finite graph Γ embedded in Mtogether with
+a colouring of M\Γ in a checkerboard manner. An atom is called
+orientable (connected ) if the surface Mis orientable (connected). Here
+Γ is called the frameof the atom. By genus(Euler characteristic,
+orientation ) of the atom we mean that of the surface M. Atoms and
+theirgenerawerealsostudiedbyTuraev[24], andinconsequ entpapers
+that atom genus is also called the Turaev genus [24].
+Having a chord diagram, we can construct an atom correspondi ng
+to the chord diagram, see [5, 6].
+In fact, chord diagrams in the sense of rotating circuits wit h all
+chords having framing 0 encode all orientable atoms, see [13 ]. Chord
+26X1
+234
+5
+6Deleting x and
+changing circuit
+at 11
+234
+5
+6=
+14
+6
+2 35
+(circuit rotating at 1, and 
+transverse elsewhere)Changing 2 and 3
+to rotating circuit
+2
+32
+314
+6
+2 35
+(circuit rotating at 1,2,3, 
+and transverse elsewhere)(transverse circuit)
+14
+5
+2 36
+(circuit rotating everywhere
+except 4)Changing 5 and 6
+to rotating circuit
+=
+46
+5
+2 13
+Local complementation at 4
+(changing circuit to rotating)
+4
+4
+43
+5
+1 62
+(rotating circuit; framing is 0 for 
+all vertices)=
+Figure 20: The Two-Component Free Link
+27diagrams with all positive chords encode all atoms with one w hite
+cell: this white cell corresponds to the A-state of the virtual diagram,
+and chords show how this cell approaches itself in neighbour hoods
+of crossings (atom vertices). If we want to deal with all atom s and
+restrict ourselves for the case of one circle, we should take this circle
+to correspond to some other state of the atom, which is encode d by
+labelings of the chords.
+5 Kauffman’s Bracket Generalisation
+and Other Invariants. Minimality The-
+orems
+We have already considered some minimality theorems. Inthi s section
+we present minimality theorems which use Kauffman’s bracket g ener-
+alisation. Establishingminimalcrossingnumberofacerta inlinkisone
+of the important problems in classical knot theory. In late 1 9’s cen-
+tury, famous physicist and knot tabulator P.G. Tait [17] con jectured
+that alternating prime diagrams of classical links are mini mal with re-
+spect to the number of classical crossings. This celebrated conjecture
+was solved only in 1987, after the notions of the Jones polyno mial
+and the Kauffman bracket polynomial appeared. The first soluti on
+was obtained by Murasugi [14], then it was reproved by Thistl eth-
+waite [18], Turaev [24], and others. Later, Thistlethwaite [19] estab-
+lished the minimality for a larger class of diagrams (so-cal ledadequate
+diagrams). Itturnsoutthatmanyresultsinthesedirection sgeneralise
+for virtual links (establishing the minimal number of class ical cross-
+ings); these results were obtained by the second named autho r of the
+present paper. See [12] for the proofs and some further gener alisations
+and other results concerning virtual knots.
+Definition 5.1. The difference between the leading degree and the
+lowest degreeof non-zeroterms of apolynomial P(x) is called the span
+ofP(x) and is denoted by span P(x).
+Definition 5.2. A classical link diagram is called alternating if while
+passing along every component of it we alternate undercross ings and
+overcrossings.
+28From the ‘atomic’ point of view, alternating link diagrams a re
+those having atom genus (Turaev genus) zero (more precisely , diagram
+hasgenus zeroifit isaconnected sumofseveral alternating diagrams).
+Definition 5.3. A virtual link diagram Dis called splitif there
+is a vertex Xof the corresponding atom ( M,Γ) such that Γ \Xis
+disconnected.
+Themain reason of theseminimality theorems and furthercro ssing
+estimatescomefromthewell-knownKauffman-Murasugi-Thist lethwaite
+Theorem:
+Theorem 5.1. For a non-split classical link diagram Konncrossings
+we have span/a\}bracketle{tK/a\}bracketri}ht/lessorequalslant4n, whence for alternating non-split diagrams we
+havespan/a\}bracketle{tK/a\}bracketri}ht= 4n. Here,/a\}bracketle{tK/a\}bracketri}htis the Kauffman bracket of K.
+Note that the span of the Kauffman bracket is invariant under
+all Reidemeister moves. This theorem is generalised for vir tual dia-
+grams[12]. Theestimatespan /a\}bracketle{tK/a\}bracketri}ht/lessorequalslant4ncanbesharpenedtospan /a\}bracketle{tK/a\}bracketri}ht/lessorequalslant
+4n−4g, wheregis the genus of the corresponding atom.
+All the minimality theorems for the case of graph-links in th is
+section rely on the generalisation of the Kauffman bracket pol ynomial
+for graph-links. Now, let us generalise the notions defined a bove for
+the case of graph-links and define the Kauffman bracket polynom ial
+for a labeled graph G.
+Lets⊂V(G) beasubsetof theset V(G) of vertices of G. SetG(s)
+to be the induced subgraph of the graph Gwith the set of vertices
+V(G(s)) =sand the set of edges E(G(s)) such that {u,v} ∈E(G(s)),
+whereu,v∈s, if and only if {u,v} ∈E(G).
+Definition 5.4. We call asubsetof V(G) astateof thegraph G. The
+A-stateis the state consisting of all the vertices of Glabeled ( a,−),
+a∈ {0,1}, and no vertex labeled ( b,+),b∈ {0,1}. Analogously,
+theB-stateis the state consisting of all vertices of Glabeled ( b,+),
+b∈ {0,1}, and no vertex labeled ( a,−),a∈ {0,1}.
+Definition 5.5. TheKauffman bracket polynomial ofGis
+/a\}bracketle{tG/a\}bracketri}ht(a) =/summationdisplay
+saα(s)−β(s)(−a2−a−2)corankA(G(s)),
+where the sum is taken over all states sof the graph G,α(s) is equal
+to the sum of the vertices labeled ( a,−),a∈ {0,1}, fromsand the
+vertices labeled ( b,+),b∈ {0,1}, fromV(G)\s,β(s) =|V(G)|−α(s).
+29Theorem 5.2. The Kauffman bracket polynomial of a labeled graph is
+invariant under Ωg2−Ωg4′graph-moves and is multiplied by (−a±3)
+underΩg1graph-move.
+Definition 5.6. For a labeled graph Glet theatom genus (Tu-
+raev genus ) be 1−(k+l−n)/2, where kandlare the numbers
+of circles in the A-states1and theB-states2ofG, respectively, i.e.
+k= corank Z2A(G(s1))+1 and l= corank Z2A(G(s2))+1.
+Note that this number agrees with the atom genus in the usual
+case: we just use χ= 2−2g, whereχis the Euler characteristic, and
+countχby using the number of crossings n, number of edges 2 nand
+the number of 2-cells ( A-state circles and B-state circles).
+Definition 5.7. A labeled graph Gonnvertices is alternating if its
+atom genus is equal to 0. A labeled graph Gisnon-split if it has no
+isolated vertices.
+Theorem 5.3. An alternating non-split labeled graph is minimal.
+Example. Consider the graph BW3consisting of the 7 vertices
+with the following incidences. For i,j= 1,...,6,iis connected to jif
+and only if i−j≡ ±1 (mod 6), and 7 is connected to 2 ,4,6. Label all
+even vertices by (0 ,+), and label all odd vertices by (0 ,−), see Fig. 21.
+This graph is alternating. By Theorem 5.3, BW3is minimal. Note
+that this graph is not realisable as an intersection graph of a chord
+diagram. We conjecture that the graph-link represented by BW3is
+non-realisable. At least we know that it has no such represen tatives
+with the number of crossings less than or equal to 7.
+We conclude the present paper with the significant one questi on
+which seems to be very interesting for us: is there a graph-li nk having
+two representatives realisable by chord diagrams but they a re equiva-
+lent in the graph-link only by means of non-realisable graph s?
+References
+[1] A. Bouchet, Circle graph obstructions, J. Combinatorial Theory
+B60(1994), pp. 107–144.
+[2] S.V. Chmutov, S.V. Duzhin, S.K. Lando, Vassiliev Knot In vari-
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+arXiv:1001.0385v1  [math.AP]  3 Jan 2010Stabilities for Euler-Poisson Equations with
+Repulsive Forces in RN
+Manwai Yuen∗
+Department of Applied Mathematics,
+The Hong Kong Polytechnic University,
+Hung Hom, Kowloon, Hong Kong
+Revised 3-Jan-2010
+Abstract
+This article extends the previous paper in ”M.W. Yuen, Stabilities for Euler-Poisson Equa-
+tions in Some Special Dimensions , J. Math. Anal. Appl. 344(2008), no. 1, 145–156.”, from
+the Euler-Poisson equations for attractive forces to the re pulsive ones in RN(N≥2). The
+similar stabilities of the system are studied. Additionall y, we explain that it is impossible to
+have the density collapsing solutions with compact support to the system with repulsive forces
+forγ >1.
+Key Words: Euler-Poisson Equations, Repulsive Forces, Sta bilities, Frictional Damping,
+Second Inertia Function, Non-collapsing Solutions
+1 Introduction
+The semi-conductor models can be formulated by the isentropic Eule r-Poisson equation with re-
+pulsive forces in the following form:
+
+
+ρt+∇·(ρu) =0,
+(ρu)t+∇·(ρu⊗u)+∇P+βρu=ρ∇Φ,
+∆Φ(t,x) =α(N)ρ,(1)
+whereα(N) is a constant related to the unit ball in RN:α(1) = 2;α(2) = 2π; ForN≥3,
+α(N) =N(N−2)V(N) =N(N−2)πN/2
+Γ(N/2+1), (2)
+whereV(N) is the volume of the unit ball in RNand Γ is the Gamma function. As usual,
+ρ=ρ(t,x) andu=u(t,x)∈RNare the density and the velocity respectively. P=P(ρ) is the
+pressure and β≥0 is the frictional damping constant. In the above system, the self -repulsive
+potential field Φ = Φ( t,x) is determined by the density ρthrough the Poisson equation. The
+equations (1) 1and (1) 2are the compressible Euler equations with forcing term. The equatio n (1)3
+is the Poisson equation through which the potential with repulsive fo rces is determined by the
+density distribution of the electrons themselves. Thus, we called th e system (1) the Euler-Poisson
+equations with repulsive forces. In this case, the equations can be viewed as a semiconductor
+∗E-mail address: nevetsyuen@hotmail.com
+12 M.W. Yuen
+model. See [2], [5] for a detail about the system. For some fixed K≥0, we have a γ-law on the
+pressure P(ρ), i.e.
+P(ρ)=Kργ.=ργ
+γ, (3)
+forK >0, which is a common hypothesis. When K= 0, the pressureless system can be used as
+models in plasma physics [1]. The constant γ=cP/cv≥1, where cPandcvare the specific heats
+per unit mass under constant pressure and constant volume resp ectively, is the ratio of the specific
+heats. Additionally, the fluid is called isothermal if γ= 1.
+The Poisson equation (1) 3can be solved as
+Φ(t,x) =/integraldisplay
+RNG(x−y)ρ(t,y)dy, (4)
+whereGis the Green’s function for the Poisson equation in the N-dimensional spaces defined by
+G(x).=
+
+|x|, N= 1;
+log|x|, N= 2;
+−1
+|x|N−2, N≥3.(5)
+If we seek solutions in radial symmetry with the radius r=/parenleftBig/summationtextN
+i=1x2
+i/parenrightBig1/2
+, the Poisson equation
+(1)3is transformed to
+rN−1Φrr(t,x)+(N−1)rN−2Φr=α(N)ρrN−1, (6)
+Φr=α(N)
+rN−1/integraldisplayr
+0ρ(t,s)sN−1ds. (7)
+We can seek the radial symmetry solutions
+ρ(t,/vector x) =ρ(t,r) and/vector u=/vector x
+rV(t,r) =:/vector x
+rV. (8)
+By standard computation, the Euler-Poisson equations in radial sy mmetry can be written in the
+following form:
+
+ρt+Vρr+ρVr+N−1
+rρV= 0,
+ρ(Vt+VVr)+Pr(ρ) =ρΦr(ρ).(9)
+Perthame discovered the blowup results for 3-dimensional pressu reless system with repulsive forces
+[9]. In short, all the results above rely on the solutions with radial sy mmetry:
+Vt+VVr=α(N)
+rN−1/integraltextr
+0ρ(t,s)sN−1ds. (10)
+And theEmdenordinarydifferentialequationswerededucedonthe boundarypointofthesolutions
+with compact support:
+D2R
+Dt2=M
+RN−1, R(0,R0) =R0≥0,˙R(0,R0) = 0, (11)
+wheredR
+dt:=VandMis the mass of the solutions, along the characteristic curve. They s howed
+the blowup results for the C1solutions of the system (9).
+Very recently, Yuen showed in [14] that the classical non-trivial s olutions ( ρ,V) for the Euler or
+Euler-Poisson equations with repulsive forces, in radial symmetry, with compact support in [0 ,R],
+whereRis a positive constant (which is V(t,0) = 0;ρ(t,r) = 0,V(t,r) = 0 for r≥R) and the
+initial velocity such that:
+H0=/integraldisplayR
+0V0dr >0, (12)Stabilites of Euler-Poisson Equations 3
+blow up on or before the finite time T= 2R/H0,for pressureless fluids ( K= 0) orγ >1.
+The system with attractive forces were studied in [3], [6], [7], [8], [12].and [13]. This article extends
+the previous paper in [13], from the Euler-Poisson equations for att ractive forces with or without
+frictional damping to the repulsive ones in RN(N≥2). The similar stabilities of the system are
+studied.
+In the last section, we exclude the possibility of collapsing solutions fo r this system. The
+non-existence of collapsing solutions can be shown by the simple argu ment for the energy function:
+E(t) =/integraltext
+Ω/parenleftBig
+1
+2ρ|u|2+1
+γ−1P−1
+2ρΦ/parenrightBig
+dx,forγ >1. (13)
+Theorem 1 For the classical solutions with compact support of the Eule r-Poisson equations with
+repulsive force, (1), in RN(N≥2)withγ >1or without pressure (K= 0), there is no collapsing
+phenomenon where part of the density ρ(t,x)collapses to a point.
+2 Stabilities
+In this section, we study the stabilities of the Euler-Poisson equatio ns with repulsive forces, (1),
+inRN(N≥2). The total energy can be defined by,
+
+
+E(t) =/integraltext
+Ω/parenleftBig
+1
+2ρ|u|2+1
+γ−1P−1
+2ρΦ/parenrightBig
+dx,forγ >1,
+E(t) =/integraltext
+Ω/parenleftBig
+1
+2ρ|u|2−1
+2ρΦ/parenrightBig
+dx,for without pressure.(14)
+For the system, we have the energy estimate:
+Lemma 2 For the Euler-Poisson equations, (1), suppose the solution s(ρ,u)have compact support
+inΩ. We have,
+·
+E(t) =−β/integraldisplay
+Ωρ|u|2dx≤0. (15)
+Initially, Sideris used the second inertia function
+H(t) =/integraldisplay
+Ωρ(t,x)|x|2dx, (16)
+to study instability results for the Euler equations [10]. After that in [8], the instability result of
+the Euler-Poisson equations with attractive forces, in radial symm etry, was obtained for γ≥4/3
+andN= 3. For the corresponding cases in RN, with non-radial symmetry, were studied [3], [13].
+By the standard computation for energy method, it is clear to have the following lemma:
+Lemma 3 Consider (ρ,u)is asolution with compact support in Ωfor the Euler-Poisson equations,
+(1) with β= 0. We have
+
+
+··
+H(t) = 2/integraltext
+Ω/bracketleftBig
+(ρ|u|2+NP)dx−N−2
+2ρΦ/bracketrightBig
+dx,forN≥3;
+··
+H(t) = 2/integraltext
+Ω(ρ|u|2+2P)dx+M2,forN= 2.(17)
+By applying the above lemma, we could have:
+Theorem 4 Suppose (ρ,u)is a global classical solution in the Euler-Poisson equatio ns, (1) with
+γ >1, without frictional damping (β= 0). We have
+(1)forN≥3,
+lim
+t→∞infR(t)
+t≥/bracketleftbigginf(2,N(γ−1),N−2)E
+M/bracketrightbigg1/2
+; (18)4 M.W. Yuen
+(2)forN= 2,
+lim
+t→∞infR(t)
+t≥/radicalbigg
+1
+2M; (19)
+(3)forN≥2,
+lim
+t→∞infR(t)
+t≥/bracketleftBigg
+NKMγ−1
+|Ω|(γ−1)/bracketrightBigg1/2
+, (20)
+withR(t) = max x∈Ω(t){|x|}. Here
+M=/integraldisplay
+Ωρ(t,x)dx, (21)
+is the total mass which is constant for any classical solutio n and|Ω|is the fixed volume of Ω.
+Proof.(1)ForN≥3,we have the positive energy function E≥0. We can get from Lemma 3,
+··
+H(t) = 2/braceleftbigg/integraldisplay
+Ω/bracketleftBig
+ρ|u|2+NP/bracketrightBig
+dx−N−2
+2/integraldisplay
+ΩρΦdx/bracerightbigg
+≥2inf(2,N(γ−1),N−2)E.(22)
+That is
+H(t)≥H(0)+·
+H(0)t+inf(2,N(γ−1),N−2)Et2. (23)
+On the other hand, we obtain,
+H(0)+·
+H(0)t+inf(2,N(γ−1),N−2)Et2≤H(t)≤R(t)2M. (24)
+That is,
+O(1
+t)+inf(2,N(γ−1),N−2)E≤R(t)2M
+t2, (25)
+lim
+t→∞infR(t)
+t≥/bracketleftbigginf(2,N(γ−1),N−2)E
+M/bracketrightbigg1/2
+. (26)
+ForN= 2,we have
+··
+H(t) = 2/integraldisplay
+Ω(ρ|u|2+2P)dx+M2≥M2, (27)
+M2
+2t2+C0t+C1≤H(t)≤R(t)2M, (28)
+lim
+t→∞infR(t)
+t≥/radicalbigg
+1
+2M. (29)
+ForN≥2, we obtain
+M=/integraldisplay
+Ωρdx≤/parenleftbigg/integraldisplay
+Ωργdx/parenrightbigg1/γ
+|Ω|(γ−1)/γ, (30)
+and
+··
+H(t) = 2/braceleftbigg/integraldisplay
+Ω/bracketleftBig
+ρ|u|2+NP/bracketrightBig
+dx−N−2
+2/integraldisplay
+ΩρΦdx/bracerightbigg
+≥2/integraldisplay
+ΩNPdx. (31)
+From the inequality (30), it is clear to have
+··
+H(t)≥2NK|Ω|1−γMγ>0, (32)
+H(0)+·
+H(0)t+NK|Ω|1−γMγt2≤H(t)≤R(t)2M, (33)
+O(1
+t)+NK|Ω|1−γMγ≤R(t)2M
+t2. (34)Stabilites of Euler-Poisson Equations 5
+This gives
+lim
+t→∞infR(t)
+t≥/bracketleftBigg
+NKMγ−1
+|Ω|(γ−1)/bracketrightBigg1/2
+. (35)
+The proof is completed.
+Byapplyingthe similarmethod, in the abovesectionto the system, (1 ), with frictional damping
+constant ( β >0), the below theorem is followed:
+Theorem 5 Suppose (ρ,u)is a global classical solution with compact support in the sy stem, (1),
+with frictional damping (β >0). We have for N≥2,
+lim
+t→∞infR(t)
+t1/2≥/parenleftBigg
+2βNKMγ−1
+|Ω|(γ−1)/parenrightBigg1/2
+, (36)
+withR(t) = max x∈Ω(t){|x|}.
+Proof.ForN≥2, we get,
+·
+H(t) =/integraldisplay
+Ω2xρudx, (37)
+and
+··
+H(t) = 2/integraldisplay
+Ωx[−∇·(ρu⊗u)−∇P+ρ∇Φ−βρu]dx. (38)
+Additionally, it can be arranged as the following:
+··
+H(t) = 2/integraldisplay
+Ω[x−∇·(ρu⊗u)−∇P+ρ∇Φ]dx−β·
+H(t), (39)
+··
+H(t)+1
+β·
+H(t) = 2/braceleftbigg/integraldisplay
+Ω/bracketleftBig
+ρ|u|2+NP/bracketrightBig
+dx−N−2
+2/integraldisplay
+ΩρΦdx/bracerightbigg
+≥2/integraldisplay
+NPdx≥2NK|Ω|1−γMγ>0.
+(40)
+Therefore, we are able to obtain the inequality,
+C3+C4e−βt+2βNK|Ω|1−γMγt≤H(t)≤R(t)2M, (41)
+O(1
+t)+2βNK|Ω|1−γMγ≤R(t)2M
+t. (42)
+This gives
+lim
+t→∞infR(t)
+t1/2≥/parenleftBigg
+2βNKMγ−1
+|Ω|(γ−1)/parenrightBigg1/2
+. (43)
+The proof is completed.
+3 Non-existence of Collapsing Solution
+In this section, we explain the idea that there is no possibility to have a density collapsing solution,
+with compact support for the Euler-Poisson equations with repulsiv e forces:
+We restate their energy for γ >1 or the pressureless fluids is:
+E(t) =/integraldisplay
+Ω/parenleftbigg1
+2ρ|u|2+1
+γ−1P−1
+2ρΦ/parenrightbigg
+dx≥ −/integraldisplay
+Ω1
+2ρΦdx. (44)
+When a δ-shock exists for the density function ρ(t,x), the potential energy function, with N≥3,
+becomes to be infinite:
+−/integraldisplay
+ΩρΦdx=/integraldisplay
+Ωρ(t,x)/integraldisplay
+Ωρ(t,y)
+|x−y|N−2dydx= lim
+ǫ→0+/integraldisplay
+Ωδ(t,x)/integraldisplay
+Ωδ(t,y)
+ǫN−2dydx=∞. (45)6 M.W. Yuen
+WithN= 2, the situation is similar:
+−/integraldisplay
+ΩρΦdx=−/integraldisplay
+Ωδ(t,x)/integraldisplay
+Ωδ(t,y)ln|x−y|dydx=∞. (46)
+Therefore, the total energy of the δ-shock density solutions must be infinite. However, for the
+classical solutions with compact support, the initial energy is finite. By the energy estimate in
+Lemma 2, we have,
+E(t)≤E(0). (47)
+If the total energy is finite, it is impossible to obtain the density collap sing solutions. It is clear to
+have Theorem 1.
+References
+[1] F. Bouchut and G. Bonnaud, Numerical Simulation of Relativistic Plasmas in Hydrodyna mic
+Regime, ZAMM 76(1996), 287-290.
+[2] C. F. Chen, Introduction to Plasma Physics and Controlled Fusion , Plenum, New York (1984)
+[3] Y.B. Deng, T.P. Liu, T. Yangand Z.A. Yao, Solutions of Euler-Poisson Equations for Gaseous
+Stars, Arch. Rational Mech. Anal. 164(2002), 261-285.
+[4] R. T. Glassey, The Cauchy Problem in Kinetic Theory . Society for Industrial and Applied
+Mathematics (SIAM), Philadelphia, PA, 1996.
+[5] P.L. Lions, Mathematical Topics in Fluid Mechanics . Vols. 1, 2, 1998, Oxford: Clarendon
+Press, 1998.
+[6] T. Makino, On a Local Existence Theorem for the Evolution Equation of Ga seous Stars ,
+Patterns and waves, 459–479, Stud. Math. Appl., 18, North-Holla nd, Amsterdam, 1986.
+[7] T. Makino, Blowing up Solutions of the Euler-Poission Equation for the Evolution of the
+Gaseous Stars , Transport Theory and Statistical Physics 21(1992), 615-624.
+[8] T. Makino and B. Perthame, Sur les Solutions a symmetric spherique de lequation d’Eule r-
+poisson Pour levolution d’etoiles gazeuses,(French) [On R adially Symmetric Solutions of the
+Euler-Poisson Equation for the Evolution of Gaseous Stars] ,Japan J. Appl. Math. 7(1990),
+165-170.
+[9] B. Perthame, Nonexistence of Global Solutions to Euler-Poisson Equatio ns for Repulsive
+Forces, Japan J. Appl. Math. 7(1990), 363–367.
+[10] T.C. Sideris, Formation of Singularities in Three-dimensional Compress ible Fluids , Comm.
+Math. Phys. 101(1985), No. 4, 475-485.
+[11] M.W. Yuen, Blowup Solutions for a Class of Fluid Dynamical Equations in RN, J. Math.
+Anal. Appl. 329(2)(2007), 1064-1079.
+[12] M.W.Yuen, Analytical Blowup Solutions to the 2-dimensional Isotherm al Euler-Poisson Equa-
+tions of Gaseous Stars , J. Math. Anal. Appl. 341 (1)(2008),445-456.
+[13] M.W. Yuen, Stabilities for Euler-Poisson Equations in Some Special Di mensions , J. Math.
+Anal. Appl. 344(2008), no. 1, 145–156.
+[14] M.W. Yuen, Blowup for the Euler and Euler-Poisson Equations with Repul sive Forces , pre-
+print, arXiv:1001.0380v1.
\ No newline at end of file
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+arXiv:1001.0386v5  [math.AP]  20 Mar 2012UPPER SEMICONTINUOUS ATTRACTORS FOR 3D
+HYPERVISCOUS FLOW
+ABDELHAFID YOUNSI
+Abstract. We regularized the 3D Navier-Stokes equations by adding a hi gh-
+order viscosity term. We first prove the existence of the glob al attractors of the
+Leray-Hopf weak solutions of the regularized 3D Navier-Sto kes equations and
+then we study the upper semicontinuity, as the artificial dis sipation εgoes to
+0. We also consider applications of obtained results to the r egularized problem
+by allowing the family of forcing functions to vary with ε, forε >0.
+1.Introduction
+In this paper, we study the robustness, or upper semicontinuity o f the global
+attractors of the Leray-Hopf weak solutions of modified three dim ensional Navier-
+Stokes equations. We regularized the 3D Navier-Stokes system by adding a high
+order artificial viscosity term to the conventional system
+∂uε
+∂t+ε(−△)luε−ν△uε+(uε.∇)uε+∇p=f(x),in Ω×(0,∞)
+divuε= 0,in Ω×(0,∞),uε(x,0) =uε
+0,in Ω,
+p(x+Lei,t) =p(x,t), uε(x+Lei,t) =uε(x,t)i= 1,2,3, t∈(0,∞)(1.1)
+where Ω = (0 ,L)3with periodic boundary conditions and ( e1,...,ed) is the natural
+basis ofRd. Hereε >0 is the artificial dissipation parameter, uεis the velocity
+vector field, pis the pressure, ν >0 is the kinematic viscosity of the fluid and fis
+a given force field. For ε= 0, the model is reduced to 3D Navier–Stokes system.
+Mathematical model for such fluid motion has been used extensively in turbu-
+lence simulations (see e.g. [3, 4, 10]). For further discussion of theo retical results
+concerning (1.1 ), see [1, 2, 5, 12, 16, 17, 21, 23].
+In the work [23], the strong convergence of the solution of this pro blem to the
+solution of the conventional system as the regularization paramet er goes to zero,
+was established for each dimension d≤4.
+For the 3D Navier–Stokes system weak solutions of problem are kno wn to exist
+by a basic result by J. Leray from 1934 [11], only the uniqueness of we ak solutions
+remains as an open problem. Then the known theory of global attra ctors of infinite
+dimensional dynamical systems is not applicable to the 3D Navier–Sto kes system.
+The theory of trajectory attractors for evolution partial differ ential equations
+was developed in [15, 19], which the uniqueness theorem of solutions of the cor-
+responding initial-value problem is not proved yet, e.g. for the 3D Nav ier–Stokes
+system (see [8, 15, 18, 19]). Such trajectory attractor is a class ical global attractor
+2000Mathematics Subject Classification. 35-xx, 76Dxx, 76D05, 35D-xx, 35B41, 35B40.
+Key words and phrases. Navier Stokes equations; attractor; upper semicontinuity ;
+hyperviscosity.
+12 ABDELHAFID YOUNSI
+but in the space of weak solutions defined on [0 ,∞), with the corresponding semi-
+group being simply the translation in time of such solutions. A compact setA⋐E
+is said to be a global attractor of a semigroup {S(t),t >0}acting in a Banach or
+Hilbert space EifAis strictly invariant with respect to {S(t)}:S(t)A=A∀t≥0
+andAattracts any bounded set B⊂E:dist(S(t)B,A)→0 (t→ ∞) (see [14],
+[15], [18], [19], [21]).
+In this article, we study the upper semicontinuity, of the global att ractors of
+the Leray-Hopf weak solutions of a regularized 3D Navier-Stokes e quations, as
+the artificial dissipation εgoes to 0. While there exist other examples of such
+robustness in the literature of the Navier-Stokes equations, the specific emphasis
+on the regularized problem is new for the 3D Navier-Stokes equation s and is of
+interest. This would bean extension of the earlier work on Ou and Srit haran for
+the 2D Navier-Stokes equations, see references [16] and [17]. It is now known that
+there is a global attractor A0for the Leray-Hopf weak solutions of the 3D Navier-
+Stokes equations, see Sell [18, 19] and Chepyzhov [13].
+The main object of this paper to show that there is a global attract or, which one
+might denote by Aε, for the regularized problem (1.1 ), and that the family {Aε}is
+upper semicontinuous at ε= 0. Moreover, we can modify the argument described
+above so that the final result will have broader applicability by allowing the family
+of forcing functions fεto vary with ε, forε >0.
+The family of sets Aε, 0< ε≤1 is robust at A0, or is upper semicontinuous with
+respect to εatε0= 0, provided that, for every ε0>0, there is a neighborhood
+O(ε0) of 0∈Rand a neighborhood Nε0(A0) ofA0, such that Aε⊂Nε0(A0), for
+everyε∈O(ε0) withε >0, see (23.13) in [19].
+The paper is organized as follows. In Section 2, we present the relev ant math-
+ematical framework for the paper. In Section 3, we recall the defi nition of the
+trajectory attractor A0of the conventional 3-D Navier-Stokes equations. In Sec-
+tion 4, we study the regularized problem (see equation (1.1 )), then we show the
+existence of trajectory attractor Aε. In Section 5, we present the main result of
+this paper, that is, a theorem on the upper semicontinuity on the at tractors Aε.
+Finally, an application of our general results to the study of the rob ustness of the
+system (1.1 ) with a perturbed external force.
+2.Preliminary
+We denote by Hm
+per(Ω), the Sobolev space of L-periodic functions endowed with
+the inner product
+(u,v) =/summationtext
+|α|≤m(Dαu,Dαv)L2(Ω)and the norm /ba∇dblu/ba∇dblm=/summationtext
+|α|≤m(/ba∇dblDαu/ba∇dbl2
+L2(Ω))1
+2.
+We define the spaces Vmas completions of smooth, divergence-free, periodic, zero-
+average functions with respect to the Hm
+pernorms.V′
+mdenote the dual space of Vm
+andVdenote the space V0.
+We present the topology to be used for generating the neighborho od of robust-
+ness. Let Fany vector space. A metric d(f,g) onFis said to be invariant if
+d(f,g) =d(f−g,0) for all f,g∈F.
+A Fr´ echet space is a complete topological vector space whose top ology is induced
+by a translation invariant metric d(f,g). Given a Banach space X, with norm
+/ba∇dbl./ba∇dblXand 1≤p <∞, we denote by Lp
+loc[0,∞;X) the Fr´ echet space of mesurableUPPER SEMICONTINUOUS HYPERVISCOUS FLOW 3
+functions f: [0,∞)→Xthat are p-integrable over [0 ,T], for each 0 < T <∞,
+endow with the metric
+d(f,g) =∞/summationdisplay
+n=12−nmin(/ba∇dblf−g/ba∇dblLp(0, n;X),1).
+Wedenoteby Lp
+loc(0,∞;X)theFr´ echetspaceofmesurablefunctions f: (0,∞)→X
+that are p-integrable over [ t0,T], for each 0 < t0≤T <∞endow with the metric
+d(f,g) =∞/summationdisplay
+n=22−nmin(/ba∇dblf−g/ba∇dblLp(1
+n, n;X),1).
+Similarly for p=∞, we will let L∞
+loc(0,∞;X) denote the collection of all functions
+f: (0,∞)→Xwith the property that, for all τandTwith 0< T <∞, one has
+ess sup
+0<s<T/ba∇dblf/ba∇dblX<∞. We denote by C[0,∞;X) the space of strongly continuous
+functions from [0 ,∞) toX, endowed with the topologie of the uniform convergence
+over compact sets and by Cw[0,∞;X) the space of weakly continuous functions
+from [0,∞) toX. We denote by L∞C=L∞(R,X)∩C(R,X) the Fr´ echet space
+L∞Cendow with the L∞
+loc−topology, wich is the topology of uniform convergence
+on bounded sets.
+LetEbe a complete metric space with metric d. We write Brfor the open
+ball centre 0 ∈Eand radius r. The following quantity is called the Hausdorff
+(non-symmetric) semidistance from a set Xto a setYin a Banach space E
+distE(X,Y) = sup
+x∈Xinf
+y∈Y/ba∇dbl./ba∇dblE.
+LetMbe a subset of Eand letR+= [0,∞). A mapping σ=σ(u,t), where
+σ:M×[0,∞)→Mis said to be a semiflow on Mprovided the following hold
+1)σ(w,0) =w, for allw∈M.
+2) The semigroup property holds, i. e,
+σ((w,s),t) =σ(w,s+t) for all w∈Mands,t∈R+.
+3) The mapping σ:M×(0,∞)→Mis continuous.
+If in addition the mapping σ:M×[0,∞)→Mis continuous we will say that
+the semiflow is continuous at t= 0. Here we use t >0 in order that the Robustness
+Theorem 23.14 in [19] is valid, see Sell [19] and Hale [8]. For any u∈Mthe
+positive trajectory through uis defined as the set γ+(u) ={σ(t)u, t≥0}. For
+any setB⊂Mwe define the positive hull H+(B) and the omega limit set ω(B)
+as follows
+H+(B) =ClMγ+(B) andω(B) =∩τ≥0H+(σ(τ)B).
+IfA ⊂Eandε >0 we write
+Nε(A) ={z∈E,inf
+a∈Ad(z,a)< ε}.
+for the open ε−neighbourhood of A.
+We denote by Athe Stokes operator Au=−△uforu∈D(A).We recall that
+the operator Ais a closed positive self-adjoint unbounded operator, with
+D(A) ={u∈V0,Au∈V0}. We have in fact, D(A) =V2. The spectral theory of
+Aallows us to define the powers AlofAforl≥1,Alis an unbounded self-adjoint
+operator in V0with a domain D(Al) dense in V2⊂V0. We set here
+Alu= (−△)luforu∈D/parenleftbig
+Al/parenrightbig
+=V2l.4 ABDELHAFID YOUNSI
+The space D/parenleftbig
+Al/parenrightbig
+is endowed with the scalar product and the norm
+(u,v)D(Al)= (Alu,Alv),/ba∇dblu/ba∇dblD(Al)={(u,u)D(Al)}1
+2.
+Now define the trilinear form b(.,.,.) associated with the inertia terms
+b(u,v,w) =3/summationdisplay
+i,j=1/integraldisplay
+Ωui∂vj
+∂xiwjdx
+Recall that for usatisfying ∇.u= 0 we have
+b(u,u,u) = 0 and b(u,v,w) =−b(u,w,v). (2.1)
+Hereafter, ci∈N,will denote a dimensionless scale invariant positive constant
+which might depend on the shape of the domain. The trilinear form b(.,.,.) is
+continuous on Vm1(Ω)×Vm2+1(Ω)×Vm3(Ω),mi=1,2,3≥0
+|b(u,v,w)| ≤c0/ba∇dblu/ba∇dblm1/ba∇dblv/ba∇dblm2+1/ba∇dblw/ba∇dblm3, m3+m2+m1≥3
+2(2.2)
+see [6, 19]. The continuity property of the trilinear form enables us t o define (using
+Riesz representationtheorem) a bilinear continuous operator B(u,v);V2×V2→V′
+2
+will be defined by
+/an}b∇acketle{tB(u,v),w/an}b∇acket∇i}ht=b(u,v,w),∀w∈V2.
+We recall some inequalities that we will be using in what follows.
+Young’s inequality
+ab≤ǫ
+pap+1
+qǫq
+pbq,a,b,ǫ > 0,p >1,q=p
+p−1. (2.3)
+Poincar´ e’s inequality
+λ1/ba∇dblu/ba∇dbl2≤ /ba∇dblu/ba∇dbl2
+1for allu∈V0, (2.4)
+whereλ1is the smallest eigenvalue of the Stokes operator A.
+3.Navier-Stokes equations
+The conventional Navier-Stokes system can be written in the evolu tion form
+∂u
+∂t+νAu+B(u,u) =f, t >0,
+divu= 0,in Ω×(0,∞) andu(x,0) =u0,in Ω,(3.1)
+letf∈L∞(0,∞;V0) be given. We will say that a function uis a weak solu-
+tion of the 3D Navier-Stokes of Class LH(Leray–Hopf ) on [0 ,∞) provided that
+u(x,0) =u0(x)∈V0, and the following properties hold
+1)u∈L∞(0,∞;V0)∩L2
+loc[0,∞;V1).
+2)du
+dt∈[L4
+3
+loc0,∞;V′
+1).
+Taking the inner product of (3.1 ) with u, and using (2.3 ) we have
+d
+dt/ba∇dblu(t)/ba∇dbl2+2ν/ba∇dbl∇u/ba∇dbl2= 2/an}b∇acketle{tf,u/an}b∇acket∇i}ht. (3.2)
+by application of Young’s inequality and the Poincar´ e’s Lemma, yields
+d
+dt/ba∇dblu(t)/ba∇dbl2+ν/ba∇dbl∇u/ba∇dbl2≤/ba∇dblf/ba∇dbl2
+νλ1, (3.3)UPPER SEMICONTINUOUS HYPERVISCOUS FLOW 5
+using the Poincar´ e Lemma and Gronwall’s inequality, to get
+/ba∇dblu(t)/ba∇dbl2≤e−νλ1(t−t0)/ba∇dblu(t0)/ba∇dbl2+1
+ν2λ2
+1/ba∇dblf/ba∇dbl2/parenleftBig
+1−e−νλ1(t−t0)/parenrightBig
+,with 0< t0< t,
+3) which implies that
+/ba∇dblu(t)/ba∇dbl2≤e−νλ1(t−t0)/ba∇dblu(t0)/ba∇dbl2+1
+ν2λ2
+1/ba∇dblf/ba∇dbl2. (3.4)
+Integrating (3.2 ) over [ t0,t] we find that
+/ba∇dblu(t)/ba∇dbl2+2ν/integraldisplayt
+t0/ba∇dblA1
+2u(s)/ba∇dbl2ds≤ /ba∇dblu(t0)/ba∇dbl2+2/integraldisplayt
+t0/an}b∇acketle{tf(s),u(s)/an}b∇acket∇i}htds. (3.5)
+4) The function usatisfies the following equality
+/an}b∇acketle{tu(t)−u(t0),v/an}b∇acket∇i}ht+ν/integraldisplayt
+t0/an}b∇acketle{tA1
+2u(s),A1
+2v/an}b∇acket∇i}htds+/integraldisplayt
+t0/an}b∇acketle{tB(u(s),u(s)),v/an}b∇acket∇i}htds=/integraldisplayt
+t0/an}b∇acketle{tf,v/an}b∇acket∇i}htds,
+(3.6)
+for allv∈V1and for all t≥t0≥0.
+The proof of the following theorem is given in [12, 19, 22].
+Theorem 3.1. Letf∈V′
+1andu0∈V0be given. Then for every T >0, there
+exists a weak solution u(t)of (3.1 ) from the space L2(0,T;V1)∩L∞(0,T;V0), such
+thatu(x,0) =u0andu(t)satisfies the energy equality (3.6 ).
+Moreover (see [22]), u(.) is weakly continuous from [0 ,T] intoV0, the function
+u∈Cw([0,T];V0) and consequently u(x,0) =u0(x)∈V0.LetWis the set of
+all Leray–Hopf weak solutions u(.) of equation (3.1 ) in the space L∞(0,∞;V0)∩
+L2
+loc[0,∞;V1) that satisfy the following properties
+•du
+dt∈L4
+3
+loc(0,∞;V′
+1);
+•for almost all tandt0, witht > t0>0, inequalities (3.5 ,3.6 ) are valid.
+LetX0denote the Fr´ echet space used to define the Leray-Hopf weak s olutions.
+Thus
+ϕ∈X0=L∞(0,∞;V0)∩L2
+loc[0,∞;V1),
+whereϕ∈Cw[0,∞;V0) and we let F0denote a compact, translation invariant set
+of forcing functions fin
+L∞C=L∞/parenleftbig
+R,L2(Ω)/parenrightbig
+∩C/parenleftbig
+R,L2(Ω)/parenrightbig
+where the topology on the Fr´ echet space L∞Cis the topology of uniform conver-
+gence on bounded sets in R.
+Then, we use the Leray-Hopf solutions of the 3D Navier-Stokes eq uations with
+ε= 0 to generate a semiflow π0onF0×X0, where
+π0(τ)(f,ϕ) =/parenleftbig
+fτ,S0(f,τ)ϕ/parenrightbig
+forτ≥0,
+fτ(t) =f(τ+t) andu(t) =S0(f,t)ϕis the Leray-Hopfsolution ofthe 3DNavier-
+Stokes equations that satisfies u(0) =S0(f,0)ϕ=ϕ(0). By using the theory of
+generalized weak solutions, as in Sell [18] or [19] , we note that π0has a trajectory
+attractor A0⊂F0×X0see Theorem 65.12 in [19], and Chepyzhov [13, 14].6 ABDELHAFID YOUNSI
+4.The regularized Navier-Stokes system
+Using the operators defined in the previous section, we can write th e modified
+system (1.1 ) in the evolution form
+∂tuε+εAluε+νAuε+B(uε,uε) =f(x),in Ω×(0,∞)
+divuε= 0,in Ω×(0,∞),uε(x,0) =uε
+0,in Ω.(4.1)
+The existence and uniqueness results for initial value problem (1.1) c an be found
+in J. L. Lions [12, Remark 6.11].
+In three dimensions, the following theorem collects the main result in t his work
+Theorem 4.1. Forl≥5
+4,ε >0fixed,f∈L2(0,T;V′
+0)anduε
+0∈V0be given.
+There exists a unique weak solution of (4.1)which satisfies uε∈L2(0,T;Vl)∩
+L∞(0,T;V0),∀T >0.
+We introduce the following result of the convergence of uεas the regularized
+parameter ε→0
+Theorem 4.2. Forl≥3
+2,ε >0fixed,f∈L2(0,T;V′
+0)anduε
+0∈V0be given.
+i) There exists a unique weak solution of (4.1)which satisfies
+uε∈L2(0,T;Vl)∩L∞(0,T;V0),∀T >0.
+ii) This weak solution uεconverges strongly in L2(0,T;V0)asε→0toua weak
+solution of the Navier-Stokes equations.
+The above theorem is established directly by using of a general resu lt [23, The-
+orem 3.9.]. For ε >0, we let πεdenote the semiflow on F0×X0generated by the
+weak solutions of regularized 3D Navier-Stokes equations of (4.1 ). Thus
+πε(τ) = (fτ,Sε(f,τ)ϕ), (4.2)
+whereuε
+0=ϕand
+uε(t) =Sε(f,t)ϕ=Sε(f,t)uε
+0 (4.3)
+is the weak solution of (4.1 ) that satisfies ϕ(0) =uε
+0.
+Regarding the existence of the attractor Aεwhenε >0, we use especially the
+related papers of Chepyzhov and Vishik, such as [15, 13] to show t hat the system
+(4.1 ) possesses a global attractor. For ε >0, we consider the trajectory space Kε
+of the modified Navier-Stokes equations (4.1 ). Kεis the union of all weak solutions
+uε∈X0that satisfy (4.1 ), see (6.163) in [12].
+Using the described scheme in [15], we construct the spaces Sb
+Sb={v(.)∈L∞(0,T;V0)∩L2
+b(0,T;V1),∂tv(.)∈L2
+b(0,T;D(Al
+2)′)}
+with norm
+/ba∇dblv/ba∇dblSb=/ba∇dblv/ba∇dblL2
+b(0,T;V1)+/ba∇dblv/ba∇dblL∞(0,T;V0)+/ba∇dbl∂tv/ba∇dblL2
+b(0,T;D(Al
+2)′)
+where
+/ba∇dblv/ba∇dblL2
+b(0,T;V1)= sup
+t≥0(/integraldisplayt+1
+t/ba∇dblv(s)/ba∇dbl2
+1ds)1
+2,/ba∇dblv/ba∇dblL∞(0,T;V0)= esssup
+t≥0/ba∇dblv/ba∇dbl
+and
+/ba∇dbl∂tv/ba∇dblL2
+b(0,T;V′
+l)= sup
+t≥0(/integraldisplayt+1
+t/ba∇dblv(s)/ba∇dbl2
+V′
+lds)1
+2.UPPER SEMICONTINUOUS HYPERVISCOUS FLOW 7
+We need a topology in the space Kε. We define on X0the following sequential
+topology which we denote Γ.
+By definition, a sequence of functions {vn} ⊆X0converges to a function v∈X0
+in the topology Γ as n→ ∞if, for any T >0,vn→vweakly in L2(0,T;V1);
+vn→vweak-∗inL∞(0,T;V0) andvn→vstrongly in L2(0,T;V0), asn→ ∞.
+We consider the topology Γ on Kε. It is easy to prove that the space Kεis closed
+in Γ. From the definition of Kε, it follows that πεKε⊂ Kεfor allt≥0.
+Corollary 4.3. Ifuε(t)is a solution of (4.1 ), then the following inequalities hold
+for allt >0
+/ba∇dbluε(t)/ba∇dbl2≤e−νλ1t/ba∇dbluε
+0/ba∇dbl2+/ba∇dblf/ba∇dbl2
+ν2λ2
+1, (4.4)
+/integraldisplayt+1
+t/ba∇dbluε(s)/ba∇dbl2ds≤e−νλ1t
+νλ1/ba∇dbluε
+0/ba∇dbl2+/ba∇dblf/ba∇dbl2
+ν2λ2
+1, (4.5)
+ν/integraldisplayt+1
+t/ba∇dbluε(s)/ba∇dbl2
+1ds≤e−νλ1t
+νλ1/ba∇dbluε
+0/ba∇dbl2+/ba∇dblf/ba∇dbl2
+ν2λ2
+1+/ba∇dblf/ba∇dbl2
+νλ1. (4.6)
+Proof.Taking the inner product of (4.1 ) by uε, we obtain
+d
+dt/ba∇dbluε/ba∇dbl2+2ε/ba∇dblAl
+2uε/ba∇dbl2+2ν/ba∇dbl∇uε/ba∇dbl2= 2(f,uε). (4.7)
+Applying Young’s inequality and using the Poincar´ e Lemma, we obtain
+d
+dt/ba∇dbluε/ba∇dbl2+ν/ba∇dbl∇uε/ba∇dbl2≤/ba∇dblf/ba∇dbl2
+νλ1. (4.8)
+Using the Gronwall’s inequality over [0 , t], we obtain (4.4 ). Integrating (4.4 ) over
+[t,t+1] we find (4.5 ). Integrating (4.8 ) over [ t,t+1] we find
+ν/integraldisplayt+1
+t/ba∇dbl∇uε(s)/ba∇dbl2ds≤/ba∇dblf/ba∇dbl2
+νλ1+/ba∇dbluε(t)/ba∇dbl2. (4.9)
+Applying inequality (4.4 ), we get (4.6 ). /square
+We recall the following result
+Lemma 4.4. Letf∈L2(0,T;V′
+1), then, for any solution uε(t)of problem (1.1)
+the time derivativeduε
+dtis uniformly bounded in L2(0,T;V′
+l).
+A simple consequence of Lemma 3.6 [23] is the following Corollary
+Corollary 4.5. Letf∈L2(0,T;V′
+1), then any solution uε(t)of (4.1 ) satisfies
+/integraldisplayt+1
+t/ba∇dbl∂tuε(s)/ba∇dbl2
+D(Al
+2)′ds≤C3, (4.10)
+C3is a positive constant independent of ε.
+Moreover, due to estimates (4.4 ) and (4.10 ), we also have the unifo rm estimate.
+Proposition 4.6. Letf∈L2(0,T;V′
+1), then any solution uε(t)of (4.1 ) satisfies
+the inequality
+/ba∇dblπε(uε)/ba∇dbl2
+Sb≤c7e−νλ1t
+νλ1/ba∇dbluε(0)/ba∇dbl2+c7/ba∇dblf/ba∇dbl2
+ν2λ2
+1+C4 (4.11)
+where the positive constant C4is independent of ε.8 ABDELHAFID YOUNSI
+Proposition 4.7. Forl≥3
+2andf∈L∞Ca time independent functions, πεis a
+continuous family of semiflows on X0.
+Proof.Sinceuε∈L2(0,T;Vl) andduε
+dt∈L2(0,T;V′
+l),uεis almost everywhere
+equal to an uniform continuous function from [0 ,T] to the space V0. The continuity
+ofuεis a direct consequence of [22, Lemma 1.4. ChIII, Sec1].
+From the result of the strong convergence, there exists ε1>0, such that
+/ba∇dbluε(t)−uε0(t)/ba∇dbl ≤ǫ,∀ǫ≥0,for each ε≤ε1, (4.12)
+it follows from (4.12 ) that lim
+ε→ε0/ba∇dblπε(t)−πε0(t)/ba∇dblgoes to 0, for all 0 ≤t≤T.
+This shows that πεis continuous semiflow on X0andπεapproximates π0uni-
+formly for tin compact sets in [0 ,∞). /square
+From Proposition 4.6 it follows that Kε⊂ Sbfor allε >0 and for all τ >0.
+Also Proposition 4.6 implies that the semigroup πεhas absorbing set in Kεfor
+allε >0 and for all τ >0 (We note, that this absorbing set does not depend
+onε, since the constant C4in (4.11 ) is independent of ε), bounded in Sband
+inequality (4.11 ) implies that absorbing set is compact in Γ. The continu ity of
+πεis proved. These facts are sufficient to state that πεhas a global attractor Aε.
+Such that Aε⊂F0×X0, bounded in Sband compact in Γ. For a more detailed,
+see [13, 14, 15].
+5.Upper semicontinuity of attractors
+We now prove the robustness property for the global attractor Aε. We have
+shownin Proposition4.7the continuityofthe family ofsemiflows πεonX0. Having
+done this, We can simply invoke Theorem 23.14 in [19] to complete the pr oof of the
+robustness for the family of attractors Aεatε= 0. We denote by
+BR={uε(x,t),0≤t,0< ε <1;/ba∇dblπε(uε)/ba∇dbl2
+Sb≤R}
+withR=c7e−νλ1t
+νλ1/ba∇dbluε(0)/ba∇dbl2+c7/ba∇dblf/ba∇dbl2
+ν2λ2
+1+C4,uε(x,t)isafamilyofsolutionsofsystem
+(4.1 ), and the norms of uε(x,t) inSbare uniformly bounded, see Proposition 4.7.
+Itis sufficientto showthatasmall δ−neighbourhoodofattractor A0isanabsorbing
+set andπεapproximates π0onBRuniformly for all tin [0,∞), see [7, 19].
+Theorem 5.1. Forl≥3
+2, forε >0the family of semiflows πεgenerated by
+the weak solutions of the regularized 3D Navier-Stokes equa tions (4.1 ) admits a
+compact attractor {Aε,0< ε≤1}which attracts bounded sets of V0and is contained
+in the absorbing balls BRwhereRis independent of ε. Moreover, dX0(Aε,A0)→0,
+asε→0.
+Proof.SinceA0is a global attractor, for any bounded set BR0={u(x,0)∈
+V,/ba∇dblu(x,0)/ba∇dbl ≤R0} ⊂V, we have
+dX0/parenleftbig
+π0BR0,A0/parenrightbig
+→0, ast→ ∞. (5.1)
+Thus, there exists δ >0 such that
+dX0/parenleftbig
+π0BR0,A0/parenrightbig
+≤δ
+2,fort≥tδ. (5.2)
+Consequently
+π0(t)BR0⊂Nδ(A0), fort≥tδ, (5.3)UPPER SEMICONTINUOUS HYPERVISCOUS FLOW 9
+whereNδ(A0)be theδ-neighborhoodof A0. Thisshowsthat Nδ(A0) isanabsorbing
+set. Since πεapproximates π0uniformly for all t≥0, then for any δ >0, there are
+ε1>0 andt0≥0 such that
+πε(BR∩BR0)⊂Nδ(A0), for 0< ε < ε 1, t≥t0. (5.4)
+Since the attractor Aεis contained in BR∩BR0, we have
+πε(Aε)⊂Nδ(A0), forε≤ε1, t≥t0. (5.5)
+SinceAεis an invariant set, we deduce that
+Aε⊂Nδ(A0), for 0< ε < ε 1, t≥t0. (5.6)
+Moreover, since δis arbitrary, we obtain the upper semicontinuity of Aε, atε0= 0
+dX0(Aε,A0)→0,asε∈O(ε0). (5.7)
+/square
+One can modify the argument described above so that the final res ult will have
+broader applicability by allowing the family of forcing functions to vary withε,
+forε >0. Thus, we consider the regularized Navier-Stokes system (4.1 ) w ith a
+perturbed external force fεin place of f, forε >0. Then (4.1 ) becomes
+∂tuε+εAluε+νAuε+B(uε,uε) =fε(x),in Ω×(0,∞)
+divuε= 0,in Ω×(0,∞),uε(x,0) =uε
+0,in Ω.(5.8)
+We show that the trajectory attractor of the perturbed syste m (5.8 ) coincides with
+the trajectory attractor Aεof the unperturbed system (3.1 ). Our results rely on
+the work of Hale ([15]) who show that the limit behaviour is valid even thr oughFε,
+whereFεdenote a compact, translation invariant set of perturbed forcing functions
+to vary with ε, forε >0 and satisfy the condition
+ω(H+(fε)) =ω(H+(f)). (5.9)
+Thus we would use Fεin place of F0, forε >0. Moreover, by using a metric don
+theL∞C-toplogy, see [19] for some samples, we can note that (5.9 ) is equiva lent
+to saying that for every δ >0 there is an ε1>0 andTδ=T(δ)≥0 such that
+dX0(fε,F0)≤δ,for 0< ε≤ε2andfε∈Fε(5.10)
+for anyt≥Tδ, that is
+Fε⊂Nδ/parenleftbig
+F0/parenrightbig
+, for 0< ε≤ε2, (5.11)
+whereNδdenotes the δ-neighborhood of F0inL∞C. The resulting argument for
+robustness will then depend on two parameters λ= (ε,δ), where λ→(0,0).
+The following statement generalizes Theorem 5.1
+Theorem 5.2. Under the above conditions, the trajectory attractor of the perturbed
+3D Navier-Stokes system (5.8 ) coincides with the trajector y attractor Aεof the
+non-perturbed system (3.1 ). Moreover, the perturbed attra ctor of (5.8 ) is upper
+semicontinuous with respect to εatε= 0.
+Proof.The existence of trajectory attractor Aεis treated above. The proof follows
+from formulas (5.9 ), (5.11 ) and Theorem 5.1 . /square
+Proposition 4.7 can be used to extend the 2D result of Ou and Srithar an[16] to
+l-lplacian with l >1.10 ABDELHAFID YOUNSI
+References
+[1] J. Avrin, The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model
+of 3D Turbulent Flow. J. Dynam. Diff. Eqns. 20(2008), 1-40.
+[2] J. Avrin, Singular initial data and uniform global bound s for the hyperviscous Navier–Stokes
+equations with periodic boundary conditions. J. Diff. Eq. 19 0(2003), 330-351.
+[3] P. Bartello, O. Metais and M. Lesieur, Coherent structur es in rotating three-dimentional
+turbulence, J. Fluid Mech., 1994, vol. 273, pp. 1-29.
+[4] C. Basdevant, B. Legras, R. Sadourny, M. B´ eland, A study of barotropic model flows: inter-
+mittency, waves and predictability, J. Atmos. Sci. 38 (1981 ) 2305–2326.
+[5] M. Cannone and G. Karch, About the regularized Navier-St okes equations, Journal of Math-
+ematical Fluid Mechanics 7, No. 1 (2005), 1 - 28.
+[6] P. Constantin, C. Foias, Navier-Stokes Equations, The U niversity of Chicago Press, Chicago,
+1988.
+[7] J.K. Hale, X. B.Lin, and G. Raugel. Upper semicontinuity of attractors forapproximations of
+semigroups and partial differential equations. Mathematic s of Computation, 50(181):89-123,
+1988.
+[8] J. K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Mono-
+graphs n◦25, Amer. Math. Soc, Providence, R.I., 1988.
+[9] O. A. Ladyzhenskaya, Nonstationary Navier-Stokes equa tions. Amer. Math. Soc. Transl., Vol.
+25 (1962) pp. 151-160.
+[10] B. Legras, G. David, Dritschel, A comparison of the cont our surgery and pseudo-spectral
+methods, J. Comput. Phys. 104 (2) (1993) 287–302.
+[11] J. Leray, Sur le mouvement d’un liquide visqueux emplis sant l’espace., Acta Mathematica,
+63 (1934), pp. 193–248.
+[12] J. L. Lions, Quelques M´ ethodes de R´ esolution des Prob l` emes aux Limites Non Lin´ eaires,
+Dunod Gauthier-Villars, Paris, 1969.
+[13] V. Chepyzhov, E. S. Titi , M. Vishik On the convergence of solutions of the Leray-alpha
+model to the trajectory attractor of the 3D Navier-Stokes sy stem. Discrete and Continuous
+Dynamical Systems. 17. 2007. N.3. P.481-500.
+[14] V. V. Chepyzhov and M. I. Vishik, Attractors for Equatio ns of Mathematical Physics. AMS
+Colloquium Publications. V. 49. Providence: AMS, 2002.
+[15] V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C. R. Acad.
+Sci. Paris, Series I, 321 (1995), 1309-1314.
+[16] Y. Ou and S. S. Sritharan, Upper Semicontinuous Global A ttractors for Viscous Flow, Jour-
+nal: Dynamic Systems and Applications 5 (1996), 59-80.
+[17] Y. U. Ou and S. S. Sritharan, Analysis Of Regularized Nav ier-Stokes Equations I, Quart.
+Appl. Math. 49, 651-685 (1991).
+[18] G. Sell, Global attractors for the three-dimensional N avier–Stokes equations. J.Dyn. Diff.
+Eq.,8, 1–33 (1996).
+[19] G. Sell and Y.You, Dynamics of Evolutionary Equations, Springer-Verlag, 68, New york,
+2002.
+[20] S. S. Sritharan, Deterministic and stochastic control of Navier-Stokes equation with linear,
+monotone, and hyperviscosities, Appl. Math. Optim. 41 (2) ( 2000) 255–308.
+[21] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Ph ysics, Applied
+Mathematical Sciences Series, 68, New york, Springer-Verl ag, 2nd ed. 1997.
+[22] R. Temam, Navier-Stokes Equations . North-Holland Pub. Company, Amsterdam, 1979.
+[23] A. Younsi; Effect of hyperviscosity on the Navier-Stoke s turbulence, Electron. J. Diff. Equ.,
+Vol. 2010(2010), No. 110, pp. 1-19.
+Department of Mathematics and Computer Science , University of Djelfa , Algeria.
+E-mail address :younsihafid@gmail.com
\ No newline at end of file
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+arXiv:1001.0387v1  [math.DG]  3 Jan 2010Finitnessofthe basicintersectioncohomologyof aKilling foliation
+Martintxo Saralegi-Aranguren∗
+Universit´ e d’ArtoisRobertWolak†
+Uniwersytet Jagiellonski
+November 18,2018
+Abstract
+We prove that the basic intersection cohomology I H∗
+p(M/F), whereFis the singular foliation
+determinedbyanisometricactionofaLiegroup Gonthecompactmanifold M,isfinitedimensional.
+This paper deals with an action Φ:G×M→Mof a Lie group on a compact manifold preserving
+a riemannian metric on it. The orbits of this action define a si ngular foliationFonM. Putting together
+the orbits of the same dimension we get a stratification of M. This structure is still very regular. The
+foliationFis in fact a conical foliation and we can define the basicinter section cohomology I H∗
+p(M/F)
+(cf. [10]). Thisinvariantbecomesthebasiccohomology H∗(M/F)whentheactionΦisalmostfree,and
+theintersectioncohomology I H∗
+p(M/G)when theLiegroup Giscompact.
+The aim of this work is to prove that this cohomology I H∗
+p(M/F)is finite dimensional. This result
+generalizes [ 3](almostfreecase), [ 11](abelian case)and [ 10](compactcase).
+The paper is organized as follows. In Section 1 we present the foliationF. The basic intersection
+cohomology I H∗
+p(M/F)associatedtothisfoliationisstudiedinSectiontwo. Twis tedproductsarestudied
+inSection 3. Thefinitenessof I H∗
+p(M/F)isprovedinSection 4.
+Inthesequel Misaconnected,secondcountable,Haussdor ff,withoutboundaryandsmooth(ofclass
+C∞)manifoldofdimension m. Allthemapsareconsidered smoothunlesssomethingelseis indicated.
+1 Killingfoliationsdeterminedbyisometricactions.
+Westudyinthisworkthefoliationsinducedbyisometricact ions: theKillingfoliations . Thesefoliations
+areexamplesoftheconicalfoliationsforwhichthebasicin tersectioncohomologyhasbeendefined(see
+[10,11]). Wepresent thisgeometricalframework inthissection.
+∗Univ Lille Nord de France F-59 000 LILLE, FRANCE. UArtois, La boratoire de Math´ ematiques de Lens EA 2462.
+F´ ed´ eration CNRS Nord-Pas-de-Calais FR 2956. Facult´ e de s Sciences Jean Perrin. Rue Jean Souvraz, S.P. 18. F-62 300
+LENS,FRANCE. saralegi@euler.univ-artois.fr .
+†Instytut Matematyki. Uniwersytet Jagiellonski. Stanisla wa Lojasiewicza 6, 30 348 KRAKOW, POLAND.
+robert.wolak@im.uj.edu.pl. PartiallysupportedbytheKBN grant2 PO3A02125.
+1Finiteness ...November18, 2018˙ 2
+1.1 Killing foliations . A smooth action Φ:G×M→Mof a Lie group Gon a manifold Mis a
+isometricaction whenthereexistsariemannianmetric µonMpreserved by G.
+The connected components of the orbits of the action Φdetermine a partition FonM. In fact, this
+partition is a singular riemannian foliation that we shall c allKilling foliation (cf. [7]). Notice thatF
+is also a conical foliation in the sense of [ 10,11]. Classifying the points of Mfollowing the dimension
+of the leaves ofFone gets the stratification SFofF. It is determined by the equivalence relation
+x∼y⇔dimGx=dimGy.Theelementsof SFarecalled strata.
+Intheparticularcasewheretheclosureof Gintheisometrygroupof( M,µ)isacompactLiegroup1
+we shall say that the action Φis atame action . In fact, a smooth action Φ:G×M→Mis tame if and
+onlyifit extendsto a smoothaction Φ:K×M→MwhereKis acompact Liegroup containing G(cf.
+[6]). The group Kis not unique, but we always can choose Kin such a way that Gis dense in K. We
+shallsay that Kis atamergroup . Herethestrataof SFareK-invariant closedsubmanifoldsof M.
+Since theaim ofthis work is thestudy of Fand not theactionΦitself, we can consider that theLie
+groupGisconnected. Let usseethat.
+Proposition1.1.1 LetΦ:G×M→M is a tame action. Let G 0be the connected component of G
+containingtheunityelement. The Killingfolationdefined b ytherestrictionΦ:G0×M→M isalsoF.
+Proof.ThepartitionFisdefined by thisequivalencerelation:
+x∼y⇐⇒∃continuouspathα: [0,1]→G(x) such thatα(0)=xandα(1)=y.
+Since the map∆:G→G(x), defined by∆(g)=Φ(g,x)=g·x, is a submersion (see for example [ 2])
+then
+x∼y⇐⇒∃continuouspathβ: [0,1]→Gsuchthatβ(0)=eandβ(1)·x=y,
+and bydefinitionof G0
+x∼y⇐⇒∃continuouspathβ: [0,1]→G0such thatβ(0)=eandβ(1)·x=y.
+Thisgivestheresult. ♣
+WhenGisconnected, thetamergroup Khas richerproperties.
+Proposition1.1.2 LetG beaconnectedLiesubgroupofacompactLiegroupK. IfG i sdenseinK then
+G⊳K andthequotientgroup K /G iscommutative.
+Proof.The Lie algebragis AdG- invariant and hence, by density, Ad K-invariant. Thengis an ideal of
+k. Theconnectednessof Ggivesthat Gisanormalsubgroupof K. SinceAd Gacts triviallyonk/g, AdK
+acts trivially,too. Therefore, k/gis abelian(see forexample[ 8, pag. 628]). ♣
+1.2 Particular tame actions. Atriois a triple ( K,G,H), withKis a compact Lie group, Ga normal
+subgroup of KandHa closed subgroup of K. We present now some tame actions associated to a trio
+(K,G,H). Theyare goingto beintensivelyusedin thiswork. First of allweneed somedefinitions.
+- TheactionΦl:K×K→KisdefinedbyΦl(g,k)=g·k. Foreachelement uoftheLiealgebrakof
+K, we shall write Xuthe associated (right invariant) vector field. It is defined b yXu(k)=TeRk(u)
+whereRk:K→Kisgivenby Rk(ℓ)=ℓ·k.
+1Thisisalwaysthe casewhenthemanifold Misa compact.Finiteness ...November18, 2018˙ 3
+- The actionΦr:K×K→Kis defined byΦr(g,k)=k·g−1. For each element u∈kofK, we
+shall write Xuthe associated (left invariant) vector field. It is defined by Xu(k)=−TeLk(u) where
+Lk:K→Kisgivenby Lk(ℓ)=k·ℓ.
+- The actionΨ:K×K/H→K/His defined byΨ(g,kH)=(g·k)H. For each element u∈k,
+we shall write Yuthe associated vector field. Since the canonical projection π:K→K/His a
+K-equivariant map,thenwehave π∗Xu=Yuforeachu∈k.
+- TheactionΓ:H×H→Hisdefined byΓ(g,h)=g·h. Foreachelement uoftheLiealgebrahof
+HwewriteZutheassociated(rightinvariant)vectorfield.
+Theassociatedactionsweare goingtousearethefollowing.
+(a)TherestrictionΦl:G×K→K, whichinducestheregularKillingfoliation K.
+(b)The restrictionΦr:G×K→K, whichinduces theregularKillingfoliation K.
+SinceG⊳K, the foliationKis determined by thefamily ofvectorfields {Xu/u∈g}, wheregis theLie
+algebra of G, and also by the family {Xu/u∈g}. The orbits G(k)=Gk=kGhave the same dimension
+dimG.
+(c)TherestrictionΨ:G×K/H→K/H, which inducestheregularKillingfoliation D.
+ThefoliationDisdeterminedbythefamilyofvectorfields {Yu/u∈g}. Theorbits G(kH)havethesame
+dimensiondim G−dim(G∩H).
+(d)The restrictionΓ: (G∩H)×H→H, whichinducestheregularKillingfoliation C.
+ThefoliationCis determinedby thefamily ofvectorfields {Zu/u∈g∩h}. Theorbits( G∩H)(k) have
+thesamedimensiondim( G∩H).
+(e)TherestrictionΦr:GH×K→K, which inducestheregularKillingfoliation E.
+Notice that GHis a Lie group since Gis normal in K. The foliationEis, in fact, determined by the
+vectorfields{Xu/u∈g+h}. Theorbits( GH)(k)havethesamedimensiondim G+dimH−dim(G∩H).
+1.3 Twisted product. In order to prove the finiteness of the basic intersection coh omology we de-
+composethemanifoldinafinitenumberofsimplerpieces. The searethetwistedproductsweintroduce
+now.
+We fix a trio ( K,G,H) and a smooth action Θ:H×N→NofHon the manifold N. Thetwisted
+productis the quotient K×HNofK×Nby the equivalencerelation ( k,z)∼(k·h−1,Θ(h,z)=h·z).The
+elementof K×HNcorrespondingto( k,z)∈K×Nisdenotedby<k,z>. Thismanifoldisendowedwith
+thetameaction
+Φ:G×(K×HN)−→(K×HN),
+defined byΦ(g,<k,z>)=<g·k,z>. We denotebyWtheinducedKillingfoliation.
+Wealso usethefollowingtameaction,namely,therestricti on
+Θ: (G∩H)×N→N
+whoseinduced Killingfoliationis denotedby N.
+Thecanonical projection Π:K×N→K×HNrelatestheinvolvedfoliationsasfollows:
+(a)Π∗(K×I)=W, whereIis thepointwisefoliation(sincethemap ΠisG-equivariant).
+(b)SW={Π(K×S)/S∈SN}=Π/parenleftbig{K}×SN/parenrightbig(sinceG<k,z>=k(G∩H)zk−1).Finiteness ...November18, 2018˙ 4
+2 BasicIntersectioncohomology
+In this section we recall the definition of the basic intersec tion2cohomology and we present the main
+properties we are going to use in this work. For the rest of thi s section, we fix a conical foliation F
+defiedonamanifold M. Theassociatedstratificationis SF. Theregularstratumofisdenotedby RF. We
+shallwrite m=dimM,r=dimFands=m−r=codim MF.
+Wearegoingtodealwithdi fferentialformsonaproduct(manifold) ×[0,1[p,theyarerestrictionsof
+differentialformsdefined on (manifold) ×]−1,1[p.
+2.1 Perverse forms. Recall that a conical chart is a foliated diffeomorphismϕ: (Rm−n−1×cSn,H×
+cG)→(U,FU) where (Rm−n−1,H) is a simple foliation and ( Sn,G) is a conical foliation without 0-
+dimensional leaves. We also shall denote this chart by ( U,ϕ,S) whereSis the stratum of SFverifying
+ϕ(Rm−n−1×{ϑ})=U∩S.
+ThedifferentialcomplexΠ∗
+F(M×[0,1[p)ofperverseforms ofM×[0,1[pisintroducedbyinduction
+ondepthSF. Whenthisdepth is0 then
+Π∗
+F(M×[0,1[p)=Ω∗(M×[0,1[p).
+Consider now the generic case. A perverse form of M×[0,1[pis first of all a differential form
+ω∈Ω∗(RF×[0,1[p)suchthat,
+thepull-back ( ϕ×I[0,1[p)∗ω∈Ω∗/parenleftig
+Rm−n−1×RG×]0,1[×[0,1[p/parenrightig
+extendsto ωϕ∈Π∗
+H×cG/parenleftig
+Rm−n−1×Sn×[0,1[p+1/parenrightig
+for any conical chart ( U,ϕ), whereI•stands for the identity map. Notice that Ω∗(M)is included on
+Π∗
+F(M)3.
+2.2 Perverse degree. The amount of transversality of a perverse form ω∈Π∗
+F(M)with respect to a
+singularstratum S∈SFis measured by the perverse degree ||ω||S. We recall here the definition of local
+perversedegree||ω||U∈{−∞}∪Nofωrelativelyto aconical chart ( U,ϕ,S):
+1.||ω||U=−∞whenωϕ≡0 onRm−n−1×RG×{0},
+2.||ω||U≤p, withp∈N, whenωϕ(v0,...,vp,−)≡0 wherethevectors {v0,...,vp}are tangent to the
+fibers ofPϕ:Rm−n−1×RG×{0}−→U∩S4.
+Thisnumberdoesnotdependonthechoiceoftheconicalchart (cf. [11,Proposition1.3.1]). Finally,we
+definethe perversedegree||ω||Sby
+||ω||S=sup/braceleftig
+||ω||U/(U,ϕ,S)conical chart/bracerightig
+.
+Theperversedegreeof ω∈Ω∗(M)verifies||ω||S≤0 foranysingularstratum S∈SF(cf.2.1).
+2We referthereaderto [ 10],[11]fordetails.
+3Throughtherestriction ω/mapsto→ωRF.
+4Themap Pϕ:Rm−n−1×Sn×[0,1[−→Uisdefinedby Pϕ(x,y,t)=ϕ(x,[y,t]).Finiteness ...November18, 2018˙ 5
+2.3 Basic cohomology . The basic cohomology of the foliation Fis an important tool to study its
+transversal structure and plays the rˆ ole of the cohomology of the orbit space M/F, which can be a wild
+topologicalespace. Adi fferentialformω∈Ω∗(M)isbasicifiXω=iXdω=0,foreach vectorfield Xon
+Mtangent tothefoliation F. Forexemple,a function fisbasicifffis constanton theleaves of F. We
+shallwriteΩ∗(M/F)forthecomplexofbasicforms. Itscohomology H∗(M/F)isthebasiccohomology
+of(M,F). We alsousethe relativebasiccohomology H∗((M,F)/F), that is,thecohomologycomputed
+from thecomplexofbasicformsvanishingon thesaturatedse tF⊂M. Thebasiccohomologydoesnot
+usethestratification SF.
+2.4 Basic intersection cohomology . Aperversity is a map p:Sσ
+F→Z∪{−∞,∞}, whereSσ
+Fis the
+familyofsingularstrata. The constantperversity ιis defined byι(S)=ι, whereι∈Z∪{−∞,∞}.
+Thebasicintersectioncohomologyappearswhenoneconside rsbasicperverseformswhoseperverse
+degreeiscontrolledby aperversity. Weshallput
+Ω∗
+p(M/F)=/braceleftig
+ω∈Π∗
+F(M)/ωisbasicand max/parenleftig
+||ω||S,||dω||S/parenrightig
+≤p(S)∀S∈Sσ
+F/bracerightig
+the complexof basicperverse forms whoseperverse degree (a nd that of thetheir derivative)is bounded
+by the perversity p. The cohomology I H∗
+p(M/F)of this complex is the basic intersection cohomology5
+of(M,F)relativelyto theperversity p.
+Consideratwistedproduct K×HN. Perversities on K×HNandK×Nare determinateby perversities
+onNby theformula(cf. 1.3(b)):
+(1) p(K×S)=p(Π(K×S))=p(S).
+2.5 Mayer-Vietoris . Thisisthetechniqueweuseinordertodecomposethemanifo ldinnicerpieces.
+Anopencovering{U,V}ofMbysaturatedopensubsetsisa basiccovering . Itpossessesasubordinated
+partition of the unity made up of basic functions defined on M(see [9]). For a such covering we have
+theMayer-Vietorisshortsequence
+0→Ω∗
+p(M/F)→Ω∗
+p(U/F)⊕Ω∗
+p(V/F)→Ω∗
+p((U∩V)/F)→0,
+wherethemaparedefinedby ω/mapsto→(ω,ω)and(α,β)/mapsto→α−β. Thethirdmapisontosincetheelementsof
+thepartitionoftheunityare controlledfunctions ,idest, elementsof Ω0
+0(−)(cf.2.2). Thus,thesequence
+isexact. Thisresultisnot longertrueformoregeneral cove rings.
+Weshallusein thiswork thetwofollowinglocalcalculation s(see[11, Proposition3.5.1and Propo-
+sition3.5.2]fortheproofs).
+Proposition2.6 Let(Rk,H)be a simple foliation. Consider p a perversity on M and define the per-
+versityp onRk×M byp(Rk×S)=p(S). The canonical projection pr :Rk×M→M induces the
+isomorphism
+I H∗
+p(M/F)/simequalI H∗
+p/parenleftig
+Rk×M/H×F/parenrightig
+.
+Proposition2.7 LetGbe a conical foliation without 0-dimensional leaves on the s phereSn. A per-
+versityp on cSngives the perversity p onSndefined by p(S)=p(S×]0,1[). The canonical projection
+pr :Sn×]0,1[→Sninducestheisomorphism
+I Hi
+p(cSn/cG)=/braceleftiggI Hi
+p(Sn/G)ifi≤p({ϑ})
+0ifi>p({ϑ}).
+5BICforshort.Finiteness ...November18, 2018˙ 6
+In thenextsectionweshallneed thefollowingtechnicalLem ma.
+Lemma 2.8 LetΦ:K×M→M be a smooth action, where K is a compact Lie group, and let V be a
+fundamentalvector field of thisaction. Considera normalsu bgroupG of K and write Ftheassociated
+conicalfoliationon M. Then,theinterioroperatori V:Ω∗
+p(M/F)−→Ω∗−1
+p(M/F)iswelldefined,forany
+perversity p.
+Proof.Since the question is a local one, then it su ffices to consider where Mis a twisted product
+K×HN6. Notice that the blow up Π:K×N→K×HNis aK-equivariant map relatively to the action
+ℓ·(k,z)=(ℓ·k,z). This givesΠ∗(Xu,0)=Vfor some u∈k. From Lemma 3.1we know that it suffices
+toprovethattheoperator
+i(Xu,0):Ω∗
+p(K×N/K×N)−→Ω∗−1
+p(K×N/K×N)
+is well defined. Since G⊳Kthen the vector field Xupreserves the foliation K. So, it suffices to prove
+thattheoperator
+i(Xu,0):Ω∗
+p(K×N)−→Ω∗−1
+p(K×N)
+is well defined. This comes from the fact that Xuacts on the K-factor while the perversion conditions
+aremeasured on the N-factor (cf. ( 1)). ♣
+3 TheBICofa twistedproduct
+WecomputenowtheBIC ofatwistedproduct K×HN(cf.1.3)foraperversity p(cf. (1)).
+Lemma 3.1 Thenaturalprojection Π:K×N→K×HN induces thedifferentialmonomorphism
+(2) Π∗:Ω∗
+p/parenleftbigK×HN/W/parenrightbig−→Ω∗
+p(K×N/K×N).
+Moreover, givena di fferentialformωon K×HRW, we have:
+(3) Π∗ω∈Ω∗
+p(K×N/K×N)⇐⇒ω∈Ω∗
+p/parenleftbigK×HN/W/parenrightbig.
+Proof.Noticethattheinjectivityof Π∗comesfromthefactthat Πisasurjection. Fortherest,weproceed
+inseveralsteps.
+(a)Afoliatedatlasfor π:K→K/H.
+Sinceπ:K→K/His aH-principal bundle then it possesses an atlas A=/braceleftigϕ:π−1(U)−→U×H/bracerightig
+made up with H-equivariant charts: ϕ(k·h−1)=(π(k),h·h0) ifϕ(k)=(π(k),h0). We study the foliation
+ϕ∗K. Thisequivarianceproperty gives ϕ∗Xu=(0,Zu)for each u∈g∩h. Thus, thetrace ofthefoliation
+ϕ∗Kon the fibers of the canonical projection pr : U×H→UisC. On the otherhand, sincethe map π
+is aG-equivariant map then π∗K=D, which gives pr∗ϕ∗K=D. We concludethat ϕ∗K⊂D×C . By
+dimensionreasons weget ϕ∗K=D×C. TheatlasAisanH-equivariantfoliatedatlas of π.
+(b)A foliatedatlasfor Π:K×N→K×HN.
+6Infact,Nisaneuclideanspace RaetΘisanorthogonalaction.Finiteness ...November18, 2018˙ 7
+We claim thatA#=/braceleftigϕ:π−1(U)×HN−→U×N/(U,ϕ)∈A/bracerightig
+is a foliated atlas of K×HNwhere the
+mapϕisdefinedbyϕ(<k,z>)=(π(k),(Θ((ϕ−1(π(k),e))−1·k,z))). Thismapisadi ffeomorphismwhose
+inverseisϕ−1(u,z)=<ϕ−1(u,e),z>. It verifies
+ϕ∗W1.3(a)===ϕ∗Π∗(K×I)=ϕ∗Π∗(ϕ−1×IN)∗(D×C×I ).
+A straightforwardcalculation shows ϕ◦Π◦(ϕ−1×IN)=(IU×Θ). SinceCisdefined by theaction Γthen
+Θ∗(C×I)=N. Finallyweobtain ϕ∗W=D×N.
+(c)LastStep.
+Given(U,ϕ)∈A#,wehavethecommutativediagram
+U×H×N
+Q
+/d15/d15ϕ−1×IN/d47/d47K×N
+Π
+/d15/d15
+U×Nϕ−1
+/d47/d47K×HN
+whereQ(u,h,z)=(u,h−1·z),Π−1(Imϕ−1)=Im/parenleftigϕ−1×IN/parenrightig
+andtherowsarefoliatedimbeddings. Now,
+since(2)and(3)are local questionsthenit su ffices toprovethat
+-Q∗:Ω∗
+p(U×N/D×N)−→Ω∗
+p(U×H×N/D×C×N )iswell-defined, and
+-Q∗ω∈Ω∗
+p(U×H×N/D×C×N )⇐⇒ω∈Ω∗
+p(U×N/D×N), foranyω∈Ω∗(U×RN).
+Thiscomesfrom thefact thatthemap
+∇: (U×H×N,D×C×N )−→(U×H×N,D×C×N ),
+defined by∇(u,h,z)=(u,h,h−1·z)), is a foliated diffeomorphism7andQ=pr0◦∇, with pr0:U×H×
+N→U×Ncanonical projection(cf. Proposition 2.6). ♣
+3.2 The Lie algebra k.We suppose in this paragraph that that Gis also dense on K. Chooseνa
+bi-invariantriemannianmetricon K, whichexistsby compactness. Consider
+B=/braceleftig
+u1,...ua,ua+1,...,ub,ub+1,...,uc,uc+1,...,uf/bracerightig
+an orthonormal basis of the Lie algebra kofKwith{u1,...ub}basis of the Lie algebra gofGand
+{ua+1,...uc}basis of the Lie algebra hofH. For each indice 1 ≤i≤fwe shall write Xi≡Xuiand
+Xi≡Xui(cf.1.2).
+Letγi∈Ω1(K)be the dual form of Xi, that is,γi=iXiν. Notice thatδij=γj(Xi). These forms are
+invariant by the left action of K. Since the flow of Xjis the multiplication on the left by exp( tuj) then
+LXjγi=0 foreach 1≤j≤f.
+For the differential, we have the formula dγl=/summationdisplay
+1≤i<j≤fCl
+ijγi∧γj,where [Xi,Xj]=f/summationdisplay
+l=1Cl
+ijXl,and 1≤
+i,j,l≤f. We have several restrictions on these coe fficients. Since G⊳Kthengis an ideal ofkand
+thereforewehave
+Cl
+ij=0 fori≤b<l.
+7SinceG∩H⊳H.Finiteness ...November18, 2018˙ 8
+SinceK/Gisanabeliangroup(cf. Proposition 1.1.2)thentheinducedbracketon k/giszeroandtherefore
+wehave
+Cl
+ij=0 forb<i,j,l≤f.
+Theseequationsimplythat
+(4) dγl=0 foreach b<l.
+TheE-basic differential forms in/logicalandtext∗(γ1,...,γ f) are exactly/logicalandtext∗(γc+1,...,γ f) since they are cycles
+and thefamily{X1,...,Xc}generates thefoliation E. Thisgives
+(5) H∗/parenleftig
+K/slashig
+E/parenrightig
+=/logicalanddisplay∗
+(γc+1,...,γ f).
+3.3 Two actions of H/H0.The Lie group Hpreserves the foliation Nsince the Lie group G∩His a
+normalsubgroupof H. PutH0theconnectedcomponentof Hcontainingtheunityelement. Sinceitisa
+connected compactLiegroupthen astandardargument showst hat
+(6)/parenleftig
+I H∗
+p(N/N)/parenrightigH0=H∗/parenleftbigg/parenleftig
+Ω.
+p(N/N)/parenrightigH0/parenrightbigg
+=I H∗
+p(N/N)
+(cf. [5,TheoremI,Ch. IV,vol. II]).Weconcludethatthefinitegrou pH/H0actsnaturallyon I H∗
+p(N/N).
+SinceH0is a connected Lie subgroup of GHthen/parenleftig
+H∗(K/E)/parenrightigH0=H∗(K/E).We conclude that the
+finitegroup H/H0acts naturallyon H∗(K/E).
+Proposition3.4 Let(K,G,H)bea triowithG connected anddensein K. Then
+I H∗
+p/parenleftbigK×HN/W/parenrightbig=/parenleftig
+H∗(K/E)⊗I H∗
+p(N/N)/parenrightigH/H0.
+Proof.Usingtheblowup Π:K×N→K×HN,thecomputationof I H∗
+p/parenleftbigK×HN/W/parenrightbigcanbedonebyusing
+the complex Im/braceleftig
+Π∗:Ω∗
+p/parenleftbigK×HN/F/parenrightbig−→Ω∗
+p(K×N/K×N)/bracerightig
+(cf. Lemma 3.1). We study this complex
+inseveralsteps. We fix B=/braceleftig
+u1,...,uf/bracerightig
+an orthonormalbasis of kas in3.2.
+/a\}b∇acketle{ti/a\}b∇acket∇i}htDescriptionofΩ∗(K×RN).
+A differentialformω∈Ω∗(K×RN)isoftheform
+(7) η+/summationdisplay
+1≤i1<···<iℓ≤fγi1∧···∧γiℓ∧ηi1,...,iℓ,
+where the formsη,ηi1,...,iℓ∈Ω∗(K×RN)verifyiXjη=iXjηi1,...,iℓ=0 for each 1≤j≤fand each
+1≤i1<···<iℓ≤f.
+/a\}b∇acketle{tii/a\}b∇acket∇i}htDescriptionofΠ∗
+K×N(K×N).
+Sincethefoliation Kisregularthenwealwayscanchooseaconicalchartofthefor m(U1×U2,ϕ1×
+ϕ2) where (U1,ϕ1) is a foliated chart of ( K,K) and (U2,ϕ2) is a conical chart of ( N,N). The local blow
+up ofthechart ( U1×U2,ϕ1×ϕ2)is constructed from thesecond factorwithoutmodifyingth efirst one.
+So, the differential formsγiare always perverse forms and a di fferential formω∈Π∗
+K×N(K×N)is of
+the form ( 7) whereη,ηi1,...,iℓ∈Π∗
+K×N(K×N)verifyiXjη=iXjηi1,...,iℓ=0 for each 1≤j≤fand each
+1≤i1<···<iℓ≤f.Finiteness ...November18, 2018˙ 9
+/a\}b∇acketle{tiii/a\}b∇acket∇i}htDescriptionofΩ∗(K×RN/K×N).
+Takeω∈Ω∗(K×RN/K×N). SinceKis generated by the family {Xj/1≤j≤b}thenLXjω=0
+for any 1≤j≤b, or equivalently, R∗
+gω=ωfor each g∈GsinceGis connected. By density,
+R∗
+kω=ωfor each k∈Kand therefore LXjω=0 for any 1≤j≤fsinceKis connected. We
+conclude that LXjη=LXjηi1,...,iℓ=0 for any 1≤j≤fand each 1≤i1<···<iℓ≤f. This gives
+ω∈/logicalandtext∗(γ1,...,γ f)⊗Ω∗(RN).
+TheN-basic differential forms ofΩ∗(RN)are exactlyΩ∗(RN/N). TheK-basic differential forms of/logicalandtext∗(γ1,...,γ f)are exactly/logicalandtext∗(γb+1,...,γ f)(cf. (4)). Fromthesetwofacts, weget
+Ω∗(K×RN/K×N)=/logicalanddisplay∗(γb+1,...,γ f)⊗Ω∗(RN/N)
+as differentialgraduatecommutativealgebras.
+/a\}b∇acketle{tiv/a\}b∇acket∇i}htDescriptionofΩ∗
+p(K×N/K×N).
+From/a\}b∇acketle{tii/a\}b∇acket∇i}htand/a\}b∇acketle{tiii/a\}b∇acket∇i}htitsuffices to controltheperversedegreeoftheforms
+η+/summationdisplay
+b+1≤i1<···<iℓ≤fγi1∧···∧γiℓ∧ηi1,...,iℓ∈/logicalanddisplay∗(γb+1,...,γ f)⊗Π∗
+N(N).
+Consider Sa stratum of SN. From||γi||K×S=0 and||η||K×S=||η||S, we get||γi1∧...γiℓ∧ηi1,...,iℓ||K×S=
+||ηi1,...,iℓ||S. We concludethat
+Ω∗
+p(K×N/K×N)/simequal/logicalanddisplay∗(γb+1,...,γ f)⊗Ω∗
+p(N/N)
+(cf.1.3(b)).
+/a\}b∇acketle{tv/a\}b∇acket∇i}htDescriptionof Im/braceleftig
+Π∗:Ω∗
+p/parenleftbigK×HN/F/parenrightbig−→Ω∗
+p(K×N/K×N)/bracerightig
+.
+We denote by{Wa+1,...,Wc}the fundamental vector fields of the action Θ:H×N→Nasso-
+ciated to the basis {ua+1,...,uc}. Consider now the action Υ:H×(K×N)→(K×N) defined
+byΥ(h,(k,z))=(k·h−1,Θ(h,z)). Its fundamental vector fields associated to the basis {ua+1,...,uc}
+are{(Xa+1,Wa+1),...,(Xc,Wc)}. Given h∈H, we takeΥh:K×N→K×Nthe map defined by
+Υh(k,z)=Υ(h,(k,z)). Then, wehave
+ImΠ∗=ω∈/logicalanddisplay∗(γb+1,...,γ f)⊗Ω∗
+p(N/N)/slashigg(i)iXiω=−iWiωifa<i≤c}
+(ii)LXiω=−LWiωifa<i≤c},
+(iii)(Υh)∗ω=ωforh∈H..
+LetH0betheunityconnected componentof H. Recall thatthesubgroup H0is normalin Hand thatthe
+quotientH/H0is a finite group. Conditions (ii) gives that ωisH0-invariant. So, condition (iii) can be
+replaced by: (iv)(Υh)∗ω=ωforh∈H/H0.Therefore
+ImΠ∗=ω∈/logicalanddisplay∗(γb+1,...,γ f)⊗Ω∗
+p(N/N)/slashigg(i)iXiω=−iWiωifa<i≤c}
+(ii)LXiω=−LWiωifa<i≤c}.H/H0
+.
+Sincethegroup H/H0isafiniteone,wegetthatthecohomology H∗(ImΠ∗)isisomorphicto/parenleftig
+H∗(A·)/parenrightigH/H0,
+whereA∗isthedifferentialcomplex
+ω∈/logicalanddisplay∗(γb+1,...,γ f)⊗Ω∗
+p(N/N)/slashigg(i)iXiω=−iWiωifa<i≤c}
+(ii)LXiω=−LWiωifa<i≤c}.
+So, itremains tocompute H∗(A·). Thiscomputationcan besimplifiedbyusingthesethreefact s:Finiteness ...November18, 2018˙ 10
+-iWiω=LWiω=0 foreach a<i≤b,sincethefoliation Nisdefined by theaction of G∩H.
+-iXiγj=δijforalli,j(cf.3.2).
+-dγj=0 forb<j(cf. (4)).
+Weget that A∗isthedifferentialcomplex
+ω∈/logicalanddisplay∗(γb+1,...,γ f)⊗Ω∗
+p(N/N)/slashigg(i)iXiω=−iWiωifb<i≤c}
+(ii)0=LWiωifb<i≤c}=
+/logicalanddisplay∗(γc+1,...,γ f)⊗ω∈/logicalanddisplay∗(γb+1,...,γ c)⊗Ω∗
+p(N/N)/slashigg(i)iXiω=−iWiωifb<i≤c}
+(ii)0=LWiωifb<i≤c}
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipupright
+B∗.
+Astraightforwardcomputationgivesthatthecanonicalwri tingofaformω∈/logicalandtext∗(γb+1,...,γ c)⊗Ω∗
+p(N/N)
+verifying(i)is
+(8) ω=ω0+/summationdisplay
+b<i1<···<iℓ≤c(−1)ℓγi1∧···∧γiℓ∧(iWiℓ···iWi1ω0)
+forsomeω0∈Ω∗
+p(N/N)(cf. Lemma 2.8).
+Considernow b<i,j≤c. SinceK/Gisanabeliangroup(cf. Proposition 1.1.2)andHisaLiegroup
+then [Wi,Wj]=b/summationdisplay
+l=a+1Cl
+ijWl. Then,i[Wi,Wj]ω0=0 sincethe foliation Nis defined by the action of G∩H.
+So,thecanonicalwritingofaform ω∈B∗is(8)forsomeω0∈/braceleftig
+η∈Ω∗
+p(N/N)/LWiη=0ifb<i≤c/bracerightig
+=/parenleftig
+Ω∗
+p(N/N)/parenrightigH0.
+Then,theoperator∆:B∗−→/parenleftig
+Ω∗
+p(N/N)/parenrightigH0,definedby∆(ω)=ω0,isadifferentialisomorphism. We
+concludethatthedi fferentialcomplex A∗isisomorphicto/logicalandtext∗(γc+1,...,γ f)⊗/parenleftig
+Ω∗
+p(N/N)/parenrightigH0andtherefore
+H∗(A·)/simequalH∗(K/E)⊗I H∗
+p(N/N)(cf. (5) and (6)). Since the operator ∆is (H/H0)-equivariant (cf. 3.3)
+thenweget
+I H∗
+p/parenleftbigK×HN/W/parenrightbig=H∗(ImΠ∗)=/parenleftig
+H∗(A·)/parenrightigH/H0=/parenleftig
+H∗(K/E)⊗I H∗
+p(N/N)/parenrightigH/H0.
+Thisendstheproof. ♣
+3.5 Remarks.
+(a) When the Lie group Gis commutative then Kis also commutative. Di fferential formsγ•are
+K-invariants ontheleft and ontheright,so/parenleftig
+H∗(K/E)/parenrightigH=H∗(K/E)and therefore
+I H∗
+p/parenleftbigK×HN/W/parenrightbig=H∗(K/E)⊗/parenleftig
+I H∗
+p(N/N)/parenrightigH/H0=H∗(K/E)⊗/parenleftig
+I H∗
+p(N/N)/parenrightigH
+as ithas been provedin [ 11,Proposition3.8.4].
+(b) Sincethefoliation Eisariemannianfoliationdefined onacompactmanifoldthenw eknowthat
+thecohomology H∗(K/E)isfinite(cf. [ 4]). So,thefinitenessof I H∗
+p/parenleftbigK×HN/W/parenrightbigdependsonthefiniteness
+ofI H∗
+p(N/N).Finiteness ...November18, 2018˙ 11
+4 FinitenessoftheBIC
+We provein this section that the BIC of a Killingfoliation on a compact manifold is finite dimensional.
+First of all, we present two geometrical tools we shall use in the proof: the isotropy type stratification
+and theMolino’sblowup.
+Wefixanisometricaction Φ:G×M→Monthecompactmanifold M. WedenotebyFtheinduced
+Killing foliation. For the study of Fwe can suppose that Gis connected (see Lemma 1.1.1). We fix
+Ka tamer group. Notice that the group Gis normal in Kand the quotient K/Gis commutative (cf.
+Proposition 1.1.2).
+4.1 Isotropy type stratification. Theisotropytype stratification SK,MofMis defined by the equiva-
+lencerelation8:
+x∼y⇔Kxisconjugatedto Ky.
+When depth SK,M>0, any closed stratum S∈SK,Mis aK-invariant submanifold of Mand then it
+possesses a K-invariant tubular neighborhood ( T,τ,S,Rm) whose structural group is O(m). Recall that
+thereare thefollowingsmoothmaps associatedwiththisnei ghborhood:
++Theradius mapρ:T→[0,1[ defined fiberwise from the assignation [ x,t]/mapsto→t. Eacht/nequal0 is a
+regularvalueofthe ρ. Thepre-imageρ−1(0)isS. Thismap is K-invariant, thatis,ρ(k·z)=ρ(z).
++Thecontraction H :T×[0,1]→Tdefined fiberwisely from ([ x,t],r)/mapsto→[x,rt]. The restriction
+Ht:T→Tis an embedding for each t/nequal0 andH0≡τ. We shall write H(z,t)=t·z. This map is
+K-invariant, thatis, t·(k·z)=k·(t·z).
+Thehyper-surface D=ρ−1(1/2)isthetubeofthetubularneighborhood. It isa K-invariant submanifold
+ofT. Noticethat themap
+∇:D×[0,1[−→T,
+defined by∇(z,t)=(2t)·zisaK-equivariant smoothmap,where Kacts triviallyonthe[0 ,1[-factor. Its
+restriction∇:D×]0,1[−→T\SisaK-equivariant diffeomorphism.
+DenoteSminthe union of closed (minimal) strata and choose Tmina disjoint family of K-invariant
+tubularneighborhoodsoftheclosedstrata. Theunionofass ociatedtubesisdenotedby Dmin. Noticethat
+theinducedmap∇min:Dmin×]0,1[−→Tmin\SminisaK-equivariantdiffeomorphism.
+4.2 Molino’sblow up . TheMolino’blowup[ 7]ofthefoliationFproducesanewfoliation /hatwideFofthe
+samekindbutofsmallerdepth. Wesupposedepth SK,M>0. TheblowupofMisthecompactmanifold
+/hatwideM=/braceleftig/parenleftig
+Dmin×]−1,1[/parenrightig/coproductdisplay/parenleftig
+(M\Smin)×{−1,1}/parenrightig/bracerightig/slashig
+∼,
+where(z,t)∼(∇min(z,|t|),t/|t|),andthemapL:/hatwideM−→Mdefined by
+L(v)=∇min(z,|t|) ifv=(z,t)∈Dmin×]−1,1[
+z ifv=(z,j)∈(M\Smin)×{−1,1}.
+Notice thatLis a continuous map whose restriction L:/hatwideM\L−1(Smin)→M\Sminis aK-equivariant
+smoothtrivial2-covering.
+8FornotionsrelatedwithcompactLie groupactions,we refer thereaderto [ 1].Finiteness ...November18, 2018˙ 12
+Since the map∇minisK-equivariant thenΦinduces the action /hatwideΦ:K×/hatwideM−→/hatwideMby saying that
+the blow-upLisK-equivariant. The open submanifolds L−1(Tmin) andL−1(Tmin\Smin) are clearly K-
+diffeomorphicto Dmin×]−1,1[andDmin×(]−1,0[∪]0,1[)respectively.
+Therestriction/hatwideΦ:G×/hatwideM−→/hatwideMisanisometricactionwith Kasatamergroup. TheinducedKilling
+foliation is/hatwideF. FoliationsFand/hatwideFare related byLwhich is a foliated map. Moreover, if Sis a not
+minimalstratumof SK,Mthenthereexistsanuniquestratum S′∈SK,/hatwideMsuchthatL−1(S)⊂S′. Thefamily
+{S′/S∈SK,M}covers/hatwideMand verifies therelationship: S1≺S2⇔S′
+1≺S′
+2. We conclude theimportant
+property
+(9) depth SK,/hatwideM<depthSK,M.
+4.3 Finiteness of a tubular neighborhood . We suppose depth SK,M>0. Consider a closed stratum
+S∈SK,M. Take(T,τ,S,Rm)aK-invariant tubularneighborhood. Wefixabasepoint x∈S. Theisotropy
+subgroup Kxactsorthogonallyonthefiber Rm=τ−1(x). So,theinducedaction Λx:Gx×Rm→Rmisan
+isometricaction,itgivestheKillingfoliation NonRm.
+Proposition4.3.1 IftheBICof (Rm,N)isfinitedimensionalthentheBICof (T,F)isalsofinitedimen-
+sional.
+Proof.Weproceed in twosteps.
+(a)Ky=Kxforeach y∈S.
+The canonical projection π:S→S/Kis an homogeneous bundle with fiber K/Kx. For any open
+subsetV⊂S/Kthe pull backτ−1π−1(V) is aK-invariant subset of T, then we can apply the Mayer-
+Vietoristechnicsto thiskind ofsubsets(cf. 2.5).
+Sincethemanifold S/Kisacompactonethenwecanfindafinitegoodcovering {Ui/i∈I}ofit(cf.
+[2]). An inductive argument on the cardinality of Ireduces the proof of the Lemma to the case where
+T=τ−1π−1(V), whereVisacontractibleopensubsetof S/K.
+Here, themanifold TisK-equivalentlydiffeomorphicto V×/parenleftig
+K×KxRm/parenrightig
+,whereKdoesnotactonthe
+first factor. So, the natural retraction of Vto a point gives a K-equivariant retraction of Tto the twisted
+productK×KxRm. Nowtheresultcomesdirectly from 3.5(b)since( K,G,Kx)isa trio.
+(b)Generalcase .
+The stratum SisK-equivariantly diffeomorphic to the twisted product K×N(Kx)FwhereN(Kx) is the
+normalizer of KxonKandF=SKx. So, the tubular neighborhood TisK-equivariantly diffeomorphic
+to thetwistedproduct K×N(H)NwhereNisthe manifoldτ−1(F). The previouscase givesthat theBIC of
+(N,FN)isfinitedimensional. Nowtheresultcomesdirectly from 3.5(b)since( K,G,N(Kx))isatrio.♣
+Themainresultofthiswork isthefollowing
+Theorem 4.4 TheBICofthefoliationdeterminedbyanisometricactionon acompactmanifoldisfinite
+dimensional.
+Proof.LetFbe a Killing foliation defined on a compact manifold Minduced by an isometric action
+Φ:G×M→MwhereGis a Lie group. Without loss of generality we can suppose that the Lie group
+Gis a connected one (cf. Lemma 1.1.1). We fix a tamer group K. We know that Gis normal in Kand
+thequotientgroup K/Giscommutative(cf. Proposition 1.1.2).
+Let usconsiderthefollowingstatementFiniteness ...November18, 2018˙ 13
+A(U,F)=“TheBIC I H∗
+p(U/F)is finitedimensionalforeach perversity p,”
+whereU⊂Mis aK-invariant submanifold. We prove A(M,F) by induction on dim M. The result is
+clear when dim M=0. We supposeA(W,F) for any K-invariant compact submanifold WofMwith
+dimW<dimMand weproveA(M,F). Weproceed inseveral steps.
+Firststep: 0-depth. Letussupposedepth SK,M=0. SinceG⊳KandKxisconjugatedto KythenGx
+is conjugated to Gy,∀x,y∈M. We get that the foliation Fis a (regular) riemannian foliation (cf. [ 7]).
+ItsBIC is justthebasiccohomology(cf. 2.3). ThenA(M,F)comesfrom [ 4].
+Secondstep: Inside M.Letussupposedepth SK,M>0. Thefamily/braceleftbigM\Smin,Tmin/bracerightbigisabasiccovering
+ofMandtheweget theexact sequence(cf. 2.5)
+0→Ω∗
+p(M/F)→Ω∗
+p/parenleftbig/parenleftbigM\Smin/parenrightbig/F/parenrightbig⊕Ω∗
+p/parenleftbigTmin/F/parenrightbig→Ω∗
+p/parenleftbig/parenleftbigTmin\Smin/parenrightbig/F/parenrightbig→0.
+TheFiveLemmagives
+A(Tmin\Smin,F),A(Tmin,F) andA(M\Smin,F)=⇒A(M,F).
+SinceTmin\SminisK-diffeomorphic to Dmin×]0,1[ (cf. (cf. 4.1)) thenA(Dmin,F)=⇒A(Tmin\Smin,F).
+Theinequalitydim Dmin<dimMgives
+A(Tmin,F) andA(M\Smin,F)=⇒A(M,F).
+In order to proveA(Tmin,F) it suffices to proveA(T,F) where ( T,τ,S,Rm) aK-invariant tubular
+neighborhoodofclosed stratum SofSK,M. FollowingProposition 4.3.1wehave
+A(Rm,N)=⇒A(T,F)=⇒A(Tmin,F).
+Consider the orthogonal decomposition Rm=Rm1×Rm2, whereRm1=(Rm)Gx. The only fixed point
+of therestrictionΛx:Gx×Rm2→Rm2is theorigin. So, there existsa Killingfoliation9Gon the sphere
+Sm2−1with(Rm1×Rm2,F)=(Rm1×cSm2−1,I×cG). Propositions 2.6and2.7give:
+A(Sm2−1,G)=⇒A(Rm1×cSm2−1,I×cG)=⇒A(Rm,N).
+Finally,sincedimSm2−1<m≤dimT≤dimMwehave
+(10) A(M\Smin,F)=⇒A(M,F).
+Thirdstep: Blow-up. Letussupposedepth SK,M>0. Thefamily/braceleftig
+L−1(M\Smin),L−1(Tmin)/bracerightig
+isabasic
+coveringof/hatwideMand thewegettheexact sequence(cf. 2.5)
+0→Ω∗
+p/parenleftig/hatwideM//hatwideF/parenrightig
+→Ω∗
+p/parenleftig
+L−1(M\Smin)//hatwideF/parenrightig
+⊕Ω∗
+p/parenleftig
+L−1(Tmin)//hatwideF/parenrightig
+→Ω∗
+p/parenleftig
+L−1(Tmin\Smin)//hatwideF/parenrightig
+→0.
+Following 4.2 wehavethat
+-L−1/parenleftbigM\Smin/parenrightbigisK-diffeomorphictotwo copiesof M\Smin,
+-L−1/parenleftbigTmin/parenrightbigisK-diffeomorphicto Dmin×]−1,1[,
+-L−1/parenleftbigTmin\Smin/parenrightbigisK-diffeomorphicto Dmin×(]−1,0[∪]0,1[).
+9It isgivenbytheorthogonalaction Λx:Gx×Sm2−1→Sm2−1.Finiteness ...November18, 2018˙ 14
+Now,theFiveLemmagives
+A(Dmin,/hatwideF)andA/parenleftig/hatwideM,/hatwideF/parenrightig
+=⇒A(M\Smin,F).
+But, theinequalitydim Dmin<dimMgives
+(11) A/parenleftig/hatwideM,/hatwideF/parenrightig
+=⇒A(M\Smin,F).
+Forth step: Final blow-up. When depth SK,M=0 we getA(M,F) from the First step. Let us
+supposedepth SK,M>0. From ( 10)and(11)weget
+A/parenleftig/hatwideM,/hatwideF/parenrightig
+=⇒A(M,F).
+withdepth SK,/hatwideM<depthSK,M(cf. (9)). By iteratingthisprocedureweget
+A/parenleftig/tildewideM,/tildewideF/parenrightig
+=A/hatwider···
+/hatwiderM,/hatwide···
+/hatwideF=⇒···=⇒A/parenleftig/hatwideM,/hatwideF/parenrightig
+=⇒A(M,F),
+withdepth SK,/tildewideM=0. Wefinish theproofby applyingagaintheFirst Step. ♣
+References
+[1] G. E. Bredon: Introduction to Compact Transformation Groups - Pure Appl. Math. 46. Academic
+Press. 1972.
+[2] G.E. Bredon: Topologyand Geometry -Graduate TextinMath. 139.Springer. 1993.
+[3] A. El Kacimi and G. Hector D´ ecomposition de Hodge basique pour un feuilletage rieman nien.-
+Ann.Inst.Fourier 36(1988),207-227.
+[4] A.ElKacimi,V.Sergiescu andG.Hector Lacohomologiebasiqued’unfeuilletageriemannienest
+dedimensionfinie -Math.Z. 188(1985),593-599.
+[5] W.Greub,S.HalperinandR.Vanstone: Connections,curvatureandcohomology -PureandAppl.
+Math.,vol.II, AcademicPress, NewYork and London,1972.
+[6] S. Kobayashi: Transformation groups in di fferential geometry . Classic in Mathematics. Springer,
+1995.
+[7] P. Molino: RiemannianFoliations -Progress in Math. 73.Birkh¨ auser, 1988.
+[8] D.Poguntke: DenseLiegrouphomomorphisms - J.Algebra 169(1994),625-647.
+[9] J.I.RoyoPrieto,M.Saralegi-ArangurenandR. Wolak: Tautnessforriemannianfoliationsonnon-
+compactmanifolds .-Manus.Math. 126(2008)177-200.
+[10] M. Saralegi-Aranguren and R. Wolak: The BIC of a conical fibration. - Mat. Zametki 77(2005),
+235–257.Translationin Math.Notes 77(2005),213-231.
+[11] M.Saralegi-Aranguren and R. Wolak: The BICof asingularfoliationdefined byanabeliangroup
+ofisometries. -Annal. Pol.Math. 89(2006)203-246.
\ No newline at end of file
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+arXiv:1001.0388v3  [math.AG]  2 Oct 2013TheGysinsequence
+forS3-actionsonmanifolds∗.†.‡
+Jos´eIgnacio RoyoPrieto§
+University of the Basque Country UPV /EHUMartintxo Saralegi-Aranguren¶
+Universit´ e d’Artois
+Abstract
+Given a smooth action of S3on a manifold M, we are interested in the relationship between the
+cohomologies of MandM/S3. If the action is free, we have indeed a principal S3-bundle, and this
+relationship is described by the classical Gysin sequence, which also exists when the action is semi-
+free (i.e., fixed points are allowed) [ 2]. In this work, we obtain a Gysin sequence for the case of a
+general smooth action. An exotic term appears, and we show th at it is an obstruction for the duality
+of the second term of the deRham spectral sequence associate d tothe action.
+Let us consider a smooth action Φ:G×M→Mof a compact Lie group on a manifold M. The
+actionΦinduces naturally a filtration {FiΩ∗(M)/vextendsingle/vextendsingle/vextendsinglei∈N}of the complex of de Rham di fferential forms
+Ω∗(M), defined by:
+FiΩi+j(M)=/braceleftig
+ω∈Ωj(M)/vextendsingle/vextendsingle/vextendsingleiX0···iXjω=0 foreach family/braceleftig
+X0,...,Xj/bracerightig
+⊂XΦ(M)/bracerightig
+.
+Here, we have denoted by XΦ(M)the orbit distribution of TMformed by the vector fields of Mtangent
+to the orbits ofΦ. This filtration defines thefirst quadrant de Rham spectral sequence , which converges
+toH∗(M). TheunderlyingmotivationofthispaperisthestudyoftheP oincar´ edualityofthesecondterm
+Es,t
+2ofthisspectral sequence.
+WhenΦis free, wehavetheduality Es,t
+2/simequalEn−s,ℓ−t
+2,wheren=dimM/Gandℓ=dimG. Thisproperty
+islostwhen theaction isno longerfree.
+Inspired by the work of Goresky and MacPherson, one expects t o recover the Poincar´ e duality by
+using intersection cohomology. This is the case when the gro upGis the circleS1(see [6]). The next
+naturalgroup GtostudyisS3(ofrank 1and notabelian). Ithas beenprovedin[ 7]thatPoincar´ eduality
+stillholdswhen theaction Φissemi-free.
+What about the other S3-actions? Surprisingly, the second term of the above spectr al sequence has
+not been computedyet in this context. The mainresult ofthis paperis in thefollowingGysin sequence,
+whichcomputesthissecondterm:
+∗ThisworkhasbeenpartiallysupportedbytheUPV /EHUgrantEHU09 /04andbytheSpanishMICINNgrantMTM2010-
+15471.
+†KeyWordsandPhrases: Gysinsequence,basic cohomology, S3-actions,controlledforms.
+‡2010MathematicalSubjectClassification. Primary57R19;S econdary57R30,57S15.
+§Departamento de Matem´ atica Aplicada, University of the Ba sque Country UPV /EHU. Alameda de Urquijo s /n. 48013
+Bilbao,SPAIN. joseignacio.royo@ehu.es .
+¶Universit´ e d’Artois, Laboratoire de Math´ ematiques de Le ns EA 2462. F´ ed´ eration CNRS Nord-Pas-de-CalaisFR 2956.
+Facult´ edesSciencesJean Perrin. Rue JeanSouvraz,S.P. 18 . F-62 300Lens,France. saralegi@euler.univ-artois.fr .Gysinsequence ...August28, 2018 ˙ 2
+···/d47/d47Hi(M)2/circlecop†rt/d47/d47Hi−3/parenleftig
+M/S3,Σ/S3/parenrightig
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipupright
+Ei−3,3
+2⊕/parenleftbigg
+Hi−2/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipupright
+Ei−2,2
+23/circlecop†rt/d47/d47Hi+1/parenleftig
+M/S3/parenrightig
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipupright
+Ei+1,0
+21/circlecop†rt/d47/d471/circlecop†rt/d47/d471/circlecop†rt/d47/d471/circlecop†rt/d47/d47Hi+1(M)/d47/d47···
+withEi,1
+2=0, where
+-Σisthesubsetofpointsof Mwhoseisotropygroupisinfinite;
+- ourchoiceofmaximaltorus, S1, is/braceleftig
+a+bi/vextendsingle/vextendsingle/vextendsinglea2+b2=1/bracerightig
+≤S3;
+- theZ2-action isinduced by j∈S3,
+- (−)−Z2denotesthesubspaceofantisymmetricelements(cf. ( 6)),
+-1/circlecop†rtis inducedbythenatural projection π:M→M/S3,
+-2/circlecop†rtis inducedbytheintegrationalongthefibers of π, and
+-3/circlecop†rtinvolvesthemultiplicationbythe Eulerclass [e]∈I H4
+4/parenleftig
+M/S3/parenrightig
+(cf. [7])
+(cf. Theorem 2.4and paragraph 2.5).
+Noticethatthefirstfloor Ei,1
+2alwaysvanishes(evenforanyperversity!) whereastheseco ndfloorEi,2
+2
+may not, as we show in example 2.4. So, it follows that Poincar ´ e duality does not work in the generic
+case, evenconsideringintersectioncohomology.
+As a consequence, a new approach is needed in order to extend t he Poincar´ e duality for general
+actions, and, thus, for Singular Riemannian Foliations, as is the partition induced by the orbits of a
+generalS3-action. Results inthisdirectionarebeing explored.
+In fact, in [ 5] the duality of the spectral sequence associated to a Singul ar Riemannian Foliation is
+claimed for the special case of extremeperversities (which is tantamountto working only in the regular
+stratum or relatively to the singular strata). As the previo us counterexample shows, a new approach is
+needed in order to extend the duality result for general perv ersities if one wants to consider the usual
+cohomology( p=0)case.
+A different Gysin sequence relating the cohomologyof Mand theS3-equivariant cohomologyof M
+was constructedin[ 3], also inthecaseofageneral smooth S3-action.
+Inthesequel Misaconnected,secondcountable,Hausdor ff,withoutboundaryandsmooth(ofclass
+C∞)manifold. Wefix asmoothaction Φ:S3×M→M.
+Wewish tothankthereferees fortheindicationsgivenin ord ertoimprovethispaper.
+1. Stratifications and di fferential forms.
+We describe the stratification arising from the action. We al so introduce the controlled di fferential
+forms,defined by Verona, inorderto computethesingularcoh omologyin thiscontext.
+1.2. Thom-Mather structure.
+Therearethreepossibilitiesforthedimensionoftheisotr opysubgroup1S3
+xofapoint x∈M,namely:
+0,1and 3. So, wehavethedimension-typefiltration
+F=/braceleftig
+x∈M/vextendsingle/vextendsingle/vextendsingledimS3
+x=3/bracerightig
+⊂Σ=/braceleftig
+x∈M/vextendsingle/vextendsingle/vextendsingledimS3
+x≥1/bracerightig
+⊂M=/braceleftig
+x∈M/vextendsingle/vextendsingle/vextendsingledimS3
+x≥0/bracerightig
+.
+1We refer the reader to [ 2] for the notions related with compact Lie group actions, suc h as isotropy, invariant tubular
+neighborhoods,...Gysinsequence ...August28, 2018 ˙ 3
+In this section, we describe the geometry of the triple ( M,Σ,F). The subsetΣis not necessarily a
+manifold, but the subsets F=MS3,Σ\F=/braceleftig
+x∈M/vextendsingle/vextendsingle/vextendsingledimS3
+x=1/bracerightig
+andM\Σ=/braceleftig
+x∈M/vextendsingle/vextendsingle/vextendsingledimS3
+x=0/bracerightig
+are
+properinvariantsubmanifolds2ofM. So, wecan consider τ0:T0→Fandτ1:T1→Σ\Ftwo invariant
+tubular neighborhoods in M. Over each connected component, the structure group is the o rthogonal
+group. Associatedto thesetubularneighborhoodswe haveth efollowingmaps( k=0,1):
+/squigglerightTheradiusmapνk:Tk→[0,∞[, defined fiberwiseby u/mapsto→/bardblu/bardbl. It isan invariantsmoothmap.
+/squigglerightThedilatation map∂k: [0,∞[×Tk→Tk, defined fiberwise by ( t,u)/mapsto→t·u. It is a smooth
+equivariantmap.
+Thefamilyoftubularneighborhoods TM={T0,T1}isaThom-Mather system when:
+(TM)/braceleftiggτ0=τ0◦τ1
+ν0=ν0◦τ1/bracerightigg
+onT0∩T1=τ−1
+1(T0∩(Σ\F)).
+Lemma 1.3 Thom-Mathersystemsexist.
+Proof.We fix an invariant tubular neighborhood τ0:T0→F. It exists since Fis an invariant closed
+submanifold of M. Since the isotropy subgroup of any point of Fis the wholeS3, we can find3an atlas
+A=/braceleftigϕ:U×Rn→τ−1
+0(U)/bracerightig
+ofτ0,havingO(n)asstructuregroup,andanorthogonalaction Ψ:S3×Rn→
+Rnsuchthat
+(1) ϕ(x,Ψ(g,v))=Φ(g,ϕ(x,v))∀x∈U,∀v∈Rnand∀g∈S3.
+Wewriteτ′
+0:S0→Ftherestrictionofτ0,whereS0isthesubmanifoldν−1
+0(1). Itisafiberbundle. There-
+strictionτ′′
+0: (S0∩(Σ\F))→Fisalsoafiberbundlewhoseinducedatlasis A′′=/braceleftigϕ:U×Sn−1
+Σ→τ′′
+0−1(U)/bracerightig
+,
+whereSn−1
+Σ=/braceleftig
+w∈Sn−1/vextendsingle/vextendsingle/vextendsingledimS3
+w=1/bracerightig
+.
+The mapL0:T0\F→S0×]0,∞[, defined byL0(x)=/parenleftig
+∂0/parenleftig
+ν0(x)−1,x/parenrightig
+,ν0(x)/parenrightig
+, is an equivariant dif-
+feomorphism. Under L0:
+/squigglerightthemapτ0becomes( y,t)/mapsto→τ′
+0(y),
+/squigglerightthemapν0becomes ( y,t)/mapsto→t,and
+/squigglerightthemanifold T0∩(Σ\F) becomes( S0∩(Σ\F))×]0,∞[.
+Since the structure group of τ′
+0is a compact Lie group, condition ( 1) allows us to construct an
+invariant Riemannian metric µ0onS0such that the fibers of τ′
+0are totally geodesic submanifolds and
+(T(S0∩(Σ\F)))⊥⊂ker/parenleftig
+τ′
+0/parenrightig
+∗. Then, if we consider the associated tubular neighborhood τ′
+1:T′
+1→
+S0∩(Σ\F) wehaveτ′
+0◦τ′
+1=τ′
+0.
+Wecan constructnowan invariantRiemannian metric µonM\Fsuchthat underL0:
+/squigglerightthemetricµbecomesµ0+dr2onS0×]0,∞[.
+Weconsidertheassociatedtubularneighborhood τ1:T1→Σ\F. Verificationoftheproperty(TM)must
+bedoneon T0∩T1, whereusingL0, weget:
+/squigglerightT0∩T1becomesT′
+1×]0,∞[.
+/squigglerightτ1becomes ( y,t)/mapsto→(τ′
+1(y),t).
+A straightforwardcalculationgives(TM)and endstheproof . ♣
+We fix a such system TM. For each k∈{0,1},we shallwrite Dk⊂Mthe open subsetν−1
+k([0,1[) and
+call itthe soulofthetubularneighborhood τk. We shallwrite∆0=D0∩Σ.
+1.5. Verona’sdifferentialforms. Asitisshownin[ 8],thesingularcohomologyof Mcanbecomputed
+by using differential forms on M\Σ. This is the tool we use in this work. The complex of controlled
+2Infact,these manifoldsmayhaveconnectedcomponentswith differentdimensions.
+3Foreachconnectedcomponentof F.Gysinsequence ...August28, 2018 ˙ 4
+forms(orVerona’sforms )ofMis defined by
+Ω∗
+V(M)=ω∈Ω∗(M\Σ)/vextendsingle/vextendsingle/vextendsingle∃ω1∈Ω∗(Σ\F)andω0∈Ω∗(F)with(a)τ∗
+1ω1=ωonD1\Σ
+(b)τ∗
+0ω0=ωonD0\Σ
+(c)τ∗
+0ω0=ω1on∆0\F.
+Following[ 8] weknowthat thecohomologyofthecomplex Ω∗
+V(M)is thesingularcohomology H∗(M).
+We also usein this work the complex Ω∗
+V(Σ)=/braceleftig
+γ∈Ω∗(Σ\F)/vextendsingle/vextendsingle/vextendsingle∃γ0∈Ω∗(F)withτ∗
+0γ0=γon∆0\F/bracerightig
+and therelativecomplexes Ω∗
+V(M,Σ)={ω∈Ω∗
+V(M)/vextendsingle/vextendsingle/vextendsingleω1≡0}andΩ∗
+V(Σ,F)={γ∈Ω∗
+V(Σ)/vextendsingle/vextendsingle/vextendsingleγ0≡0}.
+SinceMis amanifold,controlledforms arein fact di fferentialformson M.
+Lemma 1.6 Anycontrolledformof M istherestrictionof adi fferentialformof M.
+Proof.First,weconstructasection σoftherestrictionρ:Ω∗
+V(M)→Ω∗
+V(Σ)definedbyρ(ω)=ω1. Letus
+considera smoothfunction f: ]0,∞[→[0,1]verifying f≡0 on [3,∞[andf≡1 on ]0,2]. Noticethat
+the compositions f◦ν0:M→[0,1] andf◦ν1:M\F→[0,1] are smooth invariant maps. So, for each
+γ∈Ω∗
+V(Σ)wehave
+(2) σ(γ)=(f◦ν0)·τ∗
+0γ0+(1−(f◦ν0))·(f◦ν1)τ∗
+1γ∈Ω∗(M).
+Thisdifferentialformis acontrolledformsince
+(a) Since ( f◦ν1)≡1 onD1, (f◦ν0)≡0 onM\T0and (TM)then wehave
+σ(γ)=(f◦ν0)·τ∗
+1τ∗
+0γ0+(1−(f◦ν0))·τ∗
+1γ=τ∗
+1/parenleftbig(f◦ν0)·τ∗
+0γ0+(1−(f◦ν0))·γ/parenrightbig
+onD1\Σ. This gives (σ(γ))1=(f◦ν0)·τ∗
+0γ0+(1−(f◦ν0))·γ. Sinceτ∗
+0γ0=γon∆0\Fthen
+(σ(γ))1=(f◦ν0)·γ+(1−(f◦ν0))·γ=γ.
+(b) Since ( f◦ν0)≡1 onD0thenwehaveσ(γ)=τ∗
+0γ0onD0\Σ. Thisgives(σ(γ))0=γ0.
+(c) We have(σ(γ))1=γ=τ∗
+0γ0=τ∗
+0(σ(γ))0on∆0\F.
+Thismapσisasection ofρsinceρ(σ(γ))=(σ(γ))1=γ.
+In particular,ρ(ω−σ(ρ(ω)))=0 for eachω∈Ω∗
+V(M). Asσ(ρ(ω))∈Ω∗(M)(cf. (2)) and coincides
+withωintheopenset ( D0∪D1)\Σweconcludethatωcan beextendedto M. ♣
+1.8. Invariantforms.
+We fix{u1,u2,u3}a basis of the Lie algebra of S3with [u1,u2]=u3, [u2,u3]=u1and [u3,u1]=u2.
+Wedenoteby Xi∈XΦ(M)thefundamentalvector field associatedto ui,i=1,2,3.
+A controlled form ωofMis aninvariant form whenLXiω=0 for each i=1,2,3. The complex
+of invariant forms is denoted by Ω∗
+V(M). The inclusionΩ∗
+V(M)֒→Ω∗
+V(M)induces an isomorphism in
+cohomology. This a standard argument based on the fact that S3is a connected compact Lie group (cf.
+[4, TheoremI, Ch. IV, vol. II]). So,
+(3) H∗/parenleftig
+Ω.
+V(M)/parenrightig
+=H∗/parenleftig
+Ω.
+V(M)/parenrightig
+=H∗(M).
+1.10. Basicforms. AcontrolledformωofMisabasicform wheniXω=iXdω=0foreach X∈XΦ(M).
+Thecomplexofthebasicformsisdenotedby Ω∗
+V/parenleftig
+M/S3/parenrightig
+. Inasimilarfashionwedefine Ω∗
+V/parenleftig
+Σ/S3/parenrightig
+. Inthis
+work, we shall use the following relative versions of these c omplexes:Ω∗
+V/parenleftig
+M/S3,Σ/S3/parenrightig
+=Ω∗
+V/parenleftig
+M/S3/parenrightig
+∩
+Ω∗
+V(M,Σ), as wellasΩ∗
+V/parenleftig
+Σ/S3,F/parenrightig
+=Ω∗
+V/parenleftig
+Σ/S3/parenrightig
+∩Ω∗
+V(Σ,F).Gysinsequence ...August28, 2018 ˙ 5
+Lemma 1.11
+H∗/parenleftig
+Ω·
+V/parenleftig
+M/S3/parenrightig/parenrightig
+=H∗/parenleftig
+M/S3/parenrightig
+and H∗/parenleftig
+Ω·
+V/parenleftig
+M/S3,Σ/S3/parenrightig/parenrightig
+=H∗/parenleftig
+M/S3,Σ/S3/parenrightig
+.
+Proof.Theorbitspace M/S3isastratifiedpseudomanifold. Thefamilyoftubularneighb orhoodsTM/S3=
+{π(T0),π(T1)}is aThom-Mather system . Here,π:M→M/S3denotes the canonical projection. Using
+thisprojection,weidentifythecomplexofcontrolledform sofM/S3withΩ·
+V/parenleftig
+M/S3/parenrightig
+,andthesameholds
+forΣ.
+SinceH∗/parenleftig
+Ω·
+V(X)/parenrightig
+=H∗(X)for any stratified pseudomanifold X, thenH∗/parenleftig
+Ω·
+V/parenleftig
+M/S3/parenrightig/parenrightig
+=H∗/parenleftig
+M/S3/parenrightig
+andH∗/parenleftig
+Ω·
+V/parenleftig
+Σ/S3/parenrightig/parenrightig
+=H∗/parenleftig
+Σ/S3/parenrightig
+(cf. [8]). In fact, the orbit spaces M/S3andΣ/S3are triangulable [ 9],
+and by [10], both of them possess good coverings. Moreover, any open co vering of M/S3(resp.Σ/S3)
+possessesasubordinatedpartitionofunitymadeupofcontr olledfunctions. So,wecanproceedasin[ 1]
+and constructacommutativediagram
+···/d47/d47Hp/parenleftig
+Ω·
+V/parenleftig
+M/S3,Σ/S3/parenrightig/parenrightig
+/d47/d47Hp/parenleftig
+Ω·
+V/parenleftig
+M/S3/parenrightig/parenrightig
+/d47/d47
+fM/d15/d15Hp/parenleftig
+Ω·
+V/parenleftig
+Σ/S3/parenrightig/parenrightig
+/d47/d47
+fΣ/d15/d15Hp+1/parenleftig
+Ω·
+V/parenleftig
+M/S3,Σ/S3/parenrightig/parenrightig
+/d47/d47···
+···/d47/d47Hp/parenleftig
+M/S3,Σ/S3/parenrightig
+/d47/d47Hp/parenleftig
+M/S3/parenrightig
+/d47/d47Hp/parenleftig
+Σ/S3/parenrightig
+/d47/d47 /d47/d47Hp+1/parenleftig
+M/S3,Σ/S3/parenrightig
+/d47/d47···
+wheretheverticalarrowsareisomorphismsandthehorizont alrowsarethelongexactsequencesassoci-
+ated tothepair( M/S3,Σ/S3). Thisgives H∗/parenleftig
+Ω·
+V/parenleftig
+M/S3,Σ/S3/parenrightig/parenrightig
+=H∗/parenleftig
+M/S3,Σ/S3/parenrightig
+4. ♣
+2. Gysinsequence.
+We construct the long exact sequence associated to the actio nΦ:S3×M→Mrelating the coho-
+mology of MandM/S3. First of all, we shall use strongly that Φisalmost free5inM\Σto get a better
+descriptionofthecontrolledformsof M.
+2.2. Decompositionofa di fferential form.
+Weendow M\ΣwithanS3-invariantRiemannianmetric µ0,whichexistsbecause S3iscompact. We
+also fix a bi-invariant Riemannian metric νon the Lie groupS3. Consider now the µ0-orthogonalS3-
+invariantdecomposition T(M\Σ)=D⊕ξ,whereDis thedistributiongenerated by Φ. Since theaction
+Φis almost free on M\Σ, for each point x∈M\Σ, the family{X1(x),X2(x),X3(x)}is a basis ofDx. We
+definetheS3-Riemannianmetric µonM\Σby putting
+µ(w1,w2)=µ0(w1,w2) ifw1,w2∈ξx
+0 if w1∈ξx,w2∈Dx
+δi,j ifw1=Xi(x),w2=Xj(x)
+We denote byχi=iXiµ∈Ω1(M\Σ)thecharacteristic form associated to ui,i=1,2,3. Since
+χj(Xi)=µ(Xi,Xj)=δij,each differentialformω∈Ω∗(M\Σ)possessesauniquewriting,
+ω=0ω+3/summationdisplay
+p=1χp∧pω+/summationdisplay
+1≤p<q≤3χp∧χq∧pqω+χ1∧χ2∧χ3∧123ω,
+4Noticethat thisis notthe fivelemma.
+5All theisotropysubgroupsarefinitegroups.Gysinsequence ...August28, 2018 ˙ 6
+where the coefficients•ωarehorizontalforms , that is, they verify iX/parenleftbig
+•ω/parenrightbig=0 for each X∈XΦ(M). This
+isthecanonicaldecomposition ofω. Forexample, dβ=0(dβ)+χ1∧LX1β+χ2∧LX2β+χ3∧LX3β,forany
+horizontal formβ(notice that this formula is no longer true if βis not horizontal). Since LXiχj=χ[ui,uj],
+with1≤i,j≤3, then
+(4)LX1χ1=LX2χ2=LX3χ3=0 LX1χ2=−LX2χ1=χ3
+LX1χ3=−LX3χ1=−χ2 LX2χ3=−LX3χ2=χ1
+and wehavethecanonical decompositions
+(5)dχ1=e1−χ2∧χ3
+dχ2=e2+χ1∧χ3
+dχ3=e3−χ1∧χ2.
+Here, theforms eiare basicfor i=1,2,3. Notice that e1−χ2∧χ3is theEuler form of theaction of the
+maximal torus with fundamental vector field X1, and that e2
+1+e2
+2+e2
+3is the Euler form of the action Φ
+(seesection( 2.8)).
+Consider U⊂M\Σan equivariant open subset. If ω∈Ω∗(M\Σ,U)then the coefficients of its
+canonical decomposition are horizontal forms of Ω∗(M\Σ,U). The following Lemma is the key for
+the construction of the Gysin sequence. Given an action of Z2on a vector space Egenerated by the
+morphism h:E→E, weshallwrite
+(6) E−Z2={e∈E/vextendsingle/vextendsingle/vextendsingleh(e)=−e},
+thesubspaceof antisymmetricelements . Noticethat j∈S3actsnaturallyon MS1.
+Lemma 2.3
+H∗Ω·
+V(M)
+Ω·
+V/parenleftig
+M/S3/parenrightig=H∗−3/parenleftig
+M/S3,Σ/S3/parenrightig
+⊕/parenleftbigg
+H∗−2/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2
+Proof.Weconsidertheintegrationoperator:
+/fIntegraltext
+:Ω∗
+V(M)
+Ω∗
+V/parenleftig
+M/S3/parenrightig−→Ω∗−3
+V/parenleftig
+M/S3,Σ/S3/parenrightig
+,
+givenby:/fIntegraltext
+(<ω>)=(−1)degωiX3iX2iX1ω.
+It isawell defined di fferentialoperatorsince
+- thetubularneighborhoodsoftheThom-Mather’sstructure Tareinvariant,
+- theoperator iX3iX2iX1vanishesonΣ, and
+-iXiX3iX2iX1ω=iXdiX3iX2iX1ω=0foreach X∈XΦ(M)6.
+6LAiB=iBLA+i[A,B],∀A,B∈X(M).Gysinsequence ...August28, 2018 ˙ 7
+Every formγ∈Ω∗−3
+V/parenleftig
+M/S3,Σ/S3/parenrightig
+vanishesin aneighborhoodof Σ. So, theproductχ1∧χ2∧χ3∧γ
+belongstoΩ∗
+V(M)(cf. (4)). SinceiX3iX2iX1(χ1∧χ2∧χ3∧γ)=γthenwe havetheshortexactsequence
+(7) 0/d47/d47Ker∗/fIntegraltext/d31 /d127/d47/d47Ω∗
+V(M)
+Ω∗
+V/parenleftig
+M/S3/parenrightig/fIntegraltext
+/d47/d47Ω∗−3
+V/parenleftig
+M/S3,Σ/S3/parenrightig
+/d47/d470
+By Lemma 1.11, itsuffices to provethefollowing:
+(a)H∗/parenleftig
+Ker∗/fIntegraltext/parenrightig
+=/parenleftbigg
+H∗−2/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2.
+(b)Theassociatedconnecting homomorphism δvanishes.
+(a)
+Forthesakeofsimplicityweput A∗(M)=Ker∗/fIntegraltext
+. InfactwehaveA∗(M)=/braceleftig
+ω∈Ω∗
+V(M)/vextendsingle/vextendsingle/vextendsingleiX3iX2iX1ω=0/bracerightig
+Ω∗
+V/parenleftig
+M/S3/parenrightig.
+Analogously,wedefine A∗(M,Σ),A∗(Σ)andA∗(Σ,F). Toget(a),itsufficestoprovethefollowingfacts:
+(a1)H∗/parenleftig
+A∗(M)/parenrightig
+=H∗/parenleftig
+A∗(Σ)/parenrightig
+.
+(a2)H∗/parenleftbigA·(Σ)/parenrightbig=H∗/parenleftbigA·(Σ,F)/parenrightbig.
+(a3)H∗/parenleftbigA·(Σ,F)/parenrightbig=/parenleftbigg
+H∗−2/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2.
+(a1)
+Consider the inclusion L:A∗(M,Σ)−→A∗(M)and the restriction R:A∗(M)→A∗(Σ),which are
+differentialmorphisms. This givestheshort sequence
+0/d47/d47A∗(M,Σ)L/d47/d47A∗(M)R/d47/d47A∗(Σ)/d47/d470.
+Noticethat R◦L=0. Thisshortsequence isexact since:
+•TheoperatorRisanontomap. Considerγ∈Ω∗
+V(Σ). Weknowthatσ(γ)∈Ω∗
+V(M)(cf. Lemma 1.6).
+Theresultcomesfrom:
+/squigglerightσ(γ)∈Ω∗
+V(M). Sinceτ0,τ1are equivariantand f◦ν0,f◦ν1areinvariant.
+/squigglerightiX3iX2iX1σ(γ)=0. Sinceτ0,τ1are equivariantand rank {X1(x),X2(x),X3(X)}≤2 forany x∈Σ.
+/squigglerightR(<σ(γ)>)=<(σ(γ))1>=<γ>.
+•KerR⊂ImL.Considerω∈Ω∗
+V(M)withiX3iX2iX1ω=0andiXjω1=0 forj∈{1,2,3}. Sinceτ0and
+τ1are equivariantand Xj=0 onFtheniXjσ(ω1)=0 forj∈{1,2,3}. This gives<σ(ω1)>=0. Finally,
+wehave<ω>=<ω−σ(ω1)>=L(<ω−σ(ω1)>)since(ω−σ(ω1))1=ω1−(σ(ω1))1=ω1−ω1=0.
+Now, we will get (a1) by proving that H∗/parenleftbigA·(M,Σ)/parenrightbig=0. By definition of Verona’s forms we have
+A∗(M,Σ)=A∗(M,D)excision===A∗(M\Σ,D\Σ),whereD=D0∪D1.A straightforwardcalculationgives:
+H∗/parenleftig
+A·(M\Σ,D\Σ)/parenrightig
+=/braceleftig
+ω∈Ω∗(M\Σ,D\Σ)/vextendsingle/vextendsingle/vextendsingleiX3iX2iX1ω=0 andiXjdω=0forj∈{1,2,3}/bracerightig
+Ω∗/parenleftig
+(M\Σ)/S3,(D\Σ)/S3/parenrightig
++/braceleftig
+dβ/vextendsingle/vextendsingle/vextendsingleβ∈Ω∗−1(M\Σ,D\Σ)andiX3iX2iX1β=0/bracerightigGysinsequence ...August28, 2018 ˙ 8
+Letωbe a differential form ofΩ∗(M\Σ,D\Σ)verifying iX3iX2iX1ω=0 andiXjdω=0 forj∈{1,2,3}.
+Then
+d/parenleftbigχ1∧iX3iX2ω−χ2∧iX3iX1ω+χ3∧iX2iX1ω/parenrightbig
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipupright
+β
+ω= +
+−e1∧iX3iX2ω+e2∧iX3iX1ω−e3∧iX2iX1ω+0ω/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipupright
+α
+(cf. (5)) withβ∈Ω∗−1(M\Σ,D), verifying iX3iX2iX1β=0, andα∈Ω∗/parenleftig
+(M\Σ)/S3,D/S3/parenrightig
+. This implies
+H∗/parenleftbigA·(M\Σ,D\Σ)/parenrightbig=0 and then H∗/parenleftbigA·(M,Σ)/parenrightbig=0.
+(a2)
+Consider the inclusion L:A∗(Σ,F)֒→A∗(Σ)which is a differential morphism. It su ffices to prove
+thatLis an ontomap.
+Let us consider a smooth function f: ]0,∞[→[0,1] verifying f≡0 on [3,∞[ andf≡1 on ]0,2].
+Notice that the composition f◦ν0:M→[0,1] is a smooth invariant map. So, for each γ∈Ω∗(F)we
+haveσ(γ)=(f◦ν0)τ∗
+0γ∈Ω∗(M). Thisdifferentialformverifies
+/squigglerightσ(γ)∈Ω∗
+V(Σ). Since (f◦ν0)≡1on∆0thenσ(γ)=τ∗
+0γon∆0\F. Thisgives(σ0(γ))0=γ.
+/squigglerightσ(γ)∈Ω∗
+V(Σ). Sinceτ0isan equivariantmapand f◦ν0isan invariantmap.
+/squigglerightiXjσ(γ)=0forj∈{1,2,3}sinceτ0isan equivariantmap and Xj=0 onF.
+Then<σ(γ)>=0onA∗(Σ).
+Let<ω>be a class ofA∗(Σ). We can write:<ω>=<ω−σ((ω)0)>=L(<ω−σ((ω)0)>)since
+(ω−σ((ω)0))0=ω0−(σ(ω0))0=ω0−ω0=0. Thisprovesthat Lis anonto map.
+(a3)
+By definitionofVerona’s di fferentialformswehave
+A∗(Σ,F)=A∗(Σ,∆0)excision===A∗(Σ\F,∆0\F)=Ω∗(Σ\F,∆0\F)
+Ω∗/parenleftig
+(Σ\F)/S3,(∆0\F)/S3/parenrightig.
+The isotropy subgroup of a point of Σ\Fis conjugated toS1orN(S1) (cf. [2, Th. 8.5, pag. 153]).
+We consider the manifold Γ=(Σ\F)S1
+.A straightforward calculation gives that Σ\FisG-equivariant
+diffeomorphicto
+S3×N(S1)Γ=/parenleftig
+S3/S1/parenrightig
+×N(S1)/S1Γ=S2×Z2Γ.
+Notice thatΓ/Z2=(Σ\F)/S3. LetΓ0be the open subset Γ∩∆0ofΓ. Analogously we have ∆0\F=
+S2×Z2Γ0andΓ0/Z2=(∆0\F)/S3.
+TheZ2-action onS2is generated by ( x0,x1,x2)/mapsto→(−x0,−x1,−x2)7. Then, theZ2-action on H0/parenleftig
+S2/parenrightig
+(resp.H2/parenleftig
+S2/parenrightig
+) is the identity Id (resp. −Id). TheZ2-action onΓis induced byΦ(j,−). The K¨ unneth
+7Thismapis inducedby j:S3→S3definedby j(u)=u·j(see [1, Example17.23]).Gysinsequence ...August28, 2018 ˙ 9
+formulagives
+H∗/parenleftig
+Ω∗(Σ\F,∆0\F)/parenrightig
+=H∗/parenleftig
+Ω·/parenleftig
+S2×Z2Γ,S2×Z2Γ0/parenrightig/parenrightig
+=H∗/parenleftbigg
+Ω·/parenleftig
+S2×Γ,S2×Γ0/parenrightigZ2/parenrightbigg
+=H∗/parenleftig
+Ω·/parenleftig
+S2×Γ,S2×Γ0/parenrightig/parenrightigZ2=/parenleftig
+H∗/parenleftig
+S2/parenrightig
+⊗H∗(Γ,Γ0)/parenrightigZ2
+=/parenleftig
+H0/parenleftig
+S2/parenrightig
+⊗H∗(Γ,Γ0)/parenrightigZ2⊕/parenleftig
+H2/parenleftig
+S2/parenrightig
+⊗H∗−2(Γ,Γ0)/parenrightigZ2=/parenleftig
+H∗(Γ,Γ0)/parenrightigZ2⊕/parenleftig
+H∗−2(Γ,Γ0)/parenrightig−Z2
+=H∗/parenleftbigΓ/Z2,Γ0/Z2/parenrightbig⊕/parenleftig
+H∗−2(Γ,Γ0)/parenrightig−Z2=H∗/parenleftig
+(Σ\F)/S3,(∆0\F)/S3/parenrightig
+⊕/parenleftig
+H∗−2(Γ,Γ0)/parenrightig−Z2,
+and then
+H∗/parenleftig
+A·(Σ\F,∆0\F)/parenrightig
+=H∗Ω·(Σ\F,∆0\F)
+Ω·/parenleftig
+(Σ\F)/S3,(∆0\F)/S3/parenrightig=/parenleftig
+H∗−2(Γ,Γ0)/parenrightig−Z2=/parenleftbigg
+H∗−2/parenleftbigg
+(Σ\F)S1
+,(∆0\F)S1/parenrightbigg/parenrightbigg−Z2
+excision===/parenleftbigg
+H∗−2/parenleftbigg
+ΣS1
+,∆S1
+0/parenrightbigg/parenrightbigg−Z2retraction===/parenleftbigg
+H∗−2/parenleftbigg
+ΣS1
+,FS1/parenrightbigg/parenrightbigg−Z2.
+Considerthelongexact sequenceassociatedto the Z2-invariantpair/parenleftbigg
+ΣS1,FS1/parenrightbigg
+:
+···→/parenleftbigg
+Hi−1/parenleftbigg
+FS1/parenrightbigg/parenrightbigg−Z2→/parenleftbigg
+Hi/parenleftbigg
+ΣS1
+,FS1/parenrightbigg/parenrightbigg−Z2→/parenleftbigg
+Hi/parenleftbigg
+ΣS1/parenrightbigg/parenrightbigg−Z2→/parenleftbigg
+Hi/parenleftbigg
+FS1/parenrightbigg/parenrightbigg−Z2→···.
+Since the action of Z2onFS1=Fis trivial, then/parenleftbigg
+Hi/parenleftbigg
+FS1/parenrightbigg/parenrightbigg−Z2=0. On the other hand, we have
+ΣS1=MS1. Thisgives/parenleftbigg
+H∗−2/parenleftbigg
+ΣS1,FS1/parenrightbigg/parenrightbigg−Z2=/parenleftbigg
+H∗−2/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2.
+(b)
+Notice that the connecting morphism δis defined byδ([ζ])=±/bracketleftbig<d(χ1∧χ2∧χ3)∧ζ>/bracketrightbig. We have
+δ≡0 sinceζ1=0 (cf (a1)). ♣
+Theorem 2.4 Givenanysmoothaction Φ:S3×M−→M wehavetheGysinsequence
+···/d47/d47Hi(M) /d47/d47Hi−3/parenleftig
+M/S3,Σ/S3/parenrightig
+⊕/parenleftbigg
+Hi−2/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2/d47/d47Hi+1/parenleftig
+M/S3/parenrightig
+/d47/d47Hi+1(M) /d47/d47···
+whereΣis the subset of points of M whose isotropygroup is infinite, t heZ2-action is induced by j ∈S3
+and(−)−Z2denotesthesubspaceof antisymmetricelements.
+Proof.Considertheshortexact sequence
+(8) 0/d47/d47Ω∗
+V/parenleftig
+M/S3/parenrightig/d47/d47Ω∗
+V(M)/d47/d47Ω∗
+V(M)
+Ω∗
+V/parenleftig
+M/S3/parenrightig/d47/d470,
+takeitsassociatedlongexact sequenceand then,applyLemm a1.11, (3)and Lemma 2.3.♣
+2.6. Example. Considertheconnectedsum M=CP2#CP2/simequal/parenleftig
+S3×[0,1]/parenrightig
+/∼, with
+((z1,z2),i)∼((z·z1,z·z2),i),i=0,1,Gysinsequence ...August28, 2018 ˙ 10
+forallz∈S1and(z1,z2)∈S3incomplexcoordinates. Theproductof S3induceson Mtheaction:
+g·[h,t]=[g·h,t],∀g,h∈S3,∀t∈[0,1].
+Forthisaction,wehave:
+Σ=/parenleftig
+S3×{0,1}/parenrightig
+/∼/simequalS2×{0,1},F=∅,
+M/S3/simequal[0,1],Σ/S3/simequal{0,1},MS1/simequal{N,S}×{0,1},
+whereNandSstandfortheNorthandSouthpolesof S2. TheZ2-actionon MS1isdeterminedby j∈S3,
+whichinducestheantipodalmapon S2,andso,interchangesitspoles. Thus,theexotictermthata ppears
+inthecentral part oftheGysinSequence isnottrivial:
+H2(M)/simequal−→/parenleftbigg
+H0/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2=/parenleftig
+H0({N,S}×{0,1})/parenrightig−Z2/simequalR⊕R.
+2.8. Morphisms. WedescribethemorphismsoftheGysinsequence.
+1/circlecop†rt:H∗/parenleftig
+M/S3/parenrightig
+−→H∗(M)
+It isthepull-backπ∗ofthecanonical projection π:M→M/S3(cf. Lemma 1.11).
+2/circlecop†rt:H∗(M)−→H∗−3/parenleftig
+M/S3,Σ/S3/parenrightig
+⊕/parenleftbigg
+H∗−2/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2
+We have already seen that the first component of this morphism is induced by/fIntegraltext
+S3[ω]=[iX3iX2iX1ω].
+For thesecond componentwe keep track of theisomorphismsgi ven by Lemma 2.3and we get that it is
+defined by: [ω]/mapsto→class/parenleftig/fIntegraltext
+S2(ω1−σ(ι∗ω1))/parenrightig
+.
+3/circlecop†rt:H∗−3/parenleftig
+M/S3,Σ/S3/parenrightig
+⊕/parenleftbigg
+H∗−2/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2−→H∗+1/parenleftig
+M/S3/parenrightig
+A straightforward calculation using sequences ( 7) and (8) gives that the connecting morphism 3/circlecop†rtof the
+Gysinsequencesends:
+•[ζ]∈H∗−3/parenleftig
+M/S3,Σ/S3/parenrightig
+to-/bracketleftig/parenleftig
+e2
+1+e2
+2+e2
+3/parenrightig
+∧ζ/bracketrightig
+, and
+•[ξ]∈/parenleftbigg
+H∗−2/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2=/parenleftbigg
+H∗−2/parenleftbigg
+ΣS1,FS1/parenrightbigg/parenrightbigg−Z2to [dσ∧ǫ∧τ∗
+1ξ] whereǫis an Euler form of the
+restrictionΦ1:S1×/parenleftbigg
+τ−1
+1/parenleftbigg
+ΣS1/parenrightbigg
+\ΣS1/parenrightbigg
+→/parenleftbigg
+τ−1
+1/parenleftbigg
+ΣS1/parenrightbigg
+\ΣS1/parenrightbigg
+ofΦ.
+Sincee2
+1+e2
+2+e2
+3is not a Verona’s form, then it does not define a class of H4/parenleftig
+M/S3/parenrightig
+. Nevertheless, it
+doesgenerate aclassin theintersectioncohomologygroup I H4
+4/parenleftig
+M/S3/parenrightig
+(as in thesemi-freecase of[ 7]).
+2.10. Remarks.
+(a) We have/parenleftbigg
+H∗/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2=H∗/parenleftbigg
+MS1/parenrightbigg/slashig
+H∗/parenleftbigg
+MS1/Z2/parenrightbigg
+. Let us see that. The correspondence ω/mapsto→
+/parenleftiggω+j∗ω
+2,ω−j∗ω
+2/parenrightigg
+establishestheisomorphism Ω∗/parenleftbigg
+MS1/parenrightbigg
+=/parenleftbigg
+Ω∗/parenleftbigg
+MS1/parenrightbigg/parenrightbiggZ2⊕/parenleftbigg
+Ω∗/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2=Ω∗/parenleftbigg
+MS1/Z2/parenrightbigg
+⊕
+/parenleftbigg
+Ω∗/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2and hence, H∗/parenleftbigg
+MS1/parenrightbigg
+=H∗/parenleftbigg
+MS1/Z2/parenrightbigg
+⊕/parenleftbigg
+H∗/parenleftbigg
+MS1/parenrightbigg/parenrightbigg−Z2.Thisgivestheclaim.Gysinsequence ...August28, 2018 ˙ 11
+(b)Letussupposethattheactionissemi-free,almostfreeo rfree. Then, jactstriviallyon MS1=F,
+and hence, wehavea longexactsequence
+···→Hi(M)→Hi−3/parenleftig
+M/S3,F/parenrightig
+→Hi+1/parenleftig
+M/S3/parenrightig
+→Hi+1(M)→···.
+(c) Let us supposethat thereis not a pointof Mwhoseisotropysubgroupis conjugated to S1. Then,
+wehavea longexactsequence
+···→Hi(M)→Hi−3/parenleftig
+M/S3,Σ/S3/parenrightig
+→Hi+1/parenleftig
+M/S3/parenrightig
+→Hi+1(M)→···.
+sincejacts triviallyon MS1=/braceleftig
+x∈M/vextendsingle/vextendsingle/vextendsingleS3
+x=S3orN(S1)/bracerightig
+.
+2.12. Actions over S1.
+Using the Gysin sequence we have constructed, we now givea li st of all the different cohomologies
+ofaS3-manifold Mhavingthecircleasorbitspace8. Bygeometricalreasons,theorbitspaceiscomposed
+byjustonestratum,thewholecircle. Followingthenatureo ftheorbits,wedistinguishfourcases.
+(a)Allorbitsareofdimension3. Wehave PM=1+t+t3+t4. Thisisthecaseofthemanifold S3×S1,
+whereS3acts bymultiplicationon theleft factor.
+(b) All orbits are isomorphicto S2. We distinguishtwo cases following wether the covering MS1→
+MS1/Z2=M/MS1is trivial or not. In the first case we have PM=1+t+t2+t3. This is the case of the
+manifoldS2×S1,whereS3actsbymultiplicationontheleftfactor. Inthesecondcase wehavePM=1+t,
+as isthecaseofthemanifold S2×
+Z2S1whereS3actsby multiplicationontheleft factor.
+(c)Allorbitsareisomorphicto RP2. Inthiscase,wehave PM=1+t. Thisisthecaseofthemanifold
+RP2×S1whereS3acts bymultiplicationon theleft factor.
+(d) All orbits are points. We have PM=1+t. This corresponds to the manifold S1whereS3acts
+ineffectively.
+References
+[1] R. Bott and L. W. Tu: Differential forms in Algebraic Topology, Graduate Texts in Mathematics,
+82,Springer-Verlag, New York-Berlin, 1982.
+[2] G. Bredon: Introduction to compact transformation groups , Pure and Applied Mathematics, Vol.
+46.AcademicPress, New York-London,1972.
+[3] H. Chen, An analytic model for S3-equivariant cohomology , Topology Hawaii (Honolulu, HI,
+1990),53–61,WorldSci. Publ. , RiverEdge, NJ,1992.
+[4] W. Greub, S. Halperin and R. Vanstone: Connections, curvature and cohomology, Vol. II: Lie
+groups,principalbundles,andcharacteristicclasses ,PureandAppliedMathematics,Vol.47-II.
+AcademicPress, NewYork and London,1972.
+[5] X. Masa: Duality and minimality for Riemannian foliations on open ma nifolds,Differential Ge-
+ometry - Proceedings of the VIII International Colloquium ( Ed.´Alvarez L´ opez & E. Garc´ ıa-R´ ıo),
+WorldScientific2009,102–103.
+8Infact,we givethe Poincar´ epolynomial PMofM.Gysinsequence ...August28, 2018 ˙ 12
+[6] J. I. Royo Prieto, Estudio Cohomol ´ogicode flujos riemannianos, Ph.D. Thesis, Universityof the
+BasqueCountryUPV /EHU, 2003.
+[7] M. Saralegi: A Gysin sequence for semi-free actions of S3,Proc. Amer. Math. Soc. 118(1993),
+no.4, 1335–1345.
+[8] A. Verona: Le th´ eor` eme de de Rham pour les pr´ estratifications abstra ites,C. R. Acad. Sci. Paris
+S´ er. A-B273(1971),886–889
+[9] A.Verona: Stratifiedmappings-structuresandtriangulability, LectureNotesinMathematics, 1102,
+Springer-Verlag, Berlin, 1984.
+[10] C. T. Yang : The triangulability of the orbit space of a di fferentiable transformation group,
+Bull.Amer.Math.Soc. 69(1963), 405–408.
\ No newline at end of file
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+arXiv:1001.0389v1  [hep-ph]  3 Jan 2010αsDetermination from τDecays: Theoretical Status∗
+Antonio Pich
+Departament de F´ ısica Te` orica, IFIC, Universitat de Val` encia - CSIC
+Apt. Correus 22085, E-46071 Val` encia, Spain
+The totalτhadronic width can be accurately calculated using analyt-
+icity and the operator product expansion. The result turns out to be very
+sensitive to the value of αs(m2
+τ), providing a precise determination of the
+strong coupling constant. The theoretical description of this obs ervable is
+updated, including the recently computed O(α4
+s) contributions. The exper-
+imental determination of αs(m2
+τ) and its actual uncertainties are discussed.
+PACS numbers: 12.38.-t, 12.38.Bx, 12.38.Qk
+1. Introduction
+The inclusive hadronic decay width of the τlepton provides one of the
+most precise measurements of the strong coupling [1–7]. Mor eover, the
+comparison of αs(m2
+τ) withαsdeterminations at higher energies constitutes
+the most accurate test of asymptotic freedom, successfully confirming the
+predicted running of the QCD coupling at the four-loop level .
+The calculation of the O(α4
+s) contribution [8] has triggered a renewed
+theoretical interest on the αs(m2
+τ) determination, since it allows to push
+the accuracy to the four-loop level [8–13]. However, the rec ent theoretical
+analysesslightly disagreeonthefinalresult, givingriset oarangeofdifferent
+values forαs(m2
+τ). The differences among these results, shown in Table 1,
+are too large compared with the claimed O(α4
+s) accuracy and originate in
+the different inputs or theoretical procedures which have bee n adopted.
+In the following I try to clarify the reasons behind these num erical dis-
+crepancies andreassesstheactual uncertainties ofthe τdecay determination
+ofαs. Using all present experimental and theoretical knowledge , I derive
+the value
+αs(m2
+τ) = 0.342±0.012. (1)
+∗Presented at the Flavianet topical workshop on Low energy constraints on extensions
+of the Standard Model , Kazimierz, Poland, 23-27 July 20092
+Reference Method δP αs(m2
+τ)
+Baikov et al. [8] CIPT, FOPT 0.1998±0.00430.332±0.016
+Davier et al. [9] CIPT 0.2066±0.00700.344±0.009
+Beneke-Jamin [10] BSR + FOPT 0.2042±0.00500.316±0.006
+Maltman-Yavin [11] PWM + CIPT — 0.321±0.013
+Menke [12] CIPT, FOPT 0.2042±0.00500.342+0.011
+−0.010
+Caprini-Fischer [13] BSR + CIPT 0.2042±0.00500.320+0.011
+−0.009
+Table 1. O(α4
+s) determinations of αs(m2
+τ).
+2. Theoretical framework
+The inclusive character of the total τhadronic width renders possible
+an accurate calculation of the ratio [1–5]
+Rτ≡Γ[τ−→ντhadrons(γ)]
+Γ[τ−→ντe−¯νe(γ)]=Rτ,V+Rτ,A+Rτ,S.(2)
+The theoretical analysis involves the two-point correlati on functions for the
+left-handed quark currents Lµ
+ij=¯ψjγµ(1−γ5)ψi(i,j=u,d,s):
+Πµν
+ij(q)≡i/integraldisplay
+d4xeiqx/an}bracketle{t0|T(Lµ
+ij(x)Lν
+ij(0)†)|0/an}bracketri}ht
+=/parenleftBig
+−gµνq2+qµqν/parenrightBig
+Π(1)
+ij(q2) +qµqνΠ(0)
+ij(q2).(3)
+Using analyticity, Rτcan be written as a contour integral in the complex
+s-plane running counter-clockwise around the circle |s|=m2
+τ:
+Rτ= 6πi/contintegraldisplay
+|s|=m2τds
+m2τ/parenleftbigg
+1−s
+m2τ/parenrightbigg2/bracketleftbigg/parenleftbigg
+1+2s
+m2τ/parenrightbigg
+Π(0+1)(s)−2s
+m2τΠ(0)(s)/bracketrightbigg
+,
+(4)
+where Π(J)(s)≡ |Vud|2Π(J)
+ud(s) +|Vus|2Π(J)
+us(s). This expression requires
+the correlators only for complex sof orderm2
+τ, which is significantly larger
+than the scale associated with non-perturbative effects. Usi ng the Opera-
+tor Product Expansion (OPE) to evaluate the contour integra l,Rτcan be
+expressed as an expansion in powers of 1 /m2
+τ. The uncertainties associated
+with the use of the OPE near the time-like axis are heavily sup pressed by
+the presence in (4) of a double zero at s=m2
+τ[1,9,14].
+The theoretical prediction for the Cabibbo-allowed decay w idth can be
+written as [1]
+Rτ,V+A=NC|Vud|2SEW{1+δP+δNP}, (5)3
+whereNC= 3 is the number of quark colours and SEW= 1.0201±0.0003
+contains the electroweak radiative corrections [15–17]. T he dominant cor-
+rection (∼20%) is the perturbativeQCDcontribution inthemassless-q uark
+limitδP, whichis already knownto O(α4
+s) [1,8]. Quarkmass effects [1,18,19]
+are tiny for the Cabibbo-allowed current and amount to a negl igible correc-
+tion smaller than 10−4[1,10].
+Non-perturbative contributions are suppressed by six powe rs of theτ
+mass [1] and, therefore, are very small. Their numerical siz e has been deter-
+mined from the invariant-mass distribution of the final hadr ons inτdecay,
+through the study of weighted integrals [20] which can be cal culated theo-
+retically in the same way as Rτ. The predicted suppression [1] of the non-
+perturbative corrections has been confirmed by ALEPH [21], C LEO [22]
+and OPAL [23]. The most recent analysis [6] gives
+δNP=−0.0059±0.0014. (6)
+The measured values of the τlifetime and leptonic branching ratios im-
+plyRτ= 3.640±0.010 [9]. Subtracting the Cabibbo-suppressed contribu-
+tionRτ,S= 0.1615±0.0040 [9], one obtains Rτ,V+A= 3.479±0.011. Using
+|Vud|= 0.97418±0.00027 [24] and (6), the pure perturbative contribution
+toRτis determined to be:
+δP= 0.2038±0.0040. (7)
+3. Perturbative contribution to R τ
+In the chiral limit ( mu=md=ms= 0), the vector and axial-vector
+currents are conserved. This implies sΠ(0)(s) = 0; therefore, only the cor-
+relator Π(0+1)(s) contributes to Eq. (4). The result is more conveniently
+expressed in terms of the logarithmic derivative of the two- point correlation
+function of the vector (axial) current, Π( s) =1
+2Π(0+1)(s), which satisfies an
+homogeneous renormalization–group equation:
+D(Q2)≡ −Q2d
+dQ2Π(Q2) =NC
+12π2/summationdisplay
+n=0Kn/parenleftBigg
+αs(Q2)
+π/parenrightBiggn
+.(8)
+With the choice of renormalization scale µ2=Q2≡ −sall logarithmic
+corrections, proportional to powers of log( −s/µ2), have been summed into
+the running coupling. The Kncoefficients are known to order α4
+s. For
+nf= 3 quark flavours, one has [8,25–29]: K0=K1= 1,K2= 1.63982,
+KMS
+3= 6.37101 andKMS
+4= 49.07570.
+The perturbative component of Rτis given by
+δP=/summationdisplay
+n=1KnA(n)(αs), (9)4
+LoopsA(1)(αs)A(2)(αs)A(3)(αs)A(4)(αs)A(5)(αs)δP
+10.13247 0.01570 0.001698 0.000169 0.0000154 0.1773
+20.13523 0.01575 0.001629 0.000151 0.0000124 0.1788
+30.13540 0.01565 0.001597 0.000145 0.0000115 0.1784
+40.13529 0.01557 0.001579 0.000142 0.0000112 0.1779
+Table 2. Exact results for A(n)(αs) (n≤5) at different β-function approximations,
+and corresponding values of δP=/summationtext4
+n=1KnA(n)(αs), forαs(m2
+τ)/π= 0.1.
+where the functions [2]
+A(n)(αs)≡1
+2πi/contintegraldisplay
+|s|=m2τds
+s/parenleftbiggαs(−s)
+π/parenrightbiggn/parenleftBigg
+1−2s
+m2τ+2s3
+m6τ−s4
+m8τ/parenrightBigg
+(10)
+are contour integrals in the complex plane which only depend on the strong
+coupling. Using the exact solution (up to unknown βn>4contributions) for
+αs(s) given by the renormalization-group β-function equation, they can be
+numerically computed with a very high accuracy [2].
+Table 2 gives the numerical values for A(n)(αs) (n≤5) obtained at the
+one-, two-, three- and four-loop approximations (i.e. βn>1= 0,βn>2= 0,
+βn>3= 0 andβn>4= 0, respectively), together with the corresponding re-
+sults forδP=/summationtext4
+n=1KnA(n)(αs), takingαs(m2
+τ)/π= 0.1. The perturbative
+convergence is very good. Theerrorinducedby thetruncatio n of theβfunc-
+tion at fourth order can be conservatively estimated throug h the variation
+of the results at five loops, assuming β5=±β2
+4/β3=∓443, i.e. a geometric
+growth of the βfunction. Higher-order contributions to the Adler functio n
+D(Q2) will be taken into account adding the fifth-order term K5A(5)(αs)
+withK5= 275±400. Including, moreover, the5-loop variation withchange s
+of the renormalization scale in the range µ2∈[0.5,1.5], one gets the final
+resultδP= 0.1810±0.0045K5±0.0013β5±0.0013µ= 0.1810±0.0049 for
+αs(m2
+τ)/π= 0.1.
+Adopting this very conservative procedure, the experiment al value ofδP
+given in Eq. (7) implies the strongcoupling determination q uoted in Eq. (1).
+4. Discussion on previous O(α4
+s) determinations of αs(m2
+τ)
+The integrals A(n)(αs) can be expanded in powers of aτ≡αs(m2
+τ)/π,
+A(n)(αs) =an
+τ+O(an+1
+τ). One recovers in this way the naive perturbative
+expansionδP=/summationtext
+n=1(Kn+gn)an
+τ≡/summationtext
+n=1rnan
+τ[2]. This approximation
+is known as fixed-order perturbation theory (FOPT), while the improved
+expression (9), keeping the non-expanded values for the int egralsA(n)(αs),5
+is usuallycalled contour-improved perturbation theory (CIPT) [2,30]. FOPT
+gives rise to a pathological non-convergent series (its rad ius of convergence
+is slightly smaller than the physical value of aτ), because the long running
+ofαs(s) along the circle |s|=m2
+τgenerates very large gncoefficients, which
+depend on Km<nandβm<n[2]:g1= 0,g2= 3.56,g3= 19.99,g4=
+78.00,g5= 307.78. These corrections are much larger than the original Kn
+contributions and lead to values of αs(m2
+τ) smaller than (1). FOPT suffers
+from a huge renormalization-scale dependence [2]. As shown in a recent
+detailed analysis [12], the actual FOPT uncertainties are m uch larger than
+usually estimated. Once this is taken properly into account , the FOPT
+results are consistent with CIPT, but their huge uncertaint ies make them
+irrelevant.
+An ad-hoc model of higher-order coefficients for the Adler fun ction has
+been recently advocated [10]. The model mixes three different types of
+renormalons ( n=−1, 2 and 3) plus a linear polynomial, trying to enforce a
+cancelation of the Knandgncoefficients in order to get a better behaviour
+of the FOPT series. It contains 5 free parameters which are de termined
+by the known coefficients of the Adler series and the chosen val ue ofK5.
+Making a Borel summation of the full renormalon series (BSR) , one gets a
+sizeable positive contribution to δPfrom higher orders, implying a smaller
+value forαs(m2
+τ) independently of the adopted FOPT [10] or CIPT [13]
+procedure. Thismodelconstitutes aninterestingexampleo fpossiblehigher-
+order corrections, makingapparent that theassociated unc ertainties have to
+becarefullyestimated. However, itcannotbeusedtodeterm inethephysical
+value ofαs, because the result is model dependent. Different assumption s
+about the unknown Kncoefficients would lead to different central values for
+αs.
+The determination of αs(m2
+τ) in Eq. (1) agrees with Refs. [6,12] and
+with the CIPT value in Ref. [8]. It is somewhat larger than the (CIPT)
+result extracted in Ref. [11] from pinched-weight moments ( PWM) of the
+hadronic distribution, which seems to correspond to a very d ifferent value of
+δNP. After evolution up to the scale MZ[31], the strong coupling in Eq. (1)
+decreases to
+αs(M2
+Z) = 0.1213±0.0014, (11)
+in excellent agreement with the direct measurements at the Z peak.
+Acknowledgements
+This work has been supported in part by the EU Contract MRTN-
+CT-2006-035482 (FLAVIAnet), by MICINN, Spain (grants FPA2 007-60323
+and Consolider-Ingenio 2010 CSD2007-00042 –CPAN–) and by G eneralitat
+Valenciana (Prometeo/2008/069, ACOMP/2009/312).6
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+arXiv:1001.0390v2  [math.DS]  2 May 2010A DIRECTIONAL UNIFORMITY OF PERIODIC
+POINT DISTRIBUTION AND MIXING
+RICHARD MILES AND THOMAS WARD
+Abstract. For mixing Zd-actions generated by commuting au-
+tomorphisms of a compact abelian group, we investigate the di-
+rectional uniformity of the rate of periodic point distribution and
+mixing. When each of these automorphismshas finite entropy, it is
+shown that directional mixing and directional convergence of the
+uniform measure supported on periodic points to Haar measure
+occurs at a uniform rate independent of the direction.
+1.Introduction
+It is well-known that, under mild hypotheses, sufficiently smooth
+functions mix at an exponential rate, and periodic point measures b e-
+comeuniformlydistributed onsmoothfunctionsatanexponential r ate,
+for dynamical systems with hyperbolic behaviour or comparable reg u-
+larity properties. For example, Bowen [2, 1.26] shows an ‘exponent ial
+cluster property’, that Anosov diffeomorphisms preserving a smoo th
+measure mix Lipschitz functions exponentially fast, and Lind [9] sho ws
+similar properties for H¨ older functions on quasihyperbolic toral au to-
+morphisms. Oncompactgroups, smoothnessconditionscanbephr ased
+in terms of how well a function can be approximated by a function
+with finitely supported Fourier transform (that is, by trigonometr ic
+polynomials). Thus for a group automorphism α:X→Xof a
+compact metric abelian group X, and an exhaustive increasing se-
+quenceH1⊂H2⊂ ···of finite subsets of the character group /hatwideX,
+the rate of mixing and the rate of uniform distribution of periodic
+points amount to the existence of functions φandψ, withφ(k)→ ∞
+andψ(k)→ ∞ask→ ∞, such that
+(i)Hk∩/hatwideαkHk={0}for|n|>φ(k) (a rate of mixing), and
+(ii)Hk∩(/hatwideαk−1)Hk={0}for|n|>ψ(k) (arateofequidistribution
+of periodic points).
+Date: Draft - November 1, 2018.
+2000Mathematics Subject Classification. 37C25, 37C40, 37C35.
+12 RICHARD MILES AND THOMAS WARD
+Statement (i) gives a class of functions C(X) with prescribed decay of
+Fourier coefficients, a rate function φ′=o(1), andC=C(f,g) for
+which
+f,g∈ C(X) =⇒/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+f(x)g(αnx)dµ(x)−/integraldisplay
+fdµ/integraldisplay
+gdµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle<Cφ′(n),
+(1)
+whereµdenotes Haar measure on X. Statement (ii) gives a rate func-
+tionψ′=o(1) and a constant C=C(f,α) for which
+f∈ C(X) =⇒/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+fdµn−/integraldisplay
+fdµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle<Cψ′(n), (2)
+whereµndenotes Haar measure on the subgroup of points fixed by αn.
+For a given group X, the class of test functions on which the mixing
+and uniform distribution may be seen depends, via the exhaustive se -
+quence, on the functions φandψ. For explicit calculations of this sort
+whenX=Tdis the torus, and C(X) is a class of H¨ older functions, see
+Lind [9, Sect. 4].
+Our interest here is in commuting automorphisms with finite en-
+tropy, together defining an algebraic Zd-actionα(anentropy rank one
+actionin the sense of Einsiedler and Lind [5]); this means that αis
+a homomorphism from Zdto the group of continuous automorphisms
+ofX. We writeαnfor the automorphism α(n). Examples include com-
+muting toral automorphisms, Ledrappier’s example [8], the (invertib le
+extension of the) ×2,×3 system, and many others (Schmidt’s mono-
+graph [13] describes many dynamical properties of these systems ).
+A non-uniform rate of mixing, or a non-uniform rate of convergenc e
+of periodic point measures, for a Zd-actionα, corresponds to the state-
+ments (1) and (2) respectively for each of the maps αnwithn/\e}atio\slash= 0. The
+uniformityof thetitleamountsto asking if, having fixed anappropria te
+exhaustion H1⊂H2⊂ ···of the character group /hatwideX, there is a uniform
+waytochoose thefunctions φandψwitnessing adirectional uniformity
+in mixing and in the distribution of periodic points. We show that the
+decay functions φ′andψ′can be chosen so that they depend only on
+the distance from the origin in Zd, by proving the following theorem.
+Theorem 1. Let(X,α)be a mixing entropy rank one Zd-action by au-
+tomorphisms of a compact abelian group Xsatisfying the Descending
+Chain Condition on closed α-invariant subgroups. Write µfor Haar
+measure on Xandµnfor Haar measure on the subgroup of points
+fixed by the automorphism αn. Then there is a class of smooth func-
+tionsC(X)strictly containing the trigonometric polynomials, and ra te3
+functionsφ′=o(1),ψ′=o(1), such that, for any f,g∈ C(X),
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+f(x)g(αnx)dµ(x)−/integraldisplay
+fdµ/integraldisplay
+gdµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle<C(f,g)φ′(/bardbln/bardbl),
+and /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+fdµn−/integraldisplay
+fdµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle<C(f)ψ′(/bardbln/bardbl).
+In addition to the motivation already given, this question arose as
+a result of our paper [12], where it is shown that for a large class of
+such systems there is a uniform lower bound to the exponential rat e of
+growth in the number of periodic points, and the papers [10] and [11 ],
+in which more subtle directionally uniform bounds and counts for peri-
+odic points are found. Because of the diversity of underlying compa ct
+groups, and our main interest in uniformity, we have not delved into
+the articulation between the growth in the exhaustive sequence (m ea-
+suring the smoothness of the function class) and the permitted gr owth
+in the control functions φandψ(measuring the rate of mixing or of
+equidistribution of periodic points).
+Our methods combine the formalism introduced by Kitchens and
+Schmidtin[7],Diophantinearguments, andEinsiedlerandLind’sadelic
+Lyapunov vectors for entropy rank one actions [5].
+2.Algebraic Zd-actions
+Following Kitchens and Schmidt, we exploit the correspondence be-
+tween analgebraic Zd-actionαby automorphisms of a compact abelian
+metric group Xand a module over the ring of Laurent polynomi-
+alsRd=Z[u±1
+1,...,u±1
+d]. This is achieved by identifying each dual
+automorphism /hatwideαnwith multiplication by un=un1
+1···und
+d, then extend-
+ing this in a natural way to polynomials. As a result, attention may
+be restricted to a fixed Rd-moduleMwith dynamical properties of α
+translated into algebraic properties of M. For example, the Descend-
+ing Chain Condition on closed α-invariant subgroups of Xcorresponds
+toMbeing Noetherian. The mixing property for αtranslates to mul-
+tiplication by uknbeing injective on Mfor allk∈Nand alln/\e}atio\slash= 0.
+These two properties will be assumed throughout. A full introduct ion
+to algebraic Zd-actions and the correspondence just described is given
+in Schmidt’s monograph [13].
+Some especially useful algebraic machinery is available when αhas
+entropy rank one (that is, when each element of the action has finit e
+topological entropy). Since Mis assumed to be Noetherian, it has a
+finite set of associated prime ideals Ass( M)⊂Spec(Rd). Furthermore,4 RICHARD MILES AND THOMAS WARD
+sinceαis mixing and of entropy rank one, for each p∈Ass(M), the
+moduleRd/phas Krull dimension one, written kdim( Rd/p) = 1 (see [5,
+Prop. 6.1] and [10, Lem. 2.3]). Therefore, the field of fractions of Rd/p
+is a global field that we denote by K(p). LetP(K(p)) denote the set
+of places of K(p),|·|vthe absolute value corresponding to the place v,
+and set
+S(p) ={v∈ P(K(p))| |Rd/p|vis an unbounded subset of R},
+which is a finite set because Rd/pis finitely generated. Following Ein-
+siedler and Lind, associate to pthe list of Lyapunov vectors
+L(p) ={ℓv= (log|π(u1)|v,...,log|π(ud)|v)|v∈S(p)},
+whereπ:Rd→Rd/pdenotes the usual quotient map.
+In what follows, our approach is to prove an algebraic version of
+Theorem 1 for a module of the form Rd/p, and then build up to a
+general Noetherian module Musing standard methods (see [14] for
+example).
+3.Two uniformities
+LetMbe anRd-module, and suppose that/parenleftbig
+HM
+k/parenrightbig
+k/greaterorequalslant1is an increasing
+sequence of finite subsets of Mwith/uniontext∞
+k=1HM
+k=M(that is, an ex-
+haustive sequence). Let φMandψMbe functions (to be chosen later)
+withφM(k),ψM(k)→ ∞ask→ ∞. We are interested in the following
+two properties of M.
+I:HM
+k∩unHM
+k={0}for alln∈Zdwith/bardbln/bardbl> φM(k) (direc-
+tional uniformity of mixing),
+II:HM
+k∩(un−1)HM
+k={0}for alln∈Zdwith/bardbln/bardbl> ψM(k)
+(directional uniformity of distribution of periodic points).
+Theorem 2. Letp⊂Rdbe a prime ideal with kdim (Rd/p) = 1, and
+suppose that θ(k)ր ∞ask→ ∞. If the module M=Rd/pcorre-
+sponds to a mixing action, then there exists an exhaustive in creasing
+sequence/parenleftbig
+HM
+k/parenrightbig
+k/greaterorequalslant1of finite subsets of M, and a constant B >0, such
+that Property Iis satisfied for
+φM(k) =Blogθ(k).
+The proof of Theorem 2 is facilitated by the following result, which
+is adapted from [12].
+Lemma 3. If the prime ideal p⊂Rdhas kdim (Rd/p) = 1, and the
+moduleRd/pcorresponds to a mixing algebraic Zd-action, then the set
+of Lyapunov vectors L(p)spansRd.5
+Proof.This can be seen using properties of the directional entropy
+functionh:Rd→R(see [3] and [6]). The proof of [12, Th. 1.1] shows
+thathis bounded away from zero when the action is mixing. If L(p)
+does not span Rd, then there exists w∈Sd−1={z∈Rd| /bardblz/bardbl= 1}
+such that
+h(w) =/summationdisplay
+v∈Vmax{ℓv·w,0}= 0,
+giving an immediate contradiction. /square
+Proof of Theorem 2. Write/hatwiden=n//bardbln/bardblfor any non-zero integer vec-
+torn. We claim that there is a constant C >0 such that given any
+non-zero vector n∈Zd, there exists w∈S(p) such that |ℓw·/hatwiden|>C. If
+this were not the case, then compactness of Sd−1would give a point z
+inSd−1such that/summationtext
+v∈S(p)|ℓv·z|= 0 (since w/mapsto→/summationtext
+v∈S(p)|ℓv·w|is
+continuous on Sd−1), meaning that L(p) would lie in the subspace or-
+thogonal to z, contradicting Lemma 3.
+Set
+HM
+k={a∈M|θ(k)−1/lessorequalslant|a|v/lessorequalslantθ(k) for allv∈S(p)}∪{0}.
+and note that/parenleftbig
+HM
+k/parenrightbig
+k/greaterorequalslant1is an increasing exhaustive sequence of finite
+subsets ofMsinceθ(k)ր ∞ask→ ∞.
+Givenn∈Zdwith/bardbln/bardbl> φM(k), there exists v∈S(p) such that
+|ℓv·/hatwiden|>C. Leta∈HM
+kbe non-zero. If ℓv·/hatwiden<0, then
+|π(un)a|v= exp(/bardbln/bardblℓv·/hatwiden)|a|v
+<exp/parenleftbigg−Clogθ(k)
+C/parenrightbigg
+θ(k)1/2
+=θ(k)−1/2.
+On the other hand, if ℓv·/hatwiden>0, then
+|π(un)a|v= exp(/bardbln/bardblℓv·/hatwiden)|a|v
+>exp/parenleftbiggClogθ(k)
+C/parenrightbigg
+θ(k)−1/2
+=θ(k)1/2.
+Hence,una/\e}atio\slash∈HM
+k, and the statement of the theorem follows by set-
+tingB=1
+C. /square
+WenowturnourattentiontoPropertyII.Foramixingactionarising
+from a Noetherian module M, an essential consequence of the entropy
+rank one assumption is that for each n∈Zd, the set of points fixed
+byαnis finite, and the cardinality of this set is equal to |M/(un−1)M|
+by duality.6 RICHARD MILES AND THOMAS WARD
+Theorem 4. Letp⊂Rdbe a prime ideal with kdim (Rd/p) = 1,
+and suppose that θ(k)ր ∞ask→ ∞. If the module M=Rd/p
+corresponds to a mixing action, then there exists an increas ing ex-
+haustive sequence H=/parenleftbig
+HM
+k/parenrightbig
+k/greaterorequalslant1of finite subsets of M, and con-
+stantsσ,A,C 1,C2>0such that Property IIis satisfied for
+ψM(k) = max/braceleftbigg
+C1,σ+1
+C2log/parenleftbiggθ(k)
+(Aσ/2)1/(σ+1)/parenrightbigg/bracerightbigg
+.
+Proof.Just as in the proof of Theorem 2, there is a constant C >0
+such that given any non-zero n∈Zd, there exists w∈S(p) such that
+|ℓw·/hatwiden|>C.
+Without loss of generality, we may always choose wsuch that
+ℓw·/hatwiden>C.
+For,givenw∈S(p)suchthat ℓw·/hatwiden<−C,wecanconsider/producttext
+v∈S(p)|π(un)|v
+asfollows: Since π(un) is a unit in M, the product formula implies that
+/productdisplay
+v∈S(p)\{w}|π(un)|v=|π(un)|−1
+w.
+Hence, for some v∈S(p)\{w}it follows that
+|π(un)|v>|π(un)|−1/σ
+w,
+whereσ=|S(p)|−1. Therefore,
+exp(/bardbln/bardblℓv·/hatwiden)>exp(−/bardbln/bardblℓw·/hatwiden/σ).
+Henceℓv·/hatwiden>C/σ, and we simply need to replace CbyC/σ.
+As before, set
+HM
+k={a∈M|θ(k)−1/lessorequalslant|a|v/lessorequalslantθ(k) for allv∈S(p)}∪{0},
+which again defines an increasing exhaustive sequence since θ(k)ր ∞
+ask→ ∞. Fixǫ >0 and let n∈Zdsatisfy/bardbln/bardbl> ψM(k), where in
+the definition of ψM,σ=|S(p)|−1,C2=C−ǫ, and the constants A
+andC1are to be specified later. Let a∈HM
+kbe non-zero and suppose
+that (un−1)a∈HM
+k. This implies that |π(un−1)a|w<θ(k), so
+|a|w<θ(k)
+|π(un)−1|w<2θ(k)
+|π(un)|w. (3)
+By the product formula/producttext
+v∈P(K(p))|a|v= 1, so
+/productdisplay
+v∈S(p)\{w}|a|v=|a|−1
+w/productdisplay
+v∈P(K(p))\S(p)|a|−1
+v/greaterorequalslant|a|−1
+w,7
+as|a|v/lessorequalslant1 for allv∈ P(K(p))\S(p). Thus at least one v∈S(p)\{w}
+satisfies
+|a|v/greaterorequalslant|a|−1/σ
+w>/parenleftbigg|π(un)|w
+2θ(k)/parenrightbigg1/σ
+, (4)
+by (3). Ifvis archimedean, then by Baker’s Theorem [1], there exist
+constantsA,B >0 such that
+|π(un)−1|v/greaterorequalslantA
+max1/lessorequalslanti/lessorequalslantd{ni}B.
+Ifvisnon-archimedean, asimilarboundholdsbyYu’sTheorem[15]. In
+boththearchimedeanandnon-archimedeancases, giventheideal p, the
+constants arising can (in principle) be computed explicitly. Combining
+these bounds with (4) gives
+|π(un−1)a|v=|π(un−1)|v|a|v/greaterorequalslantA|π(un)|1/σ
+w
+max1/lessorequalslanti/lessorequalslantd{ni}B(2θ(k))1/σ
+=Aexp(/bardbln/bardblℓw·/hatwiden/σ)
+max1/lessorequalslanti/lessorequalslantd{ni}B(2θ(k))1/σ
+/greaterorequalslantAexp(/bardbln/bardblC/σ)
+max1/lessorequalslanti/lessorequalslantd{ni}B(2θ(k))1/σ
+/greaterorequalslantA
+(2θ(k))1/σexp/parenleftbigg(C−ǫ)/bardbln/bardbl
+σ/parenrightbigg
+,(5)
+provided that /bardbln/bardblis large enough to ensure that
+max
+1/lessorequalslanti/lessorequalslantd{ni}B/lessorequalslantexp(/bardbln/bardblǫ/σ).
+We may ensure this by a suitable choice of C1=C1(ǫ) in the definition
+ofψM(k), since/bardbln/bardbl> ψM(k). Furthermore, since /bardbln/bardbl> ψM(k), the
+right-hand side of (5) is strictly greater than
+A
+(2θ(k))1/σ/parenleftbiggθ(k)
+(Aσ/2)1/(σ+1)/parenrightbigg1+1/σ
+=θ(k),
+so (un−1)a/\e}atio\slash∈HM
+k, disagreeing with our contrary assumption which
+therefore must have been false. /square
+Theorems 2 and 4 describe (in an opaque form) uniformity in rate of
+mixing and in the distribution of periodic points respectively for cyclic
+systems – those corresponding to cyclic Rd-modules. As usual, we need
+arguments from commutative algebra to build up to a more general
+picture. The next lemma allows the two properties to be inherited by
+a suitable extension of one action by another.8 RICHARD MILES AND THOMAS WARD
+Lemma 5. LetL,MbeRd-modules with L⊂M, and suppose that
+bothLandM/Lare mixing.
+(i)If Property Iholds forLand forM/L, then there is an appro-
+priate increasing exhaustive sequence/parenleftbig
+HM
+k/parenrightbig
+k/greaterorequalslant1and function
+φM(k) = max{φL(k),φM/L(k)},
+such that it also holds for M.
+(ii)If Property IIholds forLand forM/L, then there is an appro-
+priate increasing exhaustive sequence/parenleftbig
+HM
+k/parenrightbig
+k/greaterorequalslant1and function
+ψM(k) = max{ψL(k),ψM/L(k)},
+such that it also holds for M.
+Proof.For eachk∈N, let
+HM
+k=/uniondisplay
+x∈K∩π−1(HM/L
+k)x+HL
+k,
+whereπ:M→M/Lis the natural quotient map of Rd-modules,
+andKis a set of coset representatives containing 0. By construction,
+eachHk
+Mis finite and/uniontext∞
+k=1HM
+k=M.
+(i) Suppose that Property I is violated for some nwith/bardbln/bardbl>φM(k).
+Then there exist w,x∈K∩π−1(HM/L
+k) andg,h∈HL
+ksuch that
+un(x+h) =w+g/\e}atio\slash= 0. (6)
+In particular, unπ(x) =π(w), which means that π(w) = 0 since Prop-
+erty I holds for M/Lby hypothesis. Therefore, w= 0 by our choice
+ofK, andπ(x) = 0asmultiplicationby unisanautomorphismof M/L.
+It followsthat x= 0by our choice of Kandso (6) implies that unh=g
+meaning that g= 0, as Property I holds for L. So,w+g= 0, contra-
+dicting (6).
+(ii) Suppose that Property II is violated for some nwith/bardbln/bardbl>ψM(k).
+Then there exist x∈K,w∈K∩π−1(HM/L
+k),g∈HL
+kandh∈Lsuch
+that
+(un−1)(x+h) =w+g/\e}atio\slash= 0. (7)
+Thus (un−1)π(x) =π(w), which means that π(w) = 0 since Prop-
+ertyIIholdsfor M/L. Thisforces π(x) = 0asmultiplicationby( un−1)
+is injective on M/L(sinceM/Lcorresponds to a mixing system).
+Thereforex= 0 by our choice of K, and so (7) implies that ( un−1)h=
+gmeaningg= 0, as Property II holds for L. Sow+g= 0, contradict-
+ing (7). /square9
+The next lemma shows that bothproperties are inherited when pass -
+ing to factors of systems (factors of algebraic Zd-actions correspond to
+submodules under duality).
+Lemma 6. LetL,MbeRd-modules with L⊂Mand suppose that M
+is mixing.
+(i)If Property Iholds forMthen there is an appropriate increas-
+ing sequence/parenleftbig
+HL
+k/parenrightbig
+k/greaterorequalslant1and function φL(k) =φM(k)such that
+it also holds for L.
+(ii)If Property IIholds forMthen there is an appropriate increas-
+ing exhaustive sequence/parenleftbig
+HL
+k/parenrightbig
+k/greaterorequalslant1and function ψL(k) =ψM(k)
+such that it also holds for L.
+Proof.For eachk∈NletHL
+k=HM
+k∩L. Then/uniontext∞
+k=1HL
+k=Land
+eachHL
+kis finite.
+(i) Forn∈Zdwith/bardbln/bardbl>φL(k),
+HL
+k∩unHL
+k=HM
+k∩L∩un(HM
+k∩L)
+=HM
+k∩L∩unHM
+k∩unL,
+since multiplication by unis injective. Furthermore, the right-hand
+side is{0}, as Property I holds for M.
+(ii) Similarly, for n∈Zdwith/bardbln/bardbl>ψL(k),
+HL
+k∩(un−1)HL
+k=HM
+k∩L∩(un−1)(HM
+k∩L)
+=HM
+k∩L∩(un−1)HM
+k∩(un−1)L,
+as multiplication by un−1 is injective by the mixing assumption. Fur-
+thermore, the right-hand side is {0}, as Property II holds for M./square
+We are now ready to pass the two uniformity properties up from
+cyclic modules to Noetherian modules.
+Theorem 7. LetMbe a Noetherian Rd-module corresponding to a
+mixing algebraic Zd-action of entropy rank one, and suppose θ(k)ր ∞
+ask→ ∞. Then there is an increasing exhaustive sequence/parenleftbig
+HM
+k/parenrightbig
+k/greaterorequalslant1
+of finite subsets of M, and there are constants B,C >0, such that
+Property Iis satisfied for
+φM(k) =Blogθ(k),
+and Property IIis satisfied for
+ψM(k) =Clogθ(k).
+Proof.Write Ass(M) ={p1,...,pr}and note that kdim( Rd/pi) = 1
+for each 1 /lessorequalslanti/lessorequalslantrby the mixing and entropy rank one assump-
+tions. By [13, Cor. 6.3] or [14, Sect. 4], Membeds in a module of the10 RICHARD MILES AND THOMAS WARD
+formM′=/circleplustextr
+i=1M(i), where each Rd-moduleM(i) for 1/lessorequalslanti/lessorequalslantr, has
+a prime filtration of the form
+{0}=N(i)
+0⊂N(i)
+1⊂ ··· ⊂N(i)
+s(i)=M(i), (8)
+withN(i)
+j/N(i)
+j−1∼=Rd/pifor all 1/lessorequalslantj/lessorequalslants(i). Each of these modules is
+mixing by [13, Th. 6.5].
+We first consider Property I. For each module M(i), using Theo-
+rem 2, Lemma 5, and induction on the prime filtration (8), we may
+find an increasing exhaustive sequence/parenleftBig
+HM(i)
+k/parenrightBig
+k/greaterorequalslant1of finite subsets
+ofM(i) such that Property I is satisfied for
+φM(i)(k) =Bilogθ(k),
+whereBi>0 is the constant appearing in Theorem 2 (which follows
+from the proof of Lemma 5). For each k∈N, set
+HM′
+k=r/circleplusdisplay
+i=1HM(i)
+k
+and
+φM′(k) =Blogθ(k),
+whereB= max 1/lessorequalslanti/lessorequalslantr{Bi}. Therefore, Property I holds for M′and the
+required result follows by applying Lemma 6.
+Property II is obtained in an analogous way, noting that ψM(k) can
+be replaced by ψM(k) =Clogθ(k) for a suitably large choice of Cin
+Theorem 4. /square
+4.Remarks
+(1) Theorem 7 gives Theorem 1 simply by translation: the class of
+functions C(X) is defined by choosing a rate of decay for the size of
+coefficients in the Fourier expansion outside HM
+kso rapid that the sum
+overM\Hkiso(1) ink, choosingθ, and then computing φ′
+Mandψ′
+M.
+(2) Throughout, the function θcan be chosen arbitrarily. We have
+left it in the statements to facilitate future more explicit estimates f or
+specific classes of compact groups.
+(3)Itseemspossiblethattheuniformityinmixingcouldbepresentin
+higher entropy ranks. We initially attempted to prove this using adelic
+amoebas[4]in placeofLyapunov vectors. Fora mixing actionof highe r
+entropy rank, an unpublished argument due to Einsiedler enables on e
+to see that the adelic amoeba spans Rd, just as the set of Lyapunov
+vectorsdoesforanentropyrankoneaction. However, findingas uitable
+exhaustive sequence in the dual module based on this appears to be11
+rather problematic. Notably, however, one only needs to consider an
+action corresponding to a cyclic module as the method of passing up t o
+Noetherian modules used here (Lemmas 5 and 6) works for all entro py
+ranks.
+References
+[1] A. Baker. Linear forms in the logarithms of algebraic numbers. IV .Mathe-
+matika, 15:204–216, 1968.
+[2] R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomo r-
+phisms. Lecture Notes in Mathematics, Vol. 470.Springer-Verlag, Berlin, 1975.
+[3] M. Boyle and D. Lind. Expansive subdynamics. Trans. Amer. Math. Soc. ,
+349(1):55–102, 1997.
+[4] M. Einsiedler, M. Kapranov, and D. Lind. Non-Archimedean amoeb as and
+tropical varieties. J. Reine Angew. Math. , 601:139–157, 2006.
+[5] M. Einsiedler and D. Lind. Algebraic Zd-actions of entropy rank one. Trans.
+Amer. Math. Soc. , 356(5):1799–1831, 2004.
+[6] M. Einsiedler, D. Lind, R. Miles, and T. Ward. Expansive subdynamic s for
+algebraic Zd-actions. Ergodic Theory Dynam. Systems , 21(6):1695–1729,2001.
+[7] Bruce Kitchens and K. Schmidt. Automorphisms of compact grou ps.Ergodic
+Theory Dynam. Systems , 9(4):691–735, 1989.
+[8] F. Ledrappier. Un champ markovien peut ˆ etre d’entropie nulle et m´ elangeant.
+C. R. Acad. Sci. Paris S´ er. A-B , 287(7):A561–A563, 1978.
+[9] D. A. Lind. Dynamical properties of quasihyperbolic toral autom orphisms.
+Ergodic Theory Dynamical Systems , 2(1):49–68, 1982.
+[10] R. Miles. Zeta functions for elements of entropy rank-one act ions.Ergodic
+Theory Dynam. Systems , 27(2):567–582, 2007.
+[11] R. Miles and T. Ward. Periodic point data detects subdynamics in e ntropy
+rank one. Ergodic Theory Dynam. Systems , 26(6):1913–1930, 2006.
+[12] R. Miles and T. Ward. Uniform periodic point growth in entropy ran k one.
+Proc. Amer. Math. Soc. , 136(1):359–365, 2008.
+[13] K. Schmidt. Dynamical systems of algebraic origin , volume 128 of Progress in
+Mathematics . Birkh¨ auser Verlag, Basel, 1995.
+[14] T. Ward. The Bernoulli property for expansive Z2actions on compact groups.
+Israel J. Math. , 79(2-3):225–249, 1992.
+[15] K. R. Yu. Linear forms in p-adic logarithms. II. Compositio Math. , 74(1):15–
+113, 1990.
+School of Mathematics, University of East Anglia, Norwich, NR4
+7TJ, UK
+E-mail address :r.miles@uea.ac.uk
+School of Mathematics, University of East Anglia, Norwich, NR4
+7TJ, UK
+E-mail address :t.ward@uea.ac.uk
\ No newline at end of file
diff --git a/1001.0391.txt b/1001.0391.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6f77ea31edadcfd1b2d5b277f77227c440536faa
--- /dev/null
+++ b/1001.0391.txt
@@ -0,0 +1,117 @@
+Electrically stabilized magnetic vortex and antivortex states 
+ in magnetic dielectrics 
+ 
+A.P. Pyatakov*, G.A. Meshkov 
+ 
+Physics Department, M.V. Lomonosov MSU, Leninskie gori, Moscow, 119991, Russia 
+ 
+The micromagnetic distribution in a dielectric nanoparticle is theoretically considered. It is shown that the existence of 
+inhomogeneous magnetoelectric interaction in magnetic dielectrics provides the possibility to stabilize the vortex and 
+antivortex state in magnetic nanoparticle. The estimation of the critical voltage necessary for vortex/antivortex 
+nucleation in bismuth ferrite and iron garnet nanoparticles yields a value of ±150 V. This system can be considered as 
+electrically switchable two state-logic magnetic element.  
+ 
+ *) pyatakov@physics.msu.ru 
+  
+Numerous micromagnetic structures observed in magnetic media are the result of 
+competition between only a few types of interact ions that include magnetostatic and exchange 
+energy. These two provide us with magnetic vortex structures in ferromagnetic nanodiscs or 
+nanodots [1-6]. An equally fundamental though much less known is antivortex state that is a topological counterpart of the vortex . The realization of an tivortex state is a challenging task since it 
+is unfavorable for magnetostatic reasons due to the formation of the magnetic charges at the edge of 
+the particle. There are only a few reports in which the antivortex was observed either in a complicated four ferromagnetic rings cross-junction structure [7] or as a metastable state of 
+asymmetric cross-like nanostructures [8].  
+Meanwhile there is an additional interaction that should be taken in to account when we 
+consider micromagnetic structure under influence of electric field or the structures in magnetic 
+ferroelectrics (multiferroics). It is proportional to  spatial derivatives of magnetic order parameter 
+vector. This spin flexoelectricity is the variety of magnetoelectric coupling related to the magnetic 
+inhomogeneties and it is described by P
+iMj∇kMn coupling term, where P and M, are electric and 
+magnetic order parameters, resp ectively [9-18]. This type of interaction is responsible for 
+magnetically induced electric polarization in spiral multiferroics [19] and electric field driven 
+magnetic domain wall motion in ferr imagnetic iron garnet films [20]. 
+ In this short Letter we will show that the gradient electric field produced by point electrode 
+(e.g. cantilever tip of atomic force microscope) can stabilize in magnetic dielectric nanoparticle 
+either vortex or antivortex state depending on the electric polarity of the tip.  
+ 
+a) 
+   b)
+  
+Fig. 1 Magnetic dielectric nanoparticle subjected to the electric field from the cantilever tip (the electric polarity 
+depends on the sign of magnetoelectric constant in (1)) 
+a) electrically induced magnetic vortex state b) electrically induced magnetic antivortex state.  
+Magnetoelectric materials, mostly being ferrimagnets or weak ferromagnets, have rather 
+moderate value of spontaneous magnetization. This results in large exchange length 
+22s exch MA l= , where A is the exchange stiffness, MS is saturation magnetization, and large 
+magnetostatic lengths 22s MS M AK l=  (the maximum size of single domain particle, where K is 
+anisotropy constant) compared to conventional ferromagnets. The typical values for room 
+temperature magnetoelectric iron garnet films are:  m lexchμ5,0= , m lMSμ5=  and room temperature 
+multiferroic bismuth ferrite: m lexchμ5.1= , m lMSμ240= . Thus the magnetic dielectric particles of 
+nanometric size should be in homogeneous magnetic state since exchange interaction is the 
+predominant one. 
+The only thing that can compete against the exchange interaction in magnetic dielectrics on 
+this nanometer scale is the spin flexoelectricity that for isotropic media can be represented in the 
+form [10,13]:  
+() ()( )n n n nP ∇⋅−⋅∇⋅⋅⋅=γMEF     ( 1 )  
+where γ is magnetoelectric constant, P is electric polarization, n is unit vector of magnetic 
+order parameter.  
+The linear charge density corresponding to magnetic vortex can be found as in [13]:  
+e LQπγχ2±= ,        ( 2 )  
+where eχ is electric susceptibility of the medium, signs “±” correspond to the vortex and antivortex 
+state, respectively.  
+The magnetoelectric energy thus can be represented in the form of electrostatic energy of the 
+charge (2) in the electric potential φ of the point electrode:  
+2ME eWq hϕπγχ ϕ== ⋅ ⋅ ,        (3) 
+where q is the integral charge, h is the height of the particle.  
+The exchange energy can be estimated as 
+() h A h AVAk WExch2 22
+222ππ=Δ⎟⎠⎞⎜⎝⎛
+Δ==    ( 4 )  
+where k is the modulation wave vector, Δ is the lateral size of the particle.  
+Thus the critical potential of the electrode  tip in which the nucl eation of the vortex 
+(antivortex) occurs that correspond to the condition W ME  + W exch <0 can be found as 
+2
+C
+eAπϕγχ=⋅       ( 5 )  
+Assuming γ~10-6 cm erg/ (typical values for BiFeO 3 and iron garnet), dielectric 
+susceptibility χe~4 (it can be estimated using the value of BiFeO 3 dielectric constant ε=1+4πχe=50) 
+and exchange stiffness A~3 *10-7 erg/cm (typical value for room temperature magnets) we obtain 
+φc=±150V.  
+More rigorous approach should take into account the finite size of the vortex core in which 
+the spin come out of the plane. In the case of dielectric nanoparticle its diameter is determined by the balance between the exchange energy and the inhomogeneous magnetoelectric energy. The 
+estimation (5) remains valid provided that the critic al potential corresponds to the lowest voltage at 
+which the vortex/antivortex state appears at the perimeter of the particle, the charge (3) is distributed on the surface of the cylinder wrapping the vortex core and the parameter Δ in (4) stands for the core 
+diameter. Thus the electric field can stabilize the vortex state for one sign of the charge and antivortex 
+state for another (see figure). This possibility besi des fundamental interest can be considered as a 
+prototypical example of electrically switchable magnetic system with two logic states. The use of 
+ferroelectric media as a material of the tip electrode will make it possible to implement the memory cell based on this principle.   
+ 
+Authors are grateful to A. Zvezdin, D. Khomskii and M. Mostovoy for the interest to the 
+work and valuable discussion.   
+ 
+References 
+ 
+1. B. Van Waeyenberge et al., Nature (London) 444, 461 (2006)   
+2. V. S. Pribiag, , I. N. Krivorotov, G. D. Fuchs, et al., Nature Physics, Vol. 3, 498 (2007).  
+3. K.Yu. Guslienko, J. Nanosc. Nanotechn., v. 8, 2745-2760 (2008) 
+4. M. Tanase, A. K. Petford-Long, O. Heinonen, K. S. Buchanan, J. Sort, and J. Nogués, Magnetization reversal in 
+circularly exchange-biased ferromagnetic disks, Phys. Rev. B 79, 014436 (2009) 
+5. S. Prosandeev, I. Ponomareva, I. Kornev, and L. Bellaiche, Control of Vortices by Homogeneous Fields in 
+Asymmetric Ferroelectric and Ferromagnetic Rings, Phys. Rev. Lett., v.100, 047201 (2008) 
+6. K. Yamada et al., Nat. Mater. V. 6, 270 (2007)  7. K. Shigeto, T. Okuno, K. Mibu, T. Shinjo , T. Ono, Appl. Phys. Lett., 80, 4190, (2002) 
+8.  V.L. Mironov, O.L. Ermolaeva,  S.A. Gusev, A.Yu. Klimov, V.V. Ro gov, B.A. Gribkov, A.A. Fraerman, 
+O.G.Udalov, Anti-vortex state in cross-like nanomagnets, Phys. Rev. B, 81, 094436 (2010) 
+9. V. G. Bar’yakhtar, V. A. L’vov,  and D. A. Yablonskii, JETP Lett. 37, 673 (1983) 
+10. Yu. F. Popov, A. Zvezdin, G. Vorob’ev, A. Ka domtseva, V. Murashev, and D. Rakov, JETP Lett. 57, 69 (1993) 
+11. A. Sparavigna, A. Strigazzi, and A. Zvezdin , Phys. Rev. B 50, 2953 (1994) 
+12. A.A. Khalfina, M.A. Shamtsutdinov, Ferroelectrics, 279, 19 (2002) 
+13. M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006). 
+14. Logginov A.S., Meshkov G.A., Nikolaev A.V., Pyatakov A.  P., Shust V. A., Zhdanov A.G., Zvezdin A.K., J. Magn. 
+Magn. Mater., 310, 2569 (2007) 
+15. I. Dzyaloshinskii, EPL, 83, 67001 (2008) 
+16. D. Mills, I.E.Dzyaloshinskii, Phys.Rev. 78, 184422 (2008) 
+17. A.A. Mukhin, A.K. Zvezdin, JETP Lett. 89, 328 (2009) 
+18. A.P. Pyatakov and A.K. Zvezdin, Flexomagnetoelectric interaction in multiferroics, The European Physical Journal 
+B - Condensed Matter and Complex Systems, Volume 71, Number 3, p. 419 (2009) 
+19. S.-W. Cheong, M. Mostovoy, Nature Materials, 6, 13 (2007) 
+20. A.S. Logginov, G.A. Meshkov, A.V. Nikolaev, E.P. Ni kolaeva, A.P. Pyatakov, A.K. Zvezdin Room temperature 
+magnetoelectric control of micromagnetic structure in iron garnet films, Applied Physics Letters ,  v.93, p.182510 (2008) 
\ No newline at end of file
diff --git a/1001.0392.txt b/1001.0392.txt
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@@ -0,0 +1,343 @@
+arXiv:1001.0392v1  [astro-ph.EP]  3 Jan 2010Pre-Spectroscopic False Positive Elimination of KeplerPlanet
+Candidates
+Natalie M. Batalha
+Department of Physics and Astronomy, San Jose State University , San Jose, CA 95192
+Natalie.Batalha@sjsu.edu
+Jason F. Rowe
+NASA Postdoctoral Program Fellow, NASA Ames Research Center, Moffett Field, CA
+94035
+Ronald L. Gilliland
+STScI, Baltimore, MD 21218
+Jon J. Jenkins and Douglas Caldwell
+SETI Institute, Mountain View, CA 94043
+William J. Borucki, David G. Koch, and Jack J. Lissauer
+NASA Ames Research Center, Moffett Field, CA 94035
+Edward W. Dunham
+Lowell Observatory, Flagstaff, AZ 86001
+Thomas N. Gautier
+Jet Propulsion Laboratory, Pasadena, CA 91109
+Steve B. Howell
+NOAO, Tucson, AZ 85726
+David W. Latham
+Harvard-Smithsonian, CfA, Cambridge, MA 02138– 2 –
+Geoff W. Marcy
+University of California, Berkeley, CA 94720
+and
+Andrej Prsa
+Villanova University, Villanova, PA 19085
+Received ; accepted– 3 –
+ABSTRACT
+Ten days of commissioning data (Quarter 0) and thirty-three days of science
+data (Quarter 1) yield instrumental flux timeseries of ∼150,000 stars that were
+combed for transit events, termed Threshold Crossing Events (TCE), each hav-
+ing a total detection statistic above 7.1- σ. TCE light curves are modeled as
+star+planet systems. Those returning a companion radius smaller t han 2RJare
+assigned a KOI ( Kepler Object of Interest ) number. The raw flux, pixel flux,
+and flux-weighted centroids of every KOI are scrutinized to asses s the likelihood
+of being an astrophysical false-positive versus the likelihood of a be ing a plane-
+tary companion. This vetting using Keplerdata is referred to as data validation.
+Herein, we describe the data validation metrics and graphics used to identify
+viable planet candidates amongst the KOIs. Light curve modeling tes ts for a)
+the difference in depth of the odd- versus even-numbered transit s, b) evidence of
+ellipsoidal variations, and c) evidence of a secondary eclipse event a t phase=0.5.
+Flux-weighted centroids are used to test for signals correlated wit h transit events
+with a magnitude and direction indicative of a background eclipsing bina ry. Cen-
+troid timeseries are complimented by analysis of images taken in-tran sit versus
+out-of-transit,thedifferenceoftenrevealing thepixelcontribu ting themosttothe
+flux change during transit. Examples are shown to illustrate each te st. Candi-
+dates passing data validationaresubmitted to ground-based obse rvers for further
+false-positive elimination or confirmation/characterization.
+Subject headings: techniques: photometric — binaries: general — planetary sy stems– 4 –
+1. Introduction
+TheKeplerscience team is reporting on its first planet discoveries (Borucki et al.
+2010b,a; Dunham et al. 2010; Koch et al. 2010; Latham et al. 2010) in this volume –
+discoveries discerned from data taken in the first weeks of science operations. The
+progression from pixels to planet detection passes through numer ous stages, from target
+selection, data collection, and aperture photometry to transiting planet search, data
+validation, and follow-up observations. In this progression, data v alidation (DV) and
+follow-up operations (FOP) are both tasks concerned, in part, wit h the vetting of false
+positives. They are distinct, however, in that DV performs false-p ositive elimination
+from diagnostics that can be pulled out of the Keplerdata itself, while the FOP relies on
+additional observations such as moderate and high-precision spec troscopic radial velocities
+(Gautier et al. 2010). Automated DV is under development in the Scie nce Operations
+Center at NASA Ames Research Center and will soon provide pipeline g eneration of
+metrics, reports, and graphics that the science team will use to sif t through the thousands
+of transit events that Keplerexpects to detect.
+These, our first discoveries, were scrutinized in non-pipeline fashio n with some metrics
+taken from the pipeline DV, and others developed in real time to get t he job done as
+efficiently as possible to support the 2009 ground-based observing season – the first since
+launch. Many of the tools that sprang up out of this effort have fed back into the DV
+development, ensuring it will be a powerful analysis pipeline for futur e processing. A full
+description of the DV pipeline currently under development can be fo und in Jenkins et al.
+(2010a).
+This communication describes the pre-pipeline DV tools applied to this fi rst analysis
+of science data. An event flagged by the Transiting Planet Search (TPS) pipeline as having
+transit-like features with a total detection statistic greater tha n 7.1-σis termed a Threshold– 5 –
+Crossing Event (TCE). Each TCE light curve is modeled, and those returning a compa nion
+radius<2RJare assigned a Kepler Object of Interest (KOI) number. Only those that pass
+the DV tests described herein are submitted to the follow-up obser vers for spectroscopic
+vetting, confirmation, and characterization. The objective is to e liminate as many of the
+false-positives as possible to efficiently utilize the limited amount of tele scope time available
+for follow-up observations. We note that elimination of a TCE (i.e. tra nsit detection)
+as a viable planet candidate does not imply that the target itself is no lo nger observed
+or no longer subjected to transit searches. Moreover, a light cur ve may yield multiple
+threshold-crossing events (TCEs). Each is considered independe ntly with regards to the
+planet interpretation.
+The large majority of false-positives are caused by either grazing o r diluted eclipsing
+binaries ((Brown 2003)), the latter being the more likely. Dilution occ urs when light from
+a nearby star falls within the photometric aperture of the foregro und star, where “nearby”
+can be either a true physical companion to the star or a chance pro jection on the sky. We
+refer to the latter scenario as a Background Eclipsing Binary (BGEB ). The photometric
+and astrometric precision of the Keplerphotometer affords us unprecedented opportunities
+to vet out the false positives from the flux and photocenter timese ries themselves.
+Section 2 gives a brief summary of the data, from acquisition to tran sit detection.
+Section 3 describes the metrics used to identify grazing and diluted E Bs derived from light
+curve modeling, and Section 4 addresses the analysis of the photoc enter variations. The
+stars used to exemplify the various techniques will be referred to b y their KOI designation
+as well as the KeplerID archived in the Kepler Input Catalog1(KIC).
+1http://archive.stsci.edu/kepler– 6 –
+2. Observations, Light Curves, and Transit Detection
+The analysis is based on two sets of data: 1) a 9.7-day run, May 2 thr ough May
+12, 2009, during the commission period to measure the initial photom etric performance
+of the instrument, and 2) the first 33.5 days of science operations , May 13 through June
+17, 2009, the end of which is marked by a 90◦quarterly roll of the spacecraft about the
+optical axis (Haas et al. 2010). The former data set is referred to as Quarter 0 (Q0), while
+the latter is referred to as Quarter 1 (Q1). All relatively uncrowde d stars brighter than
+Kp= 13.6 (where Kpis the apparent magnitude in the Keplerpassband) make up a list of
+52,496 targets observed during Q0. The Q1 target list contains 156 ,097 stars, 93% of which
+were selected based on metrics that address the detectability of t errestrial-size planets
+(Batalha et al. 2010) – metrics dependent on stellar properties (e.g . effective temperature,
+surface gravity, and apparent magnitude). The intersection of t he two lists contains 45,742
+targets. The aperture photometry, data conditioning, and tran sit search algorithms are
+described in Jenkins et al. (2010a). A discussion of the resulting flux timeseries is given in
+Jenkins et al. (2010b).
+The transiting planet search applied to all stars yielded several tho usand TCEs,
+some of which were triggered by artifacts in the light curves and sing le strong outliers.
+This was mitigated by comparing the χ2statistic of a transit model fit to the folded
+light curve against the χ2statistic of a linear fit to the same. If the two are statistically
+indistinguishable, the TCE is disregarded. A constraint on the trans it duration (1-hour) is
+used to isolate TCEs triggered by strong outliers in the flux timeserie s.
+The transit model provides an estimate of the companion radius. TC Es with a
+companion radius larger than 2 RJare also disregarded. Such large radii are likely associated
+with late M-dwarfs and are not considered high priority planetary ca ndidates. This leaves
+approximately 103TCEs.– 7 –
+3. Light Curve Modeling
+We use the analytic expressions of Mandel & Agol (2002) to model t he transit of the
+planet. For our initial fits we use Claret (2000) V-band nonlinear limb d arkening parameters
+for the star as an approximation for the Keplerbandpass. Values for the stellar radius
+(R⋆), effective temperature ( Teff) and surface gravity (log g) are retrieved from the KIC.
+The stellar mass ( M⋆) is computed from R⋆and logg. WithM⋆andR⋆fixed to their initial
+values a transit fit is computed to determine the orbital period, pha se, orbital inclination,
+and planetary radius, Rp. The best fit is found using a Levenberg-Marquardt minimization
+algorithm (Press et al. 1992).
+In the case of an eclipsing binary where the radii and temperature o f the two
+components are slightly different, the depths of the eclipses imprint ed on the light curve will
+be different. Every other transit will have a different depth. When a t least 2 transits are
+present in the light curve, the transit model is recomputed for the odd and even numbered
+transits where only the companion radius is allowed to vary. If the co mpanion radii differ
+by more than 3- σthen the TCE is rejected.
+Figure 1 illustrates such an example. The upper panel shows the unf olded Q0+Q1
+light curve of KOI-106 (KeplerID 10489525) – a somewhat evolved F -type star according
+to the KIC. Transit modeling provides an estimate of 0 .48RJfor the companion radius,
+1.61236±0.00013 days for the orbital period, and 74.06◦for the inclination (shown in the
+lower panel). The difference between the modeled companion radii ut ilizing first the odd
+transits and then the even transits, is given as depth-sig= 10 .1σ. The folded light curve
+is shown in the bottom panel where the odd transits are plotted with asterisks and the
+even transits are plotted with plus symbols. The difference in the dep ths of the odd versus
+even transits is clear, indicating either a grazing eclipse of a binary sy stem or, more likely,
+a diluted background eclipsing binary. In this case, the true orbital period is twice that– 8 –
+identified by the TPS pipeline.
+The same type of astrophysical false-positive can be identified by s earching for
+very shallow secondary events at phase=0.5. We must proceed with caution, however,
+in the interpretation of the secondary event since the occultation of a planet can also
+produce a secondary as is the case of HAT-P-7b (Borcuki et al. 20 09). If the depth of
+the secondary eclipse has a significance greater than 2- σ, then the dayside temperature
+of the planet candidate can be estimated. The flux ratio of the plane t and star ( Fp/F∗)
+over the instrumental bandpass is given by the depth of the secon dary eclipse. The ratio
+of the planet and star radii is obtained from the transit depth. By a ssuming the star and
+companion behave as blackbodies and the flux ratio is bolometric, the dayside effective
+temperature can be estimated.
+For Kepler-5b (KOI-18; KeplerID 8191672) (Koch et al. 2010) – a c onfirmed planet
+with a 2.6- σsecondary transit – we find Fp/F∗= 3.3×10−5, which gives Teff= 1657±223
+where an error of 30% is assumed for the input stellar luminosity and r adius. This estimate
+is a lower limit as a significant fraction of the planetary flux is emitted at wavelengths
+longer than the red edge of the Keplerbandpass, but it is a useful diagnostic to determine
+whether the depth of the secondary eclipse is consistent with a str ongly irradiated planet.
+To make this comparison, we estimate the equilibrium temperature,
+Teq=T∗(R∗/2a)1/2[f(1−AB)]1/4, (1)
+for the companion, where R∗,T∗are the stellar radius and temperature, with the planet at
+distance awith a Bond albedo of AB, andfis a proxy for atmospheric thermal circulation.
+We assume AB= 0.1 for highly irradiated planets (Rowe et al. 2006) and f= 1 for
+efficient heat distribution to the night side. These choices give a roug h estimate for the
+dayside temperature of the planet assuming stellar irradiation is the primary energy source.
+Assuming a 30% error in the input stellar parameters and that star a nd planet act as– 9 –
+blackbodies we find Teq= 1868±284 K. The consistency of TeffandTeqto within 1- σ
+demonstrates that the secondary eclipse is consistent with a stro ngly irradiated planet. If
+Teffwere found to be much larger than Teq, then the companion is likely self-luminous and
+thus a stellar binary.
+Figure 2 illustrates the case of a secondary eclipse that is not consis tent with occulted
+emission from a planet. The light curve of KOI-23 (KeplerID 9071386 ), upper panel, as well
+as the phase folded transit, lower panel, is again shown. The middle pa nel shows a close-up
+of the light curve at phase=0.5 where a 16- σsecondary transit is evident. Equation 1 yields
+Teq= 1618±259 K which is inconsistent with the effective temperature of the com panion
+(2554±206 K) derived from the light curve (difference >3.5-σ). The secondary transit of
+KOI-23 does not support the planetary companion interpretation . This system is likely a
+grazing or diluted EB. We note, however, that the temperature ar gument is invalid if the
+planet is young or in an eccentric orbit.
+4. Photocenter Motion Diagnostics
+Tracking the photocenter of the photometric aperture given Kepler’s very high SNR
+and stable pointing is an effective means of identifying background ec lipsing binaries. The
+dimming of any object in the aperture will shift the photocenter of t he light distribution
+since the photocenter is determined by the combination of various d iffuse and discrete
+sources. The apparent change in the position of the target star d ue to a background
+eclipse event is dependent on the separation of the stars, their re lative brightnesses, and
+the transit/eclipse depth. A 50% eclipse from a star 5 magnitudes dim mer than the target
+star and offset by one pixel will cause a 5 millipixel shift assuming 1) the re are only two
+stars plus diffuse background in the photometric aperture and 2) a ll of the flux from the
+BGEB is included in the photometric aperture. Pre-launch simulations of the photometric– 10 –
+and astrometric stability of the instrument predict a 6.5-hour prec ision in the flux-weighted
+centroids for a 12th magnitude star of 20 µpix (80µarcsec).
+To help determine whether background eclipsing binaries in the apert ure of a KOI are
+responsible for the transit-like features identified by TPS, we corr elate changes in centroid
+location with the transit-like features in the photometry. Systema tics due primarily to focus
+and pointing changes (Jenkins et al. 2010a,b) were removed from th e flux and centroid
+timeseries. A high-pass filter was then applied to remove signatures occurring on timescales
+longer than 2 days. Isolated outliers are removed from both the flu x and centroid timeseries
+using a 5-sample wide moving median filter and rejecting points beyond 10 median absolute
+deviations from the moving median.
+Figure 3 shows the flux timeseries (upper) and the photocenter (fl ux-weighted centroid)
+timeseries of the row (middle) and column (lower) residuals for KOI-1 40 (KeplerID
+5130369). A 1 and 3 millipixel shift projected onto row and column, re spectively, and
+correlated with the transit signal is clearly seen even at the 30-minu te cadence. An alternate
+way of displaying the same information is shown in Figure 4. Here, the f ractional change
+in brightness is plotted against the residual pixel position of the flux -weighted centroids,
+with row and column values distinguished using different symbols. The “ cloud” of points
+near (0,0) represents the out-of-transit points from which we es timate the per-cadence
+uncertainty – 0.3 millipixels in the case of KOI-140. The points that rain down are the
+in-transit points. If the target star is responsible for the transit s and if the photometric
+aperture is free from contaminating flux spatially offset from the ta rget star, then the rain
+will fall vertically under the cloud corresponding to zero centroid re siduals. In the case of
+KOI-140, the rain falls diagonally, showing a −0.68 millipixel row residual and a +3 .02
+millipixel column residual. With 27 in-transit samples, this corresponds to a 12-σand 52-σ
+significance, respectively. These centroid offsets together with k nowledge of the apparent– 11 –
+brightnesses of stars in the vicinity (from the KIC) allow us to identif y the star responsible
+for the transit-like event.
+In crowded fields, the flux distribution in a photometric aperture is r arely limited to
+the target star plus diffuse background. Complex spatial flux distr ibutions imply that the
+out-of-transit photocenter is not necessarily centered on the t arget star. Consequently, the
+photocenter can shift in unexpected directions during transit eve n when it is the target star
+itself that is responsible for the signal. For this reason, we look towa rds other diagnostics
+in an effort to confirm the identity of the transiting (or eclipsing) obj ect.
+KOI-08 (KeplerID 5903312) is an example of a BGEB. The aperture is especially
+crowded, with 8 stars less than 3 magnitudes fainter than the targ et located within a
+3-pixel radius. The centroid timeseries and rain plot show a ∼0.75 millipixel centroid shift
+in both row and column. An average of the images taken at cadences occurring during
+transit is subtracted from an out-of-transit image. The upper pa nel of Figure 5 shows
+the comparison of the out-of-transit image of KOI-08 (right) with the differenced image
+(left). The brightest pixel in the difference image shows the spatial location of the pixels
+contributing most strongly to the flux change during the transit. I n this case, the position
+is offset by approximately one pixel in both row and column from the no minal target.
+This coincides with an object in the KIC that is 1.5 magnitudes fainter in theKepler
+passband than the target star (2.45 magnitudes in 2MASS J). The lo wer panel shows two
+single-pixel flux timeseries. The upper corresponds to the brighte st pixel in the direct
+image, and the lower corresponds to the brightest pixel in the differ ence image. The transit
+event disappears completely from the pixel timeseries drawn from t he target star. To the
+contrary, the fractional transit depth is larger in the flux timeser ies of the brightest pixel in
+the difference image. The centroid timeseries, the rain plot, the diffe rence image, and the
+pixel flux timeseries collectively allow us to identify the majority of BGE Bs directly from– 12 –
+Keplerdata without the need for more complex modeling or observations.
+5. Summary
+Modeling light curves under the assumption that the companion is a pla net provides
+discriminators against grazing and diluted eclipsing binaries. The metr ics used to flag likely
+false-positives are the odd/even transit depth statistics and the secondary eclipse statistic.
+The latter is complimented by a comparison of the equilibrium temperat ure of the planet
+computed from the orbial and stellar characteristics versus the d ay-side temperature of the
+planet computed from the transit/occultation depths. When the t wo are markedly different,
+the planet interpretation for the secondary is discarded.
+The centroid timeseries is a powerful discriminator against diluted ec lipsing binaries.
+KOI-140 is presented as an example of a BGEB identified via centroid a nalysis. This
+13.8-magnitude star yields a per-cadence (30-min) centroid precis ion of 0.3 millipixels, or 83
+µpix at 6.5-hours. Rain plots exemplified in Figure 4 provide an efficient vis ual assessment
+of the likelihood of a BGEB. For more complex flux distributions in the ph otometric
+aperture, the difference image (out-of-transit minus in-transit) often clearly shows the
+location of the BGEB. Individual pixel flux timeseries confirm the sou rce in that they
+isolate the flux of the transiting system, minimizing dilution, and there by diminishing (or
+augmenting) the transit depth. This is effective when the BGEB is spa tially well-separated
+from the target star and the stars are sufficiently bright.
+AllKepler Objects of Interest must pass the modeling and centroid inspections before
+being passed on for follow-up observations. In the upcoming year, these metrics will be
+folded into pipeline Data Validation tools forming a more efficient TCE-to -KOI filter.
+The authors wish to acknowledge the Kepler Science Operations Center , especially– 13 –
+Hema Chandrasekharan for her work on the transiting planet sear ch software. Funding for
+this Discovery mission is provided by NASA’s Science Mission Directorat e.– 14 –
+REFERENCES
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+Caldwell, D. A., Gilliland, R. L., Latham, D. W., Meibom, S., Monet, D. G. 201 0,
+ApJ, submitted
+Borucki, W. J. et al. 2010, Science, submitted
+Borucki, W. J., Koch, D., Jenkins, J., Sasselov, D., Gilliland, R., Batalha, N., Latham, D.
+W., Caldwell, D., Basri, G., Brown, T., Christensen-Dalsgaard, J., Coch ran, W. D.,
+DeVore, E., Dunham, E., Dupree, A. K., Gautier, T., Geary, J., Gould, A., Howell,
+S., Kjeldsen, H., Lissauer, J., Marcy, G., Meibom, S., Morrison, D., Tar ter, J. 2009,
+Science, 325, 5941
+Borucki, W. J. et al. 2010, ApJ, submitted
+Brown, T. M. 2003, ApJ, 593, L125
+Bryson, S. B. et al. 2010, ApJ, submitted
+Claret, A. 2000 A&A, 363, 1081
+Dunham, E. W. et al. 2010, ApJ, submitted
+Gautier, T. N. 2010, ApJ, submitted
+Haas, M. et al. 2010, ApJ, submitted
+Jenkins, J. M. et al. 2010a, ApJ, submitted
+Jenkins, J. M. et al. 2010b, ApJ, submitted
+Koch, D. G. et al. 2010, ApJ, submitted
+Latham, D. L. et al. 2010, ApJ, submitted– 15 –
+Mandel & Agol, ApJ, 580, L171
+Press, W.H. Teukolsky, S.A., Vetterling, W.T. Flannery, B.P. 1992, Nu merical Recipes in
+Fortran 77, Second Edition, Cambridge University Press, 678
+Rowe et al. 2006, ApJ, 646, 1241
+This manuscript was prepared with the AAS L ATEX macros v5.2.– 16 –
+Fig. 1.— The detrended light curve of KOI-106 (upper panel) is plotte d against HJD (days).
+The phased light curve isshown (lower panel) together witha model ( solidline) that assumes
+the transit features are due to a planetary companion (resulting r adius,Rp= 0.48RJ). The
+region of the light curve centered on phase=0.5 is shown in the botto m panel (circles) and
+plotted in units of ppm (parts per million) as indicated on the axis on the right. Stellar
+parameters ( Kp,Teff, log(g), andR∗are listed at the top of the figure. The metrics Rp,
+Eclip-sig, and Depth-sig are defined in Section 3. Modeling the odd (+) and even (*)
+numbered transits independently yields a significant difference in the companion radii (10- σ
+difference) suggesting that this system is likely a diluted EB at twice th e orbital period.– 17 –
+Fig. 2.— Same as Figure 1 for KOI-23 (estimated companion radius, Rp= 1.74RJ). The
+14-σ(“Eclip-sig”) feature is not consistent with a planetary occultation as evidenced by
+the difference between the equilibrium temperature, Teq, of a planet at that orbital period
+compared to the surface temperature, Teff, required for a companion to yield the observed
+transit depth. One-sigma uncertainties for TeqandTeffare given in parentheses. These
+metrics are defined in Section 3.– 18 –
+−2.5−2−1.5−1−0.500.51Relative Flux Residuals, pptA
+−1.5−1−0.500.51Row Residuals, mpixB
+4965 4970 4975 4980 4985 4990 4995−2−101234Column Residuals, mpix
+Time, JD−2450000C
+Fig. 3.— Flux (upper, plotted in units of parts-per-thousand) and c entroid (middle, lower)
+time series of KOI-140. A high-pass filter has been applied to each tim eseries to remove
+non-transit signatures on timescales longer than 2 days. Transit- like features in both the
+row and column centroid residuals are evident and highly correlated w ith the transit features
+in the flux time series suggesting this may be a diluted eclipsing binary.– 19 –
+−2−101234−2.5−2−1.5−1−0.500.51
+Centroid Shift, mpixNormalized Flux Residuals, ppt
+Fig. 4.— Flux versus residual row (x) and column (+) centroids for KO I-140. The mea-
+surements are the same as those presented in the Figure 3. This “r ain plot” shows that
+centroids are systematically shifting during each transit event. Th e extent of the shift in
+each dimension, coupled to the depth of the transit-like feature is in dicative of a BGEB.
+Had the transit originated from the central target star, the out -of-transit points would have
+rained down vertically under the cloud of out-of-transit points.– 20 –
+Difference Image
+columnrow
+2468246Direct Image
+columnrow
+2468246
+0 100 200 300 400 5008.78.88.9x 107
+0 100 200 300 400 5001.241.261.28x 107
+Fig. 5.— An average of images taken at a time when KOI-08 is not trans iting is subtracted
+from an average of images taken during transit. The brightest pixe l in the difference image
+(row=6, column=7) indicates the location of largest flux change and is not coincident with
+the position of the target star evident in the direct image (row=4, c olumn=6).The corre-
+sponding pixel flux timeseries (both taken from a direct image during transit) confirms that
+the events detected in KOI-08 are due to a BGEB (middle and lower pa nel).
\ No newline at end of file
diff --git a/1001.0393.txt b/1001.0393.txt
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+arXiv:1001.0393v2  [cs.GT]  30 Jul 2010Market Equilibrium with Transaction Costs
+Sourav Chakraborty∗Nikhil R Devanur†Chinmay Karande‡
+Abstract
+Identical products being sold at different prices in different location s is a common phenomenon.
+Price differences might occur due to various reasonssuch as shippin g costs, trade restrictions and price
+discrimination. To model such scenarios, we supplement the classica l Fisher model of a market by
+introducing transaction costs . For every buyer iand every good j, there is a transaction cost of cij; if
+the price of good jispj, then the cost to the buyer iper unit ofjispj+cij. This allows the same
+good to be sold at different (effective) prices to different buyers.
+We provide a combinatorial algorithm that computes ǫ-approximate equilibrium prices and alloca-
+tions inO/parenleftbig1
+ǫ(n+logm)mnlog(B/ǫ)/parenrightbig
+operations - where mis the number goods, nis the number of
+buyers and Bis the sum of the budgets of all the buyers.
+1 Introduction
+Identical products being sold at different prices in different l ocations is a common phenomenon. Price
+differences might occur due to different reasons such as
+•Shipping costs. Oranges produced in Florida are cheaper in F lorida than they are in Alaska, for
+example.
+•Trade restrictions. A seller with access to a wider market mi ght sustain a higher price than one
+that does not. Kakade et al[22] considered a model called graphical economies to study how price
+differences occur due to trade agreements in international tr ade. Note that this is different from
+shipping costs since two countries are either allowed to tra de (in which case they pay the same
+price) or not.
+•Price discrimination. A good might be priced differently for d ifferent people based on their respec-
+tive ability to pay. For example, conference registration f ees are typically lower for students than
+for professors.
+In order to capture all these scenarios, we supplement the cl assical Fisher model of a market (see below
+for a formal definition) by introducing transaction costs . For every buyer iand every good j, there is a
+transaction cost of cij; if the price of good jispj, then the cost to the buyer iper unit ofjispj+cij.
+This allows the same good to be sold at different (effective) pric es to different buyers. Note that apart
+∗Technion. sourav@cs.technion.ac.il
+†Microsoft Research. nikdev@microsoft.com
+‡Georgia Institute of Technology. ckarande@cc.gatech.edu . Part of the work done when the author was at Microsoft
+Research, Redmond.
+1from non-negativity, the transaction costs are not restric ted in any way and in particular, do not have
+to satisfy the triangle inequality.
+The scenarios mentioned earlier can all be modeled as follow s: Shipping costs are most naturally
+modeled as transaction costs. We can model the trade restric tions using costs that are either 0 or ∞.
+Our model can also be used to incorporate price discriminati on into the equilibrium computation by
+using the transaction costs to specify buyer specific reserv e prices.
+Alternative Models - Exogenous VsEndogenous Price Differentiation : Inthesection onrelated
+work, we will discuss other models of price differentiation th at choose an approach similar to ours, i.e.to
+augment the market model with extra costs specified for the bu yer. A fundamentally different alternative
+thatdoesnot involve suchcosts, especially whenthepriced ifferenceisduetoshippingcosts, is toconsider
+‘shipping’ or ‘transaction’ itself as another good in the ma rket. This special good has a given capacity
+and the equilibrium price also determines the ‘price’ of shi pping,i.e., the transaction costs. A buyer
+now derives utility if she gets both a good and the correspond ing transaction good. (Each buyer could
+need the two in a different ratio.) There are two reasons to choo se our approach over the one in which
+transaction costs are determined by the market: (a) In many c ases the transaction costs are exogenously
+specified, and one cannot choose the above model. (b) Even if o ne does have a choice, our results indicate
+that there are faster algorithms to compute the equilibrium in the model with given transaction costs
+than the one with given capacities. Thus computational cons iderations might induce one choose the
+model with given transaction costs over the other one.
+An important offshoot of the algorithmic study is the definitio n of new economic models, for example,
+the spending constraint model [24] was motivated mainly by c omputational considerations, but has been
+found to have interesting economic properties as well. Comp utational considerations was one of the main
+motivations that led to the definition of our model as well.
+Fisher’s Market Model with Transaction Costs
+In Fisher’s model, a market Mhasnbuyers and mdivisible goods. Every buyer ihas budget Bi.
+We consider linearutility functions, i.e., the utility of a buyer ion obtaining a bundle of goods xi=
+(xi1,xi2,...) is/summationtext
+juijxijwhereuijare given constants. Each good has an available supply of one unit
+(which is without loss of generality). In addition to its pri ce, a buyer also pays a transaction cost cijper
+unit of good j. The allocation bundle for buyer iis a vector xisuch that xijdenotes the amount of good
+jallocated to buyer i. A price vector pis an equilibrium of Mif there exists allocations xisuch that
+•ximaximizes the utility of iamong all bundles that satisfy the budget constraint, i.e.
+xi∈argmaxyi
+
+/summationdisplay
+juijyij:/summationdisplay
+j(pj+cij)yij≤Bi
+
+,
+•and every good is fully allocated or is priced at zero, i.e. ∀j, either/summationtext
+ixij= 1 orpj= 0.
+Remark : A more general model is that of Arrow-Debreu, in which the en dowments of buyers are goods,
+and the budget constraint is defined by the income obtained by the buyer by selling his goods at the
+given prices. With the introduction of transaction costs, t he money is not conserved. For this reason
+transaction costs are not so natural for the Arrow-Debreu mo del.
+Characterization of Market Equilibrium : We now characterize the equilibrium prices and alloca-
+tions in our model. In our model, the ratio uij/(pj+cij) denotes the amount of utility gained by buyer i
+2through one dollar spent on good j. At given prices, a bundle of goods that maximizes the total u tility of
+a buyer contains only goods that maximize this ratio. Let αi= max juij/(pj+cij) be the bang-per-buck
+of buyer iat given prices. We will call the set
+Di={j|uij=αi(pj+cij)}
+the demand set of buyer i. Hence, xij>0⇒j∈Di. The conditions characterizing these equilibrium
+prices and allocations appear in table A below.
+Anǫ-approximatemarket equilibriumischaracterized by relax ingthemarket clearingcondition (Equa-
+tion (3)) and optimal allocation condition (Equation (4)). Refer to equations (7) and (8) in table B.
+A: Market Equilibrium
+∀i/summationdisplay
+j(pj+cij)xij=Bi (1)
+∀j/summationdisplay
+ixij≤1 (2)
+∀j p j>0⇒/summationdisplay
+ixij= 1 (3)
+∀i,j x ij>0⇒uij=αi(pj+cij) (4).B:ǫ-Approximate Market Equilibrium
+/summationdisplay
+j(pj+cij)xij=Bi (5)
+/summationdisplay
+ixij≤1 (6)
+pj> ǫ⇒/summationdisplay
+ixij≥1/(1+ǫ) (7)
+xij>0⇒uij≥αi(pj+cij)/(1+ǫ) (8)
+The relaxation of exact equilibrium conditions can be achie ved in other ways. For example, [17] use a
+definition of ǫ-approximate market equilibrium that relaxes the budget co nstraints. Our algorithm can
+be easily adapted to this definition by simple modifications t o the termination conditions.
+Our Results and Methods
+We consider a Fisher market with linear utilities and arbitrary transaction costs: Buyer iincurs a
+transaction cost of cijper unit of good j. Our main result is a combinatorial algorithm that computes
+ǫ-approximate equilibrium prices and allocations in O/parenleftbig1
+ǫ(n+logm)mnlog(B/ǫ)/parenrightbig
+operations - where m
+is the number goods, nis the number of buyers and Bis the sum of the budgets of all the buyers. This
+algorithm is a generalization of the auction algorithm of Ga rg and Kapoor [17] to our model with the
+transaction costs. This generalization is not straight for ward; the presenceof transaction costs introduces
+new challenges. We now outline some of these difficulties and o ur approach to solving them.
+- Even the existence of an equilibrium in our model does not fo llow directly from any classical results.
+One also notices other differences right away: equilibrium pr ices might be irrational numbers, in contrast
+to the traditional model where they are guaranteed to be rati onal. An example where the equilibrium
+prices are irrational is presented in Appendix B.
+- The term ‘auction algorithm’ is used to describe ascending price algorithms (such as the one in [17])
+which maintain a feasible allocation at all times. The algor ithm makes progress by revoking a portion of
+goods currently assigned to a buyer and reallocating it to an other buyer offering a higher price. An easy
+monotonicity property that all variants of the auction algo rithm use crucially is the fact that the total
+surplus (unspent money of the buyers) decreases throughout the algorithm. However one cannot have
+such monotonically decreasing surplus in the presence of tr ansaction costs. This is because even though
+the prices are increasing, a good may be reallocated to a buye r with a lower transaction cost and thus
+3the total money spent by the buyers decreases. We get around t his difficulty by analyzing the running
+time in a way that does not rely on this property.
+- Another important property that holds in the traditional m odel is that an increase in the price of a
+good has the same effect on the bang-per-buck of that good for any buyer. Because of the transaction
+costs, this property is no longer true and hence, a key lemma i n the analysis of the auction algorithm of
+[17], that a certain directed graph is acyclic, does not hold . We provide a counter-example in Appendix
+C. We also show that due to this, a fully distributed version o f the auction algorithm and in particular
+the algorithm from [17] does not even converge. We get around this by designing a way to reallocate
+the goods in a cycle that guarantees progress. This process a lso leads to an increase in the running time
+if analyzed in the naive way, and the eventual running time is obtained by a more careful, amortized
+analysis. In spite of these difficulties, we match the running time of the auction algorithm in [17].
+- Our method of reallocating goods is similar in spirit to the path auctions used by [19]. Again however,
+the properties of monotonic decrease in surplus and acyclic ness of the demand graph hold in their case,
+whereas these properties cease to exist when transaction co sts are introduced.
+Related Work
+The computation of economic and game theoretic equilibria h as been an active area of research over the
+past decade. Hardness results [9, 10, 3, 5, 2] and algorithmi c results [11, 15, 20, 17, 24, 12, 6, 4, 21, 26, 14]
+have been delineating the boundary between what is efficientl y computable and what is not. Recently,
+there has been a lot of interest in analyzing the convergence of local, distributed processes [25, 8].
+Convex programminghas been oneof the main tools in designin g algorithms for market equilibrium. A
+simple modification of the convex program introduced by [16, 13] captures the equilibria of our problem
+as its optimal solution. (This is presented in Appendix A.) T his proves existence and uniqueness of
+equilibria. It also implies that the ellipsoid algorithm ca n be used to get a polynomial time algorithm
+to compute the equilibrium1. The auction algorithm is combinatorial, runs faster and pr ovides a simple
+alternative that can be implemented efficiently in practice. Also, the auction algorithm is more amenable
+to heuristical optimizations and modifications, such as to i mprove the running time on a specific instance
+class, or to handle situations in which one already has an equ ilibrium, an additional buyer is introduced
+into the market and we need to compute the new equilibrium. It is not clear if one can construct an
+interior point algorithm to solve the convex program. Ye [26 ] gave one such algorithm for the Eisenberg-
+Gale convex program.
+Devanur et al. [15] gave a combinatorial algorithm based on t he primal-dual schema to compute an
+exact equilibrium in the traditional model. We can generali ze this algorithm to incorporate transaction
+costs, but the best running time we can prove is exponential. We defer the description of this algorithm
+to the full version of the paper. A strongly polynomial time a lgorithm for the Fisher linear market was
+given by Orlin [23]; it does not seem like his ideas can be adap ted directly to our setting.
+Chen, Ghosh and Vassilvitskii [1] study a model similar to ou rs, in the setting of profit-maximizing
+envy-free pricing (for a single commodity but at different loc ations) and show that adding transaction
+costs that form a metric makes the algorithmic problem easier. In contrast, our model is clearly a
+generalization of the traditional Fisher model, and so the p roblem is only harder.
+The transaction costs in our model are independent of the pri ces. An alternate model is to let the
+transaction cost be a fixed fraction of the price. Such costs c an be interpreted as taxesand have been
+1Since the equilibrium could be irrational, the ellipsoid al gorithm would compute an equilibrium with precision δin time
+proportional to log(1 /δ).
+4studiedbyCodenotti et al. [7]. Taxes could beuniform, that is, dependonly onthegood, or non-uniform,
+that is, depend on the good and the buyer. In the Fisher’s mode l, all our results can be extended with
+minimum modifications to the setting where both per-dollar t axes and per-unit transaction costs are
+present in the market.
+Extensions and Open Problems
+All of our results can be easily extended to quasi-linear uti lities, that is, the buyers have utility for money
+as well, which is normalized to 1. So the utility of the bundle xiis/summationtext
+j(uij−pj)xij. Extending the results
+to other common utility functions is an open problem. In part icular, Garg, Kapoor and Vazirani [18]
+extend the auction algorithm to separable weak gross substi tute utilities. The potential function they
+use is the total surplus, and we don’t know a combinatorial bo und on the number of events in their
+algorithm. As mentioned earlier, this potential function c annot be used in the presence of transaction
+costs, and therein lies the difficulty in extending our result s to this case.
+Theauction algorithm forthetraditional modelscan bemade tobedistributedandeven asynchronous,
+with a small increase in the running time. We show that a simil ar distributed/asynchronous version of
+the algorithm may not converge in the presence of transactio n costs. An interesting open question is if
+there is some other asynchronous/distributed algorithm th at also converges fast. In particular, is there
+a tattonnement process that converges fast (like in [8])?
+Outline : The rest of this paper is structured as follows: We provide a n overview, followed by the details
+of our algorithm in Section 2. Section 3 has the proof of the co rrectness of the algorithm and the bound
+on its running time.
+2 Algorithm
+Theorem 2.1. There exists an algorithm that finds ǫ-approximate equilibrium prices and allocations in
+O/parenleftbig1
+ǫ(n+logm)mnlog(B/ǫ)/parenrightbig
+operations where B= (1+ǫ)/summationtext
+iBi.
+Overview
+Our algorithm maintains a set of prices and allocations and m odifies them progressively. To initialize,
+we set all the prices pj=ǫand all the allocations are empty. The algorithm is organize d in rounds. At
+the end of each round, we raise the price of one good by a multip licative factor of 1+ ǫ. Any allocations
+made before the price raise continue to be charged at the earl ier, lower price. Therefore at any point in
+the algorithm, a good may be allocated to buyers at two differen t prices, pjandpj/(1 +ǫ). During a
+round, we take a good away from a buyer at the lower price and al locate it to a buyer (possibly the same
+buyer) at the current, higher price. We find a sequence of such reallocations such that we find a buyer
+with positive surplus and a good in her demand set such that al l of that good is allocated at the current
+price. When we find such a buyer-good pair, we increase the pri ce of that good and end the round. The
+algorithm terminates when the budgets of all the buyers are e xhausted.
+Following invariants are maintained throughout the algori thm:
+I1: Buyers have non-negative surplus i.e. no buyer exceeds h er budget.
+I2: All prices are at least ǫ.
+I3: Every good is either priced ǫor is fully allocated.
+5I4: Any good jallocated to a buyer imust be approximately most desirable. (As in Equation (8))
+I5: A good jis allocated at price either pjorpj/(1+ǫ) wherepjis the current price.
+Invariant I3 is a tighter version of equation (7). We maintai n I3 and I5 until the end of the algorithm
+whence we merge the two price tiers. This may lead to some good s being undersold, but we prove that
+equation (7) still holds. Also note that invariant I4 holds f or any allocations, whether at the higher or
+lower price tier. Unless mentioned otherwise, the statemen ts of all the lemmas that follow are constrained
+to maintain these invariants.
+Wenowpresentthedetails ofouralgorithm. Each roundconsi sts ofroughlytwoparts: 1) Weconstruct
+ademand graph Gon the set of buyers and 2) We perform multiple iterations of a reallocation procedure
+- which we call a transfer walk . At the end of each round, we increment the price of some good. The
+sequence of rounds ends when the surplus of all the buyers red uces to zero. At the end, we readjust the
+allocations to merge the two price tiers. In what follows, we explain our algorithm in three parts: a)
+Construction and properties of the demand graph, b) Transfe r walks and c) Readjustment of allocations.
+Notation : We denote the allocations of good jto buyer iat prices pjandpj/(1+ǫ) ashijandyij
+respectively. We denote by zj= 1−/summationtext
+i(hij+yij) the amount of good junassigned at any point in the
+algorithm. Given any prices and allocations, the surplus riof buyer iis the part of her budget unspent:
+ri=Bi−/summationdisplay
+j(pj+cij)hij−/summationdisplay
+j/parenleftbiggpj
+1+ǫ+cij/parenrightbigg
+yij
+Notice that since the prices remain constant throughout a ro und except at the end, the demand sets
+of all the buyers are well defined. In each round we fix a functio nπ(i) = min{j|j∈Di}. Intuitively,
+we will attempt to allocate the good π(i) toiin this round, ignoring all the other goods in Difor the
+moment. Any choice of a good from Disuits asπ(i), but we fix a function for ease of exposition.
+Construction and properties of the demand graph
+We then construct a directed graph Gon the set of buyers. An edge exists from buyer itokif and
+only ifykπ(i)>0. A node iin this graph with (1) no out-edges ( i.e.a sink), (2) ri>0 and (3) zπ(i)= 0
+will be defined to be ‘unsatisfiable’.
+Lemma 2.2. For an unsatisfiable node i, the price of the good π(i)can be increased by a multiplicative
+factor of 1+ǫ.
+Proof.Letj=π(i). Since node iis unsatisfiable, all the allocations of good jare at the current prices
+before the price raise and they shift to the lower price tier a fter the price raise. This maintains invariant
+I5. The fact that all these allocations continue to be charge d at the earlier price maintains invariant I1.
+Invariant I3 follows from zj= 0. We now need to verify that invariant I4 is not violated.
+Letkbe any buyer such that hkj>0. We claim that j∈Dkbefore the price raise. For contradiction,
+assume otherwise. Then ukj< αk(pj+ckj). Now consider the last instance in the algorithm when any
+allocation of good jwas made to buyer k. Ifα′
+kwas the bang-per-buck of kat that instance, then
+ukj=α′
+k(pj+ckj) implying αk> α′
+k. This is impossible since by definition of bang-per-buck, th eα
+values decrease monotonically as the prices are raised.
+Hence we have ukj=αk(pj+cij). Letp′
+j= (1+ǫ)pjandα′
+kbe the bang-per-buck of kafter the price
+increase. Then
+ukj=αk(pj+cij)≥α′
+k·/parenleftbiggp′
+j
+1+ǫ+cij/parenrightbigg
+≥α′
+k·(p′
+j+cij)
+1+ǫ
+6which is exactly the statement of invariant I4.
+Note: The existence of buyer iwithri>0 is not essential to the statement of the proof, but is rather
+an artefact of the algorithm. Intuitively, we only raise the price of a good if the current price leads to
+excess demand, as evident from buyer i.
+But the graph Gmay not contain an unsatisfiable node to start with. Hence we p erform a series of
+reallocations until we create and/or find such a node.
+The reallocation involves the following step: For an edge i→kinGwithri>0, we take away the
+lower price allocation of good π(i) forkand allocate it to iat the current price. In short, we perform
+the operations ykπ(i)←ykπ(i)−δandhiπ(i)←hiπ(i)+δfor a suitably chosen value of δ. This process
+reducesri,ykπ(i)and increases rk. Ifykπ(i)reduces to zero, we drop the edge ( i,k) from the graph. When
+we make such a reallocation, we say that we transfer surplus fromitok. Note that the surplus is not
+conserved. This is because the price paid by ifor the same amount of the good, including the transaction
+costs, could even be lower than the price paid by k.
+Lemma 2.3. If the edge from itokexists in Gwithri>0, then we can transfer surplus from itok
+such that either the surplus of ibecomes zero or the edge (i,k)drops out of G.
+Proof.Refer to Section 3.
+We can repeatedly apply lemma 2.3 to transfer surplus along a path inG.
+Corollary 2.4. If there exists a path from itokinGandri>0, then we can transfer surplus from ito
+ksuch that either the surplus of all the nodes on the path excep tkbecomes zero or an edge in the path
+drops out of G.
+Proof.Refer to Section 3.
+Finally,Gmay contain cycles. Consider the edges ( i1,i2) and (i2,i3) inGand letj1=π(i1) and
+j2=π(i2). If the transaction costs are all zero, then it can be argued that the last price raise for j1
+must have taken place before the last price raise for j2. Telescoping this argument, one can preclude the
+existence of cycles in Gin absence of transaction costs. This acyclicity of Gforms a pivotal argument in
+the algorithm of Garg and Kapoor [17]. In Appendix C we provid e a sketch of how a cycle can emerge in
+Gwhen transaction costs are present. Moreover, we can show th at the algorithm of [17] will slow down
+indefinitely if Gcontains cycles. In the above example suppose i1=i3, so that Gcontains a cycle of
+length two. For simplicity, assume ri1>0 andri2= 0. One round of their algorithm may perform a
+surplus transfer from i1toi2followed by another from i2toi1. One can compute that this will change
+ri1asri1←Cri1whereCis a constant at given prices. If C= 1−δfor a very small δ >0, it can require
+an infinite number of rounds for ri1to reduce to zero. Broadly speaking, any distributed versio n of the
+auction algorithm, that is unaware of the graph structure wi ll suffer from the same problem.
+Therefore, we need to be able to transfer surplus around a cyc le.
+Lemma 2.5. If there exists a cycle in Gand exactly one node in the cycle has positive surplus, then w e
+can transfer surpluses in such a way that either all the node i n the cycle have zero surplus or an edge in
+the cycle drops out.
+Proof.Refer to Section 3
+7In a round, we use the above lemmas to perform multiple iterat ions of the transfer walk.
+Transfer Walk
+Step 1: Find a node i0with a positive surplus. If there are no such nodes, then term inate the round
+and jump to readjustment of allocations.
+Step 2: Follow a path going out of i0inGin a depth-first-search fashion. We look at the first edge in
+the adjacency list of the last visited node ion the path. Let ( i,k) be this edge. If node kis yet unvisited,
+we follow that edge to extend the path. If kis already on the path, then we have found a cycle in G.
+Finally if ihas no out-edges, then we have found a sink. Whichever the cas e, we now transfer surplus
+along the current path from i0toias in corollary 2.4. If an edge along the path drops out, we tri gger
+event 2d. Otherwise, we trigger events 2a-2c depending upon case. Since the path can visit at most n
+new nodes, the transfer walk must end in a finite number of oper ations in one the of following events:
+Event 2a - The path reaches a sink iwithzπ(i)= 0: Let j=π(i). By corollary 2.4, we must have
+transferred a positive surplus to ieven ifriwas zero at the begining of the walk. Hence iis an
+unsatisfiable node. Raise pj←(1+ǫ)pj. Terminate the walk and the round.
+Event 2b - The path reaches a sink iwithzπ(i)>0: Letj=π(i). By invariant I3, pj=ǫ. We let
+δ= min(ri/ǫ, zj). We then assign hij←hij+δ. Ifδ=ri/ǫthen the surplus of igoes to zero
+otherwise zjgoes to zero. In either case we end the this transfer walk.
+Event 2c - The path finds a cycle: Let ibe the last node visited on the path and an edge ( i,k) in
+Greaches a node kalready visited on the path. By corollary 2.4, all the nodes i n the cycle except
+ihave zero surplus. Therefore, we apply lemma 2.5 until the su rplus ofibecomes zero or an edge
+in the cycle drops out. We terminate the current walk.
+Event 2d - An edge drops out during path transfer: In this case we termi nate the current walk.
+If a transfer walk ends in event 2a, we terminate the current r ound and start the next one. Otherwise
+if events 2b-2d are triggered, we start a new transfer walk. I f the surplus of all buyers is found to be zero
+in Step 1, we move to the last phase, which is readjustment of a llocations.
+Readjustment of allocations
+At the end of the transfer walks, all the required invariants are satisfied, but the same good may be
+allocated to the same or different buyers at different prices: pjandpj/(1+ǫ). Therefore in this phase,
+we merge the two tiers of allocation for every buyer-good pai r to create the final allocations. For all i,j
+such that yij>0, we assign
+xij←hij+pj
+1+ǫ+cij
+pj+cijyij
+The final equilibrium prices are the prices at the terminatio n of the algorithm.
+Theorem 2.6. The algorithm produces ǫ-approximate equilibrium prices and allocations.
+Proof.By lemmas 2.2, 2.3, 2.4 and 2.5, the prices and allocations at the end of the last round satisfy
+invariants I1-I5. Invariant I4 implies that thefinal alloca tions satisfy the approximate optimality dictated
+by equation (8). We have for all i,
+/summationdisplay
+j(pj+cij)xij=/summationdisplay
+j(pj+cij)/parenleftbigg
+hij+yij
+1+ǫ/parenrightbigg
+=/summationdisplay
+j(pj+cij)hij+/summationdisplay
+j/parenleftbiggpj
+1+ǫ+cij/parenrightbigg
+yij=Bi
+8which proves equation (5). Equation (6) follows by definitio ns ofxij’s. Finally, we need to show that
+equation (7) holds. First observe that pj> ǫimplies that the price of good jwas raised in some round,
+which implies zj= 0 in that round. Invariant 3 then implies that zj= 0 at the end of the last round.
+From definition of xij,xij≥hij+yij/(1 +ǫ)≥(hij+yij)/(1 +ǫ) for alliandj. Summing over i,
+this gives equation (7).
+3 Analysis
+Correctness of the Algorithm
+We will now prove the lemmas used in the preceding section.
+Proof of Lemma 2.3: We choose the amount of good j=π(i) transferred from ktoiasδ=
+max/parenleftBig
+ri
+pj+cij, ykj/parenrightBig
+.Clearly, if δ=ykjthen the lower price allocation of jtokis exhausted and the edge
+(i,k) drops out. Otherwise, buyer ispends all her surplus on the new allocation.
+Proof of Corollary 2.4: Leti=v0,v1,....,vl=kbe the path. Then use lemma 2.3 to transfer surplus
+fromvqtovq+1forq= 0 tol−1 in that order. If no edge in the path drops out of G, then by the lemma,
+the surplus of all the nodes on the path except vl=kgoes to zero.
+Proof of Lemma 2.5: Letv0,v1,....,vlbe the cycle in Gwithv0=vland w.l.o.g. v0being the node
+with positive surplus. We will transfer a quantity δqof goodπ(vq) from the lower price allocation of vq+1
+to the higher price allocation of vq. We will adjust the δvalues carefully so as to maintain zero surplus
+at all nodes except v0.
+Givenδ0, the above requirement fixes all other δvalues. To maintain zero surpluses at all other nodes,
+we need to set:
+δq+1=1
+pπ(vq+1)+cvq+1π(vq+1)·/parenleftbiggpπ(vq)
+1+ǫ+cvq+1π(vq)/parenrightbigg
+·δq
+Note that this process changes the surplus of v0as
+rv0←rv0+/parenleftbiggpπ(vl−1)
+1+ǫ+cv0π(vl−1)/parenrightbigg
+δl−1−(pπ(v0)+cv0π(v0))δ0=rv0+Cδ0
+whereCis a constant determined by current prices. If Cis negative, then rv0reduces and let δ∗be the
+value ofδ0at which is reaches zero. Otherwise, let δ∗=∞.
+We setδ0to be the maximum value such that δ0≤δ∗andδq≤yvq+1π(vq)for all 0≤q < l. We
+perform the surplus transfer with these δvalues. If δ0=δ∗then the surplus of v0reduces to zero and
+hence all buyers have zero surpluses. Otherwise, there exis tsqsuch that δq=yvq+1π(vq)and hence the
+edge (vq,vq+1) drops out of the graph.
+Running Time of the Algorithm
+The major chunk of the computation in our algorithm happens i nside the transfer walks. Hence we count
+the number of transfer walks that we perform, categorized by the event that ends the walk.
+Lemma 3.1. IfRis the number of rounds in the algorithm, then the number of tr ansfer walks that end
+in an edge dropping out of Gis at most nR.
+9Proof.There are two ways in which edges can be added to Gafter a price increase.
+1. After a price increase, the function π(i) can change and hence the edges going out of ican change.
+2. When the price of a good jis raised at the end of the round, an edge ( i,k) appears in Gfor eachi
+such that j∈Diand for each kwhich has any allocation of j.
+This addition of potentially Ω( n2) edges implies a weak upper bound of n2Ron the number of such
+transfer walks in the algorithm. In what follows we define ano ther graph which mirrors Gin semantics,
+but contains a lot fewer edges.
+LetHbe a directed bipartite graph between the set of buyers Iand the set of goods J. Fori∈I
+andj∈J, an edge ( i,j)∈Hif and only if j=π(i). Similarly, ( j,i)∈Hif and only if yij>0. Clearly,
+the edge ( i,k)∈Gif and only if kcan be reached from iinHby a path of length two. Also note that
+the number of edges going from ItoJis exactly nat any point. An edge ( i,k) drops out of Gif and
+only if the edge ( π(i),k) drops out of H. But only nedges are possibly added from JtoIafter a price
+increase. Since we start with zero edges going from JtoI, an edge can drop out of Hat mostnRtimes
+throughout the algorithm, and hence the same bound applies t oG.
+Proof of Theorem 2.1 :
+Initialization and readjustment : Both the initialization and final adjustment of allocation s can be
+performed in mnoperations.
+The number of rounds : The price of exactly one good is raised by multiplicative fa ctor of 1+ ǫin
+each round except the last round. Starting at ǫ, the maximum value to which a price may be raised is
+B= (1+ǫ)/summationtext
+iBi. Therefore, there can be at most R= 1+m
+ǫlog(B
+ǫ).
+Constructing the graph : Notice that although the demand set of a buyer may contain al l them
+goods, we only need one of them at any point. For each buyer, we maintain all the goods in a balanced
+tree data structure that sorts the goods first by the bang-per -buckuij/(pj+cij) and then by the index
+j. In this manner, we can compute the function π(i) inO(logm) time. Given π(i), every node may have
+an edge to every other node. Therefore, the graph Gcan be constructed in O(n2+nlogm) operations.
+After the price increase at the end of the round, the sorted tr ees can be maintained in time O(nlogm)
+while the transfer of allocations from higher to lower price tier can be completed in O(n) operations.
+Number of transfer walks : All the remaining computation in the algorithm takes place within the
+transfer walks. We will perform an amortized analysis on the number of transfer walks that take place
+throughout the algorithm. Notice that since we follow the fir st edge going out of each vertex, the depth-
+first-search requires only O(n) operations. The surplus transfer along a path and a cycle ca n similarly
+be performed in O(n) operations. When an edge drops out, updating Ginvolves simply incrementing a
+pointer. Therefore, overall a transfer walk requires O(n) operations.
+We will now bound the number of transfer walks that happen thr oughout the algorithm, including
+all the rounds. We will classify them by the event that ends th e walk. At most Rtransfer walks can
+terminate the round. At most mwalks can end with zjgoing zero. Lemma 3.1 bounds the number of
+walks that end with an edge dropping out of the graph. The only remaining case is that the walk ends
+when the surplus of the last visited node on the path vanishes . A transfer walk ending in this case leaves
+one less node in Gwith a positive surplus. To see this, observe that a transfer walk starts with a node
+on the same path with positive surplus and by the time it ends i n this case, all the nodes on the path
+have zero surplus by corollary 2.4 and lemma 2.5.
+10Letr+be the number of nodes in Gwith positive surplus at any point in the algorithm. After
+initialization we have r+=nand the only event which may increase r+is event 2d. If an edge ( i,k)
+drops out during surplus transfer along the path, node kmay be left with some positive surplus that
+was absent at the start of the walk. Therefore r+increases by at most one in this event. Combined with
+lemma 3.1, this implies a bound of n+nRon the number of times r+reduces.
+Itisclearfromtheaboveanalysisthatthealgorithmperfor msatmost O(nR)transferwalks. Combined
+with the other computation bounds, this yields an upper boun d ofO/parenleftbig1
+ǫ(n+logm)mnlog(B/ǫ)/parenrightbig
+on the
+running time of the algorithm.
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+A Formulation as a Convex Program
+Convex Program Formulation : Consider the following convex program,
+minimize/summationdisplay
+jpj−/summationdisplay
+iBilogβi (9)
+∀i,j p j+cij≥uijβi
+∀i, βi≥0∀j, pj≥0
+12Theorem A.1. A vector (p,β)is an equilibrium of Mif and only if it minimizes Convex program (9).
+As a corollary, a Fisher market with linear utilities and tra nsaction costs has a unique set of equilibrium
+prices.
+Proof.Note that the KKT conditions that guarantee optimality of a f easible solution are as follows, with
+xijbeing the Lagrangian multiplier for the inequality pj+cij≥uijβi. There exists, for all i,j, xijsuch
+that
+∀i,j:xij>0⇒pj+cij=uijβi (10)
+∀i:/summationdisplay
+juijxij=Bi/βi. (11)
+∀j:/summationdisplay
+ixij≤1 and equality holds if pj>0. (12)
+The equivalence of the KKT conditions and market equilibriu m conditions (equations (1)-(4)) follows
+from interpreting xijas the allocation and βias 1/αi. Conditions (10) and (11) together imply (1), that
+all buyers exhaust their budget, i.e. for all i,
+/summationdisplay
+j(pj+cij)xij=/summationdisplay
+j:xij>0uijβixij=Bi.
+Conversely, equations (1) and (4) together imply (11).
+/summationdisplay
+juijxij=/summationdisplay
+j:xij>0(pj+cij)xij
+βi=Bi
+βi
+Conditions (2) and (3) are the same as (12). Condition (4) is t he same as (10). This proves existence
+of equilibrium prices.
+Uniqueness follows from the fact that the objective functio n in (9) is strictly convex. Let ( p,β) and
+(p,β) be two distinct optimal solutions to the convex program. We claim that β/ne}ationslash=β. For contradiction,
+assumeβ=β. But observe that given any β, we can determine a uniquepwhich minimizes the objective
+function by setting pj= max i(uijβi−cij). Therefore, if β=βthenp=p. Since ( p,β) and (p,β) are
+distinct, we conclude that β/ne}ationslash=β.
+Now consider a linear combination ( ˆp,ˆβ) =α(p,β) + (1−α)(p,β) for any 0 < α <1. Clearly,/summationtext
+jˆpj=/summationtext
+j/parenleftbig
+αpj+(1−α)pj/parenrightbig
+, but/summationtext
+iBilogˆβi>/summationtext
+iBi/parenleftbig
+αlogβi+(1−α)logβi/parenrightbig
+, since log is a strictly
+concave function. Therefore,
+/summationdisplay
+jˆpj−/summationdisplay
+iBilogˆβi< α
+/summationdisplay
+jpj−/summationdisplay
+iBilogβi
++(1−α)
+/summationdisplay
+jpj−/summationdisplay
+iBilogβi
+
+This contradicts the fact that ( p,β) and (p,β) were both optimal. Hence, the objective function is
+strictly convex, and the equilibrium is unique.
+13B A market with irrational equilibrium prices
+Consider a market with two buyers, iandkand two goods j,j′.
+Buyeri Buyerk
+Budgets 1 1
+Utilities uij= 1000,uij′= 1 ukj=ukj′= 1
+Transaction costs cij= 1,cij′= 1000 ckj=ckj′= 0
+It can be verified that the pj=pj′=1√
+2are equilibrium prices. At these prices, buyer ionly demands
+goodjwhereas buyer i′demands both goods. Buyer ispends her one dollar on√
+2√
+2+1units of good jat
+effective price 1+1√
+2. Buyeri′buys the remaining1√
+2+1units of good jand the entire one unit of good
+j′, both at effective price1√
+2.
+C Existence of cycles in the demand graph
+We provide a sketch of how a cycle can exist in the demand graph . Consider two buyers iandkand two
+goodsjandj′in a market with many other buyers and goods. Their utilities and transaction costs are
+tabulated below. Assume both iandjhave sufficiently large budgets and consider an instance in th e
+auction algorithm when pj=pj′= 1.
+Buyeri Buyerk
+Utilities uij= 1+ǫ
+3,uij′= 2 ukj= 2,ukj′= 1+ǫ
+3
+Transaction costs cij= 0,cij′= 1 ckj= 1,ckj′= 0whereǫ <1.
+Clearly, good jwill be allocated to buyer iand good j′to buyer k. Now if the price of both the goods
+is raised due to demand by other buyers, j′appears in the demand set of iandjappears in the demand
+set ofk. Hence, the graph Gmay contain a cycle on the two nodes iandkafter the prices are raised.
+Note: The fast version of the auction algorithm in [17] emplo ys certain tie-breaking techniques that
+suffice to ensure acyclicity in the absence of transaction cos ts. The above example holds even if those
+techniques are employed in our model. Therefore, we have cho sen to simplify our algorithm by not using
+these tie-breaking rules.
+In absence of transaction costs the following property hold s, which is used in [17]: If buyer iprefers
+goodjover good j′at prices pjandpj′, then she maintains the same preference when the prices are e ach
+raised to pj(1+ǫ) andpj′(1+ǫ). From the above example, such an assertion is false in our mo del.
+14
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+arXiv:1001.0395v2  [nucl-th]  19 Apr 2010Response functions of cold neutron matter: density fluctuat ions
+Armen Sedrakian and Jochen Keller
+Institute for Theoretical Physics, J. W. Goethe University , D-60438 Frankfurt-Main, Germany
+(Dated: October 26, 2018)
+We compute the finite temperature density response function of nonrelativistic cold fermions with
+an isotropic condensate. The pair-breaking contribution t o the response function evaluated in the
+limit of small three-momentum transfers qwithin an effective theory which exploits series expansion
+in powers of small q/pF, wherepFis the Fermi momentum. The leading order O(q2) contribution is
+universal and depends only on two fundamental scales, the Fe rmi energy and the pairing gap. The
+particle-hole Landau Fermi-liquid interaction contribut es first at the next-to-leading order O(q4).
+The scattering contribution to the polarization tensor is n onperturbative (in the above sense) and
+is evaluated numerically. The spectral functions of densit y fluctuations are constructed and the
+relevance of the q2scaling for the pair breaking neutrino emission from neutro n stars is discussed.
+I. INTRODUCTION
+The long-wavelength, low-energy dynamics of
+fermionic systems is determined by their response
+functions to soft perturbations, which are characterized
+by length scales that are large compared to the inverse
+Fermi wave vector and energies that are small compared
+to the Fermi energy. At zero temperature the response
+functions of pair-correlated nuclear matter have been
+studied long ago by Larkin and Migdal [1]. Recently, the
+response functions of pair-correlated nuclear systems at
+nonzero temperature received attention in the context of
+neutral current neutrino emission via pair breaking and
+formation in compact stars [2–5] and neutrino scattering
+in supernovae [6]. The evaluation of the response func-
+tions involves typically a resummation of infinite number
+of finite temperature ring diagrams. In the unpaired
+limit of normal Fermi liquid, this resummation scheme
+reduced to the familiar random-phase-approximation
+(RPA); for recent applications in nucleonic and neutron
+star matter, see Refs. [7–10].
+In attractive, cold, fermionic systems the gap ∆ in the
+quasiparticle spectrum is small compared to the Fermi
+energy and the hierarchy of energy scales depends on
+the magnitude of the perturbation, which can take arbi-
+trary values with respect to the pairing gap ∆. In this
+work we focus on density perturbations and show that
+the two distinct contributions to the response function
+through the scattering and pair breaking processes are
+effective below and above the energy threshold 2∆, i.e.,
+the energy needed to break a pair. The pair breaking
+processes are of special importance for applications in
+compact stars; to evaluate them, we propose a new sys-
+tematic low-transferred-momentum expansion of the re-
+sponse function which builds on the previousworkon po-
+larizationtensorsofcoldsuperfluid fermionic systems [2].
+Specifically, we show that the pair breaking contribution
+to the polarizationtensor possesa well-definedexpansion
+withrespecttothe ratioofthemomentumoftheexternal
+current to the Fermi momentum ofthe fermions. We also
+adopt more general ansatz for the driving terms in the
+integral equations of Ref. [2] by lifting the degeneracy
+among the particle-particle and particle-hole channels.We work in the nonrelativistic limit, i.e.,the ratio of the
+Fermi velocity to the speed of light is small ( vF/c≪1).
+Thedensityresponsefunctionscanbeutilizedtodeter-
+mine the spectrum of the collective modes, the stability
+of the system toward clustering, the rates of electromag-
+netic and weak radiation processes, and so on. In partic-
+ular,theratesofneutrinoreactionsinstellarinteriorscan
+be expressed through the response function of underly-
+ing matter to vector and axial-vector weak currents [11–
+14]. In nonrelativistic limit, the vector and axial-vector
+responses are mapped onto the responses to the den-
+sity and spin-density perturbations, respectively. The
+response functions in the superfluid neutron matter were
+recently computed and the neutrino emission rates were
+determined in Refs. [2–5]. Phenomenologically, these are
+important in modeling the cooling of intermediate age
+neutron stars and the superburst in accreting neutron
+stars [15–17].
+It is now well established that at zeroth order ( q= 0)
+the pair breaking density response function vanishes, as
+required by the f-sum rule for the polarization tensor,
+which is a direct consequence of the baryon number con-
+servation. Some authors found analytically the leading-
+order contribution to the polarization which arises at or-
+derq4[3, 18]. Since there is no general argument that
+requires the coefficient of the leading-order q2term to be
+zero, the answer may depend on the approximations in-
+volved in the theory. In section IIIA we study in detail a
+newsmall qexpansionofthepolarizationtensorobtained
+in Ref. [2] and find that the coefficient of the q2term in
+the series expansion of the density response function is
+indeed nonzero. Furthermore, it turns out to be univer-
+sal,i.e.,it depends only on two fundamental scales of
+the problem, the Fermi energy and pairing gap, and is
+independent of the strength of particle-hole interaction.
+This paper is organized as follows. In Sec. II we de-
+rive the vertex functions and polarization tensor in a
+more general setting that in Ref. [2] by using different
+particle-particle and particle-hole interactions and clar-
+ify the approximations that arise in the weak coupling
+Bardeen-Cooper-Schrieffer(BCS) limit. A small momen-
+tumtransferexpansionisappliedtothepairbreakingpo-
+larization tensor in Sec. IIIA. We also show in Sec. IIIB2
+the resultsofexact numericalevaluation ofthe scattering
+part of the polarization tensor. Further in Sec. IIIC we
+verify that the unpaired and uncorrelated limits are re-
+covered from the scattering part. Our conclusions are
+collected in Sec. IV. Details of calculations are pre-
+sented in Appendices A and B. We use the natural units
+/planckover2pi1=c= 1 throughout and assume that the Boltzmann
+constant kB= 1.
+II. DENSITY RESPONSE FUNCTION
+In this section we derive the general form of the
+density-response function of neutron matter at nonzero
+temperature. At densities below the saturation density
+neutron matter forms a1S0pair condensate and can be
+described by the weak coupling limit of the BCS theory.
+This affects not only the approximationsthat are applied
+to the gap equation, but also the approximate relations
+between the loop integrals, as we discuss below.
+Thecouplingsin theparticle-particle( pp)andparticle-
+hole (ph) channels, vppandvph, are assumed zero range;
+often their values are taken to be degenerate vpp=vph
+and equal to the lowest order Landau parameter f0. We
+shall lift this approximation below by assuming vpp∝ne}ationslash=
+vph. The spectrum of paired neutrons is given by
+ǫp=/radicalBig
+ξ2p+∆2(p), (1)
+whereξp=p2/2m∗−µis the quasiparticle spectrum
+in the normal state, ∆( p) is the energy gap, pis the
+three-momentum, m∗is the effective mass and µis the
+chemical potential. For contact pairing interaction the
+gap is momentum independent, ∆( p)≡∆.
+The softness of the modes implies that their wave vec-
+tor|q| ≪pF. Accordingly, we write
+ξp+q=p2
+2m∗/parenleftbigg
+1+2p·q
+p2+q2
+p2/parenrightbigg
+−µ, (2)
+and consider the second and third terms in the bracket
+as small compared to unity, since p≃pF, wherepFis
+the Fermi momentum. Thus, we may write ξp+q≃ξp+
+µ0(2yx+y2), where x= (p·q)/(|p||q|) andy=q/pF,
+whereby y≪1. Here µ0=p2
+F/2m∗is the chemical
+potential at zero temperature; we shall drop the 0 index
+in the following. Several observations are in order:
+1. If the expansion is carried out with respect to the
+small parameter δξ=ξp+q−ξp, as in Ref. [2], the
+power counting is not manifest. At the leading-
+order the terms which scale linearly in xdrop on
+angle integration in symmetrical limits. The only
+nonzero contribution proportionalto q2is then fur-
+nished by the recoil term. One needs to carry out
+theδξexpansion at least to second order to obtain
+all relevant terms that are of order q2.2. An alternative to the expansion with respect to
+δξis the expansion with respect to the ratio y=
+|q|/pF. This expansion exploits the softness of the
+modes and applies both for timelike and spacelike
+momentum transfers. It is an alternative to ex-
+pansions in the ratios ω/qvFandqvF/ω, which are
+valid in these regimes, respectively.
+3. Organizing the expansion in powers of vF/c≪1
+(nonrelativistic fermions) does not guarantee per se
+the convergence of the series. At any fixed density,
+theFermivelocity vFisconstantandforsufficiently
+large momentum transfers ( q≥pF/2) the series
+will fail to converge.
+4. Finally, the smallness of the expansion parameter
+is necessarybut not sufficient condition forthe con-
+vergence of the Taylor series. The validity of the
+expansion should be checked by an exact numerical
+computation of the loop integrals.
+Inthefollowingweshalldemonstratethatthepairbreak-
+ing part of the response function can be expanded sys-
+tematicallywith respect tothe yparameter. Such expan-
+sion is thus valid for small three-momentum transfers,
+but arbitrary energy transfers.
+A. Vertex functions
+We start with integral equations for the vertex func-
+tions and derive a (slight) generalization of their coun-
+terparts in Ref. [2] that distinguish the particle-particle
+and particle-hole interactions. These we write as sums of
+central and spin-spin interaction terms
+Vpp≃vpp+vpp(σ·σ′)+..., (3)
+Vph≃vph+vph(σ·σ′)+..., (4)
+where the ellipses stand for the tensor and spin-orbit
+terms that are subdominant at relevant densities in neu-
+tron matter.
+Theintegralequationsdefiningthescalarvertex,which
+we write in an operator form, are given by [1, 2]
+ˆΓ1= Γ0+vph(GΓ1G+ˆFˆΓ3G+GˆΓ2ˆF+ˆFΓ4ˆF),(5)
+ˆΓ2=vpp(GˆΓ2G†+ˆFΓ4G†+GΓ1ˆF+ˆFˆΓ3ˆF),(6)
+ˆΓ3=vpp(G†ˆΓ3G+ˆFΓ1G+G†Γ4ˆF+ˆFˆΓ2ˆF),(7)
+ˆΓ4= Γ0+vph(G†Γ4G†+ˆFΓ1ˆF+ˆFˆΓ2G†+G†ˆΓ3ˆF).
+(8)
+HereˆF=−iσyF,σyis the second component of the
+Pauli matrix, Γ 0= 1. When vpp=vph, Eqs. (5)-(8)
+become identical to those of Ref. [2]. Let us now define
+the “elementary loop” as
+ΠXX′(q) =g/integraldisplayd4p
+(2π)4X(p)X′(p+q),(9)3
+whereX∈ {G,G†,F,F†}andgis the degeneracy factor,
+which we omit in the intermediate equations and restore
+in the final ones. Direct calculations show that
+ΠG†F= ΠFG,ΠGF= ΠFG†,ΠG†G†= ΠGG; (10)these equalities imply that Γ 1= Γ4, which is a conse-
+quence of the time-reversal invariance of the system.
+The remaining equations read
+
+1−vph[ΠGG−ΠFF]vphΠGF vphΠFG
+−2vppΠGF [1−vppΠGG†]vppΠFF
+−2vppΠFG vppΠFF[1−vppΠG†G]
+
+Γ1
+Γ2
+Γ3
+=
+Γ0
+0
+0
+. (11)
+There are six distinct loops in Eq. (11), namely Π GG,
+ΠFF, ΠGF, ΠFG, ΠG†G, ΠGG†. In the weak coupling
+limit Π GF≃ −ΠFG, ΠG†G≃ΠGG†, an approximation
+discussed in detail in Sec. IIB. This reduces the number
+of equations from three to two
+/parenleftbigg
+1−vphAvphB
+−vppB −vppC/parenrightbigg/parenleftbigg
+Γ1
+Γ2/parenrightbigg
+=/parenleftbigg
+Γ0
+0/parenrightbigg
+,(12)
+where
+A(q) = Π GG(q)−ΠFF(q), (13)
+B(q) = 2Π FG(q), (14)
+C(q) = Π GG†(q)+ΠFF(q)−(vpp)−1,(15)
+withq= (ω,q). The solutions for the remaining two
+vertex functions reads
+Γ1(q) =Γ0C(q)
+C(q)−vph[A(q)C(q)+B(q)2],(16)
+Γ2(q) =−Γ0B(q)
+C(q)−vph[A(q)C(q)+B(q)2].(17)Whenvpp=vphthese reduce to Eqs. (16) and (17) of
+Ref. [2]. It seen that the interaction in the ppchannel is
+absorbed in the gap equation and it is the phinteraction
+that enters the renormalization of the one-loop polariza-
+tion tensor. This result could have been anticipated from
+the limiting form of the RPA polarization tensor of nor-
+mal Fermi liquids (see Sec. IIIC).
+B. Polarization tensor
+The full polarization tensor is given by Eq. (35) of
+Ref. [2] with the replacement v→vph
+ΠR(q) =A(q)C(q)+B(q)2
+C(q)−vph[A(q)C(q)+B(q)2].(18)4
+The “elementary loops” are defined explicitly as
+ΠGG(q) =/integraldisplayd3p
+(2π)3/braceleftBigg/bracketleftBigg
+u2
+pu2
+k
+iq+ǫp−ǫk−v2
+pv2
+k
+iq−ǫp+ǫk/bracketrightBigg
+[f(ǫp)−f(ǫk)]
++/bracketleftBigg
+u2
+kv2
+p
+iq−ǫp−ǫk−u2
+pv2
+k
+iq+ǫp+ǫk/bracketrightBigg
+[f(−ǫp)−f(ǫk)]/bracerightBigg
+, (19)
+ΠFG(q) =−/integraldisplayd3p
+(2π)3/braceleftBigg
+upvp/bracketleftbiggu2
+k
+iq+ǫp−ǫk+v2
+k
+iq−ǫp+ǫk/bracketrightbigg
+[f(ǫp)−f(ǫk)]
+−upvp/bracketleftbiggu2
+k
+iq−ǫp−ǫk+v2
+k
+iq+ǫp+ǫk/bracketrightbigg
+[f(−ǫp)−f(ǫk)]/bracerightBigg
+, (20)
+ΠFF(q) =/integraldisplayd3p
+(2π)3/braceleftBigg
+upukvpvk/braceleftBigg/bracketleftbigg1
+iq+ǫp−ǫk−1
+iq−ǫp+ǫk/bracketrightbigg
+[f(ǫp)−f(ǫk)]
++/bracketleftbigg1
+iq+ǫp+ǫk−1
+iq−ǫp−ǫk/bracketrightbigg
+[f(−ǫp)−f(ǫk)]/bracerightBigg/bracerightBigg
+, (21)
+ΠG†G(q) =−/integraldisplayd3p
+(2π)3/braceleftBigg/bracketleftBigg
+u2
+kv2
+p
+iq+ǫp−ǫk−u2
+pv2
+k
+iq−ǫp+ǫk/bracketrightBigg
+[f(ǫp)−f(ǫk)]
++/bracketleftBigg
+u2
+pu2
+k
+iq−ǫp−ǫk−v2
+pv2
+k
+iq+ǫp+ǫk/bracketrightBigg
+[f(−ǫp)−f(ǫk)]/bracerightBigg
+, (22)
+wherek=p+qand the coherence factors are given by
+u2
+p= (1/2)[1+ξp/ǫp] andv2
+p= 1−u2
+p.
+The expression for Π FG(q) above applies at arbitrary
+couplings; however Eq. (14) presumes weak-coupling ap-
+proximation, because Π GF(q) =−ΠFG(q) holds only
+in this limit. The weak-coupling limit for the func-
+tion Π FG(q) is obtained on substituting in square braces
+u2
+k=u2
+p=v2
+p=v2
+k= 1/2. This statement is equivalent
+to ignoring integrals of the type
+I=/integraldisplayd3p
+(2π)3∆2
+2ǫpξp
+ǫpf(ǫp)−f(ǫk)
+iq+ǫp−ǫk≃0.(23)
+The integral vanishes because ξpchanges sign for mo-
+menta above and below the Fermi momentum, while the
+remainder of the integrand is an even function in the
+vicinity of pF. Thus the loop polarization function re-
+duces to
+ΠFG(q) =−1
+2/integraldisplayd3p
+(2π)3/braceleftBigg
+upvp
+/bracketleftbigg1
+iq+ǫp−ǫk+1
+iq−ǫp+ǫk/bracketrightbigg
+[f(ǫp)−f(ǫk)]
+−upvp/bracketleftbigg1
+iq−ǫp−ǫk+1
+iq+ǫp+ǫk/bracketrightbigg
+[f(−ǫp)−f(ǫk)]/bracerightBigg
+. (24)The form of the polarization loop Π G†G(q) is valid for
+arbitrary couplings. The relation Π G†G(q) = ΠGG†(q) is
+established on noting that, for example, u2
+pv2
+k= (u2
+pv2
+k−
+u2
+kv2
+p)+u2
+kv2
+pand that the combination in braces vanishes
+in the weak coupling as it leads to an integral of the type
+(23).
+The contributions to the polarization function due to
+quasiparticle scattering and pair breaking separate. For
+the contributions from scattering the poles are located at
+±(ǫp−ǫk) and the distribution is given by the combina-
+tionf(ǫp)−f(ǫk). Forthepairbreakingcontributionsthe
+polesarelocatedat ±(ǫp+ǫk) and the distribution ispro-
+portional 1 −f(ǫp)−f(ǫk). The pair breaking contribu-
+tion vanishes in the limit T→T−
+c(Tis the temperature,
+Tcis the critical temperature of the phase transition).
+Now we write the retarded polarization functions
+ΠR(q) in terms of the functions A(q),B(q) andC(q) af-
+ter performing analytical continuation ( iq→ω+iδ) in
+functions Π GG(q), ΠFF(q), ΠFG(q) ΠG†G(q). After some
+straightforward algebraic transformations we find5
+A(q) =/integraldisplayd3p
+(2π)3/braceleftBigg
+1
+2(ǫk−ǫp)/bracketleftbigg
+1+ξpξk
+ǫpǫk−∆2
+ǫpǫk/bracketrightbiggf(ǫp)−f(ǫk)
+(ω+iδ)2−(ǫp−ǫk)2
++1
+2(ǫp+ǫk)/bracketleftbigg
+1−ξpξk
+ǫpǫk+∆2
+ǫpǫk/bracketrightbiggf(−ǫp)−f(ǫk)
+(ω+iδ)2−(ǫp+ǫk)2/bracerightBigg
+, (25)
+B(q) =−∆ω/integraldisplayd3p
+(2π)31
+ǫp/bracketleftBigg
+f(ǫp)−f(ǫk)
+(ω+iδ)2−(ǫp−ǫk)2−f(−ǫp)−f(ǫk)
+(ω+iδ)2−(ǫp+ǫk)2/bracketrightBigg
+, (26)
+C(q) =1
+2/integraldisplayd3p
+(2π)3/braceleftBigg/bracketleftbigg
+(ǫp−ǫk)/parenleftbigg
+1−ξpξk
+ǫpǫk−∆2
+ǫpǫk/parenrightbigg/bracketrightbiggf(ǫp)−f(ǫk)
+(ω+iδ)2−(ǫp−ǫk)2
+−/bracketleftbigg
+(ǫp+ǫk)/parenleftbigg
+1+ξpξk
+ǫpǫk+∆2
+ǫpǫk/parenrightbigg/bracketrightbiggf(−ǫp)−f(ǫk)
+(ω+iδ)2−(ǫp+ǫk)2−1
+ǫp[1−2f(ǫp)]/bracerightBigg
+. (27)
+Note that lim q→0,ω→0C(q) = 0, since the coupling con-
+stant in the particle-particle channel can be expressed
+as
+(vpp)−1= ΠG†G(q= 0)+Π FF(q= 0)
+=/integraldisplayd3p
+(2π)31
+2ǫp[1−2f(ǫp)],(28)
+which is the gap equation for the contact interaction vpp.
+We do not need to specify the regularization of the gap
+equation, since its divergence is eliminated in the loop
+integrals. For numerical purposes we will adopt gaps
+obtained from finite-range interactions in Ref. [19].
+III. EVALUATING RESPONSE FUNCTIONS
+Equations (25)–(27) separate into scattering and pair
+breaking contributions. We shall see that the first con-
+tributes essentially below the pair breaking threshold
+ω <2∆, whereas the second contributes for ω >2∆.
+In the following we discuss in detail the pair breaking
+part and its small momentum expansion. The scatter-
+ing part will be addressed later in this section, where we
+evaluate it numerically.
+A. pair breaking response function: small
+momentum expansion
+The small- yexpansion of the pair breaking part of the
+polarization tensor is obtained upon writing ΠR(q) =/summationtext
+nPnyn,A(q) =/summationtext
+nAnynand similarly for the func-
+tionsBandCand truncating the Taylor series at the de-
+sired order in y. The odd powers of ydo not contribute
+to the series.
+Up to order y4the coefficients for the polarization ten-sor are given by
+P0= 0, (29)
+P2=2B0B2+A2C0+A0C2
+C0, (30)
+P4=B2
+2+2B0B4+A4C0+A2C2+A0C4
+C0
+− P2/bracketleftbiggC2
+C0−vphP2/bracketrightbigg
+. (31)
+The coefficients of the expansion are given by (for details
+see Appendix A)
+A0= 2∆2/integraldisplayd3p
+(2π)31
+ǫpL0, (32)
+B0= ∆ω/integraldisplayd3p
+(2π)31
+ǫpL0, (33)
+C0=−ω2
+2/integraldisplayd3p
+(2π)31
+ǫpL0, (34)
+A2=/integraldisplayd3p
+(2π)3/bracketleftBigg
+∆2µξp
+ǫ5p(6µξpx2−ǫ2
+p)L0
+−2∆2µξp
+ǫ3pL1x2+2∆2
+ǫpL2/bracketrightBigg
+, (35)
+B2= ∆ω/integraldisplayd3p
+(2π)31
+ǫpL2, (36)
+C2=−/integraldisplayd3p
+(2π)3/bracketleftbiggµξp
+ǫpL0+2µξp
+ǫpL1x2+2ǫpL2/bracketrightbigg
+,
+(37)
+where the functions L0,L1, andL2are defined by Eqs.
+(A10)–(A12) of Appendix A. The zeroth order term P0
+vanishes as a consequence of the f-sum rule. The leading
+order nonzero term is given by
+P2= ∆2ν(pF)(I0+I1+I2), (38)6
+whereν(pF) =gm∗pF/2π2is the density of states at the
+Fermi surface, and
+I0= 4ν(pF)−1µ/integraldisplayd3p
+(2π)3ξp
+ǫpω2
+/bracketleftBig
+1+ω2
+4ǫ2p/parenleftbigg6µξp
+ǫ2px2−1/parenrightbigg/bracketrightBig
+L0, (39)
+I1= 2ν(pF)−1µ/integraldisplayd3p
+(2π)3ξp
+ǫ3p/bracketleftBigg
+4ǫ2
+p
+ω2−1/bracketrightBigg
+L1x2,(40)
+I2= 2ν(pF)−1/integraldisplayd3p
+(2π)31
+ǫp/bracketleftBigg
+4ǫ2
+p
+ω2−1/bracketrightBigg
+L2.(41)
+These integrals are evaluated in Appendix B. We obtain
+the following analytical result for the imaginary part of
+P2:
+ImP2=−πν(pF)
+3T∆2µsgn(ω)
+ω5√
+ω2−4∆2sech2/parenleftBigω
+4T/parenrightBig
+/braceleftBigg
+4µω(ω2−4∆2)−T[40∆2µ
+−3ω2(6µ+/radicalbig
+ω2−4∆2)]sinh/parenleftBigω
+2T/parenrightBig/bracerightBigg
+θ(|ω|−2∆).
+(42)
+To leading-order the imaginary part of the polarization
+tensor is then given by
+ImΠR(q) = ImP2(ω)/parenleftbiggq
+pF/parenrightbigg2
++O(q4).(43)
+The real part of the polarization tensor follows from the
+dispersion (Kramers-Kronig) relation:
+ReΠR(ω,q) =−1
+π/integraldisplay
+dω′ImΠR(ω′,q)
+ω−ω′.(44)
+Using these quantities one can construct an effective the-
+ory of collective excitations. Their (full, interacting)
+propagators are completely determined by the spectral
+function of the collective excitations
+B(ω,q) =−2ImΠR(ω,q)
+[ω2−q2−ReΠR(ω,q)]2+ImΠR(ω,q)2.
+(45)
+The dispersion relation of the collective excitations is
+read-off as ω2=q2+ ReΠR(ω,q). The finite life-time
+effects are described by the width of the spectral func-
+tion,i.e.,by the function ImΠR(ω,q).
+Figure 1 illustrates the dependence of the real and
+imaginary parts of the pair breaking polarization ten-
+sor on the transferred energy for fixed three-momentum
+transfer. The value of the Landau parameter is f0=
+−0.5 [2]. The zero temperature gap at pF= 0.1 fm−1
+is ∆(0) = 1 MeV and Tc= ∆(0)/1.76. The frequen-
+cies are normalized to the zero-temperature threshold0 1 2 3 4 5
+ω/2∆-0.006-0.004-0.0020Re Π(ω),  Im Π(ω)  [ν]
+T = 0.2 Tc
+T = 0.9 Tc
+FIG. 1: Dependence of the real (solid lines) and imaginary
+(dashed lines) parts of the pair breaking polarization ten-
+sor in units of density of states on the energy transfer in
+units of threshold energy 2∆( T) for fixed momentum trans-
+ferq= 0.1pF, withpF= 0.1 fm−1, and two tempera-
+turesT= 0.2Tc(heavy lines) and T= 0.9Tc(light lines).
+The zero temperature gap is taken to be ∆(0) = 1 .MeV,
+Tc= ∆(0)/1.76.
+frequency 2∆(0), the momentum transfer to the Fermi
+momentum. The real and imaginary parts of the polar-
+ization tensor scale as q2. Their behavior at negative en-
+ergiesfollowsfrom theireven andodd paritywith respect
+to the energy transfer, i.e.,ReΠR(−ω) = ReΠR(ω) and
+ImΠR(−ω) =−ImΠR(ω). Note that the imaginaryparts
+are identically zero below the threshold for pair breaking
+process 2∆( T).
+Figure2illustratesthe spectralfunctionsofpairbreak-
+ing density fluctuation on the energy and momentum
+transfers. The parameters are the same as in Fig. 1. The
+formofthe spectralfunction suggeststhatatlowtemper-
+atures the low-momentum-transfer contribution is con-
+centratednearthe pairbreakingthreshold; for largermo-
+mentumtransfers, modesawayfromtheenergythreshold
+becomeimportant. At highertemperatures( T≤Tc)and
+for any given momentum transfer, the main contribution
+to the spectral function comes from higher energy modes
+and the peak values are larger in the latter regime.
+B. Scattering response function: numerical
+evaluation
+The scattering part of the response function is kine-
+matically important for the space-like processes and van-
+ishes automatically in the limit q→0. Small momentum
+expansion of the previous section was found inappropri-
+ate for the scattering part of the polarization function
+anditwasevaluatednumericallybyadaptingthe method7
+q/pFω/2∆  0 0.05 0.1 0.15 0.2 0.25 0.3B(ω,q)
+ 0
+ 0.05
+ 0.1
+ 0.15
+ 0.2
+ 0.25 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7B(ω,q)
+q/pFω/2∆  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8B(ω,q)
+ 0
+ 0.05
+ 0.1
+ 0.15
+ 0.2
+ 0.25 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7B(ω,q)
+FIG. 2: The spectral function of density fluctuations at
+T= 0.2Tc(upper panel) and T= 0.9Tc(lower panel). The
+parameters are as in Fig. 1.
+described in Ref. [20]. On carrying out the angular inte-
+grals we are left with a one-dimensional integral over the
+energy. As an example, we give the expression for the
+elementary loop
+ImΠ′
+GG(q)ν(pF)−1,=−πpFT
+4qµ0/integraldisplay∞
+−µ/Tdξp
+T
+/bracketleftBigg
+u2
+pu2
+p+qǫp+q
+ξp+q/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+x=x+
+0[f(ǫp)−f(ǫp+ω)]θ(1−|x+
+0|)
+−v2
+pv2
+p+qǫp+q
+ξp+q/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+x=x−
+0[f(ǫp)−f(ǫp−ω)]θ(1−|x−
+0|)/bracketrightBigg
+,
+(46)
+whereθis the Heaviside step function, x= (p·q)/|p||q|,
+andx±
+0is thexvalue satisfying the equation ±ω+ǫp−
+ǫp+q= 0 and the prime refers to the scattering part. The
+expressions for the scattering parts of the polarization-0.2-0.10
+0 0.02 0.04 0.06 0.08 0.1
+ω/2∆-0.08-0.0400.04                                                    Re Π(ω),     Im Π(ω)  [ν(µ)]
+FIG. 3: Dependence of the real (solid lines) and imaginary
+(dashed lines) parts of the scattering polarization tensor in
+units of density of states on the energy transfer in units of
+threshold energy 2∆(0) for fixed momentum transfer q=
+0.1pF, withpF= 0.1 fm−1, and two temperatures T= 0.2Tc
+(upper panel) and T= 0.9Tc(lower panel). The zero temper-
+ature gap is taken to be ∆(0) = 1 MeV, Tc= ∆(0)/1.76.
+tensors Π′
+XX′(q) are similar to Eq. (46), but involve dif-
+ferent combinations of coherence factors. The real parts
+are obtained from the dispersion relation (44).
+Figure 3 shows the real and imaginary parts of the
+scattering polarization tensor at two temperatures and
+for fixed momentum transfer. Figure 4 shows the spec-
+tralfunctionderivedfromthescatteringpolarizationten-
+sor. It is seen that the modes with energies within the
+breaking threshold ω≤2∆(0) are relevant for scattering
+processes, as opposed to the pair breaking case where
+only the modes with ω≥2∆(0) emerge. This justifies
+the separation of the modes into two classes. The peak
+values are larger at higher temperatures T≤Tc, as is the
+case for the pair breaking response.
+C. Unpaired and uncorrelated limits
+Here we wish to obtain the unpaired ( T > T c) limit of
+the polarization function (18). This amounts to setting
+the coherence factors in Eq. (19)–(22) to their values in
+the normal state
+up= 1, vp= 0. (47)
+It follows then from Eq. (20) that Π FG= 0 = Π GFand,
+therefore, B= 0. Setting B= 0 in Eq. (18) and noting
+thatC ∝ne}ationslash= 0, since Π G†G∝ne}ationslash= 0, we obtain
+ΠRPA(q) =A(q)
+1−vphA(q), (48)8
+q/pFω/2∆  0 0.002 0.004 0.006 0.008 0.01 0.012B(ω,q)
+ 0
+ 0.05
+ 0.1
+ 0.15
+ 0.2
+ 0.25 0
+ 0.05
+ 0.1
+ 0.15
+ 0.2B(ω,q)
+q/pFω/2∆  0 0.001 0.002 0.003 0.004 0.005 0.006 0.007B(ω,q)
+ 0
+ 0.05
+ 0.1
+ 0.15
+ 0.2
+ 0.25 0
+ 0.05
+ 0.1
+ 0.15
+ 0.2B(ω,q)
+FIG. 4: The spectral function of density fluctuations at
+T= 0.2Tc(upper panel) and T= 0.9Tc(lower panel). The
+parameters are as in Fig. 1.
+where the function A(q) in the unpaired state reduces to
+A(q) = ΠGG(q,∆ = 0) = g/integraldisplayd3p
+(2π)3f(ǫp)−f(ǫk)
+ω+ǫp−ǫk+iδ.
+(49)
+Equations (48) and (49) are the standard expressions for
+the polarization tensor of a Fermi liquid in the random
+phase approximation (RPA). The free Fermi gas result
+follows on setting vph= 0 in Eq. (48):
+Πfree(q) =g/integraldisplayd3p
+(2π)3f(ǫp)−f(ǫk)
+ω+ǫp−ǫk+iδ.(50)
+Clearly, the pair-braking contribution vanishes as T→
+T(−)
+c. The scattering contribution at T≥Tcreproduces
+the unpaired and uncorrelated limits, as it should.IV. CONCLUSIONS
+In this article we have carried out several steps in the
+program aimed at understanding the polarization tensor
+of pair-correlated neutron and nuclear matter by (i) con-
+structing a low-transfered-momentum approximation to
+the pair breaking polarization tensor and (ii) by numer-
+ically evaluating the scattering part of the polarization
+tensor.
+Our main result is that the low-momentum expan-
+sion of the pair breaking polarization tensor starts at
+quadratic order in the ratio of the momentum-transfer
+to the Fermi momentum. The expansion coefficient is
+universal, i.e.,depends only the two relevant scales: the
+Fermi energy and the pairing gap. We also clarified the
+structure of the integral equations for the vertex func-
+tions when the particle-particle and particle-hole inter-
+actions do not coincide and verified explicitly that the
+unpaired and uncorrelated limits are recovered.
+Further steps will require a verification of the con-
+vergence of the series by a comparison of our analyt-
+ical results with an exact numerical evaluation of the
+response functions. The rate of series convergence can
+be checked by computing the next-to-leading-order con-
+tribution. Further refinements could include finite range
+interactions, tensor forces, and so on. The new numerical
+and analytical methods, discussed in this article, could
+be useful in the studies of the response functions of pair-
+correlated fermionic systems in general.
+The main implication of our study concerns the neu-
+trino emissivity via the vector current pair breaking
+bremsstrahlung, which are phenomenologically impor-
+tant in the physics of neutron star cooling and super-
+bursts in accreting neutron stars [15–17]. Our results
+suggest that the vector current contribution to the pro-
+cess
+{nn} → {nn}+ν+ ¯ν, (51)
+where{nn}refers to the pair correlated state, νand ¯νto
+the neutrino and antineutrino, is suppressed to a lesser
+degree, than suggested in the recent literature [3, 18].
+However, further studies indicated above are needed to
+draw a final conclusion on the relative importance of this
+vector current neutrino emission process (51).
+Acknowledgments
+We thank T.Brauner, X.-G. Huang, and D. H. Rischke
+for discussions. The work of J. K. was supported by the
+Deutsche Forschungsgemeinschaft (Grant SE 1836/1-1).
+Appendix A: Expanding the functions A(q),B(q),
+andC(q)
+We write each of these functions as a product of an ap-
+propriate coherence factor and corresponding statistical9
+factor
+A(q) =/integraldisplayd3p
+(2π)3a(p,q,ω)L(p,q,ω),(A1)
+B(q) = ∆ω/integraldisplayd3p
+(2π)31
+ǫpL(p,q,ω), (A2)
+C(q) =−/integraldisplayd3p
+(2π)3c(p,q,ω)L(p,q,ω)
+−/integraldisplayd3p
+(2π)31
+2ǫp[1−2f(ǫp)], (A3)
+where
+a(p,q,ω) =ǫp+ǫp+q
+2/parenleftbigg
+1−ξpξp+q
+ǫpǫp+q+∆2
+ǫpǫp+q/parenrightbigg
+,
+(A4)
+c(p,q,ω) =ǫp+ǫp+q
+2/parenleftbigg
+1+ξpξp+q
+ǫpǫp+q+∆2
+ǫpǫp+q/parenrightbigg
+,
+(A5)
+L(p,q,ω) =1−f(ǫp)−f(ǫp+q)
+(ω+iδ)2−(ǫp+ǫp+q)2. (A6)
+Next we expand these functions in powers of small pa-
+rameteryto order O(y2)
+a(p,q,ω) =2∆2
+ǫp−2∆2µξp
+ǫ3px y
++∆2µξp
+ǫ5p(6µξpx2−ǫ2
+p)y2,(A7)
+c(p,q,ω) = 2ǫp+2µξp
+ǫpx y+µξp
+ǫpy2,(A8)
+L(p,q,ω) =L0+L1xy+L2y2, (A9)
+where the coefficients of the expansion (A9) can be writ-
+ten using the shorthand expressions D≡ω2−4ǫ2
+pand
+ζ≡ǫp/2Tas
+L0=tanhζ
+D+iδ, (A10)
+L1= 8µξptanhζ
+(D+iδ)2+µξp
+2ǫpTsech2ζ
+(D+iδ),(A11)
+L2=4µ2x2ξp2sech2ζ
+ǫpT(D+iδ)2+µ2x2sech2ζ
+2ǫpT(D+iδ)
++µξpsech2ζ
+4ǫpT(D+iδ)−µ2x2ξp2sech2ζ
+2ǫp3T(D+iδ)
++64µ2x2ξp2tanhζ
+(D+iδ)3+8µ2x2tanhζ
+(D+iδ)2
++4µξptanhζ
+(D+iδ)2−4µ2x2ξp2tanhζ
+ǫp2(D+iδ)2
+−µ2x2ξp2tanhζsech2ζ
+2ǫp2T2(D+iδ), (A12)where sech2(x)≡1−tanh(x)2. On substituting
+Eqs. (A7) and (A9) in Eq. (A1) we obtain
+A(q) =/integraldisplayd3p
+(2π)3/bracketleftBigg
+2∆2
+ǫpL0+2∆2
+ǫpL2y2
+−2∆2µξp
+ǫ3pL1x2y2+∆2µξp
+ǫ5p(6µξpx2−ǫ2
+p)L0y2/bracketrightBigg
+=A0(q)+A2(q)y2+O(y4). (A13)
+The terms that are odd in xdrop out on integration in
+symmetrical limits. Similarly,
+C(q) =−/integraldisplayd3p
+(2π)3/bracketleftBigg
+2ǫpL0+2µξp
+ǫpL1x2y2+µξp
+ǫpL0y2
++ 2ǫpL2y2/bracketrightBigg
+−/integraldisplayd3p
+(2π)31
+2ǫp[1−2f(ǫp)].(A14)
+We further use the relation
+−/integraldisplayd3p
+(2π)32ǫpL0−/integraldisplayd3p
+(2π)31
+2ǫp[1−2f(ǫp)]
+=−ω2
+2/integraldisplayd3p
+(2π)31
+ǫpL0(A15)
+to write
+C(q) =−ω2
+2/integraldisplayd3p
+(2π)31
+ǫpL0−/integraldisplayd3p
+(2π)3/bracketleftBigg
+2µξp
+ǫpL1x2y2
++µξp
+ǫpL0y2+2ǫpL2y2/bracketrightBigg
+=C0(q)+C2(q)y2.
+(A16)
+Finally,
+B(q) = ∆ω/integraldisplayd3p
+(2π)31
+ǫp(L0+L2y2) =B0(q)+B2(q)y2,
+(A17)
+andA0/C0=−4∆2/ω2,andB0/C0=−2∆/ω.
+Appendix B: Evaluating phase-space integrals
+Here we evaluate the integrals (39)–(41), which are
+given as
+I0=ν(pF)−1µ/integraldisplayd3p
+(2π)3ξp
+ǫ3p/bracketleftBigg
+4ǫ2
+p
+ω2−1+6µξp
+ǫ2px2/bracketrightBigg
+L0,
+(B1)
+I1= 2ν(pF)−1µ/integraldisplayd3p
+(2π)3ξp
+ǫ3p/bracketleftBigg
+4ǫ2
+p
+ω2−1/bracketrightBigg
+L1x2,(B2)
+I2= 2ν(pF)−1/integraldisplayd3p
+(2π)31
+ǫp/bracketleftBigg
+4ǫ2
+p
+ω2−1/bracketrightBigg
+L2(x).(B3)10
+After carrying out the angular integrals we obtain
+I0=ν(pF)−1µ/integraldisplaydpp2
+2π2ξp
+ǫ3p/bracketleftBigg
+4ǫ2
+p
+ω2−1+2µξp
+ǫ2p/bracketrightBigg
+L0,
+(B4)
+I1=2
+3ν(pF)−1µ/integraldisplaydpp2
+2π2ξp
+ǫ3p/bracketleftBigg
+4ǫ2
+p
+ω2−1/bracketrightBigg
+L1,(B5)
+I2= 2ν(pF)−1/integraldisplaydpp2
+2π21
+ǫp/bracketleftBigg
+4ǫ2
+p
+ω2−1/bracketrightBigg
+∝an}bracketle{tL2(x)∝an}bracketri}ht,(B6)
+where∝an}bracketle{tL2∝an}bracketri}htis the angle integrated loop L2, which is ob-
+tain from (A12) via the substitution x2= 1/3. Using the
+transformation dpp2=m∗pFd(p2/2m∗) =m∗pFdξp,and
+the relation ǫpdǫp=ξpdξpwe obtain
+I0=µ/integraldisplay∞
+∆dǫp1
+ǫ2p/bracketleftBigg
+4ǫ2
+p
+ω2−1+2µξp
+ǫ2p/bracketrightBigg
+L0,(B7)
+I1=2µ
+3/integraldisplay∞
+∆dǫp1
+ǫ2p/bracketleftBigg
+4ǫ2
+p
+ω2−1/bracketrightBigg
+L1, (B8)
+I2= 2/integraldisplay∞
+∆dǫp1
+ξp/bracketleftBigg
+4ǫ2
+p
+ω2−1/bracketrightBigg
+∝an}bracketle{tL2(x)∝an}bracketri}ht.(B9)
+Tomakefurtherprogressweneedtoseparatetherealand
+imaginary parts of the integrals. We shall first compute
+the imaginary parts. They are extracted with the help
+of the identity
+1
+(D+iδ)n+1=P
+Dn+1−iπ(−1)n
+n!δ(n)(D),(B10)
+wherePdenotes the principal value and δ(n)(D) is the
+n-th derivative of the delta function.For positive ωthe integral (B1) gives [21]
+ImI0=−πµ/integraldisplay∞
+∆dǫp1
+ǫ2p/bracketleftBigg
+4ǫ2
+p
+ω2−1
++2µξp
+ǫ2p/bracketrightBigg
+tanh/parenleftBigǫp
+2T/parenrightBig
+δ(ω2−4ǫ2
+p)
+=−8πµ2√
+ω2−4∆2sgn(ω)
+ω5tanh/parenleftBigω
+4T/parenrightBig
+θ(ω−2∆).
+(B11)
+Consider the integral (B2). First, note that the term ∝
+D−1vanishes, since the delta function enforces ω2= 4ǫ2
+p.
+After dropping this term we are left with the integral
+ImI1=16πµ2
+3/integraldisplay∞
+∆dǫp1
+ǫ2p/parenleftBigg
+4ǫ2
+p
+ω2−1/parenrightBigg/radicalBig
+ǫ2p−∆2
+tanh/parenleftBigǫp
+2T/parenrightBig
+δ(1)(ω2−4ǫ2
+p). (B12)
+This and similar integrals, which contain derivatives of
+the delta function, are computed via the formula
+/integraldisplay
+f(x)δn(x−a)dx= (−)nf(n)(a).(B13)
+The result of integration is
+ImI1=−8πµ2
+3√
+ω2−4∆2sgn(ω)
+ω5tanh/parenleftBigω
+4T/parenrightBig
+θ(ω−2∆).
+(B14)
+In the integral (B3) we again omit terms ∝D−1, since
+their prefactorsare zeroafter integration. After inserting
+x2= 1/3 in the remainder we obtain
+ImI2=8π
+3/integraldisplay∞
+∆dǫp/bracketleftBigg
+4ǫ2
+p
+ω2−1/bracketrightBigg/braceleftBiggµ2/radicalBig
+ǫ2p−∆2
+ǫpTsech2/parenleftBigǫp
+2T/parenrightBig
++2/radicalBig
+ǫ2p−∆2µ2tanh/parenleftBigǫp
+2T/parenrightBig
++3µtanh/parenleftBigǫp
+2T/parenrightBig
+−µ2/radicalBig
+ǫ2p−∆2
+ǫ2ptanh/parenleftBigǫp
+2T/parenrightBig/bracerightBigg
+δ(1)(ω2−4ǫ2
+p)
+−64πµ2
+3/integraldisplay∞
+∆dǫp/bracketleftBigg
+4ǫ2
+p
+ω2−1/bracketrightBigg/radicalBig
+ǫ2p−∆2tanh/parenleftBigǫp
+2T/parenrightBig
+δ(2)(ω2−4ǫ2
+p). (B15)
+Applying Eq. (B13) we obtain
+ImI2=−πµsgn(ω)
+3Tω5√
+ω2−4∆2sech2/parenleftBigω
+4T/parenrightBig/braceleftBigg
+T[24∆2µ+ω2(2µ+3/radicalbig
+ω2−4∆2)]sinh/parenleftBigω
+2T/parenrightBig
++4µω(ω2−4∆2)/bracerightBigg
+θ(ω−2∆).
+(B16)
+Finally, adding the three integrals (B11), (B14), and (B16) we obta in Eq. (42) of the main text.11
+[1] A. B. Migdal, Theory of Finite Fermi Systems and Ap-
+plications to Atomic Nuclei (Interscience, London, 1967).
+[2] A. Sedrakian, H. M¨ uther and P. Schuck, “Vertex renor-
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+[arXiv:astro-ph/0611676].
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+the Neutrino Emissivity of Neutron Matter,” Phys. Rev.
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+ph.HE]].
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+Rev. C70, 055803 (2004) [arXiv:nucl-th/0405055].
+[7] E. Olsson, P. Haensel and C. J. Pethick, “Static response
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+[arXiv:nucl-th/0502026].
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+[arXiv:astro-ph/9804264].
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+radiation in supernovae at two loops,” Phys. Rev. D 62,
+083002 (2000) [arXiv:astro-ph/0002228].
+[14] A. Sedrakian, “The physics of dense hadronic matter and
+compact stars,” Prog. Part. Nucl. Phys. 58, 168 (2007)
+[arXiv:nucl-th/0601086].
+[15] E. F. Brown and A. Cumming, “Mapping crustal heat-
+ing with the cooling lightcurves of quasi-persistent tran-
+sients,” Astrophys. J. 698, 1020 (2009) [arXiv:0901.3115
+[astro-ph.SR]].
+[16] D. Page, J. M. Lattimer, M. Prakash and A. W. Steiner,
+“Neutrino Emission from Cooper Pairs and Minimal
+Cooling of Neutron Stars,” Astrophys. J. 707, 1131
+(2009) arXiv:0906.1621 [astro-ph.SR].
+[17] S. Gupta, E. F. Brown, H. Schatz, P. Moeller and
+K. L. Kratz, “Heating in the Accreted Neutron Star
+Ocean: Implications forSuperburstIgnition,” Astrophys.
+J.662, 1188 (2007) [arXiv:astro-ph/0609828].
+[18] L. B. Leinson and A. Perez, “Vector current conserva-
+tion and neutrino emission from singlet-paired baryons
+in neutron stars,” Phys. Lett. B 638, 114 (2006)
+[arXiv:astro-ph/0606651].
+[19] A. Sedrakian, T. T. S. Kuo, H. M¨ uther and P. Schuck,
+“Pairing in nuclear systems with effective Gogny and
+Vlow−kinteractions,” Phys. Lett. B 576, 68 (2003)
+[arXiv:nucl-th/0308068].
+[20] A. Sedrakian, “Direct Urca neutrino radiation from su-
+perfluid baryonic matter,” Phys. Lett. B 607, 27 (2005)
+[arXiv:nucl-th/0411061].
+[21] Weuse theidentity δ(x2−a2) =|2a|−1[δ(x+a)+δ(x−a)].
\ No newline at end of file
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+arXiv:1001.0396v1  [nucl-th]  3 Jan 2010The Long Journey from Ab Initio Calculations to Density Functional Theory for
+Nuclear Large Amplitude Collective Motion
+Aurel Bulgac
+Department of Physics, University of Washington, Seattle, WA 98195–1560, USA
+(Dated: August 10, 2018)
+At present there are two vastly different ab initio approaches to the description of the the many-
+body dynamics: the Density Functional Theory (DFT) and the f unctional integral (path integral)
+approaches. Onone hand, ifimplemented exactly, theDFTapp roach canallow inprinciple theexact
+evaluation of arbitrary one-body observable. However, whe n applied to Large Amplitude Collective
+Motion (LACM) this approach needs to be extended in order to a ccommodate the phenomenon
+of surface-hoping, when adiabaticity is strongly violated and the description of a system using a
+single (generalized) Slater determinant is not valid anymo re. The functional integral approach
+on the other hand does not appear to have such restrictions, b ut its implementation does not
+appear to be straightforward endeavor. However, within a fu nctional integral approach one seems
+to be able to evaluate in principle any kind of observables, s uch as the fragment mass and energy
+distributions in nuclear fission. These two radically appro aches can likely be brought brought
+together by formulating a stochastic time-dependent DFT ap proach to many-body dynamics.
+PACS numbers: 21.60.De, 21.60.Jz, 21.60.Ka, 24.10.Cn
+I. DIFFICULTIES IN IMPLEMENTING A THEORETICAL FRAMEWORK FO R THE NUCLEAR
+LARGE AMPLITUDE COLLECTIVE MOTION
+We know a great deal by now about interactions between nucleons, although one might rightly argue that the
+link to quarks and gluons through the more fundamental theory Qu antum Chromodynamics has not yet been fully
+established. It will still be a long time until this goal will be reached with the needed degree of accuracy and
+sophistication. In this respect the nuclear many-body theory is at a great disadvantage when compared to condensed
+matter and atomictheories orchemistry, wherethe dominant roleo fthe Coulomb interactionand the link to Quantum
+Electrodynamics were clarified a long time ago. There is a general con sensus by now however, that in order to describe
+nuclearmany-bodysystemsoneprobablycanget awaywith the use ofoneset oranotherofreasonablywell established
+two- and three-nucleon potentials (if only one would manage to use t hem in a convincing implementation) and that a
+non-relativistic approach is likely sufficient for now. The best example we have so far that such an ab initio approach
+can be brought to fruition is the Green Function Monte-Carlo (GFMC ) approach implemented so successfully by V.R.
+Pandharipande and his many students and collaborators J. Carlson (truly the author of GFMC), S.C. Pieper, R.
+Schiavilla, K.E. Schmidt, R.B. Wiringa, and many others [1]. One of the mo st frustrating limitations of the GFMC
+approach is the relatively low particle number this method can deal wit h. Presently one cannot envision the study of
+nuclei heavier than oxygen within a GFMC framework. The Coupled-C luster (CC) method [2] looks very promising
+in this respect, as it might be applied to medium size closed shell nuclei in the near future. The No-Core-Shell
+Model (NCSM) [3] will likely be successfully applied to light and medium size nuclei in the foreseeable future with a
+reasonable degree of accuracy.
+Heavy-nuclei seem out of reach from a direct frontal ab initio attack and the only alternative appears to be an
+indirect one, namely through the use of the Density Functional The ory (DFT) [4], and only if one would be able to
+generate a sufficiently accurate nuclear energy density functiona l. During the last few years many theorists became
+rather optimistic in this respect and the goal could prove to be within reach [5]. The strategy proclaimed by the
+UNEDF group [5] can be rather briefly formulated as follows: assumin g that a universal nuclear density functional
+exists and that it can be written down (whether in a local or non-loca l form is still somewhat a matter of debate) one
+can verify it by describing light and hopefully medium nuclei, many prope rties of which could be described within one
+or another ab initio approach (GFMC, CC, NCSM). In this way one would test one theore tical approach against the
+other. Such an ab initio inspired energy density functional could be put to further test an d validated by computing
+a range of nuclear properties throughout the whole nuclear table a nd insuring that a desired degree of accuracy has
+been reached, assuming also that the magnitude of the theoretica l errors could be also estimated.
+Within a DFT framework one can calculate only the total binding energ y and one-body properties. With the
+appropriate extension to time-dependent phenomena, when a Time -Dependent DFT (TDDFT) approach is imple-
+mented [6, 7], one can address the properties of a number of excite d states as well, basically those states which can
+be excited by a weak external probe and which can be described with in the linear response regime and even a rather
+wide class of nuclear reactions. While reaching this goals and putting o n a firm basis the description of a large number
+of nuclear properties will be a truly heroic endeavor, there are a nu mber of extremely important nuclear phenomena
+Typeset by REVT EX2
+and properties, which clearly fall outside a DFT-like description. Gen erically, any excitation which is not of small
+amplitude, and thus one which is not amenable to a description within th e linear response regime, is of this type.
+Perhaps the most prominent example of nuclear Large Amplitude Colle ctive Motion (LACM) is the nuclear fission,
+a phenomenon discovered seven decades ago and which is still waiting for its full microscopic description, and many
+would claim even its full understanding still.
+While proceedingalongthis roadoneshouldkeepin mindhoweverthat a numberoftheoreticalerrorsareintroduced
+from the start, by assuming that the many-body dynamics can be d escribed with a non-relativistic Hamiltonian, with
+instantaneous interactions, and also within the framework of quan tum mechanics and not within a quantum field
+theory framework. There exist so far no reliable estimates of such corrections to the binding energy of208Pbfor
+example, and no one can claim that these corrections are smaller or b igger than 1 MeV even. Mostly likely the latter
+is true. One can hardly make the claim that such effects are effectively included, either through the choice of the
+parameters or through the choice of the parameterization form.
+There is a wide consensus at this time in the community that the role of the three nucleon forces are crucial in nuclei
+[1, 8] and that a (the) major contribution arises from a virtual exc itation of a ∆-isobar in a three nucleon system. A
+major assumption is also made that in a large nucleus the form of the N N and NNN interactions are not modified by
+the presence of the other nucleons, and implicitly also that the prop erties of nucleons are not modified by the medium.
+While there is clear evidence from the EMC-effect that that is not the case [9], we lack any reliable estimates of how
+big the corrections to the binding energy of208Pbwould be, and by how much its properties would be altered by an
+explicit treatment of ∆-isobars in the nuclear medium and by the modifi cation of the nucleon properties in nuclear
+medium. Nobody can state that these corrections are for example either smaller or larger than 1 MeV for example.
+One cannot make the claim either that such effects are negligible, if th e binding energies of nuclei is to be calculated
+within a specific phenomenological approach with an error smaller tha n 1 MeV, which is the state of the art in the
+case of nuclear mass phenomenological fits [10, 11].
+No phenomenological approach used so far in nuclear physics has ev er been proven to be an exact theory, with
+or without a set of fitting parameters. Many argue that in order to be able to describe accurately heavy nuclei for
+example, one has to perform a fine tuning of various parameters of an energy density functional. By performing
+such a fine tuning however one clearly sweeps under the rug lots of physics. So far we simply do not know if a
+specific (semi-)phenomenological approach does indeed correspo nd to an accurate solution of the many-body non-
+relativistic Schr¨ odinger equation for a medium or heavy nucleus and we have no idea how precise is in principle the
+description of nuclei within a non-relativistic Schr¨ odinger equation approach to nuclei. In other words, so far we lack
+even an estimate of the order of magnitude of the theoretical err ors inherited by adopting one or another theoretical
+framework. Consequently, any apparent agreement achieved be tween the description of nuclei within a specific (semi-
+)phenomenological approach and data can notbe used as an argument to make any inferences about the magnitud e
+of other unaccounted effects.
+The main ideas behind the currently accepted theoretical framewo rk to LACM go back more than half a century, to
+Hill and Wheeler [12, 13], and in its most modern formulation it comes un der the name of Adiabatic Time-Dependent
+Hartree-Fock (ATDHF) theory [14]. The nuclear many-body wave f unction Ψ is represented as
+Ψ(x1,...xA) =/integraldisplay
+ΠN
+k=1dqkφ(q1,...qN)Φ(x1,...xA|q1,...qN), (I.1)
+wherex1,...xAare the (spatial and spin-isospin) nucleon coordinates and q1,...qNare the collective variables. The
+wave function Φ( x1,...xA|q1,...qN) is chosen as a generalized Slater determinant (GSD), parameteriz ed by the values
+of the collective variables, which in selfconsistent treatments are r equired to satisfy the constraints
+ql=/integraldisplay
+ΠA
+k=1dxkΦ∗(x1,...xA|q1,...qN)ˆqlΦ(x1,...xA|q1,...qN), l= 1,...,N, (I.2)
+and where ˆ qlis an one-body operator (e.g. a particular quadrupole moment). On ce a set of collective variables has
+been chosen, either guided by the magical physical intuition of a give n set of authors or by some other more formal
+set of rules [15], one has to determine for each particular value of ea ch collective coordinate the optimal GSD, which
+will minimize the total energy of a nuclear many-body Hamiltonian. This goal is typically achieved by solving a
+constrained self-consistent meanfield problem.
+All GSDs form a complete set of orthogonal many-body wave funct ions, and in principle the expansion in Eq.
+(I.1) has the potential to be exact. However, the idea put forwar d by Hill and Wheeler [12, 13] impels us to use a
+much smaller set of such GSDs, which hopefully parameterize sufficien tly accurately the sequence of nuclear shapes
+followed by a nucleus during the fissioning process for example. If on e makes a physically inspired educated guess and
+an optimal set of GSDs has been determined, the next step is to det ermine the collective wave function φ(q1,...qN).
+The ensuing equation satisfied by the collective wave function is know n as the Hill-Wheeler equation [12] and some3
+other times it comes under the name of the Generator Coordinate M ethod [14]. In practice this approach often
+appears to lead to a surprisingly accurate description of a large num ber of nuclear data [16] and this success seems
+to lend support to the hope that an accurate description of many n uclear properties within a DFT-like approach
+will eventually be reached. While physically extremely appealing and intu itive, and also apparently backed by the
+agreement with data, this theoretical approach is plagued by a long list of serious drawbacks (some mentioned above,
+others to be discussed below), which have never really been serious ly addressed. A serious drawback of this approach
+is the lack ofa reliable measureof the theoretical error. This restr icted representationofthe many-body wavefunction
+could be exact if the many-body dynamics would be exactly separable by introducing a suitable set of intrinsic and
+the collective coordinates qk[15]. Since this is never the case, the representation (I.1) is of an un quantifiable quality.
+The number and even the character of the relevant collective coordinates is not known and varies from author to
+author and often in the case of the same author from work to work , and it is essentially determined by the current
+computational capabilities of a given set of authors at a given momen t in time, see Refs. [16–19] and earlier references
+quoted therein. Apart from the fact that these collective variable s are not optimized, as discussed for example in
+Ref. [15], in the sense that they are not optimally decoupled from the intrinsic degrees of freedom, the set of GSDs
+generated in this manner is also not complete and, moreover, there is a large degree of redundancy [20] (as one can
+easily establish by computing the spectrum of the norm matrix). If c ollective degrees of freedom would be indeed well
+decoupled from the intrinsic motion, there would be no need to even d iscuss whether one should perform calculations
+with variation before or after the projection for example.
+Even if one were to admit that the choice of the collective operators made by various authors is sensible, there are
+clear inconsistencies in using commonly introduced constraining oper ators. For example, if one uses the quadrupole
+moment 2 z2−x2−y2as a constraining operator and if one were to treat it mathematically exactly, this constraint
+would introduce unphysical modifications of the GSD, which are prac tically impossible to quantify. A constraining
+fieldλ20(2z2−x2−y2) plays the role of a one-body potential which tends to + ∞in some spatial directions and to
+−∞in others. Clearly single-particle wave functions in such a one-body fi eld have a totally wrong spatial asymptotic
+behavior. If one would try to increase the size of the single-particle basis set used to diagonalize the self-consistent
+equations, the size of the errors introduced into the description w ould obviously increase. At the same time by
+decreasing the size of the single-particle basis one increases the siz e of the error as well, but for different reasons. One
+can then hope that there is an optimal basis set size for which the inc urred error is minimal. But even when this
+incurred error is minimal the size of the error is not known and there are no known ways to even estimate it. No
+one knows which is the optimal size of the single-particle basis set, ho w this size varies with the proton and neutron
+numbers, how this size depends on deformation. It is not clear even if the size of the optimal single-particle basis
+set (when the incurred error is minimal) is similar for various constrain ing operators. Thus it is absolutely unclear
+whether one can describe with any quantifiable accuracy the n uclear dynamics in the case of several collective variables
+within the commonly accepted parameterizations of the nucl ear shape or otherwise.
+The only constructive procedure that I am aware of, which in princip le could determine the relevant set of collective
+coordinates and also generate some measure of the accuracy of t he representation (I.1) was developed by Klein and
+collaborators [15]. However, this approach is extremely difficult to imp lement in practice and the mathematical
+structure of the approach is not fully understood. In this approa ch the very character of the constraints depends
+on the collective variables themselves, and they have to be determin ed concurrently with the optimal GSDs. The
+collective inertia tensor is also determined concurrently along with a r ather well defined measure of the quality of
+the collective variables, but not of the size of the incurred error. I f the quality of the collective variables is not
+satisfactory, one is required to increase the size of the collective s pace. It suffices to say that after a heroic theoretical
+effort over several decades, only recently some relatively simple ca ses have been considered within a very simplified
+nuclear structure model [17]. The progress in this direction does no t appear to be accelerating in the near future, and
+the prospects for success do not appear to be very encouraging .
+What makes matters even worse are a few additional aspects of an ATDHF-like approach. In most implementations
+ofthis schemethe collectivecoordinatesarebasicallyintroduced ba sedonthe magical physical intuition ofthe authors.
+Usually there is no objective and unambiguous measure of the “good ness” of an assumed set of collective variables,
+neither of their form nor of their number. The typical set of collect ive coordinates considered in literature are the
+quadrupole deformationsand sometimes a few highermultipoles, but even for the description of the samephenomenon
+various authors could not agree upon a unique, or even minimal (by s ome unknown yet measure) set of collective
+coordinates [10, 19]. In these approaches the decoupling of the co llective variables and optimization of the GSDs
+as required by theory [15] is never considered and thus their quality is simply unknown. One can safely state that
+as of now nobody knows how many collective coordinates are necess ary in order to describe for example the fission
+dynamics of heavy nuclei with a predetermined and satisfactory ac curacy, within the theoretical framework based on
+the representation (I.1) of the total wave function. Success is t ypically declared when agreement is achieved with a
+selected subsetofexperimentaldata (byintroducing andvarying arelativelylargenumberofparameters), andit isnot
+determined intrinsically by the theoretical approach. When one not ices differences between theoryand experiment,4
+one has the choice to claim that either one needs to increasethe num erical accuracyofthe theory, but at the same time
+one can also claim that some unaccounted physical phenomena could be responsible as well for that disagreement.
+Apart from the fact that there are more basis states than neede d on physical grounds [20], it is basically impractical
+to deal with collective spaces of large dimensions. According to Ref. [18] one needs at least five different collective
+variables and in their calculations these authors needed about five m illion different nuclear shape configurations. Only
+generatingthis number ofconstrained meanfield calculations requir esbasically a thousand times more computing time
+than for generating the entire nuclear mass table in a selfconsisten t meanfield approach. It is obvious that even with
+significantly increased computer power envisioned during the next d ecade such an approach becomes unmanageable.
+In order to determine the collective wave function φ(q1,...qN) one needs to solve the integral Hill-Wheeler equation,
+the size of which becomes quickly extremely large for large dimensiona l collective spaces. For this reason most
+practitioners choose instead to reduce the Hill-Wheller equation to a differential equation, using the Gaussian Overlap
+Approximation [14], the accuracy of which however has never been q uantified.
+All in all, the method based on the representation of the nuclear man y-body function (I.1) is not a controlled
+approximation, as it is not clear how big the theoretical error incurr ed really is. The usual approach chosen in
+practice, to fine tune the parameters of the Hamiltonian, or to increase the size of either the single-particle basis
+set or of the number of collective variables, are clearly not defensib le arguments on theoretical grounds. One cannot
+achieve a faithful description of Nature when success is determine d by more accurate fitting of the parameters of a
+clearly approximate approach, based on solving the non-relativistic Schr¨ odinger equation (which is an approximation
+to Nature from the start) and improve the agreement with experim ent, when one has no idea about the theoretical
+errorsincurred and when the approachis unfalsifiable by construc tion. Moreover,one cannot use the currentapproach
+to test for physics beyond the initial assumptions. (For example, a phenomenon similar to the role of general relativity
+corrections in the precession of the perihelion of Mercury is clearly o utside its scope of applicability.)
+Another theoretical inconsistency in how the emerging Hill-Wheler/G CM equations are solved was never really
+discussed so far. Typically, in the case of spatial deformation one r educes the Hill-Wheller/GCM integral equations to
+a differential form, using as a rule the Gaussian Overlap Approximatio n or related approximations [14, 16]. However,
+when treating other collective degrees of freedom, such as partic le number, isospin and/or angular momentum, typi-
+cally one uses the integral formulation of the theory, as the collect ive wave function has a simple form and the theory
+is equivalent to merely a projection on the correct quantum number s. In this case non-diagonal matrix elements need
+to be evaluated [21] and rather arbitrary restrictions have to be im posed in order to avoid mathematical difficulties.
+There exist no defensible theoretical arguments why different colle ctive degrees of freedom should be treated using
+non-equivalent theoretical schemes with different degrees of acc uracy.
+There is a rather subtle argument against a DFT based approach to LACM, which as far as I know, has never been
+considered in the nuclear physics literature. Related issues have be en addressed for quite some time in the description
+of the nuclear motion in molecules [22]. While the description of the intrin sic motion is based on a density matrix
+approach, the collective dynamics is described by wave functions. I t is not obvious that such a mixed approach makes
+any sense theoretically, and it has never been derived in any fashion , but it has been merely postulated implicitly.
+II. POSSIBLE RESOLUTION OF THESE DIFFICULTIES
+In order to evade the vicious circle described above one could proce ed in a slightly different manner. There will be
+a price, and for some people this price may appear too high, as agree ment between theoryand experiment (which in
+my opinion in this case is over-rated) would naturally be worse at the b eginning. But we will likely be able to achieve
+a deeper understanding, and that is by far a more valuable goal to s trive for. Instead of trying to improve the quality
+of the description of a specific approach by allowing for the fine tunin g of some parameters, one should instead fix
+such parameters before hand in a sensible manner and try to produ ce within a theoretically convincing approach the
+properties ofmedium and heavy nuclei with clearly defined and quant ifiable errors,both theoretically and numerically.
+Any inconsistencies between theory and data which lie outside the er ror bars could then be attributed to phenomena
+not accounted for, such as relativistic effects, charge symmetry breaking of nuclear forces, modification of nucleon
+properties in the medium, excitation of other degrees of freedom, such as ∆-isobars, and the potential modification
+of their properties in the medium, see for example Refs. [23, 24].
+The natural question arises, whether any sensible theoretical ap proach could be put forward in order to describe
+nuclear LACM in particular. I shall describe here an approach to LAC M which brings together two rather different
+many-body techniques, each one of them in principle exact, but at t he same time each one of them suffering from the
+fact that an exact implementation is more a matter of desire rather than fact, and each one of them still displaying
+a number of difficult unresolved aspects.5
+A. Real-Time Functional Integral Approach and some of its li mitations
+The first approach is based on the real time functional integral tr eatment of the many-body problem [25]. It can
+be easily shown that at least in principle one can represent the evolut ion operator for a many-body system governed
+by a Hamiltonian ˆHfrom an initial time tito a final time tfas follows ( /planckover2pi1= 1 in this work):
+exp[−iˆH(tf−ti)] =/integraldisplay
+ΠNt
+k=1Παβdσαβ(k)exp
+i∆t
+2Nt/summationdisplay
+k=1/summationdisplay
+αβγδσαβ(k)Vαβγδσγδ(k)
+
+×exp
+−i∆t/summationdisplay
+αβ(Kαβ+/summationdisplay
+γδVαβγδσγδ(k))ˆραβ
+, (II.1)
+wheretf−ti=Nt∆t(andNt→ ∞),Vαβγδare the matrix elements of the two-body interaction, Kαβthe matrix
+elements of the kinetic energy (up to some simple renormalization), ˆ ραβ=a†
+αaβis bilinear in the creation and
+annihilation operators, and σαβ(k) are the matrix elements of an auxiliary field. The stationary phase a pproximation
+for this evolution operator leads to a time dependent meanfield appr oximation, which however it is not uniquely
+defined [25]. It has been shown that the one-body evolution operat or (the last exponent in Eq. (II.1) can be chosen
+to have an arbitrary form [26] and that this arbitrariness is resolv ed only when one goes beyond the stationary phase
+approximation and includes the quadratic fluctuations of the auxiliar y fieldσαβ(k).
+The fact that the stationary phase approximation is not uniquely de fined is somewhat unsettling, and begs the
+question of whether the best stationary phase approximation can be introduced. Likely the answer is given within
+the TDDFT to be discussed below. At the same time, the fact that th e non-uniqueness of the stationary phase
+approximation of the real-time functional integral approach is res olved only after the fluctuations of the auxiliary field
+are considered seems to suggest that the dynamics is truly govern ed by fluctuations and as such they always have to
+be accounted for. This aspect will be discussed in more detail below.
+At the stationary phase approximation level however one can esta blish a quantization condition, which determines
+both the ground and the excited states. In Ref. [25] it was shown t hat the resolvent operator for a many body system
+can be introduced as follows:
+G(E) =−i/integraldisplay∞
+0dtTrexp(iEt−iˆHt) =−i/integraldisplay∞
+0dtA(t)exp(iEt) =−i/integraldisplay∞
+0dta(t)exp[iEt+iS(t)],(II.2)
+where the trace A(t) = Trexp( −iˆHt) of the evolution operator (II.1) has been separated into an expo nent of a (large)
+phaseS(t) and an (assumed) slowly varying function a(t) of the time t. The trace is performed over the entire Hilbert
+space of the many-body system under consideration. In the stat ionary phase approximation the phase S(t) is the
+classical action and the amplitude a(t) is further determined by evaluating the quadratic fluctuations in t he auxiliary
+fieldσαβ(k). It is a rather straightforward exercise to show that by applying the stationary phase approximation to
+Eq. (II.2) one obtains the quantization condition [25]
+(2n+1)π=i/integraldisplayT(E)
+0dt/integraldisplay
+dx/summationdisplay
+αφ∗
+α(x,t)∂
+∂tφα(x,t), (II.3)
+where the single particle wave functions satisfy the time-dependen t meanfield equations
+i∂
+∂tφα(x,t) =h(x,t)φα(x,t) = [K(x)+U(x,t)−εk(T)]φα(x,t), (II.4)
+with the periodic time boundary conditions φα(x,0) =φα(x,T). The auxiliary field σ(x,t) is determined from the
+condition
+σ(x,t) =/summationdisplay
+α|φα(x,t)|2, U(x,t) =/integraldisplay
+dyV(x−y)σ(y,t) (II.5)
+and the period T(E) is determined from the equation
+E=−∂S(T)
+∂T=−∂
+∂T/bracketleftBigg/integraldisplayT
+0dtdxdyV (x−y)σ(x,t)σ(y,t)−T/summationdisplay
+kεk(T)/bracketrightBigg
+, (II.6)6
+(In Eq. (II.3) one should replace 2 n+1 by 2nif there are no turningpoints.) Even though here these equations have
+been written down in their simplest from, when the time-dependent m eanfield is chosen to have a Hartree form, the
+formal structure of these equations remains basically the same if o ne chooses various other possibilities (Hartree-Fock
+or Hartree-Fock-Bogoliubovfor example). The quantization cond ition (II.3) essentially amounts to requiring that the
+so-called Berry phase (obtained after removing the dynamical pha se is quantized), see Refs. [27].
+The stationaryphase approximationofthe real-timefunctional int egralapproachiswell defined and straightforward
+to implement. However, the inclusion of the fluctuations of the auxilia ry field amount to the evaluation of a functional
+integral of a complex exponential. If the amplitude of the fluctuatio ns are small, one can expect that this integral
+could be evaluated numerically. It is still a largely unresolved question however of how to evaluate such an integral
+when the size of the fluctuating terms are large, though some light of the end of the tunnel is visible, see Refs. [28 ?]
+and below.
+B. Time-dependent Density Functional Theory approach its l imitations
+The second approach is the Density Functional Theory (DFT) [4], e xtended to describe superfluid systems [31]. In
+the TDDFT extension of the approach, which has been studied for n ormal systems for quite some time [6, 7], one
+needs to extend the form the energy density functional in order t o allow for currents, which are typically absent in
+the ground states. In many cases requiring that the energy dens ity functional to satisfy Galilean invariance allows
+to extend the DFT approach to a rather large class of phenomena [3 1, 32]. As in the case of the stationary form of
+DFT, within TDDFT one can in principle evaluate exactly the one-body o bservables, when the many-body system
+under consideration is subject to an arbitrary one-body externa l potential [6, 7]. The existence of such a formal
+result does provide us with an answer to a question raised within the f unctional integral approach, namely: Could
+one define the best stationary phase approximation evolution oper ator? In order to define such an evolution operator
+one would have to introduce some criterion. The obvious choice in this regard appears to be an one-body evolution
+operator which would lead to a correct description of one-body obs ervables. It is not clear however that with such a
+choice one can also ensure the fastest convergence of the forma lism beyond the one-body level, which would be a more
+valuable requirement. Within the TDDFT approach the problems discu ssed above concerning the choice of collective
+variables becomes superfluous and it is replaced instead by the prob lem of determination of periodic trajectories and
+determination of a solution of the quantization condition (II.3).
+The implementation of DFT in the case of nuclei faces a number of rat her serious difficulties [33]. As in the case
+of electrons, there is no theoretical recipe on how to construct a nuclear energy density functional. Nuclei, unlike
+electrons, which reside in a somewhat well defined external potent ial created by the ions and for which a certain
+TDDFT formalism has been elaborated [22], are self-bound systems and there are difficulties with introducing a
+one-body density formalism. At this time it is not yet clear whether a v iable solution exists, even though a number
+of suggestions has been put forward by various authors [33]. Mos t researchers believe that this formal difficulty is
+not relevant in the study of medium and heavy nuclei, as the existenc e of single-particle excitation modes in nuclei
+is rather well established for more than half a century. A central t enet of DFT is that the correlation energy can be
+successfully incorporated into an exact energy density functiona l. In nuclei however the situation does not seem to be
+so clear. In particular, the gain in energy one obtains in constructin g a rotationally invariant solution for open shell
+nuclei, which have intrinsic deformation, is a form of correlation ener gy and, in the opinion of most, see for example
+Refs. [16], does not appearto be containedwithin anyconceivablenu clearenergydensityfunctional. The same applies
+to other collective degrees of freedom, such as pairing [21], and in th e case of relatively soft nuclei to the fluctuation
+of the shape parameters [16]. The only meaningful way to account f or this type of correlation energy in case of nuclei
+appears to be either the introduction of a projection of the intrins ic many-body wave function (typically a GSD)
+or the the need to perform a GCM calculation (which often is reduced to the solution of a collective Schr¨ odinger
+equation) on top of the DFT calculation for the intrinsic wave functio n. In principle similar difficulties should arise in
+the case of extremely floppy molecules in chemistry [22]. In the case o f projection on good quantum numbers and of
+GCM calculations based on a DFT treatment of the intrinsic nuclear mo tion one needs to introduce non-diagonal in
+collective degrees of freedom expectation values for the internal energy, for which so far only prescriptions have been
+suggested [21].
+The issues raised above are just a forerunner of much deeper pro blems one encounters when applying the meanfield
+approach to LACM. The analog of the type of problems one faces in t his case is the well known phenomenon of
+crossing of molecular terms. When two atoms approach each other they might undergo an electron excitation or
+not at some separation. When such an internal excitation occurs, in which case an electron can be transferred from
+one atom to another and thus leading to formation of ions, the mean field experienced by the ions/atoms changes.
+The rate of electron excitation depends very strongly on the ion/a tom relative speed and it is often described as a
+Landau-Zener like transition. In order to describe the random cha racter of such transitions and of their influence7
+on the motion of nuclei a stochastic extension of the meanfield appr oximation - the surface-hopping formalism - has
+been developed in chemistry by Tully [34] and this approach has been in corporated in modern versions of TDDFT.
+Another example discussed by Tully [34] is that of an atom impacting on a metal surface. Upon reflection such an
+atom will follow drastically different trajectories, depending on whet her an electron-hole pair has been excited or not
+in the metal. If the metal is not excited, the atom will reflect elastica lly at an angle of reflection equal to the angle
+in incidence. However, if an electron-hole pair is excited within the met al surface, a noticeable amount of energy and
+linear momentum is being transferred to the metal and the atom will f ollow a trajectory at a much large angle of
+reflection than the incidence angle. Because of the existence of su ch processes the time honored Born-Oppenheimer
+approximation (in which the nuclear and electronic motion are adiabat ically separated) is violated and corrections
+should be taken into account. As the work of Tully and others has de monstrated, these non-adiabatic corrections can
+be largely reduced to random transitions occurring mostly when var ious molecular terms cross.
+In the caseof nucleardynamicsthe equivalent ofmolecularterms is p layedby variouspotential energysurfaces. The
+case studied a long time ago by Negele and collaborators[35], of the “s pontaneousfission” of32S, is a great example of
+the need to modify the TDHF approach to nuclear dynamics and of th e role of the potential energy surfaces crossing
+in such processes. By artificially increasing the proton electric char ge these authors forced32Sto split into two16O
+fragments. As these authors have shown, during the splitting of t he initial nucleus the system breaks the ground
+state symmetry of32Sand develops a transitory octupole moment. This behavior can be int erpreted as the influence
+of a diabolic point and the emergence of a gauge field located at a level crossing [27]. Unlike the surface-hoping
+mechanism of Tully, there does not seem to be a need for a “stochas tic” ingredient in order to allow a system to
+perform jumps from one potential energy surface to another, if one allows for the natural emergence of non-abelian
+gauge fields [27, 36].
+In nuclear dynamics it was recognized a long time ago that one needs t o go beyond the meanfield approximation. In
+a TDHF approach, as in the TDDFT method, one can hope to describe correctly only one-body observables at most.
+In case of nuclear reactions between two heavy ions for example on e would not be able to describe adequately the
+widths of the fragment mass distribution. However, a long time ago B alian and V´ en´ eroni suggested a simple extension
+of the TDHF approximation in order to calculate second moments only of one-body observable [37], and thus allowing
+us to overcome to some extent the restraints of a one-body form alism such as TDHF or TDDFT. The need to go
+beyond the simple TDHF framework has been argued in nuclear theor y for quite sometime, see Refs. [38–40]: in the
+role of the zero-point oscillations of the nuclear shape [38], in the role of two-body collisions in affecting the meanfield
+behavior and leading to thermalization of the single-particle dynamics [39], and the need to extend in some ways
+the meanfield approach in order to allow for the fluctuations of the s ingle-particle properties [40]. While apparently
+these extensions of the meanfield approximation appear physically w ell founded and the agreement with experiment
+is most of the time satisfactory (partially due to the existence of an appropriate set of fitting parameters), the formal
+argumentation is somewhat lacking.
+C. Bringing the parts together
+On one hand the TDDFT is very appealing, as at least in principle allows an exact calculation of one-body
+observable. However, TDDFT does not seem to have an obvious ext ension if one would like to calculate many-body
+observables. One the other hand, the functional integral metho d does not seem to be limited in principle, only if one
+would be able to really implement it in practice. And also, the functional integral approach appears to substantiate in
+the stationary phase approximation the existence of a meanfield ap proach, even though this is not uniquely identified.
+On physical grounds we do not seem to need much of an argument to introduce some kind of surface-hopping within
+a TDDFT approach. A mere look at a Nilsson diagram will suffice to convin ce anyone that during a LACM a nucleus
+is bound to come across many level crossings. As it was argued in Ref s. [13] the number of such crossings is likely
+of the order of A2/3in a fissioning nucleus. The existence of so many level crossings will inv alidate at once the use a
+single energy potential surface to describe LACM. While evolving thr ough a sequence of shapes a nucleus will most
+likely perform jumps from one energy surface to the next in an esse ntially stochastic manner and there will be a very
+small chance that while trying to follow in reverse the sequence of sh apes the nucleus would likely manage to return
+to its initial state. As it was mentioned above, each such level cross ing, is not merely a source of surface-hopping,
+but also a source of a dynamically generated gauge field. Thus, the le vel crossings will lead to two rather different
+effects, both of them depending in a critical manner of the collective velocity. By enabling the surface-hopping in an
+essentially stochastic manner these will be the main source of irreve rsibility and of the generation of entropy. At the
+same time, unlike the potential forces, level crossings will generat e non-abelian “magnetic” type of forces, which will
+force the system to go around a diabolical point, as it was observed in the case of “fissioning”32S[35].
+The two ab initio approaches described above seem to come together rather natu rally now. While within the
+strict functional approach one could not identify the best station ary phase approximation, on which to further build8
+corrections due to fluctuations of the auxiliary field σαβ, the TDDFT formalism appears to suggest its existence. At
+the same time, the missingsurface-hopping element in the TDDFT, which is clearly required by th e physics of LACM
+and the need to compute many-body observables, appears rathe r naturally within the functional integral approach,
+which provides a very specific complex measure for the fluctuations of the meanfield. The fact that the m easure
+of these fluctuations is complex is certainly an worrisome fact, as th e practical implementation and the numerical
+convergenceof such a scheme is far from evident. However, the r ecent progressin implementing the complex Langevin
+method [30] and the simple argument that in the case of fluctuations of vanishing amplitude one should not encounter
+any problems, serve as a very good encouragement that the succ essful implementation of such an approach is not out
+of question. There are however a number of formal questions to b e addressed still. One of them is, what two-body
+interaction should be used to control the amplitude of the fluctuat ions, see Eq. (II.1). While the functional integral
+approach appears to suggest that the matrix elements of the bar e interaction should be used, the lessons we have
+learned from the implementation of the Boltzmann-Uhling-Uehlenbec k equation to describe nuclear kinetics [39] and
+ourphysical intuition would suggest that these matrix elements are most likely strongly re normalized and this is an
+aspect which has not been clarified yet. Since Eq. (II.1) will most likely be evaluated using a Monte-Carlo approach
+there is a need for a Metropolis importance sampling algorithm approp riate for this type of functional integrals.
+III. CONCLUDING REMARKS
+If successfully implemented, the strategy outlined above will allow us to obtain in principle an exact solution
+of the Schr¨ odiger equation for a nuclear system in particular, sinc e formally the representation of the real-time
+evolution operator (II.1) is exact. This representation is equivalen t to a Langevin type of extension of the TDDFT
+formalism, which can be implemented in a straightforward manner in pr actice. As recent progress suggest, the
+numerical implementation of the Langevin dynamics for many-body s ystems [29, 30] does not appear to be a hopeless
+endeavor anymore [30]. In this approach the transition amplitude f rom an initial state Ψ ito an arbitrary final state
+Ψfis calculated from the expression:
+/angbracketleftΨf/vextendsingle/vextendsingle/vextendsingleexp[−iˆH(tf−ti)]/vextendsingle/vextendsingle/vextendsingleΨi/angbracketright=/integraldisplay
+ΠNt
+k=1Παβdσαβ(k)exp
+i∆t
+2Nt/summationdisplay
+k=1/summationdisplay
+αβγδσαβ(k)Vαβγδσγδ(k)
+U[σ],(III.1)
+whereU[σ] is the one-body evolution operator for a specific spatio-tempora l realization of the auxiliary field σαβ(k),
+the last exponent in Eq.(II.1). The beststationary phase approximation of this transition amplitude is given b y the
+TDDFT evolution. This expression is similar in spirit to what one would see in a low intensity electron beam two-slit
+experiment, where one records one hit of the screen at a time and in terference pattern appears only after adding
+up a large number of events. In order to obtain the full solution of t he many-body problem, in particular a specific
+transition amplitude, one has to “record” many spatio-temporal r ealizations of the auxiliary filed σαβ(k). If one is
+interested in a specific transition amplitude only, the optimal station ary phase approximation can be determined by
+adjusting it to the given initial and final states [25, 41].
+A definite advantage of this approach is that it obviates the need to determine a set of collective variables. Various
+strategies devised so far in the literature to find such collective var iables are based on an approximate solution of the
+time-dependent meanfield equations of motion such as Eqs. (II.4) in the strict adiabatic limit. Adiabaticity is broken
+most prominently at level crossings and the need for surface-hop ing has been advocated for a long time in literature
+[34, 38–40]. By solving Eqs. (II.4) exactly, see Ref. [42, 43], the ne ed to construct collective variables, potential
+energy surfaces and the corresponding mass tensor is obviated. Phenomena such as dissipation and fragment mass
+and energy distributions do not seem anymore out of reach for an ab initio inspired approach. Within an ATDHF
+approach in a five-dimensional collective space, which has been advo cated by P. Moller and collaboratorsas a minimal
+one for fission dynamics [18], one would need to generated a few million d ifferent configurations. And even when such
+a goal is achieved one would still obtain only a one-body description of the dynamics at best, and not arguably the
+most accurate. The generation of an ensemble of a few million realizat ion of the auxiliary field σαβ(k) on the other
+hand will allow for a significantly more physically complete description of the nuclear dynamics of a heavy nucleus.
+Support is acknowledged from the DOE under grants DE-FG02-97E R41014 and DE-FC02-07ER41457.
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\ No newline at end of file
diff --git a/1001.0397.txt b/1001.0397.txt
new file mode 100644
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+arXiv:1001.0397v1  [quant-ph]  3 Jan 2010Decoherence as a signature of an excited state quantum phase
+transition
+A. Rela˜ no1,∗J. M. Arias2, J. Dukelsky1, J. E. Garc´ ıa-Ramos3, and P. P´ erez-Fern´ andez2
+1Instituto de Estructura de la Materia,
+CSIC, Serrano 123, E-28006 Madrid, Spain
+2Departamento de F´ ısica At´ omica, Molecular y Nuclear,
+Facultad de F´ ısica, Universidad de Sevilla,
+Apartado 1065, 41080 Sevilla, Spain
+3Departamento de F´ ısica Aplicada,
+Universidad de Huelva, 21071 Huelva, Spain
+(Dated: November 10, 2018)
+Abstract
+We analyze the decoherence induced on a single qubit by the in teraction with a two-level boson
+system with critical internal dynamics. We explore how the d ecoherence process is affected by
+the presence of quantum phase transitions in the environmen t. We conclude that the dynamics of
+the qubit changes dramatically when the environment passes through a continuous excited state
+quantum phase transition. If the system-environment coupl ing energy equals the energy at which
+the environment has a critical behavior, the decoherence in duced on the qubit is maximal and the
+fidelity tends to zero with finite size scaling obeying a power -law.
+PACS numbers: 03.65.Yz, 05.70.Fh, 64.70.Tg
+∗Electronic address: armando@iem.cfmac.csic.es
+1Real quantum systems always interact with the environment. This in teraction leads to
+decoherence, the process by which quantum information is degrad ed and purely quantum
+properties of a system are lost. Decoherence provides a theoret ical basis for the quantum-
+classical transition [1], emerging as a possible explanation of the quan tum origin of the
+classical world. It is also a major obstacle for building a quantum comp uter [2] since it
+can produce the loss of the quantum character of the computer. Therefore, a complete
+characterization of the decoherence process and its relation with the physical properties of
+the system and the environment is needed for both fundamental a nd practical purposes.
+The connection between decoherence and environmental quantu m phase transitions has
+been recently investigated [3–5]. A universal Gaussian decay regime in the fidelity of the
+system wasinitially identified, andrelatedtoa second-order quantu m phase transitioninthe
+environment [4]; as a consequence, the decoherence process was postulated as an indicator
+of a quantum phase transition in the environment. Subsequently, t his analysis was refined,
+and it was found that the universal regime is neither always Gaussian [6], nor always related
+to an environmental quantum phase transition [5].
+In this paper we analyze the relationship between decoherence and an environmental
+excited statequantum phasetransition(ESQPT). Weshow thatth efidelity ofasinglequbit,
+coupled to a two-level boson environment, becomes singular when t he system-environment
+coupling energy equals the critical energy for the occurrence of a continuous ESQPT in the
+environment. Therefore, our results establish that a critical phe nomenon in the environment
+entails a singular behavior in the decoherence induced in the central system.
+An ESQPT is a nonanalytic evolution of some excited states of a syste m as the Hamil-
+tonian control parameter is varied. It is analogous to a standard q uantum phase transition
+(QPT), but taking place in some excited state of the system, which d efines the critical en-
+ergyEcat which the transition takes place. We can distinguish between differ ent kinds
+of ESQPT. As it is stated in [7], in the thermodynamic limit a crossing of tw o levels at
+E=Ecdetermines a first order ESQPT, while if the number of interacting lev els is locally
+large atE=Ecbut without real crossings, the ESQPT is continuous. In this paper we will
+concentrate in the latter case, which usually entails a singularity in th e density of states (for
+an illustration see Fig. 1). As the entropy of a quantum system is rela ted to its density
+of states, a relationship between an ESQPT and a standard phase t ransition at a certain
+critical temperature, can be established in the thermodynamic limit [ 8].
+2These kinds of phase transitions have been identified in the Lipkin mod el [9], in the inter-
+actingbosonmodel[10],andinmoregeneralbosonorfermiontwo-le velpairingHamiltonians
+(for a complete discussion, including a semiclassical analysis, see [11]) . In all these cases,
+the ESQPT takes place beyond the critical value of the Hamiltonian co ntrol parameter,
+implying that the critical point moves from the ground state to an ex cited state.
+Here we consider an environment having both QPTs and ESQPTs coup led to a single
+qubit. The Hamiltonian of the environment, defined as a function of a control parameter α,
+presents a QPT at a critical value αc. We define a coupling between the central qubit and
+the environment that entails an effective change in the control par ameter,α→α′, making
+the environment to cross the critical point if α′> αc. Moreover, the coupling also implies an
+energy transfer to the enviroment E→E′, and therefore it can also make the environment
+to reach the critical energy Ecof an ESQPT.
+Following [4] we will consider our system composed by a spin 1 /2 particle coupled to a
+spin environment by the Hamiltonian HSE:
+HSE=IS⊗HE+|0∝angbracketright∝angbracketleft0|⊗Hλ0+|1∝angbracketright∝angbracketleft1|⊗Hλ1, (1)
+where|0∝angbracketrightand|1∝angbracketrightare the two components of the spin 1 /2 system, and λ0,λ1the couplings of
+each component to the environment. The three terms HE,Hλ0andHλ1act on the Hilbert
+space of the environment.
+With this kind of coupling, the environment evolves with an effective Ha miltonian de-
+pending on the state of the central spin Hj=HE+Hλj,j= 0,1. Assuming /planckover2pi1= 1, it gives
+rise to a decoherence factor r(t) =∝angbracketleftΨ(0)|eiH0te−iH1t|Ψ(0)∝angbracketright, where Ψ(0) represents a generic
+environmental state at t= 0. If the environment is initially in its ground state |g0∝angbracketright, the
+decoherence factor is determined, up to an irrelevant phase fact or, byH1, and its absolute
+value is equal to
+|r(t)|=/vextendsingle/vextendsingle∝angbracketleftg0|e−iH1t|g0∝angbracketright/vextendsingle/vextendsingle. (2)
+This quantity has the same form as the Loschmidt echo or the fidelity , and it contains all
+the relevant information about the decoherence process.
+To be specific, let us consider a spin environment described by the we ll known Lipkin
+model,
+HE=α/parenleftBigg
+N
+2+N/summationdisplay
+i=1Sz
+i/parenrightBigg
+−4(1−α)
+NN/summationdisplay
+i,j=1Sx
+iSx
+j, (3)
+3whereSx
+i,Sy
+i, andSz
+i, are the three components of a spin 1 /2 in the site iof a spin chain
+withNsites, and αis a control parameter given in arbitrary units of energy. Consequ ently,
+in the following energy and time are always written in the correspondin g arbirtary units.
+Using the Schwinger representation of the spin operators we tran sform the Hamiltonian
+(3) into a two level boson Hamiltonian, constructed out of scalar sandtbosons of opposite
+parity
+HE=αnt−1−α
+N(Qt)2,withQt=s†t+t†s, (4)
+wherentisthenumber of tbosonsand Nthetotalnumber ofbosons. This particular Hamil-
+tonian belongs to a more general class of two-level Hamiltonian that has been extensively
+applied to nuclear structure [12, 13] and molecular physics [14], and r ecently also proposed
+as a model for optical cavity QED [15].
+This Hamiltonian has a second order QPT at αc= 4/5 [12]. The critical point can
+be easily calculated in the thermodynamic limit assuming a condensed bo son of the form
+Γ†=/parenleftbig
+s†+βt†/parenrightbig
+//radicalbig
+1+β2. Forα >4/5 the environment is a condensate of sbosons
+(β= 0) corresponding to a ferromagnetic state in the spin represent ation. For α <4/5 the
+environment condensate mixes sandtbosons breaking the reflection symmetry ( |β|>0).
+Choosing λ0= 0 and λ1=λthe coupling Hamiltonian reduces to a very simple form
+HCoup=λnt, whichresultsintotheeffective Hamiltoniansforeachcomponentof thesystems
+H0=αnt−1−α
+N(Qt)2, (5)
+H1= (α+λ)nt−1−α
+N(Qt)2. (6)
+Therefore, the system-environment coupling parameter λmodifies the environment Hamil-
+tonian. Using the coherent state approach [12], it is straightforwa rd to show that H1goes
+through a second order QPT at λ∗= 4−5α, forα <4/5. Furthermore, a semiclassical
+calculation [11] shows that HEalso passes through an ESQPT at Ec= 0, ifλ < λ ∗. This
+phenomenon is illustrated in Fig. 1. For α < α c, a bunch of energy levels collapse at E= 0,
+and thus a singularity in the density of states arises. At α=αc, the collapse takes place in
+the ground state, transforming the ESQPT into a standard QPT. F orα > α cno singular
+behavior is observed.
+To understand how this critical energy can be reached when we cou ple this environment
+to a central qubit let us take in consideration the following. We start the evolution with the
+40 0.2 0.4 0.6 0.8 1α-50-2502550E
+λc <nt>
+FIG. 1: (Color online). Energy levels for theenvironment Ha miltonian (4) with N= 50. Thearrow
+(red online) shows the jump that the coupling with the centra l qubit produces in the environment.
+ground state of the environment |g0∝angbracketright. Att= 0 we switch on the interaction between the
+system and the environment, and let the system evolve under the c omplete Hamiltonian.
+By instantaneously switching on this interaction, the energy of the environment increases,
+and its state gets fragmented into a region with average energy eq ual toE=∝angbracketleftg0|H1(α)|g0∝angbracketright.
+Therefore, if ∝angbracketleftg0|H1(α)|g0∝angbracketright= 0, thecoupling withthe central qubit induces theenvironment
+tojumpinto a region arround the critical energy Ec. This is illustrated in Fig. 1. Starting
+from a state in the parity broken phase with α < α c, the coupling with the qubit, Hλ1=λnt
+increases theenergy oftheenvironment uptothecritical point Ec. Resorting tothecoherent
+state approach [12], we can obtain a critical value of the coupling str ength
+λc(α) =1
+2(4−5α), α <4
+5. (7)
+InFig. 2weshowthemodulusofthedecoherence factor |r(t)|forα= 0andseveral values
+ofλ(see caption). In four of the five cases we can see a similar pattern , fast oscillations
+plus a smooth decaying envelope. For λ/greaterorsimilarλ∗, this envelope is weakly dependent on λ. As
+we increase λ, the frequency of the short period oscillations increases linearly ν≈λ/3, but
+the main trend of the curve remains the same. The shape of the env elope can be fitted
+to a Gaussian decay for short times, and to a power-law decay for lo nger times; a similar
+behavior was identified in [5] if the Hamiltonian parameter is close to or la rger than the
+critical value. However, the most striking feature of Fig. 2 is the pa nel corresponding to
+λ= 2, for which |r(t)|quickly decays to zero and then randomly oscillates around a small
+value. We note that this particular case constitutes a singular point for both the shape of
+500.5100.5100.5100.5100.51
+ 0 5 10 15 20 25|r(t)|
+tλ=1
+λ=2
+λ=3
+λ=4
+λ=6
+FIG. 2:|r(t)|forα= 0 and five different values of λ. In all cases N= 10000.
+the envelope of |r(t)|and the period of its oscillating part. Making use of Eq. (7) for α= 0
+we obtain precisely λc= 2, the value at which the coherence of the system is completely
+lost. Therefore, the existence of an ESQPT in the environment has a strong influence on
+the decoherence that it induces in the central system. We can sum marize this result with
+the following conjecture:
+If the system-environment coupling drives the environment to the critical energy Ecof a
+continuous ESQPT, the decoherence induced in the coupled qu bit is maximal .
+Let us check this conjecture for different values of α < α c= 4/5 . In Fig. 3 we show
+how the decoherence process changes around the critical value λcfor different values of N
+andα. As a representative quantity, we plot rmax(α) which is the value of |r(t)|at the
+second maximum (the first maximum is |r(0)|= 1 in all cases). We can extract two main
+conclusions from Fig. 3. First, rmax(λ) is minimum at λ≈λc, as given in Eq. (7) —
+the four cases plotted in the figure correspond to λc= 0.25,0.5,1.0,and 2.0. Second, the
+behavior of this quantity is smooth and independent of the environm ent sizeN, except in
+6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
+ 0 0.5  1 1.5  2 2.5rmax (λ)
+λ
+FIG. 3: (Color online). rmax(λ) in function of the coupling λ, for different values of αandN.
+Black lines represent the case α= 0; dark grey (red online) lines, α= 0.4; grey (magenta online),
+α= 0.6; and light grey (cyan online), α= 0.7. Solid lines represent N= 10000; dotted lines,
+N= 2500; and dashed lines, N= 600.
+-0.8-0.6-0.4-0.2 0
+ 2  2.5  3  3.5  4log10 rmax (λc)
+log10 N
+FIG. 4: rmax(λc) in function of the size of the environment N, in a double logarithmic scale.
+Squares represent the case α= 0; circles, α= 0.4; upper triangles, α= 0.6; lower triangles,
+α= 0.7. The straight lines correspond to a least squares fit to the p ower law rmax(λc) =AN−1/4.
+a small region around λ≈λcwhere it becomes sharp and size dependent. Therefore, the
+decoherence factor behaves in a critical way around λ=λcwherermax(λc) undergoes a dip
+towards zero which is sharper and deeper for larger values of N.
+On the light of these results, one may wonder how rmax(λc) behaves in the thermody-
+namical limit. As it is very difficult to do exact calculations beyond size N >10000, we will
+rely on the finite size scaling analysis to extrapolate its behavior for N→ ∞. In Fig. 4 we
+show how this quantity evolves with the size Nof the environment. We plot results for the
+7same values of αas in Fig. 3 in a double logarithmic scale (see caption of Fig. 4 for details) .
+In all the cases we obtain a power law rmax(λc)∼N−γ, withγ= 1/4.
+The results displayed in figures 3 and 4 confirm that the presence of an ESQPT in the
+environment spectrum is clearly signaled by the qubit decoherence f actor. Moreover, the
+quantity rmaxbehaves in parallel way as the order parameter ntat the ESQPT [11].
+Prior to the conclusions, it is worth mentioning that the environment al Hamiltonian Eq.
+(4) is a particular case of a more generic class of two-level bosonic s ystems [12], described
+byH=αnl−(1−α)Q(ω)
+l·Q(ω)
+l/N, withQ(ω)
+l=/parenleftBig
+s†/tildewidel+l†s/parenrightBig(l)
++ω/parenleftBig
+l†/tildewidel/parenrightBig(l)
+, where the boson
+s†is a scalar and the boson l†has multipolarity l, as well as the operator Q(ω)
+ldepending
+on an extra parameter ω. As a representative case, we have presented here calculations
+using the environmental Hamiltonian with l= 0 andω= 0, characterized by a second order
+QPT. For l >1 this wide class of Hamiltonians, that have been extensively used in nu clear
+and molecular physics, present a richer structure with second ord er (ω= 0) and first order
+(ω∝negationslash= 0) QPTs. A complete analysis on the influence of the order of the ES QPT in the
+decoherence process will be given in a future publication. Moreover , as the natural state of
+a realistic environment is a thermal one —that is, a mixed state in which the environment is
+in contact with a thermal bath characterized by a temperature T— it will be also interesting
+to check to which extent the temperature affects this critical beh avior.
+In this paper, we have studied the decoherence induced on a one-q ubit system by the in-
+teraction with a two-level boson environment, which presents bot h a standard QPT and an
+ESQPT. Our main finding is that the decoherence is maximal when the s ystem-environment
+coupling introducesintheenvironment theenergyrequiredtounde rgoacontinuous ESQPT.
+The decoherence factor displays a critical behavior well described byrmax(λ), the value of
+the second local maximum of |r(t)|.rmax(λ) approaches zero for λ→λcin the thermody-
+namical limit showing a finite size scaling rmax(λc)∝N−γ, with a critical exponent γ= 1/4.
+Therefore, we conclude that the decoherence induced in a one-qu bit system gives a unique
+signature of an ESQPT in the environment. Finally, let us point out tha t quantum deco-
+herence could constitute an important tool to detect critical reg ions in the energy spectrum
+of mesoscopic systems; conversely, this knowledge could help to av oid certain environments
+that are particularly efficient to destroy quantum coherence. Mor eover, as the particular
+Hamiltonian we have studied in this paper has been broadly applied in nuc lear structure,
+molecular physics and, more recently, optical cavity QED, it is feasib le to use any of these
+8systems as the environment in a decoherence experiment.
+This work has been partially supported by the Spanish Ministerio de Ed ucaci´ on y Ciencia
+and by the European regional development fund (FEDER) under pr ojects number FIS2005-
+01105, FIS2006-12783-C03-01 FPA2006-13807-C02-02 and FP A2007-63074, by Comunidad
+de Madrid and CSIC under project 200650M012, and by Junta de An aluc´ ıa under projects
+FQM160, FQM318, P05-FQM437 and P07-FQM-02962. A.R. is suppor ted by the Spanish
+program ”Juan de la Cierva”, and P. P-F., by a grant from the Plan Pr opio of the University
+of Sevilla.
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+10
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+arXiv:1001.0398v1  [quant-ph]  3 Jan 2010Decoherence due to an excited state quantum phase transitio n in
+a two-level boson model
+P. P´ erez-Fern´ andez1, A. Rela˜ no2, J. M. Arias1, J. Dukelsky2, and J. E. Garc´ ıa-Ramos3
+1Departamento de F´ ısica At´ omica, Molecular y Nuclear,
+Facultad de F´ ısica, Universidad de Sevilla,
+Apartado 1065, 41080 Sevilla, Spain
+2Instituto de Estructura de la Materia,
+CSIC, Serrano 123, E-28006 Madrid, Spain
+3Departamento de F´ ısica Aplicada,
+Universidad de Huelva, 21071 Huelva, Spain∗
+(Dated: November 17, 2018)
+Abstract
+The decoherence induced on a single qubit by its interaction with the environment is studied.
+The environment is modelled as a scalar two-level boson syst em that can go through either first
+order or continuous excited state quantum phasetransition s, dependingon the values of the control
+parameters. A mean field method based on the Tamm-Damkoff appr oximation is worked out in
+order to understand the observed behaviour of the decoheren ce. Only the continuous excited state
+phase transition produces a noticeable effect in the decohere nce of the qubit. This is maximal when
+the system-environment coupling brings the environment to the critical point for the continuous
+phase transition. In this situation, the decoherence facto r (or the fidelity) goes to zero with a finite
+size scaling power law.
+PACS numbers: 03.65.Yz, 05.70.Fh, 64.70.Tg
+∗Electronic address: armando@iem.cfmac.csic.es
+1I. INTRODUCTION
+Decoherence is the quantum phenomenon by which the coherence o f a quantum system
+can be destroyed when it is put in contact with a large environment [1, 2]. The Schr¨ oedinger
+equation is a linear differential equation, consequently any linear com bination of solutions
+is also a solution of the problem. Thus, a general possible quantum st ate is a superposition
+of quantum states. Nevertheless, such a state does not appear in the classical macroscopic
+world. The decoherence interpretation of quantum mechanics [1] c laims that this is due to
+the interaction with the environment, which destroys the quantum correlations between the
+states of the system, making it to transite from a quantum superp osition state to a classical-
+like mixture of states. Moreover, only a small set of states take pa rt of the classical-like
+mixture; they are called pointer states [1].
+The study of decoherence is important for several reasons: i) it m ight be responsible for
+the emergence of classical properties out of the underlying quant um nature of the physical
+systems, ii) it is a major problem for the construction of a quantum c omputer since it
+will produce the loss of the necessary quantum entanglement. Thu s, both for fundamental
+reasons (i) and for practical purposes (ii) it is important to charac terize the decoherence
+process and its effects on the physical properties of a quantum sy stem.
+Along this line of study, it is important to address the issue of the effe ct produced in
+the coherence of a quantum state when the environment evolves b etween different quantum
+phases. There have been several works on the relation between d ecoherence and an environ-
+mental quantum phase transition [3–8]. Recently, we have present ed a novel phenomenon
+in which the decoherence of the system suffers drammatic changes when the environment
+crosses an excited state quantum phase transition (ESQPT)[9]. An ESQPT is a nonanalytic
+evolution of the system as the control parameters in the Hamiltonia n vary. It is similar to a
+ground state quantum phase transition but affecting to excited st ates. Correspondingly, an
+ESQPT can be classified in the thermodynamic limit as first order, when a crossing between
+two excited levels is present, or continuous, when the number of int eracting levels is locally
+very large at an excited energy but without crossings.
+In Ref. [9] we presented briefly the case of a qubit in interaction with an environment
+modelled as a two-level bosonsystem undergoing a continuous ESQP T. We used a particular
+simple Hamiltonian in terms of single control parameter to model the e nvironment in order
+2toshowthemaineffect. Herewepresent amoreextensive studyof asimilar systemincluding
+both first and second order ESQPT, and more general sets of par ameters. Together with the
+exact evolution of the system, we present a simple mean field treatm ent. We show that the
+decoherence is maximal when the interaction of the system with the environment produces
+second order ESQPT, while no noticeable effects are observed in the case of a first order
+ESQPT. For the former case, a finite-size scaling analysis allows us to postulate that the
+fidelity goes to zero as soon as the interaction between system and environment is switched
+on. We also show that mean field treatment provides a good descript ion for the decoherence
+of the small system, except around the critical points.
+The paper is structured as follows. In Sect. II, we present the mo del for the environment
+and study the phase diagram and its relation with the density of ener gy levels. We then
+discuss the interaction of the environment with the system. In Sec t. III we show results
+for the decoherence factor. Both exact numerical results for la rge boson number, and an
+analytic mean field method with simple extensions of the Tamm-Dankoff approximation are
+presented. In Sect. IV, results for the decoherence factor in t he case of continuous and first
+order ESQPT, including a finite size scaling study for the decoherenc e factor (or fidelity),
+are discussed. Finally, in Sect. V we summarize giving the main conclusio ns of this work.
+II. THE MODEL
+Following [3], we will consider our system composed by a spin 1 /2 particle coupled to a
+spin environment by the Hamiltonian HSE,
+HSE=IS⊗HE+|0/angbracketright/angbracketleft0|⊗Hλ0+|1/angbracketright/angbracketleft1|⊗Hλ1, (1)
+where|0/angbracketrightand|1/angbracketrightare the two components of the spin 1 /2 system, and λ0,λ1the couplings
+of each component to the environment. The three terms HE,Hλ0andHλ1act on the Hilbert
+space of the environment; therefore, it evolves with an effective H amiltonian depending on
+the state of the central spin Hi=HE+Hλi,i= 0,1. The term Hλimakes it possible that
+the environment crosses a critical point as a consequence of the in teraction with the central
+spin [3].
+Considering the initial state |ΨSE(0)/angbracketright= (a|0/angbracketright+b|1/angbracketright)|E(0)/angbracketright, where|E(0)/angbracketrightis the initial
+3state of the environment, the evolved reduced density matrix of t he system is
+ρS(t) = Tr E|ΨSE(t)/angbracketright/angbracketleftΨSE(t)| (2)
+=|a|2|0/angbracketright/angbracketleft0|+ab∗r(t)|0/angbracketright/angbracketleft1|
++a∗br∗(t)|1/angbracketright/angbracketleft0|+|b|2|1/angbracketright/angbracketleft1|.
+The off-diagonal terms of the density matrix are modulated by the d ecoherence factor
+r(t) which is the overlap between two states of the environment obtain ed by evolving the
+initial state |Ψ(0)/angbracketrightwith two different Hamiltonians,
+r(t) =/angbracketleftΨ(0)|eiH0te−iH1t|Ψ(0)/angbracketright. (3)
+If the environment is initially in the ground state of H0,|0,g/angbracketright, the decoherence factor,
+up to an irrelevant phase factor, is
+r(t) =/angbracketleft0,g|e−iH1t|0,g/angbracketright. (4)
+This quantity has the same form as the Loschmidt echo or the fidelity , and it contains all
+the relevant information about the decoherence process.
+Tobemorespecific, letusintroduceasanenvironment atwo-level b osonsystemdescribed
+by a generalized Lipkin Model, whose Hamiltonian is
+HE=α/hatwident−1−α
+N/hatwideQω/hatwideQω, (5)
+where the operators /hatwidentand/hatwideQωare defined as
+/hatwident=t†t,/hatwideQω=s†t+t†s+ω t†t, (6)
+in terms of two species of scalar bosons sandt.αandωare two independent control
+parameters, and the total number of bosons N=/hatwidens+/hatwidentis a conserved quantity.
+It is worth to mention that this two-level bosonic Hamiltonian is comple tely equivalent
+to an SU(2) spin Hamiltonian, with long-range spin exchange interact ion. The equivalence
+is defined by the inverse Schwinger representation of the SU(2) ge nerators
+S+=t†s= (S−)†, Sz=1
+2(t†t−s†s), (7)
+whereSrepresents the total spin of a chain of N1/2 spins.
+4A. Mean field theory for HE
+In order to study the phase diagram of the Hamiltonian (5) as a func tion of the control
+parameters αandω, it is usual to rely on a coherent state of the form
+|N,β/angbracketright=e/radicalBig
+N
+(1+β2)(s†+β t†)|0/angbracketright, (8)
+where|0/angbracketrightdenotes the boson vacuum. The corresponding energy surface a s a function of the
+variational parameter βis the expectation value of HE(5) in the coherent state (8)
+E(N,β) =/angbracketleftN,β|HE|N,β/angbracketright
+/angbracketleftN,β|N,β/angbracketright
+=Nβ2
+(1+β2)2/braceleftBig
+5α−4+4βω(α−1)+β2/bracketleftbig
+α+ω2(α−1)/bracketrightbig/bracerightBig
+.(9)
+Minimizationoftheenergy(9)withrespectto β,forgivenvaluesofthecontrolparameters
+αandω, gives the equilibrium value βedefining the phase of the system in the ground state.
+The value βe= 0 corresponds to the symmetric phase, and βe/negationslash= 0 to the broken symmetry
+phase.
+This Hamiltonian has a second order Quantum Phase Transition (QPT) along the line
+ω= 0, and a first order QPT for ω/negationslash= 0. In the latter, the critical point is defined as the
+situation in which the minimum in the symmetric phase and in the broken s ymmetry phase
+are degenerate and their energies are equal to zero. The study o f the phase diagram has
+been done in several publications [10]. Here we summarize its main fea tures.
+•β= 0 is always a stationary point. For ω= 0, the solution with β= 0 is a maximum
+forα <4/5, and becomes a minimum for α >4/5. In the case of α= 4/5,β= 0 is an
+inflection point. α= 4/5 is the point in which a minimum at β= 0 starts to develop
+and defines the antispinodal line.
+•Forω/negationslash= 0 there exists a region where two minima, one spherical and one def ormed,
+coexist. This region is defined by the point where the β= 0 minimum appears
+(antispinodal point)andthepointwherethe β/negationslash= 0minimum appears(spinodalpoint).
+The spinodal line is defined by the implicit equation,
+3α
+3α−4=A
+B/parenleftBigg
+1−/parenleftbigg
+1+B
+A/parenrightbigg3
+2/parenrightBigg
+, (10)
+5E(    )βE(    )β
+E(    )β
+E(    )βE(    )β
+4/5β
+ββ β
+β
+Symmetric Brokenω
+α
+FIG. 1: Schematic phase diagram for HE(5) as a function of the control parameters αandω.
+whereA= (4−3α+2(α−1)ω2)2andB= 36ω2(α−1)2. For example, for ω= 1/√
+2,
+α≃0.822559.
+•In the coexistence region, the critical point is defined by the condit ion that both min-
+ima (spherical anddeformed) aredegenerate. At thecritical poin t thetwo degenerated
+minima are at βe= 0 and βe=ω/2, and their energy is equal to zero. The critical
+line is therefore defined as
+αc=4+ω2
+5+ω2. (11)
+For example, for ω= 1/√
+2,αc= 9/11.
+•According to the previous analysis, for ω/negationslash= 0 there appears a first-order phase tran-
+sition, while for ω= 0 there is an isolated point of second-order phase transition at
+α= 4/5. In this case, antispinodal, spinodal and critical points collapse to a single
+point.
+In Fig. 1 we present a schematic view of the phase diagram for the en vironment Hamil-
+tonianHE(5) in the ω−αplane.
+TheHamiltonian(5)alsodisplays anExcited StateQuantumPhase Tra nsition(ESQPT),
+which is analogousto a standard quantum phase transition, but tak ing place at some excited
+critical energy Ecof the system. We can distinguish between different kinds of ESQPTs . As
+it is stated in [11], in the thermodynamic limit a crossing of two levels at E=Ecdetermines
+6-50-25 0 25 50
+ 0 0.2 0.4 0.6 0.8  1E
+α-100-75-50-25 0 25 50
+ 0 0.2 0.4 0.6 0.8  1E
+α
+(a)ω= 0 (b) ω= 1/√
+2
+FIG. 2: Energy levels of the Hamiltonian (5) as a function of αforN= 50 and two different values
+ofω.
+a first order ESQPT, while if the number of interacting levels is locally lar ge atE=Ec
+but without real crossings, the ESQPT is continuous. As the entro py of a quantum system
+is related to its density of states, a relationship between an ESQPT a nd a standard phase
+transition at a certain critical temperature can be established in th e thermodynamic limit
+[12]. These kinds of phase transitions have been identified in the Lipkin model [13], in
+the interacting boson model [14], and in more general boson or ferm ion two-level pairing
+Hamiltonians (for a complete discussion, including a semiclassical analy sis, see [15]). In all
+these cases, the ESQPT takes place beyond the critical value of th e Hamiltonian control
+parameter, implying that the critical point moves from the ground s tate to an excited state.
+In Fig. 2 we show the energy eigenvalues of the environmental Hamilt onian (5) with
+N= 50 bosons as a function of the control parameter αforω= 0 in left panel, and
+ω= 1/√
+2 in right panel. In both cases, we see for α < α ca collapse of several levels at
+E≈0. In the right panel ( ω= 1/√
+2) we can also see a second critical curve for E <0 that
+divides the level diagram in two regions: one in which levels behave smoo thly, and another
+in which the level density increases and some crossings are observe d.
+One simple way to analyze the phase diagram is by means of the density of states.
+To obtain an analytical approximation for this quantity, one can sta rt from a coherent
+state similar to (8), in which real parameter βis replaced by the complex parameter z=
+tan(φ/2)exp(iξ), intermsofwhichtheenergyisexpressed as H(φ,ξ) =/angbracketleftN,φ,ξ|H|N,φ,ξ/angbracketright.
+7 0 200 400 600 800 1000 1200 1400 1600
+-800-600-400-200 0 200 400 600ρ (E)
+E
+FIG. 3: Density of states of the Hamiltonian (5) for N= 1000, α= 0.5. The dashed line
+corresponds to ω= 0 and the dotted line to ω= 1/√
+2.
+A goodapproximation for the density of states can be obtained by c ounting how many levels
+are there in an energy window dE,
+ρ(E) =1
+N/integraldisplay
+dξdφ|J(φ,ξ)|δ(H(φ,ξ)−E), (12)
+where|J(φ,ξ)|istheJacobianofthetransformation( φ,ξ)→(p,q),pandqarethecanonical
+coordinates of the Hamiltonian, and Nis a normalization constant.
+In Fig. 3 we show the density of levels of the environmental Hamiltonia n (5), calculated
+by means of Eq. (12), for N= 1000,α= 1/2 and the same values of ωas in Fig. 2. As it
+can be seen, the collapse of levels at E= 0 gives rise to a cusp singularity of ρ(E) for both
+ω= 0 and ω= 1/√
+2. In the latter case, there also exists a jump in the density of stat es
+for a fixed value E <0 (E≈ −125 for this value of ω) consistent with the energy spectra
+of Fig. 2. Although not shown, similar results are obtained for other values of αandω. In
+particular, the jump in the density of states at a certain value E <0 only appears for ω >0.
+Therefore, two different kinds of ESQPT exist in excited spectrum o f Hamiltonian (5). If
+we keep the terminology of thermodynamical phase transitions and we take the number of
+levels up to an energy E, N(E) =/integraltext
+dEρ(E) as the analogue of the free energy F(N,T), we
+can conclude: (a) there exists a continuous λquantum phase transition at E(2)
+c= 0, for any
+value of parameter ω; (b) there also exists a first-order quantum phase transition at s ome
+critical energy E(1)
+c<0 ifω >0.
+To estimate the critical energies at which these quantum phase tra nsitions take place, we
+canrely onthe energysurface H(φ,ξ). InFig. 4 weshow H(φ,ξ)/Ninthethermodynamical
+80π/2π3π/22π0π/4π/23π/4π-0.6-0.3 0 0.3E (ξ,φ)/N
+ξφ
+FIG. 4: (Color online). Energy surface H(φ,ξ)/Nin the thermodynamical limit N→ ∞for
+α= 1/2 andω= 1/√
+2. Contour curves are drawn on the base of the figure (see text) .
+limitN→ ∞forα= 1/2 andω= 1/√
+2. The curves drawn in the base of the figure are
+contour curves for fixed values of the energy H(φ,ξ)/N=E. Gray curves (red online)
+represent different values of EaroundE(2)
+cfor the continuous phase transition. The solid
+gray (red online) line represents the critical point Ec= 0; this is the only value for which
+the contour curve is non-analytic. On the other hand, black curve s (blue online) represent
+different values of Earound the critical energy at which the first-order ESQPT takes p lace,
+E(1)
+c. In this case, the critical value is the one at which the island around ξ=πappears,
+that correspond to a local minimum in the energy surface. This enta ils the appearance of
+another region in the ( φ,ξ) plane for which the equation H(φ,ξ)/N=Ehas a solution, and
+consequently the density of states ρ(E) suddenly increases.
+B. Coupling to a single qubit
+Since we are interested in relating the phenomenon of decoherence in a single qubit with
+the structure of phases and critical regions in the environment as defined by the Hamiltonian
+(1), we propose as a coupling Hamiltonian Hλi=λi/hatwident. Choosing λ0= 0 if the qubit is on
+9state|0/angbracketrightandλ1=λif the qubit is on the state |1/angbracketright, the effective environment Hamiltonian
+for each component of the system results into
+H0=α/hatwident−1−α
+N/hatwideQt/hatwideQt, (13)
+H1= (α+λ)/hatwident−1−α
+N/hatwideQt/hatwideQt. (14)
+This means that the qubit only interacts with the environment when it is on state |1/angbracketright.
+The system-environment coupling parameter λmodifies the environment Hamiltonian.
+For certain values of αandλ, this modification entails a crossing of the critical lines. Similar
+phenomena were previously analyzed by several authors [3–5], st udying whether a quantum
+quench that drives the environment through a QPT implies some kind o f universality in the
+decoherence process.
+Using the coherent state approach [10], it is straightforward to sh ow thatH1goesthrough
+a ground state QPT at
+λ∗= (1−α)(4+ω2)−α (15)
+forα < α ∗. Therefore, if λ > λ ∗the quench makes the environment jump from one phase
+to the other.
+The main purpose of this paper to show that an ESQPT, instead of a g round state QPT,
+indeed produces dramaticconsequences inthe decoherence proc ess. Using the coherent state
+approximation, it is straightforward to obtain that the coupling bet ween the environment
+and the qubit entails an energy transfer in the former one, which is e qual to
+∆E(N,β,λ) =/angbracketleftN,β|λ/hatwident|N,β/angbracketright=λNβ2
+1+β2. (16)
+Therefore, the critical coupling λcwhich leads the environment to the critical energy Ecis
+E(N,β)+∆E(N,β,λ c) =Ec, (17)
+valid for both first order critical energy E(1)
+cand second order one E(2)
+c. In general, this is
+a trascendent equation, and therefore λ(1)
+candλ(2)
+c(theλ’s corresponding to E(1)
+candE(2)
+c,
+respectively) have to be obtained numerically.
+10III. CALCULATION OF THE DECOHERENCE FACTOR
+In order to calculate the decoherence factor (4) the expectatio n value of H1(14) in the
+groundstate |0,g/angbracketrightofH0(13) is needed. The decoupling of the complete system-environmen t
+Hamiltonian into the independent Hamiltonians H0andH1for each qubit state allows an
+exact diagonalization for large systems. In the following two subsec tions we will describe
+the exact formalism and make a comparison with mean field techniques supplemented with
+a Tamm-Dankoff approximation (TDA) treatment of the excited spe ctrum.
+A. Exact diagonalization
+A general Hamiltonian in terms of sandtbosons including up to two body terms is,
+Hst=at†t+b(t†s+s†t)+ct†ss†t
++d(t†st†s+s†ts†t)+e(t†st†t+t†ts†t)+ft†tt†t (18)
+wherea,b,c,d,e andfare arbitrary parameters.
+Both Hamiltonians, H0(13) and H1(14), are particular cases of Hst(18) with the fol-
+lowing parameters,
+a=α+λ−2α−1
+N
+b=ωα−1
+N
+c= 2α−1
+N
+d=α−1
+N
+e= 2ωα−1
+N
+f=ω2α−1
+N
+∆ =α−1, (19)
+where ∆ is an irrelevant global shift in energy.
+The exact diagonalization of the stHamiltonian (18), and consequently of H0andH1,
+reduces to the diagonalization of a tridiagonal matrix in the basis
+|Nl/angbracketright=t†ls†N−l
+/radicalbig
+l!(N−l)!|0/angbracketright, (20)
+11where|0/angbracketrightis the boson vacuum and 0 ≤l≤N. Therefore, the dimension of the Hamiltonian
+matrix is d=N+1.
+The relevant matrix elements are,
+/angbracketleftNl|Hst|Nl/angbracketright=al+fl2+cl(1+N−l), (21)
+/angbracketleftNl|Hst|Nl+1/angbracketright=b/radicalbig
+(N−l)(l+1)+e/radicalbig
+(l+1)(N−l)l, (22)
+/angbracketleftNl|Hst|Nl+2/angbracketright=d/radicalbig
+(l+2)(l+1)/radicalbig
+(N−l)(N−l−1), (23)
+being all the others equal to zero. The diagonalization of the corre sponding tridiagonal ma-
+trix can be done easily even for large Nvalues, providing the exact results for the eigenen-
+ergies and eigenfunctions of H0andH1and consequently allowing to calculate numerically
+r(t).
+B. The Tamm-Dankoff approximation
+Beforeapplyingtheexactdiagonalizationtechniques tostudytheb ehaviorofthedecoher-
+ence as a fuction of the set of model parameters and particularly in relation to the quantum
+phase transitions (first and second order) in the ground (QPT) an d excited states (ESQPT)
+of the environment, we will introduce an extension of the mean field a pproximation based
+on the TDA but including two phonon anharmonicities.
+Let us consider the condensate boson of the state (8) as a groun d state deformed boson
+in a rotated basis. Since two Hamiltonians are involved, H0andH1, let us formulate the
+approximation for both in terms of a generic Hi(i= 0,1). The variational parameter βin
+the condensate could be different for both Hamiltonians; therefor e, the notation βi(i= 0,1)
+will be used to distinguish between both cases. With this notation, th e deformed bosons ( g
+ande) forHiare related to the initial ones ( sandtbosons) by
+Γ†
+i,g=1/radicalbig
+1+β2
+i(s†+βit†), (24)
+Γ†
+i,e=1/radicalbig
+1+β2
+i(−βis†+t†). (25)
+12In terms of the deformed bosons the ground state and the first e xcited states are
+|i,g/angbracketright=1√
+N!/parenleftBig
+Γ†
+i,g/parenrightBigN
+|0/angbracketright, (26)
+|i,e/angbracketright=1/radicalbig
+(N−1)!Γ†
+i,e(Γ†
+i,g)N−1|0/angbracketright. (27)
+Inthis framework, higher excited states canbe constructed by d irectly replacing a ground
+statebosoncondensate by anexcited βboson; thisprocedure isknown astheTamm-Dancoff
+approximation (TDA) method. In addition, with this basis is possible to write a diagonal
+Hamiltonian in terms of the new bosons. If only one body terms are inc luded,
+Hi≈ /angbracketlefti,g|Hi|i,g/angbracketright+/parenleftBig
+/angbracketlefti,e|Hi|i,e/angbracketright−/angbracketlefti,g|Hi|i,g/angbracketright/parenrightBig
+Γ†
+i,eΓi,e (28)
+=Ei,0+∆eiΓ†
+i,eΓi,e,
+whereEi,0=/angbracketlefti,g|Hi|i,g/angbracketrightand ∆ ei= (/angbracketlefti,e|Hi|i,e/angbracketright−/angbracketlefti,g|Hi|i,g/angbracketright).
+The calculation for r(t) (4) involves the H0ground state and the Hamiltonian H1. Thus,
+it is necessary to relate the intrinsic bosons for H0andH1. The relation between both boson
+families is given by
+Γ†
+0,g=/summationdisplay
+pfgpÆ
+1,p, (29)
+Γ†
+0,e=/summationdisplay
+pfβpΓ†
+1,p, (30)
+where this sum is for p=gandp=e, and the coefficients of the needed transformation are,
+fgg=1/radicalbig
+1+β2
+01/radicalbig
+1+β2
+1(1+β0β1), (31)
+fge=1/radicalbig
+1+β2
+01/radicalbig
+1+β2
+1(β0−β1). (32)
+With the preceding transformation it is possible to write |0,g/angbracketrightin terms of the H1intrinsic
+bosons
+|0,g/angbracketright=1√
+N!/parenleftBig
+fggÆ
+1,g+fgeÆ
+1,e/parenrightBigN
+|0/angbracketright. (33)
+13Using the binomial expansion of (33) is then straightforward to calc ulate the decoherence
+factorr(t) using the TDA basis up to an irrelevant phase factor
+r(t) =N/summationdisplay
+k=0/parenleftbiggN
+k/parenrightbigg
+(fgg)2(N−k)(fge)2ke−i∆e1kt, (34)
+where ∆ e1= (/angbracketleft1,e|H1|1,e/angbracketright−/angbracketleft1,g|H1|1,g/angbracketright). A more compact expression for r(t) can be
+obtained using the transformation,
+e−i∆e1tk=e−i(∆e1/2)tNei(∆e1/2)t(N−k)e−i(∆e1/2)tk. (35)
+Therefore, the decoherence factor r(t) in the TDA reduces to
+r(t) =e−i(∆e1/2)tN/parenleftbig
+(fgg)2ei(∆e1/2)t+(fge)2e−i(∆e1/2)t/parenrightbigN. (36)
+The matrix elements required for calculating r(t) are
+/angbracketlefti,g|Hi|i,g/angbracketright= (a+c+f)β2
+i
+1+β2
+iN+2bβi
+1+β2
+iN
++N(N−1)
+(1+β2
+i)2/parenleftBig
+(c+2d)β2
+i+2eβ3
+i+fβ4
+i/parenrightBig
+, (37)
+and
+/angbracketlefti,e|Hi|i,e/angbracketright=1
+1+β2
+i(a+c+f−2bβi)+c(1−β2
+i)2
+(1+β2
+i)2(N−1)
++4(N−1)
+(1+β2
+i)2(fβ2
+i−2dβ2
+i+e(βi−β3
+i))
++ (a+c+f)β2
+i
+1+β2
+i(N−1)+2bβi
+1+β2
+i(N−1)
++(N−1)(N−2)
+(1+β2
+i)2/parenleftBig
+(c+2d)β2
+i+2eβ3
+i+fβ4
+i/parenrightBig
+. (38)
+A simple inspection reveals that decoherence factor r(t) (36) does not give a good ap-
+proximation of the exact results (see below and [9]). The modulus of r(t) is
+|r(t)|=/vextendsingle/vextendsingle/vextendsingle/parenleftbig
+f2
+gg/parenrightbig
+ei(∆e1/2)t+/parenleftbig
+f2
+ge/parenrightbig
+e−i(∆e1/2)t/vextendsingle/vextendsingle/vextendsingleN
+=|fge|2N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggfgg
+fge/parenrightbigg2
++ei∆e1t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN
+.(39)
+As a particular example, let us consider β1= 0 and β0/negationslash= 0, that is, the situation in which
+the coupling of the qubit to the environment forces the environmen ts to cross the phase
+14transition from the broken phase to the symmetric phase. In this s ituation
+fgg=1/radicalbig
+1+β2
+0(40)
+fge=β0/radicalbig
+1+β2
+0. (41)
+From these expressions, it is straightforward to obtain that |r(t)|oscillates between
+|r(t)|max= 1; (42)
+|r(t)|min=/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ2
+0−1
+β2
+0+1/vextendsingle/vextendsingle/vextendsingle/vextendsingleN
+−→0,forN−→ ∞. (43)
+Therefore, we can conclude that TDA approximation including just o ne phonon excita-
+tions does not account for the decay of the envelope of the decoh erence factor reported in
+[9] (see below for more details). This evidence suggests to go furth er within the spirit of
+TDA by including the anharmonicities of the two-phonon excitations. For this purpose it is
+needed to construct the states two TDA excitations as
+/vextendsingle/vextendsinglei,e2/angbracketrightbig
+=1√
+21/radicalbig
+(N−2)!(Γ†
+i,e)2(Γ†
+i,g)N−2|0/angbracketright. (44)
+From this state we derive the diagonal part of the Hamiltonian as
+Hi≈ /angbracketlefti,g|Hi|i,g/angbracketright+/parenleftBig
+/angbracketlefti,e|Hi|i,e/angbracketright−/angbracketlefti,g|Hi|i,g/angbracketright/parenrightBig
+Γ†
+i,eΓi,e
++/parenleftbigg/angbracketlefti,e2|Hi|i,e2/angbracketright
+2−/angbracketlefti,e|Hi|i,e/angbracketright+/angbracketlefti,g|Hi|i,g/angbracketright
+2/parenrightbigg
+Γ†
+i,eÆ
+i,eΓi,eΓi,e
+=Ei,0+∆eiΓ†
+i,eΓi,e+ΩeiΓ†
+i,eÆ
+i,eΓi,eΓi,e,(45)
+whereEi,0=/angbracketlefti,g|Hi|i,g/angbracketright, ∆ei= (/angbracketlefti,e|Hi|i,e/angbracketright−/angbracketlefti,g|Hi|i,g/angbracketright), and Ω ei=/parenleftBig
+/angbracketlefti,e2|Hi|i,e2/angbracketright
+2−/angbracketlefti,e|Hi|i,e/angbracketright+/angbracketlefti,g|Hi|i,g/angbracketright
+2/parenrightBig
+.
+With the preceding transformation (33) we can obtain the decoher ence factor in the
+improved approximation up to an irrelevant phase factor
+r(t) =N/summationdisplay
+k=0/parenleftbiggN
+k/parenrightbigg
+(fgg)2(N−k)(fge)2ke−i(∆e1k+Ωe1k(k−1))t. (46)
+In addition to (37) and (38), the only needed matrix element for obt ainingr(t) is Ω e1
+which follows from /angbracketlefti,g|Hi|i,g/angbracketright,/angbracketlefti,e|Hi|i,e/angbracketright, given above and
+15/angbracketlefti,e2|Hi|i,e2/angbracketright=1
+(1+β2
+i)2/parenleftBig
+(2c+4d)β2
+i+4eβ3
+i+2fβ4
+i/parenrightBig
++2
+1+β2
+i(a+c+f−2bβi)+2c(1−β2
+i)2
+(1+β2
+i)2(N−2)
++8(N−2)
+(1+β2
+i)2(fβ2
+i−2dβ2
+i+e(βi−β3
+i))
++ (a+c+f)β2
+i
+1+β2
+i(N−2)+2bβi
+1+β2
+i(N−2)
++(N−2)(N−3)
+(1+β2
+i)2/parenleftBig
+(c+2d)β2
+i+2eβ3
+i+fβ4
+i/parenrightBig
+. (47)
+Inserting (37), (38) and (47) into (46) we arrive to the final form of the decoherence factor
+r(t) within the extended TDA approximation. In this case, a semi-quant itative analysis as
+the previous one cannot be easily done. A comparison with exact num erical calculations is
+performed in next section.
+IV. RESULTS
+In this section we present the main features of evolution of the sys tem described by (1)
+under the influence of the environment given by (5). A brief report of the relationship
+between the decoherence in the qubit and the excited state quant um phase transitions in
+the environment was given in [9]. Here, we extend the analysis, compa ring the numerical
+results with the Tamm-Dankoff approximation, and also facing the ca se ofω/negationslash= 0, that was
+not considered in [9]. As two paradigmatic cases, we deal with the cas esα= 1/2, andω= 0
+andω= 1/√
+2. Different choices for the defining parameters of the model give r ise to the
+same qualitative results.
+A. Decoherence factor for the Continuous ESQPT
+All the information about the decoherence process induced by the environment (5) in
+the central qubit is encoded in the decoherence factor (4). As me ntioned above, for the
+Hamiltonian we are using there is always a continuous ESQPT independe ntly of the value
+ofω. In addition, for ω/negationslash= 0 there also appears a first order ESQPT. In this subsection we
+will analyze the effect of the continuous ESQPT on the decoherence factor, while the effect
+1600.51λ= 8
+00.51λ= 4
+00.51|r(t)|λ= 1.5
+00.51λ= 0.75
+05101520
+t00.51λ = 0.300.51λ=8
+00.51λ= 4
+00.51|r(t)|λ= 1.75
+00.51λ= 1.17
+05101520
+t00.51λ=0.3
+(a)ω= 0 (b) ω= 1/√
+2
+FIG. 5: (Color online) |r(t)|forα= 1/2 and five different values of λfor two selections of the
+coupling system-environment parameter ω,ω= 0 on the left and ω= 1/√
+2 on the right. In all
+casesN= 1000. Solid (black) lines correspond to the exact solution , and dashed (red) lines to the
+TDA calculation.
+of the first order ESQPT on the decoherence factor will be discuss ed in the next subsection.
+In Fig. 5 we show the results for the decoherence factor for N= 1000 bosons and
+α= 1/2, for two ωvalues,ω= 0 (left panels) and ω= 1/√
+2 (rigth panels). The objective
+of this figure is to show the effect of the continuous ESQPT on the de coherence factor.
+Solving Eq. (17) for the value of α= 1/2, the continuous ESQPT ( E(2)
+c= 0) takes place at
+λ(2)
+c= 0.75 forω= 0 (left panel) and at λ(2)
+c= 1.17 forω= 1/√
+2 (right panel). Several
+features deserve to be discussed. First of all, we can see that the TDA calculation works
+pretty well for small and large values of λ. In particular, the shape of the envelope, which
+remains unaffected by the increase of λforλ≫λ(2)
+c, is very well described by the TDA
+calculation (see panels for λ= 4 and λ= 8 in Fig. 5). Since this approximation mainly
+17relies on the position of the first and the second excited states of HE, we can conclude
+that the information contained in the low energy spectrum is enough to have a good idea
+about the properties of the highest excited levels of the environme ntal Hamiltonian. Note
+that switching on the interaction between the central qubit and th e environment entails an
+effective increase of the environmental energy roughly given by ∆ E=/angbracketleftg0|H1(λ)|g0/angbracketright−E0,
+and therefore a large value of λimplies that the state of the environment jumpsfrom the
+ground state to a mixed high-energy state.
+Ontheotherhand,asitisclearlyshownintheleftpanelscorrespond ingtoλ=λ(2)
+c= 0.75
+andλ=λ∗= 1.5, and the right panels λ=λ(2)
+c= 1.17 andλ=λ∗= 1.75, the TDA
+calculations fails for intermediate values of λ. These two values correspond to the critical
+couplings λ(2)
+candλ∗, corresponding to the excited state and the ground state quant um
+phase transitions, given by Eqs. (17) and (15) respectively. The r eason why the Tamm-
+Dankoff approximation does not work for these values is straightfo rward. The ESQPT
+entails a singularity in the energy spectrum far above the first excit ed state, which gives rise
+to the main contribution in the TDA calculation. On the other hand, th e ground state QPT
+does not affect the decoherence suffered by the central qubit be cause the coupling λmakes
+the environment to jump far above the critical point which entails a s ingularity in the gap
+between the ground and the first excited states. However, as th e TDA calculation for r(t)
+strongly depends on this gap, it is spuriously affected by the QPT indu ced by the critical
+coupling λ∗.
+Finally, the best agreement between the Tamm-Dankoff approximat ion and the exact
+calculation happens for λ= 0.3, far below λ(2)
+c. Not only the envelope of the decoherence
+factor is well reproduced, but also the positions of the local maximu m are well placed. In
+this case, the small coupling makes the environment to jump from th e ground state to a
+mixed low-energy state. Therefore, it is reasonable to assume tha t the description provided
+by the TDA, which only takes in consideration the first excited state and a global measure
+of the anharmonicites of the spectrum, is a better approximation f or small values of λ.
+1. Analysis of the critical behavior of the decoherence at th e continuous ESQPT
+As it is shown in Fig. 5, the decoherence of the central qubit behave s in a singular way
+for a critical coupling λ(2)
+c, which makes the environment to jump to the critical energy
+18 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
+ 0 0.5  1 1.5  2 2.5rmax (λ)
+λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
+ 0.6 0.8 1 1.2 1.4 1.6 1.8rmax (λ)
+λ
+(a)ω= 0 (b) α= 1/2
+FIG. 6: (Color online) rmaxin function of the coupling λ, for different values of α,ω, and N. In
+left panel ω= 0. Black lines represent the case α= 0; dark gray (red online) lines, α= 0.4; gray
+(magenta online), α= 0.6; and light gray (cyan online), α= 0.7. Solid lines represent N =10 000;
+dotted lines, N=2500; and dashed lines, N=600. In right pane l,α= 1/2, and lines represent the
+casesω= 0.2,ω= 1/2,ω= 1/√
+2, andω= 1, with the same color code than right panel. Arrows
+show the critical coupling λcprovided by Eq. (17).
+E(2)
+c= 0. As the density of states in both cases ω= 0 and ω/negationslash= 0 display the same critical
+behavior around this value (see Fig. 3), also the same singular behav ior for the decoherence
+is expected.
+In. Fig. 6 we show rmax(λ), defined as the second maximum of |r(t)|(the first maximum
+is trivially |r(t= 0)|= 1). The left panel displays the case ω= 0 for several values of α, and
+the right panel the case α= 1/2, for several values of ω/negationslash= 0 (see caption for details). We can
+see that the behavior of this quantity is the same for ω= 0 and ω/negationslash= 0. It evolves smoothly
+and independently of the size of the system Nfor values far from the critical coupling λ(2)
+c,
+provided by Eq. (17) and shown in Tab. I. In a small region around λ∼λ(2)
+c,rmaxbecomes
+sharp, and the value of the minimum depends on the size of the syste mN; the larger is the
+system, the smaller is rmax(λ(2)
+c). Therefore, for both ω= 0 and ω/negationslash= 0, the decoherence
+factor behaves in a critical way around λ=λ(2)
+cwherermax(λ(2)
+c) undergoes a dip towards
+zero which is sharper and deeper for larger values of N.
+We now investigate the thermodynamical limit, by performing a finite s ize scaling anal-
+ysis. The largest system that we could treat exactly has a size of ar oundN= 10000; going
+19ω= 0 α= 1/2
+α= 0α= 0.4α= 0.6α= 0.7ω= 0.2ω= 0.5ω= 1/√
+2ω= 1
+λ(2)
+c= 2λ(2)
+c= 1λ(2)
+c= 0.5λ(2)
+c= 0.25λ(2)
+c= 0.83λ(2)
+c= 1.01λ(2)
+c= 1.17λ(2)
+c= 1.45
+TABLE I: Critical couplings λ(2)
+cfor the eight cases depicted in Fig. 6
+-0.8-0.6-0.4-0.2 0
+ 2  2.5  3  3.5  4log10 rmax (λ(2)
+c)
+log10 N-1.0-0.8-0.6
+ 2  2.5  3  3.5  4log10 rmax (λ(2)
+c)
+log10 N
+(a)ω= 0 (b) α= 1/2
+FIG. 7:rmax(λ(2)
+c) in function of the size of the environment N, in a double logarithmic scale. Left
+panel represents ω= 0; right panel α= 1/2. Squares represent the case α= 0 (left) and ω= 0.2
+(right); circles, α= 0.4 (left) and ω= 0.5 (right); upper triangles, α= 0.6 (left) and ω= 1/√
+2
+(right); lower triangles, α= 0.7 (left) and ω= 1 (right). Straight lines represent the best fit to a
+power law rmax(λ(2)
+c) =AN−γ.
+beyond this value is very difficult since for a complete calculation of rmax(λ(2)
+c) all the eigen-
+values and eigenvectors of the environmental Hamiltonianare need ed. Starting with systems
+ofN= 100, we analize the finite size scaling along two orders of magnitude.
+In Fig. 7 we show how rmax(λ(2)
+c) evolves with the size Nof the environment, both for
+ω= 0 and several values of α(left panel), and α= 1/2 and several values of ω/negationslash= 0. In all
+the cases, a power-law scaling rmax(λ(2)
+c)∼N−γis observed, and therefore we can expect
+thatrmax(λ(2)
+c)→0 in the thermodynamical limit N→ ∞. Nevertheless, subtle differences
+between verying αwithω= 0 and varying ωwithα= 1/2 are observed. The results for
+the exponent γ, shown in Tab. II, are very close to the proposed γ= 1/4 [9] for ω= 0.
+20ω= 0
+α= 0 α= 0.4 α= 0.6 α= 0.7
+γ= 0.247±0.003γ= 0.248±0.003γ= 0.248±0.001γ= 0.245±0.003
+α= 1/2
+ω= 0.2 ω= 0.5 ω= 1/√
+2 ω= 1
+γ= 0.255±0.006γ= 0.259±0.003γ= 0.264±0.008γ= 0.284±0.001
+TABLE II: Finite size scaling exponents γfor the cases depicted in Fig. 7
+00.5100.5100.51
+ 0 5 10 15 20|r(t)|
+tλ=1.10
+λ=1.05
+λ=1.00
+FIG. 8: |r(t)|forα= 1/2,ω= 1/√
+2 and three different values of λ. In all cases N= 10000.
+However, the numerical estimates seem to increase for larger valu es ofω; in particular, for
+the case ω= 1, the result for exponent γis significatively larger than γ= 1/4.
+B. Decoherence factor for the first order ESQPT
+For the case ω/negationslash= 0, the Hamiltonian considered produce, in addition to the continuou s
+ESQPT studied in the preceding subsection, a first order ESQPT at e nergyE(1)
+c. This
+critical energy can be estimated calculating the local minima in the ene rgy surface H(φ,ξ),
+as it is shown in Fig. 4. Inserting this value in Eq. (17) a critical coupling λ(1)
+cis obtained.
+For the case α= 1/2 andω= 1/√
+2 the first order EQSPT is obtained at λ(1)
+c≈1.05.
+InFig. 8weshow theexact result forthedecoherence factor |r(t)|forα= 1/2,ω= 1/√
+2,
+and three different values of λaroundλ=λ(1)
+c≈1.05. The most significative result is that
+21no trace of critical phenomena are observed in |r(t)|– the shape of this magnitude is smooth
+aroundλ=λ(1)
+c. Moreover, Fig. 6 confirms that rmax(λ) also behaves in a smooth an
+size-independent way. The conclusion is, thus, that the first-ord er ESQPT does not affect
+the decoherence induced in the central qubit.
+V. SUMMARY AND CONCLUSIONS
+The decoherence induced in a single qubit by its interaction with the en vironment, mod-
+elled as a scalar two-level boson model, is studied. The environment p resents a quantum
+phase transition from symmetric to non-symetric phases at aroun dα= 4/5, which can be
+first order ( ω/negationslash= 0) or second order ( ω= 0). In the non-symmetric phase, the environment
+also presents excited state quantum phase transitions (ESQPTs) : a second order one for
+anyωvalue atE(2)
+c= 0, and also a first order one for ω/negationslash= 0 at an energy E(1)
+c<0. We have
+shown that the second order ESQPT affects dramatically the decoh erence factor which goes
+rapidly to zero. A finite size scaling study shows that in that case the decoherence factor
+goes to zero at the critical point following a power law. On the other h and, the first order
+ESQPT does not affect the decoherence of the central qubit.
+We have also shown that a mean field treatment provides a good desc ription of the
+decoherence factor r(t), except in the regions around the critical points. Therefore, mo re
+sophisticated approximations are needed to obtain an analytical de scription of the critical
+behavior of r(t), and, particulary, to estimate the critical exponent λ.
+Acknowledgements
+This work has been partially supported by the Spanish Ministerio de Ed ucaci´ on y
+Ciencia and by the European regional development fund (FEDER) un der projects num-
+ber FIS2008-04189, FIS2006-12783-C03-01 FPA2006-13807- C02-02 and FPA2007-63074, by
+CPAN-Ingenio, by Comunidad de Madrid under project 200650M012 , CSIC and by Junta
+de Analuc´ ıa under projects FQM160, FQM318, P05-FQM437 and P0 7-FQM-02962. A.R. is
+supported by the Spanish program ”Juan de la Cierva” and P. P-F. is supported by a FPU
+22grant of the Spanish Ministerio de Educaci´ on y Ciencia.
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+23
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+arXiv:1001.0399v1  [astro-ph.SR]  3 Jan 2010Draft version November 7, 2018
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+DISCOVERY OF A RED GIANT WITH SOLAR-LIKE OSCILLATIONS IN AN E CLIPSING BINARY SYSTEM
+FROMKEPLER SPACE-BASED PHOTOMETRY
+S. Hekker1, J. Debosscher2, D. Huber3, M. G. Hidas3,4,5, J. De Ridder2, C. Aerts2,6, D. Stello3, T.R. Bedding3,
+R. L. Gilliland7, J. Christensen-Dalsgaard8, T. M. Brown4, H. Kjeldsen8, W. J. Borucki9, D. Koch9, J. M.
+Jenkins10, H. Van Winckel2, P. G. Beck2, J. Blomme2, J. Southworth11, A. Pigulski12, W. J. Chaplin1, Y. P.
+Elsworth1, I. R. Stevens1, S. Dreizler13, D. W. Kurtz14, C. Maceroni15, D. Cardini16, A. Derekas3,17, M. D.
+Suran18
+Draft version November 7, 2018
+ABSTRACT
+Oscillating stars in binary systems are among the most interesting st ellar laboratories, as these can
+provide information on the stellar parameters and stellar internal s tructures. Here we present a red
+giant with solar-like oscillations in an eclipsing binary observed with the N ASAKeplersatellite. We
+compute stellar parameters of the red giant from spectra and the asteroseismic mass and radius from
+the oscillations. Although only one eclipse has been observedso far, we can alreadydetermine that the
+secondary is a main-sequence F star in an eccentric orbit with a semi- major axis larger than 0.5AU
+and orbital period longer than 75days.
+Subject headings: binaries: eclipsing - stars: oscillations - stars: individual (KIC84106 37)
+1.INTRODUCTION
+The prospects for asteroseismology of red-giant stars
+have increased significantly with the launch of Ke-
+pler(Borucki et al. 2009) in March 2009 and CoRoT
+(Baglin et al. 2006) in December 2006. These satellites
+provide uninterrupted long time series of high-precision
+photometry for a large sample of stars. Among these
+stars are many red giants, allowing for both statistical
+analyses as well as detailed individual studies. CoRoT
+saskia@bison.ph.bham.ac.uk
+1School of Physics and Astronomy, University of Birming-
+ham, Edgbaston B15 2TT, United Kingdom
+2Instituut voor Sterrenkunde, Katholieke Universiteit Leu -
+ven, Celestijnenlaan 200D, B-3001 Leuven, Belgium
+3Sydney Institute for Astronomy (SIfA), School of Physics,
+University of Sydney, NSW 2006, Australia
+4Las Cumbres Observatory Global Telescope, Goleta, CA
+93117, USA
+5Department of Physics, University of California, Santa
+Barbara, CA 93106, USA
+6IMAPP, Department of Astrophysics, Radbout University
+Nijmegen, PO Box 9010, 6500 GL Nijmegen, the Netherlands
+7Space Telescope Science Institute, 3700 San Martin Drive,
+Baltimore, MD 21218, USA
+8Department of Physics and Astronomy, Aarhus University,
+DK-8000 Aarhus C, Denmark
+9NASA Ames Research Center, MS 244-30, Moffet Field, CA
+94035, USA
+10SETI Institute/NASA Ames Research Center, MS 244-30,
+Moffet Field, CA 94035, USA
+11Astrophysics Group, Keele University Newcastle-under-
+Lyme, ST5 5BG, UK
+12Instytut Astronomiczny Uniwersytetu Wroclawskiego,
+Kopernika 11, 51-622 Wroclaw, Poland
+13Georg-August Universit¨ at, Institut f¨ ur Astrophysik,
+Friedrich-Hund-Platz 1, D-37077 G¨ ottingen
+14Jeremiah Horrocks Institute of Astrophysics, University o f
+Central Lancashire, PR1 2HE, UK
+15INAF-Osservatorio Astronomica di Roma, via Frascati 3,
+I-00040 Monteporzio C. (RM), Italy
+16IASF-Roma, INAF, V. del Fosso del Cavaliere 100, 00133
+Roma, Italy
+17Konkoly Observatory, Hungarian Academy of Sciences,
+H-1525 Budapest, P.O. Box 67, Hungary
+18Astronomical Institute of the Romanian Academy, Str.
+Cutitul de Argint 5, RO 40557, Bucharest, ROdata have already revealed the presence of nonradial
+oscillation modes in red giants (De Ridder et al. 2009)
+and have given rise to a population synthesis study
+(Miglio et al. 2009). The currently available Keplerdata
+reveal clear solar-like high radial overtone p-mode oscil-
+lations in a large sample of red giants, extending in lu-
+minosity from the red clump to the bottom of the giant
+branch (Bedding et al. 2009).
+For in-depth studies, including detailed stellar model-
+ing, accurate stellar parameters are needed. Red giants
+of different masses occupy a narrow region in the H-R
+diagram, which results in relatively large errors when es-
+timating the mass from measured classical parameters,
+such as effective temperature, surface gravity and lumi-
+nosity, and from evolutionary tracks. A better estimate
+can be obtained from the characteristic frequencies of
+solar-like oscillations in these stars and their frequency
+separations, either through modeling or through scal-
+ing relations (Kjeldsen & Bedding 1995). Here we use
+a modeling approach and scaling relations to compute
+an asteroseismic mass and radius.
+Eclipsing binary systems provide the primary method
+for measuring masses and radii of stars. However, the
+probability of observing an eclipsing system is low, be-
+cause of the geometrical restriction that the system has
+to be viewed close to the orbital plane. We report
+here the discovery of a pulsating red giant in an eclips-
+ing binary system observed by Kepler. This star was
+found using automated variability classification methods
+to the targets of the asteroseismology program of Kepler
+(Debosscher et al. 2009: Blomme et al. 2009).
+2.OBSERVATIONS
+Photometric data for KIC841063719(TYC 3130-2385)
+have been taken with Keplerduring the commissioning
+run (Q0) from 2009 May 2 through 11, and the follow-
+ing first science run (Q1) of 33.5d. We present results
+based on these combined time series of about 40d. This
+19KIC =Kepler Input Catalog , Brown et al. (2005).2 S. Hekker et al.
+TABLE 1
+Stellar parameters of KIC8410637.
+KIC Ofek (2008) spectroscopy
+Teff[K] 4680 ±150 4650 4650 ±80
+log(g) (c.g.s.) 2.8 ±0.3 2.70 ±0.15
+[Fe/H] [dex] −0.3±0.3 −0.38 0.0 ±0.1
+vsini[kms−1] 5.0 ±0.7
+ξmicro[kms−1] 1.3 ±0.1
+star has a magnitude of 10.77 mag in the Keplerband-
+pass and was selected by the KeplerAsteroseismic Sci-
+ence Consortium (KASC, Gilliland et al. 2009) for long-
+cadence observations. The time series data have a 29.4-
+min cadence and show both solar-like oscillations in the
+frequency regime expected for a red giant, and an eclipse
+(see Fig. 1).
+This star has also been part of the TrES-Lyr1 ground-
+based survey20(O’Donovan et al. 2006) from which time
+series photometry with a time span of 75d are at our
+disposal. In these data no eclipse is visible, while the
+primary and possibly the secondary eclipse (depth pre-
+dicted in Section 6) would have been detected, had they
+occurred during the time span of the observations. The
+fact thatthe primaryisnot seenin the TrESdataimplies
+that the period is longer than the duration of the TrES
+observations.
+This star was also observed photometrically by the Su-
+perWASP cameras (Pollacco et al. 2006) for 100 days in
+2007 and for 90 days in 2008. There is a suspicion of a
+secondary eclipse, but no primary eclipse has been ob-
+served during these periods, which have some gaps due
+to weather conditions. Observations for KIC8410637
+are also available from the All Sky Automated Sur-
+vey (ASAS) (Pojma´ nski 1997), in which there are three
+points that have lower flux than the majority. These
+points might be due to the eclipse, which would imply
+an orbital period of about 380d or a submultiple thereof.
+Unfortunately, there are too few points, and their uncer-
+tainty is too large to draw firm conclusions.
+Additionally, we checked whether we could put more
+constraints on the period from the time stamps of the
+data we haveat our disposal. We determined which peri-
+ods would haveled us to miss primaryeclipses because of
+a lack of observations. We would have missed all eclipses
+for an orbital period of about 135, 220, 253, 261, 270 and
+295 days. All other possible periods are longer than the
+380 days, mentioned above.
+3.STELLAR PARAMETERS
+For KIC8410637 the Kepler Input Catalogue
+(Brown et al. 2005) provides a set of stellar pa-
+rameters, as listed in Table 1. Tycho-2 (Høg et al.
+2000) and 2MASS (Skrutskie et al. 2006) photometry
+are also available. Ofek (2008) fitted these colors with
+synthetic photometry of stellar templates (Pickles 1998).
+Using two methods, Ofek (2008) classified the star as a
+metal-weak K2III star with an effective temperature of
+4650K or a K4V star with a temperature of 4350K.
+We also obtained three spectra with the HERMES
+spectrograph (Raskin & Van Winckel 2008) mounted on
+the Mercator Telescope, La Palma, Spain: two on 2009
+20http://nsted.ipac.caltech.edu/NStED/docs/holdings.h tmlFig. 1.— Top: time series data from Keplerof KIC8410637.
+Middle: a section of the light curve centered on the eclipse w ith the
+best-fit model (solid line). Bottom: the residuals after cor recting
+for the eclipse model, clearly showing the solar-like oscil lations.
+October 13 and one on 2009 October 15. These spec-
+tra have a resolution of about 85000 and a wavelength
+range of 4000 −9000˚A. We analyzed the average spec-
+trum, which has a typical signal-to-noise ratio of ∼90 at
+6000˚A, with methods described by Hekker & Mel´ endez
+(2007) and list the results in the last column of Ta-
+ble 1. This method does not directly provide uncer-
+tainties on the individual parameters. Here, we adopted
+the total of the difference and scatter between the val-
+ues of Hekker & Mel´ endez (2007) and the values of
+Luck & Heiter (2007) (see Hekker & Mel´ endez 2007, for
+more details). For vsini, we computed the uncertainty
+from a 10% uncertainty in the total FWHM of the lines,
+which is combined with the σon the macro turbulence.
+Theseresults arein agreementwith those in the KIC and
+the literature. They confirm the red-giant nature of the
+primary star.
+We investigated the spectrum for signatures of the sec-
+ondary star, in the form of additional absorption lines
+and cross-correlation profiles, but no clear signatures
+could be detected. The three spectra obtained over a
+time span of 3 days are not sufficient to see radial veloc-
+ity variations due to the binary orbit.
+4.ASTEROSEISMIC ANALYSIS
+Solar-like oscillations are visible both during and out-
+side the eclipse (Fig. 1). To analyze these we adopted
+two approaches. In the first approach, we corrected the
+light curve including the eclipse by fitting and subtract-
+ing trends for different parts of the time series using a
+linear polynomial for the commissioning data, a second-
+order polynomial for the part of the Q1 data before the
+eclipse, the mean value for the data in full eclipse, and a
+linear polynomial for the data obtained after the eclipse.
+We omitted the observations during ingress and egress.
+The resulting time series is shown in the top panel ofRed giant in an eclipsing binary system from Kepler 3
+Fig. 2.— Top: flux as a function of time, after removing the
+binary signature as described in the text. Centre: the power spec-
+trum of the red giant oscillations. The power excess due to so lar-
+like oscillations is present in the range 30 −60µHz. The excess
+below 20 µHz might be due to granulation in the primary’s at-
+mosphere. Bottom: H-R diagram with BASTI (Pietrinferni et a l.
+2004) evolutionary tracks for different masses and solar met allicity.
+The filled circle shows the position of the red giant. The shad ed
+area shows the secondary position constrained by two values de-
+rived from the light-curve fit: the radius ratio (dashed line s) and
+the bolometric luminosity ratio (dashed-dotted lines; tak ing into
+account the Keplerbandpass). The asterisk indicates the position
+of the Sun.
+Fig. 2. In the second approach, we omitted the complete
+eclipse from the time series.
+The Fourier spectrum shows a clear power excess (see
+the second panel of Fig. 2). The power excess is present
+between 30 −60µHz, which is the dominant range of
+νmax(frequency of maximum oscillation power) found
+for a large sample of red giants observed by CoRoT
+(Hekker et al. 2009b). This reflects the high population
+density of red-clump stars, compared to stars on the as-
+cending branch (Miglio et al. 2009). No power excess is
+present at frequencies between 70 µHz and the Nyquist
+frequency of ∼280µHz. We analyzed the power spec-
+trum obtained from the corrected time series with two
+methodsofthe Octavepipeline (Hekker et al.2009a: Oc-
+tave1 and Octave2) and the power spectrum of the light
+curve where the eclipse had been omitted with the auto-
+mated pipeline developed in Sydney (Huber et al. 2009).
+Octave1 computed νmaxandAℓ=0(maximum mode
+amplitude)fromabinnedpowerspectrum, and∆ ν(large
+frequency separation) using the power spectrum of thepower spectrum. Octave2 computed νmaxandAℓ=0
+from a Gaussian fitting to the oscillation envelope, and
+∆νfrom an autocorrelation of the oscillation frequen-
+cies. The latter have been determined with a Bayesian
+method, where oscillation frequencies are those with a
+posterior probability of less than 0.1% of being noise.
+The Sydney pipeline produced νmaxfrom a smoothed
+power spectrum. The current data do not yet allow us to
+measure the life times of the solar-like oscillations. For
+more details on the pipelines we refer to Hekker et al.
+(2009a) and Huber et al. (2009).
+The resulting parameters are listed in Table 2. Note
+thatAℓ=0is computed for a power spectrum with mean
+squared scaling, which is a factor√
+2 different with
+respect to the scaling relations of Kjeldsen & Bedding
+(1995). With the effective temperature and the scaling
+laws from Kjeldsen & Bedding (1995) we computed the
+radius, massand luminosity ofthe primary(see Table 2).
+We also used the RADIUS pipeline described by
+Stello et al. (2009) to compute the stellar radius. The in-
+put parameters for this pipeline are the large separation,
+Teffand metallicity, for which we used the spectroscopic
+values listed in Table 1. The results from this analysis
+are listed in Table 2 and lead to a value for log(g) of 2.6
+±0.1, which is in agreement with our spectroscopically
+derived value.
+The values computed with different methods and
+approaches are in good agreement. The weighted
+mean values of the results from the different meth-
+ods are used throughout the rest of the paper,
+i.e.,RR= 11.8 ±0.6 R ⊙,LR= 58±6 L⊙,
+MR= 1.7±0.3 M ⊙(suffix ‘R’ refers to the red gi-
+ant) together with the spectroscopically determined ef-
+fective temperature. We point out that the estimates of
+these fundamental parameters rely on the input physics
+of stellar models, through the interpretation of observ-
+ables (e.g. νmaxandAl=0) in terms of the properties of
+the convection and thermodynamics, which we assumed
+to be known and error-free. This implies that there may
+be unknown systematic uncertainties. Nevertheless, the
+radius and mass have values in good agreement with the
+expected range for red-clump stars. For the radius there
+is also agreement between the result from scaling rela-
+tions and from models (RADIUS).
+5.BINARY ANALYSIS
+The very flat bottom of the eclipse, except for the red-
+giant oscillations, seems to be indicative of a smaller ob-
+ject occulted by a considerably larger object. This is
+underlined by the fact that for an annular eclipse, i.e.,
+transit,limbdarkeningeffectsplayamoreprominentrole
+and would have caused a more gradually changing shape
+of the eclipse. Because we have only one eclipse, a full
+analysis of the binary is not yet possible, but we are able
+to put several constraints on the nature of the secondary
+and on the orbit.
+From the Keplerdata, we directly measured the depth
+of the eclipse to be about 9%. This depth is determined
+by the luminosity ratio of the two stars, i.e., l=L2/LR
+=0.099±0.005. The uncertaintyisestimated takingthe
+shape of the light curve outside the eclipse into account.
+In addition to the depth, we measured the time dur-
+ing which the secondary is fully obscured: 1 .64±0.01d.
+The ingress and egress are symmetric and each lasts for4 S. Hekker et al.
+TABLE 2
+Oscillation parameters of KIC8410637 computed with differ ent methods.
+Octave1 Octave2 Sydney RADIUS
+— measured — model
+νmax[µHz] 45.0 ±4.8 45.2 ±1.3 44.2 ±0.5
+∆ν[µHz] 4.64 ±0.23 4.5 ±0.1 4.51 ±0.37
+δν02[µHz] 0.6 ±0.1 0.69 ±0.09
+Aℓ=0[ppm] 57.6 ±7.3 61.8 ±6.2 53.4 ±1.9
+— derived —
+R[R⊙] 11.2 ±1.6 11.8 ±0.7 11.4 ±1.9 13.1 ±1.7
+L[L⊙] 53 ±16 58 ±8 55 ±19 72 ±19
+M[M⊙] 1.7 ±0.7 1.8 ±0.3 1.7 ±0.9
+0.276±0.007d. The symmetry of the ingress and egress
+indicatesthatthe relativeaccelerationofthebinarycom-
+ponentsbetweeningressandegressisequaltozero,i.e. it
+is justified to assume the sky-projected relative velocity
+of the two stars to be constant during the eclipse.
+The eclipse time depends on the radii of both starsand
+the geometry of the system. Knowing that the eclipse
+lasts only during ∼2% or less of the total orbital period,
+i.e., only a very small segment of the orbit is covered
+during the eclipse, and that the system is viewed close
+to the orbital plane, implies that the time of total eclipse
+(τtotal) can be expressed as:
+τtotal=P(RRcosδ−R2)
+4aE(ǫ), (1)
+withPthe orbital period, athe semi-major axis, Ethe
+complete elliptic integral, ǫthe eccentricity and δthe
+latitude of the eclipse on the stellar disc and RRandR2
+the radius of the red giant and the secondary, respec-
+tively. The total time of ingress, total eclipse and egress
+(τin+total+eg ) fromthe samegeometricconsiderationscan
+be expressed as:
+τin+total+eg =P(RRcosδ+R2)
+4aE(ǫ). (2)
+Dividing Eq. (1) by Eq. (2) provides an estimate of R2
+as a function of cos δ:
+τtotal
+τin+total+eg≡x=RRcosδ−R2
+RRcosδ+R2=cosδ−r
+cosδ+r,(3)
+withr=R2/RR. In this way we found that
+R2
+RR= cosδ1−x
+1+x, (4)
+wherex= 0.75±0.03. Using δ= 0 (central eclipse)
+and the value of the radius of the red-giant star derived
+in the previous section, we then find an upper limit of
+R2≤1.7±0.1R⊙.
+To confirm this geometric analysis we computed a ba-
+sic model of the eclipse signal. We neglected limb dark-
+ening and computed the fractional overlap between the
+two discs, taking into account the relative luminosity of
+the star being eclipsed. As explained above, we assumed
+the sky-projected relative velocity ( v) of the two stars
+to be constant. This model has been fitted to the por-
+tion of the light curve near the eclipse using a Markov-
+Chain Monte Carlo approach (see e.g. Torres et al. 2008,
+and references therein), which yields the joint posterior
+probability density of all the parameters, given the data.The physical parameters in our model are r,v,l, and
+b= sinδ(the impact parameter). In order to avoid
+strong correlations among these parameters (in partic-
+ular between r,v, andb), when performing the fit we
+replacevwithv2/(1−b2) andrwithr/v2. The values
+of the physical parameters resulting from this fit are b=
+0.0+0.15,r= 0.1435+0.0008
+−0.0027,v= 1.0338+0.0005
+−0.011RR/day,
+andl= 0.10060+0.00006
+−0.00006, consistent with the values ob-
+tained from the geometric analysis.
+To test our assumption that vis constant, we repeated
+the fit with allowing vto change linearly with time. The
+result was a deceleration of 0 .015+0.005
+−0.004RR/day2(i.e.∼
+3% slower at egress than at ingress). This may be a real
+asymmetry in the eclipse, implying an eccentric orbit.
+However, it could also be due to the pulsation signal
+altering the apparent shapes of the ingress and egress.
+We converted the fitted value of linto a bolometric
+luminosity ratio by folding blackbody curves with the
+Keplerspectral response curve. This is a function of the
+secondary’s temperature, as shown in Fig. 2 (dot-dashed
+lines). With the constraint from the radius ratio and
+L2
+LR=R2
+2T4
+eff,2
+R2
+RT4
+eff,R, (5)
+we findL2= 5.2±0.7 L⊙. The secondary is thus proba-
+bly an F main-sequence star. Using the mass-luminosity
+relation for stars with masses between 0.5 and 2.0M ⊙,
+i.e., (L/L⊙) = (M/M⊙)4.5, we found a mass for the sec-
+ondaryM2of 1.44±0.05M⊙, where we did not take the
+uncertainty of the mass-luminosity relation into account.
+We also computed the surface gravity for the secondary
+and found log(g 2) = 4.2±0.1.
+The binary also has to obey Kepler’s third law:
+MR+M2=4π2a3
+GP2, (6)
+in which we still have aas unknown, and we take a min-
+imum value of 75d for P. With this value we computed
+a minimum semi-major axis for the system of 0.5AU.
+On the other hand, if we assume a circular orbit, we
+cancomputethe period. Fromthe eclipsefit weknowthe
+orbital velocity and combining this with Kepler’s third
+law, we get:
+P=2πG(MR+M2)
+v3. (7)
+Using the masses of both stars, we obtain P= 32±6
+days. This is considerably shorter than the minimum
+orbital period we inferred from currently available obser-
+vations, which implies an eccentric orbit.Red giant in an eclipsing binary system from Kepler 5
+TABLE 3
+parameter overview of eclipsing binary KIC8410637.
+primary secondary
+R[R⊙] 11.8 ±0.6 1.7 ±0.1
+L[L⊙] 58 ±6 5.2 ±0.7
+M[M⊙] 1.7 ±0.3 1.44 ±0.05
+Teff[K] 4650 ±80 6700 ±200
+log(g) (c.g.s.) 2.70 ±0.15 4.2 ±0.1
+orbital parameters
+Porbit[days] >75
+a[AU] >0.5
+6.DISCUSSION
+From observations obtained with Kepler, we found a
+pulsating red giant in an eclipsing binary. Once a full
+orbit has been observed, the orbital parameters will pro-
+vide an independent measure of the stellar parameters,
+such as mass and radius, with respect to the asteroseis-
+micvalues. Theseadditionalconstraintsmakethisavery
+interesting case and the star is therefore being followed
+up by the Kepler Mission as well as with ground-based
+spectroscopy.
+Stellar parameters of the primary red giant have been
+computed from a spectroscopic analysis and from the
+solar-like oscillations. We did not find evidence of the
+secondarycomponentinthespectra. Thecurrentlyavail-
+able spectra are not yet sufficient to put constraints on
+the orbit.
+From the single eclipse observed by Keplerand addi-
+tional independent observationsfrom TrES, SuperWASP
+and ASAS, we inferred constraints on the orbit and sec-
+ondary star. The annular eclipse, i.e., the secondary
+passing in front of the red giant, would cause an eclipse
+with a depth of ∼1.7%, assuming a circular orbit and
+without taking limb-darkening into account. Such a dip
+would be observable, but has not been detected by Ke-
+pler, TrES or ASAS. Only the SuperWASP data show
+a feature that might be due to an annular eclipse. The
+SuperWASP and ASAS data have gaps due to weather
+conditions. Although there are also three gaps (4.7, 2.9
+and 1.7d) in the TrES time series data with a duration
+longer than the duration of the eclipse, we expect the
+chance that an eclipse falls exactly during one of these
+gaps to be low. Therefore, we infer that the orbit islonger than the time span of the TrES data, i.e., at least
+75d. Also, an initial inspection ofthe raw Keplerdata of
+Q2 (additional 3 months of data following the Q1 phase)
+does not seem to reveal another eclipse. In the case that
+we missed an eclipse due to a lack of observations, the
+minimum orbital period would be 135 days.
+From the orbital velocity of the secondary, we com-
+puted the period for a circular orbit, which would be
+shorter than half the total time span of the TrES data
+and we should have seen an occultation in those data.
+Therefore we concluded that the orbit is eccentric. The
+possible annular eclipse in the SuperWASP data has a
+length of ∼7days and depth of ∼0.03 mag. If confirmed
+this would indeed imply a large orbital eccentricity.
+From the eclipse times and depth we were able to com-
+pute the luminosity and the radius of the secondary. All
+values seem to be compatible with an F main-sequence
+star. The mass-luminosity relation for main-sequence
+stars then leads to a mass estimate of the secondary as
+well as its surface gravity. These results are summarized
+in Table 3 and the position of both stars are indicated in
+an H-R diagram in Fig. 2.
+KIC8410637is an extremely interesting binary for fur-
+ther follow-up. Longer time series from Keplerwill im-
+provedramaticallythedetectionthresholdforthederiva-
+tion of the properties of the oscillations of the red giant.
+Funding for this Discovery mission is provided by
+NASA’s Science Mission Directorate. We would like to
+acknowledge the entire Keplerteam for their efforts over
+manyyears. Without theseeffortsitwouldnothavebeen
+possible to obtain the results presented here. SH, WJC,
+YPE, IRS and DWK acknowledge support by the UK
+Science and Technology Facilities Council. The research
+leading to these results has received funding from the
+European Research Council under the European Com-
+munity’s Seventh Framework Programme (FP7/2007–
+2013)/ERCgrantagreementn◦227224(PROSPERITY),
+fromtheResearchCouncilofK.U.Leuven, fromtheFund
+for Scientific Research of Flanders (FWO), and from the
+BelgianFederalScience Office. DSacknowledgessupport
+from the Australian Research Council.
+Facilities: The Kepler Mission.
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+arXiv:1001.0400v1  [hep-ph]  3 Jan 2010Emissivity of neutrinos in supernova in a left-right
+symmetric model
+A. Guti´ errez-Rodr´ ıguez,1E. Torres-Lomas1, and A. Gonz´ alez-S´ anchez1
+1Facultad de F´ ısica, Universidad Aut´ onoma de Zacatecas
+Apartado Postal C-580, 98060 Zacatecas, M´ exico.
+(Dated: November 22, 2018)
+Abstract
+We calculate the emissivity due to neutrino-pair productio n ine+e−annihilation in the context
+of a left-right symmetric model in a way that can be used in sup ernova calculations. We also
+present some simple estimates which show that such process c an act as an efficient energy-loss
+mechanism in the shocked supernova core. We find that the emis sivity is dependent of the mixing
+angleφof the model in the allowed range for this parameter.
+PACS numbers: 14.60.Lm,12.15.Mm, 12.60.-i
+Keywords: Ordinary neutrinos, neutral currents, models beyon d the standard model.
+1I. INTRODUCTION
+TheexistenceofneutrinoswaspostulatedbyPauliin1932inordert oexplaintheobserved
+continuous electron spectrum accompanying nuclear beta decay. Based on the idea of Pauli,
+Fermi [1, 2] proposed the beta decay theory, while Bethe and Peier ls [3–5] predicted an
+extremely small cross-section for the interaction of neutrino with matter, and Gamow [6, 7]
+and Pontecorvo [8] were the first to recognize the important role p layed by neutrinos in the
+evolution of stars. The neutrino emission processes may affect the properties of matter at
+high temperatures, and hence affect stellar evolution.
+Neutrino emission isknown to playanimportant roleinstellar evolution, especially inthe
+late stages when the rate of evolution is almost fully dependent on en ergy loss via neutrinos.
+This refers to the stage of steady burning prior to the implosion of t he stellar core, to the
+process of catastrophic core-collapse, and to the cooling of the n eutron star which is formed.
+The explosion energy of the core-collapse is typically 1053ergwhich makes it one of the
+most impressive violent events in the universe. This energy comes fr om the explosion of
+the progenitor star, and only partly manifests itself in the shock wa ve that is launched
+somewhere at the boundary between the iron core of mass MFe= (1.2−2)M⊙and the inner
+most regions, collapsing into a neutron star. Even when the mechan ism of the core-collapse
+is not yet understood in great detail, the most distinctive feature is the enormous energy of
+(3−5)×1053erg= (10−15)%MFec2radiated in the form of neutrinos and antineutrinos
+of all flavors ( νe,νµ,ντ) during a burst of about 10 seconds. While such neutrinos were
+first observed in the supernova SN1987A, various observatories running or under design, like
+next generationlarge-sizedetectors, couldprovide uswiththelum inosity curvefromafuture
+(extra)galactic explosion and/or the observation of relic neutrino s from past supernovae. To
+disentangle the information from such neutrino signals represents a challenging task, since
+the crucial information from the explosion phenomenon and neutrin o properties such as the
+neutrino hierarchy and the third neutrino mixing angle are intertwine d.
+The detection of neutrinos from SN1987A by the Kamiokande II [9] and Irvine-Michigan-
+Brookhaven [10] detectors confirmed the standard model of cor e-collapse (type II) super-
+novae [11, 12] and provided a laboratory to study the properties o f neutrinos [13–18] and
+exotic particles such as axions [19]. The collapse of stellar iron-core in to a neutron star is
+preceded by a high-power pulse of neutrino emission. In general, a b olometric neutrino light
+2curve that includes all the neutrino and antineutrino flavors consis ts of two parts. In the
+first part ( t <5s) the non-thermal neutrino emission is dominated by the electron ne utrinos
+νeproduced by the non-thermal neutronization. For t≈0.5s, the core is transparent to
+νeemitted due to electrons captures by nuclei and free protons. By this time, the mean
+individual νeenergy becomes 10 −20MeV. Thus, the non-thermal neutrinos carry away
+approximately a small fraction of the total available energy Eνtot= (3−5)×1053erg. Nearly
+90% of this energy is emitted in the regime of thermal emission once th e innermost region
+of the core becomes opaque to all the neutrino flavors, which get d ecoupled from the stellar
+mass at a surface so-called neutrinosphere.
+Therefore, one of the crucial parameters which strongly affect t he stellar evolution is the
+cooling rate. During their lifetime, stars can emit energy in the form o f electromagnetic
+or gravitational waves, and a flux of neutrinos. However, in late st ages a star mainly
+looses energy through neutrinos, and this is quite independent of t he star mass. In fact,
+white dwarfs and supernovae, which are the evolution end points of stars formed from very
+different masses, have cooling rates largely dominated by neutrino p roduction. An accurate
+determination ofneutrino emission rates istherefore mandatoryin order toperforma careful
+study of the final branches of star evolutionary tracks. In part icular, a change in the cooling
+ratesat thevery last stageof massive star evolution could percep tibly affect theevolutionary
+time scale and the iron core configuration at the onset of the super nova explosion whose
+triggering mechanism still waits a full theoretical understanding [20 ].
+The energy loss rate due to neutrino emission receives contribution s from both weak nu-
+clear reactions and purely leptonic processes. However, for the la rge values of density and
+temperaturewhich characterize thefinal stageofstellar evolutio n, thelatterarelargelydom-
+inant, and are mainly produced by four possible interaction mechanis ms:e+e−→ν¯ν(pair
+annihilation), γe±→e±ν¯ν(ν-photoproduction), γ∗→ν¯ν(plasmon decay), e±Z→e±Zν¯ν
+(bremsstrahlung on nuclei). These mechanisms play an important ro le in astrophysics and
+cosmology and have been considered by many authors in various the ories of weak interac-
+tions.
+Actually these processes are the dominant cause of the energy los s rate in different re-
+gions in a density-temperature plane. For very large core tempera ture,T>∼109oK, and not
+excessively high values of density, pair annihilations are most efficient , whileνphotopro-
+duction gives the leading contribution for 108oK<∼T<∼109oKand relatively low density,
+3ρ<∼105g cm−3. These are the typical ranges for very massive stars in their late e volution.
+Our main objetive in this paper is to provide suitable expressions for t he emissivity of
+pair production of neutrinos via the process e+e−→ν¯νin the context of a Left-Right
+Symmetric Model (LRSM) [21–28] and in a form which can be easily incor porated into
+realistic supernova models to evaluate the energy lost in the form of neutrinos.
+The amplitude of transition Min the context of the LRSM can be written as a function
+of the mixing angle φbetween W3
+L,W3
+RandBbosons of the model to give the physical
+Z1andZ2and the photon, being φthe only extra parameter besides the standard model
+parameters. For which in this paper we choice the Left-Right symme tric model [26–28] to
+calculate the emissivity of neutrinos in supernova.
+This paper is organized as follows: In Sect. II we present the calcula tion of the transition
+amplitude of the process e+e−→ν¯νin the context of a left-right symmetric model. In Sect.
+III we calculate the emissivity and, finally, we give our results and con clusions in Sec. IV.
+II. CROSS SECTION OF THE PROCESS e++e−→ν+ ¯ν
+In this section we obtain the cross section for the Zexchange process
+e+(p1)+e−(p2)→¯ν(k1,λ1)+ν(k2,λ2), (1)
+i.e., in the limit of a four-fermion electroweak interaction no electroma gnetic radiative cor-
+rections. Here the kiandpiare the particle momenta and λis the helicity of the neutrino.
+We recall that within the context of the standard theory, a neutr ino interaction eigenstate
+(νLorνR) is a superposition of helicity ( λ) eigenstates ν±, whereλ=σ·p=±1. For
+a relativistic particle, this translates into the statement that a νLis predominantly in the
+λ=−1 state and a νRis predominantly in the λ= +1 state, with small admixtures of the
+opposite helicity of the order m/Eν.
+The amplitude of transition for the process (1) is given by
+M=g2
+Z
+2M2
+Z/bracketleftbigg
+¯u(k2,λ2)γµ1
+2(agν
+V−bgν
+Aγ5)v(k1,λ1)/bracketrightbigg/bracketleftbigg
+¯v(p1)γµ1
+2(age
+V−bge
+Aγ5)u(p2)/bracketrightbigg
+,(2)
+where the constant aandbdepend only on the parameters of the LRSM model [26–28]
+4a= cosφ−sinφ√cos2θWandb= cosφ+/radicalBig
+cos2θWsinφ, (3)
+whereφis the mixing parameter of the LRSM [26–28], uandvare the usual Dirac spinors,
+and the electron and positron helicity indexes have been suppresse d since they will be aver-
+aged over. We then write
+1
+2×1
+2|M|2=G2
+F
+8NµνEµν, (4)
+where
+Nµν=1
+4Tr[(k/2+mν)(1+γ5s/2)γµ(a−bγ5)(k/1−mν)(1+γ5s/1)γν(a−bγ5)],(5)
+Eµν=1
+4Tr[(p/2+me)γµ(age
+V−bge
+Aγ5)(p/1−me)γν(age
+V−bge
+Aγ5)]. (6)
+Heres1ands2are the spin four-vectors associated with the antineutrino and ne utrino
+respectively, while mνandmeare the neutrino and electron mass. These spin vectors satisfy
+the Lorentz invariant conditions
+si·si=−1;si·ki= 0; (7)
+and for a relativistic neutrino, the additional constraint
+si/bardblλikifori= 1,2 (8)
+holds, where
+kµ= (Eν,kˆk), (9)
+withˆkbeing a unit vector along the three-momentum of the neutrino.
+We now introduce two four-vectors associated with the neutrino p air
+Kµ
+1=kµ
+1+mνsµ
+1;Kµ
+2=kµ
+2−mνsµ
+2. (10)
+In conjunction with the properties given in Eqs. (7) and (8), these will allow us to write
+the amplitude squared for the process under consideration in a com pact and physically
+5revealing form. As a first step towards this, we note that the spin v ector may be expressed
+as
+sµ=λ
+mν(k,Eνˆk). (11)
+Using this and Eq. (10), we write
+K1=η1(1,ˆk);K2=η2(1,−ˆk); (12)
+with
+η1=Eν+(E2
+ν−m2
+ν)1/2;η2=Eν−(E2
+ν−m2
+ν)1/2. (13)
+Note that for mν<< E νwe have
+η1≈2Eν;η2≈m2
+ν
+2Eν. (14)
+We now evaluate the traces given in Eqs. (5) and (6) and the contra ctionNµνEµνis
+given by
+NµνEµν= 8 (a2+b2)/braceleftBigg/bracketleftBigg
+a2(ge
+V)2+b2(ge
+A)+4a2b2
+(a2+b2)ge
+Vge
+A/bracketrightBigg
+(p1·K1)(p2·K2)
++/bracketleftBigg
+a2(ge
+V)2+b2(ge
+A)−4a2b2
+(a2+b2)ge
+Vge
+A/bracketrightBigg
+(p1·K2)(p2·K1)
++/bracketleftbigg
+a2(ge
+V)2−b2(ge
+A)/bracketrightbigg
+m2
+e(K1·K2)/bracerightbigg
+,
+(15)
+wherege
+V=−1
+2+ 2sin2θWandge
+A=−1
+2. From this expression and Eqs. (12) and (14)
+above, we see that the amplitude of transition vanishes for massles s neutrino, as expected.
+Furthermore, Eq. (15) is taken as the usual weak pair production amplitude with the
+replacement Ki→ki.
+From Eqs. (4) and (15) the explicit form for the squared transition amplitude is
+1
+2×1
+2|M|2=G2
+F(a2+b2)/braceleftBigg/bracketleftBigg
+a2(ge
+V)2+b2(ge
+A)+4a2b2
+(a2+b2)ge
+Vge
+A/bracketrightBigg
+(p1·k1)(p2·k2)
+6+/bracketleftBigg
+a2(ge
+V)2+b2(ge
+A)−4a2b2
+(a2+b2)ge
+Vge
+A/bracketrightBigg
+(p1·k2)(p2·k1)
++/bracketleftbigg
+a2(ge
+V)2−b2(ge
+A)/bracketrightbigg
+m2
+e(k1·k2)/bracerightbigg
+, (16)
+where the contribution of the parameters of the LRSM is contained in the constants aand
+b. Upon evaluating the limit when the mixing angle φ= 0 and a=b= 1, Eq. (16) is thus
+reduced to the expression to the amplitude given in Refs. [29–35].
+III. CALCULATION OF EMISSIVITY
+In this section, we calculate the emissivity associated with neutrino p air production by
+using Eq. (16). The formula of the emissivity is given by [29–31, 33, 36 ]
+Qν¯ν=4
+(2π)8/integraldisplayd3p1
+2E1d3p2
+2E2d3k1
+2ǫ1d3k2
+2ǫ2(E1+E2)F1F2δ(4)(p1+p2−k1−k2)|M|2,(17)
+where the quantities F1,2= [1 + exp( Ee−±µe−)/T]−1are the Fermi-Dirac distribution
+functions for e±,µeis the chemical potential for the electrons and Tis the temperature (we
+takeKB= 1 for the Boltzmann constant).
+From the transition amplitude Eq. (16) and the formula of the emissiv ity Eq. (17) we
+obtain
+Q[1]
+ν¯ν=G2
+F(a2+b2)/bracketleftBigg
+a2(ge
+V)2+b2(ge
+A)2+4a2b2
+a2+b2ge
+Vge
+A/bracketrightBigg
+I1, (18)
+whereI1is explicitly given by
+I1=4
+(2π)8/integraldisplayd3p1
+2E1d3p2
+2E2d3k1
+2ǫ1d3k2
+2ǫ2(E1+E2)F1F2δ(4)(p1+p2−k1−k2)(p1·k1)(p2·k2).(19)
+The intregration can be performed by using the Lenard formula, na mely [36]
+/integraldisplayd3k1
+2ǫ1d3k2
+2ǫ2kα
+1kβ
+2δ(4)(p1+p2−k1−k2) =π
+24/bracketleftBig
+gαβ(p1+p2)2+2(pα
+1+pα
+2)(pβ
+1+pβ
+2)/bracketrightBig
+·Θ/bracketleftBig
+(p1+p2)2/bracketrightBig
+, (20)
+thus Eq. (19) takes the form
+7I1=1
+24(2π)7/integraldisplayd3p1
+E1d3p2
+E2(E1+E2)F1F2/bracketleftBig
+3m2
+e(p1·p2)+2(p1·p2)2+m4
+e/bracketrightBig
+.(21)
+In a similar way for the second and third term of Eq. (16), we obtain
+Q[2]
+ν¯ν=G2
+F(a2+b2)/bracketleftBigg
+a2(ge
+V)2+b2(ge
+A)2−4a2b2
+a2+b2ge
+Vge
+A/bracketrightBigg
+I2, (22)
+Q[3]
+ν¯ν=G2
+F(a2+b2)/bracketleftBig
+a2(ge
+V)2−b2(ge
+A)2/bracketrightBig
+m2
+eI3, (23)
+where
+I2=I1=1
+24(2π)7/integraldisplayd3p1
+E1d3p2
+E2(E1+E2)F1F2/bracketleftBig
+3m2
+e(p1·p2)+2(p1·p2)2+m4
+e/bracketrightBig
+,(24)
+I3=1
+(2π)7/integraldisplayd3p1
+2E1d3p2
+2E2(E1+E2)F1F2/bracketleftBig
+m2
+e+(p1·p2)/bracketrightBig
+. (25)
+The calculation of the emissivity can be more easily performed by expr essing the latest
+integrals in terms of the Fermi integral, which is defined as [33]
+G±
+s=1
+α3+2s/integraldisplay∞
+αx2s+1√
+x2−α2
+1+ex±βdx, (26)
+whereα=me
+KT,β=µe
+KTandx=E
+KT.
+With these definitions, Eq. (26) becomes
+G±
+s=1
+m3+2se/integraldisplay∞
+me/KTE2s+1/radicalBig
+E2−m2
+e
+1+e(E±µe)/KTdE, (27)
+therefore
+/integraldisplay∞
+me/KTEn/radicalBig
+E2−m2e
+1+e(E±µe)/KTdE=mn+2
+eG±
+n−1
+2, (28)
+/integraldisplay∞
+me/KTEn+1/radicalBig
+E2−m2e
+1+e(E±µe)/KTdE=mn+3
+eG±
+n
+2, (29)
+/integraldisplay∞
+me/KTEn+2/radicalBig
+E2−m2e
+1+e(E±µe)/KTdE=mn+4
+eG±
+n+1
+2. (30)
+From (28), (29) and (30), Eqs. (24) and (25) are expressed as
+8Inm
+1=mn+m+8
+e
+6(2π)5/bracketleftbigg
+3G−
+n
+2G+
+m
+2+2G−
+n+1
+2G+
+m+1
+2+G−
+n−1
+2G+
+m−1
+2
++4
+9/parenleftbigg
+G−
+n+1
+2−G−
+n−1
+2/parenrightbigg/parenleftbigg
+G+
+m+1
+2−G+
+m−1
+2/parenrightbigg/bracketrightbigg
+, (31)
+Inm
+3=mn+m+6
+e
+(2π)5/bracketleftbigg
+G−
+n−1
+2G+
+m−1
+2+G−
+n
+2G+
+m
+2/bracketrightbigg
+. (32)
+Therefore, Eqs. (18), (22) and (23) are explicitly
+Q[1]
+ν¯ν=G2
+F(a2+b2)/bracketleftBigg
+a2(ge
+V)2+b2(ge
+A)2+4a2b2
+a2+b2ge
+Vge
+A/bracketrightBigg/bracketleftBig
+I10
+1+I01
+1/bracketrightBig
+,(33)
+Q[2]
+ν¯ν=G2
+F(a2+b2)/bracketleftBigg
+a2(ge
+V)2+b2(ge
+A)2−4a2b2
+a2+b2ge
+Vge
+A/bracketrightBigg/bracketleftBig
+I10
+2+I01
+2/bracketrightBig
+,(34)
+Q[3]
+ν¯ν=G2
+F(a2+b2)/bracketleftBig
+a2(ge
+V)2−b2(ge
+A)2/bracketrightBig
+m2
+e/bracketleftBig
+I10
+3+I01
+3/bracketrightBig
+. (35)
+Finally, the expression for the emissivity of neutrino pair production via the process
+e+e−→ν¯νin the context of a left-right symmetric model is given by
+QLRSM
+ν¯ν(φ,β) =Q[1]
+ν¯ν(φ,β)+Q[2]
+ν¯ν(φ,β)+Q[3]
+ν¯ν(φ,β), (36)
+where the dependence of the φmixing parameter of the LRSM is contained in the constants
+aandb, while the dependence of the βdegeneration parameter is contained in the Fermi
+integrals G±
+s.
+IV. RESULTS AND CONCLUSIONS
+Thenumerical result ontheemissivity ofneutrinopairproductionvia theprocess e+e−→
+ν¯νas a function of the mixing angle φand the generation parameter βare present in
+this section. For our analysis we consider the following data: the Fer mi constant GF=
+1.166×10−5GeV−2, angle of Weinberg sin2θW= 0.223 and the electron mass me= 0.51
+MeV, thereby obtaining the emissivity of the neutrinos QLRSM
+ν¯ν=QLRSM
+ν¯ν(φ,β).
+For the mixing angle φof the left-right symmetric model, we use the reported data of A.
+Guti´ errez-Rodr´ ıguez, et al.[37]:
+−1.6×10−3≤φ≤1.1×10−3, (37)
+9with a 90% C.L. Other limits on the mixing angle φreported in the literature are given in
+Refs. [26, 28, 38, 39].
+In Figure 1 we show the emissivity as a function of the degeneration p arameter βfor
+several values representative of the mixing angle φ=−0.0016,0,0.0011. We observe that
+emissivity decreases when βincreases, which is due to the reduction in the number of
+positronsavailablenecessary to causethecollision. Alsowe seethat theemissivity isaffected
+by theφparameter. There are other effects which may change the emissivit y, for example,
+the radiative corrections at one-loop level.
+Toanalyzetheeffectsofthe φparameteroftheleft-rightsymmetric model ontheemissiv-
+ityQLRSM
+ν¯ν(φ,β) of the neutrinos, in Fig. 2 we show the ratioQLRSM
+ν¯ν(φ,β)
+QSM
+ν¯ν(β)(ratio of emissivity
+calculated in the LRSM to the emissivity calculated in the SM [34]), as a fu nction of the φ
+parameter for several values representative of degeneration p arameter β= 0,5,12.
+According to collapse theories, the full energy loss in a stellar collaps e isLν¯ν=Qν¯ν·Vstar·
+tcol≈1053erg, whereVstaris the volume of a star and tcolthe collapse time. Therefore, if we
+assume that tcol∼1−10s, the stellar collapse temperature Tcol∼(1.1−1.4)×1011oKwill
+be obtained. On the contrary, if we know the exact stellar collapse t emperature, the collapse
+time can be obtained. For instance, the value Tcol∼ ×1012oKcorresponds to tcol∼10−7s.
+In summary, we have analyzed the effects of the mixing angle φof a left-right symmetric
+model on the emissivity of the neutrinos via the process e+e−→ν¯ν. We find that the
+emissivity is dependent of the mixing angle φof the model in the allowed range for this
+parameter. As expected, in the limit of vanishing φwe recover the expression for the
+emissivity QSM
+ν¯ν(β) for the SM previously obtained in the literature [29–33]. In addition , the
+analytical and numerical results for the emissivity have never been reported in the literature
+before and could be of relevance for the scientific community.
+Acknowledgments
+This work was supported by CONACyT, SNI and PROMEP (M´ exico).
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+12FIG. 1: The emissivity for e+e−→ν¯νas a function of degeneration parameter βforφ=
+−0.0016,0,0.0011.
+FIG. 2: Plot of ratioQν¯ν(φ,β)
+Qν¯ν(β), as a function of mixing angle φforβ= 0,5,12.
+13
\ No newline at end of file
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+arXiv:1001.0401v4  [math.PR]  9 Jan 2012The Annals of Applied Probability
+2011, Vol. 21, No. 5, 1933–1964
+DOI:10.1214/10-AAP744
+c/circlecopyrtInstitute of Mathematical Statistics , 2011
+NUMERICAL SIMULATION OF BSDES WITH DRIVERS OF
+QUADRATIC GROWTH
+By Adrien Richou
+Universit´ e Rennes 1
+This article deals with the numerical resolution of Markovi an
+backward stochastic differential equations (BSDEs) with dr ivers of
+quadratic growth with respect to zand bounded terminal conditions.
+We first show some bound estimates on the process Zand we specify
+the Zhang’s path regularity theorem. Then we give a new time d is-
+cretization scheme with a nonuniform time net for such BSDEs and
+we obtain an explicit convergence rate for this scheme.
+1. Introduction. Since the early nineties, there has been an increasing
+interest for backward stochastic differential equations (BS DEs). Theseequa-
+tions have a wide range of applications in stochastic contro l, in finance or
+in partial differential equation theory. A particular class o f BSDE has been
+studied for a few years: BSDEs with drivers of quadratic grow th with re-
+spect to thevariable z. Thisclass arises, for example, inthecontext of utility
+optimization problems with exponential utility functions or alternatively in
+questions related to risk minimization for the entropic ris k measure (see,
+e.g., [13]). Many papers deal with existence and uniqueness of soluti on for
+such BSDEs; we refer the reader to [ 17,18] when the terminal condition
+is bounded and [ 3,4,9] for the unbounded case. Our concern is rather re-
+lated to the simulation of BSDEs and more precisely time disc retization of
+BSDEs coupled with a forward stochastic differential equatio n (SDE). Ac-
+tually, the design of efficient algorithms which are able to so lve BSDEs in
+any reasonable dimension has been intensely studied since t he first work of
+Chevance [ 6] (see, e.g., [ 1,11,19]). But in all these works, the driver of the
+BSDE is a Lipschitz function with respect to zand this assumption plays a
+key role in their proofs. In a recent paper, Cheridito and Sta dje [5] studied
+approximation of BSDEs by backward stochastic difference equ ations which
+are based on random walks instead of Brownian motions. They o btain a
+Received February 2010; revised September 2010.
+AMS 2000 subject classifications. Primary 60H35; secondary 65C30, 60H10.
+Key words and phrases. BSDEs,driverofquadraticgrowth, timediscretization sch eme.
+This is an electronic reprint of the original article published by the
+Institute of Mathematical Statistics inThe Annals of Applied Probability ,
+2011, Vol. 21, No. 5, 1933–1964 . This reprint differs from the original in
+pagination and typographic detail.
+12 A. RICHOU
+convergence result when the driver has a subquadratic growt h with respect
+tozand they give an example where this approximation does not co nverge
+when the driver has a quadratic growth. To the best of our know ledge, the
+only work wherethe time approximation of aBSDE with aquadra tic growth
+with respect to zis studied is the one of Imkeller and Reis [ 14]. Notice that,
+when the driver has a specific form (roughly speaking, the dri ver is a sum
+of a quadratic term z/mapsto→C|z|2and a function that has a linear growth with
+respect to z), it is possible to get around the problem by using an exponen -
+tial transformation method (see [ 15]) or by using results on fully coupled
+forward–backward differential equations (see [ 7]).
+To explain the ideas of this paper, let us introduce ( X,Y,Z) the solution
+to the forward–backward system
+Xt=x+/integraldisplayt
+0b(s,Xs)ds+/integraldisplayt
+0σ(s)dWs,
+Yt=g(XT)+/integraldisplayT
+tf(s,Xs,Ys,Zs)ds−/integraldisplayT
+tZsdWs,
+wheregis bounded, fis locally Lipschitz and has a quadratic growth with
+respect to z. A well-known result is that when gis a Lipschitz function
+with Lipschitz constant Kg, then the process Zis bounded by C(Kg+1)
+(see Theorem 3.1). So, in this case, the driver of the BSDE is a Lipschitz
+function with respect to zand we are able to use standard results about
+discretization of BSDEs. Because of the above observation, this paper will
+focus on the case that the terminal function gis not Lipschitz. To obtain
+our main results, we will assume that gis anα-H¨ older function but it is also
+possible to adapt our methods when gis notα-H¨ older; for example, Remark
+4.13deals with the case of an indicator function of a smooth domai n. Let
+us notice that the time approximation of BSDEs with an irregu lar terminal
+function has already been studied by Gobet and Makhlouf [ 12] when the
+generator is a Lipschitz function with respect to z.
+In light of previous observation, a simple idea is to do an app roximation
+of (Y,Z) by the solution ( YN,ZN) to the BSDE
+YN
+t=gN(XT)+/integraldisplayT
+tf(s,Xs,YN
+s,ZN
+s)ds−/integraldisplayT
+tZN
+sdWs,
+wheregNis a Lipschitz approximation of g. Thanks to bounded mean os-
+cillation martingale (BMO martingale in the sequel) tools, we have an error
+estimate for this approximation (see, e.g., [ 2,14] or Proposition 4.2). For ex-
+ample, if gisα-H¨ older, we are able to obtain the error bound CK−α/(1−α)
+gN
+(see Proposition 4.11). Moreover, we can have an error estimate for the time
+discretization of the approximated BSDE thanks to any numer ical scheme
+for BSDEs with Lipschitz driver. But this error estimate dep ends onKgN;NUMERICAL SIMULATION OF QUADRATIC BSDES 3
+roughly speaking, this error is CeCK2
+gNn−1withnthe number of discretiza-
+tion times. The exponential term results from the use of Gron wall’s inequal-
+ity. Finally, when gisα-H¨ older and KgN=N, the global error bound is
+C/parenleftbigg1
+Nα/(1−α)+eCN2
+n/parenrightbigg
+. (1)
+So, when Nincreases, n−1will have to become small very quickly and the
+speed of convergence turns out to be bad; if we take N=(C
+εlogn)1/2with
+0< ε <1, then the global error bound becomes Cε(logn)−α/(2(1−α)). The
+same drawback appears in the work of Imkeller and Reis [ 14]. Indeed, their
+idea is to do an approximation of ( Y,Z) by the solution ( YN,ZN) to the
+truncated BSDE
+YN
+t=g(XT)+/integraldisplayT
+tf(s,Xs,YN
+s,hN(ZN
+s))ds−/integraldisplayT
+tZN
+sdWs,
+wherehN:R1×d→R1×dis a smooth modification of the projection on the
+open Euclidean ball of radius Nabout 0. Thanks to several statements
+concerning the path regularity and stochastic smoothness o f the solution
+processes, the authors show that for any β≥1, the approximation error is
+lower than CβN−β. So they obtain the global error bound
+Cβ/parenleftbigg1
+Nβ+eCN2
+n/parenrightbigg
+(2)
+and, consequently, the speed of convergence also turns out t o be bad; if we
+takeN=(C
+εlogn)1/2with 0<ε<1, then the global error bound becomes
+Cβ,ε(logn)−β/2.
+Another idea is to use an estimate of Zthat does not depend on Kg. So
+we extend a result of [ 8] which shows
+|Zt|≤M1+M2
+(T−t)1/2,0≤t<T. (3)
+Let us notice that this type of estimation is well known in the case of drivers
+with linear growth as a consequence of the Bismut–Elworthy f ormula (see,
+e.g., [10]). But in our case, we do not need to suppose that σis invertible.
+Then, thanks to this estimation, we know that when t<T,f(t,·,·,·) is a
+Lipschitz function with respect to zand the Lipschitz constant depends
+ont. So we are able to modify the classical uniform time net to obt ain a
+convergence speed for a modified time discretization scheme for our BSDE;
+the idea is to put more discretization points near the final ti meTthan near
+0. Roughly speaking, our discretization grid is equal to
+tk:=T/parenleftbigg
+1−/parenleftbiggε
+T/parenrightbiggk/n/parenrightbigg
+,0≤k≤n,4 A. RICHOU
+withεa parameter. But due to technical reasons we need to apply thi s
+modified time discretization scheme to the approximated BSD E
+YN,ε
+t=gN(XT)+/integraldisplayT
+tfε(s,Xs,YN,ε
+s,ZN,ε
+s)ds−/integraldisplayT
+tZN,ε
+sdWs
+with
+fε(s,x,y,z):=1s≤T−εf(s,x,y,z)+1s>T−εf(s,x,y,0).
+Thanks to the estimate ( 3), we obtain a speed convergence for the time
+discretization scheme of this approximated BSDE (see Theor em4.9). More-
+over, BMO tools give us again an estimate of the approximatio n error
+(see Proposition 4.2). Finally, if we suppose that gisα-H¨ older, we prove
+that we can choose properly Nandεto obtain the global error estimate
+Cn−2α/((2−α)(2+K)−2+2α)(see Theorem 4.14) whereK >0 depends on con-
+stantM2defined in equation ( 3) and constants related to f. Let us notice
+that such a speed of convergence where constants related to f,g,bandσ
+appear in the power of nis unusual. Even if we have an error far better than
+(1) or (2), this result is not very interesting in practice because th e speed of
+convergence strongly depends on K. But, when bis bounded, we prove that
+we can take M2as small as we want in ( 3). Finally, we obtain a global error
+estimate lower than Cηn−(α−η)for allη>0 (see Theorem 4.17).
+To conclude, it could be interesting to do some comparisons b etween our
+work and the article of Gobet and Makhlouf [ 12]. We already explain that
+this paperstudies thetime approximation of Lipschitz BSDE s with irregular
+terminal functions. These authors show that the error of app roximation is
+lower than Cηn−αwhengis anα-H¨ older function and the discretization
+grid is uniform. So, our better speed of convergence is very c lose to their
+result. Nevertheless, they also show that it is possible to o btain the classical
+speed of convergence, that is to say Cn−1, when we use the nonuniform grid
+given by
+tk:=T−T/parenleftbigg
+1−k
+n/parenrightbigg1/β
+,0≤k≤n,
+withβ <α. It is interesting to notice that we both use nonuniform time dis-
+cretization points but their grid is different than our grid; t he accumulation
+speed of discretization points near the terminal time Tis not the same; it
+is faster in our case.
+The paper is organized as follows. In the introductory Secti on2we recall
+some of the well-known results concerning SDEs and BSDEs. In Section
+3we establish some estimates concerning the process Z; we show a first
+uniform bound for Z, then a time dependent bound and finally we specify
+the classical path regularity theorem. In Section 4we define a modified time
+discretization scheme for BSDEs with a nonuniform time net a nd we obtain
+an explicit error bound.NUMERICAL SIMULATION OF QUADRATIC BSDES 5
+2. Preliminaries.
+2.1.Notation. Throughoutthispaper,( Wt)t≥0willdenotea d-dimensional
+Brownian motion, defined on a probability space (Ω ,F,P). Fort≥0, letFt
+denote the σ-algebra σ(Ws;0≤s≤t), augmented with the P-null sets of
+F. The Euclidean norm on Rdwill be denoted by |·|. The operator norm
+induced by |·|on the space of linear operator is also denoted by |·|. For
+p≥2,m∈N, we denote further:
+(1)Sp(Rm) orSpwhen no confusion is possible, the space of all adapted
+processes ( Yt)t∈[0,T]with values in Rmnormed by
+/ba∇⌈blY/ba∇⌈blSp=E/bracketleftBig/parenleftBig
+sup
+t∈[0,T]|Yt|/parenrightBigp/bracketrightBig1/p
+;
+S∞(Rm) orS∞, the space of bounded measurable processes;
+(2)Mp(Rm) orMp, the space of all progressively measurable processes
+(Zt)t∈[0,T]with values in Rmnormed by
+/ba∇⌈blZ/ba∇⌈blMp=E/bracketleftbigg/parenleftbigg/integraldisplayT
+0|Zs|2ds/parenrightbiggp/2/bracketrightbigg1/p
+.
+In the following we keep the same notation Cfor all finite, nonnegative
+constants that appear in our computations; they may depend o n known
+parameters deriving from assumptions and on Tbut not on any of the
+approximation and discretization parameters. In the same s pirit, we keep
+the same notation ηfor all finite, positive constants that we can take as
+small as we want independently of the approximation and disc retization
+parameters.
+2.2.Some results on BMO martingales. In our work, the space of BMO
+martingales play akeyrolefortheaprioriestimates needed inouranalysisof
+BSDEs.Wereferthereaderto[ 16]forthetheoryofBMOmartingalesandwe
+just recall the properties that we will use in the sequel. Let Φt=/integraltextt
+0φsdWs,
+t∈[0,T],bearealsquareintegrablemartingalewithrespecttothe Brownian
+filtration. Then Φ is a BMO martingale if
+/ba∇⌈blΦ/ba∇⌈blBMO= sup
+τ∈[0,T]E[/an}b∇ack⌉tl⌉{tΦ/an}b∇ack⌉t∇i}htT−/an}b∇ack⌉tl⌉{tΦ/an}b∇ack⌉t∇i}htτ|Fτ]1/2= sup
+τ∈[0,T]E/bracketleftbigg/integraldisplayT
+τφ2
+sds|Fτ/bracketrightbigg1/2
+<+∞,
+where the supremum is taken over all stopping times in [0 ,T];/an}b∇ack⌉tl⌉{tΦ/an}b∇ack⌉t∇i}htdenotes
+the quadratic variation of Φ. In our case, the very important feature of BMO
+martingales is the following lemma.
+Lemma 2.1. LetΦbe a BMO martingale. Then we have:6 A. RICHOU
+(1)The stochastic exponential
+E(Φ)t=Et=exp/parenleftbigg/integraldisplayt
+0φsdWs−1
+2/integraldisplayt
+0|φs|2ds/parenrightbigg
+,0≤t≤T,
+is a uniformly integrable martingale.
+(2)Thanks to the reverse H¨ older inequality, there exists p>1such that
+ET∈Lp. The maximal pwith this property can be expressed in terms of the
+BMO norm of Φ.
+(3)∀n∈N∗,E[(/integraltextT
+0|φs|2ds)n]≤n!/ba∇⌈blΦ/ba∇⌈bl2n
+BMO.
+2.3.The backward–forward system. Given functions b,σ,gandf, for
+x∈Rdwe will deal withthesolution ( X,Y,Z) to thefollowing system of (de-
+coupled) backward–forward stochastic differential equatio ns: fort∈[0,T],
+Xt=x+/integraldisplayt
+0b(s,Xs)ds+/integraldisplayt
+0σ(s)dWs, (4)
+Yt=g(XT)+/integraldisplayT
+tf(s,Xs,Ys,Zs)ds−/integraldisplayT
+tZsdWs. (5)
+For the functions that appear in the above system of equation s we give some
+general assumptions.
+(HX0).b:[0,T]×Rd→Rd,σ:[0,T]→Rd×dare measurable functions.
+There exist four positive constants Mb,Kb,MσandKσsuch that ∀t,t′∈
+[0,T],∀x,x′∈Rd,
+|b(t,x)|≤Mb(1+|x|),
+|b(t,x)−b(t′,x′)|≤Kb(|x−x′|+|t−t′|1/2),
+|σ(t)|≤Mσ,
+|σ(t)−σ(t′)|≤Kσ|t−t′|.
+(HY0).f:[0,T]×Rd×R×R1×d→R,g:Rd→Rare measurable func-
+tions. There exist five positive constants Mf,Kf,x,Kf,y,Kf,zandMgsuch
+that∀t∈[0,T],∀x,x′∈Rd,∀y,y′∈R,∀z,z′∈R1×d,
+|f(t,x,y,z)|≤Mf(1+|y|+|z|2),
+|f(t,x,y,z)−f(t,x′,y′,z′)|≤Kf,x|x−x′|+Kf,y|y−y′|
++(Kf,z+Lf,z(|z|+|z′|))|z−z′|,
+|g(x)|≤Mg.
+We next recall some results on BSDEs with quadratic growth. F or their
+original version and their proof we refer to [ 2,17] and [14].NUMERICAL SIMULATION OF QUADRATIC BSDES 7
+Theorem 2.2. Under(HX0),(HY0), the system ( 4)–(5) has a unique
+solution (X,Y,Z)∈S2×S∞×M2. The martingale Z∗Wbelongs to the
+space of BMO martingales and /ba∇⌈blZ∗W/ba∇⌈blBMOonly depends on T,Mgand
+Mf. Moreover, there exists r>1such that E(Z∗W)∈Lr.
+3. Some useful estimates of Z.
+3.1.A first bound for Z.
+Theorem 3.1. Suppose that (HX0),(HY0)hold and that gis Lipschitz
+with Lipschitz constant Kg. Then, there exists a version of Zsuch that,
+∀t∈[0,T],
+|Zt|≤e(2Kb+Kf,y)TMσ(Kg+TKf,x).
+Proof. First, we suppose that b,gandfare differentiable with re-
+spect to x,yandz. Then (X,Y,Z) is differentiable with respect to xand
+(∇X,∇Y,∇Z) is the solution of
+∇Xt=Id+/integraldisplayt
+0∇b(s,Xs)∇Xsds, (6)
+∇Yt=∇g(XT)∇XT−/integraldisplayT
+t∇ZsdWs
++/integraldisplayT
+t∇xf(s,Xs,Ys,Zs)∇Xs+∇yf(s,Xs,Ys,Zs)∇Ysds (7)
++/integraldisplayT
+t∇zf(s,Xs,Ys,Zs)∇Zsds,
+where∇Xt=(∂Xi
+t/∂xj)1≤i,j≤d,∇Yt=t(∂Yt/∂xj)1≤j≤d∈R1×d,∇Zt=(∂Zi
+t/
+∂xj)1≤i,j≤dand/integraltextT
+t∇ZsdWsmeans
+/summationdisplay
+1≤i≤d/integraldisplayT
+t(∇Zs)idWi
+s
+with (∇Z)idenoting the ith line of the d×dmatrix process ∇Z. Thanks
+to usual transformations on the BSDE we obtain
+e/integraltextt
+0∇yf(s,Xs,Ys,Zs)ds∇Yt
+=e/integraltextT
+0∇yf(s,Xs,Ys,Zs)ds∇g(XT)∇XT
+−/integraldisplayT
+te/integraltexts
+0∇yf(u,Xu,Yu,Zu)du∇Zsd˜Ws
++/integraldisplayT
+te/integraltexts
+0∇yf(u,Xu,Yu,Zu)du∇xf(s,Xs,Ys,Zs)∇Xsds8 A. RICHOU
+withd˜Ws=dWs−∇zf(s,Xs,Ys,Zs)ds. We have
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay·
+0∇zf(s,Xs,Ys,Zs)dWs/vextenddouble/vextenddouble/vextenddouble/vextenddouble2
+BMO
+= sup
+τ∈[0,T]E/bracketleftbigg/integraldisplayT
+τ|∇zf(s,Xs,Ys,Zs)|2ds|Fτ/bracketrightbigg
+≤C/parenleftbigg
+1+ sup
+τ∈[0,T]E/bracketleftbigg/integraldisplayT
+τ|Zs|2ds|Fτ/bracketrightbigg/parenrightbigg
+=C(1+/ba∇⌈blZ∗W/ba∇⌈bl2
+BMO).
+SinceZ∗Wbelongs to the space of BMO martingales,
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay·
+0∇zf(s,Xs,Ys,Zs)dWs/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+BMO<+∞.
+Lemma2.1gives us that E(/integraltext·
+0∇zf(s,Xs,Ys,Zs)dWs)tis a uniformly inte-
+grable martingale so we are able to apply Girsanov’s theorem : there exists
+a probability Qunder which ( ˜W)t∈[0,T]is a Brownian motion. Then,
+e/integraltextt
+0∇yf(s,Xs,Ys,Zs)ds∇Yt
+=EQ/bracketleftbigg
+e/integraltextT
+0∇yf(s,Xs,Ys,Zs)ds∇g(XT)∇XT
++/integraldisplayT
+te/integraltexts
+0∇yf(u,Xu,Yu,Zu)du∇xf(s,Xs,Ys,Zs)∇Xsds|Ft/bracketrightbigg
+and
+|∇Yt|≤e(Kb+Kf,y)T(Kg+TKf,x), (8)
+because|∇Xt|≤eKbT. Moreover, thanks to the Malliavin calculus, it is clas-
+sical toshowthat aversion of ( Zt)t∈[0,T]is given by ( ∇Yt(∇Xt)−1σ(t))t∈[0,T].
+So we obtain
+|Zt|≤eKbTMσ|∇Yt|≤e(2Kb+Kf,y)TMσ(Kg+TKf,x) a.s. ,
+because|∇X−1
+t|≤eKbT.
+Whenb,gandfare not differentiable, we can also prove the result by a
+standard approximation and stability results for BSDEs wit h linear growth.
+/square
+Remark 3.2. Thanks to Theorem 3.1, the generator fbecomes a Lip-
+schitz function with respect to z, so we are able to use standard results
+about time discretization of BSDEs. In this case, we obtain t hat the errorNUMERICAL SIMULATION OF QUADRATIC BSDES 9
+of approximation is lower than Cn−1withnthe number of discretization
+times (see, e.g., [ 1,11]). Let us notice that, in all studies about discretization
+of BSDEs, we do not care about the constant in the error bound; we only
+consider the asymptotic speed of convergence. But, with a pr actical point of
+view, the constant could play an important role, particular ly for small n. In
+our case, the generator fmay be viewed as Lipschitz in zwith a Lipschitz
+constant Ce(2Kb+Kf,y)T. So, if we apply the standard result, the generic con-
+stant in the rate of convergence will be in the order of CeCe2(2Kb+Kf,y)T. This
+is, of course, not desirable because it blows up when Kb,Kf,yorTincrease.
+We think that it could be interesting to see if we are able to ob serve such a
+phenomena with numerical simulation.
+3.2.A time dependent estimate of Z.We will introduce two alternative
+assumptions.
+(HX1).bis differentiable with respect to xandσis differentiable with
+respect to t. There exists λ∈R+such that ∀η∈Rd
+|tησ(s)[tσ(s)t∇b(s,x)−tσ′(s)]η|≤λ|tησ(s)|2. (9)
+(HX1′).σis invertible and ∀t∈[0,T],|σ(t)−1|≤Mσ−1.
+Example. Assumption (HX1) is verified when, ∀s∈[0,T],∇b(s,·) com-
+mutes with σ(s) and∃A:[0,T]→Rd×dbounded such that σ′(t)=σ(t)A(t).
+Theorem 3.3. Suppose that (HX0),(HY0)hold and that (HX1)or
+(HX1′)holds. Moreover, suppose that gis lower (or upper) semi-continuous.
+Then there exists a version of Zand there exist two constants C,C′∈R+
+that depend only in T,Mg,Mf,Kf,x,Kf,y,Kf,zandLf,zsuch that, ∀t∈
+[0,T[,
+|Zt|≤C+C′(T−t)−1/2.
+Proof. In a first time, we will suppose that (HX1) holds and that f,
+gare differentiable with respect to x,yandz. Then (Y,Z) is differentiable
+with respect to xand (∇Y,∇Z) is the solution of the BSDE
+∇Yt=∇g(XT)∇XT−/integraldisplayT
+t∇ZsdWs
++/integraldisplayT
+t∇xf(s,Xs,Ys,Zs)∇Xs+∇yf(s,Xs,Ys,Zs)∇Ysds
++/integraldisplayT
+t∇zf(s,Xs,Ys,Zs)∇Zsds.10 A. RICHOU
+Thanks to usual transformations we obtain
+e/integraltextt
+0∇yf(s,Xs,Ys,Zs)ds∇Yt
++/integraldisplayt
+0e/integraltexts
+0∇yf(u,Xu,Yu,Zu)du∇xf(s,Xs,Ys,Zs)∇Xsds
+=e/integraltextT
+0∇yf(s,Xs,Ys,Zs)ds∇g(XT)∇XT
++/integraldisplayT
+0e/integraltexts
+0∇yf(u,Xu,Yu,Zu)du∇xf(s,Xs,Ys,Zs)∇Xsds
+−/integraldisplayT
+te/integraltexts
+0∇yf(u,Xu,Yu,Zu)du∇Zsd˜Ws
+withd˜Ws=dWs−∇zf(s,Xs,Ys,Zs)ds. We can rewrite it as
+Ft=FT−/integraldisplayT
+te/integraltexts
+0∇yf(u,Xu,Yu,Zu)du∇Zsd˜Ws (10)
+with
+Ft:=e/integraltextt
+0∇yf(s,Xs,Ys,Zs)ds∇Yt
++/integraldisplayt
+0e/integraltexts
+0∇yf(u,Xu,Yu,Zu)du∇xf(s,Xs,Ys,Zs)∇Xsds.
+Z∗Wbelongs to the space of BMO martingales so we are able to apply
+Girsanov’s theorem: there exists a probability Qunder which ( ˜W)t∈[0,T]is
+a Brownian motion. Thanks to the Malliavin calculus, it is po ssible to show
+that (∇Yt(∇Xt)−1σ(t))t∈[0,T]is a version of Z. Now we define
+αt:=/integraldisplayt
+0e/integraltexts
+0∇yf(u,Xu,Yu,Zu)du∇xf(s,Xs,Ys,Zs)∇Xsds(∇Xt)−1σ(t),
+˜Zt:=Ft(∇Xt)−1σ(t)=e/integraltextt
+0∇yf(s,Xs,Ys,Zs)dsZt+αta.s.,
+˜Ft:=eλtFt(∇Xt)−1.
+Sinced∇Xt=∇b(t,Xt)∇Xtdt,thend(∇Xt)−1=−(∇Xt)−1∇b(t,Xt)dtand
+thanks to Itˆ o’s formula,
+d˜Zt=dFt(∇Xt)−1σ(t)−Ft(∇Xt)−1∇b(t,Xt)σ(t)dt+Ft(∇Xt)−1σ′(t)dt
+and
+d(eλt˜Zt)=˜Ft(λId−∇b(t,Xt))σ(t)dt+˜Ftσ′(t)dt+eλtdFt(∇Xt)−1σ(t).
+Finally,
+d|eλt˜Zt|2=2[λ|˜Ftσ(t)|2−˜Ftσ(t)[tσ(t)t∇b(t,Xt)−tσ′(t)]t˜Ft]dt
++d/an}b∇ack⌉tl⌉{tM/an}b∇ack⌉t∇i}htt+dM∗
+tNUMERICAL SIMULATION OF QUADRATIC BSDES 11
+withMt:=/integraltextt
+0eλsdFs(∇Xs)−1σ(s) andM∗
+taQ-martingale. Thanks to the
+assumption(HX1)weareabletoconcludethat |eλt˜Zt|2isaQ-submartingale.
+Hence,
+EQ/bracketleftbigg/integraldisplayT
+te2λs|˜Zs|2ds|Ft/bracketrightbigg
+≥e2λt|˜Zt|2(T−t)
+≥e2λt|e/integraltextt
+0∇yf(s,Xs,Ys,Zs)dsZt+αt|2(T−t) a.s. ,
+which implies
+|Zt|2(T−t)=e−2λte−2/integraltextt
+0∇yf(s,Xs,Ys,Zs)dse2λt
+×|e/integraltextt
+0∇yf(s,Xs,Ys,Zs)dsZt+αt−αt|2(T−t)
+≤C(e2λt|e/integraltextt
+0∇yf(s,Xs,Ys,Zs)dsZt+αt|2+1)(T−t)
+≤C/parenleftbigg
+EQ/bracketleftbigg/integraldisplayT
+te2λs|˜Zs|2ds|Ft/bracketrightbigg
++(T−t)/parenrightbigg
+a.s.
+withCa constant that only depends on T,Kb,Mσ,Kf,x,Kf,yandλ.
+Moreover, we have, a.s.,
+EQ/bracketleftbigg/integraldisplayT
+te2λs|˜Zs|2ds|Ft/bracketrightbigg
+≤CEQ/bracketleftbigg/integraldisplayT
+t|Zs|2+|αs|2ds|Ft/bracketrightbigg
+≤C(/ba∇⌈blZ/ba∇⌈bl2
+BMO(Q)+(T−t)).
+But/ba∇⌈blZ/ba∇⌈blBMO(Q)does not depend on Kgbecause ( Y,Z) is a solution of the
+following quadratic BSDE:
+Yt=g(XT)+/integraldisplayT
+t(f(s,Xs,Ys,Zs)−Zs∇zf(s,Xs,Ys,Zs))ds
+(11)
+−/integraldisplayT
+tZsd˜Ws.
+Finally,|Zt|≤C(1+(T−t)−1/2) a.s.
+Whenσisinvertible, theinequality ( 9)isverifiedwith λ:=Mσ−1(MσKb+
+Kσ). Since this λdoes not depend on ∇bandσ′, we can prove the result
+whenb(t,·) andσare not differentiable by a standard approximation and
+stability results for BSDEs with linear growth. So, we are al lowed to replace
+assumption (HX1) by (HX1′).
+Whenfis not differentiable and gis only Lipschitz, we can prove the
+result by a standard approximation and stability results fo r linear BSDEs.
+But we notice that our estimation on Zdoes not depend on Kg. This allows12 A. RICHOU
+us to weaken the hypothesis on gfurther; when gis only lower or upper
+semi-continuous the result stays true. The proof is the same as the proof of
+Proposition 4.3 in [ 8]./square
+Remark 3.4. The previous proof gives us a more precise estimation for
+a version of Zwhenfis differentiable with respect to z:∀t∈[0,T],
+|Zt|≤C+C′EQ/bracketleftbigg/integraldisplayT
+t|Zs|2ds|Ft/bracketrightbigg1/2
+(T−t)−1/2.
+Remark 3.5. When assumption (HX1) or (HX1′) is not verified, the
+processZmay blow up before T. Zhang gives an example of such a phe-
+nomenon in dimension 1; we refer the reader to Example 1 in [ 20].
+3.3.Zhang’s path regularity theorem. Let 0 =t0<t1<···< tn=Tbe
+any given partition of [0 ,T] and denote δnthe mesh size of this partition.
+We define a set of random variables
+¯Zti=1
+ti+1−tiE/bracketleftbigg/integraldisplayti+1
+tiZsds|Fti/bracketrightbigg
+∀i∈{0,...,n−1}.
+Then we are able to give a more detailed version at Theorem 3.4 .3 in [21].
+Theorem 3.6. Suppose that (HX0),(HY0)hold and gis a Lipschitz
+function with Lipschitz constant Kg. Then we have
+n−1/summationdisplay
+i=0E/bracketleftbigg/integraldisplayti+1
+ti|Zt−¯Zti|2dt/bracketrightbigg
+≤C(1+K2
+g)δn,
+where C is a positive constant independent of δnandKg.
+Proof. We will follow the proof of Theorem 5.6 in [ 14]; we just need
+to specify how the estimate depends on Kg. First, it is not difficult to show
+that¯Ztiis the best Fti-measurable approximation of ZinM2([ti,ti+1]),
+that is,
+E/bracketleftbigg/integraldisplayti+1
+ti|Zt−¯Zti|2dt/bracketrightbigg
+= inf
+Zi∈L2(Ω,Fti)E/bracketleftbigg/integraldisplayti+1
+ti|Zt−Zi|2dt/bracketrightbigg
+.
+In particular,
+E/bracketleftbigg/integraldisplayti+1
+ti|Zt−¯Zti|2dt/bracketrightbigg
+≤E/bracketleftbigg/integraldisplayti+1
+ti|Zt−Zti|2dt/bracketrightbigg
+.
+In the same spirit as previous proofs, we suppose in a first tim e thatb,g
+andfare differentiable with respect to x,yandz. So,
+Zt−Zti=∇Yt(∇Xt)−1σ(t)−∇Yti(∇Xti)−1σ(ti)=I1+I2+I3a.s.,NUMERICAL SIMULATION OF QUADRATIC BSDES 13
+withI1=∇Yt(∇Xt)−1(σ(t)−σ(ti)),I2=∇Yt((∇Xt)−1−(∇Xti)−1)σ(ti)
+andI3=∇(Yt−Yti)(∇Xti)−1σ(ti). First, thanks to the estimation ( 8) we
+have
+|I1|2≤|∇Yt|2e2KbTK2
+σ|ti+1−ti|2≤C(1+K2
+g)δ2
+n.
+We obtain the same estimation for |I2|because
+|(∇Xt)−1−(∇Xti)−1|≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt
+ti(∇Xs)−1∇b(s,Xs)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤KbeKbT|t−ti|.
+Last,|I3|≤MσeKbT|∇Yt−∇Yti|. So,
+n−1/summationdisplay
+i=0E/bracketleftbigg/integraldisplayti+1
+ti|I3|2dt/bracketrightbigg
+≤Cδnn−1/summationdisplay
+i=0E/bracketleftBig
+ess sup
+t∈[ti,ti+1]|∇Yt−∇Yti|2/bracketrightBig
+.
+By using the BSDE ( 7), (HY0), the estimate on ∇Xsand the estimate ( 8),
+we have
+|∇Yt−∇Yti|2≤C/parenleftbigg/integraldisplayt
+ti(C(1+Kg)+|∇zf(s,Xs,Ys,Zs)||∇Zs|)ds/parenrightbigg2
++C/parenleftbigg/integraldisplayt
+ti∇ZsdWs/parenrightbigg2
+.
+The inequalities of H¨ older and Burkholder–Davis–Gundy gi ve us
+n−1/summationdisplay
+i=0E/bracketleftBig
+ess sup
+t∈[ti,ti+1]|∇Yt−∇Yti|2/bracketrightBig
+≤C(1+K2
+g)+Cn−1/summationdisplay
+i=0E/parenleftbigg/integraldisplayti+1
+ti|∇zf(s,Xs,Ys,Zs)||∇Zs|ds/parenrightbigg2
++CE/parenleftbigg/integraldisplayti+1
+ti|∇Zs|2ds/parenrightbigg
+≤C(1+K2
+g)
++CE/bracketleftbigg/parenleftbigg/integraldisplayT
+0|∇zf(s,Xs,Ys,Zs)||∇Zs|ds/parenrightbigg2
++/integraldisplayT
+0|∇Zs|2ds/bracketrightbigg
+≤C(1+K2
+g)
++CE/bracketleftbigg/parenleftbigg/integraldisplayT
+0(1+|Zs|2)ds/parenrightbigg/parenleftbigg/integraldisplayT
+0|∇Zs|2ds/parenrightbigg
++/integraldisplayT
+0|∇Zs|2ds/bracketrightbigg
+≤C(1+K2
+g)
++C/parenleftbigg
+1+E/bracketleftbigg/parenleftbigg/integraldisplayT
+0|Zs|2ds/parenrightbiggp/bracketrightbigg1/p/parenrightbigg
+E/bracketleftbigg/parenleftbigg/integraldisplayT
+0|∇Zs|2ds/parenrightbiggq/bracketrightbigg1/q14 A. RICHOU
+for allp>1 andq>1 such that 1 /p+1/q=1. But, ( ∇Y,∇Z) is the solution
+of BSDE ( 7) so, from Corollary 9 in [ 2], there exists qthat only depends on
+/ba∇⌈blZ∗W/ba∇⌈blBMOsuch that
+E/bracketleftbigg/parenleftbigg/integraldisplayT
+0|∇Zs|2ds/parenrightbiggq/bracketrightbigg1/q
+≤C(1+K2
+g).
+Moreover, we can apply Lemma 2.1to obtain the estimate
+E/bracketleftbigg/parenleftbigg/integraldisplayT
+0|Zs|2ds/parenrightbiggp/bracketrightbigg1/p
+≤C/ba∇⌈blZ/ba∇⌈bl2
+BMO≤C.
+Finally,
+n−1/summationdisplay
+i=0E/bracketleftbigg/integraldisplayti+1
+ti|I3|2dt/bracketrightbigg
+≤C(1+K2
+g)δn
+and
+n−1/summationdisplay
+i=0E/bracketleftbigg/integraldisplayti+1
+ti|Zt−¯Zti|2dt/bracketrightbigg
+≤n−1/summationdisplay
+i=0E/bracketleftbigg/integraldisplayti+1
+ti(|I1|2+|I2|2+|I3|2)dt/bracketrightbigg
+≤C(1+K2
+g)δn. /square
+4. Convergence of a modified time discretization scheme for t he BSDE.
+4.1.An approximation of the quadratic BSDE. In a first time we will
+approximate our quadratic BSDE ( 5) by another one. We set ε∈]0,T[ and
+N∈N. Let (YN,ε
+t,ZN,ε
+t) be the solution of the BSDE
+YN,ε
+t=gN(XT)+/integraldisplayT
+tfε(s,Xs,YN,ε
+s,ZN,ε
+s)ds−/integraldisplayT
+tZN,ε
+sdWs (12)
+with
+fε(s,x,y,z):=1s≤T−εf(s,x,y,z)+1s>T−εf(s,x,y,0)
+andgNa Lipschitz approximation of gwith Lipschitz constant N.fεverifies
+assumption (HY0) with the same constants as f. SincegNis a Lipschitz
+function, ZN,εhas a bounded version and the BSDE ( 12) is a BSDE with a
+linear growth. Moreover, we can apply Theorem 3.3to obtain the following
+proposition.
+Proposition 4.1. Let us assume that (HX0),(HY0)and(HX1)or
+(HX1′)hold. There exists a version of ZN,εand there exist three constants
+Mz,1,Mz,2,Mz,3∈R+that do not depend on Nandεsuch that, ∀s∈[0,T],
+|ZN,ε
+s|≤/parenleftbigg
+Mz,1+Mz,2
+(T−s)1/2/parenrightbigg
+∧(Mz,3(N+1)).NUMERICAL SIMULATION OF QUADRATIC BSDES 15
+Thanks to BMO tools we have a stability result for quadratic B SDEs (see
+[2] and [14]).
+Proposition 4.2. Let us assume that (HX0)and(HY0)hold. There
+exists a constant Cthat does not depend on Nandεsuch that
+E/bracketleftBig
+sup
+t∈[0,T]|YN,ε
+t−Yt|2/bracketrightBig
++E/bracketleftbigg/integraldisplayT
+0|ZN,ε
+t−Zt|2dt/bracketrightbigg
+≤C(e1(N)+e2(N,ε))
+with
+e1(N):=E[|gN(XT)−g(XT)|2q]1/q,
+e2(N,ε):=E/bracketleftbigg/parenleftbigg/integraldisplayT
+T−ε|f(t,Xt,YN,ε
+t,ZN,ε
+t)−f(t,Xt,YN,ε
+t,0)|dt/parenrightbigg2q/bracketrightbigg1/q
+andqdefined in Theorem 2.2.
+Remark 4.3. The authors of [ 14] obtain this result with q2instead of q.
+Nevertheless, we areable to obtain thegood resultby applyi ngtheestimates
+of [2].
+Then, in a second time, we will approximate our modified backw ard–
+forward system by a discrete-time one. We will slightly modi fy the classical
+discretization by usinga nonequidistant net with 2 n+1 discretization times.
+We define the n+1 first discretization times on [0 ,T−ε] by
+tk=T/parenleftbigg
+1−/parenleftbiggε
+T/parenrightbiggk/n/parenrightbigg
+and we use an equidistant net on [ T−ε,T] for the last ndiscretization times
+tk=T−/parenleftbigg2n−k
+n/parenrightbigg
+ε, n≤k≤2n.
+Wedenotethetimestepby( hk:=tk+1−tk)0≤k≤2n−1.Weconsider( Xn
+tk)0≤k≤2n
+the classical Euler scheme for Xgiven by
+Xn
+0=x,
+(13)
+Xn
+tk+1=Xn
+tk+hkb(tk,Xn
+tk)+σ(tk)(Wtk+1−Wtk)
+for 0≤k≤2n−1. We denote ρs:R1×d→R1×dthe projection on the ball
+B/parenleftbigg
+0,Mz,1+Mz,2
+(T−s)1/2/parenrightbigg16 A. RICHOU
+withMz,1andMz,2givenbyProposition 4.1.Finally,wedenote( YN,ε,n,ZN,ε,n)
+our time approximation of ( YN,ε,ZN,ε). This couple is obtained by a slight
+modification of the classical dynamic programming equation
+YN,ε,n
+t2n=gN(Xn
+t2n),
+ZN,ε,n
+tk=ρtk+1/parenleftbigg1
+hkEtk[YN,ε,n
+tk+1(Wtk+1−Wtk)]/parenrightbigg
+, (14)
+YN,ε,n
+tk=Etk[YN,ε,n
+tk+1]+hkEtk[fε(tk,Xn
+tk,YN,ε,n
+tk+1,ZN,ε,n
+tk)], (15)
+where 0≤k≤2n−1 andEtkstand for the conditional expectation given
+Ftk. Let usnotice that theclassical dynamicprogrammingequat ion doesnot
+useaprojection in ( 14); it is theonly differencewith ourtime approximation
+(see, e.g., [ 11] for the classical case). This projection comes directly fr om the
+estimate of Zin Proposition 4.1. The aim of our work is to study the error
+of discretization
+e(N,ε,n):= sup
+0≤k≤2nE[|YN,ε,n
+tk−Ytk|2]+2n−1/summationdisplay
+k=0E/bracketleftbigg/integraldisplaytk+1
+tk|ZN,ε,n
+tk−Zt|2dt/bracketrightbigg
+.
+It is easy to see that
+e(N,ε,n)≤C(e1(N)+e2(N,ε)+e3(N,ε,n))
+withe1(N) ande2(N,ε) defined in Proposition 4.2and
+e3(N,ε,n):= sup
+0≤k≤2nE[|YN,ε,n
+tk−YN,ε
+tk|2]+2n−1/summationdisplay
+k=0E/bracketleftbigg/integraldisplaytk+1
+tk|ZN,ε,n
+tk−ZN,ε
+t|2dt/bracketrightbigg
+.
+4.2.Study of the time approximation error e3(N,ε,n).We need an extra
+assumption.
+(HY1).There exists a positive constant Kf,tsuch that ∀t,t′∈[0,T],
+∀x∈Rd,∀y∈R,∀z∈R1×d,
+|f(t,x,y,z)−f(t′,x,y,z)|≤Kf,t|t−t′|1/2.
+Moreover, we set ε=Tn−aandN=nb, witha,b∈R+,∗two parameters.
+Before giving our error estimates, we recall two technical l emmas that we
+will prove in Appendices AandB.
+Lemma 4.4. For all constant M >0there exists a constant Cthat de-
+pends only on T,Manda, such that
+2n−1/productdisplay
+i=0(1+Mhi)≤C∀n∈N∗.NUMERICAL SIMULATION OF QUADRATIC BSDES 17
+Lemma 4.5. For all constants M1>0andM2>0there exists a constant
+Cthat depends only on T,M1,M2anda, such that
+n−1/productdisplay
+i=0/parenleftbigg
+1+M1hi+M2hi
+T−ti+1/parenrightbigg
+≤CnaM2.
+First, we give a convergence result for the Euler scheme.
+Proposition 4.6. Assume (HX0)holds. Then there exists a constant
+Cthat does not depend on n, such that
+sup
+0≤k≤2nE[|Xtk−Xn
+tk|2]≤Clnn
+n.
+Proof. We just have to copy the classical proof to obtain, thanks to
+Lemma4.4,
+sup
+0≤k≤2nE[|Xtk−Xn
+tk|2]≤Csup
+0≤i≤2n−1hi=Ch0.
+But
+h0=T(1−n−a/n)≤Clnn
+n,
+because (1 −n−a/n)∼aTlnn
+nwhenn→+∞, so the proof is complete. /square
+Now, let us treat the BSDE approximation. In a first time we wil l study
+the time approximation error on [ T−ε,T].
+Proposition 4.7. Assume that (HX0),(HY0)and(HY1)hold. Then
+there exists a constant Cthat does not depend on nand such that
+sup
+n≤k≤2nE[|YN,ε,n
+tk−YN,ε
+tk|2]+2n−1/summationdisplay
+k=nE/bracketleftbigg/integraldisplaytk+1
+tk|ZN,ε,n
+tk−ZN,ε
+t|2dt/bracketrightbigg
+≤Clnn
+n1−2b.
+Proof. The BSDE ( 12) has a linear growth with respect to zon [T−
+ε,T] so we are allowed to apply classical results which give us th at
+sup
+n≤k≤2nE[|YN,ε,n
+tk−YN,ε
+tk|2]+2n−1/summationdisplay
+k=nE/bracketleftbigg/integraldisplaytk+1
+tk|ZN,ε,n
+tk−ZN,ε
+t|2dt/bracketrightbigg
+≤C/parenleftbigg
+E[|gN(XT)−gN(Xn
+T)|2]+ε
+n/parenrightbigg
+by using the fact that gNisN-Lipschitz and by applying Proposition 4.6.
+/square18 A. RICHOU
+Remark 4.8.
+(1) When a≥1−2b, thenε=Tn−a=o(n2b−1lnn). We do not need to
+have a discretization grid on [ T−ε,T];n+2 points of discretization are
+sufficient on [0 ,T].
+(2) When a<1−2b, then it is possible to take only ⌈nc⌉discretization
+points on [ T−ε,T] witha+c=1−2b. In this case the error bound becomes
+sup
+n≤k≤2nE[|YN,ε,n
+tk−YN,ε
+tk|2]+2n−1/summationdisplay
+k=nE/bracketleftbigg/integraldisplaytk+1
+tk|ZN,ε,n
+tk−ZN,ε
+t|2dt/bracketrightbigg
+≤C/parenleftbigglnn
+n1−2b+1
+na+c/parenrightbigg
+and the Proposition 4.7stays true.
+Now, let us see what happens on [0 ,T−ε].
+Theorem 4.9. Assume that (HX0),(HY0),(HY1)and(HX1)or(HX1′)
+hold. Then for all η >0, there exists a constant Cthat does not depend on
+N,εandn, such that
+sup
+0≤k≤2nE[|YN,ε,n
+tk−YN,ε
+tk|2]+2n−1/summationdisplay
+k=0E/bracketleftbigg/integraldisplaytk+1
+tk|ZN,ε,n
+tk−ZN,ε
+t|2dt/bracketrightbigg
+≤C
+n1−2b−Ka
+withK=4(1+η)L2
+f,zM2
+z,2.
+Proof. First, we will study the error on Y. From (12) and (15) we get
+YN,ε
+tk−YN,ε,n
+tk
+=Etk[YN,ε
+tk+1−YN,ε,n
+tk+1]
++Etk/integraldisplaytk+1
+tk(f(s,Xs,YN,ε
+s,ZN,ε
+s)−f(tk,Xn
+tk,YN,ε,n
+tk+1,ZN,ε,n
+tk))ds.
+We introducea parameter γk>0 that will bechosen later. ThankstoPropo-
+sition4.1and assumption (HY0), fis Lipschitz on [ tk,tk+1] with a Lipschitz
+constant Kk:=K1+K2
+(T−tk+1)1/2whereK2=2Lf,zMz,2. A combination of
+Young’s inequality ( a+b)2≤(1+γkhk)a2+(1+1
+γkhk)b2and properties of
+fgives
+E|YN,ε
+tk−YN,ε,n
+tk|2
+≤(1+γkhk)E|Etk[YN,ε
+tk+1−YN,ε,n
+tk+1]|2NUMERICAL SIMULATION OF QUADRATIC BSDES 19
++(1+η)1/3K2
+k/parenleftbigg
+hk+1
+γk/parenrightbigg
+E/integraldisplaytk+1
+tk|ZN,ε
+s−ZN,ε,n
+tk|2ds (16)
++C/parenleftbigg
+hk+1
+γk/parenrightbigg/parenleftbigg
+h2
+k+/integraldisplaytk+1
+tkE|Xs−Xn
+tk|2ds/parenrightbigg
++C/parenleftbigg
+hk+1
+γk/parenrightbigg/parenleftbigg/integraldisplaytk+1
+tkE|YN,ε
+s−YN,ε,n
+tk+1|2ds/parenrightbigg
+.
+We define
+˜ZN,ε,n
+tk:=1
+hkEtk[YN,ε,n
+tk+1(Wtk+1−Wtk)].
+So,ZN,ε,n
+tk=ρtk+1(˜ZN,ε,n
+tk). Moreover, Proposition 4.1implies that ZN,ε
+s=
+ρtk+1(ZN,ε
+s) and, since ρtk+1is 1-Lipschitz, we have
+|ZN,ε
+s−ZN,ε,n
+tk|2=|ρtk+1(ZN,ε
+s)−ρtk+1(˜ZN,ε,n
+tk)|2≤|ZN,ε
+s−˜ZN,ε,n
+tk|2. (17)
+As in Theorem 3.6, we define ¯ZN,ε
+tkby
+hk¯ZN,ε
+tk:=Etk/integraldisplaytk+1
+tkZN,ε
+sds
+=Etk/parenleftbigg/parenleftbigg
+YN,ε
+tk+1+/integraldisplaytk+1
+tkf(s,Xs,YN,ε
+s,ZN,ε
+s)ds/parenrightbigg
+t(Wtk+1−Wtk)/parenrightbigg
+.
+Clearly,
+E/integraldisplaytk+1
+tk|ZN,ε
+s−˜ZN,ε,n
+tk|2ds
+(18)
+=E/integraldisplaytk+1
+tk|ZN,ε
+s−¯ZN,ε
+tk|2ds+hkE|¯ZN,ε
+tk−˜ZN,ε,n
+tk|2.
+The Cauchy–Schwarz inequality yields
+|Etk((YN,ε
+tk+1−YN,ε,n
+tk+1)t(Wtk+1−Wtk))|2
+≤hk{Etk(|YN,ε
+tk+1−YN,ε,n
+tk+1|2)−|Etk(YN,ε
+tk+1−YN,ε,n
+tk+1)|2}
+and consequently
+hkE|¯ZN,ε
+tk−˜ZN,ε,n
+tk|2
+≤(1+η)1/3E[Etk(|YN,ε
+tk+1−YN,ε,n
+tk+1|2)−|Etk(YN,ε
+tk+1−YN,ε,n
+tk+1)|2] (19)
++ChkE/integraldisplaytk+1
+tk|f(s,Xs,YN,ε
+s,ZN,ε
+s)|2ds.20 A. RICHOU
+Plugging ( 18) and (19) into (16), we get
+E|YN,ε
+tk−YN,ε,n
+tk|2
+≤(1+γkhk)E|Etk[YN,ε
+tk+1−YN,ε,n
+tk+1]|2
++(1+η)K2
+k/parenleftbigg
+hk+1
+γk/parenrightbigg
+E/integraldisplaytk+1
+tk|ZN,ε
+s−¯ZN,ε
+tk|2ds
++C/parenleftbigg
+hk+1
+γk/parenrightbigg/parenleftbigg
+h2
+k+/integraldisplaytk+1
+tkE|Xs−Xn
+tk|2ds
++/integraldisplaytk+1
+tkE|YN,ε
+s−YN,ε,n
+tk+1|2ds/parenrightbigg
++(1+η)2/3K2
+k/parenleftbigg
+hk+1
+γk/parenrightbigg
+E[Etk(|YN,ε
+tk+1−YN,ε,n
+tk+1|2)
+−|Etk(YN,ε
+tk+1−YN,ε,n
+tk+1)|2]
++CK2
+k/parenleftbigg
+hk+1
+γk/parenrightbigg
+hkE/integraldisplaytk+1
+tk|f(s,Xs,YN,ε
+s,ZN,ε
+s)|2ds.
+Now write
+E|YN,ε
+s−YN,ε,n
+tk+1|2≤2E|YN,ε
+s−YN,ε
+tk+1|2+2E|YN,ε
+tk+1−YN,ε,n
+tk+1|2, (20)
+E|Xs−Xn
+tk|2≤2E|Xs−Xtk|2+2E|Xtk−Xn
+tk|2(21)
+and we obtain
+E|YN,ε
+tk−YN,ε,n
+tk|2
+≤(1+γkhk)E|Etk[YN,ε
+tk+1−YN,ε,n
+tk+1]|2
++(1+η)K2
+k/parenleftbigg
+hk+1
+γk/parenrightbigg
+E/integraldisplaytk+1
+tk|ZN,ε
+s−¯ZN,ε
+tk|2ds
++C/parenleftbigg
+hk+1
+γk/parenrightbigg/parenleftbigg
+h2
+k+/integraldisplaytk+1
+tkE|Xs−Xtk|2ds+hkE|Xtk−Xn
+tk|2/parenrightbigg
++C/parenleftbigg
+hk+1
+γk/parenrightbigg/parenleftbigg/integraldisplaytk+1
+tkE|YN,ε
+s−YN,ε
+tk+1|2ds+hkE|YN,ε
+tk+1−YN,ε,n
+tk+1|2/parenrightbigg
++(1+η)2/3K2
+k/parenleftbigg
+hk+1
+γk/parenrightbigg
+E[Etk(|YN,ε
+tk+1−YN,ε,n
+tk+1|2)
+−|Etk(YN,ε
+tk+1−YN,ε,n
+tk+1)|2]
++CK2
+k/parenleftbigg
+hk+1
+γk/parenrightbigg
+hkE/integraldisplaytk+1
+tk|f(s,Xs,YN,ε
+s,ZN,ε
+s)|2ds.NUMERICAL SIMULATION OF QUADRATIC BSDES 21
+Takingγk=(1+η)2/3K2
+kand forhksmall enough, it gives
+E|YN,ε
+tk−YN,ε,n
+tk|2
+≤(1+Chk+(1+η)2/3K2
+khk)E|YN,ε
+tk+1−YN,ε,n
+tk+1|2+Ch2
+k
++Chkmax
+0≤k≤nE|Xtk−Xn
+tk|2
++CE/integraldisplaytk+1
+tk|ZN,ε
+s−¯ZN,ε
+tk|2ds+C/integraldisplaytk+1
+tkE|Xs−Xtk|2ds
++C/integraldisplaytk+1
+tkE|YN,ε
+s−YN,ε
+tk+1|2ds+ChkE/integraldisplaytk+1
+tkf(s,Xs,YN,ε
+s,ZN,ε
+s)2ds,
+because K2
+khk≤C(h0+hk(T−tk+1)−1)≤Clnn
+n. The Gronwall’s lemma
+gives us
+E|YN,ε
+tk−YN,ε,n
+tk|2
+≤Cn−1/summationdisplay
+j=0/bracketleftBiggj−1/productdisplay
+i=0(1+Chi+(1+η)2/3K2
+ihi)/bracketrightBigg
+×/bracketleftbigg
+h2
+j+hjmax
+0≤l≤nE|Xtl−Xn
+tl|2
++E/integraldisplaytj+1
+tj(|ZN,ε
+s−¯ZN,ε
+tj|2+|Xs−Xtj|2+|YN,ε
+s−YN,ε
+tj+1|2)ds
++hjE/integraldisplaytj+1
+tj|f(s,Xs,YN,ε
+s,ZN,ε
+s)|2ds/bracketrightbigg
++/bracketleftBiggn−1/productdisplay
+i=0(1+Chi+(1+η)2/3K2
+ihi)/bracketrightBigg
+E|YN,ε
+tn−YN,ε,n
+tn|2.
+Then, we apply Lemma 4.5.
+E|YN,ε
+tk−YN,ε,n
+tk|2
+≤Cn(1+η)(K2)2a
+×/bracketleftBigg
+h0+ max
+0≤l≤nE|Xtl−Xn
+tl|2
++n/summationdisplay
+j=0E/parenleftbigg/integraldisplaytj+1
+tj|ZN,ε
+s−¯ZN,ε
+tj|2+|Xs−Xtj|2+|YN,ε
+s−YN,ε
+tj+1|2ds/parenrightbigg
++h0E/integraldisplaytn
+0|f(s,Xs,YN,ε
+s,ZN,ε
+s)|2ds+E|YN,ε
+tn−YN,ε,n
+tn|2/bracketrightBigg
+.22 A. RICHOU
+A classical estimation gives us E[|Xs−Xtj|2]≤|s−tj|. Moreover, since
+ZN,εis bounded,
+E/integraldisplaytn
+0|f(s,Xs,YN,ε
+s,ZN,ε
+s)|2ds
+≤CT(1+|YN,ε|∞)+CE/bracketleftbigg/integraldisplaytn
+0|ZN,ε
+s|4ds/bracketrightbigg
+≤CT(1+|YN,ε|∞)+Cn2bE/bracketleftbigg/integraldisplayT
+0|ZN,ε
+s|2ds/bracketrightbigg
+.
+But we have an a priori estimate for E[/integraltextT
+0|ZN,ε
+s|2ds] that does not depend
+onNandε. So
+E/integraldisplaytn
+0|f(s,Xs,YN,ε
+s,ZN,ε
+s)|2ds≤Cn2b. (22)
+With the same type of argument we also have
+E|YN,ε
+s−YN,ε
+tj+1|2≤Chjn2b. (23)
+If we add Zhang’s path regularity Theorem 3.6, Propositions 4.6and4.7,
+we finally obtain
+E|YN,ε
+tk−YN,ε,n
+tk|2≤Cn(1+η)(K2)2an2blnn
+n=Clnn
+n1−2b−(1+η)(K2)2a. (24)
+Now, let us deal with the error on Z. First of all, ( 17) gives us
+n−1/summationdisplay
+k=0E/bracketleftbigg/integraldisplaytk+1
+tk|ZN,ε,n
+tk−ZN,ε
+t|2dt/bracketrightbigg
+≤n−1/summationdisplay
+k=0E/bracketleftbigg/integraldisplaytk+1
+tk|˜ZN,ε,n
+tk−ZN,ε
+t|2dt/bracketrightbigg
+.
+For 0≤k≤n−1, we can use ( 18) and (19) to obtain
+E/bracketleftbigg/integraldisplaytk+1
+tk|˜ZN,ε,n
+tk−ZN,ε
+t|2dt/bracketrightbigg
+≤E/bracketleftbigg/integraldisplaytk+1
+tk|¯ZN,ε
+tk−ZN,ε
+t|2dt/bracketrightbigg
++(1+η)2/3E[Etk(|YN,ε
+tk+1−YN,ε,n
+tk+1|2)−|Etk(YN,ε
+tk+1−YN,ε,n
+tk+1)|2]
++ChkE/bracketleftbigg/integraldisplaytk+1
+tk|f(s,Xs,YN,ε
+s,ZN,ε
+s)|2ds/bracketrightbigg
+.
+Inequality ( 22) and estimates for Zgive us
+n−1/summationdisplay
+k=0E/bracketleftbigg/integraldisplaytk+1
+tk|ZN,ε,n
+tk−ZN,ε
+t|2dt/bracketrightbiggNUMERICAL SIMULATION OF QUADRATIC BSDES 23
+≤n−1/summationdisplay
+k=0E/bracketleftbigg/integraldisplaytk+1
+tk|¯ZN,ε
+tk−ZN,ε
+t|2dt/bracketrightbigg
++(1+η)2/3n−1/summationdisplay
+k=0E[Etk(|YN,ε
+tk+1−YN,ε,n
+tk+1|2)−|Etk(YN,ε
+tk+1−YN,ε,n
+tk+1)|2]
++Ch0E/bracketleftbigg/integraldisplayT
+0|f(s,Xs,YN,ε
+s,ZN,ε
+s)|2ds/bracketrightbigg
+(25)
+≤n−1/summationdisplay
+k=0E/bracketleftbigg/integraldisplaytk+1
+tk|¯ZN,ε
+tk−ZN,ε
+t|2dt/bracketrightbigg
++(1+η)2/3n−1/summationdisplay
+k=0E[Etk(|YN,ε
+tk−YN,ε,n
+tk|2)−|Etk(YN,ε
+tk+1−YN,ε,n
+tk+1)|2]
++CE[|YN,ε
+tn−YN,ε,n
+tn|2]+Ch0n2b
+with an index change in the penultimate line. Then, by using ( 16) we get
+(1+η)2/3E[Etk(|YN,ε
+tk−YN,ε,n
+tk|2)−|Etk(YN,ε
+tk+1−YN,ε,n
+tk+1)|2]
+≤CγkhkE|Etk[YN,ε
+tk+1−YN,ε,n
+tk+1]|2
+(26)
++(1+η)K2
+k/parenleftbigg
+hk+1
+γk/parenrightbigg
+E/integraldisplaytk+1
+tk|ZN,ε
+s−ZN,ε,n
+tk|2ds
++C/parenleftbigg
+hk+1
+γk/parenrightbigg
+hk/parenleftBig
+hk+ sup
+s∈[tk,tk+1]E[|Xs−Xn
+tk|2+|YN,ε
+s−YN,ε,n
+tk+1|2]/parenrightBig
+.
+Thanks to ( 20), (21), (23) and a classical estimation on E[|Xs−Xtk|2] we
+have
+sup
+s∈[tk,tk+1]E[|Xs−Xn
+tk|2+|YN,ε
+s−YN,ε,n
+tk+1|2]
+≤C(hkn2b+E[|YN,ε
+tk+1−YN,ε,n
+tk+1|2]).
+Let us set γk=3(1+η)K2
+k. We recall that hkK2
+k≤Clnn
+n→0 whenn→0.
+So, fornbig enough, ( 26) becomes
+(1+η)2/3E[Etk(|YN,ε
+tk−YN,ε,n
+tk|2)−|Etk(YN,ε
+tk+1−YN,ε,n
+tk+1)|2]
+≤Clnn
+nE[|YN,ε
+tk+1−YN,ε,n
+tk+1|2]+1
+2E/integraldisplaytk+1
+tk|ZN,ε
+s−ZN,ε,n
+tk|2ds
++Ch0hkn2b.24 A. RICHOU
+If we inject this last estimate in ( 25) and we use Theorem 3.6, we obtain
+1
+2n−1/summationdisplay
+k=0E/bracketleftbigg/integraldisplaytk+1
+tk|ZN,ε,n
+tk−ZN,ε
+t|2dt/bracketrightbigg
+≤Ch0n2b+Clnnsup
+0≤k≤n−1E[|YN,ε
+tk+1−YN,ε,n
+tk+1|2].
+By using ( 24) and Proposition 4.7, we finally have
+sup
+0≤k≤2nE[|YN,ε,n
+tk−YN,ε
+tk|2]+2n−1/summationdisplay
+k=0E/bracketleftbigg/integraldisplaytk+1
+tk|ZN,ε,n
+tk−ZN,ε
+t|2dt/bracketrightbigg
+≤C(lnn)2
+n1−2b−Ka
+withK=4(1+η)L2
+f,zM2
+z,2. Since this estimate is true for every η >0, we
+have proved the result. /square
+4.3.Study of the global error e(N,ε,n).Let us study errors e1(N) and
+e2(N,ε).
+Proposition 4.10. Let us assume that (HX0)and(HY0)hold. There
+exists a constant C >0such that
+e2(N,ε)≤C
+n2a−4b.
+Proof. We just have to notice that
+|f(t,Xt,YN,ε
+t,ZN,ε
+t)−f(t,Xt,YN,ε
+t,0)|≤C|ZN,ε
+t|2
+and|ZN,ε
+t|is bounded by Cnb./square
+ForgNwe use the classical Lipschitz approximation
+gN(x)=inf{g(u)+N|x−u||u∈Rd}.
+Proposition 4.11. We assume that (HX0)holds and gisα-H¨ older.
+Then, there exists a constant Csuch that
+e1(N)≤C
+n2bα/(1−α).
+Proof. gNis aN-Lipschitz function and gN→gwhenN→+∞uni-
+formly on Rd. More precisely, we have
+|g−gN|∞≤C
+Nα/(1−α)./squareNUMERICAL SIMULATION OF QUADRATIC BSDES 25
+Remark 4.12. For some explicit examples, it is possible to have a bet-
+ter convergence speed. For example, let us take g(x)=(|x|α1x≥0)∧Cand
+assume that σis invertible. Then, we can use the fact that this function is
+not Lipschitz only in 0 and obtain
+e1(N)≤C
+n2αb/(1−α)P(XT∈[0,N−1/(1−α)])1/q≤C
+n(b/(1−α))(2α+1/q).
+Remark 4.13. It is also possible to obtain convergence speed when g
+is notα-H¨ older. For example, we assume that σis invertible and we set
+g(x)=/producttextd
+i=11xi>0(x). Then
+e1(N)≤C/bracketleftBiggd/summationdisplay
+i=1P((XT)i∈[0,1/N])/bracketrightBigg1/q
+≤C
+N1/q=C
+nb/q.
+Now we are able to gather all these errors.
+Theorem 4.14. We assume that (HX0),(HY0),(HY1)and(HX1)or
+(HX1′)hold. We assume also that gisα-H¨ older. Then for all η >0, there
+exists a constant C >0that does not depend on nsuch that
+e(n):=e(N,ε,n)≤C
+n2α/((2−α)(2+K)−2+2α)
+withK=4(1+η)L2
+f,zM2
+z,2.
+Proof. Thanks to Theorem 4.9, Propositions 4.10and4.11we have
+e(n)≤C
+n1−2b−Ka+C
+n2a−4b+C
+n2αb/(1−α).
+Then we only need to set a:=1+2b
+2+Kandb:=1−α
+(2−α)(2+K)−2+2αto obtain the
+result./square
+Corollary 4.15. We assume that assumptions of Theorem 4.14hold.
+Moreover, we assume that fhas a sub-quadratic growth with respect to z;
+there exists 0<β <1such that, for all t∈[0,T],x∈Rd,y∈R,z,z′∈R1×d,
+|f(t,x,y,z)−f(t,x,y,z′)|≤(Kf,z+Lf,z(|z|β+|z′|β))|z−z′|.
+Then we are allowed to take Kas small as we want. So, for all η>0, there
+exists a constant C >0that does not depend on nsuch that
+e(n)≤C
+nα−η.
+Remark 4.16. We are able to specify Remark 4.8in our case, when
+a=1+2b
+2+Kandb=1−α
+(2−α)(2+K)−2+2α.26 A. RICHOU
+(1) When K≤2−3α
+2−α, that is to say, when α<2/3 andKis sufficiently
+small, then we do not need to have a discretization grid on [ T−ε,T].
+(2) When K >2−3α
+2−α, then it is possible to take only ⌈nc⌉discretization
+points on [ T−ε,T] with
+c=1+3α−4
+(2−α)(2+K)−2+2α.
+Theorem 4.14is not interesting in practice because the speed of conver-
+gence depends strongly on K. But we see that the global error becomes
+e(n)≤C
+nα−ηwhen we are allowed to choose Kas small as we want. Under
+extra assumption we can show that we are allowed to take the co nstantMz,2
+as small as we want.
+(HX2).bis bounded on [0 ,T]×Rdby a constant Mb.
+Theorem 4.17. We assume that (HX0),(HY0),(HY1),(HX2)and
+(HX1)or(HX1′)hold. We assume also that gisα-H¨ older. Then for all
+η >0, there exists a constant C >0that does not depend on nsuch that
+e(n)≤C
+nα−η.
+Remark 4.18. With the assumptions of the previous theorem, it is
+also possible to have an estimate of the global error for exam ples given in
+Remarks 4.12and4.13. Wheng(x)=(|x|α1x≥0)∧C, we have
+e(n)≤C
+nα+(1−α)/(1+2q)−η
+and when g(x)=/producttextd
+i=11xi>0(x), we have
+e(n)≤C
+n1/(1+2q)−η.
+Proof of Theorem 4.17.First, we suppose that fis differentiable
+with respect to z. Thanks to Remark 3.4we see that it is sufficient to show
+that
+EQN,ε/bracketleftbigg/integraldisplayT
+t|ZN,ε
+s|2ds|Ft/bracketrightbigg
+is small uniformly in ω,Nandεwhentis close to T. We will obtain an
+estimation for this quantity by applying the same computati on as [2] for the
+BMO norm estimate of Z, page 831. Thus, we have
+EQN,ε/bracketleftbigg/integraldisplayT
+t|ZN,ε
+s|2ds|Ft/bracketrightbigg
+≤EQN,ε[|ϕ(YN,ε
+T)−ϕ(YN,ε
+t)||Ft]+C(T−t)NUMERICAL SIMULATION OF QUADRATIC BSDES 27
+withϕ(x)=(e2c(x+m)−2c(x+m)−1)/(2c2),m=|Y|∞andcthat depends
+on constants in assumption (HY0) but does not depend on ∇zf. Let us
+notice that m,cand soϕdo not depend on Nandε. SinceYis bounded,
+ϕis a Lipschitz function, so
+EQN,ε/bracketleftbigg/integraldisplayT
+t|ZN,ε
+s|2ds|Ft/bracketrightbigg
+≤CEQN,ε[|YN,ε
+T−YN,ε
+t||Ft]+C(T−t).
+We denote by ( YN,ε,t,x,ZN,ε,t,x) the solution of BSDE ( 12) whenXt,x
+t=x.
+Asusual,weset Xt,x
+s=xandZN,ε,t,x
+s=0fors≤tandwedefine uN,ε(t,x):=
+YN,ε,t,x
+t. Then we give a proposition that we will prove in Appendix C.
+Proposition 4.19. We assume that (HX0),(HY0),(HY1),(HX2)and
+(HX1)or(HX1′)hold. We assume also that gis uniformly continuous on
+Rd. ThenuN,εis uniformly continuous on [0,T]×Rdand there exists ωa
+concave modulus of continuity for all functions in {uN,ε|N∈N,ε>0}, that
+is,ωdoes not depend on Nandε.
+Then
+EQN,ε[|YN,ε
+T−YN,ε
+t||Ft]
+=EQN,ε[|uN,ε(T,XT)−uN,ε(t,Xt)||Ft]
+≤EQN,ε[1|/integraltextT
+tσ(s)d˜Ws|≤ν|uN,ε(T,XT)−uN,ε(t,Xt)|
++2|YN,ε|∞1|/integraltextT
+tσ(s)d˜Ws|>ν|Ft]
+≤EQN,ε[ω(|T−t|+1|/integraltextT
+tσ(s)d˜Ws|≤ν|XT−Xt|)
++2|YN,ε|∞1|/integraltextT
+tσ(s)d˜Ws|>ν|Ft]
+withd˜Ws=dWs−∇zfε(s,Xs,YN,ε
+s,ZN,ε
+s)ds. But,
+1|/integraltextT
+tσ(s)d˜Ws|≤ν|XT−Xt|
+=1|/integraltextT
+tσ(s)d˜Ws|≤ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT
+tb(s,Xs)ds
++/integraldisplayT
+t∇zfε(s,Xs,YN,ε
+s,ZN,ε
+s)ds+/integraldisplayT
+tσ(s)d˜Ws/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤Mb(T−t)+ν+C/integraldisplayT
+t(1+|ZN,ε
+s|)ds
+≤C(T−t)+ν+C(T−t)1/2/parenleftbigg/integraldisplayT
+t|ZN,ε
+s|2ds/parenrightbigg1/2
+.28 A. RICHOU
+Sinceωis concave, we have by Jensen’s inequality
+EQN,ε[ω(|T−t|+1|/integraltextT
+tσ(s)d˜Ws|≤ν|XT−Xt|)|Ft]
+≤ω/parenleftbigg
+C|T−t|+ν+C(T−t)1/2EQN,ε/bracketleftbigg/parenleftbigg/integraldisplayT
+t|ZN,ε
+s|2ds/parenrightbigg1/2/vextendsingle/vextendsingle/vextendsingleFt/bracketrightbigg/parenrightbigg
+≤ω/parenleftbigg
+C|T−t|+ν+C(T−t)1/2EQN,ε/bracketleftbigg/integraldisplayT
+t|ZN,ε
+s|2ds|Ft/bracketrightbigg1/2/parenrightbigg
+≤ω(C|T−t|+ν+C(T−t)1/2/ba∇⌈blZN,ε/ba∇⌈blBMO(Q)).
+But,/ba∇⌈blZN,ε/ba∇⌈blBMO(Q)only depends on constants in assumption (HY0), so it
+is bounded uniformly in Nandε. Moreover, |/integraltextT
+tσ(s)d˜Ws|is independent
+ofFtso we have by the Markov inequality
+EQN,ε[1|/integraltextT
+tσ(s)d˜Ws|>ν|Ft]=QN,ε/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT
+tσ(s)d˜Ws/vextendsingle/vextendsingle/vextendsingle/vextendsingle>ν/parenrightbigg
+≤C(T−t)1/2
+ν.
+Finally, we have
+EQN,ε[|YN,ε
+T−YN,ε
+t||Ft]≤ω(C|T−t|1/2+ν)+C(T−t)1/2
+ν
+≤ω(C|T−t|1/2+|T−t|1/4)+C|T−t|1/4
+by setting ν=|T−t|1/4andEQN,ε[|YN,ε
+T−YN,ε
+t||Ft]→0 uniformly in ω,N
+andεwhent→T. Whenfis not differentiable with respect to zbut is only
+locally Lipschitz, then we can prove the result by a standard approximation.
+/square
+APPENDIX A: PROOF OF LEMMA 4.4
+We have
+2n−1/productdisplay
+i=0(1+Mhi)=/parenleftBiggn−1/productdisplay
+i=0(1+Mhi)/parenrightBigg/parenleftBigg2n−1/productdisplay
+i=n(1+Mhi)/parenrightBigg
+.
+First,
+2n−1/productdisplay
+i=n(1+Mhi)≤/parenleftbigg
+1+MT
+n/parenrightbiggn
+≤C.NUMERICAL SIMULATION OF QUADRATIC BSDES 29
+Moreover, for 0 ≤i≤n−1,
+hi=ti+1−ti=Tn−ai/n(1−e−(alnn)/n)≤Tn−ai/nalnn
+n
+thanks to the convexity of the exponential function. So
+n−1/productdisplay
+i=0(1+Mhi)≤n−1/productdisplay
+i=0/parenleftbigg
+1+MTan−ai/nlnn
+n/parenrightbigg
+=exp/parenleftBiggn−1/summationdisplay
+i=0ln/parenleftbigg
+1+MTan−ai/nlnn
+n/parenrightbigg/parenrightBigg
+≤exp/parenleftBiggn−1/summationdisplay
+i=0MTa(n−a/n)ilnn
+n/parenrightBigg
+≤exp/parenleftbigg
+MTalnn
+n/parenleftbigg1−(1/na)
+1−(1/n(a/n))/parenrightbigg/parenrightbigg
+≤exp/parenleftbigg
+MTalnn
+nna/n
+na/n−1/parenrightbigg
+.
+But,
+lnn
+nna/n
+na/n−1∼lnn
+n1
+(alnn)/n∼1
+a,
+whenn→+∞. Thus, we have shown the result.
+APPENDIX B: PROOF OF LEMMA 4.5
+Thanks to Lemma 4.4, we have
+/producttextn−1
+i=0(1+M1hi+M2hi/(T−ti+1))/producttextn−1
+i=0(1+M2hi/(T−ti+1))=n−1/productdisplay
+i=0/parenleftbigg
+1+M1
+1+M2hi/(T−ti+1)hi/parenrightbigg
+≤n−1/productdisplay
+i=0(1+M1hi)≤C.
+So we just have to show that
+n−1/productdisplay
+i=0/parenleftbigg
+1+M2hi
+T−ti+1/parenrightbigg
+≤CnaM2.
+But
+1+M2hi
+T−ti+1=1+M2(na/n−1).30 A. RICHOU
+So
+n−1/productdisplay
+i=0/parenleftbigg
+1+M2hi
+T−ti+1/parenrightbigg
+=(1+M2(na/n−1))n
+=exp/parenleftbigg
+nln/parenleftbigg
+1+aM2lnn
+n+O/parenleftbiggln2n
+n2/parenrightbigg/parenrightbigg/parenrightbigg
+=exp/parenleftbigg
+aM2lnn+O/parenleftbiggln2n
+n/parenrightbigg/parenrightbigg
+∼naM2,
+whenn→+∞. Thus, we have shown the result.
+APPENDIX C: PROOF OF PROPOSITION 4.19
+Wewill provethispropositionastheauthorsof[ 9]dofortheirProposition
+4.2. In this proof we omit the superscript N,εforu,YandZto be more
+readable. Let x0,x′
+0∈Rdandt0,t′
+0∈[0,T]. By an argument of symmetry we
+are allowed to suppose that t0≤t′
+0. We have
+|u(t0,x0)−u(t′
+0,x′
+0)|≤|u(t0,x0)−u(t0,x′
+0)|+|u(t0,x′
+0)−u(t′
+0,x′
+0)|.
+Let us begin with the first term. We will use a classical argume nt of lin-
+earization:
+Yt0,x0
+t−Yt0,x′
+0
+t=gN(Xt0,x0
+T)−gN(Xt0,x′
+0
+T)
++/integraldisplayT
+tαs(Xt0,x0s−Xt0,x′
+0s)+βs(Yt0,x0s−Yt0,x′
+0s)ds
+−/integraldisplayT
+t(Zt0,x0s−Zt0,x′
+0s)d˜Ws
+with
+αs:=fε(s,Xt0,x0s,Yt0,x′
+0s,Zt0,x′
+0s)−fε(s,Xt0,x′
+0s,Yt0,x′
+0s,Zt0,x′
+0s)
+Xt0,x0s−Xt0,x′
+0s,
+ifXt0,x0s−Xt0,x′
+0s/n⌉}ationslash=0 and αs=0 elsewhere,
+βs:=fε(s,Xt0,x0s,Yt0,x0s,Zt0,x′
+0s)−fε(s,Xt0,x0s,Yt0,x′
+0s,Zt0,x′
+0s)
+Yt0,x0s−Yt0,x′
+0s,
+ifXt0,x0s−Xt0,x′
+0s/n⌉}ationslash=0 and βs=0 elsewhere,
+γs:=fε(s,Xt0,x0s,Yt0,x0s,Zt0,x0s)−fε(s,Xt0,x0s,Yt0,x0s,Zt0,x′
+0s)
+|Zt0,x0s−Zt0,x′
+0s|2
+×t(Zt0,x0s−Zt0,x′
+0s),NUMERICAL SIMULATION OF QUADRATIC BSDES 31
+ifZt0,x0s−Zt0,x′
+0s/n⌉}ationslash= 0 and γs= 0 elsewhere and d˜Ws:=dWs−γsds. By a
+BMO argument, there exists a probability Qunder which ˜Wis a Brownian
+motion. Then we apply a classical transformation to obtain
+EQ[e/integraltextt
+t0βsds(Yt0,x0
+t−Yt0,x′
+0
+t)]
+=EQ/bracketleftbigg
+e/integraltextT
+t0βsds(gN(Xt0,x0
+T)−gN(Xt0,x′
+0
+T))
++/integraldisplayT
+t0αse/integraltexts
+t0βudu(Xt0,x0s−Xt0,x′
+0s)ds/bracketrightbigg
+and
+|u(t0,x0)−u(t0,x′
+0)|
+≤C/parenleftbigg
+EQ[ω(|Xt0,x0
+T−Xt0,x′
+0
+T|)]+/integraldisplayT
+t0EQ[|Xt0,x0s−Xt0,x′
+0s|]ds/parenrightbigg
+withωa modulus of continuity of gthat is also a modulus of continuity for
+gN. We are allowed to suppose that ωis concave; indeed, there exist two
+positive constants aandbsuch that ω(x)≤ax+b, then the concave hull of
+x/mapsto→ω(x)∨(1x≥1(ax+b)) is also a modulus of continuity of g. So Jensen’s
+inequality gives us
+|u(t0,x0)−u(t0,x′
+0)|
+≤C/parenleftbigg
+ω(EQ[|Xt0,x0
+T−Xt0,x′
+0
+T|])+/integraldisplayT
+t0EQ[|Xt0,x0s−Xt0,x′
+0s|]ds/parenrightbigg
+.
+By using the fact that bis bounded we can prove the following proposition
+exactly as authors of [ 9] do for their Proposition 4.7.
+Proposition C.1. ∃C >0that does not depend on Nandεsuch that
+∀t,t′∈[0,T],∀x,x′∈Rd,∀s∈[0,T],
+EQ[|Xt,x
+s−Xt′,x′
+s|]≤C(|x−x′|+|t−t′|1/2).
+Then,
+|u(t0,x0)−u(t0,x′
+0)|≤C(ω(|x0−x′
+0|)+|x0−x′
+0|).
+Now we will study the second term,
+|u(t0,x′
+0)−u(t′
+0,x′
+0)|=|Yt0,x′
+0
+t0−Yt′
+0,x′
+0
+t′
+0|
+≤|Yt0,x′
+0
+t0−Yt′
+0,x′
+0
+t0|+|Yt′
+0,x′
+0
+t0−Yt′
+0,x′
+0
+t′
+0|.32 A. RICHOU
+First,
+|Yt′
+0,x′
+0
+t0−Yt′
+0,x′
+0
+t′
+0|≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt′
+0
+t0f(s,x′
+0,Yt′
+0,x′
+0s,0)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|t0−t′
+0|.
+Moreover, as for the first term we have
+EQ[e/integraltextt
+t0βsds(Yt0,x′
+0
+t−Yt′
+0,x′
+0
+t)]
+=EQ/bracketleftbigg
+e/integraltextT
+t0βsds(gN(Xt0,x′
+0
+T)−gN(Xt′
+0,x′
+0
+T))
++/integraldisplayT
+t0αse/integraltexts
+t0βudu(Xt0,x′
+0s−Xt′
+0,x′
+0s)ds/bracketrightbigg
+and
+|Yt0,x′
+0
+t0−Yt′
+0,x′
+0
+t′
+0|≤C(ω(|t0−t′
+0|1/2)+|t0−t′
+0|1/2).
+Finally,
+|u(t0,x′
+0)−u(t′
+0,x′
+0)|≤C(ω(|t0−t′
+0|1/2)+|t0−t′
+0|1/2)
+and
+|u(t0,x0)−u(t′
+0,x′
+0)|
+≤C(ω(|x0−x′
+0|)+ω(|t0−t′
+0|1/2)+|x0−x′
+0|+|t0−t′
+0|1/2).
+Souis uniformly continuous on [0 ,T]×Rdand this function has a modulus
+of continuity that does not depend on Nandε. Moreover, we are allowed
+to suppose that this modulus of continuity is concave.
+Acknowledgments. The author would like to thank his Ph.D. advisers,
+Philippe Briand and Ying Hu, an anonymous referee and the Ass ociate Ed-
+itor for their careful reading and helpful comments.
+REFERENCES
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+Carlo simulation of backward stochastic differential equat ions.Stochastic Pro-
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+Universit ´e de Rennes 1
+Campus de Beaulieu
+35042 Rennes Cedex
+France
+E-mail: adrien.richou@univ-rennes1.fr
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+arXiv:1001.0402v2  [math.NT]  27 Mar 2012Volume 00, Number 0, Pages 000–000
+S (XX)0000-0
+MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES
+REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+Abstract. We present a new algorithm to compute the classical modular
+polynomial Φ lin the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime land any
+positive integer m. Our approach uses the graph of l-isogenies to efficiently
+compute Φ lmodpfor many primes pof a suitable form, and then applies the
+Chinese Remainder Theorem (CRT). Under the Generalized Rie mann Hypoth-
+esis (GRH), we achieve an expected running time of O(l3(logl)3loglogl), and
+compute Φ lmodmusingO(l2(logl)2+l2logm) space. We have used the new
+algorithm to compute Φ lwithlover 5000, and Φ lmodmwithlover 20000.
+We also consider several modular functions gfor which Φg
+lis smaller than Φ l,
+allowing us to handle lover 60000.
+1.Introduction
+For a prime l, the classical modular polynomial Φ lis the minimal polynomial of
+the function j(lz) over the field C(j), wherej(z) is the modular j-function. The
+polynomial Φ lparametrizes elliptic curves Etogether with an isogeny E→E′
+of degreel. From classical results, we know that Φ llies in the ring Z[X,Y] and
+satisfies Φ l(X,Y) = Φl(Y,X), with degree l+1 in both variables [58, §69].
+The fact that the moduli interpretation of Φ lremains valid modulo primes p/n⌉}ationslash=l
+was crucial to the improvements made by Atkin and Elkies to Schoof’s point-
+counting algorithm [22, 51]. More recently, the polynomials Φ lmodphave been
+used to compute Hilbert class polynomials [2, 54], and to determine the endomor-
+phism ring of an elliptic curve over a finite field [6]. Explicitly computing Φ lis
+notoriously difficult, primarily due to its large size. As shown in [20], the lo garith-
+mic height of its largest coefficient is 6 llogl+O(l), thus its total size is
+(1) O(l3logl).
+As this bound suggests, the size of Φ lgrowsquite rapidly; the binaryrepresentation
+of Φ79already exceeds one megabyte, and Φ 659is larger than a gigabyte.
+The polynomial Φ lcan be computed by comparing coefficients in the Fourier
+expansions of j(z) andj(lz), an approach considered by several authors [8, 22, 37,
+39, 41, 45, 47]. As detailed in [8], this only requires integer arithmetic, and may
+be performed modulo pfor any prime p>2l+2. The time to compute Φ lmodp
+is thenO(l3+ε(logp)1+ε), and for a sufficiently large pthis yields an O(l4+ε) time
+algorithm to compute Φ loverZ. Alternatively (and preferably), one computes Φ l
+modulo several smaller primes and applies the Chinese Remainder Theo rem, as
+suggested in [8, 37, 45, 47].
+An alternative CRT-based approach appears in [17]. This algorithm us es isoge-
+nies between supersingular elliptic curves defined over a finite field, a nd computes
+Φlmodpin timeO(l4+ε(logp)2+ε+(logp)4+ε), under the GRH.
+2000Mathematics Subject Classification. Primary 11Y16 ; Secondary 11G15, 11G20, 14H52.
+12 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+In [24], Enge uses interpolation and fast floating-point evaluations t o compute
+Φl∈Z[X,Y] in timeO(l3(logl)4+ε), under reasonable heuristic assumptions. The
+complexity of this method is nearly optimal, quasi-linear in the size of Φ l. How-
+ever, most applications actually use Φ lin a finite field Fpn, and Φ lmodpmay be
+much smaller than Φ l. In general, Enge’s algorithm can compute Φ land reduce
+it modulopmuch faster than either of the methods above can compute Φ lmodp,
+but this may use an excessive amount of space. For large lthis approach becomes
+impractical, even when Φ lmodpis reasonably small.
+Here we present a new method to compute Φ l, either over the integers or modulo
+an arbitrary positive integer m, including m≤l. Our algorithm is both asymp-
+totically and practically faster than alternative methods, and achie ves essentially
+optimal space complexity. More precisely, we prove the following res ult.
+Theorem 1. Letldenote an odd prime and ma positive integer. Algorithm 6.1
+correctly computes Φl∈(Z/mZ)[X,Y]. Under the GRH, it runs in expected time
+O(l3log3lloglogl),
+usingO(l2loglm)expected space.
+To compute Φ loverZ, we choose a modulus mthat is large enough to uniquely
+determine the coefficients, via an explicit height bound proven in [13]. I n general,
+we may assume log m=O(llogl), since otherwise Φ land Φ lmodmare effectively
+the same (hence the time bound does not depend on m).
+Our algorithmis of the Las Vegas type, a probabilistic algorithmwhose output is
+unconditionally correct; the GRH is only used to analyze its running tim e. We have
+used it to compute Φ lfor alll <3600, and many larger lup to 5003. The largest
+previous computation of which we are aware has l= 1009. Working modulo mwe
+can go further; we have computed Φ lmodulo a 256-bit integer mwithl= 20011.
+Applications that rely on Φ lcan often improve their running times by using
+alternative modular polynomials that have smaller coefficients. Our alg orithm can
+be adapted to compute polynomials Φg
+lrelatingg(z) andg(lz), for modular func-
+tionsgthat share certain properties with j. This includes the cube root γ2ofj,
+and we are then able to compute Φ lmodmmore quickly by reconstructing it from
+Φγ2
+lmodm, capitalizing on a suggestion in [22]. Other examples include simple
+and double eta-quotients, the Atkin functions, and the Weber f-function. The last
+is especially attractive, since the modular polynomials for fare approximately 1728
+timessmallerthanthosefor j. Thishasallowedustocomputemodularpolynomials
+Φf
+lwithlas large as 60013.
+The outline of this article is as follows. In Section 2 we give a rough over view of
+our new algorithm. The theory behind the algorithm is presented in Se ctions 3–5.
+Wepresentthe algorithm, proveitscorrectnessandanalyzeitsru ntime inSection 6.
+Section 7 deals with modular polynomials for modular functions other t hanj, and
+a final Section 8 contains computational results.
+2.Overview
+Our basic strategy is a standard CRT approach: we compute Φ lmodpfor various
+primespandusetheChineseRemainderTheoremtorecoverΦ l∈Z[X,Y]. Alterna-
+tively, the explicit CRT (mod m) allows us to directly compute Φ l∈(Z/mZ)[X,Y],
+via [4, Thm. 3.1]. By applying the algorithm of [54, §6], this can be accomplished
+inO(l2loglm) space, even though the total size of all the Φ lmodpisO(l3logl).MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 3
+Ourmethod forcomputingΦ lmodpis new, andappliesonlyto certainprimes p.
+Strategic prime selection has been used effectively in other CRT-bas ed algorithms,
+suchas[54], andit is especiallyhelpful here. Workingin the finite field Fp, we select
+l+2 distinct values ji, compute Φ l(X,ji)∈Fp[X] for each, and then interpolate
+the coefficients of Φ l∈(Fp[Y])[X] as polynomials in Fp[Y]. The key lies in our
+choice ofp, which allows us to select particular interpolation points that greatly
+facilitate the computation. We are then able to compute Φ lmodpin expected time
+(2) O(l2(logp)3loglogp).
+In contrast to the methods above, this is quasi-linear in the size of Φ lmodp.
+Our algorithm exploits the structure of the l-isogeny graph Gldefined on the set
+ofj-invariants of elliptic curves over Fp. Each edge in this graph corresponds to
+anl-isogeny; the edge ( j1,j2) is present if and only if Φ l(j1,j2) = 0. As described
+in [30, 42], the ordinarycomponentsofthis graphhavea particulars tructureknown
+as anl-volcano. Depicted in Figure 1 are a set of four l-volcanoes, each with two
+levels: the surface(atthetop), andthe floor(onthebottom). Notethateachvertex
+jion the surface has l+1 neighbors, these are the roots of Φ l(X,ji)∈Fp[X], and
+there are at least l+2 suchji.
+figure1. A set of l-volcanoes arising from Theorem 4.1. In
+this example l= 7 splits into ideals of order 3 in cl( O) and we
+haveh(O) = 12 surface curves and h(R) = 72 floor curves.
+This configurationcontains enough information to compute the l+2polynomials
+Φl(X,ji) that we need to interpolate Φ l(X,Y) modp. It is not an arrangement
+that is likely to arise by chance; it is achieved by our choice of the orde rOand the
+primespthat we use. To further simplify our task, we choose pso that vertices on
+the surface correspond to curves with Fp-rationall-torsion. Our ability to obtain
+such primes is guaranteed by Theorems 4.1 and 4.4, proven in Section 4.
+The curves on the surface all have the same endomorphism ring typ e, isomorphic
+to an imaginary quadratic order O. Theirj-invariants are precisely the roots of the
+Hilbert class polynomial HO∈Z[X]. As described in [2], the roots of HOmay be
+enumerated via the action of the ideal class group cl( O). To do so efficiently, we
+use an algorithm of [54] to compute a polycyclic presentation for cl(O) that allows
+us to enumerate the roots of HOvia isogenies of low degree, typically much smaller
+thanl. We may use this presentation to determine the action of any elemen t of
+cl(O), including those that act via l-isogenies. This allows us to identify the l-
+isogeny cycles that form the surfaces of the volcanoes in Figure 1.
+Similarly, the vertices on the floor are the roots of HR, whereRis the order of
+indexlinO, and we use a polycyclic presentation of cl( R) to enumerate them. To
+identity children of a common parent (siblings), we exploit the fact th at siblings
+lie in a cycle of l2-isogenies, which we identify using our presentation of cl( R). It
+remains only to connect each parent to one of its children. This may b e achieved
+by using V´ elu’s formula [55] to compute an l-isogeny from the surface to the floor.
+By matching each parent to a group of siblings, we avoid the need to c ompute an
+l-isogeny to every child, which is critical to obtaining the complexity bo und in (2).4 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+Below is a simplified version of the algorithm to compute Φ lmodp.
+Algorithm 2.1. Letlbe an odd prime, and let Obe an imaginary quadratic order
+of discriminant Dwith class number h(O)≥l+ 2. Letp≡1 modlbe a prime
+satisfying 4 p=t2−l2v2Dfor some integers tandvwithl∤v. LetRbe the order
+of indexlinO. Compute Φ lmodpas follows:
+1. Find a root of HOoverFp.
+2. Enumerate the roots jiofHOand identify the l-isogeny cycles.
+3. For each jifind anl-isogenousjon the floor.
+4. Enumerate the roots of HRand identify the l2-isogeny cycles.
+5. For each jicompute Φ l(X,ji) =/producttext
+(ji,jk)∈Gl(X−jk).
+6. Interpolate Φ l∈(Fp[Y])[X] using the jiand the polynomials Φ l(X,ji).
+The conditions on the inputs l,O, andpsuffice to ensure that Theorem 4.1 is
+satisfied, so that we have a configuration of l-volcanoes similar to the example in
+Figure 1. We use the same Ofor eachp, so the Hilbert class polynomial HOmay
+be precomputed, but we do not need to compute HR, instead we enumerate its
+roots by applying the Galois action of cl( R) to a root obtained in Step 3.
+A more detailed version of Algorithm 2.1 appears in Section 6 together with
+Algorithm 6.1, which selects the order Oand the primes p, and performs the CRT
+computations needed to determine Φ loverZ, or modulo m.
+3.Orders in imaginary quadratic fields
+It is a classical fact that the endomorphism ring of an ordinary elliptic curve
+over a finite field is isomorphic to an imaginary quadratic order O. The orderO
+is necessarily contained in the maximal order OKof its fraction field K, but we
+quite often haveO/subsetnoteqlOK. As most textbooks on algebraic number theory focus on
+maximal orders, we first develop some useful tools for working with non-maximal
+orders. Tosimplify thepresentation, weworkthroughoutwith field sofdiscriminant
+dK<−4, ensuring that we always have the unit groups O∗=O∗
+K={±1}. We
+use/parenleftbigdK
+p/parenrightbig
+to denote the Kronecker symbol, which is −1,0, or 1 as the prime psplits,
+ramifies, or remains inert in K(respectively).
+LetObe a (not necessarilymaximal) orderin a quadraticfield Kofdiscriminant
+dK<−4. LetNbe a positive integer prime to the conductor u= [OK:O]. The
+orderR=Z+NOhas indexNinO, and its ideal class group cl( R) is an extension
+of cl(O). More precisely, as in [53, Thm. 6.7], there is an exact sequence
+(3) 1−→(O/NO)∗/(Z/NZ)∗−→cl(R)ϕ−→cl(O)−→1,
+whereϕmaps the class [ I] to the class[ IO]. ForR-ideals prime to uN, the underly-
+ing mapI/ma√sto→IOpreserves ‘norms’, that is, [ R:I] = [O:IO], as in [21, Prop. 7.20].
+We have a particular interest in the kernel of the map ϕ.
+Lemma 3.1. In the exact sequence above, if N=pnis a power of an unramified
+odd primep, thenkerϕis cyclic of order pn−1/parenleftbig
+p−/parenleftbigdK
+p/parenrightbig/parenrightbig
+.
+Proof.We compute the structure of ker ϕ∼=(O/pnO)∗/(Z/pnZ)∗. The group
+(Z/pnZ)∗is cyclic, isomorphic to the additive group (Z/(p−1)Z)×(Z/pn−1Z).
+We now apply [18, Cor. 4.2.11] to compute the structure of ( O/pnO)∗:MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 5
+(4) (O/pnO)∗∼=/braceleftBigg
+(Z/(p−1)Z)2×(Z/pn−1Z)2,ifpsplits inK;
+(Z/(p2−1)Z)×(Z/pn−1Z)2,ifpis inert inK.
+In both cases, the factor ( Z/pn−1Z) of (Z/pnZ)∗is a maximal cyclic subgroup of
+the Sylowp-subgroup of (O/pnO)∗, and must correspond to a direct summand.
+Thus thep-rank of the quotient ( O/pnO)∗/(Z/pnZ)∗is 1. The order of the factor
+(Z/(p−1)Z) of(Z/pnZ)∗isnotdivisibleby p, andmust correspondtoasubgroupof
+a cyclic factor of ( O/pnO)∗in both cases. It follows that the quotient is cyclic. The
+calculations above also show that #( O/pnO)∗/(Z/pnZ)∗=pn−1/parenleftbig
+p−/parenleftbigdK
+p/parenrightbig/parenrightbig
+./square
+Even when ker ϕis not necessarily cyclic, the size of ker ϕis as in Lemma 3.1. More
+generally, the exact sequence (3) can be used to derive the formu la
+(5) h(O) =h(OK)u/productdisplay
+p|u/parenleftbigg
+1−/parenleftbiggdK
+p/parenrightbigg
+p−1/parenrightbigg
+,
+as in [21, Thm. 7.24].
+We now describe a particular representation of ker ϕwhenN=lis prime. In
+this case ker ϕis cyclic, of order l−/parenleftbigdK
+l/parenrightbig
+; this follows from Lemma 3.1 for l >2,
+and from (5) for l= 2. LetO=Z[τ] for someτ∈Kthat is coprime to l. There are
+exactlyl+1 indexlsublattices ofO: the order R, and lattices Si=lZ+(τ+i)Z,
+forifrom 0 tol−1. EachO-ideal of norm lcorresponds to one of the Si. The
+remainingSiare fractional invertible R-ideals corresponding to proper R-ideals
+(6) Ji=lSi=l2Z+l(τ+i)Z,
+for whichR={β∈K:βJi⊂Ji}. Exactly 1+/parenleftbigdK
+l/parenrightbig
+of theSiareO-ideals, leaving
+l−1−/parenleftbigdK
+l/parenrightbig
+properR-idealsJi. These are all non-principal and inequivalent in
+cl(R), and each lies in ker ϕ, since we have JiO=lO. The invertible Jiare exactly
+the non-trivial elements of ker ϕ. We summarize with the following lemma, which
+guarantees that we can find a generator for the cyclic group ker ϕthat has norm l2.
+Lemma 3.2. IfN=lis prime in the exact sequence (3), then theR-ideallRand
+the invertible R-idealsJidefined in (6)are representatives for kerϕ. In particular,
+kerϕis generated by the class of an invertible R-ideal with norm l2.
+This representation of ker ϕhas proven useful in other settings [16]. We use it to
+obtain thel2-isogeny cycles we need in Step 4 of Algorithm 2.1.
+We conclude this section with a theorem that allows us to construct a rbitrarily
+large class groups that are generated by elements of bounded nor m.
+Theorem 3.3. LetObe an order in a quadratic field of discriminant dK<−4,
+and letp∤disc(O)be an odd prime. Let Pbe a set of primes that do not divide
+p[OK:O]. Forn∈Z≥0, letRndenote the order Z+pnO, and letGnbe the
+subgroup of cl(Rn)generated by the set Snof classes of Rn-ideals with norms in P.
+Then ifG2= cl(R2), we haveGn= cl(Rn)for everyn∈Z≥0.
+Proof.For eachRn, letϕn: cl(Rn)→cl(O) denote the corresponding map in
+the exact sequence (3), and let φn+1: cl(Rn+1)→cl(Rn) send [I] to [IRn], so
+thatϕn+1=ϕn◦φn+1. These are all surjective group homomorphisms, and the
+underlying ideal maps preserve the norms of ideals prime to p[OK:O]. We assume
+G2= cl(R2), which implies Gn= cl(Rn) forn≤2, and proceed by induction on n.6 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+For each prime q∈Pand everyn, there are exactly 1 −/parenleftbigdK
+q/parenrightbig
+ideals inRnof
+normq, andφn+1mapsSn+1ontoSnandGn+1ontoGn. By the inductive hypoth-
+esis,Gn= cl(Rn), therefore Gn+1intersects every coset of ker φn+1⊂kerϕn+1. To
+proveGn+1= cl(Rn+1), it suffices to show ker ϕn+1⊂Gn+1.
+The groups ker ϕnand kerϕn+1are cyclic, by Lemma 3.1, since pis odd and
+unramified. Let αnbe a generator for ker ϕn. Since #ker ϕnis divisible by p,
+αncannot be a pth power in ker ϕn. Expressing αnin terms of Sn, we see that
+φ−1
+n+1(αn) must intersect Gn+1. Letαn+1lie in this intersection, and note that
+αn+1∈kerϕn+1. The order of αn+1must be a multiple of |αn|= #kerϕn, and
+αn+1cannot be a pth power in ker ϕn+1. It follows that αn+1has order #ker ϕn+1,
+hence it generates ker ϕn+1, proving ker ϕn+1⊂Gn+1as desired. /square
+To see Theorem 3.3 in action, let Obe the order of discriminant D=−7, let
+p= 3, and letP={2}. The class group of the order Rnof discriminant 32nD
+happens to be generated by an ideal of norm 2 when n= 2, and the theorem then
+implies that this holds for all n. This allows us to construct arbitrarily large cyclic
+class groups, each generated by an ideal of norm 2.
+We remark that Theorem 3.3 may be extended to handle p= 2 if the condition
+G2= cl(R2) is replaced by G3= cl(R3), and easily generalizes to treat families of
+orders lying inOthat haveb-smooth conductors, for any constant b.
+4.Explicit CM theory
+4.1.The theory of complex multiplication (CM). As in Section 3, let Obe
+an order in a quadratic field Kof discriminant dK<−4. We fix an algebraic
+closure ofK. It follows from class field theory that there is a unique field KOwith
+the property that the Artin map induces an isomorphism
+Gal(KO/K)∼−→cl(O)
+between the Galois group of KO/Kand the ideal class group of O. The field KO
+is called the ring class field for the orderO. IfOis the maximal order of K, then
+KOis the Hilbert class field of K, the maximal totally unramified abelian extension
+ofK. In general, primes dividing [ OK:O] ramify in the ring class field.
+The first main theorem of complex multiplication [21, Thm. 11.1] states that
+KO=K(j(E)),
+for any complex elliptic curve Ewith endomorphism ring O. Furthermore, the
+minimal polynomial HOofj(E) overKactuallyhas coefficientsin Z, and its degree
+ish(O) =|cl(O)|. The polynomial HOis known as the Hilbert class polynomial .
+Ifpis a prime that splits completely in the extension KO/Q, thenHOsplits into
+distinct linear factors in Fp[X]. Its roots are the j-invariants of the elliptic curves
+E/Fpwith End(E)∼=O, a set we denote Ell O(Fp). LetD= disc(O). The primes
+that split completely in KOare precisely the primes p∤Dthat are the norm
+p=NK/Q/parenleftBigg
+t+v√
+D
+2/parenrightBigg
+=t2−v2D
+4
+of an element ofO. The equation 4 p=t2−v2Dis often called the norm equation .
+For a positive integer N, there is a unique extension KN,Oof the ring class field
+KOsuch that the Artin map induces an isomorphism
+Gal(KN,O,KO)∼−→(O/NO)∗/{±1}.MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 7
+The fieldKN,Ois theray class field of conductor NforO. WhenO=OK, this is
+simply the ray class field of conductor N, and forN= 1 we recover the ring class
+fieldKO=K1,O. The ring class field KRof the order R=Z+NOis a subfield of
+the ray class field KN,O. The Galois group of KR/KOis isomorphic to
+(O/NO)∗/(Z/NZ)∗,
+the kernel of the map ϕin (3).
+The second main theorem of complex multiplication [21, Thm. 11.39] sta tes that
+KN,O=KO(x(E[N])),
+wherex(E[N]) denotes the set of x-coordinatesofthe N-torsionpoints ofan elliptic
+curveEwith endomorphism ring O. The Galois invariance of the Weil pairing
+E[N]×E[N]→µNimplies that the cyclotomic field Q(ζN) is contained in the
+ray class field KN,O(a fact that also follows directly from class field theory). In
+particular,aprime pthatsplits completelyin KN,Oalsosplits completelyin Q(ζN),
+and is therefore congruent to 1 modulo N.
+4.2.Primes that split completely in the ray class field. We are specifically
+interested in primes pthat split completely in the ray class field Kl,O, wherelis
+an odd prime. For such pwe can achieve the desired setting for Algorithm 2.1, as
+depicted in Figure 1.
+Theorem 4.1. Letl >2be prime, and let O/n⌉}ationslash⊂Z[i],Z[ζ3]be an imaginary qua-
+dratic order that is maximal at l. LetR=Z+lObe the order of index linO.
+Letpbe a prime that splits completely in the ray class field Kl,O, but does not split
+completely in the ring class field for the order of index l2inO.
+(1)There are exactly h(O)different Fp-isomorphism classes of elliptic curves
+E/Fpwith endomorphism ring Othat haveE[l]⊂E(Fp).
+(2)There are exactly h(R)different Fp-isomorphism classes of elliptic curves
+E/Fpwith endomorphism ring Rthat have an Fp-rationall2-torsion point.
+Proof.The inclusions KO⊆KR⊆Kl,Oimply that both HOandHRsplit into
+linear factors in Fp. Eachj-invariant in Ell O(Fp), resp. Ell R(Fp), corresponds
+to two distinct isomorphism classes over Fp, since these curves are ordinary and
+O/n⌉}ationslash=Z[i],Z[ζ3]. We will show that exactly one of these satisfies (1), resp. (2).
+Sincepsplits completely in the ray class field Kl,O, we can factor p=πpπp∈O
+withπp≡1 modlO. LetE/Fpbe an elliptic curve with endomorphism ring O
+whose Frobenius endomorphism corresponds to πpunder one of the two isomor-
+phisms End( E)∼−→O. Sinceπp≡1 modlO, we haveE[l]⊂E(Fp). The Frobe-
+nius endomorphism of the non-isomorphic quadratic twist ˜E/Fpcorresponds to
+−πp, and we then have # ˜E(Fp) =p+1−tr(−πp)≡2 modl. Forl/n⌉}ationslash= 2 this implies
+that˜Ehas triviall-torsion over Fp, proving (1).
+To prove (2), let E′/Fpbe a curve with endomorphism ring Rthat isl-isogenous
+toE. The Frobenius endomorphism of E′also corresponds to πpunder an isomor-
+phism End( E′)∼−→R. The cardinality of E′(Fp) is thus equal to the cardinality
+ofE(Fp) and therefore divisible by l2. However, since pdoesnotsplit completely
+in the ring class field of index l2inO, we cannot have πp≡1 modlR. It follows
+thatE′[l]/n⌉}ationslash⊂E′(Fp) andE′(Fp) must contain a point of order l2. As above, the
+quadratic twist of E′must have trivial l-torsion over Fp, proving (2). /square8 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+Provided the order Oin Theorem 4.1 also satisfies h(O)≥l+2, we can achieve
+the desired setting for Algorithm 2.1. We say such an order is suitableforl.
+To determine the coefficients Φ lvia the Chinese Remainder Theorem, we need to
+compute Φ lmodpfor many primes psatisfying Theorem 4.1. We necessarily have
+p>l, sincep≡1 modl, and the height bound [20] on the coefficients of Φ limplies
+that 6l+O(1) primes suffice. We now show these primes exist and bound their
+size, assuming the GRH. For this purpose we define a suitable family of orders .
+Definition 4.2. LetSbe the set of odd primes and let Tbe the set of all imaginary
+quadratic orders. A suitable family of orders is a function F:S→Tsuch that:
+(1)for alll∈Sthe orderF(l)is suitable for l.
+(2)there exist effective constants c1,c2∈R>0such that for all l∈Sthe bounds
+l+2≤h(F(l))≤c1landl2≤|disc(F(l))|≤c2l2hold.
+Example 4.3. LetF(3) =Z[√−47], and forl >3letF(l)be the orderOof
+discriminant−7·32n, wherenis the least integer for which h(O) = 2·3n−1≥l+2.
+Lettingc1= 4andc2= 205, we see thatFis a suitable family of orders.
+Theorem 4.4. LetFbe a suitable family of orders and let c0∈R>0be an arbitrary
+constant. Then for each prime l >2the set of primes pfor whichl,O=F(l),
+andpsatisfy the conditions of Theorem 4.1 has positive density.
+Assuming the GRH, there is an effective constant c∈R>0such that at least
+c0l3(logl)3of these primes are bounded by B=cl6(logl)4, for all primes l>7.
+Proof.For a prime l>2, letO=F(l) have fraction field K, and letu= [OK:O].
+The ray class field Kl,Oand the ring class field KSfor the order S=Z+l2O
+are both invariant under the action of complex conjugation, hence both are Galois
+extensions of Q. One finds that
+#Gal(Kl,O/Q) = 2/parenleftbig
+l−1/parenrightbig/parenleftbig
+l−/parenleftbigdK
+l/parenrightbig/parenrightbig
+h(O)<2l/parenleftbig
+l−/parenleftbigdK
+l/parenrightbig/parenrightbig
+h(O) = #Gal(KS/Q),
+and the Chebotar¨ evdensity theorem [49, Thm. 13.4]yields the unc onditional claim.
+To prove the conditional claim, we apply an effective Chebotar¨ ev bo und to the
+extensionKl,O/Q, assuming the GRH for the Dedekind zeta function of Kl,O.
+The extension Kl,O/Kis abelian of conductor dividing lu, with degree nh(O),
+wheren≤2#(O/lO)∗≤2l2. TheOK-ideal disc(Kl,O/O) is a divisor of ( lu)nh(O),
+by Hasse’s F¨ uhrerdiskriminantenproduktformel [49, Thm. VII.11.9]. We then have
+|disc(Kl,O/Q)|=|NK/Q(disc(Kl,O/K))·disc(K/Q)[Kl,O:K]|
+≤(lf)2nh(O)|disc(K/Q)|nh(O)≤(c2l4)nh(O),
+where disc(O)≤c2l2. Using the bound h(O)≤c1l, Theorem 1.1 of [43] then yields
+(7)/vextendsingle/vextendsingle/vextendsingle/vextendsingleπ(x,Kl,O/Q)−Li(x)
+2nh(O)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤c3/parenleftBig
+x1/2log(lx)+l3logl/parenrightBig
+,
+whereπ(x,Kl,O/Q) counts the primes up to x∈R>0that split completely in Kl,O,
+andc3∈R>0is an effectively computable constant, independent of l.
+If we now suppose x=cl6(logl)4, and apply Li( x)∼xlogxandnh(O)≤c1l3,
+we may choose c∈R>0so that Li(x)/(2nh(O)) is greater than the RHS of (7)
+by an arbitrarily large constant factor. In particular, for any c4∈R>0there
+is an effectively computable choice of cthat ensures π(x,Kl,O/Q)≥c4l3(logl)3,
+independent of l. Moreover, for the least such cwe havec/c4→1 asc4→∞.MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 9
+We now show that most of these primes do not split completely in KS. Any
+primepthat splits completely in Kl,Omust split completely in the ring class field
+forR=Z+lO. PuttingD= disc(O), we then have
+(8) 4 p=t2−v2l2D,
+witht,v∈Z>0andt≡2 modl. Ifv/n⌉}ationslash≡0 modl, thenpcannot split completely
+inKS. Forp≤cl6(logl)4, we havev≤2c1/2l(logl)2andt≤2c1/2l3(logl)2, since
+D≥l2, hence there are at most 2 c1/2(logl)2positivev≡0 modl, and at most
+2c1/2l2(logl)2+1 positive t≡2 modl, that satisfy (8).
+It follows that no more than 4 cl2(logl)4+ 2c1/2(logl)2primesp≤cl6(logl)4
+split completely in KS. For a sufficiently large choice of c4, we can choose cso that
+π(x,Kl,O/Q) =π(cl6(logl)4,Kl,O/Q)≥c4l3(logl)3and also
+c4l3(logl)3−4cl2(logl)4−2c1/2(logl)2>c0l3(logl)3,
+providedl>7, sincel/(logl) is bounded above 4 for primes l>7. /square
+Theorem 4.4 guarantees we can obtain a sufficient number of primes pfor use
+with Algorithm 2.1. In fact, as is typical for such bounds, it provides far more than
+we need. The task of finding these primes is addressed in Section 6.1.
+4.3.Computing the CM action. The Galois action of Gal( KO/K)∼=cl(O) on
+the set Ell O(Fp) may be explicitly computed using isogenies, as described in [2].
+Let the prime psplit completely in the ring class field KO, and letE/Fpbe an
+elliptic curve with End( E)∼=O. Fixing an isomorphism End( E)∼−→O, for each
+invertibleO-idealawe define
+E[a] ={P∈E(Fp)|∀τ∈a:τ(P) = 0},
+the‘a-torsion’subgroupof E. Thesubgroup E[a]isthekernelofaseparableisogeny
+E→E/E[a] of degree [O:a], with End( E/E[a])∼=O. This yields a group action
+j(E)a=j(E/E[a]),
+in which the ideal group of Oacts on the set Ell O(Fp). This action factors through
+the class group, and the cl( O)-action is transitive and free. Equivalently, Ell O(Fp)
+is atorsorfor cl(O); for each pair ( j1,j2) of elements in Ell Othere is a unique
+element of cl(O) whose action sends j1toj2.
+Now let l0be an invertibleO-ideal of prime norm l0/n⌉}ationslash=p. The curves Eand
+E/E[l0] arel0-isogenous, hence
+Φl0(j0,jl0
+0) = 0,
+wherej0=j(E). To compute the action of l0, we need to find the corresponding
+root of Φ l0(X,j0)∈Fp[X]. We assume that Φ l0(X,Y) is known, either via one of
+the algorithmsfrom the introduction, or by a previous application of Algorithm 6.1.
+The polynomial Φ l0(X,j0)∈Fp[X] has either 1 or 2 roots that lie in Ell O(Fp),
+depending on whether l0ramifies or splits (it is not inert). These roots correspond
+to the actions of l0and its inverse l-1
+0, which coincide when l0ramifies.
+Our fixed isomorphism End( E)∼−→Omaps the Frobenius endomorphism of E
+to an element πp∈O⊂O Kwith norm p. We then have the norm equation
+(9) 4 p=t2−v2dK,
+wheret= tr(πp), andvis the index of Z[πp] inOK. Whenl0does not divide v,
+the order Z[πp] is maximal at l0and the only roots of Φ l0(X,j0) overFpare those10 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+in EllO(Fp). Otherwise Φ l0(X,j0) hasl+1 roots in Fp, and those in Ell O(Fp) lie
+on the surface of the l0-volcano containing j, as described in [30]. The roots on the
+surface can be readily distinguished, as in [54, §4], for example, but typically we
+choosepwithl0∤vso that every root of Φ l0(X,j0) inFpis on the surface.
+Whenl0splitsanddoesnotdivide v, theactionsof l0andl-1
+0maybedistinguished
+asdescribedin[11, §5]and[31,§3]. Thekernelsofthetwo l0-isogeniesaresubgroups
+ofE[l0]. A standard component of the SEA algorithm computes a polynomial
+Fl0(X), whose roots are the abscissa of the points in one of these kerne ls [22, 51].
+In our setting l0splits inZ[πp], and provided l0∤v, the action of πponE[l0] has two
+distinct eigenvalues corresponding to the two kernels. Expressing the ideal l0in the
+form (l0,c+dπp) yields the eigenvalue λ=−c/dmodl0. We may then use Fl0(X)
+to test whether πp’s action is equivalent to multiplication by λin the corresponding
+kernel. See [11] for an example and further details.
+As a practical optimization (see Section 6.6), we avoid the need to ev er make
+this distinction. The asymptotic complexity of computing the action o fl0is the
+same in any case.
+Lemma 4.5. Letl0andpbe distinct odd primes, and let O /n⌉}ationslash=Z[i],Z[ζ3]be
+an imaginary quadratic order. Let j0=j(E)∈EllO(Fp), fix an isomorphism
+End(E)∼−→O, and letπp∈Odenote the image of the Frobenius endomorphism.
+Letl0be an invertibleO-ideal of norm l0, and assume Z[πp]is maximal at l0.
+GivenΦl0∈Fp[X,Y], thej-invariantjl0
+0may be computed using an expected
+O(l2
+0+M(l0)logp)operations in Fp.
+HereM(n) denotes the complexity of multiplying two polynomials of degree less
+thann, as in [56, Def. 8.26]. Na¨ ıvely, M(n) =O(n2), Karatsuba’s algorithm yields
+M(n) =O(nlog23), and methods based on the fast Fourier transform (FFT) achiev e
+M(n) =O(nlognloglogn).
+Proof.We first compute gcd( Xp−X,Φl0(X,j0)), the product of the distinct linear
+factorsofΦ l0(X,j0) overFp. Instantiating f(X) = Φl0(X,j0) usesO(l2
+0) operations
+inFp, exponentiating XpmodfusesO(M(l0)logp) operations in Fp, and the fast
+Euclideanalgorithm[56, §11.1]obtainsgcd( Xp−X,f)usingO(M(l0)logl0) =O(l2
+0)
+operations in Fp. This gcd has degree at most 2, since Z[πp] is maximal to l0, and
+we may find its roots using an expected O(logp)Fp-operations [56, Cor. 14.16].
+The desired root jl0
+0is then distinguished as outlined above. We first compute
+the eigenvalue λ−c/dmodl0, where l0= (l0,c+dπp), usingO(l2
+0) bit opera-
+tions. Applying [9, Thm. 2.1], the kernel polynomial Fl0(X) can be computed
+usingO(M(l0)) operations in Fp. To compare ( Xp,Yp) to the scalar multiple
+λ·(X,Y), we compute Xp,Yp, and the required division polynomials ψn(X,Y),
+moduloFl0(X) and the curve equation for E, as in the SEA algorithm [7, Ch. VII].
+This usesO((logl0+logp)M(l0)) =O(l2
+0+M(l0)logp) operations in Fp./square
+5.Mapping the CM torsor
+The previoussectionmadeexplicit theGaloisactioncorrespondingto anelement
+of cl(O)∼=Gal(KO/K) represented by an ideal l0of prime norm l0. We now use
+this to enumerate the set Ell O(Fp), and at the same time compute a map that
+explicitly identifies the action of each element of cl( O). To do this efficiently it is
+critical to work with generators whose norms are small, since the co st of computing
+the action of l0increases quadratically with its norm.MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 11
+5.1.Polycyclic presentations. As a finite abelian group, each element of cl( O)
+canbeuniquelyrepresentedusingabasis. However,asnotedin[54, §5.3],thenorms
+arising in a basis may need to be much larger than those in a set of gene rators.
+Thus we are led to consider polycyclic presentations.
+Letα= (α1,...,α k) be a sequence ofgeneratorsfor G, and letGi=/an}brack⌉tl⌉{tα1,...,α i/an}brack⌉tri}ht
+denote the subgroup generated by α1,...,α i. The composition series
+1 =G0≤G1≤···≤Gk−1≤Gk=G,
+is then polycyclic, meaning that each quotient Gi+1/Giis a cyclic group. The
+sequencer(α) = (r1,...,r k) ofrelative orders forαis defined by
+ri=|Gi:Gi−1|.
+Eachrinecessarily divides |αi|, and fori >1 we typically have ri<|αi|. The
+sequences αandr(α) allow us to uniquely represent each β∈Gin the form
+(10) β=αx=αx1
+1···αxk
+k,
+wherex= (x1,...,x k) with 0≤xi< ri. The vector s(α,i) =xfor which
+αri
+i=αxhasxj= 0 forj≥i, and is called a power relation , see [38,§8.1]. A
+generic algorithm to compute r(α) and thes(α,i) can be found in [54, Alg. 2.1].
+The vector xin (10) is the discrete logarithm or exponent vector ofβ. We let
+X(α) ={x∈Zk: 0≤xi<rk},
+and note that the map x/ma√sto→αxdefines a bijection from X(α) toG.
+We now consider the case G= cl(O), whereOis an order in a quadratic field
+Kof discriminant dK<−4. LetP= (p1,p2,p3,...) be an increasing sequence of
+primes with the property that cl( O) is generated by the classes of invertible ideals
+with norms inP. By Dirichlet’s density theorem [21, Thm. 9.12], any sequence
+containing all but a finite set of primes works, and from [21, Cor. 7.17 ] we know
+that some finite prefix of Pactually suffices. For O=OKwe may takePto be
+the sequence of primes less than |dK/3|1/2, by [14, Prop. 9.5.2].
+There is a unique lexicographically minimal subsequence ( l1,...,lk) ofPthat
+corresponds to a polycyclic sequence α= (α1,...,α k) for cl(O) in which αiis
+represented by an ideal of norm liandr(α) hasri>1. Whenlisplits there are
+two possibilities for αi. To fix a choice, let αibe the ideal class represented by the
+unique binary quadratic form ax2+bxy+cy2of discriminant D= disc(O) with
+a=liandbnonnegative[14,§3.4], correspondingtotheideal li= (li,(−b+√
+D)/2).
+We callαthepolycyclic presentation of cl(O) determined by P. We usel(α)
+to denote the sequence of norms ( l1,...,lk), but note that this also depends on P;
+eachliis the least prime in Pthat is the norm of an ideal in αi.
+We may compute αby applying [54, Alg. 2.1] to an implicit sequence of gen-
+eratorsγ= (γ1,γ2,γ3,...) corresponding to the subsequence of Pfor which there
+exists an invertible O-ideal of norm pi. The algorithm computes rifor eachγiin
+turn, and if we find that ri>1, we append γito an initially empty vector α. We
+terminate when/producttextri=h(O), a value which we assume has been precomputed.
+The computation of αuses|G|=h(O) group operations in G= cl(O), and
+creates a table T:X(α)→Gthat stores|G|=h(O) group elements [54, Prop. 6].
+Using binary quadratic forms to represent cl( O), the group operation has bit-
+complexity O(log2|disc(O)|), as shown in [5], and each element may be stored in
+O(log|disc(O)|) space. Evaluating T(x), orT−1(β), has bit-complexity O(log|G|).12 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+5.2.Suitable presentations. WhenPis the sequence of all primes, the norms
+l(α) for the polycyclic presentation αof cl(O) determined by Pare as small as
+possible. However, when working in the finite field Fpwe may wish to ensure that
+each norm lidoes not divide v= [OK:Z[πp]], as noted in Section 4.3. This is
+achieved by excluding from Pprimes that divide v, and we call the corresponding
+αthe presentation of cl( O)suitableforp. This may cause us to use norms that are
+slightly larger than optimal. We now show that, provided we work with a family
+of orders that satisfies certain (easily met) constraints, the nor ms in every suitable
+presentation are quite small, assuming the GRH.
+Theorem 5.1. Letc3∈R>0be a fixed constant, and let Fbe a suitable family of
+orders with the following additional property: if Ois an order inFwhose fraction
+fieldKhas discriminant dK, andsOdenotes the square-free part of [OK:O], then
+sOis coprime to 2dkand bothsOand|dK|are bounded by c3.
+Then under the GRH, for every OinFand every prime pthat splits completely
+inKO, the presentation αofcl(O)suitable for phas normsl(α)for which
+maxl(α)≤cω(v)log(ω(v)+1),
+wherevis defined by 4p=t2−v2disc(O), the function ω(v)counts the distinct
+prime factors of v, andcis an effective constant that depends only on c3.
+Proof.LetO,p,v, andαbe as above. Let Rbe the order of index s2
+OinOK,
+and letγbe the presentation of Rdetermined by the increasing sequence of primes
+that do not divide v. It follows from Theorem 3.3 that max l(α)≤maxl(γ).
+LetKRbe the ring class field for R, and letπC(x,KR/Q) count the primes
+bounded by x∈R>0whose Frobenius symbol (under the Artin map) lies in the
+conjugacy class Cof Gal(KR/Q). We may bound [ KR:Q], #Gal(KR/Q), and
+disc(KR) by constants that depend only on c3, independent ofO. Under the GRH,
+the Chebotar¨ ev bound of [43, Thm. 1.1] then yields
+πC(x,KR/Q)≥c4x/logx,
+for some effective constant c4∈R>0and allx >2, wherec4depends only on c3.
+For an effective constant cdepending on c3, settingx=cω(v)log(ω(v)+1) yields
+πC(x,KR/Q)>ω(v). In this case the Frobenius symbol of at least one prime not
+dividingvlies inC, and this applies to every C. It follows that every class in cl( R)
+contains an element whose norm is a prime bounded by xthat does not divide v.
+We then have max l(α)≤maxl(γ)≤x=cω(v)log(ω(v)+1), as desired. /square
+The family of orders in Example 4.3 satisfies the requirements of Theo rem 5.1.
+We note that provided log p=O(logl), we haveω(v) =O(logl/loglogl), and the
+theorem then yields an O(logl) bound on the norms l(α). This is sharper than
+the more general O(log2l) bound implied by [1]. In fact, by [34, Thm. 431], one
+expectsω(v) =O(loglogp), which yields a bound of O(loglogllogloglogl).
+5.3.Realizing the CM torsor. We now consider how to explicitly map cl( O)
+to the torsor Ell O(Fp), so that we may then compute the action of any element
+or subgroup of cl( O) on any element of Ell O(Fp),without needing to compute any
+further isogenies . We use the presentation αof cl(O) suitable for p, and the table
+T:X(α)→cl(O) described in Section 5.1. As above, we have α= ([l1],...,[lk]),
+with norms l(α) = (l1,...,lk) and relative orders r(α) = (r1,...,r k). We assume
+the modular polynomials Φ l1,...,Φlkare known, since the liare small.MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 13
+To enumerate Ell O(Fp) we use [54, Alg. 1.3], but we augment this algorithm
+to also compute an explicit bijection φ: cl(O)→EllO(Fp) in which [ a]∈cl(O)
+corresponds to ja
+0. Givenj0∈EllO(Fp), we compute a path of lk-isogenies
+(11) j0lk−→j1lk−→j2lk−→···lk−→jrk−1,
+wherethejiaredistinctelementsofEll O(Fp). AsexplainedinSection4.3,eachstep
+in this path is computed by finding a root jiof Φ(X,ji−1) that lies in Ell O(Fp).
+Whenlksplits inKwe have two choices for j1, and the correct choice may be
+determined using a kernel polynomial as outlined in Section 4.3. For i>1 we use
+the polynomial Φ( X,ji−1)/(X−ji−2), which has exactly one root ji∈EllO(Fp).
+Whenk >1, the enumeration of Ell O(Fp) proceeds recursively: for each ji
+in (11) we compute a path of lk−1isogenies containing rk−1distinctj-invariants.
+Eventually, every element of Ell O(Fp) is enumerated exactly once [54, Prop. 5].
+For eachjn∈EllO(Fp) we also compute a vector x∈X(α) that describes the
+path used to reach jnfromj0, wherexiindicates the number of steps taken on an
+li-isogeny path. By correctly choosing the direction of each path, w e ensure that
+eachjnis the image of j0under the cl(O)-action of αx=αx1
+1···αxk
+k. This yields
+the desired bijection φ; sinceαxuniquely represents some β∈cl(O), we may set
+φ(αx) =jn, a process facilitated by the map T:X(α)→cl(O).
+The bijection φallows us translate any computation in the group cl( O) to the
+torsor Ell O(Fp). In particular, by enumerating the cyclic subgroup H⊆cl(O)
+generated by [ l], where lis an ideal of norm l, we obtain the l-isogeny cycle con-
+tainingj0, corresponding to the surface of one of the l-volcanoesin Figure 1. Doing
+the same for each coset of Hpartitions Ell O(Fp) intol-isogeny cycles.
+This may also be applied to the order R=Z+lO. After obtaining a bijection
+from cl(R) to Ell R(Fp), we enumerate the kernel of the map ϕ: cl(R)→cl(O) from
+the exact sequence of (3). Here we use one of the generators of norml2guaranteed
+by Lemma 3.2. Enumerating the cosets of ker ϕthen partitions Ell R(Fp) intol2-
+isogeny cycles of siblings with a common l-isogenous parent in Ell O(Fp).
+6.The algorithm
+We now present our algorithm to compute the modular polynomial Φ lusing the
+Chinese Remainder Theorem (CRT). Algorithm 6.1 follows the standar d pattern
+of a CRT-based algorithm; the details lie in Algorithm 6.2, which selects a set of
+primesS, and in Algorithm 2.1, which computes Φ lmodulo each prime p∈S.
+The computation of Φ l∈Z[X,Y] may be viewed as a special case of computing
+Φl∈(Z/mZ)[X,Y], wheremis the product of the primes in S. The choice of S
+ensures that this mis large enough to uniquely determine Φ l∈Z[X,Y].
+Algorithm6.1. Letlbeanoddprime,let mbeapositiveinteger,andlet O=F(l)
+lie in a suitable family of orders F. Compute Φ l∈(Z/mZ)[X,Y] as follows:
+1. Compute the Hilbert class polynomial HO∈Z[X].
+2. Select a set of primes Swith Algorithm 6.2, using landO.
+3. Perform CRT precomputation using S.
+4. For each prime p∈S:
+a. Compute Φ lmodpwith Algorithm 2.1, using OandHO.
+b. Update CRT data using Φ lmodp.
+5. Perform CRT postcomputation.
+6. Output Φ l∈(Z/mZ)[X,Y].14 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+The suitable family of orders Fis as defined in Section 4.2, see Definition 4.2.
+The polynomial HOcomputed in Step 1 may be obtained using any of several
+algorithms whose running time is quasi-linear in disc( O), including [11, 23, 54].
+6.1.Selecting primes. The primes pin the setSselected by Algorithm 6.1 must
+satisfy the conditions of Theorem 4.1 in order to use them in Algorithm 2.1. We
+requirepto split completely in the ray class field Kl,O, but to not split completely
+in the ring class field for the order Z+l2O. Equivalently, we need p∤Dto satisfy
+(12) 4 p=t2−v2l2D,
+witht≡2 modlandl∤v, whereD= disc(O). To apply the CRT, we also require
+(13)/productdisplay
+p∈Slogp≥4|c|,
+for every coefficient cof Φl∈Z[X,Y]. From [13], we use the explicit bound
+(14) Bl= 6llogl+18l
+on the logarithmic height of Φ lto achieve this. We then have # S=O(l), by (12).
+Heuristically, it iseasytofind primesthat satisfy(12). If D≡1 mod 8, fix v= 2,
+otherwise fix v= 1. Then, for increasing t≡2 modlwith the correct parity, test
+whetherp= (t2−v2l2D)/4 is prime. We expect to need O(llogl) primality tests,
+and each can be accomplished in time polynomial in log l, although typically pis
+small enough to make an attempted factorization more efficient. We could obtain
+slightly smaller p’s by letting vvary, but it is more convenient to fix vso that we
+can use the same presentation of cl( O) and cl(R) for everyp. This approach is easy
+to implement and very fast in practice.
+However, in order to prove Theorem 1 we must take a more cautious approach.
+Even assuming the GRH, we cannot guarantee we will find anyprimes with a fixed
+value ofv. On the other hand, Theorem 4.4 implies that if we construct random in-
+tegersp≤xsatisfying (12), for sufficiently large xwe havepprime with probability
+Ω(1/logx), and under the GRH, x=O(l6(logl)4) is large enough. Additionally,
+we would like to avoid v’s with many prime factors, so that we may more prof-
+itably apply Theorem 5.1. The restriction ω(v)≤2log(logv+ 3) eliminates an
+asymptotically negligible proportion of the integers v∈[1,x] (see Lemma 8.1).
+We now present Algorithm 6.2, emphasizing that its purpose is to facilit ate the
+proof of Theorem 1. In practice we use the heuristic procedure de scribed above.
+Algorithm 6.2. Letlbe an odd prime, let D<−4 be a discriminant, and let Bl
+be as in (14). Construct the set Sas follows:
+1. Setn←(Bl+2log2)/log(l2|D|/4) and then x←4l2|D|nlogn.
+Setb←0 andS←∅.
+2. SetT←2x1/2andV←2x1/2l−1|D|−1/2.
+3. Repeat⌈2Nlogx⌉times:
+a. Construct an integer p= (t2−v2l2D)/4 using uniformly random integers
+v∈[1,V] andt∈[1,T], subject to l∤v,t≡2 modl, andt≡vDmod 2.
+b. Ifω(v)>2log(logv+3) then go to Step 3d.
+c. Ifp /∈Sandpis prime then set S←S∪{p}andb←b+logp.
+d. Ifb>Bl+2log2 then output Sand terminate.
+4. Setx←2xand go to Step 2.MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 15
+In Step 3a, the integer tis generated as t=al+ 2 using a uniformly random
+integera∈[0,V/l−2]. The computation of ω(v) in Step 3b is performed by
+factoringv. Note that # S≤n, andp≤xfor allp∈S.
+Lemma 6.3. LetFbe a suitable family of orders, let lbe an odd prime and let
+D= disc(F(l)). Given inputs landD, the expected running time of Algorithm 6.2
+is finite. Under the GRH we also have the following:
+(1)The expected running time is O(l1+ε), for anyε∈R>0.
+(2)There is a constant c<1such that for all l>7andk∈Z>0, the algorithm
+terminates with logx≤(6+k)loglwith probability at least 1−c−klogl.
+Proof.To analyze Algorithm 6.2, we count the number of times Step 4 is execu ted,
+referring to the period between each execution as an iteration. By Theorem 4.4,
+the set of primes that satisfy Theorem 4.1, equivalently, those tha t satisfy (12), has
+positive density. Here we may use the natural density, via [40, Thm. 4.3.e]. For
+every fixed odd prime l, this implies a lower bound of Ω(1 /logx) on the proba-
+bility that a random integer in [1 ,x] is a prime that satisfies (12). Each integer p
+tested by Algorithm 6.2 necessarily satisfies (12), hence such a pis prime with
+probability Ω(1 /logx). By Lemma 8.1 in the appendix, the probability that a
+candidatepis skipped due to the test in Step 3b is o(1). This implies that for
+all sufficiently large x, the probability that Algorithm 6.2 terminates in a given
+iteration is bounded above zero, and the expected running time is fin ite.
+Now assume the GRH and let l >7. Applying Theorem 4.4 with c0= 1, there
+are at least l3(logl)3primesp≤c1l6(logl)4that satisfy (12), for some constant
+c1∈R>0that does not depend on l. Letx0be the least value of x≥c1l6(logl)4).
+Whenx=x0we haveVT/l≤8c1l3(logl)4, since|D|≥l2, and the probability that
+a given primality test succeeds is at least 8 c1/logl≥c2/logx, for some constant
+c2∈R>0. From the inequality (7) in the proof of Theorem 4.4, one finds that t his
+holds for all x≥x0, with the same constants. As above, Step 3b has negligible
+impact, and for x≥x0the probability that the algorithm terminates in a given
+iteration is at least c, for some constant c∈R>0independent of landx.
+We now consider the running time as a function of l, fixing an arbitrary ε∈R>0.
+It takesO(logl) iterations to achieve x=x0, assuming that we don’t terminate
+earlier, and we execute Steps 3b and 3c a total of O(l(logl)2) times during this
+process. The computation of ω(v) and the primality test of pcan both be achieved
+in expected time subexponential in log x, by [46], yielding an O(l1+ε) bound on the
+time to reach x=x0, since logx0=O(log(l)).
+Forx≥x0, the probability of reaching each subsequent iteration declines exp o-
+nentially, while the cost of Steps 3b and 3c grows subexponentially, im plying that
+the total expected running time is also O(l1+ε), proving (1).
+Claim (2) follows from the same analysis. We have log x0= (6+o(1))logland
+addlog2tolog xineachiteration. Once x=x0, it takesmorethan klogliterations
+to reach log x>(6+k)logl. There is a probability of at least cthat the algorithm
+terminates in each subsequent iteration, yielding the bound in (2). /square
+6.2.CRT computations. The computations involved in Steps 3, 4b, and 5 of
+Algorithm 6.1 are described in detail in [54, §6]. We summarize briefly here.
+GivenS={pi}, letM=/producttextpi,Mi=M/pi, andai≡M−1
+imodpi. Letc
+denote a coefficient of Φ l∈Z[X,Y], and letci≡cmodpidenote the corresponding
+coefficientofΦ l∈Fpi[X,Y]. Asin[56,§10.3], wecanusefastChineseremaindering16 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+to efficiently compute
+(15) c≡ciaiMimodM.
+Provided that M >2|c|, we can then lift the result from Z/MZtoZ.
+Whenmis “large,” by which we mean m≥M >2|c|, we compute cmodM,
+lift toZ, and output the integer cas its representative modulo m. In this scenario
+Step 4b simply stores the coefficients ciand Step 3 can be deferred to Step 5.
+Whenmis “small,” by which we mean M(logm) =O(log3lloglogl), we instead
+use the explicit CRT modulo m. Assuming M >4|c|, we may apply
+(16) c≡ciaiMi−rMmodm,
+whereris the closest integer to s=/summationtextciai/pi, by [4, Thm. 3.1]. In this scenario,
+we update the sum C=/summationtextciaiMimodmand an approximation to sin Step 4b as
+eachciis computed. This uses O(logm+ logl) space per coefficient, rather than
+theO(llogl) space used to compute cmodM. The postcomputation in Step 5
+determines rfrom the approximation to sand computes cmodmvia (16).
+Whenmis neither small nor large, a hybrid approach is used, see [54, §6.3].
+6.3.Computing Φl(X,Y) modp.An overview of Algorithm 2.1 was given in the
+introduction, we now fill in the details.
+Algorithm 2.1 .Letl,p, andObe as in Theorem 4.1, with h(O)≥l+2, and let
+R=Z+lO. GivenHO∈Z[X], compute Φl∈Fp[X,Y]as follows:
+1. Compute the presentations αof cl(O) andα′of cl(R) suitable for p.
+2. Find a root of j0ofHO(X) overFp.
+3. Useαto enumerate Ell O(Fp) fromj0and identify the l-isogeny cycles.
+4. For distinct j0,...,j l+1∈EllO(Fp):
+a. Construct a curve Eiwithj(Ei) =jisuch thatldivides #Ei(Fp).
+b. Generate a random point P∈Ei(Fp) of orderl.
+c. UseEiandPto compute an l-isogenous curve E′
+i/Fpvia Algorithm 6.4.
+d. Ifj(E′
+i)/n⌉}ationslash∈EllO(Fp) then setj′
+i←(E′
+i), otherwise return to Step 3b.
+5. Useα′to enumerate Ell R(Fp) fromj′
+0and identify the l2-isogeny cycles.
+6. Forifrom 0 tol+1:
+a. Letji0,...,j ilconsist of the neighbors of jiin itsl-isogeny cycle in
+EllO(Fp) together with the l2-isogeny cycle of Ell R(Fp) containing j′
+i.
+b. Compute Φ l(X,ji) =/summationtext
+kaikXkas the product/producttext
+k(X−jik).
+7. Forkfrom 0 tol+1:
+a. Interpolate φk∈Fp[Y] with degφk≤l+1 satisfying φk(ji) =aik.
+8. Output Φ l(X,Y) =/summationtext
+kφk(Y)Xk.
+Steps 2, 6, and 7 involve standard computations with polynomials ove r finite fields,
+as described in [56], for example. Step 1 is addressed in Section 5.1, an d Steps 3
+and 5 are the topic of Section 5.3. Only Step 4 merits further discuss ion here.
+The existence of the curve Eiconstructed in Step 4a is guaranteed by The-
+orem 4.1. The trace of Frobenius tof the desired curve is uniquely determined
+by the norm equation for pand the constraint t≡2 modl, as in (12). With
+k=ji/(1728−ji), the curve E/Fpdefined by y2=x3+3kx+2khasj(E) =ji,
+and we may determine whether it is Eor its quadratic twist that has trace tby
+attempting to generate a point of order lon both curves in Step 4b.MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 17
+To obtain a point Pof orderl, we generate a random point Quniformly dis-
+tributed over Ei(Fp), compute the scalar multiple P=nQ, wheren= (p+1−t)/l,
+and then check that P/n⌉}ationslash= 0. This will be true with probability 1 −1/l2, since
+Theorem 4.1 implies that the Sylow l-subgroup of Ei(Fp) isEi[l]∼=Z/lZ×Z/lZ.
+Thus we expect to succeed within 1+ O(1/l2) attempts, and the expected cost of
+generating PisO(logp) operations in Fp.
+Note thatEi(Fp) containsl+1 distinct subgroups of order l, each corresponding
+to a distinct l-isogenousj-invariant. At most 2 of these lie in Ell O(Fp). Thus in
+Step 4d we have j(E′
+i)/n⌉}ationslash∈EllO(Fp) with probability at least 1 −2/(l+1) and expect
+to need 1 + O(1/l) random points Pto obtain such an E′
+i. The curve E′
+iis the
+image of an l-isogeny whose kernel is generated by P, obtained via Algorithm 6.4.
+6.4.Isogenies from subgroups. LetE/Fpbe an elliptic curve. Given a cyclic
+subgroupH⊆E(Fp), V´ elu’s formulas construct an isogeny E→E′withHas its
+kernel [55]. In our setting His actually generated by an Fp-rationall-torsionpoint,
+allowing us to work in Fprather than an extension field. Additionally, the order l
+ofHis odd and p>3, allowing us to simplify the formulas.
+Algorithm 6.4. Letl >2 andp >3 be primes, let E/Fpbe an elliptic curve
+defined byy2=x3+Ax+B, and letP= (Px,Py) be a point on E(Fp) of orderl.
+Compute the image E′/Fpof thel-isogeny with kernel H=/an}brack⌉tl⌉{tP/an}brack⌉tri}htas follows:
+1. Sett←0,w←0, andQ←P.
+2. Repeat ( l−1)/2 times:
+a. Sets←6Q2
+x+2A, and then set u←4Q2
+y+sQx.
+b. Sett←t+s,w←w+u, andQ←Q+P.
+3. SetA′=A−5tandB′=B−7w.
+4. Output the curve E′/Fpdefined byy2=x3+A′x+B′.
+The addition Q+Pin Step 2b is performed using the group operation in E(Fp).
+The complexity of Algorithm 6.4 is O(l) operations in Fp.
+6.5.Complexity analysis. We first bound the complexity of Algorithm 2.1, as
+used by Algorithm 6.1.
+Lemma 6.5. LetFbe a suitable family of orders that satisfies the condition of
+Theorem 5.1. For an odd prime l, letO=F(l)and letD= disc(O). Letpbe a
+prime in the set Sselected by Algorithm 6.2 on input landD. Assuming the GRH,
+the expected running time of Algorithm 2.1 is O(l2(logp)3loglogp).
+Proof.We note that p=t2−v2l2D>l4, thus logl<logp, and recall that the bit-
+complexityofmultiplying twopolynomialsofdegree O(l) inFp[X] mayby bounded
+byO(M(llogp)), using Kronecker substitution, see Corollaries 8.28 and 9.8 of [56].
+In the analysis below we use O(M(llogp)) =O(l(logp)2loglogp), via the bound
+M(n) =O(nlognloglogn) for fast multiplication [50]. When computing the cost of
+multiplications in Fpwe use the weaker bound M(logp) =O((logp)2/(loglogp)c),
+wherecis any constant, which is more convenient and does not change the o verall
+bound. Weboundthecostofinversionsin FpbyO(M(logp)loglogp) =O((logp)2),
+via [56, Cor. 11.10].
+We now bound the (expected) cost of each step in Algorithm 2.1:18 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+1. We have h(O)<h(R) =O(l2). As described in Section 5.1, the cost of com-
+putingthepresentations αandα′isO(l2)operationsincl( O)andcl(R). Us-
+ing binary quadratic forms, each group operation has complexity O((logl)2),
+by [5], yielding an O(l2(logl)2) =O(l2(logp)2) bound on Step 1.
+2. Using Berlekamp’s probabilistic root-finding algorithm [3, §7] with a fast
+GCD computation [56, Alg. 11.4], the expected time to find a single root
+ofHD∈Fp[X] may be bounded by O(M(l)(logl+logp)) operations in Fp,
+since degHD=O(l). This implies an O(l(logp)3) bound on Step 2.
+3. Forp∈Swe haveω(v)≤2loglogp, yielding an O(loglogplogloglogp)
+bound on max l(α), by Theorem 5.1. From Lemma 4.5, O((loglogp)2logp)
+operations in Fpsuffice to compute the action of any element of α. We
+obtain anO(l(logp)3) bound on the time to enumerate Ell O(Fp), which
+dominates the O(llogl) time to identify the l-isogeny cycles.
+4. From the discussion in Section 6.3 and the complexity of Algorithm 6.4 , we
+expect to use O(l2+llogp) operations in Fpduring Step 4. This yields an
+O(l2(logp)2+l(logp)3) bound.
+5. Recall that the surjective map ϕ: cl(R)→cl(O) in (3) preserves the norms
+of representative ideals. The subgroup of cl( R) generated by invertible
+R-ideals with norms in l(α) containsϕ−1(cl(O)). It follows that the ele-
+ments of α′with norm at most max l(α) generate a subgroup of size at
+leasth(O)> l. All butO(l) of theO(l2) steps taken when enumerating
+EllR(Fp) involve these elements, and, as in Step 3, we obtain a total cost of
+O(l2(logp)3) for these steps. Assuming the GRH, the remaining elements
+ofα′all have norm O((log|D|)2) =O((logl)2), by [1], yielding a total cost
+ofO(l(logl)4(logp)3) for these steps, via Lemma 4.5. Thus the expected
+time to enumerate Ell R(Fp) isO(l2(logp)3), which dominates the O(l2logl)
+time to identify the l2-isogeny cycles.
+6. Usingaproducttreewemaycompute/producttext
+k(X−jik) intimeO(M(llogp)logl),
+yielding a total cost of O(l2(logp)3loglogp) for Step 6.
+7. Using a product tree and fast interpolation [56, Alg. 10.11], we also obtain
+a cost ofO(l2(logp)3loglogp) for Step 7. Here we use the O(M(llogp))
+bit-complexity of polynomial multiplication in Fp[X] to bound the cost at
+each level, rather than using the bound in [56, Cor. 10.12].
+The bound O(l2(logp)3loglogp) applies to every step, completing the proof. /square
+We are now ready to prove our main theorem, which bounds the comp lexity of
+using Algorithm 6.1 to compute Φ lmodm, wherelis an odd prime and mis any
+positive integer. Recall that that the algorithm must be given a suita ble family
+of ordersF, as defined in Definition 4.2, and to prove our complexity bound we
+additionally require that Fsatisfy the property given in Theorem 5.1. Example 4.3
+provides one such F, and there are many others that can be efficiently computed
+and may yield better performance, as discussed in Section 6.6.
+Theorem 1. LetFbe a suitable family of orders that satisfies the condition of
+Theorem 5.1. Let lbe an odd prime and let m∈Z>0. Given inputs l,m, and
+O=F(l), Algorithm 6.1 correctly computes Φl∈(Z/mZ)[X,Y]. Under the GRH,
+its expected running time is
+O(l3log3lloglogl),
+usingO(l2loglm)expected space.MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 19
+Proof.We first argue correctness. By Lemma 6.3, Algorithm 6.2 obtains a se t of
+primesSthat satisfy Theorem 4.1, with/producttext
+p∈Sp>4|c|, for every coefficient cof Φl.
+We now claim that for p∈S, Algorithm 2.1 obtains, for each of l+2 distinct j-
+invariantsji, a list ofl+1 distinct j-invariants jikofl-isogenous curves. Granting
+theclaim, wemayinvokestandardpropertiesofΦ ltoshowthatAlgorithmcorrectly
+interpolates Φ l∈Fp[X,Y], see [57, Thm. 12.19] and [44, Thm. 5.3], for example.
+The correctness of Algorithm 6.1 then follows from the CRT and/or t he explicit
+CRT modm, via [4, Thm. 3.1], as described in Section 6.2.
+As usual, letO=F(l) have fraction field K, and letR=Z+lO. The claim
+above rests on three facts: (1) the explicit CM-action described in Section 4.3
+is correct, (2) any two l2-isogenous elements of Ell R(Fp) must bel-isogenous to
+exactly one and the same element of Ell O(Fp), and (3) each l2isogeny cycle in R
+contains exactly l−/parenleftbigdK
+l/parenrightbig
+elements. We note that (1) follows from the theory of
+complex multiplication and the properties of Φ l0guaranteed by [57, Thm. 12.19],
+(2) follows from the l-volcano structure, as shown by [30, §2.2] and [42, Prop. 23],
+and (3) is explicitly proven in Lemma 3.1.
+WenowassumetheGRHandboundthecomplexityofAlgorithm6.1. Lem ma6.3
+showsthattheexpectedsizeofthelargest p∈SisO(logl), andwehave# S=O(l).
+Applying [54,§6] yields an O(l2loglm) space bound for m∈Z>0.
+By [54, Thm. 1], the expected time to compute HOin Step 1 is O(l2+ε), and
+Lemma 6.3 gives an expected time of O(l1+ε) for Step 2, for any ε∈R>0. Addi-
+tionally, we have log p >(6+k)loglfor allp∈Swith probability approaching 1
+exponentially as kincreases. The time complexity of all remaining steps in Algo-
+rithm 6.1, including calls to Algorithm 2.1, depends polynomially on log p, hence
+we may bound the expected running time assuming log p=O(logl).
+Regardless of the exact cutoff used, if M(logm) =O((logl)3loglogl) whenever
+we consider m“small”, we may apply the results of [54, §6] to obtain a bound
+ofO(l3(logl)3loglogl) on the expected time for all CRT computations, for every
+m∈Z>0. SinceFsatisfies the property of Theorem 5.1, we may apply Lemma 6.5
+with logp=O(logl) to obtain an O(l2(logl)3loglogl) bound on the expected time
+of each call to Algorithm 2.1. Applying # S=O(l) completes the proof. /square
+6.6.Selecting a suitable order. The family of orders used in Theorem 1 suffices
+toprovethecomplexitybound, butwecansimplifytheimplementation andimprove
+performance with some additional constraints on the order O. Let us fix a bound
+b < l(sayb= 256, for large l), and a small prime l0< l(typicallyl0= 2). As
+above,Ris the order of index linO, andOKis the maximal order. We seek an
+orderOfor which the following hold:
+(1) The conductor of Oisb-smooth,h(OK)≤b, andh(O)≥l+2.
+(2) The groups cl( O) and cl(R) are either generated by a single ideal with norm
+l0, or by two ideals with norms l0andl1, wherel1≤bis ramified.
+The first condition ensures that Ois suitable for land allows us to to obtain a
+root ofHO(X) using only polynomials of degree at most b. This is accomplished
+by finding a root of HOK(X) and descending to the proper level of the l′-isogeny
+volcano for each prime l′≤bdividing the conductor of O, as in [54,§4.1]. The
+second condition allows us to realize the torsors for cl( O) and cl(R) either by
+walking a single l0-isogeny cycle, or by walking two l0-isogeny cycles connected by
+a singlel1-isogeny. In the latter case we orient the two cycles by computing o ne20 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+extral1-isogeny. In both cases we avoid the need to ever distinguish the ac tion of
+an ideal and its inverse, simplifying the computation described in Sect ion 5.3.
+Subject to these conditions, we also wish to minimize h(O)≥l+ 2. To find
+such orders we enumerate fundamental discriminants dK<−4 with/parenleftbigdK
+l1/parenrightbig
+= 1 and
+h(dK)≤b, and for each dKwe selectb-smooth integers ufor whichh(u2dK) is
+slightly greater than l+2 and test whether condition (2) holds. In practice we are
+almost always able to obtain Owithh(O) within a few percent of l+2.
+7.Modular functions other than j
+Letgbe a modular function of level N, and letl∤Nbe a prime. We define the
+modular polynomial Φg
+lof levellforgas the minimal polynomial of the function
+g(lz) over the field C(g). Much of the theory for the classical modular polynomial
+of thej-function generalizes to g. In particular, if the Fourier expansion of ghas
+integercoefficients then we have Φg
+l∈Z(g)[X]. The following lemma gives us
+further information in this case.
+Lemma 7.1. Letgbe a modular function and let lbe a prime not dividing the level
+ofg. Suppose that Φg
+lhas integer coefficients. If gis invariant under the action of
+eitherS=/parenleftbig0
+1−1
+0/parenrightbig
+∈SL2(Z)orM=/parenleftbig0
+1−l
+0/parenrightbig
+∈GL2(Q), then we have
+Φg
+l(X,Y) = Φg
+l(Y,X).
+Proof.The proof follows the symmetry proof for Φ l(X,Y), see [44, Thm. 5.3]. /square
+The polynomial Φg
+lshould not be confused with the minimal polynomial of gas
+an element of C(j), which we denote Ψg(X,J). The polynomial Ψgdepends only
+ong, notl, and we assume it is known (for our purposes, it effectively defines g).
+Given Ψg, our goal is to efficiently compute Φg
+lfor a prime l∤N.
+To apply our method we require that Φg
+lhave degree l+1 (in both XandY).
+The degree of Φg
+lcan be explicitly computed using [12, §5.2], and we note that this
+degree must be l+ 1 when degJΨg= 1 (this applies to the function γ2and the
+Weberf-function considered in Sections 7.1 and 7.3), and also when degJΨg= 2
+andgis invariant under the Atkin-Lehner involution (this applies to the var ious
+modular functions considered in Section 7.4).
+We wish to adapt Algorithm 2.1 to compute Φg
+l∈Fp[X]. We may then apply
+Algorithm 6.1 to recover Φg
+lover the integers or modulo some integer mvia the
+Chinese Remainder Theorem. To simplify matters, we place some addit ional re-
+strictions on the order Othat we use in Algorithm 6.1. Specifically, we require that
+there is a generator τ∈HofO=Z[τ] with the property that
+g(τ)∈KO,
+whereKOis the ring class field for the order O. We say that gis aclass invariant
+forOin this case. If we now take a prime pthat splits completely in KOandE/Fp
+an elliptic curve with endomorphism ring O, then the polynomial
+Ψg(X,j(E))∈Fp[X]
+has at least one root in Fp. Indeed, the value h=g(τ) modp, for a prime p|p
+ofKO, satisfies Ψg(h,j(E)) = 0.
+We can analyze how many roots the polynomial Ψg(X,j(E)) has in Fpusing a
+combination of Deuring lifting and Shimura reciprocity. We refer to [10 ,§6.7] for a
+detailed description of the techniques involved and only state the re sult here. LetMODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 21
+gi:H→Cbe the roots of Ψg(X,j). The functions giare modular of level N,
+and they are permuted by the Galois group GL 2(Z/NZ) of the field of all modular
+functionsoflevel N. Tostateourresult, wewillassociateamatrix A∈GL2(Z/NZ)
+to the Frobenius morphism of Eas follows. Fix an isomorphism End( E)∼−→Oand
+letπp∈Obe the image of the Frobenius morphism. If πphas minimal polynomial
+X2−tX+pof discriminant ∆ = t2−4p, then we put
+(17) A=/parenleftbiggt−2
+2∆−1
+2
+2t+2
+2/parenrightbigg
+∈GL2(Z/NZ).
+Theorem 7.2. Letgbe a modular function with the property that Ψghas integer
+coefficients and is separable modulo a prime pthat does not divide the level of N. If
+E/Fpis an elliptic curve with endomorphism ring Oandj0=j(E), then we have
+#{x∈Fp: Ψg(x,j0) = 0}= #{gi: Ψg(gi,j) = 0andgA
+i=gi},
+whereAis the matrix in (17).
+Proof.See [10,§6.7]. /square
+To check if gA
+i=giholds is a standard computation, see [32] for example.
+Although the matrix Ahas normpand tracetand therefore depends on p, we can
+often derive a result that merely depends on a congruence conditio n onpmodN.
+Lemma 7.3 in Section 7.3 gives an example.
+7.1.Computing modular polynomials for γ2.Letγ2(z) denote the unique
+cube root of j(z) that has integral Fourier expansion. It was known to Weber
+already that γ2is a modular function of level 3, see [58, §125], andγ2is a class
+invariant forOwhenever 3 ∤disc(O). We have Ψγ2(X,j) =X3−j, and in this
+simple case there is no need to apply Theorem 7.2; if we restrict to p= 2 mod 3
+then every element of Fphas a unique cube root. Thus we can compute Φγ2
+lwith
+only minor modifications to Algorithm 6.1:
+•Use a suitable order Owith 3∤disc(O) and select only primes p≡2 mod 3.
+•AfterStep 5ofAlgorithm2.1, replaceeachelement ofEll O(Fp)andEll R(Fp)
+with its unique cube root in Fp.
+These changes suffice, but we can also improve the algorithm’s perfo rmance.
+First, Lemmas 2–3 and Corollary 9 of [13] yield the bound
+(18) Bγ2
+l= 2llogl+8l
+on the logarithmic height of Φγ2
+l(conjecturally, Bγ2
+l= 2llogl+4lfor alll>60, but
+we do not use this). Thus we can reduce the height bound Blin (14) by a factor of
+approximately 3 when computing Φγ2
+l. This reduces the number of primes p∈S,
+and the corresponding number of calls Algorithm 6.1 makes to Algorith m 2.1.
+Second, we may take advantage of the fact that Φγ2
+lissparserthan Φ l. As noted
+in [22, p. 37], the coefficient of XaYbin Φγ2
+lis zero unless
+(19) a+lb≡l+1 mod 3 .
+The proof of this relation goes back to Weber: the argument given in [58, p. 266]
+generalizes immediately to γ2. It allows us to reduce the number of points we use
+to interpolate Φγ2
+lby a factor of approximately 3.
+Letn=⌈(l+1)/3⌉+1. When selecting a suitable order Oas in Section 6.6, we
+now only require that h(O)≥n, and further modify Algorithm 2.1 as follows:22 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+•In Steps 4-6 we construct just npolynomials Φγ2
+l(X,3√ji) of degree l+1.
+•In Step 7 we interpolate l+1 polynomials φ∗
+kof degree less than nby writing
+φk=Ycφ∗
+k(Y3), withc∈{0,1,2}satisfyingc+lk≡l+1 mod 3.
+This reduces the cost of all the significant components of Algorithm 2.1 by a factor
+of approximately 3. The reduction in the cost of the interpolations in Step 7 is
+actually greater than this, since its complexity is superlinear in the de gree.
+The total size of Φγ2
+lis approximately 9 times smaller than Φ l, and with the
+optimizations above, the time to compute it is effectively reduced by t he same
+factor. A small amount of additional time is required to compute the cube roots of
+the elements in Ell O(Fp) and Ell R(Fp), but even this can be avoided.
+Provided we have already computed Φγ2
+l′for some small values of l′(specifically,
+for the primes l0andl1of Section 6.6), we may use these polynomials to directly
+enumerate sets Ellγ2
+O(Fp) and Ellγ2
+R(Fp) containingthe cube roots of the elements in
+EllO(Fp) andEll R(Fp) respectively. We need onlycompute the cube rootsof j0and
+j′
+0as starting points. This third optimization yields a small but useful imp rovement
+in the case of γ2, and plays a critical role in the examples that follow.
+7.2.Recovering ΦlfromΦγ2
+l.Having computed Φγ2
+l, we note that Φ lmay be
+computed via [22, Eq. 23]:
+(20) Φ l(X3,Y3) = Φγ2
+l(X,Y)Φγ2
+l(X,ωY)Φγ2
+l(X,ω2Y),
+whereω=e2πi/3. For computation in Z, or modulo m, it is more convenient to
+express Φγ2
+lin terms of polynomials P0,P1,P2∈Z[X,Y] satisfying
+(21) Φγ2
+l(X,Y) =P0(X3,Y3)Yb+P1(X3,Y3)XY+P2(X3,Y3)X2Y2−b,
+whereb= 2 whenl≡1 mod 3, and b= 0 whenl≡2 mod 3. We then have
+(22) Φ l=P3
+0Yb+(P3
+1−3P0P1P2)XY+P3
+2X2Y2−b.
+Using Kronecker substitution and fast multiplication, it is possible to e valuate (22)
+in timeO(l3(logl)2+ε), which is asymptotically faster than Algorithm 6.1. This
+suggests that we might more efficiently compute Φ lby recovering it from Φγ2
+l, but
+we do not find this to be true in practice: it actually takes longer to ev aluate (22)
+than it does to compute Φ ldirectly. This can be explained by two factors. First,
+theFp-operations used in Algorithm 2.1 effectively have unit cost for word- size
+primes, making it faster than Theorem 1 would suggest for all but ve ry largel.
+Secondly, the evaluation of (22) becomes extremely memory intens ive whenlis
+large. However, if we are computing Φ lmodmwith logm≪llogl, then the time
+to apply (22) modulo mis negligible. In this situation it is quite advantageous to
+derive Φ lmodmfrom Φγ2
+lmodm, as may be seen in Table 3 of Section 8.
+7.3.Computing modular polynomials for the Weber ffunction. We now
+consider the classical Weber function [58, p. 114] defined by
+f(z) =ζ−1
+48η((z+1)/2))
+η(z),
+whereζ48=eπi
+24andη(z) is the Dedekind eta function. This is a modular function
+of level 48 that satisfies γ2= (f24−16)/f8, see [58, p. 179], thus we have
+Ψf(X,j) = (X24−16)3−X24j.MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 23
+Asymptotically, we expect to be able to reduce the height bound Blby a factor of
+degXΨf/degjΨf= 72 when computing Φf
+l. One can derivean explicit bound along
+the lines of (18), but this tends to overestimate the O(l) term quite significantly,
+so in practice for large lwe use the heuristic bound
+(23) Bf
+l=1
+12llogl+1
+5l (l>2400),
+which has been verified for every prime lbetween 2400 and 10000. The modular
+polynomial Φf
+lis also sparse: the coefficient of XaYbcan be nonzero only when
+(24) la+b≡l+1 mod 24 ,
+asshownin [58, p. 266]. Thus Φf
+lis roughly72·24 = 1728times smallerthan Φ l. By
+applying the technique described above for γ2,mutatis mutandis , we can actually
+compute Φf
+lmore than 1728 times faster than Φ lfor large values of l, as may be
+seen in Table 2 of Section 8. When applying Algorithm 6.1, we now insist th at the
+orderOhave discriminant D≡1 mod 8 and 3 ∤D, since the Weber function yields
+class invariants in (at least) this case, see [32], for example.
+Since−fwill also yield class invariants for O, the polynomial Ψf(X,j0) will
+always have at least tworoots in Fp. The following lemma tells us that we can
+impose a congruence condition on pto ensure that we have exactly two roots.
+Lemma 7.3. Letp≡11 mod 12 be prime and let j0be thej-invariant of an
+elliptic curve E/FpwithEnd(E)isomorphic to an imaginary quadratic order O
+with discriminant D≡1 mod 8 and3∤D. ThenΨf(X,j0)∈Fp[X]has exactly
+two roots in Fp, and these are of the form x0and−x0.
+Note that if the lemma applies to O, it also applies to the order Rof indexlinO.
+Proof.We only have to apply Theorem 7.2. The action of Aon the roots of
+Ψf(X,j0) is computed in [10, §6.7], and this yields the lemma. /square
+Given aj-invariantj0∈Fpthat corresponds to j(τ0)∈KO, we cannot readily
+determine which of the roots x0and−x0of Ψf(X,j0) actually corresponds to f(τ0).
+The functions fand−fyield distinct class invariants, but they share the same
+modular polynomials, since Φf
+l(X,Y) = Φf
+l(−X,−Y) = Φ−f
+l(X,Y), by (24).
+Thus for the initial j0obtained in Step 2 of Algorithm 2.1, it does not matter
+whether we pick x0or−x0as a root of Ψf(X,j0), and we need not be concerned
+with making a consistent choice for each prime p. However it is critical that while
+computing Φf
+lmodpwe make a consistent choice of sign for each j-invariant we
+convert to an “ f-invariant” (a root of Ψf(X,ji) modp). This makes it impractical
+to enumerate j-invariants and convert them en masse . Instead, as described for
+γ2above, we use modular polynomials Φf
+l′for smalll′to enumerate sets Ellf
+O(Fp)
+and Ellf
+R(Fp) from starting points x0andx′
+0satisfying Ψf(x0,j0) = Ψf(x′
+0,j′
+0) = 0.
+This ensures that signs are chosen consistently within each of thes e sets; we only
+need to check that the sign choices for the two sets are consisten t with each other.
+To do so, we use the fact that the coefficient of XlYlin Φf
+l(X,Y) is−1. This
+is shown for Φ lin [58,§69], and the same argument applies to Φf
+l. We modify
+Algorithm 2.1 to compute the coefficient of XlYlin Φf
+lmodpin between Steps 5
+and6. Wedothistwice, switchingthesignsinEllf
+O(Fp)thesecondtime, andexpect
+exactly one of these computations to yield −1, thereby determining a consistent24 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+choice of signs. This test should be regarded as a heuristic, since we do not rule out
+the possibility that both choices produce −1. However, in the course of extensive
+testing this has never happened, and we suspect that it cannot. I f it does occur,
+the algorithm can detect this and simply choose a different prime p.
+7.4.Eta quotients and Atkin modular functions. For a prime N, let
+fN(z) =Ns/2/parenleftbiggη(Nz)
+η(z)/parenrightbiggs
+,
+wheres= 24/gcd(12,N−1). These are modular functions of level N, and the
+polynomials Ψ N= ΨfNthat relatefNtojare sometimes called canonical modular
+polynomials [19, p. 418]. The functions fNare closely related to the functions ws
+N
+considered in [25], and in fact ΨfN= Ψws
+N, so what follows applies to both. When
+Nis 2, 3, 5, 7, or 13, we have degjΨN= 1 and can adapt Algorithm 6.1 to compute
+polynomials ΦfN
+lfor odd primes l∤N. We assume here that Nis also odd.
+We have degXΨN=N+1, hence we can reduce the height bound Blby a factor
+of approximately N+1. When selecting a suitable order O, we require that Nis
+prime to the conductor and splits into prime ideals that are distinct in c l(O). This
+assumption is stronger than we need, but it simplifies the implementat ion. For the
+primesp∈Swe require that Nis prime to v, where 4p=t2−v2disc(O).
+As shown in [48], the polynomial Ψ N(X,j0) modphas the same splitting type
+as ΦN(X,j0) modp. In particular, for j0∈EllO(Fp) (orj0∈EllR(Fp)) it has
+exactly two roots, say x1, andx2. These correspond to N-isogenies as follows:
+thej-invariants j1andj2of the two elliptic curves that are N-isogenous to j0are
+uniquely determined by the relations Ψ N(Ns/x1,j1) = 0 and Ψ N(Ns/x2,j2). Here
+the transformation x/ma√sto→Ns/xrealizes the Atkin-Lehner involution on fN(z).
+Starting points x0andx′
+0corresponding to j0andj′
+0are chosen as follows.
+Letx′
+0be a root of Ψ N(X,j′
+0), chosen arbitrarily, and let j′
+1be determined by
+ΨN(Ns/x′
+0,j′
+1) = 0. We then use V´ elu’s formulas to obtain the j-invariantj1
+of the elliptic curve that is l-isogenous to j′
+1(there is exactly one and it lies in
+EllO(Fp), sincej′
+1∈EllR(Fp) is on the floorofits l-volcano). Finally, x0is uniquely
+determined by the constraints Ψ N(x0,j0) = 0 and Ψ N(Ns/x0,j1) = 0.
+We next consider double eta-quotients [26, 27] of composite level N=p1p2:
+ws
+p1,p2(z) =/parenleftBigg
+η(z
+p1)η(z
+p2)
+η(z
+p1p2)η(z)/parenrightBiggs
+,
+wherep1/n⌉}ationslash=p2are primes and s= 24/gcd(24,(p1−1)(p2−1)). For (p1,p2) in
+/braceleftbig
+(2,3),(2,5),(2,7),(2,13),(3,5),(3,7),(3,13),(5,7)/bracerightbig
+,
+the polynomial Ψ p1,p2= Ψws
+p1,p2has degree 2 in jand we can compute Φws
+p1,p2
+lfor
+odd primes l∤N. Our restrictions on Oare analogous to those for fNorws
+N: we
+require that Nis prime to the conductor and that both p1andp2split into distinct
+prime ideals in cl( O). Our requirements for p∈Sare as above. We can reduce the
+height bound Blby a factor of approximately ( p1+1)(p2+1)/2.
+With the double eta-quotients, the polynomial Ψ p1,p2(X,j0) has four roots, cor-
+respondingtofourdistinctisogeniesof(composite)degree N. Eachroot xiuniquely
+determines the j-invariant of a curve N-isogenous to E/Fpas the unique root of
+Ψp1,p2(xi,J)/(J−j0)∈Fp[J]. The double eta-quotients are invariant under theMODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 25
+Atkin-Lehner involution, so we need not transform xi. With this understanding,
+the procedure for selecting x0andx′
+0is as above.
+In some cases one can obtain smaller modular polynomials by considerin g suit-
+able roots of the functions defined above. For example, a sixth roo t off3also
+yields class invariants (this is shown for w2
+3in [25]), and the corresponding modular
+polynomials are sparser and of lower height (by a factor of 6).
+Our algorithm also applies to the Atkin modular functions, which we den oteAN.
+Theseare(optimal)modularfunctionsfor X+
+0(N)invariantundertheAtkin-Lehner
+involution, see [22, 47] for further details. The polynomials ΨANare known as
+Atkin modular polynomials, and are available in computer algebra syste ms such as
+Magma[15]andSage[52]. Forprimes N <32, andalso Nin theset{41,47,59,71},
+we have degjΨAN= 2 and can compute polynomials ΦAN
+lfor odd primes l/n⌉}ationslash=N.
+This is done in essentially the same way as with the double eta-quotient s, except
+that nowNis prime and ΨAN(X,j0) has just two roots, rather than four. For these
+AN, the height bound can be reduced by a factor of approximately ( N+1)/2.
+Finally, we note an alternative approach applicable to both eta-quot ients and
+the Atkin modular functions. If we choose Oso that the prime factors of Nare
+all ramified, then there is actually a unique x0∈Fpcorresponding to each j0
+in EllO(Fp) and Ell R(Fp), that is, Ψg(X,j0) has exactly one root in Fp. In this
+scenario we can simply enumerate j-invariants as usual and then replace each ji
+with a corresponding xi. This is not as efficient and places stricter requirements
+onO, but it allows us to compute Φg
+lwithout needing to know Φg
+l′for anyl′. This
+provides a convenient way to “bootstrap” the process. In fact a ll of the modular
+polynomials Φg
+lwe have considered can eventually be obtained via Algorithm 6.1,
+starting from the polynomials Ψgand Φ 2.
+8.Computational results
+We have applied our algorithm to compute polynomials Φg
+lfor all the modular
+functions discussed in Section 7 and everyapplicable lup to 1000. Forthe functions
+j,γ2, andfwe have gone further, and present details of these computations here.
+8.1.Implementation. The algorithms described in this paper were implemented
+using the GNU C/C++ compiler [29] and the GMP library [33] on a 64-bit Lin ux
+platform. Multiplication of large polynomials is handled by the znpolylibrary
+developed by Harvey [36, 35].
+The hardware platform included four 3.0 GHz AMD Phenom II process ors, each
+with four cores and 8GB of memory. Up to 16 cores were used in the la rger tests,
+with essentially linear speedup. For consistency we report total CP U times, noting
+that in a multi-threaded implementation, disk and network I/O can be overlapped
+with CPU activity so that all computations are CPU bound.
+As a practical optimization, we do not use the Hilbert class polynomial HOin
+Step 1 of Algorithm 6.1. Instead, we compute the minimal polynomial o f some
+more favorable class invariant, as described in [28], which is then used to obtain
+aj-invariant. Additionally, as noted in Section 6.6, it suffices to compute a class
+polynomial for the maximal order containing O. With these optimizations the time
+spent computing class polynomials is completely negligible (well under on e second).
+Another important optimization is the use of polynomial gcds to acce lerate root-
+finding when walking paths in the isogeny graph, a technique develope d in [28,§2].
+This greatly acceleratesthe enumeration of the sets Ell O(Fp) and Ell R(Fp) in Steps26 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+l|D|n B lblsize (MB) time (s) MB/s
+101 216407 184 6511 5751 2.65 2.25 1.18
+211 393047 369 14949 13359 27.6 14.4 1.92
+307 837407 531 22748 20483 90.5 51.0 1.78
+401 626431 725 30640 27642 211 130 1.62
+503 3076175 870 39421 35686 431 264 1.63
+601 461351 1011 48027 43542 755 485 1.56
+701 1254871 1229 56953 51731 1227 863 1.42
+809 916599 1376 66731 60743 1926 1410 1.37
+907 986855 1517 75712 69017 2759 2010 1.37
+1009 2871983 1728 85157 77653 3857 2910 1.32
+2003 91696103 3410 180941 166095 33120 31800 1.04
+3001 248329639 5122 281635 259272 117256 143000 0.82
+4001 72135279 6939 385300 355707 287783 363000 0.79
+5003 67243191 8373 491355 454429 577740 749000 0.77
+Table1. Computations of Φ loverZ.
+3 and 5 of Algorithm 6.1. As a result, most of the computation (typica lly over 75%)
+is spent interpolating polynomials in Steps 6 and 7.
+8.2.Computations over Z. Tables 1 and 2 provide performance data for com-
+putations of Φ land Φf
+lusing Algorithm 6.1. For each lwe list:
+•The discriminant Dof the suitable order O.
+•The number of CRT primes n= #Sused.
+•Theheightbound Blinbitsandtheactualbit-size blofthelargestcoefficient.
+•The total size of Φ l(resp. Φf
+l) in megabytes (1MB = 106bytes), computed
+as the sum of the coefficient sizes, with symmetric terms counted on ly once.
+•The total CPU time, in seconds. This includes the time to select O.
+•The throughput, defined as the total size divided by the total CPU time.
+In the last column of Table 1 one can see the quasilinear performance of Algo-
+rithm 6.1 as a function of the size of Φ l, and the constant factors appear to be
+advantageous relative to other algorithms. For example, computin g Φ1009with the
+evaluation/interpolation algorithm of [24] uses approximately 10000 0 CPU seconds
+(scaled to our hardware platform), while Algorithm 6.1 needs less tha n 3000.
+The first five rows of Table 2 may be compared to the corresponding rows of
+Table 1 to see the performance advantage gained when computing m odular poly-
+nomials for the Weber ffunction rather than j. As expected, these polynomials are
+approximately 1728 times smaller, and the speedup achieved by Algor ithm 6.1 is
+even better; we already achieve a speedup of around 1800 when l= 1009, and this
+increases to to over 3000 when l= 5003. This can be explained by the superlinear
+complexity of interpolation, as well as the superior cache utilization a chieved by
+condensing the sparse coefficients of Φf
+l, as described in Section 7.1.
+As noted in Section 7.3, we used a heuristic height bound for the comp utations
+in Table 2. The gap between the values of blandBlin each case gives us high
+confidence in the results (the probability of this occurring by chanc e is negligible).MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES 27
+l|D|n B lblsize (MB) time (s) MB/s
+1009 1391 33 1275 1099 2.34 1.59 1.47
+2003 37231 58 2542 2271 19.5 10.7 1.81
+3001 88879 88 3822 3611 69.6 47.7 1.46
+4001 53191 112 5201 4801 167 116 1.45
+5003 30959 136 6613 6228 339 241 1.41
+6007 463039 170 8052 7530 595 493 1.21
+7001 150631 192 9496 8876 957 701 1.37
+8009 315031 220 10979 10292 1453 1200 1.21
+9001 179159 240 12453 11974 2123 1790 1.18
+10009 207919 265 13964 13453 2953 2630 1.12
+20011 1114879 537 29485 27860 24942 27600 0.90
+30011 2890639 795 45649 43304 87660 123000 0.71
+40009 22309439 1032 62210 59439 214273 335000 0.64
+50021 37016119 1316 79116 78077 508571 677000 0.75
+60013 27334823 1594 96165 91733 747563 1150000 0.65
+Table2. Computations of Φf
+loverZ.
+8.3.Computations modulo m.Table 3 gives timings for computations of Φ l
+modulo 256-bit and 1024-bit primes m. The values of mare arbitrary, and, in par-
+ticular, they are not of a form suitable for direct computation with A lgorithm 2.1.
+Instead, Algorithm 6.1 derives Φ lmodmfrom the computations of Φ lmodp, for
+p∈S, using the explicit CRT. The same set Sis used aswhen computingΦ loverZ,
+so the running time is largely independent of m, but using the explicit CRT yields
+a noticeable speedup when log mis significantly smaller than 6 llogl. For example,
+whenl= 1009 it takes approximately 2300 seconds to compute Φ lmodm, for the
+mlisted in Table 3, versus about 2900 seconds to compute Φ loverZ.
+In addition to computing Φ lmodmdirectly, we may also obtain Φ lmodmby
+computing Φγ2
+lmodmand applying (22), as discussed in Section 7.1. The time to
+compute Φγ2
+lmodmis essentially independent of m, but the time to apply (22) is
+not. Even so, for the 256-bit and 1024-bit mthat we used, computing Φ lmodm
+in this fashion is much faster than computing Φ lmodmdirectly; for l= 1009
+we achieve times of 223 and 403 seconds, respectively. As with Φf
+l, this speedup
+improves superlinearly, and for large lit exceeds the expected factor of 9.
+When computing Φγ2
+lmodmwe used the height bound Bγ2
+l= 2llogl+8lgiven
+by (18). The timings in Table 3 would be further improved if the heurist ic bound
+Bγ2
+l= 2llogl+4lwere used instead.
+The computations listed in Tables 1 and 2 were practically limited by spac e, not
+time. The largest computations took only a day or two when run on 16 cores, but
+required nearly a terabyte of disk storage. However when comput ing Φlmodm,
+we can handle larger values of lwithout using an excessive amount of space. When
+l= 20011, for example, the total size of Φ lis over 30 terabytes, but we are able to
+compute Φ lmodulo a 256-bit integer musing less than 10 gigabytes.28 REINIER BR ¨OKER, KRISTIN LAUTER, AND ANDREW V. SUTHERLAND
+m= 2256−189 m= 21024−105
+l ΦlΦγ2
+lΦ∗
+l ΦlΦγ2
+lΦ∗
+l
+101 2.12 0.16 0.47 2.16 0.17 1.53
+211 12.4 1.64 3.26 12.7 1.68 7.95
+307 43.3 4.82 8.34 44.0 4.93 19.3
+401 109 10.9 17.9 111 11.1 38.0
+503 215 23.3 34.0 219 23.8 66.4
+601 390 40.5 55.8 395 41.4 110
+701 695 69.1 90.2 703 70.3 158
+809 1130 105 134 1150 107 222
+907 1590 158 194 1600 160 306
+1009 2300 223 267 2320 225 403
+2003 23900 2400 2590 24100 2440 3210
+3001 106000 9250 9650 107000 9360 11200
+4001 283000 25100 25900 287000 25400 28600
+5003 647000 57000 58300 653000 60200 65700
+10009 7180000 681000 687000 7320000 688000 713000
+Table3. Computations of Φ land Φγ2
+lmodulom.
+Columns Φ∗
+llist the total time to obtain Φ lby computing Φγ2
+land applying (22).
+Acknowledgments
+We thank David Harvey for the znpolylibrary, and Andreas Enge for providing
+timings for his evaluation/interpolation algorithm. We also thank Igor Shparlinski
+for his helpful comments on an early draft of this paper.
+Appendix
+Lemma8.1. Letcbe a real number greater than c0= log2e≈1.44. Letπc(x)count
+the integers n∈[3,x]for whichω(n)≥cloglogn. Thenπc(x) =O/parenleftbig
+x(logx)1−c/c0/parenrightbig
+.
+Proof.Letd(n) count the divisors of n. From [34, Thm. 320] we have
+/summationdisplay
+n≤x2ω(n)≤/summationdisplay
+n≤xd(n) =xlogx+O(x).
+At mostO/parenleftbig
+x(logx)1−c/c0/parenrightbig
+terms on the LHS have n≥√xandω(n)≥cloglogx.
+Applying√x=O/parenleftbig
+x(logx)1−c/c0/parenrightbig
+andloglog√x
+loglogx= 1+o(1) yields the lemma. /square
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\ No newline at end of file
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+arXiv:1001.0403v1  [astro-ph.EP]  3 Jan 2010WASP-19b: the shortest period transiting exoplanet yet dis covered
+L. Hebb1, A. Collier-Cameron1, A.H.M.J. Triaud2, T.A. Lister3, B. Smalley4, P.F.L. Maxted4,
+C. Hellier4, D.R. Anderson4, D. Pollacco5, M.Gillon2,6, D. Queloz2, R.G. West7, S.Bentley4,
+B. Enoch1, C.A. Haswell8, K. Horne1, M. Mayor2, F. Pepe2, D. Segransan2, I. Skillen9, S. Udry2,
+and
+P.J. Wheatley10
+ABSTRACT
+We report on the discovery of a new extremely short period tra nsiting extra-solar
+planet, WASP-19b. The planet has mass, Mpl= 1.15±0.08 MJ, radius, Rpl= 1.31±
+0.06 RJ, and orbital period, P= 0.7888399 ±0.0000008 days. Through spectroscopic
+analysis, we determine the host star to be a slightly super-s olar metallicity ([ M/H] =
+0.1±0.1 dex) G-dwarf with Teff= 5500±100 K. In addition, we detect periodic,
+sinusoidal flux variations in the light curve which are used t o derive a rotation period
+forthestar of Prot= 10.5±0.2 days. Therelatively shortstellar rotation periodsugges ts
+that either WASP-19 is somewhat young ( ∼600 Myr old) or tidal interactions between
+the two bodies have caused the planet to spiral inward over it s lifetime resulting in the
+spin-up of the star. Due to the detection of the rotation peri od, this system has the
+potential to place strong constraints on the stellar tidal q uality factor, Q′
+s, if a more
+precise age is determined.
+Subject headings: stars: planetary systems – techniques: radial velocities – techniques:
+photometric
+1School of Physics and Astronomy, University of St Andrews, N orth Haugh, St Andrews, Fife KY16 9SS, UK
+2Observatoire de Gen` eve, Universit´ e de Gen` eve, 51 Ch. des Maillettes, 1290 Sauverny, Switzerland
+3Las Cumbres Observatory, 6740 Cortona Dr. Suite 102, Santa B arbara, CA 93117, USA
+4Astrophysics Group, Keele University, Staffordshire, ST5 5 BG, UK
+5Astrophysics Research Centre, School of Mathematics & Phys ics, Queen’s University, University Road, Belfast,
+BT7 1NN, UK
+6Institut d’Astrophysique et de Gophysique, Universit de Li ge, Alle du 6 Aot, 17, Bat. B5C, Lige 1, Belgium
+7Department of Physics and Astronomy, University of Leicest er, Leicester, LE1 7RH, UK
+8Department of Physics and Astronomy, The Open University, M ilton Keynes, MK7 6AA, UK
+9Isaac Newton Group of Telescopes, Apartado de Correos 321, E -38700 Santa Cruz de la Palma, Tenerife, Spain
+10Department of Physics, University of Warwick, Coventry CV4 7AL, UK– 2 –
+1. Introduction
+Since the unexpected discovery of the first ‘hot Jupiter’, 51 Peg b (Mayor & Queloz 1995),
+exoplanets with an exceptionally wide variety of propertie s have been detected which have dra-
+matically changed our understanding of planetary physics. In particular, through the discovery of
+various transiting planets, we have learned that extra-sol ar planets can have radii much larger than
+Jupiter (e.g. Hebb et al. 2009) or densities much higher (Sat o et al. 2005). Many, but not all, ‘hot
+Jupiters’ have temperature inversions in their atmosphere s (e.g. Knutson et al. 2008), and they can
+have very low optical albedos (Rowe et al. 2008). Despite the ir short periods, not all transiting
+exoplanets have been tidally circularized (Gillon et al. 20 09a), and both rocky (e.g. CoRoT-Exo-7,
+P∼0.85 days) and gas giant (e.g. WASP-12b, P∼1.09 days) planets can exist in extremely short
+period orbits. Here, we report on the discovery of a new extre me transiting extra-solar planet with
+the shortest orbital period yet detected which is on the verg e of spiraling into its host star. This
+transiting planet can not only inform us about the propertie s and evolution of close-in planets, but
+it also has the potential to provide information about the ch aracteristics of its host star.
+In this paper, we first describe all the observations that wer e obtained to detect and analyse
+the transiting star-planet system ( §2). We describe the data analysis in §3 where we present the
+planet and its host star. Finally in §4, we discuss the implications of the planet’s short period a nd
+its future evolution.
+2. Observations
+2MASS J09534008-4539330 (hereafter WASP-19) is an apparen tly unremarkable 12th mag-
+nitude (V= 12.59), G8V star in the southern hemisphere located at α=09:53:40.08, δ=-
+45:39:33.0 (J2000). The target was observed with the WASP-S outh telescope and instrumentation
+(Pollacco et al. 2006; Wilson et al. 2008) in the winter and sp ring observing seasons from 2006 to
+2008. 1496 photometric data points were obtained between 4 M ay - 20 June 2006, 6695 measure-
+ments were made between 18 Dec 2006 - 18 May 2007, and 8968 obse rvations were taken from 18
+Dec 2007 - 22 May 2008. All data sets were processed independe ntly with the standard WASP
+datareductionpipelineandphotometrypackage (Collier Ca meron et al. 2006). Theindividualdata
+points have typical uncertainties of ∼0.02 mags including poisson noise and systematic noise. The
+resulting light curves were then run through our implementa tion of the box least squares algorithm
+(Kov´ acs et al. 2002) designed to detect periodic transit-s haped dips in brightness.
+The target was initially flagged as a transiting planet candi date because a strong periodic
+signal was detected in the the 2007 data. The phase-folded li ght curve showed a square-shaped dip
+in brightness with a depth, δ∼25 mmag and duration, τ∼1.2 hours, consistent with a planet
+sized object around a main sequence star. Further, a periodi c transit was also apparent in the
+2006 data when phase-folded with the 2007 ephemeris, and a tr ansit was subsequently detected in
+the 2008 season of data. Therefore, we classified the object a s needing follow up photometry and– 3 –
+spectroscopy to assess the planetary nature of the system. T he phase-folded light curve containing
+all WASP-South data is shown in Figure 1.
+WASP-19 was observed photometrically with the 2m Faulkes Te lescope South (FTS) on 17
+December 2008 during transit. 139 Pan-STARRS z-band1observations were made over 3.3 hours.
+The images were observed in 2 ×2 binning mode such that one binned pixel corresponds to 0 .279′′.
+They were processed in the standard way with IRAF using a stac ked bias image, dark frame, and
+sky flat. Minimal fringing was present in the z-band images du e to the deep depletion CCD in the
+camera, so no fringe correction was applied. The DAOPHOT pho tometry package (Stetson 1987)
+was used to perform object detection and aperture photometr y with an aperture size of 8 binned-
+pixels in radius. The 5′×5′field-of-view of the instrument contained 53 comparison sta rs that
+were used in deriving the differential magnitudes with a photo metric precision of 1.3 mmag. We
+measured the red noise (Pont et al. 2006) in the light curve on a 30 minute timescale to be 449 ppm
+and added this value in quadrature to the formal uncertainti es on each data point. The resulting
+light curve is shown in Figure 2.
+Thirty-four radial velocity measurements were obtained wi th the CORALIE spectrograph on
+the 1.2m Euler telescope (Baranne et al. 1996; Wilson et al. 2 008). The stable, temperature con-
+trolled, high-resolution echelle spectrograph has a resol ution ofR∼55000 over the spectral region
+from 3800 −6800˚A. The WASP-19 spectra, obtained between 29 May 2008 and 23 Ap ril 2009, were
+processed through a slightly updated version of the CORALIE data reduction pipeline. In addition
+to the standard pipeline described in Baranne et al. (1996), we corrected for the blaze function
+and scaled the fitted cross-correlation region to match the f ull-width half maximum of the object.
+The final radial velocity (RV) values were obtained by cross- correlating the spectra with a G2 tem-
+plate mask. Table 1 presents the radial velocity measuremen ts of WASP-19 at each Barycentric
+Julian date, the 1 σPoisson errors on the velocities, and the line bisector span measurements (Gray
+1988; Queloz et al. 2001). Based on our experience, we adopt u ncertainties on the line bisector
+measurements of twice the measured RV errors.
+The radial velocity of the star varies sinusoidally with the same period measured from the
+photometry (see Figure 3). In addition, the line bisector sp ans, which are used to discriminate spot
+induced velocity variations and the effects of line-of-sight binarity, show no correlation with radial
+velocity within the uncertainties. The slope of the bisecto r versus RV (Figure 4) is −0.006±0.037,
+and the bisector measurements have an r.m.s. scatter of ∼50 m s−1.
+Although all existing photometric and spectroscopic data s uggests WASP-19 is orbited by a
+short period transiting extra-solar planet, we explore the possibility of a false positive detection. In
+general, it is difficult to mimicthe photometric and spectros copic observations of a transiting planet
+without showing a visible second star in the spectrum, a sign ificant bisector trend, and/or incon-
+sistencies between the transit duration and host star spect ral type. Starspots can cause periodic
+1http://pan-starrs.ifa.hawaii.edu/public/design-feat ures/cameras.html– 4 –
+Table 1: Radial velocity measurements of WASP-19 obtained w ith the CORALIE spectrograph.
+BJD Vr σRVBisector
+km s−1km s−1km s−1
+2454616.46326 20.965 0.0218 -0.02975
+2454623.46713 20.712 0.0289 0.03735
+2454624.46191 21.019 0.0220 0.00607
+2454652.46587 20.512 0.0210 0.02176
+2454653.46543 20.770 0.0231 -0.03106
+2454654.46530 21.007 0.0219 0.03017
+2454656.47564 20.511 0.0193 -0.04093
+2454657.46805 20.902 0.0386 0.04432
+2454658.46376 21.008 0.0302 0.05792
+2454660.46652 20.604 0.0400 0.05038
+2454661.46434 20.974 0.0217 0.04052
+2454662.46549 20.922 0.0191 0.01634
+2454663.46607 20.547 0.0188 -0.02782
+2454664.46573 20.645 0.0362 -0.14578
+2454665.46709 21.077 0.0305 -0.00590
+2454827.74752 20.683 0.0188 0.02320
+2454832.74362 21.050 0.0242 -0.08001
+2454833.66844 20.860 0.0212 0.01644
+2454834.67693 20.548 0.0191 -0.04711
+2454837.67024 20.726 0.0227 0.06777
+2454838.68496 20.562 0.0244 -0.04762
+2454839.70064 20.936 0.0185 -0.08634
+2454890.61491 20.637 0.0187 0.00276
+2454894.68743 20.518 0.0290 -0.02356
+2454895.69238 20.894 0.0179 0.01631
+2454896.66180 21.041 0.0145 -0.03853
+2454897.65715 20.674 0.0175 -0.04530
+2454898.66015 20.544 0.0197 -0.02811
+2454939.53373 20.578 0.0164 -0.01384
+2454940.52747 20.642 0.0154 0.03494
+2454941.53928 20.996 0.0156 0.02525
+2454942.52421 20.899 0.0169 -0.08243
+2454943.53594 20.559 0.0191 0.05531
+2454944.52955 20.798 0.0155 -0.02327– 5 –
+low-amplitude RV variations (e.g. Hu´ elamo et al. 2008; Des ort et al. 2007) but not photometric
+transits as well, thus a single star blended with a fainter st ellar eclipsing binary (EB) is the pre-
+ferred false positive scenario for transiting planets. Unf ortunately, no comprehensive simulations
+have been performed which model the expected RV variations, eclipse shapes, and bisector slopes
+for different blended EB scenarios, and performing such simul ations is beyond the scope of this
+discovery paper. Instead, we explore the blended EB scenari o through qualitative reasoning.
+In order to produce a flat-bottomed, 2% transit, as seen for WA SP-19, the flux ratio of the
+EB compared to WASP-19 would have to be small enough that the E B was undetectable as a peak
+in the cross-correlation function. However, it could not be so small that the necessary unblended
+eclipsedepthwouldrequirenearly equalsizedEBcomponent sandtherefore, aV-shapedeclipse(i.e.
+0.05< FEB/FW19<0.2). Although the eclipsing star would have to be small (and pr esumably
+less massive) compared to its primary to create the flat-bott omed eclipse, the RV amplitude of
+the visible EB primary would still be 40-90 km s−1for all reasonable mass ratios due to the
+high inclincation angle needed to eclipse, the short orbita l period of the observed transits, and
+the relatively deep transit (i.e. brown dwarf mass eclipsin g objects would produce much shallower
+transit depths). Therefore, the RV variations of the EB coul d not be hidden within the WASP-19
+spectral features given the resolution of the CORALIE data.
+Furthermore, a short period stellar EB would almost certain ly be tidally synchronized with
+vsini∼40−80 km s−1. In the analysis of HD 41004 A (Santos et al. 2002), which exhi bits
+planet-like RV variations due to a blended M dwarf + brown dwa rf spectroscopic binary (SB), the
+bisector correlation is most dependent on the width of the vi sible SB component. This system,
+in which the SB is only 3% as bright as the single star and the wi dth of the SB cross correlation
+function is ∼8 km s−1, shows a significant bisector correlation (slope of 0.67). A ccording to their
+simulations, higher rotational broadening would produce a n even greater bisector slope. Therefore,
+we would expect WASP-19 to show a significant bisector trend i f the RV signal was caused by a
+rapidly rotating, blended EB, rather than a transiting plan et.
+Finally, we see no difference in the eclipse depth of the z-band FTS transit compared the
+SuperWASP transit taken in a bluer, V+R filter. Although this is not a strong constraint given
+the scatter in the SuperWASP photometry, an eclipse by a stel lar object could show eclipse depth
+variations in different filters which we do not see. In summary, we conclude that the existing
+photometric and radial velocity variations of WASP-19 are m ost likely due to the presence of a
+transiting extra-solar planet.
+3. Analysis
+A combined analysis of the radial velocity curve and light cu rve of a transiting planet host star
+will provide direct measurements of the mass and radius of it s orbiting planet with one additional
+constraint, e.g. the stellar mass. Below, we describe the de termination of the stellar mass and other– 6 –
+host star properties.
+3.1. Spectroscopic parameters
+The individual CORALIE spectra were co-added into a single h igher signal-to-noise (S/N)
+spectrum which was then used to measure the stellar temperat ure, gravity, metallicity, vsiniand
+elemental abundance information through comparisons with the synthetic spectra of Castelli et al.
+(1997). The results of the analysis are listed in Table 2. The quoted uncertainties directly correlate
+with the modest signal-to-noise of the co-added spectrum (S /N∼70).
+The spectral synthesis technique is described in detail in G illon et al. (2009a), therefore we
+only give a short summary here. The H αline was the primary temperature determinant, while
+the NaiD and Mg ib lines were used as surface gravity (log g) diagnostics. By measuring the
+equivalent widthof several clean andunblendedmetal lines , we derived abundancesforthe elements
+listed in Table 2. Due to the spread in abundance measurement s, we adopt an overall metallicity
+for the system of [M/H]= 0 .1±0.1. The microturbulence (1 .1±0.2 km s−1), which directly affects
+the abundance measurements, was derived from the Fe ilines using Magain’s (1984) method.
+The projected rotational velocity ( vsini) was determined by fitting the profiles of several
+unblended Fe ilines. We accounted for line broadening due to the instrumen tal FWHM (0.11 ˚A)
+which was determined from telluric lines around 6300 ˚A, and the macroturbulence (2 km s−1) which
+was based on the tabulation by Gray (2008). Finally, we do not detect Li iin the stacked spectrum,
+and can therefore only put an upper limit on the abundance of t his element of log A(Li)<1.
+3.2. Host star mass
+By comparing the effective temperature, metallicity, and mea n stellar density ( ρ∗) of WASP-19
+to theoretical stellar models, we determine its mass. The st ellar density is dependent on the shape
+of the transit and largely independent of any assumptions or models (Seager & Mall´ en-Ornelas
+2003). Thus, after deriving the spectroscopic parameters, we model the WASP-South and FTS
+transit light curves of WASP-19 using the Markov-chain Mont e Carlo (MCMC) routined described
+in Collier Cameron et al. (2007) to derive the stellar densit y and its uncertainty. The code uses
+the MCMC approach to simultaneously solve for the orbital an d physical properties of the star-
+planet system. We apply the limb darkening coefficients of Cla ret (2000, 2004) for the appropriate
+temperature of the star and wavelength of the light curves. W e note that the eccentricity value
+has an effect on the stellar density determination, thus we firs t allow the eccentricity to be a free
+parameter. Since the resulting eccentricity value gives a n on-signficant 1 .5σdetection, we solve for
+the stellar density a second time while fixing the eccentrici ty to be zero.
+InterpolatingamongtheGirardi et al.(2000)stellarevolu tiontracksasdescribedinHebb et al.– 7 –
+Table 2: Stellar properties of WASP-19 obtained from the NOM AD catalogue and derived from our
+analysis of the spectra and light curves.
+Parameter WASP-19
+RA(J2000) 09:53:40.08
+Dec(J2000) -45:39:33.0
+J 10 .911±0.026
+H 10 .602±0.022
+K 10 .481±0.023
+µRA −41.3±2.5 mas yr−1
+µDEC 16.5±1.9 mas yr−1
+U −49+21
+−15km s−1
+V −25+2
+−2km s−1
+W −13+7
+−10km s−1
+Teff 5500±100 K
+logg 4.5±0.2
+ξt 1.1±0.1 km s−1
+vsini 4±2 km s−1
+[Fe/H] 0.02 ±0.09
+[Si/H] 0.15 ±0.07
+[Ca/H] 0.12 ±0.15
+[Ti/H] 0.13 ±0.12
+[Ni/H] 0.10 ±0.08
+log A(Li) <1.
+ρ∗ 1.13±0.12ρ⊙
+Prot 10.5±0.2 days– 8 –
+(2009), we derive a mass for the host star of M∗= 0.95+0.09
+−0.10M⊙using the circular orbit solution.
+The non-circular orbit value for the stellar density result s in a mass of 0.96 M⊙, well within the
+existing error bars (adopted from the non-circular solutio n). Figure 5 shows a plot of the position
+of WASP-19 in a modified Hertzprung-Russell diagram as compa red to the theoretical tracks. The
+errors on the stellar mass are dominated by the uncertainty o n the metallicity. Here, we take
+into account uncertainties on the stellar temperature, met allicity and density. We do not include
+systematic uncertainties on the chosen stellar model which are difficult to determine, but we might
+expect this additional uncertainty to be at least ∼4 %. Southworth (2009) finds that with stellar
+parameters measured at the limit of our technological and th eoretical ability for HD 209458b, the
+variation in the mass determination using four different stel lar models is ∼4%.
+3.3. Host star age
+The isochrone fitting also allows for deriving an age estimat e for WASP-19. According to these
+stellar models, WASP-19 is a main-sequence star with an age o f 5.5+9.0
+−4.5Gyr when we adopt the zero
+eccentricity value for the stellar density. This essential ly places a weak constraint on the stellar age
+to be/greaterorsimilar1Gyr. If weadopt thevalue forthe stellar density fromtheso lution whentheeccentricity is
+a free parameter (non-significant 1 .5σresult), we are unable to place any constraints on the stella r
+age from the isochrones. The depletion of lithium in the atmo sphere of a G8V star can also be
+used as an age indicator. However, the non-detection of Li iin WASP-19 again gives only a weak
+constraint on the age suggesting the star is older than the Hy ades (0.6 Myr, Sestito & Randich
+2005).
+We, therefore, examine the three dimensional velocity of WA SP-19 as compared to theoretical
+Galactic model stars to estimate the probability that WASP- 19 is part of a young disk population.
+Using the catalogue proper motions and measured systemic ra dial velocity, we calculate its U,V,
+andWspace motion (given in Table 2) compared to the Sun. We estima te the distance of WASP-
+19 to be 250+80
+−60pc using the measured spectral type, G8V, with absolute magn itude,Mv= 5.6
+from Gray (1988) and V magnitude from the NOMAD catalogue ( V= 12.59). We adopt a ±0.2
+uncertainty on the magnitude measurement as it is not given i n the catalogue and a generous
+error of ±2 subclasses in spectral type to determine the uncertainty o n the distance estimate.
+We select a set of model main sequence stars of spectral types F7-K7 in a small volume around
+WASP-19 ( l= 273±3◦,b= 7±3◦, distance=150-400 pc) using the Besan¸ con Galactic model
+(Robin et al. 2003). We generate 100 resolutions of the simul ation which provide the heliocentric
+velocities, metallicities, and population classes for ove r 500,000 model stars. Of the model stars
+with metallicities of 0.0-0.2 dex and with space motions wit hin the errors of the calculated U,V,
+andWvalues of WASP-19, 35% of the model stars have ages of <1 Gyr (population class 1 and
+2), 59% have ages of 1-5 Gyr (population class 3-5) and 6% have ages of 5-10 Gyr. Contributions
+from the thick disk, halo and bulge populations are negligib le. This analysis suggests WASP-19
+has a 65% probability of being older than 1 Gyr.– 9 –
+In summary, we present three different age dating techniques w hich all suggest WASP-19 is
+older than ∼1 Gyr, however a precise age for the star cannot be determined from the existing
+data.
+3.4. Host star rotation period
+We search for variability in the WASP-South light curves of W ASP-19 caused by asymmetric
+starspots on the photosphere which modulate the flux. Any sta rspots will rotate in and out of
+view with the stellar surface, such that the period of the var iability gives the rotation period of
+the star. To exhibit rotational variability, the star must h ave a sufficient coverage of starspots to
+create a variable brightness signal which is detectable giv en the scatter in the photometric data.
+Furthermore, starspots evolve on timescales of weeks or mon ths causing the amplitude and phase of
+the variability to change, therefore, each season of WASP-S outh data was examined independently.
+To detect the rotational variability, we determine the impr ovement in χ2over a flat, non-
+variable model when a sine wave of the form y=a0+a1sin(ωt+a3) is fit to each season of the
+WASP-South data phase-folded at a set of trial periods, Prot= 2π/ω. We subtract all transits
+from the light curves using the model derived from the parame ters in Table 3 before fitting the
+sine curve model. We test periods between 0 .2−50 days and find a strong periodic signal in the
+2007 data with Prot= 10.5 days and amplitude, a1= 7.6 mmag. The phase-folded light curve and
+periodogram of normalized ∆ χ2values are shown in Figure 6 and Figure 7 (top), respectively . In
+the 2008 data, we also detect a weaker signal with a similar pe riod,Prot= 10.6 days, and with an
+amplitude of 3.6 mmag.
+To assess the veracity of the sinusoidal signal detected in t he 2007 data, we determine the sig-
+nificance and the false alarm probability (FAP) following Ze chmeister & K¨ urster (2009) (employing
+the residual variance normalization). The equations inclu de,N, the number of independent data
+points, and M, the number of independent frequencies, as well as the measu red peak value in
+the periodogram. According to Cumming (2004), the number of independent frequencies can be
+approximated by the duration of the time-series data times t he difference between the highest and
+lowest frequencies tested (here M= 753). The number of independent data points depends on the
+level of red noise in the light curve. We empirically determi neNby generating a new light curve
+in which the red noise is preserved, but the periodic signal i s destroyed. We randomly re-order
+the individual nights of data, so that the integer part of the light curve time values are shuffled,
+but the fractional parts (containing the red noise) remain t he same. We then run the sine fitting
+program on the shuffled light curve and find N= 664 by assuming the highest peak in the resulting
+periodogram (Figure 7 (bottom)) has a 95% probability of bei ng false ( FAP= 0.95). Using our
+calculated NandM, we then apply the equations in Zechmeister & K¨ urster (2009 ) to the results
+of the sine fitting on the original 2007 light curve and find a hi ghly significant periodic signal with
+a Prob(p >pbest) = 1.4×10−10andFAP= 1×10−7. Thus, we adopt a rotation period for
+WASP-19 of Prot= 10.5 days.– 10 –
+We calculate the error on this measurement using the formula in Horne & Baliunas (1986) and
+findσP= 0.08, but given the variation in period values we measure using two other techniques
+(Lomb-Scargle and auto-correlation), we find this to be unde restimated and suggest σP= 0.2 days
+is a more realistic uncertainty. Finally, we note that the 10 .5 day rotation period of the star
+measured via the photometry is consistent with the less prec isevsinivalue of 4 ±2 km s−1which
+corresponds to a rotational period of between 8-24 days (ass uming the spin axis of the star is
+perfectly aligned with the orbital axis).
+The rotation period of a main sequence star can also be used to estimate its age. According to
+the Barnes (2007) gyrochronology relationship, a rotation period of Prot= 10.5 days corresponds
+to an age of 500-600 Myr for a G8V star like WASP-19 with B-V=0. 74. This value is not consistent
+with the older age inferred from the isochrones, lithium, an d space motion. In §4, we discuss the
+implications of the possible discrepancy between the differe nt age indicators.
+3.5. Planet properties
+Wesolvedforthepropertiesofthestar-planetsystembyrun ninganMCMCcode(Collier Cameron et al.
+2007) using as inputs the WASP-South light curve, the FTS lig ht curve, the CORALIE radial ve-
+locity curve, and the stellar mass. Initially, we allowed th eecosωandesinωto be free parameters,
+but the low, non-zero values that resulted were not significa nt. Therefore, we also solved for the
+parameters of the system while forcing the planet to be on a ci rcular orbit. The difference in stellar
+and planet properties derived for the circular and non-circ ular orbit cases is negligible and within
+the 1σuncertainties on all parameters. However, for completenes s, we provide the resulting solu-
+tions for both cases. Table 3 gives the final model parameters for the transit and radial velocity
+curves as well as the physical properties of the star-planet system.
+Note, we do not remove the rotational variability from the li ght curves before deriving the
+final system parameters. The model fit is dominated by the FTS d ata, and the low-amplitude
+10.5 day rotational variability is negligible on the 3.3 hou r timescale of this light curve. We confirm
+this by fixing the orbital period to the value found when analy sing all the light curve data and
+fitting just the FTS light curve and radial velocity data. All the resulting light curve parameters
+and physical system parameters are well within their report ed 1σuncertainties. Therefore, we
+do not ‘pre-whiten’ the WASP-South data before performing t he final transit model fits on all
+the existing photometric data. However, it is important to n ote that we cannot determine if any
+starspots were present on WASP-19 during the FTS observatio ns which might affect the resulting
+parameter determinations because of the relatively small a mount of out-of-transit data. Additional
+high quality transit data would allow for investigating any variations in the derived parameters due
+to starspots.
+Finally, to derive the final parameters, we use the 4-coefficie nt limb darkening model by Claret
+(2004, 2000) with the ATLAS model atmospheres, an effective te mperature of 5500 K, and a log gof– 11 –
+4.5. However, we tested a range of limb darkening coefficients with different temperatures, log g
+values, andmodelatmospheres andfoundtheresultingparam eters tobehighlyrobustto changes in
+these coefficients. We suspect this is due to the fact the FTS z- band data dominates our parameter
+results, and this filter is not as susceptible to limb darkeni ng uncertainties as bluer filter data.
+3.6. Transit Timing
+We measured the heliocentric Juilian date of the mid-transi t times for WASP-19b using the
+technique described in Anderson et al. (2009) to search for v ariations which could indicate the
+presence of a outer planet (e.g. Agol et al. 2005). In general , the SuperWASP transits are not
+precise enough to provide a useful constraint on the existen ce of a third body in the system, but
+these data do rule out transit timing variations larger than ∼15 minutes.
+4. Discussion
+The most striking aspect of WASP-19b is its extremely short o rbital period. With a period,
+P= 0.7888399 ±0.0000008 days, WASP19b is the shortest period planet yet disc overed. What
+physical processes in the evolution of the planet have lead t o such a close separation? Did the initial
+migration process leave the planet in its extremely short pe riod orbit or has there been subsequent
+evolution of the orbital separation? The observations pres ented here suggest the possibility that
+WASP-19b has been spiralling into its host star throughout i ts lifetime and has spun up its host
+star in the process.
+Most transiting extrasolar planets will ultimately spiral into their host stars because there is
+insufficient total angular momentum in the systems to reach a s tate of tidal equilibrium (Hut 1980;
+Rasio et al. 1996; Jackson et al. 2009). However, the timesca le for this evolution is not well known
+becausethestellar tidal quality factor, Q′
+s, is uncertaintoat least threeordersof magnitude. Values
+between 106< Q′
+s<109are reasonable (Jackson et al. 2008, and references therein ). Furthermore,
+the planetary tidal quality factor, Q′
+pis equally uncertain, but it affects the evolution of the stell ar
+spin and planetary orbital separation to a much lesser degre e.
+For WASP-19b, the ratio of total to critical angular momentu m is well below the limiting
+value of one, thus ensuring the planet will ultimately colli de with its host star, but the lifetime of
+this evolution ranges from 4 Myr to 4 Gyr depending on the valu e used for Q′
+s. However, unlike
+most other transiting planets, we have measured the rotatio n period of WASP-19 which contributes
+additional information that can be used to place tighter con straints on the overall lifetime of the
+planet and on the tidal quality factor of the star.
+By integrating the coupled equations of orbital eccentrici ty and separation (Dobbs-Dixon et al.
+2004) while conserving the total angular momentum of the sys tem, we can model the future evo-– 12 –
+Table 3: WASP-19 system parameters and 1 σerror limits derived from the MCMC analysis.
+Parameter Symbol Value ( efixed) Value ( efree) Units
+Transit epoch (BJD) T02454775.3372+0.0001
+−0.00022454775.3372+0.0002
+−0.0002days
+Orbital period P0.7888399+0.0000008
+−0.0000008 0.7888399+0.0000008
+−0.0000008 days
+Planet/star area ratio ( Rp/Rs)20.0203+0.0004
+−0.0004 0.0203+0.0004
+−0.0004
+Transit duration tT 0.0642+0.0006
+−0.0006 0.0643+0.0006
+−0.0007 days
+Impact parameter b 0.62+0.03
+−0.03 0.62+0.03
+−0.03 R∗
+Stellar reflex velocity K1 0.256+0.005
+−0.005 0.256+0.005
+−0.005 km s−1
+Centre-of-mass velocity γ 20.78534+0.0002
+−0.0002 20.78535+0.0003
+−0.0003km s−1
+Orbital semimajor axis a 0.0165+0.0005
+−0.0006 0.0164+0.0005
+−0.0006 AU
+Orbital inclination I 80.5+0.7
+−0.7 80.8+0.8
+−0.8 degrees
+Orbital eccentricity e 0 (fixed) 0 .02+0.02
+−0.01
+Longitude of periastron ω 0 (fixed) −76+112
+−23 deg
+eccentricity ×cos(ω) ecosω 0 (fixed) 0 .004+0.009
+−0.009
+eccentricity ×sin(ω) esinω 0 (fixed) −0.02+0.02
+−0.02
+Stellar mass M∗ 0.96+0.09
+−0.10 0.95+0.10
+−0.10 M⊙
+Stellar radius R∗ 0.94+0.04
+−0.04 0.93+0.05
+−0.04 R⊙
+Stellar surface gravity log g∗ 4.47+0.03
+−0.03 4.48+0.03
+−0.03 [cgs]
+Stellar density ρ∗ 1.13+0.09
+−0.09 1.19+0.12
+−0.11 ρ⊙
+Planet radius Rp 1.31+0.06
+−0.06 1.28+0.07
+−0.07 RJ
+Planet mass Mp 1.15+0.08
+−0.08 1.14+0.07
+−0.07 MJ
+Planetary surface gravity log gp 3.19+0.03
+−0.03 3.20+0.03
+−0.03 [cgs]
+Planet density ρp 0.51+0.06
+−0.05 0.54+0.07
+−0.06 ρJ
+Planet temperature ( A= 0,F=1) Teq 2009+26
+−26 1993+32
+−33 K– 13 –
+lution of the planet as it spirals inward and further spins up the host star. In the analysis, we
+include magnetic braking which takes the form ˙Ω =−κΩ3withκ= 3.88×10−7s as evaluated
+using equation (12) of Collier Cameron & Jianke (1994). We us e the current orbital separation
+derived from the MCMC analysis (Table 3) and the measured rot ation period ( Prot= 10.5 days)
+as initial conditions. Figure 8 shows the future evolution o f the planet in orbital separation, and
+in Figure 9, we plot the corresponding rotational evolution of the star.
+The results show that in the Q′
+s= 109case, the tidal interaction is so weak that the star’s
+future evolution is dominated by magnetic braking. In this s ituation, the relatively short rotation
+period could not have been caused by prior tidal interaction s with the planet and would instead
+suggest a young stellar age similar to that of the Hyades ( ∼600 Myr). Although it is possible
+WASP-19 is as young as the Hyades, the isochrones analysis, t he non-detection of lithium, and the
+stellar velocity all favor an older age for the star. Therefo re, a lower value for Q′
+sis preferred.
+For all models with Q′
+s<108, the star’s spin period has already passed through a maximum
+and is decreasing as the planet spirals in. However, in the Q′
+s= 106case, the remaining lifetime
+of the planet is extremely short (4 Myr). Therefore, the exis ting data appears to favor values of
+Q′
+s= 107−108which are high enough to give significant life expectancy, bu t low enough to have
+reversed magnetic braking and initiated spiral-in. This sc enario would allow for the extremely
+short period of the planet to be a consequence of further evol ution in orbital separation after the
+initial migration process early in its formation and evolut ion. However, the Q′
+swould have to
+be 1-2 orders of magnitude greater than the nominal value of 1 06that is typically adopted (e.g.
+Bodenheimer et al. 2003; Jackson et al. 2008).
+It is important to note that Q′
+sis a simple parameterization of the complex physics involvi ng
+the interaction between the tides raised by the planet on its star and the turbulent viscosity in the
+stellar convection zone and inertial wave modes excited in t he stellar interior (Rasio et al. 1996;
+Sasselov 2003; Ogilvie & Lin 2007). A more detailed analysis of this system and others like it (e.g.
+WASP-18, OGLE-TR-56b, WASP-12) will hopefully lead to a bet ter understanding of the physics
+modulating tidal dissipation in stars, since currently, th ere is no comprehensive theory which is
+able to explain observations of both main sequence binary st ars and close-in extra-solar planets
+with regard to their tidal evolution (Rasio et al. 1996; Sass elov 2003; Terquem 1998; Ogilvie & Lin
+2007).
+Finally, we note that the stellar flux incident on WASP-19b at the substellar point is 3 .64×
+109ergscm−2s−1placingitintheclassofhighlyirraditedplanetslikeOGLE -TR-56b(Konacki et al.
+2003; Torres et al. 2004) and WASP-1b (Cameron et al. 2007). T herefore, we expect the planet to
+have large secondary eclipse depths in the mid-IR, evidence of an atmospheric temperature in-
+version, molecular emission features, and a large day/nigh t contrast (Fortney et al. 2008). The
+existence of a hot stratosphere can be tested using secondar y eclipse measurements that are cur-
+rently beingobtained with Spitzer. Eclipsemeasurements i nthenear-IRandoptical z-bandarealso
+possible given current technology (e.g. Gillon et al. 2009b ). Furthermore, the density of WASP-19b– 14 –
+ishalfthat ofJupiter’s( ρp= 0.51ρJ), soit isslightly bloated foritsmass, butnotextremely so . The
+high irradation, increased metallicity of the host star, an d dissipation of tidal energy are possible
+factors causing the enhanced radius (Bodenheimer et al. 200 3; Fortney et al. 2007; Burrows et al.
+2007).
+In summary, WASP-19b is the shortest period transiting plan et yet detected. It has a mass,
+Mpl=1.15 M Jand radius, Rpl=1.31 R J. The planet orbits a main sequence G-dwarf with a slightly
+super-solar metallicity and a rotation period of Prot= 10.5±0.2 days. It is likely WASP-19b
+has been spiraling into its host star over its lifetime and ha s spun up the star in the process. A
+more precise age determination of WASP-19 will allow us to co nfirm this and to place stronger
+constraints on the star’s tidal quality factor, Q′
+s.
+The SuperWASP Consortium consists of astronomers primaril y from the Queen’s University
+Belfast, St Andrews, Keele, Leicester, The Open University , Isaac Newton Group La Palma and
+Instituto de Astrof´ ısica de Canarias. The SuperWASP Camer as were constructed and operated
+with funds made available from Consortium Universities and the UK’s Science and Technology
+Facilities Council.
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+This preprint was prepared with the AAS L ATEX macros v5.2.– 17 –
+Fig. 1.— WASP-South discovery photometry of WASP-19. The da ta are phase-folded with the
+ephemeris given in Table 3.– 18 –
+Fig. 2.— FTS z-band photometry of the WASP-19 transit. The data are conver ted to phase using
+the ephemeris given in Table 3. Overplotted is the best fittin g model transit light curve using the
+formalism of Mandel & Agol (2002) applying the 4th-order lim b darkening coefficients from Claret
+(2004).– 19 –
+Fig. 3.— Radial velocity curve of WASP-19 phase-folded with the ephemeris given in Table 3.
+Overplotted is the best fitting model curve obtained from a co mbined analysis of the photometric
+and spectroscopic data.– 20 –
+Fig. 4.— Line bisector span versus radial velocity. The unce rtainties on the line bisector values
+are double the values on the radial velocity measurement. Th ere is no correlation between the line
+bisector span measurements and radial velocity which rules out star spot variations or a blended
+eclipsing binary as the cause for the velocity variations.– 21 –
+Fig. 5.— Modifed Hertzprung-Russell diagram comparing the stellar density and temperature of
+WASP-19 to theoretical stellar evolution tracks by Girardi et al. (2000) interpolated at a metallicity
+of [M/H]=+0.1. The solid circle shows the result when the ecc entricity is zero, and the asteriks
+shows the value for the stellar density if the eccentricity i s a free floating parameter. The mass
+tracks are labelled and the isochrones are 0.1 (solid), 1 (da shed), 5 (dot-dashed), and 10 (dotted)
+Gyr.– 22 –
+Fig. 6.— WASP-South light curve data from 2007 phase-folded on the rotation period detected in
+the sine fitting, Prot= 10.5 days.– 23 –
+Fig. 7.— Top: Periodogram, ∆ χ2/χ2
+bestversus frequency, resulting from fitting a sine wave to the
+2007 WASP-South light curve. The peak in the periodogram has aFAP= 1×10−7and occurs at
+Prot= 10.5 days indicating the rotation period of the star. Bottom: Pe riodogram resulting from
+fitting a sine wave to the shuffled 2007 light curve in which the p eriodic signal was destroyed, but
+the red (Pont et al. 2006) and white noise were preserved. We a dopted a FAP= 0.95 for the
+highest peak.– 24 –
+Fig. 8.—OrbitalseparationversusageofWASP-19bwhenevol ved forwardfromthecurrentstateto
+the point when theplanet overflows its Roche lobe. Due to exch ange of angular momentum through
+tides, the planet loses orbital angular momentum and spiral s into the star. The different lines which
+are labelled correspond to different values for the stellar qu ality factor, Q′
+s. The remaining lifetime
+of the planet ranges from 4 Myr to 4 Gyr depending on the value o fQ′
+s.– 25 –
+Fig. 9.— Stellar rotation versus age of WASP-19b when evolve d forward accounting for tidal
+evolution. The star gains rotational angular momentum and s pins up as the planet spirals into
+the star, losing angular momentum. The different lines are lab elled for different values of Q′
+s. For
+Q′
+s= 109, the tidal evolution is slow enough that magnetic breaking d ue to stellar winds dominates
+the angular momentum of the star causing it to slow down prior to the eventual spiral in of the
+planet after ∼4 Gyr.
\ No newline at end of file
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@@ -0,0 +1,995 @@
+arXiv:1001.0404v2  [math.AP]  7 Jan 2010Nonlinear stability of periodic traveling wave solutions o f
+systems of viscous conservation laws in the generic case
+Mathew A. Johnson∗Kevin Zumbrun†
+November 3, 2018
+Keywords : Periodic traveling waves; Bloch decomposition; modulate d waves.
+2000 MR Subject Classification : 35B35.
+Abstract
+Extending previous results of Oh–Zumbrun and Johnson–Zumbrun , we show that
+spectralstabilityimplieslinearizedandnonlinearstabilityofspatiallype riodictraveling-
+wave solutions of viscous systems of conservation laws for system s of generic type,
+removing a restrictive assumption that wave speed be constant to first order along the
+manifold of nearby periodic solutions.
+1 Introduction
+Nonclassical viscous conservation laws arising in multiph ase fluid and solid mechanics ex-
+hibit a rich variety of traveling wave phenomena, including homoclinic (pulse-type) and
+periodic solutions along with the standard heteroclinic (s hock, or front-type) solutions
+[GZ, Z6, OZ1, OZ2]. Here, we investigate stability of spatia lly periodic traveling waves:
+specifically, sufficient conditions for stability of the wave .
+In previous work [OZ4, JZ3], we showed that strong spectral s tability in the sense of
+Schneider [S1, S2, S3] implies linearized and nonlinear L1∩HK→L∞stability in all
+dimensions d≥1. However, as pointed out in [OZ1, Se1], the conditions of Sc hneider are
+nongeneric in the conservation law setting , implying the restrictive condition that wave
+speed be constant to first order along the manifold of nearby p eriodic solutions. Indeed,
+it was shown in [OZ2] that failure of this condition implies a degradation in the decay
+rates of the Green function of the linearized equations abou t the periodic wave, suggesting
+∗Indiana University, Bloomington, IN 47405; matjohn@india na.edu: Research of M.J. was partially sup-
+ported by an NSF Postdoctoral Fellowship under NSF grant DMS -0902192.
+†Indiana University, Bloomington, IN 47405; kzumbrun@indi ana.edu: Research of K.Z. was partially
+supported under NSF grants no. DMS-0300487 and DMS-0801745 .
+11 INTRODUCTION 2
+that nonlinear stability would be unlikely in the general (n onstationary wave speed) case
+in dimension d= 1.
+In this paper, we show that these difficulties are only apparen t, and that, somewhat
+surprisingly, spectral stability implies nonlinear stabi lity even if this additional condition
+on wave speeds is dropped. More precisely, we show that small L1∩Hsperturbations
+of a planar periodic solution u(x,t)≡¯u(x1) (without loss of generality taken stationary)
+converge at Gaussian rate in Lp,p≥2 to a modulation
+(1.1) ¯ u(x1−ψ(x,t))
+of the unperturbed wave, where x= (x1,˜x), ˜x= (x2,...,xd), andψis a scalar function
+whosex- andt-gradients decay at Gaussian rate in all Lp,p≥2, but which itself decays
+more slowly by a factor t1/2; in particular, ψis merely bounded in L∞for dimension d= 1.
+In proving this result, we make crucial use of the tools devel oped in [OZ4, JZ3], in
+particular, a key nonlinear cancellation argument of [JZ3] . The key new observation mak-
+ing possible the treatmen of the generic case is a careful stu dy of the Bloch perturbation
+expansion about frequency ξ= 0, motivated by relations to the Whitham averaged system
+observed in [Se1, OZ3, JZ1, JZB].
+It was shown in [Se1, OZ3] that the low-frequency dispersion relation near zero of the
+linearized operator about a periodic solution ¯ uagrees to first order with that of the lin-
+earization about a constant state of the Whitham averaged sy stem
+(1.2)∂tM+/summationdisplay
+j∂xjFj= 0,
+∂t(ΩN)+∇x(ΩS) = 0,
+whereM∈Rndenotes the average over one period, Fjthe average of an associated flux,
+Ω =|∇xΨ| ∈R1the frequency, S=−Ψt/|∇xΨ| ∈R1the speeds, andN=∇xΨ/|∇xΨ| ∈
+Rdthe normal νassociated with nearby periodic waves, with an additional c onstraint
+(1.3) curl (Ω N) = curl ∇xΨ≡0.
+As noted in [Se1, OZ3], this implies both that the eigenvalue sλj(ξ) bifurcating from λ= 0
+atξ= 0 areC1along rays through the origin, and that weak hyperbolicity ( reality of
+characteristics of (1.2)–(1.3)) is necessary for spectral or linearized stability.
+As noted in[JZB], thereis adeeperanalogy between thelow-f requency linearized disper-
+sion relation and the Whitham averaged system at the structu ral level, suggesting a useful
+rescaling of the low-frequency perturbation problem. It is this intuition that motivates our
+derivation of sharp low-frequency estimates crucial to the analysis of nonlinear stability.
+With these estimates in place, the rest of the argument goes e xactly as in [JZ3, OZ4].
+1.1 Equations and assumptions
+Consider a parabolic system of conservation laws
+(1.4) ut+/summationdisplay
+jfj(u)xj= ∆xu,1 INTRODUCTION 3
+u∈ U(open)∈Rn,fj∈Rn,x∈Rd,d≥1,t∈R+, and a periodic traveling wave solution
+(1.5) u= ¯u(x·ν−st),
+of periodX, satisfying the traveling-wave ODE ¯ u′′= (/summationtext
+jνjfj(¯u))′−s¯u′with boundary
+conditions ¯u(0) = ¯u(X) =:u0.Integrating, we obtain a first-order profile equation
+(1.6) ¯ u′=/summationdisplay
+jνjfj(¯u)−s¯u−q,
+where (u0,q,s,ν,X )≡constant. Without loss of generality take ν=e1,s= 0, so that
+¯u= ¯u(x1) represents a stationary solution depending only on x1.
+Following [Se1, OZ3, OZ4], we assume:
+(H1)fj∈CK+1,K≥[d/2]+4.
+(H2) Themap H:R×U×R×Sd−1×Rn→Rntaking (X;a,s,ν,q)/mapsto→u(X;a,s,ν,q)−a
+is full rank at ( ¯X;¯u(0),0,e1,¯q), whereu(·;·) is the solution operator of (1.6).
+Conditions (H1)–(H2) imply that the set of periodic solutio ns in the vicinity of ¯ uform
+a smooth (n+d+1)-dimensional manifold {¯ua(x·ν(a)−α−s(a)t)}, withα∈R,a∈Rn+d.
+1.1.1 Linearized equations
+Linearizing (1.4) about ¯ u(·), we obtain
+(1.7) vt=Lv:= ∆xv−/summationdisplay
+(Ajv)xj,
+where coefficients Aj:=Dfj(¯u) are now periodic functions of x1. Taking the Fourier
+transform in the transverse coordinate ˜ x= (x2,···,xd), we obtain
+(1.8)ˆvt=L˜ξˆv= ˆvx1,x1−(A1ˆv)x1−i/summationdisplay
+j/negationslash=1Ajξjˆv−/summationdisplay
+j/negationslash=1ξ2
+jˆv,
+where˜ξ= (ξ2,···,ξd) is the transverse frequency vector.
+1.1.2 Bloch–Fourier decomposition and stability conditions
+Following [G, S1, S2, S3], we define the family of operators
+(1.9) Lξ=e−iξ1x1L˜ξeiξ1x1
+operating on the class of L2periodic functions on [0 ,X]; the (L2) spectrum of L˜ξis equal
+to the union of the spectra of all Lξwithξ1real with associated eigenfunctions
+(1.10) w(x1,˜ξ,λ) :=eiξ1x1q(x1,ξ1,˜ξ,λ),
+whereq, periodic, is an eigenfunction of Lξ. By continuity of spectrum, and discreteness of
+the spectrum of the elliptic operators Lξon the compact domain [0 ,X], we have that the
+spectra ofLξmay be described as the union of countably many continuous su rfacesλj(ξ).1 INTRODUCTION 4
+Without loss of generality taking X= 1, recall now the Bloch–Fourier representation
+(1.11) u(x) =/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xˆu(ξ,x1)dξ1d˜ξ
+of anL2functionu, where ˆu(ξ,x1) :=/summationtext
+ke2πikx1ˆu(ξ1+ 2πk,˜ξ) are periodic functions of
+periodX= 1, ˆu(˜ξ) denoting with slight abuse of notation the Fourier transfo rm ofuin the
+full variable x. By Parseval’s identity, the Bloch–Fourier transform u(x)→ˆu(ξ,x1) is an
+isometry in L2:
+(1.12) /ba∇dblu/ba∇dblL2(x)=/ba∇dblˆu/ba∇dblL2(ξ;L2(x1)),
+whereL2(x1) is taken on [0 ,1] andL2(ξ) on [−π,π]×Rd−1. Moreover, it diagonalizes the
+periodic-coefficient operator L, yielding the inverse Bloch–Fourier transform representation
+(1.13) eLtu0=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xeLξtˆu0(ξ,x1)dξ1d˜ξ
+relating behavior of the linearized system to that of the dia gonal operators Lξ.
+Loosely following [OZ4], we assume along with (H1)–(H2) the strong spectral stability
+conditions:
+(D1)σ(Lξ)⊂ {Reλ<0}forξ/ne}ationslash= 0.
+(D2) Reσ(Lξ)≤ −θ|ξ|2,θ>0, forξ∈Rdand|ξ|sufficiently small.
+(D3’)λ= 0 is an eigenvalue of L0of multiplicity exactly n+1.1
+As shown in [OZ3], (H1)-(H2) and (D1)–(D3’) imply that there existn+ 1 smooth
+eigenvalues
+(1.14) λj(ξ) =−iaj(ξ)+o(|ξ|)
+ofLξbifurcating from λ= 0 atξ= 0, where −iajare homogeneous degree one functions;
+see Lemma 2.1 below.
+As in [OZ4], we make the further nondegeneracy hypothesis:
+(H3) The functions aj(ξ) in (1.14) are distinct.
+The functions ajmay be seen to be the characteristics associated with the Whi tham av-
+eraged system (1.2)–(1.3) linearized about the values of M,S,N, Ω associated with the
+background wave ¯ u; see [OZ3, OZ4]. Thus, (D1) implies weak hyperbolicity of (1 .2)–(1.3)
+(reality ofaj), while (H1) corresponds to strict hyperbolicity.
+Remark 1.1. Condition (D3’) is a weakened version of the condition (D3) o f [OZ4, JZ3]
+thatλ= 0 be a semisimple eigenvalue of L0of minimal multiplicity n+ 1, which implies
+[OZ1, OZ2, Se1] the special property that wave speed be stati onary at ¯ualong the manifold
+of nearby periodic solutions. The stronger conditions (D1) –(D3) are exactly the spectral
+assumptions of [S1, S2, S3] introduced by Schneider in the re action-diffusion case. Condi-
+tions (D1)–(D3) (resp. (D1)–(D3’)) correspond to “dissipa tivity” of the large-time behavior
+of the linearized system [S1, S2, S3].
+1The zero eigenspace of L0is at least ( n+1)-dimensional by linearized existence theory and (H2), a nd
+hencen+1 is the minimal multiplicity; see [Se1].1 INTRODUCTION 5
+1.2 Main result
+With these preliminaries, we can now state our main result.
+Theorem 1.1. Assuming (H1)–(H3) and (D1)–(D3’), for some C >0andψ∈WK,∞(x,t),
+(1.15)|˜u−¯u(·−ψ)|Lp(t)≤C(1+t)−d
+2(1−1/p)|˜u−¯u|L1∩HK|t=0,
+|˜u−¯u(·−ψ)|HK(t)≤C(1+t)−d
+4|˜u−¯u|L1∩HK|t=0,
+|(ψt,ψx)|WK+1,p≤C(1+t)−d
+2(1−1/p)|˜u−¯u|L1∩HK|t=0
+for allt≥0,p≥2,d≥1, and
+(1.16) |˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+t)−d
+2(1−1
+p)+1
+2|˜u−¯u|L1∩HK|t=0
+for allt≥0andp=∞orp≥2andd≥3, for solutions ˜uof(1.4)with|˜u−¯u|L1∩HK|t=0
+sufficiently small. In particular, ¯uis nonlinearly bounded L1∩HK→L∞stable ford≥1,
+asymptotically L1∩HK→L∞stable ford≥2, and asymptotically L1∩HK→HKstable
+ford≥3.
+Remark 1.2. In Theorem 1.1, derivatives in x∈Rdford≥2refer to total derivatives.
+Moreover, unless specified by an appropriate index, through out this paper derivatives in
+spatial variable xwill always refer to the total derivative of the function.
+In dimension one, Theorem 1.1 asserts only bounded L1∩HK→L∞stability, a very
+weak notion of stability. The bounds (1.15)–(1.16) agree fo r dimension d= 1 with those
+obtained in [JZ3] in the stationary wave speed case that (D3) holds in place of (D3’), but
+for higher dimensions are weaker by roughly factor t1/2.
+Remark 1.3. In dimension d= 1, it is straightforward to show that the results of Theorem
+1.1 extend to all 1 ≤p≤ ∞using the pointwise techniques of [OZ2]; see Remark 3.6.
+1.3 Discussion and open problems
+The proof of Theorem 1.1 largely completes the line of invest igation carried out in [OZ2,
+Se1, OZ3, OZ4, JZ3], showing that spectral stability implie s linear and nonlinear stability of
+planar spatially periodic traveling waves. The correspond ing spectral stability problem has
+been studied analyticially in [OZ1, Se1, OZ3], yielding var ious necessary conditions, and by
+numerical Evans function investigation in [OZ1]. An intere sting direction for further study
+would be more systematic numerical investigation along the lines of [BLZ, HLyZ1, HLyZ2,
+BHZ, BLZ] in the viscous shock wave case. A second interestin g open problem would be
+to extend the results for planar waves to the case of solution s with multiple periods, as
+considered in the reaction–diffusion setting in [S1, S2, S3].
+The key to the nonlinear analysis in critical dimensions d= 1,2, as in [JZ3, S1, S2, S3],
+is to subtract out a slower-decaying part of the solution des cribed by an appropriate mod-
+ulation equation and show that the residual decays sufficient ly rapidly to close a nonlinear2 SPECTRAL PREPARATION 6
+iteration. Note that the modulated approximation ¯ u(x1−ψ(x,t)) of (1.1) is not the full
+Ansatz ¯ua(Ψ(x,t)), Ψ(x,t) :=x1−ψ(x,t), associated with the Whitham averaged system
+(1.2)–(1.3), where ¯ uais the manifold of periodic solutions near ¯ uintroduced below (H2),
+but only the translational part not involving perturbation sain the profile. (See [OZ3] for
+the derivation of Ansatz and (1.2)–(1.3).) That is, we don’t need to separate out all vari-
+ations along the manifold of periodic solutions, but only th e special variations connected
+with translation invariance.
+This can be understood heuristically by the observation tha t (1.2) indicates that vari-
+ablesa,∇xΨ are roughly comparable, which would suggest, by the diffusiv e behavior
+Ψ>>∇xΨ, thatais neglible with respect to Ψ. Indeed, this heuristic argume nt translates
+rigorously to our ultimate computation of linearized behav ior leading to the final result;
+see Section 2 and Remark 2.2. In this respect, the connection to the Whitham system is
+somewhat clearer in the generic case considered here than in the quasi-Hamiltonian case
+treated previously in [OZ2, OZ4, JZ3].2
+It would be interesting to better understand the connection between the Whitham av-
+eraged system (or suitable higher-order correction) and be havior at the nonlinear level, as
+explored at the linear level in [OZ3, OZ4, JZ1, JZB]. As discu ssed further in [OZ3], another
+interesting problem would be to try to rigorously justify th e WKB expansion for the related
+vanishing viscosity problem, in the spirit of [GMWZ1, GMWZ2 ].
+2 Spectral preparation
+We begin by a careful study of the Bloch perturbation expansi on atξ= 0.
+Lemma 2.1. Assuming (H1)–(H3), (D1)–(D3’), the eigenvalues λj(ξ/|ξ|,ξ)ofLξare an-
+alytic functions of ξ/|ξ|and|ξ|. Suppose further that 0is a nonsemisimple eigenvalue of
+L0, i.e., (D3’) holds, but not (D3). Then, the Jordan structure o f the zero eigenspace of
+L0consists of an n-dimensional kernel and a single Jordan chain of height 2, where the
+left kernel of L0is then-dimensional subspace of constant functions, and ¯u′spans the right
+eigendirection lying at the base of the Jordan chain. Moreov er, for|ξ|sufficiently small,
+there exist right and left eigenfunctions qj(ξ/|ξ|,ξ,·)and˜qj(ξ/|ξ|,ξ,·)ofLξassociated with
+λjof formqj=/summationtext
+kβj,kvkand˜qj=/summationtext
+k˜βj,k˜vkwhere{vj}and{˜vj}are dual bases of the to-
+tal eigenspace of Lξassociated with sufficiently small eigenvalues, analytic in ω=ξ/|ξ|and
+|ξ|, with˜vj(ω;0)constant for j/ne}ationslash=nandvn(ω;0)≡¯u′(·);˜βj,1,...,˜βj,n−1,|˜ξ|−1˜βj,n,˜βj,n+1
+andβj,1,...,βj,n−1,|ξ|βj,n,βj,n+1are analytic in ξ/|ξ|,|ξ|; and/an}b∇acketle{t˜qj,qk/an}b∇acket∇i}ht=δk
+j.
+Proof.Recall that Lξas an elliptic second-order operator on boundeddomain has s pectrum
+consisting of isolated eigenvalues of finite multiplicity. Expanding
+(2.1) Lξ=L0+|ξ|L1
+ξ/|ξ|+|ξ|2L2
+ξ/|ξ|
+2In the degenerate case that the stronger condition (D3) hold s, i.e., wave speed is stationary at ¯ u, the
+situation is somewhat more complicated, and these relation s break down; see [JZ3] for further discussion.2 SPECTRAL PREPARATION 7
+for each fixed angle ˆξ:=ξ/|ξ|, consider the continuous family of spectral perturbation
+problems in |ξ|indexed by angle ω=ξ/|ξ|about the eigenvalue λ= 0 ofL0.
+Because 0 is an isolated eigenvalue of L0, the associated total right and left eigenpro-
+jectionsP0and˜P0perturb analytically in both ωand|ξ|, giving projection Pξand˜Pξ[K].
+These yield in standard fashion (for example, by projecting appropriately chosen fixed sub-
+spaces) locally analytic right and left bases {vj}and{˜vj}of the associated total eigenspaces
+given by the range of Pξ,˜Pξ.
+DefiningV= (v1,...,vn+1) and˜V= (˜v1,...,˜vn+1)∗,∗denoting adjoint, we may con-
+vert the infinite-dimensional perturbation problem (2.1) i nto an (n+ 1)×(n+ 1) matrix
+perturbation problem
+(2.2) Mξ=M0+|ξ|M1+|ξ|2M2,
+whereMξ(ω,|ξ|) :=/angbracketleftBig
+˜V∗
+ξ,LξVξ/angbracketrightBig
+and/an}b∇acketle{t·,·/an}b∇acket∇i}htrefers to the L2(x1) inner product on [0 ,X]. That
+is, the eigenvalues λj(ξ) lying near 0 of Lξare the eigenvalues of Mξ, and the associated
+right and left eigenfunctions of Lξare
+(2.3) fj=Vwjand˜fj= ˜wj˜V∗,
+wherewjand ˜wjare the associated right and left eigenvectors of Mξ.
+Case (i). Ifλ= 0 is a semisimple eigenvalue of L0, thenM0= 0, and (2.2) reduces
+to the simpler perturbation problem ˇMξ:=|ξ|−1Mξ=M1+|ξ|M2studied in [OZ4, JZ3],
+whichλj(ξ) =|ξ|ˇλj(ξ),ˇλj(ξ) denoting the eigenvalues of ˇMξ. Sinceˇλjare continous, λj
+are differentiable at |ξ|= 0 in the parameter |ξ|as asserted in the introduction. Moreover,
+by (H3), the eigenvalues ˇλj(0) ofM1=ˇM0are distinct, and so they perturb analytically in
+ω,|ξ|, as do the associated right and left eigenvectors.
+Case (ii) . Hereafter, assume that λ= 0 is a nonsemisimple eigenvalue of L0, so that
+M0is nilpotent but nonzero, possessing a nontrivial associat ed Jordan chain. Moreover, as
+then-dimensional subspace of constant functions by direct comp utation lie in the kernel of
+L∗
+0= (∂2
+x1+A∗
+1∂x1), whereA1(x1) :=df1(¯u(x1)), we have that the ( n+1)-dimensional zero
+eigenspace of L0is consists precisely of an n-dimensional kernel and a single Jordan chain
+of height two. Moreover, by translation-invariance (differe ntiate inx1the profile equation
+(1.6)), we have L0¯u′= 0, so that ¯ u′lies in the right kernel of L0.
+Now, recall the assumption (H2) that H:R× U ×R×Sd−1×Rn→Rntaking
+(X;a,s,ν,q)/mapsto→u(X;a,s,ν,q)−aisfullrankat( ¯X;¯u(0),0,e1,¯q), whereu(·;·) isthesolution
+operator of profile ODE (1.6). The fact that ker L0isn-dimensional implies that the
+restriction ˇHtaking (a,q)/mapsto→u(X;a,s,ν,q)−afor fixed (X,ν,s) is also full rank, i.e.,
+His full rank with respect to the specific parameters ( X,s,ν). Applying the Implicit
+Function Theorem and counting dimensions, we find that the se t of periodic solutions, i.e.,
+the inverse image of zero under map Hlocal to ¯uis a smooth ( n+d+ 1)-dimensional
+manifold {¯ua(x·ν(a)−α−s(a)t)}, withα∈R,a∈Rn+d. Moreover, d+1 dimensions may
+be parametrized by ( X,s,ν), or without loss of generality ( a1,...,ad+1) = (X,s,ν).2 SPECTRAL PREPARATION 8
+Fixing (X,ν) and (ad+2,...,an+d+1), and varying s, we find by differentiation of (1.6)
+thatf∗:=−∂s¯usatisfies3the generalized eigenfunction equation
+L0f∗= ¯u′.
+Thus, ¯u′spans the eigendirection lying at the base of the Jordan chai n, with the generalized
+zero-eigenfunction of L0corresponding to variations in speed along the manifold of p eriodic
+solutions about ¯ u. Without loss of generality, therefore, we may take ˜ v1,...˜vn−1and ˜vn+1
+to be constant at |ξ|= 0, i.e., depending only on ω=ξ/|ξ|and notx1, andvn≡¯u′at
+|ξ|= 0 independent of ω.
+Recalling from [JZ3] the fact that
+/an}b∇acketle{tc,L1¯u′/an}b∇acket∇i}ht=/an}b∇acketle{tc,(ω1(2∂x1−A1)−/summationdisplay
+j/negationslash=1ωjAj))¯u′/an}b∇acket∇i}ht=/an}b∇acketle{tc,ω1∂2
+x1¯u−/summationdisplay
+j/negationslash=1ωj∂x1fj(¯u)/an}b∇acket∇i}ht ≡0
+for any constant functions c, where again /an}b∇acketle{t·,·/an}b∇acket∇i}htdenotesL2(x1) inner product on the interval
+x1∈[0,X], andAj:=dfj(¯u(·)), we find under this normalization that (2.2) has the specia l
+structure
+(2.4) M0=
+0(n−1)×(n−1)0n−10n−1
+0 0 1
+0 0 0
+, M 1=
+∗0n−1∗
+∗ ∗ ∗
+∗0∗
+.
+Now, rescaling (2.2) as
+(2.5) ˇMξ:=|ξ|−1S(ξ)MξS(ξ)−1,
+where
+(2.6) S:=
+In−10 0
+0|ξ|0
+0 0 1
+,
+we obtain
+(2.7) ˇMξ=ˇM0+|ξ|ˇM1+O(|ξ|2),
+whereˇMj=ˇMj(ω) like the original Mjare analytic matrix-valued functions of ω, and the
+eigenvalues mj(ξ) =mj(ω;|ξ|) ofˆMξare|ξ|−1λj(ξ).
+As the eigenvalues mjofˇMξare continuous, the eigenvalues λj(ξ) =|ξ|mjare differen-
+tiable at |ξ|= 0 as asserted in the introduction. Moreover, by (H3), the ei genvalues ˇλj(0)
+ofˇM0are distinct, and so they perturb analytically in ω,|ξ|, as do the associated right and
+left eigenvectors zjand ˜zj. Undoing the rescaling (2.5), and recalling (2.3), we obtai n the
+result.
+Remark 2.2. Note that the nth coordinate of vectors w∈Cn+1in the perturbation
+problem (2.2) corresponds as the coefficient of ¯ u′to variations Ψ in displacement. Thus,
+rescaling (2.5) amounts to substituting for Ψ the variable |ξ|Ψ∼Ψxof the Whitham
+averaged system (1.2).
+3Note the function f∗isX-periodic, and hence in the domain of L0since we have fixed the period X.3 LINEARIZED STABILITY ESTIMATES 9
+3 Linearized stability estimates
+By standard spectral perturbation theory [K], the total eig enprojection P(ξ) onto the
+eigenspace of Lξassociated with the eigenvalues λj(ξ),j= 1,...,n+ 1 described in the
+previous section is well-defined and analytic in ξforξsufficiently small, since these (by dis-
+creteness of the spectra of Lξ) are separated at ξ= 0 from the rest of the spectrum of L0.
+Introducing a smooth cutoff function φ(ξ) that is identically one for |ξ| ≤εand identically
+zero for |ξ| ≥2ε,ε >0 sufficiently small, we split the solution operator S(t) :=eLtinto
+low- and high-frequency parts
+(3.1) SI(t)u0:=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xφ(ξ)P(ξ)eLξtˆu0(ξ,x1)dξ1d˜ξ
+and
+(3.2) SII(t)u0:=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·x/parenleftbig
+I−φP(ξ)/parenrightbig
+eLξtˆu0(ξ,x1)dξ1d˜ξ.
+3.1 High-frequency bounds
+By standard sectorial bounds [He, Pa] and spectral separati on ofλj(ξ) from the remaining
+spectra ofLξ, we have trivially the exponential decay bounds
+(3.3)/ba∇dbleLξt(I−φP(ξ))f/ba∇dblL2([0,X])≤Ce−θt/ba∇dblf/ba∇dblL2([0,X]),
+/ba∇dbleLξt(I−φP(ξ))∂l
+x1f/ba∇dblL2([0,X])≤Ct−l
+2e−θt/ba∇dblf/ba∇dblL2([0,X]),
+/ba∇dbl∂l
+x1eLξt(I−φP(ξ))f/ba∇dblL2([0,X])≤Ct−l
+2e−θt/ba∇dblf/ba∇dblL2([0,X]),
+forθ,C >0, and 0 ≤m≤K(Kas in (H1)). Together with (1.12), these give immediately
+the following estimates.
+Proposition 3.1 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D2), for some θ,
+C >0, and allt>0,2≤p≤ ∞,0≤l≤K+1,0≤m≤K,
+(3.4)/ba∇dbl∂l
+xSII(t)f/ba∇dblL2(x),/ba∇dblSII(t)∂l
+xf/ba∇dblL2(x)≤Ct−l
+2e−θt/ba∇dblf/ba∇dblL2(x),
+/ba∇dbl∂m
+xSII(t)f/ba∇dblLp(x),/ba∇dblSII(t)∂m
+xf/ba∇dblLp(x)≤Ct−d
+2(1
+2−1
+p)−m
+2e−θt/ba∇dblf/ba∇dblL2(x),
+where, again, derivatives in x∈Rdrefers to total derivatives.
+Proof.The first inequalities follow immediately by (1.12). The sec ond follows for p=∞,
+m= 0 by Sobolev embedding from
+/ba∇dblSII(t)f/ba∇dblL∞(˜x;L2(x1))≤Ct−d−1
+4e−θt/ba∇dblf/ba∇dblL2([0,X])
+and
+/ba∇dbl∂x1SII(t)f/ba∇dblL∞(˜x;L2(x1))≤Ct−d−1
+4−1
+2e−θt/ba∇dblf/ba∇dblL2([0,X]),3 LINEARIZED STABILITY ESTIMATES 10
+which follow by an application of (1.12) in the x1variable and the Hausdorff–Young in-
+equality /ba∇dblf/ba∇dblL∞(˜x)≤ /ba∇dblˆf/ba∇dblL1(˜ξ)in the variable ˜ x. The result for derivatives in x1and general
+2≤p≤ ∞then follows by Lpinterpolation. Finally, the result for derivatives in ˜ xfollows
+from the inverse Fourier transform, equation (3.2), and the large|ξ|bound
+|eLtf|L2(x1)≤e−θ|˜ξ|2t|f|L2(x1),|ξ|sufficiently large ,
+which easily follows from Parseval and the fact that Lξis a relatively compact perturbation
+of∂2
+x−|ξ|2. Thus, by the above estimate we have
+/ba∇dbleLt∂˜xf/ba∇dblL2(x)≤C/ba∇dbleLξt|˜ξ|ˆf/ba∇dblL2(x1,ξ)
+≤Csup/parenleftBig
+e−θ|˜ξ|2t|ξ|/parenrightBig
+/ba∇dblˆf/ba∇dblL2(x1,ξ)
+≤Ct−1/2/ba∇dblf/ba∇dblL2(x).
+A similar argument applies for 1 ≤m≤K.
+3.2 Low-frequency bounds
+Denote by
+(3.5) GI(x,t;y) :=SI(t)δy(x)
+the Green kernel associated with SI, and
+(3.6) [ GI
+ξ(x1,t;y1)] :=φ(ξ)P(ξ)eLξt[δy1(x1)]
+the corresponding kernel appearing within the Bloch–Fouri er representation of GI, where
+the brackets on [ Gξ] and [δy] denote the periodic extensions of these functions onto the
+whole line. Then, we have the following descriptions of GI, [GI
+ξ], deriving from the spectral
+expansion (1.14) of Lξnearξ= 0.
+Proposition 3.2 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D3’),
+(3.7)[GI
+ξ(x1,t;y1)] =φ(ξ)n+1/summationdisplay
+j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗,
+GI(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)[GI
+ξ(x1,t;y1)]dξ
+=/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
+j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ,
+where∗denotes matrix adjoint, or complex conjugate transpose, qj(ξ,·)and˜qj(ξ,·)are right
+and left eigenfunctions of Lξassociated with eigenvalues λj(ξ)defined in (1.14), normalized
+so that/an}b∇acketle{t˜qj,qj/an}b∇acket∇i}ht ≡1.3 LINEARIZED STABILITY ESTIMATES 11
+Proof.Relation (3.7)(i) is immediate from the spectral decomposi tion of elliptic operators
+on finite domains, and the fact that λjare distinct for |ξ|>0 sufficiently small, by (H3).
+Substituting (3.5) into (3.1) and computing
+(3.8) /hatwideδy(ξ,x1) =/summationdisplay
+ke2πikx1/hatwideδy(ξ+2πke1) =/summationdisplay
+ke2πikx1e−iξ·y−2πiky1=e−iξ·y[δy1(x1)],
+where the second and third equalities follow from the fact th at the Fourier transform either
+continuous or discrete of the delta-function is unity, we ob tain
+GI(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·xφP(ξ)eLξt/hatwideδy(ξ,x1)dξ
+=/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·(x−y)φP(ξ)eLξt[δy1(x1)]dξ,
+yielding (3.7)(ii) by (3.6)(i) and the fact that φis supported on [ −π,π].
+Proposition 3.3. Under assumptions (H1)-(H3) and (D1)-(D3’), the low-frequ ency Green
+functionGI(x,t;y)of(3.5)decomposes as GI=E+˜GI,
+(3.9) E= ¯u′(x)e(x,t;y),
+where, for some C >0, allt>0,
+(3.10)sup
+y/ba∇dbl˜GI(·,t,;y)/ba∇dblLp(x)≤C(1+t)−d
+2(1−1
+p)
+sup
+y/ba∇dbl∂r
+y˜GI(·,t,;y)/ba∇dblLp(x),sup
+y/ba∇dbl∂r
+t˜GI(·,t,;y)/ba∇dblLp(x)≤C(1+t)−d
+2(1−1
+p)−1
+2
+forp≥2,1≤r≤2,
+(3.11) sup
+y/ba∇dbl∂j
+x∂l
+t∂r
+ye(·,t,;y)/ba∇dblLp(x)≤C(1+t)−d
+2(1−1
+p)−(j+l)
+2−1
+2
+forp≥2,0≤j,k,l,j+l≤K,1≤r≤2, and
+(3.12) sup
+y/ba∇dbl˜∂j
+x∂l
+te(·,t,;y)/ba∇dblLp(x)≤C(1+t)−d
+2(1−1
+p)−(j+l)
+2
+for0≤j,k,l,j+l≤K, provided that p≥2andj+l≥1ord≥3, orp=∞andd≥1.
+Moreover,e(x,t;y)≡0fort≤1.
+Remark 3.4. In Proposition 3.3, and throughout the remainder of the paper , derivatives
+iny∈Rdrefer to total derivatives, just as with the variable x∈Rd.3 LINEARIZED STABILITY ESTIMATES 12
+Proof.In the degenerate case (D3) that 0 is a semisimple eigenvalue ofL0, these estimates
+have been established in [OZ4, JZ3]. Without loss of general ity, therefore, we hereafter
+assume that 0 is a nonsemisimple eigenvalue of L0, with the consequences described in
+Lemma 2.1. Recalling that
+(3.13)GI(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
+j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ
+=/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
+j,k,l=1eλj(ξ)tβj,kvk(ξ,x1)˜βj,l˜vl(ξ,y1)∗dξ,
+define
+(3.14) ˜e(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)/summationdisplay
+j,leλj(ξ)tβj,n˜βj,l˜vl(ξ,y1)∗dξ
+so that
+(3.15)GI(x,t;y)−¯u′(x1)˜e(x,t;y) =
+/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)/summationdisplay
+j,k/negationslash=n,leλj(ξ)tβj,k˜βj,lvk(ξ,x1)˜vl(ξ,y1)∗dξ
++/parenleftBig1
+2π/parenrightBigd/integraldisplay
+Rdeiξ·(x−y)φ(ξ)/summationdisplay
+j,leλj(ξ)tβj,n˜βj,l/parenleftBig
+vn(ξ,x1)−¯u′(x1)/parenrightBig
+˜vl(ξ,y1)∗dξ,
+where, by analyticity of vn,vn(ξ,x1)−¯u′(x1) =O(|ξ|), and so, by Lemma 2.1,
+(3.16) βj,n˜βj,l/parenleftBig
+vn(ξ,x1)−¯u′(x1)/parenrightBig
+˜vl(ξ,y1)∗=O(1)
+and
+(3.17) βj,k˜βj,lvk(ξ,x1)˜vl(ξ,y1)∗=O(1) fork/ne}ationslash=n.
+Note further that ˜ vl≡constant unless l=n, in which case ˜βjl=O(|ξ|) by Lemma 2.1;
+hence
+(3.18) ∂y1/parenleftBig
+βj,n˜βj,l/parenleftBig
+vn(ξ,x1)−¯u′(x1))˜vl(ξ,y1)∗/parenrightBig
+=O(|ξ|)
+and
+(3.19) ∂y1/parenleftBig
+βj,k˜βj,lvk(ξ,x1)˜vl(ξ,y1)∗/parenrightBig
+=O(|ξ|) fork/ne}ationslash=n.
+From representation (3.15), bounds (3.16)–(3.17), and ℜλj(ξ)≤ −θ|ξ|2, we obtain by
+the triangle inequality
+(3.20) /ba∇dblGI−¯u′˜e/ba∇dblL∞(x,y)≤C/ba∇dble−θ|ξ|2tφ(ξ)/ba∇dblL1(ξ)≤C(1+t)−d
+2.3 LINEARIZED STABILITY ESTIMATES 13
+Derivative bounds follow similarly, since x1-derivatives falling on vjkare harmless, whereas,
+by (3.18)–(3.19), y1- ort-derivatives falling on ˜ vjlor oneiξ·(x−y)bring down a factor of |ξ|
+improving the decay rate by factor (1 + t)−1/2. (Note that |ξ|is bounded because of the
+cutoff function φ, so there is no singularity at t= 0.)
+To obtain bounds for p= 2, we note that (3.10 may be viewed itself as a Bloch–
+Fourier decomposition with respect to variable z:=x−y, withyappearing as a parameter.
+Recalling (1.12), we may thus estimate
+(3.21)
+sup
+y/ba∇dblGI(·,t;y)−¯u′˜e(·,t;y)/ba∇dblL2(x)≤
+C/summationdisplay
+j,k/negationslash=n,lsup
+y/ba∇dblφ(ξ)eλj(ξ)tvk(·,z1)˜v∗
+l(·,y1)˜vl(·,y1)∗/ba∇dblL2(ξ;L2(z1∈[0,X]))
++C/summationdisplay
+j,lsup
+y/ba∇dblφ(ξ)eλj(ξ)t/parenleftBigvn(·,x1)−¯u′(x1)
+|·|/parenrightBig
+˜vl(·,y1)∗/ba∇dblL2(ξ;L2(z1∈[0,X]))
+≤C/summationdisplay
+j,k/negationslash=n,lsup
+y/ba∇dblφ(ξ)e−θ|ξ|2t/ba∇dblL2(ξ)sup
+ξ/ba∇dblvk(·,z1)/ba∇dblL2(0,X)/ba∇dbl˜vl(·,y1)∗/ba∇dblL∞(0,X)
++C/summationdisplay
+j,lsup
+y/ba∇dblφ(ξ)e−θ|ξ|2t/ba∇dblL2(ξ)sup
+ξ/ba∇dbl/parenleftBigvn(ξ,x1)−¯u′(x1)
+|ξ|/parenrightBig
+/ba∇dblL2(0,X)/ba∇dbl˜vl(·,y1)∗/ba∇dblL∞(0,X)
+≤C(1+t)−d
+4,
+wherewe have usedin acrucial way theboundednessof ˜ vlinL∞,4andalso theboundedness
+of/parenleftBigvn(ξ,x1)−¯u′(x1)
+|ξ|/parenrightBig
+∼∂|ξ|vn(ω;r)
+inL2, where 0<r <|ξ|. Derivative bounds follow similarly as above, noting that y- ort-
+derivatives bring down a factor |ξ|, whilex-derivatives are harmless, to obtain an additional
+factor of (1+ t)−1/2decay. Finally, bounds for 2 ≤p≤ ∞follow byLp-interpolation.
+Defining
+(3.22) e(x,t;y) :=χ(t)˜e(x,t;y),
+whereχis a smooth cutoff function such that χ(t)≡1 fort≥2 andχ(t)≡0 fort≤1,
+and setting ˜G:=G−¯u′(x1)e(x,t;y), we readily obtain the estimates (3.25) by combining
+the above estimates on GI−¯u˜ewith bound (3.4) on GII.
+4This is clear for ξ= 0, since vjare linear combinations of genuine and generalized eigenfu nctions, which
+are solutions of the homogeneous or inhomogeneous eigenval ue ODE. More generally, note that resolvent
+ofLξ−γgains one derivative, hence the total eigenprojection, as a contour integral of the resolvent, does
+too- now, use the one-dimensional Sobolev inequality for pe riodic boundary conditions to bound the L∞
+difference from the mean by the (bounded) H1norm, then bound the mean by the L1norm, which is
+controlled by the L2norm.3 LINEARIZED STABILITY ESTIMATES 14
+Finally, recalling, by Lemma 2.1, that ˜ vl≡constant for l/ne}ationslash=nwhile˜βj,n=O(|ξ|), we
+have
+∂y1/parenleftBig
+βj,n˜βj,l˜vl(ξ,y1)∗/parenrightBig
+=o(|ξ|).
+Bounds (3.11) thus follow from (3.14) by the argument used to prove (3.10), together with
+the observation that x- ort-derivatives bring down factors of |ξ|.
+Bounds (3.12) follow similarly for p=∞ife−θ|ξ|2t/|ξ|is integrable in Rd, and forp≥2
+ife−θ|ξ|2t/|ξ|2is integrable, thus yielding the stated results for all d≥2. In the special
+cased= 1,p=∞, (2.1) becomes a simpler one-parameter perturbation in ξ, and the
+|ξ|−1contributions become analytic multiples of ξ−1, whose principal value integrals may
+be carried out explicitly to give a sum of traveling error fun ctions that is bounded in L∞;
+see the proofof Proposition 1.5, [OZ2] in the one-dimension al case. We omit this calculation
+as largely outside our analysis. (However, note that we need this bound to conclude L∞
+bounded stability in the one-dimensional case.)
+Remark 3.5. Underlying our analysis, and that of [OZ2, JZ3], is the funda mental relation
+(3.23) G(x,t;y) =/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1eiξ·(x−y)[Gξ(x1,t;y1)]dξ.
+3.3 Final linearized bounds
+Corollary 3.1. Under assumptions (H1)–(H3), (D1)–(D3’), the Green functi onG(x,t;y)
+of(1.7)decomposes as G=E+˜G,
+(3.24) E= ¯u′(x)e(x,t;y),
+where, for some C >0, allt>0,1≤q≤2≤p≤ ∞,0≤j,k,l,j+l≤K,1≤r≤2,
+(3.25)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp(x)≤C(1+t)−d
+2(1/2−1/p)t−1
+2(1/q−1/2)|f|Lq∩L2,
+/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞∂r
+y˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp(x)≤C(1+t)−d
+2(1/2−1/p)−1
+2+r
+2
+×t−d
+2(1/q−1/2)−r
+2|f|Lq∩L2,
+/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞∂r
+t˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp(x)≤C(1+t)−d
+2(1/2−1/p)−1
+2+r
+×t−d
+2(1/q−1/2)−r|f|Lq∩L2.
+(3.26)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞∂j
+x∂k
+te(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp≤(1+t)−d
+2(1/q−1/p)−(j+k)
+2+1
+2|f|Lq,
+/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞∂j
+x∂k
+t∂r
+ye(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+Lp≤(1+t)−d
+2(1/q−1/p)−(j+k)
+2|f|Lq.
+Moreover,e(x,t;y)≡0fort≤1.3 LINEARIZED STABILITY ESTIMATES 15
+Proof.(Caseq= 1). From (3.10) and the triangle inequality we obtain
+/vextenddouble/vextenddouble/vextenddouble/integraldisplay
+Rd˜GI(x,t;y)f(y)dy/vextenddouble/vextenddouble/vextenddouble
+Lp(x)≤/integraldisplay
+Rdsup
+y/ba∇dbl˜GI(·,t;y)/ba∇dblLp|f(y)|dy≤C(1+t)−d
+2(1−1/p)/ba∇dblf/ba∇dblL1
+and similarly for y- andt-derivative estimates, which, together with (3.4), yield ( 3.25).
+Bounds (3.26) follow similarly by the triangle inequality a nd (3.11)–(3.12).
+(Caseq= 2). From (3.16)–(3.17), and analyticity of vj, ˜vj, we have boundedness from
+L2[0,X]→L2[0,X] of the projection-type operators
+(3.27) f→βj,n˜βj,l/parenleftBig
+vn(ξ,x1)−¯u′(x1)/parenrightBig
+/an}b∇acketle{t˜vl,f/an}b∇acket∇i}ht
+and
+(3.28) f→βj,k˜βj,lvk(ξ,x1)/an}b∇acketle{t˜vl,f/an}b∇acket∇i}htfork/ne}ationslash=n,
+uniformly with respect to ξ, from which we obtain by (3.15), (3.22), and (1.12) the bound
+(3.29)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞˜GI(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+L2(x)≤C/ba∇dblf/ba∇dblL2(x),
+for allt≥0, yielding together with (3.4) the result (3.25) for p= 2,r= 1. Similarly, by
+boundedness of ˜ vj,vj, ¯u′in allLp[0,X], we have
+|eλj(ξ)tβj,n˜βj,l/parenleftBig
+vn(ξ,x1)−¯u′(x1)/parenrightBig
+/an}b∇acketle{t˜vl,ˆf/an}b∇acket∇i}ht|L∞(x1)≤Ce−θ|ξ|2t|ˆf(ξ,·)|L2(x1),
+|eλj(ξ)tβj,k˜βj,lvk(ξ,x1)/an}b∇acketle{t˜vl,ˆf/an}b∇acket∇i}ht|L∞(x1)≤Ce−θ|ξ|2t|ˆf(ξ,·)|L2(x1),fork/ne}ationslash=n,
+C, θ>0, yielding by definitions (3.15), (3.22) the bound
+(3.30)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
+−∞˜GI(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
+L∞(x)≤/parenleftBig1
+2π/parenrightBigd/integraldisplayπ
+−π/integraldisplay
+Rd−1Cφ(ξ)e−θ|ξ|2t|ˆf(ξ,·)|L2(x1)dξ1d˜ξ
+≤C|φ(ξ)e−θ|ξ|2t|L2(ξ)|ˆf|L2(ξ,x1)
+=C(1+t)−d
+4/ba∇dblf/ba∇dblL2([0,X]),
+hence giving the result for p=∞,r= 0. The result for r= 0 and general 2 ≤p≤ ∞then
+follows byLpinterpolation between p= 2 andp=∞. Derivative bounds 1 ≤r≤2 follow
+by similar arguments, using (3.18)–(3.19). Bounds (3.26) f ollow similarly.
+(Case1≤q≤2). By Riesz–Thorin interpolation between the cases q= 1 andq= 2,
+we obtain the bounds asserted in the general case 1 ≤q≤2, 2≤p≤ ∞.
+Remark 3.6. The bounds on ˜G,et,exmay be recognized as the standard diffusive bounds
+satisfied for the heat equation [Z7]. For dimension d= 1, it may be shown using pointwise
+techniques as in [OZ2] that the bounds of Corollary 3.1 exten d to all 1 ≤q≤p≤ ∞.
+We note a striking analogy between the Green function decomp osition of Corollary 3.1
+and that of [MaZ3, Z4] in the viscous shock case; compare Prop osition 3.3, [Z7].4 NONLINEAR STABILITY IN DIMENSION ONE 16
+4 Nonlinear stability in dimension one
+WiththeboundsofCorollary (3.1), nonlinearstability fol lows byexactly thesameargument
+as in [JZ3], included here for completeness. We carry out the nonlinear stability analysis
+only in the most difficult, one-dimensional, case. The extens ion to the multi-dimensional
+case is straightforward [JZ3, OZ4]. (Recall that the nonlin ear iteration is easier to close in
+multi-dimensions, since the linearized behavior is faster decaying [OZ4, JZ3, S1, S2, S3].)
+Hereafter, take x∈R1, dropping the indices on fjandxjand writing ut+f(u)x=uxx.
+4.1 Nonlinear perturbation equations
+Given a solution ˜ u(x,t) of (1.4), define the nonlinear perturbation variable
+(4.1) v=u−¯u= ˜u(x+ψ(x,t))−¯u(x),
+where
+(4.2) u(x,t) := ˜u(x+ψ(x,t))
+andψ:R×R→Ris to be chosen later.
+Lemma 4.1. Forv,uas in(4.1),(4.2),
+(4.3) ut+f(u)x−uxx= (∂t−L)¯u′(x1)ψ(x,t)+∂xR+(∂t+∂2
+x)S,
+where
+R:=vψt+vψxx+(¯ux+vx)ψ2
+x
+1+ψx=O(|v|(|ψt|+|ψxx|)+/parenleftBig|¯ux|+|vx|
+1−|ψx|/parenrightBig
+|ψx|2)
+and
+S:=−vψx=O(|v|(|ψx|).
+Proof.To begin, notice from the definition of uin (4.2) we have by a straightforward
+computation
+ut(x,t) = ˜ux(x+ψ(x,t),t)ψt(x,t)+ ˜ut(x+ψ,t)
+f(u(x,t))x=df(˜u(x+ψ(x,t),t))˜ux(x+ψ,t)·(1+ψx(x,t))
+and
+uxx(x,t) = (˜ux(x+ψ(x,t),t)·(1+ψx(x,t)))x
+= ˜uxx(x+ψ(x,t),t)·(1+ψx(x,t))+(˜ux(x+ψ(x,t),t)·ψx(x,t))x.
+Using the fact that ˜ ut+df(˜u)˜ux−˜uxx= 0, it follows that
+(4.4)ut+f(u)x−uxx= ˜uxψt+df(˜u)˜uxψx−˜uxxψx−(˜uxψx)x
+= ˜uxψt−˜utψx−(˜uxψx)x4 NONLINEAR STABILITY IN DIMENSION ONE 17
+where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
+at (x+ψ(x,t),t). Moreover, by another direct calculation, using the fact t hatL(¯u′(x)) = 0
+by translation invariance, we have
+(∂t−L)¯u′(x)ψ= ¯uxψt−¯utψx−(¯uxψx)x.
+Subtracting, and using the facts that, by differentiation of ( ¯u+v)(x,t) = ˜u(x+ψ,t),
+(4.5)¯ux+vx= ˜ux(1+ψx),
+¯ut+vt= ˜ut+ ˜uxψt,
+so that
+(4.6)˜ux−¯ux−vx=−(¯ux+vx)ψx
+1+ψx,
+˜ut−¯ut−vt=−(¯ux+vx)ψt
+1+ψx,
+we obtain
+ut+f(u)x−uxx= (∂t−L)¯u′(x)ψ+vxψt−vtψx−(vxψx)x+/parenleftBig
+(¯ux+vx)ψ2
+x
+1+ψx/parenrightBig
+x,
+yielding (4.3) by vxψt−vtψx= (vψt)x−(vψx)tand (vxψx)x= (vψx)xx−(vψxx)x.
+Corollary 4.2. The nonlinear residual vdefined in (4.1)satisfies
+(4.7) vt−Lv= (∂t−L)¯u′(x1)ψ−Qx+Rx+(∂t+∂2
+x)S,
+where
+(4.8) Q:=f(˜u(x+ψ(x,t),t))−f(¯u(x))−df(¯u(x))v=O(|v|2),
+(4.9) R:=vψt+vψxx+(¯ux+vx)ψ2
+x
+1+ψx,
+and
+(4.10) S:=−vψx=O(|v|(|ψx|).
+Proof.Taylor expansion comparing (4.3) and ¯ ut+f(¯u)x−¯uxx= 0.4 NONLINEAR STABILITY IN DIMENSION ONE 18
+4.2 Cancellation estimate
+Our strategy in writing (4.7) is motivated by the following b asic cancellation principle.
+Proposition 4.3 ([HoZ]).For anyf(y,s)∈Lp∩C2withf(y,0)≡0, there holds
+(4.11)/integraldisplayt
+0/integraldisplay
+G(x,t−s;y)(∂s−Ly)f(y,s)dyds=f(x,t).
+Proof.Integrating the left hand side by parts, we obtain
+(4.12)/integraldisplay
+G(x,0;y)f(y,t)dy−/integraldisplay
+G(x,t;y)f(y,0)dy+/integraldisplayt
+0/integraldisplay
+(∂t−Ly)∗G(x,t−s;y)f(y,s)dyds.
+Noting that, by duality,
+(∂t−Ly)∗G(x,t−s;y) =δ(x−y)δ(t−s),
+δ(·) here denoting the Dirac delta-distribution, we find that th e third term on the righthand
+side vanishes in (4.12), while, because G(x,0;y) =δ(x−y), the first term is simply f(x,t).
+The second term vanishes by f(y,0)≡0.
+4.3 Nonlinear damping estimate
+Proposition 4.1. Letv0∈HK(Kas in (H1)), and suppose that for 0≤t≤T, theHK
+norm ofvand theHK(x,t)norms ofψtandψxremain bounded by a sufficiently small
+constant. There are then constants θ1,2>0so that, for all 0≤t≤T,
+(4.13) |v(t)|2
+HK≤Ce−θ1t|v(0)|2
+HK+C/integraldisplayt
+0e−θ2(t−s)/parenleftBig
+|v|2
+L2+|(ψt,ψx)|2
+HK(x,t)/parenrightBig
+(s)ds.
+Proof.Subtracting from the equation (4.4) for uthe equation for ¯ u, we may write the
+nonlinear perturbation equation as
+(4.14) vt+(df(¯u)v)x−vxx=Q(v)x+ ˜uxψt−˜utψx−(˜uxψx)x,
+where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
+at (x+ψ(x,t),t). Using (4.6) to replace ˜ uxand ˜utrespectively by ¯ ux+vx−(¯ux+vx)ψx
+1+ψx
+and ¯ut+vt−(¯ux+vx)ψt
+1+ψx, and moving the resulting vtψxterm to the lefthand side of
+(4.14), we obtain
+(4.15)(1+ψx)vt−vxx=−(df(¯u)v)x+Q(v)x+ ¯uxψt
+−((¯ux+vx)ψx)x+/parenleftBig
+(¯ux+vx)ψ2
+x
+1+ψx/parenrightBig
+x.4 NONLINEAR STABILITY IN DIMENSION ONE 19
+Taking the L2inner product in xof/summationtextK
+j=0∂2j
+xv
+1+ψxagainst (4.15), integrating by parts, and
+rearranging the resulting terms, we arrive at the inequalit y
+∂t|v|2
+HK(t)≤ −θ|∂K+1
+xv|2
+L2+C/parenleftBig
+|v|2
+HK+|(ψt,ψx)|2
+HK(x,t)/parenrightBig
+,
+for someθ >0,C >0, so long as |˜u|HKremains bounded, and |v|HKand|(ψt,ψx)|HK(x,t)
+remain sufficiently small. Using the Sobolev interpolation |v|2
+HK≤ |∂K+1
+xv|2
+L2+˜C|v|2
+L2for
+˜C >0 sufficiently large, we obtain ∂t|v|2
+HK(t)≤ −˜θ|v|2
+HK+C/parenleftBig
+|v|2
+L2+|(ψt,ψx)|2
+HK(x,t)/parenrightBig
+from which (4.13) follows by Gronwall’s inequality.
+4.4 Integral representation/ ψ-evolution scheme
+By Proposition 4.3, we have, applying Duhamel’s principle t o (4.7),
+(4.16)v(x,t) =/integraldisplay∞
+−∞G(x,t;y)v0(y)dy
++/integraldisplayt
+0/integraldisplay∞
+−∞G(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds+ψ(t)¯u′(x).
+Definingψimplicitly as
+(4.17)ψ(x,t) =−/integraldisplay∞
+−∞e(x,t;y)u0(y)dy
+−/integraldisplayt
+0/integraldisplay+∞
+−∞e(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds,
+following [ZH, Z4, MaZ2, MaZ3], where eis defined as in (3.24), and substituting in (4.16)
+the decomposition G= ¯u′(x)e+˜Gof Corollary 3.1, we obtain the integral representation
+(4.18)v(x,t) =/integraldisplay∞
+−∞˜G(x,t;y)v0(y)dy
++/integraldisplayt
+0/integraldisplay∞
+−∞˜G(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds,
+and, differentiating (4.17) with respect to t, and recalling that e(x,s;y)≡0 fors≤1,
+(4.19)∂j
+t∂k
+xψ(x,t) =−/integraldisplay∞
+−∞∂j
+t∂k
+xe(x,t;y)u0(y)dy
+−/integraldisplayt
+0/integraldisplay+∞
+−∞∂j
+t∂k
+xe(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds.
+Equations (4.18), (4.19) together form a complete system in the variables ( v,∂j
+tψ,∂k
+xψ),
+0≤j≤1, 0≤k≤K, from the solution of which we may afterward recover the shif tψvia
+(4.17). From the original differential equation (4.7) togeth er with (4.19), we readily obtain
+short-time existence and continuity with respect to tof solutions ( v,ψt,ψx)∈HKby a
+standard contraction-mapping argument based on (4.13), (4 .17), and and (3.26).4 NONLINEAR STABILITY IN DIMENSION ONE 20
+4.5 Nonlinear iteration
+Associated with the solution ( u,ψt,ψx) of integral system (4.18)–(4.19), define
+(4.20)ζ(t) := sup
+0≤s≤t|(v,ψt,ψx)|HK(s)(1+s)1/4.
+Lemma 4.2. For allt≥0for whichζ(t)is finite, some C >0, andE0:=|u0|L1∩HK,
+(4.21) ζ(t)≤C(E0+ζ(t)2).
+Proof.By (4.9)–(4.10) and definition (4.20),
+(4.22) |(Q,R,S)|L1∩L∞≤ |(v,vx,ψt,ψx)|2
+L2+|(v,vx,ψt,ψx)|2
+L∞≤Cζ(t)2(1+t)−1
+2,
+so long as |ψx| ≤ |ψx|HK≤ζ(t) remains small, and likewise (using the equation to bound t
+derivatives in terms of x-derivatives of up to two orders)
+(4.23) |(∂t+∂2
+x)S|L1∩L∞≤ |(v,ψx)|2
+H2+|(v,ψx)|2
+W2,∞≤Cζ(t)2(1+t)−1
+2.
+Applying Corollary 3.1 with q= 1,d= 1 to representations (4.18)–(4.19), we obtain for
+any 2≤p<∞
+(4.24)|v(·,t)|Lp(x)≤C(1+t)−1
+2(1−1/p)E0
++Cζ(t)2/integraldisplayt
+0(1+t−s)−1
+2(1/2−1/p)(t−s)−3
+4(1+s)−1
+2ds
+≤C(E0+ζ(t)2)(1+t)−1
+2(1−1/p)
+and
+(4.25)
+|(ψt,ψx)(·,t)|WK,p≤C(1+t)−1
+2E0+Cζ(t)2/integraldisplayt
+0(1+t−s)−1
+2(1−1/p)−1/2(1+s)−1
+2ds
+≤C(E0+ζ(t)2)(1+t)−1
+2(1−1/p).
+Using (4.13) and(4.24)–(4.25), we obtain |v(·,t)|HK(x)≤C(E0+ζ(t)2)(1+t)−1
+4. Combining
+this with (4.25), p= 2, rearranging, and recalling definition (4.20), we obtain (4.2).
+Proof of Theorem 1.1. By short-time HKexistence theory, /ba∇dbl(v,ψt,ψx)/ba∇dblHKis continuous
+so long as it remains small, hence ηremains continuous so long as it remains small. By
+(4.2), therefore, it follows by continuous induction that η(t)≤2Cη0fort≥0, ifη0<1/4C,
+yielding by (4.20) the result (1.15) for p= 2. Applying (4.24)–(4.25), we obtain (1.15) for
+2≤p≤p∗for anyp∗<∞, with uniform constant C. Takingp∗>4 and estimating
+|Q|L2,|R|L2,|S|L2(t)≤ |(v,ψt,ψx)|2
+L4≤CE0(1+t)−3
+4REFERENCES 21
+in place of the weaker (4.22), then applying Corollary 3.1 wi thq= 2,d= 1, we obtain
+finally (1.15) for 2 ≤p≤ ∞, by a computation similar (4.24)–(4.25); we omit the detail s of
+this final bootstrap argument. Estimate (1.16) then follows using (3.26) with q=d= 1, by
+(4.26)|ψ(t)|Lp≤CE0(1+t)1
+2p+Cζ(t)2/integraldisplayt
+0(1+t−s)−1
+2(1−1/p)(1+s)−1
+2ds
+≤C(1+t)1
+2p(E0+ζ(t)2),
+together with the fact that ˜ u(x,t)−¯u(x) =v(x−ψ,t)+(¯u(x)−¯u(x−ψ),so that|˜u(·,t)−¯u|
+is controlled by the sum of |v|and|¯u(x)−¯u(x−ψ)| ∼ |ψ|. This yields stability for
+|u−¯u|L1∩HK|t=0sufficiently small, as described in the final line of the theore m.
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+Large wavelength analysis , Comm. Partial Differential Equations 30 (2005), no.
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+Stokes equations, withanappendixbyHelge KristianJenssenandGregoryLyng,
+in Handbook of mathematical fluid dynamics. Vol. III, 311–53 3, North-Holland,
+Amsterdam, (2004).
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+viscosity-capillarity , SIAM J. Appl. Math. 60 (2000), 1913-1929.
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+arXiv:1001.0405v1  [cs.LG]  3 Jan 2010Optimal Query Complexity for Reconstructing Hypergraphs
+Nader H. Bshouty
+Technion, Israel
+bshouty@cs.technion.ac.ilHanna Mazzawi
+Technion, Israel
+hanna@cs.technion.ac.il
+Abstract
+In this paper we consider the problem of reconstructing a hid den weighted hypergraph of constant rank
+using additive queries. We prove the following: Let Gbe a weighted hidden hypergraph of constant rank
+withnvertices and mhyperedges. For any mthere exists a non-adaptive algorithm that finds the edges of
+the graph and their weights using
+O/parenleftbiggmlogn
+logm/parenrightbigg
+additive queries. This solves the open problem in [S. Choi, J . H. Kim. Optimal Query Complexity Bounds for
+Finding Graphs. STOC, 749–758, 2008].
+When the weights of the hypergraph are integers that are less thanO(poly(nd/m)) where dis the rank of
+the hypergraph (and therefore for unweighted hypergraphs) there exists a non-adaptive algorithm that finds
+the edges of the graph and their weights using
+O/parenleftBigg
+mlognd
+m
+logm/parenrightBigg
+.
+additive queries.
+Using the information theoretic bound the above query compl exities are tight.
+1 Introduction
+In this paper we consider the following problem of reconstructing we ighted hypergraphs of constant rank1(the
+maximal size of a hyperedge) using additive queries: Let G= (V,E,w) be a weighted hidden hypergraph where
+E⊂2V,|e|is constant for all e∈E,w:E→R, andnis the number of vertices in V. Denote by mthe size
+ofE. Suppose that the set of vertices Vis known and the set of edges Eis unknown. Given a set of vertices
+S⊆V, an additive query, QG(S), returns the sum of weights in the sub-hypergraph induced by S. That is,
+QG(S) =/summationdisplay
+e∈E∩2Sw(e).
+Our goal is to exactly reconstruct the set of edges using additive q ueries.
+One can distinguish between two types of algorithms to solve the pro blem. Adaptive algorithms are algorithms
+that take into account outcomes of previous queries while non-ada ptive algorithms make all queries in advance,
+1Sometimes called dimension.
+1Tight Upper Adaptive Non-adaptive
+Bound Poly. time Poly. time
+Loops rank = 1
+Unweighted Loops [13, 17, 14, 6] [8] OPEN
+Bounded Weighted Loops [11] OPEN OPEN
+Unbounded Weighted Loops [10] OPEN†OPEN§
+Graph rank = 2
+Unweighted Graph [11] [22] OPEN
+Bounded Weighted Graph [11, 9] OPEN OPEN
+Unbounded Weighted Graph [10] OPEN†OPEN
+Hypergraph rank >2
+Unweighted HyperGraph Ours OPEN OPEN
+Unbounded Weighted Hypergraph Ours OPEN†OPEN
+Figure 1: Results for weighted and un-weighted hypergraphs with o ptimal query complexity.†A non-optimal
+adaptive query complexity algorithm for Hypergraph can be found in [12].§A non-optimal non-adaptive query
+complexity algorithms can be found in [20] and the references within it .
+before any answer is known. In this paper, we consider non-adapt ive algorithms for the problem. Our concern is
+the query complexity, that is, the number of queries needed to be a sked in order to reconstruct the hypergraph.
+The hypergraph reconstructing problem has known a significant pr ogress in the past decade. For unweighted
+hypergraph of rank dthe information theoretic lower bound gives
+Ω/parenleftBigg
+mlognd
+m
+logm/parenrightBigg
+for the query complexity for any adaptive algorithm for this problem .
+Many independent results [13, 17, 14, 6] have proved a tight upper bound for hypergraph of rank 1, i.e., loops. A
+tight upper bound was proved for some subclasses of unweighted h ypergraphs of rank two, i,e., graphs (Hamil-
+tonian graphs, matching, stars and cliques etc.) [19, 18, 17, 7], unw eighted graphs with Ω( dn) edges where the
+degree of each vertex is bounded by d[17], graphs with Ω( n2) edges [17] and then the former was extended to
+d-degenerate unweighted graphs with Ω( dn) edges [19], i.e., graphs that their edges can be changed to directed
+edges where the out-degree of each vertex is bounded by d. A recent paper by Choi and Kim, [11], gave a tight
+upper bound for all unweighted graphs. In this paper we give a tight upper bound for all unweighted hypergraphs
+of constant rank. Our bound is tight even for weighted hypergrap hs with integer weights |w|=poly(nd/m)
+wheredis the rank of the hypergraph.
+For weighted hypergraph of constant rank with unbounded weight s the information theoretic lower bound gives
+Ω/parenleftbiggmlogn
+logm/parenrightbigg
+In [11], Choi and Kim prove a tight upper bound for loops (hypergrap h of rank 1). For weighted graphs (hyper-
+graph of rank 2) Choi and Kim, [11], proved the following: If m >(logn)αfor sufficiently large α, then, there
+2exists a non-adaptive algorithm for reconstructing a weighted gra ph where the weights are real numbers bounded
+between n−aandnbfor any positive constants aandbusing
+O/parenleftbiggmlogn
+logm/parenrightbigg
+queries.
+In [9], Bshouty and Mazzawi close the gap in mand proved that for any weighted graph where the weights are
+bounded between n−aandnbfor any positive constants aandbandanymthere exists a non-adaptive algorithm
+that reconstructs the hidden graph using
+O/parenleftbiggmlogn
+logm/parenrightbigg
+queries. Then in [10] they extended the result to any weighted gra ph with any unbounded weights.
+In this paper extend all the above results to any hypergraph of co nstant rank, i.e., the edges of the graph has
+constant size. This solves the open problems in [11, 9, 10].
+The paper is organized as follows: In Section 2, we present notation , basic tools and some background. In
+Section 3, we prove the main result.
+2 Preliminaries
+In this section we present some background, basic tools and notat ion.
+For an integer rlet [r] be the set {1,2,...,r}. ForS⊂[r] we define xS∈ {0,1}rwherexS
+i= 1 if and only if
+i∈S. The inverse operation is Sx={i|xi= 1}. We say that x1,...,x d∈ {0,1}narepairwise disjoint if for
+everyi/\e}atio\slash=j, we have xi∗xj=0where∗is component-wise product of two vectors. For a prime pand integers a
+andbwe write a=pbfora=bmodp. We will also allow p=∞. In this case aandbcan be any real numbers
+anda=∞bwill mean a=bas real numbers.
+2.1d-Dimensional Matrices
+Ad-dimensional matrix Aofsizen1×···×ndover a field Fis a map A:/producttextd
+i=1[ni]→F. We denote by Fn1×···×nd
+the set of all d-dimensional matrices Aof sizen1×···×nd. We write Ai1,...,idforA(i1,...,id).
+The zero map is denoted by 0n1×···×nd. The matrix
+B= (Ai1,i2,...,id)i1∈I1,i2∈I2,...,id∈Id,
+whereIj⊆[nj], is the|I1|×···×| Id|matrix where Bj1,...,jd=Aℓ1,...,ℓdandℓiis thejith smallest number in Ii.
+WhenIj= [nj]wejustwrite jandwhen Ij={ℓ}wejustwrite j=ℓ. Forexample,( Ai1,i2,...,id)i1,i2=ℓ,i3∈I2,...,id∈Id=
+(Ai1,i2,...,id)i1∈[n1],i2∈{ℓ},i3∈I2,...,id∈Id.
+Whenn1=n2=···=nd=nthen we denote Fn1×···×ndbyF×dnand 0n1×···×ndby 0×dn.
+We say that the entry Ai1,i2,...,idis ofdimension rif|{i1,...,id}|=r.Ford-dimensional matrix Awe denote by
+wt(A) the number of points in/producttextd
+i=1[ni] that are mapped to non-zero elements in F. We denote by wtr(A) the
+3number of points in/producttextd
+i=1[ni] of dimension rthat are mapped to non-zero elements in F. Therefore, wt(A) =
+wt1(A)+wt2(A)+···+wtd(A).We denote by Ad,mthe set of d-dimensionalmatrices A∈F×dnwherewtd(A)≤m
+andA⋆
+d,mthe set of d-dimensional matrices A∈F×dnwhere 1≤wtd(A)≤m.
+Ford-dimensional matrix Aof sizen1×···×ndandxi∈Fniwe define
+A(x1,...,x d) =n1/summationdisplay
+i1=1···nd/summationdisplay
+id=1Ai1,i2,...,idx1i1···xdid.
+The vector v=A(·,x2,...,x d) isn1-dimensional vector that its ith entry is
+n2/summationdisplay
+i2=1···nd/summationdisplay
+id=1Ai,i2,...,idx2i2···xdid.
+For a set of d-dimensional matrices B, a setS⊆({0,1}n)dis called a zero test set forBif for every A∈ B,A/\e}atio\slash= 0,
+there isx∈Ssuch that A(x)/\e}atio\slash= 0.
+Ad-dimensional matrix is called symmetric if for every i= (i1,...,id)∈[n]dand any permutation φon [d],
+we have Ai=Aφi, whereφi= (iφ(1),...,iφ(d)). Notice that for a symmetric d-dimensional matrix A∈F×dn,
+xi∈ {0,1}nand any permutation φon [d], we have A(x1,...,x d) =A(xφ(1),...,x φ(d)).
+We will be interested mainly in the fields F=Rthe field of real numbers and F=Zpthe field of integers modulo p
+and in matrices of constant d=O(1) dimension. Also p > d!. Although it seems that we are restricting the
+parameters, the final result has no restriction on the parameter s except for d=O(1). We will also abuse the
+notations Zpand = pand allow p=∞(so in this paper ∞is also prime number). In that case Z∞=Rand = ∞
+is equality in the filed of real numbers.
+2.2 Hypergraph
+Ahypergraph Gis a pair G= (V,E) whereV= [n] is a set of elements, called nodes or vertices, and Eis a set of
+non-empty subsets of 2Vcalledhyperedges oredges. Therankr(G) of a hypergraph Gis the maximum cardinality
+of any of the edges in the hypergraph. A hypergraph is called d-uniform if all of its edges are of size d.
+Aweighted hypergraph G= (V,E,w) overZpis a hypergraph ( V,E) with a weight function w:E→Zp. For
+two weighted hypergraph G1= (V,E1,w1) andG2= (V,E2,w2) we define the weighted hypergraph G1−G2=
+(V,E,w) where E={e∈E1∪E2|w1(e)/\e}atio\slash=w2(e)},and for every e∈E,w(e) =w1(e)−w2(e).Obviously,
+G1=G2if and only if G1−G2is an independent set, i.e., E=∅.
+We denote by Gdthe set of all weighted hypergraphs over Zpof rank at most d,Gd,mthe set of all weighted
+hypergraphs over Zpof rank at most dand at most medges and G⋆
+d,mthe set of all weighted hypergraphs over
+Zpof rankdand at most medges.
+Letw⋆: 2V→Zpbewextended to all possible edges where for e∈E,w⋆(e) =w(e) and for e/\e}atio\slash∈E,w⋆(e) = 0.
+Anadjacency d-dimensional matrix of a weighted hypergraph Gis ad-dimensional matrix AG
+dwhered≥r(G)
+such that for every set e={i1,i2,...,iℓ}of size at most dwe have AG
+d(j1,...,jd)=pw⋆(e)/N(d,ℓ) for allj1,...,j d
+such that {j1,j2,...,j d}={i1,...,iℓ}where
+N(d,ℓ) =ℓ/summationdisplay
+i=0(−1)i/parenleftbiggℓ
+i/parenrightbigg
+(ℓ−i)d.
+4That is, N(d,ℓ) is the number of possible sequences ( j1,...,j d) such that {j1,...,j d}={i1,...,iℓ}. Note that
+N(d,ℓ)≤d!< pand therefore N(d,ℓ)/\e}atio\slash=p0 andAG
+dis well defined.
+It is easy to see that the adjacency matrix of a weighted hypergra ph is a symmetric matrix and r(G) =rif and
+only if the adjacency matrix of Ghas an non-zero entry of dimension rand all entries of dimension greater than r
+are zero.
+2.3 Additive Model
+In the Additive Model the goal is to exactly learn a hidden hypergrap h with minimal number of additive queries.
+Given a set of vertices S⊆V, anadditive query ,QG(S), returns the sum of weights in the subgraph induces
+byS. That is, QG(S) =p/summationtext
+e∈E∩2Sw(e).Our goal is to exactly reconstruct the set of edges and find their w eights
+using additive queries. See the many applications of this problem in [7, 1 1, 12].
+We say that the set S={S1,S2,···,Sk} ⊆2Vis adetecting set forGd,mif for any hypergraph G∈ Gd,mthere
+isSisuch that QG(Si)/\e}atio\slash= 0. We say that the set S={S1,S2,···,Sk} ⊆2Vis asearch set forGd,mif for any two
+distinct hypergraphs G1,G2∈ Gd,mthere isSisuch that QG1(Si)/\e}atio\slash=QG2(Si). That is, given ( QG(Si))ione can
+uniquely determines G. We now prove the following,
+Lemma 1. IfS={S1,S2,···,Sk} ⊆2Vis a detecting set for Gd,2mthen it is a search set for Gd,m.
+Proof.LetG1,G2∈ Gd,mbe two distinct weighted hypergraphs. Let G=G1−G2. SinceG∈ Gd,2mthere must
+beSi∈ Ssuch that QG(Si)/\e}atio\slash= 0. Since QG(Si) =QG1(Si)−QG2(Si) we have QG1(Si)/\e}atio\slash=QG2(Si).
+2.4 Algebraic View of the Model
+It is easy to show that for any hypergraph Gof rankrthe adjacency d-dimensional matrix of G,AG
+d, ford≥r,
+is symmetric, contains a nonzero entry of dimension rand
+QG(S) =pAG
+d(xS,xS,d...,xS)∆=BG
+d(xS).
+For a symmetric d-dimensional matrix AletB(x) =pA(x,x,d...,x) wherex∈ {0,1}n. Whenx1,...,x d∈ {0,1}n
+are pairwise disjoint the following lemma shows that A(x1,...,x d) can be found by 2dvalues of B.
+Lemma 2. Ifx1,...,x d∈ {0,1}nare pairwise disjoint then
+A(x1,...,x d) =p1
+d!/summationdisplay
+I∈2[d](−1)d−|I|B/parenleftBigg/summationdisplay
+i∈Ixi/parenrightBigg
+.
+Proof.Since
+A(x1+x′
+1,x2,...,x d) =pA(x1,x2,...,x d)+A(x′
+1,x2,...,x d)
+and
+A(x1,x2,...,x d) =pA(xφ(1),xφ(2),...,x φ(d))
+5for any permutation φon [d], the result is analogous to the fact that
+y1y2···yd=p1
+d!/summationdisplay
+I∈2[d](−1)d−|I|/parenleftBigg/summationdisplay
+i∈Iyi/parenrightBiggd
+, (1)
+for formal variables y1,...,y d. Now notice that
+/parenleftBigg/summationdisplay
+i∈Iyi/parenrightBiggd
+=p/summationdisplay
+q1+···+qd=dχ[{i|qi/\e}atio\slash= 0} ⊆I]/parenleftbiggd
+q1q2···qd/parenrightbigg
+yq1
+1···yqd
+d,
+whereχ[L] = 1 if the statement Lis true and 0 otherwise. Therefore, the coefficient of yq1
+1···yqd
+din the right
+hand side of (1) is
+/summationdisplay
+I∈2[d](−1)d−|I|χ[{i|qi/\e}atio\slash= 0} ⊆I]/parenleftbiggd
+q1q2···qd/parenrightbigg
+=p/parenleftbiggd
+q1q2···qd/parenrightbigg/summationdisplay
+I∈2[d](−1)d−|I|χ[{i|qi/\e}atio\slash= 0} ⊆I].
+Now ifℓ=|{i|qi/\e}atio\slash= 0}|< dthen
+/summationdisplay
+I∈2[d](−1)d−|I|χ[{i|qi/\e}atio\slash= 0} ⊆I] =pd/summationdisplay
+i=ℓ(−1)d−i/parenleftbiggd−ℓ
+i−ℓ/parenrightbigg
+=pd−ℓ/summationdisplay
+i=0(−1)d−ℓ−i/parenleftbiggd−ℓ
+i/parenrightbigg
+= 0.
+Ifℓ=|{i|qi/\e}atio\slash= 0}|=dthenq1=q2=···=qd= 1 and
+/summationdisplay
+I∈2[d](−1)d−|I|χ[{i|qi/\e}atio\slash= 0} ⊆I] =p1.
+This implies the result.
+LetGbe a hypergraph of rank dandG(i),i≤d, be the sub-hypergraph of Gthat contains all the edges in Gof
+sizeithen
+Lemma 3. Ifx1,...,x d∈ {0,1}nare pairwise disjoint then, we have that AG
+d(x1,...,x d) =AG(d)
+d(x1,...,x d).
+In particular, if r(G)< dthenAG
+d(x1,...,x d) = 0.
+Proof.Sincex1,...,x d∈ {0,1}nare pairwise disjoint we have
+AG
+d(x1,...,x d) =n1/summationdisplay
+i1=1···nd/summationdisplay
+id=1w⋆({i1,i2,...,id})
+N(d,|{i1,i2,...,id}|)x1i1···xdid
+=/summationdisplay
+|{i1,...,id}|=dw⋆({i1,i2,...,id})
+N(d,d)x1i1···xdid
+=AG(d)
+d(x1,...,x d).
+6Now when r(G)< dthenG(d)is an independent set (has no edges) and AG(d)
+d= 0. Then
+AG
+d(x) =AG(d)
+d(x) = 0.
+We now prove
+Lemma 4. LetΦd={z(d)
+1,...,z(d)
+kd} ⊂({0,1}n)dwhere for every ithe vectors z(d)
+i,1,...,z(d)
+i,dare pairwise disjoint.
+IfΦdis a zero test set for A⋆
+d,(d!)mthen
+SΦd∆=
+
+SyJ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleyJ=/summationdisplay
+j∈Jz(d)
+i,j, J⊂[d]
+
+
+is a detecting set for G⋆
+d,m.
+Proof.Let Φdbe a zero test set for A⋆
+d,(d!)m. LetG∈ G⋆
+d,m. ThenAG
+d/\e}atio\slash= 0 and AG
+d∈ A⋆
+d,(d!)m. Therefore, for
+everyG∈ G⋆
+d,mthere isz(d)
+isuch that AG
+d(z(d)
+i)/\e}atio\slash= 0.By Lemma 2,
+AG
+d(z(d)
+i) =p1
+d!/summationdisplay
+J∈2[d](−1)d−|J|BG
+d
+/summationdisplay
+j∈Jz(d)
+i,j
+/\e}atio\slash= 0,
+and therefore for some J0⊂[d],
+BG
+d
+/summationdisplay
+j∈J0z(d)
+i,j
+/\e}atio\slash= 0,
+which implies that QG(SyJ0)/\e}atio\slash= 0 foryJ0=/summationtext
+j∈J0z(d)
+i,j.
+We now show
+Lemma 5. A detecting set for Gd,moverZpis a detecting set for Gd,moverR.
+Proof.Consider a detecting set S={S1,S2,···,Sk} ⊆2VforGd,moverZp. Consider a k×qmatrixMwhere
+q=d/summationdisplay
+i=0/parenleftbiggn
+i/parenrightbigg
+that its columns are labelled with sets in 2[n]of size at most dand for every S⊂[n] of size at most dwe have
+M[i,S] = 1 if S⊆Siand 0 otherwise. Consider for every graph G∈ Gd,maq-vectorvGthat its entries are
+labelled with subsets of [ n] of size at most dandvG[S] =w⋆(S). The labels in vGare in the same order as the
+labels of the columns of M. Then it is easy to see that
+MvG=p(QG(S1),...,Q G(Sk))T.
+7SinceMvG/\e}atio\slash=p0 for every vG∈Zq
+pof weight at least one and at most m, everymcolumns in Mare linearly
+independent over Zp. Since the entries of Mare zeros and ones every mcolumns in Mare linearly independent
+overR. Therefore,
+MvG= (QG(S1),...,Q G(Sk))T/\e}atio\slash= 0,
+for every vG∈Rqof weight at least 1 and at most m.
+2.5 Distributions
+In this subsection we give a distribution that will be used in this paper.
+Theuniform disjoint distribution Ωd,n(x) over ({0,1}n)dis defined as
+Ωd,n(x) =/braceleftbigg1
+(d+1)nx1,...,x dis pairwise disjoint.
+0 otherwise.
+In order to choose a random vector xaccording to the uniform disjoint distribution, one can randomly in-
+dependently uniformly choose nelements w1,w2,...,w nwherewi∈[d+ 1] and define the following vector
+x= (x1,x2,...,x d)∈({0,1}n)d:
+xji=/braceleftbigg
+1j=wiandwi∈[d]
+0 otherwise.
+We call any index k∈[n] such that xjk= 0 for all j∈[d] afree index . Let Γ d,n⊂({0,1}n)dbe the set of all
+pairwise disjoint d-tuple.
+2.6 Preliminary Results
+In this section we prove,
+Lemma 6. LetA∈F×dn\ {0×dn}be an adjacency d-dimensional matrix of a hypergraph Gof rankd. Let
+x= (x1,x2,...,x d)∈({0,1}n)dbe a randomly chosen d-tuple, that is chosen according to the distribution Ωd,n.
+Then
+Pr
+x∈Ωd,n[A(x) = 0]≤1−1
+(d+1)d.
+Proof.Lete={i1,...,id}be an edge of size |e|=dand letx′
+j= (xj,i1,...,x j,id). Consider φ(x′
+1,...,x′
+d)
+that is equal to A(x) with some fixed xj,i=ξj,i∈ {0,1}fori/\e}atio\slash∈e. SinceA(x) contains the monomial M=
+x1,i1x2,i2···xd,idand no other monomial in A(x) contains it, φcontains monomial Mand therefore
+φ(x′
+1,...,x′
+d)/\e}atio\slash≡0.
+If we substitute xj1,ij2= 0 inφfor allj1/\e}atio\slash=j2we still get a nonzero function φ′(x1,i1,x2,i2,···,xd,id) that
+contains M. Therefore, there is ξ= (ξ1i1,ξ2i2,···,ξdid)∈ {0,1}dsuch that φ′(ξ)/\e}atio\slash= 0. The probability that
+(x1,i1,x2,i2,···,xd,id) =ξandxj1,ij2= 0 for all j1/\e}atio\slash=j2is (1/d+1)d.This implies the result.
+We will also use the following two lemmas from [9, 10].
+8Lemma 7. Leta∈Zn
+pbe a non-zero vector, where p > wt(a)is a prime number. Then for a uniformly randomly
+chosen vector x∈ {0,1}nwe have
+Pr
+x[aTx=p0]≤1
+wt(a)β,
+whereβ=1
+2+log3= 0.278943···.
+Letιbe a function on non-negative integers defined as follows: ι(0) = 1 and ι(i) =ifori >0.
+Lemma 8. Letm1,m2,...,m tbe integers in [m]∪{0}such that m1+m2+···+mt=ℓ≥t.Then/producttextt
+i=0ι(mi)≥
+m⌊(ℓ−t)/(m−1)⌋.
+3 Reconstructing Hypergraphs
+In this section we prove,
+Theorem 1. There is a search set for Gd,moverRof sizek=O/parenleftBig
+mlogn
+logm/parenrightBig
+.
+Theorem 2. There is a search set for G′d,moverRof sizek=O/parenleftbigg
+mlognd
+m
+logm/parenrightbigg
+, whereG′d,mdenotes the set of
+all weighted hypergraphs over Rof rank at most d, at most medges and weights that are integers bounded by
+w=poly(nd/m).
+Proof.We give the proof of Theorem 1. The proof of Theorem 2 is similar. Mor e details in the full paper.
+Letm < p < 2mbe a prime number. Suppose there is a zero test set from Γ d,nforA⋆
+d,moverZpof sizeT(n,m,d).
+By Lemma 4, there is a detecting set for G⋆
+d,moverZpof size 2dT(n,(d!)m,d). Therefore, by Lemma 3, there is
+a detecting set for Gd,moverZpof sizeT′(n,m,d) =/summationtextd
+ℓ=12ℓT(n,(ℓ!)m,ℓ).By Lemma 5, there is a detecting set
+forGd,moverRof sizeT′(n,m,d). Finally, by Lemma 1, there is a search set for Gd,moverRof sizeT′(n,2m,d).
+Now for constant d, if
+T(n,m,d) =O/parenleftbiggmlogn
+logm/parenrightbigg
+, (2)
+thenT′(n,2m,d) =O(T(n,m,d)). Therefore it is enough to prove the following.
+Lemma 9. Letpbe a prime number such that m < p < 2m. There exists a set S={x1,x2,...,x k} ⊆({0,1}n)d
+wherexi= (xi,1,...,x i,d)∈Γd,nfori∈[k]and
+k=O/parenleftbiggmlogn
+logm/parenrightbigg
+,
+such that: for every d-dimensional matrix A∈Z×dn
+p\{0×dn}with1≤wtd(A)≤mthere exists an isuch that
+A(xi)/\e}atio\slash=p0.
+9Proof.Sincewtd(A)>1 the matrix Ahas at least one nonzero entry of dimension d. We will assume that all the
+entries of dimension less than dare zero, that is, wt(A) =wtd(A). This is because, by Lemma 3, the entries of
+dimension less than dhave no effect when the vectors xi∈Γd,n.
+We divide the set of such matrices A={A|A∈Z×dn
+p\{0×dn}andwt(A)≤m}intod+1 (non-disjoint) sets:
+• A0: The set of all non-zero matrices A∈Z×dn
+psuch that wt(A)≤m/logm.
+• Ajforj= 1,...,d: The set of all non-zero matrices A∈Z×dn
+psuch that m≥wt(A)> m/logmand there
+are at least/parenleftbiggm
+logm/parenrightbigg1/d
+non-zero elements in Ij={ij|∃(i1,i2,...,ij−1,ij+1,...,id) :Ai1,i2,...,id/\e}atio\slash= 0}.
+Note that I={(i1,i2,...,id)|Ai1,i2,...,id/\e}atio\slash= 0} ⊆I1×I2×···×Idand therefore either I=wt(A)≤m/logmor
+there isjsuch that |Ij|>(m/logm)1/d. Therefore, A=A0∪A1∪···∪A d.
+Using the probabilistic method, we give d+1 sets of pairwise disjoint tuples of vectors S0,S1,...,S dsuch that
+for every j∈ {0}∪[d] andA∈ Ajthere exists a d-tuplexinSjsuch that A(x)/\e}atio\slash= 0 and
+|S0|+|S1|+···+|Sd|=O/parenleftbiggmlogn
+logm/parenrightbigg
+.
+Case 1:A∈ A0: For a random d-tuplex, chosen according to the distribution Ω d,nwe have that
+Pr
+x[A(x) =p0]≤1−1
+(d+1)d.
+If we randomly choose
+k1=cmlogn
+logm
+d-tuples,x1,...,x k1, according to the distribution Ω d,n, then the probability that A(xi) = 0 for all i∈[k1] is
+Pr[∀i∈[k1] :A(xi) =p0]≤/parenleftbigg
+1−1
+(d+1)d/parenrightbiggk1
+.
+Therefore, by union bound, the probability that there exists a mat rixA∈ A0such that A(xi) = 0 for all i∈[k1]
+is
+Pr[∃A∈ A0,∀i∈[k1] :A(xi) =p0]≤/parenleftbiggnd
+m
+logm/parenrightbigg
+pm
+logm/parenleftbigg
+1−1
+(d+1)d/parenrightbiggcmlogn
+logm
+< ndm
+logmnm
+logmn−c′cm
+logm<1,
+for some constant c. This implies the result.
+Case2:A∈ Ajwherej= 1,...,d: We will assume w.l.o.g that j= 1. We first prove the following lemma
+10Lemma 10. LetU⊆Z×d−1n
+pbe the set of all d−1-dimensional matrices with weight smaller than md/(d+1). For
+A∈UletΥ(A)⊆[n]be following set
+Υ(A) ={j|∃Ai1,i2,...,id−1/\e}atio\slash= 0 and j/\e}atio\slash∈ {i1,i2,...,id−1}}.
+DefineQ={(A,j)|A∈Uandj∈Υ(A)}.Then, there is a constant c0such that for every C > c0and
+k2=Cmlogn
+logm
+there exists a multi-set of d−1-tuples of (0,1)-vectors Z={z1,z2,...,z k2} ⊆({0,1}n)d−1such that for every
+(A,j)∈Qthe size of the set
+Z(A,j)={i|A(zi)/\e}atio\slash= 0 and jis a free index }
+is at leastk2
+2dd.
+Proof.Letzi= (zi,1,zi,2,...,z i,d−1)∈({0,1}n)d−1be random d−1-tuple of (0 ,1)-vector chosen according to
+the distribution Ω d−1,n. For (A,j)∈Q, and by Lemma 6, we have
+Pr
+zi∈Ωd−1,n[A(zi)/\e}atio\slash= 0 and jis a free] = Pr[ jis free]Pr[ A(zi)/\e}atio\slash= 0|jis free]≥1
+d·1
+dd−1=1
+dd.
+Therefore, the expected size of Z(A,j)is greater thank2
+dd. By Chernoff bound, if we randomly choose all zi,i∈[k2]
+according to the distribution Ω d−1,n, then, we have
+Pr/bracketleftbigg
+|Z(A,j)| ≤k2
+2dd/bracketrightbigg
+≤e−k2
+8dd.
+Thus, the probability that there exists ( A,j)∈Qsuch that |Z(A,j)| ≤k2
+2ddis
+Pr/bracketleftbigg
+∃(A,j)∈Q:|Z(A,j)| ≤k2
+2dd/bracketrightbigg
+≤|Q|
+e−k2
+8dd≤|U×[n]|
+e−k2
+8dd≤n/parenleftbignd−1
+md/(d+1)/parenrightbig
+pmd/(d+1)
+eCmlogn
+8ddlogm
+≤n/parenleftbignd−1
+md/(d+1)/parenrightbig
+nmd/(d+1)
+nC(loge)m
+8ddlogm≤nO(md/(d+1))
+nCc′m
+logm<1,
+for large enough C. This implies the result.
+Now, Let UandQbe the sets we defined in Lemma 10. Let A∈ A1. Sincewt(A)≤mthere are at most m1/(d+1)
+d−1-dimensional matrices ( Ai1,i2,...,id)i1=j,i2,...,idwith weight greater than md/(d+1). Therefore, there is at least
+q=/parenleftbiggm
+logm/parenrightbigg1/d
+−m1/(d+1)
+indicesjsuch that ( Ai1,i2,...,id)i1=j,i2,...,id∈U. LetU′contain any qindices such that ( Ai1,i2,...,id)i1=j,i2,...,id∈U.
+LetAUbe the matrix
+(Ai1,i2,...,id)i1∈U′,i2,...,id.
+11Letz1,z2,...,z k2∈({0,1}n)d−1be the set we proved its existence in Lemma 10. We now choose xi∈ {0,1}n,
+i∈[k2] in the following way: Take zi. For every free index j, choose xijto be “1” with probability 1/2 and
+“0” with probability 1/2 (independently for every j). All other entries in xiare zero, that is, all entries that
+correspond to non-free index jinziare zero. Let u∈ {0,1}nbe a vector where uj= 1 ifj∈U′and zero
+otherwise. Also, for a d−1-tupleziletvi∈ {0,1}nbe the vector where vij= 1 ifjis a free index in ziand
+vij= 0 otherwise. By Lemma 7 we have that
+Pr
+x[A(xi,zi) =p0]≤/productdisplay
+i1
+ι(wt(vi∗A(·,zi)))β≤/productdisplay
+i1
+ι(wt(vi∗(u∗A(·,zi))))β. (3)
+Note that, Ais a hypergraph, thus, for every jsuch that ( Ai1,i2,...,id)i1=j,i2,...,id∈U, we have that
+((Ai1,i2,...,id)i1=j,i2,...,id,j)∈Q.
+Therefore,/summationdisplay
+iwt(vi∗(u∗A(·,zi)))≥qk2
+2dd.
+Using Lemma 8 we have
+/productdisplay
+iι(wt(vi∗(u∗A(·,zi))))≥q⌊qk2
+2dd−k2
+q−1⌋=mc1k2.
+Therefore, using (3), Pr x[A(xi,zi) =p0]≤1
+mc1βk2. Thus, the probability that there exists a matrix A∈ A1such
+that for all i∈[k2] we have A(xi,zi) = 0 is
+Pr
+x[A(xi,zi) =p0]≤|A1|
+mc1βk2≤/parenleftbignd
+m/parenrightbig
+pm
+mc1βk2≤ndmnm
+mc1βk2<1,
+for large enough constant. This implies Lemma 9.
+This completes the proof of Theorem 1.
+References
+[1] M. Aigner. Combinatorial Search. John Wiley and Sons , 1988.
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+Boolean Functions. COLT 2008, 123-134, 2008.
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+13
\ No newline at end of file
diff --git a/1001.0406.txt b/1001.0406.txt
new file mode 100644
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+++ b/1001.0406.txt
@@ -0,0 +1,1653 @@
+arXiv:1001.0406v2  [physics.atom-ph]  31 Jan 2010Theoretical energies of low-lying states of light helium-l ike ions
+Vladimir A. Yerokhin
+Center for Advanced Studies, St. Petersburg State Polytech nical University,
+Polytekhnicheskaya 29, St. Petersburg 195251, Russia
+Krzysztof Pachucki
+Institute of Theoretical Physics, University of Warsaw, Ho ˙ za 69, 00–681 Warsaw, Poland
+Rigorous quantumelectrodynamical calculation is present ed for energy levels of the 11S, 21S, 23S,
+21P1, and 23P0,1,2states of helium-like ions with the nuclear charge Z= 3...12. The calculational
+approach accounts for all relativistic, quantum electrody namical, and recoil effects up to orders mα6
+andm2/Mα5, thus advancing the previously reported theory of light hel ium-like ions by one order
+inα.
+PACS numbers: 12.20.Ds, 31.30.J-, 06.20.Jr, 31.15.-p
+I. INTRODUCTION
+Atomic helium and light helium-like ions have long
+been attractive subjects of theoretical and experimen-
+tal investigations. From the theoretical point of view,
+helium-like atoms are the simplest few-body systems. As
+such, they are traditionally used as a testing ground for
+different methods of the description of atomic structure.
+On the experimental side, small natural linewidths of
+transitions between the metastable3Pand3Sstates of
+helium-like ions permit spectroscopic measurements of
+high precision. For atomic helium, experimental investi-
+gations are nowadays carried out with the relative accu-
+racy up to 7 ×10−12[1]. An advantage of the helium-
+like ions as compared to, e.g., hydrogen-like ones, is that
+the transitionfrequency increasesslowlywith the nuclear
+charge number Z(∼Z). This feature ensures that wave-
+lenghts of a significant part of the helium isoelectronic
+sequence fall in the region suitable for accurate experi-
+mental determination.
+There are presently two main theoretical approaches
+that allow one to systematically account for the electron-
+correlation, relativistic, and quantum electrodynamical
+(QED) effects in few-electron systems. The first one,
+traditionally used for light systems, relies on an expan-
+sion of the relativistic and QED effects in terms of α
+andZα(αis the fine-structure constant) and treats the
+nonrelativisticelectron-electroninteractionnonperturba-
+tively. This approach started with the pioneering works
+of Araki [2] and Sucher [3], who derived the expression
+for the Lamb shift in many-electron systems complete
+through the order mα5. The other approach aims pri-
+marily at high- Zions. It does not use any expansion
+in the binding-strength parameter Zα(and thus is of-
+ten referred to as the all-order approach) but treats the
+electron-electron interaction within the perturbative ex-
+pansion with the parameter 1 /Z. A systematic formula-
+tion of this method is presented in Ref. [4].
+These two approaches can be considered as comple-
+mentary, the first being clearly preferable for light atoms
+and the second, for heavy ions. The intermediate regionof nuclear charges around Z= 12 is the most difficult
+one for theory, as contributions not (yet) accounted by
+either of these methods have their maximal value there.
+In order to provide accurate predictions for the whole
+isoelectronic sequence, it is necessary to combine these
+two approaches.
+For the first time a combination of the complementary
+approaches was made by Drake [5]. His results for en-
+ergies of helium-like ions comprise all effects up to order
+mα5in the low- Zregion, whereas in the high- Zregion,
+they are complete up to the next-to-the-leading order
+in 1/Zfor nonradiative effects and to the leading or-
+der, for radiative effects. Since then, significant progress
+was achieved in theoretical understanding of energy lev-
+els of atomic helium, whose description is now complete
+through order mα6[6, 7]. Also in the high- Zregion,
+theoretical energies have recently been significantly im-
+proved by a rigorous treatment of the two-electron QED
+corrections [8], which completed the O(1/Z) part of the
+radiative effects.
+In the present investigation we aim to improve theo-
+retical predictions of the n= 1 andn= 2 energy levels of
+light helium-like ions. To this end, we perform a calcula-
+tion that includes all QED and recoil effects up to orders
+mα6andm2/M α5(Mis the nuclear mass). In order to
+establish a basis for merging the current approach with
+the all-order calculations, we perform an extensive anal-
+ysis of the 1 /Zexpansion of individual corrections. This
+analysis also provides an effective test of consistency of
+our calculational results and of the 1 /Z-expansion data
+available in the literature.
+II. THEORY OF THE ENERGY LEVELS
+In this section, we present a summary of contributions
+to the energy levels of two-electron atoms complete up to
+ordersmα6andm2/M α5.
+According to QED theory, energy levels of atoms are
+represented by an expansion in powers of αof the form
+E(α) =E(2)+E(4)+E(5)+E(6)+E(7)+...,(1)2
+whereE(n)≡mαnE(n)is a contribution of order αnand
+may include powers of ln α. Each of E(n)is in turn ex-
+panded in powers of the electron-to-nucleus mass ratio
+m/M
+E(n)=E(n)
+∞+E(n)
+M+E(n)
+M2+... , (2)
+whereE(n)
+Mdenotes the correction of first order in m/M
+andE(n)
+M2is the second-order correction. To note, for the
+nonrelativistic energy, it is more natural to expand in
+mr/M(wheremris the reduced mass) rather than in
+m/M, since such expansion has smaller coefficients. For
+the relativistic corrections, however, the natural recoil
+expansion parameter is m/M, so for consistency we use
+it for the nonrelativistic energy as well.
+The terms of the double perturbation expansion (1)
+and (2) are expressed as expectation values of some effec-
+tive Hamiltonians (in some cases, of nonlocal operators)
+and as second- and higher-orderperturbation corrections
+induced by these Hamiltonians (operators). It is note-
+worthy that the expansion (1) is employed also for the
+states that are mixed by the relativistic effects, namely
+21P1and 23P1. The mixing effects are treated pertur-
+batively. (So, the leading effect due to the 21P1−23P1
+mixing appears naturally as the second-order mα6cor-
+rection, together with contributions from other interme-
+diate states.) This differs from the approach used, e.g.,
+in Ref. [5], where a two-by-two matrix was constructed
+for this pair of states and the energies were obtained by
+a diagonalization.
+The leading contribution to the energy E(2)
+∞≡ E0is the
+eigenvalue of the nonrelativistic Hamiltonian,
+H(2)≡H0=/summationdisplay
+a/parenleftbigg/vector p2
+a
+2−Z
+ra/parenrightbigg
++/summationdisplay
+a<b1
+rab.(3)
+The first- and second-order recoil corrections to the non-
+relativistic energy are given by
+E(2)
+M=−m
+ME(2)
+∞+/angbracketleftBig
+H(2)
+rec/angbracketrightBig
+, (4)
+E(2)
+M2=/parenleftBigm
+M/parenrightBig2
+E(2)
+∞−2m
+M/angbracketleftBig
+H(2)
+rec/angbracketrightBig
++/angbracketleftbigg
+H(2)
+rec1
+(E0−H0)′H(2)
+rec/angbracketrightbigg
+, (5)
+where
+H(2)
+rec=m
+M/summationdisplay
+a<b/vector pa·/vector pb (6)
+is the mass polarization operator.
+The leading relativistic correction E(4)
+∞is given by the
+expectation value of the Breit-Pauli Hamiltonian H(4)[9],
+H(4)=/summationdisplay
+a/bracketleftbigg
+−/vector p4
+a
+8+πZ
+2δ3(ra)+Z
+4/vector σa·/vector ra
+r3a×/vector pa/bracketrightbigg
++/summationdisplay
+a<b/braceleftbigg
+−πδ3(rab)−1
+2pi
+a/parenleftbiggδij
+rab+ri
+abrj
+ab
+r3
+ab/parenrightbigg
+pj
+b
+−2π
+3/vector σa·/vector σbδ3(rab)+σi
+aσj
+b
+4r3
+ab/parenleftbigg
+δij−3ri
+abrj
+ab
+r2
+ab/parenrightbigg
++1
+4r3
+ab/bracketleftbig
+2/parenleftbig
+/vector σa·/vector rab×/vector pb−/vector σb·/vector rab×/vector pa/parenrightbig
++/parenleftbig
+/vector σb·/vector rab×/vector pb−/vector σa·/vector rab×/vector pa/parenrightbig/bracketrightbig/bracerightbigg
+. (7)
+The finite nuclear mass correction to the Breit contri-
+butionE(4)
+Mis conveniently separated into the mass scal-
+ing, the mass polarization, and the operator parts. The
+mass scaling prefactor is ( mr/m)4for the first term in
+Eq. (7) and ( mr/m)3, for all the others. The mass po-
+larization part represents the first-order perturbation of
+E(4)
+∞by the mass-polarization operator (6). The opera-
+tor part is given by the expectation value of the recoil
+addition to the Breit-Pauli Hamiltonian,
+H(4)
+rec=Zm
+2M/summationdisplay
+ab/bracketleftbigg/vector ra
+r3a×/vector pb·/vector σa−pi
+a/parenleftbiggδij
+ra+ri
+arj
+a
+r3a/parenrightbigg
+pj
+b/bracketrightbigg
+.
+(8)
+E(5)
+∞is the leading QED correction [2, 3]. We divide it
+intothe logarithmicand the nonlogarithmicparts, E(5)
+∞=
+E(5)
+∞(log)+E(5)
+∞(nlog), which are given by
+E(5)
+∞(log) =14
+3ln(Zα)/summationdisplay
+a<b/angb∇acketleftδ3(rab)/angb∇acket∇ight
++4Z
+3ln/bracketleftbig
+(Zα)−2/bracketrightbig/summationdisplay
+a/angb∇acketleftδ3(ra)/angb∇acket∇ight,(9)
+and
+E(5)
+∞(nlog) =164
+15/summationdisplay
+a<b/angb∇acketleftδ3(rab)/angb∇acket∇ight−14
+3/summationdisplay
+a<b/tildewideQab
++/bracketleftbigg19
+30−ln/parenleftbiggk0
+Z2/parenrightbigg/bracketrightbigg4Z
+3/summationdisplay
+a/angb∇acketleftδ3(ra)/angb∇acket∇ight+/angb∇acketleftH(5)
+fs/angb∇acket∇ight,
+(10)
+where
+/tildewideQab=/angbracketleftbigg1
+4πr3
+ab+δ3(rab) lnZ/angbracketrightbigg
+, (11)
+and the singular operator r−3is defined by
+/angbracketleftbigg1
+r3/angbracketrightbigg
+≡lim
+a→0/integraldisplay
+d3rφ∗(/vector r)φ(/vector r)
+×/bracketleftbigg1
+r3Θ(r−a)+4πδ3(r)(γ+lna)/bracketrightbigg
+,(12)3
+whereγis the Euler constant. The Bethe logarithm is
+defined as
+ln(k0) =/angbracketleftBig/summationtext
+a/vector pa(H0−E0) ln/bracketleftbig
+2(H0−E0)/bracketrightbig/summationtext
+b/vector pb/angbracketrightBig
+2πZ/angbracketleftBig/summationtext
+cδ3(rc)/angbracketrightBig .
+(13)
+The operator H(5)
+fsis the anomalous magnetic moment
+correction to the spin-dependent part of the Breit-Pauli
+Hamiltonian. H(5)
+fsdoes not contribute to the energies
+of the singlet states and to the spin-orbit averaged levels
+but it yields the mα5contribution to the fine structure
+splitting. It is given by
+H(5)
+fs=Z
+4π/summationdisplay
+a/vector σa·/vector ra
+r3a×/vector pa
++/summationdisplay
+a<b/braceleftbigg1
+4πσi
+aσj
+b
+r3
+ab/parenleftbigg
+δij−3ri
+abrj
+ab
+r2
+ab/parenrightbigg
++1
+4πr3
+ab/bracketleftbig
+2/parenleftbig
+/vector σa·/vector rab×/vector pb−/vector σb·/vector rab×/vector pa/parenrightbig
++/parenleftbig
+/vector σb·/vector rab×/vector pb−/vector σa·/vector rab×/vector pa/parenrightbig/bracketrightbig/bracerightbigg
+.(14)
+We note that despite the presence of terms with ln Zin
+Eq. (10), the correction E(5)
+∞(nlog) does not have loga-
+rithmic terms in its 1 /Zexpansion.
+The recoil correction E(5)
+Mconsists of four parts [10],
+E(5)
+M=m
+M/parenleftbig
+E1+E2+E3/parenrightbig
++/angb∇acketleftH(5)
+fs,rec/angb∇acket∇ight,(15)
+where
+E1=−3E(5)
+∞+4Z
+3/summationdisplay
+a/angb∇acketleftδ3(ra)/angb∇acket∇ight−14
+3/summationdisplay
+a<b/angb∇acketleftδ3(rab)/angb∇acket∇ight,(16)
+E2=Z2/bracketleftBigg
+−2
+3ln(Zα)+62
+9−8
+3ln/parenleftbiggk0
+Z2/parenrightbigg/bracketrightBigg/summationdisplay
+a/angb∇acketleftδ3(ra)/angb∇acket∇ight
+−14Z2
+3/summationdisplay
+a/tildewideQa, (17)
+with/tildewideQadefined analogously to Eq. (11), and ( m/M)E3
+is the first-order perturbation of E(5)
+∞due to the mass-
+polarization operator (6). The operator H(5)
+fs,recyields anonvanishing contribution to the fine-structure splitting
+only. It is given by
+H(5)
+fs,rec=m
+MZ
+4π/summationdisplay
+ab/vector ra
+r3a×/vector pb·/vector σa.(18)
+We note that the last term in Eq. (16) was omitted in
+the original derivation of Ref. [10].
+The complete result for the mα6correction E(6)
+∞to the
+energy levels was derived by one of the authors (K.P.) in
+a series of papers [6, 7, 11, 12]
+E(6)
+∞=−ln(Zα)π/summationdisplay
+a<b/angb∇acketleftδ3(rab)/angb∇acket∇ight+Esec
++/angbracketleftBig
+H(6)
+nrad+H(6)
+R1+H(6)
+R2+H(6)
+fs+H(6)
+fs,amm/angbracketrightBig
+.
+(19)
+The first term in the above expression contains the com-
+plete logarithmic dependence of the mα6correction.
+The part of it proportional to ln αwas first obtained in
+Ref. [13]. The remaining logarithmic part proportional
+to lnZwas implicitly present in formulas reported in
+Ref. [6, 7] (it originates from the expectation value of the
+operator 1 /r3
+ab). In Eq. (19), we group all logarithmic
+terms together so that the remaining part does not have
+any logarithms in its 1 /Zexpansion.
+Theterm EsecinEq.(19)isthesecond-orderperturba-
+tion correction induced by the Breit-Pauli Hamiltonian.
+(More specifically, it is the finite residual after separat-
+ing divergent contributions that cancel out in the sum
+with the expectation value of the effective mα6Hamilto-
+nian.) The first part of the effective Hamiltonian, H(6)
+nrad,
+originates from the non-radiative part of the electron-
+nucleus and the electron-electron interaction. The next
+two terms, H(6)
+R1andH(6)
+R2, are due to the one-loop and
+two-loop radiative effects, respectively. The last two
+partsH(6)
+fsandH(6)
+fs,ammare the spin-dependent opera-
+tors first derived by Douglas and Kroll [14]. They do not
+contribute to the energies of the singlet states and to the
+spin-orbit averaged levels. Expressions for these opera-
+tors are well known and are given, e.g., by Eqs. (3) and
+(7) of Ref. [15]. The non-radiative part of the mα6effec-
+tive Hamiltonian is rather complicated. For simplicity,
+we present it specifically for a two-electron atom. The
+corresponding expression reads [6, 7]4
+H(6)
+nrad=−E3
+0
+2+/bracketleftbigg/parenleftbigg
+−E0+3
+2/vector p2
+2+1−2Z
+r2/parenrightbiggZπ
+4δ3(r1)+(1↔2)/bracketrightbigg
++/vectorP2
+6πδ3(r)−(3+/vector σ1·/vector σ2)
+24π/vector pδ3(r)/vector p−/parenleftbiggZ
+r1+Z
+r2/parenrightbiggπ
+2δ3(r)
++/parenleftbigg13
+12+8
+π2−3
+2ln(2)−39ζ(3)
+4π2/parenrightbigg
+πδ3(r)+E2
+0+2E(4)
+4r
+−E0
+r2(31+5/vector σ1·/vector σ2)
+32−E0
+2r/parenleftbiggZ
+r1+Z
+r2/parenrightbigg
++E0
+4/parenleftbiggZ
+r1+Z
+r2/parenrightbigg2
+−1
+r2/parenleftbiggZ
+r1+Z
+r2−1
+r/parenrightbigg(23+5/vector σ1·/vector σ2)
+32−1
+4r/parenleftbiggZ
+r1+Z
+r2/parenrightbigg2
++Z2
+2r1r2/parenleftbigg
+E0+Z
+r1+Z
+r2−1
+r/parenrightbigg
+−Z/parenleftbigg/vector r1
+r3
+1−/vector r2
+r3
+2/parenrightbigg
+·/vector r
+r3(13+5/vector σ1·/vector σ2)
+64
++Z
+4/parenleftbigg/vector r1
+r3
+1−/vector r2
+r3
+2/parenrightbigg
+·/vector r
+r2−Z2
+8ri
+1
+r3
+1(rirj−3δijr2)
+rrj
+2
+r3
+2
++/bracketleftbiggZ2
+81
+r2
+1/vector p2
+2+Z2
+8/vector p11
+r2
+1/vector p1+/vector p11
+r2/vector p1(47+5/vector σ1·/vector σ2)
+64+(1↔2)/bracketrightbigg
++1
+4pi
+1/parenleftbiggZ
+r1+Z
+r2/parenrightbigg(rirj+δijr2)
+r3pj
+2+Pi(3rirj−δijr2)
+r5Pj(−3+/vector σ1·/vector σ2)
+192
+−/bracketleftbiggZ
+8pk
+2ri
+1
+r3
+1/parenleftbigg
+δjkri
+r−δikrj
+r−δijrk
+r−rirjrk
+r3/parenrightbigg
+pj
+2+(1↔2)/bracketrightbigg
+−E0
+8p2
+1p2
+2−1
+4p2
+1/parenleftbiggZ
+r1+Z
+r2/parenrightbigg
+p2
+2+1
+4/vector p1×/vector p21
+r/vector p1×/vector p2
++1
+8pk
+1pl
+2/parenleftbigg
+−δjlrirk
+r3−δikrjrl
+r3+3rirjrkrl
+r5/parenrightbigg
+pi
+1pj
+2+ln(Z)πδ3(r), (20)
+where/vectorP=/vector p1+/vector p2,/vector p= (/vector p1−/vector p2)/2,/vector r=/vector r1−/vector r2. We
+note thatthe operator H(6)
+nradisdefined in sucha waythat
+its expectation values does not contain any logarithmic
+terms in the 1 /Zexpansion, as the last term of Eq. (20)
+is compensated by the corresponding contribution from
+the 1/r3operator.
+The effective Hamiltonians induced by the radiative
+effects are [6, 16, 17]
+H(6)
+R1=Z2/bracketleftbigg427
+96−2 ln(2)/bracketrightbigg
+π/bracketleftbig
+δ3(r1)+δ3(r2)/bracketrightbig
++/bracketleftbigg6ζ(3)
+π2−697
+27π2−8 ln(2)+1099
+72/bracketrightbigg
+πδ3(r),(21)
+and
+H(6)
+R2=Z/bracketleftbigg
+−9ζ(3)
+4π2−2179
+648π2+3 ln(2)
+2−10
+27/bracketrightbigg
+π
+×/bracketleftbig
+δ3(r1)+δ3(r2)/bracketrightbig
++/bracketleftbigg15ζ(3)
+2π2+631
+54π2
+−5 ln(2)+29
+27/bracketrightbigg
+πδ3(r). (22)The second-order correction can be represented as
+Esec=/angbracketleftbigg
+H(4)′
+nfs1
+(E0−H0)′H(4)′
+nfs/angbracketrightbigg
++2/angbracketleftbigg
+H(4)
+nfs1
+(E0−H0)′H(4)
+fs/angbracketrightbigg
++/angbracketleftbigg
+H(4)
+fs1
+(E0−H0)′H(4)
+fs/angbracketrightbigg
+, (23)
+whereH(4)
+nfsandH(4)
+fsare the spin-independent and spin-
+dependent parts of the Breit-Pauli Hamiltonian (7), re-
+spectively. The operator H(4)′
+nfsis obtainedfrom H(4)
+nfsby a
+transformationthateliminates divergencesinthe second-
+order matrix elements [6]. The transformed operator is
+given by
+H(4)′
+nfs=−1
+2(E0−V)2−pi
+11
+2r/parenleftbigg
+δij+rirj
+r2/parenrightbigg
+pj
+2
++1
+4/vector∇2
+1/vector∇2
+2−Z
+4/vector r1
+r3
+1·/vector∇1−Z
+4/vector r2
+r3
+2·/vector∇2,(24)
+where/vector∇2
+1/vector∇2
+2is understood as a differentiation of the
+wave function on the right hand side as a function (omit-
+tingδ3(r)) andV=−Z/r1−Z/r2+1/r.5
+An intriguing feature of the formulas presented in this
+section is that the logarithmic dependence of all of them
+enters only in the form of ln( Zα). This is not at all obvi-
+ousa priorisinceln(Zα)appearsnaturallyonlyincontri-
+butions induced by the electron-nucleus interaction. The
+effects of the electron-electron interaction usually yield
+logarithms of α, whereas logarithms of Zare implicitly
+present in matrix elements of singular operators. The
+fact that logarithms of αand logarithms of Zhave the
+coefficients that match each other comes “accidentally”
+from the derivation.
+The complete result for the corrections of order mα7
+for the helium Lamb shift is not presently available (it
+is known for the fine-structure splitting only [15, 18]).
+One can, however, easily generalize some of the hydro-
+genic results, namely those that are proportional to the
+electron density at the nucleus. These are (i) the one-
+loop radiative correction of order mα(Zα)6ln2(Zα)−2,
+(ii) the two-loopradiativecorrectionof order mα2(Zα)5,
+and (iii) the nonrelativistic correction due to the finite
+nuclear size. The first two effects yield the main contri-
+bution to the higher-orderremainder function of Sstates
+in light hydrogen-like ions. We expect that they domi-
+nate for light helium-like ions as well.
+FollowingRef. [5], we approximatethe higher-orderra-
+diative (“rad”) and the finite-nuclear-size (“fs”) correc-
+tion to the energies of helium-like ions by
+E(7+)
+rad=E(7+)
+rad,H/angbracketleftbig/summationtext
+iδ3(ri)/angbracketrightbig
+/angbracketleftbig/summationtext
+iδ3(ri)/angbracketrightbig
+H, (25)
+Efs=Efs,H/angbracketleftbig/summationtext
+iδ3(ri)/angbracketrightbig
+/angbracketleftbig/summationtext
+iδ3(ri)/angbracketrightbig
+H, (26)
+where the subscript Hcorresponds to the “hydrogenic”
+limit, i.e., the limit of the non-interacting electrons, and
+/angbracketleftbig/summationdisplay
+iδ3(ri)/angbracketrightbig
+H=Z3
+π/parenleftbigg
+1+δl,0
+n3/parenrightbigg
+.(27)
+The approximation of Eqs. (25) and (26) is exact for the
+above-mentioned correctionsproportional to the electron
+density at the nucleus. It is expected also to provide
+a meaningful estimate for contributions that weakly de-
+pend on n(such as the nonlogarithmic radiative correc-
+tion of order mα(Zα)6). Moreover, this approximation
+is exact to the leading order in the 1 /Zexpansion, thus
+providing a meaningful estimate for high- Zhelium-like
+ions as well.
+For all the states under consideration except 21P1and
+23P1, the “hydrogenic” remainder function is just the
+sum of the corresponding remainders for the two elec-
+trons in the configuration,
+E(7+)
+rad,H=E(7+)
+rad(1s)+E(7+)
+rad(nlj).(28)
+For the 21P1and 23P1states, the Dirac levels need to
+be first transformed from the jjto theLScoupling andthus [5]
+E(7+)
+rad,H(21P1) =E(7+)
+rad(1s)+2
+3E(7+)
+rad(2p3/2)+1
+3E(7+)
+rad(2p1/2),
+E(7+)
+rad,H(23P1) =E(7+)
+rad(1s)+1
+3E(7+)
+rad(2p3/2)+2
+3E(7+)
+rad(2p1/2).
+(29)
+In our calculation, the one-electron remainder func-
+tionE(7+)
+rad(nlj) includes all known contributions of order
+mα7and higher coming from (i) the one-loop radiative
+correction, (ii) the two-loop radiative correction, (iii) the
+three-loopradiativecorrection, see the review in Ref. [19]
+and Ref. [20] for an update on the two-loop remainder
+function.
+Besides the finite-nuclear-size and radiative correc-
+tions, there are also non-radiative effects, denoted as
+E(7+)
+nradand estimated within the 1 /Zexpansion. More
+specifically, we include the higher-order remainder due
+to the one-electron Dirac energy and due to the one-
+photon exchange correction. They enter at the order
+mα8only but are enhanced by factors of Z8andZ7,
+respectively. Despite this enhancement, numerical con-
+tributions of these effects are rather small for the ions
+considered in the present work.
+III. RESULTS AND DISCUSSION
+A. Numerical results
+The nonrelativisticenergiesand wavefunctions areob-
+tained by minimizing the energy functional with the ba-
+sis set constructed with the fully correlated exponential
+functions. The choice of the basis set and the general
+strategy of optimization of the nonlinear parameters fol-
+low the main lines of the approach developed by Ko-
+robov [21, 22]. The calculational scheme is described in
+previous publications [6, 7, 15] and will not be repeated
+here. Numerical values of the nonrelativistic energies of
+helium-like ions with the nuclear charge Z= 3...12 are
+presented in Table I. The results were obtained with
+N= 2000 basis functions and are accurate to about 18
+decimals (more than shown in the table). The energy
+levels of the helium atom traditionally attract special at-
+tention, so we present the corresponding results in full
+length below. Our numerical values of the upper varia-
+tional limit of the nonrelativistic energies of helium are
+E(2)
+∞(11S) =−2.903724377034119598310+0
+−2,(30)
+E(2)
+∞(21S) =−2.145974046054417415799+0
+−8,(31)
+E(2)
+∞(23S) =−2.175229378236791305738977+0
+−2,
+(32)
+E(2)
+∞(21P) =−2.123843086498101359246+0
+−2,(33)
+E(2)
+∞(23P) =−2.13316419077928320514696+0
+−10.
+(34)6
+The value for the ground state is given only for com-
+pleteness, sincemuchmoreaccuratenumericalresultsare
+available in the literature [22, 23]. The numerical results
+fortheleadingrelativisticcorrection E(4)aresummarized
+in Table II.
+The leading QED correction E(5)is given by Eqs. (9),
+(10), and (15). Computationally the most problematic
+part of it is represented by the Bethe logarithm ln( k0)
+and its mass-polarization correction ln( k0)M. Accurate
+calculationsofln( k0) wereperformedbyDrakeandGold-
+man [24] for helium-like like atoms with Z≤6 and by
+Korobov[25] for Z= 2. Calculations of the recoil correc-
+tion to the Bethe logarithm were reported by Pachucki
+and Sapirstein [10] for Z= 2 and by Drake and Gold-
+man [24] for Z≤6. In the present investigation, we per-
+form accurate evaluations of the Bethe logarithm ln( k0)
+and its recoil correction ln( k0)Mfor helium-like ions with
+Z≤12. The calculational approach is described in Ap-
+pendix A.
+Table III summarizes the numerical results obtained
+and gives a comparison with the previous calculations.
+Numerical values for the Bethe logarithm are presented
+forthedifferenceln( k0)−ln(Z2)sincethisdifferencehasa
+weakZ-dependence and does not contain any logarithms
+in its 1/Zexpansion. The table also lists the coefficients
+of the 1/Zexpansion of ln( k0/Z2). The leading coef-
+ficientc0is known from the hydrogen theory; accurate
+numerical values can be found in Ref. [26]. The higher-
+order coefficients were obtained by fitting our numerical
+data. It is interesting to compare them with the analo-
+gous results reported previously by Drake and Goldman
+[24]. For the next-to-the-leading coefficient c1, the re-
+sults agree up to about 4-5 digits for Sstates and about
+3-4 digits for Pstates. For the higher-order coefficients,
+the agreement gradually deteriorates. However, the re-
+sults for the sum of the two expansions agree very well
+with each other. More specifically, the maximal absolute
+deviation between the values of the Bethe logarithms for
+Z >12 obtained with our 1 /Z-expansion coefficients and
+with those by Drake and Goldman is 1 ×10−8for the 11S
+state, 3×10−8for the 21Sstate, 6×10−9for the 23S
+state, 1×10−7for the 21Pstate, and 6 ×10−8for the 23P
+state. So, the accuracy of these expansions is sufficient
+for most practical purposes.
+Another part of the calculation of E(5)that needs a
+separate discussion is the evaluation of the expectation
+value of the singular operator 1 /r3, which is defined by
+Eq. (12). The calculational approach is described in Ap-
+pendix B. Total results for the logarithmic and the non-
+logarithmic part of the leading QED correction are sum-
+marized in Tables IV and V, respectively. The results are
+in good agreement with the previous calculations [5].
+Table VI presents the numerical values of the mα6cor-
+rection, the main result of this investigation. The corre-
+sponding calculations for atomic helium were reported
+in Refs. [6, 7]; our present numerical values agree with
+the ones obtained previously. Calculations performed in
+this work for helium-like ions were accomplished alongthe lines described in Refs. [6, 7]. Here we only note that
+calculations for higher values of Zoften exhibit a slower
+numerical convergence (and numerical stability) than for
+helium, especially so for the second-order corrections in-
+volving singular operators. The variational optimization
+of nonlinear parameters for the symmetric second-order
+corrections was performed in several steps with increas-
+ing the number of basis functions on each step up to
+N= 1000 or 1200. The final values were obtained with
+merging several basis sets and enlarging the number of
+functions up to N= 5000−7000. The nonsymmetric
+second-order corrections were evaluated as described in
+Ref.[15]. Thecalculationswereperformedinthequadru-
+ple, sixtuple, and octuple arithmetics implemented in
+Fortran 95 by V. Korobov [27].
+Table VII presents the results for the finite nuclear size
+correction and approximate values of the higher-order
+(mα7and higher) correction to the ionization energy .
+The uncertainty of the total theoretical prediction origi-
+nates from the higher-order radiative effects; it was esti-
+matedbydividing the absolutevalue ofthis correctionby
+Z. The values for the root-mean-square radius of nuclei
+were taken from Ref. [28].
+B. Comparison with the all-order approach
+In this subsection we discuss the calculational results
+obtained for the mα6correction in more detail and
+make a comparison with the results obtained previously
+within the all-order, 1 /Z-expansion approach. The log-
+arithmic part of the correction, E(6)(log), behaves as
+mα3(Zα)3for large Zand thus corresponds to diagrams
+with three photon exchanges that have not yet been ad-
+dressed within the all-order approach. The nonlogarith-
+mic part E(6)(nlog), however, contains some pieces that
+are known and identified below.
+Forallstatesexcept21P1and23P1, theleadingtermof
+the 1/Zexpansion of E(6)(nlog) is of order m(Zα)6and
+comes from the Zαexpansion of the one-electron Dirac
+energy. For the 21P1and 23P1states, the leading term is
+ofthe previous orderin 1 /Z,m(Zα)6Z, and is due to the
+mixing of these levels. More specifically, the mixing con-
+tribution is δEmix=/vextendsingle/vextendsingle/angb∇acketleft21P1|H(4)|23P1/angb∇acket∇ight/vextendsingle/vextendsingle2/[E0(21P1)−
+E0(23P1)] for the 21P1state and that with the opposite
+sign, for 23P1. The contribution of order m(Zα)6for
+the mixing states comes from the expansion of the one-
+electron Dirac energy and from the expansion of δEmix.
+The next term of the 1 /Zexpansion is of order
+mα(Zα)5and comes from the one-electron one-loop ra-
+diativecorrectionand fromthe one-photonexchangecor-
+rection. The radiative part is well known [19]. The part
+duetotheone-photonexchangewasobtainedforthe11S,
+23S, 23P0, and 23P2states analytically in Ref. [29] and
+for the other states numerically in this work. For the
+21P1and 23P1states, there is a small additional mixing
+contribution, which we were unable to determine unam-
+biguously.7
+The first two coefficients of the 1 /Zexpansion of
+E(6)(nlog) are listed in Table VI. A fit of our numeri-
+cal data agrees well with these coefficients. The agree-
+ment observedshowsconsistency ofour numericalresults
+with independent calculations at the level of the one-
+photon effects. We now turn to the contribution of order
+mα2(Zα)4. This contribution is induced by nontrivial
+two-photon effects, so that a comparison drawn for this
+part will yield a much more stringent test of consistency
+of different approaches.
+The part of E(6)(nlog) that is of order mα2(Zα)4is
+implicitly present in the two-electron QED contribution
+calculated numerically in Ref. [8] to all orders in Zα.
+This contribution can be represented as (see Eq. (72) of
+Ref. [8])
+∆EQED
+2el=mα2(Zα)3/bracketleftBig
+a31ln(Zα)−2+a30+(Zα)G(Z)/bracketrightBig
+,
+(35)
+where the remainder function G(Z) incorporates all
+higher orders in Zα. The two-electron QED correction
+comprises the so-called screened self-energy and vacuum-
+polarizationcontributionsandthepartofthetwo-photon
+exchange correction that is beyond the Breit approxima-
+tion.
+The coefficients a31anda30in Eq. (35) correspond to
+the second termofthe 1 /Zexpansionofthe leadingQED
+correction E(5)
+∞. More specifically, a31corresponds to the
+coefficient c1from Table IV and a30, to that from Ta-
+ble V. The Z= 0 limit of the higher-order remainder
+function G(Z) corresponds to the third coefficient of the
+1/Zexpansion of the correction E(6)(nlog),G(0) =c2,
+for all states except 21P1and 23P1. The values of c2
+obtained by fitting our numerical data are listed in Ta-
+ble VI. For the 21P1and 23P1states, the coefficient c2is
+notdirectlycomparablewiththeall-orderresultsbecause
+of the mixing effects.
+The higher-order remainder function G(Z) inferred
+from the numerical results of Ref. [8] is plotted in Fig. 1,
+together with its limiting value at Z= 0 obtained by
+a fit of our numerical data. It should be stressed that
+the identification of the remainder implies a great deal
+of numerical cancellations, especially for the all-order re-
+sults. The comparison drawn in Fig. 1 provides a strin-
+gent cross-check of the two complementary approaches.
+The visual agreement between the results is very good
+for theSstates, whereas for the Pstates a slight dis-
+agreement seems to be present.
+It is tempting to merge the all-order and the Zα-
+expansion results by fitting the all-order data for G(Z)
+towards lower values of Z. However, we do not attempt
+to do this in the present work. The reasons are, first,
+that the numerical accuracy of the all-order results is
+apparently not high enough and, second, that the ex-
+pansion of the remainder function G(Z) contains terms
+(Zα)ln2(Zα) and (Zα)ln(Zα), which cannot be reason-
+ably fitted with numerical data available in the high- Z
+region only.C. Total energies
+Our total results for the ionization energy of the n= 1
+andn= 2 states of helium-like atoms with the nuclear
+chargeZ= 2...12 are listed in Table VIII. The fol-
+lowing values of fundamental constants were employed
+[19],R∞= 10973731 .568527(73) m−1andα−1=
+137.0359999679(94). The atomic masses were taken
+from Ref. [30].
+The results for atomic helium presented in Table VIII
+differ from those reported previously only because of
+the different approximate treatment of the higher-order
+(mα7and higher) contribution employed in this work.
+For theSstates of helium, the present values are prac-
+tically equivalent to those of Refs. [6, 7]. (The difference
+is that now we include some contributions of order mα8
+and higher, which are negligible for helium but become
+noticeable for higher- Zions.) However, for the helium
+Pstates, our present estimate of the higher-order contri-
+bution is by about 1 MHz higher than that of Ref. [7].
+The reason is that the one-electron radiative correction
+of thepelectron state was previously not included into
+the approximation (25). It is included now [see Eq. (29)]
+in order to recover the correct asymptotic behaviour of
+the radiative correction in the high- Zlimit.
+A selection of our theoretical results for transition en-
+ergies is compared with the theory by Drake [5, 31] and
+with experimental data in Table IX. Agreement between
+theory and experiment is excellent in all cases studied.
+We observe a distinct improvement in theoretical accu-
+racy as compared to the previous results by Drake. This
+improvementis due to the complete treatment ofthe cor-
+rections of order mα6andm2/Mα5accomplished in this
+work.
+Theoretical results for the fine structure splitting in-
+tervals 23P0−23P1and 23P1−23P2are not analyzed
+in the present work. This is because these intervals can
+nowdays be calculated more accurately (complete up to
+ordermα7), like it was recently done for helium [15]. We
+intend to perform such a calculation in a subsequent in-
+vestigation.
+Among the results listed in Table VIII for helium-like
+ions, the ground-state energy of the carbon ion is of par-
+ticular importance, because it is used in the procedure
+of the adjustment of fundamental constants [32] for the
+determination of the mass of12C4+and, consequently,
+of the proton mass from the Penning trap measurement
+by Van Dyck et al.[33]. Our result for the ground-state
+ionization energy of helium-like carbon is
+E(12C4+) =−3162423 .60(32) cm−1,(36)
+which is in agreement with the previous result by Drake
+[5] of−3162423 .34(15) cm−1. We note that despite the
+fact that our calculation is by an additional order of α
+more complete than that by Drake, our estimate of un-
+certainty is more conservative.
+In summary, significant progress has been achieved
+during last decades both in experimental technique and8
+theoretical calculations of helium-like atoms. In the
+present investigation, we performed a calculation of the
+energy levels of the n= 1 and n= 2 states of light
+helium-like atoms with the nuclear charge Z= 2...12,
+within the approach complete up to orders mα6and
+m2/Mα5. An extensive analysis of the 1 /Zexpansion
+of individual corrections was carried out and comparison
+with results of the complementary approach was made
+whenever possible. Our general conclusion is that the
+results obtained within the approaches based on the Zα
+and the 1 /Zexpansion are consistent with each other
+up to a high level of precision. However, further im-
+provement of numerical accuracy of the all-order, 1 /Z-
+expansion results and their extension into the lower- Z
+region is needed in order to safely merge the two comple-
+mentary approaches.
+Acknowledgments
+We wish to thank Tomasz Komorek for participation
+in the beginning of the project. Computations reported
+in this work were performed on the computer clusters
+of St. Petersburg State Polytechnical University and of
+Institute of Theoretical Physics, University of Warsaw.
+The research was supported by NIST through Precision
+Measurement Grant PMG 60NANB7D6153. V.A.Y. ac-
+knowledges additional support from the “Dynasty” foun-
+dation and from RFBR (grant No. 10-02-00150-a).
+Appendix A: Bethe logarithm
+Following the approach of Refs. [34, 35], the Bethe log-
+arithm (13) is expressed in terms of an integral over the
+momentum of the virtual photon,
+ln(k0) =1
+Dlim
+K→∞/bracketleftBigg
+/angbracketleftbig/vector∇2/angbracketrightbig
+K+Dln(2K)+/integraldisplayK
+0dkkJ(k)/bracketrightBigg
+,
+(A1)
+whereD= 2πZ/angbracketleftbig
+δ3(r1)+δ3(r2)/angbracketrightbig
+,/vector∇ ≡/vector∇1+/vector∇2, and
+J(k) =/angbracketleftbigg
+/vector∇1
+E0−H0−k/vector∇/angbracketrightbigg
+. (A2)
+For the purpose of numerical evaluation, the integration
+over the photon momentum kis divided into two regions
+by introducing the auxiliar parameter κ,
+ln(k0) =R(κ)+1
+D/integraldisplayκ
+0dkkJ(k)+/integraldisplay∞
+κdkw(k)
+k2,(A3)
+where the function w(k) represents the residual obtained
+fromJ(k) by removing all known terms of the large- k
+asymptotics,
+w(k) =k3
+DJ(k)+k2
+D/angbracketleftbig/vector∇2/angbracketrightbig
++k−2√
+2Zk1/2+2Z2lnk,
+(A4)andR(κ) is a simple function obtained byintegratingout
+the separated asymptotic expansion terms,
+R(κ) =κ/angbracketleftbig/vector∇2/angbracketrightbig
+D+ln(2κ)+4√
+2Z
+κ1/2−2Z2(lnκ+1)
+κ.
+(A5)
+The calculational scheme employed for the evaluation
+of Eq. (A3) is similar to that previously used [15] for the
+relativistic corrections to the Bethe logarithm. At the
+first step, a careful optimization of nonlinear basis-set
+parameters was carried out for a sequence of scales of the
+photon momentum: ki= 10iandi= 1,...,imax, with
+imax= 5 for the Sstates and imax= 4 for the Pstates.
+The optimization was performed with incrementing the
+size of the basis until the prescribed level of convergence
+is achieved for the function w(k). At the second step, the
+integrations of the photon momentum kwere performed.
+For a given value of k, the function J(k) was calculated
+with a basis obtained by merging together the optimized
+bases for the two closest kipoints, thus essentially dou-
+bling the number of the basis functions. The function
+w(k) was obtained from J(k) according to Eq. (A4).
+The integral over k∈[0,κ] was calculated analytically,
+by diagonalizing the Hamiltonian matrix and using the
+spectral representation of the propagator. The value of
+the auxiliarparameter κwasset to κ= 100. The integral
+overk∈[κ,∞) was separated into two parts, k <10imax
+andk >10imax. The first part was evaluated numeri-
+cally by using the Gauss-Legendre quadratures, after the
+change of variables t= 1/k2. The second part was ob-
+tained by integrating the asymptotic expansion of the
+function w(k). The coefficients of this expansion were
+obtained by fitting the numerical data for w(k) to the
+form
+w(k) = pol/parenleftbigg1√
+k/parenrightbigg
++lnk
+kpol/parenleftbigg1
+k/parenrightbigg
+,(A6)
+where pol( x) denotes a polynomial of x. The total num-
+ber of fitting parameters was about 9 −11. The range of
+kto be fitted and the exact form of the fitting function
+were optimized so as to yield the best possible results for
+the known asymptotic expansion terms of J(k).
+The first-order perturbation of the Bethe logarithm by
+the mass-polarizationoperator can be represented as [10]
+ln(k0)M=m
+M/bracketleftBigg
+RM(κ)+1
+D/integraldisplayκ
+0dkkJM(k)
++/integraldisplay∞
+κdkwM(k)
+k2/bracketrightBigg
+, (A7)
+where
+JM(k) = 2/angbracketleftbigg
+φ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vector∇1
+E0−H0−k/vector∇/vextendsingle/vextendsingle/vextendsingle/vextendsingleδφ/angbracketrightbigg
++/angbracketleftbigg
+φ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vector∇1
+E0−H0−k/bracketleftbig
+δE−/vector p1·/vector p2/bracketrightbig1
+E0−H0−k/vector∇/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ/angbracketrightbigg
+,
+(A8)9
+andδE=/angbracketleftbig
+/vector p1·/vector p2/angbracketrightbig
+. The perturbed wave function δφis
+defined by
+|δφ/angbracketrightbig
+=|δMφ/angbracketrightbig
+−|φ/angbracketrightbigδMD
+D, (A9)
+whereδMstands for the first-order perturbation induced
+by the operator /vector p1·/vector p2. The asymptotic expansion of
+JM(k) is much simpler than that of J(k) andwM(k) is
+just
+wM(k) =k3
+DJM(k)+k2
+D2/angbracketleftbig
+φ|/vector∇2|δφ/angbracketrightbig
+.(A10)
+Correspondingly, the function RM(κ) is
+RM(κ) =2κ
+D/angbracketleftbig
+φ|/vector∇2|δφ/angbracketrightbig
+. (A11)
+The numerical evaluation of Eq. (A7) was performed in
+a way similar to that for the plain Bethe logarithm. In
+particular, the same sets of optimized nonlinear param-
+eters were used. Since a high accuracy is not needed
+for this correction, a somewhat simplified calculational
+scheme was used in this case. The high-energy part of
+the photon-momentum integral, k∈[100,∞), was eval-
+uated by integrating the fitted asymptotic expansion for
+wM(k), which wastaken to be of the form (A6) with 6 −9
+fitting parameters.
+Appendix B: Expectation value of 1 /r3
+The definition of the expectation value of the regular-
+ized operator 1 /r3is given by Eq. (12). With the basis-
+set representation of the wave function employed in this
+work, a typical singular integral to be calculated is
+Iǫ=1
+16π2/integraldisplay
+d3r1d3r2exp(−αr1−βr2−γr)
+r3Θ(r−ǫ).
+(B1)
+The straightforward way is to evaluate this integral an-
+alytically for a finite value of the regulator ǫand thenexpand it in small ǫ. This way is possible, but we prefer
+to use a simpler procedure, which is also the closest to
+the method of evaluation of the regular integrals.
+We recall that all regularintegralsare immediately ob-
+tained from the master integral
+1
+16π2/integraldisplay
+d3r1d3r2exp(−αr1−βr2−γr)
+r1r2r
+=1
+(α+β)(β+γ)(α+γ)(B2)
+by formal differentiation or integration with respect to
+the corresponding parameters. The differentiation over
+αandβand an integration over γdelivers a result for
+the integral of the type 1 /r2. This integral is conver-
+gent, so the result is exact. The second integration over
+γ(which would yield an integral of the type 1 /r3) is di-
+vergent. The simplest way to proceed is as follows. We
+introduce a cutoff parameter for large values of γ, eval-
+uate the integral over γ, and drop all cutoff-dependent
+terms. The expression obtained in this way differs from
+the correct one by a γ-independent constant only, which
+can be proved by differentiating with respect to γ.
+The missing constant is most easily recovered by con-
+sidering the behavior of the integral Iwhenγ→ ∞. For
+very large γ, only the region of very small rcontributes
+and we have
+Iǫ= 2/integraldisplay∞
+ǫdrr/integraldisplay∞
+0dr1r1/integraldisplayr1+r
+|r1−r|dr2r2e−αr1−βr2−γr
+r3
+≈2
+(α+β)3/integraldisplay∞
+ǫdre−γr
+r. (B3)
+Therefore,
+Ireg≡lim
+ǫ→0/bracketleftbig
+Iǫ+γ+lnǫ/bracketrightbigγ→∞=−2
+(α+β)3lnγ.(B4)
+The above equation yields the necessary condition for
+determining the missing constant term in the general ex-
+pression for the regularized integral Ireg.
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+0 10 20 30 40 50 60 70 80 90 100 -0.60 -0.55 -0.50 -0.45 -0.40    
+23S
+0 10 20 30 40 50 60 70 80 90 100 -4.4 -4.2 -4.0 -3.8 -3.6 -3.4 -3.2 -3.0 
+G(Z) 
+Z ZZ Z  
+11SG(Z) 
+Z0 10 20 30 40 50 60 70 80 90 100 -1.06 -1.04 -1.02 -1.00 -0.98 -0.96 -0.94    
+21S
+0 10 20 30 40 50 60 70 80 90 100 -1.0 -0.9 -0.8 -0.7 -0.6    
+23P0
+0 10 20 30 40 50 60 70 80 90 100 -0.16 -0.15 -0.14 -0.13    
+23P2
+FIG. 1: The higher-order remainder function G(Z) from Eq. (35) inferred from the all-order numerical result s of Ref. [8] for the
+two-electron QED correction, in comparison with the Z= 0 limit obtained by fitting the 1 /Zexpansion of the mα6correction
+calculated in this work (denoted by the cross on the yaxis). The all-order results for Zsmaller than 30 (in some cases, 20)
+were left out since their numerical accuracy turns out to be n ot high enough.
+Phys.80, 633 (2008).
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+(2005).
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+P. B. Schwinberg, in Trapped Charged Particles and Fun-
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+(1994).
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+207 (1994).
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+T8, 10 (1984).
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+Rev. A57, 180 (1998).11
+TABLE I: Nonrelativistic energies of helium-like ions E(2)
+∞,E(2)
+M, andE(2)
+M2. For the helium atom, the nonrelativistic energy E(2)
+∞
+is given in the text, see Eqs. (30)-(34). For E(2)
+∞andE(2)
+M, the results of fitting the numerical data to the form E=/summationtext
+ici/Ziare
+also presented. The 1 /Zexpansion of the second-order recoil correction E(2)
+M2was not studied since this correction is relevant
+for the light atoms only. Atomic units are used.
+Z 11S 21S 23S 21P 23P
+E(2)
+∞+Z2(1+1/n2)/2
+3 1.720086587330694 0 .584123254404560 0 .514272627429260 0 .631648922219983 0 .597284318602632
+4 2.344433761576413 0 .815126104651679 0 .702833410222385 0 .889228377083556 0 .825026856929027
+5 2.969028419757218 1 .046471967118562 0 .891102651185767 1 .147716734692201 1 .051862307786520
+6 3.593753398101470 1 .277982298711029 1 .079244097692043 1 .406667686611591 1 .278289303511949
+7 4.218554851227295 1 .509584284500004 1 .267318262508963 1 .665883599383315 1 .504498255070109
+8 4.843404877242074 1 .741242692371423 1 .455352679914990 1 .925264764124369 1 .730577283616668
+9 5.468287636040509 1 .972938360878370 1 .643361670481692 2 .184755723292205 1 .956572706521290
+10 6.093193484962451 2 .204659953768328 1 .831353415926627 2 .444323260421919 2 .182511179215097
+11 6.718116223927278 2 .436400325538931 2 .019332929131276 2 .703946297095311 2 .408409121058957
+12 7.343051687353070 2 .668154743908730 2 .207303452397112 2 .963610824902483 2 .634277196002167
+1/Zexpansion coefficients
+c−1 5/8 169 /729 137 /729 1705 /6561 1481 /6561
+c0 −0.15766643 −0.11451014 −0.04740930 −0.15702866 −0.07299898
+c1 0.00869903 0 .00932761 −0.00487228 0 .02610626 −0.01658530
+c2 −0.00088869 −0.00128499 −0.00345775 0 .00578246 −0.01035367
+c3 −0.00103659 0 .00619473 −0.00203070 −0.00503312 −0.00542743
+c4 −0.00061067 −0.00147194 −0.00127808 −0.00709902 −0.00200175
+c5 −0.00038813 −0.00377551 −0.00093477 −0.00110332 0 .00010074
+E(2)
+M/(Z2m/M)
+2 0 .76569846 0 .53886948 0 .54566788 0 .54247190 0 .51714794
+3 0 .84098769 0 .56250901 0 .56981061 0 .58272590 0 .52473209
+4 0 .87975541 0 .57617876 0 .58283044 0 .60826818 0 .52929961
+5 0 .90334897 0 .58499029 0 .59090914 0 .62520360 0 .53234914
+6 0 .91921060 0 .59112383 0 .59639930 0 .63711706 0 .53451627
+7 0 .93060334 0 .59563360 0 .60037003 0 .64591388 0 .53613048
+8 0 .93918163 0 .59908696 0 .60337429 0 .65266102 0 .53737739
+9 0 .94587329 0 .60181529 0 .60572619 0 .65799392 0 .53836863
+10 0 .95123885 0 .60402479 0 .60761717 0 .66231212 0 .53917509
+11 0 .95563686 0 .60585039 0 .60917053 0 .66587855 0 .53984379
+12 0 .95930733 0 .60738402 0 .61046921 0 .66887298 0 .54040711
+1/Zexpansion coefficients
+c0 1 5 /8 5 /8 5 /8+29/385/8−29/38
+c1 −0.4917065 −0.2196812 −0.1769924 −0.4222328 −0.0825631
+c2 0.0396517 0 .1003376 0 .0306844 0 .1337270 0 .0459236
+c3 0.0129725 −0.0098744 0 .0091545 0 .1623107 0 .0099837
+c4 −0.0000226 −0.0067446 0 .0038963 −0.0004453 −0.0079454
+c5 0.0030374 0 .0089491 0 .0018358 −0.0990485 −0.0113262
+E(2)
+M2/(Z2m2/M2)
+2 −0.92306775 −0.57506528 −0.56190254 −0.59605662 −0.55223802
+3 −1.01502787 −0.62326322 −0.59180340 −0.68774668 −0.57415642
+4 −1.06176241 −0.65148997 −0.60762664 −0.74238592 −0.58333575
+5 −1.09005134 −0.66968217 −0.61736445 −0.77689285 −0.58842534
+6 −1.10901461 −0.68231257 −0.62395164 −0.80035477 −0.59168777
+7 −1.12261079 −0.69157288 −0.62870172 −0.81726102 −0.59396708
+8 −1.13283581 −0.69864559 −0.63228826 −0.82999471 −0.59565331
+9 −1.14080512 −0.70422080 −0.63509175 −0.83991981 −0.59695297
+10 −1.14719096 −0.70872706 −0.63734321 −0.84786827 −0.59798612
+11 −1.15242261 −0.71244412 −0.63919098 −0.85437466 −0.59882753
+12 −1.15678703 −0.71556217 −0.64073467 −0.85979749 −0.5995262712
+TABLE II: The leading relativistic corrections E(4)
+∞andE(4)
+Mfor helium-like atoms and their 1 /Z-expansion coefficients. The
+analytical results for the coefficient c1forE(4)
+∞were taken from Ref. [29] for the 11S, 23S, 23P0, and 23P2states. For the other
+states, this coefficient was evaluated numerically to high ac curacy in this work by the same method as in Ref. [29]. The c0
+coefficient of E(4)
+Mfor theSstates originates from the one-electron recoil effect and is well known from the hydrogen theory.
+For thePstates, it contains also the two-electron contribution, wh ich was derived in Ref. [36]. The remaining 1 /Z-expansion
+coefficients were obtained by fitting the numerical data for E(4)
+∞andE(4)
+M. Atomic units are used.
+Z 11S 21S 23S 21P1 23P0 23P1 23P2
+E(4)
+∞/Z4
+2−0.12198467 −0.12713546 −0.13527987 −0.12750160 −0.11804252 −0.12331623 −0.12372958
+3−0.14579473 −0.13188123 −0.14284030 −0.13095391 −0.12112829 −0.12660186 −0.12441072
+4−0.16326353 −0.13629775 −0.14735668 −0.13322953 −0.12666774 −0.13051205 −0.12556923
+5−0.17604864 −0.13988298 −0.15031008 −0.13478660 −0.13152271 −0.13370851 −0.12655782
+6−0.18567419 −0.14273756 −0.15238324 −0.13591790 −0.13544497 −0.13621654 −0.12734275
+7−0.19313821 −0.14502897 −0.15391610 −0.13677846 −0.13859636 −0.13819925 −0.12796582
+8−0.19907787 −0.14689544 −0.15509463 −0.13745597 −0.14115638 −0.13979329 −0.12846748
+9−0.20390905 −0.14843923 −0.15602856 −0.13800371 −0.14326617 −0.14109757 −0.12887810
+10−0.20791170 −0.14973447 −0.15678671 −0.13845596 −0.14502986 −0.14218211 −0.12921951
+11−0.21128006 −0.15083516 −0.15741435 −0.13883582 −0.14652358 −0.14309691 −0.12950741
+12−0.21415265 −0.15178123 −0.15794244 −0.13915946 −0.14780354 −0.14387826 −0.12975321
+1/Zexpansion coefficients
+c0−1/4 −21/128 −21/128 −55/384 −21/128 −59/384 −17/128
+c10.48013961 0 .16947818 0 .07693523 0 .05540303 0 .21976822 0 .13042876 0 .04063872
+c2−0.63650686 −0.28185862 −0.04277547 −0.09063215 −0.30352335 −0.16212941 −0.04731568
+c30.45631423 0 .20291921 0 .01047395 0 .15641239 0 .09174625 0 .04246890 0 .00224438
+c4−0.17117961 −0.04254210 −0.00446083 −0.17804253 −0.00884433 −0.00431944 −0.00023651
+c50.01858749 0 .01886171 −0.00156673 0 .05906831 0 .01555282 0 .00769846 0 .00369105
+E(4)
+M/(Z4m/M)
+2−0.1349607 −0.0043516 0 .0055741 −0.0036553 0 .0155968 0 .0166771 0 .0127607
+3−0.1237592 −0.0016161 0 .0114269 −0.0085744 0 .0261482 0 .0268552 0 .0192485
+4−0.1076271 0 .0023039 0 .0152883 −0.0121396 0 .0326665 0 .0321855 0 .0216508
+5−0.0937841 0 .0057924 0 .0179334 −0.0144516 0 .0375802 0 .0357681 0 .0229265
+6−0.0826257 0 .0086722 0 .0198408 −0.0159704 0 .0414077 0 .0383887 0 .0237369
+7−0.0736470 0 .0110269 0 .0212763 −0.0170051 0 .0444505 0 .0403953 0 .0243047
+8−0.0663370 0 .0129659 0 .0223938 −0.0177366 0 .0469152 0 .0419814 0 .0247276
+9−0.0602991 0 .0145811 0 .0232876 −0.0182710 0 .0489462 0 .0432663 0 .0250561
+10−0.0552416 0 .0159430 0 .0240184 −0.0186728 0 .0506455 0 .0443280 0 .0253193
+11−0.0509505 0 .0171045 0 .0246268 −0.0189822 0 .0520864 0 .0452197 0 .0255351
+12−0.0472678 0 .0181055 0 .0251411 −0.0192256 0 .0533228 0 .0459791 0 .0257155
+1/Zexpansion coefficients
+c0 0 1 /32 1 /32 −0.0207447 0 .0692059 0 .0553920 0 .0277642
+c1−0.6450402 −0.1826434 −0.0784124 0 .0025830 −0.2171136 −0.1253972 −0.0256050
+c20.9723728 0 .3148003 0 .0628343 0 .2201046 0 .3323021 0 .1576070 0 .0154532
+c3−0.4600919 −0.1884436 −0.0188669 −0.3874951 −0.1520388 −0.0962265 −0.0366639
+c4−0.0403680 −0.0482282 0 .0004138 −0.0462824 −0.2528784 −0.0301604 −0.017642513
+TABLE III: Bethe logarithm for helium-like atoms with the in finite nuclear mass, ln( k0/Z2), and its first-order perturbation by
+the mass polarization operator, ln( k0)M. Coefficients of the 1 /Zexpansion of ln( k0/Z2) are also presented. The leading term
+c0is known with a high accuracy from the hydrogen theory. The hi gher-order coefficients are obtained by fitting the numerical
+data.
+Z 11S 21S 23S 21P 23P Ref.
+ln(k0/Z2)
+2 2.9838658618(1) 2 .9801183651(1) 2 .97774245929(2) 2 .9838033824(1) 2 .9836910033(2)
+2.9838658609(1) 2 .9801183648(1) 2 .9777424592(1) 2 .983803377(1) 2 .983690995(1) [25]
+2.983865857(3) 2 .98011836(7) 2 .97774246(1) 2 .98380346(3) 2 .98369084(2) [24]
+3 2.9826245630(2) 2 .9763630630(2) 2 .97385170992(4) 2 .9831860136(2) 2 .9829587982(2)
+4 2.9825030991(3) 2 .9739769112(3) 2 .97173557890(7) 2 .9826982138(4) 2 .9824435984(3)
+5 2.9825913761(4) 2 .9723880988(4) 2 .97042496490(8) 2 .9823401149(8) 2 .9820896049(4)
+6 2.982716948(1) 2 .9712662464(5) 2 .9695370719(3) 2 .982072719(2) 2 .9818359385(6)
+2.982716948(4) 2 .97126624(4) 2 .96953707(1) 2 .98207276(2) 2 .98183592(3) [24]
+7 2.982839085(3) 2 .970435367(1) 2 .968896814(1) 2 .981867337(7) 2 .981646451(2)
+8 2.982948318(4) 2 .969796528(3) 2 .968413645(2) 2 .98170533(1) 2 .981499939(4)
+9 2.983043667(8) 2 .969290586(5) 2 .968036227(5) 2 .98157456(3) 2 .981383443(7)
+10 2 .98312646(2) 2 .96888024(1) 2 .967733341(9) 2 .98146692(5) 2 .98128868(1)
+11 2 .98319850(3) 2 .96854085(2) 2 .96748493(2) 2 .9813769(1) 2 .98121012(2)
+12 2 .98326147(5) 2 .96825557(4) 2 .96727754(3) 2 .9813004(2) 2 .98114396(3)
+Coefficients of the 1 /Zexpansion:
+c02.98412856 2 .96497759 2 .96497759 2 .98037647 2 .98037647
+c1−0.01229928 0 .04078809 0 .02775943 0 .01200383 0 .00962797
+c20.02244974 −0.01643935 −0.00142395 −0.01098208 −0.00481060
+c30.00358619 −0.01235503 −0.00596856 −0.00048219 −0.00245788
+c4−0.00250370 0 .00581330 0 .00011995 0 .00376432 −0.00023670
+ln(k0)M/(m/M)
+2 0.0943894(1) 0 .0177344(1) 0 .00478554(1) −0.0035534(2) 0 .0087095(1)
+0.09438(1) 0 .017734(1) 0 .004784(3) −0.003538(6) 0 .008701(4) [24]
+3 0.1095397(1) 0 .0342103(1) 0 .00785251(1) −0.0066023(2) 0 .0163283(1)
+4 0.1169197(1) 0 .0448768(1) 0 .00961661(1) −0.0079512(2) 0 .0201992(1)
+5 0.1213045(2) 0 .0520124(2) 0 .01075420(1) −0.0086295(2) 0 .0224790(1)
+6 0.1242129(3) 0 .0570530(3) 0 .01154792(1) −0.0090153(2) 0 .0239736(1)
+0.12421(1) 0 .057051(1) 0 .011541(1) −0.00898(1) 0 .02398(1) [24]
+7 0.1262831(4) 0 .0607830(9) 0 .01213320(1) −0.0092556(2) 0 .0250273(1)
+8 0.1278336(5) 0 .0636470(7) 0 .01258268(1) −0.0094160(2) 0 .0258095(2)
+9 0.1290371(2) 0 .065912(1) 0 .01293876(1) −0.0095287(2) 0 .0264128(3)
+10 0 .1299989(2) 0 .067746(1) 0 .01322787(1) −0.0096112(2) 0 .0268922(4)
+11 0 .1307851(2) 0 .069261(1) 0 .01346730(1) −0.0096736(2) 0 .0272822(4)
+12 0 .1314397(2) 0 .070534(2) 0 .01366886(1) −0.0097222(3) 0 .0276057(4)14
+TABLE IV: The leading logarithmic QED corrections E(5)
+∞(log) and E(5)
+M(log). For the non-recoil correction, we present the
+coefficients of the 1 /Zexpansion obtained by fitting the numerical data (except for c0which is known analytically). The recoil
+correction is very small for ions with Z >12, so its 1 /Zexpansion was not studied. Atomic units are used.
+Z 11S 21S 23S 21P 23P
+E(5)
+∞(log)/[Z4ln(Zα)−2]
+2 0 .587967740 0 .435225697 0 .440118361 0 .424690417 0 .419620202
+3 0 .661366949 0 .444453845 0 .450745298 0 .425056974 0 .418277856
+4 0 .702709966 0 .450600513 0 .456788812 0 .425087644 0 .418654616
+5 0 .729153414 0 .454883655 0 .460628756 0 .425031723 0 .419253068
+6 0 .747506286 0 .458013834 0 .463274004 0 .424963548 0 .419808283
+7 0 .760984320 0 .460392952 0 .465204144 0 .424901229 0 .420280937
+8 0 .771300325 0 .462259020 0 .466673600 0 .424848038 0 .420676712
+9 0 .779449465 0 .463760328 0 .467829292 0 .424803420 0 .421008829
+10 0 .786049205 0 .464993553 0 .468761819 0 .424766013 0 .421289730
+11 0 .791502875 0 .466024214 0 .469530027 0 .424734472 0 .421529561
+12 0 .796085045 0 .466898202 0 .470173774 0 .424707664 0 .421736263
+1/Zexpansion coefficients
+c0 8/(3π) 3 /(2π) 3 /(2π) 4 /(3π) 4 /(3π)
+c1 −0.65955048 −0.13774461 −0.08975644 0 .00315846 −0.03647876
+c2 0.33058616 0 .13645203 0 .02669363 0 .00911722 0 .05136286
+c3 −0.13276816 −0.06129750 0 .00548816 −0.06123169 0 .01113217
+c4 0.04855042 −0.00048558 0 .00197936 0 .07036756 −0.00192441
+c5 −0.00583476 −0.00041968 0 .00108209 0 .00299402 −0.01422578
+E(5)
+M(log)/[(m/M)Z4ln(Zα)−2]
+2 −1.4907878 −1.0878273 −1.0995716 −1.0480334 −1.0729421
+3 −1.5030007 −0.9998200 −1.0135256 −0.9310877 −0.9729291
+4 −1.4148151 −0.9007682 −0.9130235 −0.8203695 −0.8695494
+5 −1.2810385 −0.7952958 −0.8056243 −0.7123589 −0.7652223
+6 −1.1228525 −0.6859536 −0.6944957 −0.6055099 −0.6604148
+7 −0.9501011 −0.5741357 −0.5811375 −0.4991858 −0.5553058
+8 −0.7679688 −0.4606488 −0.4663441 −0.3931081 −0.4499907
+9 −0.5794430 −0.3459870 −0.3505736 −0.2871469 −0.3445275
+10 −0.3863652 −0.2304684 −0.2341084 −0.1812384 −0.2389542
+11 −0.1899321 −0.1143064 −0.1171318 −0.0753503 −0.1332967
+12 0 .0090452 0 .0023507 0 .0002321 0 .0305341 −0.027573615
+TABLE V: The leading nonlogarithmic QED corrections E(5)
+∞(nlog) and E(5)
+M(nlog). For the non-recoil part, we present the
+coefficients of the 1 /Zexpansion. The coefficient c0is known with a very good accuracy from the hydrogen theory. T he
+remaining coefficients were obtained by numerical fitting. Th e radiative recoil correction is very small for ions with Z >12, so
+its 1/Zexpansion was not studied. Atomic units are used.
+Z 11S 21S 23S 21P1 23P0 23P1 23P2
+E(5)
+∞(nlog)/Z4
+2−1.3902824 −1.0217567 −1.0327195 −0.9991046 −0.9864875 −0.9878248 −0.9872736
+3−1.5524234 −1.0407339 −1.0558743 −1.0000288 −0.9840145 −0.9852856 −0.9836457
+4−1.6458291 −1.0534943 −1.0689514 −0.9998389 −0.9855575 −0.9863178 −0.9838177
+5−1.7066684 −1.0625513 −1.0772303 −0.9994275 −0.9874803 −0.9877485 −0.9846170
+6−1.7494645 −1.0692748 −1.0829206 −0.9990285 −0.9891847 −0.9890409 −0.9854376
+7−1.7812119 −1.0744484 −1.0870662 −0.9986875 −0.9906094 −0.9901284 −0.9861621
+8−1.8057010 −1.0785460 −1.0902187 −0.9984047 −0.9917910 −0.9910330 −0.9867797
+9−1.8251658 −1.0818684 −1.0926959 −0.9981707 −0.9927768 −0.9917888 −0.9873036
+10−1.8410087 −1.0846148 −1.0946934 −0.9979759 −0.9936076 −0.9924260 −0.9877496
+11−1.8541545 −1.0869222 −1.0963380 −0.9978122 −0.9943150 −0.9929688 −0.9881323
+12−1.8652378 −1.0888875 −1.0977156 −0.9976733 −0.9949235 −0.9934358 −0.9884632
+1/Zexpansion coefficients
+c0−1.9954170 −1.1132781 −1.1132781 −0.9961160 −1.0027475 −0.9994318 −0.9928003
+c11.6588160 0 .3255170 0 .1911475 −0.0175598 0 .1062102 0 .0811929 0 .0594525
+c2−1.2262714 −0.4204560 −0.0516033 −0.0318538 −0.1465372 −0.1085003 −0.0858934
+c30.8258435 0 .3273250 −0.0131945 0 .2472567 −0.0166285 −0.0295276 −0.0367739
+c4−0.3730623 −0.1146272 −0.0089491 −0.3349535 0 .0068462 0 .0082013 0 .0094312
+c50.0881600 0 .0375752 0 .0109846 0 .0852990 0 .0182050 0 .0161412 0 .0155024
+E(5)
+M(nlog)/[(m/M)Z5]
+2 3.292520 2 .455583 2 .489805 2 .385181 2 .393644 2 .393984 2 .393698
+3 2.816621 1 .914328 1 .949969 1 .811294 1 .817918 1 .818182 1 .817751
+4 2.527383 1 .642421 1 .673220 1 .526466 1 .531293 1 .531426 1 .530930
+5 2.334128 1 .478228 1 .504411 1 .356337 1 .359930 1 .359969 1 .359460
+6 2.196193 1 .368117 1 .390605 1 .243181 1 .245918 1 .245898 1 .245400
+7 2.092908 1 .289066 1 .308659 1 .162444 1 .164573 1 .164516 1 .164039
+8 2.012719 1 .229525 1 .246828 1 .101918 1 .103606 1 .103526 1 .103074
+9 1.948679 1 .183048 1 .198512 1 .054847 1 .056206 1 .056113 1 .055685
+10 1.896366 1 .145753 1 .159714 1 .017186 1 .018295 1 .018194 1 .017790
+11 1.852835 1 .115160 1 .127873 0 .986367 0 .987283 0 .987177 0 .986794
+12 1.816049 1 .089607 1 .101272 0 .960678 0 .961441 0 .961334 0 .96097116
+TABLE VI: The mα6corrections E(6)
+∞(log) and E(6)
+∞(nlog) and their 1 /Zexpansion coefficients. Atomic units are used.
+Z 11S 21S 23S 21P1 23P0 23P1 23P2
+E(6)
+∞(log)/[Z3ln(Zα)−2]
+2 0 .020880865 0 .001698116 0 0 .000144350 0 0 0
+3 0 .031050719 0 .003738928 0 0 .000572351 0 0 0
+4 0 .037377475 0 .005250650 0 0 .001013004 0 0 0
+5 0 .041625375 0 .006344104 0 0 .001386910 0 0 0
+6 0 .044658932 0 .007157223 0 0 .001692317 0 0 0
+7 0 .046928852 0 .007781230 0 0 .001941610 0 0 0
+8 0 .048689367 0 .008273618 0 0 .002147092 0 0 0
+9 0 .050093825 0 .008671365 0 0 .002318555 0 0 0
+10 0 .051239918 0 .008999033 0 0 .002463398 0 0 0
+11 0 .052192721 0 .009273471 0 0 .002587159 0 0 0
+12 0 .052997208 0 .009506579 0 0 .002694007 0 0 0
+1/Zexpansion coefficients
+c0 1/16 1 /81 0 1 /243 0 0 0
+c1−0.1214680 −0.0371977 0 −0.0199971 0 0 0
+c20.0919637 0 .0387114 0 0 .0379957 0 0 0
+c3−0.0334009 −0.0142832 0 −0.0329951 0 0 0
+c40.0048044 0 .0036000 0 0 .0088653 0 0 0
+c5−0.0004874 −0.0062726 0 0 .0028677 0 0 0
+E(6)
+∞(nlog)/Z6
+2 2 .1812333(1) 1 .528981(2) 1 .5365931(1) 1 .489195(1) 1 .459456(1) 1 .460802(1) 1 .466251(1)
+3 1 .5824717(1) 1 .016337(1) 1 .0204405(1) 0 .977073(1) 0 .945154(1) 0 .937827(1) 0 .950303(1)
+4 1 .2244519(8) 0 .753467(1) 0 .7546608(1) 0 .723937(1) 0 .691625(1) 0 .676327(1) 0 .696280(1)
+5 0 .9891850(1) 0 .592746(1) 0 .5922385(1) 0 .574526(1) 0 .539381(1) 0 .517140(1) 0 .544513(1)
+6 0 .8233481(1) 0 .484094(1) 0 .4826262(4) 0 .477044(1) 0 .437493(1) 0 .408810(1) 0 .443440(1)
+7 0 .7003178(1) 0 .405658(1) 0 .4036503(1) 0 .409248(1) 0 .364415(1) 0 .329514(1) 0 .371247(1)
+8 0 .6054696(1) 0 .346341(1) 0 .3440352(1) 0 .360008(1) 0 .309399(1) 0 .268374(1) 0 .317082(1)
+9 0 .5301381(1) 0 .299896(1) 0 .2974360(1) 0 .323136(1) 0 .266468(1) 0 .219353(1) 0 .274935(1)
+10 0 .4688716(1) 0 .262537(1) 0 .2600086(1) 0 .294919(1) 0 .232024(1) 0 .178826(1) 0 .241202(1)
+11 0 .4180725(6) 0 .231831(1) 0 .2292871(1) 0 .272996(1) 0 .203772(2) 0 .144482(4) 0 .213589(1)
+12 0 .3752721(1) 0 .206144(1) 0 .2036170(4) 0 .255791(1) 0 .180178(2) 0 .114783(4) 0 .190568(1)
+1/Zexpansion coefficients
+c−1 0 0 0 729 /114688 0 −729/114688 0
+c0−1/8 −85/1024 −85/1024 −0.0792398 −85/1024 −0.0672446 −65/1024
+c16.3428979 3 .5496121 3 .4875483 3 .09980 3 .2045132 3 .11905 3 .0597402
+c2−4.2619 −1.0426 −0.5900 0 .0686 −0.6236 −0.2609 −0.1578
+c32.4241 1 .0870 0 .1557 −0.0390 0 .8334 0 .3289 0 .2714
+c4−2.4088 −0.8157 0 .0539 −0.4013 −0.2105 0 .0717 0 .101017
+TABLE VII: The finite nuclear size correction Efs(the used values of the root-mean-square nuclear charge rad ii are listed
+in Table VIII) and the higher-order correction E(7+)≡ E(7+)
+rad+E(7+)
+nrad. Contributions to the ionization energy are presented.
+Numerical values of the finite nuclear size correction are sc aled by the same factor as for the higher-order correction, i n order
+to simplify the comparison between them.
+Z 11S 21S 23S 21P1 23P0 23P1 23P2
+Efs/[mα7Z6]
+2 3 .41(1) 0 .2346(5) 0 .3039(7) 0 .00735(2) −0.0914(3) −0.0914(3) −0.0914(3)
+3 4 .5(1) 0 .393(8) 0 .47(1) 0 .0166(4) −0.110(3) −0.110(3) −0.110(3)
+4 3 .16(3) 0 .305(3) 0 .351(3) 0 .0114(1) −0.0623(6) −0.0622(6) −0.0623(6)
+5 2 .01(5) 0 .206(5) 0 .230(6) 0 .0066(2) −0.0327(8) −0.0327(8) −0.0327(8)
+6 1 .560(4) 0 .1655(5) 0 .1818(5) 0 .00455(1) −0.02150(6) −0.02144(6) −0.02150(6)
+7 1 .281(8) 0 .1396(8) 0 .1512(9) 0 .00336(2) −0.01528(9) −0.01521(9) −0.01528(9)
+8 1 .130(6) 0 .1256(7) 0 .1347(7) 0 .00268(1) −0.01187(6) −0.01180(6) −0.01187(6)
+9 1 .056(5) 0 .1191(6) 0 .1268(6) 0 .00229(1) −0.00990(5) −0.00981(5) −0.00990(5)
+10 0 .942(5) 0 .1076(6) 0 .1138(6) 0 .00188(1) −0.00797(4) −0.00788(4) −0.00797(4)
+11 0 .789(5) 0 .0911(6) 0 .0958(6) 0 .00146(1) −0.00608(4) −0.00599(4) −0.00608(4)
+12 0 .705(5) 0 .0821(6) 0 .0860(7) 0 .001219(9) −0.00499(4) −0.00490(4) −0.00499(4)
+E(7+)/Z6
+2−8.2(4.1)−0.43(22) −0.59(30) 0 .093(46) 0 .37(18) 0 .35(17) 0 .31(15)
+3−9.5(3.2)−0.72(24) −0.89(30) 0 .063(21) 0 .37(12) 0 .35(12) 0 .31(10)
+4−9.6(2.4)−0.83(21) −0.97(24) 0 .055(14) 0 .312(78) 0 .294(73) 0 .261(65)
+5−9.4(1.9)−0.86(17) −0.98(20) 0 .052(10) 0 .266(53) 0 .249(50) 0 .219(44)
+6−9.0(1.5)−0.87(14) −0.96(16) 0 .0510(85) 0 .230(38) 0 .214(36) 0 .187(31)
+7−8.7(1.2)−0.86(12) −0.94(13) 0 .0500(72) 0 .202(29) 0 .188(27) 0 .162(23)
+8−8.3(1.0)−0.84(11) −0.91(11) 0 .0490(62) 0 .181(23) 0 .167(21) 0 .144(18)
+9−8.00(89) −0.825(91) −0.886(98) 0 .0477(54) 0 .163(18) 0 .150(17) 0 .129(14)
+10−7.70(77) −0.806(80) −0.859(85) 0 .0463(48) 0 .148(15) 0 .136(14) 0 .116(12)
+11−7.43(67) −0.786(71) −0.834(75) 0 .0450(43) 0 .135(13) 0 .124(12) 0 .1066(97)
+12−7.18(60) −0.768(63) −0.811(67) 0 .0435(39) 0 .124(11) 0 .1132(99) 0 .0984(82)
+TABLE VIII: Total theoretical ionization energies of n= 1 and n= 2 states in light helium-like ions, in cm−1.Ais the nuclear
+mass number and Rchis the root-mean-square nuclear charge radius.
+Z A R ch[fm] 11S 21S 23S
+2 4 1 .676(3) −198310.6651(12) −32033.228734(63) −38454.694593(86)
+3 7 2 .43(3) −610078.549(11) −118704.79971(79) −134044.25696(98)
+4 9 2 .52(1) −1241256 .625(45) −260064.3427(38) −284740.7858(45)
+5 11 2 .41(3) −2091995 .58(13) −456261.994(12) −490434.928(14)
+6 12 2 .470(2) −3162423 .60(32) −707370.691(31) −751130.806(34)
+7 14 2 .558(7) −4452723 .93(66) −1013458 .475(66) −1066874 .012(72)
+8 16 2 .701(6) −5963074 .2(1.2) −1374588 .68(13) −1437720 .48(14)
+9 19 2 .898(2) −7693708 .5(2.1) −1790837 .48(22) −1863745 .75(24)
+10 20 3 .005(2) −9644843 .7(3.5) −2262278 .68(36) −2345025 .61(39)
+11 23 2 .994(2) −11816821 .8(5.4) −2789016 .99(57) −2881669 .22(60)
+12 24 3 .057(2) −14209915 .2(8.1) −3371143 .82(86) −3473772 .54(90)
+Z 21P1 23P0 23P1 23P2
+2 −27175.771929(13) −29222.838110(54) −29223.826028(51) −29223.902466(45)
+3 −108270.881302(69) −115812.95489(40) −115818.14868(38) −115816.05808(34)
+4 −243787.56765(25) −257876.1744(14) −257887.7324(14) −257872.8408(12)
+5 −434000.42129(74) −455041.3003(37) −455057.4990(35) −455004.8400(31)
+6 −679022.7617(18) −707232.0192(81) −707244.5283(76) −707108.7312(66)
+7 −978929.1185(39) −1014453 .504(15) −1014444 .829(14) −1014153 .830(12)
+8 −1333765 .9887(74) −1376741 .630(27) −1376682 .831(25) −1376131 .273(22)
+9 −1743582 .370(13) −1794154 .597(44) −1794003 .382(41) −1793045 .585(35)
+10 −2208407 .741(22) −2266761 .636(70) −2266460 .989(65) −2264903 .347(54)
+11 −2728303 .815(35) −2794653 .26(11) −2794130 .721(97) −2791723 .350(82)
+12 −3303295 .620(52) −3377923 .51(15) −3377091 .31(14) −3373518 .82(12)18
+TABLE IX: Comparison of theoretical and experimental trans ition energies. Units are MHz for He and Li+and cm−1for other
+ions. Results by Drake are from 2005 for He [31], from 1994 for Li+[37], and from 1988 for other ions [5].
+Z This work Drake Experiment Reference
+23P0–23S1transition:
+2 276764094.7(3.0) 276764099(17) 276764094.6788(21) [1]
+3 546560686(32) 546560627 546560683.07(42) [37]
+4 26864.6114(47) 26864.64(3) 26864.6120(4) [38]
+5 35393.628(14) 35393.70(8) 35393.627(13) [39]
+8 60978.85(14) 60979.6(5) 60978.44(52) [40]
+10 78263.98(39) 78265.9(1.2) 78265.0(1.2) [40]
+23P1–23S1transition:
+2 276734477.7(3.0) 276734476(17) 276764477.7242(20) [1]
+3 546404980(31) 546404885 546404978.80(51) [37]
+4 26853.0534(47) 26852.04(3) 26853.0534(3) [38]
+5 35377.429(14) 35377.40(8) 35377.424(13) [39]
+8 61037.65(14) 61037.7(5) 61037.62(93) [40]
+23P2–23S1transition:
+2 276732186.1(2.9) 276732183(17) 276732186.593(15) [1]
+3 546467655(31) 546467553 546467657.21(44) [37]
+4 26867.9450(47) 26867.92(3) 26867.9484(3) [38]
+5 35430.088(14) 35430.02(8) 35430.084(9) [39]
+8 61589.21(14) 61589.0(5) 61589.70(53) [40]
+10 80122.3(4) 80121.6(1.2) 80121.53(64) [41]
+21P1–21S0transition:
+4 16276.775(4) 16276.77(3) 16276.774(9) [42]
+23P1–21S0transition:
+7 986.36(7) 986.6(3) 986.3180(7) [43]
\ No newline at end of file
diff --git a/1001.0407.txt b/1001.0407.txt
new file mode 100644
index 0000000000000000000000000000000000000000..dc72c4735348309b2fcc90f907499a7dfa3af83e
--- /dev/null
+++ b/1001.0407.txt
@@ -0,0 +1,1036 @@
+arXiv:1001.0407v1  [math-ph]  3 Jan 2010Smoluсhowski problem for degenerate Bose gases
+Anatoly V. Latyshev and Alexander A. Yushkanov
+Department of Mathematical Analysis and Department of Theo retical Physics,
+Moscow State Regional University, 105005, Moscow, Radio st ., 10–A
+(Dated: 2 ноября 2018 г.)
+We construct a kinetic equation simulating the behavior of d egenerate quantum Bose gases with
+the collision rate proportional to the molecule velocity. W e obtain an analytic solution of the half–
+space boundary–value Smoluchowski problem of the temperat ure jump at the interface between the
+degenerate Bose gas and the condensed phase.
+Keywords : degenerate quantum Bose gas, Bose /emdash.cyr Einstein condensate, c ollision integral,
+temperature jump, Kapitsa resistance.
+PACS numbers: 05.30.-d, 05.70.Ce, 65.80.+n, 82.60.Qr
+1. INTRODUCTION
+In recent years, the behavior of quantum gases
+has increased interest. In particular, this is related
+to the development of experimental procedure for
+producing and studying quantum gases at extremely
+low temperatures [1]. In the majority of papers, bulk
+properties of quantum gases have been studied [2] and
+[3]. At the same time, it is obvious that it is important to
+take boundary effects on the properties of such systems
+into account. In particular, such a phenomenon as the
+temperature jump at the interface between a gas and
+a condensed (in particular, solid) body in the presence
+of a heat flux normal to the surface is important. Such
+a temperature jump is frequently called the Kapitsa
+temperature jump [4]. We also mention a paper where
+the problem of boundary conditions for the motion of
+superliquids was considered [5].
+The problem of a temperature jump in a quantum
+Fermi gas was studied in [6], where an analytic solution
+for an arbitrary degree of gas degeneracy was obtained.
+A similar problem for a Bose gas was considered in
+[7], where the gas was assumed to be nondegenerate,
+i.e., it was assumed that there was no Bose /emdash.cyr Einstein
+condensate.
+The present paper is devoted to solving the problem of
+a temperature jump in a degenerate Bose gas analytically.
+The presence of a Bose /emdash.cyr Einstein condensate [8] leads
+to a considerable modification of both the problem
+statement and its solution method. In this case, a kinetic
+equation with a model collision integral is used to
+describe kinetic processes. We assume that the boundaryconditions at the surface are purely diffusive.
+2. KINETIC EQUATION
+To describe the gas behavior, we use a kinetic equation
+with a model collision integral analogous to that used to
+describe a classical gas. In this case, we take into account
+the quantum character of the Bose gas and the presence
+of the Bose /emdash.cyr Einstein condensate.
+For a rarefied Bose gas, the evolution of the molecule
+distribution function fcan be described by the kinetic
+equation
+∂f
+∂t+∂E
+∂p∇f=I[f],
+whereEis the kinetic energy of molecules, pis the
+molecule momentum, and I[f]is the collision integral.
+In the case of the kinetic description of a degenerate
+Bose gas, we must take into account that the properties of
+the Bose /emdash.cyr Einstein condensate can change as functions
+of the space and time coordinates. In other words, we
+must consider a two–liquid model (more precisely, a two–
+fluid model, because we consider a gas rather than a
+liquid). We let ρc=ρc(r,t)andu=u(r,t)denote the
+respective density and velocity of the Bose condensate /emdash.cyr
+Einstein. We then have the expressions [9]
+j=ρcu,Q=ρcu2
+2u,
+Πik=ρcuiuk.
+for the densities jandQof the respective mass and
+energy fluxes and for the momentum flux tensor Πikof2
+the Bose condensate /emdash.cyr Einstein (under the assumption
+that the chemical potential is zero).
+The conservation laws for the number of particles,
+energy, and momentum require that the relations
+∂ρc
+∂t+∇j=−/integraldisplay
+I[f]dΩB,
+∂Ec
+∂t+∇Q=−/integraldisplayp2
+2mI[f]dΩB,
+∂(ρcu)
+∂t+∇Π =−/integraldisplay
+pI[f]dΩB
+be satisfied, where sis the molecule spin,
+dΩB=(2s+1)d3p
+(2π/planckover2pi1)3, E c=ρcu2
+2,
+where/planckover2pi1is the Planck constant.
+In what follows, we are interested in the case of
+stationary motion with small velocities (compared with
+the thermal velocities). We note that for the Bose
+condensate, the quantities QandΠik, are depend
+nonlinearly on the velocity (they are proportional to
+the respective third and second powers of the velocity).
+Therefore, in the approximation linear in the velocity
+u, the energy and momentum conservation laws can be
+written as
+/integraldisplayp2
+2mI[f]dΩB
+and
+/integraldisplay
+pI[f]dΩB= 0.
+According to the Bogolyubov theory, the relation for
+the excitation energy ε(p)[8]
+ε(p) =/bracketleftBigg
+u2p2+/parenleftbiggp2
+2m/parenrightbigg2/bracketrightBigg1/2
+, (1)
+where
+u=/parenleftbigg4π/planckover2pi12an
+m2/parenrightbigg1/2
+.
+Hereais the scattering length for gas molecules,
+nis the concentration, mis the mass, and pis the
+momentum of the gas molecule, holds for a weakly
+interacting Bose gas. The parameter acharacterizes the
+interaction force of gas molecules and can be assumed to
+be small for a weakly interacting gas.The relation u2≪kT/m , wherekis the Boltzmann
+constant and Tis the gas temperature, holds for
+sufficiently small a. In this case, we can neglect the first
+term in the brackets in (1). The expression for the energy
+E(p)takes the same form as in the case of noninteracting
+molecules:
+E(p) =p2
+2m.
+We now consider the widely used kinetic equation
+in the Boltzmann /emdash.cyr Krook /emdash.cyr Welander form with
+the molecule collision rate proportional to the molecule
+velocity [6, 7, 10]
+∂f
+∂t+(v∇)f=ν0w(f∗
+M−f). (2)
+Here,fis the distribution function, vis the molecule
+velocity, w=|v−v0|,v0is the mean–mass gas velocity,
+f∗
+Mis the Maxwell distribution function,
+f∗
+M=n∗/parenleftbiggm
+2πkT∗/parenrightbigg3/2
+exp/bracketleftbigg
+−m
+2kT∗(v−u∗)2/bracketrightbigg
+,
+andν0is a model parameter corresponding to the inverse
+mean free path lof a molecule, ν0∼1/l.
+The parameters in the formula for f∗
+M, namely, n∗,
+T∗andu∗, can be determined from the conservation
+laws for the number of molecules, momentum, and energy
+/integraldisplay
+wf dΩM=/integraldisplay
+wf∗
+MdΩM, (3a)
+/integraldisplay
+wvf dΩM=/integraldisplay
+wvf∗
+MdΩM, (3b)
+/integraldisplay
+wm
+2(v−u)2f dΩM=/integraldisplay
+wm
+2(v−u)2f∗
+MdΩM,(3c)
+where
+dΩM=d3v.
+The conservation law for the number of particles in
+the normal state is inapplicable because the transition of
+particles to the Bose /emdash.cyr Einstein condensate can occur.
+As mentioned above, the effect of the condensate on
+the energy and momentum conservation laws can be
+neglected in the approximation linear in u.
+We note that Eq. (2) corresponds to the assumption
+that the mean free path of molecules is constant (it is
+independent of molecule velocities). It hence follows that3
+Eq. (2) to a greater extent corresponds to the model in
+which the molecules are regarded as solid spheres.
+We consider a generalization of Eq. (2) to the case
+of a degenerate Bose gas. We assume that the general
+structure of Eq. (2) is preserved, but by a function f∗
+M,
+we must mean the Bose /emdash.cyr Einstein distribution (Bosean)
+with a zero chemical potential [8]
+f∗
+B=/bracketleftbigg
+exp/parenleftbiggm
+2kT∗(v−u∗)2/parenrightbigg
+−1/bracketrightbigg−1
+.
+Here, the parameters T∗andu∗, are determined by
+the second and third conditions in (3). In this case, we
+have
+dΩB=2s+1
+(2π/planckover2pi1)3d3p
+insteaddΩM.
+We assume that the mass velocity of the gas is much
+less than the mean thermal velocity of the molecules
+and the typical temperature variations along the mean
+free path of molecules are small compared with the gas
+temperature. The problem can be linearized under these
+assumptions.
+We seek the distribution function in the form
+f=fs
+B(v)+ϕ(t,r,v)g(v),
+where
+fs
+B(v) =1
+exp(βsv2)−1, β s=m
+2kTs,
+ϕis a new unknown function, Tsis the surface
+temperature, and
+g(v) =−∂
+∂εsfs
+B, ε s=βsv2.
+We introduce the notation
+C=/radicalbig
+βsv, ε ∗=m
+2kT∗(v−u∗)2.
+Taking this notation into account, we have
+f∗
+B(ε∗) =1
+exp(ε∗)−1,
+fs
+B(C) =1
+exp(C2)−1,
+g(C) =exp(C2)
+(exp(C2)−1)2.We linearize the local Bose /emdash.cyr Einstein distribution f∗
+B,
+passing to dimensionless quantities. We note that
+ε∗=Ts
+T∗/bracketleftbig
+βs(v−u∗)2/bracketrightbig
+=Ts
+T∗/bracketleftbig
+(C−W)2/bracketrightbig
+,
+where
+W=/radicalbig
+βsu∗.
+TakingT∗=Ts+δTsinto account, we obtain
+ε∗=C2−δT∗
+TsC2−2CW,
+whence we find
+δε∗=−2CW−C2δT∗
+Ts,
+where
+δε∗=ε∗−εs, ε s=C2.
+Consequently,
+f∗
+B=fs
+B+/parenleftbigg∂f∗
+B
+∂ε∗/parenrightbigg
+ε∗=εsδε∗,
+or
+f∗
+B=fs
+B+g(C)/bracketleftbigg
+2CW+C2δT∗
+Ts/bracketrightbigg
+.
+We note that the quantity win Eq. (2) can be replaced
+withvin the approximation under consideration. We
+introduce the dimensionless quantities t∗=tν0/βsand
+r∗=rν0and omit the asterisks on these quantities
+below. It is now clear that Eq. (2) (in the dimensionless
+variables) becomes
+∂ϕ
+∂t+(C∇)ϕ= 2C(CW)+C3δT∗
+Ts−Cϕ.(4)
+The parameters of this equation can be found from the
+momentum and energy conservation laws (relations (3)),
+which now become
+/integraldisplay
+(Lϕ)C2g(C)d3C= 0,
+/integraldisplay
+(Lϕ)Cg(C)d3C= 0,
+where
+Lϕ= 2CCW+C3δT∗
+Ts−Cϕ.4
+From this system, we find
+W=3
+8πg0/integraldisplay
+ϕCg(C)Cd3C,
+δT∗
+Ts=1
+4πg2/integraldisplay
+ϕg(C)C3d3C,
+where
+gn=∞/integraldisplay
+0C5+ng(C)dC, n = 0,1,2,
+In this case, we have
+g0=∞/integraldisplay
+0g(C)C5dC=∞/integraldisplay
+0exp(C2)C5dC
+/parenleftBig
+exp(C2)−1/parenrightBig2=
+=−2∞/integraldisplay
+0Cln(1−exp(−C2))dC=π2
+6= 1.64493,
+g1=∞/integraldisplay
+0g(C)C6dC= 2.22912,
+g2=∞/integraldisplay
+0g(C)C7dC= 3.60617.
+We represent Eq. (4) in the standard form
+∂ϕ
+∂t+(C∇)ϕ+Cϕ(t,r,C) =
+=C
+4π/integraldisplay
+k(C,C′)ϕ(t,r,C′)C′g(C′)d3C′,(5)
+где
+k(C,C′) =3
+g0CC′+1
+g2C2C′2.
+3. PROBLEM STATEMENT
+A degenerate Bose gas occupies the half-space x >
+0above the plane surface in the problem under
+consideration. A heat flux Qnormal to the surface
+is maintained in the gas. We let T0denote the gas
+temperature far from the surface. The quantity T0differs
+from the surface temperature Tsif there is a heat flux.
+We let∆ =T0−Tsbe the difference between thesetemperatures. The quantity ∆is called the temperature
+jump (the Kapitsa temperature jump in the case of low
+temperatures).
+The problem is to find ∆Tas a function of the heat
+fluxQ. Taking into account that the problem is linear,
+we can write
+∆T=TQQ.
+The quantity TQis called the coefficient of the
+temperature jump. Another notation can be used:
+Q=R∆T,
+whereRis called the Kapitsa resistance.
+Taking into account that the problem is stationary and
+that the function ϕis independent of the coordinates
+andz, we simplify Eq. (5):
+µ∂ϕ
+∂x+ϕ(x,µ,C) =
+=1
+21/integraldisplay
+−1∞/integraldisplay
+0k(µ,C;µ′,C′)ϕ(x,µ′,C′)dΩ(C′),(6)
+where
+µ=Cx
+C, dΩ(C′) =g(C′)C′3dµ′dc′,
+k(C,µ;C′,µ′) =3
+g0CµC′µ′+1
+g2C2C′2.
+It is easy to verify that Eq. (6) has the particular
+solutions
+ϕ1=µC, ϕ 2=C2.
+Consequently, the function
+ϕas(x,µ,C) =BµC+εTC2, (7)
+where is proportional to the heat flux, is an asymptotic
+distribution function (as x→+∞).
+Assuming that the reflection of the molecules from the
+wall is purely diffusive, we now formulate the boundary
+conditions
+ϕ(0,µ,C) = 0,0< µ <1, (8)
+ϕ(x,µ,C) =
+=ϕas(x,µ,C)+o(1), x→+∞,−1< µ <0.(9)5
+4. REDUCTION TO THE ONE–VELOCITY
+PROBLEM AND THE SEPARATION OF
+VARIABLES
+We seek the solution of problem (6)–(9) in the form
+ϕ(x,µ,C) =Ch1(x,µ)+C2h2(x,µ).(10)
+We obtain the system of equations
+µ∂h1
+∂x+h1(x,µ) =
+=3
+2µ1/integraldisplay
+−1µ′h1(x,µ′)dµ′+3g1
+2g0µ1/integraldisplay
+−1µ′h2(x,µ′)dµ′,
+µ∂h2
+∂x+h2(x,µ) =
+=g1
+2g21/integraldisplay
+−1h1(x,µ′)dµ′+1/integraldisplay
+−1h2(x,µ′)dµ′.
+We represent this system of equations with respect to
+the column vector
+h(x,µ) =/bracketleftBigg
+h1(x,µ)
+h2(x,µ)/bracketrightBigg
+in the vector form
+µ∂h
+∂x+h(x,µ) =1
+21/integraldisplay
+−1K(µ,µ′)h(x,µ′)dµ′,(11)
+whereK(µ,µ′)is the kernel of Eq. (11),
+K(µ,µ′) =
+3µµ′3g1
+g0µµ′
+g1
+g21
+.
+Using (10), we transform boundary conditions (8) and
+(9) into
+h(0,µ) =0,0< µ <1,0=/bracketleftBigg
+0
+0/bracketrightBigg
+,(12)
+h(x,µ) =
+=has(x,µ)+o(1), x→+∞,−1< µ <0,(13)wherehas(x,µ)is the asymptotic distribution function
+has(x,µ) =/bracketleftBigg
+Bµ
+εT/bracketrightBigg
+.
+We note that the equation kernel can be represented
+in the form
+K(µ,µ′) =K0+3µµ′K1,
+K0=
+0 0
+g1
+g21
+, K1=
+1g1
+g0
+0 0
+.
+The general method for the separation of variables
+yields the expression
+hη(x,µ) = exp(−x
+η)Φ(η,µ). (14)
+Substituting (14) in (11), we obtain the characteristic
+equation
+(η−µ)Φ(η,µ) =η
+2K0n(0)(η)+3
+2µηK1n(1)(η),
+where
+n(k)(η) =1/integraldisplay
+−1µkΦ(η,µ)dµ, k = 0,1.
+It is obvious, that
+n(1)(η) =η(E−K0)n(0)(η),
+whereEis the unit matrix of the second order.
+We obtain now the characteristic equation
+(η−µ)Φ(η,µ) =1
+2ηD(µη)n(η), (15)
+n(η)≡n(0)(η) =1/integraldisplay
+−1Φ(η,µ)dµ,
+where
+D(µη) =K0+3µηK1(E−K0) =
+=
+3gµη0
+g1
+g21
+,
+g= 1−g2
+1
+g0g2= 0.96288.6
+Forη∈(−1,1), we use Eqs. (15) in the class of
+generalized functions [11] to find the eigenvectors
+Φ(η,µ) =F(η,µ)n(η)
+of the continuous spectrum, where
+F(η,µ) =1
+2ηD(µη)P1
+η−µ+Λ(η)δ(η−µ),
+n(η) =/bracketleftBigg
+n1(η)
+n2(η)/bracketrightBigg
+=1/integraldisplay
+−1Φ(η,µ)dµ, (16)
+F(η,µ)is the eigen matrix–function, the symbol Px−1
+denotes the principal value of the integral in the
+integration of x−1,δ(x)is the delta function, and Λ(z)
+is the dispersion matrix,
+Λ(z) =E+z
+21/integraldisplay
+−1D(µz)
+µ−zdµ.
+The dispersion matrix has following elements
+Λ11(z) = 1+3 gz2λ0(z)≡λ1(z),
+Λ12(z)≡0,
+Λ21(z) =g1
+g2λ0(z)−g1
+g2,
+Λ22(z) =λ0(z).
+We represent the dispersion matrix in the explicit form
+Λ(z) =λ0(z)D(z2)+D0,
+where
+D0=
+1 0
+−g1
+g20
+,
+λ0(z) = 1+z
+21/integraldisplay
+−1du
+u−z.
+Its determinant, called the dispersion function, is given
+by
+λ(z) = detΛ( z) =λ0(z)λ1(z).5. EIGENVECTORS OF THE DISCRETE
+SPECTRUM
+By definition [7], the set of zeros of the dispersion
+function is called the discrete spectrum.
+The function λ0(z)has a single double zero at the
+point at infinity zi=∞. Two eigensolutions of Eq.
+(11) correspond to this zero (they coincide with the
+eigenvectors of the characteristic equation):
+h0(x,µ) =µ/bracketleftBigg
+1
+0/bracketrightBigg
+, h 1(x,µ) =/bracketleftBigg
+0
+1/bracketrightBigg
+.
+We find zeros of λ1(z)outside the cut [−1,1]. It is
+obvious that λ1(∞) = 1−g= 0.03712>0. We take the
+contour γε(see Fig. 1) that encompasses the cut [−1,1]
+at a distance εfrom it such that there are no zeros
+of the function λ1(z)inside the contour. The number
+Nof zeros of λ1(z)outside the contour is equal to the
+increment of the argument of this function according to
+the argument principle [12], i.e.,
+N= (2π)−1∆γεargλ1(z),
+where the symbol ∆γεfmeans the increment of falong
+γε. Passing to the limit as ε→0in this equality, we
+obtain
+N=1
+2π∆(−1,1)argλ+
+1(µ)
+λ−
+1(µ),
+whereλ±
+1(µ)are the boundary values of λ1(z)on the
+interval (−1,1)from above and from below,
+λ±
+1(µ) =λ1(µ)±3iπgµ3.
+Taking into account that Reω+(µ) =ω(µ)is an even
+function and Imω+(µ)is odd, we have
+N=1
+π∆(0,1)argG(µ),
+G(µ) =λ+
+1(µ)
+λ−
+1(µ).
+Letθ1(µ) = argλ+
+1(µ)be the principle value of the
+argument specified by the condition θ1(0) = 0 . Because
+λ+
+1(µ) =λ−
+1(µ),|λ+
+1(µ)|=|λ−
+1(µ)|,
+we have argG(µ) = 2θ1(µ)and therefore
+N=2
+π∆(0,1)θ1(µ).7
+xy
+R=1
+εγε
+γεDRΓR
+ΓRz
+•
+1 O −1• •
+Fig. 1.•
+Fig. 1. Dependence of the temperature jump coefficient on
+the parameter 7: the specular reflection coefficient is q = 0.3
+for curve 1, q = 0.5 for curve 2, and q = 0.8 for curve 3.
+It is easy to see that the angle θ1(µ)on the cut
+[0,1]has an increment equal to π; consequently, we
+haveN= 2(the number of zeros is equal to two). We
+let±η0(η0= 1.27573) denote these zeros. In view of
+the equality λ1(z) =λ1(¯z), these zeros are real. Two
+eigensolutions h±η0(x,µ), where
+hη0(x,µ) = Φ(η0,µ)exp(−x
+η0),(17a)
+Φ(η0,µ) =η0
+2D(µη0)
+η0−µn(η0), (17b)
+correspond to them.
+We note that the homogeneous equation
+Λ(η0)n(η0) =0has a nonzero solution because
+detΛ(η0)≡λ(η0) = 0.
+We represent Eq. (17) in the form of two scalar
+equations
+λ1(η0)n1(η0) = 0, (18)
+g1
+g2[λ0(η0)−1]n1(η0)+λ0(η0)n2(η0) = 0.(19)
+In view of the condition λ1(η0) = 0 , it follows from
+Eq. (18) that the upper element n1(η0)is arbitrary and
+nonzero. From Eq. (19), we now find the lower element
+of the vector n(η0),
+n2(η0) =−g1
+g2/parenleftBig
+1−1
+λ0(η0)/parenrightBig
+n1(η0).
+From the equation
+λ1(η0) = 1+3 gη2
+0λ0(η0) = 0
+we find
+λ0(η0) =−1
+3gη2
+0.
+Hence, the vector n(η0)has the form
+n(η0) =
+1
+−g1
+g2(1+3gη2
+0)
+n1(η0).
+We substitute this vector in the second condition in
+(17) and obtain
+Φ(η0) =1
+η0−µ
+µ
+−g1
+g2η0
+,
+for
+n1(η0) =2
+3gη2
+0=−2λ0(η0).
+6. DECOMPOSITION OF THE SOLUTION INTO
+EIGENVECTORS: THE RIEMANN /emdash.cyr HILBERT
+BOUNDARY VALUE PROBLEM
+We show that the solution of boundary value
+problem (11)–(13) can be represented in the form of a
+decomposition, namely,
+h(x,µ) =has(x,µ)+A0exp(−x
+η0)Φ(η0,µ)+8
++∞/integraldisplay
+0exp(−x
+η)F(η,µ)A(η)dη, (20)
+where the unknowns are the constants εTandA0and
+a vector function A(η)with the elements Aj(η),j=
+1,2,3. Decomposition (20) can be represented in the
+form
+h(x,µ) =has(x,µ)+A0exp(−x
+η0)Φ(η0,µ)+
++exp(−x
+µ)Λ(µ)A(µ)θ+(µ)+
++1
+21/integraldisplay
+0exp(−x
+η)ηD(µη)A(η)dη
+η−µ,(21)
+where
+θ+(µ) = 1,µ∈(0,1);θ+(µ) = 0,µ /∈(−1,0).
+Substituting decomposition (21) in boundary condition
+(12), we obtain a singular integral equation with the
+Cauchy kernel [12]
+has(0,µ)+A0Φ(η0,µ)+Λ(µ)A(µ)+
++1
+21/integraldisplay
+0ηD(µη)A(η)dη
+η−µ=0,0< µ <1.(22)
+We introduce an auxiliary vector function
+N(z) =1
+21/integraldisplay
+0ηD(zη)A(η)dη
+η−z(23)
+and the matrix
+P(z) = Λ(z)D−1(z2).
+Using the boundary values N(z),Λ(z)andP(z)
+and the corresponding Sokhotski formulas, we reduce
+Eq. (22) to an inhomogeneous vector Riemann /emdash.cyr Hilbert
+boundary value problem
+P+(µ)[N+(µ)+has(0,µ)+A0Φ(η0,µ)] =
+=P−(µ)[N−(µ)+has(0,µ)+A0Φ(η0,µ)],0< µ <1.
+(24)Let’s lead reduction to the diagonal form of the
+problem (25). For this purpose we will lead to a diagonal
+kind the matrix
+P(z) =
+λ1(z)
+3gz20
+−g1
+3gg2z2λ0(z)
+.
+The matrix bringing the matrix P(z)to a diagonal
+kind, is that
+S=
+0−1
+1g1
+g2
+,detS= 1, S−1=
+g1
+g21
+−1 0
+.
+It is obvious that
+S−1P(z)S≡Ω(z) = diag/braceleftBigλ1(z)
+3gz2, λ0(z)/bracerightBig
+.
+We first solve the homogeneous boundary value
+problem
+P+(µ)X+(µ) =P−(µ)X−(µ),0< µ <1,(25)
+corresponding to (24).
+Clearly, that it is necessary to search for matrix X(z)
+in the form
+X(z) =Sdiag/braceleftBig
+U1(z), U2(z)/bracerightBig
+S−1.
+The method for solving such problems was developed
+in [8]; therefore, we give the solution of problem (25)
+without a derivation,
+X(z) =
+U1(z) 0
+g1
+g2[U0(z)−U1(z)]U0(z)
+,
+where
+U0(z) =zexp(−V0(z)),
+U1(z) =zexp(−V1(z)),
+V0(z) =1
+π1/integraldisplay
+0ζ0(u)du
+u−z,
+V1(z) =1
+π1/integraldisplay
+0ζ1(u)du
+u−z,
+ζ0(u) =θ0(u)−π, ζ 0(0) =−π, ζ 0(1−0) = 0,9
+ζ1(u) =θ1(u)−π, ζ 1(0) =−π, ζ 1(1−0) = 0.
+In this case, the angles have the form
+ζ0(u) =−π
+2−arctg2λ0(u)
+πu,
+ζ1(u) =−π
+2−arctg2λ1(u)
+3gπu3.
+We now return to the solution of inhomogeneous
+problem (24). Substituting the matrix P+(µ)found
+using (25) in (24), we obtain the problem of determining
+an analytic vector function from its jump
+/bracketleftBig
+X+(µ)/bracketrightBig−1/bracketleftBig
+N+(µ)+has(0,µ)+A0Φ(η0,µ)/bracketrightBig
+=
+=/bracketleftBig
+X−(µ)/bracketrightBig−1/bracketleftBig
+N−(µ)+has(0,µ)+A0Φ(η0,µ)/bracketrightBig
+.(26)
+We note that
+X−1(z) =
+1
+U1(z)0
+g1
+g2/parenleftBig1
+U0(z)−1
+U1(z)/parenrightBig1
+U0(z)
+,
+N(z) =/bracketleftBigg
+N1(z)
+N2(z)/bracketrightBigg
+.
+The asymptotic behavior of these functions in a
+neighborhood of the point at infinity can be described
+as
+X−1(z)∼/bracketleftBigg
+z−10
+z−2z−1/bracketrightBigg
+, z→ ∞,
+N(z) =/bracketleftBigg
+N(0)
+1
+0/bracketrightBigg
++o(1), z→ ∞,
+where
+N(0)
+1=−3g
+21/integraldisplay
+0η2A1(η)dη
+according to (23).
+Taking the behavior of X−1(z)andN(z)at finite
+points of the complex plane and in the neighborhood of
+the point at infinity into account, we obtain a general
+solution of problem (26)
+N(z) =−has(0,z)−A0Φ(η0,z)++X(z)/bracketleftbigg/bracketleftBigC1
+0/bracketrightBig
++1
+η0−z/bracketleftBigd1
+d2/bracketrightBig/bracketrightbigg
+,
+or, in scalar form,
+N1(z) =−Bz−A0z
+η0−z+U1(z)/bracketleftBig
+C1+d1
+η0−z/bracketrightBig
+,(27)
+N2(z) =−εT+g1
+g2A0η0
+η0−z+
++g1
+g2/bracketleftBig
+U0(z)−U1(z)/bracketrightBig/bracketleftBig
+C1+d1
+η0−z/bracketrightBig
++d2U0(z)
+η0−z.(28)
+HereC1,d1andd2are constants that can be found
+from the solvability conditions for the boundary value
+problem.
+We note that
+U0(z) =z−V−1
+0+o(1), z→ ∞,
+U1(z) =z−V−1
+1+o(1), z→ ∞,
+where
+V(−1)
+1=−1
+π1/integraldisplay
+0ζ1(u)du= 0.84188,
+V(−1)
+0=−1
+π1/integraldisplay
+0ζ0(u)du= 0.71045,
+Eliminating the pole of the function N1(z)given by
+equality (27) at the point at infinity, we obtain C1=B.
+Equating the limits on the right and on the left at the
+pointz=∞in equality (27) for N1(z)and using (23),
+we obtain the equation
+3g
+21/integraldisplay
+0η2A1(η)dη=−A0+d1+BV(−1)
+1.(29)
+From the equation N2(∞) = 0 , where the function
+N2(z)is given by equality (28), we find
+εT=−d2+Bg1
+g2/bracketleftBig
+V(−1)
+1−V(−1)
+0/bracketrightBig
+.(30)
+Eliminating the poles of the functions N1(z)and
+N2(z)at the point η0, we obtain
+d1=A0η0
+U1(η0)10
+and
+d2=−g1
+g2d1=−g1
+g2A0η0
+U1(η0).
+Taking these relations into account, we can rewrite
+equality (30) as
+εT=g1
+g2/bracketleftBigA0η0
+U1(η0)+B(V(−1)
+1−V(−1)
+0)/bracketrightBig
+.(31)
+From definition (23) of the function N(z), we have
+N1(z) =3g
+2z1/integraldisplay
+0η2A1(η)dη
+η−z,
+whence
+N+
+1(µ)−N−
+1(µ) = 3gπiµ3A1(µ) (32)
+according to the Sokhotsli formulas.
+Substituting solution (27) in (32), we find
+3g
+2zη2A1(η) =
++1
+2πi/bracketleftBig/parenleftBig
+C1+d1
+η0/parenrightBig1
+η−d1
+η0(η−η0)/bracketrightBig/bracketleftBig
+U+
+1(η)−U−
+1(η)/bracketrightBig
+.(33)
+Integrating equality (33) over ηfrom0to1, we
+obtain
+3g
+21/integraldisplay
+0η2A1(η)dη=/parenleftBig
+C1+d1
+η0/parenrightBig
+J(0)−d1
+η0J(η0),(34)
+where
+J(z) =1
+2πi1/integraldisplay
+0U+
+1(τ)−U−
+1(τ)
+τ−zdτ.
+We give the integral representation of the function
+U1(z)without a derivation,
+U1(z)−z+V(−1)
+1=
+=1
+2πi1/integraldisplay
+0U+
+1(τ)−U−
+1(τ)
+τ−zdτ, z /∈[0,1].
+According to this representation, we have
+J(η0) =U1(η0)−η0+V(−1)
+1
+and
+J(0) =U1(0)+V(−1)
+1.Substituting these equalities in (34), we obtain the
+equation
+3g
+21/integraldisplay
+0η2A1(η)dη=/parenleftBig
+C1+d1
+η0/parenrightBig/parenleftBig
+U1(0)+V(−1)
+1/parenrightBig
+−
+−d1
+η0/parenleftBig
+U1(η0)−η0+V(−1)
+1/parenrightBig
+. (35)
+From Eqs. (29) and (35), we now obtain an equation
+from which we find
+A0=−BU1(η0).
+Substituting the found value A0in (31), we find that
+the temperature jump is given by
+εT=Bg1
+g2/bracketleftBig
+η0+V(−1)
+1−V(−1)
+0/bracketrightBig
+. (36)
+7. TEMPERATURE JUMP
+We express the temperature jump in terms of the heat
+flux, which is only transferred by the normal component
+and is proportional to its velocity [9]. We note that the
+mean velocity of the gas (the normal component and
+Bose condensate) is equal to zero in the gas volume. We
+calculate the heat flux using the formula
+Q=/integraldisplay
+fvmv2
+2(2s+1)d3p
+(2π/planckover2pi1)3,p=mv.(37)
+Passing to dimensionless quantities in the integral in
+(37), we obtain
+Q=(2s+1)m4
+2(2π/planckover2pi1)3β6s/integraldisplay
+[fs
+B+ϕ(x,C)g(C)]CC2d3C,
+or
+Q=(2s+1)mk3T3
+s
+2π3/planckover2pi13×
+×/integraldisplay/bracketleftBig
+Ch1(x,µ)+C2h2(x,µ)/bracketrightBig
+CC2g(C)d3C.
+Taking into account that
+C= (Cµ,Csinθcosχ,Csinθsinχ),
+d3C=C2dµdCdχ,
+we obtain
+Qx(x) =11
+=Q01/integraldisplay
+−1∞/integraldisplay
+0/bracketleftBig
+Ch1(x,µ)+C2h2(x,µ)/bracketrightBig
+µC5g(C)dµdC,
+where
+Q0=(2s+1)mk3T3
+s
+π2/planckover2pi13.
+We calculate the inner integral in (38),
+Qx=Q0/parenleftBig
+g11/integraldisplay
+−1µh1(x,µ)dµ+g21/integraldisplay
+−1µh2(x,µ)dµ/parenrightBig
+.
+Taking into account that the heat flux is conserved
+(i.e.,Qx= const ), we find
+Qx=Q0g12
+3B,
+hence
+B=3Qx
+2Q0g1. (39)
+Taking (39) into account, we represent (36) in the form
+∆T=RQx, (40)
+whereRis the Kapitsa resistance,
+R=3
+2g2/bracketleftBig
+η0+V(−1)
+1−V(−1)
+0/bracketrightBigπ2/planckover2pi13
+(2s+1)mk3T2s,
+or
+R= 0.58514π2/planckover2pi13
+(2s+1)mk3T2s=
+= 5.77510/planckover2pi13
+(2s+1)mk3T2s. (41)
+Formula (40) is the sought temperature jump (the
+Kapitsa jump) in the degenerate Bose gas. The coefficient
+Rof the temperature jump, called the Kapitsa
+resistance, is given by (41). It follows from relation
+(1) that obtained formula (41) is applicable under the
+condition
+T/greaterorequalslant4π/planckover2pi12an
+mk.
+8. ANALYSIS OF RESULTS
+It can be seen from (41) that the Kapitsa resistance
+increases without bound as the temperature Tsdecreases. At present, there are no experimental data
+on the Kapitsa resistance for a degenerate Bose gas.
+There exist only data on the Kapitsa resistance for liquid
+helium4He[4], [9]. According to these data, there exists
+a divergence of the Kapitsa resistance as Ts→0in the
+case of liquid helium. The experimental data thus agree
+qualitatively with formula (41).
+[1]L. P. Pitaevskii , Phys. Usp., 49, 333–351 (2007).
+[2]H. Spon , Kinetics of the Bose–Einstein Condensation,
+arXiv:0809.4551v1 [cond-mat.mes-hall] (2008).
+[3]Hai Pang, Wu–Sheng Dai, Mi Xie , J. Phys. A. Mat. Gen.
+39(2006), 2563–2571; arXiv:cond-mat/0603289 (2006).
+[4]Yu. V. Prokhorov , ed., Encyclopedic Dictionary
+of Physics [in Russian], Bol’shaya Rossiiskaya
+Entsiklopediya, Moscow (1955).
+[5]Y. Pomeau and D. C. Roberts , Phys. Rev. B, 77, 144508
+(2008); arXiv:cond-mat/0702669v2 (2007).
+[6]A. V. Latyshev and A. A. Yushkanov , Theor. Math.
+Phys., 134, 271-284 (2003).
+[7]A. V. Latyshev and A. A. Yushkanov , Mathematical
+Modeling, 15, No. 5, 80-94 (2003).
+[8]E. M. Lifshitz and L. P. Pitaevskii , Statistical Physics,
+Part 2 [in Russian], Nauka, Moscow (1978); English
+transl.: Statistical Physics: Part 2. Theory of the
+Condensed State (Vol. 9 of Course of Theoretical Physics,
+L. D. Landau and E. M. Lifshits, eds.), Pergamon Press,
+Oxford (1980).
+[9]I. M. Khalatnikov, An Introduction to the Theory
+of Superfluidity [in Russian], Nauka, Moscow (1965);
+English transl., Addison-Wesley, Redwood City, Calif.
+(1989).
+[10]Carlo Cercignani , Theory and Application of the
+Boltzmann Equation, Scottish Academic, Edinburgh
+(1975).
+[11]A. A. Vladimirov and V. V. Zharinov , Equations of
+Mathematical Physics, Fiz.–Mat. Lit., Moscow (2000).
+[12]F. D. Gakhov , Boundary Value Problems [in Russian],
+Nauka, Moscow (1977); English transl., Dover, New York
+(1990).
\ No newline at end of file
diff --git a/1001.0408.txt b/1001.0408.txt
new file mode 100644
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@@ -0,0 +1,405 @@
+arXiv:1001.0408v1  [cond-mat.mtrl-sci]  3 Jan 2010Electron transport through asymmetric ferroelectric tunn el junctions:
+current-voltage characteristics
+Natalya A. Zimbovskaya
+Department of Physics and Electronics, University of Puert o Rico,
+100 CUH Station, Humacao, PR 00791, and Institute for Functi onal Nanomaterials,
+University of Puerto Rico, San Juan, PR 00931
+(Dated: November 15, 2018)
+We have carried out calculations of current-voltage charac teristics for the electron tunnel current
+through a junction with a thin insulating ferroelectric bar rier assuming that interface transmissions
+for the left and right interfaces noticeably differ due to dis similarity of the interfaces. Obtained con-
+ductance vs voltage and current vs voltage curves exhibit we ll distinguishable asymmetric hysteresis.
+We show that the asymmetry in the hysteretic effects could ori ginate from the asymmetric bias volt-
+age profile inside the junction. In particular, we analyze th e hysteresis asymmetries occurring when
+the bias voltage distribution is low-sensitive to the spont aneous polarization reversal.
+PACS numbers: 73.23.-b, 85.50.-n
+Ferroelectric materials attract significant interest of
+the research community due to their potential useful-
+ness in various technological applications [1]. In the
+past decade it was repeatedly shown in the experiments
+that ferroelectricity could be maintained in films of some
+perovskite oxides with thickness of the order of a few
+nanometers [2–6]. These experimental observations were
+corroborated by first-principles computations predicting
+the critical film thickness for ferroelectricity to persist in
+perovskite films to be as thin as a few lattice parame-
+ters [7–10]. Thus, both experimental and theoretical re-
+sults indicate that ultrathin ferroelectric films exist and
+may be used as barriers in ferroelectric tunnel junctions
+(FTJ). A ferroelectric tunnel junction consists of two
+conducting electrodes (leads) separated by a nanometer-
+thick ferroelectric film. Electrons can tunnel across the
+film thus maintaining a nonzero conductance of the junc-
+tion. Specific nature of the ferroelectric barriers add new
+functional properties to FTJs [11] Such junctions could
+serve as essential components in manifold nanodevices
+such as binary data storage memories and switches [12].
+Themostimportantfeatureofferroelectricsisthepres-
+ence of spontaneous polarization which could be reversed
+by an applied electric field. The polarization switch-
+ing inherent to ferroelectric materials alters the sign of
+polarization charges at the electrode/barrier interfaces
+thuschangingtheelectrostaticpotentialprofileinsidethe
+junction. Also, the polarization switching shifts ions in
+ferroelectric unit cells and changes lattice strains in the
+film affecting the electron band structure of the junction.
+As a result, transport characteristics of a FTJ may be
+significantly modified. The effect of the polarization re-
+versal on FTJ conduction was predicted in earlier works
+(see e.g. Refs. [13–15]) and observed in experiments
+onPt/Pb(Zr0.52Ti0.48)O3/SrRuO 3FTJ [15], on junc-
+tions with La0.1Bi0.9MnO3tunnel barrier [16] and on
+Pt/SrTiO 3/Ptjunctions [17].
+Currently, theoretical and computational efforts aremostly concentrated on studies of zero bias conductance
+of FTJs. However, the theoretical analysis of current-
+bias voltage ( I−V) relationships is also important
+for it provides more thorough characterization of the
+electron transport through FTJs. As shown in Ref.
+[13], the polarization reversal in the ferroelectric films
+causes a hysteretic behavior of both current-voltage and
+conductance-voltage characteristics. Especially interest-
+ing hysteretic effects may occur in asymmetric FTJs
+where the interfaces are dissimilar. The dissimilarity
+could occurwhen the leads aremade out ofdifferent met-
+als [14, 18]. Also, it could originate from the atomic and
+electronic structure of the junction [19]. When interfaces
+differ, the effects of the polarization reversal also may
+differ depending on the specific properties of the inter-
+faces. This may result in asymmetry of tunnel electrore-
+sistance (TER) versus bias voltage curves and of hystere-
+sis loops in current-voltage characteristics. In strongly
+asymmetric FTJs both TER and hysteresis could be well
+pronounced for a certain bias voltage polarity and sig-
+nificantly reduced on the polarity reversal. The asym-
+metric TER and hysteretic effects were recently reported
+for asymmetric junctions including BaTiO 3[20, 21] and
+Pb(Zr0.2Ti0.8)O3[22] ferroelectric barriers.
+In the present work we theoretically analyze transport
+characteristics of an asymmetric FTJ. We simulate the
+FTJ by a ferroelectric film of thickness dsandwiched
+between two semi-infinite metal electrodes. We assume
+that the film is uniformly polarized in the direction per-
+pendicular to the electrode planes, which is the direction
+of “z” axis within the chosen coordinate system. At
+small values of the applied bias voltage the conductance
+of a tunnel junction per area Acould be described by
+the following expression [23]:
+G
+A=2e2
+h/integraldisplayd2k||
+(2π)2T(E,k||) (1)
+wherek||istheprojectionofthetunneling electronwave2
+vector to the electrode’s plane, and Eis the tunnel en-
+ergy, whichtakesonvaluescloseto the Fermienergy EF.
+The transmission function T(E,k||) may be factorized
+and presented in the form [24]:
+T(E,k||) =tL(E,k||)exp/braceleftbig
+−2κ(E,k||)d/bracerightbig
+tR(E,k||) (2)
+Here,tL(E,k||) andtR(E,k||) are the interface trans-
+mission functions for the left and right electrodes, re-
+spectively, and κ(E,k||) is determined by the height and
+shape of the potential barrier in the tunnel junction and
+by the energy-momentum relation of a tunneling elec-
+tron. Generally speaking, it also depends on the bias
+voltage but one may disregard this at small values of V.
+Employing a simple form for the energy-momentum re-
+lation:
+E=/planckover2pi12k2
+||
+2m+/planckover2pi12k2
+z
+2m(3)
+wheremis the effective mass of the tunneling electron,
+we may approximate:
+κ(E,k||) =1
+d/integraldisplayd
+0/radicalbigg
+2mΦ(z)
+/planckover2pi12+k2
+||dz (4)
+Here, Φ( z) is the overall potential profile seen by the
+tunneling electrons. At zero bias voltage the potential
+Φ(z) is a superposition of the potential, which deter-
+mines the positions of conduction bands in metal elec-
+trodes with respect to the Fermi energy, and the bar-
+rier potential occurring due to the presence of the insu-
+lating ferroelectric film. Also, the overall potential in-
+cludes a contribution created by surface charges on the
+surfaces of the ferroelectric film and screening charges
+appearing on the electrode surfaces. The simplest ap-
+proximation for Φ( z) appropriate for a symmetric FTJ
+is a trapezoid barrier with a thickness dand heights
+Φ(0) =U+ϕ(0),Φ(d) =U+ϕ(d) where ϕ(z) is the
+potential induced by the surface charges. The relative
+heights of the barrier at z= 0 and z=dare deter-
+mined by the potential difference ϕ(0)−ϕ(d).The dif-
+ference changes its sign when the polarization in the film
+reverses causing the hysteresis in the conduction through
+the FTJ to occur. In asymmetric FTJs the change in the
+complex band structure induced by the left-right asym-
+metry of the ferroelectric displacements in the barrier
+may noticeably change the averagebarrier heights Ufor
+two polarization states, as was theoretically shown for
+aSrRuO 3/BaTiO 3/SrRuO 3junction basing on first
+principles computations [19]. The difference in the bar-
+rier heights for different polarization directions may be
+of the same order or greater than that induced by the
+potential ϕ(z).In the following transport calculations
+we take into account both the effect of depolarizing field
+corresponding to the potential ϕ(z) and variations in
+the barrier height appearing due to the junction asym-
+metry. However, to avoid extra complications in furthercomputations we do not include in our model the effect
+of an intrinsic electric field throughout the barrier, which
+appears due to the difference in the electrostatic dipoles
+at the interfaces, as well as the strain effects due to piezo-
+electricity.
+To estimate the current through the junction we use
+the expression:
+J
+A=2e
+h/integraldisplay
+dE(fL−fR)/integraldisplayd2k||
+(2π)2T(E,k||).(5)
+Here, the transmissionfunction T(E,k||) is given by Eq.
+(2)andfL,RareFermidistributionfunctionswith chem-
+ical potentials µL,R.The chemical potentials are shifted
+with respect to EFwhen a nonzero bias voltage Vis
+applied across the junction, namely:
+µL=EF+ηeV;µR=EF−(1−η)eV.(6)
+The division parameter ηshows how the voltage Vis
+distributed in the junction. The expression (5) resem-
+bles the well known Landauer formula for tunnel current
+through molecular junctions and/or quantum dots cou-
+pled to the conducting leads [25]. It could be derived
+using surface Green’s functions formalism presented in
+the earlier work [26]. We have grounds to belive that η
+takes on values depending on the relation of the interface
+transmissions tLandtR.Calculating the current den-
+sity we hypothesize that η=tR/(tL+tR),which resem-
+bles the expression for the division parameter commonly
+used in the theory of electron tunnel transport through
+molecularbridgesand quantum dots. In that theory ηis
+described by the similar expressionwhere the parameters
+determining the coupling of a molecule/quantum dot to
+the left and right leads take on the parts of the interface
+transmissions.
+Regardless of symmetry/asymmetry of a FTJ, the
+transmission coefficients tLandtRdiffer, and the dif-
+ference in their values may be significant. For instance,
+it was reported that in a Pt/BaTiO 3/Ptjunction one
+of the transmission coefficients was approximately three
+times greater than another one [10]. One may expect
+even greater difference in the interface transmission for
+the FTJ studied by Maksymovich et al [22], where recti-
+fying current-voltage characteristics were reported. One
+reason for the disparity in the transmission functions is
+the trapezoid-like profile of the potential barrier. The
+barrier height decreases along the polarization direction,
+therefore tRshould exceed tLwhen the polarization
+points to the right and vice versa. In symmetrical FTJs
+with identical interfaces one may expect the transmis-
+sion coefficients simply interchange their values on the
+polarization reversal:
+t→
+L=t←
+R;t→
+R=t←
+L (7)
+where an arrow indicates the polarization direction. In
+such a case the bias voltage division parameter ηin the3
+Eq. (6) is to be replaced by 1 −ηwhen the polarization
+direction is reversed. Accordingly, the current-voltage
+curves associated with different polarization states (al-
+though asymmetric by themselves due to the asymmetry
+in the bias voltage distribution across the junction) are
+arrangedin the V−Iplane in such a waythat hysteretic
+features remain symmetrical. The TER versus voltage
+curve must be symmetrical upon bias voltage reversal,
+as well.
+In asymmetric FTJs the relationship between the
+transmissions tLandtRfor different polarization direc-
+tions is more complicated. Due to an asymmetric defor-
+mation of the ferroelectric potential profile the relations
+(7) donot hold anymore. Besides, dissimilarityin the in-
+terfaces may affect the transmissions due to the reasons
+unrelated to ferroelectric properties of the film linking
+the electrodes. Even within the simple one-dimensional
+model of a FTJ employed in the present work, one must
+take into account specific features of the contact between
+the electrodes and the film, which could significantly dif-
+ferontheleftanfrightsidesofthejunction. Forinstance,
+in the case when the electrodes are made out of differ-
+ent metals, this may significantly influence the interface
+transmissions. Also, FTJs studied in the recent works
+[20, 22] consisted of an epitaxial ferroelectric film grown
+on the top of conducting electrode and a sharp metal tip
+serving as another electrode. We may expect a nonepi-
+taxial contact between the tip and the film to notably
+modify the corresponding interface transmission for both
+polarization directions. Some other effects, which could
+modify interface transmissionsin asymmetric FTJs (and,
+in consequence, the bias voltage distribution across the
+junctions) are discussed elsewhere (see e.g. Ref. [22]
+and references therein). So, the bias voltage distribu-
+tion across an asymmetric FTJ for different polarization
+orientations could hardly be predicted basing on some
+general considerations. For each particular FTJ the rela-
+tion of the interface transmissions (which greatly affects
+the bias voltage distribution) must be separately estab-
+lished, and the results may significantly vary depending
+on the FTJ characteristics. For instance, we remark that
+in the asymmetric FTJ whose transport properties were
+studied by Gruverman et al [20], tL< tRfor both polar-
+ization directions. This follows from the reported shape
+of theI−Vcharacteristics.
+In summary, various asymmetric features in the hys-
+teretic behavior of the tunnel current and conductance
+in asymmetrical FTJs originate from asymmetries in the
+tunnel barrier profile for different polarization directions
+and from specifics of the bias voltage distribution (deter-
+mined by the relationship between the interface trans-
+missions). The relative effects of these two factors could
+vary but neither one should be disregarded on general
+grounds. Here, we concentrate on the analysis of trans-
+port properties of a strongly asymmetric FTJ where the
+biasvoltagedistributionexhibits arelativelylowsensibil-
+FIG. 1: (Color online) ConductanceG
+A=1
+AdI
+dVof the model
+asymmetric FTJ vs bias voltage for the paraelectric (dash-
+dotted line) and ferroelectric (solid and dashed lines) sta tes
+of the film. The curves are plotted using Eqs. (1)-(6) at T=
+30K, d= 1.8nm, U = 0.6eV(low resistance ferroelectric
+state),U= 0.8eV(high resistance ferroelectric state), for
+tL= 3tR(left panel) and tR= 3tL(right panel).
+ity to the polarization reversal. Accordingly, we accept
+thattL/tRremains the same regardless of the polariza-
+tion direction.
+Computing the conductance and current for our model
+FTJ we used the results reported in Ref. [10] for
+Pt/BaTiO 3/Ptjunctions. So, we assumed that d=
+1.8nm,and zero bias conductances per unit cell area
+take on values G/A≈17.0×10−5e2/h(for the paraelec-
+tric state of the film) and G/A≈2.9×10−5e2/h(for
+its low resistance ferroelectric state), respectively [27].
+We did carry out calculations for tR/tL= 3 (η= 0.75)
+andtL/tR= 3 (η= 0.25) for ferroelectric states of
+the film. For the film in the paraelectric state we as-
+sumedtL=tR.Using these values we estimated the
+interface transmissions for paraelectric and ferroelectric
+barriers in the junction. As shown in the Fig. 1, the
+conductance through the junction with the ferroelectric
+barriershouldexhibitratherwelldistinguishablehystere-
+sis. The hysteresis is asymmetric with respect to V= 0,
+and we observe the significant asymmetry in the TER.
+AttL= 3tRthe TER is small at positive bias volt-
+age but it significantly increases upon the reversal of the
+bias voltage polarity, as shown in the left panel of the
+figure. When tR= 3tLthe TER is much better pro-
+nounced at positive bias voltage (see the right panel of
+the Fig. 1). Basing on the experimental results of Gar-
+cia et al [21] for thin BaTiO 3films, we may roughly
+estimate the coercive voltage Vcfor our model FTJ as:
+Vc∼1.0−1.5V.Therefore, the differencein the TER val-
+ues atV=±Vc,which occurs due to the specifics of the
+bias voltage profile across the junction may reach values
+of the order of 102−103.The current-voltage charac-
+teristics are presented in the Fig. 2. We see asymmetric4
+FIG. 2: (Color online) Current-voltage characteristics fo r the
+model ferroelectric tunnel junction. The curves are plotte d
+using Eqs. (1)-(6) The parameters take on the same values
+as in the Fig. 1.
+hysteretic behavior of the tunnel current. As well as for
+conductance, the asymmetry in the hysteresis loops to a
+considerable degree originates from the bias voltage pro-
+file in the junction. We remark that the I−Vcurves
+in the right panel of the Fig. 2 resemble those recently
+reported in the work [20].
+In conclusion, we have demonstrated that current-
+voltage and conductance-voltage characteristics of an
+asymmetric FTJ may exhibit a pronounced hysteretic
+behavior of a special kind whose distinctive feature is
+a well pronounced asymmetry in the hysteresis loops for
+different bias voltage polarities. The asymmetry in the
+hysteresis largely originates from the asymmetric bias
+voltage division in the junction. In strongly asymme-
+tryc junctions the dissimilarity of the interfaces may af-
+fect interface transmissions so much that the effects of
+the polarization reversal may be surpassed. In this case,
+the relationship between the transmissions is low sen-
+sitive to the polarization revesal, and the bias voltage
+distribution across the junction rather weakly depends
+on the polarization direction. Our results give grounds
+to expect that the bias voltage reversal in the consid-
+ered strongly asymmetric FTJs should bring significant
+changes to their transport characteristics, and this could
+be used in designing of nanodevices.
+Acknowledgement: AuthorthankJ.P.Velevforhelpful
+discussion and G. M. Zimbovsky for help in manuscript.
+[1] T. M. Shaw, S. Trolier-Mckinstry, and P. C. McIntyre,
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+Jaswal, and E. Y. Tsymbal, Phys. Rev. Lett. 98, 137201
+(2007).
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+(2006).
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+Y. Tsymbal, Phys. Rev. Lett. 94, 246802 (2005).
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+Waser, C. Buchal, N. A. Pertsev, Appl. Phys. Lett. 83,
+4595 (2003).
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+(2007).
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+94, 062903 (2009).
+[18] Y. Zheng and C. H. Woo, Nanotechnology, 20, 075401
+(2009).
+[19] J. P. Velev, C.-G. Duan, J. D. Burton, A. Smogunov, M.
+K. Niranjan, E. Tosatti, S. S. Jaswal, and E. Y. Tsymbal,
+Nano Lett. 9, 427 (2009).
+[20] A. Gruverman, D. Wu, H. Lu, Y. Wang, H. W. Jang,
+C. M. Folkman, M. Ye. Zhuravlev, D. Felker, M. Rz-
+chowski, C.-B. Eom, and E. Y. Tsymbal, Nano Lett. 9,
+3539 (2009).
+[21] V. Garcia, S. Fusil, K. Bouzehouane, S. Enouz-Vedrenne ,
+N. D. Mathur, A. Barthelmy, and M. Bibes, Nature 460,
+81 (2009).
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+dorf, S. V. Kalinin, Science 324, 1421 (2009).
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+1969).
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+D. Steward, I. I. Oleinik. S. S. Jaswal, Phys. Rev. B 69,
+174408 (2004).
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+thickness of 1 .8nmcould exist in both paraelectric and
+ferroelectric states.
\ No newline at end of file
diff --git a/1001.0409.txt b/1001.0409.txt
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+arXiv:1001.0409v6  [math.HO]  15 Mar 2011COUNTING IRREDUCIBLE POLYNOMIALS OVER FINITE
+FIELDS USING THE INCLUSION-EXCLUSION PRINCIPLE
+SUNIL K. CHEBOLU AND J ´AN MIN ´AˇC
+Abstract. C. F. Gauss discovered a beautiful formula for the number of
+irreducible polynomials of a given degree over a finite field. Using just very
+basic knowledge of finite fields and the inclusion-exclusion formula, we show
+how one can see the shape of this formula and its proof almost i nstantly.
+Why there are exactly
+1
+30(230−215−210−26+25+23+22−2)
+irreducible monic polynomials of degree 30 over the field of t wo elements? In
+this note we will show how one can see the answer instantly usi ng just very basic
+knowledge of finite fields and the well-known inclusion-excl usion principle.
+To set the stage, let Fqdenote the finite field of qelements. Then in general,
+the number of monic irreducible polynomials of degree nover the finite field Fq
+is given by Gauss’s formula
+1
+n/summationdisplay
+d|nµ(n/d)qd,
+wheredruns over the set of all positive divisors of nincluding 1 and n, andµ(r)
+is the M¨ obius function. (Recall that µ(1) = 1 and µ(r) evaluated at a product
+of distinct primes is 1 or -1 according to whether the number o f factors is even
+or odd. For all other natural numbers µ(r) = 0.) This beautiful formula is
+well-known and was discovered by Gauss [2, p. 602-629] in the case when qis a
+prime.
+We will present a proof of this formula that uses only element ary facts about
+finite fields and the inclusion-exclusion principle. Our app roach offers the reader
+a new insight into this formula because our proof gives a prec ise field theoretic
+meaning to each summand in the above formula. The classical p roof [3, p. 84]
+which uses the M¨ obius’ inversion formula does not offer this i nsight. Therefore
+we hope that students and users of finite fields may find our appr oach helpful.
+It is surprising that our simple argument is not available in textbooks, although
+it must be known to some specialists.
+Date: March 17, 2011.2 SUNIL K. CHEBOLU AND J ´AN MIN ´AˇC
+Proof of Gauss’s formula Before we present our proof we collect some basic
+facts about finite fields that we will need. These facts and the ir proofs can be
+found in almost any standard algebra textbook that covers fin ite fields. See for
+example [1, Chapter 14.3], [3, Chapter 7.1, 7.2], or [4, Chap ter 20.1].
+(1) A finite field of order qexists if and only if qis a prime power. Moreover,
+such a field is unique up to isomorphism, and is denoted by Fq.
+(2)Fqnis the splitting field of any irreducible polynomial p(x) of degree n
+overFq. (This means p(x) factors into linear factors over Fqnbut not
+over any smaller subfield of Fqn.)
+(3) The roots of an irreducible polynomial over Fqare always distinct.
+(4) No two distinct irreducible polynomials over Fqcan have a common root.
+(5)Fqa⊆Fqbif and only adividesb.
+Withthesebasicfactsunderourbeltweproceedtoshowthatt hetotal number
+of irreducible monic polynomials of degree noverFqis equal to
+1
+n/summationdisplay
+d|nµ(n/d)qd.
+The case n= 1 is easy because every degree one monic polynomial is irred ucible.
+In fact, the total number of degree one monic polynomials ove rFqis equal to q,
+and this is exactly what we get from the above formula when we p lug inn= 1.
+Therefore for the rest of the proof we will assume that n >1. LetPndenote
+the collection of all irreducible monic polynomials of degr eenoverFqand let
+Rnbe the union of all the roots of all the polynomials in Pn. Note that fact (2)
+ensures that the roots thus obtained are contained in Fqn. Moreover, using facts
+(3) and (4) we conclude that Rnis the disjoint union of n-element sets, one for
+each polynomial in Pn. Thus,
+|Rn|=n|Pn|.
+Therefore, it is enough to compute |Rn|. To this end, observe that
+Rn={αinFqn|[Fq(α):Fq] =n},
+={αinFqn|αis not contained in any proper subfield of Fqn},
+={αinFqn|αis not contained in any maximal proper subfield of Fqn}
+Letn=uavbwc···be the prime factorization of nwithrdistinct prime
+factors (recall that n >1). Then the maximal subfields of Fqnby fact (4) are of
+the form
+Fu=Fqn
+u,Fv=Fqn
+v,Fw=Fqn
+w,···.
+Then by the third interpretation of Rngiven above, we have
+|Rn|=|(Fu∪Fv∪Fw···)c|COUNTING IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS 3
+where the complement is taken in the field Fqn. Fact (4) implies that the lattice
+of subfields of Fqnthat contain Fqis isomorphic to the lattice of divisors of n.
+Thus, we have Fu∩Fv=Fqn
+uvandFu∩Fv∩Fw=Fqn
+uvw, etc.
+The cardinality of Rncan now be computed using the inclusion-exclusion
+principle as follows.
+|Rn|=qn
+−qn
+u−qn
+v−qn
+w−···
++qn
+uv+qn
+uw+qn
+vw+···
+···
++(−1)rqn
+uvw···.
+Finally, the formula for |Pn|takes the desired form when we divide |Rn|byn
+and use the M¨ obius function.
+We end by pointing out that if one is interested in counting th e cardinality of
+all irreducible polynomials of degree n(not necessarily monic) over Fqthen we
+simply multiply |Pn|byq−1. This is because every such polynomial p(x) can
+be uniquely written as αr(x) whereαis a non-zero element of Fqandr(x) is an
+irreducible monic polynomial of the same degree.
+Acknowledgements. We would like to thank our students at Illinois State
+University and University of Western Ontario for their insp iration and insistence
+on penetrating mysteries of finite fields. We also would like t o thank Fusun
+Akman, Micheal Dewar, and Alan Koch for helping us improve th e exposition
+through their interesting and very encouraging comments on this paper. Finally,
+we would like to thank the anonymous referee for their sugges tions to improve
+the exposition.
+References
+[1] D.S. Dummit and R. M. Foote, Abstract algebra, third edit ion. John Wiley & Sons, Inc.,
+Hoboken, NJ, 2004.
+[2] C. F. Gauss, Untersuchungen ¨Uber H¨ ohere Arithmetik, second edition, reprinted, Chels ea
+publishing company, New York 1981.
+[3] K. Ireland and M. Rosen, A classical introduction tomode rn number theory, second edition,
+Springer-Verlag, GTM Vol 84 (second edition) 1990.
+[4] T. W. Judson, Abstract Algebra: Theory and Applications , PWS-Kent, Boston, 1994.
+Department of Mathematics, Illinois State University, Nor mal, IL 61790, USA
+E-mail address :schebol@ilstu.edu
+Department of Mathematics, University of Western Ontario, London, ON N6A
+5B7, Canada
+E-mail address :minac@uwo.ca
\ No newline at end of file
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+arXiv:1001.0410v2  [math.AP]  1 Feb 2010Nonlinear porous medium flow
+with fractional potential pressure
+Luis Caffarelli∗and Juan Luis Vazquez†
+Abstract
+We study a porous medium equation with nonlocal diffusion effect s given
+by an inverse fractional Laplacian operator:
+∂tu−∇·(u∇p) = 0, p= (−∆)−su,0< s <1.
+We pose the problem for x∈Rnandt >0 with bounded and compactly
+supported initial data, and prove existence of weak and boun ded solutions that
+propagate with finite speed, a property that is not shared by o ther fractional
+diffusion models.
+1 Introduction
+We study a nonlinear diffusion model with nonlocal effects described b y the system
+(1.1) ∂tu=∇·(u∇p), p=K(u).
+Here,uis a function of the variables ( x,t) to be thought of as a density or con-
+centration, and therefore nonnegative, while pis the pressure, which is related to u
+via a linear positive operator K, which we assume to be the inverse of a fractional
+Laplacian. To be specific, the problem is posed for x∈Rn,n≥1, andt>0, and we
+give initial conditions
+(1.2) u(x,0) =u0(x), x∈Rn,
+∗caffarel@math.utexas.edu
+†juanluis.vazquez@uam.es
+1whereu0is a nonnegative and bounded function in Rnwith compact support or fast
+decay at infinity.
+The model arises from the consideration of a continuum, say, a fluid or a population,
+represented byadensitydistribution u(x,t)thatevolves withtimefollowingavelocity
+fieldv(x,t), according to the continuity equation
+ut+∇·(uv) = 0.
+We now assume that vderives from a potential, v=−∇p, as happens for instance
+in fluids in porous media according to Darcy’s law, and in that case pdenotes the
+pressure. But potential velocity fields are also found in many other instances, like
+Hele-Shaw cells.
+Westillneedaclosurerelationtorelate uandp. Inthecaseofgasesinporousmedia,
+as modeled in the 1930’s by Leibenzon and Muskat [25, 27], the closure relation takes
+the form of a state law: p=f(u), wherefis a nondecreasing scalar function, which
+is linear when the flow is isothermal, and a higher power of uif it is adiabatic. The
+linear relationship happens also in the simplified description of water infi ltration in
+an almost horizontal soil layer according to Boussinesq. See [34] f or a description
+of these and other applications. Summing up, we get the standard p orous medium
+equation,ut=c∆(u2), or more generally, ut= ∆(um) withm>1.
+In this paper we propose to consider the case where pes related to uthrough a linear
+fractional potential operator, K= (−∆)−swith kernel K(x,y) =c|x−y|−(n−2s)(i.e.,
+a Riesz operator, cf. [24, 33], see also Appendix for precise definitio ns and some com-
+ments). The interest in using fractional Laplacians in modeling diffusiv e processes
+has a wide literature, especially when one wants to model long-range diffusive interac-
+tion, and this interest has been activated by the recent progress in the mathematical
+theory. This literature is mostly elliptic, cf. [6, 11, 31], but cf. works like [4, 28] for
+related parabolic problems.
+More generally, it could be assumed that Kis an operator of integral type defined
+by convolution on all of Rn, with the assumptions that is positive and symmetric.
+The fact the Kis a homogeneous operator of degree 2 s, 0<s<1, will be important
+in the proofs given below. An interesting variant would be K= (−∆+cI)−s. We are
+not exploring here such extensions.
+Extreme cases. (1) If we take in our model s= 0, so that K= the identity
+operator, we get the standard porous medium equation with m= 2, whose behavior
+is well-known, see [2, 34] for the mathematical theory and the applic ations.
+2(2) In the other end of the sinterval, when s= 1 and we take K= (−∆)−1, we get
+(1.3) ut=∇u·∇p−u2,−∆p=u.
+In one dimension this leads to ut=uxpx−u2,pxx=−u.In terms of v=−px=/integraltext
+udx
+we have
+vt=upx+c(t) =−vxv+c(t),
+Forc= 0 this is the Burgers equation vt+vvx= 0 which generates shocks in finite
+time if we allow for uto have two signs.
+As a related precedent we may mention the model studied by Lions an d Mas-Gallic
+[26], who are interested in the regularization of the velocity field in the standard
+porous medium equation by means of a convolution kernel. They get a system which
+is formally like ours, but there is a big difference in the study since they assume the
+kernel to be smooth and integrable, and in fact an approximation of the Dirac delta,
+in other words, a short-range interaction. Since the kernel of th e fractional operator
+(−∆)−sisk(x,y)∼ |x−y|−(n−2s), i.e., a long-range interaction, we are far away from
+that situation, but it may serve as a previous regularization step.
+A model from superconductivity arises in recent work by Ambrosio a nd Serfaty
+[1] describing the evolution of the vortex-density in superconduct or modeling. The
+system is similar to our system with K= (−∆)−s,s= 1 and their mathematical
+tools are quite different. On the other hand, the equation with s= 1/2 has been
+proposed by Head [21] as the equation of motion of a dislocation cont inuum, and
+thenuis the dislocation density and the space dimension is n= 1. The mathematical
+investigation of this case is performed by Biler et al. in [8], though in ter ms of the
+integrated equation vt+|vx|Λ(v) = 0, where Λ is the L´ evy operator of order 1, which
+is equivalent to ( −∂2
+xx)1/2.
+Models of this kind arise in other contexts. Some variants will be indica ted at the
+end of the paper.
+Organization of the paper. Section 2 derives the basic estimates in a formal
+way. The proof of existence of a weak solution proceeds by approx imation, whereby
+the degeneracy of the equation is eliminated, diffusion is added and th e kernels are
+regularized. This is technically delicate, so we first prove existence o f weak solutions
+for the approximate problems posed in bounded domains in Section 3, and at the
+time basic estimates are rigorously derived. In Section 4 weak solutio ns of the original
+problem are constructed in the whole space by passage to the limit af ter a tail control
+step based on a novel argument with so-called suitable “true upper barriers”, cf.
+3Theorem 4.1. Such barrier method in new and turns out to be well ada pted to obtain
+comparison results in the presence of nonlocal operators.
+We then establish the main properties of the solutions: Section 5 est ablishes the
+property of finite propagation, which is a main feature of porous me dia equations
+and gives rise to the appearance of a free boundary. We discuss th e persistence of
+positivity in Section 6.
+Two sections close the paper: the Appendix, Section 7, gathers so me useful def-
+initions. A final Section 8 contains comments on variants, extension s or ongoing
+work on the topic of this paper: this refers in particular to the pend ing questions of
+uniqueness, smoothness or asymptotic behaviour.
+Notation. We will use the notation Ls= (−∆)swith 0< s <1 for the frac-
+tional powers of the Laplace operator defined on smooth function s inRnby Fourier
+transform and extended in a natural way to functions in the Sobole v spaceH2s(Rn).
+Technical reasons imply that in one space dimension the restriction s <1/2 will be
+observed. The inverse operator is denoted by Ks= (−∆)−sand can be realized by
+convolution
+Ks=Ks⋆u, K s(x) =c(n,s)|x|−(n−2s).
+as described in the appendix. Ksis a positive self-adjoint operator. We will write
+Hs=K1/2
+swhich has kernel Ks/2. The subscript swill be omitted when sis fixed
+and known. For functions that depend on xandt, convolution is applied for every
+fixedtwith respect to the space variables. We then use the abbreviated n otation
+u(t) =u(·,t).
+2 Basic estimates
+The existence theory of weak solutions needs a lengthy process ba sed on several
+approximations and passage to the limit that may obscure to the rea der the main
+properties of the solutions. These are however very clear from us ual considerations
+in mathematical physics, and the authors think that it useful for t he reader to have
+them in mind as a goal. Therefore, we do at this stage formal calculat ions, assum-
+ing thatu≥0 satisfies the required smoothness and integrability assumptions a nd
+decreases fast enough as |x| → ∞. The calculations are to be justified later by the
+approximation process. We fix s∈(0,1) and put K= (−∆)−sandH=K1/2.
+4•Conservation of mass
+(2.1)d
+dt/integraldisplay
+Rnu(x,t)dx=/integraldisplay
+Rn∇·(uKu)dx= 0.
+•First energy estimate:
+(2.2)d
+dt/integraldisplay
+Rnu(x,t)logu(x,t)dx=−/integraldisplay
+Rn(∇u·∇Ku)dx=−/integraldisplay
+Rn|∇Hu|2dx,
+where we use the fact that K=H2, andHis a positive self-adjoint operator that
+commutes with the gradient.
+•Second energy estimate:
+(2.3)1
+2d
+dt/integraldisplay
+Rn|Hu(x,t)|2dx=/integraldisplay
+RnHu(Hu)tdx=
+/integraldisplay
+RnKuutdx=/integraldisplay
+Rn(Ku)∇·(u∇Ku)dx=−/integraldisplay
+Rnu|∇Ku|2dx.
+•L∞estimate. We prove that the L∞norm does not increase in time.
+Sketch of the proof. At a point of maximum of uat timet=t0, sayx= 0, we have
+ut=∇u·∇P+u∆K(u).
+The first term is zero, and for the second we have −∆K=LwhereL= (−∆)qwith
+q= 1−sso that
+∆Ku(0) =−Lu(0) =−c/integraldisplay
+Rnu(0)−u(y)
+|y|n+2(1−s)dy≤0,
+wherec(s,n)>0.
+•Conservation of positivity: u0≥0 implies that u(t)≥0 for all times. The argument
+is similar.
+•We derive next the Lpestimates, 1 <p<∞:
+d
+dt/integraldisplay
+up(x,t)dx=p/integraldisplay
+Rnup−1utdx=−p(p−1)/integraldisplay
+Rnup−1∇u·∇Kudx=
+(p−1)/integraldisplay/integraldisplay
+(u(x)p−u(y)p)∆Ks(x−y)u(y)dxdy=
+−(p−1)/integraldisplay/integraldisplay
+(u(x)p−u(y)p)∆Ks(x−y)(u(x)−u(y))dxdy
++(p−1)/integraldisplay/integraldisplay
+(u(x)p−u(y)p)∆Ks(x−y)u(x)dxdy=−I1+I2.
+5Sinceu≥0and∆Ks≥0for0<s<1, thefirst integral I1ispositive. But symmetry
+means that I2=I1/2. We conclude that/integraltext
+updxis decreasing in time. See a related
+calculation in [15] and [18].
+•Astandard comparison result for parabolicequations does not see m to work. This
+is one of the main technical difficulties in the study of this equation. In fact, we will
+find special situations where some comparison holds. Some partial c omparison allows
+us to prove two main results of the paper.
+3 Existence I. Smooth approximate solutions
+We want to solve the initial-value problem for the equation
+(3.1) ∂tu=∇(u∇Ku),K= (−∆)−s,
+posed inQ=Rn×(0,∞) or at least QT=Rn×(0,T), with parameter 0 < s <1.
+We will take initial data u0(x)≥0,u0∈L1(Rn).We assume mostly for technical
+convenience that u0is bounded, and in the next section we will also impose decay
+conditions as |x| → ∞.
+We want to obtain a suitable weak solution u(x,t) defined in Q. We approach
+this problem by a process that consists of regularization, elimination of the degener-
+acy, and reduction of the space domain. Once the approximate pro blems are solved,
+estimates are obtained that allow to pass to the limit step by step in all the approxi-
+mations to obtain in the end a weak solution of the original problem.
+Definition. We say that uis a weak solution of equation (3.1) in QT=Rn×(0,T)
+with initial data u0∈L1(Rn) ifu∈L1(QT),K(u)∈L1(0,T:W1,1
+loc(Rn)), and
+u∇K(u)∈L1(QT), and the identity
+(3.2)/integraldisplay/integraldisplay
+u(φt−∇K(u)·∇φ)dxdt+/integraldisplay
+u0(x)φ(x,0)dx= 0
+holds for all continuous test functions φinQTsuch that ∇xφis continuous, and φ
+has compact support in the space variable and vanishes near t=T.
+3.1.The modifications that we use as a starting point are as follows: regu larization is
+done by adding Laplacian term plus a kernel smoothing; the degener acy is eliminated
+by raising the u= 0 level in the diffusion coefficient. Specifically, we take small
+numbersε,δ,µ∈(0,1) and consider the equation ( E(ε,δ,µ)):
+(3.3) ut=δ∆u+∇·(d(u)∇Kε(u)),
+6posed inQT,R={x∈BR(0),0< t < T}. A simple option for d(u) isd(u) =u+µ
+with a small µ>0. Another option would be d(u) =µfor 0≤u≤µandd(u) =u
+foru≥µ(we prefer the former one). Besides, the equation is posed in the s patial
+domainBR=BR(0) for 0<t<T. We also take initial conditions
+(3.4) u(x,0) = ˆu0(x)x∈BR(0),
+where ˆu0=u0,ε,Ris a nonnegative, smooth and bounded approximation of the initial
+datau0≥0. Finally, we take boundary data
+(3.5) u(x,t) = 0 for |x|=R, t≥0.
+Let us explain now a convenient approximation of the kernel: Kεu(t) =ζε⋆u(t), and
+ζεis a smooth approximation of the Riesz kernel ks(x) =c|x|−(n−2s)corresponding
+to the inversion of the s-Laplacian on Rn. In our implementation it acts on the
+extension of u(x,t) to the whole domain x∈Rn, which is done in the natural way,
+i.e., putting u= 0 for|x| ≥Randt>0. This approximation process is a bit similar
+to the approximation of the porous media performed by Lions and Ma s-Gallic in [26],
+but their kernel was just a mollifying kernel representing short-r ange effects and the
+consequences of a long-range kernel with slow decay at infinity are quite different.
+The existence and uniqueness of a solution u(x,t) =uε,δ,µ,R(x,t) for the model is
+then more or less standard, and the solution is smooth. In the weak formulation we
+have
+(3.6)/integraldisplay/integraldisplay
+u(φt−δ∆φ)dxdt−/integraldisplay/integraldisplay
+d(u)(∇Kε(u)·∇φ)dxdt+/integraldisplay
+u0(x)φ(x,0)dx= 0
+with double integrals in QT,R=BR(0)×(0,T), and valid for every smooth φthat
+vanishes at the lateral boundary and for all large t. It is also clear that a priori
+estimates like the ones in the previous section apply to this model. In p articular we
+have
+(E1.a) An easy computation gives the decay of the ”total mass”
+(3.7)/integraldisplay
+BRu(x,t)dx≤/integraldisplay
+BRˆu0(x)dx.
+Of course, we lose the expected mass conservation of the original problems because of
+the zero Dirichlet conditions of our present problem, but the estima te is still useful.
+(E1.b) We also need conservation of nonnegativity, which is easy.
+7(E1.c) The L∞
+xbound is conserved, 0 ≤u(x,t)≤ /ba∇dblˆu0/ba∇dbl∞, and the argument is as in
+the previous section. As a consequence, the solutions u(·,t) at timetbelong to all
+Lp(BR) spaces with norm that is independent of the parameters δ,ε,µandR.
+(E1.d) We now introduce a version of the first energy inequality of pr evious section.
+We select the function Fdefined by the conditions F′′(u) = 1/d(u) andF(0) =
+F′(0) = 0. After some integrations by parts we will get
+(3.8)d
+dt/integraldisplay
+BRF(u)dx=−δ/integraldisplay
+BR|∇u|2
+d(u)dx−/integraldisplay
+BR|∇Hεu|2dx,
+whereHε=K1/2
+εand we have used the fact that F′(u) = 0 on Σ = {|x|=R}×[0,T]
+to annihilate the boundary term in the integration by parts. This for mula implies
+that for all 0 <t<T:
+(3.9)/integraldisplay
+BRF(u(t))dx+δ/integraldisplayt
+0/integraldisplay
+BR|∇u|2
+d(u)dxdt+/integraldisplayt
+0/integraldisplay
+BR|∇Hεu|2dxdt=/integraldisplay
+F(ˆu0)dx.
+This implies estimates for |∇Hεu|2andδ|∇u|2/d(u) inL1
+x,t(QT,R) and the bounds
+for such norms are independent of ε,δ,R, andT. They do depend on µ>0 through
+the value of F=Fµ. Indeed, the explicit formula for F(u) is defined for u>0 as:
+(3.10) Fµ(u) = (u+µ)log(1+(u/µ))−u, F′
+µ(u) = log(1+( u/µ)).
+This offers difficulties when µ→0. (In case we take the second option, we would
+have
+Fµ(u) = (1/2µ)u2for 0≤u≤µ, F µ(u) =ulog(u/µ)+(µ/2) otherwise ,
+which is no better)
+Note.We take as Kεthe operator obtained by convolution with a standard molli-
+fication ofksof the form ζε=ks⋆ρεwhereρε(x) =ε−nρ(x/ε) andρis aC∞
+c(Rn),
+ρ≥0,ρradially symmetric and decreasing; moreover, if ρ=σ⋆σwhereσhas the
+same properties. Then, we can write Hεas the operator with kernel ks/2⋆σε.
+3.2.We have to pass to the limit in four parameters: δ,ε,µandR. The last two
+limits are the most delicate. In order to examine the convergence ar guments, we can
+try to pass to the limit ε→0 as a next step to obtain a solution of the equation
+(3.11) ut=δ∆u+∇·(d(u)∇K(u)),
+8with same initial and boundary data. Using in a precise way the above e stimates we
+get convergence of uε→uinL∞
+t(L1
+x) weak. This is enough for the limit of the first
+integral of the formula of weak solution. The second integral cont ains the product
+u∇Kuand we need to study better the consequences of the estimates:
+(i) Since K(u) =H(H(u)) and∇K(u) =∇H(H(u)) =H(∇(Hu)), we derive from
+the bound for ∇HuinL2
+x,tthe needed estimates for K(u). Recall that H(u) andK(u)
+are defined on all of Rn. We recall that ∇H(u) is a ”derivative of order 1 −sofu”,
+and sinceuis bounded, u∈L∞
+x,t, we conclude that u∈L2
+tH1−s
+x,loc. By potential theory,
+it is then clear that K(u)∈L2
+tH1+s
+x.
+(ii) We also need some continuity in time. Using the equation, which expr essesutas
+the divergence of u∇Ku,together with the boundedness of uand the bound for K(u)
+inH1+s
+x, we conclude that ut∈L2
+t(H−1+s
+x). Hence, the family of approximations to
+uis relatively compact in the sense of the parabolic compactness theo rems by Aubin
+andSimon, cf. [7], [32]. This meansthatalimit uexists (alongsuitablesubsequences)
+andu∈C([0,T] :L2(BR)).
+All of this is used in passing to the limit of the term/integraltext/integraltext
+u(∇Ku)∇φdxdtas follows:
+we have the convergence of uεinC([0,T] :L2(BR)) together with the weak conver-
+gence ofpε=Kεuεand∇pε. So we can pass to the limit in this term. The conclusion
+is that we have obtained a weak solution of the initial value problem for the equa-
+tion we have mentioned, posed in QRwith zero Dirichlet boundary conditions. The
+regularity of u,HuandKuis as stated before. We also have the energy formula
+(3.12)/integraldisplay
+BRFµ(u(t))dx+δ/integraldisplayt
+0/integraldisplay
+BR|∇u|2
+d(u)dxdt+/integraldisplayt
+0/integraldisplay
+BR|∇Hu|2dx=/integraldisplay
+Fµ(u0)dx.
+Remark. If we pass also to the limit δ→0, which is feasible with no extra effort,
+thenwe losethe H1estimates for u, andbesides wehave aproblem withtheboundary
+data that is maybe important. Therefore, we will keep the term δ∆ufor the moment
+to avoid the problem.
+3.3We will now try to pass to the limit, either as µ→0 or asR→ ∞, the order
+depending on convenience. In that sense we recall the detailed for m of the energy
+identity
+δ/integraldisplayt
+0/integraldisplay
+BR|∇u|2
+d(u)dxdt+/integraldisplayt
+0/integraldisplay
+BR|∇Hu|2dxdt+/integraldisplay
+BR(u0−u(t))dx
+/integraldisplay
+BR(u(t)+µ)log(1+u(t)
+µ)dx=/integraldisplay
+BR(u0+µ)log(1+u0
+µ)dx.
+9(Note: we write ubut we should write uRsince the solutions changes here with the
+radius of the ball). Terms 1, 2 and 3 are positive and behave well in bot h limits. It
+remains to examine the behavior of Terms 4 and 5. Both are nonnega tive which is
+good for us in the case of Term 4, but Term 5 diverges as µ→0. Therefore, our
+choice is letting R→ ∞. Now Term 5 is bounded by hypothesis (though the bound
+depends on µ). In the limit we easily get a solution of the problem in the whole space,
+for the equation
+(3.13) ut=δ∆u+∇·((u+µ)∇K(u)), x∈Rn, t>0.
+The limitδ→0 offers then no difficulty if one wants to take it. However, the limit
+µ→0 needs some extra properties.
+3.4.One of such useful properties is the Conservation of Mass, that we establish next
+for the solutions of (3.13).
+Lemma 3.1 Under the assumption that u0∈L1(Rn)∩L∞(Rn)the constructed non-
+negative solutions of the previous problem satisfy
+(3.14)/integraldisplay
+u(x,t)dx=/integraldisplay
+u0(x)dxfor allt>0.
+Proof. Recall that we assume s <1/2 ifn= 1. We integrate against a cutoff
+function to get ϕ∈C∞(Rn) supported in R≤ |x| ≤2Rwithϕ= 1 for|x| ≤R. We
+get /integraldisplay
+utϕdx=δ/integraldisplay
+u∆ϕdx−/integraldisplay
+(u+µ)(∇Ku·∇ϕ)dx=I1+I2.
+For the typical cutoff choice we estimate the first integral as I1=O(R−2) using the
+fact thatu(t)∈L1(Rn) and then I1→0 asR→ ∞. As for the last integral, we do
+I2=/integraldisplay
+Ku∇u∇ϕdx+/integraldisplay
+uKu∆ϕdx+µ/integraldisplay
+Ku∆ϕdx.
+The latter integral can be estimated as
+I23=µ/integraldisplay
+u(K∆ϕ)dx=µ/ba∇dblu/ba∇dbl1O(R−2(1−s))
+(where we use the fact that K∆ has homogeneity 2(1 −s) as a differential operator)
+and this goes to zero as R→ ∞.
+10Before we estimate the other two integrals we split the kernel Kinto a bounded part
+K1= min{1,K∫}and the rest K2=K∫−K1which is supported in a small ball.
+Both parts are nonnegative and moreover K1∈L∞andK2∈L1. It means that
+K1∗u∈Lq∗Lpforq >q0=n/(n−2s) (recall that we always have 2 s<n) and for
+everyp≥1, henceK1u∈Lpfor allp > q0, whileK2∗u∈Lq∗Lpwithq < q0and
+p≥1 which means that K2∗u∈Lpfor allp≥1. We conclude that Ku∈Lpfor
+p>q0. We then have
+I22=/integraldisplay
+uKu∆ϕdx≤C
+R2/ba∇dblu/ba∇dblq/ba∇dblKu/ba∇dblp
+which has in particular a bound of the form |I22| ≤C/ba∇dblu/ba∇dbl1R−2that goes to zero as
+R→ ∞. Finally,
+I21=/integraldisplay
+Ku(∇u·∇ϕ)dx.
+Since∇u∈L2and∇ϕ=O(R−1) and∇ϕ∈Lpwithp > nwe only need Ku∈Lq
+with a small q<2n/(n−2) which is true since q0<2n/(n−2), i.e., 4s<n+2.
+In the limit R→ ∞whenϕ= 1, we get (3.14).
+Theorem 3.2 Letu0∈L1(Rn)∩L∞(Rn),u0≥0. Then there exists a weak solution
+u=uδ,ε,µof the approximate equation (3.11)posed inQTwith initial data u0, and
+u∈L∞(0,∞:L1(Rn)),u∈L∞(Q),∇H(u)∈L2(Q). Moreover, for all t >0we
+have
+(3.15)/integraldisplay
+Rnu(x,t)dx=/integraldisplay
+Rnu0(x)dx,
+and/ba∇dblu(t)/ba∇dbl∞≤ /ba∇dblu0/ba∇dbl∞. The first energy inequality holds, in the form
+δ/integraldisplayt
+0/integraldisplay
+Rn|∇u|2
+u+µdxdt+/integraldisplayt
+0/integraldisplay
+Rn|∇Hu|2dxdt+
+/integraldisplay
+Rnu(t)log(u(t)+µ)dx+µ/integraldisplay
+Rnlog(1+(u/µ))dx≤
+/integraldisplay
+Rnu0log(u0+µ)dx+µ/integraldisplay
+Rnlog(1+(u0/µ))dx.
+Note that the last integral is less than/integraltext
+u0dx. The parameters δ,εandµare larger
+than 0. Passing to the limits δ→0,ε→0 offers now no difficulties. But we recall
+that before taking those limits the solution is smooth in xandt, and this may be
+convenient in justifying calculations to be done below.
+114 Tail control. Existence of weak solutions
+We start the section by stating the main existence theorem that will be proved here
+by passage to the limit µ→0 in the construction of the last section. The limit that
+we want to take is not trivial because of the possible behaviour of th e constructed
+solutions for large |x|which affects the convergence of some of the resulting integrals.
+We will pass to the limit in the last formula for the energy using as an ext ra tool a
+nice decay at infinity, so that the term/integraltext
+ulog−(u+µ)dxwill be bounded uniformly
+asµ→0.
+Theorem 4.1 Letu0∈L1(Rn)∩L∞(Rn)be such that
+(4.1) 0 ≤u0(x)≤Ae−a|x|for someA,a>0.
+Then there exists a weak solution uof Equation (3.1)with initial data u0. Besides,
+u∈C([0,∞) :L1(Rn)),u∈L∞(Q),∇H(u)∈L2(Q). For allt>0we have
+(4.2)/integraldisplay
+Rnu(x,t)dx=/integraldisplay
+Rnu0(x)dx,
+and/ba∇dblu(t)/ba∇dbl∞≤ /ba∇dblu0/ba∇dbl∞. The solution decays exponentially as |x| → ∞as explained in
+the next tail control subsection. The first energy inequalit y holds in the form
+(4.3)/integraldisplayt
+0/integraldisplay
+Rn|∇Hu|2dxdt+/integraldisplay
+Rnu(t)log(u(t))dx≤/integraldisplay
+Rnu0log(u0)dx,
+while the second says that for all 0<t1<t2<∞
+(4.4)/integraldisplayt2
+t1/integraldisplay
+Rnu|∇Ku|2dxdt+1
+2/integraldisplay
+Rn|Hu(t2)|2dx≤1
+2/integraldisplay
+Rn|H(u(t1)|2dx.
+We recall that Q=Rn×(0,∞), and we take 0 <s<1 for alln≥2, 0<s<1/2
+forn= 1. The result is first proved for the solutions of the approximate e quation
+withδ,ε,µ>0 constructed in Theorem 3.2, and the information is then transfer red
+to the original equation once we show that we can pass to the limit and the estimates
+are uniform.
+124.1 Tail control
+In order to prove the above existence theorem we need to estimat e a certain rate of
+decay of our solutions as |x| → ∞. This will of course depend of a similar assumption
+that we are imposing on the initial data. In the case of the PME a poss ible proof pro-
+ceeds by constructing explicit weak solutions (or supersolutions) w ith that property
+and then using the comparison principle, that holds for that equatio n. Since we do
+not have such a general comparison principle here, we have to devis e a comparison
+method with a suitable family of barrier functions that work because they are some
+kind of “exaggerated supersolutions”. What we need is to make sur e that they do not
+admit a first contact from below. We will give to functions with such a p roperty the
+more serious name of true supersolutions . This original technique has to be adapted
+to the peculiar form of the integral kernels involved in operator K. We prove two
+kinds of results, the stronger one for small s.
+Theorem 4.2 Let0< s <1/2and assume that our solution uis bounded 0≤
+u(x,t)≤L, andu0lies below a function of the form
+(4.5) U0(x) =Ae−a|x|, A,a> 0.
+IfAis large then there is a constant C >0that depends only on (n,s,a,L,A )such
+that for any T >0we will have the comparison
+(4.6) u(x,t)≤AeCt−a|x|for allx∈Rnand all0<t≤T.
+Proof. In order to have enough regularity in the comparison argument belo w we
+make the proof for the solutions constructed in Theorem 3.2 in the w hole space with
+parameters δ,εandµ>0 and we will show that the constants in the upper estimate
+are uniform with respect to such parameters if the mentioned para meters are small.
+•Reduction. By scaling we may put a=L= 1. This is done by considering instead
+ofuthe function /tildewideudefined as
+(4.7) u(x,t) =L/tildewideu(ax,bt), b=La2−2s,
+which satisfies the equation /tildewideut=δ1∆/tildewideu+∇ ·(/tildewided(/tildewideu)∇K(/tildewideu)) withδ1=a2sδ/L.Note
+that then /tildewideu(x,0)≤A1e−|x|withA1=A/L. A simple calculation then shows that
+C(a,L,A) =La2−2sC(1,1,A/L).
+13•Contact analysis. Therefore, we assume that 0 ≤u(x,0)≤1 and also that
+u(x,0)≤Ae−rr=|x|>0,
+whereA >0 is a constant that will be chosen below, say larger than 2. Given
+constantsC,εandη>0 we consider a radially symmetric candidate to upper barrier
+function of the form
+(4.8) /hatwideU(x,t) =AeCt−r+εAeηt,
+and we take εsmall. Then Cwill have to be determined in terms of Ato satisfy a
+“true supersolution condition” which is obtained by contradiction at the first point
+(xc,tc)∈QTof possible contact of uand/hatwideU. Note that if there is contact it cannot
+happen only at |x|=∞sinceuis integrable and /hatwideUconverges to a positive constant.
+This is the reason to add the correction term in ε(use of the correction term can be
+avoided if we start the comparison argument with the solutions of th e approximate
+problem posed in a ball BR(0) with zero boundary data and then pass to the limit
+R→ ∞as done in the previous section). The contact does not happen at xc= 0
+ifAis large since, putting rc=|xc|, we have at the contact point equality /hatwideU=
+AeCtc−rc+εAeηtc=uand alsou≤1, so thatA≤AeCtc≤ercsorcis big ifAis.
+We need at least A>1. Note that we are assuming 0 <tc≤TandTfixed (in this
+argument).
+At the point and time of contact we have u=AeCtc−rc+εAeηtc. Assuming also
+thatuisC2smooth, a standard argument also gives
+∂ru=−AeCtc−rc,∆u≤AeCtc−rc, ut≥ACeCtc−rc+εηAeηtc,
+(all of them computed at the point ( xc,tc)), and the spatial derivatives of uatxcin
+directions perpendicular to the radius are zero. Next, we put p=Kuand use the
+equation in full approximate form :
+(4.9) ut=δ∆u+∇u·∇p+(u+µ)∆p,
+to get the basic inequality
+CAeCtc−rc+εAηeηtc≤δAeCtc−rc−AeCtc−rc∂rp+(AeCtc−rc+εAeηtc+µ)∆p.
+Cleaning up this expression, we get at the contact point the following condition:
+(4.10) C+εηeξ≤δ−∂rp+(1+εeξ+µ1)∆p,
+14whereξ=rc+(η−C)tc,µ1= (µ/A)erc−Ctcand the overline indicates that the values
+of∂rpand ∆pare calculated at the point of contact.
+Summing up the main ideas. In (4.9) we have written the nonlinear term of the
+equation involving the fractional operator in “non-divergence for m”, precisely as the
+sum of a transport term involving first derivatives on pand another term containing
+the Laplacian of p. When we evaluate those terms at the contact point with the
+barrier and arrive at (4.10), the latter term is not difficult in terms of global integrals
+ofuand the other two first terms in (4.9) have also simple contributions. However,
+the transport term does not have a sign and its control becomes m ore difficult. We
+proceed with the simple cases and leave the difficult case for the end.
+•In order to get a contradiction with inequality (4.10) we will estimate t he values
+of∂rpand∆p.Forn≥1, 0<s<1, we use the formula
+(4.11) p(x,t) =c/integraldisplayu(x−y,t)
+|y|n−2sdy=c/integraldisplayu(x+y,t)
+|y|n−2sdy,
+with somec=c(s)>0, which produces the singular integrals
+(4.12) pxi(x,t) =c1/integraldisplay(u(x+y,t)−u(x−y,t))yi
+2|y|n+2−2sdy,
+(i= 1,···,n) that fors<1/2 can also be written as
+(4.13) pxi(x,t) =c1/integraldisplay(u(x+y,t)−u(x,t))yi
+|y|n+2−2sdy,
+and finally,
+(4.14) ∆ p(x,t) =c2/integraldisplayu(x+y,t)+u(x−y,t)−2u(x,t)
+|y|n+2−2sdy.
+Here and in the sequel of the proof we denote by c,ci,K,K idifferent absolute positive
+constants, i.,e., constants that depend only on nands.
+•We now estimate ∆patxc. In view of inequality (4.10) we need a control from
+above. Using formula (4.14) we see that the integral on the set |y| ≥1 is absolutely
+convergent and absolutely bounded (since uis bounded by 1). Besides, to bound
+from above the part of integral on the set |y| ≤1 we will use the fact that u(tc) lies
+belowAeCtc−|x|with tangency at |x|=rc; since the numerator of the integrand is the
+15discretization of the second derivative for uatxcwe use the fact that it is bounded
+above by the same expression for /hatwideUto get
+/integraldisplay
+|y|≤1u(xc+y,tc)+u(x−y,tc)−2u(x,tc)
+|y|n+2−2sdy≤/integraldisplay
+|y|≤12AeCtc−rc|y|2
+|y|n+2−2sdy,
+which is bounded since s >0. Noting that AeCtc−rc≤1 we conclude that the term
+in (4.10) containing ∆pcontributes with at most K1to the inequality.
+•Next, we estimate the transport term involving ∂rp. We have
+∂rp=c1/integraldisplay(u(x+y,t)−u(x,t))(ˆxc·ˆy)
+|y|n+1−2sdy,
+where ˆx=x/|x|and ˆy=y/|y|. Since we want to get a contradiction in formula (4.10)
+and∂rpappears with a minus sign, we need to estimate this term from below. T his
+integral is delicate so we split the calculation into several pieces. At t his moment we
+make the further assumption s<1/2. Then the integral for |y| ≥1/2 is bounded as:
+|I{|y|>1/2}(∂rp)| ≤c/integraldisplay
+|y|>1/2|u(xc+y,tc)−u(xc−y,tc)|
+|y|n+1−2sdy≤c/integraldisplay
+y≥11
+|y|n+1−2sdy≤K2.
+Hence, this part has the desired control.
+•The integral on the ball {|y| ≤1/2}is split again into two parts. By rotation
+we may assume that xcis directed along the first axis, xc= (r,0,···,0), where
+r=|x|. Then the integral is calculated on Ω 1={y:|y| ≤1/2,y1>0}and on
+Ω2={y:|y| ≤1/2,y1<0}. The last is easily bounded since utouchesUatxcand
+lies below everywhere at time tc. Hence,
+−IΩ2(∂rp) =/integraldisplay
+Ω2(u(xc,tc)−u(xc+y,tc))y1
+|y|n+2−2sdy=/integraldisplay
+Ω1(u(xc−y,tc)−u(xc,tc))y1
+|y|n+2−2sdy
+≤/integraldisplay
+Ω1(U(xc−y,tc)−U(xc,tc))y1
+|y|n+2−2sdy≤/integraldisplay
+Ω12AeCtc−rc
+|y|n+1−2sdy≤K3.
+where in the last inequality we have used the linear approximation for U. Again, this
+is a good control.
+•Now, the difficult part. The integral on Ω 1(i.e., the “half integral looking outside
+nearxc”) is more delicate since the difference u(x,tc)−u(x+y,tc) could in principle
+dropquiteabruptlyevenatarelativelyshortdistancefrom xcandthiswouldmakethe
+integral very big. However, we have a miraculous control by combin ing the estimate
+on the Laplacian with the good part of the estimate of ∂rp.
+16Lemma 4.3 With the previous assumptions and notations we have
+(4.15) −IΩ1(∂rp)+1
+2IΩ1(∆p)≤ −1
+2IΩ2(∂rp).
+Proof.We combine the integral of −∂rpwith a part of the integral for ∆pas follows:
+Y=−/integraldisplay
+Ω1(u(xc+y,tc)−u(xc,tc))y1
+|y|n+2−2s+1
+2/integraldisplay
+Ω1u(xc+y,tc)+u(xc−y,tc)−2u(xc,tc)
+|y|n+2−2sdy
+and we study carefully the integrand of Y. We have a numerator of the form
+(u(xc,tc)−u(xc+y,tc))y1+1
+2(u(xc+y,tc)+u(xc−y,tc)−2u(xc,tc)) =
+−(1
+2−y1)(u(xc,tc)−u(xc+y,tc))+1
+2(u(xc−y,tc)−u(xc,tc))
+Luckily, the first term is negative, hence we conclude that estimate (4.15) holds.
+•Wecannowfinishthecontradictionargument. Alltheseestimates a llowtoconclude
+that−I(∂rp)≤K4. Wegonowbackto(4.10)andconcludethattheinequalityimplies
+that
+C+ε(η−K)erc+(η−C)tc≤δ+K+Kµ
+Aerc,
+This cannot happen if we choose ε≥µ
+AandC=η≥2K, where we keep the extra
+condition that µandδmust be small (at least less than 1). Then εmay go to zero
+asµ→0.
+•Finally, we worry about the regularity of the solutions. To make the a bove proof
+fully rigorous, we apply the argument to the solutions of the regular ized problem
+where we smooth the velocity field ∇pby regularizing the kernel. We have to make
+the estimates for prand∆pwhen
+p(x,t) =/integraldisplay
+Kε(y)u(x+y)dy.
+where for instance Kε(y) =K(y) forε≤ |y| ≤1/ε,Kε(y) is a parabolic cap with C1
+fit in|y| ≤ε, and finally Kε(y) = 0 for |y| ≤2/ε. The regularization mentioned in
+Section 3 will also do. The solutions uεto this problem have bounded speeds, they
+are smooth and bounded with smooth and bounded pxand the previous estimates for
+prand∆phold uniformly in ε. Passing to the limits ε→0, the previous conclusions
+hold for any weak limit solution as constructed above, cf. equation ( 3.13). The extra
+limitδ→0 offers then no difficulty.
+17Theorem 4.4 Let now1/2≤s<1. Under the assumptions of the previous theorem
+the stated tail estimate works locally in time. The global st atement must be replaced
+by the following: there exists an increasing function C(t)such that
+(4.16) u(x,t)≤AeC(t)t−a|x|for allx∈Rnand all0<t≤T.
+Proof.(1) In the previous proof we had to put s<1/2 only because of the problem
+in estimating the integral for ∂rpon an exterior domain, away from the contact point.
+Whens≥1/2 we can estimate such integral as a convolution integral between uand
+the kernelK1(y) =y1|y|−n−2+2sχ(|y| ≥1). Now, this kernel belongs to Lp(Rn) for
+allp > n/(n+ 1−2s), hence we only need to bound u(t) in anLq(Rn) norm with
+1≤q<n/(2s−1) to get
+(4.17) I∗:=|I{|y|>1/2}(∂rp)| ≤ /ba∇dblu(t)/ba∇dblq/ba∇dblK1/ba∇dblp
+Moreover, since u∈L1(Rn)∩L∞(Rn) we have
+/ba∇dblu(t)/ba∇dblq≤ /ba∇dblu(t)/ba∇dbl1/q
+1/ba∇dblu(t)/ba∇dbl(q−1)/q
+∞
+We know that /ba∇dblu(t)/ba∇dbl∞≤1 andu(x,t)≤AeCte−|x|hence/ba∇dblu(t)/ba∇dbl1≤cAeCt. Therefore
+the term contributes
+I∗≤KA1/qeCtc/q
+which allows to go back to (4.10) and get
+(4.18) C+ε(η−K)erc+(η−C)tc≤δ+K+KA1/qeCtc/q+Kµ
+Aerc,
+Thecontradictionargumentworksasbeforewithonlyonebigdiffere nce. Onceweput
+C=η=KA1/qthe contradiction is gotten if we restrict the time so that eCtc/q≤2
+which happens if
+tc≤T1=c2/C=c3K−1A−1/q=c4A−1/q.
+We do not play with Ahere, onlyAbigger than 2 or the like.
+(2) Once we prove estimate (4.6) for 0 < t < T 1we can repeat the argument
+for another time interval, but now with constant A1=AeCT1=Aec2, and then
+C1=KA1/q
+1. We get a valid time interval
+T2=c4A−1/q
+1=c4A−1/qe−c2/q
+18where a new factor will appear every time we repeat the iterations. In this way we
+can extend the bound up to a certain time that depends on the initial data through
+the value of A.
+(3) In order to prove that the time for which the estimate is valid goe s forever we
+need to improve the L1norm ofu(t) by using the fact that u≤1 together with
+u(x,t)≤AeCte−|x|. Summing the contributions of both upper bounds, we now get
+/ba∇dblu(t)/ba∇dbl1≤c0(log(AeCt))n/parenleftbig
+1+A−1e−Ct/parenrightbig
+so that
+I∗≤K(logA+Ct)n/q/parenleftbig
+1+A−1e−Ct/parenrightbig1/q
+and then Inequality (4.18) may be replaced by
+(4.19) C+...≤K+K(logA+Ct)n/q/parenleftbig
+1+A−1e−Ct/parenrightbig1/q+Kµ
+Aerc,
+where we have dropped the terms in δ,εandµthat add no novelties or problems.
+Now we may put C=K[(logA)n/q+2] andCT1=c2so that
+T1=c4(logA)−n/q
+At the new starting time, we have A1=AeCT1=Aec2and then
+T2=c4(logA1)−n/q=c4(logA+c2)−n/q
+and so on. Since for large kwe getTk∼ck−n/qand we m ay always take q >n, the
+series/summationtextTkdiverges, so that Estimate (4.16) is global in time.
+4.2 Proof of the existence result
+We may now pass to the limit in the weak formulation of the equation sat isfied by
+u=uδ,ε,µconstructed at the end of previous section. The exponential dec ay bound
+on the solutions, which is uniform in µallows to improve the consequences of the
+energy inequality that is now written as
+δ/integraldisplayt
+0/integraldisplay|∇uµ|2
+uµ+µdxdt+/integraldisplayt
+0/integraldisplay
+|∇Huµ|2dxdt+/integraldisplay
+uµ(t)log+(uµ(t)+µ)dx
+≤/integraldisplay
+u0log(u0+µ)dx+µ/integraldisplay
+log(1+(u0/µ))dx+/integraldisplay
+uµ(t)log−(uµ(t)+µ)dx.
+19where we use the notation uµfor the solution for clarity. Since the right-hand side is
+bounded uniformly in µwe get a uniform estimate for the family |∇Huµ|inL2(Q).
+We also have a uniform bound on ∇uµinL2(Q) ifδ >0. We can therefore pass
+to the weak limit in uµ→˜u, and then Huµ→ H˜uand∇Huµ→ ∇H˜u. The same
+happens to Kuµand∇Kuµ. The weak solution is obtained and we have
+δ/integraldisplayt
+0/integraldisplay|∇˜u|2
+˜udxdt+/integraldisplayt
+0/integraldisplay
+|∇H˜u|2dxdt+/integraldisplay
+˜u(t)log+(˜u(t))dx
+≤/integraldisplay
+u0log(u0)dx+/integraldisplay
+˜u(t)log−(˜u(t))dx.
+Note that the term/integraltext
+µlog(1+(u0/µ))dxdisappears in the limit by the Dominated
+Convergence Theorem since the integrand is uniformly bounded by u0. Due to the
+decay at infinity the integral/integraltext
+˜u(t)log˜u(t)dxis absolutely convergent.
+The constructed solution has the same L∞bound asu0and conservation of mass
+holds by the proof of Lemma 3.1.
+Passing to the limit δ→0 andε→0 offers no difficulty, so Theorem 4.1 is proved
+but for the second energy estimate.
+4.3 Second energy estimate
+Let us establish the second energy estimate for weak solutions with tail decay. We
+compute formally of the approximations where everything is justifie d
+1
+2d
+dt/integraldisplay
+ϕ|Hu(x,t)|2dx=/integraldisplay
+ϕHuHutdx=/integraldisplay
+H(ϕHu)utdx=
+/integraldisplay
+H(ϕHu)∇(u∇Ku) =−/integraldisplay
+u∇H(ϕHu)·∇Kudx=
+−/integraldisplay
+uH(ϕ∇Hu)·∇Kudx−/integraldisplay
+uH(∇ϕHu)·∇Kudx,
+so that, putting ψ= 1−ϕwe have
+1
+2d
+dt/integraldisplay
+ϕ|Hu(x,t)|2dx+/integraldisplay
+ϕu|∇Ku|2dx=
+/integraldisplay
+u(ψ∇Ku−H(ψ∇Hu))·∇Kudx−2/integraldisplay
+Hu∇ϕ·H(u∇Ku)dx,
+With the properties we have for uthe right-hand side must go to zero as ϕ→1 along
+the typical cutoff sequence, at least after integration in time.
+205 Finitepropagation. Solutionswithcompact sup-
+port
+One of the most important features of the porous medium equation and other related
+degenerate parabolic equations is the property of finite propagat ion, whereby com-
+pactly supported initial data u0(x) give rise to solutions u(x,t) that have the same
+property for all positive times, i.e., the support of u(·,t) is contained in a ball BR(t)(0)
+for allt>0 andR(t) is bounded on bounded intervals 0 <t<T.
+A possible proof in the case of the PME proceeds by constructing ex plicit weak
+solutions with that property (and possibly larger initial data) and th en using the
+comparison principle, which holds for that equation. Since we do not h ave such a
+general principle here, we have to devise a comparison method with a suitable family
+of upper barriers that behave as “true (exaggerate) supersolu tions”. The technique
+has been presented in whole detail in the tail analysis of the previous section and is
+here adapted to the peculiar needs of bounded support. Here is th e end result.
+Theorem 5.1 Assume that uis a bounded solution, 0≤u≤L, of equation (3.1)
+withK= (−∆)−swith0<s<1 (0<s<1/2ifn= 1), as constructed in Theorems
+4.2and4.4. Assume that u0has compact support. Then u(·,t)is compactly supported
+for allt >0. More precisely, if 0< s <1/2andu0is below the ”parabola-like”
+function
+(5.1) U0(x) =a(|x|−b)2,
+for somea,b>0, with support in the ball Bb(0), then there is Clarge enough, such
+that
+(5.2) u(x,t)≤a(Ct−(|x|−b))2
+Actually, we can take C=C0(n,s)L(1/2)+sa(1/2)−s. For1/2≤s <2a similar
+conclusion is true, but now Cis an increasing function of tand we do not obtain a
+scaling formula for its dependence of Landa.
+Proof.The application of the method is very similar to the case worked out th e
+tail control section. Therefore, we will dispense with some of the t echnicalities of
+regularization to gain space and clarity. We assume that our solution u(x,t)≥0 has
+bounded initial data u0(x) =u(x,t0)≤Land also that u0is below the parabola
+U0(x) =a(|x|−b)2, a,b>0; moreover, the support of u0in the ball of radius band
+21the graphs of u0andU0are strictly separated in that ball. We take as comparison
+functionU(x,t) =a(Ct−(|x| −b))2and argue at the first point and time where
+u(x,t) touchesUfrom below. The fact that such a first contact point happens for
+t>0 and does not happen at x=∞is justified by regularization as before. We put
+r=|x|.
+By scaling we may put a=L= 1. See detail of the reduction step below. We
+examineindetailthesituationinwhichthetouchingpoint( xc,tc)isnottheminimum,
+say,xclies at a distance from the front |xf(t)|:=b+Ct, so thatb+Ctc−|xc|=h>0.
+Note that since u≤1 we must have |h| ≤1. Assuming also that uisC2smooth, a
+standard argument gives
+u=h2, ur=−2h,∆u≤2n, u t≥2Ch,
+all of them computed at the point ( x,t) = (xc,tc). Putting p=Kuand using the
+equationut=∇u·∇p+u∆p, we get the inequality
+(5.3) 2 Ch≤ −2hpr+h2∆p,
+where the overline indicates that the values of prand∆parecalculated at the point of
+contact. Moreover, u(x,tc)≤(xf−x)2for allx∈Rn. In order to get a contradiction
+we will estimate the values of prand∆p. The formulas for p,prand ∆pin terms of
+uare given in (4.11), (4.12), (4.13), (4.14).
+•Estimating ∆pat the contact point offers no novelties. As before and in view
+of inequality (5.3) we need a control from above. Using formula (4.14 ) we see that
+the integral on |y| ≥1 is absolutely convergent and bounded (since uis bounded).
+Besides, to bound the integral for |y| ≤1 from above we will use the fact that ulies
+below a parabola, hence
+/integraldisplay
+|y|≤1u(x+y,tc)+u(x−y,tc)−2u(x,tc)
+|y|n+2−2sdy≤/integraldisplay
+|y|≤12|y|2
+|y|n+2−2sdy,
+which is bounded since s >0. We conclude that the term in (5.3) containing pxx
+contributes with at most Kh2to the inequality, where K >0 is an absolute constant.
+•Next, we estimate pr. Much of the argument is similar to the tail analysis but
+the end is more delicate. Since we want to get a contradiction in formu la (5.3) and
+the coefficient ur=−2hofpris negative, we need to estimate this term from below.
+Using formula (4.12) we see again that the integral for |y| ≥1/2 is absolutely and
+uniformly bounded by a constant K2.
+22•The integral on the ball {|y| ≤1/2}is split into several parts. We will drop for
+convenience of writing the dependence on tcin the formulas the follow. By rotation
+we may assume that xcis directed along the first axis, xc= (r,0,···,0), where
+r=|x|. Then the integral is calculated on Ω 1={y:|y| ≤1/2,y1>0}and on
+Ω2={y:|y| ≤1/2,y1<0}. The last is easily bounded since utouchesUatxcand
+lies below everywhere at time tc. As in the tail analysis we have,
+−IΩ2(∂rp) =/integraldisplay
+Ω2(u(xc)−u(xc+y))y1
+|y|n+2−2sdy≤K3.
+•The integral on Ω 1(i.e., the “half integral looking outside near xc”) is more delicate
+as we have said, since the difference u(x,tc)−u(x+y,tc) could in principle drop quite
+abruptly even at a relatively short distance from xcand this would make the integral
+very big. There is a part with |y|betweenθhand 1 for any given θ∈(0,1) that is
+easy (note that 0 ≤u(x,0)−u(x+y,t)≤u(x,0) =h2):
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1
+θhu(xc+y)−u(xc)
+|y|n+1−2sdy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤h2/integraldisplay1
+θh1
+|y|n+1−2sdy=ch2(θh)−1+2s=c3(θ)h1+2s
+and this is good even for small h.
+•The last part of the integral for pr, over the half-ball H1={0<|y|< θh}with
+y1>0, is in principle bad since u(xc+y) could drop abruptly, thus making the
+integral very negative or even divergent near y= 0. We are going to combine the
+integral of −∂rpwith a part of the integral for ∆pas follows:
+Y=−/integraldisplay
+H1(u(xc+y)−u(xc)y1
+|y|n+2−2s+h
+2/integraldisplay
+H1u(xc+y)+u(xc−y)−2u(xc)
+|y|n+2−2sdy
+and we study carefully the integrand of Y. We have a numerator of the form
+(u(xc)−u(xc+y)y1+h
+2(u(xc+y)+u(xc−y)−2u(xc)) =
+−(h
+2−y1)(u(xc)−u(xc+y))+h
+2(u(xc−y)−u(xc))
+Luckily, the first term is negative (we take 0 <θ<1/2 (a security factor), hence we
+conclude that
+−IH1(∂rp)+h
+2IH1(∆p)≤ −h
+2IΩ2(∂rp).
+•We may now sum up all the terms in the right-hand side of (5.3) and sho w that
+they are bounded above by KhwhereKis a uniform constant. Therefore, for large
+23Cinequality (5.3) is impossible, hence there cannot be a contact point w ithh/ne}ationslash= 0.
+In this way we get a minimal constant C=C0(n,s) for which such contact does not
+take place.
+•Reduction Step. We use it to get the dependence on Landa. Here, it goes as
+follows: since the equation scaling is
+/hatwideu(x,t) =Au(Bx,Tt)
+with parameters A,B,T > 0 such that T=AB2−2s, if we have done the proof for u
+that has height 1 and is below ( |x|−b0)2initially, and get a comparison with a Uas
+above with speed C0>0, then the assumptions /hatwideu(x,t)≤Land/hatwideu(x,t0)≤a(|x|−b)2
+initially, are satisfied if we put
+A=L, AB2=a, b=b0/B,
+i.e.,A=L,B= (a/L)1/2, and thenT=a1−sLs. The new speed is then
+/hatwideC=C0T
+B=C0a1/2−sL1/2+s.
+•We still have to consider the modification of the proof when 1 /2≤s<1. The only
+problem is the estimate of ∂rpon the exterior of a ball. This is done as in the tail
+control case.
+We next lemmas complete the details of the comparison proof that ha ve been left
+out in the previous lemma.
+Lemma 5.2 Under the assumptions of Prop. 5.1there is no contact either at u= 0,
+in the sense that strict separation of uandUholds for all t>0ifCis large enough
+as in the previous lemma.
+Proof.Precisely, what we want is to eliminate the possible contact of the sup ports
+at the lower part of the parabola. Instead of doing this by analyzing the possible
+contact point, we proceed by a change in the test function that we replace by
+Uε= (Ct−(|x|−b))2+ε(1+Dt) for|x| ≤b+Ct,
+andUε=ε(1+Dt) for 1x| ≥b+Ct. Hereε>0 is a small constant and D>0 will
+be suitably chosen. Assume that the solution starts at t= 0 and touches this test
+functionUεfor the first time at the time t=tc.
+24The argument for a contact at the points |x|<b+CtcwhereUεis a parabola works
+like previously.
+The argument at contact points |x| ≥b+Ctcleads to
+(5.4) Dε≤ε(1+Dtc)∆p,
+where the overline indicates as before that the value ∆ pare calculated at the point
+of contact. Moreover, u(x,tc)≤U1(x,tc) for allx∈R. If we are able to prove again
+that∆p≤Kwe will get
+Dε≤ε(1+Dtc)K
+which is contradictory if D > K, sayD= 2K, andtc<1/D= 1/(2K), This
+estimate is uniform in εand gives in the limit and upper bound of the form u(x,t)≤
+(Ct−(|x|−b))2for all 0<t<1/2K, and as a corollary, the support of ubounded
+on the right by the line |x|=Ct+bin that time interval. After this time the process
+can be repeated and the conclusion is true for all times.
+•We now reflect for a moment on the regularity requirements. Arguin g as in the tail
+control case by using the smooth solutions of the approximate equ ations, the previous
+conclusions hold for any weak limit solution.
+5.1 Consequences. Growth estimates of the support
+The following analysis is done for s <1/2 and concerns bounded solutions with
+compactlysupportedinitialdata. Byfreeboundarywemean, asus ual, thetopological
+boundary of the support of the solution S(u) :={(x,t) :u>0}.
+Corollary 5.3 Letu0be bounded above by Lwithu0(x) = 0for|x| ≤R. Then, we
+get an estimate for the free boundary points of the form |x(t)| ≤R+C2t1/(2−2s)if
+s<1/2.
+Proof:We know that the support of u(·,t) isbounded for all times, say, it is contained
+in a ball of radius r(t). We take a time t1and find a parabolic barrier as before, with
+coefficienta>0 that is initially above and separated from u(·,t1). This can be done
+by choosing first r1such thatar2
+1=Landthen putting intheformula oftheparabolic
+barrierb=r(t1)+r1+ǫ, and then we can go forever in time in the comparison. Using
+the speed estimate in Theorem 5.1 we get in the limit ǫ→0:
+r(t)−r(t1)−r1≤C(t−t1) =C0L(t1)1/2+sa1/2−s(t−t1) =C0L(t1)(t−t1)/r1−2s
+1
+25Herer1>0 is free by moving a. We can use the L∞boundL(t1)≤L(0) We get
+r(t)≤r(t1)+r1+C0L(t−t1)/r1−2s
+1
+Now taket1= 0 and optimize the right-hand expression in r1>0 fors <1/2. .
+Notice that in the limit s= 1/2 we would get linear growth, while for s= 0 we get
+the standard t1/2growth of the Porous Medium Equations under these assumptions.
+6 Persistence of positivity
+We establish another property that plays an interesting role in the t heory of porous
+medium equations toavoid degeneracy points forthesolutions. It is called persistence
+of positivity. For continuous solutions it implies non-shrinking of the s upport.
+Theorem 6.1 Letube a weak solution as constructed in Theorem (4.1)and assume
+thatu0(x)is positive in a neighborhood of a point x0. Thenu(x0,t)is positive for all
+timest>0.
+Proof.This issue allows for another use of the technique presented inthe t ail analysis,
+butthistimewithatruesubsolutions. Weassumethat u0(x)≥c>0inaballBR(x0).
+By translation and scaling we may assume that x0= 0 andc=R= 1. The idea is
+to study the contact point with a parabola that shrinks quickly in time , like
+U(x,t) =e−atF(|x|),
+withFsuitably chosen and a >0 large enough. Firstly, we choose Fto be radially
+symmetric and decreasing, with F(0) = 1/2 andF(|x|) = 0 for |x|>1/2. The
+contact point ( xc,tc) is sought in B1/2(0)×(0,∞). By approximation we may assume
+thatuis positive everywhere so no contact at the parabolic border is assu med. At a
+positive contact point we have ut≤Ut=−aU,∇u=e−atF′(|x|)erand the standard
+arguments on the equation imply
+−aF(|x|)≥F′(|x|)pr+F(|x|)∆xp
+Now we have
+p(x,t) =Ku(x,t) =e−at(KF)(r)
+26As in the tall control analysis we prove that ∆xpis bounded uniformly. The novelty
+is thatF′≤0 implies that pr≤0 so that the term F′(|x|)pr≥0. In this way the
+inequality implies
+a≤Ke−at
+which is false if a > K. Hence, under this size assumption there can be no contact
+point, andUis a true subsolution, and positive for all times in B1/2(0).
+7 Appendix. Fractional Laplacians
+We collect some data on fractional Laplacians for the reader’s conv enience
+Fractional Laplacians and potentials. According to Stein [33], Chapter V, the
+definition of ( −∆)β/2is done by means of Fourier series
+((−∆)β/2f)/hatwide(x) = (2π|x|βˆf(x)
+and can be used for positive and negative values of β. Forβ=−αnegative, with
+0<α<n, we have the equivalence with the Riesz potentials [30]
+(7.1) ( −∆)−α/2f=Iα(f) :=1
+γ(α)/integraldisplay
+Rnf(y)
+|x−y|n−αdy
+(acting on functions of the class Sfor instance) with precise constant
+γ(α) =πn/22αΓ(α/2)/Γ((n−α)/2).
+Note that γ→ ∞asα→n, butγ/(n−α) converges to a nonzero constant,
+πn/22n−1Γ(n/2). Stein also mentions the Bessel potentials which are associated t o
+the modified inverse Laplace operators
+((I−∆)−α/2f) =Iα(f)
+The Bessel potential has a kernel Gα(x) that is better behaved at infinity, though not
+given by a simple kernel (see [33], page 132).
+Fractional Laplacians via extensions
+In the case s= 1/2 it is the well-known technique of harmonic extension to the upper
+plane of the space with one more dimension and then taking the bound ary normal
+derivative, and has been used in the study or variational problems w ith thin obstacles
+as in [3], see also [9, 10]. For s/ne}ationslash= 1/2 the method has been recently developed in [12],
+it involves elliptic equations with weights and weighted normal derivativ es.
+278 Comments and extensions
+Extensions. Very different versions to the evolution process are obtained when the
+pressure is related to uin other ways. Thus, we can use a pressure-density relation of
+the formp=f(u) withfan increasing function, and then the model equation would
+be
+(8.1) ∂tu=∇·(u∇K(f(u))).
+Many of the results proved here should apply to this model. Another possibility
+consists of equations of the form ∂tu=∇·(f(u)∇K(u)).
+Relaxation. Chemotaxis models. We could also relax the relation of ptouinto
+the form
+(8.2) ∂tp+(−∆)sp=u.
+This reflection is motivated by a very important system, the Keller-S egel chemotaxis
+model, [23, 22], in which the phenomenon which is modeled by uis not diffusion but
+the concentration of a certain population and pis replaced by variable cproportional
+to the concentration of the chemical substance responsible for t he aggregation of the
+population. A suitable general system is proposed in the form
+(8.3) ut=ε∆u−∇·(u∇c), δct+K−1c=f(u,c).
+The standard chemotaxis model uses K=−∆ andf(u,c) =u−bc. In the limit case
+whereδandbare zero, and if we use as Kan integral operator as described above, we
+get amodel with aterm like oursbut notethedifferent sign, ut=ε∆u−∇·(u∇K(u)),
+which is a consequence of the fact that we are dealing with aggregat ion and not
+diffusion. The study of these equations is also of interest.
+Finite propagation. The finite propagation property is not true for other alterna-
+tive models of porous medium equation with fractional diffusion like the ones studied
+in [4], [28] The last reference deals with the model
+(8.4)∂u
+∂t+(−∆)1/2(|u|m−1u) = 0,
+For anym>m ∗= (n−1)/n, it is proved that a unique nonnegative strong solution
+of this problem exists for data in L1
++(Rn) and is strictly positive. The maximum
+principle applies to this problem.
+28Uniqueness and comparison. These are widely open issues. We have used in
+the present paper comparison with what we call “true supersolutio ns” and “true
+subsolutions”. In one space dimension, a uniqueness proof has bee n obtained in [8]
+by integrating the equation (with respect to x) and using then solutions in the sense
+of viscosity. Such trick is not available in several dimensions.
+Evolution and regularity of free boundaries. This is quite important topic
+motivated by the property of finite propagation.
+Smoothing L1intoL∞andCαregularity. These topics will be treated in [13].
+Asymptotic behaviour. This topic is under study. Let us outline the main details.
+There exists a family of self-similar solutions for this problem, in the sp irit of the fun-
+damental solution of the linear problems or the Barenblatt solutions of the standard
+Porous Medium Equation. The spatial profile of such solutions is obta ined by solving
+for the pressure pan obstacle problem with a truncated paraboloid as obstacle; the
+corresponding density uis then the mass of the negative fractional Laplacian of p,
+supported on the contact set; all this fits perfectly into the elliptic theory described
+[6]). The details of the construction, as well as the convergence of a typical solution
+to such asymptotic profiles after suitable scaling, will be established in an upcoming
+publication, [14].
+Acknowledgment. The work was started while JLV was an Oden Fellow at the
+ICES Institute, Univ. of Texas at Austin. LAC was also a guest of th e Univ.
+Aut´ onoma de Madrid. His work was supported by a National Science Foundation
+grant. JLV was partially supported by Spanish Project MTM2008-0 6326-C02 and
+by ESF Programme “Global and geometric aspects of nonlinear part ial differential
+equations”.
+References
+[1] L.Ambrosio, S.Serfaty. Agradientflowapproachtoanevolutionproblemarising
+in superconductivity, Preprint, 2007.
+[2] D. G. Aronson. The Porous Medium Equation. In “Nonlinear Diffusion Prob-
+lems”,LectureNotesinMath. 1224, A.FasanoandM. Primicerio eds., Springer-
+Verlag New York, 1986, pp. 12–46.
+29[3] I. Athanasopoulos and L. A. Caffarelli. Optimal regularity of lower dimensional
+obstacle problems . Preprint.
+[4] I. Athanasopoulos and L. A. Caffarelli. Continuity of the temperature in bound-
+ary heat control problem . Adv. Math., to appear (2009, online).
+[5] I. Athanasopoulos, L. A. Caffarelli, andS. Salsa. The structure of thefreebound-
+ary for lower dimensional obstacle problems . arXiv:math/0609031v1 [math.AP]
+[6] I. Athanasopoulos, L. A. Caffarelli, andS. Salsa. The structure of thefreebound-
+ary for lower dimensional obstacle problems. Amer. J. Math. 130(2008), no. 2,
+485–498.
+[7] J. P. Aubin. Un th´ eor` eme de compacit´ e , C. R. Acad. Sci. 256(1963), 5042–5044.
+[8] P. Biler, G. Karch, and R. Monneau. Nonlinear diffusion of dislocation density
+and self-similar solutions . Comm. Math. Phys., (2009), online.
+[9] L. A. Caffarelli. Further regularity for the Signorini problem. Comm. Partial
+Differential Equations 4 (1979), no. 9, 1067–1075.
+[10] L. A. Caffarelli. The obstacle problem revisited. J. Fourier Anal. Appl. 4(1998),
+no. 4-5, 383–402.
+[11] L. A. Caffarelli, S. Salsa, and L. Silvestre. Regularity estimates for the solu-
+tion and the free boundary to the obstacle problem for the fractio nal Laplacian ,
+Invent. Math. 171(2008), no. 2, 425–461.
+[12] L. A. Caffarelli and L. Silvestre. An extension problem related to the fractional
+laplacian , Comm. Partial Differential Equations 32(2007), 12451260.
+[13] L. A. Caffarelli, F. Soria, and J. L. Vazquez. In preparation.
+[14] L. A. Caffarelli and J. L. Vazquez. In preparation.
+[15] L. A. Caffarelli and A. Vasseur. Drift diffusion equations with fractional diffusion
+and the quasi-geostrophic equation. To appear in Annals of Math.
+[16] S. J. Chapman, A mean-field model of superconducting vortices in three dimen-
+sions.SIAM J. Appl. Math. 55 (1995), no. 5, 1259–1274.
+[17] Chapman, S. J., Rubinstein, J. Schatzman, M. A mean-field mode l of supercon-
+ducting vortices. European J. Appl. Math. 7 (1996), no. 2, 97–11 1.
+30[18] A. C´ ordoba and D. C´ ordoba. A pointwise estimate for fractionary derivatives
+with applications to partial differential equations, Proc. Natl. Acad. Sci. USA
+100 (2003), no. 26, 15316–15317.
+[19] Weinan E, Dynamics of vortex liquids in Ginsburg–Landau theories with appli-
+cation to superconductivity, Phys. Rev. B 50 (1994) 1126–1135.
+[20] D. Gilbarg and N. S. Trudinger. “Elliptic partial differential equations of second
+order”. Reprint of the 1998 edition. Classics in Mathematics. Springer, Ber lin,
+2001.
+[21] A.K. Head. Dislocationgroupdynamics II. Similarity solutions ofthecontinuum
+approximation. Phil. Mag. 26(1972), 65–72.
+[22] W. J¨ ager and S. Luckhaus, On explosions of solutions to a system of partial
+differential equations modelling chemotaxis, Trans. AMS 329 (1992), 819–824.
+[23] E. Keller, andc L. Segel, Initiation of slime mold aggregation viewed as an insta-
+bility,J. Theor. Biol, 26 (1970), pp. 399–415.
+[24] N. S. Landkof. “Foundations of modern potential theory”. Die Grundlehren der
+mathematischen Wissenschaften, Band 180. Translated from the Russian by A.
+P. Doohovskoy. Springer, New York, 1972.
+[25] L. S. Leibenzon. General problem of the movement of a compre ssible fluid in a
+porous medium, Izv. Akad. Nauk SSSR , Geography and Geophysics 9(1945),
+7–10 (Russian).
+[26] P. L. Lions and S. Mas-Gallic. Une m´ ethode particulaire d´ eterministe pour des
+´ equations diffusives non lin´ eaires, C.R. Acad. Sci. Paris t. 332 (s´ erie 1) (2001)
+369–376.
+[27] M. Muskat. The Flow of Homogeneous Fluids Through Porous Media, McGraw-
+Hill, New York, 1937.
+[28] A.dePablo, F.Quir´ os, A.Rodriguez, J.L.V´ azquez, Afractionalporousmedium
+equation, preprint.
+[29] D.Richardson. Doctoraldissertation. University ofBritishColumbia, Vancouver,
+1978.
+31[30] M. Riesz. L’int´ egrale de Riemann-Liouville et le probl` eme de Cauchy , Acta
+Mathematica 81 (1949), 1–223.
+[31] L. E. Silvestre. H¨ older estimates for solutions of integro differential equations
+like the fractional Laplace. Indiana Univ. Math. J. 55(2006), no. 3, 1155–1174.
+[32] J. Simon. Compact sets in the space Lp(0,T;B), Ann. Mat. Pura Appl. (4) 146
+(1987), 65–96.
+[33] E. M. Stein. “Singular integrals and differentiability properties of functions” .
+Princeton Mathematical Series, No. 30. Princeton University Pres s, Princeton,
+N.J., 1970.
+[34] J. L. V´ azquez. “The porous medium equation. Mathematical theory” , Oxford
+Mathematical Monographs. The Clarendon Press, Oxford Univers ity Press, Ox-
+ford (2007).
+Addresses:
+Luis A. Caffarelli
+School of Mathematics, Univ. of Texas at Austin, 1 University Stat ion, C1200,
+Austin, Texas 78712-1082.
+Second affiliation: Institute for Computational Engineering and Scie nces.
+e-mail: caffarel@math.utexas.edu
+Juan Luis V ´azquez
+Departamento de Matem´ aticas, Universidad Aut´ onoma de Madrid , 28049 Madrid,
+Spain. Second affiliation: Institute ICMAT.
+e-mail: juanluis.vazquez@uam.es
+2000Mathematics Subject Classification. 35K55, 35K65, 76S05.
+Keywords and phases. Porous medium equation, fractional Laplacian, nonlocal
+operator, finite propagation.
+32
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+arXiv:1001.0411v1  [gr-qc]  3 Jan 2010October 29, 2018 17:23 WSPC - Proceedings Trim Size: 9.75in x 6.5in hoqufo
+1
+Holographic Quantum Foam
+Y. Jack Ng
+Institute of Field Physics, Department of Physics and Astro nomy,
+University of North Carolina, Chapel Hill, NC 27599-3255, U SA
+E-mail: yjng@physics.unc.edu
+Due to quantum fluctuations, probed at small scales, spaceti me is very complicated
+—something akin incomplexity to a turbulent frothwhich the late John Wheeler dubbed
+quantum foam, aka spacetime foam. Our recent work suggests t hat (1) we may be close
+to being able to detect quantum foam with extragalactic sour ces once the Very Large
+Telescope Interferometers (VLTI) are fully operational; ( 2) dark energy is arguably a
+cosmological manifestation of quantum foam, the constitue nts of which obey infinite
+statistics; (3) in the gravitational context, turbulence i s closely related to holographic
+quantum foam, partly validating Wheeler’s picture of a turb ulent spacetime.
+Keywords : quantum foam, holography, turbulence, dark energy, infini te statistics
+1. Introduction
+At microscopic scales our world is known to obey quantum mechanics w hich is
+characterized by an indeterminism giving rise to fluctuations in measu rements. If
+spacetime, like all matter and energy, undergoes quantum fluctua tions, there will be
+an intrinsic limitation to the accuracy with which one can measure a dist ancel, for
+that distance fluctuates by δl. On fairly general grounds, we expect δl/greaterorsimilarl1−αlα
+P,1
+wherelPis the Planck length, the characteristic length scale in quantum grav ity.
+The parameter α∼1 specifies the different quantum foam models. (The canonical
+model2corresponds to α= 1 with δl∼lP.)
+Applying quantum mechanics and black hole physics (from general re lativity),
+we3find that a distance lfluctuates by an amount ∼l1/3l2/3
+P. The corresponding
+quantum foam model with α= 2/3 has become known as the holographic model
+since it can be shown1to be consistent with the holographic principle.4
+2. Turbulence and Holography
+There are deep similarities between the problem of quantum gravity a nd turbu-
+lence.5The connection between these seemingly disparate fields is provided by
+the role of diffeomorphism symmetry in classical gravity and the volum e pre-
+serving diffeomorphisms of classical fluid dynamics. Furthermore, in the case
+of irrotational fluids in three spatial dimensions, the equation for t he fluctua-
+tions of the velocity potential can be written in a geometric form6of a har-
+monic Laplace–Beltrami equation:1√−g∂a(√−ggab∂bϕ) = 0 . Here, apart from a
+conformal factor, the effective space time metric has the canonic al ADM form
+ds2=ρ0
+c[c2dt2−δij(dxi−vidt)(dxj−vjdt)], where cis the sound velocity and
+viare the components of the fluid’s velocity vector. We observe that in this ex-
+pression for the metric, the velocity of the fluid viplays the role of the shift vector
+Niwhich is the Lagrange multiplier for the spatial diffeomorphism constr aint (theOctober 29, 2018 17:23 WSPC - Proceedings Trim Size: 9.75in x 6.5in hoqufo
+2
+momentum constraint) in the canonical Dirac/ADM treatment of Ein stein gravity:
+ds2=N2dt2−hij(dxi+Nidt)(dxj+Njdt). Hence in the fluid dynamics context,
+Ni→vi, and a fluctuation of viwould imply a fluctuation of the shift vector.
+This is possible provided the metric of spacetime fluctuates, which is a very loose,
+intuitive, semi-classical definition of the quantum foam.
+But which quantum foam model? Recall δℓ∼ℓ1−αℓα
+P. If one defines the velocity
+asv∼δℓ
+tc, where the natural characteristic time scale is tc∼ℓP
+c, then it follows that
+v∼c/parenleftbigℓ
+ℓP/parenrightbig1−α. It is now obvious that a Kolmogorov-like scaling7in turbulence has
+been obtained, i.e., the velocity scales as v∼ℓ1/3and the two-point function has
+the needed two-thirds power law provided that α= 2/3. Since the velocities play
+the role of the shifts, they describe how the metric fluctuates at t he Planck scale.
+The implication is that at short distances, spacetime is a chaotic and s tochastic
+fluid in a turbulent regime8with the Kolmogorov length l. The energy cascades are
+a property of the spacetime foam. But we emphasize that this inter pretation of the
+Kolmogorov scaling in the quantum gravitational setting is valid only fo r the case
+of holographic quantum foam (corresponding to α= 2/3).a
+3. Detectability of Quantum Foam with Extragalactic Source s
+We10suggested that spacetime foam might be uncovered by looking for u nresolved
+cores in the images of distant quasars.11The point is that, due to quantum foam-
+induced fluctuations in the phase velocity of an incoming light wave fro m a distant
+point source, the wave front itself develops a small scale “foamy” s tructure. This
+results in the wave vector acquiring a cumulative random fluctuation in direction
+with an angular spread of the order of ∆ φ/2π, where ∆ φ= 2πδl/λ= 2πNl1−αlα
+P/λ
+is the fluctuation in the phase of the electromagnetic wave with wave lengthλafter
+traveling a distance lfrom the distant source. In effect, spacetime foam creates a
+“seeing disk” whose angular diameter is10∆φ/(2π)∼(l/λ)1−α(lP/λ)α.b
+For a telescope or interferometer with baseline length D, this means that dis-
+persion (on the order of ∆ φ/2πin the normal to the wave front) will be recorded
+as a spread in the angular size of a distant point source, causing a re duction in
+the Strehl ratio, and/or the fringe visibility when ∆ φ/2π∼λ/Dfor a diffraction
+limited telescope.cThus, in principle, for arbitrarilylargedistances spacetime foam
+aUpdate: Recently we9have proposed a string theory of turbulence that explains th e Kolmogorov
+scaling in 3 + 1 dimensions and the Kraichnan and Kolmogorov s calings in 2 + 1 dimensions.
+We argue that this string theory of turbulence should be unde rstood from the viewpoint of the
+AdS/CFT dictionary. We find that not only can string theory be useful i n formulating a theory
+of turbulence, but the physics of turbulence may provide som e guidance to understanding the
+quantum foam phase of strong quantum gravity.
+bThis is partly based on the intuition that cumulative fluctua tions in lengths are comparable in
+all directions. (In particular, we have assumed the samecumulative factors1for both transverse
+and longitudinal directions.) But we should keep in mind tha t this intuition, though reasonable,
+could be wrong; after all, spatial isotropy is here “spontan eously” broken with the detected light
+being from a particular direction.
+cFor example , for a quasar of 1 Gpc away, at an infrared wavelen gth of the order of 2 microns, theOctober 29, 2018 17:23 WSPC - Proceedings Trim Size: 9.75in x 6.5in hoqufo
+3
+sets a lower limit on the observable angular size of a source at a given w avelength
+λ. Furthermore, the disappearance of “point sources” will be stro ngly wavelength
+dependent happening first at short wavelengths. Interferomet er systems (like the
+VLTI when it reaches its design performance) with multiple baselines m ay have
+sufficient signal to noise to allow for the detection of quantum foam fl uctuations.
+More recently we12have considered the feasibility of using various existing or
+proposed telescopes (for wavelengths from hard X-rays down to radio waves) to test
+the different spacetime foam models. Figure 1 shows the prediction m ade for three
+different models of spacetime foam corresponding to α= 2/3,0.6,1/2 respectively,
+for the size of observed haloes produced by accumulated phase dis persion for a
+source at two redshifts, respectively z= 4 and z= 1. The labeled arrows delineate
+the diffraction limited response of various telescopes. If the arrow representing a
+telescope’s diffraction limited response lies below the halo size curve fo r a given α,
+that model may be excluded by observations.dWe notice that the VLTI appears
+to be the most promising to test the holographic model.
+4. Holographic Quantum Foam Cosmology and Infinite Statisti cs
+If there is unity of physics connecting the very small and the very la rge, then it is
+natural to apply holographic quantum foam physics to cosmology.1,13We find that,
+according to the cosmololgy inspired by holographic quantum foam (d ubbed HFC),
+the cosmic density is ρ= (3/8π)(RHlP)−2∼(H/lP)2, consistent with observation
+(His the Hubble parameter of the observable universe and RHis the Hubble ra-
+dius),eand that the cosmic entropy is given by I∼(RH/lP)2. Furthermore, with
+the aid of archived data from the Hubble Space Telescope indicating t he demise
+of theα= 1/2 model, HFC has provided another argument for the existence of
+dark energy (indepenent of other cosmological/astrophysical ob servations of recent
+years). Successes of the conventional big bang cosmology, such as nucleosynthesis,
+can also be incorporated into HFC.
+Here we concentrate on one crucial question: What is the overridin g difference
+between conventional matter and dark energy (perhaps also dar k matter) according
+toHFC? Since darkenergycarriesmost ofthe energyofthe Univer se,let us focuson
+its constituent particles. Consider a perfect gas of Nparticles obeying Boltzmann
+holographic model of spacetime foam predicts a phase fluctua tion ∆φ∼2π×10−9radians. On the
+other hand, an infrared interferometer with D∼100 meters (like the VLTI) has λ/D∼5×10−9.
+dUpdate: Recently we12have elaborated on our proposal to detect quantum foam with e xtragalac-
+tic sources and have argued that the apppropriate distance m easure for calculating the expected
+angular broadening is the total line-of-sight comoving dis tance. We then deal with recent data and
+the constraints they put on spacetime foam models. Thus far, images of high-redshift quasars from
+the Hubble Ultra-Deep Field (UDF) provide the most stringen t test of spacetime foam theories.
+Indeed, we see a slight wavelength-dependent blurring in th e UDF images selected for this study.
+Using existing data in the HST archive we find it is impossible to rule out the α= 2/3 model, but
+exclude all models with α <0.65.
+eSince critical cosmic energy density is the hallmark of the i nflatonary paradigm, HFC may sup-
+plement/implement inflation in solving some of the classic c osmological problems.October 29, 2018 17:23 WSPC - Proceedings Trim Size: 9.75in x 6.5in hoqufo
+4
+Fig. 1. Detectability of various models of foamy spacetime w ith existing and planned telescopes.
+statistics at temperature Tin a volume V. For the problem at hand, we take V∼
+R3
+H,N∼(RH/lP)2≫1andT∼R−1
+H(the averageenergycarriedby eachparticle).
+A standard calculation yields the partition function ZN= (N!)−1(V/λ3)N, where
+λ∼T−1. We get, for the entropy of the system, S=N[ln(V/Nλ3) + 5/2]. The
+important point to note is that, since V∼λ3, the entropy Sbecomes nonsensically
+negative.But the solutionis pretty obvious:the Ninside the login Ssomehowmust
+be absent. In that case, the Gibbs 1 /N! factor must be absent from the partition
+function ZN, accordingly the “particles” are distinguishable and nonidentical, an d
+the entropy becomes S=N[ln(V/λ3)+3/2].October 29, 2018 17:23 WSPC - Proceedings Trim Size: 9.75in x 6.5in hoqufo
+5
+Now the only known consistent statistics in greater than two space dimensions
+without the Gibbs factor is infinite statistics (sometimes called “quan tum Boltz-
+mann statistics”).14,15Thus the “particles” constituting dark energy obey infinite
+statistics, instead of the familiar Fermi or Bose statistics.13(Using the Matrix the-
+ory approach, Jejjala, Kavic and Minic16have also argued that dark energy quanta
+obey infinite statistics.) This is the crucial difference between the co nstituents of
+dark energy and ordinary matter, according to HFC. Since each “p article” has such
+long wavelength( ∼RH), dark energyacts like a (dynamical) cosmologicalconstant.
+But to get the correct equation of state and an appropriate tran sition from an ear-
+lier decelerating to a recent accelerating cosmic expansion, one may need to take
+into account the coupling between pressureless dark matter and h olographic dark
+energy.17Finally we note that a theory of particles obeying infinite statistics ca n-
+not be local.15But intuitively holographic theories also possess non-locality. Thus
+it is not surprising that non-locality is present in HFC, a cosmology tha t is inti-
+mately connected to holography.1,13The appearance of infinite statistics in HFC is
+suggestive of a holographic view of spacetime.
+Acknowledgments
+This work was supported in part by the US Department of Energy.
+References
+1. Y. J. Ng, Entropy 10, 441 (2008) [arXiv:0801.2962[hep-th]] and references the rein.
+2. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, San Francisco,
+1973). For the α= 1/2 model, see G. Amelino-Camelia, Nature398, 216 (1999).
+3. Y. J. Ng and H. Van Dam, Mod. Phys. Lett. A 9, 335 (1994); 10, 2801 (1995);
+F. Karolyhazy, Nuovo Cim. A 42, 390 (1966); S. Lloyd and Y. J. Ng, Sci. Am. 291,
+#5, 52 (2004).
+4. G. ’tHooft, in Salamfestschrift , (World Scientific, Singapore, 1993); L. Susskind, J.
+Math. Phys. 36, 6377 (1995).
+5. V. Jejjala, D. Minic, Y.J. Ng, and C.H. Tze, Class. Quant. Grav. 25, 225012 (2008)
+[arXiv:0806.0030[hep-th]].
+6. W. Unruh, Phys. Rev. D 51, 282 (1995).
+7. A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941); 32, 16 (1941).
+8. J. A. Wheeler, Phys. Rev. 97, 511 (1955).
+9. V. Jejjala, D. Minic, Y.J. Ng, and C.H. Tze, arXiv:0912.27 25[hep-th].
+10. W. Christiansen, Y. J. Ng and H. van Dam, Rev. Phys. Lett. 96, 051301 (2006).
+11. R. Lieu & L. W. Hillman, ApJ585, L77 (2003); R. Ragazzoni, M. Turatto, &
+W. Gaessler, ApJ587, L1 (2003); Y J. Ng, W. Christiansen & H. van Dam, ApJ
+591, L87 (2003); E. Steinbring, ApJ655, 714 (2007).
+12. W.A.Christiansen, D.J. E. Floyd, Y.J. NgandE. S.Perlma n, arXiv:0912.0535[astro-
+ph] and references therein.
+13. Y. J. Ng, Phys. Lett. B 657, 10 (2007) [arXiv:gr-qc/0703096].
+14. S. Doplicher, R. Haag, and J. Roberts, Commun. Math. Phys. 23, 199 (1971).
+15. O. W. Greenberg, Phys. Rev. Lett. 64, 705 (1990).
+16. V. Jejjala, M. Kavic, and D. Minic, arXiv:0705.4581[hep -th].
+17. W. Zimdahl and D. Pavon, arXiv:0606555[astro-ph].
\ No newline at end of file
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+arXiv:1001.0412v2  [astro-ph.GA]  5 Jan 2010Identifying Bright Stars in Crowded Environments Using Vel ocity
+Dispersion Measurements, and an Application to the Center o f
+M32
+T. J. Davidge
+Herzberg Institute of Astrophysics,
+National Research Council of Canada, 5071 West Saanich Road ,
+Victoria, B.C. Canada V9E 2E7
+email: tim.davidge@nrc.ca
+Tracy L. Beck
+Space Telescope Science Institute,
+3700 San Martin Drive,
+Baltimore, MD 21218
+email: tbeck@stsci.edu
+Peter J. McGregor
+Research School of Astronomy and Astrophysics,
+Australian National University, Cotter Road, Weston Creek ,
+ACT 2611, Australia
+email: peter@mso.anu.edu.au
+ABSTRACT
+The identification of individual stars in crowded environments using p hoto-
+metric information alone is confounded by source confusion. Howev er, with the
+addition of spectroscopic information it is possible to distinguish betw een blends
+andareaswherethelight isdominatedbyasinglestarusingthewidths ofabsorp-
+tion features. We describe a procedure for identifying locations in k inematically
+1Based on observations obtained at the Gemini Observatory, which is operated by the Association of
+Universities for Research in Astronomy, Inc., under a co-operativ e agreement with the NSF on behalf of the
+Gemini partnership: the National Science Foundation (United Stat es), the Science and Technology Facilities
+Council (United Kingdom), the National Research Council of Canad a (Canada), CONICYT (Chile), the
+AustralianResearchCouncil (Australia), the MinisteriodaCiencia eT echnologia(Brazil), and the Ministerio
+de Ciencia, Tecnologia e Innovacion Productiva (Argentina).– 2 –
+hot environments where the light is dominated by a single star, and ap ply this
+method to spectra with 0.1 arcsec angular resolution covering the 2 .1−2.3µm
+interval in the central regions of M32. Targets for detailed invest igation are se-
+lected as areas of localized brightness enhancement. Three locatio ns where at
+least 60% of the K−band light comes from a single bright star, and another with
+light that is dominated by two stars with very different velocities, are identified.
+The dominant stars are evolving near the tip of the asymptotic giant branch
+(AGB), and have M5 III spectral type. The lack of a dispersion in sp ectral-type
+suggests that the upper AGB within the central arcsec of M32 has a dispersion
+inJ−Kof only a few hundreths of a magnitude, in agreement with what is see n
+at larger radii. One star has weaker atomic absorption lines than the others,
+such that [M/H] is 0.2 dex lower. Such a difference in metallicity is consist ent
+with the metallicity dispersion inferred from the width of the AGB in M32 . The
+use of line width to distinguish between blends involving many relatively f aint
+stars, none of which dominate the light output, and areas that are dominated by
+a single intrinsically bright star could be extended to crowded environ ments in
+other nearby galaxies.
+Subject headings: Data Analysis and Techniques - Galaxies
+1. INTRODUCTION
+Stellar content studies of nearby galaxies are of fundamental impo rtance for furthering
+ourunderstanding ofgalaxyevolution. Unfortunately, sourceco nfusionpresents aformidable
+obstacle for efforts to identify individual stars in even the closest g alaxies. Stars that fall
+within a single angular resolution element will appear as a single source, with spectrophoto-
+metric properties that are a luminosity-weighted amalgam of the com ponent stars. With the
+possible exception of the outermost regions of galaxies, almost all s tars observed in external
+galaxies are blends, although in most cases these involve an intrinsica lly bright star and a
+much fainter star (or stars), such that the combined light can be c onsidered to be that of
+the brighter source. A more insidious situation occurs when source s with similar properties
+are blended. Consider the implications on a stellar content study if tw o stars with the same
+brightness and color are blended together. The blend will be brighte r than the component
+stars but have the same color, and such an object may be mis-inter preted as evidence for a
+younger and/or more metal-poor population than that of the comp onent stars. While the
+incidence of blending among stars with comparable brightness drops markedly for objects
+that are in the most advanced stages of evolution, because such o bjects are relatively rare,– 3 –
+this remains an issue for studies of the central regions of even the nearest galaxies (Renzini
+1998).
+There is a continuum in the severity of blending, and the key for stella r content studies
+is to identify locations in galaxies where the light output is dominated by one star. In
+this paper we discuss a technique for identifying such objects in cro wded, kinematically
+hot environments. The core principle is that the spectra of blends in which no single star
+dominates the light output will have line widths that are defined by the velocities of the
+componentstars, andhencethatarecomparabletothelocalvelo citydispersioninthegalaxy.
+Spectra of blends that are dominated by one star will have line widths that are considerably
+narrower. The application of the technique is demonstrated using o bservations of the center
+of the Local Group compact elliptical (cE) galaxy M32. M32 is an unsu rpassed laboratory
+for honing techniques that are used to explore the star-forming h istories of galaxies. It is
+at a favorable distance such that light from a representative samp le of stars passes through
+a spectrograph slit with a width of a few arcsec, while individual stars can also be resolved
+throughout much of the galaxy. As a result, the stellar content of M32 can be probed by
+investigating the properties of both resolved stars and integrate d light. In addition, the low
+interstellar gas content (e.g. Welch & Sage 2001) suggests that M3 2 is free of the dust that
+can complicate stellar content studies at visible wavelengths. Finally, the brightest stars in
+M32 have K∼15.5 (Davidge 2000; Davidge et al. 2000; Davidge & Jensen 2007), and h ence
+are well within the realm of being studied spectroscopically with existin g telescopes.
+Numerous studies have concluded that M32 is not a uniformly old stella r system, and
+contains a significant population of stars that formed within the pas t few Gyr, but not as
+recently as in the LMC. Indeed, the peak AGB brightness in M32 is M K≈ −9, which is ≈1
+magnitude fainter than the peak brightness seen among long period variables (LPVs) in the
+LMC (Fraser et al. 2005). Panoramic surveys have also found a num ber of AGB stars in
+the LMC that are at leastas bright as those in M32 (Nikolaev & Weinberg 2000; Cioni et
+al. 2006). With M K≈ −9, the peak AGB brightness in M32 is comparable to that near the
+center of M31 (Davidge 2001), where there are spectroscopic sig natures of an intermediate
+age population (e.g. Davidge 1997; Sil’Chenko, Burenkov, & Vlasyuk 1 998). Adopting a
+distance modulus for the Galactic Center of 14.5, then the two brigh test AGB stars in the
+sample of Galactic Bulge stars studied by Frogel & Whitford (1987), which are numbers 239
+and 181 in their target list, have M Kbetween –9 and –9.5. Many of the brightest AGB
+stars in M32 are LPVs with amplitudes of 1.0 magnitude in K(Davidge & Rigaut 2004),
+and so the non-variable AGB-tip in M32 corresponds to M K≈ −8.5. Isochrones with solar
+metallicity from the compilation discussed by Girardi et al. (2002) indic ate that this AGB-
+tip brightness is appropriate for a system with an age that is in exces s of 1 Gyr, in agreement
+with studies of the integrated spectrum of M32 (O’Connell 1980; Da vidge 1990; Bica, Alloin,– 4 –
+& Schmidt 1990; del Burgo et al. 2001; Worthey 2004; Rose et al. 20 05).
+In the present study, three locations that are dominated by light f rom one star are
+identified in the central arcsec of M32, while a fourth location that is dominated by light
+from two stars that have very different velocities is also identified. T hese are the closest-in
+individual stars to be detected near the center of M32 at visible/ne ar-infrared wavelengths.
+Whilefivestarsconstituteanobviously limitedsample, thedataaresu fficient todemonstrate
+the application of the technique. Even though the number of sourc es that are dominated by
+a single star and that can be identified at the diffraction limit of an 8 met re telescope near
+the center of nearby galaxies like M32 is modest, a 20 – 30 metre teles cope that delivers
+moderately high Strehl ratios should be able to detect roughly an or der of magnitude more
+resolved sources.
+2. OBSERVATIONS & REDUCTIONS
+The data used in this paper were discussed previously by Davidge et a l. (2008) in their
+investigation of cE galaxies. In brief, spectra of M32 were recorde d with NIFS+ALTAIR on
+Gemini North (GN) on the night of October 23, 2005 during NIFS com missioning. ALTAIR
+(Herriot et al. 2000) is the facility AO system at GN, while NIFS (McGre gor et al. 2003) is
+an integral field spectrograph for use in the 0 .9−2.5µm wavelength interval. NIFS samples
+a 3×3 arcsec2area on the sky with 29 0 .1×3 arcsec2slitlets. A spectrum with a dispersion
+5300 is recorded on the 2048 ×2048 HAWAII-2RG HgCdTe detector in one of three ( J,H,
+orK) atmospheric windows during a single exposure.
+The bright semi-stellar nucleus of M32 served as the reference sou rce for AO correction.
+Six 600 sec exposures were recorded in the K−band, and each observation of M32 was
+followed by that of a blank sky field. Each M32 + sky pair was difference d, and the results
+were divided by a flat-field frame. The flat-fielded data were wavelen gth calibrated using an
+Ar arc observation that was recorded on the same night as the M32 data. The wavelength-
+calibrated spectra were summed and then divided by the spectrum o f a telluric standard
+star, which was observed immediately following the M32 observations .
+There are a number of distinct sources with FWHM ∼0.1 arcsec in the two-dimensional
+NIFS spectra, and these define the locations that are investigate d in the remainder of the
+paper. The locations of these point sources are indicated in Figure 1 , where (X,Y) = (0,0) is
+the center of M32, while the angular offsets of the sources from th e center of M32 are listed
+in Table 1. The detection of objects along the X = 0 axis is hindered by e levated star counts
+along the major axis of M32, which runs close to X = 0 arcsec, and sca ttered light from the– 5 –
+bright nucleus of M32.
+A spectrum was extracted of each of the sources in Figure 1. The s urface brightness
+in the central arcsec of M32 exceeds 11 mag arcsec−2inK(e.g. Jarrett et al. 2003), and
+thus contributes substantial noise to the extracted spectra. I ndeed, the unresolved stellar
+background, rather than the ambient ‘sky’, is the dominant source of noise in studies of stars
+near the centers of most nearby galaxies. A mean background spe ctrum was constructed
+from galaxy light bracketing each source, and the result was subtr acted from the extracted
+spectrum. The background-subtracted spectra were binned alo ng the dispersion axis to
+increase the signal-to-noise ratio, and the final spectral resolut ion is≈1000. The continuum
+was also removed from the spectra in preparation for the velocity d ispersion analysis.
+3. VELOCITY MEASUREMENTS
+With an image quality of FWHM ≈0.1 arcsec, each angular resolution element near the
+center of M32 samples an area with an integrated magnitude M K≈ −9, which is comparable
+to that of a Galactic globular cluster. Consequently, all of the sour ces in Figure 1 are blends,
+in the sense that light from more than one star falls within each angula r resolution element.
+In addition, only objects that have an intrinsic brightness that is co mparable to or greater
+than that in each angular resolution element (i.e. M K≈ −9 in this case) will be resolved.
+The AGB-tip in M32 occurs near K = 15.5, which corresponds to M K≈ −9 (Davidge et
+al. 2000; Davidge & Jensen 2007), and so any sources identified her e are evolving near the
+AGB-tip.
+Are there locations in the NIFS dataset where the light is dominated b y a single bright
+star? Line width measurements are one way to determine if such loca tions are present. If
+the light in a blend originates from a large number of stars, with no sing le star contributing
+more than a small fraction of the light, then the absorption lines in th e composite spectrum
+will be blurred by an amount that is comparable to the local velocity dis persion,σ, which
+is∼100 km sec−1near the edges of the NIFS field. At the other extreme, if the light c omes
+from only one star then the absorption lines will be narrow, with the lin e width defined by
+the spectral resolution of the instrument. There are a number of techniques for measuring σ,
+and in this study the Tonry & Davis (1979) routine, as implemented in I RAF, is employed.
+A key issue is translating velocity dispersion into a measure of the amo unt of light that
+a bright single source contributes to the total light from a resolutio n element. Consider the
+spectrum of a location S locon the sky that contains light from a star S ∗, which contributes
+a fraction fof the light, combined with that of a population of other fainter obje cts Sother,– 6 –
+which together contribute a fraction (1 −f) of the light. The reader is reminded that S loc, S∗,
+Sother, andfare all functions of wavelength, but this is not indicated here in the in terest of
+brevity. The spectrum of this location is the luminosity-weighted sum of these components:
+Sloc=f·S∗+(1−f)·Sother
+To measure the velocity dispersion the spectrum of the target loca tion is convolved with
+that of a reference star, S ref:
+Sloc⋆Sref= (f·S∗+(1−f)·Sother)⋆Sref
+where convolution is indicated with the symbol ⋆. The Fourier transform of the cross-
+correlation function is then:
+F(Sloc⋆Sref) =F(f·S∗+(1−f)·Sother)·F(Sref)
+where the cross-correlation theorum has been applied, and Fdenotes complex conjugation.
+This reduces to:
+F(Sloc⋆Sref) =f·F(S∗)·F(Sref)+(1−f)·F(Sother)·F(Sref)
+Applying the inverse transform then:
+Sloc⋆Sref=f·(S∗⋆Sref)+(1−f)·(Sother⋆Sref)
+The cross correlation function of the composite spectrum with the reference star is thus
+the sum of the cross correlation functions of the components of t he target spectrum with
+the reference star. The properties of the final cross-correlat ion function then depend on the
+properties of the various sources that contribute to the total lig ht, their degree of similarity,
+not only among themselves but with the reference star, and the fr actional contribution that
+each component makes to the total light.
+In principle, a threshold value of the measured velocity dispersion th at corresponds to
+some pre-defined value of fcan be determined from simulations, in which a spectrum corre-
+sponding to that of the dominant light source areadded in various pr oportions to a spectrum
+representing the galaxy body that has been smeared to simulate th e impact of velocity dis-
+persion. The situation is simplified somewhat in near-infrared studies of intermediate age– 7 –
+or older systems, as a large fraction of the K−band light comes from the brightest evolved
+stars. A consequence is that the spectra of the brightest resolv ed stars are similar to that
+of the underlying integrated light, making it is easier to match the ref erence star spectrum
+with those of the integrated galaxy light and any resolved stars.
+Abasicassumptionisthattheresolutionelementsthataredominate dbyindividualstars
+have an underlying stellar component with a σthat surpasses that which can be measured
+from the data. Higher spectral resolutions will be required to dete ct single sources in fields
+with lower σ. It will thus prove more challenging to identify locations that are dom inated by
+light froma single source in crowded enviroments where the stars ha ve modest (at least when
+compared with the centers of galaxy bulges) random motions, such as globular clusters.
+4. AN APPLICATION TO M32 NIFS SPECTRA
+4.1. The Identification of Individual Stars
+The velocity dispersions of the sources are listed in the last column of Table 1. The
+M0III star HR 4884, which was also observed with NIFS, was used as the reference template
+for these measurements. σwas measured in the 2 .1−2.3µm interval, which is where the
+S/N ratio is highest. The uncertainties in the σmeasurements are ≤ ±10 km sec−1. The
+spectra of sources 10 and 14 were too noisey to allow velocity disper sions to be measured.
+There is a broad range in velocity dispersions in Table 1, and half of the objects have
+σ >90 km sec−1. Given that the velocity dispersion of M32 within the central 1.5 arcs ec
+is≥100 km sec−1, then the majority of locations in the M32 NIFS spectra that conta in
+a distinct point source are actually blends where the light originates f rom a mix of stars,
+with no single star dominating the light output. The appearance of a p oint source at these
+locations is simply the result of statistical flucuations in the stellar co ntent.
+For this study, we identify locations where at least 60% (i.e. f≥0.6) of the light comes
+from a single star. Simulations were run in which an M5III spectrum (s ee below) was added
+in various proportions to M giant spectra that were smeared to σ= 100 km sec−1. An
+M giant spectrum was selected to represent the underlying light as t his is consistent with
+the integrated K−band spectrum of M32 (Davidge 1990). These simulations indicate th at
+locations with photometrically-distinct sources that have σ <35 km sec−1are those where
+at least 60% of the light comes from a single star.
+Three of the sources in Table 1 have σ≤35 km sec−1. A fourth source is also of interest,
+as the cross-correlation function with the reference star is doub le-peaked, suggesting that it– 8 –
+is dominated by two stars that contribute equally to the light. The sp ectra of the sources
+withσ <35 km sec−1and of the two-star blend are shown in Figure 2. The spectra have lo w
+S/N ratios due to noise introduced by the bright unresolved body of fainter stars in M32.
+Still, the S/N ratio is such that the12CO band head can be identified. As by far the most
+prominent feature in each spectrum, the12CO band head dominates the velocity dispersion
+measurements. The depth of the12CO band head is a measure of velocity dispersion, and
+this is demonstrated in Figure 2, where the mean spectrum of M32 at 1 arcsec radius from
+Davidge et al. (2008) is shown. The12CO band heads in the mean M32 and Star 13 spectra
+have similar depths, as expected given their comparable velocity disp ersions. However, the
+12CO band heads in the spectra of Stars 1, 3, and 8 are much deeper, reflecting their lower
+velocity dispersions.
+The velocity dispersions of Stars 1, 3, and 8 fall well below the range of velocity dis-
+persions that are measured in integrated light throughout the cen tral regions of M32. This
+is demonstrated in Figure 3, which shows the histogram distribution o f velocity disperions
+measured in 0 .12×0.1 arcsec2sections within the 3 ×3 arcsec2NIFS field of view. The
+0.12×0.1 arcsec2area corresponds to that subtended by point sources in these da ta.
+The histogram in Figure 3 contains σmeasurements from locations that span a range
+of radii and position angles, and so the distribution is broader than e xpected from random
+measurement errors alone. In addition, the velocity dispersions me asured for Stars 1, 3, and
+8 are much smaller than what is measured in integrated light througho ut the central regions
+of M32. Hence, the small velocity dispersions measured for these s ources are not due to
+random flucuations in the velocity field.
+4.2. Spectral Types and Line Strengths
+The low S/N ratio of the extracted spectra notwithstanding, some information can be
+extracted by multiplexing the signal over a range of wavelengths, r ather than relying on
+measurements made within narrow wavelength intervals, such as line index measurements.
+A basic piece of information that can be extracted are the spectra l types of the resolved
+stars. This is relevant for stellar content studies because spectr al-type can serve as a proxy
+for color, allowing comparisons to be made with photometric observa tions at larger radii.
+Spectral types were determined by cross-correlating the M32 sp ectra with those of ref-
+erence stars from Rayner, Cushing, & Vacca (2009), which were d ownloaded from the IRTF
+website. The IRTF spectra were re-sampled and smoothed to matc h the wavelength sam-
+pling and spectral resolution of the processed NIFS spectra. The height of the central peak– 9 –
+in the cross-correlation function measures the degree of similarity between the two spectra.
+The amplitude of the cross-correlation peak can be biased by a single strong feature, such
+as the12CO (2,0) band head. Therefore, in addition to correlations in the 2 .1−2.3µm wave-
+length interval, a second set of correlations was done in the 2 .1−2.28µminterval to avoid the
+12CO band head and thereby base the spectral matching on the stre ngths of atomic and less
+prominent molecular features. Significantly, the cross-correlatio ns involving the wavelength
+intervals with and without the12CO band head produced consistent results, indicating that
+the spectral typing is not biased by a single strong feature.
+The highest degree of semblance resulted when the spectra in Figur e 2 were compared
+with the M5III star. The next best fit was for M6III, while the peak s of the cross-correlation
+functions involving types M9III and C2.2 were substantially lower tha n those involving M0
+III - M6 III stars. The degree of similarity between the near-infra red spectra of the M32
+sources and the Galactic M5 III standard star is demonstrated in F igure 4, where the mean
+spectrum of sources 1, 3, and 8 is compared with that of selected s tars from Rayner et al.
+(2009). The12CO (2,0) band head and the lines of Na I, Fe I, and Ca I in the mean M32
+spectrum are similar in strength to those in the M5 III spectrum.
+The similarity in spectral-types is significant, as it indicates that the u pper regions
+of the AGB in M32 are populated by stars with a narrow range in effect ive temperature.
+Adopting the relation between spectral type and J−Kfrom Bessell & Brett (1988) for solar
+neighborhood giants, then these sources have J−K= 1.2−1.3, which is consistent with
+the color of the upper AGB at larger radii in M32 (Davidge & Jensen 20 07).
+The amplitude of the cross-correlation function also contains infor mation about the
+strengths of absorption features, albeit multiplexed over a range of elements. Davidge et
+al. (2008) found that the integrated spectrum of M32 falls near re lations defined by solar
+neighborhood stars on the (Ca I,12CO) and (Na I,12CO) diagrams, suggesting that stars
+in M32 formed from material that experienced slow enrichment; hen ce, the use of solar
+neighborhood reference stars for abundance analysis is appropr iate. The dominant features
+inthe2.1−2.28µmwavelength interval areCa, Na, andFelines, althoughothereleme nts also
+contribute to these features (Silva, Kuntschner, & Lyubenova 2 008). When cross-correlated
+with an M5III star in this wavelength range, the peaks measured fo r stars 3, 8, and 13 are
+very similar. However, the peak of the Star 1 cross-correlation fu nction is ∼0.6×those of
+the other stars, suggesting that Star 1 has weaker atomic absor ption lines than the other
+sources. It should be emphasized that this measurement is not bas ed on any one feature in
+the spectrum, but the depth of all of the features in the 2 .1−2.28µm interval, and hence is
+best understood as an overall metallity difference, such that the m ean metallicity in Star 1 is
+≈0.2 dex lower than the other stars. This is well within the dispersion tha t is seen in other– 10 –
+spheroidal systems, such as the Galactic Bulge (e.g. Zoccali et al. 2 003), and falls within
+the metallicity dispersion that has been proposed to explain the width of the AGB at larger
+radii in M32 (Davidge & Jensen 2007).
+5. SUMMARY & DISCUSSION
+We have discussed the use of absorption line widths to identify locatio ns in crowded
+fields where the light is dominated by a single object. The use of line widt hs to identify such
+objects is most effective in kinematically hot environments, where th e line width contrast
+between obviousblendsandlocationswherethelightisdominatedbya singlestarisgreatest.
+We have examined 2 .1−2.3µm spectra with an angular resolution of ≈0.1 arcsec in the
+central few arcsec of the cE galaxy M32. An initial list of potential in dividual stars was
+defined by identifying photometrically distinct sources. The majorit y of these are found to
+have line widths that are consistent with the light originating from a mix of stars with a
+velocity dispersion that is comparable to that of the main body of the galaxy; the extracted
+spectra are thus of composite systems made up of physically unrela ted stars that, due to
+stochastic effects, appear as compact sources at this angular re solution. As for the four
+remaining sources, three have line widths indicating that the light is do minated by a single
+intrinsically luminous source, while the fourth has a double-peaked cr oss-correlation profile
+thatisindicativeoftwoobjectswithsimilar K−bandmagnitudesthatareisolatedinvelocity
+space.
+There are hints from other datasets that point sources with light c oming largely from
+one star should be found in these data. Davidge et al. (2000) used im ages with an angular
+resolution of 0.12 arcsec to investigate the brightest stars near t he center of M32. While
+their (K,H−K) CMD of the 0 – 3 arcsec interval has a conspicuous spray of bright sources
+that are obvious blends, a weak concentration of AGB-tip stars ca n be seen at K≈15.5,
+which is the AGB-tip magnitude seen at larger radii. These are point so urces close to the
+center of M32 where the light is probably dominated by a single bright A GB-tip star.
+It can be anticipated that, when compared with blends, unblended s ources (1) will be
+fainter, and (2) will be in areas of lower projected stellar density. T he brightnesses and
+local background levels of all fourteen locations were measured fr om theKimages discussed
+by Davidge et al. (2000). The locations with broad lines tend to be almo st 2×brighter
+inKthan those with narrow lines, and are in regions where the backgrou nd within 0.2
+arcsec of the source is ∼10% higher. With the caveat that a large fraction of the brightest
+stars are LPVs (Davidge & Rigaut 2004), the sources with low velocit y dispersions thus
+tend to be fainter than the high velocity-dispersion sources and ar e located in areas with– 11 –
+lower background levels, as expected if they are not blends of many objects with comparable
+brightness.
+While the sample of single stars detected here is small, modest insights can be gained
+into the bright stellar content near the center of M32. The object s that are dominated by
+one or two bright stars are almost certainly evolving near the AGB-t ip, and these all have
+an M5III spectral type. These spectral type measurements ar e not biased by a single strong
+feature, such as the (2,0)12CO band head. The uniformity in spectral type is consistent
+with the distinct red cut-off in the ( K,H−K) CMD of the central regions of M32 found by
+Davidge et al. (2000). It also suggests that the J−Kcolor of the AGB-tip near the center
+of M32 is consistent with that at large radii, indicating that the homog eneous nature of the
+AGB sequence in M32 measured by Davidge & Jensen (2007) extends into the central arcsec
+of the galaxy.
+The spectra also suggest that therearemodest star-to-starm etallicity differences among
+thebrighteststarsinM32. Whenthenarrow-linespectraarecros s-correlatedwithareference
+spectrum in the 2 .1−2.28µm interval, one source – # 1 – has a significantly lower peak in
+the correlation function, by an amount that suggests it has a meta llicity that is ≈0.2 dex
+lower than the other stars. Metallicity differences of this size fall we ll within the spread seen
+in other spheroidal systems, and are consistent with the spread in J−Kcolor seen amongst
+AGB stars in the outer region of M32 (Davidge & Jensen 2007).
+Many of the bright AGB stars in M32 are LPVs (Davidge & Rigaut 2004) , and there
+is an obvious bias to detect these objects when they are brightest . Observations at future
+epochsmayfindnewlocationsnearthecenterofM32thataredomin atedbysingleLPVsthat
+are near the peak of their light variation, but were in a part of their lig ht curve that rendered
+them too faint to be resolved in the present data. By the same toke n, the sources found
+here may not be detected if they are variable and in a faint phase of t heir light cycle at the
+time of future observation. This raises the possibility that the samp le of spectroscopically-
+identified stars near the center of M32 can be increased by observ ing at different epochs.
+This will provide a larger sample of objects from which to measure, fo r example, a more
+robust metallicity dispersion.
+Thanks are extended to an anonymous referee for suggesting th e comparison shown in
+Figure 3.– 12 –
+# R Nucσ
+(arcsec) (km sec−1)
+1 1.34 0
+2 1.46 96
+3 1.58 35
+4 1.45 65
+5 1.10 120
+6 0.96 161
+7 0.87 62
+8 1.33 31
+9 1.14 106
+10 0.95 –
+11 1.15 114
+12 1.27 80
+13 1.52 96 (Bimodal)
+14 1.53 –
+Table 1: Galactocentric Radii and Velocity Dispersion of the Sources Studied– 13 –
+REFERENCES
+Bessell, M. S., & Brett, J. M. 1988, PASP, 100, 1134
+Bica, E., Alloin, D., & Schmidt, A. A. 1990, A&A, 228, 23
+Cioni, M.-R.-L., Girardi, L., Marigo, P., & Habing, H. J. 2006, A&A, 456, 9 67
+Davidge, T. J. 1990, AJ, 99, 561
+Davidge, T. J. 1997, AJ, 113, 985
+Davidge, T. J. 2000, PASP, 112, 1177
+Davidge, T. J., & Jensen, J. B. 2007, AJ, 133, 576
+Davidge, T. J., & Rigaut, F. 2004, ApJ, 607, L25
+Davidge, T. J., Rigaut, F., Chun, M., Brandner, W., Potter, D., North cott, M., & Graves,
+J. E. 2000, ApJ, 545, L89
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+321, 227
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+McGregor, P. J., et al. 2003, Proc. SPIE, 4841, 1581
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+Renzini, A. 1998, AJ, 115, 2459
+Rose, J. A., Arimoto, N., Caldwell, N., Schiavon, R. P., Vazdekis, A., & Ya mada, Y. 2005,
+AJ, 129, 712
+Sil’Chenko, O. K., Burenkov, A. N., & Vlasyuk, V. V. 1998, A&A, 337, 3 49
+Tonry, J., & Davis, M. 1979, AJ, 84, 1511
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+Worthey, G. 2004, AJ, 128, 2826– 14 –
+Zoccali, M., et al. 2003, A&A, 399, 931
+This preprint was prepared with the AAS L ATEX macros v5.2.– 15 –
+-1 0 1-101
+Fig. 1.— The locations of point sources for which spectra were extra cted.XandYare
+offsets from the center of M32, with North at the top and East to t he left. Sources that have
+a velocity dispersion ≤35 km sec−1, indicating that a single bright object contributes at
+least 60% of the light, are marked with a box. A source with a two-pea ked cross-correlation
+function, suggesting that the light is dominated by two stars of com parable brightness, is
+indicated with the triangle.– 16 –
+Fig. 2.— Spectra of the sources near the center of M32 in which the K−band light is
+dominated by one (Sources 1, 3, and 8) or two (Source 13) bright s tars. The mean spectrum
+of M32 at 1 arcsec radius from Davidge et al. (2008) is also shown. Th e locations of various
+atomic features, and the first-overtone (2,0)12CO band head are indicated. Note that the
+Ca I and Na I lines in Star 1 appear to be weaker than in the other 3 sou rces. The arrows
+show the depth of the12CO band head in the composite M32 spectrum. The12CO band
+head in the Star 13 spectrum is almost as deep as in the M32 spectrum , while the12CO
+band head in the Star 1, 3, and 8 spectra are much deeper. These d ifferences in12CO band
+head depth are due to differences in velocity dispersion.– 17 –
+0 50 100 150 200020406080100
+Fig. 3.— The histogram distribution of velocity dispersion measuremen ts of integrated light
+throughout the 3 ×3 arcsec2NIFS field of view. Velocity dispersions were measured in
+0.12×0.1 arcsec2regions, which is the approximate area subtended by point sources in
+these data. Nresis the number of angular resolution elements that have a velocity disp ersion
+within each 10 km sec−1interval. The velocity dispersions measured for Stars 1, 3, and 8
+are also indicated. These objects have velocity dispersions that fa ll well below the range of
+velocity dispersions measured in the NIFS field of view, indicating that their low velocity
+dispersions are not a consequence of statistical flucuations in the velocity field.– 18 –
+Fig. 4.— The mean spectrum of the three stars with σ≤35 km sec−1is compared with
+spectra of various Galactic stars. A cross-correlation analysis ind icates that the spectra of
+the M32 stars have the greatest similarity with the M5 III spectrum . Note that the atomic
+features that dominate the wavelength region shortward of 2 .28µm in early to mid-M giants
+largely disappear in the M9III and C star spectra.
\ No newline at end of file
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+arXiv:1001.0413v2  [astro-ph.SR]  5 Feb 2010To appear in the Astrophysical Journal Letters
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+THE DISCOVERY OF ELLIPSOIDAL VARIATIONS IN THE KEPLER LIGHT CURVE OF HAT-P-7
+William F. Welsh, Jerome A. Orosz
+Astronomy Department, San Diego State University, San Dieg o, CA 92182 USA
+Sara Seager
+Massachusetts Institute of Technology, MA 02139 USA
+Jonathan J. Fortney
+University of California, Santa Cruz, CA 95064 USA
+Jon Jenkins
+SETI Institute, Mountain View, CA 94043, USA
+Jason F. Rowe, David Koch, and William J. Borucki
+NASA Ames Research Center, Moffett Field, CA 94035 USA
+To appear in the Astrophysical Journal Letters
+ABSTRACT
+We presentananalysisofthe early Keplerobservationsofthe previouslydiscoveredtransitingplanet
+HAT-P-7b. The light curve shows the transit of the star, the occu ltation of the planet, and the orbit
+phase-dependent light from the planet. In addition, phase-depen dent light from the star is present,
+known as “ellipsoidal variations”. The very nearby planet (only 4 ste llar radii away) gravitationally
+distortsthe starand results in a flux modulation twice per orbit. The ellipsoidal variationscan confuse
+interpretation of the planetary phase curve if not self-consisten tly included in the modeling. We fit the
+light curve using the Roche potential approximationand derive impro vedplanet and orbit parameters.
+Subject headings: binaries: eclipsing
+1.INTRODUCTION
+Keplerisareconnaissancemissiontoobtaintime-series
+optical photometry of ∼150,000 stars in order to de-
+termine characteristics of Earth-size and larger extra-
+solar planets: frequencies, sizes, orbital distributions,
+and correlations with the properties of the host stars
+(Borucki et al. 2010). To achieve these goals, Keplerre-
+quires exceptional long-term photometric stability and
+precision (Koch et al. 2010). In addition to the discov-
+ery aspect of the mission, this unprecedented photomet-
+ric capability provides exquisite observations of a host
+of astrophysical objects, including the previously known
+extrasolar planets in the Keplerfield of view. In this
+Letter, we examine the early Keplerobservations of the
+planet HAT-P-7b.
+Discovered via the HATNet project, the transiting
+planet HAT-P-7b revolves in a tight circular orbit
+(a=0.038 AU, P=2.20473 d) around a bright (V=10.5
+mag) F6 star (P´ al et al. 2008). The proximity to its
+6350 K host star (P´ al et al. 2008) means the planet
+is highly irradiated, resulting in very high tempera-
+tures (∼2140 K), making it an extreme pM-type planet
+(Fortney et al.2008). Usingobservationsofthe Rossiter-
+McLaughlin effect, Narita et al. (2009) and Winn et al.
+(2009) find that the planet’s orbital axis is extremely
+tiltedcomparedtothe star’sspinaxis, andprobablyeven
+retrograde,implyingan interestingformationand orbital
+wfw@sciences.sdsu.eduevolution history. Additionally, the radial velocities ex-
+hibit an acceleration, suggesting the presence of another
+body in the system, perhaps responsible for the tilted
+orbit (Winn et al. 2009).
+Observations of HAT-P-7 during the 10 days of
+commissioning of the Keplerphotometer revealed the
+presence of an occultation (also known as a “sec-
+ondary eclipse”) as the planet passes behind the star
+(Borucki et al. 2009). These data also show the phase
+“reflected” light from the planet, a result of both scat-
+tered and thermal emission. Additional observations of
+HAT-P-7havebeen obtained, and we presentthese in §2.
+In§3 we describe our modeling method, with particular
+emphasis on the ellipsoidal variations from the star. In
+§4 we present and discuss our findings.
+2.OBSERVATIONS
+HAT-P-7 was monitored continuously for 33.5 d dur-
+ing the 2009 May 13–Jun 15 “Quarter 1” (Q1) epoch
+in short-cadence mode. The 42 KeplerCCDs were read
+out every 6 s and co-added on-board to achieve approx-
+imately 1 min sampling cadence. The photometer has
+no shutter, so an overscan region is used to remove the
+effects of smearing during readout. In 6 s exposures
+HAT-P-7 saturates the CCD; however, because the Ke-
+plerphotometer is such a stable platform, this does not
+hamper the relative precision and superb photometry
+is possible. For details on the design and performance
+of theKeplerphotometer, see Koch et al. (2010) and
+Jenkins et al. (2010).2 Welsh et al.
+Fifteen complete transits of HAT-P-7 were observed,
+and after removing 21 cosmic rays via a +4 sigma rejec-
+tion from a 30-min wide running median, 49,023 mea-
+surements remained. The background-subtracted light
+curve exhibited a gradual drop in counts (0.13%), sus-
+pected to be due to a drift in focus coupled with a fixed
+extraction aperture. This trend was removed using a
+cubic polynomial (masking out the transits). Because
+we are interested in phenomena on the orbital timescale
+(2.2 d), we chose this modest detrending to leave in as
+much power as possible on orbital timescales. Fluxes
+were normalized at phase 0.5 (mid occultation) because
+the planet is hidden at this phase and we see starlight
+alone. Short-timescale systematic calibration features
+are present (Gilliland et al. 2010), but they does not af-
+fect the analysis other than being a noise term. Uncer-
+tainties were estimated by taking the median of the rms
+deviationsin 30-minbins, resultingin 150ppm per 1-min
+datum.
+The detrended and phase-folded light curve is shown
+in Fig. 1, where the upper panel includes our fit to the
+transit, and the lower panel highlights the eclipse of the
+planet and the phase-dependent light from the planet
+and star. Notice that the maximum does not occur just
+outside occultation as one would expect for simple “re-
+flection” from the planet. Two maxima occur 0.15-0.20
+away in phase, a result of the “ellipsoidal variation” of
+the star’s light, as discussed in §4.
+The absolute timing is still preliminary in this early
+version of the data calibration pipeline, but relative tim-
+ing precision is reliable (though without the modest BJD
+correction, which is unimportant for our investigation
+over the 33.5 d time series). So we do not report the
+value for the epoch of transit T0, though of course it is
+a parameter in the model fitting. Also, the period mea-
+sured is based only on these 33.5 d, so the precision is
+not as high as would be if other epochs many cycles away
+were included.
+3.MODELING
+We employ the ELC code of Orosz & Hauschildt
+(2000) to model the transit, occultation, and phase-
+varying light from the planet and star. The code si-
+multaneously fits the photometry along with several ob-
+servational parameters: the radial velocity K, and the
+mass and radius of the star. The K amplitude was
+taken from Winn et al. (2009), and the mass and ra-
+dius from the asteroseismology analysis of the Kepler
+data (Christensen-Dalsgaard et al. 2010). These param-
+eters are not fixed; rather the models are started and
+steered toward them via a chi-square penalty for devia-
+tions. Markovchain Monte Carlo and genetic algorithms
+were used to search parameter space, find the global chi-
+square minimum, and determine confidence intervals of
+the fitted parameters.
+The analytic model of Gim´ enez (2006) for a spheri-
+cal star and planet is not sufficient to model the phase
+variations. Therefore ELC is used in its full numeri-
+cal mode: the star and planet are tiled in a fine grid,
+and the intensity and velocity from each tile is summed
+to give the light curve and radial velocities. Limb and
+gravitydarkeningareincluded, and the gravitationaldis-
+tortions are modeled assuming a standard Roche poten-
+tial. We employed a blackbody approximation evalu-ated at a wavelength of 6000 ˚A, and a hybrid method1
+where model atmospheres are used to determine the in-
+tensities at the normal for each tile and a parameterized
+limb darkening law is used for other angles; the model is
+then filtered through the Keplerspectral response func-
+tion (spanning roughly 4250–8950 ˚A, peaking at 5890 ˚A
+with a mean wavelength of 6400 ˚A — see Koch et al.
+(2010)andVan Cleve & Caldwell(2009). Thetwometh-
+ods give essentially the same results, with the exception
+of a higher albedo from the hybrid models. Interest-
+ingly, the blackbody models yielded significantly lower
+chi-square values, so we quote the blackbody model val-
+ues in this work. This needs to be kept in mind when in-
+terpreting the temperatures and albedos estimates given
+in§4. We use a 2-parameter logarithmic limb darkening
+law and adopt an eccentricity of zero consistent with the
+radial velocities and phase of occultation. We assume
+the planet is tidally locked in synchronous rotation. For
+more details on using ELC to model exoplanet data see
+Wittenmyer et al. (2005).
+Following the prescription of Wilson (1990), the light
+from the planet is modeled as the sum of an isothermal
+component (with temperature Tpthat essentially adds a
+constant flux at all phases outside of occultation), and
+a “reflection” component on the day hemisphere. The
+local temperature is given by T4=T4
+p×/bracketleftBig
+1+AbolF∗
+Fp/bracketrightBig
+which comes from assuming the fluxes and temperature
+are coupled by the Stefan-Boltzmann law. The bolomet-
+ric albedo Abol, also known as the “heat albedo” is the
+ratio of re-radiated to incident energy, and should not
+be confused with the Bond albedo. For stars, a radiative
+atmosphere has Abol=1 (local energy conservation) while
+for convective atmospheres a value of 0.5 is appropriate
+(half the energy is radiated, half gets redistributed) —
+see Kallrath & Milone (1999) for a full description of the
+method. In our model we allow Aboland the tempera-
+ture ratio Tp/T∗to be free parameters, keeping T∗fixed
+at 6350 K (P´ al et al. 2008). The term in brackets is the
+local reflection factor R, equal to the ratio of the total
+radiated flux (internal + re-radiated) to internal flux.
+ThusR ≥1, withR= 1 on the night side. For an iso-
+lated planet, Tpis very low (e.g. Burrows et al. (2006)
+assume 50 K), but for an irradiated planet, much of the
+incident energy is eventually re-distributed, bringing the
+night-time temperature up to much highertemperatures.
+In the ELC model, the local emitted flux is completely
+dependent on the local temperature. The local tempera-
+turedependsonthemeaneffectivetemperature,thelocal
+gravity, and irradiation. The irradiation term is the sum
+over all the visible tiles on the star as seen from each tile
+on the planet, and includes the ellipsoidal shape of the
+star, gravity darkening, limb darkening, and penumbra
+correction (accounting for the fact that parts of the star
+arenot visible because they areblocked by the local hori-
+zon). Note that given the close proximity of the planet
+to the star, no part of the planet sees a full hemisphere
+of the star. There is no explicit scattering term in the
+1ELC can also fully employ stellar atmosphere intensities,
+where no parameterized limb darkening is used. But prelimin ary
+tests gave significantly worse fits unless the stellar temper ature was
+allowed to be several thousand degrees hotter. The cause see ms to
+be related to the very wide Keplerbandpass and the lack of free-
+dom to adjust the limb darkening.Ellipsoidal Variations in HAT-P-7 3
+reflection: the incident radiation heats up the planet and
+is always re-emitted locally. Scattered light is implicitly
+accounted for by allowing the global temperature Tpto
+be de-coupled from the day-side temperature.
+TheELCmodelprovidesagoodfittotheobservations.
+The chi square is 57,225 for 49,016 degrees of freedom
+(reduced χ2
+ν=1.167). Uncertainties on the parameters
+were boosted by a factor of/radicalbig
+χ2ν(=8%) to account for
+the formally high χ2
+ν, which we attribute to systematic
+non-Gaussian noise. The values of the parameters are
+listed in Table 1.
+4.RESULTS AND DISCUSSION
+The 1.8 M Jupplanet orbiting only 4.1 stellar radii from
+its host star induces a tidal distortion on the star, chang-
+ing its shape from oblate (not spherical, due to its rota-
+tion) to amoretriaxialshape, with the longestaxisalong
+the direction towardthe planet and the shortestaxis per-
+pendicular to the orbital plane.2This shape causes the
+well-known “ellipsoidal variation” effect seen in binary
+stars: a modulation in light at half the orbital period,
+with maxima at phases near 0.25 and 0.75, and unequal
+minima at phases 0.0 and 0.5. The ellipsoidal variation
+is primarily a geometrical (projected surface area) effect
+whose relative amplitude depends on the mass ratio and
+the inclinationofthe binarysystem. Given thetight con-
+straintontheinclinationfromtheeclipses, theymaypro-
+vide some limits on the mass ratio, and hence the mass
+of the planet. In the optical, where the phase-dependent
+planet-to-star flux ratio is small, the presence of the stel-
+lar ellipsoidal variation becomes significant. Neglecting
+its contribution can lead to a confused interpretation of
+thephasecurveandthusanincorrectmeasurementofthe
+albedoandphase-dependentscattered/thermalemission.
+The presence of ellipsoidal variations in exoplanetary
+systems was anticipated by Loeb & Gaudi (2003) and
+Drake (2003), and Pfahl et al. (2008) present a detailed
+theoretical investigation of the tidal force on the star by
+the planet. Figs. 1 and 3 show the first detection of
+the effect in an exoplanet system. In Fig. 3 we show
+the light curve binned to 30 min, along with the best-
+fit model decomposition. The dotted curve shows the
+stellar-only ellipsoidal light curve while the dashed curve
+istheplanet-onlylightcurve(offsetvertically). The solid
+curveis the sum of the two and equals ourbest-fit model.
+The planet-only model can match the light curve only
+very near phase 0.5; it is a very poor fit to the data at
+other phases, indicating the need for the ellipsoidal vari-
+ation component. The amplitude of the ellipsoidal com-
+ponent is 37.3 ppm, detectable only because of Kepler’s
+high-precision photometric capability.
+Ellipsoidal variations arise as a consequence of gravity
+on a luminous fluid body. Within the Roche framework,
+its amplitude is exactly known if the mass ratio, incli-
+nation, and stellar radius are known. However, since
+in exoplanet systems the star is not expected to be in
+synchronous rotation, the Roche potential is an approx-
+imation, albeit a good one since the maximum tidal dis-
+tortion ∆ R/Ris only 10−4. However, we stress that the
+model presented here is only a starting point, based on
+2More exactly, in a Roche potential there are 4 distinct radii :
+towards the companion, perpendicular to the orbit plane, al ong the
+direction of motion, and away from the companion.the well-developed and successful Roche model. Effects
+such as those discussed in Pfahl et al. (2008) based on
+the equilibrium tide approximation can be present. As
+moreKeplerdata become available it will be interest-
+ing to try to distinguish between these approximations,
+as they can lead to measuring internal properties of the
+star. For example, in the frame of the rotating binary,
+the star is spinning retrograde if Pspin< Porb, which is
+expected to be true for most hot Jupiter planets. This
+could induce a phase shift in the ellipsoidal variations, as
+the star’sspin “drags”the tide awayfrom the line ofcen-
+ters. If a phase-lag not associated with the planet light
+can be unambiguously measured, it may be possible to
+put constraints on the tidal Q factor. This asymmetry
+could also lead to interesting orbital dynamics effects.
+The detailed shape and amplitude of the ellipsoidal vari-
+ation also depends on the stellar envelope (convective vs.
+radiative – Pfahl et al. (2008)), thus potentially offering
+a probe of stellar interiors. However, for HAT-P-7, the
+situation is complicated by the fact that the spin axis
+is not aligned with the orbit axis (Winn et al. 2009), so
+these optimistic statements must be tempered with cau-
+tion.
+Fig. 2 shows model light curves that illustrate the con-
+tributionsofthestartotheopticallight. Thelowercurve
+isthe stellarlight only, showingthe ellipsoidalvariations.
+It is not constant outside of transit, and its amplitude
+can be significant compared to the depth of the occulta-
+tion. Above this are four curves that include the planet
+contribution. The lowest is for an isothermal isotropic
+planet with no phase-dependent scattered or re-radiated
+emission. To firstorder, it is essentiallyan additive offset
+to the star’s light, the amount depending on the relative
+radius and temperature ofthe planet. If the absorbed in-
+cident energy from the star is not perfectly uniformly re-
+distributed around the planet, the day side of the planet
+will be hotter than the night side, resulting in an orbit-
+phase dependent modulation peaking at the sub-stellar
+point (phase 0.5). Or, if the atmosphere scatters the
+incident radiation, again the day side will be brighter
+than the night side (though not necessarily hotter). The
+upper three curves show this phase-dependent effect for
+different values of the heat albedo, which is a proxy for
+scattered and re-radiated emission on the day side. Note
+that advection can move the peak downwind and pro-
+duce a phase shift in the planet’s emission, as seen in the
+infrared phase curve of HD189733 (Knutson et al. 2007);
+this is not included in these models.
+Given a stellar temperature estimate, the ELC model
+can yield day and night hemisphere temperatures. Be-
+cause the planet is not black it contributes light at all
+phases (other than occultation) and its flux contribution
+at each orbital phase, relative to ellipsoidal and other ef-
+fects, allows its temperature to be estimated. In partic-
+ular, the ability to measure relative temperatures arises
+fromthe requirementofgettingthe occultationand tran-
+sit depths correct given the tight geometric constraints
+(e.g. a2600Kplanetreducesthe6000 ˚Atransitdepthby
+29ppm comparedtoazerotemperatureplanet). Wefind
+the peak temperature on the planet (at the sub-stellar
+point) is 3160 K, but a more meaningful flux-weighted
+day temperature is 2885 ±100 K. This is notably higher
+than the 2560 ±100 K estimate by Borucki et al. (2009).4 Welsh et al.
+We caution that the day-sidetemperature estimate is de-
+rived from the assumption that the light is entirely ther-
+mal in origin with no scattered component; thus this is
+a maximum day-side temperature estimate for the given
+heat albedo. We find the night side temperature Tpis a
+surprisingly hot 2570 ±95 K, only 590 K less than the
+peak day temperature.
+As seen in Fig. 3, the depth of the occultation is
+85.8 ppm, and the night-side contribution of the planet
+is 22.1 ppm, giving a day-side peak flux enhancement of
+63.7 ppm. Combining these flux ratios, temperatures,
+and the relative radii, we can attempt to estimate the
+geometric and Bond albedos — see Rowe et al. (2008)
+for a discussion. The planet’s albedo is key to the en-
+ergy balance in the planetary atmosphere and its im-
+portance for the dayside emission is discussed in sev-
+eral papers (e.g. Seager et al. (2005), Burrows et al.
+(2006), Lopez-Morales & Seager (2007)). The geomet-
+ric albedo is given by the ratio of planet to star flux
+at phase 0.5, scaled by the ratio of surface areas:
+Ag= (Fp/F∗)/(Rp/a)2. Using the peak planet to star
+flux of 63.7 ppm yields Ag=0.18. This agrees with
+the<0.20 value found for the pM-class planet CoRoT-
+1b (Snellen et al. (2009), Alonso et al. (2009b)), but is
+significantly higher than some other planets, notably
+Ag<0.08 for HD 209458b (Rowe et al. 2008), and 0.06
+±0.06 for CoRoT-2b (Alonso et al. 2009a). The equilib-
+rium temperature and Bond albedo are related by the
+definition: T4
+eq=T4
+∗(R∗/2a)2[f(1−AB)] where fis the
+redistribution factor and equals 1 for complete redistri-
+bution or 2 if the incident flux is re-radiated only on the
+dayhemisphere. Setting f=1andequatingthenight-side
+temperature with the equilibrium temperature should in
+principle allow one to solve for the Bond albedo since all
+the other terms are known. However, this fails because
+the maximum equilibrium temperature possible (whenAB=0) is 2213 K, considerably lessthan the ELC esti-
+mate. It may be that our 6000 ˚A brightness tempera-
+ture estimate exceeds the equilibrium temperature sim-
+ply because the planet is not a blackbody and at other
+wavelengths lower temperatures may be measured. We
+speculate on two other possibilities, both related to the
+presence of a third body in the HAT-P-7 system. First,
+the planet may genuinely be hotter then the equilibrium
+temperature due to non-radiative heating, perhaps tidal
+heating due to an encounter with another object (recall
+the>86 degree offset between the orbit and stellar spin
+axes—Winn et al.(2009), Narita et al.(2009)). Amore
+mundane, but perhaps more likely, explanation is that
+light from a third body is contaminating the flux ratios.
+The acceleration term in the radial velocities suggests
+the presence of another body (Winn et al. 2009). When
+moreKeplertransits are available, we can check for tran-
+sit timing variations and address the issue of potential
+third light contamination.
+In closing, the Keplerlight of HAT-P-7 curve reveals
+ellipsoidalvariationswith anamplitude ofapproximately
+37 ppm. This is the first detection of ellipsoidal varia-
+tions in an exoplanet host star, and shows the precision
+Kepleris capable of producing at this early stage. For
+comparison, a transit of an Earth-analog planet around
+a Sun-like star would produce a signal depth of 84 ppm,
+a factor of 2 larger than this effect.
+We thank the anonymous referee for highly valuable
+comments. We thank Ron Gilliland for kindly provid-
+ing assistance with the Q1 time series. WFW grate-
+fully acknowledges support from Research Corporation
+forScience Advancement. The authorsacknowledgesup-
+port from the KeplerParticipating Scientists Program
+via NASA grant NNX08AR14G. Kepler was selected as
+the 10th mission of the Discovery Program. Funding for
+this mission is provided by NASA, Science Mission Di-
+rectorate.
+Facilities: TheKeplerMission
+REFERENCES
+Alonso, R. et al. 2009, A&A, 506, 353
+Alonso, R. et al. 2009, A&A, 501, L23
+Borucki, W.J. et al. 2009, Science, 325, 709
+Borucki, W.J., Koch, D., et al. 2010, Science, (submitted)
+Burrows, A., Sudarsky, D., & Hubeny, I. 2006, ApJ, 650, 1140
+Christensen-Dalsgaard, J. et al. 2010, ApJ, (submitted)
+Drake, A.J. 2003, ApJ, 589, 1020
+Fortney, J. J., Lodders, K., Marley, M., & Freedman, R. 2008,
+ApJ, 678, 1419
+Gilliland, R. et al. 2010, PASP, (submitted)
+Gim´ enez, A. 2006, ApJ, 650, 408
+Jenkins, J., et al. 2010, ApJL, (submitted)
+Kallrath, J. & Milone, E.F. 1999, Eclipsing Binary Stars,
+Springer-Verlag, NY.
+Knutson, H. A., et al. 2007, Nature, 447, 183
+Koch, D., Borucki, W.J., et al. 2010, ApJL, (submitted)
+Loeb, A. & Gaudi, B.S. 2003 ApJ, 588, 117L
+Lopez-Morales M. & Seager, S. 2007, ApJ, 667, 191Narita, N., Sato, B., Hirano, T. & Tamura, M. 2009, PASJ, 61,
+L35
+Orosz, J. A. & Hauschildt, P. H. 2000, A&A, 364, 265
+P´ al, A. et al. 2008, ApJ, 680, 1450
+Pfahl, E., Arras, P. & Paxton, B. 2008, ApJ, 679, 783
+Rowe, J. F. et al. 2008, ApJ, 689, 1345
+Rowe, J. F. et al. 2006, ApJ, 646, 1241
+Seager, S., et al. 2005, ApJ, 632, 1122
+Snellen, I.A.G, de Mooij, E.J.W. & Albrecht, S. 2009, Nature ,
+459, 543
+Van Cleve, J. & Caldwell, D. 2009, Kepler Instrument Handbook
+KSCI-10933-001, NASA Ames Research Center, Moffett Field,
+CA (http://keplergo.arc.nasa.gov/Instrumentation.sht ml)
+Wilson, R.E. 1990, ApJ, 356, 613
+Winn, J.N. et al. 2009, ApJ, 703, 99
+Wittenmyer, R.A. et al. 2005, ApJ, 632, 1157Ellipsoidal Variations in HAT-P-7 5
+0.96 0.98 1.00 1.02 1.04orbital phase
+0.9920.9940.9960.9981.000normalized flux
+0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
+orbital phase0.9999501.0000001.0000501.0001001.000150normalized flux
+Fig. 1.— upper:Detrended and phase-folded light curve at 1-min cadence alo ng with the ELC fit. For scale, the horizontal bars show the
+verticalsize of the lower panel. lower:Light curve averaged in 5 min and 75 min bins. The double-hump ed shape is due to the ellipsoidal
+variations of the star plus light from the planet.6 Welsh et al.
+0.1 0.2 0.3 0.4 0.5
+orbital phase1.0000001.0000201.0000401.0000601.0000801.0001001.000120
+normalized flux
+Fig. 2.— ELC model light curves for half an orbit. The bottom curve is t he stellar light curve exhibiting the ellipsoidal variatio ns. Above
+this are planet+star light curves with increasing bolometr ic (heat) albedos: 0.0, 0.3, 0.6 and 0.8. If the albedo is 0.0 t he planet emits a
+constant amount of light resulting in a simple offset in flux. A s the albedo increases, the light from the planet becomes mor e pronounced.Ellipsoidal Variations in HAT-P-7 7
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
+orbital phase0.9999801.0000001.0000201.0000401.0000601.0000801.0001001.000120
+normalized flux
+Fig. 3.— Phase-folded Keplerlight curve with best fit model (solid curve). Also shown are t he component stellar-only ellipsoidal model
+(dotted) and the planet-only model (offset by +1; dashed). Bo th the data and models have been cast into 30-min bins.8 Welsh et al.
+TABLE 1
+HAT-P-7 System Parameters
+Parameter Value Uncertainty Unit
+T∗a6350 — K
+M∗b1.53 0.04 M⊙
+R∗b1.98 0.02 R⊙
+K∗c212 5 m s−1
+Orbital inclination, i 83.1 0.5 degrees
+Orbital period, P 2.204733 0.000010 days
+Star-to-planet radius, R ∗/Rp 12.85 0.05
+limb dark coefficient x 0.58 0.08
+limb dark coefficient y 0.21 0.13
+Mass of planet, Mp 1.82 0.03 M Jup
+Radius of planet, Rp 1.50 0.02 R Jup
+Semimajor axis, a 8.22 0.02 R⊙
+Bolometric (heat) albedo, A bol 0.57 0.05
+Tp (night side) 2570 95 K
+Tp (average day side) 2885 100 K
+Note. —
+aHeld fixed at 6350 K; P´ al et al. (2008)
+bSteered toward 1.52 ±0.036; Christensen-Dalsgaard et al. (2010)
+bSteered toward 1.991 ±0.018; Christensen-Dalsgaard et al. (2010)
+cSteered toward 211.8 ±2.6; Winn et al. (2009)
\ No newline at end of file
diff --git a/1001.0414.txt b/1001.0414.txt
new file mode 100644
index 0000000000000000000000000000000000000000..dc1ddd9cd106c3d0fbfccc5104aa6eeb808d9bfc
--- /dev/null
+++ b/1001.0414.txt
@@ -0,0 +1,291 @@
+Photometric Variability in Kepler Target Stars: The Sun Among
+Stars { A First Look.
+Gibor Basri1, Lucianne M. Walkowicz1, Natalie Batalha2, Ronald L. Gilliland3, Jon
+Jenkins2, William J. Borucki2, David Koch2, Doug Caldwell2, Andrea K. Dupree4, David
+W. Latham4, Soeren Meibom4, Steve Howell5, Tim Brown6
+ABSTRACT
+TheKepler mission provides an exciting opportunity to study the lightcurves
+of stars with unprecedented precision and continuity of coverage. This is the
+rst look at a large sample of stars with photometric data of a quality that has
+heretofore been only available for our Sun. It provides the rst opportunity to
+compare the irradiance variations of our Sun to a large cohort of stars ranging
+from vary similar to rather dierent stellar properties, at a wide variety of ages.
+Although Kepler data is in an early phase of maturity, and we only analyze the
+rst month of coverage, it is sucient to garner the rst meaningful measurements
+of our Sun's variability in the context of a large cohort of main sequence stars in
+the solar neighborhood. We nd that nearly half of the full sample is more active
+than the active Sun, although most of them are not more than twice as active.
+The active fraction is closer to a third for the stars most similar to the Sun, and
+rises to well more than half for stars cooler than mid K spectral types.
+Subject headings: stars: activity | stars: spots | stars: statistics
+1. Introduction
+Since Galileo and Kelper noted dark spots on the face of the Sun 400 years ago, scientists
+have been aware that its output is not completely constant. It has only been in the past
+1Astronomy Department, University of California, Berkeley, CA 94720
+2NASA Ames Research Center, Moett Field, CA 94035
+3Space Telescope Science Institute, Baltimore, MD 21218
+4Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138
+5National Optical Astronomy Observatory, Tucson, AZ 85719
+6Las Cumbres Observatory Global Telescope, Goleta, CA 93117arXiv:1001.0414v1  [astro-ph.SR]  3 Jan 2010{ 2 {
+few decades that precise measurements of the solar irradiance variations have been possible
+(Fr ohlich 2006). These show that the solar output varies by a small amount (a tenth of a
+percent, or a part in a thousand at most), and that although sunspots really do cause 
ux
+to be suppressed, bright faculae counteract this eect in a general way (although not in a
+one-to-one relationship) so that the integrated irradiance of the Sun is actually larger in the
+active part of a solar cycle despite clear dips produced by big spot groups.
+It has also been obvious for a long time that some stars are quite variable. There
+are a plethora of sources of stellar variability, from eclipses to pulsations to accretion phe-
+nomena to explosions on various scales, and a host of other possible causes. In the last
+century we have also been able to identify variations due to starspots, most of which are
+much larger than sunspots and are found on stars that are considerably more magneti-
+cally active (Strassmeier 2009). Eorts have been made to evaluate what the range and
+character of brightness variations (both positive and negative) due to magnetic activity on
+solar-type7stars are, and what stellar characteristics go with them. In general, we can
+say that more rapidly rotating stars exhibit more variability due to increased magnetism
+(Pizzolato et al. 2003). Activity is also related to age, since solar-type stars spin down due
+to magnetic braking, so that young stars are more active than older ones. It also seems
+that cooler (relative to the Sun, in the spectral range F-M) stars show high variability more
+commonly than hotter stars (Eyer & Grenon 1997), in part because there is greater contrast
+and temperature sensitivity of heated regions against cooler photospheres (which also allows
+
ares to be more obvious).
+One pressing question has been the place of the Sun in the pantheon of solar-type stars.
+Suggestions that it might be quieter than typical were made based on a small sample, but
+this issue has remained unresolved (Lockwood et al. 2007). This question has gained greater
+currency as the issue of the eect of solar variability on climate change has come into the
+public consciousness. It is of interest in any case to know what the range of behavior in
+solar-type stars is. Proxies for magnetic eld (such as CaII H & K or X-ray emission) have
+given us a fairly good idea of the general landscape of magnetic activity and the Sun's place
+in it. Based on emission proxies, Batalha et al. (2002) estimated that about two-thirds of
+solar-type stars lie in the same range of magnetism as the Sun (minimum to maximum) with
+one-third more active. This result can arise naturally since by the Suns age most of the
+spindown has already been accomplished, and the Sun is nearly half the age of the Galaxy.
+It is subject to some imprecision, however, since there is a fair spread of activity at a given
+stellar rotation rate, and stars exhibit cycles of varying magnitudes. The translation of
+7By \solar-type" we mean main sequence stars with radiative cores and signicant convective envelopes.{ 3 {
+magnetism into photometric variability is also dependent on spectral type.
+Until recently, only variations in fairly active stars have been directly measurable by
+photometry. The CoRoT satellite has provided the rst opportunity for a large-scale survey
+of photometric variability that approaches the precision needed to resolve variability at
+the level of the quiet Sun. The rst results have been published on a few hundred stars
+(Aer et al. 2009); they suggest that almost half of solar-type stars exhibit enough variability
+that CoRoT can discern a rotation period over timescales up to 80 days or so. While not
+exact, the activity levels detected seem to be at least that of the active Sun.
+The Kepler mission (e.g Borucki et al. 2009) is a wide-eld 1-m space telescope whose
+primary purpose is to detect transits and to discover Earth-size exoplanets. The mission
+was launched on March 6, 2009. To accomplish its main goal, it obtains photometry of
+roughly 150,000 stars, most of which are intentionally selected to be solar-type. Although
+several months of data have been returned, the data reduction process is under renement.
+Nonetheless, the rst month of data has sucient precision even in its rough form to under-
+take an informative analysis of the activity on a very large sample of a wide variety of main
+sequence stars, and compare their gross variability to that of the Sun.
+2. Observations and Analysis
+The Kepler Quarter 1 (Q1) observations took place over 33.5 days between May 13
+and June 15 2009. 156,097 stars were observed at a cadence of 30 minutes. As the intent
+of this paper is to determine the intrinsic variability of main sequence stars, we only kept
+stars that have logg 4 in the Kepler Input Catalogue (KIC). No cut was made on eective
+temperature. Of the 156,097 stars observed, 121,432 stars survived this cut, while 24,815
+are assigned lower gravity (and therefore presumably subgiants and giants) and 9,849 are
+unclassied. The KIC is known to have some mis-identications (Koch et al. 2009), and
+some of the unclassied stars are dwarfs, but the errors introduced are thought to be a few
+percent at most. Since we are primarily interested in variability due to modulation by spots
+and other manifestations of magnetic activity (at least for stars cooler than 6500K), the
+sample was further cleaned of obvious transiting/eclipsing systems and contact binaries as
+described below. The nal sample analyzed consisted of 104,376 stars between T eff3200
+and19,000 K, with the median T effaround 5600K.{ 4 {
+2.0.1. Variability Statistics and Periodograms
+The raw 
ux time series provided by Kepler consist of the summed 
ux in the target pixel
+aperture, background subtracted and corrected for cosmic ray hits (Jenkins et al. 2009).
+This does mean that more than one star contributes to it, but the target star is generally
+the greatest contributor. We rst remove a linear slope from the raw 
ux time series, then a
+fourth-order polynomial. While this may sometimes remove physical eects, much of these
+trends are instrumental in nature (as can be inferred from geometrical correlations on the
+focal plane). These include small shifts of stars on pixels caused by focal changes due to
+thermal eects from the Sun-angle of the telescope, and changing velocity aberration over
+the rather large eld of view as the velocity vector moves during a quarter.
+The lightcurves are then converted to dierential photometry by dividing out the median
+
ux and subtracting unity. The absolute deviations from zero of the rectied lightcurves
+(boxcar smoothed on a 10-hour timescale for this study) are dened as the \range" (half-
+amplitude) of variability. This will be the primary statistic employed in this paper. A
+further set of statistics are used to characterize other measures of variability; they are only
+employed here to lter out non-solar forms of variability. These include metrics to determine
+the typical timescale of variability: the time separation between points where the dierential
+lightcurve crosses zero, and the time separation between changes in the sign of the slope in the
+lightcurve. These are easily measured on very large datasets. We calculate the mean, median
+and maximum for each of our statistics. In addition, we run Horne-Baliunas periodograms
+(Horne & Baliunas 1986) on all of our targets. From the periodograms we determine the
+signicant peaks and then store both the power and location of the individual peaks, as well
+as the total number of peaks and the power in the most signicant peak. This information
+is then combined with our other statistics to identify subsamples of stars that are likely
+pulsators and eclipsing systems. This subsample was then examined by eye to ensure that
+no spot-modulated lightcurves were lost. In future work we will return to the behavior of
+the full set of stars using all these statistics.
+2.0.2. SOHO Lightcurves
+The SOHO lightcurves provide a straightforward means of comparing the Kepler data
+with that of the Sun as a star. Like Kepler photometry, the SOHO bolometric instrument
+is dominated by a broad white light passband. We also examined the sum of the green and
+red passbands of the VIRGO instrument, which should also vary like Kepler photometry (all
+these measures vary as a blackbody in each bandpass given the same very small temperature
+perturbations). We used the 2001 active Sun VIRGO g+r lightcurves, re-binning them to{ 5 {
+the Kepler cadence ( 30 minute sampling) and transforming them into dierential 
uxes.
+The SOHO VIRGO g+r lightcurves were then broken into 33 day segments of time,
+one set oset from the other by 16 days, to provide 20 analogs to a Kepler lightcurve. The
+same statistics and periodogram analysis was performed on them. These statistics were then
+compared with the Kepler sample to place the Sun in the context of the Kepler observations.
+3. Results
+3.1. Variability by brightness and temperature
+The range in variability is our key statistic that best separates active stars from quiet
+ones { objects with a large range have large modulation/features, while those with small
+range are quiet. In Figure 1, we plot the range in millimags as a function of Kepler magnitude
+(Kepmag). The rise of minimum range with fainter magnitude is caused by the smaller stellar
+signal relative to the xed instrument noise and increase in shot noise for dimmer stars {
+hence, the minimum amount of stellar variability that can be measured is larger for fainter
+stars. Recall that the range is dened for a smoothed lightcurve; this does not matter for
+the brighter stars but calms the noise spikes in the fainter stars.
+The approximate activity level of the active Sun is shown in this Figure 1 as a red
+line { lightcurves near this line have modulations similar to the active Sun; those above are
+more active and those below are less active. The locus of this line was determined through
+extensive by-eye comparison between the SOHO and Kepler lightcurves. We determined
+at what value of the range the amplitude of apparently stellar variability for the noisier
+solar-like stars was consistent with solar-like variability, by looking at many examples with
+similar range. We will explore a more objective way of doing this later. Across the range of
+magnitudes, Kepler targets which resemble the active Sun were found, although clearly the
+boundary between these populations is not as sharp as a single line implies. As the noise
+levels increase, the range has to be bigger to reveal clear variations with the same character
+as the active Sun. This is why the red line curves up at faint magnitudes.
+Figure 2 gives a few examples of lightcurves. The upper left shows a few solar examples
+from SOHO, while the upper right has a solar example at the top and three Kepler stars we
+deem comparable below it. The lower left panel shows some quiet Kepler examples (with
+dierent noise levels). The lower right panel has a much coarser scale in order to capture the
+much bigger amplitudes in stars with larger ranges; here the active Sun looks very quiet by
+comparison at the top. The active fraction is determined by comparing the number of stars
+below the active Sun (red line) to the numbers of stars above. We nd that 46% of stars in{ 6 {
+our entire sample are more active than the active Sun. We have also calculated the active
+fraction of stars as a function of temperature and magnitude bin { these results are shown
+In Table 1. Here \bright" indicates a Kepler magnitude brighter than 13.5, while \faint"
+indicates stars dimmer than that.
+Figure 3 shows the range versus eective temperature. The minimum amplitude of
+variability is larger for cooler stars in the sample, and there is a dearth of quiet cool stars.
+This is partially, but not entirely due to the fact that cool dwarfs tend to be fainter and
+noiser. If there were many quiet ones, however, there would be a thick ridge along the noise
+boundary. The thin vertical gaps in this gure re
ect structure in the KIC, not real stellar
+eects. Figure 4 provides an alternative view of activity as a function of magnitude and
+eective temperature: each of the four panels shows the range for a dierent bin of eective
+temperatures as a function of Kepler magnitude. Clockwise from upper left it shows the
+hottest stars ( 21,000 stars, T eff>6000 K), stars with 6000K >Teff>5500 K ( 43,000
+stars corresponding roughly to late F to mid G dwarfs), stars with 5500K >Teff>4500K
+(35,000 stars from mid G to mid K dwarfs) and T eff<4500K ( 6500 stars, mid K to mid
+M dwarfs). The progression of typical range by temperature is evident in these panels, with
+the hottest stars covering the full space from quiet (low amplitude) to extremely variable
+(high amplitude), while the majority of the coolest stars (lower right panel) are more variable
+overall. The vertical features at Kepler magnitudes 13.9 and 15.5 in the two hotter panels
+are artifacts of target selection { hot stars fainter than selected magnitudes were disfavored
+to leave more capacity for smaller stars.
+We also looked at the statistics for stars that are more than twice as variable as the
+active Sun. Overall this fraction drops to 18%, showing that a great many of the active stars
+are not that much more active than our Sun. For the stars in the second (most solar-like)
+temperature bin, the more active fraction is only 10%. On the other hand, in the coolest
+bin we still nd 43% more than twice as active. This result is less sensitive to the noise
+
oor than the straight active Sun fraction, and once again reinforces the impression that the
+cooler stars are simply more photometrically variable.
+4. Conclusions and Future Work
+The rst thing to remember about these results is that they are rather preliminary. The
+data reduction pipeline is under development (Jenkins et al. 2009), and much longer time
+coverage will be available as the mission proceeds. It is also possible to separate out sub-types
+of lightcurves to a much greater extent than we have done here. This will allow statements
+not just about the general level of variability, but the separation of magnetic from other{ 7 {
+sources of variability, and the characterization of the sources of variability. Longer sampling
+time will help both with conrming short-term behaviors and with capturing variability that
+occurs on longer timescales. The current sample period is only a bit longer than one solar
+rotation (although the timescale for features to appear and disappear in the solar lightcurve
+is comfortably shorter than that). On the other hand, the patching together of data from
+successive quarterly 90deg rolls about the optical axis of the spacecraft will require better
+understanding of the secular trends in the data: which of them are instrumental, which are
+astrophysical, and how best to remove the instrumental eects.
+With these caveats in mind, our preliminary results are fairly clear. As expected, there
+are a lot of stars more variable than our Sun. In the entire sample nearly half are more active,
+and this result does not appear to be driven by the noise 
oor at faint magnitudes. It results
+from the average of late F to early G stars with active fraction of only a third (these are
+most comparable to the Sun), late G and K stars whose active fraction is roughly a half, and
+the even more variable cooler stars. The results for the most Sun-like stars are compatible
+with the expectations described in Section 1. In the moderately cool stars, the contrast of
+active areas may increase and/or they have greater magnetic 
uxes. The majority of stars
+cooler than mid-K are much more variable (active fractions 85%); this is only partly due
+to the fact that the noise 
oor is greater for these fainter stars. Stars hotter than 7000 K
+are comparably variable to the late F to early G dwarfs but more evenly spread out (Figure
+3). This is a warning that we are probably looking at more than just magnetic variability,
+since that is not expected for early F and A stars (except of course the Am and Ap stars).
+Figure 3 appears to show a general trend of increasing variability at lower temperature
+in the high density region of the plot. This is not primarily an eect of increasing noise for
+fainter stars, because Figure 4 shows that there are many faint stars in both of the mid-
+temperature panels (b and c, 5500-6000 K and 4500-5500 K). The high density bulk of stars
+is also at most three times as variable as the active Sun, and many of them are quieter than
+the active Sun. There is a thinner distribution of yet more active stars above them, which
+appears at late F, peaks around early K and declines below mid K. This group reaches a
+variability level up to 50 times the active Sun. There is a yet sparser set of stars that are
+even more active than that, which reaches a gentle peak in activity around early K.
+One goal of our future work is to determine rotation periods for as many of the Kepler
+stars as possible. Since rotation is intimately linked to the magnetic activity of stars, spots
+provide a natural way to trace the stellar rotation period (except for the quietest stars). It
+is clear that there are tens of thousands of Kepler targets which exhibit sucient variability
+to allow determination of their rotation period. As improvements in the data occur and
+the length of coverage keeps increasing, even more stars will yield their rotations. It will{ 8 {
+be possible to dene a rotation-activity relation purely from the Kepler data, since the
+amplitude of variability may be related to activity for spotted stars. Of course, there are
+many potential complications. For example, polar or widely distributed spots might not
+produce a recognizable signal, bright faculae produce a somewhat opposite eect, and the
+solar analogy might break down to various degrees. In the cooler stars, we will also be able
+to study the 
aring fraction. Beyond determining periods, we will be able to model the spot
+coverage fraction, average lifetime and growth and decay rates, and track dierential rotation
+among dierent spots on the same star in many cases. It is more dicult to constrain the
+stellar inclination, and polar spots will only show up as slow variations. Over the few years
+of the mission, it may be possible in some cases to discern the stellar activity cycle. The
+potential of the Kepler mission to contribute to the understanding of solar-type stars via
+photometric variability due to magnetic activity is extremely promising.
+The authors wish to thank the entire Kepler mission team, including the engineers and
+managers who were so pivotal in the ultimate success of the mission. LW is grateful for the
+support of the Kepler Fellowship for the Study of Planet-Bearing Stars . Funding for this
+Discovery mission is provided by NASA's Science Mission Directorate.
+REFERENCES
+Aer, L., Micela, G., Favata, R., & Flaccomio, E. 2009, American Institute of Physics
+Conference Series, 1094, 341
+Borucki, W. J. et al. Science , in press.
+Batalha, N. M., Jenkins, J., Basri, G. S., Borucki, W. J., & Koch, D. G. 2002, Stellar
+Structure and Habitable Planet Finding, 485, 35
+Table 1. Fraction of Active Stars
+Teff>6000 K 5500 K <Teff<6000 K 4500 K <Teff<5500 K T eff<4500 K
+all stars 0.36 0.37 0.57 0.87
+Nstars , total 21023 42832 33288 6522
+bright stars 0.41 0.31 0.46 0.84
+Nstars , bright 8747 6164 2613 369
+faint stars 0.33 0.38 0.57 0.88
+Nstars , faint 12276 36668 30675 6153Note:
+The dividing line between bright and faint is taken at Kepmag=13.5.{ 9 {
+Eyer, L., & Grenon, M. 1997, Hipparcos - Venice 1997, 402, 467
+Fr ohlich, C. 2006, SOHO-17. 10 Years of SOHO and Beyond, 617
+Horne, J. H., & Baliunas, S. L. 1986, ApJ, 302, 757
+Jenkins, J. et al. ApJL , in press.
+Koch, D. et al. ApJL , in press.
+Lockwood, G. W., Ski, B. A., Henry, G. W., Henry, S., Radick, R. R., Baliunas, S. L.,
+Donahue, R. A., & Soon, W. 2007, ApJS, 171, 260
+Pizzolato, N., Maggio, A., Micela, G., Sciortino, S., & Ventura, P. 2003, A&A, 397, 147
+Strassmeier, K. G. 2009, A&A Rev., 17, 251
+This preprint was prepared with the AAS L ATEX macros v5.0.{ 10 {
+Fig. 1.| Kepler magnitude versus the range of lightcurve modulation. The denition of the
+range is discussion in Section 2. The red line indicates the locus of the active Sun { stars
+lying along this line have levels of activity similar to the active Sun; those that lie below it
+are quieter, and those above more active.{ 11 {
+Fig. 2.| Example lightcurves from Kepler and SOHO { lightcurves are shown oset by
+various constants for clarity, with their zero levels indicated as dashed lines. Upper left:
+several segments of SOHO Virgo lightcurves are shown in red. In subsequent panels SOHO
+lightcurves are shown at the top in red for comparison. Upper right: Kepler lightcurves with
+activity levels similar to the active Sun. Lower left: quiet Kepler stars. Lower right: several
+examples of Kepler lightcurves of spotted stars; note that this panel has a markedly dierent
+scale than the prior three to show the amplitude of the lightcurve features.{ 12 {
+Fig. 3.| Eective temperature from the Kepler Input Catalogue versus the range of
+lightcurve variability. The red line gives the range value of the active Sun. The dearth
+of quiet stars at cool temperatures is evident in this gure; this is partly due to the faintness
+of cool dwarfs and measurement noise.{ 13 {
+Fig. 4.| Kepler magnitude versus the range of lightcurve variability in dierent temperature
+ranges, with the locus of the active Sun shown as red lines. The four panels correspond to
+dierent eective temperature bins, clockwise from upper left: stars hotter than 6000K, stars
+between 5500 and 6000 K, stars between 4500 and 5500 K, and stars cooler than 4500 K.
\ No newline at end of file
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+A new Rational Generating Function for the Frobenius Coin Problem
+Deepak Ponvel Chermakani, IEEE Member
+deepakc@pmail.ntu.edu.sg  d.chermakani@sms.ed.ac.uk deepakc@usa.com  deepakc@myfastmail.com
+Abstract: -     An important question arising from the Frobenius Coin Problem is to decide whether or not a given 
+monetary sum S can be obtained from N coin denominations. We develop a new Generating Function G(x), where the 
+coefficient of xiis equal to the number of ways in which coins from the given denominations can be arranged as a 
+stack whose total monetary worth is i. We show that the Recurrence Relation for obtaining G(x), is linear, enabling 
+G(x) to be expressed as a rational function, that is, G(x) = P(x)/Q(x), where both P(x) and Q(x) are Polynomials whose 
+degrees are bounded by the largest coin denomination.
+1.     Introduction
+In the language of mathematics, the Frobenius Coin Problem is as follows: - Let A 1, A 2, …A Nbe fixed positive integers. Find 
+the largest integer S, such that S ≠A1X1+ A 2X2+ … + A NXN, for all non-negative integers X 1, X 2, …X N[2]. An important 
+question arising from the Frobenius Coin Problem is therefore to decide whether or not, for some given target positive integer 
+S, there exist non-negative integers X 1, X 2, … X N, such that S = A 1X1+ A 2X2+ … + A NXN[1].
+     If we denote F ito be an integer sequence, such that F iis the number of ways in which i can be expressed using the given N 
+coin denominations, then one can obtain a Generating Function given by SUMMATION (F iXi, i being integers ≥0) [3]. The 
+Generating Function helps predict whether or not some monetary sum S can be obtained using the given coin denominations, 
+and is expressible as P(x)/Q(x) where both P(x) and Q(x) are Polynomials. It has been proved that if N is fixed, then the
+Generating Function for F ican be expressed as a sum of short rational functions [4]. For N ≥ 4, there is no explicit bound on 
+the number of terms in P(x) [3][5], though Q(x) is bounded [3][5].
+     In this paper, we do not aim to develop a Generating Function explicitly for the Frobenius Coin Problem, i.e. for the 
+sequence F imentioned in the above paragraph. Instead, we develop a Generating Function for the sequence of integers 
+(which we shall denote as E i) representing the number of ways that coins from the given N denominations may be arranged 
+as a stack, one coin over the other, such that the sum of the values of all coins in this stack add up to i. Clearly, the sequence 
+Fiis different from E i, for example, given 2 coin denominations of 2 units and 5 units, then F 7is 1, while E 7is 2. It is also 
+clear that F iis 0, if and only if, E iis 0. And we shall see that the Generating Function for E ican be expressed as a rational 
+function, i.e. as P(x)/Q(x), where both P(x) and Q(x) are Polynomials, and where the degrees, of both P(x) and Q(x), are 
+equal to L, the largest coin denomination.
+     In the next Section 2, we shall prove a Theorem on a Linear Integer Recurrence Relation, and demonstrate how it 
+represents the sequence E i. In Section 3, we shall show the Generating Function for E i.
+2.     The Linear Integer Sequence E i
+Theorem-1 : Let L be the largest of the given N coin denominations, and the binary vector <B 1, B2, …B j, …B L-1, BL> be 
+such that B jis 1, if j is a coin denomination, and 0 otherwise. Then the number of ways that coins from the given 
+denominations may be arranged as a stack, such that the total monetary worth of the stack is equal to i, is given by the 
+following Linear Recurrence Relation for E i
+E0= 1, and,
+Ei= B LEi-L+ B L-1Ei-L+1+ …+ B 2Ei-2+ B 1Ei-1when iis an integer greater than 0, and,
+Ei= 0 when iis an integer lesser than 0.
+Proof : Expanding the sequence E iand expressing solely in terms of the B vector, we get the following
+E1= B 1
+E2= B 2+ B 12
+E3= B 3+ B 2B1+ B 1B2+ B 13= B 3+ 2B 1B2+ B 13
+E4= B 4+ B 3B1+ B 22+ B 2B12+ B 1B3 + 2B 12B2+ B 14= B 4+ 2B 1B3+ B 22+ 3B 12B2+ B 14
+E5= B 5+ 2B 1B4+ 2B 2B3+ 3B 22B1+ 4B 2B13+ 3B 12B3+ B 15
+and so on, showing that the sum of subscripts in each of the terms generated so far, are the same as the subscript of the 
+corresponding E i, and represent all possible stack formations. Next, let us prove by induction, that if we assume that the expressions via the B vector for E i-L, Ei-L+1, … E i-2, Ei-1represent all possible stack formations with monetary worth equal to 
+their corresponding subscripts, then E i= B LEi-L+ B L-1Ei-L+1+ …+ B 2Ei-2+ B 1Ei-1will represent all possible stack formations 
+of monetary value equal to i. To show this is true, we shall verify two things.
+First, we verify that all terms of the form B 1K1B2K2B3K3 … B LKLare represented in E ifor non-negative integers K 1, 
+K2…K Lsuch that K 1+ 2K 2+…+ LK L= i. If, for all integers tless than i, one were to consider all possible ways of 
+representing terms in E t(i.e. the sum of subscripts in the product should be equal to i), then it suffices to have the element B i-t
+appended to the terms of E t, and represent that product in E i, because every term in E pis the subset of some term in E q, where 
+p< q. For example, E tneed not be multiplied by the product B t1Bt2Bt3, where t1+t2+t3 = i-t, because terms generated by this 
+product would already be coming into E ithrough other routes, namely from B t1Ei-t1, from B t2Ei-t2, and from B t3Ei-t3.
+Second, we verify the coefficient of the term B 1K1B2K2B3K3 … B LKLin E i, where K 1, K 2…K Lare positive integers. 
+Clearly, this term gets linearly additive contributions from the coefficient of B 1K1-1B2K2B3K3 … B LKLin E i-1, from the 
+coefficient of B 1K1B2K2-1B3K3 … B LKLin E i-2, …and so on, and finally from the coefficient of B 1K1B2K2B3K3 … B LKL-1in E i-L.
+So, if n! denotes the factorial of n, then the coefficient of B 1K1B2K2B3K3 … B LKLin E i, is equal to the following.
+)!1 ...(!!!)!1 ... (.....! ...)!1 ( ! !)!1 ... (
+! ...! )!1 ( !)!1 ... (
+! ...!! )!1 ()!1 ... (
+3 2 12 1
+3 2 12 1
+3 2 12 1
+3 2 12 1
+
+LL
+LL
+LL
+LL
+K KKKK K K
+K K KKK K K
+K K K KK K K
+K KK KK K K
+For each of the above L additive components, multiply and divide the jthcomponent by K j, to obtain the following.
+!...!!!)!... (
+3 2 12 1
+LL
+K KKKK K K, which verifies to be the number of ways of arranging the term B 1K1B2K2B3K3 … B LKLin a stack.
+Finally, as B jis either 1 or 0, depending on whether or not coin denomination j is available, the value of E iwill 
+always equal the number of ways in which a monetary stack worth i, may be obtained from the given coin denominations.
+Hence Proved Theorem-1
+3.     The Generating Function
+The corresponding Generating Function G(x), for the sequence E imentioned in Section 2, can be shown to be given by
+1 ...1) ( ...) ... ( ) ... ()(
+11
+11 1 2 3 1 1 3 22
+1 2 1 1 2 11
+      
+   
+   
+xB x B xBE Bx E EB E B B x E EB E B B xxGL
+LL
+LL L L LL
+L L L LL
+4.     Conclusion
+In this paper, we presented a Linear Recurrence Relation for the integer sequence E i, representing the number of ways in 
+which a stack of coins worth i, may be built from the given coin denominations. This linearity enabled its Generating 
+Function G(x) to be expressed as a rational function, i.e. as P(x)/Q(x), where both P(x) and Q(x) are Polynomials, and where 
+the degrees of both P(x) and Q(x), are bounded by the largest coin denomination. As E iis zero, if and only if, a monetary 
+amount of icannot be obtained from the coin denominations, G(x) will be of use to the Frobenius Coin Problem.
+References
+[1] Sebastian Bocker, Zsuzsanna Liptak, The Money Changing Problem Revisited: Computing the Frobenius Number in Time O( k a 1), 
+Sep-2005, Computing and Combinatorics, Volume 3595/2005, Lecture Notes in Computer Science, Springer Berlin Heidelberg, ISBN 978-
+3-540-28061-3.
+[2] Jorge L. Ramirez Alfonsin, The Diophantine Frobenius problem , Oxford University Press, Oxford, 2006.
+[3] Matthias Beck, The Coin Exchange Problem of Frobenius , San Fransisco State University, 
+http://math.sfsu.edu/beck/papers/frobeasy.slides.pdf
+[4] Alexander Barvinok, Kevin Woods, Short rational generating functions for lattice point problems , Journal of the American 
+Mathematical Society, Volume 16 pp 957-979, April-2003.
+[5] Book Chapter, The Coin-Exchange Problem of Frobenius , Chapter 1, pp 3-24, Computing the Continuous Discretely, Springer 
+Publishers New York, ISBN 978-0-387-46112-0, Nov-2007.
+About the Author
+I, Deepak Ponvel Chermakani, have written this paper, out of my own interest and initiative, during my spare time. I am currently a student 
+at the University of Edinburgh UK (www.ed.ac.uk), where since Sep-2009, I have been enrolled in a fulltime one year Master Degree
+course in Operations Research with Computational Optimization . In Jul-2003, I completed a four year fulltime four year Bachelor Degree 
+course in Electrical and Electronic Engineering , from Nanyang Technological University Singapore (www.ntu.edu.sg). I completed my 
+high schooling from the National Public School in Bangalore in India in Jul-1999.
\ No newline at end of file
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@@ -0,0 +1,678 @@
+arXiv:1001.0416v1  [astro-ph.EP]  4 Jan 2010Discovery and Rossiter-McLaughlin Effect
+of Exoplanet Kepler-8b0
+Jon M. Jenkins2, William J. Borucki1, David G. Koch1, Geoffrey W. Marcy3,
+William D. Cochran8, Gibor Basri,3Natalie M. Batalha4, Lars A. Buchhave5,6, Tim M. Brown7,
+Douglas A. Caldwell2, Edward W. Dunham9, Michael Endl8, Debra A. Fischer10,
+Thomas N. Gautier III11, John C. Geary5, Ronald L. Gilliland12, Steve B. Howell13,
+Howard Isaacson3, John Asher Johnson14, David W. Latham5, Jack J. Lissauer1,
+David G. Monet15, Jason F. Rowe1,16, Dimitar D. Sasselov5, William F.Welsh17,
+Andrew W. Howard3, Phillip MacQueen8, Hema Chandrasekaran2, Joseph D. Twicken2,
+Stephen T. Bryson1, Elisa V. Quintana2, Bruce D. Clarke2, Jie Li2, Christopher Allen18,
+Peter Tenenbaum2, Hayley Wu2, Soren Meibom5, Todd C. Klaus18, Christopher K. Middour18,
+Miles T. Cote1, Sean McCauliff18, Forrest R. Girouard18, Jay P. Gunter18, Bill Wohler18,
+Jennifer R. Hall18, Khadeejah Ibrahim18, AKM Kamal Uddin18, Michael S. Wu19,
+Paresh A. Bhavsar1, Jeffrey Van Cleve2, David L. Pletcher1, Jessie A. Dotson1, Michael R. Haas1
+0Based in part on observations obtained at the W. M. Keck Obser vatory, which is operated as a scientific part-
+nership among the California Institute of Technology, the U niversity of California, and the National Aeronautics
+and Space Administration. The Observatory was made possibl e by the generous financial support of the W.M. Keck
+Foundation.– 2 –
+ABSTRACT
+We report the discovery and the Rossiter-McLaughlin effect of Kepler-8b, a transit-
+ingplanetidentifiedbytheNASA Kepler Mission .KeplerphotometryandKeck-HIRES
+radial velocities yield the radius and mass of the planet aro und this F8IV subgiant host
+star. The planet has a radius RP= 1.419RJand a mass, MP= 0.60MJ, yielding
+a density of 0 .26gcm−3, among the lowest density planets known. The orbital pe-
+riod isP= 3.523 days and orbital semimajor axis is 0 .0483+0.0006
+−0.0012AU. The star has
+a large rotational vsiniof 10.5±0.7 kms−1and is relatively faint (V ≈13.89 mag),
+both properties deleterious to precise Doppler measuremen ts. The velocities are indeed
+noisy, with scatter of 30 ms−1, but exhibit a period and phase consistent with the
+planet implied by the photometry. We securely detect the Ros siter-McLaughlin effect,
+confirming the planet’s existence and establishing its orbi t as prograde. We measure
+an inclination between the projected planetary orbital axi s and the projected stellar
+rotation axis of λ=−26.9±4.6◦, indicating a moderate inclination of the planetary
+orbit. Rossiter-McLaughlin measurements of a large sample of transiting planets from
+Keplerwill provide a statistically robust measure of the true dist ribution of spin-orbit
+1NASA Ames Research Center, Moffett Field, CA 94035
+2SETI Institute/NASA Ames Research Center, Moffett Field, CA 94035
+3University of California, Berkeley, Berkeley, CA 94720
+4San Jose State University, San Jose, CA 95192
+5Harvard-Smithsonian Center for Astrophysics, 60 Garden St reet, Cambridge, MA 02138
+6Niels Bohr Institute, Copenhagen University, DK-2100 Cope nhagen, Denmark
+7Las Cumbres Observatory Global Telescope, Goleta, CA 93117
+8University of Texas, Austin, TX 78712
+9Lowell Observatory, Flagstaff, AZ 86001
+10Radcliffe Institute, Cambridge, MA & Yale University, New Ha ven, CT
+11Jet Propulsion Laboratory/California Institute of Techno logy, Pasadena, CA 91109
+12Space Telescope Science Institute, Baltimore, MD 21218
+13National Optical Astronomy Observatory, Tucson, AZ 85719
+14California Institute of Technology, Pasadena, CA 91109
+15US Naval Observatory, Flagstaff Station, Flagstaff, AZ 86001
+16NASA Postdoctoral Fellow Program
+17San Diego State University, San Diego, CA
+18Orbital Sciences Corporation/NASA Ames Research Center, M /S 244-30, Moffett Field, CA 94035, USA
+19Bastion Technologies/NASA Ames Research Center, M/S 244-3 0, Moffett Field, CA 94035, USA– 3 –
+orientations for hot jupiters in general.
+Subject headings: planetary systems — stars: fundamental parameters — stars: indi-
+vidual (Kepler-8, KIC 6922244, 2MASS 18450914+4227038)
+1. Introduction
+To date, 90 “hot jupiters”—gas giant planets with periods ≤10 days—have been detected
+around Sun-like stars (Torres et al. 2008). The front-runni ng formation scenario supposes that
+these planets did not form where they reside today, close the host star, because the inner regions
+of protoplanetary disks have inadequate surface densities and high temperatures (Lin et al. 1996).
+Instead hot jupiters are presumed to form several astronomi cal units (AU) from their host stars
+followed by subsequent migration inward to their current lo cations. One likely migration sce-
+nario involves tidal interactions between the planet and a r emaining gaseous disk (Lin et al. 1996;
+Moorhead & Adams 2008), causing the planet to spiral inward w hile maintaining its nearly circular
+orbit that is co-planar with the disk. Alternatively, migra tion may occur by N-body gravitational
+interactions, such as planet–planet scattering (Rasio & Fo rd 1996; Chatterjee et al. 2008), dy-
+namical relaxation (Papaloizou & Terquem2001;Adams & Laug hlin 2003), andKozai interactions
+withadistant object, anddampedlater bytidal friction (Ho lman et al. 1997; Fabrycky & Tremaine
+2007; Wu et al. 2007; Nagasawa et al. 2008). Thus, measuremen ts of both the orbital eccentricities
+and the orbital inclinations relative to the star’s equator offer diagnostics of the original migration
+process.
+We assume that planets form in protoplanetary disks with the stellar spin and planetary
+orbital axes aligned. If so, the nearly adiabatic tidal inte ractions between planets and disks would
+maintain the alignment (Ward & Hahn 1994). In contrast, few- body gravitational interactions
+would typically cause misalignments. Few-body models by Ad ams & Laughlin (2003) predict a
+final inclination distribution for dynamically relaxed pla netary systems that peaks near 20◦and
+which extends to inclinations as high as 85◦. Kozai interactions between a planet and an outer
+body (star or planet) result in a wide distribution of final or bital inclinations for the inner planet,
+including retrograde orbits (Fabrycky & Tremaine 2007; Wu e t al. 2007; Nagasawa et al. 2008).
+TheRossiter-McLaughlin (R-M) effect offers away to assessquan titatively thespin-orbitalign-
+ment of a planetary system by measuring the Doppler effect of th e star’s light during a planetary
+transit. Asthe planet blocks recedingportion of arotating star’s surface, thespectrumfrom theun-
+obscured surface has a net Doppler shift toward shorter wave lengths, and vice versa for blocking the
+approachingportionofthestar. TheR-M effecthasbeenmeasur edin18starstodate(Queloz et al.
+2000; Winn et al. 2005, 2006, 2007; Wolf et al. 2007; Narita et al. 2007, 2008; Bouchy et al. 2008;
+Cochran et al. 2008; Loeillet et al. 2008; Winn et al. 2008; Jo hnson et al. 2009; Pont et al. 2009;
+Moutou et al. 2009; Pont et al. 2009; Winn et al. 2009; Narita e t al. 2009a,b; Simpson et al. 2010;
+Anderson et al. 2009; Gillon 2010; Pont et al. 2009; Amaury, A . et al. 2010).– 4 –
+About 2/3 of the 18 planetary systems measured by the R-M effect have an orbital plane well
+aligned with the star’s equatorial plane, as projected onto the sky, giving λnear 0◦. This alignment
+is as expected from simple migration theory due to gentle los s of orbital energy to the gas in the
+protoplnaetary disk (Lin et al. 1996). However, six exoplan etary systems show a significant spin-
+orbit misalignment, namely HD 80606 (Winn et al. 2009; Mouto u et al. 2009; Pont et al. 2009;
+Gillon 2010), WASP-14b (Johnson et al. 2009; Joshi et al. 200 9), XO-3b (H´ ebrard et al. 2009;
+Winn et al. 2009), HAT-P-7b (Winn et al. 2009; Narita et al. 20 09a), CoRoT-1 (Pont et al. 2009)
+Wasp-17b (Anderson et al. 2009).
+Thevariety of alignments supportthebimodal distribution foundby Fabrycky & Winn (2009).
+Interestingly, all four misaligned systems contain quite m assive planets, above 1 MJ. This cor-
+relation may be related to the association of massive planet s with higher orbital eccentricity
+(Wright et al. 2009), as both eccentricity and inclination m ay arise from perturbations of plan-
+ets from their original circular orbits. But there are two ma ssive planets on eccentric orbits for
+whichλappears to be consistent with zero, namely HD 17156b (Cochra n et al. 2008; Barbieri et al.
+2009;Narita et al. 2009b), andHATp–2b (Winn et al. 2007; Loe illet et al. 2008). Thereiscurrently
+no dominant and secure explanation for the misaligned or ecc entric hot jupiters.
+Here, we present the firstdetection of the Rossiter-McLaugh lin effect from a planet detected by
+theKepler Mission . As this mission is expected to detect dozens of transiting h ot jupiters, Kepler
+offers an opportunity to provide a statistically robust measu re of the distribution of spin-orbit
+angles, and to correlate that angle with other physical prop erties of the systems.
+2.Kepler Photometry
+Nearly continuous photometry in a 100 square degrees field ne ar Cygnus and Lyra was carried
+out during 42 days by the Keplerspaceborne telescope, as described previously (Borucki et al.
+2010; Borucki et al. 2010; Koch et al. 2010; Jenkins et al. 201 0; Batalha et al. 2010; Gautier et al.
+2010). The star Kepler-8 (= KIC 6922244, α= 18h45m09s.15,δ= +42◦27′03′′.9, J2000, KIC
+r= 13.511mag) exhibits a repeated dimming of 9 .82±0.22millimag, obvious against uncertainties
+in each 30 minute integration of 0.1 millimag. The light curv e for Kepler-8 is plotted in Figure
+1. The numerical data are available electronically in the on line edition of the journal. A modest
+amount of detrending has been applied (Koch et al. 2010; Rowe et al. 2010) to the time series.
+We detect no systematic difference between alternating trans it events at 50 µmag levels, ruling
+out nearly equal components of an eclipsing binary star, (se e Fig.1). We also see no evidence of
+dimming at the expected times of a secondary eclipse, which w ould be visible for most eclipsing
+binary systems of unequal surface brightness. The photocen ter shows no displacement astrometri-
+cally above millipixel levels (0.5 mas) during times in and o ut of transit as would be seen if there
+were a background eclipsing binary masquerading as a transi ting planet. Thus the photometric
+and astrometric non-detections of an eclipsing binary supp ort the planet interpretation for the re-– 5 –
+peated transit signatures. Moreover, the shape of the photo metric transit is adequately fit with a
+planet-transit model further supporting the planet interp retation.
+We fit the light curves by solving for a/R Star, the density of the star, and the ratio of planet
+to stellar radius. We follow the method for estimating stell ar radii and other stellar parameters
+described by Sozzetti et al. (2007), Bakos et al. (2007), Win n et al. (2007) and Charbonneau et al.
+(2007). This method extracts physical properties directly from the light curve, geometry, and
+Newtonian physics, and it uses the greater orthogonality of parameters to yield more robust fits to
+observables (Torres et al. 2008). We fit explicitly for a/R⋆,b, andr/R⋆using a procedure developed
+by one of us (J.Rowe) and described by Koch et al. (2010); Boru cki et al. (2010).
+We begin with an LTE spectroscopic analysis (Valenti & Pisku nov 1996; Valenti & Fischer
+2005) of a high resolution template spectrum from Keck-HIRE S of Kepler-8 to derive an effective
+temperature, Teff= 6213±150 K, surface gravity, log g= 4.28±0.10(cgs), metallicity, and the
+associated errordistributionforeach ofthem. Themultitu deofYale-Yonsei stellar evolution models
+(Demarque et al. 2004) are constrained by both those LTE meas urements and by the stellar density
+that stems directly from the orbital period, the fractional dimming during transit, and measures
+of transit durations (Sozzetti et al. 2007; Brown et al. 2010 ). By Monte Carlo analysis, those
+photometric and spectroscopic constraints and their uncer tainties establish the probabiliby density
+contours among the evolutionary tracks where the star may re side. We iterate the self-consistent
+fitting of light curves, radial velocities, and evolutionar y models until a domain of stellar mass,
+radius, and age is identified. That domain encompasses a rang e of evolutionary states that satisfy
+all of the constraints within their error distributions. Th e resulting mass and radius of Kepler-8
+are given in Table 1, along with other associated stellar pro perties such as luminosity and age.
+3. A Background Eclipsing Binary: Follow-up Imaging
+We carried out extensive tests of the possibility that the ap parent photometric transit was
+actually caused by a background eclipsing binary star withi n the photometric aperture of radius
+∼8′′. We obtained images with 0.8 arcsec seeing with the Keck tele scope HIRES guider camera and
+the bg38 filter to search for stellar companions that might be eclipsing. This filter combined with
+the CCD detector have a response similar to the V plus R bands, in turn similar to the bandpass of
+theKeplerphotometer. This Keck image is shown in 2. There is one star ha ving 0.0075 the flux of
+the main star (V and R band) that resides 3.8′′northwest of Kepler-8. This background star resides
+within the Kepleraperture and could conceiveably be an eclipsing binary that is masquerading as
+a transiting planet.
+But this faint neighboring star cannot be the source of the ph otometric transit signature for
+two reasons. If the background star were the cause of the obse rved 1% dimming, the photocenter
+centroid shifts would be 3-20 millipixel on the KeplerCCD. Instead, astrometric measurements in
+and out of transit (Jenkins et al. 2010) show shifts of no more than 0.1 millipixel in and out of– 6 –
+transit. Figure 3 shows both the flux and astrometric photoce nter (centroid) of the Keplerimages
+during quarter 1 month of data. We applied a high-pass filter t o remove non-transit signatures
+on timescales longer than 2 days. At times of transits there a re no displacements in either the
+row or column direction at a level above a millipixel. This in dicates that any background eclipsing
+binary would need to be well within ∼0.1 pixels or 0.4′′of the target star in order to explain the
+photometric transit signals.
+To hunt further for background eclipsing binaries we plotte d flux versus the photocenter cen-
+troids in both the row and column directions, as shown in Figu re 4. These so-called “rain plots”
+would reveal an eclipsing binary as a “breeze” in the centroi ds to the left or right as the flux drops.
+No such breeze is detected at a level near 0.1 millipixel, rul ing out all but eclipsing binaries located
+within a few tenths of an arcsec of Kepler-8. Finally, we look ed directly at the Keplerimages taken
+both during and out of transit to detect motion of the centroi d of light, as shown in Figure 5.
+We formed the difference of the images in and out of transit to de tect astrometric displacements
+associated with the flux dimming, as would occur if a neighbor ing eclipsing binary were the cause.
+Those difference images show no shift of the photocenter. We co nclude that the transit photometry
+with its 1% dimming cannot be explained by any eclisping bina ry companions beyond 1 arcsecond
+of Kepler-8.
+Furthermore, the flux ratio of the two stars is only 0.0075, ma king it impossible for the back-
+ground star to cause the 1% photometric dimming. Even if that background star were to vanish,
+the total flux would decline by less than the observed 1%. We co nclude that the 1% photometric
+dimming is not caused by an eclipsing binary star within the Keplerphotometric aperture, from 1
+- 10 arcsec of the target star.
+To hunt for additional stars in the field within an arcsec of Ke pler-8 we used both speckle
+imagingandadaptiveopticsimaging. Kepler-8wasobserved atthePalomar Hale200-inch telescope
+on 09 Sep 2009 UT with Palomar near-infrared adaptive optics system [PHARO] (Hayward et al.
+2001). ThePHAROinstrumentwasutilized intheJ-bandfilter withthe25masperpixel (25′′FOV)
+mode. The source was observed with a 5-point dither and an int egration time of 2.8 seconds
+per frame. The dither pattern was repeated 7 times for a total on-source integration time of 98
+seconds. The average uncorrected seeing during the observa tions at J-band was 0 .65′′, and the
+average AO corrected images produced point spread function s that were 0.09′′FWHM. There are
+two faint sources within 4′′of Kepler-8; these objects are 7 and 8.4 magnitudes fainter t han the
+primary target and are too faint to produce the Kepler-observed transit and centroid shift. The
+AO imaging detected no sources at J-band down to within 0 .1′′−0.2′′of the primary target that
+are within ∆ m≈6−7 mag at J.
+Speckle observations of Kepler-8 were made on 1 October 2009 (UT) at the WIYN observatory
+located on Kitt Peak. The observations were made using the WI YN speckle camera during a night
+of very good seeing (0.62′′) and under clear conditions. We used a narrow bandpass 40 nm w ide
+centered at 692 nm. The Kepler speckle program obtains obser vations of both double and single– 7 –
+standard stars throughout the night. We use a robust backgro und estimator on the reconstructed
+images to set a limit for the level of companion star we should detect if present. The speckle
+observations show that Kepler-8 has no companion star betwe en 0.05 and 2.0 arcsec within a delta
+magnitude of <4.2 mags. These AO and speckle observations effectively rule o ut the possibility of
+an eclipsing binary star to within as close as 0.1 arcsec from Kepler-8.
+3.1. Covariance of Inclination, Limb Darkening, and Stellar Parameters
+As Kepler-8 has a large impact parameter, the solution is qui te sensitive to errors in limb-
+darkening. The derived impact parameter and inclination an gle are directly related to the assumed
+limb-darkening law, poorly known for the wide Keplerbandpass. We adopted an adhoc approach
+to estimate the limb-darkening parameters (Rowe et al. 2010 ) as follows. We noted that fits to the
+observed Rossiter-McLaughlin effect (see section 6), couple d with the measured vsini= 10.5±0.7
+kms−1offered a constraint on the inclination angle and the impact pa rameter, and hence on the
+inferred stellar radius, for a given limb-darkening law. To solve the genereal case for eccentric orbits
+see Pal et al. (2010).
+For the cases of Kepler-4 and Kepler-5 (Borucki et al. 2010; K och et al. 2010), we discovered
+that a model using tabular values (Prsa & Zwitter 2006) for li mb darkening over-predicted the
+amount of curvature in the variation of flux as a function of ti me during transit. We therefore
+modified the limb-darkening parameters by fitting the three k nown exoplanets, TRES-2, HAT-P-7,
+and HAT-P-11, and using published values for their stellar a nd planetary parameters. We linearly
+interpolated over the values of effective temperature to deri ve a superior measure of the true limb-
+darkening law for the Keplerbandpass. The photometry from new transiting planets found by
+Kepler, along with IR photometry that is less sensitive to limb dark ening, will allow us to refine
+the limb-darkening law for the Keplerbandpass in the coming months. The resulting fits to the
+photometric transits of Kepler-4, 5, 6, 7, and 8 were all impr oved with our new limb-darkening
+treatment (Rowe et al. 2010).
+For Kepler-8, the resulting impact parameter and inclinati on were b=0.724±0.020 and i=
+84.07±+0.33 both about 10% smaller than we had obtained by using the firs t-guess limb-darkening
+law. The transit duration in turn dictates the best-fit stell ar radius, and after iterating with the
+Yale-Yonsei models, the resulting stellar radius is R= 1.486+0.053
+−0.062R⊙. The stellar mass has a value
+within a narrow range, M= 1.213+0.067
+−0.063M⊙, again derived from iteration between the fit to the
+light curve and to the Yale-Yonsei models. Additional infor mation on stellar mass comes from the
+projected rapid rotation of the star, vsini= 10.5±0.7 kms−1. Thisvsiniis faster than typical
+rotation for stars at the lower end of the mass range quoted ab ove, especially as some evolution and
+associated increase in moment of inertia has occurred. Thus we marginally favor the upper half of
+the quoted mass range, i.e. M >1.213M⊙. The stellar gravity is log g= 4.174±0.026(cgs), and
+the approximate age is 3 .84±1.5Gyr, both indicating a star nearing the end of its main seque nce
+lifetime.– 8 –
+4. Radial Velocities
+We took high resolution spectra of Kepler-8 using HIRES on th e Keck I 10-m telescope
+(Vogt et al. 1994). We set up the HIRES spectrometer in the sam e manner that has been used
+consistently for 10 years with the California planet search (Marcy et al. 2008). We employed the
+red cross-disperser and used the I 2absorption cell to measure the instrumental profile and the
+wavelength scale. The slit width was set at 0 .′′87 by the “B5 decker”, giving a resolving power
+of about 60,000 at 5500 ˚A˙Between 2009 June 1 and 2009 October 31 we gathered 15 spectra of
+Kepler-8 out of transit (See Figure6). Typical exposure tim es were between 10 - 45 min, yielding
+signal-to-noise of 20 - 40 pixel−1. These are quite low S/N ratios with which to attempt precise
+Doppler measurements. We tested the Doppler precision from such low SNR spectra on standard
+stars, notably HD 182488 and HD 9407, and found that the expec ted errors are ∼5 ms−1, as ex-
+pected from Poisson statistics of the photons, with no syste matic errors above 1 ms−1, as usual for
+the iodine technique. On 2009 October 29 (UT) we obtained nin e spectra of Kepler-8 duringtransit
+(the last taken during egress). The observations were taken between airmass 1.9 and 3.8, while the
+star was setting, the last exposure occurring at hour angle 5 hr 35 min with extreme atmospheric
+refraction and dispersion. Again, tests with standard star s observed at such high airmass show no
+systematic errors in the Doppler measurements above 1 ms−1.
+We carried out careful reduction of the raw images, includin g cosmic ray elimination and
+optimal extraction of the spectra, to minimize the backgrou nd moonlight. We performed the
+Doppler analysis with the algorithm of Johnson et al. (2009) . We estimated the measurement error
+in the Doppler shift derived from a given spectrum based on th e weighted standard deviation of the
+mean among the solutions for individual 2 ˚A spectral segments. The typical internal measurement
+error was 7-10 m s−1. The resulting velocities are given in Table 2 and plotted in Figure6.
+The actual uncertainties in the velocities are certainly cl oser to 20-25ms−1for several reasons.
+The spectra have such low photon levels, ∼200 photons per pixel in the raw CCD images, that
+cosmic rays and background sky are significant noise sources . The latter was especially problematic
+as we used an entrance slit with dimensions 0.87 x 3.5 arcsec, not long enough to separate the
+wings of the star’s point spread function from the backgroun d sky. Nearly all HIRES spectra were
+taken with the moon gibbous or full, and about half of the nigh ts had moderate cirrus that scatters
+moonlight into the slit. A few observations madewith a 14 arc sec slit revealed that 1-3% of the light
+came from the moonlit sky in typical observations. Simulati ons with stellar spectra contaminated
+by moonlit sky suggested that errorsof ∼10 ms−1would accrue, nodoubtsystematic as well. Errors
+in the velocities are large because of relatively rapid rota tion,vsini= 10.5±0.7, broadening the
+absorption lines by 5x that of the slowest rotating stars. Do ppler errors increase linearly with
+line widths. Thus, while the S/N=40 might be expected to yiel d velocities of ∼5 ms−1, the high
+vsiniincreases that error to ∼25 ms−1. Indeed, the best orbital fit to the velocities, shown in
+Figure6, exhibit discrepancies of 20-30ms−1. We estimate that the true errors in velocities are
+thus∼25ms−1. We have accounted for these errors by adding a ”jitter” of 25 ms−1to the internal
+velocity errors. Those augmented uncertainties are reflect ed in Table 2.– 9 –
+We carried out a Levenberg-Marquardt least-squares fit of a K eplerian, single-planet model to
+the observed velocities. In all models the orbital period an d the time of mid-transit was constrained
+to be that found in the photometric fit. We first assumed a circu lar orbit leaving only two free
+parameters in our fit, namely the velocity amplitude and the g amma velocity of the system. The
+best-fitmodelisoverplotted inFigure6. Thebest-fitvalueo ftheamplitudeis K= 68.4±12.0ms−1.
+This, coupledwiththeadoptedstellar massof 1 .213+0.067
+−0.063M⊙, yieldsaplanetmassof 0 .603+0.13
+−0.19MJ.
+We also carried out fits in which the eccentricity was allowed to float. The best-fit model
+has an eccentricity = 0.24, K= 71.4 ms−1, yielding a planet mass of 0.62 MJ. The RMS of the
+residuals is 39 ms−1compared with 40 ms−1for the circular orbit, rendering the eccentric orbit no
+better than the circular. Indeed a bootstrap Monte Carlo est imate of the parameter uncertainties
+gives a formal error in eccentricity of 0.16 . We conclude tha t the eccentricity is consistent with
+zero, but could be as high as 0.4. This large uncertainty stem s from the high vsiniof the star and
+it’s relative faintness, V = 13.9 mag.
+A line bisector analysis showed no variation at a level of a fe w meters per second and no
+correlation with measured radial velocities. Thus, the vel ocity variation appears to represent actual
+acceleration of the center of mass of the star.
+5. Properties of the Planet Kepler-8b
+With the planet interpretation highly likely, the properti es of the host star and the depth of
+transit directly yield a planet radius of 1 .419+0.056
+−0.058RJand a mass of 0 .603+0.13
+−0.19MJ. The errors
+represent the 68%-probability domain of integrated uncert ainty from all input measurements and
+models. The planet density is 0 .261±0.071 gcm−3apparently placing it among the low density
+exoplanets. Assuming a bond albedo of 0.1, its orbital radiu s of 0.0483+0.0006
+−0.0012AU, and the best-fit
+stellar luminosity from the Yale-Yonsei models gives an equ ilibrium temperature for the planet of
+1764±200 K, assuming rapid and complete redistribution of therma l energy around the planet’s
+surface. This planet apparently is a member of the populatio n of low density, bloated hot jupiters
+having thermal histories yet to be firmly understood. The pla net properties are listed in Table 1.
+6. Rossiter-McLaughlin Effect
+The radial velocities obtained during the transit of 2009 Oc tober 29 (UT) were modeled using
+the techniques described by Cochran et al. (2008). The model s adopted the system parameters in
+Table1, and searched for the value of λthat best matched the observed velocities. In particular, w e
+assumedthe zero eccentricity orbit, as the Keplerphotometric lightcurve gives noindication of non-
+zero eccentricity. We used a Claret (2004) four parameter li mb-darkening law, although running
+similar models with both a linear and a quadratic limb-darke ning law gave essentially the same
+results. We derive a value of λ=−26.9±4.6◦for the angle between the projected stellar rotation– 10 –
+axis and the projected planetary orbital axis. The model fit t o the data gives a reduced chi-squared
+χ2
+0= 0.66 for the velocities obtained during the transit, indicati ng that the uncertainties in these
+velocities are probably slightly overestimated.
+The radial velocities measured during the transit and the Ro ssiter-McLaughlin effect model
+fit are shown in Figure7. The observed asymmetric Rossiter-M cLaughlin effect, with a positive
+deviation of larger amplitude and longer duration than the n egative amplitude is a result of the
+combination of the large impact parameter and the non-zero v alue ofλ. A central transit would
+give a symmetric Rossiter-McLaughlin deviation, with the a mplitude of the effect then depending
+onλ. However, for an off-center transit, the overall shape depend s critically on both the impact
+parameter and on λ(cf. Figure 2 of Gaudi & Winn (2007)).
+This Rossiter-McLaughlin analysis provides independent v erification of the major results of
+the transit lightcurve analysis. The lightcurve analysis a lone suggested a transit chord that is
+significantly off-center, with impact parameter, b= 0.724±0.020 and inclination, i= 84.07±+0.33.
+This large impact parameter coupled with the non-zero value ofλresults in the planet transit
+blocking mostly the approaching (blue-shifted) half of the rotating star, as indicated in 8. The
+planet crosses to the receding (red-shifted) portion of the stellar disk just before the end of the
+transit. This causes the observed asymmetry in the Rossiter McLaughlin velocity perturbation
+during the transit. The observed amplitude of the R-M effect is in excellent agreement with
+the photometrically determined stellar radius and impact p aramter, and with the spectroscopically
+determined vsini. Theduration of the photometric transit agrees with thedur ation of the observed
+R-M velocity perturbation. This consistency between the pr operties of the system as derived from
+the transit photometry and from the R-M velocity perturbati on gives a confirmation that both
+phenomena were caused by an orbiting, planet-sized compani on to the star. There is no other
+explanation of the observed R-M variations that is also cons istent with both the photometric and
+the radial velocity observations.
+Thisobservation of theRossiter-McLaughlin effect in Kepler -8 is important becauseit confirms
+that the observed lightcurve and radial velocity variation s in Kepler-8 were indeed caused by an
+orbiting planetary companion. The amount of scatter in the r adial velocity data giving the solution
+shown in Figure 6 is uncomfortably large, even given the lack of line bisector variations in these
+spectra. However, the R-M data absolutely confirm the existe nce of the planet. There is no other
+explanation of the observed R-M variations that is also cons istent with both the photometric,
+astrometric, imaging, and the radial velocity observation s.
+7. DISCUSSION
+We have carried out a wide array of observations of Kepler-8 t hat pinpoint the existence and
+properties of the exoplanet. The Keplerphotometric measurements, with ∼0.1 mmag errors in 30
+min intervals, are tightly fit by a model of a transiting exopl anet (Figure 1). This good fit to the– 11 –
+inflections of the photometric data provide immediate suppo rt for the planet interpretation, with
+few plausible alternative interpretations except for a bac kground, diluted eclipsing binary system
+having just the right brightness and radius ratio to mimic a p lanet. We note that, unlike ground-
+based transit work that has lower photometric precision, th eKeplerphotometry is so precise that
+blends of eclipsing binaries are more readily identified fro m the photometry alone.
+Nonetheless, to test the unlikely possibility of a blend, we carried out a battery of astrometric
+tests to hunt for an eclipsing binary. The resulting steadin ess of the position of the primary star,
+Kepler-8, duringandout of transit, argues against any ecli psing binaryinthephotometric aperture.
+We further carried out both adaptive optics imaging and spec kle interferometric measurments to
+hunt for faint eclipsing binaries located within an arcsec o f the Kepler-8. None was found, further
+diminishing the chance of such a masquerade. We followed wit h high resolution spectroscopy at
+both low and high signal-to-noise ratio, finding no evidence of double lines nor rapid rotation.
+Further support for the planet interpretation came from the precise radial velocities that varied in
+phase with, and had the same period as, the photometric light curve, further supporting the planet
+model and constraining the planet mass.
+Finally, the Rossiter-McLaughlin effect confirmed independe ntly the planet interpretation and
+provided further geometrical information about the orbit, notablyλ=−26.9±4.6◦. A sketch
+of the star and planet’s orbit are shown in (Figure 8). Remain ing unknown is the inclination of
+the star’s rotation axis, as indicated in 8. But continued ph otometry during the Keplermission
+lifetime may reveal a photometric periodicity caused by the rotation of spots around the star. The
+resulting rotation period, coupled with the measured rotat ionalvsini= 10.5±0.7of the star, will
+allow the star’s inclination to be measured, putting the R-M geometry on firmer ground.
+The suite of ground-based observations obtained for Kepler -8described above over-constrained
+a set of parameters that were obligated to mutually agree, in cluding such subtle issues as the limb-
+darkening, the stellar rotation rate, and stellar radius, m ass, and density. Thus, the R-M effect can,
+in general, be used to confirm the existence of a planetary com panion in the case where standard
+follow-up observing procedures may provide somewhat ambig uous or equivocal results. Examples
+would be Keplerstars that are sufficiently saturated that measurement of pho tocenter shifts during
+transits is problematic, or stars like Kepler-8with large vsinithat make orbit determination via
+precise RV measurement difficult.
+Observations of the Rossiter-McLaughlin effect in dozens of t ransiting planets, discovered from
+the ground and from Kepler, offer an excellent opportunity to determine the distributio n of orbital
+geometries of the short-period planetary systems in genera l. These planetary systems will continue
+to be studied in an extraordinarily unform and consistent ma nner with the same set of tools. The
+sample selection effects are extremely well understood and do cumented. The Keplerlightcurves
+are of unprecidented precision, allow sensitive searches f or additional planets in the system via
+transit timing variations, and significantly tighter limit s to be placed in the future on the orbital
+eccentricity. Coupling these data with R-M measurements of the orbital alignment in these systems– 12 –
+will allow us to correlate the observed properties with the d egree of orbital alignment, and thus
+to search for the physical mechanisms causing observed misa lignments. We will also have the
+information to be able to begin to back-out the physical angl e between the spin and orbital angular
+momentum vectors, not just the projection of these angles on the plane of the sky.
+Funding for this Discovery mission is provided by NASA’s Sci ence Mission Directorate. Many
+people have contributed to the success of the Kepler Mission , and it is impossible to acknowledge
+them all. Valuable advice and assistance were provided by Wi llie Torres, Riley Duren. M. Crane,
+and D. Ciardi. Special Technical help was provided by Carly C hubak, G. Mandushev, and Josh
+Winn. We thank E. Bachtel and his team at Ball Aerospace for th eir work on the Keplerpho-
+tometer and R. Thompson for key contributions to engineerin g; and C. Botosh, and J. Fanson,
+for able management. GWM thanks acknowledges support from N ASA Cooperative Agreement
+NNX06AH52G.
+Facilities: Kepler.
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+This preprint was prepared with the AAS L ATEX macros v5.2.– 17 –
+Table 1. System Parameters for Kepler-8
+Parameter Value Notes
+Transit and orbital parameters
+Orbital period P(d) 3 .52254+0.00003
+−0.00005A
+Midtransit time E(HJD) 2454954 .1182+0.0003
+−0.0004A
+Scaled semimajor axis a/R⋆ 6.97+0.20
+−0.24A
+Scaled planet radius RP/R⋆ 0.09809+0.00040
+−0.00046A
+Impact parameter b≡acosi/R⋆ 0.724±0.020 A
+Orbital inclination i(deg) 84 .07±+0.33 A
+Orbital semi-amplitude K(ms−1) 68 .4±12.0 A,B
+Orbital eccentricity e 0 (adopted) A,B
+Observed stellar parameters
+Effective temperature Teff(K) 6213 ±150 C
+Spectroscopic gravity log g(cgs) 4 .28±0.10 C
+Metallicity [Fe/H] −0.055±0.03 C
+Projected rotation Velocity vsini(kms−1) 10 .5±0.7 C
+Absolute (Helio) radial velocity (kms−1) −52.72±0.10 B
+Derived stellar parameters
+MassM⋆(M⊙) 1.213+0.067
+−0.063C,D
+RadiusR⋆(R⊙) 1.486+0.053
+−0.062C,D
+Surface gravity log g⋆(cgs) 4 .174±0.026 C,D
+Luminosity L⋆(L⊙) 4 .03+0.52
+−0.54C,D
+Absolute V magnitude MV(mag) 3 .28±0.15 D
+Age (Gyr) 3.84±1.5 C,D
+Distance (pc) 1330±180 D
+Planetary parameters
+MassMP(MJ) 0.603+0.13
+−0.19A,B,C,D
+RadiusRP(RJ) 1 .419+0.056
+−0.058A,BC,D
+Density ρP(gcm−3) 0 .261±0.071 A,B,C,D
+Surface gravity log gP(cgs) 2 .871±0.119 A,B,C,D
+Orbital semimajor axis a(AU) 0 .0483+0.0006
+−0.0012E
+Equilibrium temperature Teq(K) 1764 ±200 F
+Note. —
+A: Based on the photometry.
+B: Based on the radial velocities.
+C: Based on an SME analysis of the Keck-HIRES spectra,
+D: Based on the Yale-Yonsei stellar evolution tracks.
+E: Based on Newton’s version of Kepler’s Third Law and total m ass.
+F: Assumes Bond albedo = 0.1 and complete redistribution.– 18 –
+Table 2. Relative Doppler Velocity Measurements of Kepler- 8
+HJD RV σRV
+(-2450000) (ms−1) (ms−1)
+4984.040 -22.15 34.5
+4985.043 52.45 34.5
+4986.065 -42.39 34.0
+4987.060 -74.37 33.6
+4988.027 63.04 33.7
+4988.973 98.87 33.8
+4995.103 46.58 34.2
+5014.891 -64.61 33.7
+5015.975 -29.97 33.4
+5017.020 -28.28 33.3
+5075.788 41.65 33.9
+5109.850 -122.00 32.8
+5110.790 -67.42 31.9
+5133.839 49.58 32.8
+5133.718 58.11 32.6
+5133.732 51.13 32.5
+5133.747 71.85 32.7
+5133.761 34.51 32.8
+5133.776 24.38 32.6
+5133.790 -9.49 33.2
+5133.805 -31.04 33.2
+5133.821 -14.32 32.6
+5134.740 -23.48 31.9
+5135.790 48.50 32.6– 19 –
+Fig. 1.— Thedetrended light curve for Kepler-8. The time ser ies for the entire Keplerphotometric
+data set is plotted in the upper panel. The lower panel shows t he photometry folded by the period
+P= 3.52254days, and the model fit to the primary transit is plotted as a solid line. Data for even
+and for odd transits are denoted by ’+’s and ’*’s, respective ly. The even and odd transits do not
+exhibit significant variations in transit depth indicative of the primary and secondary eclipses of an
+eclipsing binary. The photometry at the predicted time of oc cultation of the planet is shown just
+above the phased transit curve, showing no significant dimmi ng at the level of 0.1 mmag. The lack
+of any dimming at the predicted time of occulation, along wit h the absence of both astrometric
+motion in and out of transit and the absence of detected stell ar companions, leaves the planetary
+interpretation as the only plausible model, confirmed by the Rossiter-McLaughlin detection.– 20 –
+Fig. 2.— The image of the field near Kepler-8 obtained from the Keck 1 telescope guider camera
+with a bg38 filter, making the bandpass approximately V and R c ombined, similar to that of the
+Keplerphotometer. North is up, and east to the left, and the plate sc ale is 0.397′′per pixel. The
+main star, Kepler-8 is the brightest source at the center of t he image. A faint star is located
+3.8′′to the northwest of Kepler-8, having a flux 0.75% that of Keple r-8 and residing within the
+Keplerphotometric aperture. With a flux less than 1% that of Kepler- 8, the companion cannot be
+responsible for the periodici 1% dimming, even if it were a ba ckground eclipsing binary. But this
+neighboring star is sufficiently bright to account for the ast rometric displacement of the Kepler-8
+that occurs during transit of its planet, explaining the shi ft of the photocenter of the two stars by
+0.0001Keplerpixels.– 21 –
+5 10 15 20 25 30−0.0100.01
+Time − 54964, MJDNormalized Flux ResidualsKOI 10   13.6
+5 10 15 20 25 30−101x 10−3
+Time − 54964, MJDRow Residuals
+5 10 15 20 25 30−101x 10−3
+Time − 54964, MJDColumn Residuals
+Fig. 3.— Flux(upper) and astrometric centroid (middle, low er) timeseries of Kepler-8b from Kepler
+quarter-1 data. A high-pass filter has been applied to each ti me series to remove non-transit
+signatures on timescales longer than 2 days. There is no obvi ous displacement of the centroid of
+light in either the row or column residuals. This lack of moti on indicates that any background
+eclipsing binary would need to be located within ∼0.001/.01 = 0.1 pixels corresponding to 0.4′′of
+the target star in order to explain the photometric transit s ignals but avoid showing motion.– 22 –
+−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−1−0.8−0.6−0.4−0.200.2
+Centroid Shift, millipixelsNormalized Flux Residuals, %
+  
+Fig. 4.— Flux versus residual astrometric centroids in row ( x) and column (+) for Kepler-8. The
+measurements are the same as those presented in Figure 3. Thi s “rainplot” shows no evidence that
+the centroids are systematically shifting during each tran sit event, which would be indicative of a
+blend with a background star. If this were the case, identific ation of the actual star responsible for
+the transit features in the light curve would require inspec tion of a high resolution image or catalog
+to unravel the mystery. Since the in-transit points ”rain” s traight down, rather than slanting along
+a diagonal, any blend scenario would require a very small sep aration on the sky between the two
+sources.– 23 –
+KOI 10 diff
+columns, pixelsrows, pixels
+2 4 6 812345678direct
+columns, pixelsrows, pixels
+2 4 6 812345678
+direct
+columns, pixelsrows, pixels
+2 4 6 812345678
+0246810−15−10−505x 10−3
+Time, DaysNormalized Relative Flux
+Fig. 5.— The results of a ”poor man’s” difference image analysi s are presented in this figure: (A)
+(upper left) is the result of subtracting the average of the i n-transit images from the average of a
+similar number of just out-of-transit images for one of Kepl er-8b’s transits. The pixels summed to
+form the photometric brightness measurements are indicate d by the blue circles. For comparison,
+the average of the out-of-transit images are displayed in (B – upper right) and (C – lower left). The
+raw flux light curve from Q0 is displayed panel (D – lower right ). The difference image, Panel (A)
+is remarkably similar to the direct images. Had the transits been due to a background eclipsing
+binary that were 0.5 or more pixels offset from the target star, then the difference image would
+display a noticeable shift in the location of the pixel conta ining the peak energy. Instead, both
+difference and direct images supporting a coincident localiz ation of the source of the transits to the
+target star, Kepler-8.– 24 –
+ 
+−0.5 0.0 0.5
+ Orbital Phase−150−100−50050100150   Velocity  (m s−1)
+Fig. 6.— Theorbital solution for Kepler-8. The observed rad ial velocities obtained with HIRES on
+the Keck 1 Telescope are plotted together with the best-fit ve locity curve for a circular orbit with
+the period and time of transit fixed by the photometric epheme ris. Uncertainties are large due to
+the faintness of the star (V=13.9 mag), the high rotational D oppler broadening ( vsini= 10.5±0.7),
+and contamination from background moonlight in most spectr a. The measured velocities vary high
+and low as predicted by the orbit from the photometric transi t curve, supporting the planet model.– 25 –
+Fig. 7.— The Rossiter-McLaughlin effect for Kepler-8. The obs erved radial velocities obtained
+with HIRES on the Keck 1 Telescope are plotted together with t he predicted velocities from the
+best-fit model of the Rossiter-McLaughlin effect. The high vel ocities near ingress and low velocities
+near egress imply a prograde orbit. The asymmetry in the mode l curve indicates a projected tilt
+of the orbit relative to the star’s equator, as well as a trans it with a large impact parameter on the
+star.– 26 –
+Fig. 8.— A sketch of the transit geometry for Kepler-8. The Keplerphotometry and R-M
+measurements set the impact parameter, b= 0.724±0.020, and the angle projected on the sky
+between the star’s spin axis and the normal to the orbital pla ne,λ=−27◦, indicating a prograde
+orbit with a moderate inclination.
\ No newline at end of file
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@@ -0,0 +1,1338 @@
+1 
+ 
+Improving Human-Computer Interaction by 
+Developing Culture-sensitive Applications 
+based on Common Sense Knowledge 
+ 
+Junia Coutinho Anacleto and Aparec ido Fabiano Pinatti de Carvalho 
+Advanced Interaction Lab / Computing Department / Federal University of São Carlos     
+Brazil 
+ 
+1. Introduction     
+ 
+Computing is becoming ever more present in people’s everyday life. Regarding this fact, 
+researchers started to think of ways for im proving Human-Computer Interaction (HCI) so 
+that computer applications and devices provide users with a more natural interaction, 
+considering the context which users are inserted  into, as humans can do (Harper et al., 
+2008). Developing culture-sensitive interactive sy stems seems to be a possibility for reaching 
+such goal. Bailey et al. (2001) already mentio ned the importance of considering cultural 
+issues in computer systems development. For th em, culture is a system of shared meanings 
+which forms a framework for solving problem and establishing behavior in everyday life 
+that has to be considered in interactive system development. Since cultural knowledge is often implicit, designers often have trouble even  realizing that their designs carry cultural 
+dependencies implicitly. Moreover, it is not possible to design by hand for every 
+combination of possible cultures, nor is it practical to exhaustively test for every possible 
+user culture (Bailey et al., 2001). A support is necessary for making this possibility come true 
+and information and communication technology can offer it, as it is explained further ahead. 
+This chapter presents the solutions of  the Advanced Interaction Laboratory
+1 (LIA) from the 
+Federal University of São Carlos (UFSCar), Brazil, for developing culture-sensitive 
+interactive systems. LIA’s approach re lies on using common sense knowledge for 
+developing such kind of systems. That is be cause individuals communicate with each other 
+by assigning meaning to their messages based on their prior beliefs, attitudes, and values, i.e. based on their common sense.  Previous researches developed at the Lab have shown 
+that common sense expresses cultural knowledge.  So, providing this kind of knowledge for 
+computers is a way of allowing the development of culture-sensitive computer applications 
+(Anacleto et al., 2006a). 
+One idea for providing computers with common sense is to construct a machine that could 
+learn as a child does, observing the real world.  However, this approach was discarded after 
+Minsky and Papert’s experience of building an  autonomous hand-eye robot, which should 
+perform simple tasks like building copies of children’s building-block structures. In this 
+                                                 
+1 LIA’s homepage (in Portuguese): http://lia.dc.ufscar.br  Human-Computer Interaction, New Developments 
+ 2 
+experience, they realized that numerous sh ort programs would be necessary to give 
+machines human abilities as cognition, perception  and locomotion and that it would be very 
+difficult to develop those programs (Minsky, 1988). 
+Another idea is to build a huge common sense knowledge base, store it in computers and 
+develop procedures that can work on it. This seems to be an easier approach; nevertheless 
+there are at least three big challenges that mu st be won in order to achieve it. The first 
+challenge is to build the necessary common sense knowledge base, since it is estimated that, in order to cover the human common sense, billions of pieces of knowledge such as 
+knowledge about the world, myths, beliefs, an d so on, are necessary (Liu & Singh, 2004). 
+Furthermore it is known that common sense is cu ltural and time dependent, i.e. a statement 
+that is common sense today may not be a common sense statement in the future (Anacleto et 
+al., 2006b). For instance, consider the statem ent “The Sun revolves around the Earth”. 
+Nowadays this statement is considered wrong, however, hundreds of years ago people used 
+to believe that it was right.    
+One possible idea to transpose this difficulty is to build the knowledge base collaboratively 
+by volunteers through the Web, since every ordinary people has the common sense that 
+computers lack (Liu & Singh, 2004, Anacleto et  al. 2006a). In order to make the collection 
+process as simple as possible to the volunteers,  it is kind to think of collecting the common 
+sense statements in natural language. 
+Then the second big challenge arises: to re present the knowledge collected in natural 
+language in a way that computers can make inferences over it. In order to be used by 
+computer inference mechanisms, it is still ne cessary that the knowledge be represented in 
+specifics structures such as semantic network or  ontology. So, it is necessary to process the 
+statements in natural language in order to  build a suitable knowledge representation. 
+Natural language processing is a well-known AI challenge (Vallez & Pedraza-Jimenez, 
+2007). 
+The third challenge is to generate natu ral language from the adopted knowledge 
+representation, so that computer systems ca n naturally and effectively communicate with 
+users. This is another well-known challenge of current AI researches (Vallez & Pedraza-
+Jimenez, 2007). 
+This chapter discusses LIA’s approaches for common sense knowledge acquisition, 
+representation and use, as well as for natural language processing, developed in the context 
+of the Brazilian Open Mind Common Sense (OMC S-Br) project (Anacleto et al., 2008b), in 
+order to develop applications using such ap proach. It shows how common sense knowledge 
+can be used for instantiating some issues of cult ural-sensitive systems in the area of HCI. For 
+this purpose, some applications developed at LIA are presented and the use of common 
+sense knowledge in those applications is explained. Those applications are mainly 
+developed for the domain of education, which is  extremely culture-sensitive, and one of the 
+main technological, social and economical challenges considering globalization and the 
+necessary digital inclusion of every ordinary person.  
+The chapter is organized as follows: section 2 go es over some related projects that proposes 
+to build large scale common sense knowledge ba ses such as Cyc (Lenat et al., 1990), the 
+American Open Mind Common Sense (OMCS- US) and ThoughtTreasure (Mueller, 1997); 
+section 3 presents details on OMCS-Br archit ecture for acquiring, representing, making 
+available and using common sense knowledge in  computer application; section 4 suggests 
+ways of using common sense knowledge in th e development of interactive systems and Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 3 
+exemplifies the suggestions with computer applications developed at LIA; finally, section 5 
+presents some conclusions remarks and point to some future works. 
+ 
+2. Building Up Common Sense Knowledge Bases 
+ 
+Nowadays there are several projects, like Cyc, OMCS-US and Thou ghtTreasure, which aim 
+to build up large scale common sense knowledg e bases. While those projects are mainly 
+focused on Artificial Intelligence (AI) issues, OMCS-Br is more concerned about HCI. This 
+section goes over the approach of such projects to build up large scale common sense 
+knowledge bases, contextualizing LIA’s researches. In late 1950s, the goal of providing common se nse knowledge for computers received great 
+attention from AI researches. However, in th e early 1960s that goal was abandoned by many 
+of those researches, due especi ally to the difficulties they faced with for building the 
+necessary common sense knowledge base (Minsky, 2000; Lenat, 1995). 
+Believing in the possibility of building up the knowledge base and making computer applications intelligent so that they could flex ibly answer to users input through inferences 
+mechanisms that would work on the common sense knowledge base, Douglas Lenat started 
+in 1984 the Project Cyc, founding  the Cycorp Inc. (Lenat et al., 1990). Nowadays, Cyc is one 
+of the more active projects on acquirin g and representing common sense knowledge 
+computationally and on developing inferences mechanisms for common sense reasoning, as it can be verified in its scientific production
+2. 
+At the beginning, the approach used by Cy c researchers was to construct the project 
+knowledge base by hand. Panton et al. (2006) mention that the use of natural language 
+understanding or machine learning techniques was considered by the project researchers, 
+but it was concluded that, to have good results using those techniques, computers would have to have a minimum of common sense kn owledge for understanding natural language 
+and learning something from it. Therefore,  it was defined a specific language for 
+representing the knowledge, named CycL (Lenat  et al., 1990), and, then, the corporation’s 
+knowledge engineers , trained in that language, started to codify and store pieces of knowledge 
+in Cyc knowledge base. The knowledge was gottem from ordinary people by interviews about the several subjects that are part of  p e o p l e ’ s  c o m m o n  s e n s e ,  i n  a n  e f f o r t  o f  
+approximately 900 people per year (Matuszek et al., 2006). 
+Inspired in Cyc, Eric Mueller started the Project ThoughtTreasure  in 1994, having as goal to 
+develop a platform for making natural langua ge processing and common sense reasoning 
+viable to computer applications (Mueller, 1997). As in Cyc, the approach used for building ThoughtTreasure knowledge base was manual.  
+The existence of concepts, phrases and pred icates both in English and French in 
+ThoughtTreasure knowledge base is a differential of this project. The project also does not 
+limit the knowledge representation to a logic language, as CycL does, but also allows 
+knowledge representation in the format of finite automata and scripts (Mueller, 1997). 
+Besides the flexibility of ThoughtTreasure knowle dge representation, it does not discard the 
+need of knowing a specific structure to populate  the base of the projec t. This issue, shared 
+by Cyc and ThoughtTreasure proj ects, impedes that people wh o do not know the specific 
+format that knowledge should be inserted into the knowledge base collaborate on building 
+                                                 
+2 See Cycorp Incorporation group’s publications in http://www.cyc.com/cyc/technology/pubs  Human-Computer Interaction, New Developments 
+ 4 
+it. This demands a bigger time frame for making the knowledge base rise so that robust 
+inferences can be done over it. Cyc and Thou ghtTreasure are samples of this. After more 
+than two decades, Cyc’s knowledge engineers could map only 2.2 millions of assertions 
+(statements and rules) (Matuszek et al., 200 6). ThoughtTreasure, one decade younger, has 
+only 50,000 assertions relating its 25,000 concepts (Mueller, 2007). 
+Thinking about fast building a large scale common sense knowledge base, researchers from 
+the MediaLab of the Massachusetts Institute of Technology (MIT) launched OMCS-US, taking into account that every ordinary pe ople have the common sense knowledge that 
+machines lack, so all of them are eligible  for helping build the common sense knowledge 
+base machines need to be intelligent (Singh , 2002). In order to involve every ordinary 
+people, OMCS-US collects common sense statements in natural language on a website made 
+available by its researchers, and uses natural language processing techniques to build up its semantic network, the knowledge representation  adopted in the project (Liu & Singh, 2004). 
+Therefore, volunteers need only to know how to write in a specific language – nowadays 
+there are three OMCS projects, each on e in a language: OMCS-US, in English
+3, OMCS-Br, in 
+Portuguese4 and OMCS-MX, in Spanish5 – and have access to the Internet in order to 
+collaborate in building the knowledge base. In less than five years OMCS-US got a semantic 
+network with more than 1.6 millions assertions of common sense knowledge “encompassing 
+the spatial, physical, social, temporal and ps ychological aspects of everyday life” (Liu & 
+Singh, 2004). Section 3.1 gives details on the common sense collection process used by 
+OMCS-Br, the same used by OMCS-US. 
+Being aware of the importance of including or dinary people in constructing its knowledge 
+base, Cyc also started to develop systems for populating the knowledge base from natural 
+language statements. The first try was made by developing KRAKEN (Panton et al., 2002). 
+The system was developed in order to allow subject-matter experts to add information to 
+pieces of knowledge already stored in Cy c knowledge base through natural language 
+interface. In 2003, Witbrock et al. (2003) pres ented a system for interactive dialog, which 
+used a structure similar to KRAKEN for allowing  amateurs in CycL to enter information in 
+the knowledge base. Another approach for collecting common sense knowledge is proposed by von Ahn et al. 
+(2006). Von Ahn’s team agrees that collecting  common sense knowledge in natural language 
+is a good approach. However, they criticize th e absence of fun in the collection process used 
+by Cyc and OMCS-US, blaming this absence for until nowadays no project which has 
+proposed to construct large scale common sense knowledge bases has beaten the amount of hundreds of millions facts that are supposed to be necessary to cover the whole human 
+common sense. Thus, they propose to use games for fast collecting common sense facts, 
+believing that through games it is possible to  collect million of facts in few weeks (von Ahn 
+et al., 2006). Considering this premise, the gr oup developed Verbosity, a game in which 
+common sense facts are collected as side effect of playing the game. In the same way, OMCS-US research group developed Common Co nsensus (Lieberman et al., 2007), aiming 
+to motivate volunteers to collaborate in  building the project knowledge base. 
+Last but not least, there is another approach for acquiring common sense  knowledge  under 
+                                                 
+3 OMCS-US Homepage: http://commons.media.mit.edu/en/   
+4 OMCS-Br Homepage (in Portuguese): http://www.sensocomum.ufscar.br   
+5 OMCS-Mx Homepage(in Spanish):  http://openmind.fi-p.unam.mx  Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 5 
+development at Cycorp, which aims to automate the collection process by using crawlers to  
+get common sense knowledge from web pages. The developed method has six steps (Matuszek et al., 2005): (i) to use Cyc inference engine for selecting subjects of interest from 
+Cyc knowledge base and composing query strings to be submitted to Google; (ii) to search 
+for relevant documents on the Internet using Google’s resources; (iii) to identify relevant 
+statements to the subjects of inte rest on the retrieved documents; (iv) to automatically codify 
+the statements in CycL; (v) to validate the new assertions using the knowledge already 
+stored at Cyc common sense knowledge base an d performing new searches on the Internet 
+using Google; and (vi) to send the assertions to a know ledge engineer who decides whether 
+they are going to be stored in th e project knowledge base or not. 
+Although this last approach seems to be ve ry innovative, it does not allow mapping the 
+profile of the people from whom common  sense knowledge was collected. OMCS-Br 
+researchers defend that associating common se nse knowledge to the profile of the person 
+from whom it was collected is extremely im portant to make viable the use of such 
+knowledge for HCI. This is because common sense knowledge varies from group to group 
+of people and, therefore, the feedback that a common sense-aided application should give to 
+users from different target grou ps should be different (Anacleto et al., 2006b). The variation 
+in common sense among groups of people is what makes interesting use it for supporting 
+the development of culture-sensitive computer application, as it is explained in section 4. 
+That is why all OMCS-Br applications used for collecting common sense demands that users 
+subscribe themselves and log into the system before starting interacting. In such way 
+developers can analyze how people from different target groups talk about specific things and propose different solutions for each group. 
+Furthermore, the judgment performed by know ledge engineers makes possible question if 
+the statements stored in Cyc knowledge base really represent common sense. Statements 
+such as “Avocado is red” would not be stor ed in Cyc knowledge base, according to the 
+process described by Matuszek et al. (2005), Shah et al. (2006) and Panton et al. (2006). 
+However, if the majority of people with the same cultural background say that “Avocado is 
+red”, this should be considered as common se nse for that community and should be in the 
+knowledge base despite the know ledge engineers judge it as wr ong. Because of that OMCS-
+Br does not judge the semantic of the statements  collected in its site (Anacleto et al., 2008b). 
+ 
+3. The Project OMCS-Br 
+ 
+OMCS-Br is a project based on OMCS-US, whic h has been developed by LIA/UFSCar since 
+August 2005. The project works on five work  fronts: (1) common sense collection, (2) 
+knowledge representation, (3) knowledge manipu lation, (4) access and (5) use. Figure 1 
+illustrates the project architecture.  
+First of all, common sense knowledge is collected both through a website and through 
+applications which have been developed in th e context of OMCS-Br and stored in the OMCS 
+knowledge base. After that, the knowledge is represented as semantic networks, called 
+ConceptNets. In order to buil d the semantic networks, the na tural language statements are 
+exported through an Export Module  and sent to the Semantic Network Generator . The 
+generator is composed of three modules: the Extraction, Normalization and Relaxation  
+modules. The result of these three modules is a group of binary predicates. For dealing with 
+Portuguese natural language processing, Curupi ra (Martins et al., 2006), a natural language Human-Computer Interaction, New Developments 
+ 6 
+parser for Portuguese, is used. Th e predicates are filtered according to parameters of profile, 
+such as gender, age and level of education, ge nerating different ConceptNets. In order to 
+make inferences over the ConceptNets, severa l inference mechanisms, which are grouped in 
+an API (Application Programming Interface), were developed. The access to the API 
+functions is made through a Management Server, which makes available access to instances 
+of APIs associated with differ ent ConceptNets. Through this ac cess, applications can use the 
+API inference mechanisms and can perform common sense reasoning. Details about each element of the architecture presented in Figure 1 and the OMCS-Br work fronts are 
+presented in the following sub-sections. 
+ 
+ 
+Fig. 1. OMCS-Br Project architecture 
+ 
+3.1. Common Sense Collection 
+OMCS-Br adopts template-based activities which guide users in  such a way that they can 
+contribute with different kinds of knowledge.  The templates are semi-structured statements 
+in natural language with some lacunas that should be filled out with the contributors’ 
+knowledge so that the final statement corresponds to a common sense fact. They were 
+planned to cover those kinds of knowledge pr eviously mentioned and to get pieces of 
+information that will be used further to give applications the capacity of common sense reasoning. The template-based approach makes easier to manage the knowledge acquired, 
+since the static parts are intentionally propos ed to collect statements which can be mapped 
+into first order predicates, which composes the pr oject semantic networks. In this way, it is 
+possible to generate extraction rules to identi fy the concepts present in a statement and to 
+establish the appropriate relation-type between  them. In OMCS projec ts, there are twenty 
+relation-types, used to represent the different  kinds of common sense knowledge, as it is 
+presented in (Liu & Singh, 2004). Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 7 
+Those templates have a static and a dynamic part. The dynamic part is filled out by a 
+feedback process that uses part of statements stored in the knowledge base of the project to 
+compose the new template to be presented. Fi gure 2 exemplifies how the feedback process 
+works. At the first moment the template “You usually find a ___________ in a chair ” of the 
+activity Location is presented to a contributor – the templates bold part is the one filled out 
+by the feedback system. In the example, the contributor fills ou t the sentence with the word 
+“screw”. Then, the sentence “You usually find a screw  in a chair” is stored in the OMCS 
+knowledge base. At the second moment, the template “A screw is used for __________” of 
+the activity Uses is shown to another contributo r. Note that the word “screw” entered at the 
+first moment is used to compose the template presented at the second moment. 
+ 
+ 
+Fig. 2. Example of the OMCS-Br feedback process 
+ 
+The feedback process used in OMCS-Br website was planned in order to allow varied 
+templates to be generated so that users are able  to contribute on several subjects and do not 
+get bored with always filling out the same sentence. 
+Still related to the feedback process, considerin g that the statements st ored in the knowledge 
+base will be used to compose templates that will be shown to other contributors, it is 
+important to provide a way through what it coul d be selected the statements that should be 
+used by the feedback process. Thinking in th is need, it was developed in OMCS-Br an on-
+line review system, which can be accessed just by the ones who have administrator 
+privileges, where the statements are selected to be  or not to be used by the feedback process.  
+In order to perform the review, it was defi ned some rules to assure that common sense 
+knowledge would not be discarded (Anacleto et  al., 2008b). It is worth pointing out that 
+during the review process the reviewer is not allowed to judge the semantic of a sentence. 
+That is because it does not matter if a sentence seems strange in meaning or if it has already 
+been scientifically proved as wrong. Common  sense knowledge does not match scientific 
+knowledge necessarily. Since a sentence is acce pted as true by the most people who share 
+the same cultural background, it is considered as a common sense sentence. Because of that , reviewers are not allowed to judge if a senten ce is common sense statement or not. Only 
+statements with misspelling errors are discarded, in order not to cause noisy in the semantic 
+networks generated in the next step (Anacleto et al., 2008b). 
+Besides the templates about general themes such  as those about “things” which people deal 
+with in their daily life, “locations” where th ings are usually found and the common “uses” 
+Screen shot of the activity 
+Location
+Screen shot of the 
+activit y UsesHuman-Computer Interaction, New Developments 
+ 8 
+of things, there are also, in the Brazilian project website, templates about three specific 
+domains: health, colors and sexual education. Th ey are domains of interest to the researches 
+that are under development in the research gr oup which keeps the project (Anacleto et al., 
+2008b). This approach is only used in Brazil and it was adopted taking into account the 
+necessity of making the collection of common sense knowledge related to those domains. 
+The specific-domain templates were defined with  the help of professionals of each domain. 
+They were composed with some specifics word s which instantiate the templates of general 
+themes in the domain, in order to guide users to contribute with statements related to it. 
+Table 1 shows the accomplishments that OMCS-Br has gotten with that approach. 
+ 
+domain number of contributions  period of collection 
+Healthcare 6505 about 29 months 
+Colors 8230 about 26 months 
+Sexual Education  3357 about 21 months 
+Table 1. Contributions on specific domains in OMCS-Br 
+ 
+The numbers of contributions in each doma in can seem to be irrelevant, however, 
+considering the only 2 statements about AIDS found in the knowledge base before creating 
+the theme Sexual Education, it can be notice d the importance of domain-contextualized 
+templates in order to make faster the collectio n of statements related to desired domains. 
+Another accomplishment of the OMCS-Br is rela ted to the variety of contributor profiles. 
+Nowadays there are 1499 contributors registered  in the project site of which 19.33% are 
+women and 80.67% are men. The most part of  contributors (72.80%) is from Brazil South-
+east area, followed by the South area (15.25%) . Those numbers point to the tendency proved 
+by geographic sciences, which present the So uth-east and South area as being the most 
+developed areas of Brazil. Considering that, it  is perfectly understandable that, being well 
+developed areas, their inhabitants have easier access to the Internet. Table 2 and Table 3 
+present other characteristic s of OMCS-Br contributors. 
+ 
+age group percentage  
+Younger than 12 years  0.75 % 
+13 – 17 20.51 % 
+18 – 29 67.36 % 
+30 – 45 9.88 % 
+46 – 65 1.22 % 
+Older than 65 years 0.28 % 
+Table 2. Percentage of Contributors by Age 
+Group   
+school degree percentage 
+Elementary school 2.21 % 
+High school 18.17 % 
+College 65.86 % 
+Post-Graduation 4.52 % 
+Master Degree 7.04 % 
+Doctorate Degree 2.21 % 
+Table 3. Percentage of Contributors by 
+School Degree
+ 
+Another conquest of OMCS-Br is the amount of  contributions. Within two years and a half 
+of project, it has been gotten more than 174.000 statements written in natural language. This 
+was possible thanks to the web technology and the marketing approach adopted by LIA. As 
+the project was released in Brazil in 2005, it wa s realized that the know ledge base would rise 
+up significantly just when there were an even t that put the project in evidence. Figure 3 
+demonstrates this tendency. Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 9 
+ 
+Fig. 3. OMCS-Br knowledge base tendency of growing up 
+ 
+It can be noticed in Figure 3 that the periods where the knowledge base grew up 
+significantly were from August to October 2 005, from January to March 2006, from August 
+to October 2006, from January to  February 2007 and from November to December 2007. This 
+is an interesting fact, because those jumps in the knowledge base just followed some 
+marketing appeals performed by LIA. In the first one, LIA got published some articles in 
+newspapers of national coverage telling people about the project and asking for people 
+contribution. After getting those articles prin ted, the OMCS-Br knowledge base reached the 
+number of 50.000 statements. Three months la ter, the knowledge base established and 
+passed to grow up very slowly. 
+Thinking of having another jump in the knowle dge base size, it was released in the later 
+January 2006 a challenge associated to the Brazilian carnival. In that challenge, it was offered 
+little gifts as prizes to the three first collabora tors that contributed with the most number of 
+statements in the site activities. The winners received T-Shirts of the OMCS-Br Project and 
+pens of MIT. The challenge was announced amon g the project contributors, who received an 
+e-mail telling about it. The announcement was also posted in the Ueba website 
+(www.ueba.com.br ), a site of curiosities which target public is people interested in novelties. 
+As it can be noticed, the knowledge base size had a jump as soon as the challenge was 
+launched. The same approach was used in  August 2006, January 2007 and December 2007. 
+Although the approach has gotten a good respon se from the contributors in the first three 
+challenges, it can be noticed in Figure 3 that th is approach is becoming inefficient. Thinking 
+about keeping the knowledge base growing up, it is under development some games, 
+following project contributors’ suggestions, in order to make the collection process funnier 
+and more pleasant. One example is the game framework "What is it?" (Anacleto et al., 2008a), presented in 
+section 4.3. The framework is divided in two mo dules: (1) the player’s module, a quiz game 
+where players must guess a secret word cons idering a set of common sense-based clues and 
+(2) the game editor’s module, a seven-step wizard which guides people to create game 
+instances by using common sense knowledge. In the player’s module, while the player tries 
+to find out the secret word, the system collects common sense knowledge storing the 
+relationship between the word suggested and th e clues already seen by the player. In the 
+game editor’s module common sense knowledg e is collected during the whole set up 
+process, where the editor defines: (a) the community profile whose common sense 
+statements should be considered; (b) the game  main theme; (c) the topics, i.e. specifics 
+subjects related to the main theme; and (d) the cards of the game. All possible combinations 
+among the data supplied by the system and the editor’s choice s are stored in the knowledge 
+base. For example, the secret word select by th e editor and the synonyms (s)he are stored in Human-Computer Interaction, New Developments 
+ 10 
+the knowledge base as statements like “ secret word is also known as synonym ”. Moreover, 
+statements relating the synonyms among them, i.e. statements like “ synonym-1 is also known 
+as synonym-2 ”,…, “ synonym-(n-1) is also known as synonym-n ”, are stored. 
+ 
+3.2. Knowledge Representation 
+As OMCS-Br adopts natural language approach to populate its knowledge base, it was 
+decided to pre-process the contributions stored in the knowledge base so that they could be 
+easier manipulated by inference procedures. Fr om this pre-processing, semantic networks 
+are generated with the knowledge represented as  binary relations. These semantic networks 
+are called ConceptNets in OMCS projects.  Currently twenty relation-types are used to build the OMCS-Br ConceptNets. Those 
+relation-types are organized in eight classes,  presented by Liu and Sing (2004). They were 
+defined to cover all kinds of knowledge that compose the human common sense – spatial, 
+physical, social, temporal, and ps ychological (Liu & Singh, 2004). Some of the relation-types, 
+the K-Lines, were adopted from Minsk’s theory  about the mind. In that theory, K-Line is 
+defined as a primary mechanism for context in memory (Minsky, 1988).  
+ConceptNet relations have the format: 
+(Relation-type “parameter-1” “parameter-2” “f=;i=” “id_numbers”)   
+Figure 4 depicts the graphical representation of a mini ConceptNet. That semantic network 
+was generated from a very small excerpt of  the OMCS-Br knowledge base. The nodes, 
+originally in Portuguese, was freel y translated for this chapter. 
+ 
+ 
+Fig. 4.  Sample of ConceptNet 
+ 
+One example of relation that can be mapped from Figure 4 is:   
+ 
+(UsedFor “computer” “study” “f=3;i=2” “1;55;346;550;555”) 
+ 
+The parameters f (frequency of uttered statements) and i (frequency of inferred statements) 
+are part of the semantic relations. The f represents how many times a relation was generated 
+directly from the statements stored in the knowledge base, through extraction rules. The i Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 11 
+represents how many times a relation was genera ted indirectly from th e statements. In this 
+case they are submitted to the natural language  parser and afterwards  inference heuristics 
+are applied on them. The generation of the f and i is explained further ahead. The id 
+numbers correspond to the id of the natural language stat ements in the knowledge base 
+which originated the relation. This allows comp uter programs to go back to the knowledge 
+base and retrieve natural language statements related to a certain context of interaction. 
+In OMCS-Br, differently from OMCS-US, it is possible to generate several ConceptNets considering only the statements collected from a specific profile of volunteers. This feature 
+was incorporated in the Brazilian project since it is known that common sense varies from 
+group to group of people, inserted in differ ent cultural context (Anacleto et al., 2006a; 
+Carvalho et al., 2008). The profile parameters which can be used to filter the knowledge 
+from the knowledge base and to generate diffe rent ConceptNets are: age, gender, level of 
+education and geographical location. 
+Furthermore, in OMCS-Br, each non k-line relation-type has its negative version. For 
+example, in OMCS-Br ConceptNets, it can be  found both “IsA“ and “NotIsA“ relations, 
+“PropertyOf“ and “NotPropertyOf“, and so on. The decision for the affirmative or negative 
+relation version is made in the second phase of  the ConceptNet generati on process. In total 
+there are 5 phases in the ConceptNet generation process (Carvalho, 2007): (1) Exporting,  (2) 
+Extraction,  (3) Normalization,  (4) Relaxation and (5) Filtering . Each step is presented in 
+details in the following. 
+ 
+Exporting 
+The first step to generate an OMCS-Br Conc eptNet is to export the natural language 
+statements collected on the site and stored in  the project knowledge base. This export is 
+made by a PHP script that generates a file wh ose lines are seven-slot structures with the 
+statements and information ab out them. The slots are separa ted one another through “$$”. 
+Figure 5 presents a very small excerpt of the OMCS-Br knowledge base after the export 
+phase. Although the data in Figure 5 and in  the other Figures related to the ConceptNet 
+generation process are in Portuguese, they  are explained in English in the text. 
+ 
+Um(a) computador é usado(a) para estuda r$$M$$18_29$$mestrado$$Clementina$$SP$$1 
+Um(a) computador é usado(a) para jogar$$M $$13_17$$2_incompleto$$São Carlos$$SP$$25 
+Você geralmente encontra um(a) computador em uma mesa de 
+escritório$$F$$18_29$$mestrado$$Campinas$$SP$$8 
+Pessoas usam cadernos novos quando elas co meçam a estudar$$M$$13_17$$2_incompleto$$São 
+Carlos$$SP$$45 
+Fig. 5. Sample result of the exporting phase 
+ 
+As it can be seen in Figure 5, the slot 1 stores  the natural language statement, the slots 2 to 6 
+present information about the profile of the volunteer who entered the statement in the 
+knowledge base (gender, age group, level of education, origin city and origin state, 
+respectively) and the last slot stores the id number of the sentence in the knowledge base.  
+For example, in the first stru cture presented in Figure 5: (i) the natural language statement is 
+“Um(a) computador é usado(a) para estudar”  (in English, “A(n) co mputer is used for 
+studying”); (ii) the profile parameters are “male” (M), “person between 18 and 29 years old” 
+(18_29), “Master of Science” (mestrado), “Clementina City” (Clementina), “São Paulo State” Human-Computer Interaction, New Developments 
+ 12 
+(São Paulo); and the knowledge base statement id number is “1”. The id is very important 
+for common sense-based applications to trace ba ck the natural language statements related 
+to certain interaction scenarios. This is better explained in section 3.3. 
+ 
+Extraction 
+In the extraction phase, extracti on rules, i.e. regular expression patterns (Liu & Singh, 2004), 
+are used for identifying specifics structures  in the natural langua ge statements. These 
+regular expressions are designed according to the templates of the OMCS-Br site. Based on 
+the templates, it is possible to identify the re lation-type and the parameters of a relation. For 
+example, considering the first statement of Fi gure 5, “Um(a) computador é usado(a) para 
+estudar”, through a extraction rule it is po ssible to identified the following structures 
+“computador” (“computer”), “é usado(a) para” (“is used for”), “estudar” (“studying”). By 
+these structures it is generated the relation  (UsedFor “computador” “estudar”). Figure 6 
+shows the relations generated from  the sample structures presen ted in Figure 5. It can be 
+noticed in Figure 6 that the profile paramete r are still in the relation structure. Those 
+parameters are kept until the Filtering phase,  where they are used to generate different 
+ConceptNets, according to the application needs. 
+ 
+(UsedFor “computador” “estudar” “M” “18_29” “mestrado” “Clementina” “SP” “1”) 
+(UsedFor “computador” “jogar” “M” “13_17” “2_incompleto” “São Carlos” “SP” “25”) 
+(LocationOf “computador” “mesa de escritório” “F” “18_29” “mestrado” “Campinas” “SP” “8”) 
+(MotivationOf “usam cadernos novos” “começam a es tudar” “M” “13_17” “2_incompleto” “São Carlos” 
+“SP” “265”) 
+Fig. 6. Result of the Extraction ph ase for the structures in Figure 5 
+ 
+In this phase, it is decided which version of the relation-type should be used, the affirmative 
+or the negative. The extraction  rules decide for one of th em taking into account the 
+following heuristic: “If there is a negative adverb before the structure which defines which relation-
+type should be used, then use the negative version. Use the affirmative  version instead, if no negative 
+adverb precedes the structure which defines the relation-type”.  
+For example, consider the following natura l language statement: “Você quase nunca 
+encontra um(a) mesa de escritório em um(a) rua” (“You hardly ever find a(n) office desk in 
+a(n) street”). After being processed by an ex traction rule, the structures “quase nunca 
+encontra” (“hardly ever find”), “mesa de escrit ório” (“office desk”) and “rua” (“street”) are 
+identified. In this case the verb “encontra” (“to  find”) expresses the semantics that leads to 
+the relation-type “LocationOf”. However, as the adverbial phrase “quase nunca” (“hardly 
+ever”) precedes the verb, the generation system  decides to use the negative version of the 
+relation-type. So, it is generated the relation  (NotLocationOf “mesa de escritório” “rua”).  
+The incorporation of negative version to the Co nceptNet relation-types was an initiative of 
+the OMCS-Br. Until 2007, the OMCS-US did not ta ke into account this feature, and after 
+some discussions led by the Brazilian rese arch group, the American research group 
+implemented the polarity parame ter in their ConceptNet (Havasi et al., 2007). Nonetheless 
+the Brazilian group decided to define new re lation-types instead of implement another 
+parameter in the ConceptNet relations, since this approach would cause a smaller impact in 
+the inferences procedures whic h had been already developed. 
+ Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 13 
+Normalization 
+Since the statements collected in the site ca n express the same knowledge about the human 
+aspects in several ways, such as, using synon yms, different phrase structures, and different 
+inflectional morphology, those statements shou ld be manipulated to increase the semantic 
+network quality. In order not to have infl ected concepts, which means the same word 
+varying in number, tense, etc., a normalization process is performed. 
+The normalization process of OMCS-US makes use of Montylingua (Liu & Singh, 2004) – a 
+natural language parser for English, that ta gs and strips inflectional morphology of the 
+knowledge base statements. However, Montylingua cannot be used to parse Brazilian  
+Portuguese statements, due to syntactic and lexical differences between those languages. 
+The alternative found by OMCS-Br was to adop t Curupira (Martins et al., 2003), a parser 
+developed for Brazilian Portuguese. However Curupira does not strip the sentence 
+inflectional morphology. Because of that, a modu le to normalize the statements submitted to 
+OMCS-Br generation process was developed, usin g the inflectional dictionary developed by 
+the UNITEX-PB project (UNITEX-PB, 2008), which has all in flectional forms of Brazilian 
+Portuguese morphological classes. The “normalizer ” works in 3 steps, as it is described in 
+the following (Anacleto et al., 2008): 
+1. First of all, each sentence token is tagged using Curupira. Here it was necessary to 
+develop another module which makes the bridge between the “normalizer” and the parser library, available through a dll (dynamic linked library) format. 
+2. Afterward, articles are taken off – proper names are kept in original form. Special 
+language structures that are proper of Br azilian Portuguese are treats. For instance, 
+the structure called ênclise , a case of pronominal posi tion where the pronoun goes 
+after the verb, is stripped from the statemen ts and the verb is put in the infinitive 
+form. For example, the verb “observá-la” (“observe it”) is normalized to “observar” 
+(“to observe”). 
+3. Overall, each tagged token is normalized into its normal form found in the adopted 
+inflectional dictionary. In this way, the statements that were separated by 
+morphological variations, like “compraria cadernos novos” (“would buy new notebooks”) and “comprou um caderno no vo” (“the bought a new notebook”), are 
+reconciled during the normal ization process generating the normalized expression 
+“comprar caderno novo” (“to buy new notebook”). 
+For the purpose of measuring the effects of the normalization process on the semantic 
+network, the average nodal edge-density, as we ll as the number of distinct nodes and of 
+distinct relations in OMCS-Br ConceptNet, wa s calculated. This processing was performed 
+using and not using the normalization proces s. The results of this measurement are 
+presented in Table 4. 
+ 
+ non-
+normalized normalized  normalized/ 
+non-
+normalized 
+nodes 36,219 31,423 - 13.24 % 
+relations 61,455 57,801 - 5.95 % 
+average nodal edge-
+density 4.4643 3.3929 + 31.57 % 
+Table 4. Effects of the normalization proc ess on the Brazilian ConceptNet structure Human-Computer Interaction, New Developments 
+ 14 
+It can be seen in the results that the number of nodes and relations were decreased after the 
+normalization process. This confirms the tendency that the normalization process makes reconciliations between morphological variations, and thus unifies them.  
+Another result that can be inferred examining the connectivity of semantic network is that 
+the nodal edge-density has increased more than 30%. This is the most relevant data of the 
+measurement performed, since it demonstrates that the normalization process improves the 
+connectivity of the nodes. It is worth mentioning that the OMCS-Br re search group has been facing with several 
+difficulties regarding the use of Curupira. For instance, when an ênclise  occurs, Curupira 
+assigns more than a tag to the same token. The same happens with some composite words 
+such as “ fada-madrinha ” (“fairy godmother”) and “ cavalo-marinho ” (hippocampus). 
+Moreover, some verbal phrases are tagged incorrectly. For instance, “ fazer compras ” (go 
+shopping) is tagged as “fazer/VERB“ and “com pras/VERB“ when it should be tagged as 
+“fazer/VERB“ and “compras/SUB ST“. However, some heuristics to transpose these 
+difficulties has been developed, as it  can be verified in (Carvalho, 2007). 
+ 
+Relaxation 
+Other strategy developed to improve the ConceptN et connectivity is to extract new relations 
+from the relations uttered dire ctly from the natural language statements. This is made 
+applying a set of heuristic inferences over th e relations generated in the Extraction phase. 
+The heuristics applied in this phase are based on  grammatical and semantic patterns, as it is 
+explained in the following (Carvalho, 2007).  
+The Relaxation module receives as input a file with the relations generated in the first phase 
+of the ConceptNet generation process normalized an d tagged, as it can be notice in Figure 7.  
+ 
+(UsedFor “computador/SUBST” “estudar/VERB” “M” “18_29” “mestrado” “Clementina” “SP” “1”) 
+ 
+(UsedFor “computador/SUBST” “jogar/VERB” “M” “13_17” “2_incompleto” “São Carlos” “SP” “25”) 
+ 
+(LocationOf “computador/SUBST” “mesa/SUBST de/PREP escritório/SUBST” “F” “18_29” “mestrado” 
+“Campinas” “SP” “8”) 
+ 
+(MotivationOf “usar/VERB caderno/VERB novo/ADJ” “começar/VERB  a/PREP estudar/VERB” “M” 
+“13_17” “2_incompleto” “São Carlos” “SP” “265” ) 
+Fig. 7. Sample result of the Normalization phase 
+ 
+As an example, consider the last relation in th e group. In Figure 6, that relation was like: 
+ 
+(MotivationOf “usam cadernos novos” “começam a estudar” “M” “13_17” “2_incompleto” 
+“São Carlos” “SP” “265”) 
+ 
+After the normalization process, the rela tion got the following representation: 
+ 
+(MotivationOf “usar/VERB caderno/VERB  novo/ADJ” “começar/VERB  a/PREP 
+estudar/VERB” “M” “13_17” “2_incompleto” “São Carlos” “SP” “265” ) 
+ Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 15 
+As it can be noticed, the verbs “usam” (“to use”) e “começam” (“to start”) were put in the 
+infinitive form and tagged as verb (VERB); th e noun “cadernos” (“notebooks”) was put in 
+singular and tagged as substantive (SUBST); th e adjective “novos” was put in singular, since 
+in Portuguese adjectives have also plural fo rm, and was tagged as adjective (ADJ); and the 
+preposition “a” received the preposition tag (P REP). The tags are very important in this 
+phase because they are used by the inference mech anisms as it is explained in the following. 
+The first step in the relaxation phase is to assi gn the parameters f and i to the relations. All 
+the relations receives “f=1; i=0” at the beginnin g, because they are generated just once by an 
+extraction rule and up to this  point no relations were gene rated by inference mechanisms. 
+The second step is to group all equal relation s, incrementing the f parameter and appending 
+the id number. For example, consid er the following two relations: 
+ 
+(UsedFor “computador/SUBST” “jogar/VERB” “M” “13_17” “2_incompleto” “São Carlos” 
+“SP” “25” “f=1;i=0”) 
+ 
+(UsedFor “computador/SUBST” “jogar/VERB” “M” “13_17” “2_incompleto” “São Carlos” 
+“SP” “387” “f=1;i=0”) 
+ 
+They were generated from two different natu ral language statements (the statements 
+number 25 and 387). However, they are the same relation and it was collected from people 
+with the same profile (male, between 13 and 17,  high school, São Carlos, SP). Note that 
+although the profile parameters are the same, th is does not mean that the statements 25 and 
+387 were collected from the same person, but fr om people with the same profile. After this 
+second step, the two relations are reconciled in the following relation:  
+(UsedFor “computador/SUBST” “jogar/VERB” “M” “13_17” “2_incompleto” “São Carlos” 
+“SP” “25;387” “f=2;i=0”) 
+ 
+Note that f received a new value (f=2) and the id numbers were groped (“25;387”). If ten 
+people with the same profile had entered a statement which generated that relation, f would 
+be “f=10“ and there would be 10 id numbers in the id number slot. 
+The next step of the relaxation phase is to generate new “PropertyOf“ relations. They are 
+generated from “IsA“ relations. All IsA relation  whose first parameter is a noun or a noun 
+phrase and the second parameter is an adjective, generates a new “PropertyOf“.  For 
+example the relation: 
+ 
+(IsA “computador/SUBST pessoal/ADJ” “caro/ADJ” “M” “18_29” “2_completo” “São 
+Carlos” “SP” “284” “f=1;i=0”) 
+ 
+generates the relation: 
+ 
+(PropertyOf “computador/SUBST pessoal/ADJ” “caro/ADJ” “M” “18_29” “2_completo” 
+“São Carlos” “SP” “284” “f=0;i=1”) 
+ 
+Note that the profile parameters and the id numb er are kept the same as the relation used by 
+the inference process. It is worth pointing out that for each new relation generated, it is 
+verified whether an equal relation is alread y in the ConceptNet. In case affirmative the i Human-Computer Interaction, New Developments 
+ 16 
+parameter of the existing relation is incr emented and the id number of the generated 
+relation is appended to its id numbers. Fo r instance, consider that when the relation 
+previously presented is genera ted, there is already the foll owing relation in ConceptNet: 
+ 
+(PropertyOf “computador/SUBST pessoal/ADJ” “caro/ADJ” “M” “18_29” “2_completo” 
+“São Carlos” “SP” “45;78;171” “f=3;i=0”) 
+ 
+Instead of registering the generated relation in  ConceptNet as a new relation, the existing 
+relation is updated to:  
+(PropertyOf “computador/SUBST pessoal/ADJ” “caro/ADJ” “M” “18_29” “2_completo” 
+“São Carlos” “SP” “45;78;171;284” “f=3;i=1”) 
+ 
+Notice that the parameter i is now 1 and the id number 284 is part of th e relation id numbers. 
+New relations “CapableOf“,“CapableOfRe ceiveingAction“,“ThematickKLine“ and 
+“SuperThematicKLine“ are create d by similar processes. For de tail about the other inference 
+mechanisms that generate such rela tions, see (Anacleto et al., 2008b). 
+ 
+Filtering 
+After the Relaxation phase, different Concep tNets can be generated, according to the 
+possible combination of the profile parameter va lues. This generation is made on demand, 
+as common sense based applicat ions which use the OMCS-Br ar chitecture need a certain 
+ConceptNet. This is only possible in OMCS-Br, since the OMCS-US, Cyc and 
+ThoughtTreasure do not register  their volunteers’ profile.  
+The Filtering module receives an array of arra ys as input with the profile parameter values 
+which should be considered to generate the desired ConceptNet. The first sub-array in the 
+global array has the parameters related to the volunteers’ gender; the second, to their age 
+group; the third, to their level of education; the forth, to the city they come from; and the 
+fifth, to the state they live. If an array in the global array is empty, it means that a specific 
+profile parameter does not matter for the desired ConceptNet, and then it is considered all 
+possible value for that paramete r. For example, if the array [[], [13_17, 18_29], [2_completo], 
+[], [SP, MG]]  is provided to the Filtering module, a ConceptNet will be generated, whose 
+relations were build from statements gotten from  people of both gender, since the first sub-
+array is empty; who are between  13 and 17 years old and between 18 and 29 years old, 
+whose highest level of educatio n is high school; who come fr om any city located in the 
+Brazilian São Paulo  and Minas Gerais states . 
+The first step in the Filterin g phase is to recalculate the f and i parameter values, grouping 
+the equal relations whose profiles fit to the pr ofile values previously provided. After that, 
+another heuristic is applied on the relations in order to generate new “PropertyOfs“. This 
+heuristic is applied only in this step becaus e it considers two groups of relations in the 
+inference process and these two groups shou ld have only relations which fit to the 
+considered profile. Therefore, fo r guaranteeing this constraint, it was decided to apply this 
+heuristic only in this stage. For details about this heuristic, see (Carvalho, 2007). 
+After the Filtering phase, the ConceptNet is stor ed in a ConceptNet Server so that it can be 
+accessed by the ConceptNet Server Management, as it is explained in details in section 3.5. 
+ Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 17 
+3.3. Knowledge Manipulation 
+Once the semantic network is built, it is necessary to develop procedures to manipulate it, in 
+order to make computer applications capable of common sense reasoning and, then, to use 
+these resources for developing culture-sensit ive computer applications. The procedures 
+developed in the context of OMCS-Br are integrated in its API for being used by computer 
+applications. Currently there are five basic fu nctions that simulate some sorts of human 
+reasoning ability. They are related to context, projections, analogy making, topic gisting and 
+affective sensing (Liu & Singh, 2004). The OMCS-B r applications use mainly three of them.  
+The first one is the get_context()  function, which determines the context around a certain 
+concept or around the intersection of several concepts. For example, when someone searches 
+for the concept “have lunch” using get_context(),  it returns concepts like “be hungry”, “eat 
+food”, “meet friend”, “buy food”, “take a br eak”, and “find a restaurant”. These related 
+concepts are retrieved by performing spreadin g  a c t i v a t i o n  f r o m  t h e  s o u r c e  n o d e  i n  t h e  
+semantic network that finds related concep ts by and considering the number and the 
+intensity of connected pairs of concepts. This function is very useful for semantic query 
+expansion and topic generation. For instance, in  the “What is it?” framework (Anacleto et 
+al., 2008a), it is important to expand an initial concept provided by the game editor in order 
+to bring more relations about the main concept. 
+The second function is display_node(),  that can be used for bringing relations about a 
+particular concept. This function retrieves: the relation-type, the two concepts associated 
+through it, and the id numbers of the statemen ts in knowledge base which originated the 
+relation. Therefore, the applications can us e these results to create complete natural 
+language statements. Basically, there are two ways of doing so: (1) the relations can be 
+mapped in statements such as the templates used in the OMCS-Br site; for example, the 
+relation (UsedFor “computer” “study”) enable to create a statement like “A computer is 
+used for study”; (2) the id numbers can be us ed to find the origin al statements in the 
+knowledge base. Note that in the first case th ere are still difficulties to generate natural 
+language statements grammatically correct. This  is one of the AI challenges concerning 
+natural language generation (Vallez & Pedraza-Jimenez, 2007). Other function is get_analogy() , which has been developed base d on the Gentner’s Theory of 
+Mapping of Structures (TME) (Gentner, 1983) . From the Structure-Mapping Engine (SME) 
+(Falkenhainer et al., 1990), an algorithm capabl e of generating analogies and similarities 
+from OMCS-Br ConceptNets was developed. Th is algorithm uses a ConceptNet as basis 
+domain and an ExpertNet as target domain, re turning literal similarities. Details about the 
+algorithm can be found in (Anacleto et al., 2007b). In the same way it can be found details 
+about the other common sense inference procedur es used in OMCS-Br in (Liu & Sing, 2004). 
+ 
+3.4. Access 
+The ConceptNet API is available to any co mputer application through XML-RPC (Remote 
+Procedure Call) technology (XML-RPC, 2008), wh ich allows a simple connection between a 
+client application and a server application over the Internet. This connection is established 
+through HTTP and all data are encoding in XML format. First of all, the application informs 
+the ConceptNet Management Server shown in Fi gure 1 about the profile parameters values 
+related to the desired ConceptNet. The server checks whether an API for that ConceptNet 
+has been already instantiated in a specific port  handled by it. If the desired ConceptNet API 
+has been not been instantiated yet, the server asks for the Filtering module to verify whether Human-Computer Interaction, New Developments 
+ 18 
+the desired ConceptNet has been already gene r a t e d  i n  a n o t h e r  m o m e n t  s o  t h a t  i t  c a n  
+instantiate an API for it. In case affirmative, the server allocates a port for the ConceptNet 
+and makes it available also through the XML-RP C protocol. If the ConceptNet has not been 
+generated yet, the Filtering module generates it, and then the server instantiates and makes 
+available an API for the ConceptNet so that the application can use the API inference 
+procedures. Since the application has been in formed about the port where the ConceptNet 
+API is available, it can connect to the port and use the API procedures. 
+ 
+4. Designing and Developing Culture-Sensitive Applications 
+ 
+As the complexity of computer applications grows, one way to make them more helpful and 
+capable of avoiding unwise mistakes and unnecessary misunderstandings is to make use of 
+common sense knowledge in their developmen t (Lieberman et al., 2004). LIA has been 
+experiencing that cultural differences registered in common sense knowledge bases can be 
+helpful in: (a) developing systems to support decision-making by presenting common sense knowledge of a specific group to users; (b) developing systems capable of common sense 
+reasoning, providing different feedback for pe ople from different target group; and (c) 
+helping designers who want to consider th ese differences in the interactive systems 
+development by customizing interfaces and content according to the user’s profile.  
+In the following, those three it ems are approached in details  considering the domain of 
+education and examples of cultural sensitiv e common sense-aided computer applications 
+for each of them are presented. 
+ 
+4.1. Developing systems that show cultural issues for decision-making 
+There are situations in which it is interestin g to know the common sense of a specific group 
+in order to make suitable decisions. This is especially interesting in the domains of Human-
+Human Interaction (HHI), when two or more pe ople from different cultural background are 
+interacting with each other, and of Education, where it is important for educators to know 
+how their learners talk about themes which is go ing to be taught, so that they can decide on 
+how to approach those themes during the lear ning process. This wa s the first kind of 
+common sense-aided computer application development approached by LIA’s researchers. In the following, two applications developed at LIA to illustrate how common sense 
+knowledge can be used for th is purpose are presented. 
+ 
+WIHT: a common sense-aided mail client for avoiding culture clashes 
+As the world economy becomes more and more globalized, it is common to see situations 
+where people from two or more cultural backgr ounds have to communica te with each other. 
+Developing communication systems which show cultural differences to people who want to communicate with each other, making commen taries on the differences in the grounding 
+that can lead to possible misunderstandings so that someone can correct him/herself before 
+getting into an embarrassing situation, can be co nsidered an advance in HCI that reflects on 
+HHI. This issue has been approa ched by the development of an e-mail client which shows 
+cultural difference between people from thr ee different cultures combined two by two: 
+Brazilian, Mexican and American (Anacleto et al., 2006a). 
+The application has an agent that keeps watc hing what the user is typing, while makes 
+commentaries on the differences in the grounding that can lead to possible Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 19 
+misunderstandings. The system also uses th ese differences to calculate analogies for 
+concepts that evoke the same social meaning in  those cultures. This prototype is focused on 
+the social interaction among people in the contex t of eating habits, but it could scale to other 
+domains. The system interface has three sections, as can be seen in Figure 8. The first one – 
+at the upper left – is the information for the e-mail addresses and the su bject; the second one 
+– at the upper right – is where the agent posts its commentaries about the cultural 
+differences and the third part – the lower part  – is the body of the message. The second 
+section has four subsections: the upper one sh ows the analogies that the agent found and the 
+other three show the data that are not suitable for analogy. For example, in the screen shot in 
+figure 8, the third label for the Mexican culture – Mexicans thinks that dinner is coffee and 
+cookies – and the second for American cult ure – Americans think that dinner is baked 
+chicken – cannot make a meaningful analogy even if they differ only in one term. 
+ 
+ 
+Fig. 8. WIHT screen shot (Anacleto et al., 2006a, p. 7) 
+ 
+In order to make the cultural analogies, the sy stem uses three culturally specific semantic 
+network that have knowledge about the Braz ilian, Mexican and North-American culture – 
+the ConceptNetBR, ConceptNetMX and Concep tNetUS respectively. The ConceptNetBR 
+was built from data mined from the OMCS-Br knowledge base, originally in Portuguese. 
+Specifically in this project, a small group of statements related to eating habits were selected and freely translated to English to be pars ed by the system (Anacleto et al., 2006a). 
+ 
+PACO-T: a common sense-aided framewor k for planning learning activities 
+Another application developed at LIA for suppo rting decision making is PACO-T (Carvalho 
+et al., 2008b), a computational tool for PACO ( Planning learning Activities based on 
+COmputers), a seven-step framework which ai ms to support educators in planning 
+pedagogically suitable learning activities (N eris et al., 2007). PACO seven steps are: (1) 
+define the learning activity theme, target pub lic and general goal; (2) organize the learning 
+activity topics; (3) choose a pedagogical/method ological reference; (4) plan the learning 
+tasks; (5) choose computer tools to support the tasks execution; (6) edit the learning objects 
+Dinner is for Brazilians what lunch is for Mexicans  
+Dinner is for Americans what dinner is for Brazilians  
+ 
+Brazilians think that dinner is rice and beans 
+Brazilians think that dinner is salad, meat and potatoes 
+Brazilians make dinner between 6:30 PM and 8:00 PM 
+Brazilians think dinner is soup 
+ 
+Mexicans think that dinner is quesadillas 
+Mexicans make dinner at time between 8:00 PM and 9:00 
+Mexicans think that dinner is coffee and cookies 
+Mexicans think that dinner is bread with beans 
+ 
+Americans thinks that dinner is steak and eggs 
+Americans think that dinner is bake chicken 
+Americans think that dinner is smash potatoesjunia@dc.ufscar.br jhe@media.mit.edu 
+Dinner 
+Hi Junia, 
+ I am going to have a dinner next Friday in my place. Dinner is for Brazilians what lunch is for Mexicans  
+Dinner is for Americans what dinner is for Brazilians  
+ 
+Brazilians think that dinner is rice and beans 
+Brazilians think that dinner is salad, meat and potatoes 
+Brazilians make dinner between 6:30 and 8:00 PM 
+Brazilians think dinner is soup 
+ 
+Mexicans think that dinner is quesadillas 
+Mexicans make dinner between 8:00 and 9:00 PM 
+Mexicans think that dinner is coffee and cookies 
+Mexicans think that dinner is bread with beans 
+ 
+Americans thinks that dinner is steak and eggs 
+Americans think that dinner is bake chicken 
+Americans think that di nner is smash potatoes 
+Americans make dinner between 6:00 and 7:00 PM Human-Computer Interaction, New Developments 
+ 20 
+which are going to be used in the learni ng activity; and (7) test pedagogical and 
+technological issues.  Researches developed at LIA shows that common sense knowledge can be used for 
+educators to (Carvalho et  al., 2008a) (i) identify topics of general interest to the target group; 
+(ii) fit the learning activity content to the target group’s previous knowledge; (iii) identify 
+suitable vocabulary to be used in the learning activity; and (iv) identify knowledge from the 
+target group’s domain to which new knowledge can be anchored so that meaningful learning can be achieved (this knowledge can be used, for instance, in composing metaphors 
+and analogies to be presented during the learning activity).  Those are some pedagogical 
+issues necessary to allow effective learning to  take place, according to Learning Theories 
+from renowned authors from the field of Peda gogy, such as Freire (2003), Freinet (1993), 
+Ausubel (1968) and Gagné (1977). Therefore, a case study was conducted specia lly to check the possibility of using common 
+sense knowledge during the planning of lear ning activities by us ing PACO, to support 
+teachers in answering questi ons brought up along the fram ework steps. The case study 
+allowed a requirement elicitation, which was used for designing PACO-T. In the case study, 
+two Nursing Professors from the Nursing Depa rtment of the Federal University of São 
+Carlos planned a learning activity to prepare th eir students on how to orient caregivers in 
+the community from which the common sense knowledge was collected. In the learning 
+activity, the common sense knowledge was used to call the learners’ attention to the way 
+which the population talked about requiremen ts to be a caregiver or about procedures 
+which might be taken while home caring a sick  person and to point which learners should 
+emphasize during the orientation were presen ted (Carvalho et al., 2007; Carvalho et al., 
+2008a). Table 5 summarizes the support that commo n sense knowledge can give to teachers 
+during learning activities planning using PACO, identifi ed during the case study. 
+  
+Step Support 
+To define the learning activity theme. 
+To compose the learning activity justification. 
+To define the learning ac tivity general objective. 1 
+To define the learning activity specific objectives. 
+To decide which topics should be approached  during the learning activity, so that 
+it can fit to the students’ needs. 2 
+To decide the detail degree with whic h each topic should be approached. 
+3 To reach pedagogical issues addressed in Freire’s (2003), Freinet’s (1993), 
+Ausubel’s (1968) and Gagné’s (1977) Learning Theories. 
+To fit the learning  tasks to the pedagogi cal/ methodological references adopted. 4 To know how the target group usually study. 
+5 To know with which computer tools the target group is familiar. 
+6 To compose the learning material. 
+7 - 
+Table 5. Support offered by common sense knowledge in each PACO’s Step 
+In PACO-T, the target group’s common sense know ledge is presented to educators, so that 
+they can assess how people from that target gr oup talk about topics re lated to the theme of 
+the learning activity being planned and decide  which topics they should approach in the 
+learning activity, according to  the needs they identify. Th rough common sense analysis Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 21 
+educators can become aware about the learners ’ level of knowledge, avoiding approaching 
+topics which learners already know and approa ching topics which is misunderstood by that 
+learners’ profile, since common sense registers myths, believes and misunderstandings of 
+people from whom the knowledge was collected.  
+The tool uses the semantic networks and the API provided by OMCS-Br to present 
+knowledge related to the context of the learning  activity planning, co llected from volunteers 
+with the profile of target group. For this purp ose, teachers should define the target group’s 
+profile at the beginning of the planning so th at a suitable ConceptNet can be instantiated 
+and used during it. Figure 9 presents one of  PACO-T interfaces with common sense support. 
+The common sense support box can be noticed on the right.  
+Note that the items are presented in the commo n sense support box as links. Clicking on the 
+link, the system performs a new search in ConceptNet, using as keyword the text of the link and updates the information in the common sens e support box. This allows educators to 
+navigate among the concepts previously pres ented and to analyze how people with the 
+same profile of their target group talk abou t related subjects. Concerning the content in 
+Figure 9, by analyzing it the educator can see that one of the motivations of home caring a 
+sick person is to save money. Therefore, s(he) can consider that for her/his target group it is important to know procedures of home cari ng sick person which are not expensive and 
+approach this theme during  the learning activity. 
+ 
+What is the learning activity theme? (example )
+The continuity of care of a sick person at home.
+Define or refine the learning activity’s 
+theme and/or the justification 
+analyzing how people from your target 
+group talk about related subjects. 
+(explain how )
+home care a sick person SearchEnter the search keywords*
+* Please, separate the keywords by commaGeneralizations about home care a sick 
+person :
+•home care •home
+•care •sick person
+•person
+Motivation of home care a sick person :
+•love
+•make the person happy
+•not have money
+•save money
+Property of care a sick person :
+•boring
+•tiring
+Conceptually related to:
+•relativeHide common sense support
+What is the learning activity’s justification? 
+(example )
+(i) the health professionals have to be 
+prepared to orient the population about how to home care a sick person; (ii) there is no formal approach to this theme in the Federal University of São Carlos nursing course and there is the possibility of the Nursing Department provide extra courses to complement the learning formation through a DL approach; (iii) in order to provide a suitable orientation, the health professionals might know the social context into which the caregiver is inserted.Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7
+Define the target group’s profile
+Define the theme
+Define the objectives
+Informações adicionaisDefine the learning activity’s title
+Define theme >> Cancel Interrupt Save << Define profilePlanning learning Activity supported by COmputer - ToolHelp
+Log out
+ 
+Fig. 9. PACO-T Interface – Step 1: Define the theme 
+ 
+Tools such as PACO-T, which can personalize the information that it is going to present to 
+users, according to some parameters they provide, can be considered an valuable 
+contribution to HCI, especially when the deve lopment of adjustable interfaces, which seems 
+to be one of the main issues of HCI in  a near future, is taken into account. Human-Computer Interaction, New Developments 
+ 22 
+4.2. Developing systems which infer over cultural issues 
+Imagine the possibility of developing an agen da system capable of warning someone who 
+tries to set up a meeting with an Indian in a ba rbecue place that this is not the best place to 
+take an Indian, based on the information “Ind ians does not eat cow meat, because cows are 
+considered holly in its culture”. This is undo ubtedly an advance in HCI that will allow the 
+development of systems capable of acting flexibly to different kind of situations based on 
+cultural information. 
+To support teachers contextualizing considering the educational context, the learning 
+content they are preparing by suggesting topi cs of interest for the target group through 
+inferences over a common sense knowledge ba se, an example to illustrate the use of 
+common sense knowledge for the purpose of deve loping systems which infer over cultural 
+issues. Moreover, when communiti es of people with common interests are considered like a 
+group of learners of a certain educator in ce rtain school year, it is possible to improve the 
+interaction between educator and learners allo wing educators to know about the learners’ 
+daily life and to prepare the content they w ill approach in classroom, considering the 
+learners’ cultural context. If a certain concep t is explained through metaphors and analogies 
+coming from learners’ community the chances of such concept to be understood by the 
+learners are bigger, according to pedagogical principles. Then, developing systems capable 
+of suggesting those examples by inferring over a common sense knowledge base can be 
+helpful. This section presents Cognitor, a computational framework for editing web-like 
+learning content, and a Common Sense-Based On-line Assistant for Training Employees 
+which implement those ideas.  
+ 
+Cognitor: a framework for editing culture-contextualized learning material 
+Cognitor is an authoring tool, based on the e-Learning Pattern Language Cog-Learn 
+(Talarico Neto et al., 2008). Its main objective is  to support teachers in designing and editing 
+quality learning material to be delivered electr onically to students on the Internet or on 
+CD/DVD. It aims to facilitate the development of accessible and usable learning material, 
+which complies with issues from Pattern and Le arning Theories, giving  teachers access to 
+common sense knowledge in order to make th em able to contextualize their learning 
+materials to their learners’ needs (Anaclet o et al., 2007a). About using common sense 
+knowledge in the learning process, Cognitor is  being prepared to make possible the use of 
+common sense knowledge through all the edition of the learning content, in order to 
+instantiate the Patterns implem ented in the tool, to organize the hyper document, to 
+contextualize the content, to adjust the voca bulary used in the content and to fulfill the 
+metadata of such learning objects. One of the tools available in Cognitor is th e implementation of the Cog-Learn Knowledge 
+View Pattern (Talarico Neto et al., 2008), wh ich supports the planning and the automatic 
+generation of learning material  navigational structure, by using the technique of Concept 
+Map, proposed by Novak (Novak, 1986). For that purpose, teachers shou ld: (i) enter into the 
+system the Concepts which they want to approa ch in their learning material and organize 
+them hierarchically; (ii) name the natural re lations between concepts, i.e. the relations 
+mapped from the hierarchy previously define d; and (iii) establish and name any other 
+relation between concepts, which they consider  important to the content understanding and 
+exploration. See (Anacleto et al., 2007a) for more  information on the process of generating a 
+Concept Map in Cognitor, using the Knowledge View Pattern.  Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 23 
+When the teacher chooses to use the Knowledg e View pattern, Cognitor offers the common 
+sense support to provide her/him with inform ation about the facts that are considered 
+common sense in the domain that s/he is thinking of. So the teacher can complete the 
+Concept list based on that information, decr easing the time on planning the material 
+organization. Figure 10 depicts the first step of Cog-learn Knowledge View Pattern tool, in 
+which teachers are expected to enter into th e system the concepts which they want to 
+approach in the content.  In the example, the teacher entered the concep t “Health” (“Saúde”, in Portuguese), and the 
+system suggested in the box on the right conc epts such as “sick person” (“pessoa doente”), 
+“place” (“lugar”) and “drug” (“remédio”). Teachers can select one of the suggestion from 
+common sense knowledge and click on the button “<< Include”, select a concept in the list 
+of concepts on the right and perform a sear ch in the common sense knowledge base for 
+related concepts through the button “Search >>”, or even add another concept that is not in 
+the common sense concepts suggestion list. By  using common sense suggestion educators 
+can contextualize the learning content they are preparing to their target  group´s needs in the 
+same way it  is done in PACO-T. 
+ 
+ 
+Fig. 10. Knowledge View Pattern First Step 
+ 
+On-line Help Assistant for Training Employees 
+The On-line Help Assistant is a web system  to support continuing education about 
+workplace safety issues, where the learner can ask a doubt about a certain concept in natural 
+language and the system gives him back the formal definition and some analogies coming from common sense of that employee community to explain the concept (Anacleto et al., 
+2007b). In order to build the analogie s, the system uses the function get_analogy() mentioned 
+in section 3.3 of this chapter. 
+The system is basically a common sense-aided se arch engine. Its interface is composed for 
+five areas, four of them  presented in Figure 11: 
+1. Search area , where users should ty pe the search query; 
+2. Definition area,  where definitions for the concepts in the search query are 
+presented; 
+3. Analogy area, where the analogies to explai n the concepts are shown; 
+ill person
+place medicine to have thing home eat ill clothes water to know to take care head to talk cakeHuman-Computer Interaction, New Developments 
+ 24 
+4. Relations area, where the relations among the concep ts present in th e search query 
+and the ones in the Concep tNet are presented; and 
+5. Related concept area , where related concepts retrieved from ConceptNet are 
+presented and users can continue their stud ies by exploring thos e concepts. For this 
+purpose, learner can use the links offered by the assistant and browse through the 
+concepts. 
+Users should type the concept they want to get explained and then click on the button 
+“Shown information found”. In order to identify  the morphological variations in the query, 
+the system uses two techniques. The first one is  the expansion technique. It is useful when 
+students use short expressions to perform the se arch. In this technique, the terms related to 
+the context are retrieved by the function get_context()  of the ConceptNet API. Then, terms 
+that have the lemmatized expression as a substr ing are also retrieved. For instance, when a 
+learner provides the expression “f ogo” (“fire”, in English) in th e search area the system will 
+also search for expressions like “fire alarm”, “f ire place”, “fire door” and so on. The second 
+technique is useful especially when large expressions are prov ided by the students to be 
+searched. In this case phrasal structures, such as noun phrases, verbal phrases and adjective 
+phrases are identified by the system and then the system performs a search for each 
+structure identified. For example, when a studen t asks the  system for results related to the 
+expression “prevenir acidentes no ambiente de trabalho” (in English, “preventing accidents 
+in the workplace”), the expression will be di vided in “preventing work accidents”, “work 
+accidents”, “accidents” and “accident in th e workplace”. This technique increases the 
+likelihood of getting results from the search, since the terms that ar e searched are simpler 
+and more likely to be found in the sema ntic network (Anacleto et al., 2007b). 
+ 
+Search area
+Definition area
+Analogy area
+Rel at i o n s ar ea
+ 
+Fig. 11. On-line Help Assistant for Training Employees Interface 
+ 
+4.3. Designing systems considering cultural issues 
+Attributes as attraction, dynamism, level of expe rtise, confidence, intention, location, social 
+validation and preferences have different weight s in different cultures. Consequently, user Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 25 
+interface developers face many challenges, such  as the design of the content which should 
+be presented for users with different culture pr ofiles. For example, which colors should be 
+used in the design of an interface in order to  make users feel comfortable or react in the 
+desired way? Which symbols should be used so  that the right message is transmitted?  
+Despite of the importance of these issues, deve lopers face the almost unreachable tasks of 
+researching culture-dependent requirements and having a contextualized development 
+accepted due to tight budgets, limited agenda, and scarce human resources. In this context, 
+collecting these “world views” and making th em available for everyone that wants to 
+develop culture-contextualized user interfaces, can be expensive and laborious.  
+Taking into account that this ki nd of information and that the answer for question´s such the 
+ones stated before can be found in people’s common sense. Using this kind of knowledge in 
+the system development seem´s to be a alternat ive for providing cultural contextualization. 
+Common sense-based interface content design has been tested at LIA in an educational 
+game called “What is it?” (Anacleto et al., 2 008a), a card-based quiz game that proposes to 
+work on curricular themes. The game fram ework is presented in the following.  
+ 
+“What is it?”: a framework for designing culture-contextualized educational games  
+The game framework “What is it?” is a proposal  to support teachers in contextualizing the 
+content of quiz-like educational games to the students’ local culture, in order to promote a more effective and significant learning. It is divided in two modules: a module for 
+educators, called “Editor’s Module” and another for learners, called “Player’s Module” 
+(Anacleto et al., 2008a). The cards designed by th e educators are presented for the learners in 
+the learner’s module, which automatically consid ers cultural issues when the topics to be 
+worked on are established. So, the content pres ented to learners is customized according the 
+common sense knowledge shared by people with their profile, once the educator previously 
+established the topics and stat ements from that learner’s community to set up the game 
+(Anacleto et al., 2008a). 
+ 
+Editor’s Module 
+The game editor is a seven-step wizard whic h guides the teacher towards creating game 
+instances, which fit to their pedagogical goals.  It implements the pr inciple of developing 
+systems that shows cultural issues for decision-making, presented in section 4.1. 
+During the card definition, teachers receiv e the support of common sense knowledge. For 
+that purpose, in the framework editor Step 1 te achers have to define the population profile 
+which should be considered in the search for common sense statements in the knowledge 
+base. In this way, the system guarantees that the statements which are going to be presented 
+to the teacher were gathered from people who have the desired profile to the game instance, 
+i.e. the statements are contextu alized to the target group.  
+In the two next steps the teacher must define  two items: (1) the game main theme that 
+nowadays have to be related to one of the si x transversal themes defined on the Brazilian 
+curriculum (sexual education, ethics, healthcare , environment, cultural plurality, market 
+and consumers) and (2) the topics, which are sp ecifics subjects related to the theme chosen, 
+to compose the game dice faces (Anacleto et al., 2008a).  
+The next steps consist of defining the secret words, their synonyms and the set of clues for 
+each secret word. For each secret word defined,  it is performed a search on the ConceptNet 
+instantiated at Step1, that increasing the numb er of words associated with the secret work. 
+The concepts associated with the secret word and their synonyms are presented to teachers Human-Computer Interaction, New Developments 
+ 26 
+as natural language statements and, based on  these statements, teachers can compose the 
+card clues. For example, the relation (IsA “a ids”, “sexually transmitted disease”), found in 
+the ConceptNet-Br, is presented to teachers as “Aids is a(n) sexually transmitted disease”.  
+Then the teacher can (a) select a common sense st atements as clues, adding them to the card 
+set of clue; (b) edit a statement to make it suitable to the game purpose; or (c) just ignore the 
+suggestions and compose others clues. It is wort h pointing out that the statements edited or 
+composed by the teachers are also stored in the OMCS-Br knowledge base as new common 
+sense statement collected from that teacher.  
+It is also important to point out the fail-soft approach adopted in the framework. This means 
+that the statements suggested to teachers can be valid or not and the teachers will decide for 
+accepting or not the suggestion. However, the suggestion does not bring any problem to the 
+teachers’ task performance. On the contrary, it helps the teachers to finish their task faster 
+and more efficiently. 
+ 
+Player’s Module 
+Figure 12 presents an instance of “What is it ?” in theme “Sexual Edu cation”. That is the 
+interface presented to learners after teachers finish preparing the game cards. To start the 
+game the player should click on the virtual di ce, represented in Figure 12 by the letter “S”, 
+whose faces represent the topics related to the transversal theme on which the teacher 
+intents to work. In Figure 12, the letter “S” corresponds to the topi c “Sexually transmitted 
+diseases”. Other topics which can potentially  compose the “Sexual Education” theme dice, 
+according to the teachers’ needs, are “anatomy and physiology”, “behavior” and 
+“contraceptives methods”. The letters, which represent the topics, are presented to the 
+player fast and randomly. When the player clicks on the dice it stops and say about which 
+topic the secret word, which should be guessed, is.  
+ 
+ 
+Fig. 12. Player’s Module Main Interface 
+ 
+Each topic has a set of cards associated with, which are related to different secret words. 
+These cards are defined by teachers in the ga me’s editor module, using the support of a 
+common sense knowledge base. In addition to that, it is possible to relate a list of synonyms 
+to each secret word. These synonyms are also accepted as expected answers.  Improving Human-Computer Interaction by Deve loping Culture-sensitive Applications based on 
+Common Sense Knowledge  
+ 27 
+The clues play the role of suppor ting the player to guess which the secret word is. Each card 
+can have a maximum of ten clues which can be selected by the learners by clicking on a number into the “Set of clues” area, which can be seen in Figure 12. After having the topic 
+defined by the dice, a card with clues is presented to the player and, as s(he) selects a clue, it 
+is displayed on the blue balloon. The players can select as many clues as they consider 
+necessary before trying to guess the word.  
+As the players try to find out the secret wo rd, the system collects common sense knowledge, 
+storing the relation between the word they have  suggested and clues that have been already 
+displayed. This collection process is interesting (1) for teachers, who can identify possible 
+misunderstanding by analyzing the answers that learners with the profile of their target 
+group give to a specific set of clues, and, therefore, approach thos e misunderstandings in 
+classroom to clarify them; and (2) for the OM CS-Br, which will get its knowledge base 
+increased. It is important to point out that the answers provided by the learners, which do 
+not correspond either to the secret word or to a synonym defined by the teacher, are not 
+considered incorrect by the system.  
+ 
+5. Conclusion and Future Works 
+ 
+The advent of Web 3.0, claiming for person alization in interactive systems (Lassila & 
+Hendler, 2007), and the need for systems capable of interacting in a more natural way in the 
+future society flooded with computer system s and devices (Harper et al., 2008) show that 
+great advances in HCI should be done.  
+This chapter presents some contributions of LI A for the future of HCI, defending that using 
+common sense knowledge is a possibility for improving HCI, especially because people 
+assign meaning to their messages based on th eir common sense and, therefore, the use of 
+this knowledge in developing user interfaces can make them more intuitive to the end-user. 
+Moreover, as common sense knowledge varies from  group to group of people, it can be used 
+for developing applications capable of giving different feedback for different target groups, 
+as the applications presented along this chapte r illustrate, allowing, in this way, interface 
+personalization taking into account cultural issues.  
+For the purpose of using common sense know ledge in the development and design of 
+computer systems, it is necessary to provide an architecture that allows it. This chapter 
+presents LIA’s approaches for common sense knowledge acquisition, representation and 
+use, as well as for natural language processing , contributing with those ones who intent to 
+get into this challenging world to get started.  Regarding the educational context adopted by LIA’s researchers, the approaches presented 
+here go towards one of the grand challenges for engineering in 21
+st century recent 
+announced by the National Ac ademy of Engineering (NAE) 
+(http://www.nae.edu/nae/naehome.nsf ): advance personalized learning (Advance 
+Personalized Learning, 2008). Usin g intelligent internet  systems, capable of inferring over 
+the learning contexts which educators want to approach and presen ting related themes, 
+topics and pieces of information retrieved from is one of the possible approaches mentioned 
+by NAE. 
+As future work it is proposed to perform user  tests on the applications presented along this 
+chapter, in order to check their usability and, consequently, the users’ satisfaction in using 
+such kind of application. Human-Computer Interaction, New Developments 
+ 28 
+ 
+ 
+6. Acknowledgements 
+ 
+We thank FAPESP and CAPES for financial suppo rt. We also thank all volunteers who have 
+been collaborating with the Project OMCS-Br and all LIA’s researchers, who have been 
+giving valuable contributi ons towards the project. 
+ 
+7. References 
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\ No newline at end of file
diff --git a/1001.0419.txt b/1001.0419.txt
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+arXiv:1001.0419v2  [math.DS]  2 Jan 2012COMPACT GROUP AUTOMORPHISMS, ADDITION FORMULAS
+AND FUGLEDE-KADISON DETERMINANTS
+HANFENG LI
+Abstract. For a countable amenable group Γ and an element fin the integral
+group ring ZΓ being invertible in the group von Neumann algebra of Γ, we show
+that the entropy of the shift action of Γ on the Pontryagin dual of the quotient
+ofZΓ by its left ideal generated by fis the logarithm of the Fuglede-Kadison
+determinant of f. For the proof, we establish an ℓp-version of Rufus Bowen’s def-
+inition of topological entropy, addition formulas for group extensio ns of countable
+amenable group actions, and an approximation formula for the Fugle de-Kadison
+determinant of fin termsofthedeterminantsofperturbationsofthe compression s
+off.
+1.Introduction
+There are two motivations for this paper. First, for topological or measure-
+preserving actions of countable amenable groups, one has the ent ropy defined [57,
+60]. But unlike the case of Z-actions or Zd-actions (for 2≤d <∞), not many
+examples have been calculated for nonabelian group actions. Secon d, the study of
+automorphisms of compact metrizable groups has drawn much atte ntion in the de-
+velopment of ergodic theory, because of the rich interplay betwee n dynamics and
+compact group structures. Though the Z-actions of compact metrizable groups by
+automorphisms are well understood (cf. [39, 43, 52, 81, 82]), and much is known for
+Zd-actions (cf. [21, 36, 40–42, 67, 71–73, 78]), very little has been u nderstood for
+general countable amenable group actions (cf. [3, 13, 17, 20, 54, 55]).
+In this paper, we calculate the entropy for a rich class of actions of countable
+amenable groups on compact metrizable groups by automorphisms, providing some
+steps towards understanding the entropy theory of such algebr aic actions.
+Let Γ be a countable amenable group, and let fbe an element in the integral
+group ring ZΓ. One may consider the quotient ZΓ/ZΓfofZΓ by the left ideal ZΓf
+generated by f. Then Γ acts on the abelian group ZΓ/ZΓfby automorphisms via
+left translation, andhence acts onitsPontryagin dual (a compact metrizable abelian
+Date: December 28, 2011.
+2010Mathematics Subject Classification. Primary 37B40, 37A35, 22D25.
+Key words and phrases. Entropy, amenable group, group extension, addition formula, Fug lede-
+Kadison determinant.
+Partially supported by NSF Grants DMS-0701414 and DMS-1001625 .
+12 HANFENG LI
+group)
+Xf:=/hatwiderZΓ/ZΓf
+by automorphisms. Denote the latter action by αf. Explicitly, Xfconsists of ele-
+mentshin (R/Z)Γsatisfying/summationdisplay
+γ∈Γfγhγ−1γ′= 0
+for allγ′∈Γ, and the action αfis the restriction of the right shift action of Γ
+on (R/Z)ΓtoXf, i.e., (γh)γ′=hγ′γfor allh∈Xfandγ,γ′∈Γ (see Section 3).
+The topological entropy and the measure-theoretical entropy ( with respect to the
+normalized Haar measure) of αfcoincide [13], and will be denoted by h( αf).
+When Γ = Z, one may identify ZΓ with the one-variable Laurent polynomial ring
+Z[u±1] via identifying 1 ∈Z= Γ withu. Writingf∈ZΓ asu−k(/summationtextn
+j=0cjuj) with
+n≥0 andcnc0/\e}atio\slash= 0, and denoting by λ1,...,λ nthe roots of/summationtextn
+j=0cjuj, Yuzvinski˘ ı
+[82] showed that
+h(αf) = log|cn|+n/summationdisplay
+j=1log+|λj|, (1)
+where log+t= logmax(1 ,t) fort≥0. In general, Yuzvinski ˘i calculated the entropy
+of any endomorphism of a compact metrizable group [82].
+When Γ = Zdfor some 1≤d<∞, one may identify ZΓ with the d-variable Lau-
+rentpolynomialring Z[u±1
+1,...,u±1
+d]naturally. Fornonzero f∈ZΓ =Z[u±1
+1,...,u±1
+d],
+Lind, Schmidt and Ward [42, 71] showed that
+h(αf) = logM(f), (2)
+whereM(f) is theMahler measure off[50, 51] defined as
+M(f) = exp(/integraldisplay
+Tdlog|f(s)|ds)
+forTbeing the unit circle in CandTdbeing endowed with the normalized Haar
+measure. (When f= 0, clearly h( αf) =∞.) And this is the main step in their
+calculation for the entropy of any action of Zdon a compact metrizable group by
+automorphisms [42, 71]. In the case d= 1, the calculation (2) reduces to (1) via
+Jensen’s formula.
+Several years before Mahler introduced the Mahler measure, in [2 4] Fuglede and
+Kadison introduced a determinant det Affor invertible elements fin a unital C∗-
+algebraAwith respect to a tracial state tr A. It has found wide application in the
+study ofL2-invariants [48]. For a discrete group Γ, the group ring ZΓ sits naturally
+in the left group von Neumann algebra LΓ. Furthermore, LΓ has a canonical tracial
+state tr LΓ. Thus one may consider det LΓffor invertible f∈LΓ. When Γ = Zd,
+f∈ZΓ is invertible in LΓ if and only if fhas no zero point on Td. In such case,
+detLΓfis exactly M(f) [13].ENTROPY AND DETERMINANT 3
+In [13] Deninger pointed out the possibility of h( αf) = logdet LΓffor general
+countable amenable groups Γ and f∈ZΓ, and confirmed it in the special case that
+fis invertible in ℓ1(Γ) (this is stronger than the condition that fis invertible in LΓ,
+see Appendix A) and positive in LΓ and that Γ has a log-strong Følner sequence.
+Deninger and Schmidt [17] also confirmed it in the special case that fis invertible
+inℓ1(Γ) and that Γ is (amenable and) residually finite. The connection bet ween en-
+tropy, Mahler measure and Fuglede-Kadison determinant has been further explored
+by Deninger in [14–16].
+Our main result in this paper is
+Theorem 1.1. LetΓbe a countable amenable group and let f∈ZΓbe invertible in
+LΓ. Then
+h(αf) = logdet LΓf.
+One of the dynamical consequences of Theorem 1.1 and the genera l properties of
+the Fuglede-Kadison determinant is that under the hypothesis of T heorem 1.1, the
+actionsαfandαf∗have the same entropy, where f∗is the adjoint of fdefined as
+(f∗)γ=fγ−1for allγ∈Γ. This is a very non-trivial fact, as a priori there is no
+relation between αfandαf∗unlessfis in the center of ZΓ.
+Our proof of Theorem 1.1 consists of three steps.
+In the first step, we establish Theorem 1.1 under the further assu mption that
+fis positive in LΓ. Since the invertibility of finLΓ means that f−1exists as
+a bounded linear operator on ℓ2(Γ), while Rufus Bowen’s definition of topological
+entropy is taking the maximum of distances between finite orbits of p oints and
+should be thought of an ℓ∞-distance, we develop an ℓ2-version of his definition in
+Section 4, which is of independent interest. Then we prove the posit ive case of
+Theorem 1.1 in Section 5, using an estimate of number of integral poin ts in balls
+and an approximation formula of Deninger for det LΓfin such case.
+In the second step, we prove the Yuzvinski˘ ı addition formula in Se ction 6, which
+says that the entropy of a Γ-action on a compact metrizable group by automor-
+phisms is the sum of the entropy of the restriction of the action to a n invariant
+closed subgroup and the entropy of the induced action on the quot ient group. This
+formula allows us to reduce the calculation for the entropy of one ac tion to that
+for the entropy of simpler actions. In fact, we establish addition fo rmulas for group
+extensions in both topological and measure-theoretical settings , and the formula
+in either of these settings implies the Yuzvinski˘ ı addition formula. T he proof for
+each of these addition formulas employs both topological and measu re-theoretical
+tools, using generalization of the various fibre and conditional entr opies studied in
+[19], and the addition formula for Γ-extensions in [12, 79] which in turn depends on
+Rudolph and Weiss’s orbit equivalence method in [68].
+The third step is to prove h( αf)≥logdet LΓfunder the hypothesis in Theo-
+rem 1.1. Compared to the positive case in step one, the main difficulty h ere is that
+the compression of fto a nonempty finite subset of Γ map fail to be invertible.4 HANFENG LI
+Our method of dealing with this difficulty is to perturbthe compression of fto an
+invertible linear operator. For this purpose, we establish an approx imation formula
+for logdet LΓfin terms of the determinants of the compressions, in Section 7. This
+uses an approximation formula for traces, initiated by L¨ uck in work onL2-invariants
+[47] and extended by Schick in [70]. We complete the third step in Sectio n 8, using
+Ornstein and Weiss’s theory of quasitiling in [60].
+The proof of Theorem 1.1, which uses the fact that the Fuglede-Ka dison determi-
+nants offandf∗are equal, is finished in Section 9. Some dynamical consequences
+of the theorem including the equality of h( αf) and h(αf∗) are also established there.
+We recall some background in Section 2, and give a proof of the case Γ is finite in
+Section 3, which shows clearly how the entropy and the Fuglede-Kad ison determi-
+nant are connected via several equalities. In an appendix, we comp are invertibility
+inℓ1(Γ) andLΓ.
+Recently, entropy has been defined for continuous actions of a co untable sofic
+group on compact metrizable spaces and measure-preserving act ions of a countable
+sofic group on standard probability measure spaces, with respect to a sofic approx-
+imation sequence of the sofic group [4, 34]. The class of sofic groups include all
+discrete amenable groups and residually finite groups. The sofic ent ropies coincide
+with the classical entropies when the sofic group is amenable [6, 35]. F or a count-
+able residually finite (not necessarily amenable) group Γ and an f∈ZΓ, when the
+sofic approximation sequence of Γ comes from a sequence of finite- index normal sub-
+groups of Γ, in various cases it has been shown that the sofic topolo gical entropy
+and the sofic measure entropy (for the normalized Haar measure o fXf) ofαfare
+equal to logdet LΓf[5, 7, 34].
+Throughout this paper, for a group G, we denote by eGthe identity element
+ofG. For a discrete group Γ, we write C[[Γ]],R[[Γ]] and Z[[Γ]] for CΓ,RΓand
+ZΓrespectively. For a finite set F, we write C[F] forCF, and equip it with the
+standardℓ2-norm. For a Hilbert space H, we denote by B(H) the set of bounded
+linear operators on H, and equip it with the operator norm /⌊ard⌊l·/⌊ard⌊l.
+After this paper was finished, Douglas Lind informed us that Douglas Lind and
+Klaus Schmidt have independently obtained results similar to ours in Se ction 6.
+Acknowledgements. I thank George Elliott, Wen Huang and Zhuang Niu for help-
+ful discussion, especially my colleague Jingbo Xia for discussion about Example A.1.
+I also thank Lewis Bowen, Christopher Deninger, Douglas Lind, Andr eas Thom,
+Thomas Ward and the referee for very helpful comments. Part of this work was car-
+ried out while I visited Fields Institute in Summer 2007 and Vanderbilt Un iversity
+in Fall 2009. I am grateful to Guoliang Yu for his warm hospitality.
+2.Preliminaries
+2.1.Background on Entropy Theory. In this subsection we recall some back-
+ground about the entropy theory. The reader is referred to [25 , 57, 61, 77] for detail.ENTROPY AND DETERMINANT 5
+Throughout this paper Γ will be a discrete amenable group, unless sp ecified other-
+wise. The amenability of Γ means that Γ has a (left) Følner net {Fn}n∈J, i.e., each
+Fnis a nonempty finite subset of Γ, and lim n→∞|KFn∆Fn|
+|Fn|= 0 for every finite subset
+Kof Γ [63].
+The following subadditivity result is known as the Ornstein-Weiss lemma [44,
+Theorem 6.1].
+Proposition 2.1. Ifϕis a real-valued function which is defined on the set of
+nonempty finite subsets of Γand satisfies
+(1) 0≤ϕ(F)<+∞,
+(2)ϕ(F)≤ϕ(F′)for allF⊆F′,
+(3)ϕ(Fγ) =ϕ(F)for all nonempty finite F⊆Γandγ∈Γ,
+(4)ϕ(F∪F′)≤ϕ(F)+ϕ(F′)ifF∩F′=∅,
+then1
+|F|ϕ(F)converges to some limit bas the setFbecomes more and more (left)
+invariant in the sense that for every ε>0there exist a nonempty finite set K⊆Γ
+and aδ >0such that/vextendsingle/vextendsingle1
+|F|ϕ(F)−b/vextendsingle/vextendsingle<εfor all nonempty finite sets F⊆Γsatisfying
+|KF∆F|≤δ|F|.
+LetαbeanactionofΓ onacompact Hausdorff space Xby homeomorphisms. For
+any open cover UofX, and any nonempty finite subset Fof Γ, setUF=/logicalortext
+γ∈Fγ−1U
+and denote by N(U) the minimal number of elements in Uneeded to cover X. Then
+the function F/ma√sto→logN(UF) defined on the set of nonempty finite subsets of Γ
+satisfies the conditions in Proposition 2.1, and hence1
+|F|logN(UF) converges as F
+becomes more and more (left) invariant. We denote this limit by h top(α,U). The
+topological entropy ofα, denoted by h top(α), isdefined asthe supremum of h top(α,U)
+over all finite open covers UofX.
+Letαbe an action of Γ on a probability space ( X,B,µ) by automorphisms. For
+any finite measurable partition P={P1,...,P k}ofX, and any nonempty finite
+subsetFof Γ, set PF=/logicalortext
+γ∈Fγ−1Pand H µ(P) =/summationtextk
+j=1−µ(Pj)logµ(Pj), where we
+take the convention that 0log0 = 0 so that the function t/ma√sto→tlogtis continuous for
+0≤t≤1. The function F/ma√sto→Hµ(PF) defined on the set of nonempty finite subsets
+of Γ satisfies the conditions in Proposition 2.1, and hence1
+|F|logHµ(PF) converges
+asFbecomes more and more (left) invariant. We denote this limit by h µ(α,P). The
+measure entropy orKolmogorov-Sinai entropy ofα, denoted by h µ(α), is defined as
+the supremum of h µ(α,P) over all finite measurable partitions PofX.
+A topological space is called a Polish space if it is separable and admits a compat-
+ible complete metric. A probability space ( X,B,µ) is called a standard ifBis the
+Borelσ-algebra for some Polish topology on X. Suppose that ( X,B,µ) is standard
+and that B′is a sub-σ-algebra of B. Then there is a map E(·|B′) :L1(X,B,µ)→6 HANFENG LI
+L1(X,B′,µ), called the conditional expectation , determined by
+/integraldisplay
+AE(f|B′)(x)dµ(x) =/integraldisplay
+Af(x)dµ(x)
+for everyf∈L1(X,B,µ) andA∈B′. Here one can use either complex or real
+valued functions for L1(X,B,µ) andL1(X,B′,µ). For any A∈B, one has 0≤
+E(1A|B′)(x)≤1 forµa.e.x∈X, where 1 Adenotes the characteristic function of A.
+Foranyfinitemeasurablepartition PofX,setH µ(P|B′) =/summationtext
+P∈P−/integraltext
+PlogE(1P|B′)(x)dµ(x).
+Now assume further that B′is Γ-invariant. Then the function F/ma√sto→Hµ(PF|B′) de-
+fined on the set of nonempty finite subsets of Γ satisfies the condit ions in Propo-
+sition 2.1, and hence1
+|F|Hµ(PF|B′) converges as Fbecomes more and more (left)
+invariant. We denote this limit by h µ(α,P|B′). Theconditional entropy of αgiven
+B′, denoted by h µ(α|B′), is defined as the supremum of h µ(α,P|B′) over all finite
+measurable partitions PofX.
+For a compact space X, denote by BXthe Borelσ-algebra of X. Ifαis an action
+of Γ on a compact space Xby homeomorphisms, and µis a regular Γ-invariant Borel
+probability measure on X, thenαis also an action of Γ on the probability space
+(X,BX,µ) by automorphisms.
+Note that every (continuous) automorphism of a compact group p reserves the
+normalized Haar measure. Thus if αis an action of Γ on a compact group Gby
+automorphisms, it automatically preserves the normalized Haar mea sureµofG.
+Then we have both the topological entropy h top(α) and the measure entropy h µ(α).
+It is a result of Deninger that these two entropies coincide [13, Theo rem 2.2]. (It
+was assumed in [13, Theorem 2.2] that Gis abelian; but this is not needed.) (The
+case Γ = Zwas proved by Berg [2]; the case Γ = Zdwas proved by Lind et al. [42,
+page 624] [71, Theorem 13.3].) Thus we shall denote h top(α) and h µ(α) simply by
+h(α).
+2.2.Background on Group von Neumann Algebras and Fuglede-Kadis on
+Determinants. In this subsection we recall some background about the group von
+Neumann algebra and the Fuglede-Kadison determinant.
+For a Hilbert space H, the setB(H) is a∗-algebra with T∗being the adjoint of
+T, and is equipped with the operator norm /⌊ard⌊l·/⌊ard⌊l. AC∗-algebrais a sub-∗-algebra
+ofB(H) for some Hilbert space H, closed under/⌊ard⌊l·/⌊ard⌊l. An element ainAis called
+positiveand written as a≥0 ifa=b∗bfor someb∈A. A tracial state of a unital
+C∗-algebraAis a linear functional tr A:A→Csuch that tr Atakes value 1 at the
+identity of A,|trA(a)|≤/⌊ard⌊la/⌊ard⌊land tr A(ab) = tr A(ba) for alla,b∈A. We refer the
+reader to [31, 75] for detail.
+In this paper we shall need only three classes of C∗-algebras and tracial states.
+The first class is the C∗-algebraB(ℓ2
+n) for eachn∈N. EachB(ℓ2
+n) has a unique
+tracial state tr B(ℓ2n). If we take an orthonormal basis of ℓ2
+nand identify B(ℓ2
+n) with
+Mn(C), then tr B(ℓ2)(a) =1
+n/summationtextn
+j=1aj,jfor every matrix a= (ai,j)1≤i,j≤n∈Mn(C).ENTROPY AND DETERMINANT 7
+Let Γ be a discrete amenable group. The complex group algebra CΓ consists
+of elements in CΓwith finite support. Its multiplication is defined as ( fg)γ′=/summationtext
+γ∈Γfγgγ−1γ′for allf,g∈CΓ andγ∈Γ. We shall also extend this multiplication
+to the cases like g∈C[[Γ]], orf∈ZΓ andg∈(R/Z)Γwhenever it can be defined.
+One may identify CΓ as a linear subspace of ℓ2(Γ) naturally. For each f∈CΓ, its
+left multiplication g/ma√sto→fgforg∈CΓ extends to a bounded linear map of ℓ2(Γ).
+In this way we shall identify CΓ as a subalgebra of B(ℓ2(Γ)). It is easily checked
+thatCΓ is closed under taking adjoint in B(ℓ2(Γ)). Explicitly, ( f∗)γ=fγ−1for all
+f∈CΓ andγ∈Γ. The second class of C∗-algebras we need, the left group von
+Neumann algebra LΓ, is defined as the closure of CΓ under the strong operator
+topology. Explicitly, LΓ consists of T∈B(ℓ2(Γ)) commuting with the right regular
+representation ρof Γ onℓ2(Γ), i.e., (T(hγ))γ′= (Th)γ′γfor allh∈ℓ2(Γ) and
+γ,γ′∈Γ, where (hγ)γ′′=hγ′′γfor allγ′′∈Γ. The algebra LΓ has a canonical
+tracial state tr LΓdefined as tr LΓ(a) =/a\}⌊ra⌋ketle{taeΓ,eΓ/a\}⌊ra⌋ketri}ht. The trace tr LΓis faithful in the
+sense that if a∈LΓ is positive and tr LΓ(a) = 0, then a= 0. Throughout this
+article, we fix this tracial state of LΓ, and the determinant det LΓ is calculated with
+respect to it.
+Another way to describe the elements of LΓ is that they are the elements hof
+C[[Γ]] for which the map from CΓ toℓ2(Γ) sending xtohxis well-defined and
+extends to a bounded linear operator on ℓ2(Γ). It is easy to see that if h1andh2are
+inR[[Γ]], thenh1+ih2is inLΓ if and only if both h1andh2are inLΓ. It follows
+that ifh∈R[[Γ]]∩LΓ is invertible in LΓ, then its inverse lies in R[[Γ]] and hence
+preservesℓ2
+R(Γ).
+The third class of C∗-algebras we need is the unital commutative C∗-algebras.
+They can be described as unital commutative Banach complex algebr asAwith a
+∗-operation satisfying ( a∗)∗=a, (λa+b)∗=¯λa∗+b∗, (ab)∗=b∗a∗,/⌊ard⌊la∗/⌊ard⌊l=/⌊ard⌊la/⌊ard⌊land
+/⌊ard⌊la∗a/⌊ard⌊l=/⌊ard⌊la/⌊ard⌊l2for alla,b∈Aandλ∈C.
+For a tracial state tr Aof a unitalC∗-algebraA, theFuglede-Kadison determinant
+of an invertible a∈Awith respect to tr A[24] is defined as
+(3) det Aa:= exp(tr Alog|a|) = exp(1
+2trAlog(a∗a)),
+where|a|= (a∗a)1/2is the absolute part of a. (The Fuglede-Kadison determinant
+is also defined for noninvertible elements of A, but the definition is more involved.)
+For detail and application of the Fuglede-Kadison determinant to L2-invariants, see
+[48].
+For anyn∈Nand any invertible a∈B(ℓ2
+n), one has det B(ℓ2n)(a) =|deta|1/n.
+Among many nice properties of the Fuglede-Kadison determinant, w e shall need
+the following ones:
+Theorem 2.2. [24, Lemma 1, Theorem 1] Let trbe a tracial state of a unital C∗-
+algebraA. Then8 HANFENG LI
+(1)for any invertible a∈A, one has detA(a) = det A(a∗);
+(2)for any0≤a≤binAwithabeing invertible in A, one has detAa≤detAb.
+3.Finite Group Case
+In this section we prove Theorem 1.1 for the case Γ is finite. This case is easily
+proved and appeared in [13, Section 7]. However, we choose to give a proof of this
+case here, since it reveals the essence of the equality in Theorem 1.1 .
+The following lemma is well known [74, Lemma 4]. For the convenience of t he
+reader, we give a proof.
+Lemma 3.1. Letn∈Nand letT:Cn→Cnbe an invertible linear map, preserving
+Zn. Then|detT|=|Zn/TZn|.
+Proof.Note thatTZnhas rankn. By the elementary divisor theorem [37, Theorem
+III.7.8] there are a basis e1,...,e nofZnand nonzero integers k1,...,k nsuch that
+k1e1,...,k nenis a basis of TZn. SinceTe1,...,Te nis also a basis of TZn, there
+existsQ∈GLn(Z) with (Te1,...,Te n) = (k1e1,...,k nen)Q. Then the matrix of T
+under the basis e1,...,e nis diag(k1,...,k n)·Q. Thus
+|detT|=|det(diag(k1,...,k n)·Q)|=|/productdisplay
+1≤j≤nkj|=|Zn/TZn|. (4)
+/square
+LetΓbeadiscreteamenablegroupandlet f∈ZΓ. Thecanonicalpairingbetween
+ZΓ and its Pontryagin dual /hatwiderZΓ = (R/Z)Γis given by
+/a\}⌊ra⌋ketle{tg,h/a\}⌊ra⌋ketri}ht=/summationdisplay
+γ∈Γgγhγ
+for allg∈ZΓ andh∈(R/Z)Γ. It is easy to check that
+/a\}⌊ra⌋ketle{tgf,h/a\}⌊ra⌋ketri}ht=/a\}⌊ra⌋ketle{tg,hf∗/a\}⌊ra⌋ketri}ht
+for allg∈ZΓ andh∈(R/Z)Γ. It follows that Xf={h∈(R/Z)Γ:hf∗= 0}and
+αfis the restriction of the left shift action of Γ on ( R/Z)ΓtoXf. Forh∈(R/Z)Γ,
+denote by ˜hthe “adjoint” element in ( R/Z)Γdefined as ˜hγ=hγ−1for allγ∈Γ.
+Note that the map h/ma√sto→˜his an automorphism of the compact group ( R/Z)Γ, and
+intertwines the left and right shift actions of Γ. The image of Xfunder this map is
+{h∈(R/Z)Γ:fh= 0}. In the rest of this paper, we shall write
+Xf={h∈(R/Z)Γ:fh= 0}, (5)
+and under this identification αfis the restriction of the right shift action of Γ on
+(R/Z)ΓtoXf.ENTROPY AND DETERMINANT 9
+Theorem 3.2. LetΓbe a finite group and let f∈ZΓbe invertible in LΓ. Then
+h(αf) =1
+|Γ|log|Xf|=1
+|Γ|log|ZΓ/fZΓ|=1
+|Γ|log|detf|= logdet LΓf.
+Proof.From the definition of topological entropy, we have h( αf) =1
+|Γ|log|Xf|.
+Note that both XfandZΓ/fZΓ are abelian groups. We claim that they are
+isomorphic. Writing ( R/Z)ΓasRΓ/ZΓ, we may identify Xfwith{g∈RΓ :fg∈
+ZΓ}/ZΓ. Since the left multiplication by frestricts to a group automorphism of
+RΓ and sends ZΓ ontofZΓ, the claim is proved. It follows that |Xf|=|ZΓ/fZΓ|.
+By Lemma 3.1 one has |ZΓ/fZΓ|=|detf|.
+Note that the unique tracial state of B(ℓ2(Γ)) restricts to the canonical trace of
+LΓ. Thus det LΓf= detB(ℓ2(Γ))f=|detf|1
+|Γ|. /square
+Notation 3.3. For any nonempty finite subset Fof Γ, denote by pFthe restriction
+mapC[[Γ]]→C[F], and byιFthe embedding C[F]→ℓ2(Γ). Forf∈LΓ, set
+fF:=pF◦f◦ιF∈B(C[F]).
+Now consider the case Γ is infinite countable. Let {Fn}n∈Nbe a (left) Følner
+sequence of Γ, and let f∈ZΓ be invertible in LΓ. SinceFnis the analogue of a
+finite group, the analogue of Theorem 3.2 is
+h(αf) = lim
+ε→01
+|Fn|logsFn,∞(ε) =1
+|Fn|log|Z[Fn]/fFnZ[Fn]|
+=1
+|Fn|log|detfFn|= logdet LΓf
+for eachn∈N, wheresFn,∞(ε) is the cardinality of certain set resembling Xf
+restricted to Fnand will be defined at the beginning of Section 5. On the other
+hand,Fnapproximates Γ as n→∞. Thus a more precise and reasonable analogue
+of Theorem 3.2 is
+h(αf) = lim
+ε→0lim
+n→∞1
+|Fn|logsFn,∞(ε) = lim
+n→∞1
+|Fn|log|Z[Fn]/fFnZ[Fn]| (6)
+= lim
+n→∞1
+|Fn|log|detfFn|= logdet LΓf.
+Indeed, this is the intuition behind Theorem 1.1. But there is some imme diate
+difficulty even for making sense of (6). For instance, fFnmay fail to be invertible.
+In such case,|Z[Fn]/fFnZ[Fn]|=∞and detfFn= 0.
+4.ℓp-version of R. Bowen’s Definition of Topological Entropy
+In this section we prove Theorem 4.2, providing an ℓp-version of R. Bowen’s def-
+inition of topological entropy. Throughout this section Γ is a discret e amenable
+group.10 HANFENG LI
+Letαbe an action of Γ on a compact Hausdorff space Xby homeomorphisms.
+Recallthata continuous pseudometric onXisasymmetriccontinuousmap X×X→
+R+, vanishingonthediagonalof X×Xandsatisfyingthetriangleinequality. Denote
+byMthe set of all continuous pseudometrics on X. Letϑ∈M. For a nonempty
+finite subset F⊆Γ, 1≤p≤∞andx,y∈X, denote by dϑ,F,p(x,y) the quotient
+of theℓp-norm of the function γ/ma√sto→ϑ(γx,γy) onFdivided by|F|1/p. We say that
+E⊆Xis [ϑ,F,p,ε]-separated if for anyx/\e}atio\slash=yinE,dϑ,F,p(x,y)> ε. We say
+thatE⊆Xis [ϑ,F,p,ε]-spanning if for anyx∈X, there is some y∈Ewith
+dϑ,F,p(x,y)≤ε. Denote by sϑ,F,p(ε) (rϑ,F,p(ε) resp.) the maximal (minimal resp.)
+cardinality of [ ϑ,F,p,ε]-separated ([ ϑ,F,p,ε]-spanning resp.) subsets of X.
+Lemma 4.1. Letαbe an action of Γon a compact Hausdorff space Xby home-
+omorphisms. Let ϑbe a continuous pseudometric of X. For any ε >0,λ >1,
+and1≤p <∞, there exists some ε′>0such thatλ|F|sϑ,F,p(ε′)≥sϑ,F,∞(ε)for all
+nonempty finite subsets FofΓ.
+Proof.CoverXby finitely many, say M, closedϑ-balls of radius ε/2. By Stirling’s
+formula there is some c∈(0,1/2) such that/parenleftbign
+cn/parenrightbig
+≤λn/2for alln∈N. We may
+assume that Mc≤λ1/2. Setε′=c1
+pε/2.
+LetFbe a nonempty finite subset of Γ andlet Ebe a [ϑ,F,∞,ε]-separatedsubset
+ofXwith|E|=sϑ,F,∞(ε). For each x∈Edenote byB(x,ε/2) the set of elements
+yinEsuch that|{γ∈F:ϑ(γx,γy)> ε/2}|< c|F|. Ifxandyare inEand
+y/\e}atio\slash∈B(x,ε/2), then
+dϑ,F,p(x,y)>((ε/2)pc|F|)1/p
+|F|1/p= (ε/2)c1/p=ε′.
+Take a subset E′ofEmaximal with respect to the property that for any x/\e}atio\slash=y
+inE′,y /∈B(x,ε/2). Then/uniontext
+x∈E′B(x,ε/2) =EandE′is [ϑ,F,p,ε′]-separated.
+Denote byDthe maximum of|B(x,ε/2)|over allx∈E. ThenD|E′|≥|E|. Thus
+it suffices to show that λ|F|≥D.
+Fixx∈E. For anyy∈B(x,ε/2) there is some Ky⊆Fwith|Ky|=⌊c|F|⌋
+andϑ(γx,γy)≤ε/2 for allγ∈F\Ky, where⌊t⌋denotes the largest integer no
+bigger than t. Then there are a subset B′ofB(x,ε/2) with|B′|≥|B(x,ε/2)|//parenleftbig|F|
+c|F|/parenrightbig
+and a subset KofFwith|K|=⌊c|F|⌋such thatKy=Kfor ally∈B′. Then
+ϑ(γy,γz)≤ϑ(γy,γx)+ϑ(γx,γz)≤εfor ally,z∈B′andγ∈F\K. Note that, as
+a subset of E,B′is [ϑ,F,∞,ε]-separated. It follows that for any y/\e}atio\slash=zinB′there
+is someγinKwithϑ(γy,γz)>ε. Thenγyandγzmust lie in different closed balls
+which we take at the beginning of the proof. Consequently, |B′|≤M|K|. Therefore
+|B(x,ε/2)|≤|B′|/parenleftbigg|F|
+c|F|/parenrightbigg
+≤Mc|F|λ|F|/2≤λ|F|.
+This finishes the proof of the lemma. /squareENTROPY AND DETERMINANT 11
+We say that an open subset UofXisgenerated by ϑifUis in the weakest
+topology of Xmakingϑcontinuous, i.e., Uis a union of open ϑ-balls with positive
+radii. We say that a finite open cover U={U1,...,U n}ofXisgenerated by ϑif
+eachUjis generated by ϑ. For any nonempty finite subset Fof Γ, we define ϑF∈M
+by settingϑF(x,y) = max γ∈Fϑ(γx,γy) for allx,y∈X. We say that an open subset
+UofXisgenerated by ϑunderαifUis contained in the weakest topology on X
+making all the pseudometrics ( x,y)/ma√sto→ϑ(γx,γy) continuous, equivalently, Uis a
+union of open sets UFgenerated by ϑFforFrunning over nonempty finite subsets
+of Γ. We say that the topology of Xisgenerated by ϑunderαif the topology on
+Xis exactly the weakest topology making all the pseudometrics ( x,y)/ma√sto→ϑ(γx,γy)
+continuous. Having zero ϑ-distance is an equivalence relation on X. Forx∈X
+denote by [x] its equivalence class. Denote by Xϑthe quotient space of Xconsisting
+of all such equivalence classes, equipped with the quotient topology . Thenϑinduces
+a metric on Xϑ. Equip (Xϑ)Γwith the right shift action of Γ. It is easy to see that
+the topology of Xis generated by ϑunderαif and only if the natural Γ-equivariant
+continuous map X→(Xϑ)Γsendingxtoγ/ma√sto→[γx] is an embedding, if and only if
+any two points xandyofXare equal exactly when ϑ(γx,γy) = 0 for all γ∈Γ.
+We say that a finite open cover U={U1,...,U n}ofXisgenerated by ϑunderαif
+eachUjis so.
+The casep=∞and Γ = Zdof the following theorem was proved by Schmidt [71,
+Proposition 13.7], and the case p=∞for general Γ was proved by Deninger [13,
+Proposition 2.3]. For completeness we include also a proof for the cas ep=∞here.
+Theorem 4.2. Letαbe an action of Γon a compact Hausdorff space Xby home-
+omorphisms. Let ϑbe a continuous pseudometric of X. Let{Fn}n∈Jbe a (left)
+Følner net of Γ. For any 1≤p≤∞, we have
+sup
+Uhtop(α,U) = lim
+ε→0limsup
+n→∞1
+|Fn|logsϑ,Fn,p(ε) = lim
+ε→0liminf
+n→∞1
+|Fn|logsϑ,Fn,p(ε)
+= lim
+ε→0limsup
+n→∞1
+|Fn|logrϑ,Fn,p(ε) = lim
+ε→0liminf
+n→∞1
+|Fn|logrϑ,Fn,p(ε),
+whereUruns through all finite open covers of Xgenerated by ϑunderα. In partic-
+ular, if the topology of Xis generated by ϑunderα, then we have
+htop(α) = lim
+ε→0limsup
+n→∞1
+|Fn|logsϑ,Fn,p(ε) = lim
+ε→0liminf
+n→∞1
+|Fn|logsϑ,Fn,p(ε)
+= lim
+ε→0limsup
+n→∞1
+|Fn|logrϑ,Fn,p(ε) = lim
+ε→0liminf
+n→∞1
+|Fn|logrϑ,Fn,p(ε).
+Proof.We prove first the theorem for p=∞. Note that
+rϑ,F,∞(ε)≤sϑ,F,∞(ε)≤rϑ,F,∞(ε/2).
+Thus it suffices to show that supUhtop(α,U)≥limε→0limsupn→∞1
+|Fn|logsϑ,Fn,∞(ε)
+and supUhtop(α,U)≤limε→0liminf n→∞1
+|Fn|logsϑ,Fn,∞(ε).12 HANFENG LI
+Letε>0. Take a finite open cover UofXconsisting of open ϑ-balls with radius
+ε/2. Then Uis generated by ϑ. We have sϑ,F,∞(ε)≤N(UF) for every nonempty
+finite subset Fof Γ, and hence limsupn→∞1
+|Fn|logsϑ,Fn,∞(ε)≤htop(α,U). Therefore
+limε→0limsupn→∞1
+|Fn|logsϑ,Fn,∞(ε)≤supUhtop(α,U).
+LetUbe a finite open cover of Xgenerated by ϑunderα. Then we can find
+a finite open cover VofXfiner than Usuch that Vis generated by ϑKfor some
+nonempty finite subset Kof Γ. It follows that there exists some ε >0 such that
+every open ϑK-ball with radius 3 εis contained in some element of V. CoverXby
+finitely many, say M, openϑ-balls with radius ε. We have
+M|KF\F|rϑ,F,∞(ε)≥rϑ,KF,∞(2ε)≥N(VF)≥N(UF)
+for every nonempty finite subset Fof Γ, and hence liminf n→∞1
+|Fn|logrϑ,Fn,∞(ε)≥
+htop(α,U). Therefore lim ε→0liminf n→∞1
+|Fn|logrϑ,Fn,∞(ε)≥supUhtop(α,U). This
+proves the case p=∞.
+Now the case 1≤p <∞follows from the case p=∞, the facts sϑ,F,p(ε)≤
+sϑ,F,∞(ε) andrϑ,F,p(ε)≤sϑ,F,p(ε)≤rϑ,F,p(ε/2), and Lemma 4.1. /square
+5.Positive Case
+In this section we show that the intuitive equalities (6) do hold when fis positive
+(Theorem 5.6). This proves Theorem 1.1 in such case. Throughout t his section Γ is
+a discrete amenable group.
+Denote by ϑthe metric on R/Zinduced from the standard metric on R, i.e.
+ϑ(tmodZ, t′modZ) = min m∈Z|t−t′−m|. Recall the identification (5). Via
+the projection Xf→R/ZsendingxtoxeΓ, we shall think of ϑas a continuous
+pseudometric on Xf. Clearly the topology of Xfis generated by ϑunderαf. Thus
+we can apply Theorem 4.2. We shall make use of the cases p= 2 andp=∞. We
+shall abbreviate sϑ,F,p(ε) assF,p(ε) etc.
+Thefollowingresultiscrucialforthecomparisonof sF,p(ε),rF,p(ε)and|Z[Fn]/fFnZ[Fn]|.
+Lemma 5.1. There exists some universal constant C >0such that for any λ>1,
+there is some δ >0so that for any nonempty finite set Y, any positive integer n
+with|Y|≤δn, and anyM≥1one has
+|{x∈Z[Y] :/⌊ard⌊lx/⌊ard⌊l2≤M·n1/2}|≤CλnM|Y|.
+Proof.Letδ >0 be a small number less than e−1, which we shall determine later.
+LetYbe a nonempty finite set and nbe a positive integer with |Y|≤δn. For
+eachx∈Z[Y], denote{z∈R[Y] : 0≤zy−xy≤1 for ally∈Y}byDx. Denote
+{x∈Z[Y] :/⌊ard⌊lx/⌊ard⌊l2≤M·n1/2}bySand denote the union of Dxfor allx∈S
+byDS. Then the (Euclidean) volume of DSis equal to|S|. Note that/⌊ard⌊lz/⌊ard⌊l2≤
+M·n1/2+n1/2≤2Mn1/2for everyz∈DS.
+A simple calculation shows that the function ς(t) := (n/t)t/2is increasing for
+0< t≤ne−1. The volume of the unit ball of R[Y] under/⌊ard⌊l·/⌊ard⌊l2isπ|Y|/2/(|Y|/2)!ENTROPY AND DETERMINANT 13
+[11, page 9]. By Stirling’s formula there exists some constant C′>0 such that
+m!≥C′√m(m
+e)mfor allm≥1. Thus the volume of DSis no bigger than
+(π|Y|/2(2Mn1/2)|Y|)/(|Y|/2)!≤(π|Y|/2(2Mn1/2)|Y|)/(C′/radicalbig
+|Y|/2(|Y|/(2e))|Y|/2)
+≤CC|Y|
+1(n/|Y|)|Y|/2M|Y|=CC|Y|
+1ς(|Y|)M|Y|
+≤CCδn
+1ς(δn)M|Y|=CCδn
+1δ−δn/2M|Y|,
+whereC=√
+2/C′andC1= 2√
+2eπ. Takeδ >0 so small that Cδ
+1δ−δ/2≤λ. Then
+the volume of DSis no bigger than CλnM|Y|. Consequently,|S|≤CλnM|Y|./square
+We need the following result of Deninger (note that the assumption in [13, Corol-
+lary 3.4] that Γ is finitely generated is not needed). In Corollary 7.2 we shall gen-
+eralize the equality part to non-positive elements in the presence of perturbations.
+Recall the notations pFandfFin Notation 3.3.
+Lemma 5.2. [13, Theorem 3.2, Proposition 3.3, Corollary 3.4] Let f∈ZΓbe
+invertible and positive in LΓ. ThenfFis invertible and/⌊ard⌊l(fF)−1/⌊ard⌊l≤/⌊ard⌊lf−1/⌊ard⌊lfor every
+nonempty finite subset FofΓ, and
+detLΓf= lim
+n→∞|detfFn|1/|Fn|= lim
+n→∞|Z[Fn]/fFnZ[Fn]|1/|Fn|
+for any (left) Følner net {Fn}n∈JofΓ.
+Notation 5.3. Forf∈CΓ, denote by Kfthe union of the supports of fandf∗,
+and the identity of Γ.
+Lemma 5.4. Letf∈ZΓbe invertible and positive in LΓ. Then for any λ>1and
+ε>0, there is some δ >0such that when a nonempty finite subset F⊆Γsatisfies
+|K2
+fF\F|≤δ|F|we have
+sF,2(ε)≤Cλ|F||Z[F]/fFZ[F]|,
+whereCis the universal constant in Lemma 5.1.
+Proof.WriteKforKf. Take 1> δ >0 such that (/⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊lf/⌊ard⌊l·21/2)δ≤λ1/2and
+δ1/2/⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊lf/⌊ard⌊l1≤ε, and thatδsatisfies the conclusion of Lemma 5.1 for λ′=λ1/2.
+LetFsatisfy the hypothesis.
+Take an [F,2,ε]-separated subset E⊆Xfwith|E|=sF,2(ε). For each x∈E
+denoteby ˜xtheelement in[0 ,1)Γsuchthatxistheimageof ˜ xunder thenaturalmap
+[0,1)Γ→(R/Z)Γ. Thenf˜x∈Z[[Γ]] and hence pF(f˜x)∈Z[F]. Denote by ϕFthe
+quotient map Z[F]→Z[F]/fFZ[F]. We get a map ψ:E→Z[F]/fFZ[F] sending
+xtoϕF(pF(f˜x)). It suffices to show that for any a∈Z[F]/fFZ[F], the preimage of
+aunderψhas at most Cλ|F|elements. Fix a∈Z[F]/fFZ[F] andy∈ψ−1(a).
+For eachx∈E, setx′=pKF(˜x). We shall identify C[KF] naturally as a subspace
+ofℓ2(Γ) via the embedding ιKFin Notation 3.3. Note that ψ(x) =ϕF(pF(fx′)).
+Suppose that x∈ψ−1(a). ThenpF(f(x′−y′)) lies infFZ[F], and hence
+pF(f(x′−y′)) =fF(hx) (7)14 HANFENG LI
+for somehx∈Z[F]. Setzx=f(x′−y′)−fhx. Then
+pF(zx) =pF(f(x′−y′)−fhx) =pF(f(x′−y′))−fF(fhx) = 0.
+Thuszxis inRΓ and vanishes on F, and
+f(x′−y′) =fhx+zx. (8)
+By Lemma 5.2 the linear operator fFis invertible and/⌊ard⌊l(fF)−1/⌊ard⌊l≤/⌊ard⌊lf−1/⌊ard⌊l. From (7)
+we get
+hx= (fF)−1(pF(f(x′−y′))).
+Thus
+/⌊ard⌊lhx/⌊ard⌊l2≤ /⌊ard⌊l(fF)−1/⌊ard⌊l·/⌊ard⌊lf/⌊ard⌊l·/⌊ard⌊lx′−y′/⌊ard⌊l2≤/⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊lf/⌊ard⌊l·/⌊ard⌊lx′−y′/⌊ard⌊l∞·|KF|1/2
+≤ /⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊lf/⌊ard⌊l·|KF|1/2≤/⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊lf/⌊ard⌊l·21/2·|F|1/2,
+and hence
+/⌊ard⌊lpFK\F(fhx)/⌊ard⌊l2≤/⌊ard⌊lf/⌊ard⌊l·/⌊ard⌊lhx/⌊ard⌊l2≤/⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊lf/⌊ard⌊l2·21/2·|F|1/2.
+By Lemma 5.1 one has
+|{pKF\F(fhx) :x∈ψ−1(a)}| ≤Cλ|F|/2(/⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊lf/⌊ard⌊l2·21/2)|KF\F|
+≤Cλ|F|/2(/⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊lf/⌊ard⌊l2·21/2)δ|F|
+≤Cλ|F|.
+Thus we can find a subset W⊆ψ−1(a) withCλ|F||W| ≥ |ψ−1(a)|such that
+pKF\F(fhx1) =pKF\F(fhx2) for allx1,x2∈W. Letx1,x2∈W. Applying (8)
+tox=x1andx=x2respectively, we get
+f(x′
+1−x′
+2) =f(x′
+1−y′)−f(x′
+2−y′) =f(hx1−hx2)+(zx1−zx2).
+Sincef(hx1−hx2) has support contained in F, whilezx1−zx2has support contained
+inK2F\F, one has
+/⌊ard⌊lzx1−zx2/⌊ard⌊l2=/⌊ard⌊lpK2F\F(f(x′
+1−x′
+2))/⌊ard⌊l2≤/⌊ard⌊lf(x′
+1−x′
+2)/⌊ard⌊l∞·|K2F\F|1/2
+≤ /⌊ard⌊lf/⌊ard⌊l1·/⌊ard⌊lx′
+1−x′
+2/⌊ard⌊l∞·|K2F\F|1/2≤δ1/2/⌊ard⌊lf/⌊ard⌊l1·|F|1/2,
+and hence
+/⌊ard⌊lpF(f−1(zx1−zx2))/⌊ard⌊l2≤ /⌊ard⌊lf−1(zx1−zx2)/⌊ard⌊l2≤/⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊lzx1−zx2/⌊ard⌊l2
+≤δ1/2/⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊lf/⌊ard⌊l1·|F|1/2≤ε|F|1/2.
+Ifx1/\e}atio\slash=x2, then
+/⌊ard⌊lpF(f−1(zx1−zx2))/⌊ard⌊l2=/⌊ard⌊lpF((x′
+1−x′
+2)−(hx1−hx2))/⌊ard⌊l2≥dF,2(x1,x2)|F|1/2>ε|F|1/2,
+which is a contradiction. Therefore Wcontains at most one point. Thus
+|ψ−1(a)|≤Cλ|F||W|≤Cλ|F|,
+as desired. /squareENTROPY AND DETERMINANT 15
+For an abelian group G, denote by Gtorthe subgroup of torsion elements. If
+f∈ZΓ andfFis invertible for some nonempty finite subset Fof Γ, thenfFZ[F]
+has rank|F|, and hence Z[F]/fFZ[F] is a finite group. In the case, we shall apply
+the following result.
+Lemma 5.5. Letf∈ZΓbe invertible in LΓ. Then for any λ >1, there is some
+δ >0such that for any nonempty finite subset F⊆Γsatisfying|KfF\F|≤δ|F|
+we have
+Cλ|F|sF,∞(1
+2/⌊ard⌊lf/⌊ard⌊l1)≥|(Z[F]/fFZ[F])tor|,
+whereCis the universal constant in Lemma 5.1.
+Proof.WriteKforKf. SetD= 4/⌊ard⌊lf/⌊ard⌊l1andε= 2D−1. Takeδ >0 such that
+(D·/⌊ard⌊lf/⌊ard⌊l·/⌊ard⌊lf−1/⌊ard⌊l)δ≤λ1/2, and that δsatisfies the conclusion of Lemma 5.1 for
+λ′=λ1/2. LetFsatisfy the hypothesis.
+DenoteZ[F]/fFZ[F] byG. Letx∈Gtor. Take ˜x∈Z[F] such that the image of
+˜xinGunder the quotient map Z[F]→Gis equal tox. Then
+k˜x=fFw
+for some positive integer kand somew∈Z[F]. Write1
+kwasw1+w2for some
+w1∈Z[F] andw2∈[0,1)F. Then ˜x=fFw1+fFw2and/⌊ard⌊lfFw2/⌊ard⌊l2≤/⌊ard⌊lf/⌊ard⌊l·/⌊ard⌊lw2/⌊ard⌊l2≤
+/⌊ard⌊lf/⌊ard⌊l·|F|1/2. Note that ˜ xandfFw2have the same image in G. Thus we may replace
+˜xbyfFw2and hence assume that /⌊ard⌊l˜x/⌊ard⌊l2≤/⌊ard⌊lf/⌊ard⌊l·|F|1/2.
+Denote by ϕthe quotient map R[[Γ]]→(R/Z)[[Γ]]. We identify C[F] with a
+subspace of ℓ2(Γ) naturally. For any x∈Gtor, we have
+fϕ(f−1˜x) =ϕ(f(f−1˜x)) =ϕ(˜x) = 0
+in (R/Z)[[Γ]], and hence ϕ(f−1˜x)∈Xfby (5). This defines a map ψ:Gtor→Xf
+sendingxtoϕ(f−1˜x).
+For eachx∈Gtor, pickwx∈1
+DZ[KF\F] such that
+/⌊ard⌊lwx−pKF\F(f−1˜x)/⌊ard⌊l∞≤1/D=ε/2
+and|wx(t)|≤|(f−1˜x)(t)|for allt∈KF\F. ThenDwx∈Z[KF\F] and
+/⌊ard⌊lDwx/⌊ard⌊l2≤D/⌊ard⌊lpKF\F(f−1˜x)/⌊ard⌊l2≤D·/⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊l˜x/⌊ard⌊l2≤D·/⌊ard⌊lf/⌊ard⌊l·/⌊ard⌊lf−1/⌊ard⌊l·|F|1/2.
+By Lemma 5.1 one has
+|{Dwx:x∈Gtor}| ≤Cλ|F|/2(D·/⌊ard⌊lf/⌊ard⌊l·/⌊ard⌊lf−1/⌊ard⌊l)|KF\F|
+≤Cλ|F|/2(D·/⌊ard⌊lf/⌊ard⌊l·/⌊ard⌊lf−1/⌊ard⌊l)δ|F|
+≤Cλ|F|.
+Thus we can find a subset W⊆GtorwithCλ|F||W|≥|Gtor|such thatwx=wyfor
+allx,y∈W.16 HANFENG LI
+Now it suffices to show that ψinjectsWinto an [F,∞,ε]-separated subset of Xf.
+Suppose that x/\e}atio\slash=yinWanddF,∞(ψ(x),ψ(y))≤ε. From the definition of dF,∞we
+have
+dF,∞(ψ(x),ψ(y)) = max
+γ∈Fϑ((αf)γ(ψ(x)),(αf)γ(ψ(y)))
+= max
+γ∈Fϑ((ψ(x))γ,(ψ(y))γ).
+For eachγ∈F, one gets
+min
+m∈Z|(f−1˜x)γ−(f−1˜y)γ−m|=ϑ((ψ(x))γ,(ψ(y))γ)≤ε,
+and thus there exists hγ∈Zwith|(f−1˜x)γ−(f−1˜y)γ−hγ|≤ε. Defineh∈Z[F] to
+be the element with value hγfor everyγ∈F. Set
+z=f−1˜x−f−1˜y−h∈R[[Γ]].
+Then/⌊ard⌊lz|F/⌊ard⌊l∞≤ε. Sincexandyare inW, we havewx=wyand hence
+/⌊ard⌊lz|KF\F/⌊ard⌊l∞=/⌊ard⌊lpKF\F(f−1˜x)−pKF\F(f−1˜y)/⌊ard⌊l∞
+≤ /⌊ard⌊lpKF\F(f−1˜x)−wx/⌊ard⌊l∞+/⌊ard⌊lpKF\F(f−1˜y)−wy/⌊ard⌊l∞≤ε.
+Writezasz1+z2such that the supports of z1andz2are contained in KFand
+Γ\KFrespectively. Note that pF(fz) =pF(fz1) and/⌊ard⌊lz1/⌊ard⌊l∞≤ε. Consequently,
+/⌊ard⌊lpF(fz)/⌊ard⌊l∞=/⌊ard⌊lpF(fz1)/⌊ard⌊l∞≤/⌊ard⌊lfz1/⌊ard⌊l∞≤/⌊ard⌊lf/⌊ard⌊l1·/⌊ard⌊lz1/⌊ard⌊l∞≤ε/⌊ard⌊lf/⌊ard⌊l1= 1/2.
+We have
+˜x−˜y=pF(˜x−˜y) =pF(fh)+pF(fz) =fFh+pF(fz).
+Since ˜x−˜yandfFhare both in Z[F], we must have pF(fz) = 0. Therefore ˜ x−˜y=
+fFh∈fFZ[F], contradicting the assumption x/\e}atio\slash=y. This finishes the proof of the
+lemma. /square
+Theorem 5.6. LetΓbe an infinite amenable group and let f∈ZΓbe positive and
+invertible in LΓ. Let{Fn}n∈Jbe a (left) Følner net of Γ. Then for any 1/(2/⌊ard⌊lf/⌊ard⌊l1)≥
+ε>0, one has
+h(αf) = lim
+n→∞1
+|Fn|logsFn,∞(ε) = lim
+n→∞1
+|Fn|log|Z[Fn]/fFnZ[Fn]|
+= lim
+n→∞1
+|Fn|log|detfFn|= logdet LΓf.
+Proof.By Theorem 4.2 and Lemma 5.4, one has
+h(αf)≤liminf
+n→∞1
+|Fn|log|Z[Fn]/fFnZ[Fn]|.ENTROPY AND DETERMINANT 17
+ByLemma5.2,each fFnisinvertibleandhence( Z[Fn]/fFnZ[Fn])tor=Z[Fn]/fFnZ[Fn].
+Thus by Theorem 4.2 and Lemma 5.5, one has
+h(αf)≥limsup
+n→∞1
+|Fn|logsFn,∞(ε)≥limsup
+n→∞1
+|Fn|log|Z[Fn]/fFnZ[Fn]|,
+and
+liminf
+n→∞1
+|Fn|logsFn,∞(ε)≥liminf
+n→∞1
+|Fn|log|Z[Fn]/fFnZ[Fn]|.
+Then the first two equalities of the theorem follow. The last two equa lities of the
+theorem come from Lemma 5.2. /square
+6.Addition Formulas
+In this section we establish addition formulas for the entropy of gro up extensions,
+in both topological and measure-theoretical settings (Theorems 6.1 and 6.2). From
+these formulas we deduce the Yuzvinski˘ ı addition formula (Corolla ry 6.3) and use
+it to obtain a formula for the entropy of products fg(Corollaries 6.5 and 6.6).
+Throughout this section Γ is a countable amenable group.
+LetαX,αYandαGbe actions of Γ on compact metrizable spaces X,YandG
+by homeomorphisms respectively. A factor map X→Yis a continuous surjective
+Γ-equivariant map. We say that αXis a(right)G-extension of αYif there are a
+factor map π:X→Yand a continuous map P:X×G→Xsending (x,g) to
+xgsuch thatπ−1(π(x)) =xG,xg=xg′only when g=g′, andγ(xg) =γ(x)γ(g)
+for allx∈X,g,g′∈Gandγ∈Γ. (Usually Gis a compact metrizable group,
+(xg)g′=x(gg′), and Γ acts on Gby automorphisms; but this is not necessary.) The
+case Γ = Zof the following theorem was proved by R. Bowen [8, Theorem 19].
+Theorem 6.1 (Topological Addition Formula) .LetαX,αYandαGbe actions of Γ
+on compact metrizable spaces X,YandGby homeomorphisms respectively. If αX
+is aG-extension of αY, thenhtop(αX) = htop(αY)+htop(αG).
+LetαYbe an action of Γ on a standard probability space ( Y,BY,µ) by auto-
+morphisms. Also let αGbe an action of Γ on a compact metrizable group Gas
+(continuous) automorphisms. Endow Gwith its Borel σ-algebraBGand normalized
+Haar measure ν. Note that every automorphism of Gpreservesν. AcocycleforαY
+andαGis a measurable map σ: Γ×Y→Gsuch that
+σ(γ1γ2,y) =σ(γ1,γ2y)·γ1(σ(γ2,y)) (9)
+for allγ1,γ2∈Γ andy∈Y. Given a cocycle σ, one can define a skew product
+actionαY×σαGof Γ on the standard probability space ( Y×G,BY×BG,µ×ν) by
+automorphisms, by
+γ(y,g) = (γy,σ(γ,y)·(γg)) (10)
+forγ∈Γ,y∈Yandg∈G. It is clear that the projection Y×G→Yis a
+factor map for the actions αY×σαGandαYin the sense that it is Γ-equivariant,18 HANFENG LI
+measurable andmeasure-preserving. Theaction αY×σαGis calleda group extension
+of the action αY. The case Γ = Zof the following theorem was proved by Thomas
+[76], and the case Γ = Zdfor 2≤d <∞was proved by Lind et al. [42, Theorem
+B.1].
+Theorem 6.2 (Measure-theoretical Addition Formula) .LetαYandαGbe actions
+ofΓon a standard probability space (Y,BY,µ)and a compact metrizable group G
+by automorphisms respectively. Let σbe a cocycle for αYandαG. Then
+hµ×ν(αY×σαG) = hµ(αY)+h(αG).
+As a direct consequence of Theorem 6.1 we obtain the following Yuzvin ski˘ ı ad-
+dition formula, for which the case Γ = Zwas proved by Yuzvinski˘ ı [81] and the
+case Γ = Zdfor 2≤d<∞was proved by Lind et al. [42, Corollary B.2] (see also
+[71, Theorem 14.1]). The case Γ = Z∞andGis abelian was proved by Miles [53,
+Proposition 5.1]. The case Γ is locally normal and Gis abelian and zero-dimensional
+was proved by Miles and Bj¨ orklund [54, Theorem 3.1].
+Corollary 6.3 (Yuzvinski˘ ı Addition Formula) .LetαG1,αG2andαG3be actions
+ofΓon compact metrizable groups G1,G2,G3as (continuous) automorphisms re-
+spectively. Suppose that there is a Γ-equivariant short exact sequence of compact
+groups
+1−→G1−→G2−→G3−→1.
+Thenh(αG2) = h(αG1)+h(αG3).
+One can also obtain Corollary 6.3 from Theorem 6.2 via a standard proc edure, as
+follows.
+Proof of Corollary 6.3 using Theorem 6.2. We may identify G1with its image in
+G2. Denote by πthe mapG2→G3. Every continuous open surjective map between
+compact metrizable spaces has a Borel cross section [1, Theorem 3 .4.1]. Thus we
+can find a Borel map ψ:G3→G2such thatπ◦ψis the identity map on G3. It
+is easily verified that the map φ:G3×G1→G2sending (g3,g1) toψ(g3)g1is an
+isomorphism from the measurable space ( G3×G1,BG3×BG1) onto the measurable
+space (G2,BG2). Furthermore, denoting the normalized Haar measure on Gjbyνj,
+one sees that φ(ν3×ν1) is left-translation invariant and hence φ(ν3×ν1) =ν2. It is
+also readily checked that the map σ: Γ×G3→G1defined byσ(γ,g3) = (ψ(γg3))−1·
+γ(ψ(g3)) is a cocycle for the actions αG3andαG1, and thatφintertwines the actions
+αG3×σαG1andαG2. Thus h(αG2) = hν3×ν1(αG3×σαG1). Theorem 6.2 implies that
+hν3×ν1(αG3×σαG1) = hν3(αG3) + h(αG1). Therefore, h( αG2) = h(αG3) + h(αG1) as
+desired. /square
+Now we use Corollary 6.3 to obtain a formula for the entropy of fg. Recall that
+an element bof a ringRis called a right zero divisor ifab= 0 for some non-zero
+elementaofR. The following result was pointed out by Deninger [13, page 757].
+For the convenience of the reader, we give a proof here.ENTROPY AND DETERMINANT 19
+Lemma 6.4. Letf,g∈ZΓ. Then one has a Γ-equivariantshort sequenceof compact
+groups
+1−→Xg−→Xfg−→Xf→1,
+where the homomorphism Xfg→Xfis given by left multiplication by g. It is exact
+atXgandXfg. If furthermore gis not a right zero divisor of ZΓ, then the above
+sequence is exact.
+Proof.The dual sequence of the above one is the following
+0←−ZΓ/ZΓg←−ZΓ/ZΓfg←−ZΓ/ZΓf←−0, (11)
+where the homomorphism ZΓ/ZΓfg←ZΓ/ZΓfis given by right multiplication by
+g. By the Pontryagin duality it suffices to show that (11) is exact at th e correspond-
+ing places. Clearly it is exact at ZΓ/ZΓgandZΓ/ZΓfg. Now assume that gis not
+a right zero divisor of ZΓ. Suppose that x∈ZΓ/ZΓfandxg= 0 inZΓ/ZΓfg. Say,
+xis represented by ˜ xinZΓ. Then ˜xg= ˜zfginZΓ for some ˜ z∈ZΓ. Sincegis not
+a right zero divisor in ZΓ, we have ˜ x= ˜zfinZΓ. Consequently, x= 0 and hence
+(11) is also exact at ZΓ/ZΓf. /square
+Ifαis an action of Γ on a compact Hausdorff space Xby homeomorphisms, and
+Yis a closed invariant subspace of X, thenαrestricts to an action βof Γ onY, and
+from the definition of topological entropy one can see easily that h top(α)≥htop(β).
+Combining this fact with Corollary 6.3 and Lemma 6.4 we obtain the followin g
+product formula.
+Corollary 6.5. Letf,g∈ZΓ. Thenh(αfg)≤h(αf)+ h(αg). If furthermore gis
+not a right zero divisor in ZΓ, thenh(αfg) = h(αf)+h(αg).
+Thezero divisor conjecture states that for any torsion-free group H, the group
+ringZHhas no nontrivial right zero divisors. See [48, page 376–379] and [5 6, page
+62–63] for relation between the zero divisor conjecture and othe r conjectures such
+as the (strong) Atiyah conjecture and the embedding conjectur e. Recall that the
+class ofelementary amenable groups is the smallest class of groups containing all
+cyclic and all finite groups and being closed under taking group exten sions and
+direct unions. Because of Linnell’s work on the strong Atiyah conjec ture [45] (see
+also [18, 69]), we know that the zero divisor conjecture holds for all torsion-free
+groups in the smallest class of groups containing all free groups and being closed
+under extensions with elementary amenable quotients and under dir ect unions. In
+particular, the zero divisor conjecture holds for all torsion-free elementary amenable
+groups. See also [62, Chapter 13] for work on the zero divisor pro blem ofKHfor a
+fieldKand a group H.
+Iff= 0 inZΓ, thenαfis the full shift action of Γ on TΓand hence h( αf) =∞.
+Thus we have
+Corollary 6.6. Suppose that Γis torsion-free and satisfies the zero divisor conjec-
+ture. Then for any f,g∈ZΓ, one has h(αfg) = h(αf)+h(αg).20 HANFENG LI
+R. Bowen’s proof of Theorems 6.1 in the case Γ = Zis purely topological, while
+the proofs of Thomas and Lind et al. for Theorem 6.2 in the case Γ = Zdis
+purely using ergodic theory and depends on a technique of Yuzvinsk i˘ ı reducing G
+to simpler compact groups. Our proof for these addition formulas, in each setting,
+employ both topological and measure-theoretical tools. There ar e two main tools
+used in our proof. One is Ward and Zhang’s addition formula [79, Theo rem 4.4] (see
+also [12, Theorem 0.2]), a generalization of the Abramov-Rohlin additio n formula.
+Anotheristhevariouskindsoffibreentropyfortopologicalexten sions. Inparticular,
+our proof of Theorem 6.2, even in the case Γ = Zd, is completely different from that
+of Thomas and Lind et al.
+The rest of this section is devoted to the proofs of Theorems 6.1 an d 6.2. Fix a
+(left) Følner sequence {Fn}n∈Nof Γ.
+A systematic study of various fibre and conditional entropies was c arried out in
+[19] for dynamical systems of continuous maps on compact Hausdo rff spaces. It
+will be interesting to see to what extent the results in [19] generaliz e to actions of
+discrete amenable groups. Here we confine ourselves to extend a f ew definitions and
+results in [19] to Γ-actions, needed for the proofs of Theorems 6.1 and 6.2.
+LetαXbe an action of Γ on a compact metrizable space Xby homeomorphisms.
+Denote by MΓ(X) the set of all Γ-invariant Borel probability measures on X. For
+any finite open cover UofXand any subset Z⊆X, denote by N(U|Z) the minimal
+number of elements in Uneeded to cover Z. SetUF:=/logicalortext
+γ∈Fγ−1Ufor a nonempty
+finite subset Fof Γ, and
+htop(αX,U|Z) := limsup
+n→∞1
+|Fn|logN(UFn|Z).
+LetαYbe an action of Γ on another compact metrizable space Yby homeomor-
+phisms. Consider a factor map π:X→Y. Given a finite open cover UofX, note
+that the function y/ma√sto→N(U|π−1(y)) fory∈Yis upper semicontinuous and hence is
+a Borel function. Let ν∈MΓ(Y). Set
+H(U|ν) :=/integraldisplay
+YlogN(U|π−1(y))dν(y).
+It is easy to verify that the function F/ma√sto→H(UF|ν) defined on the set of nonempty fi-
+nitesubsetsofΓsatisfiesthehypothesisinProposition2.1andhenc elimn→∞1
+|Fn|H(UFn|ν)
+exists and does not depend on the choice of the Følner sequence {Fn}n∈N.
+Definition 6.7. LetUbe a finite open cover of X. Fory∈Y, we define the topolog-
+ical fibre entropy of Ugivenyas htop(αX,U|π−1(y)) and denote it by h top(αX,U|y).
+Foranyν∈MΓ(Y),wedefinethe topologicalfibre entropyof Ugivenνaslimn→∞1
+|Fn|H(UFn|ν)
+and denote it by h top(α,U|ν). We define the topological fibre entropy of αXgiveny,
+andgivenν, respectively, as supUhtop(αX,U|y) and supUhtop(αX,U|ν) respectively
+for the supremum being taken over all finite open covers of X, and denote them by
+htop(αX|y) and h top(αX|ν) respectively.ENTROPY AND DETERMINANT 21
+The following result is the analogue of part of [19, Theorem 3].
+Lemma 6.8. LetαXandαYbe actions of Γon compact metrizable spaces Xand
+Yrespectively. Let π:X→Ybe a factor map. Then we have
+sup
+y∈Yhtop(αX|y)≥sup
+ν∈MΓ(Y)htop(αX|ν).
+Proof.It suffices to prove supy∈Yhtop(αX,U|y)≥htop(αX,U|ν) for every finite open
+coverUofXandeveryν∈MΓ(Y). Since the function y/ma√sto→N(UF|π−1(y))is a Borel
+function on Yfor any nonempty finite subset Fof Γ, the function y/ma√sto→htop(αX,U|y)
+is also Borel. Note that
+1
+|F|logN(UF|π−1(y))≤1
+|F|logN(UF)≤logN(U) (12)
+for any nonempty finite subset Fof Γ andy∈Y. Thus
+sup
+y∈Yhtop(αX,U|y)≥/integraldisplay
+Yhtop(αX,U|y)dν(y)
+=/integraldisplay
+Ylim
+n→∞sup
+m≥n1
+|Fm|logN(UFm|π−1(y))dν(y)
+= lim
+n→∞/integraldisplay
+Ysup
+m≥n1
+|Fm|logN(UFm|π−1(y))dν(y)
+≥lim
+n→∞sup
+m≥n1
+|Fm|/integraldisplay
+YlogN(UFm|π−1(y))dν(y)
+= lim
+n→∞sup
+m≥n1
+|Fm|H(UFm|ν)
+= htop(αX,U|ν),
+where the third lines comes from Lebesgue’s monotone convergenc e theorem [66,
+Theorem 1.26] and the uniform upper bound in (12). /square
+The factor map π:X→Yinduces a surjective continuous affine map from the
+spaceM(X) of Borel probability measures on XtoM(Y). For any ν∈MΓ(Y),
+takeµ′∈M(X) withπ(µ′) =νand letµbe a limit point of the sequence
+{1
+|Fn|/summationtext
+γ∈Fnγµ′}n∈Ninthecompactspace M(X). ThenµisinMΓ(X)andπ(µ) =ν.
+Thus
+π(MΓ(X)) =MΓ(Y). (13)
+Note thatπ−1(BY) is a Γ-invariant sub- σ-algebra of BX. We shall identify BY
+withπ−1(BY), and write H µ(·|π−1(BY)) and h µ(·|π−1(BY)) simply as H µ(·|BY) and
+hµ(·|BY) respectively.
+The following result is the analogue of part of [19, Theorem 4].22 HANFENG LI
+Lemma 6.9. Let the assumptions be as in Lemma 6.8. For any ν∈MΓ(Y), we
+have
+htop(αX|ν)≥sup
+µ∈MΓ(X),πµ=νhµ(αX|BY).
+Proof.We combine the ideas in the proofs of [19, Theorem 4] and [57, Theore m
+5.2.8]. Let µ∈MΓ(X) withπ(µ) =ν. LetP={P1,...,P k}be a finite Borel
+partition of Xand letε>0. It suffices to show that there exists a finite open cover
+UofXsuch that h µ(αX,P|BY)≤htop(αX,U|ν)+ε.
+We may assume that min 1≤i≤kµ(Pi)>0. Letδbe a small positive constant which
+we shall determine later. Since µis regular [32, Theorem 17.11], we may find an
+open setUi⊇Pifor each 1≤i≤ksuch thatµ(Ui\Pi)<δ. ThenU={U1,...,U k}
+is an open cover of X.
+LetFbe a nonempty finite subset of Γ. Define an equivalence relation ∼onYas
+y∼y′wheneverπ−1(y) andπ−1(y′) are covered by exactly the same subfamilies of
+UF. Denote by βthe finite partition of Yinto the equivalence classes. It is readily
+verified that each item of βis the intersection of a closed set and an open set, and
+hence is Borel. For each D∈βwe can find some VD⊆UFsuch that VDcovers
+π−1(D) and|VD|=N(UF|π−1(y)) for every y∈D. It is easy to construct a Borel
+partition QD={QD,R:R∈VD}ofπ−1(D) withQD,R⊆Rfor eachR∈VD. Set
+QR:=/uniontext
+D∈βQD,RforR∈/uniontext
+D∈βVD. ThenQ:={QR:R∈/uniontext
+D∈βVD}is a Borel
+partition of X. For any finite Borel partition P′ofX, denote by /hatwideP′theσ-algebra
+generated by the items of P′. Note that for any m-item Borel partition P′ofX, one
+has Hµ(P′)≤logm[77, page 80]. Thus
+Hµ(Q|ˆβ)≤/summationdisplay
+D∈βν(D)log|VD|=/integraldisplay
+YlogN(UF|π−1(y))dν(y) = H(UF|ν). (14)
+We say that a finite partition P′ofXisadaptedto a finite open cover U′ofX
+if there is an injective (not necessarily surjective) map ψfromP′toU′such that
+eachP∈P′is contained in ψ(P). Denote by Rµ(U′) the supremum of H µ(P′|/hatwideQ′)
+for all Borel partitions P′andQ′ofXadapted to U′. By [57, Prop. 5.2.11] one has
+Rµ(U′∨V′)≤Rµ(U′)+Rµ(V′) for all finite open covers U′andV′ofX. Note that
+bothPFandQare adapted to UFand hence
+Hµ(PF|ˆQ)≤Rµ(UF)≤|F|Rµ(U). (15)
+For two sub- σ-algebras B1andB2ofBX, denote by B1∨B2the sub-σ-algebra of
+BXgenerated by B1andB2. We have
+Hµ(PF|BY)≤Hµ(PF∨Q|BY)
+= H µ(Q|BY)+Hµ(PF|ˆQ∨BY)
+≤Hµ(Q|ˆβ)+Hµ(PF|ˆQ)ENTROPY AND DETERMINANT 23
+(14),(15)
+≤H(UF|ν)+|F|Rµ(U).
+Dividebothsidesoftheaboveinequality by |F|, replaceFbyFnandtakelimits. We
+obtain h µ(αX,P|BY)≤htop(αX,U|ν)+Rµ(U). It remains to show that Rµ(U)≤ε
+whenδis small enough.
+We may assume that δ <1
+kmin1≤i≤kµ(Pi). Then the sum of the µ-measures of
+the elements in any proper subset of Uis strictly less than 1. It follows that every
+Borel partition of Xadapted to Uhas exactly kitems. Let P′={P′
+1,...,P′
+k}and
+Q′={Q′
+1,...,Q′
+k}be Borel partitions of Xadapted to UwithP′
+i,Q′
+i⊆Uifor each
+1≤i≤k. By [57, Lemma 4.3.9] one has H µ(P′|/hatwideQ′)≤2k2ξ(2d(P′,Q′)/k2), where
+d(P′,Q′) :=1
+2/summationtext
+1≤i≤kµ(P′
+i△Q′
+i) andξ(t) := max 0≤s≤t(−slogs) for 0≤t≤1. Note
+that/summationdisplay
+1≤i≤kµ(P′
+i\Q′
+i)≤/summationdisplay
+1≤i≤kµ(Ui\Q′
+i) =/summationdisplay
+1≤i≤k(µ(Ui)−µ(Q′
+i))
+=/summationdisplay
+1≤i≤kµ(Ui)−1 =/summationdisplay
+1≤i≤k(µ(Ui)−µ(Pi))
+=/summationdisplay
+1≤i≤kµ(Ui\Pi)<kδ.
+Similarly,/summationtext
+1≤i≤kµ(Q′
+i\P′
+i)< kδ. It follows that d(P′,Q′)< kδ. ThusRµ(U)≤
+2k2ξ(2δ/k). Therefore it suffices to require further ξ(2δ/k)≤ε/(2k2). /square
+The case Γ = Zof the next theorem was proved by R. Bowen [8, Theorem 17].
+Our proof for the general case takes the approach in [19].
+Theorem 6.10. Let the assumptions be as in Lemma 6.8. We have
+htop(αX)≤htop(αY)+sup
+y∈Yhtop(αX|y).
+Proof.By Theorem 0.2 of [12], when Γ is infinite, we have
+hµ(αX) = hπµ(αY)+hµ(αX|BY) (16)
+for everyµ∈MΓ(X). If Γ is finite and αZis an action of Γ on a standard
+probability space ( Z,BZ,µZ) by automorphisms and Dis a Γ-invariant sub- σ-
+algebra of BZ, then clearly h µZ(αZ|D) =H(µZ|D)
+|Γ|, whereH(µZ|D) denotes the
+supremum of HµZ(P|D) forPrunning over all finite measurable partitions of Z.
+IfµZis purely atomic in the sense that/summationtext
+z∈ZµZ({z}) = 1, then H(µZ|{∅,Z}) =/summationtext
+z∈Z−µZ({z})logµZ({z}). IfµZis not purely atomic, then there is some Z′∈BZ
+withµZ(Z′)>0 such that Z′equipped with the restriction of BZandµZis iso-
+morphic to the interval [0 ,µZ(Z′)] equipped with the Borel structure of its canonical
+topology and the Lebesgue measure [32, Theorem 17.41], and hence H(µZ|{∅,Z}) =
+∞. It follows easily that the formula (16) holds also when Γ is finite.24 HANFENG LI
+By the variational principle [57, page 76] we have h top(αX) = supµ∈MΓ(X)hµ(αX)
+and h top(αY) = supν∈MΓ(Y)hν(αY). Thus Theorem 6.10 follows from Lemmas 6.8
+and 6.9, and (13). /square
+Fix a compatible matric donX. For anyε>0 and any nonempty finite subset
+F⊆Γ, we saythataset E⊆Xis(F,ε)-separated ifforanyx/\e}atio\slash=yinEthereissome
+γ∈Fwithd(γx,γy)>εand we say that a set E′⊆X(F,ε)-spansanother subset
+Z⊆Xif for anyx∈Zthere is some y∈E′withd(γx,γy)≤εfor allγ∈F.
+For anyZ⊆X, denote by rF(ε,Z) the smallest cardinality of any set Ewhich
+(F,ε)-spansZand denote by sF(ε,Z) the largest cardinality of any ( F,ε)-separated
+setEcontained in Z.
+It is routine to prove the following lemma (cf. [8, Lemma 1] [13, Prop 2.1]).
+Lemma 6.11. LetαXbe an action of Γon a compact metrizable space Xby home-
+omorphisms. For any Z⊆X, we have
+sup
+Uhtop(αX,U|Z) = lim
+ε→0limsup
+n→∞1
+|Fn|logrFn(ε,Z) = lim
+ε→0limsup
+n→∞1
+|Fn|logsFn(ε,Z),
+where the supremum is taken over all finite open covers of X.
+We are ready to prove Theorem 6.1.
+Proof of Theorem 6.1. If Γ is finite and αZis an action of Γ on a compact Hausdorff
+spaceZby homeomorphisms, then clearly h top(αZ) =log|Z|
+|Γ|whenZis finite while
+htop(αZ) =∞whenZis infinite. It follows that Theorem 6.1 holds when Γ is finite.
+Thus we may assume that Γ is infinite. We follow the proof of [8, Theore m 19], but
+using Theorem 6.10 and Lemma 6.11.
+Fix compatible metrics dX,dYanddGforX,YandGrespectively. To show
+htop(αX)≤htop(αY)+ htop(αG), by Theorem 6.10 it suffices to show h top(αX|y)≤
+htop(αG) for every y∈Y. Takez∈π−1(y). Givenε >0, takeδ >0 such
+thatdX(xg1,xg2)≤εfor anyx∈Xandg1,g2∈GwithdG(g1,g2)≤δ. Let
+Fbe a nonempty finite subset of Γ. If a subset EofG(F,δ)-spansG, thenzE
+(F,ε)-spanszG=π−1(y). ThusrF(ε,π−1(y))≤rF(δ,G). By Lemma 6.11 we get
+htop(αX|y)≤htop(αG) as desired.
+Next we show h top(αX)≥htop(αY)+htop(αG). Givenε >0, sinceXandGare
+compact and xg=xg′only wheng=g′, we can find δ>0 such that dX(x1,x2)>δ
+for anyx1,x2∈XwithdY(π(x1),π(x2))> ε, and that dX(xg1,xg2)> δfor any
+x∈Xandg1,g2∈GwithdG(g1,g2)> ε. LetFbe a nonempty finite subset of
+Γ. LetEYandEGbe subsets of YandGbeing (F,δ)-separated respectively. Take
+EX⊆Xsuch that the restriction of πonEXmapsEXbijectively to EY. We
+claim that|EXEG|=|EX|·|EG|and thatEXEGis (F,δ)-separated. If x1,x2are
+distinct points in EXandg1,g2∈EG, thenπ(x1),π(x2)∈EYare distinct, thus
+for someγ∈Fone hasdY(π(γ(x1g1)),π(γ(x2g2))) =dY(γπ(x1),γπ(x2))> εand
+hencedX(γ(x1g1),γ(x2g2))> δ. Ifg1,g2are distinct points in EGandx∈EX,ENTROPY AND DETERMINANT 25
+then for some γ∈Fone hasdG(γ(g1),γ(g2))> εand hencedX(γ(xg1),γ(xg2)) =
+dX(γ(x)γ(g1),γ(x)γ(g2))>δ. Thisprovestheclaim. Thus sF(δ,X)≥sF(ε,Y)sF(ε,G).
+By Lemma 6.11 we get h top(αX)≥htop(αY)+htop(αG) as desired. /square
+LetXbeaG-extensionof Y. InthesecondparagraphoftheproofofTheorem6.1,
+we have proved that h top(αX|y)≤htop(αG) for every y∈Y. The argument in the
+third paragraph of the proof also shows that h top(αX|y)≥htop(αG) for everyy∈Y.
+For later use, we record this as
+Lemma 6.12. Let the assumptions be as in Theorem 6.1. Then htop(αX|y) =
+htop(αG)for everyy∈Y.
+Next we consider group extensions constructed out of continuou s cocycles.
+Lemma 6.13. LetαYandαGbe actions of Γon a compact metrizable space Yand
+a compact metrizable group Gby homeomorphisms and (continuous) automorphisms
+respectively. Let σ: Γ×Y→Gbe a continuous cocycle, i.e. a continuous map
+satisfying (9). Consider the action αY×σαGofΓon the compact metrizable space
+Y×Gby homeomorphisms, defined by (10). For any µ∈MΓ(Y), denoting by νthe
+normalized Haar measure of G, we have
+hµ×ν(αY×σαG|BY) = h(αG).
+Proof.Note thatαY×σαGis aG-extension of αYandπ(µ×ν) =µ, whereπdenotes
+the projection Y×G→Y. From Lemmas 6.8, 6.9 and 6.12 we have
+hµ×ν(αY×σαG|BY)≤htop(αY×σαG|µ)≤sup
+y∈Yhtop(αY×σαG|y) = h(αG).
+Thus it suffices to show h µ×ν(αY×σαG|BY)≥h(αG).
+Take compatible metrics dYanddGonYandGrespectively. Replacing dG(·,·) by/integraltext
+GdG(g·,g·)dν(g) if necessary, we may assume that dGis left-translation invariant.
+WeendowY×Gwiththemetric dY×G((y1,g1),(y2,g2)) = max(dY(y1,y2),dG(g1,g2)).
+Letε>0 andFbe a nonempty finite subset of Γ. Let Ebe an (F,ε)-separated
+subset ofGwith|E|=sF(ε,G). SetV={g∈G: maxγ∈FdG(γg,eG)≤ε/2}, where
+eGdenotestheidentityelement of G. ThenVisaclosedsubset of G, andthesets gV
+forg∈Eare pairwise disjoint. Thus 1 ≥ν(/uniontext
+g∈EgV) =/summationtext
+g∈Eν(gV) =|E|ν(V).
+Thereforeν(V)≤|E|−1.
+LetPbe a finite Borel partition of Y×Gwith each item having diameter no
+bigger than ε/2, underdY×G. LetPbe an item of PF, and let (y,g1),(y,g2)∈P.
+Then for each γ∈F, one has
+ε/2≥dY×G(γ(y,g1),γ(y,g2))
+=dY×G((γy,σ(γ,y)(γg1)),(γy,σ(γ,y)(γg2)))
+=dG(σ(γ,y)(γg1),σ(γ,y)(γg2))
+=dG(γg1,γg2)) =dG(γ(g−1
+1g2),eG),26 HANFENG LI
+where the last two equalities come from the left-translation invarian ce ofdG. Thus
+g−1
+1g2∈Vand henceg2∈g1V. It follows that
+E(1P|BY)(x) =/integraldisplay
+G1P(π(x),g′)dν(g′)≤ν(V)≤|E|−1
+forµ×νa.e.x∈Y×G, where1 Pdenotesthecharacteristic functionof P. Therefore
+Hµ×ν(PF|BY) =/summationdisplay
+P∈PF/integraldisplay
+Y×G−1P(x)logE(1P|BY)(x)d(µ×ν)(x)
+≥/summationdisplay
+P∈PF/integraldisplay
+Y×G−1P(x)log|E|−1d(µ×ν)(x) = log|E|= logsF(ε,G).
+Itfollowsthath µ×ν(αX×σαG,P|BY)≥limsupn→∞1
+|Fn|logsFn(ε,G). ByLemma6.11
+we get h µ×ν(αY×σαG|BY)≥h(αG) as desired. /square
+Now we show that every measure-theoretical group extension ha s a topological
+model.
+Lemma 6.14. Let the assumptions be as in Theorem 6.2. Then there exists a
+compact metrizable space Y′containingYsuch thatYis a dense Borel subset of Y′,
+BYis the restriction of BY′onY, the action of ΓonYextends to an action of Γ
+onY′by homeomorphisms, the measure µextends to a Γ-invariant Borel probability
+measure on Y′, andσextends to a continuous cocycle Γ×Y′→G.
+Proof.Denote by B(Y) the set of bounded C-valued Borel functions on Y. It is
+complete under the supremum norm /⌊ard⌊l·/⌊ard⌊l, and is a unital algebra under the pointwise
+addition and multiplication. Furthermore, it is a ∗-algebra with the ∗-operation
+defined byf∗(y) =f(y) forf∈B(Y) andy∈Y. It is clear that/⌊ard⌊lf∗f/⌊ard⌊l=/⌊ard⌊lf/⌊ard⌊l2for
+everyf∈B(Y). Thus B(Y) is a unital commutative C∗-algebra (see Section 2.2).
+NotethattheactionofΓon Yinduces anactionofΓon B(Y) asisometric∗-algebra
+automorphisms naturally.
+SinceBYis the Borel σ-algebra for some Polish topology on Y, we can find a
+countable subset WofBYseparating the points of Y. That is, for any distinct
+y1,y2inY, we can find A∈Wsuch that 1 A(y1)/\e}atio\slash= 1A(y2), where 1 Adenotes the
+characteristic function of A. SetV1={1A∈B(Y) :A∈W}.
+Note that the algebra C(G) of continuous C-valued functions on Gis also a
+normed space under the supremum norm. Since Gis compact metrizable, C(G)
+is separable. Write σasσγ:Y→Gforγ∈Γ. That is, σγ(y) =σ(γ,y) for
+γ∈Γ andy∈Y. Thenf◦σγis inB(Y) for every f∈C(G) andγ∈Γ. Set
+V2={f◦σγ∈B(Y) :f∈C(G),γ∈Γ}. SinceC(G) is separable and Γ is
+countable,V2is a separable subset of B(Y).
+Denote by Athe closed Γ-invariant sub- ∗-algebra of B(Y) generated by V1∪V2.
+Then Ais separable and contains the constant functions. Denote by Y′theGelfand
+spectrum ofA, i.e., the set of all unital algebra homomorphisms A→C[10, pageENTROPY AND DETERMINANT 27
+219]. Note that Y′is contained in the unit ball of the Banach space dual A′ofA
+[10, PropositionVII.8.4]. Endowed withtherelativeweak∗-topology,Y′isacompact
+Hausdorff space [10, Proposition VII.8.6]. Since Ais separable, Y′is metrizable.
+Clearly the action of Γ on Ainduces an action of Γ on Y′by homeomorphisms.
+For eachy∈Y, the evaluation at ygives rise to an element ψ(y) ofY′. SinceW
+separates the points of Y, the mapψ:Y→Y′is injective. Consider the Gelfand
+transformϕ:A→C(Y′) defined by ϕ(a)(y′) =y′(a) fora∈Aandy′∈Y′
+[10, page 220]. Note that Ais a unital commutative C∗-algebra. Thus ϕis an
+isometric∗-isomorphism of AontoC(Y′) [10, Theorem VIII.2.1]. Also note that
+ϕ(f)◦ψ=ffor everyf∈A. It follows that ψis measurable and Γ-equivariant.
+Recall that a measurable space ( X,BX) is called a standard Borel space ifBXis the
+Borelσ-algebra for some Polish topology on X. The Lusin-Souslin theorem says
+that for any injective measurable map ζfrom one standard Borel space ( X,BX) to
+another standard Borel space ( Z,BZ), the image ζ(X) is measurable and ζis an
+isomorphism from ( X,BX) to (ζ(X),BZ|ζ(X)) [32, page 89]. Thus, identifying Y
+withψ(Y), we haveY∈BY′andBYis the restriction of BY′onY. Thenµcan be
+though of as a Borel probability measure on Y′via settingµ(Y′\Y) = 0. Clearly µ
+is still Γ-invariant. Note that Yseparatesϕ(A) =C(Y′). By the Urysohn lemma
+[33, page 115], for any disjoint nonempty closed subsets Z1andZ2ofY′, there exists
+f∈C(Y′) withf|Z1= 1 andf|Z2= 0. It follows that Yis dense in Y′.
+Eachγ∈Γ and each y′∈Y′give rise to a unital algebra homomorphism C(G)→
+Csendingftoy′(f◦σγ). Note that every unital algebra homomorphism C(G)→C
+is given by the evaluation at a unique point of G[32, Theorem VII.8.7]. Thus there
+is a unique point in G, denoted by σ′
+γ(y′), such that f(σ′
+γ(y′)) =y′(f◦σγ) for every
+f∈C(G). Clearly the map σ′
+γ:Y′→Gis continuous and extends σγfor every
+γ∈Γ. Writeσ′(γ,y′) forσ′
+γ(y′). SinceYis dense in Y′, by continuity σ′also
+satisfies the cocycle condition (9). /square
+We are ready to prove Theorem 6.2.
+Proof of Theorem 6.2. By Lemmas 6.14 and 6.13 we have h µ×ν(αY×σαG|BY) =
+h(G). Thus the desired formula follows from Ward and Zhang’s addition fo rmula
+hµ×ν(αY×σαG) = hµ(αY) + hµ×ν(αY×σαG|BY) [79, Theorem 4.4] [12, Theorem
+0.2]. /square
+7.Approximation of Fuglede-Kadison Determinant
+Throughout thissection Γwill beadiscrete amenablegroup. Aswepo intedoutat
+the end of Section 3, one of the main difficulties to establish the intuitiv e equalities
+(6) is thatfFnmay fail to be invertible even when fis invertible in LΓ. Our method
+of dealing with this difficulty is to “perturb” fFnto make it invertible. Here the
+meaning of Sn∈B(C[Fn]) being a perturbation of fFnis that rank( Sn−fFn) is
+small compared to |Fn|. Our task in the section is to calculate det LΓfin terms of28 HANFENG LI
+the determinants of Sn. Though Corollary 7.2 gives a precise formula for such a
+calculation, for some technical reason which will be explained in Remar k 8.2, we
+have to get an approximate formula as follows:
+Theorem 7.1. Letf∈CΓbe invertible in LΓ. For any C1>0andε >0,
+there exists δ >0such that if{Fn}n∈Jis a (left) Følner net of ΓandSn∈
+B(C[Fn])is invertible for each n∈Jsuch that supn∈Jmax(/⌊ard⌊lSn/⌊ard⌊l,/⌊ard⌊lS−1
+n/⌊ard⌊l)≤C1
+andlimsupn→∞rank(Sn−fFn)
+|Fn|≤δ, then
+limsup
+n→∞/vextendsingle/vextendsinglelogdet LΓf−1
+|Fn|log|detSn|/vextendsingle/vextendsingle<ε.
+Proof.Letδ >0 be a small number whose value will be determined later. Let
+{Fn}n∈Jand{Sn}n∈Jsatisfy the hypothesis.
+Note that supn∈Jmax(/⌊ard⌊lS∗
+nSn/⌊ard⌊l,/⌊ard⌊l(S∗
+nSn)−1/⌊ard⌊l)≤C2
+1. Thus there is a closed finite
+intervalIinRdepending only on C1such thatIdoes not contain 0 and the spectra
+off∗fandS∗
+nSnare contained in Ifor eachn∈J.
+Letn∈J. From
+S∗
+nSn−(fFn)∗fFn= (Sn−fFn)∗Sn+(fFn)∗(Sn−fFn)
+we have
+rank(S∗
+nSn−(fFn)∗fFn)≤rank((Sn−fFn)∗Sn)+rank((fFn)∗(Sn−fFn))
+≤rank((Sn−fFn)∗)+rank(Sn−fFn)
+= 2rank(Sn−fFn).
+Recall the operators pFnandιFnin Notation 3.3 and the set Kfin Notation 5.3.
+When restricted on ℓ2(Γ), one has pFn= (ιFn)∗. Thus
+(fFn)∗fFn−(f∗f)Fn= (fFn)∗fFn−pFnf∗fιFn
+= (fFn)∗fFn−(fιFn)∗fιFn
+= (fFn−fιFn)∗fFn+(fιFn)∗(fFn−fιFn),
+and hence
+rank((fFn)∗fFn−(f∗f)Fn)≤rank((fFn−fιFn)∗fFn)+rank((fιFn)∗(fFn−fιFn))
+≤rank((fFn−fιFn)∗)+rank(fFn−fιFn)
+= 2rank(fFn−fιFn)
+≤2|KfFn\Fn|.
+Therefore
+rank(S∗
+nSn−(f∗f)Fn)≤rank(S∗
+nSn−(fFn)∗fFn)+rank((fFn)∗fFn−(f∗f)Fn)
+≤2rank(Sn−fFn)+2|KfFn\Fn|.ENTROPY AND DETERMINANT 29
+It follows that
+limsup
+n→∞rank(S∗
+nSn−(f∗f)Fn)
+|Fn|≤limsup
+n→∞2rank(Sn−fFn)
+|Fn|≤2δ.
+Denote by tr the trace of B(C[Fn]) taking value 1 on minimal projections. By the
+Weierstrass approximation theorem [66, page 312] we can find a rea l polynomial Q
+such that|Q(x)−logx|≤ε/2 for allx∈I. Then
+1
+|Fn||tr(Q(S))−tr(logS)|≤/⌊ard⌊lQ(S)−logS/⌊ard⌊l≤ε/2 (17)
+for all self-adjoint S∈B(C[Fn]) with spectrum contained in I, and
+/⌊ard⌊ltrLΓ(Q(T))−trLΓ(logT)/⌊ard⌊l≤/⌊ard⌊lQ(T)−logT/⌊ard⌊l≤ε/2 (18)
+for all self-adjoint T∈LΓ with spectrum contained in I.
+Fornoncommutativevariables XandY,wehaveQ(X+Y) =Q(X)+/summationtextk
+j=1Qj(X,Y)
+for some two-variable noncommutative monomials QjwithYappearing in Qj. Fix
+1≤j≤k. Then supn∈J/⌊ard⌊lQj((f∗f)Fn,S∗
+nSn−(f∗f)Fn)/⌊ard⌊l≤Djfor some constant Dj
+depending only on Qj,/⌊ard⌊lf/⌊ard⌊landC1. Furthermore,
+limsup
+n→∞rank(Qj((f∗f)Fn,S∗
+nSn−(f∗f)Fn))
+|Fn|≤limsup
+n→∞rank(S∗
+nSn−(f∗f)Fn)
+|Fn|≤2δ.
+For anyS∈B(C[Fn]), extending an orthonormal basis e1,...,e rank(S)of the range
+ofSto an orthonormal basis e1,...,e |Fn|ofC[Fn], one sees that erank(S)+1,...,e |Fn|
+are orthogonal to the range of S, and hence
+|tr(S)|=||Fn|/summationdisplay
+j=1/a\}⌊ra⌋ketle{tSej,ej/a\}⌊ra⌋ketri}ht|=|rank(S)/summationdisplay
+j=1/a\}⌊ra⌋ketle{tSej,ej/a\}⌊ra⌋ketri}ht|≤rank(S)/summationdisplay
+j=1|/a\}⌊ra⌋ketle{tSej,ej/a\}⌊ra⌋ketri}ht|≤rank(S)·/⌊ard⌊lS/⌊ard⌊l.
+It follows that limsupn→∞|tr(Qj((f∗f)Fn,S∗
+nSn−(f∗f)Fn))|
+|Fn|≤2δDj, and hence
+limsup
+n→∞1
+|Fn||tr(Q(S∗
+nSn))−tr(Q((f∗f)Fn))|≤2δD,
+whereD=/summationtextk
+j=1Dj. By a result of L¨ uck and Schick [47] [70, Lemma 4.6] [48,
+Lemma 13.42] [13, page 745], for any T∈LΓ one has
+trLΓ(Q(T)) = lim
+n→∞1
+|Fn|tr(Q(TFn)).
+Thus
+limsup
+n→∞/vextendsingle/vextendsingletrLΓ(Q(f∗f))−1
+|Fn|tr(Q(S∗
+nSn))/vextendsingle/vextendsingle≤2δD. (19)
+Combining (17), (18) and (19) together, we get
+limsup
+n→∞/vextendsingle/vextendsingletrLΓ(log(f∗f))−1
+|Fn|tr(log(S∗
+nSn))/vextendsingle/vextendsingle≤ε+2δD.30 HANFENG LI
+That is,
+limsup
+n→∞/vextendsingle/vextendsinglelogdet LΓ(f∗f)−1
+|Fn|logdet(S∗
+nSn)/vextendsingle/vextendsingle≤ε+2δD.
+As logdet LΓ(f∗f) = 2logdet LΓfand det(S∗
+nSn) =|detSn|2, we get
+limsup
+n→∞/vextendsingle/vextendsinglelogdet LΓf−1
+|Fn|log|detSn|/vextendsingle/vextendsingle≤ε/2+δD.
+Now we just need to take δ <ε/(2D). /square
+Corollary 7.2. Letf∈CΓbe invertible in LΓ. Let{Fn}n∈Jbe a (left) Følner net of
+ΓandSn∈B(C[Fn])beinvertiblefor each n∈Jsuch that supn∈Jmax(/⌊ard⌊lSn/⌊ard⌊l,/⌊ard⌊lS−1
+n/⌊ard⌊l)<
+∞andlimn→∞rank(Sn−fFn)/|Fn|= 0. Then
+logdet LΓf= lim
+n→∞1
+|Fn|log|detSn|.
+8.Proof of h(αf)≥logdet LΓf
+In this section we show h( αf)≥logdet LΓffor anyf∈ZΓ invertible in LΓ
+(Lemma 8.5). Throughout this section Γ is a discrete amenable group .
+Forf∈CΓ, recall that Kfdenotes the union of the supports of fandf∗, and
+the identity of Γ. For a finite subset Fof Γ, we identify C[F] with a subspace of
+ℓ2(Γ) naturally. In particular, if F′⊆Fare finite subsets of Γ, then C[F] is the
+direct sum of C[F′] andC[F\F′].
+Lemma 8.1. Letf∈ZΓbe invertible in LΓ. Then for any λ >1andC1≥1,
+there is some δ >0such that, for any M≥1and any nonempty finite subsets
+F′⊆FofΓsatisfying|KfF\F|≤δ|F|and|F\F′|≤δ|F|, ifTFis a linear map
+C[F\F′]→C[F]withMTF(Z[F\F′])⊆Z[F]and/⌊ard⌊lTF/⌊ard⌊l≤C1so that the linear
+mapSF:C[F]→C[F]defined asfFonC[F′]andTFonC[F\F′]is invertible in
+B(C[F]), then
+Cλ|F|M|KfF\F|rF,∞(1
+8/⌊ard⌊lf/⌊ard⌊l1)≥|detSF|,
+whereCis the universal constant in Lemma 5.1.
+Proof.The proof is similar to that of Lemma 5.5. Write KforKf. SetD= 8/⌊ard⌊lf/⌊ard⌊l1
+andε=D−1. Take 1> δ >0 such that (2 D(/⌊ard⌊lf/⌊ard⌊l+C1)/⌊ard⌊lf−1/⌊ard⌊l)2δ≤λ1/2, and
+δ1/2≤/⌊ard⌊lf−1/⌊ard⌊l, and thatδ′= 2δsatisfies the conclusion of Lemma 5.1 for λ′=λ1/2.
+LetF,F′, andTFsatisfy the hypothesis.
+ConsiderS′
+F∈B(C[F]) defined as fFonC[F′] andMTFonC[F\F′]. Then
+detS′
+F=M|F\F′|detSFand/⌊ard⌊lS′
+F/⌊ard⌊l ≤ /⌊ard⌊lf/⌊ard⌊l+MC1≤(/⌊ard⌊lf/⌊ard⌊l+C1)M. Note that
+S′
+F(Z[F])⊆Z[F], and hence det S′
+F=|Z[F]/S′
+FZ[F]|by Lemma 3.1. Thus it
+suffices to show
+Cλ|F|M|KF\F′|rF,∞(ε)≥|Z[F]/S′
+FZ[F]|.ENTROPY AND DETERMINANT 31
+Letx∈Z[F]/S′
+FZ[F]. Take ˜x∈Z[F] such that the image of ˜ xinZ[F]/S′
+FZ[F]
+under the quotient map Z[F]→Z[F]/S′
+FZ[F] is equal to x. SinceS′
+Fis invertible,
+one has
+˜x=S′
+Fw
+for somew∈R[F]. Writewasw1+w2for somew1∈Z[F] andw2∈[0,1)F. Then
+˜x=S′
+Fw1+S′
+Fw2and
+/⌊ard⌊lS′
+Fw2/⌊ard⌊l2≤/⌊ard⌊lS′
+F/⌊ard⌊l·/⌊ard⌊lw2/⌊ard⌊l2≤(/⌊ard⌊lf/⌊ard⌊l+C1)M|F|1/2.
+Note that ˜xandS′
+Fw2have the same image in Z[F]/S′
+FZ[F]. Thus we may replace
+˜xbyS′
+Fw2and hence assume that /⌊ard⌊l˜x/⌊ard⌊l2≤(/⌊ard⌊lf/⌊ard⌊l+C1)M|F|1/2.
+Denote by ϕthe quotient map R[[Γ]]→(R/Z)[[Γ]]. For each x∈Z[F]/S′
+FZ[F],
+one has
+fϕ(f−1˜x) =ϕ(f(f−1˜x)) =ϕ(˜x) = 0
+in(R/Z)[[Γ]], andhence ϕ(f−1˜x)∈Xfby(5). Thisdefinesamap ψ:Z[F]/S′
+FZ[F]→
+Xfsendingxtoϕ(f−1˜x).
+For eachx∈Z[F]/S′
+FZ[F], pickwx∈1
+DZ[KF\F] such that
+/⌊ard⌊lwx−pKF\F(f−1˜x)/⌊ard⌊l∞≤1/D=ε
+and|wx(t)|≤|(f−1˜x)(t)|for allt∈KF\F. ThenDwx∈Z[KF\F] and
+/⌊ard⌊lDwx/⌊ard⌊l2≤D/⌊ard⌊lpKF\F(f−1˜x)/⌊ard⌊l2≤D·/⌊ard⌊lf−1/⌊ard⌊l·/⌊ard⌊l˜x/⌊ard⌊l2≤D(/⌊ard⌊lf/⌊ard⌊l+C1)M/⌊ard⌊lf−1/⌊ard⌊l·|F|1/2.
+Take an [F,∞,ε]-spanning subset E⊆Xfwith|E|=rF,∞(ε). For each v∈E
+setWv={x∈Z[F]/S′
+FZ[F] :dF,∞(ψ(x),v)≤ε}. Then/uniontext
+v∈EWv=Z[F]/S′
+FZ[F].
+Now it suffices to show that
+|Wv|≤Cλ|F|M|KF\F′|
+for eachv∈E. Fixv∈Eandy∈Wv.
+Letx∈Wv. Then
+max
+γ∈Fϑ((ψ(x))γ,(ψ(y))γ) =dF,∞(ψ(x),ψ(y))≤dF,∞(ψ(x),v)+dF,∞(ψ(y),v)≤2ε.
+Foreachγ∈F′, take(hx)γ∈Zsuchthat|(f−1˜x)γ−(f−1˜y)γ−(hx)γ|≤2ε. Similarly,
+for eachγ∈F\F′, take (θx)γ∈Zsuch that|(f−1˜x)γ−(f−1˜y)γ−(θx)γ|≤2ε.
+Definehx∈Z[F′] to be the element taking value ( hx)γat eachγ∈F′. Also define
+θx∈Z[F\F′] to be the element taking value ( θx)γat eachγ∈F\F′. Set
+zx=f−1˜x−f−1˜y−hx−θx−wx+wy∈R[[Γ]]. (20)
+Then
+/⌊ard⌊lzx|F′/⌊ard⌊l∞=/⌊ard⌊l(f−1˜x−f−1˜y−hx)|F′/⌊ard⌊l∞≤2ε,
+and
+/⌊ard⌊lzx|F\F′/⌊ard⌊l∞=/⌊ard⌊l(f−1˜x−f−1˜y−θx)|F\F′/⌊ard⌊l∞≤2ε,32 HANFENG LI
+and
+/⌊ard⌊lzx|KF\F/⌊ard⌊l∞=/⌊ard⌊l(f−1˜x−f−1˜y−wx+wy)|KF\F/⌊ard⌊l∞
+≤/⌊ard⌊l(f−1˜x−wx)|KF\F/⌊ard⌊l∞+/⌊ard⌊l(f−1˜y−wy)|KF\F/⌊ard⌊l∞≤2ε,
+and thus
+/⌊ard⌊lzx|KF/⌊ard⌊l∞≤2ε.
+It follows that
+/⌊ard⌊lθx/⌊ard⌊l2≤ /⌊ard⌊lf−1˜x/⌊ard⌊l2+/⌊ard⌊lf−1˜y/⌊ard⌊l2+/⌊ard⌊lpF\F′(zx)/⌊ard⌊l2
+≤2(/⌊ard⌊lf/⌊ard⌊l+C1)M/⌊ard⌊lf−1/⌊ard⌊l·|F|1/2+(2ε)δ1/2|F|1/2
+≤D(/⌊ard⌊lf/⌊ard⌊l+C1)M/⌊ard⌊lf−1/⌊ard⌊l·|F|1/2.
+Notethatθx+Dwx∈Z[KF\F′]with/⌊ard⌊lθx+Dwx/⌊ard⌊l2≤2D(/⌊ard⌊lf/⌊ard⌊l+C1)M/⌊ard⌊lf−1/⌊ard⌊l·|F|1/2
+and|KF\F′|≤2δ|F|=δ′|F|. By Lemma 5.1 one has
+|{θx+Dwx:x∈Wv}| ≤Cλ|F|/2(2D(/⌊ard⌊lf/⌊ard⌊l+C1)M/⌊ard⌊lf−1/⌊ard⌊l)|KF\F′|
+≤Cλ|F|/2(2D(/⌊ard⌊lf/⌊ard⌊l+C1)/⌊ard⌊lf−1/⌊ard⌊l)2δ|F|M|KF\F′|
+≤Cλ|F|M|KF\F′|.
+Thus we can find a subset W′
+v⊆WvwithCλ|F|M|KF\F′||W′
+v|≥|Wv|such that
+θx1+Dwx1=θx2+Dwx2forallx1,x2∈W′
+v. Sinceθx∈R[F\F′]andwx∈R[KF\F]
+for allx∈W′
+v, we haveθx1=θx2andwx1=wx2forx1,x2∈W′
+v.
+Now it suffices to show that |W′
+v|≤1. Suppose that x1/\e}atio\slash=x2inW′
+v. Applying
+(20) tox=x1andx=x2respectively, one gets
+f−1/tildewidex1−f−1/tildewidex2=hx1−hx2+zx1−zx2.
+Writezx1−zx2asz1+z2such that the supports of z1andz2are contained in KF
+and Γ\KFrespectively. Note that pF(f(zx1−zx2)) =pF(fz1) and/⌊ard⌊lz1/⌊ard⌊l∞≤4ε.
+Consequently,
+/⌊ard⌊lpF(f(zx1−zx2))/⌊ard⌊l∞=/⌊ard⌊lpF(fz1)/⌊ard⌊l∞≤/⌊ard⌊lfz1/⌊ard⌊l∞≤/⌊ard⌊lf/⌊ard⌊l1·/⌊ard⌊lz1/⌊ard⌊l∞≤4ε/⌊ard⌊lf/⌊ard⌊l1= 1/2.
+We have
+/tildewidex1−/tildewidex2=pF(/tildewidex1−/tildewidex2) =pF(f(hx1−hx2))+pF(f(zx1−zx2))
+=S′
+F(hx1−hx2)+pF(f(zx1−zx2)).
+Since/tildewidex1−/tildewidex2andS′
+F(hx1−hx2) are both in Z[F], we must have pF(f(zx1−zx2)) = 0.
+Therefore /tildewidex1−/tildewidex2=S′
+F(hx1−hx2)∈S′
+FZ[F], contradicting the assumption x1/\e}atio\slash=x2.
+This finishes the proof of the lemma. /square
+Remark 8.2. Note that in Lemma 8.1 the operator SFmay fail to preserve Z[F],
+while the norm/⌊ard⌊lS′
+F/⌊ard⌊lof the operator S′
+Fdefined in the 2nd paragraph of the proof
+of Lemma 8.1 may be large when Mgets large. Indeed, this is what happens when
+we construct Snin the proof of Lemma 8.5 below. A modification of the proofs of
+Lemmas 5.4, 5.5 and 8.1 shows that if f∈ZΓ is invertible in LΓ, and there areENTROPY AND DETERMINANT 33
+a (left) Følner net {Fn}n∈Jof Γ and an invertible Sn∈B(C[Fn]) preserving Z[Fn]
+for eachn∈Jsuch that supn∈Jmax(/⌊ard⌊lSn/⌊ard⌊l,/⌊ard⌊lS−1
+n/⌊ard⌊l)<∞and lim n→∞rank(Sn−
+fFn)/|Fn|= 0, then h( αf) = lim n→∞1
+|Fn|log|detSn|. Combined with Corollary 7.2,
+this proves Theorem 1.1 for such case, without using Theorem 5.6 an d Corollary 6.5.
+When Γ is also residually finite, Weiss showed that there are a net {Γn}n∈Jof finite
+index normal subgroups of Γ and a (left) Følner net {Fn}n∈Jof Γ such that the
+quotient map Γ→Γ/ΓnmapsFnbijectively to Γ /Γnfor eachn∈J[80, Section 2]
+(see also [17, Corollary 5.6]). Via taking Snto be the image of finC(Γ/Γn) and
+identifying B(ℓ2(Γ/Γn)) andB(C[Fn]), it is easily checked that when Γ is residually
+finite,{Fn}n∈Jand{Sn}n∈Jsatisfying the above conditions do exist. However, we
+have not been able to show the existence of such {Fn}n∈Jand{Sn}n∈Jin general.
+This is why we have to use Theorem 7.1, Lemma 8.5 and Ornstein and Weis s’s
+theory of quasitiling.
+Forε>0, we say that a family of finite subsets {F1,...,F m}of Γ areε-disjoint
+if there are F′
+j⊆Fjfor all 1≤j≤msuch thatF′
+1,...,F′
+mare pairwise disjoint,
+and|F′
+j|≥(1−ε)|Fj|for all 1≤j≤m. We need the following theorem of Ornstein
+and Weiss:
+Theorem 8.3. [60, page 24, Theorem 6] Let ε>0and letKbe a nonempty finite
+subset of Γ. Then there exist δ >0and nonempty finite subsets K′,F1,...,F mof
+Γ, such that
+(1)|{g∈Fj:Kg⊆Fj}|≥(1−ε)|Fj|for each 1≤j≤m;
+(2)for any nonempty finite subset FofΓsatisfying|K′F\F|≤δ|F|, there
+are finite subsets D1,...,D mofΓsuch that/uniontext
+1≤j≤mFjDj⊆F, the family
+{Fjc: 1≤j≤m,c∈Dj}of subsets of Γisε-disjoint, and|/uniontext
+1≤j≤mFjDj|≥
+(1−ε)|F|.
+Remark 8.4. In Theorem 8.3, choosing Fc,j⊆Fjfor every 1≤j≤mandc∈Dj
+such that|Fc,j|≥(1−ε)|Fj|for all 1≤j≤mandc∈Dj, and that the family
+{Fc,jc: 1≤j≤m,c∈Dj}of subsets of Γ is pairwise disjoint, and noticing that
+Fc,jis one element in the finite set {W⊆Fj:|W|≥(1−ε)|Fj|}, we see that we
+can actually require the family {Fjc: 1≤j≤m,c∈Dj}to be pairwise disjoint.
+Lemma 8.5. LetΓbe an infinite amenable group and let f∈ZΓbe invertible in
+LΓ. For any (left) Følner net {Fn}n∈JofΓ, one has
+liminf
+n→∞1
+|Fn|logrFn,∞(1
+8/⌊ard⌊lf/⌊ard⌊l1)≥logdetf.
+Proof.SetC1= max(/⌊ard⌊lf/⌊ard⌊l,/⌊ard⌊lf−1/⌊ard⌊l) + 2. Let λ >1 andε >0. Takeδ >0 working
+for both Theorem 7.1 and Lemma 8.1. Denote by Kthe union of the supports of f
+andf∗, and the identity of Γ.
+By Theorem 8.3 and Remark 8.4, there exist nonempty finite subsets W1,...,W m
+of Γ andN∈J, such that34 HANFENG LI
+(I)|W′
+j|≥(1−δ
+2)|Wj|for each 1≤j≤m, whereW′
+j={g∈Wj:Kg⊆Wj};
+(II) forany n≥N,therearefinitesubsets Dn,1,...,D n,mofΓsuchthat/uniontext
+1≤j≤mWjDn,j⊆
+Fn, the family{Wjc: 1≤j≤m,c∈Dn,j}of subsets of Γ is pairwise dis-
+joint, and|/uniontext
+1≤j≤mWjDn,j|≥(1−δ
+2)|Fn|.
+We may also assume that
+(III) for any n≥N, one has|KFn\Fn|≤δ|Fn|.
+We shall construct Tnfor eachn≥Nsatisfying the hypothesis in Lemma 8.1 for
+someMnot depending on nsuch that the associated Snsatisfies the hypothesis in
+Theorem 7.1. For this purpose, we shall construct TnonC[Wj] first, then transfer
+them toC[Fn].
+Fix 1≤j≤m. SinceKW′
+j⊆Wj, we have fWj=fonC[W′
+j]. Write
+(fC[W′
+j])⊥for the orthogonal complement of fC[W′
+j] inC[Wj]. Note that the
+dimension of ( fC[W′
+j])⊥is equal to|Wj\W′
+j|, and that ( fC[W′
+j])⊥is the linear
+span of (fC[W′
+j])⊥∩Q[Wj]. Identify Wjwith the standard orthonormal basis of
+C[Wj]. Take an orthonormal basis {eg:g∈Wj\W′
+j}of (fC[W′
+j])⊥, consisting
+of elements in R[Wj]. Takinge′
+g∈(fC[W′
+j])⊥∩Q[Wj] close enough to egfor all
+g∈Wj\W′
+j, we find that the linear map /tildewideTj:C[Wj\W′
+j]→(fC[W′
+j])⊥sendingg
+toe′
+gis bijective and max( /⌊ard⌊l/tildewideTj/⌊ard⌊l,/⌊ard⌊l/tildewideT−1
+j/⌊ard⌊l)≤2. Then there exists Mj∈Nsuch that
+Mj/tildewideTj(Z[Wj\W′
+j])⊆Z[Wj]. Note that the linear map /tildewideSj:C[Wj]→C[Wj] defined
+asfWjonC[W′
+j] and/tildewideTjonC[Wj\W′
+j] is invertible, and /⌊ard⌊l/tildewideS−1
+j/⌊ard⌊l≤/⌊ard⌊lf−1/⌊ard⌊l+2.
+SetM=/producttext
+1≤j≤mMj.
+Now letn≥N. LetDn,1,...,D n,mbe as in (II) above. Set F′
+n=/uniontext
+1≤j≤mW′
+jDn,j.
+Then|Fn\F′
+n|≤δ|Fn|. Next we define the desired linear map Tn:C[Fn\F′
+n]→
+C[Fn]. OnC[Fn\(/uniontext
+1≤j≤mWjDn,j)], the map Tnis the identity map. On C[(Wj\
+W′
+j)c] for 1≤j≤mandc∈Dn,j, the mapTnis the same as /tildewideTjonC[Wj\W′
+j], if we
+identifyC[Wj\W′
+j] andC[Wj] withC[(Wj\W′
+j)c] andC[Wjc] respectively via the
+right multiplication by c. ThenMTn(Z[Fn\F′
+n])⊆Z[Fn], and/⌊ard⌊lTn/⌊ard⌊l≤2. Denote
+bySnthe linear map C[Fn]→C[Fn] which is equal to fFnonC[F′
+n] and equal to Tn
+onC[Fn\F′
+n]. Clearly/⌊ard⌊lSn/⌊ard⌊l≤/⌊ard⌊lf/⌊ard⌊l+2. Note that the restriction of SnonC[Wjc]
+for each 1≤j≤mandc∈Dn,j, or onC[Fn\(/uniontext
+1≤j≤mWjDn,j)] is an isomorphism,
+and the norm of the inverse of this restriction is bounded above by /⌊ard⌊lf−1/⌊ard⌊l+2. Thus
+Snis invertible with /⌊ard⌊lS−1
+n/⌊ard⌊l≤/⌊ard⌊lf−1/⌊ard⌊l+2. By Lemma 8.1 we have
+Cλ|Fn|M|KFn\Fn|rFn,∞(1
+8/⌊ard⌊lf/⌊ard⌊l1)≥|detSn|,
+whereCis the universal constant in Lemma 5.1. Therefore
+liminf
+n→∞(1
+|Fn|logrFn,∞(1
+8/⌊ard⌊lf/⌊ard⌊l1)−1
+|Fn|log|detSn|)≥−logλ. (21)ENTROPY AND DETERMINANT 35
+SinceSnandfFncoincide on C[F′
+n], we have rank( Sn−fFn)≤|Fn\F′
+n|≤δ|Fn|.
+By Theorem 7.1 we get
+limsup
+n→∞/vextendsingle/vextendsinglelogdet LΓf−1
+|Fn|log|detSn|/vextendsingle/vextendsingle<ε. (22)
+Combining (21) and (22) we get
+liminf
+n→∞(1
+|Fn|logrFn,∞(1
+8/⌊ard⌊lf/⌊ard⌊l1)−logdet LΓf)≥−logλ−ε.
+Sinceλ>1 andε>0 are arbitrary, the lemma is proved. /square
+9.Proof of Theorem 1.1 and Consequences
+We are ready to prove Theorem 1.1.
+Proof of Theorem 1.1. By Theorem 3.2 we may assume that Γ is infinite. Let
+{Fn}n∈Nbe a (left) Følner sequence of Γ. By Theorem 4.2 and Lemma 8.5 we
+have
+h(αf)≥liminf
+n→∞1
+|Fn|logrFn,∞(1
+8/⌊ard⌊lf/⌊ard⌊l1)≥logdet LΓf.
+Applying the inequality to f∗, we also have
+h(αf∗)≥logdet LΓf∗.
+Then we have
+h(αf∗f) = h(αf∗)+h(αf)≥logdet LΓf∗+logdet LΓf= 2logdet LΓf
+= logdet LΓ(f∗f) = h(αf∗f),
+where the first equality comes from Corollary 6.5, the second one co mes from The-
+orem 2.2, the third one comes from the definition of det LΓf, and the last one comes
+from Theorem 5.6. Thus h( αf) = logdet LΓf. /square
+SincesF,∞(ε)≥rF,∞(ε) for any nonempty finite subset Fof Γ andε>0, in the
+proof of Theorem 1.1 we actually have proved the following result.
+Corollary 9.1. LetΓbe a countable amenable group and let f∈ZΓbe invertible
+inLΓ. For any1
+8/bardblf/bardbl1≥ε>0and any (left) Følner sequence {Fn}n∈NofΓ, one has
+h(αf) = lim
+n→∞1
+|Fn|logrFn,∞(ε) = lim
+n→∞1
+|Fn|logsFn,∞(ε).
+The follow result is a consequence of Theorems 1.1 and 2.2.
+Corollary 9.2. LetΓbe a countable amenable group and let f∈ZΓbe invertible
+inLΓ. Thenh(αf) = h(αf∗).
+The following result is a generalization of [17, Corollary 6.6], whose proo f we
+follow.36 HANFENG LI
+Corollary 9.3. LetΓbe a countable amenable group and let f,g∈ZΓbe invertible
+inLΓwith0≤f≤g. Thenh(αf)≤h(αg). Furthermore, h(αf) = h(αg)if and
+only iff=g.
+Proof.The inequality follows from Theorems 1.1 and 2.2. Suppose that h( αf) =
+h(αg). Seth= logg−logf. Then
+trLΓ(h) = trLΓlogg−trLΓlogf= logdet LΓg−logdet LΓf= h(αg)−h(αf) = 0.
+Note that the function log is operator monotone in the sense that for any invertible
+bounded positive operators T,Son a Hilbert space HwithT≤S, one has log T≤
+logS[46] [64, Page 10]. Thus h≥0. Since tr LΓis faithful and tr LΓh= 0, we get
+h= 0. Thusf=g. /square
+Appendix A.Comparison of Invertibility in ℓ1(Γ)andLΓ
+A Banach complex algebra Awith an operation ∗is called a Banach∗-algebra
+if (a∗)∗=a, (λa+b)∗=¯λa∗+b∗, (ab)∗=b∗a∗, and/⌊ard⌊la∗/⌊ard⌊l=/⌊ard⌊la/⌊ard⌊lfor alla,b∈A
+andλ∈C. Arepresentation of a Banach∗-algebraAon a Hilbert space His a
+∗-homomorphism π:A→B(H). A Banach∗-algebraAis called an A∗-algebra
+if it has an injective representation π. For anA∗-algebraA, there is a C∗-algebra
+C∗(A) and an injective ∗-homomorphism A ֒→C∗(A) with dense image such that
+every∗-homomorphism A→BofAinto aC∗-algebra Bextends to a unique ∗-
+homomorphism C∗(A)→B. TheC∗-algebraC∗(A) is unique up to isomorphism,
+and is called the enveloping C∗-algebra ofA[75, page 42]. Explicitly, the norm
+/⌊ard⌊l·/⌊ard⌊lC∗(A)ofC∗(A) is given by/⌊ard⌊la/⌊ard⌊lC∗(A)= supπ/⌊ard⌊lπ(a)/⌊ard⌊lfora∈A, whereπruns over
+all representations of A.
+A unital Banach∗-algebraAis called symmetric if for eacha∈A, the spectrum
+ofa∗ainAis contained in R≥0. It is well known that a unital A∗-algebraAis
+symmetric if and only if for each a∈A, the spectra of ainAandC∗(A) are the
+same. We recall briefly the reason here. The “if” part follows from t he fact that
+everyC∗-algebra is symmetric. Assume that Ais symmetric. By a result of Raikov
+[65] [58, page 308, Corollary 4], for every a∈Awitha∗=a, the spectral radius of a
+inAis equal to/⌊ard⌊la/⌊ard⌊lC∗(A). According to an observation of Hulanicki [27, Proposition
+2.5] (see also [22, Proposition 6.1]), for every a∈Awitha∗=a, the spectra of ain
+AandC∗(A) are the same. It follows that for every a∈A, the spectra of ainA
+andC∗(A) are the same (see for example [22, page 804]).
+Let Γ be a discrete (not necessarily amenable) group. Then ℓ1(Γ) is a unital Ba-
+nach∗-algebra with the algebraic operations extending those of CΓ. The embedding
+CΓ֒→LΓ extends to an injective representation ℓ1(Γ)֒→LΓ. Thusℓ1(Γ) is anA∗-
+algebra, and for every a∈ℓ1(Γ), ifais invertible in ℓ1(Γ), then it is invertible in LΓ.
+The enveloping C∗-algebra of ℓ1(Γ) is denoted by C∗(Γ) and is called the (maximal)
+groupC∗-algebraof Γ. The embedding ℓ1(Γ)֒→LΓ extends to a∗-homomorphism
+ψ:C∗(Γ)→LΓ. The group Γ is amenable if and only if ψis injective [63, TheoremENTROPY AND DETERMINANT 37
+4.21]. Thus, when Γ is amenable, ℓ1(Γ) is symmetric if and only if for any a∈ℓ1(Γ),
+the spectra of ainℓ1(Γ) andLΓ are the same.
+If Γ is a finite extensions of a discrete nilpotent group, then ℓ1(Γ) is symmetric
+[38, 49], and hence for any a∈ℓ1(Γ),ais invertible in ℓ1(Γ) if and only if it is
+invertible in LΓ.
+Nica showed that if Γ is a finitely generated group of subexponential growth, then
+for anya∈CΓ,ais invertible in ℓ1(Γ) if and only if it is invertible in LΓ [59, page
+3309].
+Jenkins [29, 30] showed thatif Γisa discrete groupcontaining two ele ments gener-
+ating afreesubsemigroup, then ℓ1(Γ)isnot symmetric. Under thesameassumption,
+Nica showed that there exist a∈CΓ which are invertible in LΓ but not invertible
+inℓ1(Γ) [59, Proposition 52]. In fact, in such case there exist a∈ZΓ which are
+invertible in C∗(Γ) (in particular, invertible in LΓ) but not invertible in ℓ1(Γ), as
+the following example shows. This example is inspired by the ideas in [30]. I am
+grateful to Jingbo Xia for very helpful discussion leading to this exa mple.
+Example A.1. Let Γ be a discrete group with elements γ1,γ2∈Γ generating a free
+subsemigroup. We claim that for every λ∈Cwith|λ|= 3, the element
+a=λeΓ−(eΓ+γ1−γ2
+1)γ2
+is invertible in C∗(Γ) but not invertible in ℓ1(Γ). Taking λ=±3, we geta∈ZΓ.
+The spectrum of γ1inC∗(Γ) is contained in the unit circle TofC. By the spectral
+theorem for unitaries,
+/⌊ard⌊leΓ+γ1−γ2
+1/⌊ard⌊lC∗(Γ)≤max
+z∈T|1+z−z2|<3.
+Then
+/⌊ard⌊l(eΓ+γ1−γ2
+1)γ2/⌊ard⌊lC∗(Γ)≤ /⌊ard⌊leΓ+γ1−γ2
+1/⌊ard⌊lC∗(Γ)·/⌊ard⌊lγ2/⌊ard⌊lC∗(Γ)
+=/⌊ard⌊leΓ+γ1−γ2
+1/⌊ard⌊lC∗(Γ)<3.
+It follows that ais invertible in C∗(Γ), and its inverse is given by
+λ−1∞/summationdisplay
+k=0λ−k((eΓ+γ1−γ2
+1)γ2)k.
+From the natural homomorphism C∗(Γ)→LΓ, we see that ais also invertible
+inLΓ with inverse bgiven by the above formula. Under the natural embedding
+LΓ→ℓ2(Γ),b=λ−1/summationtext∞
+k=0λ−k((eΓ+γ1−γ2
+1)γ2)k∈ℓ2(Γ). Sinceγ1andγ2generate
+a free subsemigroup, it is easily checked that the supports of (( eΓ+γ1−γ2
+1)γ2)kfor
+k≥0 are pairwise disjoint and /⌊ard⌊l((eΓ+γ1−γ2
+1)γ2)k/⌊ard⌊l1= 3kfor eachk≥0. It follows
+thatλ−1/summationtext∞
+k=0λ−k((eΓ+γ1−γ2
+1)γ2)k/\e}atio\slash∈ℓ1(Γ). Ifawere invertible in ℓ1(Γ), then its
+inverse inℓ1(Γ) would be band henceb∈ℓ1(Γ), which is a contradiction. Therefore
+ais not invertible in ℓ1(Γ).38 HANFENG LI
+There are discrete amenable groups which contain two elements gen erating free
+subsemigroups [26, 28]. Actually Frey showed that every discrete a menable group
+with nonamenable subsemigroups has such elements [23]. Also, Chou s howed that
+if a finitely generated elementary amenable group has no finite-index nilpotent sub-
+groups, then it contains such elements [9, Theorem 3.2’]. We recall th e examples in
+[28]. Consider the action of the multiplicative group R∗=R\{0}on the additive
+groupRby multiplication. One has the semi-direct product group R⋊R∗, which is
+R×R∗as a set and has multiplication ( s1,t1)·(s2,t2) = (s1+t1s2,t1t2). For any
+0≤a≤1/2, the subgroup Γ aofR ⋊ R∗generated by (1 ,a) and (1,−a) is 2-step
+solvable (and hence amenable), and (1 ,a) and (1,−a) generate a free subsemigroup
+in Γa.
+References
+[1] W.Arveson. An Invitation to C∗-algebras .GraduateTextsinMathematics, No.39.Springer-
+Verlag, New York, reprinted 1998.
+[2] K. R. Berg. Convolution of invariant measures, maximal entropy .Math. Systems Theory 3
+(1969), 146–150.
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+Department of Mathematics, SUNY at Buffalo, Buffalo, NY 142 60-2900, U.S.A.
+E-mail address :hfli@math.buffalo.edu
\ No newline at end of file
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+2 
+ 
+ The Role of Head-Up Display in Computer-
+Assisted Instruction 
+ 
+Kikuo Asai 
+National Institute of Multimedia Education 
+Japan 
+ 
+1. Introduction 
+ 
+The head-up display (HUD) creates a new form  of presenting information by enabling a 
+user to simultaneously view a real scene and superimposed info rmation without large 
+movements of the head or eye scans (New man, 1995; Weintraub and Ensing, 1992). HUDs 
+have been used for various applications such  as flight manipulation, vehicle driving, 
+machine maintenance, and sports, so that the users improve situational comprehension with 
+the real-time information. Recent downsizing of the display devices will expand the HUD 
+utilization into more new areas.  
+The head-mounted display (HMD) has been us ed as a head-mounted type of HUDs for 
+wearable computing (Mann, 1997) that gives a user situational information by wearing a 
+portable computer like clothes, a bag, and a wristwatch. A computer has come to interact intelligently with people based on the context of the situation with sensing and wireless communication systems.  
+One promising application is in computer-assisted instruction (CAI) (Feiner, et al., 1993) that supports the works such as equipment op eration, product assembly, and machine 
+maintenance. These works have witnessed th e introduction of increasingly complex 
+platforms and sophisticated procedures, and have required the instructional support. HUD-based CAI applications are characterized by real-time presentation of instructional 
+information related to what a user is looking at. It is commonly thought that HUD-based 
+CAI will increase productivity in instruction ta sks and reduce errors by  properly presenting 
+task information based on a user’s viewpoint.  
+However, there are not enough empirical studie s that show which factors of HUDs improve 
+user performance. A considerable amount of systematic research must be carried out in 
+order for HUD-based CAI to fulfill its potential to use the scene augmentation to improve 
+human-computer interaction. 
+We have developed a HUD-based CAI system that enables non-technical staff to operate the transportable earth station (Asai, et al., 2006). Although we observed that users of the HUD-
+based CAI system performed slightly better than users of conventional PCs and paper 
+manuals, it was not clear which factors signif icantly affected performance in operating the 
+system. We here conducted a laboratory experi ment in which participants performed a task 
+of reading articles and answering questions, in order to evaluate how readable the display Human-Computer Interaction, New Developments   32 
+of the HUD is, how easy it is to search information using the system, and how it affects the 
+work efficiency. We then disc uss the characteristics of HUD- based CAI, comparing the task 
+performance between the HUD-based CAI and conventional media.  
+Thus, this chapter is a study on the information processing behavior at an HUD, focusing on its role in CAI. Our aim is to introduce the basic concept and human factors of an HUD, 
+explain the features of HUD-based CAI, and show user performance with our HUD-based 
+CAI system. 
+ 
+2. HUD Technology 
+ 
+The HUD basically has an optical mechanism th at superimposes synthetic information on a 
+user’s field of view. Although the HUD is desi gned to allow a user to concurrently view a 
+real scene and superimposed information, its type depends on the application. We here categorize HUDs into three design types: he ad-mounted or ground-r eferenced, optical see-
+through or video see-through, and si ngle-sided or two-sided types.  
+ 
+2.1 Head-mounted and Ground-referenced Types 
+HUDs are categorized into the head-mounted  and ground-referenced types in terms of 
+spatial relationship between the head and HUD, as shown in Fig. 1.   
+In the head-mounted type (Fig. 1 (a)), an HUD is mounted on the head, being attached to a 
+helmet or a head band. It is generally calle d a head-mounted display (HMD). Since the HUD 
+is fixated to the head, a user can see visual information, even though moving the head. The 
+h ead-m ou n ted ty pe of  HUD  is u s ed  at th e  en vironment where users have to look around 
+them, such as building construction, surgical operation, and sports activities. The head-
+mounted HUD should be light in weight, because the user has to support its weight.  
+In the ground-referenced type (F ig. 1 (b)), an HUD is grounded to a desktop, wall, or floor. 
+Since the relation between the head and HUD is not fixated spatially, visual information can be viewed just in case that a user directs th e head to the HUD. The ground-referenced type 
+of HUD is used at the environment where users almost look at the same direction, such as 
+flight manipulation and vehicle driving. The user  does not need to support the weight in the 
+ground-referenced HUD.  
+ 
+        
+  
+(a) Head-mounted type  (b ) Ground-referenced type 
+Fig. 1. Spatial relation between the head and HUD  
+  The Role of Head-Up Display in Computer-Assisted Instruction  33 
+ 
+2.2 Optical See-through and Video See-through Types 
+HUDs are categorized into the optical see-thro ugh and video see-through types in terms of 
+optical mechanism, as shown in Fig. 2.  
+In the optical see-through type (Fig. 2 (a)), a user sees the re al scene through a half mirror 
+(transparent display) on which the synthetic im ages including graphics or text are overlaid. 
+The optical see-through HUD has advantages of  seeing the real scene without degradation 
+of the resolution and delay of the presentation. In addition, eye accommodation and 
+convergence responses work for the real scen e. However, the responses do not work for 
+virtual objects. That is, the real scene and the synthetic images are at different distances from 
+the user. Therefore, the user’s ey es need to alternately adjust to these distances in order to 
+perceive information in the both contexts. Freq uent gaze shifting to different depths may 
+result in eye strain (Neveu, et al., 1998). The optical see-through HUD does not also 
+represent occlusion correctly because the real sce ne goes through the half mirror at the pixel 
+area of the front virtual objects. One more pr oblem is of difficulty in use under a bright 
+illumination condition such as an outdoor field because of low luminance of the display.  
+In the video see-through type (Fig. 2 (b)), a real scene is captured by a camera. The user sees 
+the real scene images, in which information such as graphics or text is superimposed, at a 
+display monitor. The video see-through HUD has the following advantages; (1) the real 
+scene can be flexibly processed at the pixe l unit, making brightness control and color 
+correction, (2) there is no temporal deviation between the real scene and virtual objects because of their synchronous presentation in  the display image, and (3) the additional 
+information is obtained by using the captured scene, deriving depth information from parallax images and user’s position from th e geometric features. According to (1), the 
+occlusion is achieved by coveri ng the real scene with the virtual objects or culling the back 
+pixels out of the virtual objects. While, the video see-through HUD has shortcomings due to 
+losing rich information on the real scene. Lo w temporal and spatial resolution of the HUD 
+decreases the realistic and immersive sense of the real scene. The inconsistent focus-depth 
+information may result in high physiological load during the use. Despite (2), the video see-
+through HUD has presentation delay due to the image composition and rendering, which 
+may sometimes lead to a critical accident at the environment such as a construction site.  
+ 
+     
+  
+(a) Optical see-through type   (b) Video see-through type 
+Fig. 2. Optical difference between the HUDs  
+ 
+ Human-Computer Interaction, New Developments   34 
+2.3 Single-sided and Two-sided Types 
+HUDs are categorized into the si ngle-sided and two-sided types in  terms of the field of view 
+based on relation of the eyes and HUD, as show n in Fig. 3. Whether presenting the synthetic 
+images to one eye or two eyes is an important factor that dominates the range of applicable 
+areas.  
+In the single-sided type of HUD (Fig. 3 (a)), the real scene is  viewed by two eyes, and the 
+synthetic images are presented to one eye using an optical se e-through or small video see-
+through display. The real scene images captured by a video camera have a time lag to be displayed at the video see-through display. A single-sided HUD is used at the environment 
+where a user works looking at the peripheral situations or experience of the real world 
+proceeds acquisition of the complementary information. For example, the single-sided type 
+is usually used in a construction site due to sa fety reasons. When the synthetic images or the 
+device frames interfere largely with the user ’s field of view, vital accidents may occur 
+during the work.  
+In the two-sided type of HUD (F ig. 3 (b)), the real scene and sy nthetic images are viewed by 
+two eyes using an optical see-through or video see-through display. A two-sided HUD is used at the situation where safety of the us er is ensured without looking around, because 
+the cost of visual interference would be high  at the two-sided HMD, in which the overlaid 
+information interferes with the view of the wo rkspace. For example, the two-sided type is 
+often used at an entertainment situation becaus e of producing the feeling of being there.  
+There is a tradeoff relationship between the si ngle-sided and two-sided types in readability 
+of documents on the HUD and visibility of th e real scene via the HUD. The single-sided 
+HUD enables a user to easily see real objects using one eye with no occlusion, though the 
+user has to read documents using only one eye. On the other hand, the two-sided HUD 
+enables the user to read documents with both ey es, though the user has to view real objects 
+through the display on which the documents ar e presented. The single-sided HUD is more 
+convenient for acquiring information on real objects, and the two-sided HUD is more 
+convenient for acquiring information on the display.  
+In the head-mounted type, the weight of the HU D is an important factor for user’s comfort. 
+A single-sided HUD, in general, weighs le ss than the two-side d HUD. Although the 
+difference in weight is only 150 g for the HUDs, it turns out to be significant because the 
+device is attached to the head (Asai, et al ., 2005). The heavier the HUD is, the tighter the 
+HUD has to be placed on the head without being shifted, which may result in difficulty for a long-time use.  
+ 
+                 
+  
+(a) Single-sided type   (b) Two-sided type 
+Fig. 3. Relation between the eyes and HUD  
+  The Role of Head-Up Display in Computer-Assisted Instruction  35 
+ 
+3. Human Factors of HUDs 
+ 
+HUD systems have developed for improving performance of multiple tasks in aircraft 
+landing and vehicular driving. In the aircraft landing, the HUD system supports pilots to keep operation performance in navigating th rough a crowded airspace. In the vehicular 
+driving, the HUD supports drivers to keep dr iving performance in accessing information 
+from multiple sources such as speed, navigation, and accidents. Although numerous 
+information and communication tools have pr ovided a user with a large amount of 
+information, the user’s capacity to process information does not change.  
+There are many researches regarding the costs and benefits of HUDs compared with head-
+down displays (HDDs). The benefits of HUDs are mainly characterized by visual scanning 
+and re-accommodation. In the visual scanning, HUDs reduces the amount of eye scans and head movements required to monitor information and view the outside world (Haines, et al., 
+1980; Tufano, 1997). The traditional HDD causes  time sharing between the tasks. For 
+example, drivers must take their eyes off the ro ad ahead in order to read the status at the 
+control panel, which affects driving safety. The HUD degrades the problem because of 
+simultaneous viewing of the monitor inform ation and real scene. In the visual re-
+accommodation, HUDs reduces the adjustment time of refocusing the eyes required to monitor information and view the outside worl d (Larry and Elworth, 1972; Okabayashi, et 
+al., 1989). The HDD makes the user refocus the eyes frequently for viewing the closer and 
+far domains, which may cause fatigue. The HUD degrades the re-accommodation problem 
+by allowing the user to read the status without shifting focus largely in case being optically focused farther.  
+However, use of HUDs did not always improve us er performance in aviation safety studies, 
+especially when unexpected even ts occurred (Fischer, et al., 1980; Weintraub, et al., 1985). 
+The HUD users had a shorter response time than the HDD users to detect unexpected 
+events only in conditions of low workload. The benefits of HUDs, however, were reduced or even reversed in conditions of high workload  (Larish and Wickens, 19 91; Wickens, et al., 
+1993; Fadden, et al., 2001). Measurement of the time required to mentally switch between 
+the superimposed HUD symbology and the moving  real-world scene revealed that it took 
+longer to switch when there was differential motion between superimposed symbology in a 
+fixed place on the HUD and movement in th e real-world scene behind the symbology 
+caused by motion of the aircraft (McCann, et al., 1993). As a result, conformal imagery 
+(Wickens and Long, 1995) or scene-linked symbol ogy (Foyle, et al., 1995) that moved as the 
+real objects moved was configured on the HUD to reduce the time it takes to switch attention. The HUD can depict virtual information reconciled with physically viewable parts 
+in the real world. This conformal imagery or scene-linked symbology is based on the same 
+concept as spatial registration of virtual and real objects in augmented reality (AR) 
+technology (e.g., Milgram and Kishino, 1994; Azuma, 1997).  
+ 
+4. HUD-based CAI 
+ 
+CAI has been applied to maintenance and manufa cturing instructions in engineering (Dede, 
+1986), in which complex tasks must be performe d. CAI systems have helped new users learn 
+how to use devices by illustrating a range of functional capabilities of the device with 
+ Human-Computer Interaction, New Developments   36 
+multimedia content. However, the human-comput er interfaces were inadequate for eliciting 
+the potential of human performance, due to the limitations of the input/output devices, 
+including inconvenience of holding a CAI system while working and mismatch of 
+information processing between computer an d human. An HUD environment may make it 
+possible to improve human-computer interaction in CAI by allowing information to be 
+integrated into the real scene. 
+ 
+4.1 Applications 
+Typical examples of HUD-based CAI applications  are operation of equipment, assembly of 
+products, and maintenance of ma chines. Many systems have been developed as applications 
+of AR technology, including assistance and tr aining on new systems, assembly of complex 
+systems, and service work on plants and systems in industri al context (Friedrich, 2002; 
+Schwald and Laval, 2003).  
+In the service and maintenance, the work is expected to improve e fficiency by accessing 
+databases on-line and reduce human errors by  augmenting the real objects with visual 
+information such as annotations, data maps, and virtual models (Navab, 2004). As a solution, an online guided maintenance approach was ta ken for reducing necessity and dependency 
+on trained workers facing increasingly complex platforms and sophisticated maintenance procedures (Lipson, et al., 1998;  Bian, et a., 2006). It has potential to create a new quality of 
+remote maintenance by conditional instructions adjusting automatically to conditions at the 
+maintenance site, according to input informat ion from the machine and updated knowledge 
+at the manufacturer. The service and maintenance of nuclear power plants also require workers to minimize the time for diagnostics (troubleshooting and repair) and comply with 
+safety regulations for inspection of critical  subsystems. The context-based statistical pre-
+fetching component was implem ented by using document history as context information 
+(Klinker, et al., 2001). The pre-fetching compon ent records each document request that was 
+made by the application, and stores the identi fier of the requested document in a database. 
+The database entries and the time dependencies  are analyzed for prediction of documents 
+suitable for the current location and work of the mobile workers.  
+Early work at Boeing in the assembly proces s indicated the advantages of HUD technology 
+in assembling cable harnesses (Caudell and Mizell, 1992). Larg e and complex assemblies are 
+composed of parts, some of which are linked to gether to form subassemb lies. To identify the 
+efficient assembly sequence, en gineers evaluate whether the assembly operation is feasible 
+or difficult and edit the assembly plan. An  interactive evaluation tool using AR was 
+developed to attempt various sequencing alternatives of the manufacturing design process 
+(Sharma, et al., 1997; Raghavan, et al., 1999). On the other hand, an AR-based furniture 
+assembly tool was introduced for assemblers to be guided step-by-step in a very intuitive 
+and proactive way (Zauner, et al., 2003).  The authoring tool was also developed offering 
+flexible and re-configurable instructions. An AR environment allows engineers to design 
+and plan assembly process through manipulating  virtual prototypes at the real workplace, 
+which is important to identify the drawbacks an d revise the process. However, the revision 
+of the design and planning is time-consuming in the large-scale assembly process. Hierarchical feature-based models were applied updating the related feature models in 
+stead of the entire model. This results in computational simplicity offering a real-time 
+environment (Pang, et al., 2006)  
+  The Role of Head-Up Display in Computer-Assisted Instruction  37 
+ 
+4.2 Effects 
+Compared to conventional printed manuals, HUD-based CAI using an HMD has the 
+following advantages:  
+1) Hands-free presentation 
+An HMD presents information with a displa y mounted on the user's head. Therefore, 
+although cables are attached to supply electr ic power and data, both hands can freely be 
+used for a task. 
+2) Reduction of head and eye movements 
+Superimposing virtual objects onto real-world scenes enables users to view both virtual 
+objects and real scenes with less movement of the eyes and the head. Spatial separation of 
+the display beyond 20 deg. involves progressively larger head movements to access of visual 
+information (Previc, 2000), and information ac cess costs (effort required to access 
+information) increase as spatial se paration increases (Wickens, 1992).  
+3) Viewpoint-based interaction 
+Information related to the scenes detected by a camera attached to an HMD is presented to 
+the user. Unlike printed media, there is no n eed for the user to search for the information 
+required for a specific task. The user simply l ooks at an object, and th e pertinent information 
+is presented at the display. This triggering effect enables efficient retrieval of information with little effort by the user (Neumann and Majoros, 1998). 
+While many systems have been designed based on implicit assumptions that HUDs improve 
+user performance, little direct empirical evid ence concerning their effectiveness has been 
+collected. An inspection scenario was examined in three different conditions: an optical see-
+through AR, a web browser, and a traditional paper-based manual (Jackson, et al., 2001). 
+They found that the condition of the paper ma nual outperformed those of the others. In a 
+car door assembly, the experimental results showed that the task performance depended on 
+degree of difficulty on the assembly tasks (Wiedenmaier, et al., 2003). The AR condition 
+wearing an HMD was more suitable for the difficult tasks than the paper manual condition, whereas the performance had no significant di fference for the easy tasks between the two 
+conditions.  
+There has been an investigation of how effectively information is accessed in annotated assembly domains. The effectiveness of spatia lly-registered AR instructions was compared 
+to the other three instructions: a printed manu al, CAI on an LCD monitor, and CAI on a see-
+through HMD, in experiments on a Duplo-bloc k assembly (Tang, et al., 2003). The results 
+showed that the spatially-registered AR in structions improved task performance and 
+relieved mental workload on assembly tasks by overlaying and registering information to the workspace in a spatially meaningful way. 
+ 
+5. Case Study 
+ 
+We applied a HUD-based CAI to a support system for the operation of a transportable earth 
+station containing many pieces of equipment used in satellite communications (Tanaka and 
+Kondo, 1999). The transportable earth station was designed so that non-technical staff could 
+manage the equipment in cooperation with a technician at a hub station. However, 
+operating unfamiliar equipment was not easy for them, even though a detailed instruction 
+manual was available. One of the basic problems staff has during the operation was to 
+ Human-Computer Interaction, New Developments   38 
+understand what part of the instruction ma nual related to which equipment and then 
+figuring out the sequence of steps to carry out a procedure. Another problem was that the 
+transmission at a session leaves little room for error in operating the equipment because 
+mistakes of the transmission operation may give serious damage to the communication 
+devices of the satellite. 
+ 
+5.1 Transportable Earth Station 
+A transportable earth station has been construc ted as an extension of the inter-university 
+satellite network that is used for remote lectures, academic meetings, and symposia in 
+higher education, to exchange audio and video signals. The network now links 150 stations at universities and institutes. The transportable earth station has the same functionality as 
+the original stations on campus but can be transported throughout Japan. 
+Figure 4 (a) shows a photograph of the tr ansportable earth station. The van carries 
+transmitting-receiving devices, video-coding machines, a GPS-based automatic satellite-acquisition system, and various instruments fo r measurements. The operator has to manage 
+these pieces of equipment with the appropri ate procedures and perform the adjustments 
+and settings required for satellite communicati ons. The uplink access test involves the 
+operation of the transmitters and receivers show n in Figure 4 (b), and this requires some 
+specialized expertise and error-free performance. 
+ 
+    
+(a) Van               (b) Equipment 
+Fig. 4. Transportable earth station  
+ 
+5.2 HUD-based CAI System 
+Our HUD-based CAI system was originally designed to improve operation of the 
+transportable earth station. Here, the outline of the prototype system assumes that the 
+system will be used to operate the pieces of  equipment in the transportable earth station, 
+though the experiment described in the next section was done under laboratory conditions. 
+Figure 5 shows a schematic configuration of our prototype system. A compact camera attached to the user’s head captures images from the user’s viewpo int. These images are 
+  The Role of Head-Up Display in Computer-Assisted Instruction  39 
+transferred to a PC through a DV format. Identi fication (ID) patterns registered in advance 
+are stuck on the equipment, and each marker  is detected in the video images using 
+ARToolkit (Kato, et al., 2000), which is a C/OpenGL-based open-source library that detects 
+and tracks objects using square frames. 
+ 
+ 
+Fig. 5. System configuration  
+ 
+When the marker pattern is found in the regist ered list, ARToolkit identifies the piece of 
+equipment and the appropriate instructions are presented to the user via an HMD. At the 
+same time, the name of the equipment is stated audibly by a recorded voice to alert the user 
+and make sure he or she works on the right pi ece of equipment. A square marker centered 
+in or close to the center of the scene is dete cted, as several markers are present in the same 
+scene. A trackball is used to control pages by, for example, scrolling pages, sticking a page, 
+and turning pages. 
+ 
+5.3 Software Architecture 
+Figure 6 shows the software architecture of the prototype system. The software consists of 
+two parts that have a server-client relationship: image processing and display application. 
+The server and client exchange data using so cket communications with UDP/IP. The server-
+client architecture enables the load to be dist ributed to two processors, though the prototype 
+system was implemented on one PC.  
+It is common for graphical signs or simple icons to be presented with spatial registration to 
+real objects in assembly tasks. In our case, however, using such graphical metaphors is insufficient to represent the amount of inform ation because detailed operating instructions 
+for the equipment should be provided based on conditions. Such documents contain too 
+much information to be spatially registered with a single piece of equipment. Unfortunately, 
+the resolution and field of view are current ly limited in commercially available HMDs. 
+Therefore, we displayed the manual docume nts with large fonts, sacrificing spatial 
+registration to the real equipment. The lack of  spatial registration may not be a problem for 
+us because, unlike aircraft landing simulation, there is no differential motion in the 
+manipulation of the equipment and the background scene is stable. 
+ Human-Computer Interaction, New Developments   40 
+ 
+ 
+Fig. 6. Software architecture  
+ 
+Pages presented on the HMD are formatted in  HTML, which enables a Web browser to be 
+used to display multimedia data and makes in struction writing easier. Writing content for 
+this kind of application, which has usually required programming skills and graphics 
+expertise, is often costly and time consuming. Instructions must be customized to each piece of equipment and sometimes need to be revise d, which may greatly increase the workload. 
+The use of a Web browser enables fast implemen tation and flexible reconfiguration of the 
+component elements in  the instructions. 
+ 
+5.4 Implementation 
+The prototype system was implemented on a 2.8-GHz Pentium 4 PC with a 512-MB memory. 
+The video frame rate of the image processing  was roughly 30 frames per second. A single-
+sided HMD (SV-6, produced by Micro Optical) wa s installed as shown in Fig. 7, attaching a 
+compact camera. The HMD has a viewing angle of roughly 20 degrees in the diagonal and 
+the pinhole camera has a viewing angle of 43 degrees. The HMD, including the camera, 
+weighs 80 g. 
+ 
+   
+  
+Fig. 7. HMD with a camera     Fig. 8. Sample ID marker  
+ 
+  The Role of Head-Up Display in Computer-Assisted Instruction  41 
+Figure 8 shows a sample ID marker. When the surroundings are too dark, the camera 
+images become less clear, degrading marker detection and recognition accuracy. Although 
+the recognition rate depends on illumination, no recognition error ha s been observed at a 
+viewing distance within 80 cm when the illum ination environment is the same as that in 
+which the marker patterns were registered. 
+ 
+6. Experiment 
+ 
+We conducted an experiment that compared the performance of participants using a HUD 
+system, a laptop PC, and a paper manual. We  hypothesized that the HUD system would 
+make users receive the HUD profits (hands-fr ee environments, reduction of head and eye 
+movements, and awareness of real objects) an d difficulty viewing information on an HMD. 
+We expected that these would affect the time  required to perform tasks. Figure 9 shows 
+photos of the experiment being carried out: a) HMD system, b) laptop PC, and c) paper 
+manual, respectively. 
+ 
+   
+  
+Fig. 9. Experimental tasks (a: HMD system, b: laptop PC, and c: paper manual).  
+ 
+6.1 Method 
+Thirty-five people participated in the experi ment. The participants were undergraduate and 
+graduate students who had normal vision and no previous experience of HUD-based CAI. 
+The task was to read articles and answer qu estions about the articles. The questions were 
+multiple choices and had three possible answers each. Participants were instructed to read 
+20 articles with each instruction media. For each  article, participants were required to write 
+down the answer and the times when they found the answer possibilities and when they 
+finished answering the questions. The participan ts were instructed to complete the task as 
+quickly and accurately as possible. 
+ Human-Computer Interaction, New Developments   42 
+In the experiment, slips of paper with possible answers were taped to the wall in front of the 
+participant. In the HUD system, each possible answer slip has an ID  marker, and articles 
+and questions are presented on  the HMD. In the PC and th e paper manual, articles and 
+questions are presented on the PC mo nitor and paper sheets, respectively. 
+All three media used the same format to pres ent the articles, but there were differences 
+among the media in how the questions and an swers were presented. In the HUD system 
+and the PC, the title and marker number were di splayed in blue at the top of the page, and 
+the headings of the article were displayed in  black below the title. These headings were 
+presented in larger fonts on the HMD (24 pt) than on the PC monitor (12 pt). When the 
+participant centers the marker on  the display, the article is id entified, and the article’s top 
+page is presented on the HMD. While reading an  article, a participant presses a button of the 
+trackball to hold the page. 
+The time required to find and complete each article were recorded for all 20 articles. Finding 
+time is defined as the time it took the particip ant to find the possible  answers for a question, 
+and completion time is defined as the time it  took the participant to finish answering the 
+question. These times were recorded by the pa rticipants themselves using a stopwatch, and 
+participants were asked at a preference test about which media was the best for performing 
+the task. The questions ar e listed in Table 1. 
+ 
+ Question 
+1 The instruction medium was easy to use.  
+2 Practice was not necessary. 
+3 Looking for possible answers was easy. 
+4 Identifying the answer to the question was easy. 
+5 The task was not tiring. 
+6 The medium was enjoyable. 
+Table 1. Preference test questions 
+ 
+The participants were divided into three groups  and each group started performing the task 
+using one of the three different instruction medi a. There were three tr ials with each group 
+using each medium once. For example, if a pa rticipant began with the HUD system, he or 
+she would use the laptop PC in the second tria l and the paper manual in the third trial. The 
+preference test was conducted immediately afte r participants had finished all three trials. 
+For the HUD system to work properly, the po sition of the HMD and the direction of the 
+compact camera needed to be calibrated. The calibration took approximately two minutes. 
+The participants practiced with the HUD system , and any questions were answered at that 
+time. The trials started when th e participants reported feeling comfortable in using the HUD 
+system. 
+ 
+  The Role of Head-Up Display in Computer-Assisted Instruction  43 
+6.2 Results 
+Figure 10 shows the results of the experiment. The bars indicate average times required by 
+participants to complete trials. The black, sh aded, and unshaded bars represent times for the 
+HUD system, the laptop PC, and the paper manu al, respectively. The error bar on each bar 
+represents the standard deviation. 
+Analysis of Variance (ANOVA) was carried out on the task time data. Presentation media 
+significantly affected finding time (F[2,32]=39. 6, p<0.01). Post hoc analysis for all possible 
+pairs of presentation media showed that the trial with the HUD system was significantly 
+shorter than those with the others. Presentation  media also significantly affected work time 
+(the time required to finish the article after the possible answers had been found) 
+(F[2,32]=22.4, p<0.01). Post hoc analysis sh owed the trial with the HUD system took 
+significantly longer than those with the othe r media but no significant difference between 
+the PC system and the paper manual. Presen tation media also significantly affected 
+completion time (F[2,32]=6.8, p<0.01). Post hoc analysis showed that the trial with the PC 
+took significantly longer than those with th e other media, but there was no difference in 
+completion time between the HU D system and the paper manual. 
+ 
+05101520253035404550
+123
+Itemtime [sec]
+ 
+Fig. 10. Experimental results (black: HUD, shaded: laptop PC, unshaded: paper manual) 
+ 
+Figure 11 shows the results of th e preference test. Scores for qu estions by number, as listed 
+in Table 1, are ranged along the horizontal axis . The bars indicate the average score reported 
+by participants. The black, sh aded, and unshaded bars repres ent scores for the HUD system, 
+the laptop PC, and the paper manual, respective ly. The error bar on each bar represents the 
+standard deviation. 
+ 
+ Human-Computer Interaction, New Developments   44 
+0123456
+123456
+Question NumberScore
+ 
+Fig. 11. Preference test results (black: HUD, shaded: laptop PC, unshaded: paper manual) 
+ 
+In answers to questions 1 and 2, related to ea se of use and practice, the same tendency was 
+observed: the paper manual was the most preferred instruction medium, the PC was ranked 
+second, and the HUD system was third. In an swer to question 5, which asked about how 
+tiring the task was, participants reported that  they preferred the PC and the paper manual to 
+the HUD system, but neither of these two was clea rly preferred over the other. In answers to 
+question 4, about mental switching, all the scor es were comparable. In answers to question 6, 
+participants reported that the HUD system was the most enjoyable, the PC was next, and the 
+paper manual was boring. In answers to question 3, the HUD system was reported to be the most helpful in searching articles, and the other media had comparable scores. 
+ 
+6.3 Discussion 
+Overall, the performance and preference test results did not show that the HUD system was 
+clearly superior to the other media over the whole course of the task. As expected, the 
+participants using the HUD system took less time finding the possible answers and more 
+time reading the articles and the question on the HMD than participants using the other 
+media. The results showed that the HUD system  excelled at finding the place of the answer 
+possibilities but seemed to spoil the excellence by careful reading of the articles and questions, which affected the task time. This  suggests that the HUD-based CAI system is 
+good at indicating which equipment the user  needs to treat but not so suitable for 
+presenting instructional information, because the HMD requires the user to see letters and 
+characters at the limited resolution and field of view. 
+We also observed that the ID on the answer po ssibility sheet made it easy to identify the 
+location of the answer on the wall and the article on the HMD. That is, the ID worked as a 
+sign guiding the task procedure. 
+  The Role of Head-Up Display in Computer-Assisted Instruction  45 
+We found that there was a difference between the experimental results and those obtained 
+in the actual operation of the transportable earth station. In the laboratory experiment, the 
+task completion time was comparable among the three media. In the actual operation, 
+however, people using the HUD-based CAI sy stem performed better than those using the 
+other two media. This was interpreted as a difference of the task or the experimental 
+condition that worked against the HUD system or in favor of the paper manual in this 
+experiment. It was important for the people to  check if they were reading the appropriate 
+instructions during the actual operation, because the pieces of equipment were not familiar 
+to them. This situation coul d work for the HUD system.  
+ 
+7. Conclusion 
+ 
+We investigated the role of HUDs in CAI. HUDs have been used in various situations in 
+daily lives by recent downsizing and cost down  of the display devices. CAI is one of the 
+promising applications for HUDs. We have developed an HUD-based CAI system for 
+effectively presenting instructions of the equi pment in the transportable earth station. This 
+chapter described HUDs in CAI from a viewpo int of human-computer interaction based on 
+the development experience.  
+First, the basic concept of an HUD was introduced by briefly describing general HUD technology and its relevant applications. An HUD is basically a display medium on which information is presented, allowing a user to simultaneously view a real scene and 
+superimposed information without large movements of the head or eye scans. The HUD has 
+been incorporated into various applications, on which its type depends. We described HUD 
+design types: head-mounted or ground-reference d, optical see-through or video see-through, 
+and single-sided or two-sided types, and discussed their characteristics by comparing each HUD design type. 
+Second, the features of HUD-based CAI were ex plained by describing its applications, such 
+as equipment operation, product assembly, and machine maintenance. These HUD-based 
+CAI applications have witnessed the introduction of increasingly complex platforms and sophisticated procedures and are characterized by the real-time presentation of instructional information related to what the user is viewing. Common thought is that HUD-based CAI 
+will increase the efficiency of instructions and reduce errors by properly presenting 
+instructional information based on a user’s viewpoint. We discussed the advantages of 
+HUD-based CAI, such as a hands-free environm ent, reduction of head and eye movements, 
+and awareness of real objects, compared  to conventional printed manuals.  
+Third, a user study with our HUD-based CAI system was reported. Our system provides 
+information using a head-mounted HUD, on whic h a piece of equipment is identified with 
+identification markers, as the user looks at  the piece of equipment that she tries to 
+manipulate. User performance with the syst em was evaluated during a task in which 
+participants read articles and answered questions about the articles, and this performance 
+was compared to performance with a laptop  PC and paper manual. The experimental 
+results for the performance of the HUD-based CAI system showed less time finding pairs of questions and the possible answers and more ti me selecting one of the possibilities after 
+reading the articles. This suggested that the us er would receive the advantages of an HUD, 
+but also have difficulty viewing the informat ion because of the narrow field of view and 
+insufficient resolution in the HUD.  
+ Human-Computer Interaction, New Developments   46 
+We did not here deal with eye strain and a sa fety measure. These i ssues become important 
+when HUDs are used for CAI in actual situations  such as a construction site. In general, the 
+display surface and the real objects are different  in distance from the user’s eyes. This may 
+cause eye strain for a long term use, resulting in visually-induced sickness. Besides, the 
+mechanics of the eyes’ protection must work in an accident of bumps between the display 
+device and the periphery, especially for the head-mounted type. These issues have to be 
+addressed before the practical uses of HUD-based CAI.  
+ 
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+Weintraub, D. J., Ensing, M. (1992). Human factors issues in head-up display design: the book of 
+HUD  (CSERIAC State-of-the-art Report), Wr ight-Patterson AFB, Dayton, OH: Crew 
+System Ergonomics Information Analysis Center 
+Wickens, C. D. (1992). Computationa l models of human performance, Technical Report No. 
+ARL-92-4/NASA-A3I-92-1 , University of Illinois, Aviation Research Laboratory 
+Wickens, C. D., and Long, J. (1995). Object- vs . space-based models of visual attention: 
+Imprecations for the design of head-up displays, Journal of Experimental Psychology: 
+Applied, vol. 1, pp. 179-194 
+Wickens, C. D., Martin-Emerson, R., and Lari sh, I. (1993). Attentional tunneling and the 
+head-up display, Proc. The 7th International Symposium on Aviation Psychology , 
+pp.865-870, OH: The Ohio State University 
+Wiedenmaier, S., Oehme, O., Schmidt, L., and Luczak, H. (2003). Augmented Reality (AR) 
+for assembly processes design and experimental evaluation, International Journal of 
+Human-Computer Interaction , vol. 16, no. 3, pp. 497-514 
+Zauner, J., Haller, M., Brandl, A. (2003). Authoring of a mi xed reality assembly instructor for 
+hierarchical structures, Proc International Symposium on Mixed and Augmented Reality , 
+pp. 237-246 
+ 
+ 
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+arXiv:1001.0421v1  [cs.DC]  4 Jan 2010MAPREDUCE FOR INTEGER FACTORIZATION
+JAVIER TORDABLE
+Abstract. Integer factorization is a very hard computational problem . Cur-
+rently no efficient algorithm for integer factorization is pu blicly known. How-
+ever, this is an important problem on which it relies the secu rity of many real
+world cryptographic systems.
+I present an implementation of a fast factorization algorit hm on MapRe-
+duce. MapReduce is a programming model for high performance applications
+developed originally at Google. The quadratic sieve algori thm is split into the
+different MapReduce phases and compared against a standard i mplementation.
+1.Introduction
+The security of many cryptographic algorithms relies on the fact that factoring
+large integers is a very computationally intensive task. In particular RSA [1] would
+be vulnerable if there was an efficient algorithm to factor sem iprimes (products of
+two primes). This could have severe consequences, as RSA is o ne of the most widely
+used algorithms in electronic commerce applications [2].
+There are many algorithms for integer factorization [3]. Fr om the trivial trial
+division to the classical Fermat’s factorization method [4 ] and Euler’s factoring
+method [5] to the modern algorithms, the quadratic sieve [6] and the number field
+sieve [7]. In particular the number field sieve algorithm was used in 1996 to factor
+a 512 bit integer [8], the lowest integer length used in comme rcial RSA implemen-
+tations. There have been several other big integers factore d over the course of the
+last decade. I would like to point out that in those cases the f eat was accomplished
+with tremendous effort developing the software and a very con siderable investment
+in hardware [9],[10].
+In what follows I will expose how MapReduce, a distributed co mputational
+framework, can be used for integer factorization. As an exam ple I will show an
+implementation of the quadratic sieve algorithm. I will als o compare in terms of
+performance and cost a conventional implementation with th e MapReduce imple-
+mentation.
+2.MapReduce
+I claim no participation in the development of the MapReduce framework. This
+section is basically a short extract of the original MapRedu ce paper by Jeff Dean
+and Sanjay Ghemawat [11]. MapReduce is a programming model i nspired in com-
+putational programming. Users can specify two functions, mapandreduce . The
+mapfunction processes a series of (key, value) pairs, and outpu ts intermediate (key,
+value) pairs. The system automatically orders and groups al l (key, value) pairs for
+a particular key, and passes them to the reduce function. The reduce function re-
+ceives a series of values for a single key, and produces its ou tput, which is sometimes
+a synthesis or aggregation of the intermediate values.
+1MAPREDUCE FOR INTEGER FACTORIZATION 2
+The canonical example of a MapReduce computation is the cons truction of an
+inverted index. Let’s take a collection of documents D={D0,D1,...,DN}which
+are composed of words D0= (d0,0,d0,1,...,d0,L0),D1= (d1,0,d1,1,...,d1,L1)and so
+on. We define a map function the following way:
+map: (i,Di)→ {(di,0,(i,0)),(di,1,(i,1)),...,(di,Li,(i,Li))}
+that is, for a given document it processes each word in the doc ument and outputs
+an intermediate pair. The key is the word itself, and the valu e is the location in
+the corpus, indicated as (document, position). The reduce f unction is defined as:
+reduce:{(d,(i1,j1)),...,(d,(iL,jL))} →(d,{(i1,j1),...,(iL,jL)})
+For a collection of pairs with the same key (the same word), it outputs a new pair,
+in which the key is the same, and the value is the aggregation o f the intermediate
+values. In this case, the set of locations (document and posi tion in the document)
+in which the word can be found in the corpus.
+The MapReduce implementation automatically takes care of t he parallel exe-
+cution in a distributed system, data transmission, fault to lerance, load balancing
+and many other aspects of a high performance parallel comput ation. The MapRe-
+duce model escales seamlessly to thousands of machines. It i s used continously for
+a multitude of real world applications, from machine learni ng to graph computa-
+tions. And most importantly the effort required to develop a h igh performance
+parallel application with MapReduce is much lower than usin g other models, like
+for example MPI [12].
+3.Quadratic Sieve
+The Quadratic Sieve algorithm was conceived by Carl Pomeran ce in 1981. A
+detailed explanation of the algorithm can be found in [13]. H ere we will just re-
+view the basic steps. Let Nbe the integer that we are trying to factor. We
+will attempt to find a,bsuch that: N|/parenleftbig
+a2−b2/parenrightbig
+⇒N|(a+b)(a−b). If
+{(a+b,N),(a−b,N)} /ne}ationslash={1,N}then we will have a factorization of N.
+Lets define:
+Q(x) =x2−N
+if we find x1,x2,...xKsuch that/producttextK
+i=1Q(xi)is a perfect square, then:
+N|K/productdisplay
+i=1Q(xi)−/parenleftiggK/productdisplay
+i=1xi/parenrightigg2
+=K/productdisplay
+i=1/parenleftbig
+x2
+i−N/parenrightbig
+−x2
+1x2
+2...x2
+K
+3.1.Finding Squares. Let’s take a set of integers x1,...,xLwhich are B-smooth
+(allxifactor completely into primes ≤B). One way to look for i1,i2,...,iMsuch
+that/producttextM
+j=1xijis a square is as follows. Let’s denote pithe i-th prime number./producttextM
+j=1xij=pa1
+j1pa2
+j2...paL
+jLis a square if and only if 2|akfor allk⇔ak≡0mod(2).
+For each xiwe will obtain a vector vi=v(xi)wherevi
+j=max/braceleftbig
+k:pk
+j|xi/bracerightbig
+mod(2).
+That is, each component jofviis the exponent of pjin the factorization of ximod-
+ulo2. For example, for B= 4:MAPREDUCE FOR INTEGER FACTORIZATION 3
+x1= 6,v1= (1,1,0,0)
+x2= 45,v2= (0,0,1,0)
+x3= 75,v3= (0,1,0,0)
+It is immediate that:
+v
+M/productdisplay
+j=1xij
+=M/summationdisplay
+j=1v/parenleftbig
+xij/parenrightbig
+Then
+M/productdisplay
+j=1xijis a square ⇔v
+M/productdisplay
+j=1xij
+=− →0
+In conclussion, in order to find a subset of x1,...,xLwhich is a perfect square,
+we just need to solve the linear system:
+/parenleftbigv1|v2|...|vL/parenrightbig
+e1
+e2
+...
+eL
+≡− →0mod(2)
+3.2.Sieving for smooth numbers. Back to the original problem, we just need
+to find a convenient set {x1,x2,...,xL}such that {Q(x1),Q(x2),...,Q(xL)}are
+B-smooth numbers for a particular B. First of all, lets notice that we don’t need to
+consider every prime number ≤B. If a prime pverifies:p|Q(x)for some xthen:
+p|Q(x)⇔p|x2−N⇔x2≡N mod(p)⇔/parenleftbiggN
+p/parenrightbigg
+= 1
+Because Nis a quadratic residue modulo p if and only if the Legendre sym bol
+of n over p is 1. We will take a set of primes which verifies that p roperty and we
+will call it factor base .
+In order to consider smaller values of Q(x)we will take values of xaround√
+N,
+i.e.x∈/bracketleftig
+⌊√
+N⌋−M,⌊√
+N⌋+M/bracketrightig
+for some M.BothBabove and Mhere are
+chosen as indicated in [13].
+In order to factor all the Q(xi)we will use a method called sieving which is
+what gives the quadratic sieve its name. Notice that p|Q(x)⇒p|Q(x+kp) =
+x2+2kpx+k2p2−N=/parenleftbig
+x2−N/parenrightbig
++p/parenleftbig
+2kx+k2p/parenrightbig
+. Then
+Q(x)≡0mod(p)⇒ ∀k∈N,Q(x+kp)≡0mod(p)
+We can solve the equation Q(x)≡0mod(p)⇔x2−N≡0mod(p)efficiently
+and obtain two solutions s1,s2[14]. If we take:
+zp,{1,2}=min/braceleftig
+x∈/bracketleftig
+⌊√
+N⌋−M,⌊√
+N⌋+M/bracketrightig
+:x≡s{1,2}mod(p)/bracerightig
+then allQ/parenleftbig
+zp,{1,2}+kp/parenrightbig
+,k∈[0,K]are divisible by p. We can divide each one of
+them by the highest power of ppossible. For example:MAPREDUCE FOR INTEGER FACTORIZATION 4
+(xi) = ( ...,6,7,8,9,10,...)
+(Q(xi)) = ( ...,−41,−28,−13,4,23,...)/parenleftbig77
+2/parenrightbig
+= 1as77≡1≡12mod(2)
+x2−77≡0mod(2)yields1,3,5,7,9,...
+(...,−41,−7,−13,1,23,...) after sieving by 2
+After sieving for every appropriate p, all theQ(z)that are equal to 1are smooth
+over the factor base.
+4.Method
+I developed a basic implementation of the Quadratic Sieve Ma pReduce which
+runs on Hadoop [15]. Hadoop is an open source implementation of the MapReduce
+framework. It is made in Java and it has been used effectively i n configurations
+ranging from one to a few thousand computers. It is also avail able as a commercial
+cloud service [16].
+This implementation is simply a proof of concept. It relies t oo heavily on the
+MapReduce framework and it is severy bound by IO. However the size and complex-
+ity of the implementation are several orders of manitude low er than many competing
+alternatives.
+The 3 parts of the program are :
+•Controller : Is the master job executed by the platform. It runs before
+spawning any worker job. It has two basic functions: first it g enerates the
+factor base. The factor base is serialized and passed to the w orkers as a
+counter. Second it generates the full interval to sieve. All the data is stored
+in a single file in the distributed Hadoop file system [17]. It t hen relies on
+the MapReduce framework to automatically split it in an adeq uate number
+of shards and distribute it to the workers
+•Mapper : The mappers perform the sieve. Each one of them receives an
+interval to sieve, and they return a subset of the elements in that input
+sieve which are smooth over the factor base. All output eleme nts of all
+mappers share the same key
+•Reducer : The reducer receives the set of smooth numbers and attempts
+to find a subset of them whose product is a square by solving the system
+modulo 2 using direct bit manipulation. If it finds a suitable subset, it tries
+to factor the original number, N. In general there will be man y subsets
+to choose from. In case that the factorization is not succesf ul with one of
+them, it proceeds to use another one. The single output is the factorization
+In order to compare performance I developed another impleme ntations of the Qua-
+dratic Sieve algorithm in Maple. Both implementations are b asic in the sense that
+they implement the basic algorithm described above and the c ode has not been heav-
+ily optimized for performance. There are many differences be tween the two frame-
+works used that could impact performance. Because of that a d irect comparison of
+running times or memory space may not be meaningful. However it is interesting to
+notice how each of the implementations scales depending on t he size of the problem.
+The source code is available online at http://www.javierto rdable.com/research.MAPREDUCE FOR INTEGER FACTORIZATION 5
+Decimal Sieve MapReduce Maple
+Digits Size Time (s)Memory ( MB)Time (s)Memory ( MB)
+10 5832 2.0 149.6 0.1 7.5
+15 85184 3.0 397.1 3.5 15.5
+20970299 35.0 463.1 116.0 100.8
+257529536 495.0 670.0 3413.7 894.0
+Table 1. Absolute performance of the MapReduce and Maple implementa tions
+Decimal Sieve MapReduce Maple
+Digits Size Time Memory Time Memory
+10 1.01.01.01.01.0
+15 14.61.52.735.02.1
+20166.417.53.11160.013.4
+251291.1247.54.534137.0119.2
+Table 2. Normalized performance of the MapReduce and Maple implemen tations
+Decimal Absolute Sieve Relative Sieve Absolute Relative
+Digits Size Size Disk (MB)Disk (MB)
+10 5832 1.0 0.1 1.0
+15 85184 14.6 2.1 14.6
+20 970299 166.4 29.4 166.4
+25 7529536 1291.1 275.3 1291.1
+Table 3. Disk usage of the MapReduce implementation
+5.Results
+Figures 1 and 2 show the results both in absolute terms and nor malized. Figure
+3 shows the disk usage of the MapReduce implementation. To te st both implemen-
+tations I took a set of numbers of different sizes1. The number of decimal digits d
+is indicated in the first column of each table. In order to cont ruct those numbers I
+took two factors close to 10d
+2, with their product slightly over 10d.
+In each table sieve size indicates the number of elements tha t the algorithm
+analyzed in the sieve phase. For the MapReduce application t he time result is
+taken from the logs, and the memory result is obtained as the m aximum memory
+used by the process. For the Maple implementation both time a nd memory data
+are taken from the on screen information in the Maple environ ment. Finally disk
+usage data for the MapReduce is taken as the size of the file tha t contains the list of
+numbers to sieve. The Maple program runs completely in memor y for the samples
+analyzed.
+11164656837, 117375210056563, 10446257742110057983, 110 0472550655106750000029MAPREDUCE FOR INTEGER FACTORIZATION 6
+6.Discussion
+The MapReduce implementation has a relatively big setup cos t in time and
+memory when compared with an application in a conventional m athematical envi-
+ronment. However it scales better with respect to the size of the input data.
+MapReduce is optimized to split and distribute data form dis k. If an application
+handles a significant volume of data, IO capacity and perform ance can be a limiting
+factor. In our case disk usage is directly proportional to th e size of the sieve set,
+which grows exponentially on the number of digits.
+Both MapReduce and Maple implementations are similar in ter ms of develop-
+ment effort. The Maple implementation seems more adequate fo r small-sized prob-
+lems while the MapReduce application is more efficient for med ium-sized problems.
+Also it will be easier to scale in order to solve harder proble ms.
+References
+[1] Rivest, R.; A. Shamir; L. Adleman. 1978. A Method for Obta ining Digital Signatures and
+Public-Key Cryptosystems. Communications of the ACM 21 (2) : 120–126.
+[2] Nash, A., Duane, W., and Joseph, C. 2001. Pki: Implementi ng and Managing E-Security.
+McGraw-Hill, Inc.
+[3] Donald Knuth. 1997. The Art of Computer Programming, Vol ume 2: Seminumerical Algo-
+rithms, Third Edition. Addison-Wesley. ISBN 0-201-89684- 2. Section 4.5.4: Factoring into
+Primes, pp. 379–417
+[4] Israel Kleiner. 2005. Fermat: The Founder of Modern Numb er Theory. Mathematics Maga-
+zine, Vol. 78, No. 1 (Feb., 2005), pp. 3-14
+[5] McKee, James. 1996. Turning Euler’s Factoring Method in to a Factoring Algorithm"; in
+Bulletin of the London Mathematical Society; issue 28 (volu me 4); pp. 351-355
+[6] Pomerance, C. 1985. The quadratic sieve factoring algor ithm. In Proc. of the EUROCRYPT
+84 Workshop on Advances in Cryptology: theory and Applicati on of Cryptographic Tech-
+niques. Springer-Verlag New York. 169-182.
+[7] Lenstra, A. K., Lenstra, H. W., Manasse, M. S., and Pollar d, J. M. 1990. The number
+field sieve. In Proceedings of the Twenty-Second Annual ACM S ymposium on theory of
+Computing. ACM, New York, NY, 564-572.
+[8] Cowie, J., Dodson, B., et al. 1996. A World Wide Number Fie ld Sieve Factoring Record: On
+to 512 Bits. In Proceedings of the international Conference on the theory and Applications of
+Cryptology and information Security. Lecture Notes In Comp uter Science, vol. 1163. Springer-
+Verlag, London, 382-394.
+[9] Golliver, R. A., Lenstra, A. K., and McCurley, K. S. 1994. Lattice sieving and trial division.
+In Proceedings of the First international Symposium on Algo rithmic Number theory. L. M.
+Adleman and M. A. Huang, Eds. Lecture Notes In Computer Scien ce, vol. 877. Springer-
+Verlag, London, 18-27.
+[10] S. Cavallar and W. M. Lioen and H. J. J. Te Riele and B. Dods on and A. K. Lenstra and P.
+L. Montgomery and B. Murphy Et Al and Mathematisch Centrum. 2 000. Factorization of a
+512-bit RSA modulus. Proceedings of Eurocrypt 2000. Spring er-Verlag. 1-18.
+[11] Dean, J. and Ghemawat, S. 2004. MapReduce: Simplified Da ta Processing on Large Clusters.
+OSDI’04: Sixth Symposium on Operating System Design and Imp lementation, San Francisco,
+CA, December, 2004, 137-150
+[12] Gropp, W., Lusk, E., and Skjellum, A. 1994. Using Mpi: Po rtable Parallel Programming
+with the Message-Passing Interface. MIT Press. 257-260
+[13] Carl Pomerance. 1996. A Tale of Two Sieves, Notices of th e AMS, 1473-1485
+[14] Niven, I. and Zuckerman, H.S. and Montgomery, H.L. 1960 . An introduction to the theory
+of numbers. John Wiley and Sons, Inc. 110-115
+[15] http://hadoop.apache.org/
+[16] http://aws.amazon.com/elasticmapreduce/
+[17] Borthakur, D. 2007. The hadoop distributed file system: Architecture and design.
+http://svn.apache.org/repos/asf/hadoop/core/tags/re lease-0.15.3/docs/hdfs_design.pdf
\ No newline at end of file
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+arXiv:1001.0422v3  [cond-mat.mes-hall]  20 Aug 2010Plasmon-mediated superradiance near metal nanostructure s
+Vitaliy N. Pustovit1,2and Tigran V. Shahbazyan1
+1Department of Physics, Jackson State University, Jackson, MS 39217, USA and
+2Centre de Recherche Paul Pascal, CNRS, Avenue A. Schweitzer , 33600 Pessac, France
+We develop a theory of cooperative emission of light by an ens emble of emitters, such as fluo-
+rescing molecules or semiconductor quantum dots, located n ear a metal nanostructure supporting
+surface plasmon. The primary mechanism of cooperative emis sion in such systems is resonant energy
+transfer between emitters and plasmons rather than the Dick e radiative coupling between emitters.
+We identify two types of plasmonic coupling between the emit ters, (i) plasmon-enhanced radiative
+coupling and (ii) plasmon-assisted nonradiative energy tr ansfer, the competition between them gov-
+erning the structure of system eigenstates. Specifically, w hen emitters are removed by more than
+several nm from the metal surface, the emission is dominated by three superradiant states with the
+same quantum yield as a single emitter, resulting in a drasti c reduction of ensemble radiated en-
+ergy, while at smaller distances cooperative behavior is de stroyed by nonradiative transitions. The
+crossover between two regimes can be observed in distance de pendence of ensemble quantum effi-
+ciency. Our numerical calculations incorporating direct a nd plasmon-assisted interactions between
+the emitters indicate that they do not destroy the plasmonic Dicke effect.
+I. INTRODUCTION
+Superradiance of an ensemble of dipoles confined
+within a limited region in space has been discovered in
+the pioneering work by Dicke.1The underlying physical
+mechanism can be described as follows. Suppose that a
+large number, N, of dipoles with frequency ω0are con-
+fined in a volume with characteristic size Lmuch smaller
+than the radiation wavelength λ0= 2π/ω0. Then radia-
+tion of an ensemble is a cooperative process in which the
+emission of a photon is accompanied by virtual photon
+exchange between individual emitters. This near field ra-
+diativecouplingbetweenthedipolesleadstoformationof
+new system eigenstates, each comprised of all individual
+dipoles. The eigenstates with angular momentum l= 1
+aresuperradiant , i.e., their radiative lifetimes are very
+short,∼τ/N, whereτis radiative lifetime of an indi-
+vidual dipole; the remaining states are subradiant with
+much longer decay times, ∼τ(λ0/L)2≫τ.
+Since the appearance of Dicke paper, cooperative ef-
+fects based on Dicke radiative coupling mechanism have
+been extensively studied in atomic and semiconductor
+systems (see, e.g., reviews in Refs. 2–4). Two differ-
+ent decay times corresponding to superradiant and sub-
+radiant states were observed in a system of two laser-
+trapped ions5and, more recently, in laterally arranged
+quantum dots.6Other examples of cooperative behavior
+analogous to the Dicke effect include, e.g., electron tun-
+neling through a system of quantum dots7,8and sponta-
+neous phonon emission by coupled quantum dots.4,9
+Recently, we extended the Dicke effect to plasmonic
+systems comprised of Ndipoles located in the vicinity
+of a metal nanostructure, e.g., metal nanoparticle (NP),
+supporting localized surface plasmon (SP).10In such sys-
+tems, the dominant coupling mechanism between dipoles
+isplasmonic rather than radiative, i.e., it is based on
+virtual plasmon exchange (see Fig. 1). This plasmonic
+coupling leads to formation of collective states, simi-
+lar to Dicke superradiant states, which dominate pho-/s101
+/s106
+/s101
+/s107/s83/s80 
+/s106/s107 
+/s112/s104 
+/s106/s107 
+/s101
+/s107/s101
+/s106
+/s40/s98/s41 /s40/s97/s41
+FIG. 1: (Color online) Radiative coupling of emitters in fre e
+space (a), and plasmonic coupling of emitters near a metal
+nanoparticle (b).
+ton emission. Furthermore, the nanostructure acts as a
+hub that couples nearby and remote dipoles with about
+equal strength and hence provides a more efficient hy-
+bridization of dipoles compared to radiative coupling.
+In general, as dipoles orientations in space are non-
+uniform, there are three superradiant states with total
+angular momentum l= 1, each having radiative decay
+rate∼NΓr/3, where Γris radiative decay rate of a
+singledipole near a nanostructure (i.e., withplasmon
+enhancement).10
+The principal difference between plasmonic and usual
+(photonic) Dicke effects stems from non-radiative energy
+transfer between the dipoles and the nanostructure. Let
+us first outline its role for the case of a singledipole
+near metal NP. When an excited emitter is located close
+to metal surface, its energy can be transferred to op-
+tically inactive excitations in the metal and eventually
+dissipated (Ohmic losses). This is described by the non-
+radiative decay rate, Γnr∝d−3, wheredis the dipole–
+surface separation.11Note that very close to metal sur-
+face (∼1 nm), this dependence changes to ∝d−4due
+to surface-assisted generation of electron-hole pairs out
+of the Fermi sea.12,13As a result, the radiation of a cou-
+pled dipole-NP system is governed by a competition be-
+tween non-radiative losses and plasmon enhancement142
+that determines system quantum efficiency, Q= Γr/Γ,
+where Γ = Γr+ Γnris thefulldecay rate. Indeed, the
+radiated energy is W= (/planckover2pi1kc/2)Q,kandcbeing wave
+vectorandspeedoflight, anditsdistancedependencefol-
+lows that of Q. Namely, with decreasing d, the emission
+first increases due to plasmon enhancement, and then, at
+several nm from metal surface, it is quenched due to sup-
+pression of Qby non-radiative losses. Both enhancement
+and quenching were observed in recent experiments on
+fluorescing molecules attached to a metal NP,15–19and,
+not too close to NP surface, the distance dependence of
+single-molecule fluorescence17,18was found in excellent
+agreement with single dipole-NP models.20–23
+When radiation takes place from an ensemble of emit-
+ters near a metal nanostructure, there are twodistinct
+types of plasmon-induced couplings between the emit-
+ters. The first is plasmon-enhanced radiative coupling ,
+described by radiative decay matrix Γr
+jk, where indexes
+j,k= 1,...,Nreferto emitters, that is astraightforward
+extension of Dicke radiative coupling obtained by incor-
+porating SP local field into the common radiation field.
+Correspondingly, the eigenstates of Γr
+jkare superradiant
+and subradiant states characterized by the strength of
+their coupling to radiation field. In the ideal case of
+”point sample,” i.e., kL≪1, the subradiant decay rates
+are negligibly small and Γr
+jkessentially has just three
+non-zero eigenvalues, corresponding to superradiant de-
+cay rates, each scaling with Nas∼NΓr/3.10
+The second coupling mechanism is non-radiative en-
+ergy transfer between dipoles that takes place in two
+steps: an excited dipole first transfers its energy to plas-
+mons in nanostructure via its electric field, and then this
+energy is transferred to another dipole. This process in-
+volves plasmons with allangular momenta l, and it is
+described by non-radiative decay matrix, Γnr
+jk. Impor-
+tantly, plasmons with l >1 couple to bothsuperradiant
+andsubradiantstates,sothattheeigenstatesof full decay
+matrix, Γjk= Γr
+jk+Γnr
+jk, are not superradiant and subra-
+diantstates, but their admixtures. Closetometalsurface
+where non-radiativeprocesses are dominant, Γnr
+jkprevails
+over Γr
+jk, and no cooperative behavior is expected. How-
+ever, when dipoles are removed from the surface by more
+than several nm, the energy transfer occurs primarily via
+optically active dipole surface SP and therefore no signif-
+icant mixing of superradiant and subradiant states takes
+place and superradiance is intact.
+This observation was confirmed by numerical calcula-
+tion of eigenvalues of Γ jk, i.e., full decay rates of sys-
+tem eigenstates, for ensemble of Ndipoles randomly
+distributed in a solid angle around a NP.10Namely, in
+a wide range of dipole-NP distances, three eigenvalues
+corresponding to superradiant states are well separated
+from the rest and scale with Naccording to ∼NΓ/3.
+Since the superradiantstatesarethe opticallyactiveones
+with radiative decay rate also scaling as ∼NΓr/3, their
+quantum efficiencies essentially coincide withQof single
+dipole-NP system. Therefore, in the cooperative regime,
+the ensemble quantum efficiency, Qens, is thrice that ofthe single dipole-NP system,
+Qens≃3Q (1)
+regardless of the ensemble size. Thus, the total radiated
+energy of the ensemble, Wens= (/planckover2pi1kc/2)Qens, is reduced
+to just 3W. The remaining energy is trapped by N−3
+subradiant states and eventually dissipated in the metal
+rather than being emitted with a much slower rate, as
+it would be the case in free space. On the other hand,
+at several nm from metal surface, the non-radiative cou-
+pling is dominated by higher lplasmons causing strong
+mixing of superradiant and subradiant states, i.e., all
+system eigenstates have comparable quantum efficien-
+cies andQens∝N. Therefore, with increasing distance,
+Qens, should first exhibit a sharp rise with its slope ∝N,
+and then switch to a more slower 3 Qdependence.
+Indications of such behavior were reported in the re-
+cent experiment by Dulkeith et al.,16where a systematic
+study of the ensemble fluorescence vs. distance to metal
+surface was performed for Cy5 fluorophores attached to
+Au NP in water. The distance was controlled by vary-
+ing fluorophores concentration; with increasing concen-
+tration, the linker molecules stretched outwards to ac-
+commodate repulsive dipole-dipole interaction between
+fluorophores. Therefore, within isolated dipole-NP pic-
+ture, one would expect that for largerdistances, at which
+fluorophores concentration was higher than average, the
+measured quantum efficiency, normalized to some aver-
+age fluorophores number, should have exceeded the cal-
+culated efficiency in single dipole-NP models.20Instead,
+with increasing distance, the normalized Q, exhibited
+rapid saturation and, at large distances, was consider-
+ably smaller in magnitude than the calculated one.
+In the above discussion, we completely ignored inter-
+actions between the dipoles. In fact, the role of dipole-
+dipole interactions in cooperative emission is highly non-
+trivial since they introduce a disorder into the energy
+spectrum by causing frequency shifts among randomly
+distributed in space but otherwise identical emitters.24
+Mesoscopic cooperative emission from a disordered sys-
+tem, i.e., for particular disorder realizations rather than
+its averaged effect, was considered in Ref. 25 (see also
+Ref. 4). It was found that frequency shifts due to
+dipole-dipole interactions lift the degeneracy of subradi-
+ant states without having significant impact on superra-
+diant states. It was also shown that interactions between
+collective eigenstates are much weaker than those be-
+tween individual dipoles, including typical nearest neigh-
+bors (i.e., separated by ∼LN−1/3), due to cancellations
+between dipole-dipole terms among individual pairs with
+theirconstituentsbelongingtodifferenteigenstates.25On
+the other hand, for completely random distribution and
+high concentration of dipoles, the rare instances of ex-
+tremely close dipoles (i.e., with separation ≪LN−1/3)
+can prevent the formation of superradiant states.26
+The role of interactions in plasmon-mediated coop-
+erative emission is characterized by several distinctive
+features. First, since the decay matrices Γ jkcontain3
+plasmon pole, the relativestrength of dipole-dipole in-
+teractions is effectively reduced as compared to purely
+photonic case. Second, there are additional corrections
+to emitters’ frequencies, one originating from plasmon-
+enhanced radiative coupling and another from nonradia-
+tive coupling. While neither of those corrections diverges
+as dipoles approach each other, the latter becomes very
+large as dipoles approachthe metal surface due to contri-
+bution of high- lplasmons. Therefore, the actual system
+eigenstates are determined by boththe interactions and
+the energy exchange, and their frequencies and decays
+rates must be found simultaneously. It is precisely the
+goal of this paper to calculate the full spectrum of inter-
+actingemitters near a metal NP.
+Specifically, we consider a common situation when
+emitters, e.g., fluorescent molecules or quantum dots, are
+attached to NP surface via flexible linkers. Typically, flu-
+orophoresbound to linkermoleculeshavecertainorienta-
+tion of their dipole moments with respect to NP surface
+and, due to repulsiveinteractions, their angularpositions
+are ordered rather then random.16Therefore, we assume
+here that angular positions of emitters coincide with the
+sites of spherical lattice, such as fullerenes. Specifically,
+we perform our numerical simulations for C20, C60, and
+C80 configurations for respective number Nof dipoles;
+we also study the effect of deviations from ideal lattice.
+We find that not too close to metal surface, the system
+eigenstates fall into three groups, each dominated by a
+particularcouplingmechanism: threesuperradiantstates
+dominated by plasmon-enhanced radiative coupling, one
+state dominated by direct dipole-dipole interactions, and
+the rest dominated by non-radiative coupling via NP.
+Importantly, superradiant states are notsignificantly af-
+fected by dipole-dipoleinteractionswhosemaineffect is a
+largefrequencyshift ofsubradiantstatewiththe smallest
+decay rate.
+We also address the effect of individual emitters’ in-
+ternal nonradiative processes on the ensemble quantum
+efficiency. Internal relaxation is known to inhibit the
+photonic Dicke effect if the corresponding decay rate,
+Γnr
+0, is sufficiently high, i.e., relaxation time is shorter
+than the lifetime of superradiant state.3However, near
+metalnanostructure, the plasmon-enhancedradiativede-
+cay rate, Γr, is significantly larger than Γnr
+0, so that
+plasmon-mediatedsuperradianceislesssensitivetoquan-
+tum yield of individual emitters. Specifically, we perform
+numerical calculations for high-yield and low-yield emit-
+ters to show that Eq. (1) holds for both types. More
+precisely, in cooperative regime, it holds almost exactly
+for high-yield emitters while for low-yield emitters Qens
+is somewhat larger than 3 Qdue to the effective N-fold
+suppression of Γnr
+0by superradiant states’ decay rate.
+An obvious application of plasmon-mediated coopera-
+tive emission is related to fluorescence of a large but un-
+certain number of molecules at some average distances
+from metal nanostructure. For single-molecule case, flu-
+orescence intensity variation with distance was proposed
+to serve as nanoscopic ruler,27owing to the excellentagreementofmeasureddistancedependences with single-
+dipolemodels.20–23Inthecaseofmolecularlayer,theam-
+biguitiescausedbyuncertainmoleculesnumberandtheir
+separationfrom the metal surface prevent, in general, de-
+termination of system characteristics from fluorescence
+variations. However, in cooperative regime, the ambigu-
+ity related to molecules number is removed, and fluores-
+cence intensity is essentially determined by Eq. (1) with
+distance-averaged single-molecule quantum efficiency.
+The paper is organized as follows. In Sec. II, the
+derivation of plasmonic coupling for an ensemble of
+dipoles distributed near metal NP is given. In Sec. III,
+the general expression for radiated energy is obtained.
+Our numerical results and discussion are presented in
+Sec. IV. Conclusions and some technical details are pro-
+vided, respectively, in Sec. V and in the Appendix.
+II. PLASMONIC COUPLING OF RADIATING
+DIPOLES
+We consider a system of Nemiters, such as fluorescing
+molecules or quantum dots, with dipole moments dj=
+djej, wheredjandejare magnitude and orientation, re-
+spectively, located at positions rjnear a metal NP with
+radiusR. Throughout the paper, we assume that charac-
+teristic size of the system (NP+dipoles) is much smaller
+thanthe radiationwavelength, |rj−rk| ≡ |rjk| ≪λ0. We
+also assume that emission events by individual molecules
+are uncorrelated, i.e., after excitation each molecule re-
+laxes through its own internal nonradiative transitions
+before emitting a photon. Then the ensemble emission
+can be described within classical approach by consid-
+ering dipoles as identical Lorentz oscillators with ran-
+dom initial phases driven by common electric field, i.e.,
+one created by all dipoles in the presence of metal NP.
+The frequency-dependent electric field, E(r,ω), satisfies
+Maxwell’s equation
+ǫ(r,ω)ω2
+c2E(r,ω)−∇×∇×E(r,ω) =−4πiω
+c2j(r,ω),(2)
+where dielectric permittivity ǫ(r,ω) is that of the metal
+inside NP,ǫ(ω) forr<Rand that of outside dielectric ǫ0
+forr > R. Herej(r,ω) =−i∞/integraltext
+0eiωtj(t)dt, is the Laplace
+transform of dipole current
+j(t) =q/summationdisplay
+j˙dj(t)ejδ(r−rj), (3)
+where dipole displacements dj(t) are driven by the com-
+mon electric field at dipoles’ positions
+¨dj+ω2
+0dj=q
+mE(rj,t)·ej (4)
+with the initial conditions (at t= 0):dj=d0ejsinϕj,
+˙dj=ω0d0ejcosϕj, andE= 0 (dot stands for time-
+derivative). Hereafter, ω0,q,m, andϕjare oscillators’4
+excitation frequency, charge, mass, and initial phase, re-
+spectively ( ω0=/planckover2pi1/md2
+0). Closed equations for dj(ω)
+can be obtained by using Laplace transform of Eq. (4)
+with the above initial conditions and eliminating Efrom
+Eqs. (2) and (4). Laplace transform of Eq. (2) has the
+form
+ǫ(r,ω)ω2
+c2E(r,ω)−∇×∇×E(r,ω)
+=4πq
+c2/summationdisplay
+jδ(r−rj)/bracketleftbig
+iω0d0ejcosϕj
+−ω2dj(ω)ej+ωd0ejsinϕj/bracketrightbig
+.(5)
+At this point, it is convenient to introduce normalized
+displacements
+vj(ω) =dj(ω)/d0−i/parenleftbig
+ω0/ω2/parenrightbig
+cosϕj−ω−1sinϕj(6)
+and the solution of Eq. (5) reads
+E(r,ω) =4πd0qω2
+c2/summationdisplay
+jG(r,rj,ω)·ejvj,(7)
+whereG(r,r′,ω) is the electric field Green diadic in the
+presence of NP. From Eq. (4), for photon frequency close
+to dipoles frequency, ω≈ω0, we obtain a coupled system
+of equations for normalized displacements,
+/summationdisplay
+k/bracketleftBig
+(ω0−ω)δjk+Σjk/bracketrightBig
+vk=−i
+2e−iϕj,(8)
+where Σ jk= ∆jk−i
+2Γjkis the complex self-energy ma-
+trix, given by
+Σjk(ω) =−2πq2ω0
+mc2ej·G(rj,rk;ω)·ek.(9)
+The system Eq. (8) determines eigenstates of the ensem-
+ble ofNdipoles in common radiation and plasmon field,
+while complex eigenvalues of the self-energy matrix give
+eigenstates frequency shifts with respect to ω0and their
+decay rates. In the absence of NP, real and imaginary
+parts of the self-energy matrix are dipole-dipole inter-
+action and radiation coupling between dipoles jandk,
+given by (in lowest order in krjk)
+∆0
+jk=3Γr
+0
+4(krjk)3/bracketleftbigg
+(ej·ek)−3(ej·rjk)(ek·rjk)
+r2
+jk/bracketrightbigg
+,
+Γ0
+jk= Γr
+0ej·ek, (10)
+where
+Γr
+0=2kq2ω0
+3mc2=2µ2k3
+3/planckover2pi1ǫ0, (11)
+is the radiative decay rate of a dipole in a dielectric
+medium,µ=qd0is the dipole moment, and k=√ǫ0ω/c
+is the wave vector. The eigenstates of photonic decay
+matrix Γ0
+jkare superradiant and subradiant states, theformer having decay rate ∼NΓr
+0. In the case when all
+dipoles are aligned, there is only one superradiant state
+that couples to radiation field, while for general dipole
+orientations there are three such states with angular mo-
+mentuml= 1. Note that in the longwave approximation
+used here, the decay rates of subradiant states vanish
+(”point sample”).
+In the presence of metal nanostructure, the system
+eigenstates are determined by the full Green diadic in
+self-energy matrix Eq. (9). In the case of spherical NP,
+the longwave approximation for G(rj,rk;ω) can be eas-
+ily found.21,23,28The details are given in the Appendix,
+and the result reads
+Σjk(ω) =∆0
+jk−3Γr
+0
+4k3/summationdisplay
+lαlT(l)
+jk(12)
+−i
+2Γr
+0/bracketleftBig
+(ej·ek)−α1/bracketleftbig
+K(1)
+jk+h.c./bracketrightbig
++|α1|2T(1)
+jk/bracketrightBig
+,
+where
+αl(ω) =R2l+1[ǫ(ω)−ǫ0]
+ǫ(ω)+(1+1/l)ǫ0(13)
+is NPl-pole polarizability. The matrices K(l)
+jkandT(l)
+jk
+are defined as
+K(l)
+jk=4π
+2l+1l/summationdisplay
+m=−l[ej·ψlm(rj)][ek·χ∗
+lm(rk)],(14)
+T(l)
+jk=4π
+2l+1l/summationdisplay
+m=−l[ej·ψlm(rj)][ek·ψ∗
+lm(rk)],(15)
+where
+ψlm(r) =∇/bracketleftbiggYlm(ˆr)
+rl+1/bracketrightbigg
+,χlm(r) =∇/bracketleftbig
+rlYlm(ˆr)/bracketrightbig
+,(16)
+andYlm(ˆr) are spherical harmonics. For l= 1, these
+matrices can be evaluated as
+K(1)
+jk=1
+r3
+j/bracketleftBig
+(ej·ek)−3(ej·ˆrj)(ˆrj·ek)/bracketrightBig
+, (17)
+T(1)
+jk=1
+r3
+jr3
+k/bracketleftBig
+(ej·ek)−3(ej·ˆrj)(ek·ˆrj) (18)
+−3(ej·ˆrk)(ek·ˆrk)+9(ek·ˆrk)(ej·ˆrj)(ˆrj·ˆrk)/bracketrightBig
+,
+where we have used identities
+1/summationdisplay
+m=−1ψ1m(rk)Y∗
+1m(ˆr) =3
+4πr3
+k/bracketleftbigˆr−3ˆrk(ˆr·ˆrk)/bracketrightbig
+,
+1/summationdisplay
+m=−1χ1m(rk)Y∗
+1m(ˆr) =3
+4πˆr. (19)
+For dipoles oriented normally with respect to NP surface
+(ej=ˆ rj), we obtain
+K(1)
+jk=−2
+r3
+j(ej·ek), T(1)
+jk=4
+r3
+jr3
+k(ej·ek),(20)5
+and for parallel orientation ( ej·ˆ rj= 0), we similarly get
+K(1)
+jk=1
+r3
+j(ej·ek), T(1)
+jk=1
+r3
+jr3
+k(ej·ek).(21)
+Thedecay matrix , Γjk=−2ImΣjk, can be decomposed
+into radiative and nonradiative terms, Γ jk= Γr
+jk+Γnr
+jk,
+as follows
+Γr
+jk= Γr
+0/bracketleftBig
+(ej·ek)−α′
+1/bracketleftbig
+K(1)
+jk+h.c./bracketrightbig
++|α1|2T(1)
+jk/bracketrightBig
+,
+Γnr
+jk=3Γr
+0
+2k3/summationdisplay
+lα′′
+lT(l)
+jk. (22)
+ThediagonalelementsΓr
+jjand Γnr
+jjdescribe, respectively,
+plasmon-enhanced radiative decay rate of an isolated
+dipole near a NP, and nonradiative transfer of its en-
+ergy to electronic excitations in metal. The non-diagonal
+elements ofΓr
+jkdescribe plasmon-enhanced radiative cou-
+plingthat generalizes the Dicke mechanism responsible
+for cooperative emission by incorporating local field en-
+hancement into near field radiative coupling. On the
+other hand, the non-diagonal terms in Γnr
+jkdescribe non-
+radiative energy transfer between dipoles mediated by
+NP plasmons. The latter coupling is absent in the pho-
+tonic Dicke effect but it plays important role in the plas-
+monic Dicke effect, as we will see below.
+Since the numerical calculations below are carried for
+normal dipoles orientations, we provide here the corre-
+sponding expressions for self-energy matrix Σ jk. The de-
+cay matrix has the form
+Γr
+jk= Γr
+0/bracketleftBigg
+1+2α′
+1/parenleftBigg
+1
+r3
+j+1
+r3
+k/parenrightBigg
++4|α1|2
+r3
+jr3
+k/bracketrightBigg
+cosγjk,
+Γnr
+jk=3Γr
+0
+2k3/summationdisplay
+lα′′
+l(l+1)2
+rl+2
+jrl+2
+kPl(cosγjk), (23)
+wherePlis Legendre polynomial and γjkis the angle
+between positions of dipoles jandkmeasured from NP
+center(herecos γjk=ej·ek). The realpartofself-energy
+matrix has the form
+∆jk= ∆0
+jk+Γr
+0/bracketleftbigg
+α′′
+1/parenleftBigg
+1
+r3
+j+1
+r3
+k/parenrightBigg
+cosγjk
+−3
+4k3/summationdisplay
+lα′
+l(l+1)2
+rl+2
+jrl+2
+kPl(cosγjk)/bracketrightbigg
+,(24)
+where the second term describes NP-induced interac-
+tions. The latter in turn consists of two terms, first
+coming from plasmon-enhanced radiative coupling and
+second coming from non-radiative coupling.
+Note that both NP-induced terms are weaker than
+their counterparts in Γ jkwhile having same symmetry
+and therefore is not expected to significantly alter the
+eigenstates. On the other hand, the dipole-dipole inter-
+action term, Eq. (10), has different symmetry and can
+become large for two dipoles in a close proximity to each
+other. The effect of interactions on cooperative emission
+is studied in Sec. IV.III. RADIATED ENERGY
+In this section, we derive general expression for total
+energy radiated by an ensemble of dipoles near a metal
+nanostructure. Theradiatedenergyintheunit frequency
+interval is obtained by integrating the far field ( r→ ∞)
+spectral intensity over solid angle29
+dW(ω)
+dω=cǫ0
+4π2/integraldisplay
+|E(r,ω)|2r2dΩ, (25)
+and then averaging the result over initial random phases
+of individual dipoles, ϕj. The electric field E(r,ω) is
+given by Eq. (7), where vjis the solution of Eq. (4).
+Then the energy density takes the form
+dW
+dω=4r2ǫ0µ2ω4
+0
+c3(26)
+×/summationdisplay
+jk/integraldisplay
+dΩvjv∗
+k/bracketleftbig
+G(r,rj)·ej/bracketrightbig
+·/bracketleftbig
+G∗(r,rk)·ek/bracketrightbig
+,
+withG(r,rj,ω) =G0(r,rj,ω)+Gs(r,rj,ω) replaced by
+its far field asymptotics (see Appendix),
+Gµν(r,rj) =eikr
+4πr/bracketleftBig
+δµν−4π
+3/summationdisplay
+mˆrµY1m(ˆr)χν∗
+1m(rj)
+−4π
+3˜α1r/summationdisplay
+m/bracketleftbig
+∇µY1m(ˆr)/bracketrightbig
+ψν∗
+1m(rj)/bracketrightBig
+,(27)
+where first two terms come from the free space part, G0,
+and the last term comes from the scattered part, Gs, of
+the Green diadic. The angular integral in Eq. (25) can
+be performed using the relations Eq. (19). The free space
+contribution yields
+4πr2/integraldisplay
+dΩ/bracketleftbig
+G0(r,rj)·ej/bracketrightbig
+·/bracketleftbig
+G0∗(r,rk)·ek/bracketrightbig
+=2
+3(ej·ek).
+(28)
+The other integrals in the product Eq. (26) are evaluated
+using the relations
+/parenleftbigg4πr
+3/parenrightbigg2/summationdisplay
+mm′/bracketleftbig
+ψ∗
+1m(rj)·ej/bracketrightbig/bracketleftbig
+ψ1m′(rk)·ek/bracketrightbig
+(29)
+×/integraldisplaydΩ
+4π∇Y1m(ˆr)·∇Y∗
+1m′(ˆr) =2
+3T(1)
+jk,
+and
+4πr
+3/summationdisplay
+m/integraldisplaydΩ
+4π/bracketleftbig
+ek·∇Y1m(ˆr)/bracketrightbig/bracketleftbig
+ψ∗
+1m(rj)·ej/bracketrightbig
+=2
+3K(1)
+jk,
+(30)
+yielding
+/integraldisplaydΩ
+4π/bracketleftbig
+G(r,rj)·ej/bracketrightbig
+·/bracketleftbig
+G∗(r,rk)·ek/bracketrightbig
+(31)
+=1
+(4πr)22
+3/bracketleftBig
+(ej·ek)−α1K(1)
+jk−α∗
+1K(1)
+kj+|α1|2T(1)
+jk/bracketrightBig
+.6
+The energy density then takes the form
+dW(ω)
+dω=√ǫ0/planckover2pi1ω0
+π/summationdisplay
+jkvjv∗
+kAjk, (32)
+whereAjk= Γr
+0/bracketleftBig
+(ej·ek)−˜α1K(1)
+jk−˜α∗
+1K(1)
+kj+|˜α1|2T(1)
+jk/bracketrightBig
+.
+MatrixAjkis not symmetrical, howeveronly its symmet-
+rical part, equal to Γr
+jk, contributes to the final expres-
+sion. The solution of Eq. (8) can be presented as
+vj=−i
+2/summationdisplay
+k/bracketleftbigg1
+ω0−ω+ˆΣ/bracketrightbigg
+jke−iϕk, (33)
+and after averaging out over the initial random phases
+ϕj, we finally obtain
+dW(ω)
+dω=/planckover2pi1kc
+4πTr/bracketleftbigg1
+ω−ω0−ˆΣ†ˆΓr1
+ω−ω0−ˆΣ/bracketrightbigg
+,(34)
+where trace is taken over the indexes ( jk). The analysis
+of this expression and the results of numerical calcula-
+tions will be presented in the next section.
+IV. DISCUSSION AND NUMERICAL RESULTS
+Let us start with a singledipole located at r0near a
+metal NP. In this case, the self-energy is a complex num-
+ber, Σ = ∆ −i
+2Γ, where ∆ = ∆ jjand Γ = Γr+Γnr= Γjj
+aresingle-dipoleenergyshiftanddecayrate,respectively.
+For normal ( s=⊥) and parallel ( s=∝ba∇dbl) dipole orienta-
+tions, using Eqs. (20) and (21), these are given by20
+Γr= Γr
+0/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+asα1
+r3
+0/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+,Γnr=3Γr
+0
+2k3/summationdisplay
+lb(l)
+sα′′
+l
+r2l+4
+0,
+∆ = Γr
+0/parenleftBigg
+asα′′
+1
+r3
+0−3
+4k3/summationdisplay
+lb(l)
+sα′
+l
+r2l+4
+0/parenrightBigg
+, (35)
+witha⊥= 2,b(l)
+⊥= (l+1)2anda/bardbl=−1,b(l)
+/bardbl=l(l+1)/2.
+Note that both terms in ∆ are smaller than their coun-
+terpartsΓrandΓnrdue toplasmonpoleinthe imaginary
+part of NP polarizability α′′
+l. Radiated energy of single
+dipole-NP system, obtained by frequency integration of
+Eq. (34), is given by
+W=/planckover2pi1kc
+2Γr
+Γ+Γnr
+0=/planckover2pi1kc
+2Q, (36)
+where we included internal molecular relaxation rate,
+Γnr
+0, intoquantumefficiency Q. ForNuncoupled dipoles,
+i.e., for purely diagonal Σ jk=δjk/parenleftbig
+∆−i
+2Γ/parenrightbig
+, Eq. (34)
+decouples into sum of Nindependent terms, yielding
+Wens=NW.
+In the presence of inter-dipole coupling, the system
+eigenstates, |J∝angb∇acket∇ight, are those of the self-energy matrix,
+Eq. (12). The corresponding eigenvalues are complex,ˆΣ|J∝angb∇acket∇ight= (∆J−i
+2ΓJ)|J∝angb∇acket∇ight, where ∆ Jis frequency shift of
+collective eigenstate |J∝angb∇acket∇ightrelative toω0and ΓJis its decay
+rate. The molecular relaxation can be accounted for by
+adding to Σ jka diagonal term, −i
+2δjkΓnr
+0. Then, after
+frequency integration of Eq. (34), the ensemble radiated
+energy takes the form
+Wens=/planckover2pi1kc
+2Qens, Qens=/summationdisplay
+JΓr
+J
+ΓJ+Γnr
+0,(37)
+where Γr
+J=∝angb∇acketleftJ|ˆΓr|J∝angb∇acket∇ightis radiative decay rate of state |J∝angb∇acket∇ight.
+In the photonic Dicke effect, superradiant and subra-
+diant states are eigenstates of the radiative decay matrix
+Γ0
+jkobtained from the free space Green diadic. Similarly,
+in the plasmonic Dicke effect, superradiant states are
+eigenstates of plasmon-enhanced radiative decay matrix
+Γr
+jk. Letusillustratethe emergenceofplasmon-mediated
+superradianceforasimplecasewhenalldipolesareatthe
+same distance from NP surface and are oriented normal
+or parallel to it. Then it is easy to see that the corre-
+sponding decay matrix, Γr
+jk= Γrej·ekwith Γrgiven by
+Eq. (35), has just three nonzero eigenvalues. Indeed, let
+us introduce new decay matrices as
+γr
+µν=NΓr
+3Bµν, (38)
+where
+Bµν=3
+N/summationdisplay
+jeµ
+jeν
+j (39)
+is 3×3 matrix in coordinate space with Tr ˆB= 3. It is
+easy to see that Tr/bracketleftbig
+(ˆΓr)n/bracketrightbig
+= Tr/bracketleftbig
+(ˆγr)n/bracketrightbig
+for any integer
+n, i.e., theN×Nmatrix Γr
+jkhas only three non-zero
+eigenvalues coinciding with those of matrix γr
+µν
+Γr
+µ=NΓr
+3λµ, (40)
+whereλµ∼1areeigenvaluesof Bµν. Notethatthedecay
+rates of the remaining N−3 subradiant states vanish
+in the long wave approximation used here; they acquire
+finite values in the next order in ( kr0)2.
+Let us turn to non-radiative coupling, described by
+matrix Γnr
+jk. Its diagonal elements, Γnr
+jj, describe nonra-
+diative energy exchange between excited dipole and NP
+plasmon modes with all angular momenta, as indicated
+by polarizabilities α′′
+lin Eq. (23). The non-diagonal el-
+ements of Γnr
+jkdescribe a process by which a plasmon
+nonradiatively excited in the NP by dipole ktransfers its
+energyto another dipole j. In general, due to high- lplas-
+mons involved in nonradiative coupling, the eigenstates
+of Γnr
+jkaredifferent from those of plasmon-enhanced ra-
+diative coupling Γr
+jkwhich contains only dipole ( l= 1)
+plasmon mode. Therefore, the eigenstates of full decay
+matrix Γ jk= Γnr
+jk+ Γnr
+jkarenotpure superradiant and
+subradiant states but their admixtures. However, the
+high-lplasmon contribution to Γnr
+jkis significant only at7
+very small d[see, e.g., Eq. (23)] while for dlarger than
+severalnmΓnr
+jkisdominatedbythe l= 1term. Infact, in
+a wide range d, nonradiative coupling between dipoles is
+mainly through the optically active dipoleplasmon mode
+that does not cause mixing between superradiant and
+subradiant states. Namely, it can be easily seen from
+Eq. (23) that the eigenstates of the l= 1 term in Γnr
+jkare
+the same as those of Γr
+jkso the corresponding eigenval-
+ues are similarly given by Γnr
+µ= (NΓnr/3)λµ. Thus, the
+superradiant quantum efficiency
+Qµ=Γr
+µ
+Γµ+Γnr
+0=Γr
+Γ+3Γnr
+0/Nλµ(41)
+only weakly depends on N. Therefore, the sum in
+Eq. (37) includes just three terms, yielding
+Wens=/planckover2pi1kc
+2Qens=/planckover2pi1kc
+23/summationdisplay
+µ=1Γr
+Γ+3Γnr
+0/Nλµ.(42)
+For high-yield (small Γnr
+0) emitters, we obtain Eq. (1)
+and henceWens≈3W. In contrast, the radiated power,
+Pens=/planckover2pi1kc
+23/summationdisplay
+µ=1QµΓr
+µ≃N/parenleftbigg/planckover2pi1kc
+2QΓr/parenrightbigg
+=NP,(43)
+scales with the ensemble size due to shorter (by factor
+N/3) radiative lifetime of superradiant states.
+For low-yield emitters (large Γnr
+0), the relation Eq. (1)
+holds only approximately. However, it is evident from
+comparison of Eqs. (36) and (42) that here the relative
+effect ofinternalrelaxationis much weakerthan for usual
+cooperative emission. Numerical results for both high-
+yield and low-yield emitters are presented below.
+Let us now turn to the role of interactions between
+dipoles in the ensemble, which is the main subject of this
+paper. Interactions play critical role in cooperative emis-
+sion since they introduce a disorder into system energy
+spectrum by causing random shifts of individual dipole
+frequencies.24–26In the conventional cooperative emis-
+sion, the main disorder effect is to split the narrow sub-
+radiant peak in the ensemble emission spectra.25In the
+presence of metal nanostructure, radiation of subradiant
+states is expected to be quenched by much faster non-
+radiative losses in the metal. The crucial question is,
+however, whether interactions between closely spaced in-
+dividual dipoles can significantly alter the structure of
+collective eigenstates. In the remaining part of the pa-
+per, we present the results of our numerical simulations
+of cooperative emission fully incorporating both direct
+and plasmon-mediated interactions.
+We consideran ensemble of Nmoleculardyes attached
+to an Ag spherical particle with radius R= 20 nm via
+molecular linkers with approximately same length. The
+system is embedded in aqueous solution with dielectric
+constantǫ0= 1.77, and two types of dyes with quantum
+efficiencies q= 0.3 andq= 0.95 are used in the calcula-
+tions. A distinguishing feature of this system is a strong/s45/s50/s48/s48 /s48 /s50/s48/s48/s48/s50/s52/s54/s56/s49/s48
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s49/s48/s32/s110/s109
+/s32/s32/s69/s105/s103/s101/s110/s118/s97/s108/s117/s101/s115/s32/s99/s111/s117/s110/s116/s115
+/s47
+/s48/s114/s40/s97/s41
+/s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s49/s50/s51/s52/s53
+/s47
+/s48/s114/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s50/s48/s32/s110/s109
+/s32/s32
+/s40/s98/s41
+/s45/s50/s53 /s48 /s50/s53 /s53/s48 /s55/s53/s49/s50/s51/s52/s53
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s51/s48/s32/s110/s109
+/s47
+/s48/s114
+/s32/s32
+/s40/s99/s41
+FIG. 2: (Color online) Distribution of energy shifts for 20
+dipoles in C20 configuration around Ag NP at several dis-
+tances to its surface.
+/s48 /s50/s48/s48 /s52/s48/s48/s49/s50/s51/s52/s53/s54/s55/s56
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s49/s48/s32/s110/s109
+/s32/s32/s69/s105/s103/s101/s110/s118/s97/s108/s117/s101/s115/s32/s99/s111/s117/s110/s116/s115
+/s47
+/s48/s114/s40/s97/s41
+/s48 /s52/s48 /s56/s48/s48/s49/s50/s51/s52/s53/s54/s55/s56
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s50/s48/s32/s110/s109
+/s32/s32
+/s47
+/s48/s114/s40/s98/s41
+/s48 /s49/s48 /s50/s48 /s51/s48/s49/s50/s51/s52/s53/s54/s55/s56
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s51/s48/s32/s110/s109
+/s32/s32
+/s47
+/s48/s114/s40/s99/s41
+FIG. 3: (Color online) Distribution of decay rates for 20
+dipoles in C20 configuration around Ag NP at several dis-
+tances to its surface.
+effectofinteractionsonitsgeometry.16The flexiblelinker
+molecules hold the attacheddyes with certain orientation
+of their dipole moments, so that repulsive inter-molecule
+interactions compel the dyes to form a spatially ordered
+structure on spherical surface. In our simulations, the
+dyes with normal dipole orientations were located at the
+sites of spherical lattice, specifically, fullerenes C20, C32,
+C60, and C80, and, in some calculations, we included
+random deviations from the ideal lattice positions. The
+system eigenstates are found by numerical diagonaliza-
+tion of self-energy matrix, Σ jk= ∆jk−i
+2Γjk, with its
+real and imaginary parts given by Eqs. (24) and (23), re-
+spectively. Calculations were carried at the SP energy of
+3.0 eV, the size-dependent Landau damping was incor-
+porated for all plasmon modes, and NP polarizabilities,
+Eq. (13), with angularmomenta up to l= 30, werecalcu-
+lated using the experimental bulk Ag complex dielectric
+function.
+Figures 2 and 3 show distribution of real and imag-8
+/s45/s50/s48/s48 /s48 /s50/s48/s48/s49/s50/s51/s52
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s49/s48/s32/s110/s109
+/s32/s32/s69/s105/s103/s101/s110/s118/s97/s108/s117/s101/s115/s32/s99/s111/s117/s110/s116/s115
+/s47
+/s48/s114/s40/s97/s41
+/s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s49/s50/s51/s52
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s50/s48/s32/s110/s109
+/s47
+/s48/s114
+/s32/s32
+/s40/s98/s41
+/s45/s50/s53 /s48 /s50/s53 /s53/s48 /s55/s53/s49/s50/s51/s52
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s51/s48/s32/s110/s109
+/s47
+/s48/s114
+/s32/s32
+/s40/s99/s41
+FIG. 4: (Color online) Distribution of energy shifts for 20
+dipoles in C20 configuration around Ag NP at several average
+(with 10% fluctuations) distances to its surface.
+inary parts of complex eigenvalues of Σ jkforN= 20
+molecules at the sites of C20 fullerene at three different
+molecule-surface distances. The system spectrum repre-
+sents several sets of degenerate eigenvalues indicating a
+high degree of lattice symmetry. For all distances, the
+histograms show a singleeigenvalue with a large posi-
+tive energy shift (Fig. 2), which corresponds to the di-
+rect dipole-dipole interaction between nearest-neighbor
+molecules. On the other hand, there are threedegenerate
+eigenvalues with the largest decay rate, corresponding to
+predominantly superradiant states while the smaller de-
+cay rates are those of predominantly subradiant states
+(Fig. 3). With increasing distance, the mixing between
+superradiant and subradiant states decreases and, for
+d= 30 nm, decay rates of all but three eigenstates nearly
+vanish; note that in our approximation, pure subradiant
+states should have zero decay rate. Such behavior is due
+todiminishingcontributionofhigher- lplasmonsatlarger
+distances [see Eqs. ((23) and 24)]. Importantly, direct in-
+teractions between close molecules result only in energy
+shift of subradiant states, without affecting superradiant
+states. We therefore conclude that dipole-dipole interac-
+tions do notdestroy cooperative emission in plasmonic
+systems.
+This main conclusion remains unchanged when fluctu-
+ations (up to 10%) of molecules positions in radial direc-
+tion are included into simulations (Figs. 4 and 5). The
+spatial disorder lifts lattice symmetry, so that superradi-
+ant states now have different, however close, decay rates.
+Note that without interactions, i.e., when molecules an-
+gular positions are completely random, the spread of su-
+perradiant decay rates is considerably higher.10
+To elucidate the structure of collective states, we cal-
+culate the distance dependences of complex eigenvalues,
+∆J−i
+2ΓJ, for C20, C60, and C80 configurations of dyes,
+as shown in Figs. 6, 7, and 8, respectively. For C20 con-
+figuration (Fig. 6), there are five sets of eigenvalues with
+3, 4, 4, 7, and 1-fold degeneracies, in descending order/s48 /s50/s48/s48 /s52/s48/s48/s49/s50/s51
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s49/s48/s32/s110/s109
+/s32/s32/s69/s105/s103/s101/s110/s118/s97/s108/s117/s101/s115/s32/s99/s111/s117/s110/s116/s115
+/s47
+/s48/s114/s40/s97/s41
+/s48 /s52/s48 /s56/s48/s49/s50/s51
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s50/s48/s32/s110/s109
+/s32/s32
+/s47
+/s48/s114/s40/s98/s41
+/s48 /s49/s48 /s50/s48 /s51/s48/s49/s50/s51
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100/s32/s61/s32/s50/s48/s32/s110/s109
+/s32/s32
+/s47
+/s48/s114/s40/s99/s41
+FIG. 5: (Color online) Distribution of decay rates for 20
+dipoles in C20 configuration around Ag NP at several average
+(with 10% fluctuations) distances to its surface.
+/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s49/s48/s48/s48
+/s82/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s50/s48
+/s100 /s32/s40/s110 /s109/s41/s47
+/s48/s114
+/s32/s32
+/s32/s40/s97/s41
+/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s52/s48/s48/s45/s50/s48/s48/s48/s50/s48/s48/s47
+/s48/s114
+/s32/s32
+/s40/s98/s41
+FIG. 6: (Color online) (a) Decay rates and (b) energy shifts
+vs. distance for 20 dipoles in C20 configuration around Ag
+NP. Each line corresponds to a system eigenstate and is sim-
+ilarly marked in both graphs. The dash-dotted line corre-
+sponds to three degenerate superradiant states, the dashed
+line is the darkest subradiant states dominated by dipole-
+dipole interactions, and solid lines correspond to the rest of
+subradiant states.
+of ΓJmagnitudes. Down to the distance of d= 5 nm,
+the largest decay rates, corresponding to three predom-
+inantly superradiant states, are well separated from the
+rest. The steep rise of Γ Jat small distances is due to
+increasing contribution of high- lplasmon modes close to
+NP surface [see Eqs. (23) and (24)]. The interplay be-
+tweenvariouscouplingmechanismsisespeciallyrevealing9
+/s48/s49/s48/s48/s48/s50/s48/s48/s48/s51/s48/s48/s48
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s54/s48/s47
+/s48/s114
+/s32/s32
+/s32/s40/s97/s41
+/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s51/s48/s48/s48/s45/s50/s48/s48/s48/s45/s49/s48/s48/s48/s48/s49/s48/s48/s48/s50/s48/s48/s48/s47
+/s48/s114
+/s32/s32
+/s100 /s32/s40/s110/s109 /s41/s40/s98/s41
+FIG. 7: (Color online) Same as in Fig. 6, but for 60 dipoles
+in C60 configuration.
+whencomparingtheplotsfor∆ JandΓJ(curvesforsame
+eigenvalue sets have similar patterns). By their ddepen-
+dence, the eigenvalues fall into three main groups. The
+superradiant states have the largest decay rate Γ Jfor
+alldand relatively small mainly positivefrequency shift
+ford/greaterorsimilarR/2; these states are dominated by plasmon-
+enhanced radiative coupling . The non-degenerate state
+with large positiveenergy shift and smallest decay rate is
+dominatedbydirectnearest-neighbordipolesinteraction;
+this state is least affected by the presence of NP and does
+not participate in the emission. The third group of states
+with mostly negative ∆Jand small Γ Jis dominated by
+nonradiative plasmon coupling . Closer to NP surface,
+the coupling becomes dominant due to high- lplasmons
+and all states develop large decay rates and negative en-
+ergy shifts. Note that down to d/greaterorsimilarR/4, the admix-
+ture between superradiant and subradiant modes is still
+relatively weak; below R/4, the non-radiative coupling
+dominates the spectrum and the admixture is strong.
+For larger ensembles, the eigenstates have similar
+structure, as illustrated in Figs. 7 and 8 which show cal-
+culated eigenvalues for dipoles in C60 and C80 config-
+urations, respectively. Importantly, even with decreas-
+ing distance between the emitters in large ensembles, the
+dipole-dipole interactions still do not destroy coopera-
+tive emission. This can be understood from the follow-
+ing argument.25Mixing of superradiant and subradiant
+states takes place if the interactions between them are
+sufficiently strong. The latter requires that the electric
+field of a collective state is strongly inhomogeneous in
+space since, e.g., subradiant states couple only weakly
+to homogeneous field. On the other hand, such a field/s48/s49/s48/s48/s48/s50/s48/s48/s48/s51/s48/s48/s48/s52/s48/s48/s48
+/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s78/s32/s61/s32/s56/s48/s32/s47
+/s48/s114
+/s32/s32
+/s32/s40/s97/s41
+/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s52/s48/s48/s48/s45/s50/s48/s48/s48/s48/s50/s48/s48/s48/s52/s48/s48/s48/s47
+/s48/s114
+/s32/s32
+/s100 /s32/s40/s110/s109 /s41/s40/s98/s41
+FIG. 8: (Color online) Same as in Fig. 6, but for 80 dipoles
+in C80 configuration.
+is comprised of individual fields of all the constituent
+dipoles so the resulting field’s spatial fluctuations are
+weak if no two dipoles approach too close to each other,
+i.e., deviations of nearest-neighborseparationsfrom their
+average, ¯s=LN−1/3,Lbeing characteristic system
+size, are small. However, if deviations from ¯ sare large,
+i.e., two dipoles can be separated by a much closer dis-
+tance,s≪LN−1/3, causingastrongspatialfieldfluctua-
+tion, then the eigenstates are no longer superradiant and
+subradiant states and cooperative emission is destroyed.
+Thisargumentwasconfirmednumericallyherebyfinding
+system eigenstates for both cases – dipoles on a spher-
+ical lattice with some fluctuations (see Figs. 4 and 5),
+and a completely random angular distribution with fixed
+dipole-NP distance with no minimal separation between
+two dipoles (not shown). In the latter case, no superra-
+diant states were formed and the reason was traced to
+configurations with extremely close dipoles. Note, how-
+ever, that with both radial and angular distributions be-
+ingrandom, thesearerareevents. In thecaseofrepulsive
+interactions between individual dipoles, considered here,
+deviations from the average dipole-dipole separation ¯ s
+are exceedingly small and cooperative emission survives
+the interactions.
+Another sharp contrast between plasmonic and pho-
+tonic Dicke effects is the fate of subradiant states. In the
+latter, the energy trapped in subradiant states is eventu-
+ally radiated, albeit with a much slower rate, resulting in
+sharp spectral features of emission spectrum.2,3Instead,
+in plasmonic systems, the trapped energy is dissipated in
+the NP and only a small fraction of total energy leaves
+the system via superradoant states. Thus, the net effect10
+/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s49/s50/s51
+/s32/s32/s78/s32/s61/s32/s49
+/s32/s32/s78/s32/s61/s32/s50/s48
+/s32/s32/s78/s32/s61/s32/s51/s50
+/s32/s32/s78/s32/s61/s32/s54/s48
+/s32/s32/s78/s32/s61/s32/s56/s48
+/s32/s32/s32
+/s32/s100 /s32/s40/s110/s109/s41/s81
+/s101/s110/s115/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s113/s32/s61/s32/s48/s46/s57/s53
+FIG. 9: (Color online) Fluorescence quantum efficiency vs.
+distance for several ensembles of high-yield emitters on sp her-
+ical lattices around AG NP.
+/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s49/s50/s51
+/s32/s32/s78/s32/s61/s32/s49
+/s32/s32/s78/s32/s61/s32/s50/s48
+/s32/s32/s78/s32/s61/s32/s51/s50
+/s32/s32/s78/s32/s61/s32/s54/s48
+/s32/s32/s78/s32/s61/s32/s56/s48/s65/s103
+/s82/s32/s61/s32/s50/s48/s32/s110/s109
+/s113/s32/s61/s32/s48/s46/s51
+/s32/s32/s32
+/s32/s100 /s32/s40/s110/s109/s41/s81
+/s101/s110/s115
+FIG. 10: (Color online) Fluorescence quantum efficiency vs.
+distance for several ensembles of low-yield emitters on sph er-
+ical lattices around AG NP.
+ofplasmonicDickeeffectis todrastically reducethe emis-
+sion as compared to same number of individual dipoles.
+Remarkably, as the eigenvalues scale uniformly withN,
+thequantum efficiencies ofsuperradiantstatesarenearly
+independent of the ensemble size, leading to the simple
+relation (1) that holds in the cooperative regime.
+This is illustrated in Figs. 9 and 10, which show en-
+semblequantum efficiencies Qens[see Eq. (37)] for two
+types of dyes with quantum yields q= 0.95 andq= 0.3,
+respectively. Two regimes can be clearly distinguished in
+the distance dependence of Qens: it first shows a sharp
+rise with its slope proportional to N(non-cooperative
+regime) followed by a slower ddependence (cooperative
+regime). The crossover between two regimes takes place
+atd≃5 nm due to diminished high- lplasmons con-
+tribution to nonradiative coupling for larger distances.
+In the cooperative regime, the precise behavior of Qens
+is affected by molecules’ quantum yield. For high- q
+molecules, all Qensdependences collapse onto a single
+curve,Qens= 3Q, while for low- qmolecules,Qensshowsa weak dependence on N. In both cases, this behavior
+can be easily understood from Eq. (42). Indeed, for de-
+generate superradiant eigenvalues we have λµ= 1, and
+for large distances, as Γ →Γr
+0, we obtain
+Qens≃3
+1+3(q−1−1)/N, (44)
+i.e., for large ensembles, the role of molecular quantum
+yield is diminished.
+V. CONCLUSIONS
+Westudiedhereplasmon-mediatedsuperradiancefrom
+an ensemble of dipoles near metal nanoparticle support-
+ing localized surface plasmon, thereby extending the
+Dicke effect to plasmonic systems. Our main conclu-
+sion is that the plasmonic Dicke effect is a robust phe-
+nomenon, more so than the usual photonic Dicke effect,
+because of a more efficient hybridization of individual
+dipoles via nanoparticle plasmon. We have established
+that hybridization takes place through two types of plas-
+monic coupling mechanisms – plasmon-enhanced radia-
+tive coupling and nonradiativeplasmon coupling, the lat-
+ter havingnoanaloguein the usual Dicke effect and caus-
+ing demise of cooperative emission at very close dipole-
+nanoparticle distances.
+While we considered a specific nanostructure – spheri-
+cal metal particle - the plasmon-mediated superradiance
+is a quite general phenomenon that should take place
+in any plasmonic system tuned into resonance with emit-
+ters,andEq.(1)shouldapplyprovidedthattheusualcri-
+teriaforcooperativeemissionaremet. Infact, ourtheory
+remains unchanged for any nanostructure with spherical
+symmetry, for example metal nanoshells with dielectric
+core, upon simple replacement of NP polarizabilities in
+self-energy matrix Eq. (12) and elsewhere with appropri-
+ate expressions. In fact, one expects that in nanoshells
+the nonradiatice losses would be smaller and so the plas-
+monic Dicke effect would be more robust than for solid
+nanoparticles.
+This work was supported in part by the NSF under
+Grant Nos. DMR-0906945and HRD-0833178,and under
+the EPSCOR Program.
+Appendix
+Here we collect relevant some formulas for the elec-
+tric field Green dyadic in the presence of metal NP. The
+Green dyadic satisfies Maxwell equation
+∇×∇׈G−k2ǫ(r)ˆG=ˆI, (45)
+whereǫ(r) =ǫ(ω)θ(R−r)+ǫ0θ(r−R) is local dielectric
+function (θ(x) is the step-function). The Green dyadic
+can be split into free space and Mie-scattered parts,11
+Gµν(r,r′) =G0
+µν(r,r′)+Gs
+µν(r,r′), where the free-space
+Green dyadic is
+G0
+µν(r−r′) =/parenleftbigg
+δµν−∇µ∇′
+ν
+k2/parenrightbigg
+g(r−r′),(46)
+with
+g(r) =eikr
+4πr(47)
+satisfying a scalar equation
+(△+k2)g(r) =−δ(r). (48)
+Considerfirstthefreespacepart. Itsnearfieldexpression
+can be obtained in the long wave approximation, i.e. by
+expanding in kr≪1. In the first order,
+G0
+µν(r) =1
+4πk2r3/bracketleftBig3rµrν
+r2−δµν/bracketrightBig
++ik
+6πδµν.(49)
+In the far field limit, i.e., kr≫1 andkr′≪1, the free-
+space part can be expanded via Bessel functions,
+eik|r−r′|
+4π|r−r′|=ik/summationdisplay
+lmjl(kr′)hl(kr)Ylm(r)Y∗
+lm(r′),(50)
+which are approximated as
+jl(kr′) =(kr′)l
+(2l+1)!!, hl(kr) = (−i)l+1eikr
+kr,(51)
+yielding
+G0
+µν(r,r′) =/parenleftBig
+δµν−1
+k2∇µ∇′
+ν/parenrightBigeikr
+r/bracketleftbigg1
+4π
+−ikr′
+3/summationdisplay
+mY1m(ˆr)Y∗
+1m(ˆr′)/bracketrightbigg
+.(52)
+After differentiation, the far field asymptotics takes the
+form
+G0
+µν(r,r′) =eikr
+4πr/bracketleftbigg
+δµν−4π
+3/summationdisplay
+mˆrµY1m(ˆr)χν∗
+1m(r′)/bracketrightbigg
+,
+(53)
+whereweintroduced χµ
+lm(r) =∇µ[rlYlm(ˆr)andψµ
+lm(r) =
+∇µ[r−l−1Ylm(ˆr)].
+Now turn to the scattered part of the Green dyadic
+derived from solution of Mie problem for electromagnetic
+wave scattered on single sphere,
+Gs
+µν(r,r′,k) =ik/summationdisplay
+lm/bracketleftbig
+alNµ
+lm(r)Nν
+lm(r′)+
+blMµ
+lm(r)Mν
+lm(r′)/bracketrightbig
+,(54)
+where the first and second terms are electric and mag-
+netic contributions and alandblare the Mie coefficients.
+In the long wave approximation, kR≪1, the magneticcontribution in Eq.(54) can be neglected as bl≪1.20–23
+The Mie coefficient alhas a form
+al=ǫ0jl(ρ0)[ρjl(ρ)]′−ǫjl(ρ)[ρ0jl(ρ0)]′
+ǫ0hl(ρ0)[ρjl(ρ)]′−ǫjl(ρ)[ρ0hl(ρ0)]′(55)
+whereρi=kiR,ki=ω
+c√ǫi, andi= (ǫ,ǫ0). ForkR≪1,
+it becomes
+al=−isl˜αlk2l+1, sl=l+1
+l(2l+1)[(2l−1)!!]2,(56)
+where
+˜αl=αl
+1−islk2l+1αl, (57)
+is NP multipolarpolarizabilitythat accountsfor plasmon
+radiative decay, and
+αl=R2l+1l(ǫ−ǫ0)
+lǫ+(l+1)ǫ0, (58)
+is the standard NP polarizability. The function Nlm(r)
+is given by
+Nlm(r) =1
+k/radicalbig
+l(l+1)∇×/bracketleftbig
+h(1)
+l(kr)LYlm(ˆr)/bracketrightbig
+,(59)
+whereL=−i(r×∇) is angular momentum operator.
+Using the following identity,
+∇×/bracketleftbig
+h(1)
+l(kr)LYlm(ˆr)/bracketrightbig
+=irk2h(1)
+l(kr)Ylm(ˆr)
++i∇/bracketleftbig
+[krh(1)′
+l(kr)+h(1)
+l(kr)]Ylm(ˆr)/bracketrightbig
+,(60)
+prime standing for derivative, and expanding h(1)
+l(kr) =
+jl(kr)+inl(kr) inkras
+h(1)
+l(kr) =(kr)l
+(2l+1)!!−i(2l−1)!!
+(kr)l+1, (61)
+we obtain
+Nlm(kr) =−1
+k/radicalbig
+sl(2l+1)∇[ϕl(kr)Ylm(ˆr)],(62)
+where
+ϕl(kr) =1
+(kr)l+1−isl(kr)l. (63)
+Thus, forkr≪1 andkr′≪1, the scattered part of the
+Green dyadic has the form
+Gs
+µν(r,r′,k)≈ik/summationdisplay
+lm/bracketleftbig
+alNµ
+lm(kr)Nν
+lm(kr′)/bracketrightbig
+(64)
+≈/summationdisplay
+lmk2l˜αl
+2l+1∇µ/bracketleftbig
+ϕl(kr)Ylm(ˆr)/bracketrightbig
+∇′
+ν/bracketleftbig
+ϕl(kr′)Y∗
+lm(ˆr′)/bracketrightbig
+.12
+This expression can be further simplified by substituting
+˜αl= ¯αl+islk2l+1|˜αl|2, where ¯αl=αl|1−islk2l+1αl|−2,
+and keeping the first two powers of k
+Gs
+µν(r,r′) =1
+k2/summationdisplay
+lm¯αl
+(2l+1)ψµ
+lm(r)ψν∗
+lm(r′)
+−iks1
+31/summationdisplay
+m=−1/bracketleftBig
+˜α1/bracketleftbig
+ψµ
+1m(r)χν∗
+1m(r′)+χµ
+1m(r)ψν∗
+1m(r′)/bracketrightbig
+−|˜α1|2ψµ
+1m(r)ψν∗
+1m(r′)/bracketrightBig
+, (65)
+which, after adding the free-space part of the Green
+dyadic and neglecting plasmon radiative decay, leads to
+Eq. (12).Forkr≫1, with help of Eqs. (51), (59), and (60), we
+easily obtain
+Nlm(r) =−(−i)l+1eikr
+k/radicalbig
+l(l+1)∇Ylm(ˆr), (66)
+and combining this expression with Eq. (62), we obtain
+the far field Green dyadic (i.e., kr≫1 andkr′≪1)
+Gs
+µν(r,r′) =−˜α1
+31/summationdisplay
+m=−1eikr/bracketleftbig
+∇µY1m(ˆr)/bracketrightbig
+ψν∗
+1m(r′),(67)
+where we set l= 1.
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+arXiv:1001.0423v1  [cond-mat.stat-mech]  4 Jan 2010Novel Features Arising in Maximally Random Jammed Packings
+of Superballs
+Y. Jiao1, F. H. Stillinger2and S. Torquato2,3,4,5
+1Department of Mechanical and Aerospace Engineering,
+Princeton University, Princeton New Jersey 08544, USA
+2Department of Chemistry, Princeton University,
+Princeton New Jersey 08544, USA
+3Program in Applied and Computational Mathematics,
+Princeton University, Princeton New Jersey 08544, USA
+5Princeton Center for Theoretical Physics,
+Princeton University, Princeton New Jersey 08544, USA and
+4School of Natural Sciences, Institute for Advanced Study, P rinceton NJ 08540
+1Abstract
+Dense random packings of hard particles are useful models of granular media and are closely
+relatedtothestructureofnonequilibriumlow-temperatur eamorphousphasesofmatter. Mostwork
+has been done for random jammed packings of spheres, and it is only recently that corresponding
+packings of nonspherical particles (e.g., ellipsoids) hav e received attention. Here we report a
+study of the maximally random jammed (MRJ) packings of binar y superdisks and monodispersed
+superballs whose shapes are defined by |x1|2p+···+|xd|2p≤1 withd= 2 and 3, respectively,
+wherepis the deformation parameter with values in the interval (0 ,∞). Aspincreases from zero,
+one can get a family of both concave (0 < p <0.5) and convex ( p≥0.5) particles with square
+symmetry ( d= 2), or octahedral and cubic symmetry ( d= 3). In particular, for p= 1 the particle
+is a perfect sphere (circular disk) and for p→ ∞the particle is a perfect cube (square). We find
+that the MRJ densities of such packings increase dramatical ly and nonanalytically as one moves
+away from the circular-disk and sphere point ( p= 1). Moreover, the disordered packings are
+hypostatic, i.e., the average number of contacting neighbo rs is less than twice the total number of
+degrees of freedom per particle, and the packings are mechan ically stable. As a result, the local
+arrangements of particles are necessarily nontrivially co rrelated to achieve jamming. We term such
+correlated structures “nongeneric”. The degree of “nongen ericity” of the packings is quantitatively
+characterized by determining the fraction of local coordin ation structures in which the central
+particles have fewer contacting neighborsthan average. We also show that such seemingly “special”
+packing configurations are counterintuitively not rare. As the anisotropy of the particles increases,
+the fraction of rattlers decreases while the minimal orient ational order as measured by the cubatic
+order metric increases. These novel characteristics resul t from the unique rotational symmetry
+breaking manner of the particles, which also makes the super disk and superball packings distinctly
+different from other known nonspherical hard-particle packi ngs.
+PACS numbers: 61.50.Ah, 05.20.Jj
+2I. INTRODUCTION
+Particle packing problems, such as how to fill a volume with given solid ob jects as densely
+aspossible, areamongthemost ancient andpersistent problems ins cience andmathematics.
+Apackingis a large collection of non-overlapping solid objects (particles) in d-dimensional
+Euclidean space Rd. The packing density φis defined as the fraction of space Rdcovered by
+the particles. Dense ordered and random packings of nonoverlapp ing (hard) particles have
+been employed to understand the equilibrium and non-equilibrium stru cture of a variety
+many-particle systems, including crystals, glasses, heterogeneo us materials and granular
+media [1–4]. Packing problems in dimensions higher than three attract current interest for
+retrieving stored data transmitted through a noisy channel [5–8 ].
+The packings of congruent hard spheres in R3have been intensively studied since despite
+the simplicity they exhibit rich packing characteristics. It is only rece ntly that the densest
+packings with φmax=π/√
+18≈0.74, realized by the face-centered cubic lattice and its
+stacking variants, have been proved [9]. In addition, three-dimen sional random packings
+can be prepared both experimentally and numerically with a relatively r obust density φ≈
+0.64 [10, 11]. The term random close packing (RCP) [12], widely used to designate the
+“random” packing with the highest achievable density, is ill-defined sin ce random packings
+can be obtained as the system becomes more ordered and a definitio n of randomness has
+been lacking. A more recent concept that has been suggested to r eplace RCP is that of
+the maximally random jammed (MRJ) state [11], corresponding to the most disordered
+among all jammed (mechanically stable) packings. A jammed packing is one in which the
+particle positions andorientations arefixed by the impenetrability co nstraints andboundary
+conditions [13]. It has been established that the MRJ state for sphe res inR3has a density
+ofφ≈0.637, as obtained by a variety of different order metrics [13, 14]. This density value
+is consistent with what has traditionally asscoiated with RCP in three d imensions.
+It has been argued in the granular materials literature that large dis ordered jammed
+(MRJ) packings of hard frictionless spheres are isostatic [15, 16], meaning that the total
+number of interparticle contacts (constraints) equals the total number of degrees of freedom
+of system and that all of the constraints are (linearly) independen t. This implies that
+the average number of contacts per particle Zis equal to twice the number of degrees of
+freedom per particle f(i.e.,Z= 2f), in the limit as the number of particles gets large. This
+3prediction has been verified computationally with very high accuracy [17, 18]. On the other
+hand, a packing is hypostatic if it is mechanically stable (i.e., jammed) while the number
+of constraints is smaller than the number of degrees of freedom. F or large packings, this is
+equivalent to the inequality Z <2f. It has been shown that a jammed sphere packing can
+be not hypostatic [17].
+It is also of great practical and fundamental interest to underst and the organizing prin-
+ciples of dense packings of nonspherical particles [19–28]. The effec t of asphericity is an
+important feature to include on the way to characterizing more com pletely real dense granu-
+lar media aswell as low-temperaturestates of matter. Another imp ortant applicationrelates
+to supramolecular chemistry [29] of organic compounds whose molec ular constituents can
+possess many different types of group symmetries [30]. Such syste ms can be approximated
+by nonspherical hard particles with the same group symmetries.
+Recently, MRJ packings of three-dimensional ellipsoids [31, 32] have been studied. In
+particular, it was found that the density φand the average coordination number Z(the
+average number of touching neighbors per particle) increase rapid ly, in a cusp-like manner,
+as asphericity is introduced from the sphere point. The density φreaches a maximum at a
+critical aspect ratio α∗[33] and then begins to decrease; while Zis increasing monotonically
+until it attains the plateau value for all αbeyondα∗. In addition, Zis always smaller than
+twice the number of degrees of freedom per particle fwith its plateau value slightly below
+2f(for an ellipse f= 3 and for an ellipsoid f= 6). In other words, the packings are
+hypostatic [32]. The characteristics of MRJ ellipsoid packings are distinctly differe nt from
+their densest crystalline counterpart [21], in which φincreases smoothly as one moves away
+from the sphere point, and reaches a plateau value of 0 .7707...forα >√
+3 (oblate spheroids)
+andα <1/√
+3 (prolate spheroids).
+In Refs. [25] and [26], we studied dense and maximally dense packings o fsuperballs ,
+a family of nonspherical particles with versatile shapes. In particula r, ad-dimensional
+superball is a centrally symmetric body in Rdoccupying the region
+|x1|2p+|x2|2p+···+|xd|2p≤1, (1)
+wherexi(i= 1,...,d) are Cartesian coordinates and p≥0 is thedeformation parameter ,
+which indicates to what extent the particle shape has deformed fro m that of a d-dimensional
+sphere (p= 1). Henceforth, the terms superdisk andsuperball will be our designations for
+4FIG. 1: (color online). Superdisks with different values of th e deformation parameter p.
+FIG. 2: (color online). Superballs with different values of th e deformation parameter p.
+the two-dimensional ( d= 2) and three-dimensional ( d= 3) cases, respectively. A superdisk
+possesses square symmetry, as pmoves away from unity, two families of superdisks can
+be obtained, with the symmetry axes rotated 45 degrees with resp ect to each other; when
+p <0.5, the superdisk is concave (see Fig. 1). A superball can possess t wo types of shape
+anisotropy: cube-like shapes (for p >1) and octahedron-like shapes (for 0 < p <1) with a
+shape change from convexity to concavity as ppasses downward through 0 .5 (see Fig. 2).
+Optimal packings of congruent superdisks and superballs apparen tly are realized by cer-
+tain Bravais lattices possessing symmetries consistent with those o f the particles [25, 26, 28].
+Even these crystalline packings exhibit rich characteristics that ar e distinctly different from
+other known packings of nonspherical particles. For example, we f ound that the maximal
+densityφmaxas a function of patp= 1 (the sphere or circular-disk point) is nonanalytic
+and increases dramatically as pmoves away from unity. In addition, we have discovered
+two-fold degenerate maximal density states for square-like supe rdisks, and both cube-like
+and octahedron-like superballs.
+In this paper, we generate both packings of binary superdisks in R2and monodisperse
+superballs in R3that represent the maximally random jammed (MRJ) state of these parti-
+cles, using a novel event-driven molecular dynamics algorithm [34, 35 ] and investigate their
+5characteristics. For both superdisks and superballs, we find that the corresponding density
+φand the average contact number Zincrease rapidly, in a cusp-like manner, as the particles
+deviate from perfect circular disks and spheres, respectively. In particular, we find that the
+MRJ packing density φincreases monotonically as pmoves away from unity, and shows no
+signs of a plateau even for large pvalues. This is to be contrasted with the case of ellipsoids
+for which the packing density reaches a maximum as the aspect ratio ratio increases from
+its sphere-point value and then begins to decrease as the aspect r atio grows beyond that
+associated with the density-maximum value.
+Moreover, wefindthat Zforsuperdisk andsuperballpackings reachitsassociatedplateau
+value at relatively small asphericity deviations (i.e., |p−1|) and the packings remain hypo-
+staticfor all values of pexamined. By “hypostatic”, we mean that the Zis smaller than
+twice the number of degrees of freedom per particle, compared to random ellipsoid packings
+where the plateau value of Zis only slightly below 2 f. Therefore, to achieve jamming,
+the local particle arrangements are necessarily correlated in a non trivial way. We call such
+correlated structures “nongeneric” [36]. We quantify the degree of “nongenericity” of the
+packings by determining the fraction of local coordination configur ations in which the cen-
+tral particles have fewer contacting neighbors than average Z. We also show that such
+“nongeneric” configurations are not rare, which is a rather count erintuitive conclusion. In
+addition, we find that although the rapid increase of density is unrela ted to any observable
+translational order, the orientational order (e.g., the cubatic or der parameter [37]) increases
+aspmoves away from unity. These packing characteristics, which are d istinctly different
+from that of the MRJ packings of ellipsoids, are due to the unique way in which rotational
+symmetry is broken in superdisk and superball packings.
+The rest of the paper is organized as follows: In Sec. II, we briefly d escribe the simulation
+techniques andthe obtained packings. In Sec. III, we provide a de tailed analysis of the novel
+packing characteristics. In Sec. IV, we make concluding remarks.
+II. MAXIMALLY RANDOM JAMMED PACKINGS VIA COMPUTER SIMULA-
+TION
+We use an event-driven molecular dynamics packing algorithm recent ly developed by
+Donev, Torquato and Stillinger [34, 35] (henceforth, referred to as the DTS algorithm) to
+6generate MRJ packings of convex superdisks in two dimensions and s uperballs in three
+dimensions. The DTS algorithm generalizes the Lubachevsky-Stillinge r (LS) sphere-packing
+algorithm [38] to the case of other centrally symmetric convex bodie s (e.g., ellipsoids and
+superballs). Initially, small particles are randomly distributed and ra ndomly oriented in
+the simulation box (fundamental cell) with periodic boundary conditio ns and without any
+overlap. The particles are then given translational and rotational velocities randomly and
+their motionfollowed asthey collide elastically andalso expand uniformly with anexpansion
+rateγ, while the fundamental cell deforms to better accommodate the c onfiguration. After
+sometime, ajammedstatewithadiverging collisionrateisreached and thedensity reaches a
+local maximum value. To generate random jammed packings, initially lar geγare employed
+to prevent the system following the equilibrium branch of the phase d iagram that leads to
+crystallization. Near the jamming point, sufficiently small expansion r ate is necessary for
+the particles to establish contacting neighbor networks and to for m a truly jammed packing.
+On the basis of our experience with spheres [17] and ellipsoids [31], we b elieve that our
+algorithm with rapid particle expansion produces final states that r epresent the MRJ state
+well. Here we use the largest possible initial γ∈(0.1−0.5) that is numerically feasible
+to ensure the generated superdisk and superball packings are ma ximally random jammed.
+We mainly focus on superdisks and superballs with deformation param eterspwithin the
+range 0.85−3.0, since extreme values of passociated with polyhedron-like shapes present
+numerical difficulties.
+All of the generated packings used in the subsequent analyses are verified to be at least
+collectively jammed using an “infinitesimal shrinkage” method [13], i.e., t he particles in the
+packing areshrunk by avery small amount [39] andgiven randomve locities. If no significant
+structural changes occur after the system “relaxes” after a s ufficiently long enough time, the
+packing is considered to be collectively jammed. It is well established t hat the “infinitesimal
+shrinkage” methodisrobust, i.e., italwaysgives thesameresultsfor spherepackings asthose
+obtained from a rigorous linear programming jamming test algorithm p rovided the amount
+ofshrinkagefromthejammedstateissufficiently small foragiven nu mber ofparticleswithin
+the periodic cell [40].
+7(a)p= 0.85 (b) p= 1.5
+FIG. 3: (color online). Typical configurations of MRJ packin gs of binary superdisks with different
+valuesofthedeformationparameter p. Thechordsshowoneofthesymmetryaxesofthesuperdisks.
+1 1.5 2 2.5 3p0.840.860.880.90.92
+φ
+11.5 22.5 3
+p44.24.44.64.8
+Z
+FIG. 4: The density of MRJ packings of binary superdisks as a f unction of p. Insert: the average
+contact number Zas a function of p.
+A. Binary Mixtures of MRJ Superdisks
+In two dimensions, we study MRJ packings of a specific family of binary superdisk mix-
+tures in which the size ratio is κ= 1.4 and the molar ratio is β= 1/3. The size ratio κis
+defined as the ratio of the diameter of large superdisks over that o f the small superdisks;
+and the molar ratio βis defined as the number large superdisks over the number of small
+8(a)p= 0.85 (b)p= 1.5
+FIG. 5: (color online). Typical configurations of MRJ packin gs of superballs with different values
+of the deformation parameter p.
+superdisks. We do not use monodispersed superdisk systems here because they are easily
+crystallized into ordered packings [25]. For p= 1, one obtains the binary circular-disk sys-
+tem which has been intensively studied as a prototypical glass forme r [41]. Typical jammed
+packing configurations are shown in Fig. 3. The density φand the average contact number
+per particle Zas a function of pare shown in Fig. 4, which reveals that the initial rapid in-
+creases of φandZare linear in |p−1|[42]. The density φincreases monotonically as pmoves
+away from unity and shows no signs of a plateau, even for relatively la rgep. In addition,
+φquickly surpasses the density of the optimal binary circular-disk pa cking associated with
+the size and molar ratios employed here, which contains phase-sepa rate regions of triangular
+lattice packings of different sized circular disks [41]. The quantity Zquickly reaches its
+plateau value Z∗≈4.7 atp≈1.3, which is smaller than 2 f= 6, indicating the packings are
+hypostatic. The cubatic order parameter P4is defined as P4=/angbracketleft(35cos4θ−30cos2θ+3)/8/angbracketright,
+whereθis the angle between the particle axis and the director, along which th e principle
+axes of the particles have maximum mutual alignment [37]. The measu red cubatic order
+parameter is P4≈0.06 to 0.32 with the tendency to increase as |p−1|grows.
+91 1.5 2 2.5 3p0.6250.650.6750.70.7250.75
+φ
+11.5 22.5 3
+p66.577.58
+Z
+FIG. 6: The density of MRJ packings of superballs as a functio n ofp. Insert: the average contact
+numberZas a function of p.
+B. MRJ Packings of Monodisperse Superballs
+In three dimensions, monodispersed superballs can be easily compre ssed into a jammed
+random packing due to geometrical frustration (i.e., the densest lo cal particle arrangement
+cannot tile space). Typical jammed packing configurations are sho wn in Fig. 5. The quan-
+titiesφandZas a function of pare shown in Fig. 6, respectively. As in two dimensions,
+φandZincrease rapidly, in a cusp-like manner [43], as the particles deviate fr om perfect
+sphere.φincreases monotonically as pmoves away from unity, quickly goes beyond the
+optimal sphere packing density and shows no signs of plateau, even for relatively large p
+values. The contact number per particle Zreaches its plateau value Z∗≈8.15 atp≈1.4,
+which is significantly smaller than 2 f= 12, indicating that the packings are hypostatic. The
+measured cubatic order parameter is P4≈0.03 to 0.21, which increases with |p−1|.
+III. PACKING CHARACTERISTICS
+A. Rattlers
+MRJ packings generated in both two and three dimensions contain a s mall fraction of
+rattlers, i.e., particles that can wander freely within cages formed b y their jammed non-
+10rattling neighbors. When pis close to unity, the fraction of rattlers is approximately 2 .6%
+and 1.2% for two and three dimensions, respectively. As pmoves away from unity, the
+fraction of rattlers decreases quickly and practically vanishes for largep(e.g.,p >2.75).
+This behavior results from the increasing protuberance of the par ticle shape, which makes it
+more difficult to form isotropic cages and also requires more average contacts per particle to
+achieve jamming. Note that rattlers are excluded when reporting a verage contact numbers
+in the following discussion.
+B. Packing Density
+The rapid increase of the density is mainly due to the broken rotation al symmetry of
+the particles. In particular, the cubic-like (square-like) and octah edral-like particles are
+more efficient to cover the space than spheres (circular disks), i.e., near the jamming point
+the particles can rotate to accommodate the neighbors by orientin g the “far corners” to
+fill the available gaps and thus cover more space. For small values of p, the increase in
+φis also attributed to the expected increase in the number of contac ting neighbors per
+particle, which means locally more particles can be packed in a given volu me. The manner
+in which rotational symmetry is broken in superball packings is distinc tly different from that
+in ellipsoid packings. For example, the asphericity γ[27, 28], defined as the ratio of the radii
+of circumsphere andinsphere of a nonspherical particle, is always b ounded andclose to unity
+for all values of pfor superballs, while it can increase without limit as the largest aspect ratio
+αgrows for ellipsoids. For very elongated or flake-like ellipsoids with larg e aspect ratios,
+the effect of a very anisotropic exclusion volume becomes dominant a nd causes the density
+of random ellipsoid packings to decrease. By contrast, the shape o f superballs becomes
+more efficient in filling space as the deformation parameter deviates m ore from unity and
+thus results in a monotonically increasing density. The nonanalyticity ofφatp= 1 is
+also associated with the broken symmetry of superdisks and super balls. This nonanalytical
+behavior has also been observed in the optimal packings of these pa rticles realized by various
+Bravais lattices [25, 26]. This stands in contrast to the densest kn own ellipsoid packings,
+which are periodic packings with a two-particle basis possessing a smo oth initial increase of
+φmaxas the aspect ratio moves away from unity [21].
+11(a) (b)
+FIG. 7: (color online). (a) Distribution of contact numbers for different pvalues for MRJ packings
+of superdisks (upper panel) and superballs (lower panel). ( b) Local packing structures with more
+contacts than average (shown in blue) and those with less con tacts than average (shown in pink)
+in two-dimensional superdisk packings for different pvalues.
+C. Hypostaticity and Nongeneric Local Structures
+There have been conjectures [15, 16] that frictionless random pa ckings have just enough
+constraints to completely statically define the system (i.e., it is isostatic), i.e., for large
+packings, one has Z= 2f. It has been shown both experimentally and computationally
+that although the isostatic conjecture [15] holds for large sphere packings [40, 44], it is
+generally not applicable to nonspherical particles, such as ellipsoids [3 1, 32]. It was found
+that even for ellipsoids with large aspect ratios, Zis still slightly below 2 f[31].
+Here we observe that in MRJ packings of superdisks and superballs Zis significantly
+smaller than 2 ffor all values of pexamined, i.e., the packings are significantly hypostatic .
+The hypostatic packings result from the competition between fTtranslational and fRro-
+tational degrees of freedom of the particles ( f=fR+fT) in developing the contacting
+networks close to the jamming point. In particular, although it is tru e that to constrain
+the translational degrees of freedom each particle needs at least 2fTcontacts, rotational
+degrees of freedom can be blocked with less than 2 fRadditional contacts per particle if
+12the curvatures at the contacting points are sufficiently small [32]. I n addition, due to the
+relatively small asphericity γof superdisks and superballs, there is little reason to expect the
+rotational motions of these particles (especially those with pclose to unity) would be frozen
+even when they are translational trapped and may only rattle inside small “cages” formed
+by their neighbors. Near the jamming point, it is expected that the p articles can rotate
+significantly [45] until the actual jamming point is reached, at which r otational jamming
+will also come into play, and rotational degrees of freedom are froz en with the number of
+additional contacts much less than 2 fR. This is in contrast to hypostatic MRJ packings
+of ellipsoids with large aspect ratios, for which the translational and rotational degrees of
+freedom are on the same footing and, thus, the average contact number per particle is only
+slightly below twice the number of total degrees of freedom.
+Furthermore, thelocalgeometryoftheMRJpackingsisnecessar ilynontriviallycorrelated
+(nongeneric), i.e., allthenormalvectorsatthepointsofcontact foraparticleshouldintersect
+at a common point to achieve torque balance and block rotations. In light of the isostatic
+conjecture, the local packing structures are less nongeneric wh en they possess larger contact
+numbers so that the constraining neighbors are less correlated. T he truly generic local
+packing structures should have Z= 2fper particle, for which the constraining neighbors
+could be completely uncorrelated. To characterize the “nongener icity” of the packings, we
+compute Gng, the fraction of local structures composed of particles with less c ontactsZlocal
+than average Zaverage, i.e.,
+Gng=N(Zlocal≤Zaverage)
+Ntotal. (2)
+A larger Gngindicates a larger degree of nongenericity. We find Gngis approximately
+0.65 in two dimensions and 0.78 in three dimensions when pis close to unity, which quickly
+decreases and plateaus at 0.6 and 0.68, respectively as pincreases. Figure 7 shows the
+distribution of contact numbers for different pvalues and the topologyof the local structures
+contributingto Gng. Itcanbeseenthatas pmovesawayfromunity, thedistributionsbecome
+moreskewed asthemeans shift to larger Z. Moreover, thesubset ofparticles associated with
+the nongeneric structures do not percolate. We do not observe a ny tendency of increasing
+Zeven for the largest pvalues that are computationally feasible and we expect that MRJ
+packings in the cubic limit are also hypostatic. It is noteworthy that is ostatic random
+packings of superdisks and superballs are difficult to construct, sin ce achieving isostaticity
+13(a) (b) (c)
+FIG. 8: (color online). Nongeneric locally jammed configura tions associated with three fixed su-
+perdisks (pink) and the trapped one (blue). In each configura tion, the central superdisk is approx-
+imately aligned with one of its fixed neighbors to form contac ts associated with small curvatures
+to block rotation.
+requiresZ= 2f= 12 which is necessarily associated with translational crystallization [46].
+We note that the aforementioned nongeneric structures (see Fig . 7(b)) are not rare. In
+particular, a nonspherical particle can be rotationally jammed if it ha s neighbors that can
+translationally jamthe particle [32]. To illustrate this point, we will cons ider a small packing
+composed of four superdisks in two dimensions. Now we show that on e can locally jam a
+superdisk by three contacting neighbor superdisks. Translationa l jamming requires that
+the centroids of the neighbors cannot lie in the same semi-circle arou nd the centroid of
+the central superdisk. Suppose a superdisk is translationally trap ped (not jammed) by its
+three neighbors, whose positions and orientations are fixed. This f our-particle configuration
+has four degrees of freedom: two translational and one rotation al degrees of freedom of the
+trapped particle as well as the expansion of the particles. To obtain a jammed configuration,
+the four degrees of freedom need to be completely constrained. T his can be achieved by the
+three contact conditions for the jammed particles and its neighbor s and the requirement
+that the three inward normal vectors at the contacting points me et at a common point, a
+sufficient condition for torque balance [32]. Thus, one has four indep endent equations for
+the four degrees of freedom; see the Appendix for details.
+Figure 8 shows the nongeneric jammed configurations associated w ith three specific fixed
+trapping superdisks. The multiplicity ofthe configurations is dueto t he multiple solutions of
+the equations. The jamming configurations can be also obtained usin g the DTS algorithm,
+14which allows the trapped particle to translate and rotateand allows a ll the particles to grow.
+Although the above analysis is for local jamming (i.e., the neighbors of the central particle
+are fixed), it is reasonable to expect that collective particle rearra ngements further facilitate
+the formation of nontrivial orientational correlations and thus, e nable a larger number of
+nongeneric jamming configurations. Indeed, the numerous hypos tatic jammed packings that
+we found from our simulations strengthen our argument that nong eneric structures are not
+rare.
+D. Nonvanishing Orientational Order
+We also observe the increase of the orientational order (measure d byP4) associated with
+the increasing pvalues, although the largest possible expansion rate γhas been used to
+suppress the formation of orders (i.e., to maintain the maximal degr ee of randomness) [47].
+Aspdeviates from unity, the particle shape develops “edges” and “cor ners” with large
+curvatures, which may not be able to block rotational unjamming mo tions if contacts occur
+at significantly curved regions of the particle surface. On the othe r hand, low-curvature
+contacts aremorefavorable, which isassociated with partialalignm ents oftheparticles. The
+tendency of particle alignments to form low-curvature surface co ntacts required by jamming
+becomes stronger as the particle moves further away from the sp here point. Thus, there is
+also a competition between orientational disorder and jamming for p ackings of superdisks
+and superballs, resulting from their unique symmetry-breaking man ner, which has not been
+observed in randompackings ofellipsoids. Dueto numerical difficulties , we could not use the
+DTS algorithm to study the random jammed packings of particles with extreme shapes, i.e.,
+in the limit p→0.5 andp→ ∞. However, it reasonable to expect considerable orientational
+ordering in such packings.
+IV. CONCLUSIONS
+In this paper, we studied the maximally random jammed packings of su perdisks and su-
+perballs. Thepacking densities increase dramaticallyandnonanalytic ally asonemoves away
+from the circular-disk and sphere point ( p= 1) and the packings are hypostatic. To achieve
+jamming, the local arrangements of particles are necessarily non- trivially correlated and we
+15termthesestructures“nongeneric”inlightofthecorrelations. T hedegreeof“nongenericity”
+of the packings is quantitatively characterized by the fraction of lo cal structures composed
+of particles with less contacts than average. Moreover, we showe d that such seemingly “spe-
+cial” packing configurations are not rare. As the anisotropy of the particles increases, the
+fraction of rattlers decreases while the minimal orientational orde r increases. The novel fea-
+tures arising in MRJ packings of superdisks and superballs result fro m the unique manner
+in which rotational symmetry is broken. This makes such packings dis tinctly different from
+other known MRJ packings of nonspherical particles, such ellipsoids and ellipses.
+The ability to produce dense random packings using superballs casts new lights on sev-
+eral industrial processes, such as sintering and ceramic formatio n, where interest exists in
+increasing the density of powder particles to be fused. If superba ll-like particles instead
+of spherical particles are used, the packing density of a randomly p oured and compacted
+powder could be increased to a value surpassing that of the maximal sphere-packing density.
+We note that superdisks and superballs can be experimentally mass p roduced using current
+lithography techniques. Understanding the statistical thermody namics of the jamming tran-
+sition of superdisks and superballs, especially the role of rotational and translational degrees
+of freedom for different deformation parameters is a subject tha t merits future investiga-
+tion. Such studies could may our understanding of the nature of gla ss transitions, since the
+preponderance of previous investigations have focused on spher ical particles.
+APPENDIX: EQUATIONS FOR LOCALLY JAMMED FOUR-SUPERDISK
+CONFIGURATIONs
+In this section, we provide the equations that determine locally jamm ed four-superdisk
+configurations composed of a trapped central particle and three fixed contacting neighbors
+(see Fig. 8). In particular, the boundary of a superdisk with radius Ris define by
+/vextendsingle/vextendsingle/vextendsinglex1
+R/vextendsingle/vextendsingle/vextendsingle2p
++/vextendsingle/vextendsingle/vextendsinglex2
+R/vextendsingle/vextendsingle/vextendsingle2p
+= 1, (A-1)
+which can be also expressed by the parametric equations
+x1(θ) =|cosθ|1
+p·R·sign(cosθ),
+x2(θ) =|sinθ|1
+p·R·sign(sinθ),(A-2)
+16wheresign(x) gives the sign of argument x. Let the centroids of the three fixed neighbors
+be (ai,bi) (i= 1,2,3), and the orientations be θi. Their boundaries are then given by
+x(i)
+1(θ) =ai+cosθi|cosθ|1
+p·R·sign(cosθ)+sinθi|sinθ|1
+p·R·sign(sinθ),
+x(i)
+2(θ) =bi−sinθi|cosθ|1
+p·R·sign(cosθ)+cosθi|sinθ|1
+p·R·sign(sinθ).(A-3)
+Similarly, ifthecentroidofthecentralparticleisat( ao,bo)anditsorientationischaracterized
+byθo, its boundary is specified by
+xo
+1(θ) =ao+cosθo|cosθ|1
+p·R·sign(cosθ)+sinθo|sinθ|1
+p·R·sign(sinθ),
+xo
+2(θ) =bo−sinθo|cosθ|1
+p·R·sign(cosθ)+cosθo|sinθ|1
+p·R·sign(sinθ).(A-4)
+Since the positions and orientations of the three neighbors are fixe d, the four-particle system
+has four degrees of freedom, namely the position ( ao,bo) and orientation θoof the central
+particle, as well as the radius Rof all particles, as discussed in Sec. III.C.
+In the jammed configuration, the central particle contacts all its three neighbors. From
+Eqs. (A-1)and(A-4), thecontact point ( x(i)
+1c,x(i)
+2c)between neighbor particle iandthecentral
+particle can be expressed in terms of ( ao,bo,θo,R), which must also lie on the boundary of
+the neighboring particle i, i.e.,
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex(i)
+1c(ao,bo,θo,R)−ai
+R/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2p
++/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex(i)
+2c(ao,bo,θo,R)−bi
+R/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2p
+= 1, (A-5)
+fori= 1,2,3. This leads to three equations in the variables ao,bo,θo, andR. In addition,
+to achieve jamming the three normals at contacts must meet at a co mmon point, which
+guarantees torque balance. The normals at contacts are along th e lines
+(x2−x(i)
+2c) =−dx(i)
+1/dθ
+dx(i)
+2/dθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+(x(i)
+1c,x(i)
+2c)(x1−x(i)
+1c). (A-6)
+The aforementioned torque balance condition requires that the th ree lines given by Eq.(A-6)
+must intersect at a common point. This leads to another equation in t he variables ao,bo,θo,
+andR. Thus, there are four independent equations for the four degre es of freedom and
+(ao,bo,θo,R) can be completely determined for a locally jammed four-particle con figuration.
+17ACKNOWLEDGMENTS
+The authors thank Robert Batten and Aleksandar Donev for valua ble discussions. S. T.
+thanks the Institute for Advanced Study for its hospitality during his stay there. This work
+was supported by the Division of Mathematical Sciences at the Natio nal Science Foundation
+under Award Number DMS-0804431 and by the MRSEC Program of th e National Science
+Foundation under Award Number DMR-0820341.
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+conditions up to the same order of magnitude as the relative a mount of shrinkage. To imple-
+ment this method numerically for the system sizes that have b een examined, we shrink the
+19particles by a finite small amount such that the sizes of inter -particle gaps increase from 10−6
+to 10−4of the particle diameter.
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+[42] The numerically computed right and left slope of φ(p) for binary superdisks at p= 1 are
+a+≈0.135 and a−≈ −0.083, respectively.
+[43] The numerically computed right and left slope of φ(p) for monodispersed superballs at p= 1
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+[45] The fact that the superdisks and superballs can signific antly explore their configurational
+space associated with the rotational degrees of freedom, i. e., they can rotate significantly,
+near the jamming point is manifested as long-period oscilla tions of the pressure of the system,
+which is also observed for ellipsoid systems.
+[46] In this sense, it is similar to the ellipse packings in tw o dimensions, i.e., the isostatic packing
+requiressixcontacts perparticles, whichcanonlybereali zed withtranslational crystallization.
+[47] Expansion rates γwithin the range 0 .1−0.5 were employed to suppress the formation of order
+in the superdisk and superball packings; larger γwould cause numerical instability of the DTS
+algorithm. Note that for ellipsoids, γ∼0.05 is sufficient to produce random packings with
+vanishing orientational order [31, 32].
+20
\ No newline at end of file
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+arXiv:1001.0424v1  [math.OA]  4 Jan 2010Families of Type III KMS States on a Class of C∗-Algebras containing OnandQN
+A.L. Careya, J. Phillipsb,c, I.F. Putnamb, A. Renniea
+aMathematical Sciences Institute, Australian National Uni versity, Canberra, ACT, AUSTRALIA
+bDepartment of Mathematics and Statistics, University of Vi ctoria, Victoria, BC, CANADA
+cCorresponding Author. email: johnphil@uvic.ca; telephon e: 250-721-7450; fax: 250-721-8962
+Abstract
+We construct a family of purely infinite C∗-algebras,Qλforλ∈(0,1) that are classified by their
+K-groups. There is an action of the circle Twith a unique KMS state ψon eachQλ.Forλ= 1/n,
+Q1/n∼=On, with its usual Taction and KMS state. For λ=p/q,rational in lowest terms, Qλ∼=On
+(n=q−p+1) with UHF fixed point algebra of type ( pq)∞.For anyn >0,Qλ∼=Onfor infinitely
+manyλwith distinct KMS states and UHF fixed-point algebras. For an yλ∈(0,1),Qλ/\e}atio\slash=O∞.Forλ
+irrational the fixed point algebras, are NOT AF and the Qλare usually NOT Cuntz algebras. For λ
+transcendental, K1∼=K0∼=Z∞, so thatQλis Cuntz’QN, [Cu1]. Ifλ±1are both algebraic integers,
+theonlyOnwhich appear satisfy n≡3(mod4).For eachλ, the representation of Qλdefined by the
+KMS state ψgenerates a type III λfactor. These algebras fit into the framework of modular inde x
+(twisted cyclic) theory of [CPR2, CRT] and [CNNR].
+Keywords: KMS state, IIIλfactor, modular index, twisted cyclic theory, K-Theory.
+AMS Classification codes: 46L80, 58J22, 58J30.
+1.Introduction
+In this paper we introduce some new examples of KMS states on a large class of purely infinite C∗-
+algebras that were motivated by the ‘modular index theory’ o f [CPR2, CNNR]. We were aiming to
+find examples of algebras that were not Cuntz-Krieger algebr as (or the CAR algebra) and were not
+previously known in order to explore the possibilities open ed by [CNNR]. These algebras, denoted by
+Qλfor 0<λ<1, are not constructed as graph algebras, but as “corner alge bras” of certain crossed
+productC∗-algebras. TheQλhave similar structural properties to the Cuntz algebras, h owever there
+are important new features, such as
+1) whenλ=p/qis rational in lowest terms, then Qλ∼=Oq−p+1as mentioned in the Abstract,
+2) whenλis algebraic, the K-groups depend on the minimal polynomial (and its coefficient s) ofλ,
+3) whenλis transcendental, Qλ∼=QN, Cuntz’ algebra, [Cu1].
+We prove in Section 3 that the Qλare purely infinite, simple, separable, nuclear C∗-algebras, so there
+is no nontrivial trace on them. Also in Section 3 we determine in many cases the K-groups of these
+algebras and use classification theory to identify them when these algebras have the same K-groups
+as others found previously (these facts are summarised in th e Abstract). As each Qλcomes equipped
+with a gauge action of the circle, our results thus give an unc ountable family of distinct circle actions
+onQN, each with its own unique KMS state. Indeed, for all 0 <λ<1, we find a unique KMS state,
+[BR2], for this gauge action, and we prove in Section 4 that th e GNS representation of Qλassociated
+to our KMS state generates a type III λvon Neumann algebra. The result of [CPR2] that motivated
+this paper was the construction of a ‘modular spectral tripl e’ with which one may compute an index
+pairing using the KMS state. In [CNNR] it was shown how modula r spectral triples arise naturally for
+12
+KMS states of circle actions and lead to ‘twisted residue coc ycles’ using a variation on the semifinite
+residue cocycle of [CPRS2]. It is well known that such twiste d cocycles can not pair with ordinary
+K1. In [CPR2, CRT] a substitute was introduced which is called ‘ modularK1’. The correct definition
+of modular K1was found in [CNNR], and there is a general spectral flow formu la which defines the
+pairing of modular K1with our ‘twisted residue cocycle’.
+There is a strong analogy with the local index formula of nonc ommutative geometry in the L1,∞-
+summable case, however, there are important differences: the usual residue cocycle is replaced by
+a twisted residue cocycle and the Dixmier trace arising in th e standard situation is replaced by a
+KMS-Dixmier functional. The common ground with [CPRS2] ste ms from the use of the spectral
+flow formula of [CP2] to derive the twisted residue cocycle an d this has the corollary that we have a
+homotopy invariant. To illustrate the theory for these exam ples we compute, for particular modular
+unitaries in matrix algebras over the algebras Qλ, the precise numerical values arising from the general
+formalism.
+2.The algebrasQλfor0<λ<1.
+2.1.TheC∗-algebras C∗(Γλ) =C(ˆΓλ)and their K-theory. We will construct our algebras Qλ
+as “corner” algebras in certain crossed product C∗-algebras but first we need some preliminaries. For
+0<λ<1, let Γλbe the countable additive abelian subgroup of Rdefined by:
+Γλ=/braceleftBiggk=N/summationdisplay
+k=−Nnkλk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN≥0 andnk∈Z/bracerightBigg
+.
+Loosely speaking, Γ λconsists of Laurent polynomials in λandλ−1with integer coefficients. It is not
+only a dense subgroup of R, but is clearly a unital subring ofR.
+Proposition 2.1. Let0<λ<1.
+(1)Ifλ=p/qwhere0<p<qare integers in lowest terms, then Γλ=Z[1/n],wheren=pq.
+(2)Ifλandλ−1are both algebraic integers, then Γλ=Z+Zλ+···+Zλd−1is an internal direct
+sum where d≥2is the degree of the minimal (monic) polynomial in Z[x]satisfied by λ.
+(3)Ifλis transcendental then, Γλ=/circleplustext
+k∈ZZλkis an internal direct sum.
+(4)Ifλ= 1/√nwithn≥2a square-free positive integer, then Γλ=Z[1/n] +Z[1/n]·√nis an
+internal direct sum.
+(5)In general, if λis algebraic with minimal polynomial, nλd+···+m= 0overZ,then
+Z⊕Zλ⊕···⊕Zλd−1⊆Γλ⊆Z[1
+mn]⊕Z[1
+mn]λ⊕···⊕Z[1
+mn]λd−1.
+Hence,rank(Γλ) := dim Q(Γλ⊗ZQ) =d.
+Proof.In case (1), since gcd(p,q) = 1,there exista,b∈Zso that 1 = ap+bq.Therefore,1
+q=ap+bq
+q=
+aλ+b∈Γλ; and similarly,1
+p∈Γλ.Since, Γλis a commutative ring, for any k,m∈Zwithk≥1 we
+have:m
+nk=m
+(pq)kis in Γλ.That is, Z[1/n]⊆Γλ.On the other hand, for k≥1 we have
+λk=pk
+qk=p2k1
+(pq)k=p2k1
+nk∈Z[1/n] andλ−k=qk
+pk=q2k1
+(pq)k=q2k1
+nk∈Z[1/n].
+That is, Z[1/n] = Γλ.
+In case (2), it is not hard to see the minimal polynomial of λinZ[x] is not only monic, but also
+has constant term = ±1; say,p(λ) =λd+aλd−1+···±1 = 0.Clearly,λ∈Z+Zλ+···+Zλd−1.
+Sinceλ−1p(λ) = 0,we also have λ−1∈Z+Zλ+···+Zλd−1.By an easy induction, we have λk∈3
+Z+Zλ+···+Zλd−1,for allk∈Z.Hence, Γ λ=Z+Zλ+···+Zλd−1.The sum is direct by the
+minimality of the degree of the minimal polynomial.
+In case (3) the sum is direct because if λsatisfied a Laurent polynomial over Z, then by multipling by
+a high power of λit would also satisfy a genuine polynomial over Z.
+Case (4) is an easy calculation which we leave to the reader. C ase (5) is proved by similar methods
+used in case (2). Again, the sum Z[1
+mn]+Z[1
+mn]λ+···+Z[1
+mn]λd−1is direct by the minimality of the
+degree of the minimal polynomial. /square
+Proposition 2.2. Let0<λ<1.
+(1)Ifλ=p/qis rational in lowest terms so that Γλ=Z[1/n],wheren=pq,then
+K0(C(ˆΓλ)) =Z[1ˆΓλ)] andK1(C(ˆΓλ)) =Z[1/n].
+(2)Ifλandλ−1are both algebraic integers, so that Γλ=Z+Zλ+···+Zλd−1is an internal direct
+sum as above, then
+K0(C(ˆΓλ)) =even/logicalanddisplay
+(Γλ) =d/circleplusdisplay
+k=0,kevenk/logicalanddisplay
+(Γλ) andK1(C(ˆΓλ)) =odd/logicalanddisplay
+(Γλ) =d/circleplusdisplay
+k=1,koddk/logicalanddisplay
+(Γλ).
+(3)Ifλis transcendental then,
+K0(C(ˆΓλ)) =even/logicalanddisplay
+(Γλ) =∞/circleplusdisplay
+k=0,kevenk/logicalanddisplay
+(Γλ) andK1(C(ˆΓλ)) =odd/logicalanddisplay
+(Γλ) =∞/circleplusdisplay
+k=1,koddk/logicalanddisplay
+(Γλ).
+(4)Ifλ= 1/√nwithn≥2a square-free positive integer, then
+K0(C(ˆΓλ))∼=Z⊕Z[1/n] andK1(C(ˆΓλ))∼=Z[1/n]⊕Z[1/n]
+(5)In general, if λis algebraic with nλd+···+m= 0overZthen the composition of the inclusions
+Z⊕Zλ⊕···⊕Zλd−1⊆Γλ⊆Z[1
+mn]⊕Z[1
+mn]λ⊕···⊕Z[1
+mn]λd−1
+induces an inclusion on K-Theory, so that both of the following maps are one-to-one
+even/logicalanddisplay
+(Zd)∼=K0(C∗(Z⊕···Zλd−1))֒→K0(C(ˆΓλ)) andodd/logicalanddisplay
+(Zd)∼=K1(C∗(Z⊕···Zλd−1))֒→K1(C(ˆΓλ)).
+Proof.In case (1), Γ λ= lim−→Zwhere each map is multiplication by n,so thatˆΓλ= lim←−T.Since
+K0(C(T)) =Z[1] is generated by multiples of the trivial rank one bundle, the maps in the direct limit
+K0(C(ˆΓλ) = lim−→K0(C(T)) are the identity map in each case, so that K0(C(ˆΓλ)) =Z[1].On the other
+hand,K1(C(T)) is generated by the maps on C(T)z/ma√sto→zk, and each map in the direct limit is the
+same map induced by z/ma√sto→zn.Thus,K1(C(ˆΓλ)) =Z[1/n].
+Cases (2) and (3) are well-known facts about the K-Theory of tori.
+Case (4): first one uses item (4) of the previous Proposition, then the proof of case (1) above in order
+to apply Proposition 2.11 of [Sc]. The proof is finished off wit h the easily proved observation that
+Z[1/n]⊗Z[1/n] =Z[1/n].
+Case (5) the composed embedding is just containment:
+Z⊕Zλ⊕···⊕Zλd−1⊆Z[1
+mn]⊕Z[1
+mn]λ⊕···⊕Z[1
+mn]λd−1.Since we know that K∗(C∗(Z))→4
+K∗(C∗(Z[1/mn])) is one-to-one (even an isomorphism after tensoring with Q), an application of C.
+Schochet’s K¨ unneth Theorem, [Sc] shows that the induced ma p onK-Theory:
+K∗(C∗(Z⊕Zλ⊕···⊕Zλd−1))−→K∗(C∗(Z[1
+mn]⊕Z[1
+mn]λ⊕···⊕Z[1
+mn]λd−1))
+is one-to-one (even an isomorphism after tensoring with Q). /square
+Corollary 2.3. Ifλis algebraic with minimal polynomial of degree dso thatrank(Γλ) =dthen
+rank(K0(C(ˆΓλ))) = rank(even/logicalanddisplay
+(Zd)) = 2d−1= rank(odd/logicalanddisplay
+(Zd)) = rank(K1(C(ˆΓλ))).
+Proof.For eachN≥d−1,let ΓN=Zλ−N+···+ZλN⊆Γλ.Then each Γ Nis a finitely generated
+torsion free (and hence free abelian) subgroup of Γ λ.Moreover,
+Z⊕Zλ⊕···⊕Zλd−1⊆ΓN⊆Γλ⊆Z[1
+mn]⊕Z[1
+mn]λ⊕···⊕Z[1
+mn]λd−1,
+so that by tensoring with Qthe induced inclusions are all equalities, and hence all are Q-vector spaces
+of dimension d.Since ΓNis free abelian, Γ N∼=Zd.Now,
+K0(C∗(ΓN))∼=K0(C(Td))∼=even/logicalanddisplay
+(Zd)∼=Z2d−1andK1(C∗(ΓN))∼=K1(C(Td))∼=odd/logicalanddisplay
+(Zd)∼=Z2d−1.
+So, eachKi(C∗(ΓN))⊗ZQis aQ-vector space of dimension 2d−1and the map:
+K∗(C∗(Z⊕Zλ⊕···⊕Zλd−1))⊗ZQ−→K∗(C∗(ΓN))⊗ZQ
+is one-to-one and hence an isomorphism of Q-vector spaces. Since the corresponding isomorphism
+ontoK∗(C∗(ΓN+1))⊗ZQfactors through K∗(C∗(ΓN))⊗ZQthe maps
+K∗(C∗(ΓN))⊗ZQ→K∗(C∗(ΓN+1))⊗ZQ
+are all isomorphisms. Now, C∗(Γλ) = limNC∗(ΓN) and soKi(C∗(Γλ)) = lim NKi(C∗(ΓN)),and
+therefore,
+Ki(C∗(Γλ))⊗ZQ= lim
+NKi(C∗(ΓN))⊗ZQ∼=Q2d−1
+for eachi= 1,2. /square
+Now, letGλ⊃G0
+λbe the following countable discrete groups of matrices:
+Gλ=/braceleftbigg/parenleftbigg
+λna
+0 1/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglea∈Γλ, n∈Z/bracerightbigg
+⊃G0
+λ=/braceleftbigg/parenleftbigg
+1a
+0 1/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglea∈Γλ/bracerightbigg
+.
+Of course, G0
+λis isomorphic to the additive group Γ λ, andGλis semidirect product of Zacting on
+G0
+λ∼=Γλ.We letGλact onRas an “ax+b” group, noting that the action leaves Γ λinvariant. That
+is,
+fort∈Randg=/parenleftbigg
+λna
+0 1/parenrightbigg
+∈Gλdefineg·t:=λnt+a.
+Notation. For such an element g∈Gλwe will use the notation g:= [λn:a] in place of the matrix
+forgand|g|:= det(g) =λnfor the determinant of g.Note:G0
+λ={g∈Gλ||g|= 1}⊳Gλ.
+We use this action on Rto define the transpose action αofGλonL∞(R) :
+αg(f)(t) =f(g−1t) forf∈L∞(R) andt∈R.
+Now letCλ
+0(R) be the separable C∗-subalgebra ofL∞(R) generated by the countable family of pro-
+jectionsX[a,b)wherea,b∈Γλ.That is,5
+Cλ
+0(R) = closure/parenleftBigg/braceleftBiggn/summationdisplay
+k=1ckX[ak,bk)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleck∈C;ak,bk∈Γλ/bracerightBigg/parenrightBigg
+.
+We observe that Cλ
+0(R) is a commutative AF-algebra. Clearly, C0(R)⊂Cλ
+0(R) and since αg(X[a,b)) =
+X[g(a),g(b))both are invariant under the action αofGλ.We define the separable C∗-algebrasAλ⊃Aλ
+0
+as the crossed products:
+Aλ=Gλ⋊αCλ
+0(R) =Z⋊(G0
+λ⋊αCλ
+0(R))⊃Aλ
+0=G0
+λ⋊αCλ
+0(R).
+SinceGλandG0
+λare amenable these equal the reduced crossed products by [Pe d, Theorem 7.7.7 ].
+LetCλ
+00(R) denote the dense ∗-subalgebra of Cλ
+0(R) consisting of finite linear combinations of the
+generating projections, X[a,b),and letAλ
+c⊂l1
+α(Gλ,Cλ
+0(R))⊂Aλdenote the dense ∗-subalgebra of Aλ
+consisting of finitely supported functions x:Gλ→Cλ
+00(R).Similarly we define Aλ
+0,c⊂Aλ
+0.
+Proposition 2.4. For anyλ∈(0,1)Aλ
+0andAλare in the bootstrap class Nnuc.
+Proof.SinceAλ=Z⋊Aλ
+0,it suffices to see that Aλ
+0is inNnuc.By the proof of the previous Corollary,
+we can write Γ λas an increasing union of finitely generated torsion-free ab elian groups Γ Nwhich are
+free abelian group of finite rank so that Aλ
+0is the direct limit of crossed products of the separable
+commutative C∗-algebraCλ
+0(R) byZmiand hence is in Nnuc. /square
+Notation: We remind the reader of the crossed product operations in our setting (Definition 7.6.1 of
+[Ped]) together with some particular notations we use. To th is end, letx,y∈l1
+α(Gλ,Cλ
+0(R)) then we
+have the product and adjoint formulas:
+(x·y)(g) =/summationdisplay
+h∈Gλx(h)αh(y(h−1g)) forg∈Gλ;
+x∗(g) =αg((x(g−1))∗) forg∈Gλ.
+Ifx∈l1
+α(Gλ,Cλ
+0(R)) is supported on the single element g∈Gλandx(g) =f∈Cλ
+0(R), then we write
+x=f·δg. SinceAλ
+c(respectively, Aλ
+0,c) is dense in Aλ(respectively, Aλ
+0) we often do our calculations
+with these elements and we have the following easily verified calculus for them.
+Lemma 2.5. Letf1·δg1,f2·δg2,f·δg∈Aλ
+c,then:
+(1) (f1·δg1)·(f2·δg2) =f1αg1(f2)·δg1g2
+(2) (f·δg)∗=αg−1(¯f)·δg−1.
+(3)f·δgis self-adjoint if and only if fis self-adjoint and g= 1.
+(4)f·δgis a projection if and only if fis a projection and g= 1.
+(5)f·δgis a partial isometry if and only if |f|is a projection.
+(6)The product of partial isometries of the form X[a,b)·δgis a partial isometry of the same form.
+(7)Consider the partial isometry, v=X[a,b)·δg.Any two of the following: vv∗, v∗v, gcompletely
+determine the interval [a,b)and the element g.
+Definition 2.6. Lete∈Aλ
+0,cbe the projection e=X[0,1)·δ1.We define the separable unital C∗-algebras:
+Qλ:=eAλe⊃eAλ
+0e=:Fλ.
+We will also have occasion to use the dense subalgebras Qλ
+c:=eAλ
+ce,andFλ
+c:=eAλ
+0,ce.
+Proposition 2.7. The orthogonal family of projections en=X[n,n+1)·δ1∈Aλ
+0forn∈Zare mutually
+equivalent by partial isometries in Aλ
+0of the form Vn,k:=X[n,n+1)·δgn−kwheregn−k= [1 : (n−k)].6
+Moreover, the finite sums EN:=/summationtextN−1
+n=−Nen=X[−N,N)·δ1form an approximate identity for Aλso
+that
+Aλ∼=Qλ⊗K(l2(Z)) andAλ
+0∼=Fλ⊗K(l2(Z)).
+Proof.By Lemma 2.5, one easily calculates that:
+for each pair n,k∈Z, Vn,kV∗
+n,k=enandV∗
+n,kVn,k=ek.
+Now for each positive integer Nif we havey∈Aλ
+cthat satisfies supp( yh)⊆[−N,N) for allh,then
+using Lemma 2.5 again we see that EN·y=y.Since the collection of all such elements y∈Aλ
+cis
+dense inAλ, we see that the increasing sequence of projections {EN}form an approximate identity
+forAλ. /square
+Corollary 2.8. It follows from Proposition 2.4.7 of [RS]and Proposition 2.4 that for any λ∈(0,1),
+QλandFλare both in Nnuc.
+Lemma 2.9. (cf.[PhR, Proposition 3.1, Lemma 3.6] ) The algebra Cλ
+0(R)is a commutative separable
+AF algebra consisting of all functions f:R→Cwhich vanish at∞and: are right continuous at
+eachx∈Γλ; have a finite left-hand limit at each x∈Γλ;and are continuous at each x∈(R\Γλ).
+Moreover, if φ∈/hatwiderCλ
+0(R),(the space of all nonzero ∗-homomorphisms: Cλ
+0(R)→C) then there exists a
+uniquex0∈Rsuch that:
+(1)ifx0∈(R\Γλ)thenφ(f) =f(x0)for allf∈Cλ
+0(R),
+(2)ifx0∈Γλthen either/braceleftbiggφ(f) =f(x0)for allf∈Cλ
+0(R),or
+φ(f) =f−(x0) = limx→x−
+0f(x)for allf∈Cλ
+0(R).
+Proof.Since generating functions for Cλ
+0(R) satisfy each of the properties above which are clearly
+preserved by passing to uniform limits, we see that any funct ion inCλ
+0(R) satisfies these properties.
+Conversely, it is easy to show that any function satisfying t hese properties can be uniformly approxi-
+mated by a finite linear combination of the generators. The re mainder of the proof is given in [PhR,
+Lemma 3.6]. /square
+Notation. We denote the dual space,/hatwiderCλ
+0(R) byRλand endow it with the relative weak- ∗topology,
+that is the topology of pointwise convergence on Cλ
+0(R).Of course, Rλis a locally compact Hausdorff
+space, and Cλ
+0(R)∼=C0(Rλ).
+Proposition 2.10. The algebras AλandAλ
+0(and henceQλandFλ) are simple C∗-algebras. More-
+over,Aλis purely infinite and hence so is Qλ.
+Proof.Now, both GλandG0
+λact onCλ
+0(R) as countable discrete groups of outer automorphisms.
+Thus, we can apply Theorem 3.2 of [E] once we check that neithe r action has any nontrivial invariant
+ideals inCλ
+0(R) and that the actions are properly outer in the sense of Definition 2.1 of [E].
+To do this we look at the induced action of GλandG0
+λonRλ.So, forg∈Gλwe havegacting on Rλ
+viag(φ) =φ◦α−1
+gso that forφ=φxgiven by evaluation at x∈R,we have as expected g(φx) =φg(x).
+Now, ifx∈Γλwe use the notation φx−to denote the∗-homomorphism φx−(f) =f−(x) =f(x−) =
+limy→x−f(y).One easily checks that since g(x)∈Γλ,we haveg(φx−) =φg(x)−.
+Next we claim that each of the sets {φm|m∈Γλ}and{φm−|m∈Γλ}is dense in Rλin the relative
+weak-∗topology. For example, we show that the second set is dense. T o approximate φxfor some
+x∈Rwe let{mn}be a sequence in Γ λconverging to xfrom the right in R.Letf∈Cλ
+0(R) so thatf
+is right continuous at x.One easily shows that |φmn−(f)−φx(f)|→0; that is, the sequence {φmn−}
+converges to φxin the relative weak- ∗topology.7
+It is easy to see that the action of G0
+λonRλhas dense orbits, and so, of course, the action of Gλhas
+dense orbits also. This implies that the actions of G0
+λandGλonCλ
+0(R) have no nontrivial invariant
+ideals since the induced action on Rλhas no nontrivial invariant closed sets. We complete the pro of
+by showing that the action is properly outer in the sense of Definition 2.1 of [E]. Since there are
+no nontrivial α-invariant ideals and Cλ
+0(R) is commutative this is the condition that for each g/\e}atio\slash= 1
+and each nonzero closed two sided ideal Iinvariant under αgwe have/bardbl(αg−Id)|I/bardbl= 2.SinceIis
+nonzero there is a nonempty open subset, OofRλso thatˆI=O.But sinceg/\e}atio\slash= 1 andOis not finite
+there exists y∈Osuch thatg(y)/\e}atio\slash=yandg(y)∈O.Letx=g(y)∈Oso thatg−1(x) =y∈Oand
+x/\e}atio\slash=g−1(x).So we can choose a continuous compactly supported real-valu ed function fonOwith
+f(x) = 1, f(g−1(x)) =−1 and/bardblf/bardbl= 1.But thenf∈Iand
+2≥/bardbl(αg−Id)|I/bardbl|≥/bardbl(αg−Id)(f)/bardbl=/bardblαg(f)−f/bardbl≥|f(g−1(x))−f(x)|= 2.
+Now that we know Aλis simple, we can easily apply Theorem 9 of [LS] to conclude th atAλsatisfies
+hypothesis(v)ofProposition 4.1.1 (page66) of[RS]. For si mpleC∗-algebras, thisis equivalent tobeing
+purely infinite by Definition 4.1.2 of [RS]: the authors of [LS ] had used one of the earlier definitions of
+purely infinite in their paper (namely, hypothesis (v)). By P roposition 4.1.8 of [RS] Qλis also purely
+infinite. /square
+Corollary 2.11. It follows from Corollaries 8.2.2 and 8.4.1 (Kirchberg-Phill ips) of[RS]and the fact
+thatAλis stable that for any λ∈(0,1), Aλis classified up to isomorphism (among Kirchberg algebras
+inNnuc) by itsK-theory.
+Since we need to calculate with elements of QλandFλ,we make the following observations.
+Lemma 2.12. Now,Qλ(respectively, Fλ) is the norm closure of finite linear combinations of the ele-
+ments of the form e(X[a,b)·δg)e,whereg∈Gλ(respectively, g∈G0
+λ), henceforth called the generators .
+Thus, we calculate
+(1) Iff·δg∈Aλ,(respectively, f·δg∈Aλ
+0) wheref∈Cλ
+0(R), then
+e(f·δg)e=X[a,b)f·δgwhere[a,b) = [0,1)∩[g(0),g(1)).
+(2) Thus, for g∈Gλ,(respectively, g∈G0
+λ)f·δgis inQλ(respectively, Fλ) iffsupp(f)⊆[0,1)∩
+[g(0),g(1)).In particular, for g∈Gλ,(respectively, g∈G0
+λ)X[a,b)·δgis inQλ(respectively, Fλ) iff
+[a,b)⊆[0,1)∩[g(0),g(1)).
+Proof.The first item is an easy calculation using part (1) of Lemma 2. 5 and the fact that αg(X[a,b)) =
+X[g(a),g(b)).The second item follows easily from the first. /square
+Proposition 2.13. Ifλis rational, then Aλ
+0andFλare AF-algebras. In particular, if λ=p/qwhere
+0<p<q are in lowest terms, then Fλis the UHF algebra n∞wheren=pq.Moreover, the minimal
+projections in the finite-dimensional subalgebras can all b e chosen from the canonical commutative
+subalgebraCλ
+0(R)·δI.
+Proof.We have shown in Proposition 2.1 that if λ=p/qwhere 0< p < q are in lowest terms, then
+Γλ=Z[1/n],wheren=pq.Now, any element in Z[1/n] has the form m/nk=m(1/nk) wherek≥1.
+Therefore any of the generating partial isometries X[a,b)·δ[1:c]∈Aλ
+0can (by bringing a,bandcto a
+common denominator) be written (assuming c>0) as a finite linear combination of partial isometries
+of theformX[l/nk,(l+1)/nk)·δ[1:1/nk].For partial isometries in Fλwewould have to restrict 0 ≤l≤nk−1
+and such partial isometries generate an nkbynkmatrix subalgebra of Fλ.It should now be clear that
+Fλis a UHF algebra of type n∞. /square8
+At this point we define some special elements in Qλwhich behave very much like the isometries
+Sµ∈On,except for the fact that someof them are notisometries.
+Definition 2.14. Fix0<λ<1and letkbe a positive integer. Define mkto be the unique positive
+integer satisfying: mkλk<1≤(mk+1)λk.For0≤m≤mkdefinepartial isometries Sk,m∈Qλ
+via:
+Sk,m=X[mλk,(m+1)λk)·δgk,mwheregk,m= [λk:mλk].
+Note: form<mktheSk,mare actually isometries , andSk,mkis an isometry iff 1 = (mk+1)λk.
+Remarks. The defining inequalities mkλk<1≤(mk+1)λkfor the positive integer mkare equivalent
+to: 0< λ−k−mk≤1.In particular, these differences are positive and bounded abo ve by 1. In the
+case ofQ1/nwe havemk=nk−1.Generally we have mk
+1≤mk<1≤(mk+1)≤(m1+1)k.
+Lemma 2.15. With the previously defined elements we have:
+S∗
+k,m=X[0,1)·δg−1
+k,mandS∗
+k,mk=X[0,λ−k−mk)·δg−1
+k,mkwhere for all m, g−1
+k,m= [λ−k:−m].
+Moreover, for 0≤m<mk, S∗
+k,mSk,m=X[0,1)·δ1=ewhileS∗
+k,mkSk,mk=X[0,λ−k−mk)·δ1.
+Finally, for 0≤m<mk, Sk,mS∗
+k,m=X[mλk,(m+1)λk)·δ1whileSk,mkS∗
+k,mk=X[mkλk,1)·δ1,so that
+mk/summationdisplay
+m=0Sk,mS∗
+k,m=X[0,1)·δ1=e.
+Proof.These are just straightforward calculations based on Lemma 2.5 which we leave to the reader.
+/square
+Theorem 2.16. For eachλwith0<λ<1,consider the partial isometries S1,mform= 0,1,...,m1
+wherem1λ<1≤(m1+1)λ.Form<m 1,S1,mis an isometry and/summationtextm1
+m=0S1,mS∗
+1,m= 1.Forλ= 1/n,
+m1=n−1, S1,m1is also an isometry, and Q1/n∼=On,the usual Cuntz algebra.
+Proof.The first statement is clear. With λ= 1/nwe have insideQ1/n, nisometries one for each
+m= 0,1,...,(n−1) defined by:
+Sm=X[m
+n,m+1
+n)·δgmwheregm= [1/n:m/n] and soS∗
+m=X[0,1)·δg−1
+mwhereg−1
+m= [n:−m].
+Using Lemma 2.12, we easily see that for each m, Sm∈Q1/n.
+Then, using item (1) of Lemma 2.5 we calculate:
+S∗
+mSm=X[0,1)·δ1=eandSmS∗
+m=X[m
+n,m+1
+n)·δ1and son−1/summationdisplay
+m=0SmS∗
+m=X[0,1)·δ1=e.
+Sinceeis the identity ofQ1/n,we have constructed a unital copy of OninsideQ1/n.Now one shows by
+induction that for each k>0 the product of exactly kof thesenisometries has the form Sk,mwhere
+Sk,mhas the same defining equation as Smabove but with nkin place ofnandm= 0,1,...,(nk−1).
+These new isometries have range projections Sk,mS∗
+k,m=X[m
+nk,m+1
+nk)·δ1which therefore lie in this copy
+ofOn.By adding up some of these projections, we can get any project ion of the formX[a,b)·δ1where
+0≤a<b≤1 and both a,bhave the form m/nk.But any element a∈Γ1/ncan be written as a=m
+nk
+for a sufficiently large k≥0 and some m∈Zdepending on k,and any pair a,bcan be brought to a
+common denominator nk.Hence any projection of the form X[a,b)·δ1inQ1/nis in this copy of On.9
+Now, a straightforward calculation gives us:
+nk−1/summationdisplay
+m=1Sk,mS∗
+k,m−1=nk−1/summationdisplay
+m=1X[m+1
+nk,m+2
+nk)·δ[1 : 1/nk]=X[1/nk,1)·δ[1 : 1/nk]∈On. (1)
+Finally, letX[a,b)·δg∈Q1/nbe an arbitrary generator. By taking adjoints if necessary w e can assume
+thatghas the form g= [nk:∗] wherek≥0.SinceSk,0is an isometry in Onit suffices to prove that
+Sk,0(X[a,b)·δg)∈On.That is, we are reduced to the case g= [1 :c] and again by taking adjoints
+if necessary we can assume that c≥0.The casec= 0 is done and so we can assume that c >0.So
+(with possibly new a,b) we haveX[a,b)·δ[1 :c]where 0< c≤1 and [a,b)⊆[0,1)∩[c,c+1) = [c,1).
+But,X[a,b)·δ[1 :c]=X[a,b)X[c,1)·δ[1 :c]=X[a,b)·δ1X[c,1)·δ[1 :c]and we already know that X[a,b)·δ1∈On.
+Therefore it suffices to see that X[c,1)·δ[1 :c]∈On.However,c=l/nkfor some 0<l<nkand so:
+X[c,1)·δ[1 :c]=X[l/nk,1)·δ[1 :l/nk]=/parenleftBig
+X[1/nk,1)·δ[1 : 1/nk]/parenrightBigl
+which is in Onby Equation 1. Since all generators for Q1/nare inOnwe’re done. /square
+2.2.K-Theory ofQλforλrational. : SinceAλ
+0is stable and stably isomorphic to the UHF
+algebraFλ,each of its projections is equivalent to one in some finite-di mensional subalgebra and
+hence to some projection in Cλ
+0(R), and in this case the trace induces an isomorphism from K0(Aλ
+0)
+onto Γλ=Z[1/(pq)]⊂R.This isomorphism carries the projection e=X[0,1)·δ1which is the identity
+ofQλandFλonto 1∈Z[1/(pq)].Now, since Aλ
+0is AF,K1(Aλ
+0) ={0},and sinceAλ=Z⋊λAλ
+0we
+can use the Pimsner-Voiculescu exact sequence to calculate K∗(Aλ) =K∗(Qλ).When we do this we
+get:
+K1(Qλ) ={0},andK0(Qλ) =Z[1/(pq)]/(1−λ)Z[1/(pq)].
+Proposition 2.17. Forλrational with λ=p/qin lowest terms, we have
+K1(Qλ) ={0},andK0(Qλ)∼=Z[1/(pq)]/(1−λ)Z[1/(pq)]∼=Z(q−p).
+Proof.By Proposition 2.1, Γ λ=Z[1/(pq)],so we must show that
+Z[1/(pq)]/(1−(1/(pq))Z[1/(pq)]∼=Z(q−p).
+Since (q−p) = (1−p/q)qand every element of Z[1/(pq)] is of the form m/(pq)N,it is easy to see that
+(q−p)Z[1/(pq)] = (1−p/q)Z[1/(pq)].Now, (q−p) and (pq)Nare relatively prime for any Nand so
+there exist a,b∈Zso that 1 = a(q−p)+b(pq)Nand hencem/(pq)N= (q−p)am/(pq)N+mb.That
+is,m/(pq)Nandmbrepresent the same element in the quotient. So, every elemen t in the quotient has
+an integer representative. Two integers c,drepresent the same element in the quotient if and only if
+c−d= (p−q)n/(pq)N,or (c−d)(pq)N=n(q−p).But then:
+(c−d) = (c−d)[a(q−p)+b(pq)N] = (c−d)a(q−p)+b(c−d)(pq)N= [(c−d)a+bn](q−p).
+That is,c,drepresent the same element in Z/(q−p)Z=Z(q−p).On the other hand if ( c−d) is in
+(q−p)Zthen clearly, [ c] = [d] inZ[1/(pq)]/(1−(1/(pq))Z[1/(pq)] and we are done. /square
+Corollary 2.18. Ifλ=p/qin lowest terms, then
+Fλ=Fp/q∼=UHF((pq)∞) andQλ=Qp/q∼=O(q−p+1).
+In particular, if λ=k
+k+1then
+Fλ∼=UHF((k(k+1))∞) andQλ∼=O2.10
+Proof.Since eachQλis separable, nuclear, simple, purely infinite and in the boo tstrap categoryNnuc
+once we show that the class of the identity e∈Qλis a generator for K0(Qλ) =Z/(q−p)Z,the
+Kirchberg-Phillips Classification Theorem, Theorem 8.4.1 of [RS], shows that Qλ∼=O(q−p+1).To this
+end we observe that since eis mapped to 1 in Z[1/pq],we must show that [1] is a generator for
+K0(Qλ) =Z[1/pq]/(1−(p/q))Z[1/pq].Now, by the proof of the previous proposition, k[1] = [k·1] =
+0∈Z[1/pq]/(1−(p/q))Z[1/pq] if and only if [ k·1] = 0∈Z/(q−p)Zif and only if k−0 =m(q−p)
+for somem∈Zif and only if kis a multiple of ( q−p).That is, [1],[2·1],...,[(q−p−1)·1] are all
+nonzero in K0(Qλ) =Z/(q−p)Zand hence [1] is a generator. /square
+2.3.TheK-Theory of the Algebras Aλ
+0forλirrational. The caseλrational is much simpler,
+and while it does fit into the following scheme, it does not nee d this deeper machinery. Initially, we
+(and others) believed that the algebras Aλ
+0were AF algebras when λis irrational. In fact we will
+show thatAλ
+0isneverAF whenλis irrational. We will set up our examples to fit the situation on
+page 1487 of [Put2] so that we can apply the six-term exact seq uence of Theorem 2.1 on page 1489 of
+[Put2].
+We let Γ = Γ λ∼=G0
+λ.Thus, Γ⊂Ris a countable dense subgroup of Rwhich acts on Rby translations.
+Before looking at the crossed product of Γ acting on Cλ
+0(R) =C0(Rλ) (which gives us Aλ
+0) we first
+consider the crossed product of Γ acting on C0(R).Since Γ acts on Rby translation we can Fourier
+transform to get an isomorphism:
+Γ⋊C0(R)∼=ˆR⋊C(ˆΓ).
+Then, by Connes’ Thom isomorphism we get for i= 0,1:
+Ki(Γ⋊C0(R))∼=Ki(ˆR⋊C(ˆΓ))∼=Ki+1(C(ˆΓ)).
+Proposition 2.19. The composition:
+K1(C0(R))i∗−→K1(Γ⋊C0(R))/hatwide−→K1(ˆR⋊C(ˆΓ))∼=−→K0(C(ˆΓ))
+takes the generator [u]∈K1(C0(R)) =Z·[u]; whereuis the Bott element in C0(R)1defined by
+u(t) =1+it
+1−it; to[1ˆΓ]where1ˆΓis the identity function in C(ˆΓ).
+Proof.We first work on the right hand side of this sequence of maps. Le tu(t) = 1+ε(t),then by the
+proof of Connes’ Thom isomorphism from
+K0(C(ˆΓ))⊗ZK1(C0(R))−→K1(R⋊C(ˆΓ))
+we see that [1 ˆΓ]⊗[u] gets mapped to the class [1 + ( convolution by ˆε·1ˆΓ)].Now in this displayed
+equation,K0(C(ˆΓ))⊗ZK1(C0(R)) =K0(C(ˆΓ))⊗ZZ·[u] =K0(C(ˆΓ))·[u]∼=K0(C(ˆΓ)).Thus, [1 ˆΓ] in
+K0(C(ˆΓ)) gets mapped to the class [1+( convolutionby ˆε·1ˆΓ)] by the Thom isomorphism.
+On the other hand, the map K1((C0(R)1)−→K1((Γ⋊C0(R))1) takes [u]/ma√sto−→[δ0·ε+1] and by the
+Fourier transform this goes to [( convolutionby ˆε·1ˆΓ)+1] inK1(R⋊C(ˆΓ)).Combining these we get:
+1∈Z/ma√sto−→[u]∈Z·[u] =K1((C0(R))1) =K1(C0(R))/ma√sto−→[1ˆΓ]∈K0((C(ˆΓ)).
+/square
+Now, by Proposition 2.1 we know Γ in many cases so that these la st groups are quite computable. In
+the notation of [Put2] we define the transformation groupoid s:
+G:=Rλ⋊Γ, G′:=R⋊Γ,andH:= Γ⋊Γ.
+Then,Aλ
+0=C∗
+r(G) is the reduced C∗-algebra of G; Γ⋊C0(R) =C∗
+r(G′) is the reduced C∗-algebra of
+G′; andK(l2(Γ)) is the reduced C∗-algebra ofH. By theproof of Proposition 2.10 thereis acontinuous11
+proper surjective map: Rλ→R,where points in Rwhich are not in Γ each have a single pre-image,
+while points γ∈Γ have exactly two pre-images in Rλ,which we denote by γ−andγ+.Thus, there
+are two disjoint embeddings of Γ in Rλ:
+i0,i1: Γ→Rλ:i0(γ) =γ−, i1(γ) =γ+.
+Now in order to mesh with the notation of [Put2], we let Y:= Γ with the equivalence relation,
+“=”;X:=Rλ,with the equivalence relation ( i0(γ)∼i1(γ)); and quotient π:X→X′:=Rwhere
+X′=X/(i0(γ)∼i1(γ)) =R;whilethe“factorgroupoid”of G=Rλ×Γ =X×ΓisG′:=R×Γ =X′×Γ.
+We represent each of these three C∗-algebras onH:=l2(Γ+)⊕l2(Γ−) where Γ±={γ±|γ∈Γ}in
+the following way. First we denote the natural orthonormal b asis elements ofHbyδa+andδa−for
+eacha∈Γ.Now the unitary representation Uof Γ onHisUγ(δa±) =δ(a−γ)±.The actions of C0(Rλ),
+C0(R), andC0(Γ) onHare as follows for f1∈C0(Rλ), f2∈C0(R),f3∈C0(Γ), andδa±∈H
+π1(f1)(δa±) =f1(a±)δa±π2(f2)(δa±) =f2(a)δa±π3(f3)(δa±) =f3(a)δa±.
+These three covariant pairs of representations, ( π1,U), (π2,U), and (π3,U) define representations of
+C∗
+r(G) =Aλ
+0,C∗
+r(G′) = Γ⋊C0(R), andC∗
+r(H) =K(l2(Γ)) respectively on H.Since each of these
+C∗-algebras is simple these representations are faithful.
+Now, one checks that the hypotheses of Theorem 2.1 of [Put2] a re satisfied. As in [Put1, Put2] one
+shows that the two mapping cone algebras of the inclusions:
+C∗
+r(G′) = Γ⋊C0(R)−→Aλ
+0=C∗
+r(G) andC∗
+r(H)−→C∗
+r(H)⊕C∗
+r(H) : (x/ma√sto→(x,x) )
+have isomorphic K-Theory. One then pastes these isomorphisms into the mappin g cone long exact
+sequence for C∗
+r(G′) = Γ⋊C0(R)−→Aλ
+0=C∗
+r(G).Next one observes that for any C∗-algebra,B
+the diagonal embedding B−→B⊕Binduces the diagonal embedding K∗(B)−→K∗(B)⊕K∗(B)
+with quotient isomorphic to K∗(B) (this is true for any abelian group). This implies that K∗(B)∼=
+K∗+1(M(B,B⊕B)) so that we get the six-term exact sequence from [Put2]:
+K1(C∗
+r(H)) /d47/d47K0(C∗
+r(G′)) /d47/d47K0(C∗
+r(G))
+/d15/d15
+K1(C∗
+r(G))/d79/d79
+K1(C∗
+r(G′)) /d111/d111 K0(C∗
+r(H)) /d111/d111
+In our set-up this becomes:
+{0} /d47/d47K0(Γ⋊C0(R)) /d47/d47K0(Γ⋊C0(Rλ))
+/d15/d15
+K1(Γ⋊C0(Rλ))/d79/d79
+K1(Γ⋊C0(R)) /d111/d111Z/d111/d111
+Which by Connes’ Thom isomorphism becomes:
+{0} /d47/d47K1(C(ˆΓ))/d47/d47K0(Aλ
+0)
+/d15/d15
+K1(Aλ
+0)/d79/d79
+K0(C(ˆΓ))/d111/d111Z/d111/d111
+By Proposition 2.19, the nonzero element [1 ˆΓ] inK0(C0(ˆΓ))∼=K1(Γ⋊C0(R)) is mapped to the
+image of the class [ u] inK1(Γ⋊C0(R)) by Connes’ Thom isomorphism, and then the image of [1 ˆΓ] in
+K1(Γ⋊C0(Rλ)) is the same as the image of [ u] under the inclusion K1(Γ⋊C0(R))−→K1(Γ⋊C0(Rλ)).
+However, this is clearly the same as the image of [ u] under the inclusion K1(C0(R))→K1(C0(Rλ))→
+K1(Γ⋊C0(Rλ)).This composition is 0 since C0(Rλ) is an AF-algebra. That is, the element [1 ˆΓ] in12
+K0(C0(ˆΓ)) is mapped to 0 in K1(Aλ
+0) and hence is in the image of the map Z−→K0(C0(ˆΓ)).Since
+[1ˆΓ] generates a copy of ZinK0(C0(ˆΓ)), we have a nonzero homomorphism from ZtoZ[1ˆΓ] which is
+onto and hence one-to-one. By the exactness, the map K0(Aλ
+0)−→Zis the zero map.
+CONCLUSION : K0(Aλ
+0)∼=K1(C(ˆΓλ)) andK1(Aλ
+0)∼=K0(C(ˆΓλ))/[1ˆΓλ]Z.
+Proposition 2.20. Ifλis irrational, then K1(Aλ
+0)/\e}atio\slash={0}so thatAλ
+0is not an AF-algebra.
+Proof.By items (3) and (5) of Proposition 2.2 we see that when λis irrational, K0(C(ˆΓλ)) is not
+singly generated so that K1(Aλ
+0)∼=K0(C(ˆΓλ))/[1ˆΓλ]Z/\e}atio\slash={0}. /square
+2.4.K-theory computations of particular Qλforλirrational. Example(s) λ= 1/√n:for
+n>1 a square-free integer. Using Proposition 2.1, we get:
+K0(Fλ) =K0(Aλ
+0) =K1(C(ˆΓλ)) =Z[1/n]⊕Z[1/n]
+K1(Fλ) =K1(Aλ
+0) = (K0(C(ˆΓλ)))/slashbig
+Z[1] = (Z[1]⊕Z[1/n])/slashbig
+Z[1] =Z[1/n].
+To compute the K-theory ofQλin this case using the Pimsner-Voiculescu exact sequence, o ne must
+first compute the induced automorphism λ∗onK1(C(ˆΓλ)) and onK0(C(ˆΓλ)) by a more detailed
+analysis of the proof of [Sc, Proposition 2.11]. In the case o fK1(C(ˆΓλ)) we get a copy of the group
+Γλ=Z[1/n] +Z[1/n]√nand the action on Γ λis just multiplication by λ= 1/√n.As an action
+translated to the abstract group Z[1/n]⊕Z[1/n],the automorphism becomes λ∗(a,b) = (b,a/n).
+Therefore, id∗−λ∗onK0(Aλ
+0) =Z[1/n]⊕Z[1/n] to itself is clearly 1 : 1. Now it is an instructive
+exercise to show that the kernel of the homomorphism
+(a,b)∈Z[1/n]⊕Z[1/n]/ma√sto→[a+b]∈Z[1/n]/slashbig
+(1−1/n)Z[1/n]
+is exactly the range of the homomorphism
+id∗−λ∗:Z[1/n]⊕Z[1/n]−→Z[1/n]⊕Z[1/n].
+Hence, we have the isomorphisms:
+(Z[1/n]⊕Z[1/n])/slashbig
+(id∗−λ∗)(Z[1/n]⊕Z[1/n])∼=Z[1/n]/slashbig
+(1−1/n)Z[1/n]∼=Z/slashbig
+(n−1)Z.
+where the last isomorphism follows from the proof of Proposi tion 2.17 with p= 1 andq=n.
+Once we have computed the action of λ∗onK1(Aλ
+0) =Z[1/n] we will be ready to compute K∗(Qλ).
+Now, by Proposition 2.11 of [Sc] we have the isomorphism:
+K0(C(ˆΓλ))∼=(Z[1]⊗ZZ[1])⊕(Z[1/n]⊗ZZ[1/n]) =Z[1]⊕(Z[1/n]⊗ZZ[1/n]).
+The action of λ∗onZ[1] is of course the identity. However, the action of λ∗on (Z[1/n]⊗ZZ[1/n])
+is justx⊗y/ma√sto→y⊗x/n.If one combines this with the multiplication isomorphism x⊗y/ma√sto→xy:
+Z[1/n]⊗ZZ[1/n]−→Z[1/n] we see that λ∗acts as multiplication by 1 /nonZ[1/n] =Z[1/n]⊗ZZ[1/n].
+Thus,λ∗on the quotient K1(Aλ
+0) =Z[1/n] is just multiplication by 1 /n.Therefore,id∗−λ∗becomes
+multiplication by (1 −1/n) onZ[1/n] which is clearly 1 : 1 .Applying the Pimsner-Voiculescu exact
+sequence and recalling that Ki(Qλ) =Ki(Aλ) we get the isomorphisms:
+K0(Qλ)∼=Z/slashbig
+(n−1)Z,andK1(Qλ)∼=Z/slashbig
+(n−1)Z,forλ= 1/√n.
+Forn>2 we getK1/\e}atio\slash= 0 and so these are not Cuntz algebras, in fact not even Cuntz- Krieger algebras
+sinceK1has nonzero torsion. For λ= 1/√
+2 however we get K0= 0 =K1and by classification theory,
+we must haveQ1/√
+2∼=O2! However, even in this case the fixed point algebra, is NOT AF s ince it has
+K1=Z[1/2], the tape-measure group. So for the simplest irrational n umber 1/√
+2 we get the Cuntz
+algebra,O2with a strange gauge action of T.13
+Remarks. In the examples below it is important to note that any polynom ial of the form f(x) =
+xn+axn−1+···+bx±1 has at most n−1 roots in the open interval (0 ,1) because the product of
+all the roots of fmust equal±1.
+Example(s) quadratic integers and an algorithm: If bothλandλ−1are quadratic integers
+withλ∈(0,1),thenλ2+aλ±1 = 0 where the integer polynomial f(x) =x2+ax±1 is irreducible
+overQ.With these restrictions there are two cases, either f(x) =x2+ax−1 wherea >0 and
+λ= 1/2·(√
+a2+4−a)∈(0,1); orf(x) =x2+ax+1wherea≤−3andλ= 1/2·(−√
+a2−4−a)∈(0,1).
+In the first case, λ2+aλ−1 = 0,witha>0,so thatλ+a−λ−1= 0 andλ−1=a+λ.For this case
+we outline an algorithm using the ideas of the Smith Normal Fo rm and the Pimsner-Voiculescu exact
+sequence to calculate the K-Theory. By Proposition 2.2, and the CONCLUSION before Prop osition
+2.20, Γλ=Z+ZλandK0(Aλ
+0)∼=K1(C(ˆΓλ)) =/logicalandtext1(Γλ) = Γλ∼=Z2.Giving Γ λitsZ-basis{1,λ}we
+see that the action of the automorphism λ∗onK0(Aλ
+0)∼=Γλhas matrix:/bracketleftbigg
+0 1
+1−a/bracketrightbigg
+.So, (id−λ∗) =
+/bracketleftbigg
+1−1
+−1 (a+1)/bracketrightbigg
+:=M.To compute the kernel and cokernel of this matrix mapping Z2→Z2we
+row and column-reduce MoverZto obtain matrices P,Q∈GL(2,Z) so thatPMQ=Dwhere
+Dis diagonal over Z.Then ker(M)∼=ker(D) andZ2/M(Z2)∼=Z2/D(Z2).In this case, we get
+D=/bracketleftbigg
+1 0
+0a/bracketrightbigg
+.Hence, onK0(Aλ
+0) we have
+ker(id−λ∗) = ker(M)∼=ker(D) ={0}and coker(id−λ∗)∼=coker(D) =Z/aZ.
+Now we compute ( id−λ∗) on
+K1(Aλ
+0)∼=K0(C(ˆΓλ))/Z·1o= (Z·1o⊕Z(1∧λ))/Z·1o=Z(1∧λ).
+Now,λ∗(1∧λ) =λ∧λ2=λ∧(1−aλ) =λ∧1 = (−1)1∧λ.Thatis,λ∗=−idonK1(Aλ
+0)∼=Z.Therefore,
+(id−λ∗) = multiplication by 2 on Z(1∧λ) which has ker( id−λ∗) ={0}and cokernel( id−λ∗)∼=Z/2Z.
+Applying these results to the Pimsner-Voiculescu exact seq uence we obtain:
+K0(Qλ) =Z/aZandK1(Qλ) =Z/2Z,forλ2+aλ−1 = 0, n≥1.
+None of these examples are Cuntz-Krieger algebras since K1is not torsion-free. In particular, when
+λ= (1/2)(√
+5−1) is the inverse of the golden mean, we get K0={0}andK1=Z/2Z.
+In the second case, λ2+aλ+ 1 = 0,we have as above, K0(Aλ
+0)∼=Γλ=Z+ZλwithZ-basis
+{1,λ}; the diagonal version of ( id−λ∗) isD= diag[1,(a+ 2)] so that ker( id−λ∗) ={0}and
+coker(id−λ∗)∼=Z/(a+2)Z.On the other hand, K1(Aλ
+0)∼=Z(1∧λ) only now, λ∗=idhere so that
+(id−λ∗) = 0 and hence ker( id−λ∗)∼=Zwhile coker( id−λ∗)∼=Z.By Pimsner-Voiculescu we get
+K0(Qλ) =Z⊕(Z/(a+2)Z) andK1(Qλ) =Z,forλ2+aλ+1 = 0, a≤−3.
+We note that in this case, Qλhas the correct K-theory to be a Cuntz-Krieger algebra (and is therefore
+stably isomorphic to one), and that in the case a=−3 (i.e.,λ= (1/2)(3−√
+5)) we have K0=Z=K1.
+Example cubic integers: Ifλandλ−1are cubic integers with λ∈(0,1),thenλ3+aλ2+bλ±1 = 0
+where the integer polynomial f(x) =x3+ax2+bx±1 is irreducible over Q.Such anfis irreducible
+if and only if f(1)/\e}atio\slash= 0/\e}atio\slash=f(−1).There are two cases depending on the constant, ±1.
+First, consider f(x) =x3+ax2+bx−1 = 0 with f(1) =a+b/\e}atio\slash= 0 andf(−1) =a−b−2/\e}atio\slash= 0 so that
+fis irreducible. Now assume a+bispositive (buta/\e}atio\slash=b+2). Then f(0) =−1 andf(1) =a+b>0
+so thatfhas auniqueroot in (0,1) since it is a cubic.14
+Next consider the same polynomial, f(x) =x3+ax2+bx−1 = 0,witha+bnegative (buta/\e}atio\slash=b+2).
+Since both f(0) andf(1) are negative, in order to have a solution the function fmust have a local
+maximum on (0 ,1).There are examples with no solutions in (0 ,1); for example, f(x) =x3−3x−1.
+In order to have a unique solution , then considering f′(x), one would need 4 a2−12b= 0 : while
+this has many solutions, they all satisfy |a|≤band so we can not have a+b<0.So solutions are not
+unique in this case. But, there are infinitely many cubics wit htwo distinct solutions in (0 ,1); eg.,
+f(x) =x3−(a+k)x2+ax−1 fora≥k+4 andk≥1 has two solutions in (0 ,1),sincef(.5)>0.
+We now calculate the K-theory ofQλassuming that λsatisfiesf(x) =x3+ax2+bx−1 = 0,where
+a+b/\e}atio\slash= 0,anda−b/\e}atio\slash= 2.Now,λ3+aλ2+bλ−1 = 0,sothatλ3= 1−aλ2−bλandλ−1=λ2+aλ+b.Then,
+Γλ=Z+Zλ+Zλ2andK0(Aλ
+0)∼=K1(C(ˆΓλ)) =/logicalandtextodd(Γλ) = Γλ⊕(Γλ∧Γλ∧Γλ) = Γλ⊕(Z(1∧λ∧λ2))∼=
+Z4.Giving Γ λits natural Z-basis{1,λ,λ2}the induced homomorphism ( id−λ∗) onK0(Aλ
+0)∼=Z4
+yields the diagonal matrix, D= diag[1,1,(a+b),0] so that on K0(Aλ
+0) we have
+ker(id−λ∗)∼=ker(D)∼=Zand coker(id−λ∗)∼=coker(D) = (Z/(a+b)Z)⊕Z.
+Now,K1(Aλ
+0)∼=K0(C(ˆΓλ))/Z·1o=/logicalandtext2(Γλ) =Z(1∧λ) +Z(1∧λ2) +Z(λ∧λ2)∼=Z3.By similar
+computations we get for K1(Aλ
+0)∼=Z3; the matrix D= diag[1,1,(a+b)].Hence, onK1(Aλ
+0) we have
+ker(id−λ∗)∼=ker(D) ={0}and coker(id−λ∗)∼=coker(D) =Z/(a+b)Z.
+Applying these results to the Pimsner-Voiculescu exact seq uence we obtain:
+K0(Qλ) =Z⊕(Z/(a+b)Z) andK1(Qλ) =Z⊕(Z/(a+b)Z) forλ3+aλ2+bλ−1 = 0.
+Remarks. In casea+b= 1 (which has infinitely many solutions corresponding to infi nitely many
+distinct invertible cubic integers λ∈(0,1)) we get K0(Qλ) =Z=K1(Qλ),which as noted above
+is also true for the invertible quadratic integer, λ= (1/2)(3−√
+5).In the general cubic case with
+constant term−1 we always have non-torsion elements in both K0andK1: this is the opposite of
+the case where the constant term is +1 ,where we see below that K0andK1are both torsion groups.
+A similar phenomenon occurs in the quadratic case above, exc ept that there we get torsion in the
+−1 case and non-torsion in the +1 case! That this may be a periodic phenomenon is suppo rted by
+a calculation of two quartic examples: first, the unique solu tionλ∈(0,1) to the irreducible quartic
+f(x) =x4−3x3+ 1 gives us K0=ZandK1=Z⊕(Z/3Z)⊕(Z/3Z); while, second, the unique
+solutionλ∈(0,1) to the irreducible quartic f(x) =x4+3x3−1 gives usK0= (Z/3Z)⊕(Z/3Z) and
+K1= (Z/9Z)⊕(Z/2Z),similar to the quadratic case. Proposition 2.21 is further e vidence.
+When an irreducible polynomial f(x) =xn+axn−1+···±1 has two roots, λ1,λ2∈(0,1),then
+Γλ1∼=Γλ2asrings(butnotas ordered rings, for that would imply equality). Still, Qλ1∼=Qλ2(at
+least stably) since the calculation of their K-groups are identical. Their KMS states are not equivalent
+since the type III factors that they generate are not isomorp hic, as we will see below.
+Proposition 2.21. Supposeλsatisfies the irreducible (over Z) polynomial, f(x) =xn+···±1 = 0.
+(1)Fornodd, iff(x) =xn+···+1thenK0(Qλ)hasZ/2Zas a summand.
+While, iff(x) =xn+···−1thenK0(Qλ)hasZas a summand (so, by the next Proposition, rank(K0) =
+rank(K1)≥1in this case).
+(2)Forneven, iff(x) =xn+···+1thenK1(Qλ)hasZas a summand (so, by the next Proposition,
+rank(K0) = rank(K1)≥1in this case).
+While, iff(x) =xn+···−1thenK1(Qλ)hasZ/2Zas a summand.
+Proof.InK∗(Aλ
+0) there is a λ∗-invariant summand, (1 ∧λ∧λ2∧···∧λn−1)Z.Depending on n(mod 2)
+and the constant term ±1,the action of λ∗on this summand is ±id.Hence, (id−λ∗) here is either 0
+or 2(id).Applying Pimsner-Voiculescu gives a summand in K∗(Qλ) of either ZorZ/2Z. /square15
+Proposition 2.22. Supposeλis algebraic.
+(1)Then,rank(K0(Qλ)) =rank(K1(Qλ))so thatQλis not stably isomorphic to O∞.
+(2)Ifλandλ−1are both algebraic integers and Qλis stably isomorphic to a Cuntz algebra On,then the
+minimal polynomial of λhas odd degree and constant term +1.Moreover,nis congruent to 3(mod4)
+and all such Cuntz algebras appear this way.
+Proof.To see part (1) we tensor the Pimsner-Voiculescu exact seque nce byQ(which preserves exact-
+ness) to obtain an exact hexagon of Q-vector spaces:
+V1θ1/d47/d47V1τ1/d47/d47KQ
+1
+µ1
+/d15/d15
+KQ
+0µ0/d79/d79
+V0τ0/d111/d111 V0θ0/d111/d111
+whereVi=Ki(Aλ
+0)⊗Q,andKQ
+i=Ki(Aλ)⊗Q.Then
+dim(KQ
+0) = rank(µ0)+nullity(µ0) = rank(µ0)+rank(τ0) and dim( KQ
+1) = rank(µ1)+rank(τ1)
+and
+rank(τ0)+rank(θ0) = dim(V0) = rank(θ0)+rank(µ1) so that rank( τ0) = rank(µ1).
+Similarly, rank( τ1) = rank(µ0),so that:
+dim(KQ
+0) = rank(µ0)+rank(τ0) = rank(µ1)+rank(τ1) = dim(KQ
+1).
+That is, rank( K0(Qλ)) = dimK0(Qλ)⊗ZQ= dimK0(Aλ)⊗ZQ=···rank(K1(Qλ)).By Proposition
+2.21, if the minimal polynomial of λhas even degree, then K1(Qλ)/\e}atio\slash={0},and soQλcannot be stably
+isomorphic to a Cuntz algebra. If Qλis stably isomorphic to Onthennis finite by part (1)and by
+Proposition 2.21, the order of K0(Qλ) must be even and therefore nmust be odd. Furthermore, the
+minimal polynomial must have constant term +1 .In order for K0(Qλ) to be a finite cyclic group of
+even order, it must be of the form Z/mZ⊕Z/2ZwheremisoddsinceZ/2Zis a summand. Let
+m= 2k+1 then
+n=♯[Z/mZ⊕Z/2Z]+1 = 2m+1 = 4k+3
+as claimed.
+In the examples below where λ3+aλ2+bλ+1 = 0,and eithera−b= 1 andb≤−2 ORa=b=−1
+andb≤−1, we obtain (stably, at least) all the Cuntz algebras Onwheren≡3(mod4). /square
+Now considerthecaseofirreduciblecubicsoftheform f(x) =x3+mx2+nx+1;sof(1) =m+n+2/\e}atio\slash= 0
+andf(−1) =m−n/\e}atio\slash= 0.Sincef(0) = 1,if we havef(1) =m+n+2<0,then we have as above a
+uniqueroot in (0,1).
+Iff(1) =m+n+ 2>0,we can have distinct roots. For example, if n=−4 andm= 3, then,
+f(x) =x3+3x2−4x+1 has two roots in (0 ,1).Ifn<<0,we get several solutions mfor eachn: eg.,
+n=−7 implies that any mwith 6≤m≤9 will yield a polynomial with two roots in (0 ,1).
+We now calculate the K-Theory ofQλassumingλsatisfiesf(λ) =λ3+mλ2+nλ+ 1 = 0.Again,
+K0(Aλ
+0)∼=Z4,but now the diagonal matrix D= diag[1,1,(m+n+2),2].OnK1(Aλ
+0)∼=Z3,the matrix
+D= diag[1,1,(m−n)].Both matrices are 1 : 1 since m+n+2/\e}atio\slash= 0/\e}atio\slash=m−n.We get:
+K0(Qλ) =Z/(n+m+2)Z⊕Z/2ZandK1(Qλ) =Z/(m−n)Zforλ3+mλ2+nλ+1 = 0.
+To obtain Cuntz algebras, we need m−n=±1.It turns out f(1)>0 can not occur, so we must
+havef(1) =m+n+ 2<0 hence there is a unique root λin (0,1).Combining this inequality with
+m−n=±1 we get exactly two infinite families of solutions; m=n+1 forn≤−2, ORm=n−116
+forn≤−1.In either case, the sequence of numbers {|m+n+2|}is the same:{2k+1|k≥0}.For
+this sequence we get the K0groups:Z/(2k+1)Z⊕Z/2Z∼=Z/(4k+2)Z.Since theK1groups are all
+{0}, by construction, the algebras Qλare (at least stably) the Cuntz algebras, O4k+3fork≥0.That
+is,O3,O7,O11, etc.
+Example, λtranscendental:
+Lemma 2.23. Letϕ:/circleplustext
+n∈ZZ−→Zbe the surjective homomorphism, φ({an}) :=/summationtext
+n∈Zan;and
+letS∈Aut(/circleplustext
+n∈ZZ)be the shift S({an}n∈Z) :={an−1}n∈Z.Then,(id−S)is1 : 1andker(ϕ) =
+Im(id−S),so that(/circleplustext
+n∈ZZ)/Im(id−S)∼=Z.
+Proof.As a model for/circleplustext
+n∈ZZwe useZ[x,x−1] =/circleplustext
+n∈ZZxn,the ring of Laurent polynomials over
+Z(i.e., the group ring over Zof the group{xn|n∈Z}). Here,ϕis the augmentation map, Sis
+multiplication by x,and (id−S) is multiplication by (1 −x) which is 1 : 1 .Now,
+N/summationdisplay
+n=−Nanxn∈ker(ϕ)⇔N/summationdisplay
+n=−Nan= 0⇔N/summationdisplay
+n=−Nanxn+N∈ker(ϕ).
+Letp(x) =/summationtextN
+n=−Nanxn+N∈Z[x] sop(1) =/summationtextN
+n=−Nan= 0.Hence,p(x) factors:p(x) = (1−x)q(x)
+where initially q(x)∈Q[x].Sincep(x)∈Z[x] it is easy to see that in fact, q(x)∈Z[x] also. Then,
+N/summationdisplay
+n=−Nanxn=x−Np(x) = (1−x)x−Nq(x)∈(1−x)Z[x,x−1] = Im(id−S).
+That is, ker( ϕ)⊆Im(id−S),and the other containment is immediate. /square
+Proposition 2.24. Ifλis transcendental then
+K0(Qλ)∼=∞/circleplusdisplay
+n=1Z∼=K1(Qλ).
+Proof.In this case, by Proposition 2.2 and the CONCLUSION before Pr oposition 2.20 we have:
+K0(Aλ
+0) =∞/circleplusdisplay
+k=12k−1/logicalanddisplay
+(Γλ) andK1(Aλ
+0) =∞/circleplusdisplay
+k=12k/logicalanddisplay
+(Γλ) where Γ λ=∞/circleplusdisplay
+n=−∞Zλn.
+Now, each individual summand/logicalandtextm(Γλ) is invariant under λ∗and yields (for m>1) an infinite direct
+sum of (λ∗-invariant) examples of the previous lemma where the action ofλ∗is just the shift. The
+general case is notation-heavy, so we do the examples,/logicalandtext2and/logicalandtext3.LettingZ+denote the positive
+integers, we have:
+2/logicalanddisplay
+(Γλ) =/circleplusdisplay
+k∈Z+/parenleftBigg/circleplusdisplay
+n∈Z(λn∧λn+k)Z/parenrightBigg
+and3/logicalanddisplay
+(Γλ) =/circleplusdisplay
+(k1,k2)∈Z2
++/parenleftBigg/circleplusdisplay
+n∈Z(λn∧λn+k1∧λn+k1+k2)Z/parenrightBigg
+.
+The casem= 1 is just the group Γ λ=/circleplustext
+n∈ZZλnwhich yields a single instance of the lemma.
+Applying the lemma we see that ( id−λ∗) is 1 : 1 on both K0(Aλ
+0) andK1(Aλ
+0) and that both
+K0(Aλ
+0)/(id−λ∗)(K0(Aλ
+0)) andK1(Aλ
+0)/(id−λ∗)(K1(Aλ
+0)) are isomorphic to a countable direct sum
+of copies of Z.An application of the Pimsner-Voiculescu exact sequence co mpletes the proof. /square
+Remark . The classification theory of Kirchberg algebras implies th at forλtranscendental we have a
+new realisation of the algebras found in [Cu1] and denoted QNthere.17
+2.5.The dual action of T1onAλand its restriction to the gauge action on Qλ.Recall,
+G0
+λ={g∈Gλ||g|= 1}is a normal subgroup of Gλ.The subgroup of Gλof elements of the form
+[λn: 0] is isomorphic to Zand acts on G0
+λby conjugacy:
+[λn: 0][1 :b][λ−n: 0] = [1 :λnb].
+ThusGλ=Z⋊G0
+λis a semidirect product and we can write Aλas an iterated crossed product:
+Aλ=Gλ⋊αCλ
+0(R) =Z⋊(G0
+λ⋊αCλ
+0(R)) =Z⋊Aλ
+0.
+The dual action γofT1onAλis relative to this latter crossed product so that for each z∈T1andx
+in the Banach∗-algebra,l1
+α(Gλ,Cλ
+0(R)) we have:
+γz(x)(g) =znx(g) ifx∈l1
+α(Gλ,Cλ
+0(R));g∈Gλand|g|=λn.
+SinceAλis defined to be the completion of this Banach ∗-algebra in its universal representation, the
+actionγextends uniquely to an action (also denoted by γ) ofT1as automorphisms of Aλ.The fixed
+point subalgebra of the dual action is, of course, exactly Aλ
+0=G0
+λ⋊αCλ
+0(R).
+Since the projection eis inAλ
+0,the actionγrestricts to an action of T1onQλ=eAλe,which we will
+also denote by γ.We call this the gauge action ofT1onQλ.Now,γis clearly a strongly continuous
+action of T1onQλ. Averaging over γwith respect to normalised Haar measure gives a positive,
+faithful expectation Φ of Qλonto the fixed-point algebra which is clearly Fλ:
+Φ(a) :=1
+2π/integraldisplay
+T1γz(a)dθfora∈Qλ,andz=eiθ.
+Proposition 2.25. The fixed point algebra, Fλ=eAλ
+0eis the norm closure of finite linear combina-
+tions of elements of the form:
+X[a,b)·δgwhereg= [1 :c]and[a,b)⊆[0,1)∩[c,1+c),
+fora,b,c∈Γλ.Recall,Aλ
+0∼=K(l2(Z))⊗Fλ.
+Proof.Applying the integral formula for Φ to a finite linear combina tion of the generators for Qλwe
+see that the only terms that survive are those where |g|= 1 : that is, ghas the above form. Then we
+apply item (2) of Lemma 2.12 to obtain the condition on the int erval [a,b). /square
+Corollary 2.26. The stabilised algebra Qλ⊗Kis a crossed product of the stabilised fixed-point algebra
+Fλ⊗Kby an action of Z.Forλ= 1/nthis is a theorem of J. Cuntz.
+Proof.By Proposition 2.7, Aλ
+0∼=Fλ⊗K,andAλ∼=Qλ⊗K.By the discussion at the beginning of
+subsection 3.1, Aλ∼=Z⋊Aλ
+0and the proof is complete. See [Cu, Section 2]. /square
+Remarks. If we combine the previous observation that Fλis the fixed point subalgebra of Qλunder
+the gauge action with Corollary 2.18 we get, for example, O2∼=Q2/3with a gauge action whose fixed
+point subalgebra F2/3is a UHF algebra of type 6∞.Interestingly, F3/4is UHF of type 12∞= 6∞
+which is therefore isomorphic to F2/3.So we have two gauge actions on O2with isomorphic UHF
+fixed point subalgebras, with distinct, inequivalent KMS st ates: one where β= log(3/2) and the other
+whereβ= log(4/3) by Proposition 2.30 below. Moreover, the two von Neumann a lgebras generated
+by the GNS representations of O2are not isomorphic as they are type III λfactors forλequalling 2/3
+and 3/4,respectively, by Theorem 2.35 below.18
+2.6.Theγ-invariant semifinite weight on Aλand its restriction to Qλ.The aim of this
+subsection is to exhibit the unique KMS states for the gauge a ction onQλ. We first recall the
+definition of KMS states.
+Definition 2.27. LetAbe aC∗-algebra with a continuous action γ:R→Aut(A).Letψbe a state
+onAandβ∈Ra real number. We define ψto be aKMSβstate for the action γif
+ψ(xγiβ(y)) =ψ(yx)
+for allx,y∈Aa denseγ-invariant∗-subalgebra of Aγ,the subalgebra of analytic elements for the
+actionγ.We refer to [BR1, Section 2.5] for basic information on the subalgebra of analytic element s,
+Aγand to[BR2, Section 5.3] for all the basic information on KMSstates.
+SinceGλis discrete it is well-known that the map
+x/ma√sto→x(1) :l1
+α(Gλ,Cλ
+0(R))→Cλ
+0(R)
+extends uniquely to afaithful conditional expectation E:Aλ→Cλ
+0(R).Composing Ewith thedensely
+defined (norm) lower semicontinuous weight on Cλ
+0(R) given by integration, gives us a densely defined
+(norm) lower semicontinuous weight on Aλwhich we denote by ¯ψ. In particular, for x∈l1
+α(Gλ,Cλ
+0(R))
+we have:
+¯ψ(xx∗) =/integraldisplay
+Rxx∗(1)(t)dt=/integraldisplay
+R
+/summationdisplay
+h∈Gλx(h)x(h)
+(t)dt=/summationdisplay
+h∈Gλ/parenleftbigg/integraldisplay
+R|x(h)(t)|2dt/parenrightbigg
+.
+So that ¯ψis faithful. We observe that ¯ψis not a trace , since¯ψ(x∗x) =/summationtext
+h∈Gλ|h−1|/integraltext
+R|x(h)(t)|2dt.
+Proposition 2.28. The weight ¯ψonAλrestricts to a faithful semifinite trace ¯τonAλ
+0and also
+restricts to a state denoted by ψonQλsatisfying:
+(1) The gauge action γofT1onQλleaves the state ψinvariant.
+(2) The state ψrestricted to the fixed point algebra, Fλis a faithful (finite) trace denoted by τ;which
+is, of course, the restriction of ¯τonAλ
+0toFλ.
+(3) With Φ :Qλ→Fλthe canonical expectation, we have ψ=τ◦Φ.
+Proof.Since¯ψ(e) =/integraltext
+RX[0,1)(t)dt= 1,we see that ¯ψrestricted toQλis a faithful state. To see item
+(1), it suffices to see that the gauge action on l1
+α(Gλ,Cλ
+0(R)) leaves ¯ψinvariant. To this end, let x≥0
+be inl1
+α(Gλ,Cλ
+0(R)), and letz∈T1. Then
+E(γz(x)) =γz(x)(1) =z0x(1) =x(1) =E(x)
+and so
+¯ψ(γz(x)) =/integraldisplay
+Rγz(x)(1)(t)dt=/integraldisplay
+RE(x)(t)dt=¯ψ(x).
+To see item (2) we use Proposition 2.25 and the above computat ion that shows that while ¯ψis not
+generally a trace, to see that it is a trace when the group elements all have determinant 1.
+To see item (3), it suffices to see that for any x∈Qλwe haveE(x) =E(Φ(x)), but this is the same
+asx(1) = Φ(x)(1) which is clear since det(1) = 1 . /square
+Now, since the state ψis invariant under the action γ, this action is unitarily implemented on
+L2(Qλ,ψ).Forz∈T1andx∈Qλ
+cwe define:
+(uz(x))h=znxhforh∈Gλwith|h|=λn.19
+We define the spectral subspaces of this unitary group on L2(Qλ,ψ) in the usual way. For each k∈Z
+let Φkbe the operator on L2(Qλ,ψ) :
+Φk(x) =1
+2π/integraldisplay
+T1z−kuz(x)dθ, z=eiθ, x∈L2(Qλ,ψ).
+We observe that if x=f·δgis a typical generator of Qλconsidered as a vector in L2(Qλ,ψ) then we
+have:
+Φk(f·δg) =/braceleftbigg
+f·δgif|g|=λk
+0 otherwise
+More generally, on H:=L2(Qλ,ψ),we have Φ k(H) ={x∈H|uz(x) =zkxfor allz∈T1}.
+Lemma 2.29. For eachk∈Zthe subspace span {f·δg∈Qλ||g|=λk}is dense in the range of
+Φk.The operators Φkare mutually orthogonal projections on Hwhich sum to the identity operator
+1 =π(e).
+Proof.The proof of the first statement is similar to the proof of Prop osition 2.25. The mutual orthog-
+onality of the Φ kfollows from the fact that /a\}bracketle{tf1·δg1|f2·δg2/a\}bracketri}htψ= 0 unlessg1=g2. /square
+Proposition 2.30. The dense∗-subalgebra ofQλconsisting of finite linear combinations of the partial
+isometriesX[a,b)·δgis contained in the subset of entire elements, Qλ
+γ,for the action γconsidered as
+an action of R:t/ma√sto→γeit.Moreover,ψis aKMSβstate for this action where β= log(λ−1).In fact,ψ
+is theuniqueKMSstate for this action (regardless of β).
+Proof.Lety=X[a,b)·δg∈Qλwhere det(g) =λk.Then,t/ma√sto→γeit(y) =eikty;t∈Robviously extends
+to the entire function w/ma√sto→γeiw(y) =eikwy;w∈C. Forw= log(λ−1)i, this equation becomes
+γeiw(y) =γλ(y) =λky. Lettingβ= log(λ−1), we haveγβi(y) =λky. Now, letx=X[c,d)·hso we want
+to see that: λkψ(xy) =ψ(xγβi(y)) =ψ(yx).That is, we wantλkψ(xy) =ψ(yx).Now both sides of
+this equation are zero unless h=g−1.But, whenh=g−1,we have
+xy=X[c,d)·X[g−1(a),g−1(b))·δIwhileyx=X[a,b)·X[g(c),g(d))·δI.
+Moreover,
+s∈[c,d)∩[g−1(a),g−1(b))⇐⇒g(s)∈[g(c),g(d))∩[a,b).
+Since det(g) =λkthe transformation gincreases the measure by a factor of λkand the result follows.
+That is,ψis a KMS βstate for the action γofRforβ= log(λ−1).
+Now letφbe a KMS state on Qλfor the action γ.SinceQλis purely infinite it has no nontrivial traces
+and soφmust be KMS for some nonzero β.Hence by [BR2, Proposition 5.3.3], φis invariant under
+the action of γ. Now, ifX[a,b)·δg∈Qλwith det(g) =λk, then we have for all z∈T:
+φ(X[a,b)·δg) =φ(γz(X[a,b)·δg)) =zkφ(X[a,b)·δg).
+That is, if det( g)/\e}atio\slash= 1 we must have φ(X[a,b)·δg) = 0,and soφis supported on Fλ.SinceFλis
+γ-invariant and φis KMS for some nonzero β,φis a trace on Fλby [BR2, 5.3.28].
+Now, ifx=X[a,b)·δg∈Fλandg/\e}atio\slash=I, then we claim that φ(x) = 0. For suppose g= [1 :c] with
+c>0. Then there is a positive integer nsuch thata+nc<b≤a+(n+1)cand so
+x=X[a,b)·δg=X[a,c)·δg+X[a+c,a+2c)·δg+···+X[a+nc,b)·δg:=v0+v1+···+vn.
+Now each of these partial isometries vksatisfiesv2
+k= 0,and soφ(vk) =φ(vkv∗
+kvk) =φ(v2
+kv∗
+k) = 0 since
+φis a trace on Fλ.Thus,φ(x) = 0 as claimed.20
+Henceφis supported on the commutative subalgebra
+C:=span{f·δI|f∈Cλ
+0(R) andsupp(f)⊆[0,1)}.
+Morever, if f1,f2are characteristic functions of subintervals of [0 ,1) with endpoints in Γ λand having
+the same length they give equivalent elements fi·δIinFλand therefore have the same value under φ.
+Now, since Aλ
+0∼=Fλ⊗Kwe can define a lower semicontinuous, densely defined trace, ˜TronAλ
+0via
+˜Tr=φ⊗Tr,whereTris the trace onK.So, forX[a,b)·δI∈Fλwe have ˜Tr(X[a,b)·δI) =φ(X[a,b)·δI).
+Then, fork1<k2∈Zthe elementX[k1,k2)·δIis the sum of ( k2−k1) projections in Aλ
+0each equivalent
+toX[0,1)·δIwhich has trace equal to 1; that is, ˜Tr(X[k1,k2)·δI) = (k2−k1).Now, for any a<bin Γλ,
+we haveX[a,b)·δI∼X[0,b−a)·δIand so˜Tr(X[a,b)·δI) =˜Tr(X[0,b−a)·δI),and these values are finite
+since both these projections are dominated by X[−N,N)·δIfor a sufficiently large integer N.It now
+suffices to prove the following. Claim: ˜Tr(X[a,b)·δI) =b−afora<b∈Γλ.
+By the previous discussion we can assume that a= 0 so that b >0.Givenε >0 we choose positive
+integersm,nsuch that
+1
+m≤εandn−1
+m≤b<n
+m,
+so that (n−1)≤bm<nand (n−1),bm,n∈Γλ.Hence
+(n−1) =˜Tr(X[0,(n−1))·δI)≤˜Tr(X[0,bm)·δI)≤˜Tr(X[0,n)·δI) =n.
+But,
+X[0,bm)·δI=X[0,b)·δI+X[b,2b)·δI+···+X[(m−1)b,bm)·δI
+and these projections are mutually equivalent in Aλ
+0.That is,
+(n−1)≤m˜Tr(X[0,b)·δI)≤nso thatn−1
+m≤˜Tr(X[0,b)·δI)≤n
+m.
+Hence,|˜Tr(X[0,b)·δI)−b|<1
+m≤ε.That is,b=˜Tr(X[0,b)·δI) =φ(X[0,b)·δI) andφagrees with the
+given trace τonFλand therefore φagrees with ψonQλ. /square
+Remarks. The above proof shows that the algebra Fλhas a unique (faithful) tracial state τ,and
+thatAλ
+0has a unique (faithful) lower semicontinuous, densely defin ed trace normalized so that it has
+value 1 ate=X[0,1)·δI.
+2.7.The von Neumann algebra π(Aλ)−woacting onL2(Aλ,¯ψ)is a type IIIλfactor.To prove
+this we will show that it is unitarily equivalent to a version of the Murray-von Neumann “group-
+measure space” construction of type III factors on l2(Gλ)⊗L2(R) : see [D, Chapter 1, Section 9]. We
+conclude that it is a III λfactor by an appeal to Connes’ thesis [C0]. In order to be cons istent with
+our use of right C∗-modules later, we will do our GNS constructions so that our i nner products are
+linear in the secondvariable.
+Proposition 2.31. The∗-algebraAλ
+cis a Tomita algebra with the inner product:
+/a\}bracketle{ty|x/a\}bracketri}ht¯ψ=¯ψ(y∗x) =/summationdisplay
+h∈Gλ|h|−1/a\}bracketle{txh|yh/a\}bracketri}htL2(R).
+Here we denote xhin place of x(h)to simplify notation. In this setting we have for x∈Aλ
+c:
+(1)Sharp:S(x)h=αh(xh−1);
+(2)Flat:F(x)h=|h|αh(xh−1);
+(3)Delta:∆(x)h=|h|xh.21
+Proof.Wereferto[Ta]forTakesaki’s versionoftheaxiomsforaTom itaalgebra. Since Sharpisdefined
+to be the adjoint operation on the algebra, item (1) is immedi ate. A straightforward calculation shows
+that for all x,y∈Aλ
+cwe have that the defining equation for Flatholds, namely:/a\}bracketle{tS(y)|x/a\}bracketri}ht¯ψ=/a\}bracketle{tF(x)|y/a\}bracketri}ht¯ψ
+so that item (2) holds. By definition, ∆ = FSand so a simple calculation shows that ∆( x)h=|h|xh
+and (3) holds. From this formula for ∆ we see that for each z∈Cwe have ∆z(x)h=|h|zxhand a
+straightforward calculation shows that ∆z(x·y) = (∆z(x))·(∆z(y)) so that each ∆zis an algebra
+homomorphism of Aλ
+cas required. That each left multiplication π(x) is bounded when xis supported
+on a single group element is straightforward and the general ization to finitely supported elements is
+then trivial. The fact that it is a ∗-representation holds as it does for the GNS representation for any
+weight.
+In order to see that products are dense we recall that we have l ocal units. That is, for each positive
+integerNwe have defined EN=X[−N,N)·δ1,and have noted that for each y∈Aλ
+cthat satisfies
+supp(yh)⊆[−N,N) for allh,we haveEN·y=y.Axioms IV, V, VI in [Ta] are simple calculations
+involving the definitions of S,F, and ∆ which we leave to the reader.
+Since our inner products are linear in the second variable, w e modify Tomita’s Axiom VIII to
+read:z/ma√sto→/a\}bracketle{tx|∆x(y)/a\}bracketri}ht¯ψis analytic on Cfor allx,y∈Aλ
+c.We easily calculate that /a\}bracketle{tx|∆z(y)/a\}bracketri}ht¯ψ=/summationtext
+h|h|z−1/a\}bracketle{tyh|xh/a\}bracketri}htL2(R).This function is analytic since the sum is finite. /square
+Lemma 2.32. The representation of Aλ
+conL2(Aλ
+c,¯ψ)decomposes as the integrated form of a covariant
+pair of representations:
+(1) ˜π:Cλ
+00(R)→B(L2(Aλ
+c,¯ψ)),where : (˜π(f)(y))h=f·yhforf∈Cλ
+00(R) andy∈Aλ
+c;
+(2)U:Gλ→U(L2(Aλ
+c,¯ψ)) where : ( Ug(y))h=αg(yg−1h) forg∈Gλandy∈Aλ
+c.
+Proof.It is straightforward to verify that Uis a unitary representation of Gλand that ˜πis a∗-
+representation of Cλ
+00(R).To see the covariance condition:
+(Ug˜π(f)Ug−1(y))h=···=αg(f·αg−1(ygg−1)) =αg(f)·yh= (˜π(αg(f))y)h.
+That is,Ug˜π(f)Ug−1= ˜π(αg(f)).
+Now, by Proposition 7.6.4 of [Ped] the integrated form of thi s covariant pair is the representation:
+(˜π×U)(y) =/summationdisplay
+h˜π(yh)Uhfory∈Aλ
+c.
+Now, we evaluate this operator on the vector x∈Aλ
+c:
+[((˜π×U)(y))(x)]k=/summationdisplay
+h[˜π(yh)Uh(x)]k=/summationdisplay
+hyhαh(xh−1k) = (y·x)(k) = (π(y)(x))k.
+That is, (˜π×U)(y) =π(y) the operator left multiplication by y. /square
+2.7.1.A representation of Aλonl2(Gλ)⊗L2(R).We define a covariant pair of representations of
+Cλ
+0(R) andGλonl2(Gλ)⊗L2(R) as follows:
+(1) forf∈Cλ
+0(R) letπ(f) = 1⊗Mf,and (2) for g∈GλletUg= Λ(g)⊗Vg
+where Λ is the left regular representation of Gλonl2(Gλ) :
+(Λ(g)ξ)(h) =ξ(g−1h) forξ∈l2(Gλ);
+andVis the unitary action of GλonL2(R) induced by the action of GλonR:
+(Vg(f))(t) =|g|−1/2f(g−1t); forf∈L2(R).22
+Using these equations one easily checks the covariance cond ition forg∈Gλandf∈Cλ
+0(R) :
+Ugπ(f)U∗
+g=π(αg(f)).
+Clearly the representation πextends uniquely by weak-operator continuity to the usual r epresentation
+1⊗MofL∞(R) onl2(Gλ)⊗L2(R) and is covariant with the unitary representation UofGλfor the
+actionαofGλonL∞(R).Clearly, the von Neumann algebra on l2(Gλ)⊗L2(R) generated by the
+unitariesUgand the operators 1 ⊗Mfforg∈Gλandf∈Cλ
+00(R), is the same as the von Neumann
+algebra generated by the unitaries Ugand the operators 1 ⊗Mfforg∈Gλandf∈L∞(R). The
+second item of the following Proposition is clear.
+Proposition 2.33. (1) The representation π= (˜π×U)ofAλonL2(Aλ
+c,¯ψ)is unitarily equivalent to
+the representation (π×U)ofAλonl2(Gλ)⊗L2(R).
+(2)(π×U)(Aλ)′′is the von Neumann crossed product (in the sense of the group- measure space con-
+struction of Chapter 1 Section 9 of [D])Gλ⋊αL∞(R).
+(3) This von Neumann algebra is a type IIIfactor.
+Proof.To see item (3) we use the proof of [D, Theorem 2, Section 9, Cha pter 1] where instead of the
+ax+bgroupGwitha,b∈Qanda>0 and its subgroup G0(witha= 1), we use Gλand its subgroup
+G0
+λ(with|g|= 1), to conclude that our von Neumann algebra is a type III fac tor.
+To see item (1), we first define a unitary W:L2(Aλ
+c,¯ψ)→l2(Gλ)⊗L2(R) as follows:
+W/parenleftBiggm/summationdisplay
+i=1fi·δhi/parenrightBigg
+=m/summationdisplay
+i=1|hi|−1/2δhi⊗fi.
+On the left side of this equation we are using the formalism f·δhfor singly supported elements in Aλ
+c
+withf∈Cλ
+00(R) andh∈Gλ.On the right of this equation we are using δhto denote the canonical
+orthonormal basis elements in l2(Gλ) and regarding f∈Cλ
+00(R)⊂L2(R).Clearly,Wis well-defined
+and linear with dense range. One easily checks that: for all x,y∈Aλ
+cwe have
+/a\}bracketle{ty|x/a\}bracketri}ht¯ψ=/a\}bracketle{tW(x)|W(y)/a\}bracketri}htl2⊗L2
+recalling that the inner product on Aλ
+cis linear in the second coordinate. Thus Wis a unitary and its
+inverse (adjoint) defined at first on the elements in l2(Gλ)⊗L2(R) of the form/summationtextm
+i=1δhi⊗fiwith the
+fi∈Cλ
+00(R), is given by:
+W∗/parenleftBiggm/summationdisplay
+i=1δhi⊗fi/parenrightBigg
+=m/summationdisplay
+i=1|hi|1/2fi·δhi.
+One then verifies the following two equations for f∈Cλ
+00(R) andg∈Gλ:
+(1)W˜π(f)W∗= 1⊗Mf=π(f) and (2) WUgW∗= Λ(g)⊗Vg=Ug.
+The second equation is more subtle and requires the observat ion:Ug(f·δh) =|g|1/2Vg(f)·δgh.
+This completes the proof of the proposition. /square
+2.7.2.The factorπ(Aλ)′′acting onL2(Aλ
+c,¯ψ)is typeIIIλ.We work in the unitarily equivalent setting
+of (π×U)(Aλ)′′acting onl2(Gλ)⊗L2(R) afforded by Proposition 2.33. Recall that the subgroup of Gλ
+of matrices of the form [ λn: 0] is isomorphic to Zand acts on the normal subgroup G0
+λby conjugacy,
+and soGλ=Z⋊G0
+λis a semidirect product and we can write a canonical right cos et decomposition
+ofGλ:
+Gλ=/uniondisplay
+n∈ZG0
+λ·[λn: 0].23
+This gives us an internal orthogonal decomposition of l2(Gλ) :
+l2(Gλ) =/summationdisplay
+n∈Z⊕l2/parenleftbig
+G0
+λ·[λn: 0]/parenrightbig∼=l2(Z)⊗l2(G0
+λ).
+Here the latter isomorphism is given explicitly on basis ele ments by the map which takes the δ-function
+atg·[λn: 0] toδn⊗δgforn∈Zandg∈G0
+λ.
+One checks that the restriction of the representation ( π×U) ofAλ=Gλ⋊Cλ
+0(R) toAλ
+0:=G0
+λ⋊Cλ
+0(R)
+onl2(Gλ)⊗L2(R)isunitarilyequivalenttotherepresentationon l2(Z)⊗l2(G0
+λ)⊗L2(R)viathecovariant
+pair:
+1Z⊗Λ(h)⊗Vh= 1Z⊗Uhforh∈G0
+λand
+1Z⊗1⊗Mf= 1Z⊗π(f) forf∈Cλ
+0(R).
+Therefore, the von Neumann subalgebra of ( π×U)(Aλ)′′generated by ( π×U)(Aλ
+0) is isomorphic to
+the von Neumann algebra on l2(G0
+λ)⊗L2(R) generated by the operators Λ( h)⊗Vhforh∈G0
+λand
+1⊗Mfforf∈Cλ
+0(R).This is clearly the same as the von Neumann algebra generated by the operators
+Λ(h)⊗Vhforh∈G0
+λand 1⊗Mfforf∈L∞(R),and this von Neumann algebra is a factor of type
+II∞by the methods of [D, Chapter 1, Section 9]. Thus, ( π×U)(Aλ
+0)′′is a typeII∞subfactor of the
+type III factor, ( π×U)(Aλ)′′.Moreover, the faithful normal semifinite trace on ( π×U)(Aλ
+0)′′is given
+by the restriction of ¯ψ.
+Finally, conjugation by the unitary, Ugforg= [λ: 0], which lies in our type IIIfactor, defines an
+automorphism βof the type II∞subfactor which scales the trace by λ.IfN0is our type II∞factor
+acting onl2(G0
+λ)⊗L2(R) then our type III factor, say Aλacting onl2(Z)⊗l2(G0
+λ)⊗L2(R) is unitarily
+equivalent to the von Neumann crossed product Aλ∼=Z⋊βN0and hence is a type III λfactor by [C0,
+Theorem 4.4.1]. We have proved the following Proposition.
+Proposition 2.34. The von Neumann algebra π(Aλ)′′acting onL2(Aλ
+c,¯ψ)is a type IIIλfactor.
+Moreover, it is unitarily equivalent to (π×U)(Aλ)′′acting onl2(Gλ)⊗L2(R).The von Neumann
+subalgebra of (π×U)(Aλ)′′generated by (π×U)(Aλ
+0)is a typeII∞factor. The space l2(Gλ)⊗L2(R)
+factors asl2(Z)⊗l2(G0
+λ)⊗L2(R)and with this factorization, our II∞factor has the form N0= 1Z⊗˜N0
+where˜N0acts onl2(G0
+λ)⊗L2(R).Thus, our type IIIλfactor is unitarily equivalent to the von Neumann
+crossed product Z⋊βN0where the automorphism βofN0is given by β=Ad(Ug)whereg= [λ: 0].
+2.8.The von Neumann algebra, π0(Qλ)−woacting onL2(Qλ,ψ)is type IIIλ.
+Theorem 2.35. The von Neumann algebra, π0(Qλ)−woacting onL2(Qλ,ψ)is typeIIIλ.Moreover,
+the von Neumann subalgebra, π0(Fλ)−wois a typeII1factor with unique faithful normal state given
+by the restriction of the vector state, ψwhich is the same as τonFλ.By the general theory of type
+IIIfactors,π0(Qλ)−wois isomorphic to π(Aλ)−woacting onL2(Aλ
+c,¯ψ).
+Proof.Recall thatQλ=eAλewheree=X[0,1)·δ1∈Aλ.Then
+π(e)(π(Aλ)−wo)π(e) = (π(e)π(Aλ)π(e))−wo=π(Qλ)−wo
+and the cut-down of the type III factor π(Aλ)−wo(on its separable Hilbert space) by the nonzero
+projection π(e) is isomorphic to π(Aλ)−wosinceπ(e) is Murray-von Neumann equivalent to the
+identity operator. Of course the cut-down mapping byπ(e) isnotan isomorphism. Moreover,
+by left Hilbert algebra theory, the operator rightmultiplication by ewhich is denoted by π′(e)
+is in the commutant of π(Aλ)−woacting onL2(Qλ,¯ψ) and since we are in a factor the mapping
+π(Aλ)−wo→π′(e)π(Aλ)−wois an isomorphism by [D, Chapter 1, Section 2, Prop. 2]. Restricting
+this isomorphism to π(Qλ)−wogives us an isomorphism π(Qλ)−wo→π′(e)π(Qλ)−wowhich acts on the24
+Hilbertspace π′(e)π(e)(L2(Aλ,¯ψ)),whichhas asadensesubspace π′(e)π(e)(Aλ
+c) =eAλ
+ce⊂eAλe=Qλ
+with the inner product given by ¯ψwhich is the same as the inner product on eAλ
+cegiven by the state
+ψ.The completion of this space is, of course, L2(Qλ,ψ) with the action of Qλbeing the GNS represen-
+tation afforded by the state ψ.We denote this representation of QλonL2(Qλ,ψ) byπ0to distinguish
+it from the representation πofAλon the larger space, L2(Aλ
+c,¯ψ).
+Similar considerations applied to the type II∞subfactor,π(Aλ
+0)−wo⊂π(Aλ)−woonL2(Aλ
+c,¯ψ),show
+that:
+π(e)(π(Aλ
+0)−wo)π(e) = (π(e)π(Aλ
+0)π(e))−wo=π(Fλ)−wo.
+Now the projection π(e) is actually in the type II∞subfactorπ(Aλ
+0)−woofπ(Aλ)−woand has finite ( ψ)
+trace = 1 there. Therefore, π(Fλ)−wois a typeII1factor onL2(Qλ,ψ) with trace given by the vector
+stateψ.We remark that this is clearly a larger space than the subspac e,L2(Fλ,τ)⊂L2(Qλ,ψ)./square
+Proposition 2.36. The∗-algebraQλ
+cis a Tomita algebra with the inner product: /a\}bracketle{ty|x/a\}bracketri}htψ=ψ(y∗x).
+Again we denote xhin place of x(h)to simplify notation. In this setting we have for x∈Qλ
+c:
+(1)Sharp:S(x)h=αh(xh−1);
+(2)Flat:F(x)h=|h|αh(xh−1);
+(3)Delta:∆(x)h=|h|xh.
+Proof.This is really a corollary of Proposition 2.7, as Qλ
+cis just a Tomita-subalgebra of Aλ
+c./square
+3.The modular spectral triple of the algebra Qλ
+Having introduced the main features of the algebras Qλ, we now turn briefly to the modular index
+theory of [CNNR, CPR2, CRT]. We begin with some semifinite pre liminaries.
+3.1.Semifinite noncommutative geometry. We need to explain some semifinite versions of stan-
+dard definitions and results following [CPRS2]. Let φbe a fixed faithful, normal, semifinite trace on
+a von Neumann algebra N. LetKNbe theφ-compact operators in N(that is the norm closed ideal
+generated by the projections E∈Nwithφ(E)<∞).
+Definition 3.1. Asemifinite spectral triple (A,H,D)is given by a Hilbert space H, a∗-algebra
+A⊂NwhereNis a semifinite von Neumann algebra acting on H, and a densely defined unbounded
+self-adjoint operator Daffiliated toNsuch that [D,a]is densely defined and extends to a bounded
+operator inNfor alla∈Aand(λ−D)−1∈KNfor allλ/\e}atio\slash∈R.The triple is said to be evenif there
+isΓ∈Nsuch that Γ∗= Γ,Γ2= 1,aΓ = Γafor alla∈AandDΓ+ΓD= 0. Otherwise it is odd.
+Note that if T∈Nand [D,T] is bounded, then [ D,T]∈N.
+We recall from [FK] that if S∈N, thet-th generalized singular value ofSfor each real t>0 is
+given by
+µt(S) = inf{/bardblSE/bardbl:Eis a projection in Nwithφ(1−E)≤t}.
+The idealL1(N,φ) consists of those operators T∈Nsuch that/bardblT/bardbl1:=φ(|T|)<∞where|T|=√
+T∗T. In the Type I setting this is the usual trace class ideal. We w ill denote the norm on L1(N,φ)
+by/bardbl·/bardbl1. An alternative definition in terms of singular values is tha tT∈L1(N,φ) if/bardblT/bardbl1:=/integraltext∞
+0µt(T)dt<∞.WhenN/\e}atio\slash=B(H),L1(N,φ) need not be complete in this norm but it is complete in
+the norm/bardbl·/bardbl1+/bardbl·/bardbl∞. (where/bardbl·/bardbl∞is the uniform norm). We use the notation
+L(1,∞)(N,φ) =/braceleftbigg
+T∈N:/bardblT/bardbl
+L(1,∞):= sup
+t>01
+log(1+t)/integraldisplayt
+0µs(T)ds<∞/bracerightbigg
+.25
+The reader should note that L(1,∞)(N,φ) is often taken to mean an ideal in the algebra /tildewideNofφ-
+measurable operators affiliated to N. Our notation is however consistent with that of [C] in the
+special caseN=B(H). With this convention the ideal of φ-compact operators, K(N), consists of
+thoseT∈N(as opposed to /tildewideN) such that µ∞(T) := limt→∞µt(T) = 0.
+Definition 3.2. A semifinite spectral triple (A,H,D)relative to (N,φ)withAunital is (1,∞)-
+summable if (D−λ)−1∈L(1,∞)(N,φ)for allλ∈C\R.
+It follows that if ( A,H,D) is (1,∞)-summable then it is n-summable (with respect to the trace φ) for
+alln>1. We next need to briefly discuss Dixmier traces. For more inf ormation on semifinite Dixmier
+traces, see [CPS2, CRSS]. For T∈L(1,∞)(N,φ),T≥0, the function
+FT:t→1
+log(1+t)/integraldisplayt
+0µs(T)ds
+is bounded. There are certain ω∈L∞(R+
+∗)∗, [CPS2, C], which define(Dixmier) traces on L(1,∞)(N,φ)
+by setting
+φω(T) =ω(FT), T≥0
+and extending to all of L(1,∞)(N,φ) by linearity. For each such ωwe writeφωfor the associated
+Dixmier trace. Each Dixmier trace φωvanishes on the ideal of trace class operators. Whenever the
+functionFThas a limit at infinity, all Dixmier traces return that limit a s their value. This leads to
+the notion of a measurable operator [C, LSS], that is, one on w hich all Dixmier traces take the same
+value.
+3.2.The Kasparov module and modular spectral triple. We have seen that the algebras Qλ
+do not possess a faithful gauge invariant trace but that ther e is a KMS βwhereβ=−log(λ) for the
+gauge action, γ,namelyψ:=τ◦Φ :Qλ→C, where Φ :Qλ→Fλis the expectation and τ:Fλ→C
+is a faithful normalised trace. In fact, ψis the only KMS state for the gauge action (for any β),
+by Proposition 2.30. We show below that the generator of the g auge actionDacting on a suitable
+C∗-Fλ-moduleXgives us a Kasparov module ( X,D) whose class lies in KK1,T(Qλ,Fλ). In some
+examples, including the case λ∈Q, we haveK1(Qλ) ={0}and so pairing with ordinary K1would be
+fruitless. However, following [CPR2, CNNR] we may compute a numerical pairing using a ‘modular
+spectral triple’ constructed from the Kasparov module.
+We now review this construction adapted to the present situa tion. LetH=L2(Qλ) be the GNS
+Hilbert space given by the faithful state ψwith the inner product on Qλdefined by/a\}bracketle{ta,b/a\}bracketri}ht=ψ(a∗b) =
+(τ◦Φ)(a∗b).ThenDis a self-adjoint unbounded operator on H, [CPR2]. The representation of Qλ
+onHby left multiplication (which we now denote by πin place ofπ0) is bounded and nondegenerate:
+the left action of an element a∈Qλbyπ(a) satisfiesπ(a)b=abfor allb∈Qλ.This distinction
+between elements of Qλas vectors inL2(Qλ) and operators on L2(Qλ) is sometimes crucial. The
+dense subalgebra Qλ
+c:=eAλ
+cewhich is the finite span of elements in Qλof the formX[a,b)·δgis in the
+smooth domain of the derivation δ= ad(|D|). We remind the reader that the KMS condition on the
+modular automorphism group of the state ψ, [Ta], (for t=i) is:ψ(xy) =ψ(σi(π(y))x) =ψ(σ(y)x)
+forx,y∈π(Qλ),whereσ(y) = ∆−1(y).
+Lemma 3.3. The group of modular automorphisms of the von Neumann algebra π(Qλ)′′is given on
+the generators by
+(2)σt(π(f·δg)) := ∆itπ(f·δg)∆−it=π(∆it(f·δg)) =|g|itπ(f·δg) = det(g)itπ(f·δg).
+Proof.This is immediate from Lemma 2.8 if we note that |g|= det(g). /square26
+Corollary 3.4. WithQλacting onH:=L2(Qλ)and withDthe generator of the natural unitary
+implementation of the gauge action of T1onQλ,we have ∆ =λDoreitD= ∆it/logλ.
+To simplify notation, we let A=QλandF=Fλ=Aγ, the fixed point algebra for the T1gauge
+action,γ.For convenience we will suppress the notations D⊗1kand so on. The algebras Ac,Fcare
+defined as the finite linear span of the generators. Right mult iplication makes Ainto a right F-module,
+and similarly Acis a right module over Fc. We define an F-valued inner product ( ·|·)Ron both these
+modules by ( a|b)R:= Φ(a∗b).
+Definition 3.5. LetXbe the right F C∗-module obtained by completing A(orAc) in the norm
+/bardblx/bardbl2
+X:=/bardbl(x|x)R/bardblF=/bardblΦ(x∗x)/bardblF.
+Thealgebra Aactingbyleftmultiplication on Xprovidesarepresentationof Aasadjointableoperators
+onX. LetXcbe the copy of Ac⊂X. TheT1action onXcis unitary and extends to X, [CNNR, PR].
+For allk∈Z, the projection operator onto the k-th spectral subspace of the T1action is also denoted
+(somewhat carelessly) Φ konX:
+Φk(x) =1
+2π/integraldisplay
+T1z−kuz(x)dθ, z=eiθ, x∈X.
+Observe that Φ 0restricts to Φ on Aand on generators of Qλwe have
+(3) Φ k(f·δg) =/braceleftbigg
+f·δgif|g|=λk
+0 otherwise
+Of courseL2(Qλ) andXhave a common dense subspace Qλ
+con which these projections are identical.
+LetAk= Φk(A) and observe from (3) that A∗
+kAk=F=AkA∗
+kso that the gauge action γonQλhas
+full spectral subspaces .
+We quote the following result from [PR], the proof in our case is the same.
+Lemma 3.6. The operators Φkare adjointable endomorphisms of the F-moduleXsuch that Φ∗
+k=
+Φk= Φ2
+kandΦkΦl=δk,lΦk. IfK⊂Zthen the sum/summationtext
+k∈KΦkconverges strictly to a projection in the
+endomorphism algebra. The sum/summationtext
+k∈ZΦkconverges to the identity operator on X. For allx∈X,
+the sumx=/summationtext
+k∈ZΦkx=/summationtext
+k∈Zxkconverges in X.
+The unbounded operator of the next proposition is of course t he generator of the T1action onX. We
+refer to Lance’s book, [L, Chapters 9,10], for information o n unbounded operators on C∗-modules.
+Proposition 3.7. [PR]LetXbe the right C∗-F-module of Definition 3.5. Define D:XD⊂Xto be
+the linear space
+XD={x=/summationdisplay
+k∈Zxk∈X:/bardbl/summationdisplay
+k∈Zk2(xk|xk)R/bardbl<∞}.
+Forx∈XDdefineD(x) =/summationtext
+k∈Zkxk.ThenD:XD→Xis a is self-adjoint, regular operator on X.
+This should be compared to the following Hilbert space versi on.
+Proposition 3.8. The generatorDof the one-parameter unitary group {uz|z∈T1}onL2(Qλ,ψ)
+has eigenspaces given by the ranges of the ΦkandD(x) =kxiffΦk(x) =x.In particular
+dom(D) ={x=/summationdisplay
+kxk|Φk(xk) =xkand/summationdisplay
+kk2/bardblxk/bardbl2<∞},
+andD(/summationtext
+kxk) =/summationtext
+kkxk.27
+Remark. On generators in Qλregarded as elements of either XorL2(Qλ,ψ) we haveD(f·δg) =
+(logλ(|g|))f·δg.
+To continue, we recall the underlying right C∗-Fλ-module,X, which is the completion of Qλfor the
+norm/bardblx/bardbl2
+X=/bardblΦ(x∗x)/bardblFλ. Introduce the rank one operators on X: ΘR
+x,yby ΘR
+x,yz=x(y|z)R. Then
+using the operators Sk,mdefined above, we obtain formulas for the projections Φ ksimilar to those of
+[PR, Lemma 4.7] with some important differences. First recall [CPR2, Lemma 3.5].
+Lemma 3.9. AnyFλ-linear endomorphism Tof the module Xwhich preserves the copy of Qλinside
+X, extends uniquely to a bounded operator on the Hilbert space H=L2(Qλ).
+In particular, the finite rank endomorphisms of the pre- C∗moduleQλ
+c(acting on the left) satisfy this
+condition, and we denote the algebra of all these endomorphi sms byEnd00
+F(Qλ
+c).
+Lemma 3.10. Compare [PR, Lemma 4.7] . The following formulas hold in both L(X)and inB(H).
+(1)Fork≥0,we have
+Φ0= ΘR
+e,ewhile fork>0,Φk=mk/summationdisplay
+m=0ΘR
+Sk,m,Sk,m.
+(2)For−k<0,we have
+Φ−k= ΘR
+S∗
+k,m,S∗
+k,mfor anym= 0,1,...,mk−1 and also for mkifλ−k=mk+1.
+Proof.Since both Φ kand the finite rank endomorphismssatisfy the hypotheses of t he previous lemma,
+the first statement of this lemma will follow from calculatio ns done on generators. The following
+calculations are based on the formulas in Lemma 2.15.
+(1) Letk>0 and letx=/summationtext
+lxlbe a finite sum of generators, xlsatisfying Φ l(xl) =xl.Then
+mk/summationdisplay
+m=0ΘR
+Sk,m,Sk,m(x) =/summationdisplay
+lmk/summationdisplay
+m=0ΘR
+Sk,m,Sk,m(xl) =/summationdisplay
+lmk/summationdisplay
+m=0Sk,mΦ(S∗
+k,mxl) =mk/summationdisplay
+m=0Sk,mΦ(S∗
+k,mxk)
+=mk/summationdisplay
+m=0Sk,mS∗
+k,mxk=exk=xk= Φk(x).
+Fork= 0 this is a similar but far easier calculation.
+(2) Let−k<0 and letx=/summationtext
+lxlbe a finite sum of generators as above. Then, for 0 ≤m<mk
+ΘR
+S∗
+k,m,S∗
+k,m(x) =/summationdisplay
+lΘR
+S∗
+k,m,S∗
+k,m(xl) =/summationdisplay
+lS∗
+k,mΦ(Sk,mxl) =S∗
+k,mΦ(Sk,mx−k)
+=S∗
+k,mSk,mx−k=ex−k=x−k= Φ−k(x).
+/square
+We recall the following result discussed in Section 3 of [CNN R] (a ‘bare hands’ proof can be given by
+the method in [CPR2]).
+Proposition 3.11. LetNbe the von Neumann algebra N= (End00
+F(Qλ
+c))′′,where we take the commu-
+tant insideB(H). ThenNis semifinite, and there exists a faithful, semifinite, norma l trace˜τ:N→C
+such that for all rank one endomorphisms ΘR
+x,yofQλ
+c,
+˜τ(ΘR
+x,y) = (τ◦Φ)(y∗x), x,y∈Qλ
+c.
+In addition,Dis affiliated toNandπ(Qλ)is a subalgebra of N.28
+The fact that ˜ τ(Φk) =λ−kimplies that with respect to the trace ˜ τwe can not expect Dto satisfy a
+finite summability criterion. We solve this problem exactly as in [CPR2].
+Definition 3.12. We define a new weight on N+: letT∈N+thenτ∆(T) := supN˜τ(∆NT)where
+∆N= ∆(/summationtext
+|k|≤NΦk).
+Remarks . Since ∆ Nis ˜τ-trace-class, we see that T/ma√sto→˜τ(∆NT) is a normal positive linear functional
+onNand henceτ∆is a normal weight on N+which is easily seen to be faithful and semifinite.
+As in [CPR2], we now give another way to define τ∆which is not only conceptually useful but also
+makes a number of important properties straightforward to v erify. Many proofs require only trivial
+notation changes and the substitution of n±withλ∓.
+Notation . LetMbe the relative commutant in Nof the operator ∆. Equivalently, Mis the relative
+commutant of the set of spectral projections {Φk|k∈Z}ofD.Clearly,M=/summationtext
+k∈ZΦkNΦk.
+Definition 3.13. As˜τrestricted to each ΦkNΦkis a faithful finite trace with ˜τ(Φk) =λ−kwe define
+/hatwideτkonΦkNΦkto beλktimes the restriction of ˜τ.Then,/hatwideτ:=/summationtext
+k/hatwideτkonM=/summationtext
+k∈ZΦkNΦkis a faithful
+normal semifinite trace /hatwideτwith/hatwideτ(Φk) = 1for allk.
+We use/hatwideτto give an alternative expression for τ∆below
+Lemma 3.14. An element m∈Nis inMif and only if it is in the fixed point algebra of the action,
+στ∆
+tonNdefined for T∈Nbyστ∆
+t(T) = ∆itT∆−it.Bothπ(Fλ)and the projections Φkbelong to
+M. The map Ψ :N →M defined by Ψ(T) =/summationtext
+kΦkTΦkis a conditional expectation onto Mand
+τ∆(T) =/hatwideτ(Ψ(T))for allT∈N+.That is,τ∆=/hatwideτ◦Ψso that/hatwideτ(T) =τ∆(T)for allT∈M+.Finally,
+if one ofA,B∈Mis/hatwideτ-trace-class and T∈Nthenτ∆(ATB) =τ∆(AΨ(T)B) =/hatwideτ(AΨ(T)B).
+Proof.The proof is the same as the proof of [CPR2, Lemma 3.9] with λkin place ofn−k. /square
+Lemma 3.15. The modular automorphism group στ∆
+tofτ∆is inner and given by στ∆
+t(T) = ∆itT∆−it.
+The weight τ∆is aKMSweight for the group στ∆
+t, andστ∆
+t|Qλ=στ◦Φ
+t.
+Proof.This follows from: [KR, Thm 9.2.38], which gives us the KMS pr operties of τ∆: the modular
+group is inner since ∆ is affiliated to N.The final statement about the restriction of the modular
+group toQλis clear. /square
+We now have the key lemma:
+Lemma 3.16. Supposegis a function on Rsuch thatg(D)isτ∆trace-class inM, then for all f∈Fλ
+we have
+τ∆(π(f)g(D)) =τ∆(g(D))τ(f) =τ(f)/summationdisplay
+k∈Zg(k).
+Proof.First note that τ∆(g(D)) =/hatwideτ(/summationtext
+k∈Zg(k)Φk) =/summationtext
+k∈Zg(k)/hatwideτ(Φk) =/summationtext
+k∈Zg(k).We first do the
+computation for f∈Fλ
+cso that all the sums are finite. Now,
+τ∆(π(f)g(D)) =/hatwideτ(π(f)/summationdisplay
+k∈Zg(k)Φk) =/summationdisplay
+k∈Zg(k)/hatwideτ(π(f)Φk)
+=/summationdisplay
+k∈Zg(k)/hatwideτk(π(f)Φk) =/summationdisplay
+k∈Zg(k)λk˜τ(π(f)Φk).
+So it suffices to see for each k∈Z, we have ˜τ(π(f)Φk) =λ−kτ(f).29
+Now, by Theorem 2.35 π(Fλ)′′is a typeII1factor onHwhose unique trace say Tr(with norm
+one) extends the trace τonFλin the sense that Tr(π(f)) =τ(f).Since the projection Φ kis in the
+commutant of the factor π(Fλ)′′the map
+T∈π(Fλ)′′/ma√sto→TΦk= ΦkTΦk
+is a normal isomorphism by [D, Chapter 1, section 2, Prop. 2] a nd so it has a unique normalised trace
+also given by Trace(TΦk) =Tr(T).But ˜τ(TΦk) is a trace on Φ kπ(Fλ)′′Φk=π(Fλ)′′Φkand so must
+be ˜τ(Φk) =λ−ktimes the unique norm one trace. That is, we get the required f ormula:
+˜τ(π(f)Φk) =λ−kTrace(π(f)Φk) =λ−kTr(π(f)) =λ−kτ(f).
+So forf∈Fλ
+c,we have the formula:
+τ∆(π(f)g(D)) =τ∆(g(D))τ(f) =/summationdisplay
+k∈Zg(k)τ(f).
+Now, therighthandsideisanorm-continuousfunctionof f. Toseethattheleftsideisnorm-continuous
+we do it in more generality. Let T∈N, then since/hatwideτis a trace onMwe get:
+|τ∆(Tg(D))|=|/hatwideτ(Ψ(Tg(D))|=|/hatwideτ(Ψ(T)g(D))|≤/bardblΨ(T)/bardbl/hatwideτ(|g(D)|)≤/bardblT/bardbl/hatwideτ((|g(D)|) =/bardblT/bardblτ∆(|g(D)|).
+That is the left hand side is norm-continuous in Tand so we have the formula:
+τ∆(π(f)g(D)) =τ∆(g(D))τ(f) =/summationdisplay
+k∈Zg(k)τ(f)
+for allf∈Fλ. /square
+Proposition 3.17. (i) We have (1+D2)−1/2∈L(1,∞)(M,τ∆). That is,τ∆((1 +D2)−s/2)<∞for
+alls>1.Moreover, for all f∈Fλ
+lim
+s→1+(s−1)τ∆(π(f)(1+D2)−s/2) = 2τ(f)
+so thatπ(f)(1+D2)−1/2is a measurable operator in the sense of [C].
+(ii) Forπ(a)∈π(Qλ)⊂Nthe following (ordinary) limit exists and
+/hatwideτω(π(a)) =1
+2lim
+s→1+(s−1)τ∆(π(a)(1+D2)−s/2) =τ◦Φ(a),
+the original KMSstateψ=τ◦ΦonQλ.
+Proof.(i) This proof is identical to [CPR2, Proposition 3.12].
+(ii) This proof is the same as [CPR2, Proposition 3.14] with Qλ,FλreplacingOn,F. /square
+Definition 3.18. The triple (A,H,D)along withγ, ψ,N, τ∆satisfying properties (0) to (3) below
+is called the modular spectral triple of the dynamical system (Qλ,γ,ψ)
+0) The∗-subalgebraA=Qλ
+cof the algebraQλis faithfully represented in Nwith the latter acting on
+the Hilbert space H=L2(Qλ,ψ),
+1) there is a faithful normal semifinite weight τ∆onNsuch that the modular automorphism group of
+τ∆is an inner automorphism group σt(fort∈C) of (the Tomita algebra of) Nwithσi|A=σin the
+sense thatσi(π(a)) =π(σ(a)),whereσis the automorphism σ(a) = ∆−1(a)onA,
+2)τ∆restricts to a faithful semifinite trace /hatwideτonM=Nσ, with a faithful normal projection Ψ :N→
+Msatisfyingτ∆=/hatwideτ◦ΨonN,
+3) withDthe generator of the one parameter group implementing the ga uge action of TonHwe
+have:[D,π(a)]extends to a bounded operator (in N) for alla∈Aand forλin the resolvent set of
+D,(λ−D)−1∈K(M,τ∆), whereK(M,τ∆)is the ideal of compact operators in Mrelative toτ∆. In
+particular,Dis affiliated toM.30
+For matrix algebras A=Qλ
+c⊗MkoverQλ
+c, (Qλ
+c⊗Mk,H⊗Mk,D⊗Idk) is also a modular spectral
+triple in the obvious fashion.
+We need some technical lemmas for the discussion in the next S ection. A function ffrom a complex
+domain Ω into a Banach space Xis called holomorphic if it is complex differentiable in norm on Ω .
+The following is proved in [CPR2, Lemma 3.15].
+Lemma 3.19. (1) LetBbe aC∗-algebra and let T∈B+.The mapping z/ma√sto→Tzis holomorphic (in
+operator norm) in the half-plane Re(z)>0.
+(2) LetBbe a von Neumann algebra with faithful normal semifinite trac eφand letT∈B+be in
+L(1,∞)(B,φ).Then, the mapping z/ma√sto→Tzis holomorphic (in trace norm) in the half-plane Re(z)>1.
+(3) LetB, andTbe as in item (2) and let A∈Bthen the mapping z/ma√sto→φ(ATz)is holomorphic for
+Re(z)>1.
+Lemma 3.20. In these modular spectral triples (A,H,D)for matrices over the algebras Qλwe have
+(1+D2)−s/2∈L1(M,τ∆)for alls>1and forx∈N,τ∆(x(1+D2)−r/2)is holomorphic for Re(r)>1
+and we have for a∈Qλ
+c,τ∆([D,π(a)](1+D2)−r/2) = 0,forRe(r)>1.
+Proof.We include a brief proof since there are some small but import ant differences from [CPR2,
+Lemma 3.16]. Since the eigenvalues for Dare precisely the set of integers, and the projection Φ kon
+the eigenspace with eigenvalue ksatisfiesτ∆(Φk) = 1,it is clear that (1+ D2)−s/2∈L1(M,τ∆).Now,
+τ∆(x(1 +D2)−r/2) =/hatwideτ(Ψ(x)(1 +D2)−r/2) is holomorphic for Re(r)>1 by item (3) of the previous
+lemma.
+To see the last statement, we observe that τ∆([D,π(a)](1+D2)−r/2) =τ∆(Ψ([D,π(a)])(1+D2)−r/2),
+so it suffices to see that Ψ([ D,π(a)]) = 0 fora∈A=Qλ
+c.To this end, let a=f·δgwheredet(g) =λn
+is one of the linear generators of Qλ
+c.Then by calculating the action of the operator Dπ(f·δg) on the
+linear generators fi·δhiof the Hilbert space, H, we obtain:
+Dπ(f·δg) =nπ(f·δg)+π(f·δg)Dthat is [D,π(f·δg)] = logλ(|g|)π(f·δg).
+More generally,
+[D,π(m/summationdisplay
+i=1cifi·δhi)] =m/summationdisplay
+i=1ci(logλ(|hi|))π(fi·δhi).
+If we apply Ψ to this equation, we see that Ψ( π(fi·δhi)) =π(Φ(fi·δhi)) = 0 whenever logλ(|hi|)/\e}atio\slash= 0,
+and so the whole sum is 0. We also observe that [ D,π(a)]∈π(Qλ
+c) for alla∈Qλ
+c.This is not too
+surprising sinceDis the generator of the action γofTonQλ. /square
+3.3.ModularK1.We now make appropriate modifications to [CPR2, Section 4]) u sing [CNNR]
+introducing elements of these modular spectral triples ( A,H,D) (whereAis a matrix algebra over
+Qλ
+c) that will have a well defined pairing with our Dixmier functi onal/hatwideτω. LetA=Qλ. Following [HR]
+we say that a unitary (invertible, projection,...) in the n×nmatrices overQλfor somenis a unitary
+(invertible, projection,...) over Qλ. We write σtfor the automorphism σt⊗IdnofA.
+Definition 3.21. Letvbe a partial isometry in the ∗-algebraA. We say that vsatisfies the modular
+condition with respect to σif the operators vσt(v∗)andv∗σt(v)are in the fixed point algebra F⊂A
+for allt∈R. Of course, any partial isometry in Fis a modular partial isometry.
+Lemma 3.22. ([CPR2, Lemma 4.8] ) Letv∈Abe a modular partial isometry. Then we have
+uv=/parenleftbigg
+1−v∗v v∗
+v1−vv∗/parenrightbigg
+is a modular unitary over A. Moreover there is a modular homotopy uv∼uv∗.31
+Note that in [CPR2] we used a different approach which is implie d by the one given here. In [CPR2]
+we defined modular unitaries in terms of the regular automorp hism:
+π(σ(a)) =π(∆−1(a)) = ∆−1π(a)∆ =σi(π(a)).
+That is we said that a unitary in Ais modular if uσ(u∗) andu∗σ(u) are in the fixed point algebra.
+Examples .
+(1) Fork,j >0 recallSk,m∈Qλ
+cwithm < mk(see Definition 2.14) we write Pk,m=Sk,mS∗
+k,m=
+X[mλk,(m+1)λk)·δ1which is in clearly Fλ. Then for each{k,m},{j,n}we have a unitary
+u{k,m},{j,n}=/parenleftbigg1−Pk,mSk,mS∗
+j,n
+Sj,nS∗
+k,m1−Pj,n/parenrightbigg
+.
+It is simple to check that this a self-adjoint unitary satisf ying the modular condition, and that
+τ(Pk,m) =λkandτ(Pj,n) =λj.These examples behave very much like the SµS∗
+νexamples of [CPR2].
+(2) Fork,j >0 consider the “leftover” partial isometries Sk,mkandSj,mjof Definition 3.13 which we
+will denote by SkandSjto lighten the notation. We let vj,k=SjS∗
+kand calculate its range and initial
+projections which are both in Fλ:
+Pj=SjS∗
+kSkS∗
+j=X[mjλj,mjλj+λj(λ−k−mk))·δ1,and
+Pk=SkS∗
+jSjS∗
+k=X[mkλk,mkλk+λk(λ−j−mj))·δ1.
+We note for future reference that:
+τ(Pj) =λj(λ−k−mk) andτ(Pk) =λk(λ−j−mj).
+We also note that we have a modular unitary uj,k:
+uj,k=/parenleftbigg1−PkSkS∗
+j
+SjS∗
+k1−Pj/parenrightbigg
+.
+Define the modular K1group as follows.
+Definition 3.23. LetK1(A,σ)be the abelian group with one generator [v]for each partial isometry
+voverAsatisfying the modular condition and with the following rel ations:
+1) [v] = 0ifvis overF,
+2) [v]+[w] = [v⊕w],
+3)ifvt, t∈[0,1],is a continuous path of modular partial isometries in some ma trix algebra over A
+then[v0] = [v1].
+One could use modular unitaries as in [CPR2] in place of these modular partial isometries.
+The following can now be seen to hold.
+Lemma 3.24. (Compare [CPR2, Lemma 4.9] ) Let(A,H,D)be our modular spectral triple relative
+to(N,τ∆)and setF=Aσandσ:A→A. LetL∞(∆) =L∞(D)be the von Neumann algebra
+generated by the spectral projections of ∆thenL∞(∆)⊂Z(M). Letv∈Abe a partial isometry with
+vv∗, v∗v∈F. Thenπ(v)Qπ(v∗)∈Mandπ(v∗)Qπ(v)∈Mfor all spectral projections QofD, if and
+only ifvis modular. That is, π(v)∆π(v∗)andπ(v∗)∆π(v)(orπ(v)Dπ(v∗)andπ(v∗)Dπ(v)) are both
+affiliated toMif and only if vis modular.32
+Thus we see that modular partial isometries conjugate ∆ to an operator affiliated to M, and sov∆v∗
+commutes with ∆ (and vDv∗commutes withD).
+We will nextshow that thereis an analytic pairingbetween (p art of) modular K1and modularspectral
+triples. To do this, we are going to use the analytic formulae for spectral flow in [CP2].
+3.4.The mapping cone algebra. Ouraimintheremainderistocalculateanindexpairingexpl icitly
+for the matrix algebras Aover the smooth subalgebra Qλ
+cofQλ. In the following few pages we will
+sometimes abuse notation and write ain place ofπ(a) fora∈Ain order to make our formulae more
+readable. Whenever we do this, however, we will use σi(·) = ∆−1(·)∆ the spatial version of the algebra
+homomorphism, σ. We will generally use the spatial version σiwhen in the presence of operators not
+inπ(A).
+We briefly review some results from [CNNR], that provide an in terpretation of the modular index
+pairing given by the spectral flow.
+IfF⊂Ais a sub-C∗-algebra of the C∗-algebraA, then the mapping cone algebra for the inclusion is
+M(F,A) ={f:R+= [0,∞)→A:fis continuous and vanishes at infinity , f(0)∈F}.
+WhenFis an ideal in Ait is known that K0(M(F,A))∼=K0(A/F), [Put1]. In general, K0(M(F,A))
+is the set of homotopy classes of partial isometries v∈Mk(A) with range and source projections
+vv∗, v∗vinMk(F), with operation the direct sum and inverse −[v] = [v∗]. All this is proved in [Put1].
+Itis shownin[CNNR] that thereis anatural mapthat injects K1(A,σ) intoKT
+0(M,F), theequivariant
+K-theory of the mapping cone algebra. Note that the Taction onAlifts in the obvious way to the
+mapping cone. Now, it was shown in [CPR1] that certain Kaspar ovA,F-modules extend to Kasparov
+M(F,A),F-modules, andthiswasextendedtotheequivariant casein[C NNR]. Importantlythetheory
+applies to the equivariant Kasparov module coming from a cir cle action. The extension is explicit,
+namely there is a pair ( ˆX,ˆD) which is a graded unbounded Kasparov module for the mapping cone
+algebraM(F,A) constructed using a generalised APS construction, [APS3] .
+Ifvis a partial isometry in Mk(A), setting
+ev(t) =/parenleftBigg
+1−vv∗
+1+t2−ivt
+1+t2
+iv∗t
+1+t2v∗v
+1+t2/parenrightBigg
+,
+definesevas a projection over M(F,A).
+Then in [CNNR] we showed that if v∈Ais a modular partial isometry we have
+/a\}bracketle{t[ev]−/bracketleftbigg/parenleftbigg
+1 0
+0 0/parenrightbigg/bracketrightbigg
+,[(ˆX,ˆD)]/a\}bracketri}ht=−Index(PvP:v∗vP(X)→vv∗P(X))∈K0(F)
+= Index(Pv∗P:vv∗P(X)→v∗vP(X))∈KT
+0(F). (4)
+We thus obtain an index map K1(A,σ)→KT
+0(F). The latter may be thought of as the ring of
+Laurent polynomials K0(F)(χ,χ−1) where we think of χ,χ−1as generating the representation ring of
+T. We may obtain a real valued invariant from this map by evalua tingχate−βwhereβis the inverse
+temperature of our KMS state and applying the trace to the res ultant element of K0(F). Then one of
+the main results of [CNNR] is that the real valued invariant s o obtained is identical with the spectral
+flow invariant of the next subsection. However the general th eory of [CNNR] does not tell us the range
+of this index map and it is the latter that is of interest for th ese explicit calculations.33
+3.5.A local index formula for the algebras Qλ.Usingthefactthatwehavefullspectralsubspaces
+we know from [CNNR] that there is a formula for spectral flow wh ich is analogous to the local index
+formula in noncommutative geometry. We remind the reader th atτ∆=/hatwideτ◦Ψ where Ψ :N→M is
+the canonical expectation, so that τ∆restricted toMis/hatwideτ.
+Theorem 3.25. (Compare [CPR2, Theorem 5.5] ) Let(A,H,D)be the(1,∞)-summable, modular
+spectral triple for the algebra Qλwe have constructed previously. Then for any modular partial isometry
+vand for any Dixmier trace /hatwideτ˜ωassociated to/hatwideτ, we have spectral flow as an actual limit
+sf/hatwideτ(vv∗D,vDv∗) =1
+2lim
+s→1+(s−1)/hatwideτ(v[D,v∗](1+D2)−s/2) =1
+2/hatwideτ˜ω(v[D,v∗](1+D2)−1/2) =τ◦Φ(v[D,v∗]).
+The functional on A⊗Adefined bya0⊗a1/ma√sto→1
+2lims→1+(s−1)τ∆(a0[D,a1](1+D2)−s/2)is aσ-twisted
+b,B-cocycle (see the proof below for the definition).
+Remark . Spectral flow in this setting is independent of the path join ing the endpoints of unbounded
+self adjoint operators affiliated to Mhowever it is not obvious that this is enough to show that it is
+constant on homotopy classes of modular unitaries. This lat ter fact is true but the proof is lengthy so
+we refer to [CNNR].
+Theorem 3.26. We let(Qλ
+c⊗M2,H⊗C2,D⊗12)be the modular spectral triple of (Qλ
+c⊗M2).
+(1)Letube a modular unitary defined in Section 5 of the form
+u{k,m},{j,n}=/parenleftbigg1−Pk,mSk,mS∗
+j,n
+Sj,nS∗
+k,m1−Pj,n/parenrightbigg
+.
+Then the spectral flow is positive being given by
+sfτ∆(D,uDu∗) = (k−j)(λj−λk)∈Z[λ]⊂Γλ.
+(2)Letube a modular unitary defined in Section 5 of the form:
+uj,k=/parenleftbigg1−PkSkS∗
+j
+SjS∗
+k1−Pj/parenrightbigg
+,
+whereSkS∗
+j=Sk,mkS∗
+j,mjandPkandPjare its range and initial projections, respectively. Then th e
+spectral flow is given by
+sfτ∆(D,uDu∗) = (k−j)[λj(λ−k−mk)−λk(λ−j−mj)]∈Γλ.
+Proof.We have already observed that these are, in fact modular unit aries. For the computations we
+use a calculation from the proof of Lemma 3.20 to get in exampl e (1):
+u[D⊗12,u] =/parenleftbigg1−Pk,mSk,mS∗
+j,n
+Sj,nS∗
+k,m1−Pj,n/parenrightbigg/parenleftbigg0 [D,Sk,mS∗
+j,n]
+[D,Sj,nS∗
+k,m] 0/parenrightbigg
+=/parenleftbigg1−Pk,mSk,mS∗
+j,n
+Sj,nS∗
+k,m1−Pj,n/parenrightbigg/parenleftbigg0 ( k−j)Sk,mS∗
+j,n
+(j−k)Sj,nS∗
+k,m0/parenrightbigg
+= (k−j)/parenleftbigg
+−Pk,m0
+0Pj,n/parenrightbigg
+.
+So using Theorem 3.25 and our previous computation of the Dix mier trace, Proposition 3.17, and the
+fact thatPk,m=Sk,mS∗
+k,m=X[mλk,(m+1)λk)·δ1so thatτ(Pk,m) =λkwe have
+sfτ∆(D,uk,mDuk,m) = (k−j)τ(Pj,n−Pk,m) = (k−j)(λj−λk).
+This number is always positive as the reader may check, and is contained in Z[λ].
+The computations in example (2) are similar and use the fact t hatPk=X[mkλk,mkλk+λk(λ−j−mj))·δ1,
+so thatτ(Pk) =λk(λ−j−mj)∈Γλ.In these examples, the spectral flow is notcontained in the
+smaller polynomial ring, Z[λ]. /square34
+Remarks. The observation of [CPR2] that the twisted residue cocycle f ormula for spectral flow
+is calculating Araki’s relative entropy of two KMS states [A r] also applies to the examples in this
+subsection.
+Acknowledgements We would like to thank Nigel Higson, Ryszard Nest, Sergey Nes hveyev, Marcelo
+Laca, Iain Raeburn and Peter Dukes for advice and comments. T he first and fourth named authors
+weresupportedby theAustralian Research Council. Theseco ndandthirdnamedauthorsacknowledge
+the support of NSERC (Canada).
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+rennie@maths.anu.edu.au
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+arXiv:1001.0425v1  [nlin.PS]  4 Jan 2010Solitons in combined linear and nonlinear lattice
+potentials
+Hidetsugu Sakaguchi1and Boris A. Malomed2
+1Department of Applied Science for Electronics and Materials,
+Interdisciplinary Graduate School of Engineering Sciences,
+Kyushu University, Kasuga, Fukuoka 816-8580, Japan
+2Department of Physical Electronics,
+School of Electrical Engineering, Faculty of Engineering,
+Tel Aviv University, Tel Aviv 69978, Israel
+November 1, 2018
+abstract
+We study ordinary solitons and gap solitons (GSs) in the fram ework of the one-
+dimensional Gross-Pitaevskii equation (GPE) with a combin ation of linear and nonlin-
+earlattice potentials. Themainpointsoftheanalysisaree ffectsof(in)commensurability
+between the lattices, development of analytical methods, viz., the variational approxi-
+mation (VA) for narrow ordinary solitons, and various forms of the averaging method
+for broad solitons of both types, and also the study of mobili ty of the solitons. Under
+the direct commensurability (equal periods of the lattices ,Llin=Lnonlin), the family of
+ordinary solitons is similar to its counterpart in the GPE wi thout external potentials.
+In the case of the subharmonic commensurability, with Llin= (1/2)Lnonlin, or incom-
+mensurability, there is an existence threshold for the ordi nary solitons, and the scaling
+relation between their amplitude and width is different from that in the absence of the
+potentials. GS families demonstrate a bistability, unless the direct commensurability
+takes place. Specific scaling relations are found for them to o. Ordinary solitons can
+be readily set in motion by kicking. GSs are mobile too, featu ring inelastic collisions.
+The analytical approximations are shown to be quite accurat e, predicting correct scal-
+ing relations for the soliton families in different cases. Th e stability of the ordinary
+solitons is fully determined by the VK (Vakhitov-Kolokolov ) criterion, i.e., a negative
+slope in the dependence between the solitons’s chemical pot entialµand norm N. The
+stability of GS families obeys an inverted (“anti-VK”) crit erion,dµ/dN > 0, which is
+explained by the approximation based on the averaging metho d. The present system
+provides for a unique possibility to check the anti-VK crite rion, asµ(N) dependences
+for GSs feature turning points, except for the case of the dir ect commensurability.
+11 Introduction
+It is well known that periodic potentials, induced by optica l lattices (OLs), provide
+for a versatile tool for controlling dynamics of Bose-Einst ein condensates (BECs).
+This tool is especially efficient for the creation and stabili zation of solitons – both
+ordinary ones and gap solitons (GSs), which are supported by the interplay of the
+OL potential and self-repulsive nonlinearity. Many result s obtained in theoretical
+and experimental studies of this topic were summarized in re views focused on one-
+dimensional(1D)[1]andmultidimensional[2]matter-wave dynamics. Earlier, asimilar
+model was introduced in optics for the description of spatia l solitons in nonlinear
+waveguides with a periodic transverse modulation of the ref ractive index [3]. In the
+experiment, lattices controlling the transmission of opti cal beams were implemented in
+the form of photoinduced gratings in photorefractive cryst als, which made it possible
+to create various species of 1D and 2D spatial solitons [4]. G ratings were also created
+as permanent structures written by femtosecond laser beams in silica [5]. Recently,
+this topic was reviewed in Ref. [6]; a closely related topic i s the study of discrete
+solitons in optics, which was a subject of another comprehen sive review in Ref. [7].
+A differentpossibility, which has drawn muchattention in st udies of BEC (thusfar,
+primarilyat thetheoretical level), is theuseofaspatiall y profiledeffectivenonlinearity,
+that may be implemented by means of properly designed configu rations of external
+fields via the Feshbach-resonance effect. In terms of the cond ensed-matter theory,
+the nonuniform nonlinearity coefficient induces an effective pseudopotential [8, 9], that
+can be used to control the dynamics of localized modes. Vario us problems of this sort
+were considered in 1D settings, with periodic pseudopotent ials in the form of nonlinear
+lattices (NLs) [10, 11, 12], as well as with spatial modulati ons of the nonlinearity
+coefficient represented by one [13] or two [9] delta-function s (actually, the model with
+the self-attractive nonlinearity concentrated at a single delta-function was introduced
+long ago as a model for tunneling of interacting particles th rough a junction [14]).
+In particular, the configuration with two delta-functions m akes it possible to study
+spontaneous symmetry breaking of matter waves trapped by a s ymmetric double-well
+pseudopotential [9]. Specially chosen profiles of the nonli nearity coefficient may also
+be employed to design a pulse-generating atomic-wave laser [15], as well as traps and
+barriers for such pulses [16]. The analysis of 1D matter-wav e solitons in NLs was
+further extended for two-component models [17, 18], and for some spatiotemporal
+patterns of the nonlinearity modulation [19]. Certain resu lts for solitons supported
+by 2D nonlinear pseudopotentials were reported too, althou gh the stabilization of 2D
+solitons in this setting is a tricky problem [20].
+In addition to the BEC, a periodic modulation of the nonlinea rity is possible in
+optics, where it was analyzed in terms of temporal [21] and sp atial [22, 23] solitons. A
+discussion of practical possibilities to create NLs in opti cal media in the “pure form”
+(without affecting the linear properties, i.e., the refract ive index) can be found in
+recent paper [18]. An experimental observation of NL-suppo rted optical solitons (in
+the form of surface solitons at an interface between lattice s) was reported in Ref. [24].
+A natural generalization of the study of NL pseudopotential s is to consider atomic
+and optical media equipped with a combination of nonlinear a nd linear lattices, in
+1D [25, 26, 12] and 2D [27] settings. In the experiment, this m ay be implemented
+by applying the above-mentioned techniques simultaneousl y – for instance, combining
+the OL, which induces the linear periodic potential in the BE C, and the patterned
+magnetic field, which gives rise to the nonlinear pseudopote ntial via the Feshbach
+resonance. Actually, photonic-crystal fibers, where vario us species of spatial solitons
+2have been predicted [28], also belong to this type of media. U nder special conditions,
+the combined models admit analytical solutions, see Ref. [1 2] and references therein.
+In particular, exact solutions were elaborated in detail fo r the lattices of the Kronig-
+Penney (piecewise-constant) type [23].
+The objective of this work is to investigate the existence, s tability, and mobility of
+ordinary 1D solitons and their GS counterparts in the combin ed NL-OL model, with
+emphasis on effects of the commensurability (spatial resonance) between the nonlinear
+and linear lattices. In the general case, the solitons are fo und in a numerical form. For
+narrow ordinary solitons (those whose chemical potential, µ, falls into the semi-infinite
+gap of the linearized version of themodel), we also developa variational approximation
+(VA). For broad solitons, both ordinary ones and GSs with µbelonging to the first
+finite bandgap, we elaborate analytical approximations bas ed on different versions of
+the averaging technique.
+The stability of the solitons is investigated below by means of systematic simula-
+tions of their perturbed evolution. We conclude that the wel l-known VK (Vakhitov-
+Kolokolov) criterion [29] completely determinestheactua l stabilityof theordinarysoli-
+tons (the criterion states that a necessary stability condi tion is a negative slope in the
+dependence of the solitons’ chemical potential, µ, on their norm, N, i.e.,dµ/dN < 0).
+For the GS families, our analysis leads to a different conclus ion: their stability fully
+obeys an “anti-VK” criterion, viz.,dµ/dN > 0. As a matter of fact, all stable GS fam-
+ilies in previously studied models had only the positive slo pe ofµ(N), thus satisfying
+the latter criterion automatically. However, the present m odel offers a rather unique
+chance to test it in a nontrivial situation, when µ(N) curves for the GSs may have
+portions with both positive and negative slope, separated b y turning points. Using
+the averaging method, we also produce a justification for the anti-VK criterion, which
+is relevant, at least, for subfamilies of broad GSs.
+The rest of the paper is organized as follows. The model is for mulated in the next
+section, which also reports basic numerical results. In the case of the direct com-
+mensurability between the nonlinear and linear lattices (f or those with equal periods,
+Llin=Lnonlin), it is concluded that ordinary solitons are similar to soli ton solutions
+of the nonlinear Schr¨ odinger equation (NLSE) in the free sp ace (without an external
+potential), in the sense that there is no threshold for their existence, the entire soliton
+family is stable, and the amplitude and width of the soliton, AandW, obey the usual
+scaling relation, A∝1/W. On the contrary to that, there is an existence threshold
+for ordinary solitons in the case of the incommensurability , or if the commensurability
+(spatial resonance) between thelattices is subharmonic , withLlin=Lnonlin/2. Inthose
+cases, the scaling relation between the amplitude and width is different too – featuring,
+in particular, A∝1/W2for the subharmonic resonance. The same commensurability-
+depending change in the scaling is observed in GS families, w hich also demonstrate a
+bistability in cases different from the direct commensurabi lity.
+Analytical results, which mayexplain aconsiderable part o f thenumerical findings,
+are collected in Section III. First, we develop the VA for nar row ordinary solitons, and
+then the averaging method is reported, in different forms for different cases. It is
+demonstrated that both the VA and averaging produce results which are in good
+agreement with numerical observations, in relevant region s of the parameter space. In
+particular, as briefly mentioned above, the averaging lends an explanation to the “anti-
+VK” stability criterion for GSs. In the case of Llin=Lnonlin/2, an average equation
+for the envelope amplitude includes a quintic nonlinear ter m, rather than the cubic
+one, which accounts for the above-mentioned change in the sc aling relation between
+AandW. The paper is concluded by Section IV.
+32 Numerical results
+2.1 The model
+The model combining the linear OL potential and nonlinear NL pseudopotential is
+based on the known effectively one-dimensional Gross-Pitae vskii equation for the
+mean-field wave function, φ(x,t) [23, 25, 26, 12]. In the scaled form, the equation
+is
+iφt=−(1/2)φxx−/bracketleftbig
+ǫcos(2πx)+gcos(πqx)|φ|2/bracketrightbig
+φ, (1)
+whereǫis the strength of the linear OL, and g=±1 is fixed by the normalization.
+The NL wavenumber is q, the above-mentioned periods of the linear and nonlinear
+lattices being Llin≡1 andLnonlin= 2/q. Forq= 0 and g=−1, Eq. (1) amounts to
+the NLSE with the OL potential and constant coefficient in fron t of the self-defocusing
+cubic term. As is well known, this equation supports familie s of stable GS solutions
+[1, 30]. On the other hand, in the absence of the OL, ǫ= 0, the NL supports ordinary
+solitons, but not GSs [10, 11, 12].
+Stationarysolutions withchemicalpotential µarelookedforintheformof φ(x,t) =
+u(x)exp(−iµt), where u(x) is a real function. The substitution of this expression int o
+Eq. (1) yields an ordinary differential equation,
+µu=−(1/2)u′′−/bracketleftbig
+ǫcos(2πx)+gcos(πqx)u2/bracketrightbig
+u, (2)
+which can be derived from the corresponding Lagrangian,
+2L=/integraldisplay+∞
+−∞/bracketleftBig
+µu2−(1/2)/parenleftbig
+u′/parenrightbig2+ǫcos(2πx)u2+(g/2)cos(πqx)u4/bracketrightBig
+dx.(3)
+We have constructed numerical solutions for localized stat ionary modes by means of
+the shooting method applied to Eq. (2). The stability of the s o found solutions against
+small perturbations was tested through direct simulations of Eq. (1). The results were
+also compared to predictions of the VK criterion for the ordi nary solitons, and to the
+above-mentioned “anti-VK” criterion for GSs. Numerical re sults reported below are
+obtained for the OL strength ǫ= 5, which adequately represents the generic situation,
+for the ordinary solitons and GSs alike.
+Figures 1(a) and (b) display, respectively, examples of ord inary solitons (for g=
++1, q= 1) and GSs (for g=−1, q= 1) with equal amplitudes. The choice of q= 1
+corresponds to the above-mentioned case of the subharmonic resonance between the
+OL and NL, Llin= (1/2)Lnonlin. In this case, stable ordinary solitons and GSs coexist
+for either sign of g. If, for instance, g= +1, ordinary solitons are located around even
+sites of the NL, x= 2n(with integer n), while GSs may be centered at odd sites,
+x= 2n+1.
+2.2 Ordinary solitons
+Figure 2(a) represents a family of the ordinary solitons, by means of the relation be-
+tween their norm, N=/integraltext+∞
+−∞u2(x)dx, and chemical potential µ, at three characteristic
+values of the NL’s wavenumber, q= 1,√
+5−1, and 2, for g= +1. These values are cho-
+sen because q= 2 corresponds to the direct commensurability between the l inear and
+nonlinear lattices ( Llin=Lnonlin),q= 1 represents, as said above, the subharmonic
+commensurability [ Llin= (1/2)Lnonlin], andq=√
+5−1 corresponds to incommensu-
+rate lattices. All values of µfor the ordinary solitons fall into the semi-infinite gap of
+the spectrum induced by the OL potential in the linearized ve rsion of Eq. (1).
+40
0.5
1
1.5
+30
 40
 50
 60
 70
+x
/j124/j131/j211/j124/j40/j97/j41
+0
0.5
1
1.5
+30
 40
 50
 60
 70
+x
/j124/j131/j211/j124/j40/j98/j41
+Figure 1: Typical examples of a stable broad ordinary soliton (a) and stable
+gap soliton (b), at g= +1 and −1, respectively, in the case of the nonlinear
+lattice with period Lnonlin= 2 (i.e., q= 1). The amplitudes of both solitons are
+A= 1.5. The chemical potential and norm are, respectively, µ=−1.198 and
+N= 1.628 for the ordinary soliton, and µ= 2.629 and N= 2.176 for the gap
+soliton.
+Atq= 2, direct simulations demonstrate that the entire soliton family is stable,
+precisely as suggested by the VK criterion, dµ/dN < 0, see Fig. 2(a). For q= 1
+andq=√
+5−1, the results are different, featuring non-monotonous rela tionsµ(N).
+Accordingly (again in the agreement with the prediction of t he VK criterion), nar-
+row solitons with larger amplitudes, which correspond to th e branches of µ(N) with
+dµ/dN < 0 in Fig. 2(a), are stable, while their loosely bound (broad) counterparts,
+corresponding to the branches with dµ/dN > 0, are unstable. Another difference from
+the case of the direct commensurability is that there is a min imum value of the norm,
+Nmin(the threshold), which is necessary for the existence of the ordinary solitons at
+q= 1 and√
+5−1, while there is no threshold at q= 2.
+In fact, the situation in the case of q= 1 and√
+5−1 – the existence of the
+threshold value, Nmin, which separates stable and unstable branches of the ordina ry-
+soliton solutions – is qualitatively similar to what is know n in the model with the NL
+but no linear potential [10]. On the other hand, the situatio n in the case of q= 2
+– the absence of Nminand the existence of the single branch of the soliton solutio ns,
+which is entirely stable – resembles well-known properties of soliton solutions of the
+NLSE without any lattice, linear or nonlinear.
+The evolution of those ordinary solitons which are unstable is illustrated in Fig.
+3 forq= 1 and g= +1. It is observed that the unstable soliton, with initial a mpli-
+tudeA= 0.9, rearranges itself into a narrower persistent breather, w ith time-average
+amplitude Abr≈1.1, while the norm is kept constant. The transformation of the
+unstable ordinary solitons into stable breathers is also si milar to what was reported in
+the model without the OL [10].
+Properties of the ordinary solitons in the same three famili es, with q= 1,√
+5−1
+and 2 (and g= +1), are further illustrated in Fig. 2(b) through relation s between the
+soliton’s amplitude, A, and its width, W, which we define by
+W2=N−1/integraldisplay+∞
+−∞|φ(x)|2(x−L/2)2dx (4)
+5/j131/j202 /j113/j61/j50
+/j113/j61/j49/j113/j61/j129/j227/j45/j49/j129/j64/j53
+0.1110100
+0.1 1W
+A/j113/j61/j50/j113/j61/j49
+/j113/j61/j129/j227/j45/j49/j129/j64/j53/j40/j97/j41 /j40/j98/j41/j40/j99/j41
+/j87/j129/j229/j49/j47/j65/j87/j129/j229/j49/j47/j65/j50
+0
0.5
1
1.5
2
2.5
+0
 0.5
 1
 1.5
 2
 2.5
N
+q
-2
-1.5
-1
-0.5
+0
0.5
 1
1.5
 2
2.5
 3
3.5
+N
+Figure 2: (a) Chemical potential µversus norm Nfor families of ordinary soli-
+tons at different values of qandg= +1. Portions of the curves with dµ/dN < 0
+anddµ/dN > 0 represent, respectively, stable and unstable (sub)families, in
+agreement with the VK criterion. (b) The log-log plot of amplitude Aversus
+widthW[the latter is defined as per Eq. (4)]. Bold dashed lines correspond to
+unstable branches of the soliton families, for q= 1 and√
+5−1. Thin dashed ref-
+erence lines designate scalings which different families obey at small va lues ofA,
+namely,W∝1/AandW∝1/A2. (c) The stability boundary for the ordinary
+solitons, which is defined, as per the VK criterion, by condition dN/dµ= 0.
+As predicted by the criterion and verified in direct simulations, the so litons are
+stable above the boundary.
+/j124/j131/j211/j124/j40/j97/j41/j40/j98/j41
+0.911.11.21.3
+0100 200 300 400 500 600A
+t0
0.2
0.4
0.6
0.8
1
1.2
1.4
+50
 60
 70
 80
 90
 100
+x
0
0.2
0.4
0.6
0.8
1
1.2
1.4
+50
 60
 70
 80
 90
 100
+x
/j124/j131/j211/j124/j40/j99/j41
+Figure3: An example ofthe spontaneousrearrangementofan uns table ordinary
+soliton, with amplitude A= 0.9 and norm N= 1.32,into a robust breather,
+atq= 1 and g= +1. (a) The evolution of the soliton’s amplitude. (b) The
+field profile, |φ(x,t)|, att= 110. (c) The profile at t= 140. Panels (b) and (c)
+display the shape of the breather at points where its width is close, r espectively,
+to the minimum and maximum values.
+6(x=L/2 is the central point of the integration domain). At q= 2, relation W(A)
+features scaling W∝1/Afor relatively small values of A. This is the same scaling as
+featured by exact soliton solutions of the NLSE in the free sp ace, which is in line with
+the above observation that the soliton family at q= 2 is similar to that in the NLSE
+without any lattice. However, the scaling is different in the other families, featuring
+W(A)∝1/A2forq= 1, and W(A)∝1/A1.8forq=√
+5−1. The two latter scaling
+relations imply that the width of the respective solitons is essentially larger than in
+their free-space counterparts, therefore we call them broa d solitons.
+Figure 2(c) summarizes the results by means of the stability boundary for the
+ordinary solitons in the plane of ( q,N). The boundary is identified as a VK-critical
+curve, along which dµ/dNvanishes, the stability area (with dµ/dN < 0) being located
+above the curve. Systematic simulations performed in regio ns below and above the
+boundary have confirmed that the solitons in these regions ar e, respectively, unstable
+and stable (unstable solitons transform themselves into br eathers, as shown in Fig. 3).
+The stability area reaches the limit of N= 0 (very broad solitons with a vanishingly
+small amplitude) in the form of the cusps in Fig. 2(c) at q= 0 and q= 2. Recall
+thatq= 0 with g= +1 corresponds to the constant coefficient of the self-attra ctive
+nonlinearity in Eq. (1), while q= 2 corresponds to the direct commensurability
+between the linear and nonlinear lattices.
+2.3 Static gap solitons
+In this work, the consideration of GS families was confined to the first finite bandgap
+induced by the OL potential, in terms of the linearized versi on of Eq. (1). The N(µ)
+andW(A) curves for these families are displayed in Figs. 4(a) and (b ), for the same
+three cases as above, viz.,q= 1,√
+5−1 and 2, fixing g=−1 [width Wis again defined
+as per Eq. (7)]. The VK criterion does not apply to GSs. Nevert heless, the results
+strongly suggest that the stability of all GS families follo ws an “anti-VK ” condition,
+dµ/dN > 0. For subfamilies of broad GSs, this condition is derived be low from an
+effective envelope equation for broad GSs which amounts to an “inverted” NLSE, with
+the self-repulsive nonlinear term and negative effective ma ss, see Eqs. (15). In that
+approximation, GSs reduce to ordinary solitons if the wave f unction is subjected to
+the complex conjugation, which implies the inversion of the sign ofµ, and of dµ/dN
+as well.
+In accordance with what is said above, the entire GS family fo rq= 2, which
+satisfies the “anti-VK” condition everywhere in Fig. 4(a), i s found to be completely
+stable in direct simulations. On the other hand, the µ(N) curves for q= 1 and
+q=√
+5−1 feature two folds, and direct simulations corroborate the instability of
+portions of the GS families with dµ/dN < 0. To the best of our knowledge, the
+present model produces the first example of µ(N) characteristics for GSs with turning
+points, which makes the anti-VK criterion amenable to the ac tual verification. In
+the standard model with the constant nonlinearity coefficien t [32, 1], as well as in
+its version with the quasiperiodic OL potential [33], the mo notonous character of the
+curves does not allow the verification of the criterion.
+The stability diagram in the plane of ( q,N), as predicted by the “anti-VK” cri-
+terion, is displayed in Fig. 4(c). It includes two critical c urves with dN/dµ= 0. As
+suggested by Fig. 4(b), and completely confirmed by systemat ic simulations, in the
+regions above the top curve and below the bottom one there is a single solution, which
+is stable. Between the curves, there are three solutions, tw o of which are stable, i.e.,
+this is a bistability region. The critical curves feature cusps near q= 0 and q= 2,
+7/j131/j202/j40/j97/j41
+/j113/j61/j50/j113/j61/j129/j227 /j45/j49 /j129/j64 /j53
+110100
+0.1 1W
+A/j113/j61/j50/j113/j61/j49/j113/j61/j129/j227 /j45/j49 /j129/j64/j53/j40/j98/j41/j40/j99/j41
+/j113/j61/j49
+/j87/j129/j229/j49/j47/j65/j87/j129/j229/j49/j47/j65/j50
+02468
+0 0.5 1 1.5 2 2.5N
+q2.5
3
3.5
+2
 2.5
 3
 3.5
+N
+Figure 4: (a) The relation between norm Nand chemical potential µof gap
+solitons for q= 1,√
+5−1, and 2, for g=−1. Portions of the curves with
+dµ/dN > 0 anddµ/dN < 0 represent stable and unstable (sub)families, re-
+spectively, obeying the “anti-VK” criterion (see the text). (b) Th e log-log plot
+of amplitude Aversus width W, for the gap solitons. Bold dashed portions of
+the curves correspond to unstable solutions, with dµ/dN < 0. Thin dashed
+reference lines designate scalings W∝1/AandW∝1/A2. (c) Critical lines
+dN/dµ= 0 in the plane of ( q,N). There is a single stable gap soliton above the
+upper line and beneath the lower one, and three solutions – two stab le and one
+unstable – in the bistability region between the two lines.
+similar to the situation displayed in Fig. 2(c).
+From Fig. 4(b) we conclude that the scaling relations betwee n the GS’s width and
+amplitude for broad solitons (with small values of A) take the following form: For
+q= 2,W∼1/A; forq=√
+5−1,W∝1/A1.85; and for q= 1,W∝1/A2, i.e.,
+almost exactly the same as their counterparts for the ordina ry solitons, see above. All
+portions of the GS families obeying these scaling relations are stable. On the other
+hand, the simulations demonstrate that unstable GSs (those withdµ/dN < 0) are
+not transformed into breathers, unlike unstable ordinary s olitons, but rather suffer a
+gradual decay into quasi-linear waves (not shown here).
+3 Analytical methods
+3.1 The variational approximation for ordinary solitons
+Narrow stationary solitons of the ordinary type (correspon ding to g= +1) can be
+naturally approximated by means of the VA, using the simples t Gaussian ansatz [3],
+u(x) =Aexp/bracketleftbig
+−x2//parenleftbig
+2W2/parenrightbig/bracketrightbig
+, (5)
+with norm N=√πA2W.The substitution of ansatz (5) in Lagrangian (3) yields the
+effective Lagrangian, written in terms of the norm, instead o f amplitude A:
+2Leff=µN−N
+4W2+ǫNe−π2W2+N2
+2√
+2We−π2q2W2/8. (6)
+8-2-1.5-1-0.5
+00.5 11.5 22.5 33.5
+N/j131/j202
+/j113/j61/j49/j113/j61/j50
+Figure 5: Comparison of the variational (dashed lined) and numerica lly found
+(chains of symbols) curves µ(N) for ordinary solitons ( g= +1).
+Variational equations following from Eq. (6), ∂Leff/∂W= 0 and ∂Leff/∂N= 0, take
+the form of
+4π2ǫW4e−π2W2+/parenleftBig
+1/√
+2π/parenrightBig/parenleftbig
+1+π2k2W2/parenrightbig
+NWe−(πqW)2/8= 1,(7)
+/parenleftbig
+4W2/parenrightbig−1−ǫe−(πW)2−/parenleftBig
+1/√
+2π/parenrightBig
+(N/W)e−(πqW)2/8=µ.(8)
+Figure 5 compares the µ(N) curves produced by Eqs. (7) and ( 8) to their numeri-
+cally found counterparts, for the NL wavenumbers q= 1 and 2. It is seen that the VA
+provides for a good approximation for narrow solitons with v alues ofµwhich are not
+too close to the edge of the semi-infinite gap. Portions of the curves corresponding to
+broad solitons, which are located near the gap’s edge, are no t captured by the VA, as
+the actual shape of these solitons is different from the Gauss ian, see, e.g., Fig. 1(a).
+3.2 The averaging method
+3.2.1 Ordinary solitons
+The approximation based on averaging can be applied to broad solitons, which have
+a small amplitude and large norm. In the case of the ordinary s olitons ( g= +1),
+this is the situation opposite to that (narrow localized mod es) for which the VA was
+presented in the previous section. To develop the averaging approach for ordinary
+solitons, we adopt the ansatz
+φ(x,t) = Φ(x,t)[1+2αcos(2πx)], (9)
+where the slowly varyingamplitude function, Φ, multiplies the simplest approximation
+for the Bloch wave function which may be used near the edge of t he semi-infinite gap,
+withα=/parenleftBig/radicalbig
+π4+ǫ2/2−π2/parenrightBig
+/ǫ(this approximation is obtained bydintof the analysis
+presented in Ref. [30]). The substitution of ansatz (9) into Eq. (1) and averaging,
+also performed along the lines of Ref. [30], lead to the asymp totic NLSE for the slowly
+varying envelope function,
+i∂Φ
+∂t=−1
+2m(ord)
+eff∂2Φ
+∂x2+g(ord)
+eff|Φ|2Φ, (10)
+900.020.040.060.080.1
+-200 -100 0 100 200
+x/j124/j131/j211/j124/j40/j97/j41
+/j124/j131/j211/j124/j40/j98/j41
+0
0.02
0.04
0.06
0.08
0.1
+-200
 -100
 0
 100
 200
+x
+Figure 6: (a) The numerical obtained profile of an ordinary soliton wit h am-
+plitudeA= 0.1 forg= +1 and q= 2. (b) The profile predicted for the same
+soliton by means of the averaging method, see Eqs. (9), (13), (11 ), and (12).
+where the calculations yield the following coefficients:
+m(ord)
+eff=2π4+ǫ2+π2√
+4π4+2ǫ2
+10π4+ǫ2−3π2√
+4π4+2ǫ2, (11)
+g(ord)
+eff=−/angbracketleftbig
+[1+2αcos(2πx)]4cos(qπx)/angbracketrightbig
+1+2α2, (12)
+with∝angbracketleft...∝angbracketrightstanding for the spatial average. Obviously, the effective n onlinearity coeffi-
+cient given by expression (12) vanishes unless qtakes values 0 ,2,4,6,or 8. In particu-
+lar,g(ord)
+eff=−(1+12α2+6α4)/(1+2α2) forq= 0, and g(ord)
+eff=−4α(1+3α2)/(1+2α2)
+forq= 2. Actually, g(ord)
+effdoes not vanish at q= 2nwith any integer n, if higher-
+order harmonics are kept in the expansion of the Bloch functi on at the edge of the
+semi-infinite gap, 1+ 2/summationtext
+nαncos(2πnx), cf. the lowest-order approximation used in
+Eq. (9). With the value of the OL strength adopted in the numer ical simulations
+reported above, ǫ= 5, Eq. (11) yields m(ord)
+eff≈1.128, which is virtually identical to
+the numerically found effective mass, see Eq. ( ??).
+The soliton solution to Eq. (10) with an arbitrary amplitude ,A, is
+Φ =Aexp/parenleftbiggi
+2g(ord)
+effA2t/parenrightbigg
+sech/parenleftbigg/radicalBig
+g(ord)
+effm(ord)
+effAx/parenrightbigg
+. (13)
+This solution explains scaling W∝1/A, which is observed in Fig. 2(b) for broad
+ordinary solitons in the case of q= 2. Figure 6 displays a direct comparison of profiles
+of a typical broad ordinary soliton, as obtained in the numer ical form and produced
+by the averaging method. Good agreement between the two profi les is obvious [note
+that the figure displays the full wave function, |φ(x)|, rather than the envelope, Φ( x),
+see Eq. (9)].
+103.3 Gap solitons
+3.3.1 Direct commensurability between the nonlinear and li near lat-
+tices
+To apply the averaging approximation to GSs, which correspo nd tog=−1 in Eq. (1),
+we follow Ref. [30] and adopt the simplest ansatz which is rel evant in this case,
+φ(x,t) = Φ(x,t)cos(πx), (14)
+with a slowly varying amplitude Φ( x,t). The difference in the form of the carrier wave
+in this expression, in comparison to Eq. (9), is due to the fac t that, near the edge
+of the first finite bandgap, the Bloch function is close to a per iodic function, whose
+period is twice as large as that of the underlying OL potentia l. On the contrary, near
+the edge of the semi-infinite gap the period of the Bloch funct ion coincides with the
+OL’s period.
+The substitution of ansatz (14) in Eq. (1) and the applicatio n of the averaging
+method yields the respective asymptotic NLSE equation for t he amplitude function,
+i∂Φ
+∂t=−1
+2m(gap)
+eff∂2Φ
+∂x2+g(gap)
+eff|Φ|2Φ, (15)
+cf. Eq. (10). The effective mass and interaction coefficients i n Eq. (15) are found to
+be
+m(gap)
+eff=ǫ//parenleftbig
+ǫ−2π2/parenrightbig
+, (16)
+g(gap)
+eff= 2/angbracketleftbig
+cos(πx))4cos(qπx)/angbracketrightbig
+. (17)
+Notethatcoefficient (17)is differentfrom zeroonlyfor three valuesof q,viz.,g(gap)
+eff(q=
+0) = 3/4, g(gap)
+eff(q= 2) = 1 /2,andg(gap)
+eff(q= 4) = 1 /8. Together with
+m(gap)
+eff≈ −0.339, (18)
+which Eq. (16) yields for ǫ= 5 (recall this value of the OL strength is fixed in the
+present work), the complex conjugate form of Eq. (15) gives rise to the usual soliton
+solutions for Φ∗, cf. solutions (13) for Φ. This fact gives the explanation to the scaling
+W∝1/Afor the broad GSs, as observed in Fig. 4(b) for q= 2. For stationary
+solutions, the complex conjugation implies, as mentioned a bove, the reversal of the
+sign of the chemical potential. This explains why the stabil ity of the broad GSs obeys
+the “anti-VK” criterion, dµ/dN > 0, which is simply the reverse of the ordinary
+negative-slope VK condition for the stable solutions of the equation for Φ∗.
+The above description is also relevant for moving GSs. In par ticular, the compar-
+ison of the analytically predicted effective mass (18) to the empirical dynamical mass
+(??), drawn for the moving soliton from numerical data at q= 2, clearly demonstrates
+the high accuracy of the averaging approximation for the bro ad GSs, both static and
+moving ones.
+3.3.2 Subharmonic commensurabilitybetween the nonlinear and lin-
+ear lattices
+The above approximation for GSs does not produce any definite result for q= 1, when
+the period of the NL in Eq. (1) is twice as large as OL’s period ( the subharmonic
+resonance between the nonlinear and linear lattices, as defi ned above). To derive an
+11effective envelope equation in this case, we notice that the s ubstitution of original
+ansatz (14) in Eq. (1) and making use of the same effective mass as given by Eq. (16),
+but without averaging the nonlinear term, gives rise to the f ollowing equation:
+i∂Φ
+∂t=−1
+2m(gap)
+eff∂2Φ
+∂x2+cos3(πx)|Φ|2Φ, (19)
+where it is taken into regard that q= 1. This equation suggests that ansatz (14)
+should be replaced by the following one, for stationary solu tions:
+φ(x,t) =e−iµt/bracketleftbig
+Φ1(x)cos(πx)+Φ4(x)cos4(πx)/bracketrightbig
+, (20)
+where both functions Φ 1and Φ 4are slowly varying ones. The substitution of ansatz
+(20) into Eq. (1) and straightforward trigonometric expans ions make it possible to
+eliminate Φ 4in favor of Φ 1:
+Φ4(x) =/parenleftBigg
+µ−π2
+2m(gap)
+eff/parenrightBigg−1
+|Φ1(x)|2Φ1(x). (21)
+The remaining equation for Φ 1(x) is the NLSE with the quinticself-focusing nonlinear
+term:
+µΦ1=−1
+2m(gap)
+eff∂2Φ1
+∂x2−15m(gap)
+eff
+8/parenleftBig
+π2−2m(gap)
+effµ/parenrightBig|Φ1|4Φ1. (22)
+An obvious soliton solution to Eq. (22) with arbitrary ampli tudeAis
+Φ1=A/radicalbig
+sech(kx), k=√
+5m(gap)
+eff/parenleftBig
+π2−2m(gap)
+effµ/parenrightBig−1/2
+A2, µ=−/parenleftBig
+8m(gap)
+eff/parenrightBig−1
+k2.
+(23)
+Solution (23) yields the scaling relation W∝1/A2,which explains the same scaling
+that was observed, at q= 1, for broad GSs in Fig. 4(b).
+It is necessary to mention that the nonstationary version of Eq. (22), with µΦ1re-
+placedby i∂Φ1/∂t, corresponds totheone-dimensional NLSEwiththecritical (quintic,
+in the 1D case) self-focusing nonlinearity, whose solution s, tantamount to those given
+byEq. (23), are the so-called “one-dimensional Townes soli tons”. It is well known that
+the entire family of such solitons is unstable (see, e.g., Re f. [34]). Nevertheless, simu-
+lations of Eq. (1) demonstrate that the GSs approximated by t he asymptotic solution
+(23) form a stablefamily, as long as the solitons remain broad, see the corresp onding
+stable branch in Fig. 4(b). Thus, the asymptotic descriptio n of the broad GSs by
+means of Eq. (22) is valid, at q= 1, only for static solutions, while their dynamical
+behavior does not obey the straightforward time-dependent version of this equation.
+4 Conclusion
+We have investigated the existence, stability and mobility of ordinary solitons and
+GSs (gap solitons) in the 1D model combining nonlinear and li near periodic lattices in
+the Gross-Pitaevskii equation for the nearly one-dimensio nal BEC. The emphasis was
+made on the study of effects of the commensurability and incom mensurability between
+the lattices, as well as on the development of analytical met hods – the VA (variational
+approximation) for narrow ordinary solitons, and averagin g method for broad solitons
+12of both types. We have demonstrated that, in the case of the di rect commensura-
+bility between the lattices (equal periods), the ordinary s olitons are similar to their
+counterparts in the free space. In the case of the subharmoni c commensurability and
+incommensurability, the situation is different, featuring the existence threshold for the
+solitons, and a different scaling relation between their amp litude and width. Simi-
+lar scaling relations are found for GS families, which demon strate the bistability in
+cases different from the direct commensurability. Ordinary solitons may travel long
+distances, if kicked. Broad GSs are mobile too, although col lisions between them are
+inelastic.
+The analytical approximations demonstrate good accuracy i n appropriate param-
+eter regions. In particular, they correctly explain differe nt scaling relations for the
+soliton families at different commensurability orders. As c oncerns the stability of the
+ordinary solitons, it is accurately predicted by the VK crit erion. Simultaneously, the
+stability of GSs obeys the “anti-VK” criterion, an explanat ion to which was given by
+means of the effective equation produced through the averagi ng method. A notable
+feature of the present model is that it gives rise to characte risticsµ(N) for the GSs
+that feature turning points (except for the case of the direc t commensurability). Un-
+like previously studied models with the constant nonlinear ity coefficient, the presence
+of the turning points has made it possible to test the anti-VK stability criterion for
+the GS families.
+The analysis reported in this work may be naturally extended by studying GSs
+in higher finite bandgaps; in particular, a challenging issu e is to verify the “anti-
+VK” stability criterion in the higher gaps. It may also be int eresting to develop a
+systematic analysis of the commensurability for solitons a nd solitary vortices in the
+2D model based on the combination of linear and nonlinear lat tices.
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+arXiv:1001.0426v2  [cond-mat.quant-gas]  25 Mar 2010Thermodynamic properties of a dipolar Fermi gas
+J.-N. Zhang and S. Yi
+Key Laboratory of Frontiers in Theoretical Physics, Instit ute of Theoretical Physics,
+Chinese Academy of Sciences, Beijing 100190, China
+(Dated: November 16, 2018)
+Based on the semiclassical theory, we investigate the therm odynamic properties of a dipolar
+Fermi gas. Through a self-consistent procedure, we numeric ally obtain the phase-space distribution
+function at finite temperature. We show that the deformation s in both momentum and real space
+become smaller and smaller as the temperature is increased. For the homogeneous case, we also
+calculate pressure, entropy, and heat capacity. In particu lar, at the low-temperature limit and in the
+weak interaction regime, we obtain an analytic expression f or the entropy which agrees qualitatively
+with our numerical result. The stability of a trapped gas at fi nite temperature is also explored.
+PACS numbers: 03.75.Ss, 74.20.Rp, 67.30.H-, 05.30.Fk
+I. INTRODUCTION
+The experimental success in creating ultracold
+40K87Rb molecular gas near quantum degeneracy [1–4]
+has drawn considerable attention to in studying the fun-
+damental properties of degenerate dipolar Fermi gases.
+Within the framework of the semi-classical theory, the
+groundstate properties, the collectiveexcitation, and the
+free expansion dynamics of a normal state dipolar Fermi
+gas were studied theoretically [5–7]. A recent theoret-
+ical work based on a variational approach reveals that,
+due to the Fock exchange interaction, the momentum
+distribution is stretched along the direction of dipole mo-
+mentsuchthattheFermisurfacebecomesanellipsoid[8].
+This result was confirmed numerically for both homoge-
+neous [9] and trapped [10] systems. Taking into account
+the effect of the exchange interaction, further theoretical
+work regarding the normal state of the zero tempera-
+ture dipolar Fermi gas includes studying the free expan-
+sion [7, 11], collective excitation [6, 11], zero sound [9],
+and the Fermi liquid properties [12].
+Another interesting feature of the dipole-dipole inter-
+action is that the partially attractive dipolar force is
+responsible for the formation of anisotropic BCS pair-
+ing [13–16]. With the recent experimental development
+on control the hyperfine states of40K87Rb molecules [3,
+4], the BCS pairings in a mixture of fermionic polar
+molecules with two different hyperfine states are also
+studied theoretically [17–19]. In particular, the effects
+of the Fock exchange interaction to pairing was consid-
+ered in Ref. [16, 18].
+In this paper, we extend our previous work on the
+ground state properties of the dipolar Fermi gases to fi-
+nite temperature case. Employing the semi-classical the-
+ory, we numerically obtain the phase-space distribution
+function through a self-consistent procedure. We show
+that thedeformationsofthe distributionfunction in both
+momentum and real space become smaller and smaller as
+the temperature is increased. For homogeneous system,
+we also calculate the thermodynamic quantities such as
+pressure,entropy,andheatcapacity. Inparticular,atlow
+temperature limit and in weak interaction regime, we de-rive an analytic expression for the entropy, which agrees
+qualitatively with our numerical result. For the trapped
+gases, we also explore the temperature dependence of the
+stability.
+The remainder of this paper is organized as follows.
+In Sec. II, we introduce our model and briefly outline
+the semi-classical theory for a dipolar Fermi gas at finite
+temperature. The numerical and analytical results are
+presented in Sec. III. Finally, we conclude in Sec. IV.
+II. THEORY
+Here we consider a system of Nspin polarized dipolar
+fermions interacting via dipole-dipole interaction
+Vd(r) =cd
+r3/parenleftbigg
+1−3z2
+r2/parenrightbigg
+, (1)
+wherecd=d2/(4πε0) withdbeing the electric dipole
+moment. For simplicity, we have assumed that the
+dipole moments of all fermions are orientated along z-
+axis. Additionally, we shall consider both homogeneous
+and trapped systems. In the former case, Uextcan be
+regarded as a box of volume V; while for the latter, the
+trap is assumed to be a harmonic potential with axial
+symmetry, i.e.,
+Uext(r) =1
+2mω2λ−2/3(x2+y2+λ2z2),(2)
+whereωis the geometric average of the trap frequencies
+andλis trap aspect ratio.
+Within the framework of the semiclassical theory, the
+thermodynamic properties are completely characterized
+by the phase-space distribution function f(r,k), which,
+at finite temperature T, satisfies the Fermi-Dirac statis-
+tics
+f(r,k) =1
+e(ε(r,k)−µ)/kBT+1, (3)
+whereµis the chemical potential introduced to fix the2
+01200.51
+kρ/k0
+Ff(kρ,0) (a)
+012300.51
+kz/k0
+F
+f(0,kz) (b)
+0 1 2 30.40.60.81
+kBT/ǫ0
+Fα
+  
+(c)
+Dh= 0.1
+Dh= 0.2
+Dh= 0.3
+FIG. 1: (Color online). (a) and (b) represent, respectively ,
+thephase-spacedistributionfunctions f(kρ,0)andf(0,kz)for
+Dh= 0.3. The corresponding temperatures are kBT/ǫ0
+F= 1
+(solid lines), 0 .5 (dashed lines), 1 /3 (dash-dotted lines), and
+0.1 (dotted line). (c) The temperature dependence of αfor
+various dipolar interaction strengths. The horizontal lin es
+denote the values of αat zero temperature.
+total number of particles such that
+/integraldisplaydrdk
+(2π)3f(r,k) =N. (4)
+The dispersion relation of the quasi-particle takes the
+form
+ε(r,k) =/planckover2pi12k2
+2m+Uext(r)+/integraldisplaydr′dk′
+(2π)3f(r′,k′)Vd(r−r′)
+−/integraldisplaydk′
+(2π)3f(r,k′)/tildewideVd(k−k′), (5)
+where/tildewideVd(k) =cd4π
+3/parenleftBig
+3k2
+z
+k2−1/parenrightBig
+is the Fourier transform
+ofVd(r) and the last two terms represent the mean-field
+potentials originating from Hartree direct and Fock ex-
+change interactions, respectively. We remark that the lo-
+cal density approximation has been employed to obtain
+Eq. (3) for the trapped system.
+Equations (3)-(5) form a closed system of equations
+whichcanbesolvednumericallythroughaniterativepro-
+cedure to obtain the phase-space distribution function
+f(r,k). Within semi-classical approximation, the total
+energy at the finite temperature takes the same form as
+that at zero temperature, i.e.,
+E=1
+(2π)3/integraldisplay
+drdk/bracketleftbigg/planckover2pi12k2
+2m+Uext(r)/bracketrightbigg
+f(r,k)
++1
+2(2π)6/integraldisplay
+drdkdr′dk′f(r,k)f(r′,k′)Vd(r−r′)
+−1
+2(2π)6/integraldisplay
+drdkdk′f(r,k)f(r,k′)/tildewideVd(k−k′),123Ekin/(Nǫ0F)
+(a)
+−0.4−0.20Eexc/(Nǫ0F)
+(b)
+00.5 11.5 2−2−101
+kBT/ǫ0
+Fµ/ǫ0F(c)
+FIG. 2: (Color online). The temperature dependence of Ekin
+(a),Eexc(b), and µ(c). The dimensionless dipolar interaction
+strength are Dh= 0 (dash-dotted lines), 0 .2 (dashed lines),
+and 0.3 (solid lines).
+where the first line represents the kinetic ( Ekin) and po-
+tential (Epot) energies, and last two lines are, respec-
+tively, the direct ( Edir) and exchange ( Eexc) interaction
+energies. Finally, we point out that, for axially sym-
+metric system, the phase-space distribution function re-
+duces to f(r,k) =f(ρ,z,kρ,kz), where ρ=/radicalbig
+x2+y2
+andkρ=/radicalBig
+k2x+k2y. This fact can be used to simplify
+the numerical integration [10].
+III. RESULTS
+In this section, we present our results on the thermo-
+dynamic properties of a dipolar Fermi gas. For a homo-
+geneous gas, we first obtain the phase-space distribution
+function numerically, which allows us to calculate other
+thermodynamicquantities,suchaspressure,entropy,and
+heat capacity. At low temperature limit and in weak in-
+teraction regime, we also derive an analytic expression
+for the entropy, which qualitatively agrees with our nu-
+merical result. In the second part of this section, we
+present the numerical results on the thermodynamics of
+the trapped systems.
+A. Homogeneous case
+For a homogeneous gas, the phase-space distribu-
+tion function reduces to a function of momentum, i.e.,
+f(r,k) =f(k). To present our results, it is convenient3
+012PV/(Nǫ0F)(a)
+0123S/(NkB)(b)
+00.5 11.5 200.511.5
+kBT/ǫ0
+FCV/(NkB)(c)
+FIG. 3: (Color online). The temperature dependence of the
+pressure (a), entropy (b), and heat capacity (c). The dimen-
+sionless dipolar interaction strength are Dh= 0 (dash-dotted
+lines), 0.2 (dashed lines), and 0 .3 (solid lines).
+to introduce a dimensionless dipolar interaction strength
+Dh=ncd
+ǫ0
+F,
+wheren=N/Vis the number density of the system and
+ǫ0
+F=/planckover2pi12(k0
+F)2/(2m) withk0
+F= (6π2n)1/3. We note that
+k0
+Fandǫ0
+Fare, respectively, the Fermi wavevector and
+Fermi energy of an ideal Fermi gas.
+We plot the phase-space distribution function f(k)
+in Fig. 1 (a) and (b) for various temperatures and
+Dh= 0.3. At low temperature, the momentum distri-
+bution is clearly stretched along z-axis. However, as one
+increases the temperature, the momentum distribution
+becomes less anisotropic. Such behavior can be most
+easily visualized by calculating the aspect ratio of the
+cloud in momentum space α≡/radicalbig
+/angbracketleftk2x/angbracketright//angbracketleftk2z/angbracketright. Figure 1
+(c) shows the temperature dependence of αcorrespond-
+ing to various dipolar interaction strengths. For small
+T,αapproaches to the value at zero temperature; as one
+increases the temperature, αgoes to unit asymptotically.
+Due to the spatial homogeneity of the system, the di-
+rect dipolar interaction energy vanishes. Therefore, the
+total energy only contains the contributions from kinetic
+energy and Fock exchange interaction. In Fig. 2 (a) and
+(b), we plot, respectively, EkinandEexcas functions
+of the temperature. The kinetic energy corresponding
+to stronger dipolar interaction is larger than that with
+weaker dipolar interaction, since the stronger dipolar in-
+teraction results in larger momentum space deformation.
+As to the exchange interaction energy, we see that Eexc
+vanishesas αapproaches1athightemperaturelimit. Wealsoplot the temperature dependence ofthe chemical po-
+tential in Fig. 2 (c). Clearly, µdecreases as one increases
+T, in analogy to the ideal Fermi gas. In addition, the
+chemical potential is also a decreasing function of Dh.
+This can be most easily understood at zero temperature
+limit, where the chemical potential becomes [20]
+µ0
+ǫ0
+F≃/parenleftbigg
+1−8π2
+15D2
+h/parenrightbigg2/3
+(6)
+in weak interaction regime ( Dh≪1). Equation (6) in-
+dicates that, for given density n, the chemical potential
+decreases as the dipolar interaction strength grows.
+To calculate other thermodynamic quantities, we work
+with the thermodynamic potential
+Ω(T,V,µ) =E−TS−µN, (7)
+where
+S
+VkB=−/integraldisplaydk
+(2π)3/braceleftBig
+f(k)lnf(k)+[1−f(k)]ln[1−f(k)]/bracerightBig
+.
+isthe entropyofthesystem. Thepressuredirectlyrelates
+to the thermodynamics potential as PV=−Ω(T,V,µ)
+and the heat capacity can also be evaluated using the
+definition CV=T/parenleftbig∂S
+∂T/parenrightbig
+VN. In Fig. 3, we present the
+temperature dependence of P,S, andCVcorresponding
+to different dipolar interaction strengths. As a compar-
+ison, we also plot the results for an ideal Fermi gas. In
+general, the results for dipolar Fermi gases approach to
+those corresponding to the ideal gas at high temperature
+limit.
+Figure 3 (a) also indicates that, at a given tempera-
+ture,Pis a decreasing function of Dh, in agreement with
+the zero temperature result [11]. This can be understood
+since the overall dipolar interaction is attractive for ho-
+mogeneous gas. In addition, even though Svanishes at
+zero temperature, the entropy for given T/negationslash= 0 also ex-
+hibits a similar dependence on Dhas the pressure [Fig. 3
+(b)]. Furthermore, The temperature and interaction de-
+pendence of the heat capacity, shown in Fig. 3 (c), can
+be naturally deduced from the behavior of th entropy.
+To gain more insight into the entropy, we shall derive
+an analytical expression for Sat low temperature limit.
+To this end, we note that when T→0, the derivative of
+Swith respect to Tat constant Vandµcan be approx-
+imately expressed as [21]
+T/parenleftbigg∂S
+∂T/parenrightbigg
+Vµ=V/integraldisplaydk
+(2π)3(ε0(k)−µ)∂f(k,T)
+∂T,
+where we have explicitly expressed fas a function of
+the temperature Tand the zero temperature dispersion
+relation for the quasi-particle is [12]
+ε0(k)≃/planckover2pi12k2
+2m−2ǫ0
+FDhP2(cosθ)I/parenleftbiggk
+k0
+F/parenrightbigg
+,4
+FIG. 4: (Color online). The phase-space distribution func-
+tionf(ρ,z,0,0) of a trapped gas in spherical potential for
+kBT/(N1/3/planckover2pi1¯ω) = 0.2 (a), 0.5 (b), and 1 (c). The dimension-
+less dipolar interaction strength is Dt= 1.
+withP2(·) is the second order Legendre function, θis the
+polar angle of k, and
+I(x) =π
+12/braceleftBigg
+3x2+8−3
+x2+3/parenleftbig
+1−x2/parenrightbig3
+2x3ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+x
+1−x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightBigg
+.
+Followingthe same procedure asthat in Ref. [21], we find
+S(T,V,µ)≃T/parenleftbigg∂S
+∂T/parenrightbigg
+Vµ≃V
+12k2
+BTA(µ,cd),(8)
+where
+A(µ,cd) =/integraldisplay/bracketleftbigg
+k2dk
+dε0(k,θ)/bracketrightbigg
+ε0(k,θ)=µsinθdθ.
+We note that Eq. (8) is not the entropy in usual sense
+as it is a function of µ. To proceed further, we change
+variable from µtoNby using Eq. (6), which yields, at
+low temperature limit and in weak interaction regime,
+S(T,V,N)
+NkB≃π2
+2kBT
+ǫ0
+F/parenleftbigg
+1−8π2
+15D2
+h/parenrightbigg1/3/parenleftbigg
+1+7π2
+45D2
+h/parenrightbigg
+.
+This expression clearly indicates that the entropy is de-
+creasing function of Dh, in qualitative agreement with
+our numerical results.
+B. Trapped case
+Now we turn to study the trapped case, for which we
+shall adopt a set of dimensionless units based on the har-
+monic oscillator length ¯ a=/radicalbig
+/planckover2pi1/(mω):N1/6¯afor length,
+N1/6¯a−1for wave vector, and N1/3/planckover2pi1ωfor energy. Un-
+der these choices, we may define a dimensionless dipolar
+interaction strength
+Dt=N1/6cd
+/planckover2pi1ω¯a3.
+In addition, the normalization condition for phase-space
+distribution function becomes (2 π)−3/integraltext
+drdkf(r,k) = 1.
+In Fig. 4, we present the typical results for phase-
+space distribution function f(ρ,z,0,0) for a gas trapped0.80.91α
+(a)
+00.5 11.5 20.80.91
+kBT/(N1/3¯hω)β
+(b)
+FIG. 5: (Color online). The temperature dependence of the
+deformation parameters α(a) and β(b) forλ= 0.1 (solid
+lines), 1 (dashed lines), and 10 (dash-dotted lines). The di -
+mensionless dipolar interaction strength are Dt= 1.
+00.20.40.60.81246810
+kBT/(N1/3¯hω)D∗
+t
+  
+λ= 0.1
+λ= 1
+λ= 10
+FIG. 6: (Color online). The temperature dependence of the
+critical dipolar interaction strengths for various trap ge ome-
+tries.
+in a spherical potential. Similar to the momentum space
+distribution for a homogeneous gas (Fig. 1), the distri-
+bution function in real space becomes elongated along
+z-axis at low temperature. However, the deformation
+becomes smaller and smaller as the temperature is in-
+creased. To characterize the real space deformation,
+we define the deformation parameter in real space as
+β≡λ−1/radicalbig
+/angbracketleftx2/angbracketright//angbracketleftz2/angbracketright, where the inclusion of the factor
+λ−1is to eliminate the deformation caused by trapping
+potential such that β= 1 for a noninteracting gas. Fig-
+ure 5 (a) and (b) plot, respectively, the temperature de-
+pendence of the deformation parameters in momentum
+and real spaces. As expected, stronger dipolar interac-
+tion induces larger deformations in momentum and real
+spaces. However, both αandβapproach unit at high
+temperature limit.
+Finally, we explore the stability of the trapped gases
+at finite temperature. Similar to the zero temperature
+case [8, 10], due to the partially attractive feature of the
+dipolar interaction, there also exists a critical dipolar in-
+teraction strength D∗
+tfor a given temperature T, beyond5
+which the system becomes unstable. In Fig. 6, we plot
+the temperature dependence of the critical dipolar inter-
+action strength for various trap geometries. It can be
+seen that, independent of the trap aspect ratio λ,D∗
+t
+is an increasing function of T. This can be understood
+ashigher temperature correspondsto smallermomentum
+and real space deformations, resulting in smaller dipolar
+interaction. In addition, higher temperature also yields
+higher kinetic energy which stabilizes the system.
+IV. CONCLUSIONS
+To conclude, based on the semi-classical theory, we
+have studied the thermodynamics of a dipolar Fermi gasby numerically finding the phase-space distribution func-
+tion. We show that the deformations in both momentum
+and real space becomes smaller and smaller as one in-
+creasesthe temperature. Forhomogeneousgases, we also
+calculate pressure, entropy, and heat capacity. In partic-
+ular, at low temperature limit and in weak interaction
+regime, we obtain an analytic expression for the entropy,
+whichagreesqualitativelywithournumericalresult. The
+stability diagram for a trapped gas at finite temperature
+is also obtained.
+This work was supported by NSFC through grants
+10974209 and 10935010, by the National 973 program
+(Grant No. 2006CB921205), and by the “Bairen” pro-
+gram of Chinese Academy of Sciences.
+[1] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er,
+B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne,
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+[11] T. Sogo, L. He, T. Miyakawa, S. Yi, H. Lu, and H. Pu,
+New. J. Phys. 11, 055017 (2009).
+[12] C.-K. Chan, C. Wu, W. Lee, and S. Das Sarma,
+arXiv:0906.4403 (2009).
+[13] L. You and M. Marinescu, Phys. Rev. A 60, 2324 (1999).
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\ No newline at end of file
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+arXiv:1001.0427v1  [math.RA]  4 Jan 2010Filtration, automorphisms and classification of the
+infinite dimensional odd Contact superalgebras
+Jixia Yuan1,∗†and Wende Liu2,‡
+1Department of Mathematics ,Harbin Institute of Technology
+Harbin 150006, China
+2School of Mathematical Sciences ,Harbin Normal University
+Harbin 150025, China
+Abstract : The principal filtration of the infinite-dimensional odd Contact
+Lie superalgebra over a field of characteristic p >2 is proved to be invari-
+ant under the automorphism group by investigating ad-nilpotent ele ments
+and determining certain invariants such as subalgebras generated by some
+ad-nilpotent elements. Then, it is proved that two automorphisms c oincide
+if and only if they coincide on the −1 component with respect to the princi-
+pal grading. Finally, all the odd Contact superalgebras are classifie d up to
+isomorphisms.
+Mathematics Subject Classification 2000 : 17B50, 17B40, 17B65
+Keywords : Lie superalgebras, filtration, automorphisms, classification
+0. Introduction
+As is well known, filtration techniques are of great importance in the structure and clas-
+sification theories of Lie (super)algebras. A descending filtration o f a Lie superalgebra
+Lis a sequence of Z2-graded spaces L=L0⊃L1⊃ ···for which [Li,Lj]⊂Li+jholds
+for alli,j.In the situation that the Killing form on a simple Lie (super)algebra is de -
+generate, the filtration structure plays a particular role. A filtrat ionL=L0⊃L1⊃ ···
+of a Lie superalgebra Lis said to be invariant provided that ϕ(Li)⊂Lifor alliand
+all automorphisms ϕofL. We know that the simple Lie (super)algebras of Cartan
+type possess various natural filtration structures, for which th e invariance may be used
+to make an insight for the intrinsic properties and the automorphism groups of those
+Lie (super)algebras. The filtration structures have been studied for the finite dimen-
+sional Lie superalgebras of Cartan type, for example, in [4, 7, 9] th e invariance of the
+natural filtrations was determined for the generalized Witt supera lgebras, the special
+superalgebras, the Hamiltonian superalgbras and the odd Hamiltonia n superalgebras.
+∗Correspondence: jxy@hrbnu.edu.cn (J. Yuan), wendeliu@us tc.edu.cn
+†Supported by the NSF of HLJ Province (A200903)
+‡Supported by the NSF (10871057) of China and the NSF (A200802 ) of HLJ Province, China
+1Infinite-dimensional odd Contact superalgebras 2
+The reader is also refereed to [5, 8] for the similar work on certain in finite dimensional
+modular Lie superalgebras of Cartan type. Let us state the main re sults of this paper.
+WriteKO(n,n+ 1) for the infinite dimensional odd Contact Lie superalgebras over a
+field of prime characteristic (see Sec.1 for a definition and more deta ils).KO(n,n+1)
+has a canonical filtration structure known as principal. By means of characterizing
+ad-nilpotent elements of KO(n,n+1) we first obtain in this paper that:
+•(Theorem 4.1) The principal filtration of the odd Contact Lie superalgebra is invariant
+under the automorphism group of the Lie superalgebra.
+As a consequence the automorphisms can be characterized as follo ws:
+•(Theorem 4.4) Two automorphisms of the odd Contact Lie superalgebra coinc ide if
+and only if they coincide on the −1component with respect to the principal grading.
+Finally we classify all the infinite dimensional odd Contact Lie superalge bras up to
+isomorphisms:
+•(Theorem 4.5) KO(n,n+1)∼=KO(m,m+1)if and only if n=m.
+1. Preliminaries
+Throughout Fis a field of characteristic p >2;Z2:={¯0,¯1}is the additive group
+of two elements; NandN0are the sets of positive integers and nonnegative integers,
+respectively. Let O(n) be the divided power algebra with F-basis{x(α)|α∈Nn
+0}. Note
+thatx(0):= 1∈ O(n), where 0 = (0 ,...,0)∈Nn
+0.Forεi:= (δi1,δi2,...,δ in)∈Nn
+0,
+writexiforx(εi), wherei= 1,...,n.Let Λ(m) be the exterior superalgebra over Fin
+mvariablesxn+1,xn+2,...,x n+m. Set
+B(m) :=/braceleftbig
+/a\}bracketle{ti1,i2,...,ik/a\}bracketri}ht |n+1≤i1<i2<···<ik≤n+m, k∈0,m/bracerightbig
+.
+Foru:=/a\}bracketle{ti1,i2,...,ik/a\}bracketri}ht ∈B(m),write|u|:=kandxu:=xi1xi2···xik.Notice that we
+alsodenotethe index set {i1,i2,...,ik}byuitself. Foru,v∈B(m) withu∩v=∅,define
+u+vto be the uniquely determined element w∈B(m) such that w=u∪v.Ifv⊂u,
+defineu−vto bew∈B(m) such that w=u\v.Clearly, the associative superalgebra
+O(n,m) :=O(n)⊗Λ(m) has a so-called standard F-basis{x(α)xu|(α,u)∈Nn
+0×B(m)}.
+Let∂rbe the superderivations of O(n,m) defined by ∂r(x(α)) =x(α−εr)forr∈1,n
+and∂r(xs) =δrsforr,s∈1,n+m.Here1,nis the set of integers 1 ,2,...,n.The
+generalized Witt superalgebra W(n,m) isF-spanned by all fr∂r,wherefr∈ O(n,m),
+r∈1,n+m.NotethatW(n,m)isafree O(n,m)-modulewithbasis {∂r|r∈1,n+m}.
+In particular, W(n,m) has astandard F-basis{x(α)xu∂r|(α,u,r)∈Nn
+0×B(m)×
+1,n+m}.
+For ann-tupleα:= (α1,...,α n)∈Nn
+0, put|α|:=/summationtextn
+i=1αi.Whenm=n+ 1, we
+usually write O:=O(n,n+ 1), W:=W(n,n+ 1),A:=Nn
+0andB:=B(n+ 1).If
+u:=/a\}bracketle{ti1,...,ir/a\}bracketri}ht ∈B,put
+/bardblu/bardbl:=/braceleftbigg|u|+1,if 2n+1∈u
+|u|,if 2n+1/∈u.
+Recall the standard Z-grading, O=⊕i≥0Os,[i],where
+Os,[i]:= spanF{x(α)xu| |α|+|u|=i,α∈A,u∈B}.Infinite-dimensional odd Contact superalgebras 3
+It induces naturally the standard grading W=⊕i≥−1Ws,[i],where
+Ws,[i]:= spanF{f∂j|f∈ Os,[i+1],j∈1,2n+1}.
+Thestandardgradingsof OandWareoftype(1 ,...,1|1,...,1).LetWs,i=/summationtext
+j≥iWs,[j].
+Then (Ws,i)i≥−1is called the standard filtration of W.
+We shall also use the principal grading O=⊕i≥0Op,[i],where
+Op,[i]:= spanF{x(α)xu| |α|+/bardblu/bardbl=i,α∈A, u∈B},
+and the principal grading W=⊕i≥−2Wp,[i],where
+Wp,[i]:= spanF{f∂j|f∈ Op,[i+1+δj,2n+1],j∈1,2n+1}
+(cf. [1]). The principal gradings of OandWare of type (1 ,...,1|1,...,1,2).
+For a vector superspace V=V¯0⊕V¯1,we write p( x) :=θfor the parity of a Z2-
+homogeneous element x∈Vθ,θ∈Z2.Once the symbol p( x) appears, it will imply that
+xis aZ2-homogeneous element.
+The odd Contact superalgebra, which is a subalgebra of W,is defined as follows (see
+[1] for more details):
+KO(n,n+1) :={DKO(a)|a∈ O},
+where
+DKO(a) :=TH(a)+(−1)p(a)∂2n+1(a)E+(E(a)−2a)∂2n+1,
+E:=2n/summationdisplay
+i=1xi∂i, TH(a) :=2n/summationdisplay
+i=1(−1)µ(i′)p(a)∂i′(a)∂i,
+i′:=/braceleftbigg
+i+n,ifi∈1,n
+i−n,ifi∈n+1,2n,µ(i) :=/braceleftbigg¯0,ifi∈1,n
+¯1,ifi∈n+1,2n+1.
+For the operator THand further information, the reader is referred to [3]. Note that for
+a,b∈ O(see [1]),
+[DKO(a),DKO(b)] =DKO(DKO(a)(b)−(−1)p(a)2∂2n+1(a)b). (1.1)
+For simplicity, we usually write KOforKO(n,n+1). Note that KOhas a so-called
+principal Z-grading structure denoted by
+KO=⊕i≥−2KOp,[i],whereKOp,[i]:=KO∩Wp,[i].
+In particular,
+KOp,[−2]=F·DKO(1),
+KOp,[−1]= spanF{DKO(xi)|i∈1,2n},
+KOp,[0]= spanF{DKO(x2n+1),DKO(xixj)|1≤i≤j≤2n}.(1.2)
+Let
+Xp,i:=/summationdisplay
+j≥iXp,[j],whereX=WorKO.
+Then(Xp,i)i≥−2iscalledtheprincipalfiltrationof X.Recallthattheinfinite-dimensional
+generalized Witt superalgebra W(n,n) contains the following Lie superalgebra as a sub-
+algebra (see [1, 2]):
+SHO′(n,n) :={TH(a)|a∈ O(n,n),∆(a) = 0},where ∆ :=/summationtextn
+i=1∂i∂i′.
+Convention : In the sequel we shall write KO[i]andKOiforKOp,[i]andKOp,i,
+respectively.Infinite-dimensional odd Contact superalgebras 4
+2. Ad-nilpotent elements
+LetLbe a Lie superalgebra. An element y∈Lis ad-nilpotent if as a transformation
+adyis nilpotent on L. LetRbe a subalgebra of L. Put
+nil(R) := the set of the elements in Rwhich are ad-nilpotent on L,
+spanFnil(R) := the subspace spanned by nil( R),
+Nil(R) := the subalgebra of Lgenerated by nil( R).
+Lemma 2.1. Supposea∈ Ois ofZ-degree2with respect to the standard grading. If
+∂2n+1(a) = 0, that is, if a∈ O(n,n), then
+[DKO(a),DKO(b)] =DKO(TH(a)(b))for allb∈ O.
+Proof.It follows from (1.1) .
+A nonempty subset Sof a Lie superalgebra Lis called a Lie-super subset if Sis
+closed under the multiplication of Land it spans a sub-superspace (and then is a Lie
+superalgebra). A slight modification of [6, Theorem 1.3.1] yields the fo llowing lemma.
+Lemma 2.2. SupposeVis a vector superspace over FandSa Lie-super subset of Lie
+superalgebra gl(V). IfSconsists of nilpotent linear transformations spanFSis of finite
+dimension, then spanFSis strictly triangulable on V,that is, there is a finite sequence
+(Vi)0≤i≤mof sub-superspaces such that
+0 =V0⊂V1⊂ ··· ⊂Vm=V;x(Vi)⊂Vi−1for allx∈spanFS.
+Lemma 2.3. Wp,1⊆nil(W).
+Proof.There exists a sub-superspace V⊆Ws,1such that
+Wp,1= spanF{x2n+1∂i|i∈1,2n}+V.
+ForE∈Wp,1,writeE=E0+E1,whereE0∈spanF{x2n+1∂i|i∈1,2n},E1∈V.Put
+E2:= [E0,E1].Then there exists an n-tupletof positive integers such that E1,E2∈
+Ws,1(n,n+1;t) :=W(n,n+1;t)∩Ws,1[see [3] for a definition of W(n,n+1;t)]. Note
+that
+S:= spanF{ad(x2n+1∂i)|i∈1,2n}∪adWs,1(n,n+1;t)
+is a Lie-super subset of the general linear Lie superalgebra gl(W).A direct computation
+shows that spanF{x2n+1∂i|i∈1,2n} ⊆nil(W).By virtue of [8, Theorem 2.5], we have
+Ws,1(n,n+1;t)⊆nil(W).Then Lemma 2.2 ensures that E∈nil(W).
+Lemma 2.4. (1)KO1⊆nil(KO).
+(2)Supposey=y[i]+yi+1∈nil(KOi),wherey[i]∈KO[i]andyi+1∈KOi+1.Then
+y[i]∈nil(KO[i]).
+(3)Supposey=y[−1]+y0∈nil(KO¯0),wherey[−1]∈KO[−1]∩KO¯0andy0∈
+KO0∩KO¯0.Theny[−1]= 0.
+(4) Nil(KO¯0) = Nil(KO[0]∩KO¯0)+KO1∩KO¯0.Infinite-dimensional odd Contact superalgebras 5
+Proof.(1) SinceKO1⊆Wp,1,by virtue of Lemma 2.3 we get KO1⊆nil(KO).
+(2) Similar to [8, Lemma 2.7], one may prove (2).
+(3) Suppose y[−1]=/summationtextn
+i=1aiDKO(xi′),whereai∈F.Note thatDKO(x((k+1)εj))∈
+KOfork∈N.Ifaj/\e}atio\slash= 0 for some j∈1,n,theny[−1]is not ad-nilpotent, since
+(ady[−1])k(DKO(x((k+1)εj))) =ak
+jDKO(xj)/\e}atio\slash= 0. This contradicts (2). Therefore, aj= 0
+for allj∈1,n,that is,y[−1]= 0.
+(4) It follows from (1) and (3).
+Lemma 2.5. Ifi/\e}atio\slash=j′∈1,2n,then(TH(xixj))2p= 0.
+Proof.Note that
+TH(xixj) = (−1)µ(i)+µ(i)µ(j)xj∂i′+(−1)µ(j)xi∂j′, (2.1)
+(xj∂i′)p= (xi∂j′)p= 0 and [xj∂i′,xi∂j′] = 0 for all i/\e}atio\slash=j′∈1,2n.In combination with
+(2.1), we have ( TH(xixj))2p= 0.
+Lemma 2.6. Fori,j∈1,2n,the following statements hold.
+(1)Iff∈ O, DKO(f)∈KO[0]and∂2n+1(f)/\e}atio\slash= 0,thenDKO(f)/∈nil(KO[0]).
+(2)Supposei/\e}atio\slash=j′andai∈Ffor alli∈1,n.Then
+DKO(xixj)∈nil(KO[0]),n/summationdisplay
+i=1aiDKO(xixi′)/∈nil(KO[0])or is0,
+DKO(xixi′)/∈Nil(KO[0]), D KO(x2n+1)/∈Nil(KO[0]),
+DKO(xixi′−xjxj′)∈Nil(KO[0]).
+Proof.(1) Suppose f∈ O, DKO(f)∈KO[0]and∂2n+1(f)/\e}atio\slash= 0.Then there exist
+0/\e}atio\slash=a∈Fandf0∈ O(n,n) such that f=ax2n+1+f0.Since
+[DKO(ax2n+1+f0),DKO(1)] = 2aDKO(1),
+we haveDKO(f)/∈nil(KO[0]).
+(2) Applying Lemma 2.1 we obtain by induction on kthat
+(adDKO(xixj))k(DKO(f)) =DKO(TH(xixj)k(f)) for allk∈N,
+wheref∈ O.Since Ker(DKO) = 0,one sees that DKO(xixj) is ad-nilpotent if and only
+ifTH(xixj) is a nilpotent transformation of O.Then by Lemma 2.5, we know that
+DKO(xixj)∈nil(KO[0]) for alli/\e}atio\slash=j′∈1,2n.
+Note that
+/bracketleftBign/summationdisplay
+i=1aiDKO(xixi′),DKO(xjx2n+1)/bracketrightBig
+=−ajDKO(xjx2n+1), j∈1,n.
+If/summationtextn
+i=1aiDKO(xixi′)/\e}atio\slash= 0 then
+n/summationdisplay
+i=1aiDKO(xixi′)/∈nil(KO[0]).Infinite-dimensional odd Contact superalgebras 6
+It follows from (1) that nil( KO[0])⊆SHO′(n,n).Therefore, Nil( KO[0])⊆SHO′(n,n).
+ButDKO(xixi′), DKO(x2n+1)/∈SHO′(n,n),thus
+DKO(xixi′),DKO(x2n+1)/∈Nil(KO[0]).
+By virtue of the fact that
+[DKO(xixj),DKO(xi′xj′)] =−DKO(xixi′−xjxj′),
+we haveDKO(xixi′−xjxj′)∈Nil(KO[0]).
+3. Invariant subalgebras
+Let
+T:= Nor KO¯0(Nil(KO¯0)),
+Q:={y∈KO¯1|[y,KO ¯1]⊆T},
+M:={y∈KO¯1|[y,Q]⊆Nil(KO¯0)}.
+It is easy to see that Tis an invariant subspace under the automorphisms of KOand so
+areQandM.
+Proposition 3.1. T=KO0∩KO¯0.In particular, KO0∩KO¯0is an invariant subalgebra
+ofKO.
+Proof.Let
+y=n/summationdisplay
+i=1aiDKO(xi′)+y′′∈T,
+whereai∈Ffor alli∈1,n, y′′∈KO0∩KO¯0.Assume that aj/\e}atio\slash= 0 for some j∈1,n.
+Takej/\e}atio\slash=k∈1,n.ByLemma2.4(4), wehave DKO(xjxkxk′)∈KO1∩KO¯0⊆Nil(KO¯0).
+Then
+−ajDKO(xkxk′)−akDKO(xjxk′)+h
+=/bracketleftBign/summationdisplay
+i=1aiDKO(xi′)+y′′,DKO(xjxkxk′)/bracketrightBig
+∈Nil(KO¯0),
+whereh∈KO1∩KO¯0.This contradicts Lemma 2.6(2) and then T⊆KO0∩KO¯0.On
+the other hand, by (1.2) and Lemma 2.6(2), we have
+KO0∩KO¯0= spanFnil(KO¯0)+Y,
+whereY:= spanF{DKO(xixi′),DKO(x2n+1)|i∈1,n}.By (1.1), we have
+[Y,Nil(KO¯0)]⊆Nil(KO¯0).
+Then
+[KO0∩KO¯0,Nil(KO¯0)] = [spanFnil(KO¯0)+Y,Nil(KO¯0)]⊆Nil(KO¯0).
+The proof is complete.Infinite-dimensional odd Contact superalgebras 7
+Lemma 3.2. Q⊆spanF{DKO(xixj)|1≤i≤j≤n}+KO1∩KO¯1.
+Proof.Fory∈Q,we may write
+y=DKO(a)+y′,
+wherea∈F, y′∈KO−1∩KO¯1.Note thatDKO(x1′x2n+1)∈KO¯1.Then
+−2aDKO(x1′)+h= [DKO(a)+y′,DKO(x1′x2n+1)]∈KO0∩KO¯0,
+whereh∈KO0∩KO¯0.Thena= 0.Thus we may write
+y=n/summationdisplay
+i=1aiDKO(xi)+y′′,
+whereai∈Ffor alli∈1,n, y′′∈KO0∩KO¯1.Ifaj/\e}atio\slash= 0 for some j∈1,n,take
+j/\e}atio\slash=k∈1,n.Note thatDKO(xj′xk′)∈KO¯1.We have
+DKO(ajxk′−akxj′)+h=/bracketleftBign/summationdisplay
+i=1aiDKO(xi)+y′′,DKO(xj′xk′)/bracketrightBig
+∈KO0∩KO¯0,
+whereh∈KO0∩KO¯0,contradicting that aj/\e}atio\slash= 0. Thus we may write
+y=/summationdisplay
+1≤i≤j≤naijDKO(xixj)+/summationdisplay
+1≤i<j≤nbijDKO(xi′xj′)+y′′,
+whereaij,bij∈Ffor all 1≤i≤j≤n,y′′∈KO1∩KO¯1.Assume that bkl/\e}atio\slash= 0 for some
+1≤k<l≤n.SinceDKO(xk)∈KO¯1,we have
+k−1/summationdisplay
+i=1−bikDKO(xi′)+n/summationdisplay
+i=k+1bkiDKO(xi′)+h= [y,DKO(xk)]∈KO0∩KO¯0,
+whereh∈KO0∩KO¯0,contradicting the assumption that bkl/\e}atio\slash= 0. Then
+Q⊆spanF{DKO(xixj)|1≤i≤j≤n}+KO1∩KO¯1
+and this completes the proof.
+Remark 3.3. Fori/\e}atio\slash=j∈1,n,we haveDKO(xixi′x2n+1)∈Q,DKO(xi′xjxj′)∈Q.
+Proposition 3.4. M=KO0∩KO¯1.In particular, KO0∩KO¯1is an invariant subal-
+gebra ofKO.
+Proof.By (1.2), Lemmas 2.4(4), 2.6(2) and 3.2, we have
+[KO0∩KO¯1,Q]⊆[KO0∩KO¯1,spanF{DKO(xixj)|1≤i≤j≤n}+KO1∩KO¯1]
+⊆Nil(KO¯0).
+HenceKO0∩KO¯1⊆M.Conversely, for y∈M,we may write
+y=DKO(a)+y′,Infinite-dimensional odd Contact superalgebras 8
+wherea∈F, y′∈KO−1∩KO¯1.By Remark 3.3, we have
+2aDKO(xixi′)+h= [DKO(a)+y′,DKO(xixi′x2n+1)]∈Nil(KO¯0),
+whereh∈KO1∩KO¯0.By Lemma 2.6(2), we have a= 0.Thus we may write
+y=n/summationdisplay
+i=1aiDKO(xi)+y′′,
+whereai∈Ffor alli∈1,n, y′′∈KO0∩KO¯1.Assume that aj/\e}atio\slash= 0 for some j∈1,n.
+Takej/\e}atio\slash=k∈1,n. By Remark 3.3, we have
+ajDKO(xkxk′)−akDKO(xj′xk)+h
+=/bracketleftBign/summationdisplay
+i=1aiDKO(xi)+y′′,DKO(xj′xkxk′)/bracketrightBig
+∈Nil(KO¯0),
+whereh∈KO1∩KO¯0.By Lemma 2.6(2), we have aj= 0.This contradicts the assump-
+tion thataj/\e}atio\slash= 0.ThusM⊆KO0∩KO¯1and the proof is complete.
+The key in this paper is the following proposition.
+Proposition 3.5. KO0is an invariant subalgebra of KO.
+Proof.It follows from Propositions 3.1 and 3.4.
+4. Filtration, automorphisms and classification
+One of the main results is as follows, which is a direct consequence of P roposition 3.5
+and the following Lemmas 4.2 and 4.3.
+Theorem 4.1. The principal filtration of KOis invariant under the automorphisms of
+KO,that is,ϕ(KOi) =KOifor alli≥ −2and all automorphisms ϕofKO.
+Lemma 4.2. KO−1/KO0is the unique irreducible KO0-submodule of KO/KO 0. In
+particular,KO−1is an invariant subalgebra of KO.
+Proof.Clearly,KO−1/KO0is an irreducible KO0-submodule. To show the uniqueness,
+supposeM/KO 0is a nonzero KO0-submodule of KO/KO 0,whereM⊃KO0is aKO0-
+submodule of KO.For 0/\e}atio\slash=y∈M,one may write y=DKO(1)+y′,wherey′∈KO−1.
+Then
+[DKO(xix2n+1),DKO(1)+y′]∈Mfor alli∈1,2n.
+Note that [DKO(xix2n+1),y′]∈M.We have
+2DKO(xi) = [DKO(xix2n+1),DKO(1)]∈M
+for alli∈1,2n.Therefore,KO−1/KO0⊆M/KO 0.The proof is complete.
+Lemma 4.3. KOi={y∈KOi−1|[y,KO −1]⊆KOi−1}for alli≥1.Infinite-dimensional odd Contact superalgebras 9
+Proof.Put
+hi:={y∈KOi−1|[y,KO −1]⊆KOi−1}.
+Clearly,KOi⊆hi.For ally∈hi,we may write y:=/summationtext
+j≥i−1yj,whereyj∈KO[j].By
+the definition of hi,[yi−1,KO−1] = 0.Letyi−1:=/summationtext
+α,ubα,uDKO(x(α)xu),whereα∈A,
+u∈B, DKO(x(α)xu)∈KO[i−1]⊆KO0, bα,u∈F.For any fixed β/\e}atio\slash= 0 withβk≥1 for
+somek∈1,n, we have
+0 =/bracketleftBig/summationdisplay
+α,ubα,uDKO(x(α)xu),DKO(xk′)/bracketrightBig
+=/summationdisplay
+α,ubα,uDKO(∂k(x(α)xu)−(−1)p(xu)∂2n+1(x(α)xu)xk′).
+Consequently, bα,u= 0 whenever α/\e}atio\slash= 0. It remains to consider the case α= 0. Fix any
+v/\e}atio\slash=∅.If there isl∈n+1,2nsuch thatl∈v.Then
+0 =/bracketleftBig/summationdisplay
+0,ub0,uDKO(xu),DKO(xl′)/bracketrightBig
+=/summationdisplay
+0,ub0,uDKO((−1)p(xu)∂l(xu)−(−1)p(xu)∂2n+1(xu)xl′).
+It follows that b0,v= 0,where/a\}bracketle{t2n+1/a\}bracketri}ht /\e}atio\slash=v∈B.Takingi∈1,2n,since
+0 = [b0,/angbracketleft2n+1/angbracketrightDKO(x2n+1),DKO(xi)] =b0,/angbracketleft2n+1/angbracketrightDKO(xi),
+we haveb0,/angbracketleft2n+1/angbracketright= 0.Therefore,yi−1= 0 and then hi⊂KOi.
+As a corollary we give a characterization of the automorphisms of KO:
+Theorem 4.4. Two automorphisms of KOcoincide if and only if they coincide on the
+−1component KO[−1].
+Proof.Letφandψbe the automorphisms of KO.It suffices to prove that φ|KO[−1]=
+ψ|KO[−1]impliesφ=ψ.As in [5, Corollary 18], using Theorem 4.1 one can prove that
+φ|KO−1=ψ|KO−1.Since
+φ(DKO(1)) =φ([DKO(x1),DKO(x1′)]) =ψ([DKO(x1),DKO(x1′)]) =ψ(DKO(1)),
+we haveφ|KO[−2]=ψ|KO[−2].Thenφ=ψand the proof is complete.
+We are now in position to state the final main result in this paper, which says that the
+parameterndefining the odd Contact superalgebra KO(n,n+1) is intrinsic and then
+all the infinite-dimensional odd Contact superalgebrasare classifie d up to isomorphisms.
+Theorem 4.5. KO(n,n+1)∼=KO(m,m+1)if and only if n=m.
+Proof of Theorem 4.5. One direction is obvious. Assume that σ:KO(n,n+ 1)−→
+KO(m,m+1) is an isomorphism of Lie superalgebras. Clearly,
+σ(NorKO(n,n+1)¯0(Nil(KO(n,n+1)¯0))) = Nor KO(m,m+1)¯0(Nil(KO(m,m+1)¯0)).
+In view of the proof of Proposition 3.1, we have
+KO(n,n+1)0∩KO(n,n+1)¯0= Nor KO(n,n+1)¯0(Nil(KO(n,n+1)¯0)).Infinite-dimensional odd Contact superalgebras 10
+Therefore,
+σ(KO(n,n+1)0∩KO(n,n+1)¯0) =KO(m,m+1)0∩KO(m,m+1)¯0.(4.1)
+Recall that
+Q={y∈KO(n,n+1)¯1|[y,KO(n,n+1)¯1]⊆KO(n,n+1)0∩KO(n,n+1)¯0}
+and
+M={y∈KO(n,n+1)¯1|[y,Q]⊆Nil(KO(n,n+1)¯0)}.
+By Lemma 3.2 and Proposition 3.4, we have
+σ(KO(n,n+1)0∩KO(n,n+1)¯1) =KO(m,m+1)0∩KO(m,m+1)¯1.(4.2)
+By (4.1) and (4.2), we have
+σ(KO(n,n+1)0) =KO(m,m+1)0.
+Thereforeσinduces an isomorphism of Z2-graded vector spaces
+σ′:KO(n,n+1)/KO(n,n+1)0−→KO(m,m+1)/KO(m,m+1)0.
+SinceKO(n,n+1)/KO(n,n+1)0∼=KO(n,n+1)[−2]⊕KO(n,n+1)[−1]asZ2-graded
+vector spaces, we have
+dim(KO(n,n+1)[−2]⊕KO(n,n+1)[−1]) = dim(KO(m,m+1)[−2]⊕KO(m,m+1)[−1]).
+This implies that n+1 =m+1,that is,n=m.The proof is complete.
+References
+[1] V. G. Kac. Classification of infinite-dimensional simple linearly compa ct Lie super-
+algebras. Adv. Math. 139(1998): 1–55.
+[2] W.-D. Liu and Y.-H. He. Finite-dimensional special odd Hamiltonian su peralgebras
+in prime characteristic. Commun. Contemp. Math. 11(4) (2009): 523–546.
+[3] W.-D. Liu, Y.-Z. Zhang, andX.-L.Wang.The derivationalgebraoft heCartan-type
+Lie superalgebra HO.J. Algebra 273(2004): 176–205.
+[4] W.-D. LiuandY.-Z.Zhang.Finite-dimensionalodd Hamiltoniansuper algebrasover
+a field of prime characteristic. J. Aust. Math. Soc. 79(2005): 113–130.
+[5] W.-D. Liu and Y.-Z. Zhang. Infinite-dimensional modular odd Hamilto nian Lie
+superalgebras. Commun. Algebra. 32(2004): 2341–3257.
+[6] H. Strade and R. Farnsteiner. Modular Lie Algebras and Their Representations .
+New York: Marcel Dekker, 1988.
+[7] Y.-Z. Zhangand H.-C. Fu. Finite-dimensionalHamiltonian Liesupera lgebras.Com-
+mun. Algebra 30(2002): 2651–2673.
+[8] Y.-Z. Zhang and W.-D. Liu. Infinite-dimensional modular Lie Supera lgebrasWand
+Sof Cartan Type. Algebra Colloq. 132(2006): 197–210.
+[9] Y.-Z. Zhang and J.-Z. Nan. Finite-dimensional Lie superalgebras W(m,n,t) and
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\ No newline at end of file
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+arXiv:1001.0428v2  [hep-ph]  11 Jun 2010Symmetry, Integrability and Geometry: Methods and Applica tions SIGMA 6(2010), 049, 29 pages
+Finite Unification: Theory and Predictions⋆
+Sven HEINEMEYER†1, Myriam MONDRAG ´ON†2and George ZOUPANOS†3†4
+†1Instituto de F´ ısica de Cantabria (CSIC-UC), Santander, Spain
+E-mail:Sven.Heinemeyer@cern.ch
+†2Instituto de F´ ısica, Universidad Nacional Aut´ onoma de M´ e xico,
+Apdo. Postal 20-364, M´ exico 01000, M´ exico
+E-mail:myriam@fisica.unam.mx
+†3Theory Group, Physics Department, CERN, Geneva, Switzerland
+E-mail:George.Zoupanos@cern.ch
+†4Physics Department, National Technical University, 157 80 Zografou, Athens, Greece
+Received January 03, 2010, in final form May 25, 2010; Published on line June 11, 2010
+doi:10.3842/SIGMA.2010.049
+Abstract. All-loop Finite Unified Theories (FUTs) are very interesting N= 1 supersym-
+metric Grand Unified Theories (GUTs) which not only realise an old field t heoretic dream
+but also have a remarkable predictive power due to the required red uction of couplings.
+The reduction of the dimensionless couplings in N= 1 GUTs is achieved by searching for
+renormalizationgroup invariant (RGI) relations among them holding b eyond the unification
+scale. Finiteness results from the fact that there exist RGI relatio ns among dimensionless
+couplings that guarantee the vanishing of all beta-functions in cer tainN= 1 GUTs even to
+all orders. Furthermore developments in the soft supersymmetr y breaking sector of N= 1
+GUTs and FUTs lead to exact RGI relations, i.e. reduction of couplings , in this dimension-
+ful sector of the theory too. Based on the above theoretical fr amework phenomenologically
+consistent FUTS have been constructed. Here we present FUT mo dels based on the SU(5)
+andSU(3)3gauge groups and their predictions. Of particular interest is the Hig gs mass
+prediction of one of the models which is expected to be tested at the LHC.
+Key words: unification; gauge theories; finiteness; reduction of couplings
+2010 Mathematics Subject Classification: 81T60; 81V22
+1 Introduction
+A large and sustained effort has been done in the recent years aiming to achieve a unified
+description of all interactions. Out of this endeavor two ma in directions have emerged as the
+most promising to attack the problem, namely, the superstri ng theories and non-commutative
+geometry. The two approaches, although at a different stage of development, have common
+unification targets and share similar hopes for exhibiting improved renormalization properties
+in the ultraviolet (UV) as compared to ordinary field theori es. Moreover the two frameworks
+came closer by the observation that a natural realization of non-commutativity of space appears
+in the string theory context of D-branes in the presence of a c onstant background antisymmetric
+field [1]. Among the numerous important developments in bot h frameworks, it is worth noting
+two conjectures of utmost importancethat signal the develo pments in certain directions in string
+theory and not only, related to the main theme of the present r eview. The conjectures refer to
+⋆This paper is a contribution to the Proceedings of the Eighth International Conference “Symmetry in
+Nonlinear Mathematical Physics” (June 21–27, 2009, Kyiv, U kraine). The full collection is available at
+http://www.emis.de/journals/SIGMA/symmetry2009.html2 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+(i) the duality among the 4-dimensional N= 4 supersymmetric Yang–Mills theory and the type
+IIB string theory on AdS 5×S5[2]; the former being the maximal N= 4 supersymmetric Yang–
+Mills theory is known to be UV all-loop finite theory [3, 4], ( ii) the possibility of “miraculous”
+UV divergence cancellations in 4-dimensional maximal N= 8 supergravity leading to a finite
+theory, as has been recently confirmed in a remarkable 4-loo p calculation [5, 6, 7, 8, 9]. However,
+despite the importance of having frameworks to discuss quan tum gravity in a self-consistent way
+and possibly to construct there finite theories, it is very i nteresting to search for the minimal
+realistic framework in which finiteness can take place. In a ddition, the main goal expected
+from a unified description of interactions by the particle p hysics community is to understand
+the present day large number of free parameters of the Standa rd Model (SM) in terms of a few
+fundamental ones. In other words, to achieve reduction of couplings at a more fundamental level.
+To reduce the number of free parameters of a theory, and thus r ender it more predictive, one
+is usually led to introduce a symmetry. Grand Unified Theori es (GUTs) are very good examples
+of such a procedure [10, 11, 12, 13, 14]. For instance, in the c ase of minimal SU(5), because of
+(approximate) gauge coupling unification, it was possible to reduce the gauge couplings by one
+and give a prediction for one of them. In fact, LEP data [15] se em to suggest that a further
+symmetry, namely N= 1 global supersymmetry [16, 17] should also be required to m ake the
+prediction viable. GUTs can also relate the Yukawa coupling s among themselves, again SU(5)
+provided an example of this by predicting the ratio Mτ/Mb[18] in the SM. Unfortunately, requi-
+ring more gauge symmetry does not seem to help, since additio nal complications are introduced
+due to new degrees of freedom, in the ways and channels of brea king the symmetry, and so on.
+A natural extension of the GUT idea is to find a way to relate th e gauge and Yukawa sectors
+of a theory, that is to achieve gauge-Yukawa Unification (GY U) [19, 20, 21]. A symmetry which
+naturally relates the two sectors is supersymmetry, in part icularN= 2 supersymmetry [22].
+It turns out, however, that N= 2 supersymmetric theories have serious phenomenological
+problems due to light mirror fermions. Also in superstring t heories and in composite models
+there exist relations among the gauge and Yukawa couplings, but both kind of theories have
+phenomenological problems, which we are not going to addres s here.
+There have been other attempts to relate the gauge and Yukawa sectors. One was proposed
+by Decker, Pestieau, and Veltman [23, 24]. By requiring the a bsence of quadratic divergences in
+the SM, they found a relationship between the squared masses appearing in the Yukawa and in
+the gauge sectors of the theory. A very similar relation is ob tained by applying naively in the
+SM the general formula derived from demanding spontaneous s upersymmetry breaking via F-
+terms [25]. In both cases a prediction for the top quark was po ssible only when it was permitted
+experimentally to assume the MH≪MW,Zwith the result Mt= 69 GeV. Otherwise there is
+only a quadratic relation among MtandMH. Using this relationship in the former case and
+a version of naturalness into account, i.e. that the quadratic corrections to the Hig gs mass be
+at most equal to the physical mass, the Higgs mass is found to b e∼260 GeV, for a top quark
+mass of around 176 GeV [26]. This value is already excluded fr om the precision data [27].
+A well known relation among gauge and Yukawa couplings is the Pendleton–Ross (P-R)
+infraredfixed point [28]. The P-R proposal, involving the Yukawa cou pling of the top quark gt
+and the strong gauge coupling α3, was that the ratio αt/α3, whereαt=g2
+t/4π, has an infrared
+fixed point. This assumption predicted Mt∼100 GeV. In addition, it has been shown [29]
+that the P-R conjecture is not justified at two-loops, since then the ratio αt/α3diverges in the
+infrared.
+Another interesting conjecture, made by Hill [30], is that αtitself develops a quasi-infrared
+fixed point, leading to the prediction Mt∼280 GeV.
+The P-R and Hill conjectures have been done in the framework o f the SM. The same conjec-
+tures within the Minimal Supersymmetric SM (MSSM) lead to th e following relations:
+Mt≃140 GeV sin β(P-R), M t≃200 GeV sin β(Hill),Finite Unification: Theory and Predictions 3
+where tanβ=vu/vdis the ratio of the two VEV of the Higgs fields of the MSSM. From
+theoretical considerations one can expect
+1<∼tanβ<∼50⇔1√
+2<∼sinβ<∼1.
+This corresponds to
+100 GeV<∼Mt<∼140 GeV (P-R) ,140 GeV<∼Mt<∼200 GeV (Hill) .
+Thus, the MSSM P-R conjecture is ruled out, while within the M SSM, the Hill conjecture
+does not give a prediction for Mt, since the value of sin βis not fixed by other considerations.
+The Hill model can accommodate the correct value of Mt≈173 GeV [31] for sin β≈0.865
+corresponding to tan β≈1.7. Such small values, however, are stongly challenged by the SUSY
+Higgs boson searches at LEP [32]. Only a very heavy scalar top spectrum with large mixing
+could accommodate such a small tan βvalue.
+In our studies [19, 20, 21, 33, 34, 35, 36, 37, 38] we have devel oped a complementary strategy
+in searching for a more fundamental theory possibly at the Pl anck scale, whose basic ingredients
+are GUTs and supersymmetry, but its consequences certainly go beyond the known ones. Our
+methodconsistsofhuntingforrenormalization groupinvar iant (RGI)relations holdingbelowthe
+Planck scale, whichinturnarepreserveddowntotheGUT scal e. Thisprogramme, called gauge-
+Yukawa unification scheme, applied in the dimensionless co uplings of supersymmetric GUTs,
+such as gauge and Yukawa couplings, had already noticable su ccesses by predicting correctly,
+among others, the top quark mass in the finite and in the minim alN= 1 supersymmetric
+SU(5) GUTs [33, 34, 35]. An impressive aspect of the RGI relatio ns is that one can guarantee
+their validity to all-orders in perturbation theory by stud ying the uniqueness of the resulting
+relations at one-loop, as was proven in the early days of the p rogramme of reduction of couplings
+[39, 40, 41, 42]. Even more remarkable is the fact that it is po ssible to find RGI relations among
+couplings that guarantee finiteness to all-orders in pertu rbation theory [43, 44, 45, 46, 47].
+It is worth noting that the above principles have only been ap plied in supersymmetric GUTs
+for reasons that will be transparent in the following sectio ns. We should also stress that our
+conjecture for GYU is by no means in conflict with the interes ting proposals mentioned before
+(see also [48, 49, 50]), but it rather uses all of them, hopefu lly in a more successful perspective.
+For instance, the use of susy GUTs comprises the demand of the cancellation of quadratic di-
+vergences in the SM. Similarly, the very interesting conjec tures about the infrared fixed points
+are generalized in our proposal, since searching for RGI rel ations among various couplings cor-
+responds to searching for fixed points of the coupled differential equations obeyed by the various
+couplings of a theory.
+Although supersymmetry seems to be an essential feature for a successful realization of the
+above programme, its breaking has to be understood too, sinc e it has the ambition to supply
+the SM with predictions for several of its free parameters. I ndeed, the search for RGI relations
+has been extended to the soft supersymmetry breaking sector (SSB) of these theories [38, 51],
+which involves parameters of dimension one and two. Then a ve ry interesting progress has been
+made [52, 53, 54, 55, 56, 57, 58] concerning the renormalizat ion properties of the SSBparameters
+based conceptually and technically on the work of [59]. In [5 9] the powerful supergraph method
+[60, 61, 62, 63] for studying supersymmetric theories has be en applied to the softly broken ones
+by using the “spurion” external space-time independent sup erfields [64]. In the latter method
+a softly broken supersymmetric gauge theory is considered a s a supersymmetric one in which
+the various parameters such as couplings and masses have bee n promoted to external superfields
+that acquire “vacuum expectation values”. Based on this met hod the relations among the soft
+term renormalization and that of an unbroken supersymmetri c theory have been derived. In
+particular the β-functions of the parameters of the softly broken theory are expressed in terms4 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+of partial differential operators involving the dimension less parameters of the unbroken theory.
+Thekey point in the strategy of [55, 56, 57, 58] in solving the set of coupled differential equations
+so as to be able to express all parameters in a RGI way, was to tr ansform the partial differential
+operators involved to total derivative operators. This is i ndeed possible to be done on the RGI
+surface which is defined by the solution of the reduction equ ations.
+On the phenomenological side there exist some serious devel opments too. Previously an
+appealing“universal”set ofsoftscalar masseswas asummed intheSSBsector ofsupersymmetric
+theories, given that apart from economy and simplicity (1) t hey are part of the constraints
+that preserve finiteness up to two-loops [65, 66], (2) they a re RGI up to two-loops in more
+general supersymmetric gauge theories, subject to the cond ition known as P= 1/3Q[51]
+and (3) they appear in the attractive dilaton dominated supe rsymmetry breaking superstring
+scenarios [67, 68, 69]. However, further studies have exhib ited a number of problems all due to
+the restrictive nature of the “universality” assumption fo r the soft scalar masses. For instance,
+(a) in finite unified theories the universality predicts th at the lightest supersymmetricparticle is
+a charged particle, namely the superpartner of the τlepton ˜τ, (b) the MSSM with universal soft
+scalar masses is inconsistent with the attractive radiativ e electroweak symmetry breaking [69],
+and (c) which is the worst of all, the universal soft scalar ma sses lead to charge and/or colour
+breaking minima deeper than the standard vacuum [70]. There fore, there have been attempts
+to relax this constraint without loosing its attractive fea tures. First an interesting observation
+was made that in N= 1 gauge-Yukawa unified theories there exists a RGI sum rule for the
+soft scalar masses at lower orders; at one-loop for the non-f inite case [71] and at two-loops for
+the finite case [72]. The sum rule manages to overcome the abo ve unpleasant phenomenological
+consequences. Moreover it was proven [58] that the sum rule f or the soft scalar massses is RGI
+to all-orders for both the general as well as for the finite ca se. Finally, the exact β-function for
+the soft scalar masses in the Novikov–Shifman–Vainstein–Z akharov (NSVZ) scheme [73, 74, 75]
+for the softly broken supersymmetric QCD has been obtained [ 58]. Armed with the above tools
+and results we are in a position to study the spectrum of the fu ll finite models in terms of few
+free parameters with emphasis on the predictions for the lig htest Higgs mass, which is expected
+to be tested at LHC.
+2 Unification of couplings by the RGI method
+Let us next briefly outline the idea of reduction of coupling s. Any RGI relation among couplings
+(which does not depend on the renormalization scale µexplicitly) can be expressed, in the
+implicit form Φ( g1,...,gA) = const, which has to satisfy the partial differential equa tion (PDE)
+µdΦ
+dµ=/vector∇·/vectorβ=A/summationdisplay
+a=1βa∂Φ
+∂ga= 0,
+whereβais theβ-function of ga. This PDE is equivalent to a set of ordinary differential
+equations, the so-called reduction equations (REs) [39, 40 , 76],
+βgdga
+dg=βa, a= 1,...,A, (2.1)
+wheregandβgare the primary coupling and its β-function, and the counting on adoes not
+includeg. Since maximally ( A−1) independent RGI “constraints” in the A-dimensional space
+of couplings can be imposed by the Φ a’s, one could in principle express all the couplings in terms
+of a single coupling g. The strongest requirement is to demand power series soluti ons to the
+REs,
+ga=/summationdisplay
+nρ(n)
+ag2n+1, (2.2)Finite Unification: Theory and Predictions 5
+which formally preserve perturbative renormalizability. Remarkably, the uniqueness of such
+power series solutions can be decided already at the one-loo p level [39, 40, 76]. To illustrate
+this, let us assume that the β-functions have the form
+βa=1
+16π2
+/summationdisplay
+b,c,d/negationslash=gβ(1)bcd
+agbgcgd+/summationdisplay
+b/negationslash=gβ(1)b
+agbg2
++···,
+βg=1
+16π2β(1)
+gg3+···,
+where···stands for higher order terms, and β(1)bcd
+a’s are symmetric in b,c,d. We then assume
+that theρ(n)
+a’s withn≤rhave been uniquely determined. To obtain ρ(r+1)
+a’s, we insert the
+power series (2.2) into the REs (2.1) and collect terms of O(g2r+3) and find
+/summationdisplay
+d/negationslash=gM(r)d
+aρ(r+1)
+d= lower order quantities ,
+where the r.h.s. is known by assumption, and
+M(r)d
+a= 3/summationdisplay
+b,c/negationslash=gβ(1)bcd
+aρ(1)
+bρ(1)
+c+β(1)d
+a−(2r+1)β(1)
+gδd
+a,
+0 =/summationdisplay
+b,c,d/negationslash=gβ(1)bcd
+aρ(1)
+bρ(1)
+cρ(1)
+d+/summationdisplay
+d/negationslash=gβ(1)d
+aρ(1)
+d−β(1)
+gρ(1)
+a.
+Therefore, the ρ(n)
+a’s for alln >1 for a given set of ρ(1)
+a’s can be uniquely determined if
+detM(n)d
+a∝ne}ationslash= 0 for alln≥0.
+As it will beclear later by examiningspecific examples,the various couplings in supersymmet-
+ric theories have easily the same asymptotic behaviour. The refore searching for a power series
+solution oftheform(2.2) totheREs(2.1)isjustified. This isnotthecaseinnon-supersymmetric
+theories, although the deeper reason for this fact is not ful ly understood.
+Thepossibilityofcouplingunificationdescribedinthiss ectioniswithoutanydoubtattractive
+because the “completely reduced” theory contains only one i ndependent coupling, but it can be
+unrealistic. Therefore, one often would like to impose fewe r RGI constraints, and this is the idea
+of partial reduction [77, 78].
+3 Reduction of dimensionful parameters
+The reduction of couplings was originally formulated for ma ssless theories on the basis of the
+Callan–Symanzik equation [39, 40, 76]. The extension to the ories with massive parameters is
+not straightforward if one wants to keep the generality and t he rigor on the same level as for the
+massless case; one has to fulfill a set of requirements comin g from the renormalization group
+equations, theCallan–Symanzikequations, etc. alongwith thenormalization conditionsimposed
+on irreducible Green’s functions [79]. See [80] for interes ting results in this direction. Here, to
+simplify the situation, we would like to assume that a mass-i ndependent renormalization scheme
+has been employed so that all the RG functions have only trivi al dependencies of dimensional
+parameters.
+To be general, we consider a renormalizable theory which con tain a set of ( N+1) dimension-
+zero couplings, {ˆg0,ˆg1,...,ˆgN}, a set ofLparameters with dimension one, {ˆh1,...,ˆhL}, and
+a set ofMparameters with dimension two, {ˆm2
+1,...,ˆm2
+M}. The renormalized irreducible vertex
+function satisfies the RG equation
+0 =DΓ/bracketleftbig
+Φ′s;ˆg0,ˆg1,...,ˆgN;ˆh1,...,ˆhL; ˆm2
+1,...,ˆm2
+M;µ/bracketrightbig
+, (3.1)6 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+D=µ∂
+∂µ+N/summationdisplay
+i=0βi∂
+∂ˆgi+L/summationdisplay
+a=1γh
+a∂
+∂ˆha+M/summationdisplay
+α=1γm2
+α∂
+∂ˆm2α+/summationdisplay
+JΦIγφI
+Jδ
+δΦJ.
+Since we assume a mass-independent renormalization scheme , theγ’s have the form
+γh
+a=L/summationdisplay
+b=1γh,b
+a(g0,...,gN)ˆhb,
+γm2
+α=M/summationdisplay
+β=1γm2,β
+α(g0,...,gN)ˆm2
+β+L/summationdisplay
+a,b=1γm2,ab
+α(g0,...,gN)ˆhaˆhb,
+whereγh,b
+a,γm2,β
+αandγm2,ab
+aare power series of the dimension-zero couplings g’s in perturbation
+theory.
+As in the massless case, we then look for conditions under whi ch the reduction of parameters,
+ˆgi= ˆgi(g), i= 1,...,N, (3.2)
+ˆha=P/summationdisplay
+b=1fb
+a(g)hb, a=P+1,...,L, (3.3)
+ˆm2
+α=Q/summationdisplay
+β=1eβ
+α(g)m2
+β+P/summationdisplay
+a,b=1kab
+α(g)hahb, α=Q+1,...,M, (3.4)
+is consistent with the RG equation (3.1), where we assume tha tg≡g0,ha≡ˆha(1≤a≤P)
+andm2
+α≡ˆm2
+α(1≤α≤Q) are independent parameters of the reduced theory. We find t hat
+the following set of equations has to be satisfied:
+βg∂ˆgi
+∂g=βi, i= 1,...,N, (3.5)
+βg∂ˆha
+∂g+P/summationdisplay
+b=1γh
+b∂ˆha
+∂hb=γh
+a, a=P+1,...,L, (3.6)
+βg∂ˆm2
+α
+∂g+P/summationdisplay
+a=1γh
+a∂ˆm2
+α
+∂ha+Q/summationdisplay
+β=1γm2
+β∂ˆm2
+α
+∂m2
+β=γm2
+α, α=Q+1,...,M. (3.7)
+Using equation (3.1) for γ’s, one finds that equations (3.5)–(3.7) reduce to
+βgdfb
+a
+dg+P/summationdisplay
+c=1fc
+a/bracketleftBigg
+γh,b
+c+L/summationdisplay
+d=P+1γh,d
+cfb
+d/bracketrightBigg
+−γh,b
+a−L/summationdisplay
+d=P+1γh,d
+afb
+d= 0, (3.8)
+a=P+1,...,L, b = 1,...,P,
+βgdeβ
+α
+dg+Q/summationdisplay
+γ=1eγ
+α
+γm2,β
+γ+M/summationdisplay
+δ=Q+1γm2,δ
+γeβ
+δ
+−γm2,β
+α−M/summationdisplay
+δ=Q+1γm2,δ
+αeβ
+δ= 0, (3.9)
+α=Q+1,...,M, β = 1,...,Q,
+βgdkab
+α
+dg+2P/summationdisplay
+c=1/parenleftBigg
+γh,a
+c+L/summationdisplay
+d=P+1γh,d
+cfa
+d/parenrightBigg
+kcb
+α+Q/summationdisplay
+β=1eβ
+α
+γm2,ab
+β+L/summationdisplay
+c,d=P+1γm2,cd
+βfa
+cfb
+d
++2L/summationdisplay
+c=P+1γm2,cb
+βfa
+c+M/summationdisplay
+δ=Q+1γm2,δ
+βkab
+δ
+−
+γm2,ab
+α+L/summationdisplay
+c,d=P+1γm2,cd
+αfa
+cfb
+dFinite Unification: Theory and Predictions 7
++2L/summationdisplay
+c=P+1γm2,cb
+αfa
+c+M/summationdisplay
+δ=Q+1γm2,δ
+αkab
+δ
+= 0, (3.10)
+α=Q+1,...,M, a,b = 1,...,P.
+If these equations are satisfied, the irreducible vertex fu nction of the reduced theory
+ΓR[Φ′s;g;h1,...,h P;m2
+1,...,ˆm2
+Q;µ]
+≡Γ[Φ′s;g,ˆg1(g),...,ˆgN(g);h1,...,h P,ˆhP+1(g,h),...,ˆhL(g,h);
+m2
+1,...,ˆm2
+Q,ˆm2
+Q+1(g,h,m2),...,ˆm2
+M(g,h,m2);µ]
+has the same renormalization group flow as the original one.
+The requirement for the reduced theory to be perturbative re normalizable means that the
+functions ˆgi,fb
+a,eβ
+αandkab
+α, defined in equations (3.2)–(3.4), should have a power seri es expan-
+sion in the primary coupling g:
+ˆgi=g∞/summationdisplay
+n=0ρ(n)
+ign, fb
+a=g∞/summationdisplay
+n=0ηb(n)
+agn,
+eβ
+α=∞/summationdisplay
+n=0ξβ(n)
+αgn, kab
+α=∞/summationdisplay
+n=0χab(n)
+αgn.
+Toobtain theexpansioncoefficients, weinsertthepowerse riesansatzaboveintoequations (3.5),
+(3.8)–(3.10) and require that the equations are satisfied a t each order in g. Note that the
+existence of a unique power series solution is a non-trivial matter: It depends on the theory as
+well as on the choice of the set of independent parameters.
+4 Finiteness in N= 1 supersymmetric gauge theories
+Let us consider a chiral, anomaly free, N= 1 globally supersymmetric gauge theory based on
+a groupGwith gauge coupling constant g. The superpotential of the theory is given by
+W=1
+2mijφiφj+1
+6Cijkφiφjφk, (4.1)
+wheremijandCijkare gauge invariant tensors and the matter field φitransforms according to
+theirreduciblerepresentation Riofthegaugegroup G. Therenormalization constantsassociated
+with the superpotential (4.1), assuming that supersymmetr y is preserved, are
+φ0
+i= (Zj
+i)(1/2)φj,
+m0
+ij=Zi′j′
+ijmi′j′,
+C0
+ijk=Zi′j′k′
+ijkCi′j′k′.
+TheN= 1 non-renormalization theorem [81, 82, 62] ensures that th ere are no mass and cubic-
+interaction-term infinities and therefore
+Zi′j′k′
+ijkZ1/2i′′
+i′Z1/2j′′
+j′Z1/2k′′
+k′=δi′′
+(iδj′′
+jδk′′
+k),
+Zi′j′
+ijZ1/2i′′
+i′Z1/2j′′
+j′=δi′′
+(iδj′′
+j).8 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+As a result the only surviving possible infinities are the wa ve-function renormalization con-
+stantsZj
+i, i.e., one infinity for each field. The one-loop β-function of the gauge coupling gis
+given by [83]
+β(1)
+g=dg
+dt=g3
+16π2/bracketleftBigg/summationdisplay
+il(Ri)−3C2(G)/bracketrightBigg
+, (4.2)
+wherel(Ri) is the Dynkin index of RiandC2(G) is the quadratic Casimir of the adjoint repre-
+sentation of the gauge group G. Theβ-functions of Cijk, by virtue of the non-renormalization
+theorem, are related to the anomalous dimension matrix γijof the matter fields φias:
+βijk=dCijk
+dt=Cijlγl
+k+Ciklγl
+j+Cjklγl
+i. (4.3)
+At one-loop level γijis [83]
+γi(1)
+j=1
+32π2/bracketleftbig
+CiklCjkl−2g2C2(Ri)δ1
+j/bracketrightbig
+, (4.4)
+whereC2(Ri) is the quadratic Casimir of the representation Ri, andCijk=C∗
+ijk. Since di-
+mensional coupling parameters such as masses and couplings of cubic scalar field terms do not
+influence the asymptotic properties of a theory on which we a re interested here, it is sufficient
+to take into account only the dimensionless supersymmetric couplings such as gandCijk. So
+we neglect the existence of dimensional parameters, and ass ume furthermore that Cijkare real
+so thatC2
+ijkalways are positive numbers.
+As one can see from equations (4.2) and (4.4), all the one-loo pβ-functions of the theory
+vanish ifβ(1)
+gandγ(1)
+ijvanish, i.e.
+/summationdisplay
+iℓ(Ri) = 3C2(G), (4.5)
+CiklCjkl= 2δi
+jg2C2(Ri). (4.6)
+The conditions for finiteness for N= 1 field theories with SU(N) gauge symmetry are dis-
+cussed in [84], and the analysis of the anomaly-free and no-c harge renormalization requirements
+for these theories can be found in [85]. A very interesting re sult is that the conditions (4.5),
+(4.6) are necessary and sufficient for finiteness at the two -loop level [83, 86, 87, 88, 89].
+In case supersymmetry is broken by soft terms, the requireme nt of finiteness in the one-loop
+soft breaking terms imposes further constraints among them selves [65]. In addition, the same
+set of conditions that are sufficient for one-loop finitene ss of the soft breaking terms render the
+soft sector of the theory two-loop finite [65].
+The one- and two-loop finiteness conditions (4.5), (4.6) re strict considerably the possible
+choices of the irreps. Rifor a given group Gas well as the Yukawa couplings in the superpo-
+tential (4.1). Note in particular that the finiteness condi tions cannot be applied to the minimal
+supersymmetric standard model (MSSM), since the presence o f aU(1) gauge group is incom-
+patible with the condition (4.5), due to C2[U(1)] = 0. This naturally leads to the expectation
+that finiteness should be attained at the grand unified leve l only, the MSSM being just the
+corresponding, low-energy, effective theory.
+Another important consequence of one- and two-loop finiten ess is that supersymmetry (most
+probably) can only be broken due to the soft breaking terms. I ndeed, due to the unacceptability
+of gauge singlets, F-type spontaneous symmetry breaking [9 0] terms are incompatible with
+finiteness, as well as D-type [91] spontaneous breaking whi ch requires the existence of a U(1)
+gauge group.Finite Unification: Theory and Predictions 9
+A natural question to ask is what happens at higher loop order s. The answer is contained
+in a theorem [44, 43] which states the necessary and sufficie nt conditions to achieve finiteness
+at all orders. Before we discuss the theorem let us make some i ntroductory remarks. The
+finiteness conditions impose relations between gauge and Y ukawa couplings. To require such
+relations which render the couplings mutually dependent at a given renormalization point is
+trivial. What is not trivial is to guarantee that relations l eading to a reduction of the couplings
+hold at any renormalization point. As we have seen, the neces sary and also sufficient, condition
+for this to happen is to require that such relations are solut ions to the REs
+βgdCijk
+dg=βijk (4.7)
+and hold at all orders. Remarkably, the existence of all-ord er power series solutions to (4.7) can
+be decided at one-loop level, as already mentioned.
+Let us now turn to the all-order finiteness theorem [44, 43], which states that if a N= 1 su-
+persymmetric gauge theory can become finite to all orders in the sense of vanishing β-functions,
+that is of physical scale invariance. It is based on (a) the st ructure of the supercurrent in N= 1
+supersymmetric gauge theory [92, 93, 94], and on (b) the non- renormalization properties of
+N= 1 chiral anomalies [44, 43, 95, 96, 97]. Details on the proof can be found in [44, 43] and
+further discussion in [95, 96, 97, 45, 98]. Here, following m ostly [98] we present a comprehensible
+sketch of the proof.
+Consider a N= 1 supersymmetric gauge theory, with simple Lie group G. The content of
+this theory is given at the classical level by the matter supe rmultiplets Si, which contain a scalar
+fieldφiand a Weyl spinor ψia, and the vector supermultiplet Va, which contains a gauge vector
+fieldAa
+µand a gaugino Weyl spinor λa
+α.
+Let us first recall certain facts about the theory:
+(1) A massless N= 1 supersymmetric theory is invariant under a U(1) chiral transforma-
+tionRunder which the various fields transform as follows
+A′
+µ=Aµ, λ′
+α= exp(−iθ)λα,
+φ′= exp/parenleftbigg
+−i2
+3θ/parenrightbigg
+φ, ψ′
+α= exp/parenleftbigg
+−i1
+3θ/parenrightbigg
+ψα, ....
+The corresponding axial Noether current Jµ
+R(x) is
+Jµ
+R(x) =¯λγµγ5λ+··· (4.8)
+is conserved classically, while in the quantum case is viola ted by the axial anomaly
+∂µJµ
+R=r(ǫµνσρFµνFσρ+···). (4.9)
+From its known topological origin in ordinary gauge theorie s [99, 100, 101], one would expect
+that the axial vector current Jµ
+Rto satisfy the Adler–Bardeen theorem and receive correctio ns
+only at the one-loop level. Indeed it has been shown that the s ame non-renormalization theorem
+holds also in supersymmetric theories [95, 96, 97]. Therefo re
+r=/planckover2pi1β(1)
+g. (4.10)
+(2) The massless theory we consider is scale invariant at the classical level and, in general,
+there is a scale anomaly due to radiative corrections. The sc ale anomaly appears in the trace of
+the energy momentum tensor Tµν, which is traceless classically. It has the form
+Tµ
+µ=βgFµνFµν+···. (4.11)10 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+(3) Massless, N= 1 supersymmetric gauge theories are classically invarian t under the su-
+persymmetric extension of the conformal group – the superco nformal group. Examining the
+superconformal algebra, it can be seen that the subset of sup erconformal transformations con-
+sisting of translations, supersymmetry transformations, and axialRtransformations is closed
+under supersymmetry, i.e. these transformations form a rep resentation of supersymmetry. It
+follows that the conserved currents corresponding to these transformations make up a super-
+multiplet represented by an axial vector superfield called supercurrent J,
+J≡ {J′µ
+R,Qµ
+α,Tµ
+ν,...}, (4.12)
+whereJ′µ
+Ris the current associated to Rinvariance, Qµ
+αis the one associated to supersymmetry
+invariance, and Tµ
+νthe one associated to translational invariance (energy-mo mentum tensor).
+The anomalies of the RcurrentJ′µ
+R, the trace anomalies of the supersymmetry current, and
+the energy-momentum tensor, form also a second supermultip let, called the supertrace anomaly
+S={ReS,ImS,Sα}=/braceleftbig
+Tµ
+µ, ∂µJ′µ
+R, σµ
+α˙β¯Q˙β
+µ+···/bracerightbig
+,
+whereTµ
+µin equation (4.11) and
+∂µJ′µ
+R=βgǫµνσρFµνFσρ+···, σµ
+α˙β¯Q˙β
+µ=βgλβσµν
+αβFµν+···.
+(4) It is very important to note that the Noether current defi ned in (4.8) is not the same as
+the current associated to Rinvariance that appears in the supercurrent Jin (4.12), but they
+coincide in the tree approximation. So starting from a uniqu e classical Noether current Jµ
+R(class),
+the Noether current Jµ
+Ris defined as the quantum extension of Jµ
+R(class)which allows for the
+validity of the non-renormalization theorem. On the other h andJ′µ
+R, is defined to belong to the
+supercurrent J, together with the energy-momentum tensor. The two require ments cannot be
+fulfilled by a single current operator at the same time.
+Although the Noether current Jµ
+Rwhich obeys (4.9) and the current J′µ
+Rbelonging to the su-
+percurrent multiplet Jare not the same, there is a relation [44, 43] between quantit ies associated
+with them
+r=βg(1+xg)+βijkxijk−γArA, (4.13)
+whererwas given in equation (4.10). The rAare the non-renormalized coefficients of the
+anomalies of the Noether currents associated to the chiral i nvariances of the superpotential,
+and – liker– are strictly one-loop quantities. The γA’s are linear combinations of the anoma-
+lous dimensions of the matter fields, and xg, andxijkare radiative correction quantities. The
+structure of equality (4.13) is independent of the renormal ization scheme.
+One-loop finiteness, i.e. vanishing of the β-functions at one-loop, implies that the Yukawa
+couplingsλijkmust be functions of the gauge coupling g. To find a similar condition to all
+orders it is necessary and sufficient for the Yukawa couplin gs to be a formal power series in g,
+which is solution of the REs (4.7).
+We can now state the theorem for all-order vanishing β-functions.
+Theorem 1. Consider an N= 1supersymmetric Yang–Mills theory, with simple gauge group .
+If the following conditions are satisfied
+1)there is no gauge anomaly;
+2)the gaugeβ-function vanishes at one-loop
+β(1)
+g= 0 =/summationdisplay
+il(Ri)−3C2(G);Finite Unification: Theory and Predictions 11
+3)There exist solutions of the form
+Cijk=ρijkg, ρ ijk∈C (4.14)
+to the conditions of vanishing one-loop matter fields anomal ous dimensions
+γi(1)
+j= 0
+=1
+32π2/bracketleftbig
+CiklCjkl−2g2C2(Ri)δij/bracketrightbig
+;
+4)these solutions are isolated and non-degenerate when consi dered as solutions of vanishing
+one-loop Yukawa β-functions:
+βijk= 0.
+Then, each of the solutions (4.14)can be uniquely extended to a formal power series in g, and the
+associated super Yang–Mills models depend on the single cou pling constant gwith aβ-function
+which vanishes at all-orders.
+Itisimportant tonoteafewthings: Therequirementof isola ted andnon-degeneratesolutions
+guarantees the existence of a unique formal power series sol ution to the reduction equations.
+The vanishing of the gauge β-function at one-loop, β(1)
+g, is equivalent to the vanishing of the
+Rcurrent anomaly (4.9). The vanishing of the anomalous dimen sions at one-loop implies the
+vanishing of the Yukawa couplings β-functions at that order. It also implies the vanishing of th e
+chiral anomaly coefficients rA. Thislast property is a necessary condition for having β-functions
+vanishing at all orders1.
+Proof.Insertβijkas given by the REs into the relationship (4.13) between the a xial anoma-
+lies coefficients and the β-functions. Since these chiral anomalies vanish, we get for βgan
+homogeneous equation of the form
+0 =βg(1+O(/planckover2pi1)).
+The solution of this equation in the sense of a formal power se ries in/planckover2pi1isβg= 0, order by order.
+Therefore, due to the REs (4.7), βijk= 0 too. /squaresolid
+Thus we see that finiteness and reduction of couplings are in timately related. Since an
+equation like equation (4.13) is lacking in non-supersymme tric theories, one cannot extend the
+validity of a similar theorem in such theories.
+5 Sum rule for SB terms in N= 1 supersymmetric
+and finite theories: all-loop results
+The method of reducing the dimensionless couplings has been extended [38], to the soft su-
+persymmetry breaking (SSB) dimensionful parameters of N= 1 supersymmetric theories. In
+addition it was found [71] that RGI SSB scalar masses in gauge -Yukawa unified models satisfy
+a universal sum rule. Here we will describe first how the use o f the available two-loop RG
+functions and the requirement of finiteness of the SSB param eters up to this order leads to the
+soft scalar-mass sum rule [72].
+1There is an alternative way to find finite theories [102].12 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+Consider the superpotential given by (4.1) along with the La grangian for SSB terms
+−LSB=1
+6hijkφiφjφk+1
+2bijφiφj+1
+2/parenleftbig
+m2/parenrightbigj
+iφ∗iφj+1
+2Mλλ+h.c.,
+where theφiare the scalar parts of the chiral superfields Φ i,λare the gauginos and Mtheir
+unified mass. Since we would like to consider only finite the ories here, we assume that the gauge
+group is a simple group and the one-loop β-function of the gauge coupling gvanishes. We also
+assume that the reduction equations admit power series solu tions of the form
+Cijk=g/summationdisplay
+nρijk
+(n)g2n.
+According to the finiteness theorem of [43, 44], the theory i s then finite to all orders in pertur-
+bation theory, if, among others, the one-loop anomalous dim ensionsγj(1)
+ivanish. The one- and
+two-loop finiteness for hijkcan be achieved by [66]
+hijk=−MCijk+···=−Mρijk
+(0)g+O/parenleftbig
+g5/parenrightbig
+, (5.1)
+where···stand for higher order terms.
+Now, to obtain the two-loop sum rule for soft scalar masses, w e assume that the lowest order
+coefficients ρijk
+(0)and also (m2)i
+jsatisfy the diagonality relations
+ρipq(0)ρjpq
+(0)∝δj
+ifor allpandqand/parenleftbig
+m2/parenrightbigi
+j=m2
+jδi
+j,
+respectively. Then we find the following soft scalar-mass s um rule [72, 21, 103]
+/parenleftbig
+m2
+i+m2
+j+m2
+k/parenrightbig
+/MM†= 1+g2
+16π2∆(2)+O/parenleftbig
+g4/parenrightbig
+(5.2)
+fori,j,kwithρijk
+(0)∝ne}ationslash= 0, where ∆(2)is the two-loop correction
+∆(2)=−2/summationdisplay
+l/bracketleftbig/parenleftbig
+m2
+l/MM†/parenrightbig
+−(1/3)/bracketrightbig
+T(Rl),
+which vanishes for the universal choice in accordance with t he previous findings [66].
+If we know higher-loop β-functions explicitly, we can follow the same procedure and find
+higher-loop RGI relations among SSB terms. However, the β-functions of the soft scalar masses
+are explicitly known only up to two loops. In order to obtain h igher-loop results some relations
+amongβ-functions are needed.
+Making use of the spurion technique [60, 61, 62, 63, 64], it is possible to find the following
+all-loop relations among SSB β-functions, [52, 53, 54, 55, 56, 57]
+βM= 2O/parenleftbiggβg
+g/parenrightbigg
+,
+βijk
+h=γilhljk+γjlhilk+γklhijl−2γi
+1lCljk−2γj
+1lCilk−2γk
+1lCijl,
+(βm2)ij=/bracketleftbigg
+∆+X∂
+∂g/bracketrightbigg
+γij,
+O=/parenleftbigg
+Mg2∂
+∂g2−hlmn∂
+∂Clmn/parenrightbigg
+,
+∆ = 2OO∗+2|M|2g2∂
+∂g2+˜Clmn∂
+∂Clmn+˜Clmn∂
+∂Clmn,Finite Unification: Theory and Predictions 13
+where (γ1)ij=Oγij,Clmn= (Clmn)∗, and
+˜Cijk=/parenleftbig
+m2/parenrightbigi
+lCljk+/parenleftbig
+m2/parenrightbigj
+lCilk+/parenleftbig
+m2/parenrightbigk
+lCijl.
+It was also found [53] that the relation
+hijk=−M(Cijk)′≡ −MdCijk(g)
+dlng,
+among couplings is all-loop RGI. Furthermore, using the all -loop gauge β-function of Novikov
+et al. [73, 74, 75] given by
+βNSVZ
+g=g3
+16π2/bracketleftbigg/summationtext
+lT(Rl)(1−γl/2)−3C(G)
+1−g2C(G)/8π2/bracketrightbigg
+,
+it was found the all-loop RGI sum rule [58],
+m2
+i+m2
+j+m2
+k=|M|2/braceleftbigg1
+1−g2C(G)/(8π2)dlnCijk
+dlng+1
+2d2lnCijk
+d(lng)2/bracerightbigg
++/summationdisplay
+lm2
+lT(Rl)
+C(G)−8π2/g2dlnCijk
+dlng. (5.3)
+In addition the exact- β-function for m2in the NSVZ scheme has been obtained [58] for the first
+time and is given by
+βNSVZ
+m2
+i=/bracketleftbigg
+|M|2/braceleftbigg1
+1−g2C(G)/(8π2)d
+dlng+1
+2d2
+d(lng)2/bracerightbigg
++/summationdisplay
+lm2
+lT(Rl)
+C(G)−8π2/g2d
+dlng/bracketrightBigg
+γNSVZ
+i.
+Surprisingly enough, the all-loop result (5.3) coincides w ith the superstring result for the finite
+case in a certain class of orbifold models [72] if dlnCijk/dlng= 1.
+6 Finite SU(5) Unified Theories
+Finite Unified Theories (FUTs) have always attracted inter est for their intriguing mathematical
+properties and their predictive power. One very important r esult is that the one-loop finiteness
+conditions (4.3), (4.4) are sufficient to guarantee two-lo op finiteness [83]. A classification of
+possible one-loop finite models was done by two groups [104, 105, 106]. The first one and
+two-loop finite SU(5) model was presented in [107], and shortly afterwards the conditions for
+finiteness in the soft SUSY-breaking sector at one-loop [88 ] were given. In [108] a one and two-
+loop finiteSU(5) model was presented where the rotation of the Higgs secto r was proposed as a
+way of making it realistic. The first all-loop finite theory was studied in [33, 34], without taking
+into account the soft breaking terms. Finite soft breaking t erms and the proof that one-loop
+finiteness in the soft terms also implies two-loop finitene ss was done in [66]. The inclusion of
+soft breaking terms in a realistic model was done in [109] and their finiteness to all-loops studied
+in [56], although the universality of the soft breaking term s lead to a charged LSP. This fact was
+also noticed in [110], where the inclusion of an extra parame ter in the boundary condition of the
+Higgs mixing mass parameter was introduced to alleviate it. The derivation of the sum-rule in
+the soft supersymmetry breaking sector and the proof that it can be made all-loop finite were
+done in [72] and [58] respectively, allowing thus for the con struction of all-loop finite realistic
+models.14 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+From the classification of theories with vanishing one-loo p gaugeβ-function [104], one can
+easily see that there exist only two candidate possibilitie s to construct SU(5) GUTs with three
+generations. These possibilities require that the theory s hould contain as matter fields the chiral
+supermultiplets 5,5,10,5,24with the multiplicities (6 ,9,4,1,0) and (4,7,3,0,1), respectively.
+Only the second one contains a 24-plet which can be used to provide the spontaneous symmetry
+breaking(SB)of SU(5) downto SU(3)×SU(2)×U(1). Forthefirstmodelonehastoincorporate
+another way, such as the Wilson flux breaking mechanism to ac hieve the desired SB of SU(5)
+[33, 34]. Therefore, for a self-consistent field theory dis cussion we would like to concentrate only
+on the second possibility.
+The particle content of the models we will study consists of t he following supermultiplets:
+three (5+10), needed for each of the three generations of quarks and lept ons, four ( 5+5) and
+one24considered as Higgs supermultiplets. When the gauge group o f the finite GUT is broken
+the theory is no longer finite, and we will assume that we are l eft with the MSSM.
+Therefore, a predictive gauge-Yukawa unified SU(5) model which is finite to all orders, in
+addition to the requirements mentioned already, should als o have the following properties:
+1. One-loop anomalous dimensions are diagonal, i.e., γ(1)j
+i∝δj
+i.
+2. The three fermion generations, in the irreducible repres entations 5i,10i(i= 1,2,3),
+should not couple to the adjoint 24.
+3. The two Higgs doublets of the MSSM should mostly be made out of a pair of Higgs quintet
+and anti-quintet, which couple to the third generation.
+In the following we discuss two versions of the all-order fin ite model. The model [33, 34],
+which will be labeled A, and a slight variation of this model (labeled B), which can also be
+obtained from the class of the models suggested in [54, 55] wi th a modification to suppress
+non-diagonal anomalous dimensions [72].
+The superpotential which describes the two models before th e reduction of couplings takes
+places is of the form [33, 34, 72, 107, 108]
+W=3/summationdisplay
+i=1/bracketleftbigg1
+2gu
+i10i10iHi+gd
+i10i5iHi/bracketrightbigg
++gu
+23102103H4
++gd
+2310253H4+gd
+3210352H4+4/summationdisplay
+a=1gf
+aHa24Ha+gλ
+3(24)3, (6.1)
+whereHaandHa(a= 1,...,4) stand for the Higgs quintets and anti-quintets.
+The main difference between model Aand model Bis that two pairs of Higgs quintets and
+anti-quintets couple to the 24inB, so that it is not necessary to mix them with H4andH4in
+order to achieve the triplet-doublet splitting after the sy mmetry breaking of SU(5) [72]. Thus,
+although the particle content is the same, the solutions to e quations (4.3), (4.4) and the sum
+rules are different, which will reflect in the phenomenolog y, as we will see.
+6.1 FUTA
+After the reduction of couplings the symmetry of the superpo tentialW(6.1) is enhanced. For
+modelAone finds that the superpotential has the Z7×Z3×Z2discrete symmetry with the
+charge assignment as shown in Table 1, and with the following superpotential
+WA=3/summationdisplay
+i=1/bracketleftbigg1
+2gu
+i10i10iHi+gd
+i10i5iHi/bracketrightbigg
++gf
+4H424H4+gλ
+3(24)3.Finite Unification: Theory and Predictions 15
+Table 1. Charges of the Z7×Z3×Z2symmetry for model FUTA.
+515253101102103H1H2H3H4H1H2H3H424
+Z7412124536−5−3−6000
+Z3000120120−1−20000
+Z2111111000000000
+Table 2. Charges of the Z4×Z4×Z4symmetry for model FUTB.
+515253101102103H1H2H3H4H1H2H3H424
+Z41001002000−20000
+Z401001002030−20−30
+Z4001001002300−2−30
+The non-degenerate and isolated solutions to γ(1)
+i= 0 for model FUTA, which are the
+boundary conditions for the Yukawa couplings at the GUT scal e, are:
+(gu
+1)2=8
+5g2,/parenleftbig
+gd
+1/parenrightbig2=6
+5g2,(gu
+2)2= (gu
+3)2=8
+5g2,
+/parenleftbig
+gd
+2/parenrightbig2=/parenleftbig
+gd
+3/parenrightbig2=6
+5g2,(gu
+23)2= 0,/parenleftbig
+gd
+23/parenrightbig2=/parenleftbig
+gd
+32/parenrightbig2= 0,
+/parenleftbig
+gλ/parenrightbig2=15
+7g2,/parenleftbig
+gf
+2/parenrightbig2=/parenleftbig
+gf
+3/parenrightbig2= 0,/parenleftbig
+gf
+1/parenrightbig2= 0,/parenleftbig
+gf
+4/parenrightbig2=g2. (6.2)
+In the dimensionful sector, the sum rule gives us the followi ng boundary conditions at the GUT
+scale for this model [72]:
+m2
+Hu+2m2
+10=m2
+Hd+m2
+5+m2
+10=M2,
+and thus we are left with only three free parameters, namely m5≡m53,m10≡m103andM.
+6.2 FUTB
+Also in the case of FUTBthe symmetry is enhanced after the reduction of couplings. T he
+superpotential has now a Z4×Z4×Z4symmetry with charges as shown in Table 2 and with
+the following superpotential
+WB=3/summationdisplay
+i=1/bracketleftbigg1
+2gu
+i10i10iHi+gd
+i10i5iHi/bracketrightbigg
++gu
+23102103H4
++gd
+2310253H4+gd
+3210352H4+gf
+2H224H2+gf
+3H324H3+gλ
+3(24)3.
+For this model the non-degenerate and isolated solutions to γ(1)
+i= 0 give us:
+(gu
+1)2=8
+5g2,/parenleftbig
+gd
+1/parenrightbig2=6
+5g2,(gu
+2)2= (gu
+3)2=4
+5g2,
+/parenleftbig
+gd
+2/parenrightbig2=/parenleftbig
+gd
+3/parenrightbig2=3
+5g2,(gu
+23)2=4
+5g2,/parenleftbig
+gd
+23/parenrightbig2=/parenleftbig
+gd
+32/parenrightbig2=3
+5g2,
+/parenleftbig
+gλ/parenrightbig2=15
+7g2,/parenleftbig
+gf
+2/parenrightbig2=/parenleftbig
+gf
+3/parenrightbig2=1
+2g2,/parenleftbig
+gf
+1/parenrightbig2= 0,/parenleftbig
+gf
+4/parenrightbig2= 0, (6.3)16 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+and from the sum rule we obtain [72]:
+m2
+Hu+2m2
+10=M2, m2
+Hd−2m2
+10=−M2
+3, m2
+5+3m2
+10=4M2
+3,
+i.e., in this case we have only two free parameters m10≡m103andMfor the dimensionful
+sector.
+As already mentioned, after the SU(5) gauge symmetry breaking we assume we have the
+MSSM, i.e. only two Higgs doublets. This can be achieved by in troducing appropriate mass
+terms that allow to perform a rotation of the Higgs sector [10 8, 33, 34, 111, 107], in such
+a way that only one pair of Higgs doublets, coupled mostly to t he third family, remains light
+and acquire vacuum expectation values. To avoid fast proton decay the usual fine tuning to
+achieve doublet-triplet splitting is performed. Notice th at, although similar, the mechanism is
+not identical to minimal SU(5), since we have an extended Higgs sector.
+Thus, after the gauge symmetry of the GUT theory is broken we a re left with the MSSM,
+with the boundary conditions for the third family given by th e finiteness conditions, while the
+other two families are basically decoupled.
+We will now examine the phenomenology of such all-loop Finit e Unified Theories with SU(5)
+gauge group and, for the reasons expressed above, we will con centrate only on the third genera-
+tion of quarks and leptons. An extension to three families, a nd the generation of quark mixing
+angles andmasses inFinite Unified Theories hasbeen addres sedin[112], whereseveral examples
+are given. These extensions are not considered here.
+6.3 Restrictions from low-energy observables
+Since the gauge symmetry is spontaneously broken below MGUT, the finiteness conditions do
+not restrict the renormalization properties at low energie s, and all it remains are boundary
+conditions on the gauge and Yukawa couplings (6.2) or (6.3), theh=−MCrelation (5.1), and
+the soft scalar-mass sum rule (5.2) at MGUT, as applied in the two models. Thus we examine
+the evolution of these parameters according to their RGEs up to two-loops for dimensionless
+parameters and at one-loop for dimensionful ones with the re levant boundary conditions. Below
+MGUTtheir evolution is assumed to be governed by the MSSM. We furt her assume a unique
+supersymmetry breaking scale MSUSY(which we define as the geometrical average of the stop
+masses)and thereforebelow that scale theeffective theory is justtheSM. Thisallows to evaluate
+observables at or below the electroweak scale.
+In the following, we briefly describe the low-energy observ ables used in our analysis. We
+discuss the current precision of the experimental results a nd the theoretical predictions. We also
+give relevant details of the higher-order perturbative cor rections that we include. We do not
+discuss theoretical uncertainties from the RG running betw een the high-scale parameters and
+the weak scale. At present, these uncertainties are expecte d to be less important than the
+experimental and theoretical uncertainties of the precisi on observables.
+As precision observables we first discuss the 3rd generatio n quark masses that are leading
+to the strongest constraints on the models under investigat ion. Next we apply Bphysics and
+Higgs-boson mass constraints. We also briefly discuss the a nomalous magnetic moment of the
+muon.
+6.4 Predictions
+We now present the comparison of the predictions of the four m odels with the experimental
+data, see [113] for more details, starting with the heavy qua rk masses. In Fig. 1 we show the
+FUTAandFUTBpredictions for the top pole mass, Mtop, and the running bottom mass at
+the scaleMZ,mbot(MZ), as a function of the unified gaugino mass M, for the two cases µ<0Finite Unification: Theory and Predictions 17
+0 1000 2000 3000 4000 5000 6000 7000
+M [GeV]0123456
+2.83mbot(MZ)µ > 0
+µ < 0FUTA
+FUTB
+FUTAFUTB
+0 1000 2000 3000 4000 5000 6000 7000
+M [GeV]170175180185190
+173.1
+171.8174.4Mtop [GeV]FUTA
+FUTBµ < 0µ > 0
+µ < 0
+µ > 0
+Figure 1. The bottom quark mass at the Zboson scale (upper) and top quark pole mass (lower) are
+shown as function of Mfor both models.
+andµ >0. The running bottom mass is used to avoid the large QCD uncer tainties inherent
+for the pole mass. In the evaluation of the bottom mass mbot, we have included the corrections
+coming from bottom squark-gluino loops and top squark-char gino loops [114]. We compare
+the predictions for the running bottom quark mass with the ex perimental value, mb(MZ) =
+2.83±0.10 GeV [115]. One can see that the value of mbotdepends strongly on the sign of µ
+due to the above mentioned radiative corrections involving SUSY particles. For both models A
+andBthe values for µ >0 are above the central experimental value, with mbot(MZ)∼4.0–
+5.0GeV. For µ<0, ontheotherhand, model Bshowsoverlap withtheexperimentally measured
+values,mbot(MZ)∼2.5–2.8 GeV. For model Awe findmbot(MZ)∼1.5–2.6 GeV, and there is
+only a small region of allowed parameter space at large Mwhere we find agreement with the
+experimental value at thetwo σlevel. Insummary, theexperimental determination of mbot(MZ)
+clearly selects the negative sign of µ.
+Now we turn to the top quark mass. The predictions for the top q uark mass Mtopare
+∼183 and ∼172 GeV in the models AandBrespectively, as shown in the lower plot of
+Fig. 1. Comparing these predictions with the most recent exp erimental value mexp
+t= (173.1±
+1.3) GeV [31], and recalling that the theoretical values for Mtopmay suffer from a correction of
+∼4% [20, 116, 103], we see that clearly model Bis singled out. In addition the value of tan βis18 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+0 2000 4000 6000 8000
+M [GeV]120122124126128130Mh [GeV]
+Figure 2. The lightest Higgs mass, Mh, as function of Mfor the model FUTBwithµ<0, see text.
+found to be tan β∼54 and∼48 for models AandB, respectively. Thus from the comparison
+of the predictions of the two models with experimental data o nlyFUTBwithµ<0 survives.
+We now analyze the impact of further low-energy observables on the model FUTBwith
+µ <0. As additional constraints we consider the following obse rvables: the rare bdecays
+BR(b→sγ) and BR(Bs→µ+µ−), the lightest Higgs boson mass as well as the density of
+cold dark matter in the Universe, assuming it consists mainl y of neutralinos. More details and
+a complete set of references can be found in [113].
+For the branching ratio BR( b→sγ), we take the experimental value estimated by the Heavy
+Flavour Averaging Group (HFAG) is [117, 118, 119]
+BR(b→sγ) =/parenleftbig
+3.55±0.24+0.09
+−0.10±0.03/parenrightbig
+·10−4,
+where the first error is the combined statistical and uncorr elated systematic uncertainty, the
+latter two errors are correlated systematic theoretical un certainties and corrections respectively.
+For the branching ratio BR( Bs→µ+µ−), the SM prediction is at the level of 10−9, while
+the present experimental upper limit from the Tevatron is 4 .7·10−8at the 95% C.L. [120], still
+providing the possibility for the MSSM to dominate the SM con tribution.
+Concerning the lightest Higgs boson mass, Mh, the SM bound of 114 .4 GeV [27, 32] can be
+applied, since the main SM search channels are not suppresse d in FUTB. For the prediction we
+use the code FeynHiggs [121, 122, 123, 124].
+The prediction of the lightest Higgs boson mass as a function ofMis shown in Fig. 2. The
+light (green) points shown are in agreement with the two B-physics observables listed above.
+The lightest Higgs mass ranges in
+Mh∼121−126 GeV, (6.4)
+where the uncertainty comes from variations of the soft scal ar masses, and from finite (i.e. not
+logarithmically divergent) corrections in changing renor malization scheme. To this value one
+has to add ±3 GeV coming from unknown higher order corrections [123]. We have also included
+a small variation, due to threshold corrections at the GUT sc ale, of up to 5% of the FUT
+boundary conditions. Thus, taking into account the Bphysics constraints results naturally in
+a light Higgs boson that fulfills the LEP bounds [27, 32].
+In the same way the whole SUSY particle spectrum can be derive d. The resulting SUSY
+masses for FUTBwithµ <0 are rather large. The lightest SUSY particle starts aroundFinite Unification: Theory and Predictions 19
+500 GeV, with the rest of the spectrum being very heavy. The ob servation of SUSY particles
+at the LHC or the ILC will only be possible in very favorable pa rts of the parameter space. For
+most parameter combination only a SM-like light Higgs boson in the range of equation (6.4) can
+be observed.
+We note that with such a heavy SUSY spectrum the anomalous mag netic moment of the
+muon, (g−2)µ(withaµ≡(g−2)µ/2), gives only a negligible correction to the SM prediction.
+The comparison of the experimental result and the SM value (b ased on the latest combination
+usinge+e−data) [125]
+aexp
+µ−atheo
+µ= (24.6±8.0)·10−10
+would disfavor FUTBwithµ <0 by about 3 σ. However, since the SM is not regarded as
+excluded by ( g−2)µ, we still see FUTBwithµ<0 as the only surviving model.
+Further restrictions on the parameter space can arise from t he requirement that the lightest
+SUSY particle (LSP) should give the right amount of cold dark matter (CDM) abundance. The
+LSP should be color neutral, and the lightest neutralino app ears to be a suitable candidate [126,
+127]. In the case where all the soft scalar masses are univers al at the unfication scale, there is
+no region of MbelowO(few TeV) in which m˜τ> mχ0is satisfied, where m˜τis the lightest ˜ τ
+mass, andmχ0the lightest neutralino mass. An electrically charged LSP, however, is not in
+agreement with CDM searches. But once the universality cond ition is relaxed this problem can
+be solved naturally, thanks to the sum rule (5.2). Using this equation a comfortable parameter
+space is found for FUTBwithµ <0 (and also for FUTAand both signs of µ). that fulfills
+the conditions of (a) successful radiative electroweak sym metry breaking, (b) m˜τ>mχ0.
+Calculating the CDM abundance in these FUT models one finds t hat usually it is very large,
+thus a mechanism is needed in our model to reduce it. This issu e could, for instance, be related
+to another problem, that of neutrino masses. This type of mas ses cannot be generated naturally
+within the class of finite unified theories that we are consi dering in this paper, although a non-
+zero value for neutrino masses has clearly been established [115]. However, the class of FUTs
+discussed here can, in principle, be easily extended by intr oducing bilinear R-parity violating
+terms that preserve finiteness and introduce the desired ne utrino masses [128]. R-parity vio-
+lation [129, 130, 131, 132] would have a small impact on the co llider phenomenology discussed
+here (apart from fact the SUSY search strategies could not re ly on a “missing energy” signa-
+ture), but remove the CDM bound completely. The details of su ch a possibility in the present
+framework attempting to provide the models with realistic n eutrino masses will be discussed
+elsewhere. Other mechanisms, not involving R-parity violation (and keeping the “missing en-
+ergy” signature), that could be invoked if the amount of CDM a ppears to be too large, concern
+the cosmology of theearly universe. For instance, “thermal inflation” [133] or “late time entropy
+injection” [134] could bring the CDM density into agreement with the WMAP measurements.
+This kind of modifications of the physics scenario neither c oncerns the theory basis nor the
+collider phenomenology, but could have a strong impact on th e CDM derived bounds.
+Therefore, in order to get an impression of the possibleimpact of the CDM abundance on
+the collider phenomenology in our models under investigati on, we will analyze the case that the
+LSP does contribute to the CDM density, and apply a more loose bound of
+ΩCDMh2<0.3. (6.5)
+(Lower values than the ones permitted by (6.5) are naturally allowed if another particle than the
+lightest neutralinoconstitutesCDM.)Forourevaluationw ehaveusedthecode MicroMegas [135,
+136]. Theprediction for the lightest Higgs mass, Mhas function of Mfor the model FUTBwith
+µ<0 is shown in Fig. 2. The dark (red) dots are the points that pas s the constraints in (6.5)
+(and that have the lightest neutralinoas LSP), which favors relatively light values of M. Thefull20 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+1001000masses [GeV]
+h  H   A   H±  stop1,2  sbot1,2  stau1,2  cha1,2 neu1,2,3,4
+Figure 3. The particle spectrum of model FUTBwithµ<0, where the points shown are in agreement
+with the quark mass constraints, the B-physics observables and the loose CDM constraint. The light
+(green) points on the left are the various Higgs boson masses. The dark (blue) points following are
+the two scalar top and bottom masses, followed by lighter (beige) sc alar tau masses. The darker (red)
+points to the right are the two chargino masses followed by the lighte r shaded (pink) points indicating
+the neutralino masses.
+Figure 4. The particle spectrum of model FUTBwithµ<0, the points shown are in agreement with
+the quark mass constraints, the B-physics observables, but the loose CDM constraint has been omitt ed
+(motivated by possible R-parity violation, see text). The color coding is as in Fig. 3.
+particle spectrum of model FUTBwithµ <0, again compliant with quark mass constraints,
+theBphysics observables (and with the loose CDM constraint), is shown in Fig. 3. The masses
+of the particles increase with increasing values of the unif ied gaugino mass M. One can see thatFinite Unification: Theory and Predictions 21
+Table 3. A representative spectrum of a light FUTB,µ<0 spectrum.
+Mbot(MZ)2.71 GeV Mtop 172.2 GeV
+Mh 123.1 GeV MA 680 GeV
+MH 679 GeV MH±685 GeV
+Stop1 1876 GeV Stop2 2146 GeV
+Sbot1 1849 GeV Sbot2 2117 GeV
+Mstau1 635 GeV Mstau2 867 GeV
+Char1 1072 GeV Char2 1597 GeV
+Neu1 579 GeV Neu2 1072 GeV
+Neu3 1591 GeV Neu4 1596 GeV
+M1 580 GeV M2 1077 GeV
+Mgluino 2754 GeV
+large parts of the spectrum are in the kinematic reach of the L HC. A numerical example of such
+a light spectrum is shown in Table 3. The colored part of this s pectrum as well as the lightest
+Higgs boson should be (relatively easily) accessible at the LHC.
+Finally for the model FUTBwithµ <0 we show the particle spectrum, where only the
+quark mass constraints and the Bphysics observables are taken into account. The loose CDM
+constraint, on the other hand, has been omitted, motivated b y the possible R-parity violation
+as discussed above. Consequently, following Fig. 2, larger values ofMare allowed, resulting in
+a heavier spectrum, as can be seen in Fig. 4. In this case only p art of the parameter space would
+be in the kinematic reach of the LHC. Only the SM-like light Hi ggs boson remains observable
+for the whole parameter range.
+A more detailed analysis can be found in [113].
+7 Finite SU(3)3model
+We now examine the possibility of constructing realistic FU Ts based on product gauge groups.
+Consider an N= 1 supersymmetric theory, with gauge group SU(N)1×SU(N)2× ··· ×
+SU(N)k, withnfcopies(numberoffamilies)ofthesupersymmetricmultiple ts(N,N∗,1,...,1)+
+(1,N,N∗,...,1)+···+(N∗,1,1,...,N). The one-loop β-function coefficient in the renormali-
+zation-group equation of each SU(N) gauge coupling is simply given by
+b=/parenleftbigg
+−11
+3+2
+3/parenrightbigg
+N+nf/parenleftbigg2
+3+1
+3/parenrightbigg/parenleftbigg1
+2/parenrightbigg
+2N=−3N+nfN. (7.1)
+This means that nf= 3 is the only solution of equation (7.1) that yields b= 0. Since b= 0
+is a necessary condition for a finite field theory, the exist ence of three families of quarks and
+leptons is natural in such models, provided the matter conte nt is exactly as given above.
+The model of this type with best phenomenology is the SU(3)3model discussed in [137],
+where the details of the model are given. It corresponds to th e well-known example of SU(3)C×
+SU(3)L×SU(3)R[138, 139, 140, 141], with quarks transforming as
+q=
+d u h
+d u h
+d u h
+∼(3,3∗,1), qc=
+dcdcdc
+ucucuc
+hchchc
+∼(3∗,1,3), (7.2)
+and leptons transforming as
+λ=
+N Ecν
+E Nce
+νcecS
+∼(1,3,3∗). (7.3)22 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+Switching the first and third rows of qctogether with the first and third columns of λ, we obtain
+the alternative left-right model first proposed in [141] in the context of superstring-inspired E6.
+In order for all the gauge couplings to be equal at MGUT, as is suggested by the LEP
+results [15], the cyclic symmetry Z3must be imposed, i.e.
+q→λ→qc→q,
+whereqandqcare given in equation (7.2) and λin equation (7.3). Then, the first of the
+finiteness conditions (4.5) for one-loop finiteness, name ly the vanishing of the gauge β-function
+is satisfied.
+Next let us consider the second condition, i.e. the vanishin g of the anomalous dimensions of
+all superfields, equation (4.6). To do that first we have to w rite down the superpotential. If
+there is just one family, then there are only two trilinear in variants, which can be constructed
+respecting the symmetries of the theory, and therefore can b e used in the superpotential as
+follows
+fTr(λqcq)+1
+6f′ǫijkǫabc(λiaλjbλkc+qc
+iaqc
+jbqc
+kc+qiaqjbqkc),
+wherefandf′are the Yukawa couplings associated to each invariant. Quar k and leptons obtain
+masseswhenthescalar partsofthesuperfields( ˜N,˜Nc) obtainvacuumexpectation values(vevs),
+md=f∝an}b∇acketle{t˜N∝an}b∇acket∇i}ht, m u=f∝an}b∇acketle{t˜Nc∝an}b∇acket∇i}ht, m e=f′∝an}b∇acketle{t˜N∝an}b∇acket∇i}ht, m ν=f′∝an}b∇acketle{t˜Nc∝an}b∇acket∇i}ht.
+With three families, the most general superpotential conta ins 11fcouplings, and 10 f′
+couplings, subject to 9 conditions, due to the vanishing of t he anomalous dimensions of each
+superfield. The conditions are the following
+/summationdisplay
+j,kfijk(fljk)∗+2
+3/summationdisplay
+j,kf′
+ijk(f′
+ljk)∗=16
+9g2δil, (7.4)
+where
+fijk=fjki=fkij, f′
+ijk=f′
+jki=f′
+kij=f′
+ikj=f′
+kji=f′
+jik.
+Quarks and leptons receive masses when the scalar part of the superfields ˜N1,2,3and˜Nc
+1,2,3
+obtain vevs as follows
+(Md)ij=/summationdisplay
+kfkij∝an}b∇acketle{t˜Nk∝an}b∇acket∇i}ht,(Mu)ij=/summationdisplay
+kfkij∝an}b∇acketle{t˜Nc
+k∝an}b∇acket∇i}ht,
+(Me)ij=/summationdisplay
+kf′
+kij∝an}b∇acketle{t˜Nk∝an}b∇acket∇i}ht,(Mν)ij=/summationdisplay
+kf′
+kij∝an}b∇acketle{t˜Nc
+k∝an}b∇acket∇i}ht.
+We will assume that the below MGUTwe have the usual MSSM, with the two Higgs doublets
+coupled maximally to the third generation. Therefore we hav e to choose the linear combinations
+˜Nc=/summationtext
+iai˜Nc
+iand˜N=/summationtext
+ibi˜Nito play the role of the two Higgs doublets, which will be
+responsible for the electroweak symmetry breaking. This ca n be done by choosing appropriately
+themassesinthesuperpotential[108], sincetheyarenotco nstrainedbythefinitenessconditions.
+We choose that the two Higgs doublets are predominately coup led to the third generation. Then
+these two Higgs doublets couple to the three families differ ently, thus providing the freedom to
+understand their different masses and mixings. The remnant s of theSU(3)3FUT are the
+boundary conditions on the gauge and Yukawa couplings, i.e. (7.4), theh=−MCrelation,Finite Unification: Theory and Predictions 23
+and the soft scalar-mass sum rule equation (5.2) at MGUT, which, when applied to the present
+model, takes the form
+m2
+Hu+m2
+˜tc+m2
+˜q=M2=m2
+Hd+m2
+˜bc+m2
+˜q,
+where˜tc,˜bc, and ˜qare the scalar parts of the corresponding
+Concerning the solution to equation (7.4) we consider two ve rsions of the model:
+I) An all-loop finite model with a unique and isolated soluti on, in which f′vanishes, which
+leads to the following relation
+f2=f2
+111=f2
+222=f2
+333=16
+9g2.
+As for the lepton masses, because all f′couplings have been fixed to be zero at this order,
+in principle they would be expected to appear radiatively in duced by the scalar lepton masses
+appearing in the SSB sector of the theory. However, due to the finiteness conditions they cannot
+appear radiatively and remain as a problem for further study .
+II)Atwo-loop finitesolution, inwhichwekeep f′non-vanishingandweuseittointroducethe
+lepton masses. The model in turn becomes finite only up to two -loops since the corresponding
+solution of equation (7.4) is not an isolated one any more, i. e. it is a parametric one. In this
+case we have the following boundary conditions for the Yukaw a couplings
+f2=r/parenleftbigg16
+9/parenrightbigg
+g2, f′2= (1−r)/parenleftbigg8
+3/parenrightbigg
+g2,
+whererisafreeparameterwhichparametrizesthedifferentsoluti onstothefinitenessconditions.
+As for the boundary conditions of the soft scalars, we have th e universal case.
+7.1 Predictions for SU(3)3
+BelowMGUTall couplings and masses of the theory run according to the RG Es of the MSSM.
+Thus we examine the evolution of these parameters according to their RGEs up to two-loops
+for dimensionless parameters and at one-loop for dimension ful ones imposing the corresponding
+boundary conditions. We further assume a unique supersymme try breaking scale MSUSYand
+below that scale the effective theory is just the SM.
+We compare our predictions with the most recent experimenta l valuemexp
+t= (173.1±
+1.3) GeV [31], and recall that the theoretical values for mtsuffer from a correction of ∼4% [20,
+116, 103]. In the case of the bottom quark, we take again the va lue evaluated at MZ,mb(MZ) =
+2.83±0.10 GeV [115]. In the case of model I, the predictions for the to p quark mass (in this
+casembis an input) mtare∼183 GeV for µ<0, which is above the experimental value, and
+there are no solutions for µ>0.
+For the two-loop model II, we look for the values of the parame terrwhich comply with the
+experimental limits given above for top and bottom quarks ma sses. In the case of µ >0, for
+the bottom quark, the values of rlie in the range 0 .15/lessorsimilarr/lessorsimilar0.32. For the top mass, the range
+of values for r is 0 .35/lessorsimilarr/lessorsimilar0.6. From these values we can see that there is a very small regio n
+where both top and bottom quark masses are in the experimenta l range for the same value of r.
+In the case of µ <0 the situation is similar, although slightly better, with t he range of values
+0.62/lessorsimilarr/lessorsimilar0.77 for the bottom mass, and 0 .4/lessorsimilarr/lessorsimilar0.62 for the top quark mass. So far in
+the analysis, the masses of the new particles h’s andE’s of all families were taken to be at the
+MGUTscale. Taking into account new thresholds for these exotic p articles below MGUTwe hope
+to find a wider phenomenologically viable parameter space. The details of the predictions of
+theSU(3)3are currently under a careful re-analysis in view of the new v alue of the top-quark
+mass, the possible new thresholds for the exotic particles, as well as different intermediate gauge
+symmetry breaking into SU(3)c×SU(2)L×SU(2)R×U(1) [142].24 S. Heinemeyer, M. Mondrag´ on and G. Zoupanos
+8 Conclusions
+A number of proposals and ideas have matured with time and hav e survived after careful the-
+oretical studies and confrontation with experimental data . These include part of the original
+GUTs ideas, mainly the unification of gauge couplings and, s eparately, the unification of the
+Yukawa couplings, a version of fixed point behaviour of coup lings, and certainly the necessity
+of supersymmetry as a way to take care of the technical part of the hierarchy problem. On the
+other hand, a very serious theoretical problem, namely the p resence of divergencies in Quantum
+Field Theories (QFT), although challenged by the founders o f QFT [143, 144, 145], was mostly
+forgotten in the course of developments of the field partly d ue to the spectacular successes of
+renormalizable field theories, in particular of the SM. How ever, as it was already mentioned
+in the Introduction, fundamental developments in Theoreti cal Particle Physics are based in re-
+considerations of the problem of divergencies and serious a ttempts to solve it. These include
+the motivation and construction of string and non-commutat ive theories, as well as N= 4
+supersymmetric field theories [3, 4], N= 8 supergravity [5, 6, 7, 8, 9] and the AdS/CFT corre-
+spondence [2]. It is a thoroughly fascinating fact that many interesting ideas that have survived
+various theoretical and phenomenological tests, as well as the solution to the UV divergencies
+problem, find a common ground in the framework of N= 1 Finite Unified Theories, which we
+have described in the previous sections. From the theoretic al side they solve the problem of
+UV divergencies in a minimal way. On the phenomenological si de, since they are based on the
+principle of reduction of couplings (expressed via RGI rela tions among couplings and masses),
+they provide strict selection rules in choosing realistic m odels which lead to testable predictions.
+The celebrated success of predicting the top-quark mass [33 , 34, 35, 36, 37, 38] is now extented
+to the predictions of the Higgs masses and the supersymmetri c spectrum of the MSSM. At least
+the prediction of the lightest Higgs sector is expected to be tested in the next couple of years at
+LHC.
+Acknowledgements
+It is a pleasure for one of us (G.Z.) to thank the Organizing Co mmittee for the very warm
+hospitality offered. This work is partially supported by th e NTUA’s basic research support
+programme 2008 and 2009, and the European Union’s RTN progra mme under contract MRTN-
+CT-2006-035505. Supported also by a mexican PAPIIT grant IN 112709, and by Conacyt grants
+82291 and 51554-F.
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\ No newline at end of file
diff --git a/1001.0429.txt b/1001.0429.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e243dd9bf83f675ce65f8da27aeedb5a4ccecc1f
--- /dev/null
+++ b/1001.0429.txt
@@ -0,0 +1,587 @@
+arXiv:1001.0429v1  [hep-th]  4 Jan 2010KEK-TH-1346
+January 2010
+Matrix Model and Elliptic Curve
+Hirotaka Sugawara∗
+JSPS Washington Office, 1800 K Street NW #920, Washington, DC 2 0006,
+Department of Physics and Astronomy, Johns Hopkins Univers ity,
+and
+High Energy Accelerator Research Organization (KEK), Tsuku ba, Ibaraki, Japan
+Abstract
+Solution to the reduced matrix model of IKKT type is studied with non -zero fermion fields. A
+suggestion is made that our universe is made of rational numbers ra ther than being a continuum. To
+substantiatethisproposal,thereducedYang-Millsequationiswritt enintheformofanellipticcurve.
+The normalization of the solution can be expressed in terms of the We ierstrass function generically
+or in terms of the Dedekind function in the case of 3-brane. A way to define the gravitational field
+in the matrix model is proposed with some new interpretation of the c osmological constant. The
+(first) quantization of the system is done within the framework of n on-commutative geometry.
+∗e-mail: hirotaka.sugawara@kek.jp
+11 Introduction
+Some time ago, two groups, independently of each other, (ref .[1, 2]) made a proposal to formulate the
+string theory in terms of matrix models. One[1] deals with th e M-theory in the light cone gauge and
+the other[2]) defines the IIB string theory in terms of the red uced matrix model[3]. Here, I follow the
+latter formulation but without being restricted to the IIB m odel.
+The purpose of this paper is to point out a simple fact that the matrix model can be considered
+within the framework of number theory in a rather elementary way. The motivation of the study is
+the following.
+First, the number of the matrix elements of the matrix models will be infinite but obviously it is
+not continuous. A conventional wisdom is to take some contin uous limit of the matrix model such as to
+takeN→ ∞andR→ ∞withN/Rfixed to obtain the desired space time continuum. If we take th e
+matrix model as it is as the fundamental theory of space-time without taking the continuum limit, a
+new insight into the problem emerges. Space-time is compose d of integers or at most rational numbers
+if it is a dense set, as is usually conceived. To give a substan ce to this view, we give a new meaning to
+the reduced Yang-Mills equation. Crudely speaking, the equ ation reads,
+A⊗A⊗A=ψ×ψ. (1)
+This is reminiscent of the so-called elliptic curve[4] in th e number theory which can be written in
+general as
+ax3+bx2+cx+d=y2. (2)
+It is important that we retain the fermion term (classical De Broglie field) in (1) to make a comparison.
+SinceAandψin (1) are infinite matrices with the product symbols having a specific meaning, we are
+treating herean infiniteset of coupled elliptic curves as wi ll beshownlater inthis paper. We assert that
+the coefficients of these equations are integers or rational n umbers and the solutions are also rational
+numbers. This view enables us to utilize the whole machineri es of number theory developed over the
+last centuries.
+For example, we are interested in the number of solutions to e quation (2). A zeta function corre-
+spondingto equation (2) can bedefined to study this problem. In the simple case of integer coefficients,
+this can be defined as;
+ζ(z) =/productdisplay
+p/parenleftbig
+1−app−z+p1−2z/parenrightbig−1, (3)
+whereap=p−NpwithNpthe number of solutions of (2) between 0 and p. The product is over all the
+prime numbers pexcept for those pwhich give rise to a multiple root when (2) is considered modu lop.
+In the case of equation (1), we assume that the infinite matrix A is composed of clusters of finite
+matrices and we define zeta function for each of them. If the ze ta function for a certain cluster satisfies
+ζ(1) = 0, we see that we have infinite number of rational number s olutions to this cluster due to Birch
+and Swinnerton-Dyer conjecture[4]. This is almost needed s ince we have to solve equation (1) together
+with the other equation of motion which looks symbolically a sA⊗ψ= 0. As is well known the elliptic
+2curve (2) is nothing but a two torus and the relation of ( x, y) to the complex coordinate on the upper
+half plane is given by;
+x=℘(z), y=1
+2℘′(z),with℘′(z)2= 4℘(z)3−g2℘(z)−g3, (4)
+where℘(z) stands for the Weierstrass function. The elliptic curve ap peared in physics in different
+context. Notably, in the analysis of modular space of N= 2 supersymmetric gauge theory (Seiberg
+and Witten[5]), the vacuum values of the fields are defined on t he elliptic curve with coordinate of the
+moduli space u being the zero of ℘′(z). There exist many papers which deal with elliptic curves in
+various contexts of physics but these are not referred to her e since they are not relevant to the present
+work[6]. In obtaining the zeta function for the equation (1) the Shimura conjecture (or Taniyama-
+Shimura conjecture)[7] might be helpful. This famous conje cture was proven by A. Wiles[8] in a
+special case which is needed for the proof of Fermat’s theore m and in its full generality by Ch. Breuil,
+B. Conrad, F. Diamond and R. taylor in 1999[9]. The conjectur e says that the zeta function (3) is
+modular.
+In this context, what I am trying to propose in this paper is to assert that equation (1) should
+be considered as modular curve. This naturally leads to the s pace-time which is composed of field of
+rational numbers or its algebraic extension.
+One of the main research goals of number theory is to generali ze the Shimura conjecture to the zeta
+functions of higher degrees and with an extended automorphi sm. This so-called Langlands program[9]
+has a geometric counter part and is being extensively studie d in connection with string theory or chiral
+theory[10, 11].
+The organization of the paper is the following: Section2 pre pares some necessary machinery such as
+the description of the fermion de Broglie fields. The cluster of the matrix will also be defined. Section3
+is devoted to the analysis of equation (1) with the proof that the scale of the classical fields is given by
+the Weierstrass function or by the Dedekind function. In sec tion4, a method is proposed to introduce
+the gravitation into the matrix models. Section5 is devoted to the problem of (first) quantization. The
+conjugate variables are introduced as the Dirac operators o f the non-commutative geometry. Section6
+is for the brief concluding remarks.
+2 Some Preparations
+The action of the matrix model is written in such a way to accom modate the non-associative case:
+L=−1
+4Tr{[Aµ,Aν][Aµ,Aν]}−g
+2Tr/braceleftbig
+Ψ(AµΓµΨ)−(AµΨΓµ)Ψ/bracerightbig
+, (5)
+whereAµis a 10-dimensional Minkowski vector and Ψ is a 10-dimension al 32-component spinor. De-
+pending on whether we take the IIB case or the IIA case, the exp ression for the Ψ is different.
+For the IIB case, we have,
+Ψ =/braceleftig
+(x+xa,bΓa+Γb++yB1)+Γ0+(xaΓa++yaΓa+B1)/bracerightig
+ζ. (6)
+3Here the notation is that of Polchinski[12]:
+Γ0±=1
+2(±Γ0+Γ1),Γa±=1
+2(Γ2a±Γ2a+1),(a= 1,2,3,4)
+B1= Γ3Γ5Γ7Γ9,
+Γa+are creation operators and Γa−are annihilation operators including a= 0,
+ζis a ground state with Γa+ζ= 0 fora= 0,1,2,3,4,
+x, xa,b, y, xaandyaare all complex numbers.
+This case corresponds to two 16-component Majorana-Weyl sp inors of same chirality.
+For the IIA case we have similarity:
+Ψ =/braceleftbig
+(x+xa,bΓa+Γb++yB1)+Γ0+(xaΓa++yaΓa+B1)
++ Γ0+(x′+x′
+a,bΓa+Γb++y′B1)+(x′
+aΓa++y′
+aΓa+B)/bracerightbig
+ζ . (7)
+The Majorana condition reads,
+x∗=y, x∗
+a,b=−εabcdyc,d, x∗
+a=ya,
+x′∗=y′, x′∗
+a,b=−εabcdy′
+c,d, x′∗
+a=y′
+aandζ∗=ζ.
+We have two Majorana-Weyl spinors of opposite chirality in t his case. The variables x′sandy′sare
+all infinite compoment matrices as is Aµ.
+The next task is to define the cluster and to express the infinit e matrix as:
+Aµ=
+A(1)
+µ
+A(2)
+µ
+A(3)
+µ
+···
+···
+(8)
+We assume that each finite dimensional cluster matrix ( Aµ, x, y,... ) to be expanded as (suppressing
+the cluster index):
+Aµ=Ai,µλi,
+whereλisatisfies the following multiplication rules:
+λiλj= (ifijk+dijk)λk+(gij+ieij)I . (9)
+The only conditions for the real parameters f,d,gandeare thatfandeare antisymmetric with
+respect to the interchange of the first two suffices and dandgare symmetric. In fact, the reality of
+ΨΓµΨ imposes further condition that either of the following two cases be satisfied,
+(1)eij= 0,dijkis cyclic symmetric in i,j,k,
+(2)e/ne}ationslash= 0, d= 0.
+4Coexistence of fande(case (2)) corresponds to the non-associative case. In this case, the ma-
+trix representation exists using the non-associative alge bra itself as the representation space but
+the multiplication rule (9) should be applied rather than th e regular matrix rule. We also assume,
+Tr(λi) = 0, Tτ(I) =a(some finite number).
+The motivation to generalize the matrix model to the non-ass ociative case stems from the recent
+interest in the non-associative algebra which arose in conn ection with the M2-brane analysis initiated
+by J. Bagger and N. Lambert[13].
+The simplest non-associative example of equation (9) is giv en by the following “generalized Pauli
+spin”:
+σiσj=iεijkσk+(δij+iκεij)I. (10)
+Forx=aiσi+a0I, y=biσi+b0I, z=ciσi+c0I, we have,
+(1) Jordan condition: ( xy)z−x(yz) = 0,
+(2) symmetrized associator: [ x, y, z] = 4iκ/braceleftig
+(/vector a×/vectorb·/vectorI)/vector c+(/vectorb×/vector c·/vectorI)/vector a+(/vector c×/vector a·/vectorI)/vectorb/bracerightig
+/vector σ,
+where/vectorIis a vector with a unit length of all three components and /vector a= (a1, a2, a3) etc.
+Sincewe regard theeach cluster to correspondto asmall port ion of space-time coordinate, the order
+of assigning each solution to a cluster should not matter. Th is will be regarded as the transformation
+which replaces the general coordinate transformation as wi ll be discussed in section4.
+3 Equations of Motion
+From the action (5) we get the following equations of motion f or each cluster:
+4Gij;i′j′Aν
+jAµ
+i′Aj′ν−g
+2FjikΨjΓµΨk= 0, (11)
+FijkAjµΓµΨk= 0, (12)
+where
+Gij;i′j′=fijkfi′j′k′gkk′+κ2eijei′j′, (13)
+Fjik= (ifjkl+djkl)(gil+iκeil)−(ifjil+djil)(glk+iκelk), (14)
+We can write
+Fijk=i˜fijk+˜dijk, (15)
+Then the condition that the fermion term in the equation (11) be real is satisfied by dorfbeing
+symmetric or anti-symmetric respectively when we exchange the first and the last suffices and this
+leads to the condition stated under the equation (9). We defin e following variables:
+Aj,2c+iAj,2c+1≡αj,c=|αc|βj,c, (16)
+5with
+/summationdisplay
+|βj,c|2= 1, (17)
+and
+Aj,1±Aj,0=α±
+j,0=|α0|β±
+j,0, (18)
+with
+β+
+j,0β−
+j,0=±1 (+for space-like and −for time-like vectot Aµ). (19)
+Then the equation (12) can be reduced to the following five equ ations (we write down the IIB case):
+Fijk(αj,axk,a−α+
+j,0xk) = 0, (20)
+Fijk(α∗
+j,ayk,a−α+
+j,0yk) = 0, (21)
+Fijk/braceleftbig1
+2(α∗
+j,ayk,b−α∗
+j,bxk,a)−α+
+j,0xk;a,b−εabcdαj,cyk,d/bracerightbig
+= 0, (22)
+Fijk(αj,ayk+α−
+j,0yk,a+εabcdα∗
+j,bxk;c,d) = 0, (23)
+Fijk(α∗
+j,axk−2αj,bxk;a,b+α−
+j,0xk,a) = 0, (24)
+Here the repeated suffices must be summed over. (11) can be writ ten as (again IIB case)
+4Gij;i′j′Aν
+jAµ
+i′Aj′ν−g
+2Fjik/an}b∇acketle{tΓµ/an}b∇acket∇i}htj,k= 0, (25)
+with
+/an}b∇acketle{tΓ0/an}b∇acket∇i}htj,k=x∗
+jxk+2x∗
+j;a,bxk;a,b+y∗
+jyk+x∗
+j,axk,a+y∗
+j,ayk,a, (26)
+/an}b∇acketle{tΓ1/an}b∇acket∇i}htj,k=x∗
+jxk+2x∗
+j;a,bxk;a,b−/parenleftbig
+y∗
+jyk+x∗
+j,axk,a+y∗
+j,ayk,a/parenrightbig
+, (27)
+/an}b∇acketle{tΓ2a/an}b∇acket∇i}htj,k=−x∗
+jxk,a−x∗
+j,axk−y∗
+jyk,a−y∗
+j,ayk
+−2x∗
+j;a,bxk,b−2x∗
+j,bxk;a,b−εabcd/parenleftbig
+x∗
+j;c,byk,d+y∗
+j,dxk;c,b/parenrightbig
+, (28)
+/an}b∇acketle{tΓ2a+1/an}b∇acket∇i}htj,k=−i/braceleftbig
+x∗
+jxk,a−x∗
+j,axk−y∗
+jyk,a+y∗
+j,ayk
+−2x∗
+j;a,bxk,b+2x∗
+j,bxk;a,b−εabcd/parenleftbig
+−x∗
+j;c,byk,d+y∗
+j,dxk;c,b/parenrightbig/bracerightbig
+, (29)
+In writing down these equations, the Grassmannian factor ζ∗ζorζTζ(in case of IIA) is absorbed in the
+coupling constant g. In fact, the easiest way to get rid of Grassmannian factor is to assume that all Aµ
+contain a factor ( ζ∗ζ)2/3(IIB) or (ζTζ)2/3(IIA). Finally the equation (25) will become the following
+two sets of equations:
+ρ±
+i|α0|3+/parenleftbig/summationdisplay
+a|αa|2σ±
+i,a/parenrightbig
+|α0|=g
+4γ0±
+i|λ|2, (30)
+σi;c,c|αc|3+/parenleftbig/summationdisplay
+a/negationslash=cσi;a,c|αa|2+ρi,c|α0|2/parenrightbig
+|αc|=g
+4γc
+i|λ|2, (31)
+where
+γ0±
+i= ˜γ1
+i±˜γ0
+i, (32)
+γc
+i= ˜γ2c
+i+i˜γ2c+1
+i, (33)
+6with
+Fjik/an}b∇acketle{tΓµ
+j,k/an}b∇acket∇i}ht= ˜γµ
+i|λ|2, (34)
+and
+ρ±
+i=Gij;i′j′/parenleftbig
+β+
+j,0β−
+j′,0+β−
+j,0β+
+j′,0/parenrightbig
+β±
+i′,0, (35)
+σ±
+i,a=Gij;i′j′/parenleftbig
+βj,aβ∗
+j′,a+β∗
+j,aβj′,a/parenrightbig
+β±
+i′,0, (36)
+ρi,c=Gij;i′j′/parenleftbig
+β+
+j,0β−
+j′,0+β−
+j,0β+
+j′,0/parenrightbig
+βi′,c, (37)
+σi;a,c=Gij;i′j′/parenleftbig
+βj,aβ∗
+j′,a+β∗
+j,aβj′,a/parenrightbig
+βi′,c, (38)
+The equations (30) and (31) have the desired form of elliptic curves. They must give the same value of
+|α0|or|α0|for thei-dependent coefficients. The consistency condition can be ob tained in the following
+way: We first solve the equations for some value of iand substitute this back to equations (30) and
+(31). Then we have
+|α0|=℘(ω) with℘′(ω) = 0, (39)
+and
+|αc|=℘c(ωc) with℘′(ωc) = 0, (40)
+where℘and℘care Weierstrass functions. The equations (30) and (31) beco me respectively,
+4℘3(ω)ρ±
+i+4/summationdisplay
+c℘2
+c(ωc)σ±
+i,c℘(ω) =gγ±
+i, (41)
+4/summationdisplay
+c℘3
+c(ωc)σi;c,a+4℘2(ω)℘a(ωa)ρi,a=gγa
+i, (42)
+These two equations are regarded as the consistency conditi ons but they themselves are a kind of
+coupled elliptic curves. My proposal is to define all these eq uations (coefficients and the solutions) on
+the field of rational numbers. The complex numbers must have r eal and imaginary parts which are
+rational respectively.
+The 3-brane case can be further reduced to a simpler form as fo llows: This case is defined by
+Aµ/ne}ationslash= 0 only for µ= 0,1,2,3,
+xj;a,b= 0,
+xj,a/ne}ationslash= 0 only for a= 1,
+yj,a/ne}ationslash= 0 only for a= 1,
+xj/ne}ationslash= 0, yj/ne}ationslash= 0
+The equations (20) to (24) become,
+Fijk(αj,1xk,1−α+
+j,0xk) = 0, (43)
+Fijk(α∗
+j,1yk,1−α+
+j,0yk) = 0, (44)
+Fijk(αj,1yk+α−
+j,0yk,1) = 0, (45)
+Fijk(α∗
+j,1xk+α−
+j,0xk,1) = 0, (46)
+7The symmetry of the equation under;
+(xj, xj,1)→(y∗
+j, y∗
+j,1),
+makes it possible to set ( y∗
+j, y∗
+j,1) = (θxj, θxj,1). Then we have,
+γ0+
+i= 2Fjik(1+|θ|2)xjx∗
+k,
+γ0−
+i= 2Fjik(1+|θ|2)xj,1x∗
+k,1,
+γi,1= 2Fjik(1+|θ|2)x∗
+j,1xk,
+It can be shown in this case that,
+|α0|= 2π2η(τ)4,
+|α1|= 2π2η(τ1)4,
+whereη(τ) is the Dedekind modular form. Instead of (41) and (42) we hav e,
+4η(τ)12ρ±
+i+4η(τ1)8η(τ)4σ±
+i,1=1
+8π6gγ±
+i, (47)
+4η(τ1)12σi;1,1+4η(τ)8η(τ1)4ρi,1=1
+8π6gγi,1, (48)
+One might think that it may be possible to get a one-brane equa tion by putting
+A2=A3= 0 andxj,1=yj,1= 0 in the above equations.
+But, unfortunately, we can easily prove that the resulting s et of equations do not have a solution.
+One-brane case must be defined in a different way. Another case i s that of M2-brane. We write down
+the equations when we have the extended Pauli spin.
+/vector α0+×/vector x+/vectorA11×/vector¯x= 0, (49)
+/vector α0+×/vector¯x+/vectorA11×/vector x= 0, (50)
+/vector α0+×/vector xa,b+/vectorA11×/vector¯xa,b= 0, (51)
+/vector α0+×/vector¯xa,b+/vectorA11×/vector xa,b= 0, (52)
+2/vector α0+×(/vector α0+×/vector α0−)+4/vectorA11×(/vector α0+×/vectorA11)
++2κ2/braceleftbig
+(/vector α0+×/vector α0−·/vectorI)/vector α0+×/vectorI+(/vector α0+×/vectorA11·/vectorI)/vectorA11×/vectorI/bracerightbig
+= 4ig(/vector x∗
+a,b×/vector xa,b), (53)
+2/vector α0−×(/vector α0−×/vector α0+)+4/vectorA11×(/vector α0+×/vectorA11)
++2κ2/braceleftbig
+(/vector α0−×/vector α0+·/vectorI)/vector α0−×/vectorI+(/vector α0−×/vectorA11·/vectorI)/vectorA11×/vectorI/bracerightbig
+=−4ig(/vector¯x∗
+a,b×/vector¯xa,b), (54)
+2/vector α0−×(/vectorA11×/vector α0+)−2/vector α0+×(/vector α0−×/vectorA11)
++2κ2/braceleftbig
+(/vector α0−×/vector α0+·/vectorI)/vector α0−×/vectorI+(/vector α0−×/vectorA11·/vectorI)/vectorA11×/vectorI/bracerightbig
+=ig(/vector¯x∗
+a,b×/vector xa,b+/vector x∗
+a,b×/vector¯xa,b).(55)
+Equations (49)-(52) should be coupled with the equations (5 3)-(55).
+84 Gravity
+We might define the gravitational field (metric tensor) as a fu nctional of the space-time coordinate.
+But by doing so we must introduce a new action to determine the form of the functional. This is in a
+way inappropriate since we want the original matrix action t o be able to determine everything.
+An elementary way to define the metric tensor is the following : First we define the operator ∇
+which operates on any function defined on the space of cluster s by,
+∇f(n) =f(n)−f(n−1). (56)
+Then,
+ds(n)2=gv
+µ,ν(n)Tr/parenleftbig
+∇Aµ(n)∇Aν(n)/parenrightbig
+=Tr/parenleftbig
+gµ,ν(n)∇Av,µ(n)∇Av,ν(n)/parenrightbig
+. (57)
+Here the notation v stands for the vacuum. The vacuum solutio nAv,µ(n) which corresponds to the
+solutionAµ(n) can be defined by taking the limit g→0 andfijk→0 in the solution Aµ(n). The latter
+limit can be taken by multiplying a small parameter to fijkand taking it to zero.
+The general covariance is defined as the transformation prop erty ofgunder the interchange of
+clusters: For n′=n′(n), we have,
+Tr/parenleftbig
+g′
+µ,ν(n′)∇Av,µ(n′)∇Av,ν(n′)/parenrightbig
+=Tr/parenleftbig
+gµ,ν(n)∇Av,µ(n)∇Av,ν(n)/parenrightbig
+. (58)
+The vacuum metric gv
+µ,νcould be just the flat metric ηµ,ν, but it could also be a metric of the deSitter
+or the anti-deSitter space. Either way, we assume it is propo rtional to the unit matrix.
+The connection is defined as
+Aµν,ρκ=1
+4/bracketleftbig/braceleftbig
+gµν(n)∇gρκ(n)−gρν(n)∇gµκ(n)/bracerightbig
+−/braceleftbig
+µ,ν↔ρ,κ/bracerightbig/bracketrightbig
+. (59)
+The justification comes from the idea of non-commutative geo metry[14]. In this particular case, we
+treat∇as the Dirac operator defined on the functional space of gµν. We have the following identities,
+Aµν,ρκ(n) =Aρκ,µν(n), Aµν,ρκ(n) =−Aρν,µκ(n) =Aµκ,ρν(n).
+The curvature is defined again loosely following non-commut ative geometry,
+Rµν,ρκ(n)≡ ∇Aµρ,νκ(n)+1
+2/parenleftbig
+Aµρ,ν′κ′(n)Aν′κ′
+νκ(n)−Aνρ,ν′κ′(n)Aν′κ′
+µκ(n)/parenrightbig
+. (60)
+We have the following identities,
+Rµν,ρκ(n) =−Rνµ,ρκ(n) =−Rµν,κρ(n).
+We do not have the cyclic identity for Rµν,ρκ(n). We can, therefore, define the Ricci tensor uniquely
+but it is not necessarily symmetric under the interchange of suffices. The vacuum curvature is assumed
+to satisfy (neglecting the symbol v),
+Rνκ(n) =gµρ(n)Rµν,ρκ(n) =λgνκ(n). (61)
+9This equation can be easily solved by the following anzatz,
+∇gµν(n) =gµν(n)−gµν(n−1) =δgµν(n),
+whereδis a certain rational number. The gµν(n) can easily be solved,
+gµν(n) =1
+(1−δ)n−1gµν(1). (62)
+Equation (61) is reduced to,
+λ=N−1
+8/bracketleftbig
+(N+6)δ2−4δ3/bracketrightbig
+. (63)
+Nis the dimension of the space-time.
+These equations give a new interpretation to the cosmologic al constant λas follows: When λis
+small we have,
+gµν(n) = exp{(n−1)δ}gµν(1). (64)
+Assuming that δis a negative number, this equation shows that 1 /|δ|is the effective number of vacuum
+space-time points. The distance between the points with the cluster number larger than 1 /|δ|becomes
+small compared to the Planck scale and so the points become un distinguishable.
+In the case of our universe, λ≈10−120which gives 1 /|δ| ≈1060. The curvature size of the universe
+is 1028cmand if we divide this by the Planck length 10−33cm, we get 1061. Since our space-time
+is 4-dimensional we have 10244as the number of points of the current universe. The correspo nding
+vacuum seems to have only one-dimensional degrees of freedo m.
+5 (First) Quantization and the Non-commutative Geometry
+The matrix model is equipped with its algebra and the Hilbert space as the representation space of the
+algebra. The (first) quantization can be done utilizing this Hilbert space. The missing operator is the
+canonical conjugate variable PµtoAµandQαto Ψαwith the commutation relation,
+[Pµ, Aν] =−iδν
+µ,
+and
+{Ψα,Qβ}={Ψα, Qβ}=iδαβ.
+If we interpret the algebra and the Hibert space to be that of n on-commutative geometry, the missing
+operator to complete the Conne’s triple spectra[13] is the D irac operator. It is natural to assume that
+they are also given by PµandQβ. In this interpretation, the solution of the equations we ha ve been
+discussingis not a classical solution but a quantum one with Aµand Ψαdiagonalized up to the clusters.
+We can define the gauge fields in the space of matrix algebra usi ng the standard prescription of
+non-commutative geometry[13]. The relation of gravity defi ned in the last section and the one defined
+in this way is not clear at this time.
+10PµandQβhave the other roles. The former is the operator for the trans lational invariance of the
+Lagrangian (5),
+Aµ−→Aµ+aµI,
+Ψ−→Ψ.
+The latter is the operator for one of the supersymmetry trans formaions defined in[2],
+Ψ−→Ψ+ξI,
+Aµ−→Aµ.
+This corresponds to the second supersymmetry defined by IKKT asδ(2)[2].
+The supersymmetry operator corresponding to δ(1)of IKKT can be defined as follows,
+Q(1)
+α= (ΓµΨ)αPµ−1
+2[Aµ, Aν](ΓµνQ)α,
+Q(1)
+α=−(ΨΓµ)αPµ−1
+2[Aµ, Aν](QΓµν)α.
+We the get the following anti-commutation relations for the se operators,
+{Q(1)
+α, Q(1)
+β}={Q(1)
+α,Q(1)
+β}= 0,
+{Q(1)
+α, Q(1)
+β}=Pρ[Aµ, Aν](δν
+ρPµ−δµ
+ρPν).
+Defining the gauge fields including the gravity using Dirac op eratorsPµandQαwill be discussed in a
+future publication.
+6 Conclusions
+The proposal is made that the space-time is constructed out o f integers or rational numbersrather than
+being a continuum. This is based on taking the matrix model as the fundamental theory of space-time
+as it is. The proposal is substantiated by observing that the reduced matrix model equations can be
+thought of as the elliptic curves which play the central role in number theory. The whole machinery of
+the number theory developed over the centuries could be inco rporated into the understanding of our
+space-time.
+The space-time coordinates Aµand Ψαare mutually non-commuting but we assume that they can
+be partially diagonalized up to certain clusters. We are the n left with the equations of finite sized
+matrices and they form a system of coupled elliptic curves.
+The metric tensor for this space-time can be calculated usin g the solution and the prescription
+described in section4 The gravity is completely determined by the original reduced matrix model
+equations in this sense. It is necessary to define the vacuum m etric for this purpose. It could be just
+a flat metric but it could also have a finite cosmological const ant when the number of effective space-
+time points is finite. The latter is given by the inverse of the former thus providing a new meaning to
+the cosmological constant. The fact that our space-time see ms to have a finite cosmological constant
+corresponds to the fact that it is composed of finite number of effective space-time points.
+11The (first) quantized system of the matrix model can be unders tood within the framework of
+non-commutative geometry. The fact that the matrix model is equipped with the algebra and the
+representation space allows us to introduce a canonical var iable as the Dirac operator in the non-
+commutative geometry. This possibly provides us an alterna tive way to define the gravity but it starts
+with the level of a connection or a gauge field. Our naive defini tion of the metric tensor must be related
+to the latter definition in a certain way.
+Non-commutative geometry itself is very much related to the number theory but it is outside of the
+scope of the present work.
+Finally I will list some of the problems for the future invest igations:
+(1) Can the non-commutative definition of gauge theory lead t o theE(8)×E(8) theory?
+(2) The discussions of section2 suggest that we may need the n on-abelian Galois extension of the
+field. What is the physical meaning of the Galois group?
+(3) Is it possible to understand our assertion that equation (1) should be modular within the frame-
+work of Langlands philosophy?
+(4) Is it possible to incorporate all the geometrical concep t related to the string theory into the
+number theory formulation of the matrix model?
+Acknowledgment
+I would like to thank H. Hagura for the technical assistance.
+References
+[1] T.Banks, W.Fischler, S.H.ShenkerandL.Susskind, “M theory as a matrix model: A conjecture,”
+Phys. Rev. D55(1997) 5112; hep-th/9610043.
+[2] N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, Nucl . Phys.B498(1997) 467.
+[3] T. Eguchi and H. Kawai, “Reduction of Dynamical Degrees of Freedom in the Large N Gau ge
+Theory”, Phys. Rev. Lett. B48(1982) 1063
+[4] See, for example, Yu. I. Manin and A. A. Panchishkin, “Introduction to Modern Number Theory” ,
+Second corrected printing, Springer 2000, Page 321.
+[5] N. Seiberg and E. Witten, “Electric-magnetic duality, monopole condensation and co nfinement in
+N= 2supersymmetric Yang-Mills theory” , Nucl.Phys. B426, 19 (1994); [Erratum-ibid. B430,
+485 (1994)]; arXiv:hep-th/9407087.
+12[6] More than three dozen papers are listed when a key word “el liptic curve” is put into the search
+box in the section of high energy theory in arXiv e-print libr ary.
+[7] G. Shimura and Y. Taniyama, “Complex Multiplication of Abelian Varieties and Its Applica tion
+to Number Theory” , Tokyo: Mathematical Society of Japan, 1961; G. Shimura, “Introduction
+to the Arithmetic Theory of Automorphic Functions” , Princeton University Press, 1971. See also
+reference[4], page 320,330,334.
+[8] A. Wiles, “Modular Elliptic-Curves and Fermat’s Last Theorem” , Ann.Math. 141 (1995) 443-551;
+R. Taylor. and A. Wiles, “Ring-Theoretic Properties of Certain Hecke Algebras” , Ann.Math. 141
+(1995) 553-572.
+[9] F. Conrad Diamond and R. Taylor, “Modularity of Certain Potentially Barsotti-Tate Galois Re p-
+resentations” , J. Amer. Math. Soc. 12 (1999) 521-567;
+H. Darmon, “A Proof of the Full Shimura-Taniyama-Weil Conjecture is Ann ounced”, Not. Amer.
+Math. Soc. 46 (1999) 1397-1406.
+[10] A. Kapustin and E. Witten, “Electric-Magnetic Duality and the Geometric Langlands Pr ogram”,
+Communications in Number Theory and Physics, Vol.1, No1. (2 007) 1-236.
+[11] E. Frenkel, “Lectures on the Langlands Program and Conformal Field Theory ”,
+arXiv.hep-th/0512172.
+[12] J. Polchinski, String Theory , Cambridge University Press,1998, Volume II, Appendix B.
+[13] J. Bagger and N. Lambert, “Modeling Multiple M2’s” , Phys.Rev. D75(2007) 045020,
+arXiv.hep-th/0611108.
+[14] A. Conne, Noncommutative Geometry , Academic Press,1994; M. Marcolli, Lectures on Arithmetic
+Noncommutative Geometry, ArXiv:math/0409520.
+13
\ No newline at end of file
diff --git a/1001.0430.txt b/1001.0430.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4883aa13b22f559aab591d9904b5e84afc66b5b6
--- /dev/null
+++ b/1001.0430.txt
@@ -0,0 +1,335 @@
+Coupled fiber taper extraction of 1.53 um 
+photoluminescence from erbium doped silicon 
+nitride photonic crystal cavities 
+Gary Shambat 1,* , Yiyang Gong 1, Jesse Lu 1, Selçuk Yerci 2, Rui Li 2, Luca Dal Negro 2, and 
+Jelena Vu čkovi ć1 
+1Department of Electrical Engineering, Stanford Univ ersity, Stanford 94305 
+2Department of Electrical and Computer Engineering, Boston University, Boston, MA 02215 
+*email:gshambat@stanford.edu  
+Abstract:  Optical fiber tapers are used to collect photolumi nescence 
+emission at ~1.5 um from photonic crystal cavities fabricated in erbium 
+doped silicon nitride on silicon. Photoluminescence  collection via fiber 
+taper is enhanced 2.5 times relative to free space,  with a total taper 
+collection efficiency of 53%. By varying the fiber taper offset from the 
+cavity, a broad tuning range of coupling strength i s obtained. This material 
+system combined with fiber taper collection is prom ising for building on-
+chip optical amplifiers.   
+___________________________________________________ ________________________ 
+References and links 
+ 
+1.  D. A. B. Miller, “Device requirements for optical i nterconnects to silicon chips,” Proceedings of the IEEE 
+97 (7) , 1166-1185 (2009). 
+2.  L. Liao, A. Liu, D. Rubin, J. Basak, Y. Chetrit, H.  Nguyen, R. Cohen, N. Izhaky, and M. Paniccia, “40 
+Gbit/s silicon optical modulator for high speed app lications,” Electron. Lett. 43 (22), 1196–1197 (2007). 
+3.  P. Dong, S. Liao, D. Feng, H. Liang, D. Zheng, R. S hafiha, C. Kung, W. Qian, G. Li, X. Zheng, A. 
+Krishnamoorthy, and M. Asghari., “Low V pp , ultralow-energy, compact, high-speed silicon elec tro-optic 
+modulator,” Opt. Express 17 (25), 22484-22490 (2009). 
+4.  L. Colace, M. Balbi, G. Masini, G. Assanto, H. C. L uan, and L. C. Kimerling, “Ge on Si p-i-n photodiod es 
+operating at 10 Gb/s,” Appl. Phys. Lett. 88 , 101111 (2006).  
+5.  O. Fidaner, A. K. Okyay, J. E. Roth, R. K. Schaevit z, Y.-H. Kuo, K. C. Saraswat, J. S. Harris, Jr., an d D. A. 
+B.Miller, “Ge-SiGe quantum-well waveguide photodete ctors on silicon for the near-infrared,” IEEE Photon. 
+        Technol. Lett. 19 , 1631–1633 (2007).  
+6.  M. Makarova, Y. Gong, S-L. Cheng, Y. Nishi, S. Yerc i, R. Li, L. Dal Negro, J. Vuckovic, “Photonic crys tal 
+and plasmonic silicon based light sources," accepte d by JSTQE (2010).  
+7.  Y. Gong, M. Makarova, S. Yerci, R. Li, M.J. Stevens , B. Baek, S.W. Nam, R.H. Hadfield, S. N. Dorenbos,  
+V. Zwiller, J. Vuckovic, and L. Dal Negro, “Linewid th Narrowing and Purcell enhancement in photonic 
+crystal cavities on an Er-doped silicon nitride pla tform,” submitted to Optics Express (2010). 
+8.  K. Srinivasan, P.E. Barclay, M. Borselli, & O. J. P ainter, “An optical-fiber-based probe for photonic crystal 
+microcavities, IEEE J. Sel. Area Comm. 23 (7), 1321-1329 (2005). 
+9.  C. Grillet, C. Monat, C. Smith, B. Eggleton, D. Mos s, S. Frederick, D. Dalacu, P. Poole, J. Lapointe, G. 
+Aers, and R. Williams, “Nanowire coupling to photon ic crystal nanocavities for single photon sources,”  Opt. 
+Express, 15 (3), 466-470 (2007). 
+10.  I. Hwang, S. K. Kim, J. Yang, S. H Kim, S. Lee, and  Y. Lee, “Curved-microfiber photon coupling for 
+photonic crystal light emitter,” Appl. Phys. Lett. 87 , 131107 (2005). 
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+___________________________________________________ ________________________ 
+1. Introduction 
+Optical interconnects have the potential to improve  data communication in computer systems 
+by providing a high bandwidth, low power alternativ e to present electrical interconnects. 
+Particularly for off-chip links, where traditional metal interconnects suffer from large 
+capacitive delays and high energy consumption, impl ementing optical interconnects may be 
+very advantageous [1]. While much progress has been  made with CMOS-compatible 
+modulators and detectors [2-5], a suitable light so urce that is well integrated with a silicon 
+platform remains to be demonstrated. Recently, it w as shown that photonic crystal (PC) 
+cavities in erbium-doped silicon nitride (Er:SiN x) exhibited linewidth narrowing with 
+increased optical pump power [6, 7], demonstrating that 31% of Er ions can be excited under 
+optical pumping. Such a material may be suitable fo r an on-chip amplifier or laser source in 
+the future.  
+To connect a nanoscale photonic crystal cavity to t he off-chip environment of an optical 
+interconnect system, silica fiber tapers may be use d to in- and out-couple light efficiently. 
+Fiber tapers have been shown to be useful not only in probing and characterizing cavities 
+[8,9], but also in providing optical pumping for a laser and extracting photoluminescence [10]. 
+Because fiber tapers rely on resonant evanescent co upling, they can inject and extract light 
+with much higher efficiency than with free space te chniques. In this study, we report the 
+extraction of photoluminescence from Er:SiN x PC cavities into fiber tapers with much higher  
+efficiency than outcoupling through free space. We also demonstrate that a broad tuning range 
+of quality factor (up to 98% of the intrinsic Q val ue) is attainable by altering the strength of 
+the coupling and thus the coupling efficiency.  
+ 
+2. Fabrication  
+2.1 Taper fabrication: 
+Fiber tapers were fabricated using the well known “ flame brushing” technique [11] in which a 
+standard single mode communication fiber is simulta neously heated by a torch and pulled 
+outward by motorized stages. A hydrogen-oxygen torc h was used to provide a small flame 
+that was quickly scanned back and forth along the d esignated “hot zone” of the fiber. In order 
+to provide the best mechanical stability of the tap er probe, the pull length was kept to only ~3 
+mm, while the taper diameter was ~1 um. This diamet er was chosen because when the taper is 
+below approximately 1.1 um the waveguide becomes si ngle mode [12]. This single mode 
+behavior cleans up coupling spectra and removes err oneous fringing in the transmission and 
+photoluminescence spectra. After the fiber tapers w ere drawn, they were carefully bent such 
+that the radius of curvature was small enough for t he tapers to interact only with photonic 
+crystal cavities and not the nearby substrate. An o ptical image of a fabricated taper is shown 
+in Fig. 1(a). The whole fiber was mounted to a glas s slide and was fusion spliced into 
+connector cables. Finally, the slide was mounted in  a measurement rig where the taper could 
+be positioned onto photonic crystal cavities via bo th motorized and manual nanopositioning 
+stages. The transmission of the mounted fiber taper  was 16%, with most of the loss coming 
+from scattering within the short taper transition r egions.    2.2 Cavity fabrication: 
+Photonic crystal structures were fabricated in a tw o layer membrane consisting of a bottom 
+silicon layer and a top, light emitting, erbium-dop ed silicon nitride (Er:SiNx) layer, same as in 
+our previous work [7]. Approximately 110 nm of erbi um-doped amorphous silicon nitride was 
+deposited on a SOI wafer with a silicon thickness o f 250 nm by N 2 reactive magnetron co-
+sputtering from Si and Er targets in a Denton Disco very 18 confocal-target sputtering system 
+[13].  
+Photonic crystal L3 cavities [14] were patterned wi th an electron beam lithography system 
+using ZEP-520A as the resist. A Cl 2:HBr dry etch chemistry was used to etch the air ho les and 
+the sacrificial oxide layer in the SOI was undercut  at the end using a 6:1 Buffered oxide etch 
+to form the suspended PC membrane. The lattice peri odicity was set to a = 410 nm, and hole 
+radius was r = 125 nm. Additional outward shifts of  0.15a and 0.075a for the on-axis holes 
+surrounding the cavity were made to increase the in trinsic Q of the cavity. Fig. 1(b) shows a 
+scanning electron microscope (SEM) image of the tes ted cavity. 
+ 
+ 
+Fig. 1. (a) Optical microscope image of the fiber t aper. (b) SEM picture of the fabricated 
+SiN x/Si photonic crystal cavity. (c) Calculated E y field of the fundamental mode of the photonic 
+crystal. 
+ 
+3. Theoretical analysis 
+3.1 – Free space collection efficiency 
+Finite difference time domain (FDTD) calculations w ere performed on the cavity structure 
+with the size parameters of the SEM image above. Th e computational grid contained 10 
+mirror periods on each side of the cavity in the y direction and 12 mirror periods on each side 
+in the x direction. Q-factor calculations were perf ormed by integrating the power flux through 
+the appropriate boundaries: top and bottom half spa ces for the vertical Q’s and slab walls for 
+the parallel Q. The fundamental mode profile is sho wn in Fig. 1(c).  
+In order to estimate the percentage of light collec ted by our microscope objective, a two-
+dimensional Fourier Transform was performed on the field components of the cavity mode 
+just above the slab. From the k-space distribution of the tangential electric and magnetic 
+fields, the total power radiated within a 0.5 numer ical aperture (NA) objective was  
+determined [15] to be 10% of the total emission int o the half space above the slab. The low 
+value is not surprising if one examines the k-space  map for the cavity E y field shown in Figure 
+2. The transverse electric-like (TE-like) fundament al mode is predominantly E y in character 
+and from the k-space pattern in Fig. 2(b), we see t hat the magnitude of the components 
+radiated within the NA cone is much smaller than th e sum inside the light cone. Similar 
+observations were made for the other tangential fie lds. Additionally, since the bottom half of 
+the slab radiates approximately 1.4 times more than  the top half, as indicated by the top and 
+bottom Q factors of the FDTD simulation (this is a result of the PC membrane asymmetry), 
+the total efficiency of collection of light into th e objective is approximately 4.2%.  kx(π/a) ky(π/a) 
+kx(π/a) ky(π/a) a) b) 
+ 
+Fig. 2. (a) K-space map for the E y component of the fundamental mode, taken just abov e the 
+PC slab. The white circle is the light cone while t he green circle is the NA = 0.5 cone. (b) 
+Zoomed in image of the previous plot highlighting t he difference between the radiated 
+components captured and not captured by the objecti ve lens. The magnitude of the k 
+components inside the NA = 0.5 cone is clearly much  lower than those outside the cone.  
+ 
+3.2 – Coupled cavity Q-factor 
+ 
+The intrinsic cavity Q for the fundamental mode was  found to be 16,000 and is dominated by 
+the in-plane Q which was only 26,000. Due to asymme try of the PC in the vertical direction, 
+the confined TE-like cavity mode couples to TM-like  propagating Bloch states, which suffer 
+from an incomplete bandgap and hence leak out of th e cavity [16]. In contrast, a similar PC 
+structure composed of just the symmetric silicon la yer has an in-plane Q of nearly 900,000 
+and a net Q of 61,000.  
+Fig. 3 shows a plot of the various FDTD simulated Q -factors as a fiber taper, represented 
+as a cylinder of silica (n = 1.45, diameter = 1 um) , is scanned across the surface of the L3 
+cavity along the y-axis. For zero offset, we see an  expected drop in the top Q due to the 
+additional decay into the fiber. The in-plane Q is also reduced by a factor of two due to the 
+increased vertical asymmetry of the structure. This  parasitic decay reduces the overall Q 
+beyond the intrinsic limit of just additional decay  into the fiber.  
+As the taper is shifted to the side of the cavity, several trends can be observed. First, the 
+top Q is seen to reach a minimum for an offset of 2 50 nm, before increasing monotonically 
+towards the intrinsic limit of 104,000. The decreas e in coupling strength due to a smaller field 
+overlap for the offset taper causes the vertical de cay to decrease as expected. The bottom Q is 
+mostly unaffected by the taper but does show some i ncreased decay when the taper is in close 
+proximity to the cavity. The in-plane Q returns to its original value as the taper is moved 
+farther away from the cavity as well. Finally, we s ee that even for large offsets, the fiber taper 
+can still couple with the cavity without inducing p arasitic decay. In particular, at 1.3 um 
+offset, the in-plane and bottom Q are at their intr insic limits while the top Q is 75,000, below 
+the intrinsic value of 104,000. As a rough estimate  of the collection efficiency when the taper 
+offset is zero, if the top Q is composed of only th e intrinsic vertical decay and the new fiber-
+induced decay, then the fiber Q is ~16,000. Meanwhi le the total Q for zero offset is ~7,000 
+and thus the collection efficiency, or ratio of dec ay into fiber versus total decay, is 
+7,000/16,000 ≈ 44%. Therefore simulations predict that a much lar ger (~10-fold) coupling 
+efficiency can be achieved into the fiber relative to the coupling to free space and a collection 
+lens of NA = 0.5.  
+Fig. 3. Variation of the top, in-plane, bottom, and  total Q factors as a function of taper 
+displacement, d, along the y axis. When the taper i s directly over the cavity, the vertical Q goes 
+down expectedly and the in-plane Q is parasitically  reduced by a factor of two. For large taper 
+offsets, the Q-values approach their intrinsic limi ts. Inset shows the cross-section of the FDTD 
+simulated structure. 
+ 
+4. Experiment 
+4.1 – Free space photoluminescence  
+ 
+Photoluminescence (PL) from fabricated photonic cry stal cavities was collected in free space, 
+in the direction perpendicular to the PC plane, thr ough a 100x (NA = 0.5) objective lens, 
+directed toward a monochromator, and detected with a liquid nitrogen cooled linear InGaAs 
+CCD array, same as in our earlier work [7]. PL from  the cavity region was spatially filtered 
+with an iris in order to minimize the background si gnal and optimized for greatest collection 
+efficiency. Samples were pumped with a 980 nm laser  diode at 6 mW. Fig. 4 displays the PL 
+of the L3 cavity showing the fundamental resonance at approximately 1535.2 nm at the 
+maximum of erbium’s typical emission spectrum. This  particular cavity was chosen for testing 
+because its cavity resonance overlaps with the peak  emission of erbium. A higher order mode 
+is observed at 1480 nm and the features at 1560 nm are likely band edge modes [17]. The 
+measured Q value of the intrinsic, unloaded cavity was found to be 12,630 as seen by the fit to 
+a Lorentzian in the inset of Fig. 4.   
+Fig. 4. PL spectrum of the Er:SiNx on Si PC cavity,  collected from free space in the direction 
+perpendicular to the PC plane (through an objective  lens with NA = 0.5). The fundamental 
+cavity mode corresponds to the highest peak near 15 30nm, and the background emission has 
+the expected erbium profile. The inset shows the fu ndamental peak zoomed in with a 
+Lorentzian fit.   
+ 
+4.2 – Passive (transmission) cavity measurements vi a fiber taper 
+Transmission measurements were performed by couplin g a broadband source into a fiber 
+taper that was well aligned with an L3 cavity main axis, and by measuring the spectrum at the 
+output of the fiber with an optical spectrum analyz er (OSA) (Fig. 5(a)). Light was not 
+polarized at the input so that all the modes of the  cavity could be excited, thus facilitating the 
+identification of the fundamental mode. The taper r ested on the sample surface.  
+Fig. 5(b) shows a full transmission spectrum of the  cavity whose PL is shown in Fig. 4. 
+The fundamental resonance is clearly seen at about 1530 nm, as in Fig. 4, as well as two 
+higher order modes at shorter wavelengths. Four add itional modes were found at wavelengths 
+below 1400 nm. The background loss of 2-5 dB comes primarily from TM loss in the PC slab 
+and characteristic absorption of the erbium-doped n itride material.    
+ 
+Fig. 5. (a) Experimental setup for measurement of t ransmission through the Er:SiNx on Si PC 
+cavities via fiber taper. The fiber taper is aligne d and in contact with the L3 cavity. The output 
+spectrum of a broadband source at the input of the fiber is measured by an OSA. Inset shows 
+optical picture of aligned taper. (b) Full transmis sion spectrum of the L3 cavity. (c) Multiple 
+transmission spectra for the same cavity but with d ifferent fiber taper offsets along the y axis.   
+ 
+In Fig. 5(c), multiple transmission scans are taken  for the taper as it is progressively offset 
+from the main axis in of the cavity in the y-direct ion. The short pull length of the taper was 
+necessary for the tension of the fiber to be large enough to allow this seamless translation, 
+which is otherwise prevented by the Van der Waals s ticking attraction between the fiber and 
+semiconductor. As can be seen in the plot, as the f iber taper is dragged away from the cavity 
+center, the transmission coupling magnitude drops a nd the Q factor increases. This is not 
+surprising since the coupling coefficient is given by the overlap between the cavity and 
+waveguide modes [18]. As the taper shifts, this ove rlap decreases along with the transmission 
+coupling. Likewise, since this secondary decay path  via the taper decreases with taper offset, 
+the measured Q value approaches the intrinsic limit . Also worth noting is the red shift induced 
+by the taper as it moves over the cavity which is d ue to the increased effective index of the 
+cavity.    
+   
+4.3 – Fiber-outcoupled photoluminescence   
+Fiber taper coupling provides a convenient means to  extract and collect photoluminescence 
+from a cavity mode that would ordinarily radiate no n-uniformly into free space. A well 
+designed cavity (such as the described L3 type) is meant to have few wave vector components 
+in the light cone in order to maximize the Q (see F ig. 2) [14]. The fiber taper can, on the other 
+hand, extract those components that are outside the  light cone through evanescent coupling. 
+PL measurements with a coupled fiber taper were tak en under similar conditions as those 
+for the free space measurement. To do so, a 980 nm laser diode pump with power 6 mW was 
+focused to a small spot where the fiber taper reste d on the cavity. The output from one end of 
+the fiber was then sent to the same spectrometer se tup as before (Fig. 6(a)). 
+ 
+a)  b)  
+c)   
+Fig. 6. (a) Experimental setup for outcoupling PL f rom Er:SiNx on Si PC cavities via fiber 
+tapers. Pump light is focused on the cavity while P L emission is collected by the same fiber 
+taper and sent to a spectrometer. (b) Full PL spect rum for fiber-coupled emission from the PC 
+cavity. The integrated intensity is 2.5x larger tha n for the free space measurement and the peak 
+has a Q of 4,300. (c) A different data point for wh ich the taper was offset from the cavity so 
+that the cavity was minimally loaded. 
+ 
+When the taper is positioned over the center of the  cavity, the PL extraction is maximized 
+as can be seen in Fig. 6(b). The non-resonant backg round PL of erbium is minimized in this 
+configuration while the integrated intensity of the  fundamental mode is 2.5 times that 
+collected through free space (in the vertical direc tion) and an NA = 0.5 objective lens (Fig. 4). 
+Accounting for the loss in this particular taper an d adding up the intensity in both arms, the 
+total collection into the fiber is 12.5 times the c ollection through the NA = 0.5 lens in the 
+direction perpendicular to the PC slab. Since the r adiation calculation revealed that collection 
+through free space is 4.2%, the total fiber couplin g efficiency is 53%. This value is pretty 
+close to the result predicted by simulation and add itional discrepancies could be due to 
+differences in the actual taper size and exact tape r positioning. Such a high extraction 
+efficiency is made possible because the fiber taper  relies on evanescent coupling from the 
+cavity and is not limited geometrically by the fini te extent of an objective lens. It is interesting 
+to note that a large coupling ratio can be obtained  despite the phase matching between the 
+taper and cavity not being fully optimized [19]. Th is is a result of the fact that a strongly 
+localized cavity resonance spans a number of k-vect ors, which permits coupling to a taper 
+mode. Additionally, the curved taper itself spans a  wide k-space [20] as was found out from 
+separate coupling experiments with waveguides. 
+Fig. 6(c) shows the PL of the cavity resonance for the case when the taper is displaced 
+from the cavity main axis. The Q in this case is fo und to be approximately 12,350, which 
+corresponds to 98% of the intrinsic Q value. Coupli ng to fiber tapers in direct contact 
+measurements is usually associated with high loadin g of the cavities due to either direct 
+coupling into the fiber or parasitic loss into othe r decay paths [19]. As expected, we confirm 
+that by reducing the coupling strength (by decreasi ng the mode field overlap), the cavity Q 
+can remain very high. This agrees with simulation r esults in Fig. 3. The ability to have a 
+control over the cavity Q is important for PC cavit y-based lasers or amplifiers which need to 
+have as low loss as possible.    
+a)  b)  
+c)  To more clearly see the relationship between collec tion efficiency and measured Q value, 
+a series of spectra were taken for various taper of fsets from the cavity as seen in Fig. 7(a). The 
+fiber taper was slowly displaced along y-direction and as expected, the measured cavity Q 
+increased while the collected intensity decreased. Fig. 7(b) shows a few traces at different 
+coupling strengths where the intensity-Q relationsh ip is seen clearly. This behavior can be 
+described by an efficiency relationship under the c ondition that no parasitic loading of the 
+cavity takes place [9]: 
+ 
+        
+01QQt−=η                 (1) 
+ 
+Here η is the collection efficiency given as the decay ra te into the fiber divided by the total 
+decay rate, Qt is the loaded cavity Q, and Q0 is the intrinsic Q. The straight line in Fig. 7(a)  is 
+a fit to the data using Eq. (1). As can be seen in the plot, the fit is only good for the region of 
+low coupling (high Q) and deviates significantly wh ere taper loading is strong. This is 
+expected from simulations, since for high coupling the parasitic Q grows and the denominator 
+in equation (1) would have to be modified by adding  a parasitic Qp term. The additional 
+degradation is due to the newly generated decay fro m the parasitic TM coupling. 
+ 
+Fig.7. (a) Collected PL intensity and the total cav ity Q-factor as the fiber taper is offset from 
+the cavity along the y-direction. The blue line is a fit to the data. (b) Several spectra for the data  
+in (a). As the taper moves away from the cavity the  integrated intensity decreases and the Q 
+increases. The peaks are centered together for easi er visualization. 
+ 
+5. Conclusion 
+In summary, we demonstrate fiber taper coupling and  efficient PL extraction from a CMOS-
+compatible light-emitting material, Er:SiNx embedde d in a Si PC cavity. The collection 
+efficiency through the fiber taper was 2.5 times th e collection through free space and could be 
+improved up to 6.25 times through better taper fabr ication and even 12.5 times with an 
+interferometric taper design [21]. Larger collectio n enhancements are also possible by better 
+phase matching of the fiber and cavity modes. The l oaded quality factor of the coupled cavity 
+can remain very high and in the limit of weak coupl ing can be as high as 98% of its intrinsic 
+value. A broad tuning range of collection efficienc ies and quality factors is made possible by a 
+variable coupling strength inversely proportional t o the taper displacement from the cavity. 
+The fiber taper provides a convenient platform for linking the nano-scale on-chip environment 
+with the off-chip interconnect network, which is im portant for building devices such as on-
+chip optical amplifiers. For the promising erbium-d oped silicon nitride material, fiber coupled 
+emission is an easy and efficient way to extract li ght and connect to outside systems.     
+ 
+Acknowledgements 
+The authors would like to acknowledge the MARCO Int erconnect Focus Center and the U.S. 
+Air Force MURI program under Award No. FA9550-06-1- 0470 for funding. Gary Shambat 
+and Yiyang Gong would also like to thank the NSF GR F for support. The fabrication has been 
+performed in the Stanford Nanofabrication Facilitie s.  
+ 
\ No newline at end of file
diff --git a/1001.0431.txt b/1001.0431.txt
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index 0000000000000000000000000000000000000000..714b718ab8139e0aadf105f57bdfd7fcc9f7e1d4
--- /dev/null
+++ b/1001.0431.txt
@@ -0,0 +1,361 @@
+arXiv:1001.0431v1  [cond-mat.supr-con]  4 Jan 2010The single-ion anisotropy in LaFeAsO
+Ren-Gui Zhu
+College of physics and electronic information, Anhui Normal Univers ity, Wuhu,
+241000, P. R. China
+E-mail:rgzhu@mail.ahnu.edu.cn
+Abstract. We use Green’s function method to study the Heisenberg model of
+LaFeAsO with the striped antiferromagnetic collinear spin structur e. In addition to
+the intra-layer spin couplings J1a,J1b,J2and the inter-layer coupling Jc, we further
+consider the contributions of the single-ion anisotropy Js. The analytical expressions
+for the magnetic phase-transition temperature TNand the spin spectrum gap ∆ are
+obtained. According to the experimental temperature TN= 138K and the previous
+estimations of the coupling interactions, we make a further discuss ion about the
+magnitude and the effects of the single-ion anisotropy Js. We find that the magnitudes
+ofJsandJccan compete. The dependences of the transition temperature TN, the
+zero-temperature average spin and the spin spectrum gap on the single-ion anisotropy
+are investigated. We find they both increase as Jsincreases. The spin spectrum gap
+at low temperature T→0 is calculated as a function of Js, the result of which is a
+useful reference for the future experimental researches.R G Zhu 2
+1. Introduction
+It was recently discovered that aniron-basedmaterial LaFeAsO s hows high-temperature
+superconductivity when O atomsarepartially substituted by F atom s[1]. This discovery
+has triggered great research interest on the FeAs-based pnictid es superconductors and
+their undoped compounds. It has been theoretically and experimen tally confirmed
+that these pure FeAs-based compounds have a ground state with collinear stripe-like
+antiferromagnetic(AF) spin order formed by Fe atoms[2, 3, 4, 5, 6 , 7, 8, 9, 10, 11]. Thus
+to establish an effective spin Hamiltonian for them and to elucidate the corresponding
+antiferromagnetism are helpful in understanding the underlying me chanism to make
+them superconducting upon doping.
+For undoped LaFeAsO and other similar parent compounds, a Heisen berg exchange
+model was suggested to explain their AF structure[2, 4, 12, 13], an d was used to explore
+theirmagneticproperties[14,15,16]. Figure1showstheunitcellof theorthorhombicAF
+spin structure of the Fe lattice. This orthorhombic structure exis ts below a structure
+transition temperature TS, which is 15 ∼20K higher than the magnetic transition
+temperature TN[6, 17]. Usually the nearest neighbor (NN) coupling J1(including J1a
+andJ1b), and the next-nearest neighbor (NNN) coupling J2in FeAs layers are dominant
+and must be considered. The NN coupling Jcbetween spins on neighboring layers is
+regarded to be much smaller than the in-plane couplings[2, 4]. Howeve r,Jcwas found
+to be essential for the existence of a non-zero magnetic transitio n temperature TN[15].
+A further consideration can include the single-ion anisotropy Js. It was estimated to
+be even much smaller than Jcin the model of SrFe 2As2, but a spin spectrum gap was
+found to be produced by it[14]. For LaFeAsO, so far there is no res earch report about
+the magnitude or the effects of the single-ion anisotropy.
+In this paper, we use Green’s function method[18] to study the Heis enberg model
+of FeAs-based pure parent compounds. The Hamiltonian of this mod el in a detailed
+form is
+H=1
+2J1a/summationdisplay
+/angbracketleftij/angbracketrightS1i·S1j+1
+2J1a/summationdisplay
+/angbracketleftij/angbracketrightS2i·S2j+J1b/summationdisplay
+/angbracketleftij/angbracketrightS1i·S2j
++J2/summationdisplay
+/angbracketleft/angbracketleftij/angbracketright/angbracketrightS1i·S2j+Jc/summationdisplay
+/angbracketleftij/angbracketrightS1i·S2j−Js/summationdisplay
+i[(Sz
+1i)2+(Sz
+2i)2],(1)
+where the spin coupling Jcbetween layers and the single-ion anisotropy Jsare both
+considered. The subscripts 1 and 2 mean the sublattices 1 and 2 res pectively. /angbracketleftij/angbracketright
+means NN spin pairs, and /angbracketleft/angbracketleftij/angbracketright/angbracketrightmeans NNN spin pairs. The self-consistent equations
+for the average sublattice spin will be derived. An analytical expres sion for the magnetic
+transition temperature TNwill be obtained. For LaFeAsO, according to the recent
+estimations of the strengths J1a,J1b,J2andJc[15] with the experimental temperature
+TN= 138K[6, 17], we shall make afurther estimation ofthe single-ionanis otropyJs. We
+find that the magnitude of Jscan compete with Jc, and in some situations even bigger
+thanJc. The effects of the single-ion anisotropy on the transition tempera tureTN, the
+zero-temperature average spin /angbracketleftSz/angbracketright0and the spin spectrum gap are investigated. WeR G Zhu 3
+xyz
+abc
+aJ1
+bJ12JcJ
+Figure 1. A unit cell of the orthorhombic Fe spin lattice. a,b,care the three base
+vectors. The lattice consists of two sublattices (distinguished by t he color gray and
+black).
+find they all increase as Jsincreases. In section 2, we shall give our analytical results
+derived from the Green’s function method. In section 3, we shall pr esent our numerical
+results. Finally a conlusion is given in section 4.
+2. Green’s function derivation
+According to the general scheme of Green’s function method to so lve an
+antiferromagnetic spin model with two sublattices, we construct t he following Green’s
+functions:
+G1k,1l(ω) =/angbracketleft/angbracketleftS+
+1k;S−
+1l/angbracketright/angbracketright, G2k,1l(ω) =/angbracketleft/angbracketleftS+
+2k;S−
+1l/angbracketright/angbracketright. (2)
+The equation of motion is
+ω/angbracketleft/angbracketleftA;S−
+1l/angbracketright/angbracketright=/angbracketleft[A,S−
+1l]/angbracketright+/angbracketleft/angbracketleft[A,H];S−
+1l/angbracketright/angbracketright, (3)
+whereArepresents the spin operator S+
+1korS+
+2k. The commutator [ A,H] can be derived
+using Hamiltonian (1) and the basic commutation relations of spin oper ators:[S+
+i,S−
+j] =
+2Sz
+iδij, [Sz
+i,S±
+j] =∓S±
+iδij, whereS±
+i=Sx
+i±iSy
+i.
+In order to close the system of equations, the so-called PRA or Tya blikov
+decoupling[18] is adopted for the terms stemming from the exchang e couplings:
+/angbracketleft/angbracketleftSz
+iS+
+j;S−
+l/angbracketright/angbracketright ≈ /angbracketleftSz
+i/angbracketright/angbracketleft/angbracketleftS+
+j;S−
+l/angbracketright/angbracketright, i/negationslash=j. (4)
+While for the terms stemming from the single-ion anisotropy, we adop t the Anderson-
+Callen(AC) decoupling[19]:
+/angbracketleft/angbracketleftSz
+iS+
+i+S+
+iSz
+i;S−
+j/angbracketright/angbracketright ≈2/angbracketleftSz
+i/angbracketrightΘ(z)
+i/angbracketleft/angbracketleftS+
+i;S−
+j/angbracketright/angbracketright, (5)
+where
+Θ(z)
+i= 1−1
+2S2[S(S+1)−/angbracketleftSz
+iSz
+i/angbracketright]. (6)
+The AC decoupling has been demonstrated to be most adequate for the single-ion
+anisotropy much small compared to the exchange interactions[20 , 21].R G Zhu 4
+In order to write the decoupled equations of motion in the kspace, we take the
+following Fourier transformation:
+Gk,l(ω) =1
+N/summationdisplay
+kG(k,ω)eik·(Rk−Rl), (7)
+whereNis the number of sites in either sublattice, and the summation over kis
+restricted to the first Brillouin zone of the sublattice. At the same t ime, the equation
+δij=1
+N/summationtext
+keik·(Ri−Rj)is also used.
+Because of the translation invariant, we have /angbracketleftSz
+1k/angbracketright=/angbracketleftSz/angbracketright,/angbracketleftSz
+2k/angbracketright=−/angbracketleftSz/angbracketrightand
+Θ(z)
+1k= Θ(z)
+2k= Θ(z)= 1−1
+2S2[S(S+ 1)− /angbracketleftSzSz/angbracketright]. Finally, we obtain the decoupled
+equations of the two Green’s function in kspace:
+[ω−/angbracketleftSz/angbracketrightAk]G11(k,ω)−/angbracketleftSz/angbracketrightBkG21(k,ω) = 2/angbracketleftSz/angbracketright, (8)
+and
+[ω+/angbracketleftSz/angbracketrightAk]G21(k,ω)+/angbracketleftSz/angbracketrightBkG11(k,ω) = 0, (9)
+where
+Ak= 2J1acos(kxa)−2J1a+2J1b+4J2+2Jc+2JsΘ(z), (10)
+and
+Bk= 2J1bcos(kyb)+4J2cos(kxa)cos(kyb)+2Jccos(kzc), (11)
+in which a,b,care the three lattice constants. Solving equations (8) and (9), we obtain
+the Green’s function:
+G11(k,ω) =/angbracketleftSz/angbracketright
+ωk/bracketleftBiggAk/angbracketleftSz/angbracketright+ωk
+ω−ωk−Ak/angbracketleftSz/angbracketright−ωk
+ω+ωk/bracketrightBigg
+, (12)
+and the spin spectrum:
+ωk=/angbracketleftSz/angbracketright/radicalBig
+A2
+k−B2
+k. (13)
+Whenk→0, we obtain a expression for the spectrum gap:
+∆ = 2/angbracketleftSz/angbracketright/radicalBig
+JsΘ(z)[2J1b+4J2+2Jc+JsΘ(z)], (14)
+which is similar with the expression given in ref[14] derived from the spin -wave theory,
+expect for the factor Θ(z). From this expression for the gap, one see that the single-ion
+anisotropy is essential for the existence of the spectrum gap.
+Then following the process of solving the average spin, we derive the correlation
+function /angbracketleftS−S+/angbracketrightusing the spectrum theorem:
+/angbracketleftS−S+/angbracketright=−1
+Nπ/summationdisplay
+k/integraldisplay∞
+−∞dωImG11(k,ω+iǫ)
+eβω−1
+=/angbracketleftSz/angbracketright
+N/summationdisplay
+k
+Ak/radicalBig
+A2
+k−B2
+kcothβωk
+2−1
+, (15)
+in which the equation1
+x+iǫ=P(1
+x)−iπδ(x)(P(···) means taking the principle value)
+has been used to obtain the imaginary part of G11(ω+iǫ), andβ=1
+kBT,kBis the
+Boltzmann constant, Tis the temperature.R G Zhu 5
+According to the theory of Callen[22], the average spin for arbitrar yScan be
+calculated using the following equation:
+/angbracketleftSz/angbracketright=(S−Φ)(1+Φ)2S+1+(S+1+Φ)Φ2S+1
+(1+Φ)2S+1−Φ2S+1, (16)
+where
+Φ =/angbracketleftS−S+/angbracketright
+2/angbracketleftSz/angbracketright
+=1
+2N/summationdisplay
+k
+Ak/radicalBig
+A2
+k−B2
+kcothβωk
+2−1
+. (17)
+On the other hand, the correlation function /angbracketleftSzSz/angbracketrightcan be calculated from the equation
+/angbracketleftSzSz/angbracketright=S(S+1)−(1+2Φ)/angbracketleftSz/angbracketright. Using equation (6), we can relate Θ(z)to Φ by
+Θ(z)= 1−/angbracketleftSz/angbracketright
+2S2(1+2Φ). (18)
+Now the equations (16)(17)(18) can be solved self-consistently t o obtain the average
+spin at any given temperature, provided we know the values of the e xchange couplings
+J1a,J1b,J2,Jcand the single-ion anisotropy Js.
+When the temperature Tapproaches zero, we obtain coth(βωk
+2)→1. The equation
+(17) is reduced to
+Φ|T→0=1
+2N/summationdisplay
+k
+Ak/radicalBig
+A2
+k−B2
+k−1
+. (19)
+The zero-temperature average spin /angbracketleftSz/angbracketright0can be obtained by self-consistently solving
+the equations (16)(18)(19).
+When the temperature Tapproaches the magnetic transition temperature TN, the
+average spin /angbracketleftSz/angbracketrightas well as the spectrum ωkwill approach zero. Expanding coth(βωk
+2)
+in the equation (17), we obtain
+Φ|T→TN≈Γ
+β/angbracketleftSz/angbracketright−1
+2, (20)
+where Γ =1
+N/summationtext
+kAk
+A2
+k−B2
+k. Inserting (20) into (16), and expanding the terms in the
+denominator and the numerator as the series of /angbracketleftSz/angbracketright, we finally derive
+/angbracketleftSz/angbracketright ≈/radicaltp/radicalvertex/radicalvertex/radicalbt12(ΓkBTN)2
+S(2S−1)/parenleftbigg
+1−T
+TN/parenrightbigg
+, (21)
+where
+TN=S(S+1)
+3kBΓ. (22)
+On the other hand, inserting (20) into (18), and using the equation (22), we obtain the
+reduced expression for Θ(z)near the temperature TN:
+Θ(z)|T→TN≈2S−1
+3S. (23)R G Zhu 6
+00.0005 0.001 0.0015 0.002 0.0025 0.003
+JS/Slash1J1 b0.2150.220.2250.230.2350.240.245TN/Slash1/LParen1J1 b/Slash1kB/RParen1
+Figure 2. The transition temperature TNas a function of the single-ion anisotropy
+JsforJ1a= 0.98,J2= 0.52 andJc= 0.0004 in the unit of J1b= 50 meV. The black
+point correspond to the experimental temperature TN= 0.238 in the unit of J1b/kB.
+3. Numerical results and discussions
+So far there is no consensus on the magnitudes of the exchange co uplingsJ1a,J1b,J2and
+Jc, because of the unclear microscopic origin of the observed AF spin s tructure. Here
+we prefer the estimations in ref[15], which gave J1b= 50±10 meV, J1a= 49±10 meV,
+J2= 26±5 meV and Jc= 0.020±0.015 meV by using the experimental transition
+temperature TN= 138K of pure LaFeAsO. The main purpose of this paper is to
+investigate the magnitude and the effects of the single-ion anisotro pyJsin LaFeAsO.
+Through out our numerical calculation, we take J1b= 50 meV, J1a= 49 meV, J2= 26
+meV and the spin S= 1. The result J1b∼50meV are obtained from the first-principle
+calculating[4, 23]. In the present systems of units, the Boltzmann c onstant is taken as
+kB= 0.086 meV/K.
+Figure 2 shows the effect of the single-ion anisotropy Json the transition
+temperature TN. We can see that TNincreases as Jsincreases. This means that the
+single-ion anisotropy term is in favor of the AF spin structure. It ca n be understood
+from the expression of the single-ion anisotropy term in the Hamilton ian (1). Increasing
+the magnitude of Jswill make the spins incline to alignalong the zaxis, andgive a lower
+total energy, which make the system more stable. To one’s surpris e, the magnitude of Js
+corresponding to theexperimental transitiontemperature TNisabout 0 .00143J1b, which
+is much bigger than the magnitude of the exchange coupling Jc= 0.0004J1bestimated
+in ref[15]. Furthermore, we see from figure 3 that the variation ran ge of/angbracketleftSz/angbracketright0withJs
+varying from 0 to 0.1 is almost the same as the one produced by Jcin ref[15]. All these
+results imply that the magnitude of Jsis probably not much small compared with Jc.
+So the estimation of Jcmaybe need to be adjusted if the single-ion anisotropy term is
+considered. As to the estimations of the other exchange couplings J1a,J1bandJ2, we
+think there are still reasonable.R G Zhu 7
+0 0.02 0.04 0.06 0.08 0.1
+JS/Slash1J1 b0.70.720.740.760.78/LessSz/Greater0
+Figure 3. The zero-temperatureaveragespin asafunction ofthe single-ion anisotropy
+JsforJ1a= 0.98,J2= 0.52 andJc= 0.0004 in the unit of J1b= 50 meV.
+0 0.0002 0.0004 0.0006 0.0008 0.001
+Jc/Slash1J1 b0.00050.0010.00150.0020.0025JS/Slash1J1 b
+Figure 4. The competing relation of JcandJs. The curve is depicted by using the
+equation (22) for TN= 0.238(J1b/kB),J1a= 0.98J1bandJ2= 0.52J1b.
+Figure4 shows the competing relation of JcandJswhen thetransition temperature
+is fixed at the experimental value. The increase of Jcis accompanied by the decrease
+ofJs, and vice vera. From figure 4, we can see that the ranges of their c orresponding
+variations are at the same magnitude, which implies they probably hav e the same status
+in the viewpoint of theoretical study. As to revealing the actual ma gnitudes of the two
+parameters JcandJs, we think it is not enough to use only the experimental transition
+temperature TN.
+Figure 5 shows the effect of the single-ion anisotropy Json the spectrum gap at
+low temperature. The gap vanishes as Jsvanishes, and increases as Jsincreases. We
+find the effect of Jcon the gap is very trivial. The curves for different values of Jc
+between (0 .0001,0.01) are almost the same, while the single-ion anisotropy affects the
+gap apparently. Considering the gap can be obtained from inelastic n eutron-scatteringR G Zhu 8
+0 0.02 0.04 0.06 0.08 0.1
+Js/Slash1J1 b00.10.20.30.40.50.6/CapDelta/Slash1J1 bJc= 0.01
+❅❅ ❘
+Jc= 0.0001❅❅ ■
+Figure 5. The spin spectrum gaps at low temperature T→0 as functions of the
+single-ion anisotropy JsforJ1a= 0.98J1b,J2= 0.52J1bandJc= 0.0001J1b(the below
+curve),Jc= 0.01J1b(the above curve).
+experiment[14], we suggest that the magnitude of the single-ion anis otropy be estimated
+fromthefutureexperimentalresultsofthespectrumgap. Fore xample, if Jc= 0.0004J1b,
+we obtain Js= 0.00143J1bfrom the experimental transition temperature TN= 138K.
+Then calculating the spectrum gap with Js= 0.00143J1b, we obtain the magnitude of
+the gap ∆ ≈3.4 meV, which can be compared with the future experimental result.
+4. Conclusion
+We use Green’s function method to study the Heisenberg model (1) of LaFeAsO with
+the striped AF spin structure as shown in figure 1. The main purpose of this paper is to
+investigate the magnitude and the effects of the single-ion anisotro pyJs. We derive the
+self-consistent equations for the average spin, and obtained the analytical expressions
+for the spin spectrum gap ∆, and the magnetic transition temperat ureTN. We find
+that the transition temperature TN, the zero-temperature average spin /angbracketleftSz/angbracketright0and the
+spin spectrum gap ∆ are all increasing functions of the single-ion anis otropyJs. From
+our numerical results by using TN= 138K and the previous estimations of J1a,J1b,J2
+andJcin ref[15], we find that the magnitude of Jsis probably not much small compared
+withJc. Because the single-ion anisotropy is essential for the existence o f the spin
+spectrum gap, we suggest using the experimental result of the sp in spectrum gap to fix
+the magnitude of the single-ion anisotropy Jsin the future.
+References
+[1] Kamihara Y, Watanabe T, Hirano M and Hosono H 2008 J. Am. Chem. Soc. 1303296
+[2] Yildirim T 2008 Phys. Rev. Lett. 101057010
+[3] Cao C, Hirschfeld P J and Cheng H P 2008 Phys. Rev. B77220506(R)
+[4] Ma F and Lu Z Y 2008 Phys. Rev. B78033111R G Zhu 9
+[5] Dong J et al2008Europhys. Lett. 8327006
+[6] de la Cruz C et al2008Nature453899
+[7] Chen Y et al2008Phys. Rev. B78064515
+[8] Huang Q et al2008Phys. Rev. Lett. 101257003
+[9] Zhao J et al2008Phys. Rev. B78140504(R)
+[10] Goldman A I et al2008Phys. Rev. B78106506(R)
+[11] McGuire M A et al2008Phys. Rev. B78094517; arXiv:0804.0796
+[12] Si Q and Abrahams E 2008 Phys. Rev. Lett. 101076401
+[13] Fang C et al2008Phys. Rev. B77224509
+[14] Zhao J et al2008Phys. Rev. Lett. 101167203
+[15] Liu G B and Liu B G 2009 J. Phys.:Condens. Matter 21195701
+[16] Yao D X and Carlson E W 2008 Phys. Rev. B72052507
+[17] Klauss H H et al2008Phys. Rev. Lett. 101077005
+[18] Tyablikov S V 1959 Ukr. Mat. Zh. 11289; S. V. Tyablikov 1967 Methods in the Quantum Theory
+of Magnetism (New York: Plenum Press)
+[19] Anderson F B and Callen H B 1964 Phys. Rev. 136A1068
+[20] Fr¨ obrich P, Jensen P J and Kuntz P J 2000 Eur. Phys. J. B13477
+[21] Henelius P, Fr¨ obrich P, Kuntz P J et al2002Phys. Rev. B66094407
+[22] Callen H B 1963 Phys. Rev. 130890
+[23] Yin Z P et al2008Phys. Rev. Lett. 101047001
\ No newline at end of file
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+arXiv:1001.0432v4  [math.RT]  19 Apr 2010LECTURE NOTES ON CHEREDNIK ALGEBRAS
+PAVEL ETINGOF AND XIAOGUANG MA
+Contents
+1. Introduction 4
+2. Classical and quantum Olshanetsky-Perelomov systems for finit e Coxeter groups 6
+2.1. The rational quantum Calogero-Moser system 6
+2.2. Complex reflection groups 6
+2.3. Parabolic subgroups 7
+2.4. Olshanetsky-Perelomov operators 7
+2.5. Dunkl operators 8
+2.6. Proof of Theorem 2.9 9
+2.7. Uniqueness of the operators Lj 11
+2.8. Classical Dunkl operators and Olshanetsky-Perelomov Hamilto nians 11
+2.9. Rees algebras 12
+2.10. Proof of Theorem 2.24 12
+2.11. Notes 13
+3. The rational Cherednik algebra 14
+3.1. Definition and examples 14
+3.2. The PBW theorem for the rational Cherednik algebra 15
+3.3. The spherical subalgebra 15
+3.4. The localization lemma 16
+3.5. CategoryOfor rational Cherednik algebras 16
+3.6. The grading element 17
+3.7. Standard modules 18
+3.8. Finite length 19
+3.9. Characters 19
+3.10. Irreducible modules 20
+3.11. The contragredient module 20
+3.12. The contravariant form 20
+3.13. The matrix of multiplicities 21
+3.14. Example: the rank 1 case 21
+3.15. The Frobenius property 22
+3.16. Representations of H1,cof typeA 23
+3.17. Notes 25
+4. The Macdonald-Mehta integral 26
+4.1. Finite Coxeter groups and the Macdonald-Mehta integral 26
+4.2. Proof of Theorem 4.1 26
+4.3. Application: the supports of Lc(C) 31
+4.4. Notes 35
+15. Parabolic induction and restriction functors for rational Chere dnik algebras 36
+5.1. A geometric approach to rational Cherednik algebras 36
+5.2. Completion of rational Cherednik algebras 36
+5.3. The duality functor 37
+5.4. Generalized Jacquet functors 37
+5.5. The centralizer construction 37
+5.6. Completion of rational Cherednik algebras at arbitrary points o fh/G 38
+5.7. The completion functor 39
+5.8. Parabolic induction and restriction functors for rational Cher ednik algebras 40
+5.9. Some evaluations of the parabolic induction and restriction func tors 41
+5.10. Dependence of the functor Res bonb 41
+5.11. Supports of modules 43
+5.12. Notes 44
+6. The Knizhnik-Zamolodchikov functor 45
+6.1. Braid groups and Hecke algebras 45
+6.2. KZ functors 45
+6.3. The image of the KZ functor 47
+6.4. Example: the symmetric group Sn 48
+6.5. Notes 48
+7. Rational Cherednik algebras and Hecke algebras for varieties wit h group actions 49
+7.1. Twisted differential operators 49
+7.2. Some algebraic geometry preliminaries 49
+7.3. The Cherednik algebra of a variety with a finite group action 49
+7.4. Globalization 51
+7.5. Modified Cherednik algebra 52
+7.6. Orbifold Hecke algebras 52
+7.7. Hecke algebras attached to Fuchsian groups 53
+7.8. Hecke algebras of wallpaper groups and del Pezzo surfaces 55
+7.9. The Knizhnik-Zamolodchikov functor 55
+7.10. Proof of Theorem 7.15 56
+7.11. Example: the simplest case of double affine Hecke algebras 57
+7.12. Affine and extended affine Weyl groups 58
+7.13. Cherednik’s double affine Hecke algebra of a root system 58
+7.14. Algebraic flatness of Hecke algebras of polygonal Fuchsian gr oups 59
+7.15. Notes 61
+8. Symplectic reflection algebras 62
+8.1. The definition of symplectic reflection algebras 62
+8.2. The PBW theorem for symplectic reflection algebras 62
+8.3. Koszul algebras 63
+8.4. Proof of Theorem 8.6 63
+8.5. The spherical subalgebra of the symplectic reflection algebra 6 4
+8.6. The center of the symplectic reflection algebra Ht,c 65
+8.7. A review of deformation theory 66
+8.8. Deformation-theoretic interpretation of symplectic reflectio n algebras 69
+8.9. Finite dimensional representations of H0,c. 71
+28.10. Azumaya algebras 72
+8.11. Cohen-Macaulay property and homological dimension 73
+8.12. Proof of Theorem 8.32 74
+8.13. Notes 75
+9. Calogero-Moser spaces 76
+9.1. Hamiltonian reduction along an orbit 76
+9.2. The Calogero-Moser space 76
+9.3. The Calogero-Moser integrable system 77
+9.4. Proof of Wilson’s theorem 79
+9.5. The Gan-Ginzburg theorem 80
+9.6. The space McforSnand the Calogero-Moser space. 81
+9.7. The Hilbert scheme Hilb n(C2) and the Calogero-Moser space 82
+9.8. The cohomology of Cn 83
+9.9. Notes 84
+10. Quantization of Claogero-Moser spaces 85
+10.1. Quantum moment maps and quantum Hamiltonian reduction 85
+10.2. The Levasseur-Stafford theorem 85
+10.3. Corollaries of Theorem 10.1 87
+10.4. The deformed Harish-Chandra homomorphism 88
+10.5. Notes 89
+References 90
+31.Introduction
+Double affine Hecke algebras, also called Cherednik algebras, were int roduced by Chered-
+nikin1993asatoolinhisproofofMacdonald’sconjecturesaboutor thogonalpolynomialsfor
+root systems. Since then, it has been realized that Cherednik algeb ras are of great indepen-
+dent interest; they appeared in many different mathematical cont exts and found numerous
+applications.
+The present notes are based on a course on Cherednik algebras giv en by the first author
+at MIT in the Fall of 2009. Their goal is to give an introduction to Cher ednik algebras, and
+to review the web of connections between them and other mathema tical objects. For this
+reason, the notes consist of many parts that are relatively indepe ndent of each other. Also,
+to keep the notes within the bounds of a one-semester course, we had to limit the discussion
+of many important topics to a very brief outline, or to skip them altog ether. For a more
+in-depth discussion of Cherednik algebras, we refer the reader to research articles dedicated
+to this subject.
+The notes do not contain any original material. In each section, the sources of the expo-
+sition are listed in the notes at the end of the section.
+The organization of the notes is as follows.
+In Section 2, we define the classical and quantum Calogero-Moser s ystems, and their
+analogs for any Coxeter groups introduced by Olshanetsky and Pe relomov. Then we intro-
+duce Dunkl operators, prove the fundamental result of their co mmutativity, and use them to
+establish integrability of the Calogero-Moser and Olshanetsky-Per elomov systems. We also
+prove the uniqueness of the first integrals for these systems.
+In Section 3, we conceptualize the commutation relations between D unkl operators and
+coordinate operators by introducing the main object of these not es - the rational Cherednik
+algebra. We develop the basic theory of rational Cherednik algebra s (proving the PBW
+theorem), and then pass to the representation theory of ration al Cherednik algebras, more
+precisely, study the structure of category O. After developing the basic theory (parallel to
+the case of semisimple Lie algebras), we completely work out the repr esentations in the rank
+1 case, and prove a number of results about finite dimensional repr esentations and about
+representations of the rational Cherednik algebra attached to t he symmetric group.
+In Section 4, we evaluate the Macdonald-Mehta integral, and then u se it to find the sup-
+ports of irrieducible modules over the rational Cherednik algebras w ith the trivial lowest
+weight, in particular giving a simple proof of the theorem of Varagnolo and Vasserot, classi-
+fying such representations which are finite dimensional.
+In Section 5, we describe the theory of parabolic induction and rest riction functors for
+rational Cherednik algebras, developed in [BE], and give some applicat ions of this theory,
+such as the description of the category of Whittaker modules and o f possible supports of
+modules lying in category O.
+In Section 6, we define Hecke algebras of complex reflection groups , and the Knizhnik-
+Zamolodchikov (KZ) functor from the category Oof a rational Cherednik algebra to the
+category of finite dimensional representations of the correspon ding Hecke algebra. We use
+this functor to prove the formal flatness of Hecke algebras of co mplex reflection groups (a
+theorem of Brou´ e, Malle, and Rouquier), and state the theorem o f Ginzburg-Guay-Opdam-
+Rouquier that the KZ functor is an equivalence from the category Omodulo its torsion part
+to the category of representations of the Hecke algebra.
+4In Section 7, we define rational Cherednik algebras for orbifolds. W e also define the
+corresponding Hecke algebras, and define the KZ functor from th e category of modules over
+the former to that over the latter. This generalizes to the “curve d” case the KZ functor
+for rational Cherednik algebras of complex reflection groups, defi ned in Section 6. We then
+apply the KZ functor to showing that if the universal cover of the o rbifold in question has
+a trivialH2(with complex coefficients), then the orbifold Hecke algebra is forma lly flat,
+and explain why the condition of trivial H2cannot be dropped. Next, we list examples of
+orbifold Hecke algebras which satisfy the condition of vanishing H2(and hence are formally
+flat). These include usual, affine, and double affine Hecke algebras, a s well as Hecke algebras
+attached to Fuchsian groups, which include quantizations of del Pe zzo surfaces and their
+Hilbert schemes; we work these examples out in some detail, highlightin g connections with
+other subjects. Finally, we discuss the issue of algebraic flatness, and prove it in the case of
+algebras of rank 1 attached to Fuchsian groups, using the theory of deformations of group
+algebras of Coxeter groups developed in [ER].
+In Section 8, we define symplectic reflection algebras (which inlude ra tional Cherednik al-
+gebras as a special case), and generalize to them some of the theo ry of Section 3. Namely, we
+use the theory of deformations of Koszul algebras to prove the P BW theorem for symplectic
+reflection algebras. We also determine the center of symplectic refl ection algebras, showing
+that it is trivial when the parameter tis nonzero, and is isomorphic to the shperical subal-
+gebra ift= 0. Next, we give a deformation-theoretic interpretation of symp lectic reflection
+algebras as universal deformations of Weyl algebras smashed with finite groups. Finally, we
+discuss finite dimensional representations of symplectic reflection algebras for t= 0, show-
+ing that the Azumaya locus on the space of such representations c oincides with the smooth
+locus. This uses the theory of Cohen-Macaulay modules and of homo logical dimension in
+commutative algebra. In particular, we show that for Cherednik alg ebras of type An−1, the
+whole representation space is smooth and coincides with the spectr um of the center.
+In Section 9, we give another description of the spectrum of the ce nter of the rational
+Cherednik algebra of type An−1(fort= 0), as a certain space of conjugacy classes of pairs
+of matrices, introduced by Kazhdan, Kostant, and Sternberg, a nd called the Calogero-Moser
+space (this space is obtained by classical hamiltonian reduction, and is a special case of a
+quiver variety). This yields a new construction of the Calogero-Mos er integrable system.
+We also sketch a proof of the Gan-Ginzburg theorem claiming that th e quotient of the
+commuting scheme by conjugation is reduced, and hence isomorphic toC2n/Sn. Finally, we
+explain that the Calogero-Moser space is a topologically trivial defor mation of the Hilbert
+scheme of the plane, we use the theory of Cherednik algebras to co mpute the cohomology
+ring of this space.
+In Section 10, we generalize the results of Section 9 to the quantum case. Namely, we
+prove the quantum analog of the Gan-Ginzburg theorem (the Haris h Chandra-Levasseur-
+Stafford theorem), and explain how to quantize the Calogero-Mose r space using quantum
+Hamiltonianreduction. Notsurprisingly, thisgives thesame quantiza tion aswasconstructed
+in the previous sections, namely, the spherical subalgebra of the r ational Cherednik algebra.
+Acknowledgements. We are grateful to the participants of the course on Cherednik
+algebras at MIT, especially to Roman Travkin and Aleksander Tsymba lyuk, for many useful
+comments and corrections. This work was partially supported by th e NSF grants DMS-
+0504847 and DMS-0854764.
+52.Classical and quantum Olshanetsky-Perelomov systems for f inite
+Coxeter groups
+2.1.The rational quantum Calogero-Moser system. Consider thedifferential operator
+H=n/summationdisplay
+i=1∂2
+∂x2
+i−c(c+1)/summationdisplay
+i/\e}atio\slash=j1
+(xi−xj)2.
+This is the quantum Hamiltonian for a system of nparticles on the line of unit mass and the
+interaction potential (between particle 1 and 2) c(c+ 1)/(x1−x2)2. This system is called
+the rational quantum Calogero-Moser system .
+It turns out that the rational quantum Calogero-Moser system is completely integrable.
+Namely, we have the following theorem.
+Theorem 2.1. There exist differential operators Ljwith rational coefficients of the form
+Lj=n/summationdisplay
+i=1(∂
+∂xi)j+lower order terms , j= 1,...,n,
+which are invariant under the symmetric group Sn, homogeneous of degree −j, and such
+thatL2=Hand[Lj,Lk] = 0,∀j,k= 1,...,n.
+We will prove this theorem later.
+Remark 2.2. L1=/summationtext
+i∂
+∂xi.
+2.2.Complex reflection groups. Theorem 2.1 can be generalized to the case of any finite
+Coxeter group. To describe this generalization, let us recall the ba sic theory of finite Coxeter
+groups and, more generally, complex reflection groups.
+Lethbe a finite-dimensional complex vector space. We say that a semisimp le element
+s∈GL(h) is a(complex) reflection if rank(1−s) = 1. This means that sis conjugate to
+the diagonal matrix diag( λ,1,...,1) whereλ/\e}atio\slash= 1.
+Now assume hcarries a nondegenerate inner product ( ·,·). We say that a semisimple
+elements∈O(h) is areal reflection if rank(1−s) = 1; equivalently, sis conjugate to
+diag(−1,1,...,1).
+Now letG⊂GL(h) be a finite subgroup.
+Definition 2.3. (i) We say that Gis acomplex reflection group if it is generated by
+complex reflections.
+(ii) Ifhcarries an inner product, then a finite subgroup G⊂O(h) is areal reflection
+group(or afinite Coxeter group ) ifGis generated by real reflections.
+For the complex reflection groups, we have the following important t heorem.
+Theorem 2.4 (The Chevalley-Shepard-Todd theorem, [Che]) .A finite subgroup GofGL(h)
+is a complex reflection group if and only if the algebra (Sh)Gis a polynomial (i.e., free)
+algebra.
+By the Chevalley-Shepard-Todd theorem, the algebra ( Sh)Ghas algebraically independent
+generators Pi, homogeneous of some degrees difori= 1,...,dimh. The numbers diare
+uniquely determined, and are called the degrees ofG.
+6Example 2.5. IfG=Sn,h=Cn−1(the space of vectors in Cnwith zero sum of co-
+ordinates), then one can take Pi(p1,...,pn) =pi+1
+1+···+pi+1
+n,i= 1,...,n−1 (where/summationtext
+ipi= 0).
+2.3.Parabolic subgroups. LetG⊂GL(h) be a finite subgroup.
+Definition 2.6. A parabolic subgroup of Gis the stabilizer Gaof a pointa∈h.
+Note that by Chevalley’s theorem, a parabolic subgroup of a complex (respectively, real)
+reflection group is itself a complex (respectively, real) reflection gr oup.
+Also, ifWis a real reflection group, then it can be shown that a subgroup W′⊂Wis
+parabolicifandonlyifitisconjugatetoasubgroupgeneratedbyasu bsetofsimplereflections
+ofW. In this case, the rank of W′, i.e. the number of generating simple reflections, equals
+the codimension of the space hW′.
+Example 2.7. Consider the Coxeter group of type E8. It has the Dynkin diagram:
+• • • • • • ••
+The parabolic subgroups will be Coxeter groups whose Dynkin diagra ms are obtained by
+deleting vertices from the above graph. In particular, the maximal parabolic subgroups are
+D7,A7,A1×A6,A2×A1×A4,A4×A3,D5×A2,E6×A1,E7.
+SupposeG′⊂Gis a parabolic subgroup, and b∈his such that Gb=G′. In this case,
+we have a natural G′-invariant decomposition h=hG′⊕(h∗G′)⊥, andb∈hG′. Thus we have
+a nonempty open set hG′
+regof alla∈hG′for whichGa=G′; this set is nonempty because it
+containsb. We also have a G′-invariant decomposition h∗=h∗G′⊕(hG′)⊥, and we can define
+the open set h∗G′
+regof allλ∈hG′for whichGλ=G′. It is clear that this set is nonempty. This
+implies, in particular, that one can make an alternative definition of a p arabolic subgroup
+ofGas the stabilizer of a point in h∗.
+2.4.Olshanetsky-Perelomov operators. Lets⊂GL(h) be a complex reflection. Denote
+byαs∈h∗an eigenvector in h∗ofswith nontrivial eigenvalue.
+LetW⊂O(h) be a real reflection group and S⊂Wthe set of reflections. Clearly, W
+acts onSby conjugation. Let c:S→Cbe a conjugation invariant function.
+Definition 2.8. [OP] The quantum Olshanetsky-Perelomov Hamiltonian attached to Wis
+the second order differential operator
+H:= ∆h−/summationdisplay
+s∈Scs(cs+1)(αs,αs)
+α2
+s,
+where ∆ his the Laplace operator on h.
+Here we use the inner product on h∗which is dual to the inner product on h.
+Let us assume that his an irreducible representation of W(i.e.Wis an irreducible finite
+Coxeter group, and his its reflection representation.) In this case, we can take P1(p) =p2.
+Theorem 2.9. The system defined by the Olshanetsky-Perelomov operator His completely
+integrable. Namely, there exist differential operators Ljonhwith rational coefficients and
+symbolsPj, such that Ljare homogeneous (of degree −dj),L1=H, and[Lj,Lk] = 0,∀j,k.
+7This theorem is obviously a generalization of Theorem 2.1 about W=Sn.
+To prove Theorem 2.9, one needs to develop the theory of Dunkl op erators.
+Remark 2.10. 1. We will show later that the operators Ljare unique.
+2. Theorem 2.9 for classical root systems was proved by Olshanets ky and Perelomov
+(see [OP]), following earlier work of Calogero, Sutherland, and Moser in type A. For a
+general Weyl group, this theorem (in fact, its stronger trigonom etric version) was proved by
+analytic methods in the series of papers [HO],[He3],[Op3],[Op4]. A few years later, a simple
+algebraic proof using Dunkl operators, which works for any finite C oxeter group, was found
+by Heckman, [He1]; this is the proof we will give below.
+For the trigonometric version, Heckman also gave an algebraic proo f in [He2], which used
+non-commuting trigonometric counterparts of Dunkl operators . This proof was later im-
+proved by Cherednik ([Ch1]), who defined commuting (although not W eyl group invariant)
+versions of Heckman’s trigonometric Dunkl operators, now called D unkl-Cherednik opera-
+tors.
+2.5.Dunkl operators. LetG⊂GL(h) be a finite subgroup. Let Sbe the set of reflections
+inG. For any reflection s∈S, letλsbe the eigenvalue of sonαs∈h∗(i.e.sαs=λsαs),
+and letα∨
+s∈hbe an eigenvector such that sα∨
+s=λ−1
+sα∨
+s. We normalize them in such a way
+that/a\}b∇acketle{tαs,α∨
+s/a\}b∇acket∇i}ht= 2.
+Letc:S→Cbe a function invariant with respect to conjugation. Let a∈h.
+The following definition was made by Dunkl for real reflection groups , and by Dunkl and
+Opdam for complex reflection groups.
+Definition 2.11. The Dunkl operator Da=Da(c) onC(h) is defined by the formula
+Da=Da(c) :=∂a−/summationdisplay
+s∈S2csαs(a)
+(1−λs)αs(1−s).
+Clearly,Da∈CG⋉D(hreg), where hregis the set of regular points of h(i.e. not preserved
+by any reflection), and D(hreg) denotes the algebra of differential operators on hreg.
+Example 2.12. LetG=Z2,h=C. Then there is only one Dunkl operator up to scaling,
+and it equals to
+D=∂x−c
+x(1−s),
+where the operator sis given by the formula ( sf)(x) =f(−x).
+Remark 2.13. The Dunkl operators Damap the space of polynomials C[h] to itself.
+Proposition 2.14. (i)For anyx∈h∗, one has
+[Da,x] = (a,x)−/summationdisplay
+s∈Scs(a,αs)(x,α∨
+s)s.
+(ii)Ifg∈GthengDag−1=Dga.
+Proof.(i) The proof follows immediately from the identity
+x−sx=1−λs
+2(x,α∨
+s)αs.
+(ii) The identity is obvious from the invariance of the function c. /square
+8The main result about Dunkl operators, on which all their application s are based, is the
+following theorem.
+Theorem 2.15 (C. Dunkl, [Du1]) .The Dunkl operators commute:
+[Da,Db] = 0for anya,b∈h.
+Proof.Letx∈h∗. We have
+[[Da,Db],x] = [[Da,x],Db]−[[Db,x],Da].
+Now, using Proposition 2.14, we obtain:
+[[Da,x],Db] =−[/summationdisplay
+s∈Scs(a,αs)(x,α∨
+s)s,Db]
+=−/summationdisplay
+s∈Scs(a,αs)(x,α∨
+s)(b,αs)sDα∨s·1−λ−1
+s
+2.
+Sinceaandboccur symmetrically, we obtain that [[ Da,Db],x] = 0. This means that for any
+f∈C[h], [Da,Db]f=f[Da,Db]1 = 0. So for f,g∈C[h],g·[Da,Db]f
+g= [Da,Db]f= 0. Thus
+[Da,Db]f
+g= 0 which implies [ Da,Db] = 0 in the algebra CG⋉D(hreg) (since this algebra
+acts faithfully on C(h)). /square
+2.6.Proof of Theorem 2.9. For any element B∈CW⋉D(hreg), definem(B) to be the
+differential operator C(h)W→C(h),defined byB. That is, if B=/summationtext
+g∈WBgg,Bg∈D(hreg),
+thenm(B) =/summationtext
+g∈WBg. It is clear that if BisW-invariant, then∀A∈CW⋉D(hreg),
+m(AB) =m(A)m(B).
+Proposition 2.16 ([Du1], [He1]) .Let{y1,...,yr}be an orthonormal basis of h. Then we
+have
+m(r/summationdisplay
+i=1D2
+yi) =H,
+whereH= ∆h−/summationtext
+s∈Scs(αs,αs)
+αs∂α∨s.
+Proof.For anyy∈h, we havem(D2
+y) =m(Dy∂y). A simple computation shows that
+Dy∂y=∂2
+y−/summationdisplay
+s∈Scsαs(y)
+αs(1−s)∂y
+=∂2
+y−/summationdisplay
+s∈Scsαs(y)
+αs(∂y(1−s)+αs(y)∂α∨ss).
+This means that
+m(D2
+y) =∂2
+y−/summationdisplay
+s∈Scsαs(y)2
+αs∂α∨s.
+9So we get
+m(r/summationdisplay
+i=1D2
+yi) =r/summationdisplay
+i=1∂2
+yi−/summationdisplay
+s∈Scsr/summationdisplay
+i=1αs(yi)2
+αs∂α∨s=H,
+since/summationtextr
+i=1αs(yi)2= (αs,αs). /square
+Recall that by the Chevalley-Shepard-Todd theorem, the algebra (Sh)Wis free. Let P1=
+p2,P2,...,Prbe homogeneous generators of ( Sh)W.
+Corollary 2.17. The differential operators Lj=m(Pj(Dy1,...,Dyr))are pairwise commu-
+tative, have symbols Pj, homogeneity degree −dj, andL1=H.
+Proof.Since Dunkl operators commute, the operators Ljare well defined. Since m(AB) =
+m(A)m(B) whenBis invariant, the operators Ljare pairwise commutative. The rest is
+clear. /square
+Now to prove Theorem 2.9, we will show that the operators HandHare conjugate to
+each other by a certain function; this will complete the proof.
+Proposition 2.18. Letδc(x) :=/producttext
+s∈Sαs(x)cs. Then we have
+δ−1
+c◦H◦δc=H.
+Remark 2.19. The function δc(x) is not rational. It is a multivalued analytic function.
+Nevertheless, it is easy to see that for any differential operator Lwith rational coefficients,
+δ−1
+c◦L◦δcalso has rational coefficients.
+Proof of Proposition 2.18. We have
+r/summationdisplay
+i=1∂yi(logδc)∂yi=/summationdisplay
+s∈Scs(αs,αs)
+2αs∂α∨s.
+Therefore, we have
+δc◦H◦δ−1
+c= ∆h−/summationdisplay
+s∈Scs(αs,αs)
+αs∂α∨s+U,
+where
+U=δc(∆hδ−1
+c)−/summationdisplay
+s∈Scs(cs+1)(αs,αs)
+α2s.
+Let us compute U. We have
+δc(∆hδ−1
+c) =/summationdisplay
+s∈Scs(cs+1)(αs,αs)
+α2
+s+/summationdisplay
+s/\e}atio\slash=u∈Scscu(αs,αu)
+αsαu.
+Weclaimthatthelast sumΣisactuallyzero. Indeed, thissumisinvarian tunder theCoxeter
+group, so/producttext
+s∈Sαs·Σ is a regular anti-invariant function of degree |S|−2. But the smallest
+degree of a nonzero anti-invariant is |S|, so Σ = 0, U= 0, and we are done (Proposition 2.18
+and Theorem 2.9 are proved). /square
+Remark 2.20. A similar method works for any complex reflection group G. Namely, the
+operatorsLi=m(Pi(Dy1,...,Dyr)) form a quantum integrable system. However, if Gis not
+a real reflection group, this system does not have a quadratic Ham iltonian in momentum
+variables (so it does not have a physical meaning).
+102.7.Uniqueness of the operators Lj.
+Proposition 2.21. The operators Ljare unique.
+Proof.Assume that we have two choices for Lj:LjandL′
+j. DenoteLj−L′
+jbyM.
+AssumeM/\e}atio\slash= 0. We have
+(i)Mis a differential operator on hwith rational coefficients, of order smaller than dj
+and homogeneity degree −dj;
+(ii) [M,H] = 0.
+LetM0be the symbol of M. ThenM0is a polynomial of p∈h∗with coefficients in C(h).
+We have, from (ii),
+{M0,p2}= 0,∀p∈h∗,
+and from (i) we see that the coefficients of M0are not polynomial (as they have negative
+degree).
+However, we have the following lemma.
+Lemma 2.22. Lethbe a finite dimensional vector space. Let ψ: (x,p)/mapsto→ψ(x,p)be a
+rational function on h⊕h∗which is a polynomial in p∈h∗. Letf:h∗→Cbe a polynomial
+such that the differentials df(p)forp∈h∗spanh(e.g.,f(p) =p2). Suppose that the
+Poisson bracket of fandψvanishes:{ψ,f}= 0. Thenψis a polynomial.
+Proof.(R. Raj) Let Z⊂hbe the pole divisor of ψ. Letx0∈hbe a generic point in Z. Then
+ψ−1is regular and vanishes at ( x0,p) for generic p∈h∗. Also from{ψ−1,f}= 0, we have
+ψ−1vanishes along the entire flowline of the Hamiltonian flow defined by fand starting at
+x0. This flowline is defined by the formula
+x(t) =x0+tdf(p),p(t) =p,
+and it must be contained in the pole divisor of ψnearx0. This implies that d f(p) must be
+inTx0Zfor almost every, hence for every p∈h∗. This is a contradiction with the assumption
+onf, which implies that in fact ψhas no poles. /square
+/square
+2.8.Classical Dunkl operatorsand Olshanetsky-Perelomov Hami ltonians. Wecon-
+tinue to use the notations in Section 2.4.
+Definition 2.23. The classical Olshanetsky-Perelomov Hamiltonian corresponding to Wis
+the following classical Hamiltonian on hreg×h∗=T∗hreg:
+H0(x,p) =p2−/summationdisplay
+s∈Sc2
+s(αs,αs)
+α2s(x).
+Theorem 2.24 ([OP],[HO, He3, Op3, Op4],[He1]) .The Hamiltonian H0defines a classical
+integrable system. Namely, there exist unique regular func tionsL0
+jonhreg×h∗, where highest
+terms in parePj, such that L0
+jare homogeneous of degree −dj(underx/mapsto→λx,x∈h∗,p/mapsto→
+λ−1p,p∈h), and such that L0
+1=H0and{L0
+j,L0
+k}= 0,∀j,k.
+Proof.The proof is given in the next subsection. /square
+11Example 2.25. LetW=Sn,h=Cn−1. Then
+H0=n/summationdisplay
+i=1p2
+i−c2/summationdisplay
+i/\e}atio\slash=j1
+(xi−xj)2( the classical Calogero-Moser Hamiltonian) .
+So the theorem says that there are functions L0
+j,j= 1,...,n−1,
+L0
+j=/summationdisplay
+ipj+1
+i+ lower terms,
+homogeneous of degree zero, such that L0
+1=H0and{L0
+j,L0
+k}= 0.
+2.9.Rees algebras. LetAbe a filtered algebra over a field k:k=F0A⊂F1A⊂···,
+∪iFiA=A. Then the Rees algebra A= Rees(A) is defined by the formula A=⊕∞
+n=0FnA.
+This is an algebra over k[/pla⋉ckover2pi1], where/pla⋉ckover2pi1is the element 1 of the summand F1A.
+2.10.Proof of Theorem 2.24. The proof of Theorem 2.24 is similar to the proof of its
+quantum analog. Namely, to construct the functions L0
+j, we need to introduce classical
+Dunkl operators. To do so, we introduce a parameter /pla⋉ckover2pi1(Planck’s constant) and define
+Dunkl operators Da(/pla⋉ckover2pi1) =Da(c,/pla⋉ckover2pi1) with/pla⋉ckover2pi1:
+Da(c,/pla⋉ckover2pi1) =/pla⋉ckover2pi1Da(c//pla⋉ckover2pi1) =/pla⋉ckover2pi1∂a−/summationdisplay
+s∈S2csαs(a)
+(1−λs)αs(1−s),wherea∈h.
+These operators can be regarded as elements of the Rees algebra A= Rees(CW⋉D(hreg)),
+where the filtration is by order of differential operators (and Wsits in degree 0). Reducing
+these operators modulo /pla⋉ckover2pi1, we get classical Dunkl operators D0
+a(c)∈A0:=A//pla⋉ckover2pi1A=CW⋉
+O(T∗hreg). They are given by the formula
+D0
+a(c) =pa−/summationdisplay
+s∈S2csαs(a)
+(1−λs)αs(1−s),
+wherepais the classical momentum (the linear function on h∗corresponding to a∈h).
+It follows from the commutativity of the quantum Dunkl operators Da(c) that the Dunkl
+operatorsDa(c,/pla⋉ckover2pi1) commute. Hence, so do the classical Dunkl operators D0
+a:
+[D0
+a,D0
+b] = 0.
+We also have the following analog of Proposition 2.14:
+Proposition 2.26. (i)For anyx∈h∗, one has
+[D0
+a,x] =−/summationdisplay
+s∈Scs(a,αs)(x,α∨
+s)s.
+(ii)Ifg∈WthengD0
+ag−1=D0
+ga.
+Now let us construct the classical Olshanetsky-Perelomov Hamilton ians. As in the quan-
+tum case, we have the operation m(·), which is given by the formula/summationtext
+g∈WBg·g/mapsto→/summationtextBg,
+B∈O(T∗hreg). We define the Hamiltonian
+H0:=m(r/summationdisplay
+i=1(D0
+yi)2).
+12By taking the limit of quantum situation, we find
+H0=p2−/summationdisplay
+s∈Scs(αs,αs)
+αs(x)pα∨s.
+Unfortunately, this is no longer conjugate to H0. However, consider the (outer) automor-
+phismθcof the algebra CW⋉O(T∗hreg) defined by the formulas
+θc(x) =x, θc(s) =s, θc(pa) =pa+∂alogδc,
+forx∈h∗,a∈h,s∈W. It is easy to see that if b0∈A0andb∈Ais a deformation of
+b0thenθc(b0) = lim /pla⋉ckover2pi1→0δ−1
+c//pla⋉ckover2pi1bδc//pla⋉ckover2pi1. Therefore, taking the limit /pla⋉ckover2pi1→0 in Proposition 2.16, we
+find thatH0=θc(H0).
+Now setL0
+j=m(θc(Pj(D0
+y1,...,D0
+yr))). These functions are well defined since D0
+acom-
+mute, are homogeneous of degree zero, and L0
+1=H0.
+Moreover, we can define the operators Lj(/pla⋉ckover2pi1) in Rees(D(hreg)W) in the same way as Lj, but
+using the Dunkl operators Dyi(/pla⋉ckover2pi1) instead of Dyi. Then [Lj(/pla⋉ckover2pi1),Lk(/pla⋉ckover2pi1)] = 0, and Lj(/pla⋉ckover2pi1)|/pla⋉ckover2pi1=0=
+L0
+j. This implies that L0
+jPoisson commute: {L0
+j,L0
+k}= 0.
+Theorem 2.24 is proved.
+Remark 2.27. As in the quantum situation, Theorem 2.24 can be generalized to comp lex
+reflection groups, giving integrable systems with Hamiltonians which a re non-quadratic in
+momentum variables.
+2.11.Notes.Section 2.1 follows Section 5.4 of [E4]; the definition of complex reflect ion
+groups and their basic properties can be found in [GM]; the definition o f parabolic subgroups
+and the notations are borrowed from Section 3.1 of [BE]; the remainin g parts of this section
+follow Section 6 of [E4].
+133.The rational Cherednik algebra
+3.1.Definition and examples. Above we have made essential use of the commutation
+relations between operators x∈h∗,g∈G, andDa,a∈h. This makes it natural to consider
+the algebra generated by these operators.
+Definition 3.1. The rational Cherednik algebra associated to ( G,h) is the algebra Hc(G,h)
+generated inside A= Rees(CG⋉D(hreg)) by the elements x∈h∗,g∈G, andDa(c,/pla⋉ckover2pi1),a∈h.
+Ift∈C, then the algebra Ht,c(G,h) is the specialization of Hc(G,h) at/pla⋉ckover2pi1=t.
+Proposition 3.2. The algebra Hcis the quotient of the algebra CG⋉T(h⊕h∗)[/pla⋉ckover2pi1](where
+Tdenotes the tensor algebra) by the ideal generated by the rel ations
+[x,x′] = 0,[y,y′] = 0,[y,x] =/pla⋉ckover2pi1(y,x)−/summationdisplay
+s∈Scs(y,αs)(x,α∨
+s)s,
+wherex,x′∈h∗,y,y′∈h.
+Proof.Let us denote the algebra defined in the proposition by H′
+c=H′
+c(G,h). Then accord-
+ing to the results of the previous sections, we have a surjective ho momorphism φ:H′
+c→Hc
+defined by the formula φ(x) =x,φ(g) =g,φ(y) =Dy(c,/pla⋉ckover2pi1).
+Let us show that this homomorphism is injective. For this purpose as sume thatyiis a
+basis ofh, andxiis the dual basis of h∗. Then it is clear from the relations of H′
+cthatH′
+cis
+spanned over C[/pla⋉ckover2pi1] by the elements
+(3.1) gr/productdisplay
+i=1ymi
+ir/productdisplay
+i=1xni
+i.
+Thus it remains to show that the images of the elements (3.1) under t he mapφ, i.e. the
+elements
+gr/productdisplay
+i=1Dyi(c,/pla⋉ckover2pi1)mir/productdisplay
+i=1xni
+i.
+are linearly independent. But this follows from the obvious fact that the symbols of these
+elements in CG⋉C[h∗×hreg][/pla⋉ckover2pi1] are linearly independent. The proposition is proved. /square
+Remark 3.3. 1. Similarly, one can define the universal algebra H(G,h), in which both /pla⋉ckover2pi1
+andcare variables. (So this is an algebra over C[/pla⋉ckover2pi1,c].) It has two equivalent definitions
+similar to the above.
+2. It is more convenient to work with algebras defined by generator s and relations than
+with subalgebras of a given algebra generated by a given set of eleme nts. Therefore, from
+now on we will use the statement of Proposition 3.2 as a definition of th e rational Cherednik
+algebraHc. According to Proposition 3.2, this algebra comes with a natural emb edding
+Θc:Hc→Rees(CG⋉D(hreg)), defined by the formula x→x,g→g,y→Dy(c,/pla⋉ckover2pi1). This
+embedding is called the Dunkl operator embedding .
+Example 3.4. 1. LetG=Z2,h=C. In this case creduces to one parameter, and the
+algebraHt,cis generated by elements x,y,swith defining relations
+s2= 1, sx=−xs, sy=−ys,[y,x] =t−2cs.
+142. LetG=Sn,h=Cn. In this case there is also only one complex parameter c, and the
+algebraHt,cis the quotient of Sn⋉C/a\}b∇acketle{tx1,...,xn,y1,...,yn/a\}b∇acket∇i}htby the relations
+[xi,xj] = [yi,yj] = 0,[yi,xj] =csij,[yi,xi] =t−c/summationdisplay
+j/\e}atio\slash=isij.
+HereC/a\}b∇acketle{tE/a\}b∇acket∇i}htdenotes the free algebra on a set E, andsijis the transposition of iandj.
+3.2.The PBW theorem for the rational Cherednik algebra. Let us put a filtration
+onHcby setting deg y= 1 fory∈hand degx= degg= 0 forx∈h∗,g∈G. Let gr(Hc)
+denote the associated graded algebra of Hcunder this filtration, and similarly for Ht,c. We
+have a natural surjective homomorphism
+ξ:CG⋉C[h⊕h∗][/pla⋉ckover2pi1]→gr(Hc).
+Fort∈C, it specializes to surjective homomorphisms
+ξt:CG⋉C[h⊕h∗]→gr(Ht,c).
+Proposition 3.5 (The PBW theorem for rational Cherednik algebras) .The mapsξandξt
+are isomorphisms.
+Proof.The statement is equivalent to the claim that the elements (3.1) are a basis ofHt,c,
+which follows from the proof of Proposition 3.2. /square
+Remark 3.6. 1. We have
+H0,0=CG⋉C[h⊕h∗] andH1,0=CG⋉D(h).
+2. For any λ∈C∗, the algebra Ht,cis naturally isomorphic to Hλt,λc.
+3. The Dunkl operator embedding Θ cspecializes to embeddings
+Θ0,c:H0,c֒→CG⋉C[h∗×hreg],
+given byx/mapsto→x,g/mapsto→g,y/mapsto→D0
+a, and
+Θ1,c:H1,c֒→CG⋉D(hreg),
+given byx/mapsto→x,g/mapsto→g,y/mapsto→Da. SoH0,cis generated by x,g,D0
+a, andH1,cis generated by
+x,g,Da.
+Since Dunkl operators map polynomials to polynomials, the map Θ 1,cdefines a represen-
+tation ofH1,conC[h]. This representation is called the polynomial representation ofH1,c.
+3.3.The spherical subalgebra. Lete∈CGbe the symmetrizer, e=|G|−1/summationtext
+g∈Gg. We
+havee2=e.
+Definition 3.7. Bc:=eHceiscalledthe sphericalsubalgebra ofHc. Thesphericalsubalgebra
+ofHt,cisBt,c:=Bc/(/pla⋉ckover2pi1−t) =eHt,ce.
+Note that
+e(CG⋉D(hreg))e=D(hreg)G,e(CG⋉C[hreg×h∗])e=C[hreg×h∗]G.
+Therefore, the restriction gives the embeddings: Θ 1,c:B1,c֒→D(hreg)G, and Θ 0,c:B0,c֒→
+C[h∗×hreg]G. In particular, we have
+Proposition 3.8. The spherical subalgebra B0,cis commutative and does not have zero
+divisors. Also B0,cis finitely generated.
+15Proof.The first statement is clear from the above. The second statemen t follows from the
+fact that gr( B0,c) =B0,0=C[h×h∗]G, which is finitely generated by Hilbert’s theorem. /square
+Corollary 3.9. Mc= SpecB0,cis an irreducible affine algebraic variety.
+Proof.Directly from the definition and the proposition. /square
+We also obtain
+Proposition 3.10. Bcis a flat quantization (non-commutative deformation) of B0,cover
+C[/pla⋉ckover2pi1].
+SoB0,ccarries a Poisson bracket {·,·}(thusMcis a Poisson variety), and Bcis a quanti-
+zation of the Poisson bracket, i.e. if a,b∈Bcanda0,b0are the corresponding elements in
+B0,c, then
+[a,b]//pla⋉ckover2pi1≡{a0,b0}(mod/pla⋉ckover2pi1).
+Definition 3.11. The Poisson variety Mcis called the Calogero-Moser space ofG,hwith
+parameterc.
+3.4.The localization lemma. LetHloc
+t,c=Ht,c[δ−1] be the localization of Ht,cas a module
+overC[h] with respect to the discriminant δ(a polynomial vanishing to the first order on
+each reflection plane). Define also Bloc
+t,c=eHloc
+t,ce.
+Proposition 3.12. (i)Fort/\e}atio\slash= 0the map Θt,cinduces an isomorphism of algebras
+Hloc
+t,c→CG⋉D(hreg), which restricts to an isomorphism Bloc
+t,c→D(hreg)G.
+(ii)The map Θ0,cinduces an isomorphism of algebras Hloc
+0,c→CG⋉C[h∗×hreg], which
+restricts to an isomorphism Bloc
+0,c→C[h∗×hreg]G.
+Proof.This follows immediately from the fact that the Dunkl operators hav e poles only on
+the reflection hyperplanes. /square
+Since gr(B0,c) =B0,0=C[h∗⊕h]G, we get the following geometric corollary.
+Corollary 3.13. (i)The family of Poisson varieties Mcis a flat deformation of the
+Poisson variety M0:= (h×h∗)/G. In particular, Mcis smooth outside of a subset of
+codimension 2.
+(ii)We have a natural map βc:Mc→h/G, such that β−1
+c(hreg/G)is isomorphic to
+(hreg×h∗)/G. The Poisson structure on Mcis obtained by extension of the symplectic
+Poisson structure on (hreg×h∗)/G.
+Example 3.14. LetW=Z2,h=C. ThenB0,c=/a\}b∇acketle{tx2,xp,p2−c2/x2/a\}b∇acket∇i}ht. LetX:=x2,Z:=xp
+andY:=p2−c2/x2. ThenZ2−XY=c2. SoMcis isomorphic to the quadric Z2−XY=c2
+in the 3-dimensional space and it is smooth for c/\e}atio\slash= 0.
+3.5.CategoryOfor rational Cherednik algebras. FromthePBWtheorem, we seethat
+H1,c=Sh∗⊗CG⊗Sh. It is similar to the structure of the universal enveloping algebra of a
+simple Lie algebra: U(g) =U(n−)⊗U(h)⊗U(n+). Namely, the subalgebra CGplays the role
+of the Cartan subalgebra, and the subalgebras Sh∗andShplay the role of the positive and
+negative nilpotent subalgebras. This similarity allows one to define and study the category
+Oanalogous to the Bernstein-Gelfand-Gelfand category Ofor simple Lie algebras.
+16Definition 3.15. The categoryOc(G,h) is the category of modules over H1,c(G,h) which
+are finitely generated over Sh∗and locally finite under Sh(i.e., forM∈Oc(G,h),∀v∈M,
+(Sh)vis finite dimensional).
+IfMis a locally finite ( Sh)G-module, then
+M=⊕λ∈h∗/GMλ,
+where
+Mλ={v∈M|∀p∈(Sh)G,∃Ns.t.(p−λ(p))Nv= 0},
+(notice that h∗/G= Specm(Sh)G).
+Proposition 3.16. MλareH1,c-submodules.
+Proof.Note first that we have an isomorphism µ:H1,c(G,h)∼=H1,c(G,h∗), which is given
+byxa/mapsto→ya,yb/mapsto→−xb,g/mapsto→g. Now letx1,...,xrbe a basis of h∗andy1,...,yra basis of h.
+SupposeP=P(x1,...,xr)∈(Sh∗)G. Then we have
+[y,P] =∂
+∂yP∈Sh∗,wherey∈h,
+(this follows from the fact that both sides act in the same way in the p olynomial represen-
+tation, which is faithful). So using the isomorphism µ, we conclude that if Q∈(Sh)G,Q=
+Q(y1,...,yr), then [x,Q] =−∂xQforx∈h∗.
+Now, to prove the proposition, the only thing we need to check is tha tMλis invariant
+underx∈h∗. For anyv∈Mλ, we have (Q−λ(Q))Nv= 0 for some N. Then
+(Q−λ(Q))N+1xv= (N+1)∂xQ·(Q−λ(Q))Nv= 0.
+Soxv∈Mλ.
+/square
+Corollary 3.17. We have the following decomposition:
+Oc(G,h) =/circleplusdisplay
+λ∈h∗/GOc(G,h)λ,
+whereOc(G,h)λis the subcategory of modules where (Sh)Gacts with generalized eigenvalue
+λ.
+Proof.Directly from the definition and the proposition. /square
+Note thatOc(G,h)λis an abelian category closed under taking subquotients and exten-
+sions.
+3.6.The grading element. Let
+(3.2) h=/summationdisplay
+ixiyi+1
+2dimh−/summationdisplay
+s∈S2cs
+1−λss.
+Proposition 3.18. We have
+[h,x] =x, x∈h∗,[h,y] =−y, y∈h.
+17Proof.Let us prove the first relation; the second one is proved similarly. We have
+[h,x] =/summationdisplay
+ixi[yi,x]−/summationdisplay
+s∈S2cs
+1−λs·λs−1
+2(α∨
+s,x)αs·s
+=/summationdisplay
+ixi(yi,x)−/summationdisplay
+ixi/summationdisplay
+s∈Scs(α∨
+s,x)(αs,yi)s+/summationdisplay
+s∈Scs(α∨
+s,x)αs·s.
+The last two terms cancel since/summationtext
+ixi(αs,yi) =αs, so we get/summationtext
+ixi(yi,x) =x. /square
+Proposition 3.19. LetG=Wbe a real reflection group. Let
+h=/summationdisplay
+ixiyi+1
+2dimh−/summationdisplay
+s∈Scss,E=−1
+2/summationdisplay
+ix2
+i,F=1
+2/summationdisplay
+iy2
+i.
+Then
+(i)h=/summationtext
+i(xiyi+yixi)/2;
+(ii)h,E,Fform an sl2-triple.
+Proof.A direct calculation. /square
+Theorem 3.20. LetMbe a module over H1,c(G,h).
+(i)Ifhacts locally nilpotently on M, thenhacts locally finitely on M.
+(ii)IfMis finitely generated over Sh∗, thenM∈Oc(G,h)0if and only if hacts locally
+finitely onM.
+Proof.(i) Assume that Shacts locally nilpotently on M. Letv∈M. ThenSh·vis a finite
+dimensional vector space and let d= dimSh·v. We prove that vish-finite by induction
+in dimension d. We can use d= 0 as base, so only need to do the induction step. The
+spaceSh·vmust contain a nonzero vector usuch thaty·u= 0,∀y∈h. LetU⊂Mbe
+the subspace of vectors with this property. hacts onUby an element of CG, hence locally
+finitely. So it is sufficient to show that the image of vinM//a\}b∇acketle{tU/a\}b∇acket∇i}htish-finite (where/a\}b∇acketle{tU/a\}b∇acket∇i}htis
+the submodule generated by U). But this is true by the induction assumption, as u= 0 in
+M//a\}b∇acketle{tU/a\}b∇acket∇i}ht.
+(ii) We need to show that if hacts locally finitely on M, thenhacts locally nilpotently
+onM. Assume hacts locally finitely on M. ThenM=⊕β∈BM[β], whereB⊂C. SinceM
+is finitely generated over Sh∗,Bis a finite union of sets of the form z+Z≥0,z∈C. SoSh
+must act locally nilpotently on M. /square
+We can obtain the following corollary easily.
+Corollary 3.21. Any finite dimensional H1,c(G,h)-module is inOc(G,h)0.
+We see that any module M∈Oc(G,h)0has a grading by generalized eigenvalues of h:
+M=⊕βM[β].
+3.7.Standard modules. Letτbe a finite dimensional representation of G. The standard
+module over H1,c(G,h) corresponding to τ(also called the Verma module) is
+Mc(G,h,τ) =Mc(τ) =H1,c(G,h)⊗CG⋉Shτ∈Oc(G,h)0,
+whereShacts onτby zero.
+So from the PBW theorem, we have that as vector spaces, Mc(τ)∼=τ⊗Sh∗.
+18Remark 3.22. More generally,∀λ∈h∗, letGλ= Stab(λ), andτbe a finite dimensional
+representation of Gλ. Then we can define Mc,λ(G,h,τ) =H1,c(G,h)⊗CGλ⋉Shτ, whereSh
+acts onτbyλ. These modules are called the Whittaker modules .
+Letτbe irreducible, and let hc(τ) be the number given by the formula
+hc(τ) =dimh
+2−/summationdisplay
+s∈S2cs
+1−λss|τ.
+Then we see that hacts onτ⊗Smh∗by the scalar hc(τ)+m.
+Definition 3.23. A vectorvin anH1,c-moduleMis singular if yiv= 0 for alli.
+Proposition 3.24. LetUbe anH1,c(G,h)-module. Let τ⊂Ube aG-submodule consisting
+of singular vectors. Then there is a unique homomorphism φ:Mc(τ)→UofC[h]-modules
+such thatφ|τis the identity, and it is an H1,c-homomorphism.
+Proof.The first statement follows from the fact that Mc(τ) is a free module over C[h] gen-
+erated byτ. Also, it follows from the Frobenius reciprocity that there must exis t a mapφ
+which is an H1,c-homomorphism. This implies the proposition. /square
+3.8.Finite length.
+Proposition 3.25. ∃K∈Rsuch that for any M⊂NinOc(G,h)0, ifM[β] =N[β]for
+Re(β)≤K, thenM=N.
+Proof.LetK= maxτRehc(τ). Then ifM/\e}atio\slash=N,M/Nbegins in degree β0with Reβ0>K,
+which is impossible since by Proposition 3.24, β0must equal hc(τ) for someτ. /square
+Corollary 3.26. AnyM∈Oc(G,h)0has finite length.
+Proof.Directly from the proposition. /square
+3.9.Characters. ForM∈Oc(G,h)0, define the character of Mas the following formal
+series int:
+chM(g,t) =/summationdisplay
+βtβTrM[β](g) = TrM(gth), g∈G.
+Proposition 3.27. We have
+chMc(τ)(g,t) =χτ(g)thc(τ)
+deth∗(1−tg).
+Proof.We begin with the following lemma.
+Lemma 3.28 (MacMahon’s Master theorem) .LetVbe a finite dimensional space, A:V→
+Va linear operator. Then
+/summationdisplay
+n≥0tnTr(SnA) =1
+det(1−tA).
+Proof of the lemma. IfAis diagonalizable, this is obvious. The general statement follows by
+continuity. /square
+The lemma implies that Tr Sh∗(gtD) =1
+det(1−gt)whereDis the degree operator. This
+implies the required statement. /square
+193.10.Irreducible modules. Letτbe an irreducible representation of G.
+Proposition 3.29. Mc(τ)has a maximal proper submodule Jc(τ).
+Proof.The proof is standard. Jc(τ) is the sum of all proper submodules of Mc(τ), and it is
+not equal to Mc(τ) because any proper submodule has a grading by generalized eigens paces
+ofh, with eigenvalues βsuch thatβ−hc(τ)>0. /square
+We defineLc(τ) =Mc(τ)/Jc(τ), which is an irreducible module.
+Proposition 3.30. Any irreducible object of Oc(G,h)0has the form Lc(τ)for an unique τ.
+Proof.LetL∈Oc(G,h)0beirreducible, withlowest eigenspaceof hcontaining anirreducible
+G-moduleτ. ThenbyProposition3.24,wehaveanonzerohomomorphism Mc(τ)→L,which
+is surjective, since Lis irreducible. Then we must have L=Lc(τ). /square
+Remark 3.31. Letχbe a character of G. Then we have an isomorphism H1,c(G,h)→
+H1,cχ(G,h), mapping g∈Gtoχ−1(g)g. This automorphism maps Lc(τ) toLcχ(χ−1⊗τ)
+isomorphically.
+3.11.The contragredient module. Set ¯c(s) =c(s−1). We have a natural isomorphism
+γ:H1,¯c(G,h∗)op→H1,c(G,h), acting trivially on handh∗, and sending g∈Gtog−1.
+Thus ifMis anH1,c(G,h)-module, then the full dual space M∗is anH1,¯c(M,h∗)-module.
+IfM∈Oc(G,h)0, then we can define M†, which is the h-finite part of M∗.
+Proposition 3.32. M†belongs toO¯c(G,h∗)0.
+Proof.Clearly, if Lis irreducible, then so is L†. ThenL†is generated by its lowest h-
+eigenspace over H1,¯c(G,h∗), hence over Sh∗. Thus,L†∈O¯c(G,h∗)0. Now, letM∈Oc(G,h)0
+be any object. Since Mhas finite length, so does M†. Moreover, M†has a finite filtration
+with successive quotients of the form L†, whereL∈Oc(G,h)0is irreducible. This implies
+the required statement, since Oc(G,h)0is closed under taking extensions. /square
+Clearly,M††=M. Thus,M/mapsto→M†isanequivalenceofcategories Oc(G,h)→O¯c(G,h∗)op.
+3.12.The contravariant form. Letτbe an irreducible representation of G. By Propo-
+sition 3.24, we have a unique homomorphism φ:Mc(G,h,τ)→M¯c(G,h∗,τ∗)†which is the
+identity in the lowest h-eigenspace. Thus, we have a pairing
+βc:Mc(G,h,τ)×M¯c(G,h∗,τ∗)→C,
+which is called the contravariant form .
+Remark 3.33. IfG=Wis a real reflection group, then h∼=h∗,c= ¯c, andτ∼=τ∗via a
+symmetric form. So for real reflection groups, βcis a symmetric form on Mc(τ).
+Proposition 3.34. The maximal proper submodule Jc(τ)is the kernel of φ(or, equivalently,
+of the contravariant form βc).
+Proof.LetKbe the kernel of the contravariant form. It suffices to show that Mc(τ)/Kis
+irreducible. We have a diagram:
+Mc(G,h,τ)
+/d15/d15ξ
+/d40/d40/d80/d80/d80/d80/d80/d80/d80/d80/d80/d80/d80/d80φ/d47/d47Mc(G,h∗,τ∗)†
+Lc(G,h,τ)∼
+η/d47/d47Lc(G,h∗,τ∗)†/d63/d31/d79/d79
+20Indeed, a nonzero map ξexists by Proposition 3.24, and it factors through Lc(G,h,τ),
+withηbeing an isomorphism, since Lc(G,h∗,τ∗)†is irreducible. Now, by Proposition 3.24
+(uniqueness of φ), thediagrammust commuteuptoscaling, whichimpliesthestatemen t./square
+Proposition 3.35. Assume that hc(τ)−hc(τ′)never equals a positive integer for any τ,τ′∈
+IrrepG. ThenOc(G,h)0is semisimple, with simple objects Mc(τ).
+Proof.Itisclearthatinthissituation, all Mc(τ)aresimple. AlsoconsiderExt1(Mc(τ),Mc(τ′)).
+Ifhc(τ)−hc(τ′)/∈Z,itisclearly0. Otherwise, hc(τ) =hc(τ′),andagainExt1(Mc(τ),Mc(τ′)) =
+0, since for any extension
+0→Mc(τ′)→N→Mc(τ)→0,
+by Proposition 3.24 we have a splitting Mc(τ)→N. /square
+Remark 3.36. In fact, our argument shows that if Ext1(Mc(τ),Mc(τ′))/\e}atio\slash= 0, thenhc(τ)−
+hc(τ′)∈N.
+3.13.The matrix of multiplicities. Forτ,σ∈IrrepG, writeτ <σif
+Rehc(σ)−Rehc(τ)∈N.
+Proposition 3.37. There exists a matrix of integers N= (nσ,τ), withnσ,τ≥0, such that
+nτ,τ= 1,nσ,τ= 0unlessσ<τ, and
+Mc(σ) =/summationdisplay
+nσ,τLc(τ)∈K0(Oc(G,h)0).
+Proof.This follows from the Jordan-H¨ older theorem and the fact that ob jects inOc(G,h)0
+have finite length. /square
+Corollary 3.38. LetN−1= (¯nτ,σ). Then
+Lc(τ) =/summationdisplay
+¯nτ,σMc(σ).
+Corollary 3.39. We have
+chLc(τ)(g,t) =/summationtext¯nτ,σχσ(g)thc(τ)
+deth∗(1−tg).
+Both of the corollaries can be obtained from the above proposition e asily.
+One of the main problems in the representation theory of rational C herednik algebras is
+the following problem.
+Problem: Compute the multiplicities nσ,τor, equivalently, ch Lc(τ)for allτ.
+In general, this problem is open.
+3.14.Example: the rank 1case.LetG=Z/mZandλbe anm-th primitive root of 1.
+Then the algebra H1,c(G,h) is generated by x,y,swith relations
+[y,x] = 1−2m−1/summationdisplay
+j=1cjsj, sxs−1=λx, sys−1=λ−1y.
+Consider the one-dimensional space Cand letyact by 0 and g∈Gact by 1. We have
+Mc(C) =C[x]. The contravariant form βc,ConMc(C) is defined by
+βc,C(xn,xn) =an;βc,C(xn,xn′) = 0,n/\e}atio\slash=n′.
+21Recall that βc,Csatisfiesβc,C(xn,xn) =βc,C(xn−1,yxn), which gives
+an=an−1(n−bn),
+wherebnare new parameters:
+bn:= 2m−1/summationdisplay
+j=11−λjn
+1−λjcj(b0= 0,bn+m=bn).
+Thus we obtain the following proposition.
+Proposition 3.40. (i)Mc(C)is irreducible if only if n−bn/\e}atio\slash= 0for anyn≥1.
+(ii)Assume that ris the smallest positive integer such that r=br. ThenLc(C)has
+dimensionr(which can be any number not divisible by m) with basis 1,x,...,xr−1.
+Remark 3.41. According to Remark 3.31, this proposition in fact describes all the ir re-
+ducible lowest weight modules.
+Example 3.42. Consider the case m= 2. TheMc(C) is irreducible unless c∈1/2+Z≥0.
+Ifc= (2n+1)/2∈1/2+Z,n≥0, thenLc(C) has dimension 2 n+1. A similar answer is
+obtained for lowest weight C−, replacing cby−c.
+3.15.The Frobenius property. LetAbe aZ+-graded commutative algebra. The algebra
+Ais called Frobenius if the top degree A[d] ofAis 1-dimensional, and the multiplication
+mapA[m]×A[d−m]→A[d] is a nondegenerate pairing for any 0 ≤m≤d. In particular,
+the Hilbert polynomial of a Frobenius algebra Ais palindromic.
+Now, let us go back to considering modules over the rational Chered nik algebra H1,c. Any
+submodule Jof the polynomial representation Mc(C) =Mc=C[h] is an ideal in C[h], so
+the quotient A=Mc/Jis aZ+-graded commutative algebra.
+Now suppose that Gpreserves an inner product in h, i.e.,G⊆O(h).
+Theorem 3.43. IfA=Mc(C)/Jis finite dimensional, then Ais irreducible ( A=Lc(C))
+⇐⇒Ais a Frobenius algebra.
+Proof.1) Suppose Ais an irreducible H1,c-module, i.e., A=Lc(C). By Proposition 3.19, A
+is naturally a finite dimensional sl2-module (in particular, it integrates to the group SL 2(C)).
+Hence, by the representation theory of sl2, the top degree of Ais 1-dimensional. Let φ∈A∗
+denote a nonzero linear function on the top component. Let βcbe the contravariant form
+onMc(C). Consider the form
+(v1,v2)/mapsto→E(v1,v2) :=βc(v1,gv2),whereg=/parenleftbigg
+0 1
+−1 0/parenrightbigg
+∈SL2(C).
+ThenE(xv1,v2) =E(v1,xv2). So for any p,q∈Mc(C) =C[h],E(p,q) =φ(p(x)q(x)) (for a
+suitable normalization of φ).
+SinceEis a nondegenerate form, Ais a Frobenius algebra.
+2) Suppose Ais Frobenius. Then the highest component is 1-dimensional, and
+E:A⊗A→C,E(a,b) =φ(ab) is nondegenerate. We have E(xa,b) =E(a,xb). So
+setβ(a,b) =E(a,g−1b). Thenβsatisfiesβ(a,xib) =β(yia,b). Thus, for all p,q∈C[h],
+β(p(x),q(x)) =β(q(y)p(x),1). Soβ=βcup to scaling. Thus, βcis nondegenerate and Ais
+irreducible. /square
+Remark 3.44. IfG/⋉otsubseteqlO(h), this theorem is false, in general.
+22Now consider the Frobenius property of Lc(C) for anyG⊂GL(h).
+Theorem 3.45. LetU⊂Mc(C) =C[h]be aG-subrepresentation of dimension l= dimh,
+sitting in degree r, which consists of singular vectors. Let J=/a\}b∇acketle{tU/a\}b∇acket∇i}ht. Assume that A=Mc/J
+is finite dimensional. Then
+(i)Ais Frobenius.
+(ii)Aadmits a BGG type resolution:
+A←Mc(C)←Mc(U)←Mc(∧2U)←···←Mc(∧lU)←0.
+(iii)The character of Ais given by the formula
+χA(g,t) =tl
+2−/summationtext
+s∈S2cs
+1−λsdetU(1−gtr)
+deth∗(1−gt).
+In particular, dimA=rl.
+(iv)IfGpreserves an inner product, then Ais irreducible.
+Proof.(i) Since Spec Ais a complete intersection, Ais Frobenius.
+(ii) We will use the following theorem:
+Theorem 3.46 (Serre).Letf1,...,fn∈C[t1,...,tn]be homogeneous polynomials, and
+assume that C[t1,...,tn]is a finitely generated module over C[f1,...,fn]. Then this is a free
+module.
+ConsiderSU⊂Sh∗. ThenSh∗is a finitely generated SU-module (as Sh∗//a\}b∇acketle{tU/a\}b∇acket∇i}htis finite
+dimensional). By Serre’s theorem, we know that Sh∗is a freeSU-module. The rank of this
+module isrl. Consider the Koszul complex attached to this module. Since the mo dule is
+free, the Koszul complex is exact (i.e., it is a resolution of the zero fib er). At the level of
+SU-modules, it looks exactly like we want in (3.45).
+So we only need to show that the maps of the resolution are morphism s overH1,c. This
+is shown by induction. Namely, let δj:Mc(∧jU)→Mc(∧j−1U) be the corresponding
+differentials (so that δ0:Mc(C)→Ais the projection). Then δ0is anH1,c-morphism, which
+is the base of induction. If δjis anH1,c-morphism, then the kernel of δjis a submodule
+Kj⊂Mc(∧jU). Its lowest degree part is ∧j+1Usitting in degree ( j+1)rand consisting of
+singular vectors. Now, δj+1is a morphism over Sh∗which maps∧j+1Uidentically to itself.
+By Proposition 3.24, there is only one such morphism, and it must be an H1,c-morphism.
+This completes the induction step.
+(iii) follows from (ii) by the Euler-Poincar´ e formula.
+(iv) follows from Theorem 3.43.
+/square
+3.16.Representations of H1,cof typeA.Let us now apply the above results to the case
+of typeA. We will follow the paper [CE].
+LetG=Sn, andhbe its reflection representation. In this case the function creduces
+to one number. We will denote the rational Cherednik algebra H1,c(Sn) byHc(n). It is
+generated by x1,...,xn,y1,...,ynandCSnwith the following relations:
+/summationdisplay
+yi= 0,/summationdisplay
+xi= 0,[yi,xj] =−1
+n+csij,i/\e}atio\slash=j,
+23[yi,xi] =n−1
+n−c/summationdisplay
+j/\e}atio\slash=isij.
+The polynomial representation Mc(C) of this algebra is the space of C[x1,...,xn]Tof poly-
+nomials of x1,...,xn, which are invariant under simultaneous translation T:xi/mapsto→xi+a.
+In other words, it is the space of regular functions on h=Cn/∆, where ∆ is the diagonal.
+Proposition 3.47 (C. Dunkl) .Letrbe a positive integer not divisible by n, andc=r/n.
+ThenMc(C)contains a copy of the reflection representation hofSn, which consists of
+singular vectors (i.e. those killed by y∈h). This copy sits in degree rand is spanned by the
+functions
+fi(x1,...,xn) = Res ∞[(z−x1)···(z−xn)]r
+ndz
+z−xi.
+(the symbol Res∞denotes the residue at infinity).
+Remark 3.48. The space spanned by fiis (n−1)-dimensional, since/summationtext
+ifi= 0 (this sum
+is the residue of an exact differential).
+Proof.This proposition can be proved by a straightforward computation. The functions fi
+are a special case of Jack polynomials. /square
+LetIcbe the submodule of Mc(C) generated by fi. Consider the Hc(n)-moduleVc=
+Mc(C)/Ic, and regard it as a C[h]-module. We have the following results.
+Theorem 3.49. Letd= (r,n)denote the greatest common divisor of randn. Then the
+(set-theoretical) support of Vcis the union of Sn-translates of the subspaces of Cn/∆, defined
+by the equations
+x1=x2=···=xn
+d;xn
+d+1=···=x2n
+d;... x (d−1)n
+d+1=···=xn.
+In particular, the Gelfand-Kirillov dimension of Vcisd−1.
+Corollary 3.50 ([BEG]).Ifd= 1then the module Vcis finite dimensional, irreducible,
+admits a BGG type resolution, and its character is
+χVc(g,t) =t(1−r)(n−1)/2det|h(1−gtr)
+det|h(1−gt).
+Proof.Ford= 1 Theorem 3.49 says that the support of Mc(C)/Icis{0}. This implies that
+Mc(C)/Icis finite dimensional. The rest follows from Theorem 3.45. /square
+Proof of Theorem 3.49. The support of Vcis the zero set of Ic, i.e. the common zero set of
+fi. Fixx1,...,xn∈C. Thenfi(x1,...,xn) = 0 for all iiffn/summationdisplay
+i=1λifi= 0 for allλi, i.e.
+Res∞/parenleftBiggn/productdisplay
+j=1(z−xj)r
+nn/summationdisplay
+i=1λi
+z−xi/parenrightBigg
+dz= 0.
+Assumethat x1,...xntakedistinctvalues y1,...,ypwithpositivemultiplicities m1,...,mp.
+The previous equation implies that the point ( x1,...,xn) is in the zero set iff
+Res∞p/productdisplay
+j=1(z−yj)mjr
+n−1/parenleftBiggp/summationdisplay
+i=1νi(z−y1)···/hatwider(z−yi)···(z−yp)/parenrightBigg
+dz= 0∀νi.
+24Sinceνiare arbitrary, this is equivalent to the condition
+Res∞p/productdisplay
+j=1(z−yj)mjr
+n−1zidz= 0, i= 0,...,p−1.
+We will now need the following lemma.
+Lemma 3.51. Leta(z) =p/productdisplay
+j=1(z−yj)µj, whereµj∈C,/summationtext
+jµj∈Zand/summationtext
+jµj>−p. Suppose
+Res∞a(z)zidz= 0, i= 0,1,...,p−2.
+Thena(z)is a polynomial.
+Proof.Letg(z) be a polynomial. Then
+0 = Res∞d(g(z)·a(z)) = Res ∞(g′(z)a(z)+a′(z)g(z))dz,
+and hence
+Res∞/parenleftBigg
+g′(z)+/summationdisplay
+iµj
+z−yjg(z)/parenrightBigg
+a(z)dz= 0.
+Letg(z) =zl/productdisplay
+j(z−yj). Theng′(z)+/summationdisplay
+jµj
+z−yjg(z) is a polynomial of degree l+p−1
+with highest coefficient l+p+/summationtextµj/\e}atio\slash= 0 (as/summationtextµj>−p). This means that for every l≥0,
+Res∞zl+p−1a(z)dzis a linear combination of residues of zqa(z)dzwithq < l+p−1. By
+the assumption of the lemma, this implies by induction in lthat all such residues are 0 and
+henceais a polynomial. /square
+In our case/summationtext(mjr/n−1) =r−p(since/summationtextmj=n) and the conditions of the lemma are
+satisfied. Hence ( x1,...,xn) is in the zero set of Iciffp/productdisplay
+j=1(z−yj)mjr
+n−1is a polynomial. This
+is equivalent to saying that all mjare divisible by n/d.
+We have proved that ( x1,...,xn) is in the zero set of Icif and only if ( z−x1)···(z−xn)
+is the (n/d)-th power of a polynomial of degree d. This implies the theorem. /square
+Remark 3.52. Forc >0, the above representations are the only irreducible finite di-
+mensional representations of H1,c(Sn). Namely, it is proved in [BEG] that the only finite
+dimensional representations of H1,c(Sn) are multiples of Lc(C) forc=r/n, and ofLc(C−)
+(whereC−is the sign representation) for c=−r/n, whereris a positive integer relatively
+prime ton.
+3.17.Notes.The discussion of the definition of rational Cherednik algebras and t heir basic
+properties follows Section 7 of [E4]. The discussion of the category Ofor rational Cherednik
+algebras follows Section 11 of [E4]. The material in Sections 3.14-3.16is borrowed from [CE].
+254.The Macdonald-Mehta integral
+4.1.Finite Coxeter groups and the Macdonald-Mehta integral. LetWbe a finite
+Coxeter group of rank rwith real reflection representation hRequipped with a Euclidean
+W-invariant inner product ( ·,·). Denote by hthe complexification of hR. The reflection
+hyperplanes subdivide hRinto|W|chambers; let us pick one of them to be the dominant
+chamber andcall its interior D. Foreach reflection hyperplane, pick the perpendicular vector
+α∈hRwith (α,α) = 2 which has positive inner products with elements of D, and call it
+the positive root corresponding to this hyperplane. The walls of Dare then defined by the
+equations ( αi,v) = 0, where αiare simple roots. Denote by Sthe set of reflections in W,
+and for a reflection s∈Sdenote byαsthe corresponding positive root. Let
+δ(x) =/productdisplay
+s∈S(αs,x)
+be the corresponding discriminant polynomial. Let di,i= 1,...,r, be the degrees of the
+generators of the algebra C[h]W. Note that|W|=/producttext
+idi.
+LetH1,c(W,h) be the rational Cherednik algebra of W. Here we choose c=−kas a
+constant function. Let Mc=Mc(C) be the polynomial representation of H1,c(W,h), andβc
+be the contravariant form on Mcdefined in Section 3.12. We normalize it by the condition
+βc(1,1) = 1.
+Theorem 4.1. (i)(The Macdonald-Mehta integral) For Re(k)≥0, one has
+(4.1) (2 π)−r/2/integraldisplay
+hRe−(x,x)/2|δ(x)|2kdx=r/productdisplay
+i=1Γ(1+kdi)
+Γ(1+k).
+(ii)Letb(k) :=βc(δ,δ). Then
+b(k) =|W|r/productdisplay
+i=1di−1/productdisplay
+m=1(kdi+m).
+For Weyl groups, this theorem was proved by E. Opdam [Op1]. The n on-crystallographic
+cases were done by Opdam in [Op2] using a direct computation in the ra nk 2 case (reducing
+(4.1) to the beta integral by passing to polar coordinates), and a c omputer calculation by F.
+Garvan for H3andH4.
+Example 4.2. In the case W=Sn, we have the following integral (the Mehta integral):
+(2π)−(n−1)/2/integraldisplay
+{x∈Rn|/summationtext
+ixi=0}e−(x,x)/2/productdisplay
+i/\e}atio\slash=j|xi−xj|2kdx=n/productdisplay
+d=2Γ(1+kd)
+Γ(1+k).
+In the next subsection, we give a uniform proof of Theorem 4.1 which is given in [E2]. We
+emphasize that many parts of this proof are borrowed from Opdam ’s previous proof of this
+theorem.
+4.2.Proof of Theorem 4.1.
+Proposition 4.3. The function bis a polynomial of degree at most |S|, and the roots of b
+are negative rational numbers.
+26Proof.Sinceδhas degree|S|, it follows from the definition of bthat it is a polynomial of
+degree≤|S|.
+Suppose that b(k) = 0 for some k∈C. Thenβc(δ,P) = 0 for any polynomial P. Indeed,
+if there exists a Psuch thatβc(δ,P)/\e}atio\slash= 0, then there exists such a Pwhich is antisymmetric
+of degree|S|. ThenPmust be a multiple of δwhich contradicts the equality βc(δ,δ) = 0.
+Thus,Mcis reducible and hence has a singular vector, i.e. a nonzero homogene ous poly-
+nomialfof positive degree dliving in an irreducible representation τofWkilled byya.
+Applying the element h=/summationtext
+ixaiyai+r/2+k/summationtext
+s∈Sstof, we get
+k=−d
+mτ,
+wheremτis the eigenvalue of the operator T:=/summationtext
+s∈S(1−s) onτ. But it is clear (by
+computing the trace of T) thatmτ≥0 andmτ∈Q. This implies that any root of bis
+negative rational. /square
+Denote the Macdonald-Mehta integral by F(k).
+Proposition 4.4. One has
+F(k+1) =b(k)F(k).
+Proof.LetF=/summationtext
+iy2
+ai/2. Introduce the Gaussian inner product onMcas follows:
+Definition 4.5. The Gaussian inner product γconMcis given by the formula
+γc(v,v′) =βc(exp(F)v,exp(F)v′).
+This makes sense because the operator Fis locally nilpotent on Mc. Note that δis a
+nonzeroW-antisymmetric polynomial of the smallest possible degree, so (/summationtexty2
+ai)δ= 0 and
+hence
+(4.2) γc(δ,δ) =βc(δ,δ) =b(k).
+Fora∈h, letxa∈h∗⊂H1,c(W,h),ya∈h⊂H1,c(W,h) be the corresponding generators
+of the rational Cherednik algebra.
+Proposition 4.6. Up to scaling, γcis the unique W-invariant symmetric bilinear form on
+Mcsatisfying the condition
+γc((xa−ya)v,v′) =γc(v,yav′), a∈h.
+Proof.We have
+γc((xa−ya)v,v′) =βc(exp(F)(xa−ya)v,exp(F)v′) =βc(xaexp(F)v,exp(F)v′)
+=βc(exp(F)v,yaexp(F)v′) =βc(exp(F)v,exp(F)yav′) =γc(v,yav′).
+Let us now show uniqueness. If γis anyW-invariant symmetric bilinear form satisfying
+the condition of the Proposition, then let β(v,v′) =γ(exp(−F)v,exp(−F)v′). Thenβis
+contravariant, so it’s a multiple of βc, henceγis a multiple of γc. /square
+Now we will need the following known result (see [Du2], Theorem 3.10).
+27Proposition 4.7. ForRe(k)≥0we have
+(4.3) γc(f,g) =F(k)−1/integraldisplay
+hRf(x)g(x)dµc(x)
+where
+dµc(x) :=e−(x,x)/2|δ(x)|2kdx.
+Proof.It follows from Proposition 4.6 that γcis uniquely, up to scaling, determined by the
+condition that it is W-invariant, and y†
+a=xa−ya. These properties are easy to check for the
+right hand side of (4.3), using the fact that the action of yais given by Dunkl operators. /square
+Now we can complete the proof of Proposition 4.4. By Proposition 4.7, we have
+F(k+1) =F(k)γc(δ,δ),
+so by (4.2) we have
+F(k+1) =F(k)b(k).
+/square
+Let
+b(k) =b0/productdisplay
+(k+ki)ni.
+We know that ki>0, and also b0>0 (because the inner product β0on real polynomials is
+positive definite).
+Corollary 4.8. We have
+F(k) =bk
+0/productdisplay
+i/parenleftbiggΓ(k+ki)
+Γ(ki)/parenrightbiggni
+.
+Proof.Denote the right hand side by F∗(k) and letφ(k) =F(k)/F∗(k). Clearly, φ(0) = 1.
+Proposition 4.4 implies that φ(k) is a 1-periodic positive function on [0 ,∞). Also by the
+Cauchy-Schwarz inequality,
+F(k)F(k′)≥F((k+k′)/2)2,
+so logF(k) is convex for k≥0. This implies that φ= 1, since (log F∗(k))′′→0 ask→
++∞. /square
+Remark 4.9. The proof of this corollary is motivated by the standard proof of th e following
+well known characterization of the Γ function.
+Proposition 4.10. TheΓfunction is determined by three properties:
+(i) Γ(x)is positive on [1,+∞)andΓ(1) = 1;
+(ii) Γ(x+1) =xΓ(x);
+(iii) logΓ(x)is a convex function on [1,+∞).
+Proof.It is easy to see from the definition Γ( x) =/integraltext∞
+0tx−1e−tdtthat the Γ function has
+properties (i) and (ii); property (iii) follows from this definition and th e Cauchy-Schwarz
+inequality.
+Conversely, suppose we have a function F(x) satisfying the above properties, then we have
+F(x) =φ(x)Γ(x) for some 1-periodic function φ(x) withφ(x)>0. Thus, we have
+(logF)′′= (logφ)′′+(logΓ)′′.
+28Since lim x→+∞(logΓ)′′= 0, (logF)′′≥0, andφis periodic, we have (log φ)′′≥0. Since/integraltextn+1
+n(logφ)′′dx= 0, we see that (log φ)′′≡0. So we have φ(x)≡1. /square
+In particular, we see from Corollary 4.8 and the multiplication formulas for the Γ function
+that part (ii) of Theorem 4.1 implies part (i).
+It remains to establish (ii).
+Proposition 4.11. The polynomial bhas degree exactly |S|.
+Proof.By Proposition 4.3, bis a polynomial of degree at most |S|. To see that the degree is
+precisely|S|, let us make the change of variable x=k1/2yin the Macdonald-Mehta integral
+and use the steepest descent method. We find that the leading ter m of the asymptotics of
+logF(k) ask→+∞is|S|klogk. This together with the Stirling formula and Corollary 4.8
+implies the statement. /square
+Proposition 4.12. The function
+G(k) :=F(k)r/productdisplay
+j=11−e2πikdj
+1−e2πik
+analytically continues to an entire function of k.
+Proof.Letξ∈Dbe an element. Consider the real hyperplane Ct=itξ+hR,t >0. Then
+Ctdoes not intersect reflection hyperplanes, so we have a continuou s branch of δ(x)2kon
+Ctwhich tends to the positive branch in Dast→0. Then, it is easy to see that for any
+w∈W, the limit of this branch in the chamber w(D) will bee2πikℓ(w)|δ(x)|2k, whereℓ(w) is
+the length of w. Therefore, by letting t= 0, we get
+(2π)−r/2/integraldisplay
+Cte−(x,x)/2δ(x)2kdx=1
+|W|F(k)(/summationdisplay
+w∈We2πikℓ(w))
+(as this integral does not depend on tby Cauchy’s theorem). But it is well known that
+/summationdisplay
+w∈We2πikℓ(w)=r/productdisplay
+j=11−e2πikdj
+1−e2πik,
+([Hu], p.73), so
+(2π)−r/2|W|/integraldisplay
+Cte−(x,x)/2δ(x)2kdx=G(k).
+Since/integraltext
+Cte−(x,x)/2δ(x)2kdxis clearly an entire function, the statement is proved.
+/square
+Corollary 4.13. For everyk0∈[−1,0]the total multiplicity of all the roots of bof the
+formk0−p,p∈Z+, equals the number of ways to represent k0in the form−m/di,m=
+1,...,di−1. In other words, the roots of bareki,m=−m/di−pi,m,1≤m≤di−1, where
+pi,m∈Z+.
+Proof.We have
+G(k−p) =F(k)
+b(k−1)···b(k−p)r/productdisplay
+j=11−e2πikdj
+1−e2πik,
+29Now plug in k= 1 +k0and a large positive integer p. Since by Proposition 4.12 the left
+hand side is regular, so must be the right hand side, which implies the cla imed upper bound
+for the total multiplicity, as F(1 +k0)>0. The fact that the bound is actually attained
+follows from the fact that the polynomial bhas degree exactly |S|(Proposition 4.11), and
+the fact that all roots of bare negative rational (Proposition 4.3). /square
+It remains to show that in fact in Corollary 4.13, pi,m= 0 for all i,m; this would imply
+(ii) and hence (i).
+Proposition 4.14. Identity(4.1)of Theorem 4.1 is satisfied in C[k]/k2.
+Proof.Indeed, we clearly have F(0) = 1. Next, a rank 1 computation gives F′(0) =−γ|S|,
+whereγis the Euler constant (i.e. γ= limn→+∞(1+···+1/n−logn)), while the derivative
+of the right hand side of (4.1) at zero equals to
+−γr/summationdisplay
+i=1(di−1).
+But it is well known that
+r/summationdisplay
+i=1(di−1) =|S|,
+([Hu], p.62), which implies the result. /square
+Proposition 4.15. Identity(4.1)of Theorem 4.1 is satisfied in C[k]/k3.
+Notethat Proposition4.15immediately implies (ii), andhence thewhole t heorem. Indeed,
+it yields that
+(logF)′′(0) =r/summationdisplay
+i=1di−1/summationdisplay
+m=1(logΓ)′′(m/di),
+so by Corollary 4.13
+r/summationdisplay
+i=1di−1/summationdisplay
+m=1(logΓ)′′(m/di+pi,m) =r/summationdisplay
+i=1di−1/summationdisplay
+m=1(logΓ)′′(m/di),
+which implies that pi,m= 0 since (logΓ)′′is strictly decreasing on [0 ,∞).
+To prove Proposition 4.15, we will need the following result about finite Coxeter groups.
+Letψ(W) = 3|S|2−/summationtextr
+i=1(d2
+i−1).
+Lemma 4.16. One has
+(4.4) ψ(W) =/summationdisplay
+G∈Par2(W)ψ(G),
+wherePar2(W)is the set of parabolic subgroups of Wof rank 2.
+Proof.Let
+Q(q) =|W|r/productdisplay
+i=11−q
+1−qdi.
+30It follows from Chevalley’s theorem that
+Q(q) = (1−q)r/summationdisplay
+w∈Wdet(1−qw|h)−1.
+Let us subtract the terms for w= 1 andw∈Sfrom both sides of this equation, divide both
+sides by (q−1)2, and setq= 1 (cf. [Hu], p.62, formula (21)). Let W2be the set of elements
+ofWthat can be written as a product of two different reflections. Then by a straightforward
+computation we get
+1
+24ψ(W) =/summationdisplay
+w∈W21
+r−Trh(w).
+In particular, this is true for rank 2 groups. The result follows, as a ny element w∈W2
+belongs to a unique parabolic subgroup Gwof rank 2 (namely, the stabilizer of a generic
+pointhw, [Hu], p.22). /square
+Proof of Proposition 4.15. Now we are ready to prove the proposition. By Proposition 4.14,
+it suffices to show the coincidence of the second derivatives of (4.1) atk= 0. The second
+derivative of the right hand side of (4.1) at zero is equal to
+π2
+6r/summationdisplay
+i=1(d2
+i−1)+γ2|S|2.
+On the other hand, we have
+F′′(0) = (2π)−r/2/summationdisplay
+α,β∈S/integraldisplay
+hRe−(x,x)/2logα2(x)logβ2(x)dx.
+Thus, from a rank 1 computation we see that our job is to establish t he equality
+(2π)−r/2/summationdisplay
+α/\e}atio\slash=β∈S/integraldisplay
+hRe−(x,x)/2logα2(x)logβ2(x)
+α2(x)dx=π2
+6(r/summationdisplay
+i=1(d2
+i−1)−3|S|2) =−π2
+6ψ(W).
+Since this equality holds in rank 2 (as in this case (4.1) reduces to the b eta integral), in
+general it reduces to equation (4.4) (as for any α/\e}atio\slash=β∈S,sαandsβare contained in a
+unique parabolic subgroup of Wof rank 2). The proposition is proved. /square
+4.3.Application: the supports of Lc(C).In this subsection we will use the Macdonald-
+Mehta integral to computation of the support of the irreducible qu otient of the polynoamial
+representation of a rational Cherednik algebra (with equal param eters). We will follow the
+paper [E3].
+First note that the vector space hhas a stratification labeled by parabolic subgroups of W.
+Indeed, for a parabolic subgroup W′⊂W, lethW′
+regbe the set of points in hwhose stabilizer
+isW′. Then
+h=/coproductdisplay
+W′∈Par(W)hW′
+reg,
+where Par(W) is the set of parabolic subgroups in W.
+For a finitely generated module MoverC[h], denote the support of Mby supp(M).
+The following theorem is proved in [Gi1], Section 6 and in [BE] with differen t method. We
+will recall the proof from [BE] later.
+31Theorem 4.17. Consider the stratification of hwith respect to stabilizers of points in W.
+Then the support supp(M)of any object MofOc(W,h)inhis a union of strata of this
+stratification.
+This makes one wonder which strata occur in supp( Lc(τ)), for given candτ. In [VV],
+Varagnolo and Vasserot gave a partial answer for τ=C. Namely, they determined (for W
+being a Weyl group) when Lc(C) is finite dimensional, which is equivalent to supp( Lc(C)) =
+0. For the proof (which is quite complicated), they used the geomet ry affine Springer fibers.
+Here we will give a different (and simpler) proof. In fact, we will prove a more general result.
+Recall that for any Coxeter group W, we have the Poincar´ e polynomial:
+PW(q) =/summationdisplay
+w∈Wqℓ(w)=r/productdisplay
+i=11−qdi(W)
+1−q,wheredi(W) are the degrees of W.
+Lemma 4.18. IfW′⊂Wis a parabolic subgroup of W, thenPWis divisible by PW′.
+Proof.By Chevalley’s theorem, C[h] is a free module over C[h]WandC[h]W′is a direct
+summand in this module. So C[h]W′is a projective module, thus free (since it is graded).
+Hence, there exists a polynomial Q(q) such that we have
+Q(q)hC[h]W(q) =hC[h]W′(q),
+wherehV(q) denotes the Hilbert series of a graded vector space V. Notice that we have
+hC[h]W(q) =1
+PW(q)(1−q)r, so we have
+Q(q)
+PW(q)=1
+PW′(q),i.e.Q(q) =PW(q)/PW′(q).
+/square
+Corollary 4.19. Ifm≥2then we have the following inequality:
+#{i|mdividesdi(W)}≥#{i|mdividesdi(W′)}.
+Proof.This follows from Lemma 4.18 by looking at the roots of the polynomials PWand
+PW′. /square
+Our main result is the following theorem.
+Theorem 4.20. [E3]Letc≥0. Thena∈supp(Lc(C))if and only if
+PW
+PWa(e2πic)/\e}atio\slash= 0.
+We can obtain the following corollary easily.
+Corollary 4.21. (i)Lc(C)/\e}atio\slash=Mc(C)if and only if c∈Q>0and the denominator mof
+cdividesdifor somei;
+(ii)Lc(C)is finite dimensional if and only ifPW
+PW′(e2πic) = 0, i.e., iff
+#{i|mdividesdi(W)}>#{i|mdividesdi(W′)}.
+for any maximal parabolic subgroup W′⊂W.
+32Remark 4.22. Varagnolo and Vasserot prove that Lc(C) is finite dimensional if and only if
+there exists a regular elliptic element in Wof orderm. Case-by-case inspection shows that
+this condition is equivalent to the combinatorial condition of (2). Also , a uniform proof of
+this equivalence is given in the appendix to [E3], written by S. Griffeth.
+Example 4.23. For typeAn−1, i.e.,W=Sn, we get that Lc(C) is finite dimensional if and
+only if the denominator of cisn. This agrees with our previous results in type An−1.
+Example 4.24. SupposeWis the Coxeter group of type E7. Then we have the following
+list of maximal parabolic subgroups and the degrees (note that E7itself is not a maximal
+parabolic).
+Subgroups E7 D6A3×A2×A1A6
+Degrees 2,6,8,10,12,14,18 2,4,6,6,8,10 2,3,4,2,3,2 2,3,4,5,6,7
+Subgroups A4×A2 E6 D5×A1A5×A1
+Degrees 2,3,4,5,2,3 2,5,6,8,9,12 2,4,5,6,8,2 2,3,4,5,6,2
+SoLc(C) is finite dimensional if and only if the denominator of cis 2,6,14,18.
+The rest of the subsection is dedicated to the proof of Theorem 4.2 0. First we recall some
+basic facts about the Schwartz space and tempered distributions .
+LetS(Rn) be the set of Schwartz functions on Rn, i.e.
+S(Rn) ={f∈C∞(Rn)|∀α,β,sup|xα∂βf(x)|<∞}.
+This space has a natural topology.
+A tempered distribution on Rnis a continuous linear functional on S(Rn). LetS′(Rn)
+denote the space of tempered distributions.
+We will use the following well known lemma.
+Lemma 4.25. (i)C[x]e−x2/2⊂S(Rn)is a dense subspace.
+(ii)Any tempered distribution ξhas finite order, i.e., ∃N=N(ξ)such that if f∈S(Rn)
+satisfyingf= df=···= dN−1f= 0onsuppξ, then/a\}b∇acketle{tξ,f/a\}b∇acket∇i}ht= 0.
+Proof of Theorem 4.20. Recall that on Mc(C), we have the Gaussian form γcfrom Section
+4.2. We have for Re( c)≤0,
+γc(P,Q) =(2π)−r/2
+FW(−c)/integraldisplay
+hRe−x2/2|δ(x)|−2cP(x)Q(x)dx,
+whereP,Qare polynomials and
+FW(k) = (2π)−r/2/integraldisplay
+hRe−x2/2|δ(x)|2kdx
+is the Macdonald-Mehta integral.
+Consider the distribution:
+ξW
+c=(2π)−r/2
+FW(−c)|δ(x)|−2c.
+It is well-known that this distribution is meromorphic in c(Bernstein’s theorem). Moreover,
+sinceγc(P,Q) is a polynomial in cfor anyPandQ, this distribution is in fact holomorphic
+inc∈C.
+33Proposition 4.26.
+supp(ξW
+c) ={a∈hR|FWa
+FW(−c)/\e}atio\slash= 0}={a∈hR|PW
+PWa(e2πic)/\e}atio\slash= 0}
+={a∈hR|#{i|denominator of cdividesdi(W)}
+= #{i|denominator of cdividesdi(Wa)}}.
+Proof.First note that the last equality follows from the product formula fo r the Poincar´ e
+polynomial, and the second equality from the Macdonald-Mehta ident ity. Now let us prove
+the first equality.
+Look atξW
+cneara∈h. Equivalently, we can consider
+ξW
+c(x+a) =(2π)−r/2
+FW(−c)|δ(x+a)|−2c
+withxnear 0. We have
+δW(x+a) =/productdisplay
+s∈Sαs(x+a) =/productdisplay
+s∈S(αs(x)+αs(a))
+=/productdisplay
+s∈S∩Waαs(x)·/productdisplay
+s∈S\S∩Wa(αs(x)+αs(a))
+=δWa(x)·Ψ(x),
+where Ψ is a nonvanishing function near a(sinceαs(a)/\e}atio\slash= 0 ifs /∈S∩Wa).
+So neara, we have
+ξW
+c(x+a) =FWa
+FW(−c)·ξWa
+c(x)·|Ψ|−2c,
+and the last factor is well defined since Ψ is nonvanishing. Thus ξW
+c(x) is nonzero near aif
+and only ifFWa
+FW(−c)/\e}atio\slash= 0 which finishes the proof. /square
+Proposition 4.27. Forc≥0,
+supp(ξW
+c) = suppLc(C)R,
+where the right hand side stands for the real points of the sup port.
+Proof.Leta /∈suppLc(C) and assume a∈suppξW
+c. Then we can find a P∈Jc(C) = kerγc
+such thatP(a)/\e}atio\slash= 0. Pick a compactly supported test function φ∈C∞
+c(hR) such that Pdoes
+not vanish anywhere on supp φ, and/a\}b∇acketle{tξW
+c,φ/a\}b∇acket∇i}ht/\e}atio\slash= 0 (this can be done since P(a)/\e}atio\slash= 0 andξW
+c
+is nonzero near a). Then we have φ/P∈S(hR). Thus from Lemma 4.25 (i) it follows that
+there exists a sequence of polynomials Pnsuch that
+Pn(x)e−x2/2→φ
+PinS(hR), whenn→∞.
+SoPPne−x2/2→φinS(hR), whenn→∞.
+But we have/a\}b∇acketle{tξW
+c,PPne−x2/2/a\}b∇acket∇i}ht=γc(P,Pn) = 0 which is a contradiction. This implies that
+suppξW
+c⊂(suppLc(C))R.
+To show the opposite inclusion, let Pbe a polynomial on hwhich vanishes identically on
+suppξW
+c. By Lemma 4.25 (ii), there exists Nsuch that/a\}b∇acketle{tξW
+c,PN(x)Q(x)e−x2/2/a\}b∇acket∇i}ht= 0. Thus,
+34for any polynomial Q,γc(PN,Q) = 0, i.e.PN∈Kerγc. Thus,P|suppLc(C)= 0. This implies
+the required inclusion, since supp ξW
+cis a union of strata. /square
+Theorem 4.20 follows from Proposition 4.26 and Proposition 4.27. /square
+4.4.Notes.Ourexposition inSections 4.1and4.2followsthepaper [E2]; Section4.3f ollows
+the paper [E3].
+355.Parabolic induction and restriction functors for rational Cherednik
+algebras
+5.1.A geometric approach to rational Cherednik algebras. An important property
+of the rational Cherednik algebra H1,c(G,h) is that it can be sheafified, as an algebra, over
+h/G(see [E1]). More specifically, the usual sheafification of H1,c(G,h) as aOh/G-module
+is in fact a quasicoherent sheaf of algebras, H1,c,G,h. Namely, for every affine open subset
+U⊂h/G, the algebra of sections H1,c,G,h(U) is, by definition, C[U]⊗C[h]GH1,c(G,h).
+The same sheaf can be defined more geometrically as follows (see [E1], Section 2.9). Let
+/tildewideUbe the preimage of Uinh. Then the algebra H1,c,G,h(U) is the algebra of linear operators
+onO(/tildewideU) generated byO(/tildewideU), the group G, and Dunkl operators
+∂a−/summationdisplay
+s∈S2cs
+1−λsαs(a)
+αs(1−s),wherea∈h.
+5.2.Completion of rational Cherednik algebras. For anyb∈hwe can define the
+completion/hatwidestH1,c(G,h)bto be the algebra of sections of the sheaf H1,c,G,hon the formal neigh-
+borhood of the image of binh/G. Namely,/hatwidestH1,c(G,h)bis generated by regular functions on
+the formal neighborhood of the G-orbit ofb, the group G, and Dunkl operators.
+The algebra/hatwidestH1,c(G,h)binherits from H1,c(G,h) the natural filtration F•by order of dif-
+ferential operators, and each of the spaces Fn/hatwidestH1,c(G,h)bhas a projective limit topology; the
+whole algebra is then equipped with the topology of the nested union ( or inductive limit).
+Consider thecompletionoftherationalCherednikalgebraatzero, /hatwidestH1,c(G,h)0. Itnaturally
+contains the algebra C[[h]]. Define the category /hatwideOc(G,h) of representations of /hatwidestH1,c(G,h)0
+which are finitely generated over C[[h]]0=C[[h]].
+We have a completion functor /hatwide:Oc(G,h)→/hatwideOc(G,h), defined by
+/hatwiderM=/hatwidestH1,c(G,h)0⊗H1,c(G,h)M=C[[h]]⊗C[h]M.
+Also, forN∈/hatwideOc(G,h), letE(N) be the subspace spanned by generalized eigenvectors of
+hinNwherehis defined by (3.2). Then it is easy to see that E(N)∈Oc(G,h)0.
+Theorem 5.1. The restriction of the completion functor /hatwidetoOc(G,h)0is an equivalence
+of categoriesOc(G,h)0→/hatwideOc(G,h). The inverse equivalence is given by the functor E.
+Proof.It is clear that M⊂/hatwiderM, soM⊂E(/hatwiderM) (asMis spanned by generalized eigenvectors
+ofh). Let us demonstrate the opposite inclusion. Pick generators m1,...,mrofMwhich
+are generalized eigenvectors of hwith eigenvalues µ1,...,µr. Let 0/\e}atio\slash=v∈E(/hatwiderM). Thenv=/summationtext
+ifimi, wherefi∈C[[h]]. Assume that ( h−µ)Nv= 0 for some N. Thenv=/summationtext
+if(µ−µi)
+imi,
+where forf∈C[[h]] we denote by f(d)the degreedpart off. Thusv∈M, soM=E(/hatwiderM).
+It remains to show that /hatwiderE(N) =N, i.e. thatNis the closure of E(N). In other words,
+lettingmdenote the maximal ideal in C[[h]], we need to show that the natural map E(N)→
+N/mjNis surjective for every j.
+To do so, note that hpreserves the descending filtration of Nby subspaces mjN. On
+the other hand, the successive quotients of these subspaces, mjN/mj+1N, are finite dimen-
+sional, which implies that hacts locally finitely on their direct sum gr N, and moreover each
+36generalized eigenspace is finite dimensional. Now for each β∈Cdenote byNj,βthe general-
+izedβ-eigenspace of hinN/mjN. We have surjective homomorphisms Nj+1,β→Nj,β, and
+for large enough jthey are isomorphisms. This implies that the map E(N)→N/mjNis
+surjective for every j, as desired. /square
+Example. Suppose that c= 0. Then Theorem 5.1 specializes to the well known fact that
+the category of G-equivariant local systems on hwith a locally nilpotent action of partial
+differentiations is equivalent to the category of all G-equivariant local systems on the formal
+neighborhood of zero in h. In fact, both categories in this case are equivalent to the catego ry
+of finite dimensional representations of G.
+We can now define the composition functor J:Oc(G,h)→Oc(G,h)0, by the formula
+J(M) =E(/hatwiderM). The functorJis called the Jacquet functor ([Gi2]).
+5.3.The duality functor. Recall that in Section 3.11, for any H1,c(G,h)-moduleM, the
+full dual space M∗is naturally an H1,¯c(G,h∗)-module, via πM∗(a) =πM(γ(a))∗.
+It is clear that the duality functor ∗defines an equivalence between the category Oc(G,h)0
+and/hatwideO¯c(G,h∗)op, and thatM†=E(M∗) (whereM†is the contragredient, or restricted dual
+module toMdefined in Section 3.11).
+5.4.Generalized Jacquet functors.
+Proposition 5.2. For anyM∈/hatwideOc(G,h), a vectorv∈Mish-finite if and only if it is
+h-nilpotent.
+Proof.The “if” part follows from Theorem 3.20. To prove the “only if” part, assume that
+(h−µ)Nv= 0. Then for any u∈Srh·v, we have ( h−µ+r)Nu= 0. But by Theorem 5.1,
+the real parts of generalized eigenvalues of hinMare bounded below. Hence Srh·v= 0 for
+large enough r, as desired. /square
+According to Proposition 5.2, the functor Ecan be alternatively defined by setting E(M)
+to be the subspace of Mwhich is locally nilpotent under the action of h.
+This gives rise to the following generalization of E: for anyλ∈h∗we define the functor
+Eλ:/hatwideOc(G,h)→Oc(G,h)λby settingEλ(M) to be the space of generalized eigenvectors of
+C[h∗]GinMwith eigenvalue λ. This way, we have E0=E.
+We can also define the generalized Jacquet functor Jλ:Oc(G,h)→Oc(G,h)λby the
+formulaJλ(M) =Eλ(/hatwiderM). Then we haveJ0=J, and one can show that the restriction of
+JλtoOc(G,h)λis the identity functor.
+5.5.The centralizer construction. For a finite group H, leteH=|H|−1/summationtext
+g∈Hgbe the
+symmetrizer of H.
+IfG⊃Hare finite groups, and Ais an algebra containing C[H], then define the algebra
+Z(G,H,A) to be the centralizer End A(P) ofAin the right A-moduleP= FunH(G,A) of
+H-invariantA-valued functions on G, i.e. such functions f:G→Athatf(hg) =hf(g).
+Clearly,Pis a freeA-module of rank|G/H|, so the algebra Z(G,H,A) is isomorphic to
+Mat|G/H|(A), but this isomorphism is not canonical.
+The following lemma is trivial.
+Lemma 5.3. The functor N/mapsto→I(N) :=P⊗AN= FunH(G,N)defines an equivalence of
+categoriesA−mod→Z(G,H,A)−mod.
+375.6.Completion of rational Cherednik algebras at arbitrary poi nts ofh/G.The
+following result is, in essence, a consequence of the geometric appr oach to rational Cherednik
+algebras, described in Subsection 5.1. It should be regarded as a dir ect generalization to the
+case of Cherednik algebras of Theorem 8.6 of [L] for affine Hecke alg ebras.
+Letb∈h. Abusing notation, denote the restriction of cto the setSbof reflections in Gb
+also byc.
+Theorem 5.4. One has a natural isomorphism
+θ:/hatwidestH1,c(G,h)b→Z(G,Gb,/hatwidestH1,c(Gb,h)0),
+defined by the following formulas. Suppose that f∈P= FunGb(G,/hatwidestH1,c(Gb,h)0). Then
+(θ(u)f)(w) =f(wu),u∈G;
+for anyα∈h∗,
+(θ(xα)f)(w) = (x(b)
+wα+(wα,b))f(w),
+wherexα∈h∗⊂H1,c(G,h),x(b)
+α∈h∗⊂H1,c(Gb,h)are the elements corresponding to α; and
+for anya∈h,
+(5.1) ( θ(ya)f)(w) =y(b)
+waf(w)−/summationdisplay
+s∈S:s/∈Gb2cs
+1−λsαs(wa)
+x(b)
+αs+αs(b)(f(w)−f(sw)).
+whereya∈h⊂H1,c(G,h),y(b)
+a∈h⊂H1,c(Gb,h).
+Proof.The proof is by a direct computation. We note that in the last formula , the fraction
+αs(wa)/(x(b)
+αs+αs(b)) is viewed as a power series (i.e., an element of C[[h]]), and that only
+the entire sum, and not each summand separately, is in the centraliz er algebra. /square
+Remark. Let us explain how to see the existence of θwithout writing explicit formulas,
+and how to guess the formula (5.1) for θ. It is explained in [E1] (see e.g. [E1], Section
+2.9) that the sheaf of algebras obtained by sheafification of H1,c(G,h) overh/Gis generated
+(on every affine open set in h/G) by regular functions on h, elements of G, and Dunkl
+operators. Therefore, this statement holds for formal neighbo rhoods, i.e., it is true on the
+formalneighborhoodoftheimagein h/Gofanypoint b∈h. However, looking attheformula
+for Dunkl operators near b, we see that the summands corresponding to s∈S,s /∈Gbare
+actually regular at b, so they can be safely deleted without changing the generated alge bra
+(as all regular functions on the formal neighborhood of bare included into the system of
+generators). But after these terms are deleted, what remains is nothing but the Dunkl
+operators for ( Gb,h), which, together with functions on the formal neighborhood of band
+the groupGb, generate the completion of H1,c(Gb,h). This gives a construction of θwithout
+using explicit formulas.
+Also, this argument explains why θshould be defined by formula (5.1) of Theorem 5.4.
+Indeed, what this formula does is just restores the terms with s /∈Gbthat have been
+previously deleted.
+The mapθdefines an equivalence of categories
+θ∗:/hatwidestH1,c(G,h)b−mod→Z(G,Gb,/hatwidestH1,c(Gb,h)0)−mod.
+38Corollary 5.5. We have a natural equivalence of categories
+ψλ:Oc(G,h)λ→Oc(Gλ,h/hGλ)0.
+Proof.The categoryOc(G,h)λis the category of modules over H1,c(G,h) which are finitely
+generated over C[h] and extend by continuity to the completion of the algebra H1,c(G,h)
+atλ. So it follows from Theorem 5.4 that we have an equivalence Oc(G,h)λ→Oc(Gλ,h)0.
+Composingthisequivalence withtheequivalence ζ:Oc(Gλ,h)0→Oc(Gλ,h/hGλ)0, weobtain
+the desired equivalence ψλ. /square
+Remark 5.6. Note that in this proof, we take the completion of H1,c(G,h) at a point of
+λ∈h∗rather than b∈h.
+5.7.The completion functor. Let/hatwideOc(G,h)bbe the category of modules over /hatwidestH1,c(G,h)b
+which are finitely generated over/hatwidestC[h]b.
+Proposition5.7. Thedualityfunctor ∗definesananti-equivalenceofcategories Oc(G,h)λ→
+/hatwideO¯c(G,h∗)λ.
+Proof.This follows from the fact (already mentioned above) that Oc(G,h)λis the category
+of modules over H1,c(G,h) which are finitely generated over C[h] and extend by continuity
+to the completion of the algebra H1,c(G,h) atλ. /square
+Let us denote the functor inverse to ∗also by∗; it is the functor of continuous dual (in
+the formal series topology).
+We have an exact functor of completion at b,Oc(G,h)0→/hatwideOc(G,h)b,M/mapsto→/hatwiderMb. We also
+have a functor Eb:/hatwideOc(G,h)b→Oc(G,h)0in the opposite direction, sending a module Nto
+the spaceEb(N) ofh-nilpotent vectors in N.
+Proposition 5.8. The functor Ebis right adjoint to the completion functor /hatwiderb.
+Proof.We have
+Hom/hatwidestH1,c(G,h)b(/hatwiderMb,N) = Hom /hatwidestH1,c(G,h)b(/hatwidestH1,c(G,h)b⊗H1,c(G,h)M,N)
+= HomH1,c(G,h)(M,N|H1,c(G,h)) = Hom H1,c(G,h)(M,Eb(N)).
+/square
+Remark 5.9. Recall that by Theorem 5.1, if b= 0 then these functors are not only adjoint
+but also inverse to each other.
+Proposition 5.10. (i)ForM∈O¯c(G,h∗)b, one hasEb(M∗) = (/hatwiderM)∗inOc(G,h)0.
+(ii)ForM∈Oc(G,h)0,(/hatwiderMb)∗=Eb(M∗)inO¯c(G,h∗)b.
+(iii)The functors Eb,Ebare exact.
+Proof.(i),(ii) are straightforward from the definitions. (iii) follows from (i),( ii), since the
+completion functors are exact. /square
+395.8.Parabolic induction and restriction functors for rational Cherednik algebras.
+Theorem 5.4 allows us to define analogs of parabolic restriction funct ors for rational Chered-
+nik algebras.
+Namely, let b∈h, andGb=G′. Define a functor Res b:Oc(G,h)0→Oc(G′,h/hG′)0by
+the formula
+Resb(M) = (ζ◦E◦I−1◦θ∗)(/hatwiderMb).
+We can also define the parabolic induction functors in the opposite dir ection. Namely, let
+N∈Oc(G′,h/hG′)0. Then we can define the object Ind b(N)∈Oc(G,h)0by the formula
+Indb(N) = (Eb◦θ−1
+∗◦I)(/hatwiderζ−1(N)0).
+Proposition 5.11. (i)The functors Indb,Resbare exact.
+(ii)One has Indb(Resb(M)) =Eb(/hatwiderMb).
+Proof.Part (i) follows from the fact that the functor Eband the completion functor /hatwidebare
+exact (see Proposition 5.10). Part (ii) is straightforward from the definition. /square
+Theorem 5.12. The functor Indbis right adjoint to Resb.
+Proof.We have
+Hom(Res b(M),N) = Hom((ζ◦E◦I−1◦θ∗)(/hatwiderMb),N) = Hom((E◦I−1◦θ∗)(/hatwiderMb),ζ−1(N))
+= Hom((I−1◦θ∗)(/hatwiderMb),/hatwiderζ−1(N)0) = Hom(/hatwiderMb,(θ−1
+∗◦I)(/hatwiderζ−1(N)0))
+= Hom(M,(Eb◦θ−1
+∗◦I)(/hatwiderζ−1(N)0)) = Hom(M,Indb(N)).
+At the end we used Proposition 5.8. /square
+Then we can obtain the following corollary easily.
+Corollary 5.13. The functor Resbmaps projective objects to projective ones, and the functor
+Indbmaps injective objects to injective ones.
+We can also define functors res λ:Oc(G,h)0→Oc(G′,h/hG′)0and indλ:Oc(G′,h/hG′)0→
+Oc(G,h)0, attached to λ∈h∗G′
+reg, by
+resλ:=†◦Resλ◦†,indλ:=†◦Indλ◦†,
+where†is as in Subsection 5.3.
+Corollary 5.14. The functors resλ,indλare exact. The functor indλis left adjoint to resλ.
+The functor indλmaps projective objects to projective ones, and the functor resλinjective
+objects to injective ones.
+Proof.Easy to see from the definition of the functors and the Theorem 5.1 2. /square
+We also have the following proposition, whose proof is straightforwa rd.
+Proposition 5.15. We have
+indλ(N) = (J◦ψ−1
+λ)(N),andresλ(M) = (ψλ◦Eλ)(/hatwiderM),
+whereψλis defined in Corollary 5.5.
+405.9.Some evaluations of the parabolic induction and restrictio n functors. For
+genericc, the categoryOc(G,h) is semisimple, and naturally equivalent to the category
+RepGof finite dimensional representations of G, via the functor τ/mapsto→Mc(G,h,τ). (IfGis
+a Coxeter group, the exact set of such c(which are called regular) is known from [GGOR]
+and [Gy]).
+Proposition 5.16. (i)Suppose that cis generic. Upon the above identification, the
+functors Indb,indλandResb,resλgo to the usual induction and restriction functors
+between categories RepGandRepG′. In other words, we have
+Resb(Mc(G,h,τ)) =⊕ξ∈/hatwiderG′nτξMc(G′,h/hG′,ξ),
+and
+Indb(Mc(G′,h/hG′,ξ)) =⊕τ∈/hatwideGnτξMc(G,h,τ),
+wherenτξis the multiplicity of occurrence of ξinτ|G′, and similarly for resλ,indλ.
+(ii)The equations of (i) hold at the level of Grothendieck groups for allc.
+Proof.Part (i) is easy for c= 0, and is obtained for generic cby a deformation argument.
+Part (ii) is also obtained by deformation argument, taking into accou nt that the functors
+Resband Indbare exact and flat with respect to c. /square
+Example 5.17. Suppose that G′= 1. Then Res b(M) is the fiber of Matb, while Ind b(C) =
+PKZ, theobjectdefined in[GGOR], which isprojective andinjective (see Remark5.22). This
+shows that Proposition 5.16 (i) does not hold for special c, asPKZis not, in general, a direct
+sum of standard modules.
+5.10.Dependence of the functor Resbonb.LetG′⊂Gbe a parabolic subgroup. In
+the construction of the functor Res b, the point bcan be made a variable which belongs to
+the open set hG′
+reg.
+Namely, let/hatwidesthG′
+regbe the formal neighborhood of the locally closed set hG′
+reginh, and let
+π:/hatwidesthG′
+reg→h/Gbe the natural map (note that this map is an´ etale covering of the im age with
+the Galois group NG(G′)/G′, whereNG(G′) is the normalizer of G′inG). Let/hatwidestH1,c(G,h)hG′
+reg
+be the pullback of the sheaf H1,c,G,hunderπ. We can regard it as a sheaf of algebras over
+hG′
+reg. Similarly to Theorem 5.4 we have an isomorphism
+θ:/hatwidestH1,c(G,h)hG′
+reg→Z(G,G′,/hatwidestH1,c(G′,h/hG′)0)ˆ⊗D(hG′
+reg),
+whereD(hG′
+reg)isthesheafofdifferential operatorson hG′
+reg, andˆ⊗isanappropriatecompletion
+of the tensor product.
+Thus, repeating the construction of Res b, we can define the functor
+Res :Oc(G,h)0→Oc(G′,h/hG′)0⊠Loc(hG′
+reg),
+where Loc(hG′
+reg) stands for the category of local systems (i.e. O-coherentD-modules) on hG′
+reg.
+This functor has the property that Res bis the fiber of Res at b. Namely, the functor Res is
+defined by the formula
+Res(M) = (E◦I−1◦θ∗)(/hatwiderMhG′
+reg),
+where/hatwiderMhG′
+regis the restriction of the sheaf Monhto the formal neighborhood of hG′
+reg.
+41Remark 5.18. IfG′isthetrivialgroup, thefunctorResisjusttheKZfunctorfrom[G GOR],
+which we will discuss later. Thus, Res is a relative version of the KZ fun ctor.
+Remark 5.19. Note that the object R es(M) is naturally equivariant under the group
+NG(G′)/G′.
+Thus, we see that the functor Res bdoes not depend on b, up to an isomorphism. A similar
+statement is true for the functors Ind b, resλ, indλ.
+Conjecture 5.20. For anyb∈h,λ∈h∗such thatGb=Gλ, we have isomorphisms of
+functors Res b∼=resλ, Indb∼=indλ.
+Remark 5.21. Conjecture 5.20 would imply that Ind bis left adjoint to Res b, and that Res b
+maps injective objects to injective ones, while Ind bmaps projective objects to projective
+ones.
+Remark 5.22. Ifbandλare generic (i.e., Gb=Gλ= 1) then the conjecture holds. Indeed,
+in this case the conjecture reduces to showing that we have an isom orphism of functors
+Fiberb(M)∼=Fiberλ(M†)∗(M∈Oc(G,h)). Since both functors are exact functors to the
+categoryofvector spaces, itsufficestocheck thatdimF iberb(M) = dimFiberλ(M†). Butthis
+is true because both dimensions aregiven by the leading coefficient of the Hilbert polynomial
+ofM(characterizing the growth of M).
+It is important to mention, however, that although Res bis isomorphic to Res b′ifGb=Gb′,
+this isomorphism is not canonical. So let us examine the dependence of Resbonba little
+more carefully.
+Theorem 5.16 implies that if cis generic, then
+Res(Mc(G,h,τ)) =⊕ξMc(G′,h/hG′,ξ)⊗Lτξ,
+whereLτξis a local system on hG′
+regof ranknτξ. Let us characterize the local system Lτξ
+explicitly.
+Proposition 5.23. The local system Lτξis given by the connection on the trivial bundle
+given by the formula
+∇= d−/summationdisplay
+s∈S:s/∈G′2cs
+1−λsdαs
+αs(1−s).
+with values in HomG′(ξ,τ|G′).
+Proof.This follows immediately from formula (5.1). /square
+Definition 5.24. WewillcalltheconnectionofProposition5.23theparabolicKZ(Knizhn ik-
+Zamolodchikov) connection.
+Example 5.25. LetG=SnandG′=Sn1×···×Snkwithn1+···+nk=n. In this case,
+there is only one parameter c.
+Leth=Cnbe the permutation representation of G. Then
+hG′= (Cn)G′={v∈h|v= (z1,...,z 1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+n1,z2,...,z 2/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+n2,...,zk,...,zk/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+nk)}.
+42Thus, the parabolic KZ connection on the trivial bundle with fiber bein g a representation τ
+ofSnhas the form
+d−c/summationdisplay
+1≤p<q≤kn1+···+np/summationdisplay
+i=n1+···+np−1+1n1+···+nq/summationdisplay
+j=n1+···+nq−1+1dzp−dzq
+zp−zq(1−sij).
+So the differential equations for a flat section F(z) of this bundle have the form
+∂F
+∂zp=c/summationdisplay
+q/\e}atio\slash=pn1+···+np/summationdisplay
+i=n1+···+np−1+1n1+···+nq/summationdisplay
+j=n1+···+nq−1+1(1−sij)F
+zp−zq.
+SoF(z) =G(z)/producttext
+p<q(zp−zq)cnpnq, where the function Gsatisfies the differential equation
+∂G
+∂zp=−c/summationdisplay
+q/\e}atio\slash=pn1+···+np/summationdisplay
+i=n1+···+np−1+1n1+···+nq/summationdisplay
+j=n1+···+nq−1+1sijG
+zp−zq.
+Letτ=V⊗nwhereVis a finite dimensional space with dim V=N(this class of repre-
+sentations contains as summands all irreducible representations o fSn). LetVp=V⊗np, so
+thatτ=V1⊗···⊗Vk. Then the equation for Gcan be written as
+∂G
+∂zp=−c/summationdisplay
+q/\e}atio\slash=pΩpqG
+zp−zq, p= 1,...,k,
+where Ω =/summationtextN
+s,t=1Es,t⊗Et,sis the Casimir element for glN(Ei,jis theNbyNmatrix with
+the only 1 at the ( i,j)-th place, and the rest of the entries being 0).
+This is nothing but the well known Knizhnik-Zamolodchikov system of e quations of the
+WZW conformal field theory, for the Lie algebra glN, see [EFK]. (Note that the repre-
+sentationsViare “the most general” in the sense that any irreducible finite dimens ional
+representation of glNoccurs inV⊗rfor somer, up to tensoring with a character.)
+This motivates the term “parabolic KZ connection”.
+5.11.Supports of modules. The following two basic propositions are proved in [Gi1],
+Section 6. We will give different proofs of them, based on the restric tion functors.
+Proposition 5.26. Consider the stratification of hwith respect to stabilizers of points in G.
+Then the (set-theoretical) support SuppMof any object MofOc(G,h)inhis a union of
+strata of this stratification.
+Proof.This follows immediately from the existence of the flat connection alon g the set of
+pointsbwith a fixed stabilizer G′on the bundle Res b(M). /square
+Proposition 5.27. For any irreducible object MinOc(G,h),SuppM/G is an irreducible
+algebraic variety.
+Proof.LetXbe a component of Supp M/G. LetM′be the subspace of elements of M
+whose restriction to a neighborhood of a generic point of Xis zero. It is obvious that M′is
+anH1,c(G,h)-submodule in M. By definition, it is a proper submodule. Therefore, by the
+irreducibility of M, we haveM′= 0. Now let f∈C[h]Gbe a function that vanishes on X.
+Then there exists a positive integer Nsuch thatfNmapsMtoM′, hence acts by zero on
+M. This implies that Supp M/G=X, as desired. /square
+43Propositions 5.26 and 5.27 allow us to attach to every irreducible modu leM∈Oc(G,h),
+a conjugacy class of parabolic subgroups, CM∈Par(G), namely, the conjugacy class of the
+stabilizer of a generic point of the support of M. Also, for a parabolic subgroup G′⊂G,
+denote byX(G′) the set of points b∈hwhose stabilizer contains a subgroup conjugate to
+G′.
+The following proposition is immediate.
+Proposition 5.28. (i)LetM∈Oc(G,h)0be irreducible. If bis such that Gb∈CM,
+thenResb(M)is a nonzero finite dimensional module over H1,c(Gb,h/hGb).
+(ii)Conversely, let b∈h, andLbe a finite dimensional module H1,c(Gb,h/hGb). Then
+the support of Indb(L)inhisX(Gb).
+LetFD(G,h)bethesetof cforwhichH1,c(G,h)admitsafinitedimensional representation.
+Corollary 5.29. LetG′be a parabolic subgroup of G. ThenX(G′)is the support of some
+irreducible representation from Oc(G,h)0if and only if c∈FD(G′,h/hG′).
+Proof.Immediate from Proposition 5.28. /square
+Example 5.30. LetG=Sn,h=Cn−1. In this case, the set Par( G) is the set of partitions
+ofn. Assume that c=r/m, (r,m) = 1, 2≤m≤n. By a result of [BEG], finite dimensional
+representations of Hc(G,h) exist if and only if m=n. Thus the only possible classes CM
+for irreducible modules Mhave stabilizers Sm×···×Sm, i.e., correspond to partitions into
+parts, where each part is equal to mor 1. So there are [ n/m] + 1 possible supports for
+modules, where [ a] denotes the integer part of a.
+5.12.Notes.Our discussion of the geometric approach to rational Cherednik alg ebras in
+Section 5.1 follows [E1] and Section 2.2 of [BE]. Our exposition in the othe r sections follows
+the corresponding parts of the paper [BE].
+446.The Knizhnik-Zamolodchikov functor
+6.1.Braid groups and Hecke algebras. LetGbe a complex reflection group and let h
+be its reflection representation. For any reflection hyperplane H⊂h, its pointwise stabilizer
+is a cyclic group of order mH. Fix a collection of nonzero constants q1,H,...,qmH−1,Hwhich
+areG-invariant, namely, if HandH′are conjugate to each other under some element in G,
+thenqi,H=qi,H′fori= 1,...,mH−1.
+LetBG=π1(hreg/G,x0) be the braid group of G, andTH∈BGbe a representative of
+the conjugacy class defined by a small circle around the image of Hinh/Goriented in the
+counterclockwise direction.
+The following theorem follows from elementary algebraic topology.
+Proposition 6.1. The groupGis the quotient of the braid group BGby the relations
+TmH
+H= 1
+for all reflection hyperplanes H.
+Proof.See, e.g., [BMR] Proposition 2.17. /square
+Definition 6.2. The Hecke algebra of Gis defined to be
+Hq(G) =C[BG]//a\}b∇acketle{t(TH−1)mH−1/productdisplay
+j=1(TH−exp(2πij/mH)qj,H),for allH/a\}b∇acket∇i}ht.
+Thus, by Proposition 6.1 we have an isomorphism
+H1(G)∼=CG.
+SoHq(G) is a deformation of CG.
+Example 6.3 (Coxeter group case) .Now letWbe a Coxeter group. Let Sbe the set of
+reflections and let αs= 0 be the reflection hyperplane corresponding to s∈S. The Hecke
+algebraHq(W) is the quotient of C[BW] by the relations
+(Ts−1)(Ts+qs) = 0,
+for all reflections swhereTsis a small counterclockwise circle around the image of the
+hyperplane αs= 0 inh/W.
+6.2.KZ functors. For a complex reflection group G, let Loc( hreg) be the category of local
+systems (i.e.,O-coherentD-modules) on hreg, and let Loc( hreg)Gbe the category of G-
+equivariant local systems on hreg, i.e. of local systems on hreg/G.
+Suppose that G′= 1 is the trivial subgroup in G. Then the restriction functor defined in
+Section 5.10 defines a functor Res : Oc(G,h)0→Loc(hreg/G). Also, we have the monodromy
+functor Mon: Loc(hreg/G)∼=Rep(BG). The composition of these two functors is a functor
+fromOc(G,h)0to Rep(BG), which is exactly the KZ functor defined in [GGOR]. We will
+denote this functor by KZ.
+Theorem 6.4 (Ginzburg, Guay, Opdam, Rouquier, [GGOR]) .The KZ functor factors
+through
+RepHq(G), where
+qj,H= exp(2πibj,H/mH),andbj,H= 2mH−1/summationdisplay
+ℓ=1csℓ
+H(1−e2πijℓ/mH)
+1−e−2πiℓ/mH.
+45Proof.Assume first that cis generic. Then the category Oc(G,h)0is semisimple, with simple
+objectsMc(τ), so it is enough to check the statement on Mc(τ). Consider the trivial bundle
+overhregwith fiberτ. The KZ connection on it has the form
+d−/summationdisplay
+s∈S2cs
+1−λsdαs
+αs(1−s).
+The residue of the connection form of this connection on the hyper planeHon thej-th
+irreducible representation of Z/mHZis
+2mH−1/summationdisplay
+ℓ=1csℓ
+H
+1−e−2πiℓ/mH(1−e2πijℓ/mH).
+Therefore, the monodromy operator around this hyperplane is dia gonalizable, and the eigen-
+values of this operator are 1 and exp(2 πij/mH)qj,H, as desired.
+For special c, introduce the generalized Verma module
+Mc,n(τ) =Hc(G,h)⊗CG⋉Sh(τ⊗Sh/mn+1),
+wherem⊂Shis the maximal ideal of 0, n≥0. Clearly, Mc,0=Mc(τ). Moreover,
+Mc,n∈Oc(G,h)0for anyn, since it has a finite filtration whose successive quotients are
+Verma modules.
+Theorem 6.5. For large enough n,Mc,n(CG)contains a direct summand which is a projec-
+tive generator ofOc(G,h)0.
+Proof.From the definition, Mc,n=Sh∗⊗CG⊗Sh/mn+1. Let∂be the degree operator on
+Mc,n(CG) with deg h∗= 1,degh=−1, and degG= 0, i.e., we have
+[∂,x] =x,[∂,y] =−y,wherex∈h∗,y∈h.
+Soh−∂is a module endomorphism of Mc,n(CG) wherehis the operator defined in (3.2).
+Moreover, h−∂acts locally finitely. In particular, we have a decomposition of Mc,n(CG)
+into generalized eigenspaces of h−∂:
+Mc,n(CG) =/circleplusdisplay
+β∈CMβ
+c,n(CG).
+We have
+Hom(Mc,n(CG),N) ={vectors inNwhich are killed by mn+1},
+and
+Hom(Mβ
+c,n(CG),N) ={vectors inNwhich are killed by mn+1
+and are generalized eigenvectors of hwith generalized eigenvalue β}.
+Let Σ = {hc(τ)|τis a irreducible representation of G}(recall that
+hc(τ) =dimh
+2−/summationtext
+s∈S2cs
+1−λss|τ), and let
+MΣ
+c,n(CG) =/circleplusdisplay
+β∈ΣMβ
+c,n(CG).
+Claim: for largen,MΣ
+c,n(CG) is a projective generator of Oc(G,h)0.
+46Proof of the claim. First, for any β, there exists nsuch thatMΣ
+c,n(CG) is projective (since
+the condition of being killed by mn+1is vacuous for large n).
+Secondly, consider the functor Hom( MΣ
+c,n(CG),•). For any module N∈Oc(G,h)0, if
+Hom(MΣ
+c,n(CG),N) = 0, then⊕β∈ΣN[β] = 0. SoN= 0. Thus this functor does not kill
+nonzero objects, and so MΣ
+c,n(CG) is a generator. /square
+Theorem 6.5 is proved. /square
+Corollary 6.6. (i)Oc(G,h)0has enough projectives, so it is equivalent to the category
+of modules over a finite dimensional algebra.
+(ii)Any object ofOc(G,h)0is a quotient of a multiple of Mc,n(CG)for large enough n.
+Proof.Directly from the definition and the above theorem. /square
+Now we can finish the proof of Theorem 6.4. We have shown that for g enericc,
+KZ(Mc,n(CG))∈RepHq(G). Hence this is true for any c, sinceMc,n(CG) is a flat fam-
+ily of modules over Hc(G,h). Then, KZ( M) is aHq(G)-module for all M, since any Mis a
+quotient of Mc,n(CG) and the functor KZ is exact. /square
+Corollary 6.7 (Brou´ e, Malle, Rouquier, [BMR]) .Letqj,H= exp(tj,H)wheretj,H’s are
+formal parameters. Then Hq(G)is a free module over C[[tj,H]]of rank|G|.
+Proof.We have
+Hq(G)/(t) =H1(G) =CG.
+So it remains to show that Hq(G) is free. To show this, it is sufficient to show that any
+τ∈IrrepGadmits a flat deformation τqto a representation of Hq(G). We can define this
+deformation by letting τq= KZ(Mc(τ)). /square
+Remark 6.8. 1. The validity of this Corollary in characteristic zero implies that it is als o
+valid over a field positive characteristic.
+2. It is not known in general if the Corollary holds for numerical q(even generically). This
+is a conjecture of Brou´ e, Malle, and Rouquier. But it is known for ma ny cases (including all
+Coxeter groups).
+3. The proof of the Corollary is analytic (it is based on the notion of mo nodromy). There
+is no known algebraic proof, except in special cases, and in the case of Coxeter groups, which
+we’ll discuss later.
+6.3.Theimage of theKZ functor. First,letusrecallthedefinitionofaquotientcategory.
+LetAbe an abelian category and B⊂Aa full abelian subcategory.
+Definition 6.9.Bis a Serre subcategory if it is closed under subquotients and extens ions
+(i.e., if two terms in a short exact sequence are in B, so is the third one).
+IfB⊂Ais a Serre subcategory, one can define a category A/Bas follows:
+objects inA/B= objects inA,
+HomA/B(X,Y) = lim
+−→
+Y′,X/X′∈BHomA(X′,Y/Y′).
+The categoryA/Bis an abelian category with the following universal property: any exa ct
+functorF:A→Cthat killsBmust factor through A/B.
+47Now letOc(G,h)tor
+0be the full subcategory of Oc(G,h)0consisting of modules supported
+on the reflection hyperplanes. It is a Serre subcategory, and ker (KZ) =Oc(G,h)tor
+0. Thus
+we have a functor:
+KZ :Oc(G,h)0/Oc(G,h)tor
+0→RepHq(G).
+Theorem 6.10 (Ginzburg, Guay, Opdam, Rouquier, [GGOR]) .IfdimHq(G) =|G|, the
+functorKZis an equivalence of categories.
+Proof.See [GGOR], Theorem 5.14. /square
+6.4.Example: the symmetric group Sn.Leth=Cn,G=Sn. Then we have (for
+q∈C∗):
+Hq(Sn) =/a\}b∇acketle{tT1,...,Tn−1/a\}b∇acket∇i}ht//a\}b∇acketle{tthe braid relations and ( Ti−1)(Ti+q) = 0/a\}b∇acket∇i}ht.
+The following facts are known:
+(1) dimHq(Sn) =n!;
+(2)Hq(Sn) is semisimple if and only if ord( q)/\e}atio\slash= 2,3,...,n.
+Now suppose qis generic. Let λbe a partition of n. Then we can define an Hq(Sn)-
+moduleSλ, the Specht module for the Hecke algebra in the sense of [DJ]. This is a certain
+deformation of the classical irreducible Specht module for the symm etric group. The Specht
+module carries an inner product /a\}b∇acketle{t·,·/a\}b∇acket∇i}ht. DenoteDλ=Sλ/Rad/a\}b∇acketle{t·,·/a\}b∇acket∇i}ht.
+Theorem 6.11 (Dipper, James, [DJ]) .Dλis either an irreducible Hq(Sn)-module, or 0.
+Dλ/\e}atio\slash= 0if and only if λise-regular where e= ord(q)(i.e., every part of λoccurs less than
+etimes).
+Proof.See [DJ], Theorem 6.3, 6.8. /square
+Now letMc(λ) be the Verma module associated to the Specht module for SnandLc(λ)
+be its irreducible quotient. Then we have the following theorem.
+Theorem 6.12. Ifc≤0, thenKZ(Mc(λ)) =SλandKZ(Lc(λ)) =Dλ.
+Proof.See Section 6.2 of [GGOR]. /square
+Corollary 6.13. Ifc≤0, thenSuppLc(λ) =Cnif and only if λise-regular. If c >0,
+thenSuppLc(λ) =Cnif and only if λ∨ise-regular, or equivalently, λise-restricted (i.e.,
+λi−λi+1<efori= 1,...,n−1).
+Proof.Directly from the definition and the above theorem. /square
+6.5.Notes.The references for this section are [GGOR], [BMR].
+487.Rational Cherednik algebras and Hecke algebras for varietie s with
+group actions
+7.1.Twisted differential operators. Let us recall the theory of twisted differential oper-
+ators (see [BB], section 2).
+LetXbe a smooth affine algebraic variety over C. Given a closed 2-form ωonX, the
+algebraDω(X) of differential operators on Xtwisted by ωcan be defined as the algebra
+generated byOXand “Lie derivatives” Lv,v∈Vect(X), with defining relations
+fLv=Lfv,[Lv,f] =Lvf,[Lv,Lw] =L[v,w]+ω(v,w).
+This algebra depends only on the cohomology class [ ω] ofω, and equals the algebra D(X)
+of usual differential operators on Xif [ω] = 0.
+An important special case of twisted differential operators is the a lgebra of differential
+operators on a line bundle. Namely, let Lbe a line bundle on X. SinceXis affine,Ladmits
+an algebraic connection ∇with curvature ω, which is a closed 2-form on X. Then it is easy
+to show that the algebra D(X,L) of differential operators on Lis isomorphic toDω(X).
+If the variety Xis smooth but not necessarily affine, then (sheaves of) algebras of twisted
+differential operators are classified by the space H2(X,Ω≥1
+X), where Ω≥1
+Xis the two-step com-
+plex of sheaves Ω1
+X→Ω2,cl
+X, given by the De Rham differential acting from 1-forms to closed
+2-forms (sitting in degrees 1 and 2, respectively). If Xis projective then this space is
+isomorphic to H2,0(X,C)⊕H1,1(X,C). We refer the reader to [BB], Section 2, for details.
+Remark 7.1. One can show that Dω(X) is the universal deformation of D(X) (see [E1]).
+7.2.Some algebraic geometry preliminaries. LetZbe a smooth hypersurface in a
+smooth affine variety X. Leti:Z→Xbe the corresponding closed embedding. Let
+Ndenote the normal bundle of ZinX(a line bundle). Let OX(Z) denote the module
+of regular functions on X\Zwhich have a pole of at most first order at Z. Then we
+have a natural map of OX-modulesφ:OX(Z)→i∗N. Indeed, we have a natural residue
+mapη:OX(Z)⊗OXΩ1
+X→i∗OZ(where Ω1
+Xis the module of 1-forms), hence a map
+η′:OX(Z)→i∗OZ⊗OXTX=i∗(TX|Z) (whereTXis the tangent bundle). The map
+φis obtained by composing η′with the natural projection TX|Z→N.
+We have an exact sequence of OX-modules:
+0→OX→OX(Z)φ− →i∗N→0
+Thus we have a natural surjective map of OX-modulesξZ:TX→OX(Z)/OX.
+7.3.The Cherednik algebra of a variety with a finite group action. We will now
+generalize the definition of Ht,c(G,h) to the global case. Let Xbe an affine algebraic variety
+overC, andGbe a finite group of automorphisms of X. LetEbe aG-invariant subspace
+of the space of closed 2-forms on X, which projects isomorphically to H2(X,C). Consider
+the algebra G⋉OT∗X, whereT∗Xis the cotangent bundle of X. We are going to define a
+deformation Ht,c,ω(G,X) of this algebra parametrized by
+(1) complex numbers t,
+(2)G-invariant functions con the (finite) set Sof pairss= (Y,g), whereg∈G, andY
+is a connected component of the set of fixed points Xgsuch that codimY= 1, and
+(3) elements ω∈EG= H2(X,C)G.
+49If all the parameters are zero, this algebra will conicide with G⋉OT∗X.
+Lett,c={c(Y,g)},ω∈EGbe variables. Let Dω/t(X)rbe the algebra (over C[t,t−1,ω])
+of twisted (by ω/t) differential operators on Xwith rational coefficients.
+Definition 7.2. A Dunkl-Opdam operator for ( X,G) is an element of Dω/t(X)r[c] given by
+the formula
+(7.1) D:=tLv−/summationdisplay
+(Y,g)∈S2c(Y,g)
+1−λY,g·fY(x)·(1−g),
+whereλY,gis the eigenvalue of gon the conormal bundle to Y,v∈Γ(X,TX) is a vector
+field onX, andfY∈OX(Z) is an element of the coset ξY(v)∈OX(Z)/OX(recall that ξY
+is defined in Subsection 7.2).
+Definition 7.3. The algebra Ht,c,ω(X,G) is the subalgebra of G⋉Dω/t(X)r[c] generated
+(overC[t,c,ω]) by the function algebra OX, the group G, and the Dunkl-Opdam operators.
+By specializing t,c,ωto numerical values, we can define a family of algebras over C, which
+we will also denote Ht,c,ω(G,X). Note that when we set t= 0, the term tLvdoes not become
+0 but turns into the classical momentum.
+Definition 7.4. Ht,c,ω(G,X) is called the Cherednik algebra of the orbifold X/G.
+Remark 7.5. One hasH1,0,ω(G,X) =G⋉Dω(X). Also, ifλ/\e}atio\slash= 0 thenHλt,λc,λω(G,X) =
+Ht,c,ω(G,X).
+Example 7.6. X=his a vector space and Gis a subgroup in GL( h). Letvbe a constant
+vector field, and let fY(x) = (αY,v)/αY(x), whereαY∈h∗is a nonzero functional vanishing
+onY. Then the operator Dis just the usual Dunkl-Opdam operator Dvin the complex
+reflection case (see Section 2.5). This implies that all the Dunkl-Opda m operators in the
+sense of Definition 7.2 have the form/summationtextfiDyi+a, wherefi∈C[h],a∈G⋉C[h], andDyi
+are the usual Dunkl-Opdam operators (for some basis yiofh). So the algebra Ht,c(G,h) =
+Ht,c,0(G,X) is the rational Cherednik algebra for ( G,h), see Section 3.1.
+The algebra Ht,c,ω(G,X) has a filtration F•which is defined on generators by deg( OX) =
+deg(G) = 0, deg(D) = 1 for Dunkl-Opdam operators D.
+Theorem 7.7 (the PBW theorem) .We have
+grF(Ht,c,ω(G,X)) =G⋉O(T∗X)[t,c,ω].
+Proof.Suppose first that X=his a vector space and Gis a subgroup in GL( h). Then, as
+we mentioned, Ht,c,ω(G,h) =Ht,c(G,h) is the rational Cherednik algebra for G,h. So in this
+case the theorem is true.
+Now consider arbitrary X. We have a homomorphism of graded algebras
+ψ: grF(Ht,c,ω(G,X))→G⋉O(T∗X)[t,c,ω] (the principal symbol homomorphism) .
+The homomorphism ψis clearly surjective, and our job is to show that it is injective (this
+is the nontrivial part of the proof). In each degree, ψis a morphism of finitely generated
+OG
+X-modules. Therefore, to check its injectivity, it suffices to check t he injectivity on the
+formal neighborhood of each point z∈X/G.
+Letxbe a preimage of zinX, andGxbe the stabilizer of xinG. ThenGxacts on the
+formal neighborhood UxofxinX.
+50Lemma 7.8. Any action of a finite group on a formal polydisk over Cis linearizable.
+Proof.LetDbe a formal polydisk over C. Suppose we have an action of a finite group G
+onD. Then we have a group homomorphism:
+ρ:G→Aut(D) = GLn(C)⋉AutU(D),
+where Aut U(D) is the group of unipotent automorphisms of D(i.e. those whose derivative
+at the origin is 1), which is a prounipotent algebraic group.
+Our job is to show that the image of Gunderρcan be conjugated into GL n(C). The
+obstruction to this is in the cohomology group H1(G,AutU(D)), which is trivial since Gis
+finite and Aut U(D) is prounipotent over C. /square
+It follows from Lemma 7.8 that it suffices to prove the theorem in the lin ear case, which
+has been accomplished already. We are done. /square
+Remark 7.9. The following remark is meant to clarify the proof of Theorem 7.7. In t he case
+X=h, the proof of Theorem 7.7 is based, essentially, on the (fairly nontr ivial) fact that the
+usual Dunkl-Opdam operators Dvcommute with each other. It is therefore very important
+to note that in contrast with the linear case, for a general Xwe donothave any natural
+commuting family of Dunkl-Opdam operators. Instead, the operat ors (7.1) satisfy a weaker
+property, which is still sufficient to validate the PBW theorem. This pr operty says that if
+D1,D2,D3are Dunkl-Opdam operators corresponding to vector fields v1,v2,v3:= [v1,v2]
+and some choices of the functions fY, then [D1,D2]−D3∈G⋉O(X) (i.e., it has no poles).
+To prove this property, it is sufficient to consider the case when Xis a formal polydisk, with
+a linear action of G. But in this case everything follows from the commutativity of the us ual
+Dunkl operators Dv.
+Example 7.10. (1) Suppose G= 1. Then for t/\e}atio\slash= 0,Ht,0,ω(G,X) =Dω/t(X).
+(2) Suppose Gis a Weyl group and X=Hthe corresponding torus. Then H1,c,0(G,H)
+is called the trigonometric Cherednik algebra.
+7.4.Globalization. LetXbe any smooth algebraic variety, and G⊂Aut(X). Assume
+thatXadmits a cover by affine G-invariant open sets. Then the quotient variety X/G
+exists.
+For any affine open set UinX/G, letU′be the preimage of UinX. Then we can
+define the algebra Ht,c,0(G,U′) as above. If U⊂V, we have an obvious restriction map
+Ht,c,0(G,V′)→Ht,c,0(G,U′). The gluing axiom is clearly satisfied. Thus the collection of
+algebrasHt,c,0(G,U′) can be extended (by sheafification) to a sheaf of algebras on X/G. We
+are going to denote this sheaf by Ht,c,0,G,Xand call it the sheaf of Cherednik algebras on
+X/G. Thus,Ht,c,0,G,X(U) =Ht,c,0(G,U′).
+Similarly, if ψ∈H2(X,Ω≥1
+X)G, we can define the sheaf of twisted Cherednik algebras
+Ht,c,ψ,G,X. This is done similarly to the case of twisted differential operators (w hich is the
+caseG= 1).
+Remark 7.11. (1) The construction of Ht,c,ω(G,X) and the PBW theorem extend in a
+straightforwardmannertothecasewhenthegroundfieldisnot Cbutanalgebraically
+closed field kof positive characteristic, provided that the order of the group Gis
+relatively prime to the characteristic.
+51(2) The construction and main properties of the (sheaves of) Che rednik algebras of alge-
+braic varieties can be extended without significant changes to the c ase whenXis a
+complex analytic manifold, and Gis not necessarily finite but acts properly discon-
+tinuously. In the following lectures, we will often work in this generaliz ed setting.
+7.5.Modified Cherednik algebra. It will be convenient for us to use a slight modification
+of the sheaf Ht,c,ψ,G,X. Namely, let ηbe a function on the set of conjugacy classes of Ysuch
+that (Y,g)∈S. We define Ht,c,η,ψ,G,X in the same way as Ht,c,ψ,G,Xexcept that the Dunkl-
+Opdam operators are defined by the formula
+(7.2) D:=tLv+/summationdisplay
+(Y,g)∈SfY(x)2c(Y,g)
+1−λY,g(g−1)+/summationdisplay
+YfY(x)η(Y).
+Thefollowingresult showsthatthismodificationisinfacttautological. LetψYbetheclass
+in H2(X,Ω≥1
+X) defined by the line bundle OX(Y)−1, whose sections are functions vanishing
+onY.
+Proposition 7.12. One has an isomorphism
+Ht,c,η,ψ,G,X→Ht,c,ψ+/summationtext
+Yη(Y)ψY,G,X.
+Proof.Lety∈Yandzbe a function on the formal neighborhood of ysuch thatz|Y= 0
+and dzy/\e}atio\slash= 0. Extend it to a system of local formal coordinates z1=z,z2,...,zdneary. A
+Dunkl-Opdam operator near yfor the vector field∂
+∂zcan be written in the form
+D=∂
+∂z+1
+z(n−1/summationdisplay
+m=12c(Y,gm)
+1−λm
+Y,g(gm−1)+η(Y)).
+Conjugating this operator by the formal expression zη(Y):= (zm)η(Y)/m, we get
+zη(Y)◦D◦z−η(Y)=∂
+∂z+1
+zn−1/summationdisplay
+m=12c(Y,gm)
+1−λm
+Y,g(gm−1)
+This implies the required statement. /square
+We note that the sheaf H1,c,η,0,G,Xlocalizes to G⋉DXon the complement of all the
+hypersurfaces Y. This follows from the fact that the line bundle OX(Y) is trivial on the
+complement of Y.
+7.6.Orbifold Hecke algebras. LetXbe a connected and simply connected complex man-
+ifold, andGis a discrete group of automorphisms of Xwhich acts properly discontinuously.
+ThenX/Gis a complex orbifold. Let X′⊂Xbe the set of points with trivial stabilizer. Fix
+a base point x0∈X′. Then the braid group of X/Gis defined to be BG=π1(X′/G,x0).
+We have an exact sequence 1 →K→BG→G→1.
+Now letSbe the set of pairs ( Y,g) such that Yis a component of Xgof codimension 1 in
+X(suchYwill be called a reflection hypersurface). For ( Y,g)∈S, letGYbe the subgroup of
+Gwhose elements act trivially on Y. This group is obviously cyclic; let nY=|GY|. LetCY
+be the conjugacy class in BGcorresponding to a small circle going counterclockwise around
+the image of YinX/G, andTYbe a representative in CY.
+The following theorem follows from elementary topology:
+52Theorem 7.13. Kis defined by relations TnY
+Y= 1, for all reflection hypersurfaces Y(i.e.,
+Kis the intersection of all normal subgroups of BGcontaining TnY
+Y).
+Proof.See, e.g., [BMR] Proposition 2.17. /square
+For any conjugacy class of hypersurfaces Ysuch that ( Y,g)∈Swe introduce formal
+parameters τ1Y,...,τnYY. The entire collection of these parameters will be denoted by τ.
+LetA0=C[G].
+Definition 7.14. We define the Hecke algebra of ( G,X), denotedA=Hτ(G,X,x 0), to be
+the quotient of the group algebra of the braid group, C[BG][[τ]], by the relations
+(7.3)nY/productdisplay
+j=1(T−e2πji/nYeτjY) = 0, T∈CY
+(i.e., by the closed ideal in the formal series topology generated by t hese relations).
+Thus,Ais a deformation of A0.
+It is clear that up to an isomorphism this algebra is independent on the choice ofx0, so
+we will sometimes drop x0form the notation.
+The main result of this section is the following theorem.
+Theorem 7.15. Assume that H2(X,C) = 0. ThenA=Hτ(G,X)is aflatformal defor-
+mation ofA0, which means A=A0[[τ]]as a module over C[[τ]].
+Example 7.16. Lethbe a finite dimensional vector space, and Gbe a complex reflection
+group in GL( h). Then Hτ(G,h) is the Hecke algebra of Gstudied in [BMR]. It follows
+from Theorem 7.15 that this Hecke algebra is flat. This proof of flatn ess is in fact the same
+as the original proof of this result given in [BMR] (based on the Dunkl- Opdam-Cherednik
+operators, and explained above).
+Example 7.17. Lethbe a universal covering of a maximal torus of a simply connected
+simple Lie group G,Q∨be the dual root lattice, and /hatwideG=G⋉Q∨be its affine Weyl group.
+ThenHτ(h,/hatwideG) is the affine Hecke algebra. This algebra is also flat by Theorem 7.15. I n
+fact, its flatness is a well known result from representation theor y; our proof of flatness is
+essentially due to Cherednik [Ch].
+Example 7.18. LetG,h,Q∨be as in the previous example, η∈C+be a complex number
+with a positive imaginary part, and/hatwide/hatwideG=G⋉(Q∨⊕ηQ∨) be the double affine Weyl group.
+ThenHτ(h,/hatwide/hatwideG) is(oneof theversions of) thedouble affineHecke algebraof Chere dnik ([Ch]),
+and it is flat by Theorem 7.15. The fact that this algebra is flat was pro ved by Cherednik,
+Sahi, Noumi, Stokman (see [Ch],[Sa],[NoSt],[St]) using a different approach (q-deformed
+Dunkl operators).
+7.7.Hecke algebras attached to Fuchsian groups. LetHbe a simply connected com-
+plex Riemann surface (i.e., Riemann sphere, Euclidean plane, or Lobac hevsky plane), and Γ
+be a cocompact lattice in A ut(H) (i.e., a Fuchsian group). Let Σ = H/Γ. Then Σ is a com-
+pact complex Riemann surface. When Γ contains elliptic elements (i.e., n ontrivial elements
+53of finite order), we are going to regard Σ as an orbifold: it has specia l pointsPi,i= 1,...,m
+with stabilizers Zni. Then Γ is the orbifold fundamental group of Σ.1
+Letgbe the genus of Σ, and al,bl,l= 1,...,g, be thea-cycles and b-cycles of Σ. Let cj
+be the counterclockwise loops around Pj. Then Γ is generated by al,bl,cjwith relations
+cnj
+j= 1, c1c2···cm=/productdisplay
+lalbla−1
+lb−1
+l.
+For eachj, introduce formal parameters τkj,k= 1,...,nj. Define the Hecke algebra Hτ(Σ)
+of Σ to be generated over C[[τ]] by the same generators al,bl,cjwith defining relations
+nj/productdisplay
+k=1(cj−e2πji/njeτkj) = 0, c1c2···cm=/productdisplay
+lalbla−1
+lb−1
+l.
+ThusHτ(Σ) is a deformation of C[Γ].
+Thisdeformationisflatif HisaEuclideanplaneoraLobachevsky plane. Indeed, Hτ(Σ) =
+Hτ(Γ,H), so the result follows from Theorem 7.15 and the fact that H2(H,C) = 0.
+On the other hand, if His the Riemann sphere (so that the condition H2(H,C) = 0 is
+violated) and Γ/\e}atio\slash= 1 then this deformation is not flat. Indeed, let τ=τ(/pla⋉ckover2pi1) be a 1-parameter
+subdeformation of Hτ(Σ) which is flat. Let us compute the determinant of the product
+c1···cmin the regular representation of this algebra (which is finite dimension al ifHis the
+sphere). On the one hand, it is 1, as c1···cmis a product of commutators. On the other
+hand, the eigenvalues of cjin this representation are e2πji/njeτkjwith multiplicity |Γ|/nj.
+Computing determinants as products of eigenvalues, we get a nont rivial equation on τkj(/pla⋉ckover2pi1),
+which means that the deformation Hτis not flat.
+Thus, we see that Hτ(Σ) fails to be flat in the following “forbidden” cases:
+g= 0, m= 2,(n1,n2) = (n,n);
+m= 3,(n1,n2,n3) = (2,2,n),(2,3,3),(2,3,4),(2,3,5).
+Indeed, the orbifold Euler characteristic of a closed surface Σ of g enusgwithmspecial
+pointsx1,...,xmwhose orders are n1,...,nmis
+χorb(Σ,x1,...,xm) = 2−2g−m+m/summationdisplay
+i=11
+ni,
+and above solutions are the solutions of the inequality
+χorb(CP1,x1,...,xm)>0.
+(note that the solutions for m= 1 and solutions ( n1,n2) withn1/\e}atio\slash=n2don’t arise, since they
+don’t correspond to any orbifolds).
+1LetXbe a connected topological space on with a properly discontinuous a ction of a discrete group G.
+Then the orbifold fundamental group of the orbifold X/Gwith base point x∈X, denoted πorb
+1(X/G,x),
+is the set of pairs ( g,γ), where g∈Gandγis a homotopy class of paths leading from xtogx, with
+multiplication law ( g1,γ1)(g2,γ2) = (g1g2,γ), where γisγ1followed by g1(γ2). Obviously, in this situation
+we have an exact sequence
+1→π1(X,x)→πorb
+1(X/G,x)→G→1.
+547.8.Hecke algebras of wallpaper groups and del Pezzo surfaces. The case when H
+is the Euclidean plane (i.e., Γ is a wallpaper group) deserves special att ention. If there are
+elliptic elements, this reduces to the following configurations: g= 0 and
+m= 3,(n1,n2,n3) = (3,3,3),(2,4,4),(2,3,6) (casesE6,E7,E8),
+or
+m= 4,(n1,n2,n3,n4) = (2,2,2,2) (caseD4).
+In these cases, the algebra Hτ(Γ,H) (for numerical τ) has Gelfand-Kirillov dimension 2,
+so it can be interpreted in terms of the theory of noncommutative s urfaces.
+Recall that a del Pezzo surface (or a Fano surface) is a smooth pr ojective surface, whose
+anticanonical line bundle is ample. It is known that such surfaces are CP1×CP1, or a blow-
+up ofCP2at up to 8 generic points. The degree of a del Pezzo surface Xis by definition the
+self intersection number K·Kof its canonical class K. For example, a del Pezzo surface of
+degree 3 is a cubic surface in CP3, and the degree of CP2withngeneric points blown up is
+d= 9−n.
+Now suppose τis numerical. Let /pla⋉ckover2pi1=/summationtext
+j,kn−1
+jτkj. Also letnbe the largest of nj, andc
+be the corresponding cj. Lete∈C[c]⊂Hτ(Γ,H) be the projector to an eigenspace of c.
+Consider the “spherical” subalgebra Bτ(Γ,H) :=eHτ(Γ,H)e.
+Theorem 7.19 (Etingof, Oblomkov, Rains, [EOR]) .(i)If/pla⋉ckover2pi1= 0thenthe algebra Bτ(Γ,H)
+is commutative, and its spectrum is an affine del Pezzo surface . More precisely, in
+the case (2,2,2,2), it is a del Pezzo surface of degree 3 (a cubic surface) with a t ri-
+angle of lines removed; in the cases (3,3,3),(2,4,4),(2,3,6)it is a del Pezzo surface
+of degrees 3,2,1 respectively with a nodal rational curve re moved.
+(ii)The algebra Bτ(Γ,H)for/pla⋉ckover2pi1/\e}atio\slash= 0is a quantization of the unique algebraic symplectic
+structure on the surface from (i) with Planck’s constant /pla⋉ckover2pi1.
+Proof.See [EOR]. /square
+Remark 7.20. In the case (2 ,2,2,2),Hτ(Γ,Γ) is the Cherednik-Sahi algebra of rank 1; it
+controls the theory of Askey-Wilson polynomials.
+Example 7.21. This is a “multivariate” version of the Hecke algebras attached to Fu chsian
+groups, defined in the previous subsection. Namely, letting Γ ,Hbe as in the previous
+subsection, and N≥1, we consider the manifold X=HNwith the action of Γ N=SN⋉ΓN.
+IfHis a Euclidean or Lobachevsky plane, then by Theorem 7.15 Hτ(ΓN,XN) is a flat
+deformation of the group algebra C[ΓN]. IfN >1, this algebra has one more essential
+parameter than for N= 1 (corresponding to reflections in SN). In the Euclidean case, one
+expects that an appropriate “spherical” subalgebra of this algebr a is a quantization of the
+Hilbert scheme of a del Pezzo surface.
+7.9.The Knizhnik-Zamolodchikov functor. In this subsection we will define a global
+analog of the KZ functor defined in [GGOR]. This functor will be used a s a tool of proof of
+Theorem 7.15.
+LetXbe a simply connected complex manifold, and Ga discrete group of holomorphic
+transformations of Xacting onXproperly discontinuously. Let X′⊂Xbe the set of points
+with trivial stabilizer. Then we can define the sheaf of Cherednik alge brasH1,c,η,0,G,Xon
+X/G. Note that the restriction of this sheaf to X′/Gis the same as the restriction of the
+55sheafG⋉DXtoX′/G(i.e. onX′/G, the dependence of the sheaf on the parameters cand
+ηdisappears). This follows from the fact that the line bundles OX(Y) become trivial when
+restricted to X′.
+Now letMbe a module over H1,c,η,0,G,Xwhich is a locally free coherent sheaf when
+restricted to X′/G. Then the restriction of MtoX′/Gis aG-equivariantD-module on
+X′which is coherent and locally free as an O-module. Thus, Mcorresponds to a locally
+constant sheaf (local system) on X′/G, which gives rise to a monodromy representation of
+the braid group π1(X′/G,x0) (wherex0is a base point). This representation will be denoted
+by KZ(M). This defines a functor KZ, which is analogous to the one in [GGOR].
+It follows from the theory of D-modules that any OX/G-coherentH1,c,η,0,G,X-module is
+locally free when restricted to X′/G. Thus the KZ functor acts from the abelian category
+Cc,ηofOX/G-coherentH1,c,η,0,G,X-modules to the category of representations of π1(X′/G,x0).
+It is easy to see that this functor is exact.
+For anyY, letgYbe the generator of GYwhich has eigenvalue e2πi/nYin the conormal
+bundle toY. Let (c,η)→τ(c,η) be the invertible linear transformation defined by the
+formula
+τjY= 2πi(2nY−1/summationdisplay
+m=1c(Y,gm
+Y)1−e2πjmi/nY
+1−e−2πmi/nY−η(Y))/nY.
+Proposition 7.22. The functor KZmaps the category Cc,ηto the category of representations
+of the algebra Hτ(c,η)(G,X).
+Proof.The result follows from the corresponding result in the linear case (w hich we have
+already proved) by restricting Mto the union of G-translates of a neighborhood of a generic
+pointy∈Y, and then linearizing the action of GYon this neighborhood. /square
+7.10.Proof of Theorem 7.15. Consider the module M= IndG⋉DX
+DXOX. Then KZ( M)
+is the regular representation of Gwhich is denoted by reg G. We want to show that M
+deforms uniquely (up to an isomorphism) to a module over H1,c,0,η,G,Xfor formalc,η. The
+obstruction to this deformation is in Ext2
+G⋉DX(M,M) and the freedom of this deformation
+is in Ext1
+G⋉DX(M,M). Since
+Exti
+G⋉DX(M,M) = Exti
+DX(OX,ResM) = Exti
+DX(OX,OX⊗CG)
+= Exti
+DX(OX,OX)⊗CG= Hi(X,C)⊗CG,
+andXis simply connected, we have
+Ext1
+G⋉DX(M,M) = 0,and Ext2
+G⋉DX(M,M) = 0 if H2(X,C) = 0.
+Thus such deformation exists and is unique if H2(X,C) = 0.
+Now letMc,ηbe the deformation. Then KZ( Mc,η) is aHτ(c,η)(G,X)-module from Propo-
+sition 7.22 and it deforms flatly the module reg G. This implies Hτ(c,η)(G,X) is flat over
+C[[τ]].
+Remark 7.23. WhenXis not simply connected, the theorem is still true under the as-
+sumptionπ2(X)⊗C= 0 (i.e. H2(/tildewideX,C) = 0, where/tildewideXis the universal cover of X), and the
+proof is contained in [E1].
+567.11.Example: the simplest case of double affine Hecke algebras. Now letG=
+Z2⋉Z2acting on C. Then the conjugacy classes of reflection hyperplanes are four p oints:
+0,1/2,1/2 +η/2,η/2, where we suppose the lattice in CisZ⊕Zη. Correspondingly, the
+presentation of Gis as follows:
+generators: T1,T2,T3,T4;relations: T1T2T3T4= 1,T2
+i= 1.
+Thus, the corresponding orbifold Hecke algebra is the following defo rmation of CG:
+generators: T1,T2,T3,T4;relations: T1T2T3T4= 1,(Ti−pi)(Ti−qi) = 0,
+wherepi,qi(i= 1,...,4), are parameters.
+If we renormalize the Ti, these relations turn into
+(Ti−ti)(Ti+t−1
+i) = 0, T1T2T3T4=q,
+and we get the type C∨C1double affine Hecke algebra. If we set three of the four Ti’s
+satisfying the undeformed relation T2
+i= 1, we get the double affine Hecke algebra of type
+A1. More precisely, this algebra is generated by T1,...,T 4with relations
+T2
+2=T2
+3=T2
+4= 1,(T1−t)(T1+t−1) = 0, T1T2T3T4=q.
+Another presentation of this algebra (which is more widely used) is as follows. Let E=
+C/Z2, an elliptic curve with an Z2action defined by z/mapsto→−z. Definethe partial braid group
+B=πorb
+1(E\{0}/Z2,x),
+wherexis a generic point. Notice that comparing to the usual braid group, w e do not
+delete three of the four reflection points. The generators of the groupπ1(E\{0},x) (the
+fundamental groupofapunctured 2-torus)are X(corresponding tothe a-cycleonthetorus),
+Y(corresponding to the b-cycle on thetorus) and C(corresponding to the looparound0). In
+order to construct B, which is an extension of Z2byπ1(E\{0},x), we introduce an element
+Ts.t.T2=C(the half-loop around the puncture). Then X,Y,Tsatisfy the following
+relations:
+TXT=X−1, T−1YT−1=Y−1, Y−1X−1YXT2= 1.
+The Hecke algebra of the partial braid group is then defined to be th e group algebra of B
+plus an extra relation: ( T−q1)(T+q2) = 0.
+A common way to present this Hecke algebra is to renormalize the gen erators so that one
+has the following relations:
+TXT=X−1,T−1YT−1=Y−1,Y−1X−1YXT2=q,(T−t)(T+t−1) = 0.
+This is Cherednik’s definition for HH(q,t), the double affine Hecke algebra of type A1which
+depends on two parameters q,t.
+There are two degenerations of the algebra HH(q,t).
+1.The trigonometric degeneration.
+SetY=e/pla⋉ckover2pi1y,q=e/pla⋉ckover2pi1,t=e/pla⋉ckover2pi1candT=se/pla⋉ckover2pi1cs, wheres∈Z2is the reflection. Then s,X,y
+satisfy the following relations modulo /pla⋉ckover2pi1:
+s2= 1, sXs−1=X−1, sy+ys= 2c, X−1yX−y= 1−2cs.
+The algebra generated by s,X,ywith these relations is called the type A1trigonomet-
+ric Cherednik algebra. It is easy to show that it is isomorphic to the Ch erednik algebra
+H1,c(Z2,C∗), where Z2acts onC∗byz→z−1.
+572.The rational degeneration.
+In the trigonometric Cherednik algebra, set X=e/pla⋉ckover2pi1xandy= ˆy//pla⋉ckover2pi1. Thens,x,ˆysatisfy the
+following relations modulo /pla⋉ckover2pi1:
+s2= 1,sx=−xs,sˆy=−ˆys,ˆyx−xˆy= 1−2cs.
+The algebra generated by s,x,ˆywith these relations is the rational Cherednik algebra
+H1,c(Z2,C) with the action of Z2onCis given by z→−z.
+7.12.Affine and extended affine Weyl groups. LetR={α}⊂Rnbe a root system
+with respect to a nondegenerate symmetric bilinear form ( ·,·) onRn. We will assume that
+Ris reduced. Let{αi}n
+i=1⊂Rbe the set of simple roots and R+(respectively R−) be the
+set of positive (respectively negative) roots. The coroots are de noted byα∨= 2α/(α,α).
+LetQ∨=/circleplustextn
+i=1Zα∨
+ibe the coroot lattice and P∨=/circleplustextn
+i=1Zω∨
+ithe coweight lattice, where
+ω∨
+i’s are the fundamental coweights, i.e., ( ω∨
+i,αj) =δij. Letθbe the maximal positive root,
+and assume that the bilinear form is normalized by the condition ( θ,θ) = 2. Let Wbe the
+Weyl group which is generated by the reflections sα(α∈R).
+By definition, the affine root system is
+Ra={˜α= [α,j]∈Rn×R|whereα∈R,j∈Z}.
+The set of positive affine roots is Ra
++={[α,j]|j∈Z>0}∪{[α,0]|α∈R+}. Defineα0=
+[−θ,1]. We will identify α∈Rwith ˜α= [α,0]∈Ra.
+For an arbitrary affine root ˜ α= [α,j] and a vector ˜ z= [z,ζ]∈Rn×R, the corresponding
+affine reflection is defined as follows:
+s˜α(˜z) = ˜z−2(z,α)
+(α,α)˜α= ˜z−(z,α∨) ˜α.
+The affine Weyl group Wais generated by the affine reflections {s˜α|˜α∈/tildewideR+}, and we have
+an isomorphism:
+Wa∼=W⋉Q∨,
+where the translation α∨∈Q∨is naturally identified with the composition s[−α,1]sα∈Wa.
+Define the extended affine Weyl group to be Wext
+a=W⋉P∨acting on Rn+1viab(˜z) =
+[z,ζ−(b,z)] for ˜z= [z,ζ], b∈P∨. ThenWa⊂Wext
+a. Moreover, Wais a normal subgroup of
+Wext
+aandWext
+a/Wa=P∨/Q∨. The latter groupcanbeidentified withthe groupΠ = {πr}of
+the elements of Wext
+apermuting simple affine roots under their action in Rn+1. It is a normal
+commutative subgroup of Aut = Aut(Dyna) (Dynadenotes the affine Dynkin diagram).
+The quotient Aut /Π is isomorphic to the group of the automorphisms preserving α0, i.e. the
+group AutDyn of automorphisms of the finite Dynkin diagram.
+7.13.Cherednik’s double affine Hecke algebra of a root system. In this subsection,
+we will give an explicit presentation of Cherednik’s DAHA for a root sys tem, defined in
+Example 7.18. This is done by giving an explicit presentation of the corr esponding braid
+group (which is called the elliptic braid group ), and then imposing quadratic relations on the
+generators corresponding to reflections.
+For a root system R, letm= 2 ifRis of typeD2k,m= 1 ifRis of typeB2k,Ck, and
+otherwisem=|Π|. Letmijbe the number of edges between vertex iand vertex jin the
+58affine Dynkin diagram of Ra. LetXi(i= 1,...,n) be a family of pairwise commutative and
+algebraically independent elements. Set
+X[b,j]=n/productdisplay
+i=1Xℓi
+iqj,whereb=n/summationdisplay
+i=1ℓiωi∈P,j∈Z/mZ.
+For an element ˆ w∈Wext
+a, we can define an action on these X[b,j]by ˆwX[b,j]=Xˆw[b,j].
+Definition 7.24 (Cherednik) .The double affine Hecke algebra (DAHA) of the root system
+R, denoted byHH, is an algebra defined over the field Cq,t=C(q1/m,t1/2), generated by
+Ti,i= 0,...,n,Π,Xb,b∈P, subject to the following relations:
+(1)TiTjTi···=TjTiTj···,mijfactors each side;
+(2) (Ti−ti)(Ti+t−1
+i) = 0 fori= 0,...,n;
+(3)πTiπ−1=Tπ(i), forπ∈Π andi= 0,...,n;
+(4)πXbπ−1=Xπ(b), forπ∈Π,b∈P;
+(5)TiXbTi=XbX−1
+αi, ifi>0 and (b,α∨
+i) = 1;TiXb=XbTi, ifi>0 and (b,α∨
+i) = 0;
+(6)T0XbT0=Xb−α0if (b,θ) =−1;T0Xb=XbT0if (b,θ) = 0.
+Heretiare parameters attached to simple affine roots (so that roots of t he same length
+give rise to the same parameters).
+The degenerate double affine Hecke algebra (trigonometric Ch erednik algebra)HHtrigis
+generated by the group algebra of Wext
+a, Π and pairwise commutative y˜b=/summationtextn
+i=1(b,α∨)yi+u
+for˜b= [b,u]∈P×Z, with the following relations:
+siyb−ysi(b)si=−ki(b,α∨
+i),fori= 1,...,n,
+s0yb−ys0(b)s0=k0(b,θ), πrybπ−1
+r=yπr(b)forπr∈Π.
+Remark 7.25. This degeneration can be obtained from the DAHA similarly to the case of
+A1, which is described above.
+7.14.Algebraic flatness of Hecke algebras of polygonal Fuchsian g roups.LetW
+be the Coxeter group of rank rcorresponding to a Coxeter datum:
+mij(i,j= 1,...,r,i/\e}atio\slash=j),such that 2≤mij≤∞andmij=mji.
+So the group Whas generators sii= 1,...,r, and defining relations
+s2
+i= 1,(sisj)mij= 1 ifmij/\e}atio\slash=∞.
+It has a sign character ξ:W→{±1}given byξ(si) =−1. Denote by W+the kernel of ξ
+(the even subgroup of W). It is generated by aij=sisjwith relations:
+aij=a−1
+ji, aijajkaki= 1, amij
+ij= 1.
+Wecandeformthegroupalgebra C[W]asfollows. Definethealgebra A(W)withinvertible
+generators si, andtij,k,i,j= 1,...,r,k∈Zmijfor (i,j) such that mij<∞and defining
+relations
+tij,k=t−1
+ji,−k, s2
+i= 1,[tij,k,ti′j′,k′] = 0, sptij,k=tji,ksp,
+mij/productdisplay
+k=1(sisj−tij,k) = 0 ifmij<∞.
+Notice that if we set tij,k= exp(2πki/mij), we get C[W].
+59Define also the algebra A+(W) overR:=C[tij,k] (tij,k=t−1
+ji,−k) by generators aij,i/\e}atio\slash=j
+(aij=a−1
+ji), and relations
+mij/productdisplay
+k=1(aij−tij,k) = 0 ifmij<∞, aijajpapi= 1.
+Ifwis a word in letters si, letTwbe the corresponding element of A(W). Choose a
+functionw(x) which attaches to every element x∈W, a reduced word w(x) representing x
+inW.
+Theorem 7.26 (Etingof, Rains, [ER]) .(i)Theelements Tw(x),x∈W, forma spanning
+set inA(W)as a leftR-module.
+(ii)The elements Tw(x),x∈W+, form a spanning set in A+(W)as a leftR-module.
+(iii)The elements Tw(x),x∈W, are linearly independent if Whas no finite parabolic
+subgroups of rank 3.
+Proof.We only give the proof of (i). Statement (ii) follows from (i). Proof of (iii), which is
+quite nontrivial, can be found in [ER] (it uses the geometry of constr uctible sheaves on the
+Coxeter complex of W).
+Let us write the relationmij/productdisplay
+k=1(sisj−tij,k) = 0
+as a deformed braid relation:
+sjsisj...+S.L.T.=tijsisjsi...+S.L.T.,
+wheretij= (−1)mij+1tij,1···tij,mij, S.L.T. mean “smaller length terms”, and the products
+on both sides have length mij. This can be done by multiplying the relation by sisj···(mij
+factors).
+Now let us show that Tw(x)spanA(W) overR. Clearly,Twfor all words wspanA(W).
+So we just need to take any word wand express TwviaTw(x).
+It is well known from the theory of Coxeter groups (see e.g. [B]) th at using the braid
+relations, one can turn any non-reduced word into a word that is no t square free, and any
+reduced expression of a given element of Winto any other reduced expression of the same
+element. Thus, if wis non-reduced, then by using the deformed braid relations we can r educe
+Twto a linear combination of Tuwith words uof smaller length than w. On the other hand,
+ifwis a reduced expression for some element x∈W, then using the deformed braid relations
+we can reduce Twto a linear combination of Tuwithushorter than w, andTw(x). ThusTw(x)
+are a spanning set. This proves (i). /square
+Thus,A+(W) is a “deformation” of C[W+] overR, and similarly A(W) is a “twisted
+deformation” of C[W].
+Now let Γ = Γ( m1,...,mr),r≥3, be the Fuchsian group defined by generators cj,
+j= 1,...,r, with defining relations
+cmj
+j= 1,r/productdisplay
+j=1cj= 1.
+Here 2≤mj<∞.
+60Suppose Γ acts on HwhereHis a simply connected complex Riemann surface as in
+Section 7.7. We have the Hecke algebra of Γ, Hτ(H,Γ), defined by the same (invertible)
+generatorscjand relations
+/productdisplay
+k(cj−exp(2πik/nj)qjk) = 0,r/productdisplay
+j=1cj= 1,
+whereqjk= exp(τjk).
+We saw above (Theorem 7.15) that if τjk’s are formal, the algebra Hτ(Γ,H) is flat inτif
+|Γ|is infinite (i.e., His Euclidean or hyperbolic). Here is a much stronger non-formal ver sion
+of this theorem.
+Theorem 7.27. The algebra Hτ(Γ,H)is free as a left module over R:=C[q±1
+jk]if and only
+if/summationtext
+j(1−1/mj)≥2(i.e.,His Euclidean or hyperbolic).
+Proof.Let us consider the Coxeter datum: mij,i,j= 1,...,r, such that mi,i+1:=mi
+(i∈Z/rZ), andmij=∞otherwise. Suppose the corresponding Coxeter group is W. Then
+we can see that Γ = W+. Notice that the algebra Hτ(Γ,H) for genus 0 orbifolds is the
+algebraA+(W), i.e., we have Hτ(Γ,H) =A+(W).
+The condition/summationtext
+j(1−1/mj)≥2 is equivalent to the condition that Whas no finite
+parabolic subgroups of rank 3. From Theorem 7.26 (ii) and Theorem 7 .15, we can see that
+A+(W) is free as a left module over R. We are done. /square
+7.15.Notes.Section 7.8 follows Section 6 of the paper [EOR]; Cherednik’s definition o f the
+double affine Hecke algebra of a root system is from Cherednik’s book [Ch]; Sections 7.7 and
+7.14 follow the paper [ER]; The other parts of this section follow the p aper [E1].
+618.Symplectic reflection algebras
+8.1.The definition of symplectic reflection algebras. Rational Cherednik algebras for
+finiteCoxeter groupsareaspecial caseof awider class ofalgebras called symplectic reflection
+algebras. To define them, let Vbe a finite dimensional symplectic vector space over Cwith
+a symplectic form ω, andGbe a finite group acting symplectically (linearly) on V. For
+simplicity let us assume that ( ∧2V∗)G=Cω(i.e.,Vis symplectically irreducible) and that
+Gacts faithfully on V(these assumptions are not important, and essentially not restric tive).
+Definition 8.1. A symplectic reflection in Gis an element gsuch that the rank of the
+operator 1−gonVis 2.
+Ifsis a symplectic reflection, then let ωs(x,y) be the form ωapplied to the projections of
+x,yto the image of 1−salong the kernel of 1 −s; thusωsis a skewsymmetric form of rank
+2 onV.
+LetS⊂Gbe the set of symplectic reflections, and c:S→Cbe a function which is
+invariant under the action of G. Lett∈C.
+Definition 8.2. The symplectic reflection algebra Ht,c=Ht,c[G,V] is the quotient of the
+algebraC[G]⋉T(V) by the ideal generated by the relation
+(8.1) [ x,y] =tω(x,y)−2/summationdisplay
+s∈Scsωs(x,y)s.
+Example 8.3. LetWbe a finite Coxeter group with reflection representation h. We can set
+V=h⊕h∗,ω(x,x′) =ω(y,y′) = 0,ω(y,x) = (y,x), forx,x′∈h∗andy,y′∈h. In this case
+(1) symplectic reflections are the usual reflections in W;
+(2)ωs(x,x′) =ωs(y,y′) = 0,ωs(y,x) = (y,αs)(α∨
+s,x)/2.
+Thus,Ht,c[G,h⊕h∗] coincides with the rational Cherednik algebra Ht,c(G,h) defined in
+Section 3.
+Example 8.4. Let Γ be a finite subgroup of SL2(C), andV=C2be the tautological
+representation, with its standard symplectic form. Then all nontr ivial elements of Γ are
+symplectic reflections, and for any symplectic reflection s,ωs=ω. So the main commutation
+relation of Ht,c[Γ,V] takes the form
+[y,x] =t−/summationdisplay
+g∈Γ,g/\e}atio\slash=12cgg.
+Example 8.5. (Wreath products) Let Γ be as in the previous example, G=Sn⋉Γn, and
+V= (C2)n. Then symplectic reflections are conjugates ( g,1,...,1),g∈Γ,g/\e}atio\slash= 1, and also
+conmjugates of transpositions in Sn(so there is one more conjugacy class of reflections than
+in the previous example).
+Note also that for any V,G,H0,0[G,V] =G⋉SV, andH1,0[G,V] =G⋉Weyl(V), where
+Weyl(V) is the Weyl algebra of V, i.e. the quotient of the tensor algebra T(V) by the
+relationxy−yx=ω(x,y),x,y∈V.
+8.2.The PBW theorem for symplectic reflection algebras. To ensure that the sym-
+plectic reflection algebras Ht,chave good properties, we need to prove a PBW theorem for
+them, which is an analog of Proposition 3.5. This is done in the following th eorem, which
+also explains the special role played by symplectic reflections.
+62Theorem 8.6. Letκ:∧2V→C[G]be a linear G-equivariant function. Define the algebra
+Hκto be the quotient of the algebra C[G]⋉T(V)by the relation [x,y] =κ(x,y),x,y∈V. Put
+an increasing filtration on Hκby setting deg(V) = 1,deg(G) = 0, and define ξ:CG⋉SV→
+grHκto be the natural surjective homomorphism. Then ξis an isomorphism if and only if κ
+has the form
+κ(x,y) =tω(x,y)−2/summationdisplay
+s∈Scsωs(x,y)s,
+for somet∈CandG-invariant function c:S→C.
+Unfortunately, for a general symplectic reflection algebra we don ’t have a Dunkl operator
+representation, so the proof of the more difficult “if” part of this T heorem is not as easy
+as the proof of Proposition 3.5. Instead of explicit computations wit h Dunkl operators, it
+makes use of the deformation theory of Koszul algebras, which we will now discuss.
+8.3.Koszul algebras. LetRbe a finite dimensional semisimple algebra (over C). LetA
+be aZ+-graded algebra, such that A[0] =R, and assume that the graded components of A
+are finite dimensional.
+Definition 8.7. (i) The algebra Ais said to be quadratic if it is generated over Rby
+A[1], and has defining relations in degree 2.
+(ii)Ais Koszul if all elements of Exti(R,R) (whereRis the augmentation module over
+A) have grade degree precisely i.
+Remark 8.8. (1) Thus, in a quadratic algebra, A[2] =A[1]⊗RA[1]/E, whereEis the
+subspace (R-subbimodule) of relations.
+(2) It is easy to show that a Koszul algebra is quadratic, since the c ondition to be
+quadratic is just the Koszulity condition for i= 1,2.
+Now letA0be a quadratic algebra, A0[0] =R. LetE0be the space of relations for A0. Let
+E⊂A0[1]⊗RA0[1][[/pla⋉ckover2pi1]] be a free (over C[[/pla⋉ckover2pi1]])R-subbimodule which reduces to E0modulo/pla⋉ckover2pi1
+(“deformation of the relations”). Let Abe the (/pla⋉ckover2pi1-adically complete) algebra generated over
+R[[/pla⋉ckover2pi1]] byA[1] =A0[1][[/pla⋉ckover2pi1]] with the space of defining relations E. ThusAis aZ+-graded
+algebra.
+The following very important theorem is due to Beilinson, Ginzburg, an d Soergel, [BGS]
+(less general versions appeared earlier in the works of Drinfeld [Dr], Polishchuk-Positselski
+[PP], Braverman-Gaitsgory [BG]). We will not give its proof.
+Theorem 8.9 (Koszul deformation principle) .IfA0is Koszul then Ais a topologically free
+C[[/pla⋉ckover2pi1]]module if and only if so is A[3].
+Remark. Note thatA[i] fori<3 are obviously topologically free.
+We will now apply this theorem to the proof of Theorem 8.6.
+8.4.Proof of Theorem 8.6. Letκ:∧2V→C[G] be a linear G-equivariant map. We write
+κ(x,y) =/summationtext
+g∈Gκg(x,y)g, whereκg(x,y)∈∧2V∗. To apply Theorem 8.9, let us homogenize
+our algebras. Namely, let A0= (CG⋉SV)⊗C[u] (whereuhas degree 1). Also let /pla⋉ckover2pi1be a
+formal parameter, and consider the deformation A=H/pla⋉ckover2pi1u2κofA0. That is,Ais the quotient
+ofG⋉T(V)[u][[/pla⋉ckover2pi1]] by the relations [ x,y] =/pla⋉ckover2pi1u2κ(x,y). This is a deformation of the type
+considered in Theorem 8.9, and it is easy to see that its flatness in /pla⋉ckover2pi1is equivalent to Theorem
+638.6. Also, the algebra A0is Koszul, because the polynomial algebra SVis a Koszul algebra.
+Thus by Theorem 8.9, it suffices to show that Ais flat in degree 3.
+The flatness condition in degree 3 is “the Jacobi identity”
+[κ(x,y),z]+[κ(y,z),x]+[κ(z,x),y] = 0,
+which must be satisfied in CG⋉V. In components, this equation transforms into the system
+of equations
+κg(x,y)(z−zg)+κg(y,z)(x−xg)+κg(z,x)(y−yg) = 0
+for everyg∈G(herezgdenotes the result of the action of gonz).
+This equation, in particular, implies that if x,y,gare such that κg(x,y)/\e}atio\slash= 0 then for any
+z∈V z−zgis a linear combination of x−xgandy−yg. Thusκg(x,y) is identically zero
+unless the rank of (1 −g)|Vis at most 2, i.e. g= 1 orgis a symplectic reflection.
+Ifg= 1 thenκg(x,y) has to be G-invariant, so it must be of the form tω(x,y), where
+t∈C.
+Ifgis a symplectic reflection, then κg(x,y) must be zero for any xsuch thatx−xg= 0.
+Indeed, if for such an xthere had existed ywithκg(x,y)/\e}atio\slash= 0 thenz−zgfor anyzwould
+be a multiple of y−yg, which is impossible since Im(1 −g)|Vis 2-dimensional. This implies
+thatκg(x,y) = 2cgωg(x,y), andcgmust be invariant.
+Thus we have shown that if Ais flat (in degree 3) then κmust have the form given in
+Theorem 8.6. Conversely, it is easy to see that if κdoes have such form, then the Jacobi
+identity holds. So Theorem 8.6 is proved.
+8.5.The spherical subalgebra of the symplectic reflection algeb ra.Thepropertiesof
+symplectic reflection algebras are similar to the properties of ration al Cherednik algebras we
+have studied before. The main difference is that we no longer have th e Dunkl representation
+andlocalizationresults, sosomeproofsarebasedondifferent ideas andaremorecomplicated.
+The spherical subalgebra of the symplectic reflection algebra is defi ned in the same way
+as in the Cherednik algebra case. Namely, let e=|G|−1/summationtext
+g∈Gg, andBt,c=eHt,ce.
+Proposition 8.10. Bt,cis commutative if and only if t= 0.
+Proof.LetAbe aZ+-filtered algebra. If Ais not commutative, then we can define a nonzero
+Poisson bracket on gr Ain the following way. Let mbe the minimum of deg( a)+deg(b)−
+deg([a,b]) (overa,b∈Asuch that [a,b]/\e}atio\slash= 0). Then for homogeneous elements a0,b0∈A0of
+degreesp,q, we can define{a0,b0}to be the image in A0[p+q−m] of [a,b], wherea,bare
+any lifts of a0,b0toA. It is easy to check that {·,·}is a Poisson bracket on A0of degree
+−m.
+Let us now apply this construction to the filtered algebra A=Bt,c. We have gr( A) =
+A0= (SV)G.
+Lemma 8.11. A0has a unique, up to scaling, Poisson bracket of degree −2, and no nonzero
+Poisson brackets of degrees <−2.
+Proof.A Poisson bracket on ( SV)Gis the same thing as a Poisson bracket on the variety
+V∗/G. On the smooth part ( V∗/G)sofV∗/G, it is simply a bivector field, and we can lift
+it to a bivector field on the preimage V∗
+sof (V∗/G)sinV∗, which is the set of points in V
+with trivial stabilizers. But the codimension on V∗\V∗
+sinV∗is 2 (asV∗\V∗
+sis a union
+of symplectic subspaces), so the bivector on V∗
+sextends to a regular bivector on V∗. So if
+64this bivector is homogeneous, it must have degree ≥−2, and if it has degree −2 then it
+must be with constant coefficients, so being G-invariant, it is a multiple of ω. The lemma is
+proved. /square
+Now, for each t,cwe have a natural Poisson bracket on A0of degree−2, which depends
+linearly on t,c. So by the lemma, this bracket has to be of the form f(t,c)Π, where Π is the
+unique up to scaling Poisson bracket of degree −2, andfa homogeneous linear function.
+Thus the algebra A=Bt,cis not commutative unless f(t,c) = 0. On the other hand, if
+f(t,c) = 0, and Bt,cis not commutative, then, as we’ve shown, A0has a nonzero Poisson
+bracket of degree <−2. But By Lemma 8.11, there is no such brackets. So Bt,cmust be
+commutative if f(t,c) = 0.
+It remains toshow that f(t,c)is infacta nonzero multiple of t. First notethat f(1,0)/\e}atio\slash= 0,
+sinceB1,0is definitely noncommutative. Next, let us take a point ( t,c) such that Bt,cis
+commutative. Look at the Ht,c-module Ht,ce, which has a commuting action of Bt,cfrom the
+right. Its associated graded is SVas an (CG⋉SV,(SV)G)-bimodule, which implies that
+the generic fiber of Ht,ceas aBt,c-module is the regular representation of G. So we have a
+family of finite dimensional representations of Ht,con the fibers of Ht,ce, all realized in the
+regular representation of G. Computing the trace of the main commutation relation (8.1) of
+Ht,cin this representation, we obtain that t= 0 (since Tr( s) = 0 for any reflection s). The
+proposition is proved. /square
+Note that B0,chas no zero divisors, since its associated graded algebra ( SV)Gdoes not.
+Thus, like in the Cherednik algebra case, we can define a Poisson varie tyMc, the spectrum
+ofB0,c, called the Calogero-Moser space of G,V. Moreover, the algebra Bc:=B/pla⋉ckover2pi1,coverC[/pla⋉ckover2pi1]
+is an algebraic quantization of Mc.
+8.6.The center of the symplectic reflection algebra Ht,c.Consider thebimodule Ht,ce,
+which has a left action of Ht,cand a right commuting action of Bt,c. It is obvious that
+EndHt,cHt,ce=Bt,c(with opposite product). The following theorem shows that the bimo dule
+Ht,cehas the double centralizer property (i.e., End Bt,cHt,ce=Ht,c).
+Note that we have a natural map ξt,c:Ht,c→EndBt,cHt,ce.
+Theorem 8.12. ξt,cis an isomorphism for any t,c.
+Proof.The complete proof is given [EG]. We will give the main ideas of the proof skipping
+straightforward technical details. The first step is to show that t he result is true in the
+graded case, ( t,c) = (0,0). To do so, note the following easy lemma:
+Lemma 8.13. IfXis an affine complex algebraic variety with algebra of functio nsOXand
+Ga finite group acting freely on Xthen the natural map ξX:G⋉OX→EndOG
+XOXis an
+isomorphism.
+Therefore, the map ξ0,0:G⋉SV→End(SV)G(SV) is injective, and moreover becomes an
+isomorphism after localization to the field of quotients C(V∗)G. To show it’s surjective, take
+a∈End(SV)G(SV). There exists a′∈G⋉ C(V∗) which maps to a. Moreover, by Lemma
+8.13,a′can have poles only at fixed points of GonV∗. But these fixed points form a subset
+of codimension≥2, so there can be no poles and we are done in the case ( t,c) = (0,0).
+Now note that the algebra End Bt,cHt,cehas an increasing integer filtration (bounded be-
+low) induced by the filtration on Ht,c. This is due to the fact that Ht,ceis a finitely generated
+65eHt,ce-module (since it is true in the associated graded situation, by Hilbert ’s theorem about
+invariants). Also, the natural map grEnd Bt,cHt,ce→EndgrBt,cgrHt,ceis clearly injective.
+Therefore, our result in the case ( t,c) = (0,0) implies that this map is actually an iso-
+morphism (as so is its composition with the associated graded of ξt,c). Identifying the two
+algebras by this isomorphism, we find that gr( ξt,c) =ξ0,0. Sinceξ0,0is an isomorphism, ξt,c
+is an isomorphism for all t,c, as desired.2/square
+Denote by Zt,cthe center of the symplectic reflection algebra Ht,c. We have the following
+theorem.
+Theorem 8.14. Ift/\e}atio\slash= 0, the center of Ht,cis trivial. If t= 0, we have grZ0,c=Z0,0. In
+particular, H0,cis finitely generated over its center.
+Proof.Thet/\e}atio\slash= 0 case was proved by Brown and Gordon [BGo] as follows. If t/\e}atio\slash= 0, we have
+grZt,c⊆Z0,0= (SV)G. Also, we have a map
+τt,c:Zt,c→Bt,c=eHt,ce, z/mapsto→ze=eze.
+The mapτt,cis injective since gr( τt,c) is injective. In particular, the image of gr( τt,c) is
+contained in Z(Bt,c), the center of Bt,c. Thus it is enough to show that Z(Bt,c) is trivial. To
+show this, note that gr Z(Bt,c) is contained in the Poisson center of B0,0which is trivial. So
+Z(Bt,c) is trivial.
+Now suppose t= 0. We need to show that gr( τ0,c) : gr(Z0,c)→Z0,0is an isomorphism. It
+suffices to show that τ0,cis an isomorphism. To show this, we construct τ−1
+0,c:B0,c→Z0,cas
+follows.
+For anyb∈B0,c, sinceB0,cis commutative, we have an element ˜b∈EndB0,c(H0,ce) which
+is defined as the right multiplication by b. From Theorem 8.12, ˜b∈H0,c. Moreover, ˜b∈Z0,c
+since it commutes with H0,cas a linear operator onthefaithful H0,c-moduleH0,ce. So˜b∈Z0,c.
+It is easy to see that ˜be=b. So we can set ˜b=τ−1
+0,c(b) which defines the inverse map to
+τ0,c. /square
+8.7.A review of deformation theory. Now we would like to explain that symplectic
+reflection algebras are the most general deformations of algebra s of the from G⋉Weyl(V).
+Before we do so, we give a brief review of classical deformation theo ry of associative algebras.
+8.7.1.Formal deformations of associative algebras. LetA0be an associative algebra with
+unit over C. Denote by µ0the multiplication in A0.
+Definition 8.15. A (flat) formal n-parameter deformation of A0is an algebra Aover
+C[[/pla⋉ckover2pi1]] =C[[/pla⋉ckover2pi11,...,/pla⋉ckover2pi1n]] which is topologically free as a C[[/pla⋉ckover2pi1]]-module, together with an
+algebra isomorphism η0:A/mA→A0wherem=/a\}b∇acketle{t/pla⋉ckover2pi11,...,/pla⋉ckover2pi1n/a\}b∇acket∇i}htis the maximal ideal in C[[/pla⋉ckover2pi1]].
+When no confusion is possible, we will call Aa deformation of A0(omitting “formal”).
+Let us restrict ourselves to one-parameter deformations with pa rameter/pla⋉ckover2pi1. Let us choose
+an identification η:A→A0[[/pla⋉ckover2pi1]] asC[[/pla⋉ckover2pi1]]-modules, such that η=η0modulo/pla⋉ckover2pi1. Then the
+2Here we use the fact that the filtration is bounded from below. In th e case of an unbounded filtration,
+it is possible for a map not to be an isomorphism if its associated graded is an isomorphism. An example of
+this is the operator of multiplication by 1+ t−1in the space of Laurent polynomials in t, filtered by degree.
+66product inAis completely determined by the product of elements of A0, which has the form
+of a “star-product”
+µ(a,b) =a∗b=µ0(a,b)+/pla⋉ckover2pi1µ1(a,b)+/pla⋉ckover2pi12µ2(a,b)+···,
+whereµi:A0⊗A0→A0are linear maps, and µ0(a,b) =ab.
+8.7.2.Hochschild cohomology. The main tool in deformation theory of associative algebras
+is Hochschild cohomology. Let us recall its definition.
+LetAbe an associative algebra. Let Mbe a bimodule over A. A Hochschild n-cochain of
+Awith coefficients in Mis a linear map A⊗n→M. The space of such cochains is denoted
+byCn(A,M). The differential d : Cn(A,M)→Cn+1(A,M) is defined by the formula
+df(a1,...,an+1) =f(a1,...,an)an+1−f(a1,...,anan+1)+f(a1,...,an−1an,an+1)
+−···+(−1)nf(a1a2,...,an+1)+(−1)n+1a1f(a2,...,an+1).
+It is easy to show that d2= 0.
+Definition 8.16. The Hochschild cohomology HH•(A,M) is defined to be the cohomology
+of the complex ( C•(A,M),d).
+Proposition 8.17. One has a natural isomorphism
+HHi(A,M)→Exti
+A−bimod(A,M),
+whereA−bimoddenotes the category of A-bimodules.
+Proof.The proof is obtained immediately by considering the bar resolution of the bimodule
+A:
+···→A⊗A⊗A→A⊗A→A,
+where the bimodule structure on A⊗nis given by
+b(a1⊗a2⊗···⊗an)c=ba1⊗a2⊗···⊗anc,
+and the map ∂n:A⊗n→A⊗n−1is given by the formula
+∂n(a1⊗a2⊗...⊗an) =a1a2⊗···⊗an−···+(−1)na1⊗···⊗an−1an.
+/square
+Note that we have the associative Yoneda product
+HHi(A,M)⊗HHj(A,N)→HHi+j(A,M⊗AN),
+induced by tensoring of cochains.
+IfM=A, the algebra itself, then we will denote HH•(A,M) by HH•(A). For example,
+HH0(A) is the center of A, and HH1(A) is the quotient of the Lie algebra of derivations of A
+by inner derivations. The Yoneda product induces a graded algebra structure on HH•(A); it
+can be shown that this algebra is supercommutative.
+678.7.3.Hochschild cohomology and deformations. LetA0be an algebra, and let us look for
+1-parameter deformations A=A0[[/pla⋉ckover2pi1]] ofA0. Thus, we look for such series µwhich satisfy
+the associativity equation, modulo the automorphisms of the C[[/pla⋉ckover2pi1]]-moduleA0[[/pla⋉ckover2pi1]] which are
+the identity modulo /pla⋉ckover2pi1.3
+The associativity equation µ◦(µ⊗Id) =µ◦(Id⊗µ) reduces to a hierarchy of linear
+equations:
+N/summationdisplay
+s=0µs(µN−s(a,b),c) =N/summationdisplay
+s=0µs(a,µN−s(b,c)).
+(These equations are linear in µNifµi,i<N, are known).
+To study these equations, one can use Hochschild cohomology. Nam ely, we have the
+following standard facts (due to Gerstenhaber, [Ge]), which can be checked directly.
+(1) The linear equation for µ1says thatµ1is a Hochschild 2-cocycle. Thus algebra struc-
+tures onA0[/pla⋉ckover2pi1]//pla⋉ckover2pi12deformingµ0are parametrized by the space Z2(A0) of Hochschild
+2-cocycles of A0with values in M=A0.
+(2) Ifµ1,µ′
+1are two 2-cocycles such that µ1−µ′
+1is a coboundary, then the algebra struc-
+tures onA0[/pla⋉ckover2pi1]//pla⋉ckover2pi12corresponding to µ1andµ′
+1are equivalent by a transformation of
+A0[/pla⋉ckover2pi1]//pla⋉ckover2pi12that equals the identity modulo /pla⋉ckover2pi1, and vice versa. Thus equivalence classes
+of multiplications on A0[/pla⋉ckover2pi1]//pla⋉ckover2pi12deformingµ0are parametrized by the cohomology
+HH2(A0).
+(3) The linear equation for µNsays that d µNis a certain quadratic expression bNin
+µ1,...,µN−1. This expression is always a Hochschild 3-cocycle, and the equation is
+solvableif andonlyif itis acoboundary. Thus thecohomologyclass of bNinHH3(A0)
+is the only obstruction to solving this equation.
+8.7.4.Universal deformation. In particular, if HH3(A0) = 0 then the equation for µncan be
+solved for all n, and for each nthe freedom in choosing the solution, modulo equivalences,
+is the space H= HH2(A0). Thus there exists an algebra structure over C[[H]] on the space
+Au:=A0[[H]] of formal functions from HtoA0,a,b/mapsto→µu(a,b)∈A0[[H]], (a,b∈A0), such
+thatµu(a,b)(0) =ab∈A0, and every 1-parameter flat formal deformation AofA0is given
+by the formula µ(a,b)(/pla⋉ckover2pi1) =µu(a,b)(γ(/pla⋉ckover2pi1)) for a unique formal series γ∈/pla⋉ckover2pi1H[[/pla⋉ckover2pi1]], with the
+property that γ′(0) is the cohomology class of the cocycle µ1.
+Such an algebra Auis called a universal deformation ofA0. It is unique up to an isomor-
+phism (which may involve an automorphism of C[[H]]).4
+Thus in the case HH3(A0) = 0, deformation theory allows us to completely classify 1-
+parameter flat formal deformations of A0. In particular, we see that the “moduli space”
+parametrizing formal deformations of A0is a smooth space – it is the formal neighborhood
+of zero inH.
+If HH3(A0) is nonzero then in general the universal deformation parametriz ed byHdoes
+not exist, as there are obstructions to deformations. In this cas e, the moduli space of
+3Note that we don’t have to worry about the existence of a unit in Asince a formal deformation of an
+algebra with unit always has a unit.
+4In spite of the universalpropertyof Au, it may happen that there is an isomorphism between the algebras
+A1andA2corresponding to different paths γ1(/pla⋉ckover2pi1),γ2(/pla⋉ckover2pi1) (of course, reducing to a nontrivial automorphism of
+A0modulo/pla⋉ckover2pi1). For this reason, sometimes Auis called a semiuniversal , rather than universal, deformation
+ofA0.
+68deformations will be a closed subscheme of the formal neighborhoo d of zero in H, which
+is often singular. On the other hand, even when HH3(A0)/\e}atio\slash= 0, the universal deformation
+parametrized by (the formal neighborhood of zero in) Hmay exist (although its existence
+may be more difficult to prove than in the vanishing case). In this case one says that the
+deformations of A0are unobstructed (since all obstructions vanish even though the space of
+obstructions doesn’t).
+8.8.Deformation-theoretic interpretation of symplectic refle ction algebras. LetV
+be a symplectic vector space (over C) and Weyl(V) the Weyl algebra of V. LetGbe a finite
+group acting symplectically on V. Then from the definition, we have
+A0:=H1,0[G,V] =G⋉Weyl(V).
+Let us calculate the Hochschild cohomology of this algebra.
+Theorem 8.18 (Alev, Farinati, Lambre, Solotar, [AFLS]) .The cohomology space
+HHi(G⋉Weyl(V))is naturally isomorphic to the space of conjugation invaria nt functions
+on the setSiof elements g∈Gsuch that rank(1−g)|V=i.
+Corollary 8.19. The odd cohomology of G⋉Weyl(V)vanishes, and HH2(G⋉Weyl(V))
+is the space C[S]Gof conjugation invariant functions on the set of symplectic reflections. In
+particular, there exists a universal deformation AofA0=G⋉Weyl(V)parametrized by
+C[S]G.
+Proof.Directly from the theorem. /square
+Proof of Theorem 8.18.
+Lemma 8.20. LetBbe aC-algebra together with an action of a finite group G. Then
+HH∗(G⋉B,G⋉B) = (/circleplusdisplay
+g∈GHH∗(B,Bg))G,
+whereBgis the bimodule isomorphic to Bas a space where the left action of Bis the usual
+one and the right action is the usual action twisted by g.
+Proof.The algebra G⋉Bis a projective B-module. Therefore, using the Shapiro lemma,
+we get
+HH∗(G⋉B,G⋉B) = Ext∗
+(G×G)⋉(B⊗Bop)(G⋉B,G⋉B)
+= Ext∗
+Gdiagonal⋉(B⊗Bop)(B,G⋉B) = Ext∗
+B⊗Bop(B,G⋉B)G
+= (/circleplusdisplay
+g∈GExt∗
+B⊗Bop(B,Bg))G= (/circleplusdisplay
+g∈GHH∗(B,Bg))G,
+as desired. /square
+Now apply the lemma to B= Weyl(V). For this we need to calculate HH∗(B,Bg),
+wheregis any element of G. We may assume that gis diagonal in some symplectic basis:
+g= diag(λ1,λ−1
+1,...,λn,λ−1
+n). Then by the K¨ unneth formula we find that
+HH∗(B,Bg) =n/circlemultiplydisplay
+i=1HH∗(A1,A1gi),
+69whereA1is the Weyl algebra of the 2-dimensional space, (generated by x,ywith defining
+relationxy−yx= 1), andgi= diag(λi,λ−1
+i).
+Thus we need to calculate HH∗(B,Bg),B=A1,g= diag(λ,λ−1).
+Proposition 8.21. HH∗(B,Bg)is 1-dimensional, concentrated in degree 0ifλ= 1and in
+degree2otherwise.
+Proof.IfB=A1thenBhas the following Koszul resolution as a B-bimodule:
+B⊗B→B⊗C2⊗B→B⊗B→B.
+Here the first map is given by the formula
+b1⊗b2/mapsto→b1⊗x⊗yb2−b1⊗y⊗xb2−b1y⊗x⊗b2+b1x⊗y⊗b2,
+the second map is given by
+b1⊗x⊗b2/mapsto→b1x⊗b2−b1⊗xb2, b1⊗y⊗b2/mapsto→b1y⊗b2−b1⊗yb2,
+and the third map is the multiplication.
+Thus the cohomology of Bwith coefficients in Bgcan be computed by mapping this
+resolution into Bgand taking the cohomology. This yields the following complex C•:
+(8.2) 0 →Bg→Bg⊕Bg→Bg→0,
+where the first nontrivial map is given by bg/mapsto→[bg,y]⊗x−[bg,x]⊗y, and the second
+nontrivial map is given by bg⊗x/mapsto→[x,bg],bg⊗y/mapsto→[y,bg].
+Considerfirstthecase g= 1. Equipthecomplex C•withtheBernsteinfiltration(deg( x) =
+deg(y) = 1), starting with 0 ,1,2, forC0,C1,C2, respectively (this makes the differential
+preserve the filtration). Consider the associated graded complex C•
+gr. In this complex,
+brackets are replaced with Poisson brackets, and thus it is easy to see thatC•
+gris the De
+Rham complex for the affine plane, so its cohomology is Cin degree 0 and 0 in other degrees.
+Therefore, the cohomology of C•is the same.
+Now consider g/\e}atio\slash= 1. In this case, declare that C0,C1,C2start in degrees 2,1,0 respectively
+(which makes the differential preserve the filtration), and again co nsider the graded complex
+C•
+gr. The graded Euler characteristic of this complex is ( t2−2t+1)(1−t)−2= 1.
+The cohomology in the C0
+grterm is the set of b∈C[x,y] such that ab=bagfor alla. This
+means that HH0= 0.
+The cohomology of the C2
+grterm is the quotient of C[x,y] by the ideal generated by a−ag,
+a∈C[x,y]. Thus the cohomology HH2of the rightmost term is 1-dimensional, in degree 0.
+By the Euler characteristic argument, this implies that HH1= 0. The cohomology of the
+filtered complex C•is therefore the same, and we are done. /square
+The proposition implies that in the n-dimensional case HH∗(B,Bg) is 1-dimensional, con-
+centrated in degree rank(1 −g). It is not hard to check that the group Gacts on the sum
+of these 1-dimensional spaces by simply permuting the basis vector s. Thus the theorem is
+proved. /square
+Remark 8.22. Another proof of Theorem 8.18 is given in [Pi].
+Theorem 8.23. The algebra H1,c[G,V], with formal c, is the universal deformation of
+H1,0[G,V] =G⋉Weyl(V). More specifically, the map f:C[S]G→HH2(G⋉Weyl(V))
+induced by this deformation coincides with the isomorphism of Corollary 8.19.
+70Proof.The proof (which we will not give) can be obtained by a direct computa tion with the
+Koszul resolution for G⋉Weyl(V). Such a proof is given in [Pi]. The paper [EG] proves
+a slightly weaker statement that the map fis an isomorphism, which suffices to show that
+H1,c(G,V) is the universal deformation of H1,0[G,V]. /square
+8.9.Finite dimensional representations of H0,c.LetMc= SpecZ0,c. We can regard
+H0,c=H0,c[G,V] as a finitely generated module over Z0,c=O(Mc). Letχ∈Mcbe a central
+character,χ:Z0,c→C. Denote by/a\}b∇acketle{tχ/a\}b∇acket∇i}htthe ideal in H0,cgenerated by the kernel of χ.
+Proposition 8.24. Ifχis generic then H0,c//a\}b∇acketle{tχ/a\}b∇acket∇i}htis the matrix algebra of size |G|. In par-
+ticular,H0,chas a unique irreducible representation Vχwith central character χ. This repre-
+sentation is isomorphic to CGas aG-module.
+Proof.It is shown by a standard argument (which we will skip) that it is sufficie nt to check
+the statement in the associated graded case c= 0. In this case, for generic χ,G⋉SV//a\}b∇acketle{tχ/a\}b∇acket∇i}ht=
+G⋉Fun(Oχ), whereOχis the (free) orbit of Gconsisting of the points of V∗that map to
+χ∈V∗/G, and Fun(Oχ) is the algebra of functions on Oχ. It is easy to see that this algebra
+is isomorphic to a matrix algebra, and has a unique irreducible represe ntation, Fun(Oχ),
+which is a regular representation of G. /square
+Corollary 8.25. Any irreducible representation of H0,chas dimension≤|G|.
+Proof.We will use the following lemma.
+Lemma 8.26 (The Amitsur-Levitzki identity) .For anyN×NmatricesX1,...,X 2Nwith
+entries in a commutative ring A,
+/summationdisplay
+σ∈S2n(−1)σXσ(1)···Xσ(2N)= 0.
+Proof.Consider the ring Mat N(A)⊗∧(ξ1,...,ξ 2n). LetX=/summationtext
+iXiξi∈R. So we have
+X2=/summationdisplay
+i<j[Xi,Xj]ξiξj∈MatN(A⊗∧even(ξ1,...,ξ 2n)).
+It is obvious that Tr X2= 0. Similarly, one can easily show that Tr X4= 0,...,TrX2N= 0.
+Since the ring A⊗∧even(ξ1,...,ξ 2n) is commutative, from the Cayley-Hamilton theorem, we
+know thatX2N= 0 which implies the lemma. /square
+Since for generic χthe algebra H0,c//a\}b∇acketle{tχ/a\}b∇acket∇i}htis a matrix algebra, the algebra H0,csatisfies the
+Amitsur-Levitzki identity. Next, note that since H0,cis a finitely generated Z0,c-module (by
+passing to the associated graded and using Hilbert’s theorem), eve ry irreducible representa-
+tion ofH0,cis finite dimensional. If H0,chad an irreducible representation Eof dimension
+m >|G|, then by the density theorem the matrix algebra Mat mwould be a quotient of
+H0,c. But one can show that the Amitsur-Levitzki identity of degree |G|is not satisfied for
+matrices of bigger size than |G|. Contradiction. Thus, dim E≤|G|, as desired. /square
+In general, for special central characters there are represen tations of H0,cof dimension less
+than|G|. However, in some cases one can show that all irreducible represen tations have
+dimension exactly |G|. For example, we have the following result.
+71Theorem 8.27. LetG=Sn,V=h⊕h∗,h=Cn(the rational Cherednik algebra for Sn).
+Then forc/\e}atio\slash= 0, every irreducible representation of H0,chas dimension n!and is isomorphic
+to the regular representation of Sn.
+Proof.LetEbe an irreducible representation of H0,c. Let us calculate the trace in Eof any
+permutation σ/\e}atio\slash= 1. Letjbe an index such that σ(j) =i/\e}atio\slash=j. Thensijσ(j) =j. Hence in
+H0,cwe have
+[yj,xisijσ] = [yj,xi]sijσ=cs2
+ijσ=cσ.
+Hence Tr E(σ) = 0, and thus Eis a multiple of the regular representation of Sn. But by
+Theorem 8.25, dim E≤n!, so we get that Eis the regular representation, as desired. /square
+8.10.Azumaya algebras. LetZbe a finitely generated commutative algebra over C,M=
+SpecZthe corresponding affine scheme, and Aa finitely generated Z-algebra.
+Definition 8.28. Ais said to be an Azumaya algebra of degree Nif the completion ˆAχof
+Aat every maximal ideal χinZis the matrix algebra of size Nover the completion ˆZχof
+Z.
+Thus, an Azumaya algebra should be thought of as a bundle of matrix algebras on M.5
+For example, if Eis an algebraic vector bundle on Mthen End(E) is an Azumaya algebra.
+However, not all Azumaya algebras are of this form.
+Example 8.29. Forq∈C∗, consider the quantum torus
+Tq=C/a\}b∇acketle{tX±1,Y±1/a\}b∇acket∇i}ht//a\}b∇acketle{tXY−qYX/a\}b∇acket∇i}ht.
+Ifqis a root of unity of order N, then the center of Tqis/a\}b∇acketle{tX±N,Y±N/a\}b∇acket∇i}ht=C[M] where
+M= (C∗)2. It is not difficult to show that Tqis an Azumaya algebra of degree N, but
+Tq⊗C[M]C(M)/\e}atio\slash∼=MatN(C(M)), soTqis not the endomorphism algebra of a vector bundle.
+Example 8.30. LetXbe a smooth irreducible variety over a field of characteristic p. Then
+D(X), the algebra of differential operators on X, is an Azumaya algebra with rank pdimX,
+which is not an endomorphism algebra of a vector bundle. Its center isZ=O(T∗X)F, the
+Frobenius twisted functions on T∗X.
+It is clear that if Ais an Azumaya algebra (say, over C) then for every central character
+χofA,A//a\}b∇acketle{tχ/a\}b∇acket∇i}htis the algebra Mat N(C) of complex NbyNmatrices, and every irreducible
+representation of Ahas dimension N.
+The following important result is due to M. Artin.
+Theorem 8.31. LetAbe a finitely generated (over C) polynomial identity (PI) algebra of
+degreeN(i.e. all the polynomial relations of the matrix algebra of s izeNare satisfied in
+A). ThenAis an Azumaya algebra if and only if every irreducible repres entation of Ahas
+dimension exactly N.
+Proof.See [Ar] Theorem 8.3. /square
+5IfMis not affine, one can define, in a standard manner, the notion of a sh eaf of Azumaya algebras on
+M.
+72Thus, by Theorem 8.27, for G=Sn, the rationalCherednik algebra H0,c(Sn,Cn) forc/\e}atio\slash= 0
+is an Azumaya algebra of degree n!. Indeed, this algebra is PI of degree n! because the clas-
+sical Dunkl representation embeds it into matrices of size n! overC(x1,...,xn,p1,...,pn)Sn.
+Let us say that χ∈Mis an Azumaya point if for some affine neighborhood Uofχthe
+localization of AtoUis an Azumaya algebra. Obviously, the set Az( M) of Azumaya points
+ofMis open.
+Now we come back to the study the space Mccorresponding to a symplectic reflection
+algebraH0,c.
+Theorem 8.32. The setAz(Mc)coincides with the set of smooth points of Mc.
+The proof of this theorem is given in the following two subsections.
+Corollary 8.33. IfG=SnandV=h⊕h∗,h=Cn(the rational Cherednik algebra case)
+thenMcis a smooth algebraic variety for c/\e}atio\slash= 0.
+Proof.Directly from the above theorem. /square
+8.11.Cohen-Macaulay property and homological dimension. ToproveTheorem8.32,
+we will need some commutative algebra tools. Let Zbe a finitely generated commutative
+algebra over Cwithout zero divisors. By Noether’s normalization lemma, there exist ele-
+mentsz1,...,zn∈Zwhich are algebraically independent, such that Zis a finitely generated
+module over C[z1,...,zn].
+Definition 8.34. The algebra Z(or the variety Spec Z) is said to be Cohen-Macaulay if Z
+is a locally free (=projective) module over C[z1,...,zn].6
+Remark 8.35. It was shown by Serre that if Zis locally free over C[z1,...,zn] for some
+choice ofz1,...,zn, then it happens for any choice of them (such that Zis finitely generated
+as a module over C[z1,...,zn]).
+Remark 8.36. Another definition of the Cohen-Macaulay property is that the dua lizing
+complexω•
+ZofZis concentrated in degree zero. We will not discuss this definition her e.
+It canbe shown that the Cohen-Macaulay property is stable under localization. Therefore,
+it makes sense to make the following definition.
+Definition 8.37. An algebraic variety Xis Cohen-Macaulay if the algebra of functions on
+every affine open set in Xis Cohen-Macaulay.
+LetZbe a finitely generated commutative algebra over Cwithout zero divisors, and let
+Mbe a finitely generated module over Z.
+Definition 8.38. Mis said to be Cohen-Macaulay if for some algebraically independent
+z1,...,zn∈Zsuch thatZis finitely generated over C[z1,...,zn],Mis locally free over
+C[z1,...,zn].
+Again, if this happens for some z1,...,zn, then it happens for any of them. We also
+note thatMcan be Cohen-Macaulay without Zbeing Cohen-Macaulay, and that Zis a
+Cohen-Macaulay algebra iff it is a Cohen-Macaulay module over itself.
+We will need the following standard properties of Cohen-Macaulay alg ebras and modules.
+6It was proved by Quillen that a locally free module over a polynomial alge bra is free; this is a difficult
+theorem, which will not be needed here.
+73Theorem 8.39. (i)LetZ1⊂Z2be a finite extension of finitely generated commutative
+C-algebras, without zero divisors, and Mbe a finitely generated module over Z2. Then
+Mis Cohen-Macaulay over Z2iff it is Cohen-Macaulay over Z1.
+(ii)Suppose that Zis the algebra of functions on a smooth affine variety. Then a Z-
+moduleMis Cohen-Macaulay if and only if it is projective.
+Proof.The proof can be found in the text book [Ei]. /square
+In particular, this shows that the algebra of functions on a smooth affine variety is Cohen-
+Macaulay. Algebras of functions on many singular varieties are also C ohen-Macaulay.
+Example 8.40. The algebra of regular functions on the cone xy=z2is Cohen-Macaulay.
+This algebra can be identified as C[a,b]Z2by lettingx=a2,y=b2andz=ab, where the Z2
+action is defined by a/mapsto→−a,b/mapsto→−b. It contains a subalgebra C[a2,b2], and as a module
+over this subalgebra, it is free of rank 2 with generators 1, ab.
+Example 8.41. Any irreducible affine algebraic curve is Cohen-Macaulay. For example , the
+algebra of regular functions on y2=x3is isomorphic to the subalgebra of C[t] spanned by
+1,t2,t3,.... It contains a subalgebra C[t2] and as a module over this subalgebra, it is free of
+rank 2 with generators 1, t3.
+Example 8.42. Consider thesubalgebra in C[x,y]spannedby1and xiyjwithi+j≥2. Itis
+a finite generated module over C[x2,y2], but not free. So this algebra is not Cohen-Macaulay.
+Another tool we will need is homological dimension. We will say that an a lgebraAhas
+homological dimension ≤dif any (left) A-moduleMhas a projective resolution of length
+≤d. The homological dimension of Ais the smallest integer having this property. If such
+an integer does not exist, Ais said to have infinite homological dimension.
+It is easy to show that the homological dimension of Ais≤dif and only if for any A-
+modulesM,None has Exti(M,N) = 0 fori > d. Also, the homological dimension clearly
+does not decrease under taking associated graded of the algebra under a positive filtration
+(this is clear from considering the spectral sequence attached to the filtration).
+It follows immediately from this definition that homological dimension is M orita invariant.
+Namely, recall that a Morita equivalence between algebras AandBis an equivalence of
+categoriesA-mod→B-mod. Such an equivalence maps projective modules to projective
+ones, since projectivity is a categorical property ( Pis projective if and only if the functor
+Hom(P,·)isexact). Thisimpliesthatif AandBareMoritaequivalentthentheirhomological
+dimensions are the same.
+Then we have the following important theorem.
+Theorem 8.43. The homological dimension of a commutative finitely generat edC-algebra
+Zis finite if and only if Zis regular, i.e. is the algebra of functions on a smooth affine
+variety.
+8.12.Proof of Theorem 8.32. First let us show that any smooth point χofMcis an
+Azumaya point. Since H0,c= End B0,cH0,ce= End Z0,c(H0,ce), it is sufficient to show that
+the coherent sheaf on Mccorresponding to the module H0,ceis a vector bundle near χ. By
+Theorem 8.39 (ii), for this it suffices to show that H0,ceis a Cohen-Macaulay Z0,c-module.
+To do so, first note that the statement is true for c= 0. Indeed, in this case the claim is
+thatSVis a Cohen-Macaulay module over ( SV)G. ButSVis a polynomial algebra, which
+is Cohen-Macaulay, so the result follows from Theorem 8.39, (i).
+74Now, we claim that if Z,Mare positively filtered and gr Mis a Cohen-Macaulay gr Z-
+module then Mis a Cohen-Macaulay Z-module. Indeed, let z1,...,znbe homogeneous
+algebraically independent elements of gr Zsuch that gr Zis a finite module over the subalge-
+bra generated by them. Let z′
+1,...,z′
+nbe their liftings to Z. Thenz′
+1,...,z′
+nare algebraically
+independent, andthemodule MoverC[z′
+1,...,z′
+n]isfinitelygeneratedand(locally)freesince
+so is the module gr MoverC[z1,...,zn].
+Recall now that gr H0,ce=SV, grZ0,c= (SV)G. Thus the c= 0 case implies the general
+case, and we are done.
+Now let us show that any Azumaya point of Mcis smooth. Let Ube an affine open set
+inMcconsisting of Azumaya points. Then the localization H0,c(U) :=H0,c⊗Z0,cOUis an
+Azumaya algebra. Moreover, for any χ∈U, the unique irreducible representation of H0,c(U)
+with central character χis the regular representation of G(since this holds for generic χby
+Proposition 8.24). This means that eis a rank 1 idempotent in H0,c(U)//a\}b∇acketle{tχ/a\}b∇acket∇i}htfor allχ. In
+particular, H0,c(U)eis a vector bundle on U. Thus the functor F:OU-mod→H0,c(U)-mod
+given by the formula F(Y) =H0,c(U)e⊗OUYis an equivalence of categories (the quasi-
+inverse functor is given by the formula F−1(N) =eN). ThusH0,c(U) is Morita equivalent
+toOU, and therefore their homological dimensions are the same.
+On the other hand, the homological dimension of H0,cis finite (in fact, it equals to dim V).
+To show this, note that by the Hilbert syzygies theorem, the homolo gical dimension of
+SVis dimV. Hence, so is the homological dimension of G⋉SV(as Ext∗
+G⋉SV(M,N) =
+Ext∗
+SV(M,N)G). Thus, since gr H0,c=G⋉SV, we get that H0,chas homological dimension
+≤dimV. Hence, the homological dimension of H0,c(U) is also≤dimV(as the homological
+dimension clearly does not increase under the localization). But H0,c(U) is Morita equivalent
+toOU, soOUhas a finite homological dimension. By Theorem 8.43, this implies that U
+consists of smooth points.
+Corollary 8.44. Az(Mc)is also the set of points at which the Poisson structure of Mcis
+symplectic.
+Proof.The variety Mcis symplectic outside of a subset of codimension 2, because so is M0.
+Thus the set Sof smooth points of Mcwhere the top exterior power of the Poisson bivector
+vanishes is of codimension ≥2. Since the top exterior power of the Poisson bivector is locally
+a regular function, this implies that Sis empty. Thus, every smooth point is symplectic, and
+the corollary follows from the theorem. /square
+8.13.Notes.Our exposition in this section follows Section 8 – Section 10 of [E4].
+759.Calogero-Moser spaces
+9.1.Hamiltonian reduction along an orbit. LetMbe an affine algebraic variety and
+Ga reductive algebraic group. Suppose Mis Poisson and the action of Gpreserves the
+Poisson structure. Let gbe the Lie algebra of Gandg∗the dual of g. Letµ:M→g∗be a
+moment map for this action (we assume it exists). It induces a map µ∗:Sg→C[M].
+LetObe a closed coadjoint orbit of G,IObe the ideal in Sgcorresponding to O, and let
+JObe the ideal in C[M] generated by µ∗(IO). ThenJG
+Ois a Poisson ideal in C[M]G, and
+A=C[M]G/JG
+Ois a Poisson algebra.
+Geometrically, Spec( A) =µ−1(O)/G(categorical quotient). It can also be written as
+µ−1(z)/Gz, wherez∈OandGzis the stabilizer of zinG.
+Definition 9.1. Thescheme µ−1(O)/Giscalledthe Hamiltonian reduction of Mwith respect
+toGalongO. We will denote by R(M,G,O).
+The following proposition is standard.
+Proposition 9.2. IfMis a symplectic variety and the action of Gonµ−1(O)is free, then
+R(M,G,O)is a symplectic variety, of dimension dim(M)−2dim(G)+dim(O).
+9.2.The Calogero-Moser space. LetM=T∗Matn(C), andG= PGLn(C) (sog=
+sln(C)). Using the trace form we can identify g∗withg, andMwith Matn(C)⊕Matn(C).
+Then a moment map is given by the formula µ(X,Y) = [X,Y], forX,Y∈Matn(C).
+LetObe the orbit of the matrix diag( −1,−1,...,−1,n−1), i.e. the set of traceless
+matricesTsuch thatT+1 has rank 1.
+Definition 9.3 (Kazhdan, Kostant, Sternberg, [KKS]) .The schemeCn:=R(M,G,O) is
+calledthe Calogero-Moser space .
+Proposition 9.4. The action of Gonµ−1(O)is free, and thus (by Proposition 9.2) Cnis a
+smooth symplectic variety (of dimension 2n).
+Proof.It suffices to show that if X,Yare such that XY−YX+1 has rank 1, then ( X,Y)
+is an irreducible set of matrices. Indeed, in this case, by Schur’s lemm a, ifB∈GLnis such
+thatBX=XBandBY=YBthenBis a scalar, so the stabilizer of ( X,Y) in PGL nis
+trivial.
+To show this, assume that W /\e}atio\slash= 0 is an invariant subspace of X,Y. In this case, the
+eigenvalues of [ X,Y] onWare a subcollection of the collection of n−1 copies of−1 and
+one copy of n−1. The sum of the elements of this subcollection must be zero, since it is the
+trace of [X,Y] onW. But then the subcollection must be the entire collection, so W=Cn,
+as desired. /square
+Thus,Cnis the space of conjugacy classes of pairs of n×nmatrices (X,Y) such that the
+matrixXY−YX+1 has rank 1.
+In fact, one also has the following more complicated theorem.
+Theorem 9.5 (G. Wilson, [Wi]) .The Calogero-Moser space is connected.
+We will give a proof of this theorem later, in Subsection 9.4.
+769.3.The Calogero-Moser integrable system. LetMbe a symplectic variety, and let
+H1,...,Hnbe regular functions on Msuch that{Hi,Hj}= 0 andHi’s are algebraically
+independent everywhere. Assume that Mcarries a symplectic action of a reductive algebraic
+groupGwith moment map µ:M→g∗, which preserves the functions Hi, and let Obe
+a coadjoint orbit of G. Assume that Gacts freely on µ−1(O), and so the Calogero-Moser
+spaceR(M,G,O) is symplectic. The functions Hidescend to R(M,G,O). If they are still
+algebraically independent and n= dimR(M,G,O)/2, then we get an integrable system on
+R(M,G,O).
+A vivid example of this is the Kazhdan-Kostant-Sternberg constru ction of the Calogero-
+Moser system. In this case M=T∗Matn(C) (regarded as the set of pairs of matrices ( X,Y)
+as in Section 9.2), with the usual symplectic form ω= Tr(dY∧dX). LetHi= Tr(Yi),
+i= 1,...,n. LetG= PGLn(C) act onMby conjugation, and let Obe the coadjoint
+orbit ofGconsidered in Subsection 9.2. Then the system H1,...,Hndescends to a system
+of functions in involution on R(M,G,O), which is the Calogero-Moser space Cn. Since
+this space is 2 n-dimensional, H1,...,Hnform an integrable system on Cn. It is called the
+(rational) Calogero-Moser system .
+The Calogero-Moser flow is, by definition, the Hamiltonian flow on Cndefined by the
+Hamiltonian H=H2= Tr(Y2). Thus this flow is integrable, in the sense that it can be
+included in an integrable system. In particular, its solutions can be fo und in quadratures
+using the inductive procedure of reduction of order. However (as often happens with systems
+obtained by reduction), solutions can also be found by a much simpler procedure, since
+they can be found already on the “non-reduced” space M: indeed, onMthe Calogero-
+Moser flow is just the motion of a free particle in the space of matrice s, so it has the form
+gt(X,Y) = (X+ 2Yt,Y). The same formula is valid on Cn. In fact, we can use the same
+method to compute the flows corresponding to all the Hamiltonians Hi= Tr(Yi),i∈N:
+these flows are given by the formulas
+g(i)
+t(X,Y) = (X+iYi−1t,Y).
+Let us write the Calogero-Moser system explicitly in coordinates. To do so, we first need
+to introduce local coordinates on the Calogero-Moser space Cn.
+To this end, let us restrict our attention to the open set Un⊂ Cnwhich consists of
+conjugacyclasses ofthosepairs( X,Y)forwhich thematrix Xisdiagonalizable, withdistinct
+eigenvalues; by Wilson’s Theorem 9.5, this open set is dense in Cn.
+A pointP∈Unmay be represented by a pair ( X,Y) such that X= diag(x1,...,xn),
+xi/\e}atio\slash=xj. In this case, the entries of T:=XY−YXare (xi−xj)yij. In particular, the
+diagonal entries are zero. Since the matrix T+ 1 has rank 1, its entries κijhave the form
+aibjfor some numbers ai,bj. On the other hand, κii= 1, sobj=a−1
+jand henceκij=aia−1
+j.
+By conjugating ( X,Y) by the matrix diag( a1,...,an), we can reduce to the situation when
+ai= 1, soκij= 1. Hence the matrix Thas entries 1−δij(zeros on the diagonal, ones off
+the diagonal). Moreover, the representative of Pwith diagonal XandTas above is unique
+up to the action of the symmetric group Sn. Finally, we have ( xi−xj)yij= 1 fori/\e}atio\slash=j, so
+the entries of the matrix Yareyij= 1/(xi−xj) ifi/\e}atio\slash=j. On the other hand, the diagonal
+entriesyiiofYare unconstrained. Thus we have obtained the following result.
+Proposition 9.6. LetCn
+regbe the open set of (x1,...,xn)∈Cnsuch thatxi/\e}atio\slash=xjfori/\e}atio\slash=j.
+Then there exists an isomorphism of algebraic varieties ξ:T∗(Cn
+reg/Sn)→Ungiven by the
+77formula
+(x1,...,xn,p1,...,pn)/mapsto→(X,Y),
+whereX= diag(x1,...,xn), andY=Y(x,p) := (yij),
+yij=1
+xi−xj,i/\e}atio\slash=j, yii=pi.
+In fact, we have a stronger result:
+Proposition 9.7. ξis an isomorphism of symplectic varieties (where the cotang ent bundle
+is equipped with the usual symplectic structure).
+For the proof of Proposition 9.7, we will need the following general an d important but
+easy theorem.
+Theorem 9.8 (The necklace bracket formula) .Leta1,...,arandb1,...,bsbe eitherXor
+Y. Then onMwe have
+{Tr(a1···ar),Tr(b1···bs)}=/summationdisplay
+(i,j):ai=Y,bj=XTr(ai+1···ara1···ai−1bj+1···bsb1···bj−1)−
+/summationdisplay
+(i,j):ai=X,bj=YTr(ai+1···ara1···ai−1bj+1···bsb1···bj−1).
+Proof of Proposition 9.7. Letak= Tr(Xk),bk= Tr(XkY). It is easy to check using the
+necklace bracket formula that on Mwe have
+{am,ak}= 0,{bm,ak}=kam+k−1,{bm,bk}= (k−m)bm+k−1.
+On the other hand, ξ∗ak=/summationtextxk
+i,ξ∗bk=/summationtextxk
+ipi. Thus we see that
+{f,g}={ξ∗f,ξ∗g},
+wheref,gare eitherakorbk. But the functions ak,bk,k= 0,...,n−1, form a local
+coordinate system near a generic point of Un, so we are done. /square
+Now let us write the Hamiltonian of the Calogero-Moser system in coor dinates. It has the
+form
+(9.1) H= Tr(Y(x,p)2) =/summationdisplay
+ip2
+i−/summationdisplay
+i/\e}atio\slash=j1
+(xi−xj)2.
+Thus the Calogero-Moser Hamiltonian describes the motion of a syst em ofnparticles on the
+line with interaction potential −1/x2, which we considered in Section 2.
+Now we finally see the usefulness of the Hamiltonian reduction proced ure. The point is
+that it is not clear at all from formula (9.1) why the Calogero-Moser H amiltonian should be
+completely integrable. However, our reduction procedure implies th e complete integrability
+ofH, and gives an explicit formula for the first integrals:7
+Hi= Tr(Y(x,p)i).
+Moreover, this procedure immediately gives us an explicit solution of t he system. Namely,
+assume that x(t),p(t) is the solution with initial condition x(0),p(0). Let ( X0,Y0) =
+7Thus, for type Awe have two methods of proving the integrability of the Calogero-Mo ser system - one
+using Dunkl operators and one using Hamiltonian reduction.
+78ξ(x(0),p(0)). Then xi(t) are the eigenvalues of the matrix Xt:=X0+ 2tY0, andpi(t) =
+x′
+i(t)/2.
+9.4.Proof of Wilson’s theorem. Let us now give a proof of Theorem 9.5.
+We have already shown that all components of Cnare smooth and have dimension 2 n.
+Also, we know that there is at least one component (the closure of Un), and that the other
+components, if they exist, do not contain pairs ( X,Y) in which Xis regular semisimple.
+This means that these components are contained in the hypersurf ace ∆(X) = 0, where
+∆(X) stands for the discriminant of X(i.e., ∆(X) :=/producttext
+i/\e}atio\slash=j(xi−xj), wherexiare the
+eigenvalues of X).
+Thus, to show that such additional components don’t in fact exist, it suffices to show that
+the dimension of the subscheme Cn(0) cut out inCnby the equation ∆( X) = 0 is 2n−1.
+To do so, first notice that the condition rank([ X,Y] +1) = 1 is equivalent to the equa-
+tion∧2([X,Y] + 1) = 0; thus, the latter can be used as the equation defining Cninside
+T∗Matn/PGLn.
+DefineC0
+n:= Spec(grO(Cn)) to be the degeneration of Cn(we use the filtration on O(Cn)
+defined by deg( X) = 0, deg( Y) = 1). ThenC0
+nis a closed subscheme in the scheme /tildewideC0
+ncut
+out by the equations ∧2([X,Y]) = 0 inT∗Matn/PGLn.
+Let (/tildewideC0
+n)redbe the reduced part of /tildewideC0
+n. Then (/tildewideC0
+n)redcoincides with the categorical quotient
+{(X,Y)|rank([X,Y])≤1}/PGLn.
+Our proof is based on the following proposition.
+Proposition 9.9. The categorical quotient {(X,Y)|rank([X,Y])≤1}/PGLncoincides with
+the categorical quotient {(X,Y)|[X,Y] = 0}/PGLn.
+Proof.It is clear that{(X,Y)|[X,Y] = 0}/PGLnis contained in{(X,Y)|rank([X,Y])≤
+1}/PGLn. For the proof of the opposite inclusion we need to show that any re gular function
+on{(X,Y)|rank([X,Y])≤1}/PGLniscompletelydeterminedbyitsvaluesonthesubvariety
+{(X,Y)|[X,Y] = 0}/PGLn, i.e. that any invariant polynomial on the set of pairs of matrices
+with commutator of rank at most 1 is completely determined by its valu es on pairs of
+commuting matrices. To this end, we need the following lemma from linea r algebra.
+Lemma 9.10. IfA,Bare square matrices such that [A,B]has rank≤1, then there exists
+a basis in which both A,Bare upper triangular.
+Proof.Without loss of generality, we can assume ker A/\e}atio\slash= 0 (by replacing AwithA−λif
+needed) and that A/\e}atio\slash= 0. It suffices to show that there exists a proper nonzero subspa ce
+invariant under A,B; then the statement will follow by induction in dimension.
+LetC= [A,B] and suppose rank C= 1 (since the case rank C= 0 is trivial). If ker A⊂
+kerC, then kerAisB-invariant: if Av= 0 thenABv=BAv+Cv= 0. Thus ker Ais the
+requiredsubspace. Ifker A/⋉otsubseteqlkerC,thenthereexistsavector vsuchthatAv= 0butCv/\e}atio\slash= 0.
+SoABv=Cv/\e}atio\slash= 0. Thus Im C⊂ImA. So ImAisB-invariant: BAv=ABv+Cv∈ImA.
+So ImAis the required subspace.
+This proves the lemma. /square
+Now we are ready to prove Proposition 9.9. By the fundamental the orem of invariant
+theory, the ring of invariants of XandYis generated by traces of words of XandY:
+Tr(w(X,Y)). IfXandYare upper triangular with eigenvalues xi,yi, then any such trace
+79has the form/summationtextxm
+iyr
+i, i.e. coincides with the value of the corresponding invariant on the
+diagonal parts Xdiag,YdiagofXandY, which commute. The proposition is proved. /square
+We will also need the following proposition:
+Proposition 9.11. The categorical quotient {(X,Y)|[X,Y] = 0}/PGLnis isomorphic to
+(Cn×Cn)/Sn, i.e. its function algebra is C[x1,...,xn,y1,...,yn]Sn.
+Proof.Restriction to diagonal matrices defines a homomorphism
+ξ:O({(X,Y)|[X,Y] = 0}/PGLn)→C[x1,...,xn,y1,...,yn]Sn.
+Since (as explained in the proof of Proposition 9.9), any invariant poly nomial of entries of
+commuting matrices is determined by its values on diagonal matrices, this map is injective.
+Also,ξ(Tr(XmYr)) =/summationtextxm
+iyr
+i, wherexi,yiare the eigenvalues of XandY.
+Now we use the following well known theorem of H. Weyl (from his book “Classical
+groups”).
+Theorem 9.12. LetBbe an algebra over C. Then the algebra SnBis generated by elements
+of the form
+b⊗1⊗···⊗1+1⊗b⊗···⊗1+···+1⊗1⊗···⊗b.
+Proof.SinceSnBis linearly spanned by elements of the form x⊗···⊗x,x∈B, it suffices
+to prove the theorem in the special case B=C[x]. In this case, the result is simply the fact
+that the ring of symmetric functions is generated by power sums, w hich is well known. /square
+Corollary 9.13. The ring C[x1,...,xn,y1,...,yn]Snis generated by the polynomials/summationtextxm
+iyr
+i
+form,r≥0,m+r>0.
+Proof.Apply Theorem 9.12 in the case B=C[x,y]. /square
+Corollary 9.13 implies that ξis surjective. Proposition 9.11 is proved. /square
+Now we are ready to prove Wilson’s theorem. Let Cn(0)0be the degeneration of Cn(0), i.e.
+the subscheme of C0
+ncut out by the equation ∆( X) = 0. According to Propositions 9.9 and
+9.11, the reduced part ( Cn(0)0)redis contained in the hypersurface in ( Cn×Cn)/Sncut out
+by the equation/producttext
+i<j(xi−xj) = 0. This hypersurface has dimension 2 n−1, so we are done.
+9.5.The Gan-Ginzburg theorem. Let Comm( n) bethe commuting scheme defined in
+T∗Matn= Matn×Matnby the equations [ X,Y] = 0,X,Y∈Matn. LetG= PGLn, and
+consider the categorical quotient Comm( n)/G(i.e., the Hamiltonian reduction µ−1(0)/Gof
+T∗Matnby the action of G), whose algebra of regular functions is A=C[Comm(n)]G.
+It is not known whether the commuting scheme Comm( n) is reduced (i.e. whether the
+corresponding ideal is a radical ideal); this is a well known open proble m. The underlying
+variety is irreducible (as was shown by Gerstenhaber [Ge1]), but ve ry singular, and has a
+very complicated structure. However, we have the following result .
+Theorem 9.14 (Gan, Ginzburg, [GG]) .Comm(n)/Gis reduced, and isomorphic to C2n/Sn.
+ThusA=C[x1,...,xn,y1,...,yn]Sn. The Poisson bracket on this algebra is induced from
+the standard symplectic structure on C2n.
+80Sketch of the proof. Look at the almost commuting variety Mn⊂gln×gln×Cn×(Cn)∗
+defined by
+Mn={(X,Y,v,f)|[X,Y]+v⊗f= 0}.
+Gan and Ginzburg proved the following result.
+Theorem 9.15.Mnis a complete intersection. It has n+1irreducible components denoted
+byMi
+n, labeled by i= dimC/a\}b∇acketle{tX,Y/a\}b∇acket∇i}htv. Also,Mnis generically reduced.
+SinceMnis generically reduced and is a complete intersection, by a standard r esult of
+commutative algebra it is reduced. Thus C[Mn] has no nonzero nilpotents. This implies
+C[Mn]Ghas no nonzero nilpotents.
+However, it is easy to show that the algebra C[Mn]Gis isomorphic to the algebra of
+invariant polynomials of entries of XandYmodulo the “rank 1” relation ∧2[X,Y] = 0. By
+a scheme-theoretic version of Proposition 9.9 (proved in [EG]), the la tter is isomorphic to
+A. This implies the theorem (the statement about Poisson structure s is checked directly in
+coordinates on the open part where Xis regular semisimple). /square
+9.6.The space McforSnand the Calogero-Moser space. LetH0,c=H0,c[Sn,V] be
+the symplectic reflection algebra of the symmetric group Snand spaceV=h⊕h∗, where
+h=Cn(i.e., the rational Cherednik algebra H0,c(Sn,h)). LetMc= SpecB0,c[Sn,V] be the
+Calogero-Moser space defined in Section 8.5. It is a symplectic variet y forc/\e}atio\slash= 0.
+Theorem 9.16. Forc/\e}atio\slash= 0the space Mcis isomorphic to the Calogero-Moser space Cnas a
+symplectic variety.
+Proof.To prove the theorem, we will first construct a map f:Mc→Cn, and then prove that
+fis an isomorphism.
+Without loss of generality, we may assume that c= 1. As we have shown before, the
+algebraH0,cis an Azumaya algebra. Therefore, Mccan be regarded as the moduli space of
+irreducible representations of H0,c.
+LetE∈Mcbe an irreducible representation of H0,c. We have seen before that Ehas
+dimensionn! and is isomorphic to the regular representation as a representat ion ofSn. Let
+Sn−1⊂Snbe the subgroup which preserves the element 1. Then the space of invariants
+ESn−1has dimension n. On this space we have operators X,Y:ESn−1→ESn−1obtained
+by restriction of the operators x1,y1onEto the subspace of invariants. We have
+[X,Y] =T:=n/summationdisplay
+i=2s1i.
+Let us now calculate the right hand side of this equation explicitly. Let ebe the symmetrizer
+ofSn−1. Let us realize the regular representation EofSnasC[Sn] with action of Snby
+left multiplication. Then v1=e,v2=es12,...,vn=es1nis a basis of ESn−1. The element
+Tcommutes with e, so we have
+Tvi=/summationdisplay
+j/\e}atio\slash=ivj.
+This means that T+1 has rank 1, and hence the pair ( X,Y) defines a point on the Calogero-
+Moser spaceCn.8
+8Note that the pair ( X,Y) is well defined only up to conjugation, because the representatio nEis well
+defined only up to an isomorphism.
+81We now set ( X,Y) =f(E). It is clear that f:Mc→Cnis a regular map. So it remains to
+show thatfis an isomorphism. This is equivalent to showing that the correspondin g map
+of function algebras f∗:O(Cn)→B0,cis an isomorphism.
+Letuscalculate fandf∗moreexplicitly. Todoso, consider theopenset UinMcconsisting
+of representations in which xi−xjacts invertibly. These are exactly the representations that
+are obtained by restricting representations of Sn⋉C[x1,...,xn,p1,...,pn,δ(x)−1] using the
+classical Dunkl embedding. Thus representations E∈Uareof theform E=Eλ,µ(λ,µ∈Cn,
+andλhas distinct coordinates), where Eλ,µis the space of complex valued functions on the
+orbitOλ,µ⊂C2n, with the following action of H0,c:
+(xiF)(a,b) =aiF(a,b),(yiF)(a,b) =biF(a,b)+/summationdisplay
+j/\e}atio\slash=i(sijF)(a,b)
+ai−aj.
+(the group Snacts by permutations).
+Now let us consider the space ESn−1
+λ,µ. A basis of this space is formed by characteristic
+functions of Sn−1-orbits on Oλ,µ. Using the above presentation, it is straightforward to
+calculate the matrices of the operators XandYin this basis:
+X= diag(λ1,...,λn),
+and
+Yij=µiifj=i, Yij=1
+λi−λjifj/\e}atio\slash=i.
+This shows that finduces an isomorphism f|U:U→Un, whereUnis the subset ofCn
+consisting of pairs ( X,Y) for which Xhas distinct eigenvalues.
+From this presentation, it is straigtforward that f∗(Tr(Xp)) =xp
+1+···+xp
+nfor every
+positive integer p. Also,fcommutes with the natural SL 2(C)-action on McandCn(by
+(X,Y)→(aX+bY,cX+dY)), so we also get f∗(Tr(Yp)) =yp
+1+···+yp
+n, and
+f∗(Tr(XpY)) =1
+p+1p/summationdisplay
+m=0/summationdisplay
+ixm
+iyixp−m
+i.
+Now, using the necklace bracket formula on Cnand the commutation relations of H0,c, we
+find, by a direct computation, that f∗preserves Poisson bracket on the elements Tr( Xp),
+Tr(XqY). But these elements are alocal coordinate system near a generic point, so it follows
+thatfis a Poisson map. Since the algebra B0,cis Poisson generated by/summationtextxp
+iand/summationtextyp
+ifor
+allp, we get that f∗is a surjective map.
+Also,f∗is injective. Indeed, by Wilson’s theorem the Calogero-Moser space is connected,
+and hence the algebra O(Cn) has no zero divisors, while Cnhas the same dimension as Mc.
+This proves that f∗is an isomorphism, so fis an isomorphism. /square
+9.7.The Hilbert scheme Hilbn(C2)and the Calogero-Moser space. The Hilbert
+scheme Hilb n(C2) is defined to be
+Hilbn(C2) ={idealsI⊂C[x,y]|codimI=n}
+={(E,v)|Eis aC[x,y]-module of dimension n, vis a cyclic vector of E}.
+The second equality can be easily seen from the short exact sequen ce
+0→I→C[x,y]→E→0.
+82LetS(n)C2=C2×···×C2
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+ntimes/Sn, whereSnacts by permutation. We have a natural map
+Hilbn(C2)→S(n)C2which sends every ideal Ito its zero set (with multiplicities). This map
+is called the Hilbert-Chow map .
+Theorem 9.17 (Fogarty, [F]) .(i) Hilbn(C2)is a smooth quasiprojective variety.
+(ii)The Hilbert-Chow map Hilbn(C2)→S(n)C2is projective. It is a resolution of singu-
+larities.
+Proof.Proof can be found in [Na]. /square
+Now consider the Calogero-Moser space Cndefined in Section 9.2.
+Theorem 9.18 (see [Na]) .The Hilbert Scheme Hilbn(C2)isC∞-diffeomorphic to Cn.
+Remark 9.19. More precisely there exists a family of algebraic varieties over A1, sayXt,
+t∈A1, such that Xtis isomorphic toCnift/\e}atio\slash= 0 andX0is the Hilbert scheme; and also if we
+denote byXtthe deformation of C2n/Sninto the Calogero-Moser space, then there exists
+a mapft:Xt։Xt, such that for t/\e}atio\slash= 0,ftis an isomorphism and f0is the Hilbert-Chow
+map.
+Remark 9.20. Consider the action of G= PGLnonT∗Matn. As we have discussed, the
+corresponding moment map is µ(X,Y) = [X,Y], soµ−1(0) ={(X,Y)|[X,Y] = 0}is the
+commuting variety. We can consider two kinds of quotient µ−1(0)/G(i.e., of Hamiltonian
+reduction):
+(1) The categorical quotient, i.e.,
+Spec(C[xij,yij]//a\}b∇acketle{t[X,Y] = 0/a\}b∇acket∇i}ht)G∼=(Cn×Cn)/Sn.
+It is a reduced (by Gan-Ginzburg Theorem 9.14), affine but singular v ariety.
+(2) The GIT quotient, in which the stability condition is that there exis ts a cyclic vector
+forX,Y. This quotient is Hilb n(C2), which is smooth but not affine.
+Both of these reductions are degenerations of the reduction alon g the orbit of matrices T
+such thatT+1 has rank 1, which yields the space Cn. This explains why Theorem 9.18 and
+the results mentioned in Remark 9.19 are natural to expect.
+9.8.The cohomology of Cn.We also have the following result describing the cohomology
+ofCn(and hence, by Theorem 9.18, of Hilb n(C2)). Define the age filtration for the symmetric
+groupSnby setting
+age(transposition) = 1 .
+Then one can show that for any σ∈Sn, age(σ) = rank(1−σ)|reflection representation . It is easy
+to see that 0≤age≤n−1. Notice also that the age filtration can be defined for any Coxeter
+group.
+Theorem 9.21 (Lehn-Sorger,Vasserot) .Thecohomologyring H∗(Cn,C)livesin evendegrees
+only and is isomorphic to gr(Center(C[Sn]))under the age filtration (with doubled degrees).
+Proof.Let us sketch a noncommutative-algebraic proof of this theorem, given in [EG]. This
+proof is based on the following result.
+83Theorem 9.22 (Nest-Tsygan, [NT]) .IfMis an affine symplectic variety and Ais a quan-
+tization ofM, then
+HH∗(A[/pla⋉ckover2pi1−1],A[/pla⋉ckover2pi1−1])∼=H∗(M,C((/pla⋉ckover2pi1)))
+as an algebra over C((/pla⋉ckover2pi1)).
+Now, we know that the algebra Bt,cis a quantization of Cn. Therefore by above theorem,
+the cohomology algebra of Cnis the cohomology of Bt,c(for generic t). ButBt,cis Morita
+equivalent to Ht,c, so this cohomology is the same as the Hochschild cohomology of Ht,c.
+However, the latter can be computed by using that Ht,cis given by generators and relations
+(by producing explicit representatives of cohomology classes and c omputing their product),
+which gives the result. /square
+9.9.Notes.Sections 9.1–9.6 follow Section 1, 2, 4 of [E4]; the parts about the Hilbe rt
+scheme and its relation to Calogero-Moser spaces follow the book [Na ] (see also [GS]); the
+other parts follow the paper [EG].
+8410.Quantization of Claogero-Moser spaces
+10.1.Quantum moment maps and quantum Hamiltonian reduction. Now we would
+like to quantize the notion of a moment map. Let gbe a Lie algebra, and Abe an associative
+algebra equipped with a g-action, i.e. a Lie algebra map φ:g→DerA. A quantum moment
+map for (A,φ) is an associative algebra homomorphism µ:U(g)→Asuch that for any
+a∈g,b∈Aone has [µ(a),b] =φ(a)b.
+The space of g-invariants Ag, i.e. elements b∈Asuch that [µ(a),b] = 0 for all a∈g, is
+a subalgebra of A. LetJ⊂Abe the left ideal generated by µ(a),a∈g. ThenJis not
+a 2-sided ideal, but Jg:=J∩Agis a 2-sided ideal in Ag. Indeed, let c∈Ag, andb∈Jg,
+b=/summationtext
+ibiµ(ai),bi∈A,ai∈g. Thenbc=/summationtextbiµ(ai)c=/summationtextbicµ(ai)∈Jg.
+Thus, the algebra A//g:=Ag/Jgis an associative algebra, which is called the quantum
+Hamiltonian reduction of Awith respect to the quantum moment map µ.
+10.2.The Levasseur-Stafford theorem. In general, similarly to the classical case, it is
+rather difficult to compute the quantum reduction A//g. For example, in this subsection
+we will describe A//gin the case when A=D(g) is the algebra of differential operators
+on a reductive Lie algebra g, andgacts onAthrough the adjoint action on itself. This
+description is a very nontrivial result of Levasseur and Stafford.
+Lethbe a Cartan subalgebra of g, andWthe Weyl group of ( g,h). Lethregdenote the set
+ofregularpointsin h,i.e. thecomplement ofthereflectionhyperplanes. Todescribe D(g)//g,
+we will construct a homomorphism HC : D(g)g→D(h)W, called the Harish-Chandra homo-
+morphism (as it was first constructed by Harish-Chandra). Recall that we have the classical
+Harish-Chandra isomorphism ζ:C[g]g→C[h]W, defined simply by restricting g-invariant
+functions on gto the Cartan subalgebra h. Using this isomorphism, we can define an action
+ofD(g)gonC[h]W, which is clearly given by W-invariant differential operators. However,
+these operators will, in general, have poles on the reflection hyperp lanes. Thus we get a
+homomorphism HC′:D(g)g→D(hreg)W.
+The homomorphism HC′is called the radial part homomorphism, as for example for
+g=su(2) it computes the radial parts of rotationally invariant differentia l operators on R3
+in spherical coordinates. This homomorphism is not yet what we want , since it does not
+actually land inD(h)W(the radial parts have poles).
+Thus we define the Harish-Chandra homomorphism by twisting HC′by the discriminant
+δ(x) =/producttext
+α>0(α,x) (x∈h, andαruns over positive roots of g):
+HC(D) :=δ◦HC′(D)◦δ−1∈D(hreg)W.
+Theorem 10.1. (i) (Harish-Chandra, [HC])For any reductive g,HClands inD(h)W⊂
+D(hreg)W.
+(ii) (Levasseur-Stafford [LS])The homomorphism HCdefines an isomorphism D(g)//g=
+D(h)W.
+Remark 10.2. (1) Part (i) of the theorem says that the poles magically disappear a fter
+conjugation by δ.
+(2) Both parts of this theorem are quite nontrivial. The first part w as proved by Harish-
+Chandra using analytic methods, and the second part by Levasseu r and Stafford
+using the theory of D-modules.
+85In the case g=gln, Theorem 10.1 is a quantum analog of Theorem 9.14. The remaining
+part of this subsection is devoted to the proof of Theorem 10.1 in th is special case, using
+Theorem 9.14.
+We start the proof with the following proposition, valid for any reduc tive Lie algebra.
+Proposition 10.3. IfD∈(Sg)gis a differential operator with constant coefficients, then
+HC(D)is theW-invariant differential operator with constant coefficients onh, obtained from
+Dvia the classical Harish-Chandra isomorphism η: (Sg)g→(Sh)W.
+Proof.Without loss of generality, we may assume that gis simple.
+Lemma 10.4. LetDbe the Laplacian ∆gofg, corresponding to an invariant form. Then
+HC(D)is the Laplacian ∆h.
+Proof.Let us calculate HC′(D). We have
+D=r/summationdisplay
+i=1∂2
+xi+2/summationdisplay
+α>0∂fα∂eα,
+wherexiis an orthonormal basis of h, andeα,fαare root elements such that ( eα,fα) = 1.
+Thus ifF(x) is ag-invariant function on g, then we get
+(DF)|h=r/summationdisplay
+i=1∂2
+xi(F|h)+2/summationdisplay
+α>0(∂fα∂eαF)|h.
+Now letx∈h, and consider ( ∂fα∂eαF)(x). We have
+(∂fα∂eαF)(x) =∂s∂t|s=t=0F(x+tfα+seα).
+On the other hand, we have
+Ad(esα(x)−1eα)(x+tfα+seα) =x+tfα+tsα(x)−1hα+···,
+wherehα= [eα,fα]. Hence,∂s∂t|s=t=0F(x+tfα+seα) =α(x)−1(∂hαF)(x).This implies that
+HC′(D)F(x) = ∆hF(x)+2/summationdisplay
+α>0α(x)−1∂hαF(x).
+Nowthestatement oftheLemmareducestotheidentity δ−1◦∆h◦δ= ∆h+2/summationtext
+α>0α(x)−1∂hα.
+This identity follows immediately from the identity ∆ hδ= 0.To prove the latter, it suffices
+to note that δis the lowest degree nonzero polynomial on h, which is antisymmetric under
+the action of W. The lemma is proved. /square
+Now letDbe any element of ( Sg)g⊂D(g)gof degreed(operator with constant coeffi-
+cients). It is obvious that the leading order part of the operator H C(D) is the operator η(D)
+with constant coefficients, whose symbol is just the restriction of the symbol of Dfromg∗
+toh∗. Our job is to show that in fact HC( D) =η(D). To do so, denote by Ythe difference
+HC(D)−η(D). Assume Y/\e}atio\slash= 0. By Lemma 10.4, the operator HC( D) commutes with ∆ h.
+Therefore, so does Y. AlsoYhas homogeneity degree dbut orderm≤d−1. LetS(x,p) be
+the symbol of Y(x∈h,p∈h∗). ThenSis a homogeneous function of homogeneity degree
+dunder the transformations x→t−1x,p→tp, polynomial in pof degreem. From these
+properties of Sit is clear that Sis not a polynomial (its degree in xism−d <0). On
+the other hand, since Ycommutes with ∆ h, the Poisson bracket of Swithp2is zero. Thus
+Proposition 10.3 follows from Lemma 2.22. /square
+86Now we continue the proof of Theorem 10.1. Consider the filtration o nD(g) in which
+deg(g) = 1deg( g∗) = 0 (the order filtration), and the associated graded map grHC : C[g×
+g∗]g→C[hreg×h∗]W, which attaches to every differential operator the symbol of its r adial
+part. It is easy to see that this map is just the restriction map to h⊕h∗⊂g⊕g∗, so it
+actually lands in C[h⊕h∗]W.
+Moreover, grHC is a map ontoC[h⊕h∗]W. Indeed, grHC is a Poisson map, so the
+surjectivity follows from the following Lemma.
+Lemma 10.5. C[h⊕h∗]Wis generated as a Poisson algebra by C[h]WandC[h∗]W, i.e. by
+functionsfm=/summationtextxm
+iandf∗
+m=/summationtextpm
+i,m≥1.
+Proof.We have{f∗
+m,fr}=mr/summationtextxr−1
+ipm−1
+i. Thus the result follows from Corollary 9.13. /square
+LetK0be the kernel of grHC. Then by Theorem 9.14, K0is the ideal of the commuting
+scheme Comm( g)/G.
+Now consider the kernel Kof the homomorphism HC. It is easy to see that K⊃Jg,
+so gr(K)⊃gr(J)g. On the other hand, since K0is the ideal of the commuting scheme,
+we clearly have gr( J)g⊃K0, andK0⊃grK. This implies that K0= grK= gr(J)g, and
+K=Jg.
+ItremainstoshowthatImHC = D(h)W. SincegrK=K0, wehavegrImHC = C[h⊕h∗]W.
+Therefore, to finish the proof of the Harish-Chandra and Levass eur-Stafford theorems, it
+suffices to prove the following proposition.
+Proposition 10.6. ImHC⊃D(h)W.
+Proof.We will use the following Lemma.
+Lemma 10.7 (N. Wallach, [Wa]) .D(h)Wis generated as an algebra by W-invariant func-
+tions andW-invariant differential operators with constant coefficient s.
+Proof.The lemma follows by taking associated graded algebras from Lemma 1 0.5. /square
+Remark 10.8. Levasseur and Stafford showed [LS] that this lemma is valid for any fin ite
+groupWacting on a finite dimensional vector space h. However, the above proof does not
+apply, since, as explained in [Wa], Lemma 10.5 fails for many groups W, including Weyl
+groups of exceptional Lie algebras E6,E7,E8(in fact it even fails for the cyclic group of
+order>2 acting on a 1-dimensional space!). The general proof is more comp licated and uses
+some results in noncommutative algebra.
+Lemma 10.7 and Proposition 10.3 imply Proposition 10.6. /square
+Thus, Theorem 10.1 is proved.
+10.3.Corollaries of Theorem 10.1. LetgRbe the compact form of g, andOa regular
+coadjoint orbit in g∗
+R. Consider the map
+ψO:h→C, ψO(x) =/integraldisplay
+Oe(b,x)db,x∈h,
+where dbis the measure on the orbit coming from the Kirillov-Kostant symplect ic structure.
+87Theorem 10.9 (Harish-Chandra formula) .For a regular element x∈h, we have
+ψO(x) =δ−1(x)/summationdisplay
+w∈W(−1)ℓ(w)e(wλ,x),
+whereλis the intersection of Owith the dominant chamber in the dual Cartan subalgebra
+h∗
+R⊂g∗
+R, andℓ(w)is the length of an element w∈W.
+Proof.TakeD∈(Sg)g. Thenδ(x)ψOis an eigenfunction of HC( D) =η(D)∈(Sh)Wwith
+eigenvalueχO(D), whereχO(D) is the value of the invariant polynomial Dat the orbit O.
+Since the solutions of the equation η(D)ϕ=χO(D)ϕhave a basis e(wλ,x)wherew∈W,
+we have
+δ(x)ψO(x) =/summationdisplay
+w∈Wcw·e(wλ,x).
+Since it is antisymmetric, we have cw=c·(−1)ℓ(w), wherecis a constant. The fact that
+c= 1 can be shown by comparing the asymptotics of both sides as x→∞in the regular
+chamber (using the stationary phase approximation for the integr al). /square
+From Theorem 10.9 and the Weyl Character formula, we have the fo llowing corollary.
+Corollary 10.10 (Kirillov character formula for finite dimensional representations, [Ki]).If
+λis a dominant integral weight, and Lλis the corresponding representation of G, then
+TrLλ(ex) =δ(x)
+δTr(x)/integraldisplay
+Oλ+ρe(b,x)db,
+whereδTr(x)is the trigonometric version of δ(x), i.e. the Weyl denominator/producttext
+α>0(eα(x)/2−e−α(x)/2), andOµdenotes the coadjoint orbit passing through µ.
+10.4.The deformed Harish-Chandra homomorphism. Finally, we would like to ex-
+plain how to quantize the Calogero-Moser space Cn, using the procedure of quantum Hamil-
+tonian reduction.
+Letg=gln,A=D(g)asabove. Let kbeacomplexnumber, and Wkbetherepresentation
+ofslnon the space of functions of the form ( x1···xn)kf(x1,...,xn), wherefis a Laurent
+polynomial of degree 0. We regard Wkas ag-module by pulling it back to gunder the
+natural projection g→sln. LetIkbe the annihilator of WkinU(g). The ideal Ikis the
+quantum counterpart of the coadjoint orbit of matrices Tsuch thatT+1 has rank 1.
+LetBk=D(g)g/(D(g)µ(Ik))gwhereµ:U(g)→Ais the quantum momentum map (the
+quantum Hamiltonian reduction with respect to the ideal Ik). ThenBkhas a filtration
+induced from the order filtration of D(g)g.
+Let HCk:D(g)g→Bkbe the natural homomorphism, and K(k) be the kernel of HC k.
+Theorem 10.11 (Etingof-Ginzburg, [EG]) .(i)K(0) =K,B0=D(h)W,HC0= HC.
+(ii) grK(k) = Ker(grHC k) =K0for all complex k. Thus, HCkis a flat family of
+homomorphisms.
+(iii)The algebra grBkis commutative and isomorphic to C[h⊕h∗]Was a Poisson algebra.
+Because of this theorem, the homomorphism HC kis called the deformed Harish-Chandra
+homomorphism.
+Theorem 10.11 implies that Bkis a quantization of the Calogero-Moser space Cn(with
+deformation parameter 1 /k). But we already know one such quantization - the spherical
+88Cherednik algebra B1,kfor the symmetric group. Therefore, the following theorem comes as
+no surprize.
+Theorem 10.12 ([EG]).The algebra Bkis isomorphic to the spherical rational Cherednik
+algebraB1,k(Sn,Cn).
+Thus, quantum Hamiltonian reduction provides a Lie-theoretic cons truction of the spher-
+ical rational Cherednik algebra for the symmetric group. A similar (b ut more complicated)
+Lie theoretic construction exists for symplectic reflection algebra s for wreath product groups
+defined in Example 8.5 (see [EGGO]).
+10.5.Notes.Our exposition in this section follows Section 4, Section 5 of [E4].
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+positio Math. 85 (1993), 333–373.
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+[Pi] G. Pinczon, On two Theorems about Symplectic Reflection Algebra s, Lett.Math.Phys., 22, issues 2-3,
+2007, pp. 237-253.
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+ical Society, Providence, RI, 2005.
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+267–282.
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+1–41.
+P.E.: Department of Mathematics, Massachusetts Institute of Te chnology, Cambridge,
+MA 02139, USA
+E-mail address :etingof@math.mit.edu
+X.M.: Department of Mathematics, Massachusetts Institute of Te chnology, Cambridge,
+MA 02139, USA
+E-mail address :xma@math.mit.edu
+92
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+arXiv:1001.0433v1  [math-ph]  4 Jan 2010TEMPERATURE JUMP IN DEGENERATE QUANTUM GASES IN THE
+PRESENCE OF A BOSE /emdash.cyr EINSTEIN CONDENSATE
+Anatoly V. Latyshev and Alexander Yushkanov
+Department of Mathematical Analysis and Department of Theo retical Physics,
+Moscow State Regional University, 105005, Moscow, Radio st ., 10–A
+(Dated: 10 ноября 2018 г.)
+We construct a kinetic equation modeling the behavior of deg enerate quantum Bose gases whose
+collision rate depends on the momentum of elementary excita tions. We consider the case where the
+phonon component is the decisive factor in the elementary ex citations. We analytically solve the
+half-space boundary value problem of the temperature jump a t the boundary of the degenerate
+Bose gas in the presence of a Bose /emdash.cyr Einstein condensate.
+Keywords : degenerate quantum Bose gas, collision integral, Bose /emdash.cyr Ei nstein condensate,
+phonon component, temperature jump, Kapitsa resistance.
+PACS numbers: 05.20.Dd Kinetic theory, 05.30.Jp Boson syst ems, 05.60.-k Transport processes, 03.75.Nt
+Other Bose /emdash.cyr Einstein condensation phenomena
+1. INTRODUCTION
+The behavior of quantum gases has aroused increased
+interest in recent years. In particular, this is related
+to the development of experimental procedures for
+producing and studying quantum gases at extremely low
+temperatures [1]. The bulk properties of quantum gases
+have been studied in the majority of papers [2] and [3].
+At the same time, it is obviously important to take
+boundary effects on the properties of such systems into
+account. We mention a paper where the thermodynamic
+equilibrium properties of quantum gases in a half–space
+were considered [4].
+Along with the equilibrium properties, the
+nonequilibrium properties of quantum gases bounded in
+space attract interest. In particular, such a phenomenon
+as the temperature jump [5] at the interface between
+a gas and a condensed (in particular, solid) body in
+the presence of a heat flux normal to the surface is
+important. Such a temperature jump is frequently called
+the Kapitsa temperature jump [6].
+We note that up to now, the Kapitsa jump has been
+calculated in the regime where only phonon scattering at
+the interface between two media was taken into account
+and phonon scattering in the bulk was neglected [7]. Here,
+we take namely the effect of scattering of elementary
+excitations of the Bose gas (phonons) in the bulk into
+account.We take the character of phonon scattering by the
+surface into account by introducing a phenomenological
+coefficient of specular phonon scattering by the surface.
+This approach is thus additional to that proposed in [7].
+In [8], we considered the temperature jump problem in
+a quantum Fermi gas. We obtained an analytic solution
+for an arbitrary degree of gas degeneracy. In [9], we
+considered a similar problem for a Bose gas. But the
+gas was assumed to be nondegenerate, i.e., we did not
+take the presence of a Bose /emdash.cyr Einstein condensate into
+account.
+This paper is devoted to analyzing the temperature
+jump problem in a degenerate Bose gas. The presence of
+a Bose /emdash.cyr Einstein condensate leads to a considerable
+modification of both the problem statement and the
+solution method. In this case, to describe kinetic
+processes near the surface, we use a kinetic equation with
+a model collision integral. We assume that the boundary
+conditions at the surface are specular–diffuse.
+2. DERIVATION OF THE KINETIC EQUATION
+To describe the gas behavior, we use a kinetic equation
+with a model collision integral analogous to that used to
+describe a classical gas. We take the quantum character
+of the Bose gas and the presence of the Bose /emdash.cyr Einstein
+condensate into account. For a rarefied Bose gas, the
+evolution of the gas particle distribution function fcan2
+be described by the kinetic equation [5]
+∂f
+∂t+∂E
+∂p∇f=I[f], (1)
+whereEis the kinetic energy of gas particles, pis the
+gas particle momentum, and I[f]is the collision integral.
+In the case of the kinetic description of a degenerate
+Bose gas, we must take into account that the properties of
+the Bose /emdash.cyr Einstein condensate can change as functions
+of the space and time coordinates, i.e., we must consider
+a two–liquid model (more precisely, a "two-fluid" model
+because we consider a gas rather than a liquid). We let
+ρc=ρc(r,t)anduc=uc(r,t)denote the density and
+velocity of the Bose /emdash.cyr Einstein condensate. We then can
+write the expressions [7]
+j=ρcuc,Q=ρcu2
+c
+2uc,
+Πik=ρcuciuck.
+for the densities j,Q, andΠikof the respective mass,
+energy, and momentum fluxes of the Bose /emdash.cyr Einstein
+condensate (under the assumption that the chemical
+potential is zero). The conservation laws for the number
+of particles, energy, and momentum require that the
+relations
+∇j=−/integraldisplay
+I[f]dΩB,
+∇Q=−/integraldisplay
+ε(p)I[f]dΩB,
+∇Π =−/integraldisplay
+pI[f]dΩB
+be satisfied in the stationary case. Here,
+dΩB=(2s+1)d3p
+(2π/planckover2pi1)3,
+sis the molecule spin, E(p)is the energy, /planckover2pi1is the
+Planck constant and I(f)is the collision integral in Eq.
+(1).
+In what follows, we are interested in the case of
+motion with small velocities (compared with thermal
+velocities). We note that for the Bose /emdash.cyr Einstein
+condensate, the quantities QandΠikdepend on the
+velocity nonlinearly (they are proportional to the third
+and second powers of the velocity). Therefore, in theapproximation linear in the velocity uc, the energy and
+momentum conservation laws can be written as
+/integraldisplay
+ε(p)I[f]dΩB= 0,
+/integraldisplay
+pI[f]dΩB= 0.
+According to the Bogolyubov theory, the relation for
+the excitation energy E(p)holds for a weakly interacting
+Bose gas [5]
+ε(p) =/bracketleftBigg
+u2
+0p2+/parenleftbiggp2
+2m/parenrightbigg2/bracketrightBigg1/2
+, (2)
+where
+u0=/parenleftbigg4π/planckover2pi12an
+m2/parenrightbigg1/2
+,
+ais the scattering length for gas molecules, nis the
+concentration, mis the mass, and pis the momentum
+of elementary excitations. The parameter acharacterizes
+the interaction force of gas molecules and can be assumed
+to be small for a weakly interacting gas.
+In our previous paper [11], we considered the case
+where the relation
+u2
+0≪kT
+m
+holds for sufficiently small a, wherekis the Boltzmann
+constant and Tis the temperature. In that case, the
+first term in the brackets in (2) can be neglected. The
+expression for the energy E(p)takes the same form as in
+the case of noninteracting molecules:
+E(p) =p2
+2m.
+We now consider the case where the phonon component
+dominates in (2), i.e., where
+T≪mu2
+0
+k.
+In this case, we obtain
+E(p) =u0|p|=u0p.
+according to relation (2). Consequently,
+∂E(p)
+∂p=u0p
+p.
+When considering kinetic equation (1), by gas
+molecules, we must understand the elementary3
+excitations of the Bose gas with energy spectrum
+(2). The character of the elementary excitations is
+manifested in the properties of the collision integral.
+As a collision integral in Eq. (1), we take its τ–
+approximation. Then the character of the elementary
+excitations is manifested in the dependence of the
+collision rate on the excitation momentum [9], [10],
+[11]–[13]
+∂f
+∂t+u0p
+p∂f
+∂r=ν(p)(f∗
+B−f). (3)
+Here,fis the distribution function,
+ν(p) =ν0|p−p0|γ
+is the dependence of the collision rate on the excitation
+momentum, and γis a constant. In the case where
+the phonon component dominates in the elementary
+excitations, γ/greaterorequalslant3[5]. We have p0=mv0, where v0
+is the velocity of the normal component of the Bose gas,
+f∗
+Bis the equilibrium function of the Bose /emdash.cyr Einstein
+distribution
+f∗
+B=/bracketleftbigg
+exp/parenleftbiggu0|p−p∗|
+kT∗/parenrightbigg
+−1/bracketrightbigg−1
+,
+andν0is a model parameter having the meaning of the
+inverse mean free path l,ν0∼1/l.
+The parameters in f∗
+B/g, namely, T∗andu∗, can
+be determined from the requirement that the energy and
+momentum conservation laws
+/integraldisplay
+ν(p)p/bracketleftBig
+f−f∗
+B/bracketrightBig
+d3p= 0,
+/integraldisplay
+ν(p)ε(p)/bracketleftBig
+f−f∗
+B/bracketrightBig
+d3p= 0
+be applicable. The conservation law for the number of
+particles is inapplicable here because of the transition of
+a fraction of particles to the Bose /emdash.cyr Einstein condensate.
+We now assume that the gas velocity is much less than
+the mean thermal velocity and the typical temperature
+variations along the mean free path lare small compared
+with the gas temperature. Under these assumptions, the
+problem can be linearized.
+We seek the distribution function in the form
+f=fs
+B(C)+ϕ(t,r,C)g(C),
+where
+fs
+B(C) =1
+expC−1,ϕis a new unknown function, Tsis the surface
+temperature, g(v) =−∂
+∂εsfs
+B, εs=C,and we
+introduce the notation
+C=u0p
+kTs,
+and
+ε∗=u0|p−p∗|
+kT∗.
+We then have
+f∗
+B(ε∗) =1
+exp(ε∗)−1,
+fs
+B(C) =1
+expC−1,
+g(C) =expC
+(expC−1)2.
+We linearize the local Bose /emdash.cyr Einstein distribution
+(Bosean) f∗
+B, passing to dimensionless quantities. We
+note that
+ε∗=Ts
+T∗·u0|p−p∗|
+kTs=Ts
+Ts+δT∗·u0
+kTs/radicalbig
+(p−p∗)2=
+=/parenleftBig
+1−δT∗
+Ts/parenrightBigu0
+kTs/parenleftBig
+p−pp∗
+p/parenrightBig
+,
+or
+ε∗=/parenleftBig
+1−δT∗
+Ts/parenrightBig/parenleftBig
+C−CC∗
+C/parenrightBig
+=C−CC∗
+C−CδT∗
+Ts.
+Therefore
+δε∗=ε∗−εs=−CC∗
+C−CδT∗
+Ts,
+because εs=C.
+Consequently,
+f∗
+B=fs
+B+g(C)/parenleftBigCC∗
+C+CδT∗
+Ts/parenrightBig
+.
+We note that the quantity ν(p)in Eq. (3) can
+be replaced with ν0pγin the approximation under
+consideration.
+We introduce the dimensionless time
+τ=ν0/parenleftBigkTs
+u0/parenrightBigγ
+t
+and coordinate
+r1=ν0/parenleftBigkTs
+u0/parenrightBigγ
+r.4
+It is now clear that Eq. (3) (in the dimensionless
+variables) becomes
+∂ϕ
+∂τ+C
+C∂ϕ
+∂r1=Cγ/bracketleftBigCC∗
+C+CδT∗
+Ts−ϕ/bracketrightBig
+.(4)
+From the energy and momentum conservation laws, we
+find
+δT∗
+Ts=1
+4πgγ+4/integraldisplay
+Cγ+1ϕ(τ,r1,C)g(C)d3C,
+C∗=3
+4πgγ+3/integraldisplay
+CγCϕ(τ,r1,C)g(C)d3C,
+where
+gγ+n=∞/integraldisplay
+0Cγ+ng(C)dC, n = 0,1,2,···.
+We represent Eq. (4) in the form that is standard in
+transport theory:
+∂ϕ
+∂τ+C
+C∂ϕ
+∂r1+Cγϕ(τ,r1,C) =
+=1
+4π/integraldisplay
+k(C,C′)ϕ(τ,r1,C′)g(C′)d3C,(5)
+wherek(C,C′)is the kernel of Eq. (5),
+k(C,C′) =3
+gγ+3Cγ−1C′γCC′+1
+gγ+4Cγ+1C′γ+1.
+Equation (5) can be represented in the equivalent form
+∂ϕ
+∂τ+C
+C∂ϕ
+∂r1+Cγϕ(τ,r1,C) =
+=3Cγ−1C
+4πgγ+3/integraldisplay
+C′γC′ϕg(C′)d3C′+
++Cγ+1
+4πgγ+4/integraldisplay
+C′γ+1ϕg(C′)d3C′.
+3. PROBLEM STATEMENT
+In the problem under consideration, a degenerate Bose
+gas occupies the half–space x >0above a planar surface
+where the heat exchange between the condensed phase
+and the gas occurs. Therefore, the function ϕcan be
+regarded as
+ϕ(τ,r1,C) =h(x,µ,C)in what follows. Such a function satisfies the equation
+µ∂h
+∂x+Cγh(x,µ,C) =
+=1
+21/integraldisplay
+−1∞/integraldisplay
+0K(µ,C;µ′,C′)h(x,µ′,C′)g(C′)dµ′dC′,(6)
+whereK(µ,C;µ′,C′)is the kernel of Eq. (6),
+K(µ,C;µ′,C′) =3Cγ−1µC′γ+3µ′
+gγ+3+Cγ+1C′γ+3
+gγ+4.
+The problem is to find the value of the relative
+temperature jump
+εT=∆T
+Ts,
+where
+∆T=Ts−T,
+as a function of Qx, which is the projection of the heat
+flux on the xaxis. Taking linearity of the problem into
+account, we can write
+εT=RQx.
+The dimensionless coefficient Rof the temperature
+jump is called the Kapitsa resistance.
+It is obvious that Eq. (6) has the particular solutions
+h1(x,µ,C) =µ andh2(x,µ,C) =C,
+and the Chapman /emdash.cyr Enskog distribution function is
+has(x,µ,C) =B+µ−εTC,
+where the quantity B+is proportional to the heat flux
+Qx.
+Assuming that the reflection of the elementary
+excitations from the wall is specular–diffuse, we now
+formulate the boundary conditions
+h(0,µ,C) =qh(0,−µ,C),0< µ <1,(7)
+h(x,µ,C) =
+=B+µ−εTC+o(1), x→+∞,−1< µ <0,(8)
+whereqis the specular reflection coefficient.
+The problem is to solve Eq. (6) with boundary
+conditions (7) and (8). Finding the value of the
+temperature jump εTis of special interest.5
+4. REDUCTION TO THE INTEGRAL EQUATION
+We continue the function h to the half–space x <0
+symmetrically:
+h(x,µ,C) =h(−x,−µ,C), x < 0.
+Forx <0, we then have
+has(x,µ,C) =B−µ−εTC,
+withB+=−B−.
+We now separate the Chapman /emdash.cyr Enskog distribution
+from the function hassuming that
+h(x,µ,C) =B±µ−εTC+hc(x,µ,C).
+For the function hc(x,µ,C), we formulate the
+boundary conditions for the lower and upper half–spaces:
+hc(+0,µ,C) =
+=−(1+q)B+µ+(1−q)εTC+qhc(+0,−µ,C),
+where 0< µ <1,
+hc(−0,µ,C) =
+=−(1+q)B−µ+(1−q)εTC+qhc(−0,−µ,C),
+where −1< µ <0, and
+hc(+∞,µ,C) = 0, h c(−∞,µ,C) = 0.
+We include these boundary conditions in the kinetic
+equation. We obtain the equation
+µ
+Cγ∂h
+∂x+hc(x,µ,C) =C
+2gγ+4W0(x)+3µ
+2gγ+3W1(x)+
++|µ|{−(1+q)B+|µ|+
++(1−q)εTC−(1−q)hc(∓0,µ,C)}δ(x).(9)
+Here,δ(x)is the Dirac delta function, and
+Wm(x) =
++1/integraldisplay
+−1∞/integraldisplay
+0µmCγ+3h(x,µ,C)g(C)dµdC, m = 0,1.(10)Equation (9) actually combines two equations. The
+point is that the term hc(−0,µ,C)corresponds to
+positive µ: 0< µ < 1and the term hc(+0,µ,C)
+corresponds to negative µ:−1< µ <0.
+We seek the solution of Eqs. (9) in the form of Fourier
+integrals:
+hc(x,µ,C) =1
+2π∞/integraldisplay
+−∞eikxΦ(k,µ,C)dk,
+δ(x) =1
+2π∞/integraldisplay
+−∞eikxdk,
+W0(x) =1
+2π∞/integraldisplay
+−∞eikxE0(k)dk,
+W1(x) =1
+2π∞/integraldisplay
+−∞eikxE1(k)dk.
+Solving Eq. (10) with x >0andµ <0, assuming
+that the right–hand side of this equation is known, and
+assuming that the boundary conditions are satisfied far
+from the wall, we obtain
+h+
+c(x,µ,C) =
+=−Cγ
+µexp/parenleftBig
+−x
+µCγ/parenrightBig+∞/integraldisplay
+xexp/parenleftBigt
+µCγ/parenrightBig
+W(t,µ,C)dt,
+where
+W(t,µ,C) =C
+2gγ+4W0(t,µ,C)+3µ
+2gγ+3W1(t,µ,C).
+After simple calculations, we obtain
+h+
+c(x,µ,C) =Cγ
+2π+∞/integraldisplay
+−∞eikxE(k,µ,C)dk
+Cγ+ikµ,
+where
+E(k,µ,C) =C
+2gγ+4E0(k)+3µ
+2gγ+3E1(k).
+We can similarly show that
+h−
+c(x,µ,C) =Cγ
+2π+∞/integraldisplay
+−∞eikxE(k,µ,C)dk
+Cγ+ikµ.6
+From the last two expressions, taking the evenness of
+the function E(k,µ,C)with respect to the variable k
+into account, we obtain
+h±
+c(0,µ,C) =
+=Cγ
+2π+∞/integraldisplay
+−∞(Cγ−ikµ)E(k,µ,C)dk
+C2γ+k2µ2=
+=C2γ
+π+∞/integraldisplay
+0E(k,µ,C)dk
+C2γ+k2µ2. (11)
+Hence, we see that two Eqs. (9) can be combined into
+one:
+µ
+Cγ∂h
+∂x+hc(x,µ,C) =
+=C
+2gγ+4W0(x)+3µ
+2gγ+3W1(x)+
++|µ|{−(1+q)B+|µ|+
++(1−q)εTC−(1−q)h±
+c(0,µ,C)}δ(x).(12)
+We pass to the Fourier integrals in Eq. (12) and obtain
+the equation
+(Cγ+ikµ)Φ(k,µ,C) =Cγ+1
+2gγ+4E0(k)+3Cγµ
+2gγ+3E1(k)+
++Cγ|µ|/bracketleftBigg
+−(1+q)B+|µ|+(1−q)εTC−C2γ×
+×1−q
+π∞/integraldisplay
+0/bracketleftBigg
+CE0(k1)
+2gγ+4+3µE1(k1)
+2gγ+3/bracketrightBigg
+dk1
+C2γ+k2
+1µ2/bracketrightBigg
+.(13)
+We consider equality (10). Rewriting it in terms of the
+Fourier integrals, we obtain
+Ej(k) =
++1/integraldisplay
+−1∞/integraldisplay
+0µjΦ(k,µ,C)Cγ+3g(C)dµdC, j = 0,1.(14)
+We solve Eq. (13) for Φ(k,µ,C)and substitute it
+in (14) for j= 0,1. We obtain a system consisting oftwo characteristic equations. Without writing them, we
+combine them into one vector characteristic equation
+Λ(k)E(k) =−2(1+q)B+T1(k)+2(1−q)εTT2(k)−
+−1−q
+π∞/integraldisplay
+0J(k,k1)E(k1)dk1. (15)
+Here
+Tm,l(k) =1/integraldisplay
+0∞/integraldisplay
+0µlCmg(C)dµdC
+C2γ+k2µ2,
+Λ(k)is the dispersion matrix
+Λ(k) =
+1−1
+gγ+4T3γ+4,0(k)3ik
+gγ+3T2γ+3,2(k)
+ik
+gγ+4T2γ+4,2(k) 1−3
+gγ+3T3γ+3,2(k)
+,
+E(k)is an unknown column vector
+E(k) =/bracketleftBigg
+E0(k)
+E1(k)/bracketrightBigg
+,
+T1(k)andT2(k)are the column vectors of constant
+terms
+T1(k) =/bracketleftBigg
+T3γ+3,2(k)
+−ikT2γ+3,4(k)/bracketrightBigg
+,
+T2(k) =/bracketleftBigg
+T3γ+4,1(k)
+−ikT2γ+4,3(k)/bracketrightBigg
+,
+andJ(k,k1)is the matrix function
+J(k,k1) =
+1
+gγ+4J4γ+4,1(k,k1)−3ik
+gγ+3J3γ+3,3(k,k1)
+−ik
+gγ+4J3γ+4,3(k,k1)3
+gγ+3J4γ+3,3(k,k1)
+
+which we call the Neumann matrix.
+The integrals
+Jm,l(k,k1) =1/integraldisplay
+0∞/integraldisplay
+0µlCmg(C)dµdC
+(C2γ+k2µ2)(C2γ+k2
+1µ2)
+are introduced in the Neumann matrix.7
+5. METHOD OF SUCCESSIVE
+APPROXIMATIONS
+We seek the solution of Eq. (15) in the form
+εT=1+q
+1−q/bracketleftBig
+ε0+ε1(1−q)+ε2(1−q)2+···/bracketrightBig
+(16)
+and
+E(k) = 2(1+ q)×
+×/bracketleftBig
+E(0)(k)+E(1)(k)(1−q)+E(2)(k)(1−q)2+···/bracketrightBig
+.(17)
+We substitute these equalities in the characteristic
+equation. We obtain the countable system of equations
+Λ(k)E(0)(k) =−B+T1(k)+ε0T2(k),(18)
+Λ(k)E(1)(k) =ε1T2(k)−1
+π∞/integraldisplay
+0J(k,k1)E0(k1)dk1,(19)
+Λ(k)E(2)(k) =
++ε2T2(k)−1
+π∞/integraldisplay
+0J(k,k1)E1(k1)dk1,···.(20)
+We transform the elements on the principal diagonal
+of the dispersion matrix:
+1−1
+gγ+4T3γ+4,0(k) =k2
+gγ+4Tγ+4,2(k),
+1−1
+gγ+3T3γ+4,2(k) =3k2
+gγ+3Tγ+3,4(k).
+The dispersion matrix function now becomes
+Λ(k) =
+k2
+gγ+4Tγ+4,2(k)3ik
+gγ+3T2γ+3,2(k)
+ik
+gγ+4T2γ+4,2(k)3k2
+gγ+3Tγ+3,4(k)
+.
+We call the determinant of the dispersion matrix the
+dispersion function:
+λ(z)≡detΛ(z) =k2ω(k),
+where
+ω(k) =3k2
+gγ+3gγ+4××/bracketleftBig
+k2Tγ+4,2(k)Tγ+3,4(k)+T2γ+3,2(k)T2γ+4,2(k)/bracketrightBig
+.
+The matrix inverse to the dispersion matrix is
+Λ−1(k) =D(k)
+λ(k),
+, where
+D(k) =
+3k2
+gn+3Tn+3,4(k)−3ik
+gn+3T2n+3,2(k)
+−ik
+gn+4T2n+4,2(k)k2
+gn+4Tn+4,2(k)
+.
+We consider the construction of series (17). We set
+E(0)(k) =/bracketleftBigg
+E(0)
+0(k)
+E(0)
+1(k)/bracketrightBigg
+.
+We consider the construction of series (17). We set
+E(0)(k) = 2D(k)
+λ(k)×
+×/bracketleftBigg
+−B+T3γ+3,2(k)+ε0T3γ+4,1(k)
+ik(B+T2γ+3,4(k)+ε0T2γ+4,3(k))/bracketrightBigg
+.(21)
+Substituting expression (21) in Eq. (19), we find
+E(1)(k). Substituting E(1)(k)in Eq. (20), we then find
+E(2)(k). Continuing this process without bound, we
+construct all terms of series (17).
+We now show how to construct the terms of series (16).
+For this, we write vector equality (21) in the form of two
+scalar equalities:
+E(0)
+0=1
+k2ω(k)/braceleftBigg
+−3k2
+gγ+3Tγ+3,3(k)×
+×/bracketleftBig
+B+T3γ+3,2(k)−ε0T3γ+4,1(k)/bracketrightBig
++
++3k2
+gγ+3T2γ+3,2(k)/bracketleftBig
+B+T2γ+3,4(k)+ε0T2γ+4,3(k)/bracketrightBig/bracerightBigg
+(22)
+E(0)
+1=1
+k2ω(k)×
+×/braceleftBig
+ikT2γ+4,2(k)/bracketleftBig
+B+T3γ+3,2(k)−ε0T3γ+4,1(k)/bracketrightBig
++
++ik3Tγ+4,2(k)/bracketleftBig
+B+T2γ+3,4(k)+ε0T2γ+4,3(k)/bracketrightbig/bracerightBig
+.(23)8
+It can be seen from expression (22) that the function
+E(0)
+0(k)has no singularities at zero, while the function
+E(0)
+1(k)according to (23) has a second–order pole at k=
+0. Eliminating the singularity at zero, we obtain
+ε0=B+T3γ+3,2(0)
+T3γ+4,1(0)=B+2gγ+3
+3gγ+4.
+The quantity B+is proportional to the heat flux:
+Q=/integraldisplay
+f(x,p)∂ε(p)
+∂pε(p)dΩB.
+We transform this expression as
+Q=(2s+1)u2
+0
+(2π/planckover2pi1)3/integraldisplay
+h(x,p)g(p)pd3p.
+Integrating in this expression over the dimensionless
+momentum, we obtain
+Q=(2s+1)(kTs)4
+(2π/planckover2pi1)3u2
+0/integraldisplay
+h(x,C)g(C)Cd3C.
+We replace here the function h(x,C)with its
+Chapman /emdash.cyr Enskog expansion has(x,C). As a result,
+we have
+Qx=(2s+1)(kTs)4
+(2π/planckover2pi1)3u2
+01/integraldisplay
+−1∞/integraldisplay
+02π/integraldisplay
+0/bracketleftBig
+εTC+B+µ/bracketrightBig
+×
+×g(C)C3µdµdCdχ =(2s+1)(kTs)4
+(2π/planckover2pi1)3u2
+0·4π
+3g3B+
+for thexcomponent of the heat flux.
+We hence have
+B+=6π2
+g3·/planckover2pi13u2
+0Qx
+(2s+1)(kTs)4.
+The quantity ε0is therefore
+ε0=4π2gγ+3
+g3gγ+4·/planckover2pi13u2
+0
+(2s+1)(kTs)4Qx.(24)
+Returning to the formula for the temperature jump
+∆T=RQx,
+we find from expression (24) that the Kapitsa resistance
+in the zeroth approximation is
+R=4π2gγ+3
+g3gγ+4·1+q
+1−q·/planckover2pi13u2
+0
+(2s+1)k4T3s.
+We rewrite this equation in the form
+R=C(γ,q)·/planckover2pi13u2
+0
+(2s+1)k4T3s,/s48 /s53 /s49/s48/s55/s49/s52/s32
+/s32/s49/s50/s51/s67/s40 /s44/s113/s41
+Fig. 1. Dependence of the temperature jump coefficient on
+the parameter γ: the specular reflection coefficient is
+q= 0.3for curve 1,q= 0.5for curve 2, andq= 0.8for
+curve3.
+/s48/s46/s53 /s49/s46/s48/s49/s49/s48/s49/s48/s48
+/s67/s40 /s44/s113/s41 /s32
+/s32/s49/s50
+/s51
+/s113
+Fig. 2. Dependence of the temperature jump coefficient on
+the specular reflection coefficient q: the collision paramete r
+isγ= 3for curve 1,γ= 4for curve 2, andγ= 5for
+curve3.
+where
+C(γ,q) =4π2gγ+3
+g3gγ+4·1+q
+1−q(25)
+is the (dimensionless) coefficient of the temperature
+jump.
+The plus sign in formula (25) indicates that the
+wall temperature is higher that the phonon component
+temperature.9
+The graphs of the behavior of the temperature jump
+coefficient as a function of the parameter γand the
+specular reflection coefficient qare shown in Figs. 1 and
+2.
+It can be seen from these graphs that the quantity
+C(γ,q)decreases monotonically as the parameter γ
+increases. As the specular reflection coefficient qtends
+to unity, the quantity C(γ,q)increases without bound
+because the heat exchange between the wall and the gas
+adjoining it becomes impossible in this limit.
+6. CONCLUSIONS
+We have constructed a kinetic equation for a
+degenerate quantum Bose gas whose collision rate
+depends on the momentum of elementary excitations of
+the Bose gas. We considered the case where the phonon
+component is the key factor in the elementary excitations.
+The boundary conditions were assumed to be specular–
+diffuse. We solved the half–space boundary value problem
+of the temperature jump at the boundary of a degenerate
+gas in the presence of a Bose /emdash.cyr Einstein condensate. We
+derived a formula for finding the temperature jump and
+calculating the Kapitsa resistance.
+We have developed a sufficiently general method
+for solving kinetic equations with specular–diffusive
+boundary conditions; this method was first proposed in
+[14], where the problem of the skin effect was considered.
+[1]L. P. Pitaevskii , Phys. Usp., 49, 333–351 (2007).[2]H. Spon , Kinetics of the Bose–Einstein Condensation,
+arXiv:0809.4551v1 [cond-mat.mes-hall] (2008).
+[3]Hai Pang, Wu–Sheng Dai, Mi Xie , J. Phys. A. Mat. Gen.
+39(2006), 2563–2571; arXiv:cond-mat/0603289 (2006).
+[4]L. Samaj and B. Jancovici ,ArXiv : cond–mat/0701773.
+[5]E. M. Lifshitz and L. P. Pitaevskii , Statistical Physics,
+Part 2 [in Russian], Nauka, Moscow (1978); English
+transl.: Statistical Physics: Part 2. Theory of the
+Condensed State (Vol. 9 of Course of Theoretical Physics,
+L. D. Landau and E. M. Lifshits, eds.), Pergamon Press,
+Oxford (1980).
+[6]Yu. V. Prokhorov , ed., Encyclopedic Dictionary
+of Physics [in Russian], Bol’shaya Rossiiskaya
+Entsiklopediya, Moscow (1955).
+[7]I. M. Khalatnikov, An Introduction to the Theory
+of Superfluidity [in Russian], Nauka, Moscow (1965);
+English transl., Addison-Wesley, Redwood City, Calif.
+(1989).
+[8]A. V. Latyshev and A. A. Yushkanov , Theor. Math.
+Phys., 134(2), 271-284, (2003).
+[9]A. V. Latyshev and A. A. Yushkanov , Mathematical
+Modeling, 15(5), 80-94, (2003).
+[10]Carlo Cercignani , Theory and Application of the
+Boltzmann Equation, Scottish Academic, Edinburgh
+(1975).
+[11]A. V. Latyshev and A. A. Yushkanov , Theor. Math.
+Phys., 155(3), 498–511, (2008).
+[12]A. V. Latyshev and A. A. Yushkanov , Theor. Math.
+Phys., 162(1), 95–105, (2010).
+[13]A. V. Latyshev and A. A. Yushkanov , Williams–
+type Kinetic Equations and Their Exact Solutions [in
+Russian], Moscow State Regional Univ., 2004.
+[14]A. V. Latyshev and A. A. Yushkanov , Comp. Mathem.
+and Mathem. Phys., 49(1), 131–145, (2009).
\ No newline at end of file
diff --git a/1001.0434.txt b/1001.0434.txt
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@@ -0,0 +1,1219 @@
+arXiv:1001.0434v3  [hep-ph]  13 Jun 2010Lepton flavour violating Higgs and τ→µγ
+Sacha Davidson∗and Gerald Grenier†
+IPNL, Universit´ e de Lyon, Universit´ e Lyon 1, CNRS/IN2P3, 4 rue E. Fermi 69622 Villeurbanne cedex,
+France
+Abstract
+We update phenomenological constraints on a Two Higgs Doubl et Model with lepton flavour non-conserving
+Yukawa couplings. We review that tan βis ambiguous in such “Type III” models, and define it from
+theτYukawa coupling. The neutral scalars φcould be searched for at hadron colliders in φ→τ¯µ, and
+are constrained by the rare decay τ→µγ. The Feynman diagrams for the collider process, with Higgs
+production via gluon fusion, are similar to the two-loop “Ba rr-Zee” diagrams which contribute to τ→µγ.
+Some “tuning” is required to obtain a collider cross-sectio n of order the Standard Model expectation for
+σ(gg→hSM→τ+τ−), while agreeing with the current bound from τ→µγ.
+1 Introduction
+The Two Higgs Doublet Model (2HDM) may be the low energy effective t heory for many models of Beyond-the
+Standard Model (BSM) physics at the TeV scale. Various Higgses co uld be the first signals of BSM physics
+discovered at hadron colliders. The aim of this paper is therefore to study the implications for collider searches,
+of precision physics bounds on a generic 2HDM with lepton flavour viola ting couplings. Similar analyses have
+previously been performed in [1, 2, 3, 4, 5, 6]. We assume that the ad ditional Higgses are the only BSM particles
+with masses <∼400GeV, and considerconstraintson the Higgs parameterswhich a recomparativelyindependent
+of additional New Physics at higher scales.
+The 2HDM [7] (see [8] for an introduction) consists in adding a second Higgs doublet to the Standard Model.
+Despite being a fairly minimal extension1, it has many variants. In particular, a discrete symmetry [12] can
+be imposed on the Higgs plus fermion Lagrangian, to avoid tree level fl avour changing neutral interactions.
+We focus here on “Type III” models, meaning that no discrete symm etries are present, so there is no unique
+definition oftan β. We neglectnon-renormalisableoperators[13, 14] in the potential. In the absenceofadiscrete
+symmetry, the fermions can couple to both Higgs doublets with gene ric Yukawa matrices, which allows flavour-
+changing tree-level couplings of the physical Higgses. For simplicity , we make the (unrealistic) assumption that
+our Higgses only have leptonflavour violating interactions2. We are particularily interested in the Higgs −τ−µ
+interaction.
+We emphasize that, for a generic neutral Higgs φ, the search for φ→τ¯µat hadron colliders[16, 3, 4] should
+take into account the bounds on τ→µγ[5]. In figure 1, on the left is shown a fermion loop diagram of “Barr-
+Zee” type [17, 18, 19, 20, 21, 22] which contributes to τ→µγ. Beside it is the similar diagram for Higgs
+production and decay to τ¯µat a hadron collider. In the absence of cancellations between the diff erent neutral
+Higgses in the Barr-Zee, it is clear that an upper bound on BR(τ→µγ) sets a bound on σ(gg→φ→τ¯µ).
+There is a large literature on flavour changing observables in the Typ e III 2HDM. Various textures for the
+Yukawa matrices are discussed in [23] (see also references therein ). Recently, the concept of Minimal Flavour
+Violation has been extended to multi-Higgs models [24]. See also [25] for a clear discussion of CP violation in
+a Type III 2HDM that has no tree-level flavour changing neutral c ouplings. In the Minimal Supersymmetric
+Standard Model (MSSM), Higgs decay to τ±µ∓[26], and its relation to one loop rare τdecays [27] have been
+extensively studied (see also citations of [26]). Our analysis differs fr om [1, 3, 5] in that we have included the
+Barr-Zee diagram in our calculation of τ→µγ. This can be relevant if the flavour-changing Higgs has O(1)
+coupling to the top (“small tan β′′). The current bound on BR(τ→µγ) is also slightly stronger that the
+∗E-mail address: s.davidson@ipnl.in2p3.fr
+†E-mail address: g.grenier@ipnl.in2p3.fr
+1The Higgs sector of the Minimal Supersymmetric Standard Mod el [9] is a 2HDM. Larger Higgs sectors with multiple doublets
+and singlets[10, 11], and/or triplets, can also be consider ed.
+2See, for instance, [15] and references thereto, for a discus sion of quark flavour changing interactions in type III model s.
+1value used by [5], who performed a more complete study of τdecay bounds in 2005. Finally, we are attentive
+to the definition of tan β; we allow the scalar potential of the Higgs to take its most general r enormalisable
+CP-conserving form, which does not allow a Lagrangian-basis-indep endent definition of tan β. The unphysical
+character of tan βin type III models has been treated in various ways by previous auth ors, which makes it
+difficult to compare results. We introduce a “physical” definition of ta nβ, via the Yukawa coupling of the τ, to
+facilitate comparaison with “almost type II” models such as the MSSM .
+Figure 1: On the left, the “Barr-Zee” diagram which contributes to τ→µγ. On the right, the production/decay
+diagram for a neutral Higgs φat hadron collider.
+In section 2, we introduce our “basis independent” [28, 29, 30, 31] notation for the 2HDM. We discuss
+electroweak precision bounds on the Higgs masses in section 3, and r apidly review bounds (for instance, on the
+Higgs potential) that are secondary in our analysis. Section 4 discus ses the contraints from precision flavour
+observables, such as ( g−2)µandτ→µγ, and section 5 studies the sensitivity to the φ−τ−µYukawa coupling
+ofφ→τ¯µat hadron colliders, in the light of the rare decay data.
+2 Notation
+We consider a 2HDM of Type III, in the classification3used, for instance, in [28]. Contrary to “type I” and
+“type II” models, the type III scalar potential has no discrete sy mmetry that distinguishes doublets. When
+Yukawa couplings are included, tree-level flavour changing coupling s for the neutral Higgses are possible. In an
+arbitrary choice of basis in doublet Higgs space, the potential can b e written [33]
+V=m2
+11Φ†
+1Φ1+m2
+22Φ†
+2Φ2−[m2
+12Φ†
+1Φ2+h.c.]
++1
+2λ1(Φ†
+1Φ1)2+1
+2λ2(Φ†
+2Φ2)2+λ3(Φ†
+1Φ1)(Φ†
+2Φ2)+λ4(Φ†
+1Φ2)(Φ†
+2Φ1)
++/braceleftBig
+1
+2λ5(Φ†
+1Φ2)2+/bracketleftbig
+λ6(Φ†
+1Φ1)+λ7(Φ†
+2Φ2)/bracketrightbig
+Φ†
+1Φ2+h.c./bracerightBig
+, (1)
+wherem2
+11,m2
+22, andλ1,···,λ4are real parameters. In general, m2
+12,λ5,λ6andλ7are complex, but we neglect
+CP violation in this paper for simplicity, and take them ∈ ℜ. A clear discussion of CP violation can be found
+in [34, 30]. A translation dictionary to the form of potential used, fo r instance, in [8], can be found in [33].
+The scalar fields will develop non-zero vacuum expectation values (v evs) if the mass matrix m2
+ijhas at least
+one negative eigenvalue. Then, the scalar field vacuum expectation s values are of the form
+∝an}bracketle{tΦ1∝an}bracketri}ht=1√
+2/parenleftbigg0
+v1/parenrightbigg
+,∝an}bracketle{tΦ2∝an}bracketri}ht=1√
+2/parenleftbigg0
+v2/parenrightbigg
+, (2)
+wherev1andv2are real and non-negative, and
+v2≡v2
+1+v2
+2=4m2
+W
+g2= (246 GeV)2=1√
+2GFtanβ≡v2
+v1(3)
+3This definition differs from the classification given in [32], where there are 4 types of 2HDM, all of which avoid tree level fl avour
+changing neutral currents.
+2The scalar potential of the type III 2HDM, with real couplings, has three parameters in m2
+ij, four in λ1..4
+and three in λ5..7. One of these parameters can be set to zero by a basis choice (for instance m2
+12= 0), leaving
+nine independent parameters. The minimisation conditions give v, which is measured, so 8 inputs are required.
+Ideally, one would like to express these parameters in terms of obse rvables, such as masses and 3 or 4 point
+functions. This would avoid confusion stemming from the arbitrary b asis choice in Higgs space, and clarifies
+the measure on parameter space to use in numerical scans.
+However, to study the current bounds on “light” Higgses, we only n eed their masses and couplings to SM
+particles(controlledby( β−α) —seethe discussionaftereqn(5)). Sowedonotneedacompletep arametrisation
+in terms of observables; we relate constraints on the scalar poten tial to the masses in the basis independent
+notation of [28], and otherwise our parameters are m2
+h,m2
+H,m2
+A,m2
+H+, sin(β−α) and the flavour changing
+Yukawa coupling that controls p¯p→φ→τ¯µandτ→µγ.
+2.1 Basis choice in Higgs space and tanβτ
+It is always possible to choose a basis in the Higgs space, such that on ly one doublet acquires a vev [34]. This
+is known as the “Higgs basis”, defined such that ∝an}bracketle{tH1∝an}bracketri}ht ∝ne}ationslash= 0, and ∝an}bracketle{tH2∝an}bracketri}ht= 0, and in this basis all the potential
+parameters are written in upper case:
+V=M2
+11H†
+1H1+M2
+22H†
+2H2−[M2
+12H†
+1H2+h.c.] (in Higgs basis)
++1
+2Λ1(H†
+1H1)2+1
+2Λ2(H†
+2H2)2+Λ3(H†
+1H1)(H†
+2H2)+Λ4(H†
+1H2)(H†
+2H1)
++/braceleftBig
+1
+2Λ5(H†
+1H2)2+/bracketleftbig
+Λ6(H†
+1H1)+Λ7(H†
+2H2)/bracketrightbig
+H†
+1H2+h.c./bracerightBig
+, (4)
+This follows the notation of [28].
+In the Higgs basis, the angle αHBrotates to the mass basis of the CP even Higgses h,H:
+h=−(√
+2 ReH0
+1−v)sinαHB+(√
+2 ReH0
+2)cosαHB,
+H= (√
+2 ReH0
+1−v)cosαHB+(√
+2 ReH0
+2)sinαHB. (5)
+In a 2HDM of type I or II, there is also a choice of basis where the disc rete symmetry Φ i↔Φi, or Φj↔ −Φj
+is manifest. Usually, the lagrangian is written in the basis where this sy mmetry is manifest4, the angle βis
+defined between this “symmetry eigenbasis” and the Higgs basis, an d the angle αrotates between the symmetry
+basis and the CP-even mass basis. In which case
+αHB=β−α (6)
+and we shall write it as such in this paper. Then the Higgs- W+W−couplings are igmWCφWWgµνwith
+ChWW=sβ−α, CHWW=cβ−α, CAWW= 0 (7)
+where
+sβ−α≡sin(β−α), cβ−α≡cos(β−α). (8)
+The trilinear couplings between a neutral Higgs and a pair of charged Higgses are −ivCφH+H−, with [33]
+ChH+H−= Λ3sβ−α−Λ7cβ−α, CHH+H−= Λ3cβ−α+Λ7sβ−α, CAH+H−= 0 (9)
+In a type III model, there is no “symmetry basis”, so tan βcan not be defined from the scalar potential. To
+obtain a type III potential, it is not sufficient to write down a Lagrang ian with λ6,λ7∝ne}ationslash= 0; one must also check
+that it is not a type II or type I model, in a basis rotated with respect to the symmetry basis. For this there
+are basis independent “invariants”, discussed for instance in [28], w hich vanish in the presence of symmetries.
+In phenomenological calculations, tan βusually appears in Yukawa interactions, where it parametrises the
+relativesizeoftheYukawacouplingsandthefermionmasses. Wheth eritcanbe definedfromthe scalarpotential
+of the Higgses is secondary. Various “definitions” of tan β, involving fermion masses in the Higgs basis, can
+be envisaged [35, 28, 36, 37]. To maintain the intuition of tan βas the relative size of the τYukawa coupling
+4For a discussion of Lagrangian basis dependence in the 2HDM, and a formalismthat is “basis independent”, see, e.g.[28, 29, 30].
+3and√
+2mτ/v, we define Hτto be the Higgs that couples to the τ, andβτas the angle in Higgs doublet space
+between H1(the vev) and Hτ:
+Hu=/tildewideH1sinβτ+/tildewideH2cosβτ
+Hτ=H1cosβτ−H2sinβτ (10)
+We choose the τYukawa, because we are interested in τflavour violation. This reduces to the usual definition
+in a Type II (SUSY) model, where /tildewideH=iσ2H∗.
+Finally, notice that we pursue a “bottom-up” approach, where we t reat tanβτas a physical parameter, and
+calculate processes which are finite. This allows us to neglect issues r elated to the renormalisation of tan βτ
+[35, 38, 39], such as its numerical stability, and additional factors o f tanβτthat may appear.
+2.2 The leptonic Yukawa couplings
+In the Higgs basis for the Higgses, and the mass eigenstate basis fo r the{uR,dR,eR,dL,eL}, the Yukawa
+interactions of a type III 2HDM can be written
+−LY=√
+2/parenleftBig
+qLj/tildewideH1K∗
+ijmU
+i
+vuRi+qLiH1mD
+i
+vdRi+ℓLiH1mE
+i
+veRi/parenrightBig
++qLi/tildewideH2[ρU]ijuRj+qLiH2[ρD]ijdRj+ℓLiH2[ρE]ijeRj+h.c., (11)
+whereKis the CKM matrix, ℓH1= ¯νH+
+1+¯eH0
+1,/tildewideHi=iσ2H∗
+i, and the generation indices are written explicitly.
+We are principally interested in the leptons, so we drop the superscr ipt ofρE→ρ.
+If the neutral CP-even Higgses are defined to be handH, withmh≤mH, ifAis the CP-odd Higgs,
+and if flavour violating neutral couplings are allowed in the lepton sect or only, this gives the following Yukawa
+couplings of leptons in the Higgs mass basis
+−Lleptons
+Y=ei/bracketleftbiggmi
+vδijsβ−α+1√
+2([ρ]ijPR+[ρ†]ijPL)cβ−α/bracketrightbigg
+ejh
++ei/bracketleftbiggmi
+vδijcβ−α−1√
+2([ρ]ijPR+[ρ†]ijPL)sβ−α/bracketrightbigg
+eiH
++i√
+2ei([ρ]ijPR−[ρ†]ijPL)eA
++/braceleftBig
+¯ui/bracketleftbig
+[KρD]ijPR−[ρU†K]ijPL/bracketrightbig
+dH++νi[U†ρ]ijPReH++h.c./bracerightBig
+(12)
+where we included the charged Higgs interactions with the quarks. Uis the PMNS matrix, which we henceforth
+drop, assuming that the neutrinos are in the “flavour” basis. For t he quarks with flavour diagonal Yukawa
+couplings the Lagrangian can be obtained by taking [ ρ] diagonal, and substituting {ρE,e} → {ρD,d}for down
+quarks, and {ρE,e} → {ρU,u}for the ups.
+From eqn (12), The neutral Higgs couplings to fermions are :
+gh,ff′=gmf
+2mWsβ−αδf,f′+ρff′√
+2cβ−α
+gH,ff′=gmf
+2mWcβ−αδf,f′−ρff′√
+2sβ−α
+gA,ff′=iρff′√
+2, (13)
+wheregA,ff′appears in the Feynman rule with a γ5. For simplicity, we will take [ ρE] hermitian, and normalise
+it
+[ρ]ij=−˜κij/radicalbigg
+2mimj
+v2=−κijtanβτ/radicalbigg
+2mimj
+v2(14)
+It is tempting to expect |˜κij| ∼1 [40]. However, recall the definition of βτfrom the end of section 2.1: in the
+Higgs basis for the scalar doublets, and the mass eigenstate basis o f the charged leptons, the τYukawa coupling
+is
+ℓL[√
+2mτ
+vH1+ρττH2]τR+h.c.≡yτℓL[cosβτH1−sinβτH2]τR+h.c. (15)
+4This gives ˜ κττ= tanβτso we factor this out and “expect” that κij∼1. Recall that αHB= (β−α) is a physical
+mixing angle defined from the scalar potential, not the difference of t wo angles. In particular, the βin (β−α)
+is unrelated to tan βτ.
+We are interested in τ−µlepton flavour violation, so we allow an arbitrary κτµ, and assume that κτe∼
+κeµ∼0. To perform a general analysis, we should treat the ρφttandρφbbas free parameters, because the angle
+βf, defined for a fermion fin anology with eqn (15), could be different for each f. However, we attribute Type
+II values:
+[ρD]ij=−√
+2tanβτmD
+i
+vδij,[ρU]ij=√
+2cotβτK†mU
+i
+vδij,[ρE]ττ=−√
+2tanβτmτ
+v.(16)
+to all elements of the [ ρ] matrices (except ρτµ), because this parametrisation is adequately representative of t he
+cases we are interested in (see the discussion at the end of section 4.4). The expressions of eqn (16) apply in
+the mass eigenstate bases of {dL,eL,dR,uR,eR}. For a Type III model which has Type II couplings plus small
+corrections (as can arise in Supersymmetry[41, 26]), the ρ−coupling to the top quark is suppressed at large
+tanβ. If in addition, either sβ−αorcβ−αis small, eqn (13) shows that the CP-even Higgs with larger flavour
+-violating coupling, will be weakly coupled to the top. This suppresses both the diagrams in figure 1.
+There are many notations for Type III Yukawa couplings. Our κbears no relation to the one in [28], but is
+proportional to the κof [5](KOT), who define:
+mτ
+vcos2β(κL
+τµ+κR
+µτ)/vextendsingle/vextendsingle/vextendsingle
+KOT=−κτµtanβτ/radicalbiggmτmµ
+v2/vextendsingle/vextendsingle/vextendsingle
+this paper(17)
+The additional power of 1 /cosβin the LFV Lagrangian of KOT causes their LFV rates to scale as tan6β,
+rather than tan4βτas we find. Such differences must be taken into account in comparing plots. To ensure that
+our results are as “physical” as possible, we plot bounds on ( κτµtanβτ), as a function of Higgs masses, sβ−α
+from eqns (6,8) and tan βτdefined from eqn (15).
+3 Higgs mass bounds
+In this section, we list bounds on the parameters of the scalar pote ntial of eqn (1). These bounds were recently
+presented in basis-independent notation, allowing for CP-violation, in [42], where references to the earlier
+literature can also be found. The bounds can be divided into two class es: firstly, those which apply directly to
+the physical masses and coupling constants. These arise from pre cision electroweak analyses (the Tparameter),
+and are the most stringent. Secondly, there are bounds on the λicouplings, which follow from imposing that
+WW→WWis unitary at tree level, and from various considerations about the H iggs potential (positive,
+perturbative...). These must be re-expressed as bounds on the ma sses. In our “phenomenological” approach,
+where we allow arbitrary New Physics at the TeV scale, these bounds are less important. We review them
+briefly anyway.
+3.1 Bounds on the potential
+We are interested in bounds on the various Higgs masses. In the Higg s basis, these are related to potential
+parameters as [28] :
+m2
+H+=M2
+22+v2
+2Λ3 (18)
+m2
+A−m2
+H+=−v2
+2(Λ5−Λ4) (19)
+m2
+H+m2
+h−m2
+A= +v2(Λ1+Λ5) (20)
+(m2
+H−m2
+h)2= [m2
+A+(Λ5−Λ1)v2]2+4Λ2
+6v4(21)
+sin[2(β−α)] =−2Λ6v2
+m2
+H−m2
+h(22)
+TheH±mass can be raised high enough to respect bphysics bounds (see section 3.2) by increasing M2
+22(the
+decoupling limit [33]). To keep H,Alight in this limit requires large Λs.
+53.1.1 vacuum stability/bounded from below (lower bound on mφ)
+The requirement that the electroweak vacuum be stable, or that t he Higgs potential be bounded from below,
+gives a lower bound on mh. See [43, 44] for a review. The constraint is imposed at tree level; on e can also
+include radiative corrections and check that potential remains bou nded from below.
+Neccessary and sufficient conditions to obtain V(vi→ ∞)>0, in the softly broken 2HDM type II (in the
+basis where λ6=λ7= 0) were given in [33]. A basis-independent analytic discussion of Type II and Type III
+can be found in [45]. However, the Type III bounds do not give simple a nalytic formulae; a more extensive and
+numerical analysis was performed by [46, 47], who find a lower bound
+mH>121 GeV . (23)
+3.1.2 triviality/perturbativity (upper bound on mφ)
+An upper bound on the Higgs masses (which are ∝λi) can be obtained from requiring that the λicouplings
+remain perturbative at higher energy scales. One can, for instanc e, impose that the Landau Pole of the Higgs
+couplings be at some sufficiently high scale. The analysis of [46] finds no upper bound on m2
+H,A,H+for a softly
+broken Type II, or Type III.
+3.1.3 Unitarity
+In the spontaneously broken electroweak theory, there are delic ate cancellations in the tree-level amplitude
+for longitudinal Wscattering, between diagrams with Higgs or gauge boson exchange . In addition, at scales√s≫the Higgs masses, the various scattering amplitudes are proportio nal to combinations of the λicouplings.
+S-matrix unitarity gives an upper bound on the S-matrix elements. I f the tree level calculation is a good
+approximationto the full S-matrix, then this bound translatesto b ounds on the Higgsmasses. These constraints
+are most interesting for the J= 0 partial wave. This is discussed for the Standard Model in [8, 48], a nd in
+[49, 50] for the 2HDM (see also the appendix of [42]).
+S-matrix unitarity implies that 1 = SS†= (1+iT)(1−iT†). Writing the S-matrix between states labelled
+by angular momentum J,K,..and all other quantum numbers called j,k..., this gives
+−2Im{∝an}bracketle{tj,J|T|k,K∝an}bracketri}ht}=∝an}bracketle{tj,J|TT†|k,K∝an}bracketri}ht (24)
+which, applied to the J= 0 component of an amplitude:
+a0=1
+16πs/integraldisplay0
+−sdtA
+implies 2 |Im{a0}|>|a0|2.
+There are two limits in which unitarity bounds are usually calculated. Th e standard calculation assumes
+m2
+W≪m2
+φ≪s, neglects all masses( φis an arbitrary Higgs), describes the longitudinal gauge bosons as H iggses
+via the Equivalence Theorem(see [51] for a pedagogical discussion) , and obtains bounds on linear combinations
+of the Λ i(see [50, 42] for a basis-independent calculation in the 2HDM). Notic e that in the strict Type II model
+(m2
+12= 0 in the basis where λ6=λ7= 0), these bounds are quite sensitive to tan β[49]; this may be related to
+the appearance of a massless Higgs, in the limit where one of the vevs vi→0. This is no longer the case when
+m2
+12∝ne}ationslash= 0 is allowed [52, 50].
+Here, we are more interested in the limit m2
+W≪s≪m2
+φ, whereφis some of the Higgses. That is, we allow
+arbitrary New Physics at the TeV scale to preserve S-matrix unitar ity in the m2
+W≪m2
+h≪slimit, but we
+would like to know the bounds arising on mass splitting among the Higgse s, at√s <TeV, when some Higgs
+are heavier than√s, and some are lighter.
+The gauge contribution to longitudinal Wscattering (no Higgses exchanged in sortchannel), is [48]
+A(WLWL→WLWL) =ig2
+4m2
+W(s+t)+... no Higgs . (25)
+Requiring a0<1 implies the scale of Higgs masses should be <∼TeV. In the case where some Higgses are light,
+for instance mh< s, including them in the amplitude reduces the coefficient of s+t, and raises the upper bound
+on the masses of the remaining Higgs. We conclude that unitarity con straints do not give us relevant bounds
+on the mass differences among the Higgses.
+63.1.4 Electroweak precision tests
+New physics that couples weakly to Standard Model fermions, but h as electroweak gauge interactions, can be
+constrained by the measured values of the “oblique parameters” [5 3, 54]. If the vacuum polarisation tensor
+between gauge bosons iandjis defined as
+Πµν
+ij(q) =gµνAij(q2)+qµqνBij(q2) (26)
+then the parameters SandTcan be defined as [11]
+¯S=α
+4s2
+Wc2
+WS=AZZ(m2
+Z)−AZZ(0)
+m2
+Z+∂Aγγ
+∂q2/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+q2=0+c2
+W−s2
+W
+sWcW∂AγZ
+∂q2/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+q2=0(27)
+¯T=αT=AWW(0)
+m2
+W−AZZ(0)
+m2
+Z(28)
+wheresW= sinθW. The Standard Model contributionsto these parameters, includin g that ofthe SM Higgs, are
+assumed to be subtracted out. The contributions to S,T,andU, (as well as V,WandX), due to an arbitrary
+number of Higgs doublets and singlets have recently been calculated in [11].S,TandUwere calculated in the
+CP violating 2HDM in [42]. We use here the formulae of [11], which verify th e previous calculations of [55, 56]
+for the 2HDM.
+As discussed in [42], the 2HDM contributions to SandUtend to be small enough, but the contribution
+toTcan exceed the value allowed for New Physics ( −0.15< T <0.20 at one σ[53]). We impose the bound
+−0.05< T−S <0.10 at one σ[53]. The calculation of Tin multi-Higgs models is presented in detail in the
+first paper of [11]. They give
+T=1
+16πs2
+Wm2
+W/braceleftbig
+F(m2
+A,m2
+H+)+c2
+β−α[F(m2
+H+,m2
+h)−F(m2
+A,m2
+h)]
++s2
+β−α[F(m2
+H+,m2
+H)−F(m2
+A,m2
+H)]
+−3c2
+β−α/bracketleftbig
+F(m2
+Z,m2
+h)−F(m2
+W,m2
+h)+F(m2
+W,m2
+H)−F(m2
+Z,m2
+H)/bracketrightbig/bracerightbig
+(29)
+where
+F(x,y) =x+y
+2−xy
+x−ylnx
+y(30)
+is a positive function, that vanishes for degenerate masses. So as masses in the loop split, the contribution
+increases.
+It is well known, and clear by inspection, that Tbecomes small in various limits, such as mA→mH+, or
+mH→mH+whensβ−α→1. When studying handHproduction and decay at colliders, we will assure the
+precision constraint by imposing mA≃mH+±10 GeV. For hproduction, we do not consider the parameters
+mH→mH+andsβ−α→1, because σ(gg→h→τ±µ∓)∝s2
+β−αc2
+β−α. The collider cross-section for the
+pseudo-scalar Ais not suppressed by sβ−αorcβ−α, so when we study σ(gg→A→τ±µ∓), we can ensure the
+precision constraint by requiring mH≃mH+±10 GeV and sβ−α→1. FormA>100 GeV, T−Sis within
+the 1σallowed range for π/3< β−α≤π/2.
+3.2 Flavour physics bounds
+The charged Higgs H+neccessarily has tree level flavour-changing couplings, like the W+. Its mass is therefore
+constrained by various flavour-changing observables, such as b→sγandB+→τ+ν. The various bounds have
+been discussed in [57], who find mH+>250GeVfor 2<tanβ <20, andmH+>300GeVfor tanβ <50.
+This limit is lower than that of Misiak et al.fromb→sγ[58], which is mH+>300 GeV (at 2 σ), because of
+differences in the procedure of extraction of the bound.
+The charged Higgs contributes at tree level to the decay of pseud oscalar mesons M. Since the SM W-
+mediated amplitude is helicity suppressed, the additional suppressio n of the Higgs amplitude is only a factor
+∼(mM/mH+)2[59]. Charged meson decays, such as B+→τ+ν, therefore constrain the mass and couplings
+ofH+in type I and type II models. In addition, they can constrain the flav our-violating couplings of type III
+models. It was shown in [60], that a precise determination of RK= Γ(K−→e¯ν)/Γ(K−→µ¯ν), by the NA62
+experiment, could be a sensitive test of the ρτµcoupling.
+73.3 Summary
+We retain two constraints from this section: mH+>∼300 GeV from B physics (as discussed in section 3.2), and
+theTparameter will be small enough in two cases: either mA≃mH+( we take numerically mA=mH+±10
+GeV, or mH≃mH+, withβ−α>∼π/3 to ensure cβ−α→0.
+4 Constraints on flavour changing Yukawa couplings to lepton s
+In this section, we include the lepton flavour-changing Yukawa coup lings of the Higgses, see eqn (12). We
+consider mass ranges for the neutral Higgs which are consistant w ith the constraints discussed in the previous
+sections, and study the sensitivity of ( g−2)µ,τ→µγandτ→ηµto the the flavour-changing coupling ρτµ,
+introduced in eqn (14).
+A systematic study of bounds on a 2HDM (Type II), with lepton flavo ur violating Yukawa couplings, has
+been performed in [5]. They impose constraints on ρτµarising from tree and one-loop contributions to rare
+decays, then study the φ→τ¯µdecay rate at colliders. As this is a Type II analysis, tan βappears in the
+formulae. In the context of supersymmetric models [27], it was show n in [68] that the bounds from one and
+two loop contributions to τ→ℓγare more stringent than those from τ→3ℓ(except if the Higgses are very
+degenerate).
+Theanomalousmagneticmomentofthemuoncanalsobesensitivetoa 2HDMwithflavour-violatingYukawa
+couplings [3, 4, 69, 71]. For ρτµ∼10√mτmµ/v, the 2HDM can fit the current discrepancy in ( g−2)µ. However,
+as we will show, the constraint from τ→µγprecludes the explanation of the ( g−2)µdiscrepancy, for large
+areas of parameter space.
+4.1 The decay τ→ηµ
+It was shown in [5], that τ→ηµgives a relevant constraint on κτµfor light pseudoscalars A. We briefly review
+here their discussion, using the updated bound [61] BR(τ→ηµ)<6.5×10−8.
+The branching ratio is
+BR(τ→ηµ)
+BR(τ→µν¯ν)≃54π2/parenleftbiggmη
+mA/parenrightbigg4f2
+η
+m2τ|κτµ|2tan2βτmµ(mucotβτ+(md−2ms)tanβτ)2
+(mu+md+4ms)2(31)
+whereBR(τ→µν¯ν) =.17, andfη≃fπfrom [62]. The current experimental upper bound gives
+|κτµ|tan2βτ<70/parenleftBigmA
+100GeV/parenrightBig2
+(32)
+where we approximate the final fraction of (31) as tan2βτ/4. The bound (32) agrees with the result given in [5].
+This is comparable to the bound from τ→µγ(see figure 3); we see that a light Ais allowed for small tan βτ.
+4.2 The dipole effective operator
+Bounds on a large class of effective operators that change τ−µflavour are presented in [62]. Several bounds
+arise from the dipole operators, which can be included in an effective L agrangian as [63]
+Cij
+Λ2
+NP∝an}bracketle{tH∝an}bracketri}hteiσαβPRejFαβ+h.c. (33)
+wherei,jare the flavours of the external leptons. The chirality flipping oper ator must contain an odd number
+of Yukawa couplings, including ρτµand a flavour diagonal Yukawa coupling of the external Higgs vev. T he
+model-dependent coefficient Cij/Λ2
+NP, can be related to a New Physics contribution δaµto the anomalous
+magnetic moment of the muon, and to the AL,Rfactors that appear in τ→µγ[63]:
+eδaµ
+4mµ=Re{Cµµ}
+Λ2
+NPv√
+2emτAτµ
+R
+2=Cτµ
+Λ2
+NPv√
+2emτAτµ
+L
+2=Cµτ∗
+Λ2
+NPv√
+2(34)
+TheAL,Rappear in the τ→µγbranching ratio as:
+Γ(τ→µγ)
+Γ(τ→µν¯ν)=48π3α
+G2
+F/parenleftbig
+A2
+L+A2
+R/parenrightbig
+<4.4×10−8
+.17(35)
+8where we used the current τ→µγbound [64] (very similar to the BR <4.5×10−8of [65]) to obtain
+|AL|=|AR|<∼10−4GF.
+The current experimental and theoretical determinations of ( g−2)µare[53]:
+(g−2)µ
+2=aµ=/braceleftbigg
+11659208 .0(5.4)(3.3)×10−10expt
+11658471 .810(0.016)+691 .6(4.6)(3.5)+15.4(.2)(.1)×10−10QED+hadronic +EW
+(36)
+where the first experimental uncertainty is statistical and the se cond systematic. The hadronic uncertainties
+are from lowest and higher order. The electroweak contribution, w hich at one loop is
+aEW,1−loop
+µ ≃GFm2
+µ
+8√
+2π2/bracketleftbigg5
+3+1
+3(1−4s2
+W)/bracketrightbigg
+(37)
+includes the two loop effects (a ∼25% correction), and the electroweak uncertainties are from the unknown
+Higgs mass and quark loop effects. The difference between experime nt and theory is
+∆aµ=aexp
+µ−aSM
+µ= 29.2(6.3)(5.8)×10−10(38)
+sowhenweaskourmodel to“fitthe ( g−2)µdiscrepancy”, wewillaskit tocontribute δaµ∼aEW
+µ∼+15×10−10
+to the theoretical calculation of aµ.
+We now calculate the bounds from a selection of one and two loop diagr ams contributing to ( g−2)µand
+toτ→µγ. We aim to describe the leading constraints. The model we consider h as many unknown Yukawa
+interactions, so which diagramsgivethe most “interesting”limits dep ends onwhat is assumedabout the pattern
+of Yukawa couplings. For instance, a two-loop diagram that is linear in a small Yukawa coupling can give a
+more significant bound than a one loop diagram that is quadratic in tha t same Yukawa coupling [17]. Various
+theoretically motivated patterns for the Yukawas have been discu ssed in the literature [40, 41, 66], and many
+analyses concentrate on the supersymmetric case, where the fla vour-violating Yukawas are loop induced, and
+ρtt∝cotβ,ρτµ∝tanβ.
+We only include diagrams that contain the flavour changing couplings5ρτµ. It appears squared in the
+lepton flavour conserving amplitude aµ, and in the lepton flavour changing rate Γ( τ→µγ).
+At one loop, three Yukawa couplings are required on the fermion line o f the dipole operator, so we expect
+Cµµ∝ |gφτµ|2mτandCµτ∝gφττgφτµmτ. We do not consider possible large logs that could arise from
+electroweak corrections to these diagrams. We do consider two loo p contributions to Cτµwhich have only one
+Yukawa coupling on the lepton line. The other end of the φpropagatormust attach somewhere, and an external
+Higgs vev is required for the dipole operator. If this pair of Higgs cou plings can be large (for instance gauge,
+third generations Yukawas, or scalar self-interactions) they cou ld compensate the 1 /(16π2) relative to the one-
+loop diagram. Therefore we include the two-loop “Barr-Zee” diagra ms (figure 1) with third generation quarks,
+and with Wbosons. We neglect “Barr-Zee” diagrams with charged Higgses in th e second loop for simplicity,
+and because these loops are relatively suppressed6.
+Our formulae are incomplete, possibly gauge dependent (the Wloop diagrams of figure 2), and we have not
+excluded large logs in higher loop corrections (as in the SM electrowea k contribution to g−2 [67]). Due to
+these uncertainties, we will be reluctant to exploit cancellations bet ween diagrams.
+4.3 g-2
+There are one loop (and higher order) contributions to aµmediated by the flavour conserving couplings of the
+2HDM. The possibility of explaining the ( g−2)µdiscrepancy with these terms was discussed in [19], who show
+that there is little parameter space consistent with other constra ints on the 2HDM. We assume that ( g−2)µ
+does not constrain the flavour-conserving parameters of our mo del[57].
+In the presence of the flavour changing ρτµcoupling, there is a one-loop contribution to ( g−2)µ, illustrated
+on the left in figure 2. As discussed above, it should be of order the o ne-loop electroweak gauge contribution, or
+5So we neglect the flavour diagonal contributions to ( g−2) — see the references at the beginning of section 4.3.
+6Such diagrams could be expected to give significant contribu tions, because they are proportional to quartic Higgs coupl ings,
+which can be ∼1 because we have mass2differences in our Higgs spectrum ∼v2. However, the amplitudes are suppressed by
+m2
+φ/m2
+H+, and there is a partial cancellation between the H+loops with both photons attached in the same place and with th e
+two photons attached separately [22]. The H+loops are therefore an order of magnitude smaller than the to p loop, for quartic
+Higgs couplings ∼1. Secondly, the relevant Higgs couplings (see eqn (9)) Λ 3and Λ 7, are independent of the Higgs masses (see
+eqns (18) to (22)), so are free parameters which just add more confusion to the analysis. We are reluctant to allow |Λ3|,|Λ7|>∼3,
+which would allow to (partially) cancel the top loop.
+9Figure 2: One and two loop diagrams, which contribute to the dipole ope rator coefficient Cij
+eγ(see eqn (33)). A mass
+insertion is required on the fermion line of the one loop diag ram. For k=τandj=i=µ, these diagrams depend only
+on the Yukawa couplings ρτµ.
+less. The chirality flip is provided by the τmass, which increases this diagram with respect to flavour diagonal
+ones. Neglecting the lepton masses in the kinematics, the one loop co ntribution gives [3]
+a2HDM,LFV
+µ ≃/summationdisplay
+φg2
+φτµmµmτ
+8π2/integraldisplay1
+0dxx2
+m2
+φ−x(m2
+φ−m2τ)
+≃/summationdisplay
+φg2
+φτµmµmτ
+8π2m2
+φ/parenleftBigg
+lnm2
+φ
+m2τ−3
+2/parenrightBigg
+(39)
+whereφ=h,H,A, andgφff′is from eqn (13).
+In the absence of cancellation between the opposite sign CP-even a nd CP-odd Higgs diagrams,
+κτµtanβτ<∼32mφ
+100GeV(40)
+contributes at most the experiment-theory discrepancy in ∆ aµ∼2aEW
+µ[3]. This bound agrees with the naive
+power counting expectation thatmτ
+mµ|ρτµ|2/m2
+φ<∼GF.
+4.4τ→µγ
+The two body decay τ→µγcan arise via loops, and the current bound on the Branching Ratio is g iven in
+eqn (35). Since the Standard Model decay τ→ℓν¯νis three body, the bound on New Physics in a loop is
+particularily restrictive, because the |1/16π2|2loop factor is partially compensated by the two to three body
+phase space ratio.
+To estimate BR(τ→µγ) in our model, we assume that AL=AR≡A, and neglect the lepton masses
+in the kinematics. We include the one-loop diagram of figure 2 (the firs t sum below), and a subset of two-
+loop diagrams. Following [22], we include the two-loop diagram of figure 1 , with an internal photon and a third
+generation quark. This is the second sum below, with f=t,b, and is gauge invariant on its own. The remainder
+of eqn (41) is a subset of the two loop Wdiagrams (as in figure 2) [22]. The amplitude is:
+A≃1
+16π2
+√
+2/summationdisplay
+φgφµτgφττ
+m2
+φ/parenleftBigg
+lnm2
+φ
+m2τ−3
+2/parenrightBigg
++2/summationdisplay
+φ,fgφµτgφffNcQ2
+fα
+π1
+mτmffφ(m2
+f
+m2
+φ)
+−/summationdisplay
+φ=h,HgφµτCφWWgα
+2πmτmW
+3fφ(m2
+W
+m2
+φ)+23
+4g(m2
+W
+m2
+φ)+3
+4h(m2
+W
+m2
+φ)+m2
+φfφ(m2
+W
+m2
+φ)−g(m2
+W
+m2
+φ)
+2m2
+W
+
+(41)
+whereφ=h,H,A,f=t,b, the coupling gφff′of the internal loop fermion to the scalar φis given in eqn
+(13), and the scalar- W+W−couplings CφWWare given in eqn (7). There is a factor mτin the denominator
+10of the two-loop expressions because it appears in the definition (34 ), and a factor of the loop mass because the
+functions f(z),g(z),h(z) are proportional to this mass2, whereas the loop has a single mass insertion.
+The various functions are [22]:
+fA(z)≡g(z) =z
+2/integraldisplay1
+0dx1
+x(1−x)−zlnx(1−x)
+zpseudoscalar (42)
+fh,H(z) =z
+2/integraldisplay1
+0dx(1−2x(1−x))
+x(1−x)−zlnx(1−x)
+zscalars, (43)
+and
+h(z) =−z
+2/integraldisplay1
+0dx
+x(1−x)−z/bracketleftbigg
+1−z
+x(1−x)−zlnx(1−x)
+z/bracketrightbigg
+. (44)
+They are ∼zfor arguments zof order 1, and for small z,fφ(z)∼z
+2(lnz)2.
+Theτ→µγamplitude Adepends on tan βτ, (β−α), and the Higgs masses. It is useful to estimate the
+relative size of the one-loop, two-loop fermion, and two-loop Wcontributions in various cases.
+1. We start by estimating the one-loop diagrams contributing to τ→µγ. In the absence of cancellations
+between the opposite sign CP-even and CP-odd Higgs diagrams, the approximate τ→µγbound|A|<
+10−4GFgives
+κµτκττtan2βτ
+8π2/radicalbiggmµ
+mτm2
+τ
+m2
+φlnm2
+φ
+m2τ<10−4(45)
+or, forκττ≃1:
+κµτtan2βτ<∼160/parenleftbigg100GeV
+mφ/parenrightbigg2
+. (46)
+which is weaker than the estimated ( g−2) bound of eqn (40) for small tan βτ, and more restrictive as
+tanβτgrows. In the case φ=A, the bound (32) from τ→ηµis more restrictive.
+2. Consider now the two-loop contributions to τ→µγ, starting with the top loop. For small tan βτ, or large
+mixingsβ−α∼cβ−αbetween the Higgses, this can be the dominant contribution to τ→µγ. The ratio of
+the top-loop to one loop amplitudes, induced by a particular Higgs φ, is
+2loop top
+1loop∼αQ2
+t(cβ−αsβ−α+cotβτ)m2
+t
+mτvm2
+φm2
+φv
+mτκττtanβτln(mφ
+mτ)
+∼αQ2
+t(cβ−αsβ−α+cotβτ)m2
+t
+m2τ1
+κττtanβτln(mφ
+mτ)(47)
+wheregφff′is from eqn (13). So for κττ∼1, andcβ−αsβ−α∼1, the 2-loop top diagram dominates over
+the one loop for tan βτ<∼mt/mb. This is because all the Higgses have an O(mt/v) coupling to the top,
+independently of tan βτ. However, the cβ−αsβ−αterms of the handHamplitudes have opposite sign
+(recall that gAff′is independent of β−α). Imposing the approximate τ→µγbound|A|<10−4GFon
+the top amplitude, for case 1: mh≃mH≪mA, and for case 2: mh≃mA≪mHwithsβ−a∼1, implies
+10>∼/braceleftBigg /parenleftBig
+m2
+H−m2
+h
+m2
+hsin2(β−α)+1
+tanβτ/parenrightBig
+κτµtanβτcase1
+κτµ case2(48)
+which is somewhat more restrictive that the estimated ( g−2) bound of eqn (40), and one-loop τ→µγ
+bound of eqn (45). However, for large tan βτ, smallcβ−αsβ−α<cotβτandκττ∼1, the one-loop
+contribution is larger..
+3. Superficial inspection suggests that the two loop diagrams with a n internal borWloop, never provide a
+dominant contribution (although they are included in our plots). Par ametrically, the two loop contribu-
+tions of the t,Wandbgive the following three terms
+κτµtanβτ
+m2
+φ/parenleftbigg
+NcQ2
+tm2
+t[cβ−αsβ−α+cotβτ]+cβ−αsβ−αm2
+W+NcQ2
+bm2
+btanβτln(mφ
+mb)/parenrightbigg
+(49)
+11so one sees that the Wloop can be neglected with respect to the tops. Notice from eqns (1 3) and (7), that
+whensβ−αorcβ−αis small, one of horHis “SM-like”, meaning that it has coupling ∼gmWtoW+W−,
+flavour-diagonal couplings ∼mf/vto the fermions, and suppressed flavour-changing interactions. So the
+effective interaction of W+W−τ¯µ, induced by the neutral Higgses, is ∝sin2(β−α). Similarly, the Higgs-
+induced t¯tτ¯µis∝sin2(β−α)+O(ρφtt). This is in accordance with [22], who observe that the Wloop
+contribution vanishes in the decoupling limit due to unitarity argument s. It disagrees with the MSSM
+analysis of [68], where the Wcontribution is included but the tops are not.
+Comparing the one loop amplitude to the two loop bottom amplitude, fo r large tan βτ<1/(cβ−αsβ−α)
+when the bloop could dominate the top, one obtains:
+1loop
+2loop bottom∼κττtanβτmτln(mφ
+mτ)
+vm2
+φmτvm2
+φ
+αQ2
+bm2
+btanβτln(mφ
+mb)∼κττm2
+τ
+αQ2
+bm2
+b(50)
+Thissuggeststhatunless κττ≪1,theone-loopcontributionislargerthanthetwo-loopwhen1 /(cβ−αsβ−α)>
+tanβτ>∼3.
+4. Finally, recall that cancellations can occur in the sum. There is a re lative negative sign between the
+CP-even and CP-odd Higgs diagrams (which has little effect in many of o ur plots because we set mA≃
+mH+≃300 GeV, to minimise the Tparameter), and also between the handHinduced two loop top
+amplitudes at negligeable cot βτ, which would cancel for m2
+h=m2
+H.
+The bound on κτµtanβτfromτ→µγis plotted in figure3, for mh= 117GeV, mH= 130GeV, mH+= 300
+GeV,|mA−mH+|= 10 GeV, and κττ= 1. As expected, for tan βτ>∼few and small sin( β−α), the amplitude
+is dominated by the one-loop contribution, and the constraint on κτµtanβτscales as 1 /tanβτfor fixed β−α.
+ tan  beta5
+10
+15
+20
+25
+30
+35 sin (b-a)
+0.10.20.30.40.50.60.7110t mbound on tan(beta) kappa
+Figure 3: Bound on κτµtanβτfrom the experimental limit on BR(τ→µγ), as a function of tan βτand sin(β−α), for
+mh= 117 GeV, mH= 130 GeV, mH+= 300 GeV and |mA−mH+|<20GeV. We take κττ= 1.
+The relative importance of the ( g−2) andτ→µγconstraints is illustrated in figure 4, for a particular
+choice of Higgs masses. The two loop contributions to τ→µγ(discussed above) are also included in this plot.
+Forκτµ= 1, the double ratio of the predicted a2HDM,LFV
+µ from eqn (39) over the ( g−2) discrepancy (taken to
+be 15×10−10), is divided by the predicted τ→µγbranching ratio over the current bound:
+a2HDM,LFV
+µ
+∆aµ
+BR2HDM,LFV (τ→µγ)
+BRdata(τ→µγ)(51)
+Since both predictions scale as |ρτµ|2, this cancels in the ratio, which therefore quantifies the relative sig -
+nificance of the bounds. Since the ratio <1, we conclude that for generic mass choices that give a small T
+12parameter, it is not possible to take ρτµlarge enough to fit ( g−2)µ, without exceeding the bound from τ→µγ.
+The plot is obtained by a grid scan. We consider β−α: 0→π/4, because the range π/4→π/2 is equivalent
+if one simultaneously exchanges7mhandmH.
+ tan  beta
+5101520253035 sin (b-a)
+0.10.20.30.40.50.60.700.10.20.30.40.5g-2/tmg
+Figure4: The double ratio, givenin eqn(51), ofthe contributionto ( g−2)µdividedbyhalf theexperimental discrepancy,
+over the contribution to τ→µγdivided by its experimental limit. It is plotted as a functio n of tanβτand sin(β−α), for
+mh= 117 GeV, mH= 130 GeV, mH+= 300 GeV, and |mH+−mA|= 10 GeV. Both contributions are ∝(κτµtanβτ)2,
+which cancels in the ratio. Assuming that our mass choices ar e representative, this shows that the current experimental
+bound on τ→µγis more restrictive than ( g−2)µ.
+4.5 Assumptions about the ρff
+As previously mentioned, the bounds on ρτµdepends on the assumptions made for the other model parameter s,
+such asρττ,ρtt,ρbb, and the Higgs masses. We focus on three limits.
+The most conservative approach to setting bounds on ρτµ, is to neglect all other couplings. In this case, the
+(g−2) bound will hold, and vary as a function of the neutral Higgs masse s, but the τ→µγbound plotted
+in figure 3 does not apply because it assumed ρmatrices as given in eqn (16). Neglecting the flavour diagonal
+Yukawa couplings of the flavour violating φ, is equivalent to setting them to zero. So this case corresponds to
+cβ−α= 0, where one CP-even Higgs ( h) is SM-like, and both HandAhave the flavour-violating ρτµcoupling,
+butnotree-levelinteractionswith ts,bsorWs. SuchadditionalHiggseswouldbedifficult toproduceatcolliders,
+so we do not consider further this case.
+We are interested in a Higgs that could be copiously produced at hadr on colliders in ggfusion. This can
+be obtained by allowing the flavour-violating Higgs an O(1) Yukawa coupling to the top. This is realised for
+sin2(β−α)∼1, in the parametrisation of Yukawa couplings given in eqns (13) and ( 16). In this case, the top
+loop contribution to τ→µγ(see figure 1) is significant at small tan βτ, as discussed in section 4.4 and the
+diagram for the gg→τ¯µprocess at colliders is closely related to the τ→µγdiagrams, as expected from the
+diagrams in figure 1.
+Finally, consider the large tan βτlimit. Recall that we defined tan βτin our type III model using the tau
+Yukawa coupling (see eqn (15)). Then we assumed that all the othe r fermions shared this definition of tan βτ,
+and had type II Yukawa interactions. Such “almost Type II” couplin gs could arise in Supersymmetry, where
+the flavour violating Yukawas grow in the large tan βlimit [26]. For large tan βτ, the bound on κτµtanβτarises
+from the one loop contributions to τ→µγ, and is (approximately) independent of sin2( β−α). However, the
+Higgs-induced ¯ttτ¯µinteraction is ∝sin2(β−α), so, as we will see in the next section, gg→φ→τ±µ∓is
+suppressed at small sin2( β−α). This can be seen by comparing figures 5 and 3 at small8sβ−α.
+7The Higgs couplings are unchanged if one interchanges hwithH, and simultaneously ( β−α)→(β−α)−π/2. And since h
+andHare summed in the amplitude A:A(cβ−α→0) =A(sβ−α→0)+ corrections due to m2
+H∝negationslash=m2
+h.
+8Small sin2( β−α) can be obtained because either sβ−αorcβ−αis small. Both cases are discussed in the supersymmetric
+134.6 Summary
+Formuchoftheparameterspaceofthismodel, theboundfrom τ→µγprecludesexplainingthe g−2discrepancy
+(see figure 4). We will see in the following section, that generic 2HDM p arameters giving a detectable rate for
+p¯p→φ→τ±µ∓in current Tevatron data are already excluded by precision bounds . However, a signal at the
+Tevatron could arise if there are cancellations in the τ→µγamplitude.
+5 Colliders
+The leading order production cross-section of Higgses, by gluon-g luon fusion at a p¯pcollider, is [9]
+σLO(p¯p→φ) =GFα2
+sm2
+φ
+288√
+2πs/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle3
+4/summationdisplay
+q=t,bgφqqAφ(zq)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/integraldisplay1
+zφdx
+xg(x)g(x/zφ) (52)
+wherezφ=m2
+φ/s,zq= (4m2
+q)/m2
+φ, the gluon density in the proton (or anti-proton) is g(x),
+Ah,H(z) = 2z[1+(1−z)f(z)] (53)
+AA(z) = 2zf(z) (54)
+and
+f(z) =
+
+arcsin2/radicalBig
+1
+zz≥1
+−1
+4/bracketleftBig
+log1+√1−z
+1−√1−z−iπ/bracketrightBig2
+z <1(55)
+The next order QCD corrections to the production cross section a re∼20−90%, and can be mimicked by a K
+factor [9]. Finally, to obtain the cross-section for p¯p→τ¯µ, the cross-section (52) should be multiplied by the
+branching ratio BR(φ→τ¯µ).
+The Tevatron searches for SM Higgses, decaying to τ¯τ, in the mass range 105- 145 GeV [72, 73]. A recent
+D0 analysis [73] obtains limits on the cross-section ×branching ratio of order 20-90 ×the SM expectation. If
+we imagine that the τ¯µfinal state is detectable with similar efficiencies to τ¯τ, then the parameters to which the
+Tevatron is sensitive can be estimated as follows. We normalise the φproduction cross-section and branching
+ratio to the SM expectation for hSM→τ+τ−:
+Rσ
+φ≡σLO(p¯p→φ)
+σLO(p¯p→hSM)=/vextendsingle/vextendsingle/vextendsingle3
+4/summationtext
+q=t,bgφqqAφ(zq)/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingle3
+4gmt
+2mWAh(zt)/vextendsingle/vextendsingle/vextendsingle2(56)
+RBR
+φ≡BR(φ→τ¯µ,µ¯τ)
+BR(hSM→τ¯τ)
+≃2κ2
+τµtan2βτmµ/mτ/vextendsingle/vextendsingle/vextendsinglev
+mτgφττ/vextendsingle/vextendsingle/vextendsingle2
+(1−B2W)+|cφWW|2B2W×
+
+s2
+β−αφ=h
+c2
+β−αφ=H
+1φ=A(57)
+wherehSMis the Standard Model higgs, mhSMis taken to be equal to mφandB2W=BR(hSM→W+W−)
+which varies with mh. We estimate that
+Rσ
+φRBR
+φ<∼30 (58)
+because the Tevatron limit on σ(p¯p→hSM→τ¯τ) is∼20−90×the SM expectation9. In the large mA
+scenario, where the Tevatron would be looking for a CP-even φ, our lepton flavour violating rate is maximised
+for large sin2( β−α)∼1 and small tan βτ= 2. The Tevatron bound would be of order
+κτµtanβτ<∼/radicalbiggmτ
+mµ(20to90)∼20 (59)
+which is on the border of the exclusion estimate from τ→µγ, given in eqns(45) and (48). In the large mH
+scenario, where sβ−α→1, the Tevatron could look also for A→τ±µ∓. At small tan βτ, the production rate is
+14 tan  beta5101520253035 sin (b-a)
+0.10.20.30.40.50.60.7
+-1210-1110-1010-910-810-710-610-510-410-310-210-1101sigma/tmg
+Figure 5: Estimate of the relative sensitivity of σ(p¯p→h→τ±µ∓) andτ→µγtoρτµfrom current data. The plot
+is the double ratio of Rσ
+hRBR
+h(see eqns 57 and 56) over BR(τ→µγ), divided by current experimental limits (see eqns
+(35) and (58)) , as a function of tan βτand sin(β−α) (κτµtanβτcancels in the ratio). The plot is for hproduction at
+colliders, with mh= 115 GeV, mH= 130 GeV, mH+= 300 GeV and |mA−mH+|= 10GeV. Since the ratio <1,τ→µγ
+is more sensitive. Alternatively, this plot indicates the a mount of tuning required in the τ→µγrate to accomodate a
+signal for h→τµat the TeVatron in the near future.
+similar to the standard model, so the bound should be of order eqn (5 9); for large tan βτ,Aproduction in gg
+fusion is suppressed.
+A more credible estimate of the relative sensitivity of colliders and τ→µγto the parameter κτµtanβτ,
+could be obtained from a double ratio, as discussed for ( g−2) andτ→µγat the end of section 4.4. We plot in
+figure 5 the double ratio of Rσ
+hRBR
+h/[the current bound on hSM→τ¯τ], divided by the predicted BR(τ→µγ)
+normalised to its current experimental bound. Both rates are ∝(κτµtanβτ)2, so this cancels out of the ratio.
+This plot is made for the production, and decay to τ±µ∓, of a CP even Higgs hof mass 115 GeV, with
+mH= 130 GeV, mH+= 300 GeV and |mA−mH+|= 10 GeV. If mA≃mH+>∼250 GeV for B physics and
+the T parameter, then the current bound from τ→µγmakes it difficult to detect h→τ¯µat the TeVatron10.
+The sensitivity to H→τ¯µis worse, in the interesting sin( β−a)∼cos(β−a)∼tanβτregion, due to sums and
+differences among these angles all of the same orders. However, a hadron collider able to detect hSM→τ¯τ(an
+improvement of ∼30 with respect to the current Tevatron limit), would be more sensit ive than τ→µγin the
+sin2(β−a)>∼1/2 and small tan βτregion. Also, cancellations are possible in the τ→µγamplitude, which
+would increase the sensitivity of colliders relative to rare decays.
+There are dubious approximations in obtaining this plot. We neglect th e contribution of the φ→τ¯µ
+decay in computing the total φdecay rate (this approximation was also used to obtain eqn (59)). W e make
+this overestimate of the branching ratio, so as to obtain a formula w hich is∝ |gφτµ|2, so easy to compare to
+BR(τ→µγ). It is reasonable where τ→µγimposes κτµtanβτ<∼3 (because then Γ( φ→τ¯µ)<∼Γ(h→τ¯τ),
+and the contribution to the total decay rate is not to important); however, in the interesting small tan βτand
+large sin2( β−α) region, it could overestimate σ(p¯p→h→τ±µ∓by a factor ∼2 (see figure 3). A second
+doubtful approximation is that the higher order QCD corrections c ancel in eqn (57). This may be acceptable
+whenφandhSMare emitted from a top loop, but in the large tan βτlimit, the bloop can also be important for
+φproduction, and its NLO corrections are different. Finally, the expe rimental bound we use from the search for
+hSM→τ¯τ[73], allows for several Higgs production mechanisms: associated pr oduction with a WorZ, vector
+boson fusion, or gluon fusion, and assumes a final state of τ¯τ+ 2 jets. Whereas in (57), we assume production
+by gluon fusion.
+analysis of [68]. Recall that cβ−α→0 in the decoupling limit [33], where A,HandH+are “heavy” ( >∼2mW), and the light higgs
+hhas almost SM couplings.
+9we use figures from table VI of [73], obtained with ∼5fb−1of data
+10To rescale this plot for the LHC is straightforward: multipl y by the TeVatron limit of ≃30×SM expectation for hSM→τ+τ−,
+and divide by the LHC limit on σ(gg→φ→τ±µ∓)/σ(gg→hSM→τ+τ−).
+15Wecanalsocompareourpredictiontobounds[74]onaMSSMHiggspro ducedviagluonfusion, anddecaying
+toτ+τ−. This analysis assumed one τdecaying hadronically and the other one to a muon and neutrinos. Th e
+cross section limits on σ×BR(φ→ττ) displayed in figure 5 of [74] indicate the current TeVatron limit on
+gg→φ→τµ. Inφ→τ±µ∓, the muon is directly produced in the decay of the Higgs, rather tha n from a
+τ, so the signal would have a global detection efficiency roughly 5 times higher than the signal of [74], due to
+the absence of the factor 0.17 coming from the decay rate of τinµ. So the y-axis in figure 5 of [74] could
+roughly be labelled as σ×BR(φ→τµ)/0.17. For a Higgs of 115 GeV, this gives a limit around 30 times the
+SM expectation, which is similar to the limit quoted in [73] with twice the lum inosity.
+For experimental bounds, we have extrapolated the potential re ach ofh→τµsearches from the published
+searches for h→ττ. In [3], it has been shown that by using appropriate cuts, it was poss ible to distinguish
+between h→τµandh→ττsignals (see figures 13 and 14 in [3]). Various cuts can help extract a h→τµ
+signal from the standard model backgrounds that are still select ed by ah→ττanalysis. Increasing the muon
+Pt threshold is a first option. In [3], this threshold is at 20 GeV, alread y above the thresholds used in h→ττ
+searches(10 GeV for CdF and 15 GeV fo D0 [75]). With Higgs mass abov e100 GeV, increasingthe threshold up
+to 30 GeV can safely be done. Figure 1 of [74] shows that threshold in crease would greatly reduce the amount
+of standard model backgrounds remaining in a h→ττanalysis. The dominant remaining backgrounds would
+be Z→ττ, Z→µµ11and W+jets.
+Another potentially efficient variable to isolate h→τµsignal is the effective transverse mass of the tau mu
+system as defined in equation (30) of [3]. For W and Z background, th is variable would peak at the W/Z mass
+while for higgs signal, it would peak at the higgs mass (see figure 11 of [3 ]). Hence, a cut on this variable could
+further reduce the background especially for higher higgs masses .
+Compared to the h→ττanalysis, a dedicated h→τµanalysis could then have fewer background events
+for a similar selection efficiency leading to sensitivity and limits which would be better than the one crudely
+extrapolated from current experimental limits on h→ττsearches [73, 74].
+6 Summary
+At hadron colliders, an interesting signature of New Physics would be the production of a neutral Higgs φ,
+followed by its decay to τ±µ∓. Formφ≪2mW, the branching ratio can be large, and the process could arise
+in a wide variety of models. Such New Physics is constrained by precisio n observables, and the upper bounds
+on rare decays such as τ→µγ. In this paper, we are interested in the implications for collider searc hes of these
+loop contributions.
+We parametrise the New Physics at mass scales <∼400 GeV, as a (CP- conserving) 2 Higgs Doublet Model
+(2HDM) of Type III, meaning that we allow our neutral Higgses to ha ve the tree-level lepton flavour changing
+couplings of eqn (12):
+∼κτµtanβτ/radicalbiggmµmτ
+v2(cβ−αh−sβ−αH+iA)τµ
+Our choices of notation, basis in Higgs space, tan βτ, and parametrisation are discussed in section 2. We will
+quote and plot bounds on κτµtanβτ, to avoid artificially strong bounds on κτµat large tan βτ. We study the
+phenomenological constraints on this model: we admit arbitrary New Physics at scales ∼TeV, and retain the
+contraints on the 2HDM arising from the Tparameter, b→sγ, (g−2)µandτ→µγ.
+TheTparameter and b→sγrestrict the flavour-independent parameters (masses) of the m odel. The
+charged Higgs of the 2HDM gives a significant same-sign contribution to the Standard Model (SM) amplitude
+forb→sγ(see sect 3.2). For mH+>∼250−300 GeV, this contribution is within current experimental error. In
+this paper, we assume such a heavy H+; this has little effect on the neutral Higgses we are interested in. Th e
+Tparameter, discussed in section 3.1.4, is the sum of many terms of diff erent sign. One way to ensure that it
+is small enough is to take mA∼mH+, as we assume for most of the plots. However, other cancellations are
+possible within the 2HDM (such as mH∼mH+with sin( β−α)→1), or additional new light particles (such
+as arise in supersymmetric models) could contribute to T.
+The decay τ→µγand the anomalous magnetic moment of the muon ( g−2)µconstrain the flavour-changing
+coupling κτµtanβτ, and are discussed in sections 4.3 and 4.4. The neutral Higgses h,HandAcontribute to
+(g−2)µat one loop, with a flavour-changing coupling at both vertices, and c hirality-flip via a τmass insertion.
+One could hope to fit the ( g−2)µdiscrepancy, with a 2HDM containing the κτµcoupling, but this would require
+cancellations in the generically more restrictive τ→µγbound (see figure 4 and eqns (40), (45) and (48)).
+11not considered in [3]
+16There are relevant contributions to τ→µγat one and two-loop; some diagrams are shown in figures 1 and
+2. The one loop amplitude scales12asm2
+τ
+v2tan2βτ(κτµtanβτ)2, and does not vanish for sin2( β−α)→0. The
+amplitude due to Aexchange is of opposite sign from the handHamplitudes, so the one-loop contribution to
+τ→µγcan be suppressed by tuning similar masses for h,HandA. Notice however, that the tuning required
+becomes finer with increasing tan βτ, so at large tan βτ,τdecays are a more sensitive probe [5] of κτµtanβτ
+than hadron colliders. See [76] for a discussion of the promising pros pects at an e−γcollider.
+The two-loop “Barr-Zee” contributions to τ→µγcan be of the same order as the one loop. The effective
+dipole interaction of τ,µandγ, must contain at least three Yukawa couplings at any loop order; if t he Higgs has
+O(1) couplingsto the tops, then two ofthe three Yukawa couplingsin the two-loopdiagramcan be top Yukawas.
+At large tan βτ, theφ-induced effective interaction ¯tt¯τµvanishes with sin2( β−α). In this sin2( β−α)→0
+limit, one of handHbecomes “Standard-Model-like”, with large couplings to the tandW, whileAand the
+other CP-even scalar have unsuppressed flavour-changing inter actions. However, as illustrated in figure 1, the
+“Barr-Zee” diagrams are closely related to Higgs production by gluo n fusion. So if the contribution to τ→µγ
+is suppressed in this way, the collider cross-section is suppressed a s well. The two loop top diagrams tend to
+dominate the τ→µγamplitude at small tan βτand large sin2( β−α). Their contribution can be suppressed
+by cancellations between diagrams — for instance the hamplitude cancels against the Hamplitude when
+1/tanβτ→0 andmh→mH— but these cancellations do not arise for the same parameter choic es as suppress
+the one loop amplitude.
+So in practise, the model is described by mH+(taken≃300 GeV to satisfy b→sγ),mh(≃114 GeV), mA
+andmH(one of which should be “light” to avoid the decoupling limit, and the the Tparameters constraint can
+be satisfied if the other is heavy), tan βτ(∼few, to maximise Higgs production by gluon fusion at the Tevatron
+and the LHC), and sin( β−α) (which is unconstrained if mA≃mH+, and is constrained to be ∼1 by the T
+parameter if mH≃mH+). Despite the freedom to vary sin( β−α), we find it difficult to generate a detectable
+cross-section at the Tevatron with parameters that satisfy the precision constraints.
+In the absence of cancellations, τ→µγis a more sensitive probe of the κτµcoupling, than φ→τ±µ∓at
+the TeVatron with ∼5/fb data. This can be seen from figure 5. However, it is possible to e vade constraints
+that arise from loops by tuning parameters, or by adding (more) Ne w Physics. We have not studied this, for
+reasons of principle and practise. Our approach is phenomenologica l, so tuning masses and mixing angles to
+cancel one and two loop contributions appears ad hoc. In a model, c ancellations can be justified by symmetries.
+More practically, we do not use complete gauge invariant two loop for mulae, and do not have an estimate for
+higher order effects, so cancellations we find might not persist in a be tter analysis.
+Direct collider searches can give clearer bounds, being less depende nt than loop processes on the vagaries of
+cancellations. Multiplying figure 5 by the current Tevatron bound on σ(p¯p→hSM→τ+τ−) which is ∼30σSM,
+shows that σ(gg→h→τ±µ∓) can be of order σ(gg→hSM→τ+τ−), and consistent with τ→µγ(for small
+tanβτ). So observing φ→τ±µ∓in the near future would require “tuning” by the ratio plotted in figur e 5: a
+factor∼.1 for small tan βτ.
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+arXiv:1001.0435v1  [math.AG]  4 Jan 2010ON DONALDSON-THOMAS INVARIANTS OF THREEFOLD
+STACKS AND GERBES
+AMIN GHOLAMPOURAND HSIAN-HUATSENG
+ABSTRACT . WepresentaconstructionofDonaldson-Thomasinvariants
+for three-dimensional projective Calabi-Yau Deligne-Mum ford stacks.
+We alsostudythestructureoftheseinvariantsfor ´ etalege rbesoversuch
+stacks.
+1. INTRODUCTION
+We work over the field of complex numbers throughout the paper . Let
+XbeasmoothproperDeligne-Mumford(DM)stackwithprojecti vecoarse
+moduli space X. Gromov-Witten (GW) theory, which roughly speaking
+concerns integrations against virtual fundamental classe s of moduli spaces
+oftwistedstablemaps,isbynowwell-established[6],[1], [2],andhasbeen
+an areaofactiveresearch recently.
+For3-dimensional smooth projective varieties the so-called Do naldson-
+Thomas (DT) theory is constructed in [19]. An important spec ial case is
+whenDTtheoryconcernsintegrationagainstvirtualfundam entalclassesof
+the moduli spaces of torsion free, rank 1sheaves with trivial determinants.
+It has been conjectured [13, 14], and proven in some cases, th at this gives
+an equivalenttheorytotheGW theoryoftheambient3-fold.
+The first goal of thispaper is to extend theconstruction ofDT invariants
+to DM stacks. This is done in Section 2. Our construction is pa rallel to
+that of [19], and uses the modulispaces of stablesheaves on a smoothpro-
+jective DM stack X, recently constructed by Nironi [16]. More precisely,
+weshowthatmodulispacesofstablesheavesona 3-dimensionalDMstack
+Xwith trivial canonical bundle (i.e. Calabi-Yau) admit natu ral perfect ob-
+struction theories, which we use to define invariants in case there are no
+strictlysemistablesheaves.
+Foragiventorsionelement [c]∈H2
+et(X,Gm)representedbya 2-cocycle
+c, we will use the moduli space of stable c-twisted sheaves on X(see
+[5, 12, 16]). This is a connected component of the moduli spac e of stable
+sheaves on a Gm-gerbe defined on Xrepresenting the element [c](see [16,
+Appendix A]). The perfect obstruction theory on the moduli s pace of the
+Date:October29,2018.
+12 AMIN GHOLAMPOURANDHSIAN-HUATSENG
+stablesheaves(ifexists)inducesaperfectobstructionth eoryonthemoduli
+spaceofstable c-twistedsheaves.
+As an application of the construction in this paper, we study the DT in-
+variants of a G-gerbeYover a3-dimensional Calabi-Yau stack X, where
+Gis afinitegroup. Thisis thecontentof Section 3. Ourstudyis motivated
+bythephysicalconjecturein[8]whichstatesthat“conform alfieldtheories
+ontheG-gerbeYareequivalenttoconformalfieldtheoriesonadualspace
+/hatwideYtwistedby aB-fieldc”. Theconstructionofthedualspace /hatwideYisexplained
+in Definition 3.1.2. Mathematically, the B-field cis aGm-valued cocycle
+on/hatwideY. Its definition is explained in Section 3.1. The theme of Sect ion 3 is
+a comparison between DT invariants of the gerbe Yand DT invariants of
+the dual(/hatwideY,c). Our Proposition 3.4.4 can be interpreted as DT-theoretic
+versionofthephysicsconjecture.
+In the presence of strictly semistable sheaves, moduli spac es of stable
+sheavesarenotproper. Inthissituationitisdesirabletoe xtendtheconstruc-
+tionofgeneralizedDTinvariants [10]tooursetting. Weplantopresentthis
+inalaterrevision.
+Acknowledgment. We thank A. Caldararu and M. Lieblich for discussion
+ontwistedsheavesandgerbes,andX.Tangforhelpfulcommen ts. H.-H.T.
+issupportedinpart by NSFgrant DMS-0757722.
+2. DTINVARIANTS FOR DM-STACKS
+2.1.Review of Nironi’s construction. In this section we construct a per-
+fectobstructiontheoryforthemodulispaceofstablesheav esoveraspecial
+class of 3-dimensional DM stacks that is used in this paper. T he construc-
+tion is similarto the case of smooth varieties (see [19]). Th is will allow us
+to define DT invariants for such stacks in cases where the modu li schemes
+areprojective.
+LetXbe aDM stack witha projectivemodulischeme X. We denoteby
+c :X →Xthe“coarsening map”(i.e. thenatural map from thestack Xto
+itscoarsemodulischeme X). Wefurtherassumethat Xisequippedwitha
+generatingsheaf Ein thesenseof[17, 16]. By definition Eis alocally free
+sheaf on Xwhose fiber over any geometric point of x∈ Xcontains the
+regularrepresentationofthestabilizergroupat x. Throughoutthepaperwe
+fix a choice of a generating sheaf EofXand a polarization OX(1)onX.
+Following[11]and [16, Definition 2.20], we call Xprojective if it satisfies
+theseconditions1.
+1A DM stack Xis projectiveif and onlyif it is a tame separatedglobalquot ientwith a
+projectivemodulischeme(see[11]and[16,Theorem2.21]).ON DONALDSON-THOMAS INVARIANTS OF THREEFOLD STACKS AND GER BES 3
+In[16]Nironiconstructsthemodulispaceofsemistablecoh erentsheaves
+onprojectivesmoothDMstacks. Wereviewpartofhisconstru ctionbriefly
+andreferthereaderto[16]fordetails. Themaindifference withthecaseof
+coherent sheaves on schemes is that stability condition now depends on E
+as well as OX(1). More precisely, for a coherent sheaf FonXstability is
+defined withrespect totheHilbertpolynomial
+PF(m) :=χ(X,F ⊗E∨⊗c∗OX(1)⊗m).
+LetpFbethemonicpolynomialobtainedbydividing PFbythecoefficient
+of the leading term. pFis thereducedHilbert polynomial of F. A pure
+sheafFonXis called semistable ifpF′≤pFfor any proper subsheaf
+F′⊂ F.Fiscalledstableiftheinequalityisalways strict.
+LetP∈Q[z], and let
+M(X,P) =M(X,E,OX(1),P)
+be the moduli stack of stable torsion free sheaves FonXwithPF=P.
+Nironi constructs M(X,P)as a quotient stack [Q/GL(N)]whereQis an
+appropriate subscheme of a Quot scheme on X(see [?]). He shows that
+M(X,P)is aGm-gerbe over a quasi projectivemoduli scheme M(X,P).
+Moreover M(X,P)is shown to be a geometric quotient of Q, and GIT
+techniquesprovidesanaturalcompactificationof M(X,P),parameterizing
+theS-equivalence classes of semistable sheaves on X(see [16, Theorems
+6.20-23]).
+2.2.Obstruction theory and DTinariants. LetLbealinebundleon X,
+anddenoteby
+M(X,P,L)⊂ M(X,P)
+the moduli substack of stable torsion free sheaves with fixed determinant
+L. We denote by M(X,P,L)the corresponding coarse moduli scheme.
+BytheconstructionofNironianddiscussionabove, M(X,P,L)isthefine
+modulistack of DM type2. By the followingpropositionthere existperfect
+obstructiontheorieson M(X,P,L)andM(X,P),in thesenseof[4]:
+Proposition 2.2.1. SupposeXis a smooth projective DM stack of dimen-
+sion 3 satisfying ωX∼=OX. Then there exist natural perfect obstruction
+theories on M(X,P,L)andM(X,P). Moreover, these obstruction theo-
+riesaresymmetricin thesenseof [3].
+Proof.Wefirst treat thecase ML=M(X,P,L). Let
+U → X ×M L
+2In factM(X,P,L)is aµr-gerbe over M(X,P,L)whereris the rank of the objects
+parameterizedby M(X,P).4 AMIN GHOLAMPOURANDHSIAN-HUATSENG
+betheuniversalstablesheafover X ×M L. Foraclosedpoint m∈ M,let
+Um→ Xbe the stable sheaf parameterized by m. Note that we have the
+followequationon trace-less ExtgroupsExt•
+X(−,−)0:
+Ext3
+X(Um,Um)0= Ext0
+X(Um,Um)0= 0. (2.1)
+The first equality follows from Serre Duality for DM stacks (s ee [15, The-
+orem 1.32]) and the assumption that ωXis trivial. The second equality is
+becauseofthestabilityof Um.
+The construction of the obstruction theory for MLis similar to the case
+ofthemodulispaceofsheavesonsmoothCalabi-Yauthreefol ds(see[19]).
+Inwhatfollows, L/squaredenotesthecotangentcomplexofaDMstack(see[9]).
+Letπ:X ×M L→ MLbe the projection. Composing the Atiyah class
+(see[9, IV 2.3.6.2])
+U →LX×M L⊗U[1]
+withthenatural projection LX×M L→π∗LMLgives
+U →π∗LML[1]⊗U.
+Thisgivesamorphism
+RHom(U,U)→π∗LML[1].
+Sinceπis smooth of relative dimension 3, tensoring both sides by O[2]
+(notethat ωπ=π∗ωX∼=O)yieldsamorphism
+RHom(U,U[2])→π!LML.
+By duality theorem (see [15, Corollary 1.22 and Theorem 1.32 ]) this gives
+amorphism
+Rπ∗RHom(U,U[2])→LML
+which, after restricting the left hand side to its traceless part, gives a mor-
+phism
+φ:E=Rπ∗RHom(U,U[2])0→LML.
+Inwhat followsweshowthat (E,φ)isa perfect obstructiontheory.
+Firstnotethat Eisperfectofperfectamplitudecontainedin [−1,0]. This
+istruebecauseof(2.1)(see[20, Lemma4.2]).
+Next we need to show that (E,φ)is an obstruction theory. Suppose g:
+T→ Mis a morphism from a scheme TandT→Tis an extension by a
+square-zero ideal I, then we need to showthat theobstructionto extending
+gtoTisaclass w∈Ext1
+T(Lg∗E,I)obtainedby composing Lg∗φwiththe
+natural maps Lg∗LML→LT→I[1]. To show (E,φ)is an obstruction
+theoryitsuffices to check thefollowingcriterion (see[4, T heorem4.5]):
+Claim .w= 0if and only if there exists an extension g:T→ M, and if
+nonemptythesetof allsuch gmakes atorsorover Ext0
+T(Lg∗E,I).ON DONALDSON-THOMAS INVARIANTS OF THREEFOLD STACKS AND GER BES 5
+We now provethis Claim. Let f= (id,g)andf= (id,g). Consider the
+followingdiagram:
+X ×Tf− −− → X ×M Lq− −− → X
+p/arrowbt π/arrowbt
+Tg− −− → M L.
+By standardargumentsoneobtainsanatural identification
+Exti
+T(Lg∗E,I)∼=Exti+1
+X×T(f∗U,f∗U ⊗p∗I)0. (2.2)
+BecauseMLis a fine moduli space, deforming gtogis equivalent to de-
+formingf∗Utof∗U. The obstruction to the latter, denoted by w′, is ob-
+tainedby thecomposition
+f∗U →LX×T⊗f∗U[1]→p∗I⊗f∗U[2]
+and then restricting to the traceless part. The first map is th e Atiyah class
+at(f∗U)and the second one is induced from the natural map LX×T→
+p∗I[1]. Thisistruebecause of[9, PropositionIV.3.1.8]andthefa ct thatwe
+have fixed the determinant. The reason for restricting to the traceless part
+is thatlinebundles on Xare unobstructed,and as in the caseof sheaves on
+schemes(see[19]),onecan showthatthetraceoftheobstruc tionclassofa
+sheafFon asmoothDM stackistheobstructionclass of det(F).
+For a similar reason and by using (2.2), if w′= 0the set of all defor-
+mations is a torsor over Ext0
+T(Lg∗E,I).So it remains to show that wis
+mapped to w′under (2.2). We showed that φ:E→LMLarises from the
+Atiyah class at(U). Following exactly the same steps one can show that
+Lg∗E→Lg∗LMLarises from the Atiyah class at(f∗U). This means that
+theclassw,whichis thecomposition
+Lg∗Eφ−→Lg∗LML→LT→I[1],
+givesriseto
+f∗U →LX×T⊗f∗U[1]→p∗I⊗f∗U[2],
+which is what we need. This finishes the proof of the Claim, and the con-
+structionoftheperfect obstructiontheory on M(X,P,L).
+To construct an obstruction theory on M=M(X,P), we just make
+the following modifications to the construction given above . Firstly, Uis
+replaced by the universal twisted sheaf on M× Xdenoted by U(see [5,
+Section3.3]). Secondly,thenaturalcandidate
+Rπ∗RHom(U,U[2])∼=(Rπ∗RHom(U,U))∨[−1]6 AMIN GHOLAMPOURANDHSIAN-HUATSENG
+fortheobstructiontheoryisnotperfect,wherenow π:X ×M→Misthe
+projection. However,repeatingtheargumentsin[20,Secti on4.4],itcanbe
+shownthatthetrimmedcomplex
+(τ[1,2]Rπ∗RHom(U,U))∨[−1]
+givesrisetoaperfect obstructiontheoryon M(X,P).
+The symmetry of the obstructiontheories on M(X,P,L)andM(X,P)
+follows easily from Serre duality and the Calabi-Yau condit ionωX∼=OX.
+/square
+IfXis as in Proposition 2.2.1, then by the symmetry property of t he
+obstructiontheoriestheexpecteddimensionsof M(X,P,L)andM(X,P)
+are0. By Proposition2.2.1and [4]wehave
+Proposition 2.2.2. SupposeXis a smooth projective DM stack of dimen-
+sion 3 satisfying ωX∼=OX. ThenM(X,P,L)andM(X,P)carry virtual
+0-cycles denotedby [M(X,P,L)]virand[M(X,P)]vir.
+LetνM(respectively, νML)betheBehrend’sfunction([3,10])definedon
+M(X,P)(respectively, M(X,P,L)). Then we define the corresponding
+DTinvariantsas follows
+Definition 2.2.3. LetXbe as in Proposition 2.2.1, P∈Q[z], andLa
+line bundle on X. Then we define the Donaldson-Thomas invariants3ofX
+correspondingto P(andL)astheweighted Eulercharacteristics
+DT(X,P) =−χna(M(X,P),νM).
+DT(X,P,L) =χ(M(X,P,L),νML).
+Hereχnais thena¨ıveEuler characteristicdefined forArtinstacks (see [10,
+Definition2.3] andχdenotestheEuler characteristicof DMstacks.
+Remark2.2.1.
+(1) Letrbetherankoftheobjectsparameterized by M(X,P),andlet
+νM(respectively νML) be the Behrend’s function for M(X,P)(re-
+spectively,for M(X,P,L))thenby theproperties oftheBehrend’s
+functionandWeighted Eulercharacteristic(see [3, 10]), w ehave
+DT(X,P) =χ(M(X,P),νM),
+and
+DT(X,P,L) =1
+rχ(M(X,P,L),νML).
+3These invariants depend on the choices of EandOX(1), however this dependence is
+suppressedinournotation.ON DONALDSON-THOMAS INVARIANTS OF THREEFOLD STACKS AND GER BES 7
+(2) Suppose that there are no strictly semistable sheaves FonXsat-
+isfyingPF=P. It is known [16] that in this case M(X,P)and
+M(X,P,L)are proper, and hence thevirtualclasses [M(X,P)]vir
+and[M(X,P,L)]vircan be integrated. By [3, Theorem 4.18] and
+[10,Remark 5.14]
+DT(X,P,L) = deg/parenleftbig
+[M(X,P,L)]vir/parenrightbig
+,
+and
+DT(X,P) = deg/parenleftbig
+[M(X,P)]vir/parenrightbig
+.
+(3) Iftherearestrictlysemistablesheaves,then DT(X,P)andDT(X,P,L)
+arenotingeneraldeformationinvariant. Onewaytofixthisi stoex-
+tend the construction of generalized Donaldson-Thomas invariants
+[10]tothissetting. Thiswillbeexploredin alaterrevisio n.
+3. ADECOMPOSITION RESULT FOR DTINVARIANTS ON GERBES
+ThepurposeofthisSection isto studyDT invariantsof ´ etal egerbes.
+3.1.´Etale gerbes. We begin with a review of some basic notions of ´ etale
+gerbes and the construction of their duals. Let Gbe a finite group4. LetX
+beasmoothprojectiveDeligne-Mumfordstackwithcoarsemo dulischeme
+X. LetBGdenotethat stackof G-torsors.
+Definition 3.1.1 (see e.g. [7], Definition 3.1) .AG-gerbe over Xis a
+Deligne-Mumford stack Ytogether with a morphism Y → X such that
+there exists a faithfully flat, locally of finite presentatio n, mapX′→ X
+suchthat Y ×XX′≃BG×X′.
+In thisway onecan view BGas aG-gerbeoverapoint.
+LetOut(G)denote the group of outer automorphisms of G. By defini-
+tion,Out(G)is the quotient of the group Aut(G)of automorphisms of G
+bythenormalsubgroup Inn(G)ofinnerautomorphismsof G,
+Out(G) =Aut(G)/Inn(G).
+Given aG-gerbeY → X, there is a naturally defined Out(G)-bundle
+Y → X, called the band. See [7], Definition 3.3 for a detailed definition.
+We say that the G-gerbeY → X hastrivial band if theOut(G)-bundle
+Y → Xisendowedwitha section(henceistrivializedbythissecti on).
+Let/hatwideGdenote the setof isomorphism classes of irreducible representa-
+tions ofG. Note that /hatwideGis a finite set, the cardinality of /hatwideGcoincides with
+the number of conjugacy classes of G. We may also view /hatwideGas a disjoint
+unionofpoints.
+4Gis viewedasa finitegroupschemeoverSpec C.8 AMIN GHOLAMPOURANDHSIAN-HUATSENG
+Letρ:G→End(Vρ)be an irreducible representation of G, andφ∈
+Aut(G). Then thecomposite
+ρ◦φ−1:G→End(Vρ)
+is an irreducible representation of G. It is easy to see that this induces
+an action of Out(G)on/hatwideG. Note that the isomorphism class [1tr]of the
+1-dimensionaltrivialrepresentationof Gisfixed by this Out(G)action.
+Definition3.1.2 (see [8]).Define
+/hatwideY:=Y ×Out(G)/hatwideG.
+Thereisanatural map /hatwideY → Xinducedfrom themap Y → X.
+Remark3.1.3.
+(1) The morphism /hatwideY → Xis finite and ´ etale. The stack /hatwideYisdiscon-
+nected.
+(2) If the G-gerbe/hatwideY → Xhas trivial band, then /hatwideYis a disjoint union
+ofseveralcopiesof X, and themap /hatwideY → Xrestricts totheidentity
+oneach copy.
+For each isomorphism class [ρ]∈/hatwideGwe fix a representation ρ:G→
+End(Vρ)in this class. To each (x,[ρ])∈/hatwideYwe assign the vector space Vρ.
+This defines a family of vector spaces over /hatwideY, which is in general nota
+vector bundle over /hatwideY. The obstruction to find a vector bundle over /hatwideYwith
+fiber over (x,[ρ])beingVρis aGm-valued2-cocycle on /hatwideY, whoseinverse
+isdenotedby c.
+As observed in [8], the cocycle cis locally constant, and crepresents a
+torsionclassin thecohomology H2
+et(/hatwideY,Gm).
+Anotherwaytounderstandthecocycle cisthefollowing. Thefailurefor
+Vρ’stoformavectorbundleisduetothefactthattheyglue uptoscalars . In
+otherwords Vρ’s glueto a twisted sheaf (see e.g. [5]). This twistedsheaf is
+equivalent(see[12])toasheafona Gm-gerbeover /hatwideY. ThisGm-gerbeturns
+outtobeflatandtheinverseofitsclass,whichisanelementi nH2
+et(/hatwideY,Gm),
+isrepresented by the 2-cocyclec.
+3.2.Equivalence. In what follows we will be concerned with sheaves on
+gerbes and a decomposition statement about DT invariants fo r Calabi-Yau
+gerbes,which isinspiredby [8].
+We continue to use the notation in the previous section. Let Y → Xbe
+aG-gerbe over a smooth projective DM stack Xand let/hatwideY → Xandcbe
+as constructed before. Note that Yis also a smooth projective DM stack,
+and the coarse modulispace of YisX. LetcY:Y →Xbethe coarsening
+map. By construction, /hatwideYis also smooth and projective. Let c/hatwideY:/hatwideY →/hatwideYON DONALDSON-THOMAS INVARIANTS OF THREEFOLD STACKS AND GER BES 9
+denoteitscoarseningmap,andlet π/hatwideY:/hatwideY→Xbethemapbetweencoarse
+modulispaces inducedby /hatwideY → X.
+Fix an amplelinebundle OX(1)ofX. Notethat thepull-back π∗
+/hatwideYOX(1)
+isan amplelinebundleof /hatwideY.
+LetCoh(Y)denote the abelian category of coherent sheaves on Y, and
+Coh(/hatwideY,c)theabelian category of coherent c-twistedsheaves on /hatwideY. We re-
+ferto[5]and[12]fordetaileddiscussionsonthetheoryoft wistedsheaves.
+Thefollowingresult isprovenin [18].
+Theorem 3.2.1 (X. Tang-H.-H. Tseng) .There isnaturalfunctor
+F:Coh(Y)→Coh(/hatwideY,c), (3.1)
+whichisan equivalenceofabeliancategories.
+Theconstructionofthisfunctor Fisratherinvolved. Detailscanbefound
+in [18]. Roughly speaking, the inverse functor Coh(/hatwideY,c)→Coh(Y)can
+be understood as taking (−)⊗ Vρ, whereVρis the aforementioned c−1-
+twistedsheafwithfibers Vρ.
+As noted above, /hatwideYis disconnected. Let /hatwideY=/coproducttext
+i∈I/hatwideYibe the decompo-
+sition of /hatwideYinto connected components, and let cibe the2-cocycle on /hatwideYi
+obtainedbyrestrictionof c. By definition,wehave
+Coh(/hatwideY,c) =⊕i∈ICoh(/hatwideYi,ci).
+Consequentlythere isadecompositionof K-groups
+K(Coh(/hatwideY,c)) =⊕i∈IK(Coh(/hatwideYi,ci)).
+On the other hand, K(Y) =K(Coh(Y)) =K(Coh(/hatwideY,c)), therefore we
+getadecompositionof K(Y):
+K(Y) =⊕i∈IKi, Ki:=K(Coh(/hatwideYi,ci)). (3.2)
+GivenF ∈Coh(Y),wewrite
+F(F) =⊕i∈IF(F)i, F(F)i∈Coh(/hatwideYi,ci).
+Since (3.1) is an equivalence of abelian categories, it pres erves exact se-
+quences. Hence for F ∈Coh(Y)and a subsheaf F′⊂ F, the components
+F(F′)iare subsheaves of F(F)i. Also for F1,F2∈Coh(Y)we have the
+equalityon Homspaces
+HomCoh(Y)(F1,F2) =⊕i∈IHomCoh(/hatwideYi,ci)(F(F1)i,F(F2)i).(3.3)
+By the construction of the equivalence (3.1), it is easy to ch eck that if V
+is a generating sheaf of Y, thenF(V)is a generating c-twisted sheaf of /hatwideY.
+HenceF(V)i∈Coh(/hatwideYi,ci)is agenerating ci-twisted sheaf of /hatwideYi. SinceG10 AMIN GHOLAMPOURANDHSIAN-HUATSENG
+acts triviallyon c∗
+YOX(1), theconstructionoftheequivalence(3.1)implies
+that
+F(F ⊗c∗
+YOX(1)⊗m) =F(F)⊗c∗
+/hatwideYπ∗
+/hatwideYOX(1)⊗m.
+From now on, fix a generating sheaf EonXand an ample line bundle
+OX(1)onX. Also fix the generating c-twisted sheaf F(E)on/hatwideYand an
+ample line bundle π∗
+/hatwideYOX(1)on/hatwideY. With these choices it follows that the
+Hilbert polynomial PFofFcoincides with the Hilbert polynomial PF(F)
+ofF(F). Moreprecisely,
+PF=PF(F)=/summationdisplay
+i∈IPF(F)i. (3.4)
+3.3.Invariants. Let
+C(Y) ={[F]∈K(Y)|0/ne}ationslash=F ∈Coh(Y)}
+be the positive cone in K(Y). ThenC(Y) =⊕i∈ICicorresponding to
+the decomposition (3.2). Let k∈C(Y), and let M(Y,k)be the mod-
+uli stack of stable torsion free sheaves on Yof classk. It is evident that
+M(Y,k)is a component of the moduli stack of stable torsion free shea ves
+with fixed Hilbert polynomials. Suppose further that k∈Ciin the de-
+composition above. Note that for any F ∈Coh(Y)of classk, we have
+F(F) =F(F)i∈Coh(/hatwideY,ci), namely
+F(F)j= 0forj/ne}ationslash=i. (3.5)
+By(3.4)and(3.5)wehavethefollowingrelationsbetween(r educed)Hilbert
+polynomials:
+PF=PF(F)i, pF=pF(F)i.
+Consequently Fis(semi)stableifand onlyif F(F)iis(semi)stable.
+Therefore theequivalence(3.1)yieldsaset-theoreticbij ection
+M(Y,k)→ M((/hatwideYi,ci),k),[F]/mapsto→[F(F)i].
+HereM((/hatwideYi,ci),k)denotesthemoduliofsemistable ci-twistedsheaveson
+/hatwideYiof classk. As mentioned in Section 1, M((/hatwideYi,ci),k)is realized as a
+connected component of the certain moduli space of stable sh eaves on a
+µN-gerbe over /hatwideYiforN≫0, and hence Nironi’s constructionapplies (see
+[16, AppendixA]).
+Proposition3.3.1. There isan isomorphismof stacks
+M(Y,k)≃ M((/hatwideYi,ci),k).ON DONALDSON-THOMAS INVARIANTS OF THREEFOLD STACKS AND GER BES 11
+Proof.This is proven by checking that deformation theory on both si des
+agree to all order, using (3.3). Alternatively the isomorph ism can be con-
+structed as follows. Clearly the product M(Y,k)× Yis aG-gerbe over
+M(Y,k)×X. By constructionweseethatthedualofthis G-gerbeis
+/hatwiderM(Y,k)×Y=M(Y,k)×/hatwideY.
+Moreover the 2-cocycle in this case is the pull-back of cvia the projection
+M(Y,k)×/hatwideY →/hatwideY. LetU → M(Y,k)×/hatwideYbe the universalstablesheaf.
+Notethat there existsan equivalence(3.1) for any G-gerbe. Applyingsuch
+an equivalenceto the sheaf UovertheG-gerbeM(Y,k)×Y, we obtain a
+twisted sheaf F(U)overM(Y,k)×/hatwideY. It is easy to check that F(U)is a
+familyover M(Y,k)ofci-twistedstablesheaveswithclass k. Thisdefines
+a morphism M(Y,k)→ M((/hatwideYi,ci),k). The inverse morphism can be
+defined inasimilarfashionby usingan inverseof(3.1). /square
+Nowsupposeinadditionthat XisaCalabi-Yau Deligne-Mumfordstack
+ofdimension 3. Hence both Yand/hatwideYare Calabi-Yau ofdimension 3. Then
+we can define DT invariants DT(Y,k)andDT((/hatwideYi,ci),ki)as in Section
+3.3, and byProposition3.3.1wehave
+Proposition3.3.2.
+DT(Y,k) = DT(( /hatwideYi,ci),k).
+Remark3.3.3.Onecan seedirectly, using(3.3), that the 2-term perfect ob-
+structiontheoryassociatedto M(Y,k)ismappedtotheoneon M((/hatwideYi,ci),k).
+This gives an alternative proof for the proposition above in cases where
+k∈C(Y)is such that for sheaves of class ksemistability and stability
+coincide.
+3.4.Decomposition. UsingthenotationinSections 3.1-3.2, let YbeaG-
+gerbe over a smooth projective DM stack X, and Let k∈C(Y). Suppose
+kiis theCi-componentof kin thedecomposition C(Y) =⊕i∈ICiinduced
+from(3.2),
+k=/summationdisplay
+iki, ki∈Ci.
+LetM(Y,k)be the moduli of stable sheaves on Yof classk. It is evident
+thatM(Y,k)is a component of the moduli of stable sheaves with fixed
+Hilbertpolynomials. Wemakethefollowingassumption
+Assumption3.4.1. A sheafF ∈Coh(Y)ofclasskis (semi)stableifand
+onlyifF(F)i∈Coh(/hatwideYi,ci)is(semi)stableforall i∈ I.12 AMIN GHOLAMPOURANDHSIAN-HUATSENG
+Remark3.4.2.Naively Assumption 3.4.1 should follow from the equality
+(3.4) of Hilbert polynomials. We are unable to deduce Assump tion 3.4.1
+from(3.4)becausethenotionof(semi)stabilityisdefinedu singthereduced
+Hilbertpolynomial,and itisnotcleartouswhether(3.4)ho ldsforreduced
+Hilbertpolynomialsornot.
+In view of Assumption 3.4.1 the equivalence (3.1) yields a se t-theoretic
+bijection
+M(Y,k)→/productdisplay
+i∈IM((/hatwideYi,ci),ki),[F]/mapsto→([F(F)i])i∈I.
+HereM((/hatwideYi,ci),ki)denotes the moduli of semistable ci-twisted sheaves
+on/hatwideYiofclasski.
+Proposition3.4.3. AssumeAssumption3.4.1,thenthereisanisomorphism
+ofstacks
+M(Y,k)≃/productdisplay
+i∈IM((/hatwideYi,ci),ki).
+Proof.Thisis provedby thesame argumentsas in theproof ofProposi tion
+3.3.1 /square
+Now supposein additionthat Xis a Calabi-Yau DM stack of dimension
+3. ThenProposition3.4.3andthemultiplicativityoftheBeh rend’sfunction
+[3,Proposition1.5]wehavethefollowingrelationamongth eDTinvariants:
+Proposition3.4.4. AssumeAssumption3.4.1, then
+DT(Y,k) =/productdisplay
+i∈IDT((/hatwideYi,ci),ki).
+ThisisourdecompositionstatementfortheDTinvariantsof thegerbe Y.
+Remark3.4.5.BythesamediscussionasinSection3.3,theperfectobstruc -
+tiontheoryassociatedto M(Y,k)ismappedtothaton/producttext
+i∈IM((/hatwideYi,ci),ki).
+ThisobservationgivesanalternativeproofofProposition 3.4.4inthecases
+semistability and stability coincide for all sheaves of cla ssk, and for all
+ci-twistedsheavesofclass ki,i∈ I.
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+DEPARTMENT OF MATHEMATICS ,CALIFORNIA INSTITUTE OF TECHNOLOGY , PASADENA ,
+CA 91125, USA
+E-mailaddress :agholamp@caltech.edu
+DEPARTMENT OF MATHEMATICS , OHIOSTATEUNIVERSITY , 100 M ATHTOWER,
+231 WEST18THAVE., COLUMBUS , OH 43210, USA
+E-mailaddress :hhtseng@math.ohio-state.edu
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+arXiv:1001.0436v5  [cs.GT]  16 Apr 2011Truthful Assignment without Money
+Shaddin Dughmi∗
+Stanford University
+shaddin@cs.stanford.eduArpita Ghosh
+Yahoo! Research
+arpita@yahoo-inc.com
+August 27, 2018
+Abstract
+We study the design of truthful mechanisms that do not use payme nts for the generalized
+assignment problem (GAP) and its variants. An instance of the GAP c onsists of a bipartite
+graph with jobs on one side and machines on the other. Machines hav ecapacitiesand edges have
+values and sizes; the goal is to construct a welfare maximizing feasib le assignment. In our model
+of private valuations, motivated by impossibility results, the value an d sizes on all job-machine
+pairs are public information; however, whether an edge existsor not in the bipartite graph is a
+job’s private information. That is, the selfish agents in our model ar e the jobs, and their private
+information is their edge set. We want to design mechanisms that are truthful without money
+(henceforth strategyproof ), and produce assignments whose welfare is a good approximation t o
+the optimal omniscient welfare.
+WestudyseveralvariantsoftheGAPstartingwithmatching. Fort heunweightedversion,we
+give an optimal strategyproof mechanism. For maximum weight bipar tite matching, we show
+that no strategyproof mechanism, deterministic or randomized, c an be optimal, and present
+a 2-approximate strategyproof mechanism along with a matching low erbound. Next we study
+knapsack-like problems, which, unlike matching, are NP-hard. For t hese problems, we develop a
+general LP-based technique that extends the ideas of Lavi and S wamy [14] to reduce designing a
+truthfulapproximatemechanismwithoutmoneytodesigningsucha mechanismforthe fractional
+version of the problem. We design strategyproof approximate mec hanisms for the fractional
+relaxations of multiple knapsack, size-invariant GAP, and value-inva riant GAP, and use this
+technique to obtain, respectively, 2, 4 and 4-approximate strate gyproof mechanisms for these
+problems. We then design an O(logn)-approximate strategyproof mechanism for the GAP by
+reducing, with logarithmic loss in the approximation, to our solution fo r the value-invariant
+GAP. Our technique may be of independent interest for designing tr uthful mechanisms without
+money for other LP-based problems.
+∗Supported by a Yahoo! Research internship, a Siebel Foundat ion Scholarship, and NSF grant CCF-0448664.1 Introduction
+The design of truthful mechanisms, where selfish utility max imizing agents have no incentive to
+lie about their true preferences, has been studied in innume rable settings. The vast majority
+of these mechanisms, however, assume the existence of money — a carefully designed payment
+scheme incentivizes agents to report their preferences tru thfully. However, there are settings where
+monetary transfers are not feasible, either because of ethi cal or legal issues [16], or because of
+practical issues with enforcing and collecting payments [1 7]. This observation has led to a growing
+literature on designing mechanisms that incentivize agent s to report their true preferences without
+using payments: for convenience, we refer to such mechanism s asstrategyproof .
+In this paper, we focus on the design of strategyproof mechan isms forassignment problems. An
+instance of an assignment problem consists of a bipartite gr aph with items, or jobs, on one side, and
+bins, or machines, on the other; associated with each bin is a capacity, and with each edge a value
+and a size. A feasible assignment is a (partial) mapping from items to bins, where no bin’s capacity
+is exceeded by the sizes of the items assigned to it. The goal i s to compute a feasible assignment
+that maximizes welfare: the sum of the values of the jobs for machines they are assign ed to.
+The most general version of this problem, where both the size and value of a job can differ for
+different machines, is referred to as the generalized assignm ent problem (GAP). A number of well-
+known algorithmic assignment problems are special cases of the GAP. For instance, the problem
+with just one bin is the knapsack problem, and the problem wit h unit-sized items and bins is
+the maximum weight bipartite matching problem. Assignment problems are ubiquitous, and have
+been extensively studied due to their vast applicability, b oth from an algorithmic and mechanism
+design perspective. However, studying these problems in a s ettingwithout money , as we do, adds
+additional difficulties, since properties like monotonicit y and weak monotonicity (see [15, 3, 13, 16])
+no longer suffice for truthfulness. Moreover, the popular VCG mechanism, as well as the maximal
+in range paradigm, require payments for truthfulness.
+There are a number of settings where part of the input to an ass ignment problem is held by
+selfish agents, and the problem must be solved without the use of money. Unfortunately, as we
+discuss in §2.3, not much can be done if jobs hold their real-number value s for the various machines
+private; this is consistent with many impossibility result s from social choice theory. In this paper,
+we therefore focus on a restricted, yet natural, setting tha t admits interesting results— for each
+pair (i,j), it is public knowledge that the value of job ifor machine jis either vijor 0, but job i
+holds private which of those is the case for each j. This models situations where the private data
+encodes a compatibility relation between jobs and machines; the public vijvalues arise in situations
+where the value derived from an assignment materializes ove r a public channel (for instance via a
+verifiable financial transaction). As a result, a job cannot h ide its true value for any machine it
+anticipates being assigned to, although it can misreport a n on-zero value as 0, a lie that will not be
+discovered since the job will not be allocated to this machin e. We will see that, if the mechanism is
+not chosen carefully, such strategic manipulations can be b eneficial to a selfish agent even in very
+simple instances of the GAP. We also note that this is a multi-parameter problem, where each job
+holds one bit of private information for each of the bins.
+There are several natural settings that correspond to our mo del of private values. Suppose,
+for instance, a group of people are to split up a collection of tasks. Each task requires different
+skills, and how well each person can perform a task is public k nowledge. Each task also has
+different time and location constraints, and whether or not th e constraint is feasible for a person
+is only known privately to her. This problem is an instance of weighted bipartite matching; while
+1findingan optimal solution is computationally easy, we are i nterested in algorithms that also ensure
+strategyproofness. In another example, consider a resourc e allocation problem, such as scheduling
+jobs on a collection of non-identical machines. The value of running a job on a machine, as well as
+the time it takes, is public knowledge. However, each job req uires specific hardware and software
+that is available on only some of the machines. Moreover, onl y the owner of the job knows which
+machines are compatible with the job. Here, the algorithm us ed to assign jobs to machines must
+ensure that the jobs do not have an incentive to lie about whic h machines are compatible.
+As in [17], we are interested in strategyproof mechanisms th at achieve a good approximation to
+the welfare of the optimal omniscient solution. The need for approximations arises for two reasons
+in our work: first, the GAP is NP-hard, as are several of its spe cial cases; i.e., approximation is
+necessitated by computational intractability. The second reason is much more interesting— unlike
+in settings with payments, solving the allocation problem o ptimally does not necessarily lead to a
+truthfulmechanism, and we need to sacrifice approximation i n order to obtain truthfulness. We will
+see both factors playing a role as we seek strategyproof mech anisms that are good approximations
+to the various special cases of GAP. For example, we will see t hat no strategy proof mechanism
+for maximum-weight matching can be optimal, and we must sacr ifice an approximation factor of 2
+for truthfulness. In contrast, a (non-polynomial time) alg orithm that returns an optimal solution
+to the multiple knapsack problem while breaking ties consis tently is strategyproof; however, since
+we are interested in polynomial-time strategyproof algori thms, we must resort to an approximately
+optimal mechanism that is both truthful and polynomial-tim e implementable.
+Why would an agent benefit from lying in assignment problems w hen there are no payments?
+Consider, for instance, the weighted bipartite matching pr oblem (§3.2). Consider the following
+instance: job a1has edges with weights 1+ ǫand 1 to machines b1andb2, and job a2has an edge to
+b1with weight 1. An algorithm that simply chooses themaximum w eight matching according to the
+reports incentivizes job a1to simply claim that the second edge does not exist: in the firs t case, the
+assignment chosen is ( a1,b2),(a2,b1), whereas in the second case, the assignment chosen is ( a1,b1),
+which suits a1better, with value 1 + ǫ. This example makes it clear that that the optimal (and
+obvious) algorithms are not necessarily truthful— not surp risingly, a carefully designed algorithm
+is essential to ensure that no agent has an incentive to lie, e xactly as in mechanism design with
+money.
+1.1 Our Results
+We study the design of approximation mechanisms that are tru thful without money for several
+variants of the GAP. We begin in §3 withmatching , which can be solved optimally in polyno-
+mial time from a purely computational perspective. We show t hat for the maximum matching
+problem, where all edge values are equal, simply returning t he optimal solution while breaking
+ties consistently leads to a strategyproof mechanism. Howe ver, when a job’s value depends on the
+machine, as in weighted bipartite matching, no determinist ic strategyproof mechanism can achieve
+an approximation better than 2; we provide such a mechanism.
+Next, we examine knapsack-like variants of the GAP. Instead of specially tailored combinatorial
+algorithms for each variant, we extend the techniques in [14 ] to reduce designing a truthful mech-
+anism without money to designing such a mechanism for the fractional version of the problem: if
+the strategyproof mechanism for the fractional version yie lds anαapproximation to the optimal
+fractional solution, and the corresponding LP has integral ity gapβ, we derive a strategyproof ran-
+domized mechanism for the original problem with approximat ion ratio α·β. This technique applies
+2to a large class of packing problems, and may of independent i nterest.
+The GAP has integrality gap 2, so a fractional strategyproof mechanism with approximation
+ratioαyields a 2 αstrategyproof (in expectation) mechanism for each of the kn apsack-like variants
+of the GAP that we study. In §4.2, we show, usingnetwork flows, that solving thefractiona l version
+of the multiple knapsack problem (MKP) optimally, while bre aking ties consistently independent
+of the reported edges, gives an optimal strategyproof (frac tional) mechanism. For size-invariant
+GAP (SIGAP), there is no optimal truthful (fractional) mech anism without money— in §4.3, we
+design a strategyproof, 2-approximate greedy algorithm fo r fractional SIGAP. Using our extension
+of [14] gives, respectively, 2 and 4-approximate strategyp roof mechanisms for MKP and SIGAP. In
+§4.4, we sketch the construction of a 4-approximate strategy proof mechanism for value-invariant
+GAP (VIGAP), as well as a O(logn)-approximate strategyproof mechanism for the GAP.
+We point out that without the polynomial time restriction, t here exist optimal strategyproof
+mechanisms for all variants of GAP where a node has the same va lue for each of its neighbors.
+That is, for maximum matching, MKP and VIGAP, simply solving the problem optimally, while
+breaking ties consistently independent of the private valu es (edges), leads to a truthful-without-
+money mechanism. For these problems, it is only computation al intractability which causes us to
+lose an approximation factor. This is in contrast to the vari ants where a node has different values
+for different edges such as maximum weight matching and its gen eralizations. There, as we show
+in Theorem 3.3, strategyproofness and optimality cannot be achieved simultaneously.
+1.2 Related Work
+Assignment problems have been studied extensively in the al gorithms literature. Shmoys and
+Tardos [18] presented a 2-approximation for a minimization version of the GAP, and Chekuri and
+Khanna [10] observed that a 2-approximation to the maximiza tion version – the version considered
+in this paper– is implicit in [18]. Moreover, it was shown in [ 10] that the multiple knapsack problem
+– a special case of the GAP – admits a PTAS1, yet most generalizations of MKP – including the
+GAP– are APX hard. Fleischer et al [12] obtained ae
+e−1approximation for the GAP, and showed
+that this is optimal for a slight generalization of the GAP. H owever, Feige and Vondrak [11] then
+showed that the GAP admits a constant approximation slightl y better thane
+e−1, and this is the
+best currently known.
+A number of results for the mechanism design version of assig nment problems are known,
+although these are all in settings with money. In all of these results, the items hold their values
+private, and the rest of the instance is public. A 2-approxim ate truthful-in-expectation mechanism
+follows immediately fromtheframeworkofLavi andSwamy[14 ]. Moreover, Briest etal [7]deviseda
+truthful FPTAS for the knapsack problem, as well as a truthfu l PTAS for VIGAP when the number
+of bins is fixed. Recently, Azar and Gamzu [5] obtained a truth ful 11-approximate mechanism for a
+variant of MKP, and Chekuri and Gamzu obtained a 2+ ǫapproximation for a variant of VIGAP.
+We note that all the above mechanisms use money, and moreover the mechanisms in [7, 5, 9]
+consider an incomparable setting to ours: our model is multi -parameter whereas theirs is single-
+parameter, but we consider a binary private value for each it em and bin as opposed to an arbitrary
+real number.
+1However, we note that this PTAS is not applicable to the gener alization of MKP that we consider, where
+assignments are constrained by a bipartite graph over items and bins. It follows from [10, Theorem 3.2] that this is
+APX-hard.
+3Mechanisms without money have a rich history in the social ch oice literature; for a survey, see
+[16]. Interest in approximate mechanisms without money has been sparked by the recent work
+of Procaccia and Tennenholtz [17], which introduces the ide a of using approximation to enable
+truthfulness in settings where solving optimally does not a dmit truthfulness without money. Ap-
+proximatemechanismswithoutmoneyhave beendevelopedfor facility location [17,1], andselecting
+influential nodes in a social graph [2]. In very recent work, A shlagi et al. [4] study strategyproof
+mechanisms for matching motivated by kidney exchange. Whil e we also study (bipartite) matching
+as a special case of the GAP, their model is very different from o urs: each agent owns a set of
+vertices in a graph, and reports the existence of vertices to maximize the number of her vertices
+matched by the mechanism, plus the number that she can match a mongst her hidden vertices and
+the vertices unmatched by the mechanism. Also related is the work of [8], which studies a very gen-
+eral combinatorial assignment problem without money, and d esigns a mechanism which sacrifices
+efficiency, as well as weakens the notion of incentive compati bility, to achieve fairness.
+2 Model
+We describe the optimization version of the Generalized Ass ignment Problem (GAP), as well as
+its various special cases that we consider, in §2.1. We then review truthfulness in §2.2, and discuss
+the limitations of truthfulness without money for the GAP in §2.3. These limitations motivate our
+model of private valuations, which we then introduce in §2.4.
+2.1 The Generalized Assignment Problem
+IntheGAP, thereare njobsand mmachines. We denotetheset of jobsby [ n] ={1,...,n}, andthe
+set ofmachines by [ m] ={1,...,m}. Machine jhascapacity cj∈R+. For each job iandmachine j,
+we associate a value vij∈R+and a size sij∈R+. Anassignment is a function x: [n]→[m]/uniontext{∗}
+partially mapping jobs to machines, where ∗indicates that a job is left unassigned. We use x(i)
+to denote the machine (or ∗) that job iis assigned to. Moreover, the binary variable xijindicates
+whether x(i) =j. A feasible assignment may allocate to machine ja set of jobs of total size at
+mostcj. The GAP can be written as an integer program with decision va riables{xij}ij; the LP
+relaxation obtained by relaxing the constraint xij∈ {0,1}is given below.
+maximize/summationtext
+i,jvijxij
+subject to/summationtextm
+j=1xij≤1,fori= 1,...,n./summationtextn
+i=1sijxij≤cj,forj= 1,...,m.
+0≤xij≤1(GAP LP)
+The above LP is known to have an integrality gap of 2, and the ro unding can be done in
+polynomial time [18, 10].
+As detailed in §2.4, we consider a setting where the private data is a biparti te graph specifying
+job-machine compatibility. That is, the private data are no t the values vijor the sizes sij, but
+rather the existence of the edge ( i,j). Note that this does not change the GAP from an algorithmic
+point of view, since one can encode the compatibility relati on in the values {vij}ij. We define
+GAP[E] for a bipartite graph E⊆[n]×[m] as the problem of computing the welfare-maximizing
+assignment using only edges in E. The LP relaxation of GAP[E] is given below.
+4GAP≡GAP[E]
+SIGAP≡SIGAP[E] VIGAP≡VIGAP[E]
+MKP[E]
+MKPMWBM≡MWBM[E]
+MBM[E]
+Figure 1: Relationships between Assignment Problems
+Arrow indicates the second problem is a special case of the fir st.
+The≡symbol indicates computationally equivalent problems.
+maximize/summationtext
+i,jvijxij
+subject to/summationtextm
+j=1xij≤1,fori= 1,...,n./summationtextn
+i=1sijxij≤cj,forj= 1,...,m.
+0≤xij≤1
+xij= 0, for (i,j)/∈E.(GAP[E] LP)
+We distinguish several variants of the GAP and their respect ive bipartite-graph versions. The
+Size-Invariant Generalized Assignment Problem (henceforth SIGAP) is the problem where the size
+of a job idoes not depend on the machine – we denote this size by si. Similarly, the Value-
+Invariant Generalized Assignment Problem (henceforth VIGAP) is the problem where the value
+of a job idoes not depend on the machine – we denote the value by vi. TheMultiple Knapsack
+Problem (henceforth MKP) is the problem where neither size nor value depend on the machine.
+Theknapsack problem (henceforth KP) is MKP with m= 1.
+Inadditiontoknapsack-typeproblemslikethosedicusseda bove, theGAPalsogeneralizesbipar-
+tite matching problems. The maximum weight bipartite matching problem (henceforth MWBM) is
+the problem where all capacities and sizes are 1. The maximum bipartite matching problem (hence-
+forth MBM) is the special case of MWBM where all values are 1. T he latter is only interesting
+when constrained by a graph; that is, when we consider MBM[E] for some E⊆[n]×[m].
+When discussing a special case of GAP, say MKP, we refer to the bipartite-graph constrained
+version as MKP[E]. Moreover, we refer to (GAP LP) and (GAP[E] LP) as (MKP LP) and (MKP[E]
+LP), respectively. We also use similar notation for other sp ecial cases of the GAP.
+Figure1illustratestheorderingofthevariousassignment problemsbygenerality. Wesummarize
+the known algorithmic results, both upper and lower bounds, in Table 1.
+5Problem Upper Lower Integrality
+Bound Bound Gap
+GAP/ GAP[E]e
+e−1−δAPX-hard 2
+SIGAP/ SIGAP[E]e
+e−1−δAPX-hard 2
+VIGAP/ VIGAP[E]e
+e−1−δAPX-hard 2
+MKP[E]e
+e−1−δAPX-hard 2
+MKP PTAS Strongly NP-hard 2
+MWBM/ MWBM[E] 1 1 1
+MBM[E] 1 1 1
+Table 1: Computational Upper and Lower Bounds
+δis a small, positive, fixed real number.
+The integrality gap is given for the standard LP relaxation.
+2.2 Truthfulness
+We consider the setting where jobs are selfish agents, and the ir private types encode information
+about their value for being assigned on different machines. Ot her information, such as n,m,{sij}ij
+and{cj}m
+j=1is considered public2.
+We assume that player ihas a type vi={vij}m
+j=1, specifying his value for the different machines.
+We assume the possible types of player iare restricted to some public set Vi⊆Rm, and use Vto
+denoteV1×...Vn. For example, when working in the multiple knapsack problem , we require that
+for each job ithere is a real number visuch that vij=vifor allj. Moreover, as we will see in §2.3,
+no interesting results without money are possible if possib le values are unbounded. Therefore, our
+positive results will assume a restricted, discrete set of p ossible types described in §2.4.
+Amechanism without money for the GAP is simply an algorithm that takes in all the proble m
+data, public and private, and outputs an assignment of the jo bs to the machines. We allow our
+mechanisms to be randomized. We use A(I,v) to denote the output of mechanism Aon public
+dataIand private data v∈ V. Each mechanism Aand instance of the public data Iinduces a
+social choice rule : a function A(I,∗) mapping private valuations to assignments.
+We now state truthfulness without money generally.
+Definition 2.1. Fix a mechanism without money A, letxdenote the assignment A(I,v), and let
+x′denote the assignment A(I,v−i/uniontextv′
+i)when player ichanges his report to v′
+i. Mechanism Ais
+truthful if and only if the following always holds for for eac hI,v∈ V,i, andv′
+i∈ Vi.
+m/summationdisplay
+j=1vijxij≥m/summationdisplay
+j=1vijx′
+ij. (1)
+2It is conceivable that the job sizes sijare private information as well. However, if jobs can lie abo ut their sizes,
+we need to define the utility to a job when it is fractionally as signed— a job can either derive fractional utility from
+a fractional assignment, or zero utility if it is not fully as signed to a machine. Both models are reasonable; for the
+first, a nontrivial proof shows that no reasonable approxima tion can be obtained by any randomized strategyproof
+mechanism. For the second model , where jobs derive no utilit y from partial assignments, it turns out that jobs have
+no incentive to lie about their sizes in any of the algorithms we design.
+6Similarly, a randomized mechanism Ais truthful in expectation without money if and only if
+E[m/summationdisplay
+j=1vijxij]≥E[m/summationdisplay
+j=1vijx′
+ij]. (2)
+In other words, a mechanism Ais truthful if a job never benefits from misreporting its priv ate
+data. A randomized mechanism Ais truthfulin expectation if no (risk-neutral) job has an in centive
+to misreport its private data.
+2.3 Limits of Truthfulness without Money
+Here, we will justify considering a discrete valuation mode l by observing that, assuming general
+valuations, no interesting results are possible. Indeed, w e assume a restricted setting: the knapsack
+problem, with a knapsack of capacity 1, and jobs of size 1. If Vi=R+for each i, then it is easy
+to see that this is equivalent to the classical problem of a si ngle item auction [19]. It is well known
+that no non-trivial guarantees are possible for a single-it em auction if the mechanism is required
+to be truthful-in-expectation without using money. In fact , it is easy to see that no mechanism
+can outperform the trivial one which allocates the item (in o ur case, the entire capacity of the
+knapsack) uniformly at random, achieving an approximation ratio ofn.
+2.4 The Private Graph Valuation Model
+Given that no nontrivial upperbounds are possible when play ers can arbitrarily misrepresent their
+values, we consider a restricted model of the valuations: on e of a discrete nature as is characteristic
+of many problems for which truthfulness without money is pos sible. We assume job ihas a value
+ofδijvijfor being assigned to machine j, wherevijis public and δij∈ {0,1}is private. In other
+words, jobs may not lie about their potential value vij(that is,vijare publicly known or verifiable ),
+yet they may lie about which machines they are compatible with. This compatibility relation is
+encoded via a bipartite graph on the jobs and machines. Each j ob’s private data is the set of its
+outgoing edges, i.e. the machines with which it is compatibl e. As we discuss in §1, this situation
+arises in many natural settings.
+Aninstance of theGAP on a private bipartite graph is a pair ( I,E), where Iis a tuple
+({vij}ij,{sij}ij,{cj}j) of public information, and E⊆[n]×[m] is private information solicited
+from the jobs. Eis a set of edgessummarizing the compatibility of jobs and machines, where j ob
+i’s private data is the set Ei⊆Eof edges in Eincident on i. A jobireceives value vijfrom being
+assigned to machine jonly if (i,j)∈E, else it receives value 0. Our goal is to maximize welfare
+via a mechanism that, without using money, incentivizes ito report her set Eiof edges truthfully.
+We can now restate truthfulness as it applies to our model. We useA(I,E) to denote the
+assignment computed by mechanism Aon instance ( I,E). Moreover, for an edge set Ewe use
+Ei⊆Eto denote the set of edges with one endpoint at job i, and use E−ito denote E\Ei.
+Definition 2.2. For a mechanism without money A, letxdenote allocation A(I,E), and let x′
+denote allocation A(I,E−i/uniontextE′
+i).Ais truthful if and only if the following holds for each I,E,i,
+andE′
+i./summationdisplay
+j:(i,j)∈Eivijxij≥/summationdisplay
+j:(i,j)∈Eivijx′
+ij (3)
+7Similarly, a randomized mechanism Ais truthful in expectation if and only if
+E[/summationdisplay
+j:(i,j)∈Eivijxij]≥E[/summationdisplay
+j:(i,j)∈Eivijx′
+ij] (4)
+Note that the summation on both sides of the inequality is ove r the set of trueedgesEi: that
+is,Ais truthful [in expectation] if when a job misreports its inc ident edges, its [expected] utility
+from the new assignment x′does not increase on its true edges Ei.
+3 Combinatorial Mechanisms for Matching
+We will show that optimally solving the maximum matching pro blem, with some careful tiebreak-
+ing, yields a strategyproof polynomial-time mechanism. Fo r maximum weight matching, we show
+matching upper and lower bounds of 2 for truthful mechanisms without money, and a constant
+lowerbound for truthful-in-expectation mechanisms.
+3.1 Warmup: Maximum Bipartite Matching
+We consider the Maximum Bipartite Matching problem, constr ained by a privatebipartite graph E.
+We observe that simply finding the maximum matching, using co nsistent tiebreaking, immediately
+gives a strategyproof mechanism.
+Proposition 3.1. Fix a total order ≺on matchings in the complete bipartite graph. For a set of
+edgesE, letM(E)denote the set of matchings on edge set E. LetAbe the mechanism that, on
+input(I,E), finds the ≺-minimal matching in the set argmaxx∈M(E)/summationtext
+ijxij. ThenAis truthful.
+Proof.Assume for a contradiction that Ais not truthful. Then, there exists IandE= (E−i,Ei)
+andE′
+iviolating (3). Let x=A(I,E) andx′=A(I,E′), where E′=E−i/uniontextE′
+i. Jobiis not
+matched by an edge in Eiinx, yet is matched by an edge in Eiinx′. SinceAonly uses reported
+edges,iis not matched at all in x, yet is matched by an edge e∈Ei/intersectiontextE′
+iinx′. This implies that
+bothxandx′are inM(E)/intersectiontextM(E′). Observe that/summationtext
+ijxij=/summationtext
+ijx′
+ij, otherwise either xis not
+optimal in M(E), orx′is not optimal in M(E′), contradicting the definition of A. Therefore, both
+xandx′are optimal in both M(E) andM(E′). Recalling that algorithm Abreaks ties consistently,
+this yields a contradiction, as needed.
+Therefore, it suffices to define ≺so that the ≺-minimal maximum-matching on Ecan be com-
+puted in polynomial time. This gives the following proposit ion.
+Proposition 3.2. There is a polynomial-time, without-money mechanism for max imum bipartite
+matching that is optimal, and truthful in the private graph m odel.
+Proof.We represent each matching as a binary vector ( x11,x12,...,x 21,x22,...,x nm) in{0,1}nm,
+and let≺be the lexicographic order on these vectors. Then we proceed to find the ≺-minimal
+maximum matching as follows. We compute the size OPTof the maximum matching (this can
+be done in polynomial time). We then process edges in the orde r they appear in the vector
+representation, while maintaining a working set of edges Xinitialized to E. When processing an
+edgee, we check if removing edecreases the size of the maximum matching in Xby solving the
+problem on edges X\e. If so we keep einX, else we discard eby setting X=X\e. When
+finished, Xis a maximum matching; it is easy to see that Xis≺-minimal among all maximum
+matchings.
+81
+2 baγ
+γ
+11
+2 baγ
+γ
+1
+1
+(a) (b)
+Figure 2: Maximum Weight Matching Lowerbound
+Circles represent jobs, and squares represent machines.
+3.2 Maximum Weight Bipartite Matching
+Next, we consider the MWBM problem, constrained by a private bipartite graph E. Unlike MBM,
+we show constant lower bounds on the approximation ratio of t ruthful and truthful in expectation
+mechanisms.
+Theorem 3.3. No deterministic truthful mechanism without money for MWBM has approximation
+ratio better than 2. Moreover, no truthful-in-expectation randomized mechan ism gives better than
+a2
+2√
+2−1≈1.0938approximation.
+Proof.First, consider a deterministic mechanism A. Assume for a contradiction that Aattains an
+approximation α <2. Consider Figure 2(a), where γ >1. Both jobs 1 and 2 prefer machine ato
+machine b. If we let γbe sufficiently close to 1, Acannot leave either job unassigned. Without
+loss of generality, we assume job 1 is assigned to aand job 2 is assigned to b. Now consider job 2
+changing his bid as in Figure 2(b). Acannot assign job 2 at all under these new bids, as that would
+violate truthfulness. Therefore, when given the bids in Fig ure 2(b), the welfare of the solution is
+at mostγ, whereas the optimal is γ+1. Letting γbe sufficiently close to 1 gives the contradiction.
+Now, we consider an arbitrary truthful-in-expectation mec hanismAon Figure 2(a). At least
+one of the jobs must be assigned to the preferred machine awith probability no more than 1 /2;
+without loss of generality this is job 2. Job 2 derives value a t most (γ+1)/2. Now, consider job
+2 changing his bid as in Figure 2(b). By truthfulness, now Acan assign job 2 with probability at
+mostp= (γ+1)/2γ. Therefore, the welfare of the assignment returned on the bi ds of Figure 2(b)
+is at most γ+1−(1−p)·1 =γ+p. However, the optimum is still γ+1, so the approximation
+ratio is at leastγ+1
+γ+p=γ+1
+γ+(γ+1)/2γ
+Usingelementary calculus, wecan choose γtomaximize this expression andcomplete theproof.
+Therefore, wecannot hopeforbetter than aconstant factor a pproximation (specifically aPTAS,
+randomized or deterministic, is not possible). We will show a factor 2 deterministic truthful mech-
+anism, matching Theorem 3.3.
+Consider the greedy Algorithm 1. Notice that step (1) does not depend on the reported edges
+E.
+9Algorithm 1 Mechanism for MWBM on Private Bipartite Graph
+1:Order pairs ( i,j)∈[n]×[m] in decreasing order of vij, breaking ties arbitrarily.
+2:LetX=∅.
+3:for alle∈Ein the order defined above do
+4:ifX/uniontext{e}is a matching then
+5:LetX=X/uniontext{e}
+6:end if
+7:end for
+8:return X
+Theorem 3.4. Algorithm 1 is a polynomial-time, 2approximate, no-money truthful mechanism
+for maximum weight bipartite matching in the private graph m odel.
+Proof.The approximation ratio immediately follows from a standar d charging argument against
+the optimal solution.
+For truthfulness, consider a job imisrepresenting his true edges EiasE′
+i. LetE′=E−i/uniontextE′
+i.
+LetXbethematching returnedbythealgorithm onreports E, andlet X′bethematchingreturned
+on reports E′. IfX=X′, thenidoes not improve his value. Assume X/\e}atio\slash=X′, and let e′∈E′
+be the first edge in X′\Xaccording to the order of step (1). Since the algorithm proce sses edges
+in the bid-independent order of step (1), it is easy to see tha te′∈E′\E=E′
+i\Ei. Thus, iis
+matched to e′/\e}atio\slash=Ei, an edge from which he derives no value, when he reports E′
+i. This completes
+the proof.
+4 LP-Based Mechanisms for Knapsack type Problems
+The bipartite matching problems studied in the previous sec tion can be solved in polynomial time;
+there, theneedforapproximation isaresultpurelyof there quirement ofstrategyproofness. We now
+investigate knapsack-like variants of the GAP — unlike matc hing, these problems are NP-hard (in
+fact, they are APX-hard). The mechanisms we design have the f ollowing common structure: first,
+we design a truthful mechanism without money for the fractional LP relaxation of the problem.
+Then, as in [14], we use a randomized procedure to obtain a fea sible integral assignment from the
+fractional solution— this composition leads to a truthful- in-expectation mechanism without money.
+We introduce this technique for designing truthful mechani sms without money in §4.1, and then
+use it to design mechanisms without money for knapsack type a ssignment problems in §4.2-§2.1.
+Some of our analyses will use notions from network flow theory , which we recap in Appendix B.
+4.1 A Reduction to Fractional Truthfulness
+The construction of Lavi and Swamy allows us to reduce constr ucting a truthful-in-expectation
+mechanism (with money) for the integral problem to construc ting a truthful mechanism for a
+fractional version of the problem. We redevelop their const ruction, in a form convenient for our
+purposes, in Appendix A. While the truthful mechanism they c onstruct for the fractional welfare
+maximization problem (defined in Appendix A) simply solves the problem optimally a nd uses VCG
+payments, we observe that this need not be the case. Indeed, t his is crucial for our purposes; for
+10some of the assignment problems we are interested in, the opt imal algorithm for the fractional prob-
+lem requires non-zero payments for truthfulness. Instead, we sacrifice optimality in the fractional
+solution to get a truthful fractional mechanism with zero payments . Then we use the fractional
+mechanism to get a scaled down truthful-in-expectation mec hanism – also with zero payments –
+for the combinatorial problem as in Theorem A.5.
+By examining the proof of Theorem A.5, we notice that the assu mption that the fractional
+mechanism solves the LP exactly is not used. In fact, an arbit rary truthful mechanism Mfrac
+for the fractional problem can be converted to a truthful-in -expectation mechanism Mexpfor the
+combinatorial problem. If Mfracis aβ-approximation algorithm for the fractional problem, then
+Mexpis anα·βapproximation algorithm for the combinatorial problem, wh ereαis the integrality
+gap of the LP relaxation. Moreover, by examing the proof of Th eorem A.5, we notice that the
+payment scheme pexpof mechanism Mexpis simply a scaled down copy of the payment scheme
+pfracofMfrac. In particular, if Mfracis a truthful mechanism without money for the fractional
+problem, then Mexpis a truthful-in-expectation mechanism without money for t he combinatorial
+problem. We sum up these observations in the following Lemma .
+Lemma 4.1. Assume the fractional welfare maximization problem over po lytopePand valuation
+classV(as defined in Appendix A) admits a β-approximate mechanism that is truthful without
+money. Moreover, assume Psatisfies the conditions of Lemma A.4 with integrality gap α. Then
+there exists an efficient truthful-in-expectation α·β-approximate mechanism for the welfare maxi-
+mization problem over PandVthat does not use money.
+Inotherwords, wereducetheproblemofdesigningatruthful -in-expectation mechanismwithout
+money to that of designing such a mechanism for its fractiona l relaxation. This will prove partic-
+ularly useful, since arguing about truthfulness of a contin uous fractional assignment algorithm is
+more tractable than designing a combinatorial algorithm di rectly. Moreover, since the integrality
+gap of (GAP LP) is 2, an α-approximate mechanism for a fractional assignment proble m gives a
+2α-approximate mechanism for the integral problem. (Recall t hat the algorithm of [18, 10] shows
+an integrality gap of 2 for GAP as needed for Lemma A.4.)
+Corollary 4.2. Consider any special case of the GAP. If the fractional version of the problem ad-
+mits aβ-approximate mechanism that is truthful without money, the n there exists a 2β-approximate
+truthful-in-expectation mechanism for the combinatorial problem without money.
+A note is in order on the LP relaxations of the GAP and various c ases used in this reduction.
+Observe that the LP’s for the variants of the GAP on a bipartit e graphdo notfit the framework
+of [14]. This is because the valuation – i.e. the edges – are en coded explicitly in the polytope and
+not in the objective. However, this is not a problem for us, si nce the equivalent GAP LP does
+fit the framework. Therefore, after getting a fractional sol ution to, say, MKP[E], we can simply
+re-interpret it as a fractional solution to (GAP LP) and perf orm the reduction of Lavi and Swamy.
+4.2 The Multiple Knapsack Problem
+We consider the multiple knapsack problem on a private bipar tite graph E. First, we make the
+simple observation that, if we ignore computational constr aints, there exists a truthful optimal
+mechanism for the multiple knapsack problem in the private g raph model. As in maximum (un-
+weighted) bipartitematching, simplyreturningan optimal solution, breakingties consistently, leads
+to a strategyproof mechanism.
+11Proposition 4.3. Consider the without-money mechanism that, on reports E⊆[n]×[m], finds
+the optimal integral solution to MKP[E] LP, breaking ties con sistently via an arbitrary total order
+/precedesequalon the set of assignments ([m]/uniontext{∗})[n]. This mechanism is truthful in the private graph model.
+Proof.Fix a player iwith true edges Ei, and fix the reported edges E−iof the other players. Let x
+bethe assignment on reports E= (Ei,E−i), and let x′bethe assignment on reports E′= (E′
+i,E−i).
+Assume for a contradiction that truthfulness is violated, i n particular that x(i) =∗, yetx′(i) =j
+for some machine jwhere (i,j)∈Ei.
+Recall that xis the optimal feasible solution for MKP[ E], andx′is the optimal feasible solution
+for MKP[ E′]. Sinceiis unassigned in x, we know that xis feasible for MKP[ E′] as well. Moreover,
+sincex′assignsiusing an edge in E, we know x′is feasible for MKP[ E]. We conclude that each
+ofxandx′is feasible and optimal for MKP[ E] and MKP[ E′]. This contradicts the consistent
+tie-breaking of the mechanism.
+The above implies that, unlike maximum weight matching and i ts various generalizations, MKP
+isnotfundamentallyincompatiblewithtruthfulnesswitho utmoney–atleast whenignoringcompu-
+tational constraints. Nevertheless, MKP on a bipartite gra ph is APX-hard. Therefore, we consider
+the problem of findinga truthfulconstant-factor approxima tion. Though a simple greedy algorithm
+gives a deterministic, truthful 2+ ǫapproximation, we will instead illustrate our techniques f rom
+Section 4.1 – which will also come in handy for other generali zations of the GAP – by designing a
+randomized, 2-approximate, truthful-in-expectation mec hanism for MKP in our model.
+By the discussion in §4.1, it suffices to devise a truthful fractional algorithm in t he sense of
+Equation (6) of Appendix A. In this section, we show that solv ing (MKP[E] LP) optimally,
+with careful tiebreaking, yields such an algorithm. By Coro llary 4.2, this yields a 2-approximate
+truthful-in-expectation mechanism for MKP in the private g raph model.
+Algorithm 2 Fractional Mechanism for MKP on Private Bipartite Graph
+Input:Public instance of MKP, and reported edges E.
+Output: A feasible optimal solution xfor MKP[E] LP
+1:Fix an arbitrary order on the edges of the complete bipartite graph [n]×[m]. Let≺be the
+lexicographic order on [ R][n]×[m]corresponding to the order on edges.
+2:Find the optimal solution xto MKP[E] LP, breaking ties according to ≺.
+3:return x
+Consider Algorithm 2 for the fractional multiple knapsack p roblem on a bipartite graph. First
+of all, it is easy to see that Algorithm 2 can be implemented in polynomial time by solving a
+sequence of linear programs in step (2). In order to show trut hfulness, we will use a flow-based
+interpretation of Algorithm 2. For reported edges E, we define graph G[E], seen in Figure 3, as
+follows. There is a node for each job, and a node for each machi ne. We connect job ito machine
+jif (i,j)∈E, with weight w(i,j)= 0 and capacity c(i,j)=∞. Next, we include a source node s,
+and create an edge ( s,i) for each job iwith weight w(s,i)=vi/siand capacity c(s,i)=si. We then
+create a sink tand create an edge ( j,t) for each machine j, with weight w(j,t)= 0 and capacity
+c(j,t)=cj. Finally, we connect the sink to the source via an edge ( t,s) withw(t,s)= 0 and capacity
+c(t,s)=∞.
+Observe that fractional assignments [0 ,1][n]×[m]are in one to one correspondence with feasible
+circulations in the complete bipartite graph G[[n]×[m]]. Moreover, feasible solutions of MKP[E]
+12Job Machine Source/Sinks ti
+j0,∞
+vi
+si, si 0, c
+j0,∞
+Figure 3: Flow Interpretation of the Multiple Knapsack Prob lem
+Edges are labeled with (weight,capacity) pairs.
+LP are in one-to-one correspondence with feasible circulat ions inG[E]. In particular, assignment
+xfeasible for MKP[E] LP maps to the unique feasible circulati onfxonG[E] satisfying fx((i,j)) =
+xijsifor each job iand machine j. Notice, also, that the value (i.e. the welfare) of assignme ntx
+is the same as the weight of the circulation fx. Moreover, we define a total order on circulations
+inG[[n]×[m]] that corresponds to the lexicographic order ≺defined on [ R][n]×[m]. We abuse
+notation and use ≺to refer to both total orders, and let fx≺fyif and only if x≺y. Notice that
+≺also orders feasible circulations lexicographically. The refore, we can interpret Algorithm 2 as
+finding the ≺-minimal maximum-weight feasible circulation in G[E], and then converting it to the
+corresponding assignment.
+Next, we show that Algorithm 2 is a truthful fractional mecha nism, that is, a player icannot
+benefit by misrepresenting his edges Eias some E′
+i. For this, if the algorithm returns assignment
+xwhen reported edges are E, and assignment x′when reported edges are E′=E−i/uniontextE′
+i, then we
+must have x(Ei)≥x′(Ei) (note that this is because vij=viin MKP). Equivalently, we need to
+show/summationtext
+e∈Eifx(e)≥/summationtext
+e∈Eifx′(e), where we use the convention f(e) = 0 when fis a flow on G[E]
+ande /∈E(and the same for G[E′]).
+We will show that any increase in job i’s utility after lying implies that one of fxorfx′is
+suboptimal, yieldingacontradiction. Wewill usenotionsf romnetwork flow, developed inAppendix
+B. We begin with the following lemma.
+Lemma 4.4. Let circulation ∆be the difference between fx′andfx; i.e.∆ =fx′−fx. If/summationtext
+e∈Eifx′(e)>/summationtext
+e∈Eifx(e)then∆can be conformally decomposed into {C,∆−C}where: (1)
+Cis a flow cycle, and (2) Csends positive flow on both (s,i)and some e∈Ei/intersectiontextE′
+i.
+Proof.Let/braceleftbig
+C1,...,Ck/bracerightbig
+be the conformal decomposition of ∆ into cycles as in theorem B.6. It
+suffices to show that some Cjsatisfies conditions (1) and (2).
+Sincex′sends more flow on edges in Eithanx, someCjentersithrough an edge e′/∈Ei,
+13and exits through an edge e∈Ei. By conformality and the fact that x′sends no flow on Ei\E′
+i,
+we know that e∈Ei/intersectiontextE′
+i. Moreover, by conformality and the fact that xsends no flow on edges
+inE′
+i\Ei, it is easy to see that e′/∈E′
+i\E. Therefore, the only remaining possibility is that
+e′= (s,i).
+This yields truthfulness of the algorithm.
+Lemma 4.5. Algorithm 2 is a truthful fractional mechanism
+Proof.Fix an instance ( I,E), and assume for a contradiction that a player iwith true edges Ei
+benefits by reporting E′
+iinstead. Let xandx′be the assignments computed by the algorithm on
+reportsEandE′=E−i/uniontextE′
+i. By assumption,/summationtext
+e∈Eifx′(e)>/summationtext
+e∈Eifx(e). Let ∆ and Cbe as
+in Lemma 4.4. Observe that Cdoes not send flow on any edges in the symmetric difference of Ei
+andE′
+i. Therefore, by Lemma B.7 fx+Cis a feasible circulation in G[E], and moreoever fx′−C
+is a feasible circulation in G[E′]. If the weight w(C) of circulation Cis non-zero, then one of fxor
+fx′is non-optimal. Therefore, w(C) = 0. Now notice that, by definition of ≺, eitherfx+C≺fx
+orfx′−C≺fx′. Therefore, one of fxorfx′is not a ≺-minimal optimal solution, yielding the
+contradiction.
+Combining with Corollary 4.2, we get the following theorem.
+Theorem 4.6. There is a polynomial-time, 2-approximate, without-money mechanism for the mul-
+tiple knapsack problem that is truthful-in-expectation in the private graph model.
+4.3 Size-Invariant GAP
+We consider the size-invariant generalized assignment pro blem on a private bipartite graph E.
+Since SIGAP generalizes MWBM, by Theorem 3.3 no determinist ic truthful approximation can
+achieve better than a factor 2 approximation, and moreover n o truthful in expectation PTAS is
+possible. In this section, we devise a 4-approximate withou t-money mechanism for SIGAP that
+is truthful-in-expectation in the private graph model. Eve n though solving SIGAP[E] LP is not
+fractionally truthful(again, by Theorem3.3), we showthat asimplegreedy algorithm is fractionally
+truthful and yields a 2-approximate solution the LP. Combin ing this with Corollary 4.2, we get a
+4-approximate, without-money mechanism for SIGAP that is t ruthful in expectation in the private
+graph model. Consider the following algorithm.
+Algorithm 3 Fractional Mechanism for SIGAP on Private Bipartite Graph
+Input:Public instance of SIGAP, and solicited private edges E.
+Output: A feasible solution xfor SIGAP[E] LP
+1:Order[n]×[m]indecreasingorderofvaluedensity de,whered(i,j)=vij
+si,breakingtiesarbitrarily.
+2:for all(i,j)∈E, in the order defined above do
+3:Fractionally assign as much of job ion machine j, until the job is exhausted or the machine
+is full.
+4:end for
+5:return the resulting assignment x.
+First, we bound the approximation factor of Algorithm 3.
+14Lemma 4.7. Algorithm 3 returns a 2-approximate solution to SIGAP[E] LP.
+Proof.This can be shown by a charging argument, best formalized by c onstructing a feasible
+solution to a dual of SIGAP[E] LP of value at most twice the val ue attained by the algorithm. This
+dual is shown below, and has decision variables u∈Rnandz∈Rm:
+minimize/summationtextn
+i=1ui+/summationtextm
+j=1cjzj
+subject to ui+sizj≥vij,for (i,j)∈E.
+ui≥0, fori= 1,...,n.
+zj≥0, forj= 1,...,m.(SIGAP[E] LPD)
+First, we make a simple observation about the algorithm that will be useful in the proof.
+Observation 4.8. For a job i, edges incident on iare examined in decreasing order of vij(since
+sizesiis independent of the machine). For a machine j, edges incident on jare examined in
+decreasing order of vij/si.
+We now construct the dual solution u,zin parallel with the execution of the algorithm as
+follows. Begin with u= 0 and z= 0. Consider the iteration of Algorithm 3 corresponding to
+edgee= (i,j). If job iis exhausted on this iteration, set ui=vij. If the capacity on machine j
+is exhausted on this iteration, set zj=vij/si. Notice that, in both cases, this satisfies the dual
+constraint corresponding to edge ( i,j). If no assignment is made on this iteration – i.e. either i
+orjwas exhausted in a previous iteration – then we do not update t he dual variables. Indeed,
+there is no need to do so for feasibility: By Observation 4.8, ifiis already exhausted then already
+ui≥vij, and ifjis already exhausted then already zj≥vij/si, and either suffices to satisfy the
+dual constraint for edge e.
+It remains to bound the value of the dual solution as compared to the primal solution. First,
+we write twice the value of the primal in a convenient form:
+2/summationdisplay
+i,jvijxij=/summationdisplay
+i/summationdisplay
+jvijxij+/summationdisplay
+j/summationdisplay
+ivij
+si(sixij)
+Observe that, by Observation 4.8, uilower bounds the value of any edge on which any part of
+jobiis assigned, and zjlower-bounds the density of any job assigned to machine j. Moreover, ui
+is non-zero only if iis fully assigned, and zjis non-zero only if jis full. Therefore, we get
+/summationdisplay
+i/summationdisplay
+jvijxij+/summationdisplay
+j/summationdisplay
+ivij
+si(sixij)≥/summationdisplay
+iui/summationdisplay
+jxij+/summationdisplay
+jzj/summationdisplay
+i(sixij)
+=/summationdisplay
+iui+/summationdisplay
+jzjcj
+The final term is precisely the value of the dual. Invoking wea k LP duality completes the
+proof.
+It remains to show that Algorithm 3 is fractionally truthful . We begin with an observation.
+Observation 4.9. Lete1,...,enmbe the ordering [n]×[m]in decreasing order of density vij/si.
+Let≺denote the lexicographic ordering on R[n]×[m]. Algorithm 3 returns the ≺-maximal feasible
+solution of SIGAP[E] LP.
+15Job Machine Source/Sinks ti
+jvij
+si,∞
+0, si 0, c
+j0,∞
+Figure 4: Flow Interpretation of the Size-invariant Genera lized Assignment Problem
+Edges are labeled with (weight,capacity) pairs.
+As in section 4.2, we use a network flow interpretation. We defi ne graph G[E] forE⊆[n]×[m].
+This construction is similar to that of §4.2, and the graph is shown in Figure 4. As in §4.2, feasible
+assignments for SIGAP[E] LP are in one-to-one corresponden ce with feasible circulations in G[E].
+We can similarly define an order ≺on flows in G[[n]×[m]] by setting fx≺fywhenx≺y.
+Therefore, we can interpret Algorithm 3 as finding the ≺-maximal feasible circulation in G[E].
+Next, we establish a decomposition lemma similar to, yet mor e involved than, Lemma 4.4.
+Lemma 4.10. Letxbe the output assignment of Algorithm 3 with declarations E, and let x′be
+the output assignment with declarations E′=E−i/uniontextE′
+i. Let circulation ∆be the difference between
+fx′andfx; i.e.∆ =fx′−fx. If/summationtext
+e∈Eiwefx′(e)>/summationtext
+e∈Eiwefx(e)then∆can be conformally
+decomposed into {C,∆−C}where:
+1.Cis a flow cycle
+2.C≺0. In other words, adding Cto any other circulation worsens the lexicographic order.
+3.Centersithrough some e∈Ei\E′
+i, withC(e)<0.
+4.Cexitsithrough some e′∈Ei/intersectiontextE′
+i, withC(e′)>0.
+5.we′> we.
+Proof.Consider the conformal decomposition of ∆ into cycles C1,...,Ck. It suffices to show that
+someCjsatisfies the conditions above. By assumption, there is a cyc leCin the decomposition with/summationtext
+e∈EiweC(e)>0. By conformality, cycle Cmust exit ithrough an edge e′∈Ei/intersectiontextE′
+i. Moreover
+Ccannot enter through an edge in both graphs G[E] andG[E′]: if it did, then both fx+Cand
+fx′−Care feasible in G[E] andG[E′] respectively, and thus one of them is not lexigographicall y
+maximal, contradicting observation 4.9. Therefore, by con formality, Cmust remove flow from an
+16edgee∈Ei\E′
+i, and add flow to e′∈Ei/intersectiontextE′
+i, withwe′> we. Thus,Csatisfies conditions 1, 3, 4,
+and 5.
+It remains to establish condition 2. Observe that fx+Cis a feasible circulation on G[E]. Since
+fxis a lexicographically maximal feasible circulation on G[E], we deduce C≺0.
+We are now ready to show truthfulness.
+Lemma 4.11. Algorithm 3 is a truthful fractional mechanism for SIGAP.
+Proof.Assume, for a contradiction, that a player ibenefits by reporting E′
+iinstead of his true edges
+Ei. Letxandx′be the assignments computed by the algorithm on reports EandE′=E−i/uniontextE′
+i.
+By assumption,/summationtext
+e∈Eiwefx′(e)>/summationtext
+e∈Eiwefx(e). Let ∆ and Cbe as in Lemma 4.10.
+Recall that C≺0. Thus, if fx′−Cwere feasible in G[E′] we would be done, as we would
+contradict Observation 4.9. However, this is not the case, a sfx′−Csends positive flow on edge
+e∈Ei\E′
+i. We remedy this by simply zeroing out the flow on edge e= (i,j), as follows: Let Dbe
+the flow cycle through s,i,j,tsuch that fx′−C−Dsends no flow on e. It is clear that fx′−C−D
+is a feasible circulation on G[E′]. We claim that still fx′−C−D≻fx′. To see this, notice that by
+condition 5 of Lemma 4.10, edge eis not the greatest weight edge with non-zero flow in −C. Thus,
+since−C≻0, it is easy to see that also −C−D≻0. Therefore fx′−C−D≻fx′, as needed.
+Combining with Corollary 4.2, we get the theorem.
+Theorem 4.12. There is a polynomial-time, 4approximate, without-money mechanism for the
+size-invariant generalized assignment problem that is tru thful-in-expectation in the private graph
+model.
+4.4 VIGAP and GAP
+Inthis section, weoverview theresults that we obtain for VI GAP and GAP, utilizing thetechniques
+developed in §4.1-4.3. However, since these differ from our results so far on ly technically, we defer
+details to the full version of the paper.
+Consider the VIGAP. We observe that Proposition 4.3 holds es sentially unchanged; that is,
+solving VIGAP optimally gives a truthful mechanism in our mo del. Next, we observe that greedy
+Algorithm 3, when adapted to the fractional VIGAP (namely, d ensityd(i,j)is now defined as
+vi/sij), still provides a 2-approximation by essentially the same analysis. A more involved inductive
+argument is needed to show that this fractional algorithm is truthful; We defer this technical, yet
+simple, proof to the full version of the paper. We get the foll owing theorem.
+Theorem 4.13. There is a polynomial-time, 4approximate, without-money mechanism for the
+value-invariant generalized assignment problem that is tr uthful-in-expectation in the private graph
+model.
+We now turn to the GAP. First, we make an assumption – to be remo ved later – that the maxi-
+mum value vmaxof an edge in Eis publicly known up-front. Under this assumption, we reduc e de-
+signinga truthfulmechanism for GAP to the truthfulmechani sm for VIGAP with a loss of O(logn)
+in the approximation ratio. In particular, we randomly pick v∈ {vmax,vmax
+2,vmax
+4,...,vmax
+O(n2))}, and
+define a new value /hatwidevijfor each item iand bin jas follows: If vij≥vthen/hatwidevij=v, else/hatwidevij= 0
+(equivalently, we discard edge ( i,j)). This gives an instance of VIGAP that we can solve using the
+17mechanism of Theorem 4.13. It is easy to verify that, in expec tation, this reduction results in a loss
+of at most O(logn) in the approximation ratio. However, to ensure truthfulne ss we need to guar-
+antee that edge ( i,j) actually results in value exactly /hatwidevijif chosen; we do so by positing up-front
+that, whenever job iis assigned to machine jby the subroutine that solves VIGAP, we cancel i’s
+assignment with probability 1 −/hatwidevij/vij. It is now easy to see that, under our original assumption
+thatvmaxis known up-front, this mechanism is truthful-in-expectat ion and has an approximation
+ratio ofO(logn).
+We now remove the assumption that vmaxis public knowledge by appropriately incentivizing
+the job with the maximum value edge. In particular, after rec eiving the reported edges E, we
+flip a fair coin. If the coin turns up heads, we assign the job wi th the maximum value edge on
+that edge ( i.e., to his favorite machine, with value vmax), and leave all other jobs unassigned. If
+the coin turns up tails, we discard the job with the maximum va lue edge, and proceed with the
+algorithm described above using this value of vmax. It is easy to see that this is still an O(logn)
+approximation algorithm. Moreover, the job with the maximu m value edge can do no better than
+report his true edges, and no job has incentive to falsely cla im a maximum value edge. This gives
+the following theorem.
+Theorem 4.14. There is a polynomial-time, O(logn)approximate, without-money mechanism for
+the generalized assignment problem that is truthful-in-ex pectation in the private graph model.
+We leave open the question of whether there exists a constant factor truthful [in expectation]
+mechanism without money for the GAP in our model.
+5 Acknowledgements
+We thank Preston McAfee, Serge Plotkin, Ariel Procaccia, Ti m Roughgarden, Michael Schwarz,
+and Mukund Sundararajan for helpful discussions and commen ts.
+References
+[1] Noga Alon, Michal Feldman, Ariel D. Procaccia, and Moshe Tennenholtz. Strategyproof
+approximation mechanisms for location on networks. CoRR, abs/0907.2049, 2009.
+[2] Noga Alon, Felix Fischer, Ariel D. Procaccia, and Moshe T ennenholtz. Sum of us: Strate-
+gyproof selection from the selectors. Working paper , 2009.
+[3] A. Archer and ´E. Tardos. Truthful mechanisms for one-parameter agents. I nFOCS, pages
+482–491, 2001.
+[4] Itai Ashlagi, Felix Fischer, Ian Kash, and Ariel D. Proca ccia. Mix and match. In ACM
+Conference on Electronic Commerce , 2010.
+[5] Yossi Azar and Iftah Gamzu. Truthful unification framewo rk for packing integer programs
+with choices. In ICALP (1) , pages 833–844, 2008.
+[6] Dimitri Bertsekas. Network Optimization: Continuous and Discrete Models . Athena Scientific,
+1998.
+18[7] Patrick Briest, Piotr Krysta, and Berthold V¨ ocking. Ap proximation techniques for utilitarian
+mechanism design. In STOC, 2005.
+[8] Eric Budish. The combinatorial assignment problem: App roximate competitive equilibrium
+from equal incomes. Working paper , 2009.
+[9] Chandra Chekuri and Iftah Gamzu. Truthful mechanisms vi a greedy iterative packing. In
+APPROX-RANDOM , pages 56–69, 2009.
+[10] ChandraChekuri and Sanjeev Khanna. A PTAS for the multi ple knapsack problem. In SODA,
+pages 213–222, 2000.
+[11] Uriel Feige and Jan Vondr´ ak. Approximation algorithm s for allocation problems: Improving
+the factor of 1 - 1/e. In FOCS, pages 667–676, 2006.
+[12] Lisa Fleischer, Michel X. Goemans, Vahab S. Mirrokni, a nd Maxim Sviridenko. Tight ap-
+proximation algorithms for maximum general assignment pro blems. In SODA, pages 611–620,
+2006.
+[13] R. Lavi, A. Mu’alem, and N. Nisan. Towards a characteriz ation of truthful combinatorial
+auctions. In FOCS, pages 574–583, 2003.
+[14] Ron Lavi and Chaitanya Swamy. Truthful and near-optima l mechanism design via linear
+programming. In FOCS, 2005.
+[15] R. Myerson. Optimal auction design. Mathematics of Operations Research , 6:58–73, 1981.
+[16] Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Va zirani.Algorithmic Game The-
+ory. Cambridge University Press, New York, NY, USA, 2007.
+[17] Ariel D. Procaccia and Moshe Tennenholtz. Approximate mechanism design without money.
+InACM Conference on Electronic Commerce , pages 177–186, 2009.
+[18] David B. Shmoys and ´Eva Tardos. An approximation algorithm for the generalized assignment
+problem. Math. Program. , 62:461–474, 1993.
+[19] William Vickrey. Counterspeculation, auctions, and c ompetitive sealed tenders. Journal of
+Finance, 16(1):8–37, 1961.
+19A The Construction of Lavi and Swamy
+First, we recall some basic concepts from combinatorial opt imization, and state the key lemma in
+the construction of Lavi and Swamy.
+Definition A.1. LetP⊆Rdbe a polytope. We define the Integer hull ofP, which we denote by
+I(P), as the convex hull of all integer points in P. Equivalently,
+I(P) =hull(P/intersectiondisplay
+Zd)
+Definition A.2. LetPbe a polytope. The integrality gap ofPis defined as:
+max
+c∈Rdmax{cTx:x∈P}
+max{cTx:x∈I(P)}
+Note that the integrality gap of a polytope does not depend on any particular set of objectives
+c. In fact, even in problems where the only objectives of inter est are of a certain class – say
+submodularvaluations as used in combinatorial auctions, o r objectives cin the nonnegative orthant
+–theconstruction ofLavi andSwamy can only performas well a s theintegrality gap of thepolytope
+as a whole, as defined above.
+Definition A.3. We say an algorithm shows an integrality gap of αfor a polytope P⊆Rdif it
+takes as input an arbitrary vector c∈Rd, and outputs a point z∈P/intersectiontextZdwith the guarantee that:
+cz≥1
+αmax{cx:x∈P}
+The construction of Lavi and Swamy concerns packing polytopes . A polytope P⊆Rdis a
+packing polytope if it is contained in the positive orthant – i.e.P⊆R+d–, and moreover it is
+downwards closed : ify∈Pandx∈R+dis such that x/precedesequaly(component-wise) then x∈P.
+Now, we come to the key lemma. We consider a packing polytope P, and use P/αto denote
+the scaled down copy of P– i.e.P/α={y/α:y∈P}. Using an algorithm showing the integrality
+gap ofαforP, we can explicitly construct for every x∈P/αa distribution over integer points of
+Pthat evaluates to xin expectation.
+Lemma A.4 ([14]).LetPbe a packing polytope of integrality gap at most α. Then, for every
+x∈P/αthere exists a distribution DxoverP/intersectiontextZdsuch that Ey∼Dxy=x. Moreover, if there exists
+a polynomial-time algorithm Bthat shows an integrality gap of αforP, then, for any x∈P/αwe
+can compute Dxin polynomial time.
+Armed with the above lemma, Lavi and Swamy reducedesigning a truthful-in-expectation
+mechanism (with money) to designing a truthful fractional mechanism (also with money), to be
+defined later.
+We illustrate the technique of Lavi and Swamy by considering welfare maximization problems
+in a very general form. Let P⊆R+dbe a packing polytope representing the set of feasible
+solutions. Moreover, assume that there are nplayers [n], and the valuation function vi:Rd→R
+of player iis required to lie in some valid set of linear valuations Vi⊆RRd. We denote V=
+V1×V2...×Vn. Thecombinatorial welfare maximization problem (henceforth CWMP) over Pand
+Vis the problem of finding an integer point in Pmaximizing the sum of values of the players. This
+20gives the following LP relaxation, which we refer to as the fractional welfare maximization problem
+(henceforth FWMP):
+maximize/summationtext
+ivi(x)
+subject to x∈P
+We definea truthful-in-expectation mechanism for thecombinatorial welfare maximization prob-
+lem as a pair M= (f,p) wheref:V →P/intersectiontextZdis arandomized allocation rule , andp:V →Rnis
+arandomized payment scheme . As usual, we say Mistruthful in expectation if the following holds
+for each/vector v∈ Vandvi∈ Vi
+E[vi(f(v))−(p(v))i]≥E[vi(f(v−i,v′
+i))−(p(v,v′
+i))i]. (5)
+Moreover, we definea fractional mechanism as a pair M= (f,p) wheref:V →Pis afractional
+allocation rule , andp:V →Rnis apayment scheme . We say Mis atruthful fractional mechanism
+if the following holds for each /vector v∈ Vandvi∈ Vi
+vi(f(v))−(p(v))i≥vi(f(v−i,v′
+i))−(p(v,v′
+i))i. (6)
+It is easy to see that we can use the VCG mechanism to obtain a tr uthful fractional mechanism
+Mfracfor the fractional welfare maximization problem. Moreover , when optimizing objectives in V
+overPcan be done efficiently, Mfracis a polynomial time mechanism. Lavi and Swamy observed
+that, given an efficient approximation algorithm that shows t he integrality gap of Pin the sense
+defined above, one can obtain a polynomial-time, truthful-i n-expectation mechanism Mexpthat
+looks simply like a scaled down version of Mfrac. This follows from lemma A.4. We state the main
+theorem of Lavi and Swamy and sketch its proof below.
+Theorem A.5 ([14]).Assume the fractional welfare maximization problem can be s olved efficiently
+for any valuations in V. Moreover, assume Psatisfies the conditions of Lemma A.4 with integrality
+gapαand algorithm Bshowing the integrality gap. Then there exists an efficient tru thful-in-
+expectation α-approximate mechanism for the welfare maximization probl em over PandV.
+Proof Sketch. Consider the efficient, truthful fractional mechanism Mfrac= (ffrac,pfrac) con-
+structed using VCG. We will define Mexp= (fexp,pexp) as follows. We let pexp(v) =p(v)
+α, and
+letfexp(v) be drawn from the distribution Dffrac(v)/α, as defined in Lemma A.4. The random func-
+tionfexpcan beevaluated efficiently using B, as in the lemma. Now, usinglinearity of expectations,
+the linearity of the valuation functions, and Lemma A.4, it i s easy to show truthfulness of Mexpby
+showing that the expected valuation less the payment of a pla yer matches that of Mfracup to a
+scaling factor of α.
+E[vi(fexp(v′))−(pexp(v′))i] =vi(E[fexp(v′)])−(pexp(v′))i
+=vi(1
+α·ffrac(v′))−1
+α·(pfrac(v′))i
+=1
+α·[vi(ffrac(v′))−(pfrac(v′))i]
+Applying this equivalence to both sides of inequality (6) yi elds truthfulness of Mexp.
+21It is worth noting that a simple modification to pallows us to guarantee individual rationality,
+while preserving truthfulness-in-expectation. However, this will not be relevant for our purposes,
+as all our mechanisms will utilize no payments.
+B Network Flow Preliminaries
+Here we recall some notions from network flow theory, and defin e simple concepts that we will need
+in analysis of our algorithms for MKP and SIGAP.
+We recall that a weighted-capacitated directed graph is a directed graph G= (V,E), with a
+weightwe∈R+and capacity ce∈R+on an edge e. We define a flow simply as follows.
+Definition B.1. A flow is a vector f∈RE(G).
+We distinguish circulations on Gas follows.
+Definition B.2. A flowf∈RE(G)is acirculation if it conserves flow on each vertex. In particular,
+for each v∈Vwith incoming edges Γ−(v)and outgoing edges Γ+(v), we have:
+/summationdisplay
+e∈Γ−(v)f(e) =/summationdisplay
+e∈Γ+(v)f(e)
+We define the weightof circulation fas follows: w(f) =/summationtext
+e∈E(G)wef(e). Note that we allow a
+circulation to send negative flow on an edge, as well as overflo w the capacity of an edge. A feasible
+circulation is better behaved.
+Definition B.3. A vector f∈RE(G)is afeasible circulation if it is a circulation, and moreover it
+satisfies nonnegativity and capacity constraints. In partic ular, for each edge ewe have:
+0≤f(e)≤ce
+We distinguish the simplest circulations, or flow cycles , and recall that every circulation can be
+decomposed into cycles that are oriented consistently.
+Definition B.4 (Flow Cycle) .A circulation ConGis aflow cycle if there exists a simple undi-
+rected cycle Hsuch that Csends non-zero flow only on edges of H.
+Definition B.5 (Conformal Decomposition) .Fix a circulation f. We say a set of circulations/braceleftbig
+g1,...,gk/bracerightbig
+is aconformal decomposition offif the following hold
+1. (Decomposition) f=/summationtextk
+i=1gi
+2. (Conformal) For each e∈E(G),f(e)·gi(e)≥0.
+Note that every giin the conformal decomposition must send flow in the same dire ction as f
+on each edge.
+Theorem B.6 ([6]).Every circulation fcan be conformally decomposed into flow cycles.
+Conformal decomposition immediately yields a useful prope rty of feasible circulations.
+Lemma B.7. Letfandf′be feasible circulations on G. LetC1,...,C kbe a conformal decompo-
+sition into cycles of f′−f. Then, for every L⊆ {1,...,k}the circulation f+/summationtext
+i∈LCiis feasible.
+Equivalently, for every L⊆ {1,...,k}the circulation f′−/summationtext
+i∈LCiis feasible.
+In other words, the cycles of the conformal decomposition of f′−fmay be added to f(or
+subtracted from f′) in any order, maintaining feasibility along the way.
+22
\ No newline at end of file
diff --git a/1001.0437.txt b/1001.0437.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4b1b2b453ab1ebf47d68150ea71a20d0e37f1288
--- /dev/null
+++ b/1001.0437.txt
@@ -0,0 +1,329 @@
+1  
+ 
+ 
+Kepler  Science Operations  
+ 
+Michael R. Haas1,7, Natalie M. Batalha2, Steve T. Bryson1, Douglas A. Caldwell3, Jessie L. Dotson1, 
+Jennifer Hall4, Jon M. Jenkins3, Todd C. Klaus4, David G. Koch1, Jeffrey Kolodziejczak5, Chris 
+Middour4, Marcie Smith1, Charles K. Sobeck1, Jeremy Stober6, Richard S. Thompson4, and  Jeffrey E. 
+Van Cleve3 
+ 
+1NASA -Ames Research Center, MS 244 -30, Moffett Field, CA  94035  
+2San Jose State University /NASA -Ames Research Center, MS 244 -30, Moffett Field, CA  94035  
+3SETI Institute/NASA -Ames Research Center, MS 244 -30, Moffett Field, CA  94035  
+4Orbital Sciences Corporation/NASA -Ames Research Center, MS 244 -30, Moffett Field, CA  94035  
+5Space Science Office, VP62, NASA -Marshall Space Flight Center, Huntsville, AL  35805  
+6Ball Aerospace and Technology Company, Boulder, CO  80301  
+ 
+7E-mail: Michael.R.Haas@nasa.gov  
+ 
+Abstract.  Kepler’s  primary  mission is  a search for earth -size e xoplanets in the habitable zone  of late -type 
+stars using the transit method.  To effectively accomplish this mission, Kepler  orbits  the Sun  and stares 
+nearly continuously at one field -of-view  which was carefully selected to provide an appropriate density  of 
+target stars.   The data transmission rates, operational cycles , and target management requirements implied 
+by this mission design  have been  optimized and  integrated into a comprehensive plan for  science 
+operations.  The c ommissioning phase completed all critical  tasks a nd accomplished all objective s within a 
+week of the pre -launch plan .  Since starting  science , the nominal  data collection timeline has been 
+interrupted by two safemode events, several losses of fine point, and some small pointing  adjustments.  The 
+most important anomalies are understood and mitigated , so Kepler’s  technical performance metrics have 
+improved significantly over this period and the prognosis for mission success is excellent.  The Kepler  data 
+archive is  established and hosting data for the science team , guest observers, and public. The first data sets 
+to become publicly  available include the monthly full -frame images, dropped targets, and individual 
+source s as they are  publis hed.  Data are released through  the archive on a  quarterly basis ; the Kepler 
+Results Catalog  will be  released annually starting in 2011.  
+ 
+Subject headings: space vehicles — instruments: telescopes  
+Submitted  to: Astrophysical Journal Letters  
+  2 1.0 INTRODUCTION  
+ The Kepler  spacecraft  was launched into a 372.5 -day, earth-trailing , heliocentric orbit from Cape 
+Canaveral Air Force Station aboard a Delta II 7925 -10L on March 7 , 2009  (UTC) .  In this orbit, t he 
+photometer must be  rolled 90° about its axis every 93 days to keep the solar arrays illuminated and the 
+focal -plane radiator pointed away from the Sun.   The nominal  roll dates shown in Table 1 determine the 
+primary  cycle fo r science operations.  
+ Data collection is interrupted  once a month to earth -point  the h igh-gain antenna and download the 
+stored science and engineering data.   A download takes about 6 hours over Ka -band. The duration of the 
+break s is significantly longer  because  calibration data are collected before and after each roll, and the 
+photometer  must  thermal ly re-equilibrate before returning to fine point .  The on -board solid state recorder 
+(SSR) holds approx imately two months data, so data which are unsuccessfully downloaded one month are 
+saved  and downloaded  during the next month ly contact .  To date, the Ka -band performance has been 
+excellent with less than 0.1% of the data requiring re transmission.  
+ Kepler  can simultaneously observe up to 170,000 targets at its long -cadence (LC) sampling  interval 
+of 29.4 min  and up to 512 targets at its short -cadence (SC) sampling interval of 58.8 sec .  Every 48 LC s, 
+about once per day, a small subset of the targets are  extracted  from one LC  and stored separately as 
+reference pixels (RP) .  These RP and a  small amount of engineering data are downloaded biweekly via X -
+band  to monitor  spacecraft health and performance.  No data breaks are introduced because the  requisite  
+low-gain antenna is omni -directional.   
+ Every 3.0 days, one or more reaction wheels approach their maxi mum operating angular velocity as 
+they absorb angular momentum due to  solar radiation torques on the spacecraft.  In order to “desaturate”  
+the wheels, the spacecraft is transition ed from fine point , hydrazine thrusters  are fired  to dump the excess 
+momentum , and the spacecraft is commanded back to  fine point  a few minutes  later.  Each “desat ” causes  
+significant degradation of the photometric precision for one LC and several SC  intervals . 
+ 
+2.0 COMMISSIONING ACTIVITIES  
+ The commissioning activities began immediat ely after launch and lasted 67 days as summarized in 
+Table 2 .  Before dust cover eject (DCE), dark data was collected at several different temperatures for use 
+in photometric calib ration; both forward -clocked full -frame images (FFI s) and reverse -clocked long -
+cadences (RC LCs) were obtained.  After recovering  from  a safe-mode event on Day-of-Year (DOY) 83  
+and conducting a series of reviews, DCE occurred on DOY 98 . The first images showed that the pre -
+launch focus met the  encircled -energy requirem ents.  However, the LC target set is pixel -limited, so  small  
+piston and tip/tilt adjustments were used to sharpen focus and minimize  target loss.  A mean 95% 
+encircled -energy diameter of 4.3  pixels  was achieved .  After the focal -plane geometry (FPG) was 
+measured, target tables were defined for use in mapping the pixel response function (PRF; Bryson et al. 
+2010 ).  While  the FPG and PRF data were being processed  and the Quarter 1 (Q1) target table s were 
+being prepar ed, eight FFIs were collected  at convenient times  over several days  to estimate  stellar 
+variability.   The last major c ommissioning  activity was 9.7 days of cadence data collection (referred to as  
+“Quarter 0” or Q0)  to estimate the combined differential photometric performance (CDPP; Jenkins et al. 
+2010 a) and  identify  unclassifi ed or misclassified stars.   Most relatively isolated  stars in the Kepler  field-
+of-view ( FOV ) with magnitudes brighter than  13.6 were observed , independent of spectral type.  
+ 
+ 
+3.0 TARGET TABLES  3  A full-frame image (FFI ) is collected each month to confirm the  target apertures, monitor 
+photometer health and performance, and provide important  calibration data.  These FFIs include  all 96.5 
+million pixels from  the 42 CCDs  (x 2 amplifiers /CCD = 84 channels) .  Each science  pixel views  3.98 x 
+3.98 arcsecs  on the sky , so the active  FOV is 115.6 sq degrees.  In contrast, routine  data collection saves 
+and downloads  less than 6% of the pixels  to minimize on -board processing and storage and to limit the 
+duration of download breaks .   
+ To collect the desired pixels, a n aperture is  defined for each target  star using a n entry in a  target 
+table .  New LC, SC, and RP target tables are required every quarter  because  the 90° rolls move the stars 
+to new positions in the focal plane.  One LC target table is flown each quarter, but the SC target table is 
+changed month ly to provide greater flexibility in target selection for this limited, but valuable resou rce.  
+The focal plane was designed to be as four -fold symmetric as possible  in order  to maximize the number of 
+LC targets that can be observed continuously throughout  the mission.  A small central asymmetry and ±3 
+pixel errors in the CCD positions force  1% of the target  stars to vary by quarter . 
+ An extensive stellar classification program measured magnitudes, effective temperatures, and 
+surface gravities for millions of stars in the Kepler  FOV .  The resulting Keple r Input Catalog  
+(http://archive.stsci.edu/kepler/kepler_fov/search.php ) was used to identify those stars for which  transits 
+by earth -size planets in the  habitable zone  are most readil y detectable  (Batalha et al. 2010 a).  The target 
+set also includes stars  for asteroseismology ( Gilliland et al. 2010 ), astrometry ( Monet et al. 2010 ), cluster 
+studies , a control group, and the guest observer program.  Each quarter these various target lists are 
+individu ally prioritized, assigned pixel allocations, and combined into a master target  list.  For each 
+target, an optimal aperture is defined  that includes all pixels that improve the signal -to-noise ratio (SNR ) 
+for simple aperture photometry.  This calculation uses the measured pixel response function (PRF; Bryson 
+et al. 2010 ), on-orbit noise characteristics of the focal plane ( Caldwell et al. 2010 ), Kepler  magnitude  
+(Koch  et al. 2010) , and contamination by nearby stars.  The optimal apertures are enlarged to account for 
+differential velocity aberration, which varies  annually  by 1.5 pixels , and an additional halo of pixels is 
+added to accommodate pointi ng errors, changes in focus, etc.  Finally, one of the 1024  unique  aperture 
+mask s is assigned to efficiently capture the specified  pixels.   Since the individual -pixel data  are 
+downloaded, subsequent photometric processing  can include  those  pixels  which maximize the SNR . 
+ The LC and SC target table s are limited  to 5.44 million and 43,520 pixels , respectively, by the 
+Kepler  design .  Consequently, t he Q0 data collected during commissioning was limited to 52,496 LC 
+targets and 310 SC targets because double h alos were used to compensate for early uncertainties in focal -
+plane geometry and PRF (Bryson et al. 2010 ) and because this target set included many  bright stars , 
+which require more pixels on average .  During science operations, the number of  LC target s remains 
+pixel -limited, but has increased from 156, 097 in Q1 to 165,716 in Q3 due to improvements in defining 
+and assigning aperture masks.  T he number of SC targ ets has been  512 for every month except month 2 of 
+Q2, which was limited  to 479 targets because brighter stars were observed .     
+ The RP target t able is limited to 96000 pixels , or about 320 targets .  The  RP targets are distributed 
+over all 84 channels, but the ir density is larger  on the focal -plane periphery to i ncreas e the  precision of 
+the attitude  determination .  RP aperture definitions r equire more pixels per target than cadence targets  
+because  they have double halos and collect their own collateral data and background pixels . 
+ 
+ 
+ 
+4.0 DATA COLLECTION, FLOW, AND PROCESSING  4  New  target tables are generated and uploaded  a few weeks before the end of each quarter in 
+anticipation of the upcoming roll.  Each month‟s data collection begins with 3 reverse -clock (RC)  LCs for 
+calibration , includes about 1500  LCs and 45000 SCs, and ends with 1 FFI and 3 RC LCs .  These data and 
+the associated engineering data are downloaded from the flight segment using the Deep Space Network 
+(DSN) stations at Canberra, Madrid, and Goldstone .  The telemetry stream is packetized and encoded  on-
+board the spacecraft, transmitted to earth via Ka -band at 4.33 M bps, unencoded by the DSN , and 
+unpacked by the Mission Operations Center (MOC)  at the Laboratory for Atmospheric and Space Physics 
+(LASP) in Boulder, CO.  Ball Aerospace  & Technology Company (BATC)  is responsible for Kepler  
+Mission Operations, providing engineering support and directing the Kepler  operations team at LASP.  
+From the  MOC , the data flows to the Data Management Center (DMC)  at the Space Telescope Science 
+Institute ( STScI ) in Baltimore, MD, where the science data are uncompressed, correlated with the 
+appropriate  engineering data, packaged into  Flexible Image Transport System ( FITS ) files, and sent to the 
+Science Operations Center (SOC) at the NASA -Ames Research Center, Moffett Field, CA.   
+ The SOC is responsible for calibrating the raw -pixel data, rejecting cosmic rays, generating light 
+curves via aperture ph otometry, detrending against relevant  engineering data, searching for threshold -
+crossing events (TCEs), and validating these  TCEs to distinguish  possible planetary candidates from non -
+astrophysical false positives (Jenkins et al. 2010 b).  The processed data are evaluated and approved  by the 
+Data Analysis Working Group (DAWG), which is comprised of SOC, Science Office, an d Science Team 
+members, and  then sent to the DMC  for distribution through  the Multi -mission Archive at STScI 
+(MAST).  The processed data are also provided  to the TCE Review Team (Batalha et al. 2010 b), who 
+evaluate and prioritize the TCEs for ground -based follow -up observations (Gautier et al. 2010 ). 
+ RPs are downloaded from the flight segment biweekly  via X -band at rates decreasing from 8 Kbps 
+to less than  1 Kbps as Kepler  drifts away from the Earth .  These data flow directly from the MOC to the 
+SOC, where they are processed and  used to monitor pointing, brightness, plate scale, CDPP, and other 
+photometer health  metrics  (Caldwell et al. 2010 ).  RPs are also collected after each quarterly roll and each  
+monthly download to verify the science attitude (i.e., RA = 19h22m40s and Dec = 44°30‟ 00”) and perform 
+small pointing adjust ments  as required.  Additional performance metrics are computed on a monthly basis 
+using 200 un saturated, relatively isolated  stars per channel  once the LC science data are downloaded.  
+ 
+5.0 SCIENCE MISSION TIMELINE  
+ Science operations beg an on May 1 3, 2009  (UTC ).  Figure 1  compares the nominal science mission 
+timeline with the on -orbit performa nce.  Safemode events occurred  on DOY 166  and 183  with nearly 
+identical signatures .  Both  events  turned the photometer  off, which creat ed several -day gaps in the 
+cadence data and introduced  significant  thermal transient s for the first few days after the resumption of  
+science.  A root cause investigation identified two radiation -susceptible components in the 
+microprocessor power -on reset circuitry and confirmed them with  radiation t esting.  In order to minimize 
+the impact of similar events in the future, the safemode -recovery procedure  has been modified to keep  the 
+photometer powered -on following  such commanded reset s. 
+ The spacecraft experienced a loss of fine point (LOFP) on DOY 22 4, 256, 268, and 282  for periods 
+of 4 to 22 hrs, but fine-point was readily restored once  commanded .  These  anomalies appear to involve 
+star-tracker interaction s with the fine-guidance -sensor s.  Root cause is identified and fixed for the  first 
+LOFP.  Root cause for the others is under investigation , but mitigating actions have been identified and 
+deployed.    5  Figure 1  shows that th e science data collection prior to the first quarterly ro ll (DOY 169) was 
+shortened b ecause commissioning ran long.  To compensate, the monthly download on DOY 141  was 
+canceled  and a single SC target table was used for Q1.  Data collection was terminated after 33.5 days by 
+the safemode on DOY 166 . Note that Q0 and Q1 have the  same roll angle, b ut used substantially  different 
+target  tables . The first full observing quarter  is Q2, which started and ended on schedule.  Since d ata was 
+downloaded while  recovering  from the DOY -183 safemode , the mid -July monthly was cancel ed, but the 
+SC target table was chang ed per plan  on DOY 201 .  The August monthly was nominal  and Q3 has been 
+nominal to dat e, except for the two LOFP  indicated . 
+ The nominal timeline  has science  breaks  of 42 hours at the quarterly roll s and 26 hours  for the 
+intervening monthly downloads .  The quarterly break s require  extra time because  a small post-roll 
+pointing adjustment  is required to achieve the precise science attitude assumed when generating  the target 
+apertures.  The quarterly science break s are expected to decrease  in duration  after the first year when all 
+roll angles are calibrated.   The achieved data  completeness , including th e safemodes and desaturations, is 
+92%, which is consistent with  the pre -launch plan. 
+ 
+6.0 PERFORMANCE METRICS  
+ To determine pointing drift, the daily RP data are used to measure  the RA, Dec, and roll errors 
+relative to the nominal science attitude.  These quantities predict  the maximum deviation of any pixel 
+from its nominal location assuming rigid-body motion (i.e., no change in plate scale or other distortion of 
+the focal plane).   This metric, known as the maximum attitude residual (MAR), is plotted in Figure 2 .  Q1 
+and Q2 show significant pointing drift, which was controlled by performing small pointing adjustments to 
+protect the integrity of the target aperture definitions.  Aft er Q2, the fine -guidance -sensor pixel -of-interest 
+blocks were individually centered on their stars and a 3 -DN electronic bias was removed as unwanted 
+background signal.  These changes reduced the pointing drift (i.e., MAR) to a few mpix per month in Q3.  
+The root -sum-square drift on time scales of hours to days (i.e., the observed pointing detrended against a 
+5-day moving median of the pointing) decreased from 1.25 mpix in month 3 of Q2 to 0.22 mpix in month 
+1 of Q3.   Another significant change was the remo val of  an eclipsing binary  star with a 1.7-day period 
+and a variable  star with a 3.3-day period  from the ensemble of 40 fine -guidance stars.  Though not evident 
+in Figure 2, th e signature of both stars is apparent  in the raw light curves and centroid data for Q2 
+(Jenkins et al.  2010a ).  
+ Kepler  data is highly compressed to minimize on -board storage requirements and data download 
+times.  Both are important  in minimizing  the frequency and duration of the required science breaks.  Data 
+compression involves requantization  of the analog -to-digital converted values , baseline subtrac tion, and 
+Huffman encoding.  In order to requantize all  CCD channels and both cadences  with one requantizaton 
+table , a channel -specific mean black is subtr acted from each data value and a common  fixed offset is 
+added to eliminate negative numbers.   Requantization reduces the original 23 -bit pixel data  value s to 16 
+bits, while adding no  more than one -fourth  of the intrinsic measurement uncertainty.  The se requantized 
+values are differenced from a daily  baseline cadence before Huffman encoding .  Figure 3 show s the LC 
+data compression for Q1 and the first month of Q3.  The minima, which correspond to maximum data 
+compression, occur  immediately after the dail y baseline images.  The gradual increase in bits/pixel over 
+each day -long period results from cosmic ray strikes, pointing drift, stellar variability, eclipsing binaries , 
+transiting planets, and other phenomena  which further change  each subsequent image .  During Q3, the 
+compression is poor on DOY 26 2 because of  thermal transits associated with the just -completed monthly 
+download  and on DOY 26 6 because of a small, spurious increase in background during the baseline  6 image  (i.e., „Argabrightening,‟ Van Cleve &  Caldwell, 2009) .  The gap s beginn ing on DOY  268 and 282 
+are due to LOFP .  Excluding these events, the variations in compression decre ased from Q1 to Q3 and the 
+mean compression  improved  from 4.8  to 4.5 bits/pix  as a result of improved  pointing (cf Figure 2). 
+ 
+7.0 ARCHIVAL PRODUCTS  
+ The Kepler Input Catalog (KIC),  which contains roughly 4.5 million stars on Kepler  silicon  and 13 
+million stars overall , was  recently released to the public and is available for guest -observer target 
+selection.  The Kepler Results Catalog  (KRC)  will compile  the results of the planetary search effort and 
+the associated ancillary  investigations.  Stellar parameters such as parallax , age, radius , and metal licity 
+will be determined and reported for many stars, whether they have transiting planets or not .  The KRC  is 
+scheduled for release on an  annual basis starting in mid -2011, with a final delivery one -year after science 
+data collection  ends. 
+ The activities for each quarter  begin  with final target selection  about  two months prior to  the three -
+month observing  period  and end with the  data being  hosted at MAST  (http://archive.stsci.edu/kepler ) four 
+months after the last monthly download .  This nine -month cycle is repeated  for each observing quarter .  
+The data products hosted by MAST include  the raw and calibrated target -level pixel data and the 
+associated flux time series.  The  latter contain light curves and row/column centroid s as a function of 
+cadence .  Each target for each period has well-defined proprietary periods for access by the Science  
+Team, guest observers, and public.  The associated engineering data, calibration files, and focal -plane 
+characterization models used in SOC post -processing are hosted to support  archival research.  After all 
+proprietary rights  have expired  for a given quarter , the raw and calibrated pixel data will be  made  
+available by module, not just  by target .   
+ Some Kepler  data are  scheduled for rapid public release .  The monthly FFIs are calibrated and 
+delivered to MAST for public release four months after the observing period ends .   The data from 
+planetary -search targets  are released  60 days after the Kepler  Science Team  formally drops them .  These  
+dropped  targets include  stars that were originally classified as dwarfs, but found to be giants (Koch et al. 
+2010 ), stars that are too variable for transit detection , and “stars” that are identified as  KIC artifacts.  The 
+8441 targets dropped before Q2 are available at MAST.  Kepler  anticipates dropping thousand s of 
+additional targets  as the telemetry rates decrease .   
+ Planetary -search data are scheduled for public release as follows: Q0 and Q1 in June, 2010; Q2 in 
+June, 2011; Q3/Q4 in June, 2012; Q5/Q6 in June, 2013 ; and the remainder at end of mission in 
+November, 2013.  Data collected by the science team for ancillary  studies are subject to a one -year 
+proprietary period which starts upon MAST ingest.  The proprietary period for each guest -observer cycle 
+is 12 months  from MAST ingest or 6 months after the last data are ingested, whichever comes later.   The 
+first guest -observer  cycle commenced observations  in June, 2009 .  Targets that are exclusively for 
+engineering purposes have no proprietary period; most  of these targets are collections of pixels required 
+for calibration or instrument -health, but a few astrophysical  sources  may be included  as the mission 
+proceeds .  The data on individual sources are publically released once publis hed in a scientific journa l. 
+ Periodic  reprocessing will occur as the data is better understood  and the SOC pipeli ne matures.  A 
+new pipeline release is scheduled about once a year and will trig ger a complete reprocessing of all science 
+data.  Each official data release is accompanied by detailed release notes and  identified by populating the 
+keywords DATA_REL and QUARTER in e ach FITS file .  Additional information can be found in the 
+Kepler Instrument Handbook  (Van Cleve & Caldwell, 2009) , Kepler Archive Manual , and Kepler Data 
+Analysis Handbook  at http://archive.stsci.edu/kepler .  7 8.0 CONCLUSIONS  
+ Kepler  has successfully completed its commissioning phase, entered science operations, and 
+collected 8 months of science  data.  Some data are already available through the public  archive.  Given 
+Kepler’s  excellent on -orbit performance  and 10 -yr supply of expendables , an extension beyond the 
+nominal 3.5-yr mission is worthy of serious consideration.  Funding for this Discove ry mission is 
+provided by NASA's Science Mission Directorate.  
+ 
+Facilities: Kepler.  
+ 
+9.0 REFERENCES  
+Batalha, N  B, et al.  2010a, ApJ, submitted .     
+Batalha , N B, et al. 2010b , ApJ, submitted .     
+Bryson, S  T, et al.  2010, ApJ, submitted . 
+Caldwell, D  A, et al. 2010, ApJ, submitted . 
+Gautier, N  T, et al.  2010, ApJ, submitted . 
+Gilliland , R L , et al. 2010 , ApJ, submitted .       
+Jenkins, J M , et al.  2010a, ApJ, submitted .    
+Jenkins, J M , et al.  2010b, ApJ, submitted .     
+Koch, D G , et al.  2010, Science , submitted . 
+Monet, D, et al.  2010, ApJ, submitted . 
+Van Cleve, J E  and Caldwell, D A 2009, Kepler Instrument Handbook , KSCI -19033.  
+ 8 FIGURES  
+ 
+Figure 1. The planned science mi ssion timeline is compared with the actual timeline from  launch through 
+the end of 2009.  Dates and Day -of-Year (DOY) are UTC.  
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 2. The maximum attitude residual, a sensitive measure of Kepler’s  pointing drift, is plotted for the 
+period spanning Quarters 1, 2, and part of 3.  Safe mode events occurred on day -of-year 166 and 183, and 
+pointing adjustments occurred on DOY 201, 232, and 247.  The last pointing adjustment 
+overcompensated, so MAR decre ased for the last 12 days of Q2.  
+ 
+9  
+Figure 3. The mean data compression efficiency for all 84 channels as a function of Day -of-Year for 
+Quarter 1 (upper panel) and month 1 of Quarter 3 (lower panel).  
+  
+10 Table 1.  Quarterly Roll Dates  
+Year  Spring  Summer  Autumn  Winter  
+2009  --- 6/18 9/17 12/17  
+2010  3/19 6/23 9/23 12/22  
+2011  3/24 6/27 9/29 12/29  
+2012  3/29 6/28 10/1 --- 
+ 
+ 
+ 
+ 
+ 
+Table 2.  Commissioning Timeline  
+DOY (UTC)  Activity  
+66 Launch, orbit injection, separation, detumble  
+67 Transition to standby; initialize solid state recorder  
+67, 68, 70  Check -out x -band over multiple stations  
+68 Calibrate coarse sun sensor  
+68 - 70 Initialize photometer  
+70 - 71 Calibrate high -gain antenna  
+70 - 80 Collect dark FFIs and RC LCs  
+72, 93, 109  Calibrate Ka -band over multiple stations  
+75, 79  Measure cosmic ray distribution  
+83 - 94 Enter safe mode and recover  
+95 Enable sun avoidance  
+98 Eject dust cover  
+98 Collect first light image  
+98 - 100 Calibrate fine -guidance sensors  
+99 - 101 Collect calibration data during cool -down  
+103 - 112 Optimize encircled energy (focus)  
+112 Determine focal plane geometry  
+112 - 114 Measure scattered light and ghosting  
+114 - 116 Measure stellar variability (golden FFIs)  
+116 - 119 Map pixel response function  
+120 - 121 Measure  gain and linearity  
+121 - 131 Measure combined differential photometric 
+performance (CDPP; Q0)  
+132 Adjust science attitude  
+133 Start science data collection (Q1)  
+ 
\ No newline at end of file
diff --git a/1001.0438.txt b/1001.0438.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e520d4d6e86eace0c2ec5be8cb23ca34ca75ec18
--- /dev/null
+++ b/1001.0438.txt
@@ -0,0 +1,18 @@
+Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 1 of 12  Self-Assembly in the Growth of Precious Opal   A. M. Stewarta, Lewis T. Chaddertonb* and Brian R. Seniorc    aEmeritus Faculty, The Australian National University, Canberra, ACT 0200, Australia  bGonville and Caius College, Trinity Street, Cambridge CB2 1TA, United Kingdom  cSenior & Associates, 303 Shingle Hill Way, Bungendore, NSW 2621, Australia  *On leave from The Atomic and Molecular Physics Laboratories, School of Physics and Engineering,  The Australian National University, Canberra, ACT 0200, Australia.     Abstract   It is proposed that primary nucleation of amorphous microspherulites of hydrated silica in natural proto-precious-opal can be followed by a long range superlattice ordering process by means of electrostatic self-assembly.  Necessary conditions in the thermodynamics are a high surface charge density on microspherulite surfaces, a long Debye length and an appropriate number density of nucleation centres.  A further chemical requirement is a high alkaline environmental pH ~ 9–10.  It is also proposed that the characteristic concentric spherical shell-like structure of spherulites, centred on primary nuclei, are due to sequential deposition of intrinsic salts which precipitate out when the corresponding solubility limits in the liquid are successively exceeded.  It can be that the better-known sedimentation of microspherulites under gravity only plays part in the final stabilization period of overall growth.     1.   Introduction   It is well known that microspherulites of amorphous hydrated silica (SiO2!27H2O) in natural opal form a characteristic superlattice [1] with lattice translation parameters in the range ~150 - 450 nm (Fig. 1), and that it is simple Bragg diffraction of light from such nanostructures that gives precious opal its singular display of colour or ‘fire’ [2-3].  A simple model for the slow and prolonged growth of the individual spherulites in the natural geological environment, hereafter referred to as I, has recently been proposed [4].  Structural characteristics of a single spherulite strongly suggest that there is nucleation on a central core which, it is conjectured in I, carries the heavy common salts of uranium, thorium and lead.  These specific compounds are identified by strong signals in SIMS (secondary ion mass-spectroscopy) analyses in precious opal. A final proof of this hypothesis, however, probably lies in part in successful fission track mapping experiments of natural stones.  Yet this is neither a trivial nor simple experiment when concentrations of active compounds are only a few parts per million, which suggests that the more exceptional uraniferous opal might be a more quickly rewarding target.  For common opal ‘potch’, on the other hand, in which central nuclei are most frequently absent (see I), the corresponding SIMS signals are very low.  It is also the case that when clear central nuclei are present there are, in addition, up to three concentric sharp spheres that are centred upon individual nuclei.  These, and the overall superlattice structures, are clearly and consistently seen in TEM micrographs of shadowed extraction replicas, and in SEM images from polished surfaces.     There is an anomalously high gamma radioactivity measured in drill holes in the vicinity of opal deposits in the field.  It is very likely that this has its overall origin in natural spontaneous fission of the cited isotopes of thorium and uranium, and from their corresponding daughter product decay chains.  And certainly there is a clear observed Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 2 of 12  association of natural radioactivity with the abundance of precious opal which led directly to the extraordinary successful new exploration method for this important mineral described in I.    In a more macroscopic and geological picture it is found that amorphous silica cement is very abundant in deeply weathered profiles in which precious opals are found.  This evidently appears to have been derived from the breakdown of former litho-feldspathic sedimentary rocks and smectitic clays, into kaolinite. And this process took place in the late Oligocene to early Miocene periods during a deep weathering event which defined the Canaway profile in Australia and was accompanied by strong silicification episodes forming porcellanite and silcrete [5].  Opals formed in the uppermost 35 m of this profile over times of up to ~ 20 My, and at various depths that accord with porosity and permeability barriers located at sandstone/claystone interfaces.  Fractures within the profile acted as conduits for movement of silica-laden water and stress-related dilation zones and cavities formed by leaching of former soluble minerals or organic fragments, formed spaces in which siliceous water accumulated [6-8].     The detailed specific physics and chemistry of siliceous groundwater, the nucleation of individual spherulites with internal structure, and the corresponding long term sol-to-gel transformations, which lead to formation of both stable precious opal and potch, and which appear to take place simultaneously in a changing physical-chemical environment, are still poorly understood.  It is generally assumed that it is sedimentation of growing spherulites under gravity which is responsible for ordering of the long-range superlattice.  In this paper we critically consider nucleation and growth anew, present an explanation for the origin of the concentric shell structure in spherulites and, through thermodynamics, successfully describe a long-range interaction between them, which can be the chemico-physical and thermodynamic origin of the intrinsic precious opal superlattice (Fig. 1).  2. Mechanism   There exists a detailed literature on the growth of both precious and artificial opal [9-19] in which processes of self-assembly similar to those which operate for C60 (fullerene) and corresponding nanotubes, or other quasi one- dimensional structures, are qualitatively considered.  In general these approaches are based upon earlier studies on the fundamentals of colloid crystallization [20-27] and we have found no satisfactory physical outline of the critical processes governing nucleation and what immediately follows.  A useful overview of the demands asked of the basic physics can be found in Norris et al. [9] who hypothesize that the initial ordering process is in some way due to solvent flow and that thereafter, as in many publications, it is sedimentation under gravity which is the dominating feature of growth.    Our own quite different model for the growth of natural [1, 2] and artificial [28] opal is based on electrostatic self-assembly and is, at this stage, essentially semi-quantitative at best.  The system we consider is that of a silica-water metallic salt fluid comprising, for present purposes, sodium chloride (monovalent ions) or barium sulphate (divalent ions), and the ionic silicate species. When either the temperature is lowered or water is lost by evaporation or diffusion through the surrounding rock, such a solution tends to solidify into a solid hydrated silicate [29].  We adopt heterogeneous nucleation as the base mechanism from chemical reaction rate theory [30-32], and assume that the nucleation centres are situated randomly in the solution.  The many body aggregation process is then closely akin to that described for self-assembly of the C60 fullerene molecule [33-34] - either homogeneously or heterogeneously. As soon as thermodynamic conditions become favourable, growth of spherical solid objects (microspherulites of hydrated silica) starts simultaneously at these pre-Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 3 of 12  existing randomly positioned nucleation centres and proceeds according to the standard mechanisms of crystal growth in supersaturated solutions.  Our model invokes electrostatic interaction between the microspherulites.    A colloidal system of this type–with an electric double layer–has been earlier described in the theory of Derjaguin, Landau, Verwey and Overbeek (DLVO) [35].  Because of the discontinuity of the chemical and electrical properties at the solid-liquid boundary, the surfaces of the microspherulites acquires an electric charge density !, due to the charge dissociation of silanol groups [29]. This charge is compensated by an oppositely charged Debye cloud in the electrolyte close to the surface which falls away exponentially with distance and establishes the double layer. Each microspherulite has a surface charge:         Q=4!R2"         (1)   where R is the radius of the microspherulite.  Accordingly, there is a repulsive screened Coulomb force between any two microspherulites (Fig. 2) separated by a distance D (D > 2R) giving rise to electrostatic interaction energy U. We take this to be of the qualitative form  [29, 37]:         U = Q24!"0"re-(D#2R)/LDD     (2)   giving:     U = 4!R4"2#0#re-(D$2R)/LDD    (3)   where LD is the Debye length.  For water of laboratory purity the Debye length is not greater than around 150 nm [35, 41]. The greater impurity level of natural waters is expected to lead to Debye lengths significantly less, so that the system will be in the regime LD < D, as shown in Fig. 2.  If U is much less than kT we have a system of solid microspherulites randomly positioned in the liquid, the influence of the interaction being swamped by thermal fluctuations.  Conversely, however, if U is much greater than kT then it will be energetically favourable for the system to order into a regular array of small solid microspherulites situated within the liquid.  This is of course, by definition, a many body problem.  For no Debye screening (LD " infinity) calculations show that the structure with minimum energy is fcc [36], with a neighbouring hcp structure with only very slightly greater energy.  An fcc structure with stacking faults is therefore to be expected (Fig. 1).  As the radius of the microspherulites grows, the prefactor of U/kT (4!"2R4/D#0#rkT) rises steeply as R4, and the interaction energy increases rapidly as the Debye shells overlap.    In addition to calculating the effects of the fundamental many-body physics there is the requirement to calculate accurately the energy of interaction between the microspherulites for a given separation distance.  This can only be achieved through complete solution of the full non-linear Poisson-Boltzmann equation including charge regulation and the effects of the finite radius of curvature of the microspherulites [35, 37].  Such a thorough and complex analysis is beyond the scope of this paper, which means that the results we do present cannot be taken to be more than an order-of-magnitude estimate.  However, because the system is macroscopic, a sharp phase change is likely to take place from a positionally disordered to an ordered phase as the tendency to crystallisation increases by cooling and/or loss of water.   Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 4 of 12  Growth of the solid microspherulites continues after their positions transform into a regular array, and all growth is arrested through exhaustion of the solvent when they make   neighbour contact.  At this point they become bound together by attractive van der Waals forces [38-39] into a true molecular solid.  It is clear that it is easier to form a periodic array in the liquid than in the solid.  Yet it is to be remarked upon that sedimentation under gravity, although possibly helpful in forming a consolidated structure, is neither a necessary nor a major feature of this model. We note that, because during the later stages of growth the radius of the spherulites will be proportional to the square root of the time from genesis (see Appendix A), there will be a tendency to narrow the relative size distribution of the spherulites as the later starters will be able to catch up with those nucleated earlier.   It is normal to suppose that because of the attractive van der Waals force the true ground state of any colloidal system is always the coagulated one, the colloidal state being at best metastable [35].  For present purposes we set this general notion aside and accept that a regular structure may be favoured thermodynamically, and that kinematic factors will be of importance in determining whether the microspherulites transform into a regular structure, giving precious opal, or quench into a random array, giving potch.  In contrast to simpler glassy metal systems [40] where the equilibration time (the time needed for the system to transform from a positionally disordered into an ordered state) is of the order of milliseconds, equilibration times for opaline systems could be days and, more likely, centuries or millennia, depending on sustained environmental conditions.  If the Debye length is too short the microspherulites will have grown too large to slip past each other easily into the form of a regular array by the time that the Debye shells overlap to give a high electrostatic energy.  If the time needed for self-assembly, limited by kinetic factors, is longer than the time available the system will not form the ordered state.  Kinematic and, more specifically, molecular dynamical calculations are needed to investigate this matter in detail.    The superlattice constant of the ordered array depends on the number density of nucleation sites initially present.  It is also possible that regular opal structures with lattice constants lying outside the specific wave-length range needed to give electromagnetic activity in the visible spectrum (and photonics) do exist, but have not yet been identified in the field because they are not seen to be optically active.  Some material identified as potch may be in this category.  It should also be kept in mind that we in fact approach the point where control over the conditions required for the laboratory growth of tailored artificial opal will be realised.    Briefly, then, the conditions favourable for the formation of optically active precious opal are (a) a high surface charge density, (b) a long Debye length and (c) an appropriate number density of nucleation centres. A schematic diagram of the proposed process of growth is shown in Fig. 4.    3.  Calculation of the energy of interaction between microspherulites   The surface charge density of silica in aqueous solution is anomalously high; much larger than that on the perfect crystalline surface of mica [41], for example, possibly due to the fractal nature of the silica surface which allows charges to be trapped underneath the surface as well as on the outside [29].  Measurements by Bolt [42-43] of the surface charge density of silica in solutions of NaCl of concentration c, dissolved in water whose pH is made alkaline by the addition of NaOH, are reproduced in Fig. 3.  This data suggests a characteristic surface charge density ! ~ - 0.1 C/m2 for alkaline solutions. The distance D between the centres of microspherulites is most normally around 250 nm for precious opal [2] Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 5 of 12  and in our model is fixed for a given system by the number of nucleation centres per unit volume.    The electrostatic energies can be high compared to kT.  For an infinite Debye length U/kT becomes greater than unity by R around 2 nm.  A finite Debye length reduces this ratio greatly according to equation (3). The Debye length LD in equations (2) and (3) depends directly on the electrolyte concentration as given in equation (4).  It will always be shorter than 150 nm, the value for de-ionized water in equilibrium with the atmosphere [41]. Some idea of the electrolyte concentration likely to be encountered is obtained from the impurities reported in [3].  It is not, of course, established that all the metallic elements will be in solution, but, for example, if we suppose divalent magnesium to be present at a weight fraction of 7x10-3 [3] we find a corresponding concentration of 0.3x10-3 Moles /per litre.  And a standard calculation [44] then yields a Debye length ~10 nm.    The dependence of U/kT as a function of R for several Debye lengths is shown in  Fig. 5.  We assume that, for ordering to take place before contact between the microspherulites at R = D/2, U/kT has to be greater than unity. With T = 300 K, D = 250 nm, a surface charge density of - 0.1 C/m2 and a Debye length of 10 nm it is seen that U/kT >> 1 for R = 60 nm.  Under these conditions it may then just be possible for the microspherulites (radius 60 nm, average separation 250 nm) to squeeze past each other to form an ordered array.  Longer Debye lengths will make this process easier.  Shorter Debye lengths make it harder.  Only kinematic calculations can determine this more exactly.  Clearly, however, it is to be expected that the system will take a shorter time to self-assemble when R is small compared to D.    4.  Calculation of the Debye length   For the system examined by Bolt [42-43], namely silica in an alkaline NaCl solution, there is an intrinsic competition between the two features needed for the growth of precious opal.  Fig. 3 shows that in order to achieve a high surface charge density it is necessary to have a high salt concentration, or a high alkaline pH.  Both these factors increase the charge density of the electrolyte and this leads to a reduced Debye length, which makes the formation of a regular lattice unfavourable on kinematic grounds. Can these competing factors be reconciled?    We calculate the Debye lengths of the NaCl system considered by Bolt [42-43] in order to determine whether any regime exists for which the formation of precious opal can be considered plausible.  The Debye length LD in an electrolyte [44] is given by:         LD=!0!rRT2"L2e2I=0.305Inm   (4)  Where " is the density of the solvent, R the Gas constant, L is Avogadro's number and I is the ionic strength:           I=12i![i]zi2      (5) [i] Being the molar concentration (moles/litre, M/L) and zi the charge on species i.    Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 6 of 12   Since the system consists of NaCl of concentration c dissolved in water whose pH is made alkaline by the addition of NaOH it is clear that the modes of chemical dissociation and their corresponding equations of dissociation are:       H2O!H++OH-[H+][OH-]=K0    (6)  Where K0 = 10- 14 (M/L) 2 is the dissociation constant of water at 300 K and the pH is given by [H+]=10!pH    and    NaOH!Na++OH-[Na+][OH-]=K2[NaOH]  (7)   and   NaCl!Na++Cl-[Cl-]=c,      (8)  K2 being the dissociation constant of NaOH, the assumption being made that NaCl is fully dissociated. The equation of electrical neutrality is:        [H+]+[Na+]=[OH-]+[Cl-].       (9)   Since all the ions are monovalent I is given by:        I=[H+]+[Na+]      (10)   From (9) we immediately find that        I=K0/[H+]+c=10pH!14+c.    (11)   Whence, combining (4) and (11), we can calculate the Debye lengths for the same salt concentrations used by Bolt [42-43] as a function of pH (Fig. 6).    It is immediately clear that the only concentrations of NaCl in which the Debye length is longer than 10 nm are the weakest, with c = 10-4 and c = 10-3.  It then follows by examination of Fig. 3 that for the largest surface charge density with these salt concentrations the pH must be high – in fact in the interval between 9 and 10.  And, accordingly, optically active opal will only be able to form in this system within this narrow range of parameters.  It is also the case, of course, that simultaneously the volume density of nucleation centres must be appropriate.  This restricted combination of growth conditions itself suggests why natural precious opal is so rare despite the relative high abundance of hydrated silica.    5.  Shell structure of the microspherulites   A notable feature of the structure of the individual microspherulites that has been identified by TEM ([3], reproduced in [4]) is the existence of prominent nuclei surrounded by concentric shells. The nuclei have been suggested to contain heavy radioactive elements [4]. We now further suggest that the shells are due to precipitated salt. We assume that the salt solubility limit in the solid phase is much smaller than that in the liquid phase and that the salt concentration in the liquid phase is below its solubility limit. As the growth of the solid microspherulites progresses, the salt that is in the volume of the solid phase is expelled into the liquid. Accordingly, the salt concentration ahead of the solidification front grows continuously larger. At some stage it reaches the solubility limit of the liquid and the salt Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 7 of 12  rapidly precipitates onto the surface of the microspherulite. This produces a shell of the most abundant solid salt in the liquid at that point.  The process is essentially a means of separating the salts out radially – it can repeat itself larger radii – thereby providing for a multiple spheroidal shell structure. A proper microchemical analysis of the shell structure would resolve this.    6.  Conclusions   We have outlined a basic model of the growth conditions of ordered precious opal based upon electrostatic self-assembly.  Conditions favourable for growth are those which produce a high surface charge and a long Debye length.  In addition, the number density of nucleation sites must be appropriate.  This restricted combination of growth conditions does itself, by implication, suggest why precious opal is so rare in nature.  For silica growing in a sodium chloride solution a high alkaline pH and a low salt concentration are necessary conditions.  In post-nucleation growth of individual microspherulites it is proposed that sequential deposition of abundant intrinsic salts is responsible for a concentric spherical shell structure about central nuclei.  Sedimentation of spherulites under gravity is a quite separate physical process which may take place later.   Appendix A   We consider a solid spherulite of radius R which is growing in an infinitely large supersaturated solution as tiny moieties diffuse from the outer reaches of the solution towards the spherulite, finally condensing on its surface. The driving force for diffusion is the gradient of the chemical potential µ(r) of the moieties in the liquid at a distance r (r ! R) from the centre of the spherulite. The number flux density J(r) of flow of the moieties is given by Fick's law:          J(r)=!D"µ(r)                (A1)   where D is a diffusion constant. The steady state is characterised by !2µ(r)=0 whose solution is [45]:        µ(r)=µ(!)"#µRr                (A2)   where !µ=µ(")#µ(R). The quantity µ(#) is the chemical potential of the moieties in the unperturbed solution and µ(R) is the chemical potential at the surface of a spherulite which, assuming quasi-equilibrium and the absence of surface effects, is equal to the chemical potential in the solid and so is independent of R.    The gradient of the chemical potential at r = R is !µ/R and the diffusive flux density is  - D"µ/R. The rate of increase of the volume of the spherulite is given by multiplying the magnitude of this by 4#r2v, where v is the volume of a moiety and so:         !R!t=vD"µR                  (A3)   or     t=1vD!µR'dR'R0R"                (A4)  Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 8 of 12   where R0 is the initial radius of the nucleation centre. When integrated, this gives R=R02+2vD!µt, showing that at long times the radius of the spherulite increases with the square root of the time. As a result, if a second spherulite starts to grow at a time $ after an earlier first, then the relative radii r, areas a and volumes V at large time will proceed as "r/r = $/2t, "a/a = $/t, "V/V = 3$/2t respectively, and the spherulites become effectively equal in size after a time t >> $. It is of course accepted that, during the process of growth, !µ may change with alterations in the environmental conditions, for example the periodic daily or annual variations of temperature and rainfall.    Acknowledgements   Support from the CSIRO (Australia) is acknowledged by one of us (LTC), as is the generous hospitality of the Master and Fellows, Gonville and Caius College, Cambridge.   References   [1] M. Weibel in Opal The Phenomenal Gemstone, Litographie Press,  East Hampton, Connecticut, 2007. [2]  J. V. Sanders, Nature 204 (1964) 1151. [3]  D. M. Stone, R. A. Butt, A Guide to Australian Precious Opal, Periwinkle Books, Lansdowne Press, Melbourne, 1972.    [4]  B. R. Senior, L. T. Chadderton, The Australian Gemmologist 23 (2007) 160.  [5]  B. R. Senior, AGSO Journal of Australian Geology and Geophysics 17 (1998) 225. [6] B. R. Senior, D. H. McColl, B. E. Long, R. J. Whiteley, Journal of Australian Geology and Geophysics. 2 (1977) 241.  [7] J. J. Watkins, N.S.W. Geo. Survey, (1985) Report GS 1085/118.  [8] R. W. Wise, Q. J. Gemm. Inst. Am. 29 (1993) 4.    [9] D. J. Norris, E. G. Arlinghaus, L. Meng, R. Heiny, L. E. Scriven, Adv. Mater. 16 (2004) 1393.   [10] V. N. Astratov, V. N. Bogomolov, A. A. Kaplyanskii, A. V. Prokofiev, L.A.  Samoilovich, Yu. A. Vlasov, Nuovo Cimento Soc. Ital. Fis. D 17 (1995) 1349. ![11] A. Imhof, D. J. Pine, Nature 389 (1997) 948.  [12] O. D. Velev, T. A. Jede, R. F. Lobo, A. M. Lenhoff, Nature 389 (1997) 447.  [13] B. T. Holland, C. F. Blandford, A. Stein, Science 281 (1998) 538.  [14] J. E. G. J. Wijnhoven, W. L. Voss, Science 281 (1998) 802.  [15] A. A. Zakhidov, R. H. Baughman, Z. Iqbal, C. Cui, I. Khayrullin, S. O. Dantas, J. Marti, V. G. Ralchenko, Science 282 (1998) 897.  [16] Yu. A. Vlasov, N. Yao, D. J. Norris, Adv. Mater. 11 (1999) 165.  [17] G. Subramanian, K. Constant, R. Biswas, M. M. Sigalas, K. -M. Ho, Appl. Phys. Lett. 74 (1999) 3933.  [18] G. Subramanian, V. N. Manoharan, J. D. Thorne, D. J. Pine, Adv. Mater. 11 (1999) 1261.  [19] P. Jiang, J. F. Bertone, K. S. Hwang, V. L. Colvin, Chem. Mater. 11 (1999) 2132.   [20] P. Pieranski, Contemp. Phys. 24 (1983) 25.   [21] R. J. Carlson, S. A. Asher, Appl. Spectrosc. 38 (1984) 297.  [22] P. N. Pusey, W. van Megan, Nature 320 (1986) 340.   [23] P. N. Pusey, W. van Megan, P. Bartlett, B. J. Ackerson, J. G. Rarity, S. M.  Underwood, Phys. Rev. Lett. 63 (1989) 2753.  [24] K. E. Davis, W. B. Russel, W. J. Glantschnig, Science 245 (1989) 507.  Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 9 of 12  [25] A. D. Dinsmore, J. C. Crocker, A. G. Yodh, Curr. Opin. Colloid Interface Sci. 3 (1998) 5.   [26] A. P. Gast, W. B. Russel, Phys. Today, Dec. (1998) 24.   [27] D. G. Grier, MRS Bull. 23 (1998) 21.   [28]  P. Gilson, Journal of Gemmology 16 (1979) 494.   [29]  R. K. Iler, The Chemistry of Silica, Wiley, New York, 1979.  [30]  M. V. Smoluchowski, S. Physik Chem. 92 (1917) 192. [31]  T. R. Waite, Phys. Rev. 107 (1957) 463. [32]  K. H. Henisch, Crystal Growth in Gels, Penn. State University Press, 1970.  [33]  E. G. Gamaly, L. T. Chadderton. Proc. Roy. Soc. London A449 (1995) 381. [34]  L. T. Chadderton, E. G. Gamaly. Nucl. Instrum. Meth. B117 (1996) 375-386.  [35]  J. N. Israelachvili, Intermolecular and Surface Forces, Academic Press, 1985.   [36] T. Aste, D. L. Weaire, The Pursuit of Perfect Packing, Taylor and Francis, New York, 2008.   [37]  S. H. Behrens, D. G. Grier, J. Chem. Phys. 115, (2001) 6716.   [38]  A. M. Stewart, V. V. Yaminsky, S. Ohnishi, Langmuir 18 (2002) 1453.   [39]  V. V. Yaminsky, A. M. Stewart, Langmuir 19 (2003) 4037.   [40]  A. M. Stewart, Metals Forum 6 (1983) 122.   [41]  A. M. Stewart, Measurement Science and Technology 11 (2000) 298.  [42]  G. H. Bolt, Journal of Physical Chemistry 61 (1957) 1166. [43] G. H. Bolt, Journal of Physical Chemistry 71 (1957) 550. [44]  P. W. Atkins, Physical Chemistry, Oxford, 1983.   [45] N. H. Fletcher, The Physics of Rainclouds, Cambridge University Press, 1966.      
+   Fig. 1.  Scanning electron micrograph of a sliver of precious opal from a mine around Dubnik, Slovakia, taken by Professor Max Weibel and co-workers at the Swiss Institute of Technology in Zurich [1].  The spherulites are ~ 0.5 µm in diameter and are situated on a periodic lattice.    Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 10 of 12                  Fig. 2.   Schematic diagram of two solid microspherulites of hydrated silica (shaded) surrounded by their Debye shells (dashed circles).    
+  Fig. 3.  Data of Bolt [42-43] for the surface charge density on colloidal silica surfaces with a specific surface area of 180 m2g-2 in an aqueous solution of NaCl as a function of pH for various concentrations of NaCl.  The pH is varied by adding NaOH. One electron per nm2 is equivalent to a charge density of 0.16 C/m2.  (Courtesy of the Editors, Journal of Physical Chemistry)   D R L D D - 2R Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 11 of 12   Fig. 4.  Schematic diagram of the proposed growth mechanism for precious opal. (a) Growth of the opal spherulites starts at nucleation centres positioned randomly in the solution. Four of them are shown here. When the spherulites reach a critical size (b) electrostatic forces overwhelm entropic effects causing the spherulites to line up in an ordered array (c). Growth continues until the solute is exhausted (d). The distance between the centers of the spherulites in optically active opal corresponds to the wavelength of visible light of a few hundred nm.    
+Fig. 5.   Ratio of electrostatic potential energy U of interaction to kT as a function of R between two charged spheres of radii R whose centres are separated by a distance D, for given surface charge density !  for several Debye lengths, following equation (3).  Values of the parameters used in the calculation are D = 250 nm, ! = - 0.1 C/m2, %0 = 8.85x10-12 C2/Nm2,  %r = 80.  The Debye lengths used are 5 nm, 10 nm, 15 nm, 25 nm.  The vertical dashed line shows the value of R (125 nm) at which the two spheres make contact.   
+Stewart, Chadderton, Senior  Growth of opal  
+ArXiv 4/1/10 Page 12 of 12   
+   Fig. 6.  Calculated Debye lengths at a temperature of 300 K against pH, following equation (11) for the different concentrations c of NaCl in water used by Bolt [42-43]. The top curve is for pure water, the next one down for c = .0001 etc. in order as {c = 0, 0.0001, 0.001, 0.004, 0.01, 0.04, 0.1, 0.4, 1, 4}.     
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+arXiv:1001.0439v2  [math.PR]  12 Oct 2012The Annals of Probability
+2012, Vol. 40, No. 5, 2264–2297
+DOI:10.1214/11-AOP679
+c/circlecopyrtInstitute of Mathematical Statistics , 2012
+EXISTENCE, UNIQUENESS AND COMPARISONS FOR BSDES
+IN GENERAL SPACES
+By Samuel N. Cohen1and Robert J. Elliott
+University of Adelaide, University of Adelaide and Univers ity of Calgary
+We present a theory of backward stochastic differential equa -
+tions in continuous time with an arbitrary filtered probabil ity space.
+No assumptions are made regarding the left continuity of the fil-
+tration, of the predictable quadratic variations of martin gales or of
+the measure integrating the driver. We present conditions f or exis-
+tence and uniqueness of square-integrable solutions, usin g Lipschitz
+continuity of the driver. These conditions unite the requir ements for
+existence in continuous and discrete time and allow discret e processes
+to be embedded with continuous ones. We also present conditi ons for
+a comparison theorem and hence construct time consistent no nlinear
+expectations in these general spaces.
+1. Introduction. The theory of backward stochastic differential equa-
+tions (BSDEs) has been extensively studied. Typically, res ults have been
+obtained only in the context of a filtration generated by a Bro wnian motion,
+possibly with the addition of Poisson jumps. Specifically, a ttention has been
+given to equations of the form
+dYt=F(ω,t,Yt−,Zt)dt−Z∗
+tdMt, Y T=Q,
+whereMis the martingale generating the filtration (typically Brow nian
+motion), Tisafixedfiniteterminaltime, Q∈L2(FT)isastochastic terminal
+value,Fis a progressively measurable function, [ ·]∗denotes matrix/vector
+transposition(andhence A∗Bdenotestheinnerproductof AandB)andthe
+solution is a square integrable pair of processes ( Y,Z), where Yis adapted
+andZis predictable.
+Received January 2010; revised February 2011.
+1Samuel Cohen is now at the University of Oxford; this work was completed at the
+University of Adelaide.
+AMS 2000 subject classifications. Primary 60H20; secondary 60H10, 91B16.
+Key words and phrases. BSDE, comparison theorem, general filtration, separable pr ob-
+ability space, Gr¨ onwall inequality, nonlinear expectati on.
+This is an electronic reprint of the original article published by the
+Institute of Mathematical Statistics inThe Annals of Probability ,
+2012, Vol. 40, No. 5, 2264–2297 . This reprint differs from the original in
+pagination and typographic detail.
+12 S. N. COHEN AND R. J. ELLIOTT
+Anotableexception tothisistheworkofElKarouiandHuang[ 12], where
+a general probability space is considered. In the case consi dered in [ 12], the
+martingale Mis specified a priori, and the equation considered is
+dYt=F(ω,t,Yt−,Zt)dCt−Z∗
+tdMt−dNt;YT=Q, (1)
+where each term is as above, the filtration is quasi-left cont inuous,Cis
+a continuous process such that d/an}bracketle{tM/an}bracketri}htis absolutely continuous with respect
+todCandNis a martingale strongly orthogonal to M, that is, /an}bracketle{tM,N/an}bracketri}ht=0,
+where/an}bracketle{t·,·/an}bracketri}htdenotes the predictable quadratic covariation process.
+These equations depend heavily on the continuity of Cand, therefore, are
+unable to deal with any situation where martingales may jump at a point
+with positive probability. However, these situations may a rise in various ap-
+plications. For example, when using BSDEs in modeling divid end paying
+assets, the martingales involved may jump at the time of the d ividend an-
+nouncement. Similarly, if we consider embedding a discrete time process in
+continuous time, we obtain processes which jump with positi ve probability
+at every integer.
+Asignificantuseoftheseequationsistogenerate“nonlinea rexpectations”
+or “nonlinear evaluations,” in the sense of [ 18]. These are operators
+E(·|Ft):L2(FT)→L2(Ft),
+satisfying certain basic properties. They have important a pplications in
+mathematical financeandstochasticcontrol. Giventheresu ltsof[9]and[15],
+it is known that in the Brownian setting, under certain condi tions, these op-
+erators are completely described by BSDEs. Furthermore, it is clear, given
+the comparison theorem in [ 8], BSDEs of the form of ( 1) in arbitrary spaces,
+under some conditions, also describe nonlinear expectatio ns. However, it is
+not known how large a class of nonlinear expectations in a gen eral space is
+given by a BSDE.
+To establish such a result for BSDEs of the form of ( 1), one faces a signif-
+icant problem. If E(Q|Ft)=Ytis given as the solution to ( 1) for some Fnot
+dependent on Yt−, onceMis fixed, for any martingale Northogonal to M
+withN0=0, we have the property
+E(Q+NT|Ft)=E(Q|Ft).
+This property is clearly not true for most nonlinear expecta tions, whenever
+there are nontrivial examples of such processes N, which is not the case
+in the Brownian setting (as a martingale representation the orem holds).
+It follows that these equations cannot describe any nonline ar expectations
+which do not possess this property.
+Furthermore, the fact that the martingale Mmust be specified a pri-
+ori is arguably unsatisfying. Conceptually, it may be prefe rable if, in some
+sense, the probability space itself dictated what martinga les are needed for
+the BSDE. In this case, one could proceed either by specifyin g the proba-BSDES IN GENERAL SPACES 3
+bility space using a collection of martingales (which, give n a representation
+theorem holds, will then describe all martingales in the spa ce), or vice versa.
+In this paper we establish such a general result. We show that there is
+a sense in which the original BSDE can be interpreted in a gene ral space,
+using only a separability assumption on L2(FT). We establish conditions on
+the existence and uniqueness of BSDEs in this setting, where the driver is
+integrated with respect to an arbitrary deterministic Stie ltjes measure (The-
+orem6.1). We also prove a comparison theorem for these solutions, wh ich
+shows under which conditions they do indeed describe nonlin ear expecta-
+tions and evaluations.
+A similar approach is taken in [ 14], where a form of BSDE is considered
+using generic maps from a space of semimartingales to the spa ces of square-
+integrable martingales and of finite-variation processes i ntegrable with re-
+spect to a given continuous increasing process. Using Browd er’s theorem,
+they demonstrate the existence of solutions to these equati ons on an infinite
+horizon. Our approach differs from theirs by considering a cla ssical form of
+BSDEonafinitehorizonandderivinganexistenceresultusin gacontraction
+mapping technique. Because of this, our conditions for exis tence are a more
+straightforward extension of those in the classical case. M ore significantly,
+our approach does not require the driver of the BSDE to be inte grated with
+respect to a continuous measure, which allows a unification o f the discrete
+and continuous time theory of BSDEs.
+2. Martingale representations. The key result used in the construction
+ofBSDEsistheMartingale representation theorem.IntheBr owniansetting,
+thisresultiswell known(see, e.g., [ 20],ChapterV.3, or[ 13], Theorem12.33).
+In other cases, for example, when dealing with martingales g enerated by
+Markov chains, a similar result is available (see [ 4]); however it is also known
+that there exist probability spaces in which no finite-dimen sional martingale
+representation theorem exists.
+Consider a probability space (Ω ,F,P) with a filtration {Ft},t∈[0,T],
+satisfying the usual conditions of completeness and right c ontinuity. The
+time-interval [0 ,T] is given the Borel σ-fieldB([0,T]).
+Definition 2.1. For any nondecreasing process of finite variation µ, we
+define the measure induced by µto be the measure over Ω ×[0,T] given by
+A/mapsto→E/bracketleftbigg/integraldisplay
+[0,T]IA(ω,t)dµ/bracketrightbigg
+.
+HereA∈F⊗B([0,T]), andtheintegral is taken pathwise inaStieltjes sense.
+Remark 2.1. Ifµis a deterministic process, then this definition gives
+the product measure µ×P. We can also consider these as measures on the
+space (Ω ×[0,T],P), where Pis the predictable σ-algebra.4 S. N. COHEN AND R. J. ELLIOTT
+Under theassumption that the Hilbert space L2(FT) is separable, a paper
+of Davis and Varaiya [ 10] gives the following result (see also Malamud [ 17]).
+Theorem 2.1 (Martingale representation theorem; [ 10]).Suppose that
+L2(FT)is a separable Hilbert space, with an inner product (X,Y)=E[XY].
+Then there exists a finite or countable sequence of square-int egrable{Ft}-
+martingales M1,M2,...such that every square integrable {Ft}-martingale N
+has a representation
+Nt=N0+∞/summationdisplay
+i=1/integraldisplay
+]0,t]Zi
+udMi
+u
+for some sequence of predictable processes Zi. This sequence satisfies
+E/bracketleftBigg∞/summationdisplay
+i=0/integraldisplay
+[0,T](Zi
+u)2d/an}bracketle{tMi/an}bracketri}htu/bracketrightBigg
+<+∞. (2)
+These martingales are orthogonal (i.e., E[Mi
+TMj
+T]=0for alli/ne}ationslash=j), and
+the predictable quadratic variation processes /an}bracketle{tMi/an}bracketri}htsatisfy
+/an}bracketle{tM1/an}bracketri}ht≻/an}bracketle{tM2/an}bracketri}ht≻···,
+where≻denotes absolute continuity of the induced measures (Defini tion2.1).
+Furthermore, these martingales are unique, in that if Niis another such
+sequence, then /an}bracketle{tNi/an}bracketri}ht ∼ /an}bracketle{tMi/an}bracketri}ht, where ∼denotes equivalence of the induced
+measures.
+Corollary 2.1.1. For any predictable processes Zisatisfying ( 2), the
+process/summationtext
+i/integraltext
+]0,t]Zi
+udMi
+uis well defined and is a square-integrable martingale.
+Remark 2.2. When a finite-dimensional martingale representation the-
+orem holds, as when the space is generated by a Brownian motio n, then all
+but finitely many of the martingales Migiven by Theorem 2.1will be zero.
+We shall not, in general, assume that this is the case, but ack nowledge that,
+in this situation, significant simplification of the equatio ns considered is pos-
+sible.
+We shall usethis resulttoconstruct aformof BSDE onthis gen eral space.
+Definition 2.2. We denote by RK×∞the space of infinite RK-valued
+sequences. We note that the predictable processes Ziin Theorem 2.1can be
+written as a vector process Z, which takes values in R1×∞.
+3. BSDEs in general spaces: A definition. We seek to construct BSDEs,
+assumingonly the usual properties of the filtration and that L2(FT) is a sep-
+arable Hilbert space. For simplicity, we shall also assume t hatF0is trivial,
+which, by right continuity, ensures that, almost surely, no martingale has
+a jump at t=0.BSDES IN GENERAL SPACES 5
+Definition 3.1. Letµbe a deterministic signed Stieltjes measure. For
+K∈N, a BSDE is an equation of the form
+Q=Yt−/integraldisplay
+]t,T]F(ω,u,Yu−,Zu)dµu+∞/summationdisplay
+i=1/integraldisplay
+]t,T]Zi
+udMi
+u, (3)
+whereZt(ω) is the (countably infinite) vector with entries {Zi
+t(ω)∈RK}i∈N.
+For a terminal value Q∈L2(RK;FT), a predictable driverfunction F:Ω×
+[0,T]×RK×RK×∞→RK, a solution is a pair of processes ( Y,Z) taking
+values in RK×RK×∞, whereZis predictable, and Yis adapted. We shall
+restrict our attention to the case when Yis square integrable, and Zsatis-
+fies (2).
+Remark 3.1. We note that this type of equation encompasses most
+previously studied forms of BSDEs. When the filtration is Bro wnian, we can
+takeMito be the ith component of the generating Brownian motion, µ=t,
+and the equation is standard. When the filtration is generate d by a Poisson
+random measure over a separable space and a Brownian motion, as in [2,22]
+and others, or by a Markov chain, as in [ 4,5], we have a similar reduction.
+When we consider the analogous equations in discrete time, w e can form the
+discrete-time filtration embedded in this continuous time c ontext (see [ 16],
+Chapter 1f) and hence obtain the backward stochastic differen ce equations
+considered in [ 6] and [7].
+Comparingwiththeworkof[ 12], weseethat if Fdependsonlyonthepro-
+jection of Zinto a finite-dimensional subspace of RK×∞, then it is possible
+to reduce the equation to a form similar to ( 1).
+We shall present a result (Theorem 6.1) demonstrating conditions under
+which there exists a unique solution to such an equation.
+4. Inequalities for Stieltjes integrals. To give conditions under which so-
+lutions to a BSDE exist, we must first establish the following results regard-
+ing integrals with respect to Stieltjes measures. These res ults are standard
+whenever the measures are continuous.
+4.1.Stieltjes exponentials.
+Definition 4.1. Foranyc` adl` agfunctionoffinitevariation ν:[0,∞[→R,
+we write
+E(νt):=eνt/productdisplay
+0≤s≤t(1+∆νs)e−∆νs,
+and call this the Stieltjes exponential ofν. Note that this is also a c` adl` ag
+function.
+Note that E(νt) should be more properly written as E(ν(·);t), as it is
+a function of {νs;s≤t}not just of νt. We use the former notation purely6 S. N. COHEN AND R. J. ELLIOTT
+for compactness, whenever this does not lead to confusion. W e note the
+following useful bound.
+Lemma 4.1. Ifνis a c` adl` ag function, then E(νt)≤eνt, whereeνtis the
+classical exponential of νt.
+Proof. Asex≥1+x, it is clear that (1+∆ νt)e−∆νt≤1 for all t. The
+result follows. /square
+Lemma 4.2. For any c` adl` ag function of finite variation, the Stieltjes ex-
+ponential is well defined. Furthermore, if ∆νt≥−1, thenE(νt)≥0. If∆νs>
+−1, thenE(νt)>0, andE(νt)−1is well defined. In this case, the process ut=
+usE(νt)E(νs)−1is the solution to the Lebesgue–Stieltjes integral equatio n,
+ut=us+/integraldisplay
+]s,t]ur−dνr.
+Proof. As the process νtis c` adl` ag and of finite variation, it is a (de-
+terministic) semimartingale. E(νt) is then the standard Dol´ eans–Dade expo-
+nential of this process, and so its existence and basic prope rties can be seen
+in[13],Theorem13.5ff. Thisguarantees theconvergence of theinfin iteprod-
+ucts considered and solves the desired integral equation. T he nonnegativity
+result is clear by inspection.
+For the positivity result, we need only show that/producttext
+0≤s≤t(1+∆νs)>0.
+By continuity of the logarithm, this is equivalent to showin g that
+−/summationdisplay
+0≤s≤tlog(1+∆ νs)<∞.
+We then note that we can consider three cases. First, if ∆ νs≥0, then
+−log(1+∆ νs)≤0, and hence
+−/parenleftbigg/summationdisplay
+{0≤s≤t}∩{∆νs≥0}log(1+∆ νs)/parenrightbigg
+≤0<∞.
+Second, we note that/summationtext
+0<s≤t|∆νs|is finite, as νis of finite variation, and
+hence there are only finitely many ssuch that ∆ νs≤−0.7. Therefore
+−/parenleftbigg/summationdisplay
+{0≤s≤t}∩{∆νs≤−0.7}log(1+∆ νs)/parenrightbigg
+<∞.
+Finally, we know that 2 x<log(1+x)<0 for−0.7<x<0. Hence, we have
+−/parenleftbigg/summationdisplay
+{0≤s≤t}∩{−0.7<∆νs<0}log(1+∆ νs)/parenrightbigg
+</parenleftbigg/summationdisplay
+{0≤s≤t}∩{−0.7<∆νs<0}2|∆νs|/parenrightbigg
+<∞.
+Combining these three sums gives the desired constraint on t he logarithm,
+and hence the strict positivity of the desired product. /squareBSDES IN GENERAL SPACES 7
+Lemma 4.3. Forνa c` adl` ag function of finite variation with ∆νt>−1,
+we have the stronger result
+inf
+0≤t≤T/braceleftbigg/productdisplay
+0≤s≤t(1+∆νs)/bracerightbigg
+>0.
+Proof. By the same argument as in Lemma 4.2, we have
+−/parenleftbigg/summationdisplay
+{0≤s≤T}∩{∆νs<0}log(1+∆ νs)/parenrightbigg
+<∞.
+It follows that
+−/summationdisplay
+0≤s≤tlog(1+∆ νs)<−/parenleftbigg/summationdisplay
+{0≤s≤T}∩{∆νs<0}log(1+∆ νs)/parenrightbigg
+<∞
+for allt. Hence
+inf
+0≤t≤T/braceleftbigg/productdisplay
+0≤s≤t(1+∆νs)/bracerightbigg
+>/parenleftbigg/productdisplay
+{0≤s≤T}∩{∆νs<0}(1+∆νs)/parenrightbigg
+>0.
+/square
+Definition 4.2. Letνbe a c` adl` ag function of finite variation with
+∆νt>−1 for all t. Then the left-jump inversion ofνis defined by
+¯νt=νt−/summationdisplay
+0≤s≤t(∆νs)2
+1+∆νs.
+Similarly if ∆ νt<1 for all t, theright-jump inversion is defined by
+˜νt=νt+/summationdisplay
+0≤s≤t(∆νs)2
+1−∆νs.
+Lemma 4.4. Forνa function as in Definition 4.2, the left- and right-
+jump inversions are finite (whenever they are defined), and sa tisfy
+E(νt)−1=E(−¯νt)
+and
+E(−νt)=E(˜νt)−1.
+Proof. Consider first the left-jump-inversion. We know that ∆ νs>−1
+and/summationtext|∆νs|<∞. Hence it follows that ∆ νshas only finitely many values
+in any neighborhood not containing zero and hence is bounded away from
+−1. That is, there exists some ε>0 such that ∆ νs>ε−1 for alls. To show
+finiteness, write
+/summationdisplay
+{0≤s≤t}∩{∆νs≥0}(∆νs)2
+1+∆νs≤/summationdisplay
+{0≤s≤t}∩{∆νs≥0}|∆νs|<∞8 S. N. COHEN AND R. J. ELLIOTT
+and
+/summationdisplay
+{0≤s≤t}∩{∆νs<0}(∆νs)2
+1+∆νs≤ε−1/parenleftbigg/summationdisplay
+{0≤s≤t}∩{∆νs<0}(∆νs)2/parenrightbigg
+<ε−1/parenleftbigg/summationdisplay
+{0≤s≤t}∩{∆νs<0}|∆νs|/parenrightbigg
+<∞.
+Combining these sums gives the desired finiteness result.
+We now note that, algebraically,
+(1−∆¯νs)−1=/parenleftbigg
+1−∆νs+(∆νs)2
+1+∆νs/parenrightbigg−1
+=1+∆νs.
+Hence
+E(νt)−1=e−νt/productdisplay
+0≤s≤t(1+∆νs)−1e∆νs
+=e−νt+/summationtext
+0<s≤t((∆νs)2/(1+∆νs))/productdisplay
+0≤s≤t(1+∆νs)−1e∆νs−(∆νs)2/(1+∆νs)
+=e−¯νt/productdisplay
+0≤s≤t(1−∆¯νs)e∆¯νs
+=E(−¯νt).
+The proof for the right-jump inversion follows in the same wa y, where
+finiteness is because
+/summationdisplay
+0≤s≤t(∆νs)2
+1−∆νs=/summationdisplay
+0≤s≤t(−∆νs)2
+1+(−∆νs),
+and−νssatisfies the requirements given above for the left-jump inv ersion.
+The algebraic result is then that
+(1+∆˜νs)−1=/parenleftbigg
+1+∆νs+(∆νs)2
+1−∆νs/parenrightbigg−1
+=1−∆νs,
+and the result is as given. /square
+Lemma 4.5. Forνa c` adl` ag function of bounded variation with ∆νs>
+−1, the right-jump inversion of the left-jump inversion of νis the original
+function, that is,
+˜¯νt=νt.
+Similarly, if ∆νs<−1, then¯˜νt=νt.BSDES IN GENERAL SPACES 9
+Proof. For simplicity, we decompose νinto a discontinuous part νd
+t:=/summationtext
+0≤s≤t∆νsand a continuous part νc
+t=νt−νd. Clearly, taking either the
+left- or right-jump inversion will not alter the continuous partνc, and
+so it is sufficient to show that the discontinuous parts are equ al, that is,
+∆˜¯νt= ∆¯˜νt= ∆νtfor allt, whenever these terms are well defined. From
+Definition 4.2we have
+∆¯νt=∆νt
+1+∆νt,∆˜νt=∆νt
+1−∆νt
+and hence
+∆˜¯νt=∆¯νt
+1−∆¯νt=∆νt/(1+∆νt)
+1−∆νt/(1+∆νt)=∆νt,
+and similarly ∆ ¯˜νt=∆νt, as desired. /square
+4.2.Integrating factors. It is useful to have some results relating to the
+solutions of equations of the form dut−ut−dνt=···.These are similar
+the the classical results on the use of integrating factors a nd Gr¨ onwall’s
+inequality in the study of ordinary differential equations.
+Definition 4.3. Letu,vbe two measures on a σ-algebra A. We write
+du≤dvif, for any A∈A,u(A)≤v(A).
+Remark 4.1. Whenvis a nonnegative measure, and uis absolutely
+continuous with respect to v, this definition is equivalent to requiring that
+the Radon–Nikodym derivative satisfies du/dv≤1,dv-a.e.
+Lemma 4.6. Letu,νandwbe signed Stieltjes measures on B([0,T]),
+such that ∆νt<1for allt, and
+dut≥−ut−dνt+dwt,
+then
+d(utE(˜νt))≥(1−∆νt)−1E(˜νt−)dwt,
+where˜νis the right-jump inversion of ν.
+Proof. Applying the product rule for Stieltjes integrals we have
+d(utE(˜νt))
+E(˜νt−)=dut+ut−d˜νt+∆ut∆˜νt.
+Asd˜νt=dν/(1−∆νt) and ∆ut∆νt=(∆νt)dut, this gives
+d(utE(˜νt))
+E(˜νt−)=dut+ut−dνt
+1−∆νt+∆νt
+1−∆νtdut10 S. N. COHEN AND R. J. ELLIOTT
+=/parenleftbigg
+1+∆νt
+1−∆νt/parenrightbigg
+dut+ut−dνt
+1−∆νt
+=(1−∆νt)−1(dut+ut−dνt)
+≥(1−∆νt)−1dwt. /square
+Lemma 4.7 (Backward Gr¨ onwall inequality). Letube a process such
+that, for νa nonnegative Stieltjes measure with ∆νt<1andαa˜ν-integrable
+process,uisν-integrable and
+ut≤αt+/integraldisplay
+]t,T]usdνs,
+then
+ut≤αt+E(−νt)/integraldisplay
+]t,T]E(˜νs)αsd˜νs.
+Ifαt=αis constant, this simplifies to
+ut≤αE(˜νT)E(˜νt)−1=αE(−νt)E(−νT)−1.
+Proof. Firstnotethat dνt=d˜ν
+1+∆˜νtandthat∆˜ νt∆νt=∆˜νtdνt.Thenlet
+wt:=E(˜νt)/integraldisplay
+]t,T]usdνs.
+From the product rule for stochastic integrals, as νis of finite variation,
+dwt
+E(˜νt−)=/parenleftbigg/integraldisplay
+]t,T]usdνs/parenrightbigg
+d˜νs−utdνt−ut∆νt∆˜νt
+=−ut(1+∆˜νt)dνt+/parenleftbigg/integraldisplay
+]t,T]usdνs/parenrightbigg
+d˜νt
+=−utd˜νt+/parenleftbigg/integraldisplay
+]t,T]usdνs/parenrightbigg
+d˜νt
+=/parenleftbigg
+−ut+/integraldisplay
+]t,T]usdνs/parenrightbigg
+d˜νt
+≥−αtd˜νt.
+Note that d˜νtandE(˜νt−) are both nonnegative. Therefore, by integration,
+wt=E(˜νt)/integraldisplay
+]t,T]usdνs≤/integraldisplay
+]t,T]E(˜νs−)αsd˜νs.
+Substitution yields
+ut≤αt+E(˜νt)−1/integraldisplay
+]t,T]E(˜νs−)αsd˜νs,BSDES IN GENERAL SPACES 11
+and the desired inequalities follow from E(˜νt)−1=E(−νt). Ifαt=α, then
+this simplifies to
+ut−≤α/bracketleftbigg
+1+E(˜νt)−1/integraldisplay
+]t,T]E(˜νs−)d˜νs/bracketrightbigg
+=α[1+E(˜νt)−1(E(˜νT)−E(˜νt))]
+=αE(˜νT)E(˜νt)−1. /square
+Lemma 4.8 (Forward Gr¨ onwall inequality). Letube a function such
+that, for νa nonnegative Stieltjes measure and αa¯ν-integrable process, uis
+ν-integrable and
+ut≤αt+/integraldisplay
+]0,t]usdνs,
+then
+ut≤αt+E(νt)/integraldisplay
+]0,t]E(−¯νs)αsd¯νs.
+Ifαt=αis constant, this simplifies to
+ut≤αE(νt).
+Proof. This result follows in an almost identical fashion to Lemma 4.7,
+and the proof is therefore omitted. /square
+5. Existence of BSDE solutions: Fundamental results. In this section
+we shall establish the existence of solutions to BSDEs when t he process µ
+satisfies particular properties.
+Definition 5.1. Letµbeadeterministicnondecreasingright-continuous
+function µ:[0,T]→R+. The measure dµwill serve in the place of the
+Lebesgue measure dtin our BSDE.
+Asµis of finite variation, its discontinuities ∆ µare bounded. We assume
+thatµassigns positive measure to any nonempty open interval in [0 ,T].
+Unless otherwise indicated, all (in-)equalities should be read as “up to
+evanescence.”
+Definition 5.2. Wedenoteby /bardbl·/bardblthestandardEuclideannormon RK,
+and note that /bardbly/bardbl2=y∗y, where [ ·]∗denotes vector transposition.
+Definition 5.3. For agiven µand fixed K∈N, wedefinethestochastic
+seminorm /bardbl·/bardblMtonRK×∞as follows. For each i∈N, consider /an}bracketle{tMi/an}bracketri}htas
+a measure on the predictable σ-algebra; cf. Remark 2.1. Let/an}bracketle{tMi/an}bracketri}hthave the
+Lebesgue decomposition
+/an}bracketle{tMi/an}bracketri}htt=mi,1
+t+mi,2
+t,12 S. N. COHEN AND R. J. ELLIOTT
+wheremi,1
+tis absolutely continuous with respect to µ×P, andmi,2
+tis or-
+thogonal to µ×P. As they represent bounded measures on the predictable
+σ-algebra, both mi,1
+tandmi,2
+twill be nondecreasing predictable processes.
+We define, for zt∈RK×∞,
+/bardblzt/bardbl2
+Mt:=/summationdisplay
+i/bracketleftbigg
+/bardblzi
+t/bardbl2dmi,1
+t
+d(µ×P)/bracketrightbigg
+,
+wherezi
+t∈RKis theith element in zt, considered as a series of values in RK.
+We note that, for any predictable, progressively measurabl e process Z
+taking values in RK×∞, and, in particular, for processes satisfying ( 2) in
+each of their Kcomponents, we have the inequality
+E/bracketleftbigg/integraldisplay
+A/bardblZt/bardbl2
+Mtdµ/bracketrightbigg
+≤E/bracketleftbigg/summationdisplay
+i/integraldisplay
+A/bardblZi
+t/bardbl2d/an}bracketle{tMi
+t/an}bracketri}ht/bracketrightbigg
+=E/bracketleftbigg/summationdisplay
+i/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay
+AZi
+tdMi
+t/vextenddouble/vextenddouble/vextenddouble/vextenddouble2/bracketrightbigg
+(4)
+=E/bracketleftbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+i/integraldisplay
+AZi
+tdMi
+t/vextenddouble/vextenddouble/vextenddouble/vextenddouble2/bracketrightbigg
+for any predictable set A⊆Ω×[0,T]. (Note the latter equalities are simply
+the standard isometry used in the construction of the stocha stic integral, by
+the orthogonality of the Mi.)
+Definition 5.4. We define the following spaces of equivalence classes:
+H2
+M=/braceleftbigg
+Z:Ω×[0,T]→RK×∞, predictable,
+E/bracketleftbigg/summationdisplay
+i/integraldisplay
+[0,T]/bardblZi
+t/bardbl2d/an}bracketle{tMi/an}bracketri}htt/bracketrightbigg
+<+∞/bracerightbigg
+,
+S2=/braceleftBig
+Y:Ω×[0,T]→RK, adapted, E/bracketleftBig
+sup
+t∈[0,T]/bardblYt/bardbl2/bracketrightBig
+<+∞/bracerightBig
+,
+H2
+µ=/braceleftbigg
+Y:Ω×[0,T]→RK, progressive,/integraldisplay
+]0,T]E[/bardblYt/bardbl2]dµt<+∞/bracerightbigg
+,
+where two elements Z,¯ZofH2
+Mare deemed equivalent if
+E/bracketleftbigg/summationdisplay
+i/integraldisplay
+[0,T]/bardblZi
+t−¯Zi
+t/bardbl2d/an}bracketle{tMi/an}bracketri}htt/bracketrightbigg
+=0,
+two elements of S2are deemed equivalent if they are indistinguishable and
+two elements of H2
+µare equivalent if they are equal µ×P-a.s. Note that K
+is here taken as fixed.BSDES IN GENERAL SPACES 13
+Remark 5.1. We note that H2
+Mis itself a complete metric space, with
+norm given by Z/mapsto→E[/summationtext
+i/integraltext
+[0,T]/bardblZi
+t/bardbl2d/an}bracketle{tMi/an}bracketri}htt]; similarly for H2
+µ. Note also
+that the martingale representations constructed in Theore m2.1are unique
+inH2
+M.
+A key assumption in the study of BSDEs is the continuity of the driver
+function F. When the measure µis continuous, we shall show that it is
+sufficient that Fis uniformly Lipschitz continuous for the BSDE ( 3) to have
+a solution. On the other hand, as is clear in discrete time (cf . [7]), when µ
+is not continuous, a stronger condition is needed on F. We shall call this
+afirmLipschitz bound on F, as is defined in the following theorem.
+Theorem 5.1. Forµas in Definition 5.1, assume µT≤1. LetF:Ω×
+[0,T]×RK×RK×∞→RKbe a predictable, progressively measurable function
+such that:
+•E[/integraltext
+]0,T]/bardblF(ω,t,0,0)/bardbl2dµt]<+∞;
+•there exists a linear firm Lipschitz bound on F, that is, a measurable
+deterministic function ctuniformly bounded by some c∈R, such that, for
+anyyt,y′
+t∈RK,zt,z′
+t∈RK×∞,
+/bardblF(ω,t,yt,zt)−F(ω,t,y′
+t,z′
+t)/bardbl2
+≤ct/bardblyt−y′
+t/bardbl2+c/bardblzt−z′
+t/bardbl2
+Mt, dµ ×dP-a.s.
+and
+ct∆µt<1.
+Note that the variable bound ctneed only apply to the behavior of Fwith
+respect to y.
+A function satisfying these conditions will be called stand ard. Then for any
+Q∈L2(RK;FT), the BSDE ( 3) with driver Fhas a unique solution (Y,Z)∈
+S2×H2
+M. (S2andH2
+Mare defined in Definition 5.4.)
+To prove this theorem, we first establish the following resul ts.
+Lemma 5.1. Ifµassigns positive measure to every nonempty open in-
+terval, then two c` adl` ag processes in H2
+µare indistinguishable if and only
+if they are equivalent in H2
+µ. Similarly, two c` adl` ag processes are equivalent
+inH2
+µif and only if their left limits are equivalent in H2
+µ.
+Proof. Clearlyindistinguishabilityimpliesequivalenceofthep rocesses,
+and their left limits, in H2
+µ. By right continuity (resp., left continuity), if on
+some nonnull set A, two processes (resp., their left limits) differ at any point,
+they must differ on some nonempty openinterval. As µassigns positive mea-14 S. N. COHEN AND R. J. ELLIOTT
+sure to such an interval, it follows that the processes will n ot be equivalent
+inH2
+µ./square
+Lemma 5.2. Let(Y,Z)be the solution to a BSDE with data (F,Q). IfF
+is standard, Q∈L2(RK;FT)andZ∈H2
+M, thenY∈S2if and only if the
+left limit process Yt−∈H2
+µ.
+Proof. Clearly, if Y∈S2, then as Yis c` adl` ag and adapted, and hence
+progressive, Y∈H2
+M. For the converse, write
+sup
+t∈[0,T]/bardblYt/bardbl2≤2/bardblQ/bardbl2+4 sup
+t∈[0,T]/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+i/integraldisplay
+]t,T]Zi
+udMi
+u/vextenddouble/vextenddouble/vextenddouble/vextenddouble2
++4 sup
+t∈[0,T]/braceleftbigg/integraldisplay
+]t,T]/bardblF(ω,u,Yu−,Zu)/bardbl2dµ/bracerightbigg
+≤2/bardblQ/bardbl2+4 sup
+t∈[0,T]/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+i/integraldisplay
+]t,T]Zi
+udMi
+u/vextenddouble/vextenddouble/vextenddouble/vextenddouble2
++8/integraldisplay
+]0,T]/bardblF(ω,u,0,0)/bardbl2dµt
++8/integraldisplay
+]0,T][ct/bardblYu−/bardbl2+c/bardblZu/bardbl2
+Mu]dµt,
+and by the assumptions of the lemma, as Z∈H2
+M, and so/summationtext
+i/integraltext
+]0,t]Zi
+udMi
+u
+is a square integrable martingale, by Doob’s inequality [ 16], Theorem 1.43,
+this quantity is finite in expectation. /square
+The following lemma provides the key bounds on BSDE solution s, which
+we shall use to prove existence and uniqueness of solutions.
+Lemma 5.3. Let(Y,Z)and(¯Y,¯Z)be the solutions to two BSDEs with
+standard parameters (F,Q)and(¯F,¯Q). Define
+δY:=Y−¯Y, δZ :=Z−¯Z,
+δ2ft:=F(ω,t,¯Yt−,¯Zt)−¯F(ω,t,¯Yt−,¯Zt),
+υt:=/integraldisplay
+]0,t][(x−1
+s−∆µs)(1+ws)cs+xs]dµs, (5)
+πt:=/integraldisplay
+]0,t][(x−1
+s−∆µs)(1+w−1
+s)](1−∆υs)−1dµs,
+ρi
+t:=/integraldisplay
+]0,t][1−(x−1
+s−∆µs)(1+wt)c](1−∆υs)−1d/an}bracketle{tMi/an}bracketri}htt,BSDES IN GENERAL SPACES 15
+wherecsandcare the Lipschitz constants of F, andxt,wtare any nonneg-
+ative measurable functions such that ∆µt≤x−1
+tand∆υt<1for allt, and
+the integrands defining υ,πandρiare uniformly bounded.
+Then
+E[/bardblδYt/bardbl2]E(˜υt)+E/bracketleftbigg/summationdisplay
+i/integraldisplay
+]t,T]E(˜υs−)/bardblδZi
+s/bardbl2dρi
+s/bracketrightbigg
+(6)
+≤E[/bardblδQ/bardbl2]E(˜υT)+/integraldisplay
+]t,T]E[/bardblδ2fs/bardbl2]E(˜υs−)dπs
+and/integraldisplay
+]0,T]E[/bardblδYt−/bardbl2]E(˜υt−)dµt+E/bracketleftbigg/summationdisplay
+i/integraldisplay
+]0,T]µsE(˜υs−)/bardblδZi
+s/bardbl2dρi
+s/bracketrightbigg
+(7)
+≤µTE[/bardblδQ/bardbl2]E(˜υT)+/integraldisplay
+]0,T]µsE[/bardblδ2fs/bardbl2]E(˜υs−)dπs.
+Proof. LetδF=F(ω,t,Yt−,Zt)−¯F(ω,t,¯Yt−,¯Zt).Byapplication ofthe
+differentiation rule for stochastic integrals, we have
+d[/bardblδYt/bardbl2]=−2(δYt−)∗(δFt)dµt+2/summationdisplay
+i(δYt−)∗(δZi
+t)dMi
+t
++/summationdisplay
+i,j(δZi
+t)∗(δZj
+t)d[Mi,Mj]t−2(δFt)(∆µt)/summationdisplay
+i(δZi
+t)∆Mi
+t (8)
++/bardblδFt/bardbl2(∆µt)2.
+AsδY∈S2, by the BDG inequality it is clear that/integraltext
+]0,t]/summationtext
+i(δYs−)∗(δZi
+s)dMi
+s
+is a martingale. Similarly the process
+/summationdisplay
+s∈]0,t]/bracketleftbigg
+(δFs)(∆µs)/summationdisplay
+i(δZi
+s)∆Mi
+s/bracketrightbigg
+is a countable sum of integrable martingale differences and so is also a mar-
+tingale. Also, δZ∈H2
+Mand so, by orthogonality of the Mi,
+/summationdisplay
+i,j(δZi
+t)∗(δZj
+t)d[Mi,Mj]t−/summationdisplay
+i/bardblδZi
+t/bardbl2d/an}bracketle{tMi/an}bracketri}htt
+is a martingale.
+For anyA∈B([0,T]), integrating on Aandtakingan expectation through
+(8) then yields
+/integraldisplay
+AdE[/bardblδYt/bardbl2]=−2/integraldisplay
+AE[(δYt−)∗(δFt)]dµt+E/bracketleftbigg/summationdisplay
+i/integraldisplay
+A/bardblδZi
+t/bardbl2d/an}bracketle{tMi/an}bracketri}htt/bracketrightbigg
++/summationdisplay
+t∈AE[/bardblδFt/bardbl2](∆µt)2.16 S. N. COHEN AND R. J. ELLIOTT
+Using thefact that (∆ µt)2=(∆µt)(dµt) andthat for any x≥0, anya,b∈R,
+±2ab≤xa2+x−1b2, we have, for any measurable function xt≥0,/integraldisplay
+AdE[/bardblδYt/bardbl2]≥−/integraldisplay
+AxtE[/bardblδYt−/bardbl2]dµt−/integraldisplay
+Ax−1
+tE[/bardblδFt/bardbl2]dµt
++E/bracketleftbigg/summationdisplay
+i/integraldisplay
+A/bardblδZi
+t/bardbl2d/an}bracketle{tMi/an}bracketri}htt/bracketrightbigg
++/integraldisplay
+AE[/bardblδFt/bardbl2](∆µt)dµt (9)
+=−/integraldisplay
+AxtE[/bardblδYt−/bardbl2]dµt−/integraldisplay
+A(x−1
+t−∆µt)E[/bardblδFt/bardbl2]dµt
++E/bracketleftbigg/summationdisplay
+i/integraldisplay
+A/bardblδZi
+t/bardbl2d/an}bracketle{tMi/an}bracketri}htt/bracketrightbigg
+.
+We now note that, for any measurable wt≥0, as (a+b)2≤(1+w)a2+
+(1+w−1)b2for allw≥0,
+/bardblδFt/bardbl2≤(1+wt)/bardblF(ω,t,Yt−,Zt)−F(ω,t,¯Yt−,¯Zt)/bardbl2
++(1+w−1
+t)/bardblF(ω,t,¯Yt−,¯Zt)−¯F(ω,t,¯Yt−,¯Zt)/bardbl2
+≤(1+wt)ct/bardblδYt−/bardbl2+(1+wt)c/bardblδZt/bardbl2
+Mt+(1+w−1
+t)/bardblδ2ft/bardbl2.
+Hence, as x−1
+t−∆µt≥0,/integraldisplay
+A(x−1
+t−∆µt)E[/bardblδFt/bardbl2]dµt
+≤/integraldisplay
+A(x−1
+t−∆µt)(1+wt)ctE[/bardblδYt−/bardbl2]dµt
++/integraldisplay
+A(x−1
+t−∆µt)(1+wt)cE[/bardblδZt/bardbl2
+Mt]dµt
++/integraldisplay
+A(x−1
+t−∆µt)(1+w−1
+t)E[/bardblδ2ft/bardbl2]dµt (10)
+≤/integraldisplay
+A(x−1
+t−∆µt)(1+wt)ctE[/bardblδYt−/bardbl2]dµt
++E/bracketleftbigg/summationdisplay
+i/integraldisplay
+A(x−1
+t−∆µt)(1+wt)c/bardblδZi
+t/bardbl2d/an}bracketle{tMi/an}bracketri}htt/bracketrightbigg
++/integraldisplay
+A(x−1
+t−∆µt)(1+w−1
+t)E[/bardblδ2ft/bardbl2]dµt.
+Combining ( 9) and (10) gives
+/integraldisplay
+AdE[/bardblδYt/bardbl2]≥−/integraldisplay
+AE[/bardblδYt−/bardbl2]dυt+E/bracketleftbigg/summationdisplay
+i/integraldisplay
+A(1−∆υt)/bardblδZi
+t/bardbl2dρi
+t/bracketrightbigg
+−/integraldisplay
+AE[/bardblδ2ft/bardbl2](1−∆υt)dπt.BSDES IN GENERAL SPACES 17
+Letφbe the signed measure on B([0,T]) defined by
+φ(A)=E/bracketleftbigg/summationdisplay
+i/integraldisplay
+A(1−∆υt)/bardblδZi
+t/bardbl2dρi
+t/bracketrightbigg
+−/integraldisplay
+AE[/bardblδ2ft/bardbl2](1−∆υt)dπt.
+Asdπ/dµis bounded, dρi/d/an}bracketle{tMi/an}bracketri}htis bounded, /bardblδ2f/bardbl2isµ-integrable and
+δZt∈H2
+M, it follows that φ(A) is bounded. We see then that φis a signed
+Stieltjes measure, and we equate it with its distribution fu nctionφt:=
+φ([0,t]).
+Therefore, as ∆ υt<1, ∆µ−x−1≤0, an application of Lemma 4.6yields/integraldisplay
+Ad[E[/bardblδYt/bardbl2]E(˜υt)]≥/integraldisplay
+A(1−∆υt)−1E(˜υt−)dφt
+=E/bracketleftbigg/summationdisplay
+i/integraldisplay
+AE(˜υt−)/bardblδZi
+t/bardbl2dρi
+t/bracketrightbigg
+−/integraldisplay
+AE(˜υt−)E[/bardblδ2ft/bardbl2]dπt.
+ForA= ]t,T], it follows that
+E[/bardblδYt/bardbl2]E(˜υt)+E/bracketleftbigg/summationdisplay
+i/integraldisplay
+]t,T]E(˜υs−)/bardblδZi
+s/bardbl2dρi
+s/bracketrightbigg
+≤E[/bardblδQ/bardbl2]E(˜υT)+/integraldisplay
+]t,T]E[/bardblδ2fs/bardbl2]E(˜υs−)dπs,
+which is the desired inequality ( 6). Taking a left-limit in tgives, by the
+dominated convergence theorem,
+E[/bardblδYt−/bardbl2]E(˜υt−)+E/bracketleftbigg/summationdisplay
+i/integraldisplay
+[t,T]E(˜υs−)/bardblδZi
+s/bardbl2dρi
+s/bracketrightbigg
+≤E[/bardblδQ/bardbl2]E(˜υT)+/integraldisplay
+[t,T]E[/bardblδ2fs/bardbl2]E(˜υs−)dπs,
+and so by integration and Fubini’s theorem, we have that
+/integraldisplay
+]0,T]E[/bardblδYt−/bardbl2]E(˜υt−)dµt+E/bracketleftbigg/summationdisplay
+i/integraldisplay
+]0,T]µsE(˜υs−)/bardblδZi
+s/bardbl2dρi
+s/bracketrightbigg
+≤µTE[/bardblδQ/bardbl2]E(˜υT)+/integraldisplay
+]0,T]µsE[/bardblδ2fs/bardbl2]E(˜υs−)dπs./square
+Lemma 5.4. LetF:Ω×[0,T]→RKbe a predictable progressively mea-
+surable function such that
+E/bracketleftbigg/integraldisplay
+]0,T]/bardblF(ω,t)/bardbl2dµ/bracketrightbigg
+<+∞.18 S. N. COHEN AND R. J. ELLIOTT
+Then the BSDE
+Yt−/integraldisplay
+]t,T]F(ω,u)dµ+/summationdisplay
+i/integraldisplay
+]t,T]Zi
+udMi
+u=Q
+has a unique solution in S2×H2
+Mfor anyQ∈L2(RK;FT). (Note here that F
+does not depend on YorZ.)
+Proof. Using Theorem 2.1, we first construct the processes Ziwhich
+give a representation of the square integrable martingale
+/summationdisplay
+i/integraldisplay
+]0,t]Zi
+udMi
+u=E/bracketleftbigg
+Q+/integraldisplay
+]0,T]F(ω,u)dµ/vextendsingle/vextendsingle/vextendsingleFt/bracketrightbigg
+.
+This can clearly be done componentwise, and so we obtain a uni que process
+Z∈H2
+M, that is, Zi
+s(ω)∈RK. It follows that
+/summationdisplay
+i/integraldisplay
+]t,T]Zi
+udMi
+u=Q+/integraldisplay
+]0,T]F(ω,u)dµ−E/bracketleftbigg
+Q+/integraldisplay
+]0,T]F(ω,u)dµ/vextendsingle/vextendsingle/vextendsingleFt/bracketrightbigg
+=Q+/integraldisplay
+]t,T]F(ω,u)dµ−E/bracketleftbigg
+Q+/integraldisplay
+]t,T]F(ω,u)dµ/vextendsingle/vextendsingle/vextendsingleFt/bracketrightbigg
+,
+and so there is an adapted process
+Yt:=E/bracketleftbigg
+Q+/integraldisplay
+]t,T]F(ω,u)dµ/vextendsingle/vextendsingle/vextendsingleFt/bracketrightbigg
+(11)
+=Q+/integraldisplay
+]t,T]F(ω,u)dµ−/summationdisplay
+i/integraldisplay
+]t,T]Zi
+udMi
+u,
+which satisfies the BSDE. By uniqueness of the right-hand sid e of (11), this
+process is unique up to indistinguishability and hence in S2./square
+Lemma 5.5. Letν:[0,T]→Rbe a nondecreasing c` adl` ag function of
+finite variation and c(·):[0,T]→Rbe a nonnegative bounded measurable
+function. Then ct∆νt=sups∈[0,T]{cs∆νs}for some t; that is ct∆νtattains
+its maximum. Consequently, if, for some k∈R,ct∆νt< kfor allt, then
+there exists an ε>0such that ct∆νt<k−εfor allt.
+Proof. Ifct∆νt≡0, then the result is trivial. Let cbe the upper bound
+ofc(·). Asνis right-continuous, it has at most countably many jumps.
+Then, as νis nondecreasing,/summationtext
+tct∆νt≤c(/summationtext
+t∆νt)≤cνT<∞. Therefore,
+ct∆νtis a summable sequence, and hence has finitely many values gre ater
+than or equal to δ, for any δ >0. Letδ∈]0,ct∆νt] for some t, and so
+{ct∆νt:ct∆νt≥δ}is a finite nonempty set, and therefore has a maximum.BSDES IN GENERAL SPACES 19
+Now suppose ct∆νt< kfor allt. Lett∗be the value at which ct∆νt
+attains its maximum, hence ct∗∆νt∗<k. For any ε<k−ct∗∆νt∗the result
+then holds. /square
+Proof of Theorem 5.1.We consider constructing a sequence of ap-
+proximations in the usual way. For a BSDE with driver Fand terminal
+condition Q, we fix an initial approximation ( Y0,Z(0))∈S2×H2
+M. (Note
+that we denote by Z(n)thenth approximation of the infinite-dimensional
+processZ, to distinguish it from Zi, theith component of Z.) We shall first
+allow the Zcomponent of the solution to converge, then allow the Ycompo-
+nent to do likewise. This two-stage approach is needed due to the difference
+in the Lipschitz coefficients of Fwith respect to YandZ. We shall assume,
+without loss of generality, that the Lipschitz coefficient of F(with respect
+toZ) satisfies c>0 uniformly.
+Step 1: BSDEs where the driver has Yfixed.To construct the Zsolu-
+tions, we first fix some c` adl` ag process ˜Y∈H2
+µ. We wish to define a sequence
+of approximations of solutions to the BSDE with driver F(·,·,˜Y·−,·).
+Foranyapproximation Z(n),wefixthedriver Fn(ω,t)=F(ω,t,˜Yt−,Z(n)
+t).
+Using Lemma 5.4, we obtain a new approximation ( Yn+1,Z(n+1)). We shall
+show that the induced map Z(n)/mapsto→Z(n+1)is a contraction, and hence that
+a unique limit exists.
+Suppose at the nth stage we have two approximations ( Yn,1,Z(n,1)) and
+(Yn,2,Z(n,2)) of the solution of a BSDE with terminal value Qand driver
+F(·,·,˜Y·−,·). We can hence construct new approximations ( Yn+1,1,Z(n+1,1))
+and (Yn+1,2,Z(n+1,2)). We consider the difference
+(δYn+1,δZ(n+1))=(Yn+1,1−Yn+1,2,Z(n+1,1)−Z(n+1,2)).
+Notethat( Yn+1,1,Z(n+1,1))comesfromaBSDEwithdriver F(·,·,˜Y·−,Z(n,1)
+·)
+which does not depend on the solutions ( Yn+1,1,Z(n+1,1)). Hence, for ap-
+propriate functions x·andw·, the differences ( δYn+1,δZ(n+1)) satisfy our
+estimate ( 6), with
+δ2fs=F(ω,s,˜Ys−,Z(n,1)
+s)−F(ω,s,˜Ys−,Z(n,2)
+s)
+andδQ=0, and when defining υandρiin (5) we can take cs=c=0.
+We take the values wt=1,x−1
+t=1
+4c+∆µt, and so we see that ∆ µt−x−1
+t≤0,
+υt=/integraldisplay
+]0,t]xsdµs=/integraldisplay
+]0,t]4c
+1+4c∆µsdµs≤4cµt
+is nondecreasing and bounded (and hence of finite variation) and
+∆υt=4c∆µt
+1+4c∆µt≤1−1
+1+4c<1.20 S. N. COHEN AND R. J. ELLIOTT
+It follows that the integrands in ( 5) are bounded, our estimate ( 6) holds and
+E(˜υs−)(1−∆υs)−1is strictly positive and bounded. Hence
+Z/mapsto→E/bracketleftbigg/summationdisplay
+i/integraldisplay
+]0,T]/bardblZi
+s/bardbl2E(˜υs−)(1−∆υs)−1d/an}bracketle{tMi/an}bracketri}hts/bracketrightbigg
+is an equivalent norm on H2
+M.
+As we can take c=0 in (5), we have the simplification
+dρi
+t=(1−∆υt)−1d/an}bracketle{tMi/an}bracketri}htt=(1−xt∆µt)−1d/an}bracketle{tMi/an}bracketri}htt.
+From (6) we obtain
+E/bracketleftbigg/summationdisplay
+i/integraldisplay
+]t,T]/bardbl(δZ(n+1))i
+s/bardbl2E(˜υs−)(1−∆υs)−1d/an}bracketle{tMi/an}bracketri}htt/bracketrightbigg
+≤/integraldisplay
+]t,T]E[/bardblδ2fs/bardbl2][(x−1
+s−∆µs)(1+w−1
+s)]E(˜υs−)(1−∆υs)−1dµs
+=/integraldisplay
+]t,T]E[/bardblδ2fs/bardbl2]/bracketleftbigg1
+2c/bracketrightbigg
+E(˜υs−)(1−∆υs)−1dµs.
+By the Lipschitz continuity of the original driver, we have
+E[/bardblδ2fs/bardbl2]≤cE[/bardblδZ(n)
+s/bardbl2
+Ms],
+and so, for our chosen values of wtandxt, using inequality ( 4),
+E/bracketleftbigg/summationdisplay
+i/integraldisplay
+]0,T]/bardbl(δZ(n+1))i
+s/bardbl2E(˜υs−)(1−∆υs)−1d/an}bracketle{tMi/an}bracketri}htt/bracketrightbigg
+≤1
+2/integraldisplay
+]0,T]E[/bardblδZ(n)
+s/bardbl2
+Ms]E(˜υs−)(1−∆υs)−1dµs
+≤1
+2E/bracketleftbigg/summationdisplay
+i/integraldisplay
+]0,T]/bardbl(δZ(n))i
+s/bardbl2E(˜υs−)(1−∆υs)−1d/an}bracketle{tMi/an}bracketri}htt/bracketrightbigg
+.
+By completeness, the contraction mapping principle gives t he existence
+of a unique limit Z∈H2
+Msolving the BSDE with driver F(·,·,˜Y·−,·) and
+terminal value Q. (The solution Yprocess can, of course, be found using
+Lemma5.4, fixing the Zprocess at the constructed limit.)
+Step 2: BSDEs with general drivers. We now construct a convergent
+sequence of approximations in Yfor a general driver. Consider the Lipschitz
+bounds of the original driver F. Without loss of generality, we assume that
+c>0 uniformly. As cs∆µs<1,µis nondecreasing and of finite variation,
+andcsis bounded, Lemma 5.5yields a fixed ε>0 such that cs∆µs<1−ε.BSDES IN GENERAL SPACES 21
+Let
+x−1
+t=1
+c(1+2ε−1)+∆µt,
+w−1
+t=ε
+2+ε2
+8−4ε.
+As
+x−1
+t−∆µt=1
+c(1+2ε−1)<1
+c(1+wt),
+it is clear that
+dρi
+t
+d/an}bracketle{tMi/an}bracketri}htt=[1−(x−1
+t−∆µt)(1+wt)c](1−xt∆µt)−1>0,
+soρiis a nonnegative measure for each i.
+For any terminal value Q, consider an approximation Yn∈S2. We can
+thenconstructasolution( Yn+1,Z(n+1))totheBSDEwithdriver Fn(ω,t,z)=
+F(ω,t,Yn
+t−,z), using the above result. Again we shall show that Yn/mapsto→Yn+1
+is a contraction, and hence that a unique limit exists.
+As above, we consider the sequence of differences ( δYn,δZ(n)) from two
+initialapproximations.As Yn+1,1isdefinedusingthedriver Fn=F(ω,t,Yn,1
+t−,
+z), which does not depend on Yn+1,1, we can take cs=0 when defining ρi
+in (5).
+Hence, for our chosen values of xtandwt, we can again easily verify
+that the integrands in ( 5) are bounded, and the resulting υis nonnegative,
+bounded and ∆ υ<1. It follows that E(˜υs) is strictly positive and bounded.
+Considering the difference of any two approximations δYn, by the Lips-
+chitz continuity of the original driver, we have
+E[/bardblδ2fs/bardbl2]≤csE[/bardblδYn
+s−/bardbl2],
+so, asρiis a family of nonnegative measures, our estimate ( 7) gives
+/integraldisplay
+]0,T]E[/bardblδYn+1
+t−/bardbl2]E(˜υt−)dµt
+≤/integraldisplay
+]0,T]µsE[/bardblδ2fs/bardbl2]E(˜υs−)dπs
+≤/integraldisplay
+]0,T]E[/bardblδYn
+s−/bardbl2]E(˜υs−)µscs[(x−1
+s−∆µs)(1+w−1
+s)]
+×(1−xs∆µs)−1dµs.
+By construction we have
+µscs[(x−1
+s−∆µs)(1+w−1
+s)](1−xs∆µs)−1
+=µscsx−1
+s(1+w−1
+s)22 S. N. COHEN AND R. J. ELLIOTT
+=µscs/parenleftbigg1
+c(1+2ε−1)+∆µs/parenrightbigg
+(1+w−1
+s)
+≤µs/parenleftbiggcs
+c(1+2ε−1)+1−ε/parenrightbigg/parenleftbigg
+1+ε
+2+ε2
+8−4ε/parenrightbigg
+≤/parenleftbigg
+1−ε
+2/parenrightbigg/parenleftbigg
+1+ε
+2+ε2
+8−4ε/parenrightbigg
+=1−ε2
+8,
+where the fifth line is because µs≤µT≤1 and
+cs
+c≤1<1+ε
+2=ε
+2(1+2ε−1).
+We then have/integraldisplay
+]0,T]E[/bardblδYn+1
+t−/bardbl2]E(˜υt−)dµt≤/parenleftbigg
+1−ε2
+8/parenrightbigg/integraldisplay
+]0,T]E[/bardblδYn
+s−/bardbl2]E(˜υs−)dµs.
+AsE(˜υs−) is strictly positive and bounded,/integraltext
+]0,T]E[/bardbl·/bardbl2]E(˜υs−)dµsis an
+equivalent norm on H2
+µ. By completeness, the contraction mappingprinciple
+gives the existence of a limit Y∞
+t=limn→∞Yn
+t−, which is unique in H2
+µ. We
+also have the existence of a limit Z, asdρi
+t/d/an}bracketle{tMi/an}bracketri}httis strictly positive, and
+from (7),
+lim
+n→∞E/bracketleftbigg/summationdisplay
+i/integraldisplay
+]0,T]µsE(˜υs−)/bardbl(δZ(n))i
+s/bardbl2dρi
+s/bracketrightbigg
+≤lim
+n→∞/integraldisplay
+]0,T]µsE[/bardblδ2fn
+s/bardbl2]E(˜υs−)dπs=0;
+that is,δZ(n)also converges to zero in H2
+M.
+We take the right limits of a left-continuous version of the p rocessY∞,
+namely
+Yt=E/bracketleftbigg
+Q+/integraldisplay
+]t,T]F(ω,s,Y∞
+s,Zs)dµs/vextendsingle/vextendsingle/vextendsingleFt/bracketrightbigg
+=E/bracketleftbigg
+Q+/integraldisplay
+]t,T]F(ω,s,Ys−,Zs)dµs/vextendsingle/vextendsingle/vextendsingleFt/bracketrightbigg
+.
+By Lemma 5.2,Y∈S2and by Lemma 5.1it is unique in S2. This pair
+(Y,Z) will solve the BSDE with driver Fand terminal value Q. This limit is
+unique for Z∈H2
+M, as can by seen by fixing Yand using our earlier result.
+/square
+Remark 5.2. In discrete time, we have shown in [ 6] that a necessary
+and sufficient condition for the existence of a solution to the discrete BSDEBSDES IN GENERAL SPACES 23
+is thatFis invariant with respect to equivalent Zin/bardbl·/bardblMtnorm, and that
+y→y−F(ω,t,y,z) is a bijection in yfor allz,tand almost all ω. The
+requirement that Fis firmly Lipschitz is sufficient, but not necessary, to
+guarantee that these conditions hold.
+6. Existence of BSDEsolutions: General results. We nowwishto extend
+our above solution to allow µto be any Stieltjes measure, by relaxing the
+condition that µT≤1. In so doing, we shall also weaken slightly the firm
+Lipschitz requirement.
+Lemma 6.1. Letνbe a nonnegative Stieltjes measure with ∆ν <1. Then
+there exists an η>0and a finite sequence {0=t0<t1<···<tB=T}such
+thatν(]ti,ti+1])≤1−ηfor alli.
+Proof. By Lemma 5.5withct≡1, there exists an η >0 with ∆ ν <
+1−η. LettK=Tfor some large K. Define recursively for integers j <K,
+tj=sup{t:νt<νtj+1−1+η}∨0. By right continuity, ν(]tj,tj+1])=νtj+1−
+νtj≤1−η. For any j, it is also easy to show that νtj+2−νtj>1−η. Hence,
+asνTis finite, the sequence tjhas only finitely many nonzero terms. Let
+k= max{j:tk= 0}, letB=K−kand rescale the index of our sequence
+accordingly. We then have a sequence with the desired proper ties./square
+Theorem 6.1. Letµbe any deterministic Stieltjes measure assigning
+positive measure to every open interval. (Note /bardbl·/bardblMis still well defined
+in relation to µ.) LetF:Ω×[0,T]×RK×RK×∞→RKbe a predictable,
+progressively measurable function such that:
+•E[/integraltext
+]0,T]/bardblF(ω,t,0,0)/bardbl2dµt]<+∞.
+•There exists a quadratic firm Lipschitz bound on F, that is, a measurable
+deterministic function ctuniformly bounded by some c∈R, such that, for
+anyyt,y′
+t∈RK,zt,z′
+t∈RK×∞,
+/bardblF(ω,t,yt,zt)−F(ω,t,y′
+t,z′
+t)/bardbl2
+≤ct/bardblyt−y′
+t/bardbl2+c/bardblzt−z′
+t/bardbl2
+Mt, dµ ×dP-a.s.
+and
+ct(∆µt)2<1.
+Note that the variable bound ctneed only apply to the behavior of Fwith
+respect to y.
+A function satisfying these conditions will be called stand ard. Then for any
+Q∈L2(RK;FT), the BSDE ( 3) with driver Fhas a unique solution (Y,Z)∈
+S2×H2
+M. (S2andH2
+Mare defined in Definition 5.4.)
+Proof. Weassume,withoutlossofgenerality, that c≥1.ByLemma 5.5,
+as(µt)2isanondecreasingc` adl` agfunctionoffinitevariationzan dct(∆µt)2<24 S. N. COHEN AND R. J. ELLIOTT
+1 for all t, there exists an ε>0 such that ct(∆µt)2≤1−ε. Let
+νt=/integraldisplay
+]0,t]2(1+ε−1)c
+ε+2(1+ε−1)c∆µtdµt=:/integraldisplay
+]0,t]λ−1
+tdµt.
+Thenν∼µ, and ∆νt=λ−1
+t∆µt<1. Asνtis right continuous, deterministic
+and has no jumps of size equal to or greater than one, by Lemma 6.1there
+exists afinitesequence {t0=0<t1<···<tB=T}suchthat ν(]tj,tj+1])≤1
+for allj.
+We now note that, omitting the ωandtarguments, our BSDE ( 3) can be
+written
+Q=Yt−/integraldisplay
+]t,T]λuF(Yu−,Zu)dνu+∞/summationdisplay
+i=1/integraldisplay
+]t,T]Zi
+udMi
+u, (12)
+which is a BSDE in νwith Lipschitz property
+/bardblλtF(yt,zt)−λtF(y′
+t,z′
+t)/bardbl2≤λ2
+tct/bardblyt−y′
+t/bardbl2+λ2
+tc/bardblzt−z′
+t/bardbl2
+Mt, dν×dP-a.s.
+We write
+¯c=sup
+t{λ2
+tc}≤/parenleftbiggε
+2(1+ε−1)c+µT/parenrightbigg2
+c<∞
+and ¯ct=λ2
+tct. Note that as ε<1,ct/c<1,
+¯ct∆νt=/parenleftbiggε+2(1+ε−1)c∆µt
+2(1+ε−1)c/parenrightbigg2
+ct∆νt
+≤/parenleftbiggε
+2(1+ε−1)c+∆µt/parenrightbigg2
+ct
+≤(1+ε−1)ε2ct
+4(1+ε−1)2c2+(1+ε)ct(∆µt)2
+(13)
+≤ε2
+4+(1+ε)(1−ε)
+≤1−3ε2
+4
+<1.
+Finally, we define the measures
+νk
+t=/integraldisplay
+]0,t∧tk+1]/parenleftbiggη
+νtk+/parenleftbigg
+1−η
+νtk/parenrightbigg
+It>tk/parenrightbigg
+dνt.
+It is easy then to show that νk
+tk+1≤1 for all k. Furthermore, as
+dνk
+dν=η
+νtkIt<tk+It∈[tk,tk+1]>0,BSDES IN GENERAL SPACES 25
+we seeνkassigns positive measure to every interval in ]0 ,tk+1]. Hence, on
+]0,tk+1],νkis a measure of the type considered in Theorem 5.1. Also, ∆ νk
+t≤
+∆νt<1, andνkagrees with νfor all subsets of ] tk,tk+1].
+We now consider the sequence of BSDEs
+Yk+1
+tk+1=Yk
+t−/integraldisplay
+]t,tk+1]λtF(Yk
+u−,Zk
+u)dνk
+u+∞/summationdisplay
+i=1/integraldisplay
+]t,tk+1](Zk)i
+udMi
+u (14)
+withYB
+T=Q. For each k, (14) is a standard BSDE with a driver λtF, which
+has Lipshitz coefficients of ¯ ctand ¯c, and hence is (linearly) firmly Lipschitz
+by (13). Hence, the existence of a unique solution for each kis guaranteed
+by Theorem 5.1.
+Fork=B−1,(14)agreeswith( 12),andhencewiththeoriginalBSDE( 3),
+for allt∈[tk,tk+1]. It follows that the solution YB−1
+tis a solution to our
+original BSDE on the interval [ tB−1,tB]. Similarly, for k=B−2, this ar-
+gument then implies that YB−2
+tis a solution to our original BSDE on the
+interval [ tB−2,tB−1], etc.
+Wenowpiecetogetherthesesolutionstodefine Yt=Yk
+twheret∈]tk,tk+1],
+and similarly for Z. By an inductiveargument, wecan seethat this will solve
+the desired BSDE. Furthermore, this solution will be unique , as the solution
+is unique on each subsection ] tk,tk+1]./square
+Remark 6.1. We note that, even when µT≤1, the conditions of The-
+orem6.1are strictly weaker than those of Theorem 5.1. In this case, the
+jumps of µsatisfy ∆ µ≤1, and it follows that a quadratic firm Lipschitz
+bound is weaker than a linear firm Lipschitz bound.
+Remark 6.2. Clearly if ∆ µ=0, then the requirement that Fis firmly
+Lipschitz degenerates into the classical requirement that Fis uniformly Lip-
+schitz. It is to be expected that many of the generalisations of the Lipschitz
+conditions which are known in the case where our filtration is generated by
+a Brownian motion, that is, to drivers with a stochastic Lips chitz bound, to
+drivers with quadratic growth, to drivers with linear growt h and a mono-
+tonicity condition, etc., will also be possible in this situ ation. There is, how-
+ever, considerable difficulty involved in obtaining these re sults in the simple
+continuous case, and it is to be expected that this difficulty w ill be increased
+by the discontinuities present here.
+Remark 6.3. The situation where Fhas stochastic Lipschitz bounds is
+of particular interest here, as it would then be possible to c onsider replac-
+ingµwith a general predictable process of finitevariation, andc onsequently,
+with any square integrable special semimartingale. Such a g eneral situation
+is arguably as general as can be expected within the context o f stochastic
+integration.26 S. N. COHEN AND R. J. ELLIOTT
+7. A comparison theorem. Given we have now established the existence
+of solutions to these equations, we now wish to prove a compar ison theo-
+rem for them. This is based on the theorem in [ 8], for BSDEs of the type
+of (1).
+Theorem 7.1 (Comparisontheorem). Suppose we have two BSDEs cor-
+responding to standard coefficients and terminal values (F,Q)and(¯F,¯Q).
+Let(Y,Z)and(¯Y,¯Z)be the associated solutions. Suppose that for some s,
+the following conditions hold:
+(i)Q≥¯QP-a.s.;
+(ii)µ×P-a.s. on[s,T]×Ω,
+F(ω,u,¯Yu−,¯Zu)≥¯F(ω,u,¯Yu−,¯Zu);
+(iii)for each j, there exists a measure ˜Pjequivalent to Psuch that the
+jth component of X, as defined for r≥sby
+e∗
+jXr:=−/integraldisplay
+]s,r]e∗
+j[F(ω,u,¯Yu−,Zu)−F(ω,u,¯Yu−,¯Zu)]dµu
++/summationdisplay
+i/integraldisplay
+]s,r]e∗
+j[Zi
+u−¯Zi
+u]dMi
+u
+is a˜Pjsupermartingale on [s,T];
+(iv)if, for all r∈[s,T],
+e∗
+iYr−E˜Pi/bracketleftbigg/integraldisplay
+]r,t]e∗
+iF(ω,u,Yu−,Zu)dµu/vextendsingle/vextendsingle/vextendsingleFr/bracketrightbigg
+≥e∗
+i¯Yr−E˜Pi/bracketleftbigg/integraldisplay
+]r,t]e∗
+iF(ω,u,¯Yu−,Zu)dµu/vextendsingle/vextendsingle/vextendsingleFr/bracketrightbigg
+for alli, thenYr≥¯Yrfor allr∈[s,t]componentwise.
+It is then true that Y≥¯Yon[s,T], except possibly on some evanescent set.
+Proof. We omit the ωandtarguments of Ffor clarity.
+Then, for r∈[s,T]
+Yr−¯Yr−/integraldisplay
+]r,T][F(Yu−,Zu)−¯F(¯Yu−,¯Zu)]dµu
++/summationdisplay
+i/integraldisplay
+]r,T][Zi
+u−¯Zi
+u]dMi
+u (15)
+=Yτ−¯Yτ≥0.BSDES IN GENERAL SPACES 27
+This can be rearranged to give
+Yr−¯Yr−/integraldisplay
+]r,T][F(Yu−,Zu)−F(¯Yu−,Zu)]dµu
+≥/integraldisplay
+]r,T][F(¯Yu−,¯Zu)−¯F(¯Yu−,¯Zu)]dµu
+(16)
++/integraldisplay
+]r,T][F(¯Yu−,Zu)−F(¯Yu−,¯Zu)]dµu
+−/summationdisplay
+i/integraldisplay
+]r,T][Zi
+u−¯Zi
+u]dMi
+u.
+We have that/integraldisplay
+]r,T][F(¯Yu−,¯Zu)−¯F(¯Yu−,¯Zu)]dµu≥0
+by assumption (ii). As e∗
+jXris a˜Pjsupermartingale, we know that the
+process given by
+e∗
+j˜Xr:=e∗
+jXr−E˜Pi[e∗
+jXT|Fr]
+=E˜Pj/bracketleftbigg/integraldisplay
+]r,T]e∗
+j[F(¯Yu−,Zu)−F(¯Yu−,¯Zu)]dµu (17)
+−/summationdisplay
+i/integraldisplay
+]r,T]e∗
+j[Zi
+u−¯Zi
+u]dMi
+u/vextendsingle/vextendsingle/vextendsingleFr/bracketrightbigg
+is also a ˜Pj-supermartingale, with e∗
+j˜XT=0˜Pj-a.s. Hence e∗
+j˜Xr≥0.
+For each j, taking a ˜Pj|Frconditional expectation throughout ( 16) and
+premultiplying by e∗
+jgives
+e∗
+jYr−e∗
+j¯Yr−E˜Pj/bracketleftbigg/integraldisplay
+]r,T]e∗
+j[F(Yu−,Zu)−F(¯Yu−,Zu)]dµu/vextendsingle/vextendsingle/vextendsingleFr/bracketrightbigg
+≥0.
+This must hold for all r∈[s,T] and almost all ω. By assumption (iv), for
+almost all ω, it follows that the comparison Yr≥¯Yrmust hold for all r∈
+[s,T].
+AsY−¯Yis c` adl` ag, we have that Y−Yis indistinguishable from a non-
+negative process and, therefore, the inequality holds up to evanescence. /square
+Remark 7.1. Assumption (iv) is clearly trivial whenever Fdoes not
+depend on Y.
+Remark 7.2. Assumption (iii) is very closely related to the Fundamen-
+tal theorem of asset pricing (see [11]), as it relates an inequality in current
+values to the existence of an equivalent (super-)martingal e measure.28 S. N. COHEN AND R. J. ELLIOTT
+Corollary 7.1.1. If assumption (iv)holds for any Twhenever s≥
+T−εfor some fixed ε, then the comparison also holds.
+Proof. In this case, we can show that the comparison holds on [ T−ε,
+T]. We can then replace TwithT−εthroughout the theorem, replacing Q
+and¯QwithYT−εand¯YT−εinassumption(i). Itis clear that assumptions(ii)
+and (iii) will continue to hold, with the same choice of measu res˜Pi. By the
+statement of the corollary, assumption (iv) will then hold o n the interval
+[T−2ε,T−ε]. By induction, it follows that the comparison holds on [ T−
+nε,T] foralln∈N. Fornsufficiently large, this implies thecomparison holds
+on [s,T] as desired. /square
+Definition 7.1. Astandarddriver Fsuchthatassumptions(iii)and(iv)
+of Theorem 7.1hold on [0 ,T] for allY,¯Y∈S2andZ,¯Z∈H2
+Mwill be called
+balanced.
+Theorem 7.2. In the scalar ( K= 1) case, assumption (iv)of Theo-
+rem7.1holds for any standard F.
+Proof. As we are in the scalar case, we can omit the eifrom the state-
+ment of the assumption. Hence, we wish to show that, given for allr∈[s,T]
+Yr−E˜P/bracketleftbigg/integraldisplay
+]r,T]F(ω,u,Yu−,Zu)dµu/vextendsingle/vextendsingle/vextendsingleFr/bracketrightbigg
+≥¯Yr−E˜P/bracketleftbigg/integraldisplay
+]r,T]F(ω,u,¯Yu−,Zu)dµu/vextendsingle/vextendsingle/vextendsingleFr/bracketrightbigg
+,
+we must have Yr≥¯Yr. For simplicity, let δY:=Y−¯Y.
+It is clear from the problem and the recursivity of BSDE solut ions that we
+can replace Twith any stopping time τ≤Tsuch that δYτ≥0. By applying
+Lemma6.1, we can also assume that sis such that/integraltext
+]s,T]cudµu<1, and
+simply piece together the result for general s.
+Suppose on some nonnull set A∈F,δYu<0 for some u∈[s,T]. AsδYis
+adapted and right continuous, this implies that there are st opping times σ,τ
+such that δYu<0 for all u∈[σ,τ[, ands≤σ < τonA. Without loss of
+generality, let τbe the largest such upper bound. Then, as δYT≥0 and
+τ≤T, it follows that δYτ≥0. Replacing Twithτin the above inequality,
+we know that
+E˜P[Ir∈[σ,τ[|δYr|]
+=E˜P[−Ir∈[σ,τ[δYr]
+≤E˜P/bracketleftbigg
+−Ir∈[σ,τ[/integraldisplay
+]r,τ]F1(ω,u,Y1
+u−,Z1
+u)−F1(ω,u,Y2
+u−,Z1
+u)dµu/bracketrightbiggBSDES IN GENERAL SPACES 29
+≤E˜P/bracketleftbigg
+Ir∈[σ,τ[/integraldisplay
+]r,τ]cu|δYu−|dµu/bracketrightbigg
+≤/integraldisplay
+]r,T]E˜P[Iu∈[σ,τ[|δYu−|]cudµu.
+Taking a left limit in r, we see
+E˜P[Ir∈[σ,τ[|δYr−|]≤/integraldisplay
+[r,T]E˜P[Iu∈[σ,τ[|δYu−|]cudµu.
+By assumption, this quantity is strictly positive. Integra tion on ] t,T] and
+Fubini’s theorem gives, for t>s,
+/integraldisplay
+]t,T]E˜P[Ir∈[σ,τ[|δYr−|]crdµr≤/integraldisplay
+]t,T]/parenleftbigg/integraldisplay
+[r,T]E˜P[Iu∈[σ,τ[|δYu−|]cudµu/parenrightbigg
+crdµr
+=/integraldisplay
+]t,T]/parenleftbigg/integraldisplay
+]t,u]crdµr/parenrightbigg
+E˜P[Iu∈[σ,τ[|δYu−|]cudµu
+</integraldisplay
+]t,T]E˜P[Iu∈[σ,τ[|δYu−|]cudµu,
+where the last line is due to our assumption that/integraltext
+]s,T]ctdµt<1. This con-
+tradicts our assumption that this quantity is strictly posi tive. Therefore, A
+is a null set, that is, δYu≥0 for all u∈[s,t]./square
+Definition 7.2. The comparison between Yand¯Ywill be called strict
+on [s,T] if the conditions of Theorem 7.1hold, and, for any A∈Fssuch
+thatYs=¯YsP-a.s. onA, we have Yu=¯Yuon [s,T]×A, up to evanescence.
+Lemma 7.1. If the comparison is strict on [s,T], then for any A∈Fs
+such that Ys=¯YsP-a.s. onA, it follows that:
+•Q=¯QP-a.s. onA;
+•F(ω,u,¯Yu−,¯Zu)=¯F(ω,u,¯Yu−,¯Zu)µ×P-a.s. on[s,T]×A;
+•Z≡¯ZinH2
+Mon[s,T]×A.
+Proof. We omit the ωandtarguments of Fand¯Ffor clarity. Let ˜X
+be as in ( 17), and let Sbe the process defined by
+e∗
+jSr:=e∗
+jE˜Pi[Q−¯Q|Fr]
+(18)
++e∗
+jE˜Pi/bracketleftbigg/integraldisplay
+]r,T][F(¯Yu−,¯Zu)−¯F(¯Yu−,¯Zu)]dµu/vextendsingle/vextendsingle/vextendsingleFr/bracketrightbigg
++e∗
+j˜Xr.
+Thene∗
+jSis a˜Pj-supermartingale, as the first term is a ˜Pj-martingale, the
+second is nonincreasing in rby assumption (ii) of Theorem 7.1and the third30 S. N. COHEN AND R. J. ELLIOTT
+is a˜Pj-supermartingale by assumption (iii) of Theorem 7.1. Furthermore,
+each of these terms is nonnegative.
+Taking a ˜Pj|Frconditional expectation through ( 3), we have that, for all
+r∈[s,T],
+e∗
+j(Yr−¯Yr)=e∗
+jSr+E˜P/bracketleftbigg/integraldisplay
+]r,t]e∗
+j[F(Yu−,Zu)−F(¯Yu−,Zu)]dµu/vextendsingle/vextendsingle/vextendsingleFr/bracketrightbigg
+. (19)
+IfYr=¯Yron [s,T]×Aup to evanescence, then it is clear from ( 19) that
+Sr=0P-a.s. on [ s,T]×A. Hence, by nonnegativity, each of the terms on
+the right-hand side of ( 18) must be zero. The first two points of the lemma
+immediately follow.
+Considerthe BSDE ( 3) satisfied by ¯Y. AsF(¯Yu−,Zu)=¯F(Yu−,Zu)µ×P-
+a.s. on [s,T]×AandQ=¯QP-a.s. onA, we know that
+¯Yr−/integraldisplay
+]r,T]¯F(¯Yu−,¯Zu)dµu+/summationdisplay
+i/integraldisplay
+]r,T]¯Zi
+udMi
+u=¯Q
+isP-a.s. equal to
+¯Yr−/integraldisplay
+]r,T]F(¯Yu−,¯Zu)dµu+/summationdisplay
+i/integraldisplay
+]r,T]¯Zi
+udMi
+u=Q.
+Hence, in A, (¯Y,¯Z) is a solution at time rto the BSDE defining ( Y,Z).
+As the solution to this BSDE is unique, it follows that, on [ s,T]×A,
+¯Z≡ZinH2
+Mt./square
+Theorem 7.3 (Strict comparison). Consider the scalar ( K=1) case,
+whereFis balanced. Then the comparison is strict on [s,T]for alls.
+Proof. Again, as K=1 we can omit ejfrom all equations, and we omit
+theωandtarguments of Fand¯Ffor clarity. Let Srbe as defined in ( 18),
+and note that Sis a nonnegative ˜P-supermartingale.
+Taking a ˜P|Fsconditional expectation of ( 19) gives
+E˜P[Yr−¯Yr|Fs]=E˜P/bracketleftbigg
+Sr+/integraldisplay
+]s,t][F(Yu−,Zu)−F(¯Yu−,Zu)]dµu/vextendsingle/vextendsingle/vextendsingleFs/bracketrightbigg
+−E˜P/bracketleftbigg/integraldisplay
+]s,r][F(Yu−,Zu)−F(¯Yu−,Zu)]dµu/vextendsingle/vextendsingle/vextendsingleFs/bracketrightbigg
+≤Ss+E˜P/bracketleftbigg/integraldisplay
+]s,t][F(Yu−,Zu)−F(¯Yu−,Zu)]dµu/vextendsingle/vextendsingle/vextendsingleFs/bracketrightbigg
+(20)
++/integraldisplay
+]s,r]E˜P[|F(Yu−,Zu)−F(¯Yu−,Zu)||Fs]duBSDES IN GENERAL SPACES 31
+≤Ss+E˜P/bracketleftbigg/integraldisplay
+]s,t][F(Yu−,Zu)−F(¯Yu−,Zu)]dµu/vextendsingle/vextendsingle/vextendsingleFs/bracketrightbigg
++c/integraldisplay
+]s,r]E˜P[|Yu−−¯Yu−||Fs]dµu.
+We know from ( 19) and the assumption Ys−¯Ys=0 onAthat
+IASs+IAE˜P/bracketleftbigg/integraldisplay
+]s,t][F(Yu−,Zu)−F(¯Yu−,Zu)]dµu/vextendsingle/vextendsingle/vextendsingleFs/bracketrightbigg
+=IA(Ys−¯Ys)=0,
+and so, as Y−¯Yis nonnegative by Theorem 7.1, premultiplication of ( 20)
+byIAand then taking an expectation gives
+E˜P[IA(Yr−¯Yr)]≤c/integraldisplay
+]s,r]E˜P[IA(Yu−−¯Yu−)]dµu.
+As all quantities are nonnegative, taking a limit from below yields
+E˜P[IA(Yr−−¯Yr−)]≤c/integraldisplay
+]s,r]E˜P[IA(Yu−−¯Yu−)]dµu,
+and an application of (the forward version of) Gr¨ onwall’s l emma implies
+E˜P[IA(Yr−¯Yr)]≤0.
+By nonnegativity, it follows that Yr=¯Yr,˜P-a.s. onA. Again, as Y−¯Yis
+c` adl` ag, this shows that Y=¯Yon [s,t]×A, up to evanescence. /square
+Corollary7.3.1. If theith component of F(ω,t,y,z)depends only on the
+ith component of y(as well as on ω,tandz), then the comparison is strict.
+Proof. As theith component of Fdepends only on the ith component
+ofy, we can repeat the construction of Theorem 7.3in each component. The
+result follows. /square
+Remark7.3. Inthescalarcase,withasimpleBrownianfiltration( M1=
+W,Mi=0 fori≥2) anddµ=dt, we can use Girsanov’s transformation to
+constructthemeasurerequiredforassumption(iii) ofTheo rem7.1.Wewrite
+Λt:=1+/integraldisplay
+]0,t]Λu−F(ω,u,¯Yu−,Zu)−F(ω,u,¯Yu−,¯Zu)
+Zu−¯ZudWu,
+thend˜P/dP=ΛT. It is then easy to verify that Xis a martingale. In this
+case, using Theorem 7.2we can see that any Lipschitz continuous Fis
+balanced.
+8. Nonlinear expectations. We are now in a position to explicitly con-
+struct nonlinear expectations in a general probability spa ce. We shall not
+here consider the more general theory of nonlinear evaluati ons. An ap-
+proach without these restrictions can be seen in [ 8]. These operators, dis-32 S. N. COHEN AND R. J. ELLIOTT
+cussed in [ 19], are closely related to the theory of dynamic risk measures ,
+as in [1,3,21] and others, as each concave nonlinear expectation E(·|Ft)
+corresponds to a dynamic convex risk measure through the rel ationship
+ρt(Q)=−E(Q|Ft).
+A further discussion of this relationship can be found in [ 21].
+Definition 8.1. A family of operators
+E(·|Ft):L2(FT)→L2(Ft),0≤t≤T,
+is called an Ft-consistent nonlinear expectation ifE(·|Ft) satisfies the fol-
+lowing properties:
+(1) IfQ≥¯QP-a.s. componentwise
+E(Q|Ft)≥E(¯Q|Ft),P-a.s. componentwise
+with equality iff Q=¯QP-a.s.
+(2) ForQ∈L2(Ft),E(Q|Ft)=QP-a.s.
+(3) For any s≤t,
+E(E(Q|Ft)|Fs)=E(Q|Fs),P-a.s.
+(4) For any A∈Ft,
+IAE(Q|Ft)=E(IAQ|Ft),P-a.s.
+Theorem 8.1. LetFbe a balanced driver which does not depend on Y
+(i.e.,ct≡0) and satisfies F(ω,t,y,0)=0µ×P-a.s. Then the operator de-
+fined by
+E(Q|Ft)=Yt,
+whereYis the solution to a BSDE ( 3) with driver F, is a nonlinear expec-
+tation.
+Proof. (1) AsFis balanced, this result follows directly from the com-
+parison theorem (Theorem 7.1). AsFdoes not depend on Y, the strict
+comparison will also hold, by Corollary 7.3.1.
+(2) Consider the BSDE ( 3) on [t,T]
+Ys−/integraldisplay
+]s,T]F(ω,u,Yu−,Zu)dµu+/summationdisplay
+i/integraldisplay
+]s,T]Zi
+udMi
+u=Q.
+This has a solution Ys=Q,Zs=0. AsQ∈L2(Ft), this solution is adapted
+and, by Theorem 5.1, unique. Therefore E(Q|Ft)=Yt=Qas desired.
+(3) By definition the BSDE with terminal condition Qat time Thas
+solution Ytat timet. Simple manipulation of the BSDE ( 3) at time sshows
+thatYsis also the time ssolution to the BSDE with terminal condition Yt
+at timet. Hence, by property 2, Yssolves both the BSDE with terminal
+condition Yt=E(Q|Ft) and the BSDE with terminal condition Q.BSDES IN GENERAL SPACES 33
+(4) Consider the BSDE with driver Fand terminal condition Q. Multi-
+plying by IA, asIAF(ω,t,y,z) =F(ω,t,IAy,IAz), we see that ( IAY,IAZ)
+is the solution to the BSDE with driver Fand terminal condition IAQ, as
+desired. /square
+Remark 8.1. It is known in discrete time [ 6], and under some condi-
+tions in continuous time [ 9], that BSDEs describe all nonlinear expectations,
+subject to some boundedness conditions. It is likely that a s imilar result will
+hold in this setting. However, obtaining such a result is bey ond the scope of
+this paper.
+9. Conclusions. We have constructed BSDEs in a general filtered proba-
+bility space, using only basic properties of the filtration. We have presented
+conditions for the existence of unique solutions to these eq uations, and seen
+how these are related to the conditions in both the classical setting, and the
+discrete time setting. We have given a comparison theorem fo r these solu-
+tions,whichallowstheconstructionofnonlinearexpectat ions in these spaces.
+These results are significantly more general than those prev iously avail-
+able,astheymakeveryfewassumptionsontheunderlyingpro babilityspace.
+A consequence of this is that a possibly infinite-dimensiona l martingale rep-
+resentation theorem is required. In full generality, they a lso make no as-
+sumptions regarding the relationship of the integrator of t he driver and the
+quadratic variations of the martingale terms. At the same ti me, this general
+setting provides an approach unifying the theory of BSDEs in discrete and
+continuous time.
+Acknowledgment. Robert Elliott wishes to thank the Australian Re-
+search Council for support.
+REFERENCES
+[1]Artzner, P. ,Delbaen, F. ,Eber, J.-M. ,Heath, D. andKu, H.(2007). Coherent
+multiperiod risk adjusted values and Bellman’s principle. Ann. Oper. Res. 152
+5–22.MR2303124
+[2]Barles, G. ,Buckdahn, R. andPardoux, E. (1997). Backward stochastic differen-
+tial equations and integral-partial differential equation s.Stochastics Stochastics
+Rep.6057–83.MR1436432
+[3]Barrieu, P. andEl Karoui, N. (2004). Optimal derivatives design under dynamic
+risk measures. In Mathematics of Finance .Contemporary Mathematics 35113–
+25. Amer. Math. Soc., Providence, RI. MR2076287
+[4]Cohen, S. N. andElliott, R. J. (2008). Solutions of backward stochastic differen-
+tial equations on Markov chains. Commun. Stoch. Anal. 2251–262. MR2446692
+[5]Cohen, S. N. andElliott, R. J. (2010). Comparisons for backward stochastic dif-
+ferential equations on Markov chains and related no-arbitr age conditions. Ann.
+Appl. Probab. 20267–311. MR2582649
+[6]Cohen, S. N. andElliott, R. J. (2010). A general theory of finite state back-
+ward stochastic difference equations. Stochastic Process. Appl. 120442–466.
+MR259436634 S. N. COHEN AND R. J. ELLIOTT
+[7]Cohen, S. N. andElliott, R. J. (2011). Backward stochastic difference equations
+and nearly time-consistent nonlinear expectations. SIAM J. Control Optim. 49
+125–139. MR2765659
+[8]Cohen, S. N. ,Elliott, R. J. andPearce, C. E. M. (2010). A general comparison
+theorem for backward stochastic differential equations. Adv. in Appl. Probab. 42
+878–898. MR2779563
+[9]Coquet, F. ,Hu, Y.,M´emin, J. andPeng, S. (2002). Filtration-consistent nonlinear
+expectations and related g-expectations. Probab. Theory Related Fields 1231–
+27.MR1906435
+[10]Davis, M. H. A. andVaraiya, P. (1974). The multiplicity of an increasing family
+ofσ-fields.Ann. Probab. 2958–963. MR0370754
+[11]Delbaen, F. andSchachermayer, W. (2006).The Mathematics of Arbitrage .
+Springer, Berlin. MR2200584
+[12]El Karoui, N. andHuang, S. J. (1997). Ageneral resultofexistenceanduniqueness
+ofbackwardstochastic differentialequations.In Backward Stochastic Differential
+Equations (Paris, 1995–1996) .Pitman Research Notes in Mathematics Series
+36427–36. Longman, Harlow. MR1752673
+[13]Elliott, R. J. (1982).Stochastic Calculus and Applications .Applications of Math-
+ematics (New York) 18. Springer, New York. MR0678919
+[14]Hassani, M. andOuknine, Y. (2002). On a general result for backward stochastic
+differential equations. Stochastics Stochastics Rep. 73219–240. MR1932160
+[15]Hu, Y.,Ma, J.,Peng, S. andYao, S.(2008). Representation theorems for quadratic
+F-consistent nonlinear expectations. Stochastic Process. Appl. 1181518–1551.
+MR2442369
+[16]Jacod, J. andShiryaev, A. N. (2003).Limit Theorems for Stochastic Processes , 2nd
+ed.Grundlehren der Mathematischen Wissenschaften [Fundamen tal Principles
+of Mathematical Sciences] 288. Springer, Berlin. MR1943877
+[17]Malamud, S. (2007). The Davis–Varaiya multiplicity and branching numb ers of
+a filtration. Available at http://ssrn.com/abstract=970541 .
+[18]Peng, S. (2004). Nonlinear expectations, nonlinear evaluations an d risk measures. In
+Stochastic Methods in Finance .Lecture Notes in Math. 1856165–253. Springer,
+Berlin.MR2113723
+[19]Peng, S. (2005). Dynamically consistent nonlinear evaluations and expectations.
+Preprint No. 2004-1, Institute of Mathematics, Shandong Un iv. Available at
+http://arxiv.org/abs/math/0501415 .
+[20]Revuz, D. andYor, M. (1999).Continuous Martingales and Brownian Motion , 3rd
+ed.Grundlehren der Mathematischen Wissenschaften [Fundamen tal Principles
+of Mathematical Sciences] 293. Springer, Berlin. MR1725357
+[21]Rosazza Gianin, E. (2006). Risk measures via g-expectations. Insurance Math.
+Econom. 3919–34.MR2241848
+[22]Royer, M. (2006). Backward stochastic differential equations with ju mps andrelated
+non-linear expectations. Stochastic Process. Appl. 1161358–1376. MR2260739
+Mathematical Institute
+University of Oxford
+OX1 3LB, Oxford
+United Kingdom
+E-mail: samuel.cohen@maths.ox.ac.uk
+URL:http://people.maths.ox.ac.uk/cohens/School of Mathematical Sciences
+University of Adelaide
+Adelaide, South Australia, 5005
+E-mail: relliott@ucalgary.ca
+URL:http://people.ucalgary.ca/˜relliott/
\ No newline at end of file
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+IEEE International Advance Computing Conference (IACC’09) Patiala, IndiaTutoring System for Dance Learning
+Rajkumar Kannan
+Department of Computer Science
+Bishop Heber College 
+Trichy 620017, TN, INDIA
+rajkumarkannan@yahoo.co.inFrederic Andres
+National Institute of Informatics
+University of Advanced Studies
+Tokyo, JAPAN
+andres@nii.ac.jpBalakrishnan Ramadoss
+Department of Computer Applns
+National Institute of Tech
+Trichy 620015, TN, INDIA
+brama@nitt.edu
+Abstract - Recent advances in hardware sophistication related to 
+graphics display, audio and video devices made available a large 
+number of multimedia and hypermedia applications. These 
+multimedia applications need to store and retrieve the different 
+forms of media like text, hypertext, graphics, still images, 
+animations, audio and video. Dance is one of the important 
+cultural forms of a nation and dance video is one such 
+multimedia types. Archiving and retrieving the required 
+semantics from these dance media collections is a crucial and 
+demanding multimedia application. This paper summarizes the 
+difference dance video archival techniques and systems.
+Keywords: Multimedia, Culture Media, Metadata archival and 
+retrieval systems, MPEG-7, XML.
+I. INTRODUCTION
+Recent advances in hardware sophistication related to 
+graphics display, audio and video devices made available a 
+large number of multimedia and hypermedia applications. In a 
+scenario, one may want to digitize and store one’s favorite 
+dance presentations and later on view, say for instance, all solo 
+dance pieces in which a particular dancerX was involved [16].
+These multimedia applications need to store and retrieve 
+the different forms of media like text, hypertext, graphics, still 
+images, animations, audio and video. Multimedia database can 
+be viewed as a controlled collection of multimedia data items. 
+Multimedia database system, for instance dance video 
+database, would then provide the support for multimedia data 
+types, besides facilities for the creation, authoring, storage, 
+access, query and control of the multimedia database.
+Dance generally refers to human movements either used as 
+a form of expression or presented in a social, spiritual or 
+performance setting. Dance is one of the important cultural 
+forms of a nation. The other cultural forms are art, music, and 
+sculptures. Preserving this cultural heritage is very important 
+for the future generations of dance community. This paper 
+surveys the various dance video archival and retrieval systems. 
+The organization of the paper is as follows: Section 2 deals 
+with the dance video archival methods. The various dance 
+analysis and retrieval systems are discussed in section 3. 
+Section 4 deals with the challenges of Dance Analysis and 
+Retrieval Systems (DARS). Finally section 5 concludes the 
+paper.II. DANCE VIDEO ARCHIVAL METHODS
+A dance can be archived using human memories, dance 
+notations and digital mediums. In ancient times dancers passed 
+the knowledge of the dance verbally to the following 
+generations. However, such knowledge was limited to the 
+memory of the dancers and hence many dance productions had 
+been lost due to this fact unfortunately.
+Encyclopedia Britannica defines a dance notation as 
+the recording of dance movement through the use of written 
+symbols .  Since 16thcentury, dance notations are used to 
+archive choreographies and to resurrect the dance by dancers 
+and choreographers. A dance notation, similar to musical notes, 
+is a symbolic form of representing the movements of the 
+dancers using various graphical symbols such as lines, circles, 
+rectangles, squares, bars etc. 
+The primary use of a dance notation is the documentation, 
+analysis and reconstruction of choreography. Many different 
+forms of dance notations such as stick figure have been created, 
+but the two main systems used in Western culture are:
+Labanotation
+Benesh notation
+Labanotation [1] is a standardized system for analyzing and 
+recording any human movement. The original inventor is 
+Rudolf von Laban (1879-1958), an important figure in 
+European modern dance. He published this notation first in 
+1928 as Kinetographie in the first issue of Schrifttanz . Several 
+people continued the development of the notation. In 
+Labanotation, it is possible to record every kind of human 
+motion. Labanotation is not connected to a singular, specific 
+style of dance. The basis is the natural human motion, and 
+every change from this natural human motion (e.g. turned-out 
+legs) has to be specifically written down in the notation as 
+shown in Figs. 1-3.
+Benesh [1] movement notation invented by Benesh is 
+particularly prominent in ballet dance and was designed 
+to write the whereabouts of a dancer on the stage, the 
+direction the dancer facing, the positions of the limbs, and 
+the details of the head, hand and foot. Since dancing has 
+movement, Benesh also notates the movement by recording the 
+paths traced by the moving limbs. Figure 4 shows an example 
+of Benesh notation for the human avatar shown in Fig. 3.IEEE International Advance Computing Conference (IACC’09) Patiala, IndiaFig.1-4. Laban and Banesh notations
+A fundamental problem with Labanotation and Benesh 
+notations is that very few people understand these notations 
+and are experts in this field. Here the annotation process 
+involves the expert to visualize the representation of the 
+notation and to translate the movement to the dance learners. 
+Further, these annotations exist in paper form only and no 
+visual form exists.
+III. DANCE VIDEO RETRIEVAL SYSTEMS
+Recording mediums [2] such as VHS and DVD can provide 
+a quicker and cheaper means of acquiring and storing a dance. 
+However, to capture the complete stage requires a camera to be 
+either set back, making the dancers very small or include close-
+ups resulting parts of the dancers or choreography being 
+missed or obscured. Professionally made recordings for 
+televisions, video and DVD can capture many great 
+productions with world-class dancers; however, the 2D camera 
+viewpoint is subject to the director’s interpretation. Also, 
+searching these collections of digital data to learn the dance 
+movements is tedious because of the huge volume of data.
+Current research in performing arts such as dance can be 
+broadly divided into two categories:
+Dance composition and Visualization
+Dance Analysis and Retrieval
+A. Dance composition and Visualization
+Some researches have been conducted concerning the 
+composition of dance using the notations especially 
+Labanotation and Benesh [3, 4, 5, 6]. Hachimura’s system [7] 
+uses markers which are placed on the body parts of the dancers 
+and records 3D motion data. Hattori et al [8] developed key-
+frame based dance animation software. In his approach certain 
+key poses in the dance are identified and are coded in 
+Labanotation.
+Several graphical editors have been developed such as 
+LabanEditor [7], LabanWriter for Mac OS, Calaban [9] for 
+Windows, NUNTIUS [10], LED and LINTER [11] for UNIX 
+systems. Further, graphical editors for Benesh notation are 
+available such as MacBenesh and Benesh Notation Editor [12]. 
+MacBenesh is a Macintosh application that lets us create high 
+quality single dancer Benesh movement notation scores that can be saved in a document. Benesh Notation Editor is a 
+Windows based application for writing benesh movement 
+notation scores. It resembles a word processor thereby the 
+Benesh scores can be saved in a file. Several commercial 
+softwares such as LifeForms and DanceForms [13] are also 
+available for composing the dance interactively. These tools 
+use virtual reality features to model the movements of the 
+dancers.
+B. Dance Analysis and Retrieval Systems
+Dance analysis and retrieval systems perform the semantic 
+interpretation of the dance movements in the context of culture, 
+action, gestures and emotions. A system that directly calculates 
+the similarities among the body motion data is described in [7] 
+(Hachimura, 2006). It considers the issue of how a similarity of 
+identical body motion should be defined. 
+Kalajdziski and Davcev [14] developed a system for the 
+Macedonian dance videos to annotate the shots with keywords 
+from the controlled vocabulary. The system consists of 
+modules for segmentation, annotation, 3D dancer animation 
+generation and Laban score generation.
+Forouzan et al [15] designed a multimedia information 
+system for the Macedonian dance annotation and the analysis 
+of the dance features. Macedonian dance exists in the form of 
+videos, images, audios and written commentaries. In order to 
+process all these dance forms, the system includes several tools 
+to extract the low level features automatically from these 
+mediums. The visual tool extracts the visual features such as 
+color, texture, shape, sketch, objects and their spatial 
+relationships. The 3D motion tool extracts the motion 
+histogram of the dancers using 3D Vicon motion capture 
+system. This system captures data from the cameras and 
+generates 3D motion files. Motion data are generated by 
+interpolating the cameras’ traces of translucent markers placed 
+on the body parts of the dancers. Audio tool extracts audio 
+features like pitch, loudness and rhythm from the audio 
+streams. These low level multimedia features are automatically 
+annotated by the dance experts through the annotation tool.
+COSMOS-7 [16] is a multimedia annotation and retrieval 
+system based on MPEG-7 MDS. It includes a multimedia 
+model that integrates both low level and high level features of 
+the video by using multimedia frames. COSMOS-7 provides a 
+conversion tool that translates M-frames into the corresponding 
+MPEG-7 descriptions. 
+Rajkumar et al [17,18,19,20] presented a semantic approach 
+to evolve a dance video retrieval system for the Indian and 
+generic dance videos. A central contribution of their research is 
+the development of semantic models, annotation and authoring 
+frameworks and query processing engines for the Indian and 
+generic dance videos to process spatio-temporal queries. Their 
+MPEG-7 based system allows the annotation experts to 
+describe the metadata of the dance presentations, automatically 
+generates the MPEG-7 instances and processes generic dance 
+video queries. The query processor handles the dance video 
+queries with the tree embedding technique. Also, XML based 
+authoring and retrieval framework facilitates semantic 
+annotation of the dance media descriptions, semi-automatic 
+XML instance generation and query processing. The query 
+IEEE International Advance Computing Conference (IACC’09) Patiala, Indiaprocessor allows the user to submit the knowledge based dance 
+video queries and handles them using tree embedding 
+technique that is supported with the domain specific ontology.
+IV. CHALLENGES OF DANCE VIDEO RETRIEVAL SYSTEMS
+The dance analysis and retrieval systems are best regarded 
+as a platform that demonstrates the feasibility of choreography 
+design, dance learning, self-paced dance learning, culture 
+learning and dance training. However, there are a number of 
+problems yet to be solved. The strongest qualitative drawback 
+of these systems is the manual annotation of the dance videos, 
+thus possibly reducing the effectiveness of DARS.
+Dance retrieval models have been designed with a view of 
+the classical and folk dance types to describe the semantics. 
+However, there are other types of dance videos such as festival 
+dance videos, street dance videos etc, for which some of the 
+current semantics may not be applicable and the new semantics 
+have to be introduced.
+Semiotic data such as mood are expressed in all dance 
+retrieval systems using the controlled vocabulary of text. 
+Another important drawback of these systems is the absence of 
+sound. Nevertheless, sound would enhance DARS’s 
+effectiveness involving the semiotic queries.
+V. CONCLUSION
+Dance generally refers to human movements either used as 
+a form of expression or presented in a social, spiritual or 
+performance setting. Dance video is one of the important types 
+of multimedia resource. Since dance is the ancient cultural 
+heritage, which is to be preserved for the future generations, 
+archiving the dance presentations of various dance types 
+becomes a necessary task. Further, adding a search, retrieval 
+and faceted browsing facilities to these dance media collections 
+by way of tools certainly provide the present and future dance 
+trainers and learners a better eLearning environment.
+REFERENCES
+[1] Ann Hutchinson, G. (1984). Dance Notation: Process of recording 
+movemen t. London: Dance Books.
+[2] Chitra, D., Manthe, A., Nack, F., Rutledge, L., Sikora, T & Zettl, H. 
+(2002). Media Semantics: Who needs it and why?. Proceedings of ACM 
+Multimedia , 580-583[3] Don Herbison & Evans. (1988). Dance, Video, Notation and Computers. 
+Leonardo , 21(1), 45-50.
+[4] George, P. (1990). Computers and Dance: A bibliography, Leonardo , 
+23(1), 87-90
+[5] Calfert, T.W & Chapman, J. (1978). Notation of movement with 
+computer assistance. Proceedings of ACM annual conference , 731-736
+[6] Hatol, J & Kumar, V. (2005). Semantic representation and interaction of 
+dance objects. Proceedings of LORNET conference , Poster
+[7] Hachimura, K. (2006). Digital archiving of dancing. Review of the 
+National Center for Digitization , 8, 51-66
+[8] Hattori, M & Takamori, T. (2002). The description of human movement 
+in computer based on movement score.  Proceedings of 41stSICE , 2370-
+2371
+[9] Calaban. (2002). Homepage: http://www.bham.ac.uk/calaban/frame.htm
+[10] Bimas, U., Simon, W & Peter, R. (1992). NUNTIUS: A computer 
+system for the interactive composition and analysis of music and dance. 
+Leonardo , 25(1), 59-68
+[11] LED & LINTER. (2006). An X-Windows Editor / Interpreter for 
+Labanotation. http://wwwstaff.it.uts.edu.au/ don/pubs/led.html
+[12] MacBenesh. (2004). Behesh notation editor for Apple Macintosh. 
+http://members.rogers.com /dancewrite/macbenesh/macbenesh.htm
+[13] Ilene, F. (2000). Documentation Technology for the 21st Century. 
+Proceedings of World Dance Academic Conference . 137-142.
+[14] Kalajdziski, S & Davcev, D. (2004). Augmented reality system interface 
+for dance analysis and presentation based on MPEG-7. Proceedings of 
+IASTED conference on Visualization, Imaging, and Image Processing , 
+725-730
+[15] Forouzan, G., Pegge, V & Park, Y.C. (2004). A multimedia information 
+repository for cross cultural dance studies. Multimedia Tools and 
+Applications , 24, 89-103
+[16] Athanasios, C., Gkoritsas & Marios, C.A. (2005). COSMOS-7: A video 
+content modeling framework for MPEG-7. Proceedings of IEEE 
+MultiMediaModeling , 123-130
+[17] Rajkumar, K., Ramadoss, B & Ilango, K. (2004). An approach to 
+Conceptual Modeling and Extracting Expressive Semantics of Dance 
+Videos. Lecture Notes in Computer Science (LNCS) , 3356, 1-10. 
+Springer Verlag, Heidelberg, Germany
+[18] Rajkumar, K., & Ramadoss, B (2006). Modeling and Querying the 
+Expressive Semantics of Dance Videos. International Journal of 
+Information and Knowledge Management (JIKM) , 5(3), 193-210. World 
+Scientific, Singapore
+[19] K.Rajkumar, B.Ramadoss (2007). Semi-automated annotation and 
+retrieval of dance media objects. Cybernetics and Systems, Taylor and 
+Francis, 38(4):349-379, PA, USA.
+[20] K.Rajkumar, B.Ramadoss (2008). Modeling the dance video semantics 
+using Regular Tree Automata. Fundamenta Informaticae, published 
+under EATCS, IOS Press, The Netherlands. 86(1-2):175-189
\ No newline at end of file
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+Semantic Modeling and Retrieval of Dance Video Annotations
+BALAKRISHNAN RAMADOSS1
+KANNAN RAJKUMAR2
+Department of Computer Applications
+National Institute of Technology
+Tiruchirappalli 620015, TN, India
+1brama@nitt.edu
+2rajkumarkannan@yahoo.co.in
+Abstract . Dance video is one of the important types of narrative videos with semantic rich content. This paper 
+proposes a new meta model, Dance Video Content Model ( DVCM ) to represent the expressive semantics of the 
+dance videos at multiple granularity levels. The DVCM  is designed based on the concepts such as video, shot, 
+segment, event and object, which are the components of MPEG-7 MDS. This paper introduces a new relationship 
+type called Temporal Semantic Relationship  to infer the semantic relationships between the dance video objects. 
+Inverted file based index is created to reduce the search time of the dance queries. The effectiveness of containment 
+queries using precision and recall is depicted.
+Keywords : Dance Video Annotations, Effectiveness Metrics, Metamodeling, Temporal Semantic Relationships.
+(Received May 12, 2006 / Accepted January 03, 2007)
+1. Introduction
+Dances are archived in many ways. In ancient times, 
+dancers passed the knowledge of the dance verbally to 
+the following generations [13]. However, such 
+knowledge was limited to the memory of the dancers 
+and many dance productions have been lost due to this 
+fact.
+Traditionally, dance notations are used to archive the 
+choreographies and to resurrect the dance production by 
+the dancers and the choreographers. The two most 
+common dance notations that succeed are Labanotation
+[2] and Banesh [9], which are developed by Rodolph 
+Laban and Rudolf Banesh respectively. These two 
+notational systems provide an abstract view of a three 
+dimensional dancer in space. The fundamental problem 
+with these systems is that very few people understand 
+these notations and are experts in this field. The notation 
+process involves the expert to visualize the 
+representation of the notation and to translate the 
+notation into the corresponding movement to the 
+dancers.
+Recording mediums such as CD, VCD, DVD and VHS, 
+provide a quicker and easier way of acquiring and 
+storing the dance performance. However, searching these collections of digital data is tedious [4] because of 
+the huge volume of data.
+Modeling the different semantics of the dance, archived 
+in any of the above three mediums, is a large step. Apart 
+from modeling, the semantic querying facilities to these 
+archives are needed for the dancers and the dance 
+learners. So a system with these semantic modeling and 
+querying facilities will aid the present and future 
+generation of dance teachers and students with better 
+dance training and learning methods. With such system, 
+dance students can understand and learn the dance 
+pieces, emotions to be expressed while performing the 
+dance pieces and also culture-specific features. Hence, 
+this kind of dance video information system is highly 
+desirable and more useful to the nation, as a cultural 
+heritage .
+This paper proposes a new dance video data model, 
+temporal semantic relationships, inverted file based 
+index structure and a query processor. The dance video 
+data model abstracts the different dance video 
+semantics, offers dance learners and choreographers 
+semantic querying capabilities and captures the video 
+structure as shot, scene and compound scene. The video 
+structure characterizes the dance steps of Pallavi , Anu 
+Pallavi, Saranam  and Chores [13] of a song. Some of the temporal operators are follows , repeats , observe , 
+perform_same  and perform_different  steps.
+The rest of the paper is organized as follows. Section 2 
+presents some related works on dance video data models. 
+The basics of dance video semantics are explained in 
+Section 3. The dance video data model for the dance 
+annotations is introduced in Section 4. Section 5 
+discusses the modeling of the temporal relationships. 
+Section 6 depicts the class design using DVCM . Section 
+7 and section 8 present the algorithms for containment 
+query processing and experimental results respectively. 
+Finally, Section 9 concludes the paper.
+2. Related Work
+This section briefly reviews some of the existing video 
+modeling and annotation proposals and discusses their 
+applicability to dance videos.
+The Extended DISIMA [10] model expresses the events 
+and concepts based on the spatiotemporal relationships 
+among the salient objects. However, in dance video 
+systems, the model has to consider not only salient 
+objects, but all objects such as instruments, costume, 
+background etc.
+A graph based video model [11] records the appearance 
+and disappearance of salient objects and stores the 
+trajectory details efficiently without any redundancy. 
+This model is more suitable for sports and traffic 
+surveillance videos, in which the system keeps track of 
+ball movements and human movements. But it is not 
+suitable for dance videos where we do not track the 
+movements of the dancers or any other objects.
+Forouzan et al [6] provide a system to extract some of 
+the low level features such as color, silhouette of the 
+dancers and sound. Also, the spatiotemporal features of 
+the dancers in Macedonian dance are manually 
+annotated for further query processing. The main focus 
+of their system is to study the historical changes of a 
+dance that undergoes over centuries .
+COSMOS7 [3] models the objects along with a set of 
+events in which the objects participate, as well as events 
+along with a set of objects and temporal relationships 
+between the objects. This model represents the events at 
+a higher level only like speak, play, listen  and not at the 
+level of actions, gestures and movements. Also, 
+emotions expressed by the body parts of the dancers, are 
+not considered.3. Dance Video Semantics
+Dance videos contain several low-level and high-level 
+features and provide ample scope for the efficient 
+semantic retrieval and dance video mining. Low level 
+features of the dance videos, such as color, texture and 
+sound [12], do not unfortunately provide any 
+meaningful cue to the dance video model. But on the 
+other hand, dance videos provide very interesting high 
+level features that can be captured in the video data 
+model. These features include:
+accompanying song
+dance types such as solo, duet, trio  or group dance
+dynamics such as slow, graceful  and staccato
+movements 
+accompanying music
+These features reveal the structural characteristics of the 
+dance videos.
+3.1 Song Granularity and Song Types
+Accompanying song of a dance is an important 
+structural feature of the dance video and can be used as 
+a cue for grouping the dance steps together. Hence, 
+dance is viewed as a collection of steps that are 
+portrayed to the audience in temporal order, according 
+to the lyrics of the song. 
+Basically, a song [13] is composed of parts such as 
+Pallavi (PA), Anu Pallavi (AP), Saranam (SA) and 
+Chores (CH). Pallavi  is the introduction of the song and 
+is usually one or two lines in the lyrics, occurring in the 
+beginning of the song. Anu Pallavi  is the additional 
+introduction to the song. Saranam  is a group of lines, 
+usually 2 to 8 lines in the lyrics representing a stanza. 
+The Chores  is also few lines with some interesting 
+pieces. Generally, song components are group of 
+rhyming lines forming a unit in the song. However, 
+some of these components may be optional as described 
+in song types shortly. Every song component groups a 
+set of dance steps corresponding to the lines of lyrics of 
+the song. Therefore, the song granularity corresponding 
+to the dance steps is Song, Song Components and Dance 
+Steps. 
+The different types of song can be enumerated as a set 
+of regular expressions [8]. The song used in dance video 
+retrieval system can be any of the six types.
+(PA AP  SA+)
+(PA SA+)
+(SA+)
+(PA AP  SA  (CH  SA)+)(PA SA  (CH  SA)+)
+(SA  (CH  SA)+)
+The first regular expression type (PA AP SA+) specifies 
+that a song can have dance steps for one Pallavi , one 
+Anu Pallavi  and one or more Saranam s. Here, the 
+operator + denotes one or more occurrences of SA. 
+Similarly, the regular expression for the remaining song 
+types can also be illustrated.
+3.2 Dance Video Granularity
+Each shot contains one dance step either 
+Bharathanatyam or a casual step for each dancer. That 
+is, a shot can contain several dancers performing dance 
+steps. Group of lines of steps are known as either, 
+Pallavi , Anu Pallavi , Saranam  or Chores . The scene is 
+an abstraction of any of the four song components. 
+Similarly, compound scene is the abstraction of a 
+collection of song components, according to the song 
+type defined as a regular expression representing a song. 
+Finally, dance video is a collection of songs of a dance 
+presentation or a movie.
+4. Dance Video Content Model
+The aim of multimedia content modeling is to build a 
+high level model of the multimedia domain based on the 
+content description requirements. The model applies 
+notions form both Entity-Relationship and UML 
+methodologies in modeling the dance video content. 
+The proposed model integrates the structural 
+characteristics of the video such as video, compound 
+scene, scene and shots, abstracting the corresponding 
+granularity of the song. Moreover, it considers the 
+different dance semantics such as dancers, costume, 
+dance steps, background etc, apart from the spatio-
+temporal relationships among the dancers. This section 
+presents the DVCM  that efficiently describes the dance 
+steps of the songs.
+The dance video is composed of one or more shots. 
+Each shot represents a dance step. This model does not 
+consider frames to represent the dance steps. Each step 
+has starting position of the dancers, intermediate 
+movements of body parts and the final position. So a 
+dance step cannot be abstracted in a frame. Further, the 
+dance video can be modeled as segments. For instance, 
+the set of dance steps constituting a song can be called 
+as segments. Segments can be decomposed into sub 
+segments. Therefore, a shot is a segment at the bottom 
+level.A segment depicts one or more events. Here dance steps 
+are events that occur in the dance video. Objects are part 
+of the events and dancers are objects who participate in 
+the events. Graphical representation of DVCM  is 
+depicted in Fig.1. The entity classes and relationships of 
+the model are formally defined as below:
+4.1 Objects
+Object can be distinctly identified by studying the 
+entities in a domain. A class is used to describe the set 
+of objects which have the same characteristics. 
+Examples of structural objects are video, shot and 
+segment and semantic objects such as object and event.
+Fig.1.Graphical representation of DVCM.
+4.2 Attributes
+Attributes are descriptive properties of objects. There 
+are three types of attributes.
+Identifiers : which uniquely identify objects like 
+dancer ID.
+Descriptors : which describe objects such as age and 
+sex of the dancers.
+Temporal Attributes : whose values change over 
+time. There are two types of time elements [14]: 
+Transaction time  and Existence time . For instance, 
+the attributes background sceneries and location are 
+temporal attributes with existence time. 
+4.3 Relationships
+An association among one or more objects is described 
+as follows:
+Generalization (G) : It partitions entity classes into 
+mutually exclusive sub classes. For example, in the 
+dance video domain a shot is a segment.
+Aggregation(AG)  and Composition(C) : 
+Aggregation is called as part-of  relationship. For 
+instance, in dance video domain dancer is part of 
+the dance step event.Association (A) : Association is a casual semantic 
+relationship that relates objects, without exhibiting 
+existence dependency. For example, in dance video 
+domain, a dancer performs a dance step. Here, the 
+association is performs  which relates the semantic 
+objects dancer and dance step.
+Spatiotemporal relationships : It is a relationship 
+along the spatiotemporal dimension in which one 
+object relates to another, spatially, temporally and 
+spatiotemporally. For instance, dancerA is in front 
+of dancerB facing each other and they move 
+forward. Here, the spatio-temporal relationship are
+in front of  and move  (i.e. meet).
+5. Temporal Semantic Relationships
+The basic 13 temporal relationships for video modeling
+are defined by Allen [1]. Some more temporal 
+relationships can be exclusively defined for dance 
+videos, which are quite useful for query retrieval. These 
+relationships correspond to the relationships among the 
+objects that need not be stored, but can be inferred from 
+the attribute values. Temporal relationships between the 
+dancers, who are performing dance steps, capture 
+temporal characteristics of the dancers. Some of the 
+temporal characteristics of the dancers are:
+a dancer following a dance step of another dancer 
+immediately.
+a dancer repeating a dance step of another some 
+time later in the same song or another song.
+a dancer observing a dance step of another dancer.
+a dancer performing the same or different step of 
+another dancer simultaneously.
+Temporal relationships become the basis for any spatial 
+relationship. For any two dancers who appear in a shot, 
+the different semantic relationships abstracting the 
+temporal characteristics can be defined as follows:
+Definition 5.1 . Let I be the time interval. Given two 
+time intervals I 1 and I 2, I1 is the sub interval  of I 2 , I1 « I2
+, if and only if I 2.Ts≤  I 1.Ts and I 2.Te≥  I 1.Te, where T s
+and T e denote the start and end times of the time 
+interval. 
+Definition 5.2 . Let D i and D j be the dancers performing 
+dance steps S 1 and S 2 in shots SH m and SH n of the scene 
+SC k respectively. Let I 1 = [Ti
+s, Ti
+e] and I 2 = [Tj
+s, Tj
+e] be 
+the time intervals associated with D i and D j respectively. 
+Then D jfollows a step  of D i, denoted as, D j »→ D i iff
+Ti
+e = Tj
+s
+SH m.Di.S1 = SH n.Dj.S2I1 « SC k.I and
+I2 « SC k.I
+Definition 5.3 . Let D i and D j be the dancers performing 
+dance steps S 1 and S 2 in the shots in shots SH m and SH n
+of the scene SC k respectively. Let I 1 = [Ti
+s, Ti
+e] and I 2 = 
+[Tj
+s, Tj
+e] be the time intervals associated with D i and D j
+respectively. Then D jrepeats  a step of D i, denoted as, D j
+«→ D i iff
+Ti
+e < Tj
+s
+SH m.Di.S1 = SH n.Dj.S2
+I1 « SC k.I and
+I2 « SC k.I
+Note: Follows  and Repeats  relationships are very
+interesting in a dance. Here, one dancer challenges the 
+other with a dance step and the second dancer performs 
+that dance step correctly.
+Definition 5.4.  Let D i and D j be the dancers performing 
+dance steps in scene SC k, associated with the intervals I 1
+= [Ti
+s, Ti
+e] and I 2 = [Tj
+s, Tj
+e] respectively. Let SHS be 
+the sequence of shots (in partial order) containing D i
+and D j . Then D jfollows steps  of D i, denoted as, D j»»→ 
+Di iff
+Ti
+e= Tj
+s
+Temporal relation D j «→ D i holds for every shot s, 
+where s ε SHS
+I1 « SC k.I and
+I2 « SC k.I
+Definition 5.5.  Djrepeats steps  of D i, denoted as, D j
+««→ D i. It is similar to Definition 5.4, except the 
+condition Ti
+e= Tj
+s is replaced with Ti
+e< Tj
+s.
+Definition 5.6.  Let D i and D j be the dancers performing 
+dance steps S 1 and S 2 in the shot SH k in the interval I 1 = 
+[Ts, T e]. Then D jperforms same step  together with D i, 
+denoted as, D j≡ D i iff 
+SH k.Di.S1 = SH k.Dj.S2 
+I1 « SH k.I 
+Definition 5.7.  Let D i and D j be the dancers performing 
+dance steps S 1 and S 2 in the shot SH k in interval I 1 = [T s, 
+Te]. Then D jperforms different step  together with D i, 
+denoted as D j≈ D i iff 
+SH k.Di.S1≠ SH k.Dj.S2 
+I1 « SH k.I 
+Definition 5.8.  Let D i and D j be the dancers performing 
+dance steps S 1 and S 2 in the shot SC k in the interval I 1 = 
+[Ts, T e]. Let SHS be the sequence of shots (in partial order) containing D i and D j. Then D jperforms same 
+steps together  with D i, denoted as, D j≡≡ D i iff 
+Dj≡ D i holds for each shot s, where s ε SHS
+I1 « SC k.I 
+Definition 5.9.  Let D i and D j be the dancers performing 
+dance steps S 1 and S 2 in the shot SC k in the interval I 1 = 
+[Ts, T e]. Let SHS be the sequence of shots(in partial 
+order) containing D i and D j. Then D jperforms different 
+steps together  with D i, denoted as, D j≈≈ D i iff 
+Dj≈ D i holds for each shot s, where s ε SHS
+I1 « SC k.I 
+Definition 5.10.  Let D i and D j be the dancers in a shot 
+SH of the scene SC k in the interval I 1 = [T s, Te]. Let S be 
+the dance step performed by D j. Then D iobserves  step 
+of D j, denoted as, D j∆ D i iff 
+SH.D i.S = NIL and I 1 « SC k.I
+6. Class Design based of DVCM
+The following set of classes is designed in order to 
+represent the different object types. They include Video, 
+Compound Scene, Scene, Shot, Song, Musician, 
+Background, Costume, Dancer, Dance Step, 
+SpatialTriplet, Peytham(PY), Adavu(AD), Ashamyutha 
+Hashtam(ASHA), Shamyutha Hashtam(SHA) and 
+Casual Step(CS). PY is a Bharathanatyam step that is 
+rendered with head, eye, eyebrow, nose, lips, neck, 
+chest and sides only. AD is a Bharathanatyam step that 
+is performed with legs and hands. ASHA is performed 
+using fingers of either hand, while SHA with fingers of 
+both hands simultaneously.
+The class Video has attributes ID, life span, date of 
+recording, video description and compound scene 
+sequence IDs. The class Compound Scene includes 
+attributes such as ID, videoID, songID, description and 
+sequence of sceneIDs. The class Song has attributes 
+such as ID, name of the song, lyrics and musicianID. 
+The Musician class has ID, name, address, sex and 
+phone attributes.
+The class Scene includes the following attributes: ID, 
+life span, compound sceneID, backgroundID, 
+costumeID and sequence of shotIDs. Also, it has 
+mapping oc: O i→CO which maps each dancer to a set 
+of costumes. The class Background has ID, background 
+name, location and description attributes. The class 
+Costume has ID, name and description attributes.
+The class Shot has the following attributes: ID, life 
+span, sceneID, set of dancerIDs, dance eventID, mapping oa:O i→S i which maps each dancer to a dance 
+step and description. The class Dance event has the 
+attributes step type, posture, reflexion, spatial triplet list 
+and a mapping oi:O i→IN i which maps each dancer to an 
+instrument. The class Instrument has ID, instrument 
+name and description attributes. Posture is an 
+enumeration of front, left, right, back side. Similarly, 
+Reflexion is also an enumeration with values sad, 
+happy, delighted and excited.  Spatial Triple is the list 
+with values dancer1, dancer2 and spatial relation 
+between them.
+The class Peytham has ID, peytham name and 
+movement attributes. Similarly the class Ashamyutha 
+Hashtam has ID, name and movement attributes. The 
+classes Adavu and Shamyutha Hashtam have attributes:  
+ID, name and movement of two body parts.
+7. Dance Video Query Processing
+Semantic queries that are related to dance videos can be 
+classified into four types as below:
+Containment Queries: In containment queries, a dance 
+learner is interested to obtain video shots in which a 
+dancer performs a step, a particular body part is used, a 
+specific mood is reflected or an instrument is kept by a 
+dancer and so on. For instance, a dance learner may 
+submit a query, Give me all video shots in which dancer 
+A performs step S , in order to learn the specific step S.
+Temporal Queries: Temporal queries involve temporal 
+relationships among the dancers. Temporal operators 
+can be selected from Allen’s algebra and our temporal 
+semantic relationships, in order to facilitate the search 
+and learning. For example a query, Give me all video 
+shots in which a step S done by dancer A is repeated by 
+dancer B . From this, one can conclude that this dance 
+step may represent a competition dance , or is a repeated 
+step which is an important one in a dance. 
+Spatial Queries: In these queries, choreographers and 
+learners can express directional relationships between 
+the dancers who perform same or different steps. These 
+queries may be useful when the users want to know the 
+spatial arrangement between two dancers or group of 
+dancers. For instance, a query that the user may submit 
+to the dance video retrieval system could be: Find all 
+video shots in which dancer A is to the left of dancer B
+performing steps . 
+Spatiotemporal Queries: In these types of queries, the 
+dance learners are concerned with analyzing the spatiotemporal relationships between the dancers. 
+Consider a query, Give me all video shots in which 
+dancer A observes dancer B who performs step S and B
+is to the left of A . The temporal operator observes
+indicates dancer A, is simply watching the step of 
+dancer B without performing any step. 
+The choreographers, teachers and students can query the 
+system to find shots, scenes or compound scenes which 
+contain the specified dancers(D), body parts(AD), 
+posture(PO), mood(RX), costume(CO), instrument(IN), 
+background(BG) or dance step(ST) types such as PY, 
+AD, ASHA, SHA or CS without any regard to the 
+temporal or spatial relations among the dancers. Dance 
+video queries can be composed using conjunctive and 
+disjunctive operators. Containment query components 
+are classified into three types as shown below:
+Type1: D, AG, PO, RX, IN, BG, CO
+Type2: PY, AD, ASHA, SHA, CS
+Type3: DPY, DAD, DASHA, DSHA, DCS, 
+DPO, DRX
+Due to the page limitations, this paper considers only 
+containment query types.
+Algorithm1 : Type1 and Type2 cqueries
+Input:  query qc, video granularity vg, Song song
+Output:  IDs of shots, scenes or compound scenes
+1.for each ID of qc
+2. for each shot in song
+3.  ifqc exists in object record of shot then
+4.   cshots i = cshots i U shot.ID
+5.   endif
+6. endfor
+7. ifvg = scene or vg = compound scene then
+8.  for each shot ID in cshots
+9.   get the scene ID from object record of shot ID
+10.    cscenes i = cscenes i U scene ID
+11.  endfor
+12. endif
+13. ifvg = compound scene then
+14.  for each shot ID in cscenes
+15.    get compound scene ID from object record of 
+scene ID
+16.    ccscenes i = ccscenes i U compound scene ID
+17.  endfor
+18. endif
+19. endfor
+20.resultset i is cshots, cscenes or ccscenes 
+depending on vg21. return resultset
+Algorithm2 : Type3 cqueries
+Input:  dancer D, dance step S, Song song
+Output:  IDs of shots, scenes or compound scenes
+1.find shot IDs of D, say SHOTS1  using 
+Algorithm1
+2.find shot IDs of S, say SHOTS2  using 
+Algorithm1
+3.cshots = SHOTS1 ∩ SHOTS2
+4.filter shot IDs from cshots which do not have D
+performing S
+5.process  vg of scene and compound scene as in 
+Algorithm1
+6.return resultset
+Fig.2.(a) & (b). Sequential Search Algorithms.
+7.1 Sequential Search
+Sequential search retrieves the required video 
+granularity by searching through the shot objects for the 
+specified video query. For example, the query “find all 
+video shots in which dancer a performs step s” returns 
+all shot IDs where s is choreographed by a in a song. 
+Algorithms for Sequential search of the containment 
+query are depicted in Fig.2.
+7.2 Inverted File Index Search
+The inverted file index [7] speeds up the computation of 
+the containment queries, by eliminating the sequential 
+scan of the object records. The inverted file is the 
+collection of inverted lists where each list defines an 
+entry or key and a set of values for the entry. The 
+following inverted files are created to facilitate 
+processing of the containment queries.
+Dancer inverted file (DA_IF)
+Agent inverted file (AG_IF)
+Posture inverted file (PO_IF)
+Reflexion inverted file (RX_IF)
+Instrument inverted file (IN_IF)
+Background inverted file (BG_IF)
+Costume inverted file (CO_IF)
+Step inverted file (ST_IF)Fig.3. Dancer Inverted File.
+Algorithm3 . Type-1  and Type-2 containment queries
+Input : query component qc, video granularity vg
+Output : IDs of shots, scenes or compound scenes 
+1.for each ID of qc
+2. obtain a set of step IDs, say STEPS , from the 
+respective inverted file if ID belongs to Type-
+1      query or directly from the object record 
+of Step in video data model if ID belongs to 
+Type-2  query
+3.   for each step ID in STEPS
+4. obtain the corresponding shot IDs of the 
+step ID, say SHOTS , from the step inverted file ST_IF
+5.     cshots i = cshots iU SHOTS
+6.   endfor
+7. process vg of scene and compound scene as in      
+Algorithm1
+8.endfor
+9.  return resultset.
+Algorithm4:  Type3 queries
+Input:  dancer D, dance step S, Song song
+Output:  IDs of shots, scenes or compound scenes
+1.find shot IDs of D, say SHOTS1  using 
+Algorithm3
+2.find shot IDs of S, say SHOTS2  using 
+Algorithm3
+3.cshots = SHOTS1 ∩ SHOTS2
+4.filter shot IDs from cshots which do not have D
+performing S
+5.process  vg of scene and compound scene as in 
+Algorithm1
+6.return resultset
+Fig.4.(a) & (b). Inverted Index Search.
+Fig.5. Screen shot of the Query Processor.For instance, the structure of DA_IF is depicted in 
+Fig.3. In the figure, each dancer is linked to an inverted 
+list which contains a set of dance steps performed by the 
+dancer in a song. Each node in the inverted list contains 
+ID of the dance step. The inverted files AG_IF, PO_IF, 
+RX_IF and IN_IF share the same structure as DA_IF. 
+The BG_IF stores scene IDs for each background. That 
+is, a set of steps in which the background appears is 
+recorded in BG_IF. Similarly, CO_IF describes a set of 
+scenes in which the specific costume is used by the 
+dancers. The ST_IF describes a set of shot IDs for each 
+dance step that belong to a song. Algorithms 3 and 4 
+(shown in Fig.4) present the query processing steps with 
+inverted file index.
+8. Performance
+The experiments were run on Dell workstation with 
+512 MB DRAM, 80 GB HD under Windows XP 
+environment. Fig.5 depicts the screen shot of the query 
+processor which has been implemented using J2SE1.5
+and JMF. The dance video users can select the query 
+components such as dance steps and the query 
+processor retrieves either shot, scene or compound 
+scene based on the video granularity. Fig.6 shows the 
+running time (RTIME) with shot groups 10, 100, 1000 
+and 10000 shots. As the number of shots increase, the 
+Inverted file index greatly reduces the RTIME of the 
+queries, compared to the sequential search. Due to lack 
+of sufficiently annotated dance data, synthetic data 
+have been generated representing the contents of the 
+shots.
+Further, the effectiveness of the containment queries 
+(see Appendix-A) are studied using Precision and 
+Recall [5] and are depicted in Table1. In this table, the 
+average precision and recall works out to be 100% and 
+83.3% respectively. For the query Q5, the precision is 
+less. The reason for this is that the system also retrieves 
+the shots that match “joy”, besides the romantic shots 
+and hence the precision is less. Similarly, for the query 
+Q2, the system retrieves the shots matching both left 
+eye and right eye, thus making the precision less. For 
+the remaining queries, the system achieves 100% 
+because the retrieved shots are exactly same as the 
+required shots.
+Fig.6. RTIME of Containment Queries.
+Query Recall Precision
+Q1 100 100
+Q2 100 50
+Q3 100 100
+Q4 100 100
+Q5 100 66.66
+Table1. Effectiveness of Containment Queries.
+9. Conclusion
+Dance videos possess rich semantics compared to other 
+video types such as sports, news and movie. Dance 
+video is multimedia comprising of textual, audio and 
+video materials. This paper has proposed a dance video 
+content model (DVCM) to incorporate the different 
+dance video semantics such as dancer, steps, mood etc 
+and video granularity. Set of temporal semantic 
+relationships are introduced for spatial and temporal 
+queries. The inverted file index speeds up the 
+computation of the dance queries elegantly. The 
+performance of containment queries under sequential 
+and inverted file index search is also discussed.
+Acknowledgments
+This work is fully funded by the University Grants 
+Commission (UGC), Government of India Grant:
+XTFTNBD065.References
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+Appendix-A
+Q1:Show Anitha performing Samathristy
+Q2:Show the dance steps of left eye performed by Lisa
+Q3:Show Anitha performing dance steps denoting 
+flowers
+Q4:Show all dance steps of a song, Kanna Varuvaaya
+Q5:Show all romantic dance steps
\ No newline at end of file
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+International Journal "Information Technologies and Knowledge" Vol.1 / 2007 
+ 
+ 137
+[Calejo2002] Calejo, M. et al. 2002. Web Application Make r - A model based approach to web database development. 6th 
+International Conference on Enterprise Information Systems.  
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+ [Schmidt2000] Schmidt, D., et al., 2000. POSA2: Pattern -Oriented Software Architecture: Patterns for Concurrent and 
+Networked Objects. 
+Authors' Information 
+Hector Garcia - Contrated Professor. Technical University of Madrid. E.U. Informática. Ctra. de Valencia Km. 7. 
+E28031 Madrid. e-mail: hgarcia@eui.upm.es 
+Carlos del Cuvillo - Associate Professor. Technical University of Ma drid. E.U. Informática. Ctra. de Valencia Km. 
+7. E28031 Madrid. e-mail: ccuvillo@eui.upm.es 
+Diego Perez - . Consultant. Technical University of Madrid. E.U. Informática. Ctra. de Valencia Km. 7. E28031 
+Madrid. e-mail: dperez@tdi.eui.upm.es 
+Borja Lazaro - Technician, Group leader. Technical University of Madrid. E.U. Informática. Ctra. de Valencia Km. 
+7. E28031 Madrid. e-mail: blazaro@eui.upm.es 
+ 
+  
+MODELING AND ANNOTATING THE EXPR ESSIVE SEMANTICS OF DANCE VIDEOS 
+Balakrishnan Ramadoss,  Kannan Rajkumar 
+Abstract: Dance videos are interesting and semantics-intensive.  At the same time, they are the complex type of 
+videos compared to all other types such as sports, news and movie videos. In fact, dance video is the one which 
+is less explored by the researchers across the globe. Dance videos exhibit rich semantics such as macro features and micro features and can be classified into several types. Hence, the conceptual modeling of the expressive semantics of the dance videos is very crucial and complex. This paper presents a generic Dance Video 
+Semantics Model (DVSM) in order to represent the semantics of the dance videos at different granularity levels, 
+identified by the components of the accompanying song. Th is model incorporates both syntactic and semantic 
+features of the videos and introduces a new entity type ca lled, Agent, to specify the micro features of the dance 
+videos. The instantiations of the model are expressed as graphs. The model is implemented as a tool using J2SE 
+and JMF to annotate the macro and micro features of th e dance videos. Finally examples and evaluation results 
+are provided to depict the effectiveness of the proposed dance video model.  
+ACM Classification : J.5 [Arts and Humanities]: Performing Arts; H.5.4 [Information Interface and Presentation]: 
+Hypertext/Hypermedia. 
+Keywords:  Agents, Dance videos, Macro features, Micro features, Video annotation, Video semantics . 
+1. Introduction 
+Dance data is essentially multimedia by nature consisting of visual, audio and textual materials. Dance video 
+modeling and mining depends significantly on our ability to recognize the releva nt information in each of these 
+data streams. One of the most challenging problems here is  the modeling of the dance video semantics such that 
+the relevant semantics are consistent with the perception of the real world. 
+The classical and folk dances are the real cultural wealth of a nation. In India, the most important classical dances 
+are Bharathanatyam, Kadak, Kadakali, Kuchipudi  and Manipuri  (Saraswathi, 1994). Traditionally, dance learners 
+perform dance steps by observing the natural language verbal descriptions and by emulating the steps of the 
+choreographers. Therefore, the properly annotated dance vi deos will help the present and future generations to 
+learn dance themselves and minimize the physical presence of the choreographers.  International Journal "Information Technologies and Knowledge" Vol.1 / 2007 
+ 
+ 138 
+Notations are used everywhere and are most important for the dancers to communicate the ideas to the learners. 
+They use graphical symbols such as vertical lines, horizontal lines, dots, triangle, rectangle etc, to denote body parts’ actions on paper. Labanotation (Hutchinson, 1954) and Banesh (Ann, 1984) have been the frontier notational systems to record the dance movements or danc e steps. Many western dances are using Labanotation 
+to describe dance steps. However, many  choreographers still follow the traditi onal way of training their students 
+using natural language descriptions, because of the very few recording experts  and inherent complexity of reading and understanding the symbols. Moreover, all Indian dances have unique structure and no common notational structure exists, apart from wire-frame sti ck diagram representing a dance step. Due to lack of 
+notations, it is evident that the complexity of modeling the dance video semantics is relatively high. 
+Since the dance steps were archived in paper form and many classical dances lack notations, this kind of archival 
+of dance becomes impossible even today. With the advances in digital technologies (Dorai, 2002) nowadays, magnetic tapes and disks record dance presentations efficiently. But, searching a dance sequence from these 
+collections is not efficient, because of the huge volume of video data. The solution is to build a dance video 
+information system so as to preserve and query the different dance semantics like, dance steps, beyond the 
+spatio-temporal characteristics of the dancers and their dance steps. 
+The dance video database system requires an efficient video data model to abstract the semantics of the dance 
+videos. To be more precise, the dance video data model should: 
+• abstract the different dance video semantics such as dancers, dance steps, agents (i.e., body parts of the 
+dancers), posture, speed of dance steps, mood, music,  beat, instrument used, background sceneries and the 
+costume. More importantly, the spatio-temporal charac teristics of the dancers must be incorporated in the 
+model; 
+• capture the structure of the dance videos such as s hot, scene and compound scene abstracting the different 
+components of the accompanying song. 
+This paper addresses two related issues: modeling the semantics of the dance videos and annotating the dance 
+steps from the real dance videos. The dance video sema ntics model represents the different types of dance 
+semantics in a simple, efficient and flexible way. The annotation tool manually annotates the semantics (as macro and micro features) for further query processing and vide o mining. The main contributions of this paper are as 
+follows: 
+• We propose a generic video data model to describe the dance steps as video events; 
+• We introduce the Actor entity in order to store the event  specific roles of a video object. That is, an actor 
+entity describes the context dependent role of the video object; 
+• We introduce the Agent entity to describe the context dependent action that is associated with the actor 
+entity; 
+• We develop a tool that implements the dance video model in order to annotate the different dance semantics. 
+The rest of the paper is organized as follows: Secti on 2 presents some related works on video data models. 
+Section 3 describes the different semantics of the dance videos. The DVSM for the dance video is introduced in 
+Section 4. Section 5 illustrates the implementation of the DVSM using Java technologies. The proposed video 
+model is evaluated against a set of conceptual and semantic quality factors in Section 6. Finally, Section 7 
+concludes the paper. 
+2. Related Work 
+Video data modeling is an import ant component of the dance video database system, as it abstracts the 
+underlying semantics of the dance. This section briefly reviews some of the existing video modeling proposals 
+and discusses the applicability to dance videos. 
+Colombo(1999) classifies the content-based search as semantic level search (e.g. objects, events and 
+relationships) and low level search (e.g. color, texture and motion). They call the corresponding systems as first and second generation visual information systems. Seve ral key word based techniques are applied to semantic 
+search models, such as OVID (Oomoto, 1993), AVIS (Adali, 1996), Layered model (Koh, 1999) and Schema less 
+semantic model (Al Safadi, 2000). Se cond generation systems pr ovide automatic tools to extract low level 
+features and subsequently semantic search is perform ed. Some of these systems include, but not limited to 
+QBIC, Virage, VisualSEEK, VideoQ, VIOLONE, MARS, PhotoBook, ViBE, and PictHunter (Smeulders, 2000; International Journal "Information Technologies and Knowledge" Vol.1 / 2007 
+ 
+ 139
+Antani, 2002). However, thes e systems are either based on textual annota tions or purely low level features, but 
+not incorporating the other one. 
+In (Shu, 2000), Augmented Transition Network based sem antic data model is proposed. The ATN models the 
+video based on scenes, shots and key frames using st rings as a sequence of characters. The string 
+representation is used to model the spatial and temporal re lationships of each object (moving and static) in a shot 
+of the traffic video. Since the semantic features of da nce videos are complex, the entire scene or shot cannot be 
+abstracted in a single string. 
+Translucent markers, reflector costumes, special sensors and specialized cameras are used to capture and track 
+human body parts’ movements in some applications such as aerobics, traffic surveillance, sign language, news 
+and sports videos (Vendrig, 2002). In order to record and analyze the dance steps of a dancer, based on this 
+technique requires a special translucent markers or reflector costumes for the dancers. However, dancers do not prefer to use these costumes as these costumes hide the dancer’s make-ups and costumes. Moreover, these markers and reflectors prohibit the realism, affect dancer’s  comfort as well as reduce the focus or concentration of 
+the dancers. Hence, automatic analysis of dance steps to  extract the semantics of the dance steps is very 
+complex. 
+Recently, the extended DISIMA(Lei Chen, 2003) model expresses events and concepts based on spatio-temporal 
+relationships among salient objects. However, the required dance video database model has to consider not only salient objects, but all objects such as in struments, costumes, background and so on. 
+Event based syntactic-semantic vid eo model (we call it as, ESSVM) (Ahme t, 2004) proposes Actor entity to 
+specify the context dependent role of a player in soccer sports. This model represents the events such as free kick, goal, penalty etc, in which player assumes different  roles such as scorer, assist-maker etc. But in dance 
+videos, the contextual information of the dance events has to be described at multiple levels like actor and agent, 
+rather than at a single gr anularity of actor entity. 
+COSMOS7 (Athanasios, 2005) models objects along with a set of events in which they participate, events along 
+with a set of objects and temporal relationships between the objects. This model does not model the temporal 
+relationships between events and the contextual roles. It models the events at a higher level only like speak, play, listen etc, whereas dance video model needs more detailed level of event representation such as agents, their action, speed of action, associated song and so on. 
+3. The Semantics of Dance Videos 
+Generally, dance information is dominated by visual content such as steps, posture and costume and the accompanying audio such as song and music. Hence, dance videos are rich in semantics and provide ample scope for the efficient semantic retrieval and dance video mining. This section illustrates the song that accompanies the dance performance, the different dance video types and the features of the dance videos in 
+detail. 
+ 
+3.1. Song Granularity 
+Dance video contains several dance steps representing each song. In the case of classical dance, it is simply a 
+collection of songs choreographed on the stage or theatre with a single start-stop (Cheng, 2003) camera 
+operation. On the other hand, in a movie dance, a movie contains several songs and for each song dance steps are choreographed by the dancers. A song in a movie may be recorded with multiple start-stop camera 
+operations. For instance, an Indian movie will normally contain about five to six songs. Here, each dance step 
+may represent a step from any of the Indian dances or  a new step innovated by the choreographer. Further, 
+it includes the presentation aesthetics such as mood, feelings, emotion and so on. 
+Song is composed of four parts: Introduction, Additi onal Introduction, Chores and Stanzas (Web of Indian 
+Classical Dances, 2003). Depending on the type of a song, Additional Introduction and Chores may be optional. 
+Each part has few lines of lyrics for which dance steps are choreographed. In the dance video hierarchy, a shot represents a dance step, a scene represents dance steps of any of the song parts which are recorded in the 
+same location and a video clip represents dance steps of a song. Our DVSM will represent the semantics of one 
+dance step as a dance event. Dance step is the unit of analysis in this paper. 
+ International Journal "Information Technologies and Knowledge" Vol.1 / 2007 
+ 
+ 140 
+3.2. Features of Dance Videos 
+There are two types of dance video features- macro feat ures and micro features and are annotated by the human 
+annotators at macro and micro levels (Forouszan, 2004) acco rdingly. Macro features are general properties of the 
+dance that are event independent and micro features are th e properties of the dance st ep. That is, micro features 
+are spatio-temporal characteristics of the dancers while rendering the dance steps. Micro features can also be 
+called as event dependent features. 
+Macro features(or Bib liographic features):  Date of recording, time of recording, geographic origin of the dance, 
+geographic origin of the dancers, sex, age, number of da ncers in a dance, type of performance venue (such as 
+theatre, open-air, etc), type of the accompanying song, type of accompaniment, type of musical instrument used 
+and types of dance videos. The different dance videos ar e movie dance video, theatre dance video, folk dance 
+video, classical dance vi deo, street dance vid eo and festival dance video. These macro features are independent 
+of the dance steps and are common to all dances. 
+Micro features (Dance step specific features):  Spatio-temporal features classify dance movement behavior 
+which include: movement of one dancer in relation to an other dancer, movement of a specific body part (such as 
+eye, leg etc. Refer Appendix-A for a complete list) of a dancer in relation to another part of the body, movement 
+path of the dance (such as circular, linear, serpentine and zigzag), distance between body parts of a dancer while 
+performing a dance step and distance between dancers. 
+Hence, the proposed video model has to characterize a set of macro features and micro features that exist in the 
+dance videos. 
+4. The Dance Video Semantics Model 
+Conceptual model abstracts the dance video data into a structure for later querying its contents and mining some 
+interesting patterns. For efficient conceptual modeling, one should know how choreographers demonstrate a dance to the learners. They are the experts in describ ing the rhythmic steps to the audience. This section 
+presents a generic dance video model that efficiently describes the dance steps. Every dance step is called as an 
+event and the model represents dance events by a set of mi cro features. The model is generic in the sense that it 
+is applicable to any type of dance videos. DVSM is an extension of ER (Chen, 1976) with object oriented 
+features. The goal of the model is to describe a dance step as an event. 
+The main entities of the model are events, objects that participate in these events, actor entities that describe 
+contextual roles of objects in the events, agent entities that represent the action of the actor and concept entities 
+that model the cognitive and affe ctive features of the dancers. 
+For example, consider a dancer object with name, age, address and all other event independent attributes. The 
+same dancer assumes different roles throughout the danc e video. That is, he becomes hero in one dance step, 
+lover in another dance step and so on. Roles are defined as attributes of Actors. Some other examples of actors are heroine, leader, follower, group dancer, friend etc. These context specific object roles form separate actor 
+entities, which all refer to the same dancer object. Although one would say that actor performs the action in an 
+event, finer granularity is necessary as far as dance vi deos are concerned. Therefore, contextual data of the 
+dancers have to be described in two levels. A particular dance step is characterized by the actions of the agents who belong to the actors. Spatio-temporal characteristics are part of the actors as well as agents. Hence, they are 
+described as attributes of actors and agents. 
+Apart from the dancer object, DVSM may also represent the ordinary objects with a standard UML class diagram. 
+Some of them are: speed of the action of an agent, in strument used and the posture of the actor. The graph 
+meta-schema of the DVSM is depicted in Figure1.  
+The graphical notations used in DVSM are described as follow s. A rectangle node refers to an entity or an object. 
+A round rectangle node refers to a concept. A dotted rectangle node denotes an actor entity. A thick rectangle 
+node shows an agent object. Event entity is modeled with a trapezoid. Attributes of entities and relationships are 
+represented with oval nodes. Relationships are denoted with directed lines on which the name of the relationship is denoted. Relationships without their names represent the containment type. 
+The model is instantiated as a directed acyclic graph. The reason for choosing graphs is that it elegantly models 
+repetition of dance steps and has matured as a graph database. If a dance step repeats after some time, it just requires another edge to point to the same node. A graph is formally defined as follows: Let G = (V, E) be a International Journal "Information Technologies and Knowledge" Vol.1 / 2007 
+ 
+ 141
+directed acyclic graph, where V denotes set of vertices and E denotes set of directed edges. The different entity 
+classes, events, actors, agents, concepts, and other basic classes become vertices of the graph. Similarly, the 
+set of interaction relationships will be denoted as directed edges of the graph.  
+Concept
+ActorEvent
+Agent
+ObjectCC C,T,SEM
+S,T,SEMS,T,SEM
+Figure1: Graphical representation of DVSMS-Spatial
+T-Temporal
+SEM-Semantic
+C-ComposedOf
+ RaiseHeroJoy Dance step
+Left HandHeroineJoy
+Right hand
+Slow SpeedShow Flower
+Figure2: Graph of dance step containing actors and agents.SittingLeft, East 
+Approach
+Diverge
+ 
+The conceptual representation of an event highlighting a da nce step as an instance of the graph, is depicted in 
+Figure2. In this figure2, the dance event consists of two actors whose roles are hero and heroine. Hero is 
+standing left to the heroine initially and facing east. Heroine is sitting and facing west. Now hero approaches the heroine. These spatio-temporal semantics are stored as re lations. Event independent characteristics of the actor 
+are stored as video objects separately (not shown in the figure). The actors express joy and it is initialized as 
+emotion. Hero raises his left hand to chest level with medium speed and displays a flower to the heroine with his right hand. Heroine remains idle without performing any ac tion. This dance step is choreographed as part of one 
+line of lyrics of a song. Due to overflowing of nodes, a ttribute nodes are not shown in this figure.  The entity 
+classes and relationships of the model are formally defined as shown below: 
+ 
+4.1. Event Entity Class 
+Dance step of a song is known as a dance event. For instance, consider a Bharathanatyam step Samathristy 
+(Saraswathi, 1994). It is performed with the eyes by keeping them static without blinking. This step represents a 
+thought, firmness, surprise or an image of an angel. Also, dance events can be combined to form a composite 
+dance event. As an illustration, consider a dance step, Chandran. This step represents a moon and is a 
+combination of two other steps: Pathagam and Lola pathm am and should be rendered concurrently. Pathagam is 
+performed by keeping the thumb closed and the other four  fingers straight and denotes clouds, air, sword and 
+blessing. Similarly, Lola pathmam is performed by keep ing all the fingers open and stretched and represents a 
+sun. Composite dance event many represent events which are rendered concurrently or sequentially by a dancer. 
+Dance events are composed of actors, posture of the actors , cognitive state of the actors and the interactions in 
+space and time between agents and actors. Formally, a dance event is described as a tuple, 
+Event = { EID, D, AL, ND, ML } 
+where, EID is a unique identifier of the dance step, D is  the description of the dance step, AL denotes the list of 
+actors, ND is the number of dancers who are performing steps in the event and ML is the media locator of the 
+video clip.  Here, this Event tuple co rresponds to Event object of Figure1. 
+ 
+4.2. Basic Entity Class 
+A dance video object refers to a meaningful semantic entity of a dance video database. It can be described using 
+attributes which can represent macro and micro features. Formally, it is defined as shown below: 
+Object = { OID, V, TY } International Journal "Information Technologies and Knowledge" Vol.1 / 2007 
+ 
+ 142 
+where OID is a unique object id, V = {a1:v1, ..., an:vn } is n event independent or dependent attribute value pairs 
+and TY = { AID, AGID } denotes the dependency of the object, either actor or agent (to be defined later). 
+For example, hero is showing a flower in his right hand. Here, ShowFlower  is the object in which V denotes 
+attributes action and instrument with values show and flower respectively. TY holds ID of the agent, Right Hand 
+which belongs to the actor Hero. In this case, V represents event dependent (i.e., dance step dependent) values. 
+Similarly, object may also represent any of the macro features. 
+ 
+4.3. Actor Entity Class 
+Actor is a spatiotemporal entity in dance videos. So the existence time (Vijay, 2004) can be associated with the 
+entity and it represents the life span of it. Actor is also a spatial entity. Therefore actor’s displacement in space is 
+modeled using Trajectory Points as in MPEG-7 (Marti nez, 2003). Hence, actors are spatio-temporal entities 
+playing context dependent roles in the events. Actors can have spatial, temporal and event specific semantic 
+attributes describing their roles. The roles can be linguistic roles (Martinez, 2003) as in MPEG-7 or any semantic roles, such as loves. 
+The existence time predicate ΦACTOR, which is associated with the actor entity class, defines life span of the 
+actor in terms of the existence time granularity (e.g. min and sec). ΦACTOR: S(ACTOR) × Z → B. This predicate 
+takes a particular actor entity and a particular granule (denoted by an integer; say sec) and evaluates to a Boolean. If it is true, then that actor exists in the modeled reality at that granule (sec). 
+Constraint.1:  Life span of an actor can exist only within the defined lifespan of the event to which it belong . 
+Formally, an actor entity ca n be described as follows: 
+Actor = { AID, EID, DID, R, L, T, P } 
+where EID is the event id, DID is the corresponding dancer id, R denotes either semantic or linguistic roles of an 
+actor, L is the existence time or lifespan, T represents t he trajectory points(Point Set) as in Mpeg7 and P is the 
+posture of the actor, which is a basic entity. 
+ 
+4.4. Agent Entity Class 
+Agent entity class represents the finer spatio-temporal semantics of the actions. The agent entity is the one which 
+is most important in dance videos. The essence of a danc e step is the actions done by the actors and it is the 
+agent that performs the action. This is an exclusive feat ure of the dance videos. All other video types possess just 
+one or two agents, which are fixed and do not play any sign ificant role at all. For example, legs are agents in 
+soccer sport videos, bat and ball are agents in cricket sport videos. Agent entity elegantly models the action of the agent which belongs to an actor. For instance, left eye and right eye of a heroine are agents. Formally, it is defined as: 
+Agent = { AGID, AID, EID, L, T, X, S, I } 
+where AID and EID denote the actor id and event id res pectively, X is the action agent performs, S denotes 
+speed of X and I is the instrument held by the agent. Also, L and T depict the lifespan and spatial trajectory, 
+similar to actor objects. Here, X, S and I are all basic entity types as defined earlier. 
+ 
+4.5. Concept Entity Class 
+The cognitive and affective content of an actor is mode led as a concept object. The concept is modeled as a 
+separate entity type because of its ontological nature, thereby improving the semantic search. Formally, a 
+concept entity can be defined as 
+Concept = { CID, AID, EID, T, D } 
+where T = { Emotion, Feeling, Mood } and D denote type of  the concept and description as a string using natural 
+language respectively.   
+ 
+4.6. Interaction Relationships 
+An interaction relationship relates members of an entity set to those of one or more entity sets. The DVSM 
+employs the following set of relationships between the differ ent entity sets. They are Composition (C), Spatial(S): 
+which are topological (Egenhofer, 1994) and directional (Li, 1996), Temporal(T): Allen’s interval algebra (Allan, 1983), Spatio-temporal, Motion(M): such as approach, di verge, stationary which are defined over the basic 
+temporal relations (Athanasios, 2005), Semantic(SE) and Ontological(O). International Journal "Information Technologies and Knowledge" Vol.1 / 2007 
+ 
+ 143
+The following section describes the set of relationships that occur between the various dance video entity sets. 
+ 
+4.6.1. Event Relationship 
+The relations between events are composition and temporal. Intuitively, a dance step of an actor may be followed 
+by another actor immediately. Similarly, a dance step of an actor may be repeated by another dancer some time 
+later. These follows and repeats relations are cues for later retrieval and mining operations. For example the query, find the set of dance steps done by a dancer, that is repeated by another dancer, can be processed by 
+checking the life spans of the corresponding events. 
+Suppose E1, E2, ..., En are dance events participating in a tempor al relationship. Let a1 and a2 be the actors, x1 
+and x2 be the actions of agents present in E1 and E2 respectively. Then, the predicate 
+ΦREPEATS:S(X)×S(A) →E can take an actor and action and can return a set of events in which the action is 
+performed. There is a constraint on the REPEATS predicate.  
+Constraint.2 . Let LS1 and LS2 be the lifespan of E1 and E2 respectively. Then  
+(x1 = x2 ) V (LS1 < LS1) ═> (E1 = E2) 
+Similarly, the other predicates such as performSameStep, performDifferentStep , and observe  can be formulated, 
+apart from follows  and repeats  predicates. Event relationships are formally defined as follows: 
+EE = { SRC, TAR, LST } 
+where SRC and TAR denote the source and target event ids and LST is the set of composition and temporal 
+relationships which hold betwee n source and target events. 
+ 
+4.6.2. Object Relationship 
+Objects can be composed of other objects. For example, consider Figure2 where hero holds a flower in his right 
+hand. Here, flower is an example of an object. Formally, the relationship between objects can be represented 
+similar to event relationships with a restriction that the SRC and TAR can be basic entities and LST will contain only composition relations.  
+ 
+4.6.3. Actor Relationship 
+Actor relationship represents the relationship between the roles of the objects, such as relation between hero and 
+heroine who are dancer objects. Spatial, temporal and semantic relationships exist between the actors in a 
+particular dance event. For instance, hero standing left to the heroine initially, may approach the heroine. This 
+dance semantic contains spatial and motion relationships left and approach respectively. The actor relationship is formally defined as shown below: 
+AA = { AID1, AID2, O1, O2, LST } 
+where AID1 and AID2 are roles of the dancers O1 and O2 respectively and LST is now the set containing spatial, 
+temporal and semantic relationships. Note that O1 and O2 are basic entity types. 
+ 
+4.6.4. Agent Relationship 
+Agent relationship is a second level semantic relation that describes the spatial and temporal characteristics of 
+the agents. That is, agent relationship represents the finer semantics between the body parts of an actor. For instance, heroine is touching her left cheek with the index finger of her right hand. So, left cheek and right hand 
+are the agents and finger can be the instrument used in the semantic relationship touch. Agent relationship is 
+formally defined as, 
+AGAG = { AGID1, AGID2, AID, LST } 
+where AGID1 and AGID2 are agentIDs of an actor AID and LST is similar to actor relationships. 
+ 
+4.6.5. Concept Relationship Concept relationship is an ontological relationship (O) between concept entities. Typical ontological relationships 
+(Guiness, 2004) are subClassOf, cardinality, intersection and union. This relationship is similar to event 
+relationship with a modification that the source and target ids represent concepts and LST holds only ontological relations. All other types of relationships between the different dance video entities are either semantic relationships or composition relationships such as partOf, composedOf, memberOf and so on. Table 1 
+summarizes the semantics of the kinds of relation ships that exist between the dance video entities. International Journal "Information Technologies and Knowledge" Vol.1 / 2007 
+ 
+ 144 
+Table 1. Semantics of Relationships . 
+      Event Object  Actor Agent Concept 
+Event C,T,SE  C  C 
+Object  C C C  
+Actor C C S,T,SE C C 
+Agent  C C S,T,SE  
+Concept C  C  O 
+ 
+5. Implementation of DVSM 
+We have implemented the model in order to annotate the macro and micro features that are associated with the 
+dance video. The tool has been developed using J2SE1.5 and JMF2.1.1 under Dell workstation. The tool is 
+interactive as it minimizes the hard coding. 
+ 
+   
+Fig.3 and Fig.4. Macro annotation and Mi cro annotation of the event semantics. 
+ 
+The dance video can be annotated by looking at the vide o clips that is running. Macro features can be annotated 
+initially. The details of the dancers, musician, music,  song, background, tempo, dance origin, context (whether 
+live, rehearsal, professional play, competition etc), date and time of recording, type of performance venue and 
+type of dance video are annotated. The screen shot de picting the rendering of the dance and interactive 
+annotation of macro features is shown in Figure 3. 
+Then, micro features of every dance step of a song have  to be annotated. The screen shot depicted in Figure 4 
+represents events, actors, agents and concepts. The annota tor, by looking at the video, annotates the different 
+information pertaining to these entity types in the order: even t, actors of this event, agents of the actors, concepts 
+revealed by the actors. But, one can swap the annotation of agents and concepts depending on his interest. 
+The user interface has been carefully designed such a way that it minimizes the hard coding, as many of the graphical components will be populated automatically. 
+The second part of the micro features annotation involves the description of the various relationships between the 
+entity types. For instance, event relationships, actor relationships, agent relationships and concept relationships describe the spatial, temporal, motion and semantic relations that exist between the entity types. The annotated data are stored in a backend database. 
+6. Evaluation 
+Batini(1996) posits that conceptual model should possess the basic qualities: expressiveness, simplicity, minimality and formality. Additionally, Harry(2001) outlin es other semantic qualities for video types. They are: 
+explicit media structure(M), ability to describe objects (O), events (E), spatial relationships(S), temporal 
+relationships(T) and integration of syntactic and semantic  information (I). This paper introduces another factor, 
+contextual description (Actor(A) and Agent(G)) to evaluate the proposed model. 
+The DVSM satisfies all the semantic quality requirements. Moreover, DVSM is unique in modeling the finer 
+spatio-temporal contextual semantics of the events at finer  granularity, with the help of agent entity type. Table 2 
+International Journal "Information Technologies and Knowledge" Vol.1 / 2007 
+ 
+ 145
+contrasts the existing semantic content based models agai nst the semantic quality factors. The table illustrates 
+that some models lack semantic features, some lack syntactic features and only few models integrate both 
+syntactic and semantic relationships. Some applications, li ke soccer sports video, require the model to represent 
+the contextual features of objects (called actors). T he ESSVM proposed by Ahmet et al, describes contextual 
+information at actor level. However, dance videos require contextual description at multiple granularities (called 
+agent), beyond the actor level. Our proposed semantic model possesses both contextual abstractions-actor and agent, apart from the other semantic qualities. Hence, wi th the agent based approach, the paper claims to have 
+achieved conceptual, semantic and contextual qualities in dance video data modeling.      
+Table 2. Comparison of semantic video models.   
+Legend: M-Media structure, O-Object, E-Event, S-Spatial, T- Temporal, I-Syntactic-semantic info, A-Actor, G-Agent. 
+Semantic Qualities Model 
+M O E S T I A G 
+AVIS × ×  × × ×   
+OVID × ×  × × ×   
+QBIC × ×  ×     
+DISIMA × × × × × ×   
+COSMOS7  × × × × ×   
+ATN × ×  × × ×   
+ESSVM × × × × × × ×  
+DVSM × × × × × × × × 
+7. Conclusion 
+Data semantics provides a connection from a database to the real world outside the database and the conceptual 
+model provides a mechanism to capture the data semantics (Vijay, 2004). The task of conceptual modeling is 
+crucial and important, because of the vast amount of semant ics that exist in multimedia applications. In particular, 
+dance videos possess several interesting semantics for modeling and mining. This paper described as agent based approach for elicitation of the semantics such as macro and micro features of the dance videos. An 
+interactive annotation tool has been developed based on the DVSM for annotating the dance video semantics at 
+syntactic, semantic and contextual levels. Since dance steps are annotated manually, it is somewhat tedious to annotate dances by the dance expert. 
+Further work would be useful in many areas. It would be interesting to explore how DVSM can be used as a video 
+model for exact and approximate query processing. As MPEG-7 is used to document the video semantics 
+recently, it is valuable to employ MPEG-7 for repres enting dance semantics to enable better interoperability.  
+Finally, it will be useful to explore how video mi ning techniques can be applied to dance videos. 
+Acknowledgements 
+This work is supported fully by the University Grants Commission (UGC), Government of India grant 
+XTFTNBD065 . The authors thank the editors and the anonymous reviewers for their insightful comments. 
+Bibliography 
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+retrieval of images and video. Pattern Recognition, 35(4), 945-965. 
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+Knowledge and Data Engineering, 5(4), 629-643. 
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+Information Management, 3(1), 9-25. 
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+days. IEEE Transaction on. Pattern Analysis and Machine Intelligence. 22(12),.1349-1380. 
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+Web of Indian Classical Dances, http:// www.narthaki.com/index.htm. 
+Appendix-A: List of Agents 
+Head  Hand  Knee  Leg  Foot  Arm 
+Finger  Ankle  Elbow  Heel  Lower Leg Wrist 
+Toe  Hip  Shoulder Waist  Back  Torso 
+Forearm  Palm  Pelvis  Thigh  Ball of Foot Chest 
+Authors’ Information 
+Balakrishnan Ramadoss - Assistant Professor in the Department of Co mputer Applications, National Institute of 
+Technology, Tiruchirappalli, In dia (Deemed University).  
+email: brama@nitt.edu . 
+Kannan Rajkumar - M.Phil, Lecturer of Computer Science at Bishop  Heber College, Tiruchirappalli. Currently he 
+is pursuing Ph.D. in Computer Applications at National Institute of Technology, Tiruchirappalli. 
+e-mail: rajkumarkannan@yahoo.co.in   
\ No newline at end of file
diff --git a/1001.0443.txt b/1001.0443.txt
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@@ -0,0 +1,236 @@
+Discovering Knowledge from Multi-modal Lecture Recordings
+Rajkumar. K 1, Christian Gütl 2,3, 1 Department of Computer Science, Bishop Heber College, Tiruchirappalli 620017, India2 Institute for Information Systems and Computer Media (IICM), TU-Graz, Austria3 Infodelio Information Systems and GÜTL IT Research & Consulting, Austria1 rajkumarkannan@{yahoo.co.in, demfoundation.org}, 2 cguetl@iicm.edu
+Abstract
+Educational media mining is the process of converting  
+raw  media  data  from  educational  systems  to  useful  
+information  that  can  be  used  to  design  learning  
+systems,  answer  research  questions  and  allow  
+personalized  learning  experiences.  Knowledge  
+discovery  encompasses  a  wide  range  of  techniques  
+ranging  from  database  queries  to  more  recent  
+developments  in  machine  learning  and  language  
+technology. Educational media mining techniques are  
+now  being  used in  IT Services  research  worldwide.  
+Multi-modal  Lecture  Recordings  is  one  of  the  
+important types of educational media and this paper  
+explores  the  research  challenges  for  mining  lecture  
+recordings  for  the  efficient  personalized  learning  
+experiences.
+Keywords:  Educational  Media  Mining;  Lecture  
+Recordings,  Multimodal  Information  System,  
+Personalized  Learning;  Online  Course  Ware;  Skills  
+and Competences;
+1. Introduction
+The  rapid  evolution  of  communication  technologies,  
+advanced  data  networks  and  the semi-automation of  
+multimedia  information  processing  have  greatly  
+impacted  learning  experiences.  Multimedia  resources  
+include not only entertainment media such as movie,  
+cartoon, advertisement, computer games and so on, but  
+also  educational  media  such  as  available  in  digital  
+library,  museum, digital archives, e-learning systems  
+etc. Lecture recording, like courses and presentations,  
+is  one  of  the  important  educational  media  types.  
+Further information of multi-media resources and their  
+application  in  e-Learning  can  be  found  in  literature  
+such as [6].
+Within  the  last  years  universities  have  increasingly  
+started  to record  lectures  and  offer  these  recordings  
+students  for  their  learning  procedures.  Some  
+universities offer even complete series of recordings of  
+hundreds of courses available for public access such as  
+MIT OCW [8] and Berkeley courses on YouTube [2]. 
+Most  of  the  aforementioned  lecture  recordings  are  
+indeed multi-modal in nature.  Most of them include  
+audio and video stream(s) as well as an information  
+channel  for  synchronized  presentation  slides.  Additionally, information about the interaction with the  
+presenter PC (e.g. next slide, mouse movements and  
+like) and interaction with other media can add further  
+useful information. Moreover, the linkage with lecture  
+notes, course outline and background information are  
+further  information  which  should  be  taken  into  
+account. Also teacher-student interaction and student  
+inputs  are  sources  of  information  to  be  kept  and  
+analyzed.  Further  information  about  multimodal  
+information systems can be found for example in [4].
+Such media recordings provide a wide range of useful  
+information acquisition within the learning process (see  
+for example [9]) but also for analyzing purposes (see  
+for example [1]). Therefore Educational Media Mining  
+is a vital area of research because it facilitates
+•Detecting affect and disengagement
+•Detecting  attempts  to  circumvent  learning  
+called "gaming the system" 
+•Guiding student learning efforts 
+•Developing or refining student models 
+•Measuring  the  effect  of  individual  
+interventions 
+•Improving teaching support 
+•Predicting student performance and behavior 
+However, we need to develop standard data formats  
+and ontology, so that researchers can more easily share  
+data and conduct meta-analysis across systems, and use  
+pre-existing systems for storing, managing, accessing,  
+searching and visualizing media recordings and linked  
+metadata.  Furthermore,  we need  to determine  which  
+data mining techniques are  most appropriate  for  the  
+specific  features  of  educational  data, and how these  
+techniques can be used on a large scale.
+In this paper, we explore the foremost challenges for  
+educational  media  mining  in  particular  lecture  
+recordings  for  the  efficient  personalized  learning  
+experiences.
+2. Challenges for Mining Lecture Recordings
+Recently, there is a growing interest in capturing the  
+live lecture presentations for subsequent distribution to  
+students  at  distant  locations,  besides  organizing  the  
+course content as a collection of web pages [5] [7].  
+Hence,  incorporating  knowledge  and  discovery  in  distance learning environments becomes important and  
+unavoidable. Therefore, the foremost challenges for the  
+mining of lecture recordings are to enable the effective
+•Creation
+•Integration
+•Exploration
+•Analysis
+•Presentation  of  knowledge  from  web  based  
+content,  so  that  the  users  can  enjoy  the  
+facilities. 
+For this, we need an intelligent system that will capable  
+of  organizing  lecture  recordings  automatically  into  
+technically  exploitable  knowledge  source  and  
+providing users with information access environment  
+to utilize the multimedia content.
+3.  Phases  in  Mining  Multi-modal  Lecture  
+Recordings
+Personalized learning is a demanding requirement for  
+any learning systems, see for example [5] and [3]. For  
+a learner, it customizes the learning material based on  
+their  learning  preferences  in  the  context  of  culture,  
+content  and  so  on.  For  an  efficient  personalized  
+learning experience, one should know:
+•What sort of information should be extracted  
+from lecture media.
+•What  knowledge  can  be  derived  from  this  
+information.
+•How to put the information into a form that is  
+suitable for students’ learning experience.
+By focusing to lecture recordings, students want either  
+to  view  lectures  they  have  missed,  view  again  
+problematic parts or to retrieve just segments which are  
+in the context the current learning task. Therefore, a  
+modern system must be able deal with these different  
+situations.
+The  above  requirements  lead  us  into  the  different  
+phases of the mining process. The mining process can  
+be broadly categorized as shown below:
+•Annotation
+•Knowledge discovery
+•E-learning environment
+•Knowledge repository
+Annotation
+Instructional repositories archive enormous amount of  
+multimedia data as lecture recordings with the support  
+of advanced digital technology. Since lecture media is  
+highly  unstructured,  an  efficient  access  cannot  be  
+possible without the proper indexing scheme. Lack of  
+such  indexing  scheme  results  inefficient  search  and  
+browse  of  the  lecture  notes  and  recordings.  Since  
+manual indexing is tedious, automatic indexing scheme  is  highly  important.  Techniques  for  transforming  
+lecture  recordings  into  meaningful  structured  
+information can be done by
+•Classifying  lecture  video  content  
+automatically
+•Recognizing key events in the delivery of the  
+lecture
+•Summarization.
+Knowledge Discovery
+Educational media mining is fundamentally different  
+from  other  general  data  mining  methods.  It  should  
+integrate the existing data mining technologies with the  
+educational theories to improve the performance of our  
+intelligent system. During educational media mining,  
+one  should  consider  the  following  questions,  which  
+will guide the discovery process. They are,
+•What techniques are especially useful for data  
+mining 
+•How can data mining improve trainer support
+•How can data mining build better student  
+models
+•How can data mining dynamically alter  
+instruction more effectively for personalized  
+learning 
+E-learning Environment
+The intelligent system should possess a good learning  
+environment  for  personalized  content  learning  upon  
+request.   It  should  have  state-of-the-art  human-
+computer  interaction  methods  and  visualization  
+techniques  to  interface  with  the  system.  The  user  
+interface can contain a PC in intranet or internet set up.  
+It can also integrate mobile devices such as touch pad,  
+audio  and  visual  display.  Besides,  hardware  based  
+dedicated  chip with the necessary  sensors and  other  
+devices  would  be  noteworthy  environment  for  the  
+intelligent system. 
+Knowledge Repositories
+In order to make use of the discovered structures from  
+the  media  recordings,  the  management  of  these  
+structures  is  an  essential  technological  goal.  The  
+management includes addition, editing and deleting of  
+ontological  elements  and  relations  between  those  
+elements.
+3. ConclusionEducational media mining is emerged as a new way of  
+discovering facts from the diverse educational media  
+objects more recently. In this paper, particular attention  
+has  been  paid  to  lecture  recordings  and  their  
+application  for  learning  purposes.  Based  on  that,  
+challenges  and  the  different  phases  in  the  mining  
+process have been discusses. Currently, we have started  
+to build on a prototype dealing with lecture recordings.
+References
+[1] Anderson, R.J., Anderson, R., Vandegrift, T.,  
+Wolfman, S., Yasuhara, K., Designing for Change in  
+Networked Learning Environments, Proceedings of the  
+International Conference on Computer Support for  
+Collaborative Learning 2003, Series: Computer-
+Supported Collaborative Learning Series , Vol. 2,  
+Wasson, B.; Ludvigsen, Sten; Hoppe, Ulrich (Eds.),  
+2003, 119-123
+[2] Berkeley, Campus launches YouTube channel,  
+Universtiy Berkely Press release, last edited Oct. 3rd, 
+2007, last retrieved Jan. 20th, 2008 from 
+http://www.berkeley.edu/news/media/releases/2007/10/
+03_YouTube.shtml
+[3] Gütl, C., Moving Towards a Generic, Service-based  
+Architecture for Flexible Teaching and Learning  
+Activities. In C. Pahl (Ed.) Architecture Solutions for  
+E-Learning Systems (peer-reviewed), Peer-reviewed  
+book chapter, Idea Group Inc., Hershey, USA, Nov.  
+2007.[4] Gütl, C., Automatic Extraction, Indexing, Retrieval  
+and Visualization of Multimodal Meeting Recordings  
+for Knowledge Management Activities, in M.  
+Granitzer, M. Lux, and M. Spaniol (Eds.) Multimedia  
+Semantics – The Role of Metadata (peer reviewed),  
+Springer, 2008 (not yet published. available: April 3,  
+2008)
+[5] Fiehn, T., Lauer, T., Lienhard, J., Ottmann, T.,  
+Trahasch, S., Zupancic, B., From Lecture Recordings  
+towards Personalized Collaborative Learning,  
+Proceedings of CSCL 2003.
+[6] Huang, W., Eze, E., Webster, D.,  Towards  
+integrating semantics of multi-media resources and  
+processes in e-Learning, Multimedia Systems, Springer  
+Berlin / Heidelberg, Vo. 11, No. 3, März 2006, 203-
+215.
+[7] Lampi, F., Kopf, S., Benz, M., Effelsberg, W., An 
+automatic cameraman in a lecture recording system,  
+Proceedings of the international workshop on  
+Educational multimedia and multimedia education,  
+2007, 11 - 18
+[8] MIT OCW, About OCW, MIT OpenCourseWare  
+site, last retrieved Jan. 20th, 2008 from 
+http://ocw.mit.edu/OcwWeb/web/about/about/index.ht
+m
+[9] Zupancic, B., Horz, H., Lecture recording and its  
+use in a traditional university course, ACM SIGCSE  
+Bulletin  archive, Vo. 34 , Issue 3, September 2002, 24  
+– 28.
\ No newline at end of file
diff --git a/1001.0444.txt b/1001.0444.txt
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@@ -0,0 +1,537 @@
+arXiv:1001.0444v1  [gr-qc]  4 Jan 2010Nonminimal Inflation on the Randall-Sundrum II
+Brane with Induced Gravity
+Kourosh Nozaria,bandM. Shoukrania
+aDepartment of Physics, Faculty of Basic Sciences,
+University of Mazandaran,
+P. O. Box 47416-95447, Babolsar, IRAN
+bResearch Institute for Astronomy and Astrophysics of Maragh a,
+P. O. Box 55134-441, Maragha, IRAN
+knozari@umz.ac.ir
+Abstract
+We study an inflation model that inflaton field is non-minimall y coupled to the
+induced scalar curvature on the Randall-Sundrum (RS) II bra ne. We investigate the
+effects of the non-minimal coupling on the inflationary dynami cs of this braneworld
+model. Our study shows that the number of e-folds decreases b y increasing the value
+of the non-minimal coupling. We compare our model parameter s with the minimal
+case and also with recent observational data. In comparison with recent observation,
+we obtain a constraint on the values that the non-minimal cou pling attains.
+PACS:98.80.Cq, 98.80.-k, 04.50.-h
+Key Words: Inflation, Braneworld Scenario, Scalar-Tensor Theories
+11 Introduction
+Itiswellknownthatinflationprovidesanaturalexplanationforsom elong-standingproblems
+of the standard hot big bang cosmology. Inflation is also a successf ul scenario for production
+and evolution of the perturbations in primary stages of the univers e evolution [1]. Although
+inflation paradigm is successful in these respects, there is a proble m for realization of this
+scenariothatwedonotknowhowtointegrateitwithideasofthepar ticlephysics[2,3,4]. One
+importantfeatureoftheinflationaryparadigmisthefactthatinfla toncaninteractwithother
+fields suchasgravitationalsector ofthetheory. This interaction isshownbythenon-minimal
+coupling of the inflaton field and Ricci scalar in the spirit of the scalar- tensor theories. In
+fact, there are several compelling reasons for inclusion of an explic it non-minimal coupling
+of the inflaton field and gravity in the action. For instance, non-minim al coupling arises at
+the quantum level when quantum corrections to the scalar field the ory are considered. Even
+if for the classical, unperturbed theory this non-minimal coupling va nishes, it is necessary
+for the renormalizability of the scalar field theory in curved space. I n most theories used
+to describe inflationary scenarios, it turns out that a non-vanishin g value of the coupling
+constant cannot be avoided. Also, in general relativity, and in all ot her metric theories of
+gravity in which the scalar field is not part of the gravitational secto r, the coupling constant
+necessarily assumes the value of1
+6( see [5,6] for more details). Thus, it is natural to
+study an extension of the inflation proposal that contains explicit n on-minimal coupling of
+the scalar field and gravity. Currently, theories of extra spatial d imensions, in which the
+observed universe is realized as a brane embedded in a higher dimensio nal spacetime, have
+attracted a lot of attention. In this framework, ordinary matter s are trapped on the brane,
+but gravitation propagates through the entire spacetime [7-15]. T he cosmological evolution
+on the brane is given by an effective Friedmann equation that incorpo rates the effects of
+the bulk in a non-trivial manner [16,17,18,19]. The chaotic inflation on th e RSII brane has
+been studied for first time in [20]. Then extension of this study to war m inflation has been
+preformed in [21].
+With these preliminaries, the goal of the present study is to investig ate a braneworld
+viewpoint of inflation with an explicit non-minimal coupling of the scalar fi eld and Ricci
+curvature on the brane. We study possible impact of the non-minima l coupling on the dy-
+namics of this braneworld-inspired inflation. Our setup is based on th e RS II braneworld
+model andweassume thatbraneisstableandinflationdynamics onth ebraneisindependent
+of the bulk dynamics. We study modifications of this slow-roll inflation due to non-minimal
+coupling of the inflaton field and gravity on the brane. We study also p arameter space of the
+2model numerically to obtain constraints imposed on this model by rec ent observations re-
+leased by WMAP5+SDSS+SNIa datasets and we compare our results with the minimal case
+studied in Ref. [20]. Through this paper a dot on a quantity marks its t ime differentiation
+while a prime denotes differentiation with respect to the scalar field, ϕ.
+2 The Setup
+We start with the following action to construct a braneworld non-min imal inflation scenario
+S=1
+2κ2
+5/integraldisplay
+bulkd5X/radicalBig
+−g(5)(R(5)−2Λ5)+/integraldisplay
+braned4x√−g/bracketleftbiggR
+2κ2
+4−1
+2gµν∂µϕ∂νϕ−1
+2ξRϕ2−V(ϕ)−λ/bracketrightbigg
+(1)
+In this action, which is written in the Jordan frame, Xis coordinate in the bulk, while x
+shows induced coordinate on the brane. κ2
+5is 5-dimensional gravitational constant, R(5)is
+5-dimensional Ricci scalar and ξis the nonminimal coupling (NMC) of the scalar field ϕand
+Ricci curvature on the brane,Λ 5is the 5-dimensional cosmological constant in the bulk and
+λis the brane tension. We have chosen a conformal coupling of the infl aton and gravity on
+the brane for simplicity. In other words, non-minimal coupling of the scalar field and gravity
+on the brane is α(ϕ) =1
+2(1−ξϕ2) withκ2
+4= 8πG= 1. We note that there are two critical
+values of ϕgiven by ϕc=±1√
+ξthat should be avoided to have well-defined field equations.
+Variation of the action with respect to ϕleads to the equation of dynamics for scalar field
+on the brane
+¨ϕ+3H˙ϕ+ξRϕ+dV
+dϕ= 0, (2)
+whereR= 6(¨a
+a+˙a2
+a2) for spatially flat FRW geometry on the brane and H=˙a
+ais the brane
+Hubble parameter. In the slow-roll approximation where ¨ ϕ≪V(ϕ), equation of motion for
+the scalar field takes the following form
+˙ϕ=−ξRϕ+V′(ϕ)
+3H. (3)
+The energy density and pressure of the non-minimally coupled scalar field are given by (see
+Ref. [5,6] for discussion on the various representations of the ene rgy-momentum tensor of a
+non-minimally coupled scalar field)
+ρϕ=1
+2˙ϕ2+V(ϕ)+6ξHϕ˙ϕ+3ξH2ϕ2(4)
+and
+Pϕ=1
+2˙ϕ2−V(ϕ)−2ξϕ¨ϕ−2ξ˙ϕ2−4ξHϕ˙ϕ−ξ(2˙H+3H2)ϕ2(5)
+3respectively. The conservation equation for scalar field energy de nsity in this setup is given
+by
+˙ρϕ+3H(ρϕ+Pϕ) = ˙ϕV′+ ˙ϕ¨ϕ+3H˙ϕ2+ξRϕ˙ϕ (6)
+where from equation (2) we find
+˙ρϕ+3H(ρϕ+Pϕ) = 0 (7)
+Note that the authors of Ref. [22] have treated this conservatio n on the brane in a relatively
+different way. They have defined a total energy-momentum tenso r which consists of two
+parts: a pure (canonical) scalar field energy-momentum tensor an d a non-minimal coupling-
+dependent part. The total energy density defined in this manner is then conserved. In our
+case, we have included all possible terms in equations (4) and (5) fro m the beginning and
+therefore, total energy density defined in this manner is conserv ed too.
+Now, in a cosmological scenario based on the Randall-Sundrum II bra neworld, the effec-
+tive Friedmann equation on the brane can be written as follows [16,17,1 8,19]
+H2=8π
+3M2ρ(1+ρ
+2λ), (8)
+where we have assumed that metric on the brane is spatially flat FRW t ype and the effective
+cosmological constant on the brane is negligible during inflation phase .Mis the four-
+dimensional Plank scale, ρis the total energy density on the brane, namely ρ=ρϕ+ρm
+whereρmis energy density of ordinary matter on the brane and λis 3-brane tension. In
+the low energy limit where λ≫ρ, one recovers the standard Friedmann cosmology. In the
+high energy limit where ρ≫λ, the braneworld effects are dominant. In which follows we
+setρm= 0 for simplicity.
+In the slow-roll approximation, ρϕattains the following approximate form
+ρϕ≈V(ϕ)+6ξHϕ˙ϕ+3ξH2ϕ2, (9)
+We use this approximate form in equation (10) to find cosmological dy namics of the model.
+Now the Friedmann equation of the model takes the following form(s ee[23,24] for more
+details)
+H2+k
+a2=/bracketleftBigg
+rcα(ϕ)(H2+k
+a2)−κ2
+5
+6(ρ+λ)/bracketrightBigg2
+. (10)
+where we assume that the metric on the brane is spatially flat FRW typ e (k= 0).rc=κ2
+5
+2κ2
+4,
+is the induced -gravity crossover length scale.
+H2=/bracketleftBigg
+rcα(ϕ)H2−κ2
+5
+6/parenleftbigg
+V(ϕ)−2ξ2Rϕ2−2ξϕV′+3ξH2ϕ2+λ/parenrightbigg/bracketrightBigg2
+(11)
+4Using the definition of ˙ ϕas given by equation (3), this leads to a forth order equation for
+Hubble parameter of the model
+H4/parenleftBig
+rcα(ϕ)−ξϕ2κ2
+5
+2/parenrightBig2−H2/bracketleftBigg
+1+2/parenleftBig
+rcα(ϕ)−ξϕ2κ2
+5
+2/parenrightBig/parenleftBiggκ2
+5
+6/parenleftBig
+V(ϕ)−2ξ2Rϕ2−2ξϕV′+λ/parenrightBig/parenrightBigg/bracketrightBigg
++/bracketleftBiggκ2
+5
+6/parenleftBig
+V(ϕ)−2ξ2Rϕ2−2ξϕV′+λ/parenrightBig/bracketrightBigg2
+= 0 (12)
+With a minimally coupled scalar field ( ξ= 0)and in the absence of induced gravity on the
+brane (the pure Randall-Sandrum case with rc= 0) we find
+H2≃8π
+3M2V(ϕ)/parenleftBig
+1+V(ϕ)
+2λ/parenrightBig
+,
+˙ϕ≃ −V′
+3H
+which are appropriate Friedmann and scalar field equations for this c ase [20]. By assuming
+a non-vanishing value of the non-minimal coupling ξ, we can solve equation (12) for H2to
+find the following the following solutions
+H2=λ
+b2/bracketleftBig8π
+m2p+8πrc
+3m2pb(1+a
+λ)/bracketrightBig
+±2
+b/radicalBigg
+1
+b2+rc
+b(1+a
+λ) (13)
+whereaandbare defined as follows
+a≡V−2ξϕ(ξRϕ+V′), b≡2rc−3ξϕ2κ2
+4. (14)
+To be more specific, in which follows we consider just the plus sign in equ ation (13). We
+note that reality of the solution for H2requires that
+λ≥−βa
+1+β
+whereβ≡rcb. Now, we define the slow-roll parameters of our model as follows
+ε≡ −˙H
+H2,
+η≡ −¨H
+H˙H
+and
+γ2≡2εη−dη
+dt.
+5We need to calculate these parameters in our non-minimal branewor ld model. For εwe find
+ε=2˙AB−2˙BA−˙CC−1A
+4H3A, (15)
+where we have defined
+A=/parenleftBig
+2rc−3ξκ2
+4ϕ2/parenrightBig2
+λ
+B=8π
+m2p+8πrc
+3m2pb(1+a
+λ),
+and
+C=4
+b4+4rc
+b3(1+a
+λ).
+The second slow-roll parameter, η, is calculated as follows
+η=/bracketleftbigg
+4A2H3(2˙BA−2˙B˙A+˙CC−1A)/bracketrightbigg−1/bracketleftbigg
+(8˙AAH2+2˙BA−2˙AB+˙CC−1A)(2˙BA−2˙AB+˙CC−1A)
+−(2¨BA−2¨AB+¨CC−1A−˙C2C−2A+˙CC−1˙A)4A2H2/bracketrightbigg
+. (16)
+And finally, the third slow-roll parameter, γ2, in our model takes the following form
+γ2=−˙ϕV′′′
+3H3. (17)
+Inflation occurs when the condition ε <1 ( or equivalently |η|<1) is fulfilled. With this
+condition and using (15) we find
+ρϕ< V+3ξH2ϕ2. (18)
+This is the condition for realization of the non-minimal inflation on the b rane. Therefore, to
+have an inflationary period on the brane, the condition (18) should b e fulfilled. Inflationary
+phase will terminates when the universe heats up so that the condit ionε= 1 ( or |η|= 1)
+is satisfied. We should stress here that in our non-minimal setup, de pending on the values
+of the non-minimal coupling ξ, it is possible to fulfill |η|= 1 condition even before fulfilling
+ǫ= 1. In other words, for some values of the non-minimal coupling, infl ation terminates if
+the condition |η|= 1 is fulfilled. Figure 1 shows the the natural exit from inflation phase
+when the condition |η|= 1 is fulfilled before occurrence of ǫ= 1. In this case we have
+ρϕ≃V+3ξH2ϕ2. (19)
+By comparison with the minimal case as has been studied in Ref. [20], we see that non-
+minimal coupling of the scalar field and gravity on the brane plays an imp ortant role on
+6the natural exit from inflationary phase. On the other hand, it is po ssible to achieve the
+condition ε= 1 before occurrence of |η|= 1, so that inflation phase terminates when the
+condition ε= 1 is fulfilled. This feature is shown in figure 2 for different values of th e non-
+minimal coupling. We stress that in this figure the minimal case with ξ= 0 accounts for
+natural exit without additional mechanism too. This is possible in our s etup because of the
+braneworld effect. In the standard 4-dimensional inflationary sce nario with one minimally
+coupled inflaton field, it is impossible to exit inflationary phase naturally . We note also that
+the values of the non-minimal coupling ξused in the figures lie within the acceptable range
+forξas has been obtained in Ref. [25] by confrontation of a non-minimally c oupled phantom
+cosmology with the recent observations (see also [26]).
+x=1
+10x=1
+20f0.05 0.10 0.15eta
+0.00.20.40.60.81.0
+Figure 1: Natural exit from inflation phase when the condition |η|= 1 is fulfilled before occurrence
+ofǫ= 1.
+Nowwefocusonthenumber ofe-foldsasanotherimportantquant ityinatypicalinflation
+scenario. The number of e-folds is defined as
+N≡/integraldisplaytend
+tHdt=/integraldisplayϕend
+ϕH
+˙ϕdϕ
+In our setup, the number of e-folds at the end of the inflation pase is given by
+N=−/integraldisplayϕe
+ϕi/parenleftbigg3H2
+ξRϕ+V′/parenrightbigg
+dϕ. (20)
+Using the explicit form of H2with positive sign as given by equation (13), the number of
+7x= 0x=1
+20x=1
+10x=1
+6f0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20e
+0.00.20.40.60.81.01.21.4
+x= 0x=1
+20x=1
+10x=1
+6f0.1 0.2 0.3 0.4 0.5e
+0.00.20.40.60.81.01.2
+Figure 2: Natural exit from inflationary phase with various values of t he non-minimal coupling
+forV(ϕ) =V0ϕ2(left) and V(ϕ) =V0ϕ4(right) in the case that hen the condition ε= 1 is
+fulfilled before occurrence of |η|= 1. The minimal case is given by ξ= 0. As these figure show,
+incorporation of the non-minimal coupling causes consider able difference in the variation of the
+slow-roll parameter ε, but there is graceful exit as for the minimal case.
+e-folds in our non-minimal setup takes the following form
+N=−/integraldisplayϕe
+ϕi/parenleftbigg3
+ξRϕ+V′/parenrightbigg/braceleftBiggλ
+b2/bracketleftBig8π
+m2
+p+8πrc
+3m2
+pb(1+a
+λ)/bracketrightBig
++2
+b/radicalBigg
+1
+b2+rc
+b(1+a
+λ)/bracerightBigg
+dϕ(21)
+withaandbas defined in (14). ϕidenotes the value of the scalar field ϕwhen universe
+scale observed today crosses the Hubble horizon during inflation, w hileϕeis the value of
+the scalar field when the universe exits the inflationary phase. To st udy the effect of the
+non-minimal coupling on the number of e-folds, we define
+g≡/parenleftbigg3
+ξRϕ+V′/parenrightbigg/braceleftBiggλ
+b2/bracketleftBig8π
+m2
+p+8πrc
+3m2
+pb(1+a
+λ)/bracketrightBig
++2
+b/radicalBigg
+1
+b2+rc
+b(1+a
+λ)/bracerightBigg
+which is the integrand of equation (21). The number of e-folds is pro portional to the area
+enclosed between the g-curve and horizontal axis from ϕitoϕe. Figure 3 shows the variation
+ofgversusϕfor different values of the non-minimal coupling and for two different energy
+scales. As the value of the non-minimal coupling increases, the enclo sed area between g-
+curve and horizontal axis decreases leading to the conclusion that the number of e-folds
+reduces by increasing the values of the non-minimal coupling. Compa rison with the minimal
+case shows also that the number of e-folds reduces by inclusion of a positive non-minimal
+8x=1
+20x=1
+10x=3
+20f0.10 0.15 0.20 0.25 0.30g
+0.10.20.30.40.50.60.70.80.9
+x=1
+20x=1
+10x=3
+20f0.10 0.12 0.14 0.16 0.18 0.20 0.22g
+0.00.10.20.30.40.50.60.7
+Figure 3: gversusϕandξfor two different regimes: λ≫1 andλ≪1. Number of e-folds is given
+by the area bounded between g-curve and the horizontal axis from ϕitoϕeing−ϕplot. The
+number of e-folds decreases by increasing ξ.
+coupling. Now, we study scalar and tensor perturbations in this non -minimal model. These
+perturbations are supposed to be adiabatic on the brane. We defin e the scalar curvature
+perturbation amplitude of a given mode when re-enters the Hubble r adius as follows
+As=2
+5H
+˙ϕδϕ. (22)
+The scalar spectrum index is defined as follows
+ns= 1+dlnA2
+s
+dlnk(23)
+The interval in wave number is related to the number of e-folds by th e relation dlnk(ϕ) =
+−dN(ϕ). Substituting (3) into the relation (22) we find for As
+A2
+s=36
+25H4
+(ξRϕ+V′)2δϕ2(24)
+Using the slow-roll parameters, we have
+ns= 1−7
+2ε+3
+2η. (25)
+The running of the spectral index which is defined as
+αs=dns
+dlnk(26)
+9in our model takes the following form
+αs=9
+2(2εη−4ε2)+3
+2/parenleftBig
+2εη−γ2/parenrightBig
+. (27)
+Tensor perturbations are bounded to the brane at long-waveleng th and up to the first
+order these perturbations are decoupled from matter on the bra ne [21]. Therefore, on the
+large scale we can use the classical expression for these amplitudes [20]. These amplitudes
+at the Hubble crossing are given by
+A2
+T=4
+25π/parenleftbiggH
+M/parenrightbigg2/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+k=aH(28)
+whereinournon-minimalcaseandwithintheslow-roll approximation Hisgivenbyequation
+(13)with positive sign, explicitly depended on the non-minimal coupling ,ξ. The tensor
+spectral index is given by
+nT≡dlnA2
+T
+dlnk≈ −2ε (29)
+whereεis given by equation (15). The ratio between the amplitude of tensor and scalar
+perturbations is given by
+A2
+T
+A2s≈4π/parenleftBig
+ξRϕ+V′(ϕ)/parenrightBig2
+9M2H4. (30)
+After constructing the basic formalism, we study inflationary dyna mics of this non-minimal
+model numerically. For this goal, in which follows and in all of our numeric al calculations,
+we consider the well-known large-field inflationary potential V(ϕ) =V0ϕnwithn= 2,4 to
+investigate outcomes of our model. As has been shown previously in fi gures 1 and 2, this
+non-minimal model accounts for natural exit from inflationary pha se without any additional
+mechanism. In fact, non-minimal coupling of the inflaton field and gra vity on the brane
+provides a suitable parameter space for natural exit from inflation ary phase. Figure 4 shows
+the variation of nsfor different values of the non-minimal coupling versus the inflaton fi eld.
+Combining WMAP5 with SDSS and SNIa data, the spectral index is given by ( see for
+instance [27] and references therein)
+ns= 0.960+0.014+0.026
+−0.013−0.027(1σ,2σ CL). (31)
+This result shows that a red power spectrum is favored and ns>1 is disfavored even when
+gravitational waves are included, which constrains the models of infl ation that can produce
+significant gravitational waves, such as chaotic or power-law inflat ion models, or a blue
+spectrum, such as hybrid inflation models. As we have shown, our no n-minimal scenario on
+10x= 0f1 2 3 4ns
+0.40.50.60.70.80.9
+x=1
+6x=1
+10f0.00 0.02 0.04 0.06 0.08 0.10ns
+0.60.70.80.91.0
+Figure 4: Variation of nsversus the inflaton field for three values of the non-minimal c oupling.
+the brane gives also a red and nearly scale invariant power spectrum which excludes blue
+spectrum. Since, 0 .92≤ns≤1, our numerical calculation shows that we can constraint ξ
+to the following range
+0≤ξ≤0.114, (32)
+whichlieswithintheacceptablerangefor ξashasbeenobtainedinRef. [25]byconfrontation
+of a non-minimally coupled phantom cosmology with the recent observ ations. Note that
+negative values of ξcorresponding to anti-gravitation are excluded from our consider ations.
+If there is running of the spectral index, the constraint on this ru nning from combined
+data of WMAP5+SDSS+SNIa is given bydns
+dlnk=−0.032+0.021
+−0.020within the 1 σCL [27]. The
+special case with ns= 1 anddns
+dllnk= 0 results in the scale invariant spectrum. The signifi-
+cance of nsanddns
+dllnkis that different inflation models motivated by different physics make
+specific, testable predictions for the values of these quantities. F igure 5 shows the running of
+thespectral index versus theinflatonfieldfordifferent values oft he non-minimal coupling for
+V(ϕ) =V0ϕ2. Evidently, our non-minimal setup agrees with this dataset and aga in we can
+use this observational data to constraint the values of the non-m inimal coupling. In other
+words, since −0.052≤αs≤ −0.011, we can constraint ξso that 0 ≤ξ≤0.106. Comparing
+this range of ξwith constraint (32), we see that confrontation of this non-minima l brane
+inflation with recent observations leads to the result that 0 ≤ξ≤0.106 which is acceptable
+from other studies such as Ref. [25].
+One important point in our study is the issue of frame. There is a conf ormal transforma-
+tionwhich transforms the action(1) ( which is written inthe Jordanf rame) to corresponding
+action in the Einstein frame [5,6]. In the Einstein frame, the gravitatio nal sector is expressed
+11x=1
+6x=1
+10f0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10as 
+K0.8K0.7K0.6K0.5K0.4K0.3K0.2K0.10.0
+x= 0f1 2 3 4as 
+K0.6K0.5K0.4K0.3K0.2K0.10.0
+Figure 5: αsversusϕfor different values of ξ.
+in terms of a re-scaled scalar field which is minimally coupled to gravity an d evolves in a
+re-scaled potential, thereby simplifying the formalism. However, we should keep in mind
+that the matter sector is strongly affected by such a conformal t ransformation since all of
+the matter fields are now non-minimally coupled to the re-scaled metr ic: in particular, stress
+tensor conservation in the matter sector is no longer ensured [28,2 9]. On the other hand, as
+Makino and Sasaki [30] and Fakir et al[31] have shown, the amplitude of scalar perturbation
+in the Jordan frame exactly coincides with that in the Einstein frame. This proof (for details
+see [32]) allows us to calculate the scalar power spectrum in the Jorda n and Einstein frame.
+As a result, the scalar power spectrum has no dependence on the c hoice of frames, i.e., it
+is conformally invariant. So, our results can be compared to observ ations directly without
+any ambiguities. This is an important point since one has to check validit y of non-gravity
+experiments in Einstein frame. Forinstance, validity of electro-mag netic related experiments
+such as CMB experiment should be checked in Einstein frame. As Koma tsu and Futamase
+have shown, the scalar power spectrum is independent on the choic e of frames [32].
+3 Summary and Conclusions
+In the inflationary paradigm, inflaton field can interact with other fie lds such as Ricci cur-
+vature. This interaction is shown by the non-minimal coupling of the in flaton and gravity
+in the action of the scenario. There are many compelling reasons to in clude an explicit non-
+minimal coupling in the action of the theory. So, naturally we should inc lude non-minimal
+12coupling of gravity and inflaton field in the action of an inflation scenar io. Usually by incor-
+poration of the non-minimal coupling it is harder to realize inflation eve n with potential that
+are known to be inflationary in the minimal case. However, inclusion of the non-minimal
+coupling is inevitable from field theoretical viewpoint especially their re normalizability. In
+this paper we have studied an inflation model that there is an explicit n on-minimal coupling
+of the inflaton field and gravity on the Randall-Sundrum II brane. Th is is an extension of
+the study presented in Ref. [20] to scalar-tensor type theories. We have studied impact
+of the non-minimal coupling on the dynamics of this braneworld-inspir ed inflation. As we
+have shown, this model provides a natural exit from the inflationar y phase without adopting
+any additional mechanism for appropriate values of the conformal coupling. The number of
+e-folds decreases by increasing the values of the conformal coup ling,ξ. A confrontation with
+recent WMAP5+SDSS+ SNIa combined data shows that this model giv es a red and nearly
+scaleinvariant power spectrum for V(ϕ) =V0ϕ2andV(ϕ) =V0ϕ4. Fromanother viewpoint,
+comparison of this non-minimal inflation model with observational da ta could provide severe
+constraints on the values of the non-minimal coupling. In this paper our numerical analysis
+shows that 0 ≤ξ≤0.106 is favored by non-minimal inflation on the Randall-Sundrum II
+brane. Since we have calculated the first order contributions in slow -roll parameters, our
+results are valid in both Jordan and Einstein frames. However, highe r order corrections
+could lead to different results in these two frames. Finally, we note th at inclusion of the
+non-minimal coupling of gravity and inflaton field produces a wider par ameter space rela-
+tive to minimal case. This wider parameter space provides much more freedom than single
+self-interacting scalar field to fit with the observational data.
+Acknowledgments
+This work has been supported partially by Research Institute for A stronomy and Astro-
+physics of Maragha, Iran.
+References
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+15
\ No newline at end of file
diff --git a/1001.0445.txt b/1001.0445.txt
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+arXiv:1001.0445v2  [quant-ph]  12 Nov 2010Adiabatic Creation of Atomic Squeezing in Dark States vs. De coherences
+Z. R. Gong),1Xiaoguang Wang,2and C. P. Sun1
+1Institute of Theoretical Physics, Chinese Academy of Scien ces, Beijing 100190, China
+2Zhejiang Institute of Modern Physics, Department of Physic s, Zhejiang University, Hangzhou 310027, China
+We study the multipartite correlations of the multi-atom da rk states, which are characterized by
+the atomic squeezing beyond the pairwise entanglement. It i s shown that, in the photon storage
+process with atomic ensemble via electromagnetically indu ced transparency (EIT) mechanism, the
+atomic squeezing and the pairwise entanglement can be creat ed by adiabatically manipulating the
+Rabi frequency of the classical light field on the atomic ense mble. We also consider the sudden
+death for the atomic squeezing and the pairwise entanglemen t under various decoherence channels.
+An optimal time for generating the greatest atomic squeezin g and pairwise entanglement is obtained
+by studying in details the competition between the adiabati c creation of quantum correlation in the
+atomic ensemble and the decoherence that we describe with th ree typical decoherence channels.
+PACS numbers: 03.67.Mn, 03.65.Ud, 03.65.Yz
+I. INTRODUCTION
+Atomic ensemble can serve as a quantum memory for
+storing the quantum information of photons [1–4] where
+the memory elements can be described as the quasi-spin
+wave excitations in atomic ensemble [5], or Dicke type
+collective states [6]. Recent experiments have demon-
+strated that such storable and quantum memory could
+have long lifetime for the use in long-distance quantum
+communication [7]. When using the magnetically insen-
+sitive clock transition in atomic rubidium confined in a
+one-dimensional optical lattice, quantum memory life-
+time can exceed 3 ms [8].
+The relevant quantum storage process could be im-
+plemented through the many-particle enhancement of
+the absorption cross section with the adiabatic-passage
+techniques [9]. Through the electromagnetically induced
+transparency (EIT) mechanism, the adiabatic manipula-
+tion utilizes the dark polariton |dn(θ)/an}bracketri}ht(a photon-dressed
+collective atomic state). With adiabatically changing
+parameter θ∈[0,π/2], the quantum state of photon
+|P/an}bracketri}ht=/summationtext
+ncn|n/an}bracketri}ht(a superposition of photon Fock states
+|n/an}bracketri}ht) can be adiabatically converted to a state of the col-
+lective atom excitation |M/an}bracketri}ht=/summationtext
+ncn|Mn/an}bracketri}ht, which is a
+superposition of the collective atomic states |Mn/an}bracketri}htwith
+the same coefficients cnas that in the photon state |P/an}bracketri}ht.
+Mathematically, this coherent conversion of the photon
+state is an associated mapping
+|P/an}bracketri}ht⊗|d′/an}bracketri}ht=/summationdisplay
+cn|dn(0)/an}bracketri}ht →/summationdisplay
+cn|dn(π
+2)/an}bracketri}ht=|p′/an}bracketri}ht⊗|M/an}bracketri}ht,
+(1)
+where|d′/an}bracketri}htand|p′/an}bracketri}htare the initial state of the memory
+and the final state of photon respectively. Obviously,
+here only used are the macroscopically coherent proper-
+ties of each collective excitation, which is described by
+individual collective state |Mn/an}bracketri}ht,rather than the inter-
+nal entanglement and quantum correlation in the single
+collective state |Mn/an}bracketri}htas well as |dn(θ)/an}bracketri}ht.
+In this paper we will pay attention to the latter by
+considering the multipartite correlation measured by spinsqueezing in |Mn/an}bracketri}ht, which is obtained from |dn(θ)/an}bracketri}htby adi-
+abatic manipulation for the dark state |dn(θ)/an}bracketri}htas
+|n/an}bracketri}ht⊗|0/an}bracketri}ht=|dn(0)/an}bracketri}ht → |dn(π
+2)/an}bracketri}ht=|0/an}bracketri}ht⊗|Mn/an}bracketri}ht.(2)
+We study in details the dynamic competition between the
+adiabatic creation of quantum correlation in the atomic
+ensemble and the decoherence that we describe with three
+typical decoherence channels. There are two time scales
+t1andt2depicting such competition, where t1repre-
+sents the adiabatic time limited by the adiabatic condi-
+tions and t2represents the decoherence time. Actually, if
+we regard the atomic ensemble as a macroscopic object
+in large N limit, quantum decoherence scenario seems
+to result in the quantum disappearance due to noisy
+ambient environments [10] or the internal random mo-
+tions [11, 12]. A surprising discovery about the pairwise
+entanglement is that it suddenly die due to influence of
+environment [13, 14]. In this paper, we also consider this
+sudden death phenomena of the pairwise entanglement
+as well as the atomic squeezing for quantum correlation
+for more than two particles. We compare the time scales
+of various sudden death processes with the speed of adi-
+abatic manipulations for creating such squeezing as well
+as the pairwise entanglement.
+Actually, creating spin squeezing in a single Dicke like
+state is the crucial step for the precision measurements
+based on many-atom spectroscopy. With such controlled
+production of complex entangled states of matter and
+light, squeezing the fluctuations via entanglement be-
+tween 2-level atoms can improve the precision of sens-
+ing, clocks, metrology, and spectroscopy [15]. This paper
+seems to provide a simple scheme for adiabatic creation
+of atomic squeezing. This adiabatic passage manipula-
+tion seems feasible in creating the spin squeezing, but it
+requires an ideal technique of single photon source, which
+can initially produce the single Fock states |n/an}bracketri}ht.Therefore,
+our present theoretical proposal may meet the difficulty
+for physical implementation based on the existing tech-
+nology.
+This paper is arranged as follows. In Sec. II, we study2
+FIG. 1: (Color online) Schematic illustration of the atomic
+ensemble interacting with two light fields: the classical li ght
+field (denoted by the blue arrow) with Rabi frequency Ωand
+the wave vector Kmeand the quantum light field (denoted
+by the red arrow) with the coupling constant gand the wave
+vectorKge. EachΛ−type atom confined in the rectangular
+container has the same excited state |e/angbracketright, the relative ground
+state|g/angbracketrightand the metastable state |m/angbracketright.
+the multipartite correlations characterized by the atomic
+squeezing beyond the pairwise entanglement in the dark
+states. In Sec. II, we study the relationship between the
+concurrence and the spin squeezing under adiabatic ma-
+nipulation, and indicate that the photon statistics varies
+in the same way of the spin squeezing. In Sec. IV, the
+sudden death of the atomic squeezing and the pairwise
+entanglement are considered by introducing three typical
+decoherence channels. The optimal times for generating
+the atomic squeezing and the pairwise entanglement are
+obtained in Sec. V. We conclude in Sec. V.
+II. ADIABATICALLY CREATING SQUEEZING
+IN DARK STATES
+A. Dark states
+As shown in Fig. 1, the model we consider consists
+of atomic ensemble with NΛ−type atoms, which have
+the excited state |e/an}bracketri}ht, the relative ground state |g/an}bracketri}htand
+the metastable state |m/an}bracketri}ht. For convenience we assume all
+the atoms have the same energy spacings for the above
+three atomic states. For the cold atomic ensemble, the
+Doppler width broadening of the atoms can be depressed
+effectively. These atoms interact with two single-mode
+light fields: one is quantized radiation mode (with cou-
+pling constant gand annihilation operator a) to lead theapproximately resonant transition from |e/an}bracketri}htto|g/an}bracketri}ht; the
+other is an exact classical field with Rabi frequency Ωto
+lead the transition from |e/an}bracketri}htto|m/an}bracketri}ht. The quantum dy-
+namics of the total system is described by the following
+Hamiltonian in the interaction picture [5]
+H=gaN/summationdisplay
+j=1exp(iKge·j)σj
+ge+
+ΩN/summationdisplay
+j=1exp(iKme·j)σj
+me+h.c., (3)
+whereKgeandKmeare, respectively, the wave vectors
+of the quantum and the classical light fields. We have in-
+troduced the quasi-spin operators σj
+αβ≡ |α/an}bracketri}htjj/an}bracketle{tβ|(α,β=
+e,g,m)forα/ne}ationslash=βin above Hamiltonian to describe the
+transitions among the levels of |e/an}bracketri}ht,|g/an}bracketri}htand|m/an}bracketri}ht.
+For EIT, to find the degenerate class of the eigenstates
+|dn(θ)/an}bracketri}htsatisfying
+H|dn(θ)/an}bracketri}ht= 0, (4)
+we introduce the dark state polariton operator
+D†(θ) =a†cosθ−C†sinθ, (5)
+which obviously mix the electromagnetic field with cre-
+ation operator a†and the quasi-spin wave with corre-
+sponding operator
+C†=1√
+NN/summationdisplay
+j=1exp(iK·j)σj
+mg. (6)
+The angle tanθ=g√n/Ωcan be adiabatically manip-
+ulated in the quantum storage process, and K=Kge−
+Kmeis the wave vector difference between the quantum
+and the classical fields. We construct the degenerate class
+of the eigenstates |dn(θ)/an}bracketri}htas
+|dn(θ)/an}bracketri}ht=A(n,N,θ)√
+n!/bracketleftbig
+D†(θ)/bracketrightbign|0/an}bracketri}ht, (7)
+where the normalization constant
+A(n,N,θ) =/radicalbigg
+(N−n)!Nn
+N!sin2n2θ×
+/bracketleftbig
+1F1(−n;N−n+1;−Ncot2θ)/bracketrightbig−1
+2(8)
+reduces to 1when the atom number Nis much larger
+thann. Here,nis the excitation number of the atomic
+ensemble, the vacuum state
+|0/an}bracketri}ht=|0/an}bracketri}ht⊗|↓/an}bracketri}ht (9)
+is the direct product of the photonic vacuum state
+|0/an}bracketri}htand the quasi-spin ground state |↓/an}bracketri}ht ≡/producttextN
+j=1|g/an}bracketri}htj.3
+Here, 1F1(a;b;c)is the Kummer hypergeometric func-
+tion [16]. Eq. (4) indicates that |dn(θ)/an}bracketri}htis totally can-
+celled by the interaction Hamiltonian, and thus is called
+a dark state or a dark state polariton.
+Obviously, the dark state |dn(θ)/an}bracketri}htis photon Fock state
+|n/an}bracketri}htwhenθ= 0and quasi-spin wave state when θ=
+π/2. In the low excitation limit as n≪N,C†is an
+approximately bosonic operator as well as D†(θ).
+With the help of the dark states, by adiabatically
+manipulating the classical field the quantum informa-
+tion of the photon with quantum state |P/an}bracketri}ht=/summationtext
+ncn|n/an}bracketri}ht
+can be coherently converted into the atomic ensemble as
+|M/an}bracketri}ht=/summationtext
+ncn|Mn/an}bracketri}ht, where|n/an}bracketri}htis the Fock state of photon
+and
+|Mn/an}bracketri}ht=/vextendsingle/vextendsingle/vextendsingledn/parenleftBig
+θ=π
+2/parenrightBig/angbracketrightBig
+= (−1)n/radicalbigg
+Nn(N−n)!
+N!n!/bracketleftbig
+C†/bracketrightbign|0/an}bracketri}ht(10)
+is the collective atomic state. This coherent conversion
+is an associated mapping
+|P/an}bracketri}ht⊗| ↓/an}bracketri}ht=/summationdisplay
+cn|dn(0)/an}bracketri}ht →/summationdisplay
+cn|dn(π
+2)/an}bracketri}ht=|0/an}bracketri}ht⊗|M/an}bracketri}ht,
+(11)
+which only use the macroscopically coherent properties
+of each collective excitation described by individual col-
+lective state |Mn/an}bracketri}ht.We notice that |Mn/an}bracketri}htis actual multi-
+atom state with multipartite quantum correlations. For
+two-particle case, we call it internal entanglement.
+The internal entanglement and the quantum correla-
+tion in the single collective state |Mn/an}bracketri}htas well as |dn(θ)/an}bracketri}ht
+also play important role in quantum storage process.
+Now we consider the quantum correlation measured by
+spin squeezing of those quasi-spins in |Mn/an}bracketri}ht, which is ob-
+tained from |dn(θ)/an}bracketri}htby adiabatic manipulation
+|n/an}bracketri}ht⊗|0/an}bracketri}ht=|dn(0)/an}bracketri}ht → |dn(π
+2)/an}bracketri}ht=|0/an}bracketri}ht⊗|Mn/an}bracketri}ht.(12)
+B. Atomic squeezing in dark states without
+decoherence
+The above dark state mixes the electromagnetic field
+and the quasi-spins defined by the two levels |g/an}bracketri}htand|m/an}bracketri}ht
+of the atomic ensemble. The collective operators
+Jα=1
+2N/summationdisplay
+j=1σj
+α,(α=x,y,z) (13)
+can be generally used to demonstrate the exchange sym-
+metry and dynamic properties of the quasi-spin system.
+To calculate the averages of the collective operators, the
+reduced density matrix of the quasi-spins is obtained by
+tracing over the degree of freedom of the photons asρr
+a= Trp|dn(θ)/an}bracketri}ht/an}bracketle{tdn(θ)|or
+ρr
+a=n/summationdisplay
+k=0n!(cosθ)2k(sinθ)2n−2k
+k!(n−k)!|A(n,N,θ)|2N!
+Nk(N−n+k)!×
+U/vextendsingle/vextendsingle/vextendsingle/vextendsinglen−k−N
+2/angbracketrightbigg/angbracketleftbigg
+n−k−N
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingleU†, (14)
+where
+/vextendsingle/vextendsingle/vextendsingle/vextendsinglek−N
+2/angbracketrightbigg
+≡/radicalbigg
+(N−k)!
+N!k!(J+)k|↓/an}bracketri}ht (15)
+is a symmetric Dicke state and the collective operator
+J+=√
+NU†C†U (16)
+is a unitary transformation of the quai-spin wave opera-
+tor with the unitary matrix
+U= exp
+−iN/summationdisplay
+j=1K·j/parenleftbig
+σj
+z+1/parenrightbig
+2
+. (17)
+Since only is the z−component quasi-spin operator
+contained in the unitary transformation, which will never
+change the quantum number of the symmetric Dicke
+bases, the density matrix in Eq. (14) only has the diag-
+onal elements. Thus the averages of JxandJyvanish as
+/an}bracketle{tJx/an}bracketri}ht=/an}bracketle{tJy/an}bracketri}ht= 0.In this situation, only the z−component
+collective operator Jzsurvives as
+/an}bracketle{tJz/an}bracketri}ht=/an}bracketle{tJz/an}bracketri}ht|θ=π
+2+δJz, (18)
+where/an}bracketle{tJz/an}bracketri}ht|θ=π/2=n−N/2is meanz−component spin
+of the Dicke state |n−N/2/an}bracketri}htand
+δJz=−nNcot2θ1F1(1−n;N−n+2;−Ncot2θ)
+(N−n+1)1F1(−n;N−n+1;−Ncot2θ)
+(19)
+is the deviation of the z−component spin. The fact that
+/an}bracketle{tJz/an}bracketri}htdoes not vanish means that the mean spin is along
+thez−direction. Since δJzvanishes when θ=π/2, its
+value can measure the mixture of the electromagnetic
+field and the quasi-spin wave.
+The spin squeezing has several measures, and we only
+list three typical and related ones as follows:[14, 17–19]
+ξ2
+1=4(∆J⊥)2
+min
+N, (20a)
+ξ2
+2=N2
+4/an}bracketle{tJ2/an}bracketri}htξ2
+1, (20b)
+ξ2
+3=λmin
+/an}bracketle{tJ2/an}bracketri}ht−N
+2. (20c)
+In Eq.(20a), the minimization is over all the direction
+denoted by ⊥, which is perpendicular to the mean spin
+direction /an}bracketle{tJ/an}bracketri}ht//angbracketleftbig
+J2/angbracketrightbig
+.For the quasi-spins in the dark state,4
+theξ2
+1andξ2
+2actually measure the spin squeezing of the
+atomic ensemble in the x−yplane. In Eq.(20c), λminis
+the minimal eigenvalue of matrix
+Γ = (N−1)γ+C, (21)
+where
+γkl=Ckl−/an}bracketle{tJk/an}bracketri}ht/an}bracketle{tJl/an}bracketri}ht(k,l∈ {x,y,z}) (22)
+is the covariance matrix and
+Ckl=1
+2/an}bracketle{tJlJk+JkJl/an}bracketri}ht (23)
+is the global correlation matrix. Parameters ξ2
+1,ξ2
+2,and
+ξ2
+3were defined by Kitagawa and Ueda, Wineland et al.,
+and Toth et al., respectively. If ξ2
+2<1 (ξ2
+3<1),the spin
+squeezing occurs, and we can safely say that the multi-
+partite state is entangled [19, 20]. Although we cannot
+say that the squeezed state according to ξ2
+1<1is entan-
+gled, it is indeed closely related to quantum entanglement
+[21].
+For the quasi-spins in the dark state, since the averages
+/an}bracketle{tJxJz/an}bracketri}ht=/an}bracketle{tJyJz/an}bracketri}ht= 0vanish as the same reason of the
+vanishing of /an}bracketle{tJx/an}bracketri}htand/an}bracketle{tJy/an}bracketri}ht, theξ2
+1andξ2
+3are simplified
+as [14, 21]
+ξ2
+1=2
+N/parenleftbig/angbracketleftbig
+J2
+x+J2
+y/angbracketrightbig
+−/vextendsingle/vextendsingle/angbracketleftbig
+J2
+−/angbracketrightbig/vextendsingle/vextendsingle/parenrightbig
+, (24a)
+ξ2
+3=min{ξ2
+1,ς2}
+4
+N2/an}bracketle{tJ2/an}bracketri}ht−2
+N, (24b)
+where
+ς2=4
+N2/bracketleftBig
+N(∆Jz)2+/an}bracketle{tJz/an}bracketri}ht2/bracketrightBig
+(25)
+characterizes the spin squeezing along the z−direction.
+Because/angbracketleftbig
+J2
+−/angbracketrightbig
+= 0for the quasi-spins in the dark state,
+ξ2
+1andξ2
+3are always greater than 1and thus there is no
+spin squeezing of the atomic ensemble in the x−yplane.
+In the following discussions, we only use
+ξ2
+3=ς2
+4
+N2/an}bracketle{tJ2/an}bracketri}ht−2
+N(26)
+to characterize the z−component spin squeezing of the
+atomic ensemble in the dark states.
+We consider the special case that when the wave vector
+difference Kbetween the quantum and the classical field
+is zero. Since U|K=0is identity matrix, the quai-spin
+wave operator in Eq. (16) actually is proportional to the
+collective operator and the reduced density matrix in Eq.
+(14) is spanned by the symmetric Dicke states with total
+quasi-spin J=N/2. Thus
+/angbracketleftbig
+J2/angbracketrightbig/vextendsingle/vextendsingle
+K=0=N
+2/parenleftbiggN
+2+1/parenrightbigg
+, (27)
+FIG. 2: (Color online) The squeezing parameter ζ2
+3versus
+the excitation number nforθ= 0(blue dashed line), θ=π/4
+(orange dotdashed line) and θ=π/2(red solid line). The
+total number of the atoms Nare chosen as 20. The horizontal
+purple dotted line as the base line represents the upper limi t of
+the spin-squeezing. The dark state is always squeezed excep t
+without excitation ( n= 0) or fully excited ( n=N).
+and the straightforward calculation gives
+ξ2
+3/vextendsingle/vextendsingle
+K=0=ς2=4
+N2/bracketleftBig
+N/angbracketleftbig
+J2
+z/angbracketrightbig
++(1−N)/an}bracketle{tJz/an}bracketri}ht2/bracketrightBig
+,(28a)
+/angbracketleftbig
+J2
+z/angbracketrightbig
+=/angbracketleftbig
+J2
+z/angbracketrightbig/vextendsingle/vextendsingle
+θ=π
+2+δJ2
+z, (28b)
+δJ2
+z=Ncot2θ/bracketleftbig
+2n+Ncot2θ
+−(N+1)(n+Ncot2θ)
+(N−n+1)Γ(n,N,θ)/bracketrightbigg
+,(28c)
+where
+/angbracketleftbig
+J2
+z/angbracketrightbig/vextendsingle/vextendsingle
+θ=π
+2=/parenleftbigg
+n−N
+2/parenrightbigg2
+(29)
+is mean z−component spin square of the Dicke state
+|n−N/2/an}bracketri}htand
+Γ(n,N,θ) =1F1(−n;N−n+2;−Ncot2θ)
+1F1(−n;N−n+1;−Ncot2θ).(30)
+For convenience we define equivalent squeezing parame-
+ter [14]
+ζ2
+3= max{1−ξ2
+3,0}, (31)
+which characterizes the spin squeezing when 0< ζ2
+3≤1.
+C. Numerical pictures of the atomic squeezing in
+dark states
+The squeezing parameters of the dark state |dn(θ)/an}bracketri}ht
+without wave vector difference ( K= 0) are illustrated in
+Fig. 2, 3 and 4. In all three figures, the total number of
+the atoms Nare chosen as 20. Since the reduced density
+matrix in Eq. (14) is symmetric when θis replaced by
+θ+π, the range of the θis chosen as [0,π). Fig. 2 shows5
+FIG. 3: (Color online) The polar plot of the squeezing pa-
+rameter ξ2
+3versus the parameter θfor the excitation number
+n= 0(blue dotted line), n= 10 (orange dashed line) and
+n= 20(red solid line). The total number of the atoms Nare
+chosen as 20. The greatest spin-squeezing is obtained for the
+symmetric Dicke state when the half of the quasi-spins are
+excited ( n=N/2) andθ=π/2.
+FIG. 4: (Color online) The contour plot of the squeezing pa-
+rameter ξ2
+3versus the excitation number nand parameter θ.
+The total number of the atoms Nare chosen as 20. The
+greatest spin-squeezing ( ζ2
+3= 1) is obtained for the symmet-
+ric Dicke state when the half of the quasi-spins are excited
+(n=N/2) andθ=π/2.
+that the squeezing parameter ζ2
+3varies with the excita-
+tion number nfor different θ. The dark state is always
+squeezed except without excitation ( n= 0) or fully ex-
+cited (n=N). It is noticed that when θ/ne}ationslash= 0,π/2, the
+dark state is squeezed even when the excitation number
+nis equal to the total number of the atoms N. Because
+the dark state actually mix the quasi-spins wave state
+FIG. 5: (Color online) The polar plot of the squeezing pa-
+rameter ξ2
+3versus the vector difference K. The total number
+of the atoms N, the excitation number nand the parameter
+θare chosen as 20,4andπ/2. Here, ξ2
+3= 1is the base-
+line to measure whether the dark state is squeezed or not.
+Obviously, the dark state is squeezed only when the vector
+difference K∈[−Kc,Kc].
+and the photon Fock state, this non-vanishing squeez-
+ing mainly results from the sub-Poisson distribution of
+the photons, which will be explicitly demonstrated in
+the last subsection of the Sec. II. Fig. 3 is the polar
+plot of the squeezing parameter ρ=ξ2
+3(θ)versusθfor
+different excitation numbers n. For the state without
+excitation ( n= 0), the atomic ensemble always has no
+spin-squeezing because all of the quasi-spins are in their
+ground states. The greatest spin-squeezing is obtained
+for the symmetric Dicke state when the half of the quasi-
+spins are excited ( n=N/2) andθ=π/2. Fig. 4 is
+the contour plot of the squeezing parameter ζ2
+3versus
+the excitation number nand parameter θ. Obviously,
+the greatest spin-squeezing ( ζ2
+3= 1drawn by red color)
+can be obtained by adiabatically manipulating the pa-
+rameterθtoπ/2at the half excitation of the quasi-spins
+(n=N/2).
+If the wave vectors between the quantum and the clas-
+sical field are different, the nonzero Kresults in the non-
+symmetrical Dicke bases in Eq. (14) and the spin squeez-
+ing decreases rapidly as the Kdeviates from 0.Fig. 5 is
+the polar plot of the squeezing parameter ρ=ξ2
+3(θ)ver-
+susK. The parameters are chosen as N= 20,n= 4,
+andθ=π/2. Sinceξ2
+3= 1is the baseline to measure the
+atomic squeezing, the dark state is squeezed only when
+the vector difference K∈[−Kc,Kc], whereKcis shown
+in Fig. 5. Fig. 6 is the contour plot of the squeezing pa-
+rameterζ2
+3versus the vector difference Kand parameter
+θ. The total number of the atoms Nand the excitation
+numbernare chosen as 20,4. In Fig. 6, [−Kc,Kc]is still6
+0.64 
+0.64 
+FIG. 6: (Color online) The contour plot of the squeezing pa-
+rameterξ2
+3versus the vector difference Kand the parameter θ.
+The total number of the atoms Nand the excitation number
+nare chosen as 20,4. Obviously, the dark state is squeezed
+only when the vector difference K∈[−Kc,Kc].
+the boundary for the vector difference Kwhen the dark
+states are squeezed.
+III. INTERNAL ENTANGLEMENT IN DARK
+STATES AND PHOTON STATISTICS
+A. Concurrence in dark states without decoherence
+As the physical observable to quantify the pairwise en-
+tanglement of spin-1/2, the concurrence is closely related
+to spin squeezing [21]. The concurrence is defined as [22]
+C= max{0,λ1−λ2−λ3−λ4}, (32)
+where the quantities λi(i= 1,2,3,4)are the square roots
+of the eigen-values in descending order of the matrix
+product
+̺12=ρ12(σ1y⊗σ2y)ρ∗
+12(σ1y⊗σ2y). (33)
+Here,ρ12is the two-spin reduced density matrix and ρ∗
+12
+is the complex conjugate of the ρ12.
+For the dark states, the two-quasi-spin reduced density
+matrix [23]
+ρ12=
+v+0 0u∗
+0w y0
+0y w0
+u0 0v−
+ (34)
+can be explicitly obtained by tracing out the degrees of
+freedom of all the other quasi-spins. It is clear to verify
+that for the dark states only the elements v±,u,w andyof theρ12survive as ( φ=K·l)
+v±=v±|θ=π
+2+δv±, (35a)
+w=w|θ=π
+2+δw, (35b)
+u=−wisinφ, (35c)
+y=wcosφ, (35d)
+where [23]
+v+|θ=π
+2=1
+4/parenleftbig
+1+2/angbracketleftbig
+σ1
+z/angbracketrightbig
++/angbracketleftbig
+σ1
+zσ2
+z/angbracketrightbig/parenrightbig
+=n(n−1)
+N(N−1), (36a)
+v−|θ=π
+2=1
+4/parenleftbig
+1−2/angbracketleftbig
+σ1
+z/angbracketrightbig
++/angbracketleftbig
+σ1
+zσ2
+z/angbracketrightbig/parenrightbig
+=(n−N)(n−N+1)
+N(N−1), (36b)
+w|θ=π
+2=1
+4/parenleftbig
+1−/angbracketleftbig
+σ1
+zσ2
+z/angbracketrightbig/parenrightbig
+=nN−n2
+N(N−1)(36c)
+are the corresponding elements of the reduced density
+matrix over the Dicke state |n−N/2/an}bracketri}htand
+δv+=B(N,θ)/bracketleftbig
+n+(Ncsc2θ+n−1)Λ(n,N,θ)/bracketrightbig
+,
+(37a)
+δv−=B(N,θ)/bracketleftbig
+n+(Ncot2θ−N+2n+1)Λ(n,N,θ)/bracketrightbig
+,
+(37b)
+δw=B(N,θ)/bracketleftbig
+−n−/parenleftbig
+n+Ncot2θ/parenrightbig
+Λ(n,N,θ)/bracketrightbig
+(37c)
+withB(N,θ) = cot2θ/(N−1)and
+Λ(n,N,θ) = 1−(N+1)
+(N−n+1)Γ(n,N,θ), (38)
+whereΓ(n,N,θ)is defined in Eq. (30).
+Thus the concurrence of the dark state is given by
+C= max/braceleftbig
+0,2/parenleftbig
+w|cosφ|−√v+v−/parenrightbig/bracerightbig
+. (39)
+Fig. 7 is the contour plot of the concurrence Cversus the
+vector difference Kand the parameter θ. In compari-
+son with the one region [−Kc,Kc]for generating atomic
+squeezing, there are three regions for generating pairwise
+entanglement. This is also shown in Fig. 8, which is the
+sections of the atomic squeezing and the concurrence.
+Obviously, in the vicinity of the K=π, the concurrence
+appears but atomic squeezing does not. In Fig. 7 and 8,
+the total number of the atoms Nand the excitation num-
+bernare chosen as 20,4. In the next section, the evolve-
+ment of the concurrence as well as the spin-squeezing
+under various decoherence channels will be considered.
+B. Sub-Poisson distribution of the photons
+We have shown that the atomic squeezing for infor-
+mation storage can be generated through the adiabat-
+ically manipulating the angle θ. In another aspect, as7
+FIG. 7: (Color online) The contour plot of the concurrence
+Cversus the vector difference Kand the parameter θ. The
+total number of the atoms Nand the excitation number nare
+chosen as 20and4. Here we find three regions for generat-
+ing concurrence rather than one region for generating atomi c
+squeezing.
+FIG. 8: (Color online) The sections of the atomic squeezing ζ2
+3
+(blue solid blue) and concurrence C(red dashed line) versus
+the vector difference K. The total number of the atoms N,
+the excitation number nand the parameter θare chosen as
+20,4andπ/2. Obviously, in the vicinity of the K=π, the
+concurrence appears but atomic squeezing does not.
+the photon dressed collective atom state the dark states
+mix the atomic ensemble state and the photon states.
+In our setup, the conservative quantity is the summa-
+tion of the z−component collective operator Jzand the
+number operator n=a†aof the photons. The multi-
+partite quantum correlation in dark states actually is the
+z−component spin squeezing for the dark states, which
+would result from the quantum correlation of the pho-
+tons. Usually, such photonic quantum correlation can be
+measured by the sub-Poisson distribution of the photons
+FIG. 9: (Color online) The sections of the atomic squeezing ζ2
+3
+(blue solid blue) and sub-Poisson distribution sp(red dashed
+line) versus the parameter θ. The total number of the atoms
+N, the excitation number nand the vector difference Kare
+chosen as 20,4and0. When the θincreases from 0toπ/2, the
+atomic squeezing increases and the sub-Poisson distributi on
+decreases simultaneously and vise vesa when the θincreases
+fromπ/2toπ.
+defined as
+sp=1
+N/parenleftbig/angbracketleftbig
+∆n2/angbracketrightbig
+−/an}bracketle{tn/an}bracketri}ht/parenrightbig
+=1
+N/parenleftBig/angbracketleftbig
+a†a†aa/angbracketrightbig
+−/angbracketleftbig
+a†a/angbracketrightbig2/parenrightBig
+. (40)
+If the photonic number fluctuation
+/angbracketleftbig
+∆n2/angbracketrightbig
+=/angbracketleftbig
+n2/angbracketrightbig
+−/an}bracketle{tn/an}bracketri}ht2(41)
+is smaller than the average of the particle number /an}bracketle{tn/an}bracketri}htof
+the photons ( sp<0), the dark states actually has the
+sub-Poisson distribution.
+The straightforward calculation gives
+/angbracketleftbig
+a†a†aa/angbracketrightbig
+=(n−1)nN2cot4θ
+(N−n+2)(N−n+1)×
+1F1(2−n;N−n+3;−Ncot2θ)
+1F1(−n;N−n+1;−Ncot2θ),(42a)
+/angbracketleftbig
+a†a/angbracketrightbig
+=nNcot2θ
+(N−n+1)×
+1F1(1−n;N−n+2;−Ncot2θ)
+1F1(−n;N−n+1;−Ncot2θ).(42b)
+Substituting the relevant expectation values into Eqs.
+(40) leads to the explicit expression of the sub-Poisson
+distribution of the photons sp, which always follow the
+atomic squeezing ζ2
+3. Fig. 9 illustrates the numerical rela-
+tionship between sub-Poisson distribution and the atomic
+squeezing. When the θincreases from 0toπ/2, the
+atomic squeezing increases and the sub-Poisson distri-
+bution decreases simultaneously and vise vesa when the
+θincreases from π/2toπ. From the above discussion,
+the greatest atomic squeezing and the concurrence can
+be generated when the parameter θis manipulated to
+π/2, which means all the sub-Poisson distribution of the
+photons converts to the atomic squeezing.8
+In this section, we just consider the adiabatic manip-
+ulation without any decoherence process. In the next
+section, we will take the typical three atomic decoher-
+ence channels into consideration to estimate the sudden
+death of the atomic squeezing and concurrence. Since
+the coherence time of photons is much larger than atomic
+one, the photonic decoherence such as the dephasing due
+to the imperfect single-photon indistinguishability [24] is
+not considered below.
+IV. ATOMIC SQUEEZING AND PAIRWISE
+ENTANGLEMENT UNDER DECOHERENCE
+CHANNELS
+A. Decoherence channels
+Through the adiabatically manipulating the angle θ,
+the spin-squeezing and the concurrence are generated for
+information storage. However, due to the existence of the
+environment, they are irreversible depressed and vanish
+eventually. Such decoherence processes are usually de-
+scribed by three typical decoherence channels: the ampli-
+tude damping channel (ADC), the phase damping chan-
+nel (PDC) and the depolarizing channel (DPC) [21, 25].
+They are defined by some maps of the elements of the
+density matrix |i/an}bracketri}ht/an}bracketle{tj|. Since the ADC is an energy-losing
+process and thus it forces the initial state to dissipate
+into the ground state, the map is defined as
+|i/an}bracketri}ht/an}bracketle{ti| →(1−p)|i/an}bracketri}ht/an}bracketle{ti|+p|0/an}bracketri}ht/an}bracketle{t0|, (43a)
+|i/an}bracketri}ht/an}bracketle{tj| →(1−p)1
+2|i/an}bracketri}ht/an}bracketle{tj|,(i/ne}ationslash=j). (43b)
+The PDC is phase-losing process and the corresponding
+map is described by
+|i/an}bracketri}ht/an}bracketle{tj| →(1−p+pδi,j)|i/an}bracketri}ht/an}bracketle{tj|. (44)
+The DPC is polarization-losing process and the corre-
+sponding map is represented by
+|i/an}bracketri}ht/an}bracketle{tj| →(1−p)|i/an}bracketri}ht/an}bracketle{tj|+pδi,jI
+2. (45)
+The behaviors of the spin-squeezing and the concur-
+rence under these decoherence channels will be present
+in the following sub sections. Here, pis the decoherence
+strength whose range is [0,1].Whenp= 0there is no
+decoherence and when p= 1the decoherence processes
+are completed.
+If we fix the initial values of the spin-squeezing and the
+concurrence of the dark state, the dark state degrades to
+a product state with single quasi-spin state as a compo-
+nent when the decoherence strength pincreases from 0.
+Thus the spin-squeezing and the concurrence would have
+sudden death. Before the sudden death happens, the
+information storage can be in progress safely. In other
+word, the longer time the spin-squeezing and the concur-
+rence survive, the better the information storage can be
+achieved.From the above expression of the spin-squeezing and
+the concurrence, we notice that if we know the expec-
+tations/an}bracketle{tJz/an}bracketri}ht,/angbracketleftbig
+J2
+z/angbracketrightbig
+and/angbracketleftbig
+σ1
+z/angbracketrightbig
+, and the correlations/angbracketleftbig
+σ1
+zσ2
+z/angbracketrightbig
+,
+all the spin-squeezing and the concurrence can be deter-
+mined. We will give explicit analytical expressions for
+them under three decoherence channels.
+B. Amplitude-damping channel
+Based on the map in Eq. (43) for ADC, one can find
+the following relations for single quasi-spin operators [1 4]
+/angbracketleftbig
+σ1
+z(p)/angbracketrightbig
+=s/angbracketleftbig
+σ1
+z/angbracketrightbig
+0−p, (46a)
+/angbracketleftbig
+σ1
+zσ2
+z(p)/angbracketrightbig
+=s2/angbracketleftbig
+σ1
+zσ2
+z/angbracketrightbig
+0−2sp/angbracketleftbig
+σ1
+z/angbracketrightbig
+0+p2,(46b)
+/angbracketleftbig
+σ1
+ασ2
+β(p)/angbracketrightbig
+=s/angbracketleftbig
+σ1
+ασ2
+β/angbracketrightbig
+0(α,β=x,y). (46c)
+Hereafter, we define s= 1−p. Then the corresponding
+relations for the collective operators are
+/angbracketleftbig
+J2
+z(p)/angbracketrightbig
+=s2/angbracketleftbig
+J2
+z/angbracketrightbig
+0+(N−1)sp/an}bracketle{tJz/an}bracketri}ht0
++N2
+4p/parenleftbigg
+p+2
+Ns/parenrightbigg
+, (47a)
+/an}bracketle{tJz(p)/an}bracketri}ht=s/an}bracketle{tJz/an}bracketri}ht0−N
+2p, (47b)
+/angbracketleftbig
+J2(p)/angbracketrightbig
+=s/angbracketleftbig
+J2/angbracketrightbig
+0−sp/angbracketleftbig
+J2
+z/angbracketrightbig
+0+(N−1)sp/an}bracketle{tJz/an}bracketri}ht0
++N
+4p(Np+2s). (47c)
+Substituting the relevant expectation values and the cor-
+relation function into Eqs. (25,26) leads to the explicit
+expression of the spin-squeezing parameters
+ξ2
+3(p)A=ς2(p)
+4
+N2/an}bracketle{tJ2(p)/an}bracketri}ht−2
+N(48)
+with
+ς2(p) =s2ς2
+0+24
+N(N−1)sp/an}bracketle{tJz/an}bracketri}ht0+p(1+s),(49)
+whereς2
+0is the initial spin-squeezing parameter defined
+in Eq. (25).
+In the same case, in order to investigate the evolution
+of the concurrence, the two-qubit density matrix is shown
+as
+ρA
+12=
+vA
++0 0uA,∗
+0wAyA0
+0yAwA0
+uA0 0 vA
+−
+(50)
+withvA
++=s2v+, yA=sy, uA=suand
+vA
+−=s2v+−s/an}bracketle{tσz/an}bracketri}ht0+p, (51a)
+wA=s2w+1
+2sp/angbracketleftbig
+σ1
+z/angbracketrightbig
+0+1
+2sp, (51b)9
+(a) (b) 
+ (a) (b) 
+(c) 
+FIG. 10: (Color online) The sudden death of the atomic
+squeezing ζ2
+3(blue solid line) and the concurrence C(red
+dashed line) under ADC. (a)Usually, the atomic squeezing
+ζ2
+3disappear later than the concurrence C. (b)The atomic
+squeezing ζ2
+3can disappear earlier than the concurrence C.
+(c)The atomic squeezing ζ2
+3may reach its maximum and then
+disappear near the maximum decoherence strength p= 1.
+The parameters are chosen as N= 20,n= 16,K= 0and
+(a)θ=π/3,(b)θ=π/2,(c)θ= 0.673π.
+wherev±,y,uandware initial elements defined in Eq.
+(35). Thus we obtain the time evolution of the concur-
+rence as
+CA= max/braceleftbig
+0,CA
+1,CA
+2/bracerightbig
+, (52a)
+CA
+1= 2/parenleftbigg
+sw|cosφ|−/radicalBig
+vA
++vA
+−/parenrightbigg
+, (52b)
+CA
+2= 2/parenleftbig
+sw|sinφ|−wA/parenrightbig
+. (52c)
+Hereafter, we only consider the case at the vector differ-
+enceK = 0 because the quantum correlations decrease
+rapidly when the vector difference deviates from 0shown
+in Fig. 5.
+The subindex Ain the above Eqs. (48,50,51,52) rep-
+resents that the evolutions of the spin-squeezing and the
+concurrence are taken under ADC. The sudden death of
+the spin-squeezing and the concurrence implied in Eq.
+(48,52) are shown in Fig. 10. Usually, the atomic squeez-
+ingζ2
+3disappear later than the concurrence Cas shown
+in Fig. 10(a). However for ADC, the atomic squeez-
+ingζ2
+3can disappear earlier than the concurrence C
+(Fig. 10(b)). With appropriate parameters, the atomic
+squeezing ζ2
+3may reach its maximum and then disap-
+pear near the maximum decoherence strength p= 1
+(Fig. 10(c)). The parameters are chosen as N= 20,n=
+16,K= 0and (a)θ=π/3,(b)θ=π/2,(c)θ= 0.673π.
+C. Phase-damping channel
+For PDC described by the map in Eq. (44), one can
+0.64 0.05 
+0 0
+(a) (b) 
+FIG. 11: (Color online) The sudden death of (a) the atomic
+squeezing ζ2
+3and (b) the concurrence Cunder PDC. The pa-
+rameters are chosen as N= 20,n= 16 andK= 0. The
+atomic squeezing ζ2
+3always disappear later than the concur-
+renceC.
+find the following relations for single quasi-spin opera-
+tors [14]
+/an}bracketle{tσz(p)/an}bracketri}ht=/an}bracketle{tσz/an}bracketri}ht0, (53a)/angbracketleftbig
+σ1
+zσ2
+z(p)/angbracketrightbig
+=/angbracketleftbig
+σ1
+zσ2
+z/angbracketrightbig
+0, (53b)
+/angbracketleftbig
+σ1
+ασ2
+β(p)/angbracketrightbig
+=s2/angbracketleftbig
+σ1
+ασ2
+β/angbracketrightbig
+0(α,β=x,y).(53c)
+Then the corresponding relations for the collective oper-
+ators are/angbracketleftbig
+J2
+z(p)/angbracketrightbig
+=/angbracketleftbig
+J2
+z/angbracketrightbig
+0,/an}bracketle{tJz(p)/an}bracketri}ht=/an}bracketle{tJz/an}bracketri}ht0and
+/angbracketleftbig
+J2(p)/angbracketrightbig
+=s2/angbracketleftbig
+J2/angbracketrightbig
+0+(1−s2)N
+2. (54)
+Substituting the relevant expectation values and the cor-
+relation function into Eqs. (25,26) leads to the explicit
+expression of the spin-squeezing parameters
+ξ2
+3(p)P=ς2
+0
+4
+N2/an}bracketle{tJ2(p)/an}bracketri}ht−2
+N. (55)
+In the same sense to investigate the evolution of the con-
+currence, the two-qubit density matrix is present as
+ρP
+12=
+vP
++0 0uP,∗
+0wPyP0
+0yPwP0
+uP0 0 vP
+−
+(56)
+with relations vP
+±=v±, wP=w, yP=s2y,anduP=
+s2u. Thus we obtain the evolution of the concurrence as
+CP= max/braceleftbig
+0,2/parenleftbig
+s2w|cosφ|−√v+v−/parenrightbig/bracerightbig
+. (57)
+The subindex Pin the above Eqs. (55,56,57) represents
+that the time evolutions of the spin-squeezing and the
+concurrence are taken under PDC. The sudden death of
+the spin-squeezing and the concurrence implied in Eq.
+(55,57) are shown in Fig. 11. In contrast of the ADC
+case, the atomic squeezing ζ2
+3always disappear later than
+the concurrence C.10
+0.64 0.05 
+0 0
+(a) (b) 
+FIG. 12: (Color online) The sudden death of (a) the atomic
+squeezing ζ2
+3and (b) the concurrence Cunder DPC. The pa-
+rameters are chosen as N= 20,n= 4andK= 0. The atomic
+squeezing ζ2
+3always disappear later than the concurrence C.
+D. Depolarizing channel
+Based on the map in Eq. (45) for DPC, one can find
+the following relations for single quasi-spin operators [1 4]
+/an}bracketle{tσz(p)/an}bracketri}ht=s/an}bracketle{tσz/an}bracketri}ht0, (58a)/angbracketleftbig
+σ1
+ασ2
+β(p)/angbracketrightbig
+=s2/angbracketleftbig
+σ1
+ασ2
+β/angbracketrightbig
+0(α,β=x,y,z).(58b)
+Then the corresponding relations for the collective oper-
+ators are /an}bracketle{tJz(p)/an}bracketri}ht=s/an}bracketle{tJz/an}bracketri}ht0and
+/angbracketleftbig
+J2
+z(p)/angbracketrightbig
+=s2/angbracketleftbig
+J2
+z/angbracketrightbig
+0+/parenleftbig
+1−s2/parenrightbigN
+4, (59a)
+/angbracketleftbig
+J2(p)/angbracketrightbig
+=s2/angbracketleftbig
+J2/angbracketrightbig
+0+(1−s2)3N
+4.(59b)
+Substituting the relevant expectation values and the cor-
+relation function into Eqs. (25,26) leads to the explicit
+expression of the spin-squeezing parameters
+ξ2
+3(p)D=s2ς2
+0+1−s2
+4
+N2/an}bracketle{tJ2(p)/an}bracketri}ht−2
+N. (60)
+In the same sense to investigate the evolution of the con-
+currence, the two-qubit density matrix is present as
+ρD
+12=
+vD
++0 0uD,∗
+0wDyD0
+0yDwD0
+uD0 0 vD
+−
+(61)
+with relations yD=s2y, uD=s2uand
+vD
+±=s2+s
+2v±+s2−s
+2v∓+1−s2
+4, (62a)
+wD=s2w0+1−s2
+4. (62b)
+Thus we obtain the evolution of the concurrence as
+CD= max/braceleftbigg
+0,2/parenleftbigg
+s2w|cosφ|−/radicalBig
+vD
++vD
+−/parenrightbigg/bracerightbigg
+.(63)The subindex Din the above Eqs. (60,61,63,63) rep-
+resents that the evolutions of the spin-squeezing and the
+concurrence are taken under DPC. The sudden death of
+the spin-squeezing and the concurrence implied in Eq.
+(60,63) are shown in Fig. 12. In contrast of the ADC
+case, the atomic squeezing ζ2
+3always disappear later than
+the concurrence C.
+V. OPTIMAL TIME FOR GENERATING
+ATOMIC SQUEEZING
+Now, two competitive processes dominate generating
+the spin-squeezing. One is adiabatically manipulating
+the parameter θfrom0toπ/2, which increases the spin-
+squeezing and the concurrence for storing the information
+of phonons into the atomic ensemble. However, the other
+process resulting from the various decoherence channels
+decreases the spin-squeezing and the concurrence, which
+means that the system loses information continuously.
+Therefore, the competition between this two processes
+leads to the existence of an optimal time for storing infor-
+mation, after which the information stored in the atomic
+ensemble is always losing. We can define two time scales:
+if only considering the adiabatic manipulation, one time
+scale ist1representing the time when the spin squeezing
+increases to the half maximum value; if only considering
+the decoherence processes, the other time scale is t2rep-
+resenting the time when the spin squeezing decreases to
+the half maximum value. Only when t1> t2the optimal
+time exists.
+We would like to demonstrate such competition in
+a typical quantum memory based on the cold87Rb
+atomic ensemble, which is released from a magneto-
+optical trap at a temperature of about 100µK[7].
+In such quantum memory, the ground state |g/an}bracketri}ht, the
+metastable state |m/an}bracketri}htand the excited state |e/an}bracketri}htare chosen
+as/vextendsingle/vextendsingle5S1/2,F= 1,mF= 1/angbracketrightbig
+,/vextendsingle/vextendsingle5S1/2,F= 2,mF=−1/angbracketrightbig
+and/vextendsingle/vextendsingle5S1/2,F= 2,mF= 0/angbracketrightbig
+, respectively. An off-resonant
+σ−-polarized write pulse contributes to the transition
+from|g/an}bracketri}htto|e/an}bracketri}htand the Stokes photon with σ−polar-
+ization associates with the transition from |e/an}bracketri}htto|m/an}bracketri}ht.
+The dark state for storing the photonic information is
+described in Eq. (7).
+To realize the adiabatic creation of the non-classical
+correlation, we adopt the hyperbolic tangent type pulse
+sequence for both Rabi frequencies [26] as
+g(t) = Ω m/bracketleftbigg
+1−tanh(a
+t+a
+t−τ)/bracketrightbigg
+,(64a)
+Ω(t) = Ω m/bracketleftbigg
+1+tanh(a
+t+a
+t−τ)/bracketrightbigg
+(64b)
+instead of the usual gaussian type ones [27] or solitonary
+type ones [28]. Here, Ωmis the maximum Rabi frequency,
+τis the pulse length and acorresponds to the half width
+of the pulse. The pulse sequences of both Rabi frequen-
+cies are depicted in Fig. 13. Only at the time interval11
+FIG. 13: (Color online) Optical pulse sequences for both Rab i
+frequencies. The red dashed line and black solid line repre-
+sent the Rabi frequencies Ω(t)andg(t), respectively. The
+adiabatic following condition is Ωmτ≫1.
+[0,τ]are both sequences applied to adiabatically manip-
+ulate the dark state in Eq. (7). Two Rabi frequencies
+respectively reach their maximum value Ωmat the initial
+timet= 0and the terminal time t=τ, which guarantees
+that the angle θvaries from 0toπ/2and thus the infor-
+mation of the photons is stored into the atomic ensemble
+at the time interval [0,τ].
+Actually, due to the adiabatic manipulation, the time
+scalet1for generating spin squeezing is limited by the
+adiabatic condition
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig
+n(t)|˙dn(t)/angbracketrightBig
+En−Ed/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1 (65)
+with eigenstates |n(t)/an}bracketri}ht(the corresponding eigen-energy
+En) and the time-dependent dark states |dn(t)/an}bracketri}ht(the cor-
+responding eigen-energy Ed). Here,
+/angbracketleftBig
+n(t)|˙dn(t)/angbracketrightBig
+≡/an}bracketle{tn(t)|∂
+∂tH(t)|dn(t)/an}bracketri}ht
+En−Ed(66)
+describes the variation due to the time-dependent Hamil-
+tonian. Thus the condition for the adiabatic following is
+Ωmτ≫1.The typical pulse length τis about150µs[26],
+which means the maximum value of the Rabi frequencies
+Ωmshould much larger than 6.67×103Hz. According to
+the pulse sequence in Eq. (64), the time scale t1can be
+determined when θ(t1)is tuned to π/6as
+t1=2a+τr−√
+4a2+τ2r2
+2r, (67)
+wherer= tanh−1[(6−π)/(6+π)].Whenatends to in-
+finity, the longest time tL
+1≃1/2τ= 75µsis obtained.
+In the other hand, we also assume the exponential de-
+cays for the decoherence processes with the decoherence
+strength
+p(t) = 1−e−γt, (68)
+which leads to the time scale
+t2=1
+γ. (69)
+FIG. 14: (Color online) The evolution of the atomic squeez-
+ingζ2
+3(blue solid line) and the concurrence C(red dashed
+line). The parameters are chosen as N= 20,n= 4,K= 0.
+The maximum of the atomic squeezing and the concurrence
+are respectively obtained at ts≈0.12γ−1= 120µsand
+tc≈0.09γ−1= 90µs.
+For example, for the cold trapped87Rbatomic ensemble,
+the typical dephasing damping rate γ= 103Hzand the
+corresponding dephasing time t2= 1ms[7, 29].
+The evolution of the spin-squeezing and the concur-
+rence is shown in Fig. 14. Obviously, The maximum of
+the atomic squeezing and the concurrence are respec-
+tively obtained at ts≈0.12γ−1= 120µsandtc≈
+0.09γ−1= 90µs. Though the Fig. 14 is drawn under
+PDC, the optimal times can also be obtained under ADC
+and DPC.
+Such competition can also be observed by measuring
+the cross-correlation [7, 30]
+gS,AS= 1+CΓ(t) (70)
+as a function of the time, where Cis a fitting parameter
+determined by the excitation probability together with
+the background noise in the anti-Stokes channel. Here,
+Γ(t) =|/an}bracketle{tΨmax|Ψ(t)/an}bracketri}ht|2(71)
+is the retrieval efficiency, which actually characterizes
+the complete time-dependence properties of the cross-
+correlation. For the dark state we discuss above, to esti-
+mate the adiabatical creation and the decoherence chan-
+nels at the same time, we choose |Ψmax/an}bracketri}htas the collective
+atomic state |Mn/an}bracketri}htand thus the retrieval efficiency is ob-
+tained as
+Γ(t) =n!(N−n)!
+N!2F1(n−N,−n;1;e−2γt)
+1F1(−n;N−n+1;−Ncot2θ(t)).
+(72)
+Here,2F1(a,b;c;z)is generalized Kummer hypergeomet-
+ric function [16]. The retrieval efficiency is depicted in
+Fig. 15, in which the maximum of the retrieval efficiency
+exists when the same condition t1> t2is satisfied.12
+FIG. 15: (Color online). The retrieval efficiency as a functio n
+of the time. The parameters are chosen as N= 20,n=
+4,K= 0. The maximum retrieval efficiency is also obtained
+around the 0.12γ−1= 120µs.
+VI. CONCLUSION
+We investigate the multipartite correlations character-
+ized by the atomic squeezing beyond the pairwise en-
+tanglement in the dark states, which are proposed for
+the quantum information storage based on electromag-
+netically induced transparency mechanism. The atomic
+squeezing and the pairwise entanglement can be created
+by adiabatically manipulating the Rabi frequency of the
+the classical light field on the atomic ensemble. The
+atomic squeezing in dark states converts from the sub-Poisson distribution of the photons. The greatest atomic
+squeezing and the concurrence can be generated when
+the parameter θis manipulated to π/2, which means all
+the sub-Poisson distribution of the photons converts to
+the atomic squeezing.
+We also consider the sudden death for the atomic
+squeezing and the pairwise entanglement under various
+decoherence channels. For the three typical decoherence
+channels, the sudden deaths of the atomic squeezing hap-
+pens later than the sudden death of the concurrence.
+According to the above investigation, an optimal time
+for generating the greatest atomic squeezing and pair-
+wise entanglement is obtained by studying in details the
+competition between the adiabatic creation of quantum
+correlation in the atomic ensemble and the decoherences
+we describe with decoherence channels.
+Acknowledgments
+The work is supported by National Natural Science
+Foundation of China under Grant Nos. 10547101 and
+10604002, the National Fundamental Research Program
+of China under Grant No. 2006CB921200. X. Wang is
+supported by NSFC with grant No.10874151, 10935010,
+NFRPC with grant No. 2006CB921205; Program for
+New Century Excellent Talents in University (NCET),
+and Science Fundation of Chinese University.
+[1] C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature
+(London) 409, 490 (2001).
+[2] M. D. Lukin, Rev. Mod. Phys. 75, 457 (2003).
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+022314 (2002).
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+arXiv:1001.0447v1  [math.AG]  4 Jan 2010OPEN STRING INVARIANTS AND MIRROR CURVE OF THE
+RESOLVED CONIFOLD
+JIAN ZHOU
+Abstract. Forthe resolvedconifold withone outer D-braneinarbitrar yfram-
+ing, we present some results for the open string partition fu nctions obtained by
+some operator manipulations. We prove some conjectures by A ganagic-Vafa
+and Aganagic-Klemm-Vafa that relates such invariants to th e mirror curve of
+the resolved conifold. This establishes local mirror symme try for the resolved
+confolds for holomorphic disc invariants. We also verify an integrality conjec-
+ture of such invariants by Ooguri-Vafa in this case and prese nt closed formulas
+for some Ooguri-Vafa type invariants in genus 0 and arbitrar y genera.
+1.Introduction
+The resolved conifold is probably the most studied noncompact Calab i-Yau 3-
+fold, comparable as its compact counterpart the quintic in P4. It provides testing
+ground for new developments and is a rich source of many new discov eries. In
+this work we will study it as the first nontrivial local Calabi-Yau geome tries for
+which we try to establish local mirror symmetry in the new formalism of the local
+B-theory [25, 8], which involves open string invariants.
+The localA-theory of the resolved conifold has played a crucial role in the de-
+velopment of duality between topological string theory and Chern- Simons theory.
+Witten [32] proposed that the large N expansion of Chern-Simons link invariants
+can be identified with the genus expansion of open string invariants o f the deformed
+conifoldT∗S3, which counts holomorphic maps from Riemann surfaces boundaries
+toT∗S3, withhboundary components mapped to the Lagrangian submanifold S3.
+Gopakumar-Vafa [15] proposed that after summing over the numb erhof boundary
+components, one gets from the large Nexpansion of the 3-manifold invariant of S3
+the genus expansion of closed string invariants of the resolved con ifold. Ooguri and
+Vafa [31] extended this and proposed that the large Nexpansion of link invariants
+corresponds to open string invariants of the resolved conifold, as sociated with some
+Lagrangiansubmanifolds correspondingto the link. Mari˜ no-Vafa [26] extended this
+further by considering framed knots. The duality has been extend ed to other local
+Calabi-Yaugeometries[4] and a formalismcalled the topologicalverte x [2] has been
+developed to compute open and closed string invariants of local tor ic Calabi-Yau
+geometries.
+Much of these developments in the physics literature has been made mathemat-
+ically rigorous, using the method of localization. This method reduces the calcula-
+tions to Hodge integrals which are intersection numbers on the Delign e-Mumford
+moduli spaces. A byproduct in [26] is a conjectural closed formula f or some Hodge
+integrals, by comparing with the formal localization calculations perf ormed for the
+resolved conifold by Katz and Liu [18]. This has played no role in later phy sical
+12 JIAN ZHOU
+development, but it turns out to provide the crucial clue for the ma thematical de-
+velopment. The formula for Hodge integrals conjectured by Mari˜ no and Vafa has
+been proved [21, 28] and generalized [33, 22, 20], and all such formu las have played
+an important role in the mathematical development of the calculation s of open and
+closed string invariants of local Calabi-Yau geometries [34, 20]. Note mathemati-
+cally the open string invariants is very difficult to define in symplectic ge ometry.
+Li and Song [23] notice that in the case of the resolved conifold one c an use the
+relative moduli space in algebraic geometry to define open string inva riant. This
+is also the approach taken in the mathematical theory of the topolo gical vertex
+[20] to bypass the difficulties in the definition of open string invariants for general
+local Calabi-Yau geometries. So now we have a mathematically rigorou s way to
+define and compute certain open string invariants of local Calabi-Ya u geometries
+in algebraic geometry, which should have a parallel theory in symplect ic geometry
+as shown in the resolved conifold case in [23].
+Local B-theory and local mirror symmetry in genus 0 have also been studied
+extensively in physics literature. For closed strings, see e.g. [11]. Fo r open string,
+AganagicandVafa[5]conjecturedarelationshipbetweengenus0o penstringinvari-
+antsandtheequationforthemirrorcurvegivenbytheHori-Iqbal- Vafaconstruction
+[17]. This was extended by Aganagic, Klemm and Vafa [3] to include the e ffect of
+framing, and the case of the resolved conifold was treated in [26]. K lemm and
+Zaslow [19] studied local B-theory from the point of view of holomorp hic anomaly
+equation [6]. For another approach see [1]. Recently there has appe ared a new
+formalism of the local B-theory in arbitrary genera. Extending res ults in the the-
+ory of matrix models, Eynard-Orantin [13] developed a formalism tha t recursively
+defines some differentials from a plane algebraic curve. See also their more recent
+work [14]. These differentials correspond to n-point open string partition functions
+in arbitrary genus. Mari˜ no [25] proposed that this formalism can b e used to define
+unambiguously the local B-theory for local toric Calabi-Yau geomet ries, where the
+algebraic curves are taken to be the mirror curves. Bouchard, Kle mm, Mari˜ no,
+Pasquetti [8] extended this proposal to more general setting ba sed on [3], including
+some conjectures on the framing transformation and the effects of phase changes
+of the D-branes. Bouchard and Mari˜ no [9] made the proposal mo re explicit in the
+case ofC3with one D-brane, incorporating the ideas from [3]. Mathematically,
+this involves one-partition Hodge integrals in the Mari˜ no-Vafa for mula [26, 21, 28].
+They also derived a related conjecture for Hurwitz number in the sa me paper.
+Borot-Eynard-Safnuk-Mulase [7] and Eynard-Safnuk-Mulase [12 ] have proved
+Bouchard-Mari˜ noConjectureforHurwitznumbersbytwodiffer entmethods. Based
+on the method in [12], two proofs, one by Chen [10] and one by the aut hor [37],
+haveappearedforBouchard-Mari˜ noConjectureforthecase ofC3withoneD-brane.
+The proof for the case of C3with three D-branes, i.e., the case of topological vertex
+[2, 20], has been given by the author in a more recent work [38]. Our s trategy for
+the study the case of more general local Calabi-Yau geometries is t o generalize the
+earlier results. It has three steps:
+Step 1. Compute the genus zero one point functions of open string invariants and
+relate it to the mirror curve.
+Step 2. Compute the genus zero two point functions of open string invariants and
+relate it to the Bergman kernel of the mirror curve.OPEN STRING INVARIANTS AND MIRROR CURVE 3
+Step 3. Establish the Eynard-Orantin type recursion relations, fa vorably by the
+cut-and-join equations.
+In this work we focus on the first step for the resolved conifold, lea ving the other
+two steps to subsequent work in progress. We will start with the ca lculation of the
+open string invariants by topological vertex. Such calculations invo lve summations
+over partitions and so are often very complicated. A key technique we use in
+this work is to transform thecalculations to vacuum expectation va lues of some
+operators developed by Okounkov-Pandharipande [29, 30] origin ally for Hurwitz
+numbers. This was first proposed in an earlier work [36] for the case of inner
+brane, whereas in this work we deal with the case of out brane. How ever we go
+one step further by finding simple closed formula for one-point func tions of open
+string invariant and hence establish the conjectured relation with t he mirror curve
+proposed in [5, 3]. This establishes local mirror symmetry for the res olved conifold
+in the case of holomorphic disc invariants. The case of inner brane ca n be treated
+similarly with details to be presented in a separate paper. As another application,
+we also check the Ooguri-Vafa Integrality Conjecture [31] for one -point functions in
+arbitrary framing. We present a closed formula for holomorphic disc numbers and
+a closed formula for some integral invariants in arbitrary genera.
+Another application of our results is that it leads to a proof of the fu ll Mari˜ no-
+Vafa Conjecture that identifies open string invariants of the reso lved conifold with
+the link invariants of the unknot, i.e., the quantum dimensions. The de tails will
+appear in a forthcoming paper.
+ThenextstepinestablishingthelocalmirrorsymmetryintheBKMPCo njecture
+is to compute the two-point functions in genus zero and relate it to t he Bergman
+kernel of the mirror curve. Our method in this paper can also be use d to evaluate
+the two-point functions, but the combinatorics is much more complic ated. We will
+address this in a subsequent work.
+Note in this paper an important simplifying trick we have exploited is tha t after
+theconversionfromthe topologicalverticestoHodgeintegrals,o necanchoosesome
+parameterssuitablysothatmostofthetermsvanishandforthen onvanishingterms
+one can use the Mumford’s relations. For more general local Calabi- Yau geometries
+this may not work anymore.
+The rest of the paper is arranged as follows. In Section 2 we start w ith the
+topological vertex computations of open string invariants of the r esolved conifold
+in general framing, and show that after conversion to 2-partition Hodge integrals,
+some results in [33] can be used to simplify the calculations. In Section 3 we
+adapt the proposal in the inner brane case in [36] to the outer bra ne case, and
+reformulate the partition functions in terms of operators introdu ced in [29]. We
+indicate the method of evaluating the n-point functions by operator manipulations,
+and carry out in Section 4 the calculation of one-point functions by t his method.
+Closed formulas in genus 0 and in general genera are presented in th is section.
+As an application, we discuss in Section 5 the Ooguri-Vafa integrality p roperties
+of disc invariants of the resolved conifold in general framing. In the final Section
+6, we establish the Aganagic-Vafa Conjecture that relates count ing of open string
+disc invariants to the mirror curve, and the Aganagic-Klemm-Vafa C onjecture on
+framing transformation, both in the resolved conifold case.
+Acknowledgements . This research is partially supported by two NSFC grants
+(10425101 and 10631050) and a 973 project grant NKBRPC (2006 cB805905).4 JIAN ZHOU
+2.Open String Amplitude of the Resolved Conifold with One oute r
+Brane
+In this section we will start with the calculation by topological vertex [2, 20] of
+the open string amplitude of the resolved conifold. We reformulate it by a formula
+that relates the relevant topologicalvertexto two-partitionHod ge integrals[33, 22].
+We then show that by choosing the parameters in the Hodge integra ls suitably the
+computations can be greatly simplified.
+2.1.Open string amplitudes by the topological vertex. By the theory of the
+topological vertex [2, 20], the open string amplitude for the resolv ed conifold with
+one outer brane and framing acan be computed as follows:
+(1)Z(a)(λ;t;p) =/summationdisplay
+µ,ν,ηqaκµ/2C(0),µ,νt(q)·Q|ν|·Cν,(0),(0)(q)·χµ(η)
+zη√−1l(η)pη(x),
+whereq=e√−1λandQ=−e−t. The topological vertices here can be rewritten as
+follows (cf [35]):
+C(0),µ,νt(q) =q−κν/2Wµ,ν(q), C ν,(0),(0)(q) =Wν(q) =qκν/2Wνt(q), (2)
+so we have
+(3)Z(a)(λ;t;p) =/summationdisplay
+µ,ν,ηqaκµ/2Wµν(q)Q|ν|Wνt(q)·χµ(η)
+zη√−1l(η)pη(x).
+We will not recall the complicated definitions of Wµ,ν(q) here, since we will not use
+it belowtocarryout the computations. Instead, we willrewriteit in t ermsofHodge
+integrals and use a trick that greatly simplifies the summations over p artitions.
+2.2.Open string amplitudes by two-partition Hodge integrals. For a pair
+of partitions ( µ+,µ−)∈ P2
++(one of which might be empty), consider the following
+Hodge integrals:
+Gµ+,µ−(x,y,−x−y)
+=−√−1l(µ+)+l(µ−)
+zµ+·zµ−·/summationdisplay
+g≥0λ2g−2+l(µ+)+l(µ−)
+·/integraldisplay
+Mg,l(µ+)+l(µ−)Λ∨
+g(x)Λ∨
+g(y)Λ∨
+g(−x−y)
+/producttextl(µ+)
+i=1x
+µ+
+i/parenleftBig
+x
+µ+
+i−ψi/parenrightBig/producttextl(µ−)
+j=1y
+µ−
+i/parenleftbigg
+y
+µ−
+j−ψj+l(µ+)/parenrightbigg
+·[xy(x+y)]l(µ+)+l(µ−)−1·l(µ+)/productdisplay
+i=1/producttextµ+
+i−1
+a=1/parenleftbig
+µ+
+iy+ax/parenrightbig
+µ+
+i!xµ+
+i−1·l(µ−)/productdisplay
+i=1/producttextµ−
+i−1
+a=1/parenleftbig
+µ−
+ix+ay/parenrightbig
+µ−
+i!yµ−
+i−1.
+The following formula was conjectured by the author [33] and prove d in joint work
+with C.-C. Liu and K. Liu [22]:
+G•(λ;x,y,−x−y;p+,p−) =R•(λ;x,y,−x−y;p+,p−), (4)OPEN STRING INVARIANTS AND MIRROR CURVE 5
+where
+G•(λ;x,y,−x−y;p+,p−) = exp
+/summationdisplay
+(µ+,µ−)∈P2
++Gµ+,µ−(λ;x,y,−x−y)p+
+µ+p−
+µ−
+,
+R•(λ;x,y,−x−y;p+,p−) =/summationdisplay
+µ±,ν±χν+(µ+)
+zµ+χν−(µ−)
+zµ−q(κν+y
+x+κν−x
+y)λ/2Wν+,ν−(q)p+
+µ+p−
+µ−.
+Write
+(5)G•(λ;x,y,−x,y;p+,p−) =/summationdisplay
+(µ+,µ−)∈P2
++G•
+µ+,µ−(λ;x,y,−x−y)p+
+µ+p−
+µ−,
+then by (4) we get:
+(6)Wν1,ν2(q) =/summationdisplay
+|µi|=|νi|2/productdisplay
+i=1χνi(µi)·q−(κν1w2/w1+κν2w1/w2)/2G•
+µ1,µ2(λ;w1,w2,w3),
+wherew3=−w1−w2. One can use this formula to rewrite (3) as follows:
+(7)Z(a)(λ;t;p) =/summationdisplay
+µ,ξ,η,ǫq(a−w1/w2)κµ/2χµ(ǫ)G•
+ξǫ(λ;w1,w2,w3)·zξ
+·Q|ξ|(−1)|ξ|−l(ξ)·G•
+ξ(λ;−w1,−w2,−w3)·χµ(η)
+zη√−1l(η)pη(x).
+Indeed,
+Z(a)(λ;t;p)
+=/summationdisplay
+µ,ν,ξ,η,ǫ,φqaκµ/2χν(ξ)χµ(ǫ)q−(κνw2/w1+κµw1/w2)/2G•
+ξǫ(λ;w1,w2,w3)
+·Q|ν|χνt(φ)q−(κνtw2/w1)/2G•
+φ(λ;−w1,−w2,−w3)·χµ(η)
+zη√−1l(η)pη(x)
+=/summationdisplay
+µ,ξ,η,ǫ,φq(a−w1/w2)κµ/2χµ(ǫ)G•
+ξǫ(λ;w1,w2,w3)
+·Q|ξ|/summationdisplay
+νχνt(φ)χν(ξ)·G•
+φ(λ;−w1,−w2,−w3)·χµ(η)
+zη√−1l(η)pη(x)
+=/summationdisplay
+µ,ξ,η,ǫq(a−w1/w2)κµ/2χµ(ǫ)G•
+ξǫ(λ;w1,w2,w3)·zξ
+·Q|ξ|(−1)|ξ|−l(ξ)·G•
+ξ(λ;−w1,−w2,−w3)·χµ(η)
+zη√−1l(η)pη(x).
+In the above we have used the following facts:
+κνt=−κν, χ νt(φ) = (−1)|φ|−l(φ)χν(φ), (8)
+and the following orthogonality relations for characters:
+(9)/summationdisplay
+νχν(ξ)χν(φ) =δξ,φzξ.6 JIAN ZHOU
+2.3.Hodge integrals at special parameters. In general it is very hard to eval-
+uateG•
+ξǫ(λ;w1,w2,−w1−w2). Fortunately in [33] we notice that G•
+ξǫ(λ;w1,−w1,0)
+can be easily evaluated. In fact, when y=−x,Gµ+,µ−(λ,x,y,−x−y) vanishes
+except for the following three cases. Case 1.
+G(n),(0)(λ;x,−x,0)
+=−√−1
+n/summationdisplay
+g≥0λ2g−1/integraldisplay
+Mg,1Λ∨
+g(x)Λ∨
+g(−x)Λ∨
+g(0)
+x
+n/parenleftbigx
+n−ψ1/parenrightbig·/producttextn−1
+a=1(−nx+ax)
+n!xn−1
+=(−1)n−1
+√−1λn2/summationdisplay
+g≥0(nλ)2g/integraldisplay
+Mg,1ψ2g−2
+1λ
+=(−1)n−1
+n1
+2√−1sin(nλ/2)=(−1)n−1
+n·[n],
+where [n] =qn/2−q−n/2. Here in the second equality we have used Mumford’s
+relations:
+(10) Λ∨
+g(x)Λ∨
+g(−x) = (−x)g.
+Similarly,
+G(0),(n)(λ;x,−x,0) =(−1)n−1
+n·[n].
+Another case where the Gµ,ν(λ;x,−x,0) may be nonvanishing is
+G(n),(n)(λ;x,−x) =1
+n.
+Therefore,
+(11)G•(λ;x,−x,0;p+,p−)
+= exp∞/summationdisplay
+n=1/parenleftbigg(−1)n−1
+n·[n]pn(x+)+(−1)(n−1)
+n·[n]pn(x−)+1
+npn(x+)pn(x−)/parenrightbigg
+.
+One can then easily find
+G•
+(1),(0)(λ;x,−x,0) =G•
+(0),(1)(λ;x,−x,0) =1
+[1], (12)
+G•
+(2),(0)(λ;x,−x,0) =G•
+(0),(2)(λ;x,−x,0) =−1
+2·[2], (13)
+G•
+(12),(0)(λ;x,−x,0) =G•
+(0),(12)(λ;x,−x,0) =1
+21
+[1]2, (14)
+G•
+(1),(1)(λ;x,−x,0) = (1
+[1]2+1), (15)
+etc. Now we take w2=−w1and sow3= 0 in (7):
+(16)Z(a)(λ;t;p) =/summationdisplay
+µ,ξ,η,ǫq(a+1)κµ/2χµ(ǫ)G•
+ξǫ(λ;w1,−w1,0)·zξ
+·Q|ξ|(−1)|ξ|−l(ξ)·G•
+ξ(λ;−w1,w1,0)·χµ(η)
+zη√−1l(η)pη(x).
+Hence one can apply (11) to compute Z(a)(λ;t;p).OPEN STRING INVARIANTS AND MIRROR CURVE 7
+3.Open String Amplitude of the Resolved Conifold by Operator
+manipulations
+In last section we have seen that the computation of the open strin g amplitude
+of the resolved conifold involves complicated summations over partit ions. In this
+section we will follow the treatment of the inner brane case in [36] to r eformulate
+it using the operators in [29].
+3.1.Reformulation in terms of symmetric functions. We now rewrite the
+openstringamplitude asthe vacuumexpectation valueon the space Λofsymmetric
+functions [24]. Recall the Newton functions {pµ}and the Schur functions {sν}form
+additive bases of Λ, and they are related by the character values o f the symmetric
+groups:
+pµ=/summationdisplay
+νχν(µ)sν, s ν=/summationdisplay
+µχν(µ)
+zµpµ. (17)
+Under the natural scalar product on Λ one has:
+/an}bracketle{tpµ,pν/an}bracketri}ht=δµ,νzµ, /an}bracketle{tsµ,sν/an}bracketri}ht=δµ,ν. (18)
+In particular,
+(19) /an}bracketle{tpµ,sν/an}bracketri}ht=χν(µ).
+On Λ one can introduce the following operators for nonzero integer sn:
+(20) βn: Λ→Λ, β n(f) =/braceleftBigg
+p−n·f, n<0,
+n∂
+∂pnf, n> 0.
+These operators satisfy the following commutation relations:
+(21) [ βm,βn] =mδm,−nIdΛ.
+Another useful operator on Λ is the cut-and-join operator
+K: =1
+2∞/summationdisplay
+i,j=1/parenleftbig
+pi+jij∂2
+∂pi∂pj+pipj(i+j)∂
+∂pi+j/parenrightbig
+(22)
+=1
+2∞/summationdisplay
+i,j=1(β−(i+j)βiβj+β−iβ−jβi+j).
+The Schur functions are eigenvectors of this operator:
+(23) Ksµ=1
+2κµsµ.
+For an linear operator A: Λ→Λ, It vacuum expectation value is
+(24) /an}bracketle{tA/an}bracketri}ht=/an}bracketle{t0|A|0/an}bracketri}ht,
+where|0/an}bracketri}ht= 1∈Λ. One clearly has
+(25) βn|0/an}bracketri}ht= 0, n> 0.
+For a partition µ= (µ1,...,µ l), we write
+βµ=l/productdisplay
+i=1βµi, β −µ=l/productdisplay
+i=1β−µi. (26)8 JIAN ZHOU
+It is easy to see that
+(27) /an}bracketle{tβµβ−ν/an}bracketri}ht=δµ,νzµ.
+It follows from this identity that
+(28) /an}bracketle{texp(/summationdisplay
+n=1an
+nβn)·exp(∞/summationdisplay
+n=1bn
+nβ−n)/an}bracketri}ht= exp∞/summationdisplay
+n=1anbn
+n.
+Using an idea from [34], we introduce two sets Λ and ˜Λ of symmetric functions,
+˜Λ for the internal edge in Figure 1, Λ for the external edge where t he outer brane
+is located in Figure 1. Hence we operators {βn}and{˜βn}acting on Λ and ˜Λ
+respectively. In physical terminology, we introduce two systems o f free bosons.
+With the above preparations, we now reformulate (16) in terms of t he vacuum
+expectation value of an operator on Λ ⊗˜Λ:
+Z(a)(λ;t;p) =/an}bracketle{texp(∞/summationdisplay
+n=1pn(x)
+niβn)q(a+1)K/summationdisplay
+ξ,ǫG•
+ξǫ(λ;w1,−w1,0)β−ǫ˜βξ
+·/summationdisplay
+ηQ|η|(−1)|η|−l(η)·G•
+η(λ;−w1,w1,0)˜β−η/an}bracketri}ht.
+Now by definition of G•and (11) we have:
+/summationdisplay
+ξ,ǫG•
+ξǫ(λ;w1,−w1,0)β−ǫ˜βξ= exp∞/summationdisplay
+n=1/parenleftbigg(−1)n−1
+n·[n]β−n+(−1)(n−1)
+n·[n]˜βn+1
+nβ−n˜βn/parenrightbigg
+,
+/summationdisplay
+ηQ|η|(−1)|η|−l(η)·G•
+η(λ;−w1,w1,0)˜β−η= exp/parenleftbigg∞/summationdisplay
+n=1Qn
+n·[n]˜β−n/parenrightbigg
+,
+so we get:
+Z(a)(λ;t;p) =/an}bracketle{texp(∞/summationdisplay
+n=1pn(x)
+niβn)q(a+1)K
+·exp∞/summationdisplay
+n=1/parenleftbigg(−1)n−1
+n·[n]β−n+(−1)(n−1)
+n·[n]˜βn+1
+nβ−n˜βn/parenrightbigg
+·exp/parenleftbigg∞/summationdisplay
+n=1Qn
+n·[n]˜β−n/parenrightbigg
+/an}bracketri}ht
+=/an}bracketle{texp(∞/summationdisplay
+n=1pn(x)
+niβn)qKexp∞/summationdisplay
+n=1/parenleftbigg(−1)n−1
+n·[n]+Qn
+n·[n]/parenrightbigg
+β−n/an}bracketri}ht·exp/parenleftbigg∞/summationdisplay
+n=1(−1)n−1Qn
+n·[n]2/parenrightbigg
+.
+When allpn(x) = 0, one gets the closed string partition function:
+(29) Z(λ;t) :=Z(a)(λ;t;p)|pn=0= exp/parenleftbigg∞/summationdisplay
+n=1(−1)n−1Qn
+n·[n]2/parenrightbigg
+.
+The normalized open string amplitude is defined by:
+(30) ˆZ(a)(λ;t;p) =Z(a1)(λ;t;p)
+Z(λ;t).
+To summarize, we have the following:OPEN STRING INVARIANTS AND MIRROR CURVE 9
+Proposition 3.1. The normalized open string amplitude of the resolved conifo ld
+with one outer brane and framing a∈Zis given by:
+(31)ˆZ(a)(λ;t;p) =/an}bracketle{texp(∞/summationdisplay
+n=1pn(x)
+niβn)q(a+1)Kexp/parenleftbigg∞/summationdisplay
+n=1/parenleftbig(−1)n−1
+[n]+Qn
+[n]/parenrightbigβ−n
+n/parenrightbigg
+/an}bracketri}ht.
+Corollary 3.1. The normalized open string amplitude of the resolved conifo ld with
+one outer brane and framing −1is given by:
+(32) ˆZ(−1)(λ;t;p) = exp/parenleftbigg∞/summationdisplay
+n=1/parenleftbig(−1)n−1
+[n]+Qn
+[n]/parenrightbigpn(x)
+ni/parenrightbigg
+.
+From now on we will only consider the case when a/ne}ationslash=−1.
+3.2.Reformulation in terms of other operators. When the framing a/ne}ationslash=−1,
+we will follow the approach in [36] for the inner brane case to evaluate the vacuum
+expectation values. First notice that,
+ˆZ(a)(λ;t;p)
+=/an}bracketle{texp(∞/summationdisplay
+n=1pn(x)
+niβn)ei(a+1)λKexp/parenleftbigg∞/summationdisplay
+n=1/parenleftbig(−1)n−1
+n·[n]+Qn
+n·[n]/parenrightbig
+β−n/parenrightbigg
+e−i(a+1)λK/an}bracketri}ht.
+=/an}bracketle{texp(∞/summationdisplay
+n=1pn(x)
+niβn)exp/parenleftbigg∞/summationdisplay
+n=1/parenleftbig(−1)n−1
+n·[n]+Qn
+n·[n]/parenrightbig
+ei(a+1)λKβ−ne−i(a+1)λK/parenrightbigg
+/an}bracketri}ht.
+By [30, (2.14)],
+(33) euKβ−me−uK=E−m(um).
+The operators Er(z) were defined in [29] as follows:
+(34) Er(z) =/summationdisplay
+k∈Z+1
+2ez(k−r
+2)Ek−r,k+δr,0
+ς(z),
+where the function ς(z) is defined by
+(35) ς(z) =ez/2−e−z/2.
+The operators Ersatisfy
+(36) Er(z)∗=E−r(z)
+and
+(37) [ Ea(z),Eb(w)] =ς(aw−bz)Ea+b(z+w).
+These operators were originally defined in the fermionic picture, and by the boson-
+fermion correspondence[27], they alsoact on the bosonic Fock spa ce, which we take
+as the space Λ of symmetric functions. In the fermionic picture the cut-and-join
+operator is the operator F2defined by
+(38) F2=1
+2/summationdisplay
+k∈Z+1
+2k2Ekk.10 JIAN ZHOU
+Now we can rewrite (31) as follows:
+(39)ˆZ(a)(λ;t;p) =/an}bracketle{texp/parenleftbig∞/summationdisplay
+n=1pn(x)
+inβn/parenrightbig
+·exp/parenleftbig∞/summationdisplay
+n=11
+n((−1)n−1
+[n]+Qn
+[n])E−n(i(a+1)nλ)/parenrightbig
+/an}bracketri}ht.
+3.3.Operator manipulations. Fork/ne}ationslash= 0,
+(40) βk=Ek(0).
+The commutation relation (37) specializes to
+(41) [ βk,El(z)] =ς(kz)Ek+l(z).
+Starting from (41), one can easily prove by induction:
+βm
+kE−n(z) =m/summationdisplay
+j=0/parenleftbiggm
+j/parenrightbigg
+ς(kz)jE−n+jk(z)βm−j
+k. (42)
+It then follows that
+exp(ak
+kβk)·E−n(z) =∞/summationdisplay
+j=0aj
+k
+j!kjς(kz)jE−n+kj(z)·exp(ak
+kβk). (43)
+Repeating these formulas for all k, one gets the following identity [36]:
+(44) exp(∞/summationdisplay
+k=1ak
+kβk)·E−n(z)·exp(∞/summationdisplay
+k=1ak
+kβk) =/summationdisplay
+µ˜aµ(z)
+zµ+E−n+|µ|(z),
+where
+(45) ˜ aµ(z) =/productdisplay
+k(akς(kz))mk(µ).
+By (39) we have
+(46)ˆZ(a)(λ;t;p) =/an}bracketle{texp/parenleftbig∞/summationdisplay
+n=11
+n((−1)n−1
+[n]+Qn
+[n])
+·exp/parenleftbig∞/summationdisplay
+m=1pm(x)
+imβm/parenrightbig
+·E−n(i(a+1)nλ)·exp/parenleftbig
+−∞/summationdisplay
+m=1pm(x)
+imβm/parenrightbig/parenrightbig
+/an}bracketri}ht.
+Hence one can reduce the calculation of ˆZ(a)(λ;t;p) to the computations of the
+correlators:
+(47) /an}bracketle{tEa1(b1)···Ean(bn)/an}bracketri}ht.
+This approach was proposed for the inner brane case in [36].
+To evaluate the correlators (47), notice that
+(48) E0(z)|0/an}bracketri}ht=ς(z)|0/an}bracketri}ht,En(z)|0/an}bracketri}ht= 0, n> 0.
+Therefore,
+(49) /an}bracketle{tEa1(b1)···Ean(bn)/an}bracketri}ht= 0OPEN STRING INVARIANTS AND MIRROR CURVE 11
+foran>0 ora1<0. One uses (37) to reduce the correlators to the correlators of
+the form:
+(50) /an}bracketle{tE0(b1)···E0(bn)/an}bracketri}ht=ς(b1)···ς(bn).
+For example, one can get in this fashion:
+Lemma 3.1. Supposem1,...,m lis a partition of n>0, anda1,...,a lare arbi-
+trary numbers, then the following identity holds:
+(51) /an}bracketle{tβnE−m1(a1iλ)···E−ml(aliλ)/an}bracketri}ht=1
+[/summationtextl
+j=1aj]l/productdisplay
+j=1[dj],
+whered1=na1, and forj >1,
+(52) dj=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen−/summationtextj−1
+k=1mj−mj/summationtextk−1
+k=1akaj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
+Corollary 3.2. Suppose that m1,...,m lis a partition of n>0, then the following
+identity holds:
+(53)/an}bracketle{tβnE−m1((a+1)m1iλ)···E−ml((a+1)mliλ)/an}bracketri}ht=1
+[(a+1)n]l/productdisplay
+j=1[(a+1)nmj].
+3.4.The open string free energy and the n-point functions. Write
+(54) F(a)(λ;t;p) := logˆZ(a)(λ;t;p).
+It will be referred to as the open string free energy of the resolve d conifold with
+framea. Write
+(55) F(a)(λ;t;p) =∞/summationdisplay
+n=11
+n!/summationdisplay
+m1,...,m n≥1F(a)
+m1,...,m n(λ;t)pm1···pmn,
+wherethecoefficients F(a)
+m1,...,m n(λ;t), whicharesymmetricin m1,...,m n, arecalled
+then-point functions. Write
+(56) F(a)
+m1,...,m n(λ;t) =∞/summationdisplay
+g=0λ2g−2+nF(a)
+g;m1,...,m n(t),
+and
+(57) Φ(a)
+g,n(p) :=1
+n!/summationdisplay
+m1,...,m n≥1F(a)
+g;m1,...,m n(t)pm1···pmn.
+The symmetrization of Φ(a)
+g,n(p) is defined by:
+(58) Ψ(a)
+g,n(t;x1,...,x n) :=/summationdisplay
+m1,...,m n≥1F(a)
+g;m1,...,m n(t)xm1
+1···xmn
+n.
+We also set
+(59) Ψ(a)
+n(λ;t;x1,...,x n) =/summationdisplay
+g≥0λ2g−2+nΨ(a)
+g,n(t;x1,...,x n).
+4.One-Point Functions
+In this section we will use the method in last section to find closed form ula for
+one-point functions for open string invariants.12 JIAN ZHOU
+4.1.Preliminary results for one-point functions. It is easy to see that the
+one-point functions are given by:
+∞/summationdisplay
+n=1F(a)
+n(λ;t)pn(x)
+=/an}bracketle{t0|∞/summationdisplay
+n=1pn(x)
+niβn·exp/parenleftbig∞/summationdisplay
+n=11
+n((−1)n−1
+[n]+Qn
+[n])E−n(i(a+1)nλ)/parenrightbig
+|0/an}bracketri}ht
+=∞/summationdisplay
+n=1pn(x)
+ni/summationdisplay
+m1·1+···+ml·l=nl/productdisplay
+j=1((−1)j−1
+[j]+Qj
+[j])mj
+jmjmj!·/an}bracketle{tβnl/productdisplay
+j=1E−j(i(a+1)jλ)mj/an}bracketri}ht.
+Now by (53), we get:
+(60)/summationdisplay
+n≥1F(a)
+n(λ;t)pn(x)
+=∞/summationdisplay
+n=1pn(x)
+ni/summationdisplay
+m1·1+···+ml·l=nl/productdisplay
+j=1((−1)j−1
+[j]+Qj
+[j])mj
+jmjmj!·1
+[(a+1)n]l/productdisplay
+j=1[(a+1)jn]mj.
+After symmetrization we get:
+(61) Ψ(a)
+1(λ;t;x) =∞/summationdisplay
+n=1xn
+ni·[(a+1)n]
+·/summationdisplay
+/summationtextl
+j=1jmj=nl/productdisplay
+j=1/parenleftbig
+((−1)j−1
+[j]+Qj
+[j])·[(a+1)jn]/parenrightbigmj
+jmjmj!.
+The following are the first few terms:
+F(a)
+1(λ;t) =1
+[1]+Q
+[1].
+F(a)
+2(λ;t) =1
+4(−1
+[2]+Q2
+[2])[4(a+1)]
+[2(a+1)]+1
+4(1
+[1]+Q
+[1])2[2(a+1)].
+F(a)
+3(λ;t) =1
+3·(1
+3[3]+Q3
+3[3])[9(a+1)]
+[3(a+1)]
++1
+3·1
+2(1
+[1]+Q
+[1])(−1
+[2]+Q2
+[2])[6(a+1)][3(a+1)]
+[3(a+1)]+1
+3·1
+3!(1
+[1]+Q
+[1])3[3(a+1)]3
+[3(a+1)].
+In particular, when a= 0, we have
+F(0)
+1(λ;t) =1
+[1]+Q
+[1],
+F(0)
+2(λ;t) =1
+2[2]+[2]
+2[1]2Q+[3]
+2[1][2]Q2,
+F(0)
+3(λ;t) =1
+3[3]+[3]
+3[1]2Q+[3][4]
+3[1]2[2]Q2+[4][5]
+3[1][2][3]Q3.
+We will present below a method to take the summations on the right-h and side of
+(60). It will provide us explicit expressions for genera F(a)
+n(λ;t).OPEN STRING INVARIANTS AND MIRROR CURVE 13
+4.2.The genus 0case.To motivate our method, we will first treat the g= 0
+case, where our method is elementary. By (60), we get:
+(62) Ψ(a)
+0,1(t;x) =−∞/summationdisplay
+n=1xn
+n2(a+1)/summationdisplay
+m1·1+···+ml·l=nl/productdisplay
+j=1[n(a+1)((−1)j−1+Qj)]mj
+jmjmj!.
+The following are the first few terms:
+−Ψ(a)
+0,1(t;x) = (1+ Q)x+1
+4((2a+1)+4(a+1)Q+(2a+3)Q2)x2
++1
+9(1+9a(a+1)
+2+9(a+1)(3a+2)
+2Q+9(a+1)(3a+4)
+2Q2
++(1+9(a+1)(a+2)
+2Q3)x3+···.
+In particular, when a= 0,
+−Ψ(0)
+0,1(t;x) = (1+ Q)x+1
+4(1+4Q+3Q2)x2+1
+9(1+9Q+18Q2+10Q3)x3
++1
+16(1+16Q+60Q2+80Q3+35Q4)x4
++1
+25(1+25Q+150Q2+350Q3+350Q4+126Q5)x5+···.
+To evaluate the summation
+(63)/summationdisplay
+m1·1+···+ml·l=nl/productdisplay
+j=1[n(a+1)((−1)j−1+Qj)]mj
+jmjmj!,
+we regard it as the coefficient of znof the following series:
+(64)∞/summationdisplay
+N=0zN/summationdisplay
+m1·1+···+ml·l=Nl/productdisplay
+j=1[n(a+1)((−1)j−1+Qj)]mj
+jmjmj!.
+This can be easily rewritten as follows:
+(65) exp∞/summationdisplay
+j=1n(a+1)((−1)j−1+Qj)
+jzj=(1+z)n(a+1)
+(1−Qz)n(a+1).
+Now we have the binomial expansions:
+(66)1
+(1−Qz)n(a+1)=∞/summationdisplay
+j=0/producttextj−1
+k=0(n(a+1)+k)
+j!Qjzj
+and
+(67) (1+ z)n(a+1)=/summationdisplay
+m≥0/producttextm−1
+k=0(n(a+1)−k)
+m!zm,
+it follows that the coefficient of znof the series in (64) is
+n/summationdisplay
+j=0/producttextj−1
+k=0(n(a+1)+k)
+j!Qj·/producttextn−j−1
+k=0(n(a+1)−k)
+(n−j)!
+=n(a+1)·n/summationdisplay
+j=0/producttextn−1
+k=1(na+j+k)
+j!(n−j)!Qj.14 JIAN ZHOU
+So we have proved our first main result:
+Theorem 4.1. For the resolved confiold we have
+(68) Ψ(a)
+0,1(t;x) =−∞/summationdisplay
+n=1xn
+nn/summationdisplay
+j=0/producttextn−1
+k=1(na+j+k)
+j!(n−j)!Qj.
+4.3.The case of arbitrary genera. Similarly, we understand the summation
+(69)/summationdisplay
+/summationtextl
+j=1jmj=nl/productdisplay
+j=1/parenleftbig
+((−1)j−1
+[j]+Qj
+[j])·[(a+1)jn]/parenrightbigmj
+jmjmj!
+in (60) as the coefficient of znin
+f(a)
+n(z;t) =/summationdisplay
+N≥0zN/summationdisplay
+m1·1+···+ml·l=Nl/productdisplay
+j=1(((−1)j−1
+[j]+Qj
+[j])mj
+jmjmj!)·l/productdisplay
+j=1[(a+1)jn]mj
+= exp∞/summationdisplay
+j=1(−1)j−1
+[j]+Qj
+[j]
+j[(a+1)jn]zj.
+Because
+(70)[j(a+1)n]
+[j]=(a+1)n/summationdisplay
+k=1qj((a+1)n−2k+1)/2
+we have
+f(a)
+n(z;t)
+= exp(a+1)n/summationdisplay
+k=1∞/summationdisplay
+j=1(−1)j−1
+j(qj((a+1)n−2k+1)/2−(−1)jQjqj((a+1)n−2k+1)/2)zj
+=(a+1)n/productdisplay
+k=11+q((a+1)n−2k+1)/2z
+1−Qq((a+1)n−2k+1)/2z.
+We have the quantum binomial formula:
+(71)(a+1)n/productdisplay
+k=1(1+q((a+1)n−2k+1)/2z) =(a+1)n/summationdisplay
+j=0/bracketleftbigg
+(a+1)n
+j/bracketrightbigg
+zj,
+and
+(72)(a+1)n/productdisplay
+k=11
+1−Qq((a+1)n−2k+1)/2z=∞/summationdisplay
+j=0/bracketleftbigg
+(a+1)n+j−1
+j/bracketrightbigg
+Qjzj,
+where the quantum binomial coefficients are defined by:
+(73)/bracketleftbiggn
+j/bracketrightbigg
+=[n][n−1]···[n−j+1]
+[j][j−1]···[1].OPEN STRING INVARIANTS AND MIRROR CURVE 15
+Indeed, using the definition of elementary symmetric functions one has
+(a+1)n/productdisplay
+k=1(1+q((a+1)n−2k+1)/2z)
+=n/summationdisplay
+j=0ej(1,q···,q(a+1)n−1)q−j((a+1)n−1)/2zj=(a+1)n/summationdisplay
+j=0/bracketleftbigg
+(a+1)n
+j/bracketrightbigg
+zj,
+where in the second equality we have used the following equality [24, p. 26]:
+ej(1,q,...,q(a+1)n−1) =qj(j−1)/2(1−q(a+1)n)(1−q(a+1)n−1)···(1−q(a+1)n−j+1)
+(1−q)(1−q2)···(1−qj);
+Similarly, using the definition of complete symmetric functions one has
+(a+1)n/productdisplay
+k=11
+1−Qq((a+1)n−2k+1)/2x
+=∞/summationdisplay
+j=0hj(1,q,...,q(a+1)n−1)q−j((a+1)n−1)/2Qjzj
+=∞/summationdisplay
+j=0/bracketleftbigg(a+1)n+j−1
+j/bracketrightbigg
+Qjzj,
+where in the second equality we have used the following equality [24, p. 26]:
+hj(1,q,...,q(a+1)n−1) =(1−q(a+1)n+j−1)(1−q(a+1)n+j−2)···(1−q(a+1)n)
+(1−q)(1−q2)···(1−qj).
+It follows that we have
+f(a)
+n(z;t) =∞/summationdisplay
+j=0/bracketleftbigg(a+1)n+j−1
+j/bracketrightbigg
+Qjzj·(a+1)n/summationdisplay
+j=0/bracketleftbigg(a+1)n
+j/bracketrightbigg
+zj,
+hence its coefficient of znis:
+(74)n/summationdisplay
+j=0/bracketleftbigg
+(a+1)n+j−1
+j/bracketrightbigg
+·/bracketleftbigg
+(a+1)n
+n−j/bracketrightbigg
+Qj= [(a+1)n]·n/summationdisplay
+j=0/producttextn−1
+k=1[an+j+k]
+[j]![n−j]!Qj.
+Therefore, the following formula for a≥0:
+(75) Ψ(a)
+1(λ;t;x) =∞/summationdisplay
+n=1xn
+nin/summationdisplay
+j=0/producttextn−1
+k=1[an+j+k]
+[j]![n−j]!Qj.16 JIAN ZHOU
+This also holds when a+1 =−b≤ −1. In this case f(a)
+n(z;t) is equal to:
+/summationdisplay
+N≥0zN/summationdisplay
+m1·1+···+ml·l=Nl/productdisplay
+j=1(((−1)j−1
+[j]+Qj
+[j])mj
+jmjmj!)·l/productdisplay
+j=1[−bjn]mj
+= exp∞/summationdisplay
+j=1−(−1)j−1
+[j]+Qj
+[j]
+j[bjn]zj
+= exp−bn/summationdisplay
+k=1∞/summationdisplay
+j=1(−1)j−1
+j(qj(bn−2k+1)/2−(−1)jQjqj(bn−2k+1)/2)zj
+=bn/productdisplay
+k=11−Qq(bn−2k+1)/2x
+1+q(bn−2k+1)/2z=bn/summationdisplay
+j=0/bracketleftbiggbn
+j/bracketrightbigg
+(−Qz)j·∞/summationdisplay
+j=0/bracketleftbiggbn+j−1
+j/bracketrightbigg
+(−z)j.
+Its coefficient of znis:
+n/summationdisplay
+j=0/bracketleftbigg
+bn
+j/bracketrightbigg
+(−Q)j·/bracketleftbigg
+bn+n−j−1
+n−j/bracketrightbigg
+(−1)n−j
+= (−1)n[bn]·/producttextn−1
+k=1[bn−j+(n−k)]
+[j]!·[n−j]!]
+=[(a+1)n]·/producttextn−1
+k=1[an+j+k]
+[j]!·[n−j]!].
+So we have proved the following generalization of Theorem 4.1:
+Theorem 4.2. For the resolved confiold we have
+(76) Ψ(a)
+1(λ;t;x) =∞/summationdisplay
+n=1xn
+nin/summationdisplay
+j=0/producttextn−1
+k=1[an+j+k]
+[j]![n−j]!Qj.
+5.Ooguri-Vafa Integrality for One-Point Functions
+In this section we check the Ooguri-Vafa Integrality Conjecture f or the one-point
+function of the resolved conifold in general framing. This recovers and generalizes
+some results by Aganagic-Vafa [5], Aganagic-Klemm-Vaf [3] and Mari ˜ no-Vafa [26].
+5.1.Genus0Ooguri-Vafa integral invariants. Ooguri and Vafa [31] conjec-
+tured an integral property of open string invariant of Calabi-Yau 3 -folds, similar to
+the Gopakumar-Vafa integrality [16] for closed string invariants. F or the resolved
+conifold with one outer brane, it takes the following form for genus 0 one-point
+functions [5]:
+(77) x∂
+∂xΨ(a)
+0,1(t;x) =/summationdisplay
+m,n≥1md(a)
+k,mlog(1−e−ktxm)
+for some integers d(a)
+k,m=/summationtext
+sN(a)
+k,m,s. Aganagic and Vafa [5] checked this for a= 0,
+and a table for dk,m:=d(0)
+k,m,0can be found in their paper. We now establish the
+general case.OPEN STRING INVARIANTS AND MIRROR CURVE 17
+By (68),
+(78) x∂
+∂xΨ(a)
+0,1(t;x) =−∞/summationdisplay
+n=1xnn/summationdisplay
+k=0/producttextn−1
+j=1(na+j+k)
+k!(n−k)!(−1)ke−kt.
+The right-hand side of (77) can be written as:
+(79) −∞/summationdisplay
+k=0∞/summationdisplay
+m=1md(a)
+k,m∞/summationdisplay
+j=11
+je−jktxjm.
+The first few terms are
+−d(a)
+0,1(x+1
+2x2+1
+3x3+···)−2d(a)
+0,2(x2+1
+2x4+1
+3x6+···)
+−3d(a)
+0,3(x3+1
+2x6+1
+3x9+···)−······
+−d(a)
+1,1(e−tx+1
+2e−2tx2+1
+3e−3tx3+···)−2d(a)
+1,2(e−tx2+1
+2e−2tx4+1
+3e−3tx6+···)
+−3d(a)
+1,3(e−tx3+1
+2e−2tx6+1
+3e−3tx9+···)−······
+−d(a)
+2,1(e−2tx+1
+2e−4tx2+1
+3e−6tx3+···)−2d(a)
+2,2(e−2tx2+1
+2e−4tx4+1
+3e−6tx6+···)
+−3d(a)
+2,3(e−2tx3+1
+2e−4tx6+1
+3e−6tx9+···)−······
+Putting similar terms together
+−d(a)
+0,1x−(2d(a)
+0,2+1
+2d(a)
+0,1)x2−(3d(a)
+0,3+1
+3d(a)
+0,1)x3−(4d(a)
+0,4+d(a)
+0,2+1
+4d(a)
+0,1)x4−···
+−d(a)
+1,1e−tx−2d(a)
+1,2e−tx2−3d(a)
+1,3e−tx3−4d(a)
+1,4e−tx4−···
+−d(a)
+2,1e−2tx−(2d(a)
+2,2+1
+2d(a)
+1,1)e−2tx2−3d(a)
+2,3e−2tx3−(4d(a)
+2,4+d(a)
+1,2)e−2tx4−···
+−d(a)
+3,1e−3tx−2d(a)
+3,2e−3tx2−(3d(a)
+3,3+1
+3d(a)
+1,1)e−3tx3−d(a)
+3,4e−3tx4−···.
+The coefficient of e−ktxmis of the form:
+(80) −md(a)
+k,m−/summationdisplay
+j|(k,m),j>1m
+j2d(a)
+k/j,m/j.
+Therefore, by comparing with the right-hand side of (78):
+(81) d(a)
+k,m+/summationdisplay
+j|(k,m),j>11
+j2d(a)
+k/j,m/j= (−1)k/producttextm−1
+j=1(ma+j+k)
+m·k!(m−k)!.
+From this one can recursively determine all d(a)
+k,m. Letµ:N→ {0,1,−1}be the
+M¨ obius mu functions. I.e,
+(82)µ(n) =/braceleftBigg
+(−1)r, n=p1···pr,p1,...,p rare distinct primes ,
+0,otherwise.18 JIAN ZHOU
+Then
+(83) d(a)
+k,m=/summationdisplay
+n|(k,m)µ(n)
+n2(−1)k/n/producttextm/n−1
+j=1(ma/n+j+k/n)
+(m/n)·(k/n)!(m/n−k/n)!.
+Indeed, (81) has a unique solution, and by the following property of the M¨bius mu
+function
+(84)/summationdisplay
+l|nµ(l) =δn,1,
+one can check that (83) is a solution.
+For example, when ( k,m) = 1,
+(85) d(a)
+k,m=(−1)k
+m/producttextm−1
+j=1(ma+j+k)
+k!·(m−k)!, m> 1.
+Ifkis a prime number p,
+d(a)
+p,pn=(−1)p
+m/producttextm−1
+j=1(ma+j+p)
+p!·(pn−p)!−1
+p2d(a)
+1,n,
+therefore,
+(86) d(a)
+p,pn=(−1)p
+m/producttextm−1
+j=1(ma+j+p)
+p!·(pn−p)!+/producttextn
+j=2(na+j)
+p2·n!.
+Ifkis a square of a prime p,m=pn, (k,n) = 1, then
+(87)d(a)
+p2,pn=(−1)p
+pn/producttextpn−1
+j=1(pna+j+p2)
+p2!·(pn−p2)!+(−1)p−1/producttextn−1
+i=1(na+i+p)
+p2n·p!(n−p)!.
+Whenais taken to be 0, the numbers d(0)
+k,mmatch with the numbers dk,min
+Table 1 in [5]. The above formula for d(a)
+k,mnow takes the following simplified forms:
+(88) dk,m=/summationdisplay
+n|(k,m)µ(n)
+n2(−1)k/n (m/n+k/n−1)!
+(m/n)·(k/n)!2(m/n−k/n)!.
+When (k,m) = 1,
+(89) dk,m=(−1)k
+m(m+k−1)!
+k!·k!·(m−k)!.
+Ifkis a prime number p,
+(90) dp,pn=(−1)p
+pn(pn+p−1)!
+p!·p!·(pn−p)!+1
+p2.
+Ifkis a square of a prime p,m=pnand (p,n) = 1, then
+(91)dp2,pn=(−1)p
+pn(pn+p2−1)!
+p2!·p2!·(pn−p2)!+(−1)p−1(n+p−1)!
+p2n·p!·p!(n−p)!.OPEN STRING INVARIANTS AND MIRROR CURVE 19
+It is easy to see that
+dm,m=/summationdisplay
+n|mµ(n)
+n2(−1)m/n (2m/n−1)!
+(m/n)·(m/n)!2(m/n−m/n)!
+=1
+2m2/summationdisplay
+l|m(−1)lµ(m/l)/parenleftbigg2l
+l/parenrightbigg
+.
+The first few terms of the sequence ( |dm,m|)m≥1are given by
+1,1,1,2,5,13,100,300,925,2911,9386,30771,102347,344705,1173960,...,
+matches with the sequence number A131868 of the AT&T On-Line En cyclopedia of
+IntegerSequences. It hasthe followingcombinatorialmeaning: na(n) isthe number
+ofn-member subsets of {1,2,3,...,2n−1}that sum to 1 (mod n). Similarly,
+(92) dk,k+1= (−1)k(2k)!
+(k+1)·k!2.
+The first few terms of the sequence ( |dk,k+1|)k≥1are given by
+1,1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,267440,
+9694845,35357670,...,
+This is the sequence A000108 which are the Catalan numbers (also ca lled Segner
+numbers):
+(93) C(n) =1
+n+1/parenleftbigg2n
+n/parenrightbigg
+=(2n)!
+n!(n+1)!.
+These numbers have numerous combinatorial meanings. It will be int eresting to
+find out combinatorial meanings for the sequences ( dk,k+n)k≥1for other fixed n.
+5.2.Another related half integral property. One can also do the following
+expansion:
+x∂
+∂xΨ(a)
+0,1(t;x) = log(1 −x)+alog(1−x2)+3a(a+1)
+2log(1−x3)
++4a(a+1)(2a+1)
+3log(1−x4)+25a(a+1)(5a2+5a+2)
+24log(1−x5)
++3a(a+1)(36a3+54a2+31a+4)
+10log(1−x6)
++ log(1 −Qx)+2(a+1)log(1 −Qx2)+···
+This also has some integral properties.
+Conjecture 5.1. There are half integers e(a)
+k,msuch that
+(94) x∂
+∂xΨ(a)
+0,1(t;x) =/summationdisplay
+m,n≥1me(a)
+k,mlog(1−Qkxm)
+whereQ=−e−t.
+Similar to (81) we have
+(95) me(a)
+k,m+/summationdisplay
+j|(k,m),j>1m
+j2e(a)
+k/j,m/j=/producttextm−1
+j=1(ma+j+k)
+k!(m−k)!.20 JIAN ZHOU
+Then
+(96) e(a)
+k,m=/summationdisplay
+n|(k,m)µ(n)
+n2/producttextm/n−1
+j=1(ma/n+j+k/n)
+(m/n)·(k/n)!(m/n−k/n)!.
+For example, when ( k,m) = 1,
+(97) e(a)
+k,m=1
+m/producttextm+k−1
+j=k+1(ma+j)
+k!·(m−k)!, m> 1,
+in particular, when a= 0,
+(98) e(0)
+k,m=1
+m(m+k−1)!
+k!·k!·(m−k)!.
+Clearly,
+(99) e(a)
+k,m=|d(a)
+k,m|
+for (k,m) = 1. For ( k,m)/ne}ationslash= 1, they also coincide for many cases. Take a= 0 case
+for example. We have the following table for ek,m:=e(0)
+k,m:
+me1,me2,me3,me4,me5,me6,m
+11 0 0 0 0 0
+211/2 0 0 0 0
+31 2 1 0 0 0
+417/2 5 2 0 0
+51 6 14 14 5 0
+6117/2 31 52 42 25/2
+71 12 60 150 198 132
+8131/2 105 360 693 1499/2
+91 20 171 770 2002 3114
+10149/2 264 1500 5045 10507
+111 30 390 2730 11466 30576
+12171/2 556 4690 24024 158809/2
+131 42 770 7700 47124 188496
+14197/21040 12152 87516 415686
+151 56 1375 18564 155195 862194
+161127/2 1785 27552 264537 3394839/2
+Comparing with the table of dk,min [5], we notice that ek,m=|dk,m|for most
+of the pairs ( k,m). (The places where they are different are put in boxes.)
+5.3.Some integral invariants in arbitrary genera. By the integrality predic-
+tion of Ooguri-Vafa [31], there should be integers N(a)
+m,k,ssuch that
+(100) Ψ(a)
+1(λ;t;x) =∞/summationdisplay
+n=1/summationdisplay
+m,k,sNm,k,s
+in[n]en(−kt+isλ)((−a)ax)nm,OPEN STRING INVARIANTS AND MIRROR CURVE 21
+wherem,k∈Z,s∈1
+2Z. By Theorem 4.2, we should have:
+(101)∞/summationdisplay
+n=1/summationdisplay
+m,k,sN(a)
+m,k,s
+i[n]en(−kt+isλ)xnm=∞/summationdisplay
+n=1((−1)ax)n
+in/summationdisplay
+j=0/producttextn−1
+k=1[an+j+k]
+[j]![n−j]!Qj.
+For fixedm,k,N(a)
+m,k,s= 0 fors≫0, so that
+N(a)
+m,k(q1/2) =/summationdisplay
+s∈1
+2ZN(a)
+k,m,sqs∈Z[q1/2,q−1/2]
+is a Laurent polynomial in q1/2. Comparing the coefficients of xme−kton both sides
+of (101), one gets:
+(102)/summationdisplay
+j|(k,m)N(a)
+m/j,k/j(qj/2)
+[j]= (−1)ma+k/producttextm−1
+j=1[am+j+k]
+[k]![m−k]!.
+Hence
+(103)N(a)
+m,k(q1/2)
+[1]=/summationdisplay
+n|(k,m)µ(n)(−1)(ma+k)/n/producttextm/n−1
+j=1[am+nj+k]
+/producttextk/n
+j=1[nj]·/producttext(m−k)/n
+j=1[nj].
+In particular, when ( m,k) = 1,
+(104) N(a)
+m,k(q1/2) = (−1)ma+k[1]/producttextm−1
+j=1[am+j+k]
+[k]![m−k]!.
+Whenkis a primepandm=knfor some integer n,
+(105)N(a)
+pn,p= (−1)pna+p[1]/producttextpn−1
+j=1[apn+j+p]
+[p]![pn−p]!−(−1)na+1[1]/producttextn−1
+j=1[an+pj+1]
+[p]·/producttextn−1
+j=1[pj]!.
+Whenk=p2for some prime, m=pnand (p,n) = 1,
+(106)
+N(a)
+pn,p2= (−1)pna+p2[1]/producttextpn−1
+j=1[apn+j+p2]
+[p2]![pn−p2]!−(−1)na+p[1]/producttextn−1
+j=1[anp+pj+p2]
+/producttextp
+j=1[pj]·/producttextn−p
+j=1[pj].22 JIAN ZHOU
+For example, when a= 0,itimes the right-hand side of (101) is:
+(1
+[1]−1
+[1]e−t)x+(1
+[2]−[2]
+[1]2e−t+[3]
+[2][1]e−2t)x2
++ (1
+[3]−[3]
+[1]2e−t+[3][4]
+[2][1]2e−2t−[4][5]
+[3][2][1]e−3t)x3+···
+= (1
+[1]x+1
+[2]x2+1
+[3]x3+···)
+−(1
+[1]xe−t+1
+[2]x2e−2t+1
+[3]x3e−3t+···)
+−(q1/2+q−1/2
+[1]x2e−t+q+q−1
+[2]x2e−2t+q3/2+q−3/2
+[3]x3e−3t+···)
++ (q1/2+q−1/2
+[1]x2e−2t+q+q−1
+[2]x4e−4t+q3/2+q−3/2
+[3]x6e−6t+···)
+−(q+1+q−1
+[1]x3e−t+q2+1+q−2
+[2]x6e−2t+q3+1+q−3
+[3]x9e−3t+···)
++ (q2+q+2+q−1+q−2
+[1]x3e−2t+···)
+−(q2+1+q−2
+[1]x3e−3t+···)+···.
+In other words,
+N(0)
+1,0= 1, N(0)
+1,1=−1, N(0)
+2,0= 0,
+N(0)
+2,1=−(q1/2+q−1/2), N(0)
+2,2=q1/2+q−1/2, N(0)
+3,0= 0,
+N(0)
+3,1=−(q+1+q−1), N(0)
+3,2=q2+q+2+q−1+q−2, N(0)
+3,3=−(q2+1+q−2).
+Whena= 1,itimes the right-hand side of (101) becomes:
+(1
+[1]−e−t
+[1])(−x)+([3]
+[2][1]−[4]e−t
+[1]2+[5]e−2t
+[2][1])x2
++ ([4][5]
+[3]!−[5][6]
+[2][1]2e−t+[6][7]
+[2][1]2e−2t−[7][8]
+[3][2][1]e−3t)(−x)3+···
+=−(1
+[1]x+1
+[2]x2+1
+[3](−x)3+···)
++ (1
+[1]xe−t+1
+[2]x2e−2t+1
+[3]x3e−3t+···)
++ (q1/2+q−1/2
+[1]x2+q+q−1
+[2]x4+···)
+−(q3/2+q1/2+q−1/2+q−3/2
+[1]x2e−t+q3+q+q−1+q−3
+[2]x4e−2t+···)
++ (q3/2+q−3/2
+[1]x2e−2t+q3+q−3
+[2]x4e−4t+···)
+−(q2−q+1−q−1+q−2
+[1]x3+···)+···.OPEN STRING INVARIANTS AND MIRROR CURVE 23
+In other words,
+N(1)
+1,0= 1, N(1)
+1,1=−1, N(1)
+2,0=q1/2+q−1/2,
+N(1)
+2,1=−(q3/2+q1/2+q−1/2+q−3/2), N(1)
+2,2=q3/2+q−3/2,
+N(1)
+3,0=q2−q+1−q−1+q−2, N(1)
+3,1=−2/summationdisplay
+k=−2qk·(q2+1+q−2),
+N(1)
+3,2=3/summationdisplay
+k=−3qk·(q2+1+q−2),
+N(1)
+3,3=−(q+1+q−1)(q−1+q−1)(q3+1+q−3).
+6.The Mirror Curve of the Resolved Conifold
+We apply results on open string one-point functions above to relate them to the
+mirror curve of the resolved conifold.
+6.1.The proof of the Conjecture of Aganagic-Vafa. Fora= 0, we have
+proved the following identity:
+(107) Ψ(0)
+0,1(t;x) =−∞/summationdisplay
+n=1xn
+nn/summationdisplay
+j=0(n+j−1)!
+j!j!(n−j)!Qj.
+This is exactly the prediction by Aganagic-Vafa [5]. From this one can r ecover the
+equation of the mirror curve of the resolved conifold with the outer brane in zero
+framing as follows. Aganagic-Vafa [5] conjectured that the mirror curve when the
+outer brane is in framing ais described by a plane curve
+h(x,y) = 0,
+which determines near x= 0 a function y=y(x;t) so that:
+(108) xd
+dxΨ(a)
+0,1(t;x) = logy(x;t).
+Now we have
+x∂
+∂xΨ(0)
+0,1(t;x) =−∞/summationdisplay
+n=1xnn/summationdisplay
+j=0(n+j−1)!
+j!j!(n−j)!Qj
+=−∞/summationdisplay
+j,k≥0,j+k>0xj+k(k+2j−1)!
+j!j!k!Qj.
+One can take the summation in kfirst to get:
+x∂
+∂xΨ(0)
+0,1(t;x) = ln(1 −x)−∞/summationdisplay
+j=1(2j−1)!
+j!j!(Qx)j
+(1−x)2j.
+By the well-known series:
+(109) ln(1
+2+1
+2/radicalbig
+1+t2) =−∞/summationdisplay
+j=1(−1)j(2j−1)!
+j!j!(t
+2)2j,
+we then get:
+(110) x∂
+∂xΨ(0)
+0,1(t;x) = ln(1
+2(1−x)+1
+2/radicalbig
+(1−x)2+4xe−t).24 JIAN ZHOU
+It follows that
+(111) y:=y(x;t) =1
+2(1−x)+1
+2/radicalbig
+(1−x)2+4xe−t,
+and so
+(112) y2−(1−x)y−xe−t= 0,
+or equivalently,
+(113) x+y−1−xy−1e−t= 0.
+This is exactly the equation for the mirror curve. So we have proved
+Theorem 6.1. Aganagic-Vafa’s Conjectures holds for the case of the resol ved coni-
+fold[5,§5.1]when the outer brane is in zero framing.
+6.2.Proof of the Aganagic-Klemm-Vafa conjecture for framing tr ansfor-
+mation. The mirror curve for the nonzero framing case is more difficult to find .
+It might be possible to generalize the summation method in last subsec tion by the
+theory of hypergeometric series for a/ne}ationslash= 0, but we do not know how to do so. We
+will take a more elementary approach.
+In§4.2 we have seen that the sum in (63) is equal to the coefficient of znin/parenleftbig1+z
+1−Qz/parenrightbig(a+1)n, i.e., it is equal to
+(114) res z=01
+zn+1/parenleftbig1+z
+1−Qz/parenrightbig(a+1)n=1
+2πi/integraldisplay
+|z|=ǫ1
+zn+1/parenleftbig1+z
+1−Qz/parenrightbig(a+1)ndz.
+for sufficiently small ǫ>0 (sayǫ<1
+|Q|), Therefore, (62) can be rewritten as follows
+(we assume |x|is sufficiently small to ensure uniform convergence):
+(x∂
+∂x)2Ψ(a)
+0,1(t;x) =−∞/summationdisplay
+n=1xn
+a+11
+2πi/integraldisplay
+|z|=ǫ1
+zn+1/parenleftbig1+z
+1−Qz/parenrightbig(a+1)ndz
+=−1
+2π(a+1)i/integraldisplay
+|z|=ǫ1
+z∞/summationdisplay
+n=1xn
+zn/parenleftbig1+z
+1−Qz/parenrightbig(a+1)ndz
+=−1
+2π(a+1)i/integraldisplay
+|z|=ǫ/parenleftbigg1
+z−x/parenleftbig1+z
+1−Qz/parenrightbiga+1−1
+z/parenrightbigg
+dz
+=−1
+2π(a+1)i/integraldisplay
+|z|=ǫ/parenleftbigg1
+z−x/parenleftbig1+z
+1−Qz/parenrightbiga+1/parenrightbigg
+dz+1
+a+1.
+We use Cauchy’s residue theorem to evaluate the contour integral of the analytic
+function
+f(z) =1
+z−x/parenleftbig1+z
+1−Qz/parenrightbiga+1=(1−Qz)a+1
+z(1−Qz)a+1−x(1+z)a+1.
+BecauseQandxare very small, so the denominator is close to z, therefore by
+Rouch´ e’stheorem,thereisonlyoneroot z0=z0(x0;Q)forz(1−Qz)a+1−x(1+z)a+1
+inside the disc |z|< ǫ. This determines z0as an analytic function in xby the
+Lagrange inversion formula, i.e., by finding the inverse series of
+(115) x=z0(1−Qz0)a+1
+(1+z0)a+1.OPEN STRING INVARIANTS AND MIRROR CURVE 25
+The first couple of terms are given by:
+(116) z=x+(a+1)(1+Q)x2+···.
+So we get:
+(117) (x∂
+∂x)2Ψ(a)
+0,1(t;x)
+=−1
+a+1(1−Qz0)a+1
+(1−Qz0)a+1−(a+1)Qz0(1−Qz0)a−(a+1)x(1+z0)a+1
+a+1.
+One can use the equation
+(118) z0(1−Qz0)a+1−x(1+z0)a+1= 0.
+to rewrite the right-hand side of (117). Differentiate this equation on both sides
+with respect to x, we get:
+(119)∂z0
+∂x=(1+z0)a+1
+(1−Qz0)a+1−(a+1)Qz0(1−Qz0)a−(a+1)x(1+z0)a.
+By (118) and (119), (117) becomes:
+(120) ( x∂
+∂x)2Ψ(a)
+0,1(t;x) =−x
+(a+1)z0∂z0
+∂x+1
+a+1.
+Integrating once:
+(121) x∂
+∂xΨ(a)
+0,1(t;x) =1
+a+1log(x
+z0) = log1+z0
+1−Qz0.
+According to Aganagic-Vafa [5], the mirror curve is determined by th e following
+equation:
+(122) y:=y(x;t;a) =1+z0
+1−Qz0= (x
+z0)1/(a+1).
+Theorem 6.2. The mirror curve for the resolved conifold with an outer bran e and
+framingais given by the following equation:
+(123) y+xy−a−1−e−txy−a−1= 0.
+Proof.By (122) we get:
+(124) z0=xy−a−1.
+One can rewrite (118) as
+(125) (x
+z0)1/(a+1)=1−Qz0
+1+z0.
+Plug in (122) and (124):
+(126) y=1−Qxy−a−1
+1+xy−a−1.
+This is equivalent to (123). /square
+In [3], it was conjectured that the equation of the mirror curve in ze ro framing
+and theaframing are related by the following framing transformation:
+x/ma√sto→xy−a, y /ma√sto→y. (127)26 JIAN ZHOU
+Note (113) is transformed to (123). So we have established the Ag anagic-Klemm-
+Vafa conjecture for framing transformation in the case of the re solved conifold with
+one outer brane.
+6.3.The quantum mirror curve. Because
+(128) λΨ(a)
+1(λ;t;x1) =/summationdisplay
+g≥0λ2gΨ(a)
+g,1(t;x)
+is a deformation of Ψ(a)
+0,1(t;x), with the parameter λserving as the Planck constant,
+if we set
+(129) y(λ;t,x) = exp(λ·xd
+dxΨ(a)
+1(λ;t;x)),
+theny(λ;t;x) is a deformation of y(t;x). The family of curves y=y(λ;t;x) can be
+understood as the “quantum mirror curve” of the resolved conifo ld.
+In§4.3 we have seen that the sum in (69) is equal to the coefficient of znin
+f(a)
+n(z;t) =/braceleftBigg/producttext(a+1)n
+k=11+q((a+1)n−2k+1)/2z
+1−Qq((a+1)n−2k+1)/2z, a≥0,
+/producttextbn
+k=11−Qq(bn−2k+1)/2z
+1+q(bn−2k+1)/2z, a +1 =−b≤ −1.
+We will treat the case of a≥0 here, the a≤ −2 case is similar. The coefficient is
+equal to
+resz=0(a+1)n/productdisplay
+k=11+q((a+1)n−2k+1)/2z
+1−Qq((a+1)n−2k+1)/2z
+=1
+2πi/integraldisplay
+|z|=ǫ1
+zn+1(a+1)n/productdisplay
+k=11+q((a+1)n−2k+1)/2z
+1−Qq((a+1)n−2k+1)/2zdz,
+for sufficiently small ǫ >0, now depending on n. To avoid difficulties in analysis,
+we will understand the residue formally as taking the coefficient of z−1. Now (76)
+can be rewritten as follows:
+(130)x∂
+∂xΨ(a)
+1(λ;t;x) = res z=0∞/summationdisplay
+n=1xn
+i·[(a+1)n](a+1)n/productdisplay
+k=11+q((a+1)n−2k+1)/2z
+1−Qq((a+1)n−2k+1)/2z.
+We do not know how to take the summation on the right-hand side at p resent.
+References
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+hierarchies , Commun.Math.Phys. 261(2006), 451-516, arXiv:hep-th/0312085.
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+recursion , arXiv:0906.1206.OPEN STRING INVARIANTS AND MIRROR CURVE 27
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+arXiv:0709.1453.
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+interpretations , Adv.Theor.Math.Phys. 3(1999) 495-565, arXiv:hep-th/9903053.
+[12] B. Eynard, M. Mulase, B. Safnuk The Laplace transform of the cut-and-join equation and
+the Bouchard-Marino conjecture on Hurwitz numbers , arXiv:0907.5224.
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+arXiv:math-ph/0702045.
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+string theories , arXiv:0911.5096.
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+Phys.3(1999), no. 5, 1415-1443, arXiv:hep-th/9811131.
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+tions and multiple covers of the disc, Adv. Theor. Math. Phys . 5 (2001), 1-49.
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+math.AG/0408426.
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+Differential Geom. 65(2003), 289-340.
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+149-184, math.AG/0310272.
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+Phys.5(2001), no. 1, 67C91.
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+1995.
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+arXiv:hep-th/0612127.
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+WI, 2001), 185-204, Contemp. Math., 310, Amer. Math. Soc., P rovidence, RI, 2002.
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+dimensional algebras , Cambridge Tracts in Mathematics, 135, Cambridge Universi ty Press,
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+Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China
+E-mail address :jzhou@math.tsinghua.edu.cn
\ No newline at end of file
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+arXiv:1001.0448v3  [math.AG]  2 Apr 2011GENERATORS OF MODULES IN TROPICAL
+GEOMETRY
+SHUHEI YOSHITOMI
+Contents
+1. Introduction 1
+1.1. Results 1
+1.2. Background 3
+2. Definitions and theorems 4
+3. Tropical algebra 8
+3.1. Tropical semigroups, semirings, and semifields 8
+3.2. Modules over a tropical semifield 10
+3.3. Basis and extremal rays 12
+3.4. Locators 13
+3.5. Straight modules 16
+3.6. Existence of inversions 20
+3.7. Straight reflexive modules 22
+3.8. Free modules 24
+4. Polytopes in a tropical projective space 27
+5. Square matrices over a tropical semifield 28
+6. Tropical curves 30
+7. Tropical plane curves 34
+7.1. Tropicalization 34
+7.2. Examples 36
+References 36
+1.Introduction
+1.1.Results. A tropical curve is a geometric object over the tropical
+semifield of real numbers T= (R∪ {−∞},⊕,⊙), where the addition
+⊕is the max-operation in the real field R, and the multiplication ⊙
+is the addition of R. For a tropical curve Cand a divisor DonC,
+2000Mathematics Subject Classification. Primary 14T05; Secondary 12K10,
+52B20.
+Key words and phrases. tropical geometry, semifields, lattice polytopes.
+12
+the setM=H0(C,OC(D)) of the sections of Dhas the structure of a
+T-module that is defined as follows.
+AT-moduleMisdefinedasamoduleoverasemifield. ( M,⊕,⊙,−∞)
+is said to be a T-module if ( M,⊕,−∞) is a tropical semigroup, and ⊙
+is an additive semigroup action on MbyT. A tropical semigroup is
+a commutative semigroup with unity such that any element vsatisfies
+the idempotent condition v⊕v=v.
+AT-moduleMis analogous to a module over a field. A subset
+S⊂Mis said to be a basis if it is a minimal system of generators.
+But the number of elements of a basis of Mis not necessarily equal
+to the topological dimension of it. We introduce straight T-modules in
+section2. Thisclassisageneralizationoflattice-preservingsubmod ules
+of the free T-module Tn, where a lattice-preserving submodule is a
+submodule preserving the infimum of any two elements with respect t o
+the canonical partial order relation on Tn.
+Theorem 1.1. LetMbe a finitely generated straight submodule of the
+freeT-moduleTn. ThenMis generated by nelements.
+We have four corollaries (Theorem 2.1, 2.2, 2.3, 2.4). The semifield
+Tis generalized to a quasi-complete totally ordered rational tropical
+semifieldk. We find a sufficient condition to the existence of a left-
+inversion of an injective homomorphism of k-modules (Theorem 2.1).
+The dimension of a straight reflexive k-module is defined to be the
+number of elements of a basis. We show the inequality dim( M)≤
+dim(N) for a pair of straight reflexive k-modulesM⊂N(Theorem
+2.2). We show that a finitely generated straight pre-reflexive k-module
+is reflexive (Theorem 2.3). Also we consider finiteness of a submodule
+of ak-module (Theorem 2.4). The proofs are given in section 3.7.
+This result has an application to polytopes in a tropical projective
+spaceTPn. By Joswig and Kulas [3], a polytrope (it means a polytope
+inTPnthat is real convex) is a tropical simplex, and therefore it is the
+tropically convex hull of at most n+1 points. We show a generalization
+of this result (Theorem 2.5). A polytope Pis the tropically convex hull
+of at most n+ 1 points if the corresponding submodule M⊂Tn+1is
+straight reflexive. Also Mis straight reflexive if Pis a polytrope.
+Also we have an application to tropical curves. A Riemann-Roch
+theorem for tropical curves is proved by Gathmann and Kerber [1].
+Thistheoremstatesanequality foraninvariant r(D)ofthedivisor. We
+see thatr(D) is not an invariant of the T-moduleM=H0(C,OC(D))
+(Example 6.5), and show the inequality r(D)≤dim(M)−1 (Theorem
+2.7).3
+1.2.Background. Surveys of tropical mathematics are found in [4],
+[7]. Early studies of tropical curves are found in [1], [5], [6]. Tropical
+varieties are introduced as follows. Let K=C[[R]] be the group alge-
+bra of power series defined by the group R. We have a multiplicative
+seminorm
+||·||:K→R≥0
+defined by
+||x||= exp(−val(x)),
+where val means the canonical valuation on K. This seminorm induces
+the amoeba map
+A: (K×)n−→Rn
+defined by
+A(x1,...,x n) = (log||x1||,...,log||xn||).
+The image A(V) of a variety Vin the algebraic torus ( K×)nis said to
+be a tropical variety in the tropical torus Rn.
+Tropical algebra is introduced by the map
+π:K−→R∪{−∞}
+defined by
+π(x) = log||x||.
+This map induces a hyperfield homomorphism
+π:K−→X,
+whereXis the tropical hyperfield with underlying set R∪ {−∞} ,
+introduced in [8]. The power set 2Xis a semiring with operations
+induced by multi-operations of X.
+Now we have the lower-saturation map
+ν:X−→2X
+defined by
+ν(a) ={c∈X|c≤a}.
+The power set 2Xhas a subsemiring
+I=X∪ν(X),
+whichisisomorphictoIzhakian’sextendedtropicalsemiringintroduc ed
+in [2]. The lower-saturation map νmeans the ghost map in [2]. The
+imageν(X) means the ghost part, which is isomorphic to the tropi-
+cal semifield of real numbers ( R∪{−∞},⊕,⊙), where operations are
+defined as follows.
+a⊕b= max{a,b},
+a⊙b=a+b.4
+In this paper, the symbol Tmeans the tropical semifield of real num-
+bers. Under the identification T=ν(X), the canonical homomorphism
+ν:I→Tis the lower-saturation map.
+Section 2 contains definitions and theorems. Section 3 and 4 contain
+foundation of tropical modules, and the proof of Theorem 2.1, 2.2, 2.3,
+2.4, and 2.5. Section 5 and 6 contain foundation of tropical matrices
+and tropical curves, and the proof of Theorem 2.7. Section 7 is an
+appendix for tropical plane curves.
+Acknowledgements. The author thanks Professor Yujiro Kawama ta
+for helpful advice.
+2.Definitions and theorems
+Asemigroup (M,⊕) is a setMwith an associative operation ⊕.
+Definition. (M,⊕,−∞) is atropical semigroup if it satisfies the fol-
+lowing axioms.
+(i) (M,⊕) is a semigroup.
+(ii)v⊕w=w⊕v.
+(iii)v⊕−∞=v.
+(iv)v⊕v=v.
+The element −∞is called the zero element of M.
+There is a unique partial order relation ‘ ≤’ onMsuch that for any
+v,w∈Mit implies
+sup{v,w}=v⊕w.
+The proof is given in section 3.1.
+Definition. A tropical semigroup Misquasi-complete if any non-
+empty subset S⊂Madmits the infimum inf( S) (i.e. it admits the
+maximum element of the lower-bounds of S).
+Definition. (A,⊕,⊙,−∞,0) is atropical semiring if it satisfies the
+following axioms.
+(i) (A,⊕,−∞) is a tropical semigroup.
+(ii) (A,⊙) is a semigroup.
+(iii)a⊙b=b⊙a.
+(iv)a⊙(b⊕c) =a⊙b⊕a⊙c.
+(v)a⊙0 =a.
+(vi)a⊙−∞=−∞.
+The element −∞is called the zero element of A. The element 0 is
+called the unity of A.5
+Definition. (k,⊕,⊙,−∞,0) is atropical semifield if it satisfies the
+following axioms.
+(i) (k,⊕,⊙,−∞,0) is a tropical semiring.
+(ii) For any a∈k\ {−∞} there is an element ⊘a∈ksuch that
+a⊙(⊘a) = 0.
+Definition. A tropical semifield kisrationalif it satisfies the following
+conditions.
+(i)a∈k,m∈N⇒∃b∈k,a=b⊙m.
+(ii)khas no maximum element.
+Thetropical semifield of real numbers (T,⊕,⊙,−∞,0) is the set
+T=R∪{−∞}
+equipped with addition
+a⊕b= max{a,b}
+and multiplication
+a⊙b=a+b
+andzeroelement −∞andunity0. Tisaquasi-completetotallyordered
+rational tropical semifield.
+Letkbe a quasi-complete totally ordered rational tropical semifield.
+Definition. (M,⊕,⊙,−∞) is ak-moduleif it satisfies the following
+axioms.
+(i) (M,⊕,−∞) is a tropical semigroup.
+(ii)⊙is a semigroup action k×M∋(a,v)/mapsto→a⊙v∈M, i.e.
+i) (a⊙b)⊙v=a⊙(b⊙v).
+ii) 0⊙v=v.
+(iii) (a⊕b)⊙v= (a⊙v)⊕(b⊙v).
+(iv)a⊙(v⊕w) = (a⊙v)⊕(a⊙w).
+(v)−∞⊙v=−∞.
+(vi)a⊙−∞=−∞.
+Definition. Ahomomorphism α:M→Nofk-modules is a map with
+the following conditions.
+(i)α(−∞) =−∞.
+(ii)α(v⊕w) =α(v)⊕α(w).
+(iii)α(a⊙v) =a⊙α(v).
+Let Hom(M,N) denote the k-module of homomorphisms from Mto
+N.6
+The dual module M∨is defined by M∨= Hom(M,k). We have the
+pairing map /a\}bracketle{t·,·/a\}bracketri}ht:M×M∨→kdefined by
+/a\}bracketle{tv,ξ/a\}bracketri}ht=ξ(v).
+Definition. Mispre-reflexive if the homomorphism ιM:M→(M∨)∨
+is injective. Misreflexive ifιMis an isomorphism.
+Definition. Ak-moduleMisstraightif it is a finitely distributive
+ordered lattice, i.e. it satisfies the following conditions.
+(i) Any two elements v,w∈Madmit the infimum inf M{v,w}.
+(ii)v1,v2,w∈M⇒infM{v1⊕v2,w}= infM{v1,w}⊕infM{v2,w}.
+(iii)v1,v2,w∈M⇒infM{v1,v2}⊕w= infM{v1⊕w,v2⊕w}.
+Definition. A homomorphism α:M→Nislightly surjective if for
+anyw∈Nthere isv∈Msuch thatw≤α(v).
+A homomorphism β:N→Mis said to be a left-inversion of αif
+β◦α= idM.
+Theorem 2.1. Letα:M→Nbe an injective lightly surjective homo-
+morphism of k-modules such that Mis straight reflexive. Then αhas
+a left-inversion.
+Definition. Abasis{eλ|λ∈Λ}of ak-moduleMis a minimal system
+of generators (i.e. there is no λ0∈Λ such that the elements {eλ|λ∈
+Λ\{λ0}}generateM). A subset S⊂MgenerateMif any element of
+Mis written as a linear combination
+a1⊙v1⊕···⊕ar⊙vr
+of elements of Soverk.
+Definition. Anelement e∈M\{−∞}isextremal ifforanyv1,v2∈M
+such thatv1⊕v2=eit impliesv1=eorv2=e.Misextremally
+generated ifMis generated by extremal elements. An extremal ray of
+Mis the submodule generated by an extremal element of M.
+Definition. Thedimension of a straight reflexive k-moduleMis the
+number of extremal rays.
+Thenumber ofextremal raysof Misequal tothenumber ofelements
+of any basis of M. The proof is given in section 3.3.
+Theorem 2.2. Letα:M→Nbe an injective homomorphism of
+finitely generated straight reflexive k-modules. Then
+(1) dim(M)≤dim(N).
+(2)Ifdim(M) = dim(N), thenαis lightly surjective.7
+Theorem 2.3. LetMbe a finitely generated straight pre-reflexive k-
+module. Then Mis reflexive.
+Theorem 2.4. Letα:M→Nbe an injective homomorphism of
+straight pre-reflexive k-modules. Suppose that Mhas a basis, and that
+Nis finitely generated. Then Mis finitely generated.
+LetPbea polytopein TPn.Pisthe tropicallyconvex hull of finitely
+many points p1,...,p r. Let
+ϕ:Tn+1\{−∞} −→ TPn
+be the canonical projection. Then the subset
+M=ϕ−1(P)∪{−∞} ⊂ Tn+1
+is a submodule generated by elements v1,...,v rsuch thatϕ(vi) =pi
+(1≤i≤r). Also we have an injection
+ι:Tn−→TPn
+defined by ( a1,...,a n)/mapsto→(0,a1,...,a n). This map induces an embed-
+dingRn⊂Tn⊂TPn. A polytope P⊂TPnis said to be a polytrope if
+it is a real convex subset of Rn.
+Theorem 2.5. LetPbe a polytope in TPnwith the corresponding
+submoduleM⊂Tn+1.
+(1)IfPis a polytrope, then Mis straight reflexive.
+(2)IfMis straight reflexive, then Pis the tropically convex hull of
+at mostn+1points.
+LetCbeatropicalcurve. Let Dbeadivisoron C. LetH0(C,OC(D))
+be the set of the sections of D. (A section of Dis a rational function
+f:C→Tsuch that either f=−∞or (f)+D≥0.) Forr∈Z≥0, let
+U(D,r) =Cr\S(D,r),
+S(D,r) ={(P1,...,P r)∈Cr|H0(C,OC(D−/summationdisplay
+1≤i≤rPi))/\e}atio\slash=−∞}.
+LetU(D,r) =∅ifr=−1. The following theorem is known.
+Theorem 2.6 (GathmannandKerber[1]) .LetCbe a compact tropical
+curve with first Betti number b1(C). LetDbe a divisor on C. LetK
+be the canonical divisor on C. Then
+r(D)−r(K−D) = 1−b1(C)+deg(D),
+where
+r(D) = max{r∈Z≥−1|U(D,r) =∅}.8
+The setM=H0(C,OC(D)) is aT-module with addition
+(f⊕g)(P) =f(P)⊕g(P)
+and scalar multiplication
+(a⊙f)(P) =a⊙f(P).
+The dimension of Mis defined as follows.
+Definition. Thedimension of ak-moduleMis the maximum dimen-
+sion of the straight reflexive submodules of M.
+This definition is compatible with the previous one. If Mis straight
+reflexive, then the maximum dimension of the straight reflexive sub-
+modules of Mequals the dimension of Mby Theorem 2.2.
+Theorem 2.7. LetCbe a tropical curve. Let Dbe a divisor on C.
+Then the inequality
+r(D)≤dimH0(C,OC(D))−1
+is fulfilled.
+3.Tropical algebra
+3.1.Tropical semigroups, semirings, and semifields.
+Proposition 3.1. LetMbe a tropical semigroup. Then there is a
+unique partial order relation ‘ ≤’ such that for any v,w∈Mit implies
+sup{v,w}=v⊕w.
+Proof.We define a relation ‘ ≤’ onMas follows.
+v≤w⇐⇒v⊕w=w.
+This is a partial order relation, because v⊕v=v. The element v⊕w
+is the minimum element of the upper bounds of {v,w}. /square
+LetAbe a tropical semiring.
+Example 3.2.Thesemiring of polynomials B=A[x1,...,x n] is the set
+of polynomials
+f=/circleplusdisplay
+iai⊙x⊙i
+=/circleplusdisplay
+i1,...,in≥0ai1...in⊙x⊙i1
+1⊙···⊙x⊙in
+n9
+with coefficients ai∈A, equipped with addition and multiplication of
+polynomials. Bis a tropical semiring. An element f∈Bis said to be
+atropical polynomial over A. The induced map
+f:An−→A
+(a1,...,a n)/mapsto→f(a1,...,a n)
+is said to be a tropical polynomial function.
+Remark3.3.We use the notation maby the meaning of tropical m-th
+powera⊙m. For example, 2( a⊕b) means the second power of ( a⊕b),
+so we have
+2(a⊕b) = 2a⊕a⊙b⊕a⊙b⊕2b
+= 2a⊕a⊙b⊕2b.
+Also a tropical polynomial is written as
+f=/circleplusdisplay
+iai⊙ix.
+Proposition 3.4. LetAbe a tropical semiring. Let f∈A[x1,...,x n].
+Then for any v,w∈An,
+f(v⊕w)≥f(v)⊕f(w).
+Proof.Assume that
+f=i1x1⊙···⊙inxn,
+v= (a1,...,a n),
+w= (b1,...,b n).
+Then
+f(v⊕w) =i1(a1⊕b1)⊙···⊙in(an⊕bn)
+≥(i1a1⊙···⊙inan)⊕(i1b1⊙···⊙inbn)
+=f(v)⊕f(w).
+/square
+Letkbe a tropical semifield. Recall that kis said to be rational if
+it satisfies the following conditions.
+(i)a∈k,m∈N⇒∃b∈k,a=b⊙m.
+(ii)khas no maximum element.
+Proposition 3.5. Letkbe a rational tropical semifield. Then for any
+a∈kit implies
+inf
+k{b∈k|a<b}=a.10
+Proof.The case of a=−∞. Suppose that there is an element c∈
+k\{−∞} such thatk≥(c) =k\{−∞}. Then the element 0 ⊘cis the
+maximum element of k, which is contradiction.
+The case of a/\e}atio\slash=−∞. The condition a < bis fulfilled if and only if
+0<b⊘a. So we may assume a= 0. Suppose that there is an element
+c/notlessequal0 such that cis a lower-bound of the set {b∈k|0<b}. There is
+an element c′∈ksuch thatc= (c′)⊙2= 2c′. Since 0<0⊕c′, we have
+c≤0⊕c′. So we have
+2(0⊕c′) = 0⊕c′⊕2c′
+= 0⊕c′⊕c
+= 0⊕c′,
+0⊕c′= 0.
+So we have c≤0, which is contradiction. /square
+3.2.Modules over a tropical semifield. Letkbe a tropical semi-
+field. LetMbe ak-module.
+Definition. AsubmoduleNofMis a subset with the following con-
+ditions.
+(i)−∞ ∈N.
+(ii) Ifv,w∈Nthenv⊕w∈N.
+(iii) Ifv∈Nanda∈kthena⊙v∈N.
+Example 3.6.Suppose that kis totally ordered. Let q∈k[x1,...,x n]
+be a homogeneous polynomial of degree m. Letp:kn→kbe a homo-
+morphism of k-modules. Then the subset
+M={v∈kn|mp(v)≤q(v)}
+is a submodule of kn. Indeed, for v,w∈Manda∈k,
+mp(a⊙v) =m(a⊙p(v))
+=ma⊙mp(v)
+≤ma⊙q(v)
+=q(a⊙v),
+mp(v⊕w) =m(p(v)⊕p(w))
+= max{mp(v),mp(w)}
+=mp(v)⊕mp(w)
+≤q(v)⊕q(w).
+By Proposition 3.4, we have q(v)⊕q(w)≤q(v⊕w).11
+Example 3.7.A free module M=knof finite rank is reflexive. Indeed
+there is a pairing map /a\}bracketle{t·,·/a\}bracketri}ht:kn×kn→kdefined by
+/a\}bracketle{t(a1,...,a n),(b1,...,b n)/a\}bracketri}ht=a1⊙b1⊕···⊕an⊙bn.
+So we have ( kn)∨∼=kn.
+Recallthat Missaidtobepre-reflexiveifthehomomorphism ιM:M→
+(M∨)∨is injective.
+Proposition 3.8. Mis pre-reflexive if and only if there is an injection
+M→Ffor some direct product F=/producttext
+λ∈Λk.
+Proof.There is an injection ( M∨)∨→/producttext
+λ∈Λk, where Λ is the set M∨.
+Conversely, if there is an injection M→Ffor some direct product F,
+thenMis pre-reflexive, because Fis pre-reflexive. /square
+Lemma 3.9. Suppose that kis rational. Let Mbe a pre-reflexive k-
+module. Then for any v∈Mand anya∈kit implies
+inf
+M{b⊙v|b∈k,a<b}=a⊙v.
+Proof.Letw∈Mbe a lower-bound of the subset {b⊙v|b∈k,a<b}.
+Forξ∈M∨andb∈ksuch thata<b, we have
+ξ(w)≤b⊙ξ(v).
+By Proposition 3.5, we have
+ξ(w)≤a⊙ξ(v).
+SinceMis pre-reflexive, we have w≤a⊙v. /square
+Lemma 3.10. Suppose that kis totally ordered. Let Mbe a pre-
+reflexivek-module. Then for any v,w∈Mand anya∈k,
+v/notlessequalw,a<0⇒v/notlessequalw⊕a⊙v.
+Proof.SinceMis pre-reflexive, there is an element ξ∈M∨such that
+ξ(v)/notlessequalξ(w). Sincekis totally ordered, we have ξ(w)<ξ(v). So
+max{ξ(w),a⊙ξ(v)}<ξ(v).
+So we have the conclusion. /square
+Example 3.11.LetGbe a tropical semigroup with at least two ele-
+ments. Let M= (G×R)∪{−∞} be theT-module with addition
+(v,a)⊕(w,b) = (v⊕w,a⊕b)
+and scalar multiplication
+c⊙(v,a) =/braceleftBigg
+(v,c⊙a) ifc∈R
+−∞ ifc=−∞.12
+Mis aT-module generated by the subset G× {0}.Mis not pre-
+reflexive, because it does not satisfy Lemma 3.10. Let v,w∈Gbe
+elements such that v/notlessequalw. Then
+(v,0)/notlessequal(w,0),
+(v,0)≤(w,0)⊕(−1)⊙(v,0).
+3.3.Basis and extremal rays. Letkbe a totally ordered tropical
+semifield. Let Mbe ak-module. Recall that anelement e∈M\{−∞}
+is said to be extremal if for any v1,v2∈Msuch thatv1⊕v2=eit
+impliesv1=eorv2=e.
+Proposition 3.12. LetMbe a pre-reflexive k-module. Then the fol-
+lowing are equivalent.
+(i)There is a basis of M.
+(ii)Mis extremally generated.
+More precisely, a system of generators E={eλ|λ∈Λ}is a basis if
+and only if each eλis extremal and it satisfies k⊙eλ/\e}atio\slash=k⊙eµ(λ/\e}atio\slash=µ).
+Proof.Suppose that there is a basis EofM. Lete1be an element of
+the basisE. Letv1,v2∈Mbe elements such that v1⊕v2=e1. There
+are elements e2,e3,...,e rof the basis Eand elements ai,bi∈ksuch
+that
+v1=a1⊙e1⊕a2⊙e2⊕···⊕ar⊙er,
+v2=b1⊙e1⊕b2⊙e2⊕···⊕br⊙er.
+Sincekis totally ordered, we may assume a1≤b1. Then
+e1=b1⊙e1⊕w,
+where
+w= (a2⊕b2)⊙e2⊕···⊕(ar⊕br)⊙er.
+SinceEis a basis, we have w/\e}atio\slash=e1. By Lemma 3.10, we have b1= 0.
+It meansv2≥e1. So we have v2=e1. Thuse1is extremal.
+Conversely, let Ebe a system of generators that consists of extremal
+elements with different extremal rays. Suppose that Eis not a basis.
+There are elements e1,e2,...,e rofEand elements ai∈ksuch that
+e1=a2⊙e2⊕···⊕ar⊙er.
+Sincee1is extremal, there is a number isuch thate1=ai⊙ei, which
+is contradiction. /square
+Proposition 3.13. Letα:M→Nbe a homomorphism of k-modules.
+Letw∈Nbe an extremal element. Then any minimal element of the
+subsetα−1(w)is extremal.13
+Proof.Lete∈Mbe a minimal element of α−1(w). Letv1,v2∈Mbe
+elements such that v1⊕v2=e. Thenα(v1)⊕α(v2) =w. Sincewis
+extremal, we may assume α(v1) =w. Thenv1is a lower-bound of ein
+α−1(w). Sinceeis minimal, we have v1=e. /square
+3.4.Locators. Letkbe a totally ordered tropical semifield. Let M
+be ak-module. For a subset S⊂M, the lower-saturation M≤(S) is
+defined by
+M≤(S) =/uniondisplay
+w∈S{v∈M|v≤w}.
+The set of the lower-bounds Low M(S) is defined by
+LowM(S) =/intersectiondisplay
+w∈S{v∈M|v≤w}.
+A subsetS⊂Mis said to be lower-saturated if M≤(S) =S.
+Definition. AlocatorSofMis a lower-saturated subsemigroup of the
+semigroup ( M,⊕) that generates the k-moduleM.
+LetLoc(M)denotethesetofthelocatorsofa k-moduleM, equipped
+with addition
+S∨⊕T=S∩T
+and scalar multiplication
+a∨⊙S=/braceleftBigg
+(⊘a)⊙Sifa∈k\{−∞}
+M ifa=−∞.
+Proposition 3.14. (Loc(M),∨⊕,∨⊙)is ak-module with zero element
+M. There is a homomorphism
+i:M∨−→Loc(M)
+defined by
+i(ξ) ={v∈M|/a\}bracketle{tv,ξ/a\}bracketri}ht ≤0}.
+Proof.Loc(M) is a tropical semigroup. Indeed,
+S∨⊕S=S∩S=S.
+Loc(M) is ak-module. Indeed, for a,b∈ksuch thata≤b, sinceSis
+lower-saturated, we have
+⊘b⊙S⊂ ⊘a⊙S.14
+So we have
+(a⊕b)∨⊙S=⊘b⊙S
+= (⊘a⊙S)∩(⊘b⊙S)
+=a∨⊙S⊕b∨⊙S.
+iis a homomorphism. Indeed, for v∈M,
+v∈i(ξ1⊕ξ2)⇐⇒ /a\}bracketle{tv,ξ1⊕ξ2/a\}bracketri}ht ≤0
+⇐⇒ /a\}bracketle{tv,ξ1/a\}bracketri}ht⊕/a\}bracketle{tv,ξ2/a\}bracketri}ht ≤0
+⇐⇒v∈i(ξ1)∩i(ξ2)
+⇐⇒v∈i(ξ1)∨⊕i(ξ2).
+Soi(ξ1⊕ξ2) =i(ξ1)∨⊕i(ξ2).
+v∈i(a⊙ξ)⇐⇒ /a\}bracketle{tv,a⊙ξ/a\}bracketri}ht ≤0
+⇐⇒ /a\}bracketle{ta⊙v,ξ/a\}bracketri}ht ≤0
+⇐⇒a⊙v∈i(ξ)
+⇐⇒v∈a∨⊙i(ξ).
+Soi(a⊙ξ) =a∨⊙i(ξ). /square
+Lemma 3.15. Suppose that kis quasi-complete and rational.
+(1)For any locator S∈Loc(M)there is a unique element ξ∈M∨
+that satisfies the following conditions.
+/a\}bracketle{tv,ξ/a\}bracketri}ht ≤0 (v∈S),
+/a\}bracketle{tv,ξ/a\}bracketri}ht ≥0 (v∈M\S).
+(2)The mapping S/mapsto→ξinduces a homomorphism
+p: Loc(M)−→M∨
+which satisfies p◦i= idM∨.
+Proof.(1) Letξ:M→kbe the map defined as follows.
+ξ(v) = inf
+k{a∈k|v∈a⊙S}.
+The set in right side is non-empty. (Since Sgenerates the k-module
+M, there aresi∈Sandai∈ksuch that
+v=a1⊙s1⊕···⊕ar⊙sr.15
+Letabethemaximumelement of a1,...,a r. SinceSislower-saturated,
+there ares′
+i∈Ssuch that
+v=a⊙(s′
+1⊕···⊕s′
+r).
+SinceSis a subsemigroup, we have v∈a⊙S.) For anyv∈M\Swe
+haveξ(v)≥0, because Sis lower-saturated. For any v∈S, we have
+ξ(v)≤0.
+We show that ξis a homomorphism. Since Sis lower-saturated, we
+have
+ξ(v)⊕ξ(w)≤ξ(v⊕w).
+Suppose that ξ(v)⊕ξ(w)< ξ(v⊕w). There are a,b∈ksuch that
+a⊕b<ξ(v⊕w) andv∈a⊙Sandw∈b⊙S. Thenv⊕w∈(a⊕b)⊙S.
+So we have ξ(v⊕w)≤a⊕b, which is contradiction.
+We prove uniqueness. Let ξ∈M∨be an element that satisfies the
+following conditions.
+/a\}bracketle{tv,ξ/a\}bracketri}ht ≤0 (v∈S),
+/a\}bracketle{tv,ξ/a\}bracketri}ht ≥0 (v∈M\S).
+Then
+/a\}bracketle{tv,ξ/a\}bracketri}ht ≤inf
+k{a∈k|v∈a⊙S}
+≤inf
+k{a∈k|/a\}bracketle{tv,ξ/a\}bracketri}ht<a}.
+By Proposition 3.5,
+inf
+k{a∈k|/a\}bracketle{tv,ξ/a\}bracketri}ht<a}=/a\}bracketle{tv,ξ/a\}bracketri}ht.
+So we have
+/a\}bracketle{tv,ξ/a\}bracketri}ht= inf
+k{a∈k|v∈a⊙S}.
+(2) We have
+/a\}bracketle{tv,p(S)⊕p(T)/a\}bracketri}ht ≤0 (v∈S∩T),
+/a\}bracketle{tv,p(S)⊕p(T)/a\}bracketri}ht ≥0 (v∈M\(S∩T)).
+It meansp(S)⊕p(T) =p(S∨⊕T). Sopis a homomorphism. For
+ξ∈M∨, let
+S={v∈M|/a\}bracketle{tv,ξ/a\}bracketri}ht ≤0}.
+Then
+/a\}bracketle{tv,ξ/a\}bracketri}ht ≤0 (v∈S),
+/a\}bracketle{tv,ξ/a\}bracketri}ht ≥0 (v∈M\S).
+It meansξ=p(S). /square16
+3.5.Straight modules. Letkbe a totally ordered tropical semifield.
+Recall that a k-moduleMis said to be straight if it satisfies the fol-
+lowing conditions.
+(i) Any two elements v,w∈Madmit the infimum inf M{v,w}.
+(ii)v1,v2,w∈M⇒infM{v1⊕v2,w}= infM{v1,w}⊕infM{v2,w}.
+(iii)v1,v2,w∈M⇒infM{v1,v2}⊕w= infM{v1⊕w,v2⊕w}.
+Proposition 3.16. The above conditions (ii),(iii)are equivalent.
+Proof.(ii)⇒(iii).
+inf
+M{v1⊕w,v2⊕w}= inf
+M{v1,v2}⊕inf
+M{v1,w}⊕inf
+M{w,v2}⊕w
+= inf
+M{v1,v2}⊕w.
+(iii)⇒(ii) is similar. /square
+Definition. A homomorphism α:M→Nofk-modules is lattice-
+preserving ifforanyv,w∈Mandanylower-bound x∈LowN(α(v),α(w))
+there is a lower-bound y∈LowM(v,w) such that x≤α(y).
+IfM,Nare ordered lattices, αis lattice-preserving if and only if it
+preserves the infimum of any two elements.
+Proposition 3.17. Letα:M→Nbe a lattice-preserving injective ho-
+momorphism of k-modules such that Nis straight. Then Mis straight.
+Proof.Forv,w∈M, letx= infN{α(v),α(w)}. There isalower-bound
+yof{v,w}such thatx≤α(y). Theny= infM{v,w}. (Lety′∈Mbe
+a lower-bound of {v,w}. Thenα(y′)≤x≤α(y). Sinceαis injective,
+we havey′≤y.)α(y) is a lower-bound of {α(v),α(w)}. So we have
+x=α(y).Mis finitely distributive, because αpreserves the infimum
+of any two elements. /square
+Proposition 3.18. Suppose that kis quasi-complete and rational. Let
+Mbe a straight k-module. Then M∨andLoc(M)are straight.
+Proof.We show that Loc( M) is straight. For S,T∈Loc(M), let
+U=S⊕T={s⊕t|s∈S,t∈T}.
+Uis lower-saturated. (Let v∈Mands∈Sandt∈Tbe elements
+such thatv≤s⊕t. Then
+v= inf
+M{v,s⊕t}
+= inf
+M{v,s}⊕inf
+M{v,t}.17
+So we have v∈U.)Uis a locator of M, and we have
+U= inf
+Loc(M){S,T}.
+Loc(M) is finitely distributive. Indeed,
+(S1∩S2)⊕T= (S1⊕T)∩(S2⊕T).
+(Letvbe an element of right side. There are s1∈S1ands2∈S2and
+t1,t2∈Tsuch that
+v=s1⊕t1=s2⊕t2.
+Then
+v= inf
+M{s1⊕t1,s2⊕t2}
+= inf
+M{s1,s2}⊕inf
+M{s1,t2}⊕inf
+M{t1,s2}⊕inf
+M{t1,t2}.
+So we have v∈(S1∩S2)⊕T.)
+We show that M∨is straight. For ξ1,ξ2∈M∨, letS1,S2∈Loc(M)
+be the induced element. There is a unique element η∈M∨that
+satisfies the following conditions (Lemma 3.15).
+/a\}bracketle{tv,η/a\}bracketri}ht ≤0 (v∈S1⊕S2),
+/a\}bracketle{tv,η/a\}bracketri}ht ≥0 (v∈M\(S1⊕S2)).
+We haveη= inf M∨{ξ1,ξ2}. So the canonical injection i:M∨→
+Loc(M) is lattice-preserving. Since Loc( M) is straight, M∨is straight
+(Proposition 3.17). /square
+Proposition 3.19. LetMbe ak-module. Let η:M→kbe a lattice-
+preserving homomorphism. Then ηis an extremal element of M∨.
+Proof.Suppose that ηis not extremal. There are elements ξ1,ξ2∈M∨
+and elements v1,v2∈Msuch thatξ1⊕ξ2=ηand/a\}bracketle{tv1,ξ1/a\}bracketri}ht</a\}bracketle{tv1,η/a\}bracketri}ht
+and/a\}bracketle{tv2,ξ2/a\}bracketri}ht</a\}bracketle{tv2,η/a\}bracketri}ht. We may assume /a\}bracketle{tv1,η/a\}bracketri}ht=/a\}bracketle{tv2,η/a\}bracketri}ht= 0. Since
+ηis lattice-preserving, there is a lower-bound wof{v1,v2}such that
+/a\}bracketle{tw,η/a\}bracketri}ht= 0. Then
+0 =/a\}bracketle{tw,η/a\}bracketri}ht
+=/a\}bracketle{tw,ξ1⊕ξ2/a\}bracketri}ht
+≤ /a\}bracketle{tv1,ξ1/a\}bracketri}ht⊕/a\}bracketle{tv2,ξ2/a\}bracketri}ht
+</a\}bracketle{tv1,η/a\}bracketri}ht⊕/a\}bracketle{tv2,η/a\}bracketri}ht
+= 0,
+which is contradiction. /square
+Definition. Adual element η∈M∨ofanelement e∈Mis anelement
+with the following conditions.18
+(i)/a\}bracketle{te,η/a\}bracketri}ht= 0.
+(ii)v∈M,ξ∈M∨⇒ /a\}bracketle{tv,η/a\}bracketri}ht⊙/a\}bracketle{te,ξ/a\}bracketri}ht ≤ /a\}bracketle{tv,ξ/a\}bracketri}ht.
+Proposition 3.20. The dual element of an element e∈Mis unique.
+Proof.Letηbe a dual element of e. Then
+η= min{ξ∈M∨|/a\}bracketle{te,ξ/a\}bracketri}ht= 0},
+because
+η⊙/a\}bracketle{te,ξ/a\}bracketri}ht ≤ξ
+for anyξ∈M∨. /square
+Proposition 3.21. Lete∈Mbe an element of a pre-reflexive k-
+moduleM. Suppose that ehas the dual element η∈M∨. Then
+(1)eis an extremal element.
+(2)η:M→kis a lattice-preserving homomorphism (therefore is
+an extremal element of M∨).
+Proof.(1) Letv1,v2∈Mbe elements such that v1⊕v2=e. Then
+/a\}bracketle{tv1,η/a\}bracketri}ht⊕/a\}bracketle{tv2,η/a\}bracketri}ht=/a\}bracketle{te,η/a\}bracketri}ht= 0.
+We may assume /a\}bracketle{tv1,η/a\}bracketri}ht= 0. Forξ∈M∨,
+/a\}bracketle{te,ξ/a\}bracketri}ht=/a\}bracketle{tv1,η/a\}bracketri}ht⊙/a\}bracketle{te,ξ/a\}bracketri}ht
+≤ /a\}bracketle{tv1,ξ/a\}bracketri}ht.
+SinceMis pre-reflexive, we have e≤v1. So we have e=v1.
+(2) Letv1,v2∈Mbe elements such that /a\}bracketle{tv1,η/a\}bracketri}ht ≤ /a\}bracketle{tv2,η/a\}bracketri}ht. Let
+w=/a\}bracketle{tv1,η/a\}bracketri}ht ⊙e. Forξ∈M∨, we have /a\}bracketle{tw,ξ/a\}bracketri}ht ≤ /a\}bracketle{tvi,ξ/a\}bracketri}ht. SinceMis
+pre-reflexive, we have w≤vi. Sowis a lower-bound of {v1,v2}such
+that/a\}bracketle{tw,η/a\}bracketri}ht=/a\}bracketle{tv1,η/a\}bracketri}ht. Thusηis lattice-preserving. By Proposition 3.19,
+ηis an extremal element of M∨. /square
+Lemma 3.22. Suppose that kis quasi-complete and rational. Let M
+be a straight pre-reflexive k-module. Then any extremal element of M
+has the dual element.
+Proof.Lete∈Mbe an extremal element. The subset
+S={v∈M|e/notlessequalv}
+is a subsemigroup. (Let v1,v2∈Mbe elements such that e≤v1⊕v2.
+Then
+e= inf
+M{e,v1⊕v2}
+= inf
+M{e,v1}⊕inf
+M{e,v2}.19
+We may assume e= infM{e,v1}. Thene≤v1.) AlsoSgenerates the
+kmoduleM. (Letv∈Mbe any element. By Lemma 3.9, we have
+inf
+M{b⊙v|b∈k\{−∞}} =−∞.
+So there is b∈k\{−∞} such thate/notlessequalb⊙v.) ThusSis a locator of
+M. By Lemma 3.15, there is a unique element η∈M∨that satisfies
+the following conditions.
+/a\}bracketle{tv,η/a\}bracketri}ht ≤0 (v∈S),
+/a\}bracketle{tv,η/a\}bracketri}ht ≥0 (v∈M\S).
+Then
+/a\}bracketle{tv,η/a\}bracketri}ht ≤inf
+k{a∈k|a⊙e/notlessequalv}
+≤inf
+k{a∈k|/a\}bracketle{tv,η/a\}bracketri}ht<a}
+=/a\}bracketle{tv,η/a\}bracketri}ht.
+So we have
+/a\}bracketle{tv,η/a\}bracketri}ht= inf
+k{a∈k|a⊙e/notlessequalv}.
+So
+/a\}bracketle{te,η/a\}bracketri}ht= inf
+k{a∈k|0<a}
+= 0.
+Also, for any a∈ksuch that 0 <a, we have
+(/a\}bracketle{tv,η/a\}bracketri}ht⊘a)⊙e≤v.
+By Lemma 3.9, we have
+/a\}bracketle{tv,η/a\}bracketri}ht⊙e≤v.
+Thusηis the dual element of e. /square
+Lemma 3.23. LetMbe a finitely generated pre-reflexive k-module.
+Letβ:kn→Mbe the surjection defined by a basis {e1,...,e n}ofM.
+Suppose that eihas the dual element ηi(1≤i≤n). Then
+(1)Mis straight.
+(2)Thehomomorphism α:M→kndefinedbythe elements η1,...,η n
+is a right-inversion of β, i.e.β◦α= idM.
+(3)αis the unique right-inversion of β.
+Proof.Forv∈M, we have
+v≥ /a\}bracketle{tv,η1/a\}bracketri}ht⊙e1⊕···⊕/a\}bracketle{tv,ηn/a\}bracketri}ht⊙en.20
+It meansβ◦α≤idM. Also, for 1 ≤i≤n, we have
+β◦α(ei)≥ /a\}bracketle{tei,ηi/a\}bracketri}ht⊙ei=ei.
+SinceMis generated by {e1,...,e n}, we haveβ◦α= idM. Soαis
+injective. Also αis lattice-preserving (Proposition 3.21). Since knis
+straight,Mis straight (Proposition 3.17).
+We prove uniqueness. Let η′
+1,...,η′
+n∈M∨be elements such that
+the induced homomorphism M→knis a right-inversion of β. Then
+we have
+v=/a\}bracketle{tv,η′
+1/a\}bracketri}ht⊙e1⊕···⊕/a\}bracketle{tv,η′
+n/a\}bracketri}ht⊙en.
+So
+ei=/a\}bracketle{tei,η′
+i/a\}bracketri}ht⊙ei⊕wi,
+where
+wi=/circleplusdisplay
+j/negationslash=i/a\}bracketle{tei,η′
+j/a\}bracketri}ht⊙ej.
+Since{e1,...,e n}is a basis, we have wi/\e}atio\slash=eiand
+ei=/a\}bracketle{tei,η′
+i/a\}bracketri}ht⊙ei
+(Proposition 3.12). So we have /a\}bracketle{tei,η′
+i/a\}bracketri}ht= 0. Thusη′
+iis the dual element
+ofei. /square
+3.6.Existence of inversions. Letkbe a totally ordered tropical
+semifield. Let α:M→Nbe a homomorphism of k-modules.
+Definition. An element ξ∈M∨dominates an element w∈Nif there
+is an element v∈Msuch that /a\}bracketle{tv,ξ/a\}bracketri}ht ≤0 andw≤α(v).
+Proposition 3.24. Letξi∈M∨be an element that dominates wi∈N
+(i= 1,2). Then any lower-bound ξ∈LowM∨(ξ1,ξ2)dominatesw1⊕w2.
+Proof.There are elements v1,v2∈Msuch that /a\}bracketle{tvi,ξi/a\}bracketri}ht ≤0 andwi≤
+α(vi). Then
+/a\}bracketle{tv1⊕v2,ξ/a\}bracketri}ht ≤ /a\}bracketle{tv1,ξ1/a\}bracketri}ht⊕/a\}bracketle{tv2,ξ2/a\}bracketri}ht
+≤0.
+Also we have w1⊕w2≤α(v1⊕v2). /square
+Recall that a homomorphism α:M→Nis said to be lightly surjec-
+tive if for any w∈Nthere isv∈Msuch thatw≤α(v).
+Lemma 3.25. Letα:M→Nbe an injective lightly surjective homo-
+morphism of k-modules. Suppose that M∨is straight.21
+(1)There is a homomorphism γ:N→Loc(M∨)that satisfies the
+following condition. For any w∈Nthe locator γ(w)is the
+subsemigroup of M∨generated by the elements that dominates
+the element w.
+(2)The diagram
+Mα/d47/d47
+/d15/d15N
+γ
+/d15/d15
+(M∨)∨
+i/d47/d47Loc(M∨)
+commutes, i.e. for any v∈Mand anyξ∈M∨the condition
+/a\}bracketle{tv,ξ/a\}bracketri}ht ≤0is fulfilled if and only if ξ∈γ(α(v)).
+Proof.(1) Forw∈N, letγ(w)⊂M∨be the subsemigroup of M∨
+generated by the elements that dominates the element w.γ(w) is
+lower-saturated. (Let ξ∈Mandξ′∈γ(w) be elements such that
+ξ≤ξ′. There are elements ξ1,...,ξ r∈M∨such thatξidominatesw
+and
+ξ′=ξ1⊕···⊕ξr.
+Then
+ξ= inf
+M∨{ξ,ξ1⊕···⊕ξr}
+= inf
+M∨{ξ,ξ1}⊕···⊕ inf
+M∨{ξ,ξr}.
+Soξ∈γ(w).) Alsoγ(w) generates the k-moduleM∨. (Letξ∈M∨
+be any element. Since αis lightly surjective, there is v∈Msuch that
+w≤α(v). Leta∈k\{−∞}be an element such that /a\}bracketle{tv,ξ/a\}bracketri}ht ≤a. Then
+⊘a⊙ξdominatesw.) Soγ(w) is a locator of M∨.
+We show that γis a homomorphism. For w1,w2∈N, we have
+γ(w1⊕w2)⊂γ(w1)∩γ(w2).
+Letξbe an element of right side. There are elements ξi,j∈M∨(1≤
+i≤2, 1≤j≤r) such that ξi,jdominateswiand
+ξ=ξ1,1⊕···⊕ξ1,r=ξ2,1⊕···⊕ξ2,r.
+Then
+ξ= inf
+M∨{ξ1,1⊕···⊕ξ1,r,ξ2,1⊕···⊕ξ2,r}
+=/circleplusdisplay
+i,jηi,j,
+where
+ηi,j= inf
+M∨{ξ1,i,ξ2,j}.22
+ηi,jdominatesw1⊕w2(Proposition 3.24). So we have ξ∈γ(w1⊕w2).
+(2) Letξ∈M∨be an element that dominates α(v). There is an
+elementv′∈Msuch that /a\}bracketle{tv′,ξ/a\}bracketri}ht ≤0 andα(v)≤α(v′). Sinceαis
+injective, we have v≤v′. So we have
+/a\}bracketle{tv,ξ/a\}bracketri}ht ≤ /a\}bracketle{tv′,ξ/a\}bracketri}ht ≤0.
+Let
+T={ξ∈M∨|/a\}bracketle{tv,ξ/a\}bracketri}ht ≤0}.
+Now we have ξ∈T. SinceTis a subsemigroup, we have γ(α(v)) =
+T. /square
+3.7.Straight reflexive modules. Letkbe a quasi-complete totally
+ordered rational tropical semifield. Recall that the dimension of a
+straight reflexive k-moduleMis the number of extremal rays. By
+Proposition 3.12, the number of elements of any basis of Mis dim(M).
+Proof of Theorem 2.1. We have an isomorphism ιM:M→(M∨)∨and
+a homomorphism γ:N→Loc(M∨) defined in Lemma 3.25. There is a
+left-inversion pof the homomorphism i: (M∨)∨→Loc(M∨) (Lemma
+3.15). By the commutative diagram
+Mα/d47/d47
+ιM
+/d15/d15N
+γ
+/d15/d15
+(M∨)∨
+i/d47/d47Loc(M∨)
+we haveιM−1◦p◦γ◦α= idM. /square
+Proof of Theorem 2.2. By Lemma 3.22 and Lemma 3.23, there is an
+injectionN→kn, wheren= dim(N). LetN′be the lower-saturation
+of the image of M→kn.N′is a free module of rank n′≤n. If
+n′=n, thenαis lightly surjective. Now we may assume that N=kn
+and thatαis lightly surjective. By Theorem 2.1, αhas a left-inversion
+β:N→M. Sinceβis surjective, we have dim( M)≤dim(N)./square
+Proof of Theorem 2.3. By Lemma 3.22 and Lemma 3.23, there is a
+right-inversion α:M→knof the surjection β:kn→M. By the
+commutative diagram
+knβ/d47/d47∼
+/d15/d15M
+ιM
+/d15/d15
+kn
+(β∨)∨/d47/d47(M∨)∨23
+we haveιM−1=β◦(α∨)∨. /square
+Proof of Theorem 2.4. By Theorem 2.3, Nis reflexive. Similarly to
+the proof of Theorem 2.2, we may assume that N=knand thatα
+is lightly surjective. We have a homomorphism γ:N→Loc(M∨)
+defined in Lemma 3.25. There is a left-inversion pof the homomor-
+phismi: (M∨)∨→Loc(M∨) (Lemma 3.15). There isa homomorphism
+δ:M∨→N∨such that for any w∈Nand anyξ∈M∨it implies
+/a\}bracketle{tw,δ(ξ)/a\}bracketri}ht=/a\}bracketle{tp(γ(w)),ξ/a\}bracketri}ht.
+By the commutative diagram
+Mα/d47/d47
+ιM
+/d15/d15N
+γ
+/d15/d15
+(M∨)∨
+i/d47/d47Loc(M∨)
+for anyv∈Mwe have
+/a\}bracketle{tα(v),δ(ξ)/a\}bracketri}ht=/a\}bracketle{tv,ξ/a\}bracketri}ht.
+Soα∨◦δ= idM∨. So we have
+dim(M∨)≤dim(N∨) =n.
+By Lemma 3.22 and Proposition 3.21, there is an injection from the
+set of the extremal rays of Mto the set of the extremal rays of M∨.
+So we have dim( M)≤n. /square
+Example 3.26.There is anexample ofstraight submodule M⊂T2that
+is not finitely generated. Let
+M={(a,b)∈T2|b/\e}atio\slash=−∞}∪{−∞} .
+Mis a submodule of T2.Mis straight, because it is lattice-preserving.
+Example 3.27.There is an example of extremally generated submodule
+M⊂T3that is not finitely generated. Let
+M=/braceleftbigg
+(a,b,c)∈T3|(−1)⊙a⊕c≤b,
+2b≤a⊙c/bracerightbigg
+.
+Mis a submodule of T3(Example 3.6). For 0 ≤t≤1, let
+e(t) = (2t,t,0)∈M.
+e(t) is extremal. (Proposition 3.13. Indeed e(t) is a minimal element
+of the subset
+St={(a,b,c)∈M|b=t}.24
+So it is extremal.) So Mis not finitely generated. {e(t)|0≤t≤1}is
+a basis ofM. Indeed, for any ( a,b,c)∈M,
+(a,b,c) =c⊙e(b⊘c)⊕(2b⊘a)⊙e(a⊘b).
+Mis not straight. Indeed, let
+v1= (1
+2,1
+2,1
+2),
+v2= (1,1
+2,0),
+w= (1,0,0).
+Then we have
+inf
+M{v1,v2}= (1
+2,1
+4,0),
+inf
+M{v1,v2}⊕w= (1,1
+4,0),
+inf
+M{v1⊕w,v2⊕w}= (1,1
+2,0).
+SoMis not straight.
+3.8.Free modules. Letkbe a totally ordered tropical semifield. Let
+F=knbe the free module with the basis {e1,...,e n}. LetF∗be
+the set of the linear combinations of {e1,...,e n}with coefficients in
+k∗=k\{−∞}. Let{e∨
+1,...,e∨
+n}be the dual basis in F∨. We have a
+bijective map
+ψ:F∗−→(F∨)∗
+defined by
+ψ(a1⊙e1⊕···⊕an⊙en) = (⊘a1)⊙e∨
+1⊕···⊕(⊘an)⊙e∨
+n.
+Forv,w∈F∗, the condition v≤wis fulfilled if and only if
+/a\}bracketle{tv,ψ(w)/a\}bracketri}ht ≤0.
+Forw∈F∗and 1≤i≤n, let
+M(w,i) ={v∈F|∀j,/a\}bracketle{tv,e∨
+j/a\}bracketri}ht⊙/a\}bracketle{tej,ψ(w)/a\}bracketri}ht ≥ /a\}bracketle{tv,e∨
+i/a\}bracketri}ht⊙/a\}bracketle{tei,ψ(w)/a\}bracketri}ht}.
+M(w,i) is a submodule of F(Example 3.6). It is easy to see that
+M(w,i) is lattice-preserving in F, i.e. the inclusion M(w,i)→F
+preserves the infimum of any two elements. For η∈F∨and 1≤i≤n,
+let
+N(η,i) ={v∈F|/a\}bracketle{tv,η/a\}bracketri}ht=/a\}bracketle{tv,e∨
+i/a\}bracketri}ht⊙/a\}bracketle{tei,η/a\}bracketri}ht}
+={v∈F|∀j,/a\}bracketle{tv,e∨
+j/a\}bracketri}ht⊙/a\}bracketle{tej,η/a\}bracketri}ht ≤ /a\}bracketle{tv,e∨
+i/a\}bracketri}ht⊙/a\}bracketle{tei,η/a\}bracketri}ht}.
+N(η,i) is also a lattice-preserving submodule of F.25
+Proposition3.28. LetMbe a submoduleof Fwith abasis {w1,...,w r}.
+Suppose that wh∈F∗(1≤h≤r). Then the following are equivalent.
+(i)Mis lattice-preserving in F.
+(ii)For anyi∈ {1,...,n}, there is the minimum element of M∩Vi,
+where
+Vi={v∈F|/a\}bracketle{tv,e∨
+i/a\}bracketri}ht= 0}.
+(iii)There is a surjective map
+s:{1,...,n} −→ {1,...,r}
+such that
+M=/intersectiondisplay
+1≤i≤nM(ws(i),i).
+(iv)There is a surjective map
+s:{1,...,n} −→ {1,...,r}
+such that
+M=/intersectiondisplay
+1≤i≤nN(ηs(i),i),
+whereηhis the dual element of wh.
+Proof.(iii)⇒(i) and (iv) ⇒(i) are easy.
+SinceF∨is also a free module, for η∈(F∨)∗and 1≤i≤nwe have
+the lattice-preserving submodule M(η,i) ofF∨. The bijective map
+ψ:F∗−→(F∨)∗
+induces bijective maps
+ψ′:M\{−∞} −→ M∨\{−∞},
+ψ′′:N(η,i)\{−∞} −→ M(η,i)\{−∞}.
+So we have only to prove that conditions (i), (ii), (iii) are equivalent.
+(i)⇒(ii). Let
+vi= inf
+F{⊘ah,i⊙wh|1≤h≤r},
+where
+ah,i=/a\}bracketle{twh,e∨
+i/a\}bracketri}ht.
+Thenviis the minimum element of M∩Vi.
+(ii)⇒(iii). Letvibe the minimum element of M∩Vi.viis an
+extremal element of M. The extremal ray k⊙viis generated by an
+element ofthebasis {w1,...,w r}(Proposition3.12). Thereisanumber
+s(i) such that k⊙vi=k⊙ws(i).
+We show that sis surjective. For h∈ {1,...,r}, we have
+wh=ah,1⊙v1⊕···⊕ah,n⊙vn.26
+Sincewhis extremal (Proposition 3.12), there is a number isuch that
+wh=ah,i⊙vi.
+So we have h=s(i).
+We show the equality
+M=/intersectiondisplay
+1≤i≤nM(ws(i),i).
+Forv∈F, letxi=/a\}bracketle{tv,e∨
+i/a\}bracketri}ht. The condition v∈M(ws(i),i) is fulfilled if
+and only if for any jit implies
+⊘as(i),j⊙xj≥ ⊘as(i),i⊙xi.
+For 1≤h≤rand 1≤i≤n, we have
+⊘as(i),i⊙ws(i)=vi
+≤ ⊘ah,i⊙wh.
+For 1≤j≤n, we have
+⊘as(i),i⊙as(i),j≤ ⊘ah,i⊙ah,j.
+It meanswh∈M(ws(i),i). SinceMis generated by {w1,...,w r}, we
+have
+M⊂/intersectiondisplay
+1≤i≤nM(ws(i),i).
+Letvbe an element of right side. Then
+v=/circleplusdisplay
+i⊘as(i),i⊙xi⊙ws(i).
+(Indeed,
+/a\}bracketle{tv,e∨
+i/a\}bracketri}ht=xi
+=/a\}bracketle{t⊘as(i),i⊙xi⊙ws(i),e∨
+i/a\}bracketri}ht.
+So
+v≤/circleplusdisplay
+i⊘as(i),i⊙xi⊙ws(i).
+The converse is easy.) So we have v∈M. /square27
+4.Polytopes in a tropical projective space
+LetF=Tn+1be the free module with coordinates ( x1,...,x n+1)
+overT=R∪{−∞}. LetF∗=Rn+1.
+Proposition 4.1. LetMbe a submodule of Fgenerated by finitely
+many elements of F∗. Then the following are equivalent.
+(i)Mis lattice-preserving in F.
+(ii)M\{−∞} is a real convex subset of Rn+1.
+Proof.(i)⇒(ii). By Proposition 3.28, Mis defined by inequalities
+xj≥xi−ci,j(i,j∈ {1,...,n+1})
+for someci,j∈R. SoM\{−∞} is real convex.
+(ii)⇒(i). Letπ1:F→Tnandπ2:F→Tbe projections defined
+as follows.
+π1(x1,...,x n+1) = (x1,...,x n),
+π2(x1,...,x n+1) =xn+1.
+Fora∈R, letNi(a)⊂Fbe the submodule defined as follows.
+Ni(a) ={v= (x1,...,x n+1)∈F|xn+1=xi+a}.
+By induction on n, we may assume that modules π1(M),π2(M),M∩
+Ni(a) are lattice-preserving. Suppose that Mis not lattice-preserving.
+By Proposition 3.28, there is a number isuch that there is no minimum
+element ofM∩Vi, where
+Vi={v= (x1,...,x n+1)∈F|xi= 0}.
+We may assume i≤n. Letw1,w2be minimal elements of M∩Visuch
+thatπ1(w1) is the minimum element of π1(M∩Vi) and thatπ2(w2) is
+the minimum element of π2(M∩Vi). Leta∈Rbe an element such
+that
+π2(w2)<a<π 2(w1).
+There is the minimum element v(a) ofM∩Ni(a)∩Vi. SinceM∩Vi
+is real convex, v(a) is a minimal element of M∩Vi. (Letv′∈M∩Vi
+be an element such that v′< v(a). The real line segment combining
+v′andw1contains an element v′′∈M∩Ni(a)∩Visuch thatv′′/\e}atio\slash=v′.
+Sinceπ1(w1)< π1(v′)≤π1(v(a)), we have v′′< v(a). ) SoMhas
+infinitely many extremal rays, which is contradiction. /square
+Let
+ϕ:Tn+1\{−∞} −→ TPn28
+be the canonical projection to the tropical projective space TPn. We
+identifyϕ(Rn+1) withRn. A subset P⊂TPnis said to be tropically
+convex if the subset
+M=ϕ−1(P)∪{−∞} ⊂ Tn+1
+is a submodule. A subset P⊂TPnis said to be a tropical polytope if
+it is the tropically convex hull of finitely many points of Rn.
+Proof of Theorem 2.5. (1) Suppose that Pis a polytrope. Then Pis
+real convex. By Proposition 4.1, Mis lattice-preserving in Tn+1. So
+Mis straight. By Theorem 2.3, Mis reflexive.
+(2) Suppose that Mis straight reflexive. Let {v1,...,v r}be a basis
+ofM. By Theorem 2.2, we have r≤n+1. Letpi=ϕ(vi). ThenPis
+the tropically convex hull of {p1,...,p r}. /square
+5.Square matrices over a tropical semifield
+Letkbeatotallyorderedrationaltropicalsemifield. Asquarematrix
+of ordernoverkis a homomorphism A:kn→kn. Let{e1,...,e n}be
+the basis of kn. The coefficient /a\}bracketle{tA⊙ej,ei∨/a\}bracketri}htis simply written as Aij.
+LetEn:kn→knbe the identity.
+Let ∆(A),∆(A) be square matrices of order ndefined as follows.
+∆(A)ij=δij⊙Aij,
+∆(A)ij=δij⊙Aij,
+where
+δij=/braceleftBigg
+0 ifi=j
+−∞ifi/\e}atio\slash=j
+δij=/braceleftBigg
+−∞ifi=j
+0 ifi/\e}atio\slash=j.
+The determinant det( A) is the sum of elements A1s(1)⊙···⊙Ans(n)for
+all permutations s∈S(n).
+Lemma 5.1. LetAbe a square matrix of order noverk. Suppose that
+∆(A) =Enanddet(A) = 0. ThenA⊙n=A⊙n−1.
+Proof.SinceEn≤A, we haveA⊙r≤A⊙r+1for anyr≥0. (A⊙n)ijis
+the sum of elements
+b=Ah(0)h(1)⊙Ah(1)h(2)⊙···⊙Ah(n−1)h(n)29
+for all maps h:{0,...,n} → {1,...,n}such thath(0) =iandh(n) =
+j.hisnotinjective. Sotherearenumbers l,mandacyclicpermutation
+s∈S(n) such that
+s:h(l)/mapsto→h(l+1)/mapsto→ ··· /mapsto→h(m−1)/mapsto→h(m) =h(l).
+Since ∆(A) =En, we have
+Ah(l)h(l+1)⊙···⊙Ah(m−1)h(m)≤det(A).
+So we have
+A⊙n≤det(A)⊙A⊙n−1.
+Since det(A) = 0, we have the conclusion. /square
+Lemma 5.2. LetAbe a square matrix of order noverk. Then either
+(i)or(ii)is fulfilled.
+(i)There are an element v∈(k\{−∞})nand an element ε >0
+such that
+(A⊕ε⊙∆(A))⊙v= ∆(A)⊙v.
+(ii)There is an element v∈kn\{−∞} such that
+A⊙v=∆(A)⊙v.
+Proof.Lete(A) be the sum of elements A1s(1)⊙ ··· ⊙Ans(n)for all
+s∈S(n)\{id}. Let
+c(A) = det(∆(A)) =A11⊙···⊙Ann.
+We show that the condition (i) is fulfilled if e(A)<c(A). Replacing
+Aby⊘(∆(A))⊙A, we may assume ∆( A) =En. There is an element
+ε∈ksuch thatε>0 and
+e(A)⊙nε≤c(A).
+Let
+B=A⊕ε⊙∆(A).
+Then we have e(B)≤c(B). By Lemma 5.1, we have B⊙n=B⊙n−1.
+Letw∈(k\{−∞})nbe any element. Let v=B⊙n−1⊙w. Then we
+haveB⊙v=v.
+We show that the condition (ii) is fulfilled if c(A)≤e(A). We may
+assume ∆(A) =En. (IfAii=−∞, then the element v=eisatisfies
+the conclusion.) There is a cyclic permutation s∈S(n)\ {id}and a
+maph:{0,...,l} → {1,...,n}such that
+s:h(0)/mapsto→h(1)/mapsto→ ··· /mapsto→h(l−1)/mapsto→h(l) =h(0),
+Ah(0)h(1)⊙···⊙Ah(l−1)h(l)≥0.30
+Let
+v=/circleplusdisplay
+1≤m≤l(Ah(m)h(m+1)⊙···⊙Ah(l−1)h(l))⊙eh(m).
+Then
+∆(A)⊙v≥v.
+So we have the conclusion. /square
+6.Tropical curves
+LetA=T[x1,−x1,...,x n,−xn] be the semiring of Laurent polyno-
+mials over T=R∪{−∞} (where−ximeans⊘xi). Let
+f=/circleplusdisplay
+i1...in∈Zci1...in⊙i1x1⊙···⊙inxn
+be any element of A. The induced map
+f:Rn−→T
+(a1,...,a n)/mapsto→f(a1,...,a n)
+is said to be a Laurent polynomial function over T. Iffis a monomial,
+thenfis aZ-affine function, i.e. there are c∈Randi1,...,i n∈Z
+such that
+f=c+i1x1+···+inxn.
+In general case, fis the supremum of finitely many Z-affine functions,
+which is a locally convex piecewise- Z-affine function.
+Let Γn⊂Rnbe the subset defined as follows.
+Γn=E0∪E1∪···∪En,
+E0={(a1,...,a n)∈Rn|∀i,∀j,ai=aj≥0},
+for 1≤i≤n,
+Ei={(a1,...,a n)∈Rn|ai≤0,∀j/\e}atio\slash=i,aj= 0}.
+Γnhas a (n+1)-valent vertex P= (0,...,0). Also Γ nis equipped with
+Euclidean topology on Rn.
+Definition. A function f: Γn→Tisregularif it is induced by a
+locally Laurent polynomial function f:Rn→T.
+LetOΓnbe the sheaf of the regular functions on Γ n.OΓnis a sheaf
+of semirings. Let Rbe the stalk of OΓnat the vertex P.
+Proposition 6.1. Letf∈R\{−∞} be any element. Then there are
+a unique number r∈Z≥0and a unique Laurent monomial h∈Rsuch
+that
+f=h⊙r(x1⊕0).31
+Proof.fis the sum of Laurent monomials f1,...,f m. Iffj(P)<f(P),
+thenfj<fon a neighborhood of P. So we may assume fj(P) =f(P).
+ThenfisZ-affine onEi(1≤i≤n). So there is ai∈Zsuch that
+f=f(P)⊙aixionEi. Let
+h=f(P)⊙a1x1⊙···⊙anxn.
+Thenf=honE1∪···∪En.f⊘his the sum of monomials g1,...,g m
+such thatgj(P) = 0. There are bij∈Z≥0such that
+gj=b1jx1⊙···⊙bnjxn.
+Then
+gj= (b1j+···+bnj)x1
+onE0. So we have f⊘h=rx1onE0, where
+r=/circleplusdisplay
+1≤j≤m/summationdisplay
+1≤i≤nbij.
+/square
+The number rin the above statement is called the order of fatP,
+and denoted by ord( f,P).
+For 0≤i≤n, letXifbe the partial differential of fatPwith
+directionEi. (i.e.Xif=aif and only if f=f(P)−axionEi
+(1≤i≤n).X0f=aif and only if f=f(P)+ax1onE0.)
+Proposition 6.2. Letf∈R\{−∞} be any element. Then
+ord(f,P) =/summationdisplay
+0≤i≤nXif.
+Proof.Lethbe a Laurent monomial written as follows.
+h=c⊙a1x1⊙···⊙anxn.
+Then
+Xih=−ai(1≤i≤n),
+X0h=a1+···+an.
+So we have /summationdisplay
+0≤i≤nXih= 0.
+Also we have /summationdisplay
+0≤i≤nXi(x1⊕0) = 1.
+So we have the conclusion. /square
+Proposition 6.3. Letf,g∈R\{−∞} be any elements.
+(1) ord(f⊙g,P) = ord(f,P)+ord(g,P).32
+(2)Ifg(P)≤f(P), thenord(f,P)≤ord(f⊕g,P).
+Proof.(1) is easy.
+(2) Ifg(P)<f(P), thenf⊕g=f. So we may assume g(P) =f(P).
+Then we have
+Xi(f⊕g) =Xif⊕Xig.
+By Proposition 6.2, we have the conclusion. /square
+A function f: Γn→Tis said to be rational if locally
+f=g1−g2=g1⊘g2
+for regular functions g1,g2. By Proposition 6.1, there is a number
+m≥0 such that the function m(x1⊕0)⊙fis regular at P. The order
+offatPis defined as follows.
+ord(f,P) = ord(m(x1⊕0)⊙f,P)−m.
+LetQ∈Γnbe a point such that Q/\e}atio\slash=P. Then a neighborhood of Q
+is embedded in Γ 1=R. So we can define the order of fatQsimilarly.
+Definition. (C,OC) is atropical curve if for anyP∈Cthere are
+a neighborhood UofPand a number n≥1 such that ( U,OU) is
+embedded in (Γ n,OΓn).
+A divisorDon a tropical curve Cis an element of the free abelian
+group Div(C) generated by all the points of C. For a rational function
+f:C→T, the divisor ( f)∈Div(C) is defined as follows.
+(f) =/summationdisplay
+P∈Cord(f,P)P.
+fis said to be a section of Dif eitherf=−∞or (f) +D≥0. Let
+OC(D) be the sheaf of the sections of D.
+Proposition 6.4. The setM=H0(C,OC(D))is aT-module.
+Proof.Letf,g∈M\{−∞} be any elements. By Proposition 6.3, for
+P∈Cwe have
+ord(f⊕g,P)≥min{ord(f,P),ord(g,P)}.
+So
+(f⊕g)+D≥inf
+Div(C){(f),(g)}+D≥0.
+So we have f⊕g∈M. /square
+Recall that
+r(D) = max{r∈Z≥−1|U(D,r) =∅}.33
+Proof of Theorem 2.7. Note thatr(D) =s(D)−1, where
+s(D) = min{r∈Z≥0|U(D,r)/\e}atio\slash=∅}.
+Letm=s(D). We show that there is a straight reflexive submodule
+N⊂M=H0(C,OC(D)) with dimension m. LetP1,...,P m∈Cbe
+points such that
+H0(C,OC(D−E)) =−∞,
+where
+E=P1+···+Pm.
+There is an element
+fi∈H0(C,OC(D−E+Pi))
+such thatfi/\e}atio\slash=−∞. Let
+α:Tm−→M
+be the homomorphism defined by α(ei) =fi. Let
+β:M−→Tm
+be the homomorphism defined by
+β(g) =g(P1)⊙e1⊕···⊕g(Pm)⊙em.
+LetAbe the square matrix induced by β◦α:Tm→Tm.
+Now we suppose that there is an element
+v=a1⊙e1⊕···⊕am⊙em∈Tm\{−∞}
+such thatA⊙v=∆(A)⊙v. Then there is a map h:{1,...,m} →
+{1,...,m}such thath(i)/\e}atio\slash=iand
+α(v)(Pi) =ah(i)⊙fh(i)(Pi).
+Then
+ord(α(v),Pi)≥ord(fh(i),Pi)
+(Proposition 6.3). So α(v) is a section of D−Esuch thatα(v)/\e}atio\slash=−∞,
+which is contradiction.
+So there is no element v∈Tm\{−∞} such thatA⊙v=∆(A)⊙v.
+By Lemma 5.2, there are an element v∈Rmand an element ε >0
+such that
+(A⊕ε⊙∆(A))⊙v= ∆(A)⊙v.
+LetL(v,ε)⊂Tmbe the submodule defined as follows.
+L(v,ε) =T⊙{w∈Tm|v≤w≤ε⊙v}.
+L(v,ε) is a straight reflexive T-module with dimension m. We have
+A|L(v,ε)= ∆(A)|L(v,ε).34
+Soαis injective on L(v,ε). The image N=α(L(v,ε)) is a submodule
+ofMsuch thatN∼=L(v,ε). /square
+Example 6.5.The mapping D/mapsto→r(D) is not an invariant of a T-
+module. We show that there are tropical curves C,C′and divisors
+D,D′such that
+H0(C,OC(D))∼=H0(C′,OC′(D′)),
+r(D)/\e}atio\slash=r(D′).
+LetCbeatropicalcurvewithgenus1withavertex Vandanedge E.
+LetPbe an interior point of E. LetD=V+P. ThenH0(C,OC(D))
+is isomorphic to the submodule of T2generated by (0 ,0) and (0,a
+2),
+whereais the lattice length of E. We haver(D) = 1.
+LetC′be a tropical curve with genus 2 with vertices V1,V2and edges
+E1,E2,E3such that the boundary of Eiis{V1,V2}(1≤i≤3). Let
+Pbe an interior point of E1. LetD′=V1+P. Then for any interior
+pointQofE2∪E3we have
+H0(C′,OC′(D′−Q)) =−∞.
+SoH0(C′,OC′(D′)) is isomorphic to the submodule of T2generated by
+(0,0) and (0,b
+2), wherebis the lattice length of the path from V1toP
+contained in E1. We haver(D′) = 0. In the case of a=b, the required
+condition is fulfilled.
+7.Tropical plane curves
+7.1.Tropicalization. It is well known that some example of tropical
+curve is given by tropicalization of a family of affine complex curves.
+First, we define tropical plane curves. Let f∈T[x,−x,y,−y] be a
+Laurent polynomial over T=R∪{−∞}. The subset
+V(f) ={(a,b)∈R2| −fis not locally convex at ( a,b)}
+is called the algebraic subset defined by f. The morphism Cf→R2
+parametrizing V(f) with a tropical curve Cfis called the tropical plane
+curve defined by f. The genus of Cfis defined to be the first Betti
+numberb1(Cf).
+A tropical plane curve is a dequantization of complex amoebas in
+following way. For t>1, let
+At: (C×)2−→R2
+be the homomorphism of groups defined by
+At(a,b) = (log|a|
+log(t),log|b|
+log(t)).35
+Atis called the complex amoeba map. Let
+gt∈C[z1,z1−1,z2,z2−1] (t>1)
+be a family of complex Laurent polynomials such that each coefficient
+is a Laurent polynomial of t−1. This family is written as an element
+of a valuation field K. We use the group algebra K=C[[R]] of power
+series defined by the group R. The indeterminate is denoted by t−1,
+and the valuation is defined to be the maximum index of tmultiplied
+by−1. So, val( ta) =−a. The family {gt|t >1}is written as an
+element
+g∈K[z1,z1−1,z2,z2−1].
+The amoeba map over K
+A: (K×)2−→R2
+is defined as follows.
+A(a,b) = (−val(a),−val(b)).
+The affine curve V(g)⊂(K×)2is the family of affine complex curves
+V(gt)⊂(C×)2. Takingt→+∞, the family of complex amoebas
+At(V(gt)) converges to the amoeba A(V(g)) overK. Also, the amoeba
+overKisthealgebraicsubset definedbyatropicalLaurent polynomial.
+Let
+A:K[z1,z1−1,z2,z2−1]−→T[x,−x,y,−y]
+be the map defined as follows.
+A(g) =f,
+g=/summationdisplay
+i,j∈Zcijzi
+1zj
+2,
+f=/circleplusdisplay
+i,j∈Z−val(cij)⊙ix⊙jy.
+Then we have
+A(V(g)) =V(f).
+This construction is called the tropicalization of a family of affine com-
+plex curves.36
+7.2.Examples.
+Example 7.1.Fora,b,c∈C×, let
+g=a+bz1+cz2.
+Then
+f=A(g) = 0⊕x⊕y.
+The tropical plane curve Cfis said to be a tropical projective line. We
+haveb1(Cf) = 0.
+Example 7.2.Forr,s∈Nandai,bj∈R, let
+f=f1⊙f2,
+f1=a0⊕a1⊙x⊕a2⊙2x⊕···⊕ar⊙rx,
+f2=b0⊕b1⊙y⊕b2⊙2y⊕···⊕bs⊙sy.
+Assume that
+2ai>ai−1+ai+1,
+2bj>bj−1+bj+1.
+Thenb1(Cf) = (r−1)(s−1).
+References
+[1] A. Gathmann and M. Kerber. A Riemann-Roch theorem in tropical geometry.
+Mathematische Zeitschrift 259, Number 1 (2008), 217-230.
+[2] Zur Izhakian. Tropical Algebraic Sets, Ideals and An Algebraic Nu llstellensatz.
+International Journal of Algebra and Computation 18 (2008), 10 67-1098.
+[3] M. Joswig and K. Kulas. Tropical and ordinary convexity combined . Preprint,
+arXiv:0801.4835.
+[4] Grigori Litvinov. The Maslov dequantization, idempotent and tro pical mathe-
+matics: a very brief introduction. Preprint, arXiv:math.GM/050103 8.
+[5] Grigory Mikhalkin. Enumerative tropical algebraic geometry in R2. J. Amer.
+Math. Soc. 18 (2005), 313-377.
+[6] G. Mikhalkin and I. Zharkov. Tropical curves, their Jacobians an d Theta func-
+tions. Preprint, arXiv:math.AG/0612267.
+[7] Yen-lung Tsai. Working with Tropical Meromorphic Functions of On e Variable.
+Preprint, arXiv:1101.2703.
+[8] OlegViro. Hyperfields for Tropical GeometryI. Hyperfields and d equantization.
+Preprint, arXiv:1006.3034.
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+arXiv:1001.0449v1  [math.AG]  4 Jan 2010POLYNOMIALS REPRESENTING EYNARD-ORANTIN
+INVARIANTS
+PAUL NORBURY AND NICK SCOTT
+Abstract. The Eynard-Orantin invariants of a plane curve are multilin ear
+differentials on the curve. For aparticular classof genus ze ro plane curves these
+invariants can be equivalently expressed in terms of simple r expressions given
+by polynomials obtained from an expansion of the Eynard-Ora ntin invariants
+around a point on the curve. This class of curves contains man y interesting
+examples.
+1.Introduction
+As a tool for studying enumerative problems in geometry Eynard an d Orantin
+[3] associate multilinear differentials to any compact Riemann surface Cequipped
+with two meromorphic functions xandywith the property that the branch points
+ofxare simple and the map
+C→C2
+p/ma√sto→(x(p),y(p))
+is an immersion. For every ( g,n)∈Z2withg≥0 andn >0 they define a
+multilineardifferential,i.e. atensorproductofmeromorphic1-forms ontheproduct
+Cn, notated by ωg
+n(p1,...,pn) forpi∈C. When 2 g−2 +n >0,ωg
+n(p1,...,pn)
+is defined recursively in terms of local information around branch po ints ofxof
+ωg′
+n′(p1,...,pn) for 2g′−2+n′<2g−2+n. A dilaton relation between ωg
+n+1and
+ωg
+ncan be applied in the n= 0 case to define Fg(= “ωg
+0”) forg≥2, known as the
+symplectic invariants of the curve. The invariant Fgrecursively uses all ωg′
+nwith
+g′+n≤g+1.
+In principle Fgandωg
+ncan be calculated explicitly from the recursion relations
+defining them, and implemented on a computer. In practice, the exp ressions ob-
+tained this way are unwieldy and computable only for small gandn. The main
+aim of this paper is to express ωg
+nin a simpler form—essentially its inverse discrete
+Laplace transform—and to develop methods to calculate general f ormulae for Fg
+in some examples.
+We consider the Eynard-Orantin invariants of the class of genus ze ro curves for
+whichxis two-to-one and has two branch points. The case of one branch p oint
+x=z2is dealt with in [3]. One can parametrise the domain so that the curve c an
+be written:
+(1) C=/braceleftigg
+x=a+b(z+1/z)
+y=y(z)
+1991Mathematics Subject Classification. MSC (2010) 32G15; 30F30; 05A15.
+12 PAUL NORBURY AND NICK SCOTT
+for constants a,band any rational function y(z) withy′(±1)/ne}ationslash= 0. The definition
+ofωg
+nand its properties requires the Riemann surface to be compact. Ne vertheless,
+by taking sequences of compact Riemann surfaces one can extend the definition to
+allowy(z) to be any analytic function defined on a domain in Ccontaining z=±1,
+e.g.y(z) = ln(z) defined on the complement of the negative imaginary axis.
+Theorem 1. For the plane curve Cdefined in (1) and 2g−2+n >0,ωg
+n(z1,...,zn)
+has an expansion around {zi= 0}given by
+(2)ωg
+n(z1,..,zn) =d
+dz1...d
+dzn/summationdisplay
+bi>0Ng
+n(b1,...,b n)zb1
+1...zbn
+ndz1...dzn
+whereNg
+nis a quasi-polynomial in the b2
+iof homogeneous degree 3g−3+n, dependent
+on the parity of the biand is symmetric in all variables of the same parity.
+The parity dependence means that there exists polynomials Ng
+n,k(b1,...,bn) for
+k= 1,...,nsuch that Ng
+n(b1,...,bn) decomposes
+Ng
+n(b1,...,bn) =Ng
+n,k(b1,...,bn), k= number of odd bi.
+The polynomials representing Ng
+nare simpler expressions than ωg
+n(z1,...,zn) and
+can have meaning themselves. When x=z+1/zandy=z,Ng
+narises as a solution
+of a Hurwitz problem [7, 8], with typical expression N0
+4,0=1
+4(b2
+1+b2
+2+b2
+3+b2
+4)−1
+(while the much larger expression ω0
+4can be expressed as the sum of 32 rational
+functions.) The case x=z+1/zandy= lnzarises when studying partitions and
+the Plancherel measure [2], with N0
+4,0=1
+4/parenleftbig
+b2
+1+b2
+2+b2
+3+b2
+4/parenrightbig
+.
+Theorem 2. The coefficients of the top homogeneous degree terms in the pol yno-
+mialNg
+n,k(b1,...,bn), defined above, can be expressed in terms of intersection num -
+bers of tautological line bundles over the moduli space Li→ Mg,n. For/summationtext
+iβi=
+3g−3+n, the coefficient vβof/producttextb2βi
+iis
+vβ=y′(1)2−2g−n+(−1)ky′(−1)2−2g−n
+x′′(1)2g−2+n23g−3+nβ!/integraldisplay
+Mg,nc1(L1)β1...c1(Ln)βn
+The invariants ωg
+nsatisfy string and dilaton equations—see Section 4—which
+enable one to express Ng
+n+1recursively in terms of Ng
+n. They are most easily
+expressed when y(z) is a monomial. The following two theorems apply to the
+polynomials Ng
+n,kalthough we abuse notation and simply write Ng
+n.
+Theorem 3. The polynomials associated to the plane curves
+x=z+1/z, y=zm/m, m = 1,2,...
+satisfy the following recursion relations.
+(3) Ng
+n+1(m,b1,...bn) =n/summationdisplay
+j=1bj/summationdisplay
+k=1kNg
+n(b1,...,b n)|bj=k
+(m+1)Ng
+n+1(m+1,b1,...,b n)+(m−1)Ng
+n+1(m−1,b1,...,b n) (4)
+= 2mn/summationdisplay
+j=1bj/summationdisplay
+k=1kNg
+n(b1,...,b n)|bj=k−mn/summationdisplay
+j=1bjNg
+n(b1,...,b n)POLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 3
+(m−2)Ng
+n+1(m−2,b1,...,b n)−2mNg
+n+1(m,b1,...,b n) (5)
++(m+2)Ng
+n+1(m+2,b1,...,b n) =mn/summationdisplay
+j=1/summationdisplay
+k=1±bjkNg
+n(b1,...,b n)|bj=k
+(6)Ng
+n+1(m+1,b1,...,bn)−Ng
+n+1(m−1,b1,...,bn) =m(2g−2+n)Ng
+n(b1,...,bn)
+Corollary 1. The symplectic invariants of the curve x=z+1/z,y=zm/msatisfy
+Fg=1
+m(2g−2)(Ng
+1(m+1)−Ng
+1(m−1))
+forg≥2.
+The polynomials corresponding to y= lnzsatisfy recursions obtained by setting
+m= 0 into (3) and ((4) −(6))/m.
+Theorem 3 is a special case of the following more general theorem th at applies to
+any analytic function y(z) defined on a domain in Ccontaining z=±1 expanded
+asy(z)∼/summationtext(ak+zbk)(1−z2)k. For example y(z) = lnz∼/summationtext(1−z2)k
+−2k.
+First we need the following notation. Define the operator Don functions by
+Df(n) =f(n+1).
+Further, define D{f(n)}n=a=f(a+1). As usual, for a polynomial p(z) =/summationtextpizi,
+definep(D){f(n)}n=a=/summationtextpif(a+i). Note that D−Iis a discrete derivative and
+y(D)∼/summationtext(ak+zbk)(1−D2)kis a sum overpowers of the discrete derivative D2−I.
+PutbS= (b1,...,bn) and|bS|=b1+...+bn.
+Theorem 4. For the plane curve x=z+1/z,y=y(z)
+Dy(D)/braceleftbig
+mNg
+n+1(m,bS)/bracerightbig
+m=−1=n/summationdisplay
+j=1bj/summationdisplay
+k= 1
+k/negationslash≡bj(2)kNg
+n(bS)|bj=k (7)
+(1+D2)y(D)/braceleftbig
+mNg
+n+1(m,bS)/bracerightbig
+m=−1= 2n/summationdisplay
+j=1bj/summationdisplay
+k= 1
+k≡bj(2)kNg
+n(bS)|bj=k−|bS|Ng
+n(bS) (8)
+(1−D2)2y(D)/braceleftbig
+mNg
+n+1(m,bS)/bracerightbig
+m=−2=n/summationdisplay
+j=1/summationdisplay
+k=1±bjkNg
+n(bS)|bj=k (9)
+(1−D2)y(D)/braceleftbig
+Ng
+n+1(m,bS)/bracerightbig
+m=−1= (2−2g−n)Ng
+n(bS) (10)
+Although ymay not be a polynomial the left hand sides of (7) - (10) are fi-
+nite sums, since large enough powers of a discrete derivative vanish on the quasi-
+polynomial Ng
+n+1. See Section 2.
+Corollary 2. The symplectic invariants of the curve x=z+1/z,y=y(z)satisfy
+Fg=1
+2−2g(1−D2)y(D){Ng
+1(m)}m=−1
+forg≥2.
+The definition of the Eynard-Orantininvariants is given in Section 2. T he proofs
+of Theorems 1 and 2 are in Section 3 and the proof of Theorems 3 and 4 is in
+Section 3. Section 5 contains examples.4 PAUL NORBURY AND NICK SCOTT
+2.Eynard-Orantin invariants.
+For every ( g,n)∈Z2withg≥0 andn >0 the Eynard-Orantin invariant of
+a plane curve Cis a multilinear differential ωg
+n(p1,...,pn), i.e. a tensor product of
+meromorphic 1-forms on the product Cn, wherepi∈C. When 2 g−2 +n >0,
+ωg
+n(p1,...,pn) is defined recursively in terms of local information around the poles
+ofωg′
+n′(p1,...,pn) for 2g′+2−n′<2g−2+n. Equivalently, the ωg′
+n′(p1,...,pn) are
+used as kernels on the Riemann surface to integrate against. This is a familiar idea,
+the main example being the Cauchy kernel which gives the derivative o f a function
+in terms of the bilinear differential dwdz/(w−z)2as follows
+f′(z)dz=Res
+w=zf(w)dwdz
+(w−z)2=−/summationdisplay
+αRes
+w=αf(w)dwdz
+(w−z)2
+where the sum is over all poles αoff(w).
+The Cauchy kernel generalises to a bilinear differential B(w,z) on any Riemann
+surfaceCgiven by the meromorphic differential ηw(z)dzunique up to scale which
+has a double pole at w∈Cand allA-periods vanishing. The scale factor can be
+chosen so that ηw(z)dzvaries holomorphically in w, and transforms as a 1-form in
+wand hence it is naturally expressed as the unique bilinear differential o nC
+B(w,z) =ηw(z)dwdz,/contintegraldisplay
+AiB= 0, B(w,z)∼dwdz
+(w−z)2nearw=z.
+It is symmetric in wandz. We will call B(w,z) theBergmann Kernel , following
+[3]. It is called the fundamental normalised differential of the second kind onCin
+[4]. Recall that a differential is normalised if itsA-periods vanish and it is of the
+second kind if its residues vanish. It is used to express a normalised differential o f
+the second kind in terms of local information around its poles.
+For 2g−2+n >0, the poles of ωg
+n(p1,...,pn) occur at the branch points of x, and
+they are of order 6 g−4+2n. Since each branch point αofxis simple, for any point
+p∈Cclose toαthere is a unique point ˆ p/ne}ationslash=pclose toαsuch that x(ˆp) =x(p).
+The recursive definition of ωg
+n(p1,...,pn) uses only local information around branch
+points of xand makes use of the well-defined map p/ma√sto→ˆpthere. The invariants are
+defined as follows.
+ω0
+1=ydx
+ω0
+2=B(w,z) (11)
+For 2g−2+n >0,
+(12)
+ωg
+n+1(z0,zS) =/summationdisplay
+αRes
+z=αK(z0,z)/bracketleftbigg
+ωg−1
+n+2(z,ˆz,zS)+/summationdisplay
+g1+g2=g
+I⊔J=Sωg1
+|I|+1(z,zI)ωg2
+|J|+1(ˆz,zJ)/bracketrightbigg
+where the sum is over branch points αofx,S={1,...,n},IandJare non-empty
+and
+K(z0,z) =−/integraltextz
+ˆzB(z0,z′)
+2(y(z)−y(ˆz))dx(z)POLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 5
+is well-defined in the vicinity of each branch point of x. Note that the quotient
+of a differential by the differential dx(z) is a meromorphic function. The recursion
+(12) depends only on the meromorphic differential ydxand the map p/ma√sto→ˆparound
+branch points of x.
+2.1.Rational curves with two branch points. The simplestexampleofaplane
+curve with non-trivial Eynard-Orantininvariants is the rationalcu rvey2=xwhere
+the meromorphic function xdefines a two-to-one branched cover with a single
+branch point. It is known as the Airy curve since the Eynard-Orant in invariants
+reproduce Kontsevichsgeneratingfunction [6] for intersection n umbers on the mod-
+uli space. In this paper we study rational curves such that xdefines a two-to-one
+branched cover with two branch points.
+Lemma 2.1. Ifxis a two-to-one rational map on P1with two branch points then
+we can parametrise the domain so that x=a+b(z+1/z).
+Proof.Using a conformal map we can arrange that the two branch points o fxare
+z=±1. The conformal map z/ma√sto→(z+λ)/(λz+1) which fixes z=±1 can be used
+to further arrange that x(∞) =∞. Sincex(z)−x(1) has a double root at z= 1
+we have
+x(z)−x(1) =(z−1)2
+q2z2+q1z+q0, x(∞) =∞ ⇒q2= 0, x′(−1) = 0⇒q0= 0
+so puta=x(1)−2/q1andb= 1/q1, and the result follows. /square
+Thusx(z) =a+b(z+1/z) andy(z) is any analytic function defined on a domain
+inCcontaining z=±1 and satisfying y′(±1)/ne}ationslash= 0. When yis not polynomial, for
+example yis rational or transcendental, we expand it as a series of polynomials in
+the following non-standard way. Given such y(z), define the partial sums
+y(N)(z) =N/summationdisplay
+k=0(ak+zbk)(1−z2)k
+to agree with y(z) atz=±1 up to the Nth derivatives. One can achieve this
+by expressing y(z) =y+(z) +y−(z) wherey±(z) = 1/2(y(z)±y(−z)) and define
+y(N)
++(z) =/summationtextN
+k=0ak(1−z2)kwhereakare determined by the property
+dky+
+dzk(±1) =dky(N)
++
+dzk(±1), k= 0,...,N.
+Similarly define bkfromy−(z)/z.
+The partial sums {y(N)(z)}do not necessarily converge to y(z). For example,
+y(z) = lnz∼/summationdisplay(1−z2)k
+−2k
+is a divergent asymptotic expansion for ln( z) atz= 0 in the region Re(z2)>0.
+The partial sums y(N)(z) are used in the recursions defining ωg
+nin place of y(z)
+since they contain the same local information around z=±1 up to order N. More
+precisely, to define ωg
+nfor the curve ( x(z),y(z)) it is sufficient to use ( x(z),y(N)(z))
+for anyN≥6g−6+2n.
+There are various ways to express a transcendental function as a limit of rational
+functions. The main benefit of the approach used here is the fact t hat the expres-
+sions which appear in the string and dilaton equations x(z)my(z)ωg
+nhave poles only6 PAUL NORBURY AND NICK SCOTT
+atz=±1 and 0 and ∞, allowing one to translate properties of ωg
+nnearz=±1 to
+properties of ωg
+nnearz=∞, which is encoded by Ng
+n.
+Theorem 4 involves the expression y(D) whereDis defined in the introduction,
+and in particular I− Dis a discrete derivative. It is an easy fact (proved by
+induction) that for any degree dpolynomial p(n), high enough discrete derivatives
+vanish: (1 − D)kp(n)≡0 fork > d. Similarly, 1 − D2is a discrete derivative,
+and for a parity dependent quasi-polynomial p(n) (sop(n) =p+(n) forneven and
+p(n) =p−(n) fornodd, where p±(n) are polynomials)
+(1−D2)kp(n)≡0, ksufficiently large
+in factk >maximum degreeof p±(n). To make sense of(7) - (10), one must replace
+y(D) withy(N)(D) for large enough Nso that the left hand sides of (7) - (10) have
+only finitely many terms. This procedure is well-defined, since the van ishing of
+discrete derivatives ensures that the left hand sides of (7) - (10) are independent of
+the choice of Nwhen it is large enough.
+3.Proofs
+Proof of Theorem 1. Theorem 1 reflects three main properties of the multilinear
+differential ωg
+n(z1,...,zn) proven in [3]—it is meromorphic, with poles at zi=±1 of
+order 6g−4+2n=: 2d+2 and residue 0, and possesses symmetry under zi/ma√sto→1/zi.
+Since all residues of ωg
+nvanish, the integral
+Fg
+n(z1,...,zn) =/integraldisplayz1
+0.../integraldisplayzn
+0ωg
+n(z′
+1,...,z′
+n)
+is a well-defined meromorphic function that vanishes when any zi= 0 and has poles
+of order 2 d+1 atzi=±1. Write this rational function as
+Fg
+n(z1,...,zn) =/summationtext
+0<ki<4d+2pkzk1
+1...zknn/producttextn
+i=1(1−z2
+i)2d+1
+where the pk=pk1,...,kn∈Cand the degree of the numerator is small enough to
+avoid a pole at infinity.
+The Taylor expansion
+1
+(1−z2)2d+1=1
+(2d)!d2d
+d(z2)2d∞/summationdisplay
+m=0z2m=∞/summationdisplay
+m=0/parenleftbiggm+2d
+2d/parenrightbigg
+z2m
+has quasi-polynomial coefficients, meaning that the coefficients of zbare described
+by two polynomials in b—whenbis odd the coefficient of zbis the zero polynomial
+and when bis even the coefficient of zbis a degree 2 dpolynomial in b. More
+generally, the Taylor expansion of Fg
+n(z1,...,zn) aboutzi= 0 has quasi-polynomial
+coefficients, depending on parity. When n= 1,
+/summationtextpkzk
+(1−z2)2d+1=/summationdisplay
+k,mpk/parenleftbiggm+2d
+2d/parenrightbigg
+z2m+k=/summationdisplay
+b>0Ng
+1(b)zb.
+The coefficient of zbconsists of all terms where 2 m+k=b, hence the odd part
+ofp(z) =/summationtextpkzkgives rise to a degree 2 dpolynomial representing Ng
+1(b) whenbPOLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 7
+is odd, and the even part of p(z) gives rise to a degree 2 dpolynomial representing
+Ng
+1(b) whenbis even. Similarly,
+Fg
+n(z1,...,zn) =4d+2/summationdisplay
+k1,...,kn=0pkzk1
+1...zkn
+nn/productdisplay
+i=1∞/summationdisplay
+mi=0/parenleftbiggmi+2d
+2d/parenrightbigg
+z2mi
+i
+=4d+2/summationdisplay
+k1,...,kn=0/summationdisplay
+mi≥0pkn/productdisplay
+i=1/parenleftbiggmi+2d
+2d/parenrightbigg
+z2mi+ki
+i
+=:∞/summationdisplay
+bi>0Ng
+n(b1,...,b n)zb1
+1...zbn
+n
+expresses Ng
+n(b1,...,b n) as the sum over the terms with 2 mi+ki=biwhich is a
+quasi-polynomial depending on the parity of the bi. By symmetry of the zi,Ng
+n
+does not depend on which biare odd but only how many. Hence we write
+Ng
+n(b1,...,bn) =Ng
+n,k(b1,...,bn), k= number of odd bi.
+Each binomial coefficient and hence each polynomial Ng
+n,k(b1,...,b n) is a polyno-
+mial of degree 2 din eachbi. The stronger fact that they have homogeneous degree
+2din thebiis a consequence of Theorem 2.
+It remains to show that Ng
+nis a quasi-polynomial in the b2
+i. Equivalently, we will
+show that b1...bnNg
+n(b1,...,b n) is odd in each biusing symmetries of
+ωg
+n=∞/summationdisplay
+bi>0b1...bnNg
+n(b1,...,b n)zb1−1
+1...zbn−1
+ndz1...dzn.
+Lemma 3.1. A meromorphic 1-form on P1with poles at z=±1has the following
+related expansions around z= 0
+(13) ω(z) =∞/summationdisplay
+n=1p(n)zn−1dz⇔ω/parenleftbigg1
+z/parenrightbigg
+=∞/summationdisplay
+n=1p(−n)zn−1dz
+wherep(n)is a quasi-polynomial depending on the parity of n.
+Proof.We can express ωas a rational function with numerator a polynomial of
+degree small enough so that there are no poles at infinity. In partic ular, by linearity
+it is enough to prove the lemma when the numerator is a monomial so
+ω(z) =zkdz
+(1−z2)m+1andω/parenleftbigg1
+z/parenrightbigg
+= (−1)mz2m−kdz
+(1−z2)m+1.
+From the expansion
+1
+(1−z2)m+1=/summationdisplay/parenleftbiggn+m
+m/parenrightbigg
+z2n
+one gets
+zkdz
+(1−z2)m+1=/summationdisplay/parenleftbiggn+m
+m/parenrightbigg
+zk+2ndz=/summationdisplay
+pk(b)zb−1dz
+where
+pk(b) =/braceleftbigg0, b ≡k(mod 2)/parenleftbig(b−k−1)/2+m
+m/parenrightbig
+, b/ne}ationslash≡k(mod 2).8 PAUL NORBURY AND NICK SCOTT
+Also,
+p2m−k(b) =/parenleftbigg(b−2m+k−1)/2+m
+m/parenrightbigg
+=/parenleftbigg(b+k−1)/2
+m/parenrightbigg
+= (−1)mpk(−b)
+(forb/ne}ationslash≡k(mod 2) and p2m−k(b) = 0 for b≡k(mod 2) so the above equation holds
+for allb.) Hence (13) holds when ωhas a monomial numerator and hence for all
+rationalωand the lemma is proven. /square
+An immediate corollary of the lemma is that if ω(z) =ω(1/z) then the quasi-
+polynomial p(n) is even in nand ifω(z) =−ω(1/z) thenp(n) is odd in n. A
+consequence of a more general result in [3] is the symmetry
+ωg
+n(z1,...,z n) =−ωg
+n(1/z1,...,z n)
+and similarly for each variable zi. Hence b1...bnNg
+n(b1,...,b n) is an odd quasi-
+polynomial in each bias required. /square
+Remark. The expansion (2) of ωg
+n(z1,...,zn) around ( z1,...,zn) = (0,...,0) defines
+Ng
+n(b1,...,bn) only for ( b1,...,bn)∈Zn
++. One can make sense of bi= 0 using the
+polynomial representation Ng
+n,k(b1,...,bn) ofNg
+n(b1,...,bn) fork= number of odd
+bi. In terms of ωg
+none has the following
+Ng
+1(0) =/integraldisplay0
+∞ωg
+1(z)
+and more generally, b1...bkNg
+n(b1,...,bk,0,...,0) is the coefficient of zb1−1
+1...,zbk−1
+kin
+the expansion around ( z1,...,zk) = (0,...,0) of/integraltext0
+zk+1=∞.../integraltext0
+zn=∞ωg
+n(z1,...,zn).
+Proof of Theorem 2. The proof uses the behaviour of ωg
+nnear the branch points
+zi=±1. Express ωg
+nas a rational function
+ωg
+n=/summationtext4d+2
+k1,...,kn=0ckzk1
+1...zknn/producttextn
+i=1(1−z2
+i)2d+2dz1...dzn
+forck∈C, andd= 3g−3+n. Consider the change of variables zi=ǫi+sxiwhere
+ǫi∈ {±1},s∈Ris small and xiis a local coordinate on the spectral curve. The
+asymptotic behaviour of ωg
+nnearzi=±1 corresponds to s→0 for all combinations
+of theǫi. This change gives:
+ωg
+n=/summationtext4d+2
+k1,...,kn=0ck(ǫ1+sx1)k1...(ǫn+sxn)kn
+s(2d+2)n/producttextn
+i=1x2d+2
+i(2ǫi+sxi)2d+2snΠn
+i=1dxi,
+and we must find a minimal q=q(ǫi)∈ {0,1,...4d+2}so that the coefficient of
+sqin the numerator is the first non vanishing. For example, if q >0 this tells us
+4d+2/summationdisplay
+k1,...,kn=0ckn/productdisplay
+i=1ǫki
+i= 0,
+and ifq >1, then the coefficient of s1= 0. That is:
+4d+2/summationdisplay
+k1,...,kn=0ckn/productdisplay
+i=1ǫki
+i/parenleftbigg/parenleftbiggk1
+1/parenrightbiggx1
+ǫ1+···+/parenleftbiggkn
+1/parenrightbiggxn
+ǫn/parenrightbigg
+= 0POLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 9
+and by equating coefficients of xj, for 1≤j≤n:
+4d+2/summationdisplay
+k1,...,kn=0ckn/productdisplay
+i=1ǫki
+ikj
+ǫj= 0.
+For a general qandα= (α1,...,α n)
+4d+2/summationdisplay
+k1,...,kn=0ck/parenleftbiggk1
+α1/parenrightbigg
+.../parenleftbiggkn
+αn/parenrightbigg
+ǫk1−α1
+1...ǫkn−αn
+nxα1
+1...xαn
+n= 0 if|α|< q.
+Thus inductively one gets
+(14)4d+2/summationdisplay
+k1,...,kn=0ckǫk1−α1
+1...ǫkn−αn
+nkα1
+1...kαn
+n= 0 if|α|< q.
+This means that the dominant asymptotic term as s→0 will look like:
+ωg
+n∼1
+s(2d+1)n−q/producttextn
+i=1x2d+2
+i4d+2/summationdisplay
+k1,...,kn=0/summationdisplay
+|α|=qck
+2(2d+2)nα!Πn
+i=1ǫki−αi
+ikαi
+ixαi
+idxi.
+In [2] it is shown that as z1,...,z ntends to the branch points ǫ1,...,ǫ n,ǫj=±1
+ωg
+n∼/braceleftigg
+s6−6g−3n[1
+2x′′(ǫi)y′(ǫi)]2−2g−nωg
+n[Airy],for allǫithe same
+lower order asymptotics, for mixed ǫi
+where the Airy curve is given by y2=x. Thusq= 2d(n−1) if allǫiare the same
+andq >2d(n−1) for all other combinations.
+From [3] there is a relationship between ωg
+n[Airy] and intersection numbers of
+tautological line bundles over the moduli space.
+(15) ωg
+n[Airy](zS) =x′′(0)2−2g−n
+23g−3+n/summationdisplay
+|β|=dn/productdisplay
+i=1(2βi+1)!
+βi!dzi
+z2βi+2
+i/an}bracketle{tτβ1...τβn/an}bracketri}ht,
+where we have used Witten’s [11] notation /an}bracketle{tτβ1...τβn/an}bracketri}ht=/integraltext
+Mg,nc1(L1)β1...c1(Ln)βn
+andx′′(0) = 2. From this we discover that if all ǫi= 1:
+1/producttextn
+i=1x2d+2
+i4d+2/summationdisplay
+k1,...,kn=0/summationdisplay
+|α|=2d(n−1)ck
+2(2d+2)nα!n/productdisplay
+i=1kαi
+ixαi
+idxi
+=4d+2/summationdisplay
+k1,...,kn=0/summationdisplay
+|α|=2d(n−1)ck
+2(2d+2)nn/productdisplay
+i=1kαi
+ixαi−2d−2
+i
+αi!dxi
+={x′′(1)y′(1)}2−2g−n
+23g−3+n/summationdisplay
+|β|=dn/productdisplay
+i=1(2βi+1)!
+βi!dxi
+x2βi+2
+i/an}bracketle{tτβ1...τβn/an}bracketri}ht.
+and so each αimust be even and 0 ≤αi≤2d. By equating powers of xi(that is,
+extracting the partition where αi= 2d−2βi) one gets the relation
+(16)
+4d+2/summationdisplay
+k1,...,kn=0ck
+2(2d+2)nn/productdisplay
+i=1k2d−2βi
+i
+(2d−2βi)!={x′′(1)y′(1)}2−2g−n
+23g−3+nn/productdisplay
+i=1(2βi+1)!
+βi!/an}bracketle{tτβ1...τβn/an}bracketri}ht.10 PAUL NORBURY AND NICK SCOTT
+Similarly, when all ǫi=−1 one gets
+4d+2/summationdisplay
+k1,...,kn=0ck
+2(2d+2)nn/productdisplay
+i=1(−1)kik2d−2βi
+i
+(2d−2βi)!(17)
+={x′′(−1)y′(−1)}2−2g−n
+23g−3+nn/productdisplay
+i=1(2βi+1)!
+βi!/an}bracketle{tτβ1...τβn/an}bracketri}ht.
+and when there is a mix of ǫi’s,qwill be greater, introducing more vanishing:
+(18)4d+2/summationdisplay
+k1,...,kn=0ck
+2(2d+2)nn/productdisplay
+i=1ǫki
+ik2d−2βi
+i
+(2d−2βi)!= 0 for |β|=d.
+Equations (16), (17) and (18) translate to an analogous equation (21) for coef-
+ficients of the polynomials Ng
+n,k. To show this we now study the polynomials in
+terms of the coefficients ck. As in the proof of Theorem 1, the Taylor expansion of
+ωg
+naboutzi= 0 can be written:
+ωg
+n=4d+2/summationdisplay
+k1,...,kn=0∞/summationdisplay
+b1,...,bn=0ckn/productdisplay
+i=1/parenleftbigg(bi+1−ki)/2+2d
+2d+1/parenrightbigg
+zbi−1
+idzi
+=2−(2d+1)n
+(2d+1)!n4d+2/summationdisplay
+k1,...,kn=0∞/summationdisplay
+b1,...,bn=0ckn/productdisplay
+i=1
+2d+1/summationdisplay
+j=0σj(ki)b2d+1−j
+i
+zbi−1
+idzi
+=∞/summationdisplay
+bi>0b1...bnNg
+n(b1,...,b n)zb1−1
+1...zbn−1
+ndz1...dzn
+where for bieven, respectively odd, we only sum over the odd, respectively eve n,ki
+andσj(ki) (= coefficient of b2d+1−j
+iin (bi+4d+1−ki)...(bi+3−ki)(bi+1−ki))
+is a degree jpolynomial in ki.
+Thus the homogeneous degree 2 qterms of the quasi-polynomial Ng
+nare
+(19) (n/productdisplay
+i=1bi)Ng
+n[degree 2 q] =2−(2d+1)n
+(2d+1)!n4d+2/summationdisplay
+k1,...,kn=0/summationdisplay
+|β|=qckn/productdisplay
+i=1σ2d−2βi(ki)b2βi+1
+i
+where we are still summing over parity dependent k.
+The equations (14), (16), (17) and (18) give identities for sums ov erallkiof
+cktimes monomials in kiand we wish to apply these to (19) which consists of
+coefficients that sum over only some of the ki, depending on parity. To remedy
+this, we add together the different polynomials representing Ng
+nfor every possible
+parity. This removes the restriction on the ksummand.
+For{i1,...,ik} ⊂ {1,...,n}, definev{i1,...,ik}
+βto be the coefficient of b2β1
+1...b2βnn
+inNg
+n,kwithbi1,...,b ikodd. For ǫ= (ǫ1,...,ǫn)∈ {±1}ndefineǫI=/producttext
+k∈Iǫk.
+Sinceσj(ki) is a polynomial of degree jinki, if|β|=q > d, then the homogeneous
+degreein k1,k2,...,knofthe productof σ2d−2βi(ki) issmallenoughthat(14)implies
+the vanishing of each of the coefficients. In other words we have sh own that
+(20)/summationdisplay
+I⊂{1,...,n}ǫIvI
+β= 0,|β|> d.
+If|β|=q=d, then the only non zero sums resulting from the σ2d−2βi(ki) are the
+top powers of the ki, that is/producttextn
+i=1(−ki)2d−2βi, since any component with a smallerPOLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 11
+power of ki’s will vanish when summed again by (14). There will be/producttextn
+i=1/parenleftbig2d+1
+2d−2βi/parenrightbig
+of these. This leaves/summationdisplay
+paritiesNg
+n[degree 2 d]
+=2−(2d+1)n
+(2d+1)!n4d+2/summationdisplay
+k1,...,kn=0/summationdisplay
+|β|=dckn/productdisplay
+i=1/parenleftbigg2d+1
+2d−2βi/parenrightbigg
+(−1)2d−2βik2d−2βi
+ib2βi
+i
+=1
+2(2d+1)n4d+2/summationdisplay
+k1,...,kn=0/summationdisplay
+|β|=dckn/productdisplay
+i=1k2d−2βi
+i
+(2d−2βi)!b2βi
+i
+(2βi+1)!
+thus
+/summationdisplay
+I⊂{1,...,n}vI
+β=1
+2(2d+1)n4d+2/summationdisplay
+k1,...,kn=0ckn/productdisplay
+i=1k2d−2βi
+i
+(2d−2βi)!(2βi+1)!
+where we are again extending to the full sum over all k’s. More generally
+/summationdisplay
+I⊂{1,...,n}ǫIvI
+β=1
+2(2d+1)n4d+2/summationdisplay
+k1,...,kn=0ckn/productdisplay
+i=1ǫki+1
+ik2d−2βi
+i
+(2d−2βi)!(2βi+1)!
+Now we can see that these are (up to simple combinatorial factors) the asymp-
+totic formulas (16), (17) and (18)! We now have expressions for t hese in terms of
+intersection numbers.
+(21)
+/summationdisplay
+I⊂{1,...,n}ǫIvI
+β=
+
+{x′′(1)y′(1)}2−2g−n
+23g−3β!/an}bracketle{tτβ1...τβn/an}bracketri}ht, ǫi= 1 for all i
+(−1)n{x′′(−1)y′(−1)}2−2g−n
+23g−3β!/an}bracketle{tτβ1...τβn/an}bracketri}ht, ǫi=−1 for all i,
+0 otherwise.|β|=d.
+By varying ǫ= (ǫ1,...,ǫn)∈ {±1}n, (20) and (21) are sets of 2nequations with
+2nunknowns vI
+βthat can be uniquely solved. When n= 1, the two equations use
+the matrix J=/parenleftbigg
+1 1
+1−1/parenrightbigg
+, and more generally the 2nequations use the 2n×2n
+matrix given by the tensor product M=J⊗n—equivalently Mis the linear map
+induced by Jon the tensor product ( C2)⊗n. The matrix Mis orthogonal (up to a
+2nscaling factor)
+MMT=/parenleftbig
+JJT/parenrightbig⊗n= (2I)⊗n= 2nI.
+Assemble vI
+βinto a 2n-vectorvβ. Thus (20) becomes
+(20′) Mvβ= 0,|β|> d.
+and (21) becomes Mvβ= the right hand side of (21) or more explicitly
+(21′)Mvβ=
+{x′′(1)y′(1)}2−2g−n
+0
+...
+(−1)n{x′′(−1)y′(−1)}2−2g−n
+/an}bracketle{tτβ1...τβn/an}bracketri}ht
+23g−3β!,|β|=d
+wherethe first andlast rowsof Marethe 2n-vectorse0= (1,1,..)ande1={(−1)I}
+corresponding to ǫi= 1 for all i, respectively ǫi=−1 for all i. In particular, M12 PAUL NORBURY AND NICK SCOTT
+is invertible so equations (20′) and (21′) have the unique solutions vβ= 0 when
+|β|> dand when |β|=d,vβlies in the plane spanned by e0ande1. Explicitly
+vβ=/an}bracketle{tτβ1...τβn/an}bracketri}ht
+23g−3+nβ!/parenleftig
+{x′′(1)y′(1)}2−2g−ne0+(−1)n{x′′(−1)y′(−1)}2−2g−ne1/parenrightig
+=/an}bracketle{tτβ1...τβn/an}bracketri}ht
+x′′(1)2g−2+n23g−3+nβ!/parenleftbig
+y′(1)2−2g−ne0+y′(−1)2−2g−ne1/parenrightbig
+where we have used x′′(−1) =−x′′(1). Equivalently
+vI
+β=y′(1)2−2g−n+(−1)|I|y′(−1)2−2g−n
+x′′(1)2g−2+n23g−3+nβ!/an}bracketle{tτβ1...τβn/an}bracketri}ht.
+In particular, the maximal homogeneous degree of each polynomial Ng
+n,k(b1,...,bn)
+is 2d= 6g−6+2nand the top coefficents aregiven in terms of intersection numbers
+as claimed. /square
+4.String and dilaton equations
+The Eynard-Orantin invariants ωg
+nsatisfy the following string equations [3].
+(22)/summationdisplay
+αRes
+z=αxmyωg
+n+1(z,zS) =−n/summationdisplay
+j=1dzj∂
+∂zj/parenleftbigxm(zj)ωg
+n(zS)
+dx(zj)/parenrightbig
+form= 0,1 or 2,αthe poles of ydxandzS={z1,...,z n}. Note that the m= 2
+case only works for ybeing a sum of monomials, since the proof of the above
+equation in [3] requiresxm(z)
+dx(z)to not have a pole at any of the poles of ydx(z).
+4.1.Proof of Theorem 4. Ify(z) is a polynomial, represent it as a finite sum
+y(z) =y0+y1z+y2z2+.... More generally, as described in Section 2, we can
+approximate any analytic y(z) by a polynomial y(N)(z) which agrees with y(z)
+atz=±1 up to the Nth derivatives. Express the polynomial as a finite sum
+y(N)(z) =y0+y1z+y2z2+.... Note that when y(z) is not a polynomial then
+the finite sum y0+y1z+y2z2+...is not a partial sum for a Taylor series of y(z)
+atz= 0. In the following, we choose N≥6g−6 +2(n+ 1) so that y(z) can be
+replaced in the left hand side of (22) by y(N)(z) in the residue calculations.
+m= 0
+/summationdisplay
+α=±1Res
+z=αy(z)ωg
+n+1(z,zS) =−/summationdisplay
+α=0,∞Res
+z=α(y1z+y2z2+...)ωg
+n+1(z,zS)
+=−Res
+z=∞(y1z+y2z2+...)ωg
+n+1(z,zS)
+=−Res
+z=0(y1
+z+y2
+z2+...)(−ωg
+n+1(z,zS))POLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 13
+The coefficient of/producttextn
+i=1bizbi−1
+iin the expansion about zero is
+y1Ng
+n+1(b1,...,b n,1)+2y2Ng
+n+1(b1,...,b n,2)+···=∞/summationdisplay
+k=1kykNg
+n+1(b1,...,b n,k)
+which can be expressed as Dy(D)/braceleftbig
+mNg
+n+1(m,bS)/bracerightbig
+m=−1as required.
+The right hand side of (22), as shown in [8], expands as
+(23)n/summationdisplay
+j=1∂
+∂zj[(z2
+j+z4
+j+z6
+j+...)ωg
+n(zS)]
+and the coefficient of/producttextn
+i=1bizbi−1
+iin the expansion about zero is given by
+n/summationdisplay
+j=1bj/summationdisplay
+k= 1
+k/negationslash≡bj(2)kNg
+n(b1,...,b n)|bj=k
+m= 1
+/summationdisplay
+α=±1Res
+z=αx(z)y(z)ωg
+n+1(z,zS)
+=−/summationdisplay
+α=0,∞Res
+z=α(y1+y2z+(y1+y3)z2+(y2+y4)z3+...)ωg
+n+1(z,zS)
+=−Res
+z=∞(y1+y2z+(y1+y3)z2+(y2+y4)z3+...)ωg
+n+1(z,zS)
+=Res
+z=0(y1+y2
+z+y1+y3
+z2+y2+y4
+z3+...)ωg
+n+1(z,zS).
+The coefficient of/producttextn
+i=1bizbi−1
+iin the expansion about zero is
+y2Ng
+n+1(1,b1...,bn)+∞/summationdisplay
+k=2k(yk−1+yk+1)Ng
+n+1(k,b1,...,b n)
+and if we add −y0Ng
+n+1(−1,b1,...,b n)+y0Ng
+n+1(1,b1...,bn) = 0 this can be ex-
+pressed as (1+ D2)y(D)/braceleftbig
+mNg
+n+1(m,bS)/bracerightbig
+m=−1.
+The right hand side of (22) can be expanded as
+(24)n/summationdisplay
+j=1∂
+∂zj[(zj+2z3
+j+2z5
+j+...)ωg
+n(zS)]
+and the coefficient of/producttextn
+i=1bizbi−1
+iin the expansion about zero is
+2n/summationdisplay
+j=1bj/summationdisplay
+k= 1
+k≡bj(2)kNg
+n(b1,...,b n)|bj=k+bjNg
+n(b1,...,b n).14 PAUL NORBURY AND NICK SCOTT
+m= 2
+/summationdisplay
+α=±1Res
+z=αx2(z)y(z)ωg
+n+1(z,zS)
+=−/summationdisplay
+α=0,∞Res
+z=α(y1
+z+y2+(y3+2y1)z+...)ωg
+n+1(z,zS)
+=−Res
+z=0y1
+zωg
+n+1(z,zS)−Res
+z=∞(y2+(y3+2y1)z+...)ωg
+n+1(z,zS)
+=Res
+z=0/parenleftbigg
+−y1
+z+y3+2y1
+z+y4+2y2
+z2+y5+2y3+y1
+z3+.../parenrightbigg
+ωg
+n+1(z,zS)
+and the right hand side expands as:
+(25)n/summationdisplay
+j=1∂
+∂zj[(1+3z2
+j+4z4
+j+4z6
+j+...)ωg
+n(zS)].
+In this case it is convenient to subtract four times the m= 0 case (equivalently
+we putx2−4 in place of x2in (22).) Once again, collecting the coefficient of/producttextn
+i=1bizbi−1
+iin the expansion about zero of both sides gives the result.
+4.2.Dilaton. The Eynard-Orantin invariants also satisfy the dilaton equation.
+(26)/summationdisplay
+α=±1Res
+z=αΦ(z)ωg
+n+1(z,zS) = (2g−2+n)ωg
+n(zS)
+wheredΦ =ydx. ThefunctionΦiswell-defineduptoaconstantinaneighbourhood
+of each branch point and the left hand side of (26) is independent of the choice of
+constant.
+Proof of equation (10). Starting from the Dilaton equation proven in [3], we let
+Φ(z) be an arbitrary anti derivative of ydx. That is, dΦ(z) =ydx(z). Then
+(27)/summationdisplay
+α=±1Res
+z=αΦ(zn+1)ωg
+n+1(z,zS) = (2g−2+n)ωg
+n(zS)POLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 15
+We can manipulate this, using integration by parts to rewrite the left hand side as:
+/summationdisplay
+α=±1Res
+z=αΦ(zn+1)ωg
+n+1(z,zS) =−/summationdisplay
+α=±1Res
+z=αdΦ(z)/integraldisplayz
+0ωg
+n+1(z′,zS)
+=−/summationdisplay
+α=±1Res
+z=α(−y1
+z−y2+(y1−y3)z+(y2−y4)z2+...)dz/integraldisplayz
+0ωg
+n+1(z′,zS)
+=Res
+z=∞(−y1
+z−y2+(y1−y3)z+(y2−y4)z2+...)dz/integraldisplayz
+0ωg
+n+1(z′,zS)
+=−Res
+z=∞(−y2z+y1−y3
+2z2+...)ωg
+n+1(z,zS)−Res
+z=∞y1dz
+z/integraldisplayz
+0ωg
+n+1(z′,zS)
+= Res
+z=0(−y2
+z+y1−y3
+2z2+...)ωg
+n+1(z,zS)−Res
+z=∞y1
+z/integraldisplayz
+0ωg
+n+1(z′,zS)
+where lines two to three uses the sum of residues of a meromorphic f unction is zero,
+and lines three to four uses integration by parts. The last residue a bout the simple
+polez=∞can be computed, since/integraltextz
+0ωg
+n+1(z′,zS) is analytic there. Thus
+Res
+z=∞y1dz
+z/integraldisplayz
+0ωg
+n+1(z′,zS) =−y1/integraldisplay∞
+0ωg
+n+1(z′,zS)
+and as in the remark after the proof of Theorem 1 we can extract t he coefficient of/producttextn
+i=1bizbi−1
+iin the expansion about zero to get y1Ng
+n+1(0,b1,...,b n). Hence the
+total coefficient of/producttextn
+i=1bizbi−1
+iin the expansion about zero is
+−y1Ng
+n+1(0,b1,...,b n)−y2Ng
+n+1(1,b1...,bn)+∞/summationdisplay
+k=2(yk−1−yk+1)Ng
+n+1(k,b1,...,b n)
+and if we add y0Ng
+n+1(1,b1,...,b n)−y0Ng
+n+1(−1,b1...,bn) = 0 this can be ex-
+pressed as −(1−D2)y(D)/braceleftbig
+Ng
+n+1(m,bS)/bracerightbig
+m=−1as required. /square
+Remark. It was necessary in the proof of Theorem 4 that the bi>0. The equations
+immediately extend to allow all bi, since the left hand side and right hand side
+are polynomials that agree at infinitely many values in each variable hen ce they
+coincide. For example, when x=z+1/zandy=zthe dilaton equation yields
+Ng
+n+1,k(2,b1,...,bn)−Ng
+n+1,k(0,b1,...,bn) = (2g−2+n)Ng
+n,k(b1,...,bn).
+Ifbj= 0 in the string equation then the sum on the right hand side corresp onding
+tojis empty as in the following n= 0 case.
+4.3.n= 0case.Wecanset n= 0inthestringequationsandthedilatonequation
+to get interesting results. The string equations give vanishing resu lts forNg
+1which
+are useful to help calculate Ng
+1and in practice to check calculations.
+Proposition 4.1. Forg >1, the recursions (3, 4, 5) still hold when n= 0and
+the right hand side is set to zero, leading to vanishing resul ts.16 PAUL NORBURY AND NICK SCOTT
+Proof.This comes directly from the proof of the string equation in [3], applied to
+the case of n= 0. For αthe poles of ydx(z),m= 0, 1 or 2 and Pg
+kdefined by
+theorem 4.5 in [2] we have
+/summationdisplay
+αRes
+z=αx(z)my(z)ωg
+1(z)
+=−1
+2/summationdisplay
+αRes
+z=αx(z)m
+dx(z)(−2y(z)dx(z)ωg
+1(z)+g/summationdisplay
+h=0ωh
+1(z)ωg−h
+1(z)+ωg−1
+2(z,z))
+=1
+2/summationdisplay
+a=±1Res
+z=ax(z)m
+dx(z)(−2y(z)dx(z)ωg
+1(z)+g/summationdisplay
+h=0ωh
+1(z)ωg−h
+1(z)+ωg−1
+2(z,z))
+=1
+4/summationdisplay
+a=±1Res
+z=ax(z)m
+dx(z)(−2y(z)dx(z)ωg
+1(z)+g/summationdisplay
+h=0ωh
+1(z)ωg−h
+1(z)+ωg−1
+2(z,z))
++1
+4/summationdisplay
+a=±1Res
+z=ax(z)m
+dx(z)(−2y(1
+z)dx(z)ωg
+1(1
+z)+g/summationdisplay
+h=0ωh
+1(1
+z)ωg−h
+1(1
+z)+ωg−1
+2(1
+z,1
+z))
+=1
+4/summationdisplay
+a=±1Res
+z=ax(z)m
+dx(z)(Pg
+0(x(z))dx2(z))
+= 0
+Note that in the general proof for arbitrary n, the terms added in line two are
+more complex recursions with Bergmann kernel terms present, co ntributing extra
+residues. /square
+Corollary 3. For the plane curve x=z+1/z,y=y(z)
+Dy(D){mNg
+1(m)}m=−1= 0
+(1+D2)y(D){mNg
+1(m)}m=−1= 0
+(1−D2)2y(D){mNg
+1(m)}m=−2= 0
+Whenn= 0 the dilaton equation is used to define Fg.
+Proof of Corollary 2. We need to show that
+Fg=1
+2−2g(1−D2)y(D){Ng
+1(m)}m=−1
+forg≥2. The definition of the symplectic invariant uses the n= 0 version of the
+dilaton equation
+(28)/summationdisplay
+αRes
+z=αΦ(z)ωg
+1(z) =: (2g−2)Fg
+so we can simply apply the proof of equation (10) to the left hand side of this and
+the result follows. /squarePOLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 17
+5.Examples
+This section serves a few purposes. It describes different interes ting examples
+and gives small ( g,n) polynomials in each case. These examples led to some of the
+general theorems in this paper and give checks of all of the theore ms. It is difficult
+to calculate infinite families of invariants so this section also gives exam ples of
+symplectic invariants Fg(corresponding to n= 0) which are known for all g.
+5.1.y(z) =zBranched covers and discrete surfaces. The curve
+(29) C=/braceleftigg
+x=z+1/z
+y=z
+gives rise to two rather different counts.
+Expand the invariants ωg
+nof the curve ( x,y) = (z+1/z,z) inxiaroundxi=∞
+fori= 1,...,n. Eynard and Orantin [2] show that this gives a generating function
+forcountingconnected orientablediscretesurfacesofgenus gwithnpolygonalfaces
+and a marked edge on each face. The coefficient of/producttextx−(li+1)
+iin the expansion of
+ωg
+ncounts the surfaces consisting of li-sided polygons, i= 1,...,n.
+The associated polynomials Ng
+n, obtained by expanding ωg
+ninziaroundzi= 0,
+were shown in [8] to count connected topologically inequivalent genus gbranched
+covers of S2branched over 0, 1 and ∞with ramification ( b1,...,bn) over∞, ram-
+ification (2 ,2,...,2) over 1 and ramification greater than 1 at all points above 0.
+They are counted in such a way that each branched cover contribu tes one divided
+by the order of its group of automorphisms. Equivalently, they cou nt surfaces with
+npolygonal faces of lengthsb1,...,bn. Although this resembles the count above, it
+is quite different. The number Ng
+n(b1,...,bn) is presented in [7] in terms of counting
+lattice points inside integral convex polytopes depending on ( b1,...,bn) which make
+up a cell decomposition of Mg,n, the moduli space of genus gcurves with nlabeled
+points. A brief description follows.
+LetMg,nbe the moduli space of genus gcurves with nlabeled points. The
+decorated moduli space Mg,n×Rn
++equips the labeled points with positive numbers
+(b1,...,bn) [10]. It has a cell decomposition due to Penner, Harer, Mumford an d
+Thurston
+(30) Mg,n×Rn
++∼=/uniondisplay
+Γ∈Fatg,nPΓ
+where the indexing set Fatg,nis the space of labeled fatgraphs of genus gand
+nboundary components. A fatgraph is a graph Γ with vertices of valency >2
+equipped with a cyclic ordering of edges at each vertex. The cell dec omposition
+(30) arises by the existence and uniqueness of meromorphic quadr atic differentials
+with foliations having compact leaves, known as Strebel differentials which can be
+described via labeled fatgraphs with lengths on edges. Restricting t his homeomor-
+phismtoafixed n-tupleofpositivenumbers( b1,...,bn)yieldsaspacehomeomorphic
+toMg,ndecomposed into compact convex polytopes
+PΓ(b1,...,bn) ={x∈RE(Γ)
++|AΓx=b}
+whereb= (b1,...,bn) andAΓ:RE(Γ)→Rnis the incidence matrix that maps an
+edge to the sum of its two incident boundary components.18 PAUL NORBURY AND NICK SCOTT
+When the biare positive integers the polytope PΓ(b1,...,bn) is an integral poly-
+tope and we define NΓ(b1,...,bn) to be its number of positive integer points. The
+weighted sum of NΓover all labeled fatgraphs of genus gandnboundary compo-
+nents is the piecewise polynomial [7]
+Ng
+n(b1,...,bn) =/summationdisplay
+Γ∈Fatg,n1
+|AutΓ|NΓ(b1,...,bn)
+The top homogeneous degree terms of the polynomials Ng
+n,k(keven) representing
+Ng
+ncoincides with Kontsevich’s volume polynomial [6]
+Vg
+n(b1,...,bn) =/summationdisplay
+Γ∈Fatg,n1
+|AutΓ|VΓ(b1,...,bn)
+whereVΓ(b1,...,bn) is the volume of PΓ(b1,...,bn) induced from the Euclidean vol-
+umes on RE(Γ)andRn. Kontsevich showed that the volume polynomial is a gen-
+erating function for intersection numbers over the moduli space, so this gives an
+alterative proof of Theorem 2 in this case.
+Each integral point in the polytope PΓ(b1,...,bn) corresponds to a Dessin d’en-
+fants defined by Grothendieck [5] which is a branched cover of S2branched over 0,
+1 and∞with ramification ( b1,...,bn) over∞, ramification (2 ,2,...,2) over 1 and
+ramification greater than 1 at all points above 0 as defined above.
+Table 1. y=z
+gn# oddbi Ng
+n(b1,...,bn)
+030,2 1
+1101
+48/parenleftbig
+b2
+1−4/parenrightbig
+040,41
+4/parenleftbig
+b2
+1+b2
+2+b2
+3+b2
+4−4/parenrightbig
+0421
+4/parenleftbig
+b2
+1+b2
+2+b2
+3+b2
+4−2/parenrightbig
+1201
+384/parenleftbig
+b2
+1+b2
+2−4/parenrightbig/parenleftbig
+b2
+1+b2
+2−8/parenrightbig
+1221
+384/parenleftbig
+b2
+1+b2
+2−2/parenrightbig/parenleftbig
+b2
+1+b2
+2−10/parenrightbig
+2101
+216335/parenleftbig
+b2
+1−4/parenrightbig/parenleftbig
+b2
+1−16/parenrightbig/parenleftbig
+b2
+1−36/parenrightbig/parenleftbig
+5b2
+1−32/parenrightbig
+The description of Ng
+nas counting lattice points inside cells of the moduli space
+enables one to prove Ng
+n(0,...,0) =χ(Mg,n) the orbifold Euler characteristic of the
+moduli space of genus gcurves with nlabeled points [7]. It was further shown
+in [8] that the dilaton equation together with vanishing properties of Ng
+nproves
+Ng,n+1(0,...,0) =−(2g−2+n)Ng,n(0,...,0). In particular, when n= 0 this gives
+χ(Mg,1) = (2−2g)Fg. Hence the symplectic invariants for g >1 are given by the
+orbifold Euler characteristic of the moduli space of genus gcurves
+Fg=χ(Mg).POLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 19
+5.2.y=1
+mzmMonomials. The example y=zis the first in the family of ex-
+amples
+(31) C=/braceleftigg
+x=z+1/z
+y=1
+mzm
+In the following we demonstrate similar behaviour between the polyno mials as-
+sociated to the curve y=1
+mzmform >1 and the m= 1 case, suggesting there may
+be an underlying enumeration problem. Unlike the other examples in th is paper,
+there is not yet an interpretation of the polynomials coming from an e numeration
+problem.
+Table 2. y=1
+mzmfor oddm
+gn# oddbi Ng
+n(b1,...,bn)
+030,2 1
+031,3 0
+1101
+48(b2
+1−m2−3)
+111 0
+040,41
+4(b2
+1+b2
+2+b2
+3+b2
+4−m2−3)
+041,3 0
+0421
+4(b2
+1+b2
+2+b2
+3+b2
+4−m2−1)
+1201
+384((b2
+1+b2
+2)2−4(m2+2)(b2
+1+b2
+2)+3m4+10m2+19)
+121 0
+1221
+384(b2
+1+b2
+2−m2−1)(b2
+1+b2
+2−3m2−7)
+2101
+216335(5b8
+1−(116m2+196)b6
+1+(834m4+2476m2+2402)b4
+1
+−(2028m6+7908m4+13556m2+13116) b2
+1
++1305m8+5628m6+13494m4+24636m2+28665)
+211 0
+Theorem 3 gives the following recursions relations which generalise th em= 1
+case. They apply to all genus and can be used to determine the genu s 0 and genus
+1 polynomials.
+Ng
+n+1(m,b1,...bn) =n/summationdisplay
+j=1bj/summationdisplay
+k=1kNg
+n(b1,...,b n)|bj=k
+(m+1)Ng
+n+1(m+1,b1,...,b n)+(m−1)Ng
+n+1(m−1,b1,...,b n)
+= 2mn/summationdisplay
+j=1bj/summationdisplay
+k=1kNg
+n(b1,...,b n)|bj=k−mn/summationdisplay
+j=1bjNg
+n(b1,...,b n)20 PAUL NORBURY AND NICK SCOTT
+(m−2)Ng
+n+1(m−2,b1,...,b n)−2mNg
+n+1(m,b1,...,b n)
++(m+2)Ng
+n+1(m+2,b1,...,b n) =mn/summationdisplay
+j=1/summationdisplay
+k=1±bjkNg
+n(b1,...,b n)|bj=k
+Ng
+n+1(m+1,b1,...,bn)−Ng
+n+1(m−1,b1,...,bn) =m(2g−2+n)Ng
+n(b1,...,bn)
+Table 3. y=1
+mzmfor even m
+gn# oddbi Ng
+n(b1,...,bn)
+030,2 0
+031,3 1
+110 0
+1111
+48(b2
+1−m2−3)
+040,41
+4(b2
+1+b2
+2+b2
+3+b2
+4−m2)
+041,3 0
+0421
+4(b2
+1+b2
+2+b2
+3+b2
+4−m2−2)
+1201
+384(b2
+1+b2
+2−m2)(b2
+1+b2
+2−3m2−8)
+121 0
+1221
+384((b2
+1+b2
+2)2−4(m2+2)(b2
+1+b2
+2)+3m4+8m2+12)
+210 0
+2111
+216335(5b8
+1−(116m2+196)b6
+1+(834m4+2356m2+1982)b4
+1
+−(2028m6+7428m4+10796m2+3396)b2
+1
++1305m8+5268m6+10914m4+11436m2+1605)
+The following vanishing result is proved by considering the Euler chara cteristic
+of connected branched covers of the two-sphere.
+Proposition 5.1 ([8]).Forx=z+1/zandy=z, if
+n/summationdisplay
+i=1bi≤ −2χ= 4g−4+2n, bi>0
+thenNg
+n(b1,...,bn) = 0.
+In particular, Ng
+n(1,1,...,1) = 0 which generalises as follows.
+Proposition 5.2. Forx=z+1/zandy=zm
+m,Ng
+n(m,1,1,...,1) = 0.
+Proof.We have
+Ng
+n(m,b1,...,bn) =n/summationdisplay
+j=1bj/summationdisplay
+k= 1
+k/negationslash≡bj(2)kNg
+n(b1,...,b n)|bj=kPOLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 21
+As in Section 4.3 which makes sense of the string equation for n= 0, setting
+b1,...,bn= 1 leaves an empty sum on the right hand side, hence it is zero. /square
+5.3.y= ln(z)Partitions with Plancherel measure. For a partition λ1≥
+λ2≥...≥λN≥0 define |λ|=/summationtextλiandn(λ) = #{i:λi/ne}ationslash= 0}. The Plancherel
+measure on partitions of size |λ|=Nuses the dimensions of irreducible repre-
+sentations of SN, labeled by partitions λ, satisfying/summationtext
+|λ|=Ndim(λ)2=N!. The
+partition function
+ZN(Q) =/summationdisplay
+n(λ)≤N/parenleftbiggdimλ
+|λ|!/parenrightbigg2
+Q2|λ|
+is related to the symplectic invariants of the curve [1]
+(32) C=/braceleftigg
+x=z+1/z
+y= lnz
+via the asymptotic expansion as Q→ ∞
+lnZN(Q) =/summationdisplay
+gQ2−2gFg.
+ForN→ ∞, exp(−Q2)ZN(Q)→1 so
+Fg=δg,0.
+Table 4. y= lnz
+gn# oddbi Ng
+n(b1,...,bn)
+030,2 0
+031,3 1
+110 0
+1111
+48(b2
+1−3)
+040,41
+4(b2
+1+b2
+2+b2
+3+b2
+4)
+041,3 0
+0421
+4(b2
+1+b2
+2+b2
+3+b2
+4−2)
+1201
+384(b2
+1+b2
+2−8)(b2
+1+b2
+2)
+121 0
+1221
+384(b2
+1+b2
+2−6)(b2
+1+b2
+2−2)
+210 0
+2111
+216335(b2
+1−1)2(5b4
+1−186b2
+1+1605)22 PAUL NORBURY AND NICK SCOTT
+The curve (32) can be seen as the m= 0 case of the previous family of examples
+(31). The polynomials satisfy the following recursions relations.
+Ng
+n+1(0,b1,...bn) =n/summationdisplay
+j=1bj/summationdisplay
+k=1kNg
+n(b1,...,b n)|bj=k
+Ng
+n+1(1,b1,...,b n) =n/summationdisplay
+j=1bj/summationdisplay
+k=1kNg
+n(b1,...,b n)|bj=k+χ−|b|
+2Ng
+n(b1,...,b n)
+forχ= 2−2g−nand|b|=/summationtextn
+j=1bj.
+In all calculated cases, for x=z+1/zandy=zm/mory= lnz, the genus 0
+invariants N0
+ntake integral values. When m= 1, the genus 0 invariants are proven
+to be integral, and based on observation it seems reasonable to con jecture that
+the genus 0 invariants are integral for m >1. This lends further evidence that
+there may be an underlying enumeration problem. In general for Gr omov-Witten
+or moduli space calculations, the invariants are rational, and integr al in genus 0.
+5.3.1.y= ln(z)+czExpectation values of functions on partitions. The following
+natural function on partitions
+Ck(λ) =N/summationdisplay
+i=1/parenleftbigg
+λi−i+1
+2/parenrightbiggk
+−/parenleftbigg
+−i+1
+2/parenrightbiggk
++(1−2−k)ζ(−k)
+is called a Casimir in [1], and a shifted symmetric polynomial, notated by pk, in [9].
+The expectation values of C2with respect to the Plancherel measure on partitions
+areencoded in a generatingfunction whichgivesriseto the followingc urve[1] (after
+making some coordinate changes.)
+(33) C=/braceleftigg
+x=z+1/z
+y= lnz+cz, cis a constant
+The curve has a symmetry
+(z,c)/ma√sto→(−z,−c)
+sincex/ma√sto→ −xandy/ma√sto→y+ln(−1) which implies that when n+kis evenNg
+n,kis a
+function of c2and when n+kis oddcNg
+n,kis a function of c2. In particular, Fgis
+a function of c2.
+In the limit c→ ∞,y(z)/c→z(uniformly in neighbourhoods of ±1) and this
+allows us to deduce that the symplectic invariants for g >1 are asymptotic to the
+symplectic invariants for y=z, i.e.
+Fg∼χ(Mg)c2−2g, c→ ∞.
+The polynomials interpolate between the y= lnzandy=zcases. More precisely,
+lim
+c→0Ng
+n=Ng
+n[y(z) = lnz],lim
+c→∞c2g−2+nNg
+n=Ng
+n[y(z) =z].POLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 23
+Table 5. y= lnz+cz,D=y′(1)y′(−1) =c2−1
+gn# oddbi Ng
+n(b1,...,bn)
+030,2c
+D
+031,3−1
+D
+110c
+48D2(Db2
+1−4c2+2)
+1111
+48D2(−Db2
+1+5c2−3)
+040,41
+4D3((c4−1)(b2
+1+b2
+2+b2
+3+b2
+4)−4c4)
+041,3−c
+2D3(D(b2
+1+b2
+2+b2
+3+b2
+4)−3c2+1)
+0421
+4D3((c4−1)(b2
+1+b2
+2+b2
+3+b2
+4)−2c4−4c2+2)
+1201
+384D4((c2+1)D2(b2
+1+b2
+2)2+32c6
+−4D(3c4+3c2−2)(b2
+1+b2
+2))
+210c
+216335D7(5(c2+3)D4b8
+1−8(39c4+136c2−59)D3b6
+1
++16(357c6+1417c4−1020c2+254)D2b4
+1
+−64(572c8+2192c6−1739c4+806c2−151)Db2
+1
++6144c7(12c4+21c2+2))
+5.3.2.q-deformed partition. The Plancherel weights P(λ) = (dim λ/|λ|!)2used in
+Section 5.3 can be replaced by Pq(λ) = (dim qλ/[|λ|]!)2,q-deformed Plancherel
+weights where q-numbers are used in place of integer combinatorial expressions.
+We will not give the details here since we will only use the spectral curv e calculated
+in [1] for this case.
+(34) C=
+
+x= (1−z
+z0)(1−1
+zz0)
+y=1
+x(z)/parenleftbigg
+−lnz+1
+2ln(1−z
+z0
+1−1
+zz0)/parenrightbigg,
+The curve has a symmetry
+(z,z0)/ma√sto→(−z,−z0)
+which implies that when kis evenNg
+n,kis a function of z2
+0and when kis oddz0Ng
+n,k
+is a function of z2
+0. In particular, Fgis a function of z2
+0. Putz2
+0= 1−etsoFgis a
+function of et.
+It was proven in [1] that
+Fg=cg+/summationdisplay
+d>0|χ(Mg)|d2g−3
+(2g−3)!e−td
+for a constant cg. The Euler characteristic χ(Mg) arises in some sense by coinci-
+dence as an expression involving Bernoulli numbers, with no geometr ic relation to
+the moduli space. The asymptotics of Fgast→0 can explain the appearance of
+χ(Mg) geometrically. As t→0, or equivalently z0→0,
+x∼1
+z2
+0−1
+z0/parenleftbigg
+z+1
+z/parenrightbigg
+≡ −1
+z0/parenleftbigg
+z+1
+z/parenrightbigg
+, y∼z3
+0
+2/parenleftbigg
+z−1
+z/parenrightbigg
+≡z3
+0z24 PAUL NORBURY AND NICK SCOTT
+wherexhas been shifted by a constant and yhas been adjusted by a linear function
+inxwhich does not affect the invariants. Under this scaling the invariant s scale by
+ωg
+n∼(−1)2−2g−nz4−4g−2n
+0ωg
+n[y=z]
+and in particular
+Fg∼z4−4g
+0Fg[y=z] =|χ(Mg)|t2−2g, t→0
+whereFg[y=z] =χ(Mg) follows from the relation of the case x=z+1/z,y=z
+to counting lattice points in the moduli space of curves described in S ection 5.1.
+Table 6. q-deformed partitions
+gn# oddbi Ng
+n(b1,...,bn)
+030,2−1
+z2
+0(3z2
+0+1)
+031,31
+z0(z2
+0+3)
+110−1
+48z2
+0((3z2
+0+1)b2
+1−2z2
+0−4)
+1111
+48z0((z2
+0+3)b2
+1−3z2
+0−3)
+040,41
+4z4
+0((1+z2
+0)(z4
+0+14z2
+0+1)(b2
+1+b2
+2+b2
+3+b2
+4)−4)
+041,3−1
+2z3
+0((z2
+0+3)(3z2
+0+1)(b2
+1+b2
+2+b2
+3+b2
+4)−z4
+0+4z2
+0−3)
+1201
+384z4
+0((z2
+0+1)(z4
+0+14z2
+0+1)(b2
+1+b2
+2)2−32z2
+0+32
+−4(2z6
+0+3z4
+0+8z2
+0+3)(b2
+1+b2
+2))
+2101
+216335z6
+0(−5(3z2
+0+1)(3z6
+0+27z4
+0+33z2
+0+1)b8
+1
++(952z8
+0+224z6
+0+336z4
+0+3808z2
+0+312)b6
+1
++(−3680z8
+0+11360z6
+0+20688z4
+0−13952z2
+0−5712)b4
+1
++(−5696z8
+0+5120z6
+0−38592z4
+0−21248z2
+0+36608) b2
+1
+−36864(z2
+0−2)(z2
+0−1))
+5.4.y=t
+t−γzDiscrete surfaces. The example in Section 5.1 produced a gen-
+erating function for counting connected orientable discrete surf aces of genus gwith
+npolygonal faces and a marked edge on each face. That is a special c ase of more
+general connected orientable discrete surfaces of genus gwithnmarked polygonal
+faces—each containing a marked edge—together with unmarked fa ces of perimeter
+greater than 2, with niof perimeter ifori≥3. A spectral curve for this problem is
+constructed in [2], where the expansion of ωg
+ninxaroundx=∞gives a generating
+function that counts these discrete surfaces. The coefficient of/producttextx−(li+1)
+itn3
+3...tnd
+d
+in the expansion of ωg
+ncounts the surfaces consisting of t3(unmarked) triangles,
+t4squares, ... and li-sided marked polygons, i= 1,...,n. If we set tk=−1 for all
+k≥3 then the spectral curve is given by
+(35) C=/braceleftigg
+x=−2t+γ(z+1/z)
+y=t
+t−γz, γ2=t(t+1)POLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 25
+andthegeneratingfunctiongivesanalternatingsumofnumbersof discretesurfaces.
+The curve has a symmetry
+(z,γ)/ma√sto→(−z,−γ)
+which implies that when kis evenNg
+n,kis a function of γ2and when kis oddγNg
+n,k
+is a function of γ2. In particular, Fgis a function of γ2, hence a function of tas
+shown below.
+The curve also has a less obvious symmetry
+(t,γ)/ma√sto→(−t−1,−γ)
+which takes x/ma√sto→2−x, and
+y/ma√sto→t+1
+t+1−γz=γ2
+γ2−tγz= 1+tz
+γ−tz∼1+tz−1
+γ−tz−1= 1−y
+wherethe ∼signcorrespondstoreparametrisingthecurveby z/ma√sto→z−1. Thisimplies
+that under this symmetry Ng
+nis invariant, respectively skew invariant, when nis
+even, respectively nis odd. In particular, Fgis invariant under t/ma√sto→ −t−1.
+The symplectic invariants of this curve conjecturally use the n= 0 case of
+the enumerative problem above—an alternating count of surfaces consisting of t3
+triangles, t4squares,... andnomarkedpolygons. Thesolutionofthe n= 0problem
+is
+/summationdisplay
+k>0χ(Mg,k)tk
+k!
+whereχ(Mg,k) is the orbifold Euler characteristic of the moduli space of genus g
+curves with klabeled points. This uses a cell decomposition of the moduli space
+where each polygon labels a cell. The alternating sum over all cells gives the Euler
+characteristic χ(Mg,k)/k! of the moduli space of genus gcurves with k(unlabeled)
+points.
+Ast→0,
+x∼t1/2(z+1/z), y∼ −t1/2/z
+hence the curve behaves asymptotically like x=z+1/z,y=−1/z, or equivalently
+y=z(=−1/z+x) which is the example in Section 5.1 with Fg[y=z] =χ(Mg).
+HenceFg∼χ(Mg)t2−2gast→0. Adding the two contributions gives an asymp-
+totic formula which is conjecturally exact:
+Fg=χ(Mg)t2−2g+/summationdisplay
+k>0χ(Mg,k)tk
+k!
+=χ(Mg)/parenleftbigg
+t2−2g+(−1)k/parenleftbigg2g+k−3
+2g−3/parenrightbigg
+tk/parenrightbigg
+=χ/parenleftbig
+Mg)(t2−2g+(t+1)2−2g/parenrightbig
+whereχ(Mg) =ζ(1−2g)/(2−2g).
+It is interesting that the symmetry t/ma√sto→ −t−1 ofFg(proveda priori) suggests
+that the asymptotic part t2−2gdetermines, and is determined by, the solution of
+then= 0 enumerative problem, ( t+1)2−2g.26 PAUL NORBURY AND NICK SCOTT
+5.4.1.Relation to counting surfaces. As discussed above, the expansion in xofωg
+n
+aroundx=∞gives a generating function for counting discrete surfaces, wher e the
+coefficient of/producttextx−(li+1)
+itn3
+3...tnd
+din the expansion counts surfaces consisting of t3
+(unmarked) triangles, t4squares, ... and li-sided marked polygons, i= 1,...,n. We
+can find this coefficient explicitly by evaluating
+Tg
+l1,...,ln:= (−1)nRes
+x1=∞...Res
+xn=∞xl1
+1...xln
+nωg
+n
+= (−1)nn/productdisplay
+i=1Res
+zi=∞γli(zi+1
+zi)li∞/summationdisplay
+b1,...,bn=1Ng
+n(b1,...,bn)n/productdisplay
+i=1bizbi−1
+idzi
+= (−1)nn/productdisplay
+i=1Res
+zi=0γli(1
+zi+zi)li∞/summationdisplay
+b1,...,bn=1Ng
+n(b1,...,bn)n/productdisplay
+i=1bizbi−1
+idzi
+= (−1)nn/productdisplay
+i=1Res
+zi=0γlili/summationdisplay
+k1,...,kn=0∞/summationdisplay
+b1,...,bn=1Ng
+n(b1,...,bn)n/productdisplay
+i=1bi/parenleftbiggli
+ki/parenrightbigg
+zli−2ki+bi−1
+i dzi
+= (−1)nγ/summationtextlili/summationdisplay
+ki>li
+2Ng
+n(2k1−l1,...,2kn−ln)n/productdisplay
+i=1(2ki−li)/parenleftbiggli
+ki/parenrightbigg
+where the Ng
+n’s associate to the spectral curve computed for particular choice s of
+tk. In our example, we have set t3=t4=...=−1, so that Tg
+l1,...,lncounts the bias
+of discrete surfaces having even or odd numbers of faces:
+Tg
+l1,...,ln=∞/summationdisplay
+v=1tv/summationdisplay
+S∈Mg
+n(v,l1,...,ln)(−1)# unmarked polygons in S
+|Aut(S)|
+whereMg
+n(v,l1,...,ln) is the set of connected, orientable, discrete surfaces of genus
+g, constructedby connecting nmarkedpolygonsofperimeter l1,...,lnand anyfinite
+number of unmarked polygons using only vvertices. From [2] this is a finite set.
+For example,
+T(2)
+l=
+
+0 0 ≤l≤4
+−8t−8t2l= 5
+36t+108t2+72t3l= 6
+−49t−490t2−882t3−441t4l= 7
+Remark. It seems that Tg
+l1,...,lnis polynomial in t. Equivalently, most of the coeffi-
+cients of powers of tvanish. This means that besides finitely many exceptions for a
+small number of vertices, there is a duality between discrete surfa ces with an even
+and odd number of faces. Furthermore, the following vanishing res ult shows that
+there are no exceptions when the biare positive and small enough.
+Ng
+n(b1,...,bn) = 0 forn/summationdisplay
+i=1bi<2g+n, bi>0.POLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 27
+The vanishing result is equivalent to the fact that ωg
+n(z1,...,zn) vanishes at zi= 0
+with homogeneous degree 2 gwhich can be proved inductively from the recursive
+definition. A similar vanishing result holds for the y=zcase—see Proposition 5.1.
+Table 7. Discrete surfaces
+gn# oddbi Ng
+n(b1,...,bn)
+030,21+2t
+γ2
+031,3−2
+γ
+1101+2t
+48γ2(b2
+1−4)
+111−1
+24γ(b2
+1−1)
+040,41
+4γ4((8γ2+1)(b2
+1+b2
+2+b2
+3+b2
+4)−8γ2−4)
+041,3−1
+γ(1+2t)(b2
+1+b2
+2+b2
+3+b2
+4−1)
+0421
+4γ4((8γ2+1)(b2
+1+b2
+2+b2
+3+b2
+4)−8γ2−2)
+1201
+384γ4(b2
+1+b2
+2−4)((8γ2+1)(b2
+1+b2
+2)−16γ2−8)
+121−1
+96γ3(1+2t)(b2
+1+b2
+2−1)(b2
+1+b2
+2−5)
+1221
+384γ4(b2
+1+b2
+2−2)((8γ2+1)(b2
+1+b2
+2)−32γ2−10)
+2101+2t
+216335γ6(b2
+1−4)(b2
+1−16)((80γ2+5)b4
+1−(608γ2+212)b2
+1
++1152γ2+1152)
+211−1
+215335γ5(b2
+1−1)(b2
+1−9)((80γ2+15)b4
+1−(1408γ2+438)b2
+1
++3632γ2+1575)
+5.4.2.Quadrangulations. One can enumerate discrete surfaces consisting of quadri-
+laterals by setting tk= 0 fork/ne}ationslash= 4 in the enumeration of discrete surfaces defined
+in Section 5.4 counts. (There we set tk=−1 fork≥3.) The spectral curve for
+this problem is given in [2]
+(36) C=/braceleftigg
+x=z+1/z
+y=tz−t4γ4z3, γ2=1−√1−12tt4
+6t4
+In this case the recursion relations between polynomials are
+tNg
+n+1(1,bS)−3t4γ4Ng
+n+1(3,bS) =n/summationdisplay
+j=1bj/summationdisplay
+k=1kNg
+n(bS)|bj=k
+2(t−t4γ4)Ng
+n+1(2,bS)−4t4γ4Ng
+n+1(4,bS)
+= 2n/summationdisplay
+j=1bj/summationdisplay
+k=1kNg
+n(bS)|bj=k−n/summationdisplay
+j=1bjNg
+n(bS)28 PAUL NORBURY AND NICK SCOTT
+(t−t4γ4)Ng
+n+1(1,bS)+3(t−2t4γ4)Ng
+n+1(3,bS)
++5t4γ4Ng
+n+1(5,bS) =n/summationdisplay
+j=1/summationdisplay
+k=1±bjkNg
+n(bS)|bj=k
+−tNg
+n+1(0,bS)+(t+t4γ4)Ng
+n+1(2,bS)−t4γ4Ng
+n+1(4,bS) = (2g−2+n)Ng
+n(bS)
+wherebS= (b1,...,b n).
+In the following table y′(1) =t−3t4γ4.
+Table 8. Quadrangulations
+gn# oddbi Ng
+n(b1,...,bn)
+030,21
+y′(1)
+031,3 0
+1101
+48y′(1)2(y′(1)b2
+1+36t4γ4−4t)
+111 0
+040,41
+4y′(1)3(y′(1)(b2
+1+b2
+2+b2
+3+b2
+4)+36t4γ4−4t)
+041,3 0
+0421
+4y′(1)3(y′(1)(b2
+1+b2
+2+b2
+3+b2
+4)+30t4γ4−2t)
+2101
+215335y′(1)7(5y′(1)4b8
+1−24y′(1)3(13t−155t4γ4)b6
+1
++48y′(1)2(119t2−2090t4γ4t+17295t2
+4γ8)b4
+1
+−256y′(1)(143t3−2793t2t4γ4+25857tt4γ8−237735t3
+4γ12)b2
+1
++73728t4−1548288t4γ4t3+13934592 t2
+4γ8t2
+−36495360 tt3
+4γ12+1134673920 t4
+4γ16)
+211 0POLYNOMIALS REPRESENTING EYNARD-ORANTIN INVARIANTS 29
+References
+[1] Eynard, Bertrand All orders asymptotic expansion of large partitions. J. Stat. Mech. (2008)
+P07023.
+[2] Eynard, Bertrand and Orantin, Nicolas Algebraic methods in random matrices and enumer-
+ative geometry. arXiv:0811.353
+[3] Eynard, Bertrand and Orantin, Nicolas Invariants of algebraic curves and topological expan-
+sion.Communications in Number Theory and Physics 1(2007), 347452.
+[4] Fay, J.D. Theta functions on Riemann surfaces. Springer Verlag, 1973.
+[5] Grothendieck, Alexandre Esquisse d’un Programme. London Math. Soc. Lecture Note Ser.
+242, Geometric Galois actions, 1, 5-48, Cambridge Univ. Press, Cambridge, 1997.
+[6] Kontsevich, Maxim Intersection theory on the moduli space of curves and the mat rix Airy
+function. Comm. Math. Phys. 147(1992), 1-23.
+[7] Norbury, Paul Counting lattice points in the moduli space of curves. arXiv:0801.4590
+[8] Norbury, Paul String and dilaton equations for counting lattice points in the moduli space of
+curves.arXiv:0905.4141
+[9] Okounkov, A and Pandharipande, R. Gromov-Witten theory, Hurwitz theory, and completed
+cycles.Ann. of Math. (2) 163(2006), 517-560.
+[10] Penner, R. C. The decorated Teichm¨ uller space of punctured surfaces. Comm. Math. Phys.
+113(1987), 299-339.
+[11] Witten, Edward Two-dimensional gravity and intersection theory on moduli space.Surveys
+in differential geometry (Cambridge, MA, 1990), 243–310, Le high Univ., Bethlehem, PA,
+1991.
+Department of Mathematics and Statistics, University of Mel bourne, Australia 3010
+E-mail address :pnorbury@ms.unimelb.edu.au, N.Scott@ms.unimelb.edu.a u
\ No newline at end of file
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+arXiv:1001.0450v1  [math.AT]  4 Jan 2010ORBIT SPACES OF FREE INVOLUTIONS ON THE
+PRODUCT OF TWO PROJECTIVE SPACES
+MAHENDER SINGH
+Abstract. LetXbe a finitistic space having the mod 2 cohomol-
+ogy algebra of the product of two projective spaces. We study free
+involutions on Xand determine the possible mod 2 cohomology al-
+gebra of orbit space of any free involution, using the Leray s pectral
+sequence associated to the Borel fibration X ֒→XZ2−→BZ2. We
+also give an application of our result to show that if Xhas the mod
+2 cohomology algebra of the product of two real projective sp aces
+(respectively complex projective spaces), then there does not ex-
+ist anyZ2-equivariant map from Sk→Xfork≥2 (respectively
+k≥3), where Skis equipped with the antipodal involution.
+1.Introduction
+A finitistic space is a paracompact Hausdorff space whose ever y open
+covering has a finite dimensional open refinement, where the d imension
+of a covering is one less than the maximum number of members of the
+covering which intersect non-trivially. This notion was in troduced by
+Swan [16] in his study of fixed point theory. It is a large class of spaces
+including all compact Hausdorff spaces and all paracompact s paces of
+finite covering dimension.
+It is well known that the class of finitistic spaces is the most suitable
+for studying topological transformation groups. An excell ent account of
+results in this direction can be seen in [1, 2]. Finitistic sp aces behave
+nicely under group actions. More precisely, if Gis a compact Lie group
+acting continuously on a space X, then the space Xis finitistic if and
+only if the orbit space X/Gis finitistic [6, 7].
+LetGbe a group acting continuously on a space X. Determining
+the orbit space up to topological or homotopy type is often di fficult
+and hence we try to determine its (co)homological type. For s pheres the
+orbit spaces of free actions of finite groups have been studie d extensively
+by Livesay [9], Rice [12], Ritter [13], Rubinstein [15] and m any others.
+However, very little is known if the space is a compact manifo ld other
+than a sphere. Myers [11] determined the orbit spaces of free involutions
+on three dimensional lens spaces. Tao [17] determined the or bit spaces
+1991Mathematics Subject Classification. Primary 57S17; Secondary 55R20,
+55M20.
+Key words and phrases. Cohomology algebra, finitistic space, Index of involution,
+Leray spectral sequence.
+12 MAHENDER SINGH
+of free involutions on S1×S2. Ritter [14] extended the results of Tao to
+free actions of cyclic groups of order 2n. Recently Dotzel and others [8]
+determined completely the cohomology algebra of orbit spac es of free Zp
+(pprime) and S1actions on cohomology product of two spheres. This
+paper is concerned with the orbit spaces of free involutions on finitistic
+spacesXhaving the mod 2 cohomology algebra of the product of two
+projective spaces.
+We write X≃2RPn×RPmifXis a space having the mod 2 coho-
+mology algebra of RPn×RPm. Similarly, we write X≃2CPn×CPm
+ifXis a space having the mod 2 cohomology algebra of CPn×CPm.
+The spaces RPn×RPmandCPn×CPmare compact Hausdorff spaces
+and hence are finitistic. Involutions on spaces X≃2RPn×RPmhave
+been studied by Chang and Su [4], where the cohomology struct ures of
+the fixed point sets were determined. We study free involutio ns on such
+spaces and determine the possible mod 2 cohomology algebra o f orbit
+spaces. If X/Gdenotes the orbit space, then we obtain the following
+results.
+Theorem 1.1. LetG=Z2act freely on a finitistic space X≃2RPn×
+RPm,1≤n≤m. Then H∗(X/G;Z2)is isomorphic to one of the
+following graded algebras:
+(1)Z2[x,y,z]//an}bracketle{tx2,yn+1
+2,zm+1/an}bracketri}ht,
+wheredeg(x) = 1,deg(y) = 2,deg(z) = 1andnis odd.
+(2)Z2[x,y,z]//an}bracketle{tx2,yn+1,zm+1
+2/an}bracketri}ht,
+wheredeg(x) = 1,deg(y) = 1,deg(z) = 2andmis odd.
+(3)Z2[x,y,z,w]//an}bracketle{tx2,yn+1
+2,zm+1
+2,w2−αxw−βy−γz/an}bracketri}ht,
+wheredeg(x) = 1,deg(y) = 2,deg(z) = 2,deg(w) = 1and
+α,β,γ∈Z2andn,mare odd.
+Theorem 1.2. LetG=Z2act freely on a finitistic space X≃2CPn×
+CPm,1≤n≤m. Then H∗(X/G;Z2)is isomorphic to one of the
+following graded algebras:
+(1)Z2[x,y,z]//an}bracketle{tx3,yn+1
+2,zm+1/an}bracketri}ht,
+wheredeg(x) = 1,deg(y) = 4,deg(z) = 2andnis odd.
+(2)Z2[x,y,z]//an}bracketle{tx3,yn+1,zm+1
+2/an}bracketri}ht,
+wheredeg(x) = 1,deg(y) = 2,deg(z) = 4andmis odd.
+(3)Z2[x,y,z,w]//an}bracketle{tx3,yn+1
+2,zm+1
+2,w2−αx2w−βy−γz/an}bracketri}ht,
+wheredeg(x) = 1,deg(y) = 4,deg(z) = 4,deg(w) = 2and
+α,β,γ∈Z2andn,mare odd.
+2.Free involutions on the product of two projective
+spaces
+Recall that an involution on a space Xis a continuous action of the
+groupG=Z2onX. We note that the odd dimensional real projective
+space admits a free involution. Let m= 2n−1 withn≥1. Recall thatINVOLUTIONS ON THE PRODUCT OF TWO PROJECTIVE SPACES 3
+RPmis the orbit space of the antipodal involution on Smgiven by
+(x1,x2,...,x2n−1,x2n)/mapsto→(−x1,−x2,...,−x2n−1,−x2n).
+If we denote an element of RPmby [x1,x2,...,x2n−1,x2n], then the map
+RPm→RPmgiven by
+[x1,x2,...,x2n−1,x2n]/mapsto→[−x2,x1,...,−x2n,x2n−1]
+defines an involution. It is easy to check that the involution is free.
+Similarly, the complex projective space CPmadmits a free involution
+whenm≥1 is odd. Recall that CPmis the orbit space of the free
+S1-action on S2m+1given by
+(z1,z2,...,zm,zm+1)/mapsto→(ζz1,ζz2,...,ζzm,ζzm+1) forζ∈S1.
+Denote an element of CPmby [z1,z2,...,zm,zm+1]. Then the map
+[z1,z2,...,zm,zm+1]/mapsto→[−z2,z1,...,−zm+1,zm]
+defines an involution. It is easy to see that the involution is free.
+Taking a free involution on RPmformodd (respectively CPmform
+odd) and any involution on RPn(respectively CPn), the diagonal action
+gives a free involution on RPn×RPm(respectively CPn×CPm).
+3.Preliminaries for proofs of theorems
+In this section, we recall some facts that we will be using in t he paper
+without mentioning explicitly. For details of the content i n this section
+we refer to [2, 10]. Throughout we will use ˇCech cohomology with Z2
+coefficients and we will suppress it from the cohomology notat ion. Let
+the group G=Z2act on a space X. Let
+G ֒→EG−→BG
+be the universal principal G-bundle. Consider the diagonal action of G
+onX×EG. Let
+XG= (X×EG)/G
+betheorbit spaceof thediagonal action on X×EG. Thentheprojection
+X×EG→EGisG-equivariant and gives a fibration
+X ֒→XG−→BG
+called the Borel fibration [3, Chapter IV]. Recall that,
+H∗(BG;Z2) =Z2[t],
+wheretis a homogeneous element of degree 1. We will exploit the Lera y
+spectral sequence associated to the Borel fibration X ֒→XG−→BG.
+Proposition 3.1. [10, Theorem 5.2] LetXi֒→XGρ−→BGbe the Borel
+fibration. Then there is a first quadrant spectral sequence of a lgebras
+{E∗,∗
+r,dr}, converging to H∗(XG)as an algebra, with
+Ek,l
+2=Hk(BG;Hl(X)),4 MAHENDER SINGH
+the cohomology of the base BGwith local coefficients in the cohomology
+of the fiber of ρ.
+Proposition 3.2. [10, Theorem 5.9] LetXi֒→XGρ−→BGbe the Borel
+fibration. Suppose that the system of local coefficients on BGis simple,
+then the edge homomorphisms
+Hk(BG) =Ek,0
+2−→Ek,0
+3−→ ···
+−→Ek,0
+k−→Ek,0
+k+1=Ek,0
+∞⊂Hk(XG)
+and
+Hl(XG)−→E0,l
+∞=E0,l
+l+1⊂E0,l
+l⊂ ··· ⊂E0,l
+2=Hl(X)
+are the homomorphisms
+ρ∗:Hk(BG)→Hk(XG)andi∗:Hl(XG)→Hl(X).
+The product in E∗,∗
+r+1is induced by the product in E∗,∗
+rand the dif-
+ferentials are derivations. The graded commutative algebr aH∗(XG) is
+isomorphictoTot E∗,∗
+∞, thetotal complexof E∗,∗
+∞. Notealsothat H∗(XG)
+is aH∗(BG)-module with the multiplication given by
+(b,x)/mapsto→ρ∗(b)∪x
+forb∈H∗(BG) andx∈H∗(XG).
+IfG=Z2act on a space X≃2RPn×RPmorCPn×CPmsuch that
+the induced action on H∗(X) is trivial, then, by the universal coefficient
+theorem,
+Ek,l
+2∼=Ek,0
+2⊗E0,l
+2.
+SinceE0,l
+2=H0(BG;Hl(X)) =Hl(X)G=Hl(X), we have
+Ek,l
+2∼=Hk(BG)⊗Hl(X).
+We now recall some results regarding Z2-actions on finitistic spaces.
+Proposition 3.3. [2, Chapter VII, Theorem 1.5] LetG=Z2act freely
+on a finitistic space X. Suppose that Hj(X) = 0for allj > n, then
+Hj(XG) = 0for allj > n.
+Leth:XG→X/Gbe the map induced by the G-equivariant projec-
+tionX×EG→X. Then the following is true.
+Proposition 3.4. [2, Chapter VII, Proposition 1.1] LetG=Z2act
+freely on a finitistic space X. Then
+h∗:H∗(X/G)∼=−→H∗(XG).
+In factX/GandXGhave the same homotopy type.
+Proposition 3.5. [2, Chapter VII, Theorem 1.6.] LetG=Z2act
+freely on a finitistic space X. Suppose that/summationtext
+i≥0rk/parenleftbig
+Hi(X)/parenrightbig
+<∞and
+the induced action on H∗(X)is trivial, then the Leray spectral sequence
+associated to X ֒→XG−→BGdoes not degenerate.INVOLUTIONS ON THE PRODUCT OF TWO PROJECTIVE SPACES 5
+Proposition 3.6. [2, Chapter VII, Theorem 7.4] LetG=Z2=/an}bracketle{tT/an}bracketri}htact
+on a finitistic space Xand letHi(X) = 0for alli >2nandH2n(X) =
+Z2. Suppose that c∈Hn(X)is an element such that cT∗(c)/ne}ationslash= 0, then
+the fixed point set is non-empty.
+4.Proofs of theorems
+LetX≃2RPn×RPmorCPn×CPmbe a finitistic space. We first
+observeforwhatvalues of nandm, thespace Xadmitsafreeinvolution.
+More precisely, we prove the following for the real case.
+Lemma 4.1. LetG=Z2act freely on a finitistic space X≃2RPn×
+RPm. Then both nandmcannot be even.
+Proof.Suppose both nandmare even. First consider the case when
+n < m. Note that when l≤n, a basis of Hl(X) consists of
+{al,al−1b,...,abl−1,bl}.
+Whenn < l≤m, a basis of Hl(X) is
+{anbl−n,an−1bl−n+1,...,abl−1,bl}.
+And when m < l≤n+m, a basis consists of
+{anbl−n,an−1bl−n+1,...,al−m+1bm−1,al−mbm}.
+Therefore the Euler characteristic
+χ(X) =n/summationdisplay
+l=0(−1)lrk/parenleftbig
+Hl(X)/parenrightbig
++m/summationdisplay
+l=n+1(−1)lrk/parenleftbig
+Hl(X)/parenrightbig
++n+m/summationdisplay
+l=m+1(−1)lrk/parenleftbig
+Hl(X)/parenrightbig
+=n/summationdisplay
+l=0(−1)l(l+1)+m/summationdisplay
+l=n+1(−1)l(n+1)+n+m/summationdisplay
+l=m+1(−1)l(n+m+1−l)
+=n+2
+2+0−n
+2= 1
+For the case n=m, we have either l≤norn < l≤2n. A similar
+computation gives χ(X) = 1. By Floyd’s Euler characteristic formula
+[2, p.145], we have
+χ(X)+χ(XG) = 2χ(X/G).
+But in case of a free involution χ(XG) = 0 gives a contradiction. Hence
+at least one of them must be odd. /square
+Similarly, for the complex case, we prove the following lemm a.
+Lemma 4.2. LetG=Z2act freely on a finitistic space X≃2CPn×
+CPm. Then both nandmcannot be even.6 MAHENDER SINGH
+Note that the same argument shows that Z2cannot act freely on a
+finitistic space X≃2RPnorCPnwhennis even.
+When a group Gacts on a space X, there is an action on H∗(X)
+given by the automorphisms induced by g−1:X→Xforg∈G. Note
+that it is often difficult to find the induced map on the cohomolo gy even
+in the case of nice spaces such as spheres. In our context we pr ove the
+following.
+Proposition 4.3. LetG=Z2act freely on a finitistic space X≃2
+RPn×RPm. Then the induced action on H∗(X)is trivial.
+Proof.First consider the case when n=m. LetTbe the generator of G
+and leta, b∈H1(X) be generators of the cohomology algebra H∗(X).
+Note that by the naturality of cup product
+T∗(aibj) = (T∗(a))i(T∗(b))j
+for alli, j≥0. Therefore it is enough to consider
+T∗:H1(X)→H1(X).
+Suppose T∗is not identity, then it cannot preserve both aandb. As-
+suming that T∗(a)/ne}ationslash=a, we have T∗(a) =borT∗(a) =a+b. This gives
+anT∗(an) =anbn/ne}ationslash= 0.Thus taking c=an, we have cT∗(c)/ne}ationslash= 0.There-
+fore by Proposition 3.6 the fixed point set of Tis non-empty, which is a
+contradiction. Hence T∗must be identity.
+Whenn/ne}ationslash=m, we have that orders of a,banda+baren+1,m+1
+andn+m+1 respectively. This gives T∗(a) =aandT∗(b) =b. Hence
+in this case also T∗is identity. /square
+Similarly, for the complex case we have the following.
+Proposition 4.4. LetG=Z2act freely on a finitistic space X≃2
+CPn×CPm. Then the induced action on H∗(X)is trivial.
+We now prove the theorems.
+Proof of Theorem 1.1.
+LetG=Z2act freely ona finitistic space X≃2RPn×RPm, 1≤n≤m.
+By Proposition 4.3, the induced action on the cohomology is t rivial. Let
+a,b∈H1(X) be generators of the cohomology algebra
+H∗(X)∼=Z2[a,b]//an}bracketle{tan+1,bm+1/an}bracketri}ht.
+The Leray spectral sequence does not degenerate at the E2term and we
+haved2/ne}ationslash= 0 with one of the following:
+(i)d2(1⊗a) =t2⊗1 andd2(1⊗b) = 0.
+(ii)d2(1⊗a) = 0 and d2(1⊗b) =t2⊗1.
+(iii)d2(1⊗a) =t2⊗1 andd2(1⊗b) =t2⊗1.
+We deal with each case separately.INVOLUTIONS ON THE PRODUCT OF TWO PROJECTIVE SPACES 7
+The case (i)
+Letd2(1⊗a) =t2⊗1 andd2(1⊗b) = 0. Note that by the derivation
+property of the differential we have
+d2(1⊗apbq) =/braceleftbigg
+t2⊗ap−1bqifpis odd
+0 if pis even.
+Ifnis even, then an+1= 0 gives 0 = d2(1⊗an+1) =t2⊗an, a contra-
+diction. Hence nmust be odd. Assume that mis also odd. It suffices
+to compute
+d2:E0,l
+2→E2,l−1
+2.
+Whenl≤n, a basis of E0,l
+2∼=Hl(X) consists of
+{al,al−1b,...,abl−1,bl}.
+Iflis odd, then rk( kerd2) =l+1
+2= rk(imd2). And if lis even, then
+rk(kerd2) =l
+2+1 and rk( imd2) =l
+2.
+Whenn < l≤m, a basis consists of
+{anbl−n,an−1bl−n+1,...,abl−1,bl}.
+In this case rk( kerd2) =n+1
+2= rk(imd2).
+Whenm < l≤n+m, a basis consists of
+{anbl−n,an−1bl−n+1,...,al−m+1bm−1,al−mbm}.
+Iflis odd, then rk( kerd2) =n+m+1−l
+2= rk(imd2). And if lis even, then
+rk(kerd2) =n+m−l
+2and rk(imd2) =n+m+2−l
+2.
+Since
+Ek,l
+3=ker{d2:Ek,l
+2→Ek+2,l−1
+2}/im{d2:Ek−2,l+1
+2→Ek,l
+2},
+it is clear from the observation above that rk( Ek,l
+3) = 0 and hence Ek,l
+3=
+0 for all k≥2 and for all l.
+AndEk,l
+3=ker{d2:Ek,l
+2→Ek+2,l−1
+2}fork= 0, 1 and for all l. Note
+that
+dr:Ek,l
+r→Ek+r,l−r+1
+r
+is zero for all r≥3 asEk+r,l−r+1
+r = 0. Hence E∗,∗
+∞=E∗,∗
+3. Since
+H∗(XG)∼=TotE∗,∗
+∞, we have
+Hp(XG)∼=/circleplusdisplay
+i+j=pEi,j
+∞=E0,p
+∞⊕E1,p−1
+∞
+for all 0 ≤p≤n+m. Letx=ρ∗(t)∈E1,0
+∞be determined by t⊗1∈
+E1,0
+2. AsE2,0
+∞= 0 we have x2= 0. Note that 1 ⊗a2∈E0,2
+2is a
+permanent cocycle and therefore it determines an element u∈E0,2
+∞.
+Choosey∈H2(XG) such that i∗(y) =a2. Thenydetermines uand
+satisfiesyn+1
+2= 0. Similarly, 1 ⊗b∈E0,1
+2is a permanent cocycle and
+determines an element v∈E0,1
+∞. Again we choose z∈H1(XG) such
+thati∗(z) =b. Thenzdetermines vandzm+1= 0. Therefore
+H∗(XG)∼=Z2[x,y,z]//an}bracketle{tx2,yn+1
+2,zm+1/an}bracketri}ht,8 MAHENDER SINGH
+wheredeg(x) = 1,deg(y) = 2 and deg(z) = 1. As the action of Gis
+free,H∗(X/G)∼=H∗(XG). The same argument works when mis even.
+This gives Theorem 1.1 (1).
+The case (ii)
+Letd2(1⊗a) = 0 and d2(1⊗b) =t2⊗1. As above we see that mmust
+be odd. Rest of the proof is same and we obtain
+H∗(X/G)∼=Z2[x,y,z]//an}bracketle{tx2,yn+1,zm+1
+2/an}bracketri}ht,
+wheredeg(x) = 1,deg(y) = 1 and deg(z) = 2. This gives Theorem 1.1
+(2).
+The case (iii)
+Letd2(1⊗a) =t2⊗1 andd2(1⊗b) =t2⊗1. One can see that
+d2(1⊗apbq) =
+
+t2⊗ap−1bq+t2⊗apbq−1ifpandqare odd
+t2⊗ap−1bqifpis odd and qis even
+t2⊗apbq−1ifpis even and qis odd
+0 if pandqare even.
+Note that in this case both nandmare odd. As in case (i), by looking
+at the ranks of kerd2andimd2for various values of l, we get Ek,l
+3= 0
+for allk≥2 andEk,l
+3=ker{d2:Ek,l
+2→Ek+2,l−1
+2}fork= 0, 1. Again
+dr= 0 for all r≥3 and hence E∗,∗
+∞=E∗,∗
+3. We have
+Hp(XG)∼=/circleplusdisplay
+i+j=pEi,j
+∞=E0,p
+∞⊕E1,p−1
+∞
+forall 0≤p≤n+m. Letx=ρ∗(t)∈E1,0
+∞bedeterminedby t⊗1∈E1,0
+2.
+AsE2,0
+∞= 0 we have x2= 0. Note that 1 ⊗a2∈E0,2
+2is a permanent
+cocycle and therefore yields an element u∈E0,2
+∞. Choose y∈H2(XG)
+such that i∗(y) =a2. Then ydetermines uand satisfies yn+1
+2= 0.
+Similarly, 1 ⊗b2∈E0,2
+2isapermanentcocycleanddeterminesanelement
+v∈E0,2
+∞. Again we choose z∈H2(XG) such that i∗(z) =b2. Thenz
+determines vandzm+1
+2= 0. Also 1 ⊗(a+b)∈E0,1
+2is a permanent
+cocycle and yields an element s∈E0,1
+∞. Letw∈H1(XG) such that
+i∗(w) =a+b. Thenwdetermines s. Note that
+w2=αxw+βy+γz
+for some α,β,γ∈Z2. Hence
+H∗(XG)∼=Z2[x,y,z,w]//an}bracketle{tx2,yn+1
+2,zm+1
+2,w2−αxw−βy−γz/an}bracketri}ht,
+wheredeg(x) = 1,deg(y) = 2,deg(z) = 2 and deg(w) = 1. As the
+action of Gis free,H∗(X/G)∼=H∗(XG) which gives Theorem 1.1(3). /square
+Corollary 4.5. LetG=Z2act freely on a finitistic space X≃2RPn,
+wheren≥1is odd. Then
+H∗(X/G)∼=Z2[x,y]//an}bracketle{tx2,yn+1
+2/an}bracketri}ht,INVOLUTIONS ON THE PRODUCT OF TWO PROJECTIVE SPACES 9
+wheredeg(x) = 1anddeg(y) = 2.
+Proof.The proof follows by taking b= 0 in the proof of Theorem 1.1.
+We have d2(1⊗a) =t2⊗1 and
+d2(1⊗ap) =/braceleftbigg
+t2⊗ap−1ifpis odd
+0 if pis even.
+This gives
+d2:Ek,l
+2→Ek+2,l−1
+2
+is an isomorphism for lodd and zero for leven. Hence Ek,l
+3= 0 for all
+k≥2. Also
+dr:Ek,l
+r→Ek+r,l−r+1
+r
+is zero for all r≥3 asEk+r,l−r+1
+r = 0 for all r≥3. Imitating the above
+proof we get
+H∗(X/G;Z2)∼=Z2[x,y]//an}bracketle{tx2,yn+1
+2/an}bracketri}ht,
+wheredeg(x) = 1 and deg(y) = 2. /square
+Remark 4.6.Takingn=m= 1, we have X≃2RPn×RPm=S1×S1
+and hence
+H∗(X/G)∼=Z2[x,w]//an}bracketle{tx2,w2−αxw/an}bracketri}ht,
+wheredeg(x) = 1,deg(w) = 1 and α∈Z2, which is consistent with [8,
+Theorem 2 (iii)].
+Proof of Theorem 1.2.
+Sincetheproofis analogous to that of Theorem1.1, wedescri beit rather
+briefly. Let G=Z2act freely on a finitistic space X≃2CPn×CPm,
+1≤n≤m. By Proposition 4.4, the induced action on the cohomology
+is trivial. Note that Ek,l
+2= 0 forlodd. This gives
+d2:Ek,l
+2→Ek+2,l−1
+2
+is zero and hence Ek,l
+3=Ek,l
+2for allk,l. Leta,b∈H2(X) be generators
+of the cohomology algebra H∗(X). Since the Leray spectral sequence
+does not degenerate, we have d3/ne}ationslash= 0 with one of the following:
+(i)d3(1⊗a) =t3⊗1 andd3(1⊗b) = 0.
+(ii)d3(1⊗a) = 0 and d3(1⊗b) =t3⊗1.
+(iii)d3(1⊗a) =t3⊗1 andd3(1⊗b) =t3⊗1.
+Again we proceed case by case.
+The case (i)
+Letd3(1⊗a) =t3⊗1 andd3(1⊗b) = 0, then
+d3(1⊗apbq) =/braceleftbigg
+t3⊗ap−1bqifpis odd
+0 if pis even.
+Note that nmust be odd. For various values of lwe consider the differ-
+entials
+d3:E0,2l
+3→E3,2l−2
+310 MAHENDER SINGH
+as in the proof of Theorem 1.1 and compute the ranks of kerd3and
+imd3. Since
+Ek,2l
+4=ker{d3:Ek,2l
+3→Ek+3,2l−2
+3}/im{d3:Ek−3,2l+2
+3→Ek,2l
+3},
+it is clear that rk( Ek,2l
+4) = 0 and hence Ek,2l
+4= 0 for all k≥3.
+AndEk,2l
+4=ker{d3:Ek,2l
+3→Ek+3,2l−2
+3}fork= 0, 1, 2. Note that
+Ek,l
+4= 0 for all kand for all odd l. Observe that
+dr:Ek,l
+r→Ek+r,l−r+1
+r
+is zero for all r≥4 asEk+r,l−r+1
+r = 0 for all r≥4. Hence E∗,∗
+∞=E∗,∗
+4.
+SinceH∗(XG)∼=TotE∗,∗
+∞, have
+Hp(XG)∼=/circleplusdisplay
+i+j=pEi,j
+∞=E0,p
+∞⊕E1,p−1
+∞⊕E2,p−2
+∞
+for all 0 ≤p≤2(n+m). Letx=ρ∗(t)∈E1,0
+∞be determined by
+t⊗1∈E1,0
+2. AsE3,0
+∞= 0 we have x3= 0. Note that 1 ⊗a2∈E0,4
+2
+is a permanent cocycle and determines an element u∈E0,4
+∞. Choose
+y∈H4(XG) such that i∗(y) =a2. Thenydetermines uand satisfies
+yn+1
+2= 0. Similarly, 1 ⊗b∈E0,2
+2is a permanent cocycle and gives an
+element v∈E0,2
+∞. Again we choose z∈H2(XG) such that i∗(z) =b.
+Thenzdetermines vandzm+1= 0. Therefore
+H∗(XG)∼=Z2[x,y,z]//an}bracketle{tx3,yn+1
+2,zm+1/an}bracketri}ht,
+wheredeg(x) = 1,deg(y) = 4 and deg(z) = 2. As the action of Gis
+free,H∗(X/G)∼=H∗(XG), which gives Theorem 1.2 (1).
+The case (ii)
+The proof is similar to that of case (i) and we get Theorem 1.2 ( 2).
+The case (iii)
+Letd3(1⊗a) =t3⊗1 andd3(1⊗b) =t3⊗1. One can see that
+d3(1⊗apbq) =
+
+t3⊗ap−1bq+t3⊗apbq−1ifpandqare odd
+t3⊗ap−1bqifpis odd and qis even
+t3⊗apbq−1ifpis even and qis odd
+0 if pandqare even.
+Again in this case both nandmare odd. As in case (i) we see that
+Ek,2l
+4= 0 for all k≥3 andEk,2l
+4=ker{d3:Ek,2l
+3→Ek+3,2l−2
+3}fork=
+0, 1, 2. Note that Ek,l
+4= 0 for all kand for all odd l. Thusdr= 0 for
+allr≥4 and hence E∗,∗
+∞=E∗,∗
+4. Again we have
+Hp(XG)∼=/circleplusdisplay
+i+j=pEi,j
+∞=E0,p
+∞⊕E1,p−1
+∞⊕E2,p−2
+∞
+for all 0 ≤p≤2(n+m). Letx=ρ∗(t)∈E1,0
+∞be determined by
+t⊗1∈E1,0
+2. AsE3,0
+∞= 0 we have x3= 0. Note that 1 ⊗a2∈E0,4
+2isINVOLUTIONS ON THE PRODUCT OF TWO PROJECTIVE SPACES 11
+a permanent cocycle and gives an element u∈E0,4
+∞. Lety∈H4(XG)
+such that i∗(y) =a2. Thenydetermines uandyn+1
+2= 0. Similarly,
+1⊗b2∈E0,4
+2is a permanent cocycle and yields an element v∈E0,4
+∞.
+Choosez∈H4(XG) such that i∗(z) =b2. Thenzdetermines vand
+satisfieszm+1
+2= 0. Also 1 ⊗(a+b)∈E0,2
+2is a permanent cocycle and
+gives an element s∈E0,2
+∞. Choose w∈H2(XG) such that i∗(w) =a+b.
+Thenwdetermines s. We note that
+w2=αx2w+βy+γz
+for some α,β,γ∈Z2. Hence
+H∗(XG)∼=Z2[x,y,z,w]//an}bracketle{tx3,yn+1
+2,zm+1
+2,w2−αx2w−βy−γz/an}bracketri}ht,
+wheredeg(x) = 1,deg(y) = 4,deg(z) = 4 and deg(w) = 2. Since
+H∗(X/G)∼=H∗(XG), we get Theorem 1.2 (3). /square
+Corollary 4.7. LetG=Z2act freely on a finitistic space X≃2CPn,
+wheren≥1is odd. Then
+H∗(X/G)∼=Z2[x,y]//an}bracketle{tx3,yn+1
+2/an}bracketri}ht,
+wheredeg(x) = 1anddeg(y) = 4
+Proof.The proof follows by taking b= 0 in the proof of Theorem 1.2.
+We have d2= 0 and d3(1⊗a) =t3⊗1 and
+d3(1⊗ap) =/braceleftbigg
+t3⊗ap−1ifpis odd
+0 if pis even.
+This gives
+d3:Ek,2l
+3→Ek+3,2l−2
+3
+is an isomorphism for lodd and zero for leven. Hence Ek,l
+4= 0 for all
+k≥3 and
+dr:Ek,l
+r→Ek+r,l−r+1
+r
+is zero for all r≥4 asEk+r,l−r+1
+r = 0 for all r≥4. Imitating the above
+proof we get
+H∗(X/G;Z2)∼=Z2[x,y]//an}bracketle{tx3,yn+1
+2/an}bracketri}ht,
+wheredeg(x) = 1 and deg(y) = 4. /square
+Remark 4.8.Takingn=m= 1, we have X≃2CPn×CPm=S2×S2
+and hence
+H∗(X/G)∼=Z2[x,w]//an}bracketle{tx3,w2−αx2w/an}bracketri}ht,
+wheredeg(x) = 1,deg(w) = 2 and α∈Z2, which is consistent with [8,
+Theorem 2 (iii)].
+Example 4.9. For an odd integer n, take a free involution on RPnand
+for any positive integer mtake the trivial action on RPm. Then the
+diagonal action on X=RPn×RPmis a free involution. By definition
+RPn=Sn/Z2, whereSnis equipped with the antipodal action of Z2.
+Now given the free involution on RPn, lifting the action gives a free and12 MAHENDER SINGH
+orthogonal action of a group Hof order 4 on Sn. It is well known that
+Z2⊕Z2cannot act freely on Sn[2, Chapter III, Theorem 8.1]. Hence
+Hmust be the cyclic group of order 4 acting freely and orthogon ally on
+Snand therefore RPn/Z2=Sn/H=Ln(4,1), then-dimensional Lens
+space whose cohomology algebra is known to be
+Z2[x,y]//an}bracketle{tx2,yn+1
+2/an}bracketri}ht,
+wheredeg(x) = 1 and deg(y) = 2. Hence the cohomology algebra of the
+orbit space X/Z2=Ln(4,1)×RPmis
+Z2[x,y,z]//an}bracketle{tx2,yn+1
+2,zm+1/an}bracketri}ht,
+wheredeg(x) = 1,deg(y) = 2 and deg(z) = 1. This realizes Theorem
+1.1(1). Interchanging nandmrealizes Theorem 1.1(2).
+Example 4.10. For the complex case, if we take n= 1, then CP1=S2.
+The orbit space of any free involution on S2isRP2, whose cohomology
+algebra is Z2[x]//an}bracketle{tx3/an}bracketri}ht. For any positive integer m, the trivial action on
+CPmgives a free involution on X=CP1×CPmwhose orbit space is
+X/Z2=RP2×CPmand has cohomology algebra
+Z2[x,z]//an}bracketle{tx3,zm+1/an}bracketri}ht,
+wheredeg(x) = 1 and deg(z) = 2. This realizes Theorem 1.2 (1). Simi-
+larly, one can realize Theorem 1.2(2).
+5.Application to equivariant maps
+LetSkbe the unit k-sphere equipped with the antipodal involution
+andXbe a paracompact Hausdorff space with a fixed free involution.
+We give an application of our results to non-existence of Z2-equivariant
+maps from Sk→X. Conner and Floyd defined the index of the involu-
+tion onXas
+ind(X) =max{k|there exist a Z2-equivariant map Sk→X}.
+It is natural to consider a purely cohomological criteria to study the
+above invariant. The best known and most easily managed coho mology
+classes are the characteristic classes with coefficients in Z2. Generalizing
+the Yang’s index [18], Conner and Floyd defined
+co-indZ2(X) = largest integer ksuch that wk/ne}ationslash= 0,
+wherew∈H1(X/G;Z2) is the Whitney class of the principal G-bundle
+X→X/G.
+Since co-ind Z2(Sk) =k, by [5, (4.5)], we have
+ind(X)≤co-indZ2(X).
+Also, since Xis paracompact Hausdorff, we can take a classifying map
+c:X/G→BGINVOLUTIONS ON THE PRODUCT OF TWO PROJECTIVE SPACES 13
+for the principal G-bundleX→X/G. Ifh:X/G→XGis a homotopy
+equivalence, then ρh:X/G→BGalso classifies the principal G-bundle
+X→X/Gand hence it is homotopic to c. Therefore it suffices to
+consider the map
+ρ∗:H1(BG)→H1(XG).
+The image of the Whitney class of the universal principal G-bundle
+G ֒→EG−→BGis the Whitney class of X→X/G.
+Now, when X≃2RPn×RPmis a finitistic space, from the proof
+of Theorem 1.1, we have that x∈H1(X/G) is the Whitney class with
+x/ne}ationslash= 0 and x2= 0. This gives co-ind Z2(X) = 1 and ind( X)≤1. Hence
+there is no Z2-equivariant map from Sk→Xfork≥2.
+Similarly, when X≃2CPn×CPmis a finitistic space, from the proof
+of Theorem 1.2, x∈H1(X/G) is the Whitney class with x2/ne}ationslash= 0 and
+x3= 0. This gives co-ind Z2(X) = 2 and ind( X)≤2. Hence there is no
+Z2-equivariant map from Sk→Xfork≥3.
+Acknowledgment. The author thanks the referee for many valuable
+suggestions which improved the presentation and expositio n of the pa-
+per.
+References
+1. C. Allday and V. Puppe, Cohomological Methods in Transformation Groups ,
+Cambridge Studies in Advanced Mathematics 32, Cambridge Un iversity Press,
+Cambridge, 1993.
+2. G. E. Bredon, Introduction to Compact Transformation Groups , Academic Press,
+New York, 1972.
+3. A. Borel et al., Seminar on Transformation Groups , Annals of Math. Studies 46,
+Princeton University Press, 1960.
+4. C. N. Chang and J. C. Su, Group actions on a product of two projective spaces ,
+Amer. J. Math. 101 (1979), 1063-1081.
+5. P. E. Conner and E. E. Floyd Fixed point free involutions and equivariant maps-I ,
+Bull. Amer. Math. Soc. 66 (1960), 416-441.
+6. S. Deo and H. S. Tripathi, Compact Lie group actions on finitistic spaces , Topol-
+ogy 21 (1982), 393-399.
+7. S. Deo and T. B. Singh, On the converse of some theorems about orbit spaces , J.
+London Math. Soc. 25 (1982), 162-170.
+8. R. M. Dotzel, T. B. Singh and S. P. Tripathi The cohomology rings of the orbit
+spaces of free transformation groups on the product of two sp heres, Proc. Amer.
+Math. Soc. 129 (2000), 921-930.
+9. G. R. Livesay, Fixed point free involutions on the 3-sphere, Ann. of Math. 72
+(1960), 603-611.
+10. J. McCleary, A User’s Guide to Spectral Sequences , Cambridge Studies in Ad-
+vanced Mathematics 58, Second Edition, Cambridge Universi ty Press, Cam-
+bridge, 2001.
+11. R. Myers, Free involutions on lens spaces , Topology 20 (1981), 313-318.
+12. P. M. Rice, Free actions of Z4onS3, Duke Math. J. 36 (1969), 749-751.
+13. G. X. Ritter, FreeZ8actions on S3, Trans. Amer. Math. Soc. 181 (1973),195-212.
+14. G. X. Ritter, Free actions of cyclic groups of order 2nonS1×S2, Proc. Amer.
+Math. Soc. 46 (1974), 137-140.14 MAHENDER SINGH
+15. J. H. Rubinstein, Free actions of some finite groups on S3- I, Math. Ann. 240
+(1979), 165-175.
+16. R.G. Swan, A new method in fixed point theory , Comment. Math. Helv.34(1960),
+1-16.
+17. Y. Tao, On fixed point free involutions on S1×S2, Osaka J. Math. 14 (1962)
+145-152.
+18. C. T.Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobo and Dy son-I,
+Ann. of Math. 60 (1954), 262-282.
+Institute of Mathematical Sciences, C I T Campus, Taramani, Chennai
+600113, India
+E-mail address :mahender@imsc.res.in
\ No newline at end of file
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+arXiv:1001.0451v1  [math.FA]  4 Jan 2010Maps of several variables of finite total variation
+and Helly-type selection principles✩
+Vyacheslav V. Chistyakov∗,a, Yuliya V. Tretyachenkoa
+aDepartment of Applied Mathematics and Informatics, State U niversity Higher School
+of Economics, Bol’shaya Pech¨ erskaya Street 25/12, Nizhny Novgorod 603155, Russia
+Abstract
+Given two points a= (a1,...,a n) andb= (b1,...,b n) fromRnwitha < b
+componentwise andamap ffromtherectangle Ib
+a= [a1,b1]×···×[an,bn]into
+a metric semigroup M= (M,d,+), we study properties of the total variation
+TV(f,Ib
+a) offonIb
+aintroduced by the first author in [V.V. Chistyakov, A
+selection principle for mappings of bounded variation of several var iables, in:
+Real Analysis Exchange 27th Summer Symposium, 2003, 217–222], w hich
+extends the classical notion of C. Jordan’s total variation ( n= 1) and the
+corresponding notions in the sense of [T.H. Hildebrandt, Introduc tion to the
+Theory of Integration, Academic Press, 1963] ( n= 2) and [A.S. Leonov,
+On the total variation for functions of several variables and a mult idimen-
+sional analog of Helly’s selection principle, Math. Notes 63 (1998) 61– 71]
+(n∈N) for real valued functions of nvariables. The following Helly-type
+pointwise selection principle is proved: If a sequence {fj}j∈Nof maps from
+Ib
+aintoMis such that the closure in Mof the set {fj(x)}j∈Nis compact
+for each x∈Ib
+aandC≡supj∈NTV(fj,Ib
+a)is finite, then there exists a sub-
+sequence of {fj}j∈N, which converges pointwise on Ib
+ato a map fsuch that
+TV(f,Ib
+a)≤C.A variant of this result is established concerning the weak
+pointwise convergence when values of maps lie in a reflexive Banach sp ace
+(M,/⌊ard⌊l·/⌊ard⌊l) with separable dual M∗.
+Key words: maps of several variables, total variation, selection principle,
+✩The work of V.V. Chistyakov was supported by the State Universit y Higher School of
+Economics Grant No. 09-08-0012(Priority thematics “Investigat ion of Function Spaces”).
+∗Corresponding author; tel.: +7 831 4169649; fax: +7 831 4169650 .
+Email addresses: czeslaw@mail.ru (Vyacheslav V. Chistyakov),
+tretyachenko yv@mail.ru (Yuliya V. Tretyachenko)
+Preprint submitted to arXiv October 11, 2018metric semigroup, pointwise convergence, weak convergence
+MSC Classification: 26B30, 20M15, 28A20
+1. Introduction
+TheclassicalHellyselectionprinciple([27])statesthat a boundedsequence
+of real valued functions on the closed interval, which is of u niformly bounded
+(Jordan)variation, contains a pointwise convergent subsequence wh ose limit
+is a function of bounded variation . This theorem and its recent general-
+izations for real valued functions and metric space valued maps of o ne real
+variable ([7, 10, 15, 18, 19, 20, 21]) have numerous applications in diff erent
+branches of Analysis (e.g., [6, 15, 25, 28, 33] and references ther ein).
+Extensions of the Helly theorem for functions and maps of several real
+variables heavily depend upon notions of (bounded) variation used f or these
+maps, which generalize different aspects of the classical Jordan va riation
+of univariate functions and which are known to be quite numerous in t he
+literature (e.g., [3, 8, 22, 24, 28, 30, 35, 38, 39, 41], and these ref erences
+are far from being exhaustive on the topic). Under some approach es to the
+multidimensional variation([2,8,34])involving integrationprocedure sHelly-
+type theorems are rather concerned with the almost everywhere convergence
+of extracted subsequences, and no stronger convergence can be expected in
+this case, but this convergence is far too weak for certain applicat ions (such
+as those from [15]). On the other hand, there are definitions of the notion of
+variationfor real valued functionsofseveral variables([28,32]), whichgoback
+to Vitali [38], Hardy [26] and Krause [1, 22], such that a complete ana logue
+of the Helly theorem holds with respect to the pointwise convergence of
+extracted subsequences. These counterparts of Helly’s theore m are based on
+the notionof a (totally) monotone real valued function ofseveral variables [9,
+28, 40] and an appropriate generalization of Jordan’s decompositio n theorem
+when a function of bounded variation is represented as the differen ce of two
+monotone functions.
+The aim of this paper is twofold. First, we study properties of the total
+variation of metric semigroup valued maps of several variables in the ap-
+proach of Vitali, Hardy and Krause introduced by the first author in [14],
+which extends the classical notion of Jordan’s total variation for m aps of one
+variable and the notions of the total variationin the sense of Hildebr andt [28]
+forreal valued functions of two variablesand Leonov[32] for real valued func-
+tions of any finite number of variables. Second, we present two var iants of
+2a Helly-type pointwise selection principle for metric semigroup valued maps
+and maps with values in a reflexive separable Banach space. The main d iffi-
+culty that we overcome is that for metric semigroup valued maps the re is no
+counterpart of Jordan’s decomposition theorem, and we have to d evelop a
+completely different technique, whose two-dimesional variant is give n in [5].
+The paper is organized as follows. In Section 2 we present necessar y
+definitions and our two main results, Theorems 1 and 2. In order to g et to
+theproofsof these results asquick aspossible, inSection 3we collec t allmain
+ingredients and auxiliary known facts needed for their proofs. In S ection 4
+we prove Theorems 1 and 2. The remaining Sections 5–8 contain proo fs of
+the results exposed in Section 3 and used in the proofs of the main th eorems.
+2. Definitions and main results
+Throughout the paper we adopt and follow the Vitali-Hardy-Krause ap-
+proach to the notion of variation for maps of several variables in th e multiin-
+dex notation initiated in [12, 14] and developed in detail in [17] (equivale nt
+approaches in different notation for real functions can be found in [31, 32]).
+LetNandN0stand for the sets of positive and nonnegative integers,
+respectively, and n∈N. Given x,y∈Rn, we write x= (x1,...,x n) =
+(xi:i∈ {1,...,n}) for the coordinate representation of x, and set x+y=
+(x1+y1,...,x n+yn), andx−yis defined similarly. The inequality x < y
+will be understood componentwise, i.e., xi< yifor alli∈ {1,...,n}, and
+a similar meaning applies to x=y,x≤y,y≥xandy > x. Ifx < yor
+x≤y, we denote by Iy
+xthe rectangle/producttextn
+i=1[xi,yi] = [x1,y1]× ···×[xn,yn].
+Elements of the set Nn
+0are as usual said to be multiindices and denoted by
+Greek letters and, given θ= (θ1,...,θ n)∈Nn
+0andx∈Rn, we set |θ|=
+θ1+···+θn(the order of θ) andθx= (θ1x1,...,θ nxn). Then-dimensional
+multiindices 0 n= (0,...,0) and 1 n= (1,...,1) will be denoted simply by 0
+and 1, respectively (actually, the dimension of 0 and 1 will be clear fro m the
+context). We also put E(n) ={θ∈Nn
+0:θ≤1 and|θ|is even}(the set of
+‘even’ multiindices) and O(n) ={θ∈Nn
+0:θ≤1 and|θ|is odd}(the set of
+‘odd’ multiindices). For elements from the set A(n) ={α∈Nn
+0: 0/\e}atio\slash=α≤1}
+we simply write 0 /\e}atio\slash=α≤1.
+The domain of (almost) all maps under consideration will be a rectang le
+Ib
+awith fixed a,b∈Rn,a < b, called the basic rectangle . The range of
+maps will be a metric semigroup (M,d,+), i.e., ( M,d) is a metric space,
+(M,+) is an Abelian semigroup with the operation of addition +, and d
+3is translation invariant: d(u,v) =d(u+w,v+w) for all u,v,w∈M. A
+nontrivial example of a metric semigroup is as follows ([23, 36]): Let ( X,/⌊ard⌊l·/⌊ard⌊l)
+be a real normed space and Mbe the family of all nonempty closed bounded
+convex subsets of Xequipped withtheHausdorff metric dgivenbyd(U,V) =
+max{e(U,V),e(V,U)}, whereU,V∈Mande(U,V) = supu∈Uinfv∈V/⌊ard⌊lu−v/⌊ard⌊l.
+GivenU,V∈M, defining U⊕Vas the closure in Xof the Minkowski sum
+{u+v:u∈U,v∈V}we find that the triple ( M,d,⊕) is a metric semigroup.
+Givenf:Ib
+a→(M,d,+), we define the Vitali-type n-th mixed ‘difference’
+offon a subrectangle Iy
+x⊂Ib
+a, wherex,y∈Ib
+aandx < y, by (cf. [14])
+mdn(f,Iy
+x) =d/parenleftBig/summationdisplay
+θ∈E(n)f/parenleftbig
+x+θ(y−x)/parenrightbig
+,/summationdisplay
+η∈O(n)f/parenleftbig
+x+η(y−x)/parenrightbig/parenrightBig
+.(2.1)
+For example, for the first three dimensions we have: if n= 1, then
+E(1) ={0}andO(1) ={1}, and so, md 1(f,Iy
+x) =d(f(x),f(y)); ifn= 2,
+thenE(2) ={(0,0),(1,1)}andO(2) ={(0,1),(1,0)}, and so,
+md2(f,Iy
+x) =d/parenleftbig
+f(x1,x2)+f(y1,y2),f(x1,y2)+f(y1,x2)/parenrightbig
+;
+ifn= 3, then E(3)={(0,0,0),(1,1,0),(1,0,1),(0,1,1)}andO(3)={(1,1,1),
+(1,0,0),(0,1,0),(0,0,1)}, and so,
+md3(f,Iy
+x) =d/parenleftBig
+f(x1,x2,x3)+f(y1,y2,x3)+f(y1,x2,y3)+f(x1,y2,y3),
+f(y1,y2,y3)+f(y1,x2,x3)+f(x1,y2,x3)+f(x1,x2,y3)/parenrightBig
+(one may draw corresponding pictures to see the points where fis evaluated
+at the left and right hand places of d(‘left’,‘right’)).
+Remark 2.1. Formally, the value md n(f,Iy
+x) from (2.1) is defined for x < y.
+Now ifx,y∈Ib
+a,x≤yandx/\e}atio\slash< y, then the right-hand side in (2.1) is equal
+to zero for any map f:Ib
+a→M. In fact, if xi=yifor some i∈{1,...,n}, then
+/summationdisplay
+θ∈E(n)f(x+θ(y−x)) =/summationdisplay
+θ∈O(n)f(x+θ(y−x)).
+In order to see this, given θ= (θ1,...,θ n)∈ E(n), we set θ= (θ1,...,θn) =
+(θ1,...,θ i−1,1−θi,θi+1,...,θ n) and note that θ∈ O(n) and, moreover, the
+mapθ/mapsto→θis a bijection between E(n) andO(n). It remains to take into
+account that x+θ(y−x) =x+θ(y−x) for allθ∈ E(n), because
+xi+θi(yi−xi) =xi=xi+(1−θi)(yi−xi) =xi+θi(yi−xi).
+4TheVitali-type n-th variation ([17, 32, 38]) of f:Ib
+a→Mis defined by
+Vn(f,Ib
+a) = sup
+P/summationdisplay
+1≤σ≤κmdn/parenleftbig
+f,Ix[σ]
+x[σ−1]/parenrightbig
+, (2.2)
+the supremum being taken over all multiindices κ∈Nnand allnet partitions
+ofIb
+aof the form P={x[σ]}κ
+σ=0, where points x[σ] = (x1(σ1),...,x n(σn))
+fromIb
+aare indexed by σ= (σ1,...,σ n)∈Nn
+0withσ≤κand satisfy the
+conditions: x[0] =a,x[κ] =bandx[σ−1]< x[σ] for all 1 ≤σ≤κ(in other
+words, a net partition Pis the Cartesian product of ordinary partitions of
+closed intervals [ ai,bi],i= 1,...,n). Note that all rectangles Ix[σ]
+x[σ−1]of a net
+partition are non-degenerated, non-overlapping and their union is Ib
+a.
+In order to define the notion of the total variation of a map f:Ib
+a→M
+we need the notion of variation of fof order less than n. Following [17], we
+define the truncation of a point x∈Rnby a multiindex 0/\e}atio\slash=α≤1 byx⌊α=
+(xi:i∈ {1,...,n}, αi= 1), and set Ib
+a⌊α=Ib⌊α
+a⌊α. Clearly, x⌊1 =xand
+Ib
+a⌊1 =Ib
+a, and ifx∈Ib
+a, thenx⌊α∈Ib
+a⌊α. For example, if x= (x1,x2,x3,x4)
+andα= (1,0,0,1), we have x⌊α= (x1,x4) andIb
+a⌊α= [a1,b1]×[a4,b4].
+Givenf:Ib
+a→Mandz∈Ib
+a, we define the truncated map fz
+α:Ib
+a⌊α→M
+with the base at zbyfz
+α(x⌊α) =f/parenleftbig
+z+α(x−z)/parenrightbig
+for allx∈Ib
+a. It follows that
+fz
+αdepends only on |α|variables xi∈[ai,bi], for which αi= 1, and the other
+variables remain fixed and equal to zjwhenαj= 0. In the above example
+we getfz
+α(x1,x4) =fz
+α(x⌊α) =f(x1,z2,z3,x4) for (x1,x4)∈[a1,b1]×[a4,b4].
+Now, given f:Ib
+a→Mand 0/\e}atio\slash=α≤1, the function fa
+α:Ib
+a⌊α→Mwith
+the base at z=adepends only on |α|variables, and so, making use of the
+definitions (2.2) and (2.1) with nreplaced by |α|,freplaced by fa
+αandIb
+a
+replaced by Ib
+a⌊α, we get the notion of the ( Hardy-Krause-type [1, 22, 26])
+|α|-th variation off, which is denoted by V|α|(fa
+α,Ib
+a⌊α).
+Thetotal variation off:Ib
+a→Min the sense of Hildebrandt ([13, 16],
+[28, III.6.3], [29] if n= 2) and Leonov ([12, 14, 17, 32] if n∈N) is defined by
+TV(f,Ib
+a) =/summationdisplay
+0/negationslash=α≤1V|α|(fa
+α,Ib
+a⌊α)≡n/summationdisplay
+i=1/summationdisplay
+α≤1,|α|=iVi(fa
+α,Ib
+a⌊α),(2.3)
+the summations here and throughout the paper being taken over n-dimen-
+sionalmultiindices in the ranges specified under the summation sign.
+5For the first three dimensions n= 1,2,3 we have, respectively,
+TV(f,Ib
+a) =Vb
+a(f),the usual Jordan variation on the interval [ a,b],
+TV(f,Ib
+a) =Vb1
+a1/parenleftbig
+f(·,a2)/parenrightbig
++Vb2
+a2/parenleftbig
+f(a1,·)/parenrightbig
++V2(f,Ib1,b2
+a1,a2),
+TV(f,Ib
+a) =Vb1
+a1/parenleftbig
+f(·,a2,a3)/parenrightbig
++Vb2
+a2/parenleftbig
+f(a1,·,a3)/parenrightbig
++Vb3
+a3/parenleftbig
+f(a1,a2,·)/parenrightbig
++V2/parenleftbig
+f(·,·,a3),Ib1,b2
+a1,a2/parenrightbig
++V2/parenleftbig
+f(·,a2,·),Ib1,b3
+a1,a3/parenrightbig
++V2/parenleftbig
+f(a1,·,·),Ib2,b3
+a2,a3/parenrightbig
++V3(f,Ib1,b2,b3
+a1,a2,a3).
+We denote by BV( Ib
+a;M) the space of all maps f:Ib
+a→Moffinite(or
+bounded) total variation (2.3).
+Recall that a sequence {fj} ≡ {fj}j∈Nof maps from Ib
+aintoMis said:
+(a) toconverge pointwise onIb
+ato a map f:Ib
+a→Mifd(fj(x),f(x))→0
+asj→ ∞for allx∈Ib
+a; (b) to be pointwise precompact (onIb
+a) provided the
+closure in Mof the set {fj(x)}j∈Nis compact for all x∈Ib
+a.
+Ourfirst mainresult, tobeproved inSection4, isthefollowing Helly-ty pe
+pointwise selection principle in the space BV( Ib
+a;M):
+Theorem 1. A pointwise precompact sequence {fj}of maps from the rect-
+angleIb
+ainto a metric semigroup (M,d,+)such that
+C≡sup
+j∈NTV(fj,Ib
+a)is finite (2.4)
+containsa subsequencewhichconvergespointwiseon Ib
+ato a map f∈BV(Ib
+a;M)
+such that TV(f,Ib
+a)≤C.
+This result was announced in [14]. It contains as particular cases the
+results of [28, III.6.5] and [29] ( n= 2 andM=R), [32] (n∈NandM=R)
+and [5] (n= 2 and Mis a metric semigroup).
+Our second main result (Theorem 2 below) is concerned with a weak
+analogue of Theorem 1 taking into account certain specific feature s when the
+values of maps under consideration lie in a reflexive separable Banach space.
+Let (M,/⌊ard⌊l·/⌊ard⌊l) be a normed linear space over the field K=RorCandM∗
+be itsdual, i.e.,M∗= L(M;K), the space of all continuous linear functionals
+onM. It is well-known that M∗is a Banach space under the norm /⌊ard⌊lu∗/⌊ard⌊l=
+sup{|u∗(u)|:u∈Mand/⌊ard⌊lu/⌊ard⌊l ≤1},u∗∈M∗. The natural duality between
+MandM∗is determined by the bilinear functional /a\}⌊racketle{t·,·/a\}⌊racketri}ht:M×M∗→K
+defined by /a\}⌊racketle{tu,u∗/a\}⌊racketri}ht=u∗(u) for allu∈Mandu∗∈M∗, so that |/a\}⌊racketle{tu,u∗/a\}⌊racketri}ht| ≤
+6/⌊ard⌊lu/⌊ard⌊l · /⌊ard⌊lu∗/⌊ard⌊l, where| · |is the absolute value in K. Recall that a sequence
+{uj} ⊂Mconverges weakly in Mto an element u∈M(in symbols, ujw→u
+inM) if/a\}⌊racketle{tuj,u∗/a\}⌊racketri}ht → /a\}⌊racketle{tu,u∗/a\}⌊racketri}htinKasj→ ∞for allu∗∈M∗; if this is the case
+then it is known that /⌊ard⌊lu/⌊ard⌊l ≤liminf j→∞/⌊ard⌊luj/⌊ard⌊l.
+Since a normed linear space ( M,/⌊ard⌊l·/⌊ard⌊l) is a metric semigroup, the notions
+of the Vitali-type n-th variation, |α|-th variation for 0 /\e}atio\slash=α≤1 and the total
+variation of a map f:Ib
+a→Mare introduced as above with respect to the
+induced metric d(u,v) =/⌊ard⌊lu−v/⌊ard⌊l,u,v∈M.
+Theorem 2. Suppose (M,/⌊ard⌊l · /⌊ard⌊l)is a reflexive separable Banach space with
+separable dual M∗and{fj}is a sequence of maps from Ib
+aintoM. If{fj}
+satisfies condition (2.4)from Theorem 1and
+c(x)≡sup
+j∈N/⌊ard⌊lfj(x)/⌊ard⌊lis finite for all x∈Ib
+a, (2.5)
+then there exists a subsequence of {fj}, again denoted by {fj}, and a map
+f∈BV(Ib
+a;M)satisfying TV(f,Ib
+a)≤Csuch that
+fj(x)w→f(x)inMfor allx∈Ib
+a. (2.6)
+This theorem will be proved in Section 4. It is an extension of a weak
+selection principle from[6, Chapter 1, Theorem 3.5]formaps ofboun dedJor-
+dan variationof one real variable. More comments and remarks on T heorems
+1 and 2 can be found in Section 4.
+3. Properties of mixed differences and the total variation
+In this section we collect main ingredients of the proof of Theorem 1.
+These are relations between mixed differences of all orders and pro perties of
+the total variation (2.3). For real valued functions of nvariables the main
+properties of mixed differences of all orders were elaborated in [1, 1 1, 17,
+22, 28, 32, 38] and for metric semigroup valued maps of two variable s—in
+[5, 13, 16, 35]. For our purposes we need their variants in the multiind ex
+notation, as presented in [17] with M=R, for maps of nvariables with
+values in a metric semigroup.
+First, we recall several definitions and results for real valued fun ctions.
+A function g:Ib
+a→Ris said to be totally monotone (cf., e.g., [17, Part II,
+Section 3], [32]) if, given 0 /\e}atio\slash=α≤1 andx,y∈Ib
+awithx≤y, we have:
+(−1)|α|/summationdisplay
+0≤θ≤α(−1)|θ|g/parenleftbig
+x+θ(y−x)/parenrightbig
+≥0. (3.1)
+7Forrealvaluedfunctionsthesumin(3.1)(withnofactor( −1)|α|)iscalledthe
+|α|-th mixed difference (inthesense of Vitali, Hardy andKrause) of gx
+αonthe
+rectangle Iy
+x⌊αand denoted by md |α|(gx
+α,Iy
+x⌊α) (however, note the difference
+with (3.4) in the general case). In this case the Vitalin-th variation Vn(g,Ib
+a)
+ofgonIb
+ais defined as in (2.2) with the mixed difference at the right-hand
+side of (2.2) replaced by/vextendsingle/vextendsinglemdn(g,Ix[σ]
+x[σ−1])/vextendsingle/vextendsingle. The other definitions related to
+the bounded variation context remain the same as above, and so, w e keep
+the same notation for real valued functions as well.
+DenotebyMon( Ib
+a;R)theset ofalltotallymonotonerealvaluedfunctions
+onIb
+a. It is known (e.g., from the references above) that if g∈Mon(Ib
+a;R),
+theng∈BV(Ib
+a;R), the value at the left-hand side of (3.1) is equal to
+V|α|(gx
+α,Iy
+x⌊α),g(x)≤g(y) and TV( g,Iy
+x) =g(y)−g(x) for all x,y∈Ib
+a
+withx≤y.
+The following Helly selection principle in the class Mon( Ib
+a;R) is due to
+Leonov [32, Lemma 3] (for totally monotone functions of two variab les it was
+established in [28, III.6.5] and [29, Theorem 3.1]):
+Theorem A. An infinite uniformly bounded family of totally monotone fun c-
+tions onIb
+acontains a sequence, which converges pointwise on Ib
+ato a function
+fromMon(Ib
+a;R).
+It was shown in [32, Corollary 2] that the linear space BV( Ib
+a;R) equipped
+with the norm /⌊ard⌊lg/⌊ard⌊l=|g(a)|+ TV(g,Ib
+a),g∈BV(Ib
+a;R), is a Banach space.
+This assertion was refined in [17, Part I, Theorem 1]: the space BV( Ib
+a;R) is
+a Banach algebra with respect to the norm /⌊ard⌊l·/⌊ard⌊l, and/⌊ard⌊lg·g′/⌊ard⌊l ≤2n/⌊ard⌊lg/⌊ard⌊l·/⌊ard⌊lg′/⌊ard⌊l
+for allg,g′∈BV(Ib
+a;R).
+Theorem A implies Helly’s selection principle in the space BV( Ib
+a;R) [32,
+Theorem 4]: an infinite family of functions from BV(Ib
+a;R), which is bounded
+under the norm /⌊ard⌊l·/⌊ard⌊l, contains a pointwise convergent sequence, whose point-
+wise limit belongs to BV(Ib
+a;R). The crucial observation in the proof of
+this result is that, given g∈BV(Ib
+a;R), if we set νg(x) = TV( g,Ix
+a) and
+πg(x) =νg(x)−g(x),x∈Ib
+a, then ([32, Theorem 3]) νg,πg∈Mon(Ib
+a;R),
+and Jordan’s decomposition holds: g=νg−πgonIb
+a; then Theorem A ap-
+plies to the uniformly bounded families of functions {νg}and{πg}in the
+standard way.
+Now let us consider the case of maps f:Ib
+a→Mof finite total variation
+valued in a metric semigroup ( M,d,+). Clearly, there is no counterpart of
+Jordan’s decomposition for these maps, and so, in order to prove T heorem 1,
+8we ought to argue in a completely different way. It will be seen later th at,
+along with Theorem A, the following four Theorems B through E are th e
+main ingredients in the proof of Theorem 1 (in a certain sense replacin g the
+arguments involving Jordan’s decomposition).
+Theorem B. Iff∈BV(Ib
+a;M),x,y∈Ib
+aandx≤y, then
+d(f(x),f(y))≤/summationdisplay
+0/negationslash=α≤1md|α|(fx
+α,Iy
+x⌊α)≤TV(f,Iy
+x).
+This theorem will be proved in Section 5. It is a generalization of the
+well-known property of maps of bounded Jordan variation of one va riable
+and a counterpart of Leonov’s (in)equalities established in [32, Theo rem 2
+and Corollary 5] for real valued functions of nvariables (cf. also [17, Part I,
+Lemma 6 and (3.5)]). The inequalities in Theorem B are also known for
+metric semigroup valued maps of two variables [5, 16]. However, in the
+general case Theorem B needs a different proof as compared to th e cases of
+maps of one or two variable(s) or M=R.
+Theorem C. Iff∈BV(Ib
+a;M),x,y∈Ib
+a,x≤y, and0/\e}atio\slash=γ≤1, then
+/summationdisplay
+0/negationslash=α≤γV|α|(fx
+α,Iy
+x⌊α) = TV/parenleftbig
+f,Ix+γ(y−x)
+x/parenrightbig
+≤TV(f,Ix+γ(y−x)
+a)−TV(f,Ix
+a).(3.2)
+Theorem D. Iff∈BV(Ib
+a;M)and if we set νf(x) = TV( f,Ix
+a),x∈Ib
+a,
+then for νf:Ib
+a→R, called the total variation function of f, we have :
+νf∈Mon(Ib
+a;R)andTV(νf,Ib
+a) = TV(f,Ib
+a).
+These two theorems are extensions of two more properties of the Jordan
+variation for maps of one variable; in this case (3.2) is actually the equ ality
+known as the additivity of Jordan’s variation (e.g., [37, Theorem 83 2]). On
+the other hand, Theorem C is a counterpart of Chistyakov’s inequa lity [17,
+Part II, Lemma 8] and Theorem D is a generalization of Theorem 3 fro m [32]
+and Corollary 11 from [17, Part II] given for M=R. For metric semigroup
+valued maps of two variables cf. [5, inequalities (11), (13) and Theor em 1].
+The proof of Theorem C is identical with the proof of Lemma 8 from [17 ,
+Part II] and the proof of Theorem D is identical with the proofs of L emma 9
+and Corollaries 10 and 11 from [17, Part II] when M=R, and so, they are
+9omitted. However, itistobenotedthatthese proofsrelyon(1)eq uality (3.2)
+from [17, Part I, Lemma 5], (2) Lemma 7 from [17, Part I], and (3) th e well-
+known property of the additivity of|α|-th variation V|α|for each 0 /\e}atio\slash=α≤1
+for real valued functions of nvariables. For metric semigroup valued maps
+assertions (1), (2) and (3) need a proper interpretation and diffe rent, more
+subtle and hard proofs. Their respective counterparts are pres ented below
+as Lemmas 1, 2 and 3.
+In the first lemma and throughout the paper we use the following sho rt
+notations: given 0 /\e}atio\slash=α≤1, the sum over ‘ev θ≤α’ denotes the sum over
+‘θ∈ E(n) s.t.θ≤α’, where ‘s.t.’ is the usual abbreviation for ‘such that’,
+and a similar convention applies to the sum over ‘od θ≤α’.
+Lemma 1. Iff:Ib
+a→M,x,y∈Ib
+a,x≤y,z∈Ib
+aand0/\e}atio\slash=α≤1, then
+md|α|(fz
+α,Iy
+x⌊α) =d/parenleftbigg/summationdisplay
+evθ≤αf/parenleftbig
+z+α(x−z)+θ(y−x)/parenrightbig
+,
+/summationdisplay
+odθ≤αf/parenleftbig
+z+α(x−z)+θ(y−x)/parenrightbig/parenrightbigg
+.(3.3)
+In particular, if z=aorz=x, we have, respectively,
+md|α|(fa
+α,Iy
+x⌊α) = md |α|(fa+α(x−a)
+α,Iy
+a+α(x−a)⌊α),
+md|α|(fx
+α,Iy
+x⌊α) =d/parenleftbigg/summationdisplay
+evθ≤αf/parenleftbig
+x+θ(y−x)/parenrightbig
+,/summationdisplay
+odθ≤αf/parenleftbig
+x+θ(y−x)/parenrightbig/parenrightbigg
+.(3.4)
+The proof of Lemma 1 is the same as in [17, Part I, Lemma 5] (details
+are omitted): we have to note only that θ′∈N|α|
+0and|θ′|is even (odd) if and
+only if there exists a unique θ∈Nn
+0s.t.θ≤α,|θ|is even (odd, respectively)
+andθ′=θ⌊α, and apply definition (2.1) where nis replaced by |α|.
+Since the total variation (2.3) is defined via truncated maps with the base
+at the point a, in our next lemma we present a counterpart of Chistyakov’s
+equality [17, Part I, Lemma 7] exhibiting the relation between the mixe d
+difference md |α|(fx
+α,Iy
+x⌊α) and certain mixed differences of maps fa
+βwith the
+base atafor some 0 /\e}atio\slash=β≤1.
+Lemma 2. Iff:Ib
+a→M,0/\e}atio\slash=α≤1andx,y∈Ib
+awithx≤y, then
+md|α|(fx
+α,Iy
+x⌊α)≤/summationdisplay
+α≤β≤1md|β|/parenleftbig
+fa
+β,Ix+α(y−x)
+a+α(x−a)⌊β/parenrightbig
+.
+10The proof of Lemma 2 is postponed until Section 6.
+Theadditivity property of |α|-th variation V|α|for each 0 /\e}atio\slash=α≤1, to be
+proved in Section 7, is expressed in the following
+Lemma 3. Givenf:Ib
+a→M,x,y∈Ib
+awithx < y,z∈Ib
+aand0/\e}atio\slash=α≤1,
+if{x[σ]}κ
+σ=0is a net partition of Iy
+x, then
+V|α|(fz
+α,Iy
+x⌊α) =/summationdisplay
+1⌊α≤σ⌊α≤κ⌊αV|α|(fz
+α,Ix[σ]
+x[σ−1]⌊α), (3.5)
+where the summation is taken only over those σiin the range 1≤σi≤κi
+withi∈ {1,...,n}, for which αi= 1.
+The final ingredient in the proof of Theorem 1 is the sequential lower
+semicontinuity of the total variation TV( ·,Ib
+a) to be established in Section 8:
+Theorem E. If a sequence of maps {fj}fromIb
+aintoMconverges pointwise
+onIb
+ato a map f:Ib
+a→M, thenTV(f,Ib
+a)≤liminf j→∞TV(fj,Ib
+a).
+Now we are in a position to prove Theorems 1 and 2.
+4. Proofs of Theorems 1 and 2
+Proof of Theorem 1. We divide the proof into four steps for clarity.
+1. We apply the induction argument on the dimension nof the basic
+rectangle Ib
+a⊂Rn. Forn= 1 Theorem 1 was established in[10, Theorem 5.1]
+(andrefined in [7, Theorem 1] and [15, Theorem 1.3]) in thecase when ( M,d)
+is an arbitrary metric space, and for n= 2 it was proved in [5, Theorem 2].
+Now, suppose that n≥3 and Theorem 1 is already established for domain
+rectangles of dimension ≤n−1.
+Givenj∈N, we letνjbe the total variation function of fjonIb
+a, i.e.,
+νj(x) = TV(fj,Ix
+a) for allx∈Ib
+a. By Theorem D and condition (2.4), the
+sequence {νj} ⊂Mon(Ib
+a;R) is uniformly bounded (by C), and so, by Theo-
+rem A, there exist a subsequence of {νj}and the corresponding subsequence
+of{fj}, again denoted as the whole sequences {νj}and{fj}, respectively,
+and a function ν∈Mon(Ib
+a;R) s.t.
+lim
+j→∞νj(x) =ν(x) for all x∈Ib
+a. (4.1)
+It is known ([4], [28, III.5.4], [40]) that the set of discontinuity points o f
+any totally monotone function on Ib
+a⊂Rnlies on at most a countable set
+11of hyperplanes of dimension n−1 parallel to the coordinate axes. Given
+i∈ {1,...,n}, denote by Zithe union of the set of all rational points of
+the interval [ ai,bi], the two-point set {ai,bi}and the set of those points
+zi∈[ai,bi], for which the hyperplane
+Hi(zi) = [a1,b1]×···×[ai−1,bi−1]×{zi}×[ai+1,bi+1]×···×[an,bn] (4.2)
+contains points of discontinuity of ν. Clearly, the sets Zi⊂[ai,bi] are count-
+able and dense in [ ai,bi], and so, we may assume that Zi={zi(k)}∞
+k=1.
+2. In order to apply the induction hypothesis, we need an estimate o n the
+(n−1)-dimensional total variation of any function f=fjfrom the sequence
+{fj}‘over the hyperplane’ (4.2) in the sense to be made precise below. Th is
+is done as follows.
+Let us fix i∈ {1,...,n}and set 1i= (1,...,1,0,1,...,1), where 0 is the
+i-th coordinate of 1iand the other coordinates of 1iare equal to 1. Note that
+|1i|=n−1. Given zi∈Zi, we put
+a≡a(zi) = (a1,...,a i−1,zi,ai+1,...,a n). (4.3)
+The map fa
+1i:Ib
+a⌊1i→Mwith the base at a, truncated by 1i, is defined on
+the (n−1)-dimensional rectangle Ib
+a⌊1i⊂Rn−1and given by: if x∈Ib
+a, then
+x⌊1i∈Ib
+a⌊1iand
+fa
+1i(x⌊1i) =f(a+1i(x−a)) =f(x1,...,x i−1,zi,xi+1,...,x n).(4.4)
+The (n−1)-dimensional total variation of fa
+1ionIb
+a⌊1iis equal to
+TVn−1(fa
+1i,Ib
+a⌊1i) =/summationdisplay
+0/negationslash=α≤1V|α|/parenleftbig
+(fa
+1i)a⌊1i
+α,(Ib
+a⌊1i)⌊α/parenrightbig
+, (4.5)
+where the summation is taken over ( n−1)-dimensional multiindices α=
+(α1,...,α n−1) s.t. 0 n−1/\e}atio\slash=α≤1n−1, i.e.,α∈ A(n−1) (this is the only
+instance and exception when the summation is over ( n−1)-dimensional mul-
+tiindices). Given α∈ A(n−1), we set α= (α1,...,α i−1,0,αi,...,α n−1),
+where 0 occupies the i-th place, and note that α=α⌊1i. We have
+(fa
+1i)a⌊1i
+α=fa
+αon (Ib
+a⌊1i)⌊α=Ib
+a⌊α=Ib
+a⌊α.
+12In fact, given x∈Ib
+a, we find x⌊α= (x⌊1i)⌊αand
+(fa
+1i)a⌊1i
+α(x⌊α) = (fa
+1i)a⌊1i
+α⌊1i((x⌊1i)⌊α)
+=fa
+1i/parenleftbig
+(a⌊1i)+(α⌊1i)[(x⌊1i)−(a⌊1i)]/parenrightbig
+=fa
+1i/parenleftbig
+[a+α(x−a)]⌊1i/parenrightbig
+=f/parenleftbig
+a+1i[a+α(x−a)−a]/parenrightbig
+. (4.6)
+Sincea+1i(a−a) =aand 1iα=α, we get
+a+1i[a+α(x−a)−a] =a+1i(a−a)+1iα(x−a)
+=a+α(x−a) =a+α(x−a),
+and so, the value (4.6) is equal to
+f(a+α(x−a)) =fa
+α(x⌊α).
+It follows that the |α|-th variation at the right-hand side of (4.5) is equal to
+V|α|/parenleftbig
+(fa
+1i)a⌊1i
+α,(Ib
+a⌊1i)⌊α/parenrightbig
+=V|α|(fa
+α,Ib
+a⌊α).
+Noting that the set A(n−1) is bijective to the set of those α∈ A(n), for
+which 0/\e}atio\slash=α≤1i, and applying Theorem C with x=a,y=bandγ= 1i,
+we get:
+TVn−1(fa
+1i,Ib
+a⌊1i) =/summationdisplay
+0/negationslash=α≤1iV|α|(fa
+α,Ib
+a⌊α) = TV(f,Ia+1i(b−a)
+a)
+≤TV(f,Ia+1i(b−a)
+a)−TV(f,Ia
+a)≤TV(f,Ib
+a).(4.7)
+Thus, given j∈Nandi∈ {1,...,n}, setting back f=fj, by virtue of (4.3),
+(4.7) and (2.4), we find, for all zi∈Zianda=a(zi):
+TVn−1/parenleftbig
+(fj)a(zi)
+1i,Ib
+a⌊1i/parenrightbig
+≤C <∞. (4.8)
+3. Now, we make use of the diagonal processes. For i= 1 and z1=
+zi(1) =z1(1)∈Z1the sequence {(fj)a(zi(1))
+1i}∞
+j=1={(fj)a(z1(1))
+11}∞
+j=1satisfies
+theuniformestimate(4.8)ontherectangle Ib
+a⌊11ofdimension n−1and, since
+each map from this sequence is of the form (4.4) with zi=z1=z1(1), then it
+follows from the assumptions of Theorem 1 that the sequence unde r consid-
+eration is pointwise precompact on Ib
+a⌊11. By the induction hypothesis, the
+13sequence {fj}contains a subsequence, denoted by {f1
+j}, s.t. (f1
+j)a(z1(1))
+11con-
+verges pointwise on Ib
+a⌊11to a map from Ib
+a⌊11intoMof (n−1)-dimensional
+finite total variation on Ib
+a⌊11. Since, by (4.4),
+(f1
+j)a(z1(1))
+11(x2,...,x n) = (f1
+j)a(z1(1))
+11(x⌊11) =f1
+j(z1(1),x2,...,x n)
+withx= (x1,...,x n)∈Ib
+aandxi∈[ai,bi] fori∈ {2,...,n}, then the point-
+wise convergence above means, actually, that the sequence {f1
+j}converges
+pointwise on the hyperplane H1(z1(1)) ={z1(1)}×[a2,b2]×···×[an,bn].
+Inductively, if k≥2 and a subsequence {fk−1
+j}∞
+j=1of{fj}, which is
+pointwise convergent on/uniontextk−1
+l=1H1(z1(l)), is already chosen, then the sequence
+{(fk−1
+j)a(z1(k))
+11}∞
+j=1satisfies the uniform estimate (4.8) on the rectangle Ib
+a⌊11,
+wherefjis replaced by fk−1
+janda(zi)—bya(z1(k)). Moreover, since, as
+above, the sequence is pointwise precompact on Ib
+a⌊11, then, by the induction
+hypothesis, there exists a subsequence {fk
+j}∞
+j=1of{fk−1
+j}∞
+j=1s.t. (fk
+j)a(z1(k))
+11
+converges pointwise on Ib
+a⌊11asj→ ∞to a map from Ib
+a⌊11intoMof
+(n−1)-dimensional finite total variation on Ib
+a⌊11. Again, as above, this
+pointwise convergence means that the sequence {fk
+j}∞
+j=1converges pointwise
+on the hyperplane H1(z1(k)) and, as a consequence, on the set/uniontextk
+l=1H1(z1(l))
+as well. We infer that the diagonal sequence {fj
+j}∞
+j=1, which is a subse-
+quence oftheoriginalsequence {fj}, converges pointwise ontheset H1(Z1) =/uniontext
+z1∈Z1H1(z1) =/uniontext∞
+l=1H1(z1(l)); in fact, given ( z1,x2,...,x n)∈H1(Z1), we
+havez1=z1(k)∈Z1for some k∈Nand (x2,...,x n)∈Ib
+a⌊11, and so, noting
+that{fj
+j}∞
+j=kis a subsequence of {fk
+j}∞
+j=1, we find that
+fj
+j(z1,x2,...,x n) = (fj
+j)a(z1(k))
+11(x2,...,x n)
+converges in Masj→ ∞.
+Let us denote the diagonal sequence {fj
+j}∞
+j=1extracted in the last para-
+graph again by {fj}. Then we let i= 2,z2=zi(1) =z2(1)∈Z2and,
+beginning with the sequence {(fj)a(zi(1))
+1i}∞
+j=1={(fj)a(z2(1))
+12}∞
+j=1, apply the
+above arguments of this step. Doing this, we will end up with a diago-
+nal sequence, a subsequence of the original sequence {fj}, again denoted
+by{fj}, which converges pointwise on H1(Z1)∪H2(Z2). Now suppose that
+for some i∈ {2,...,n−1}we have already extracted a (diagonal) subse-
+quence of {fj}, again denoted by {fj}, which converges pointwise on the set
+H1(Z1)∪···∪Hi−1(Zi−1). Then we let zi=zi(1)∈Ziand apply the above
+14arguments of this step to the sequence {(fj)a(zi(1))
+1i}∞
+j=1: a subsequence of the
+original sequence {fj}converges pointwise on the set H1(Z1)∪···∪Hi(Zi).
+In this way after finitely many steps we obtain a subsequence of the original
+sequence {fj}, again denoted by {fj}, which converges pointwise on the set
+H(Z) =/uniontextn
+i=1Hi(Zi).
+4. Finally, let us show that the sequence {fj}from the end of Step 3
+converges at each point y∈Ib
+a\H(Z). Note that yis a point of continuity
+of the function νfrom (4.1) s.t. its coordinates ai< yi< biare irrational
+for alli∈ {1,...,n}. Due to the density of H(Z) inIb
+a, the continuity of ν
+atyand properties of totally monotone functions, given ε >0, there exists
+x=x(ε)∈H(Z) withx < ys.t. 0≤ν(y)−ν(x)≤ε. By virtue of (4.1),
+choose a number j0(ε)∈Ns.t.|νj(y)−ν(y)| ≤εand|ν(x)−νj(x)| ≤εfor
+allj≥j0(ε). By Theorems B and C with γ= 1, for all j≥j0(ε) we have:
+d(fj(x),fj(y))≤TV(fj,Iy
+x)≤TV(fj,Iy
+a)−TV(fj,Ix
+a) =νj(y)−νj(x)
+≤ |νj(y)−ν(y)|+(ν(y)−ν(x))+|ν(x)−νj(x)| ≤3ε.
+Sincex∈H(Z) and, as it was shown in Step 3, the sequence {fj(x)}∞
+j=1
+is convergent in M, it is Cauchy, and so, there exists a number j1(ε)∈N
+s.t.d(fj(x),fj′(x))≤εfor allj≥j1(ε) andj′≥j1(ε). It follows that if
+J(ε) = max{j0(ε),j1(ε)},j≥J(ε) andj′≥J(ε), then we have:
+d(fj(y),fj′(y))≤d(fj(y),fj(x))+d(fj(x),fj′(x))+d(fj′(x),fj′(y))
+≤3ε+ε+3ε= 7ε.
+Thus, the sequence {fj(y)}∞
+j=1is Cauchy in the metric space M, and so, since
+it is also precompact by the assumption, it is convergent in M.
+It follows from here and the end of Step 3 that the sequence {fj(y)}∞
+j=1
+converges in Mat each point y∈(Ib
+a\H(Z))∪H(Z) =Ib
+a, i.e., the sequence
+{fj}, which is a subsequence of the original sequence {fj}, converges point-
+wise onIb
+a. Let us denote the pointwise limit of {fj}byf:Ib
+a→M. Then,
+by virtue of Theorem E and assumption (2.4), we find
+TV(f,Ib
+a)≤liminf
+j→∞TV(fj,Ib
+a)≤C,
+and so,f∈BV(Ib
+a;M).
+This completes the proof of Theorem 1.
+15Remark 4.1. In Theorem 1 the precompactness of the sets {fj(x)}∞
+j=1at
+all points x∈Ib
+acannot be replaced by the closedness and boundedness even
+at a single point of Ib
+a. The corresponding examples for maps of one variable
+are constructed in [7, Section 3], [10, Section 5] and [15, Section 1] a nd can
+be easily adapted for maps of several variables.
+Proof of Theorem 2. The proof is adapted for the situation under con-
+sideration from the proof of Theorem 7 from [18].
+1. In this step we show that, given j∈Nandu∗∈M∗, we have:
+TV(/a\}⌊racketle{tfj(·),u∗/a\}⌊racketri}ht,Ib
+a)≤TV(fj,Ib
+a)/⌊ard⌊lu∗/⌊ard⌊l ≤C/⌊ard⌊lu∗/⌊ard⌊l, (4.9)
+where the function /a\}⌊racketle{tfj(·),u∗/a\}⌊racketri}ht:Ib
+a→Kis given by /a\}⌊racketle{tfj(·),u∗/a\}⌊racketri}ht(x) =/a\}⌊racketle{tfj(x),u∗/a\}⌊racketri}ht,
+x∈Ib
+a, andCis the constant from (2.4).
+In fact, given 0 /\e}atio\slash=α≤1 andx,y∈Ib
+awithx < y, by virtue of (3.3)
+whered(u,v) is replaced by the absolute value |u−v|inKand later on—by
+the norm in M, we get:
+md|α|(/a\}⌊racketle{tfj(·),u∗/a\}⌊racketri}hta
+α,Iy
+x⌊α) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+0≤θ≤α(−1)|θ|/a\}⌊racketle{tfj(a+α(x−a)+θ(y−x)),u∗/a\}⌊racketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig/summationdisplay
+0≤θ≤α(−1)|θ|fj(a+α(x−a)+θ(y−x)),u∗/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+0≤θ≤α(−1)|θ|fj(a+α(x−a)+θ(y−x))/vextenddouble/vextenddouble/vextenddouble/vextenddouble·/⌊ard⌊lu∗/⌊ard⌊l
+= md |α|((fj)a
+α,Iy
+x⌊α)/⌊ard⌊lu∗/⌊ard⌊l.
+It follows that if P={x[σ]}κ
+σ=0is a net partition of Ib
+a, thenP⌊α=
+{x[σ]⌊α}κ⌊α
+σ⌊α=0is a net partition of Ib
+a⌊α, and so, setting x=x[σ−1] and
+y=x[σ] in the calculations above, we find
+/summationdisplay
+1⌊α≤σ⌊α≤κ⌊αmd|α|/parenleftbig
+/a\}⌊racketle{tfj(·),u∗/a\}⌊racketri}hta
+α,Ix[σ]
+x[σ−1]⌊α/parenrightbig
+≤/summationdisplay
+1⌊α≤σ⌊α≤κ⌊αmd|α|/parenleftbig
+(fj)a
+α,Ix[σ]
+x[σ−1]⌊α/parenrightbig
+/⌊ard⌊lu∗/⌊ard⌊l
+≤V|α|((fj)a
+α,Ib
+a⌊α)/⌊ard⌊lu∗/⌊ard⌊l,
+the summation over σ⌊αbeing taken only over those coordinates σiin the
+range 1≤σi≤κiwithi∈ {1,...,n}, for which αi= 1. Since Pis an
+arbitrary partition of Ib
+a, we get:
+V|α|/parenleftbig
+/a\}⌊racketle{tfj(·),u∗/a\}⌊racketri}hta
+α,Ib
+a⌊α/parenrightbig
+≤V|α|/parenleftbig
+(fj)a
+α,Ib
+a⌊α/parenrightbig
+/⌊ard⌊lu∗/⌊ard⌊l,
+16and so, inequality (4.9) follows from the definition of the total variat ion.
+Moreover, by virtue of (2.5), we have:
+|/a\}⌊racketle{tfj(x),u∗/a\}⌊racketri}ht| ≤ /⌊ard⌊lfj(x)/⌊ard⌊l·/⌊ard⌊lu∗/⌊ard⌊l ≤c(x)/⌊ard⌊lu∗/⌊ard⌊l, x∈Ib
+a, u∗∈M∗,(4.10)
+and so, the sequence {/a\}⌊racketle{tfj(·),u∗/a\}⌊racketri}ht}∞
+j=1of functions from Ib
+ainto (metric semi-
+group)Kis pointwise bounded on Ib
+aand, hence, pointwise precompact for
+eachu∗∈M∗.
+Taking this and (4.9) into account and applying Theorem 1 to the se-
+quence{/a\}⌊racketle{tfj(·),u∗/a\}⌊racketri}ht}∞
+j=1for any given u∗∈M∗, we extract a subsequence of
+{fj}, denotedby {fj,u∗}(whichdepends on u∗ingeneral), andfindafunction
+gu∗∈BV(Ib
+a;K) satisfying TV( gu∗,Ib
+a)≤C/⌊ard⌊lu∗/⌊ard⌊ls.t./a\}⌊racketle{tfj,u∗(x),u∗/a\}⌊racketri}ht →gu∗(x)
+inKasj→ ∞for allx∈Ib
+a.
+2. Making use of the diagonal process and the separability of M∗, let
+us get rid of the dependence of {fj,u∗}onu∗∈M∗. Let{u∗
+k}∞
+k=1be a
+countable dense subset of M∗. By Step 1, for u∗=u∗
+1we get a sub-
+sequence {f(1)
+j}={fj,u∗
+1}∞
+j=1of the original sequence {fj}and a function
+gu∗
+1∈BV(Ib
+a;K) satisfying TV( gu∗
+1,Ib
+a)≤C/⌊ard⌊lu∗
+1/⌊ard⌊ls.t./a\}⌊racketle{tf(1)
+j(x),u∗
+1/a\}⌊racketri}ht →gu∗
+1(x) in
+Kfor allx∈Ib
+a. Inductively, if k≥2 and a subsequence {f(k−1)
+j}∞
+j=1of{fj}
+is already chosen, then by virtue of (4.9) and (4.10), we have:
+TV(/a\}⌊racketle{tf(k−1)
+j(·),u∗
+k/a\}⌊racketri}ht,Ib
+a)≤C/⌊ard⌊lu∗
+k/⌊ard⌊l
+and|/a\}⌊racketle{tf(k−1)
+j(x),u∗
+k/a\}⌊racketri}ht| ≤c(x)/⌊ard⌊lu∗
+k/⌊ard⌊l,x∈Ib
+a, forallj∈N. ByTheorem1, applied
+to the sequence {/a\}⌊racketle{tf(k−1)
+j(·),u∗
+k/a\}⌊racketri}ht}∞
+j=1, there exist a subsequence {f(k)
+j}∞
+j=1of
+{f(k−1)
+j}∞
+j=1and a function gu∗
+k∈BV(Ib
+a;K) satisfying TV( gu∗
+k,Ib
+a)≤C/⌊ard⌊lu∗
+k/⌊ard⌊l
+s.t./a\}⌊racketle{tf(k)
+j(x),u∗
+k/a\}⌊racketri}ht →gu∗
+k(x) inKasj→ ∞for allx∈Ib
+a. Then the diagonal
+sequence {f(j)
+j}∞
+j=1, again denoted by {fj}, is a subsequence of the original
+sequence and satisfies the condition:
+/a\}⌊racketle{tfj(x),u∗
+k/a\}⌊racketri}ht →gu∗
+k(x) asj→ ∞for allx∈Ib
+aandk∈N. (4.11)
+3. Now, given u∗∈M∗andx∈Ib
+a, let us show that the sequence
+{/a\}⌊racketle{tfj(x),u∗/a\}⌊racketri}ht}∞
+j=1is Cauchy in K. Taking into account (4.11) we may assume
+thatu∗/\e}atio\slash=u∗
+kfor allk∈N. Letε >0 be arbitrary. By the density of {u∗
+k}∞
+k=1
+inM∗, there exists k=k(ε)∈Ns.t./⌊ard⌊lu∗−u∗
+k/⌊ard⌊l ≤ε/(1+4c(x)). By (4.11),
+17there exists j0=j0(ε)∈Ns.t.|/a\}⌊racketle{tfj(x),u∗
+k/a\}⌊racketri}ht−/a\}⌊racketle{tfj′(x),u∗
+k/a\}⌊racketri}ht| ≤ε/2 for allj≥j0
+andj′≥j0. It follows that for such jandj′we have:
+|/a\}⌊racketle{tfj(x),u∗/a\}⌊racketri}ht−/a\}⌊racketle{tfj′(x),u∗/a\}⌊racketri}ht| ≤ |/a\}⌊racketle{t fj(x)−fj′(x),u∗−u∗
+k/a\}⌊racketri}ht|
++|/a\}⌊racketle{tfj(x),u∗
+k/a\}⌊racketri}ht−/a\}⌊racketle{tfj′(x),u∗
+k/a\}⌊racketri}ht|
+≤ /⌊ard⌊lfj(x)−fj′(x)/⌊ard⌊l·/⌊ard⌊lu∗−u∗
+k/⌊ard⌊l+ε
+2
+≤2c(x)ε
+1+4c(x)+ε
+2≤ε.
+Thus,{/a\}⌊racketle{tfj(x),u∗/a\}⌊racketri}ht}∞
+j=1is Cauchy in Kand, hence, there exists an element of
+Kdenoted by gu∗(x) s.t./a\}⌊racketle{tfj(x),u∗/a\}⌊racketri}ht →gu∗(x) inKasj→ ∞. In other words,
+we have shown that for each u∗∈M∗there exists a function gu∗:Ib
+a→K
+satisfying (cf. Theorem E and (4.9))
+TV(gu∗,Ib
+a)≤liminf
+j→∞TV(/a\}⌊racketle{tfj(·),u∗/a\}⌊racketri}ht,Ib
+a)≤C/⌊ard⌊lu∗/⌊ard⌊l
+(and so, gu∗∈BV(Ib
+a;K)) and
+lim
+j→∞/a\}⌊racketle{tfj(x),u∗/a\}⌊racketri}ht=gu∗(x) inKfor allx∈Ib
+aandu∗∈M∗. (4.12)
+4. Let us prove (2.6), i.e., fj(x) converges weakly in Masj→ ∞for all
+x∈Ib
+a. By the reflexivity of M, we have fj(x)∈M=M∗∗≡L(M∗;K) for
+allj∈N. Define the functional Gx:M∗→KbyGx(u∗) =gu∗(x) for all
+u∗∈M∗. By virtue of (4.12), we get
+lim
+j→∞/a\}⌊racketle{tfj(x),u∗/a\}⌊racketri}ht=gu∗(x) =Gx(u∗) for all u∗∈M∗,
+i.e., the sequence {fj(x)}∞
+j=1⊂L(M∗;K) converges pointwise on M∗to the
+operator Gx:M∗→K. By the Banach-Steinhaus uniform boundedness
+principle, Gx∈L(M∗;K) =Mand/⌊ard⌊lGx/⌊ard⌊l ≤liminf j→∞/⌊ard⌊lfj(x)/⌊ard⌊l. Setting
+f(x) =Gxfor allx∈Ib
+a, we find that f:Ib
+a→Mand
+lim
+j→∞/a\}⌊racketle{tfj(x),u∗/a\}⌊racketri}ht=Gx(u∗) =/a\}⌊racketle{tGx,u∗/a\}⌊racketri}ht=/a\}⌊racketle{tf(x),u∗/a\}⌊racketri}htinK(4.13)
+for allu∗∈M∗andx∈Ib
+a, and so, fj(x)w→f(x) inMasj→ ∞for all
+x∈Ib
+a, which proves (2.6).
+185. It remains to show that f∈BV(Ib
+a;M) and TV( f,Ib
+a)≤C. By (4.13),
+we have: if x,y∈Ib
+awithx < yand 0/\e}atio\slash=α≤1, then
+/summationdisplay
+0≤θ≤α(−1)|θ|fj/parenleftbig
+a+α(x−a)+θ(y−x)/parenrightbigw→/summationdisplay
+0≤θ≤α(−1)|θ|f/parenleftbig
+a+α(x−a)+θ(y−x)/parenrightbig
+inMasj→ ∞, and so, by virtue of (3.3) and the remarks preceding
+Theorem 2,
+md|α|(fa
+α,Iy
+x⌊α) =/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+0≤θ≤α(−1)|θ|f/parenleftbig
+a+α(x−a)+θ(y−x)/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+≤liminf
+j→∞/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+0≤θ≤α(−1)|θ|fj/parenleftbig
+a+α(x−a)+θ(y−x)/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+= liminf
+j→∞md|α|((fj)a
+α,Iy
+x⌊α). (4.14)
+Arguing as in Step 2 of the proof of Theorem E, making use of the ineq uality
+(4.14), which coincides with (8.2) (see p. 44), and taking into accoun t (2.4),
+we get:
+TV(f,Ib
+a)≤liminf
+j→∞TV(fj,Ib
+a)≤C.
+This completes the proof of Theorem 2.
+Remark 4.2. Note that instead of condition (2.5) in Theorem 2 we may
+assume only that the value c(a) = supj∈N/⌊ard⌊lfj(a)/⌊ard⌊lis finite. In fact, it follows
+from Theorem B and condition (2.4) that, given x∈Ib
+aandj∈N,
+/⌊ard⌊lfj(x)/⌊ard⌊l ≤ /⌊ard⌊lfj(a)/⌊ard⌊l+/⌊ard⌊lfj(x)−fj(a)/⌊ard⌊l ≤c(a)+TV(fj,Ix
+a)≤c(a)+C.
+5. Proof of Theorem B
+In order to prove Theorem B, we need three more Lemmas 4, 5 and 6 .
+The following equality will be needed in Lemma 4 (cf. [17, Part I, equality
+(3.4)]): given two multiindices 0 ≤β≤γ≤1, we have:
+|{α:β≤α≤γand|α|=i}|=Ci−|β|
+|γ|−|β|for all |β| ≤i≤ |γ|,(5.1)
+where|A|denotes the number of elements in the set Aand, given 0 ≤j≤m,
+Cj
+m=/parenleftbigm
+j/parenrightbig
+=m!
+j!(m−j)!is the usual binomial coefficient (with 0! = 1). Also,
+recall (cf. Section 2) that a multiindex αis said to be even (odd) if α∈ E(n)
+(α∈ O(n), respectively).
+19Lemma 4. (a)Givenm∈Nand integer 0≤k≤m−1, we have :
+≤m/2/summationdisplay
+i≥k/2C2i−k
+m−k=≤(m+1)/2/summationdisplay
+i≥(k+1)/2C2i−1−k
+m−k= 2m−k−1,
+where the summations are taken over integeriin the ranges specified.
+(b)Given two multiindices 0≤β≤γ≤1withβ/\e}atio\slash=γ, we have :
+|{evenα:β≤α≤γ}|=|{oddα:β≤α≤γ}|.
+Proof. (a) Since the case m= 1 is clear, we suppose that m≥2. By the
+binomial formula, 0 = (1 −1)m−k=/summationtextm−k
+j=0(−1)jCj
+m−k, which is equal to
+(m−k−1)/2/summationdisplay
+j=0C2j
+m−k−(m−k+1)/2/summationdisplay
+j=1C2j−1
+m−kifmandkare of different evenness,
+and
+(m−k)/2/summationdisplay
+j=0C2j
+m−k−(m−k)/2/summationdisplay
+j=1C2j−1
+m−kifmandkare of the same evenness.
+Ifkis even, we change the summation index j/mapsto→i= (2j+k)/2 in these
+sums and get, respectively:
+0 =(m−1)/2/summationdisplay
+i=k/2C2i−k
+m−k−(m+1)/2/summationdisplay
+i=(k+2)/2C2i−1−k
+m−kifmis odd,
+0 =m/2/summationdisplay
+i=k/2C2i−k
+m−k−m/2/summationdisplay
+i=(k+2)/2C2i−1−k
+m−kifmis even,
+while ifkis odd, we change the summation index j/mapsto→i= (2j+ 1 +k)/2
+in the first sum and j/mapsto→i= (2j−1 +k)/2 in the second sum and get,
+respectively:
+0 =m/2/summationdisplay
+i=(k+1)/2C2i−1−k
+m−k−m/2/summationdisplay
+i=(k+1)/2C2i−k
+m−kifmis even,
+200 =(m+1)/2/summationdisplay
+i=(k+1)/2C2i−1−k
+m−k−(m−1)/2/summationdisplay
+i=(k+1)/2C2i−k
+m−kifmis odd.
+Thesecondequalityin(a)followsfromtheequality(1+1)m−k=/summationtextm−k
+j=0Cj
+m−k.
+Remark. The first equality in Lemma 4(a) can be written also as
+[(m+1)/2]/summationdisplay
+i=[(k+2)/2]C2i−1−k
+m−k=[m/2]/summationdisplay
+i=[(k+1)/2]C2i−k
+m−k, m∈N,0≤k≤m−1,
+where [r] designates the integer part of r∈R. However, we prefer the
+equality in Lemma 4(a) since it is more simple and suggestive.
+(b) By virtue of (5.1), the left-hand side of the equality is equal to
+/vextendsingle/vextendsingle{α:β≤α≤γand|α|= 2ifor allis.t.|β| ≤2i≤ |γ|}/vextendsingle/vextendsingle=≤|γ|/2/summationdisplay
+i≥|β|/2C2i−|β|
+|γ|−|β|,
+and the right-hand side of the equality is equal to
+/vextendsingle/vextendsingle{α:β≤α≤γand|α|= 2i−1 for allis.t.|β| ≤2i−1≤ |γ|}/vextendsingle/vextendsingle
+=≤(|γ|+1)/2/summationdisplay
+i≥(|β|+1)/2C2i−1−|β|
+|γ|−|β|.
+It remains to put m=|γ|andk=|β|, note that k < mand apply the
+equality from the previous assertion (a).
+If (M,d,+) is a metric semigroup, then, by virtue of the triangle inequal-
+ity fordand the translation invariance of metric donM, we have, for all
+u,v,u′,v′∈M:
+d(u,v)≤d(u′,v′)+d(u+u′,v+v′),
+d(u+u′,v+v′)≤d(u,v)+d(u′,v′). (5.2)
+Inequality (5.2) yields that the addition operation ( u,v)/mapsto→u+vis a con-
+tinuous map from M×MintoM. More generally, if uj→u,vj→v,
+u′
+j→u′andv′
+j→v′asj→ ∞(convergence of sequences in M), then
+limj→∞d(uj+vj,u′
+j+v′
+j) =d(u+v,u′+v′).
+21Lemma 5. Ifm∈N,u,v∈M,{uj}m
+j=1,{vj}m
+j=1⊂Mand
+≤m/2/summationdisplay
+i=1u2i+u+≤(m+1)/2/summationdisplay
+i=1v2i−1=≤m/2/summationdisplay
+i=1v2i+v+≤(m+1)/2/summationdisplay
+i=1u2i−1,(5.3)
+then
+d(u,v)≤m/summationdisplay
+j=1d(uj,vj). (5.4)
+Proof. Observe that if u+ℓ1+···+ℓk=v+r1+···+rkfor some k∈N
+and{ℓi,ri}k
+i=1⊂M, thend(u,v)≤/summationtextk
+i=1d(ℓi,ri). In fact, by the translation
+invariance of dand inequality (5.2), we have:
+d(u,v) =d/parenleftbigg
+u+k/summationdisplay
+i=1ℓi,v+k/summationdisplay
+i=1ℓi/parenrightbigg
+=d/parenleftbigg
+v+k/summationdisplay
+i=1ri,v+k/summationdisplay
+i=1ℓi/parenrightbigg
+=d/parenleftbiggk/summationdisplay
+i=1ri,k/summationdisplay
+i=1ℓi/parenrightbigg
+≤k/summationdisplay
+i=1d(ri,ℓi).
+Applying this observation and equality (5.3), we get:
+d(u,v)≤≤m/2/summationdisplay
+i=1d(u2i,v2i)+≤(m+1)/2/summationdisplay
+i=1d(v2i−1,u2i−1) =m/summationdisplay
+j=1d(uj,vj)./square
+Remark 5.1. In particular, (in)equalities (5.3) and (5.4) hold for oddmif
+u+(m−1)/2/summationdisplay
+i=1u2i=(m+1)/2/summationdisplay
+i=1u2i−1andv+(m−1)/2/summationdisplay
+i=1v2i=(m+1)/2/summationdisplay
+i=1v2i−1,(5.5)
+and forevenmif either
+u+m/2/summationdisplay
+i=1u2i=v+m/2/summationdisplay
+i=1u2i−1andm/2/summationdisplay
+i=1v2i=m/2/summationdisplay
+i=1v2i−1,(5.6)
+or
+m/2/summationdisplay
+i=1u2i=v+m/2/summationdisplay
+i=1u2i−1andm/2/summationdisplay
+i=1v2i=u+m/2/summationdisplay
+i=1v2i−1.(5.7)
+22In the next lemma we set A0≡ A0(n) ={θ∈Nn
+0:θ≤1}. Also, we
+stick to the following conventions: ‘ u.= 0’ will mean that uis omitted in the
+formula under consideration (especially in a metric semigroup with no z ero),
+and a sum over the empty set is also omitted in any context (i.e.,/summationtext
+∅.= 0).
+Lemma 6. Given a map h:A0→Mand a multiindex γ∈ A0, we have :
+/summationdisplay
+evα≤γ/summationdisplay
+evθ≤αh(θ) =cγ+/summationdisplay
+odα≤γ/summationdisplay
+evθ≤αh(θ), (5.8)
+wherecγ.= 0ifγis odd, and cγ=h(γ)ifγis even, and
+/summationdisplay
+odα≤γ/summationdisplay
+odθ≤αh(θ) =dγ+/summationdisplay
+evα≤γ/summationdisplay
+odθ≤αh(θ), (5.9)
+wheredγ=h(γ)ifγis odd, and dγ.= 0ifγis even.
+Proof. 0. Denote by L(byR) the set of all ‘admissible’ θ’s at the left
+(right) hand side of the equality under consideration and, given θ∈ L(and
+θ∈ R), byL(θ) (and by R(θ))—the multiplicity of the term h(θ) at the left
+(and right) hand sum(s). Then the equality can be rewritten as
+/summationdisplay
+θ∈LL(θ)h(θ) =/summationdisplay
+θ∈RR(θ)h(θ), (5.10)
+whereL(θ)h(θ) denotes the sum of terms of the form h(θ) takenL(θ) times
+(and likewise for R(θ)h(θ)). In what follows in order to prove (5.10), we show
+thatL=RandL(θ) =R(θ) for allθ∈ L=R.
+We divide the proof into four steps for clarity.
+In the first two steps we let γbe odd (i.e., 0 ≤γ≤1 and|γ|is odd).
+1. Letusestablish(5.8). Wehave L={evenθ:∃evenα≤γs.t.θ≤α},
+i.e.,L={evenθ:θ≤γ}, andR={evenθ:∃oddα≤γs.t.θ≤α}. The
+setsLandRare nonempty (0 ∈ Land 0∈ R) andL=R. In fact, the
+inclusion L ⊃ Ris clear, and so, we let θ∈ L. Sinceθis even,γis odd
+andθ≤γ, there exists i∈ {1,...,n}s.t.θi= 0 and γi= 1. We set
+α= (θ1,...,θ i−1,1,θi+1,...,θ n). It follows that α≤γ,|α|=|θ|+1 is odd
+andθ≤α, and so, θ∈ R.
+Givenθ∈ L=R, we find θ/\e}atio\slash=γ,L(θ) =|{evenα:θ≤α≤γ}|and
+R(θ) =|{oddα:θ≤α≤γ}|. By Lemma 4(b), L(θ) =R(θ), and so, (5.10)
+holds implying (5.8) with cγ.= 0.
+232. Let us prove (5.9). If |γ|= 1, then the equality is immediate: the
+left-hand side is equal to h(γ) =dγ, while the double sum at the right is
+omitted (in fact, even α≤γimpliesα= 0, and so, no odd θs.t.θ≤0
+exists). Now, if |γ|>1, thenL={oddθ:θ≤γ}andR={oddθ:
+∃evenα≤γs.t.θ≤α}∪{γ}(disjoint union), and L=R. Letθ∈ L=R.
+Ifθ/\e}atio\slash=γ, thenL(θ) =|{oddα:θ≤α≤γ}|andR(θ) =|{evenα:
+θ≤α≤γ}|, and so, by Lemma 4(b), L(θ) =R(θ). Now if θ=γ, then
+L(γ) =|{oddα:γ≤α≤γ}|= 1, and since dγ=h(γ), thenR(γ) = 1 as
+well. The conclusion follows as in Step 1.
+Suppose that γis even.
+3. In order to prove (5.8), we first note that if γ= 0, then the double sum
+at the right is omitted and the double sum at the left is equal to h(0) =c0.
+Assume that γ/\e}atio\slash= 0. Then L={evenθ:θ≤γ}andR={γ} ∪{evenθ:
+∃oddα≤γs.t.θ≤α}(disjoint union), and L=R. Letθ∈ L=R. Then
+L(θ) =|{evenα:θ≤α≤γ}|and, in particular, L(γ) = 1. If θ=γ, then,
+sincecγ=h(γ), we have R(γ) = 1, and if θ/\e}atio\slash=γ, thenR(θ) =|{oddα:
+θ≤α≤γ}|, and so, by Lemma 4(b), L(θ) =R(θ).
+4. Finally, we prove (5.9). Since the equality is clear for γ= 0 (i.e.,
+‘empty’ equality), we assume that |γ|>0. We have L={oddθ:θ≤γ},
+R={oddθ:∃evenα≤γs.t.θ≤α}andL=R. Givenθ∈ L=R, we find
+θ/\e}atio\slash=γ,L(θ) =|{oddα:θ≤α≤γ}|andR(θ) =|{evenα:θ≤α≤γ}|,
+and so, by Lemma 4(b), L(θ) =R(θ).
+Now we are in a position to prove Theorem B.
+Proof of Theorem B. It suffices to prove only the first inequality: the
+second one follows from the first inequality, (2.2) and (2.3). Setting u=f(x)
+andv=f(y) and taking into account (3.4), the first inequality in Theorem B
+can be rewritten equivalently as
+d(u,v)≤/summationdisplay
+0/negationslash=α≤1d(u(α),v(α)) =n/summationdisplay
+j=1/summationdisplay
+|α|=jd(u(α),v(α)) (5.11)
+(the sum over |α|=jdesignates the sum over 0 /\e}atio\slash=α≤1 s.t.|α|=j), where,
+givenα,θ∈ A0, we seth(θ) =f(x+θ(y−x)),
+u(α) =/summationdisplay
+evθ≤αh(θ),andv(α) =/summationdisplay
+odθ≤αh(θ) ifα/\e}atio\slash= 0 and v(0).= 0.
+24In order to establish (5.11), given integer 0 ≤j≤n, we also set
+uj=/summationdisplay
+|α|=ju(α) and vj=/summationdisplay
+|α|=jv(α)
+and note that
+u0=u(0) =h(0) =u,v0=v(0).= 0,v=h(1),un=u(1) andvn=v(1).
+Suppose that we have already verified equalities (5.5) if m=nis odd and
+(5.6) ifm=nis even. Applying Lemma 5, we get inequality (5.4), where,
+by virtue of (5.2), we have:
+d(uj,vj) =d/parenleftbigg/summationdisplay
+|α|=ju(α),/summationdisplay
+|α|=jv(α)/parenrightbigg
+≤/summationdisplay
+|α|=jd(u(α),v(α)).
+Now, (5.11) follows if we sum these inequalities over j= 1,...,nand take
+into account (5.4).
+It remains toverify equalities (5.5) and(5.6). Forthis, we applyLemm a 6
+withγ= 1 and note that m=n=|γ|=|1|. Suppose that n=|1|is odd.
+By virtue of (5.8), we have:
+u+(m−1)/2/summationdisplay
+i=1u2i=(n−1)/2/summationdisplay
+i=0u2i=(n−1)/2/summationdisplay
+i=0/summationdisplay
+|α|=2iu(α) =/summationdisplay
+evα≤1u(α)
+=/summationdisplay
+odα≤1u(α) =(n+1)/2/summationdisplay
+i=1/summationdisplay
+|α|=2i−1u(α) =(m+1)/2/summationdisplay
+i=1u2i−1,
+and by virtue of (5.9), we get:
+v+(m−1)/2/summationdisplay
+i=1v2i=h(1)+(n−1)/2/summationdisplay
+i=0v2i=h(1)+(n−1)/2/summationdisplay
+i=0/summationdisplay
+|α|=2iv(α) =h(1)+/summationdisplay
+evα≤1v(α)
+=/summationdisplay
+odα≤1v(α) =(n+1)/2/summationdisplay
+i=1/summationdisplay
+|α|=2i−1v(α) =(m+1)/2/summationdisplay
+i=1v2i−1,
+25which establishes (5.5). Now suppose that n=|1|is even. By (5.8), we get:
+u+m/2/summationdisplay
+i=1u2i=n/2/summationdisplay
+i=0u2i=n/2/summationdisplay
+i=0/summationdisplay
+|α|=2iu(α) =/summationdisplay
+evα≤1u(α)
+=h(1)+/summationdisplay
+odα≤1u(α) =v+n/2/summationdisplay
+i=1/summationdisplay
+|α|=2i−1u(α) =v+m/2/summationdisplay
+i=1u2i−1,
+and by virtue of (5.9), we have:
+m/2/summationdisplay
+i=1v2i=n/2/summationdisplay
+i=0v2i=n/2/summationdisplay
+i=0/summationdisplay
+|α|=2iv(α) =/summationdisplay
+evα≤1v(α)
+=/summationdisplay
+odα≤1v(α) =n/2/summationdisplay
+i=1/summationdisplay
+|α|=2i−1v(α) =m/2/summationdisplay
+i=1v2i−1,
+which establishes (5.6) and completes the proof of Theorem B.
+Remark 5.2. Theleft-handside inequality inTheoremB isofinterest when
+x < y. However, if x≤yandx/\e}atio\slash< y, it can be refined in the following way
+(cf. [17, Part I, Lemma 6]): given x,y∈Ib
+a,x < y, and 0/\e}atio\slash=γ≤1, we have:
+d/parenleftbig
+f(x),f(x+γ(y−x))/parenrightbig
+≤/summationdisplay
+0/negationslash=α≤γmd|α|(fx
+α,Iy
+x⌊α).
+In fact, by Theorem B, we find
+d/parenleftbig
+f(x),f(x+γ(y−x))/parenrightbig
+≤/summationdisplay
+0/negationslash=α≤1md|α|(fx
+α,Ix+γ(y−x)
+x⌊α),
+where, by virtue of (3.4), the mixed difference at the right is equal t o
+d/parenleftbigg/summationdisplay
+evθ≤αf(x+θγ(y−x)),/summationdisplay
+odθ≤αf(x+θγ(y−x))/parenrightbigg
+.(5.12)
+Ifα/\e}atio\slash≤γ, thenαi= 1 and γi= 0 for some i∈ {1,...,n}, and so, arguing
+as in Remark 2.1 we find x+θγ(y−x) =x+θγ(y−x) for all even θwith
+θ≤αimplying that (5.12) is equal to zero. Now if α≤γ, thenθγ=θfor
+anyθ≤α, and so, (5.12) coincides with the right-hand side of (3.4).
+266. Proof of Lemma 2
+In order to prove Lemma 2, we need an auxiliary Lemma 7, which plays
+the same role as Lemma 6 above.
+Lemma 7. Given a map h:A0→Mand a multiindex α∈ A0, we have :
+if1−αis even, then the following two equalities hold :
+/summationdisplay
+evθ≤αh(1−α+θ)+/summationdisplay
+odβ≤1−α/summationdisplay
+evθ≤α+βh(θ) =/summationdisplay
+evβ≤1−α/summationdisplay
+evθ≤α+βh(θ),(6.1)
+/summationdisplay
+odθ≤αh(1−α+θ)+/summationdisplay
+odβ≤1−α/summationdisplay
+odθ≤α+βh(θ) =/summationdisplay
+evβ≤1−α/summationdisplay
+odθ≤α+βh(θ),(6.2)
+and if1−αis odd, then the following two equalities hold :
+/summationdisplay
+1−α≤evθ≤1h(θ)+/summationdisplay
+evβ≤1−α/summationdisplay
+evθ≤α+βh(θ) =/summationdisplay
+odβ≤1−α/summationdisplay
+evθ≤α+βh(θ),(6.3)
+/summationdisplay
+1−α≤odθ≤1h(θ)+/summationdisplay
+evβ≤1−α/summationdisplay
+odθ≤α+βh(θ) =/summationdisplay
+odβ≤1−α/summationdisplay
+odθ≤α+βh(θ).(6.4)
+Proof. As in the proof of Lemma 6, the main idea is to establish equality
+(5.10). We divide the proof into four steps.
+Suppose that 1 −αis even.
+1. Let us prove (6.1). If α= 1, then 1 −α= 0 is even, and equality (6.1)
+is equivalent to the identity/summationtext
+evθ≤1h(θ)+0 =/summationtext
+evθ≤1h(θ). Ifα= 0 and if
+1−α= 1 is even, then (6.1) can be written as
+h(1)+/summationdisplay
+odβ≤1/summationdisplay
+evθ≤βh(θ) =/summationdisplay
+evβ≤1/summationdisplay
+evθ≤βh(θ),
+which was established in (5.8) for even γ= 1. Thus, in what follows we
+assume that α/\e}atio\slash= 0,1, i.e., 0<|α|< n.
+We have L=L1∪ L2, whereL1={1−α+θ′:∃evenθ′≤α}(so
+that 1−α∈ L1) andL2={evenθ:∃oddβ≤1−αs.t.θ≤α+β}(so
+that 0∈ L2), andR={evenθ:∃evenβ≤1−αs.t.θ≤α+β}, i.e.,
+R={evenθ:θ≤1}. We are going to show that L=R. This equality
+followsimmediately fromthedefinition of Randthefollowingtwo assertions:
+θ∈ L1⇐⇒θis even and α∨θ= 1, (6.5)
+θ∈ L2⇐⇒θis even and α∨θ/\e}atio\slash= 1, (6.6)
+27whereα∨θ≡max{α,θ}=α+θ−αθ; in particular, (6.5) and (6.6) imply
+thatL1andL2are disjoint. Let us prove (6.5). If θ∈ L1, thenθ= 1−α+θ′
+for some even θ′≤αand, since 1 −αis even and |θ|=|1−α|+|θ′|, thenθ
+is even,θ≤(1−α)+α= 1 and
+α∨θ=α+(1−α+θ′)−α(1−α+θ′) = 1+θ′−αθ′= 1.
+Conversely, if θisevenand α∨θ= 1, then α+θ−αθ= 1orα+θ= 1+αθ≥1.
+Settingθ′=α+θ−1, we find θ= 1−α+θ′, where|θ′|=|α|+|θ|−n=
+|θ|−|1−α|is even and θ′≤α, and so, θ∈ L1. Now we establish (6.6). If
+θ∈ L2, thenθis even and there exists odd β≤1−αs.t.θ≤α+β, and so,
+α≤α+βandθ≤α+βimplyα∨θ≤α+β. Sinceβis odd, 1 −αis even
+andβ≤1−α, we have |β|<|1−α|=n−|α|. It follows that
+|α∨θ| ≤ |α+β|=|α|+|β|<|α|+(n−|α|) =n,
+and so,α∨θ/\e}atio\slash= 1. Conversely, if θis even and α∨θ/\e}atio\slash= 1, then there exists
+i∈ {1,...,n}s.t.αi= 0 and θi= 0. Setting β= (β1,...,β n) withβi= 0
+andβj= 1−αjifj/\e}atio\slash=i, we find β≤1−α,|β|=|1−α| −1 is odd and
+θ≤α+β, and so, θ∈ L2.
+In order to calculate the values L(θ) andR(θ) forθ∈ L=R, we note
+that, given 0 ≤β≤1−α, we have:
+θ≤α+βis equivalent to (1 −α)θ≤β. (6.7)
+In fact, condition 0 ≤β≤1−αis equivalent to condition αβ= 0:
+0≤β≤1−α⇐⇒β(1−α) =β⇐⇒β−αβ=β⇐⇒αβ= 0,
+and so, if θ≤α+β, then (1−α)θ≤(1−α)(α+β) = (1−α)α+β−αβ=β,
+and if (1 −α)θ≤β, thenθ−αθ≤β, and so, θ≤αθ+β≤α+β.
+Givenθ∈ R, by virtue of (6.7), we find
+R(θ) =|{evenβ:β≤1−αandθ≤α+β}|=|{evenβ: (1−α)θ≤β≤1−α}|.
+Ifθ∈ L1, then there exists a unique even θ′≤αs.t.θ= 1−α+θ′, and so,
+sinceθ /∈ L2, thenL(θ) = 1. At the same time,
+(1−α)θ= (1−α)(1−α+θ′) = (1−α)2+(1−α)θ′= 1−α,
+28and so, by the above, R(θ) = 1 as well. Suppose now that θ∈ L2. Then, by
+(6.6), 1/\e}atio\slash=α∨θ=α+θ−αθ=α+(1−α)θor (1−α)θ/\e}atio\slash= 1−α, and so,
+taking into account (6.7) and Lemma 4(b) we find that
+L(θ) =|{oddβ:β≤1−αandθ≤α+β}|=|{oddβ: (1−α)θ≤β≤1−α}|
+is equal to R(θ).
+In the rest of the proof we exhibit only the essential ingredients an d
+differences.
+2. Let us establish (6.2). If α= 1, we get an identity, and if α= 0 and
+1 = 1−αis even, we get equality (5.9) with even γ= 1, and so, we suppose
+that0<|α|< n. Wehave L=L1∪L2, whereL1={1−α+θ′:∃oddθ′≤α}
+andL2={oddθ:∃oddβ≤1−αs.t.θ≤α+β}, andR={oddθ:
+∃evenβ≤1−αs.t.θ≤α+β}, which, actually, is R={oddθ:θ≤1}.
+We need to verify only that L1andL2are nonempty: the rest of the proof
+of (6.2) (including (6.5) and (6.6)) is the same as in Step 1 where ‘even θ’ is
+replaced by ‘odd θ’.
+Sinceα/\e}atio\slash= 0, there exists i∈ {1,...,n}s.t.αi= 1, and so, if we set
+θ′= (θ′
+1,...,θ′
+n) withθ′
+i= 1 and θ′
+j= 0 ifj/\e}atio\slash=i, then|θ′|= 1 is odd and
+θ′≤α. It follows that 1 −α+θ′∈ L1.
+Sinceα/\e}atio\slash= 1, there exists i∈ {1,...,n}s.t.αi= 0, and so if we set
+β= (β1,...,β n) withβi= 0 and βj= 1−αjifj/\e}atio\slash=i, then|β|=|1−α|−1
+is odd and β≤1−α. Givenk∈ {1,...,n},k/\e}atio\slash=i, setting θ= (θ1,...,θ n)
+withθk= 1 and θj= 0 ifj/\e}atio\slash=k, we find |θ|= 1 is odd and θ≤α+β, and
+so,θ∈ L2.
+Assume now that 1 −αis odd. Note that α/\e}atio\slash= 1.
+3. Let us prove (6.3). If α= 0 and 1 = 1 −αis odd, then (since ev θ=1
+cannot hold in the first sum at the left of (6.3)) equality (6.3) is equiva lent
+to (5.8) with odd γ= 1. Thus, we assume that |α|>0.
+We have L=L1∪ L2, where L1={evenθ: 1−α≤θ≤1}and
+L2={evenθ:∃evenβ≤1−αs.t.θ≤α+β}, andR={evenθ:
+∃oddβ≤1−αs.t.θ≤α+β}, and so, R={evenθ:θ≤1}. We have to
+show that L=R.
+First, weshowthat L1andL2arenonempty. Since α/\e}atio\slash= 0,αi= 1forsome
+i∈ {1,...,n}, and so, setting θ= (θ1,...,θ n) withθi= 1 and θj= 1−αj
+ifj/\e}atio\slash=i, we find that 1 −α≤θ≤1 and|θ|=|1−α|+ 1 is even, whence
+θ∈ L1. Now, since α/\e}atio\slash= 1,αi= 0 for some i∈ {1,...,n}, and if we set
+29β= (β1,...,β n) withβi= 0 and βj= 1−αjifj/\e}atio\slash=i, then we find that
+|β|=|1−α|−1 is even, θ= 0 is even and 0 ≤α+β, and so, 0 ∈ L2.
+Second, we assert that (6.5) and (6.6) hold; this will imply that L1and
+L2are disjoint and L=R. In order to prove (6.5), we let θ∈ L1. Thenθis
+even and 1 −α≤θ≤1, and so,
+α∨θ=α+θ−αθ=α+(1−α)θ=α+(1−α) = 1.
+Conversely, if θis even and α∨θ= 1, then α+θ−αθ= 1, and so, (1 −α)θ=
+1−αimplying 1 −α≤θandθ∈ L1. The proof of (6.6) follows the same
+lines as in Step 1 if ‘odd β’ is replaced by ‘even β’.
+Givenθ∈ R, taking into account (6.7), we have R(θ) =|{oddβ:
+(1−α)θ≤β≤1−α}|. Ifθ∈ L1, thenθ /∈ L2, and so, L(θ) = 1; in
+this case 1 −α≤θ, and so, (1 −α)θ= 1−αandR(θ) = 1. Now if θ∈ L2,
+thenα∨θ/\e}atio\slash= 1, and so, (1 −α)θ/\e}atio\slash= 1−αand, by virtue of Lemma 4(b), the
+valueL(θ) =|{evenβ: (1−α)θ≤β≤1−α}|is equal to R(θ).
+4. Finally, we establish (6.4). If α= 0 and 1 = 1 −αis odd, we get
+equality (5.9) with odd γ= 1. Assume that |α|>0. We have L=L1∪L2,
+whereL1={oddθ: 1−α≤θ≤1}(and so, 1 −α∈ L1) andL2={oddθ:
+∃evenβ≤1−αs.t.θ≤α+β}, andR={oddθ:∃oddβ≤1−αs.t.θ≤
+α+β}, and so, R={oddθ:θ≤1}. ThatL2is nonempty can be seen
+as follows. Since α/\e}atio\slash= 1,αi= 0 for some i∈ {1,...,n}, and so, if we set
+β= (β1,...,β n) withβi= 0 and βj= 1−αjifj/\e}atio\slash=i, thenβ≤1−αand
+|β|=|1−α| −1 is even. Now, since α/\e}atio\slash= 0,αk= 1 for some k/\e}atio\slash=i. If we
+setθ= (θ1,...,θ n) withθk= 1 and θj= 0 ifj/\e}atio\slash=k, then|θ|= 1 is odd and
+θ≤α+β, and so, θ∈ L2. Assertion (6.5) with ‘ θis even’ replaced by ‘ θ
+is odd’ is established as in Step 3, while the proof of (6.6) follows the sa me
+lines as in Step 1 with ‘odd β’ replaced by ‘even β’. It follows that L=R.
+The proof completes with the last paragraph of Step 3.
+Proof of Lemma 2. The inequality (actually, equality) is clear if α= 1,
+and so, we assume that α/\e}atio\slash= 1. The mixed difference at the left-hand side of
+the inequality is given by (3.4), while given α≤β≤1, noting that αβ=α
+and applying equality (3.3) we get the following expression for the mixe d
+difference at the right-hand side (cf. [17, Part I, expression (3.7) ]):
+md|β|/parenleftbig
+fa
+β,Ix+α(y−x)
+a+α(x−a)⌊β/parenrightbig
+=d/parenleftbigg/summationdisplay
+evθ≤βh(θ),/summationdisplay
+odθ≤βh(θ)/parenrightbigg
+,
+30whereh(θ) =f/parenleftbig
+a+ (α∨θ)(x−a) +αθ(y−x)/parenrightbig
+andα∨θ=α+θ−αθ.
+Changing the summation multiindex β/mapsto→β−αin the sum at the right of
+the inequality in Lemma 2, we find that it is equivalent to
+d(u,v)≤/summationdisplay
+0≤β≤1−αd/parenleftbigg/summationdisplay
+evθ≤α+βh(θ),/summationdisplay
+odθ≤α+βh(θ)/parenrightbigg
+,
+where
+u=/summationdisplay
+evθ≤αf(x+θ(y−x)) and v=/summationdisplay
+odθ≤αf(x+θ(y−x)).
+Setting
+u(β) =/summationdisplay
+evθ≤α+βh(θ) and v(β) =/summationdisplay
+odθ≤α+βh(θ) if 0≤β≤1−α,
+the last inequality can be rewritten as
+d(u,v)≤/summationdisplay
+0≤β≤1−αd(u(β),v(β)) =|1−α|/summationdisplay
+j=0/summationdisplay
+|β|=jd(u(β),v(β)).(6.8)
+In order to establish (6.8), we will apply Lemma 5 with m=|1−α|+1 =
+n−|α|+1 and
+uj=/summationdisplay
+|β|=j−1u(β) and vj=/summationdisplay
+|β|=j−1v(β) if 1≤j≤m.
+Suppose that we have already verified equalities (5.5) and (5.7). The n by
+Lemma 5, we get inequality (5.4), where, by virtue of (5.2),
+d(uj,vj) =d/parenleftbigg/summationdisplay
+|β|=j−1u(β),/summationdisplay
+|β|=j−1v(β)/parenrightbigg
+≤/summationdisplay
+|β|=j−1d(u(β),v(β)),1≤j≤m.
+Summing over j= 1,...,mand taking into account (5.4), we arrive at (6.8):
+d(u,v)≤m/summationdisplay
+j=1d(uj,vj)≤|1−α|+1/summationdisplay
+j=1/summationdisplay
+|β|=j−1d(u(β),v(β)).
+31Assume that 1 −αis even; then mis odd. Let us verify the first equality
+in (5.5). For this, we apply equality (6.1) and calculate the first sum at the
+left-hand side of (6.1). Given even θ≤α, we have 1 −α+θ∈ L1(cf.
+Step 1 in the proof of Lemma 7), and so, by (6.5), α∨(1−α+θ) = 1 and
+α(1−α+θ) =θ, so that the definition of h(1−α+θ) implies
+/summationdisplay
+evθ≤αh(1−α+θ) =/summationdisplay
+evθ≤αf(x+θ(y−x)) =u.
+Applying equality (6.1), we get:
+u+(m−1)/2/summationdisplay
+i=1u2i=u+|1−α|/2/summationdisplay
+i=1/summationdisplay
+|β|=2i−1u(β) =u+/summationdisplay
+odβ≤1−αu(β)
+=/summationdisplay
+evβ≤1−αu(β) =|1−α|/2/summationdisplay
+i=0/summationdisplay
+|β|=2iu(β) =|1−α|/2/summationdisplay
+i=0u2i+1
+=(|1−α|+2)/2/summationdisplay
+i=1u2i−1=(m+1)/2/summationdisplay
+i=1u2i−1,
+and the first equality in (5.5) follows. In a similar manner we find that th e
+first sum at the left-hand side of (6.2) is equal to v, and, by virtue of (6.2),
+the calculations above show that the second equality in (5.5) holds as well.
+Now suppose that 1 −αis odd, and so, m(defined above) is even. In
+order to verify the first equality in (5.7), we calculate the first sum a t the
+left-hand side of (6.3). Given even θwith 1−α≤θ≤1, we have (cf. Step 3
+in the proof of Lemma 7) θ∈ L1andα∨θ= 1. Moreover (cf. [17, Part I,
+assertion (3.9)]), there exists a unique θ′∈ A0s.t.θ′≤αandθ= 1−α+θ′
+(defineθ′byθ′=α+θ−1). Since |θ′|=|α|+|θ|−n=|θ| −|1−α|and
+1−αis odd, then θ′is odd, and αθ=α(1−α+θ′) =θ′. It follows that
+h(θ) =f(x+θ′(y−x)). Changing the summation multiindex θ/mapsto→θ′in the
+first sum at the left of (6.3), we get:
+/summationdisplay
+1−α≤evθ≤1h(θ) =/summationdisplay
+odθ′≤αf(x+θ′(y−x)) =v.
+32Applying equality (6.3), we find
+m/2/summationdisplay
+i=1u2i=(|1−α|+1)/2/summationdisplay
+i=1/summationdisplay
+|β|=2i−1u(β) =/summationdisplay
+odβ≤1−αu(β)
+=v+/summationdisplay
+evβ≤1−αu(β) =v+(|1−α|−1)/2/summationdisplay
+i=0/summationdisplay
+|β|=2iu(β) =v+(|1−α|−1)/2/summationdisplay
+i=0u2i+1
+=v+(|1−α|+1)/2/summationdisplay
+i=1u2i−1=v+m/2/summationdisplay
+i=1u2i−1,
+which proves the first equality in (5.7). Similarly, the first sum at the le ft-
+hand side of (6.4) is equal to u, and, by virtue of (6.4), the calculations above
+prove the second equality in (5.7).
+This completes the proof of Lemma 2.
+7. Proof of Lemma 3
+Note that if P={x[σ]}κ
+σ=0is a net partition of Ib
+a, then
+Ib
+a=/uniondisplay
+1≤σ≤κIx[σ]
+x[σ−1]=/uniondisplay
+1≤σ≤κn/productdisplay
+i=1[xi(σi−1),xi(σi)] =n/productdisplay
+i=1/parenleftbiggκi/uniondisplay
+l=1Ixi(l)
+xi(l−1)/parenrightbigg
+(7.1)
+is a union of non-overlapping non-degenerated rectangles Ix[σ]
+x[σ−1]with the
+sides parallel to the coordinate axes. In this section it will be conven ient and
+brief to term the union as in (7.1) also a partition ofIb
+a(by non-overlapping
+non-degenerated rectangles).
+IfP={x[σ]}κ
+σ=0andP′={x′[σ′]}κ′
+σ′=0are two net partitions of Ib
+a, we
+say that P′is arefinement ofPifP ⊂ P′. Also, for the sake of convenience
+we define the n-th prevariation off:Ib
+a→M, corresponding to P, by
+vn(f;P) =/summationdisplay
+1≤σ≤κmdn(f,Ix[σ]
+x[σ−1]).
+It follows that the Vitali-type n-th variation of fis given by Vn(f,Ib
+a) =
+supPvn(f;P), where the supremum is taken over all net partitions PofIb
+a.
+The basic ingredient in the proof of Lemma 3 is the following
+33Lemma 8. Givenf:Ib
+a→M, ifPandP′are two net partitions of Ib
+as.t.
+P ⊂ P′, thenvn(f;P)≤vn(f;P′).
+In order to prove this lemma we need three more Lemmas 9–11. In wh at
+follows we fix a map f:Ib
+a→M.
+Lemma 9. Givenx,y∈Ib
+awithx < yandx′∈Ib
+a, we have the following
+partition of Iy
+x, induced by the point x′:
+Iy
+x=/uniondisplay
+1−ξ≤α≤1Ix′+α(y−x′)
+x+αξ(x′−x), (7.2)
+where the multiindex ξ≡ξ(x,x′,y) = (ξ1,...,ξ n)is given by
+ξi≡ξi(x,x′,y) =/braceleftbigg1ifxi< x′
+i< yi,
+0ifx′
+i≤xiorx′
+i≥yi,i∈ {1,...,n},(7.3)
+and
+mdn(f,Iy
+x)≤/summationdisplay
+1−ξ≤α≤1mdn/parenleftbig
+f,Ix′+α(y−x′)
+x+αξ(x′−x)/parenrightbig
+. (7.4)
+Before we prove Lemma 9, let us establish two of its particular varian ts
+as Lemmas 10 and 11 (note that in Lemma 10 the rectangles in the unio n
+may degenerate).
+Lemma 10. Ifx,y∈Ib
+awithx < yandx′∈Iy
+x, then we have the following
+union of non-overlapping (possibly, degenerated )rectangles
+Iy
+x=/uniondisplay
+0≤α≤1Ix′+α(y−x′)
+x+α(x′−x), (7.5)
+and the following inequality holds :
+mdn(f,Iy
+x)≤/summationdisplay
+0≤α≤1mdn/parenleftbig
+f,Ix′+α(y−x′)
+x+α(x′−x)/parenrightbig
+. (7.6)
+Proof. Sincexi≤x′
+i≤yifor alli∈ {1,...,n}, we have:
+Iyi
+xi= [xi,yi] = [xi,x′
+i]∪[x′
+i,yi] =Ix′
+ixi∪Iyi
+x′
+i=1/uniondisplay
+αi=0Ix′
+i+αi(yi−x′
+i)
+xi+αi(x′
+i−xi),
+34and so (cf. equation (2.5) in [17, Part II]),
+Iy
+x=n/productdisplay
+i=1Iyi
+xi=n/productdisplay
+i=1/parenleftbigg1/uniondisplay
+αi=0Ix′
+i+αi(yi−x′
+i)
+xi+αi(x′
+i−xi)/parenrightbigg
+=/uniondisplay
+0≤α≤1Ix′+α(y−x′)
+x+α(x′−x).
+The mixed difference at the left-hand side of (7.6) is given by (2.1), an d
+again by virtue of (2.1), the mixed difference at the right-hand side o f (7.6)
+is equal to
+mdn(f,Ix′+α(y−x′)
+x+α(x′−x)) =d/parenleftbigg/summationdisplay
+evβ≤1h(α,β),/summationdisplay
+odβ≤1h(α,β)/parenrightbigg
+,
+whereh(α,β) =f(x+(α∨β)(x′−x)+αβ(y−x′)) andα∨β=α+β−αβ.
+Noting that if α=β, thenα∨β=βandαβ=β, we find
+/summationdisplay
+0≤α≤1/summationdisplay
+evβ≤1h(α,β) =/summationdisplay
+evβ≤1/summationdisplay
+0≤α≤1,
+α/negationslash=βh(α,β)+/summationdisplay
+evβ≤1h(β,β)
+=/summationdisplay
+evβ≤1/summationdisplay
+0≤α≤1,
+α/negationslash=βh(α,β)+/summationdisplay
+evβ≤1f(x+β(y−x))
+≡U+u.
+Let us show that the double sum Ucan be represented as
+U=/summationdisplay
+0/negationslash=γ≤1/summationdisplay
+0≤δ≤γ,
+δ/negationslash=γcγδf/parenleftbig
+x+γ(x′−x)+δ(y−x′)/parenrightbig
+withcertianintegerfactors cγδtobeevaluatedbelow. Infact,given0 /\e}atio\slash=γ≤1
+and 0≤δ≤γwithδ/\e}atio\slash=γ, there exist even β≤1 and 0≤α≤1,α/\e}atio\slash=β, s.t.
+α∨β=γandαβ=δ. In order to see this, if γis even or δis even, we may
+setβ=γandα=δ, orβ=δandα=γ, respectively. Now, if γandδare
+odd, then since δ/\e}atio\slash=γ, we can find i∈ {1,...,n}s.t.δi= 0 and γi= 1, and
+so, if we set β= (δ1,...,δ i−1,1,δi+1,...,δ n), thenδ≤β≤γ,δ/\e}atio\slash=β/\e}atio\slash=γand
+|β|=|δ|+1 is even, and it remains to put α=γ+δ−β.
+Givenγandδasabove, let usevaluate cγδ. Sinceδ=αβ≤β≤α∨β=γ
+and, given even β, the multiindex 0 ≤α≤1,α/\e}atio\slash=β, s.t.α∨β=γand
+αβ=δ, is determined uniquely by α=γ+δ−β, we have cγδ=|{evenβ:
+δ≤β≤γ}|.
+35In a similar manner, we find
+/summationdisplay
+0≤α≤1/summationdisplay
+odβ≤1h(α,β) =/summationdisplay
+odβ≤1/summationdisplay
+0≤α≤1,
+α/negationslash=βh(α,β)+/summationdisplay
+odβ≤1f(x+β(y−x))≡V+v,
+where
+V=/summationdisplay
+0/negationslash=γ≤1/summationdisplay
+0≤δ≤γ,
+δ/negationslash=γdγδf/parenleftbig
+x+γ(x′−x)+δ(y−x′)/parenrightbig
+withdγδ=|{oddβ:δ≤β≤γ}|. By Lemma 4(b), cγδ=dγδ, and so,
+U=V. Applying the translation invariance of dand inequality (5.2), we
+obtain inequality (7.6):
+d(u,v) =d(U+u,V+v) =d/parenleftbigg/summationdisplay
+0≤α≤1/summationdisplay
+evβ≤1h(α,β),/summationdisplay
+0≤α≤1/summationdisplay
+odβ≤1h(α,β)/parenrightbigg
+≤/summationdisplay
+0≤α≤1d/parenleftbigg/summationdisplay
+evβ≤1h(α,β),/summationdisplay
+odβ≤1h(α,β)/parenrightbigg
+. /square
+Remark 7.1. Ifx < x′< yin Lemma 10, then all rectangles at the right-
+hand side of (7.5) are non-degenerated, i.e., x+α(x′−x)< x′+α(y−x′) for
+all 0≤α≤1. Moreover, the point x′gives rise to a net partition {x[σ]}κ
+σ=0
+ofIy
+xwithx[σ] = (x1(σ1),...,x n(σn)) and 0 ≤σ≤κas follows: we put
+κ= 2 = 1+1 ∈Nnand, given i∈ {1,...,n}, we setxi(0) =xi,xi(1) =x′
+i
+andxi(2) =yi. We note that if 0 ≤σ≤1, thenx[σ] =x+σ(x′−x), and if
+1≤σ≤2, thenx[σ] =x′+(σ−1)(y−x′). It follows that
+Iy
+x=/uniondisplay
+0≤α≤1Ix′+α(y−x′)
+x+α(x′−x)=/uniondisplay
+1≤σ≤2Ix′+(σ−1)(y−x′)
+x+(σ−1)(x′−x)=/uniondisplay
+1≤σ≤κIx[σ]
+x[σ−1].
+However, in the general case x≤x′≤ywe may also have x/\e}atio\slash< x′orx′/\e}atio\slash< y,
+and so, there exists i∈ {1,...,n}s.t.xi=x′
+iorx′
+i=yi. Thus, since some
+coordinatesof x′maybeequaltothecorrespondingcoordinatesof xand/ory,
+certain rectangles at the right-hand side of (7.5) may degenerate into lower-
+dimensional rectangles, and so, by Remark 2.1, the mixed difference mdn
+over these rectangles is equal to zero. In order to exclude these degenerated
+rectangles from the consideration, we establish the following lemma.
+36Lemma 11. Givenx,y∈Ib
+awithx < yandx′∈Iy
+x, we have the following
+partition ofIy
+x, induced by x′:
+Iy
+x=/uniondisplay
+λ≤α≤µIx′+α(y−x′)
+x+α(x′−x), (7.7)
+where the multiindices λ≡λ(x,x′) = (λ1,...,λ n)andµ≡µ(x′,y) =
+(µ1,...,µ n)are defined for i∈ {1,...,n}by
+λi≡λi(x,x′) =/braceleftbigg1ifxi=x′
+i,
+0ifxi< x′
+i,andµi≡µi(x′,y) =/braceleftbigg0ifx′
+i=yi,
+1ifx′
+i< yi,
+and the following inequality holds :
+mdn(f,Iy
+x)≤/summationdisplay
+λ≤α≤µmdn/parenleftbig
+f,Ix′+α(y−x′)
+x+α(x′−x)/parenrightbig
+. (7.8)
+Proof. First, we note that, since xi< yifor alli∈ {1,...,n}, thenλ≤µ.
+In particular, if x < x′< y, thenλ= 0 and µ= 1, and we get (7.5) as a
+consequence of (7.7); cf. Remark 7.1.
+In order to prove (7.7), given i∈ {1,...,n}, consider the following possi-
+bilities: (i) x′
+i=xiandx′
+i< yi; (ii)xi< x′
+iandx′
+i=yi; and (iii) xi< x′
+iand
+x′
+i< yi. We have, respectively:
+(i)λi= 1 and µi= 1, and so, if λi≤αi≤µi, thenαi= 1 and
+Iyi
+xi=Iyi
+x′
+i=/uniondisplay
+αi=1Ix′
+i+αi(yi−x′
+i)
+xi+αi(x′
+i−xi);
+(ii)λi= 0 and µi= 0, and so, if λi≤αi≤µi, thenαi= 0 and
+Iyi
+xi=Ix′
+ixi=/uniondisplay
+αi=0Ix′
+i+αi(yi−x′
+i)
+xi+αi(x′
+i−xi);
+(iii)λi= 0 and µi= 1, and so, if λi≤αi≤µi, thenαi∈ {0,1}and
+Iyi
+xi=Ix′
+ixi∪Iyi
+x′
+i=1/uniondisplay
+αi=0Ix′
+i+αi(yi−x′
+i)
+xi+αi(x′
+i−xi).
+Moreover, in all the cases (i)–(iii) the left endpoint xi+αi(x′
+i−xi) is less
+than the right endpoint x′
+i+αi(yi−x′
+i), and so, all the closed intervals above
+are non-degenerated. It follows that
+Iy
+x=n/productdisplay
+i=1Iyi
+xi=n/productdisplay
+i=1/parenleftbigg/uniondisplay
+λi≤αi≤µiIx′
+i+αi(yi−x′
+i)
+xi+αi(x′
+i−xi)/parenrightbigg
+=/uniondisplay
+λ≤α≤µIx′+α(y−x′)
+x+α(x′−x).
+37The point x′gives rise to a net partition {x[σ]}κ
+σ=0ofIy
+xas follows: we
+putκ=µ−λ+1 and, given i∈ {1,...,n}, we setxi(0) =xiandxi(1) =yi
+ifκi= 1, and xi(0) =xi,xi(1) =x′
+iandxi(2) =yiifκi= 2. We note that if
+0≤σ≤µ−λ, thenx[σ] =x+(σ+λ)(x′−x), and if 1 ≤σ≤κ=µ−λ+1,
+thenx[σ] =x′+(σ−1+λ)(y−x′). Also, note that x+λ(x′−x) =xand
+x′+µ(y−x′) =y. It follows that
+Iy
+x=/uniondisplay
+λ≤α≤µIx′+α(y−x′)
+x+α(x′−x)=/uniondisplay
+1≤σ≤µ−λ+1Ix′+(σ−1+λ)(y−x′)
+x+(σ−1+λ)(x′−x)=/uniondisplay
+1≤σ≤κIx[σ]
+x[σ−1].
+Now, we turn to the proof of (7.8). By Lemma 10, inequality (7.6) hold s.
+Clearly, if λ= 0 and µ= 1 (i.e., x < x′< y), then (7.6) implies (7.8).
+Assume that λ/\e}atio\slash= 0 (i.e., x/\e}atio\slash< x′) and suppose that 0 ≤α≤1 is s.t.λ/\e}atio\slash≤α.
+Then there exists i∈ {1,...,n}s.t.λi= 1 and αi= 0, and so, xi=x′
+i,
+which implies xi+αi(x′
+i−xi) =xi=x′
+i=x′
+i+αi(yi−x′
+i). It follows
+from Remark 2.1 that md n(f,Ix′+α(y−x′)
+x+α(x′−x)) = 0. Similarly, if we assume that
+µ/\e}atio\slash= 1 (i.e., x′/\e}atio\slash< y) and suppose that 0 ≤α≤1 is s.t.α/\e}atio\slash≤µ, then there
+existsi∈ {1,...,n}s.t.αi= 1 and µi= 0, and so, x′
+i=yi. Noting that
+xi+αi(x′
+i−xi) =x′
+i=yi=x′
+i+αi(yi−x′
+i), we find md n(f,Ix′+α(y−x′)
+x+α(x′−x)) = 0.
+In this way inequality (7.8) follows.
+Proof of Lemma 9. Suppose that x,y∈Ib
+a,x < yandx′∈Ib
+a. We set
+x′′=x+ξ(x′−x), where ξis defined in (7.3) (the point x′′will play the
+role ofx′from (7.7)). We have x≤x′′< y; in fact, given i∈ {1,...,n}, we
+find: ifξi= 1, then xi< x′
+i< yiandx′′
+i=x′
+iimplying xi< x′′
+i< yi, and if
+ξi= 0, then x′
+i≤xiorx′
+i≥yi, andx′′
+i=xiimplying xi=x′′
+i< yi. Applying
+(7.7) with x′replaced by x′′, we get the following partition of Iy
+xinduced by
+x′′and, hence, by x′:
+Iy
+x=/uniondisplay
+λ′′≤α≤µ′′Ix′′+α(y−x′′)
+x+α(x′′−x), (7.9)
+whereλ′′=λ(x,x′′) andµ′′=µ(x′′,y) are defined in Lemma 11, i.e., given
+i∈ {1,...,n}, we have:
+λ′′
+i=/braceleftbigg1 ifxi=x′′
+i,
+0 ifxi< x′′
+i,andµ′′
+i=/braceleftbigg0 ifx′′
+i=yi,
+1 ifx′′
+i< yi.
+We assert that λ′′= 1−ξandµ′′= 1. In fact, since x′′< y, then
+µ′′= 1. In order to see that λ′′= 1−ξ, leti∈ {1,...,n}. Ifxi< x′
+i< yi,
+38thenξi= 1, and so, x′′
+i=xi+ξi(x′
+i−xi) =x′
+i, which implies xi< x′′
+iand
+λ′′
+i= 0 = 1 −ξi. Now if x′
+i≤xiorx′
+i≥yi, thenξi= 0, and so, x′′
+i=xi,
+which gives λ′′
+i= 1 = 1−ξi.
+Now, let us calculate the lower and upper indices in (7.9). We have:
+x+α(x′′−x) =x+αξ(x′−x) and
+x′′+α(y−x′′) =x+(1−α)ξ(x′−x)+α(y−x).
+Noting that the union in (7.9) is taken over α≤1 s.t. 1−ξ≤α, we get
+1−α≤ξ, and so, (1 −α)ξ= 1−αimplying
+x′′+α(y−x′′) =x+(1−α)(x′−x)+α(y−x) =x′+α(y−x′).
+These calculations and observations above prove equality (7.2).
+Let us show that partition (7.2) is actually induced by x′. Sincex′∈Ib
+a,
+by Lemma 11, the point x′induces a partition of Ib
+aof the form (7.7):
+Ib
+a=/uniondisplay
+λ′≤β≤µ′Ix′+β(b−x′)
+a+β(x′−a),
+where the multiindices λ′=λ(a,x′) andµ′=µ(x′,b) are defined in Lemma 11,
+i.e., given i∈ {1,...,n}, we have:
+λ′
+i=/braceleftbigg1 ifai=x′
+i,
+0 ifai< x′
+i,andµ′
+i=/braceleftbigg0 ifx′
+i=bi,
+1 ifx′
+i< bi.
+We assert that for each αwith 1−ξ≤α≤1 there exists a unique
+β≡β(α) withλ′≤β≤µ′s.t.
+Ix′+α(y−x′)
+x+αξ(x′−x)=Iy
+x∩Ix′+β(b−x′)
+a+β(x′−a). (7.10)
+In order to prove (7.10), we define β=β(α) = (β1,...,β n) by
+βi≡βi(α) =/braceleftbiggαiifx′
+i< yi,
+0 ifx′
+i≥yi,i∈ {1,...,n},
+and establish equality (7.10) componentwise. Given i∈ {1,...,n}, we con-
+sider the following two cases: (a) x′
+i< yi, and (b) x′
+i≥yi.
+In case (a) we have βi=αi. First, assume that xi< x′
+i, and so, ξi= 1.
+It follows that if 1 −ξi≤αi≤1, thenαi= 0 orαi= 1. Ifαi= 0, then we
+find (for βi=αi= 0)
+Ix′
+ixi= [xi,x′
+i] = [xi,yi]∩[ai,x′
+i] =Iyi
+xi∩Ix′
+i+βi(bi−x′
+i)
+ai+βi(x′
+i−ai),
+39and ifαi= 1, then we find (for βi=αi= 1)
+Iyi
+x′
+i= [x′
+i,yi] = [xi,yi]∩[x′
+i,bi] =Iyi
+xi∩Ix′
+i+βi(bi−x′
+i)
+ai+βi(x′
+i−ai).
+Now, assume that x′
+i≤xi, and so, ξi= 0 and x′
+i≤xi< yi≤bi. It follows
+that if 1−ξi≤αi≤1, thenβi=αi= 1 implying
+Iyi
+xi= [xi,yi] = [xi,yi]∩[x′
+i,bi] =Iyi
+xi∩Ix′
+i+βi(bi−x′
+i)
+ai+βi(x′
+i−ai).
+In case (b) we have ξi= 0,βi= 0 and ai≤xi< yi≤x′
+i, and so, if
+1−ξi≤αi≤1, thenαi= 1 and
+Iyi
+xi= [xi,yi] = [xi,yi]∩[ai,x′
+i] =Iyi
+xi∩Ix′
+i+βi(bi−x′
+i)
+ai+βi(x′
+i−ai).
+Let us show that λ′≤β≤µ′. Leti∈ {1,...,n}. Ifai=x′
+i, then
+λ′
+i= 1 =µ′
+iand, since x′
+i< yi, thenβi=αi. By (7.3), ξi= 0, and so, since
+1−ξi≤αi≤1, thenαi= 1, which implies λ′
+i=βi=µ′
+i. Now, if x′
+i=bi,
+thenλ′
+i= 0 =µ′
+iand, since x′
+i≥yi, thenβi= 0 (and ξi= 0), and so,
+λ′
+i=βi=µ′
+i. Finally, if ai< x′
+i< bi, thenλ′
+i= 0 and µ′
+i= 1, and so, since
+βi∈ {0,1}, thenλ′
+i≤βi≤µ′
+i.
+The uniqueness of β(α), for each 1 −ξ≤α≤1, is a consequence of the
+following: if λ′≤β≤µ′andβ/\e}atio\slash=β(α), then there exists i∈ {1,...,n}s.t.
+βi= 1−βi(α). Arguing as in (a) and (b) above, we find that the equality
+(7.10) cannot hold for this β.
+Now, inequality (7.4) readily follows from Lemma 11, (7.9) and (7.2).
+Remark 7.2. (a) Ifx′∈Iy
+xin Lemma 9, then it is easily seen that ξ=µ−λ,
+and so, equality (7.2) assumes the form:
+Iy
+x=/uniondisplay
+1−(µ−λ)≤α≤1Ix′+α(y−x′)
+x+α(µ−λ)(x′−x).
+Although this equality looks different from (7.7), the two equalities ar e the
+same: this is verified as in (i)–(iii) of the proof of Lemma 11.
+(b) Ifx < x′< y, thenξ= 1,λ= 0 and µ= 1, and so, (7.2), (7.7) and
+(7.5) are identical.
+(c) Here we consider a certain particular case of (7.10) and establis h
+conditions on x′, under which x′does not induce a (further) partition of Iy
+x.
+In view of (7.10), we have:
+Ix′+α(y−x′)
+x+αξ(x′−x)=Iy
+xif and only if ξ= 0 and α= 1, (7.11)
+40which is also equivalent to
+a+β(x′−a)≤xandy≤x′+β(b−x′) with β=β(1).(7.12)
+Clearly, if ξ= 0 and α= 1, then the left-hand side equality in (7.11) holds.
+Conversely, if the left-handside equality in(7.11) holds forsome 1 −ξ≤α≤1,
+thenx+αξ(x′−x) =xandx′+α(y−x′) =y, and so, if we suppose that
+ξi= 1 for some i∈ {1,...,n}, then, by (7.3), xi< x′
+i< yi, and so, αi= 0
+andx′
+i=yi, which is a contradiction. Thus, ξ= 0 and α= 1.
+Now, ifξ= 0 and α= 1, then, by (7.10) and (7.11),
+Iy
+x=Iy
+x∩Ix′+β(b−x′)
+a+β(x′−a)withβ=β(1), (7.13)
+which implies (7.12). Conversely, (7.12) implies (7.13), and so, the left -hand
+side equality in (7.11) holds, i.e., ξ= 0 and α= 1.
+This observation also shows that a point x′∈Ib
+ainduces a ‘true’ partition
+ofIy
+xprovided that, for all βwithλ′≤β≤µ′, we have:
+a+β(x′−a)/\e}atio\slash≤xory/\e}atio\slash≤x′+β(b−x′),
+which is also equivalent to ξ/\e}atio\slash= 0.
+Proof of Lemma 8. LetP={x[σ]}κ
+σ=0for some κ∈Nnandx′∈ P′.
+Given 1≤σ≤κ, we setxσ=x[σ−1],yσ=x[σ] andξσ(x′) =ξ(xσ,x′,yσ),
+whereξis defined in (7.3), and note that xσ< yσ. The point x′induces a
+partition of Iyσxσ=Ix[σ]
+x[σ−1]of the form (7.2) with x=xσandy=yσ, and so,
+by virtue of (7.1), we get the following partition of Ib
+a, induced by x′:
+Ib
+a=/uniondisplay
+1≤σ≤κ/uniondisplay
+1−ξσ(x′)≤α≤1Ix′+α(yσ−x′)
+xσ+αξσ(x′)(x′−xσ). (7.14)
+We denote by P1the net partition of Ib
+acorresponding to (7.14). Moreover,
+by (7.4), for each 1 ≤σ≤κwe have the inequality:
+mdn(f,Iyσ
+xσ)≤/summationdisplay
+1−ξσ(x′)≤α≤1mdn/parenleftbig
+f,Ix′+α(yσ−x′)
+xσ+αξσ(x′)(x′−xσ)/parenrightbig
+.(7.15)
+With no loss of generality we may assume that x′/∈ P: ifx′∈ P, i.e.,
+x′=x[σ′] for some 1 ≤σ′≤κ, thenx′does not affect the partition P
+ofIb
+ain the sense that P1=P, and so, v n(f;P1) = vn(f;P). In order to
+41see this, we note that (7.3) implies ξσ(x′) =ξ(x[σ−1],x[σ′],x[σ]) = 0, and
+so, by Remark 7.2(c), conditions (7.11) and (7.12) hold with β=β(1) =
+(β1,...,β n) s.t.
+βi=/braceleftbigg1 ifxi(σ′
+i)< xi(σi),
+0 ifxi(σ′
+i)≥xi(σi),=/braceleftbigg1 ifσ′
+i< σi,
+0 ifσ′
+i≥σi.
+Summing over 1 ≤σ≤κin (7.15) and taking into account (7.1) and
+(7.14), we obtain the inequality
+vn(f;P)≤vn(f;P1).
+Replacing PbyP1in the arguments above, taking x′∈ P′\P1and de-
+noting by P2the partition of Ib
+ainduced from P1byx′, we get v n(f;P1)≤
+vn(f;P2). Since P′\ Pis a finite set, we exhaust it by points x′in a fi-
+nite number of steps, arrive at the partition P′ofIb
+aand prove the desired
+inequality v n(f;P)≤vn(f;P′).
+Proof of Lemma 3. 1. First, we establish (3.5) for α= 1 = 1 n, i.e.,
+Vn(f,Iy
+x) =/summationdisplay
+1≤σ≤κVn(f,Ix[σ]
+x[σ−1]). (7.16)
+Modulo the notation, there is no loss of generality if we assume that x=a
+andy=b, so that {x[σ]}κ
+σ=0is a net partition of Ib
+a.
+LetPbe an arbitrary net partition of Ib
+a. Denote by P′the net partition
+ofIb
+ainduced from Pby points {x[σ]}κ
+σ=0, so that P′is a refinement of P.
+Given 1≤σ≤κ, setPσ=P′∩Ix[σ]
+x[σ−1]and note that Pσis a net partition of
+Ix[σ]
+x[σ−1], andP′=/uniontext
+1≤σ≤κPσ. Then by virtue of Lemma 8, we have:
+vn(f;P)≤vn(f;P′) =/summationdisplay
+1≤σ≤κvn(f;Pσ)≤/summationdisplay
+1≤σ≤κVn(f,Ix[σ]
+x[σ−1]).
+SincePis arbitrary, the left-hand side in (7.16) is not greater than the righ t-
+hand side.
+Let us prove the reverse inequality. If Vn(f,Ix[σ]
+x[σ−1]) is infinite for some
+1≤σ≤κ, then since Ix[σ]
+x[σ−1]⊂Ib
+a=Iy
+x, the value Vn(f,Iy
+x) is infinite as well.
+Thus, we suppose that the right-hand side of (7.16) is finite. Let ε >0 be
+42arbitrary. Given 1 ≤σ≤κ, by the definition of Vn(f,Ix[σ]
+x[σ−1]), there exists a
+net partition of Ix[σ]
+x[σ−1], denoted by Pσ(ε), s.t.
+vn(f;Pσ(ε))≥Vn(f,Ix[σ]
+x[σ−1])−(ε/c),
+wherec=|{σ: 1≤σ≤κ}|. We denote by P(ε) the net partition of Ib
+a
+induced from {x[σ]}κ
+σ=0by points from/uniontext
+1≤σ≤κPσ(ε). Given 1 ≤σ≤κ, we
+setP′
+σ(ε) =P(ε)∩Ix[σ]
+x[σ−1]and note that P′
+σ(ε) is a refinement of Pσ(ε), and
+P(ε) =/uniontext
+1≤σ≤κP′
+σ(ε). By virtue of Lemma 8, we find
+Vn(f,Ib
+a)≥vn(f;P(ε)) =/summationdisplay
+1≤σ≤κvn(f;P′
+σ(ε))≥/summationdisplay
+1≤σ≤κvn(f;Pσ(ε))
+≥/summationdisplay
+1≤σ≤κVn(f,Ix[σ]
+x[σ−1])−ε/parenleftbigg/summationdisplay
+1≤σ≤κ1/parenrightbigg/slashBig
+c,
+where the factor by εis, actually, equal to 1. The desired inequality follows
+if we take into account the arbitrariness of ε >0.
+2. Now, suppose that 0 /\e}atio\slash=α≤1 andα/\e}atio\slash= 1. Note that x⌊α,y⌊α∈
+Ib⌊α
+a⌊αandx⌊α < y⌊α, and that {x[σ]⌊α}κ⌊α
+σ⌊α=0is a net partition of Ib⌊α
+a⌊α. So,
+replacing 1 = 1 nby 1⌊α(so that|1⌊α|=|α|) andf—byfz
+αin (7.16), we get:
+V|α|(fz
+α,Iy
+x⌊α) =V|1⌊α|(fz
+α,Iy⌊α
+x⌊α)
+=/summationdisplay
+1⌊α≤σ⌊α≤κ⌊αV|1⌊α|/parenleftbig
+fz
+α,Ix[σ]⌊α
+x[σ−1]⌊α/parenrightbig
+,
+which is equal to the right-hand side of (3.5).
+This completes the proof of Lemma 3.
+8. Proof of Theorem E
+Proof of Theorem E. 1. First, we show that if x,y∈Ib
+a,x < y, and
+0/\e}atio\slash=α≤1, then
+md|α|(fa
+α,Iy
+x⌊α) = lim
+j→∞md|α|((fj)a
+α,Iy
+x⌊α). (8.1)
+By virtue of (3.3), we have:
+md|α|(fa
+α,Iy
+x⌊α) =d/parenleftbigg/summationdisplay
+evθ≤αf(a+α(x−a)+θ(y−x)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright
+(···)),/summationdisplay
+odθ≤αf(···)/parenrightbigg
+,
+43and a similar equality holds for fjin place of f. Applying the inequalities
+|d(u,v)−d(u′,v′)| ≤d(u,u′)+d(v,v′),u,v,u′,v′∈M, and (5.2) and taking
+into account the pointwise convergence of fjtof, we find
+/vextendsingle/vextendsinglemd|α|((fj)a
+α,Iy
+x⌊α)−md|α|(fa
+α,Iy
+x⌊α)/vextendsingle/vextendsingle
+≤d/parenleftbigg/summationdisplay
+evθ≤αfj(···),/summationdisplay
+evθ≤αf(···)/parenrightbigg
++d/parenleftbigg/summationdisplay
+odθ≤αfj(···),/summationdisplay
+odθ≤αf(···)/parenrightbigg
+≤/summationdisplay
+evθ≤αd(fj(···),f(···))+/summationdisplay
+odθ≤αd(fj(···),f(···))
+=/summationdisplay
+0≤θ≤αd(fj(···),f(···))→0 asj→ ∞.
+2. In the rest of this proof we need only the inequality
+md|α|(fa
+α,Iy
+x⌊α)≤liminf
+j→∞md|α|((fj)a
+α,Iy
+x⌊α), (8.2)
+which readily follows from (8.1) and is applied one more time in the proof
+of Theorem 2 (Step 5). If P={x[σ]}κ
+σ=0is a net partition of Ib
+a, then
+P⌊α={x[σ]⌊α}κ⌊α
+σ⌊αis net partition of Ib
+a⌊α, and so, given 1 ≤σ≤κ, setting
+x=x[σ−1] andy=x[σ] in (8.2), we find
+/summationdisplay
+1⌊α≤σ⌊α≤κ⌊αmd|α|(fa
+α,Ix[σ]
+x[σ−1]⌊α)≤/summationdisplay
+1⌊α≤σ⌊α≤κ⌊αliminf
+j→∞md|α|((fj)a
+α,Ix[σ]
+x[σ−1]⌊α)
+≤liminf
+j→∞/summationdisplay
+1⌊α≤σ⌊α≤κ⌊αmd|α|((fj)a
+α,Ix[σ]
+x[σ−1]⌊α)
+≤liminf
+j→∞V|α|((fj)a
+α,Ix[σ]
+x[σ−1]⌊α).
+By the arbitrariness of P, we infer that
+V|α|(fa
+α,Ix[σ]
+x[σ−1]⌊α)≤liminf
+j→∞V|α|((fj)a
+α,Ix[σ]
+x[σ−1]⌊α).
+We conclude that
+TV(f,Ib
+a) =/summationdisplay
+0/negationslash=α≤1V|α|(fa
+α,Ib
+a⌊α)≤/summationdisplay
+0/negationslash=α≤1liminf
+j→∞V|α|((fj)a
+α,Ib
+a⌊α)
+≤liminf
+j→∞/summationdisplay
+0/negationslash=α≤1V|α|((fj)a
+α,Ib
+a⌊α) = liminf
+j→∞TV(fj,Ib
+a). /square
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+47
\ No newline at end of file
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+arXiv:1001.0452v1  [hep-ex]  4 Jan 2010Plans forsuper-beams inJapan∗
+Takuya Hasegawa
+High Energy Accelerator Research Organization (KEK),Tsuk uba, Ibaraki 305-0801, Japan
+Abstract
+In Japan, as the first experiment utilizes J-PARC (Japan Prot on Accelerator
+Research Complex) neutrino facility, T2K (Tokai to Kamioka Long Baseline
+Neutrino Experiment) starts operation. T2K is supposed to g ive critical in-
+formation, which guides the future direction of the neutrin o physics. Possible
+newgeneration discoveryexperiment basedonT2Koutcomeis discussed. Es-
+pecially, description of J-PARC neutrino beam upgrade plan and discussion
+on far detector options to maximize potential of the researc h are focused. Eu-
+ropean participation and CERN commitment on Japanese accel erator based
+neutrino experiment isalso reported.
+1J-PARC andMainRing Synchrotron
+J-PARC (Figure 1) is a KEK-JAEA joint facility for MW-class h igh intensity proton accelerator. It
+provides unprecedented high fluxof various secondary parti cles, such as neutrons, muons, pions, kaons,
+and neutrinos, which areutilized for elementary particle p hysics and material and life science.
+Fig.1:J-PARC acceleratorandexperimentalfacility
+In the accelerator complex, H−ions are accelerated to 181 MeV with LINAC, fed into Rapid
+Cycling Synchrotron (RCS) with stripping out electrons and are accelerated to 3 GeV. At final stage,
+∗Contribution to the Workshop “European Strategy for Future Neutrino Physics”, CERN, Oct.
+2009, to appear in the Proceedings. Manuscript with high res olution figures is available at
+http://jnusrv01.kek.jp/ ∼hasegawa/pub/cernreport.pdf .proton beam goes into MainRing Synchrotron (MR) and acceler ated to30 GeV.For the neutrino exper-
+iment, accelerated protons arekicked inward toneutrino be am facility bysingle turn withfast extraction
+devices. Main characteristics of MRis described inFigure 2 .
+Fig.2:OverviewofMR
+2J-PARC neutrino beam facility
+The proton beam from MR run through J-PARC neutrino beam faci lity and producing intense muon
+neutrinos toward the west direction. J-PARCneutrino beam f acility iscomposed of following parts with
+their functionalities (Figure 3).
+– Preparation section: Match the beam optics to the arcsecti on.
+– Arcsection: Bendthebeam ∼90◦towardthewestdirectionwithsuperconducting combinedfu nc-
+tion magnet.
+– Final focus section: Match the beam optics to target both in position and in profile. Level of mm
+control isnecessary whichcorresponds to1mrad νdirection difference, alsonot todestroy target.
+– Graphite target andhornmagnet: Produceintense secondar yπ’sandfocusthem tothewest direc-
+tion. (3horns system with320 kApulse operation)
+– Muonmonitor: Monitor µdirection( =νdirection), pulsetopulse,withmeasuringcenterofmuon
+profile.
+– On-axis neutrino monitor (INGRID):Monitor νdirection and intensity.
+This facility is designed to be tolerable up to ∼2MW beam power. The limitation is due to temperature
+riseandthermalshockforthecomponentssuchasAlhorn,gra phitetarget,andTivacuumwindow. Since
+everywheresuffersfromhighradiation,carefultreatment ofradioactivewaterandair( ∼10GBq/3weeks)
+isrequired. Moreover, maintenance scenario of radio activ e components has tobe seriously planned.
+On 23rd April 2009, commissioning of the facility started wi th the real proton beam which was
+deliveredbyMR.Theveryfirstshotoftheprotonbeam,aftera llthebeamlinemagnetturnedon,steered
+2Fig.3:J-PARC neutrinobeamfacility
+intotarget station andmuonmonitor clearly indicated prod uction of intense muons whichcertified asso-
+ciated neutrino production. After 9 shots of tuning, the bea m is centered on the target. With subsequent
+tuning and measurement, followings areachieved.
+– Stability of the extraction beam orbit from MRisconfirmed. It istuned within0.3mmin position
+and 0.04 mrad indirection w.r.t. design orbit.
+– Functionality of the superconducting combined function m agnet is confirmed.
+– Beam is lead to the target center without significant beam lo ss. Beam trajectory is tuned within
+3mm level accuracy w.r.t. design orbit.
+– Functionality of the beam monitors (beam position, beam pr ofile, beam intensity and beam loss)
+are confirmed.
+– Response function of various magnets are measured.
+– Muon signal is observed which confirms neutrino production . (Muon direction corresponds to
+neutrino direction and muon yield corresponds to neutrino y ield.)
+– The effect of pion focusing with horn magnet is confirmed. ( ×2 which is consistent with horn
+configuration at that time.)
+– The information transfer from Tokai to Kamioka on the absol ute beam time information is con-
+firmed.
+– J-PARCneutrino facility isapproved by the government on r adiation safety.
+ThenextrunningofthefacilityisforeseenfromOctober200 9withfollowingMRintensityimprovement.
+Production data taking for neutrino experiment isforeseen tostart in January 2010.
+3T2K
+T2K[1]isthefirstexperimentwithJ-PARCneutrinobeam. Wit hthecombinationofunprecedented high
+intensityneutrinosourceandawellestablished neutrinod etector, Super-Kamiokande(SK),asafarmain
+3detector, T2K will seeks for νµtoνeconversion phenomenon and, as a consequence, measures an fin ite
+valueofoneoftheneutrino mixingangle, θ13,withanorder ofmagnitude better sensitivity compared to
+theprior experiments at anatmospheric neutrino anomaly re gime. T2Kalso conduct precision measure-
+ment of another neutrino mixing angle, θ23. Moreover, something unexpected in neutrino physics may
+be revealed byT2K.
+The baseline of 295 km and off-axis angle of 2.5◦are optimized, 1) to maximize neutrino flux
+at the neutrino energy of the first oscillation maximum, 2) to avoid severe π0background originated
+from high energy neutrino interaction, and 3) to tune neutri no energy range to be optimum for a water
+cherenkov detector (sub GeVenergy region, low multiplicit y and quasi elastic interaction dominant).
+The properties of the produced neutrino beam are measured by a system of near detectors at J-
+PARC which consists of two major parts, one is on-axis neutri no monitor (INGRID) which monitors
+neutrino direction, intensity and its stability, and the ot her is an assembly of detectors located 2.5◦off-
+axis direction as SK (ND280), which measures not only neutri no flux as is expected at SK but also sub
+GeV neutrino interaction which gives important informatio n for neutrino oscillation analysis in T2K.
+Themost outer part of the ND280 is the UA1/NOMADmagnet, whic h provides the magnetic field used
+todeterminethemomentumofchargedparticlesoriginated f romneutrinointeraction. Insidethemagnet,
+Fine Grain Detector (FGD) which is an active neutrino target , Time Projection Chamber (TPC) which
+measuresanychargedparticlesemergedfromneutrinointer action,π0detector(POD)whichisoptimized
+for measuring the rate of neutral current π0production, Electromagnetic Calorimeter (ECAL) which
+reconstructs any electromagnetic energy produced, and Sid e Muon Range Detector (SMRD) which is
+instrumented inthemagnet yokes toidentify muons fromneut rino interaction, arelocated. Thestatus of
+detectors asof October 2009 isshown in Figure 4.
+Fig.4:Statusofthe T2Knearsite neutrinodetectorsasofOctober2 009
+As a first milestone, T2K is aiming for the first results in 2010 with 100 kw ×107seconds
+integratedprotonpowerontargettounveilbelowtheCHOOZe xperimental limit[2]with νeappearance.
+44EuropeanandCERNcommitmentonJapaneseacceleratorbased neutrinoexperiment
+EuropeanparticipationinJapaneseacceleratorbasedneut rinoexperimentbeganatK2K(KEKtoKamioka
+Long Baseline Neutrino Experiment). France, Italy, Poland , Russia, Spain and Switzerland joined this
+world first accelerator based long baseline neutrino experi ment. When T2K project started, Germany
+and United Kingdom also participated. As of October 2009, T2 K collaboration consists of 477 mem-
+bers from 62 institutes spread out 12 coutries. Composition is, 240 (50.3 %) members from Europe, 84
+(17.6%) members from Japan, 77 (16.1 %) members from USA, 68 (14.3 %) members from Canada and
+8(1.7%) members from South Korea, asshown inFigure 5.
+Fig.5:ParticipantsforT2K
+T2K is registered as Recognized Experiment at CERN (RE13) an d CERN extensively supports
+T2K.Followings are the list of CERNsupport for T2K.
+– CERN experiment NA61: This experiment is indispensable pa rt of T2K to understand neutrino
+fluxfor the experiment.
+– CERNtest beam for detectors.
+– Donation of UA1/NOMADmagnet.
+– Micromegas production and its test conducted by CERNTS/DE Mgroup.
+– Various technical, administrative support on detector pr eparation, especially for UA1/NOMAD
+magnet related issues.
+– Infrastructure for detector preparation.
+– CERN-KEKcooperation on super conducting magnet for neutr ino beam line.
+KEKfeels grateful to CERNfor all the aspect of support provi ded byCERN.
+5New generation accelerator basedneutrino experiment inJa pan
+The primary motivation of T2K is to improve the sensitivity t o theνµ→νeconversion phenomenon
+in the atmospheric regime. The final goal for T2K is to accumul ate an integrated proton power on
+target of 0.75MW×5×107seconds. As is shown in Figure 6, within a few years of run, cri tical
+information, which will guide the future direction of the ne utrino physics, will be obtained based on
+the data corresponding to about 1 to 2 MW ×107seconds integrated proton power on target (roughly
+corresponding toa3 σdiscovery at sin22θ13>0.05 and 0.03, respectively).
+If a significant νµ→νecoversion signal were to be observed at T2K, an immediate ste p forward
+toanextgeneration experiment aimedatthediscoveryofCPv iolation intheleptonsector wouldberec-
+5
  
+     	 
+  
          
+  
+   !" # $ %& ' ( )* + , -. / 0 1sin 22θ13 2 3 4 5 6 7 8 9 : ; < = > ? @A B C DEF G H I JK L MN O P Q R S TU V WExcluded by CHOOZ X Y Z [ \ ] ^ _ ` a b c d e f g h ijk l m n op q rLine:   3σDiscovery 
+Dot :   90 ％C.L. Upper Bound s
+t u v wx y z {
+| } ~ €  ‚ ƒ „ … † ‡ ˆ ‰ Š ‹ Œ  Ž  
+Integrated Power (10 7Mw ・sec ：～ 1Mw ×Effective 1 Year Experimental Period) 
+‘ ’ “ ” • – — ˜ ™ š › œ  ž Ÿ   ¡ ¢ £¤¥ ¦ § ¨ ©ª « ¬
+2010 0.5MW ­2years ®0.33MW ¯3years 
+° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á ÂÃÄ Å Æ Ç ÈÉ Ê Ë
+1 2
+T2K 
+Full Proposal 
+Fig.6:T2Kdicoverypotentialon νµ→νeasafunctionofintegratedprotonpowerontarget
+ommended with high priority. Compared with T2K experimenta l conditions, lepton sector CP violation
+discovery requires
+– an improved J-PARCneutrino beam intensity;
+– an improved main far neutrino detector.
+Detector improvements include
+– detector technology;
+– its volume;
+– its baseline and off-axis angle with respect tothe neutrin o source.
+Naturally, next generation far neutrino detectors for lept on sector CP violation discovery will be very
+massive and huge. As a consequence, the same detector will gi ve us the rare and important opportunity
+to discover proton decay. A total research subject would be, to address a long standing puzzle of our
+physical world, the“Quest for theOrigin ofMatter Dominate d Universe" (see e.g.[3]),withexploration
+of
+– the Lepton Sector CPViolation by precise testing of the neu trino oscillation processes;
+– measure precisely the CPphase in lepton sector ( δ) and themixing angle θ13;
+– examine matter effect in neutrino oscillation process and possibly conclude the mass hierar-
+chy of neutrinos.
+– Proton Decay:
+– Search for p →νK+and p→eπ0inthe life timerange 1034to1035years,
+withassuming non-equilibrium environment inthe evolutio n of universe.
+6Evenincasethatsin22θ13isbelowT2Ksensitivity,itisstillworthwhiletryingtoim proveJ-PARC
+neutrino beam intensity and far detector performance to ope n the way to explore νµ→νeconversion
+phenomenon with byan order of magnitude better sensitivity [5].
+This direction of research is endorsed by KEK Roadmap defined in 2008, in which J-PARC neu-
+trinointensityimprovementandR &Dtorealizehugedetectorforneutrinoandprotondecayexpe riments
+are the two of the main subject. KEK has started R &D to realize huge liquid Argon time projection
+chamber.
+5.1 J-PARCneutrinobeam upgradeplan
+As for the neutrino beam intensity improvement, MR power imp rovement scenario toward MW-class
+power frontier machine, KEKRoadmap plan, is analyzed and pr oposed by theJ-PARCaccelerator team
+asshown in Table1.
+Table 1:MRpowerimprovementscenariotowardMW-classpowerfronti ermachine(KEKRoadmap)
+Start Up Next Step KEKRoadmap Ultimate
+Power(MW) 0.1 0.45 1.66 ?
+Energy(GeV) 30 30 30
+Rep. Cycle (sec.) 3.5 3-2 1.92
+No. of Bunches 6 8 8
+Particles/Bunch 1.2×1013<4.1×10138.3×1013
+Particles/Ring 7.2×1013<3.3×10146.7×1014
+LINAC(MeV) 181 181 400
+RCSah=2 h=2 or 1 h=1
+aHarmonic number of RCS
+Items tobe modified from start uptoward high intensity areli sted asfollowing.
+– Number of bunches in MR should be increased from 6 to 8. For th is purpose, fast rise time
+extraction kicker magnet have tobe prepared. Itsinstallat ion isforeseen in 2010 summer.
+– Repetition cycleofMRhastobeimprovedfrom3.5seconds to 1.92seconds. Forthispurpose RF
+and magnet power supply improvement is necessary.
+– RCS operation wih harmonic number 1 has to be conducted. Thi s is to make the beam bunch
+to be longer in time domain to decrease space charge effect. F or this purpose RF improvement
+is necessary. When RCS is operated with harmonic number 2, be am is injected to MR with
+2 bunches ×4 cycles. On the other hand, when RCS is operated with harmoni c number 1, beam
+is injected to MRwithsingle bunch with doubled number of pro tons×8cycles.
+– LINAC400 MeV operation isrequired toavoid severe space ch arge effect at RCSinjection. Con-
+struction of necessary component is already approved and st arted.
+5.2 Fardetector options: How to approach Lepton Sector CPVi olation
+Theeffects of CPphase δappear either
+– as a difference between νand¯νbehaviors (this method is sensitive to the CP-odd term which
+vanishes for δ= 0or 180◦);
+– in the energy spectrum shape of the appearance oscillated νecharged current events (sensitive to
+all the non-vanishing δvalues including 180◦).
+7Itshouldbenotedthatifonepreciselymeasuresthe νeappearance energyspectrumshape(peakposition
+and height for 1st and 2nd oscillation maximum and minimum)w ith high resolution, CPeffect could be
+investigated with neutrino run only. Antineutrino beam con ditions are known to be more difficult than
+those for neutrinos (lower beam flux due to leading charge eff ect in proton collisions on target, small
+antineutrinos cross-section at low energy, etc.).
+Figure7(left)showsneutrinofluxforvaiousoff-axisangle s. Ifoneselectson-axissetting,1)wide
+OA3 °OA0 °
+OA2 °
+OA2.5 °νµflux 
+δ=90 
+δ=270 δ=0 
+Δm31 2= 2.5x10 -3 eV 2
+sin 22θ13 = 0.1 
+No matter effects Oscillation probability first maximum second maximum 
+E(GeV)/L(km) 
+Fig. 7:Neutrinoflux for variousoff-axisangle (left)and Probabil ityforνµ→νeoscillationsas a functionof the
+E(GeV)/L(km)forvarious δ. (right)
+energy coverage is foreseen which is necessary to cover the 1 st and 2nd maximum simultaneously, and
+2)measurement suffersfromsevere π0background originated fromhighenergyneutrino whichrequ ires
+thedetector withhighperformance discrimination ability betweenπ0andelectron. Ontheother hand, if
+oneselectsoff-axissetting,1)requirement for π0background discrimination issoft,and2)measurement
+isessentially counting experiment at the1st oscillation m aximum.
+Figure 7 (right) shows the oscillation probability as a func tion of the E(GeV)/ L(km). If the
+distance between source anddetector isfixed, thecurves can beeasily translated to that for the expected
+neutrino energy spectrum of the oscillated events. As can be seen, if the neutrino energy spectrum of
+the oscillated events could be reconstructed with sufficien tly good resolution in order to distinguish first
+and second maximum, useful information to extract the CP pha se would be available even only with a
+neutrino run. If baseline is set to be long, 1) energy of 2nd os cillation maximum gets measurable. 2)
+statisticalsignificancemaygetworse,and3)measurementi saffectedbylargemattereffect. Ontheother
+hand, ifbaseline isset tobeshort, 1) itisimpossible toext ract 2ndoscillation maximum information, 2)
+statistical significance mayget better, and 3) measurement isless affected bymatter effect.
+To define far detector option, discovery potential for proto n decay and reality to realize huge one
+are also the essential issues to betaken into account.
+5.3 Possible scenarios fornew generation discovery experi ment with J-PARCneutrinobeam
+The study of possible new generation discovery experiments with J-PARC neutrino beam was initiated
+atthe4thInternational Workshop onNuclear andParticlePh ysicsatJ-PARC(NP08)[4]. Withthesame
+configuration asT2K(2.5◦off-axis angle), the center of the neutrino beam will go thro ugh underground
+beneath SK (295 km baseline), and will automatically reach t he Okinoshima island region (658 km
+baseline) with an off-axis angle 0.8◦(almost on-axis) and eventually the sea level east of the Kor ean
+shore (1,000 km baseline) withan off-axis angle ∼1◦.
+85.3.1 Scenario 1: J-PARCtoOkinoshima LongBaseline Neutri noExperiment
+Thefirstscenariois“J-PARCtoOkinoshimaLongBaselineNeu trinoExperiment”asshowninFig.8[5].
+Inorder tocoverawiderenergy range,detector location whi chisnearon-axis isfavored. Ifoneassumes
+Fig.8:Scenario1: J-PARC toOkinoshimaLongBaselineNeutrinoExp eriment
+that thesecond oscillation maximumhastobelocated atanen ergy larger thanabout 400MeV,thebase-
+lineshouldbelongerthanabout600km. Inaddition,inorder tocollectenoughstatistics, baselineshould
+notbetoomuchlongerthanabovestated. Takingintoaccount alloftheabovementionedconsiderations,
+the Okinoshima region (658 km baseline and almost on-axis (0 .8◦off-axis) configuration) turns out to
+be ideal.
+Analysisherebasedontheassumptionusinganeutrinorunon lyduringfiveyearstobereasonable
+time duration for the single experiment ( 107seconds running period/year is assumed), under the best J-
+PARCbeamassumption. Ananti-neutrinobeam(oppositehorn polarity)mightbeconsideredinasecond
+stage in order to cross-check the results obtained with the n eutrino run (in particular for mass hierarchy
+problem). Detector is assumed to be a 100 kton liquid Argon ti me projection chamber. This type of
+detector is supposed to provide higher precision than other huge detectors to separate the two peaks in
+energy spectrum. In addition, the π0background is expected to be highly suppressed thanks to the fine
+granularity of the readout, hence the main irreducible back ground will be the intrinsic νecomponent
+of the beam. The right hand side plot in Figure 8 shows the ener gy spectra of electron neutrino at the
+cases ofδequal 0◦, 90◦, 180◦, 270◦, respectively. Shaded region is common for all plots and it s hows
+the background from beam νe. Here perfect resolution is assumed. As shown, the value of δvaries the
+energy spectrum, especially the first and the second oscilla tion peaks (heights and positions), therefore
+comparison of the peaks determine the value δ, while the value of sin22θ13changes number of events
+predominantly. Allowed regions in the perfect resolution c ase are shown in left hand side of Figure 8.
+Twelve allowed regions are overlaid for twelve true values, sin22θ13=0.1, 0.05, 0.02, and δ=0◦, 90◦,
+180◦,270◦, respectively. The δsensitivity is20 ∼30◦depending onthe true δvalue.
+95.3.2 Scenario 2: J-PARCtoKamioka LongBaseline NeutrinoE xperiment
+Second scenario is “J-PARC to Kamioka Long Baseline Neutrin o Experiment” as shown in Fig. 9 [4].
+The concept is same as T2K except for huge detector size whose fiducial volume is assumed to be
+Fig.9:Scenario2: J-PARC toKamiokaLongBaseline NeutrinoExperi ment
+570 kt. The baseline of 295 km and off-axis angle of 2.5◦are optimum for the experimental sensitivity
+with a water cherenkov detector as they are for T2K. With this configuration, on the other hand, it is
+only possible to cover 1st oscillation maximum. In order to i nvestigate difference between neutrino and
+anti-neutrino behavior with sufficient statistics, 2.2 yea rs (107seconds running period/year is assumed)
+neutrino runand 7.8years anti-neutrino runisrequired. Si nce thecancellation of systematic uncertainty
+betweenneutrinorunandanti-neutrinorunisnotmuchexpec ted,thewaytodealwithdelicatesystematic
+uncertainty isaimportant issue tobe seriously considered .
+5.3.3 Scenario3: J-PARCto Kamioka andKorea LongBaseline N eutrino Experiment
+Third scenario is “J-PARC to Kamioka and Korea Long Baseline Neutrino Experiment” as shown in
+Fig. 10 [4]. This plan is partially same as scenario 2. In orde r to obtain 1st and 2nd oscillation maxima
+information, in addition to neutrino and anti-neutrino dif ference, two 270 kt water cherenkov detectors,
+one at Kamioka (295 km baseline) and the other in Korea (1,000 km baseline) are utilized. It would
+allowtostudyE/Lregionscorresponding tothe1stoscillat ion maximumatKamiokaand2ndoscillation
+maximum at Korea, at the suitable energy regime for the measu rement with a water cherenkov detector.
+It requires fiveyears each for neutrino and anti-neutrino ru n.
+Comparison of eachscenario isshowninTable2. Studyiscont inuing toseekfor optimum choice
+tomaximize potential of the research.
+6Accelerator based neutrino project inJapan
+Table3summarizesacceleratorbasedneutrinoprojectinJa pan. WhenK2KandT2Kprojectsstarted,the
+existence of high performance far main detector, SK, made it possible to concentrate on neutrino beam
+10Fig.10:Scenario3: J-PARC toKamiokaandKoreaLongBaseline Neutri noExperiment
+Table 2:Comparisonofpossiblescenariosfornewgenerationdiscov eryexperimentwithJ-PARC neutrinobeam
+Scenario 1 Scenario 2 Scenario 3
+Okinoshima Kamioka Kamiokaand Korea
+Baseline (km) 658 295 295 and 1000
+Off-Axis Angle (◦) 0.8(almost on-axis) 2.5 2.5and 1
+Method νeSpectrum Shape Ratio between νeand¯νeRatio between 1st and 2nd Max.
+Ratio between νeand¯νe
+Beam 5yearsνµ 2.2years νµ 5yearsνµ
+then Decide Next and 7.8years ¯νµ and 5years ¯νµ
+Detector Technology Liq. ArTPC Water Cherenkov Water Cherenkov
+Detector Mass (kt) 100 2 ×270 270+270
+sourcerelatedpreparation. Asforthe3rdgeneration exper iment, theexistenceofJ-PARCneutrinobeam
+make it possible toconcentrate on far detector issues after T2Kstarts up.
+To complete present project T2K successfully and realize ne w generation discovery experiment,
+following issues are important.
+– Deliver high quality experimental output from T2K assoon a s possible.
+– Realize quick improvement of accelerator power toward MW- class power frontier machine.
+– Validate beam line components tolerance (especially pion production target related issues) toward
+MW proton beam.
+– Conduct intensive R &D on realization of huge liquid Argon time projection chambe r and water
+cherenkov detector.
+Healthy scientific competition and cooperation in the world is key to promote high energy physics. It is
+11Table 3:Acceleratorbasedneutrinoprojectin Japan
+K2K T2K 3rd Generation Experiment
+High Power KEKPS J-PARCMR J-PARCMR
+Proton 12GeV 0.005MW 30GeV 0.75MW 30GeV 1.66MW
+Synchrotron Existing BrandNew Technically Feasible Upgrade
+NeutrinoBeamline K2K J-PARC J-PARC
+Neutrino Beamline Neutrino Beamline NeutrinoBeamline
+Brand New BrandNew Existing
+FarDetector Super Kamiokande Super Kamiokand BrandNew
+Existing at Existing at - Detector Technology ?
+KAMIOKA KAMIOKA - Place (AngleandBaseline) ?
+1st Priority Physics Case Neutrino Oscillation Neutrino Oscillation Lepton Sector CPViolation
+νµDisappearance νµ→νe Proton Decay
+welcomed to cooperate inany aspects.
+In Japan wewill proceed asfollowing.
+– Short term
+– Beam commissioning of J-PARCMR hasstarted May-2008.
+– Commissioning of J-PARCneutrino beam facility hasstarte d inApril-2009.
+– T2K is aiming for the first results in 2010 with 100 kw ×107seconds integrated proton
+power on target tounveil below CHOOZexperimental limit wit hνeappearance.
+– Middle term
+– T2Kdatawith1-2MW ×107secondsintegrated protonpowerontarget willprovidecrit ical
+information on θ13, which guides the future direction of the neutrino physics. (In any case,
+complete T2K proposal of 3.75 MW ×107seconds.)
+– Achieve MR power improvement scenario toward MW-class pow er frontier machine (KEK
+Roadmap).
+– Submit proposal "J-PARC to Somewhere Long Baseline Neutri no Experiment and Nucleon
+Decay Experiment withHuge Detector" and construct huge det ector.
+– Long term
+– Discover CP violation in lepton sector and proton decay, an d solve “Quest for the Origin of
+Matter Dominated Universe"
+References
+[1] Y.Itow et al., “TheJHF-Kamioka neutrino project,” arXiv:hep-ex/01060 19.
+[2] M. Apollonio et al.[CHOOZ Collaboration], “Limits on neutrino oscillations f rom the CHOOZ
+experiment,” Phys. Lett. B 466, 415 (1999)
+[3] G.Steigman Ann.Rev.Astron.Astrophys, 14(1976)339; A .D.Sakharov PismaZh.ETF5(1967)32.
+[4] Talk given at the 4th International Workshop on Nuclear a nd Particle Physics at J-PARC (NP08).
+Slides available at http://j-parc.jp/NP08/
+12[5] A. Badertscher, T. Hasegawa, T. Kobayashi, A. Marchionn i, A. Meregaglia, T. Maruyama,
+K. Nishikawa and A. Rubbia, “ A Possible Future Long Baseline Neutrino and Nucleon Decay
+Experiment with a 100 kton Liquid Argon TPC at Okinoshima usi ng J-PARC Neutrino Facility,”
+arXiv:0804.2111
+13
\ No newline at end of file
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+RFID et nouvelles technologies de communication; enjeux 
+économiques incontournables et problèmes d’éthique.  
+ 
+  André Thomas  
+Centre de Recherche en Automatique de Nancy 
+ENSTIB – B.P. 1041 88051 EPINAL Cedex. France  
+Andre.Thomas@cran.uhp-nancy.fr
+  
+ 
+RÉSUMÉ : 
+Actuellement, le code à barres est progressi vement remplacé par des puces RFID (pour 
+Radio-Frequency Identification). Des apports dans des domaines tels que la traçabilité, 
+la lutte contre les « contre-façons », le p ilotage de la production, la domotique, etc… 
+ont été démontrés. Mais on a vu même des puces sous-cutanées qui peuvent identifier 
+des humains. Il y a donc des questions d’éthique  sérieuses quant à leur application dans 
+le monde actuel.  L’article s’attachera donc, dans une première partie, à présenter des éléments de techniques de la RFID, puis s’appuyant sur une description globale des applications 
+actuelles, il en dégagera des apports et de s perspectives intéressantes pour le monde 
+économique. Enfin la deuxième partie, tentera de poser les risques relatifs aux individus 
+et à la société en général, pour ensuite  poser quelques questionnements relatifs aux 
+problèmes d’éthique nécessitant des normalisations et réglementations.  
+ 
+ 
+ MOTS-CLÉS
+ : RFID, Identification, Produits intelligents, Environnements ambiants 
+KEYWORDS: RFID, Identification, Intelligent products, Ambient environment 
+ 2     Revue. Volume X – n° x/année 
+ 
+ 
+INTRODUCTION 
+   
+Actuellement, le code à barres est progressi vement remplacé par des puces RFID (pour 
+Radio-Frequency Identification) ou « étique ttes» communicantes ayant la possibilité de 
+stocker de l’information de manière dynami que et de communiquer sans fil avec leur 
+l’environnement ambiant. Or depuis quel ques années, bon nombre de chercheurs et 
+industriels ont montré que l’on pouvait tirer  parti de cette « instrumentation des 
+produits » via ces technologies. Des apports da ns des domaines tels que la traçabilité, la 
+lutte contre les « contre-façons », le pilotage de la production, la domotique, etc… ont été démontrés.  La prolifération de ces puces semble auj ourd’hui très importante. On cherche à les 
+attacher à des produits usuels, tels les passeports pour augmenter la sécurité, par exemple. Il suffit de présenter un « passe  » à une borne de métro pour valider (et 
+compter) ses trajets. Il n’est presque plus nécessaire de patienter pour valider son identité dans bon nombre d’endroits. On a vu même des puces sous-cutanées qui 
+peuvent identifier des humains. Il y a donc des questions d’éthique sérieuses quant à 
+leur application dans le monde actuel. T echniquement, on peut lire désormais les puces 
+incorporées dans des vêtements et produits or dinaires sans que celui qui porte avec lui 
+ces puces le sente. Serait-il possible de « traquer » des gens grâce à cette technologie ?!  L’article s’attachera donc, dans une première partie, à présenter des éléments de techniques de la RFID, puis s’appuyant sur une description globale des applications 
+actuelles, il en dégagera des apports et de s perspectives intéressantes pour le monde 
+économique. Enfin la deuxième partie, tentera de poser les risques relatifs aux individus 
+et à la société en général, pour ensuite  poser quelques questionnements relatifs aux 
+problèmes d’éthique nécessitant des normalisations et réglementations.  
+LA TECHNOLOGIE 
+ 
+La première expérience de puces radio pourrait être en 1939 lorsque l’armée 
+britannique équipa ses avions d’un systèm e (IFF pour Identification Friend or Foe). 
+Sans vouloir faire toute l’histoire de cette  technologie ni la décrire précisément car 
+aujourd’hui beaucoup d’ouvrages ou d’article s l’ont déjà fait, nous pouvons juste faire 
+remarquer que nous avons évolué vers plus de miniaturisation et de distance de 
+lecture/écriture. Aujourd’hui ces technologies se décrivent par les fréquences et les 
+performances qui les caractérisent. Par ailleurs et pour des raisons évidentes d’interopérabilité, nous avons aussi évolué vers plus de normalisation, à l’initiative de l’Auto-ID center du MIT (1999) qui préconi se le standard EPC Global (créé par 
+l’Uniform Code Council et EAN International).   Titre courant de l’article     3 
+ 1/ Dans quelles bandes de fréquence fonc tionnent les étiquettes électroniques ?   
+Différentes bandes de fréquences aujourd’ hui existent couramment. Pour certaines 
+d’entre elles, les fréquences disponibles va rient selon les régions du monde. De manière 
+très schématique (cf. tableau 1 ci-dessous) on distingue trois grandes zones (la réalité 
+est un peu plus compliquée, notamment en Asie et Océanie).  
+Tableau 1 - répartition des fréquences selon la zone géographique   
+Fréquences Europe et Afrique Amér iques Nord et Sud Asie et Océanie 
+BF  
+(125-135 kHz)  
+ISO 18000-2   125 kHz 125 kHz 125 kHz 
+HF  
+(13,56 MHz)  
+ISO 18000-3   13,56 MHz 13,56 MHz 13,56 MHz 
+UHF  
+(860-960 MHz)  
+ISO 18000-6   865 - 868 MHz  
+( 2W ERP (1) - 
+LBT) 902 - 928 MHz  
+(4W - EIRP) 902 - 928 MHz (2)  
+(Japon 952 - 954 MHz) 
+SHF  (2,45 GHz)  
+ISO 18000-4   2,446 - 2,454 GHz 2,427 - 2,47 GHz 2,4 - 2,4835 GHz 
+1)1W ERP (Effective Radiated Power) = 1,62W EIRP (Equivalent Isotropic Radiated Power)  
+2) Pas d’harmonisation. Par exemple : 952-954 MHz au Japon, 910-914 MHz en Corée   
+2/ Quelles sont les performances des étiquettes électroniques ?   
+On distingue en particulier :  
+ la rapidité de lecture (de manière générale, elle augmente avec la fréquence) ; 
+ la possibilité de lire plusieurs étiquettes de manière quasi-simultanée (par 
+exemple lors d’un passage sur tapis roulant) ; 
+ la distance de lecture.  
+La distance de lecture d’une étiquette varie selon plusieurs paramètres :  
+ la bande de fréquences utilisée ; 
+ les paramètres réglementaires relatifs à cette bande de fréquences, en 
+particulier :  puissance du lecteur ;  modulation et protocole d’émission. 
+ les performances des composants silicium utilisés et la sensibilité du lecteur ; 
+ le recours éventuel à une batterie (qui peut porter cette distance à 100 m) ; 4     Revue. Volume X – n° x/année 
+ l’environnement d’utilisation et le déploiement effectué (niveau de bruit 
+électromagnétique ; présence d’humidité ; matériaux de l’objet étiqueté ; réflexions multiples ; interférences entre lecteurs, géométrie, etc...). 
+Chaque bande de fréquences a ses avanta ges et ses inconvénients par rapport au 
+contexte et aux finalités d’utilisation. Les principales caractéristiques liées aux bandes 
+de fréquences sont présentées dans le tableau 2 ci-dessous.  
+Tableau 2 - principales caractéristiques liées aux bandes de fréquences   
+  BF  HF  UHF  2,45 GHz  
+Sensibilité à 
+l’eau, l’humidité      Relativement sensible 
+de préférence produits 
+secs / atmosphère sèche Forte 
+sensibilité  
+Environnement métallique    déploiement 
+susceptible d’être plus difficile  Potentiellement néfaste, 
+s’il n’est pas géré    
+Caractéristiques 
+des champs  Couplage 
+inductif (*)  Couplage 
+inductif *  Propagation d’onde très 
+sensible aux matériaux de l’environnement (**)  
+Débit  Quelques 
+kbits/s  De l’ordre de 
+100 kb/s  De l’ordre de quelques 
+centaines de kbits/s    
+Portée type 
+pratique(***) De l’ordre du 
+m ; pénètre légèrement le métal  De l’ordre du m De l’ordre de 4-5 m 
+(Europe, US)    
+* les caractéristiques du champ électromagnétique sont relativement maîtrisables.  
+** sensible aux réflexions et à l’absorption - les caractéristiques du champ sont moins faciles à maîtriser.  *** fortement liée à la réglementation applicable.  
+Ainsi ces tags peuvent être passifs ou actifs,  en lecture seule ou en lecture/écriture, à 
+positionnement discret ou géolocalisés, de ta ille de quelques centimètres ou de quelques 
+centièmes de millimètres, etc… 
+ 
+LES APPORTS 
+ 
+Ces technologies proliférant, nous sommes en train de passer à une échelle 
+complètement différente au regard de la communication : nous entrons dans l’ère de 
+« l’internet des objets ». Le réseau mondial  (la toile) des ordinateurs n’est désormais 
+plus suffisant, chaque objet, chaque personne portant un tag pourrait en faire partie ! Le 
+système est prêt, offrant un potentiel d’expl oitation impressionnant. Dans ce contexte, 
+deux sphères sont concernées : celles de la vi e professionnelle et celle de la vie privée. 
+C’est le concept d’ubiquité de la connectivité dans le temps et l’espace.  Titre courant de l’article     5 
+Il est évident qu’une multitude d’usages peut être imaginé dès lors que les objets portent 
+de l’information, peuvent en capter, peuvent en  distribuer à d’autres de leur entourage, 
+voire peuvent participer à des décisions localement… Un produit pourrait tout dire de lui… Nous imaginons encore très difficilement les 
+conséquences d’une telle fonction. Cela conduirait à mettre à disposition des milliards de milliards de données, ce qui est à l’heur e actuelle encore une des plus grosses 
+faiblesses, mais qui ouvre la possibilité de concevoir des « espaces intelligents ». On 
+parle aujourd’hui « d’intelligence ambiante ».  Cette technologie peut améliorer le confort sous diverses formes. L’instrumentation 
+d’une maison pourrait ainsi permettre d’optimiser la consommation d’énergie et la température ambiante en fonction des m ouvements, des actions observés dans les 
+pièces, par exemple. La sécurité des biens et des personnes pourrait aussi être améliorée. Il en est de même de la santé des individus, ne serait-ce que une meilleure gestion des informations qui leur sont propres. Un système d’étiquettes RFID collées su r des kanbans dans une entreprise de 
+l’ameublement a été prototypé et a pu mont rer qu’en assistant l’opérateur dans ses 
+décisions de gestion des priorités entre produ its à réaliser sur un même poste de charge, 
+il a conduit à des gains en productivité, fluidité et niveaux de stocks très significatifs.  
+LES RISQUES 
+ 
+Mais si bon nombre de bienfaits sont déjà visibles et sont aussi prévisibles, nous 
+pouvons mettre en évidence déjà de sérieux ris ques tant pour la santé que pour la liberté 
+individuelle ou collective. - L’impact de la prolifération des ondes radio n’a pas encore été évalué sur la santé. - Les infrastructures mises en place permettent de surveiller les mouvements, voire les activités, des « porteurs » de tags. En effet, (le téléphone portable en est un bon exemple) au-delà des mouvements, la surv eillance de l’évolution des informations 
+contenues dans les tags permet d’avoir une  trace et un suivi des activités qui les ont 
+générées. - Numéroter des objets et des personnes condui ra à faire évoluer l’interprétation de 
+l’identité et de l’individualité, conduisant ains i à une évolution des relations sociales, à 
+une modification des règles de la vie communa utaire et même à une minimisation de la 
+sphère d’intimité. Par ailleurs, des menaces diverses pourraient survenir : - Menaces d’actions inopportunes : dès obser vation, via des tags, que plusieurs objets 
+sont en train de sortir simultanément d’un magasin, mise en fonctionnement d’un 
+système anti-vol (vidéo ou autre) qui pourrait être « brutal » ou mettre les acteurs en 
+situation délicate, alors le fait est normal. - Menaces d’exploitation commerciale : l’identification personnelle des clients associée à la surveillance de leurs consommations dans un magasin pourraient permettre à un logiciel de CRM de « forcer » des publicité s ou incitations à l’achat à l’insu de la 
+clientèle. 6     Revue. Volume X – n° x/année 
+- Menaces de détermination des goûts : concept identique au précédemment. 
+- Menaces de pertes de discrétion sur l’iden tité et localisation : Si on multiplie les tags 
+et les lecteurs et que l’on suit en pe rmanence et de manière géolocalisée les 
+déplacements, on peut reconstruire des trajets, des arrêts réguliers et finir par localiser adresse, identité, habitudes, etc…des personnes (ce système fait déjà partie de projets de recherche pour des applications sur des tr ajets de produits). En élargissant, les 
+personnes rencontrées pourraient de même être déterminées et ainsi le comportement, les actions des personnes seraient déduites. - Menaces d’effets indirects : Si les données  personnelles sont connues dans une base de 
+données, elles pourraient être modifiées et alors si quelqu’un d’autre y accède, des 
+informations fausses des personnes pourraient être diffusées. Tout ceci n’est pas exhaustif, mais n’est pas non plus irréaliste.   Face aux diverses volontés des gouvernements, dans certaines circonstances, des dangers nouveaux pourraient effectivement a pparaître. Par exemple, le Japon s’est 
+donné comme objectif pour 2010 de devenir la « société du réseau ubique », l’idée est 
+d’évoluer vers du contrôle (au sens de l’au tomatique) temps réel de plus en plus de 
+choses grâce à la prolifération de dizaines et  de dizaines de milliards de puces. Diverses 
+applications ont déjà donné de bons résu ltats (gestion de certains animaux, des 
+médicaments dans les hôpitaux, etc…). Cette évolution se fera naturellement par des 
+phases différentes permettant la vulgarisa tion, le retour sur investissements, la 
+crédibilité… On peut citer : 
+ La traçabilité et le suivi d’activité 
+ Le partage d’information (entre lieux, entre sociétés, entre systèmes) 
+ L’auto-organisation (vers le produit intelligent, vers les systèmes ambiants) 
+Nous allons vers un monde où les choses vont être animées … Mais comme dit Michel Alberganti dans son livre, animées « de quelles intentions » ?!  Paradoxalement, les personnes ne sont actuelle ment pas réellement contre. Une enquête 
+réalisée en 2005 a montré que les personnes in terrogées étaient plutôt favorables à la 
+traçabilité car elle permettait plus de contrôle (92%), plus de sécurité (91%), plus de 
+qualité (90%), plus de confiance (89%), plus de responsabilité (83%), plus de 
+ré »activité (72%) …   
+LES PROBLEMES D’ETHIQUE 
+ 
+La loi informatique et libertés reconnaît aux citoyens des droits spécifiques pour 
+préserver leur vie privée et leurs liber tés dans un monde numérique. La CNIL 
+(Commission Nationale Informatique et Li berté) a été créée en 1978 pour aider les 
+personnes à contrôler la bonne application de cette loi. Le système de protection est 
+là… Mais à l’échelle de cette prolifération en aura-t-il les moyens ?  Titre courant de l’article     7 
+L’interprétation des messages et/ou informa tions échangés entre individus n’est pas 
+nouvelle chose. Aujourd’hui, une tranche infime de la population en a le pouvoir 
+(police, gouvernement, …). Avec la généralisa tion de la RFID, il n’en sera plus de 
+même, les sociétés, les privés, toute personne ayant les moyens de lire, de décrypter des 
+informations pourra faire de même. Le pl us notable est encore de savoir qu’elles 
+pourront le faire à l’insu de tous. Tout sera donc sous contrôle … de ceux qui en auront les moyens financiers et/ou techniques ! On peut même imaginer qu’ils pourront transgresser les règles inscrites dans le système global d’information. Seuls, les citoyens « ordinaires »  n’auront plus qu’à les suivre.  Il nous paraît intéressant de classer les effe ts pervers d’une évolution mal contrôlée de 
+cette technologie (ou de toute autre relativ e à l’identification), selon les critères 
+suivants : 
+ Automatisation. 
+L’installation dans un quelconque endroit (entreprise, lieu public, …) d’un ou de 
+plusieurs lecteurs conduit à la mise sous contrôle permanent et passif des objets et 
+personnes tagés. Un lecteur peut ainsi contrôler toute une zone.  
+ Identification 
+Au passage sous le lecteur ou dans le champ de lecture pour les technologies de 
+géolocalisation, il est possible d’identifier de  manière statique, voire évolutive, les 
+entités. Les porteurs de tags ont ainsi « l’ob ligation de collaborer », dans la mesure où 
+des informations peuvent être « forcées » sur les tags à l’insu des porteurs.  
+ Intégration 
+Les tags peuvent aujourd’hui être plus ou moins intégrés aux porteurs (cousus dans un col de chemise, collé sur un objet, voire intégr é dans la matière !). Ils sont donc plus ou 
+moins visibles, plus ou moins à la connaissance des porteurs.  
+ Authentification 
+Les tags peuvent alors servir à l’authentif ication, via des informations privées, des 
+porteurs, ceci pour un usage de gestion de la santé, pénal ou de contrôle de la vie privée.  Ainsi la lecture de ces tags peut induire des problèmes d’interrogation clandestine s’il 
+n’y a pas consentement de l’individu (voire  même de connaissance du fait). On peut 
+imaginer qu’au passage à proximité d’un panneau publicitaire, une propagande apparaisse en fonction de la catégorie socioprofessionnelle du passant !  Le fait d’être aussi sous contrôle, ou « su r écoute », en permanence relèverait plus de 
+l’espionnage que du contrôle de traçabilité fonctionnelle.  Enfin les informations véhiculées pourraient être captées par d’autres acteurs que ceux 
+pour lesquelles elles ont été initialement mises en place et ainsi, dans un monde 
+ubiquitaire, se retrouver dans les mains de  personnes peu scrupuleuses. De la même 8     Revue. Volume X – n° x/année 
+manière, des informations non souhaitées pourraie nt être mises sur les tags et ainsi se 
+déployer dans un système sans la volonté du porteur.  On peut ainsi imaginer des scénarii qui « ju stifieraient » de tels agissements aux yeux 
+de certains, pour l’exemple citons en laissant le lecteur les imaginer : 
+ Scénario de propagande 
+ Scénario criminel 
+ Scénario de surveillance dans un contexte de crise 
+ Scénario de guerre 
+  
+CONCLUSION 
+ 
+Il est ainsi de nos jours évident que si les technologies d’identification, et en particulier 
+les technologies RFID, vont se déployer de manière extrêmement importante et apporteront beaucoup de gains en productiv ité, en sécurité, en réactivité par une 
+meilleure traçabilité, il nous faut rester vi gilant sur les dangers qui peuvent 
+l’accompagner. Le phénomène n’est pas exceptionnel. T oute modernisation, toute innovation est 
+toujours sujette à ce genre de risque. La particularité de cette technologie est la 
+dimension du déploiement qui lui est inhére nt, d’une part, et les domaines mêmes qui 
+seront concernés, d’autre part. A ce jour, ces derniers ne nous sont pas encore entièrement connus. Et c’est une des raisons pour lesquelles nous en avons peur. Il nous faut donc être vigilants et la 
+meilleure manière de l’être est d’en étudier  toutes les possibilités afin de les rendre 
+publiques et à la portée du maximum de personnes. Titre courant de l’article     9 
+ 
+ANNEXE 
+ 
+ 
+ 
+ 
+BIBLIOGRAPHIE  
+ 
+- Alberganti M.  « Sous l’œil des puces » - Ed Actes Sud – 2007. - Mission économique « La RFID à Singapour » - Mars 2007. - Le Monde « Des puces et des ordinateurs » - Fév 2009. - Melski, A. Muller, J. Zeier, A. Schum ann, M. “Improving supply chain visibility 
+through RFID data” - Data Engin eering Workshop, 2008. ICDEW 2008. IEEE 24th 
+International – 2008 - Steven Dorrestijn –« Michel Foucault et l’éthique des techniques » - Mémoire de 
+master Paris X - David H Nguyen and Alfred Kobsa “Better RFID Privacy Is Good for Consumers, and Manufacturers, and Distributors, and Retailers » - Revue PEP – 2006 - Juels, A. « RFID security and privacy: a research survey” - Selected Areas in 
+Communications, IEEE Journal on – 2006. - Juels A. Attack on a cryptographic RFID device. Guest Column in RFID Journal, February 28, 2005. Available from: www.rfidjournal.com/article/articleview/1415/1/39/. - Mark Roberti. Big brother’s enemy. RFID Journal, July 2003. Available from:www.rfidjournal.com/article/articleview/509/1/1/.  10     Revue. Volume X – n° x/année 
+- K. Sachs, P. Guerrero*, «  Winter RFID seminar » - Technische Universit¨at 
+Darmstadt Dept. of Computer Science – 2006 - Marc Langheinrich: RFID and Privacy.
+ In: Milan Petkovic, Willem Jonker (Eds.): 
+Security, Privacy, and Trust in Modern Data Management. Springer, ISBN 978-3-540-69860-9, pp. 433-450, Berlin Heidelberg New York, July 2007 - Association for Automatic Identification and Mobility. The ROI of privacy invasion. RFID Connections Webzine, January 2004. Available from: www.aimglobal.org/technologies/rfid/resources/articles/jan04/ 0401-roispy.htm. - Auto-ID Center/EPCglobal, Cambridge, MA, USA. 900 MHz Class 0 Radio Frequency (RF) Identification Tag Specification, 2003. Available from: www.epcglobalinc.org/standards_t echnology/Secure/v1.0/UHF-class0.pdf. 
+- Resolution on radio frequency identification. 25th International Conference of Data Protection and Privacy Commissioners, November 2003. Available from:www.privacyconference2003.org/commissioners.asp. - Klaus Finkenzeller. RFID Handbook: Fundamentals and Applications in Contactless Smart Cards and Identification. John Wiley & Sons, Ltd, 2003. 
\ No newline at end of file
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+arXiv:1001.0454v1  [cond-mat.str-el]  4 Jan 2010APS/123-QED
+Evaluations of pressure-transmitting media for cryogenic experiments
+with diamond anvil cell∗
+Naoyuki Tateiwa†and Yoshinori Haga
+Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai, Naka, Ibaraki 319-1195, Japan
+(Dated: October 13, 2018)
+The fourteenkindsofpressure-transmittingmediawere eva luatedbytherubyfluorescence method
+at room temperature, 77 K using the diamond anvil cell (DAC) u p to 10 GPa in order to find ap-
+propriate media for use in low temperature physics. The inve stigated media are a 1:1 mixture by
+volume of Fluorinert FC-70 and FC-77, Daphne 7373 and 7474, N aCl, silicon oil (polydimethyl-
+siloxane), Vaseline, 2-propanol, glycerin, a 1:1 mixture b y volume of n-pentane and isopentane, a
+4:1 mixture by volume of methanol and ethanol, petroleum eth er, nitrogen, argon and helium. The
+nonhydrostaticity of the pressure is discussed from the vie wpoint of the broadening effect of the
+rubyR1fluorescence line. The R1line basically broadens above the liquid-solid transition pressure
+at room temperature. However, the nonhydrostatic effects do constantly develop in all the media
+from the low-pressure region at low temperature. The relati ve strength of the nonhydrostatic effects
+in the media at the low temperature region is discussed. The b roadening effect of the ruby R1line
+in the nitrogen, argon and helium media are significantly sma ll at 77 K, suggesting that the media
+are more appropriate for cryogenic experiments under high p ressure up to 10 GPa with the DAC.
+The availability of the three media was also confirmed at 4.2 K .
+PACS numbers:
+I. INTRODUCTION
+The development of the diamond anvil cell (DAC) has
+revolutionized high-pressure research in various scientific
+fields such as physics, chemistry, geophysics, materials
+science and biology [1, 2]. Various types of experiments
+can be performed with the DAC. For instance, X-ray
+diffraction experiments, optical absorption and reflectiv-
+ity measurements, Raman and Brillouin scattering stud-
+ies, and transport and thermal measurements under high
+pressure above 10 GPa have become possible. These
+methods have been successfully applied in low temper-
+ature physics where many interesting pressure-induced
+phenomena such as superconductivity have been exten-
+sively studied [3, 4].
+In high-pressure experiments, nonhydrostatic effects
+such as inhomogeneous pressure distribution (pressure
+gradient) and uniaxial (deviatoric) stress need to be re-
+duced becausethe effects canstronglyinfluence the phys-
+ical state of a sample. It is important to use a pressure-
+tranmistting medium in a pressure region where it is in a
+liquid state. However, a liquid medium solidifies through
+a liquid-solid transition on cooling from room tempera-
+ture. The shear stress of a solid medium causes nonhy-
+drostatic pressure in the low temperature region.
+The purpose of the present study is to evaluate the
+pressure-qualities of media used in low temperature
+physics, and in particularfor the stronglycorrelatedelec-
+tron system in which novel types of physical phenomena
+such as non-Fermi liquid behavior or unconventional su-
+∗Review of Scientific Instruments 80, 123901 (2009)
+†Electronic address: tateiwa.naoyuki@jaea.go.jpperconductivity havebeen extensively studied [5, 6]. The
+electronic state of the system is generally sensitive to
+small amounts of impurities or nonhydrostatic effects. A
+recent example is the case of the “FeAs-based supercon-
+ductor” CaFe 2As2in which superconductivity was dis-
+covered above 0.5 GPa in high-pressure studies using or-
+ganic media [7, 8], while it was not observed with a he-
+lium medium [9]. This discrepancy may result from sen-
+sitive structural instability to the uniaxial stress of the
+compound. nonhydrostatic effects cannot be avoided in
+the low temperature region. It is necessary to compare
+quantitatively the strength of the nonhydrostatic effects
+of various kinds of pressure-media.
+Most of previous studies on pressure-media have con-
+cerned the hydrostaticity of the pressure at room tem-
+perature [10–13]. Very few studies have been carried out
+on the pressure-qualitiesof media in the low temperature
+region up to 3 GPa [14, 15]. In this study, therefore, the
+fourteen kinds of pressure-transmitting media were stud-
+ied by the ruby fluorescence technique up to 10 GPa at
+room temperature, 77 and 4.2 K.
+II. EXPERIMENTAL
+The fourteen kinds of investigated media are a 1:1
+mixture by volume of Flourinert FC-70 and FC-77
+(Flourinert FC70/77, Sumitomo 3M), Daphne 7373
+and 7474 (Idemitsu Kosan), NaCl, Vaselin, silicon oil
+(polydimethylsiloxane with a kinematic viscosity of 1
+mm2·s−1, Shin-Etsu Chemical), 2-propanol, glycerin, 1:1
+mixture by volume of n-pentane and isopentane (pen-
+tane mixture), a 4:1 mixture by volume of methanol
+and ethanol (4:1 M-E mixture), petroleum ether (Wako
+Chemical Industries), nitrogen, argon and helium. The2
+strength of the nonhydrostatic effects in the media
+was evaluated by the ruby (Al 2O3: Cr3+) fluorescence
+method using a clamp-type diamond anvil cell originally
+designed by Dunstan and Spain [16–18]. Figure 1 shows
+a schematic illustration of the DAC setup and the fluo-
+rescencespectraof the ruby Rlines for the representative
+media of NaCl, glycerin, 4:1 M-E mixture and argon at
+77 K and around 10 GPa. The culet of the diamond
+was 800 µm. A sample chamber of 400 µm in diame-
+ter was prepared in a 304 stainless steel gasket. Small
+ruby chips, containing 0.5 wt% of Cr 2O3, were uniformly
+placed in the chamber filled with a pressure-transmitting
+medium. The diameter of the chips was less than 10 µm.
+It was confirmed by X-ray diffraction analysis that the
+chips were single crystals.
+The ruby fluorescence was measured using a charged
+coupled device (CCD) spectrometer. More details on
+the optical system are given in Reference 19. A diode
+pumped solid-state (DPSS) green laser light (532 nm),
+introduced into the sample chamber through fiber op-
+tics, excited the ruby chips in the chamber. The diam-
+eter of the beam line was 600 µm, larger than that of
+the sample chamber. The present work reveals spatially
+averagedinformation on the nonhydrostaticeffects in the
+whole sample chamber.
+When the pressure becomes nonhydrostatic, the two
+R1andR2lines broaden. The pressure shift of the ruby
+R1line is sensitive to the uniaxial stress [20–22]. In the
+presentstudy, anumberofsmallrubysinglecrystalswere
+uniformly placed inside the chamber with the directions
+of the single crystals being random. The broadening of
+the ruby R1line reflects both the inhomogeneous pres-
+sure distribution and uniaxial stress pressure.
+The peak positions and line widths of the ruby R1and
+R2fluorescence lines were determined by deconvoluting
+measured R-line spectra into a pair of pseudo-Voigtfunc-
+tions that represent the contribution of the ruby R1and
+R2lines with a linear background. The pressure was
+determined using the hydrostatic ruby pressure scale by
+Zha, Mao and Hemley [23, 24].
+The pressure-transmittingmedium wasloadedinto the
+sample chamber at room temperature except for nitro-
+gen, argon and helium done using a purpose built cryo-
+genic device at 77 or 1.4 K [17, 18]. The pressure was
+applied and changed at room temperature. The ruby flu-
+orescence spectra were measured after the pressure and
+the width of the ruby R1line had been stabilized. It took
+several hours for the quantities to be stabilized when a
+medium was in a solid state under high-pressure. The
+DAC wasslowly cooleddown to 77K and 4.2K using liq-
+uid nitrogen and helium, respectively. It should be noted
+here that the pressure did not significantly change dur-
+ing the cooling process with the present DAC where load
+was maintained using springs. The difference in pres-
+sure between at 4.2 K and 300 K was in the order of a
+few % when organic and argon media were used. The
+pressure-change with the nitrogen and helium media will
+be discussed later.   Ar laser
+λ = 532 nmRuby chips Pressure medium
+Ruby fluorescenceIntensity (a.u.)
+700 695
+Wavelength (nm) 10.1 GPa
+      9.6  Glycerin
+ Argon            4:1 
+methanol-ethanol NaCl
+    10.6
+      9.577 K R1R2
+FIG. 1: (Color online) Schematic illustration of the DAC set
+upand the fluorescence spectraof the ruby Rlines with repre-
+sentative media of NaCl, glycerin, 4:1 M-E mixture and argon
+at 77 K and around 10 GPa
+Here we note differences between the present work and
+the previous studies on pressure-media from a technical
+point of view. The references 10 and 13 reported the
+inhomogeneous pressure distribution inside the sample
+chamber at room temperature [10, 13]. The previous
+studies in references 11, 12, 14 and 15 revealed spatially
+local information on the nonhydrostatic effects around
+the center of the chamber as the size of the “sensors”
+used to detect the effects, for example the ruby chips,
+Cu2O or NaCl, was very small compared with the sam-
+ple space [11, 12, 14, 15]. This work shows spatially aver-
+aged information on the nonhydrostatic effects inside the
+sample chamber. The observed broadening effect of the
+rubyR1line reflects both the inhomogeneous pressure
+distribution and uniaxial stress pressure. The purpose of
+the present study is to clarify the relative strength of the
+nonhydrostatic effects of the media in the low tempera-
+ture region.
+As shown in Figure 1, the broadening effect on the
+R1line depends on the media at 77 K and around 10
+GPa. A clear broadening effect was observed in the ruby
+R1lines for NaCl and glycerin. However, the lines for
+the 4:1 M-E mixture and argon are comparably sharper.
+Therefore, it is possible to discuss the relative strength
+of the nonhydrostatic effects of the media from the width
+of the ruby R1line.
+III. RESULTS
+In this section, weshow the pressuredependence ofthe
+full-width at half maximum (FWHM) of the ruby R1line3
+with the media. The pressure dependence was studied
+several times with independent settings for each of the
+media. The guide lines in the figures are fitted curves
+of the pressure dependences of the FWHMs with poly-
+nomial functions. The open circles and squares indicate
+the FWHMs at room temperature in the first and sec-
+ond runs, respectively. The open up and down-pointing
+triangles indicate the FWHMs at 77 K in the first and
+second runs, respectively. The solid triangles indicate
+the FWHMs at 4.2 K. The nonhydrostatic effects are
+discussed from the value of ∆FWHM( P) (= FWHM( P)-
+FWHM(0)). Here, FWHM( P) and FWHM(0) are the
+line widths of the ruby R1line at high pressure and am-
+bient pressure, respectively, obtained by the analyses of
+the ruby profile with the Voigt function where the instru-
+mental resolution was taken into account. Depending on
+the size of the additional broadening ∆FWHM at 5 GPa
+and 77 K, the media are classified into three groups (I,
+II and III).
+A. Group I: 1:1 mixture of Fluorinert FC-70 and
+FC-77, Daphne 7373, NaCl, silicon oil
+(polydimethylsiloxane) and Vaseline
+Fluorinerts and Daphne 7373 have been widely used
+in high pressure studies using piston cylinder type high-
+pressure cells because the media have the advantage of
+being chemically inert [25–27]. The hydrostatic-limit
+pressures(solidification pressures)ofFlourinert FC70/77
+and Daphne 7373 at room temperature were determined
+to be 1.2 and 2.4 GPa, respectively.
+Figures 2 (a) and (b) show the pressuredependences of
+the FWHMs for Fluorinert FC70/77 and Daphne 7373.
+At room temperature, the FWHMs start to increase
+above the solidification pressures. The value of FWHMs
+for the media is about 1.1 nm at 10 GPa, indicating
+strong nonhydrostatic effects. A strong broadening ef-
+fect was also observed at 77 K. The value of ∆FWHMs
+for the media is about 0.7 nm at 10 GPa. The present
+results suggest that they are not suitable media in the
+higher-pressure region.
+Figure 2 (c) shows the pressure dependences of the
+FWHMs for the solid medium of NaCl at room temper-
+ature and 77 K. The FWHMs at both temperatures in-
+crease non-linearly in the lower pressure region below 1
+GPa and show a tendency to saturate at the high pres-
+sure region above 5 GPa. The pressure dependence is
+quite different from those of liquid media. The value of
+∆FWHM at 10 GPa is about 0.5 nm at 77 K. It is sug-
+gested that the hydrostatic pressure is realized in a very
+small pressure region even at room temperature.
+The results of the NaCl medium are compared with
+those of previous studies that reported the pressure dis-
+tribution inside a sample chamber filled with the NaCl
+medium becomes spatially inhomogeneous above 4 GPa
+at room temperature [10]. It should be noted, however,
+that the observed broadening effect in the present study1.0
+0.5
+0.0Fluorinert FC70/77R.T
+ 77K (a) 1st.
+ 2nd.
+ 
+ 1st.
+2nd.
+1.0
+0.5
+0.0FWHM (nm)Daphne7373
+R.T
+77 K (b)
+1.0
+0.5
+0.010 5 0
+Pressure (GPa) NaCl
+R.T
+77 K (c)
+FIG. 2: (Color online)Pressure dependence of full-width at
+half maximum (FWHM) of the ruby R1fluorescence line at
+room temperature and 77 K for (a)Fluorinert FC70/77, (b)
+Daphne 7373, and (c) NaCl. Guide lines in the figures are
+fitted curves of the pressure dependences of the FWHMs with
+polynomial functions.
+reflects both the uniaxial stress and inhomogeneous pres-
+sure distribution inside the chamber, as mentioned in the
+previous section. This suggests that uniaxial stress is the
+main cause of the broadening effect in the low-pressure
+region.
+We show the experimental results with the silicon
+oil and Vaseline. Silicon oils such as polydimethylsil-
+ioxane, polyethylsiloxane and hexamethyldisiloxane have
+been accepted as good media for a long time [28–
+30]. Polydimethylsilioxane ((CH 3)3SiO-[SiO(CH 3)2]n-
+(CH3)3Si) with a kinematic viscosity of 1 mm2·s−1was
+tested. Vaseline(petroleum jellyorpetrolatum)hasbeen
+occasionally used in high-pressure experiments [31, 32].4
+1.0
+0.5
+0.0 FWHM(nm)
+10 5 0
+Pressure (GPa)(b) Vaselin
+R.T
+ 77 K1.0
+0.5
+0.0(a) Silicon oil
+R.T
+77 K1st.
+2nd.
+1st.
+2nd.
+ 
+FIG. 3: (Color online)Pressure dependence of full-width at
+half maximum (FWHM) of the ruby R1luminescence line at
+room temperature and 77 K for (a)silicon oil (polydimethyl-
+siloxane) and (b) Vaseline.
+However, there have been no reports on its suitability as
+a pressure-medium as far as is known.
+Figure 3 (a) and (b) show the pressure dependences
+of the FWHMs with silicon oil and Vaselin at room tem-
+perature and 77 K. The FWHMs start to weakly increase
+above 3 GPa. The ruby R1lines for the media do not
+show a strong broadening effect at 10 GPa, thus indicat-
+ing weaker nonhydrostatic effects of the media compared
+with those of the former media. Previous studies have
+pointed out that silicon oil is as good a pressure medium
+as the 4:1 mixture of methanol and ethanol up to 20 GPa
+at room temperature [33, 34]. However, a small but fi-
+nite broadening effect does appear above 3 GPa, thus
+suggesting the pressure-quality of the silicon oil to be
+lower than that of the 4:1 methanol-ethanol mixture. At
+77 K, the FWHMs for the media increase rapidly with
+increasing pressure above about 1 GPa. The value of
+the ∆FWHMs for the media at 10 GPa is about 0.5 nm,
+comparable with that of NaCl.
+B. Group II: 2-propanol, glycerin and Daphne 7474
+The secondary alcohol of 2-propanol (Isopropanol:
+CH3CH(OH)CH 3) has been accepted as a good pressure-transmitting medium in high pressure studies. The
+hydrostatic-limit pressure of 2-propanol was determined
+to be 4.2 GPa by the ruby fluorescence method at
+room temperature [10]. The trivalent alcohol of glyc-
+erin (glycerol, C 3H5(OH)3) has been used mainly for
+medical purposes but also in the high-pressure studies
+for a long time [35, 36]. Osakabe and Kakurai pointed
+out that glycerin is the most suitable pressure-medium
+for use in neutron scattering experiments under high
+pressure up to 7 GPa [12]. There have been no re-
+ports on the hydrostatic-limit pressure of the glycerin
+medium. Daphne 7474 is a transparent and chemically
+non-reactive medium recently developed by Murata et
+al.[37]. The solidification pressure of Daphne 7474 was
+reported to be 3.7 GPa at room temperature.
+Figures 4 (a), (b) and (c) show the pressure de-
+pendences of the FWHMs for 2-propanol, glycerin and
+Daphne 7474 at room temperature and 77 K. At room
+temperature, the FWHMs for 2-propanol and Daphne
+7474 start to increase roughly above the solidification
+pressures [10, 37]. The FWHM for the glycerin medium
+starts to increase above 5 GPa, which roughly corre-
+sponds to the solidification pressure at room tempera-
+ture [38]. This suggests the hydrostatic-limit pressure
+to be about 5 GPa at room temperature. At 77 K,
+the FWHMs of the three types of media show differ-
+ent pressure dependences. The FWHMs for 2-propanol
+and Daphne 7474 weakly increase in the low-pressure re-
+gion below 5 GPa but then strongly increase above the
+pressure. The pressure dependence of the FWHM for
+the glycerin medium shows a complex behavior: it in-
+creases non-linearly below 1 GPa, shows a tendency to
+saturate above 2 GPa, and then strongly increases again
+with increasing pressure above 5 GPa. The values of the
+∆FWHMs for the three media are less than 0.2 nm at
+5.0 GPa, significantly smaller than those of the media in
+GroupI. However,the valuesof the ∆FWHMs at 10GPa
+are comparable with those of the media in Group I.
+The value of the FWHM with glycerin is larger than
+those of some of the media in Group I, for example,
+Daphne 7373 and silicon oil, below 2 GPa, thus sug-
+gesting the strength of the nonhydrostatic effects with
+the glycerin medium to be larger than those of the two
+media. This is consistent with a previous evaluation of
+pressure-media by the63Cu-NQR spectra of Cu 2O at 4.2
+K [15].
+C. Group III: 1:1 mixture of n-pentane and
+isopentan, 4:1 mixture of methanol and ethanol,
+petroleum ether, nitrogen, argon and helium.
+We firstly show the results of the 1:1 mixture by vol-
+ume ofn-pentane and isopentane (pentane mixture), the
+4:1 mixture by volume of methanol and ethanol (4:1 M-
+E mixture) and the petroleum ether. The pentane and
+4:1 M-E mixtures have been widely used in high-pressure
+studiesforalongperiodoftime[10,39]. Thehydrostatic-5
+limit pressures at room temperature were determined by
+the ruby fluorescence method to be 7.3 and 10.5 GPa
+for the pentane and 4:1 M-E mixtures, respectively [10].
+Petroleum ether (benzine or petroleum naphtha) is a
+group of liquid hydrocarbon mixtures whose main com-
+ponents are isohexan and n-pentane. It has been mainly
+used as an industrial washing solution but also in high-
+pressure studies for a long time [40, 41]. However, there
+have been no reports on the hydrostatic-limit pressure
+as far as is known. Figures 5 (a), (b) and (c) show the
+pressuredependences ofthe FWMHs ofthe three organic
+media at room temperature and 77 K.
+At room temperature, the FWHMs with the three me-
+dia do not show any strong increase. The FWHM for
+1.0
+0.5
+0.0 (a)2-propanol R.T
+77 K1st.
+2nd.
+3rd.
+  1st.
+ 2nd.
+ 3rd.
+1.0
+0.5
+0.0FWHM(nm) (b) Glycerin
+R.T
+77 K
+1.0
+0.5
+0.0
+10 5 0Pressure (GPa) (c) Daphne 7474
+ 
+R.T
+77 K
+FIG. 4: (Color online)Pressure dependence of full-width at
+half maximum (FWHM) of the ruby R1luminescence line at
+room temperature and 77 K for (a) 2-propanol, (b) glycerin
+and (c) Daphne 7474.1.0
+0.5
+0 FWHM (nm) 4:1 methanol-ethanol
+R.T
+77 K (b)
+1.0
+0.5
+010 5 0
+Pressure (GPa)R.T
+77 K Petroleum ether (c)1.0
+0.5
+0 1:1 n-pentane and isopentane
+R.T
+77 K (a)
+ 1st.
+ 2nd. 1st.
+ 2nd.
+FIG. 5: (Color online)Pressure dependence of full-width at
+half maximum (FWHM) of the ruby R1luminescence line at
+room temperature and 77 K for (a)1:1 n-pentane and isopen-
+tane, (b) 4:1 methanol-ethanol, and (c) petroleum ether.
+the pentane mixture starts to weakly increase above the
+hydrostatic pressure of 7.3 GPa. The FWHM for the
+4:1 M-E mixture starts to weakly increase above 8 GPa,
+smaller than the hydrostatic-limit pressure of 10.5 GPa
+determined at 297 K [10]. It has been reported that the
+liquid-glass transition pressure Pglassis sensitive to tem-
+perature [42]. The value of Pglassin the present exper-
+iments at T= 283±3 K is estimated to be 8.5 ±0.5
+GPa, which is consistent with the present result. The
+FWHM for petroleum ether does not show any clear in-
+crease around Pglassof 6 GPa [43]. It is suggested that
+the mechanical strength (yield and shear strength) of
+solid petroleum ether is not strong enough to distort the
+ruby crystal. At 77 K, the FWHMs for the three media6
+0.5
+010 5 0
+Pressure (GPa) 2nd run at 4.2 K1.0
+0.5
+0FWHM (nm)Nitrogen        1st  run
+        2nd  run 
+         1st run
+         2nd runR.T
+ 77 K (a)
+1.0
+0.5
+0FWHM (nm)Argon
+R.T
+77 K (b)
+0.5
+0
+10 5 0
+ Pressure (GPa) 1st run at 4.2 K
+ 
+ 1.0
+0.5
+0.0FWHM (nm)Helium (c)
+R.T
+77 K
+0.5
+0.0
+10 5 0
+Pressure (GPa) 1st run at 4.2 K
+FIG. 6: (Color online) Pressure dependence of full-width at half maximum (FWHM) of the ruby R1luminescence line at room
+temperature and 77 and 4.2 K for (a) nitrogen, (b) argon, and ( c) helium.
+weakly increase with increasing pressure below 5 GPa.
+The values of the ∆FWHMs at 5 GPa are less than 0.1
+nm. The FWHM for the pentane mixture shows a com-
+parably stronger increase above 8 GPa. The values of
+the ∆FWHMs for the media are less than 0.3 nm at 10
+GPa.
+Finally, we show the results of nitrogen, argon and he-
+lium. It is well-known that these elements produce good
+hydrostatic pressure at room temperature [44–47]. The
+hydrostatic limit pressures at room temperature were de-
+termined to be 13, 9 and 30 GPa for nitrogen, argon
+and helium, respectively. The previous work carried out
+up to 3 GPa revealed that helium does not provide any
+nonhydrostaticity even at 4.2 K and that argon provides
+slightly less hydrostaticity but with a slight improvement
+over the 4:1 mixture of methanol and ethanol [14].
+Figures 6 (a), (b) and (c) show the pressure depen-
+dences of FWHMs for nitrogen, argon and helium, re-
+spectively, at room temperature, 77 and 4.2 K. At room
+temperature, the FWHMs for the media weakly decrease
+with increasing pressure, which is a characteristic fea-
+ture when the pressure is hydrostatic [10]. At 77 K, the
+FWHMs for the media veryweakly increasewith increas-
+ing pressure. The value of the ∆FWHMs for the helium
+medium in particular is less than 0.1 nm below 10 GPa.
+The values of FWHMs for the media at 4.2 K are the
+same as those at 77 K within experimental errors.
+The present work reveals the suitability of nitrogen,
+argon and helium as pressure-media for cryogenic exper-
+iments with DACs. The helium medium has been fre-
+quently used in X-ray diffraction experiments. But the
+application of the medium in electrical transport mea-
+surement has not remarkably advanced, apart from ina few studies [48, 49]. This is due to the large com-
+pressibility of helium in the low-pressure region [50].
+Meanwhile, the compressibility of nitrogen and argon are
+smaller than that of helium[51, 52]. The two media can
+be safely used in electrical transport measurements with
+DACs [53, 54].
+It is interesting to introduce a recent work by Klotz
+et al.in which a weak deviation from the homogenous
+pressure in the sample space of the DAC was thoroughly
+studied using eleven kinds of pressure-media at room
+temperature[13]. They pointed out that the pressure
+quality of the argon medium is worse than that of the
+nitrogen medium above 2 GPa, even at room tempera-
+ture. Unfortunately, the small pressure gradient could
+not be detected in this study as the broadening effect
+reflects the spatially averaged information on the non-
+hydrostatic effects in the sample chamber. It is noted,
+however, that the FWHM (or ∆FWHM) with the argon
+medium is slightlylargerthan with nitrogenabove8GPa
+at 77 K.
+Thepressurechangeduringthecoolingprocess(∆ P/P
+= (P(77K)−P(R.T))/P(R.T)) is in the order of few %
+for the organic and argon media with the present DAC.
+However,the valuesof∆ P/Pforthe heliumandnitrogen
+mediaareabout30and40%, respectively,at2GPa. The
+pressure change decreases with increasing pressure. The
+change is in the order of a few % above 5 GPa with the
+nitrogen medium and is 10 % at 10 GPa with the helium
+medium. This pressure change could be a disadvantage
+when the temperature dependence of a physical quantity
+is measured in a wide temperature region. Also, a large
+change of pressure in the low temperature region could
+cause the nonhydrostatic pressure, making a damage to7
+1.0
+0.5∆FWHM (nm) Fluorinert FC70/77
+ Daphne7373
+ 
+T  = 77 K
+ 
+0.5
+10 5 0 Pressure (GPa) NaCl
+ Silicon oil
+ Vaselin
+ 
+0.5
+0
+10 5 0 Pressure (GPa) Nitrogen
+ Argon
+ Helium
+ 1.0
+0.5∆FWHM (nm) 1:1 n-pentane and 
+                isopentane 
+ 4:1 methanol and ethanol
+ Petroleum ether
+ 
+T  = 77 K
+ 
+1.0
+0.5
+0∆FWHM (nm)
+10 5 0 Pressure (GPa) 2-propanol
+ Glycerin
+ Daphne7474
+T  = 77 K
+    0
+1.0   0
+1.0 Group I
+ Group II Group III
+FIG. 7: (Color online)Pressure dependence of ∆FWHM (= FWHM( P)-FWHM(0)) of the ruby R1line for fourteen pressure-
+media at 77 K. Here, FWHM( P) and FWHM(0) are the values of the fitted function for the FWHM under high pressure and
+ambient pressure, respectively. Error bars indicate avera ged deviations of the experimental data of the FWHM from the fi tted
+line at each pressure.
+a sample even if the helium medium is used [55].
+IV. DISCUSSION AND CONCLUSION
+In this section, the relative strength of the nonhy-
+drostatic effects in the media is discussed from the line
+widths of the ruby R1line. For the sake of simplicity,
+the individual data points areomitted and only the guide
+(fitted) lines for the FWHMs at 77 K in Figures 2 - 6 are
+compared. To discuss the pressure effect clearly, the am-
+bient pressure width FWHM(0) is subtracted from the
+guide line: ∆FWHM = FWHM( P) - FWHM(0). Here
+FWHM(0) was determined for each of the media. Figure
+7 shows the pressure dependences of ∆FWHMs for all
+the media.
+The ∆FWHMs for the media in Group I show the
+large value of above 0.3 nm at 5.0 GPa, indicating the
+strong nonhydrostatic effects of the media. Although
+some of the media in the group have advantages for use
+in high-pressure experiments, they are not suitable me-
+dia in cryogenicexperimentswith the DAC atthe higher-
+pressure region. The nonhydrostatic effects of the media
+in Group II are smaller than those of the media in Group
+I below 5 GPa. However, the effects become stronger
+above the pressure and are comparable with those of the
+media in the former group around 10 GPa. The effects
+of the media in Group III are smaller than those of the
+media in the other groups, even in the low temperature
+region, up to 10 GPa. In particular, the broadening ef-fect of the ruby R1line in the nitrogen, argonand helium
+media are significantly small, suggesting that the media
+are more appropriate for cryogenic experiments with the
+DAC.
+Finally, we mention some notes for interpretation of
+the present results from different points of views.
+(1) The present results are applicable in high-pressure
+studies using opposed anvil type high-pressure cells
+(DAC and Bridgman cells) but not simply in studies us-
+ing multi-anvil type high-pressure cells such as a cubic
+anvil cell or Kawai-type (6-8) multi anvil high-pressure
+cell where a sample with a medium is pressed by six or
+eight anvils[56–58]. Pressurizing a sample highly sym-
+metrically generally reduces the nonhydrostaticity of the
+pressurecaused bya solidpressuremedium and therefore
+achieves a good hydrostatic pressure.
+(2) Ruby crystals are relatively stiff material with a
+bulk modulus of 253 GPa [16]. If the material to be
+studied is strain sensitive or has mechanical strength
+smaller than the ruby crystal, the nonhydrostatic effects
+could influence the physical properties of the material,
+even though the broadening of the ruby R1line would
+be small or absent. There are usually discrepancies be-
+tween hydrostatic-limit pressures determined using dif-
+ferent methods for one pressure-medium. An extreme
+example is the nonhydrostatic effects on elastically soft
+quartz (SiO 2) with a bulk modulus of 37.1 GPa [11].
+(3) Some of the organic media in Groups II and III
+could chemically react with organic materials. There is a
+possibility that small argon and helium atoms could pen-8
+etrate into samples with a cage structure, like fullerenes
+of C60and C 70[59].
+V. ACKNOWLEDGMENTS
+This work was financially supported by a Grant-in-Aid
+for Scientific Research on Innovative Areas ”Heavy Elec-trons” (No. 20102002), Scientific Research of Priority
+Area, Creative Scientific Research (15GS0213) and Sci-
+entific Research (S and B) made by the Ministry of Edu-
+cation,Culture,Sports,ScienceandTechnology(MEXT)
+and Japan Society of the Promotion of Science (JSPS).
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\ No newline at end of file
diff --git a/1001.0455.txt b/1001.0455.txt
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+arXiv:1001.0455v2  [cond-mat.mes-hall]  25 Mar 2010Vibrationally Mediated Control of Single Electron Transmi ssion in Weakly Coupled
+Molecule-Metal Junctions
+Thomas Olsen∗and Jakob Schiøtz
+Danish National Research Foundation’s Center for Individu al Nanoparticle Functionality (CINF),
+Department of Physics, Technical University of Denmark, DK –2800 Kongens Lyngby, Denmark
+(Dated: November 15, 2018)
+We propose amechanism which allows one tocontrol thetransm ission of single electrons througha
+molecular junction. The principle utilizes the emergence o f transmission sidebands when molecular
+vibrational modes are coupled to the electronic state media ting the transmission. We will show
+that if a molecule-metal junction is biased just below a mole cular resonance one may induce the
+transmission of a single electron by externally exciting a v ibrational mode of the molecule. The
+analysis is quite general but requires that the molecular or bital does not hybridize strongly with the
+metallic states. As an example we perform a density function al theory (DFT) analysis of a benzene
+molecule between two Au(111) contacts and show that excitin g a particular vibrational mode can
+give rise to transmission of a single electron.
+PACS numbers: 82.53.St, 34.35.+a
+Several experiments have established that vibrational
+excitations can have a significant effect on the I-Vchar-
+acteristics of single-molecule junctions1–5. In particular,
+the emergence of peaks in the differential conductance
+correspondingto vibrationalfrequencies, shows that tun-
+neling electrons interact with certain vibrational states
+of the molecule, possibly providing a means for con-
+trolling the transmission of electrons. A considerable
+amount of theoretical work have been dedicated to elu-
+cidating the effect of phonons on electronic transport in
+mesoscopic systems. Analysis of model Hamiltonians6–9
+have given qualitative insight into the interaction of tun-
+neling electrons with molecular vibrations, while DFT
+based studies in conjunction with a non-equilibrium
+Green function approach show quantitative agreement
+with experiments10–12. Most efforts so far have been di-
+rectedtowardstheinfluenceofvibrationsontransmission
+functions and I-Vcharacteristics. In the present paper
+we will take a slightly different point of view and parti-
+tion the electronic transmission function into pieces that
+involve different vibrational excitations. We then show
+that the molecular junction may be put in a configura-
+tion where the vibrationally excited molecule allows the
+transmission of a single electron, while transmission is
+forbidden in the vibrational ground state. Controlling
+the vibrational state of the molecule, e.g. by means of a
+laser, then implies controlof single electron transmission.
+The system under consideration is a molecule sand-
+wichedbetweentwometallicleads. Weassumethatthere
+is a single unoccupied molecular state which obtains a fi-
+nite lifetime due to hybridization with metallic states.
+This state will be referred to as the resonance and its
+position may be tuned by applying a gate voltage. If a
+bias voltage is applied to the contacts and the resonance
+is positioned in the bias window, electrons may tunnel
+through the resonance into the downstream contact and
+onewillobserveacurrent. Ifthemoleculeisweaklyinter-
+acting with the metal such that the resonance is well lo-
+calized in energy, one can apply a gate voltage which sit-uates the resonance above the upstream chemical poten-
+tial and no current will be observed. However, if the res-
+onance couples to molecular vibrations, an off-resonant
+enhancementoftransmissionknownastransmissionside-
+bands may be observed6. In particular, if the molecule is
+initially vibrationally excited, off-resonant electrons be-
+low the resonance may tunnel though the molecule by
+absorbing a vibrational quantum of energy. This is il-
+lustrated in Fig. 1. Since transmission is only allowed
+combined with a downward vibrational transition, only
+one or a few electrons may tunnel through the contact
+and the transmission channel will be closed once the vi-
+brational mode reaches the ground state.
+Inspired by these considerations, we perform a quanti-
+tative analysis based on the model Hamiltonian6,13
+H=ε0c†
+aca+/summationdisplay
+i/planckover2pi1ωib†
+ibi+/summationdisplay
+iλic†
+aca(b†
+i+bi) (1)
++/summationdisplay
+kǫLkc†
+LkcLk+/summationdisplay
+k/parenleftBig
+VLkc†
+acLk+V∗
+Lkc†
+Lkca/parenrightBig
++/summationdisplay
+kǫRkc†
+RkcRk+/summationdisplay
+k/parenleftBig
+VRkc†
+acRk+V∗
+Rkc†
+Rkca/parenrightBig
+,
+wherec†
+ais the creation operator for the lowest unoccu-
+pied molecular orbital(LUMO), c†
+Lkandc†
+Rkarecreation
+operators for metallic states in the left and right lead
+respectively and b†
+iare creation operators for the vibra-
+tional normal modes of the molecule with frequencies ωi.
+Thus, the electronic states of the left and right contacts
+are coupled through the molecular resonance which is
+coupled to molecular vibrations with coupling strengths
+λi. We impose the wide band limit in which the contact
+densityofstatesisconstantinthe regionofthe resonance
+and the resonance hybridization with metallic states is
+determined by the parameters
+ΓL= 2π/summationdisplay
+k|VLk|2δ(ε0−ǫLk), (2)2
+E
+ee
+−−
+FIG. 1: A molecule between two metal contacts is represented
+bya resonant state (for example the lowest unoccupied molec -
+ular orbital) andavibrational potential. Ifthemolecule i s ini-
+tially vibrationally excited, an electron below the resona nce
+may tunnel through the molecule by absorbing a quantum of
+vibration. When the molecule is initially in its vibrationa l
+ground state, transmission is not possible.
+and a similar expression for the coupling to the right
+lead Γ R. Without the vibrational coupling, the reso-
+nance spectral function would then be a Lorentzian with
+full width at half maximum given by Γ = Γ L+ΓR. We
+will be interested in the regime Γ , kBT≪/planckover2pi1ωi, but we
+will not restrict ourselves to the classical limit Γ ≪kBT
+where the contact current can be expressed in terms of
+rate equations8,9,14. Instead, we consider the transmis-
+sion matrix Tn(εi,εf) for an electron with initial state
+energyεiand final state energy εf. Within scattering
+theory,thetransmissionmatrixcanbeexpressedinterms
+of a two-particle Green function6which can be evaluated
+exactly in the wide band limit15. The subscript nrefers
+totheinitialstateoftheoscillatorandintegratingoutthe
+final state energy, one obtains the transmission probabil-
+ityTnm(εi) that an electron with initial state energy εi
+is transmitted while the molecule makes the vibrational
+transition n→m.
+In the appendix we have calculated the transmission
+matrix in the ground and first excited state and in Fig.
+2 we show the transmission probability corresponding to
+four different vibrational transitions of the molecule. It
+showsthatincomingelectronswithenergiesbelow ε0−/planckover2pi1ω
+have a vanishing probability of transmission (/summationtext
+nT0n)
+when the molecule is in its vibrationalgroundstate. This
+meansthatifabiasvoltageisappliedsuchthattheFermi
+leveloftheupstreamcontactisat εF=ε0−/planckover2pi1ωnocurrent
+will be observed when the molecule is in its vibrational
+ground state. If the molecule is vibrationally excited,
+e.g. by means of a IR laser, transmission becomes possi-
+ble through the low lying vibrational sideband ( T11and
+T10). However, the first electron which is transmitted
+through the T10channel induces a transition to the vi-
+brational ground state and thus closes the transmission
+channelcompletely. Hence, the neteffect isthat applying
+a short laser pulse to the molecular contact can induce
+the transmission of a few electrons.
+The distribution of the number of electrons being
+transmitted as a result of a vibrational excitation de-FIG. 2: Transmission probabilities calculated from (1) as a
+function of incoming electron energy with Γ L= ΓR= 0.04/planckover2pi1ω
+andλ= 0.4/planckover2pi1ω. Below the resonance energy ε0the ground
+state transmission functions T00(εi) andT01(εi) essentially
+vanish. The insert shows the lower sideband, where stimu-
+lated emission of a vibrational quantum T10(εi) is the domi-
+nating transmission channel.
+pends on the ratio T11/T10shown in Fig. 3 as a function
+of the vibrational coupling. For small coupling param-
+eters the ratio approaches zero and the first electron to
+be transmitted is therefore highly likely to induce a vi-
+brational decay and close the channel. For λ≪/planckover2pi1ωwe
+thus have T11≪T10which is needed for single electron
+transmission. One might worry that the inelastic trans-
+mission amplitude may become too small for anything to
+happen in this case. However, the absolute amplitude of
+sideband transmission can easily be small if the vibra-
+tional lifetime of the molecule is long. For physisorbed
+molecules the lifetime of a vibrational state is typically
+on the order of nanoseconds, whereas for example with
+Au(111), we can use the density of states to estimate
+that∼40 electron hit each surface atom per picosecond
+within 0.1eVof the Fermi level. Thus, as long as T10is
+ontheorderof ∼1×10−4therewillbe plentyofattempts
+to result in a single transmission event.
+FIG. 3: Ratio of elastic and inelastic transmission probabi li-
+ties in the first vibrationally excited state. At weak coupli ng
+the elastic transmission T11goes to zero indicating that a
+vibrational excitation results in only a single electron be ing
+transmitted.3
+hω
+µRLµ
+BeV
+GV
+FIG. 4: The principle of single electron transmission. A sma ll
+bias is applied such that the bias window just covers the res-
+onance: eVB∼Γ and a gate voltage is tuned such that the
+resonance is located at: ε0∼µL−eVB/2+/planckover2pi1ω. In the vibra-
+tional ground state the transmission function T00(solid line)
+is zero in the bias window. Exciting the first vibrational sta te
+changes the transmission function which is dominated by the
+inelastic part T10(dashed line) in the bias window. A vibra-
+tional excitation of the molecule will thus result in a singl e
+electron being transmitted.
+The setup is illustrated in Fig. 4 where a small bias
+voltageVB∼Γ/ehas been applied and a gate voltage
+VGhas been tuned such that the position of the reso-
+nance is located /planckover2pi1ωabove the bias window. It is crucial
+that the electronic resonance has a width much smaller
+than the quantum of vibration (Γ ≪/planckover2pi1ω), since otherwise
+there will be a small but constant transmission probabil-
+ity when the molecule is in the vibrational ground state.
+As an example we have performed a density func-
+tional theory (DFT) study of a benzene molecule sand-
+wiched between two Au(111) contacts. The calculations
+were performed with the code gpaw16,17which is a real-
+space DFT code using the projector augmented wave
+method18. The contact were simulated by a three layer
+Au(111) slab where the the top layer has been relaxed.
+We used a supercell with 12 Au atoms in each slab layer
+whichweresampledbya4 ×4gridofK-pointsand12 .2A
+of vacuum. Benzene were added with its plane parallel
+to the surface and the adsorption energy was calculated
+as a function of distance to the slab. This is shown in
+Fig. 5for PBE19, revPBE20and vdW21functionals. The
+PBE and revPBE functional shows weak and no bonding
+respectively whereas the vdW functional shows a 0 .3eV
+minimum at 3 .6A. The weak van der Waals bonding
+indicates that benzene is physisorbed on Au(111) which
+means that the molecular orbitals are weakly hybridized
+with metallic states as required by vibrationally medi-
+ated transmission of single electrons. This can also be
+seen explicitly from the Kohn-Sham projected density of
+states from which we estimate the width of HOMO and
+LUMO resonances to be Γ L∼0.01eV. It should beFIG. 5: Adsorption energy of benzene on Au(111) as a func-
+tion of distance to the surface, calculated with three differ ent
+functionals. The molecular states are weakly hybridized wi th
+the metallic states and only the functional including the va n
+der Waals interaction gives the correct adsorption well.
+noted that we assumed a LUMO state in Eq. (1) and
+Fig. 2, but the analysis is equally valid for transmis-
+sion of hole states mediated by the HOMO and in the
+following we will consider both types of resonance. Ben-
+zenehastwodegenerateHOMOsandtwodegenerateLU-
+MOs which have the potential to mediate transmission
+of electrons through the molecule. One of the LUMOs
+is shown in Fig. 6 and it is expected that a transient
+occupation of the orbital may induce internal forces in
+the molecule and thus couple to the vibrational modes of
+the molecule. We have performed a DFT-based normal
+mode analysis of the benzene molecule which has 36 vi-
+brationalmodes. Thereare 6 degeneratehighly energetic
+modes with /planckover2pi1ωi∼0.39eVand the rest of the modes are
+evenly distributed in the interval /planckover2pi1ωi∼0−0.20eV. The
+high energy modes involve the hydrogen atoms oscillat-
+ing in the plane of the molecule along the individual H-C
+bonds as shown in Fig. 6. The HOMO and LUMO states
+are expected to couple to several of the molecular vibra-
+tional states but we will focus on the high energy modes
+which have highly separated vibrational sidebands in the
+weak coupling limit. The coupling constants λican be
+related to the excited state potential energy surface Va
+associatedwith the LUMO being occupied or the HOMO
+being emptied22:
+λi=li√
+2∂Va
+∂ui/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ui=u0
+i, l i=/radicalbigg
+/planckover2pi1
+miωi,(3)
+whereuiis the coordinate of the i’th normal mode, u0
+i
+its equilibrium position in the electronic ground state,
+miits effective mass and Vathe excited state potential
+energy surface associated with the resonance22,23. To
+obtainVawe have used the method of linear expansion
+∆ self-consistent field DFT23which allows us to calcu-
+late the excited state energies while moving the atoms
+along the high energy mode where the 6 hydrogen atoms
+move in phase (see Fig. 6). The results for this mode4
+FIG. 6: Left: The lowest unoccupied orbital (LUMO) of ben-
+zene. The red and blue surfaces are positive and negative
+isosurfaces of the wavefunction. Right: The vibrational mo de
+of highest energy which can be excited by a transient occu-
+pation of the LUMO.
+areλHOMO= 27meVandλLUMO= 9meVwhich
+should be compared with the quantum of oscillation
+/planckover2pi1ωi∼0.39eV. The coupling is thus rather weak and we
+obtain maximum probabilities of inelastic transmission
+of 5×10−3and 5×10−4respectively (with Γ = 0 .01eV)
+at the lower vibrational sideband. However, assuming
+the vibrational lifetime to be on the order of nanosec-
+onds, the probabilities are most likely large enough that
+an elastic transmission event will occur. Furthermore,
+a small coupling constant means that the ratio T11/T10
+becomes very small and vibrational excitation will thus
+nearly always result in only a single electron being trans-
+mitted.
+In summary, we have presented a method, which al-
+lows one to control the transmission of single electrons
+in weakly coupled molecule-metal junctions. The trans-
+mission is mediated by exciting a vibrational mode of the
+molecule while a gate voltage is tuned such that the reso-
+nant state is kept a quantum of vibrational energy above
+the bias window. It is assumed that such an excitation
+can be obtained with an external perturbation, for exam-
+ple a short laser pulse. The requirement of weak metallic
+coupling (Γ ≪/planckover2pi1ω) is essential since it excludes elastic
+transmission in the vibrational ground state. For small
+vibrational coupling ( λ≪/planckover2pi1ω) the junction will then be
+highly reliable and always give rise to one electron being
+transmitted. For large vibrational coupling ( λ >/planckover2pi1ω) a
+vibrational excitation will typically result in a few elec-
+trons being transmitted due to a non-vanishing elastic
+transmission. Toillustrateaquantitativeapproachtoob-
+tain the parameters of a real system, we have used DFT
+to calculate coupling parameters for a benzene molecule
+interacting with two gold contacts. Since benzene is
+bound by van der Waals forces and only show a weak
+hybridization with metallic states it satisfies the mini-mum requirement for the principle to work. However,
+there may be others reasons why this system is not well
+suited for experiments of this kind and it would be very
+interesting to investigate the principle in systems where
+transmissionthrough single molecules with significant vi-
+brational coupling has been observed1–5.
+We are grateful to K. S. Thygesen for advice and com-
+ments on the manuscript. This work was supported by
+the Danish Center for Scientific Computing. The Cen-
+ter for Individual Nanoparticle Functionality (CINF) is
+sponsored by the Danish National Research Foundation.
+Appendix A
+In this appendix we will show the details of the cal-
+culations leading to Fig. 2 for a single vibrational mode
+of frequency ω0and coupling λ0. Within scattering the-
+ory the transmission matrix Tn(εi,εf) for a vibrationally
+excited state ncan be expressed as6
+Tn(εi,εf) =ΓLΓR/integraldisplaydτdtds
+2π/planckover2pi13ei[(εi−εf)τ+εft−εis]//planckover2pi1
+×Gn(τ,s,t), (A1)
+where
+Gn(τ,s,t) =θ(s)θ(t)/angbracketleftn|ca(τ−s)c†
+a(τ)ca(t)c†
+a|n/angbracketright(A2)
+is the two particle Green function of the vibrational state
+n. The Green function can be evaluated exactly in the
+wide band limit and the result is15
+Gn(τ,s,t) =G0
+R(t)G0∗
+R(t)eigω0(t−s)
+×e−gfτ,s,tLn[g(fτ,s,t+f∗
+τ,s,t)],(A3)
+whereLnis then’th Laguerre polynomial, g=
+λ2
+0/(/planckover2pi1ω0)2,
+G0
+R(t) =−iθ(t)e−i(ε0−iΓ/2)t//planckover2pi1,
+and
+fτ,s,t= 2−e−iω0t−eiω0s+e−iω0τ(1−eiω0t)(1−eiω0s).
+An explicit result for Tncan be obtained by perfoming
+the integrals in Eq. (A1) after a Taylor expansion of
+e−gfτ,s,t.
+The result for the vibrational ground state T0involves
+L0(x) = 1 and has been calculated previously.6Here we
+simply state the result which is
+T0(εi,εf) = ΓLΓRe−2g∞/summationdisplay
+m=0gm
+m!δ(εi−εf−m/planckover2pi1ω0)/vextendsingle/vextendsingle/vextendsingle/vextendsinglem/summationdisplay
+j=0(−1)j/parenleftbiggm
+j/parenrightbigg∞/summationdisplay
+l=0gl
+l!/parenleftBig1
+εi−ε0−(j+l−g)/planckover2pi1ω0+iΓ/2/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+.(A4)5
+It is clear that integrating over final state energies simply produce s a sum over vibrational transitions such that the
+n’th term in Eq. (A4) represents T0n. Calculating T1(εi,εf) is a bit more involved since the integrand in (A1) now
+includes the first Laguerre polynomial L1(x) = 1−x. We start by writing
+T1(εi,εf) =T0(εi,εf)+/tildewideT(εi,εf), (A5)
+where
+/tildewideT(εi,εf) = ΓLΓRe−2gg/integraldisplay∞
+0dsei(ε0−εi−gω0+iΓ/2)s//planckover2pi1exp(geiω0s)
+×/integraldisplay∞
+0dte−i(ε0−εi−gω0−iΓ/2)t//planckover2pi1exp(ge−iω0t)e−i(εi−εf)t//planckover2pi1
+×/integraldisplay∞
+−∞dτ
+2π/planckover2pi13ei(εi−εf)τ//planckover2pi1exp/bracketleftBig
+ge−iω0τeiω0t(1−e−iω0t)(1−eiω0s)/bracketrightBig
+×/bracketleftBig
+(1−eiω0t)(e−iω0t−1)+(1−eiω0s)(e−iω0s−1)
+−e−iω0τ(1−eiω0t)(1−eiω0s)−eiω0τ(1−e−iω0t)(1−e−iω0s)/bracketrightBig
+.
+Taylor expanding the second exponential in the τintegral and performing the integration gives
+/tildewideT(εi,εf) = ΓLΓRe−2gg
+/planckover2pi12/integraldisplay∞
+0dsei(ε0−εi−gω0+iΓ/2)s//planckover2pi1exp(geiω0s)
+×/integraldisplay∞
+0dte−i(ε0−εi−gω0−iΓ/2)t//planckover2pi1exp(ge−iω0t)
+×/bracketleftbigg∞/summationdisplay
+m=0gm
+m!eiω0t(1−e−iω0t)m+2(1−eiω0s)mδ(εi−εf−m/planckover2pi1ω0)
++∞/summationdisplay
+m=0gm
+m!e−iω0s(1−e−iω0t)m(1−eiω0s)m+2δ(εi−εf−m/planckover2pi1ω0)
++∞/summationdisplay
+m=0gm
+m!(1−e−iω0t)m+1(1−eiω0s)m+1δ(εi−εf−(m+1)/planckover2pi1ω0)
++∞/summationdisplay
+m=0gm
+m!e−iω0(s−t)(1−e−iω0t)m+1(1−eiω0s)m+1δ(εi−εf−(m−1)/planckover2pi1ω0)/bracketrightbigg
+.
+The first two terms are each others complex conjugated and the la st two terms factorizes (s and t integrals) into
+complex conjugated and the integrals can then be performed. The final result is rather complicated but consist of an
+infinite number of terms, each of which involves a delta function δ(εi−εf−m/planckover2pi1ω0), where mruns from -1 to infinity.
+We can thus obtain T10andT11by collecting terms involving m=−1 andm= 0 respectively. The results are
+T10(εi) = ΓLΓRe−2gg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+l=0gl
+l!1
+εi−ε0−(l−1−g)/planckover2pi1ω0+iΓ/2−∞/summationdisplay
+l=0gl
+l!1
+εi−ε0−(l−g)/planckover2pi1ω0+iΓ/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+,
+and
+T11(εi) =ΓLΓRe−2g/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay
+l=0gl
+l!1
+εi−ε0−(l−g)/planckover2pi1ω0+iΓ/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
++2gΓLΓRe−2gRe/parenleftbigg2/summationdisplay
+j=0(−1)j/parenleftbigg2
+j/parenrightbigg∞/summationdisplay
+l=0gl
+l!1
+εi−ε0−(j+l−1−g)/planckover2pi1ω0+iΓ/2∞/summationdisplay
+l=0gl
+l!1
+εi−ε0−(l−g)/planckover2pi1ω0−iΓ/2/parenrightbigg
++g2ΓLΓRe−2g/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/summationdisplay
+j=0(−1)j/parenleftbigg2
+j/parenrightbigg∞/summationdisplay
+l=0gl
+l!1
+εi−ε0−(j+l−1−g)/planckover2pi1ω0+iΓ/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+.
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\ No newline at end of file
diff --git a/1001.0456.txt b/1001.0456.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1d907a38d2a9696bac2170b4467c8cb08afd0e50
--- /dev/null
+++ b/1001.0456.txt
@@ -0,0 +1,788 @@
+Design of microcavities in diamond-based photonic
+crystals by Fourier- and real-space analysis of cavity
+elds
+Janine Riedrich-M oller, Elke Neu, Christoph Becher
+Fachrichtung 7.3 (Technische Physik), Universit at des Saarlandes, Campus E 2.6
+66123 Saarbr ucken, Germany
+Abstract
+We present the design of two-dimensional photonic crystal microcavities in
+thin diamond membranes well suited for coupling of color centers in diamond.
+By comparing simulated and ideal eld distributions in Fourier and real space
+and by according modication of air hole positions and size, we optimize the
+cavity structure yielding high quality factors up to Q= 320000 with a modal
+volume ofVe= 0:35(=n)3. Using the very same approach we also improve
+previous designs of a small modal volume microcavity in silicon, gaining a
+factor of 3 in cavity Q. In view of practical realization of photonic crystals
+in synthetic diamond lms, it is necessary to investigate the in
uence of
+material absorption on the quality factor. We show that this in
uence can
+be predicted by a simple model, replacing time consuming simulations.
+Keywords: Photonic crystals, Microcavities, Qfactor, Color centers,
+Diamond
+PACS: 42.70.Qs, 42.79.Gn, 61.72.jn, 32.80.-t
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+Corresponding author. Tel.: +49 (0)681 302 2466; Fax: +49 (0)681 302 4676
+Email address: christoph.becher@physik.uni-saarland.de (Christoph Becher )
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+1. Introduction
+Microcavities in two-dimensional photonic crystal slabs allow to strongly
+conne light in volumes smaller than one cubic wavelength. They are ex-
+pected to enable the realization of single photon emitters [1], low threshold
+nanolasers [2{4], ultra small lters [5] and highly-ecient emitters in which
+the spontaneous emission of light from single emitters is controlled at the
+quantum level [6{8]. Due to highly developed processing techniques for semi-
+conductor materials most of these quantum information devices have already
+been demonstrated in silicon or GaAs. Besides the use of semiconductor ma-
+terials for quantum information processing, diamond has attracted signicant
+interest in recent years due to the extraordinary properties of optical active
+6defect centers that can be controlled at room-temperature [9, 10]. There have
+been a number of recent proposals for employment of color centers in dia-
+mond for cavity enhanced single photon sources [11], for cavity enhanced spin
+measurements [12] or as optical qubits in quantum networks [13], quantum
+gates [14] and measurement-based quantum computing [15, 16].
+All these proposals require direct coupling of an emitter to a cavity mode
+with high quality factor Qand small modal volume Ve. In such a cavity the
+spontaneous emission of an emitter, placed in the maximum of the electric
+eld of the cavity mode, is enhanced by the Purcell-Factor F[17]:
+F=3
+42Q
+Ve
+n3
+(1)
+In this weak coupling regime the cavity gure of merit scales as Q=V e,
+whereas the dynamics of strong emitter-photon coupling scales as Q=pVe
+[18].
+In recent years, several photonic crystal cavity designs have been pro-
+posed, e.g. modied line- or point defects in two-dimensional photonic crystal
+membranes. In silicon, modied waveguides structures have been fabricated
+yielding very high quality factor of 105- 2:5
106[19{21] with modal volumes
+of 1.2 - 2(=n)3. For diamond-based modied waveguide structures similar
+theoretical Q-factors with a modal volume of about Ve1:7(=n)3have
+been predicted [22{24]. In practice, the cavity Q-factor in diamond might
+strongly be limited by optical loss in the diamond material [25].
+In order to obtain large enhancement of the spontaneous emission rate
+for cavities with modest Q-factors, point defect cavity structures with even
+smaller modal volumes as compared to waveguide section cavities are re-
+quired. Dierent point defect structures with zero, one or more missing holes
+have been proposed [26]. As shown by Akahane et al. [27] the quality factor
+of a point defect cavity can be improved further by optimizing the surround-
+ing photonic crystal structure. They have demonstrated that the Q-factor
+of a photonic crystal in silicon with three missing holes (M3-cavity) can be
+increased up to 4 :5
104[27] by optimizing the next neighboring holes ac-
+cording to the method of \gentle connement" and up to 105by additionally
+shifting the third neighboring holes outwards [28]. The same approach has
+also been used to increase the Q-factor of a simple M1-cavity [26] in diamond
+with one missing hole in the center from 200 to 3
104[29] by optimizing
+the next neighboring holes in the x- andy-direction and up to 7 
104[18]
+7by ne-tuning the holes farther outwards. Together with a modal volume
+ofVe= 1.1 (=n)3a theoretical Purcell-Factor of 4580 can be achieved by
+coupling a color center in diamond to the optimized cavity structure.
+In order to increase the Purcell-Factor even further, here we consider a
+cavity design with very small modal volume: By shifting two adjacent holes
+outwards a so-called M0-cavity is introduced [26]. Extending the design ap-
+proach of \gentle connement" to remote holes and by comparing simulated
+and ideal eld distributions in Fourier and real space, we show that the cavity
+Q-factor is improved signicantly by optimizing the surrounding holes not
+only in the close vicinity of the point defect but also farther away. Exem-
+plarily, we consider in detail the design process of a M0-cavity in diamond
+and in silicon.
+Due to signicant progress in diamond processing techniques, rst exper-
+imental demonstrations of cavity modes in photonic crystal defect cavities in
+diamond have been achieved [25]. However, the measured quality factors in
+diamond are more than one order of magnitude smaller than the theoretically
+predicted values. This limitation might be due to scattering losses from the
+grain boundaries and subsequent emission from a nonsmooth surface. An-
+other loss mechanism is material absorption of the nano-crystalline diamond
+slab. In our simulations, we investigate the in
uence of material absorption
+on the quality factor. We show that this in
uence can be approximated by
+two simple models, replacing time-consuming simulations.
+One possibility to fabricate thin lms with enhanced optical quality is
+the use of single-crystal diamond membranes. Recently, single-crystal free-
+standing membranes [30{32] and waveguide structures [33, 34] in diamond
+have been produced using focused ion beam milling. The use of such single-
+crystal diamond membranes with small absorption losses might pave the way
+for realization of high- Qphotonic crystal microcavities.
+2. Cavity design
+Our work is focused on two dimensional photonic crystals consisting of a
+triangular lattice of air holes with a lattice constant ain a thin membrane
+of diamond suspended in air. The periodic structure gives rise to a photonic
+band gap for TE-like modes [18, 22]. By introducing a defect, light can
+be localized in three dimensions within small volumes: in the horizontal
+plane, light is localized due to distributed Bragg re
ection, and in the vertical
+direction due to total internal re
ection.
+82.1. Computation method
+The calculation of the near eld patterns of the cavity mode was per-
+formed with a nite dierence time domain algorithm (FDTD) [35], using a
+freely available software package with subpixel smoothing for increased accu-
+racy [36]. The simulated PC structure is a 27 a27asuper cell with height
+8a, surrounded by perfectly absorbing boundary conditions (PML) [37]. To
+extract the resonance frequency as well as the cavity Q-factor, we use a lter
+diagonalization method [38]. The mode volume Veis calculated by [39]:
+Ve=R
+(~ r)j~E(~ r)j2d3r
+max(~ r)j~E(~ r)j2(2)
+The integral of the electric energy is taken over the whole computational cell.
+To reduce the simulation time and the amount of stored data we implement
+mirror boundary conditions in the x,yandzdirections. To assure that the
+employment of mirror boundary conditions does not eect the calculated Q-
+factors, we reanalyze the structure at the beginning and at the end of our
+design process by applying an even mirror symmetry to the z= 0 plane only
+in order to select TE-like modes. Calculations are initially performed with
+a resolution of 32 points per lattice constant. In order to check for errors
+due to numerical discretization we also use a resolution of 40 and 50 points
+per lattice constant for all optimized structures of Sec. 2.3 but nd similar
+Q-factors.
+TheQ-factor calculated by the FDTD algorithm might vary with the
+chosen resolution of the simulation. Therefore, it is interesting to compare
+the FDTD results with an alternative method which calculates the Q-factor
+via the stored energy in the cavity and the radiated power [35, 39{41]. This
+second method permits to draw conclusions about further improvement of the
+cavity geometry by analyzing the wave vector components inside the \light
+cone" (see Sec. 2.3). If the defect cavity is surrounded by a suciently large
+number of air holes, the cavity lifetime is mainly limited by radiation losses
+in the vertical direction [39]. The radiated power in the far eld is then
+determined by the 2D Fourier transforms FT 2of the near eld components
+Ex,Ey,Hx,Hyin a planeSat a distance 
 zabove the photonic crystal slab
+9[35, 41]:
+P=
+82k2Z
+j~kkjkIdkxdky (3)
+withI=jFT2(Hy) +FT2(Ex)j2+jFT2(Hx)FT2(Ey)j2;
+whereisp
+0=0andIis the radiated intensity. The integral runs over all
+~k-vectors inside the \light cone" j~kkjk. The cavity near eld is computed
+using FDTD simulations after 400 time-steps. After that time transient
+eects, resulting from a cuto of our excitation pulse, have died away. The
+2D Fourier transforms of the near eld pattern are taken at a plane Sin
+a distance 
 z==2 [41] above the photonic crystal slab. To compare the
+radiation losses of dierent cavity designs, we normalize the radiated intensity
+Ito the stored energy Uin the cavity where Uis given by [40]:
+U=1
+2Z
+(0(~ r)j~E(~ r)j2+0j~H(~ r)j2)d3r (4)
+The quality factor Qis dened by the stored energy Uin the cavity divided
+by the total radiated Power Pper cycle [40]:
+Q=!U
+P; (5)
+where!denotes the angular frequency of the cavity mode. To calculate the
+Q-factor, the integral (4) of the energy stored between two planes Sabove and
+below the photonic slab as well as the integral in equation (4) are performed
+as a discrete sum. Finally we take the time average of the radiated power P
+and the stored energy Uover one period. This nal time-averaging can be
+avoided by calculating the imaginary part as well as the real part of the elds
+instead of the real part only. It is sucient to evaluate the complex elds at
+one instant in time, since the energy of complex elds does not change over
+one period.
+2.2. Starting point: simple M0-cavity
+As a basic cavity design, we choose a point defect structure with a very
+small modal volume: By shifting two adjacent holes along the x-direction
+by a distance d, a so called M0-cavity is created [26]. Figure 1(a) shows the
+10(a) M0-cavity design
+ (b)Ey-eld
+ (c)Hz-eld
+Figure 1: Design and eld components of the M0-cavity: (a) Nomenclature of the holes
+surrounding the defect. (b) Ey- and (c)Hz-component of the M0A-cavity mode.
+nomenclature of the M0-cavity design, which will be used in the following.
+As a rst example, we consider a M0-cavity in a lossless diamond membrane
+with a refractive index of n= 2:4 [42]. The cavity design supports only
+one resonant mode, whose EyandHz-components are shown in Fig. 1(b)-
+1(c), respectively. The quality factor of this simple M0-cavity depends on
+three parameters: the background radius Rof air holes, the slab thickness
+h, and the shift dof the two holes creating the defect. By successively
+modifying these parameters in steps of 0 :01awithin ranges R2[0:25a;0:29a],
+h2[0:9a;0:95a] andd2[0:13a;0:17a], we nd Parameter Set M0A, listed
+in Tab. 1, yielding the highest quality factor of Q24000 of all simple M0-
+cavities with a modal volume Ve= 0:390(=n)3at a resonance frequency
+!= 0:3652 (2c=a). The quality factor seems to be independent of the slab
+thicknesshin this parameter range. Therefore, we keep h= 0:91afor the
+remainder of the paper.
+R(a)h(a)d(a)Q!(2c=a)Ve(=n)3
+0.26 0.91 0.15 24000 0.3652 0.390
+Table 1: Starting point for the design optimization of the M0-cavity (Parameter Set M0A)
+2.3. Optimization of quality factor
+TheQ-factor of the simple M0Acavity is more than a factor of four smaller
+than equivalent designs in silicon membranes [26] due to the lower refractive
+index of diamond. In order to achieve comparable Q-factors in diamond,
+11the cavity design has to be optimized, without delocalizing the cavity mode,
+i.e. without increasing the modal volume. In two dimensional photonic
+crystals light can be localized in three dimensions due to Bragg re
ection
+in the horizontal plane and by total internal re
ection (TIR) in the vertical
+direction by the air cladding. However, only plane wave components with
+in-plane wave vectors jkkj> k 0are guided within the slab by TIR whereas
+all modes with wave vectors inside the \light cone" jkkjk0are radiated
+into the air cladding according to Snell's law [27] . These vertical radiation
+losses are crucial for photonic crystals in diamond because of the relatively
+low refractive index. The lower refractive index results in a higher mid-gap
+frequency of the band gap compared to silicon-based photonic crystals and
+therefore results in a larger corresponding light cone [22]. A suitable approach
+to minimize these radiation losses in the vertical direction is the method of
+\gentle connement" [27, 28]: Fourier- and real-space analysis reveals that
+radiation losses can be reduced by tailoring the cavity mode prole to resume
+a Gaussian envelope function. In the case of photonic crystals, such \gentle
+connement" can be obtained by adjusting the neighboring holes around the
+introduced defect, e.g. by reducing the radii or shifting some holes outwards.
+The design process of the M0-cavity is structured as follows:
+Step 1 Optimization for background radius R= 0:26a
+(a) As a rst optimization step, we ne-tune the holes c,e,kandb,g,
+fin the vicinity of the defect highlighted by the shaded region in
+Fig. 1(a), for a background radius R= 0:26a, such that the cavity
+mode prole along the y- andx-axis, respectively, ts a Gaussian
+envelope.
+(b) Thereafter, we adjust the holes m,pands,v,tfarther out along
+they- andx-axis, respectively.
+Step 2 Optimization for background radii R2[0:27a;0:29a]
+(a) As a next step, we check whether the background radius Rand
+thus the in-plane localization of the modied mode is still optimal.
+Therefore, we repeat step 1(a) for changed background radii R2
+[0:27a;0:29a].
+(b) Finally, we repeat step 1(b) for a background radius R= 0:28a.
+12(a) M0Acavity mode
+ (b) M0Bcavity mode
+ (c) M0Ccavity mode
+Figure 2: Comparison between the radiated intensity Inormalized to the stored energy U
+of (a) a simple M0Amode (b) a M0Bmode, where the holes along the y-axis are optimized
+and (c) a M0Ccavity mode where additionally the holes along the x-axis are ne-tuned.
+Step 1.a: Optimization of the next-neighboring holes for background radius
+R= 0:26a
+The modest quality factor Qof a simple M0Acavity is explained by con-
+sidering the radiation losses in the vertical direction. The radiation intensity
+Iof the M0Acavity calculated according to equation (4) is shown in Fig.
+2(a). In order to properly compare radiation losses of dierent cavity de-
+signs, we normalize all plots of radiation intensity Ito the energy Ustored
+in the cavity. The white circle denotes the light cone boundary jkkjk0.
+Note the logarithmic color scale to visualize weak eld components. The
+red area inside the light cone in Fig. 2(a) indicates large leaky components:
+The radiated power inside the light cone divided by the stored energy is
+P=U7:9
105(c=a). We start our design process by optimizing the cavity
+structure along the y-axis. In order to identify which holes have to be ad-
+justed, we analyze the eld distribution along the y-axis and compare it to
+an ideal Gaussian envelope function. Figure 3(a) shows the Ex-distribution
+of the M0Acavity mode (black rectangles). The ideal mode prole, shown
+in red, is given by a sinusoidal fundamental wave multiplied by a Gaussian
+envelope function (green curve in Fig. 3(a)) [27]. The shaded regions indi-
+cate deviations of the cavity mode from the ideal prole. These deviations
+already start in the immediate vicinity of the cavity center, thus indicating
+the need for optimization of the holes c,e,k(see Fig. 1(a)).
+As a rst step, we vary the radii Rc,ReandRkof the holes c,eandksuch
+that the cavity mode in the vicinity of the y-axis ts a Gaussian envelope.
+The optimal choice Rc= 0:23a,Re= 0:24a,Rk= 0:25ayields a quality
+13(a) M0A-cavity mode
+ (b) M0B-cavity mode
+Figure 3: Field amplitude of the M0A- and M0B-cavity mode along the yandx-axis,
+respectively: (a) Ex-eld of the M0A-cavity along the y-axis: The gray regions at the
+hole positions c,e,kindicate deviations of the calculated cavity mode (black) from the
+ideal mode prole (red) with a Gaussian envelope function (green). (b) Ey-distribution of
+the M0B-cavity along the x-axis: The deviations from the ideal prole are important at
+positionsb,sandt, whereas the deviations at position gare small.
+factorQ= 71400 with a modal volume of Ve= 0:390(=n)3(Parameter Set
+M0Bin Tab. 2). The 2D Fourier transform of the M0B-cavity is shown in Fig.
+2(b): By adjusting the holes in the immediate vicinity of the defect the leaky
+components inside the light cone have been reduced to P=U2:8
105(c=a).
+R(a)d(a)Rc(a)Re(a)Rk(a)Q!(2c=a)Ve(=n)3
+0.26 0.15 0.23 0.24 0.25 71400 0.3622 0.390
+Table 2: Optimization of rst, second and third next-neighbor holes along the y-axis
+(Parameter Set M0B).
+As a next step, we repeat the procedure above for the x-direction by
+analyzing the Ey-distribution along the x-axis as shown in Fig. 3(b): The
+deviations from the ideal eld distribution are important at the hole positions
+b,sandt, whereas the deviations at position gare quite small. The mismatch
+at the position of the next-neighbor holes bindicates, that the displacement
+dof the holes along the x-axis is no longer optimal, after we have optimized
+the holes in the vicinity of the y-direction.
+By slightly increasing the shift d= 0:16a, keeping all other parameters
+of the M0Bstructure xed, the quality factor can be improved signicantly
+up toQ= 121500 with a modal volume Ve= 0:388(=n)3. An additional
+14modication of the radius Rbof the next-neighbors does not lead to a further
+improvement of the cavity lifetime. As indicated in Fig. 3(b) the eld distri-
+bution at position gseems to be already optimal. Indeed neither changing
+the radiusRgnor shifting the holes galong thex-axis leads to an increase of
+the quality factor. Almost the same holds for the holes ffarther away from
+the defect. By reducing the radius Rfthe quality factor can be improved
+up toQ= 132000. The cavity design optimized in the x- andy-direction
+for background radius R= 0:26ais summarized in Parameter Set M0Cin
+Tab. 3. The radiation intensity Iis shown in Fig. 2(c): The increase of the
+Q-factor compared to the M0Bstructure is attributed to a reduction of the
+radiated power in the vertical direction to P=U1:6
105(c=a).
+R[a]d[a]Rc[a]Re[a]Rk[a]Rf[a]Q![2c=a]Ve[(=n)3]
+0.26 0.16 0.23 0.24 0.25 0.25 132000 0.3606 0.388
+Table 3: Optimization of the holes along the x- andy-axis for background radius R= 0:26a
+(Parameter Set M0C). The radii not listed here correspond to the background radius R.
+Step 1.b: Optimization of remote holes for R= 0:26a
+After the ne-tuning of the holes in the vicinity of the defect (see shaded
+region in Fig. 1(a)), we consider the holes m,pands,v,tat larger distances
+along they- andx-axis, respectively. Judging from the radiation intensity I
+of the M0Ccavity (Fig. 2(c)), the wave vector components along the ky-axis
+are already very small. They represent about 10% of the components in the
+light cone. Indeed, neither changing the radii RmnorRpof the holes m,p
+along they-axis yields further improvement of the cavity Q.
+As a second step, we consider the holes s,v,talong thex-axis. In Fig.
+3(b) deviations from the Gaussian envelope are visible at positions sandt.
+By slightly reducing the radii Rs= 0:25aandRt= 0:25athe quality factor
+can be improved up to Q= 146500 with a modal volume Ve= 0:390(=n)3.
+Additional modication of the holes vdoes not lead to further improvement
+of the cavity Q. The optimal choice of the hole radii Rm,RpandRs,Rv,
+Rtin they- andx-direction, respectively, are summarized in Parameter Set
+M0Din Tab. 4. The other parameters not listed in Tab. 4 correspond to the
+parameters of the M0C-cavity.
+Step 2.a: Optimization of the next-neighbors for dierent background radii R
+At the beginning of our design process we adjusted the background ra-
+diusRfor a simple M0-cavity. We now want to verify whether the choice
+15Rm[a]Rp[a]Rs[a]Rv[a]Rt[a]Q![2c=a]Ve[(=n)3]
+0.26 0.26 0.25 0.26 0.25 146500 0.3604 0.390
+Table 4: Optimization of the holes at larger distances from the defect for background
+radiusR= 0:26a(Parameter Set M0D). The other parameters are the same as in Set
+M0C.
+ofR= 0:26ais still optimal. On the one hand, as discussed in Ref. [18],
+larger background radii Rlead to an increase of the width of the band gap
+and hence to a better in-plane mode connement. On the other hand, an
+increase of the radius Ralso leads to larger wave vector components inside
+the light cone and thus to larger radiation losses in the vertical direction.
+Therefore, the choice of Ris a tradeo between these two opposed contri-
+butions. By optimizing the holes around the defect, vertical radiation losses
+have been reduced signicantly. We now want to check whether the in-plane
+connement of the mode can be improved as well, by increasing the back-
+ground radius R2[0:27a;0:29a] in steps of 0.01 a. Using the very same
+design process as described in step 1.a, we ne-tuned the holes c,e,kand
+b,g,fin they- andx-direction, respectively, for every background radius
+R2[0:27a;0:29a]. The highest quality factor of an optimized cavity geom-
+etry is obtained for background radius R= 0:28a(Parameter Set M0Ein
+Tab. 5). For an optimized cavity structure, the connement of the mode
+in the in-plane direction is considerably improved for a background radius
+R= 0:28aleading to larger quality factor Q= 226600 and a smaller mode
+volumeVe= 0:350(=n)3.
+R[a]d[a]Rc[a]Re[a]Rk[a]Rf[a]Q![2c=a]Ve[(=n)3]
+0.28 0.16 0.22 0.24 0.26 0.26 226600 0.3673 0.350
+Table 5: Optimized M0-cavity for background radii R= 0:28a(Parameter Set M0E).
+Step 2.b: Optimization of remote holes for background radius R= 0:28a
+After the optimization of the holes in the vicinity of the defect (see shaded
+region in Fig. 1(a)), we again investigate the in
uence of the remote holes m,
+pands,v,t. Fig. 4(a) shows the normalized radiated intensity of the M0E
+cavity mode: Wave vector components are still left inside the light cone. The
+radiated Power Pnormalized to the stored energy is P=U1
105(c=a).
+16(a) M0E-cavity mode
+ (b) M0F-cavity mode
+ (c) M0F-cavity mode
+Figure 4: Comparison between the radiated intensity normalized to the stored energy for
+background radius R= 0:28aof (a) a M0Emode (b) a M0Fmode, where the holes both
+along thex- andy-axis are optimized. (c) Ey-eld of the M0F-cavity mode along x-axis:
+The mode prole ts well to a Gaussian envelope.
+This suggests, that further improvement of the high- QM0E-cavity can be
+obtained by adjusting the holes at larger distances from the defect.
+Like before, we start to optimize the holes m,palong they-axis, which
+leads to only slight improvement of the Q-factor to 228500 for radii Rm=
+0:31a,Rp= 0:34a. As a nal step, we consider the holes s,v,talong thex-
+axis (see Fig. 1(a)). The optimal choice of the radii Rs= 0:29a,Rv= 0:27a,
+Rt= 0:28ayields another signicant decrease of the leaky components inside
+the light cone, shown in Fig. 4(b). The reduction of radiation losses to
+P=U6:9
106(c=a) leads to a considerable increase of the quality factor
+up toQ= 320000 with a modal volume Ve= 0:350(=n)3(see Parameter
+Set M0Fin Tab. 6). Coupling a color center to such an optimized cavity
+leads to a theoretical Purcell-Factor of F= 69500. The Ey-component of the
+M0F-cavity along the x-axis is shown in Fig. 4(c): After carefully ne-tuning
+the holes around the defect the mode prole of the optimized structure ts
+well to a Gaussian envelope.
+Rm[a]Rp[a]Rs[a]Rv[a]Rt[a]Q![2c=a]Ve[(=n)3]
+0.31 0.34 0.29 0.27 0.28 320000 0.3672 0.350
+Table 6: Optimization of the holes along the x-axis (Parameter Set M0F). The other
+parameters are the same as in Set M0E.
+17Figure 5: Fabrication tolerance test: Dependence of (a) the Q-factor and (b) the modal
+volumeVeof the optimized M0Fcavity on the radius Rcand the displacement d.
+2.4. Fabrication tolerance tests
+In view of experimental realization of photonic crystals in diamond, it is
+interesting to investigate the in
uence of the cavity Qon fabrication toler-
+ances. As discussed in section 2.3, the quality factor is signicantly enhanced
+up toQ= 320000 by modication of the holes surrounding the defect. In
+this section, we investigate how sensitive the high- Qmodes are to variations
+of the neighboring holes. Exemplarily, the next-neighboring holes bandcare
+considered. Starting from the optimized M0F-cavity (see Tab. 6), we vary
+the displacement d2[0:15a;0:17a] and the radius Rc2[0:21a;0:23a] in
+steps of 0.01 a. The dependence of the quality factor and the modal volume
+on the surrounding structural parameters are shown in Fig. 5: The displace-
+mentdseems to be a crucial parameter. If dchanges from 0 :16ato 0:15a, the
+Qfactor drops down to 4 6
104, for all radii Rc. In contrast, if we keep the
+shiftd= 0:16aof the holes bxed and vary the radius Rc2[0:21a;0:23a]
+only, theQ-factor of the M0Fcavity always exceeds 105. This is favorable
+considering that it is easier to x the central position of the air holes than
+to determine their radius using common etching techniques. If we envisage
+a central wavelength = 637 nm of the M0F-cavity equal to the emission
+wavelength of the NV-center, the lattice constant is a= 234 nm. To achieve
+large quality factors in practice, high precision structuring techniques with
+fabrication tolerances smaller than 3 nm are required. One available tech-
+nique is to use a focused beam of gallium ions (energy 30 keV) to pattern the
+diamond membrane. The spot size of the gallium ion beam is in the order of
+several nanometers, so structures can be fabricated with a resolution of 6 nm
+18[43]. Another technique which has already been used to fabricate photonic
+crystals in diamond [25] is reactive ion etching achieving radii of 80 nm 
+5 nm [44]. The radii of the milled air holes are sensitive to the dose used in
+the e-beam lithography to pattern the etch mask and to the applied etching
+parameters.
+In contrast to the cavity Q, the modal volume seems to be nearly inde-
+pendent of the ne-tuning of the surrounding holes. As shown in Fig. 5(b),
+the modal volume varies between Ve= (0:3360:358)(=n)3. This is a huge
+advantage of the M0 cavity design in view of achieving large enhancement of
+the spontaneous emission. Even if the quality factor is reduced to Q4
104
+due to fabrication tolerances, theoretical Purcell factors of F8500 can be
+obtained by coupling an emitter to the M0F-cavity.
+2.5. M0 cavity design in a silicon slab
+As discussed above, we have optimized the M0-cavity structure in dia-
+mond such that vertical radiation losses have been reduced signicantly yield-
+ing aQ-factor even larger than predicted for M0-cavity structures in silicon
+[26]. Because of the larger refractive index of silicon, this suggest that further
+improvement of the M0-cavity geometry can also be obtained in a silicon slab
+using the same design process as described above. As a starting point for
+the optimization in silicon we choose the M0-cavity structure presented in
+Ref. [26] yielding a quality factor of Q135000 with a modal volume of
+Ve= 0:288(=n)3. After carefully ne-tuning the background radius Rand
+the thickness hof the slab as well as the surrounding holes according to the
+procedure described in section 2.3, we nd the optimal M0Si-structure listed
+in Tab. 7. The other radii not listed in Tab. 7 correspond to the background
+radiusR= 0:26a. As a result, we gain an improvement in cavity Qby more
+then a factor of three with Q458000 and Ve= 0:260(=n)3. This result
+conrms that adaption of air holes even far away from the defect generally
+improves the cavity lifetime signicantly.
+R[a]h[a]d[a]Rc[a]Rk[a]Rm[a]Rs[a]Rt[a]Rv[a]Q
+0.26 0.8 0.14 0.24 0.24 0.25 0.24 0.28 0.25 458000
+Table 7: Optimization of the M0-cavity in a silicon slab (Parameter Set M0Si). The radii
+of the air holes not listed here are equal to the background radius R.
+19Figure 6: Comparison between the Q-factors calculated from the radiated power according
+to equation (5) and the FDTD simulations. On the one hand the radiated power is
+determined by the Poynting vector (red points) and on the other hand by integrating over
+all components inside the light cone using equation (4) (green triangles).
+3. Radiation Q-factors versus FDTD results
+In section 2.3, we calculate the radiated intensity from the 2D spatial
+Fourier transforms of the cavity near eld. The leaky components inside
+the light cone indicate in which direction further improvement of the cavity
+geometry is necessary. Furthermore, the radiated intensity Igives not only
+qualitative information about the radiation losses but also permits to draw
+quantitative conclusions. By summing over all wave vector components in-
+side the light cone according to equation (4), we can calculate the radiated
+powerP. This oers in alternative method to determine the cavity Q-factor
+according to equation (5): dividing the stored energy Uin the cavity by the
+vertical radiated power Pand multiplying by the resonance frequency !. In
+this section, we compare the radiation Q-factors with the FDTD results. The
+Q-factors calculated by the FDTD algorithm might depend on the chosen res-
+olution (black rectangles in Fig. 6). The errorbars in Fig. 6 (exemplarily for
+structures M0A, M0E, M0F) show the variation of the FDTD results with
+the change of resolution from 32 to 50 points per lattice constant. Moreover,
+the application of mirror symmetries to reduce the computational cell, might
+also cause an overestimation of the Q-factors [39]. Therefore, it is interesting
+to have a simple alternative method to determine the Q-factors in order to
+identify numerical artifacts.
+We calculate the near eld patterns as well as the stored energy Uin the
+20cavity using FDTD simulations. The vertical radiation losses are determined
+via the Fourier transforms of the near eld (see Sec. 2.1). The spatial Fourier
+transforms are calculated in a plane Sat position z==2. To determine the
+vertical radiated power, the integral (4) over all wave vector components in-
+side the light cone is performed as a discrete sum. Fig. 6 shows a comparison
+between the Q-factors calculated via radiated power (green triangles) and the
+FDTD results (black rectangles) for the cavity designs in diamond presented
+in sections 2.2 and 2.3. This simple calculation procedure reproduces well
+the FDTD results for dierent cavity designs. For structures M0A, M0B,
+M0Cand M0Dwith small background radius R= 0:26a, deviations from the
+FDTD results are maximally 20% (structure M0D). These deviations are
+mainly due to the fact, that the resonance frequency is close to the upper
+edge of the photonic band gap leading to imperfect lateral connement of
+the cavity mode. Albeit, the radiation Q-factor, taking into account verti-
+cal radiation losses only, re
ects well the improvement of the cavity lifetime
+by optimizing the photonic crystal geometry. For structures M0Eand M0F
+with enlarged background radii R= 0:28a, the deviations of the radiation
+Q-factor from the FDTD results are smaller than 6%. With increasing back-
+ground radii, the photonic band gap increases and the resonance frequency
+of the cavity mode lies deeper inside the photonic band gap resulting in a
+better lateral mode connement. Therefore, by calculating the vertical ra-
+diation losses via Fourier transform of the near eld, the radiation Q-factor
+of photonic crystal geometries with enlarged background radii is a good ap-
+proximation to the FDTD results.
+A third approach for determining the Q-factor uses the radiated power
+as well as the stored energy obtained from FDTD simulations. The total
+radiated power in this case is computed by the integral of the Poynting
+vector Re(~E~H) in all three directions through planes at the edge of the
+simulation cell. The associated Q-factors for cavity geometries in diamond
+are shown by the red points in Fig. 6: The agreement with the FDTD results
+is very well.
+Calculating the stored energy as well as the radiated power, either using
+FDTD simulations or via spatial Fourier transforms of the near eld, oers
+an alternative way to determine the cavity Q-factor. This method can be
+used to check the results of the FDTD simulations, usually obtained from
+the analysis of the cavity temporal decay, and to easily identify possible
+numerical artifacts.
+214. Absorption
+For the practical realization of photonic crystal microcavities in diamond
+it is essential to investigate the in
uence of the material absorption on the
+cavityQ-factor. To achieve eective optical coupling in practice, ultrapure,
+high-quality diamond lms are required. Actually, our measurements show
+[45], that nano-crystalline diamond lms strongly absorb light at the emis-
+sion wavelengths of the NV, SiV and NE8 color center ( 630 nm - 800 nm)
+due to graphite or amorphous carbon in the grain boundaries [46]. The ab-
+sorption coecient is determined by measuring the transmission through a
+nano-crystalline diamond membrane ( -BeSt, Innsbruck, Austria). The re-
+duction of the transmission signal due to surface roughness (15 nm rms) is
+estimated to 2%, while the material absorption leads to a reduction of the
+signal of 11% (for a lm of 300 nm thickness). Therefore, we assume that ma-
+terial absorption of the dielectric background is the primary loss mechanism
+of microcavities in photonic crystals in nano-crystalline diamond lms.
+To investigate the in
uence of material absorption on the cavity Q-factor,
+we implement articial resonances in the dielectric function such that the
+imaginary part iyields the desired resonant absorption whereas the real
+partrremains almost constant [18]. The results of the FDTD simulations
+(black rectangles in Fig. 7) reveal that the Q-factor strongly depends on
+the absorption coecient of the diamond material. For our ultra-nano-
+crystalline diamond lms we measure maximum absorption coecients 
+4000 cm1at the emission wavelength = 637 nm of the NV center [45].
+Starting from a theoretical quality factor Q50000 of a M0-cavity with
+optimized next neighboring holes ( R= 0:26a,d= 0:15a,Rc=Re= 0:24a,
+h= 0:91a), the material absorption of the diamond lm would reduce the
+quality factor to only Q < 100. Nevertheless, coupling an emitter to a
+M0-cavity, a Purcell-Factor of about F= 15 is still achievable because of
+the extraordinary small modal volume. Due to recent progress in growing
+high-quality diamond lms by chemical-vapor deposition [47], absorption co-
+ecients of 200 cm1[48] can be obtained. With such low loss diamond
+membranes an experimental cavity Q-factor of 1150 can be achieved, yielding
+a Purcell-Factor of F= 220.
+Calculation of the quality factors for dierent absorption coecients 
+using FDTD simulations is a very time-consuming task. Therefore, an ana-
+lytical approximation to evaluate the Q-factor for lossy materials would be
+very helpful. The dependence of Qoncan be described by a simple model:
+22Figure 7: In
uence of the absorption coecient of the diamond material on the Q-factor
+of the M0-cavity ( R= 0:26a,d= 0:15a,Rc=Re= 0:24a,h= 0:91a): The FDTD results
+are shown in black. The dependance can be described by a simple model of a linear cavity
+lled with a homogenous diamond slab or lled with the actual photonic crystal structure.
+In the case of a linear cavity with a homogenous diamond slab between two
+mirrors the total quality factor Qis given by [49, 50]:
+Q1=Q01+Q1
+abs; (6)
+whereQ0denotes the quality factor of an ideal, lossless cavity and Qabsthe
+quality factor due to material absorption. As described in Refs. [18, 50],
+for weakly absorbing material rithe absorption quality factor can be
+written as Qabs=k=, where the absorption coecient is given by =
+k0i=pr. The wave vector in the dielectric is k=prk0, wherek0denotes
+the wave vector in vacuum. Under the assumption that most of the electric
+eld is concentrated in the dielectric, the material losses are given by:
+Qabs=k==r=i (7)
+The blue curve in Fig. 7 shows that this rst approximation predicts well the
+dependance of the cavity Q-factor. The deviations from the FDTD results
+are smaller than 13%. Actually, equation (7) denotes a lower limit to the
+FDTD results for lossy materials, because the electric eld is not completely
+concentrated within the dielectric but also leaks into the air holes.
+In a more general approach, we want to take into account the actual pho-
+tonic crystal structure. For weakly absorbing material rithe imaginary
+partican be considered a small perturbation 
 =iiof the dielectric
+23function(~ r). This perturbation 
 results in a rst-order correction of the
+resonance frequency 
 !=i!i[51, 52]:
+
!=!r
+2R
+
(~ r)j~E(~ r)j2d3rR
+r(~ r)j~E(~ r)j2d3r; (8)
+where!rdenotes the resonance frequency of the ideal, lossless cavity struc-
+ture. Considering that the Q-factor is dened as Q=!r
+2!i, the absorption
+quality factor can be written as:
+Q0
+abs=R
+r(~ r)j~E(~ r)j2d3rR
+i(~ r)j~E(~ r)j2d3r(9)
+The integral in equation (9) is taken over the entire photonic crystal slab
+with the electric eld ~E(~ r) of the ideal, lossless photonic crystal structure
+as a weighting factor. Therefore the electric eld has to be calculated only
+once for an ideal cavity using numerical methods such as FDTD simulations.
+Afterwards it can be reused to calculate Q0
+absfor dierent lossy materials. The
+overlap integral in equation (9) of the electric eld and the dielectric function
+is performed as a discrete sum. The red curve in Fig. 7 shows the absorption
+quality factor Q0
+abscalculated for dierent absorption coecients of the
+dielectric background. The deviations from the FDTD results are mostly
+due to dierent procedures used to calculate the overlap integral in equation
+(9). The discrete summation leads to a slightly higher Q-factor of about
+11% compared to an exact integration. We estimate the error by calculating
+the eective dielectric constantR
+(~ r)j~E(~ r)j2d3r=R
+j~E(~ r)j2d3rperforming the
+integrals as a discrete sum and comparing it to FDTD simulations.
+Both simple models obtained above predict well the dependence of the
+quality factor for lossy materials. These simple equations permit to calculate
+an upper and a lower limit to the FDTD results without relying on time-
+consuming simulations.
+5. Conclusion
+We have presented an optimized M0-microcavity design in a diamond-
+based photonic crystal. The small defect is introduced by shifting two
+holes outwards along the x-axis yielding an extremely small modal volume
+ofVe= 0:35(=n)3. Using Fourier- and real-space analysis, we improved
+24the cavity structure by systematically varying the radii of the neighboring
+holes around the defect. The improved design yields a quality factor of
+Q= 320000. Coupling a color center to such an optimized cavity structure
+leads to a theoretical Purcell-Factor of F= 69500, which is one of the largest
+enhancement of the spontaneous emission that has been predicted so far for
+photonic crystal point defects in diamond. Using the same design process, we
+have additionally optimized the M0-cavity in silicon gaining a factor of three
+in cavityQ. The signicant improvement of the cavity lifetime by \gentle
+connement" shows, that it is worth looking closer at the surrounding holes
+even far away from the defect.
+In view of the practical realization of photonic crystals in nano-crystalline
+diamond lms, material absorption of the dielectric background has to be
+taken into account. The in
uence of absorption losses on the cavity Q-factor
+can be described by a simple model of a linear cavity: In rst approximation,
+we consider a homogenous diamond slab between two perfect mirrors. This
+simple description denotes a lower limit to the FDTD results, because the
+electric eld is not completely concentrated within the dielectric, but also
+leaks into the air holes. The second approximation takes into account the
+actual photonic crystal structure. These two simple models oer an ecient
+way to predict quality factors of photonic crystal cavities for lossy materials
+without relying on time-consuming FDTD simulations. Whether this simple
+model of a linear cavity is valid for a larger range of defect geometries in
+photonic crystals is subject of current work. Preliminary results show that
+equation (7) predicts the absorption quality factor Qabseven better for large
+photonic crystal point defects.
+One possibility to overcome the limitation of the Q-factor by material
+absorption is the use of single crystal diamond lms. Recent progress in
+fabrication of thin free-standing single crystal diamond membranes, might
+pave the way for ecient direct coupling of color centers to high- Qdiamond-
+based photonic crystal microcavities.
+6. Acknowledgements
+The authors thank C. Hepp for the absorption measurements of our di-
+amond lms and D. Steinmetz for valuable help with the computing infras-
+tructure. E. Neu acknowledges support from the Stiftung der Deutschen
+Wirtschaft (SDW). This work is funded by the Deutsche Forschungsgemein-
+schaft (DFG).
+25
\ No newline at end of file
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@@ -0,0 +1,1550 @@
+arXiv:1001.0457v3  [math.AG]  19 Sep 2010HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS
+MICHELE ROSSI
+Abstract. The present paper gives an account and quantifies the change i n
+topology induced by small and type II geometric transitions , by introducing
+the notion of the homological type of a geometric transition. The obtained
+results agree with, and go further than, most results and est imates, given to
+date by several authors, both in mathematical and physical l iterature.
+Contents
+Introduction 1
+1. Geometric transitions 5
+2. Local analysis of an isolated singularity 7
+3. Homological type of a conifold transition 10
+4. Homological type of a small geometric transition 11
+4.1. Homology change induced by a small geometric transition 13
+5. Homological type of type II geometric transition 19
+5.1. Normal exceptional divisor 20
+5.2. Non–normal exceptional divisor 21
+5.3. Homology change induced by a type II geometric transition 22
+5.4. The normal and rational case 23
+5.5. The elliptic case 30
+5.6. The non-normal case 34
+References 34
+Introduction
+Ageometric transition (g.t.) between two Calabi–Yau threefolds Yand/tildewideY(see
+Definition 1.1) is the process T(Y,Y,/tildewideY) obtained by “composing” a birational con-
+tractionφ:Y→Y, to a normal threefold Y, with a complex smoothing (see
+Definition 1.4). A g.t. is a useful tool for connecting to each other topologically
+distinctCalabi–Yau threefolds. This feature is probably the main source of in terest
+in the study of g.t.’s both in mathematics and physics.
+In mathematics, the story goes back to deep speculations due to H . Clemens [6],
+R. Friedman [10], F. Hirzebruch [19], J. Werner [53] and M. Reid [41]. In fact, the
+This work has been developed despite the effects of the Italia n law 133/08
+(http://groups.google.it/group/scienceaction ). Thisl aw drastically reduces public funds to public
+Italian universities, which is particularly dangerous for free scientific research, and it will prevent
+young researchers from getting a position, either temporar y or tenured, in Italy. The author is
+protesting against this law to obtain its repeal.
+12 MICHELE ROSSI
+huge multitude of known topologically distinct Calabi–Yau threefolds m akes any
+conceptof moduli space for them immediately wildly reducible . This unpleasantfact
+dramatically clashes with the well known irreducibility of moduli spaces of both
+elliptic curves and K3 surfaces, which are the lower dimensional analo gues of Ca-
+labi–Yau threefolds. Actually, Reid (op. cit.) underlines that, at the beginning (in
+the forties), the moduli space of(algebraic)K3 surfacesseemed to F. Enriquesto be
+a 19–dimensional variety admitting a countable number of irreducible components,
+Mg, one for each value of the sectional genus, g≥3 [9]. Twenty years later
+K. Kodaira [24] was able to recover a 20–dimensional irreducible modu li space, M,
+for K3 surfaces by leaving the algebraic category to work in the larg er category of
+analytic, compact, K¨ ahler varieties, and discovering that the genericK3 surface is
+an analytic non–algebraic complex variety: in particular the moduli sp aceMalg=/uniontext
+gMgof algebraic K3’s embeds in Mas a dense closed subset. The so–called
+Reid’s fantasy suggests that an analogous situation could happen for Calabi–Yau
+threefoldswherebirationaltransformationsandg.t.’scouldbethe rightinstruments
+to reduce the parameterization of birational classes of Calabi–Yau threefolds to an
+irreducible moduli space of complex structures over suitable conne cted sums of
+complex hypertori.
+In physics, Calabi–Yau threefolds made their appearance in the eigh ties [5], as
+the space spanned by the so–called internal degrees of freedom of a certain (1+1)–
+dimensionalworld–sheetfieldtheorydescribingsuperstrings’pro pagationin(3+1)–
+dimensional Minkowski space–time. The main observables of the sup erstring model
+are then typically determined by this internal structure (the Calabi–Yau vacuum).
+Therefore, in spite of the almost uniqueness, via dualities (and, late r,M–theory),
+of consistent superstring theories, the superstring model rema ins undetermined due
+to the unavoidable uncertainty on the topology of the Calabi–Yau va cuum: this is
+the so called vacuum degeneracy problem . A solution to this problem was firstly
+proposedby P.S.Greenand T. H¨ ubsch[12], [13], whoconjectured, motivated bythe
+contemporaryReid’s fantasy, that topologicallydistinct Calabi–Yau vacua could be
+connected to each other by means of conifold transitions (see Definition 1.5) which
+should induce a phase transition between corresponding superstring models. This
+latter fact, which is the physical counterpart of the mathematica l process given by
+a conifold transition, was actually understood by A. Strominger as a condensation
+of massive black holes to massless ones [49].
+In this context it seems, then, ofprimary importance to understa nd and quantify
+the change in topology induced by a geometric transition. This is what H. Clemens
+did for conifold transitions some time ago in [6]. For more general g.t’s, a lot of
+computations have been carried out by many authors, in the last th irty years, but I
+wasn’t able to find in the literature a clear and/or complete statemen t about these.
+Moreover, many results are obtained by invoking arguments which a re not strictly
+topological, like, e.g., complex moduli and the Bogomolov–Tian–Todoro vTheorem.
+InthepresentpaperIwilltrytoorganizethisproblembyintroducin gtheconcept
+of thehomological type of a geometric transition . Precisely,
+Definition 1. A g.t.T=T(Y,Y,/tildewideY) is said to admit homological type
+h[T] = (k′,k′′,c′,c′′)HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 3
+wherek′,k′′are non–negative integers and c′,c′′∈Zwithc′≡c′′mod 2, ifT
+induces the following change on
+(a)Betti numbers :bi(Y) =bi(Y) =bi(/tildewideY) fori/\e}atio\slash= 2,3,4 and
+b2(Y) =b2(Y)+k′+k′′=b2(/tildewideY)+k′+k′′,
+b3(Y) =b3(Y)−c′=b3(/tildewideY)−c′−c′′,
+b4(Y) =b4(Y)+k′′=b4(/tildewideY)+k′+k′′.
+Then in particular Tinduces the following changes on
+(b)Hodge numbers :
+h1,1(Y) =h1,1(/tildewideY)+k
+h2,1(Y) =h2,1(/tildewideY)−c
+wherek:=k′+k′′andc:= (c′+c′′)/2 ,
+(c)Euler characteristics :
+χ(Y)−χ(Y) =k+c′+k′′
+χ(Y)−χ(/tildewideY) =k′+c′′/bracerightbigg
+⇒χ(Y)−χ(/tildewideY) = 2(k+c).
+Moreover, it turns out that k′= 0if and only if YisQ–factorial (see Remark 4.4
+and Lemma 5.4).
+This notation is motivated by equations (b) on Hodge numbers allowing us to
+conclude that a g.t.T(Y,Y,/tildewideY)admitting homological type h(T) = (k′,k′′,c′,c′′)
+increases complex moduli (in physics: hypermultiplets )bycand decreases K¨ alher
+moduli(in physics: vector multiplets )byk, when passing from the Calabi–Yau
+3–foldYto the Calabi–Yau 3–fold /tildewideY.
+Main Results. Let us first of all observe that a general g.t. T(Y,Y,/tildewideY)does
+not admit a homological type since the homology change induced by Tcannot be
+summarized by a string of 4 integers: in fact in general b2(Y)/\e}atio\slash=b2(/tildewideY) requires the
+introduction of at least one further integer gauging this last discre pancy. By the
+way, quantifying the topological change given by a general g.t. is ac tually a hope-
+less problem since the exceptional locus E:= Exc(φ) of the associated birational
+contraction φ:Y→Ymay be very wild! On the other, hand what is observed
+in the present paper is that a homological type as in Definition 1 may su ffice to
+describe what happens for some classes of (understandable) g.t’s . Precisely, it is
+proven that:
+Theorem 4.3 A small g.t. T(Y,Y,/tildewideY) (see Definition 1.11) admits homological
+type
+h[T] = (k,0,c′,c′′)
+wherekis the number of homologically independent exceptional rat ional curves
+composing E= Exc(φ)⊂Y,c′is the number of independent relations linking
+the homology classes of exceptional rational curves in Yandc′′is the number of
+homologically independent vanishing cycles in /tildewideY. Moreover,
+(i)the total number of irreducible components of Eis
+n:=/summationdisplay
+p∈Sing(Y)np=k+c′,4 MICHELE ROSSI
+wherenpis the numberof irreducible (rational) components of Ep:=φ−1(p),
+for anyp∈Sing(Y),
+(ii)theglobal Milnor number of Yis
+m:=/summationdisplay
+p∈Sing(Y)mp=k+c′′,
+wherempis the Milnor number of the singular point p∈Sing(Y). Hence,
+kturns out to be also the maximal number of independent relati ons linking
+the homology classes of vanishing cycles in /tildewideY.
+In particular, Tis a type I g.t. (see Definition 1.10) if and only if k′= 1 =kand
+ifTis a conifold t., then c′=c′′=c.
+Theorem 5.2 A type II g.t. T(Y,Y,/tildewideY) (see Definition 1.10) admits homological
+type
+h[T] = (0,1,c′,c′′)
+given by
+(i)c′=−1+b2(E)−b3(E) ,
+(ii)c′′= 1−χ(/tildewideB) =mp−b2(/tildewideB) ,
+where/tildewideBis the Milnor fiber of the smoothing /tildewideYandmp:=b3(/tildewideB)is the Milnor
+number of the unique singular point p=φ(E)∈Y. In particular b1(/tildewideB) = 0 .
+Let us say a few words about the previous results.
+WhenTis a conifold t., Theorem 4.3gives preciselythe Clemens’ results in [6], he re
+reported in Theorem 3.1. Moreover, relations (ii) and part of relatio ns in Definition
+1.(a), withk′andc′′as given in the statement of Theorem 4.3, were already proved
+by Y. Namikawa and J.H.M. Steenbrink in [36] Theorem (3.2) and Example (3.8):
+as far as I know, these were the most complete known evaluations o f homology
+change induced by a small g.t. until now.
+Theorem 5.2 can be improved by specializing the assumptions. In fact , it is well
+known that the exceptional locus E= Exc(φ) of the birational contraction Yφ→Y,
+associated with a type II g.t. T(Y,Y,/tildewideY), is an irreducible generalized del Pezzo
+surface ([39], Proposition (2.13)). This means that one of the follow ing happens
+•Eis normal and rational then Theorem 5.2 can be improved as follows (see
+Theorem 5.3):
+(i)c′= 9−d−nE,whered= deg(E) :=ω2
+EandnEis the number of ra-
+tional exceptional curves composing the exceptional locus of a minimal
+resolution /hatwideEπ→E,
+(ii)ifd≤4thenc′′=mp.
+•Eadmits an elliptic singular point (in the sense of [39], Definition (2.4))
+then Theorem 5.2 can be improved as follows (see Theorem 5.7):
+(i)b2(E) = 1andb3(E) = 2, givingc′=−2,
+(ii)c′′=mpand in particular it must be even.
+•Eis a non–normal del Pezzo surface (in the sense of [42]) then Theorem
+5.2 can be improved as follows (see Theorem 5.13):
+(i)b2(E)∈ {1,2}andb3(E) = 0, givingc′∈ {0,1}, respectively ,
+(ii)χ(/tildewideB)≡b2(E) mod 2 .HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 5
+The first equation in Definition 1.(b) in the case of a type II g.t., hence giving
+k= 1, was already proved in [22] Proposition 3.1. In particular, if E= Exc(φ)is
+smooththen a comparison between the equations in Definition 1.(b) and Theo rem
+3.3andRemark3.6in[22], givesconditionsontheEulercharacteristic χ(/tildewideB)andthe
+Milnornumber mpandthenthelistinRemark5.6. Inparticular,thisshowsthatwe
+cannot expect the Milnor fiber near p∈Yto be a bouquet of3–dimensionalspheres
+whend:= deg(E) = 6 or 7, the contrary of what happens for d≤4. Moreover,
+a further comparison with [33] §7.1 and [32] §3 gives an interesting interpretation
+of the homological type of a type II g.t. in terms of (dual) Coxeter n umbers of
+suitable Weyl groups, as in (59).
+Employed techniques are almost completely topological: constructio n of strong
+deformationretractions,Mayer–Vietorisandrelativehomologylon gexactsequences,
+dualities in homology, Leray spectral sequences and associated low er terms exact
+sequences. Transcendental methods (exponential sequence o n a Calabi–Yau 3–fold
+and its pushed forward by birational contraction on a normal and s ingular 3–fold)
+are used simply to focus on how the Picard group is actually a topologic al invariant
+of all the 3–folds involved in a g.t. .
+The present paper is organized as follows. §1 reviews what a g.t. is, the related
+nomenclature, and Wilson’s classification of birational contractions of Calabi–Yau
+3–folds. In §2 Milnor’s and Looijenga’s local analysis of isolated singularities are
+reviewed and the strong deformation retractions useful in the fo llowing are intro-
+duced (Propositions 2.4 and 2.7). In §3 we review a result proved in [43], rewritten
+in terms of the homological type of conifold transitions. Finally §4 and§5 state and
+prove the main results, the former for small g.t’s and the latter for type II g.t’s .
+Acknowledgments. I would like to thank Alberto Collino for useful suggestions.
+1.Geometric transitions
+Definition 1.1 (Calabi–Yau 3–folds) .A smooth, complex, projective 3–fold Yis
+calledCalabi–Yau if
+(a)KY∼=OY,
+(b)h1,0(Y) =h2,0(Y) = 0 .
+Remark 1.2.There are a lot of more or less equivalent definitions of Calabi–Yau 3–
+folds e.g.: a K¨ ahlercomplex, compact 3–fold admitting either (1) a Ricci flat metric
+(Calabi conjecture and Yau theorem), or (2) a flat, non–degene rate, holomorphic3–
+form,or(3)holonomygroupasubgroupofSU(3)(see[21]foraco mpletedescription
+of equivalences and implications).
+In the algebraic context, the given definition gives the 3–dimensiona l analogous of
+smooth elliptic curves and smooth K3 surfaces.
+Examples 1.3.(a) Smooth hypersurfaces of degree 5 in P4,
+(b) the general element of the anti–canonical system of a sufficiently good 4–
+dimensional toric Fano variety (see [4]),
+(c) suitable complete intersections.... (iterate the previous example s),
+(d) the double covering of P3ramified along a smooth surface of degree 8 in P3
+(octic double solid).6 MICHELE ROSSI
+Definition 1.4 (Geometric transitions) .(cfr. [31], [7], [11], [43]) Let Ybe a Ca-
+labi–Yau 3–fold and φ:Y→Ybe abirational contraction onto anormalvariety.
+If there exists a complex deformation ( smoothing ) ofYto a Calabi–Yau 3–fold /tildewideY,
+then the process of going from Yto/tildewideYis called a geometric transition (for short
+transition org.t.) and denoted by T(Y,Y,/tildewideY) or by the diagram
+Y
+T/d57/d57φ/d47/d47Y/d111/d111 /d47/d47/d47/d111/d47/d111/d47/d111/tildewideY.
+A transition T(Y,Y,/tildewideY) is called trivialif/tildewideYis a deformation of Y.
+Definition 1.5 (Conifold transitions) .A g.t.
+Y /d57/d57φ/d47/d47Y/d111/d111 /d47/d47/d47/d111/d47/d111/d47/d111/tildewideY
+is called conifold (c.g.t.for short) and denoted CT(Y,Y,/tildewideY), ifYadmits only
+ordinary double points (nodes or o.d.p.) as singularities.
+Example 1.6 (cfr. [14]) .The following is a non–trivial c.g.t.. For details see [43],
+1.3.
+LetY⊂P4be thegeneric quintic 3–fold containing the plane π:x3=x4= 0. Its
+equation is
+x3g(x0,...,x 4)+x4h(x0,...,x 4) = 0
+wheregandhare generic homogeneous polynomials of degree 4. Yis then singular
+and
+Sing(Y) ={[x]∈P4|x3=x4=g(x) =h(x) = 0}={16 nodes }.
+Blow up P4along the plane πand consider the proper transform YofY. Then:
+•Yis a smooth, Calabi–Yau 3–fold,
+•the restriction to Yof the blow up morphism gives a crepant resolution
+φ:Y→Y.
+The obvious smoothing of Ygiven by the generic quintic 3–fold /tildewideYcompletes the
+c.g.t.CT(Y,Y,/tildewideY).
+Definition 1.7 (Primitive contractions and transitions) .A birational contraction
+from a Calabi–Yau3–fold to a normal one is called primitive if it cannot be factored
+into birational morphisms of normal varieties. A g.t.
+Y
+T/d57/d57φ/d47/d47Y/d111/d111 /d47/d47/d47/d111/d47/d111/d47/d111/tildewideY
+is called primitive if the associated birational contraction φis primitive.
+Proposition 1.8. LetT(Y,Y,/tildewideY)be a g.t. and φ:Y→Ythe associated birational
+contraction. Then φcan always be factored into a composite of a finitenumber of
+primitive contractions.
+Proof.The statement follows from the fact that any primitive contraction reduces
+by 1 the Picard number ρ= rk(Pic(Y)) =h1,1(Y). /square
+Theorem 1.9 (Classification of primitive contraction [54], [55]) .Letφ:Y→Y
+be a primitive contraction from a Calabi–Yau threefold to a n ormal variety and let
+Ebe the exceptional locus of φ. Then one of the following is true:HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 7
+type I:φissmallwhich means that Eis composed of finitely many rational
+curves;
+type II:φcontracts a divisor down to a point; in this case Eis irreducible
+and in particular it is a (generalized) del Pezzo surface (see[39])
+type III:φcontracts a divisor down to a curve C; in this case Eis still
+irreducible and it is a conic bundle over a smooth curve C.
+Definition 1.10 (Classificationofprimitivetransitions) .Atransition T(Y,Y,/tildewideY) is
+calledof type I, II or III ifit isprimitive and ifthe associatedbirationalcontraction
+φ:Y→Yis of type I, II or III, respectively.
+Definition 1.11 (Small geometric transition) .A transition
+Y
+T/d57/d57φ/d47/d47Y/d111/d111 /d47/d47/d47/d111/d47/d111/d47/d111/tildewideY
+will be called smallifφis the composition of primitive contractions of type I.
+2.Local analysis of an isolated singularity
+Letp∈Ybean–dimensional isolated singularity i.e. thereexistsa n–dimensional
+local analytic neighborhood Uofpbi-holomorphic to the zero locus of a polynomial
+map
+f= (f1,...,fM) :CN→CM
+whereN−M≤nand such that fhas an isolated critical point in 0 ∈CNandf
+is a submersion over CN\{0}. IfN−M=nthenp∈Yis anisolated complete
+intersection singularity (i.c.i.s.). Moreover, if M= 1, thenN=n+1, andp∈Yis
+anisolated hypersurface singularity (i.h.s). Let Dεdenote the closed ε–ballcentered
+in 0∈CNwhose boundary is the (2 N−1)–dimensional ε–sphereSε.
+Definition 2.1 (Good representatives and Milnor fibers, [27] Section 2.B) .Let
+Tmbe am–dimensional contractible neighborhood of 0 ∈CM, withm≤M, and
+consider Um:=f−1(Tm). Then
+f:Um−→Tm
+is called a good representative of p. We will omit the dimension mif unnecessary.
+Set/tildewideU:=Utfort∈T\{0}. Then, for small ε>0, the intersection
+/tildewideB=/tildewideU∩Dε
+is called the Milnor fiber of p.
+Definition 2.2. LetX⊂CNbe a subset. The following subset of CN
+Cnε
+0(X) :={tx| ∀t∈[0,ε]⊂R,∀x∈X}
+will be called the ε–cone projecting X. The 1–cone will be simply called the cone
+projectingXand denoted by Cn 0(X). Moreover, the ∞–cone Cn∞
+0(X) will be also
+called the unbounded cone projectingX.
+Theorem 2.3 (Local topology of an isolated singularity, [27] Proposition (2.4),
+[28] Theorem 2.10, Theorem 5.2) .Sincep∈Uis the unique critical point of f, for
+anyε>0there exists a homeomorphism ψεbetween the intersection
+B:=U∩Dε8 MICHELE ROSSI
+and theε–cone projecting K:=U∩Sε. Moreover,1
+εψεgives a homeomorphism
+B∼=Cn0(K).
+Kis called the link ofp: it only depends on the abstract analytic germ ( Y,p) (see
+[27], Corollary (2.6)).
+Proposition 2.4. In the same notation as before, Band/tildewideBare strong deformation
+retract ofUand/tildewideU, respectively.
+Proof.Let us first of all observe that, by direct limit construction, there exists an
+homoeomorphism
+(1) ϕ:U∼=/d47/d47Cn∞
+0(K)
+whereK:=U∩Sεfor a sufficiently small ε >0. In fact, for any 0 < ε′< ε′′,
+Theorem 2.3 gives the following commutative diagram
+U∩Dε′′ψε′′
+∼=/d47/d47Cnε′′
+0(K)
+U∩Dε′ψε′
+∼=/d47/d47/d63/d31ε′′
+ε′/d79/d79
+Cnε′
+0(K)/d63/d31ε′′
+ε′/d79/d79
+which, passing to direct limits Uand Cn∞
+0(K), gives the claimed homeomorphism
+ϕ. Moreover, by setting ψ:=1
+εψε, we get a homeomorphism
+(2) ψ:B∼=/d47/d47Cn0(K).
+Let us start by showing the strong retraction U≈B: callingi:B ֒→Uthe natural
+inclusion, we will write explicitly the retraction r:U→Band the homotopy
+F:U×[0,1]→Usuch that idU∼Fi◦r, since they will be useful in the following.
+Let
+r: Cn∞
+0(K) /d47/d47Cn0(K)
+the obvious retraction defined by setting r(x) :=/braceleftbiggxifx∈Cn0(K)
+x/|x|otherwise.
+Consider the straight line homotopy
+H: Cn∞
+0(K)×[0,1] /d47/d47Cn∞
+0(K)
+defined by setting H(x,t) :=/braceleftbigg
+x ifx∈Cn0(K)
+(1−t)x+tx/|x|otherwise.
+His clearly continuous by the gluing lemma over Kwhose points have unitary
+modulus. Moreover:
+•H(x,0) = id Cn∞
+0(K)(x) =xandH(x,1) =r(x), which means that Cn 0(K)
+is adeformation retract of Cn∞
+0(K),
+•H(x,t) =xfor everyx∈Cn0(K), which means that Cn 0(K) is astrong
+deformation retract of Cn∞
+0(K).
+We are now able to construct the retraction rand the homotopy Fas follows
+r:=ψ−1◦r◦ϕ
+F(z,t) :=ϕ−1(H(ϕ(z),t)),∀(z,t)∈U×[0,1] (3)
+(notice that ψ=ϕ◦i⇒i◦ψ−1=ϕ−1|Cn0(K)).HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 9
+To prove the strong retraction /tildewideU≈/tildewideBsetBε:=Uε∩Dε,Kε:=Uε∩Sεand consider
+the restricted fibration
+f/rightanglesw:Uε\Bε∪Kε/d47/d47T .
+Up to shrink T, the mapf/rightangleswgives a fibration in 2 n–dimensional C∞–manifolds with
+boundary whose fibers are diffeomorphic to each other by the Ehre smann fibration
+theorem. Let then
+η:/tildewideU\/tildewideB∪/tildewideK∼=/d47/d47U\B∪K
+be the induced diffeomorphism between the fibers f−1
+/rightanglesw(t)∼=f−1
+/rightanglesw(0). We can then
+construct a retraction /tildewider:/tildewideU→/tildewideBand a homotopy /tildewideF:/tildewideU×[0,1]→/tildewideUas follows
+/tildewider(z) :=/braceleftbigg
+z ifz∈/tildewideB
+η−1◦r◦η(z) otherwise
+/tildewideF(z,t) :=/braceleftbigg
+z ifz∈/tildewideB
+η−1/parenleftbig
+F(η(z),t)/parenrightbig
+otherwise
+Both/tildewiderand/tildewideFare continuous by the gluing lemma over /tildewideKand the proof concludes
+immediately by verifying that id /tildewideU∼/tildewideF/tildewidei◦/tildewider, where/tildewidei:/tildewideB ֒→/tildewideUis the inclusion. /square
+Theorem 2.5 (Local homotopy type of the smoothing, [27] (5.6), (5.8) and Cor ol-
+lary(5.10), [28]Theorems5.11,6.5,7.2) .The Milnor fiber /tildewideBis homotopy equivalent
+to a finite cell complex of dimension ≤n. Moreover, if 0∈f−1(0)is an i.c.i.s. then
+/tildewideBis(n−1)-connected and has the homotopy type of a finite bouquet of n–spheres.
+In the following the n-th Betti number bn(/tildewideB) of the Milnor fiber will be called
+the Milnor number of p∈U⊆Yand denoted by mp.
+Remark2.6.Ifp∈Uisan–dimensionali.h.s. then theMilnornumber mpcoincides
+with the multiplicity ofpas a critical point of the polynomial map f, which is the
+multiplicity of 0 ∈Cn+1as solution to the collection of polynomial equations
+∂f
+∂x1=···=∂f
+∂xn+1= 0.
+The Milnor number is then given by
+mp= dim C/parenleftbigg
+C{x1,...,xn+1}/slashbigg/parenleftbigg∂f
+∂x1,···,∂f
+∂xn+1/parenrightbigg/parenrightbigg
+whereC{x1,...,xn+1}is theC–algebra of converging power series.
+Proposition 2.7 (Local topology of the resolution) .Letφ:U→Ube a birational
+resolution of the i.s. p∈Uand setE:= Exc(φ). The pull back by φof the strong
+deformation retraction U≈B≈ {p}gives rise to a strong deformation retraction
+U≈B≈E, whereB:=φ−1(B).
+Proof of Lemma 2.7. Consider the homeomorphisms ϕ,ψgiven by (1) and (2), re-
+spectively. Then the straight line homotopy giving the contraction B≈ {p}is the
+following
+G:B×[0,1]/d47/d47B
+(z,t)/d31 /d47/d47ψ−1((1−t)ψ(z))10 MICHELE ROSSI
+Therefore the contraction U≈ {p}is realized by the composition G∗FwhereF
+is the homotopy defined in (3). Pulling back by the resolution φ:U→Uwe get
+retractions and homotopies
+r:U /d47/d47E
+G∗F:U×[0,1] /d47/d47U
+defined by setting
+r(z) :=/braceleftbiggzifz∈E
+/hatwidelφ(z)∩Eotherwise
+G∗F(z,t) :=
+
+z ifz∈E
+φ−1/parenleftbig
+F(φ(z),2t)/parenrightbig
+if (z,t)∈(U\E)×[0,1
+2]
+φ−1/parenleftbig
+G/parenleftbig
+F(φ(z),1),2t−1/parenrightbig/parenrightbig
+if (z,t)∈(U\E)×[1
+2,1)
+/hatwidelφ(z)∩E if (z,t)∈(U\E)×{1}
+where/hatwidelφ(z)is thestrict transform by φof the segment
+lφ(z)={ϕ−1((1−t)ϕ(φ(z)))|t∈[0,1]}
+i.e. it is the closure of the subset {Φ−1((1−t)Φ(z))|t∈[0,1)} ⊂U, where
+Φ :=ϕ◦φ. Then/hatwidelφ(z)∩Eis its unique closure point. This gives the continuity of
+rand ofG∗Fby the gluing lemma over Eand overU×{1
+2,1}. Moreover,
+•G∗F(z,0) =zandG∗F(z,1) =r(z), for every z∈U, which means that
+Eis a deformation retract of U,
+•G∗F(z,t) =zfor every (z,t)∈E×[0,1], which means that Eis a strong
+deformation retract of U.
+/square
+3.Homological type of a conifold transition
+Consider the following c.g.t.
+(4) Y
+CT/d57/d57φ/d47/d47Y/d111/d111 /d47/d47/d47/d111/d47/d111/d47/d111/tildewideY
+Then, by definition, Sing( Y) is entirely composed by a finite number of nodes. The
+following theorem can be obtained by summarizing several results of many authors.
+It gives a complete account of changing in topology induced by a c.g.t.
+Theorem 3.1 (Global changing in topology for a conifold transition [6], [41], [52],
+[51], [36], [32], ...) .Consider the c.g.t. (4). Then
+(5) h[CT] = (k,0,c,c)
+where
+•kis the number of homologically independent exceptional P1s inY,
+•cis the number of homologically independent vanishing cycle s in/tildewideY.
+Moreover, if N:= Sing(Y)is the number of nodes in Y, thenN=k+c.
+Adetailed topologicalproofofthis theorem, based onthe localClem ens’ analysis
+described in [6], is given in [43], section 3.HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 11
+Remark 3.2.Note that point (a) in Definition 1, with kandcas in Theorem
+3.1, implies that the conifold Ydo not satisfy Poincar´ e Duality. The difference
+b4(Y)−b2(Y) =kis called the defect of Y[36]. See also the following remark 4.4.
+This fact can be deeply understood by means of homology of intersections spaces
+andintersection homology , as recently introduced and explained by M. Banagl in
+[2]§5, giving a nice account of relations with type II string theories and m irror
+symmetry in physics (see also the following remark 3.3).
+Remark 3.3.Point (b) in Definition 1, with kandcas in Theorem 3.1, admits
+the following geometric (and physical) interpretation: a c.g.t. increases complex
+moduli(in phisics: hypermultiplets )by the number cof homologically independent
+vanishing cycles and decreases K¨ ahler moduli (in phisics: vector multiplets )by the
+numberkof homologically independent exceptional rational curves .
+Remark 3.4 (Example 1.6 continued) .If Theorem 3.1 is applied to the c.g.t. in
+Example 1.6 one finds that k′=k= 1,k′′= 0, since Yφ−→Yis primitive, as
+induced by a blow up. Then c=N−k= 16−1 = 15 =c′=c′′and recalling that
+/tildewideYis a generic quintic 3–fold in P4one gets the following table of Betti numbers:
+b2b3b4
+Y2 174 2
+Y1 189 2
+/tildewideY1 204 1
+ThenYhas defect 1, as observed in Remark 3.2.
+4.Homological type of a small geometric transition
+Let
+(6) Y
+T/d57/d57φ/d47/d47Y/d111/d111 /d47/d47/d47/d111/d47/d111/d47/d111/tildewideY
+be asmallg.t.. Then Sing( Y) is composed of a finite number of canonical singular-
+ities(see [39], section 1, for the definition) since dim φ−1(p) = 1, for any singular
+pointp∈Y.
+Recall that a compound Du Val (cDV) singularity is a 3–fold point psuch that, for
+a hyperplane section Hthroughp,p∈His a Du Val surface singularity i.e. an
+A–D–E singular point (see [39], sections 0 and 2, and [3], chapter III) . The singular
+locusP:= Sing(Y) and the exceptional locus E:=φ−1(P) have then a well known
+geometry reviewed by the following statement.
+Theorem 4.1 ([40], [26], [38], [30], [10]) .Given a small g.t. as in (6) then:
+(a)anyp∈Pis a cDV singularity,
+(b)for anyp∈P,Ep:=φ−1(p)is a connected union of rational curves meeting
+transversally, whose configuration is dually represented e ither by one of the12 MICHELE ROSSI
+following graphs
+An: ••••• (n≥1vertices)
+•
+/d61/d61/d61/d61/d61/d61/d61/d61
+Dn: ••• (n≥4vertices)
+•/d1/d1/d1/d1/d1/d1/d1/d1
+En:••••••(n= 6,7,8vertices).
+•
+or, ifpis a non–planar singularity, by one the following graphs
+•
+/d61/d61/d61/d61/d61/d61/d61/d61
+/tildewideDn: ••• (n≥3vertices)
+•/d1/d1/d1/d1/d1/d1/d1/d1
+•
+/d1/d1/d1/d1/d1/d1/d1/d1
+/d61/d61/d61/d61/d61/d61/d61/d61
+/tildewideEn:•• •••(n= 5,6,7vertices).
+where triangles are dual graphs representing the transvers e intersection of
+three rational curves at a single point.
+Clearly every conifold transition is a small g.t. admitting exceptional t rees ofA1
+type. The following example presents a small non–conifold transition .
+Example 4.2 (A small non–conifold g.t., [35] Example 1.11 and Remark 2.8, [44]) .
+LetSbe the rational elliptic surface with sections obtained as the Weierst rass
+fibration associated with the bundles homomorphism
+β:E=OP1(3)⊕OP1(2)⊕OP1 /d47/d47OP1(6) (7)
+(x,y,z)/d31 /d47/d47−x2z+y3+B(λ)z3
+for a generic B∈H0(P1,OP1(6)) i.e.Sis the zero locus of βin the projectivized
+bundleP(E). Then:
+•the natural fibration S→P1has generic smooth fiber and 6 distinct (since
+Bis generic) cuspidal fibers ,
+•the fiber product Y:=S×P1Sis a threefold admitting 6 distinct singularities
+of typeII×II, in the standard Kodaira notation [24], whose local equation
+inC4is given by
+(8) x2−u2=y3−v3,HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 13
+•Yis a special fiber of the family of fiber products S1×P1S2of rational elliptic
+surfaces with sections [46]: in particular every rational elliptic surface can
+be thought as the blow up of P2at the base locus of a bi–cubic rational map
+P2[a:b]/axisshort/axisshort/arrowaxisrightP1([34] Prop. 6.1) implying that Ycan be thought as a special fiber
+of the family of smooth resolutions of bi–cubic hypersurfac es([44] Remark
+2.2)
+(9) {a(x)b′(x′) =a′(x′)b(x)} ⊂P2[x]×P2[x′], a,b,a′,b′∈H0(OP2(3)),
+•Yadmits a smallresolutionYφ−→Ywhose exceptional locus is composed
+by 6 disjoint couples of rational curves intersecting in one point i.e. 6
+disjointA2exceptional trees in the notation of Theorem 4.1 ([35]§0.1, [44]
+Proposition 3.1).
+Let us say a few words about the construction of the resolution Yφ−→Y. The
+fibred product Y:=S×P1Scan be thought as embedded in P:=P(E)×P(E)×P1
+as follows
+(10) P:=P(E)×P(E)×P1[λ]⊃Y:/braceleftbiggx2z=y3+B(λ)z3
+u2w=v3+B(λ)z3.
+Consider the following cyclic map on P
+τ:P(E)×P(E)×P1 /d47/d47P(E)×P(E)×P1
+(x:y:z)×(u:v:w)×λ/d31 /d47/d47(x:y:z)×(−u:ǫv:w)×λ ,
+whereǫis a primitive cubic root of unity. The second equation in (10) ensures
+thatτY=Y. Moreover, for any p∈Sing(Y),τ(p) =p, by (8). Consider the
+codimension 2 diagonal locus ∆ := {(x,x′,λ)∈P|x=x′}. Clearly Sing( Y)⊂∆,
+implying that Sing( Y) =Y∩∆∩τ∆. Let then Ybe the strict transform of Yin
+the successive blow up
+/hatwidePτφτ∆/d47/d47/hatwidePφ∆/d47/d47P
+Y/d63/d31/d79/d79
+φ/d47/d47Y/d63/d31/d79/d79
+whereφ∆is the blow up of Palong ∆ and φτ∆is the blow up of /hatwidePalong the strict
+transform /hatwiderτ∆⊂/hatwidePofτ∆. Sinceφturns out to be a small, crepant resolution, Yis
+a Calabi–Yau threefold and
+(11) /hatwideY
+T/d57/d57φ/d47/d47Y/d111/d111 /d47/d47/d47/d111/d47/d111/d47/d111/tildewideY
+is a small non–conifold g.t. with 6 disjoint exceptional trees of type A2, where/tildewideY
+is the smooth small resolution of a bi–cubic hypersurface of type (9 ).
+4.1.Homology change induced by a small geometric transition. We are
+now in a position to write down, for small g.t.’s, a statement analogous to Theorem
+3.1. The following theorem is actually a revised version of results of Y. Namikawa
+andJ.H.M.Steenbrink(see[36], Theorem(3.2)andExample(3.8)). Af ter replacing
+Clemens’localanalysisbyMilnor’sone,theproofgivenhereiscomplet elyanalogous
+to the proof of Theorem 3.1 given in [43].14 MICHELE ROSSI
+Theorem 4.3. Consider the small g.t. (6). Then
+h[T] = (k,0,c′,c′′)
+where
+•kis the number of homologically independent exceptional rat ional curves
+composingE= Exc(φ)⊂Y,
+•c′is the number of independent relations linking the homology classes of
+exceptional rational curves in Y,
+•c′′is the number of homologically independent vanishing cycle s in/tildewideY.
+Moreover,
+(i)the total number of irreducible components of Eis
+n:=/summationdisplay
+p∈Pnp=k+c′,
+wherenpis the numberof irreducible (rational) components of Ep:=φ−1(p),
+for anyp∈P:= Sing(Y),
+(ii)theglobal Milnor number of Yis
+m:=/summationdisplay
+p∈Pmp=k+c′′,
+wherempis the Milnor number of the singular point p∈P. Hencekturns
+out to be also the number of independent relations linking th e homology
+classes of vanishing cycles in /tildewideY.
+In particular, Tis a type I g.t. if and only if k′= 1 =kand ifTis a conifold t.
+thenc′=c′′=c.
+Remark4.4.As in Remark 3.2, point (b) in Definition 1 with k,c′,c′′as in Theorem
+4.3, implies that Yhas defectk.
+On the other hand, by (a) of Theorem 4.1, any p∈Pis a rational i.h.s.. Moreover,
+Yis normal and H2(Y,OY) = 0, since Yis a Calabi–Yau 3–fold. Under all these
+conditions, Lemmas (3.3) and (3.5) in [36] apply to give that
+(12) k= rk/parenleftbig/angbracketleftbig
+Weil divisors of Y/angbracketrightbig
+Z//angbracketleftbig
+Cartan divisors of Y/angbracketrightbig
+Z/parenrightbig
+Recall now that a variety is called Q–factorial if any Weil divisor is a Q–Cartier
+divisor. Then:
+•given a small non trivial g.t. T(Y,Y,/tildewideY), the singular 3–fold Yis never
+Q–factorial.
+In fact, any primitive extremal transition reduces by 1 the rank of Pic(Y)∼=
+H2(Y,Z). By Poinacar´ e duality on Ythe exceptional cycle of a small transition
+can never be homologically trivial.
+Remark 4.5 (Example 4.2 continued) .If Theorem 4.3 is applied to the small g.t.
+in Example 4.2, one finds that k′=k= 2,k′′= 0, sinceYφ−→Yis induced by
+two successive blow-ups. Then c′=n−k= 12−2 = 10. On the other hand the
+singular point (8) has Milnor number mp= 4, by the Milnor–Orlik Theorem ([29]
+Theorem 1). Then the global Milnor number of Yis given by m= 6·4 = 24 givingHOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 15
+thatc′′=m−k= 24−2 = 22. Therefore one gets the following table of Betti
+numbers (see [44], Theorem 3.8):
+b2b3b4
+Y21 8 21
+Y19 18 21
+/tildewideY19 40 19
+ThenYhas defect 2.
+Proof of Theorem 4.3. Given the small g.t. (6), for any p∈PconstructUp,Bp
+and/tildewideUp,/tildewideBplike in Definition 2.1 and Theorem 2.3. Set
+Up:=φ−1(Up), Bp:=φ−1(Bp).
+We have then the following localization, near to p, of the small g.t. (6):
+(13) YT
+/d39/d39
+φ/d47/d47Y/d111/d111 /d47/d47/d47/d111/d47/d111/d47/d111/tildewideY
+Up/d63/d31/d79/d79
+ϕp/d47/d47Up/d63/d31/d79/d79
+/d111/d111 /d47/d47/d47/d111/d47/d111/d47/d111/tildewideUp/d63/d31/d79/d79
+Bp/d63/d31/d79/d79
+ϕp/d47/d47Bp/d63/d31/d79/d79
+/d111/d111 /d47/d47/d47/d111/d47/d111/d47/d111/tildewideBp/d63/d31/d79/d79
+Let us denote:
+U:=/uniondisplay
+p∈PUp, B:=/uniondisplay
+p∈PBp, Y∗:=Y\B , U∗:=U\B;
+U:=/uniondisplay
+p∈PUp,B:=/uniondisplay
+p∈PBp,Y∗:=Y\B ,U∗:=U\B;
+/tildewideU:=/uniondisplay
+p∈P/tildewideUp,/tildewideB:=/uniondisplay
+p∈P/tildewideBp,/tildewideY∗:=/tildewideY\/tildewideB ,/tildewideU∗:=/tildewideU\/tildewideB .
+We get then the following Mayer–Vietoris couples:
+C:= (Y=Y∗∪U , U∗=Y∗∩U),
+C:= (Y=Y∗∪U ,U∗=Y∗∩U),
+/tildewideC:= (/tildewideY=/tildewideY∗∪/tildewideU ,/tildewideU∗=/tildewideY∗∩/tildewideU).
+Step I. ∀i/\e}atio\slash= 2,3bi(Y) =bi(Y) and
+b2(Y)−b2(Y) =k⇔b3(Y)−b3(Y) =n−k .
+Compare the singular homology long exact sequences associated wit hCandC:
+(14)
+··· /d47/d47Hq(U∗) /d47/d47Hq(Y∗)⊕Hq(U) /d47/d47Hq(Y) /d47/d47Hq−1(U∗) /d47/d47···
+(15)
+··· /d47/d47Hq(U∗)/d47/d47Hq(Y∗)⊕Hq(U)/d47/d47Hq(Y) /d47/d47Hq−1(U∗)/d47/d47···16 MICHELE ROSSI
+Sinceφis an isomorphism outside of the exceptional locus Ewe get
+∀q Hq(Y∗)∼=Hq(Y∗) (16)
+Hq(U∗)∼=Hq(U∗)
+Moreover, by proposition 2.4, Bturns out to be a strong deformation retract of U
+and, by theorem 2.3, Bis a union of cones which can be contracted, by straight
+line homotopy, to P. Then
+(17) Hq(U)∼=Hq(P)∼=/braceleftbigg
+Z|P|ifq= 0
+0 otherwise.
+On the other hand, recalling Proposition 2.7, we get the following
+(18)
+∀q Hq(U)∼=Hq(E) =/circleplusdisplay
+p∈PHq(Ep)∼=/circleplusdisplay
+p∈PHq(S2)⊕np∼=
+
+Z|P|forq= 0
+Znforq= 2
+0 otherwise.
+In fact, let us proceed by induction. If np= 1 thenEp∼=S2. Point (b) in Theorem
+4.1 allows us to think of Epas the union
+Ep=C∪E′
+whereC∼=P1
+C,E′isaconnectedunionof np−1rationalcurves,whoseconfiguration
+is still represented by one of the listed graphs, and C∩E′is a single point y. Then
+(E′,C) gives a Mayer–Vietoris couple. Since
+Hq(C∩E′)∼=Hq({y}) = 0∀q≥1,
+the singular homology long exact sequence of ( E′,C) allows us to conclude that
+Hq(Ep)∼=Hq(C)⊕Hq(E′)∀q≥2
+giving (18), for q≥2, by induction hypothesis. We have then the following exact
+sequence
+0→H1(C)⊕H1(E′)→H1(Ep)→H0({y})→H0(C)⊕H0(E′)→H0(Ep)→0
+Since{y},C,E′andEpare all connected, (18) follows for q= 1, too.
+Let us now conclude to prove Step I. By (14) and (15) and formula ( 17) we get that
+∀q/\e}atio\slash= 2,3bq(Y) =bq(Y)
+Moreover, the gluing of sequences (14) and (15), by identification of isomorphic
+poles, reduces to the following diagram
+(19)
+H3(Y)
+/d36/d36/d74/d74/d74/d74/d74/d74/d74/d74/d74H2(Y∗)⊕Zn /d47/d47H2(Y)
+/d36/d36/d74/d74/d74/d74/d74/d74/d74/d74/d74
+H3(Y∗)/d58/d58/d116/d116/d116/d116/d116/d116/d116/d116/d116
+/d36/d36/d73/d73/d73/d73/d73/d73/d73/d73/d73H2(U∗)/d56/d56/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112
+/d38/d38/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78H1(U∗)
+H3(Y)/d58/d58/d117/d117/d117/d117/d117/d117/d117/d117/d117
+H2(Y∗)/d47/d47H2(Y)/d58/d58/d117/d117/d117/d117/d117/d117/d117/d117/d117
+which gives the following relations between Betti numbers
+b2(Y)−b2(Y) =b3(Y)−b3(Y)+n .HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 17
+Step II. ∀q/\e}atio\slash= 3,4bq(/tildewideY) =bq(Y) and
+b3(/tildewideY)−b3(Y) =c′′⇔b4(Y)−b4(/tildewideY) =m−c′′.
+Compare the following singular homology long exact sequence associa ted with
+the couple /tildewideC
+(20)
+··· /d47/d47Hq(/tildewideU∗)/d47/d47Hq(/tildewideY∗)⊕Hq(/tildewideU)/d47/d47Hq(/tildewideY)/d47/d47Hq−1(/tildewideU∗)/d47/d47···
+and the Mayer–Vietoris sequence (15) associated with C.
+Recall that, by Proposition 2.4, /tildewideBis a strong deformation retract of /tildewideU. Moreover,
+Theorem2.5assertsthat, forany p∈P, the Milnorfiber /tildewideBphas the samehomology
+type of a bouquet of mp3–dimensional spheres. Then
+(21) Hq(/tildewideU)∼=Hq(/tildewideB)∼=/circleplusdisplay
+p∈PHq(/tildewideBp)∼=
+
+Z|P|ifq= 0
+Zmifq= 3
+0 otherwise.
+The localization (13) and the Ehresmann fibration theorem allow us to assert that
+there are diffeomorphisms
+Y∗∼=/tildewideY∗andU∗∼=/tildewideU∗
+Then, recalling formulas (17) and (21), we can conclude that
+∀q/\e}atio\slash= 3,4bq(/tildewideY) =bq(Y).
+Moreover, the gluing of sequences (15) and (20), by identification of isomorphic
+poles, reduces to the following diagram
+(22)
+H4(/tildewideY)
+/d36/d36/d73/d73/d73/d73/d73/d73/d73/d73/d73H3(/tildewideY∗)⊕Zm /d47/d47H3(/tildewideY)
+/d36/d36/d73/d73/d73/d73/d73/d73/d73/d73/d73
+H4(/tildewideY∗)/d58/d58/d117/d117/d117/d117/d117/d117/d117/d117/d117
+/d36/d36/d73/d73/d73/d73/d73/d73/d73/d73/d73H3(/tildewideU∗)/d56/d56/d113/d113/d113/d113/d113/d113/d113/d113/d113/d113
+/d38/d38/d77/d77/d77/d77/d77/d77/d77/d77/d77/d77H2(/tildewideU∗)
+H4(Y)/d58/d58/d117/d117/d117/d117/d117/d117/d117/d117/d117
+H3(Y∗)/d47/d47H3(Y)/d58/d58/d117/d117/d117/d117/d117/d117/d117/d117/d117
+which gives the following relations between Betti numbers
+b3(/tildewideY)−b3(Y) =b4(/tildewideY)−b4(Y)+m .
+Step III. Letkandc′′be the same parameters defined in Steps I and II respec-
+tively. Then
+m=k+c′′.
+By Poincar´ e duality
+b2(Y) =b4(Y)
+b4(/tildewideY) =b2(/tildewideY)18 MICHELE ROSSI
+Recall then Steps I and II to get
+b2(Y) =b4(Y) =b4(Y) =b4(/tildewideY)+m−c′′
+=b2(/tildewideY)+m−c′′=b2(Y)+m−c′′=b2(Y)−k+m−c′′
+Thenm−k−c′′= 0.
+Step IV.kis the maximal number of homologically independent exceptional ra-
+tional curves in Ywhilec′′is the maximal number of homologically independent
+vanishing cycles in /tildewideY.
+Since the birational contraction φis an isomorphism outside of the exceptional
+locusEand by the Ehresmann fibration theorem, we get the following compo sition
+of diffeomorphisms
+Y∗∼=Y∗∼=/tildewideY∗
+and, by Lefschetz duality,
+(23) Hi(Y,B)∼=H6−i(Y\B)∼=H6−i(/tildewideY\/tildewideB)∼=Hi(/tildewideY,/tildewideB)
+Consider the long exact relative homology sequences of the couples (Y,B) and
+(/tildewideY,/tildewideB) and the vertical isomorphisms given by (23):
+(24) ···Hi+1(Y,B) /d47/d47
+∼=
+/d15/d15Hi(B) /d47/d47Hi(Y) /d47/d47Hi(Y,B)···
+∼=
+/d15/d15
+···Hi+1(/tildewideY,/tildewideB)/d47/d47Hi(/tildewideB)/d47/d47Hi(/tildewideY)/d47/d47Hi(/tildewideY,/tildewideB)···
+By identifying the isomorphic poles and recalling (18) and (21) the pre vious long
+exact sequences reduce to the following diagram:
+(25)
+0
+/d15/d15
+H3(Y)
+/d15/d15
+0/d47/d47H4(/tildewideY)/d47/d47H4(Y) /d47/d47H3(/tildewideB)γ/d47/d47
+/bardblH3(/tildewideY)/d47/d47H3(/tildewideY,/tildewideB)/d47/d47
+/d15/d150
+Zm H2(B)=
+κ
+/d15/d15Zn
+H2(Y)
+/d15/d15
+H2(/tildewideY)
+/d15/d15
+0HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 19
+Set
+I:= Im[κ:Zn=H2(B)−→H2(Y)]
+Thenk:= rk(I) is the number of linear independent classes of exceptional curves
+inH2(Y). Since
+0 /d47/d47I /d47/d47H2(Y) /d47/d47H2(/tildewideY)/d47/d470
+is a short exact sequence, it follows that
+b2(Y) =b2(/tildewideY)+k
+On the other hand set
+K:= ker[γ:Zm∼=H3(/tildewideB)−→H3(/tildewideY)]
+Thenm−c′′:= rk(K) is the number of linear independent relations on the classes
+of vanishing cycles in H3(/tildewideY). Since
+0 /d47/d47H4(/tildewideY)/d47/d47H4(Y) /d47/d47K /d47/d470
+is a short exact sequence, it follows that
+b4(Y) =b4(/tildewideY)+m−c′′
+Conclusion. (i) follows from Step IV and the definition of nandc′.
+(ii) follows from Steps III and IV.
+Finally (a), (b) and (c) of Definition 1 follow from Steps I, II, III and IV.
+The last assertion of the statement follows from the fact that, if pis a node, then
+mp=np= 1.
+/square
+Remark 4.6.Notice that Step IV in the proof of Theorem 4.3 gives the following
+homological interpretation of k,c′,c′′
+k= rk(Imκ), c′= rk(kerκ), c′′= rk(Imγ),
+whereκandγare the homonymous maps in diagram (25).
+Compare with the case of a type II g.t. whose exceptional locus is a d el Pezzo
+surface of degree d≤4, treated in the following Remark 5.5.
+5.Homological type of type II geometric transition
+In the present section we will consider a type IIg.t.
+(26) Y
+T/d57/d57φ/d47/d47Y/d111/d111 /d47/d47/d47/d111/d47/d111/d47/d111/tildewideY
+RecallingTheorem1.9andDefinition 1.10, φturnsouttobea primitive contraction
+of an irreducible divisor E⊂Ydown to a point p∈Y. Let us summarize what is
+known.20 MICHELE ROSSI
+5.0.1.About the singularity p=φ(E).It is a canonical singularity and in par-
+ticular it is a rational Gorenstein singular point . TheReid invariant ̺ofp(see
+[39]) can assume every value 1 ≤̺≤8 and
+•for̺≤3,pis a i.h.s.;
+•for̺≥4,phas multiplicity ̺and minimal embedding dimension
+dim(mp/m2
+p) =̺+1 ;
+in particular, for ̺= 4,pis a complete intersection singularity and, for
+̺≥5,pis never a complete intersection singular point (see 5.1.4 below);
+5.0.2.About the exceptional locus Eofφ.It is ageneralized del Pezzo surface
+(see [39], Proposition (2.13)) which is:
+•eitherEis anormaldel Pezzo surface of degree 1 ≤d=K2
+E≤8; in
+particular the degree dequals the Reid invariant ̺ofp=φ(E);
+•orEis anon–normal del Pezzo surface (see [42]).
+Observe that the values 0 and 9 cannot be assumed by ̺=d: the former since E
+has ample anti–canonical bundle, the latter because (26) is a trans ition while the
+contraction of a normal del Pezzo surface of degree 9 down to a p oint do not admit
+any smoothing: in this case E∼=P2and (Y,p) is rigid (see [1] and [45]).
+Remark 5.1.The contraction φ:Y→Yis the (weighted, for d≤2) blow up of
+the singular point φ(p) (see [39], Theorem (2.11)). Then Yis always Q–factorial,
+by [23], Proposition 5-1-6.
+5.1.Normal exceptional divisor. A normal del Pezzo surface Eoccurring as
+exceptional locus of φin (26) is a normal projective Gorenstein surface with ample
+anti–canonical bundle . Letπ:/hatwideE→Ebe a minimal resolution of E. The following
+results are essentially due to F. Hidaka and K.–I. Watanabe [18], M. Re id [39],
+H. Pinkham [37] and C.Tam´ as [50].
+5.1.1. ([18] Proposition 2.1 and Theorem 2.2) Eis birationally equivalent to a
+ruled surface and
+•eitherEisrational,
+•or/hatwideEis aP1–bundle over an elliptic curve; under the notation introduced
+in [18], we will say that Eiselliptic.
+In particular H1(E,OE) = 0 ([18], Corollary 2.5).
+5.1.2. Let Eberational. ThenEcanassumeatworstisolatedDuValsingularities.
+Moreover (see [18], Theorem 3.4):
+•ifd= 8 then either E∼=P1×P1orEis the cone over a conic in P2; in
+the latter case /hatwideE∼=P(O⊕O(−2)) andπis the contraction of the minimal
+section of /hatwideE(we have excluded the case E∼=P(O ⊕ O(−1)) since the
+contraction φyields a rigid singularity ( Y,φ(E)), contradicting diagram
+(26));
+•if 1≤d≤7 then there exists a set Σ of points on P2inalmost general
+position (see [18], Definition 3.2) such that |Σ|= 9−dand/hatwideEis the blow up
+ofP2along Σ;πis the contraction of all curves on /hatwideEwith self–intersection
+number -2.HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 21
+5.1.3. IfEis rational then |Sing(E)| ≤6 (see [18] Theorem 4.9, [37] and [50]
+subsection 3.2).
+5.1.4. Ifd≥4 thenEis rational ([15] Thm. (5.2)). Moreover, the anti-canonical
+map embeds EinPdas a surface of degree dobtained by intersecting d(d−3)/2
+hyperquadrics ([18], Theorem 4.4(i) and Corollary 4.5(i)).
+5.1.5. Let Ebeelliptic. ThenEcan assume at worst one elliptic singular point
+(see [39], Definition (2.4)), /hatwideE∼=P(OC⊕L), whereCis a smooth elliptic curve and
+Lis a positive line bundle on C, andπis the contraction of the minimal section of
+/hatwideE([18], Theorem 2.2). In particular, by 5.1.4, d= degE≤3.
+5.1.6. Ifd= 3 then the canonical map embeds Eas a cubic surface in P3([18]
+Theorem 4.4(ii) and Corollary 4.5(i), [39] Proposition (2.3)). Then Eis elliptic if
+and only if it is a cone over a plain cubic curve.
+5.1.7. Ifd= 2 thenEis isomorphic to a degree 4 hypersurface in the weighted
+projective space P(1,1,1,2). The divisor −2KEis very ample and the associated
+morphismembeds Easadegree8subvarietyof P6. Moreover, Ecanbedescribedas
+a double coveringof P2ramified along a quartic curve without multiple components
+(see 5.4.1 for d= 2). Then Eis elliptic if and only if the ramification divisor is
+given by four lines meeting in a point([18] Theorem 4.4(iii), Corollary 4.5( ii) and
+Proposition 4.6(i), [39] Proposition (2.3)).
+5.1.8. Ifd= 1 thenEis isomorphic to a degree 6 hypersurface in the weighted
+projective space P(1,1,2,3). The divisor −3KEis very ample and the associated
+morphism embeds Eas a degree 9 subvariety of P6. Moreover, Ecan be described
+as a double covering of a quadratic cone C ⊂P3, ramified along the intersection
+C ∩S, whereSis a cubic surface without multiple components and not containing
+the vertex of the cone (see 5.4.1 for d= 1). Then Eis elliptic if and only if the
+cubic surface Sis given by three planes meeting in a line which is tangent to C([18]
+Theorem 4.4(iv), Corollary 4.5(iii) and Proposition 4.6(ii), [39] Propos ition (2.3)).
+5.2.Non–normal exceptional divisor. A non–normal del Pezzo surface Eoc-
+curringas exceptionallocus of φin (26) has to satisfy the classificationgivenin [42].
+On the other hand Eis an irreducible surface embedded in the smooth Calabi–Yau
+3-foldY, meaning that Ecannot admit non–hypersurface singularities. M. Gross
+proved that these conditions imply one and only one of the following st atements
+(i) Consider the Segre–del Pezzo scroll Fa:=P(OP1⊕OP1(−a)) embedded in
+Pa+5by means of the very ample linear system |C0+(a+2)f|, whereC0is
+the class of a section and fthe class of a fiber. Then Eis the projection of
+FaintoPa+4from a point in the plane spanned by the conic C0but not on
+C0. This projection exhibits C0as a double covering of a line and makes
+no other identifications.
+(ii) Consider Faembedded in Pa+3by means of the very ample linear system
+|C0+(a+1)f|. ThenEis the projection of FaintoPa+2from a point in
+the plane spanned by the line C0and one fiber f. This projection identifies
+C0andf.
+In particular, a non–normal Ecan occur only if deg( E), or equivalently the Reid
+invariant̺ofp=φ(E), is equal to 7 ([15], Theorem 5.2).22 MICHELE ROSSI
+5.3.Homologychange induced by a type II geometric transition. Consider
+a type II g.t. (26). Then φis a primitive (in case weighted) blow up of an isolated
+3–dimensional rational Gorenstein singular point p∈Y, whose exceptional locus
+is an irreducible generalized del Pezzo surface E⊂Y. We get then the following
+result, which is the type II analogue of Theorems 3.1 and 4.3.
+Theorem 5.2. The type II g.t. (26) admits homological type
+h[T] = (0,1,c′,c′′)
+given by
+(i)c′=−1+b2(E)−b3(E),
+(ii)c′′= 1−χ(/tildewideB) =mp−b2(/tildewideB),
+where/tildewideBis the Milnor fiber of the smoothing /tildewideYandmp:=b3(/tildewideB)is the Milnor
+number of the unique singular point p=φ(E)∈Y. In particular b1(/tildewideB) = 0.
+Proof.Let us start by proving (c) in Definition 1, with
+(27) k= 1 =k′′, c′=χ(E)−3, c′′= 1−χ(/tildewideB).
+Consider the same notation introduced in the proof of Theorem 4.3, but observe
+that nowP:= Sing(Y) ={p}. Then the singular homology long exact exact
+sequences of the Mayer–Vietoris couples CandCare given by
+(28)
+··· /d47/d47Hq(U∗) /d47/d47Hq(Y∗)⊕Hq(Up) /d47/d47Hq(Y) /d47/d47Hq−1(U∗) /d47/d47···
+(29)
+··· /d47/d47Hq(U∗)/d47/d47Hq(Y∗)⊕Hq(Up)/d47/d47Hq(Y)/d47/d47Hq−1(U∗)/d47/d47···
+Then Theorem 2.3 and Proposition 2.4 give that
+(30) Hq(Up)∼=Hq(p)∼=/braceleftbigg
+Zifq= 0
+0 otherwise.
+On the other hand Proposition 2.7 gives a strong deformation retraction Up≈B≈
+E, giving
+(31) ∀q Hq(Up)∼=Hq(B)∼=Hq(E).
+Then the first equation in Definition 1.(c) follows by comparing (28) an d (29).
+Consider now the singular homology long exact sequence of the Maye r–Vietoris
+couple/tildewideC
+(32)
+··· /d47/d47Hq(/tildewideU∗)/d47/d47Hq(/tildewideY∗)⊕Hq(/tildewideUp)/d47/d47Hq(/tildewideY)/d47/d47Hq−1(/tildewideU∗)/d47/d47···
+Proposition 2.4 and Theorem 2.5 then guarantee that
+(33) ∀q Hq(/tildewideUp)∼=Hq(/tildewideB)⇒ ∀q≥4bq(/tildewideUp) = 0.
+Comparing (29) and (32) one gets the second relation in Definition 1.( c).
+Assume now the first equation of (a) in Definition 1 giving b2(Y) =b2(/tildewideY) and
+notice that b1(/tildewideY) =b1(Y) = 0, since Yand/tildewideYare Calabi–Yau 3–folds. The latter
+gives in particular that b1(Y) = 0, as a consequence of the Leray spectral sequence
+of the sheaf φ∗Z. Then the comparison of (29) to (32) and the Five Lemma give
+H1(/tildewideB)∼=H1(P) = 0, proving (ii).HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 23
+The proof of (a) and (b) in Definition 1, with k= 1 andc′,c′′as in the statement,
+are postponed to subsections 5.4, 5.5 and 5.6 where these relations will be analyzed,
+case by case, under stronger assumptions. In particular, (i) follo ws by (34), (65)
+and (69). /square
+5.4.The normal and rational case. Given the type II g.t. (26) let us now
+consider the case in which Eis assumed to be a normal and rational del Pezzo
+surface. This hypothesis includes all the cases in which Eis smooth and it is not
+so restrictive since it leaves out only the following two further cases :
+•Eis anormal and elliptic del Pezzo surface,
+•Eis anon-normal del Pezzo surface,
+which will be discussed in the following subsections 5.5 and 5.6, respect ively.
+Theorem 5.3. Assume that the type II g.t. (26) admits a normal and rational del
+Pezzo surface as exceptional divisor E= Exc(φ)and letπ:/hatwideE→Ebe a minimal
+resolution of E, withL:= Exc(π)which is composed by nErational curves organized
+in at most six A–D–E trees, by 5.1.2 and 5.1.3. Then it admits h omological type
+h[T] = (0,1,c′,c′′)as in Theorem 5.2 and moreover
+(i)c′= 9−d−nE,
+(ii)ifd≤4thenc′′=mp.
+Let us postpone the proof of the previous theorem to recall that YisQ-factorial,
+by Remark 5.1. This fact allows us to conclude the following
+Lemma 5.4. Letp∈Ybe the 3–dimensional i.s. obtained as image of a (gen-
+eralized) del Pezzo surface E⊂Yunder a type II birational contraction from a
+Calabi–Yau threefold Y, as above. Then
+b4(Y) =b2(Y).
+Proof.Ifp∈Yis ani.h.s. then the statementis aconsequenceofLemmas(3.3) and
+(3.5) in [36]. Actually we will observe that Namikawa–Steenbrink consid erations
+applies to the more general case of p∈Y, too.
+Following their notation, notice first of all that, since p∈Yis a rational singular
+point,Weil(Y)/Cart(Y) is a finitely generated abelian group and let σ(Y) denote
+its rank. Moreover, the rationality of pgivesh2(OY) = 0, by Leray spectral se-
+quence. Then Lemma (3.5) in [36] applies to p∈Y: in fact arguments proving this
+Lemmaarelocalcohomologyexactsequence, Lerayspectralseq uenceandGoresky–
+MacPherson theorem [47] Theorem (1.11) holding for an i.s. By applyin g Lemma
+(3.3) in [36] we get that σ(Y) =b2(Y)−b4(Y). Then Q–factoriality of Yends up
+the proof. /square
+Proof of Theorem 5.3. By 5.1.2 and 5.1.3, E= Exc(φ) admits at worst six isolated
+Du Val singularities. Moreover, a minimal resolution π:/hatwideE→Eis the contraction
+of all curves in /hatwideEwith self–intersection number −2 where /hatwideEis either P1×P1
+(whend= 8) or P2blown up in 9 −dpoints in almost general position. Define
+Q:= Sing(E) and consider the exceptional tree Lq:=π−1(q), for anyq∈Q.
+Sayingnqthe number of irreducible components of Lqconsider the global number
+nE:=/summationtext
+q∈Qnq. The same induction argument proving (18) gives nE=b2(L).
+Consider the relative homology of the couple ( /hatwideE,L) and define κE:H2(L,Q)→24 MICHELE ROSSI
+H2(/hatwideE,Q). Call
+kE:= rk[Im(κE)], cE:= rk[ker(κE)].
+ThennE=kE+cEand we claim that
+(34)b0(E) =b4(E) = 1, b1(E) = 0, b3(E) =cE, b2(E) = 10−d−kE.
+In fact, first equalities on the left of (34) are clearly obvious. Then apply the local
+analysis in §2 to anyq∈Q⊂Eand define
+UQ:=/uniondisplay
+q∈QUq, BQ:=/uniondisplay
+q∈QBq,/hatwideE∗:=/hatwideE\BQ, U∗
+Q:=UQ\BQ;
+UQ:=/uniondisplay
+q∈QUq,BQ:=/uniondisplay
+q∈QBq, E∗:=E\BQ,U∗
+Q:=UQ\BQ.
+Then compare the associated Mayer–Vietoris homology sequences :
+(35)
+··· /d47/d47Hi(U∗
+Q) /d47/d47Hi(/hatwideE∗)⊕Hi(UQ)/d47/d47Hi(/hatwideE)/d47/d47Hi−1(U∗
+Q) /d47/d47···
+(36)
+··· /d47/d47Hi(U∗
+Q) /d47/d47Hi(E∗)⊕Hi(UQ)/d47/d47Hi(E) /d47/d47Hi−1(U∗
+Q) /d47/d47···
+Going on precisely as in Step I of the Proof of the Theorem 4.3 and rec alling that
+/hatwideEis either P1×P1or the blow up of P2in 9−din almost general position, one
+gets the following relation
+(37) b4(/hatwideE)−nE+10−d=b4(E)−b3(E)+b2(E)−b1(E).
+On the other hand, compare the relative homology sequences of th e couples (E,Q)
+and (/hatwideE,L) as follows
+(38)
+0
+/d15/d15
+H4(/hatwideE)
+/d15/d15H3(/hatwideE)
+/d15/d15
+0/d47/d47H4(E) /d47/d47H4(E,Q)
+/d15/d15/d47/d47H3(Q) /d47/d47H3(E) /d47/d47H3(E,Q)
+/d15/d15/d47/d47H2(Q)
+/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123
+/d125/d125/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123H3(L)/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d54/d54/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108
+H2(L)
+κE/d15/d15
+H2(/hatwideE)
+/d15/d15
+H2(E) /d47/d47H2(E,Q)
+/d15/d15/d47/d47H1(Q)
+H1(L)
+Observe that b3(L) =b1(L) =b3(Q) =b2(Q) =b1(Q) = 0. Hence H4(E)∼=
+H4(E,Q)∼=H4(/hatwideE), giving
+(39) b4(/hatwideE) =b4(E),HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 25
+andHi(E)∼=Hi(E,Q) fori= 2,3. Then the right vertical sequence in (38)
+translates immediately in the following exact sequence
+(40)
+0/d47/d47H3(/hatwideE)/d47/d47H3(E) /d47/d47H2(L)κE/d47/d47H2(/hatwideE)/d47/d47H2(E) /d47/d470.
+Recalling that b2(L) =nE,b3(/hatwideE) = 0 andb2(/hatwideE) = 10−dwe get
+(41) b3(E)−b2(E) =nE−10+d.
+The comparison of (37),(39) and (41) then gives b1(E) = 0.
+Moreover,b3(/hatwideE) = 0 in (40) gives that b3(E) = rk[ker(κE)] =cE.
+The last equality in (34) is then obtained by recalling that nE=kE+cE.
+Let us now prove relations in Definition 1.(a). First of all recall Mayer –Vietoris
+exact sequences (28), (29) and (32) of couples C,Cand/tildewideC. Relations (34) and the
+Five Lemma prove that
+(42) b1(Y) =b1(Y), b5(Y) =b5(/tildewideY).
+Butb1(Y) =b1(/tildewideY) = 0, since Yand/tildewideYare Calabi–Yau threefolds. Then (42) and
+Poincar´ e duality prove the first line in Definition 1.(a).
+For the second line in Definition 1.(a), observe that Lefschetz dualit y and Ehres-
+mann Fibration Theorem give
+Hi(Y,B)∼=H6−i(Y∗)∼=H6−i(/tildewideY∗)∼=Hi(/tildewideY,/tildewideB).
+Then isomorphisms (31) and the vanishing in (34) reduce relative hom ology long
+exact sequences of couples ( Y,B) and (/tildewideY,/tildewideB) to give the following type II version
+of diagram (25), where vertical sequences are given by relative ho mology of couple
+(Y,B), while horizontal ones are obtained by relative homology of ( /tildewideY,/tildewideB):
+(43)
+H4(E)
+λ/d15/d15
+H4(Y)
+/d15/d15H3(Y)
+/d15/d15
+0/d47/d47H4(/tildewideY)/d47/d47H4(/tildewideY,/tildewideB)
+/d15/d15/d47/d47H3(/tildewideB)γ/d47/d47H3(/tildewideY)/d47/d47H3(/tildewideY,/tildewideB)
+/d15/d15/d47/d47H2(/tildewideB)
+/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123
+/d125/d125/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123/d123H3(E)/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d54/d54/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108/d108
+H2(E)
+κ/d15/d15
+H2(Y)
+/d15/d15
+H2(/tildewideY)/d47/d47H2(/tildewideY,/tildewideB)/d47/d47
+/d15/d15H1(/tildewideB)
+0
+The vertical sequence on the left of this diagram gives the following s hort exact
+sequence
+0 /d47/d47Im(λ) /d47/d47H4(Y) /d47/d47H4(/tildewideY,/tildewideB)/d47/d470.26 MICHELE ROSSI
+Observe that b4(E) = 1 andE⊂Yis the exceptional divisor of a blow up, giving
+rk(Imλ) = 1. Therefore
+(44) b4(Y)−b4(/tildewideY,/tildewideB) = 1.
+Since/tildewideYis a Calabi–Yau threefolds, h1(O/tildewideY) =h2(O/tildewideY) = 0 and the exponential
+sequence gives Pic( /tildewideY)∼=H2(/tildewideY,Z). On the other hand, by [25] (12.2.1.3) and
+(12.2.1.4.2), we are in a position to apply Proposition 3.1 in [16] implying tha t the
+Picard number remains invariant when smoothing Y, which is
+(45) ρ(Y) =ρ(/tildewideY) =b4(/tildewideY).
+Sincep∈Yis a rational singularity, pushing forward the exponential sequenc e for
+Yinduces the following exact sequence on Y
+(46) 0 /d47/d47Z /d47/d47OY/d47/d47O∗
+Y/d47/d47R1φ∗Z /d47/d470
+On the other hand, the Leray spectral sequence converging to Hi(Y,Z) gives the
+following lower terms exact sequence
+0 /d47/d47H1(Y,Z)/d47/d47H1(Y,Z) /d47/d47H0(Y,R1φ∗Z)/d47/d47H2(Y,Z)/d31/d127/d47/d47H2(Y,Z)
+where the latter injection is given by the fact that φ:Y→Yis a blow up centered
+in a point. Moreover, Yis a Calabi–Yau threefold, giving
+H1(Y,Z) = 0 =⇒H1(Y,Z) = 0.
+Notice that R1φ∗Zis a skyscraper sheaf supported on p∈Y, then
+H0(R1φ∗Z) = 0⇐⇒R1φ∗Z= 0,
+implying that (46) actually gives the following exponential sequence for Y
+(47) 0 /d47/d47Z /d47/d47OY/d47/d47O∗
+Y/d47/d470.
+On the other hand, the Leray spectral sequence converging to Hi(Y,OY) gives the
+following lower terms exact sequence
+0 /d47/d47H1(OY) /d47/d47H1(OY) /d47/d47H0(Y,R1φ∗OY)/d47/d47H2(OY) /d47/d47H2(OY)
+whereh1(OY) =h2(OY) = 0 andR1φ∗OY= 0, sinceYis a Calabi–Yau threefold
+andp∈Yis a rational singular point. Therefore
+h1(OY) =h2(OY) = 0
+giving, by (47), that
+(48) Pic( Y)∼=H1(O∗
+Y)∼=H2(Y,Z).
+Recall that φis a primitive contraction, which is ρ(Y)−ρ(Y) = 1. Then (45), (48),
+Lemma 5.4 and Poincar´ eDuality provethe second and the fourth line s in Definition
+1.(a) withk′= 0 andk=k′′= 1.
+Therefore (44) and diagram (43) give that H4(/tildewideY)∼=H4(/tildewideY,/tildewideB) (guaranteeing the
+injectivity of γ).
+Recall Theorem 5.2 and in particular equations in Definition 1.(c) with k,c′,c′′as
+in (27). By (34), these equations can be now rewritten as follows
+b3(Y) =b3(Y)+1−b2(E)+b3(E) =b3(Y)−9+d+nE (49)
+b3(Y) =b3(/tildewideY)+χ(/tildewideB)−1,HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 27
+proving relations in the third line of Definition 1.(a) with c′= 9−d−nE, as in
+point (i) of the statement. Equations in Definition 1.(b) then follow imm ediately
+by those in Definition 1.(a) and Calabi–Yau conditions on Yand/tildewideY.
+Let us now assume that d≤4: then (ii) is a consequence of Theorem 2.5, since the
+Milnor fiber /tildewideBhas the homotopy type of a bouquet of 3–spheres and
+(50) b0(/tildewideB) = 1, b1(/tildewideB) =b2(/tildewideB) = 0, b3(/tildewideB) =mp.
+/square
+Remark 5.5. If the exceptional del Pezzo surface Ehas degreed≤4 thenk,c′,c′′
+admitthe same homological interpretation given in Remark 4.6 for a small g.t.
+In fact, (50) applied to diagram (43) gives H2(/tildewideY)∼=H2(/tildewideY,/tildewideB) and the following
+short exact sequence
+0 /d47/d47Im(κ) /d47/d47H2(Y) /d47/d47H2(/tildewideY)/d47/d470.
+On the other hand, the second equation in Definition 1.(a) with k= 1 gives
+k= 1 =b2(Y)−b2(/tildewideY) = rk(Imκ).
+Recalling then (i) and (34) we get k+c′= 10−d−kE=b2(E) giving necessarily
+that
+c′=b2(E)−rk(Imκ) = rk(kerκ).
+Finally (ii) in Theorem 5.3 and the injectivity of γin diagram (43) imply that
+c′′= rk(Im(γ)).
+Remark 5.6. If the exceptional del Pezzo surface Eis smooth then, comparing
+relations (i) in Theorem 5.3 and (ii) in Theorem 5.2, with Theorem 3.3 and R emark
+3.6 in [22], we get the following conditions on the Milnor number mpof the singular
+pointp∈Y:
+ifd= 1 then mp= 50, (51)
+ifd= 2 then mp= 27, (52)
+ifd= 3 then mp= 16, (53)
+ifd= 4 then mp= 9, (54)
+ifd= 5 then mp=b2(/tildewideB)+4, (55)
+ifd= 6 then mp=/braceleftigg
+b2(/tildewideB)−1
+b2(/tildewideB)+1, (56)
+ifd= 7 then mp=b2(/tildewideB) (57)
+ifd= 8 then mp=b2(/tildewideB)+1 (58)
+Sincep∈Yis a singular point, (57) and the first case in (56) prove that b2(/tildewideB)>0,
+showing that the Milnor fiber /tildewideBcannot admit the homotopy type of a bouquet of
+3–dimensional spheres when d= 7and whend= 6withc= 1.28 MICHELE ROSSI
+Notice that c+k=c+1 gives the following (dual) Coxeter numbers
+(59) c+k=
+
+30 = Coxeter( E8) ford= 1
+18 = Coxeter( E7) ford= 2
+12 = Coxeter( E6) ford= 3
+8 = Coxeter( D5) ford= 4
+5 = Coxeter( A4) ford= 5
+2 = Coxeter( A1) ford= 8
+as argued in [33] §7.1 and in [32] §3. Unfortunately this is not true for d= 6 and
+d= 7, suggesting a relation between the vanishing of b2(/tildewideB)and the existence of a
+Coxeter group whose number gives c+k.
+5.4.1.Examples. The Reid and Hidaka–Watanabe results ([39] Proposition (2.13)
+and [18] Theorem 4.4), here reported in 5.1.4, 5.1.6, 5.1.7 and 5.1.8, allow s us to
+easily construct examples of type II g.t’s admitting Milnor numbers as in (54), (53),
+(52) and (51), respectively (see also Theorem 4.5 in [22] for a compa rison).
+d= 3. Consider the generic quintic 3-fold in P4admitting a triple point in
+p:= [1 : 0 : ···: 0]∈P4
+Y:=/braceleftbig
+x2
+0f3(x1,...,x 4)+x0f4(x1,...,x 4)+f5(x1,...,x 4) = 0/bracerightbig
+,
+wherefkare homogeneous polynomials of degree k.
+Clearly the generic quintic 3-fold /tildewideY⊆P4is a smoothing for Y.
+Blow up P4inpand letP3[l1:...:l4] be the exceptional divisor. Then the strict
+transformYofYadmits exceptional locus given by the following cubic surface
+E:={f3(l1,...,l3) = 0} ⊆P3[l1:...:l4], which is smooth for generic f3(compare
+with 5.1.6). Observe that Yis a Calabi–Yau threefold since hi(OY) =hi(OY)
+([3] Theorem I.(9.1)) and Y→Yis a crepant resolution ([39] Theorem (2.11) and
+Corollary (2.12)). Therefore
+•T(Y,Y,/tildewideY)is generically a type II g.t. whose exceptional divisor Eis a
+smooth del Pezzo surface of degree 3 .
+Sincef3= 0 is the local equation of the triple point p∈Yand it is a homogeneous
+polynomial, the Milnor–Orlik Theorem ([29] Theorem 1) allows us to conc lude that
+the Milnor number of pis given by mp= (3−1)4= 16.
+Since/tildewideYis a projectivehypersurface, its Betti numbers arewellknown an dTheorem
+5.3 givesc′= 6,c′′= 16 and the following table
+b2b3b4
+Y2 182 2
+Y1 188 1
+/tildewideY1 204 1
+d= 2. Consider the degree 6 hypersurface of P(1,1,1,1,2)[x0,x1,x2,x3,y] given
+by
+Y:=/braceleftbig
+x2
+0f4(x1,...,x 3,y)+x0f5(x1,...,x 3,y)+f6(x1,...,x 3,y) = 0/bracerightbig
+,
+wherefkis a generic weighted homogeneous polynomials of degree k. Clearly
+p:= [1 : 0 : ···: 0] is an i.h.s. for Ywhose local equation is given by the w.h.
+polynomial f4= 0. Then, again by Milnor–Orlik theorem, mp= (4−1)3= 27.HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 29
+Thestricttransform YofYunderthe weightedblowupof P(1,1,1,1,2)inpadmits
+exceptional locus given by the degree 4 surface E:={f4= 0} ⊆P(1,1,1,2) which
+is smooth for f4sufficiently general.
+Finally the generic degree 6 hypersurface of /tildewideY⊂P(1,1,1,1,2) is smooth since the
+latter has a unique isolated singular point. Then
+•T(Y,Y,/tildewideY)is generically a type II g.t. whose exceptional divisor Eis a
+smooth del Pezzo surface of degree 2 .
+Observe that f4(x1,x2,x3,y) =y2+yg2(x) +g4(x) withglgeneric homogeneous
+polynomials of degree linx=x1,x2,x3. ThenEturns out to be a double covering
+ofP2[x] ramified along the discriminant quartic plane curve ∆ = {g2
+2−4g4= 0}
+(compare with 5.1.7).
+Since/tildewideYis a weighted projective hypersurface, its Betti numbers are well known ([8]
+4.3.2, [48], [20] Thm. 7.2) and Theorem 5.3 gives c′= 7,c′′= 27 and the following
+table
+(60)b2b3b4
+Y2 174 2
+Y1 181 1
+/tildewideY1 208 1
+d= 1. Consider the degree 8 hypersurface of P(1,1,1,2,3)[x0,x1,x2,y,z] given
+by
+Y:=/braceleftbig
+x2
+0f6(x1,x2,y,z)+x0f7(x1,x2,y,z)+f8(x1,x2,y,z) = 0/bracerightbig
+,
+wherefkare generic weighted homogeneous polynomials of degree k. As before
+p:= [1 : 0 : ···: 0] is an i.h.s. for Ywhose local equation is given by the w.h.
+polynomial f6= 0. Thenmp= (3−1)(6−1)2= 50.
+Thestricttransform YofYunderthe weightedblowupof P(1,1,1,2,3)inpadmits
+exceptional locus given by the degree 6 surface E:={f6= 0} ⊆P(1,1,2,3) which
+is smooth for f6sufficiently general.
+The generic degree 8 hypersurface of /tildewideY⊂P(1,1,1,2,3) is still smooth since the
+latter has only isolated singular points. Then
+•T(Y,Y,/tildewideY)is generically a type II g.t. whose exceptional divisor Eis a
+smooth del Pezzo surface of degree 1 .
+Observe that f6(x1,x2,y,z) =z2+zg3(x,y) +g6(x,y) withglgeneric weighted
+homogeneous polynomial of degree linx=x1,x2andy. ThenEturns out to be
+a double covering of P[1,1,2] ramified along the discriminant degree 6 plane curve
+∆ ={g2
+3−4g6= 0}. Let us first of all observe that
+g3(x,y) =h1(x)·y+h3(x), g6(x,y) =ay3+h2(x)·y2+h4(x)·y+h6(x)
+wherehlis a generic homogeneous polynomials of degree l. Then the discriminant
+curve∆donotpassthroughtheuniquesingularpointof P[1,1,2],whichis[0 : 0 : 1].
+Moreover, the linear system associated with O(2) embeds P(1,1,2) as a quadratic
+coneC ⊂P3, translating the degree 6 equation of ∆ in the equation of a cubic
+surface of P3cutting ∆ on Coutside of its vertex (compare with 5.1.8).
+Since/tildewideYis a weighted projective hypersurface, its Betti numbers are well known ([8]
+4.3.2, [48], [20] Thm. 7.2) and Theorem 5.3 gives c′= 8,c′′= 50 and the following30 MICHELE ROSSI
+table
+b2b3b4
+Y2 156 2
+Y1 164 1
+/tildewideY1 214 1
+d= 4. Finally let us consider the case of a type II g.t. whose exceptiona l divisor
+is a del Pezzo surface of degree 4, meaning that Sing( Y) is composed by a unique
+complete intersection and non–hypersurface singularity . At this purpose consider
+the degree 6 complete intersection in P5given by
+Y:=X1∩X2withX1:={x0f2(x)+f3(x) = 0}
+X2:={x0g2(x)+g3(x) = 0}
+wherefk,gkare generic homogeneous polynomials of degree kinx=x1,...,x 5.
+As before Sing( Y) ={p}, wherep:= [1 : 0 :...: 0]. In particular, pis aquadratic
+3–dimensional c.i.s. , meaning that it is locally described by the germ of singularity
+given by the zero locus in C5of two homogeneous quadratic polynomials, namely
+f2andg2. Then, recalling [27] Example 1 in (5.11), the Milnor number of pis
+mp= 2·3+3 = 9.
+Blow up P5inpand letP4[l] be the exceptional divisor. Then the strict transform
+YofYadmits exceptional locus given by quartic complete intersection sur face
+E:={f2(l) =g2(l) = 0} ⊂P4[l], which is smooth for f2,g2sufficiently general
+(compare with 5.1.4).
+On the other hand the generic complete intersection /tildewideYof two cubic hypersurfaces
+inP5is a Calabi–Yau 3–fold. Then
+•T(Y,Y,/tildewideY)is generically a type II g.t. whose exceptional divisor Eis a
+smooth del Pezzo surface of degree 4 .
+Since/tildewideYis a projective complete intersection, its Betti numbers are well kn own and
+Theorem 5.3 gives c′= 5,c′′= 9 and the following table
+b2b3b4
+Y2 134 2
+Y1 139 1
+/tildewideY1 148 1
+5.5.The elliptic case. Let us now assume that the type II g.t. (26) admits
+exceptional locus E= Exc(φ) given by a normal and ellipticdel Pezzo surface
+(recall 5.1.1 and 5.1.5). Then Theorem 5.2 can be rewritten as follows:
+Theorem 5.7. Assume that the type II g.t. (26) admits a normal and elliptic del
+Pezzo surface as exceptional divisor E= Exc(φ). Then it admits homological type
+h[T] = (0,1,c′,c′′)as in Theorem 5.2 and moreover
+(i)b2(E) = 1andb3(E) = 2, givingc′=−2,
+(ii)c′′=mpand in particular it must be even.
+The previous (ii) gives immediately the followingHOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 31
+Corollary 5.8. Ifφ:Y→Yis a primitive type II smooth resolution such that
+E=Exc(φ)admits an elliptic singular point and Yis smoothable then the Milnor
+numbermpofp=φ(E)is even.
+Example 5.9.The present example is aimed to give an account of how the Milnor
+number of p=φ(E) may jump when specializing to an elliptic exceptional del
+Pezzo surface E, respecting Corollary 5.8. Let us, in fact, consider the case d= 2
+in Examples 5.4.1, whose general case gives Milnor number mp= 27. Specialize to
+consider
+f4:=y2+g2(x1,x2)y+g4(x1,x2)
+f5=ax5
+3+g5(x1,x2,y)
+f6=bx6
+3+g6(x1,x2,y)
+wheregkis a generic (weighted, for k= 5,6) homogeneous polynomial of degree k
+and (a,b)/\e}atio\slash= (0,0): this last condition is necessary to guarantee that
+p:= [1 : 0 : 0 : 0 : 0] ∈Y⊂P:=P(1,1,1,1,2)
+is anisolatedsingular point. Its local equation in C4is given by
+y2+g2(x1,x2)y+g4(x1,x2)+ax5
+3= 0 ifa/\e}atio\slash= 0
+y2+g2(x1,x2)y+g4(x1,x2)+bx6
+3= 0 ifa= 0 andb/\e}atio\slash= 0.
+Milnor–Orlik Theorem ([29] Theorem 1) then gives
+mp= 32·4 = 36 ifa/\e}atio\slash= 0
+mp= 32·5 = 45 ifa= 0 andb/\e}atio\slash= 0.
+The strict transform YofY, under the weighted blow up of Pinp, admits as
+exceptional locus the elliptic del Pezzo surface
+E:={m2+g2(l1,l2)m+g4(l1,l2) = 0} ⊂P(1,1,1,2)[l1,l2,l3,m]
+admitting an elliptic singular point in q:= [0 : 0 : 1 : 0] ∈P(1,1,1,2). By Corollary
+5.8 we see that Ycan be a smooth resolution of Yonly ifa/\e}atio\slash= 0, since only in this
+casempis even. In fact, dividing by l3, the equations of Yin the affine open subset
+A0,3:={([x0:···:x3:y],[l1,l2,l3,m])∈P(1,1,1,1,2)×P(1,1,1,2)|(x0,l3)/\e}atio\slash=0}
+are given by h= (h1,...,h 4) =0where
+h1:=x1−l1x3
+h2:=x2−l2x3
+h3:=y−mx2
+3
+h4:=m2+g2(l1,l2)m+g4(l1,l2)+ax3+x3g5(l1,l2,m)+bx2
+3+x2
+3g6(l1,l2,m)
+whose jacobian in 0∈ A0,3∼=C7gives
+J0(h) :=∂h(0)
+∂(x1,...,x 3,y,l1,l2,m)=
+1 0 0 0 0 0 0
+0 1 0 0 0 0 0
+0 0 0 1 0 0 0
+0 0a0 0 0 0
+.32 MICHELE ROSSI
+Clearly rk(J0(h)) = 4 if and only if a/\e}atio\slash= 0, as expected by Corollary 5.8. Observe
+that ifa/\e}atio\slash= 0 thenT(Y,Y,/tildewideY) is a g.t. whose Betti number can be obtained by
+Theorem 5.13 giving c′=−2,c′′= 36 and
+b2b3b4
+Y2 174 2
+Y1 172 1
+/tildewideY1 208 1
+Compare with table (60).
+5.5.1.Local analysis of the elliptic singularity. The proof of Theorem 5.7 come
+from the study of a minimal resolution, of a complex smoothing and of the Milnor
+number of the elliptic singular point of E.
+Lemma 5.10 (The resolution) .Letπ:/hatwideE→Ebe a minimal resolution of the
+elliptic del Pezzo surface E. Then
+(61)b0(/hatwideE) =b4(/hatwideE) = 1, b1(/hatwideE) =b3(/hatwideE) = 2, b2(/hatwideE) =h1,1(/hatwideE) = 2.
+Proof.First equations are clearly obvious. By 5.1.5, /hatwideE∼=P(OC⊕L) is a geomet-
+rically ruled surface over an elliptic curve C. Then
+∀i>0hi(O/hatwideE) =hi(OC) ([17] Lemma V.2.4)
+givingh1(O/hatwideE) = 1 and h2(O/hatwideE) = 0. These relations and Kodaira–Serre duality
+suffice to end up the proof. /square
+Lemma 5.11 (The smoothing) .Let/tildewideEbe a smoothing of the elliptic del Pezzo
+surfaceE. Then
+(62) b0(/tildewideE) =b4(/tildewideE) = 1, b1(/tildewideE) =b3(/tildewideE) = 0, b2(/tildewideE) = 10−d
+whered= deg(E).
+Proof.Recall that 1 ≤d≤3, by 5.1.4. Moreover:
+•ifd= 3 thenEis described in 5.1.6 and /tildewideEis a generic cubic surface in P3,
+•ifd= 2 thenEis described in 5.1.7 and /tildewideEis a generic quartic surface in
+P(1,1,1,2),
+•ifd= 1 thenEis described in 5.1.8 and /tildewideEis a generic degree 6 surface in
+P(1,1,2,3).
+Therefore /tildewideEturns out to be isomorphic to a smooth degree ddel Pezzo surface.
+Then (62) follow by (34) with kE=cE= 0. /square
+Lemma 5.12 (The Milnor number) .Letqbe the unique (elliptic) singular point
+of an elliptic del Pezzo surface Eof degree deg(E) =d. Then the Milnor number
+ofq∈Eis given by mq= 11−d.
+Proof.By 5.1.6, if d= 3 thenqis the vertex of a cone over a cubic plane curve
+f3(x1,x2,x3) = 0. Then mq= 23= 8 = 11 −3.
+By 5.1.7, if d= 2 thenEis a double covering of P2ramified along four distinct
+lines meeting in the elliptic singular point q∈E, whose local equation in C2is then
+given the degree 4 homogeneous polynomial in two variables giving the four lines.
+Thenmq= 32= 9 = 11 −2.HOMOLOGICAL TYPE OF GEOMETRIC TRANSITIONS 33
+By 5.1.8, if d= 1 thenEis the double covering of P(1,1,2)[x0,x1,y] ramified along
+the reducible degree 6 curve {/producttext3
+i=1(y−qi(x1) = 0}, where deg qi= 2. The elliptic
+singular point qis then given by [1 : 0 : 0] ∈P(1,1,2), whose Milnor number is
+mq= 2·5 = 10 = 11 −1, by the Milnor–Orlik Theorem ([29] Theorem 1). /square
+5.5.2.Proof of Theorem 5.7. Let us recall that π:/hatwideE→Eis the contraction of the
+minimal section of /hatwideE∼=P(OC⊕L) (see 5.1.5). Then setting {q}=Q:= Sing(E)
+andLq:=π−1(q) one has
+(63) L= Exc(π) =Lq∼=C
+and the total number of irreducible components of Lis given by nE=nq= 1 =
+b2(L) (notation as in the Proof of theorem 5.2). In the present case, ( 37) has to be
+rewritten as follows
+b3(/hatwideE)+b2(L)−b2(/hatwideE)−b1(L)+b1(/hatwideE)+b0(L) =b3(E)−b2(E)+b1(E)+|Q|
+which, by (61) and (63), gives
+(64) b3(E)−b2(E)+b1(E) = 1.
+On the other hand, calling /tildewideAthe Milnor fiber near q∈E, it has the homotopy type
+of a bouquet of 2–spheres, by Theorem 2.5 and the fact that q∈Eis an isolated
+hypersurface singularity. By Lefschetz Duality, Ehresmann diffeo morphism and
+relative homology of the couple ( E,Q), one has
+∀i≥2Hi(/tildewideE,/tildewideA)∼=Hi(/tildewideE\/tildewideA)∼=Hi(E\Q)∼=Hi(E,Q)∼=Hi(E).
+Then the relative homology long exact sequence of the couple ( /tildewideE,/tildewideA) gives the
+following exact sequence
+0/d47/d47H3(/tildewideE)/d47/d47H3(E) /d47/d47H2(/tildewideA)/d47/d47H2(/tildewideE)/d47/d47H2(E) /d47/d470.
+Apply relations (62) and Lemma 5.12 to such a sequence to get
+b3(E)−b2(E) =mq−10+d= 1.
+Recall (64) to get then:
+(65) b0(E) =b4(E) = 1, b1(E) = 0, b3(E)−b2(E) = 1.
+Then point (i) in the statement follows by (65) and observing that th e birational
+contraction π:/hatwideE→Eis obtained as the contraction of the minimal section of /hatwideE
+([18], Theorem 2.2), whose class is a generator of H2(/hatwideE,Q)∼=Q2.
+Let us now observe that, by replacing (34) with relations (65), it is s till possible to
+write down equations (42) in the present case, proving equations in the first line
+in Definition 1.(a). On the other hand, arguments proving (45) and ( 48) still hold
+in the present case, proving the second and the fourth line in Definit ion 1.(a) with
+k′= 0 andk=k′′= 1. Since deg E≤3 we can apply relations (50) to equations
+(49), to get immediately (ii) in the statement and
+b3(Y) =b3(Y)+1−b2(E)+b3(E) =b3(Y)+2 (66)
+b3(Y) =b3(/tildewideY)+χ(/tildewideB)−1 =b3(/tildewideY)−mp,
+proving the third line in (a), and consequently equations in (b), of De finition 1,
+withc′,c′′as in (i) and (ii), respectively. ✷34 MICHELE ROSSI
+5.6.The non-normal case. Let us now assume that the type II g.t. (26) admits
+exceptional locus E= Exc(φ) given by a non-normal del Pezzo surface (recall 5.2,
+cases (i) and (ii)). Then Theorem 5.2 can be rewritten as follows:
+Theorem 5.13. Assume that the type II g.t. (26) admits a non-normal del Pezzo
+surface as exceptional divisor E= Exc(φ). Then it admits homological type h[T] =
+(0,1,c′,c′′)as in Theorem 5.2 and moreover
+(i)b2(E)∈ {1,2}andb3(E) = 0, giving c′∈ {0,1}, respectively ,
+(ii)χ(/tildewideB)≡b2(E) mod 2 .
+Corollary 5.14. If (26) is a type II g.t. with E= Exc(φ)a non-normal del Pezzo
+surface, then Eis like in 5.2.(i) (resp. 5.2.(ii)) if and only if χ(/tildewideB)is even (resp.
+odd).
+Proof of Theorem 5.13. Let us first of all observe that the normalization Fa=
+P(O⊕O P1(−a)) ofEhas the following Betti numbers:
+(67)b0(Fa) =b4(Fa) = 1, b1(Fa) =b3(Fa) = 0, b2(Fa) =h1,1(Fa) = 2.
+In fact, the first equalities are obvious. Moreover,
+∀i>0hi(OFa) =hi(OP1) = 0
+giving the middle equalities in (67) and b2(Fa) =h1,1(Fa). In particular, χ(OFa) =
+1 and the Noether formula ([3] Theorem I.(5.4)) gives
+(68) 2+ h1,1(Fa) =χ(Fa) = 12χ(OFa)−K2
+Fa= 4 =⇒h1,1(Fa) = 2,
+whereKFa=−2C0−(a+ 2)fis the canonical divisor of Fa, givingK2
+Fa= 8
+(notation as in 5.2; see [17] Corollary V.2.11). Then (i) follows immediate ly by
+(67) recalling the construction of the normalization morphism Fa→Edescribed
+in 5.2.(i) and (ii). Moreover, we get
+(69)b0(E) =b4(E) = 1, b1(E) =b3(E) = 0, b2(E)∈ {1,2},
+which replaced to (34) guarantee that (42), (45) and (48) still ho ld, by the same
+arguments. Then equations in Definition 1.(c), with k,c′,c′′as in Theorem 5.2, give
+χ(Y)−χ(Y) = 2 −b3(Y)+b3(Y) = 1+b2(E) =χ(E)−1
+χ(Y)−χ(/tildewideY) =−b3(Y)+b3(/tildewideY) = 1−χ(/tildewideB)
+which is enough to prove (a) and (b) in Definition 1 with k′= 0,k=k′′= 1 and
+c′,c′′assigned by (i) and (ii). /square
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+E-mail address :michele.rossi@unito.it
\ No newline at end of file
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+arXiv:1001.0458v1  [math.DG]  4 Jan 2010k−type partially null and pseudo null slant
+helices in Minkowski 4-space
+Ahmad T. Ali, Rafael L´ opez∗and Melih Turgut†
+Abstract
+We introduce the notion of k-type slant helix in Minkowski space E4
+1. For
+partially null and pseudo null curves in E4
+1, we express some characterizations
+in terms of their curvature and torsion functions.
+Mathematics Subject Classification: 53C40, 53C50.
+Key words and phrases: Minkowski space, k-type slant helix, partially null curve,
+pseudo null curve.
+1 Introduction
+The notion of a slant helix is due to Izumiya and Takeuchi [10] . A curve αwith
+non-vanishing curvature is called a slant helix in Euclidea n spaceE3if the principal
+normal lines of αmake a constant angle with a fixed direction of the ambient spa ce.
+Later, spherical images, the tangent and the binormal indic atrix and some charac-
+terizations of such curves were presented in [12]. Recently , further characterizations
+and position vectors of such curves are given in [3, 13, 15].
+In recent years, by the coming of theory of relativity, resea rchers extended some
+topics of classical differential geometry to Lorentzian mani folds and there exists an
+extensiveliteratureonthissubject. Forinstance, aboutg eneralhelicesonLorentzian
+geometry, we refer [1, 2, 6, 7, 8, 9, 11, 16].
+Now we focus in Minkowski 4-dimensional space E4
+1, that is, the real vector
+4-dimensional space R4equipped with the standard flat metric given by
+g=−dx2
+1+dx2
+2+dx2
+3+dx2
+4,
+where (x1,x2,x3,x4) is a rectangular coordinate system in R4. A spacelike curve
+α:I⊂R→E4
+1is called spacelike if the inducedmetricis Riemannian. For spacelike
+curves parameterized by the length-arc, one has defined a Fre net frame {V1,...,V 4},
+whereV1(s) =α′(s). In Minkowski 4-dimensional space E4
+1, we extend the concept
+of slant helix as follows:
+∗Partially supported by MEC-FEDER grant no. MTM2007-61775 a nd Junta de Andaluc´ ıa grant
+no. P06-FQM-01642.
+†Corresponding author.
+1Definition 1.1. Letα:I→E4
+1beaspacelikecurve, withFrenetframe {V1,V2,V3,V4}.
+We say that αis ak-type slant helix if there exists a (non-zero) constant vect or field
+U∈E4
+1such that g(Vk+1,U) is constant, for 0 ≤k≤3. The vector Uis called an
+axisof the curve.
+In particular, 0-type slant helices are general helices and 1-type slant helices are
+slant helices. In this work we consider k-type slant helices for partially null curves
+and pseudonull curves. Recall that a partially null curveis a(spacelike) curve where
+V2is spacelike and V3is a lightlike vector. On the other hand, a pseudo null curve
+is a (spacelike) curve if V2is a lightlike vector. In all cases, we characterize k-type
+slant helices in terms of the curvatures of the curve and we de termine the axis of
+the curve. Next we focus on k-type pseudo null slant helices in hyperbolic space.
+Finally, we point out that type-3 slant helices in E4
+1have studied in [16] for those
+curves where the Frenet frame are non-lightlike vectors.
+2 Preliminaries
+The Lorentzian metric gin Minkowski space E4
+1is indefinite. Therefore a vector
+v∈E4
+1can have one of the three causal characters. We say that vis spacelike if
+g(v,v)>0 orv= 0, timelike if g(v,v)<0 and lightlike (or null) if g(v,v) = 0 and
+v/ne}ationslash= 0. Similarly, an arbitrary curve α=α(s) inE4
+1is called spacelike, timelike
+or lightlike, if all of its velocity vectors α′(s) are spacelike, timelike or lightlike,
+respectively. The norm of a vector v∈E4
+1is given by /bardblv/bardbl=/radicalbig
+|g(v,v)|. Therefore,
+vis a unit vector if g(v,v) =±1. A (spacelike or timelike) curve is parametrized by
+the arclength is α′(s) is a unit vector for any s. Also, we say that the vectors v,w
+inE4
+1are orthogonal if g(v,w) = 0.
+Consider α=α(s) a spacelike curve where sis the length-arc parameter. Denote
+by{T(s),N(s),B1(s),B2(s)}the moving Frenet frame along the curve α(s). Then
+T,N,B 1,B2are called the tangent, the principal normal, the first binor mal and
+the second binormal vector fields of α, respectively. The fact that the metric gis
+indefinite causes that the vector N,B1andB2have different causal characters (here
+Tis a spacelike vector, since αis a spacelike curve).
+In this paper we are interesting for partially null curves an d pseudo null curves
+(see [17]). A partially null curve is a spacelike curve where Nis spacelike and B1is
+lightlike. In such case, the vector B2is the unique lightlike vector orthogonal to T
+andNsuch that g(B1,B2) = 1. The Frenet equations are given by ([5, 17]):
+
+T′
+N′
+B′
+1
+B′
+2
+=
+0κ0 0
+−κ0τ0
+0 0 σ0
+0−τ0−σ
+
+T
+N
+B1
+B2
+. (1)
+Here,κ, τandσare first, second and third curvature of the curve α, respectively.
+In fact, one can prove that after a null rotation of the ambien t space, the curvature
+σcan be chosen to be zero (and τis determined up to a constant). This means that
+2any partially null curve lies in a three dimensional lightli ke subespace (orthogonal
+toB1).
+A spacelike curve α(s) is called a pseudo null curve ifα′′(s) is a lightlike vector
+for anys. Thenthenormal vector is N=T′. IfN′is lightlike, then αisincludedina
+lightlike plane: we discard this trivial case. In the rest of cases,B1is a unit spacelike
+vector orthogonal to {T,N}andB2is the unique lightlike vector orthogonal to T
+andB1such that g(N,B2) = 1. The Frenet equations are ([5, 17]):
+
+T′
+N′
+B′
+1
+B′
+2
+=
+0κ0 0
+0 0 τ0
+0σ0−τ
+−κ0−σ0
+
+T
+N
+B1
+B2
+(2)
+Here the first curvature κcan take only two values: κ≡0 when the curve is a
+straight-line or κ≡1 in all other cases. We will assume non-trivial cases, that i s,
+thatκ≡1, as well as, σ,τ/ne}ationslash= 0. The classification of k-type slant helices that we
+present in this paper corresponds to partially null curves ( Section 3) and pseudo
+null curves (Section 4). In the last setting, we also conside r pseudo null curves in
+pseudo-hyperbolic space.
+We end this section about some of notation. If Uis an axis of an k-type slant
+helix, we decompose Uwith respect to the Frenet frame {T,N,B 1,B2}asU=
+u1T+u2N+u3B1+u4B2, whereui=ui(s) are differentiable functions on s.
+3k-type partially null slant helices
+In this section we study k-type partially null slant helices. Recall that in the Frene t
+equations (1), the value of σis zero. We will assume that κ,τ/ne}ationslash= 0. In particular,
+B1is a constant vector. If we take U=B1, then,g(T,U) =g(N,U) = (B1,U) = 0
+andg(B2,U) = 1. This means that any partially null curve is a k-type slant helix
+for anyk∈ {0,...,3}. In this section, we discard the trivial case that U=B1.
+Theorem 3.1. Letαbe a partially null curve in E4
+1. Thenαis a0-type slant helix
+(or general helix) if and only if
+τ
+κ=constant. (3)
+Moreover, αis also a k-type slant helix, for k∈ {1,2,3}.
+Proof.Assume that αis a 0-type slant helix. Then for a constant vector field U,
+the function g(T,U) =cis constant. Differentiating this equation and from Frenet
+equations, we obtain that κg(N,U) = 0. So Uis orthogonal to Nand we decompose
+Uas
+U=cT+u3B1+u4B2. (4)
+3Differentiating (4) and using the Frenet equations (1), one ar rives to
+
+
+cκ−τ u4= 0,
+u′
+3= 0,
+u′
+4= 0.
+Thusu3andu4are constant. From the first equation, we obtain (3).
+Conversely, suppose that the relation (3) holds. We define th e vector field
+U=τ
+κT+B2. (5)
+Differentiating (5) and considering (3) and (1), we have U′= 0, that is, Uis a
+constant vector field. Because g(T,U) =τ/κis constant, we have that αis a 0-type
+slant helix.
+Finally, and because the coefficients of Uare constant, we have g(Vk+1,U) = 0,
+for any 0 ≤k≤3, that is, αis ak-type slant helix.
+From the above theorem, we have
+Corollary 3.2. Ifαis a0-type partially null slant helix in E4
+1, an axis of αis
+D=τ
+κT+B1+B2.
+As a particular case of 0-type partially null slant helices a re those with κandτ
+constant. In such case, αparametrizes as (see [4])
+α(s) = (cs,1
+κcos(κs),1
+κsin(κs),cs).
+Theorem 3.3. Letαbe a partially null curve in E4
+1. Thenαis1-type slant helix
+if and only if, there exists a constant Csuch that
+τ(s)
+κ(s)−C/integraldisplays
+0κ(t)dt= 0. (6)
+Moreover, αis a2-type slant helix.
+Proof.Assume that αis a 1-type slant helix. There exists a constant vector field U
+such that g(N,U) =cis constant. We decompose UasU=u1T+cN+u3B1+u4B2.
+Differentiation this equation, and using(1), we have the foll owing system of ordinary
+differential equations
+
+u′
+1−cκ= 0,
+κu1−τ u4= 0,
+u′
+3+cτ= 0,
+u′
+4= 0.(7)
+In particular, u4is constant. Then we obtain
+/braceleftbiggu1=c/integraltexts
+0κ(t)dt,
+u3=−c/integraltexts
+0τ(t)dt,(8)
+4Using the second equation in (7) together (8), and letting C=c
+u4we conclude (6).
+Conversely, assume that the relation (6) holds. Then, let us consider the following
+vector
+U= (/integraldisplays
+0κ(t)dt)T+N−1
+C(/integraldisplays
+0τ(t)dt)B1+B2,
+for the real number Cgiven in (6). Differentiating Uand using the Frenet equations
+(1), we have U′= 0, that is, Uis a constant vector field. Moreover, g(N,U) = 1
+which means that αis a 1-type slant helix.
+Because the coefficient u4is constant, then g(B1,U) is constant and then αis a
+2-type slant helix.
+As a consequence of the above theorem, we have
+Corollary 3.4. Letαbe a partially null curve in E4
+1. Ifαis a1-type slant helix,
+then
+D= (s/integraldisplay
+0κ(t)dt)T+N−1
+C(s/integraldisplay
+0τ(t)dt)B1+B2,
+is a constant vector.
+We now consider 2-type slant helices.
+Theorem 3.5. Letαbe a2-type partially null slant helix. Then an axis of αis
+U=v1T+v′
+1
+κN+/parenleftBig
+c3−/integraldisplayτ
+κv′
+1ds/parenrightBig
+B1+B2, (9)
+where
+v1(s) = cos/bracketleftBig/integraltext
+κds/bracketrightBig/parenleftBig
+c1−/integraltext
+τsin/bracketleftBig/integraltext
+κds/bracketrightBig
+ds/parenrightBig
++sin/bracketleftBig/integraltext
+κds/bracketrightBig/parenleftBig
+c2+/integraltext
+τcos/bracketleftBig/integraltext
+κds/bracketrightBig
+ds/parenrightBig
+.(10)
+Proof.We know that there exists a constant vector field Usuch that g(B1,U) =c,
+cis a constant. We decompose Uas
+U=u1T+u2N+u3B1+cB2.
+By differentiating Uwe have the following system of ordinary differential equatio ns
+
+
+u′
+1−κu2= 0,
+u′
+2+κu1−cτ= 0,
+u′
+3+τ u2= 0,(11)
+From the first equation, we have
+u2=u′
+1
+κ=u′(θ), (12)
+5whereθ=/integraltext
+κ(s)dsis a new variable. Substituting (12) in the second equation o f
+the system (11), we have the following ordinary differential e quation
+u′′
+1(θ)+u1(θ) =cf(θ),
+wheref(θ) =τ
+κ(θ). By solving the above equation we obtain
+
+
+u1=c/bracketleftBig
+cosθ/parenleftBig
+c1−/integraltext
+f(θ)sinθdθ/parenrightBig
++sinθ/parenleftBig
+c2+/integraltext
+f(θ)cosθdθ/parenrightBig/bracketrightBig
+,
+u2=c/bracketleftBig
+cosθ/parenleftBig
+c2+/integraltext
+f(θ)cosθdθ/parenrightBig
+−sinθ/parenleftBig
+c1−/integraltext
+f(θ)sinθdθ/parenrightBig/bracketrightBig
+,
+u3=c/bracketleftBigg
+c3−/integraltext/bracketleftBig
+cosθ/parenleftBig
+c2+/integraltext
+f(θ)cosθdθ/parenrightBig
+−sinθ/parenleftBig
+c1−/integraltext
+f(θ)sinθdθ/parenrightBig/bracketrightBig
+dθ/bracketrightBigg
+.
+This gives (9) and (10).
+Now we study 3-type partially null slant helices. Recall tha t for partially null
+curves, the vector B1is constant.
+Theorem 3.6. Letαbe a partially null curve in E4
+1. Ifαis a3-type slant helix,
+thenαis ak-type slant helix, for k∈ {0,1,2}.
+Proof.LetUbe the constant vector field such that g(B2,U) =cis constant. A
+differentiation of g(B2,U) =cleads to τ g(N,U) = 0, that is, g(N,U) = 0. We
+writeUasU=u1T+cB1+u4B2. We differentiate Uobtaining
+
+
+u′
+1= 0,
+u1κ−τu4= 0,
+u′
+4= 0.
+This implies that u1andu4are constant, and from the second equation, the quotient
+τ/κis constant too. By using Theorem 3.1, αis a 0-type slant helix.
+Moreover, the coefficients of Uwith respect to the Frenet frame are all constant,
+which means that αis ak-type slant helix for 0 ≤k≤3.
+As a consequence of the proof of Theorems 3.1, 3.3 and 3.6, we o btain the
+following
+Corollary 3.7. Letαbe a partially null curve in E4
+1. Thenαis a0-type slant helix
+if and only if αis a3-type slant helix.
+Corollary 3.8. Letαbe a partially null curve in E4
+1. Assume that there exists a
+constant vector field Usuch that g(Vk+1,U) = 0for some k∈ {0,1,3}. Thenαis a
+k-type slant helix for any k∈ {0,1,3}.
+Corollary 3.9. Letαbe a partially null curve in E4
+1. Assume that the curvatures
+κandτare constant. Then αis ak-type slant helix for k∈ {0,1,3}.
+As consequence of our study, we show a set of relations betwee nk-type partially
+null slant curves:
+601
+2 3
+Figure 1: The arrows mean the implication between the differen t ofk-type partially
+null slant helices
+4k-type pseudo null slant helices
+In this section we study pseudo null curves that are k-type slant helices. Recall that
+we assume κ≡1 in (2). We begin for general helices.
+Theorem 4.1. There does not exist 0-type pseudo null slant helices in E4
+1.
+Proof.If such curve does exist, there exists a constant vector field Usuch that
+g(T,U) =cis constant. Differentiating this equation and using the Fren et equations
+(2), we obtain, g(N,U) = 0. Then the decomposition of Uin terms of the Frenet
+frame is U=cT+u2N+u3B1. AsU′= 0, by using the Frenet equations (2), we
+have 
+
+u′
+2+σu3+c= 0,
+u′
+3+τ u2= 0,
+τ u3= 0
+From thethird equation, u3= 0, and thus, u2=c= 0, that is, U= 0: contradiction.
+We characterize 1-type pseudo null slant helices as follows :
+Theorem 4.2. Letαbe a pseudo null curve in E4
+1. Thenαis1-type slant helix if
+and only if
+σ(s)
+τ(s)=−s2
+2+as+b. (13)
+for some constants a,b∈R. Moreover αis also a 2-type slant helix.
+Proof.Assumethat αis a 1-type slant helix and let Ubeaconstant vector field such
+thatg(N,U) =cis constant. Differentiation this equation and by Frenet equa tions,
+τg(B1,U) = 0, and so, g(B1,U) = 0. If we write Uas linear combination of the
+7Frenet frame, we have U=u1T+u2N+cB2. Differentiating Uwe obtain the
+following system of ordinary differential equations
+
+
+u′
+1−c= 0,
+u′
+2+u1= 0,
+τ u2−cσ= 0.
+whereuiare the coefficients of Uin the decomposition with respect to the Frenet
+frame. If c= 0,u1=u2= 0, that is, U= 0: contradiction. Thus, c/ne}ationslash= 0. From the
+last equation, u2=c/parenleftBig
+σ
+τ/parenrightBig
+and from the first two equations, u2satisfiesu′′
+2+c= 0.
+The solution of this equation is
+u2=−cs2
+2+λs+µ, λ,µ∈R.
+Thusσ/τsatisfies the condition (13), with a=λ/candb=µ/c.
+Conversely, let usassumethattherelation (13) holds. Defin ethefollowing vector
+field
+U=−/parenleftBigσ
+τ/parenrightBig′
+T+/parenleftBigσ
+τ/parenrightBig
+N+B2. (14)
+Differentiating (14), and using (2), we easily have U′= 0, that is, Uis a constant
+vector. Moreover, g(N,U) = 1, showing that αis a 1-type slant helix. Finally, as
+g(B1,U) = 0, then αis a 2-type slant helix.
+Corollary 4.3. Letαbe a1-type pseudo null slant helix. Then an axis of αis
+D=−/parenleftBigσ
+τ/parenrightBig′
+T+/parenleftBigσ
+τ/parenrightBig
+N+B2.
+Theorem 4.4. Letαbe a pseudo null curve in E4
+1. Thenαis a2-type slant helix
+if and only if/integraldisplay
+τds+d
+ds/bracketleftbigg
+σ+d
+ds/parenleftbiggσ
+τ/integraldisplay
+τds/parenrightbigg/bracketrightbigg
+= 0. (15)
+Proof.Ifαis a 2-type slant helix, there exists a constant vector Usuch that
+g(B1,U) =cis constant. We write UasU=u1T+u2N+cB1+u4B2. Dif-
+ferentiating Uwe obtain
+
+u′
+1−u4= 0,
+u′
+2+u1+cσ= 0,
+τ u2−σu4= 0,
+u′
+4−cτ= 0.(16)
+Remark that c/ne}ationslash= 0. By solving this system, we conclude from the last equatio n that
+
+
+u1=−c/bracketleftbigg
+σ+d
+ds/parenleftBigσ
+τ/integraltext
+τ ds/parenrightBig/bracketrightbigg
+,
+u2=cσ
+τ/integraltext
+τ ds,
+u4=c/integraltext
+τ ds.
+8Hence together the first two equations of(16), we conclude (1 5).
+Conversely, assume that (15) holds. Define
+U=−/bracketleftbigg
+σ+d
+ds/parenleftbiggσ/integraltext
+τ ds
+τ/parenrightbigg/bracketrightbigg
+T+/parenleftbiggσ/integraltext
+τ ds
+τ/parenrightbigg
+N+B1+/bracketleftbigg/integraldisplay
+τ ds/bracketrightbigg
+B2.(17)
+Considering (15) and differentiating (17), we have by using (2 ), thatU′= 0, that
+is,Uis constant. Moreover, g(B1,U) = 1, which shows that αis a 2-type slant
+helix.
+Corollary 4.5. An axis of a 2-type pseudo null slant helix is the vector given by
+D=−/bracketleftbigg
+σ+d
+ds/parenleftbiggσ/integraltext
+τ ds
+τ/parenrightbigg/bracketrightbigg
+T+/parenleftbiggσ/integraltext
+τ ds
+τ/parenrightbigg
+N+B1+/bracketleftbigg/integraldisplay
+τ ds/bracketrightbigg
+B2.
+Theorem 4.6. Letαbe a pseudo null curve in E4
+1. Thenαis3-type slant helix if
+and only if
+τ√
+1+σ2+d
+ds/bracketleftBig√
+1+σ2/parenleftBig
+στ′(1+σ2)+τσ′(2−σ2)/parenrightBig
+τ(1+σ2)2−3ττ′2+σ′′(1+σ2)/bracketrightBig
+= 0. (18)
+Proof.Suppose that αis a 3-type slant helix and let Ube the constant vector field
+such that g(B2,U) =cis constant. If we write U=u1T+cN+u3B1+u4B2, a
+differentiation of this expression leads to
+
+
+u′
+1−u4= 0,
+u1+σu3= 0,
+u′
+3+cτ−σu4= 0,
+u′
+4−τ u3= 0.(19)
+The first and second equation of (19) give
+/braceleftbiggu4=u′
+1,
+u3=−u1
+σ,
+Substituting u3andu4into the two last equations of (19), we have
+/braceleftBigg
+u′
+1=σ′
+σ(1+σ2)u1+τσ
+1+σ2c,
+u′′
+1+τ
+σu1= 0.
+By the last two expressions of u1, we obtain (18). From the above system of equa-
+tions, we have the following two equations
+u1=cσ/parenleftBig
+στ′(1+σ2)+τσ′(2−σ2)/parenrightBig
+τ(1+σ2)2−3σσ′+σ′′(1+σ2).
+9and
+u1=cσ√
+1+σ2/integraldisplayτ√
+1+σ2ds.
+Conversely, assume that (18) holds. Then we define the follow ing vector
+U=σ√
+1+σ2ǫT−1√
+1+σ2ǫB1+/parenleftBig
+τσ−σ′
+√
+1+σ2ǫ/parenrightBig
+u2B2, (20)
+whereǫ=/integraltextτ√
+1+σ2ds. Differentiating of (20), we easily have U′= 0. Moreover one
+obtainsg(B2,U) = 0 which means that αis a 3-type slant helix.
+From the above theorem, one concludes
+Corollary 4.7. Ifαis a3-type pseudo null slant helix in E4
+1, an axis of αis the
+vector
+D=σ√
+1+σ2ǫT−1√
+1+σ2ǫB1+/parenleftBig
+τσ−σ′
+√
+1+σ2ǫ/parenrightBig
+u2B2,
+whereǫ=/integraltextτ√
+1+σ2ds.
+We end this section focusing into k-type pseudo null slant helices in the pseu-
+dohyperbolic space. Recall that the pseudohyperbolic spac e ofE4
+1of radius rand
+centered at x0∈E4
+1is defines by H3
+0={x∈E4
+1;g(x−x0,x−x0=−r2}. The metric
+ginduced into H3
+0is Riemannian with constant negative intrinsic curvature. In [5]
+it is proved that a pseudo null curve αlies inH3
+0if and only if the quotient σ/τis
+a negative constant. We particularize our results for pseud o null curves of H3
+0. As
+consequence of Theorem 4.2, we have
+Corollary 4.8. There does not exist a 1-type pseudo null slant helix in pseudohy-
+perbolic spaces of E4
+1.
+As a consequence of Theorem 4.4 we have
+Corollary 4.9. Ifαis a2-type pseudo null slant helix in H3
+0, then
+τ=λes√−2c+µe−s√−2c, λ,µ∈R (21)
+wherecis the negative constant given by σ=cτ.
+Proof.From ([5]) we know that σ=cτfor some real number c <0. If we put
+this into (15), we have that τsatisfies 2 cτ′′+τ= 0, whose solutions are given by
+(21).
+If we put σ=cτfor pseudo null curves in H3
+0, Theorem 4.6 implies
+Corollary 4.10. Ifαis a3-type pseudo null slant helix in H3
+0, the second curvature
+function τ(s)satisfies
+2c2ττ′τ′′′=cτ′′/bracketleftBig
+5τ2(1+c2τ2)+c(3τ′2+4ττ′′)/bracketrightBig
++c2τ5(2+c2τ2)+τ3(1−15c3τ′2).
+Acknowledgements. The third author would like to thank T¨ ubitak-Bideb for
+their financial support during his Ph.D. studies.
+10References
+[1] A. Ali and R. L´ opez, Slant helices in Minkowski space E3
+1, preprint 2008: arXiv:
+0810.1464v1 [math.DG].
+[2] A.AliandR.L´ opez, TimelikeB 2-slanthelices inMinkowski space E4
+1, toappear
+in Archivum Math.
+[3] A. Ali, Position vectors of slant helices in Euclidean 3- space, preprint 2009:
+arXiv: 0907.0750v1 [math.DG].
+[4] W.B. Bonnor, Curves with null normals in Minkowski space time, A random
+walk in relativity and cosmology, Eiley Easten Ltd, 1985, 33 –47.
+[5] C ¸. Camcı, K. ˙Ilarslan and E. ˇSu´ curovi´ c, On pseudohyperbolical curves in
+Minkowski spacetime, Turk. J. Math. 27 (2003), 315–328.
+[6] M. Erdo˘ gan and G. Yılmaz, Null generalized and slant hel ices in 4-dimensional
+Lorentz-Minkowski space, Int. J. Contemp. Math. Sci., 3 (20 08), 1113–1120.
+[7] A. Ferr´ andez, A. Gim´ enez and P. Lucas, Null Generalize d Helices in Lorentz-
+Minkowski Spaces, J. Phys. A: Math. Gen. 35 (2002), 8243–825 1.
+[8] K.˙Ilarslan and ¨O. Boyacıo˘ glu, Position vectors of a spacelike W-curve in
+Minkowski space E3
+1, Bull. Korean Math. Soc., 44 (2007), 429–438.
+[9] K.˙Ilarslan and ¨O. Boyacıo˘ glu, Position vectors of a timelike and a null hel ix in
+Minkowski 3-space, Chaos Solitons Fractals, 38 (2008), 138 3–1389.
+[10] S. Izumiya and N. Takeuchi, New special curves and devel opable surfaces, Turk.
+J. Math., 28 (2004), 531–537.
+[11] H.B. Karada˘ g and M. Karada˘ g, Null generalized slant h elices in 4-dimensional
+Lorentzian dpace, Differ. Geom. Dyn. Syst. 10 (2008), 178–185 .
+[12] L. Kula and Y. Yayli On slant helix and its spherical indi catrix, Appl. Math.
+Comput., 169 (2005), 600-607.
+[13] L. Kula, N. Ekmek¸ ci, Y. Yaylı and K. ˙Ilarslan, Characterizations of slant helices
+in Euclidean 3-space, Turk. J. Math., 33 (2009), 1-13.
+[14] B. O’Neill, Semi-Riemannian Geometry, Academic Press , New York, 1983.
+[15] M. Turgut and S. Yilmaz, Characterizations of some spec ial helices in E4, Sci.
+Magna., 4 (2008), 51–55.
+[16] M. Turgut and S. Yilmaz, Some characterizations of type −3 slant helices in
+Minkowski space-time, Involve J. Math., 2 (2009), 115–120.
+11[17] J. Walrave, Curves and Surfaces in Minkowski Space, Dis sertation, K. U. Leu-
+ven, Fac. of Science, Leuven, 1995.
+Ahmad T. Ali :
+Present address: King Abdul aziz University, Faculty of Science, Department
+of Mathematics, PO Box 80203, Jeddah, 21589, Saudi Arabia.
+Permanent address: Mathematics Department, Faculty of Science, Al-Azhar
+University, Nasr City, 11448, Cairo, Egypt.
+e-mail address: atali71@yahoo.com andaabdullwhaab@kau.edu.sa
+Rafael L´ opez :
+DepartamentodeGeometr´ ıayTopolog´ ıa, UniversidaddeGr anada, 18071Granada,
+Spain.
+e-mail address: rcamino@ugr.es
+Melih Turgut :
+Dokuz Eyl¨ ul University, Buca Educational Faculty, Depart ment of Mathematics,
+35160, Buca-Izmir, Turkey
+e-mail address: Melih.Turgut@gmail.com
+12
\ No newline at end of file
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+arXiv:1001.0459v1  [nucl-th]  4 Jan 2010Above threshold s-wave resonances illustrated by the 1/2+states in9Be and9B
+E. Garrido
+Instituto de Estructura de la Materia, CSIC, Serrano 123, E- 28006 Madrid, Spain
+D.V. Fedorov and A.S. Jensen
+Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark
+(Dated: July 6, 2018)
+We solve the persistent problem of the structure of the lowes t 1/2+resonance in9Be which
+is important to bridge the A= 8 gap in nucleosynthesis in stars. We show that the state is a
+genuine three-body resonance even though it decays entirel y into neutron-8Be relative s-waves. The
+necessary barrier is created by “dynamical” evolution of th e wave function as the short-distance
+α-5He structure is changed into the large-distance n-8Be structure. This decay mechanism leads
+to a width about two times smaller than table values. The prev ious interpretations as a virtual
+state or a two-body resonance are incorrect. The isobaric an alog 1/2+state in9B is found to have
+energy and width in the vicinity of 2.0 MeV and 1.5 MeV, respec tively. We also predict another
+1/2+resonance in9B with similar energy and width.
+PACS numbers: 21.45.-v, 21.60.Gx, 27.20.+n
+Introduction Boundstates, resonancesandothercon-
+tinuum states are well understood and described for two
+particles interacting through a potential. By weakening
+the attraction of the potential bound states are pushed
+upwards into the continuum as resonances or virtual
+states when a barrier is present or absent, respectively
+[1]. For neutral particles virtual states arise for s-waves
+whereas resonances emerge for higher partial waves. De-
+creasingthe attractionfurther until the resonanceenergy
+is above the potential barrier leads to an increase of the
+resonance width. Correspondingly the related S-matrix
+pole moves in the complex energy plane as a resonance
+with non-vanishing imaginary part.
+For three particles interacting via two- and three-body
+potentials the continuum structures can be much more
+complicated due to combinations of the different struc-
+tures for the three two-body subsystems [2, 3]. One in-
+triguing possibility arises when one of the two-body sub-
+systems has a low-lying s-wave resonance produced by a
+confining Coulomb barrier, and the third neutral parti-
+cle has dominating s-wave attractions from the first two
+particles. Even when all higher partial waves are vanish-
+ingly small, the structure of the three-body continuum
+state is a priori not easily determined or described.
+Let usassumethatthe two-bodyresonanceisverynar-
+row (long-lived) and the three-body energy is above zero
+but less than the two-body resonance energy. Then the
+three-body continuum state resembles a two-body bound
+state of the third particle and the composite resonanceof
+the two first particles. The lifetime would be determined
+by the lifetime of the two-body resonance. When the
+three-bodyenergyispushedupwardsabovethetwo-body
+thresholdbydecreasingthe attraction, thecorresponding
+structure can be described as a two or three-body virtual
+state or resonance [1, 2].
+The purpose of the present letter is first to determinein general which structure arises, and second specifically
+to solve the long-standing controversy of the9Be(1/2+)
+continuum state. This state is important in the nuclear
+synthesis of light nuclei in stars [4–6], and has therefore
+received lots of attention both theoretically [6–9] and ex-
+perimentally [10–15]. In a measurement of photo disinte-
+gration the cross section is interpreted and parametrized
+viaR-matrix analysis as one neutron and the8Be ground
+state in a two-body s-wave resonance [11]. This seems to
+be againstthe two-bodyquantum mechanicaldescription
+as such a state cannot survive as a resonance. In another
+interpretation the same neutron-8Be system is described
+as a virtual state [9] but the resulting cross section does
+not reproduce the measurement [11].
+The9Be(1/2+) structure is most often assumed to be
+one neutron and the8Be ground state [10] but some-
+times also the α+α+nrecombination reaction is as-
+sumed to proceed by α-capture on the5He ground state
+[6]. It is apparently very difficult to avoid assumptions of
+two-body sequential structures and processes via subsys-
+tems of either8Be or5He. Interestingly a two-center
+Born-Oppenheimer model based on symmetries alone
+may combine these structures as in [7] where the 1 /2+
+is lowest at large distance whereas a 3 /2−state is low-
+est at small distance. We shall allow an entirely general
+three-bodystructurewithoutaprioriassumptionsofsub-
+structures or decay mechanisms.
+Formulation Let us consider three composite struc-
+tures as point-like particles denoted n,α1andα2. The
+two mass scaled Jacobi vector coordinates, ( x,y), can be
+substituted by hyperspherical coordinates {ρ,α,Ωx,Ωy},
+where (Ω x,Ωy) describe the directions of ( x,y),ρ=/radicalbig
+x2+y2andα= arctan(x/y), see [3]. We solve this
+three-body problem by use of adiabatic hyperspherical
+expansion of the Faddeev equations, i.e. the angular
+equations are solved for each ρ, providing a set of an-2
+gular eigenfunctions φnand their corresponding eigen-
+valuesλn. The three body wave function is then written
+asψ=1
+ρ5/2/summationtext
+nfn(ρ)φn(x,y), wherenlabels each of the
+adiabatic terms. The radial functions fn(ρ) are obtained
+after solving a coupled set of radial equations where the
+λnangular eigenvalues enter as effective adiabatic poten-
+tials. The coupling between the different adiabatic terms
+appears through the functions Pnn′(ρ) andQnn′(ρ) de-
+fined for instance in [3].
+Each adiabatic potential describes a specific relative
+structure of the three particles for a given root mean
+square radius, ρ, with wave function φand eigenvalue
+λ. When only one adiabatic potential is considered, the
+coupled set of radial equations reduces to
+/bracketleftbigg
+−d2
+dρ2+λ(ρ)+15/4
+ρ2−Q(ρ)−2m(E−V3b(ρ))
+/planckover2pi12/bracketrightbigg
+f(ρ) = 0,(1)
+whereQis the diagonal coupling term Qnn, which is in
+general not zero (contrary to what happens with Pnn′,
+whose diagonal terms are zero). V3b(ρ) is the three-body
+potential usually used in three-body calculations to take
+into account all those effects that go beyond the two-
+body interactions. The total wave function, ψ, andQ
+are given by
+ψ=1
+ρ5/2f(ρ)φ(x,y), Q(ρ) =/an}bracketle{tφ|∂2
+∂ρ2|φ/an}bracketri}htΩ.(2)
+The expectation value is over angular coordinates, Ω, ex-
+cluding only ρ. When only s-waves contribute for both
+xandythe angular wave function φonly depends on ρ
+andα. For short-range attractive two-body interactions
+the angular eigenvalue λ(ρ) would be monotonously in-
+creasing towards a constant asymptotic value.
+Narrow two-body resonance. When theα1−α2two-
+bodys-wave interaction supports a bound state, the adi-
+abatic potential approaches the bound state energy at
+large distance. This reflects a two-body structure corre-
+sponding to particle nfar away from the α1−α2bound
+state. For less α1−α2attraction this state moves into
+the continuum. With a confining Coulomb barrier from
+repelling charges on the particles, the state would appear
+as a resonance at low energy E2. The adiabatic poten-
+tial would then approach the positive value equal to E2
+corresponding to the large-distance two-body structure
+ofnfar away from the resonance α1−α2. We illustrate
+in Fig.1 by the specific examples,9Be(α+α+n) and
+9B(α+α+p).
+This descriptionis onlycorrectwhen the two-bodyres-
+onance width Γ 2is very small and the coupling to the
+three-body continuum states can be ignored. In general
+the first adiabatic potential is crossed by numerous oth-
+ers whileρincreases. However, the couplings are neg-
+ligibly small for three-body energies Euntil distances
+ρ≃9√E−E2/Γ2where the energies are in MeV and ρ
+in fm. Thus for small Γ 2, say eV or keV, the couplingsfor moderate energies below 1 MeV can be neglected far
+outside the distance where the short-range n−(α1−α2)
+interactionhasvanished. Then the system caneffectively
+be described as a two-body system until the two-body
+resonance eventually decays.
+Let us now consider the three-body system with inter-
+actions leading to a three-body state of positive energy
+Erbut belowE2. Effectively this is a bound n−(α1−α2)
+two-body state, or rather a resonance decaying precisely
+with the width Γ r= Γ2of theα1−α2resonance. For
+lessn−(α1−α2)s-wave attraction this two-body bound
+state moves into the continuum above E2. The expecta-
+tion is that the proper description is as a virtual state
+with no width in contrast to a resonance [9]. However,
+this is not necessarily true.
+B9Li+α5 Be+p8+1
+2
+−3
+22.0 MeV~
+threshold0.092 MeV0.277 MeV1.67 MeV(b)Be9Be+n8He+α5−3
+2+1
+2threshold
+−1.5735 MeV0.092 MeV0.87 MeV
+0.110 MeV(a)
+FIG. 1: The two lowest energy levels for9Be (a), and9B
+(b), andtheresonance energies ofthecorresponding two-bo dy
+subsystems [17]. For9B the quoted 1 /2+state corresponds
+to the estimation obtained in this work. The widths of the
+resonances are represented by the shadowed regions.
+The9Be(1/2+) example. The low-lying states of9Be
+areexpectedtobewelldescribedasclusterstatesconsist-
+ing of one neutron and two α-particles [16]. The α−αin-
+teraction, including short range attraction and Coulomb
+repulsion, produces a low-lying s-wave two-body reso-
+nance at 0.0918 MeV with a width around 9 eV. Adding
+one neutron in s-waves leads to angular momentum and
+parity 1/2+of the resulting three-body system. Such a
+state is listed in the tables of energies [17] at an excita-
+tion energy of 1 .684 MeV, or 0 .110MeV above threshold,
+with a width of0 .217MeV (see the upper part ofFig.1a).
+Thus the state is above the two-body resonance energy
+by0.018MeVand s-wavesaremostlikelythedominating
+composition.
+Thelistedwidthismuchlargerthanthedistancetothe
+two-body resonance threshold and even about two times
+larger than the three-body energy itself. It is a peculiar
+resonance structure which apparently extends into the
+bound state region below the threshold. These values are3
+consistent with photodissociation cross section measure-
+ments and the entangled R-matrix analysisof an a priory
+assumed resonance [11]. The fitting parameters are posi-
+tion and energy dependent width of a two-body neutron-
+8Be resonance of s-wave character. Several years prior
+to this analysis it was suggested that the state should be
+understood as a virtual state and the photodissociation
+cross section correspondingly analyzed [9]. However, this
+does not reproduce the measurements in [11].
+10 20 30 40 50
+ρ (fm)-2-10123Veff(ρ) (MeV)9Be effective potentials
+9B effective potentials
+10 20 30-303 9Be: Veff
+9Be: Veff− Q11
+FIG. 2: Lowest adiabatic potentials for9Be and9B as a func-
+tion of the hyperradius. The inset shows the9Be lowest po-
+tential with and without the rearrangement coupling term Q.
+Three-body results for9Be(1/2+).The shortcomings
+of many previous methods are the initial assumption of
+two-body character of this structure. Since the final de-
+cay products are three particles we proceed to treat the
+system as a three-body system. We use the well estab-
+lished nucleon-nucleon and α-nucleon interactions from
+[18, 19].
+The adiabatic potentials produced through the adia-
+batic hyperspherical expansion method for these quan-
+tum numbers are shown in Fig.2. The higher ones show
+pronounced peaks at about 15–18 fm. The origin of the
+peaks is in the crossing between different angular eigen-
+values. In particular, the Q-functions in the effective
+potentials, (see Eqs.(1) and (2)), are responsible for the
+behaviour shown in the figure in the vicinity of the cross-
+ings. These couplings involve second derivatives of the
+adiabatic eigenfunctions, and they therefore reflect re-
+structuring of these functions. The lowest potential has
+a dominating attractive pocket at small distance. After
+hyperradiiρlarger than 12 −14 fm the potential stabi-
+lizes at the resonance energy of 0 .091 keV for the8Be
+ground state. This stable region continues until inter-
+rupted by the crossings with the (infinitely many) higher
+potentials. The first of these crossings occurs at about
+ρ= 130 fm.An attractive pocket and a constant large distance po-
+tential without any barrier in the transition region is not
+able to support a resonance of finite width at energies
+above the large distance asymptotic value. However, in-
+clusion of the coupling term Q(see Eq.(1)), provides an
+otherwise totally absent barrier as seen in the inset of
+Fig.2. This all decisive barrier then arises from a strong
+ρdependence of the intrinsic (angular) wave function φ,
+as seen from the definition in Eq.(2).
+0 25 50 75
+ρ (fm)00.20.40.60.81Partial wave contentlx=ly=0, L=0
+lx=ly=1, L=0
+lx=ly=2, L=0
+lx=ly=3, L=0Thick curves: 9Be
+Thin curves: 9B
+αα
+Nxy
+FIG. 3: The partial wave decomposition of the lowest adia-
+batic angular wave function for9Be (thick) and9B (thin) as
+function of hyperradius ρ. The partial angular momenta lx
+andlycorrespond to the coordinates indicated in the figure.
+Forlx=ly= 2 and lx=ly= 3 the curves for9Be and9B
+can not be distinguished.
+The three-body restructuring can be seen from Fig.3
+where the small distance structure of ρless than 9 fm is
+p-waves between neutron and α-particles. This is due to
+thep3/2-resonance which provides the main part of the
+attraction. As ρincreases above 10 fm this partial wave
+is rapidly substituted by s-waves which are energetically
+more advantageous at larger distance where the attrac-
+tion vanishes and the centrifugal barrier dominates. At
+much larger distances an increasing number of partial
+waves contribute corresponding to the neutron far away
+from the two-body system of two α-particles in the spa-
+tially much smaller s-wave resonance, i.e.8Be in the
+ground state.
+The combined result of the lowest adiabatic poten-
+tial and the diagonal coupling is able to support a res-
+onance. We compute the energy and width numeri-
+cally as the S-matrix pole found by complex scaling
+[20]. By adding a structureless short-range potential
+(V3b=V0exp(−ρ2/ρ2
+0), withρ0= 5 fm), which con-
+tributes only in the pocket region, we can move the res-
+onance energy by modifying the V0strength but without
+disturbing the resonance structure. This is useful both4
+because the three-body computation can not place the
+resonance at precisely the correct measured energy, and
+because the width is strongly dependent on the height
+and thickness of the barrier at the correct energy.
+0 0.5 1 1.5 2
+E (MeV)10-410-310-210-1100Γ (MeV)9Be: WKB estimate
+9Be: [17]
+9Be: R-matrix [11]
+9Be: Cross section fit
+9B: WKB estimate
+9B: Complex Scaling
+9B: Expected resonance
+FIG. 4: Width of the resonances for9Be and9B as function
+of the energy which is varied through the strength V0of the
+three-body potential. The solid and dashed curves are the
+WKB results with a knocking rate corresponding to Γ 0=
+0.6 MeV (Γ = Γ 0e−2S) for both nuclei [21]. The dot-dashed
+curve results from complex scaling for9B. The square and the
+circle are from the R-matrix analysis in [11] and the table in
+[17], respectively. The down triangle is obtained by direct fit
+of the cross section in [11]. The triangles at about 2 MeV are
+the first and second resonances of9B.
+In Fig.4 we show the width as a function of the res-
+onance energy. The solid line shows the WKB estimate
+for9Be. For an energy just above the threshold energy of
+the8Begroundstate(0 .0918MeV) thewidth isabout0.1
+MeV. This agrees with the result obtained by fitting the
+measured peak in the photodissociation cross section [11]
+(triangle down). The width increases slowly up to about
+0.7 MeV at the top of the barrier. At smaller energies
+the width is vanishingly small due to the thick barrier
+provided by the8Be structure.
+The complex scaling computations for9Be present nu-
+merical difficulties due to the fast increase of the width
+for energies above 0 .0918 MeV. To get an estimate we
+haveinsteadattachedaunit chargetotheneutron, which
+immediately leads to the results shown in Fig.4 for the
+1/2+analog state in9B. A continuous decrease of the
+charge leads us back to9Be which is approached and
+finally reached by extrapolation. Unfortunately the ac-
+curacy only allows the conclusion that the width at an
+energy of 0 .11 MeV is larger than 0 .06 MeV and most
+likely around 0 .1 MeV but 0 .2 MeV is not numerically
+excluded. The inaccuracy is due to the very large width
+compared to the distance to the threshold. Then the
+poles computed by the complex scaling method are verydifficult to distinguish from the continuum background.
+This becomesincreasinglyworseasthe chargeofthe neu-
+tron is decreased from unity to zero where our present
+techniques prohibits a clean result.
+Comparison to widths obtained in previous works re-
+quires precision in the definitions of a resonance. The
+ambiguities arise when the resonance is broad and then
+necessarily asymmetric. To account for the energy de-
+pendence of the decay probability the R-matrix theory
+employs an energy dependent width. This immediately
+implies that any number claimed to describe the width
+must be an average of some kind. The “observed” width
+inR-matrix theory is in [22] most directly related to the
+full width at half maximum, or the S-matrix pole, and
+then adopted as the width in tables of resonance proper-
+ties [17]. The results quoted in [11] and [22] are shown
+in Fig.4 by the square and the circle, respectively.
+Our computed three-body resonance width is from the
+WKB tunneling and crude extrapolations from the imag-
+inary value of the pole of the S-matrix. Our estimate is
+very inaccurate but we expect a value of about 0 .1 MeV
+which is only about half the “observed” table value from
+R-matrix theory. This rather large discrepancy can be
+due to inaccuracies in the complex scaling extrapolation
+to zero charge of the neutron, or the WKB approxima-
+tion of only one adiabatic potential combined with the
+uncertainty in the knocking rate estimate. However, we
+believe that the main reason is that our three-body bar-
+rier, which entirely is responsible for the width, arises
+from three-body restructuring effects inherently impossi-
+ble to include in the R-matrix analysis. An unambiguous
+settlement of this accuracy issue would require the use of
+anothermethod dedicated to precise width computations
+near threshold.
+Attempts to settle the issue experimentally face the
+problems inherent to relatively very broad resonances.
+Population in a reaction or beta-decay provide informa-
+tionaboutlifetime, whichnecessarilyisanaverage,orde-
+tails about decay probability as function of energy, which
+requires a model for analyzing the data. The direct mea-
+surements in photo dissociation [11] reveal an asymmet-
+ric peak with a width of about 0 .1 MeV. This is con-
+sistent with corresponding computations in the present
+model with a method to compute strength functions by
+discretization of the three-body continuum [23].
+Three-body results for9B(1/2+).The isobaric analog
+stateshouldexistin9Bbutsofarithasneverbeenfound.
+The influence of the additional Coulomb interaction in
+the present fragile case could be substantial and perhaps
+even destructive. The corresponding low energy three-
+bodyandtwo-bodylevelsareshowninFig.1b. Todiscuss
+thisproblemwealsoshowinFig.2the correspondinglow-
+est adiabatic potential for9B. The additional Coulomb
+repulsion due to the proton in9B has a relatively small
+effect. Atsmalldistancesthepocketisaspronouncedbut
+almost entirely above zero energy. The barrier is higher5
+and thicker at small energy where the adiabatic potential
+itself already exhibits a small barrier. The same constant
+energyis approachedat largerenergiesagaincorrespond-
+ing to8Be in addition to the Coulomb tail from the pro-
+ton. The partial wave decomposition in Fig.3 resembles
+the9Be results with a tendency towards an increase of
+higher partial waves due to the additional Coulomb po-
+tential.
+In Fig.4 we show the WKB estimate (dashed line) and
+the resonancewidth obtained after a complex scalingcal-
+culation (dot-dashed line). In both cases the width is
+small even above the8Be ground state energy due to the
+additional adiabatic barrier and the extended Coulomb
+tail. However, the precise value of the energy depends
+on the three-body potential used in Eq.(1). A minimum
+value for the attractive strength of the three-body force
+can be obtained by placing the9Be resonance at the8Be
+resonance energy threshold. Using this three-body force
+we canthen estimate a lowerlimit forthe 1/2+resonance
+in9B, which is found to be slightly below 2 MeV with a
+width of 1.3 MeV (left triangle up in the figure).
+This resonance is accompanied by a second state at
+2.05 MeV with a width of 1.6 MeV (right triangle up in
+the figure). This second resonance is very stable, basi-
+cally independent of the structureless three-body force.
+Therefore, further decrease of the three-body attraction
+inV3bwould eventually make this second resonance at
+2.05 MeV the first 1/2+excited state. The two reso-
+nances are dominated by sandp-waves between proton
+andα-particle, respectively. Crudely speaking these two
+structures correspond to8Be+proton and5Li+α. Since
+the large-distance p-wave properties essentially are unaf-
+fected by the three-body potential the second resonance
+remains close to the same energy. Thus the unobserved
+resonance in9B may in fact be either two or a combina-
+tion of two resonances. In any case we expect genuine
+three-body structures with an energy around 2 MeV and
+a width in the vicinity of 1.5 MeV.
+Discussion and Conclusions. The 1/2+continuum
+structures of9Be and9B are computed as genuine three-
+body resonances. The energies are above the threshold
+for forming the ground state8Be-resonance and the par-
+tialwavedecompositionsaredominatedby s-wavesinthe
+Jacobi coordinates connecting the two α-particles and
+their center-of-mass and the nucleon. Nevertheless the
+Faddeev component corresponding to the other Jacobi
+coordinate presents a very different partial wave decom-
+position where the α-nucleonp3/2attraction maintain p-
+waves at hyperradii smaller than 9 fm rapidly changing
+intos-waves at 10 fm. This “dynamic evolution” from
+α−5He at small distance to neutron-8Be at intermediate
+and large distance reconciles the two limits for the reso-
+nance structure. The restructuring of the wave function
+results in a potential barrier similar to an above barrier
+reflection at a discontinuity.
+The structuresare then genuine three-body resonancesmixingsandp-waves at small distances while at large
+distances turning into a two-body system entirely of s-
+wavesforanucleonand8Beintheunboundgroundstate.
+The deceiving appearance as a two-body virtual state
+is incorrect. The interpretation as a two-body nucleon-
+8Be resonance is also incorrect since the small-distance
+structure is of genuine three-body character and this is
+the very reason for the existence of a barrier allowing
+the appearance of resonance features. Furthermore the
+parameters from R-matrix analysis of the experimental
+data is misleading because of the incorrect but crucial
+assumption of the existence of a two-body nucleon-8Be
+resonance.
+The astrophysical nααrecombination rate is proba-
+bly unaffected provided the corresponding cross section
+is obtained by precisely the same parametrization for the
+measured inverse process of photodisintegration. The
+problem only seems to arise when a different procedure
+is applied in these mutually inverse processes. However,
+all resonance decays do not necessarily proceed through
+two-body channels.
+Theimplicationisthatthedecaymechanismisentirely
+through the8Be ground state as assumed in most previ-
+ouspublications. However,thepresenttabulatedvalueof
+the resonance width emerges through an averaging pro-
+cedureofdataanalysisandparametrizationofadecaying
+two-body structure. The energy dependent width arising
+fromtheR-matrixinterpretationasatwo-bodystructure
+is suspicious since the three-body effects responsible for
+the barrier and the width are not included in the analy-
+sis. The true resonance lifetime should be related to the
+tunneling probabilitythrough the barrierarisingfrom re-
+structuring the three-body wave function. The width is
+more likely directly found from the peak in the measured
+cross section. We conclude that the controversy over the
+1/2+continuum structures of9Be and9B is resolved in
+a full three-body model.
+Acknowledgments. We are grateful for many clarify-
+ing discussions with H.O.U. Fynbo and K. Riisager. This
+work was partly supported by DGI of MEC (Spain), con-
+tract No. FIS2008-01301.
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\ No newline at end of file
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+arXiv:1001.0460v1  [gr-qc]  4 Jan 2010Non-uniqueness of the Dirac theory
+in a curved spacetime
+Mayeul Arminjon1and Frank Reifler2
+1CNRS (Section of Theoretical Physics),
+Laboratory “Soils, Solids, Structures, Risks”, Grenoble, France
+e-mail: arminjon@hmg.inpg.fr
+2Lockheed Martin Corporation, MS2 Division, Moorestown, New Jersey, USA
+Abstract
+We summarize a recent work on the subject title. The Dirac equ ation
+in a curved spacetime depends on a field of coefficients (essent ially
+the Dirac matrices), for which a continuum of different choice s are
+possible. We study the conditions under which a change of the coef-
+ficient fields leads to an equivalent Hamiltonian operator H, or to an
+equivalent energy operator E. In this paper, we focus on the s tandard
+version of the gravitational Dirac equation, but the non-un iqueness
+applies also to our alternative versions. We find that the cha nges
+which lead to an equivalent operator H, or respectively to an equiv-
+alent operator E, are determined by initial data, or respect ively have
+to make some point-dependent antihermitian matrix vanish. Thus,
+the vast majority of the possible coefficient changes lead nei ther to an
+equivalent operator H, nor to an equivalent operator E, when ce a lack
+of uniqueness. We show that even the Dirac energy spectrum is not
+unique.
+1 Introduction
+This paper summarizes a recent work [1] on the subject title. Thus, rather
+than to give detailed arguments and to present all relevant results , our aim
+here is to make the origin of this surprising result more easily visible.
+1Quantum effects in the classical gravitational field are observed, e.g. on
+neutrons [2, 3, 4], which are spin1
+2particles. This motivates work on the
+curved spacetime Dirac equation. The standard version of the latt er (see
+e.g. Refs. [5, 6]) is due to Fock and to Weyl, and will be referred to as the
+Dirac-Fock-Weyl (DFW)equation. Ontheotherhand, two alternativeDirac
+equationsinacurvedspacetimewerederivedrecently[7],byapplying directly
+the “classical-quantum correspondence”, i.e., the corresponden ce between a
+classical Hamiltonian and a quantum wave operator. Thus, togethe r with
+the standard (DFW) version, we have three distinct versions of th e Dirac
+equation in a curved spacetime. The basic quantum mechanics was st udied
+for each of those three equations [8] (see Ref. [9] for a summary) :
+•the Dirac probability current was defined for any possible field ( γµ) of
+Dirac matrices. We derived the condition which has to be satisfied by
+(γµ) in order that the current conservation be true for any solution o f
+the Dirac equation (for DFW, this condition is always satisfied);
+•theprecise formoftherelevant scalarproduct wasshowntobeimp osed
+by the axioms of quantum mechanics (it turns out that, for DFW, th is
+is the form used by Leclerc [10]);
+•theHamiltonian operatorwas written andthe condition ofits hermitic -
+ity w.r.t. the relevant scalar product was derived in a general settin g
+(for DFW, this condition coincides, in the particular case envisaged b y
+Leclerc [10], with that derived by him).
+That foregoing work showed, in particular, that the hermiticity of t he Hamil-
+tonianisnotstableunder alladmissiblechangesofthefieldofDiracma trices.
+This implies that there is a non-uniqueness problem for the curved-s pacetime
+Dirac equation.
+Theaimof the presentwork [1], therefore,wastostudythe(non-)uniqueness
+of the Hamiltonian and energy operators, including the (non-)uniqu eness of
+the energy spectrum. The qualitative conclusions are the same for the three
+versions, i.e., the non-uniqueness applies to the alternative equatio ns of Ref.
+[7], too [1]. Thus, finding this non-uniqueness was disappointing for us as
+well. In this paper, we will focus on the standard equation (DFW) and the
+non-uniqueness results for it, because these results could be mor e directly of
+interest to many physicists.
+22 Three Dirac equations in a curved space-
+time
+The three versions of the gravitational Dirac equation have the sa me form:
+γµDµψ=−imψ, (1)
+whereγµ=γµ(X)(µ= 0,...,3)isafieldof4 ×4complexmatrices, depending
+on the point X∈V in the curved spacetime (V ,gµν), such that
+γµγν+γνγµ= 2gµν14, µ,ν∈ {0,...,3} (2)
+[here (gµν)(X) is the inverse of the matrix ( gµν)(X) and14≡diag(1,1,1,1)];
+whereψis abispinorfield for the standard equation (DFW), but is a 4-vector
+field for the two alternative equations, based on the tensor representation of
+the Dirac fields (TRD);
+and whereDµis a covariant derivative, associated with a specific connection .
+For DFW, this is the “spin connection”, which depends on the ( γµ) field
+[5, 6].
+3 Definition of the field of Dirac matrices
+For DFW, one defines
+γµ(X) =aµ
+α(X)γ♯α, (3)
+withuα=aµ
+α(X)∂µan orthonormal tetrad field ( ∂µis the natural basis
+associated with the coordinates xµ) and (γ♯α) a set of “flat” Dirac matrices,
+i.e., a constant solution of the anticommutation relation (2) with ( gµν) =
+(ηµν)≡diag(1,−1,−1,−1). One should be able to use anypossible choice
+of (γ♯α). One should study the influence of both choices: ( γ♯α) and (uα).
+For TRD, a tetrad field can also be used, though other possibilities also
+exist, e.g. parallel transport [7].
+In order to be able to use any possible field ( γµ) of Dirac matrices, we
+must have recourse to the hermitizing matrix Aof Bargmann and Pauli [12].
+This is a 4 ×4 complex matrix such that
+A†=A,(Aγµ)†=Aγµµ= 0,...,3, (4)
+3withM†≡M∗T= Hermitian conjugate of matrix M. We proved [11] the
+existence of A, and also that of B: the latter is a positive-definite hermitizing
+matrix for the αµmatrices which are defined by Eq. (10) below. In general,
+A=A(X) is also a field. However, when we use the definition from a tetrad
+field (3), we may take [8]
+A≡A♯, (5)
+whereA♯is a (constant) hermitizing matrix for the constant set ( γ♯α). More-
+over, in practice for DFW, one considers only sets ( γ♯α) for which γ♯0is a
+hermitizing matrix. (This includes the usual sets: Dirac’s, “chiral”, M ajo-
+rana’s, and any set deduced therefrom by a unitary transformat ion [13].)
+Thus, in practice,
+A=γ♯0for DFW. (6)
+4 Local similarities
+In a curved spacetime (V ,gµν), the Dirac matrices γµand the hermitizing
+matrixAare fields: they depend on X∈V. If one changes from one field
+(γµ) to another one (˜ γµ), also satisfying the anticommutation relation (2)
+and with the samemetricgµν, then the new field obtains by a local similarity
+transformation [12], which is unique up to a complex factor [11]:
+∃S=S(X)∈GL(4,C) : ˜γµ(X) =S−1γµ(X)S, µ= 0,...,3.(7)
+For the standard Dirac equation (DFW), the similarities are restrict ed to the
+spin group Spin(1,3), i.e., they result from a change of the tetrad field by
+a (proper) local Lorentz transform L(X)∈SO(1,3), depending arbitrarily
+onX∈V [5]. The corresponding local similarity transformation S(X)∈
+Spin(1,3)isdeduced[8]from L(X)throughthespinorrepresentation(defined
+only up to a sign), L/ma√sto→S=±S(L).
+5 The general Dirac Hamiltonian
+Rewriting the Dirac equation (1) in the “Schr¨ odinger” form:
+i∂ψ
+∂t= Hψ,(t≡x0), (8)
+4gives the Hamiltonian operator:
+H≡mα0−iαjDj−i(D0−∂0), (9)
+with
+α0≡γ0/g00, αj≡γ0γj/g00(j= 1,2,3). (10)
+One should realize that, in a given spacetime (V ,gµν) and with a given field
+of Dirac matrices ( γµ), this operator still depends on the coordinate system,
+or, more exactly, on the reference frame [8]—the latter being understood
+here as an equivalence class F of local coordinate systems (charts ) on the
+spacetime V, modulo the purely spatial transformations
+x′0=x0, x′j=fj((xk)). (11)
+In the present work, we fix the reference frame F and even the ch art (the
+latter being, however, an arbitrary one), and we study the depen dence of the
+Hamiltonian and energy operators on the field ( γµ).
+6 Scalar product and equivalent operators
+The Hilbert scalar product is fixed by the following result [8]:
+Theorem. Anecessary condition for the scalar productof time-independent
+wave functions to be time independent and for the Hamiltonia nHto be a
+Hermitian operator, is that the scalar product should be
+(ψ|ϕ)≡/integraldisplay
+ψ†Aγ0ϕ√−gd3x. (12)
+For DFW, we have in practice A=γ♯0, Eq. (6). Then (12) is the scalar
+product used by Leclerc [10].
+With each of any two possible “coefficient fields”: ( γµ,A) and (˜γµ,˜A),
+corresponds thus a unique scalar product. [For DFW, we have alway s˜A=
+A=A♯, Eq. (5), so the coefficient fields reduce to the field of Dirac matri-
+cesγµ.] These two scalar products are isometrically equivalent through the
+mappingψ/ma√sto→˜ψ≡S−1ψ: one shows easily [1] that
+(˜ψ˜|˜ϕ) = (ψ|ϕ). (13)
+5Moreover, H is fully determined by the set of the products (H ψ|ϕ), for
+ψ,ϕ∈ D ≡Dom(H). The same is true for the Hamiltonian ˜H after the
+similarity (7), ˜H being obtained by substituting ˜ γµforγµin Eq. (10), and
+by using the covariant derivatives ˜Dµ≡∂µ+˜Γµin (9), with ˜Γµthe new
+connection matrices [see Eq. (15) below]. It follows [1] that, in order that H
+and˜H be equivalent operators, it is necessary and sufficient that
+˜H =S−1HS, (14)
+whereSis the relevant similarity in Eq. (7). This, of course, is no surprise.
+Of course also, the same condition defines the equivalence of the en ergy
+operators E and ˜E, E being defined by Eq. (17) below.
+7 InvarianceconditionoftheHamiltonian un-
+der a local similarity (DFW)
+When does a local similarity S(X), applied to the field of Dirac matrices
+(γµ), leave the Hamiltonian operator H [equation (9)] invariant? I.e., whe n
+do we have Eq. (14)?
+The Hamiltonian (9) involves the connection matrices, Γ µ≡Dµ−∂µ. For
+DFW, these matrices change after a similarity according to [6]:
+˜Γµ=S−1ΓµS+S−1(∂µS). (15)
+A straightforward calculation shows then that, for DFW, we have ( 14) iff
+S(X) is time-independent,
+∂0S= 0. (16)
+But, in the general case, we have gµν,0/ne}ationslash= 0, hence any possible field ( γµ)
+depends on t≡x0. Thus, there is no way of finding a class of fields ( γµ)
+exchanging with ∂0S= 0. I.e.: The Dirac Hamiltonian is not unique [1].
+68 Invariance condition of the energy operator
+(DFW)
+When the Hamiltonian H is not Hermitian, one should use the Hermitian
+part of H:
+E = H+i
+2√−gB−1∂0/parenleftbig√−gB/parenrightbig
+=1
+2/parenleftbig
+H+H‡/parenrightbig
+, B≡Aγ0,(17)
+where the Hermitian conjugate operator H‡is w.r.t. the unique scalar prod-
+uct (12). This Hermitian-symmetrized operatorE coincides, inthep articular
+case envisaged by Leclerc [10], with what is called the energy operator, which
+is derived from the field Lagrangian. In the general case (thus for TRD and
+for DFW as well, and with any possible ( γµ) field), the Dirac equation (1)
+also derives from a Lagrangian [1]. We can show generally that the exp ected
+value of the energy operator E equals the classical field energy as d erived
+from that Lagrangian.
+Again a straightforward calculation gives the invariance condition of E
+(for DFW):
+B(∂0S)S−1−/bracketleftbig
+B(∂0S)S−1/bracketrightbig†≡2/bracketleftbig
+B(∂0S)S−1/bracketrightbiga= 0. (18)
+Only very particular local similarities S(X) do verify (18). Thus, there is
+a serious non-uniqueness problem for DFW (and for the alternative , TRD
+equations as well). Could even the spectrum of E be non-unique? In what
+follows, we investigate this question.
+9 Explicit expression of the energy operator
+(DFW)
+The general expression of the change of E after a local similarity S(X) is
+easily found to be [1]:
+δE≡S˜ES−1−E =−iB−1/bracketleftbig
+B(∂0S)S−1/bracketrightbiga. (19)
+In very general coordinates, the matrix ( aµ
+α) of that tetrad which defines the
+starting (γµ) field may be chosen to satisfy a0
+j= 0 [1]. Then a0
+0=/radicalbig
+g00
+7from the orthonormality of the tetrad. Let us take for “flat” mat ricesγ♯α
+any usual Dirac matrices, for which A=γ♯0[Eq. (6)]. Thus
+B≡Aγ0=γ♯0(a0
+0γ♯0) =/radicalbig
+g0014, (20)
+whence from (19):
+δE =−i/bracketleftbig
+(∂0S)S−1/bracketrightbiga. (21)
+For DFW,S(X) is an arbitrary Spin(1,3) transformation. Thus, ( ∂0S)S−1is
+the generic element of G, the Lie algebra of Spin(1,3) – whose the matrices
+sαβ≡[γ♯α,γ♯β] (α<β) (22)
+make a basis. Hence
+δE =−i/bracketleftbig
+ωαβsαβ/bracketrightbiga, (23)
+where, depending on the local Lorentz L(X) that defines S(X) =S(L(X)),
+the six coefficients ωαβ=−ωβαdepend arbitrarily on X∈V. With usual
+Dirac matrices verifying (6), we have
+(sαβ)†=γ♯0sβαγ♯0, (24)
+whence from (23) and the anticommutation formula:
+δE =−i3/summationdisplay
+j,k=1ωjksjk. (25)
+10 The case with the “chiral” Dirac matrices
+If the “flat” Dirac matrices γ♯αare the “chiral” ones {see e.g. Schulten [14],
+Eqs. (10.257) and (10.260) }, we get from (25):
+δE =−i3/summationdisplay
+j,k=1ωjksjk=/parenleftbiggN0
+0N/parenrightbigg
+, N≡ −1
+2/vectorθ./vector σ (26)
+where/vectorθ≡(θk)withθ1≡ω23(circular), andwhere /vector σ≡(σk)withσkthePauli
+matrices.
+Depending on the three real numbers ωjk,1≤j <k≤3, the matrix N
+can beanyHermitian matrix 2 ×2 with zero trace {see e.g. Schulten [14],
+Eq. (5.226) }. Any such matrix has two eigenvalues µ∈Rand−µ, and has
+an orthonormal basis of eigenvectors: respectively u∈C2forµ, andvfor
+−µ.
+811 Non-uniqueness of the energy spectrum
+(DFW)
+A small perturbation of the starting Dirac matrices γµis defined by a simi-
+larity close to the identity:
+S(ε,X) =I+ε(δS)(X)+O(ε2). (27)
+It modifies each eigenvalue of the energy operator E according to t he well-
+known formula
+δλ= (ψ|δE(ε)ψ)+O(ε2), (28)
+whereψis the eigenfunction for the unperturbed state. Inserting δE (26)
+and the expression (12) of the scalar product into Eq. (28), and d ecomposing
+the bispinor into two two-components spinors: ψ= (φ,χ), we get:
+δλ=/integraldisplay
+ψ†δEψ/radicalbig
+−gg00d3x=/integraldisplay
+(φ†Nφ+χ†Nχ) dV.(29)
+Let us fixµ>0 andt. For any spatial position x≡(xj) in our fixed chart,
+let us select the 2 ×2 Hermitian matrix with zero trace, N=N(x), in such
+a way that φ(x) be the eigenvector of N(x) for the eigenvalue µ. (The 3-
+vectorxis the coordinate expression of the position xin the space manifold
+M associated with the arbitrary given reference frame F [8].) That is:
+φ†Nφ=µ φ†φ. (30)
+SinceNhas the eigenvalues µand−µ, we have then necessarily
+χ†Nχ≥ −µ χ†χ. (31)
+In Eq. (31), the inequality becomes an equality only if χ(x) is an eigenvector
+ofN(x) with eigenvalue −µ. In that case, φ(x) is orthogonal to χ(x) inC2,
+χ(x)⊥φ(x). Returning to Eq. (29), we see that δλ>0 unless if i) χ(x)⊥φ(x)
+for almost every spatial position x,andii)/integraltext
+φ†φdV=/integraltext
+χ†χdV. This could
+occur only in extremely particular situations. In fact, i) alone implies [1 ] that
+the probability current is light-like for almost every x, and this is impossible
+ifm>0 [15]. This proves the non-uniqueness of the DFW energy spectrum ,
+at least for a massive particle.
+912 Conclusion
+Three distinct gravitational Dirac equations were envisaged: the s tandard
+equation (“Dirac-Fock-Weyl” or DFW), plus two alternative equatio ns [7]
+based on the tensor representation of the Dirac field (TRD). Using the her-
+mitizing matrix A, they were studied together, to a large extent [8, 1]. Non-
+uniqueness results have been proved for each of the three versio ns [1]. In this
+paper, we summarized the latter work, but focussed on the result s for DFW.
+The Hamiltonian operator H is not unique: it depends on the admissible
+choice of the field of Diracmatrices. The same is true for the energy operator
+E. This is true for DFW and for TRD.
+Thespectrum of E is itself non-unique. All of these results apply already
+to aflatspacetime in a non-inertial frame.
+A Appendix: The classical energy and its
+frame dependence
+One might ask if the non-uniqueness of the energy spectrum for a s pin1
+2par-
+ticle in a curved spacetime, reported here, might have something to do with
+the well-known fact that, in general relativity (GR), there is no cov ariant
+concept of local energy—already in the classical context (as oppo sed to the
+quantum context envisaged inthis paper). The latter fact may bef ormulated
+by saying that there is no covariant conservation law for the (tota l) energy-
+momentum—the energy-momentum of the gravitational field giving r ise only
+to apseudo-tensor. As is well known, the usual “conservation law” for the
+energy-momentum tensor of matter and non-gravitational fields “does not
+express the conservation of anything” [16]. However, for a classic altest par-
+ticle,in any arbitrary reference frame F, there is a well-defined Hamiltonian
+energy[7, 17, 18], though it depends on F and, in general, on the time:
+•Geodesic motion in the Lorentzian manifold (V ,gµν) derives from the
+(“super-”)Hamiltonian over the 8-dimensional phase space assoc iated
+with V:
+˜H[(pµ),(xν)]≡1
+2gµν((xρ))pµpν(c= 1). (32)
+This is because, ina (pseudo-)Riemannian manifold, geodesic motionis
+identical to free Hamiltonian motion, i.e., ˜H≡TwithT[(pµ),(xν)]≡
+101
+2gµνpµpνthe “kinetic energy”: Sect. 45 D in Arnold [19]. Note that ˜H
+does not depend on the (proper) time τ.
+•It follows from that “time”-independence that, in any given coordi-
+nate system, we deduce a “normal” Hamiltonian Hover the usual,
+6-dimensional phase space, by dimensional reduction (Sect. 45 B in
+Ref. [19]):
+H[(pj),(xj),t≡x0]≡p0 (33)
+is extracted from
+gµνpµpν−m2= 0. (34)
+{With the ( −+ ++) signature, we have H≡ −p0instead, and the
+last equation writes gµνpµpν+m2= 0 [18].}One may note that E=
+H≡p0is invariant under purely spatial coordinate changes (11), thus
+it depends only on the reference frame—which is arbitrary, but fixedin
+the present study.
+Therefore, in a fixed reference frame, a classical test particle in a given exter-
+nal gravitational field hasunique energy, inspite of theabsence of a covariant
+concept oflocalenergyinGR,wheregravitationaldynamics areals oincluded
+in the model. Similarly, the answer to the question posed at the beginn ing of
+this Appendix is no. The non-uniqueness of the energy spectrum fo r a spin-
+half particle in a given external curved spacetime, as found in this st udy, has
+nothing to do with the absence of a covariant concept of local ener gy in GR.
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+[11] Arminjon M and Reifler F 2008 Dirac equation: Representa tion
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+(Preprint arXiv:0707.1829 (quant-ph))
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+Koordinaten, Teil II: Die Diracschen Gleichungen f¨ ur die M ateriewellen Ann.
+der Phys. (5)18337–354
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+Preprint arXiv:physics/0703214
+[14] Schulten K 1999 Relativistic quantum mechanics, in Notes on Quantum Me-
+chanics, online course of the University of Illinois at Urbana-Cham paign by
+the same author
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+Theor. Phys. 441307–1324 ( Preprint arXiv:0706.1258 (gr-qc))
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+Lifshitz E M and Landau L D 1980 The Classical Theory of Fields (Oxford:
+Butterworth Heinemann)
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+arXiv:1001.0461v1  [math.CO]  4 Jan 2010Rank-width of Random Graphs∗
+Choongbum Lee†
+Department of Mathematics
+UCLA, Los Angeles, CA, 90095.
+Joonkyung Lee‡
+Department of Mathematical Sciences
+KAIST, Daejeon 305-701, Republic of Korea.
+Sang-il Oum§
+Department of Mathematical Sciences
+KAIST, Daejeon 305-701, Republic of Korea.
+January 4, 2010
+Abstract
+Rank-width of a graph G, denoted by rw(G), is a width parameter of
+graphs introduced by Oum and Seymour (2006). We investigate the asymp-
+toticbehaviorofrank-widthofarandomgraph G(n,p). Weshowthat, asymp-
+totically almost surely, (i) if p∈(0,1) is a constant, then rw(G(n,p)) =
+⌈n
+3⌉−O(1), (ii) if1
+n≪p≤1
+2, thenrw(G(n,p)) =⌈n
+3⌉−o(n), (iii) if p=c/n
+andc >1, thenrw(G(n,p))≥rnfor some r=r(c), and (iv) if p≤c/n
+andc <1, thenrw(G(n,p))≤2. As a corollary, we deduce that G(n,p) has
+linear tree-width whenever p=c/nfor each c >1, answering a question of
+Gao (2006).
+Keywords: rank-width, tree-width, clique-width, randomg raph, sharpthresh-
+old.
+1 Introduction
+Rank-width of a graph G, denoted by rw(G), is a graph width parameter introduced
+by Oum and Seymour [10] and measures the complexity of decomposin gGinto a
+∗The first author was supported in part by Samsung Scholarship. Th e second and third authors
+were supported by SRC Program of Korea Science and Engineering F oundation(KOSEF) grant
+funded by the Korea government (MOST)(No. R11-2007-035-01 002-0). The third author was also
+partially supported by TJ Park Junior Faculty Fellowship.
+†choongbum.lee@gmail.com
+‡jk87@kaist.ac.kr
+§sangil@kaist.edu
+1tree-like structure. The precise definition will be given in the following section. One
+fascinating aspect of this parameter lies in its computational applica tions, namely, if
+aclass ofgraphshasbounded rank-width, thenmanyNP-hardpro blems aresolvable
+on this class in polynomial time; for example, see [2].
+We consider the Erd˝ os-R´ enyi random graph G(n,p). In this model, a graph
+G(n,p) on a vertex set {1,2,···,n}is chosen randomly as follows: for each un-
+ordered pair of vertices, they are adjacent with probability pindependently at ran-
+dom. Given a graph property P, we say that G(n,p) possesses Pasymptotically
+almost surely , or a.a.s. for brevity, if the probability that G(n,p) possesses Pcon-
+verges to 1 as ngoes to infinity. A function f:N→[0,1] is called the sharp
+threshold ofG(n,p) with respect to having Pif the following hold: if p≥cf(n)
+for a constant c >1, thenG(n,p) a.a.s. satisfies Pand otherwise if p≤cf(n) and
+c <1, thenG(n,p) a.a.s. does not satisfy P.
+The following is our main result.
+Theorem 1.1. For a random graph G(n,p), the following holds asymptotically al-
+most surely:
+(i) ifp∈(0,1)is a constant, then rw(G(n,p)) =⌈n
+3⌉−O(1),
+(ii) if1
+n≪p≤1
+2, thenrw(G(n,p)) =⌈n
+3⌉−o(n),
+(iii) ifp=c/nandc >1, thenrw(G(n,p))≥rnfor some r=r(c), and
+(iv) ifp≤c/nandc <1, thenrw(G(n,p))≤2.
+Sincerw(G)≤ ⌈|V(G)|
+3⌉for every graph G, (i) and (ii) of this theorem give a
+narrow range of rank-width. Note that this theorem also gives a bo und when p≥1
+2,
+since the rank-width of G(n,p) in this range can be obtained from the inequality
+rw(G)≤rw(G)+1.
+Clique-width of a graph G, denoted by cw(G), is a width parameter introduced
+by Courcelle and Olariu [3]. It is strongly related to rank-width by the f ollowing
+inequality by Oum and Seymour [10].
+rw(G)≤cw(G)≤2rw(G)+1−1. (1)
+Tree-width, introduced by Robertson and Seymour [11], is a width pa rameter
+measuring how similar a graph is to a tree and is closely related to rank- width. We
+will denote the tree-width of a graph Gastw(G). The following inequality was
+proved by Oum [9]: for every graph G, we have
+rw(G)≤tw(G)+1. (2)
+There have been works on tree-width of random graphs. Kloks [8] p roved that
+G(n,p) withp=c/nhas linear tree-width whenever c >2.36. Gao [6] improved
+this constant to 2 .162 and even conjectured that ccan be improved to a constant
+less than 2. We improve the above constant to the best possible num ber, 1, by the
+following corollary, stating that there is the sharp threshold p= 1/nofG(n,p) with
+respect to having linear tree-width.
+Corollary 1.2. Letcbe a constant and let G=G(n,p)withp=c/n. Then the
+following holds asymptotically almost surely:
+2(i) Ifc >1, then rank-width, clique-width,and tree-width of Gare at least c′nfor
+some constant c′depending only on c.
+(ii) Ifc <1, then rank-width and tree-width of Gare at most 2and clique-width
+ofGis at most 5.
+Proof.(i)followsTheorem1.1with(1)and(2). (ii)followseasilyduetothethe orem
+by Erd˝ os and R´ enyi [4, 5] stating that asympototically almost sur ely, each compo-
+nent ofG(n,p) withp=c/n,c <1 has at most one cycle. It is straightforward to
+see that such graphs have small tree-width, clique-width, and ran k-width.
+2 Preliminaries
+All graphs in this paper have neither loops nor parallel edges. Let ∆( G),δ(G) be
+the maximum degree and the minimum degree of a graph Grespectively. For two
+subsetsXandYofV(G), letEG(X,Y) be the set of ordered pairs ( x,y) of adjacent
+verticesx∈Xandy∈Y. LeteG(X,Y) =|EG(X,Y)|. We will omit subscripts if
+it is not ambiguous.
+LetF2={0,1}be the binary field. For disjoint subsets V1andV2ofV(G),
+letNV1,V2be a 0-1 |V1| × |V2|matrix over F2whose rows are labeled by V1and
+columns labeled by V2, and the entry ( v1,v2) is 1 if and only if v1∈V1andv2∈V2
+are adjacent. We define the cutrankofV1andV2, denoted by ρG(V1,V2), to be
+rank(NV1,V2).
+A treeTis said to be subcubic if every vertex has degree 1 or 3. A rank-
+decomposition of a graph Gis a pair ( T,L) of a subcubic tree Tand a bijection L
+fromV(G) to the set of all leaves of T. Notice that deleting an edge uvofTcreates
+two components CuandCvcontaining uandvrespectively. Let Auv=L−1(Cu)
+andBuv=L−1(Cv). Under these notations, rank-width of a graph G, denoted by
+rw(G), is defined as
+rw(G) = min
+(T,L)max
+uv∈E(T)ρG(Auv,Buv),
+where the minimum is taken over all possible rank-decompositions. We assume
+rw(G) = 0 if|V(G)| ≤1.
+The following lemma will be used later.
+Lemma 2.1. LetG= (V,E)be a graph with at least two vertices. If rank-width of
+Gis at most k, then there exist two disjoint subsets V1,V2ofVsuch that
+|V1|=/ceilingleftBign
+2/ceilingrightBig
+,|V2|=/ceilingleftBign
+3/ceilingrightBig
+, andρG(V1,V2)≤k.
+Proof.Letk=rw(G). Let (T,L) be a rank-decomposition of width k. We claim
+that there is an edge eofTsuch that T\egives a partition( A,B) ofV(G) satisfying
+|A| ≥n/3,|B| ≥n/3 andρG(A,B)≤k. Assume the contrary. Then for each edge
+einT,T\ehas a component CeofT\econtaining less than n/3 leaves of T. Direct
+each edge e=uvfromutovifCecontains u. Since this directed tree is acyclic,
+there is a vertex tinV(T) such that every edge incident with tis directed toward
+3t. Then there are at most 3 components in T\tand each component has less than
+n/3 leaves of T, a contradiction. This proves the claim.
+Given sets A,Bas above, we may assume |A| ≥n/2. TakeV1⊆AandV2⊆B
+of size⌈n
+2⌉and⌈n
+3⌉, respectively. Then ρG(V1,V2)≤ρG(A,B)≤k.
+3 Rank-width of dense random graphs
+In this section we will show that if1
+n≪min(p,1−p), then the rank-width of G(n,p)
+isa.a.s.⌈n
+3⌉−o(n). Moreover, foraconstant p∈(0,1), rank-widthof G(n,p) isa.a.s.
+⌈n
+3⌉−O(1). This bound is achieved by investigating the rank of random matr ices.
+The following proposition provides an exponential upper bound to th e probability
+of a random vector falling into a fixed subspace.
+Proposition 3.1. For0< p <1, letη= max(p,1−p). Letv∈Fn
+2be a random
+0-1vector whose entries are 1 or 0 with probability pand1−prespectively. Then
+for each k-dimensional subspace UofFn
+2,
+P(v∈U)≤ηn−k
+Proof.LetBbe ak×nmatrix whose row vectors form a basis of U. By permut-
+ing the columns if necessary, we may assume that the first k columns are linearly
+independent. For a vector v∈Fn
+2, letv(k)be the first kentries of v, and note that
+P(v∈U) =/summationdisplay
+w∈Fk
+2P(v∈U|v(k)=w)P(v(k)=w). (3)
+Letu1,u2,···,ukbe the row vectors of B. Observe that {u(k)
+j}k
+j=1is a basis of Fk
+2.
+Thus, given v(k)=w=/summationtextk
+i=1ciu(k)
+i, we have v∈Uif and only if v=/summationtextk
+i=1ciui.
+This implies that given each first kentries of v, there is a unique choice of remaining
+entries yielding v∈U. Thus for every w∈Fk
+2,P(v∈U|v(k)=w)≤ηn−k.
+Combining with (3), we obtain
+P(v∈U)≤ηn−k/summationdisplay
+w∈Fk
+2P(v(k)=w) =ηn−k,
+and this concludes the proof.
+LetM(k1,k2;p) be a random k1×k2matrix whose entries are mutually inde-
+pendent and take value 0 or 1 with probability 1 −pandprespectively. Using
+Proposition 3.1, we can bound the probability that the rank of M(⌈n
+3⌉,⌈n
+2⌉;p) devi-
+ates from ⌈n
+3⌉.
+Lemma 3.2. For0< p <1, letη= max(p,1−p). Then for every C >0,
+P/parenleftBigg
+rank/parenleftBig
+M/parenleftBig/ceilingleftBign
+3/ceilingrightBig
+,/ceilingleftBign
+2/ceilingrightBig
+;p/parenrightBig/parenrightBig
+≤/ceilingleftBign
+3/ceilingrightBig
+−C
+log21
+η/parenrightBigg
+<2(1
+2−1
+6C)n.
+4Proof.LetM=M(⌈n
+3⌉,⌈n
+2⌉;p),α=⌈C
+log21
+η⌉, and row( M) be the linear space
+spanned by the rows of M. We may assume ⌈n
+3⌉−α≥0. Denote row vectors of M
+byv1,v2,···,v⌈n
+3⌉. Note that rank(M) is at most ⌈n
+3⌉−αif and only if there are
+⌈n
+3⌉−αrows ofMspanning row( M). Thus
+P/parenleftBig
+rank(M)≤/ceilingleftBign
+3/ceilingrightBig
+−α/parenrightBig
+≤/summationdisplay
+IP({vi}i∈Ispans row( M))
+where the sum is taken over all I⊆ {1,2,···,⌈n
+3⌉}with cardinality ⌈n
+3⌉−α. Let
+UIbe the vector space spanned by row vectors {vi}i∈I. By Proposition 3.1, we get
+P({vi}i∈Ispans row( M)) =P({vj:j /∈I} ⊆UI)≤(η⌈n
+2⌉−⌈n
+3⌉+α)α,
+since rows are mutually independent random vectors. Combining the se inequalities,
+we conclude that
+P/parenleftBig
+rank(M)≤/ceilingleftBign
+3/ceilingrightBig
+−α/parenrightBig
+≤2⌈n
+2⌉−1(ηα)⌈n
+2⌉−⌈n
+3⌉+α≤2n
+22−n
+6C= 2(1
+2−1
+6C)n
+because⌈n
+2⌉−⌈n
+3⌉+α≥n
+6and/parenleftbig⌈n
+2⌉
+k/parenrightbig
+≤2⌈n
+2⌉−1.
+Proposition 3.3. Letη= max(p,1−p)andn≥2. Then
+P/parenleftBigg
+rw(G(n,p))≤/ceilingleftBign
+3/ceilingrightBig
+−12.6
+log21
+η/parenrightBigg
+<2−0.015n.
+Proof.LetG=G(n,p),S={NV1,V2:|V1|=⌈n
+2⌉,|V2|=⌈n
+3⌉for disjoint V1,V2⊆V(G)}
+and letµ= min N∈Srank(N).By Lemma 2.1, we have µ≤rw(G). Thus it suffices
+to show that
+P/parenleftBigg
+µ≤/ceilingleftBign
+3/ceilingrightBig
+−12.6
+log21
+η/parenrightBigg
+<2−0.015n.
+For each N∈ S, letANbe the event that rank(N)≤ ⌈n
+3⌉−12.6
+log21
+η. Note that
+P/parenleftBigg
+µ≤/ceilingleftBign
+3/ceilingrightBig
+−12.6
+log21
+η/parenrightBigg
+=P(/uniondisplay
+N∈SAN)≤/summationdisplay
+N∈SP(AN).
+By Lemma 3.2, we have P(AN)≤2−1.6n. Notice also that |S| ≤3n. Therefore,
+P/parenleftBigg
+µ≤/ceilingleftBign
+3/ceilingrightBig
+−12.6
+log21
+η/parenrightBigg
+≤3n2−1.6n<2−0.015n.
+The main theorem directly follows from this proposition.
+Theorem 3.4. Asymptotically almost surely, G=G(n,p)satisfies the following:
+(i) ifp∈(0,1)is a constant, then ⌈n
+3⌉−O(1)≤rw(G)≤ ⌈n
+3⌉, and
+(ii) if1
+n≪min(p,1−p), then⌈n
+3⌉−o(n)≤rw(G)≤ ⌈n
+3⌉.
+54 Rank-width of sparse random graphs
+In this section we investigate the rank-width of G(n,p) whenp=c/nfor some
+constant c >0. Note that Proposition 3.3 does not give any information when
+p=c/nandcis close to 1. As mentioned in the introduction, the linear lower
+bound of rank-width in this range of pis closely related to a sharp threshold with
+respect to having linear tree-width. We show that, when p=c/n,
+(i) ifc <1, then rank-width is a.a.s. at most 2,
+(ii) ifc= 1, then rank-width is a.a.s. at most O(n2
+3) and,
+(iii) ifc >1, then there exists r=r(c) such that rank-width is a.a.s. at least rn.
+Erd˝ os and R´ enyi [4, 5] proved that if c <1 thenG(n,p) a.a.s. consists of trees
+and unicyclic (at most one edge added to a tree) components and if c= 1 then
+the largest component has size at most O(n2
+3). Therefore, (i) and (ii) follow easily
+because trees and unicyclic graphs have rank-width at most 2.
+Thus, (iii) is the only interesting case. When c >1,G(n,p) has a unique
+component of linear size, called the giant component . Hence, in order to prove a
+lower bound on the rank-width of G(n,p), it is enough to find a lower bound of the
+rank-width of the giant component.
+We need some definitions to describe necessary structures. Let G= (V,E)
+be a connected graph. For a non-empty proper subset SofV(G), letdG(S) =/summationtext
+v∈SdegG(v). The(edgewise) Cheeger constant of a connected graph Gis
+Φ(G) = min
+∅/n⌉gationslash=S/subsetnoteqlV(G)eG(S,V(G)\S)
+min(dG(S),dG(V(G)\S)).
+Remark. In [1], the following alternative definition of the Cheeger constant of a
+connected graph Gis used. For a vertex v, letπv=degG(v)
+2|E(G)|and for vertices vandw
+ofG, define
+pvw=/braceleftBigg
+1/degG(v) ifvandware adjacent,
+0 otherwise.
+For a subset SofV(G), letπG(S) =/summationtext
+v∈Sπv. ThusdG(S) = 2|E(G)|πG(S). In [1],
+the Cheeger constant of a graph Gis defined alternatively as
+min
+0<πG(S)≤1
+21
+πG(S)/summationdisplay
+i∈S,j/∈Sπipij.
+We can easily see that these definitions are equivalent as follows:
+Φ(G) = min
+∅/n⌉gationslash=S/subsetnoteqlV(G)eG(S,V(G)\S)
+min(dG(S),dG(V(G)\S))= min
+0<πG(S)≤1
+2eG(S,V(G)\S)
+dG(S)
+= min
+0<πG(S)≤1
+21
+πG(S)/summationdisplay
+i∈S,j/∈Sπipij,
+where the second equality follows from the fact that πG(S)+πG(V(G)\S) = 1.
+6Benjamini, Kozma and Wormald [1] proved the following theorem.
+Theorem 4.1 (Benjamini, Kozma and Wormald [1]) .Letc >1andp=c/n. Then
+there exist α,δ >0such that G(n,p)a.a.s. contains a connected subgraph Hsuch
+thatΦ(H)≥αand|V(H)| ≥δn.
+Remark. The above theorem is a consequence of [1, Theorem 4.2]. The graph H
+in Theorem 4.1 is the graph RN(G) in [1, Theorem 4.2], which proves that RN(G)
+is a.a.s. an α-strong core of G. This means that RN(G) is a subgraph of Gwith
+Φ(RN(G))≥αby the definitions given in Section 2.2 and Section 3 of [1]. The
+condition |V(H)| ≥δnis not explicit in [1, Theorem 4.2]. However this fact fol-
+lows from [1, Lemma 4.7], because RN(G) must have more vertices than its kernel
+K(RN(G)) (the definition of kernel is given in [1, Section 4]). Note that ˆ nin [1,
+Lemma 4.7] satisfies ˆ n= Ω(n) by the remark following [1, Lemma 4.1]. The proof
+of Theorem 4.2 given in [1, Section 5] also mentioned this fact explicitly.
+A graph Hwith the property as in Theorem 4.1 is called an expander graph .
+The simple restriction of Φ( H) being bounded away from 0 provides a strikingly
+rich structure to the graph as in Theorem 4.1. Interested reader s are referred to the
+survey paper [7].
+By using this expander subgraph H, we will show that G(n,p) must have large
+rank-width when p=c/nandc >1. Before proving this, we need a technical lemma
+which allows us to control the maximum degree of a random graph G(n,p).
+Lemma 4.2. Letc >1be a constant and p=c/n. Then for every ε >0, there
+existsM=M(c,ε)such that G=G(n,p)a.a.s. has the following property: Let X
+be the collection of vertices which have degree at least M. Then the number of edges
+incident with Xis at most εn.
+Proof.LetV=V(G). LetMbe a large number satisfying
+∞/summationdisplay
+k=Mkck
+(k−1)!<ε
+2. (4)
+For each v∈V, define a random variable Yv= deg(v) if deg(v)≥MandYv= 0
+otherwise. Then by (4),
+E[Y2
+v] =n−1/summationdisplay
+k=Mk2P(deg(v) =k)
+≤n−1/summationdisplay
+k=Mk2/parenleftbiggn−1
+k/parenrightbigg/parenleftBigc
+n/parenrightBigk
+≤∞/summationdisplay
+k=Mkck
+(k−1)!<ε
+2.(5)
+SinceYv≤Y2
+v, we also have E[Yv]≤ε/2. Note that the number of edges incident
+withXis at most/summationtext
+v∈VYv. Hence, it is enough to prove a.a.s. Y=/summationtext
+v∈VYv≤εn.
+Observe that E[Y]≤ε
+2n. Moreover, the variance of Ycan be computed as
+E[(Y−E[Y])2] =/summationdisplay
+v∈V/parenleftbig
+E[Y2
+v]−E[Yv]2/parenrightbig
++/summationdisplay
+v/n⌉gationslash=w∈V/parenleftbig
+E[YvYw]−E[Yv]E[Yw]/parenrightbig
+≤εn+/summationdisplay
+v/n⌉gationslash=w∈V/parenleftbig
+E[YvYw]−E[Yv]E[Yw]/parenrightbig
+,(6)
+7where for each v,w∈V,v/\⌉}atio\slash=w,
+E[YvYw]−E[Yv]E[Yw]
+=n−1/summationdisplay
+k,l=Mkl/parenleftbig
+P(deg(v) =k,deg(w) =l)−P(deg(v) =k)P(deg(w) =l)/parenrightbig
+.
+Letqk=P(deg(v) =k|vw /∈E(G)) =P(deg(v) =k+1|vw∈E(G)), for distinct
+verticesv,winG(n,p). Notice that, given either vw∈E(G) orvw /∈E(G),Yvand
+Yware independent. Thus, we deduce the following:
+E[YvYw]−E[Yv]E[Yw]
+=n−1/summationdisplay
+k,l=Mkl/parenleftbig
+pqk−1ql−1+(1−p)qkql−(pqk−1+(1−p)qk)(pql−1+(1−p)ql)/parenrightbig
+≤pn−1/summationdisplay
+k,l=Mkl(qk−1ql−1+qkql)
+≤2pn−1/summationdisplay
+k=M−1(k+1)qkn−1/summationdisplay
+l=M−1(l+1)ql≤ε2
+n.
+Last inequality follows from (4), since similarly as done in (5) we get
+n−1/summationdisplay
+k=M−1(k+1)qk=n−1/summationdisplay
+k=M−1(k+1)/parenleftbiggn−2
+k/parenrightbigg/parenleftBigc
+n/parenrightBigk
+≤∞/summationdisplay
+k=Mkck−1
+(k−1)!<ε
+2c
+andc >1. Thus, by (6), we proved that the variance σ2ofYis at most (1+ ε)εn.
+Finally, using Chebyshev’s inequality and the fact E[Yv]≤ε/2, we show that
+P/parenleftbig
+Y > εn/parenrightbig
+≤P/parenleftBig
+Y≥E[Y]+εn
+2/parenrightBig
+≤σ2
+ε2n2/4≤1+ε
+εn/4,
+which concludes the proof.
+The following lemma will be used in the proof of the main theorem.
+Lemma 4.3. LetAbe a matrix over F2with at least nnon-zero entries. If each
+row and column contains at most Mnon-zero entries, then rank(A)≥n
+M2.
+Proof.We apply induction on n. We may assume n > M2. Pick a non-zero row w
+ofA. We may assume that the first entry of wis non-zero, by permuting columns if
+necessary. Now remove all rows w′whose first entry is 1. Since the first column has
+atmostMnon-zero entries, we remove atmost Mrowsincluding witself. Hence, we
+get a submatrix A′with at least n−M2non-zero entries. By induction hypothesis,
+rank(A′)≥n−M2
+M2≥n
+M2−1.
+By construction, wdoes not belong to the row-space of A′and therefore
+rank(A)≥rank(A′)+1≥n
+M2.
+8Theorem 4.4. Forc >1, letp=c/n. Then there exists r=r(c)such that a.a.s.
+rw(G(n,p))≥rn.
+Proof.DenoteG(n,p) byG. Letα,δbe constants from Theorem 4.1, and Hbe the
+expander subgraph also given by Theorem 4.1. Let W=V(H) and let ( W1,W2) be
+an arbitrary partition of Wsuch that |W1|,|W2| ≥ |W|/3. Then since Φ( H)≥α
+andHis connected, we have
+α≤eH(W1,W2)
+min(dH(W1),dH(W2))≤eH(W1,W2)
+min(|W1|,|W2|)≤eG(W1,W2)
+|W|/3.
+ThuseG(W1,W2)≥αδ
+3n. By Lemma 4.2, there exists Msuch that the number of
+edges incident with a vertex of degree at least Mis at mostαδ
+6n. LetW′
+1=W1\X
+andW′
+2=W2\X. SinceeG(W′
+1,W′
+2)≥αδ
+6n,NW′
+1,W′
+2has at leastαδ
+6nentries with
+value 1. Moreover, NW′
+1,W′
+2has at most Mentries of value 1 in each row and column.
+Hence, we can use Lemma 4.3 to obtain
+αδ
+6M2n≤ρG(W′
+1,W′
+2)≤ρG(W1,W2).
+SinceW1,W2are arbitrary subsets satisfying |W1|,|W2| ≥ |W|/3, this implies that
+the induced subgraph G[W] has rank-width at leastαδ
+6M2nby Lemma 2.1. Therefore,
+rank-width of Gis at leastαδ
+6M2n.
+Corollary 4.5. Letc >1andp=c/n. Then there exists t=t(c)such that a.a.s.
+tw(G(n,p))≥tn.
+Acknowledgment. Part of this work was done during the IPAM Workshop Com-
+binatorics: Methods and Applications in Mathematics and Co mputer Science , 2009.
+The first author would like to thank Nick Wormald for the discussion at IPAM
+Workshop.
+References
+[1] I. Benjamini, G. Kozma, and N. Wormald. The mixing time of the giant com-
+ponent of a random graph. preprint, 2006, arxiv:math/0610459v 1.
+[2] B. Courcelle, J. A. Makowsky, and U. Rotics. Linear time solvable o ptimization
+problems ongraphs ofbounded clique-width. Theory Comput. Syst. , 33(2):125–
+150, 2000.
+[3] B.CourcelleandS.Olariu. Upperboundstothecliquewidthofgrap hs.Discrete
+Appl. Math. , 101(1-3):77–114, 2000.
+[4] P. Erd˝ os and A. R´ enyi. Onthe evolution of random graphs. Magyar Tud. Akad.
+Mat. Kutat´ o Int. K¨ ozl. , 5:17–61, 1960.
+[5] P. Erd˝ osandA. R´ enyi. Ontheevolutionofrandomgraphs. Bull. Inst. Internat.
+Statist., 38:343–347, 1961.
+9[6] Y. Gao. On the threshold of having a linear treewidth in random gra phs. In
+Computing and combinatorics , volume 4112 of Lecture Notes in Comput. Sci. ,
+pages 226–234. Springer, Berlin, 2006.
+[7] S. Hoory, N. Linial, andA. Wigderson. Expander graphs andtheir applications.
+Bull. Amer. Math. Soc. (N.S.) , 43(4):439–561 (electronic), 2006.
+[8] T. Kloks. Treewidth , volume 842 of Lecture Notes in Computer Science .
+Springer-Verlag, Berlin, 1994. Computations and approximations.
+[9] S. Oum. Rank-width is less than or equal to branch-width. J. Graph Theory ,
+57(3):239–244, 2008.
+[10] S. Oum and P. Seymour. Approximating clique-width and branch- width.J.
+Combin. Theory Ser. B , 96(4):514–528, 2006.
+[11] N.RobertsonandP.Seymour. Graphminors.III.Planartree -width.J. Combin.
+Theory Ser. B , 36(1):49–64, 1984.
+10
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+arXiv:1001.0462v1  [math.RT]  4 Jan 2010Representation Theory of Finite Groups
+Anupam Singh
+IISER, central tower, Sai Trinity building, Pashan circle, Pune 411021
+INDIA
+E-mail address :anupam@iiserpune.ac.inContents
+Chapter 1. Introduction 5
+Chapter 2. Representation of a Group 7
+2.1. Commutator Subgroup and One dimensional representati ons 10
+Chapter 3. Maschke’s Theorem 11
+Chapter 4. Schur’s Lemma 15
+Chapter 5. Representation Theory of Finite Abelian Groups o verC 17
+5.1. Example of representation over Q 19
+Chapter 6. The Group Algebra k[G] 21
+Chapter 7. Constructing New Representations 23
+7.1. Subrepresentation and Sum of Representations 23
+7.2. Adjoint Representation 23
+7.3. Tensor Product of two Representations 24
+7.4. Restriction of a Representation 25
+Chapter 8. Matrix Elements 27
+Chapter 9. Character Theory 29
+Chapter 10. Orthogonality Relations 31
+Chapter 11. Main Theorem of Character Theory 33
+11.1. Regular Representation 33
+11.2. The Number of Irreducible Representations 33
+Chapter 12. Examples 37
+12.1. Groups having Large Abelian Subgroups 38
+12.2. Character Table of Some groups 39
+Chapter 13. Characters of Index 2 Subgroups 43
+13.1. The Representation V⊗V 43
+13.2. Character Table of S5 44
+34 CONTENTS
+13.3. Index two subgroups 46
+13.4. Character Table of A5 48
+Chapter 14. Characters and Algebraic Integers 51
+Chapter 15. Burnside’s pqTheorem 53
+Chapter 16. Further Reading 55
+16.1. Representation Theory of Symmetric Group 55
+16.2. Representation Theory of GL2(Fq) andSL2(Fq) 55
+16.3. Wedderburn Structure Theorem 55
+16.4. Modular Representation Theory 55
+Bibliography 57CHAPTER 1
+Introduction
+This is class notes for the course on representation theory o f finite groups taught by
+the author at IISER Pune to undergraduate students. We study character theory of finite
+groups and illustrate how to get more information about grou ps. The Burnside’s theorem
+is one of the very good applications. It states that every gro up of order paqb, wherep,qare
+distinct primes, is solvable. We will always consider finite groups unless stated otherwise.
+All vector spaces will be considered over general fields in th e beginning but for the purpose
+of character theory we assume the field is that of complex numb ers.
+We assume knowledge of the basic group theory and linear alge bra. The point of view
+I projected to the students in the class is that we have studie d linear algebra hence we
+are familiar with the groups GL(V), the general linear group or GLn(k) in the matrix
+notation. The idea of representation theory is to compare (v ia homomorphisms) finite
+(abstract) groups with these linear groups (some what concr ete) and hope to gain better
+understanding of them.
+The students were asked to read about “linear groups” from th e book by Alperin and
+Bell (mentioned in the bibiliography) from the chapter with the same title. We also revised,
+side-by-side in the class, Sylow’s Theorem, Solvable group s and motivated ourselves for the
+Burnside’s theorem.
+The aim to start with arbitrary field was to give the feeling th at the theory is dependent
+on the base field and it gets considerably complicated if we mo ve away from characteristic
+0 algebraically closed field. This we illustrate by giving an example of higher dimensional
+irreducible representation of cyclic group over Qwhile all its irreducible representations are
+one dimensional over C. Thus this puts things in perspective why we are doing the the ory
+overCand motivates us to develop “Character Theory”.
+5CHAPTER 2
+Representation of a Group
+LetGbe a finite group. Let kbe a field. We will assume that characteristic of kis 0,
+e.g.,C,RorQthough often char(k)∤|G|is enough.
+Definition 2.1 (Representation) .Arepresentation of Goverkis a homomorphism
+ρ:G→GL(V)whereVis a vector space of finite dimension over field k. The vector
+spaceVis called a representation space ofGand its dimension the dimension of
+representation .
+Strictly speaking the pair ( ρ,V) is called representation of Gover fieldk. However if
+there is no confusion we simply call ρa representation or Va representation of G. Let us fix
+a basis{v1,v2,...,vn}ofV. Then each ρ(g) can be written in a matrix form with respect
+to this basis. This defines a map ˜ ρ:G→GLn(k) which is a group homomorphism.
+Definition 2.2 (Invariant Subspace) .Letρbe a representation of GandW⊂Vbe a
+subspace. The space Wis called a G-invariant (or G-stable) subspace if ρ(g)(w)∈
+W∀w∈Wand∀g∈G.
+Notice that once we have a G-invariant subspace Wwe can restrict the representation to
+this subspace and define another representation ρW:G→GL(W) whereρW(g) =ρ(g)|W.
+HenceWis also called a subrepresentation .
+Example 2.3 (Trivial Representation) .LetGbe a group and kand field. Let Vbe a
+vector space over k. Thenρ(g) = 1 for all g∈Gis a representation. This is called trivial
+representation. In this case every subspace of Vis an invariant subspace.
+Example 2.4. LetG=Z/mZandk=C. LetVbe a vector space of dimension n.
+(1) Suppose dim( V) = 1. Define ρr:Z/mZ→C∗by 1/ma√sto→e2πir
+mfor 1≤r≤m−1.
+(2) Defineρ:Z/mZ→GL(V) by 1/ma√sto→TwhereTm= 1. For example if dim( V) = 2
+once can take T= diag{e2πir1
+m,e2πir2
+m}. There is a general theorem in Linear
+Algebra which says that any such matrix over Cis diagonalizable.
+Example 2.5. LetG=Z/mZandk=R. LetV=R2with basis {e1,e2}. Then we have
+representations of Z/mZ:
+ρr: 1/ma√sto→/bracketleftbiggcos2πr
+m−sin2πr
+m
+sin2πr
+mcos2πr
+m/bracketrightbigg
+78 2. REPRESENTATION OF A GROUP
+where 1≤r≤m−1. Notice that we have mdistinct representations.
+Example 2.6. Letφ:G→Hbe a group homomorphism. Let ρbe a representation of H.
+Thenρ◦φis a representation of G.
+Example 2.7. LetG=Dm=/an}bracketle{ta,b|am= 1 =b2,ab=bam−1/an}bracketri}htthe dihedral group with
+2melements. We have representations ρrdefined by :
+a/ma√sto→/bracketleftbiggcos2πr
+m−sin2πr
+m
+sin2πr
+mcos2πr
+m/bracketrightbigg
+,b/ma√sto→/bracketleftbigg0 1
+1 0/bracketrightbigg
+.
+Notice that we make use of the above two examples to make this e xample as we have a map
+φ:Dm→Z/mZ.
+Example 2.8 (Permutation Representation of Sn).LetSnbe the symmetric group on
+nsymbols and kany field. Let V=knwith standard basis {e1,...,en}. We define a
+representation of Snas follows: σ(ei) =eσ(i)forσ∈Sn. Notice that while defining this
+representation we don’t need to specify any field.
+Example 2.9 (GroupAction) .LetGbeagroupand kbeafield. Let Gbeactingonafinite
+setX, i.e., we have G×X→X. We denote k[X] ={f|f:X→k}, set of all maps. Clearly
+k[X] is a vector space of dimension |X|. The elements ex:X→kdefined byex(x) = 1 and
+ex(y) = 0 ifx/ne}ationslash=yform a basis of k[X]. The action gives rise to a representation of Gon
+the spacek[X] as follows: ρ:G→GL(k[X]) given by ( ρ(g)(f))(x) =f(g−1x) forx∈X.
+In fact one can make k[X] an algebra by the following multiplication:
+(f∗f′)(t) =/summationdisplay
+xf(x)f′(x−1t).
+Note that this is convolution multiplication not the usual p oint wise multiplication. If we
+takeG=SnandX={1,2,...,n}we get back above example.
+Example 2.10 (Regular Representation) .LetGbe a group of order nandka field. Let
+V=k[G] be ann-dimensional vector space with basis as elements of the grou p itself. We
+defineL:G→GL(k[G]) byL(g)(h) =gh, called the left regular representation. Also
+R(g)(h) =hg−1, defines right regular representation of G. Prove that these representations
+are injective. Also these representations are obtained by t he action of Gon the setX=G
+by left multiplication or right multiplication.
+Example 2.11. LetG=Q8={±1,±i,±j,±k}andk=C. We define a 2-dimensional
+representation of Q8by:
+i/ma√sto→/bracketleftbigg0 1
+−1 0/bracketrightbigg
+,j/ma√sto→/bracketleftbigg0i
+i0/bracketrightbigg
+.
+Example 2.12 (Galois Theory) .LetK=Q(θ) be a finite extension of Q. LetG=
+Gal(K/Q). We take V=K, a finite dimensional vector space over Q. We have natural2. REPRESENTATION OF A GROUP 9
+representation of Gas follows: ρ:G→GL(K) defined by ρ(g)(x) =g(x). Takeθ=ζ,
+somenth root of unity and show that the cyclic groups Z/mZhave representation over field
+Qof possibly dimension more than 2. This is a reinterpretatio n of the statement of the
+Kronecker-Weber theorem.
+Definition 2.13 (Equivalence of Representations) .Let(ρ,V)and(ρ′,V′)be two represen-
+tations ofG. The representations (ρ,V)and(ρ′,V′)are calledG-equivalent (or equiv-
+alent)if there exists a linear isomorphism T:V→V′such thatρ′(g) =Tρ(g)T−1for all
+g∈G.
+Letρbe a representation. Fix a basis of V, say{e1,...,en}. Thenρgives rise to a map
+G→GLn(k) which is a group homomorphism. Notice that if we change the b asis ofV
+then we get a different map for the same ρ. However they are equivalent as representation,
+i.e. differ by conjugation with respect to a fix matrix (the bas e change matrix).
+Example 2.14. The trivial representation is irreducible if and only if it i s one dimensional.
+Example 2.15. InthecaseofPermutationrepresentationthesubspace W=<(1,1,...,1)>
+andW′={(x1,...,x n)|/summationtextxi= 0}are two irreducible Sninvariant subspaces. In fact this
+representation is direct sum of these two and hence complete ly reducible.
+Example 2.16. One dimensional representation is always irreducible. If |G| ≥2 then the
+regular representation is not irreducible.
+Exercise 2.17. LetGbe a finite group. In the definition of a representation, let us
+not assume that the vector space Vis finite dimensional. Prove that there exists a finite
+dimensional G-invariant subspace of V.
+Hint : Fixv∈Vand takeWthe subspace generated by ρ(g)(v)∀g∈G.
+Exercise 2.18. A representation of dimension 1 is a map ρ:G→k∗. There are exactly
+two one dimensional representations of SnoverC. There are exactly none dimensional
+representations of cyclic group Z/nZoverC.
+Exercise 2.19. Prove that every finite group can be embedded inside symmetri c groupSn
+fro somenas well as linear groups GLmfor somem.
+Hint: Make use of the regular representation. This represen tation is also called “God
+given” representation. Later in the course we will see why it s so.
+Exercise 2.20. Is the above exercise true if we replace SnbyAnandGLmbySLm?
+Exercise 2.21. Prove that the cyclic group Z/pZhas a representation of dimension p−1
+overQ.
+Hint: Make use of the cyclotomic field extension Q(ζp) and consider the map left mul-
+tiplication by ζp. This exercise shows that representation theory is deeply c onnected to the
+Galois Theory of field extensions.10 2. REPRESENTATION OF A GROUP
+2.1. Commutator Subgroup and One dimensional representations
+LetGbe a finite group. Consider the set of elements {xyx−1y−1|x,y∈G}andG′the
+subgroup generated by this subset. This subgroup is called t hecommutator subgroup
+ofG. We list some of the properties of this subgroup as an exercis e here.
+Exercise 2.22. (1)G′is a normal subgroup.
+(2)G/G′is Abelian.
+(3)G′is smallest subgroup of Gsuch thatG/G′is Abelian.
+(4)G′= 1 if and only if Gis Abelian.
+(5) ForG=Sn,G′=An;G=Dn=< r,s|rn= 1 =s2,srs=r−1>we have
+G′=<r>∼=Z/nZandQ′
+8=Z(Q8).
+Let/hatwideGbe the set of all one-dimensional representations of GoverC, i.e., the set of all
+group homomorphisms from GtoC∗. Forχ1,χ2∈/hatwideGwe define multiplication by:
+(χ1χ2)(g) =χ1(g)χ2(g).
+Exercise 2.23. Prove that/hatwideGis an Abelian group.
+We observe that for a χ∈/hatwideGwe haveG′⊂ker(χ). Hence we can prove,
+Exercise 2.24. Show that/hatwideG∼=G/G′.
+Exercise 2.25. Calculate directly /hatwideGforG=Z/nZ,SnandDn.
+LetGbe a group. The group Gis called simpleifGhas no proper normal subgroup.
+Exercise 2.26. (1) LetGbe an Abelian simple group. Prove that Gis isomorphic to
+Z/pZwherepis a prime.
+(2) LetGbe a simple non-Abelian group. Then G=G′.CHAPTER 3
+Maschke’s Theorem
+In the last chapter we saw a representation can have possibly a subrepresentation. This
+motivates us to define:
+Definition 3.1 (Irreducible Representation) .A representation (ρ,V)ofGis called irre-
+ducible if it has no proper invariant subspace, i.e., only invariant subspaces are 0and
+V.
+Let (ρ,V) and (ρ′,V′) be two representations of Gover fieldk. We can define direct
+sumof these two representations ( ρ⊕ρ′,V⊕V′) as follows: ρ⊕ρ′:G→GL(V⊕V′)
+such that ( ρ⊕ρ′)(g)(v,v′) = (ρ(g)(v),ρ′(g)(v′)). In the matrix notation if we have two
+representations ρ:G→GLn(k) andρ′:G→GLm(k) thenρ⊕ρ′is given by
+g/ma√sto→/bracketleftbiggρ(g) 0
+0ρ′(g)/bracketrightbigg
+.
+This motivates us to look at those nice representations whic h can be obtained by taking
+direct sum of irreducible ones.
+Definition 3.2 (Completely Reducible) .A representation (ρ,V)is called completely re-
+ducible if it is a direct sum of irreducible ones. Equivalent ly ifV=W1⊕...⊕Wr, where
+eachWiisG-invariant irreducible representations.
+This brings us to the following questions:
+(1) Is it true that every representation is direct sum of irre ducible ones?
+(2) How many irreducible representations are there for Goverk?
+Theanswertothefirstquestionisaffirmativeinthecase Gisafinitegroupand char(k)∤|G|
+which is the Maschake’s theorem proved below. (It is also tru e for Compact Groups where
+it is called Peter-Weyl Theorem). The other exceptional cas e comes under the subject
+‘Modular Representation Theory’. We will answer the second question over the field of
+complex numbers (and possible over R) which is the character theory. For the theory over
+Qthe subject is called ‘rationality questions’ (refer to the book by Serre).
+Theorem 3.3 (Maschke’s Theorem) .Letkbe a field and Gbe a finite group. Suppose
+char(k)∤|G|, i.e.|G|is invertible in the field k. Let(ρ,V)be a finite dimensional repre-
+sentation of G. LetWbe aG-invariant subspace of V. Then there exists W′aG-invariant
+1112 3. MASCHKE’S THEOREM
+subspace such that V=W⊕W′. Conversely, if char(k)| |G|then there exists a representa-
+tion, namely the regular representation, and a proper G-invariant subspace which does not
+have aG-invariant compliment.
+Proposition 3.4 (Complete Reducibility) .Letkbe a field and Gbe a finite group with
+char(k)∤|G|. Then every finite dimensional representation of Gis completely reducible.
+Now we are going to prove the above results. We need to recall n otion of projection
+from ‘Linear Algebra’. Let Vbe a finite dimensional vector space over field k.
+Definition 3.5. An endomorphisms π:V→Vis called a projection if π2=π.
+LetW⊂Vbe a subspace. A subspace W′is called a compliment of WifV=W⊕W′.
+It is a simple exercise in ‘Linear Algebra’ to show that such a compliment always exists (see
+exercise below) and there could be many of them.
+Lemma 3.6. Letπbe an endomorphism. Then πis a projection if and only if there exists
+a decomposition V=W⊕W′such thatπ(W) = 0andπ(W′) =W′andπrestricted to W′
+is identity.
+Proof. Letπ:V→Vsuch thatπ(w,w′) =w′. Then clearly π2=π.
+Now suppose πis a projection. We claim that V=ker(π)⊕Im(π). Letx∈ker(π)∩
+Im(π). Then there exists y∈Vsuch thatπ(y) =xandx=π(y) =π2(y) =π(π(x)) =
+π(0) = 0. Hence ker(π)∩Im(π) = 0. Now let v∈V. Thenv= (v−π(v))+π(v) and we
+see thatπ(v)∈Im(π) andv−π(v)∈ker(π) sinceπ(v−π(v)) =π(v)−π2(v) = 0. Now let
+x∈Im(π), sayx=π(y). Thenπ(x) =π(π(y)) =π(y) =x. This shows that πrestricted
+toIm(π) is identity map. /square
+Remark 3.7. The reader familiar with ‘Canonical Form Theory’ will recog nise the follow-
+ing. The minimal polynomial of πisX(X−1) which is a product of distinct linear factors.
+Henceπis a diagonalizable linear transformation with eigen value s 0 and 1. Hence with
+respect to some basis the matrix of πis diag{0,...,0,1,...,1}. This will give another proof
+of the Lemma.
+Proof of the Maschke’s Theorem. Letρ:G→GL(V) be a representation. Let W
+be aG-invariant subspace of V. LetW0be a compliment, i.e., V=W0⊕W. We have to
+produce a compliment which is G-invariant. Let πbe a projection corresponding to this
+decomposition, i.e., π(W0) = 0 andπ(w) =wfor allw∈W. We define an endomorphism
+π′:V→Vby ‘averaging technique’ as follows:
+π′=1
+|G|/summationdisplay
+t∈Gρ(t)−1πρ(t).
+We claim that π′is a projection. We note that π′(V)⊂Wsinceπρ(t)(V)⊂WandW
+isG-invariant. In fact, π′(w) =wfor allw∈Wsinceπ′(w) =1
+|G|/summationtext
+t∈Gρ(t)−1πρ(t)(w) =3. MASCHKE’S THEOREM 13
+1
+|G|/summationtext
+t∈Gρ(t)−1π(ρ(t)(w)) =1
+|G|/summationtext
+t∈Gρ(t)−1(ρ(t)(w)) =w(note thatρ(t)(w)∈Wandπ
+takes it to itself). Let v∈V. Thenπ′(v)∈W. Henceπ′2(v) =π′(π′(v)) =π′(v) as we
+haveπ′(v)∈Wandπ′takes any element of Wto itself. Hence π′2=π′.
+Now we write decomposition of Vwith respect to π′, sayV=W′⊕WwhereW′=
+ker(π′) andIm(π′) =W. We claim that W′isG-invariant which will prove the theorem.
+For this we observe that π′is aG-invariant homomorphism, i.e., π′(ρ(g)(v)) =ρ(g)(π′(v))
+for allg∈Gandv∈V.
+π′(ρ(g)(v)) =1
+|G|/summationdisplay
+t∈Gρ(t)−1πρ(t)(ρ(g)(v))
+=1
+|G|/summationdisplay
+t∈Gρ(g)ρ(g)−1ρ(t)−1πρ(t)ρ(g)(v)
+=ρ(g)1
+|G|/summationdisplay
+t∈Gρ(tg)−1πρ(tg)(v)
+=ρ(g)(π′(v)).
+This helps us to verify that W′isG-invariant. Let w′∈W′. To show that ρ(g)(w′)∈W′.
+For this we note that π′(ρ(g)(w′)) =ρ(g)(π′(w)) =ρ(g)(0) = 0. This way we have produced
+G-invariant compliment of W.
+For the converse let char(k)| |G|. We take the regular representation V=k[G].
+ConsiderW=/braceleftBig/summationtext
+g∈Gαgg|/summationtextαg= 0/bracerightBig
+. We claim that WisG-invariant but it has no
+G-invariant compliment. /square
+Remark 3.8. In the proof of Maschke’s theorem one can start with a symmetr ic bilinear
+form and apply the trick of averaging to it. In that case the co mpliment will be the
+orthogonal subspace. Conceptually I like that proof better however it requires familiarity
+with bilinear form to be able to appreciate that proof. Later we will do that in some other
+context.
+Proof of the Proposition (Complete Reducibility). Letρ:G→GL(V)bearep-
+resentation. We use induction on the dimension of Vto prove this result. Let dim( V) = 1.
+It is easy to verify that one-dimensional representation is always irreducible. Let Vbe of
+dimensionn≥2. IfVis irreducible we have nothing to prove. So we may assume Vhas a
+G-invariant proper subspace, say Wwith 1≤dim(W)≤n−1. By Maschke’s Theorem we
+can writeV=W⊕W′whereW′is alsoG-invariant. But now dim( W) and dim(W′) bot
+are less than n. By induction hypothesis they can be written as direct sum of irreducible
+representations. This proves the proposition. /square
+Exercise 3.9. LetVbe a finite dimensional vector space. Let W⊂Vbe a subspace. Show
+that there exists a subspace W′such thatV=W⊕W′.
+Hint: Start with a basis of Wand extend it to a basis of V.14 3. MASCHKE’S THEOREM
+Exercise 3.10. Show that when V=W⊕W′the mapπ(w+w′) =wis a projection map.
+Verify that ker(π) =W′andIm(π) =W. The map πis called a projection on W. Notice
+that this map depends on the chosen compliment W′.
+Exercise 3.11. (1) Compliment of a subspace is not unique. Let us consider V=R2.
+Take a line Lpassing through the origin. It is a one dimensional subspace . Prove
+that any other line is a compliment.
+(2) LetWbe the one dimensional subspace x-axis. Choose the complime nt space as
+y-axis and write down the projection map. What if we chose the compliment as
+the linex=y?
+The exercises below show that Maschke’s theorem may not be tr ue if we don’t have
+finite group.
+Exercise 3.12. LetG=ZandV={(a1,a2,...)|ai∈R}be the sequence space (a
+vector space of infinite dimension). Define ρ(1)(a1,a2....) = (0,a1,a2,...) andρ(n) by
+composing ρ(1)n-times. Show that this is a representation of Z. Prove that it has no
+invariant subspace.
+Exercise 3.13. Consider a two dimensional representation of Ras follows:
+a/ma√sto→/parenleftbigg1a
+0 1/parenrightbigg
+.
+It leaves one dimensional subspace fixed generated by (1 ,0) but it has no complementary
+subspace. Hence this representation is not completely redu cible.
+Exercise 3.14. Letk=Z/pZ. Consider two dimensional representation of the cyclic gro up
+G=Z/pZof orderpoverkof characteristic pdefined as in the previous example. Find a
+subspace to show that Maschke’s theorem does not hold.
+Exercise 3.15. LetVbe an irreducible representation of G. LetWbe aG-invariant
+subspace of V. Show that either W= 0 orW=V.CHAPTER 4
+Schur’s Lemma
+Definition 4.1 (G-map).Let(ρ,V)and(ρ′,V′)be two representations of Gover fieldk. A
+linear map T:V→V′is called aG-map (between two representations) if it satisfies
+the following:
+ρ′(t)T=Tρ(t)∀t∈G.
+TheG-maps are also called intertwiners .
+Exercise 4.2. Prove that two representations of Gare equivalent if and only if there exists
+an invertible G-map.
+In the case representations are irreducible the G-maps are easy to decide. In the wake
+of Maschke’s Theorem considering irreducible representat ions are enough.
+Proposition 4.3 (Schur’s Lemma) .Let(ρ,V)and(ρ′,V′)be two irreducible representa-
+tions ofG(of dimension ≥1). LetT:V→V′be aG-map. Then either T= 0orTis
+an isomorphism. Moreover if Tis nonzero then Tis an isomorphism if and only if the two
+representations are equivalent.
+Proof. Letusconsiderthesubspace ker(T). Weclaimthatitisa G-invariantsubspace
+ofV. For this let us take v∈ker(T). ThenTρ(t)(v) =ρ′(t)T(v) = 0 implies ρ(t)(v)∈
+ker(T) for allt∈G. Now applying Maschke’s theorem on the irreducible represe ntationV
+we get either ker(T) = 0 orker(T) =V. In the case ker(T) =Vthe mapT= 0.
+Hence we may assume ker(T) = 0, i.e., Tis injective. Now we consider the subspace
+Im(T)⊂V′. We claim that it is also G-invariant. For this let y=T(x)∈Im(T). Then
+ρ′(t)(y) =ρ′(t)T(x) =Tρ(t)(x)∈Im(T) for anyt∈G. HenceIm(T) isG-invariant.
+Again by applying Maschke’s theorem on the irreducible repr esentationV′we get either
+Im(T) = 0 orIm(T) =V′. SinceTis injective Im(T)/ne}ationslash= 0 and hence Im(T) =V′. Which
+proves that in this case Tis an isomorphism. /square
+Exercise 4.4. LetVbe a vector space over CandT∈End(V) be a linear transformation.
+Show that there exists a one-dimensional subspace of Vleft invariant by T. Show by
+example that this need not be true if the field is Rinstead of C.
+Hint: Show that Thas an eigen-value and the corresponding eigen-vector will do the
+job.
+1516 4. SCHUR’S LEMMA
+Corollary 4.5. Let(ρ,V)be an irreducible representation of GoverC. LetT:V→Vbe
+aG-map. Then T=λ.Idfor someλ∈CandIdis the identity map on V.
+Proof. Letλbe an eigen-value of Tcorresponding to the eigen-vector v∈V, i.e.,
+T(v) =λv. Consider the subspace W=ker(T−λ.Id). We claim that Wis aG-invariant
+subspace. Since Tand scalar multiplications are G-maps so is T−λ. Hence the kernal is
+G-invariant (as we verified in the proof of Schur’s Lemma). One can do this directly also
+see the exercise below.
+SinceW/ne}ationslash= 0 and isG-invariant we can apply Maschke’s Theorem and get W=V. This
+givesT=λ.Id. /square
+Exercise 4.6. LetTandSbetwoG-maps. Showthat ker(T+S)isaG-invariantsubspace.CHAPTER 5
+Representation Theory of Finite Abelian Groups over C
+Throughout this chapter Gdenotes a finite Abelian group.
+Proposition 5.1. Letk=CandGbe a finite Abelian group. Let (ρ,V)be an irreducible
+representation of G. Then,dim(V) = 1.
+Proof. Proof is a simple application of the Schur’s Lemma. We will br eak it in step-
+by-step exercise below. /square
+Exercise 5.2. With notation as in the proposition,
+(1) forg∈Gconsiderρ(g):V→V. Provethat ρ(g)isaG-map. (Hint: ρ(g)(ρ(h)(v)) =
+ρ(gh)(v) =ρ(hg)(v) =ρ(h)(ρ(g)(v)).)
+(2) Prove that there exists λ(depending on g) inCsuch thatρ(g) =λ.Id. (Hint: Use
+the corollary of Schur’s Lemma.)
+(3) Prove that the map ρ:G→GL(V) maps every element gto a scalar map, i.e., it
+is given by ρ(g) =λg.Idwhereλg∈C.
+(4) Prove that the dimension of Vis 1. (Hint: Take any one dimensional subspace of
+V. It isG-invariant. Use Maschke’s theorem on it as Vis irreducible.)
+Proposition 5.3. Letk=CandGbe a finite Abelian group. Let ρ:G→GL(V)be a
+representation of dimension n. Prove that we can choose a basis of Vsuch thatρ(G)is
+contained in diagonal matrices.
+Proof. SinceVis a representation of finite group we can use Maschke’s theor em to
+write it as direct sum of G-invariant irreducible ones, say V=W1⊕...⊕Wr. Now using
+Schur’s lemma we conclude that dim( Wi) = 1 for all iand hence in turn we get r=n. By
+choosing a vector in each Wiwe get the required result. /square
+Corollary 5.4. LetGbe a finite group (possibly non-commutative). Let ρ:G→GL(V)be
+a representation. Let g∈G. Then there exists a basis of Vsuch that the matrix of ρ(g)is
+diagonal.
+Proof. ConsiderH=<g >⊂Gandρ:H→GL(V) the restriction map. Since His
+Abelian, using above proposition, we can simultaneously di agonalise elements of H. This
+proves the required result. /square
+1718 5. REPRESENTATION THEORY OF FINITE ABELIAN GROUPS OVER C
+Remark 5.5. In ‘Linear Algebra’ we prove the following result: A commuti ng set of diag-
+onalizable matrices over Ccan be simultaneously diaognalised. The proposition above is a
+version of the same result. We also give a warning about corol lary above that if we have
+a finite subgroup GofGLn(C) then we can take a conjugate of Gin such a way that a
+particular element becomes diagonal.
+Nowthisleavesusthequestiontodetermineallirreducible representationsofanAbelian
+groupG. For this we need to determine all group homomorphisms ρ:G→C∗.
+Exercise 5.6. LetGbe a finite group (not necessarily Abelian). Let χ:G→C∗be a
+group homomorphism. Prove that |χ(g)|= 1 and hence χ(g) is a root of unity.
+Let/hatwideGbe the set of all group homomorphisms from Gto the multiplicative group C∗.
+Let us also denote/hatwide/hatwideGfor the group homomorphisms from /hatwideGtoC∗.
+Exercise 5.7. With the notation as above,
+(1) Prove that for G=G1×G2we have/hatwideG∼=/hatwideG1×/hatwideG2.
+(2) LetG=Z/nZ. Prove that /hatwideG={χk|0≤k≤n−1}is a group generated by χ1
+of ordernwhereχ1(r) =e2πir
+nandχk=χk
+1. Hence/hatwiderZ/nZ∼=Z/nZ.
+(3) Use the structure theorem of finite Abelian groups to prov e thatG∼=/hatwideG.
+(4) Prove that Gis naturally isomorphic to/hatwide/hatwideGgiven byg/ma√sto→egwhereeg(χ) =χ(g) for
+allχ∈G.
+Exercise 5.8 (Fourier Transform) .Forf∈C[Z/nZ] ={f|f:Z/nZ→C}we define
+ˆf∈C[Z/nZ] by,
+ˆf(q) =1
+nn−1/summationdisplay
+k=0f(k)e(−kq) =1
+nn−1/summationdisplay
+k=0f(k)χq(−k).
+Show thatf(k) =/summationtextn−1
+q=0ˆf(q)e(kq) =/summationtextn−1
+q=0ˆf(q)χq(k) and1
+n/summationtextn−1
+k=0|f(k)|2=/summationtextn−1
+q=0|ˆf(q)|2.
+Exercise 5.9. OnC[Z/nZ] let us define an inner product by /an}bracketle{tf,f′/an}bracketri}ht=1
+n/summationtextn−1
+j=0f(j)¯f′(j)
+where bar denotes complex conjugation. Prove that {χk|0≤k≤n−1}form an orthonor-
+mal basis of C[Z/nZ]. Let
+f=/summationdisplay
+χ∈/hatwiderZ/nZcχχ.
+Calculate the coefficients using the inner product and compar e this with previous exercise.
+Now we show that the converse of the Proposition 5.1 is also tr ue.
+Theorem 5.10. LetGbe a finite group. Every irreducible representation of GoverCis1
+dimensional if and only if Gis an Abelian group.5.1. EXAMPLE OF REPRESENTATION OVER Q 19
+Proof. Let all irreducible representations of GoverCbe of dimension 1. Consider
+the regular representation ρ:G→GL(V) whereV=C[G]). We know that if |G| ≥2
+this representation is reducible and is an injective map (al so called faithful representation).
+Using Maschke’s theorem we can write Vas a direct sum of irreducible ones and they are
+given to be of dimension 1. Hence there exists a basis (check w hy?){v1,...,vn}ofV
+such that subspace generated by each basis vectors are invar iant. Hence ρ(G) consists of
+diagonal matrices with respect to this basis which is an Abel ian group. Hence G∼=ρ(G) is
+an Abelian group. /square
+5.1. Example of representation over Q
+5.1.1. An Irreducible Representation of Z/pZ.ConsiderG=Z/pZwherepis an
+odd prime. Let K=Q(ζ) whereζis a primitive pth root of unity. Let us consider the left
+multiplication map lζ:K→Kgiven byx/ma√sto→ζx. Consider the basis {1,ζ,ζ2,...,ζp−2}of
+K. Thenlζ(1) =ζ,lζ(ζi) =ζi+1for 1≤i≤p−1 andlζ(ζp−2) =ζp−1=−(1+ζ+ζ2+
+...+ζp−2) and the matrix of lζis:
+
+0 0 0 ···0−1
+1 0 0 ···0−1
+0 1 0 ···0−1
+............
+0 0 0 ···1−1
+
+The map 1 /ma√sto→lζdefines a representation ρ:G→GLp−1(Q). It is an irreducible represen-
+tation ofG. For this we note that K=Q(ζ) is a simple
+5.1.2. An Irreducible Representation of the Dihedral Group D2p.Notice that
+theGaloisgroupGal( K/Q)∼=Z/(p−1)Zcomeswithanaturalrepresentationon K. Letσ∈
+Gal(K/Q). Thenσis aQ-linear map which gives representation Gal( K/Q)∼=Z/(p−1)Z→
+GLp−1(Q). If we consider slightly different basis of K, namely, {ζ,ζ2,...,ζp−1}then the
+matrixofeach σisapermutationmatrix. InfactthiswayGal( K/Q)֒→Sp−1,thesymmetric
+group. However this representation is not irreducible (the elementζ+ζ2+···+ζp−1is
+invariant and gives decomposition).
+Notice that the Galois automomrphism σ:K→Kgiven byζ/ma√sto→ζ−1is an order 2
+element. We claim that σnormalizes lζ, i.e.,σlζσ=lζ−1. Sinceσlζσ(ζi) =σlζ(ζ−i) =
+σ(ζ1−i) =ζi−1=lζ−1(ζi). Let us denote the subgroup generated by σasHand the
+subgroup generated by lζbyK. ThenHKis a group of order 2 pwhereKis a normal
+subgroup of order p. HenceHK∼=D2p. This gives representation of order p−1 of
+D2p=/an}bracketle{tr,s|rp= 1 =s2,srs=r−1/an}bracketri}htgiven byD2p→GL(K) such that r/ma√sto→lζands/ma√sto→σ.
+Exercise 5.11. Prove that the representation constructed above are irredu cible.
+Exercise 5.12. Write down the above representation concretely for D6andD10.CHAPTER 6
+The Group Algebra k[G]
+LetRbe a ring (possibly non-commutative) with 1.
+Definition 6.1. A (left)moduleMover a ring Ris an Abelian group (M,+)with a map
+(called scalar multiplication) R×M→Msatisfying the following:
+(1) (r1+r2)m=r1m+r2mfor allr1,r2∈Randm∈M.
+(2)r(m1+m2) =rm1+rm2for allr∈Randm1,m2∈M.
+(3)r1(r2m) = (r1r2)mfor allr1,r2∈Randm∈M.
+(4) 1.m=mfor allm∈M.
+Notice that this definition is same as definition of a vector sp ace over a field. Analogous
+to definitions there we can define submodules and module homom orphisms.
+Example 6.2. IfR=k(a field) or D(a division ring) then the modules are nothing but
+vector spaces over R.
+Example 6.3. LetRbe a PID (a commutative ring such as Zor polynomial ring k[X]
+etc). Then R×R...×RandR/Ifor an ideal Iare modules over R. The structure theory
+of modules over PID states that any module is a direct sum of th ese kinds. However over
+a non-PID things could be more complicated.
+Example 6.4. In the non-commutative situation the simple/semisimple ri ngs are studied.
+A moduleMover a ring Ris called simpleif it has no proper submodules. And a
+moduleMis called semisimple if every submodule of Mhas a direct compliment. It is
+also equivalent to saying that Mis a direct sum of simple modules.
+A ringRis called semisimple if every module over it is semisimple. And a ring Ris
+calledsimple1 if it has no proper two-sided ideal.
+Exercise 6.5. (1) IsZa semisimple ring or simple ring?
+(2) When Z/nZa semisimple or simple ring?
+(3) Prove that the ring Mn(D) whereDis a division ring is a simple ring and the
+moduleDnthought as one of the columns of this ring is a module over this ring.
+2122 6. THE GROUP ALGEBRA k[G]
+All of the representation theory definitions can be very neat ly interpreted in module
+theory language. Given a field kand a group G, we form the ring
+k[G] =
+
+/summationdisplay
+g∈Gαgg|αg∈k
+
+
+called the group ring of G. We can also define k[G] ={f|f:G→k}. We define following
+operations on k[G]:/parenleftBig/summationtext
+g∈Gαgg/parenrightBig
++/parenleftBig/summationtext
+g∈Gβgg/parenrightBig
+=/summationtext
+g∈G(αg+βg)g,λ/parenleftBig/summationtext
+g∈Gαgg/parenrightBig
+=/summationtext
+g∈G(λαg)gand the multiplication by
+
+/summationdisplay
+g∈Gαgg
+.
+/summationdisplay
+g∈Gβgg
+=/summationdisplay
+g∈G/parenleftBigg/summationdisplay
+t∈Gαtβt−1g/parenrightBigg
+g=/summationdisplay
+g∈G
+/summationdisplay
+ts=g∈Gαtβs
+g.
+Withaboveoperations k[G]isanalgebracalledthe group algebra ofG(clearlyit’saring).
+A representation ( ρ,V) forGis equivalent to taking a k[G]-moduleV(see the exercises
+below).
+Exercise 6.6. Prove that k[G] is a ring as well as a vector space of dimension |G|. In fact
+it is ak-algebra.
+Exercise 6.7. Letkbe a field. Let Gbe a group. Then,
+(1) (ρ,V) is a representation of Gif and only if Vis ak[G]-module.
+(2)Wis aG-invariant subspace of Vif and only if Wis ak[G]-submodule of V.
+(3) The representations VandV′are equivalent if and only if Vis isomorphic to V′
+ask[G]-module.
+(4)Vis irreducible if and only if Vis a simple k[G]-module.
+(5)Vis completely reducible if and only if Vis a semisimple module.
+We can rewrite Maschke’s Theorem and Schur’s Lemma in module s language:
+Theorem 6.8 (Maschke’s Theorem) .LetGbe a finite group and ka field. The ring k[G]
+is semisimple if and only if char(k)∤|G|.
+Proposition 6.9 (Schur’s Lemma) .LetM,M′be two non-isomorphic simple Rmodule.
+ThenHomR(M,M′) ={0}. Moreover, HomR(M,M)is a division ring.
+Proof. Proof is left as an exercise. /square
+Exercise 6.10. LetDbe a finite dimensional division algebra over CthenD=C.
+Exercise 6.11. LetRbe a finite dimensional algebra over CandMa simple module over
+R. SupposeMis a finite dimensional over C. Then Hom R(M,M)∼=C.CHAPTER 7
+Constructing New Representations
+Here we will see how we can get new representations out of the k nown ones. All of the
+representations are considered over a field k.
+7.1. Subrepresentation and Sum of Representations
+Let (ρ,V) and (ρ′,V′) be two representations of the group G. We define ( ρ⊕ρ′,V⊕
+V′) by (ρ⊕ρ′)(t)(v,v′) = (t(v),t(v′)). This is called sum of the two representations. If
+we have a representation ( ρ,V) andWis aG-invariant subspace then we can define a
+subrepresentation (˜ ρ,W) by ˜ρ(t)(w) =ρ(t)(w).
+7.2. Adjoint Representation
+LetVbe a vector space over kwith a basis {e1,...,en}. A linear map f:V→kis
+called alinear functional . We denote V∗= Hom k(V,k), the set of all linear functionals.
+We define operations on V∗: (f1+f2)(v) =f1(v) +f2(v) and (λf)(v) =λf(v) and it
+becomes a vector space. The vector space V∗is called the dual space of V.
+Exercise 7.1. With the notation as above,
+(1) Show that V∗is a vector space with basis e∗
+iwheree∗
+i(ej) =δij. Hence it has same
+dimension as V. After fixing a basis of Vwe can obtain a basis this way of V∗
+which is called dual basis with respect t the given one.
+(2) Show that Vis naturally isomorphic to V∗∗.
+LetT:V→Vbe a linear map. We define a map T∗:V∗→V∗byT∗(f)(v) =f(T(v)).
+Exercise 7.2. Fix a basis of Vand consider the dual basis of V∗with respect to that. Let
+A= (aij) be the matrix of T. Show that the matrix of T∗with respect to the dual basis is
+tT, the transpose matrix.
+Letρ:G→GL(V) be a representation. We define the adjoint representation ( ρ∗,V∗)
+as follows: ρ∗:G→GL(V∗) whereρ∗(g) =ρ(g−1)∗. In the matrix form if we have a
+repesentation τ:G→GLn(k) thenτ∗:G→GLn(k) is given by τ∗(g) =tτ(g)−1.
+Exercise 7.3. With the notation as above,
+(1) Show that ρ∗is a representation of Gof same dimension as ρ.
+2324 7. CONSTRUCTING NEW REPRESENTATIONS
+(2) Fix a basis and suppose τis the matrix form of ρthen show that the matrix form
+ofρ∗isτ∗.
+(3) Prove that if ρis irreducible then show is ρ∗.
+Thelastexercisewillbecomeeasierinthecaseofcomplexre presentationsoncewedefine
+characters as we will have a simpler criterion to test when a r epresentation is irreducible.
+7.3. Tensor Product of two Representations
+LetVandV′be two vector spaces over k. First we define tensor product of two vector
+spaces. Tensorproductof VandV′isavectorspace V⊗V′={/summationtextr
+i=1vi⊗v′
+i|vi∈V,v′
+i∈V′}
+with following properties:
+(1) (/summationtextr
+i=1vi⊗v′
+i)+(/summationtexts
+i=1wi⊗w′
+i) =v1⊗v′
+1+···+vr⊗v′
+r+w1⊗w′
+1+···+ws⊗w′
+s.
+(2) (v1+v2)⊗v′=v1⊗v′+v2⊗v′andv⊗(v′
+1+v′
+2) =v⊗v′
+1+v⊗v′
+2.
+(3)λ(/summationtextr
+i=1vi⊗v′
+i) =/summationtextr
+i=1λvi⊗v′
+i=/summationtextr
+i=1vi⊗λv′
+i.
+Let{e1,...,en}be a basis of Vand{e′
+1,...,e′
+m}be that ofV′. Then{ei⊗e′
+j|1≤i≤
+n,1≤j≤m}is a basis of the vector space V⊗V′hence of dimension nm.
+Warning: Elements of V⊗V′are not exactly of kind v⊗v′but they are finite sum of
+these ones!
+Let (ρ,V) and (ρ′,V′) be two representations of the group G. We define ( ρ⊗ρ′,V⊗V′)
+by (ρ⊗ρ′)(t)(/summationtextv⊗v′) =/summationtext(ρ(t)(v)⊗ρ′(t)(v′)).
+Exercise 7.4. Choose a basis of VandV′. LetA= (aij) be the matrix of ρ(t) and
+B= (blm) be that of ρ′(t). What is the matrix of ( ρ⊗ρ′)(t)?
+Exercise 7.5. Let (ρ,V) be a representation of G. Let{e1,...,en}be a basis of V.
+(1) Consider an automorphism θofV⊗Vdefined byθ(ei⊗ej) =ej⊗ei. Show that
+θ2=θ.
+(2) Consider the subspace Sym2(V) ={z∈V⊗V|θ(z) =z}and∧2(V) ={z∈
+V⊗V|θ(z) =−z}. Prove that V⊗V=Sym2(V)⊕∧2(V). Write down a basis
+ofSym2(V) and∧2(V) and find its dimension.
+(3) Prove that V⊗V=Sym2(V)⊕∧2(V) is aG-invariant decomposition.
+If (ρ,V) is a representation of GthenV⊗n=V⊗ ··· ⊗V,SymnV,∧n(V) are also
+representations of G. This way starting from one representation we can get many re presen-
+tations. Though even if you start from an irreducible repres entation the above constructed
+representations need not be irreducible (for example V⊗Vgiven above) but often they con-
+tain other irreducible representations. Writing down dire ct sum decompositions of tensor
+representaion is a important topic of study. Often it happen s that we need much smaller
+number of representations (called fundamental representations ) of which tensor prod-
+ucts contain all irreducible representations.7.4. RESTRICTION OF A REPRESENTATION 25
+7.4. Restriction of a Representation
+Let (ρ,V) be a representation of the group G. LetHbe a subgroup. Then ( ρ,V) is a
+representation of Halso denoted as ( ρH,V). LetNbe a normal subgroup of G. Then any
+representation of G/Ngives rise to a representation of G. Moreover if the representation
+ofG/Nis irreducible then the representation of Gremains irreducible.CHAPTER 8
+Matrix Elements
+Now onwards we assume the field k=C. We also denote S1={α∈C| |α|= 1}. LetG
+beafinite group. Then C[G] ={f|f:G→C}isavector space of dimension |G|. Letf1,f2
+be two functions from GtoC, i.e.,f1,f2∈C[G]. We define a map ( ,):C[G]×C[G]→C
+as follows,
+(f1,f2) =1
+|G|/summationdisplay
+t∈Gf1(t)f2(t−1).
+Note that ( f1,f2) = (f2,f1).
+Letρ:G→GL(V) be a representation. We can choose a basis and get a map in mat rix
+formρ:G→GLn(C) wherenis the dimesnion of the representation. This means we have,
+g/ma√sto→
+a11(g)a12(g)···a1n(g)
+a21(g)a22(g)···a2n(g)
+............
+an1(g)an2(g)···ann(g)
+
+whereaij:G→C, i.e,aij∈C[G]. The maps aij’s are called matrix elements ofρ. Thus
+to a representation ρwe can associate a subspace WofC[G] spanned by aij. In what
+follows now we will explore relations between these subspac esWassociated to irreducible
+representations of a finite group G. Letρ1,...,ρ r,···be irreducible representations of Gof
+dimensionn1,···,nr,···respectively. We don’t know yet whether there are finitely ma ny
+irreducible representations which we will prove later. Let W1,···,Wr,···be associated
+subspaces of C[G] to the irreducible representations.
+Theorem 8.1. Let(ρ,V)and(ρ′,V′)be two irreducible representations of Gof dimension
+nandn′respectively. Let aijandbijbe the corresponding matrix elements with respect to
+fixed basis of VandV′. Then,
+(1) (ail,bmj) = 0for alli,j,l,m.
+(2) (ail,amj) =1
+nδijδlm=/braceleftbigg1
+nifi=jandl=m
+0otherwise.
+Proof. LetT:V→V′be a linear map. Define
+T0=1
+|G|/summationdisplay
+t∈Gρ(t)T(ρ′(t))−1.
+2728 8. MATRIX ELEMENTS
+ThenT0is aG-linear map. Using Lemma 4.3 we get T0= 0. Let us denote the matrix of
+Tbyxlm. Thenijth entry of T0is zero for all iandj, i.e.,
+1
+|G|/summationdisplay
+t∈G/summationdisplay
+l,mail(t)xlmbmj(t−1) = 0.
+SinceTis arbitrary linear transformation the entries xlmare arbitrary complex num-
+ber hence can be treated as indeterminate. Hence coefficients ofxlmare 0. This gives
+1
+|G|/summationtext
+t∈Gail(t)bmj(t−1) = 0 hence ( ail,bmj) = 0 for all i,j,l,m.
+Now let us consider T:V→V, a linear map. Again define T0=1
+|G|/summationtext
+t∈Gρ(t)T(ρ(t))−1
+whichisaG-map. FromCorollary4.5wegetthat T0=λ.Idwhereλ=1
+ntr(T) =1
+n/summationtext
+lxll=
+1
+n/summationtext
+l,mxlmδlmsincen.λ=tr(T0) =1
+|G|/summationtext
+t∈Gtr(T) =tr(T). Now using matrix elements
+we can write ijth term ofT0:
+1
+|G|/summationdisplay
+t∈G/summationdisplay
+lmail(t)xlmamj(t−1) =λδij=1
+n/summationdisplay
+l,mxlmδlmδij.
+AgainTis an arbitrary linear map so its matrix elements xlmcan be treated as indetermi-
+nate. Comparing coefficients of xlmwe get:
+1
+|G|/summationdisplay
+t∈Gail(t)amj(t−1) =1
+nδlmδij.
+Which gives ( ail,amj) =1
+nδlmδij. /square
+Exercise 8.2. Prove that, in both cases, T0is aG-map.
+Corollary 8.3. Iff∈ Wiandf′∈ Wjwithi/ne}ationslash=jthen(f,f′) = 0, i.e.,(Wi,Wj) = 0.CHAPTER 9
+Character Theory
+We have C[G], space of all complex valued functions on Gwhich is a vector space of
+dimension |G|. We define an inner product /an}bracketle{t,/an}bracketri}ht:C[G]×C[G]→Cby
+/an}bracketle{tf1,f2/an}bracketri}ht=1
+|G|/summationdisplay
+t∈Gf1(t)f2(t).
+Exercise 9.1. With the notation as above,
+(1) Prove that /an}bracketle{t,/an}bracketri}htis an inner product on C[G].
+(2) Iff1andf2take value in S1⊂Cthen/an}bracketle{tf1,f2/an}bracketri}ht= (f1,f2).
+Definition 9.2 (Character) .Let(ρ,V)be a representation of G. Thecharacter of (cor-
+responding to) representation ρis a mapχ:G→Cdefined byχ(t) =tr(ρ(t))wheretris
+the trace of corresponding matrix.
+Strictly speaking it is χρbut for the simplicity of notation we write χonly when it is clear
+which representation it corresponds to.
+Exercise 9.3. LetA,B∈GLn(k). Prove the following:
+(1)tr(AB) =tr(BA).
+(2)tr(A) =tr(BAB−1).
+Exercise 9.4. Usually we define trace of a matrix. Show that in above definiti on character
+is a well defined function. That is, prove that χ(t) doesn’t change if we choose different basis
+and calculate trace of ρ(t). This is another way to say that trace is invariant of conjug acy
+classes ofGLn(k).
+Exercise 9.5. Ifρandρ′are two isomorphic representations, i.e., G-equivalent, then the
+corresponding characters are same. The converse of this sta tement is also true which we
+will prove later.
+Proposition 9.6. Ifρis a representation of dimension nandχis the corresponding char-
+acter then,
+(1)χ(1) =nis the dimension of the representation.
+(2)χ(t−1) =χ(t)for allt∈Gwhere bar denotes complex conjugation.
+(3)χ(tst−1) =χ(s)for allt,s∈G, i.e., character is constant on the conjugacy classes
+ofG.
+2930 9. CHARACTER THEORY
+(4)Forf∈C[G]we have (f,χ) =/an}bracketle{tf,χ/an}bracketri}ht.
+Proof. For the proof of part two we use Corollary 5.4 to calculate χ(t). From that
+corollary every ρ(t) can be diagonalised (at a time not simultaneously which is e nough for
+ourpurposes), saydiag {ω1,···,ωn}. Sincetisoffiniteorder, say d, wehaveρ(t)d= 1. That
+is eachωd
+j= 1 means the diagonal elements are dth root of unity. We know that roots of
+unity satisfy ω−1
+j=ωj. Henceχ(t−1) =tr(ρ(t)−1) =ω−1
+1+···+ω−1
+n=ω1+···+ωn=χ(t).
+We can also prove this result by upper triangulation theorem , i.e., every matrix ρ(t)
+can be conjugated to an upper triangular matrix. And the fact that trace is same for the
+conjugate matrices. /square
+Definition 9.7 (Class Function) .A function f:G→Cis called a class function iffis
+constant on the conjugacy classes of G. We denote the set of class functions on GbyH.
+Exercise 9.8. With the notation as above,
+(1) Prove that His a subspace of C[G].
+(2) The dimension of His the number of conjugacy classes of G.
+(3) Letcg=/summationtext
+x∈Gxgx−1∈C[G]. The center of the group algebra C[G] is spanned
+bycg.
+(4) Letχbe a character corresponding to some representation of G. Then,χ∈ H.
+Proposition 9.9. Let(ρ,V)and(ρ′,V′)be two representations of the group Gandχ,χ′
+be the corresponding characters. Then,
+(1)The character of the sum of two representations is equal to th e sum of characters,
+i.e.,χρ⊕ρ′=χ+χ′.
+(2)The character of the tensor product of two representations i s the product of two
+characters, i.e., χρ⊗ρ′=χχ′.
+Proof. Proof is a simple exercise involving matrices. /square
+This way we can define sum and product of characters which is ag ain a character.
+Exercise 9.10. Let (ρ,V) be a representation of the group G. ThenV⊗Vis also a
+representation of G. We look at the map θ:V⊗V−→V⊗Vdefined byθ(ei⊗ej) =ej⊗ei.
+This gives rise to the decomposition V⊗V=Sym2(V)⊕∧2(V) which is a G-decomposition.
+Calculate the characters of Sym2(V) and∧2(V).
+Exercise 9.11. Letχbe the character of a representation ρ. Show that the character of
+the adjoint representation ρ∗is given by χ.CHAPTER 10
+Orthogonality Relations
+LetGbe a finite group. Let W1,W2,...,W h,...be irreducible representations of G
+of dimension n1,n2,...,n h,...overC. Letχ1,χ2,...,χ h,...are corresponding charac-
+ters, called irreducible characters of G. We will fix this notation now onwards. We
+will prove that the number of irreducible characters and hen ce the number of irreducible
+representations if finite and equal to the number of conjugac y classes.
+In the last chapter we introduced an inner product /an}bracketle{t,/an}bracketri}htonC[G]. We also observed that
+character of any representation belongs to H, the space of class functions.
+Theorem 10.1. The set of irreducible characters {χ1,χ2,...}form an orthonormal set of
+(C[G],/an}bracketle{t,/an}bracketri}ht). That is,
+(1)Ifχis a character of an irreducible representation then /an}bracketle{tχ,χ/an}bracketri}ht= 1.
+(2)Ifχandχ′are two irreducible characters of non-isomorphic represen tations then
+/an}bracketle{tχ,χ′/an}bracketri}ht= 0.
+Proof. Letρandρ′be non-isomorphic irreducible representations and ( aij) and (bij)
+be the corresponding matrix elements. Then χ(g) =/summationtext
+iaii(g) andχ′(g) =/summationtext
+jbjj(g). Then
+/an}bracketle{tχ,χ′/an}bracketri}ht= (χ,χ′) = (/summationtext
+iaii,/summationtext
+jbjj) =/summationtext
+i,j(aii,bjj) = 0 from Theorem 8.1 and Proposi-
+tion 9.6 part 4. Using the similar argument we get /an}bracketle{tχ,χ/an}bracketri}ht= (χ,χ) = (/summationtext
+iaii,/summationtext
+jbjj) =/summationtext
+i(aii,bii) =/summationtextn
+i=11
+n= 1 wherenis the dimension of the representation ρ. /square
+Corollary 10.2. The number of irreducible characters are finite.
+Proof. Since the irreducible characters form an orthonormal set th ey are linearly in-
+dependent. Hence their number has to be less than the dimensi on ofC[G] which is |G|,
+hence finite. /square
+Once we prove that two representations are isomorphic if and only if their characters are
+same this corollary will also give that there are finitely man y non-isomorphic irreducible
+representations.
+We are going to use above results to analyse general represen tation ofGand identify
+its irreducible components.
+Theorem 10.3. Let(ρ,V)be a representation of Gwith character χ. LetVdecomposes
+into a direct sum of irreducible representations:
+V=V1⊕···⊕Vm.
+3132 10. ORTHOGONALITY RELATIONS
+Then the number of Viisomorphic to Wj(a fixed irreducible representation) is equal to the
+scalar product /an}bracketle{tχ,χj/an}bracketri}ht.
+Proof. Letφ1,...,φ mbe the characters of V1,...,V m. Thenχ=φ1+···+φm. Also
+/an}bracketle{tχ,χj/an}bracketri}ht=/summationtext
+i/an}bracketle{tφi,χj/an}bracketri}ht=/summationtext
+φi=χj/an}bracketle{tφi,χj/an}bracketri}ht= the number of Viisomorphic to Wj. /square
+Corollary 10.4. With the notation as above,
+(1)The number of Viisomorphic to a fixed Wjdoes not depend on the chosen decom-
+position.
+(2)Let(ρ,V)and(ρ′,V′)be two representations with characters χandχ′respectively.
+ThenV∼=V′if and only if χ=χ′.
+Proof. The proof of part 1 is clear from the theorem above. For the pro of of part 2 it
+is clear that if V∼=V′we getχ=χ′. Now suppose χ=χ′. LetV∼=Wn1
+1⊕···⊕Wnh
+hand
+V′∼=Wm1
+1⊕···⊕Wmh
+hbe the decomposition as direct sum of irreducible represent ations
+(we can do this using Maschke’s Theorem) where ni,mj≥0. Suppose χ1,χ2,...,χ hbe
+the irreducible characters of W1,...,W h. Thenχ=n1χ1+···+nhχhandχ′=m1χ1+
+···+mhχh. However as χi’s form an orthonormal set they are linearly independent. He nce
+χ=χ′impliesni=mifor alli. HenceV∼=V′. /square
+From this corollary it follows that the number of irreducibl e representations are same
+as the number of irreducible characters which is less then or equal to |G|. In fact later we
+will prove that this number is equal to the number of conjugac y classes. The above analysis
+also helps to identify whether a representation is irreduci ble by use of the following:
+Theorem 10.5 (Irreducibility Criteria) .Letχbe the character of a representation (ρ,V).
+Then/an}bracketle{tχ,χ/an}bracketri}htis a positive integer and /an}bracketle{tχ,χ/an}bracketri}ht= 1if and only if Vis irreducible.
+Proof. LetV∼=Wn1
+1⊕···⊕Wnh
+h. Thenχ=n1χ1+···+nhχhand
+/an}bracketle{tχ,χ/an}bracketri}ht=/an}bracketle{tn1χ1+···+nhχh,n1χ1+···+nhχh/an}bracketri}ht=/summationdisplay
+in2
+i.
+Hence/an}bracketle{tχ,χ/an}bracketri}ht= 1 if and only if one of the ni= 1, i.e,χ=χifor somei. Hence the result. /square
+Exercise 10.6. Letχbeanirreduciblecharacter. Showthat χisso. Hencearepresentation
+ρis irreducible if and only if ρ∗is so.
+Exercise 10.7. Use the above criteria to show that the “Permutations repres entation”,
+“Regular Representation” (defined in the second chater) are not irreducible if |G|>1.
+Exercise 10.8. Letρbe an irreducible representation and τbe an 1-dimensional represen-
+tation. Show that ρ⊗τis irreducible.CHAPTER 11
+Main Theorem of Character Theory
+11.1. Regular Representation
+LetGbe a finite group and χ1,...,χ hbe the irreducible characters of dimension
+n1,...,n hrespectively. Let Lbe the left regular representation of Gwith corresponding
+characterl.
+Exercise 11.1. The character lof the regular representation is given by l(1) =|G|and
+l(t) = 0 for all 1 /ne}ationslash=t∈G.
+Theorem 11.2. Every irreducible representation WiofGis contained in the regular rep-
+resentation with multiplicity equal to the dimension of Wiwhich isni. Hencel=n1χ1+
+···+nhχh.
+Proof. In the view of Theorem 10.3 the number of times χiis contained in the regular
+representation is given by /an}bracketle{tl,χi/an}bracketri}ht=1
+|G|l(1)χi(1) =1
+|G||G|ni=ni. /square
+If we are asked to construct irreducible representations of a finite group we don’t know
+wheretolookforthem. Thistheoremensuresanaturalplace, namelytheregularrepresenta-
+tion where we could find them. For this reason it is also called “God Given Representation”.
+Exercise 11.3. With the notation as above,
+(1) The degree nisatisfy/summationtexth
+i=1n2
+i=|G|.
+(2) If 1/ne}ationslash=s∈Gwe have/summationtexth
+i=1niχi(s) = 0.
+Hints: This follows from the formula for las in the theorem by evaluating at s= 1 and
+s/ne}ationslash= 1.
+These relations among characters will be useful to determin e character table of the
+groupG.
+11.2. The Number of Irreducible Representations
+Now we will prove the main theorem of the character theory.
+Theorem 11.4 (Main Theorem) .The number of irreducible representations of G(upto
+isomorphism) is equal to the number of conjugacy classes of G.
+3334 11. MAIN THEOREM OF CHARACTER THEORY
+The proof of this theorem will follow from the following prop osition. We know that the
+irreducible characters χ1,...,χ h∈ Hand they form an orthonormal set (see Theorem 10.1).
+We prove that, in fact, they form an orthonormal basis of Hand generate as an algebra
+whole of C[G].
+Proposition 11.5. The irreducible characters of Gform an orthonormal basis of H, the
+space of class functions.
+To prove this we will make use of the following:
+Lemma 11.6. Letf∈ Hbe a class function on G. Let(ρ,V)be an irreducible representa-
+tion ofGof degreenwith character χ. Let us define ρf=/summationtext
+t∈Gf(t)ρ(t)∈End(V). Then,
+ρf=λ.Idwhereλ=|G|
+n/an}bracketle{tf,χ/an}bracketri}ht.
+Proof. We claim that ρfis aG-map and use Schur’s Lemma to prove the result. For
+anyg∈Gwe have,
+ρ(g)ρfρ(g−1) =/summationdisplay
+t∈Gf(t)ρ(g)ρ(t)ρ(g−1) =/summationdisplay
+t∈Gf(t)ρ(gtg−1) =/summationdisplay
+s∈Gf(g−1sg)ρ(s) =ρf.
+Henceρfis aG-map. From Schur’s Lemma (see 4.5) we get that ρf=λ.Idfor someλ∈C.
+Now we calculate trace of both side:
+λ.n=tr(ρf) =/summationdisplay
+t∈Gf(t)tr(ρ(t)) =/summationdisplay
+t∈Gf(t)χ(t) =|G|1
+|G|/summationdisplay
+t∈Gf(t)χ(t−1) =|G|/an}bracketle{tf,χ/an}bracketri}ht.
+Hence we get λ=|G|
+n/an}bracketle{tf,χ/an}bracketri}ht. /square
+Proof of the Proposition 11.5. We need to prove that irreducible characters span
+Has being orthonormal they are already linearly independent . Letf∈ H. Supposefis
+orthogonal to each irreducible χi, i.e.,/an}bracketle{tf,χi/an}bracketri}ht= 0 for alli. Then we will prove f= 0.
+Since/an}bracketle{tf,χi/an}bracketri}ht= 0 it implies /an}bracketle{tf,χi/an}bracketri}ht= 0 for alli. Letρibe the corresponding irreducible
+representation. Then from previous lemma ρif=/parenleftBig
+|G|
+n/an}bracketle{tf,χi/an}bracketri}ht/parenrightBig
+.Id= 0 for alli. Now letρbe
+any representation of G. From Maschke’s Theorem it is direct sum of ρi’s, the irreducible
+ones. Hence ρf= 0 for any ρ.
+In particular we can take the regular representation L:G→GL(C[G]) forρand we
+getLf= 0. Hence Lf(e1) = 0 implies/summationtext
+t∈Gf(t)L(t)(e1) = 0, i.e.,/summationtext
+t∈Gf(t)et= 0 hence
+f(t) = 0 for all t∈G. Hencef= 0. /square
+Proposition 11.7. Lets∈Gand letrsbe the number of elements in the conjugacy class
+ofs. Let{χ1,...,χ h}be irreducible representations of G. Then,
+(1)We have/summationtexth
+i=1χi(s)χi(s) =|G|
+rs.
+(2)Fort∈Gnot conjugate to s, we have/summationtexth
+i=1χi(s)χi(t) = 0.11.2. THE NUMBER OF IRREDUCIBLE REPRESENTATIONS 35
+Proof. Let us define a class function ftbyft(t) = 1 andft(g) = 0 ifgis not conjugate
+tot. Since irreducible characters span H(see 11.5) we can write ft=/summationtexth
+i=1λiχiwhere
+λi=/an}bracketle{tft,χi/an}bracketri}ht=1
+|G|rtχi(t). Henceft(s) =rt
+|G|/summationtexth
+i=1χi(t)χi(s) for anys∈G. This gives the
+required result by taking sconjugate to tand not conjugate to t. /square
+Exercise 11.8. LetGbe a finite group. The elements s,t∈Gare conjugate if and only if
+χ(s) =χ(t) for all (irreducible) characters χofG.
+Exercise 11.9. A normal subgroup of Gis disjoint union of conjugacy classes.
+Exercise 11.10. Any normal subgroup can be obtained by look at the intersecti on of kernel
+=g∈G|χ(g) = 1 of some characters.CHAPTER 12
+Examples
+LetGbe a finite group. Let ρ1,...,ρ hbe irreducible representations of GoverCwith
+corresponding characters χ1,...,χ h. We know that the number his equal to the number
+of conjugacy classes in G. We can make use of this information about Gand make a
+character table ofG. The character table is a matrix of size h×hof which rows are
+labeled as characters and columns as conjugacy classes.
+r1= 1r2···rh
+g1=eg2···gh
+χ1n1= 11···1
+χ2n2
+......
+χhnh
+whereg1,g2,...denote representative of the conjugacy class and ridenotes the number
+of elements in the conjugacy class of gi. The following proposition summarizes the results
+proved about characters. We also recal the inner product on C[G] defined by /an}bracketle{tf1,f2/an}bracketri}ht=
+1
+|G|/summationtext
+t∈Gf1(t)f2(t).
+Proposition 12.1. With the notation as above we have,
+(1)The number of conjugacy classes is same as the number of irred ucible characters
+which is same as the number of non-isomorphic irreducible re presentations.
+(2)Two representations are isomorphic if and only if their char acters are equal.
+(3)A representation ρwith character χis irreducible if and only if /an}bracketle{tχ,χ/an}bracketri}ht= 1.
+(4)|G|=n2
+1+n2
+2+···+n2
+hwheren1= 1.
+(5)The characters form orthonormal basis of H, i.e.,
+/summationdisplay
+t∈Gχi(t)χj(t) =/summationdisplay
+glrlχi(gl)χj(gl) =δij|G|.
+That is, the rows of the charcater table are orthonormal.
+3738 12. EXAMPLES
+(6)The columns of the character table also form an orthogonal se t, i.e.,
+h/summationdisplay
+i=1χi(gl)χi(gl) =|G|
+rl
+and
+h/summationdisplay
+i=1χi(gl)χi(gm) = 0
+whereglandgmare representative of different conjugacy class.
+(7)The character table matrix is an invertible matrix.
+(8)The degree of irreducible representations divide the order of th group, i.e., ni| |G|.
+The proof of the last statement will be done later.
+Warning : If character table of two groups are same that does not imply t hat the
+groups are isomorphic. Look at the character tables of Q8andD4. In fact, all groups of
+orderp3have same character table (find a reference for this).
+12.1. Groups having Large Abelian Subgroups
+First we will give another proof of the Theorem 5.10 using Cha racter Theory.
+Theorem 12.2. LetGbe a finite group. Then Gis Abelian if and only if all irreducible
+representations are of dimension 1, i.e.,ni= 1for alli.
+Proof. With the notation as above let Gbe Abelian. We have
+|G|=n2
+1+···+n2
+h
+whereh=|G|. Hence the only solution to the equation is ni= 1 for all i. Now suppose
+ni= 1 for all i. Then the above equation implies h=|G|. Hence each conjugacy class is
+has size 1 and the group is Abelian. /square
+Proposition 12.3. LetGbe a group and Abe an Abelian subgroup. Then ni≤|G|
+|A|for all
+i.
+Proof. Letρ:G→GL(V) be an irreducible representation. We can restrict ρto
+Aand denote it by ρA:A→GL(V) which may not be irreducible. Let Wan invariant
+subspace of VforA. The above theorem implies dim( W) = 1. SayW=<v>wherev/ne}ationslash= 0.
+ConsiderV′=<{ρ(g)v|g∈G}>⊂V. ClearlyV′isG-invariant and as Vis irreducible
+V′=V. Notice that ρ(ga)v=λρ(g)v, i.e.,ρ(ga)vandρ(g)vare linearly dependent. Hence
+V′=<{ρ(g1)v,...,ρ(gm)v}>wheregiare representatives of the coset giAinG/A. This
+implies dim( V)≤m=|G|
+|A|. /square
+Corollary 12.4. LetG=Dnbe the dihedral group with 2nelements. Any irreducible
+representation of Dnhas dimension 1or2.12.2. CHARACTER TABLE OF SOME GROUPS 39
+12.2. Character Table of Some groups
+Example 12.5 (Cyclic Group) .LetG=Z/nZ. All representations are one dimensional
+hence give character. The characters are χ1,...,χ ngiven byχr(s) =e2πi
+nrsfor 0≤s≤n.
+Example 12.6 (S3).
+S3={1,(12),(23),(13),(123),(132)}
+We already know the one dimensional representations of S3which are trivial representation
+and the sign representation. The characters for these repre sentations are itself. Now we
+use the formula 6 = |G|=n2
+1+n2
+2+n2
+3= 1+1+n2
+3givesn3= 2. Now we use /an}bracketle{tχ3,χ1/an}bracketri}ht=
+1
+6{1.2.1+3.a.1+2.b.1}= 0 and /an}bracketle{tχ3,χ2/an}bracketri}ht=1
+6{1.2.1+3.a.(−1)+2.b.1 = 0}. Hence solving
+the above equations 3 a+2b=−2 and−3a+2b=−2 givesa= 0 andb=−1.
+r1= 1r2= 3r3= 2
+g1= 1g2= (12)g3= (123)
+χ1n1= 1 1 1
+χ2n2= 1 −1 1
+χ3n3= 2a= 0b=−1
+Example 12.7 (Q8).
+Q8={1,−1,i,−i,j,−j,k,−k}
+The commutator subgroup of Q8={1,−1}andQ8/{±1}∼=Z/2Z×Z/2Z. Hence it has
+4 one dimensional representations which are lifted from Z/2Z×Z/2Z. Notice that the
+firstZ/2Zcomponent is the image of iand second one of j. Now we use 8 = |G|=
+n2
+1+n2
+2+n2
+3+n2
+4+n2
+5= 1+1+1+1+ n2
+5givesn5= 2. Again using orthogonality of χ5
+with other known χi’s we get following equations:
+1.2.1+1.a.1+2.b.1+2.c.1+2.d.1 = 0
+1.2.1+1.a.1+2.b.1+2.c.(−1)+2.d.(−1) = 0
+1.2.1+1.a.1+2.b.(−1)+2.c.1+2.d.(−1) = 0
+1.2.1+1.a.1+2.b.(−1)+2.c.(−1)+2.d.1 = 0
+This gives the solution a=−2,b= 0,c= 0 andd= 0.
+r1= 1r2= 1r3= 2r4= 2r5= 2
+g1= 1g2=−1g3=ig4=jg5=k
+χ1n1= 1 1 111
+χ2n2= 1 1 1−1−1
+χ3n3= 1 1−11−1
+χ4n4= 1 1−1−11
+χ5n5= 2a=−2b= 0c= 0d= 040 12. EXAMPLES
+Example 12.8 (D4).
+D4={r,s|r4= 1 =s2,rs=sr−1}
+The commutator subgroup of D4is{1,r2}=Z(D4). AndD4/Z(D4)∼=Z/2Z×Z/2Zhence
+there are 4 one dimensional representations as in the case of Q8. We can also compute rest
+of it as we did in Q8. We also observe that the character table is same as Q8.
+r1= 1r2= 1r3= 2r4= 2r5= 2
+g1= 1g2=r2g3=rg4=sg5=sr
+χ1n1= 1 1 11 1
+χ2n2= 1 1 1−1−1
+χ3n3= 1 1−11−1
+χ4n4= 1 1−1−11
+χ5n5= 2a=−2b= 0c= 0d= 0
+Example 12.9 (A4).
+A4={1,(12)(34),(13)(24),(14)(23),(123),(132),(124),(142),(134),(143),(234),(243)}
+We note that the 3 cycles are not conjugate to their inverses. Here we have
+H={1,(12)(34),(13)(24),(14)(23)}∼=Z/2Z×Z/2Z
+a normal subgroup of A4andA4/H∼=Z/3Z. This way the 3 one dimensional irreducible
+representations of Z/3Zlift toA4as we have maps A4→A4/H∼=Z/3Z→GL1(C) given
+byωwhereω3= 1. We use 12+12+12+n2
+4= 12 =|A4|to getn4= 3. Now we take inner
+product of χ4with others and get the equations:
+1.3.1+3.a.1+4.b.1+4.c.1 = 0
+1.3.1+3.a.1+4.b.ω+4.c.ω2= 0
+1.3.1+3.a.1+4.b.ω2+4.c.ω= 0
+This gives us the character table.
+r1= 1r2= 3 r3= 4r4= 4
+g1= 1g2= (12)(34) g3= (123)g4= (132)
+χ1n1= 1 1 1 1
+χ2n2= 1 1 ω ω2
+χ3n3= 1 1 ω2ω
+χ4n4= 3a=−1b= 0c= 0
+Example 12.10 (S4).The groupS4has 2 one dimensional representations given by trivial
+and sign. We recall that the permutation representation (Ex ample 2.15) gives rise to the
+subspaceV={(x1,x2,...,x n)∈Cn|x1+x2+···+xn= 0}which is an n−1 dimensional12.2. CHARACTER TABLE OF SOME GROUPS 41
+irreducible representation of Sn. We will make use of this to get a 3 dimensional representa-
+tion ofS4and corresponding character χ3. A basis of the space Vis{e1−e2,e2−e3,e3−e4}
+and the action is given by:
+(12) : ( e1−e2)/ma√sto→e2−e1=−(e1−e2)
+(e2−e3)/ma√sto→e1−e3= (e1−e2)+(e2−e3)
+(e3−e4)/ma√sto→(e3−e4)
+So the matrix is
+−1 1 0
+0 1 0
+0 0 1
+andχ3((12)) = 1. The action of (12)(34) is e1−e2/ma√sto→
+e2−e1=−(e1−e2),e2−e3/ma√sto→e1−e4= (e1−e2) + (e2−e3) + (e3−e4) ande3−
+e4/ma√sto→e4−e3=−(e3−e4) and the matrix is
+−1 1 0
+0 1 0
+0 1−1
+. Soχ3((12)(34)) = −1. The
+action of (123) is e1−e2/ma√sto→e2−e3,e2−e3/ma√sto→e3−e1=−(e1−e2)−(e2−e3) and
+e3−e4/ma√sto→e1−e4= (e1−e2) + (e2−e3) + (e3−e4). So the matrix is
+0−1 1
+1−1 1
+0 0 1
+and
+χ3((123)) = 0. And the action of (1234) is e1−e2/ma√sto→e2−e3,e2−e3/ma√sto→e3−e4and
+e3−e4/ma√sto→e4−e1=−(e1−e2)−(e2−e3)−(e3−e4). So the matrix is
+0 0−1
+1 0−1
+0 1−1
+and
+χ3((1234)) = −1. This gives χ3. We can get another character χ4=χ3.χ2corresponding
+to the representation Perm⊗sgn. We can check that this is different from others so
+corresponds to a new representation and also irreducible as /an}bracketle{tχ4,χ4/an}bracketri}ht= 1.
+To findχ5we can use orthogonality relations and get equations.
+r1= 1r2= 6r3= 3 r4= 8r5= 6
+g1= 1g2= (12)g3= (12)(34) g4= (123)g5= (1234)
+χ1n1= 1 1 1 1 1
+χ2n2= 1 −1 1 1 −1
+χ3n3= 3 1 −1 0 −1
+χ4n4= 3 −1 −1 0 1
+χ5n5= 2a= 0 b= 2 c=−1d= 0
+Example 12.11. In the above example of S4we see that the representations ρ3andρ4are
+adjoint of each other. This we can see by looking at their char acters. This gives an example
+of representation which is not isomorphic to its adjoint.42 12. EXAMPLES
+Example 12.12 (Dn,neven).
+Dn={a,b|an= 1 =b2,ab=ba−1}
+and the conjugacy classes are {1},{an
+2},{aj,a−j}(1≤j≤n
+2−1),{ajb|j even}and
+{rjb|jodd}. The subgroup generatd by a2is normal and hence there are 4 one dimensional
+representations. Rest of them are two dimensional (refer to Corollary 12.4) representations
+defined in the Example 2.7. Using them we can make character ta ble.
+Example 12.13 (Dn,nodd).
+Dn={a,b|an= 1 =b2,ab=ba−1}
+and the conjugacy classes are {1},{aj,a−j}(1≤j≤n−1
+2),{ajb|0≤j≤n−1}. The
+subgroup generatd by ais normal and hence there are 2 one dimensional representati ons.
+Rest of them are two dimensional (refer to Corollary 12.4) re presentations defined in the
+Example 2.7. Using them we can make character table.CHAPTER 13
+Characters of Index 2Subgroups
+In this chapter we will see how we can use characters of a group Gto get characters of
+its index 2 subgroups. We apply this for S5and its subgroup A5.
+13.1. The Representation V⊗V
+LetGbe a finite group. Suppose that ( ρ,V) and (ρ′,V′) are representations of G. Then
+we can get a new representation of Gfrom these representations defined as follows (recall
+from section 7.3):
+ρ⊗ρ′:G→GL(V⊗V′)
+(ρ⊗ρ′)(g)(v⊗v′) =ρ(g)(v)⊗ρ′(g)(v′)
+Now suppose ρandρ′are representations over Candχandχ′are the corresponding
+characters, then the character of ρ⊗ρ′isχχ′given by (χχ′)(g) =χ(g)χ′(g). However even
+ifρandρ′are irreducible ρ⊗ρ′need not be irreducible.
+Let (ρ,V) be a representation of G. We consider ( ρ⊗ρ,V⊗V). As defined above it
+is a representation of Gwith character χ2whereχ2(g) =χ(g)2. Recall from section 7.3 it
+can be decomposed as
+V⊗V=Sym2(V)⊕Λ2(V).
+We prove below that both Sym2(V) and Λ2(V) areG-spaces.
+Theorem 13.1. Let(ρ,V)be a representation of G. ThenV⊗V=Sym2(V)⊕Λ2(V)
+where each of the subspaces Sym2(V)andΛ2(V)areG-invariant.
+Proof. Let{v1,...,vn}be a basis of V. Then{vi⊗vj|1≤i,j≤n}is a basis of
+V⊗Vwith dimension n2. Consider the linear map θdefined on the basis of V⊗Vby
+θ(vi⊗vj) =vj⊗viand extended linearly. We observe that θcan also be defined without
+the help of any basis by θ(v⊗w) =w⊗vsince ifv=/summationtextaiviandw=/summationtextbjvj, then
+θ(v⊗w) =θ(/summationtextaivi⊗/summationtextbjvj) =/summationtextaibjθ(vi⊗vj) =/summationtextaibj(vj⊗vi) =/summationtext(bjvj⊗aivi) =
+(w⊗v). Observe that θ2= 1 and we take the subspaces of V⊗Vcorresponding to eigen
+values 1 and −1:
+Sym2(V) ={x∈V⊗V|θ(x) =x}
+Λ2(V) ={x∈V⊗V|θ(x) =−x}
+4344 13. CHARACTERS OF INDEX 2 SUBGROUPS
+Note that {(vi⊗vj+vj⊗vi)|1≤i≤j≤n}is a basis for Sym2(V) and hence its dimension
+isn(n+1)
+2and{(vi⊗vj−vj⊗vi)|1≤i<j≤n}is a basis for Λ2(V) and hence dimension
+isn(n−1)
+2.
+We claim that Sym2(V) and Λ2(V) areG-invariant. Suppose that v⊗w∈Sym2(V)
+andg∈Gthen we have
+θ(ρ(g)(v⊗w)) =θ/parenleftBig
+ρ(g)/parenleftBig/summationdisplay
+λij(vi⊗vj+vj⊗vi)/parenrightBig/parenrightBig
+=θ/parenleftBig/summationdisplay
+λij(ρ(g)vi⊗ρ(g)vj+ρ(g)vj⊗ρ(g)vi)/parenrightBig
+=/summationdisplay
+λij[θ(ρ(g)vi⊗ρ(g)vj)+θ(ρ(g)vj⊗ρ(g)vi)]
+=/summationdisplay
+λij[ρ(g)vj⊗ρ(g)vi+ρ(g)vi⊗ρ(g)vj)]
+=ρ(g)/parenleftBig/summationdisplay
+λij(vj⊗vi+vi⊗vj)/parenrightBig
+=ρ(g)(v⊗w).
+We also see that Sym2(V)∩Λ2(V) ={0}. Also their individual dimensions add up to
+n(n+1)
+2+n(n−1)
+2=n2= dim(V⊗V), hence we have V⊗V=Sym2(V)⊕Λ2(V). /square
+Theorem 13.2. The characters of Sym2(V)andΛ2(V)areχSandχArespectively given
+by
+χS(g) =1
+2/parenleftbig
+χ2(g)+χ(g2)/parenrightbig
+χA(g) =1
+2/parenleftbig
+χ2(g)−χ(g2)/parenrightbig
+Proof. Suppose that |G|=d. Then for any g∈G, (ρ(g))d=I. Thusm(X), the
+minimal polynomial of ρ(g), divides the polynomial p(X) =Xd−1. Sincep(X) has distinct
+roots so will m(X) and hence ρ(g) is diagonalisable.
+Let{e1,···en}be an eigen basis for Vand let{λ1,···,λn}be the corresponding eigen
+values. Then from the proof of previous theorem it follows th at{(ei⊗ej−ej⊗ei)|i<j}
+is an eigen basis for Λ2(V) with corresponding eigen values {λiλj|i<j}. We now have
+χA(g) =Tr(ρ⊗ρ)(g) =/summationdisplay
+i<jλiλj=1
+2/parenleftbigg/parenleftBig/summationdisplay
+λi/parenrightBig2
+−/summationdisplay/parenleftbig
+λ2
+i/parenrightbig/parenrightbigg
+=1
+2/parenleftbig
+χ2(g)−χ(g2)/parenrightbig
+.
+In similar way one can calculate χS. /square
+13.2. Character Table of S5
+With the theory developed above, we are now ready to construc t the character table for
+the symmetric group S5. We record below some facts about Snand in particular about S5
+which will be the starting point to make the character table:
+(1)|S5|= 5! = 120.13.2. CHARACTER TABLE OF S5 45
+(2) Two elements σandσ′ofSnare conjugate if and only if their cycle structure is
+same when they are written as product of disjoint cycles.
+(3) The conjugacy classes in S5along with their cardinality are as mentioned below:
+|(gi)|11020 15 30 20 24
+gi1(12)(123)(12)(34) (1234)(12)(345) (12345)
+(4) For any Sn, we already know 2 one dimensional (and hence irreducible) r epresen-
+tations: One being the trivial representation and the other the sign representation
+which sends each transposition to -1.
+(5) For any Sn, we know an irreducible representation of dimension n−1 (as in the
+Example 2.15 obtained by the action of Snon the subspace VofCngiven by
+V={(x1,x2,...,x n)∈Cn|x1+x2+···+xn= 0}).
+We fill this information into the character table:
+|(gi)|11020 15 30 20 24
+gi1(12)(123)(12)(34) (1234)(12)(345) (12345)
+χ1111 1 1 1 1
+χ21−11 1−1−1 1
+χ3421 0 0 −1 −1
+We need to find 4 more irreducible characters. We now deploy th e ways described at the
+beginning to find new irreducible representations.
+Taking tensor product of the trivial representation with an y other representation ( ρ,V)
+gives a representation isomorphic to ( ρ,V) (sinceχ1χV=χV). Hence we only need to
+consider tensor product of the second and the third represen tation whose character is χ2χ3.
+/an}bracketle{tχ2χ3,χ2χ3/an}bracketri}ht=1
+120((4)2+10(−2)2)+20(1)2+20(1)2+24(−1)2)
+= 1
+=/an}bracketle{tχ3,χ3/an}bracketri}ht
+Thus this representation turns out to be irreducible. Let χ4=χ2χ3/ne}ationslash=χ3. We include
+this character χ4into the character table:
+|(gi)|11020 15 30 20 24
+gi1(12)(123)(12)(34) (1234)(12)(345) (12345)
+χ1111 1 1 1 1
+χ21−11 1−1−1 1
+χ3421 0 0 −1 −1
+χ44−21 0 0 1 −146 13. CHARACTERS OF INDEX 2 SUBGROUPS
+We now consider χ=χ3and for this we examine the representations χSandχA:
+|(gi)|11020 15 30 20 24
+gi1(12)(123)(12)(34) (1234)(12)(345) (12345)
+χA600−20 0 1
+χS1041 2 0 1 0
+We check that χAis irreducible as /an}bracketle{tχA,χA/an}bracketri}ht=1
+120((6)2+15(−2)2+24(1)2) = 1.
+Thusχ5=χAis the fifth irreducible character of S5. Suppose that χ6andχ7are the
+other 2 irreducible characters. Since every representatio n of a finite group can be written
+as a direct sum of irreducible ones we have,
+χS=m1χ1+m2χ2+···+m7χ7
+wheremi=/an}bracketle{tχS,χi/an}bracketri}ht. Calculations show that /an}bracketle{tχS,χS/an}bracketri}ht= 3,/an}bracketle{tχS,χ1/an}bracketri}ht= 1 and /an}bracketle{tχS,χ3/an}bracketri}ht= 1.
+We also have/summationtextm2
+i=/an}bracketle{tχS,χS/an}bracketri}ht= 3. Thus χS=χ1+χ3+ψ, whereψis an irreducible
+character. We rewrite ψ=χS−χ1−χ3explicitly
+|(gi)|11020 15 30 20 24
+gi1(12)(123)(12)(34) (1234)(12)(345) (12345)
+ψ51−11−1 1 0
+Sinceψis irreducible, /an}bracketle{tψ,ψ/an}bracketri}ht=1
+120(52+ 10 + 20 + 15 + 30 + 20) = 1 as expected. Let
+χ6=ψthenχ7=χ2χ6will be another new irreducible character. We have thus foun d the
+character table of S5:
+Character Table of S5
+|(gi)|11020 15 30 20 24
+gi1(12)(123)(12)(34) (1234)(12)(345) (12345)
+χ1111 1 1 1 1
+χ21−11 1−1−1 1
+χ3421 0 0 −1 −1
+χ44−21 0 0 1 −1
+χ5600−20 0 1
+χ651−11−1 1 0
+χ75−1−11 1 −1 0
+13.3. Index two subgroups
+Let (ρ,V) be a representation of G. Thenρ:G→GL(V) is a group homomorphism.
+SupposeHis a subgroup of G. Then we can restrict the map ρtoHdenoted as ρ|Hto
+obtain a representation for H. Note that even if ( ρ,V) is an irreducible representation,
+(ρ|H,V) need not be irreducible. The following theorem shows the in timate connection
+between the characters of a group Gand any of its subgroup H. Recall from Chapter 9 we13.3. INDEX TWO SUBGROUPS 47
+define an inner product on C[G] denoted as /an}bracketle{t,/an}bracketri}ht. We denote this inner product on C[H] by
+/an}bracketle{t,/an}bracketri}htHthinking of Has a group in itself.
+Proposition 13.3. LetGbe a group and Hits subgroup. Let ψbe a non zero character of
+H. Then there exists an irreducible character χofGsuch that /an}bracketle{tχ|H,ψ/an}bracketri}htH/ne}ationslash= 0where/an}bracketle{t,/an}bracketri}htH
+represents inner product on C[H]as explained above.
+Proof. Letχ1,χ2,...,χ hbe the irreducible characters of G. We have the regular
+representation of G:χreg=/summationtextχi(1)χiandχreg(1) =|G|,χreg(g) = 0,g/ne}ationslash= 1. Thus we have
+the following:
+/an}bracketle{tχreg|H,ψ/an}bracketri}htH=1
+|H|(χreg(1)ψ(1))
+=|G|ψ(1)
+|H|/ne}ationslash= 0
+∴/an}bracketle{tχreg|H,ψ/an}bracketri}htH=h/summationdisplay
+i=1/an}bracketle{tχi|H,ψ/an}bracketri}htH/ne}ationslash= 0
+Hence atleast one of the /an}bracketle{tχi,ψ/an}bracketri}htH/ne}ationslash= 0. /square
+Theorem 13.4. LetGbe a group and Ha subgroup. Let χbe an irreducible character of G.
+Supposeψ1,ψ2,...,ψ kare all irreducible characters of Handχ|H=d1ψ1+d2ψ2+···+dkψk
+for somed1,d2,...,d kintegers. Then
+k/summationdisplay
+i=1d2
+i≤[G:H]
+and equality occurs if and only if χ(g) = 0for allg /∈H.
+Proof. Thinking of Has a group we have /an}bracketle{tχ|H,χ|H/an}bracketri}htH=/summationtextd2
+i. Sinceχis an irre-
+ducible character of Gwe have,
+/an}bracketle{tχ,χ/an}bracketri}ht=1
+|G|/summationdisplay
+g∈Gχ(g)χ(g)
+1 =1
+|G|
+/summationdisplay
+h∈Hχ(h)χ(h)+/summationdisplay
+g/∈Hχ(g)χ(g)
+
+1 =|H|
+|G|/an}bracketle{tχ|H,χ|H/an}bracketri}htH+1
+|G|/summationdisplay
+g/∈Hχ(g)χ(g)48 13. CHARACTERS OF INDEX 2 SUBGROUPS
+Rewriting the above we get,
+|H|
+|G|/an}bracketle{tχ|H,χ|H/an}bracketri}htH= 1−1
+|G|/summationdisplay
+g/∈Hχ(g)χ(g)
+/summationdisplay
+d2
+i=/an}bracketle{tχ|H,χ|H/an}bracketri}htH=|G|
+|H|−1
+|H|/summationdisplay
+g/∈Hχ(g)χ(g)≤|G|
+|H|.
+Moreover equality occurs if and only if1
+|H|/summationtext
+g/∈Hχ(g)χ(g) = 0, i.e.,/summationtext
+g/∈H|χ(g)|2= 0 which
+happens if and only if |χ(g)|= 0 and hence χ(g) = 0 for all g /∈H. /square
+We will apply the above theorm for index two subgroups and get the following,
+Corollary 13.5. LetGbe a group and Ha subgroup of index 2. Letχbe an irreducible
+character of G. Then one of the following happens :
+(1)χ|H=ψis an irreducible character of H. This happens if and only if there exists
+g∈Gandg /∈Hsuch thatχ(g)/ne}ationslash= 0.
+(2)χ|H=ψ1+ψ2whereψ1andψ2are irreducible characters of H. This happens if
+and only if χ(g) = 0for allg /∈H.
+13.4. Character Table of A5
+We will write down the character table of A5. For this we will use the Corollary 13.5
+and the character table of S5derived earlier in this chapter. The conjugacy classes of A5
+and their corresponding cardinality are as follows:
+|(gi)|120 15 12 12
+gi1(123)(12)(34) (12345) (13452)
+From the character table of S5it follows that χ1|H=χ2|H,χ3|H=χ4|Handχ6|H=χ7|H
+are irreducible characters of A5. Hence we get the character table of A5partially.
+|(gi)|120 15 12 12
+gi1(123)(12)(34) (12345) (13452)
+ψ111 1 1 1
+ψ241 0 −1−1
+ψ35−11 0 0
+ψ4n4= 3a1a2a3a4
+ψ5n5= 3b1b2b3b4
+We know that 12+42+52+n42+n52= 60 and hence n42+n52= 18. The only possible
+integral solutions of this equation are n4=n5= 3.
+Now we see that the only irreducible character of S5whose restriction to A5is not
+irreducible is χ5. Sinceψ1,ψ2,ψ3are all obtained by restriction of the characters other than
+χ5it follows from Theorem 13.4 that only possibly /an}bracketle{tψ4,χ5|A5/an}bracketri}htA5/ne}ationslash= 0 and/an}bracketle{tψ5,χ5|A5/an}bracketri}htA5/ne}ationslash= 0.13.4. CHARACTER TABLE OF A5 49
+Now from Corollary 13.5 it follows that χ5|A5is sum of two characters of A5and hence
+χ5|A5=ψ4+ψ5. Thus we have a1=−b1,a2=−2−b2,a3= 1−b3anda4= 1−b4.
+Now we use orthogonality relations for characters of A5and get/an}bracketle{tψ4,ψ1/an}bracketri}ht= 3+20a1+
+15a2+12a3+12a4= 0,/an}bracketle{tψ4,ψ2/an}bracketri}ht= 12+20a1−12a3−12a4= 0, and /an}bracketle{tψ4,ψ3/an}bracketri}ht= 15−20a1+
+15a2= 0. Solving these equations we get a1= 0,a2=−1 anda3+a4= 1. Hence we have
+the following:
+|(gi)|120 15 12 12
+gi1(123)(12)(34) (12345) (13452)
+ψ111 1 1 1
+ψ241 0 −1 −1
+ψ35−11 0 0
+ψ430−1 a3 a4= 1−a3
+ψ530−1b3= 1−a3=a4b4= 1−a4=a3
+Proposition 13.6. Every element of A5is conjugate to their own inverse. Hence the
+entries of the character table are real numbers.
+Proof. Clearly it is enough to prove that the representatives of the conjugacy classes
+are conjugate to their own inverse. It is clear for the elemen t 1,(123) and (12)(34). For
+others we check that
+(12345)−1= (54321) = (15)(24)(12345)(15)(24)
+(13452)−1= (25431) = (12)(35)(13452)(12)(35)
+Now we know that χ(g−1) =χ(g) andgbeing conjugate to g−1this is also equal to
+χ()g. Henceχ(g) =χ(g) givesχ(g)∈Rfor allg∈A5. /square
+The above proposition implies that a3,a4,b3andb4are real numbers. From /an}bracketle{tχ4,χ4/an}bracketri}ht= 1
+we geta2
+3+a2
+4= 3. Substituting a4= 1−a3we geta2
+3−a3−1 = 0. And the solutions are
+a3=1+√
+5
+2=b4,a4=1−√
+5
+2=b3
+or
+a3=1−√
+5
+2=b4,a4=1+√
+5
+2=b3.
+Since the values of ψ4andψ5on other conjugacy classes are the same, both the above
+solutions would give the same set of irreducible characters . Hence without loss of generality
+we may take the first set of solutions. This gives the complete character table of A5as
+follows:
+Character Table of A550 13. CHARACTERS OF INDEX 2 SUBGROUPS
+|(gi)|120 15 12 12
+gi1(123)(12)(34) (12345) (13452)
+ψ111 1 1 1
+ψ241 0 −1−1
+ψ35−11 0 0
+ψ430−11+√
+5
+21−√
+5
+2
+ψ530−11−√
+5
+21+√
+5
+2
+Exercise 13.7. Prove that every element of Snis conjugate to its own inverse and hence
+character table consists of real numbers.
+Remark : In fact more is true that every element of Snis conjugate to all those
+powers of itself which generates same subgroup (called rati onal conjugacy). Hence it is true
+that characters of Snalways take value in integers. However this is not true for An, for
+example check A4andA5. They may not be even real valued.CHAPTER 14
+Characters and Algebraic Integers
+LetGbeafinitegroupand ρ1,...,ρ hbeallirreduciblerepresentationsof Gofdimension
+n1,···,nhwith corresponding characters χ1,···,χh.
+For any give representation ρ:G→GL(V) we can define an algebra homomorphism
+˜ρ:C[G]→End(V) by/summationtext
+gαgg/ma√sto→/summationtext
+gαgρ(g). We know that Z(C[G]), the center of C[G],
+is spanned by the elements cg1,...,cghwhere
+cgi=/summationdisplay
+{t∈G|t=sgis−1}t
+i.e., sum of all conjugates of giandgiare representatives of different conjugacy class.
+Exercise 14.1. Let/summationtext
+gαgg∈ Z(C[G]) thenαg=αsgs−1for anys∈G.
+Proposition 14.2. Letρbe an irreducible representation and z∈ Z(C[G]). Then˜ρ(z) =
+λ.Idfor someλ∈C.
+Proof. We claim that ˜ ρ(z)∈End(V) is aG-map. Letz=/summationtext
+gαggthenαg=αsgs−1
+for anys∈G. Then
+˜ρ(z)(ρ(t)v) =/summationdisplay
+gαgρ(g)(ρ(t)v) =ρ(t)/summationdisplay
+gαgρ(t−1gt)v=ρ(t)/summationdisplay
+uαtut−1ρ(u)v=ρ(t)˜ρ(z)v.
+Sinceρis irreducible Corollary 4.5 (Schur’s Lemma) implies that ˜ ρ(z) =λ.Idfor some
+λ∈C. /square
+With notation as above let us denote ˜ ρi(cgj) =λijIdniwhereλij∈CandIdnidenotes
+the identity matrix. We can take trace of both side and get,
+niλij=tr(˜ρi(cgj)) =/summationdisplay
+{t∈G|t=sgjs−1}tr(ρi(t)) =rjχi(gj)
+whererjis the number of conjugates of gj. This gives,
+λij=rjχi(gj)
+ni=rjχi(gj)
+χi(1).
+Proposition 14.3. Eachλijis an algebraic integer.
+5152 14. CHARACTERS AND ALGEBRAIC INTEGERS
+Proof. Let us consider M=cg1Z⊕···⊕cghZ⊂ Z(C[G]) =cg1C⊕···⊕cghC. Clearly
+Mis aZ-submodule. It is a subring also. Take cgj,cgk∈ Z(C[G]). Thencgjcgk∈ Z(C[G]),
+in factcgjcgk∈M. Hence we can write cgjcgk=/summationtexth
+l=1ajklcglwhereajklare integers. By
+applying ˜ρiwe get,
+(λijIdni)(λikIdni) = ˜ρi(cgj)˜ρi(cgk) = ˜ρi(cgjcgk) =h/summationdisplay
+l=1ajklλilIdni.
+This givesλijλik=/summationtexth
+l=1ajklλilwhere each ajkl∈Z.
+Now we take N=λi1Z⊕ ··· ⊕λihZ⊂Cwhich is a finitely generated Z-module and
+λijN⊂Nfor allj. This implies λijis an algebraic integer. /square
+Lemma 14.4. For anys∈Gandχcharacter of a representation, χ(s)is an algebraic
+integer.
+Proof. Lets∈Gbe of order dinG. Thenρ(s) is of order less than or equal to d.
+Since the field is Cwe can choose a basis such that the matrix of ρ(s), sayA, becomes
+diagonal (see 5.4 and also proof in 9.6). Clearly Ad= 1 implies diagonal elements are root
+of the polynomial Xd−1 hence are algebraic integer. As sum of algebraic integers i s again
+an algebraic integer we get sum of diagonals of Awhich isχ(s) is an algebraic integer. /square
+Theorem 14.5. The order of an irreducible representation divides the orde r of the group,
+i.e.,nidivides|G|for alli.
+Proof. Letρibe an irreducible representation of degree niwith character χi. From
+the orthogonality relations we have,
+1
+|G|/summationdisplay
+t∈Gχi(t)χi(t−1) = 1
+h/summationdisplay
+j=1rjχi(gj)χi(g−1
+j) =|G|
+h/summationdisplay
+j=1niλijχi(g−1
+j) =|G|
+h/summationdisplay
+j=1λijχi(g−1
+j) =|G|
+ni
+Left side of this equation is an algebraic integer (using Pro position 14.3 and Lemma 14.4)
+and right side is a rational number. Hence|G|
+niis an algebraic integer as well as algebraic
+number hence an integer. This implies nidivides|G|. /squareCHAPTER 15
+Burnside’s pqTheorem
+We continue with the notation in the last chapter and recall,
+(1)λij=rjχi(gj)
+niis an algebraic integer.
+(2)χi(t) is an algebraic integer.
+Lemma 15.1. With notation as above suppose rjandniare relatively prime. Then either
+ρi(gj)is in the center of ρi(G)ortr(ρi(gj)) =χi(gj) = 0.
+Proof. As in the proof of Proposition 9.6 and Lemma 14.4 we can choose a basis such
+that the matrix of ρi(gj) = diag{ω1,···,ωni}andχi(gj) =ω1+···+ωniwhereωk’s are
+dth root of unity ( dorder ofgj). Now|ω1+···+ωni| ≤1+···+1 =nihence|χi(gj)
+ni| ≤1.
+In the case |χi(gj)
+ni|= 1 we must have ω1=ω2=...=ωni. This implies the matrix of ρi(gj)
+is central and hence in this case ρi(gj) belongs in the center of ρi(G).
+Now suppose that |χi(gj)
+ni|<1 and letα=χi(gj)
+ni. Sincerjandniare relatively prime
+we can find integers l,m∈Zsuch thatrjl+nim= 1. Then λij=rjαgiveslλij=
+(1−nim)α=α−mχi(gj). Sinceλijandχi(gj) are algebraic integers we get αis an
+algebraic integer. Let ζbe a primitive dth root of unity and let us consider the Galois
+extensionK=Q(ζ) ofQ. Letωk=ζakthenα=1
+ni(ζa1+···+ζani). Forσ∈Gal(K/Q)
+the element σ(α) is also of the same kind hence |σ(α)| ≤1. This implies the norm of α
+defined byN(α) =/producttext
+σ∈Gal(K/Q)σ(α) has|N(α)|<1. Sinceαis an algebraic integer so are
+σ(α) and hence the product N(α) is an algebraic integer. Since N(α) is also left invariant
+by allσ∈Gal(K/Q) it is a rational number hence it must be an integer. However a s
+|N(α)|<1 this gives N(α) = 0 and which implies χi(gj) =α= 0. /square
+Proposition 15.2. LetGbe a finite group and Cbe conjugacy class of g∈G. If|C|=pr,
+wherepis a prime and r≥1, then there exist a nontrivial irreducible representation ρofG
+such thatρ(C)is contained in the center of ρ(G). In particular, Gis not a simple group.
+Proof. On contrary let us assume that ρi(C) is not contained in the center of ρi(G)
+for all irreducible representations ρiofG. Then from previous Lemma if pdoes not divide
+niwe haveχi(g) = 0 forg∈C.
+5354 15. BURNSIDE’S pqTHEOREM
+Consider the character of the regular representation χ=/summationtexth
+i=1niχi. Then for any
+1/ne}ationslash=s∈Gwe have
+0 =χ(s) =h/summationdisplay
+i=1niχi(s) = 1+h/summationdisplay
+i=2niχi(s).
+Let us take g∈C(note thatg/ne}ationslash= 1 since |C|>1). Then/summationtexth
+i=2niχi(g) =−1. On the left
+hand side we have either p|nior ifp/ne}ationslash |nithenχi(g) = 0. Rewriting the above expression we
+get/summationtexth
+i=2ni
+pχi(g) =−1
+p. Left hand side of the expression is an algebraic integer (si nceni
+pis
+an integer) and right hand side is a rational number hence it s hould be an integer which is
+a contradiction. Hence there exist a nontrivial irreducibl e representation ρisuch thatρi(C)
+is contained in the center of ρi(G).
+Now let us take the above ρ:G→GL(V) and we have ρ(C)⊂ Z(ρ(G)) whereCis a
+conjugacy class of non-identity element. If ker(ρ)/ne}ationslash= 1 it will be normal subgroup of Gand
+Gis not simple. In case ker(ρ) = 1 we have ρan injective map and Z(ρ(G))/ne}ationslash= 1. But
+center is always a normal subgroup which implies Gis not simple. /square
+Theorem 15.3 (Burnside’s Theorem) .Every group of order paqb, wherep,qare distinct
+primes, is solvable.
+Proof. We use induction on a+b. Ifa+b= 1 thenGis ap-group and hence Gis
+solvable. Now assume a+b≥2 and any group of order prqswithr+s<a+bis solvable.
+LetQbe a Sylow q-subgroup of G. IfQ={e}thenb= 0 andGis ap-group and hence
+solvable. So let us assume Qis nontrivial. Since Qis aq-group (prime power order) it has
+nontrivial center. Let 1 /ne}ationslash=t∈ Z(Q). Then
+t∈Q⊂CG(t)⊂G
+and hence |CG(t)|=plqbfor some 0 ≤l≤awhich gives [ G:CG(t)] =pa−l. We claim that
+Gis not simple. If G=CG(t) thent∈ Z(CG(t)) =Z(G), i.e.,Z(G) is nontrivial normal
+subgroup and Gis not simple. Hence we may assume |CG(t)|=plqbwithl<a. Then using
+the formula |C(t)|=|G|
+|CG(t)|we get|C(t)|=pa−l. HereC(t) denotes the conjugacy class of
+t. Using previous proposition we get Gis not simple.
+Now we know that Gis not simple so it has a proper normal subgroup, say N. The order
+ofNandG/Nboth satisfy the hypothesis hence they are solvable. Thus Gis solvable. /squareCHAPTER 16
+Further Reading
+16.1. Representation Theory of Symmetric Group
+See Fulton and Harris chapter 4.
+16.2. Representation Theory of GL2(Fq)andSL2(Fq)
+See Fulton and Harris chapter 5.
+See self-contained notes by Amritanshu Prasad on the subjec t available on Arxiv.
+16.3. Wedderburn Structure Theorem
+See “Non Commutative Algebra” by Farb and Dennis and also “Gr oups and Represen-
+tations” by Alperin and Bell.
+16.4. Modular Representation Theory
+See the notes by Amritanshu Prasad on “Representations in Po sitive Characteristic”
+available on Arxiv.
+55Bibliography
+[AB] Alperin, J. L.; Bell R. B., “Groups and Representations” , Graduate Texts in Mathe-
+matics, 162, Springer-Verlag, New York, 1995.
+[CR1] Curtis C. W.; Reiner I., “Methods of representation theory vol I, With applications to
+finite groups and orders” , Pure and Applied Mathematics, A Wiley-Interscience Publi -
+cation. John Wiley & Sons, Inc., New York, 1981.
+[CR2] Curtis C. W.; Reiner I., “Methods of representation theory vol II, With application s
+to finite groups and orders” , Pure and Applied Mathematics, A Wiley-Interscience Pub-
+lication. John Wiley & Sons, Inc., New York, 1987.
+[CR3] Curtis C. W.; Reiner I., “Representation theory of finite groups and associative al-
+gebras”, Pure and Applied Mathematics, Vol. XI Interscience Publis hers, a division of
+John Wiley & Sons, New York-London 1962.
+[D1] Dornhoff L., “Group representation theory. Part A: Ordinary representa tion theory” ,
+Pure and Applied Mathematics, 7. Marcel Dekker, Inc., New Yo rk, 1971.
+[D2] Dornhoff, L., “Group representation theory. Part B: Modular representat ion theory” .
+Pure and Applied Mathematics, 7. Marcel Dekker, Inc., New Yo rk, 1972.
+[FH] Fulton; Harris, “Representation theory: A first course” Graduate Texts in Mathemat-
+ics, 129, Readings in Mathematics, Springer-Verlag, New Yo rk, 1991.
+[Se] Serre J.P., “Linear representations of finite groups” Translated from the second French
+edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag,
+New York-Heidelberg, 1977..
+[Si] Simon, B., “Representations of finite and compact groups” Graduate Studies in Math-
+ematics, 10, American Mathematical Society, Providence, R I, 1996.
+[M] Musili C. S., “Representations of finite groups” Texts and Readings in Mathematics,
+Hindustan Book Agency, Delhi, 1993.
+[DF] Dummit D. S.; Foote R. M. “Abstract algebra” , Third edition, John Wiley & Sons,
+Inc., Hoboken, NJ, 2004.
+[L] Lang, “Algebra” , Secondedition, Addison-WesleyPublishingCompany, Adva ncedBook
+Program, Reading, MA, 1984.
+[JL] James, Gordon; Liebeck, Martin, “Representations and characters of groups” , Second
+edition, Cambridge University Press, New York, 2001.
+57
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+arXiv:1001.0463v1  [astro-ph.CO]  4 Jan 2010Mon. Not. R. Astron. Soc. 000, 1–18 (2009) Printed 1 November 2018 (MN L ATEX style file v2.2)
+Study of a homogeneous QSO sample: relations between
+the QSO and its host galaxy⋆
+Y. Letawe1, G. Letawe1†and P. Magain1
+1D´ epartement d’Astrophysique, G´ eophysique et Oc´ eanogr aphie, Ulg, All´ ee du 6 aoˆ ut, 13, 4000 Sart Tilman, Belgium
+Accepted 2009 December 15. Received 2009 December 14; in ori ginal form 2009 October 11
+ABSTRACT
+We analyse a sample of 69 QSOs which have been randomly selected in a c omplete
+sample of 104 QSOs ( R/lessorequalslant18, 0.142< z <0.198,δ <10o). 60 have been observed
+with the NTT/SUSI2 at La Silla, through two filters in the optical band (WB#655
+and V#812), and the remaining 9 are taken from archive databases . The filter V#812
+contains the redshifted Hβand forbidden [OIII] emission lines, while WB#655 covers
+a spectral region devoid of emission lines, thus measuring the QSO an d stellar con-
+tinua. The contributions of the QSO and the host are separated th anks to the MCS
+deconvolution algorithm, allowing a morphological classification of the host, and the
+computationofseveralparameterssuchasthehostandnucleus absoluteV-magnitude,
+distance between the luminosity center of the host and the QSO, an d colour of the
+host and nucleus. We define a new asymmetry coefficient, independe nt of any galaxy
+models and well suited for QSO host studies. The main results from th is study are:
+(i) 25% of the total number of QSO hosts are spirals, 51% are elliptica ls and 60%
+show signs of interaction; (ii) Highly asymmetric systems tend to hav e a higher gas
+ionization level (iii) Elliptical hosts contain a substantial amount of ion ized gas, and
+some show off-nuclear activity. These results agree with hierarchic al models merger
+driven evolution.
+Key words: quasars:general, galaxies: interactions, active, fundamental p arameters.
+1 INTRODUCTION
+Over the past decade, many observations and studies have
+been put forwards in order to reach a better understanding
+of the interrelations between the QSOs and their hosts. This
+has been made possible thanks to the availability of high
+resolution space based data, as well as ground based imag-
+ing. Several strong correlations have been found, such as th e
+famous black hole mass - bulge stellar velocity dispersion
+relation for quiescent galaxies (Ferrarese & Merritt 2000;
+Gebhardt et al. 2000; Bernardi et al. 2007), extended after-
+wards to active galaxies (McLure and Dunlop 2001, 2002;
+Marconi and Hunt 2003). The most popular global picture
+that has emerged is to place the QSO phenomenon in an
+evolutionary context, in which galaxies evolve hierarchic ally
+by successive gravitational interactions or mergers, allo w-
+ing more gas to reach the central regions and possibly trig-
+ger the QSO phase. Even if the observational basis of this
+idea remains unsecure, convincing hints of merger induced
+⋆Based on observations collected at the European Organisati on
+for Astronomical Research in the Southern Hemisphere, Chil e,
+under program IDs 77.B-0229, and 78.B-0081.
+†E-mail: gletawe@ulg.ac.beQSO activity were given using either hydrodynamical sim-
+ulations (Hopkins et al. 2005, 2006), high resolution imag-
+ing (Bahcall et al. 1997; Bennert et al. 2008; Letawe et al.
+2008b), or 2D-spectroscopy (Letawe et al. 2007). On the
+other hand, Schmitt (2001) argued, by comparing samples
+of active and non active galaxies, that the percentage of
+merging or gravitationally interacting systems is not high er
+in QSOs than in other types of active or non-active galax-
+ies. However, this study, based on low resolution images
+(1.7arcsec/pxl), does nottakeintoaccountfainttails orcom-
+pact features in the host typical of recent merging activity .
+Li et al. (2008), with a sample of 105low-redshift galaxies,
+find a strong correlation between the presence of a close
+companion and the star formation, but not between close
+companions and AGN activity. The puzzle is thus far from
+being solved.
+Inthisframework, itisimportant(1)tofindpeculiar casesi n
+specific stages of evolution, such as the famous HE0450-2958
+(Magain et al. 2005), which would allow to better assess the
+nature and creation mechanisms of the black hole-bulge co-
+evolution (Elbaz et al. 2009; Jahnke et al. 2009) and (2) to
+extract, in well resolved samples, correlations between ob -
+servables as well as the proportion of interacting systems.
+In order to bring new insights on those issues, we study a
+©2009 RAS2Y. Letawe
+sample of 69 QSOs extracted from a complete sample of 104
+QSOs drawn from different catalogues. Section 2 explains
+the construction of the sample and the observations. Sec-
+tion 3 contains a description of how the contributions of
+the host and QSO are separated. The parameters derived
+thanks to this separation (QSO and host magnitudes, cen-
+ter of luminosity, asymmetry coefficient), the correlations
+found between them, and subsequent discussion are given
+in Sections 4 and 5. Some conclusions are drawn in Section
+6. Finally, a few particularly interesting cases are analys ed
+with more scrutiny in the Appendix.
+2 SAMPLE AND OBSERVATIONS
+2.1 Sample
+We selected from the main catalogues available
+(Veron-Cetty & Veron 2006; Schneider et al. 2005;
+Hewitt & Burbidge 1993; Wisotzki et al. 2000; Green et al.
+1986) all the brightest QSOs ( R/lessorequalslant18) with δ/lessorequalslant10oand
+0.142/lessorequalslantz/lessorequalslant0.198. This redshift range was chosen to enable
+the observation of low redshift quasars through two specific
+filters, one including emission lines, the other only the QSO
+and stellar continua. This provided a sample of 104 QSOs,
+9 of which had already been observed at high resolution
+(see Table 1 for references), and 60 of which were observed
+in the context of the present study. Those 69 QSOs form
+the sample analysed here. Even if only 66% complete, we
+expect our sample to be statistically relevant and large
+enough to infer robust proportion measurments. Indeed, the
+35 remaining QSOs could not be observed only because of
+bad weather conditions during parts of the two runs, which
+is a selection independent of the QSOs characteristics, and
+thus devoid of any bias.
+2.2 Observations
+The observations of the sample was made with NTT/SUSI2,
+the Superb-Seeing Imager, a direct imaging camera opti-
+mised for periods of good seeing at the ESO/La Silla obser-
+vatory, during the run A 077.B-0229 in August 2006 and run
+B 078.B-0081 in February 2007. We observed through two
+filters, V#812 and WB#665. The complete list of targets
+is given in Table 2. The observation strategy was motivated
+by two major points in the understanding of the QSO-host
+interactions. First of all, good resolution and sampling ar e
+necessary for detecting the host and revealing its morphol-
+ogy. SUSI2, in its 2 ∗2 binning mode, offers a sampling of
+0.161 arcsec/pxl, which translates in our redshift range into
+∼0.45 kpc/pxl. Exposure times have been estimated with
+the NTT ETC in order toreach a S/N high enough to detect
+galaxies departing by as much as 3 σfrom the mean magni-
+tude relation between QSOand host. Each QSO observation
+was divided into 3 or 4 exposures in each filter in order to
+avoid saturation and to efficiently remove bad CCD pixels
+and cosmic ray hits. This set up, with a typical seeing of
+0.6, gives us observations deep enough to infer the major
+galactic morphology. Non photometric observing condition s
+were reported for 8 objects.
+Secondly, our previous study (Letawe et al. 2008b) shows
+that the distribution of ionized gas in the host does notFigure 1. VLT FORS1 spectrum of the host galaxy of the QSO
+HE1434 −1600 after the separation from the QSO, at a redshift
+ofz= 0.144 (Letawe et al. 2008a), together with the response
+curves of the two filters used for probing the gaseous and stel lar
+content.
+necesserally match the structure of the stars distribu-
+tion . For instance, the host of HE0354-5500, analysed in
+Letawe et al. (2008b) contains gas ionized by the QSO even
+in remote regions devoid of stars. This motivates to study
+the gas and stellar content separately for each host. That
+is why all the 60 SUSI2 QSOs have been observed through
+two filters: V#812 which, in our redshift range, contains the
+two forbidden [OIII] λ4959 and [OIII] λ5007 lines along with
+Hβ λ4861. Consequently, this filter allows to map the mor-
+phology of the ionized gas in the host, as we demonstrate in
+Section 5.2. The filter WB#665 contains a region devoid of
+emission lines, in which the stellar continuumshould thusb e
+the only contributor, allowing to map the morphology of the
+stellar contentof thehost. The presence ofthe stellar cont in-
+uum in the V#812 and its influence on the observed intensi-
+ties are discussed in Section 5.2. The filters response curve s
+areplottedalongwithatypicalQSOhostspectruminFig. 1.
+All observations were flatfielded and cleaned from bad pixels
+using PYRAF tools. Throughout the paper, we adopt the
+following cosmological parameters: H0= 71km s−1Mpc−1,
+Ωm= 0.27 and Ω λ= 0.73.
+3 IMAGE ANALYSIS
+3.1 NTT/SUSI2 data processing
+3.1.1 Deconvolution method
+The whole analysis of host galaxy and QSO properties re-
+lies on a good separation of those two components. We
+use a deconvolution method based on the MCS algorithm
+(Magain et al. 1998), which is known to be particularly well
+suited to separate point sources from a diffuse background
+(Letawe et al. 2007, 2008a,b; Magain et al. 2005). Its princi -
+ple is to produce images with an improved resolution with-
+out trying to reach an infinite one. Other techniques, which
+deconvolve an image with its true point spread function
+©2009 RAS, MNRAS 000, 1–18Study of a homogeneous QSO sample: relations between the QSO and its host galaxy 3
+Name RA Dec z Filter Ref. MV(QSO)MV(host) Morph. Interac?
+PHL909 00 54 32 14 29 58 0.171 F606W (1) -23.64 -21.74 Ell. N
+0205+024 02 05 14.53 02 28 42.7 0.155 F606W (1) -23.74 -19.84 S pir. N
+PKS 0736+017 07 36 42.49 01 44 00.1 0.191 F675W (2) -23.97 -22. 78 Spir Y
+PKS 1020-103 10 20 04.2 -10 22 33.6 0.197 F675W (3) ? -21.29 Ell . N
+MC 1635+119 16 35 25.88 11 55 46.4 0.146 F606W (4) -22.74 ? ? Y
+PG 2349-014 13 49 22.3 -01 25 54 0.174 F675W (3) ? -22.93 ? Y
+3C273 12 29 06.7 02 03 08 0.158 F606W (1) -26.34 -22.84 Ell. N
+HE1405-1545 14 08 24.5 -15 59 28 0.194 B (5) -24.2 -23.3 Spir. Y
+HE1434-1600 14 36 49.6 -16 13 41 0.144 F606W (6) -23.82 -22.87 Ell. Y
+Table 1. List of the QSOs already observed and main host properties. R eference: (1): Bahcall et al. (1997), (2): Dunlop et al. (200 3),
+(3): Kim et al. (2008), (4): Canalizo et al. (2007); (5): Jahn ke et al. (2004a); (6): Letawe et al. (2008b). Objects from (1 ) to (4) were
+observed with HST/ACS/WFC, (5) with NTT/EFOSC2 and (6) with HST/ACS/HRC. A question mark in MV(QSO) is when both
+absolute and apparent V-magnitudes are unknown.
+(PSF) and attempt to reach arbitrarily high resolutions,
+tend to result in artefacts, deblending problems, flux error s,
+and astrometry errors in the deconvolved image. In the MCS
+technique, the user chooses some finite value for the resolu-
+tion of the deconvolved output image by defining an output
+PSF smaller than the original image’s PSF. The user then
+chooses an appropriate pixel scale for the deconvolved out-
+put image in order to sample the chosen PSF at least at
+the Nyquist level. By fully sampling the chosen PSF, the
+MCS algorithm eliminates artefacts in the deconvolved im-
+ages, improves deblending of point sources, and reduces flux
+and position errors. For our purpose, the deconvolved im-
+age has a pixel size times smaller than in the original image
+and a PSF of Gaussian shape, with a Full Width at Half
+Maximum (FWHM) of 2 CCD pixels (4 in the deconvolved
+images). Another advantage of the method is that the point
+sources have a known shape in the deconvolved image and
+can be explicitely separated from the diffuse components,
+which allows to get an image of the host galaxy uncontam-
+inated by the QSO. A particularity of this method in the
+present context of QSO host analysis is that it introduces
+no model or prior assumption for the shape of the host, as
+the galaxy is represented by a numerical diffuse component.
+3.1.2 PSF construction
+A crucial point in the deconvolution process is the PSF con-
+struction: the more accurate the PSF at the position of the
+QSO on the CCD, the better the deconvolution. We con-
+struct the PSF as follows:
+•Some stars (between 1 and 4) are selected on the same
+frame as the QSO, as close to it as possible and with sim-
+ilar brightness. Indeed, a star as luminous as the QSO has
+the same S/N and possibly suffers from the same deviations
+from linearity. Moreover, a star far from the QSO is more
+likely affected by PSF variations across the field, especiall y
+if located near the border.
+•The kernel s(/vector x) of the deconvolution is first approxi-
+mated by a Moffat profile (better reproducing the wings of
+the PSF than a Gaussian profile). Because real PSFs have
+more complex structure than this analytic function, we add
+to it a numerical component whose sole purpose is to im-
+prove the fit with real stars. This kernel (Moffat profile +
+numerical background), after convolution with the PSF of
+Figure 2. Upper-Left: Different PSF stars are selected on the
+CCD.Upper-Right: A Moffat profile isadjusted on all the stars si-
+multaneously. Bottom-Right: Numericalbackground added t o im-
+prove the PSF. Bottom-Left: Residuals showing only faint st ruc-
+tures particular to each star.
+the deconvolved image (i.e. the chosen Gaussian profile) is
+then simultaneously fitted on the selected PSF stars. A de-
+tailed description of the way the kernel is constructed can
+be found in Magain et al. (2007).
+An example of the whole PSF construction process is given
+on Fig. 2. Only very faint structures, different for each star ,
+are left in the residuals (differences between the model and
+the data for each pixel, divided by the standard deviation in
+this pixel). It shows that the relevant available informati on
+(i.e. common to all selected stars) is taken into account in
+s(/vector x).
+3.1.3 Simultaneous deconvolution
+After the construction of the PSF for each QSO exposure,
+the simultaneous deconvolution of the different images of a
+given QSO observed through a given filter can be carried
+out. The QSO image is decomposed into its point source
+component and a numerical diffuse background identical for
+all exposures, thus containing the full available informat ion
+on the host.
+The parameters to fit are thus: the intensity and cen-
+©2009 RAS, MNRAS 000, 1–184Y. Letawe
+Name RA Dec z Run V (s) WB (s) Morphology Interaction?
+005709+144610.1 00 57 09.9 14 46 10 0.172 A 1500 1800 Ell. Y
+011110-101631.8 01 11 10.0 -10 16 32 0.179 A 750 930 Spir. Y
+011845+133327.1 01 18 45.5 13 33 27 0.189 A 2100 2625 Ell. N
+015530-085704.0 01 55 30.0 -08 57 04 0.165 A 640 800 Und. Y
+021218-073719.8 02 12 18.3 -07 37 20 0.174 A 1350 1650 Spir. N
+021360+004226.7 02 13 59.8 00 42 27 0.182 A 1350 1800 Und. Y
+025007+002525.3 02 50 07.0 00 25 25 0.198 A 1200 1600 Und. Y
+032214+005513.4 03 22 13.9 00 55 13 0.185 A 640 800 Ell. Y
+101044+004331.3 10 10 44.5 00 43 31 0.178 B 576 720 Spir. N
+113706+013947.9 11 37 06.8 01 39 48 0.193 B 1536 1720 Spir. N
+122534-024757.2 12 25 34.8 -02 47 57 0.195 B 1260 1630 Ell. N
+161532-002730.3 16 15 32.3 00 27 30 0.146 B 1260 1310 Ell. N
+205032-070131.2 20 50 32.3 -07 01 31 0.168 A 2400 2800 Spir. N
+231712-003603.6 23 17 11.8 -00 36 04 0.186 A 1940 2320 Und. Y
+232260-005359.3 23 23 00.0 -00 53 59 0.150 A 1260 1440 Spir. N
+235156-010913.3 23 51 56.1 -01 09 13 0.174 A 400 500 Ell. N
+Q 0022-2044 00 25 08.4 -20 27 35 0.170 A 240 300 Ell. N
+CT 289 01 00 39.5 -25 38 28 0.158 A 600 800 Ell. Y
+MS 01325-4151 01 34 42.7 -41 36 13 0.172 A 600 750 Und. Y
+MS 10302-2757 10 32 36.1 -28 13 26 0.148 B 600 750 Ell. N
+PKS 1241-399 12 44 29.4 -40 12 46 0.191 B 1816 2160 Und. Y
+MS 13591+0430 14 01 36.7 04 16 25 0.163 B 1272 1590 Ell. Y
+Q 1421-0013 14 24 03.8 -00 26 58 0.151 B 600 750 Spir. N
+PDS 456 17 28 19.9 -14 15 56 0.184 A 60 60 Und. Y
+MS 20078-3622 20 11 08.8 -36 13 10 0.177 A 420 520 Ell. Y
+Q 2240-2411 22 43 40.9 -23 55 16 0.184 A 1050 1200 Spir. N
+Q 2252-2434 22 55 25.0 -24 18 30 0.147 A 450 240 Und. Y
+6QZ J232927-2938 23 29 27.4 -29 38 47 0.193 A 560 720 Spir. N
+HE 0027-3118 00 29 37.3 -31 02 10 0.145 A 900 1125 Ell. Y
+HE 0108-5422 01 10 38.4 -54 06 40 0.186 A 750 900 Ell. N
+HE 0146-3755 01 48 21.1 -37 40 20 0.147 A 600 750 Spir. N
+HE 0226-5209 02 27 59.6 -51 56 32 0.145 A 1200 1530 Und. Y
+HE 0227-4123 02 29 13.3 -41 10 10 0.143 A 1200 0 Ell. Y
+HE 0250-4400 02 52 23.0 -43 47 55 0.168 A 540 600 Ell. Y
+HE 0441-2826 04 43 20.7 -28 20 52 0.155 A 150 180 Ell. Y
+HE 1101-0959 11 04 16.7 -10 16 08 0.186 B 1060 1260 Ell. N
+HE 1202-0501 12 04 53.0 -05 18 13 0.169 B 1332 1668 Und. Y
+HE 1211-1905 12 14 03.4 -19 21 43 0.148 B 1590 1935 Spir. Y
+HE 1236-2001 12 39 01.7 -20 17 30 0.196 B 1248 1500 Ell. Y
+HE 1255-0437 12 58 31.0 -04 53 49 0.172 B 1104 1380 Ell. Y
+HE 1256-2139 12 59 02.4 -21 55 38 0.146 B 1200 1500 Ell. Y
+HE 1300-0657 13 02 46.7 -07 13 55 0.181 B 1470 1830 Spir. Y
+HE 2345-3939 23 48 12.1 -39 23 07 0.196 A 1090 1125 Spir. N
+0056-363 00 56 15.8 -36 22 17 0.162 A 800 1000 Ell. Y
+0132+077 01 32 31.7 07 43 47 0.147 A 1050 1320 Spir. Y
+0213-484 02 13 52.6 -48 26 55 0.168 A 1080 1260 Ell. Y
+0357+107 03 57 27.1 10 46 48 0.182 A 544 680 Ell. Y
+1001+054 10 01 43.3 05 27 34.8 0.161 B 750 900 Ell. Y
+1012+008 10 12 20.8 00 48 33 0.185 B 684 855 Und. Y
+1023-014 10 23 03.9 -01 24 45.4 0.150 B 2060 2350 Und. Y
+1047+067 10 47 00.8 06 45 15.0 0.148 B 1176 1470 Ell. N
+1047-281 10 47 55.3 -28 07 45.0 0.190 B 576 720 Ell. N
+1151+117 11 51 15.7 11 45 10.0 0.176 B 366 456 Ell. Y
+1226+136 12 26 54.6 13 36 54.0 0.150 B 1700 2250 Ell. N
+1241+095 12 41 10.1 09 33 31.3 0.190 B 2064 2580 Spir. N
+1307+085 13 07 16.2 08 35 47 0.155 B 684 884 Ell. Y
+1325-012 13 25 59.8 -01 13 47.2 0.150 B 1032 1290 Und. Y
+1850-782 18 50 08.0 -78 15 00.0 0.162 A 180 208 Ell. N
+2140-457 21 40 10.0 -45 42 29 0.171 A 840 990 Und. Y
+Table 2. The NTT/SUSI2 sample observationnal characteristics, alo ng with host morphologies (spiral, elliptical, or undefined ) from
+visual inspection. The presence of signs of interaction is a lso indicated in the last column.
+©2009 RAS, MNRAS 000, 1–18Study of a homogeneous QSO sample: relations between the QSO and its host galaxy 5
+5"
+Figure 3. Left: One of the 3 exposures of the QSO 6QZJ2329-
+2938. Middle: The background and the point source at its fixed
+2 pixels resolution. Right: Residuals of the whole process, i.e. the
+reconvolved model minus the observation.
+ter of each point source, the common numerical background,
+whichcanbe renormalised betweendifferentexposuresif, fo r
+example, exposure times differ, and a global shift between
+images. An example of the whole process of simultaneous
+deconvolution is given Fig. 3. It clearly shows that simul-
+taneous deconvolution allows to: (1) seperate efficiently th e
+QSO and its host, as the residuals are very good; (2) reveal
+the morphology of the host.
+Among the whole observed sample, only 2 QSOs, namely
+MS13591+0430 and 231711-003604, could not be efficiently
+deconvolved, because they fall very close to the line of sigh t
+of a very bright star. They were thus removed from the pho-
+tometric analysis, even if visual inspection has given indi ca-
+tions on their host morphologies.
+We can now use the information provided by the decon-
+volution to compute some physical parameters such as the
+proportion of spiral and elliptical hosts, interacting sys tems,
+the host and QSO magnitudes in both filters, the center of
+luminosity of the host, its asymmetry coefficient, and try to
+find some correlations between those parameters.
+3.2 Archive data
+As mentioned earlier, 9 QSOs were already observed at high
+resolution, which allow to determine their host morphology
+and magnitudes. If magnitudes are given in the B or R band
+in the original paper, host V magnitudes are deduced from
+galaxy colours tables found in Fukugita et al. (1995), and
+QSO V magnitudes are deduced from apparent V magni-
+tudes taken from the NED database, if available. They are
+upper limits as they include the host luminosity. A summary
+of these additional data is given in Table 1.
+4 ANALYSIS
+Deconvolution of observations obtained in both filters pro-
+vides separated host and QSO images, thus allowing the
+computation of different parameters, which we review one
+by one.
+4.1 Magnitudes
+4.1.1 Apparent magnitudes
+The QSO flux is obtained as an output of the MCS algo-
+rithm. In order to avoid overestimation of the host flux, we
+flagged out the neighbouring objects lying in the field of
+view of the deconvolved image, by visual inspection. Thehost flux is then computed by summing the intensities of
+the pixels of the deconvolved image. The total flux of each
+component can be converted in apparent magnitudes. This
+requires the knowledge of the apparent magnitude of a stan-
+dard star. Observations of the TPHE B standard K7-star
+from the Landolt standard stars catalogue (Landolt 1992)
+allow to infer the zero-point in the V-band, but the appar-
+ent magnitude is not given for that star in the less common
+WB#665filter.However,itspansasimilar wavelengthrange
+as the usual R-band. Thus, we deduce the star WB#665
+apparent magnitude by taking the fluxes of Vega and of a
+template K-star spectrum through the R-filter and through
+the WB#665 filter:
+mR−mWB=−2.5logFlKstar∗R
+FlVega∗R
+FlKstar∗WB
+FlVega∗WB=−0.03.(1)
+Apparent magnitudes in both filters are listed in Table 3.
+8 QSOs suffer from significant flux variations between each
+exposure, indicating non-photometric observing conditio ns.
+This led us to consider the apparent magnitude taken from
+the brightest exposure in each filter as an upper limit for
+the magnitude (lower brightness limit).
+4.1.2 Absolute magnitudes
+The cosmic recession velocity of the objects we observe im-
+plies a shift in wavelength between the extragalactic emis-
+sion and the observation we make of it. The appropriate cor-
+rection, called K-correction, translate the observed magn i-
+tude measured through a filter to the intrinsic emitted mag-
+nitude, given the redshift of the object considered. As the
+host and QSO spectra are different, the conversion of appar-
+ent to absolute magnitudes uses different K-corrections for
+each component. Host galaxies are K-corrected taking into
+account their morphologies and redshift according to Pence
+(1976), Table 14, which, although rather old, proved to be
+very reliable (Kinney et al. 1996). QSOs K-corrections are
+taken from Cristiani & Vio (1990). They are only available
+in the V-band, thus WB#665 absolute magnitudes could
+not be computed. V absolute magnitudes are given in Table
+3.
+4.1.3 Errors
+Errors in the computation of magnitudes are dominated by
+two components: the photon noise inherent to the obser-
+vation, and the accuracy of the QSO-host separation. The
+photon noise is estimated by taking the square root of the
+mean QSO+host flux of our sample. That leads to a mean
+error of≃0.05%.
+To estimate the error on QSO-host separation, we first
+selected 3 representative QSOs with different quality de-
+convolution residuals: 1241+095, HE0146-3755, and Q2252-
+2434. For each of these QSOs, we varied the value of the
+QSO intensity until clear signs of bad PSF subtraction ap-
+pear (clear hole or peak at the QSO position). Signs of bad
+PSF subtraction begin to appear when we vary the QSO
+intensity by 1.1 to 7.0 per cent, suggesting that this is the
+level of uncertainty in the QSO magnitudes. Propagating
+these errors to magnitude errors leads to σM≃0.02−0.05.
+Another way to test the error bars on galactic and QSO
+©2009 RAS, MNRAS 000, 1–186Y. Letawe
+Magnitudes Asymmetry
+Name QSO Host aQSO aGal
+mVMVmWB mVMVmWB V WB V WB
+005709+144610.1 15.31 -24.59 15.26 16.95 -22.49 16.34 0.84 0.92 0.26 0.8
+011110-101631.8 15.94 -24.02 15.81 16.87 -22.84 16.29 0.7 0 .64 0.27 0.21
+011845+133327.1 17.49 -22.68 17.24 16.97 -22.97 16.28 0.13 0.12 0.14 0.1
+015530-085704.0 16.26 -23.46 16.16 17.02 -22.6 16.55 0.23 0 .91 0.38 0.53
+021218-073719.8 17.16 -22.70 17.17 17.55 -21.84 17.19 0.05 0.34 0.12 0.12
+021360+004226.7 14.55 - - 19.54 - - 1.34 1.54 0.34 0.19
+025007+002525.3 16.93 -23.44 16.87 18.69 -21.57 18.01 0.91 0.91 0.23 0.13
+032214+005513.4 15.6 -24.71 15.52 17.87 -21.92 17.27 0.64 0 .77 0.15 0.35
+101044+004331.3 15.6 -24.33 16. 17.52 -22.17 16.7 0.06 0.16 0.16 0.07
+113706+013947.9 16.22 -23.89 16.23 18.34 -21.52 17.64 0.15 0.65 0.26 0.8
+122534-024757.2 16.93 - 16.85 18.39 - 17.86 0.41 0.09 0.16 0. 22
+161532-002730.3 16.7 -23.00 16.47 16.71 -22.57 16.43 0.1 0. 53 0.08 0.08
+205032-070131.2 16.95 - 16.96 16.75 - 16.1 0.24 0.39 0.15 0.2 8
+232260-005359.3 16.73 - - 16.56 - - 0.74 0.71 0.45 0.17
+235156-010913.3 14.55 - - 19.54 - - 1.38 0.82 1.25 0.34
+Q 0022-2044 15.59 -24.19 15.18 18.59 -20.73 20.52 0.73 1.45 0 .23 0.33
+CT 289 16.41 -23.21 16.38 18.71 -20.47 17.97 0.24 0.24 0.13 0. 22
+MS 01325-4151 17.54 -22.25 17.21 17.53 -22.16 16.94 0.30 0.4 3 0.38 0.22
+MS 10302-2757 15.35 -24.24 15.40 16.79 -22.38 16.36 0.47 0.6 8 0.23 0.05
+PKS 1241-399 17.96 -22.43 17.74 17.67 -22.60 16.88 1.02 1.1 0 .65 0.64
+Q 1421-0013 15.65 -23.94 15.64 17.47 -21.91 17.64 0.54 0.29 0 .13 0.23
+PDS 456 13.42 -28.20 13.10 16.91 -24.60 16.30 1.66 2.3 1.4 0.3 6
+MS 20078-3622 16.27 -23.81 16.30 17.73 -21.85 17.48 0.42 0.6 8 0.73 0.71
+Q 2240-2411 17.49 -22.50 17.39 16.73 -23.02 16.22 0.87 0.84 0 .18 0.28
+Q 2252-2434 15.44 -24.01 15.55 16.89 -22.47 16.15 0.97 0.87 0 .33 0.14
+6QZ J232927-2938 15.85 -24.25 15.82 17.83 -22.00 16.98 0.43 0.21 0.14 0.05
+HE 0027-3118 16.10 -23.29 16.19 17.53 -21.44 16.88 0.52 0.42 0.12 0.26
+HE 0108-5422 16.70 -23.31 16.62 19.01 -20.47 17.65 0.43 0.56 0.29 0.04
+HE 0146-3755 15.98 -23.43 16.03 16.38 -22.83 15.89 0.72 0.47 0.21 0.35
+HE 0226-5209 16.64 - 17.26 16.84 - 16.74 1.12 1.21 0.66 0.58
+HE 0227-4123 18.84 - - 19.59 - - 0.24 - 0.37 -
+HE 0250-4400 15.57 -24.16 15.56 17.35 -21.92 17.06 0.7 0.34 0 .21 0.07
+HE 0441-2826 14.45 -25.16 14.32 16.04 -23.13 16.29 0.73 1.09 0.08 0.11
+HE 1101-0959 17.08 -23.03 17.04 17.69 -21.89 17.26 0.39 0.47 0.08 0.08
+HE 1202-0501 16.15 -23.67 16.06 16.5 -23.21 16.28 1.62 1.73 1 .23 1.22
+HE 1211-1905 16.45 -23.10 16.55 16.64 -22.70 16.13 1.14 1.4 0 .41 0.60
+HE 1236-2001 16.04 -24.22 15.85 17.21 -22.50 16.75 0.51 0.46 0.21 0.16
+HE 1255-0437 16.34 -23.49 16.35 16.86 -22.50 16.25 0.15 0.29 0.32 0.45
+HE 1256-2139 15.79 -23.93 15.75 18.81 -20.49 18.75 1.9 1.68 0 .43 0.61
+HE 1300-0657 16.53 -23.48 16.48 16.82 -22.94 16.22 0.32 0.27 0.13 0.04
+HE 2345-3939 16.61 -23.50 16.65 16.37 -23.47 15.83 1.17 1.14 0.47 0.38
+0056-363 14.83 -24.81 14.9 17.31 -21.9 19.27 1.49 1.53 1.55 1 .54
+0132+077 17.37 -22.18 17.4 16.96 -22.39 16.32 0.29 0.29 0.06 0.08
+0213-484 16.92 -22.85 16.92 17.67 -21.64 17.09 0.56 0.17 0.1 4 0.19
+0357+107 15.81 - -24.88 18.21 - -21.98 0.85 0.71 0.06 -
+1001+054 15.57 -24.12 15.69 17.76 -21.85 17.31 0.53 0.8 0.24 0.37
+1012+008 15.15 -24.89 15.15 16.45 -23.47 15.76 2.08 1.95 0.8 5 0.6
+1023-014 18.85 -20.73 18.89 17.49 -22 17.38 2.04 1.95 0.51 0. 18
+1047+067 16.82 -22.65 16.84 17.12 -21.82 16.49 0.09 0.34 0.0 9 0.12
+1047-281 15.27 -24.99 15.31 17.54 -22.18 16.76 0.28 0.33 0.1 0.48
+1151+117 15.61 -24.28 15.64 18.79 -21.00 17.59 1.51 1.37 0.4 0.32
+1226+136 17.43 -22.08 17.07 17.06 -22.02 16.61 0.18 0.38 0.2 6 0.22
+1241+095 16.81 -23.24 16.79 17.06 -22.74 16.73 0.74 0.65 0.1 8 0.07
+1307+085 14.79 -24.84 15.29 17.42 -21.78 17.75 1.29 0.28 0.2 9 0.15
+1325-012 16.95 - 16.96 17.24 - 16.36 1.59 1.55 0.19 0.18
+1850-782 14.99 -25.20 14.94 17.00 -22.75 16.35 0.42 0.55 0.1 3 0.1
+2140-457 15.68 -24.12 15.64 17.92 -21.77 17.47 0.45 0.39 0.1 2 0.12
+Table 3. Apparent and absolute magnitudes are given separately for t he QSO and the host in the V-band, while only apparent
+magnitudes could be computed in the WB#665-filter. For non ph otometric observations, absolute V-magnitudes are not dis played and
+apparent magnitudes must be taken as a lower limit. Asymmetr ies computed with respect to the QSO center and to the center o f
+luminosity of the host are given in both filters (see text for d etails).
+©2009 RAS, MNRAS 000, 1–18Study of a homogeneous QSO sample: relations between the QSO and its host galaxy 7
+Regular Interaction Multiple nuclei Total
+Spiral 17 8 0 25
+Elliptical 23 28 4 51
+Undef. 0 24 4 24
+Total 40 60 9
+Table 4. Occurency (in % of the whole sample) of morphology
+classes and interaction features.
+magnitudes obtained through deconvolution is to add a sim-
+ulated QSO on an observed galaxy in one of the observed
+fields and run the deconvolution process on this new image.
+The input and output fluxes may then be compared to esti-
+mate the error bars on magnitudes. After running this pro-
+cedure for several objects, we observe a deviation of at most
+20% for the host flux and 2% for the QSO flux. Not surpris-
+ingly, the uncertainty for the host is stronger when the QSO
+intensity is high relative to the host. On average, given the
+nuclear/host ratios in the sample, we can consider an un-
+certainty of 0.1 mag for the host magnitude and 0.01 for
+the QSO magnitude. The errors coming from the QSO/host
+separation thus clearly tend to dominate the photon noise.
+4.2 Morphology
+One of the first natural steps in the study of QSO-host sam-
+ples is to seek if there is a link between the host morpho-
+logical type and the QSO activity. More precisely, the cur-
+rent open questions are: Do gravitational interactions tri g-
+ger QSO activity? If yes, what are the physical processes
+at work? Are there other ways of triggering activity? Which
+ones? Once the nucleus is turned on, how does it evolve with
+its host?... One way to answer these questions is to find clas-
+sification criteria for host galaxies that enable to compare
+them with quiescent galaxies. The bimodal distribution of
+galaxy colour-magnitude diagrams (Baldry et al. 2004) al-
+lows such a comparison. Namely, Martin et al. (2007) find
+that the host galaxies of AGNs lie more often in between the
+red (corresponding to early-types) and blue (correspondin g
+to late-types) sequence of such diagrams, suggesting that
+the AGN phenomenon is a transitionary step between those
+sequences. Another possibility is to simply classify hosts by
+their morphological type and find out the frequencies of in-
+teracting, elliptical and spiral systems, in which QSOs are
+found to lie, and compare them with previous results.
+We chose to classify each host galaxy according to its mor-
+phology by visual inspection. Namely, the different selecti on
+properties are: (1) Spiral, elliptical or undefined morphol -
+ogy (exclusive), where the undefinedmorphology most often
+corresponds to cases in which the merging process is violent
+enough to totally disturb the host, preventing the deter-
+mination of its morphology, and, (2) signs of gravitational
+interaction andpresence ofmore thanonepointsource (non-
+exclusive). The percentages, calculated on the total sampl e
+(SUSI2+archive sample), can be found in Table 4.
+The proportion of ellipticals (51%) and spirals (25%)
+is essentially compatible with some previous results, such as
+Shade et al. (2000)or Letawe et al. (2007). However,wefind
+alower proportion of elliptical galaxies than in Dunlop et a l.
+(2003), Hamilton et al. (2008) and Floyd et al. (2004), butthis may be due to a selection effect in their sample, as men-
+tioned in Letawe et al. (2007). Concerning the proportion of
+hostsshowingsigns ofgravitational interaction, wejustm en-
+tion for the moment that the 60% we find are much higher
+than the 5 to 20% found for quiescent galaxies in some pre-
+vious studies (Le F` evre 2000; Bell et al. 2005; Zheng et al.
+2004), although Shi et al. (2009) argue in favour of a higher
+merger rate. One difficulty in estimating the proportion
+of merger remnants is that, as shown by simulations, the
+timescales duringwhich they show clear signs of disturbanc e
+are often shorter thanthe total merger timescale (Lotz et al .
+2008). On the other hand, we find a proportion of interac-
+tion very similar to Schmitt (2001) for both active and inac-
+tive galaxies. Namely, ≃55% of all our elliptical hosts show
+signs of gravitational interactions, and this proportion d rops
+to≃32% for spirals, while they find respectively ≃50% and
+≃25%. If we consider our undefined class as strong merg-
+ers, our percentage of 24% is very similar to Greene et al.
+(2009), who find a proportion of one quarter for highly dis-
+turbed morphologies in obscured active galaxies.
+Inconclusion, we can state that our estimates showthat
+themorphologies ofactive galaxies are not different from th e
+quiescent ones, and that the occurence of merger signs in
+QSOhosts is higher or similar to inactive galaxies, regardi ng
+previous discordant studies. If higher, we could assess the
+influence of mergers on activity. Thus detailed studies on
+the presence of merger signs in inactive galaxies at similar
+spatial resolution would be required before one can firmly
+conclude.
+All in all, the major difficulty in comparing different
+studies lies in the definition and quantification of the degre e
+of interaction in a system. It is often done by simple eye-
+check, which is always subjective. That leads us to define
+the following asymmetry coefficient.
+4.3 Asymmetry
+An idea that has been recurrent for the past 10 years is to
+quantify interactions and mergers via the asymmetry of the
+system. Several methods can be found in the literature. We
+briefly review the most relevant ones and their results. Firs t
+of all, Abraham et al. (1994); Conselice et al. (2000, 2003)
+and Shi et al. (2009) used an asymmetry index defined es-
+sentially as the subtraction of the image of a galaxy by the
+same image rotated by 180o. This simple method proved
+to be a good tracer of the degree of interaction in merging
+galaxies. With the help of two supplementary parameters,
+the concentration index and the clumpiness, Conselice et al .
+(2003) achieve to describe all usual morphological types in a
+totally quantitive way, solely from image analysis. Anothe r
+method called ZurichEstimator ofStructural Types (ZEST)
+has been proposed by Scarlata et al. (2007). It uses the Ser-
+sic index nreturned from a galaxy fit and 5 other basic
+nonparametric diagnostics (amongst which the asymmetry
+coefficient) to quantify the properties of galaxy structure,
+using Principal Component Analysis. They find strong evo-
+lutionary effects between z= 0 and z= 0.7 for faint galaxies
+(MB>−20.5).
+Another morphological parameter used as a signpost of a
+nonequilibrium global dynamical state is the so-called lop -
+sidedness. It is defined as the radially averaged m= 1 az-
+imuthal Fourier amplitude measured between radii enclos-
+©2009 RAS, MNRAS 000, 1–188Y. Letawe
+ing 50% and 90% of the galaxy light. Using this definition
+for a low-redshift sample ( z <0.06) of∼25000 galaxies,
+Reichard et al. (2009) find a trend for more powerful AGN
+to be hosted by more lopsided galaxies, and conclude that
+is due to the strong link of the age of the stellar population
+in the galaxy bulge to both the AGN luminosity and to the
+lopsidedness of the host.
+These analyses concentrate on Type 2 AGNs or quiescent
+galaxies, and do not take into account Type 1 QSOs, which
+considerably overweight the central regions of the host in
+the computation of the asymmetry index. In order to in-
+clude QSOs in that kindof analysis, Gabor et al. (2009) first
+separe QSO and host components with the GALFIT tool,
+and then compute the asymmetry index in the same way as
+Conselice et al. (2000). They find that, at 0 .3< z <1.0,
+QSO host asymmetries are no more prevalent than in qui-
+escent galaxies. Another way of computing asymmetries for
+QSO host galaxies is proposed by Kim et al. (2008), using
+an all-GALFIT computation. After separing the QSO and
+the host, Fourier components are fit to the images showing
+significant nonaxisymmetric features. Their result is that ,
+for low redshift QSOs ( z <0.35), galaxy mergers and tidal
+interactions seem to play an important role in regulating
+and fueling the nuclear activity. The apparently opposite
+results in different redshift ranges from Gabor et al. (2009)
+and Kim et al. (2008) are suggestive of a cosmological evo-
+lution of the importance of interactions. It reinforces our
+belief that the study of asymmetries is a powerful tool to
+understand QSO-host interactions.
+Here, in order to get rid of any assumption concerning the
+host morphology or the estimated center of the host, we pro-
+pose an alternative way to define the asymmetry, inspired
+by the statistical third order moment, called skewness in-
+dice. More precisely, if I(/vector x) is the intensity in the pixel /vector x,
+we define the center of luminosity of the host by
+/vector xcL=/summationtext
+/vector xI(/vector x)/vector x/summationtext
+/vector xI(/vector x). (2)
+where the sums are over the total number Nof pixels in a
+square subimage defined by the user and containing the host
+galaxy.
+Then, we define an asymmetry coefficient with respect to a
+centerclocated at some position /vector xc
+ac=/summationtext
+/vector xI(/vector x)(/vector x−/vector xc)3
+/summationtext
+/vector xI(/vector x). (3)
+This method has the advantage of being totally model
+independent, and takes into account the whole light from
+the host. It thus requires no a priori knowledge about the
+shape of a galaxy, which involves the risk to misguide any
+interpretation.
+Another advantage of this definition is that ccan be chosen
+to be either the center of luminosity cL, thus describing the
+asymmetry of the galaxy alone, or the QSO position cQSO,
+thus giving information about the asymmetry of the whole
+system (QSO+host). From now on, unless clearly stated
+otherwise, the term “asymmetry coefficient” will refer
+to the asymmetry with respect to cQSO. Indeed, a QSO
+located off-center in an otherwise seemingly symmetrical
+galaxy can be considered as a sign that something special is
+happening. For example, in the framework of binary black
+hole mergers, the resulting black hole is supposed to recoil
+2"021218−073719.8 HE1300−0657 HE0250−4400
+a(V)=0.70 a(V)=0.32 a(V)=0.05
+1325−012 PKS1241−399 1023−014
+a(V)=2.04 a(V)=1.59 a(V)=1.03
+Figure 4. Asymmetry coefficients in the V-band a(V) are dis-
+played for 6 QSO hosts which are at different levels of interac tion.
+A higher asymmetry value corresponds to a higher degree of in -
+teraction. The spatial scale is identical in all frames.
+in a direction opposite to the gravitational wave emission
+(Redmount & Rees 1989). During this recoil, the accretion
+disc may stay bound to the black hole and consequently
+shine off-centered. Simulations (Volonteri & Madau 2008)
+predict that a population of off-nuclear AGNs may already
+be detectable at low and intermediate redshifts.
+In order to minimize the importance of border effects
+linked to the choice of the square subimage where the
+asymetry coefficient is computed, we multiply the region of
+interestbyabroadGaussian function, centeredonthecente r
+of luminosity. A FWHM of 30 pixels, matching the extent
+of the galaxy, reveals to be appropriate in the majority of
+cases as it is not too narrow, which would give more weight
+to asymmetries due to shifts between cLandcQSOin com-
+parison with galactic asymmetries, and not too broad to be
+useless because including unrelated features in the field. T he
+FWHM was only changed when the galaxy was significantly
+smaller or larger (the extreme values for the sample being
+down to 15 and up to 45 pxl). If the width of the Gaussian
+is suited to the galactic extend, a difference of 2pxl FWHM
+does not induce significant changes in the asymmetry. In
+Fig. 4, six deconvolved images of QSOs and their hosts are
+shown along with their asymmetry coefficient. This illus-
+trates the relevance of using the asymmetry coefficient as a
+measure of the degree of interaction of a system.
+The global sample can be divided into two subsambles,
+one with a(V)/lessorequalslant0.7 (low asymmetry), and the other with
+a(V)>0.7 (high asymmetry). This value seems appropriate
+(see Fig. 4) for a good separation between minor interaction
+events and galactic-scale mergers involving several galax ies.
+The low-asymmetry subsample contains 60% of the sample.
+Let us mention that asymmetry coefficients were not com-
+puted for the archive data because hosts were not separated
+from their nucleus. They are thus not included in forthcom-
+ing analysis including asymmetries.
+Errors
+Errors on the asymmetry coefficient and cLhave been esti-
+mated by varying the input parameters (size of the square
+subimage in which we compute the coefficient and Gaussian
+FWHM) and see how it influences the result. First, it shows
+©2009 RAS, MNRAS 000, 1–18Study of a homogeneous QSO sample: relations between the QSO and its host galaxy 9
+thatcLis very stable (a 10% variation of the subimage size
+leads to a <1% variation of cL). Conversely, the asymmetry
+coefficient is quite sensitive to the Gaussian FWHM (a 25%
+variation in FWHM leads to a ≃15% variation in asym-
+metry coefficient). However, the restrictions for setting th e
+FWHM value explained above do not allow such large vari-
+ations, and thus we do not expect a change of more than
+>10% in the asymmetry coefficient, which is clearly suffi-
+cient for the present purpose.
+Two other sources of error must be taken into account: the
+variation in sampling due to the redshift range spanned,
+and S/N variations, which might change the influence of the
+background noise on the asymmetry.
+First of all, the exposure times were chosen to reach a com-
+parable S/N for all the hosts, thus considerably lowering th e
+risk of error due to S/N variations. Secondly, it might be ex-
+pected that higher- zhosts would show a lower asymmetry
+than their lower- zcounterparts, as the hosts substructures
+are less and less resolved with increasing redshift. In orde r to
+test the importance of this effect in our sample, we use the
+PYRAF tool “magnify” to decrease the resolution of the ob-
+servations ofthemostasymmetric hosts ( a(V)∼0.6) among
+the nearest objects ( z <0.15) from around 0.4 kpc/pxl to a
+0.52 kpc/pxl scale, corresponding to the upper limit of the
+sample,z= 0.198, and compute the asymmetry on this new
+image. The result is that the asymmetry coefficient drops
+at most by 0.02, revealing that the redshift range is small
+enough to avoid such biases in the estimates. We can thus
+neglect both of these effects. The calculated asymmetries
+with respect to cQSOandcLare given in Table 3 for both
+filters.
+5 RESULTS
+Now that the parameters have been introduced, we aim to
+find correlations between them and try to specify some as-
+pects of QSO-host interactions.
+5.1 QSO-Host absolute magnitudes relation
+A common question in QSO hosts studies is to seek
+if there is a correlation between QSO and host abso-
+lute magnitudes. Theoretically, such a trend is expected
+as a consequence of the more fundamental and well
+constrained Mass(Spheroid )-Mass(Black Hole ) relation
+(Ferrarese & Merritt 2000; Marconi and Hunt 2003). How-
+ever, it might be strongly disturbed by the variety of accre-
+tion rates found in AGN, which prevents from linking the
+black hole masses and magnitudes unequivocally. Equally,
+spheroid masses are not expected to match the galaxies
+magnitudes, as the latter contain significant supplementar y
+structures (tidal tails, spiral arms and bars, discs). Prev ious
+results (Hamilton et al. 2008; Floyd et al. 2004; Shade et al.
+2000) tend to indicate only a slight correlation, while oth-
+ers (Bahcall et al. 1997; Dunlop et al. 2003) find a relation
+compatible with no correlation at all. Our sample shows a
+weak correlation between host and QSO magnitudes (Fig.
+5, where our correlation is compared with mean relations
+from other samples, and the morphology classification is in-
+dicated). Namely, we find that, if < MV(QSO)>is theFigure 5. MV(Host) is plotted versus MV(QSO). Solid line:
+the best fit relation of our sample. Dashed line: fit from
+Hamilton et al. (2008). Dotted line: fit from Dunlop et al. (20 03).
+Filledsymbols represent non-interacting galaxies, while open ones
+are for galaxies showing signs of interaction. Triangles ar e for spi-
+rals, hexagones for ellipticals, whereas squares stand for unclassi-
+fied morphologies.
+Subclass a b p
+All 0 .21±0.10−22.20±0.11 0.28
+Spirals −0.28±0.39−22.35±0.26 0.20
+Spirals w/o inter. −0.5±0.61−22.26±0.32 0.26
+Ellipticals 0 .23±0.14−21.94±0.13 0.31
+Ellipticals w/o inter. 0 .17±0.17−22.02±0.20 0.29
+Interactions 0 .26±0.12−22.22±0.14 0.39
+Table 5. LinearMV(Host) =a(MV(QSO)−〈MV(QSO)〉) +b
+relation for different morphological subclasses. pis the Pearson’s
+correlation coefficient.
+mean value of the whole sample,
+MV(Host) = (0 .21±0.10)(MV(QSO)−< MV(QSO)>)
+−(22.20±0.11), (4)
+with a Pearson’s correlation coefficient p= 0.28 (the corre-
+lation is calculated with MV(QSO)−〈MV(QSO)〉instead
+ofMV(QSO) so that the intersection with the MV(Host)-
+axis corresponds to the average case where MV(QSO) =
+〈MV(QSO)〉).
+In Table 5, the relations found for different subclasses
+are presented with associated correlation coefficients, all
+globally weak. Brighter ellipticals most probably harbour
+brighter QSOs, as theoretically expected, but this does not
+hold for spirals, resulting in a poor correlation between
+MV(Host) andMV(QSO) for the whole sample, in agree-
+ment with the aforementioned previous studies.
+As explained above, the asymmetry coefficient is a good
+indicator of the degree of disturbance due to interaction in
+a system. Thus, we can replace the traditional morphology
+classification used in Fig. 5 by an asymmetry-based classifi-
+©2009 RAS, MNRAS 000, 1–1810Y. Letawe
+-28 -26 -24 -22 -20-25-24-23-22-21-20
+0 0.5 1 1.5 2 2.5-2-1012
+Figure 6. Top.MV(Host) is plotted versus MV(QSO) with the
+QSOs labelled according to their asymmetry subsample. Bott om.
+Deviations ∆ Mfrom the best fit are plotted against the asymme-
+try coefficient, with a linear fit on the postive and negative va lues
+of ∆Mseparately.
+cation. Figure 6 shows the MV(Host)−MV(QSO) relation,
+with this new subsample separation.
+We find that the high-asymmetry subsample shows a
+dispersion twice larger around the fitted relation. This tre nd
+is even clearer when we plot the deviations from the fit as a
+function of the asymmetry (Fig. 6, bottom). It shows a cor-
+relation between both variables indicating that the higher
+asymmetry hosts deviate more strongly from the fit.
+Moreover, the asymmetric hosts tend to lie on average below
+the fit in Fig. 6. It means that, for a given nucleus magni-
+tude, highly asymmetric systems tend to have brighter host
+galaxies. It can simply be explained by the fact that impor-
+tant mergers, corresponding to high a(V) values, contain
+two or more galaxies which boost the value of the computed
+magnitudes. More interesting are the systems which have a
+surprisingly low-luminosity host (or powerful nucleus) al ong
+with highly asymmetric features (open squares at the top of
+Fig. 6). Most probably, they are systems where interactions
+have most strongly enhanced the QSO activity and luminos-
+ity, as suggested for example in Letawe et al. (2007).
+It is also interesting to check if asymmetric hosts in a
+filter are also asymmetric in the other one. Figure 7 shows
+that for the vast majority of the QSOs observed, asymme-
+tries are, as expected, comparable in both filters. The best
+fit isa(WB) = (0.86±0.08)a(V) + (0.15±0.07), with a
+correlation coefficient p= 0.85. However, some objects lie
+way out of the mean relation. They will be discussed in-
+dividually in the Appendix. Figure 7 is useful to describe
+the link between the two classification methods (morphol-
+ogy vs asymmetry). It is clear that the most disturbed cases,
+morphologically undefined and indicated by an open square,
+also have the highest asymmetry value. Spirals and ellipti-
+cals have comparable asymmetries, whatever the filter. Ta-Subclass 〈a(V)〉σ(V)〈a(WB)〉σ(WB)
+Spirals 0.59 0.36 0.58 0.39
+Ellipticals 0.57 0.45 0.61 0.42
+Interactions 1.11 0.66 1.23 0.63
+Table 6. Average asymmetries are given in the two filters for
+spirals, ellipticals, and hosts with signs of interaction.
+Figure 7. Correlation between the asymmetry coefficients mea-
+sured in both filters. Note that all the high asymmetry sys-
+tems have also been classified as interacting systems. Fille d sym-
+bols represent non-interacting galaxies, while open ones a re for
+galaxies showing signs of interaction. Triangles are for sp irals,
+hexagones represent ellipticals, whereas squares stand fo r unclas-
+sified morphologies.
+ble 6 summarizes the mean asymmetries in both filters as a
+function of morphology.
+5.2 Colours
+As explained in Section 2, one of the goals of the study is
+to compare the stellar and gas distributions in order to seek
+for cases where there are significant differences between bot h
+components. We thus take the difference in apparent mag-
+nitudes between both filters, the colour ∆ m=mV−mWB,
+for the host and the QSO separately. As a preliminary com-
+ment, let us mention that the deconvolution process, as it
+treats both filters exactly in the same way for separating the
+QSO and host components, should have no influence on the
+derived colours. Furthermore, during the deconvolution pr o-
+cess, given the slight uncertainty on magnitudes induced by
+QSO-host separation as mentionned in Section 4.1.3, some
+flux from the QSO might be interpreted as galaxy flux. The
+QSO being bluer than galaxies, it would lead to an overes-
+timation of hosts blue colour compared to normal galaxies.
+Such a perturbation is however hard to estimate. It might
+lead to a slight shift of our hosts distribution in the his-
+togram towards the redder galaxies in both filter. It will
+nevertheless not change the conclusions obtained from the
+colour analysis.
+©2009 RAS, MNRAS 000, 1–18Study of a homogeneous QSO sample: relations between the QSO and its host galaxy 11
+Figure 8. Histogram of ∆ m(Host) with 0 .2 mag bins. Typical values for template galaxies from starbu rst to E types are indicated
+by the shaded strip. Hosts with the strongest emission lines appear to the left. Left: global sample. Middle: low and high -asymmetry
+subsamples, which have a different shape, the high asymmetry hosts being shifted to bluer colours, meaning that they disp lay both
+stronger emission lines and/or a bluer continuum. Right: el lipticals, spirals, and undefined morphology galaxies. Ell iptical hosts have the
+broadest range of colours.
+First of all, one might expect QSOs to be brighter
+in the V-filter, as it contains the strong H βand [OIII]
+emission lines. We find a mean colour for the QSOs of
+∆m(QSO) = 0.03±0.17. In order to check if this is
+not due to any processing error, the spectrum of a typi-
+cal QSO HE1302-1017 (Letawe et al. 2007) ( z= 0.278) has
+been blueshifted to z= 0.16 and integrated in both filters.
+Converting the flux ratios into magnitude differences leads
+∆m(HE1302-1017) = 0 .08, which lies well within our error
+bars.
+For the host galaxies, we find an average ∆ m(host) =
+0.41±0.58. The left panel of Fig. 8 shows the distribution
+of ∆mfor all our host galaxies. Some of these host galax-
+ies have very blue colours ∆ m <0, even bluer than the
+QSOs. To identify the origins of the blue light, we compare
+the colours of the sample with the colours of galaxy tem-
+plate spectra, estimated as for QSO colours in integrating
+spectra in both filters. We used galaxies of types E to Sc
+from Mannucci et al. (2001), and from Kinney et al. (1996)
+which constructed 6 starburst galaxies with different extin c-
+tion values (from SB1 to SB6 with increasing extinction).
+The associated colours are reported in Table 7 together with
+the colours for our sample.
+The colour is directly influenced by the strength of the
+ionization lines, but also by the slope of the continuum.
+First, we estimate an upper limit on the colour variation
+due to the continuum contribution by removing the ion-
+ization lines from the templates and compute ∆ magain.
+The values we obtained are presented in the third column
+of Table 7. It shows that the colour associated to the bluest
+slope for the continuum is estimated at 0.34. Moreover the
+slope of the continuum may be responsible for a variation
+in ∆mwhich should not go beyond 0 .5, which is the differ-
+ence between the reddest (E type) and bluest (SB1 type)
+spectra. On the other hand, placing upper limits on the
+emission lines contribution is harder, as QSO hosts might
+have stronger lines than the templates spectra. The host of
+HE1434-1600 is a good example of this as it has particu-
+larly strong emission lines with a relatively weak and red
+continuum (Letawe et al. 2004). From the spectrum of thisGalaxy ∆ m ∆m Diff. ∆ m
+Global Continuum Sample
+E 0.87 0.87 0 Ellipticals
+S0 0.86 0.86 0 0.27 ±0.78
+Sa 0.85 0.85 0
+Sb 0.76 0.77 -0.1 Spirals
+Sc 0.65 0.67 -0.02 0.60 ±0.15
+SB6 0.5 0.54 -0.04
+SB1 0.23 0.34 -0.11 Undefined
+HE1434-1600 0.27 0.63 -0.36 0.58 ±0.28
+Table 7. Colours estimated on galaxy template spectra and on
+the peculiar QSO host HE1434-1600 with a low, red continuum
+and strong emission lines (Letawe et al. 2004). Colours are g iven
+with and without the contribution of emission lines, togeth er with
+the difference in colours between both (total minus continuu m
+only). The last column gives the average values and scatter f or
+our QSO host galaxies as a function of morphology
+QSO, we compute ∆ m(Total) = 0.27 and, after removal of
+the ionization lines, ∆ m(Continuum ) = 0.63. So, the ion-
+ization lines may induce variations in ∆ mof at least 0 .36.
+In conclusion, both contributions influence the result at a
+relatively similar level. For colours lower thant 0.34, ion ized
+gas mustbepresent,andfor variations from templates large r
+than∼0.5 a bluer than expected colour cannot be explained
+only by young stellar population, also requiring the presen ce
+of ionized gas.
+In order to analyze the influence of morphology and
+asymmetry on the colour distribution, we showon Fig. 8his-
+tograms of the colours for different hosts subsamples (globa l
+sample and subsamples with different asymmetry or mor-
+phology, from left to right). We plot the numbers of hosts
+whose ∆ mis contained in bins of 0 .2 mag, from −2.to 1.5,
+andcompare themtothetemplate galaxy colours mentioned
+above.
+First, Fig. 8 shows that the range spanned in ∆ mfor
+our hosts is much larger that the range spanned by the tem-
+plates, only covering the 0 .3−0.9 region. Moreover, the ma-
+jority of the hosts seems to have colour typical of Starburst
+©2009 RAS, MNRAS 000, 1–1812Y. Letawe
+to Sc galaxies, characterized both by a large amount of ion-
+ized gas and a blue stellar continuum.
+Secondly, there is a tendency for the most asymmet-
+ric hosts to be bluer. Even if the dispersion is quite large
+for the high asymmetry sample, (∆ m(high-a(V) hosts) =
+0.17±0.81, whereas ∆ m(low-a(V) hosts) = 0 .61±0.24,
+leadingtouncertainties onthemean valuesof resp. ±0.1 and
+±0.04 given the size of these subsamples), we notice that the
+bluest host are highly asymmetric. These objects are inter-
+preted as hosts involved in major mergings, having a higher
+proportion of ionized gas and/or a bluer continuum than
+non-interacting ones. A similar tendancy has been found in
+Letawe et al. (2007, 2008b), where links between gas ionized
+by the QSO radiation and gravitational interactions are ob-
+served. Thus, it is tempting topropose that our observation s
+give a statistically more relevant proof of this kind of inte r-
+relation between QSOs and their hosts. However we cannot
+disentangle the different ionization processes, namely ion iza-
+tion by the QSO, by shocks, or by stars. Consequently, the
+higher degree of ionization in highly asymmetric hosts migh t
+also be interpreted as due to prominent HII regions typical
+of strong star formation, enhanced during the interaction
+process.
+Third, we point out the fact that all hosts that have
+∆m(Host)<0, thus the bluest ones, are ellipticals. It is
+a suprising result as ellipticals are often supposed to be
+evolved galaxies with old stellar populations and only a
+low proportion of gas and dust. This trend for elliptical
+QSO host galaxies to have bluer colours was already re-
+ported by Jahnke et al. (2004a) for similar objects, and by
+Sanchez et al. (2004) or Jahnke et al. (2004b) for samples
+at higher redshift, and was explained by a younger stel-
+lar component in the host. Spectral analysis of a sample
+of nearby QSO hosts in Letawe et al. (2007) also pointed
+to a younger-than-average stellar population. As the colou r
+variation due to the continuum (thus the stellar popula-
+tion) presented above reaches at most 0.5 mag, and as the
+bluest slope only leads to ∆ m=0.34, emission line contribu-
+tion has to be present in at least a dozen of outliers having
+∆m(Host)<0. They likely correspond to a class similar to
+the already studied QSO HE1434-1600 (Letawe et al. 2004,
+2008b), which has a seemingly normal elliptical host, but
+also filamentary structures consisting of gas ionized by the
+central nucleus on both sides of it. This system is in grav-
+itational interaction with a smaller neighbouring ellipti cal.
+Indeed, if the ionization was due to stars in those elliptica l
+outliers, it would be expected to be at least as strong in spi-
+rals, which is not the case. Thus, if an ionization source has
+to be favoured, QSO or shock ionization clearly seems to be
+preferred.
+We finally see in Fig. 8 (right panel) that spirals
+better match with classical values (inactive galaxies) tha n
+ellipticals, the latter being responsible for the broadeni ng
+of the range spanned by the whole sample.
+This hosts colour analysis reveals the differencebetween
+QSO hosts and quiescent galaxies. QSO hosts tend to have a
+bluer continuum, but also to contain more ionized gas than
+their quiescent counterparts, and this trend is stronger in
+ellipticals and highly asymmetric systems. It is consisten t
+with the previous study of Scoville et al. (2003), who find,
+by analysing CO emission of 12 low redshift ( z <0.1) PGQSOs, that elliptical hosts are gas-rich and therefore can-
+not be normal ellipticals. Our findings prove that not only
+elliptical hosts contain more gas, but that this gas has a
+significant level of ionization compared to quiescent ellip ti-
+cals. However, the available data do not allow to determine
+unambiguously the ionization source. For instance, the ion -
+ization might be due either to the QSO radiation or to shock
+waves during a merger event.
+5.3 Magnitude-Asymmetry relations
+If mergers and/or gravitational interactions are to trigge r
+or enhance QSO activity, we might expect a correlation
+between the QSO magnitude and the degree of asymme-
+try of the system. The relation is shown in Fig. 9 for el-
+lipticals and spirals separately because they have a differ-
+ent behaviour. Indeed, for the ellipticals, a reliable cor-
+relation ( p= 0.54) exists between the absolute magni-
+tude and the asymmetry coefficient (solid line in Fig. 9):
+MV(QSO) = (−1.10±0.36)a(V) + (−23.22±0.26).It is
+even more robust ( p= 0.68, dashed line) if the three higher
+asymmetry systems are removed from the fit, with a relation
+MV(QSO) = (−2.51±0.61)a(V)+(−22.67±0.30).(5)
+A similar relation is not found for the host magnitude. Con-
+versely,theasymmetryofspiralhostsdoesnotcorrelate wi th
+QSO magnitude, but does correlate with host magnitude as
+MV(Host) = (−1.09±0.34)a(V) + (−21.89±0.24),with
+p= 0.71.
+For ellipticals, in the low asymmetry range ( a(V)<
+0.7), asymmetry is mainly due to shifts between the posi-
+tion of the QSO ( cQSO) and the center of luminosity of the
+host (cL). To illustrate this statement, we plot on Fig. 10
+the asymmetry according to cQSO(aQSO(V), left plot) and
+the asymmetry according to cL(aHost(V), right plot) versus
+the distance between both centers cQSOandcL. While the
+distance is strongly correlated to aQSO(V), it is not so much
+the case for aHost(V), proving that, for elliptical hosts, the
+asymmetry is due to relatively symmetrical galaxies which
+are not centered on the nucleus position. In the framework
+of galaxy evolution via mergers, it is expected that ellipti -
+cals are created during major mergers involving galaxies of
+similar masses. Such mergers are also likely to trigger the
+activity of the nucleus. Thus, as the merger of the galaxies
+is faster than the merger of the black holes first orbiting
+around each other, it would not be surprizing to detect off-
+centered activity in apparently relaxed galaxies, where th e
+two SBMH are still sufficiently far from each other. Alterna-
+tively, an off-centered activity might also be due to a black
+hole recoil with its accretion disc after the fusion of a blac k
+holes binary. Simulations from Volonteri & Madau (2008)
+predict the existence and observability of such off-centere d
+activity.
+For spirals, the observed correlation with a(V) concerns
+MV(Host) instead of MV(QSO). Tidal disturbances in spi-
+rals are believed to arise either from minor mergers between
+different mass galaxies (4 : 1 to 10 : 1 or higher mass ratios,
+see Bournaud et al. (2004)) or in young mergers which have
+not evolved during enough time to reach their final state.
+The lack of correlation found between asymmetry and QSO
+magnitude supports the idea that young or minor merg-
+ers have not influenced the activity of the central nucleus
+©2009 RAS, MNRAS 000, 1–18Study of a homogeneous QSO sample: relations between the QSO and its host galaxy 13
+Figure 9. Asymmetry is plotted against QSO and host absolute
+magnitude in the V filter for ellipticals and spirals separat ely.
+Open symbols indicate the presence of interactions or merge rs.
+Figure 10. Left (resp. right): distance in kpc between the center
+of luminosity of the galaxy and the QSO position against the
+asymmetry coefficient computed on the QSO position (resp. the
+center of luminosity of the host), for elliptical galaxies o nly.
+yet. This idea matches well with the concluding remark in
+Reichard et al. (2009), who suggest that the period of black
+hole growth may be preferencially associated with the end
+stages of minor mergers. On the other hand, morphological
+disturbances are already visible and contribute to raise th e
+asymmetry in proportion to their luminosity, giving rise to
+the observed correlation.6 CONCLUSIONS
+We have studied a sample of 69 QSOs, amongst which 60
+have been observed with the ESO NTT and SUSI2. Our
+QSO-host separation technique, based on the MCS decon-
+volution method, proves well adapted for that kind of data.
+A morphology classification method based on the asymme-
+try of the system has enabled us to compare the properties
+of mergers and non interacting hosts. More precisely, high
+asymmetry systems tend to have a higher degree of ioniza-
+tion and a bluer continuum than low asymmetry systems.
+Another interesting finding is that the QSO hosts tend to
+contain on average more ionized gas than quiescent galaxies .
+This trend is especially strong for ellipticals. It is consi stent
+with the scenario in which AGN host galaxies lie in a transi-
+tion phase in between the red and blue modes in the colour-
+magnitude diagram (Martin et al. 2007), as these emissions
+will make the elliptical QSO hosts appear bluer than their
+quiescent counterparts. Nevertheless, our data do not allo w
+to firmly determine if the ionization source is the QSO itself
+or shocks induced during merger events.
+The relation between the QSO and host magnitudes is quite
+weak, and mainly found in ellipticals. The correlation ob-
+served is probably only a remnant of the more fundamental
+black hole-spheroid relation. We also find that high asym-
+metry systems cover a wider range of host magnitudes at a
+given QSO magnitude, which is interpreted either as lumi-
+nosity excess due to galactic fusion or to particularly pow-
+erful nuclear activity.
+At this stage, we cannot firmly determine unequivocally
+which process is responsible for QSO triggering. Our study
+enlightens links between interaction processes, QSO activ ity
+and gas ionization, but it would be risky to assess unequiv-
+ocally any hierarchical structure between those process. I n-
+deed, our study does not allow to detect explicitely gas in-
+falling to the central regions due to the interaction proces s.
+However, we find that the nuclear activity during a merg-
+ing process is different for ellipticals and spirals, reinfo rcing
+the current belief that they correspond to different stages
+of evolution in a QSO lifetime, which fits a merger-driven
+evolution scheme.
+A few ellipticals have an off-centered activity which might
+be due to a black hole recoil or to a merger induced activity
+between similar mass galaxies. Moreover, some particular
+cases are reported for the first time and described in more
+detail in the Appendix. They seem to have experienced a
+wide variety of merging scenarios, and certainly deserve fu r-
+ther investigations. Finally, we report the first image of th e
+underlying host of PDS456, and discuss the possibility that
+this system may contain a double QSO.
+As future works, thestudyof a sample of inactive galax-
+ies in similar redshift andbrightness ranges with similar s pa-
+tial resolution and filters will help clarifying several ope n
+questions, such as the frequency of mergers or the spread
+in colours compared to templates. A detailed comparison of
+thenewly introducedasymmetrycoefficient toother existing
+asymmetry measurements is also planned.
+ACKNOWLEDGMENTS
+This work was supported by PRODEX Experiment Agree-
+ment 90312 (ESA and PPS Science Policy, Belgium).
+©2009 RAS, MNRAS 000, 1–1814Y. Letawe
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+©2009 RAS, MNRAS 000, 1–18Study of a homogeneous QSO sample: relations between the QSO and its host galaxy 15
+5"
+Figure A1. Observations of QSO 1307+085 in both filters. The
+top row (resp. bottom) shows, from left to right, an observat ion,
+the deconvolved image (with the special features discussed encir-
+cled), and the residual map for the V#812 filter (resp. WB#665 ).
+Yun M., Reddy N.A., Scoville N.Z. et al., 2004, ApJ, 601,
+723
+Zheng X.Z., Hammer F., Flores H. Ass´ emat F., Pelat D.,
+2004, A& A, 421, 847
+This paperhasbeentypesetfrom aT EX/LATEXfileprepared
+by the author.
+APPENDIX A: PECULIAR CASES
+One of the goals of the present study is to find candi-
+date hosts which would represent a particular stage of evo-
+lution in a QSO’s lifetime (for example, similar to the
+now famous HE0450-2958 case studied in Magain et al.
+(2005); Letawe et al. (2008b, 2009); Elbaz et al. (2009);
+Jahnke et al. (2009)), in preparation for higher resolution
+observations which would go deeper into the central regions
+and would allow to better assess the nature of QSO-host in-
+teractions. In the present Appendix, we review some special
+cases, which are essentially outliers from the main trends
+described in the previous section. Note that all images have
+a zero position angle (North is up, and East is to the right),
+and all indicated distances are projected distances.
+A1 1307+085
+First of all, the host of this QSO lies amongst the very few
+systems which have substantially different asymmetry co-
+efficients in the two filters ( a(V) = 1.29,a(WB) = 0.28).
+Secondly, the very blue QSO colour ∆ m(QSO) =−0.5 sug-
+gests that the nucleus has strong emission lines compared
+to the continuum. Moreover, the comparison of the decon-
+volved images in both filters (Fig. A1) reveals two zones
+(encircled in the figure) in the host that show a prominent
+emission in the V-filter not seen in the continuum, which
+we thus associate to strong emission lines. Those elements
+put the host in the subclass of asymmetric ellipticals with
+substantial gas ionization, where the strong emission zone s
+increase the asymmetry coefficient and the magnitude in the
+V-filter, whereas the stellar component looks like a rather
+regular elliptical. An emission zone appears in both filters
+at≃22.5kpc West from the nucleus, close to the left edge
+of Fig. A1. However we see no obvious trace of interaction
+between this feature and the host.
+2"
+Figure A2. Observations of QSO Q0022−2044 in both filters.
+The top row (resp. bottom) shows, from left to right, an obser va-
+tion, the deconvolved image, and the residual map for the V#8 12
+filter (resp. WB#665)
+A2 Q0022-2044
+Q0022-2040 has also substantially different asymmetries in
+both filters ( a(V) = 0.73,a(WB) = 1.45). The high asym-
+metry in the WB#665 filter is due to a shift of 2 pixels = 0 .9
+kpc between the center of luminosity of the galaxy and the
+position of thenucleus,as shownin Fig. A2.This shift, alon g
+with the very blue host colour ∆ m(Host) =−1.94, indicate
+that Q0022-2044 is a typical example of the class of ellipti-
+cals with strong gas emission lines harbouring off-centered
+activity already discussed in Section 5.3. Let us also men-
+tion that the QSO is particularly bright in the WB#665
+filter (∆m(QSO) = 0.41), suggesting a very red continuum
+and weak emission lines.
+A3 PDS456
+PDS456 is known as the most powerful QSO in the local
+Universe ( z <0.3). It is thought to be a radio-quiet ana-
+logue of the famous 3 C273 (Schmidt 1963). Spectra have al-
+ready been obtained in basically all wavelength ranges and
+K-band, radio and CO(1-0) images are also available (see
+O’Brien et al. (2005) or Yun et al. (2004) for a complete re-
+view and references). It was suggested that it is in a crit-
+ical transition phase between an Ultra Luminous InfraRed
+Galaxy (ULIRG) and a QSO (Sanders & Mirabel 1996), as
+it shows an optical spectrum dominated by broad emission
+lines, large IR and X-ray luminosities, as well as a large
+dust/cold gas content. Moreover, UV and X-ray spectra re-
+veal the presence of decelerating cooling outflows probably
+driven by radiation or magnetic field. The deconvolved im-
+age (Fig. A3) shows two compact sources indicated by E1
+and E2,situated a sim2 ercseconds S-W of the QSO, and
+compatible with the three blended sources of the K-band
+image in Yun et al. (2004). Deconvolution allows to disen-
+tangle the underlying host, which seems elliptically shape d.
+It is, to our knowkedge, the first image of the galaxy host-
+ingthisQSOinthevisible. Magnitudes andasymmetries are
+given in Table 3 for the whole system (E1+E2+Host), and
+decomposition in the different components is given in Table
+A1. The difference in magnitude between the host and the
+©2009 RAS, MNRAS 000, 1–1816Y. Letawe
+m(QSO)m(Host)m(E1)m(E2)
+V 13.42 17.56 19.35 19.27
+WB 13.10 16.43 18.53 18.34
+m(S1)m(S2)
+V 14.17 14.17 17.56 19.35 19.27
+WB 13.31 14.99 17.10 18.73 18.54
+Table A1. Magnitudes of the different components of the decon-
+volution of PDS456 with one (top) or two (bottom) point sourc es
+at the center of the host.
+QSO (−4.14) is the highest of the sample, which indicates a
+particularly strong activity of the nucleus.
+Let us now focus on the residuals map shown in Fig. A3 for
+the V filter. It has a structure typical of the presence of two
+blended point sources with similar intensities. Indeed, if a
+point source is to be fit where in fact there are two closely
+blended ones, the two peaks will not be correctly fit and this
+will result in a double hole in the residual map (model minus
+observation). If such a feature is real, and does not corre-
+spond to a PSF mismatch on that particular observation,
+we expect to find it in each exposure. Figure A4 shows an
+average of the four residuals resulting from the simultaneo us
+deconvolution of 4 observations. The double-hole structur e
+is clearly present. We tested the hypothesis of a double nu-
+cleus by adding in the deconvolution process a second point
+source, along with the usual diffuse background. The result
+is displayed on Fig. A5. IntheV filter, thetwopoint sources,
+of comparable magnitude MV∼ −27.45 are only separated
+by 0.51kpc and the host nearly disappears, with only a faint
+tail starting from the nucleus. For the WB filter, the separa-
+tion is only 0 .22kpc, and a faint host, whose morphology can
+hardly be determined because of its faintness and its small
+angular size, is detected. Magnitudes in both filters for the
+two point sources fit are given in the bottom part of Table
+A1.
+Let us now examine arguments in favour and against
+the double active nucleus hypothesis. The very good quality
+of the deconvolution with twopoint sources makes clear that
+there is at least a very compact source in both filters next
+to the “main” nucleus. Moreover, for the one point source
+fit, the value ∆ m(Host) = 1.13 is the highest of our sample
+and surpasses the typical elliptical value, indicating an a b-
+normally red continuum. This strong compact continuum is
+better fitted with a point source, in coherence with an hypo-
+thetical AGN activity. The presence of two separate nuclei
+might explain its high luminosity, and is not inconsistent
+with the accepted scenario of a transitory system between
+ULIRG and QSO via a merging process. It could also ac-
+count for the exceptionally broad absorption (O’Brien et al .
+2005) and emission (Yun et al. 2004) features observed.
+Conversely,ifbothsourcesweretocorrespondtoAGNactiv-
+ity, we would expect a common center in both filters, which
+is not really the case ( δc∼0.32kpc). Also, their magnitudes
+behave differently in each filter (see Table A1), their relati ve
+intensities being much more different in the WB filter than
+in the V filter. Moreover, the value ∆ m(Host) = 1.13 might
+be interpreted as due to a strong reddening of the host,
+which would match the ULIRG hypothesis. Thus, from our
+observations, it is risky to assess their true nature unequi v-
+ocally.
+2"
+Figure A3. Observations of QSO PDS456 in both filters. The
+top row (resp. bottom) shows, from left to right, an observat ion,
+the deconvolved image, and the residual map for the V#812 filt er
+(resp. WB#665).
+2"
+Figure A4. Average residuals from the simultaneous deconvolu-
+tion of 4 observations of the QSO PDS456 in the V filter. The
+two holes in the residuals suggest the presence of a second po int
+source.
+All in all, it is clear that PDS456 is an exceptional object,
+but maybe it is exceptional in a presently unexpected way.
+Deep high resolution optical imaging, or 3D integral field
+spectroscopy, processed with a similar method as the one
+used in Letawe et al. (2008b), might help clarifying its sta-
+tus.
+A4 PG 1012+008
+From Fig. A6, it is clear that PG1012+008 is involved in a
+merger containing at least 2 galaxies separated by ≃10kpc,
+and a third one 20kpc N-W from the QSO, which might
+also be interacting gravitationally. The best fit is obtaine d
+by using 3 point sources in the deconvolution, located at
+each galactic center position. However, Bahcall et al. (199 7)
+have already observed this system with HST and WFPC2,
+and no other point source was found. This difference may
+be explained in the following way. Given the difference in
+resolution between both observations (0 .1”/pxl for WFPC2
+compared to 0 .161”/pxl for SUSI2), a very compact and in-
+tense stellar emission just resolved by WFPC2 might look
+unresolved in our SUSI2 observations. At the QSO redshift,
+the range of sizes in which an emission zone would be re-
+solved with WFPC2 but not with SUSI2 is 0 .2−0.5kpc.
+We analysed the morphology of those two putative point-
+©2009 RAS, MNRAS 000, 1–18Study of a homogeneous QSO sample: relations between the QSO and its host galaxy 17
+V#812
+WB#665
+2"
+Figure A5. Deconvolved image in both filters of PDS456, with
+two point sources very near from each other. The average resi d-
+uals are considerably improved, without any significant str ucture
+remaining.
+5"
+Figure A6. Deconvolution process for PG1012 + 008, where 3
+point sources are included, which don’t necessarily corres pond to
+any AGN-activity.
+like sources using the WFPC2 archives. Figure A7 reveals
+that the point-like source nearest from the QSO looks like a
+compact emission zone whose FWHM is 4 about pixels. The
+QSOFWHM in that image beingonly 2 pixels, this emission
+is clearly not point-like. However, this 4 pxl width convert s
+in SUSI2 pixels to 2.5 pxl FWHM, which is very close to
+the resolution fixed in the deconvolution process (2 pixels
+FWHM), making it look very much like a point-source at
+this resolution. Moreover, the flux ratio between the QSO
+and the compact emission is the same ( ≃0.045) as the flux
+ratio between the QSO and the point source in the SUSI2
+images, reinforcing the hypothesis that the compact emis-
+sion inWFPC2 andthesecondpointsource in SUSI2are the
+same object. The same arguments hold for the third point
+source. Another indication that the additional compact re-
+gions are not related to any AGN activity is that the colours
+computed for the nearest galaxy center and the other one
+are ∆m= 0.57 and 0 .62. Those values are higher than for
+any QSO in our the whole sample, revealing an important
+contribution of the continuum compared to emission lines.
+This fact tends to favour the view that the 2 added point
+Figure A7. PG1012 + 008 observed with HST/WFPC2 and a
+zoom on the companion galaxy containing a compact, but not
+point-like, emission zone.
+5"
+Figure A8. Deconvolution of 1151+117, with 3 point sources.
+The S-W one is probably a foreground star, whereas the two oth -
+ers are hosted by interacting galaxies.
+sources do not correspond to AGN activity, but rather to
+compact luminous stellar regions, characterized by a promi -
+nent continuum.
+A5 1151+117
+Apart from the central QSO, the observation of 1151+117
+(Fig. A8) reveals the presence of two emitting regions
+20.6kpc S-W, and 11 .5kpc N-E of the QSO. Both need
+a point source component to achieve good residuals. De-
+convolution reveals that the S-E one is well fit by a sin-
+gle point source, suggesting it might be a foreground star.
+On the other hand, the N-E one needs a smooth back-
+ground to achieve a good fit. This point source is quite weak
+(MV=−19.32), and has a dominant continuum emission
+∆m= 0.33. Similarly to PG1012+008, this high value of
+∆mmight indicate that the point-like emission is due to a
+compact stellar region, even if it is still within the range o f
+reasonable value for an active nucleus.
+The background needed around the second point source
+shows clear signs of gravitational interactions with the ho st,
+especially on the WB#665 deconvolved image, where the
+galaxies seem to share atidal tail, suggesting a configurati on
+similar to the well known M51 (Salo & Laurikainen 2000;
+Toomre & Toomre 1972), where two galaxies of mass ratio
+3 : 1, viewed nearly face-on, probably encountered for the
+first time 300 ±100 Myr ago. Two additional faint emissions
+are also present in both filters (N-W and N), and may also
+be linked to the merging process.
+©2009 RAS, MNRAS 000, 1–1818Y. Letawe
+5"
+Figure A9. Deconvolution process for HE1202 −0501. The very
+atypical morphology may be created by the collision of two ga lax-
+ies of different disc mass. The rather poor quality of the resi duals
+is likely due to significant short-scale structures near the center.
+A6 HE1202-0501
+The host of HE1202-0501 (Fig. A9) is very atypical, with a
+45kpcnearly straight tail extendingfrom thecenter toward s
+the S, and containing a few minor substructures. Another
+much weaker tail of ∼27 kpc length extends from the center
+to the N-W. Such a special configuration is very similar to
+the shape of a 1 .2−1.3 Gyr old major merger of nearly equal
+mass disc galaxies (Springel et al. (2005),Fig. 7).These si m-
+ulations also show very complex intense features next to the
+center. If such structures are prensent in HE1202-0501, the y
+might not be resolved by SUSI2, but could substantially
+contribute to the observed total central nucleus luminosit y,
+leading to PSF mismatches and thus decreasing the quality
+of the deconvolution around the center, as can be seen in
+the residuals shown in Fig. A9. HE1202 −0501 also looks
+very similar to the famous “Mice” merger, NGC4746, which
+is also well modelled (Toomre & Toomre 1972; Mihos et al.
+1993; Barnes 2004) by a two identical mass disc galaxies
+merger event, where one of the two discs is viewed edge-on
+and mimics an elongated bar. They are a few major differ-
+ences between NGC4746 and HE1202-0501. In “The Mice”,
+the galactic centers are separated by 20kpc, and both tails
+haveapproximatelythesame lengthandluminosity,whereas
+HE1202-0501 shows onlyone nucleus,anddifferentsize tails .
+Different tail size and luminosities are suggestive of differ -
+ent mass galaxies. We thus conclude that HE1202-0501 is
+most probably a merger between two disc galaxies of differ-
+ent masses containing a sufficient amount of gas to induce
+AGN activity, one of the two discs being viewed edge-on.
+The poor residuals near the center may result from intense
+star formation activity around the nucleus.
+A7 1023-014
+The QSO 1023 −014 is involved in a major merger involv-
+ing two galaxies of comparable size. The best deconvolu-
+tion is achieved with 3 point sources, labelled A, B and
+C on Fig. A10. While A and B have are very similar in
+both filters ( mV(A) = 19.54, mV(B) = 19.67, ∆m(A) =
+−0.06, ∆m(B) =−0.01), C has a quite different behaviour,
+withmV(C) = 20.55 and ∆ m(C) = 0.51, which deviates
+5"A
+B
+C
+Figure A10. Deconvolution process for the major merger of the
+QSO 1023 −014. 3 point sources are necessary to achieve satis-
+factory residuals.
+strongly from the mean ∆ m(QSO). As in the case of PG
+1012 +008, it indicates that the source C might not be re-
+lated to AGNactivity butrather tostrong unresolved stella r
+emission. Moreover, this system lies significantly below th e
+MV(QSO)-MV(Host) relation, having an underluminous
+AGN compared to its host. MV(QSO)−MV(Host) = 1.27
+is the highest value of the whole sample, the nuclear magni-
+tude being indeed below the standard delimitation between
+QSO and Seyfert. All in all, and given the fact that the
+QSO and total host are more powerful in the WB#655 fil-
+ter (the one containing no emission line), 1023 −014 could
+be a rather dry merger, with no sufficient amount of gas to
+create powerful AGN activity, but possibly one or two low
+luminosity AGNs. Further studies, includingspectroscopy of
+the nucleus, would be necessary to test this hypothesis.
+A8 HE1211-1905
+The QSO HE1211-1905 is surrounded by a spiral galaxy
+which shows clear signs of gravitational interaction via ex -
+tended tails N and E of the nucleus, corresponding to the
+T1 and T2 boxes in Fig. A11. Given the significant area of
+those tails ( ∼10kpc2), it makes sense to check differences
+in magnitudes between both filters in those regions in order
+to specify their nature.
+The analysis reveals that T1 and T2 are very similar, as
+they both have ∆ m=−0.05. This negative value is much
+lower than the value computed for the whole host (0 .5). It
+indicates that ionized gas is more prominent in the faraway
+regions T1andT2thaninthehost’s central regions. Thetail
+T1 goes even beyond the extracted image. Fig. A12 shows
+a larger area around the QSO, where the size of T1 can be
+estimated to ∼40 kpc. The cause for such disturbances is
+unclear because HE1211-1905 appears quite lonely, appart
+from a tiny galaxy, probably elliptical, lying 35 kpc from it .
+Let us mention that there are also a few spots in the over-
+all direction of that companion galaxy, which may also be
+linked to the system. Another possibility is that the system
+is a late-stage merger, where the two galaxies nuclei have
+rapidly merged, leaving a host not yet relaxed.
+©2009 RAS, MNRAS 000, 1–18Study of a homogeneous QSO sample: relations between the QSO and its host galaxy 19
+5"
+Figure A11. Deconvolution process for the strongly disturbed
+host of the QSO HE1211-1905.
+Figure A12. Surrounding of HE1211 −1905. The tidal tail of
+∼40kpc is surprisingly long. A companion galaxy and a few spot s
+which are present on the same side may be linked to the host
+disturbances.
+©2009 RAS, MNRAS 000, 1–18
\ No newline at end of file
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@@ -0,0 +1,904 @@
+arXiv:1001.0464v3  [cs.CC]  8 Aug 2011HOLANT PROBLEMS FOR REGULAR GRAPHS WITH COMPLEX
+EDGE FUNCTIONS
+MICHAEL KOWALCZYK1AND JIN-YI CAI2
+1Department of Mathematics and Computer Science, Northern M ichigan University
+Marquette, MI 49855, USA
+E-mail address :mkowalcz@nmu.edu
+2Computer Sciences Department, University of Wisconsin, Ma dison, WI 53706, USA
+E-mail address :jyc@cs.wisc.edu
+Abstract. We prove a complexity dichotomy theorem for Holant Problems on 3-regular
+graphs with an arbitrary complex-valued edge function. Thr ee new techniques are intro-
+duced: (1) higher dimensional iterations in interpolation ; (2) Eigenvalue Shifted Pairs,
+which allow us to prove that a pair of combinatorial gadgets in combination succeed in
+proving #P-hardness; and (3) algebraic symmetrization, wh ich significantly lowers the
+symbolic complexity of the proof for computational complexity. With holographic reduc-
+tionsthe classification theorem also applies to problems beyond t he basic model.
+1. Introduction
+In this paper we consider the following subclass of Holant Pr oblems [5, 6]. An in-
+put regular graph G= (V,E) is given, where every e∈Eis labeled with a (sym-
+metric) edge function g. The function gtakes 0-1 inputs from its incident nodes and
+outputs arbitrary values in C. The problem is to compute the quantity Holant( G) =/summationtext
+σ:V→{0,1}/producttext
+{u,v}∈Eg({σ(u),σ(v)}).
+Holant Problems are a natural class of counting problems. As introduced in [5, 6],
+the general Holant Problem framework can encode all Countin g Constraint Satisfaction
+Problems (#CSP). This includes special cases such as weight edVertex Cover ,Graph
+Colorings ,Matchings , andPerfect Matchings . The subclass of Holant Problems
+in this paper can also be considered as (weighted) H-homomorphism (or H-coloring) prob-
+lems[2,3,7,8,9,10]withanarbitrary2 ×2symmetriccomplexmatrix H, however restricted
+toregular graphs Gas input. E.g., Vertex Cover is the case when H=/bracketleftbigg
+0 1
+1 1/bracketrightbigg
+. When
+the matrix His a 0-1 matrix, it is called unweighted. Dichotomy theorems (i.e., the prob-
+lem is either in P or #P-hard, depending on H) for unweighted H-homomorphisms with
+undirected graphs Hand directed acyclic graphs Hare given in [8] and [7] respectively. A
+dichotomy theorem for any symmetric matrix Hwith non-negative real entries is proved
+1998 ACM Subject Classification: F.2.1.
+Key words and phrases: Computational complexity.
+The second author is supported by NSF CCF-0830488 and CCF-09 14969.
+c/circlecopyrtM. KowalczykandJ-Y. Cai
+CC/circlecopyrtCreativeCommonsAttribution-NoDerivsLicense2 M. KOWALCZYK AND J-Y. CAI
+in [2]. Goldberg et al. [9] proved a dichotomy theorem for all real symmetric matrices H.
+Finally, Cai, Chen, and Lu have proved a dichotomy theorem fo r all complex symmetric
+matrices H[3].
+Thecrucial difference between Holant Problems and #CSPis tha t in #CSP, Equality
+functions of arbitrary arity are presumed to bepresent. In terms of H-homomorphism prob-
+lems, this means that the input graph is allowed to have verti ces of arbitrarily high degrees.
+This may appear to be a minor distinction; in fact it has a majo r impact on complexity. It
+turns out that if Equality gates of arbitrary arity are freely available in possible in puts
+then it is technically easier to prove #P-hardness. Proofs o f previous dichotomy theorems
+make extensive use of constructions called thickening and s tretching. These constructions
+require the availability of Equality gates of arbitrary arity (equivalently, vertices of ar-
+bitrarily high degrees) to carry out. Proving #P-hardness b ecomes more challenging in
+the degree restricted case. Furthermore there are indeed ca ses within this class of count-
+ing problems where the problem is #P-hard for general graphs , but solvable in P when
+restricted to 3-regular graphs.
+We denote the (symmetric) edge function gby [x,y,z], wherex=g(0,0),y=g(0,1) =
+g(1,0) andz=g(1,1). Functions will also be called gates or signatures. (For Vertex
+Cover, the function corresponding to His theOrgate, and is denoted by the signature
+[0,1,1].) In this paper we give a dichotomy theorem for the complex ity of Holant Problems
+on 3-regular graphs with arbitrary signature g= [x,y,z], where x,y,z∈C. First, if y= 0,
+the Holant Problem is easily solvable in P. Assuming y/ne}ationslash= 0 we may normalize gand assume
+y= 1. Our main theorem is as follows:
+Theorem 1.1. Suppose a,b∈C, and let X=ab,Z= (a3+b3
+2)2. Then the Holant Problem
+on 3-regular graphs with g= [a,1,b]is #P-hard except in the following cases, for which the
+problem is in P.
+(1)X= 1.
+(2)X=Z= 0.
+(3)X=−1andZ= 0.
+(4)X=−1andZ=−1.
+If we restrict the input to planar 3-regular graphs, then thes e four categories are solvable in
+P, as well as a fifth category X3=Z. The problem remains # P-hard in all other cases.1
+These results can be extended to k-regular graphs (we detail how this is accomplished
+in a forthcoming work). Onecan also useholographic reducti ons [16] to extend this theorem
+to more general Holant Problems.
+In order to achieve this result, some new proof techniques ar e introduced. To discuss
+this we first take a look at some previous results. Valiant [14 , 15] introduced the powerful
+technique of interpolation , which was further developed by many others. In [5] a dichoto my
+theorem is proved for the case when gis a Boolean function. The technique from [5] is to
+provide certain algebraic criteria which ensure that interpolation succeeds, and then apply
+these criteria to prove that (a large number yet) finitely man y individual problems are #P-
+hard. This involves (a small number of) gadget construction s, and the algebraic criteria
+are powerful enough to show that they succeed in each case. No netheless this involves a
+1Technically, computational complexity involving complex or real numbers should, in the Turing model,
+be restricted to computable numbers. In other models such as the Blum-Shub-Smale model [1] no such
+restrictions are needed. Our results are not sensitive to th e exact model of computation.HOLANT PROBLEMS FOR REGULAR GRAPHS WITH COMPLEX EDGE FUNCTI ONS 3
+case-by-case verification. In [6] this theorem is extended t o all real-valued aandb, and
+we have to deal with infinitely many problems. So instead of fo cusing on one problem, we
+devised (a large number of) recursive gadgets and analyzed t he regions of ( a,b)∈R2where
+they fail to prove #P-hardness. The algebraic criteria from [5] are not suitable (Galois
+theoretic) for general aandb, and so we formulated weaker but simpler criteria. Using
+these criteria, the analysis of the failure set becomes expr essible as containment of semi-
+algebraic sets. As semi-algebraic sets are decidable, this offers the ultimate possibility that
+ifwe found enough gadgets to prove #P-hardness, thenthere is a computational proof (of
+computational intractability) in a finite number of steps. H owever this turned out to be a
+tremendous undertaking in symbolic computation, and many a dditional ideas were needed
+to finally carry out this plan. In particular, it would seem ho peless to extend that approach
+to all complex aandb.
+In this paper, we introduce three new ideas. (1) We introduce a method to construct
+gadgets that carry out iterations at a higher dimension, and then collapse to a lower di-
+mension for the purpose of constructing unary signatures. T his involves a starter gadget, a
+recursive iteration gadget, and a finisher gadget. We prove a lemma that guarantees that
+among polynomially many iterations, some subset of them sat isfies properties sufficient for
+interpolation to succeed (it may not be known a prioriwhich subset worked, but that does
+not matter). (2) Eigenvalue Shifted Pairs are coupled pairs of gadgets whose transition ma-
+trices differ by δIwhereδ/ne}ationslash= 0. They have shifted eigenvalues, and by analyzing their fa ilure
+conditions, we can show that except on very rare points, one o r the other gadget succeeds.
+(3) Algebraic symmetrization. We derive a new expression of the Holant polynomial over
+3-regular graphs, with a crucially reduced degree. This sim plification of the Holant and
+related polynomials condenses the problem of proving #P-ha rdness to the point where all
+remaining cases can be handled by symbolic computation. We a lso use the same expression
+to prove tractability.
+The rest of this paper is organized as follows. In Section 2 we discuss notation and
+background information. In Section 3 we cover interpolatio n techniques, including how to
+collapse higher dimensional iterations to interpolate una ry signatures. In Section 4 we show
+how to perform algebraic symmetrization of the Holant, and i ntroduce Eigenvalue Shifted
+Pairs (ESP) of gadgets. Then we combine the new techniques to prove Theorem 1.1.
+2. Notations and Background
+Westate thecountingframework moreformally. A signature grid Ω = (G,F,π) consists
+of a labeled graph G= (V,E) whereπlabels each vertex v∈Vwith a function fv∈ F. We
+consider all edge assignments ξ:E→ {0,1};fvtakes inputs from its incident edges E(v)
+atvand outputs values in C. The counting problem on the instance Ω is to compute2
+Holant Ω=/summationdisplay
+ξ/productdisplay
+v∈Vfv(ξ|E(v)).
+Suppose Gisabipartitegraph( U,V,E) suchthateach u∈Uhasdegree2. Furthermore
+suppose each v∈Vis labeled by an Equality gate = kwherek= deg(v). Then any non-
+zero term in Holant Ωcorresponds to a 0-1 assignment σ:V→ {0,1}. In fact, we can merge
+the two incident edges at u∈Uinto one edge eu, and label this edge euby the function
+fu. This gives an edge-labeled graph ( V,E′) whereE′={eu:u∈U}. For an edge-labeled
+2The term Holant was first introduced by Valiant in [16] to deno te a related exponential sum.4 M. KOWALCZYK AND J-Y. CAI
+graph (V,E′) wheree∈E′has label ge, Holant Ω=/summationtext
+σ:V→{0,1}/producttext
+e=(v,w)∈E′ge(σ(v),σ(w)).
+If eachgeis the same function g(but assignments σ:V→[q] take values in a finite set [ q])
+this is exactly the H-coloring problem (for undirected graphs gis a symmetric function).
+In particular, if ( U,V,E) is a (2,k)-regular bipartite graph, equivalently G′= (V,E′) is a
+k-regular graph, then this is the H-coloring problem restricted to k-regular graphs. In this
+paper we will discuss 3-regular graphs, where each geis the same symmetric complex-valued
+function. We also remark that for general bipartite graphs ( U,V,E), giving Equality (of
+various arities) to all vertices on one side Vdefines #CSP as a special case of Holant
+Problems. But whether Equality of various arities are present has a major impact on
+complexity, thus Holant Problems are a refinement of #CSP.
+A symmetric function g:{0,1}k→Ccan be denoted as [ g0,g1,...,gk], where giis
+the value of gon inputs of Hamming weight i. They are also called signatures . Frequently
+we will revert back to the bipartite view: for (2 ,3)-regular bipartite graphs ( U,V,E), if
+everyu∈Uis labeled g= [g0,g1,g2] and every v∈Vis labeled r= [r0,r1,r2,r3],
+then we also use #[ g0,g1,g2]|[r0,r1,r2,r3] to denote the Holant Problem. Note that
+[1,0,1] and [1 ,0,0,1] areEquality gates = 2and =3respectively, and the main dichotomy
+theorem in this paper is about #[ x,y,z]|[1,0,0,1], for all x,y,z∈C. We will also denote
+Hol(a,b) = #[a,1,b]|[1,0,0,1]. More generally, If GandRare sets of signatures, and
+vertices of U(resp.V) are labeled by signatures from G(resp.R), then we also use
+#G | Rto denote the bipartite Holant Problem. Signatures in Gare called generators and
+signatures in Rare called recognizers . This notation is particularly convenient when we
+perform holographic transformations. Throughout this pap er, all (2 ,3)-regular bipartite
+graphs are arranged with generators on the degree 2 side and r ecognizers on the degree 3
+side.
+WeuseArgtodenotetheprincipalvalueofthecomplexargume nt; i.e., Arg( c)∈(−π,π]
+for all nonzero c∈C.
+2.1.F-Gate
+Any signature from Fis available at a vertex as part of an input graph. Instead of a
+single vertex, we can use graph fragments to generalize this notion. An F-gate Γ is a pair
+(H,F), where H= (V,E,D) is a graph with some dangling edges D(Figure 1 contains
+some examples). Other than these dangling edges, an F-gate is the same as a signature
+grid. The role of dangling edges is similar to that of externa l nodes in Valiant’s notion [17],
+however we allow more than one dangling edge for a node. In H= (V,E,D) each node is
+assigned a function in F(we do not consider “dangling” leaf nodes at the end of a dangl ing
+edge among these), Eare the regular edges, and Dare the dangling edges. Then we can
+define a function for this F-gate Γ = ( H,F),
+Γ(y1,y2,...,yq) =/summationdisplay
+(x1,x2,...,xp)∈{0,1}pH(x1,x2,...,x p,y1,y2,...,yq),
+wherep=|E|,q=|D|, (y1,y2,...,yq)∈ {0,1}qdenotes an assignment on the dangling
+edges, and H(x1,x2,...,x p,y1,y2,...,yq) denotes the value of the partial signature grid
+on an assignment of all edges, i.e., the product of evaluatio ns at every vertex of H, for
+(x1,x2,...,x p,y1,y2,...,yq)∈ {0,1}p+q. We will also call this function the signature of the
+F-gate Γ. An F-gate can be used in a signature grid as if it is just a single no de with theHOLANT PROBLEMS FOR REGULAR GRAPHS WITH COMPLEX EDGE FUNCTI ONS 5
+(a) A starter gadget (b) A recursive gadget (c) A finisher gadget (d) A planar embedding
+of a single iteration
+Figure 1: Examples of binary starter, recursive, and finishe r gadgets
+same signature. We note that even for a very simple signature setF, the signatures for all
+F-gates can bequitecomplicated andexpressive. Matchgate s ignatures are an example[17].
+The dangling edges of an F-gate are considered as input or output variables. Any
+m-inputn-outputF-gate can be viewed as a 2nby 2mmatrixMwhich transforms arity- m
+signatures into arity- nsignatures (this is true even if mornare 0). Our construction will
+transform symmetric signatures to symmetric signatures. T his implies that there exists an
+equivalent n+1 bym+1 matrix /tildewiderMwhich operates directly on column vectors written in
+symmetric signature notation. We will henceforth identify the matrix /tildewiderMwith the F-gate
+itself. The constructions in this paper are based upon three different types of bipartite
+F-gates which we call starter gadgets ,recursive gadgets , andfinisher gadgets . Anarity-r
+starter gadget is anF-gate with no input but routput edges. If an F-gate has rinput
+androutput edges then it is called an arity-rrecursive gadget . Finally, an F-gate is an
+arity-rfinisher gadget if it hasrinput edges 1 output edge. As a matter of convention, we
+consider any dangling edge incident with a generator as an ou tput edge and any dangling
+edge incident with a recognizer as an input edge; see Figure 1 .
+3. Interpolation Techniques
+3.1. Binary recursive construction
+In this section, we develop our new technique of higher dimen sional iterations for in-
+terpolation of unary signatures.
+Lemma 3.1. Suppose M∈C3×3is a nonsingular matrix, s∈C3is a nonzero vector,
+and for all integers k≥1,sis not a column eigenvector of Mk. LetFi∈C2×3be three
+matrices, where rank(Fi) = 2for1≤i≤3, and the intersection of the row spaces of
+Fiis trivial {0}. Then for every n, there exists some F∈ {Fi: 1≤i≤3}, and some
+S⊆ {FMks: 0≤k≤n3}, such that |S| ≥nand vectors in Sarepairwise linearly
+independent.
+Proof.Letk > j≥0 be integers. Then MksandMjsare nonzero and also linearly
+independent, since otherwise sis an eigenvector of Mk−j. LetN= [Mjs,Mks]∈C3×2,
+then rank( N) = 2, and ker( NT) is a 1-dimensional linear subspace. It follows that there
+exists an F∈ {Fi: 1≤i≤3}such that the row space of Fdoes not contain ker( NT),
+and hence has trivial intersection with ker( NT). In other words, ker( NTFT) ={0}. We
+conclude that FN∈C2×2has rank 2, and FMjsandFMksare linearly independent.6 M. KOWALCZYK AND J-Y. CAI
+EachFi, where 1 ≤i≤3, defines a coloring of the set K={0,1,...,n3}as follows:
+colork∈Kwith the linear subspace spanned by FiMks. Thus,Fidefines an equivalence
+relation≈iwherek≈ik′iff they receive the same color. Assume for a contradiction th at
+for each Fi, where 1 ≤i≤3, there are not npairwise linearly independent vectors among
+{FiMks:k∈K}. Then, including possibly the 0-dimensional space {0}, there can be at
+mostndistinct colors assigned by Fi. By the pigeonhole principle, some kandk′with
+0≤k < k′≤n3must receive the same color for all Fi, where 1 ≤i≤3. This is a
+contradiction and we are done.
+The next lemma says that under suitable conditions we can con struct all unary sig-
+natures [ x,y]. The method will be interpolation at a higher dimensional i teration, and
+finishing up with a suitable finishergadget. The crucial new technique here is that when
+iterating at a higher dimension, we can guarantee the existe nce ofonefinisher gadget that
+succeeds on polynomially many steps, which results in overa ll success. Different finisher
+gadgets may work for different initial signatures and different input size n, but these need
+not be known in advance and have no impact on the final success o f the reduction.
+Lemma 3.2. Suppose that the following gadgets can be built using comple x-valued signatures
+from a finite generator set Gand a finite recognizer set R.
+(1)A binary starter gadget with nonzero signature [z0,z1,z2].
+(2)A binary recursive gadget with nonsingular recurrence matr ixM, for which [z0,z1,z2]T
+is not a column eigenvector of Mkfor any positive integer k.
+(3)Three binary finisher gadgets with rank 2 matrices F1,F2,F3∈C2×3, where the
+intersection of the row spaces of F1,F2, andF3is the zero vector.
+Then for any x,y∈C,#G ∪{[x,y]} | R ≤ T#G | R.
+Proof.The construction begins with the binary starter gadget with signature [ z0,z1,z2],
+which we call N0. LetF=G ∪R. Recursively, F-gateNk+1is defined to be Nkconnected
+to the binary recursive gadget in such a way that the input edg es of the binary recursive
+gadget are merged with the output edges of Nk. Then F-gateGkis defined to be Nk
+connected to one of the finisher gadgets, with the input edges of the finisher gadget merged
+with the output edges of Nk(see Figure 1(d)). Herein we analyze the construction with
+respect to a given bipartite signature grid Ω for the Holant P roblem # G∪{[x,y]} | R, with
+underlying graph G= (V,E). LetQ⊆Vbe the set of vertices with [ x,y] signatures, and
+letn=|Q|. By Lemma 3.1 fix jso that at least n+ 2 of the first ( n+ 2)3+ 1 vectors
+of the form FjMk[z0,z1,z2]Tare pairwise linearly independent. We use finisher gadget Fj
+in the recursive construction, so that the signature of GkisFjMk[z0,z1,z2]T, which we
+denote by [ Xk,Yk]. We note that there exists a subset Sof these signatures for which each
+Ykis nonzero and |S|=n+1. We will argue using only the existence of S, so there is no
+need to algorithmically “find” such a set, and for that matter , one can try out all three
+finisher gadgets without any need to determine which finisher gadget is “the correct one”
+beforehand. If we replace every element of Qwith a copy of Gk, we obtain an instance of
+#G | R(note that the correct bipartite signature structure is pre served), and we denote
+this new signature grid by Ω k. Then
+Holant Ωk=/summationdisplay
+0≤i≤nciXi
+kYn−i
+k
+whereci=/summationtext
+σ∈Ji/producttext
+v∈V\Qfv(σ|E(v)),Jiis the set of {0,1}edge assignments where the
+number of 0s assigned to the edges incident to the copies of Gkisi,fvis the signature at v,HOLANT PROBLEMS FOR REGULAR GRAPHS WITH COMPLEX EDGE FUNCTI ONS 7
+andE(v) is the set of edges incident to v. The important point is that the civalues do not
+depend on XkorYk. Since each signature grid Ω kis an instance of # G | R, Holant Ωkcan be
+solved exactly using the oracle. Carrying out this process f or every k∈ {0,1,...,(n+2)3},
+we arrive at a linear system where the civalues are the unknowns.
+
+Holant Ω0
+Holant Ω1...
+Holant Ω(n+2)3
+=
+X0
+0Yn
+0 X1
+0Yn−1
+0 ··· Xn
+0Y0
+0
+X0
+1Yn
+1 X1
+1Yn−1
+1 ··· Xn
+1Y0
+1............
+X0
+(n+2)3Yn
+(n+2)3X1
+(n+2)3Yn−1
+(n+2)3···Xn
+(n+2)3Y0
+(n+2)3
+
+c0
+c1
+...
+cn
+.
+For 0≤i≤n, letkisuch that S={[Xk0,Yk0],[Xk1,Yk1],...,[Xkn,Ykn]}, and let [ xi,yi] =
+[Xki,Yki]. Then we have a subsystem
+
+y−n
+0·Holant Ωk0
+y−n
+1·Holant Ωk1...
+y−n
+n·Holant Ωkn
+=
+x0
+0y0
+0x1
+0y−1
+0···xn
+0y−n
+0
+x0
+1y0
+1x1
+1y−1
+1···xn
+1y−n
+1............
+x0
+ny0
+nx1
+ny−1
+n···xn
+ny−n
+n
+
+c0
+c1
+...
+cn
+.
+The matrix above has entry ( xr/yr)cat index ( r,c). Due to pairwise linear independence
+of [xr,yr],xr/yris pairwise distinct for 0 ≤r≤n. Hence this is a Vandermonde system
+of full rank. Therefore the initial feasible linear system h as full rank and we can solve it
+for thecivalues. With these values in hand, we can calculate Holant Ω=/summationtext
+0≤i≤ncixiyn−i
+directly, completing the reduction.
+The ability to simulate all unary signatures will allow us to prove #P-hardness. The
+next lemma says that, if Rcontains the Equality gate = 3, then other than on a 1-
+dimensional curve ab= 1 and an isolated point ( a,b) = (0,0), the ability to simulate unary
+signatures gives a reduction from Vertex Cover . Note that counting Vertex Cover on
+3-regular graphs is just #[0 ,1,1]|[1,0,0,1]. Xia et al. showed that this is #P-hard even
+when the input is restricted to 3-regular planar graphs [18] . We will see shortly that on the
+curveab= 1 and at ( a,b) = (0,0), the problem Hol( a,b) is tractable.
+Lemma 3.3. Suppose that (a,b)∈C2−{(a,b) :ab= 1}−{(0,0)}and letGandRbe finite
+signature sets where [a,1,b]∈ Gand[1,0,0,1]∈ R. Further assume that #G∪{[xi,yi] : 0≤
+i < m} | R ≤ T#G | Rfor anyxi,yi∈Candm∈Z+. Then#G∪{[0,1,1]} | R ≤ T#G | R,
+and#G | Ris#P-hard.
+Proof.Assumeab/ne}ationslash= 1 and ( a,b)/ne}ationslash= (0,0). Since Hol(0 ,1) (which is the same as #[0 ,1,1]|
+[1,0,0,1], or counting vertex covers on 3-regular graphs) is #P-har d, we only need to show
+how to simulate the generator signature [0 ,1,1]. We split this into three cases, and use a
+chain of three reductions, each involving a gadget in Figure 2 (each gadget has [1 ,0,0,1]
+assigned to the degree 3 vertices and [ a,1,b] assigned to the degree 2 vertices).
+(1)ab/ne}ationslash= 0 and ab/ne}ationslash=−1
+(2)ab= 0
+(3)ab=−1
+Ifab/ne}ationslash= 0 and ab/ne}ationslash=−1, then we use Gadget 3, and we set its unary signatures to
+beθ= [(ab+ 1)/(1−ab),−a2(ab+ 1)/(1−ab)],γ= [−a−2,b−1(1 +ab)−1], andρ=
+[−b/(ab−1),a/(ab−1)]. Calculating the resulting signature of Gadget 3, we find that it is
+[0,1,1] as desired.8 M. KOWALCZYK AND J-Y. CAI
+Ifab= 0 then assume without loss of generality that a= 0 andb/ne}ationslash= 0. This time we use
+Gadget 1, setting θ= [b,b−1]. Then Gadget 1 simulates a [ b−1,1,2b] generator signature,
+but since this signature fits the criteria of case 1 above, we a re done by reduction from that
+case.
+Similarly, if ab=−1,thenGadget2exhibitsageneratorsignatureoftheform[0 ,1,5/(2a)]
+under the signatures θ= [1/(6a),−a/24] andγ= [−3/a,a]. Since 5 /(2a) is nonzero, we are
+done by reduction from case 2.
+θ
+(a) Gadget 1γ θ γ
+(b) Gadget 2ρ γ θ γ ρ
+(c) Gadget 3
+Figure 2: Gadgets used to simulate the [0,1,1] signature
+It will be helpful to have conditions that are easier to check than those in Lemma
+3.2. To this end, we establish condition 2 in terms of eigenva lues, and we build general-
+purpose finisher gadgets to eliminate condition 3. Let M4,M5, andFbe the recurrence
+matrices for Gadget 4, Gadget 5, and the simplest possible bi nary finisher gadget (each
+built using generator signature [ a,1,b] and recognizer signature [1 ,0,0,1]; see Figures 3(a),
+3(b), and 1(c)). Provided that ab/ne}ationslash= 1 and a3/ne}ationslash=b3, it turns out that the finisher gadget sets
+{F,FM 4,FM2
+4}and{F,FM 4,FM5}satisfy condition 3 of Lemma 3.2 when ab/ne}ationslash= 0 and
+ab= 0, respectively. Together with Lemma 3.3, these observati ons yield the following.
+Theorem 3.4. If the following gadgets can be built using generator [a,1,b]and recognizer
+[1,0,0,1]wherea,b∈C,ab/ne}ationslash= 1, anda3/ne}ationslash=b3, then the problem Hol(a,b)is#P-hard.
+(1)A binary recursive gadget with nonsingular recurrence matr ixMwhich has eigen-
+valuesαandβsuch thatα
+βis not a root of unity.
+(2)A binary starter gadget with signature swhich is not orthogonal to any row eigen-
+vector of M.
+Proof.First we show how to build general-purpose binary finisher ga dgets for the main
+construction using the assumed generator and recognizer, s tarting first with the case where
+ab/ne}ationslash= 0. Using the simplest possible choice for a finisher gadget F(Figure 1(c)), we get
+F=/bracketleftbigg
+a0 1
+1 0b/bracketrightbigg
+. LetM4be the recurrence matrix for binary recursive Gadget 4 (Figu re
+3(a)), and we calculate that
+M4=
+a32a b
+a2ab+1b2
+a2b b3
+.
+We build two more finisher gadgets F′andF′′using Gadget 4 so that F′=FM4and
+F′′=FM2
+4. SinceFandM4both have full rank (note det( M4) =ab(ab−1)3), it follows
+thatF′andF′′also have full rank. Now we will show that the row spaces of F,F′and
+F′′have trivial intersection, and it suffices to verify that the c ross products of the row
+vectors of F,F′, andF′′(denoted respectively by v,v′, andv′′) are linearly independent.
+(To see this, suppose uis a complex vector in the intersection of the row spaces of F1,HOLANT PROBLEMS FOR REGULAR GRAPHS WITH COMPLEX EDGE FUNCTI ONS 9
+F′
+1, andF′′
+1. Then v,v′, andv′′are all orthogonal to u, but since v,v′, andv′′are
+linearly independent, they span the conjugate vector uwhich is then also orthogonal to
+u. This means |u|2=uu= 0, and that u= 0.) The cross products of the row vectors
+ofF,F′, andF′′are [0,1−ab,0], (ab−1)2[2b2,−ab(ab+1),2a2], and (ab−1)3[2b(a2b3+
+a2+ab2+b4),−ab(a3b3+ 2a3+ 2a2b2+ab+ 2b3),2a(a4+a3b2+a2b+b2)] respectively.
+Then to see that these 3 vectors are linearly independent, it suffices to verify that the
+matrix/bracketleftbigg
+2b22a2
+2b(a2b3+a2+ab2+b4) 2a(a4+a3b2+a2b+b2)/bracketrightbigg
+is nonsingular. Since a/ne}ationslash= 0,
+b/ne}ationslash= 0, and ab/ne}ationslash= 1, we just check det/bracketleftbigg
+b a
+a2b3+a2+ab2+b4a4+a3b2+a2b+b2/bracketrightbigg
+=
+(ab−1)(a3−b3)/ne}ationslash= 0, so the row spaces of F,F′, andF′′have trivial intersection when
+ab/ne}ationslash= 0.
+Now suppose ab= 0. Since a3/ne}ationslash=b3, by symmetry, if ab= 0 we may assume without loss
+of generality that a/ne}ationslash= 0 and b= 0. Let M5be the recurrence matrix for binary recursive
+Gadget 5 (Figure 3(b)).
+M5=
+a6+2a3+1 2a4+2a a2
+a5+a22a3+1a
+a42a21
+.
+Composing FwithM5, we get a finisher gadget with matrix FM5, which has full rank since
+Fhas full rank and det( M5) = 1. It is also straightforward to see that F′has full rank, as
+F′=/bracketleftbigg
+a4+a2a20
+a32a0/bracketrightbigg
+. The cross products of the rows of F,F′, andFM5are [0,1,0],
+[0,0,2a2], and [−2a,2a3+1,−2a2(1+a)(a2−a+1)] respectively. Then the matrix of cross
+products is clearly nonsingular, and we conclude that for an ya,b∈C, we have 3 finisher
+gadgets satisfying item 3 of Lemma 3.2 unless ab= 1 ora3=b3.
+Now we want to show that sis not a column eigenvector of Mkfor any positive integer
+k(note that sis nonzero by assumption). Writing out the Jordan Normal For m forM, we
+haveMks=T−1DkTs, whereDkhas the form
+αk0 0
+0βk0
+0∗ ∗
+, and where αandβare
+eigenvalues of Mfor whichα
+βis not a root of unity. Let t=Tsand write t= [c,d,e]T.
+By hypothesis, sis not orthogonal to the first two rows of T, thusc,d/ne}ationslash= 0. Ifswere an
+eigenvector of Mkfor some positive integer k, thenT−1DkTs=Mks=λsfor some nonzero
+complex value λ, andDkt=T(λs) =λt. But then cαk=λcanddβk=λd, which means
+αk
+βk= 1, contradicting the fact thatα
+βis not a root of unity.
+We have now met all the criteria for Lemma 3.2, so the reductio n #S∪{[a,1,b],[x,y]} |
+[1,0,0,1]≤T#S ∪{[a,1,b]} |[1,0,0,1] holds for any x,y∈Cand any finite signature set
+S. By Lemma 3.3 the problem Hol( a,b) is #P-hard.
+3.2. Unary recursive construction
+Now we consider the unary case. The following lemma arrives f rom [13] and is stated
+explicitly in [6]. It can be viewed as a unary version of Lemma 3.2 without finisher gadgets.
+Lemma 3.5. Suppose there is a unary recursive gadget with nonsingular m atrixMand a
+unary starter gadget with nonzero signature vector s. If the ratio of the eigenvalues of Mis10 M. KOWALCZYK AND J-Y. CAI
+not a root of unity and sis not a column eigenvector of M, then these gadgets can be used
+to interpolate all unary signatures.
+Surprisingly, a set of general-purpose starter gadgets can be made for this construction
+as long as ab/ne}ationslash= 1 and a3/ne}ationslash=b3, so we refine this lemma by eliminating the starter gadget
+requirement. The starter gadgets are Fs,FM4s, andFM6swhereM6is Gadget 6 and sis
+the single-vertex starter gadget (see Figures 3(c) and 1(a) ).
+Theorem 3.6. Suppose there is a unary recursive gadget with nonsingular m atrixM, and
+the ratio of the eigenvalues of Mis not a root of unity. Then for any a,b∈Cwhereab/ne}ationslash= 1
+anda3/ne}ationslash=b3, there is a starter gadget built using generator [a,1,b]and recognizer [1,0,0,1]
+for which the resulting construction can be used to interpol ate all unary signatures.
+Proof.LetM4,M6,F, andsbe Gadget 4, Gadget 6, the binary finisher gadget in Figure
+1(c), and single-vertex binary starter gadget (Figure 1(a) ), respectively. Note s= [a,1,b]T
+and
+M6=
+a3+1 0 a2+b
+a2+b0a+b2
+a+b20b3+1
+.
+Using MathematicaTM, we verify that the block matrices [ FM4s Fs], [FM6s Fs], and
+[FM4s FM6s] are all nonsingular provided that ab/ne}ationslash= 1 and a3/ne}ationslash=b3, so the vectors Fs,
+FM4s, andFM6sare pairwise linearly independent. If the ratio of the eigen values of Mis
+not a root of unity, then the eigenvalues of Mare distinct, each eigenvalue correspondsto an
+eigenspace of dimension 1, and at least one element of {FM4s,FM6s,Fs}is not a column
+eigenvector of M. The corresponding starter gadget can be used with Min a recursive
+construction and the result follows from Lemma 3.5.
+(a) Gadget 4 (b) Gadget 5 (c) Gadget 6 (d) Gadget 7 (e) Gadget 8 (f) Gadget 9
+Figure 3: Binary recursive gadgets
+4. Complex Signatures
+Now we aim to characterize Hol( a,b) wherea,b∈C. The next lemma introduces the
+technique of algebraic symmetrization. We show that over 3- regular graphs, the Holant
+value is expressible as an integer polynomial P(X,Y), where X=abandY=a3+b3.
+This change of variable, from ( a,b) to (X,Y), is crucial in two ways. First, it allows us to
+derive tractability results easily, drawing connections b etween problems that may appear
+unrelated, and the tractability of one implies the other. Se cond, it facilitates the proof of
+hardness for those ( a,b) where the problem is indeed #P-hard by reducing the degree o f
+the polynomials involved. Once this transformation is made , four binary recursive gadgetsHOLANT PROBLEMS FOR REGULAR GRAPHS WITH COMPLEX EDGE FUNCTI ONS 11
+easily cover all of the #P-hard problems where XandYare real-valued, with a straightfor-
+ward symbolic computation using CylindricalDecomposition in MathematicaTM. All
+gadget constructions in this section use [ a,1,b] and [1,0,0,1] signatures exclusively, and we
+henceforth denote X=abandY=a3+b3for the remainder of this paper.
+Lemma 4.1. LetGbe a 3-regular graph. Then there exists a polynomial P(·,·)with two
+variables and integer coefficients such that for any signatur e gridΩhaving underlying graph
+Gand every edge labeled [a,1,b], the Holant value is Holant Ω=P(ab,a3+b3).
+Proof.Consider any {0,1}vertex assignment σwith a non-zero valuation. If σ′is the
+complement assignment switching all 0’s and 1’s in σ, then for σandσ′, we have the
+sum of valuations aibj+ajbifor some iandj. Herei(resp.j) is the number of edges
+connecting two degree 3 vertices both assigned 0 (resp. 1) by σ. We note that aibj+ajbi=
+(ab)min(i,j)(a|i−j|+b|i−j|).
+We prove i≡j(mod 3) inductively. For the all-0 assignment, this is clear since every
+edge contributes a factor aand the number of edges is divisible by 3 for a 3-regular graph .
+Now starting from any assignment σ, if we switch the assignment on one vertex from 0 to 1,
+it is easy to verify that it changes the valuation from aibjtoai′bj′, wherei−j=i′−j′+3.
+As every {0,1}assignment isobtainable fromtheall-0 assignment by asequ enceof switches,
+the conclusion i≡j(mod 3) follows.
+Nowaibj+ajbi= (ab)min(i,j)(a3k+b3k), for some k≥0 and a simple induction a3(k+1)+
+b3(k+1)= (a3k+b3k)(a3+b3)−(ab)3(a3(k−1)+b3(k−1)) shows that the Holant is a polynomial
+P(ab,a3+b3) with integer coefficients.
+Corollary 4.2. IfX=−1andY∈ {0,±2i}, thenHol(a,b)is inP.
+Proof.The problems Hol(1 ,−1), Hol(−i,−i), and Hol( i,i) are all solvable in P (these fall
+within the families F1,F2, andF3in [3]);X=−1 for each, whereas the value of Yfor
+these problems is 0, 2 i, and−2irespectively. Since the value of any 3-regular signature gr id
+is completely determined by X,Y, and the polynomial P(·,·) (which in turn depends only
+on the underlying graph G), anyaandbsuch that ab=−1 anda3+b3∈ {0,±2i}(i.e.
+ab=−1 anda12= 1) is computable in polynomial time.
+We now list all of the cases where Hol( a,b) is computable in polynomial time.
+Theorem 4.3. If any of the following four conditions is true, then Hol(a,b)is solvable in
+P:
+(1)X= 1,
+(2)X=Y= 0,
+(3)X=−1andY∈ {0,±2i}
+(4) 4X3=Y2and the input is restricted to planar graphs.
+Proof.IfX= 1 then the signature [ a,1,b] is degenerate and the Holant can be computed
+in polynomial time. If X=Y= 0 then a=b= 0, and a 2-coloring algorithm can be
+employed on the edges. If X=−1 andY∈ {0,±2i}then we are done by Corollary 4.2. If
+we restrict the input to planar graphs and 4 X3=Y2(equivalently, a3=b3), holographic
+algorithms can be applied [4].
+Our main task in this paper is to prove that all remaining prob lems are #P-hard. The
+following two lemmas provide sufficient conditions to satisf y the eigenvalue requirement of
+the recursive constructions.12 M. KOWALCZYK AND J-Y. CAI
+(a) Gadget 10 (b) Gadget 11 (c) Gadget 12 (d) Gadget 13 (e) Gadget 14 (f) Gadget 15 (g) Gadget 16
+Figure 4: Unary recursive gadgets
+Lemma 4.4. If both roots of the complex polynomial x2+Bx+Chave the same norm,
+thenB|C|=BCandB2C=B2C. If further B/ne}ationslash= 0andC/ne}ationslash= 0, thenArg(B2) = Arg( C).
+Proof.If the roots have equal norm, then for some a,b∈Cand nonnegative r∈Rand we
+can write x2+Bx+C= (x−ra)(x−rb), where |a|=|b|= 1, soB|C|=−r(a+b)r2=
+−r(a−1+b−1)r2ab=BC. Squaring both sides and dividing by C, we have B2C=B2C
+(note this is justified since this equality still holds when C= 0). Multiplying B|C|=BC
+byBwe getB2|C|=|B2|C, and ifBandCare both nonzero thenB2
+|B2|=C
+|C|, that is,
+Arg(B2) = Arg( C).
+Lemma 4.5. If all roots of the complex polynomial x3+Bx2+Cx+Dhave the same norm,
+thenC|C|2=B|B|2D.
+Proof.If the roots have equal norm, then for some a,b,c∈Cand nonnegative r∈Rwe
+can write x3+Bx2+Cx+D= (x−ra)(x−rb)(x−rc), where |a|=|b|=|c|= 1, so
+B=−r(a+b+c),C=r2(ab+bc+ca), andD=−r3abc. Then
+C|C|2=r2(ab+bc+ca)r4|ab+bc+ca|2=r(a+b+c)r2|a+b+c|2r3abc=B|B|2D,
+where we used the fact that |ab+bc+ca|=|ab+bc+ca|·|a−1b−1c−1|=|a−1+b−1+c−1|=
+|a+b+c|=|a+b+c|.
+Now we introduce a powerful new technique called Eigenvalue Shifted Pairs .
+Definition 4.6. A pair of nonsingular square matrices MandM′is called an Eigenvalue
+Shifted Pair (ESP) ifM′=M+δIfor some non-zero δ∈C, andMhas distinct eigenvalues.
+Clearly for such a pair, M′also has distinct eigenvalues. The recurrence matrices of
+Gadgets 10and11(Figure4)differonlyby ab−1alongthediagonal, andformanEigenvalue
+Shifted Pair for nearly all a,b∈C. We will make significant use of such Eigenvalue Shifted
+Pairs, but first we state a technical lemma.
+Lemma 4.7. Suppose α,β,δ∈C,|α|=|β|,α/ne}ationslash=β,δ/ne}ationslash= 0, and|α+δ|=|β+δ|. Then
+there exists r,s∈Rsuch that rδ=α+βandsδ2=αβ.
+Proof.After a rotation in the complex plane, we can assume α=β, and then since α+
+β,αβ∈Rwe just need to prove δ∈R. Then ( α+δ)(α+δ) =|α+δ|2=|β+δ|2=
+(β+δ)(β+δ) = (α+δ)(α+δ) andwe distributeto get αα+δδ+αδ+αδ=αα+δδ+αδ+αδ.
+Canceling repeated terms and factoring, we have ( α−α)(δ−δ) = 0, and since α/ne}ationslash=β=α
+we know δ=δtherefore δ∈R.HOLANT PROBLEMS FOR REGULAR GRAPHS WITH COMPLEX EDGE FUNCTI ONS 13
+Corollary 4.8. LetMandM′be an Eigenvalue Shifted Pair of 2by2matrices. If both
+MandM′have eigenvalues of equal norm, then there exists r,s∈Rsuch that tr(M) =rδ
+(possibly 0) anddet(M) =sδ2.
+Proof.Letαandβbe the eigenvalues of M, soα+δandβ+δare the eigenvalues of M′.
+Suppose that |α|=|β|and|α+δ|=|β+δ|. Then by Lemma 4.7, there exists r,s∈Rsuch
+that tr(M) =α+β=rδand det(M) =αβ=sδ2.
+We now apply an ESP to prove that most settings of Hol( a,b) are #P-hard.
+Lemma 4.9. Suppose X/ne}ationslash=±1,X2+X+Y/ne}ationslash= 0, and4(X−1)2(X+1)/ne}ationslash= (Y+2)2. Then
+either unary Gadget 10 or unary Gadget 11 has nonzero eigenva lues with distinct norm,
+unlessXandYare both real numbers.
+Proof.Gadgets 10 and 11 have M10=/bracketleftbigg
+a3+1a+b2
+a2+b b3+1/bracketrightbigg
+andM11=/bracketleftbigg
+a3+ab a+b2
+a2+b ab+b3/bracketrightbigg
+as their recurrence matrices, so M11=M10+(X−1)I, and the eigenvalue shift is nonzero.
+Checking the determinants, det( M10) = (X−1)2(X+1)/ne}ationslash= 0 and det( M11) = (X−1)(X2+
+X+Y)/ne}ationslash= 0. Also, tr( M10)2−4det(M10) = (Y+ 2)2−4(X−1)2(X+ 1)/ne}ationslash= 0, so the
+eigenvalues of M10are distinct. Therefore by Corollary 4.8, either M10orM11has nonzero
+eigenvalues of distinct norm unless tr( M10) =r(X−1) and det( M10) =s(X−1)2for
+somer,s∈R. Then we would have ( X−1)2(X+1) =s(X−1)2soX=s−1∈Rand
+Y+2 =r(X−1) soY=r(X−1)−2∈R.
+Now we will deal with the following exceptional cases from Le mma 4.9 ( X= 1 is
+tractable by Theorem 4.3).
+0.X∈RandY∈R
+1.X2+X+Y= 0
+2.X=−1
+3. 4(X−1)2(X+1) = (Y+2)2
+The case where XandYare both real is dealt with using the tools developed in
+Section 3, and some symbolic computation. This includes the case where aandbare both
+real as a subcase. When aandbare both real, a dichotomy theorem for the complexity of
+Hol(a,b) has been proved in [6] with a significant effort. With the new to ols developed, we
+offer a simpler proof. This also covers some cases where aorbis complex. Working with
+real-valued XandYis a significant advantage, since the failure condition give n by Lemma
+4.5 is simplified by the disappearance of norms and conjugate s. This brings the problem of
+proving #P-hardness within reach of symbolic computation v ia cylindrical decomposition.
+We apply Theorem 3.4 to Gadgets 4, 7, 8, and 9 (Figure 3) togeth er with a starter gadget
+(Figure 1(a)) to prove that these problems are #P-hard. Cond itions 1 and 2 of Theorem
+3.4 are encoded directly into a query for CylindricalDecomposition in MathematicaTM,
+but first we give a lemma to show how to encode condition 2 of Lem ma 3.4.
+Lemma 4.10. Suppose M∈Cn×nands∈Cn×1. Ifdet([s,Ms,M2s,...,Mn−1s])/ne}ationslash= 0
+thensis not orthogonal to any row eigenvector of M.
+Proof.Suppose sis orthogonal to a row eigenvector vofMwith eigenvalue λ. Then
+v[s,Ms,...,Mn−1s] = 0, since vMis=λivs= 0. Since v/ne}ationslash= 0 this is a contradiction.14 M. KOWALCZYK AND J-Y. CAI
+Theorem 4.11. Suppose a,b∈C,X,Y∈R,X/ne}ationslash= 1,4X3/ne}ationslash=Y2, and it is not the case that
+bothX=−1andY= 0. Then the problem Hol(a,b)is#P-hard.
+Proof.We will use binary recursive Gadgets 4, 7, 8, and 9 together wi th the single-vertex
+starter gadget given in Figure 1(a) (denote the respective m atrices by M4,M7,M8,M9,
+ands). Calculating the recurrence matrices of these gadgets, we get
+M7=
+a6+a4b+a3+a2b22a4+4a2b+2ab3a2+ab2+b4+b
+a5+a3b+a2+ab2a4b+a3+2a2b2+ab4+2ab+b3a2b+ab3+b5+b2
+a4+a2b+a+b22a3b+4ab2+2b4a2b2+ab4+b6+b3
+,
+M8=
+a6+2a3+1 2 a4+4a2b+2b2a2+2ab2+b4
+a5+a3b+a2+b2a3+2a2b2+2ab+2b3ab3+a+b5+b2
+a4+2a2b+b22a2+4ab2+2b4b6+2b3+1
+,
+M9=
+a6+2a3+a2b22a4+2a2b+2ab3+2a a2+b4+2b
+a5+2a2+ab2a4b+a3+a2b2+ab4+2ab+b3+1a2b+b5+2b2
+a4+2a+b22a3b+2ab2+2b4+2b a2b2+b6+2b3
+.
+Calculating the characteristic polynomials x3+Bx2+Cx+Dof Gadgets 4, 7, 8, and 9,
+we get
+B4=−(X+Y+1)
+C4= (X2+X+Y)(X−1)
+D4=−X(X−1)3
+B7=−(−2X3+4X2+2XY+2X+Y2+2Y)
+C7= (X−1)·
+(X5−4X4−X3Y+6X3+7X2Y+4X2+4XY2+5XY+X+Y3+2Y2+Y)
+D7=−(X−1)3(2X+Y)(X4−X3+X2Y+3X2+2XY+X+Y2+Y)
+B8=−(−2X3+2X2+2X+Y2+4Y+2)
+C8= (X−1)2(X4−2X3+2X2+4XY+6X+2Y2+4Y+1)
+D8=−2(X−1)6X(X+1)
+B9=−(3X2+XY+2X+Y2+3Y+1)
+C9= (X−1)·
+(X5−3X4−2X3Y−X3+4X2Y+7X2+2XY2+6XY+4X+Y3+4Y2+4Y)
+D9=−(X−1)3(X+Y+1)(X4−2X3+X2+2XY+4X+Y2+2Y)
+Suppose X/ne}ationslash= 1, 4X3/ne}ationslash=Y2(equivalently, a3/ne}ationslash=b3), and it is not the case that both X=−1
+andY= 0. For any real-valued setting of XandYcompatible with these constraints, we
+will see that at least one of these four binary recursive gadg ets satisfies the requirements of
+Theorem 3.4 (the only exception is ( X,Y) = (0,−1), but by Lemma 4.1 any such problem
+is equivalent to Hol(0 ,−1) which is known to be #P-hard [6, 11]). To verify that Gadget j
+satisfies condition1ofTheorem3.4, weapplyLemma4.5andch eck thatDj(B3
+jDj−C3
+j)/ne}ationslash= 0
+(note that thenorm andconjugate disappearfrom thetest sin cewe are only consideringreal
+valuedXandY). ByLemma4.10,Gadget4satisfiescondition2becausedet[ s,M4s,M2
+4s] =
+(X−1)4(b3−a3)/ne}ationslash= 0. However, det[ s,M7s,M2
+7s] = (X−1)5(b3−a3)(X2+X+Y)(X+Y+1),
+det[s,M8s,M2
+8s] = (X−1)5(b3−a3)(X2Y+4X2+2XY+Y2+Y), and det[ s,M9s,M2
+9s] =HOLANT PROBLEMS FOR REGULAR GRAPHS WITH COMPLEX EDGE FUNCTI ONS 15
+(X−1)6(b3−a3)(X+ 1)(Y+ 2), so these are zero for some settings of XandY. We
+summarize the essential observations in terms of ( X,Y) coordinates as follows.
+X= 1⇐⇒ab= 1
+4X3=Y2⇐⇒a3=b3
+X= 0∧Y=−1⇐⇒(a= 0∧b3=−1)∨
+(a3=−1∧b= 0)
+X=−1∧Y= 0⇐⇒a6= 1∧ab=−1
+D4(B3
+4D4−C3
+4)/ne}ationslash= 0 =⇒Gadget 4 fulfills Theorem 3.4
+D7(B3
+7D7−C3
+7)(X2+X+Y)(X+Y+1)/ne}ationslash= 0 =⇒Gadget 7 fulfills Theorem 3.4
+D8(B3
+8D8−C3
+8)(X2Y+4X2+2XY+Y2+Y)/ne}ationslash= 0 =⇒Gadget 8 fulfills Theorem 3.4
+D9(B3
+9D9−C3
+9)(X+1)(Y+2)/ne}ationslash= 0 =⇒Gadget 9 fulfills Theorem 3.4
+If we can verify that at least one of the 8 conditions on the lef t hand side holds for any
+real-valued setting of XandYthen we are done. Note that a disjunction of the left hand
+sides is a semi-algebraic set, and as such, is decidable by Ta rski’s Theorem [12]. Using sym-
+bolic computation via the CylindricalDecomposition function from MathematicaTM,
+we verify that for any X,Y∈R, at least one of the eight conditions above is true.
+Now we can assume that X /∈RorY /∈R, and we deal with the remaining three
+conditions. Note that if X2+X+Y= 0 then X∈RimpliesY∈R. So in the following
+lemma, the assumption that XandYare not both real numbers amounts to X /∈R.
+Lemma 4.12. IfX2+X+Y= 0andX /∈Rthen the recurrence matrix of unary Gadget
+12 has nonzero eigenvalues with distinct norm.
+Proof.LetM12be the recurrence matrix for unary Gadget 12.
+M12=/bracketleftbigg
+a6+2a4b+a3+3a2b2+ab4a4+3a2b+2ab3+b5+b2
+a5+2a3b+a2+3ab2+b4a4b+3a2b2+2ab4+b6+b3/bracketrightbigg
+Then the determinant is the polynomial X6−6X5−X4Y+ 16X4+ 11X3Y−10X3+
+5X2Y2−7X2Y−X2+XY3−4XY2−3XY−Y3−Y2.Amazingly , with the condition
+X2+X+Y= 0, this polynomial factors into −X2(X−1)5. Similarly, the trace, which is
+−2X3+6X2+3XY+Y2+Y, also factors into X(X−1)3. Since det( M12)/ne}ationslash= 0, tr(M12)/ne}ationslash= 0,
+and (1−X)det(M12) = tr(M12)2, we know Arg(det( M12))/ne}ationslash= Arg(tr( M12)2) and conclude
+by Lemma 4.4 that the eigenvalues of M12(which are nonzero) have distinct norm.
+Similarly, Gadgets 11 and 13 can be used to deal with the X=−1 condition. Recall
+that any setting of aandbsuch that X=−1 andY=±2iis tractable by Theorem 4.3.
+Lemma 4.13. IfX=−1,Y/ne}ationslash=±2i, andY /∈R, then either Gadget 11 or Gadget 13 has
+a recurrence matrix with nonzero eigenvalues with distinct norm.
+Proof.Suppose |Y| /ne}ationslash= 2,Y /∈R, and let M11be the recurrence matrix for unary Gadget
+11. Well, det( M11) =−2Y/ne}ationslash= 0 and tr( M11) =Y−2, sotr(M11)·det(M11)−tr(M11)·
+|det(M11)|=−(Y−2)(2Y)−(Y−2)·|−2Y|= 4Y−2|Y|2−2Y·|Y|+4|Y|=−2(|Y|−
+2)(|Y|+Y)/ne}ationslash= 0. Thus tr(M11)·det(M11)/ne}ationslash= tr(M11)·|det(M11)|and by Lemma 4.4, M11
+has (nonzero) eigenvalues with distinct norm.16 M. KOWALCZYK AND J-Y. CAI
+Now suppose |Y|= 2, but Y/ne}ationslash=±2iandY /∈R. LetM13be the recurrence matrix for
+unary Gadget 13.
+M13=/bracketleftbigg
+a6+3a3+3ab+b3a4+2a2b+ab3+a+b5+2b2
+a5+a3b+2a2+2ab2+b4+b a3+3ab+b6+3b3/bracketrightbigg
+.
+Then det( M13) =−16Y/ne}ationslash= 0. Using the substitution Y= 4/Y,
+tr(M13)2det(M13)−
+tr(M13)2det(M13) =−16(Y−Y)·
+(−16+8YY+8Y2Y+Y3Y+8YY2+Y2Y2+YY3)
+=−64(Y−Y)(4+Y2)(4+8Y+Y2)
+Y2
+/ne}ationslash= 0.
+Hencetr( M13)2det(M13)/ne}ationslash=tr(M13)2det(M13)andtheeigenvaluesof M13(whicharenonzero)
+have distinct norm by Lemma 4.4.
+The condition 4( X−1)2(X+1) = (Y+2)2is somewhat resilient to individual unary
+recursive gadgets, but by usingasecond Eigenvalue Shifted Pair, we can reduceit to simpler
+conditions.
+Lemma 4.14. Suppose4(X−1)2(X+1) = (Y+2)2. Then either unary Gadget 13 or unary
+Gadget 14 has nonzero eigenvalues with distinct norm, unles s eitherX3+2X2+X+2Y= 0,
+orX3+4X2+2Y−1 = 0, or both X,Y∈R.
+Proof.Assume that X3+ 2X2+X+2Y/ne}ationslash= 0,X3+ 4X2+2Y−1/ne}ationslash= 0, and it is not the
+case that both X,Y∈R. Note that X /∈ {0,1}since otherwise Y∈Rand we know that
+XandYare not both real. The recurrence matrix for Gadget 14 is
+M14=/bracketleftbigg
+a6+3a3+a2b2+ab+b3+1a4+2a2b+ab3+a+b5+2b2
+a5+a3b+2a2+2ab2+b4+b a3+a2b2+ab+b6+3b3+1/bracketrightbigg
+soM14=M13+ (X−1)2I, and the eigenvalue shift is nonzero. Now, det( M13) = (X−
+1)3(X3+2X2+X+2Y)/ne}ationslash= 0 and note that tr( M13) =−2X3+6X+Y2+4Ysimplifies to
+tr(M13) =−2X3+6X+Y2+4Y−(Y+2)2+4(X−1)2(X+1) = 2X(X−1)2using the
+fact that 4( X−1)2(X+1) = (Y+2)2.
+Similarly, det( M14) = det(M14)+(X−1)2(4(X−1)2(X+1)−(Y+2)2) = (X−1)3(X3+
+4X2+2Y−1)/ne}ationslash= 0. Furthermoretr[ M13]2−4det(M13) = 4X2(X−1)4−4(X−1)3(X3+2X2+
+X+2Y) =−4(X−1)3(3X2+X+2Y). If this is zero, then substituting Y= (−3X2−X)/2
+into (Y+ 2)2−4(X−1)2(X+ 1) = 0 we get X(X−1)2(9X+ 8) = 0 and X∈R, with
+Y∈Ras a direct consequence. Corollary 4.8 implies that either G adget 13 or Gadget
+14 has nonzero eigenvalues of distinct norm, unless tr( M13) =r(X−1)2and det( M13) =
+s(X−1)4for some r,s∈R. But then 2 X(X−1)2=r(X−1)2henceX=r/2∈R, and
+(X−1)3(X3+2X2+X+2Y) =s(X−1)4henceY= (−X3−2X2−X+s(X−1))/2∈R.
+A contradiction.HOLANT PROBLEMS FOR REGULAR GRAPHS WITH COMPLEX EDGE FUNCTI ONS 17
+Now we take advantage of another interesting coincidence; t wo gadgets with recurrence
+matrices that have identical trace.
+Lemma 4.15. IfX2+X+Y/ne}ationslash= 0,4(X−1)2(X+1) = (Y+2)2, and either X3+2X2+
+X+2Y= 0orX3+4X2+2Y−1 = 0, then the recurrence matrix of unary Gadget 15 or
+unary Gadget 16 has nonzero eigenvalues with distinct norm, unless both X,Y∈R.
+Proof.The recurrence matrices for Gadget 15 and Gadget 16 are
+M15=/bracketleftbigg
+a6+a4b+2a3+a2b2+2ab+b3a4+3a2b+2ab3+b5+b2
+a5+2a3b+a2+3ab2+b4a3+a2b2+ab4+2ab+b6+2b3/bracketrightbigg
+,
+M16=/bracketleftbigg
+a6+a4b+2a3+a2b2+2ab+b3a4+a3b2+a2b+2ab3+a+b5+b2
+a5+2a3b+a2b3+a2+ab2+b4+b a3+a2b2+ab4+2ab+b6+2b3/bracketrightbigg
+.
+LetT=X3+2X2+X+2Y,U=X3+4X2+2Y−1, and let Rdenote (Y+2)2−4(X−
+1)2(X+1). Note that regardless of whether T= 0 orU= 0,X∈RimpliesY∈R, so we
+willassume X /∈R. Themaindiagonals of M15andM16areidentical, sotr( M15) = tr(M16).
+Furthermore, if T= 0 then tr( M15) = tr(M15)−R−(X−1)T/2 =−X(X−1)3/2/ne}ationslash= 0. If
+U= 0 then tr( M15) = tr(M15)−R−(X−1)U/2 =−(X−1)(X3−1)/2, and we claim this
+is nonzero as well. Otherwise, X3= 1 then since U= 0,Y=−2X2and using ( Y+2)2=
+4(X−1)2(X+1) we get ( X2−1)2= (X−1)2(X+1) i.e. ( X−1)2(X+1)2= (X−1)2(X+1)
+together with X /∈Rwe get acontradiction. Next, det( M16) = (X−1)3(X+1)(X2+X+Y)
+and det(M15) = det(M15)−R(X−1)2= (X−1)3(X+4)(X2+X+Y), so these are both
+nonzero. If both M15andM16have eigenvalues with equal norm, then applying Lemma 4.4
+twice, Arg(det( M15)) = Arg(tr( M15)2) = Arg(tr( M16)2) = Arg(det( M16)). However, this
+would imply Arg( X+4) = Arg( X+1) and X∈R, so we conclude that either M15orM16
+has nonzero eigenvalues with distinct norm.
+Now we sum up the result of these lemmas.
+Theorem 4.16. Suppose a,b∈Csuch that X/ne}ationslash= 1,4X3/ne}ationslash=Y2, and(X,Y)/ne}ationslash= (−1,0).
+Then the problem Hol(a,b)is#P-hard.
+Proof.Under these assumptions, if XandYare both real then Hol( a,b) is #P-hard by
+Lemma 4.11, so assume either XorYis not real. For any such aandb, we know by Lemma
+4.9 that either Gadget 10 or 11 has a recurrence matrix with no nzero eigenvalues of distinct
+norm, except in the following cases, where we will use other g adgets to fill this requirement.
+(1)X2+X+Y= 0.
+(2)X=−1.
+(3) 4(X−1)2(X+1) = (Y+2)2.
+IfX2+X+Y= 0 then X /∈R, lestXandYbe real, so Lemma 4.12 implies that unary
+Gadget 12 has a recurrence matrix of the required form. If X=−1, then Lemma 4.13
+indicates that either Gadget 11or Gadget 13satisfies thereq uirement, unless Y=±2i. Now
+we may assume X2+X+Y/ne}ationslash= 0, so by Lemmas 4.14 and 4.15 if 4( X−1)2(X+1) = (Y+2)2
+then either unary Gadget 13, 14, 15, or 16 has a suitable recur rence matrix. In any case, we
+have a unary recursive gadget whose recurrence matrix has no nzero eigenvalues of distinct
+norm. Hence we are done by Theorem 3.6 and Lemma 3.3.18 M. KOWALCZYK AND J-Y. CAI
+RecallVertex Cover is#P-hardon3-regular planargraphs, andnotethatall gadg ets
+discussed are planar (in the case of Gadget 8, each iteration can be redrawn in a planar way
+by “going around”theprevious iterations; see Figure1(d)) . Thus, all of thehardnessresults
+proved so far still apply even when the input graphs are restr icted to planar graphs. There
+are, however, some problems that are #P-hard in general, yet polynomial time computable
+when the input is restricted to planar graphs. This class of p roblems corresponds exactly
+with the problems we still need to resolve at this point, i.e. when 4X2=Y3butX /∈
+{0,±1}. The relevant interpolation results can be obtained with Ga dget 4 and holographic
+reductions, using a technique demonstrated in [5].
+Lemma 4.17. The problem #[a,1,a]|[1,0,0,1]is#P-hard, unless a∈ {0,±1,±i}, in
+which case it is in P.
+Proof.Ifa∈Rthen this is already known [6], and a polynomial time algorit hm fora=±i
+is in [3]. Now assume a /∈Randa/ne}ationslash=±i. Since these problems have an extra degree
+of symmetry, we use a 2 by 2 recurrence matrix to describe the r ecursive construction
+which consists of a single-vertex starter gadget (Figure 1( a)) followed by some number of
+applications of binaryrecursiveGadget 4(nofinishergadge t isusedhere). Thatis, if F-gate
+Nihas signature [ ai,bi,ai], then the signature of Ni+1is given by [ ai+1,bi+1,ai+1] where
+[ai+1,bi+1]T=M[ai,bi]T, andM=/bracketleftbigg
+a(a2+1) 2 a
+2a2a2+1/bracketrightbigg
+. Now, det( M) =a(a−1)2(a+1)2
+and tr(M) = (a+ 1)(a2+ 1) are both nonzero under our assumptions. It can be verified
+(using the Resolve function of MathematicaTM) that tr( M)|det(M)| /ne}ationslash=tr(M)det(M)
+provided that a /∈R, so by Lemma 4.4, the eigenvalues of Mhave distinct norm. Also,
+M/bracketleftbigg
+a
+1/bracketrightbigg
+= (a+1)/bracketleftbigg
+a(a2−a+2)
+(2a2−a+1)/bracketrightbigg
+, so the starter gadget is not an eigenvector of M. We
+conclude by an analogous version of Lemma 3.5 that we can inte rpolate all signatures of
+the form [ c,1,c], and since #[0 ,1,0]|[0,1,1,0] is known to be #P-hard [5] and equivalent
+to #[−ω(ω2−2)/3,1,−ω(ω2−2)/3]|[1,0,0,1] by holographic reduction (where ωis the
+principal 12throot of unity), we conclude that #[ a,1,a]|[1,0,0,1] is #P-hard.
+Lemma 4.18. If4X3=Y2, thenHol(a,b)is#P-hard unless X∈ {0,±1}, in which case
+it is in P.
+Proof.IfX= 0 then X=Y= 0 and the problem is in P by Theorem 4.3. Otherwise,
+X/ne}ationslash= 0, letω=ba−1, and applying a holographic reduction to #[ a,1,b]|[1,0,0,1] under the
+basis/bracketleftbigg
+ω0
+0ω2/bracketrightbigg
+we see that the problem #[ a,1,b]|[1,0,0,1] is equivalent to #[ ω2a,1,ωb]|
+[1,0,0,1], because ω3=b3a−3= 1. Since ω2a=ωb, we can apply Lemma 4.17 and the
+problem #[ a,1,b]|[1,0,0,1] is in P if ab=ω2a·ωb=±1 and #P-hard otherwise.
+Given this, we have the following result.
+Theorem 4.19. The problem Hol(a,b)is#P-hard for all a,b∈Cexcept in the following
+cases, for which the problem is in P.
+(1)X= 1
+(2)X=Y= 0
+(3)X=−1andY∈ {0,±2i}
+If we restrict the input to planar graphs, then these three cat egories are tractable in P, as
+well as a fourth category 4X3=Y2, and the problem remains #P-hard in all other cases.HOLANT PROBLEMS FOR REGULAR GRAPHS WITH COMPLEX EDGE FUNCTI ONS 19
+A simple coordinate change from ( X,Y) to (X,(Y
+2)2) translates this into Theorem 1.1.
+References
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+arXiv:1001.0465v1  [hep-ph]  4 Jan 2010CPC(HEP & NP), 2009, 33(X): 1—4 Chinese Physics C Vol. 33, No. X, Xxx, 2009
+The lowest-lying spin-1/2 and spin-3/2 baryon magnetic
+moments in chiral perturbation theory
+L.S. Geng1;1)J. Martin-Camalich1;2)L. Alvarez-Ruso2;3)M. J. Vicente-Vacas1;4)
+1 ( Departamento de F´ ısica Te´ orica and IFIC, Universidad d e Valencia-CSIC, E-46071 Valencia, Spain)
+2 (Centro de F´ ısica Computacional, Departamento de F´ ısic a, Universidade de Coimbra, P3004-516 Coimbra, Portugal)
+Abstract We review some recent progress in our understanding of the lo west-lying spin-1/2 and spin-3/2
+baryon magnetic moments (MMs) in terms of Chiral Perturbati on Theory (ChPT). In particular, we show that
+at next-to-leading-order ChPT can describe the MMs of the oc tet baryons quite well. We also make predictions
+for the decuplet MMs at the same chiral order. Among them, the MMs of the ∆++and ∆+are found to agree
+well with data within the experimental uncertainties.
+Key words magnetic moments, octet baryons, decuplet baryons, chiral perturbation theory
+PACS 13.40.Em, 14.20.-c, 12.39.Fe
+1 Introduction
+The magnetic moments (MMs) of the octet
+baryons have long been related to those of the proton
+and neutron, i.e., the celebrated Coleman-Glashow
+(CG) relations[1]. These relations are a result of (ap-
+proximate) global SU(3) flavor symmetry. Of course,
+weknowthatSU(3)flavorsymmetryisbroken,ascan
+also be clearly seen by comparing the predicted MMs
+with the corresponding experimental values (see Ta-
+ble 1). How to implement SU(3) breaking in a model-
+independent and systematic way has been pursued
+since then.
+Chiralsymmetryanditsbreakingpattern,incom-
+bination with the concept ofeffective field theory first
+systematically put forwardby Weinberg[2], has led to
+a low-energy effective theory of QCD–Chiral Pertur-
+bation Theory (ChPT)[3–10]. It has long been re-
+alized that ChPT may be employed to study SU(3)
+breakingeffects on the MMs of the baryonoctet. The
+first effort was undertaken by Caldi and Pagels in
+1974[11], evenbeforeChPTasweknowtodaywasfor-
+mulated. It was found that at next-to-leading-order
+(NLO), SU(3) breaking effects are so large that the
+description of the octet baryon MMs by the CG rela-tions tends to deteriorate, which was later confirmed
+by the calculations performed in Heavy Baryon (HB)
+ChPT[12–15]andInfrared(IR) ChPT[16]. Thisappar-
+ent failure has often been used to question the valid-
+ity of SU(3) ChPT in the one-baryon sector. In order
+to solve this problem, different approaches have been
+suggested, including reorderingthe chiral series[17]or
+using a cutoff to reduce the loop contributions, i.e.,
+the so-called long-range regularization[18].
+We will show in this talk that the above-
+mentioned apparent failure of baryon SU(3) ChPT is
+causedby the power-counting-restoration(PCR) pro-
+cedureused in removingthe power-counting-breaking
+(PCB) terms, which are due to the large non-zero
+baryon masses in the chiral limit[5]. The HB, the IR,
+and the extended-on-mass-shell(EOMS)∗approaches
+all remove the PCB terms, but the HB and IR ap-
+proaches achieve this by also removing nominally
+higher-order terms in such a way that relativity and
+analyticity of the loop results are lost. These ques-
+tions have been discussed quite extensively in the lit-
+erature, e.g., see Refs.[10, 19, 20]. Once the relativity
+and analyticity of the loop results are properly con-
+served, e.g., in the EOMS regularization scheme, it
+wasfound that baryonSU(3) ChPTat NLOimproves
+Received 15 December 2009
+1)E-mail:lsgeng@ific.uv.es
+2)E-mail:camalich@ific.uv.es
+3)E-mail:alvarez@teor.fis.uc.pt
+4)E-mail:vicente@ific.uv.es
+∗It is necessary to stress that the EOMS regularization schem e is nothing but a MSprocedure with an additional subtraction
+removing the finite power-counting-breaking pieces, if the y exist.
+c/circlecopyrt2009Chinese Physical Society and the Institute of High Ener gy Physics of the ChineseAcademy of Sciences and the Institu te
+of Modern Physics of the Chinese Academy of Sciences and IOP P ublishing LtdNo. X L. S. Geng et al: The lowest-lying spin-1/2 and spin-3/2 baryon magnetic moments in chiral perturbation theory 2
+the CG relations[20], contrary to the conclusions of
+most previous ChPT studies performed at this order.
+B B B B' BB' B'M M Mγ γ
+Bγ
+(c) (a) (b)B
+Fig. 1. Feynman diagrams contributing to the
+octet baryon magnetic moments up to NLO.
+It has often been argued that different regular-
+ization schemes should yield the same results since
+the difference between them is of nominal higher or-
+der. One must notice, however, that in order for this
+to be true, the regularization procedure should not
+break the analyticity of the loop results, which is cer-
+tainly not true in certain cases for the HB and IR
+schemes, asdemonstratedin Refs.[10, 19]. Atacertain
+order, one scheme can converge faster than the other
+schemes. From a practical point of view, one should
+better choose the one that conserves the analyticity
+of the loop results, is covariant, and, meanwhile, con-
+vergesfaster. Amongthe HB,IR,andEOMSregular-
+ization schemes, the EOMS scheme has been found to
+satisfy the above criteria. Therefore, we have chosen
+the EOMS regularization scheme in all the calcula-
+tions presented in this work.
+2 The octet baryon magnetic mo-
+ments
+2.1 Dynamical octet baryon contributions
+In the following, we discuss the results for the
+octet baryon magnetic moments at NLO without
+considering the contributions of dynamical decuplet
+baryons,whichwillbestudiedinthenextsub-section.
+We will not show the detailed formalism here,
+which can be found in Ref.[20]. Up to NLO, one has
+the diagrams shown in Fig. 1. The tree-level coupling
+(a)gives the leading-order (LO) result
+κ(2)
+B=αBbD
+6+βBbF
+6, (1)
+where the coefficients αBandβBfor each of the
+octet baryons are listed in Table I of Ref.[20]. This
+lowest-order contribution is nothing but the SU(3)-
+symmetric prediction leading to the CG relations
+[1, 12].
+TheO(p3) diagrams (b)and(c)account for
+the leading SU(3)-breaking corrections that are in-
+duced by the corresponding degeneracy breaking inthe masses of the pseudoscalar meson octet. Their
+contributions to the anomalous magnetic moment of
+a given member of the octet Bcan be written as
+κ(3)
+B=1
+8π2F2
+φ/parenleftBigg/summationdisplay
+M=π,Kξ(b)
+BMH(b)(mM)
++/summationdisplay
+M=π,K,ηξ(c)
+BMH(c)(mM)/parenrightBigg
+(2)
+with the coefficients ξ(b,c)
+BMlisted in Table I of Ref.[20].
+The loop-functions, which are convergent, read
+H(b)(m) =−M2
+B+2m2+m2
+M2
+B(2M2
+B−m2)log/parenleftbiggm2
+M2
+B/parenrightbigg
++2m(m4−4m2M2
+B+2M4
+B)
+M2
+B/radicalbig
+4M2
+B−m2arccos/parenleftbiggm
+2MB/parenrightbigg
+,
+H(c)(m) =M2
+B+2m2+m2
+M2
+B(M2
+B−m2)log/parenleftbiggm2
+M2
+B/parenrightbigg
++2m3(m2−3M2
+B)
+M2
+B/radicalbig
+4M2
+B−m2arccos/parenleftbiggm
+2MB/parenrightbigg
+.(3)
+One immediately notices that they contain pieces
+∼M2
+Bthat contribute at O(p2) to the MMs, which
+break the naive PC.
+Different regularizationschemesdiffer inhowthey
+remove the PCB terms: HB performs a dual expan-
+sionwhile IRsubtractsfromthe full resultthe regular
+part. The underlying reason that one can perform a
+regularization on the results shown in Eq. (3) lies in
+the fact that ChPT includes all symmetry allowed
+terms such that the PCB terms can be absorbed by
+the correspondinglow-energy-constants(LECs). One
+can easily see that the PCB terms ( ∼M2
+B) can be
+absorbed by redefining bD
+6andbF
+6. This is how one
+performs the regularization in the EOMS scheme. In
+the HB and IR schemes, one also removes higher or-
+der analytic terms while those LECs corresponding
+to these nominally higher-order terms are not explic-
+itly taken into account in the NLO calculation. One
+should also notice that in order for the EOMS argu-
+ment to be totally true, one has to use a common
+decay constant Fφfor pions, kaons, and etas, because
+one has only two LECs at his disposal at this order,
+whichcan nottakecareofhigher-ordereffects leading
+to different values for Fφ.
+In Table 1, we show the LO and NLO results ob-
+tained in the EOMS scheme[20]. For the sake of com-
+parison, we also show the NLO results obtained by
+using the HB and IR schemes. To compare with the
+results of earlier studies, we define
+˜χ2=/summationdisplay
+(µth−µexp)2, (4)No. X L. S. Geng et al: The lowest-lying spin-1/2 and spin-3/2 baryon magnetic moments in chiral perturbation theory 3
+whileµthandµexpare theoretical and experimental
+MMs of the octet baryons. The results shown in Ta-
+ble 1 are obtained by minimizing ˜ χ2with respect to
+the two LECs ˜bD
+6and˜bF
+6, renormalized bD
+6andbF
+6. Itis clear that the HB and IR results spoil the CG rela-
+tions, as found in previous studies, while the EOMS
+results improve them.
+Table 1. The baryon-octet magnetic moments (in nuclear magn etons) up to O(p3) obtained in different χPT
+approaches in comparison with data.
+p n Λ Σ−Σ+Σ0Ξ−Ξ0ΛΣ0˜χ2
+O(p2)
+Tree level 2.56 -1.60 -0.80 -0.97 2.56 0.80 -1.60 -0.97 1.38 0 .46
+O(p3)
+HB 3.01 -2.62 -0.42 -1.35 2.18 0.42 -0.70 -0.52 1.68 1.01
+IR 2.08 -2.74 -0.64 -1.13 2.41 0.64 -1.17 -1.45 1.89 1.86
+EOMS 2.58 -2.10 -0.66 -1.10 2.43 0.66 -0.95 -1.27 1.58 0.18
+Exp. 2.793(0) -1.913(0) -0.613(4) -1.160(25) 2.458(10) — - 0.651(3) -1.250(14) ±1.61(8)
+ 0 0.5 1 1.5 2 2.5
+ 0  0.2  0.4  0.6  0.8  1χ ~ 2
+xEOMS
+HB
+IR
+Fig. 2. SU(3)-breaking evolution of the mini-
+mal ˜χ2in theO(p3)χPT approaches under
+study. The shaded bands are produced by
+varyingMBfrom 0.8 to 1.1 GeV.
+The difference between the EOMS, HB, and IR
+approaches can also be seen from Fig. 2, where we
+show the evolution of the minimal ˜ χ2as a function of
+x=MM/MM,phys, whileMM,MM,physare the masses
+ofthe pion, kaon, etausedinthe calculationand their
+physical values. It is clear that at x= 0, the chiral
+limit, all the results are identical to the CG relations.
+Asxapproaches 1, where the meson masses equal to
+the physical values, only the EOMS results show a
+proper behavior, while both the HB and IR results
+rise sharply. This shows clearly that relativity and
+analyticity of the loop results play an important role
+in the present case.
+2.2 Dynamical decuplet baryon contribu-
+tions
+Chiral perturbation theory relies on the assump-
+tion that there exists a natural cutoff such that high-
+energy degreesof freedom can be integrated out, with
+their effects approximated by the LECs. In the case
+of baryon SU(3) ChPT, the average mass gap be-
+tween the baryon octet and the baryon decuplet isonly 0.231 GeV, similar to the pion mass and even
+smaller than the kaon mass. Therefore, in baryon
+SU(3) ChPT, one has to be careful about the contri-
+butions of the decuplet baryons.
+It must be pointed out that description of spin-
+3/2 baryons in a fully consistent quantum field the-
+ory framework is an unsolved problem, see e.g.
+Refs.[21, 22]and references therein.
+Using the “consistent coupling” scheme to de-
+scribe the self-interaction of spin-3/2 baryons and
+their interaction with spin-1/2 baryons, we have
+showninRef.[22]thattheinclusionofdynamicalspin-
+3/2 baryons has only a small effect on our description
+oftheoctetbaryonMMsasdescribedabove. It isalso
+shown that this conclusion is stable with respect to
+all of the model parameters within their uncertain-
+ties[22].
+3 The decuplet baryon magnetic mo-
+ments
+In recent years, there have been increasing inter-
+est from both the experimental side and the lattice
+QCD community to study the magnetic moments of
+the lowest-lying decuplet baryons, particularly those
+of the ∆(1232)’s. Encouraged by the success of the
+baryonChPTindescribingtheoctetbaryonmagnetic
+moments, we have extended the same framework to
+study the decuplet baryon magnetic moments. De-
+tails of this study can be found in Ref.[23].
+In Table 2, we show our EOMS ChPT NLO re-
+sults[23]in comparison with those of a number of
+theoretical models and available data. We have fitted
+the only LEC at this order by reproducing the MM
+of the Ω. It can be clearly seen that our results for
+the ∆++and ∆+agree quite well with data within
+the experimental uncertainties.No. X L. S. Geng et al: The lowest-lying spin-1/2 and spin-3/2 baryon magnetic moments in chiral perturbation theory 4
+Table 2. Decuplet magnetic dipole moments (in nuclear magne tons) obtained in covariant ChPT up to O(p3),
+in comparison with those obtained in other theoretical appr oaches and data.
+∆++∆+∆0∆−Σ∗+Σ∗0Σ∗−Ξ∗0Ξ∗−Ω−
+SU(3)-symm. 4.04 2.02 0 -2.02 2.02 0 -2.02 0 -2.02 -2.02
+NQM[24]5.56 2.73 -0.09 -2.92 3.09 0.27 -2.56 0.63 -2.2 -1.84
+RQM[25]4.76 2.38 0 -2.38 1.82 -0.27 -2.36 -0.60 -2.41 -2.35
+χQM[26]6.93 3.47 0 -3.47 4.12 0.53 -3.06 1.10 -2.61 -2.13
+χQSM[27]4.85 2.35 -0.14 -2.63 2.47 -0.02 -2.52 0.09 -2.40 -2.29
+QCD-SR[28]4.1(1.3) 2.07(65) 0 -2.07(65) 2.13(82) -0.32(15) -1.66(73 ) -0.69(29) -1.51(52) -1.49(45)
+lQCD[29]6.09(88) 3.05(44) 0 -3.05(44) 3.16(40) 0.329(67) -2.50(29 ) 0.58(10) -2.08(24) -1.73(22)
+lQCD[30]5.24(18) 0.97(8) -0.035(2) -2.98(19) 1.27(6) 0.33(5) -1.8 8(4) 0.16(4) -0.62(1) —
+largeNc[31]5.9(4) 2.9(2) — -2.9(2) 3.3(2) 0.3(1) -2.8(3) 0.65(20) -2.3 0(15) -1.94
+HBχPT[32]4.0(4) 2.1(2) -0.17(4) -2.25(19) 2.0(2) -0.07(2) -2.2(2) 0 .10(4) -2.0(2) -1.94
+ChPT[23]6.04(13) 2.84(2) -0.36(9) -3.56(20) 3.07(12) 0 -3.07(12) 0 .36(9) -2.56(6) -2.02
+Expt.[33]5.6±1.9 2 .7+1.0
+−1.3±1.5±3 — — — — — — — -2.02 ±0.05
+4 Summary and conclusions
+EOMS SU(3) baryon ChPT provides a successful
+description of the SU(3) breaking effects on the octet
+baryon MMs. It has been found in this particular
+case that the relativity and analyticity of the loop re-
+sults play an important role. We have also studied
+the dynamical decuplet contributions and found that
+their inclusion only has a small effect on the SU(3)
+breaking effects on the MMs of the octet baryons.
+Encouraged by the success of the EOMS ap-
+proach, we have studied the decuplet baryon MMs.
+Fitting our only LEC at this order to reproduce the
+MM of the Ω, we have been able to predict the MMs
+of the other members of the baryon decuplet. In par-
+ticular, those of the ∆++and ∆+seem to agree well
+with data within the experimental uncertainties.
+This approach has also been employed to study
+the SU(3) breaking corrections to the hyperon vec-
+tor coupling f1(0)[34], which plays a decisive role in
+the extraction of VUSfrom hyperon semi-leptonic de-
+cay (HSD) data. It will also be interesting to apply
+the same approach to study the hyperon axial-vector
+couplings, which could provide us vital information
+about the spin structure of the baryons.
+This work was partially supported by the MEC
+grant FIS2006-03438 and the European Community-
+Research Infrastructure Integrating Activity Study of
+Strongly Interacting Matter (Hadron-Physics2, Grant
+Agreement 227431) under the Seventh Framework
+Programme of EU. L.S.G. acknowledges support from
+the MICINN in the Program “Juan de la Cierva.”
+J.M.C. acknowledges the same institution for a FPU
+grant.
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+arXiv:1001.0466v2  [math.AG]  25 Oct 2012PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES
+NIELS BORNE AND ANGELO VISTOLI
+Abstract. We show how the natural context for the definition of paraboli c
+sheaves on a scheme is that of logarithmic geometry. The key p oint is a refor-
+mulation of the concept of logarithmic structure in the lang uage of symmetric
+monoidal categories, which might be of independent interes t. Our main result
+states that parabolic sheaves can be interpreted as quasi-c oherent sheaves on
+certain stacks of roots.
+1.Introduction
+The notion of parabolic bundle on a curve was introduced by Mehta and Seshadri
+(see [MS80] and [Ses82]), and subsequently generalized to higher dim ension by
+Maruyama and Yokogawa ([MY92]); this latter definition was later impr oved by
+Mochizuki ([Moc06]), Iyer–Simpson ([IS07]) and the first author ([B or09]). Another
+important insight is due to Biswas, who connected rational parabolic bundles with
+bundles on orbifolds ([Bis97]). The first author refined Biswas’ idea in [Bor07] and
+[Bor09]; in the latter paper he proved that, given nsmooth effective divisors D1,
+. . . ,Dnintersecting transversally on a normal variety X, there is an equivalence
+between the category of rational parabolic bundles and the limit of t he category
+of vector bundles on the fibered product of dthroot stacks of the ( X,Di), asd
+becomes very divisible.
+This left two main questions open.
+(1) What about divisors that are not simple normal crossing? If, fo r example, a
+normal crossing divisor has singular components, it seems clear tha t one should
+use sheaves of weights.
+(2) What is the correct definition of parabolic coherent sheaf? [MY9 2] and [IS07]
+contain definitions of torsion-free parabolic coherent sheaves; b ut the definition
+of a general coherent sheaf has to be essentially different.
+The key point to solving these problems is the introduction of logarith mic struc-
+tures. The main purpose of this paper is to give a definition of parabo lic quasi-
+coherent sheaf with fixed rational weights on a logarithmic scheme, and to show
+the equivalence of category of such sheaves with the category of sheaves on a root
+stack.
+More precisely, suppose that ρ:M→ OXis a logarithmic structure on a scheme
+X; denote, as usual, by Mthe quotient sheaf M/O∗
+X. The denominators are taken
+in an appropriate sheaf of monoids Bcontaining M; then we define a category
+of quasi-coherent parabolic sheaves on a fine logarithmic scheme ( X,M,ρ ) with
+weights in B.
+Date : May 29, 2018.
+12 BORNE AND VISTOLI
+Also, we define a root stack XB/M; this is a tame Artin stack over X. The idea
+of the construction is essentially due to Martin Olsson, who defined it in several
+particular cases, from whom it was easy to extract the general de finition (([MO05],
+[Ols07])). If the logarithmic structure is generated by a single effect ive Cartier
+divisorD⊆X, so that Mis the constant sheaf NDonD, and we take Bto be
+1
+dND, thenXB/Mis the root stackd/radicalbig
+(X,D ) (see [AGV08] or [Cad07]). Our main
+result (Theorem 6.1) is that the category of quasi-coherent para bolic sheaves on a
+fine logarithmic scheme ( X,M,ρ ) with weights in Bis equivalent to the category
+of quasi-coherent sheaves on the stack XB/M. This represents a vast generalization
+of the correspondence of [Bor09].
+In order to do this we need to interpret the logarithmic structure ( M,ρ) as a
+symmetric monoidal functor M→DivX, whereDivX´ etis the symmetric monoidal
+stack over the small ´ etale site X´ etofXwhose objects are invertible sheaves with
+sections. We call this a Deligne–Faltings structure . The fact that a Deligne–Faltings
+structure defines a logarithmic structure is somehow implicit in the or iginal con-
+struction of the logarithmic structure associated with a homomorp hism of monoids
+P→ O (X), as in [Kat89]; going in the other direction, the construction is con tained
+in Lorenzon’s paper [Lor00].
+We find that this point of view has some advantages, and in this paper we make
+an effort to develop the theory of Deligne–Faltings structures sys tematically, with-
+out referring to known results on logarithmic structures. We are p articularly fond
+of our treatment of charts, in 3.3, which we find somewhat more tra nsparent than
+the classical one. The resulting notion of fine logarithmic structure is equivalent to
+the classical one.
+There is much left to do in the direction that we point out. Suppose th at (M,ρ)
+is a saturated logarithmic structure. Then we can associate with it a tower of
+stacksXddef=X1
+dM/M, letting drange over all positive integer. This tower seems
+to control much of the geometry of the logarithmic scheme ( X,M,ρ ); for example,
+the limit of the small ´ etale sites of the Xd(appropriately restricted when not in
+characteristic 0) is the Kummer-´ etale site of ( X,M,ρ ), and one can use the Xd
+to investigate many questions concerning this site; for example, th e K-theory of
+(X,M,ρ ), as defined by Hagihara and Nizio/suppress l (see [Hag03] and [Niz08]) is natu rally
+interpreted in this language. In subsequent papers we plan to prov e Nori’s theorem
+for logarithmic schemes, in the style of [Bor09], define real parabolic sheaves, and
+connections on them (as was pointed out to us by Arthur Ogus, this is important to
+study the Riemann–Hilbert correspondence for logarithmic scheme s, as in [Ogu03]),
+and in general apply this construction to other foundational ques tions in the theory
+of logarithmic schemes.
+Description of contents. Section 2 contains several preliminary notions, mostly
+known, concerning monoids, sheaves of monoids, symmetric monoid al categories
+and fibered symmetric monoidal categories. We also define one of ou r basic notions,
+that ofDeligne–Faltings object : a symmetric monoidal functor from a monoid to a
+symmetric monoidal category with trivial kernel.
+The main definition, that of Deligne–Faltings structure (Definition 3.1 ), is con-
+tained in Section 3. Here we also show the equivalence of the notion of Deligne–
+Faltings structure with that of quasi-integral logarithmic structu re (Theorem 3.6).PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 3
+In (3.2) we define direct and inverse images of a Deligne–Faltings stru cture, without
+going through the associated logarithmic structure.
+Our treatment of charts is contained in (3.3); we compare it with Kat o’s treat-
+ment in (3.4). Some of the results are strictly related with those in [O ls03, Section 2];
+we do not refer to this paper, but prefer to reconstruct the the ory independently.
+Proposition 3.28 implies that our resulting notion of fine structure co incides with
+the classical one.
+Section 4 contains the notion of systems of denominators and the d efinition of
+root stacks. In Section 5 we define parabolic sheaves and prove th eir basic proper-
+ties. Finally, our main result, giving an equivalence between parabolic s heaves and
+sheaves on a root stack is in Section 6.
+Acknowledgments. We would like to thank Luc Illusie, Arthur Ogus, and Martin
+Olsson for useful conversations.
+2.Definitions and preliminary results
+2.1.Conventions. The class of objects of a category Cwill be denoted by Obj C.
+IfFis a presheaf (of sets, monoids, groups, . . . ) on a site, we denote byFshthe
+associated sheaf.
+IfF,F′:C → D andG:D → E are functors and α:F→F′is a natural
+transformation, we denote by G◦α, or simply Gα, the natural transformation
+GF→GF′defined by the obvious rule ( Gα)C=G(αC). Analogously, if F:C → D ,
+G,G′:D → E are functors and α:G→G′is a natural transformation, we denote
+byα◦ForαFthe natural transformation GF→G′Fdefined by ( αF)C=αF(C).
+2.2.Monoids. All monoids considered will be commutative; we will use additive
+notation. We denote by (ComMon) the category of (commutative) monoids.
+IfAis a monoid, we denote the associated group by Agp, and byιA:A→Agpthe
+canonical homomorphism of monoids. Any element of Agpis of the form ιAa−ιAb
+for some a,b∈A; furthermore, two elements aandbofAhave the same image in
+Agpif and only if there exists c∈Asuch that a+c=b+c. A homomorphism of
+monoids f:A→Binduces a group homomorphism fgp:Agp→Bgp.
+A monoid is integral if the cancellation law holds; equivalently, a monoid is
+integral if ιA:A→Agpis injective. A monoid Aistorsion-free if it is integral and
+Agpis torsion-free. If f:A→Bis an injective homomorphism of monoids and B
+is integral, then fgp:Agp→Bgpis also injective.
+A monoid is sharp if the only invertible element is the identity. Notice that a
+sharp monoid has no non-zero element of finite order; however, th e associated group
+Agpis not necessarily torsion-free.
+Thekernel of a homomorphism of monoids f:A→Bisf−1(0)⊆A. In contrast
+with the case of groups, the kernel of fmay be trivial without fbeing injective
+(for example, look at the homomorphism N2→Ndefined by ( x,y)/maps⊔o→x+y). An
+arbitrary submonoid S⊆Ais not necessarily a kernel. The following condition is
+necessary and sufficient for Sto be a kernel: if a∈A,s∈Sanda+s∈S, then
+a∈S.
+IfSis a submonoid of A, we denote by A/S the cokernel of the inclusion S⊆A.
+This is the quotient of Aby the equivalence relation ∼Sdefined by a∼bwhen
+there exist s∈Sandt∈Ssuch that a+s=b+t. The kernel of the projection4 BORNE AND VISTOLI
+A→A/S is the set of a∈Asuch that there exists s∈Switha+s∈S. This is
+the smallest kernel that contains S, and we call it the kernel closure ofS.
+A homomorphism of monoids A→Bis called a cokernel if it is the cokernel
+of a homomorphism C⊆A. Any cokernel is surjective, but not every surjective
+homomorphism is a cokernel. A necessary and sufficient condition for f:A→Bto
+be a cokernel is that if Kis the kernel of f, the induced homomorphism A/K→B
+is an isomorphism.
+2.3.Sheaves of monoids. Many of the notions above extend to sheaves and
+presheaves of monoids on a site C. IfAis such a sheaf, we define Agpas the
+sheafification of the presheaf sending U∈ObjCintoA(U)gp. The obvious homo-
+morphism of sheaves of monoids ιA:A→Agpis universal among homomorphism
+of sheaves of monoids from Ato a sheaf of groups.
+A presheaf of monoids is called integral if eachA(U) is integral. If Ais integral,
+so is the associated sheaf Ash. It issharp if eachA(U) is sharp.
+IfKis a sub-presheaf of monoids of a presheaf of monoids A, we can define the
+presheaf quotient A/K by the rule ( A/K )(U) =A(U)/K(U). It is the cokernel
+in the category of presheaves of monoids of the inclusion K⊆A. In general, if
+C→Ais a homomorphism of presheaves, its cokernel is A/K , whereKis the
+image presheaf in A.
+If we substitute presheaves with sheaves, the quotient A/K is the sheafification
+of the presheaf quotient; all cokernels in the category of sheave s of monoids are of
+this type.
+2.4.Symmetric monoidal categories. Our treatment of logarithmic structures
+is centered around the notion of symmetric monoidal category . We freely use the
+notation and the results of [ML98, ch. VII and XI], which will be our ma in reference.
+This concept was introduced in [DMOS82, II, Definition 2.1], under the nametensor
+category .
+LetMa symmetric monoidal category. We denote the operation (the “te nsor
+product”) by ⊗:M×M → M , its action on objects and arrows by ( x,y)/maps⊔o→x⊗y,
+the neutral element of Mby1, the associativity isomorphisms x⊗(y⊗z)≃(x⊗y)⊗z
+byα, orαx,y,z, the isomorphism 1⊗x≃xbyλorλx, the isomorphism x⊗y≃y⊗x
+byσ, orσx,y. Occasionally we will use the subscript M(as in⊗M,1M, and so
+on) to distinguish among such objects relative to different symmetr ic monoidal
+categories.
+IfMandNare symmetric monoidal categories, a symmetric monoidal functor
+F:M → N will be is a strong braided monoidal functor M → N ([ML98, ch. IX,
+§2]). All natural transformations between symmetric monoidal fun ctors will be
+assumed to be monoidal.
+We denote by (SymMonCat) the 2-category of symmetric monoidal categories.
+The objects are small symmetric monoidal categories, the 1-arro ws are symmetric
+monoids functors, and the 2-arrows are monoidal natural trans formations.
+IfF:M → N is a symmetric monoidal functor, which is an equivalence, when
+viewed as a functor of plain categories, then any quasi-inverse G:N → M has a
+unique structure of a symmetric monoidal functor, such that the given isomorphisms
+FG≃idNandGF≃idMare monoidal isomorphisms.
+Any monoid Awill be considered as a discrete symmetric monoidal category:
+the arrows are all identities, while the tensor product is the operat ion inA.PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 5
+For the convenience of the reader, we make the notion of a symmet ric monoidal
+functor explicit in the case that we will use the most.
+Definition 2.1. LetAbe a monoid, Ma symmetric monoidal category. A sym-
+metric monoidal functor L:A→ M consists of the following data.
+(a) A function L:A→ObjM.
+(b) An isomorphism ǫL:1≃L(0) inM.
+(c) For each aandb∈A, an isomorphism µL
+a,b:L(a)⊗L(b)≃L(a+b) inM.
+We require that for any a,b,c∈A, the diagrams
+L(a)⊗/parenleftbig
+L(b)⊗L(c)/parenrightbigid⊗µL/d47/d47
+α
+/d15/d15L(a)⊗L(b+c)
+µL
+/d40/d40❘❘❘❘❘❘❘❘❘❘❘❘❘
+L(a+b+c),
+/parenleftbig
+L(a)⊗L(b)/parenrightbig
+⊗L(c)µL⊗id/d47/d47L(a+b)⊗L(c)µL/d54/d54❧❧❧❧❧❧❧❧❧❧❧❧❧
+L(a+b)µL/d47/d47L(a)⊗L(b)
+σ
+/d15/d15
+L(b+a)µL/d47/d47L(b)⊗L(a)
+and
+L(a)
+=
+/d15/d151⊗L(a)λ/d111/d111
+ǫL⊗id
+/d15/d15
+L(0 +a)L(0)⊗L(a)µL/d111/d111
+be commutative.
+IfL:A→ M andM:A→ M are symmetric monoidal functors, a morphism
+Φ:L→Mis collection of natural transformations Φ a:L(a)→M(a), one for each
+a∈A, such that for any a,b∈A, the diagram
+L(a)⊗L(b)µL/d47/d47
+Φa⊗Φb
+/d15/d15L(a+b)
+Φa+b
+/d15/d15
+M(a)⊗M(b)µM/d47/d47M(a+b)
+commutes.
+Iff:A→Bis a homomorphism of monoids, and L:B→ M is a symmetric
+monoidal functor, the composite L◦f:A→ObjMhas an obvious structure of
+a symmetric monoidal functor. We will use this fact, together with s ome evident
+properties, without comments.
+Definition 2.2. LetMbe a symmetric monoidal category M. AnM-valued
+Deligne–Faltings object is a pair ( A,L), where Ais a monoid and L:A→ M is a
+symmetric monoidal functor L:A→ M .6 BORNE AND VISTOLI
+There is a category of M-valued Deligne–Faltings objects. An arrow from ( A,L)
+to (B,M ) (which we call a morphism of Deligne–Faltings objects ) is a pair ( φ,Φ),
+whereφ:A→Bis a homomorphism of monoids and Φ: L→M◦φis a monoidal
+natural transformation. The composition is defined in the obvious w ay.
+Definition 2.3. LetMbe a symmetric monoidal category, L:A→ M be a
+Deligne–Faltings object. The kernel kerLis the set of elements a∈Asuch that
+L(a) is isomorphic to the neutral element 1.
+One checks immediately that ker Lis a sub-monoid of A.
+Proposition 2.4. LetL:A→ Mbe a Deligne–Faltings object. Assume that the
+monoid of endomorphisms of the neutral element 1ofMis trivial.
+Then there exists a cokernel π:A→Aand a symmetric monoidal functor
+L:A→ Mwith trivial kernel, with an isomorphism of symmetric monoi dal func-
+torsL◦π≃L. Furthermore the Deligne–Faltings object (A,L)is unique, up to a
+unique isomorphism, and is universal among morphism from (A,L)to a Deligne–
+Faltings object (B,M )with trivial kernel.
+Proof. It is clear that if ( A,L) and the isomorphism exist, the kernel of πmust be
+the kernel KofL; therefore there is a unique isomorphism of AwithA/K . So it
+is enough to show that there exists a unique factorization
+Aπ−→A/KL−→ M,
+up to a unique isomorphism, and that it has the required universal pr operty.
+Notice that if k∈K, the isomorphism 1≃L(k), which exists by hypothesis,
+must be unique, because Aut( 1) is trivial. Hence for any ainAwe get canonical
+isomorphisms
+L(a)λ≃1⊗L(a)≃L(k)⊗L(a)µL
+≃L(k+a).
+Denote by νk,athe resulting canonical isomorphism L(a)≃L(k+a). This isomor-
+phism is easily shown to have the property that
+νl+k,a=νl,a+k◦νk,a:L(a)≃L(l+k+a)
+for anyk,l∈K.
+Now, suppose that aandb∈Ahave the same image in A/K . There exist k,
+l∈Ksuch that k+a=l+b; so we have an isomorphism τa,b:L(a)≃L(b) defined
+as the composite
+L(a)νk,a≃L(k+a) =L(l+b)ν−1
+l,b≃L(b).
+It is easy to check that τa,bis independent of the choice of kandl. Ifa,band
+chave the same image in A/K , then we can find k,landm∈Ksuch that
+k+a=l+b=m+c, and then
+τa,c=τb,c◦τa,b:L(a)≃L(b).
+Now consider the category AK, whose object are the elements of A, and in which,
+given two elements a,b∈A, there exists exactly one arrow a→bifaandbhave the
+same image in A/K , and none otherwise. The category AKis a strict symmetric
+monoidal category, with the tensor product given by the operatio n inA. ThePARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 7
+projection π:A→A/K factors through AK, and the projection AK→A/K is an
+equivalence of monoidal categories. Hence it is enough to produce a factorization
+A−→AK/hatwideL−→ M,
+and then the desired functor L:A/K→ M will be obtained by composing /hatwideLwith
+a quasi-inverse A/K→AK. This factorization is obtained by defining /hatwideLto be the
+same function as Lon the elements of A; ifa→bis an arrow in AK, then we take
+as its image in Mthe isomorphism τa,b:L(a)≃L(b). We leave it to the reader to
+check that this functor is monoidal, and gives the desired factoriza tion.
+We have left to check that ( A,L) has the desired universal property. Suppose
+that (φ,Φ): (A,L)→(B,M ) is a morphism of Deligne–Faltings objects. Let K′
+be the kernel of M: clearly φsendsKtoK′, thus there is a natural commutative
+diagram:
+(A,L)
+/d15/d15/d47/d47(AK,/hatwideL)
+/d15/d15
+(B,M ) /d47/d47(BK′,/hatwiderM)
+If the kernel K′of (B,M ) is trivial, the bottom morphism is an isomorphism, and
+this shows existence in the universal property. We leave it to the re ader to check
+uniqueness. ♠
+The following two examples play a key role in this paper.
+Examples 2.5. LetXbe a scheme.
+(a) We denote by DivXthe groupoid of line bundles with sections. We consider
+DivXas a category of “generalized effective Cartier divisors”: effective Cartier
+divisors on Xform a monoid, which is, however, not functorial in X, since
+one can’t pull back Cartier divisors along arbitrary maps. Line bundle s with
+sections don’t have this problem; there is a price to pay, however, w hich is to
+have to deal with a symmetric monoidal category instead of a monoid .
+The objects of DivXare pairs ( L,s), whereLis an invertible sheaf on Xand
+s∈L(X). An arrow from ( L,s) to (L′,s′) is an isomorphism of OX-modules
+fromLtoL′carrying sintos′. The category DivXalso has a symmetric
+monoidal structure given by tensor product, defined as ( L,s)⊗(L′,s′)def= (L⊗
+L′,s⊗s′). The neutral element is ( OX,1).
+Notice that DivXhas the property that the monoid of endomorphisms of
+the neutral element ( OX,1) is trivial.
+(b) We denote by PicXthe category of invertible sheaves on X, with the monoidal
+structure given by tensor product. We notice that, in contrast w ith standard
+usage, and with the example above, the arrows in PicXwill be arbitrary
+homomorphisms of OX-modules, and not only isomorphisms. Thus, PicX
+is not a groupoid. Tensor product makes PicXinto a symmetric monoidal
+category, with neutral element OX.
+(c) The category of invertible sheaves on X, in which the only arrows are the
+isomorphisms, will be denoted by BGm(X).
+There is a natural strict symmetric monoidal functor DivX→PicXsending
+(L,s) toL.8 BORNE AND VISTOLI
+IfAis a monoid and Mis a symmetric monoidal category, we denote by
+Hom(A,M) the category of symmetric monoidal functors A→ M . Given a homo-
+morphism of monoids f:A→B, there is an induced functor f∗: Hom(B,M)→
+Hom(A,M) sending L:B→ M into the composite L◦f:A→ M .
+2.5.Monoidal fibered categories. Here we will freely use the language of fibered
+categories, for which we refer to [FGI+05, Chapter 3].
+Definition 2.6. LetCbe a category. A symmetric monoidal fibered category M →
+Cis a fibered category, together with a cartesian functor
+⊗=⊗M:M×CM −→ M ,
+a section 1M:C → M , and base-preserving natural isomorphisms
+α:⊗◦(idM×⊗)≃ ⊗◦ (⊗×idM) of functors M×CM×CM −→ M ,
+λ:⊗◦(1M×idM)≃idMof functors M ≃ C × CM −→ M ,
+and
+σ:⊗ ≃ ⊗◦ ΣMof functors M×CM −→ M ,
+where by Σ M:M ×CM → M × CMwe mean the functor exchanging the two
+terms, such that for any object UofCthe restrictions of ⊗and of the natural
+transformations above yield a structure of symmetric monoidal ca tegory on M(U).
+IfM → C andN → C are symmetric monoidal fibered categories, a symmetric
+monoidal functor F:M → N is a cartesian functor, together with an isomorphism
+µL:⊗N◦(F×F)≃F◦⊗Mof functors M×CM −→ N
+and
+ǫF:1N≃F◦1Mof functors C −→ N,
+such that the restrictions of these data to each M(U) andN(U) givesFU:M(U)→
+N(U) the structure of a symmetric monoidal functor.
+Morphisms of symmetric monoidal functors are base-preserving n atural trans-
+formation, whose restriction to each fiber is monoidal.
+IfM → C is a symmetric monoidal fibered category and we choose a cleavage
+for it, we obtain a pseudo-functor (i.e., a lax 2-functor) from Copthe 2-category
+(SymMonCat) of symmetric monoidal categories. A different choice of a cleavage
+yields a canonically isomorphic pseudo-functor.
+Conversely, given a pseudo-functor Cop→(SymMonCat), the usual construction
+yields a symmetric monoidal fibered category over Cwith a cleavage.
+In particular, if A:Cop→(ComMon) is a presheaf of monoids on a category
+C, we consider the associated fibered category ( C/A)→ C. The objects of ( C/A)
+are pairs ( U,a), where Uis an object of Canda∈A(U→X). The arrows from
+(U,a) to (V,b) are arrows f:U→Vsuch that f∗b=a. Because of the customary
+identification of categories fibered in sets on Cand functors Cop→(Set), we will
+usually write this simply as A→ C. Such a category has a canonical structure of
+strict symmetric monoidal fibered category.
+Definition 2.7. LetCbe a site. A symmetric monoidal stack over Cis a symmetric
+monoidal fibered category over Cthat is a stack.
+Examples 2.8. LetXbe a scheme; denote by (Sch /X) the category of schemes
+overX.PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 9
+(a) The symmetric monoidal stack DivX→(Sch/X) is the category associated
+with the pseudo-functor that sends each U→Xinto the category DivU.
+(b) Analogously, one defines the symmetric monoidal stack PicXwhose fiber over
+U→XisPicU.
+Remark 2.9. The stack DivXcan be described using the language of algebraic
+stacks as the quotient [ A1
+X/Gm,X].
+The stack PicXis not an algebraic stack, because it is not a stack in groupoids.
+The underlying stack in groupoids (obtained by deleting all the arrow s that are not
+cartesian) is the usual Picard stack of X, and can be described as the classifying
+stackBXGmof the group scheme Gm,X, or, in other words, as the stack quotient
+[X/Gm,X] for the trivial action of GmonX.
+We will need the following extension result. Let Cbe a site, and let A:Cop→
+(ComMon) be a presheaf of monoids on C. Denote by Ashthe associated sheaf of
+monoids, sA:A→Ashthe canonical homomorphism.
+Proposition 2.10. LetMbe a symmetric monoidal stack over C,L:A→ M
+a symmetric monoidal functor. Then there exists a symmetric monoidal functor
+Lsh:Ash→ Mand an isomorphism of the composite sA◦Lsh:A→ MwithL.
+Furthermore, the pair (Ash,Lsh)has the following universal property: any morphism
+of Deligne–Faltings objects (A,L)→(B,M ), whereBis a sheaf, factors uniquely
+through (Ash,Lsh). IfLhas trivial kernel, so has Lsh.
+This could be considered as obvious, as it says that the sheafificatio n of a presheaf
+coincides with its stackification. However, we don’t know a reasonab le reference,
+so we sketch a proof.
+Proof. Let{Ui→U}be a cover in C. There is a canonical isomorphism
+A(U)
+/d15/d15L(U)/d47/d47M(U)
+/d15/d15
+A({Ui→U})
+L({Ui→U})/d47/d47/d49/d57❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥
+M({Ui→U})
+where for a fibered category F → C ,F({Ui→U}) denotes the category of descent
+data ofFwith respect to {Ui→U}(see [FGI+05]). Since Mis a stack, the
+right-hand map is an equivalence. Since the diagram above is compatib le with
+refinements of covers, we can take the inductive limit and get a fact orization:
+A(U)L(U)/d47/d47
+/d15/d15M(U)
+Asep(U)Lsep(U)/d57/d57ttttttttt/d63/d71✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞
+whereAsep(U)def= lim− →{Ui→U}A({Ui→U}). This is compatible with restriction, so
+that we obtain a factorization Lsep:Asep→ M ofL:A→ M , and iterating this
+process we get the wished factorization Lsh:Ash→ M ofL:A→ M .
+This factorization is functorial in A, and the existence in the universal property
+follows, moreover the uniqueness is obvious.10 BORNE AND VISTOLI
+For the last assertion, it is enough to notice that Lsephas trivial kernel if L
+has. ♠
+IfXis a scheme, we denote by X´ etthe small ´ etale site of X, whose objects are
+´ etale morphisms U→X. When we mention a sheaf on X, we will always mean a
+sheaf on X´ et. Thus, for example, by OXwe mean the sheaf on X´ etsending each
+U→XintoO(U). The Zariski site of Xwill be hardly used. We will often indicate
+an object U→XofX´ etsimply by U. The restriction of the stacks DivXandPicX
+toX´ etdefined above will be denoted by DivX´ etandPicX´ et.
+3.Deligne–Faltings structures
+3.1.Deligne–Faltings structures and logarithmic structures. Now we re-
+formulate the classical notion of a logarithmic structure on a schem e in a form that
+is more suitable to define parabolic sheaves.
+Definition 3.1. LetXbe a scheme. A pre-Deligne–Faltings structure (A,L) on
+Xconsists of the following data:
+(a) a presheaf of monoids A:Xop
+´ et→(ComMon) on X´ et, and
+(b) a symmetric monoidal functor L:A→DivX´ et.
+ADeligne–Faltings structure onXis a pre-Deligne–Faltings structure ( A,L)
+such that Ais a sheaf, and Lhas trivial kernel.
+A morphism of pre-Deligne–Faltings structures from ( A,L) to (B,M ) is a pair
+(φ,Φ), where φ:A→Bis a morphism of sheaves of monoids and Φ: L→M◦φ
+is a morphism of symmetric monoidal cartesian functors. A morphism of Deligne–
+Faltings structures is a morphism of pre-Deligne–Faltings structur es between Deligne–
+Faltings structures.
+Given a sheaf of monoids AonX´ et, we will sometimes say that a cartesian
+symmetric monoidal functor L:A→DivXis a Deligne–Faltings structure to mean
+that (A,L) is a Deligne–Faltings structure, that is, that Lhas a trivial kernel.
+The composition of morphisms of pre-Deligne–Faltings structures o nXis defined
+in the obvious way. This defines the categories of pre-Deligne–Faltin gs structures
+and of Deligne–Faltings structures on X.
+Notice that, since DivXis fibered in groupoids, a morphism ( φ,Φ): (A,L)→
+(B,M ) of pre-Deligne–Faltings structures is an isomorphism if and only if φ:A→B
+is an isomorphism.
+Remark 3.2. To compute with symmetric monoidal functors L:A→DivXthe
+following convention is useful. If a∈A(U), we denote the image of ainDivU
+byL(a) = (La,σL
+a). Then σL
+0∈L0is nowhere vanishing; the isomorphism ǫL:
+(OU,1)≃L(0) is uniquely determined by the condition that it carries σL
+0into 1.
+The embedding of the category of Deligne–Faltings structures on Xinto the
+category of pre-Deligne–Faltings structures has a left adjoint.
+Proposition 3.3. Let(A,L)be a pre-Deligne–Faltings structure on a scheme X.
+There exists a Deligne–Faltings structure (Aa,La), together with a homomorphism
+of pre-Deligne–Faltings structures (A,L)→(Aa,La), that is universal among ho-
+momorphism from (A,L)to Deligne–Faltings structures.
+Furthermore, if Kis the kernel of L:A→DivX, thenAais the sheafification
+of the presheaf quotient A/K.PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 11
+Proof. Thanks to Proposition 2.4, we construct a morphism ( A,L)→(Apa,Lpa),
+such that Apais the presheaf quotient A/K , andLpahas trivial kernel, and show
+that it is universal among morphisms from ( A,L) to pre-Deligne–Faltings structures
+(B,M ) such that the kernel of Mis trivial. Then we apply Proposition 2.10 to
+sheafify. ♠
+Next we connect our notion of logarithmic structure with the classic al one.
+Definition 3.4 ([Kat89]) .Alog structure on a scheme Xis a pair ( M,ρ), where
+Mis a sheaf of monoids on X´ etandρ:M→ OXis a morphism, where OXdenotes
+the multiplicative monoid of the ring OX, with the property that the induced
+homomorphism ρ−1O∗
+X→ O∗
+Xis an isomorphism.
+A morphism ( M,ρ)→(M′,ρ′) is a homomorphism of sheaves of monoids f:M→
+M′, such that ρ′◦f=ρ. This defines the category of log structures.
+This is much too large, and one imposes various conditions on the stru cture, the
+first of them usually being that Mis an integral sheaf of monoids. An even weaker
+condition is the following.
+Definition 3.5 ([Ogu]).A log structure ( M,ρ) on a scheme Xisquasi-integral if
+the action of ρ−1O∗
+X≃ O∗
+XonMis free.
+In other words, ( M,ρ) is quasi-integral if whenever U→Xis an ´ etale morphism
+andα∈ρ−1O∗
+X(U) andm∈M(U) are such that α+m=m, thenα= 0.
+Theorem 3.6. The category of Deligne–Faltings structures on Xis equivalent to
+the category of quasi-integral log structures on X.
+Proof. Consider the morphism of symmetric monoidal fibered categories OX→
+DivX´ etsending a section f∈ OX(U) into (OU,f). This is a torsor under O∗
+X. There
+is a section X´ et→DivX´ etsendingU→Xinto (OU,1); this gives an equivalence of
+X´ etwith the full fibered subcategory of DivX´ etof invertible sheaves with a nowhere
+vanishing section, and so is represented by Zariski open embedding s. The inverse
+image of X´ etinOXisO∗
+X.
+If (A,L) is a Deligne–Faltings structure on X, defineMas the fibered product
+A×DivX´ etOX. This is a O∗
+X-torsor over A, hence it is equivalent to a sheaf. The
+symmetric monoidal structures of A,OXandDivX´ etinduce a symmetric monoidal
+structure on M, that is, Macquires a structure of a sheaf of monoids. By hy-
+pothesis, the inverse image of X´ et⊆DivX´ etviaL:A→DivX´ etis the zero-section
+X´ et⊆A; hence by base change to OXthe inverse image of O∗
+X⊆ OXinMco-
+incides with the inverse image of X´ et⊆A, which is again O∗
+X. This shows that
+M→ OXis in fact a quasi-integral log structure.
+This construction gives a functor from Deligne–Faltings structure s onXto quasi-
+integral log structures (the action of the functor on arrows is ea sy to construct). In
+the other direction, let ρ:M→ OXbe a quasi-integral log structure; the free action
+ofO∗
+X≃ρ−1O∗
+XmakesMinto aO∗
+X-torsor over Mdef=M/O∗
+X. The homomorphism
+ρ:M→ OXisO∗
+X-equivariant; the stack-theoretic quotient [ OX/O∗
+X] isDivX´ et,
+so we obtain a symmetric monoidal functor L:M→DivX´ et. It is immediate to
+see that the kernel of L:M→DivX´ etis trivial; so ( M,L ) is a Deligne–Faltings
+structure on X. This is the action on objects of a functor from quasi-integral log
+structures to Deligne–Faltings structures, which is easily seen a qu asi-inverse to the
+previous functor. ♠12 BORNE AND VISTOLI
+3.2.Direct and inverse images of Deligne–Faltings structures. Letf:
+X′→Xbe a morphism of schemes. The stack f∗DivX′
+´ etonX´ etis defined as
+the fibered product DivX′
+´ et×X′
+´ etX´ et, where the functor X´ et→X′
+´ etis given by
+fibered product. In other words, if U→Xis an ´ etale morphism, we have
+f∗DivX′
+´ et(U) =Div(X′×XU).
+There is a natural morphism of stacks: DivX´ et→f∗DivX′
+´ etonX´ et, given by
+pullback1.
+Let us begin with the definition of the direct image of a Deligne–Faltings struc-
+ture (A′,L′) onX′.
+Lemma 3.7. The symmetric monoidal stack f∗A′×f∗DivX′
+´ etOX, fibered product
+off∗L′:f∗A′→f∗DivX′
+´ etand of the composite OX→DivX´ et→f∗DivX′
+´ et, is
+equivalent to a sheaf of monoids on X.
+Proof. SetM′=A′×DivX′
+´ etOX′. The proof of Theorem 3.6 shows that M′
+is (equivalent to) a sheaf of monoids on X′, and clearly f∗A′×f∗DivX′
+´ etOX≃
+f∗M′×f∗OX′OX. ♠
+Definition 3.8. Letf:X′→Xbe a scheme morphism, and ( A′,L′) a Deligne–
+Faltings structure on X′. We define the direct image f∗(A′,L′) as the Deligne–
+Faltings structure associated with the pre-Deligne–Faltings struc turef∗A′×f∗DivX′
+´ et
+OX→ OX→DivX´ et2.
+We now define the pull-back of a log structure ( A,L) onXalong the scheme
+morphism f:X′→X.
+Proposition 3.9. There is up to unique isomorphism a unique pair (f∗L,α)where
+f∗L:f∗A→DivX′
+´ etis a Deligne–Faltings structure on X′andαan isomorphism of
+symmetric monoidal functors between the composites A→f∗f∗Af∗f∗L− −−− →f∗DivX′
+´ et
+andAL− →DivX´ et→f∗DivX′
+´ et.
+Proof. The uniqueness statement is easily proven. To show the existence, we can
+work with the pull-back presheaf f−1A, and then use Proposition 2.10 to sheafify.
+LetU′→X′be an ´ etale morphism, then f−1A(U′) = lim− →U′→UA(U), where U′→
+Uvaries in all X-morphisms from U′to an ´ etale scheme U→X. Since for each such
+morphism, we have a monoidal functor A(U)L(U)− −− →Div(U)→Div(U′), and these
+functors are compatible with restriction, we get a monoidal functo rf−1A(U′)→
+Div(U′), also compatible with restrictions. This defines a monoidal functor f∗L:
+f∗A→DivX′
+´ et, and the existence of αis obvious by construction. The only thing
+that is left to check is that f∗Lhas trivial kernel.
+LetU′→X′be an ´ etale morphism, and a′∈f∗A(U′), such that f∗L(a′) is
+invertible. There exists an ´ etale covering {U′
+i→U′}iofU′,X-morphisms fi:U′
+i→
+Uito an ´ etale scheme Ui→X, and sections ai∈A(Ui), such that a′
+|U′
+i=f∗
+iai|U′
+i.
+Thenf∗L(a′)|U′
+iis invertible, and f∗L(a′)|U′
+i≃f∗
+iL(ai) by the universal property.
+1And accordingly there is a natural morphism of stacks f∗DivX´ et→DivX′
+´ etonX′
+´ et, but since
+the definition of f∗DivX´ etis quite complicated, we won’t use it.
+2The morphism f∗A′×f∗DivX′
+´ etOX→ OXdefines a log structure, but it is not necessarily
+quasi-integral.PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 13
+Letx: Spec Ω →U′
+ibe a geometric point, we deduce that x∗(f∗
+i(σai)) does not
+vanish, and so σaiis invertible on a neighborhood of fi(x), and since Lhas trivial
+kernel,ai= 0 on this neighborhood, hence f∗
+iai= 0 on a neighborhood of x. Thus
+sincexis arbitrary, f∗
+iai= 0 onU′
+i, and this implies a′
+|U′
+i= 0, and finally a′= 0.♠
+Definition 3.10. Letf:X′→Xbe a scheme morphism, and ( A,L) a Deligne–
+Faltings structure on X. The Deligne–Faltings structure f∗L:f∗A→DivX′
+´ eton
+X′defined by Proposition 3.9 is called the pull-back Deligne–Faltings struc ture,
+and denoted by f∗(A,L).
+Remark 3.11. Proposition 3.9 shows in fact a bit more: there is a canonical
+adjunction between the functors ( A,L)/maps⊔o→f∗(A,L) and (A′,L′)/maps⊔o→f∗(A′,L′).
+3.3.Charts for Deligne–Faltings structures. IfPis a monoid and Xis a
+scheme, we denote by PXthe constant presheaf on X´ etsuch that PX(U) =Pfor
+allU→XinX´ et, and by Psh
+Xthe associated constant sheaf. Notice that if Ais a
+sheaf of monoids on X´ et, we have bijective correspondences between homomorphism
+of monoids P→A(X), homomorphism of presheaves of monoids PX→A, and
+homomorphism of sheaves of monoids Psh
+X→A.
+Definition 3.12. LetXbe a scheme, Abe a sheaf of monoids on X´ et. Achart
+forAconsists of a homomorphism of monoids P→A(X), such that Pis a finitely
+generated monoid, and the induced homomorphism Psh
+X→Ais a cokernel in the
+category of sheaves of monoids.
+Anatlas consist of an ´ etale covering {Xi→X}and a chart Pi→A(Xi) for each
+restriction A]Xi.
+Definition 3.13. A sheaf of monoids AonX´ etiscoherent if it is sharp and has
+an atlas. A sheaf of monoids is fineif it is coherent and integral.
+Being a chart is property that can be checked at the level of stalks at geometric
+points of X.
+Proposition 3.14. LetXbe a scheme, Abe a sheaf of monoids on X´ et. A homo-
+morphism of monoids P→A(X)is a chart if and only if Pis finitely generated, and
+for each geometric point x: Spec Ω →Xthe induced homomorphism of monoids
+P→Axis a cokernel.
+Proof. LetKbe the kernel of the induced homomorphism of presheaves of mono ids
+PX→A. The homomorphism Psh
+X→Ais a cokernel if and only if the induced
+homomorphism ( PX/K)sh→Ais an isomorphism; hence Psh
+X→Ais a cokernel
+if and only if P/Kx= (PX/K)sh
+x→Axis an isomorphism for all x. On the other
+hand the kernel of P→Axis the stalk Kx, and the stalk of ( PX/K)shatxis
+P/Kx; soP/Kx→Axis an isomorphism if and only if P→Axis a cokernel. ♠
+In what follows we are going to use R´ edei’s theorem, stating that ev ery finitely
+generated commutative monoid is finitely presented (see for examp le [R´ ed65, The-
+orem 72], or [Gri93]).
+If a sheaf of monoids is coherent, around each geometric point of Xthere exists
+a minimal chart.
+Proposition 3.15. LetAbe a coherent sheaf of monoids on X´ et, and let x:→
+Spec Ω→Xa geometric point. Then there exists an ´ etale neighborhood Spec Ω→14 BORNE AND VISTOLI
+U→Xofxand a chart P→A(U)for the restriction A]U, such that the induced
+homomorphism P→Axis an isomorphism.
+So, for example, a fine sheaf of monoids has an atlas/parenleftbig
+{Xi→X},{Pi→A(Xi)}/parenrightbig
+in which all the Piare integral and sharp.
+Proof. Let us start with a Lemma.
+Lemma 3.16. LetPbe a finitely generated monoid. Any cokernel P→Qis the
+cokernel of a homomorphism F→P, whereF≃Nris a finitely generated free
+monoid.
+Proof. WriteP→Qas the cokernel of a homomorphism F→P, whereFis a free
+monoid over a set I. For each finite subset A⊆IcallFA⊆Fthe free submonoid
+generated by AandCAthe cokernel of the composite FA⊆F→P, andRA⊆P×P
+the congruence equivalence relation determined by the projection P→CA. The
+monoidal equivalence relation Rdetermined by the homomorphism P→Qis the
+union of the RA; sinceRis finitely generated as a monoidal equivalence relation,
+by R´ edei’s theorem, there exists a finite subset A⊆Isuch that R=RA. SoQis
+the cokernel of FA→P. ♠
+By passing to an ´ etale neighborhood, we may assume that there ex ists a global
+chartP→A(X). Consider the homomorphism P→Ax; this is a cokernel, hence
+by the Lemma there exists a finite free monoid Fand a homomorphism F→P
+with cokernel Ax. By passing to an ´ etale neighborhood, we may assume that
+the composite F→P→A(X) is 0. Call Qthe cokernel of F→P; we have
+a factorization P→Q→A(X). IfKis the kernel of Psh
+X→A, then we see
+immediately that Qsh
+X→Ais the cokernel of the composite K⊆Psh
+X→Qsh
+X.
+It follows from the construction that the induced homomorphism Q→Axis an
+isomorphism. ♠
+For later use, we note the following fact, saying that charts can be chosen com-
+patibly with arbitrary homomorphisms of coherent sheaves of mono ids.
+Proposition 3.17. Letf:A→Bbe a homomorphism of coherent sheaves of
+monoids on a scheme X. Given a geometric point x: Spec Ω →X, there exists an
+´ etale neighborhood U→Xofx, two finitely generated monoids PandQ, and a
+commutative diagram
+P
+/d15/d15/d47/d47Q
+/d15/d15
+A(U)f(U)/d47/d47B(U)
+where the columns are charts for f|U:A]U→B]U. Furthermore, the columns
+can be chosen so that the induced homomorphisms P→AxandQ→Bxare
+isomorphisms.
+Proof. After passing to an ´ etale neighborhood, we may assume that ther e are charts
+P→A(X) andQ→B(X) such that the composites P→A(X)→Axand
+Q→B(X)→Bxare isomorphisms, by Proposition 3.15. The homomorphism
+fx:Ax⊆Bxinduces a homomorphism P→Q. The diagram above does not
+necessarily commute: however the images of a given finite number of generatorsPARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 15
+ofPinBxcoincide, hence after a further restriction we may assume that it d oes
+commute. ♠
+Proposition 3.18. Letf:Y→Xbe a morphism of schemes, Aa coherent sheaf
+of monoids on X´ et. Then the sheaf f∗AonY´ etis coherent.
+Proof. Being coherent is a local property in the ´ etale topology, so we may a ssume
+that there exists a chart P→A(X). We claim that the composite P→A(X)f∗
+− →
+f∗A(Y) is also a chart. According to Proposition 3.14 this can be checked at the
+level of stalks: but the stalk of f∗Aat a geometric point yofYis the stalk of Aat
+the image of y, so the statement is clear. ♠
+The condition of being coherent is local in the fppf topology. In fact we have the
+following stronger statement.
+Proposition 3.19. Letf:Y→Xbe an open surjective morphism of schemes,
+and letAbe a sheaf of monoids on X´ et. Iff∗Ais coherent as a sheaf on Y´ et, then
+Ais coherent.
+Proof. Letx: Spec Ω →Xbe a geometric point; after possibly extending Ω, we
+may assume that there exists a lifting y: Spec Ω →Y. After passing to an ´ etale
+neighborhood of yinYand replacing Xwith its image in X, by Proposition 3.15
+we may assume that there exists a chart P→f∗A(Y). From the induced ho-
+momorphism P→f∗Ayand the canonical isomorphism of stalks Ax≃f∗Aywe
+obtain a homomorphism P→Ax. Since by R´ edei’s theorem Pis finitely presented,
+after passing to an ´ etale neighborhood of xwe can assume that the homomorphism
+P→Axfactors as P→A(X)→Ax. The composite of P→A(X) with the
+canonical homomorphism A(X)→f∗A(Y) does not necessarily coincide with the
+given chart P→f∗A(Y); but since the images of the generators of Pin (f∗A)y
+through the two maps are the same, after passing to an ´ etale neig hborhood of yinY
+and further shrinking Xwe may assume that the two homomorphisms P→A(Y)
+coincide.
+We claim that P→A(X) is a chart for A. IfKis the kernel of the homo-
+morphism of presheaves of monoids PX→A, we need to check that the induced
+homomorphism ( PX/K)sh→Ais an isomorphism, or, equivalently, that for any
+geometric point ξofXthe induced homomorphism P/Kξ= (PX/K)sh
+ξ→Aξis an
+isomorphism. But if ηis a geometric point of Ylying over ξ, we have that the ker-
+nel ofPY→f∗Ais the presheaf pullback fpK; soKξ=fpKη⊆P. The induced
+homomorphism P/fpKη→f∗Aη≃Aξis an isomorphism, and this completes the
+proof. ♠
+Definition 3.20. A Deligne–Faltings structure ( A,L) on a scheme Xiscoherent if
+Ais a coherent sheaf of monoids. It is fineifAis fine (i.e., integral and coherent).
+Let (A,L) be a Deligne–Faltings structure on a scheme X,P→A(X) a chart.
+The composite L0:P→A(X)L(X)− −− →DivXcompletely determines the Deligne–
+Faltings structure.
+Proposition 3.21. LetXbe a scheme, Pa finitely generated monoid, L0:P→
+DivXa symmetric monoidal functor. Then there exists a Deligne–F altings struc-
+ture (A,L)onX, together with a homomorphism of monoids π:P→A(X)and16 BORNE AND VISTOLI
+an isomorphism ηof symmetric monoidal functors between L0and the composite
+Pπ−→A(X)L(X)− −− →DivX
+such that P→A(X)is a chart. If Kis the kernel of the symmetric monoidal
+functorPX→DivX´ et, thenAis isomorphic to (PX/K)sh.
+Furthermore, this is universal among such homomorphisms.
+More precisely, given (A′,L′)another Deligne–Faltings structure on X, a homo-
+morphism of monoids π′:P→A′(X), and an isomorphism η′ofL0with the com-
+positePπ′
+− →A′(X)L′(X)− −−− →DivX, there exists a unique morphism (φ,Φ): (A,L)→
+(A′,L′)such that φ(X)◦π=π′and the diagram
+L0η/d47/d47
+η′
+/d36/d36■■■■■■■■■■L(X)◦π
+/d15/d15
+L′(X)◦π′
+of symmetric monoidal functors P→DivX, where the vertical arrow is the homo-
+morphism induced by ΦX:L(X)→L′(X)◦φ(X), commutes.
+We call (A,L) the Deligne–Faltings structure associated with L0.
+Proof. This is a direct consequence of Proposition 3.3. ♠
+Example 3.22. LetXbe a scheme, ( L1,s1), . . . , (Lr,sr) objects of DivX. Con-
+sider the monoidal functor L0:Nr→DivXthat sends ( k1,...,k r)∈Nrinto
+(L1,s1)⊗k1⊗···⊗ (Lr,sr)⊗kr.
+We call the Deligne–Faltings structure associated with L0the Deligne–Faltings
+structure generated by (L1,s1), ..., (Lr,sr).
+Denote by ei∈Nrtheithcanonical basis vector, that is, the vector with 1 at
+theithplace and 0 everywhere else. Suppose that L′
+0:Nr→DivXis another
+symmetric monoidal functor, and for each i= 1, . . . , rwe have an isomorphism
+φi:L′
+0(ei)≃(Li,si); then it is easy to show that there exists a unique isomorphism
+φ:L′
+0≃L0whose value at eiisφi. By Proposition 3.21, this implies that the
+Deligne–Faltings structure ( A,L) generated by ( L1,s1), . . . , (Lr,sr) is, up to a
+unique isomorphism, the only Deligne–Faltings structure with a chart Nr→A(X),
+such that the composite Nr→A(X)L(X)− −− →Div(X) sendseito (Li,si).
+3.4.Charts and Kato charts. Our notion of chart can be compared with Kato’s.
+Definition 3.23 ([Kat89]) .AKato chart for the log structure ( M,ρ) is the data
+of a finitely generated monoid P, and a morphism P→M(X), such that the
+composite P→M(X)→M(X) is a chart for M.
+Definition 3.24 ([Ogu, Definition 2.1.1]) .A log structure admitting a chart locally
+onX´ etis called coherent . A log structure is fineif it is coherent and integral.
+If we are given a finitely generated monoid Pand a homomorphism P→ O (X),
+we compose with the morphism O(X)→DivXdefined in the proof of Theorem 3.6,
+sendingf∈ O(U) into (OU,f), we obtain a symmetric monoidal functor P→
+DivX, which, according to Proposition 3.21, gives us a Deligne–Faltings str ucture
+(A,L) onX. Call (M,ρ) the associated log structure: the homomorphism P→
+A(X), together with P→ O (X), yields a Kato chart P→M(X) (recall, from thePARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 17
+proof of Theorem 3.6, that Mdef=A×DivX´ etOX). So the log structure associated
+with a Deligne–Faltings structure ( A,L) is coherent if and only if there exists an
+´ etale cover {Xi→X}and charts {Pi→A(Xi)}, such that the composites Pi→
+A(Xi)L− →DivXilift to homomorphisms Pi→ O (Xi).
+Now we want to investigate the question of when a chart P→A(X) for a
+Deligne–Faltings structure ( A,L) onXlifts to a Kato chart P→M(X). In other
+words, when does a symmetric monoidal functor P→DivXlift to a homomor-
+phism of monoids P→ O (X)?
+Fix a finitely generated monoid P. We denote by Z[P] the monoid ring of
+P. Since we are using additive notation for P, it is convenient to introduce an
+indeterminate x, and write an element of Z[P] as a finite sum/summationtext
+p∈Papxp, where
+ap∈Zfor allp.
+A morphism P→ O (X) correspond to a ring homomorphism Z[P]→ O (X),
+hence to a morphism of schemes X→SpecZ[P]. Thus we think of Spec Z[P]
+as representing the functor Hom (P,A1) from schemes to monoids (the monoidal
+structure is given by multiplication in A1). Thus Spec Z[P] is the space of Kato
+charts.
+Consider the fibered category Hom (P,DivZ)→(Sch), whose objects over a
+schemeXare symmetric monoidal functors P→DivX. A morphism X→
+Hom (P,DivZ) gives a chart for a Deligne–Faltings structure on X. We think of
+Hom (P,DivZ) as the stack of charts. There is an obvious morphism
+Hom (P,A1)−→Hom (P,DivZ)
+that corresponds to the procedure of associating a chart to a Ka to chart. In
+other words, if a morphism X→Hom (P,A1) corresponds to a homomorphism
+P→ O (X), the corresponding morphism X→Hom (P,DivZ) corresponds to the
+composite P→ O (X)→DivX. The issue is: when is it possible to lift a sym-
+metric monoidal functor X→Hom (P,DivZ) to a symmetric monoidal functor
+X→SpecZ[P]?
+Set
+/hatwidePdef= Hom (P,Gm) = Hom (Pgp,Gm) ;
+then/hatwidePis a diagonalizable group scheme, acting on Spec Z[P] (the action is induced
+by the action of GmonA1by multiplication). Equivalently, we can think of /hatwidePas
+the group scheme of invertible elements of Spec Z[P].
+Since the group scheme /hatwidePis diagonalizable, with character group
+Pgp≃Hom(/hatwideP,Gm),
+by the standard description of representations of /hatwideP, a/hatwideP-torsorη:E→Tgives
+aPgp-grading on the sheaf of algebras η∗OE. The trivial torsor /hatwideP×T→T
+corresponds to the group algebra OT[Pgp]. This gives an equivalence of categories
+between the category of /hatwideP-torsors and the opposite of the groupoid of sheaves of
+Pgp-graded algebras over OT, such that each graded summand is invertible. Given
+such an algebra A, the torsor Eis the relative spectrum SpecTA; the action is
+defined by the grading.
+The action of /hatwidePon SpecZ[P] corresponds to the natural Pgp-grading
+Z[P] =/circleplusdisplay
+u∈Pgp/parenleftBig/circleplusdisplay
+p∈P
+ιP(p)=uZxp/parenrightBig
+.18 BORNE AND VISTOLI
+A morphism T→[SpecZ[P]//hatwideP] corresponds to a /hatwideP-torsorη:E→Tand a/hatwideP-
+equivariant morphism E→SpecZ[P]; this morphism gives η∗OEthe structure of
+aOT[P] algebra, which is compatible with the Pgp-grading of Z[P]. Hence, we see
+that the groupoid of objects of [Spec Z[P]//hatwideP] overTis equivalent to the opposite of
+the groupoid whose objects are sheaf of commutative Pgp-gradedOT[P]-algebras
+overT, whose grading is compatible with the grading of OT[P], that are fppf locally
+isomorphic to OT[Pgp] as graded OT-algebras.
+Proposition 3.25. LetPbe a finitely generated monoid. There is an equivalence
+of fibered categories between Hom (P,DivZ)with the quotient stack [SpecZ[P]//hatwideP]by
+the action defined above.
+Proof. LetTbe a scheme; suppose that we are given an object of [Spec Z[P]//hatwideP] over
+T, corresponding to a sheaf AofOT[P]-algebras, as above. With this, we associate
+a symmetric monoidal functor P→DivTas follows. Write A=⊕u∈PgpAu; then
+by the local description of Awe see that Auis an invertible sheaf on T. The functor
+P→DivTassociates with p∈Pthe pair ( AιP(p),xp), where by abuse of notation
+we identify the element xp∈Z[P] with its image in A(T). The symmetric monoidal
+structure on the functor is given by the algebra structure on A; we leave the easy
+details to the reader.
+For the inverse construction, we need the following. Suppose that Gis a finitely
+generated abelian group, L:G→PicTa symmetric monoidal functor. With this
+we can associate a G-graded sheaf of OT-modules
+ALdef=/circleplusdisplay
+g∈GLg.
+It is easy to see that the isomorphisms Lg⊗OTLh→Lg+hcoming from the sym-
+metric monoidal structure of LgiveALthe structure of a sheaf of commutative
+G-graded algebras.
+The trivial symmetric monoidal functor G→DivTdefines the sheaf of group
+algebras OT[G].
+Lemma 3.26. Locally in the fppf topology, the sheaf of commutative G-graded
+algebras associated with a symmetric monoidal monoidal fun ctorG→DivTis
+isomorphic to OT[G].
+Proof. Let us decompose Gas a product G1× ··· ×Grof cyclic groups. If we
+denote by Lithe restriction of LtoGi⊆G, it is immediate to see that if gi∈Gi
+for alli, thenLg1...gr≃Lg1⊗OT···⊗OTLgr, and that this induces an isomorphism
+of sheaves of OT-algebras
+AL≃AL1⊗OT···⊗OTALr;
+so we may assume that Gis cyclic.
+Callγa generator of G. IfGis infinite, after restricting Tin the Zariski topology
+we may assume that there exists an isomorphism Lγ≃ OT. The monoidal structure
+ofLgives an isomorphism Lγk≃L⊗k
+γfor any element γkinG; this induces an
+isomorphism of sheaved of G-graded modules AL≃/circleplustext
+g∈GOT=OT[G], which is
+immediately seen to be an isomorphism of OT-algebras.
+IfGhas order n, then the symmetric monoidal structure of Lgives isomorphism
+OT≃L1=Lγn≃L⊗n
+γ;PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 19
+after passing to a fppf cover of T, we may assume that there exists a nowhere
+vanishing section sofLγsuch that s⊗ncorresponds to 1 ∈ OT; this defines an
+isomorphism OT≃Lγ, sending 1 to s. Ifγk∈G, we obtain an isomorphism
+OT≃Lk
+γ≃Lγk;
+the condition on sensures that this is independent of k. As in the previous case,
+this defines an isomorphism AL≃ OT[G], which is easily seen to be an isomorphism
+of algebras. ♠
+We claim that the functor from [Spec Z[P]//hatwideP](T)→Hom(P,DivT) constructed
+above is an equivalence. Given a symmetric monoidal functor L:P→DivTwe
+first define a symmetric monoidal functor Lgp:P→PicTby the obvious formula3
+Lgp
+ιPa−ιPbdef=La⊗L∨
+b, and then construct a sheaf of algebras Adef=/circleplustext
+u∈PgpLgp
+u.
+We define a structure of OT[P]-algebra on Aby sending, for each p∈ O[P](T)
+the element xpinto the tautological section σL
+p∈Lp≃Lgp
+ιPp. We need to check
+thatAgives an object of [Spec Z[P]//hatwideP] overT; once this is done, it is straightfor-
+ward to verify that this construction gives a quasi-inverse to the f unctor defined
+above. In doing so the only difficulty is to show that Ais fppf locally isomorphic
+toOT[Pgp]; and this is the content of the Lemma 3.26. This concludes the proof
+of the Proposition. ♠
+Corollary 3.27. Let(A,L)be Deligne–Faltings structure on a scheme X, and call
+(M,ρ)the induced log structure. Then (A,L)is coherent if and only if there exists
+a fppf cover X′→Xsuch that the pullback of (M,ρ)toX′is coherent.
+Proof. Assume that ( M,ρ) becomes coherent after pulling to a fppf cover f:X′→
+X. Thenf∗M=f∗Ais coherent, and, by Proposition 3.19, ( A,L) is coherent.
+On the other hand, if ( A,L) is coherent, pick an ´ etale cover {Xi→X}and
+charts{Pi→A(Xi)}. Consider the induced morphisms Xi→Hom (Pi,DivZ). The
+pullback
+X′
+idef=Xi×Hom(Pi,DivZ)SpecZ[Pi]
+is a/hatwidePi-torsor over Xi, and the pullback of ( M,ρ) toX′
+iis coherent. We conclude
+the proof by setting X′def=⊔iX′
+i. ♠
+But in fact we can do better.
+Proposition 3.28. Let(A,L)be Deligne–Faltings structure on a scheme X, and
+call(M,ρ)the induced log structure. Then (A,L)is coherent if and only if (M,ρ)
+is coherent.
+From this and from Corollary 3.27 we obtain the following, which seems t o be
+new.
+Corollary 3.29. Let(M,ρ)be a logarithmic structure on a scheme X. If there
+exists a fppf cover X′→Xsuch that the pullback of (M,ρ)toX′is coherent, then
+(M,ρ)is coherent.
+Proof of Proposition 3.28. If (M,ρ) is coherent, then ( A,L) is coherent, by Corol-
+lary 3.27; so we may assume that ( A,L) is coherent. We need to show that ( M,ρ)
+is coherent.
+3See Proposition 5.4 for a more general construction.20 BORNE AND VISTOLI
+First of all, consider the case that Xis the spectrum of a strictly henselian local
+ringR. Then the global sections functor gives an equivalence between th e category
+of sheaves of monoids on X´ etand that of monoids; consequently, we can identify
+MandAwith their monoids of global sections.
+We will show that there exists a finitely generated submonoid P⊆M, such that
+the composite P⊆M→Ais a cokernel; then the embedding P⊆Mgives a Kato
+chart. This is done as follows.
+The kernel of the natural projection M→Ais the group R∗of units in R. The
+monoidAis finitely generated; let a1, . . . ,asbe generators, and let q1, . . . ,qsbe
+elements of Mmapping to a1, . . . ,asrespectively. Denote by Qthe submonoid of M
+generated by the qi’s. Denote by Sthe image of the induced homomorphism Qgp→
+Mgp; sinceQis a finitely generated monoid, the group Sis finitely generated, and
+so is the subgroup S∩R∗ofR∗.
+Letr1, . . . ,rtbe generators of the group S∩R∗; denote by Pthe submonoid of
+Mgenerated by q1, . . . ,qs,±r1, . . . ,±rt. We claim that P∩R∗=S∩R∗.
+The inclusion S∩R∗⊆P∩R∗is obvious. Let p∈P∩R∗; writep=q+p′,
+whereq∈Qandp′∈S∩R∗. Thenq=p−p′∈R∗, henceq∈Q∩R∗⊆S∩R∗,
+sop∈S∩R∗.
+Let us verify that the composite P⊆M→Ais a cokernel. It is clearly
+surjective, since Q, which is contained in P, surjects onto A. Now we need to
+check that if p1andp2are elements of Phaving the same image in A, then there
+existsm∈ker(P→A) =S∩R∗such that p2=p1+m. Such an mexists in R∗,
+becauseM/R∗=A. It is immediate to see that the image of PgpinMgpequalsS;
+hencem∈S, and this concludes the proof.
+In the general case, let x: Spec Ω →Xbe a geometric point of X; we need
+to construct a chart for ( M,ρ) in some ´ etale neighborhood of xinX. LetRbe
+the strict henselization of the local ring OX,x; by the previous case, the pullback
+(Mx,ρx) of (M,ρ) to Spec Ris coherent. It is easy to see that Mxis the fiber of M
+atx. LetP→Mxbe a Kato chart; since Pis finitely presented, after passing to an
+´ etale neighborhood of xwe may assume that P→Mxcomes from a homomorphism
+P→M(X). We need to check that the composite Psh
+X→M→Ais a cokernel,
+perhaps after further restricting X.
+LetKbe the kernel of Psh
+X→A; we have to show that the induced homo-
+morphism f:Psh
+X/K→Ais an isomorphism. Set Bdef=Psh
+X/K. The kernel of f
+is obviously trivial; this, together with the fact that Ais sharp, implies that Bis
+sharp. Also, Bhas a tautological chart, hence it is coherent. From Proposition 3.1 7,
+we see that after restricting to an ´ etale neighborhood of xwe may assume that there
+are charts Bx→B(X) andAx→A(X), inducing the identity on BxandAx, such
+that the diagram
+Bxfx/d47/d47
+/d15/d15Ax
+/d15/d15
+B(X)f(X)/d47/d47A(X)
+commutes. Call KBandKAthe kernels of the induced homomorphisms ( Bx)sh
+X→
+Band (Ax)sh
+X→Arespectively; since the kernel of fis trivial, we see that KB=
+f−1
+xKA. Since fxis an isomorphism, it induces an isomorphism ( Bx)sh
+X/KB≃PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 21
+(Ax)sh
+X/KA; but the induced homomorphisms ( Bx)sh
+X/KB→Band (Ax)sh
+X/KB→
+Aare isomorphisms by hypothesis, so fis an isomorphism, as claimed. ♠
+4.Systems of denominators and stacks of roots
+Several instances of the construction given in this section have be en introduced
+by Martin Olsson ([MO05], [Ols07]), and the idea should certainly be attr ibuted to
+him.
+4.1.Systems of denominators.
+Definition 4.1. LetAandBbe finitely generated monoids. A Kummer homo-
+morphism f:A→Bis an injective homomorphism of monoids, such that for each
+b∈Bthere exists a positive integer msuch that mbis in the image of f.
+Lemma 4.2. Letf:A→Bbe a Kummer homomorphism of finitely generated
+monoids. Then fgp:Agp→Bgpis injective with finite cokernel.
+Proof. SinceBgpis a finitely generated abelian group, to show that the cokernel is
+finite it is enough to show that it is torsion. This is evident.
+Let us show that fgpis injective. We write the elements of Agpas differences
+a−a′; we have a−a′= 0 inAgpif and only if there exists x∈Asuch that
+a+x=a′+xinA. Suppose that f(a)−f(a′) =fgp(a−a′) = 0. Then there exists
+y∈Bsuch that f(a) +y=f(a′) +y. Ifm > 0 is such that my=f(x), we have
+f(a+x) =f(a) +my=f(a′) +my=f(a′+x) inB, hencea+x=a′+xinB,
+anda−a′= 0 inAgp. ♠
+Definition 4.3. LetXbe a scheme, Aa coherent sheaf of monoids on X´ et. A
+system of denominators forAis an injective homomorphism of sheaves of monoids
+A→Bsuch that
+(a) For any geometric point xofX, the induced homomorphism Ax→Bxis a
+Kummer homomorphism, and
+(b)Bis coherent.
+Obviously, a system of denominators A→Bis injective, because it is injective
+stalkwise. We will normally write a system of denominators for AasB/A, and
+think of Aas a subsheaf of B.
+Example 4.4. Suppose that Ais a sharp integral torsion-free sheaf of monoids, d
+a positive integer. Then the homomorphism A→Asendingatodais a system of
+denominators for A.
+Definition 4.5. LetXbe a scheme, Aa coherent sheaf of monoids on X´ et,A⊆B
+a system of denominators. A chart forB/A is a commutative diagram of monoids
+(4.1) P
+/d15/d15/d47/d47Q
+/d15/d15
+A(X) /d47/d47B(X)
+where the bottom row is induced by the embedding A⊆B, the top row is a
+Kummer homomorphism, and the columns are charts for AandBrespectively.22 BORNE AND VISTOLI
+Proposition 4.6. LetB/Abe a system of denominators for the coherent sheaf of
+monoids AonX´ et. Given a geometric point x: Spec Ω →X, there exists an ´ etale
+neighborhood U→Xofxsuch that the restriction A]U→B]Uhas a chart
+P
+/d15/d15/d47/d47Q
+/d15/d15
+A(U) /d47/d47B(U).
+Furthermore, the chart can be chosen so that the induced homo morphisms P→Ax
+andQ→Bxare isomorphisms.
+Proof. This follows immediately from Proposition 3.17. ♠
+Lemma 4.7. LetAbe a coherent sheaf of monoids on X´ et,
+Pφ/d47/d47
+/d15/d15Q
+/d15/d15
+A(X) /d47/d47B(X)
+a chart for a system of denominators B/A. Denote by KAandKBthe kernel of
+the induced homomorphisms PX→AandQX→Brespectively. Let U→Xbe
+an ´ etale morphism. An element qofQis inKB(U)if an only if there exists a
+covering {Ui→U}and positive integers misuch that miq]Ui∈KA(Ui).
+Proof. The proof is easy and left to the reader. ♠
+Remark 4.8. As a corollary of this fact we have that the system of denominators
+B/A is uniquely determined up to a unique homomorphism by the chart P→A(X)
+and the Kummer homomorphism P→Q, sinceB≃(QX/KB)sh, andKBdoes
+not depend on B.
+On the other hand, if we are given a chart P→A(X) and a Kummer homomor-
+phismP→Q, this does not necessarily give a chart for a system of denominator s
+B/A. The problem is that if we define KB⊆QXby the formula of Proposition 4.7,
+the induced homomorphism A≃(PX/KA)sh→(QX/KB)shis not necessarily
+injective, in this generality.
+It is easy to show that A→(QX/KB)shinjective, for example, when PandQ
+are integral and saturated, which is the case of greatest interes t for the applications.
+With these hypotheses we can conclude that given a chart P→A(X) and Kummer
+homomorphism P→Q, there exists a system of denominators B/A, together
+with a chart as above. Furthermore, B/A is uniquely determined up to a unique
+homomorphism.
+4.2.Stacks of roots. Let us start by defining categories of roots for Deligne–
+Faltings objects.
+Definition 4.9. Letj:P→Qa homomorphism of monoids, and let Ma sym-
+metric monoidal category, L:P→ M a symmetric monoidal functor. Then we
+define the category of roots (L)(Q/P ) as follows.
+Its objects are pairs ( M,α ), where M:Q→ M is a symmetric monoidal functor,
+andα:L→M◦jis an isomorphism of symmetric monoidal functors from Lto
+the composite M◦j:P→ M .PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 23
+An arrow hfrom (M,α ) to (M′,α′) is an isomorphism h:M→M′of symmetric
+monoidal functors Q→ M , such that the diagram
+L
+α′
+/d34/d34❊❊❊❊❊❊❊❊
+α
+/d124/d124③③③③③③③③③
+M◦jh◦j/d47/d47M′◦j
+commutes.
+The name category of roots is only justified when jis a Kummer morphism; but
+this hypothesis is not required at this stage.
+Here is a more general definition.
+Definition 4.10. LetCbe a category, j:A→Ba homomorphism of presheaves
+of monoids Cop→(ComMon). Let M → C a symmetric monoidal fibered category,
+L:A→ M a symmetric monoidal morphism of fibered categories. Then we defin e
+thecategory of roots (C,L)(B/A) as follows.
+The objects are pairs ( M,α ), where M:B→ M is a symmetric monoidal func-
+tor, and α:L→M◦jis an isomorphism of symmetric monoidal functors from L
+to the composite M◦j:A→ M .
+An arrow hfrom (M,α ) to (M′,α′) is an isomorphism h:M→M′of symmetric
+monoidal functors B→ M , such that the diagram
+L
+α′
+/d34/d34❊❊❊❊❊❊❊❊
+α
+/d124/d124③③③③③③③③③
+M◦jh◦j/d47/d47M′◦j
+commutes.
+WhenCis the category with one object and one morphism, we recover the
+previous definition.
+Remark 4.11. Notice that categories of roots, as defined above, are groupoids .
+Next we define stacks of roots for Deligne–Faltings structures in t wo different
+contexts.
+Suppose that Xis a scheme, j:P→Qa homomorphism of monoids, L:P→
+DivXa symmetric monoidal functor. For each morphism of schemes t:T→X,
+the pullback t∗:DivX→DivTyields a symmetric monoidal functor t∗◦L:P→
+DivT, from which we obtain a category of roots ( t∗◦L)(Q/P ).
+Letf:T′→Tbe a homomorphism of X-schemes from t′:T′→Xtot:T→X.
+Suppose that ( M,α ) is an object of the category of roots ( t∗◦L)(Q/P ). Then the
+composite f∗◦M:P→DivT′is a symmetric monoidal functor. The isomorphism
+α:t∗◦L≃M◦jcan be pulled back along f, yielding an isomorphism
+f∗◦α:f∗◦t∗◦L≃f∗◦M◦j;
+by composing with the natural isomorphism t′∗◦L≃f∗◦t′∗◦Lwe obtain an
+isomorphism t′∗◦L≃f∗◦M◦j, which we still denote by f∗◦α. The pair
+(f∗◦M,f∗◦α) is an object of ( t′∗◦L)(Q/P ); there is a natural functor
+f∗: (t∗◦L)(Q/P )−→(t′∗◦L)(Q/P )
+(we leave it to the reader to define the action of f∗on arrows).24 BORNE AND VISTOLI
+This defines a pseudo-functor from (Sch /X) into the 2-category of categories.
+Definition 4.12. LetXbe a scheme, j:P→Qa homomorphism of monoids,
+L:P→DivXa symmetric monoidal functor. We define the stack of roots as-
+sociated with these data, denoted by ( X,L)Q/P, or simply XQ/P, as the fibered
+category over (Sch /X) associated with the pseudo-functor above.
+Suppose that in the definition above j:P→Qis a homomorphism of finitely
+generated monoids. From Proposition 3.25 we see that the homomor phismL:P→
+DivXcorresponds to a morphism X→[SpecZ[P]//hatwideP]. Again from Proposition 3.25
+we obtain the following useful description of XQ/P. The homomorphism Z[P]→
+Z[Q] induced by jinduces a morphism Spec Z[Q]→SpecZ[P]. This is /hatwide:/hatwideQ→/hatwideP
+equivariant, where /hatwideis the homomorphism of algebraic groups over Zinduced by
+j. This gives a morphism of algebraic stacks [Spec Z[Q]//hatwideQ]→[SpecZ[P]//hatwideP]. This
+corresponds with the morphism
+Hom (Q,DivZ)−→Hom (P,DivZ)
+induced by j. From Proposition 3.25 we immediately obtain the following.
+Proposition 4.13. The stack XQ/Pis isomorphic to the fibered product
+X×[SpecZ[P]//hatwideP][SpecZ[Q]//hatwideQ].
+Remark 4.14. This can also be stated as follows. Let L:P→DivXbe a symmet-
+ric monoidal functor, corresponding, according to Proposition 3.2 5, to a morphism
+X→[SpecZ[P]//hatwideP], i.e., to a /hatwideP-torsorη:E→Xand a/hatwideP-equivariant morphism
+E→SpecZ[P]. From the proof of Proposition 3.25 we see that the Pgp-graded
+OX[P] algebra Adef=η∗OEis canonically isomorphic to/circleplustext
+u∈PgpLgp
+u. The functor
+L:P→DivXsendsp∈Pto the pair ( AιP(p),xp).
+The fibered product X×[SpecZ[P]//hatwideP][SpecZ[Q]//hatwideQ] is the stack theoretic quotient
+[E×SpecZ[P]SpecZ[Q]//hatwideQ],
+where the action of /hatwideQon the fibered product E×SpecZ[P]SpecZ[Q] = SpecX(A⊗Z[P]
+Z[Q]) is given by the natural action on the second factor, while on the fir st factor
+/hatwideQacts through the natural homomorphism /hatwideQ→/hatwidePinduced by the embedding
+P⊆Q. In other words, X×[SpecZ[P]//hatwideP][SpecZ[Q]//hatwideQ] is the relative spectrum of
+theOX-algebra A⊗Z[P]Z[Q], with the obvious grading.
+This gives a description of the category of quasi-coherent sheave s on the stack
+XQ/Pthat will be used later. A quasi-coherent sheaf on XQ/Pcorresponds to a /hatwideQ-
+equivariant quasi-coherent sheaf on E×SpecZ[P]SpecZ[Q]; and this corresponds to
+aQgp-graded quasi-coherent sheaf of modules over the sheaf of rings Bdef=A⊗Z[P]
+Z[Q].
+Denote by π:XQ/P→Xthe projection. There is a tautological extension of the
+pullback Deligne–Faltings structure πP, which we will denote by Λ: Q→DivXQ/P;
+ifv∈Qgp, then Λgp:Qgp→PicXQ/P sendsvinto the sheaf B[v] (by which we
+denote the sheaf B, but with the grading shifted by v, i.e.,B[v]v′=Bv+v′).
+Ifπ:XQ/P→Xdenotes the projection, the pushforward operation π∗from
+quasi-coherent sheaves on XQ/Pto quasi-coherent sheaves on Xcorresponds to the
+operation that associates with each sheaf NofQgp-graded quasi-coherent sheaf of
+modules over Bthe part N0of degree 0.PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 25
+Corollary 4.15. Ifj:P→Qis a homomorphism of finitely generated monoids,
+Xis a scheme, and L:P→DivXis a symmetric monoidal functor, the stack
+XQ/Pis algebraic and finitely presented over X.
+Here is a variant of this definition. Suppose that Xis a scheme, j:A→B
+a homomorphism of sheaves of monoids on X´ et,L:A→DivX´ eta morphism of
+symmetric monoidal fibered categories. For each morphism of sche mest:T→X
+we have a symmetric monoidal functor t∗L:t∗A→DivT´ et, with which we associate
+a category ( t∗L)(t∗B/t∗A).
+Suppose that ( M,α ) is an object of ( t∗L)(t∗B/t∗A), and that f:T′→Tis
+a morphism of X-schemes. The isomorphism α:t∗L≃M◦t∗jpulls back to an
+isomorphism f∗t∗L≃f∗(M◦t∗j). By composing with the natural isomorphisms
+f∗t∗L≃t′∗Landf∗(M◦t∗j)≃f∗M◦f∗t∗j≃f∗M◦t′∗jwe obtain an iso-
+morphism f∗α:t′∗L≃f∗M◦t′∗j, and thus an object f∗(M,α )def= (f∗M,f∗α) of
+(t′∗L)(t′∗B/t′∗A). This construction extends naturally to a symmetric monoidal
+functor
+f∗: (t∗L)(t∗B/t∗A)−→(t′∗L)(t′∗B/t′∗A)
+which in turn gives a pseudo-functor from (Sch /X) to the 2-category of categories.
+Definition 4.16. LetXbe a scheme, j:A→Ba homomorphism of sheaves
+of monoids on X´ et,L:A→DivX´ eta morphism of symmetric monoidal fibered
+categories. We define the stack of roots associated with these data, denoted by
+(X,L)B/A, or simply XB/A, as the fibered category over (Sch /X) associated with
+the pseudo-functor above.
+Remark 4.17. Suppose that Xis a scheme, L:A→DivX´ eta Deligne–Faltings
+structure, B/A a system of denominators. Let t:T→XB/Abe a morphism, where
+Tis a scheme. Then the corresponding morphism M:t∗B→DivT´ etis a Deligne–
+Faltings structure (i.e., its kernel is trivial). This follows easily from th e fact that
+Bis sharp.
+Proposition 4.18. Let(A,L)be a Deligne–Faltings structure on X,j:A→Ba
+system of denominators,
+Pj0/d47/d47
+h
+/d15/d15Q
+k
+/d15/d15
+A(X)j(X)/d47/d47B(X)
+a chart for B/A. LetL0def=L(X)◦h:P→DivX. Then there is a canonical
+equivalence (X,L0)Q/P≃(X,L)B/Aof fibered categories over (Sch/X).
+Proof. It is easy to construct a cartesian functor XB/A→XQ/P. Lett:T→X
+be a morphism of schemes; we denote by t∗h:P→(t∗A)(T) the composite of
+h:P→A(X) with the natural pullback homomorphism t∗:A(X)→(t∗A)(T),
+and analogously for t∗k:Q→(t∗B)(T).
+Suppose that ( M,α ) is an object of XB/A(T) = (t∗L)(t∗B/t∗A). Then α(T) is
+an isomorphism between the functors t∗L(T) andM(T)◦t∗j(T), which go from
+t∗A(T) toDivT; thenα(T)◦t∗his an isomorphism from t∗L(T)◦t∗htoM(T)◦
+t∗j(T)◦t∗h=M(T)◦t∗k◦j0; hence ( M(T)◦t∗k,α(T)◦t∗h) is an object of
+XQ/P(T) = (t∗◦L0)(Q/P ). This construction extends naturally to a cartesian
+functorXB/A→XQ/P.26 BORNE AND VISTOLI
+To go in the other direction, start from an object ( M0,α0) ofXQ/P(T) = (t∗◦
+L0)(Q/P ). Denote by KAandKBthe presheaf kernels of the morphisms PT→A
+andQT→Binduced by t∗handt∗krespectively. From the characterization of
+Lemma 4.7, it is easy to see that KBis also the kernel of the symmetric monoidal
+functorQT→DivT´ etinduced by M0. Consider the Deligne–Faltings structure
+M:t∗B= (QT/KB)sh→DivT´ etinduced by M0(Proposition 3.21). Call MA
+the restriction of Mtot∗A, i.e., the composite t∗At∗j− − →t∗BM− →DivT´ et; then the
+composite Pj0− →Qt∗k− − →B(T)M(T)− −−− →DivTfactors through MA(T):A(T)→DivT.
+From the isomorphism t∗α0:t∗L0≃t∗j◦M0and the restriction to P(T) of the
+given isomorphism M(T)◦t∗k≃M0, we obtain an isomorphism α:t∗L≃M◦j,
+because of the functoriality statement in Proposition 3.21. The pair (M,α ) is an
+object of XB/A(T).
+We leave it to the reader to define a morphism of fibered categories XQ/P→
+XB/A that associates ( M,α ) with (M0,α0), and check that this yields a quasi-
+inverse to the morphism XB/A→XQ/Pdefined above. ♠
+Proposition 4.19. Let(A,L)be a Deligne–Faltings structure on X,j:A→B
+a system of denominators. Then the fibered category XB/Ais a finite and finitely
+presented algebraic stack over X. It is tame, in the sense of [AOV08] .
+Furthermore, assume that for each geometric point x: Spec Ω →X, the order of
+the quotient Bgp
+x/Agp
+xis prime to the characteristic of Ω. ThenXB/Ais a Deligne–
+Mumford stack.
+Proof. The fact that XB/Ais a stack follows from standard arguments of descent
+theory, and is omitted.
+To check the other conditions is a local question in the ´ etale topolog y overX;
+hence we may assume that there is a chart
+P
+/d15/d15/d47/d47Q
+/d15/d15
+A(X) /d47/d47B(X).
+Furthermore, if the order of the quotient Bgp
+x/Agp
+xis prime to the characteristic
+of Ω for each geometric point x: Spec Ω →X, we may assume that the order of
+the finite group Qgp/Pgpis everywhere prime to the characteristic of each of the
+residue fields of X.
+CallGthe kernel of the surjective homomorphism /hatwide:/hatwideQ→/hatwidePinduced by j; it
+is a finite diagonalizable group, the Cartier dual of the finite group Qgp/Pgp. It
+is smooth if the condition on the characteristic is verified. We have a c artesian
+diagram
+[SpecZ[Q]/G] /d47/d47
+/d15/d15[SpecZ[Q]//hatwideQ]
+/d15/d15
+SpecZ[P] /d47/d47[SpecZ[P]//hatwideP],
+which says that fppf locally on Xthe stack XQ/P is a quotient by an action of
+Gover a scheme which is finite over X(sinceZ[Q] is a finite extension of Z[P]).PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 27
+This shows that it is finite and tame, and Deligne–Mumford if the condit ions on
+the characteristic are verified. ♠
+5.Parabolic sheaves
+5.1.Categories of weights.
+Definition 5.1. Given a monoid A, letAwtbe the strict symmetric monoidal
+category whose objects are elements of Agp, and arrows a:u→vare elements aof
+Asuch that u+ιAa=v∈Agp. The monoidal structure is given by the operations
+inAgp(for the objects) and A(for the arrows).
+Notice that if Ais integral, Awtis a partially ordered set (that is, there is at
+most one arrow between any two objects of Awt).
+There is a natural symmetric monoidal functor A→Awt, given at the level of
+objects by the function ιA:A→Agp.
+Proposition 5.2. Given a monoid A, a scheme Xand a symmetric monoidal
+functorL:A→DivX, there exists a symmetric monoidal functor Lwt:Awt→
+PicX, and a monoidal 2-isomorphism Φ
+A
+L/d15/d15/d47/d47Awt
+Lwt/d15/d15
+DivX /d47/d47Φ/d50/d58♥♥♥♥♥♥♥♥♥♥♥♥
+PicX
+such that for all a∈Athe diagram
+LaΦ(a)/d47/d47Lwt(a)
+OXσL
+a/d57/d57ssssss
+ǫL /d37/d37❑❑❑❑❑❑
+L0Φ(0)/d47/d47Lwt(0)Lwt(a)/d79/d79
+commutes.
+Furthermore LwtandΦare unique up to a unique isomorphism.
+Proof. The uniqueness statement is easily proved, so we concentrate on c onstructing
+a solution ( Lwt,Φ).
+First, consider the category Adwtwhose objects are elements of A×Aand whose
+arrowsc: (a,b)→(a′,b′) are elements cofAsuch that ιA(a+b′+c) =ιA(a′+b).
+There is a natural functor Adwt→Awtsending the object ( a,b) to the object
+a−b, and this is clearly a monoidal equivalence. The functor A→Awtfactors as
+A→Adwt→Awt, whereA→Adwtis defined by a/maps⊔o→(a,0); thus it is enough
+to produce a functor Ldwt:Adwt→PicX, together with an isomorphism of the
+composites A→AdwtLdwt
+− −− →PicXandA→DivX→PicX, such that for each
+a∈Athe diagram
+LaΦ(a)/d47/d47Ldwt(a,0)
+OXσL
+a/d57/d57ssssss
+ǫL /d37/d37❑❑❑❑❑❑
+L0Φ(0)/d47/d47Ldwt(0,0)Ldwt(a)/d79/d7928 BORNE AND VISTOLI
+commutes. Then we can define Lwtby composing Ldwtwith a monoidal quasi-
+inverseAwt→Adwtof the equivalence Adwt→Awt.
+At the level of objects, we define Ldwtby the obvious rule Ldwt(a,b) =La⊗L∨
+b.
+Given an arrow c: (a,b)→(a′,b′), there is an element d∈Asuch that a+b′+c+d=
+a′+b+d, we get isomorphisms
+La⊗Lb′⊗Lc⊗Ld≃La+b′+c+d
+=La′+b+d
+≃La′⊗Lb⊗Ld,
+hence an isomorphism
+La⊗Lb′⊗Lc≃La′⊗Lb,
+which yields a homomorphism La⊗Lb′→La′⊗Lb, by sending a section sof
+La⊗Lb′into the section of La′⊗Lbcorresponding to s⊗σL
+c. This in turn yields
+a homomorphism
+Ldwt(c):La⊗L∨
+b−→La′⊗L∨
+b′.
+The verification that the operation of Ldwton arrows preserves composition is long
+but straightforward. Hence we obtain the desired functor Ldwt:Adwt→PicX.
+For each a∈A, the isomorphism Φ( a):La≃Ldwt(a,0)def=La⊗L∨
+0comes from
+the isomorphism ǫL:L0≃ OX. It is immediate to check that the diagram above is
+commutative. This completes the proof. ♠
+Remark 5.3. Conversely, given a symmetric monoidal functor M:Awt→PicX,
+anya∈Adefines an arrow 0 →ainAwt; this yields an arrow OX≃M(0)→M(a)
+inPicX. If we set La=M(a) and call sathe section of Lacorresponding to the
+morphism OX→Lajust defined, we obtain a monoidal functor L:A→DivX,
+such that Lwt≃M. In this way we obtain an equivalence of categories between
+symmetric monoidal functors A→DivXand symmetric monoidal functors Awt→
+PicX.
+This construction generalizes to sheaves. Let Xbe a scheme, Abe a sheaf of
+monoids on X´ et. We denote by Awt→X´ etthe fibered category defined as follows.
+The objects are pair ( U,u), where U→Xis an ´ etale morphism and u∈Agp(U).
+The arrows from ( U,u) to (V,v) are pairs ( f,a), where f:U→Vis a morphism
+ofX-schemes and ais an element of A(U) such that u+ιA(a) =f∗v∈Agp(U).
+Composition is defined by addition and pullback: if ( f,a): (U,u)→(V,v) and
+(g,b): (V,v)→(W,w ) are arrows, the composite is defined as
+(g,b)◦(f,a)def= (gf,f∗b+a).
+Proposition 5.4. Given a sheaf of monoids Aon a scheme Xand a symmetric
+monoidal functor L:A→DivX´ et, there exists a symmetric monoidal functor Lwt:
+Awt→PicX´ et, and a monoidal cartesian 2-isomorphism Φ:
+A
+L/d15/d15/d47/d47Awt
+Lwt/d15/d15
+DivX´ et/d47/d47Φ/d51/d59♥♥♥♥♥♥♥♥♥♥♥♥♥♥
+PicX´ et
+such that for all ´ etale morphism U→Xanda∈A(U)the following diagram
+commutes :PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 29
+LaΦ(a)/d47/d47Lwt(a)
+OUσL
+a/d57/d57ssssss
+ǫL
+U/d37/d37❑❑❑❑❑❑
+L0Φ(0)/d47/d47Lwt(0)Lwt(a)/d79/d79
+Furthermore LwtandΦare unique up to a unique monoidal cartesian 2-isomor-
+phism.
+Proof. To show the existence, we use Proposition 5.2 to produce, for each ´ etale mor-
+phismU→X, a solution ( Lwt(U),Φ(U)). The uniqueness statement in Proposition
+5.2 shows that these solutions are compatible with restriction, henc e define a global
+solution ( Lwt,Φ). ♠
+Remark 5.5. As before, we have an equivalence between the category of symme tric
+monoidal functors A→DivXand symmetric monoidal functors Awt→PicX.
+5.2.Parabolic sheaves. LetXbe a scheme, j:A→Ba Kummer homomor-
+phism of monoids, L:A→DivXa symmetric monoidal functor. We will always
+omitjfrom the notation, and think of Aas a submonoid of B, and of Awtas a
+subcategory of Bwt.
+Consider the extension Lwt:Awt→PicX(Proposition 5.2); if u∈Awt, for
+simplicity of notation we denote by Luthe invertible sheaf Lwt(u) image of u. If
+u=ιAafor some a∈A, thenLwt(u) is canonically isomorphic to La, and there
+should be no risk of confusion. Also, as usual when a∈Awe denote by σL
+athe
+corresponding section of La, so that L(a) = (La,σL
+a).
+Ifuandu′are inAwt, we have a given isomorphism Lu+u′≃Lu⊗Lu′, which
+we denote by µL
+u,v, or simply µ, once again dropping thewtsuperscripts from the
+notation. Similarly, we denote by ǫL, orǫ, the given isomorphism between OXand
+L0.
+Definition 5.6. Aparabolic sheaf (E,ρE) on (X,A,L ) with denominators in B/A
+consists of the following data.
+(a) A functor E:Bwt→QCohX, denoted by v→Evat the level of objects, and
+byb/maps⊔o→Ebat the level of arrows.
+(b) For any u∈Awtandv∈Bwt, an isomorphism of OX-modules
+ρE
+u,v:Eu+v≃Lu⊗OXEv,
+which we will call pseudo-period isomorphisms . Ifa∈A, we denote ρE
+ιAa,vby
+ρE
+a,v.
+These data are required to satisfy the following conditions. Let u,u′∈Awt,
+a∈A,b∈B,v∈Bwt. Then the following diagrams commute.
+(i)
+EvEa/d47/d47
+≃
+/d15/d15EιAa+v
+ρE
+a,v
+/d15/d15
+OX⊗EvσL
+a⊗idEv/d47/d47La⊗Ev30 BORNE AND VISTOLI
+(ii)
+Eu+vρE
+u,v/d47/d47
+Eb
+/d15/d15Lu⊗Ev
+idLu⊗Eb
+/d15/d15
+Eu+b+vρE
+u,b+v/d47/d47Lu⊗Eb+v
+(iii)
+Eu+u′+vρE
+u+u′,v/d47/d47
+ρE
+u,u′+v/d15/d15Lu+u′⊗Ev
+µ⊗idEv
+/d15/d15
+Lu⊗Eu′+vidLu⊗ρE
+u′,v/d47/d47Lu⊗Lu′⊗Ev
+(iv) Finally, the composite
+Ev=E0+vρE
+0,v/d47/d47L0⊗Evǫ⊗idEv/d47/d47OX⊗Ev
+is the natural isomorphism Ev≃ OX⊗Ev.
+Remark 5.7. This definition has the following high-level interpretation. There
+are natural functors +: Awt×Bwt→Bwt(given by addition) and ⊗:PicX×
+QCohX→QCohX(given by tensor product). These can be interpreted as action
+of the symmetric monoidal categories AwtonBwtand ofPicXonQCohX. Then
+the first two conditions mean that ρEis an isomorphism of the composites E◦+
+and⊗ ◦(Lwt×E). The other two ensure that Ecan be interpreted as an Awt-
+equivariant functor. It is easy to check that the data of a parabo lic sheaf on
+(X,A,L ) with denominators in B/A is equivalent to the data of a Awt-morphism
+of modules categories E:Bwt→QCohXin the sense of [Ost03], Definition 2.7.
+There is an abelian category QCohX(X,A,L )(Q/P ) whose objects are quasi-
+coherent sheaves on ( X,A,L ) with denominators in Q/P . An arrow Φ: E→E′is
+a natural transformation such that for all u∈Awtandv∈Bwtthe diagram
+Eu+vρE
+u,v/d47/d47
+Φu+v
+/d15/d15Lu⊗Ev
+idLu⊗Φv
+/d15/d15
+E′
+u+vρE′
+u,v/d47/d47Lu⊗E′
+v
+commutes.
+We will see that this category has tensor products and internal Ho ms.
+There is also a sheafified version of the definition of parabolic sheaf.
+Definition 5.8. LetXbe a scheme, ( A,L) a coherent Deligne–Faltings structure
+onX,j:A→Ba system of denominators. A parabolic sheaf on (X,A,L )with
+denominators in B/A consists of the following data.
+(a) A cartesian functor E:Bwt→QCohX´ et, denoted by v→Evat the level of
+objects, and by b/maps⊔o→Ebat the level of arrows.
+(b) For any U→XinX´ et, anyu∈Awt(U) andv∈Bwt(U), an isomorphism of
+OU-modules
+ρE
+u,v:Eu+v≃Lu⊗Ev.PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 31
+These data are required to satisfy the following conditions analogou s to those of
+Definition 5.6, and the following.
+Iff:U→Vis an arrow in X´ et,u∈Awt(V) andv∈Bwt(V), then the isomor-
+phism
+ρE
+f∗u,f∗v:Ef∗(u+v)=Ef∗u+f∗v≃Lf∗u⊗Ef∗v
+is the pullback of jE
+u,v:Eu+v≃Lu⊗Ev.
+Remark 5.9. This definition can also be interpreted as in Remark 5.9, substituting
+categories with fibered categories.
+There is an abelian category QCoh (X,A,L )(B/A) whose objects are quasi-
+coherent sheaves on ( X,A,L ) with denominators in B/A. A homomorphism of
+parabolic sheaves is defined as in the case when AandBare fixed monoids.
+Proposition 5.10. Let(A,L)be a Deligne–Faltings structure on X,j:A→Ba
+system of denominators,
+Pj0/d47/d47
+h
+/d15/d15Q
+i
+/d15/d15
+A(X)j(X)/d47/d47B(X)
+a chart for B/A. LetL0def=L(X)◦h:P→DivX. Then there is a canonical equiv-
+alence of abelian categories of QCoh (X,A,L )(B/A)withQCoh (X,P,L 0)(Q/P ).
+Proof. We begin by the obvious definition of the equivalence: at the level of o bjects,
+if (E,ρE) is a parabolic sheaf on ( X,A,L ) with denominators in B/A, we can
+associate with it a parabolic sheaf with denominators in Q/P : (E(X)◦i,ρE(X)◦
+(h×i)), and it is also clear how to define the functor at the level of morph isms. So
+we get a functor QCoh (X,A,L )(B/A)→QCoh (X,P,L 0)(Q/P ), and it is easy to
+check that it is fully faithful. So we now prove that the functor is in fa ct essentially
+surjective.
+Let (E0,ρE0) be a parabolic sheaf with denominators in Q/P with respect to L0.
+We must prove that the cartesian functor ( E0)X:Qwt
+X→QCohX´ etassociated with
+E0factors trough Bwt, and an analogous statement for ρE0. As in Definition 4.7, let
+KA(respectively KB) the kernel of the morphism PX→A(respectively QX→B)
+induced by h(respectively by i). LetBpre=QX/KBthe quotient presheaf, since
+Bwtis the stackification of ( Bpre)wt, andQCohX´ etis a stack, it is enough to show
+that (E0)Xfactors trough ( Bpre)wt.
+Let us first describe the cartesian category ( Bpre)wtonX´ et. Above an ´ etale
+morphism U→X, its objects are by definition elements of
+(Bpre)(U)gp= (QX(U)/KB(U))gp,
+that is, equivalence classes cl( u) of elements of uinQX(U)gpfor the equivalence
+relation u∼vwhen there exists elements k,linKB(U) such that u+ιQ(k) =
+v+ιQ(l), where ιQdenotes as usual the morphism QX→Qgp
+X. Maps from cl( u) to
+cl(v) are given by elements cl( q) of (Bpre)(U) =QX(U)/KB(U) such that cl( u) +
+ι(cl(q)) = cl(v).
+We now introduce a category ( Bpre)dwtthat is a quotient of the Gabriel-Zisman
+localization of Qwt
+Xwith respect to maps in KB, such that it is equivalent to
+(Bpre)wt. Above an ´ etale morphism U→Xobjects of ( Bpre)dwt(U) are elements32 BORNE AND VISTOLI
+ofQX(U)gp, and maps from utovare equivalence classes of pairs cl(( q,k)), where
+q∈QX(U) andk∈KB(U) are such that u+ιQ(q) =v+ιQ(k), for the equivalence
+relation ( q,k)∼(q′,k′) if there exists k1,k2∈KB(U) such that q+k1=q′+k2.
+The cartesian functor Qwt
+X→(Bpre)wtsending an object uto cl(u) and a mor-
+phismqto cl(q) factors trough a cartesian functor Qwt
+X→(Bpre)dwtsending an
+objectuto itself and a morphism qto cl((q,0)), and trough a cartesian functor
+(Bpre)dwt→(Bpre)wtsending an object uto cl(u) and a morphism cl(( q,k)) to
+cl(q). One checks immediately that this last functor is well defined and a c artesian
+equivalence, so this is enough to produce a factorization of ( E0)Xtrough (Bpre)dwt.
+To achieve this, we need the following lemma:
+Lemma 5.11. LetU→Xbe an ´ etale morphism. For any v∈Qgp
+X(U)andk∈
+KB(U)the morphism ((E0)X)k: ((E0)X)v→((E0)X)ιQ(k)+vis an isomorphism.
+Proof. This is a local problem in the ´ etale topology, hence by Lemma 4.7, we ca n
+assume that there exists a positive integer mand an element l∈KA(U) such that
+mk=l. Definition 5.6 ensures that the diagram:
+((E0)X)v((E0)X)l/d47/d47
+≃
+/d15/d15((E0)X)ιQl+v
+ρ(E0)X
+l,v/d15/d15
+OU⊗((E0)X)vσ(L0)X
+l⊗id/d47/d47Ll⊗((E0)X)v
+is commutative. Since l∈KA(U), the morphism σ(L0)X
+lis invertible, moreover
+the fact that mk=land the functoriality of E0show that (( E0)X)l: ((E0)X)v→
+((E0)X)ιQ(l)+vfactors trough (( E0)X)k: ((E0)X)v→((E0)X)ιQ(k)+v, hence this
+last morphism is left invertible. A similar argument shows that (( E0)X)kis right
+invertible. ♠
+Thanks to the lemma, we can define a cartesian functor ( Bpre)dwt→QCohX´ et
+that above an ´ etale morphism U→Xsends the arrow cl(( q,k)) :u→vto the
+composite of (( E0)X)q: ((E0)X)u→((E0)X)ιQ(k)+vwith the inverse of (( E0)X)k:
+((E0)X)v→((E0)X)ιQ(k)+v. The functoriality of E0shows that this is well defined,
+and produces a factorization of ( E0)Xtrough (Bpre)dwt, hence a factorization of
+(E0)Xtrough (Bpre)wt.
+The proof that ρE0does also factor trough Awt×Bwtis similar, so we omit it.
+Hence the functor QCoh (X,A,L )(B/A)→QCoh (X,P,L 0)(Q/P ) we have defined
+is essentially surjective, and so this is an equivalence. ♠
+5.3.Internal Hom and tensor product. LetE,E′be two objects of the cate-
+goryQCoh (X,P,L 0)(Q/P ). First, we define a quasi-coherent sheaf Hom (E,E′)0
+onXby the usual rule: for every ´ etale map U→X,
+Hom (E,E′)0(U) = Hom U(E|U,E′
+|U)
+where Hom Uis theO(U)-module of all homomorphisms of parabolic sheaves on U
+defined in the previous paragraph.
+If instead we start from an object Gof the category QCoh (X) and an object
+EofQCoh (X,P,L 0)(Q/P ), we can consider the external tensor product G⊗E
+as the object of QCoh (X,P,L 0)(Q/P ) given on objects by the rule: for v∈Qgp,
+(G⊗E)v=G⊗Ev.PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 33
+These two operations are related by the formula
+Hom (G⊗E,E′)0≃Hom (G,Hom (E,E′)0)
+where the second Hom is the usual internal Hom in QCoh (X).
+Now for v∈QgpandEan object of QCoh (X,P,L 0)(Q/P ), we can define the
+twistE[v] by the rule: for v′∈Qgp,E[v]v′=Ev+v′.
+If we start again from two objects E,E′ofQCoh (X,P,L 0)(Q/P ), we have for
+u∈Pgpandv∈Qgpcanonical isomorphisms
+Hom (E,E′[u+v])0≃Hom (E[−u],E′[v])0
+≃Hom (L−u⊗E,E′[v])0
+≃Hom (L−u,Hom (E,E′[v])0)
+≃Lu⊗Hom (E,E′[v])0
+This shows that the functor v→Hom (E,E′[v])0can be endowed with a structure
+of a parabolic sheaf, denoted by Hom (E,E′). Thus we have an internal Hom in
+QCoh (X,P,L 0)(Q/P ), and the definition of the tensor product follows from the
+standard formula:
+Hom (E⊗E′,E′′)≃Hom (E,Hom (E′,E′′))
+Along the same lines, we also can define an internal Hom in QCoh (X,A,L )(B/A),
+the only difference being that we can twist only locally. Thus for two ob jectsE,E′
+ofQCoh (X,A,L )(B/A), we define for U→X´ etale and v∈Bgp(U):
+Hom (E,E′)v=Hom (E|U,E′
+|U[v])0
+The tensor product is defined by the formula above.
+6.The main theorem
+In this section we will use the notion of a Deligne–Faltings structure o n an
+algebraic stack, which is the immediate generalization of the notion of Deligne–
+Faltings structure on a scheme.
+The following is the main result of this paper.
+Theorem 6.1. Let(A,L)be a coherent Deligne–Faltings structure on a scheme
+Xand letB/Abe a system of denominators. Then there is a canonical tensor
+equivalence of abelian categories between the category QCoh (X,A,L )(B/A)of par-
+abolic sheaves on (X,A,L )with denominator in Band the category QCohXB/Aof
+quasi-coherent sheaves on the stack (X,L)B/A.
+Proof. Let us construct a functor Φ: QCohXB/A→QCoh (X,A,L )(B/A). Denote
+byπ:XB/A→Xthe projection. On the stack XB/Awe have a canonical Deligne–
+Faltings structure Λ: π∗B→DivXB/A, with an isomorphism of the restriction of Λ
+toAwith the pullback of LtoXB/A. Consider the functor Λwt:Bwt→PicXB/A;
+for each ´ etale morphism U→Xand each v∈Bwt(U), we set Λ vdef= Λwt(v). If
+u∈A(U), then Λ uis canonically isomorphic to π∗Lu, whereLudef=Lwt(u).
+LetFbe a quasi-coherent sheaf on XB/A. We need to define a parabolic sheaf
+ΦFon (X,A,L ) with coefficients in B. For each ´ etale morphism U→Xand each34 BORNE AND VISTOLI
+v∈Bwt(U), we set
+(ΦF)vdef=π∗(F⊗OXB/AΛv).
+Ifb∈B(U) andv∈Bwt(U), the homomorphism (Φ F)b: (ΦF)v→(ΦF)b+vis
+induced via π∗by the homomorphism
+F⊗Λv≃F⊗Λv⊗OidF⊗Λv⊗σΛ
+b− −−−−−−− → F⊗Λv⊗Λb≃F⊗Λb+v.
+Givenu∈A(U) andv∈B(U), the isomorphism
+ρΦF
+u,v: (ΦF)u+v≃Lu⊗(ΦF)v
+is obtained via the following sequence of isomorphisms, using the proj ection formula
+for the morphism π:
+(ΦF)u+v=π∗(F⊗OXB/AΛu+v)
+≃π∗(F⊗OXB/AΛv⊗Λu)
+≃π∗(F⊗OXB/AΛv⊗π∗Lu)
+≃Lu⊗π∗(F⊗OXB/AΛv)
+=Lu⊗(ΦF)v.
+We leave it to the reader to show that the pair (Φ F,ρΦF) is a parabolic sheaf. This
+function on objects extends to an additive functor
+Φ:QCohXB/A−→QCoh (X,A,L )(B/A)
+in the obvious way.
+We claim that Φ is an equivalence. This is a local problem in the ´ etale topo logy:
+this can be proved as follows.
+First all, if U→Xis an ´ etale morphism, denote by UB/Athe stack of roots of
+the restriction LUofLtoU´ etwith respect to the restriction BUofB. Then there
+are fibered categories QCohXB/AandQCoh(X,A,L)(B/A), whose fiber categories over
+an ´ etale morphism U→XareQCohUB/AandQCoh (U,AU,LU)(BU/AU) respec-
+tively. The functor Φ defined above extends to a cartesian functo rQCohXB/A→
+QCoh(X,A,L)(B/A). Now, the point is that both QCohXB/AandQCoh(X,A,L)(B/A)
+are stacks in the ´ etale topology. This follows from straightforwar d but lengthy
+descent theory arguments, which we omit. Of course, to check th at a cartesian
+functor between stacks is an equivalence is a local problem.
+So we may assume that there exists a chart
+Pj0/d47/d47
+h
+/d15/d15Q
+k
+/d15/d15
+A(X)j(X)/d47/d47B(X)
+forB/A. SetL0=h◦L(X); according to Proposition 4.18 and Proposition 5.10,
+there are equivalences between the categories QCohXB/A andQCohXQ/P, and
+between QCoh (X,A,L )(B/A) andQCoh (X,P,L 0)(Q/P ).
+The functor QCohXQ/P→QCoh (X,P,L 0)(Q/P ), which we still denote by Φ,
+is described as follows. We still denote by π:XQ/P→Xthe projection. For each
+p∈Pwe denote by Lpthe invertible sheaf Lh(p)onX; analogously, if q∈Qwe
+denote by Λ qthe invertible sheaf on XQ/Pcorresponding to Λ k(q)onXB/A. ThePARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 35
+functor Φ: QCohXQ/P→QCoh (X,P,L 0)(Q/P ) sends a quasi-coherent sheaf F
+onXQ/Pinto (ΦF,ρΦF), where (Φ F)pdef=π∗(F⊗Lp), andρΦFis defined as above.
+We need to check that this functor Φ is an equivalence.
+We use the description of Proposition 3.25. The functor L0:P→DivXcorre-
+sponds to a morphism X→[SpecZ[P]//hatwideP], i.e., to a Pgp-torsorη:T→Xwith an
+equivariant morphism T→SpecZ[P]. Denote by Adef=η∗OTthe associated sheaf of
+Pgp-gradedOX[P]-algebras. According to Remark 4.14, the category QCohXQ/P
+is equivalent to the category of sheaves of Qgp-gradedA⊗Z[P]Z[Q]-modules. Set
+Bdef=A⊗Z[P]Z[Q]. The functor πfrom quasi-coherent sheaves on XQ/Pto quasi-
+coherent sheaves on Xsends such a sheaf Finto the part F0of degree 0. Since
+for each v∈Qgp, the sheaf Λ vcorresponds to the shifted sheaf B[v], the sheaf
+π∗(Λv⊗F) will be the part Fvof degree v.
+Then the functor Φ is interpreted to the functor that sends such a sheafFof
+Qgp-gradedA⊗Z[P]Z[Q]-modules into the parabolic sheaf Φ F:Qwt→QCohX
+sendingv∈QwttoFv. Ifq∈QandιQq+v=v′, so that qgives an arrow in the
+category Qwt, the image (Φ F)q:Fv→Fv′is given by multiplication by xq.
+Now, let u∈Pwtandv∈Qwt. The sheaf LuonXis isomorphic to the sheaf
+Au; multiplication gives an isomorphism Au⊗OXFv→Fu+v. ThenρΦF:Fu+v≃
+Au⊗OXFvis its inverse.
+With this description, Φ F:QCohXQ/P→QCoh (X,P,L 0)(Q/P ) is very easily
+seen to be an isomorphism. Let us construct a quasi-inverse
+Ψ:QCoh (X,P,L 0)(Q/P )−→QCohXQ/P.
+If (E,ρE) is a parabolic sheaf, we define the quasi-coherent sheaf Ψ EonXas
+the direct sum/circleplustext
+v∈QgpEv. The sheaf Ψ Eis in fact an A-module: since A=/circleplustext
+u∈PgpLu, we define the homomorphism
+A⊗OXΨE=/circleplusdisplay
+u∈Pgp
+v∈QgpLu⊗Ev−→/circleplusdisplay
+v∈QgpEv
+via the isomorphisms ( ρE
+u,v)−1:Lu⊗Ev≃Eu+v. The sheaf Ψ Eis also a sheaf of
+Z[Q]-algebras: for each q∈Q, we letxqact on Ψ Eby sending EvintoEιQ(q)+v
+as the homomorphism Eq. Thus,Z[P] acts on Ψ Ein two ways, by the embedding
+Z[P]⊆Z[Q] and via the morphism to Acoming from the structure of Aas a sheaf of
+OX[P]-algebra. Condition (i) in the definition of a parabolic sheaf (Definition 5.6)
+ensures that these two actions coincides, and so gives Ψ Ethe structure of a Qgp-
+gradedA⊗Z[P]Z[Q]-module, corresponding to a quasi-coherent sheaf on XQ/P.
+We leave it to the reader to define the action of Ψ on arrows, and sho w that it
+gives a quasi-inverse to Φ.
+It remains to prove that Φ is compatible with tensor products. It is e nough to
+show that given F,F′quasi-coherent sheaves on XB/A, there is a natural isomor-
+phism:
+ΦHom (F,F′)≃Hom (Φ(F),Φ(F′))
+LetU→Xbe an ´ etale map and v∈B(U)gp. On one hand we have
+ΦHom (F,F′)v≃π∗Hom (F,F′⊗Λv)
+and on the other hand
+Hom (Φ(F),Φ(F′))v≃Hom (φ(F)|U,φ(F′⊗Λv)|U)036 BORNE AND VISTOLI
+but the equivalence of categories we have just proven shows that these sheaves are
+canonically isomorphic. ♠
+Example 6.2. Let (L1,s1), . . . , (Lr,sr) be invertible sheaves with sections on a
+schemeX; letL:A→DivX´ etthe Deligne–Faltings structure that they generate
+(Definition 3.22). By definition, this Lhas a chart Nr→A(X). Letd1, . . . ,drbe
+positive integers and Qbe the monoid1
+d1N×···×1
+drN, with the natural embedding
+Nr⊆Q. The stack ( X,L)Q/Nris the fibered product
+d1/radicalbig
+(L1,s1)×X···×Xdr/radicalbig
+(Lr,sr)
+of root stacks (in the sense of [AGV08, Appendix B]). Thus we repr oved and gen-
+eralized the correspondence between parabolic sheaves and shea ves on root stacks
+of [Bor07] and [Bor09].
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+teristic , Ann. Inst. Fourier (Grenoble) 58(2008), no. 4, 1057–1091.
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+Grothendieck’s FGA explained.
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+no. 2, 75–99.
+[IS07] Jaya N. N. Iyer and Carlos T. Simpson, A relation between the parabolic Chern char-
+acters of the de Rham bundles , Math. Ann. 338(2007), no. 2, 347–383.
+[Kat89] Kazuya Kato, Logarithmic structures of Fontaine-Illusie , Algebraic analysis, geome-
+try, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore,
+MD, 1989, pp. 191–224.
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+for stacks , Math. Res. Lett. 12(2005), no. 2-3, 207–217.
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+structures , Math. Ann. 248(1980), no. 3, 205–239.PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES 37
+[MY92] M. Maruyama and K. Yokogawa, Moduli of parabolic stable sheaves , Math. Ann. 293
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+http://math.berkeley.edu/ ~ogus/preprints/log_book/ .
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+no. Extra Vol., 655–724 (electronic), Kazuya Kato’s fiftiet h birthday.
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+Math´ ematique de France, Paris, 1982, Notes written by J.-M . Drezet from a course at
+the´Ecole Normale Sup´ erieure, June 1980.
+(Borne) Laboratoire Paul Painlev ´e, Universit ´e de Lille, U.M.R. CNRS 8524, U.F.R.
+de Math ´ematiques, 59655 Villeneuve d’Ascq C ´edex, France
+E-mail address :Niels.Borne@math.univ-lille1.fr
+(Vistoli) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa , Italy
+E-mail address :angelo.vistoli@sns.it
\ No newline at end of file
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+arXiv:1001.0467v2  [cond-mat.mtrl-sci]  30 Jul 2010Thermal contraction in silicon nanowires at low temperatur es
+Jin-Wu Jiang∗and Jian-Sheng Wang
+Department of Physics and Centre for Computational Science and Engineering,
+National University of Singapore, Singapore 117542, Repub lic of Singapore
+Baowen Li
+Department of Physics and Centre for Computational Science and Engineering,
+National University of Singapore, Singapore 117542, Repub lic of Singapore and
+NUS Graduate School for Integrative Sciences and Engineeri ng,
+Singapore 117456, Republic of Singapore
+(Dated: November 6, 2018)
+Abstract
+Thethermal expansion effect of silicon nanowires (SiNW) in [1 00], [110] and [111] directions with
+different sizes is theoretically investigated. At low temper atures, all SiNW studied exhibit thermal
+contraction effect due to the lowest energy of the bending vibr ation mode which has negative effect
+on the coefficient of thermal expansion (CTE). The CTE in [110] direction is distinctly larger than
+the other two growth directions because of the anisotropy of the bendingmode in SiNW. Our study
+reveals that CTE decreases with an increase of the structure ratioγ=length/diameter , and is
+negative in whole temperature range with γ= 1.3.
+Keywords: thermal contraction, silicon nanowire, phonon mode
+∗Electronic mail: phyjj@nus.edu.sg
+1I. INTRODUCTION
+Silicon nanowires (SiNW) can be well produced in different directions wit h different
+diameters.[1–6] It is a promising candidate for nano-device, and has attracted a lot of
+interest.[7–19] The SiNW can serve as basic building blocks for electric ally based sensor[7–
+10], field effect transistor[11–16] and logic gates[17], photovoltaic d evice[18], and functional
+networks[19]. The possible stable configurations of the SiNW have be en investigated with
+density-functional tight-binding simulations by Zhang et al.in Refs. 20. They found that
+the stability of SiNW is determined by the competition between the minim ization of the
+surface energy and the minimization of the surface-to-volume rat io. As one of the important
+thermal properties in SiNW, the thermal expansion plays an importa nt role for these appli-
+cations in the electronic or thermal devices. The thermal expansio n effect can be studied in
+the classical molecular dynamics.[21]
+In this paper, we first analyze various phonon vibrational modes in S iNW from the
+Tersoff potential[22] implemented in the “General Utility Lattice Pro gram” (GULP).[23]
+Some important features of phonon modes in SiNW of different direct ions and with different
+structure ratio γ=length/diameter are discussed. We then investigate the coefficient
+of thermal expansion (CTE) of SiNW in [100], [110] and [111] growth dir ections by the
+nonequilibrium Green’s function approach[24], which is a quantum mech anical method and
+automatically includes contribution of all phonon modes. We find that at low temperatures,
+all SiNW studied have thermal contraction effect and the CTE in SiNW [1 10] is the largest
+one among the three directions. The CTE decreases rapidly with incr easing structure ratio
+γ, and is negative in whole temperature range with γ= 1.3.
+II. RESULTS AND DISCUSSION
+Fig. 1 demonstrates that a SiNW with length and diameter ( L,D) can be cut from the
+bulk Si crystal by using a virtual cylinder with structure paramete rs (L,D).[25] We can cut
+SiNW in different growth directions: [100], [110] and [111], by controlling the direction of
+the virtual cylinder. The SiNW displayed in the figure is in [100] growth d irection with ( L,
+D)=(3, 1) nm. As discussed in Refs. 26, it is not important to include H- passivation on the
+surface for thermal property of SiNW. We adopt this approximatio n which may introduce
+2some small deviation. To depress this small deviation, we do not use t hese atoms on the
+surface and boundaries. It seems that Si atoms in the figure do no t distribute uniformly in
+the figure. That is because the structure is viewed from the upper front direction, which
+is better for illustrating a three-dimensional structure. Other co nfiguration figures in this
+paper may also have this visual effect.
+The frequencies and eigen vectors of phonon modes are obtained f rom the Tersoff in-
+teratomic potential for silicon implemented in the GULP. The Tersoff p otential can well
+describe bonding in covalently bonded solids such as silicon and carbon system, and also
+includes the nonlinear interaction. Its accuracy has been confirme d by comparison with re-
+sults from ab initio density functional theory.[26] In the investigation of CTE in the SiNW ,
+we will use the clamped boundary condition, where one end (left, for example) is fixed and
+another end (right) is free. For consistency, we also apply this bou ndary condition in the
+study of phonon, which is different from Refs. 27, 28.
+Fig. 2 shows the vibrational morphology of three important phonon modes for CTE in
+SiNW. In the figure the SiNW is in [111] growth direction with ( L,D)=(3, 1) nm. The
+left-most 0.5 nm region (blue online) is fixed (clamped). Arrows (red o nline) attached onto
+each atom display the vibrational displacement for the atom.[29] Th e length of the arrow is
+not the exact value of vibrational displacement to show this displace ment more clearly. It
+has been enlarged by a constant factor to all atoms. Panels (a) an d (b) are the first and
+second order bending modes, which is a lateral mode with all atoms vib rate in the direction
+perpendicular to the axial direction. It is doubly degenerate due to two independent lateral
+vibrational directions. As the SiNW is clamped on the left, the vibratio nal displacement
+increases from left to right in both modes. In panel (a), the direct ion of the vibrational
+displacement doesnot change, while thisdirection changes inthesec ond order modeinpanel
+(b). The bending mode is actually a characteristic property for rod -like systems (eg. SiNW)
+or two-dimensional plane sheet (eg. graphene). Using its relation t o the Young’s modulus of
+the material, the bending mode has been used to detected the value of the Young’s modulus
+of the low-dimensional systems such as single-walled carbon nanotu bes[30] or graphene.[31]
+For longer SiNW, this mode will gradually turn to the standard flexure mode in a rod-
+like systems. Panels (c) and (d) are the first and second order twis ting modes. Because
+of its special rotating morphology, the vibrational displacement inc reases from inner to
+the surface. This is consistent with the vanishing surface stress c ondition in a rod.[27] In
+3panel (c), the direction of the vibrational displacement is the same for all atoms, while
+the direction changes for the second order mode shown in panel (d ). The twisting mode
+reflects the rod-like structure of the SiNW and is determined by the shear stiffness of the
+SiNW. It can serve to sense the tiny torsion movement of a nano dev ice.[32] This mode can
+be used to extract the internal information of the SiNW, as it carrie s the information of
+the rigid rotational invariance symmetry.[33–35] Panels (e) and (f ) are the first and second
+order longitudinal vibrational phonon modes. In very long SiNW, this mode is actually the
+well-known longitudinal acoustic mode.
+Fig. 3 shows frequencies of the three lowest-energy phonon mode s of SiNW in different
+directions with different sizes. Frequencies for all phonon modes de crease with increasing
+structure ratio γ=length/diameter . This result indicates that longer SiNW are more
+flexible and elastic. As a result, SiNW with large γmay be thermally more unstable, and
+possibly show a coiling-up behavior like a single-walled carbon nanotube .[36] Typically, we
+note that the frequency of bending mode in (a) is the lowest one com pared with the other
+two modes shown in panels (b) and (c). This is because the bending mo de turns to be the
+elastic flexure mode in the limit of γ→ ∞. An important feature of the flexure mode is its
+quadratic phonon spectrum: ω∝k2. While the other acoustic phonon modes are linearly
+dependent on the wave vector. So the flexure mode (also bending m ode) always has the
+lowest frequency. This figure also shows that the bending mode has lowest frequency in
+[100] direction and highest frequency in [110] direction with the same γ. So the SiNW in
+[100] direction are easiest to be bend, which may be due to its regular surface structure.
+We calculate the CTE along the axial direction of the SiNW using the non equilibrium
+Green’sfunctionmethod, which includes allphononmodesautomatic allyandcanbeapplied
+to very low temperatures as it is a quantum approach.[24] In this me thod, firstly, the average
+vibrational displacement of atom j,∝angbracketleftuj∝angbracketright, is calculated through: ∝angbracketleftuj∝angbracketright=i/planckover2pi1Gj, whereGjis
+the one-point Green’s function and can be calculated from its Feynm an diagram expansion
+in terms of the nonlinear interaction. We consider the nonlinear inter action as,
+Hn=/summationdisplay
+lmnklmn
+3ulumun, (1)
+where the nonlinear force constant is extracted from the Tersoff potential in GULP by finite
+4difference method. Then the CTE can be calculated through its defin ition,
+αj=d∝angbracketleftuj∝angbracketright
+dT×1
+xj, (2)
+wherexjis the lattice position of atom jalong the axial direction. We can get the final
+value of CTE for SiNW by averaging αjover all atoms.
+In Fig. 4, we compare the CTE in three different directions. All three SiNW have the
+same structure parameters ( L,D) = (1, 4) nm. It shows that the CTE in all directions
+are negative in very low temperature region. Then it increases with f urther increasing
+temperature and reach a saturate value in high temperature limit. T he saturate value is
+positive for these SiNW shown in the figure. We mention that before r eaching the saturate
+value, all three curves still show obvious increasing behavior at abo utT= 500 K, which
+implies that the Debye temperature in SiNW is roughly about 500 K. Fro m this figure, we
+also learn that the CTE in [110] direction is distinctly larger than the ot her two directions.
+While the CTE in [100] direction is close yet a little smaller than that in [111].
+The above behaviors for the CTE at different temperatures and in d ifferent directions
+can be understood from the various phonon modes in SiNW discussed in the previous part.
+Because of different vibrational morphology, these phonon modes have very different con-
+tribution to the CTE. The bending mode has negative effect on the CT E of the rod-like
+SiNW.[37, 38] The twisting mode only takes up some degrees of freedo m, without making
+any contribution to CTE. The longitudinal vibration mode has positive effect on the CTE.
+So the CTE is mainly determined by the competition between the bendin g mode and lon-
+gitudinal mode. Since the bending mode has the lowest energy among all phonon modes, it
+is the only mode excited at low temperature, leading to negative CTE. With the increase
+of temperature T, the longitudinal mode will be excited gradually and competes with ben d-
+ing mode. This competition slows down the decreasing of CTE, and res ults in a minimum
+point. As the longitudinal mode becomes more and more important, C TE will increase
+with further increasing temperature. In high temperature limit, CT E reaches a saturate
+value after sufficient competition between all phonon modes. The po sitive saturate value
+in these SiNW implies the leading contribution of longitudinal mode for th ese SiNW. Also
+the bending mode is anisotropic with the frequencies in [110], [111] and [100] directions as
+53.1, 41.2, 39.3 cm−1, respectively. This anisotropic property leads to the anisotropic e ffect
+of CTE, i.e. α[110]> α[111]> α[100].
+5We now turn to the discussion of the size dependence for CTE in SiNW [1 11]. There
+are similar effects for the other two directions. Fig. 5 shows the dep endence of CTE on
+the structure ratio γin whole temperature range. The value of CTE decreases quickly with
+increasing γ. This is because the frequency of bending mode decreases quickly w ith increas-
+ingγ(see Fig. 3). As a result, the negative effect from bending mode on C TE increases.
+For SiNW with γ= 0.25, the transition of CTE from negative to positive happens around
+120 K, and CTE reaches a saturate value of 10 ×10−6K−1in high temperature limit. A
+large positive saturate value of CTE indicates that the longitudinal m ode makes dominant
+contribution rather than the bending mode in this SiNW. The transitio n temperature in-
+creases from 120 K to 270 K with the change of γfrom 0.25 to 1.0. The saturate value of
+CTE for SiNW with γ= 1.0 decreases for more than 50% compared with that of γ= 0.25.
+This result is reasonable and has also been observed in single-walled ca rbon nanotubes,[24]
+which has similar one-dimensional rod-like structure. It seems to be a common phenomenon
+in one-dimensional system that the CTE is much smaller in longer and th inner structures.
+Whenγ= 1.3, CTE is negative in whole temperature range, which indicates that t he SiNW
+with these structure ratio undergoes thermal contraction at all temperatures. For these type
+of SiNW, the bending mode has played an dominant role over the longitu dinal mode.
+III. CONCLUSION
+To summarize, we have studied the thermal expansion of SiNW in differ ent growth di-
+rections by the nonequilibrium Green’s function method. We find that at low temperatures,
+all SiNW studied exhibit thermal contraction effect due to the lowest energy of the bending
+modewhich has negative effect onCTE. The CTE in SiNW[110] is larger th antheother two
+directions, because of higher frequency of bending mode in SiNW [110 ]. The CTE decreases
+quickly with increasing of the structure ratio γ, and is negative at all temperatures with
+γ= 1.3.
+IV. ACKNOWLEDGEMENTS
+The work is supported in part by a Faculty Research Grant of R-144 -000-257-112 of
+NUS, and Grant R-144-000-203-112 from Ministry of Education of Republic of Singapore,
+6and Grant R-144-000-222-646 from NUS.
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+8FIG. 1: (Color online) Use a virtual cylinder to cut a SiNW fro m bulk Silicon.
+9FIG. 2: (Color online) The morphology of different phonon mode s in SiNW. (a)/(b): the
+first/second order bending mode; (c)/(d): the first/second o rder twisting mode; (e)/(f): the
+first/second order longitudinal mode.
+10 0 10 20
+ 1  2  3  4  5  6  7ω   (cm−1)
+(a)[110]
+[111]
+[100]
+ 0 10 20 30
+ 1  2  3  4  5  6  7ω   (cm−1)(b)
+ 0 10 20 30 40
+ 1 2 3 4 5 6 7ω   (cm−1)
+length/width(c)
+FIG. 3: (Color online) The frequency of three lowest-energy phonon modes in SiNW with different
+size. (a). bending mode, (b). twisting mode, (c). longitudi al mode.
+11−3 0 3 6 9 12 15
+ 0 300  600  900 1200α  (10−6K−1)
+T   (K)[110]
+[111]
+[100]
+FIG. 4: (Color online) CTE of SiNW in three directions [100], [110], and [111].
+−10−5 0 5 10
+ 0 300  600  900 1200α  (10−6K−1)
+T   (K)γ=0.25
+γ=1.0
+γ=1.3
+FIG. 5: (Color online) CTE depends on T, in [111] SiNW with different structure ratio γ=
+length/diameter .
+12
\ No newline at end of file
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+arXiv:1001.0468v1  [quant-ph]  4 Jan 2010Wave and particle in molecular interference lithography
+Thomas Juffmann,1Stefan Truppe,1Philipp Geyer,1Andr´ as G. Major,1,∗
+Sarayut Deachapunya,1,2Hendrik Ulbricht,1,3and Markus Arndt1
+1Faculty of Physics, University of Vienna, Boltzmanngasse 5 , 1090 Vienna, Austria
+2Department of Physics, Faculty of Science, Burapha Univers ity, Chonburi 20131, Thailand
+3School of Physics and Astronomy, University of Southampton , SO17 1BJ, UK
+(Dated: May 28, 2018)
+The wave-particle duality of massive objects is a cornersto ne of quantum physics and a key prop-
+erty of many modern tools such as electron microscopy, neutr on diffraction or atom interferometry.
+Here we report on the first experimental demonstration of qua ntum interference lithography with
+complex molecules. Molecular matter-wave interference pa tterns are deposited onto a reconstructed
+Si(111)7 ×7 surface and imaged using scanning tunneling microscopy. T hereby both the particle
+and the quantum wave character of the molecules can be visual ized in one and the same image.
+This new approach to nanolithography therefore also repres ents a sensitive new detection scheme
+for quantum interference experiments.
+PACS numbers: 01.55.+b,03.65.-w, 03.65.Ta,03.75.-b
+The de Broglie wave nature of massive particles has
+always been an essential ingredient in the conceptual de-
+velopmentofquantum mechanics[1,2]. First demonstra-
+tions ofthe electron wavenature[3, 4] were soonfollowed
+byexperimentsonthe diffractionofheliumatomsandH 2
+molecules [5] as well as neutrons [6, 7]. The build-up of
+quantum interference patterns from single particles was
+shown in particularly nice demonstrations with individ-
+ualphotons[8]andelectrons[9,10]. Withtheavailability
+of laser and nanofabrication technologies, atom interfer-
+ometry and coherent lithography have become rapidly
+developing fields of research [11–14]. Recently, quan-
+tum interference experiments have also been extended
+to composite nanoparticles, such as fullerenes [15], He
+clusters [16] or large fluorinated molecules [17, 18].
+Our present demonstration complements these earlier
+studies as it represents the first realization of quantum
+interferencelithographywith molecules. It therebycloses
+two gaps that so far existed in the field of macromolecule
+interferometry: On the one hand, previous experiments
+had not been able to visualize the individual particles
+that traversed the interferometer. Further, it has also
+been discussed by several authors that ionizing detectors
+are inefficient already for medium-sized organic materi-
+als [19, 20]. Being sensitive to single molecules, our
+new lithographic interference detection scheme provides,
+for the first time, the opportunity to visualize both the
+quantum wave features and the composite particle na-
+ture of individual molecules in one and the same image
+and experiment. On the other hand, we show that near-
+field interferometry is a very natural approach to gen-
+erating surface-deposited and immobilized nanopatterns
+with particles that can, in principle, be regarded as func-
+tional entities by themselves. Our experiment thus oper-
+ates at the interface between matter-wave interferometry
+and surface nanoscience.
+A schematic illustration of the setup is shown in Fig-ure 1. The experiment is sectioned into five differentially
+pumped vacuum chambers, which comprise three logical
+compartments: the source, the interferometric nanode-
+position and the detector. The supplementary vacuum
+chambers are required for differential pumping as well as
+for the preparation and transfer of the detection surface.
+A Knudsen cell at T=1070K forms the molecular beam
+which is selected within a transmitted velocity band of
+∆v/v= 0.05(FWHM) to account for the fact that dif-
+ferent velocities correspond to different de Broglie wave-
+lengths, λdB=h/mv, and to varying interaction times
+with the gratings.
+Two alternative methods have been used for select-
+ing the molecular speed: First, a gravitational velocity
+selection scheme exploits the molecular free-fall trajecto-
+ries in the Earth’s gravitational field. This established
+method [21, 22] allows to work without any vibrating or
+rotating element. The interference fringe contrast of the
+molecular deposit varies, however, with the vertical po-
+sition on the sample and scattering at the slit edges may
+spoil the selection quality.
+Second, interferograms were also recorded using a
+home-built helical velocity selector which is a minia-
+turized version of devices known from neutron scatter-
+ing [23]. Grooves of 300 µm width were milled along he-
+lical trajectories into an aluminum cup of 40mm length.
+The aspect ratio of length and width of the grooves de-
+fines the transmitted velocity bandwidth. Their pitch
+and angular speed defines the center of the distribution.
+The selector is held by UHV compatible bearings and
+driven by a motor outside of the chamber. A magneto-
+fluidic seal allows the mechanical transduction of the ro-
+tational motion at frequencies in excess of 100Hz. An
+acceleration sensor at the mechanical base of the inter-
+ferometer quantified typical oscillation amplitudes to be
+as small as 2nm.
+This method was chosen for the experiment presented2
+here. Its advantage lies in the fact that all transmitted
+particles arrive with the same speed and the nanostruc-
+ture has the same contrast across the entire surface that
+is illuminated by the molecular beam. Being indepen-
+dent of gravitation the selection scheme can be used in
+any orientation of the experiment.
+About 110cm behind the source, the molecules en-
+counter the first diffraction grating G1of a near-field
+interferometer [22, 24, 25]. Diffraction at each of the in-
+dividual slits within G1expands the molecular coherence
+function to an extent that it covers several slits of the
+second grating G2. Diffraction at G2, coherent evolution
+and interference subsequently generate a molecular den-
+sity pattern, which we accumulate on the detector screen
+D.
+The SiN xgratings were fabricated by Dr.Savas at
+MIT, Cambridge [26] with a highly accurate period of
+d= 257.40(1)nm and with open slit windows as small
+as 75nm in G 1and 150nm in G 2. In our symmetri-
+cal setup, the separation Lbetween the two gratings is
+equal to the distance between G 2andD. The molecu-
+lar distribution on the detector is then an approximate
+self-image of the transmission function of G2, if the Tal-
+bot condition is met, i.e. if L=d2/λdBin the absence
+of any external potentials. Although the nanomechani-
+cal gratings are fabricated with a rectangular transmis-
+sion profile, interference and phase shifts in the gratings
+lead to a near-sinusoidal molecular density pattern on
+the screen. The fringe visibility can then be extracted
+asV= (Smax−Smin)/(Smax+Smin), where S(x) is the
+local molecular surface density at position x.
+The detector screen is a thermally reconstructed
+Si(111)7 ×7 surface. Silicon is known to capture and
+bind fullerenes exceptionally well [27] and it is also the
+naturalchoiceforinterfacingmoleculardepositstofuture
+electronic readout or control. The atomically clean and
+flat surface immobilizes the incident molecules and it al-
+lows us to operate the experiment at room temperature.
+It requires, however, a proper surface cleaning, preheat-
+ing and flash-heating in the read-out chamber which also
+containsthe variabletemperature scanningtunneling mi-
+croscope(RHK STM UHV700).
+For a grating separation of L= 13.2mm we expect
+the maximum interference contrast of the first Talbot
+order for a de Broglie wavelength of 5pm, i.e. for C 60
+with a mean velocity of 111m/s. The van der Waals
+interaction between the molecule and the grating wall
+reduces the effective grating opening and shifts the the-
+oretical fringe visibility. At a mean molecular velocity
+ofv= 115m/s we expect a maximum contrast of about
+V= 60%. If the molecules were classical billiard balls,
+i.e. following trajectories of Newtonian physics, theory
+would predict a nearly flat molecular distribution at the
+detector, i.e. avanishingfringevisibilityofonly V= 1%.
+This comparison of visibilities underlines the importance
+of the molecular quantum wave nature for the emergenceSource & selectionInterference & lithography
+Si(111)  7x7
+Transfer
+Analysis
+Transfer rodWobble sticksLoad lockBellow
+Motor
+Velocity selectorp~10 mbar-777K baffleG1G2
+L
+STM
+d
+STM control & displayp~10 mbar-10
+p < 10 mbar-10zx
+yD
+FIG. 1: Molecular interference lithography combines coher ent
+molecule wave propagation with surface deposition and scan -
+ning probe microscopy. The furnace emits a thermal fulleren e
+beam, velocity selected to ∆ v/v= 5%, which passes two SiN
+gratings ( d= 257.40nm) separated by L= 12.5mm. A
+prepared silicon sample is placed in distance LbehindG2.
+The molecular deposit is imaged with single molecule resolu -
+tion using scanning tunneling microscopy in a separate UHV
+chamber.
+of high-contrast patterns in this kind of lithography.
+Thetotalexperimentalsequenceisasfollows: Asilicon
+surface is thermally reconstructed, characterized in the
+STM and then transferred into the interferometer. The
+angular frequency of the helical velocity selector is set to
+transmit the velocity class v= 115±5m/s. The molec-
+ular exposure lasts about 30min and we deposit about
+0.001 molecular monolayers (ML) on the target. This
+low coverage is required to maintain the single-particle
+character in the quantum demonstration. The exposed
+sample is then transferred into the detector chamber.
+The STM surface scan of Figure2 clearly reveals the
+individual silicon substrate atoms as well as several im-
+mobilized single C 60molecules. The surface binding of
+the fullerenes is so strong that we could not observe any
+clustering, even over two weeks. High-resolution tunnel-
+ing microscopy at low temperatures even allows us to
+get a glimpse on the internal structure of the deposited
+fullerenes [28], as shown in the inset of Figure2.
+In order to see the interference pattern we probe an
+area of 2 µm2at a resolution that still allows to identify
+the individual molecules. This scan takes about thirty
+minutes and returns 106data points. We identify the
+molecules by their height: if a recorded pixel (i,j) is 0.5-
+0.9nm higher than the average of the neighboring pixels
+(i±3,j±3) it is identified as a fullerene molecule. The
+analog images are thus converted into a binary matrix,
+where the presence of a molecule is represented by the3
+2nm 5nm
+FIG. 2: STM image of a reconstructed Si (111) 7 ×7 sur-
+face, covered with immobilized individual fullerenes. Are a of
+this image: 33 ×36nm2; tunneling current I=0.2nA; sam-
+ple bias voltage U=+2V; temperature T=30K. A close up on
+two fullerenes (see inset) gives a glimpse on the internal ri ng
+structure of C 60(see also [28]).
+digit one, whereas all other pixels are set to zero. The
+result of this procedure is shown in Figure3a, where the
+bright dots represent the molecules. For a better visu-
+alization, the pixels were later binned in 5 ×5 boxes.
+The resulting image reveals five interference fringes in
+the chosen section.
+In order to quantify the fringe contrast we perform
+a vertical sum (along the y-direction) over all rows and
+arrive at the one-dimensional interference curve. Since
+the statistical fluctuations are still rather high, with only
+1166 molecules per µm2, we sum over twenty horizontal
+points (along the x-direction) to obtain Figure3b. The
+data are well represented by a numerical fit of the form
+F(x) =Asin(2πx/d+φ0)+B, from which we extract a
+fringe visibility of V=A/B= 36±3%. The statisti-
+cal error is computed from a Levenberg-Marquardt algo-
+rithm which is weighted with the Poissonian uncertainty
+of each individual column in the picture. A similar inter-
+ference contrast was found in the gravitational velocity
+selection scheme. This value is lower than the theoreti-
+cal expectation, but already a linear drift of 2.5nm per
+minute of either one grating or the surface is sufficient to
+explain this observation. In future experiments, thermal
+drifts can be further reduced by replacing the interfer-
+ometer support structure by materials of lower thermal
+expansion.
+The recorded lattice has a period of 267(3)nm which
+differs slightlyfrom the expected 257nm. The 4% period
+mismatch is consistent with a linear thermal drift rate,
+now in the STM instead ofthe interferometer, as small as
+0.4nm per minute. The experiment was repeated several
+times, yielding an interference pattern equally close to
+Position (nm)Position (µm)
+# of molecules
+0 200 400 600 800 1000 120002040608000.51.01.5
+a)
+b)
+FIG. 3: a) The processed STM image (see text) reveals both
+the particle-nature and the quantum wave-nature of the sur-
+face deposited fullerene molecules in one and the same image .
+b) A vertical sum over (a) yields the interference curve. The
+fringe visibility amounts to 36%. The error bars represent t he√
+N-noise related to the small number Nof particles per bin.
+the expected period and orientation.
+The observed fringe contrast exceeds the classical
+moir´ eexpectation in the presence of van der Waals forces
+by more than a factor of thirty, and the experiment is
+offset from the classical result by ten standard devia-
+tions. Quantumdelocalizationandinterferencearethere-
+foreneeded forcreatinghighcontrastmolecularnanopat-
+terns using non-contact mask imaging. Surface adsorp-
+tion and imaging is also an intuitive tool in experiments
+on the foundations of physics, as it allows to visualize
+the localized but random particle positions within a de-
+terministic fringe pattern that is prescribed by the free
+evolution of the delocalized matter wave (Figure 3). The
+surfaceprobe technique is capableof detecting individual
+moleculesanditiswell-suitedforfutureexperimentswith
+velocity selected, monodisperse beams of much largerob-
+jects in different interferometer configurations [26].
+Single molecules may be regarded as functional ele-
+ments. They may serve as immobilized single-photon
+emitters [29], organic switches [30], nanomachines [31,
+32], transistor components [33] or as nucleation cores for
+the catalysis of molecular growth. Positioning them on
+surfaces might thus be crucial in future applications in
+nanotechnology. Our molecule lithography scheme works
+at a de Broglie wavelength comparable to that prevail-
+ing in high-energy electron beam writing. The kinetic
+energy E kin=0.1eV and the velocity v= 100m/s can,
+however, be many orders of magnitude smaller than that
+of the electrons. It therefore combines the potential of
+high resolution with minimal damage to the surface.4
+Future experiments shall explore how to build more
+complex molecular patterns in this non-contact, con-
+structive and parallel way. High-contrast interference
+provides the possibility to exclude molecules with cer-
+tainty from some surface areas. The positioning within
+an interference maximum is, however, still affected by
+the probabilistic character of the quantum wavefunction.
+The combination of interferometric prestructuring with
+local self-organization or STM postprocessing [34] ap-
+pearsto be awaytowardsdeterministic molecularnanos-
+tructures [35]. Our present proof-of-principle demonstra-
+tion shows the interferometric generation of molecular
+lines. Using two-dimensional cross gratings and either
+electric deflectometry [36] or motorized gratings it will
+be possible to write periodic arrays of more complex pat-
+terns, too [37]. In an asymmetric Talbot-Lau interferom-
+eter the exploitation of the fractional Talbot effect shall
+further allow writing of structures smaller than the grat-
+ing period [38], eventuallysmaller than all features in the
+lithography masks.
+This work has been supported by the FWF within the
+Wittgenstein project Z149-N16. A.M. acknowledges sup-
+port by the FWF Lise-Meitner program M887-N02. We
+acknowledge support by the ESF EUROCORE Program
+EUROQUASAR MIME.
+∗Current address: Carl Zeiss SMC AG, Rudolf-Eber-
+Straße 2, 73447 Oberkochen, Germany
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+arXiv:1001.0469v1  [math.CA]  4 Jan 2010ON SOME CLASSICAL PROBLEMS CONCERNING
+L∞-EXTREMAL POLYNOMIALS WITH CONSTRAINTS
+FRANZ PEHERSTORFER1
+Abstract. First we consider the following problem which dates back
+to Chebyshev, Zolotarev and Achieser: among all trigonomet ric polyno-
+mials with given leading coefficients a0,...,al, b0,...,bl∈Rfind that one
+with least maximum norm on [0 ,2π].We show that the minimal poly-
+nomial is on [0 ,2π] asymptotically equal to a Blaschke product times a
+constant where the constant is the greatest singular value o f the Hankel
+matrix associated with the τj=aj+ibj.As a special case corresponding
+statements for algebraic polynomials follow. Finally the m inimal norm
+of certain linear functionals on the space of trigonometric polynomials
+is determined. As a consequence a conjecture by Clenshaw fro m the
+sixties on the behavior of the ratio of the truncated Fourier series and
+the minimum deviation is proved.
+1.Introduction
+In 1858 Chebyshev discovered that the polynomial
+(1) 2−n+1Tn(x) = 2−n+1cosnarccosx
+deviates least fromzerowithrespecttothemaximumnormon[ −1,1] among
+all polynomials with leading coefficient one. Then he posed th e following
+problem to his circle: Let lreal numbers A0,A1,...,Albe given. Among all
+polynomials of degree less or equal nwith leading coefficients A0,A1,...,Al,
+i. e., of the form/summationtextl
+j=0Ajxn−j+q(x),q∈Pn−l−1,find that one which has
+least max-norm on [ −1,1], that is, find the unique polynomial ˜ q∈Pn−l−1
+such that
+(2) min
+q∈Pn−l−1||l/summationdisplay
+j=0Ajxn−j+q(x)||=||l/summationdisplay
+j=0Ajxn−j+ ˜q(x)||
+where||f||= maxx∈[−1,1]|f(x)|. This was the begin of a long story. In-
+deed, ten years later Zolotarev, a student of Chebyshev, det ermined the
+minimal polynomial in terms of elliptic functions when the fi rst and second
+This work was supported by the Austrian Science Fund FWF, pro ject-number P20413-
+N18.
+1The last modifications and corrections of this manuscript we re done by the author in
+the two months preceding this passing away in November 2009. The manuscript remained
+unsubmitted and is not published elsewhere (submitted by P. Yuditskii and I. Moale).
+12 FRANZ PEHERSTORFER1
+coefficient is given. In 1930 Achieser gave a description of th e minimal poly-
+nomial in terms of automorphic functions when three leading coefficients
+are given. In the words of Bernstein [16, p. 156] “Akhieser tr eated the
+more difficult problem and arrived at three algebraic equatio ns containing
+automorphic Schottky functions, whose solutions let to the determination
+of the minimum deviation. Unfortunately these equations ar e so compli-
+cated that it seems to be quite difficult to obtain simple and su fficiently
+accurate inequalities”. Kolmogorov, Krein at al. also ment ioned in [5, p.
+233] that the solution of the problem was one of the significan t contributions
+of Akhieser. Since in the explicit representations there ap pear parameters
+given implicitly (as in Zolotarev’s representation the mod ule of the ellip-
+tic functions) even for these two cases there was (and is) sti ll a demand
+for an asymptotic description in elementary functions. Alr eady in 1913
+Bernstein himself attacked the problem. For Zolotarev’s ca se Bernstein suc-
+ceeded in finding an asymptotic solution of the minimum devia tion, that
+is, ofEn(A0xn+A1xn−1) = infq∈Pn−2||A0xn+A1xn−1+q(x)||in terms of
+elementary functions and also upper and lower bounds. In the sequel he
+[1] and later Achieser [] obtained asymptotics of the minimu m deviation for
+some other special cases, for more recent results on estimat es of the mini-
+mum deviation see Gutknecht and Trefethen [13], where the er ror function
+is studied also, and Haussman and Zeller [14]. But neither Be rnstein nor
+Achieser gave asymptotics for the minimal polynomials in co ntrast to the
+L2-norm where both were main contributors in the development o f an as-
+ymptotic theory. Interesting enough the same holds for Szeg ˝ o the other
+great master in asymptotics of orthogonal polynomials.
+In the sixties N. N. Meiman [20, 21, 22] attacked the problem t o describe
+theminimalpolynomial, called Zninthefollowing, when l,l∈N,coefficients
+are given.
+By theAlternation Theorem Znhas at least n−lalternation (abbreviated
+a-) points on [ −1,1]. Since every a-point from ( −1,1) is a critical point
+it follows that the inverse image of [ −1,1] underZnconsists of at most
+l′,1≤l′≤l+ 1,analytic arcs Γ j, one of them, say Γ 0is the interval
+[−1,1]. Denoting the endpoints of the arcs by α2,n,β2,n,...,αl′,n,βl′,nit
+followsbytheequioscillatingpropertythatthenormedZol otarevpolynomial
+˜Zn=Zn/||Zn||satisfies
+(3) ˜Z2
+n−1 =/parenleftBigg˜Z′
+n(x)/producttextl′
+j=1(x−γj,n)/parenrightBigg2l′/productdisplay
+j=1(x−αj,n)(x−βj,n)
+that is,
+(4)˜Z′2
+n
+˜Z2n−1=/producttextl′
+j=1(x−γj,n)2
+/producttextl′
+j=1(x−αj,n)(x−βj,n)EXTREMAL POLYNOMIALS 3
+where theγj,n’s are such that, j= 0,...,l,
+(5)/integraldisplayβj,n
+αj,n/producttextl′
+j=1(x−γj,n)/radicalBig/producttextl′
+j=1(x−αj,n)(x−βj,n)dx=kjπ
+n
+wherekjis the number of alternation points on the arc Γ j.
+(6) ˜Zn(t) =±cosh(n/integraldisplayt
+1/producttextl′
+j=1(x−γj,n)/radicalBig/producttextl′
+j=1(x−αj,n)(x−βj,n)dx).
+Roughly speaking Meiman investigated in detail the precise numberl′of
+arcs and gave a (more) detailed geometric description of the arcs Γj.For
+an explicit representation of the polynomial ˜Znexplicit expressions for the
+endpointsαj,n,βj,nand the zeros of the derivative γj,nwould be needed. To
+find such explicit expressions is extremely unlikely becaus e one has to solve
+the system of hyperelliptic integrals (5) and to find out how t he endpoints
+of the arcs are related to the given leading coefficients. In [2 0, 21, 22] no
+way of solution is offered to this fundamental open question.
+But let us observe that there is an interesting property of th ese points.
+Since˜Znhas a finite number of zeros, precisely at most lzeros, outside of
+(−1,1) and since ˜Znis a minimal polynomial on ∪Γj,the length of each
+arc Γj,j= 1,...,l, has to shrink to a point in the limit. Thus if we
+are interested in asymptotics the problem reduces to find the connection
+between the lgiven coefficients and the accumulation points of the l′-arcs
+or, in other words, the l′-zeros of ˜Znlying outside [ −1,1]; recall that every
+Γjis a component of ˜Z−1
+n([−1,1]),for a description of inverse polynomial
+images see [25]. To find this connection we proceed as follows . As usual
+we transform the problem by the Joukowski-map to the complex plane such
+that the interval [ −1,1] correspondsto theunit circle. Then weapproximate
+the polynomial of degree lwith the given lcoefficients by functions from
+H∞(so-called Caratheodory-Fejer approximation). This yiel ds a Blaschke
+product. Reflecting the Blaschke product at the unit circle i t turns out that
+its real part represents asymptotically the polynomial Znand, in particular,
+the zeros of the reflected Blaschke product are the limits of t he arcs.
+Roughly speaking we have shown that asymptotically there is a unique
+correspondence between polynomials with lfixed leading and of least maxi-
+mumnorm on [ −1,1] and polynomials which vanish outside[ −1,1] atlgiven
+points and are minimal on [ −1,1].Since we may expect that outside [ −1,1]
+the polynomial grows exponentially fast we may conclude tha t the minimal
+polynomial which vanishes at given lpoints represents asymptotically (up to
+a multiplication constant) every polynomial satisfying in each of the lgiven
+points any interpolation condition (not depending on n). Indeed in this way
+weobtain asymptotic representations of polynomials satis fying interpolation
+constraints including constraints on the derivative. So fa r for special cases4 FRANZ PEHERSTORFER1
+asymptotics for the minimum deviation (but not for the minim al polyno-
+mial) have been found by Bernstein [], see also [], n-th root asymptotics has
+been derived by Fekete and Walsh [11], see also [].
+Mostly it is more convenient to formulate the problem in term s of Cheby-
+shev polynomials, that is, to use the representation
+(7)l/summationdisplay
+j=0Ajxn−j=l/summationdisplay
+j=0ajTn−j(x)+q(x)
+q∈Pn−l−1,where the first lcoefficients ajare given by the Aj’s. In
+fact we will even study the more general problem of minimal tr igonomet-
+ric polynomials with fixed leading coefficients; more precise ly, denote by
+Tm={/summationtextm
+k=0akcoskϕ+bksinkϕ:ak,bk∈R}the set of trigonometric poly-
+nomials of degree less or equal m,and leta0,...,al,b0,...,bl∈Rbe given:
+find the unique trigonometric polynomial Zn(ϕ;a0,...,al,b0,...,bl) for which
+(8)min
+aj,bj
+l+1≤j≤n||n/summationdisplay
+j=0ajcos(n−j)ϕ+bjsin(n−j)ϕ||[0,2π]=
+=||Zn(ϕ;a0,...,al,b0,...,bl)||[0,2π]
+is attained; or in other words: given ¯ τ0=a0−ib0,...,¯τl=al−ibl∈Cfind
+the unique polynomial Zn(z;¯τ0,...,¯τl) of degree nfor which
+(9) min
+τj∈C
+l+1≤j≤n||Re{¯τjei(n−j)ϕ}||=||Re{Zn(eiϕ;¯τ0,...,¯τl)}||
+is attained. Naturally
+Zn(ϕ;¯τ0,...,¯τl) :=Zn(ϕ;a0,...,al,b0,...,bl) = Re{Zn(eiϕ;¯τ0,...,¯τl)}
+Obviouslywhenthe bj’s are zerothen, using(7), we areback inthe algebraic
+case.
+Inthe second part weconsider linear functionals onthe spac eof truncated
+trigonometricpolynomialsandtheirapplications, thatis , givenµ0,...,µl∈C
+howlargecanbethelinearfunctional |/summationtextl
+j=0µl−jτj|if||Re{/summationtextn
+j=0¯τjei(n−j)ϕ}||
+≤1.Note that the first l+1 leading coefficients of the trigonometric poly-
+nomial are given by ¯ τ0,¯τ1,...,¯τland the remaining n−l+1 coefficients are
+free available. We will determine the least upper bound of |/summationtextl
+j=0µl−jτj|for
+alln∈N.
+With the help of the solution of the problem just discussed we are able to
+solve an old problem, see [7, ?] whether truncated Fourier series can be used
+as a substitute for best approximations. A justification of t he method re-
+sultedin theconjectureof Clenshaw that |/summationtextl
+j=0τjcosjϕ|/En(f) isbounded
+by Landau’s constant as n→ ∞.The articles by Clenshaw, Lam and Elliot
+and Talbot [7, 17, 37] are devoted to show numerically that Cl enshaw’s con-
+jecture hold, at least for small l,i.e. forl= 1,2,3,4 respectively by giving
+algorithm for larger l,for details concerning open and solved questions of
+Clenshaw’s conjecture, see [37, p. 275]. In contrast to p. 27 5 the last butEXTREMAL POLYNOMIALS 5
+one paragraph in the introduction in [37] is misleading with this respect,
+partly incorrect, as the statement “The first published proo f of Clenshaw’s
+conjecture was given by Lam and Elliot [17] in 1972.” There is no proof in
+[17] as mentioned in [37, p. 275] also. The conjecture will fo llow as an easy
+consequence of our derivations.
+2.Main Theorem
+Theorem 2.1. Letpl(z) =zl+...be a polynomial of degree lwhich has all
+its zeros in |z|<1and the expansion at z= 0
+(10)pl(z)
+p∗
+l(z)=τ0+τ1z+τ2z2+...+τmzm+O(zm+1)
+wherem≥l.Then on [0,2π]the trigonometric Zolotarev polynomials of
+degreenwith leading coefficients ¯τ0,...,¯τk, k=l,...,m, k ≤n,are given
+asymptotically by
+(11) Zn(ϕ;¯τ0,¯τ1,...,¯τk) = Re{znp∗
+l(z)
+pl(z)}+O(˜rn),
+where˜r >r:= max{|zj|:zjzero ofpl}and the constant in the O( )term
+does not depend on n.Moreover
+||Zn(ϕ;¯τ0,¯τ1,...,¯τk)|| ∼1
+with geometric convergence.
+Next let us show that condition (10) is satisfied always, beca use of the
+following Theorem which goes back to Caratheodory and Fejer [] and Schur
+[35].
+Theorem 2.2. ([35])Letτ0,...,τm∈Cbe given. Then there exists a l∈N0,
+0≤l≤m,a polynomial pl(z)of degreeland aγ∈Csuch thatpl(z) =zl+...
+has all zeros in |z|<1and
+(12) γpl(z)
+p∗
+l(z)=τ0+...+τmzm+O(zm+1),
+whereγis the zero of largest modulus of
+Dl+1(λ) :=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleλ0···0τ0τ1···τl
+0λ···0 0τ0···τl−1
+........................
+0 0 ···λ0 0···τ0
+¯τ00···0λ0···0
+¯τ1¯τ0···0 0λ···0
+........................
+¯τl¯τl−1···¯τ00 0···λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+andDj(γ)/ne}ationslash= 0forj= 1,...,l.Ifτ0,...,τl∈Rthen
+Dl+1(λ) = ∆(λ)∆(−λ)6 FRANZ PEHERSTORFER1
+where
+∆(λ) :=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleτl−λ τl−1τl−2···τ1τ0
+τl−1τl−2−λ τl−3···τ2τ1
+...................................
+τ0 ··· −λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+is the characteristic polynomial of the associated Hankel m atrix.
+Combining Theorem 2.1 and Theorem 2.2 we obtain that for give nτ0,...,
+τm∈Cthere exists a l∈N0,0≤l≤mandγ∈Csuch that for ϕ∈[0,2π]
+and fork=l,...,m
+(13) Zn(ϕ;¯τ0,...,¯τk) =|γ|Re{eiargγznp∗
+l(z)
+pl(z)}+O(˜rn),
+where the constant in the O( ) term does not depend on n,and
+||Zn(ϕ;¯τ0,...,¯τk)|| ∼ |γ|
+where the convergence is geometric. l,plandγare determined by the above
+Theorem.
+Thustheproblemofdeterminingthetrigonometricandalgeb raicZolotarev
+polynomials with an arbitrary given number of leading coeffic ients with re-
+spect to the maximum norm is completely solved asymptotical ly by the
+above Theorems.
+We mention that by Fej´ er [] the condition τl≥τl−1≥...≥τ1≥τ0>0
+implies that l=min Theorem [ ?] and that the coefficients of pl(z) =/summationtextl
+ν=0aνzl−νsatisfya0≥a1≥...≥al≥0.
+For given τonly a few explicit values of |γ(τ)|=: ˜γ(τ) are known, for
+instance
+˜γ((1,...,1)) = 1/2sin/parenleftbiggπ
+2(2l+3)/parenrightbigg
+which gives by (13) that
+Zn(ϕ;(1,...,1)) =1
+2sin/parenleftBig
+π
+2(2l+3)/parenrightBig.
+Re/braceleftBigg
+sin(l+1)π
+2l+3+sinlπ
+2l+3z+···+sinπ
+2l+3zl
+sinπ
+2l+3+sin2π
+2l+3z+···+sin(l+1)π
+2l+3zl/bracerightBigg
+3.Proofs
+We will prove the following more general version of Theorem 2 .1:
+Theorem 3.1. Let(nν)be a subsequence of Nand letpl,nν(z) =zl+...be
+such that
+(14)pl,nν(z)
+p∗
+l,nν(z)=τ0,nν+τ1,nνz+...+τm,nνzm+O(zm+1),EXTREMAL POLYNOMIALS 7
+wheremis independent of nνandm≤l,and that
+(15)pl,nν(z)−→
+nν→∞pl(z),wherepl(z) has all zeros in |z| ≤r<1.
+Then forϕ∈[0,2π]andk=l,...,m,
+(16)Znν(ϕ;¯τ0,nν,...,¯τk,nν) = Re{znνp∗
+l,nν(z)
+pl,nν(z)}+O(˜rn)
+= Re{znνp∗
+l(z)
+pl(z)}+O(˜˜rn)
+where˜r > rand the constant in the O( )term does not depend on n.
+Furthermore,
+(17) ||Znν(ϕ;¯τ0,nν,...,¯τk,nν)|| ∼1.
+Notation: Let l,n∈Nand suppose that pl,n(z) =zl+...has no zero on
+|z|= 1. In the following let, z=eiϕ,
+(18) Rn(ϕ) = Re{znp∗
+l,n(z)
+pl,n(z)}=Re{zn−l(p∗
+l,n(z))2}
+|pl,n(z)|2
+and
+(19) Sn(ϕ) = Im{znp∗
+l,n(z)
+pl,n(z)}=Im{zn−l(p∗
+l,n(z))2}
+|pl,n(z)|2
+ObviouslyRn(ϕ) andSn(ϕ) are rational trigonometric functions with a
+trigonometric polynomial of degree n+lin the numerator and a positive
+trigonometric polynomial of degree lin the denominator. Note that
+(20) R2
+n(ϕ)+S2
+n(ϕ) = 1
+Lemma 3.2. Letl,n∈Nwithn>land suppose that pl,n(z)has all zeros
+in|z| ≤˜r <1for alln≥n0.Then the following statements hold for every
+n≥n1:a) BothRnandSnhave exactly 2(n−l)simple zeros in [0,2π)and
+their zeros strictly interlace. Furthermore, Rn(Sn)has2(n−l)a-points at
+the zeros of Sn(Rn).In particular, 0is a best approximation to RnandSn
+with respect to Tn−l−1.
+b) The numerators in (18)and(19)can be represented as follows, z=eiϕ,
+(21) Re {zn−l(p∗
+l,n(z))2}=tn−l,n(ϕ)|ˆr2l,n(eiϕ)|2
+and
+(22) Im {zn−l(p∗
+l,n(z))2}=un−l,n(ϕ)|ˆs2l,n(eiϕ)|2
+wheretn−l(ϕ),un−l(ϕ)are trigonometric polynomials of degree n−lwhich
+have all their 2(n−l)zeros in[0,2π)and their zeros strictly interlace. Fur-
+thermore ˆr2l(z)andˆs2l(z)are monic polynomials of degree 2lwhich have all
+their zeros in |z|<1.8 FRANZ PEHERSTORFER1
+Proof.For simplicity of writing we omit the index n.Concerning part a).
+Sincep∗
+l(z) has all zeros in |z| ≥1/r >1 it follows that for sufficiently
+largenthe arg(znp∗
+l(z)
+pl(z)) is strictly monotone increasing with respect to ϕ,
+where the change of the argument is 2( n−l)πwhenϕvaries from 0 to
+2π.Since Re {znp∗
+l(z)
+pl(z)}and Im{znp∗
+l(z)
+pl(z)}are zero, respectively, if and only if
+argznp∗
+l(z)/pl(z) = (2k+1)π/2 respectively kπ, wherek∈Z,the statement
+about the number of zeros and on the interlacing property fol lows.
+b) The statements follow immediately by part a) and the F´ eje r-Riesz
+representation for nonnegative trigonometric polynomial s. /square
+Lemma 3.3. Under the assumptions of Theorem 3.1 the polynomials ˆr2l,n(z)
+andˆs2l,n(z)associated with p∗
+l,n(z)by(21)and(22), respectively, satisfy on
+any compact subset of C
+(23)ˆr2l,n(z) = (pl(z))2+O(rn)
+ˆs2l,n(z) = (pl(z))2+O(rn)
+whereplis given by (15)and where 0<r<1.
+Proof.Let us consider the zeros of the polynomial
+(24) P2n+2l(z) =z2n(p∗
+l,n)2(z)+(pl,n)2(z).
+Note that,z=eiϕ,
+(25)P2n+2l(eiϕ) =zn+lRe{zn−l(p∗
+l,n(z))2}=zn−ltn−l,n(ϕ)r2l,n(z)r∗
+2l,n(z)
+Thus by Lemma 3.2 P2n+2l(z) has 2n−2lzeros on |z|= 1 and by the
+self-reciprocal property 2 lzeros in |z|<1 and in |z|>1 which are the
+zeros of ˆr2l,n(z) and ˆr∗
+2l,n(z),respectively. By assumption (15), (24) and
+Rouch´ e’s theorem it follows that for n≥n0ˆr2l,n(z) has (exactly) two zeros
+in each neighborhood of a zero of pl.More precisely, if vj,nis a zero of
+P2n+2lfrom the neighborhood of a zero zjofplwe have, by (24) again, that
+|vj,n−zj|=O(rn) for somer,0<r<1,whereris independent of jandn.
+Hence
+ˆr2l,n(z) =p2
+l(z)+O(rn).
+Analogously the statement for ˆ s2l,nis proved. /square
+Lemma 3.4. Under the assumption of Theorem 3.1 Rndefined in (18)is
+of the form
+(26) Rn(ϕ) =Vn(ϕ)+ψn(ϕ)
+whereVn∈TnwithVn(ϕ) = Re{/summationtextm
+j=0¯τj,nzn−j}+...andψn(ϕ)is a rational
+trigonometric function with
+(27) ||ψn||=O(˜rn)
+wheremax|zj|<˜r<1,thezj’s are the zeros of pl,and the constant in the
+O( )term does not depend on n.EXTREMAL POLYNOMIALS 9
+Proof.By EuclidRnfrom (18) can be written in the form
+(28) Rn(ϕ) =Vn(ϕ)|pl,n(eiϕ)|2+t(ϕ)
+|pl,n(eiϕ)|2,
+whereVn∈Tnandt∈Tl−1.Putting,z=eiϕ,
+(29) einϕVn(ϕ) =znP∗
+n(z)+Pn(z)
+we obtain by (18) and partial fraction expansion, assuming t hatpl,nhas
+simple zeros, that
+(30)z2n(p∗
+l,n)2(z)+p2
+l,n(z)
+pl,n(z)p∗
+l,n(z)=znP∗
+n(z)+Pn(z)+
++zn
+l/summationdisplay
+j=1λj,n
+z−zj,n+l/summationdisplay
+j=1βj,n
+z−1
+zj,n,
+
+where
+λj,n=zn
+j,np∗
+l,n(zj,n)
+p′
+l,n(zj,n)=O(˜rn)
+and analogously
+βj,n= ¯zn
+j,npl,n(1
+¯zj,n)
+(p∗
+l,n)′(1
+¯zj,n)=O(˜rn)
+where for the last equalities we took (15) into consideratio n. By (28) and
+(30) it follows that
+z2np∗
+l,n
+pl,n+pl,n
+p∗
+l,n=znP∗
+n+Pn+O(zn+l),
+hence by (14)
+Pn(z) =m/summationdisplay
+j=0τj,nzj+...
+which gives by (29) the assertion on the leading coefficients o fVn./square
+Notation 3.5.LetGbe a linear space of C[a,b] andf∈C[a,b] with 0 as
+a best approximation. Denote by γ(f,G) the strong unicity constant of f
+with respect to Gi. e.,
+(31)γ(f,G) = inf||f−g||−||f||
+||g||
+= inf
+||g||=1max
+y∈E(f)sgn(f(y))g(y)
+whereE(f) denotes the set of extremal points. If Gis a Haar space then
+by [?]
+(32) γ(f,G) = 1/max
+1≤k≤n||gk||10 FRANZ PEHERSTORFER1
+wheregk∈Gisthepolynomialgiven by gk(yj) = (−1)j,j= 1,...,n+1;j/ne}ationslash=
+kand theyjdenote the a-points off.
+Lemma 3.6.
+1/γ(Rn,Tn−l−1) =O(n)
+Proof.Let
+(33) G2(n−l)−1:=/braceleftbigg
+q(ϕ)|ˆr2l,n(eiϕ)|2
+|pl,n(eiϕ)|2:q∈Tn−l−1/bracerightbigg
+where ˆr2l,nis the polynomial associated with pl,nby (21). By Lemma 3.2
+tn−l,ncan be written in the form, z=eiϕ,
+(34)tn−l,n(ϕ) = Re{cnzn−l+...}=cn
+2z−(n−l)2(n−l)/productdisplay
+ν=1(z−eiψν,n)
+=|cn|
+2(2i)2(n−l)2(n−l)/productdisplay
+ν=1sinϕ−ϕν,n
+2
+where we used the fact that
+(35) eiargcn=e−i2(n−l)/summationtext
+ν=1ψν,n
+2
+Note that by (34) and (35)
+(36) tn−l,n(ϕ) =|cn|(cos((n−l)ϕ−2(n−l)/summationdisplay
+ν=1ψν,n
+2)+q(ϕ))
+whereq∈Tn−l−1.Analogously we obtain
+(37)un−l,n(ϕ) = Im{dnzn−l+...}=dn
+2z−(n−l)2(n−l)/productdisplay
+ν=0(z−eiϕν,n)
+=|dn|
+2(2i)2(n−l)2(n−l)/productdisplay
+ν=1sinϕ−ϕν,n
+2
+where
+eiargdn=ei/parenleftBigg
+π
+2−2(n−l)/summationtext
+ν=1ϕν,n
+2/parenrightBigg
+,
+moreover
+un−l,n(ϕ) =|dn|sin((n−l)ϕ+π
+2−2(n−l)/summationdisplay
+ν=1ϕν,n
+2)
+Taking into consideration (23) we have
+(38) lim
+ndn= lim
+ncn/ne}ationslash= 0EXTREMAL POLYNOMIALS 11
+Next let us put for k= 1,...,2n−2l,k/ne}ationslash= 1,
+qk,n(ϕ) =tn−l,n(ϕ)+
+|cn|
+|dn|un−l,n(ϕ)/parenleftBigg
+sin/parenleftBigg
+ϕ+2(n−l)/summationtext
+ν=1,ν/negationslash=1,kϕν,n
+2−2(n−l)/summationtext
+ν=1ψν,n
+2−π
+2/parenrightBigg
+−ek,n/parenrightBigg
+(−2)sin/parenleftBig
+ϕ−ϕ1,n
+2/parenrightBig
+sin/parenleftBigϕ−ϕk,n
+2/parenrightBig(39)
+where
+(40) ek,n= sin
+ϕ1,n+2(n−l)/summationdisplay
+ν=1,ν/negationslash=1,kϕν,n
+2−2(n−l)/summationdisplay
+ν=1ψν,n
+2−π
+2
+
+i.e. the second factor of the numerator is of the form constsin/parenleftBig
+ϕ−ϕ1,n
+2/parenrightBig
+sin/parenleftbigϕ−η
+2/parenrightbig
+.Moreover the second expression at the right hand side in (24) is
+fromTn−l.Since
+un−l,n(ϕ)
+(2i)2sin/parenleftBig
+ϕ−ϕ1,n
+2/parenrightBig
+sin/parenleftBigϕ−ϕk,n
+2/parenrightBig=|dn|Im{ei((n−l−1)ϕ+π
+2−2(n−l)/summationtext
+ν=1,ν/negationslash=1,kϕν,n
+2)
+}
+it follows by straightforward calculation that the second e xpression at the
+right hand side of (24) is of the form −|cn|cos/parenleftBigg
+(n−l)ϕ−2(n−l)/summationtext
+ν=1ψν,n
+2/parenrightBigg
++
+q,q∈Tn−l−1; hence it follows by (36) that qk,n∈Tn−l−1and therefore
+(41) gk,n(ϕ) =qk,n(ϕ)|ˆr2l,n(eiϕ)|2
+|pl,n(eiϕ)|2
+has the properties that gk,n∈G2(n−l)−1and, by (24) and Lemma 3.2 a)
+(42) ±gk,n(ϕν,n) = (−1)νν= 1,...,2(n−l),ν/ne}ationslash=k.
+Furthermore,
+(43) hk,n(ϕ) = sin/parenleftbiggϕ−ϕk,n
+2/parenrightbigg
+gk,n(ϕ)
+isuniformlyboundedon[0 ,2π] withrespectto n.Indeed,tn−l,n|ˆr2l,n|2/|pl,n|2
+=Rn(ϕ) andun−l,n|ˆs2l,n|2/|pl,n|2=Sn(ϕ) are bounded by one. Further
+|ˆr2l,n/pl,n|as well as, recall (23), |ˆr2l,n/ˆs2l,n|are uniformly bounded on |z|=
+1, sincepl(eiϕ) is, by assumption, bounded away from zero on |z|= 1,the
+boundedness of hk,nfollows by the choice (40) of ek,nin (24) and (38). Now
+hk,n(ϕ) is a trigonometric polynomial of half argument for which Be rnstein’s12 FRANZ PEHERSTORFER1
+inequality for the derivative still holds, therefore
+(44)||gk,n||=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglehk,n(ϕ)−hk,n(ϕk)
+ϕ−ϕk.ϕ−ϕk
+sin/parenleftbigϕ−ϕk
+2/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤ ||hk,n||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleϕ−ϕk
+sin/parenleftbigϕ−ϕk
+2/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤const.n
+Thus
+1/γ(Rn;G2(n−l)−1)≤const.n
+Using the simple fact that 0 is a best approximation to RnfromG2(n−l)−1
+as well as from Tn−l−1we obtain by ( ??) and (33) that
+γ(Rn;G2(n−l)−1)≤ ||ˆr2l,n
+pl,n||2γ(Rn;Tn−l−1)
+hence
+1/γ(Rn;Tn−l−1)≤/tildewiderconst.n
+/square
+4.Proof of Theorem 2.1 and Theorem 3.1
+Proposition 4.1. Letn,j(n)∈N.Let˜tj(n)(ϕ)be a best approximation to
+fn∈C2πon[0,2π)with respect to the linear subspace Gj(n)and let0be a
+best approximation from Gj(n)tohn∈C2πon[0,2π).Suppose that ||hn||= 1
+and that
+(45) fn(ϕ)−tj(n)(ϕ) =hn(ϕ)+ǫn(ϕ),
+wheretj(n)∈Gm(n).Ifγ(hn;Gj(n))||ǫn|| −→
+n→∞0then forϕ∈[0,2π]
+(46) fn(ϕ)−˜tj(n)(ϕ) =hn(ϕ)+O(||ǫn||γ(hn;Gj(n)))
+and in particular
+tj(n)(ϕ)−˜tj(n)(ϕ) =O(||ǫn||(1+γ(hn;Gj(n))))
+Proof.
+(47) |Ej(n)(fn)−1| ≤ ||ǫn||
+since
+Ej(n)(fn)≤ ||fn−tj(n)|| ≤ ||hn||+||ǫn||= 1+||ǫn||
+and
+1 =Ej(n)(hn) =||hn|| ≤ ||fn−tj(n)−ǫn−(˜tj(n)−tj(n))|| ≤Ej(n)(fn)+||ǫn||
+Next let us show that
+||˜tj(n)−tj(n)|| ≤(1+2γ(hn;Gj(n)))||ǫn||
+Indeed, by the definition (31) of the strong uniqueness const ant
+||˜tj(n)−tj(n)|| ≤γ(hn;Gj(n))/parenleftbig
+||hn−(˜tj(n)−tj(n))||−||hn||/parenrightbig
+≤γ(hn;Gj(n))(Ej(n)(fn)+||ǫn||−||hn||)≤2γ(hn;Gj(n))||ǫn||EXTREMAL POLYNOMIALS 13
+where we have used (45) and the fact that ||hn||= 1 and (47) in the second
+and third inequality, respectively. /square
+of Theorem 2.1. Putk(n) =n−k−1,Gk(n)=Tn−k−1andk∈ {l,...,m}
+fixed, and
+hn(ϕ) =Rn(ϕ) andfn(ϕ) = Re{k/summationdisplay
+j=0¯τj,nzn−j}.
+Recall that 0 is a best approximation with respect to Tn−l−1hence with
+respect to Tn−k−1fork∈ {l,...,n−1}.By Lemma 3.4,
+Vn=fn−tk(n)andRn=fn−tk(n)+ψ
+wheretk(n)∈Tn−j−1and||ψn||=O(rn).Sincefn−˜tk(n)=Zn(x;¯τ0,n,...,¯τj,n)
+Theorem 3.1 follows by Proposition 4.1 in conjunction with L emma 3.2 and
+Lemma 3.6. /square
+5.Extremal problems on coefficients and Clenshaw’s
+conjecture
+First let us recall how to solve the following classical prob lem in complex
+function theory. Let µ0,...,µl∈Cbe given. Among all f∈H∞with
+(48) f(z) =∞/summationdisplay
+j=0ajzjand|f(z)| ≤1 for|z|<1
+find
+(49) ηl:= max
+aj|µla0+µl−1a1+...+µ0al|.
+By [?,?] for givenµ0,...,µl∈Cthere exist polynomials sl−ν(z) andrν(z)
+of degreel−νandνrespectively, which have all zeros in |z|<1 such that
+(50)µ0+µ1z+...+µlzl+...= (s∗
+l−ν(z))2r∗
+ν(z)rν(z)
+=:h∗
+2l(z)
+F. Riesz [28] has shown that among all functions g∈H1withg(z) =
+µ0+µ1z+...+µlzl+...the mean modulus1
+2π/integraltext2π
+0|g(eiϕ)|dϕis minimal for
+h∗
+2l(z).With the help of this result Szasz [] derived that
+(51) ηl=1
+2π/integraldisplay2π
+0|h∗
+2l(eiϕ)|dϕ
+and that equality in (49) is attained by
+(52) f(z) =eiγsl−ν(z)
+s∗
+l−ν(z)=c0+c1z+...+clzl+...
+Wementionthat (51)holds, sinceSzasz’sproblem(48)-(49) andtheproblem
+considered by F. Riesz are dual problems, see [12].14 FRANZ PEHERSTORFER1
+The simple case ν= 0 appears if the expansion in z= 0
+/radicalbig
+µ0+µ1z+...+µlzl=∞/summationdisplay
+j=0λjzj
+is such that
+s∗
+l(z) =l/summationdisplay
+j=0λνzν
+has no zero in |z|<1.Then
+ηl=l/summationdisplay
+j=0|λj|2=1
+2π/integraldisplay2π
+0|sl(eiϕ)|2dϕ
+and
+f(z) =eiγsl(z)
+s∗
+l(z)
+is the function for which the maximum (49) is attained.
+The special case µj= 1,j= 0,...,l,i.e. the determination of the max-
+imum of the sum of coefficients (which fits into the case just con sidered),
+has been first considered and solved by an ad hoc method in the c elebrated
+paper by Landau []. Landau has shown that
+(53) |a0+a1+...+al| ≤1+l/summationdisplay
+j=1/parenleftbigg1.3. ... .(2j−1)
+2.4. ... .2j/parenrightbigg2
+and that equality holds only for
+(54) F(z) =eiγl/summationtext
+ν=1(−1)ν/parenleftbig−1
+2ν/parenrightbig
+zl−ν
+l/summationtext
+ν=1(−1)ν/parenleftbig−1
+2ν/parenrightbig
+zν
+whereγ∈R.
+The other simple case ν=lappears if the µj’s are such that
+(55) Re {µ0eilϕ+...+µl−1eiϕ+µl
+2} ≥0 on [0,2π]
+Then
+ηl=µl
+and equality is attained in (49) for f(z) =ε,|ε|= 1.
+Here we study the following problem: Let µ0,...,µl∈Cbe given. How
+large can be |/summationtextl
+j=0µl−jτj|if||Re{/summationtextn
+j=0¯τjzn−j}|| ≤1 andnis large. In
+other words: among all upper bounds Lsuch that for each n∈Nand for
+every (τ0,...,τl,...,τn)∈Cn
+(56) |l/summationdisplay
+j=0µl−jτj| ≤L||Re{n/summationdisplay
+j=0¯τjei(n−j)ϕ}||EXTREMAL POLYNOMIALS 15
+find the least upper bound.
+We point out that by (48) and (49) ηlis such an upper bound Lin (56).
+As we show in the next theorem it is even the least upper bound.
+Theorem 5.1. The least upper bound in (56)is given by ηl(µ0,...,µl).
+Proof.First we note that by (48) for every n∈N
+|l/summationdisplay
+j=0µl−jτj| ≤ηlif||Re{n/summationdisplay
+j=0¯τjei(n−ϕ)}|| ≤1
+Thus we have to show that the upper bound ηlcannot be improved. By (52)
+we know that max
+τj|/summationtextl
+j=1µl−jτj|=ηlis attained for a Blaschke product
+(57) eiκsl−ν(z)
+s∗
+l−ν(z)=c0+c1z+c2z2+...+clzl+...
+Thus it follows by Theorem 2.1
+Zn(ϕ;¯c0,...,¯cl) = Re{zneiκsl−ν(z)
+s∗
+l−ν(z)}+O(˜rn)
+with, setting c= (c0,...,cl),
+1−εn≤ ||Zn(ϕ;¯c)|| ≤1+εn,i.e.1
+1−εn≤En(¯c)≤1
+1+εn
+andεn→0 geometrically. Hence
+|/summationtextl
+j=0µl−jcj|
+||Zn(ϕ;¯c)||≤ηl
+1+εnwithεn→0 geometrically
+which proves the theorem. /square
+Corollary 5.2. Letµ0,...,µl∈Cbe given. Then for every (τ0,...,τl)∈Cl
+and for every n∈N
+(58) ||Re{l/summationdisplay
+j=0µl−jτjei(n−j)ϕ}|| ≤ηl||Re{n/summationdisplay
+j=0¯τjei(n−j)ϕ}||
+and the constant ηlcannot be improved.
+Proof.For anyψ∈[0,2π] and for every n∈N
+l/summationdisplay
+j=0µl−jτjei(n−j)ψ≤ηlminτj
+l+1≤j≤n||Re{n/summationdisplay
+j=0¯τje−i(n−j)ψei(n−j)ϕ}||
+=ηlminτj
+l+1≤j≤n||Re{n/summationdisplay
+j=0¯τjei(n−j)ϕ}||
+where in the first inequality the upper bound ηlis best possible by Theorem
+5.1 and the last equality follows by 2 π-periodicity. /square16 FRANZ PEHERSTORFER1
+As a consequence of Corollary 5.2 and Landau’s results ( ??) we obtain a
+proof of Clenshaw’s conjecture [] from the sixties of the las t century.
+Notation 5.3.For given τ= (τ0,...,τl)∈Cl+1let
+En(τ) =||Zn(ϕ;τ)||[0,2π]
+the minimum deviation.
+Theorem 5.4. Clenshaw’s conjecture holds, that is, for any τ∈Rl+1
+(59) lim
+n→∞
+||l/summationdisplay
+j=0τjcos(n−j)ϕ||/En(τ)
+≤1+l/summationdisplay
+j=1/parenleftbigg1.3.···(2j−1)
+2.4....2j/parenrightbigg2
+where in (59)equality is attained.
+Proof.Putµl=µl−1=···=µ0= 1.Then, as in the proof of Corollary 5.2,
+we have for ψ∈[−π,π] or equivalently for any −ψ∈[−π,π]
+||Re{l/summationdisplay
+j=0¯τjei(n−j)ψ}|| ≤ |l/summationdisplay
+j=0τje−i(n−j)ψ| ≤
+ηl((1,···,1)) min
+l+1≤j≤n||Re{n/summationdisplay
+j=0¯τjei(n−j)ϕ}||=ηl((1,···,1))En(¯τ)
+/square
+Theorem 5.5. For anyε>0and alln, n≥n0(ε),anyµ0,...,µn, τ0,...,τn∈
+C
+(60)4||Re{l/summationdisplay
+j=0µl−jτjei(n−j)ϕ}|| ≤
+||Re{n/summationdisplay
+j=0µl−jτjei(n−j)ϕ}||1+ε
+
+.||Re{n/summationdisplay
+j=0¯τjei(n−j)ϕ}||,
+where||f(ϕ)||1=/integraltext2π
+0|f(ϕ)|dϕ.
+If theµj’s andτj’s are real for j= 0,...,lthen the estimate (60) cannot
+be improved in the sense, that for given µ(τ) there exists a τ(µ) such that
+equality is attained n→ ∞.
+Proof.Proof of Theorem 5.5:
+Case 1.sl(z) satisfies condition ??.
+By ??
+(61) zn−2ls2
+l(z) =l/summationdisplay
+j=0µjzn−j+O(zn−l−1).EXTREMAL POLYNOMIALS 17
+We claim that
+min
+t∈Tn−l−1/integraldisplay2π
+0|Re{l/summationdisplay
+j=0µjzn−j}+t(ϕ)|dϕ=/integraldisplay2π
+0|Re{zn−2ls2
+l(z)}|dϕ
+=2
+π/integraldisplay2π
+0|sl(z)|2dϕ= 4l/summationdisplay
+j=0|λj|2(62)
+from which the assertion follows by recalling ( ??). Re{zn−2ls2
+l(z)}deviates
+least from zero with respect to L1-norm on [0 ,2π] among all trigonometric
+polynomials of the form Re {l/summationtext
+j=0µjzn−j+t(ϕ)}, t∈Tn−l−1.As it is well
+known it suffices to show that
+(63)/integraldisplay2π
+0eikϕsgn Re{ei(n−2l)ϕs2
+l(eiϕ)}dϕ= 0 fork= 0,...,n−l−1.
+Obviously
+sgn Re{ei(n−2l)ϕs2
+l(eiϕ)}= sgn Re {ei(n−l)ϕsl(eiϕ)
+s∗
+l(eiϕ)}
+= sgn cos(( n−l)ϕ+Φ(ϕ)) =4
+πRe arctanei((n−l)ϕ+Φ(ϕ))
+=4
+πRe arctanzn−lsl(z)
+s∗
+l(z)=4
+πzn−lsl(z)
+s∗
+l(z)+O(zn),
+where we used the fact that, z=eiϕ,
+4
+πRe arctanz=4
+πargi−z
+i+z= sgn cosϕ;
+hence (63) follows and thus the first equality in (62) is prove d. Next we
+observe that
+|Re{zn−2ls2
+l(z)}|=|sl(z)|2|Re{zn−lsl(z)
+s∗
+l(z)}|
+=|sl(z)|2|cos((n−l)ϕ+Φ(ϕ))|=|sl(z)|2Re{2
+π+O(z2(n−l))}
+where the last equality follows by Fourier-expansion. /square
+References
+[1] N. I. Achieser, Sur les propri´ et´ es asymptotiques de quelques polynomes , Comptes
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+lations of mathematical Monographs, Vol. 2, Amer. Math. Soc ., providence, R.I.,
+1962.18 FRANZ PEHERSTORFER1
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+and V. A. Marchenko, N. Akhiezer (on his seventieth birthday) , Russ. Math. Surv.
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+arXiv:1001.0471v1  [hep-ph]  4 Jan 2010BA-TH/624-10
+PHY-12545-TH-2009
+A microscopic study of pion condensation within Nambu–Jona -Lasinio model
+R. Anglani1,2,3,∗
+1Physics Division, Argonne National Laboratory, Argonne, I L 60439, USA
+2Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Ba ri, I-70126 Bari, Italy
+3Dipartimento di Fisica, Universit` a di Bari, I-70126 Bari, Italy
+We have studied the phenomenology of pion condensation in 2- flavor neutral quark matter at
+finite density with Nambu–Jona-Lasinio (NJL) model of QCD. W e have discussed the role of the
+bare quark mass mand the electric chemical potential µein controlling the condensation. The
+central result of this work is that the onset for π-condensed phase occurs when |µe|reaches the
+value of the in-medium pion mass Mπprovided the transition is of the second order, even for a
+composite pion system in the medium. Finally, we have shown t hat the condensation is extremely
+fragile with respect to the explicit chiral symmetry breaki ng via a finite current quark mass.
+PACS numbers: 12.38.Aw, 11.10.Wx, 11.30.Rd, 12.38.Gc
+I. INTRODUCTION
+Thepossibilityofpioncondensationinnuclearmedium
+was suggested by A. B. Migdal for the first time in the
+1970s[1, 2]. Thereafter, many studies [3] of “in-medium”
+pion properties have been performed due to the impor-
+tantconsequencesinsubnuclearphysics[4], inthephysics
+ofneutronstars[5], supernovas[6], andthe heavyioncol-
+lisions [7]. These analysis considered the pion as an ele-
+mentary object but we know that they can be considered
+as Nambu-Goldstone bosons generated by chiral symme-
+try breaking. Hence the internal structure and the mass
+of the pion can be sensitive to the Quantum Chromody-
+namics (QCD) vacuum modifications in the finite density
+environment [8]; moreover the finite baryon density, and
+eventhe isospindensityarisingfromthe neutralitycondi-
+tion, can modify the structure of the QCD ground state,
+and it can in turn produce significant modifications of
+the pion properties in the medium.
+In this work [9], we present the possibility of a charged
+pion(πc)condensationstartingfroma microscopic model
+which is built with quarks as the constituents of pions,
+and which exhibits chiral symmetry restoration at the fi-
+nite quark chemical potential µor temperature T. We
+derive an appropriate criterion for πccondensation, that
+we use to show the fragility of the πccondensation with
+respect to explicit chiral symmetry breaking via a finite
+current quark mass. The conditions for the onset of πc
+condensation at finite density are investigated using the
+Nambu–Jona-Lasinio(NJL) model of QCD. We find that
+the threshold for πccondensation for noninteracting ele-
+mentary pion gas is µe=Mπ−(−µe=Mπ+) for positive
+(negative)µe, whereµeis the electric chemical potential
+andMπthe in-medium pion mass. Finally, we clarify
+the effect of the current quark mass in the electrically
+neutral ground state, and the effect of a non-vanishing
+µein presence of a finite current quark mass. Based
+∗Electronic address: anglani@anl.govon these analyses, we conclude that πccondensation is
+forbidden in realistic neutral quark matter within this
+effective model.
+II. THE MODEL
+The NJL Lagrangian of the model is given for a two-
+flavor quark matter at a finite chemical potential [10]
+L= ¯e(iγµ∂µ+µeγ0)e+¯ψ(iγµ∂µ+ ˆµγ0−m)ψ
++G/bracketleftBig/parenleftbig¯ψψ/parenrightbig2+/parenleftbig¯ψiγ5/vector τψ/parenrightbig2/bracketrightBig
+, (1)
+Hereedenotes the electron field, and ψis the quark
+spinor with Dirac, color and flavor indices (implicitly
+summed). The bare quark mass is m=mu=mdand
+Gthe coupling constant. The electrical chemical poten-
+tialµeis necessary to keep the system electrically neu-
+tral [10], while µIserves as the isospin chemical poten-
+tial in the hadron sector, µI=−µe. The quark chemi-
+cal potential matrix ˆ µis defined in flavor-color space as
+ˆµ= diag(µ−2
+3µe,µ+1
+3µe)⊗1c, where1cdenotes the
+identitymatrixincolorspace, µisthequarkchemicalpo-
+tential related to the conserved baryon number. In the
+mean field approximation (MFA), we examine the pos-
+sibility that the ground state develops condensation in
+theσ=G/an}bracketle{t¯ψψ/an}bracketri}htand/orπ=G/an}bracketle{t¯ψiγ5τψ/an}bracketri}htchannels, where
+τ={τ1,τ2,τ3}denotes the Pauli matrices. Due to the
+absence of driving forces, we find /an}bracketle{tπ3/an}bracketri}htis always zero. For
+conveniencewedenote M=m−2σandN= 2/radicalbig
+π2
+1+π2
+2.
+In the numerical analyses, we have fixed Λ = 651 MeV
+andG= 2.12/Λ2so that the model reproduces fπ= 92
+MeV,/an}bracketle{t¯uu/an}bracketri}ht=−(250MeV)3, andmπ= 139 MeV in the
+vacuum with m= 5.5 MeV.
+III. PION CONDENSATION AT FINITE µIAND
+AT FINITE µ
+The pion condensation “in the vacuum” ( µ= 0)
+has been investigated with the chiral perturbation ap-
+proach [11] and within NJL model [12]. In both studies2
+FIG. 1: The constituent quark mass M, the pion condensate
+N, andmesonmassesasafunctionof µatT= 0intheneutral
+phase for a toy value of the current quark mass m= 10 keV.
+thethresholdfortheonsetofthecondensationisfoundto
+beµe=mπ, i.e., when the absolute value of the electric
+chemical potential equals the vacuum pion mass.
+We now consider neutral quark matter at µ/ne}ationslash= 0 and
+T= 0 and we want to study the relation between the
+threshold of πccondensation at finite density and the in-
+medium pion masses in the neutral ground state (for fur-
+ther recent studies at finite baryon and/or isospin chem-
+ical potential see also Refs. [13, 14]). In Ref. [10], it is
+shown that at the physical point m= 5.5 MeV, there is
+no room for πccondensation in the neutral phase (sim-
+ilar results obtained also in Ref. [15]). Even though the
+picture changeswhen the currentmass is lowered, for our
+discussion it is enough to state that we consider a current
+quark mass of the order of 10 keV. In Fig. 1 we plot M
+andNin the neutral phase as a function of µ. In this fig-
+ureMπ0,Mπ±denote the in-medium pionmassesdefined
+by the poles of the pion propagators in the rest frame,
+computed in the randomt phase approximation (RPA)
+to the Bethe-Salpeter (BS) equation. The positive and
+negative solutions of the BS equation in ωcorresponds to
+the excitation gaps for π+andπ−, which are ( Mπ++µe)
+and (Mπ−−µe), respectively. From Fig. 1 we notice that
+the transition to the pion condensed phase is of second
+order and it occurs at the point where Mπ−=µe. For a
+more detailed mathematical discussion of the numerical
+results shown in Fig. 1 we refer to [9].
+FIG. 2: Phase diagram of neutral matter in ( µ, mπ) plane.
+IV. THE ROLE OF THE CURRENT MASS AND
+THE PHASE DIAGRAM (µ,µe)
+In the past years, a large number of the analysis about
+πccondensation have been performed in the chiral limit.
+In this section we investigate the role of the finite current
+quark mass in πccondensation. In order to do this, we
+set the cutoff Λ and the coupling Gto the values spec-
+ified above and we treat mas a free parameter. As a
+consequence, the pion mass at µ=T= 0,mπin the fol-
+lowing,isafreeparameteraswell. In Fig.2wereportthe
+phase diagram in ( µ, mπ) plane in the neutral case. The
+solid line represents the border between the two regions
+where chiral symmetry is broken and restored. The bold
+dot is the critical endpoint of the first order transition.
+The shaded region indicates the region where πcconden-
+sation occurs. In the chiral limit ( mπ= 0) our results
+are in good agreement with those obtained in Ref. [16].
+Indeed, there exist two critical values of the quark chem-
+ical potential, µc1andµc2, corresponding to the onset
+andvanishingof πccondensation,respectively. Whenthe
+current quark mass increases, a shrinking of the shaded
+region occurs till the point µc1≡µc2formc
+π∼9 MeV,
+corresponding to a current quark mass of m∼10 keV.
+Hence the gapless πccondensation is extremely fragile
+with respect to the symmetry breaking effect of the cur-
+rent quark mass. As a final investigation, in Fig. 3 we
+report the phase diagram of quark matter in the ( µ, µe)
+plane when the current quark mass is tuned to m= 5.5
+MeV. At each value of ( µ, µe) we compute the chiral
+and pion condensates by minimization of the thermody-
+namical potential. The solid line represents the first or-
+der transition from the πccondensed phase to the chiral
+symmetry broken phase without the πccondensate. The
+bold dot is the critical endpoint for the first order transi-
+tion, after which the second order transition sets in. The
+dashed line indicates the first order transition between
+the two regions where chiral symmetry is broken and
+restored, respectively. The dot-dashed line is the neu-
+trality line µneut
+e=µe(µ) which is obtained by requiring
+the global electrical neutrality condition, ∂Ω/∂µe= 0.3
+FIG. 3: Phase diagram in ( µ, µe) plane at m= 5.5 MeV.
+The neutrality line manifestly shows the impossibility of
+finding aπccondensate, in this physical situation.
+V. CONCLUSIONS
+We have studied the pion condensation in two-flavor
+neutral quark matter using the Nambu–Jona-Lasinio
+model of QCD at finite density. We have investigated
+the role of electric charge neutrality, and explicit sym-
+metry breaking via quark mass, both of which controlthe onset of the charged pion condensation. We show
+that the equality between the electric chemical poten-
+tial and the “in-medium” pion mass, |µe|=Mπ, as a
+threshold, persists even for a composite pion system in
+the medium, provided the transition to the pion con-
+densed phase is of the second order (the same qualitative
+picture has been found also in a recent analysis in the
+massive Gross-Neveu model [13]). Moreover, we have
+found that the pion condensate in neutral quark matter
+is extremely fragile to the symmetry breaking effect via
+a finite current quark mass m, and is ruled out for m
+larger than the order of 10 keV.
+A final comment is in order. In this study the con-
+tribution to finite baryon density comes only from the
+constituent quarks and not from nucleons. The presence
+of nucleons may make the pion condensation even less
+favorable. The reason can be found in the Tomozawa-
+Weinberg pion-nucleon interaction, which is isospin odd,
+giving rise to a repulsive pion self-energy at the physi-
+cal situation where the neutron density is larger than the
+proton density (see Ref. [17]). For this reason an inter-
+esting investigation may consider the pion condensation
+with the presence of nucleon degrees of freedom within
+NJL model (see Ref. [18] for useful prescriptions).
+This work was supported in part by the U.S. Depart-
+ment ofEnergy, OfficeofNuclearPhysics, undercontract
+no. DE-AC02-06CH11357.
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\ No newline at end of file
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+arXiv:1001.0472v1  [math.AC]  4 Jan 2010Properties of chains of prime ideals in an
+amalgamated algebra along an ideal
+Marco D’Anna∗Carmelo A. Finocchiaro Marco Fontana
+September 10, 2018
+Abstract
+Letf:A→Bbe a ring homomorphism and let Jbe an ideal of
+B. In this paper, we study the amalgamation of AwithBalongJwith
+respect to f(denoted by A⋊ ⋉fJ), a construction that provides a general
+frame for studying the amalgamated duplication of a ring alo ng an ideal,
+introduced and studied by D’Anna and Fontana in 2007, and oth er clas-
+sical constructions (such as the A+XB[X], theA+XB[[X]] and the
+D+Mconstructions). In particular, we completely describe the prime
+spectrum of the amalgamated duplication and we give bounds f or its Krull
+dimension.
+1 Introduction
+LetAandBbe commutative rings with unity, let Jbe an ideal of Band let
+f:A−→Bbe a ring homomorphism. In this setting, we can consider the
+following subring of A×B:
+A⋊ ⋉fJ:={(a,f(a)+j)|a∈A, j∈J}
+calledthe amalgamation of AwithBalongJwith respect to f. This construc-
+tion is a generalization of the amalgamated duplication of a ring along an ideal
+(introduced and studied in [9], [6], [10] and in [20]). Moreover, severa l classical
+constructions (such as the A+XB[X], theA+XB[[X]] and the D+Mcon-
+structions) can be studied as particular cases of the amalgamation [7, Examples
+2.5 and 2.6] and other classical constructions, such as the Nagata ’s idealization
+MSC: 13A15, 13B99, 14A05.
+Key words: idealization, pullback, Zariski topology, D+Mconstruction, Krull dimension.
+∗Partially supported by a Grant MIUR-PRIN.
+1(cf. [21, page 2], [17, Chapter VI, Section 25]), also called Fossum’s t rivial ex-
+tension (cf. [15], [19] and [4]), and the CPI extensions (in the sense o f Boisen
+and Sheldon [5]) are strictly related to it [7, Example 2.7 and Remark 2.8].
+On the other hand, the amalgamation A⋊ ⋉fJis related to a construction
+proposed by D.D. Anderson in [1] and motivated by a classical constr uction
+due to Dorroh [12], concerning the embedding of a ring without identit y in
+a ring with identity. An ample introduction on the genesis of the notion of
+amalgamation is given in [7, Section 2].
+One of the key tools for studying A⋊ ⋉fJis based on the fact that the amal-
+gamation can be studied in the frame of pullback constructions [7, Se ction 4]
+(for a systematic study of this type of constructions, cf. [13]). This point of
+view allows us to deepen the study initiated in [7] and to provide an ample
+description of various properties of A⋊ ⋉fJ, in connection with the properties of
+A,Jandf.
+More precisely, in [7], we studied the basic properties of this constru ction
+(e.g., we provided characterizations for A⋊ ⋉fJto be a Noetherian ring, an inte-
+gral domain, a reduced ring) and we characterized those distinguis hed pullbacks
+that can be expressed as an amalgamation. In this paper, we pursu e the investi-
+gation on the structure of the rings of the form A⋊ ⋉fJ, with particular attention
+to the prime spectrum, to the chain properties and to the Krull dime nsion. In
+particular, after recalling (in Section 2) some basic properties prov ed in [7] and
+[8], needed in the present paper, we start our investigation by deep ening the
+study of chains of prime ideals in pullback constructions (Proposition 2.7).
+In Section 3, we study the integral closure of A⋊ ⋉fJin its total ring of
+fractions and, finally, in Section 4, we concentrate our attention t o evaluate
+its Krull dimension. In particular, we provide upper and lower bounds for
+dim(A⋊ ⋉fJ) (Proposition 4.4 and Theorem 4.9) and we show that these bounds,
+obtained in a such general setting, are so sharp that generalize, a nd possibly
+improve, analogous bounds established for the very particular cas es of integral
+domains of the form A+XB[X] [14] orA+XB[[X]] [11].
+2 Preliminaries
+Before beginning a systematic study of the ring A⋊ ⋉fJ, we recall from our
+introductory paper [7] to the subject some basic properties of th is construction.
+2.1Proposition . [7, Proposition5.1] Letf:A→Bbe a ring homomorphism,
+Jan ideal of Band setA⋊ ⋉fJ:={(a,f(a)+j)|a∈A, j∈J}.
+(1)Letι:=ιA,f,J:A→A⋊ ⋉fJbe the natural ring homomorphism defined by
+ι(a) := (a,f(a)), for alla∈A. Thenιis an embedding, making A⋊ ⋉fJa
+ring extension of A(withι(A) = Γ(f) (:={(a,f(a))|a∈A}subring of
+A⋊ ⋉fJ).
+(2)LetIbe an ideal of Aand setI⋊ ⋉fJ:={(i,f(i)+j)|i∈I,j∈J}. Then
+I⋊ ⋉fJis an ideal of A⋊ ⋉fJ, the composition of canonical homomorphisms
+2Aι֒→A⋊ ⋉fJ։A⋊ ⋉fJ/I⋊ ⋉fJis a surjective ring homomorphism and its
+kernel coincides with I.
+Hence, we have the following canonical isomorphism:
+A
+I∼=A⋊ ⋉fJ
+I⋊ ⋉fJ.
+(3)LetpA:A⋊ ⋉fJ→AandpB:A⋊ ⋉fJ→Bbe the natural projections of
+A⋊ ⋉fJ⊆A×BintoAandB, respectively. Then pAis surjective and
+Ker(pA) ={0}×J.
+Moreover, pB(A⋊ ⋉fJ) =f(A) +JandKer(pB) =f−1(J)×{0}. Hence,
+the following canonical isomorphisms hold:
+A⋊ ⋉fJ
+({0}×J)∼=AandA⋊ ⋉fJ
+f−1(J)×{0}∼=f(A)+J .
+(4)Letγ:A⋊ ⋉fJ→(f(A)+J)/Jbe the natural ring homomorphism, defined
+by(a,f(a)+j)/mapsto→f(a)+J. Thenγis surjective and Ker(γ) =f−1(J)×J.
+Thus, there exists a natural isomorphism
+A⋊ ⋉fJ
+f−1(J)×J∼=f(A)+J
+J.
+In particular, when fis surjective, we have the following natural isomor-
+phism
+A⋊ ⋉fJ
+f−1(J)×J∼=B
+J.
+Recall that, in [7] and [8], we have shown that the ring A⋊ ⋉fJcan be re-
+presented as a pullback of natural ring homomorphisms and, using t he notion of
+ring retraction, we have characterized the pullbacks that produc e exactly rings
+of the form A⋊ ⋉fJ(see also Propositions 2.3 and 2.5). Now we will make some
+pertinent remarks and prove a new result on chains of prime ideals of pullbacks,
+that will be useful for our subsequent investigation of the ring A⋊ ⋉fJ.
+2.2Definition . We recall that, if α:A→C, β:B→Care ring homomor-
+phisms, the subring D:=α×Cβ:={(a,b)∈A×B|α(a) =β(b)}ofA×Bis
+called the pullback (orfiber product ) ofαandβ. In the following, we will denote
+bypA(respectively, pB) the restriction to α×Cβof the projection of A×Bonto
+A(respectively, B).
+The followingpropositionis a straightforwardconsequenceofthe d efinitions.
+2.3Proposition . [7, Proposition4.2] Letf:A→Bbe a ring homomorphism
+andJbe an ideal of B. Ifπ:B→B/Jis the canonical projection and
+˘f:=π◦f, thenA⋊ ⋉fJ=˘f×B/Jπ.
+3Now, recall that a ring homomorphism r:B→Ais called a ring retraction if
+there exists an (injective) ring homomorphism i:A→Bsuch that r◦i=idA.
+In this case, we say also that Aisa retract of B.
+2.4Example . [7, Remark 4.6] Let f:A→Bbe a ring homomorphism and
+Jan ideal of B. ThenAis a retract of A⋊ ⋉fJand the map pA:A⋊ ⋉fJ→A,
+defined in Proposition 2.1(3), is a ring retraction. In fact, we have pA◦ι=idA,
+whereιis the ring embedding of AintoA⋊ ⋉fJ(Proposition 2.1(1)).
+The pullbacks of the form A⋊ ⋉fJform a distinguished subclass of the class
+of pullbacks of ring homomorphisms, as described in the following prop osition.
+2.5Proposition . [7, Proposition 4.7] LetA,B,C,α,β,pA,pBbe as in Defi-
+nition 2.2. Then, the following conditions are equivalent.
+(i)pA:α×Cβ→Ais a ring retraction.
+(ii)There exist an ideal JofBand a ring homomorphism f:A→Bsuch
+thatα×Cβ=A⋊ ⋉fJ. /square
+Letf:A→Bbe a ring homomorphism, and set X:= Spec( A), Y:=
+Spec(B). Recall that f∗:Y→Xdenotes the continuous map (with respect
+to the Zariski topologies) naturally associated to f(i.e.,f∗(Q) :=f−1(Q) for
+allQ∈Y). LetSbe a subset of A. Then, as usual, VX(S), or simply V(S),
+if no confusion can arise, denotes the closed subspace of X, consisting of all
+prime ideals of Acontaining S. We will denote by Jac( A) the Jacobson radical
+of a ring Aand we will call local ring a (not necessarily Noetherian) ring with
+a unique maximal ideal.
+Now, we collect some results about the structure of the prime ideals of the
+ringA⋊ ⋉fJ. The proof of the following proposition is based on well known
+properties of rings arising from pullbacks [13, Theorem 1.4] (for det ails, see [8]).
+2.6Proposition .With the notation of Proposition 2.1, set X:= Spec( A),
+Y:= Spec( B), andW:= Spec( A⋊ ⋉fJ), andJ0:={0}×J(⊆A⋊ ⋉fJ). For all
+P∈XandQ∈Y, set:
+P′f:=P⋊ ⋉fJ:={(p,f(p)+j)|p∈P, j∈J},
+Qf:={(a,f(a)+j)|a∈A, j∈J, f(a)+j∈Q}.
+Then, the following statements hold.
+(1)The map P/mapsto→P′festablishes a closed embedding of XintoW, so its
+image, which coincides with V(J0), is homeomorphic to X.
+(2)The map Q/mapsto→Qfis a homeomorphism of Y\V(J)ontoW\V(J0).
+(3)The prime ideals of A⋊ ⋉fJare of the type P′forQf, forPvarying in X
+andQinY\V(J).
+4(4)LetP∈Spec(A). Then,P′fis a maximal ideal of A⋊ ⋉fJif and only if P
+is a maximal ideal of A.
+(5)LetQbe a prime ideal of Bnot containing J. Then, Qfis a maximal
+ideal ofA⋊ ⋉fJif and only if Qis a maximal ideal of B.
+In particular:
+Max(A⋊ ⋉fJ) ={P′f|P∈Max(A)}∪{Qf|Q∈Max(B)\V(J)}./square
+The last result of this section concerns the chains of prime ideals in rin gs
+arising from pullbacks of rather general type.
+2.7Proposition .With the notation of Definition 2.2, assume βsurjective.
+LetH′andH′′be prime ideals of Dsuch that H′/subset⋉⋊teqlH′′. Assume that H′∈
+Spec(D)\V(Ker(pA)),H′′∈V(Ker(pA)), and that H′andH′′are adjacent
+prime ideals. Then, there exist two prime ideals Q′andQ′′ofB, withQ′/subset⋉⋊teqlQ′′,
+and moreover such that Q′/∈V(Ker(β)),p−1
+B(Q′) =H′, andp−1
+B(Q′′) =H′′.
+Proof. Note that the existence (and uniqueness) of a prime ideal Q′ofB
+such that Q′/∈V(Ker(β)) andp−1
+B(Q′) =H′is well known [13, Theorem 1.4,
+Statement (c) of the proof].
+On the other hand, note that p−1
+B(L+Ker(β)) =p−1
+B(L)+Ker(pA), for each
+idealLofB. Now, it is clear that the set
+S(Q′) :={Lideal ofB|Q′+Ker(β)⊆Landp−1
+B(L)⊆H′′}
+is nonempty (it contains Q′+Ker(β)) and inductive. Thus, by Zorn’s lemma,
+S(Q′) contains a maximal element Q′′, which is easy to see that is a prime ideal
+ofB. SinceH′′⊇p−1
+B(Q′′)⊇p−1
+B(Q′)+Ker(pA)/superset⋉⋊teqlH′andH′,H′′are adjacent
+prime ideals, we have p−1
+B(Q′′) =H′′. /square
+3 Integral closure of the ring A⋊ ⋉fJ
+Given a ring extension R⊆S, the integral closure of RinSwill be denoted by
+RS; the integral closure of Rin its total ring of fractions Tot( R) will be simply
+denoted by R.
+Now, we want to determine the integral closure of the ring A⋊ ⋉fJin its total
+ring of fractions. It is easy to compute Tot( A⋊ ⋉fJ) in some cases.
+3.1Proposition .Letf:A→Bbe a ring homomorphism, Jan ideal of B,
+and letA⋊ ⋉fJbe as in Proposition 2.1. Assume that Jandf−1(J)are regular
+ideals of BandA, respectively. Then Tot(A⋊ ⋉fJ)is canonically isomorphic to
+Tot(A)×Tot(B).
+Proof. Note that J1:=f−1(J)×Jis the conductor of A⋊ ⋉fJinA×B(i.e.,
+the largest ideal of A⋊ ⋉fJthat is also an ideal of A×B). Since both f−1(J) and
+5Jare regular ideals, then J1is a regular ideal of A×B. Now, the conclusion
+follows immediately by applying [16, pag. 326]. /square
+3.2Remark . Note that, in Proposition3.1, the assumption that Jandf−1(J)
+are regular ideals is essential. For example, let Abe an integral domain with
+quotient field K,Ban overring of A, and let J={0}. Then, in this situation,
+A⋊ ⋉fJ∼=A(Proposition 2.3), and thus Tot( A⋊ ⋉fJ) is isomorphic to K, but
+Tot(A)×Tot(B) =K×K.
+In the previous example, Jandf−1(J) are both the zero ideal. Another
+example, for which Jis a nonzero regular ideal, is given next. Let Abe an
+integral domain with quotient field K, setB:=A[X] andJ:= (X), and let
+f:A ֒→A[X] be the natural inclusion. In this case, from Proposition 2.3 we
+deduce that A⋊ ⋉fJ∼=A+XA[X] =A[X], and hence Tot( A⋊ ⋉fJ) =K(X).
+However, Tot( A)×Tot(B) =K×K(X). (Note that in this example f−1(J) =
+A∩J={0}.)
+Another example, for which both Jandf−1(J) are nonzero and not regular
+ideals, is the following. Let Kbe a field and set A:=K(3),B:=K(2), and
+J:={0}×K, whereK(n)is the direct product ring K×K×...×K(n–times).
+Iffis the projection defined by ( a,b,c)/mapsto→(a,b), it is immediately seen that
+A⋊ ⋉fJ∼=K(4). Then Tot( A⋊ ⋉fJ)∼=K(4), but Tot( A)×Tot(B)∼=K(5).
+We have already observed in [7, Section 5] that the ring B⋄:=f(A) +J
+(subring of B) plays a relevant role in the construction A⋊ ⋉fJ. The next result
+provides further evidence to this fact.
+3.3Lemma.Letf:A→Bbe a ring homomorphism, Jan ideal of B, and
+letA⋊ ⋉fJbe as in Proposition 2.1. The ring A×(f(A)+J), subring of A×B,
+which contains A⋊ ⋉fJis integral over A⋊ ⋉fJ. More precisely, every element of
+A×(f(A)+J)has degree at most two over A⋊ ⋉fJ.
+Proof. Let(α,f(a)+j)∈A×(f(A)+J)withα,a∈Aandj∈J. Assume that
+(α,f(a)+j)/∈A⋊ ⋉fJ, thus, in particular, α/\e}atio\slash=a. Then, the element ( α,f(a)+j)
+isarootofthemonicpolynomial( X−(α,f(α)))(X−(a,f(a)+j))∈(A⋊ ⋉fJ)[X].
+/square
+3.4Proposition .With the notation of Lemma 3.3, assume that Jandf−1(J)
+are regular ideals of BandA, respectively. Then A⋊ ⋉fJ(i.e., the integral closure
+ofA⋊ ⋉fJin its total ring of fractions) coincides with A×f(A)+J. In particular,
+iffis an integral homomorphism, then A⋊ ⋉fJ=A×B.
+Proof. Recall that, under the present hypothesis on Jandf−1(J), we have
+Tot(A⋊ ⋉fJ) = Tot(A×B), which is canonically isomorphic to Tot( A)×Tot(B)
+(Proposition 3.1). Therefore, it is easy to see that A⋊ ⋉fJ⊆A×f(A)+J. On
+the other hand, the ring A×f(A)+Jis obviously integral over A×(f(A)+J)
+andA×(f(A)+J) is integral over A⋊ ⋉fJ(Lemma 3.3). Thus A×f(A)+Jis
+integral over A⋊ ⋉fJ. The conclusion is now straightforward. /square
+63.5Remark . If we do not assume that Jandf−1(J) are regular ideals of
+BandA, respectively, then the argument used in the proof of Proposition 3.4
+shows that the integral closure of A⋊ ⋉fJin Tot(A)×Tot(B) coincides with
+A×f(A)+J.
+Now, we want to investigate when the ring A⋊ ⋉fJis integral over Γ( f)(:=
+{(a,f(a))|a∈A}).
+3.6Lemma.Letf:A−→B,J⊆B, andA⋊ ⋉fJbe as in Proposition 2.1.
+Then, the following conditions are equivalent.
+(i)f(A)+Jis integral over f(A).
+(ii)A⋊ ⋉fJis integral over Γ(f).
+In particular, if fis an integral homomorphism, then A⋊ ⋉fJis integral over
+Γ(f) (∼=A).
+Proof. (i) implies (ii). Let ( a,f(a) +j) be a nonzero element of A⋊ ⋉fJ.
+Thus, by condition (i), there exist a positive integer nanda0,a1,...,an−1∈A
+such that ( f(a) +j)n+/summationtextn−1
+i=0f(ai)(f(a) +j)i= 0. Therefore, it is easy to
+verify that ( a,f(a)+j) is a root of the monic polynomial [ X−(a,f(a))][Xn+/summationtextn−1
+i=0(ai,f(ai))Xi]∈Γ(f)[X]. Conversely, consider an element f(a) +j∈
+f(A) +J. By condition (ii), ( a,f(a) +j) is integral over Γ( f), and hence the
+equationofintegraldependence of( a,f(a)+j) overΓ(f)givesusthe equationof
+integraldependence of f(a)+joverf(A). The laststatement isstraightforward.
+/square
+4 Krull dimension of A⋊ ⋉fJ
+Now, we want to study the Krull dimension of the ring A⋊ ⋉fJ. We start with
+an easy observation.
+4.1Proposition .Letf:A→B,J, andA⋊ ⋉fJbe as in Proposition 2.1.
+Thendim(A⋊ ⋉fJ) = max {dim(A),dim(f(A) +J)}. In particular, if fis sur-
+jective, then dim(A⋊ ⋉fJ) = max{dim(A),dim(B)}= dim(A).
+Proof. By Lemma 3.3 and [18, Theorem 48], it follows immediately that
+dim(A⋊ ⋉fJ) = dim( A×(f(A)+J)). Thus, the conclusionis aneasyconsequence
+ofthe factthat Spec( A×(f(A)+J)) iscanonicallyhomeomorphicto the disjoint
+union of Spec( A) and Spec( f(A) +J). The last statement is straightforward.
+/square
+We already observed in [7, Section 5] that the kind of results as in the
+previous proposition has a moderate interest, because the Krull d imension of
+A⋊ ⋉fJis compared to the Krull dimension of f(A) +J, which is not easy to
+evaluate (moreover, if f−1(J) ={0}, we have A⋊ ⋉fJ∼=f(A)+J(Proposition
+2.1(3))).
+7An easy case for evaluating dim( A⋊ ⋉fJ) is the following.
+4.2Proposition .Letf:A→B,J, andA⋊ ⋉fJbe as in Proposition 2.1. Let
+f⋄:A→B⋄:=f(A)+Jthe ring homomorphism induced from f. If we assume
+thatf⋄is integral (e.g., fis integral), then dim(A⋊ ⋉fJ) = dim( A).
+Proof. By Lemma 3.6 and [18, Theorem 48], it follows immediately that
+dim(A⋊ ⋉fJ) = dim(Γ( f)) = dim( A). /square
+Weproceedourinvestigationlookingforupperandlowerboundsoft heKrull
+dimension of A⋊ ⋉fJ. By Proposition 2.6, we know that Spec( A⋊ ⋉fJ) =X∪U,
+whereX:= Spec( A) andU:= Spec( B)\V(J) (for the sake of simplicity, we
+identifyXandUwith their homeomorphic images in Spec( A⋊ ⋉fJ)). Further-
+more, again from Proposition 2.6, we deduce that ideals of the form Qfcan
+be contained in ideals of the form P′f, but not vice versa. Therefore, chains
+in Spec(A⋊ ⋉fJ) are obtained by juxtaposition of two types of chains, one from
+U“on the bottom” and the other one from X“on the top” (where either one
+or the other may be empty or a single element). It follows immediately t hat
+both dim( X) = dim( A) and dim( U) are lower bounds for dim( A⋊ ⋉fJ) and
+dim(A)+dim(U)+1 is an upper bound for dim( A⋊ ⋉fJ) (where, conventionally,
+we set dim( ∅) =−1).
+4.3Remark . Assume that J⊆Jac(B). By Proposition 2.6(5), we get that
+Udoes not contain maximal elements of Spec( A⋊ ⋉fJ). Hence, in this case,
+1+dim( U)≤dim(A⋊ ⋉fJ).
+Let us define the following subset of U:
+Y(f,J):=/braceleftbig
+Q∈U|f−1(Q+J) ={0}/bracerightbig
+;
+it is obvious that Y(f,J)isstable under generizations , i.e.,Q∈ Y(f,J),Q′∈
+Spec(B) andQ′⊆QimplyQ′∈ Y(f,J). Hence dim( Y(f,J)) =sup{htB(Q)|Q∈
+Y(f,J)}and we will denote this integer by δ(f,J).
+4.4Proposition .Letf:A→B,J, andA⋊ ⋉fJbe as in Proposition 2.1; let
+U= Spec(B)\V(J)andδ(f,J)= dim(Y(f,J)).
+(1)LetQ∈Spec(B), thenf−1(Q+J) ={0}if and only if Qf(= (A×Q)∩
+A⋊ ⋉fJ)is contained in J0(={0}×J).
+(2)for every Q∈ Y(f,J), the corresponding prime QfofA⋊ ⋉fJis contained
+in every prime of the form P′f.
+(3) max{dim(U),dim(A)+1+δ(f,J)} ≤dim(A⋊ ⋉fJ).
+Proof. (1) Assume that f−1(Q+J) ={0}. If (a,f(a) +j)∈Qf, with
+a∈Aandj∈J, thenf(a) +j∈Q, and so a∈f−1(Q+J) ={0}, i.e.,
+a= 0. Therefore, ( a,f(a) +j) = (0,j)∈J0. Conversely, if a∈f−1(Q+J),
+8i.e.,f(a) =q+jfor some q∈Qandj∈J, thenf(a)−j∈Q, and so
+(a,f(a)−j)∈Qf⊆J0, thusa= 0.
+(2) By Proposition 2.6(1), we have that every ideal of the form P′fcontains
+J0. The conclusion follows immediately.
+(3) By the observation preceding Remark 4.3, it is enough to show th at
+dim(A) + 1 +δ(f,J)≤dim(A⋊ ⋉fJ). IfY(f,J)=∅the statement is obvious.
+Otherwise, let Q0⊂Q1⊂...⊂Qrbe a maximal chain in Y(f,J), thusr=δ(f,J).
+LetP0⊂P1⊂...⊂Pmbe a chain realizing dim( A). By (2) we obtain that
+Qf
+0⊂...⊂Qf
+r⊂P′f
+0⊂...⊂P′fm,
+is a chain in Spec( A⋊ ⋉fJ). /square
+4.5Remark . (a)InthesituationofProposition4.4, notethat, if Jiscontained
+in the nilradical of B, i.e., if V(J) = Spec( B), thenδ(f,J)= dim(U) =−1.
+Therefore, Proposition 4.4(3) gives dim( A)≤dim(A⋊ ⋉fJ). But, in this (trivial)
+case, we can say more, precisely that Spec( A) is homeomorphic to Spec( A⋊ ⋉fJ)
+(Proposition 2.6) and so dim( A) = dim( A⋊ ⋉fJ). As a matter of fact, with
+the notation of Propositions 2.3 and 2.1, π∗
+A: Spec(A)→Spec(A⋊ ⋉fJ) is a
+homeomorphism.
+(b) Note that, if J/\e}atio\slash⊆Jac(B), the inequality 1+dim( U)≤dim(A⋊ ⋉fJ) from
+Remark 4.3 can be false, as the following Example 4.6 will show.
+(c) Letf:A→B,J, andA⋊ ⋉fJbe as in Proposition 2.1. If we assume that
+J/\e}atio\slash={0}and that A⋊ ⋉fJandBare integral domains, then, by [7, Proposition
+5.2],f−1(J) ={0}and the subset Y(f,J)of Spec(B), defined in the previous
+proposition, is nonempty, since (0) ∈ Y(f,J), and so δ(f,J)≥0. The following
+Example 4.10 will show that δ(f,J)may be arbitrarily large. Note that δ(f,J)
+may be equal to −1 even if J/\e}atio\slash={0},f−1(J) ={0}, butBis not an integral
+domain. It is sufficient to take Bequal to a local zero-dimensional ring not a
+field,Jequal to its maximal ideal, Aany subring of Bsuch that J∩A= (0),
+andfbe the natural embedding of AinB(e.g.,B:=K[X]/(X2), where Kis
+a field and Xan indeterminate over K, andAany domain contained in K). In
+this case, Spec( B) =V(J) and so δ(f,J)=−1.
+(d) Note that, in the situation ofProposition4.4(1), we canhave Qf⊆J0(=
+{0}×J) withQ/superset⋉⋊teqlJ. For instance let A:=K,B:=K[X,Y],Q:= (X,Y)B,
+J:=XB, and let f:A=K ֒→K[X,Y] =Bbe the natural embedding,
+whereKis a field and XandYtwo indeterminates over K. In this case,
+A⋊ ⋉fJ∼=A+J=K+XK[X,Y] (Proposition 2.1(3)). Clearly, f−1(Q) =
+f−1(Q+J) =f−1(J) ={0}andQf=J0∼=XK[X,Y].
+4.6Example . LetKbe a field and XandYtwo indeterminates over K. Set
+B:=K(X)[Y](Y)∩K(Y)[X](X). It is well known that Bis a one-dimensional
+semilocal domain, having two maximal ideals M:=YK(X)[Y](Y)∩Band
+N:=XK(Y)[X](X)∩B. LetJ:=M,A:=Kand letfbe the natural
+embedding of AinB. Clearly, f−1(J) =M∩K={0}. In this situation,
+N∈Spec(B)\V(J) and so dim( U) = 1. It is easy to see that A⋊ ⋉fJ∼=K+M
+9(Proposition 2.1(3)) is a one-dimensional local domain. Therefore, in this case,
+we have 2 = 1+dim( U)>1 = dim( A⋊ ⋉fJ).
+As an immediate consequence of Remark 4.3 and Proposition 4.4, we ha ve:
+4.7Corollary .With the notation of Proposition 4.4, Let f:A→B,J, and
+A⋊ ⋉fJbe as in Section 2. If we assume that J⊆Jac(B)and that δ(f,J)≥0
+(e.g.,A⋊ ⋉fJandBare integral domains). Then
+1+max{dim(A)+δ(f,J),dim(U)}≤dim(A⋊ ⋉fJ). /square
+The following observations will be useful for Remark 4.13
+4.8Remark . Letf:A→B,J, andA⋊ ⋉fJas in Proposition 2.1, and let Q
+be a prime ideal of B.
+(i) By Proposition 4.4(1), it follows immediately that Qf:= (A×Q)∩
+A⋊ ⋉fJ/subset⋉⋊teqlJ0:={0}×Jif and only if Q∈ Y(f,J)(as defined in Proposition
+4.4), i.e., f−1(Q+J) ={0}andQ/⋉⋊tsuperseteqlJ. Therefore, Y(f,J)is homeomorphic
+to{H∈Spec(A⋊ ⋉fJ)|H/subset⋉⋊teqlJ0}.
+(ii) IfA⋊ ⋉fJandBare integral domains and J/\e}atio\slash={0}then, in this situation,
+J0= (0)′f∈Spec(A⋊ ⋉fJ) andf−1(J) ={0}by [7, Proposition 5.2].
+Therefore, Q= (0)∈ Y(f,J)(/\e}atio\slash=∅) andQf=f−1(J)× {0}= (0)/subset⋉⋊teqlJ0;
+thus, if ht A⋊ ⋉fJ(J0)<∞,δ(f,J)(= dimY(f,J)) = ht A⋊ ⋉fJ(J0)−1.
+The next goalis to determine upper bounds to dim( A⋊ ⋉fJ), possibly sharper
+than dim( A)+dim(U)+1.
+4.9Theorem .Letf:A→B,J, andA⋊ ⋉fJbe as in Proposition 2.1. With
+the notation of Proposition 4.4, assume that A⋊ ⋉fJhas finite Krull dimension.
+Then
+dim(A⋊ ⋉fJ)≤max{dim(A),dim/parenleftbig
+A/f−1(J)/parenrightbig
++min{dim(B),1+dim( U)}}
+≤min{dim(A)+dim(U)+1,max{dim(A),dim/parenleftbig
+A/f−1(J)/parenrightbig
++dim(B)}}.
+Proof. We can assume that Spec( B)/\e}atio\slash=V(J), because otherwise we already
+know that dim( A⋊ ⋉fJ) = dim( A) (Remark 4.5(a)) and so the inequalities hold.
+LetH0⊂H1⊂...⊂Hnbe a chain of prime ideals of A⋊ ⋉fJrealizing
+dim(A⋊ ⋉fJ). Two extreme cases are possible.
+(1) IfH0⊇ {0}×Jthen, by Proposition 2.6(1), the chain H0⊂H1⊂...⊂
+Hninduces a chain of prime ideals of Aof length n. From Proposition 4.4(2),
+we conclude that dim( A⋊ ⋉fJ) = dim( A).
+(2) IfHn/⋉⋊tsuperseteql{0}×J. From Proposition 2.6(2), the chain H0⊂H1⊂...⊂
+Hninduces a chain of prime ideals of Uof length n. From Proposition 4.4(2),
+we conclude that dim( A⋊ ⋉fJ) = sup{ht(Q)|Q∈U}= dim(U).
+We now consider the general case.
+10(3) Lettbe the maximum index such that Ht/⋉⋊tsuperseteql{0}×J, with 0 ≤t/less⋉⋊tequaln.
+According to the notations of Proposition2.6, rewrite the given cha in as follows:
+Qf
+0⊂Qf
+1⊂...⊂Qf
+t⊂P′f
+t+1⊂P′f
+t+2⊂...⊂P′fn,
+whereQ0⊂Q1⊂...⊂Qtis an increasing chain of prime ideals of B, with
+Qt/\e}atio\slash⊇J(Proposition 2.6(2)), and Pt+1⊂Pt+2⊂...⊂Pnis an increasing
+chain of prime ideals of A(Proposition 2.6(1)). Furthermore, by Proposition
+2.7, we can find a prime ideal QinV(J) (⊆Spec(B)) such that the prime
+idealHt+1=P′f
+t+1coincides also with the restriction to A⋊ ⋉fJof the prime
+idealA×QofA×B, i.e.,Ht+1=P′f
+t+1=Qf. It follows immediately that
+Pk∈V(f−1(J)), fort+1≤k≤n. Therefore, dim( A⋊ ⋉fJ) = (1+t)+(n−t−1)
+with 1+ t≤min{1+dim( U),dim(B)}andn−t−1≤dim/parenleftbig
+A/f−1(J)/parenrightbig
+.
+Finally, it is obvious that min {dim(B),1 + dim( U)} ≤dim(B) and that
+dim/parenleftbig
+A/f−1(J)/parenrightbig
++min{dim(B),1+dim( U)} ≤dim(A)+dim(U)+1. /square
+4.10Example . LetVbe a valuation domain with maximal ideal Msuch that
+V=K+M, whereKis a field isomorphic to the residue field V/M. LetD
+be an integral domain with quotient field K, and set B:=D+M. Assume
+that dim( V) =n≥1 and that Qis a prime ideal of Vwith ht V(Q) =t+1,
+n≥t+1≥0. SetJ:=Q∩B. By the well known properties of the “ D+M
+constructions”, BM=V[16, Exercise 13(1), page 203], so Jis a prime ideal of
+Band ht B(J) =t+1. Moreprecisely, if (0) ⊂Q1⊂Q2⊂ ··· ⊂Qt⊂Qt+1=Q
+is the chain of prime ideals of Vrealizing the height of Q, thenQ0:= (0)⊂
+Q1:=Q1∩B⊂Q2:=Q2∩B⊂ ··· ⊂Qt:=Qt∩B⊂Qt+1:=Qt+1∩B=J.
+SetA:=Dand letf:A=D ֒→D+M=Bbe the canonical embedding.
+Clearly,f−1(J) ={0}and so it is easy to verify that, in the present situation,
+Y(f,J):={Q∈Spec(B)|Q /∈V(J),f−1(Q+J) ={0}}
+={Qk|0≤k≤t}= Spec(B)\V(J) =U
+(see also [16, Exercise 12(1), page 202]). Therefore, δ(f,J)=t= dim(U). More-
+over,ifm:= dim(D)(= dim( A)) then, againbythewellknownpropertiesofthe
+“D+Mconstructions”, dim( B) =m+n[16, Exercise 12(4), page 203]. Hence-
+forth, in the present example, we have max {dim(A)+1+δ(f,J),1+dim(U)}=
+dim(A)+1+δ(f,J)=m+1+t.
+On the other hand, since f−1(J) ={0}, clearly A/f−1(J) =Aand so
+max{dim(A),dim/parenleftbig
+A/f−1(J)/parenrightbig
++min{dim(B),1+dim( U)}}= dim(A) +
+min{dim(B),1 + dim( U)}=m+ min{m+n,1 +t}. Sincen≥t+ 1, then
+min{m+n,1 +t}= 1 +t. Furthermore, by the fact that f−1(J) ={0}, we
+haveA⋊ ⋉fJ∼=A+J=D+J(Proposition 2.1(3)). Therefore, from Corollary
+4.3 and Theorem 4.9, we deduce that dim( D+J) =m+1+t.
+LetA⊂Bbe an arbitrary ring extension. We will apply the previous
+results to the polynomial rings of the form A+XB[X] and we will show that
+the bounds given by Fontana, Izelgue and Kabbaj [14, Theorem 2.1] in the
+11very special case where BandAare integral domains coincide to the bounds
+obtained specializing the general setting of amalgamated algebras.
+4.11Remark . Recall that, by [7, Example 2.5], the ring A+XB[X] (re-
+spectively, A+XB[[X]]) is naturally isomorphic to A⋊ ⋉σ′XB[X] (respectively,
+A⋊ ⋉σ′′XB[[X]]), where σ′(respectively, σ′′) is the inclusion of AintoB′:=B[X]
+(respectively, into B′′:=B[[X]]).
+4.12Corollary .LetA⊆Bbe a ring extension and Xan indeterminate
+overB. Set
+δ′
+(A,B):= sup{htB[X](Q)|Q∈Spec(B[X]),X /∈Q,(Q+XB[X])∩A={0}}.
+Then
+max{dim(A)+1+δ′
+(A,B),dim(B[X,X−1])}≤dim(A+XB[X])≤
+≤dim(A)+dim(B[X]).
+Proof. LetB′:=B[X] andJ′:=XB[X]. As observed above (Remark 4.11),
+we know that A⋊ ⋉σ′J′=A+XB[X]. From the definitions, it is easy to see
+thatδ(σ′,J′)=δ′
+(A,B). Moreover, since dim( B[X,X−1]) = sup{htB[X](Q)|Q∈
+Spec(B[X]),X /∈Q}= dim(U) (where U, in this case, is homeomorphic to
+Spec(B[X])\V(J′)) andσ′−1(J′) =A∩XB[X] ={0}, the conclusion follows
+from Proposition 4.4(3) and Theorem 4.9. /square
+4.13Remark . LetA⊆Bintegral domains and and let N:=A\{0}. In [14,
+Theorem 2.1], Fontana, Izelgue and Kabbaj proved that
+max{dim(A)+dim(N−1B[X]),dim(B[X])}≤dim(A+XB[X])≤
+≤dim(A)+dim(B[X]).
+By [14, Theorem 1.2(a) and Lemma 1.3], we know that
+dim(N−1B[X]) = ht A+XB[X](XB[X]) = 1+λ′
+(A,B),
+where
+λ′
+(A,B):= sup{dim/parenleftbig
+B[X]q[X]/parenrightbig
+|q∈Spec(B),q∩A= (0)}.
+From Remark 4.8(iii) and the proof of Corollary 4.12, we deduce the eq uality
+htA+XB[X](XB[X]) = 1+δ′
+(A,B)= 1+λ′
+(A,B), henceδ′
+(A,B)=λ′
+(A,B); moreover,
+we have dim B[X] = dim B[X,X−1], by [2, Proposition 1.14]. Therefore, in
+particular, we reobtain Fontana, Izelgue and Kabbaj’s result on th e dimension
+of the integral domain A+XB[X]. This fact provides further evidence on the
+sharpness of the bounds obtained in Proposition 4.4(3) and Theore m 4.9, in the
+general setting of amalgamated algebras.
+We consider now the case of power series rings of the type A+XB[[X]] for
+arbitrary ring extensions A⊂B.
+124.14Corollary .LetA⊂Bbe a ring extension and Xan indeterminate
+overB. Set
+δ′′
+(A,B):= sup{htB[[X]](Q)|Q∈Spec(B[[X]])\V(X),(Q+XB[[X]])∩A={0}}.
+Then
+max{dim(A)+1+δ′′
+(A,B),1+ dim(B[[X]][X−1])}≤dim(A+XB[[X]])≤
+≤1+dim( A)+dim(B[[X]][X−1]).
+Proof. Keeping in mind the statements and the notation of Remark 4.11,
+it follows immediately that δ(σ′′,XB[[X]])=δ′′
+(A,B). Moreover, recalling that U,
+in this case, is homeomorphic to Spec( B[[X]])\V(X), it is easy to see that
+dim(U) = dim( B[[X]][X−1]). Finally, note that min {dim(B[[X]]),1+dim(U)}=
+1+dim( U), since every maximal ideal of B[[X]] contains X[3, Chapter 1, Ex-
+ercise 5(iv)]. The conclusion is now a straightforward consequence of Corollary
+4.3 and Theorem 4.9. /square
+4.15Remark . By applying Corollary 4.14 and Remark 4.5, it follows that, if
+Bis an integral domain, then
+1+max{dim(A)+δ′′
+(A,B),dim(B[[X]][X−1])}≤dim(A+XB[[X]])≤
+≤1+dim( A)+dim(B[[X]][X−1]).
+Now, wecancompareourlowerboundwiththatgivenbyDobbsandKh alis’s
+Theorem ([11, Theorem 11]). Setting
+λ′′
+(A,B):= sup{dim/parenleftbig
+B[[X]]q[[X]]/parenrightbig
+|q∈Spec(B),q∩A= (0)},
+they prove that
+1+max{dim(A)+λ′′
+(A,B),dim(B[[X]][X−1])}≤dim(A+XB[[X]])≤
+≤1+dim( A)+dim(B[[X]][X−1]).
+It is clear that dim/parenleftbig
+B[[X]]q[[X]]/parenrightbig
+= htB[[X]](q[[X]]). Moreover, it is immediately
+seen that, if q∈Spec(B) andq∩A= (0), then ( q[[X]] +XB[[X]])∩A= (0).
+Since the set {q[[X]]∈Spec(B[[X]])|q∈Spec(B) andq∩A={0}}is a
+subset of {Q∈Spec(B[[X]])|X /∈Qand (Q+XB[[X]])∩A={0}}, we
+haveλ′′
+(A,B)≤δ′′
+(A,B). It is natural to ask, as in the polynomial case: does
+λ′′
+(A,B)=δ′′
+(A,B)hold? At the moment, the answer to this question is open.
+However, by [11, Theorem 7], we observe that the answer could be n egative if
+htA+XB[[X]](XB[[X]]) = 1+ δ′′
+(A,B)
+andλ′′
+(A,B)/less⋉⋊tequalsup{htB[[X]](Q)|Q∈Λ(A,B)},whereΛ(A,B), as in [11, Theorem
+7],isdefinedtobe Λ(A,B)={Q∈Spec(B[[X]])|X /∈Q, Q⊂(q,X),for some q
+∈Spec(B) withq∩A= (0)}.
+134.16Example . It is possible to construct an infinite dimensional ring of the
+typeA⋊ ⋉fJ, whereAis a finite dimensional ring. In this situation, Bmust be a
+infinite dimensionalring byTheorem 4.9. Forinstance, let A:=Cbe the field of
+complex numbers, let Ybe an indeterminate over C, and let R:=C[{Y1/n|n∈
+N\{0}}]. Consider the maximal ideal MofRgenerated by the set {Y1/n|n∈
+N\{0}}. SetB:=RM, and consider the ring A+XB[[X]] (∼=A⋊ ⋉σ′′XB[[X]],
+according to notation of Remark 4.11). Then, by [11, Example 3], Bis a one-
+dimensional non-discrete valuation domain and ht A+XB[[X]](XB[[X]]) =∞, and
+thus dim( A+XB[[X]]) =∞.
+The next two examples show that the upper bound and lower bound o f
+Theorem 4.9 and Proposition 4.4(3) are “sharp”, in the sense that d im(A⋊ ⋉fJ)
+may be equal to each of the two numerical terms appearing in the fir st inequal-
+ity (respectively, in the inequality) of Theorem 4.9 (respectively, Pr oposition
+4.4(3)).
+4.17Example . LetAbe a valuation domain such that dim( A) =n≥3,
+let{0} ⊂P1⊂P2⊂ ··· ⊂ Pnbe a chain of prime ideals of Arealizing
+dim(A), and let xh∈Ph+1\Ph, with 1 ≤h≤n−2 and (xh)/\e}atio\slash=Ph+1. Since
+Ais a valuation domain, it is easily seen that V(xh) =V(Ph+1), and thus
+dim(A/(xh)) = dim( A/Ph+1) =n−(h+1). Set B:=A/(xh),f:A։Bthe
+canonical projection, Qk:=Pk/(xh) forh+1≤k≤n, andJ:=Qh+jfor some
+1≤j≤n−h. In this case, by Proposition 4.1, dim( A⋊ ⋉fJ) = dim( A×B) =
+max{dim(A),dim(B)}= dim(A) =n. Note also that dim/parenleftbig
+A/f−1(J)/parenrightbig
+=n−
+(h+j)≤n−(h+1) = dim( B), and
+dim(U) =/braceleftbigg
+−1, ifj= 1,
+j−2, if 1< j≤n−h.
+It is also easy to see that f−1(Q+J)/\e}atio\slash={0}for allQ∈Spec(B) and so in
+this case δ(f,J)=−1, for all 1 ≤j≤n−h. Moreover, in the present situation,
+A⋊ ⋉fJis a local ring, but it is not an integral domain since f−1(J)/\e}atio\slash={0}(see
+[7, Proposition 5.2]).
+Considernowachain H0⊂H1⊂...⊂Hnofprimeidealsof A⋊ ⋉fJrealizing
+dim(A⋊ ⋉fJ). Two cases are possible.
+•If 1/less⋉⋊tequalj≤n−h, then the previous chain (realizing dim( A⋊ ⋉fJ)) is of the
+type:
+((0)/\e}atio\slash=)P′f
+0⊂P′f
+1⊂...⊂P′f
+h⊂
+⊂P′f
+h+1=Qf
+h+1⊂...⊂P′f
+h+j−1=Qf
+h+j−1⊂
+⊂P′f
+h+j⊂...⊂P′fn
+(whereP′f
+k=Qf
+kalso forh+j≤k≤n, but in this case Qk⊇J);
+•Ifj= 1, then the previous chain realizing dim( A⋊ ⋉fJ) is of the type:
+((0)/\e}atio\slash=)P′f
+0⊂P′f
+1⊂...⊂P′f
+h⊂...⊂P′fn
+and none of the P′f
+kis equal to a Qf
+kforQk/\e}atio\slash⊇J.
+14In the present example, the inequality of Corollary 4.3 gives back the in-
+equality max {dim(A)+1+δ(f,J),1+dim(U)}= max{n+1−1,1+(j−2)}
+≤n= dim(A⋊ ⋉fJ). The first inequality of Theorem 4.9 gives dim( A⋊ ⋉fJ) =
+n≤max{n, n−(h+j) +min{n−(h+ 1),1+ (j−2)}}= max{dim(A),
+dim/parenleftbig
+A/f−1(J)/parenrightbig
++min{dim(B),1+dim( U)}}.
+4.18Example . LetKbe a field and let VandWbe two incomparable finite
+dimensionalvaluationdomainshavingsamefield ofquotients F. Assumethat V
+andWareK-algebras,that V=K+MandW=K+NwhereM(respectively,
+N) is the maximal ideal of V(respectively, W), and that dim( V) =m≥1 and
+dim(W) =n≥1. SetT:=V∩W. It is well known that Tis a finite
+dimensional B´ ezout domain with quotient field Fand with two maximal ideals
+M:=M∩TandN:=N∩Tsuch that TM=VandTN=W, and so
+dim(T) = max {m,n}[18, Theorem 101]. Let Dbe an integral domain of
+Krull dimension dwith quotient field K. Since Dis embedded naturally in
+V(=K+M) andW(=K+N), we have also a natural embedding ι:D ֒→T.
+In this situation, using the standard notation of the A⋊ ⋉fJconstruction,
+whenA:=D,B:=T,J:=M,andf:=ι, wehavethatthering D+M(subring
+ofT) is canonically isomorphic to D⋊ ⋉ιM, by [7, Example 2.6]. Moreover,
+f−1(J) =M∩D={0}and so dim( A/f−1(J)) = dim( D) =d, and dim( U) =
+max{m−1,n}.
+It is easy to verify that if (0) = Q0⊂Q1⊂...⊂Qm=Mare the prime
+ideals of V, then{Qk:=Qk∩B|0≤k≤m−1}coincides with the set {Q∈
+Spec(B)\V(J)|f−1(Q+J) = (Q+J)∩D={0}}. Therefore, δ(f,J)=m−1.
+On the other hand, it is easy to verify that dim( D+M) = max{m+d,n}.
+In the present example, the inequality of Proposition 4.4(3) gives ba ck the
+inequality max {dim(A)+1+δ(f,J),dim(U)}= max{d+1+m−1,max{m−
+1,n}}≤max{m+d,n}= dim(A⋊ ⋉fJ) = dim( D+M). Therefore, if n >
+m+d, then dim( A⋊ ⋉fJ) = dim( U). By the first inequality of Theorem 4.9, it
+follows that dim( A⋊ ⋉fJ) = max{m+d,n} ≤max{d, d+min{max{m,n},1+
+max{m−1,n}}}=max{dim(A),dim/parenleftbig
+A/f−1(J)/parenrightbig
++min{dim(B),1+dim(U)}}.
+Therefore, if m+d≤n, thenn= dim(D+M) = dim( D⋊ ⋉ιM) =d+
+min{max{m,n},1+max{m−1,n}}.
+Acknowledgements . The authors are very grateful to the referee for several
+suggestions and comments that greatly improved the paper.
+References
+[1] D. D. Anderson, Commutative rings , in “Multiplicative Ideal Theory in Commutative
+Algebra: A tribute to the work of Robert Gilmer” (Jim Brewer, Sarah Glaz, William
+Heinzer, and Bruce Olberding Editors),1–20, Springer, New York, 2006.
+[2] D.F. Anderson, A. Bouvier, D. Dobbs, M. Fontana and S. Kab baj,On Jaffard do-
+mains, Expo. Math. 6(1988), 145–175.
+[3] M. F. Atiyah and I. G. MacDonald, Introduction to commuta tive algebra, Addison-
+Wesley, Reading, 1969.
+15[4] C. Bakkari, S. Kabbaj, N. Mahdou, Trivial extensions defined by Pr¨ ufer conditions ,
+Preprint (arxiv.org/abs/0808.0275?context=math.AG).
+[5] M. B. Boisen and P. B. Sheldon, CPI–extension: overrings of integral domains with
+special prime spectrum , Canad. J. Math. 29(1977), 722–737.
+[6] M. D’Anna, A construction of Gorenstein rings , J. Algebra 306(2006), 507–519.
+[7] M. D’Anna, C. A. Finocchiaro, M. Fontana Amalgamated algebras along an ideal , in
+“Commutative Algebra and Applications”, Proceedings of th e Fifth International Fez
+Conference on Commutative Algebra and Applications, Fez, M orocco, 2008, W. de
+Gruyter Publisher, Berlin, 2009.
+[8] M. D’Anna, C. A. Finocchiaro, M. Fontana Algebraic and topological properties of an
+amalgamated algebra along an ideal , work in progress.
+[9] M. D’Anna and M. Fontana,, An amalgamated duplication of a ring along an ideal:
+the basic properties , J. Algebra Appl. 6(2007), 443–459.
+[10] M. D’Anna and M. Fontana, The amalgamated duplication of a ring along a
+multiplicative-canonical ideal , Arkiv Mat. 45(2007), 241–252.
+[11] D. E. Dobbs, M. Khalis, On the prime spectrum, Krull dimension and catenarity of
+integral domains of the form A+XB[[X]], J. Pure Appl. Algebra 159(2001) 5773.
+[12] J. L. Dorroh, Concerning adjunctions to algebras , Bull. Amer. Math. Soc. 38(1932),
+85–88.
+[13] M.Fontana, Topologically defined classes of commutative rings , Ann.Mat.Pura Appl.
+123(1980), 331–355.
+[14] M. Fontana, L. Izelgue and S. Kabbaj, Krull and valuative dimensions of the A+
+XB[X]rings, in “Commutative Ring Theory”(P.-J. Cahen, D.L. Costa, M. F ontana
+and S. Kabbaj Editors), Lecture Notes in Pure and Applied Mat hematics, Dekker,
+New York, 153(1994), 211–230.
+[15] R.Fossum, Commutative extensions by canonical modules are Gorenstei n rings, Proc.
+Am. Math. Soc. 40(1973), 395–400.
+[16] R. Gilmer, Multiplicative Ideal Theory, M. Dekker, New York, 1972.
+[17] J. Huckaba, Commutative rings with zero divisors, M. De kker, New York, 1988.
+[18] I. Kaplansky, Commutative Rings, The University of Chi cago Press, Chicago, 1974.
+[19] S. E. Kabbaj and N. Mahdou, Trivial Extensions Defined by Coherent-like Conditions
+Authors Comm. in Algebra, 32(2005), 3937–3953
+[20] H. R. Maimani and S. Yassemi Zero-divisor graphs of amalgamated duplication of a
+ring along an ideal J. Pure Appl. Algebra, 212(2008), 168–174
+[21] M. Nagata, Local Rings, Interscience, New York, 1962.
+16
\ No newline at end of file
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@@ -0,0 +1,465 @@
+arXiv:1001.0473v1  [hep-ph]  4 Jan 2010NeutralHiggs-pairProductionatone-loopfroma
+Generic2HDM
+DavidLópez-Val∗
+HighEnergyPhysicsGroup,Dept. ECM andInstitutdeCièncie sdelCosmos
+Universitat deBarcelona
+Av. Diagonal647,E-08028Barcelona,Catalonia,Spain
+E-mail:dlopez@ecm.ub.es
+JoanSolà
+HighEnergyPhysicsGroup,Dept. ECM andInstitutdeCièncie sdelCosmos
+Universitat deBarcelona
+Av. Diagonal647,E-08028Barcelona,Catalonia,Spain
+E-mail:sola@ecm.ub.es
+We present a one-loop analysis of the pairwise productionof neutral Higgs bosons (h0A0,H0A0)
+at linear colliders, such as the ILC and CLIC, within the gene ral Two-Higgs-Doublet Model
+(2HDM).Wesingleoutsizableradiativecorrections,which canwellreachthelevelof |δσ|/σ∼
+50%andmaybeeitherpositive(typicallyfor√s≃0.5TeV)andnegative(for√s/greaterorsimilar1TeV). These
+largequantumeffects,obtainedinfullagreementwiththe c urrentphenomenologicalboundsand
+the stringent theoretical constraints on the parameter spa ce of the model, can be traced back to
+the enhancement capabilities of the triple-Higgs self-int eractions – a trademark feature of the
+2HDM, with no counterpart in e.g. the Minimal Supersymmetri c Standard Model. In the most
+favorablescenarios,theHiggs-paircrosssectionsmaybeb oosteduptobarely30fbatthefiducial
+center-of-mass energy of 500 GeV – amounting to O(∼103)events per 500 fb−1of integrated
+luminosity. We also compare these results with several comp lementarydouble and triple Higgs-
+boson productionmechanismsat order O(α3
+ew)and leading O(α4
+ew), and we spotlight a plethora
+of potentially distinctive signatures of a Two-Higgs-Doub let structure of non-supersymmetric
+nature.
+RADCOR 2009- 9thInternationalSymposiumonRadiativeCorr ections(ApplicationsofQuantumField
+Theoryto Phenomenology)
+October25-302009
+Ascona,Switzerland
+∗Speaker.
+c/circlecopyrtCopyrightownedbytheauthor(s)underthetermsoftheCreat ive CommonsAttribution-NonCommercial-ShareAlikeLicen ce. http://pos.sissa.it/Neutral Higgs-pairProductionatone-loopfrom aGeneric2H DM DavidLópez-Val
+1. Introduction
+The Two-Higgs-Doublet Model (2HDM) is a particularly simpl e extension of the Standard
+Model (SM)whichalready encompasses outstanding newpheno menology [1]. Inaddition, aTwo-
+Higgs-Doubletstructurenaturallyemergesasthelow-ener gyrealizationofsomemorefundamental
+theories – viz. the Higgs sector of the Minimal Supersymmetr ic Standard Model (MSSM) [2].
+The 2HDM spectrum contains two neutral CP-even (h0,H0), oneCP-odd (A0) and two charged
+Higgs bosons (H±) and, as a most distinctive feature, it allows triple (3H) an d quartic (4H) Higgs
+self-interactions to be largely enhanced – in contrast to th e MSSM, wherein such couplings are
+restrained by the gauge symmetry. The phenomenological imp act of such potentially large 3H
+self-interactions has been actively investigated at linea r colliders within a manifold of processes,
+to wit: the tree-level production of triple Higgs-boson fina l states [3]; the double Higgs-strahlung
+channels hhZ0[4]; and the inclusive Higgs-pair production via gauge-bos on fusion [5]. Likewise,
+theγγmode of linac facilities has also been considered, in partic ular within the loop-induced
+single [6] and double Higgs production processes [7]. All th ese mechanisms would be capable to
+yield large Higgs production rates and furnish experimenta l signatures which, owing to the clean
+environment inherent to linear colliders, might enable for precise measurements of e.g. the Higgs
+boson masses, their couplings to fermions and gauge bosons a nd their own self-interactions; this
+means, toreconstruct theHiggspotential itself. Moreover , these largerates couldbealso revealing
+by themselves, asthey could not bedeemed e.g. toaSUSYorigi n due tothe intrinsically different
+nature of Higgs self-interactions.
+Similary, double Higgs boson (2H) production may be also ins trumental at future linac facili-
+ties [8]. Such processes cannot proceed at the tree-level in the SM,which means that, if we would
+detect asizable rateof 2Hfinalstates e.g. ofthesort e+e−→h0A0;H0A0;H+H−,thiswouldentail
+an unmistakable sign of new physics. Nonetheless, a tree-le vel analysis of these processes is most
+likely insufficient to disentangle e.g. SUSY and non-SUSY ex tensions of the Higgs sector, as the
+tree-level hA0Z0couplings ( h=h0,H0) are purely gauge, and hence both models can give rise to
+similar cross-sections at the leading-order. One-loop cor rections to these 2H production rates are
+therefore required tothiseffect. Anumber ofstudies areav ailable within theMSSM[9] whilst, on
+the 2HDM side, the efforts were first concentrated on the prod uction of charged Higgs pairs [10]
+and, only veryrecently, they have been extended tothe neutr al sector [11].
+2. Computation setup
+The general 2HDM [1] is obtained upon canonical extension of the SM Higgs sector with a
+secondSUL(2)doubletwithweakhypercharge Y=+1. Thesevenfreeparameters λiinthegeneral,
+CP-conserving, 2HDM can be sorted out as follows: the masses of the physical Higgs particles
+(Mh0,MH0,MA0,MH±), tanβ(the ratio of the two VEV’s /angbracketleftH0
+i/angbracketright), the mixing angle αbetween the
+twoCP-evenstates; andoneHiggsself-coupling λ5. Additionally, theHiggscouplings tofermions
+must be engineered so that no tree-level flavor changing neut ral currents (FCNC)are allowed: this
+gives rise to the following basic scenarios, namely Types-I and II 2HDM – see [1] for details.
+Further constraints must be imposed to assess that the SMbeh avior issufficiently well reproduced
+up to the energies explored so far, namely: i)the perturbativity and unitarity bounds [12]; ii)
+2Neutral Higgs-pairProductionatone-loopfrom aGeneric2H DM DavidLópez-Val
+the approximate SU(2)custodial symmetry, which requires |δρ2HDM|/lessorsimilar10−3[13]; and iii)the
+consistency with the low-energy radiative B-meson decays (which demands MH±/greaterorsimilar300 GeV for
+tanβ≥1 in the case of type-II 2HDM [14]). We refer the reader to [11] for further details on the
+model setup and the constraints.
+Hereafter, our endeavor will be basically threefold: i) to s ingle out the regions in the 2HDM
+parameter space for which the e+e−→A0h0/A0H0production is optimized; ii) to quantify the
+numerical impact of the associated quantum effects; and iii ) to correlate the latter with the en-
+hancement capabilities of 3H self-interactions. For defini teness, let us concentrate on the h0A0
+channel. Our starting point shall bethe e+e−→h0A0scattering amplitude at 1-loop,
+Me+e−→A0h0=/radicalBig
+ˆZh0M(0)
+e+e−→A0h0+M(1)
+e+e−→A0h0+δM(1)
+e+e−→A0h0,
+where/radicalBig
+ˆZh0h0stands for the finite WF renormalization of the external h0leg, which we shall ex-
+pand as/radicalBig
+ˆZh0h0=1−1
+2ReˆΣ′
+h0h0(M2
+h0), in order to retain only those contributions which are at
+leading-order in the triple-Higgs self-couplings. In turn ,M(1)
+e+e−→A0h0includes the entire set of
+O(α2
+ew,αewαem)one-loop contributions, to wit: i) the vacuum polarization corrections to the Z0-
+boson propagator and the Z0−γmixing; ii) the loop-induced γA0h0interaction; iii) the vertex
+corrections to the e+e−Z0and h0A0Z0interactions; and iv) box-type contributions. Finally, th e
+1-loop counterterm δM(1)
+e+e−→A0h0guarantees that the overall amplitude Mis UV-finite. Such
+counterterm ought to be anchored by a set of renormalization conditions; for the SM fields and
+parameters, we stick to the conventional on-shell scheme in the Feynman gauge, see e.g. [15].
+Concerning the renormalization of the 2HDM Higgs sector, we also employ an on-shell scheme
+whose implementation isdiscussed in full detail in [11].
+3. Numerical results
+The basic quantities of interest are: i) the predicted cross section at the Born-level σ(0)and at
+1-loopσ(0+1),inwhichweinclude thefull set of O(α3
+ew)corrections, andalsotheleading O(α4
+ew)
+which come from the squared of the scattering amplitude M; and ii) the relative size of the 1-
+loop radiative corrections, which we track through the para meterδr=σ(0+1)−σ(0)
+σ(0). Throughout the
+present work we make use of the standard computational packa ges [16]. For definiteness, we sort
+out the Higgs boson masses as follows:
+Mh0[GeV]MH0[GeV]MA0[GeV]MH±[GeV]
+SetA 130 150 200 160
+SetB 150 200 260 300
+We shall use type-I 2HDM for Set A, and type-II for Set B – as, in the latter case, MH±is suitable
+to elude B(b→sγ)constraints. The predictions for both sets are quoted in Tab le 1 for different
+values of the hA0Z0tree-level coupling and maximally enhanced 3H self interac tions – namely
+tanβ=1andλ5≃−10 (resp. −8) for Set A(resp. Set B).
+These results single out large quantum effects ( |δσ|/σ∼20−60%), which can be either
+positive (for√s≃0.5TeV) or negative (√s/greaterorsimilar1TeV) and turn out tobe fairly independent on the
+3Neutral Higgs-pairProductionatone-loopfrom aGeneric2H DM DavidLópez-Val
+h0A0H0A0
+√s=0.5TeV α=β α=β−π/6α=π/2α=β−π/2α=β−π/3α=0
+SetAσmax[fb]26.71 20.05 13.10 26.32 17.53 10.29
+δr[%]31.32 31.42 28.81 48.45 31.82 16.10
+SetBσmax[fb]11.63 9.08 6.36 5.00 3.28 1.95
+δr[%]35.17 40.68 47.86 71.81 50.22 34.15
+√s=1TeV α=β α=β−π/6α=π/2α=β−π/2α=β−π/3α=0
+SetAσmax[fb]4.08 2.70 1.56 3.59 2.52 1.55
+δr[%]-58.42 -63.28 -68.11 -62.62 -65.09 -67.75
+SetBσmax[fb]6.11 4.22 2.86 5.17 3.98 2.75
+δr[%]-30.16 -35.62 -34.58 -36.80 -35.14 -32.80
+Table1: Maximumtotalcrosssection σ(0+1)(e+e−→A0h0)andrelativeradiativecorrection( δr)at√s=0.5TeVand√s=1TeV, for Sets A and B of Higgs masses. The results are obtaine d at fixed tan β=1 and different values of α,
+withλ5≃−10(resp. −8) for SetA (resp. SetB).
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+-10-8-6-4-202
+1 2 3 4 5λ5
+tanβSet A
+−15%−15%
+−10%−15%
+−10%
+−5%−15%
+−10%
+−5%
+0%−15%
+−10%
+−5%
+0%
+5%−15%
+−10%
+−5%
+0%
+5%
+10%−15%
+−10%
+−5%
+0%
+5%
+10%
+15%−15%
+−10%
+−5%
+0%
+5%
+10%
+15%
+20%−15%
+−10%
+−5%
+0%
+5%
+10%
+15%
+20%
+25%−15%
+−10%
+−5%
+0%
+5%
+10%
+15%
+20%
+25%
+30%−15%
+−10%
+−5%
+0%
+5%
+10%
+15%
+20%
+25%
+30%
+35%−15%
+−10%
+−5%
+0%
+5%
+10%
+15%
+20%
+25%
+30%
+35%
+40%−15%
+−10%
+−5%
+0%
+5%
+10%
+15%
+20%
+25%
+30%
+35%
+40%
+45%
+-10-8-6-4-202
+1 2 3 4 5
+tanβ-10-8-6-4-202
+1 2 3 4 5
+tanβ-10-8-6-4-202
+1 2 3 4 5
+tanβ-10-8-6-4-202
+1 2 3 4 5
+tanβ-10-8-6-4-202
+1 2 3 4 5
+tanβ-10-8-6-4-202
+1 2 3 4 5
+tanβ-10-8-6-4-202
+1 2 3 4 5
+tanβ-10-8-6-4-202
+1 2 3 4 5
+tanβ-10-8-6-4-202
+1 2 3 4 5
+tanβ-10-8-6-4-202
+1 2 3 4 5
+tanβ-10-8-6-4-202
+1 2 3 4 5
+tanβ-10-8-6-4-202
+1 2 3 4 5
+tanβ−40%−40%
+−35%−40%
+−35%
+−30%−40%
+−35%
+−30%
+−25%−40%
+−35%
+−30%
+−25%
+−20%−40%
+−35%
+−30%
+−25%
+−20%
+−15%−40%
+−35%
+−30%
+−25%
+−20%
+−15%
+−10%−40%
+−35%
+−30%
+−25%
+−20%
+−15%
+−10%
+−5%−40%
+−35%
+−30%
+−25%
+−20%
+−15%
+−10%
+−5%
+0%−40%
+−35%
+−30%
+−25%
+−20%
+−15%
+−10%
+−5%
+0%
+40%
+Figure 1: Contour plots of the quantum corrections δr(in%) to the e+e−→A0h0cross-section as a function of tan β
+andλ5,forSetAofHiggs bosonmasses, α=βand√s=0.5TeV(leftpanel),√s=1.0TeV(rightpanel). Theshaded
+areas are excluded bythe vacuum stability(toppurple area) andthe unitaritybounds (bottom yellow area).
+details of the Higgs mass spectrum, the particular type of 2H DM and the specific channel under
+analysis (h0A0,H0A0). The corresponding cross-sections at 1-loop lie in the app roximate range
+of 2−30fb for√s=0.5TeV – up to barely 103−104events per 500fb−1. As for the radiative
+corrections themselves, in Figure 1 we display their detail ed behavior (for h0A0) as a function
+of tanβandλ5, as well as their interplay with the unitarity bounds (lower area, in yellow) and the
+vacuumstabilityconditions(upperarea,inpurple). Notic ethattheformerdisallowssimultaneously
+large values of tan βandλ5, whereas the latter enforces λ5/lessorsimilar0. The largest attainable quantum
+effects (positive ornegative, depending on√s) areidentified inavalley-shaped region attan β≃1
+and|λ5|∼5−10. Insuchregimes,asubsetof3Hself-couplings becomessu bstantially augmented
+– their strenght growing with ∼|λ5|– and stand as a preeminent source of radiative corrections –
+viaHiggs-boson mediated 1-loop corrections tothe hA0Z0vertex. Asaresult thelatter interaction,
+4Neutral Higgs-pairProductionatone-loopfrom aGeneric2H DM DavidLópez-Val
+which is purely gauge at the tree-level, becomes drasticall y promoted at the 1-loop order. Upon
+simple power counting arguments and educated guess we may ba rely estimate the loop-corrected
+h0A0Z0coupling as Γef f
+h0A0Z0∼Γ0
+h0A0Z0λ2
+3H/16π2s×f(M2
+h0/s,M2
+A0/s), wheref(M2
+h0/s,M2
+A0/s)is a
+dimensionless form factor.
+102030σ (e+ e- -> h0 A0) [fb]
+σ(0)σ(0+1), λ5 = -10
+σ(0+1), λ5 = -6
+σ(0+1), λ5 = 0
+50010001500
+Ecm [GeV]051015σ (e+ e- -> h0 A0) [fb]
+50010001500
+Ecm [GeV]σ(0+1), λ5 = -8
+σ(0+1), λ5 = -5
+σ(0+1), λ5 = -1
+150200250300
+Mh0Set Ba) c) e)
+b) d) f)Set Aα = β α = β sin α = 1
+Figure 2: Total cross section σ(e+e−→A0h0)(in fb) at the tree-level and at one-loop for different value s ofλ5and
+for Sets A (top panels) and B (bottom panels) of Higgs boson ma sses. Displayed is the behavior of σas a function of√s(left andcenter) and Mh0(right panels).
+All these trademark phenomenological features become also patent in Figure 2, in which we
+explore theevolution of the h0A0production cross-section for the following setups: a-b) as afunc-
+tion of√s, forα=β(maximum h0A0Z0coupling); c-d) the same, for α=π/2 (the so-called
+fermiophobic limitfortheh0bosonwithintype-I 2HDM);ande-f) asafunction of Mh0,forα=β.
+The results are presented for Sets A (upper row) and B (lower r ow). The phenomenological foot-
+print of 3H self-couplings is easily read off from the dramat ic differences in the one-loop produc-
+tion rates when the value of λ5is varied in the theoretically allowed range. As for the beha vior
+withMh0, these plots illustrate a twofold effect on σ; one is purely kinematical, and accounts for
+the decrease of the cross-section owing to the reduction of t he available phase space; the other is
+dynamical, andexplains whythe tree-level cross-sections decouple faster than their corresponding
+one-loop counterparts; indeed, several 3H self-couplings get boosted with Mh0and can partially
+counterbalance the phase space suppression.
+5Neutral Higgs-pairProductionatone-loopfrom aGeneric2H DM DavidLópez-Val
+0.5TeV 1 .0TeV 1 .5TeV 0.5TeV 1 .0TeV 1 .5TeV
+Set A SetB
+σ(h0A0)[fb]26.71 4.07 1.27 11.63 6.11 2.52
+σ(h0H0A0)[fb]0.02 5.03 3.55 below thres. 1.25 1.33
+σ(H0H+H−)[fb]0.17 11.93 8.39 below thres. 0.69 2.14
+σ(h0h0+X)[fb]1.47 17.36 38.01 0.92 9.72 23.40
+Table 2: Comparison of the predictions for the cross sections corres ponding to several Higgs-pair and triple-Higgs
+production channels, for Sets A and B of Higgs masses; tan β=1,α=β, and three different values of the center-of-
+mass energy. Thecomplementarity between the different cha nnels at different energies becomes manifest here.
+4. Discussionand conclusions
+We have undertaken a comprehensive study of the neutral 2H pr oduction from a generic
+2HDM in the context of the ILC/CLIC colliders. The upshot of o ur analysis spotlights very sig-
+nificant radiative corrections, typically up to |δσ/σ| ∼50%, which may be either positive (for√s≃0.5TeV)andnegative (for√s≃1TeVandabove), and basically concentrated intheregions
+with tanβ∼1 and|λ5|∼10, withλ5<0. Such enlarged quantum effects can be basically traced
+backtotheHiggs-mediated1-loopcorrections tothe hA0Z0interaction, thesebeingsensitivetopo-
+tentially enhanced 3H self-couplings – a genuine feature of the 2HDM, with no counterpart in the
+MSSM.Inthemostfavorable scenarios, thepredicted 2Hrate sfurnish afewdozenfbper500fb−1
+of integrated luminosity. In practice, these 2H processes w ould boil down to A0→bb/τ+τ−, and
+h0→bb/τ+τ−or h0→VV→4l,2l+mising energy, depending on the actual value of Mh0; all
+these signatures being feasible in the clean environment of linac facilities. Besides, the analysis
+of such pairwise Higgs events should be useful to enlighten t he inner structure of the underlying
+Higgs sector, in particular if combined with complementary Higgs production channels such as
+e+e−→hhh[3] and e+e−→hh+X[5]. In Table 2 we quantify the latter statement by plugging
+σ(0+1)forh0A0togetherwiththepredicted(leading-order) productionra tesforseveralcomplemen-
+tary multi-Higgs channels in e+e−collisions, assuming maximum 3H self-coupling enhancemen ts
+(as e.g. Table 1). This pattern of signatures at different√sis highly distinctive of the 2HDM and
+could not be attributed to e.g. the MSSM, as they critically d epend on the 3H self-couplings –
+whicharefully inconspicuous inthelatter model owingtosu persymmetric invariance. Additional,
+and very valuable, information can also be obtained from the study of the quantum effects on the
+production of aneutral 2HDMHiggs boson inassociation with the Z boson [17].
+Weconclude that 3H self-couplings maystamp genuine footpr ints on multi-Higgs production
+processes, either at leading-order or through quantum corr ections, and provide a plethora of com-
+plementary signatures whose detection should be perfectly feasible at future linac facilities and, if
+ever observed, might constitute a strong hint of non-standa rd, non-supersymmetric Higgs physics.
+Acknowledgments This work has been supported in part by the EU project RTN MRTN -CT-
+2006-035505 Heptools. DLV acknowledges an ESR position of t his network and also the sup-
+port of the MEC FPU grant Ref. AP2006-00357. He also wishes to thank the hospitality of the
+Theory Group at the Physikalisches Institut of the Universi ty of Bonn. JS has been supported
+in part by MEC and FEDER under project FPA2007-66665, by the S panish Consolider-Ingenio
+6Neutral Higgs-pairProductionatone-loopfrom aGeneric2H DM DavidLópez-Val
+2010 program CPANCSD2007-00042 and by DIUE/CURGeneralita t de Catalunya under project
+2009SGR502.
+References
+[1] J.F. Gunion,H.E.Haber,G.L.KaneandS. Dawson, TheHiggshunter’sguide ,Addison-Wesley,
+Menlo-Park,1990.
+[2] H.P Nilles, Phys.Rept. 110(1984)1; H.E.HaberandG.L.Kane, Phys.Rept. 117(1985)75;
+S. Ferrara,ed., Supersymmetry ,vol.1-2(NorthHolland/WorldScientific,Singapore,1987 ).
+[3] G.Ferrera,J.Guasch,D.López-ValandJ.Solà, Phys.Let t.B659(2008)297;PoS RADCOR2007,
+043(2007),arXiv:0801.3907[hep-ph].
+[4] A.Arhrib,R. BenbrikandC.-W. Chiang, Phys.Rev. D77(2008)115013.
+[5] R. N.Hodgkinson,DLópez-ValandJ.Solà, Phys.Lett. B673(2009)47.
+[6] N.Bernal,D. López-ValandJ. Solà, Phys.Lett. B677(2009)38.
+[7] F. CornetandW. Hollik, Phys.Lett. B669(2008)58;E. Asakawa,D. Harada,S. Kanemura,Y.
+OkadaandK. Tsumura, Phys.Lett. B672(2009)354;A.Arhrib,R. Benbrik,C.-H.Chen,andR.
+Santos, Phys.Rev. D80(2009)015010.
+[8] A.Djouadi,H.E.HaberandP.M.Zerwas, Phys.Lett. B375(2003)1996;A.Djouadi,W. Kilian,M.
+Mühlleitner,andP.M. Zerwas, Eur.Phys.J C10(1999)27,arXiv:hep-ph/9903229
+[9] P.Chankowski,S. PokorskiandJ.Rosiek, Nucl.Phys. B423(1994)437;V.DriesenandW. Hollik,
+Zeitsch.f.Physik C68(1995)485;V. Driesen,W. HollikandJ. Rosiek, Zeitsch.f.P hysikC71
+(1996)259;S. Heinemeyer,W. Hollik,J. RosiekandG.Weigle in, Int.J. ofMod.Phys. 19(2001)
+535;A. ArhribandG.Moultaka, Nucl.Phys. B558(1999)3; E.ConiavitisandA. Ferrari, Phys.
+Rev.D75(2007)015004.
+[10] J. Guasch,W. HollikandA. Kraft, Nucl.Phys. B558(1999)3.
+[11] D.López-ValandJ.Solà, arXiv:0908.2898[hep-ph];ac ceptedinPhys.Rev.D.
+[12] see e.g.S. Kanemura,T.KubotaandE.Takasugi, Phys.Le tt.B313(1993)155;A.Arhrib,
+arXiv:hep-ph/0012353.
+[13] C. Amsleret al.(ParticleDataGroup), Phys.Lett. B667(2008)1
+[14] M.Misiak andM. Steinhauser, Nucl.Phys. B764(2007)62;M.Misiak etal., Phys.Rev.Lett. 98
+(2007)022002.
+[15] M.Bohm,H.SpiesbergerandW. Hollik, Fortsch.Phys. G34(1986)87;A. Denner, Fortsch.Phys.
+G41(1993)307.
+[16] T.Hahn, FeynArts3.2,FormCalc andLoopTools ,http://www.feynarts.de ;T.Hahn, Comput.
+Phys.Commun. 168(2005)78.
+[17] N.Bernal,D. López-ValandJ. Solà(inpreparation).
+7
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+arXiv:1001.0475v1  [gr-qc]  4 Jan 2010November25, 2018 19:7 WSPC/INSTRUCTION FILE ppg6˙ijmpd˙DE-Gt
+International Journal of Modern Physics D
+c/circlecopyrtWorld Scientific Publishing Company
+Higher Dimensional Dark Energy Investigation with Variabl e Λ and G
+UTPAL MUKHOPADHYAY
+Satyabharati Vidyapith, North 24 Parganas, Kolkata 700 126 , West Bengal, India
+e-mail: utpal1739@gmail.com
+PARTHA PRATIM GHOSH
+Department of Physics, A. J. C. Bose Polytechnic, Date: 26.1 1.09 Berachampa, North 24
+Parganas, West Bengal, India
+e-mail: parthapapai@gmail.com
+SAIBAL RAY
+Department of Physics, Government College of Engineering a nd Ceramic Technology, Kolkata
+700 010, West Bengal, India∗
+e-mail: saibal@iucaa.ernet.in
+Received Day Month Year
+Revised Day Month Year
+Time variable Λ and Gare studied here under a phenomenological model of Λ through
+an (n+2) dimensional analysis. The relation of Zeldovich21|Λ|= 8πG2m6
+p/h4between
+Λ andGis employed here, where mpis the proton mass and his Planck’s constant. In
+the present investigation some key issues of modern cosmolo gy, viz. the age problem,
+the amount of variation of Gand the nature of expansion of the Universe have been
+addressed.
+Keywords : Higher dimension; variable Λ; variable G.
+1. Introduction
+The inception of the idea of a possible time variation of the gravitation al constant
+Gby Dirac1and subsequent supportive works, both at theoretical2,3,4,5,6,7,8and
+observational9,10,11,12,13,14,15,16,17,18level opened up a new pathway in cosmo-
+logical research. Numerous works in this direction have been done b y taking Gas a
+variable. After the discovery of the scenario of accelerating Unive rse19,20, investi-
+gations within the framework of variable Gis also not uncommon in the literature.
+One of the techniques of dark energy investigation is to include a Λ te rm in the
+field equations of Einstein. Since a constant Λ is handicapped by the w ell known
+Cosmological Constant Problem and Coincidence Problem, so it is not u nnatural
+∗Corresponding address
+1November25, 2018 19:7 WSPC/INSTRUCTION FILE ppg6˙ijmpd˙DE-Gt
+2U. Mukhopadhyay, P. P. Ghosh and S. Ray
+to consider Λ as a variable quantity. When the Λ term is placed in the rig ht hand
+side of the field equations and is considered as a part of the energy- momentum ten-
+sor, then instead of Tµν, the total energy-momentum tensor Tµν+(Λ/8πG)gµνis
+conserved. On the other hand, as an extension of Dirac’s Large Nu mber Hypothesis
+(LNH), Zeldovich21established a connection (to be shown explicitly afterwards)
+between the cosmological parameter Λ and the gravitational cons tantGby consid-
+ering Λ as the gravitational energy of the vacuum. So, variability of Λ admits the
+variability of Galso provided one considers the other quantities of the relation of
+Zeldovich21as constant.
+As a part of dark energy investigation, Ray et al.22considered Λ models in four
+dimensional space-time by taking Gas time-dependent. So, it is not unnatural to
+make an attempt for exploring new physical features by venturing into dimensions
+higher than the usual 4 D. This is the motivation behind the present work where
+a (n+ 2) dimensional study of time-dependent Λ model has been done wit hin
+the framework of variable G. We have addressed here some key issues of modern
+cosmology, viz. the age problem, the amount of variation of Gand the nature of
+expansion of the Universe. The scheme of the study is as follows: Se ction 2 deals
+with field equations and their solutions while some physical features a rising out of
+this investigation are described in different subsections of Section 3 . Finally, some
+concluding remarks are made in Section 4.
+2. Field Equations and Their Solutions
+The (n+2) dimensional metric for homogeneous and isotropic Universe is giv en by
+ds2=dt2−a2(t)[dr2+r2(dxn)2], (1)
+wherea(t) is the scale factor and
+(dxn)2=dθ2+sin2θ1dθ2
+2+......+sin2θ2
+1sin2θ2
+2....sin2dθ2
+n. (2)
+For a flat ( k= 0) Universe, the above metric yields the field equations given by
+n(n+1)
+2/parenleftbigg˙a
+a/parenrightbigg2
+= 8πGρ+Λ (3)
+and
+n¨a
+a+n(n−1)
+2/parenleftbigg˙a
+a/parenrightbigg2
+=−8πGp+Λ, (4)
+where Λ ≡Λ(t) andG≡G(t).
+The barotropic equation of state is given by
+p=ωρ, (5)
+where the barotropic index ω, assumed to be constant here, can take the values 0,
+1/3,1 and−1for pressurelessdust, electromagneticradiation,stiff (Zeldovic h) fluid
+and vacuum fluid respectively. Some other limits on ωcoming from SN Ia data23November25, 2018 19:7 WSPC/INSTRUCTION FILE ppg6˙ijmpd˙DE-Gt
+Higher Dimensional Dark Energy Investigation with Variabl eΛandG3
+and a combination ofSN Ia data with CMB anisotropyand galaxycluste ring statis-
+tics24are given by −1.67< ω <−0.62 and−1.3< ω <−0.79 respectively.
+Now, the relation of Zeldovich21between Λ and Gis given by
+|Λ|=8πG2m6
+p
+h4, (6)
+wherempis the proton mass and his Planck’s constant.
+From equation (6) we can write
+G=C√
+Λ, (7)
+whereC=h2//radicalbig
+(8π)m3
+p= constant.
+Let us use the ansatz
+Λ = 3αH2, (8)
+whereαis a parameter.
+Then from (3) we have
+ρ=/bracketleftbiggn(n+1)−6α
+16πB√α/bracketrightbigg
+H, (9)
+where ˙a/a=HandB=√
+3C.
+Using equations (9) and (5) and remembering that ¨ a/a=˙H+H2we get
+from (4),
+n˙H= (1+ω)/bracketleftbigg6α−n(n+1)
+2/bracketrightbigg
+H2. (10)
+Solving equation (10) we get our solution set as
+a(t) =C1t2n
+(1+ω)[n(n+1)−6α], (11)
+H(t) =2n
+(1+ω)[n(n+1)−6α]t−1, (12)
+ρ(t) =n
+8πB(1+ω)√αt−1, (13)
+G(t) =2nB√α
+(1+ω)[n(n+1)−6α]t−1, (14)
+Λ(t) =12n2α
+(1+ω)2[n(n+1)−6α]2t−2, (15)
+whereC1is a constant.
+It is interesting to note that in four dimensional case (i.e. n= 2), expressions
+fora(t),H(t) and Λ( t) become identical with that of Ray et al.22for the same
+Λ model and under the same assumptions, i.e. by taking both Λ and Gas time-
+dependent but without choosing any particular expression for G. Moreover, here
+G∝1/tas obtained earlier also by Dirac1as well as by Ray et al.22. Here, for
+physical validity, αcannot be equal to 1 in four dimensional case and can, in no
+way, be negative.November25, 2018 19:7 WSPC/INSTRUCTION FILE ppg6˙ijmpd˙DE-Gt
+4U. Mukhopadhyay, P. P. Ghosh and S. Ray
+3. Physical Features of the Present Model
+3.1.Age of the Universe
+From equation (12) we find that if α= 0 (i.e. Λ = 0), then in four dimensional case
+for pressureless dust,
+t=2
+3H. (16)
+It is clear from equation (16) that for the present accepted value of the Hubble
+parameter, which is (72 ±8) kms−1Mpc−125, the present age of the Universe
+becomes lessthan the ageofthe globularclusters whichlies in the ran ge9.6Gyr8to
+12.5±1.5 Gyr26,27. This means that dark energy models without Λ suffer from
+low age problem. This justifies the necessity for including the Λ term in the field
+equations as also mentioned by Sahni and Starobinsky28.
+Again, for n= 2 and α= 1/3,Ht= 1. This implies that for pressureless dust,
+we can get the exact age of the present Universe which lies in the ran ge (14±
+0.5) Gyr29,30,31,32. It may be mentioned here that Ray and Mukhopadhyay33
+obtained the exact age of the Universe for stiff fluid (i.e. ω= 1) for the same Λ
+modelwithconstant G.Butstifffluidmodelisnotanacceptedmodelforthepresent
+Universe. So, in that respect the present work has been success ful in solving the so
+called age problem ofthe Universewithout any unrealistic assumption . Moreover,it
+is easy to see that, with proper tuning of α, the exact value of the present Universe
+can be obtained from equation (12) in dust case irrespective of the dimension.
+For instance, in five dimension (i.e. n= 3), the accepted value of the age can be
+obtained for α= 1. Also for obtaining the exact age, the value of α= 1 has to be
+increased with the number of dimension and for each dimension the mo del suffers
+from low-age problem if we discard Λ, i.e. if we put α= 0.
+3.2.Rate of change of G
+From equation (14) we get,
+˙G
+G=−1
+t. (17)
+Equation (17) tells us that the rate of change of Gis of the order of t−1. If we
+take the present age of the Universe as 14 Gyr, then (since 1 Gyr is nearly 4×1017
+seconds and 1 year is about 3 ×107seconds) the value of ˙G/Gis of the order of
+10−11per year which is supported by various theoretical and observatio nal results
+(a comprehensivelist in this regardis providedin Ref. 22). Moreover ,˙G/Gdoes not
+depend on n,α, mass of proton and Planck’s Constant. This means that the rate o f
+change of Gis independent of the space-time dimension, the parameter associa ted
+with dark energy and two constants of microscopic world. This resu lt justifies the
+claim that dark energy does not exhibit any gravitational effect like c lustering and
+gravity is fundamentally different from other types of forces of na ture.November25, 2018 19:7 WSPC/INSTRUCTION FILE ppg6˙ijmpd˙DE-Gt
+Higher Dimensional Dark Energy Investigation with Variabl eΛandG5
+3.3.Calculation of the deceleration parameter
+The deceleration parameter qis given by
+q=−a¨a
+˙a2=−/parenleftBigg
+1+˙H
+H2/parenrightBigg
+. (18)
+Then, using equation (12), we get from equation (18),
+q=−/bracketleftbiggn(1+ω)+(1+ω)(6α−n2)
+2n/bracketrightbigg
+. (19)
+Then, for pressureless dust ( ω= 0) we get from equation (19),
+q=/parenleftbiggn2−n−6α
+2n/parenrightbigg
+. (20)
+It has been already shown that in four dimensional case ( n= 2) we can get the
+exact age of the Universe, so far as observationalresults are co ncerned, for α= 1/3.
+Now, for n= 2 and α= 1/3 we get q= 0 which means that the Universe expands
+with a uniform velocity. So, we are not getting an accelerating Univer se what the
+present cosmological scenario demands. Another interesting poin t is, corresponding
+to every value of αfor which we get the exact age, we get the value of qas zero
+whatever may be the spatial dimension. This means that the Univers e expands
+with uniformly in every dimension. This situation can be compared with t he so
+called ‘hesitation period’ (when the quasars were supposed to be fo rmed) of the
+Eddington-Lemaˆ ıtr´ e model. While in the Eddington-Lemaˆ ıtr´ e mod el the Universe
+behaves like de-Sitter model for large t, here the Universe expands uniformly for
+all the time. However, an accelerating Universe can be obtained fro m this model
+also under proper tuning of α. For example, in four dimensional case ( n= 2) with
+pressureless dust, qbecomes −0.1 forα= 0.4 and in that case the age of the
+Universe comes out as 14 .9 Gyr which is not an unreasonable estimate. Similar
+is the case for all other dimensions.It may be mentioned here that Kh adekar and
+Butey34also obtained q= 0 forN= 1 where under the assumption ρm=γ/RN,
+γandN(>0) being constants It is interesting to note here that for all those tuned
+values of αwhich give us the exact age of the Universe, the scale factor increa ses
+linearly. It may be mentioned here that linear expansion in a Robertso n-Walker
+Universe is shown to be possible also by Usmani et al.35for the Λ model ˙Λ∼H3.
+4. Conclusions
+In the present work, some new features of an widely used Λ-dark e nergy model has
+been explored through an ( n+2) dimensional analysis with variable Λ and variable
+G. Recently Khadekar and Butey34have studied a similar higher dimensional
+cosmologicalmodelwherebothΛand Garevariable.While KhadekarandButey34
+havechosen GbyfollowingEddington-Weinbergempiricalrelation36andSingh37,
+here in the present case a particular relation of Zeldovich’s21in connection toNovember25, 2018 19:7 WSPC/INSTRUCTION FILE ppg6˙ijmpd˙DE-Gt
+6U. Mukhopadhyay, P. P. Ghosh and S. Ray
+the cosmological parameter is used for obtaining an expression for the so-called
+gravitationalconstant.Theapproachofthepresentworkisalso completelydifferent
+from that of Khadekar and Butey34.
+In the present investigation we have been addressed, by using a we ll known Λ
+model, some key issues of modern cosmology, viz. the age problem, t he amount of
+variation of Gand the nature of expansion of the Universe. The results we have
+shown are as follows:
+(i) For any dimension, the accepted age of the present Universe ca n be attained
+under proper tuning of the parameter α.
+(ii) The amount of variation of the gravitational constant is shown t o be com-
+patible with theoretically and observationally established values witho ut any prior
+assumption on the value of the parameter α. This is an improvement over the work
+of Ray and Mukhopadhyay33where the fine-tuning of the parameters were essen-
+tial for obtaining the currently accepted values so far as the amou nt of variation of
+Gis concerned. But, here no such precondition is required.
+(iii) The present model has been successful in obtaining both negat ive and
+zero values of the deceleration parameter qwhich correspond to accelerating and
+uniformly expanding Universe respectively.
+Finally, it should be mentioned that although in the present work emph asis is
+given on the zero value of the equation of state parameter ω, but the present model
+also admits negative values of ωwith the restriction 0 < ω <−1.
+Acknowledgments
+One of the authors (SR) is thankful to the authority of Inter-Un iversity Centre
+for Astronomy and Astrophysics, Pune, India for providing Visiting Associateship
+under which a part of this work was carried out.
+References
+1. P. A. M. Dirac, Nature139(1937) 323 .
+2. C. Brans and R. H. Dicke, Phys. Lett. B 124(1961) 925.
+3. F. Hoyle and J. V. Narlikar, Mon. Not. R. Astron. Soc. 155(1972) 323.
+4. P. A. M. Dirac, Proc. R. Soc. Lond. A 333(1973) 403.
+5. W. J. Marciano, Phys. Rev. Lett. 52(1984) 489.
+6. C. M. Will, in 300 Years of Gravitation (Cambridge Univ. Press, Cambridge, p. 80,
+1987).
+7. I. Goldman, Phys. Lett. B 281(1992) 219.
+8. O. Bertolami et al., Phys. Lett. B 311(1993) 27.
+9. R. D. Reasemberg, Phil. Trans. R. Soc. Lond. A 310(1983) 227.
+10. R. W. Hellings, in Problems in Gravitation , ed Melinkov V. (Moscow State Univ.
+Press, Moscow, p. 46, 1987).
+11. V. M. Kaspi, J. H. Taylor and M. Ryba, Astrophys.J. 428(1994) 713.
+12. Z. Arzoumanian, Ph. D. thesis (Princeton University Pre ss, Princeton, New Jersey,
+USA, 1995).
+13. D. B. Guenther et al., Astrophys. J. 498(1998) 871.November25, 2018 19:7 WSPC/INSTRUCTION FILE ppg6˙ijmpd˙DE-Gt
+Higher Dimensional Dark Energy Investigation with Variabl eΛandG7
+14. E. Gaztanaga, E. Garcia-berro, J. Isern, E. Bravo and I. D ominguez, Phys. Rev. D
+65(2001) 023506.
+15. S. G. Turyshev et al., gr-qc/0311039.
+16. I. H. Stairs, Living Rev. Relativ. 6(2003) 5 .
+17. M. Biesiada and B. Malec, Mon. Not. R. Astron. Soc. 350(2004) 644.
+18. O. G. Benvenuto, E. Garcia-Berro and J. Isern, Phys. Rev. D 69(2004) 082002.
+19. S. Perlmutter et al., Nature517(1998) 565.
+20. A. G. Riess et al., Astron. J. 116(1998) 1009.
+21. Ya. B. Zeldovich, Usp. Nauk 95(1968) 209.
+22. S. Ray, U. Mukhopadhyayand S. B. DuttaChoudhury Int. J. Mod. Phys. D 16(2007)
+1791.
+23. R. Knop et al., Astrophys. J. 598(2003) 102.
+24. M. Tegmark et al., Astrophys. J. 606(2004) 70.
+25. G. Altavilla et al., Mon. Not. R. Astron. Soc. 349(2004) 1344.
+26. O. Y. Gnedin, O. Lahav and M. J. Rees, astro-ph/0108034.
+27. R. Cayrel et al., Nature409(2001) 691.
+28. V. Sahni and A. Starobinsky, Int. J. Mod. Phys. D 9(2000) 373.
+29. D. N. Spergel et al., Astrophys. J. Suppl. 148(2003) 175.
+30. R. P. Kirshner Science300(2003) 5627.
+31. M. Kunz, P. S. Corasaniti, D. Parkinson and E. J. Copeland ,Phys. Rev. D 70(2004)
+041301.
+32. M. Tegmark et al., Phys. Rev. D 69(2003) 103501.
+33. S. Ray and U. Mukhopadhyay Gravit. Cosmol. 13(2007) 46.
+34. G. S. Khadekar and B. P. Butey Int. J. Theor. Phys. 48(2009) 2618.
+35. A. A. Usmani,P. P. Ghosh, U. Mukhopadhyay, P. C. Ray and S. RayMon. Not. R.
+Astron. Soc. 386(2008) L 92.
+36. G. A. Marugan and S. Carneiro Phys. Rev. D 65(2002) 087303.
+37. C. P. Singh Int. J. Theor. Phys. 45(2006) 531.
\ No newline at end of file
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+arXiv:1001.0476v1  [physics.optics]  4 Jan 20101
+Rotational sensitivityofthe“G-Pisa” gyrolaser
+JacopoBelfi,Nicol `oBeverini,FilippoBosi,GiorgioCarelli,AngelaDiVirgil io,
+EnricoMaccioni,Marco Pizzocaro
+Abstract—G-Pisa is an experiment investigatingthe possibilityto o perate a high sensitivity laser gyroscope with area
+less than 1m2for improving the performances of the mirrors suspensions o f the gravitational wave antenna Virgo.
+The experimental set-up consists in a He-Ne ring laser with a 4 mirrors square cavity. The laser is pumped by an RF
+discharge where the RF oscillator includes the laser plasma in order to reach a better stability. The contrast of the
+Sagnacfringesistypicallyabove50%andastableregimehas beenreachedwiththelaseroperatingbothsinglemode
+or multimode. The effect of hydrogen contamination on the la ser was also checked. A low-frequencysensitivity, below
+1Hz, intherangeof 10−8(rad/s)/√
+Hzhasbeenmeasured.
+✦
+1 INTRODUCTION
+LASER gyroscopes are devices sensitive to
+inertial angular motion. They are based
+on the Sagnac effect: in a closed cavity ro-
+tating at angular velocity Ωthe two counter
+propagating beams complete the path at dif-
+ferent times. Different kinds of such devices
+have been developed mainly for navigation.
+They are only sensitive to angular velocity
+and insensitive to translational velocity. We
+distinguish between passive (fiber optic gyros)
+and active (ring lasers) Sagnac interferometers.
+Passive devices measure the phase shift be-
+tween the two beams, while the active ones
+measure the frequency difference, an inherently
+more accurate measurement. Small fiber gyros
+are typically used for navigation and have a
+resolution of 10−8rad/s, while the large ring
+laser gyros used in geophysics and geodesy
+reach the level of 10−12rad/s. In the following
+we will focus on active ring lasers and will call
+them simply gyros. One application of large
+gyros is the monitoring of the variations of the
+Earth angular velocity vector. The orientation
+with respect to the Earth axis is important since
+•J. Belfi, N. Beverini G. Carelli, E. Maccioni and M. Pizzocaro are
+with the Department of Physics “Enrico Fermi”, Universit` a di
+Pisa, and CNISM unit` a di Pisa. E-mail: carelli@df.unipi.i t
+•F. Bosi and A. Di Virgilio are with INFN sezione di Pisa.the induced signal is proportional to the scalar
+product between the normal of the gyro area
+and Earth axis, see equation 1. For horizontal
+gyros the signal is zero at the equator and
+maximum at the pole; at intermediate latitudes,
+horizontal and vertical cavities work fine. Up
+to now the reached resolution, integrating the
+signal for several hours, is 10−8of the Earth
+rotation rate [1]–[5]. The Sagnac frequency, i.e.,
+the beat signal between the two output beams,
+is:
+δφ= 4An·Ω/(λP)+φρ (1)
+whereAandPare the area and the perime-
+ter of the cavity respectively, λis the wave-
+length of the laser beam, nis the normal vector
+of the plane of the ring cavity and Ωis the
+induced vector of rotation. φρis denoting addi-
+tional, usually very small, contributions to the
+Sagnac frequency due to non-reciprocal effects
+in the laser cavity, such as Fresnel drag [6]. A
+laser gyro can monitor with high accuracy the
+orientation of the laboratory reference frame,
+but our main interest is in extending its use
+for improvements of Virgo, the gravitational
+waves interferometric antenna [7], [8]. The ma-
+jor draw-back in ring lasers is the lock-in be-
+tween the two counter-propagating modes; in
+large gyros the rate bias induced by the Earth
+rotation is enough to avoid it. The large ring
+laser gyros built by the joint ring laser working
+group in New Zealand and Germany have2
+sides of the order of several meters. “G” [9],
+[10] located at the Geodetic Observatory in
+Wettzell (Germany) is a monolithic square with
+4 m sides, “UG2”, a rectangle with 40 m by
+20 m sides, is the largest ring. In principle the
+larger the area the better the sensitivity, but
+the best gyro so far, is “G”, which is oper-
+ating very close to the quantum limit (just a
+factor 3 higher), and its measured noise power
+spectrum is of the order of 10−11rad/s/√
+Hz,
+with a duty cycle close to 100%. The Allan
+deviation has not shown any systematic effects
+on a time scale of about 3 hours, and the sensor
+drift is normally below 1.5 parts in 108of the
+Earth rotation ( 1.1·10−12rad/s). The Allan
+deviation goes down as the square root of time
+up to 2.7 hours of averaging time, then goes up
+for a while before it resumes the theoretically
+expected downward trend.
+2 SENSITIVITY LIMITS
+In large gyros the shot noise of light gives the
+fundamental limit to sensitivity:
+Ωsn=c
+2πKL/radicalBigg
+hνµT
+2P t(2)
+wherecis the speed of light, Lthe side of
+the square ring, hthe Plank Constant, νthe
+frequency of the light, µthe total cavity losses,
+Tthe transmittance, Pthe total transmitted
+power,tthe observation time, and Kis the
+scale factor of the instrument. In figure 1 the
+shot noise is shown for various values of the
+parameters µ,TandPand an observation time
+of 1 s.
+For the continuous line (good mirrors, but
+not the best ones) µ= 40 ppm,T= 0.2ppm
+andP= 10−8W, for the dotted line absorption
+has been reduced to 4 ppm, and for the point-
+dotted line the transmitted power has also been
+increased to 10−7W (top quality mirrors). In
+short, a device with Lbelow 1 m could rea-
+sonably reach a sensitivity of 10−9rad/s/√
+Hz,
+while to reach 10−11rad/s/√
+Hzdevices larger
+than 2 m are required. The lower limits to
+the size of a gyrolaser are given by the
+backscattering-induced frequency pulling and
+lock-in. As a rule, for a given set of mirrors, it
+is always possible to build a ring large enough00.511.522.533.544.555.566.577.588.599.51010−1210−1110−1010−910−8
+Size of the square cavity  (m)Shot noise [(rad/s)/Hz1/2]
+Fig. 1. Shot noise limit as a function of the
+square cavity side. The three different traces
+correspond to sets of mirrors of different quality
+(seethetext).
+that the bias induced by the Earth rotation is
+large enough to avoid lock-in. The parameter
+which sets the magnitude of the mode pulling
+is the lock-in threshold frequency l, where
+l≈csλ/(πdP), wherecis the speed of light,
+sis the fraction of the laser field amplitude
+which is scattered by each mirror, λis the
+laser wavelength, dis the beam waist and P
+is the ring perimeter. In figure 2 the dashed
+lines show the lock-in limit as a function of
+the side of a square ring, for a beam waist of
+0.5 mm and different values of of the scattering
+coefficient s. The continuous line shows the
+Sagnac frequency given by the Earth rotation
+for a device horizontally located at latitude 43◦
+as a function of the ring size.
+3 EXPERIMENTAL SET -UP
+G-Pisa is a square cavity 5.60 m in perimeter
+and 1.96 m2in area. The mechanical design
+is flexible and each side of the square can be
+scaled from 1.40 m down to 0.90 m with minor
+changes. The mechanical system is mounted
+onto an optical table and has a stainless steel
+modular structure: 4 boxes, located at the cor-
+ner of the square and containing the mirrors
+holders inside, are connected by tubes, so to
+form a ring vacuum chamber with a total
+volume of about 5·10−3m3. The vacuum3
+10−110010110−1100101102103
+Size of the square cavity (m)Frequency (Hz)
+Fig. 2. Continuous line shows the Sagnac fre-
+quency, the three dashed lines show the lock-
+in threshold frequency for three different set of
+mirrors (s=10−4, 5·10−4and 10−3). Highlighted
+intersections represent the G-Pisacase.
+chamber, is entirely filled with a mixture of
+He and a 50% isotopic mixture of20Ne and
+22Ne. The total pressure of the gas mixture is
+set to 560 Pa with a partial pressure of Neon
+of 20 Pa. In the center of one of the tubes
+there is a pyrex insertion, a capillary with 4
+mm internal diameter, approximately 15 cm
+long, which is the discharge tube. While the
+discharge in the laser medium of “G” and ana-
+log systems is excited by an RF source, where
+two coils couple the RF oscillator to the gas,
+in our system we choose a capacitive coupling.
+Two halves of a copper cylinder are used as
+electrodes and are part of the resonant circuit
+of the RF source. The laser medium in this way
+is included in the active circuit and the fluctu-
+ations in the plasma density do not affect the
+coupling of the RF source to the discharge, but
+only the oscillator frequency, about 115 MHz.
+This kind of discharge provides a very good
+passive stability and made possible to regulate
+the laser output power very close to the laser
+threshold since the first runs. The discharge is
+5 cm long, but different lengths will be tested
+in the next months to minimize the effect of
+plasma intensity fluctuations around the laser
+threshold. The discharge tube has four micro-
+metric screws used to align the pyrex capillarywith the mirrors and the optical cavity.
+Four spherical mirrors with 6 m radius cur-
+vature were chosen for the resonator, and two
+micro-metric lever arms acting on the tilts
+of each mirror, make it possible a fine tun-
+ing of the cavity alignment. Mirrors reflectiv-
+ity is optimized for the emission line around
+632.8 nm. The free spectral range of the cav-
+ity is 53.6 MHz, the horizontal beam waist
+is 0.68 mm, the sagittal beam waist 0.56 mm
+and the intra-cavity ring-down time is 20 µs.
+In order to achieve long term stability of the
+perimeter the laser gyro optical frequency will
+be locked to a reference laser. This will involve
+the measurement of the radio frequency beat
+note between the gyrolaser output and the
+reference laser: for such application the capaci-
+tive coupling will introduce less noise than the
+inductive one. The two outputs (clockwise and
+counter-clockwise sense of circulation) from
+one cavity mirror are combined by means of a
+50% intensity beam splitter and detected by a
+photodiode. The photodiode current is voltage
+converted by a trans-impedance stage with a
+gain of109and 1 ms rise time. The two single
+beam outputs are also monitored by means of
+two fiber-coupled photomultipliers. The laser
+modal structure has been detected injecting the
+output beams in a high finesse linear cavity.
+The signals are acquired and analyzed off-line.
+4 EXPERIMENTAL RESULTS
+When the laser output power is not higher
+than few tens nW only one longitudinal mode
+(TEM00) in both sense of propagation is above
+the laser threshold. The beat note at the ex-
+pected Sagnac frequency (around 111.1 Hz) can
+be observed both in the interference between
+the two counter-rotating beams and in the
+single beam. Usually both beams lase in the
+same longitudinal mode, i.e., with the same
+mode number. This behavior is the standard
+operation for a ring laser gyro, and the beat
+note between the two counter-rotating waves
+is the Sagnac frequency at the expected value.
+Unfortunately, counter-propagating beams can
+lase on different longitudinal modes and one
+gets the so called split-mode regime [11]. Their
+separation is 53.6 MHz, the free spectral range4
+of the gyro laser. Sagnac signal is lost with split
+modes. The Sagnac signal in the beat note be-
+tween the two split counter-propagating modes
+still exists, of course, but is shifted around
+53.6 MHz. The first cause of split in G-Pisa
+is mode hopping due to thermal expansion
+of the cavity. In the not controlled conditions
+of our laboratory, the typical time between
+one mode jump and the other could range
+between 5 min to 15 min. Split modes persist
+usually for some minutes, until both beams
+return on the same longitudinal mode and then
+the standard operation and Sagnac signal are
+recovered. When the gyrolaser operates close
+to threshold, as stated before, just one optical
+mode for each direction propagates inside the
+cavity, if the power increases slightly we obtain
+multimode operation, at higher power many
+different longitudinal modes are excited in both
+directions and the behavior of the gyroscope
+becomes unstable. Several measurements were
+done to learn about the behavior of the ring
+laser as a function of power and to investigate
+multimodal operation of the cavity. During the
+standard operation, G-Pisa gyro shows a stable
+Sagnac signal, even with three or four modes in
+each direction. This evidence strongly suggests
+that modes are apparently mode-locked, or,
+there is a fixed phase relation between different
+modes, and this fixed phase is the same in both
+directions. Contrast of the Sagnac signal seems
+uncorrelated to intensity of the laser beam or
+number of modes. Figure 3 shows an example,
+with number of modes ranging between one
+to four (fluctuations are probably due to hys-
+teresis effects and are not strictly reproducible).
+From 1 up to 1.1 normalized voltage it has been
+checked that the laser operates in monomode.
+We noted that the gyro sometimes does not
+mode lock. This happens especially after power
+changes and for particular distributions of the
+cavity modes under the gain curve. These last
+aspects deserve further investigation in the
+next future.
+The rotational sensitivity is obtained by re-
+constructing the phase of the beat note, and
+differentiating it, in order to obtain the angular
+velocity. The mean value of this signal gives the
+Earth rotational speed, which is usually sub-
+tracted. The power spectrum of the signal gives
+Fig. 3. Contrast of the Sagnac fringes in func-
+tion of the discharge root mean square voltage
+normalized tothe threshold value.
+10−210−110010110210−1010−910−810−710−610−510−4
+HzRad/sec/sqrt(Hz)
+  
+Fig. 4. Comparison between the best low-
+frequency measurements. Gray spectrum and
+black spectrum refer respectively to the case
+of rigid and passively air-damped optical table.
+The horizontal line is the shot noise, the trans-
+verse lines represent the minimal and typical
+request for Virgo.
+the upper limit to the low frequency sensitivity,
+which is the relevant parameter for the pos-
+sible improvements of the gravitational waves
+interferometer suspension. A typical spectrum
+is reported in fig. 4. The horizontal line is the
+shot noise limit, the two transverse lines are the
+minimal and typical requirements for the Virgo
+suspensions [12].
+Fig. 5 shows the behavior of the intensity
+of clock wise and anti-clock wise beams, mea-5
+Fig.5. Clockwiseandcounter-clockwisebeams
+in function of the voltage applied to the dis-
+charge
+sured increasing (up) and decreasing (down)
+the voltage of the discharge.
+It is possible to see that the two modes have
+different threshold and different slopes. This
+put in evidence a slight intrinsic asymmetry in
+the laser. By means of a simultaneous measure-
+ment of rotational noise, performed with the G-
+Pisa gyrolaser, and translational noise obtained
+by a three axial linear accelerometer, it was
+possible to set an upper limit to the correlation
+between detected rotations and translations to
+a level of 0.5 %. Details of such analysis can
+be found in [7]. Hydrogen contamination is
+a matter of concern in our apparatus, since
+light absorption from hydrogen can prevent
+laser action. Hydrogen is released from the
+steel vacuum tubes and from the pyrex close to
+the discharge. Fig. 6 shows the concentration
+of Hydrogen as a function of time. G-Pisa
+can now run for only three weeks before the
+gas mixture must be changed. This level of
+contamination will be reduced by introducing
+some getters inside the chamber and by baking
+the vacuum chamber before each gas refilling.
+5 CONCLUSIONS
+Our device can run continuously, interruptions
+are due to thermal expansion induced mode
+jumps (split mode regime), which makes the
+instrument blind for few minutes. This prob-
+lem will be resolved by stabilizing the perime-
+Fig.6. Hydrogencontaminationasafunctionof
+time
+ter of the ring by means of piezoelectric ac-
+tuated mirror. We characterized the different
+regimes of operation of the gyroscope and
+proved the possibility of increased power, mul-
+timode operations. The low frequency mea-
+sured power spectrum (below 1 Hz) is around
+10−8(rad/s)/√
+Hz, which looks like to be real
+motion. The system sensitivity is close to the
+limit needed to control the tilt of the suspen-
+sions of Virgo.
+REFERENCES
+[1] G. E. Stedman, “Ring-laser tests of fundamental physics
+and geophysics,” Rep. Prog. Phys. , vol. 60, pages 615–688,
+1997.
+[2] K. U. Schreiber, T. Kl ¨ ugel and G. E. Stedman,“ Earth tide
+and tilt detection by a ring laser gyroscope,” J. Geophys.
+Res. Solid Earth , vol. 108, 2003.
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+G. E. Stedman and D. Wiltshire, “Direct measurement of
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+Poulton and G.E. Stedman, “Design and operation of a
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+[6] F. Aronowitz, in “Laser applications“, vol. 1 M. Ross ed. ,
+Academic Press, New York 1971.
+[7] A. Di Virgilio et al., “The G-Pisa gyrolaser after 1 year o f
+operation and considerations about its use to improve the
+Virgo IP control”, Virgo note (to be published).
+[8] I. Fiori, S. Braccini, F. Travasso and A. Vicer´ e “Siesta
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+PIS-1390-285, 2004.6
+[9] K.U. Schreiber, A Velikoseltsev, Kl ¨ ugel T., G.E. Stedm an
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+[10] K. U. Schreiber, T. Kl ¨ ugel, G. E. Stedman, and W. Schl ¨ u ter,
+“Stabilitaetsbetrachtungen f ¨ ur grossereinglaser,” DGK Mit-
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+Phys. ,vol. 23, pages 557–565, 2005.
\ No newline at end of file
diff --git a/1001.0477.txt b/1001.0477.txt
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+arXiv:1001.0477v1  [cond-mat.str-el]  4 Jan 2010Inequivalent routes across the Mott transition in V 2O3explored by X-ray absorption
+F. Rodolakis,1,2P. Hansmann,3,4J.-P. Rueff,2,5A. Toschi,3M.W. Haverkort,4
+G. Sangiovanni,3A. Tanaka,6T. Saha-Dasgupta,7O.K. Andersen,4K. Held,3M. Sikora,8
+I. Alliot,8,9J.-P. Iti´ e,2F. Baudelet,2P. Wzietek,1P. Metcalf,10and M. Marsi1
+1Laboratoire de Physique des Solides, CNRS-UMR 8502, Univer sit´ e Paris-Sud, F-91405 Orsay, France
+2Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48, 91192 Gif-sur-Yvette Cedex, France
+3Institut for Solid State Physics, Vienna University of Tech nology, 1040 Vienna, Austria
+4Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany
+5Laboratoire de Chimie Physique–Mati` ere et Rayonnement,
+CNRS-UMR 7614, Universit´ e Pierre et Marie Curie, F-75005 P aris, France
+6Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima 739-8530, Japan
+7S.N.Bose Center for Basic Sciences, Salt Lake, Kolkata, Ind ia
+8ESRF, 6 rue Jules Horowitz, BP 220, 38043 Grenoble Cedex, Fra nce
+9CEA/DSM/INAC/NRS 17 avenue des Martyrs 38000 Grenoble
+10Department of Chemistry, Purdue University, West Lafayett e, Indiana 47907, USA
+(Dated: November 2, 2018)
+The changes in the electronic structure of V 2O3across the metal-insulator transition induced by
+temperature, doping and pressure are identified using high r esolution x-ray absorption spectroscopy
+at the V pre K-edge. Contrary to what has been taken for granted so far, the metallic phase reached
+under pressure is shown to differ from the one obtained by chan ging doping or temperature. Using
+a novel computational scheme, we relate this effect to the rol e and occupancy of the a1gorbitals.
+This finding unveils the inequivalence of different routes ac ross the Mott transition in V 2O3.
+PACS numbers: 78.70.En; 71.27.+a; 71.20.Eh
+Some materials present metal-insulator transitions
+(MIT) without any changes in crystal structure or long-
+range magnetic order. These phenomena, known as
+Mott-Hubbard transitions, constitute a fundamental sig-
+nature of strong electronic correlations. The physics
+emerging in the vicinity of these transitions is highly
+non-trivial and the properties of such materials depend
+on small changes in the electronic structure induced by
+external parameters [1, 2]. Several features of the MIT
+have been successfully clarified by resorting to realistic
+many-body calculations[3]. Yet, contraryto common as-
+sumptions, a growing number of experimental facts are
+revealing that this physical process is also strongly de-
+pendent on the route followed through the MIT. Here
+we show this to be the case for vanadium sesquioxide
+(V2O3). The isostructural MIT in Cr-doped V 2O3is
+considered as the textbook example of a Mott transition,
+which occurs between a paramagnetic insulator (PI) and
+a paramagnetic metallic (PM) phase by changing dop-
+ing level ( x), temperature (T) or pressure (P) [4]. As
+such V 2O3has served as a test bed for many theoretical
+models [5–11] and a sustained experimental effort.
+Among the different experimental methods recently
+employed to study the electronic properties of the Mott
+transition in Cr-doped V 2O3[12–15], X-ray absorption
+spectroscopy (XAS) has played a crucial role. It was the
+detailed investigation of the V L2,3absorption edges [16]
+that demonstrated the necessity of abandoning the sim-
+ple one band, S= 1/2, model to obtain a realistic de-
+scription of the changes in the electronic structure at the
+phase transition. XAS can also be performed at the VK-edge in the hard x-ray range: in this case, the pre-
+edge will carry most of the physical information we are
+interested in, as it is predominantly due to 1 s→3dtran-
+sitions. The excitations in this spectral region are influ-
+enced by the core hole and should be considered to be of
+an excitonic nature. The advantages over the L2,3edge
+studies are(i) a more straightforwardinterpretation (due
+to the simple symmetry of the s-core hole, the multiplet
+structure reveals a direct view on the d-states) and (ii)
+the possibility of applying external pressure as diamond
+anvil cells are compatible with hard X-rays.
+WeusedV K-edgeXAStoexploreextensivelytheMIT
+in V2O3by changing temperature, doping and apply-
+ing an external pressure. The pre K-edges were ana-
+lyzed by a novel computational scheme combining local
+density approximation and dynamical mean field theory
+(LDA+DMFT) [17] with configuration interaction (CI)
+full multiplet ligand field calculations to interpret subtle
+differences at the PM-PI transition. This allowed us to:
+(i) observe in detail the changes in the electronic exci-
+tations, providing also a direct estimate of the Hund’s
+coupling J; (ii) analyze the physical properties of the PI
+and PM phase on both sides of the MIT, leading to the
+main result of our work: (iii) understand the difference
+between P, T or doping-induced transitions. This dif-
+ference is mainly related to the occupancy of the a1gor-
+bitals, suggesting the existence of a new “pressure”path-
+way between PI and PM in the phase diagram.
+The experiments were performed on the inelastic x-ray
+scattering (IXS) beamlines ID-26 and BM-30 at the Eu-
+ropean Synchrotron Radiation Facility (ESRF). We used2
+high quality samples of (V 1−xCrx)2O3with various dop-
+ing in the PM ( x= 0) and PI phases ( x= 0.011, and
+0.028) at ambient conditions. The MIT was also crossed
+for the 0.011 doping by changing temperature and for
+the 0.028 doping by pressure. To obtain the best reso-
+lution, the XAS spectra were acquired in the so-called
+partial fluorescence yield (PFY) mode [18], monitoring
+the intensity of the V-K α(2p→1s) line as the incident
+energy is swept across the absorption edge. Compared to
+standard XAS, the PFY mode provides better resolved
+absorption spectra as these are partly free from core-hole
+lifetime broadening effect. The gain is particularly ap-
+preciable at the transition metal Kpre-edges as the 1 s
+core-hole is extremely short lived with respect to the 2 p
+core-hole and the dipolar tail is strongly reduced. The
+V-Kαline was measured with the help of a IXS spec-
+trometer equipped with a Ge(331) spherically bent ana-
+lyzer. For the temperature dependence, the crystalswere
+mounted in a cryostat installed on the spectrometer. On
+the other hand, measuring the PFY mode turned out to
+be difficult under pressure because of the weak fluores-
+cencesignalinthepressurecellandweoptedforstandard
+XAS measurements in transmission mode. To maximize
+the throughput, a powder sample ( x= 0.028) was loaded
+in a diamond anvil cell equipped with composite anvils
+made of a perforated diamond capped with a 500- µm
+thin diamond anvil. Pressure was measured in-situ by
+standard ruby fluorescence technique. We used silicon
+oil as a pressure transmitting medium.
+The T-dependent absorption spectra are displayed in
+Fig. 1(a) for both PM (200 K) and PI (300 K) phases
+for thex= 0.011, powder sample. The spectra have
+been normalized to an edge jump of unity. We will focus
+on the pre-edge region, where information about the V d-
+states can be easierextracted. It can be decomposed into
+threespectralfeatures(A,B,C) whichallvaryin intensity
+as the system is driven through the MIT whereas only
+C is considerably broadened (Fig. 1(a), inset). Notice
+that no feature is observed below peak A contrary to the
+early results of Ref. [19] but in agreement with the more
+recent data of Ref. [20]. Within a simplified atomic like
+picture, one could directly relate the intensity of features
+A,B and C to the unoccupied states. Starting from a V-
+t2
+2g,S= 1 configuration, one can either add an electron
+to thet2gsubshell (split into one a 1gand two eπ
+gstates
+under trigonal distortion of the V sites [21] as shown in
+Fig. 1(d)) yielding peaks A and B, or add an electron
+to theeσ
+gsub-shell which gives rise to the broader peak
+C. In this picture, one could associate peak A with a
+spin quartet ( S= 3/2) and peak B with a spin doublet
+(S= 1/2), split by the Hund’s rule exchange.
+This picture is however oversimplified as the V delec-
+trons are strongly correlated and the spectra are still
+largely influenced by the 1 score hole potential. Keeping
+that in mind, we have simulated the pre-edge by combin-
+ing CI with LDA+DMFT calculations for which the one5470 5465/s65/s66/s67
+5520 5500 5480 5460 PM
+ PI/s40/s97/s41
+(PM-PI) x 4
+5469 5465/s40/s99/s41
+/s67/s65/s76/s67/s46/s65/s66
+5469 5465/s40/s98/s41
+/s69/s88/s80/s46(T)/s65/s66
+Photon Energy (eV)Intensity (arb. units) Intensity (arb. units)
+cubic
+fieldtrigonal
+field/s40/s100/s41
+FIG. 1: (Color online) (a) V K-edge x-ray absorption spec-
+tra in (V 1−xCrx)2O3,x= 0.011 powder sample measured
+as a function of temperature (T) in the PM (200 K, solid
+line) and PI (300 K, dotted line) phases by PFY XAS; below,
+PM−PI spectral difference; (inset) pre-edge region. (c) Cal-
+culated isotropic CI V K-edge PM and PI spectra of V 2O3.
+The spectra are compared to the T-dependent experimental
+data shown in the same energy window (b). (d) From LDA
+(starting point for our LDA+DMFT and CI calculations) we
+obtain that the cubic part of the ligand field splits the V d-
+levelsint2gandegby10Dq∼2eV,andthatthesmall trigonal
+distortion further separates the t2gintoa1gandeπ
+gby ∆trig
+∼0.3 eV.
+particle part (LDA) input corresponds to the level dia-
+gram in Fig. 1(d). We concentrate our analysis to peaks
+A and B, since peak C relates mainly to the unoccupied
+eσ
+gorbitals. These hybridize much stronger with the oxy-
+gen ligands and thus lack direct information on the Mott
+transition; peak C may also be related to non-local exci-
+tations (not included here) [22] which sensitively depend
+on the metal-ligand distance. Let us also note that the V
+sites in V 2O3are non centrosymmetric which leads to an
+on-site mixing of V-3 dand V-4p-orbitalsand interference
+between dipole and quadrupole transitions [7] which has
+been included in our scheme.
+Our CI calculations confirm that for the ground state
+the occupancy ratio between the ( eπ
+g,a1g) and (eπ
+g,eπ
+g)
+states is smaller in the PI than in the PM phase [8, 16]:
+The isotropic CI-based calculated XAS spectra in the
+pre-edge region reported in Fig. 1(c) agree well with the
+experimental data for both the energy splitting of fea-
+tures Aand B and the ratiooftheir spectral weight(SW)
+which increases in the PM phase.3
+Considerable insight can be gained by comparing CI
+and LDA+DMFT calculations. Our LDA+DMFT cal-
+culations, performed using Nth-order muffin-tin orbital
+(NMTO) downfolded Hamiltonianfor the 1.1%Cr-doped
+V2O3and Hirsch-Fye Quantum Monte Carlo as impurity
+solver, confirm the above mentioned tendency (we obtain
+a mixing of 50:50 and 35:65 for the ( eπ
+g,a1g):(eπ
+g,eπ
+g) oc-
+cupation in the PM and PI phases). For LDA+DMFT
+we have chosen the same interaction parameter U= 4.2
+eV for the PI phase as in Ref. [9] and followed their sug-
+gestion to slightly decrease its value in the PM phase (we
+assumeU= 4.0 eV)[23]. Remarkably the simple struc-
+tureofthecoreholepotentialin the K-edgespectroscopy
+allows us to associate the pre-edge spectrum with the
+k−integrated spectral function above the Fermi energy
+calculated by LDA+DMFT. The electron–addition part
+of the spectral function shows three main features in PM
+phase: a coherent excitation at the Fermi level and a
+double peak associated to the incoherent electronic ex-
+citations i.e. the upper Hubbard band (UHB), similarly
+to the undoped compound. In the PI phase obviously,
+only the latter survives. Comparison with the experi-
+mental spectra clearly shows that the pre-edge features
+havetobe relatedtothe “incoherent”partofthe spectral
+function only. The physical reason is that the core hole
+potential localizes the electrons destroying the coher-
+ent quasiparticle excitations and making the XAS spec-
+trum atomic-like. All the “incoherent” LDA+DMFT,
+CI, and experimental spectra shown in Fig. 2 agree in
+many aspects, especially as for the splitting of the first
+two peaks by ≈2.0 eV (≈1.8 eV in experiment) which
+originates in LDA+DMFT from the Hund’s exchange J
+in the Kanamori Hamiltonian. This further validates
+the choice of J= 0.7 eV used in our calculations con-
+trasting with larger values assumed in previous stud-
+ies [8, 10, 23], and also clarifies the mismatch between
+XAS and LDA+DMFT spectra reported in the undoped
+V2O3compound [8]. Moreover, the ratio between A and
+B peak displays the same trend in the PM-PI transition
+as the CI (or experimental) data. The quantitative dif-
+ference between the two calculations is attributed to the
+lack of matrix elements in LDA+DMFT.
+The intensity ratio of the first two incoherent excita-
+tions peaks A and B (associated to the quartet and dou-
+blet states in the oversimplified picture) thus appears as
+the key spectral parameter to understand the differences
+between PM and PI. Even in a powder sample, this ratio
+is still sensitiveto the a1gorbital occupationof the initial
+state. Indeed, due to the trigonal distortion a consider-
+able spectral weight transfer from the peak B to higher
+energies (corresponding to final states with two a1gelec-
+trons in the limit of large ∆ trig) can take place for the
+(eπ
+g,a1g) but not for the ( eπ
+g,eπ
+g) initial state. Therefore,
+theKpre-edge XAS can serve as a direct probe of the
+a1gorbital occupation in the ground-state. As a rule of
+thumb, the larger the ratio between the SW of A and B,420/s80/s77  DMFT DOS 1g
+ CI (total)
+/s66/s65
+420/s80/s73  CI (egeg)
+ CI (ega1g)
+/s65
+/s66
+E-EF (eV) E-EF (eV)Intensity (arb. units)
+FIG. 2: (Color online) Incoherent LDA+DMFT and CI cal-
+culations in the pre-edge region; EFis the Fermi energy. Note
+the similarity in the main spectral features when crossing
+the MIT. Also shown are the different contributions of the
+CI spectrum labeled accordingly to their initial state: the
+contribution of the ( eπ
+g,a1g)→(eπ
+g,eπ
+g,a1g) transitions to the
+peak A(B) is approximately 60%(55%) in the PM phase and
+20%(15%) in the PI phase.
+the larger the a1gorbital occupation.
+Hard X-ray absorption also provides a unique way to
+explore the “pressure” pathway across the MIT, from
+which relevant information can be extracted by applying
+ourinterpretationscheme. Fig.3(a) showsthe XAS pow-
+derspectra ofthe P-induced MIT with the corresponding
+spectra for the T- and doping-driven transition (cf. the
+loci in the phase diagram, Fig. 3). To ease the com-
+parison, the pressure dataset (P) obtained in transmis-
+sion mode has been deconvolved from the 1 sLorentzian
+lifetime broadening (1 eV FWHM) using the GNXAS
+code [24] to match the improved resolution of datasets
+(x) and (T) measured by PFY-XAS. Together with the
+PM−PI differences in Fig. 3(b), Fig. 3(a) clearly evi-
+dences that, besides a rigid shift of the first two peaks of
+∼+0.13eV and contraryto the doping- or T-driven tran-
+sition, very small changes in spectral shapes and weights
+are observed in the P-driven MIT. In the light of the
+arguments discussed above, our finding proves that the
+metallic state reached by applying pressure is character-
+ized by a much lower occupation of the a1gorbitals com-
+pared to the metallic state reached just by changing tem-
+perature or doping. Importantly, the spectra measured
+through the doping induced MIT are identical within
+the experimental uncertainty to those measured through
+the T-driven transition. The temperature-doping equiv-
+alence is confirmed by our photoemission data displayed
+in Fig. 3(c,d) [15] and is consistent with the very similar
+lattice parameter changes across the transition [25]. The
+xandTequivalence is also borne out by the observa-
+tion from XAS at the L-edges in doped V 2O3[16] that
+thea1goccupation within both the PM or PI phases is
+mostly independent of the doping level. Hence, the local
+incoherent excitations probed by XAS at the V Ledge
+orKpre-edge are not directly affected by disorder [26].4
+5470 5468 5466 5464(x)
+(T)(P)/s40/s97/s41
+ PI
+ PM(P)
+Photon Energy (eV) Binding Energy (eV)-2 0/s40/s99/s41
+(T)(x)400
+300
+200
+100 T (K)
+0.020.010.00
++Cr   (V1-xMx)2O3PMPI
+AFI10 1462P (kbar)
+-2 0/s40/s100/s41
+5470 5468 5466 5464/s40/s98/s41Intensity (arb. units)
+FIG. 3: (Color online) (a) V-K XAS spectra for powder sam-
+ples of(V 1−xCrx)2O3as afunctionofpressure( ×)[x= 0.028;
+5 and 7 kbar (lower curves); 5 and 11 kbar (upper curves)]
+(P), temperature ( △) [x= 0.011; 200,300 K] (T), and dop-
+ing (◦) [x= 0, 0.011] ( x) (cf. points in the phase diagram;
+the pressure scale refers to the x= 0.028 doping). The spec-
+tral differences PM −PI (×3) shown for the three datasets (b)
+demonstrate the nonequivalence between Pand temperature-
+doping. The x-Tequivalence is confirmed by the photoemis-
+sion spectra ( hν= 9 eV) (c) and their spectral differences (d)
+(from Ref. [15]).
+Our finding clearly shows the limits of the common as-
+sumption that temperature/doping- and pressure-driven
+MIT in V 2O3can be equivalently described within the
+same phase diagram [25],[27]. Indeed, the two different
+PM electronic structures that we observed reflect differ-
+ent mechanisms driving the MIT along different path-
+ways. In the doping-driven MIT, the metallic phase
+is characterized by an increased occupation of the a1g
+electrons indicating a reduced “effective crystal-field-
+splitting” as the main driving mechanism towards metal-
+licity [8, 9], related to the jump of the lattice parameter
+c/a(1.4%) at the MIT [25]. In contrast, when pressure is
+applied the a1goccupation remains basically unchanged,
+so that this metallic phase seems to originate rather from
+an increased bandwidth, without any relevant changes of
+the orbital splitting. The smaller c/ajump observed un-
+der pressure (0.7 %) corroborates our analysis.
+In conclusion, doping, temperature and pressure are
+showntoactdifferently ontheinterplaybetweenelectron
+correlations and crystal field, so that states previouslyconsidered to be equivalent metals are actually differ-
+ent. We believe this finding may apply to many other
+strongly correlated systems presenting metal-insulator
+transitions, with pressure opening inequivalent pathways
+through their phase diagrams.
+WethankE.Tosattiforstimulatingdiscussionsandac-
+knowledge a BQR of the Universit´ e Paris-Sud, the “Tri-
+angle de la Physique”, the Austrian Science Fund (FWF,
+science college W004), and the EU researchnetwork RP7
+MONAMI for financial support.
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+[28]) need to be reduced to J∼0.7eV to reproduce
+the constrained LDA energy splitting with the DMFT
+Hamiltonian [17]. Moreover, the value of U adopted for
+LDA+U (2 .8eV) has to be assumed smaller than that for
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\ No newline at end of file
diff --git a/1001.0478.txt b/1001.0478.txt
new file mode 100644
index 0000000000000000000000000000000000000000..313b8ec1adb20c5314e69f2ecaf3fe7692ebc744
--- /dev/null
+++ b/1001.0478.txt
@@ -0,0 +1,1533 @@
+arXiv:1001.0478v1  [math.CA]  4 Jan 2010ORTHOGONAL POLYNOMIALS ON SEVERAL
+INTERVALS:
+ACCUMULATION POINTS OF RECURRENCE
+COEFFICIENTS AND OF ZEROS
+F. PEHERSTORFER1
+Abstract. LetE=∪l
+j=1[a2j−1,a2j], a1< a2< ... < a 2l, l≥2 and set
+ω(∞) = (ω1(∞),...,ωl−1(∞)), where ωj(∞) is the harmonic measure
+of [a2j−1,a2j] at infinity. Let µbe a measure which is on Eabsolutely
+continuous and satisfies Szeg˝ o’s-condition and has at most a finite num-
+ber of point measures outside E, and denote by ( Pn) and (Qn) the or-
+thonormal polynomials and their associated Weyl solutions with respect
+todµ, satisfying the recurrence relation√λ2+ny1+n= (x−α1+n)yn−√λ1+ny−1+n. We show that the recurrence coefficients have topologi-
+cally the same convergence behavior as the sequence ( nω(∞))n∈Nmod-
+ulo 1; More precisely, putting ( αl−1
+1+n,λl−1
+2+n) = (α[l−1
+2]+1+n,..., α 1+n,...,
+α−[l−2
+2]+1+n,λ[l−2
+2]+2+n,...,λ2+n,...,λ−[l−1
+2]+2+n)weprovethat( αl−1
+1+nν,
+λl−1
+2+nν)ν∈Nconverges if and only if ( nνω(∞))ν∈Nconverges modulo 1
+and we give an explicit homeomorphism between the sets of acc umu-
+lation points of ( αl−1
+1+n,λl−1
+2+n) and (nω(∞)) modulo 1. As one of the
+consequences there is a homeomorphism from the so-called ga psXl−1
+j=1
+([a2j,a2j+1]+∪[a2j,a2j+1]−) on the Riemann surface y2=/producttext2l
+j=1(x−aj)
+into the set of accumulation points of the sequence ( αl−1
+1+n,λl−1
+2+n) if the
+harmonic measures ω1(∞),...,ωl−1(∞),1 are linearly independent over
+the rational numbers Q.Furthermore it is demonstrated, loosely speak-
+ing, that the convergence behavior of the sequence of recurr ence coef-
+ficients ( αl−1
+1+n,λl−1
+2+n) and of the sequence of zeros of the orthonormal
+polynomials andWeylsolutions outsidethespectrumistopo logically the
+same. Theaboveresults areprovedbyderivingfirstcorrespo ndingstate-
+ments for the accumulation points of the vector of moments of the diago-
+nal Green’s functions, that is, of the sequence (/integraltext
+xP2
+ndµ,...,/integraltext
+xl−1P2
+ndµ,√λ2+n/integraltext
+xP1+nPndµ, ...,√λ2+n/integraltext
+xl−1P1+nPndµ)n∈N.
+This work was supported by the Austrian Science Fund FWF, pro ject no. P20413-N18.
+1The last modifications and corrections of this manuscript we re done by the author in
+the two months preceding this passing away in November 2009. The manuscript is not
+published elsewhere (submitted by P. Yuditskii and I. Moale ).
+12 F. PEHERSTORFER
+1.Introduction
+Letl∈Nandak∈Rfork= 1,...,2l,a1<a2<...<a 2l.Put
+E=l/uniondisplay
+k=1Ek,whereEk= [a2k−1,a2k],andH(x) =2l/productdisplay
+j=1(x−aj)
+and henceforth let us choose that branch of√
+Hfor which/radicalbig
+H(x)>0 for
+x∈(a2l,∞).For convenience we set
+(1)h(x) :=
+
+(√
+H)+(x)−(√
+H)−(x)
+2i= (−1)l−k/radicalbig
+|H(x)|forx∈Ek
+0 elsewhere
+where, as usual, f±(x) denote the limiting values from the upper and lower
+half plane, respectively. Note that (√
+H)−(x) =−(√
+H)+(x) forx∈E.By
+φ(z,z0) we denote a so-called complex Green’s function for C\Euniquely
+determined up to a multiplication constant of absolute valu e one (chosen
+conveniently below), that is, φ(z,z0) is a multiple valued function which is
+analytic on C\Eup to a simple pole at z=z0,has no zeros on C\Eand
+satisfies|φ(z,z0)| →1 forz→x∈Eor in other words log |φ(z,z0)|is the
+(potential theoretic) Green’s function with pole at z=z0∈C\E, as usual
+denoted by g(z,z0).In the case under consideration, as it is known [17, 34],
+a complex Green’s function may be represented as
+(2) φ(z) :=φ(z,∞) = exp/parenleftigg/integraldisplayz
+a2lr∞(x)dx/radicalbig
+H(x)/parenrightigg
+wherer∞is the unique monic polynomial of degree l−1 such that
+(3)/integraldisplaya2j+1
+a2jr∞(x)dx/radicalbig
+H(x)= 0 forj= 0,...,l−1;
+hence
+(4)r∞(x) =l−1/productdisplay
+j=1(x−cj), cj∈(a2j,a2j+1),j= 1,...,l−1.
+Recall that the so-called capacity of Eis given by
+(5) cap(E) = lim
+z→∞/vextendsingle/vextendsingle/vextendsingle/vextendsinglez
+φ(z,∞)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+The density of the equilibrium distribution of E,denoted by ρ,becomes
+(6) ρ(x) =1
+2πi/parenleftigg
+(φ′
+φ)+(x)−(φ′
+φ)−(x)/parenrightigg
+=r∞(x)
+πh(x).
+Byω(z,B,C\E) we denote the harmonic measure of B⊆Ewith respect
+toC\Eatz,which is that harmonic and bounded function on C\ElwhichACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 3
+satisfies for ξ∈Elthat lim z→ξω(z,B,C\El) =iB(ξ),whereiBdenotes the
+characteristic function of B.For abbreviation we put
+ω(z,Ek;C\E) =ωk(z).
+Recall that, k= 1,...,l,
+ωk(∞) =/integraldisplaya2k
+a2k−1ρ(x)dx.
+In the following we say that the function wis onEfrom the Szeg˝ o-class,
+writtenw∈Sz(E),if
+(7)/integraldisplay
+Elog(w(x))ρ(x)dx>−∞.
+In this paper we consider positive measures of the form
+(8) dµ(x) =w(x)dx+m/summationdisplay
+j=1µjδdj(x)
+wherew∈Sz(E),the mass points dj, j= 1,...,m,are from R\Eand
+mutually disjoint and µj∈R+.ByPnwe denote the polynomial of degree
+northonormal with respect to wi.e.,/integraldisplay
+EPm(x)Pn(x)dµ(x) =δm,n
+and the monic orthogonal polynomial are denoted by pn(x).It is well known
+that the monic orthogonal polynomials satisfy a three term r ecurrence rela-
+tion
+(9) pn(x) = (x−αn)pn−1(x)−λnpn−2(x)
+withp−1(x) := 0 and p0(x) = 1.For measures of the form (8) it follows,
+see [20, Cor. 6.1] (in fact more general sets Eand measures are considered
+there), that the recurrence coefficients are almost periodic in the limit, more
+precisely, thatthereexistrealanalyticfunctions αandλonthetorus[0 ,1]l−1
+and a constant c∈Rl−1, depending on the measure µ, such that
+(10)αn+1=α(nω(∞)−c)+o(1) and λn+2=λ(nω(∞)−c)+o(1),
+whereω(∞) = (ω1(∞),...,ωl−1(∞)); for more details see the remark follow-
+ing Proposition 2.3 below. By (10) one gets a first, quite good view into the
+convergence behavior of the αn’s andλn’s, but by far not a complete one,
+since the functions αandλand their mapping properties are not known.
+The more it is not clear whether the recurrence coefficients an d (ω(∞))n∈N
+modulo 1 have the same topological convergence behavior; th at is, that
+subsequences converge simultaneously and that there is a ho meomorphism
+between the set of accumulation points. The goal of this pape r is to show
+that, under a proper point of view, there is such a topologica l equivalence;
+also between the recurrence coefficients and the zeros of the p olynomials
+and Weyl solutions outside the spectrum. As an immediate con sequence4 F. PEHERSTORFER
+it follows that the problem of convergence of the recurrence coefficients is
+topologically equivalent to the classical problem in numbe r theory to find
+diophanticapproximationsoftheharmonicmeasures ω(∞), [22, V.Kapitel],
+[7, 9].
+Instead of continuing to study the accumulation points of th e recurrence
+coefficients with the help of (10), which looks rather hopeles s, we may also
+investigate that ones of the sequences of moments of the diag onal Green’s
+functions,thatis,(/integraltext
+xP2
+n,/radicalbig
+λn+2/integraltext
+xPn+1Pn,/integraltext
+x2P2
+n,/radicalbig
+λn+2/integraltext
+x2Pn+1Pn,...,/integraltext
+xmP2
+n,/radicalbig
+λn+2/integraltext
+xmPn+1Pn,...)n∈Nsince it can be shown (see Proposition
+5.2) that there is a unique correspondence between such vect ors of moments
+andthevectorofrecurrencecoefficients( ...,α[m−1
+2]+1+n,λ[m−2
+2]+2+n,...,α1+n,
+λ2+n,..., λ−[m−1
+2]+2+n,α−[m−2
+2]+1+n, ...).In fact it even suffices to consider
+the truncated vector sequence (/integraltext
+xP2
+n,/radicalbig
+λn+2/integraltext
+xPn+1Pn, ...,/integraltext
+xl−1P2
+n,/radicalbig
+λn+2/integraltext
+xl−1Pn+1Pn)n∈Nsince, by Corollary 2.5, it carries all information
+already. We show (Theorem 3.2; concerning the recurrence co efficients see
+Theorem 5.3) that there is a homeomorphism between the set of accumu-
+lation points of the truncated vector sequence of moments (r espectively, of
+recurrence coefficients) and of the sequence ( nω(∞)) modulo 1 and that
+convergence holds for the same subsequences. In particular this implies that
+the set of accumulation points of the truncated vector seque nce of moments
+is homeomorph to a l−1 dimensional torus if the harmonic measures are
+linearly independent over Q, where the homeomorphism is given explicitly
+even.
+Also of special interest is the fact that there is a unique cor respondence
+between the accumulation points of the discussed truncated vector sequence
+of moments and the accumulation points of zeros of the orthog onal polyno-
+mials and Weyl solutions outside the spectrum, see Theorem 4 .3.
+Next let us represent the weight function win the form
+(11) w(x) =w0(x)
+ρ(x)
+and letWo(z) be that function which is analytic on ˜Ω :=C\E,has no zeros
+and poles there, is normalized by W(∞)>0 and satisfies the boundary
+condition
+(12) w0(x) = 1/W+
+0(x)W−
+0(x).
+Forµof the form (8) with w∈Sz(E) it has been shown that Pn/φnis
+uniformly asymptotically equivalent to the solution of a ce rtain extremalACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 5
+problem, more precisely, that on compact subsets of Ω := ˜Ω\{d1,...,dm}
+(13)
+Pn(z)
+φn(z)∼UW0(z)φ′(z)
+m/producttext
+j=1φ(z;dj)
+l−1/producttext
+j=1φ(z;cj)l−1/producttext
+j=1φ(z;xj,n)δj,nl−1/producttext
+j=1(z−xj,n)
+r∞(z)
+1/2
+whereUis a constant and the points xj,nand theδj,n∈ {±1}are uniquely
+determined by the conditions that for k= 1,...,l−1
+l−1/summationdisplay
+j=1δj,nωk(xj,n) =−(2n−l+1)ωk(∞)−2
+π/integraldisplay
+Elog/parenleftbiggw0(ξ)
+ρ(ξ)/parenrightbigg∂ωk(ξ)
+∂n+
+ξdξ
++2m/summationdisplay
+j=1ωk(dj) modulo 2 ,(14)
+recall thatωk(x) :=ω(x;Ek,˜Ω) is the harmonic measure of Ekwith respect
+to˜Ω at the point xandn+
+ξis the normal vector at ξpointing in the upper
+half plane. It turns out that xj,n∈[a2j,a2j+1] forj= 1,...,l−1 and thus
+(15) ˆ g(n)(x;w) := ˆg(n)(x) :=l−1/productdisplay
+j=1(x−xj,n)
+has exactly one zero in each gap [ a2j,a2j+1],j= 1,...,l−1.The asymptotic
+representation (13) is due to Widom [34, Theorem 6.2, p. 168 a nd pp. 175-
+176] if there appear no point measures, i.e., if m= 0.The case of point
+measures has been first studied in [23] for two intervals. Asy mptotics for
+more general sets (so-called homogeneous sets) and measure s, including the
+above ones, have been given by P. Yuditskii and the author in [ 20, 21].
+The so-called Weyl solutions or functions of the second kind
+(16)Qn(y) =/integraldisplay
+Epn(x)
+y−xdµ(x),respectively ,Qn(y) =/integraldisplay
+EPn(x)
+y−xdµ(x)
+build another basis system of solutions of the recurrence re lation (9). In [20,
+Section 3] also asymptotics for Weyl solutions have been der ived, more pre-
+cisely, using the terminology of this paper, the following u niform asymptotic
+equivalence on compact subsets of Ω has been shown
+(17)
+Qn(z)∼m/producttext
+j=1φ(z;dj)
+UW0(z)φn+1(z)/parenleftigg
+ˆg(n)(z)
+r∞(z)/producttextl−1
+j=1φ(z;cj)/producttextl−1
+j=1φ(z;xj,n)δj,n/parenrightigg1/2
+whereW0Qnis denoted by hnrespectively hin [20].
+Finally we mention that other questions about finite gap Jaco bi matrices
+are under investigations by J.S. Christiansen, B. Simon and M. Zinchenko6 F. PEHERSTORFER
+[3] and that a huge part of the forthcoming book by B. Simon [27 ], see
+also [26], is devoted to this topic. The approach is based on a utomorphic
+functions similarly as in [20, 21, 28] and is different from tha t one used here.
+2.Asymptotics for principal and secondary diagonal Green’s
+function
+The diagonal matrix elements of the resolvent ( J −z)−1are the so-called
+Green’s functions G(z,n,m) =/a\}bracketle{tδn,(J −z)−1δm/a\}bracketri}ht,where as usual Jdenotes
+the Jacobi matrix associated with ( αj) and (λj).More precisely using the
+orthogonality of Pn,respectively, Pn+1one obtains
+(18) Pn(z)Qn(z) =/integraldisplayP2
+n(x)
+z−xdµ(x) =:G(z,n,n)
+(19)Pn(z)Qn+1(z) =/integraldisplayPn+1(x)Pn(x)
+z−xdµ(x) =:G(z,n+1,n)
+and
+(20)Pn+1(z)Qn(z) =/integraldisplayPn+1(x)Pn(x)
+z−xdµ(x)+1/radicalbig
+λn+2=:G(z,n,n+1).
+Recall that in the case under consideration the λn’s are bounded away
+from zero. Using Schwarz’s inequality it follows that the th ree sequences
+(PnQn),(PnQn+1) and (Pn+1Qn) are boundedand analytic on compact sub-
+sets ofR\supp(µ),hence they are so-called normal families. Combining (13)
+and (17) we obtain with the help of (2) the following asymptot ics for the
+diagonal Green’s function.
+Corollary 2.1. Letµbe given by (8)withw∈Sz(E).Then uniformly on
+compact subsets of Ωthere holds
+(21) ( PnQn)(z) =ˆg(n)(z)/radicalbig
+H(z)+o(1).
+For weight functions of the form v(x)ρ(x)dxplus a possible finite number
+of mass points outside [ a1,a2l],wherevis positive and analytic on E,the
+limit relation (21) has been derived recently in [30] indepe ndently from the
+above asymptotics (13) and (17).
+Toderivetheasymptoticrepresentationsof G(z,n+1,n)andG(z,n,n+1)
+we will make use of Abel’s Theorem and the solvability and uni queness of
+the real Jacobi inversion problem. For this reason we have to write Widom’s
+condition (14) in terms of Abelian integrals, which we will d o similarly as
+in [2, 30], see also [1].
+LetRdenote the hyperelliptic Riemann surface of genus l−1 defined by
+y2=Hwith branch cuts [ a1,a2],[a2,a3],...,[a2l−1,a2l].Points on Rare
+written in the form zandzdenotes the projection of zonCalso written
+pr(z) =z.Furthermore we also use the notation pr( x) =xand pr(y) =y.z
+andz∗denote the points which lie above each other on R,i.e. pr(z) = pr(z∗).ACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 7
+The two sheets of Rare denoted by R+andR−.To indicate that zlies on
+the first respectively second sheet we write z+andz−. OnR+the branch
+of√
+His chosen for which/radicalbig
+H(x+)>0 forx+>a2l.
+Furthermore (see e.g. [15, 29]) let the cycles {αj,βj}l−1
+j=1be the usual
+canonical homology basis on R, i.e., the curve αjlies on the upper sheet R+
+ofRand encircles thereclockwise the interval Ejand the curve βjoriginates
+ata2jarrives ata2l−1along the upper sheet and turns back to a2jalong the
+lower sheet. Let {ϕ1,...,ϕ l−1}, whereϕj=/summationtextl−1
+s=1ej,szs√
+H(z)dz,ej,s∈C, be
+a base of the normalized Abelian differential of the first kind, i.e.,
+(22)/integraldisplay
+αjϕk= 2πiδjkand/integraldisplay
+βjϕk=Bjkforj,k= 1,...,l−1
+whereδjkdenotes the Kronecker symbol here. The integrals in (22) are the
+so-called periods. Note that the ej,n’s are real since√
+His purely imaginary
+onEjand since√
+His real on R\Ethe symmetric matrix of periods ( Bj,k)
+is real also.
+Abelian differentials of third kind are differentials with simp le poles at
+given points xandyonRwith residues +1 and −1,respectively, and nor-
+malized such that integrals along the ακ−cycles vanish for κ= 1,...,l−1.
+Representing the differential dlogφ(z,c),c∈R\E,as linear combinations
+ofnormalizedAbeliandifferentials offirstandthirdkind(re call thatφ(·,c+),
+wherec=c+,has a pole at c+and a zero at c−since the analytic extension
+of the complex Green’s function to the second sheet is given b yφ(z−,c+) =
+1/φ(z+,c+)) one obtains by integrating along the αjandβjcycles that
+(23)/integraldisplayc+
+c−ϕj=l−1/summationdisplay
+κ=1ωκ(c)Bjκ.
+Similarly (see [18] for details), since the analytic extens ion of the harmonic
+measure
+(24)wk=ωk+iω∗
+konR+andwk=−ωk+iω∗
+konR−
+is just another basis of Abelian differentials of first kind, ϕjcan be repre-
+sented as linear combination of the wk’s. Integrating along the βκcycle,
+κ∈ {1,...,l−1},and recalling the fact that the integral along a βκ-cycle is
+the difference of values along the ακcycle, which is Eκ,we obtain by (24)
+that/integraldisplay
+ϕj=−1
+2l−1/summationdisplay
+κ=1wκBjκ.
+Hence, using the fact that ωk(z) = 1 onEκ, κ∈ {1,...,l},and thus by the
+Cauchy-Riemann equations dwκ(z) =i∂ωκ
+∂nds,we get
+(25)2
+πiϕ+
+j=1
+πl−1/summationdisplay
+κ=1/parenleftbigg∂ωκ
+∂n+ds/parenrightbigg
+Bjκ8 F. PEHERSTORFER
+Notation 2.2.Let [a2j,a2j+1]±denote the two copies of [ a2j,a2j+1], j=
+1,...,l−1 inR±.Note that [ a2j,a2j+1]+∪[a2j,a2j+1]−is a closed loop
+onRand thus Xl−1
+j=1([a2j,a2j+1]+∪[a2j,a2j+1]−) is topologically a l−1
+dimensional torus.
+By (23) and (25) Widom’s condition (14) becomes,
+1
+2l−1/summationdisplay
+j=1/integraldisplayxj,n
+x∗
+j,nϕk=−(n−l−1
+2)/integraldisplay∞+
+∞−ϕk−i
+π/integraldisplay
+Eϕ+
+klogw
++m/summationdisplay
+j=1/integraldisplayd+
+j
+d−
+jϕkmod periods(26)
+where pr( xj,n) =xj,nandxj,n∈Rδj,n:=R±forδj,n=±1; recall that xj,n
+andx∗
+j,nlie above each other. Furthermore pr( d±
+j) =dj.
+Next we note that (26) can be considered as so-called Jacobi i nversion
+problem, that is, for given ( η1,...,ηl−1)∈JacR,where Jac Rdenotes the
+Jacobi variety of R,(that is the quotient space Cl−1\(2πi− →n+B− →m),B=
+(Bjk) the matrix of periods,− →n,− →m∈Zl−1) findz1,...,zl−1∈Rsuch that
+l−1/summationdisplay
+j=1/integraldisplayzj
+ejϕk=ηkmod periods ,
+wheree1,...,el−1are given points on R.In this paper the ηj’s are real al-
+ways, that is, we deal with the real Jacobi inversion problem . Denoting by
+JacR/R:=Rl−1/B− →mthe Jacobi variety restricted to reals, the following
+important uniqueness property of the real Jacobi inversion problem holds
+(see e.g. [10, 17]): The restricted Abel map
+A:Xl−1
+j=1([a2j,a2j+1]+∪[a2j,a2j+1]−)→JacR/R
+(z1,...,zl−1)/ma√sto→1
+2
+l−1/summationdisplay
+j=1/integraldisplayzj
+z∗
+jϕ1,...,l−1/summationdisplay
+j=1/integraldisplayzj
+z∗
+jϕl−1
+(27)
+is a holomorphic bijection.
+Now let us show that the solutions x1,n,...,xl−1,nof (26) can be described
+uniquely with the help of two polynomials which is crucial in what follows.
+Proposition 2.3. a) Letg(n),n∈N,be given by (15). There exists a monic
+polynomial ˆf(n+1)of degreelsuch that
+(28) ˆf2
+(n+1)(x)−H(x) =Lnˆg(n+1)(x)ˆg(n)(x)
+with
+(29)
+ˆf(n+1)(xj,n) =−δj,n/radicalig
+H(xj,n) andˆf(n+1)(xj,n+1) =−δj,n+1/radicalig
+H(xj,n+1)ACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 9
+wherexj,n,xj,n+1,δj,n,δj,n+1are given by (14)andLn∈Ris given below.
+Moreover ˆf(n+1)can be represented in the form
+(30)ˆf(n+1)(x) =
+x+l−1/summationdisplay
+j=1xj,n−c1
+ˆg(n)(x)−l−1/summationdisplay
+j=1δj,n/radicalbig
+H(xj,n)ˆg(n)(x)
+ˆg′
+(n)(xj,n)(x−xj,n),
+wherec1is given by H(x) =x2l−2c1x2l−1+....
+Furthermore
+(31)(ˆf(n+1)±√
+H)2(z)
+Lnˆg(n)(z)ˆg(n+1)(z)=
+φ2(z;∞)l−1/productdisplay
+j=1φ(z;xj,n+1)δj,n+1
+φ(z;xj,n)δj,n
+±1
+and
+Ln= 4(cap(E))2l−1/productdisplay
+j=1φ(xj,n;∞)δj,n
+φ(xj,n+1;∞)δj,n+1= 4λn+2(1+o(1)) (32)
+b) To each solution (x1,n,...,xl−1,n)of(26)there corresponds a unique
+pair of polynomials (ˆg(n),ˆf(1+n))given by (15)and by(30)with/radicalbig
+H(xj,n) =
+δj,n/radicalbig
+H(xj,n).
+Proof.a) Let us write xδj,n
+j,n:=x±
+nforδj,n=±1.Considering (26) for nand
+n+1 and substracting both equations we obtain that
+l−1/summationdisplay
+j=1/integraldisplayx−δj,n+1
+j,n+1
+xδj,n+1
+j,n+1ϕk= 2/integraldisplay∞+
+∞−ϕk+l−1/summationdisplay
+j=1/integraldisplayx−δj,n
+j,n
+xδj,n
+j,nϕk, k= 1,...,l−1.
+Thus it follows from Abel’s Theorem that there is a rational f unctionRnon
+the Riemann surface with the following properties
+∞±is a double pole (zero)
+x∓δj,n+1
+j,n+1is a simple zero (pole)
+x∓δj,n+1
+j,nis a simple pole (zero) .(33)
+ThusRncan be represented in the form
+(34) Rn=(f(n+1)+√
+H)2
+Lnˆg(n)ˆg(n+1),
+wheref(n+1)is a polynomial of degree landLnis a constant, since the
+function at the RHS in (34) satisfies (33) and has no other zero s or poles
+onRas can be checked easily. Since the points x±δj,n+1
+j,n+1andx±δj,n
+j,nare real
+the rational function Rn(z) has the same properties (33) as Rn(¯z), hence
+Rn(¯z) =Rn(z) and therefore f(n+1)has real coefficients and Ln∈R.Taking10 F. PEHERSTORFER
+involution, denoted by z∗,we obtain by (33)
+(35)1
+Rn(z)=Rn(z∗) =(f(n+1)−√
+H)2
+Lnˆg(n)ˆg(n+1),
+and thus, by multiplying (35) and (34), relation (28)-(29). Concerning (29)
+note that/radicalbig
+H(xδ) =δ/radicalbig
+H(x).
+Next let us note
+−(√
+H)−(x) = (√
+H)+(x) =i(−1)l−k/radicalbig
+|H(x)|forx∈Ek
+and therefore by (34) and (28)
+|R±(x)|= 1 forx∈E.
+Thus the function
+F(z) :=Rn(z)
+φ2(z;∞)l−1/productdisplay
+j=1φ(z;xj,n)δj,n
+φ(z;xj,n+1)δj,n+1
+has neither zeros nor poles on Ω and satisfies |F±|= 1 onE.Hence log |f(z)|
+is a harmonic bounded function on Ω which has a continuous ext ension to
+Eand thusF= 1,which proves (31).
+Considering (31) at z=∞the first relation in (32) follows. Next denote
+bylc(Pn) the leading coefficient of Pn.Obviouslyλn+2=/integraltext
+p2
+n+1//integraltext
+p2
+n=
+(lc(Pn)/lc(Pn+1))2.Using the fact that lc(Pn) = lim
+z→∞Pn(z)/znit follows by
+(13), in conjunction with (5), that the last equality in (32) holds.
+Relation (30) follows by (29), note that the second term at th e RHS of
+(30) is the unique Lagrange interpolation polynomial which takes on at xj,n
+the value −δj,n/radicalbig
+H(xj,n),and by equating the first two leading coefficients
+in (28).
+b) follows immediately by a). /square
+We note in passing that (32), which may be derived from (13) al so, is the
+so-called trace formula for λn+2and that the other trace formula −αn+1=/summationtextl−1
+j=1xj,n−c1+o(1) follows immediately by (18), (21) and by equating
+coefficients of xl−1in(30). By (26)and(27)the xj,n’smay beexpressedwith
+the help of the Abel map from which representation (10) can be obtained.
+For measures from G,Gdefined in Corollary 3.4, which are so-called
+reflectionless measures (note that the set of Jacobi matrice s associated with
+the set of measures Gis the so called isospectral torus), Proposition 2.3 is
+essentially known [13, 33, 16, 32] apart from the important r elation (31).
+But the way of derivation is exactly the opposite one. First o ne shows that
+certain polynomials, which are expressions in Pad´ e approx imants, satisfy
+(28) - (30) - to find the corresponding polynomials in the gene ral case seems
+to be hopeless, if they exist at all - and then one derives the c orrespondence
+with the solutions of (26).ACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 11
+Theorem 2.4. Letµbe given by (8)withw∈Sz(E).Then uniformly on
+compact subsets of Ωthere holds
+(36) 2/radicalbig
+λn+2PnQn+1=ˆf(n+1)−√
+H√
+H+o(1)
+(37) 2/radicalbig
+λn+2Pn+1Qn=ˆf(n+1)+√
+H√
+H+o(1)
+and on compact subsets of C\[a1,a2l]
+(38) 2/radicalbig
+λn+2Pn+1
+Pn=ˆf(n+1)+√
+H
+ˆg(n)+o(1)
+(39)/integraldisplay1
+z−xdµ(n+1)(x) =/radicalbig
+λn+2Qn+1(z)
+Qn(z)=ˆf(n+1)(z)−√
+H(z)
+2ˆg(n)(z)+o(1),
+whereµ(m+1)denotes the measure associated with the m+1forwards shifted
+recurrence coefficients (αn+m+1)n∈Nand(λn+m+2)n∈N.
+Proof.Concerning relation (36). Plugging in the asymptotic value s forPn
+andQn+1from (13) and (17) and recalling the fact that φ′/φ=r∞/√
+Hthe
+asymptotic relation follows immediately by (31) and (32).
+Analogously (37) is proved. (39) and (38) follow immediatel y by (36) and
+Corollary 2.1, respectively (37) and Corollary 2.1. /square
+Corollary 2.5. In a neighborhood of x= 0let
+/radicalbig
+H∗(x) =∞/summationdisplay
+j=0hjxj
+whereH∗(x) =x2lH(1
+x)is the reciprocal polynomial of H. Then the follow-
+ing limit relations hold for m≥l
+(40) lim
+n
+m/summationdisplay
+j=0hj/integraldisplay
+tm−jP2
+n
+= 0
+and form≥l+1
+(41) lim
+n
+hm+2m−1/summationdisplay
+j=0hj/radicalbig
+λ2+n/integraldisplay
+tm−1−jP1+nPn
+= 0
+Proof.By (21) and (18) it follows that in a neighborhood of x= 0
+(42)/radicalbig
+H∗(x)
+∞/summationdisplay
+j=0(/integraldisplay
+tjP2
+n)xj
+= ˆg∗
+(n)(x)+o(1)
+Equating coefficients for m≥lrelation (40) follows.12 F. PEHERSTORFER
+Concerning the second relation we get by (36) that
+(43)/radicalbig
+H∗(x)
+1+2∞/summationdisplay
+j=0(/radicalbig
+λ2+n/integraldisplay
+tjP1+nPn)xj+1
+=ˆf∗
+(1+n)(x)+o(1)
+which proves (41). /square
+Most likely the limit relations (40) and (41) hold for measur esσwhose
+essential support is E. For the subclass of measures µ∈ Git can be shown
+by a different approach, that (40) and (41) hold for m=landm=l+1,
+respectively, without limit even, see also [13, 33] where th e relations are
+derived in terms of recurrence coefficients for l= 2,3.
+3.Accumulation points of moments of the Green’s functions
+By series expansion of 1 //radicalbig
+H(z) atz=∞
+(44)∞/summationdisplay
+j=0cjz−(l+j)=1/radicalbig
+H(z)=1
+π/integraldisplay
+E1
+z−tdt
+h(t)
+forz∈C\E,wherethelastequalityfollowsbytheSochozki-Plemelj for mula.
+In particular
+(45)/integraldisplay
+Exjdx
+h(x)= 0 forj= 0,...,l−2 andc0=/integraldisplay
+Exl−1dx
+h(x)= 1.
+Hence the following lemma holds.
+Lemma 3.1. a) LetA1,...,Al−1∈Rbe given. There exists an unique
+polynomial P(x) =l−1/summationtext
+ν=0pνxνwithpl−1= 1such that
+(46)/integraldisplay
+ExjP(x)dx
+h(x)=Ajforj= 1,...,l−1.
+b) LetB1,...,Bl−1∈Rbe given. There exists an unique polynomial Q(x)of
+degreelwith two fixed leading coefficients such that
+(47)/integraldisplay
+ExjQ(x)dx
+h(x)=Bjforj= 1,...,l−1.
+Now we are ready to state our first main result.
+Theorem 3.2. Letµbe given by (8)and letw∈Sz(E).a) The subse-
+quence of solutions (x1,nν,...,xl−1,nν)ν∈Nof(26)converges if and only if
+(/integraltext
+xP2
+nνdµ,...,/integraltext
+xl−1P2
+nνdµ,/radicalbig
+λ2+nν/integraltext
+xP1+nνPnνdµ,...,/radicalbig
+λ2+nν/integraltext
+xl−1P1+nν
+Pnνdµ)ν∈Nconverges.
+Furthermore, the map τ,given by
+(y1,...,yl−1)/ma√sto→/parenleftbigg/integraldisplay
+ExG(x)dx
+h(x),...,/integraldisplay
+Exl−1G(x)dx
+h(x),
+/integraldisplay
+ExF(x)
+2dx
+h(x),...,/integraldisplay
+Exl−1F(x)
+2dx
+h(x)/parenrightbigg (48)ACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 13
+where,
+(49)
+G(x) :=l−1/productdisplay
+j=1(x−yj), F(x) :=/parenleftig
+x+/summationdisplay
+yj−c1/parenrightig
+G(x)−l−1/summationdisplay
+j=1δj/radicalbig
+H(yj)G(x)
+G′(yj)(x−yj),
+c1as in(30)andδj/radicalbig
+H(yj) =/radicalbig
+H(yj), is a homeomorphism between the
+set of accumulation points of the sequence of solutions (x1,n,...,xl−1,n)n∈Nof
+(26)and of the sequence (/integraltext
+xP2
+ndµ,...,/integraltext
+xl−1P2
+ndµ,/radicalbig
+λ2+n/integraltext
+x P1+nPndµ,
+...,/radicalbig
+λ2+n/integraltext
+xl−1P1+nPndµ)n∈N
+b) If1,ω1(∞),...,ωl−1(∞)are linearly independent over Q,thenτis a
+homeomorphism from Xl−1
+j=1([a2j,a2j+1]+∪[a2j,a2j+1]−)into the set of ac-
+cumulation points of (/integraltext
+xP2
+ndµ,...,/integraltext
+xl−1P2
+ndµ,/radicalbig
+λ2+n/integraltext
+xP1+nPndµ, ...,/radicalbig
+λ2+n/integraltext
+xl−1P1+nPndµ)n∈N.
+Proof.At the beginning let us observe that the map τis a continuous one to
+one map from Xl−1
+j=1([a2j,a2j+1]+∪[a2j,a2j+1]−) on its range. Indeed, recall
+first the obvious fact that there is a unique correspondence b etween a point
+y∈Xl−1
+j=1([a2j,a2j+1]+∪[a2j,a2j+1]−) and (y,δ/radicalbig
+H(y)) = (pr( y),/radicalbig
+H(y)),
+whereδ=±1 ify∈R±.Since the polynomial F(x)−(x+(/summationtextyj−c1))·
+G(x) is the unique Lagrange interpolation polynomial of degree l−2 which
+takes on at the yj’s the values −δj/radicalbig
+H(yj),it follows that there is a unique
+correspondencebetween the points ( y1,...,yl−1) and the polynomials ( G,F),
+defined in (49). By Lemma 3.1 the observation is proved.
+a) Necessity of the first statement. Let ( x1,nν,...,xl−1,nν)ν∈Nbea sequence
+of solutions of (26) with limit ( y1,...,yl−1),that is,j= 1,...,l−1,
+xj,nν−→
+ν→∞yjandδj,nν/radicalig
+H(xj,nν)−→
+ν→∞δj/radicalig
+H(yj).
+By Proposition 2.3 it follows that there are unique polynomi als ˆg(nν)and
+ˆf(1+nν)such that uniformly on Ω
+(50) ˆ g(nν)(x)−→
+ν→∞G(x) andˆf(1+nν)(x)−→
+ν→∞F(x),
+where for the second relation we took (30) into account and th atGandF
+are given by (49).
+Next it follows by Corollary 2.1 and (50) that uniformly on co mpact
+subsets of Ω
+lim
+νPnνQnν(z) = lim
+ν/integraldisplayP2
+nν(x)
+z−xdµ(x) =G(z)/radicalbig
+H(z)
+=1
+π/integraldisplay
+EG(x)
+z−xdx
+h(x)(51)14 F. PEHERSTORFER
+where the last equality follows by the Sochozki-Plemelj’s f ormula using the
+fact thatG∈Pl−1.Analogously we obtain by Theorem 2.4 that
+lim
+ν/radicalbig
+λ2+nνPnνQ1+nν(z) = lim
+ν/radicalbig
+λ2+nν/integraldisplayPnν(x)P1+nν(x)
+z−xdµ(x)
+=F(z)−/radicalbig
+H(z)
+2/radicalbig
+H(z)=1
+2π/integraldisplay
+EF(x)
+z−xdx
+h(x)(52)
+where the last equality follows by Sochozki-Plemelj again a nd the fact that
+−1+(F/√
+H)(z) =O(1
+z) asz→ ∞,sincef(1+nν)has this property by (28).
+Moreover the first l−1 coefficients of the series expansions in (51) and (52)
+converge which proves the necessity part of the first stateme nt of a).
+Furthermore, by (51) and (52) we have shown also, that
+lim
+ν/parenleftbigg/integraldisplay
+xP2
+nν,...,/integraldisplay
+xl−1P2
+nν,/integraldisplay
+xP1+nνPnν,...,/integraldisplay
+xl−1P1+nνPnν/parenrightbigg
+is in the range of the restricted map τ/whose domain is the set of accumu-
+lation points of the solutions of (26).
+Sufficiency of the first statement of a). Suppose that lim
+ν/parenleftbig/integraltext
+xP2
+nν,...,
+/integraltext
+xl−1P2
+nν,/radicalbig
+λ2+nν/integraltext
+xP1+nνPnν,...,/radicalbig
+λ2+nν/integraltext
+xl−1P1+nνPnν) exists. By
+(40) and (41) and induction arguments it follows that lim ν/integraltext
+xjP2
+nνand
+limν/radicalbig
+λ2+nν/integraltext
+xjP1+nνPnνexist for every j∈Nand thus lim ν/integraltextP2
+nν
+z−xand
+limν/radicalbig
+λ2+nν/integraltextP1+nνPnν
+z−xis uniformlyconvergent at aneighborhoodof z=∞
+and thus on compact subsets of C\supp(µ).Corollary 2.1 and Theorem
+2.4 imply that the relations (50)-(52) hold, where G,Fare by Lemma 3.1
+uniquelydeterminedbylim
+ν/parenleftbig/integraltext
+xP2
+nν,...,/integraltext
+xl−1P2
+nν,/radicalbig
+λ2+nν/integraltext
+xP1+nνPnν,...,
+/radicalbig
+λ2+nν/integraltext
+xl−1P1+nνPnν/parenrightbig
+.
+Now let us take a look at the sequence of solutions of (26) ( x1,nν,...,
+xl−1,nν)ν∈Nwhichcanbeconsideredasthesequence/parenleftbig
+(x1,nν,δ1,nν/radicalbig
+H(x1,nν)),
+...,(xl−1,nν,δl−1,nν/radicalbig
+H(xl−1,nν))/parenrightbig
+ν∈N,whereδj,nν=±1 ifxj,nν∈R±.Then
+it follows byProposition 2.3 if( x1,nν,...,xl−1,nν) has two different limit points
+then the uniquely associated sequence of polynomials (ˆ g(nν),ˆf(1+nν)) has
+two different limit points. But this contradicts (50). Hence t here exists
+limν(x1,nν,...,xl−1,nν) = (y1,...,yl−1) and the first statement of a) is proved.
+Furthermore, since (51) - (52) hold, ( y1,...,yl−1) is mapped by the contin-
+uousmapτ/tolim
+ν/parenleftbig/integraltext
+xP2
+nν,...,/integraltext
+xl−1P2
+nν,/radicalbig
+λ2+nν/integraltext
+xP1+nνPnν,...,/radicalbig
+λ2+nν/integraltext
+xl−1P1+nνPnν),that is,τ/is an onto map between the two sets of accu-
+mulation points and the second statement of a) is proved.
+b) We claim that the set of accumulation points of the sequenc e of so-
+lutions of (26) is the set Xl−1
+j=1([a2j,a2j+1]+∪[a2j,a2j+1]−).Obviously it
+suffices to show that for given y= (y1,...,yl−1)∈Xl−1
+j=1([a2j,a2j+1]+∪
+[a2j,a2j+1]−) there exists a sequence xnν:= (x1,nν,...,xl−1,nν)ν∈Nof (26)
+such that lim νxnν=y.Since 1,ω1(∞),...,ωl−1(∞) is linearly independent itACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 15
+follows by Kronecker’s Theorem [7, 9] that any sequence ( −nω(∞)+c)n∈N,
+wherec= (c1,...,cl−1) is an arbitrary constant, is, modulo 1 ,dense in
+[0,1]l−1. Thus by the equivalence of (14) and (26) ( −n/integraltext∞
+−∞ϕ+˜c)n∈Nis
+modulo periods Bjk,dense in Jac R/Rwhere the constant ˜c= (˜c1,...,˜cl−1)
+is given by that part of the RHS of (26) which does not depend on nand
+where we have put ϕ= (ϕ1,...,ϕl−1),. Hence there exists a subsequence
+(nν) of natural numbers such that
+(53)/parenleftbigg
+−nν/integraldisplay∞
+−∞ϕ+˜c/parenrightbigg
+ν∈N→ A(y) modulo periods Bjk,
+whereAis the Abel map from (27). On the other hand to the sequence
+(Pnν)ν∈Noforthonormalpolynomialsthereexistpoints xnν= (x1,nν,...,xl−1,nν)
+such that (26) holds, that is, that
+A(xnν) =−nν/integraldisplay+∞
+−∞ϕ+˜cmodulo periods Bjk,
+By (53) and the bijectivity of Athe assertion follows. /square
+Remark 3.3.If one wants to get rid of the Riemann-surface the home-
+omorphism from Theorem 3.2 a) may be written also as the map fr om
+A:={((y1,δ1),...,(yl−1,δl−1)) :yj∈[a2j,a2j+1], δj∈ {−1,1}ifyj∈
+(a2j,a2j+1) andδj= 0 ifyj∈ {a2j,a2j+1}forj= 1,...,l−1}into the set of
+accumulationpointsofthesequence(/integraltext
+xP2
+n,...,/integraltext
+xl−1P2
+n,/radicalbig
+λ2+n/integraltext
+xl−1P1+n
+Pn,...,/radicalbig
+λ2+n/integraltext
+xl−1P1+nPn)n∈N,where ((y1,δ1),...,(yl−1,δl−1))/ma√sto→(/integraltext
+x
+G(x)dx
+h(x),...,/integraltext
+xl−1G(x)dx
+h(x),/integraltextxF(x)
+2dx
+h(x),...,/integraltextxl−1F(x)
+2dx
+h(x)).
+Corollary 3.4. Suppose that the harmonic measures 1,ω1(∞),...,ωl−1(∞)
+are linearly independent over Q. Then the following statements hold:
+a) The set of limit points of the associated measures {µ(n)}n∈N,see(39),
+with respect to weak convergence is the set of measures
+G:={h(t)
+2π/producttextl−1
+j=1(t−yj)dt+l−1/summationdisplay
+j=11−δj
+2/radicalbig
+H(yj)
+d
+dt(/producttextl−1
+j=1(t−yj))t=yjδ(t−yj) :
+yj∈[a2j,a2j+1],δj∈ {±1}forj= 1,...,l−1}.
+b) Letsbe the map associated with coefficient stripping, i.e. s(µ) =µ(1).
+Thens(G)⊆ Gand for every µ∈ Gthe orbit under composition {sn(µ) :
+n∈N}is dense in Gwith respect to weak convergence.
+Proof.a) By (39) on compact subsets of C\[a1,a2l]
+/integraldisplay1
+z−tdµ(1+n)=/radicalbig
+λ2+nQ1+n(z)
+Qn(z)=ˆf(1+n)(z)−/radicalbig
+H(z)
+2ˆg(n)(z)+o(1)16 F. PEHERSTORFER
+IntheproofofTheorem3.2wehaveshownthat foreach/parenleftig
+(y1,δ1/radicalbig
+H(y1),...,
+(yl−1,δl−1δ1/radicalbig
+H(yl−1))/parenrightig
+there is a sequence ( nν) such that
+ˆg(nν)−→
+ν→∞Gandˆf(1+nν)−→
+ν→∞F,
+whereGandFare given by (49). Since
+ˆf(1+n)(z)−/radicalbig
+H(z)
+2ˆg(n)(z)=1
+2π/integraldisplay
+E1
+z−th(t)
+ˆg(n)(t)dt
++/summationdisplay(1−δj,n)
+2/radicalbig
+H(xj,n)
+ˆg′
+(n)(xj,n)δ(z−xj,n)
+the assertion follows.
+b) The invariance with respect to coefficient stripping has be en proved in
+[16, Theorem 5]. Applying part a) to µ∈ Gthe assertion follows. /square
+The set of Jacobi matrices associated with the set of measure sGis called
+the isospectral torus nowadays.
+4.Accumulation points of zeros outside the support
+It is well known that polynomials orthogonal with respect to a measure
+σmay have zeros in the convex hull of supp( σ),that is, in the case under
+consideration in the gaps [ a2j,a2j+1],j∈ {1,...,l−1}.The same holds for
+the Weyl solutions Qn.
+Notation 4.1.Let (nj) be a strictly monotone increasing subsequence of the
+natural numbers and let ( fnj) be a sequence of functions. We say that a
+pointyis an accumulation point of zeros of ( fnj) if there exists a sequence
+of points (yj) such that fnj(yj) = 0 and lim
+jyj=y.As usual,yis called a
+limit point of zeros of ( yj) ifUε(y)\{y},ε>0, contains an infinite number
+ofyj’s.
+Lemma 4.2. Letσbe a positive measure with supp(σ)bounded and suppose
+that0<const≤λnforn≥n0.Then the following pairs of sequences have
+no common accumulation point of zeros on R\supp(σ) : (Pnj)and(P1+nj),
+(Qnj)and(Q1+nj),and(Pnj)and(Qnj).
+Proof.Since the three sequences ( PnQn),(PnQn+1) and (Pn+1Qn) are nor-
+mal families on R\supp(µ),see (18)-(20), they are equicontinuous. Thus the
+assertion follows by the well known relation
+PnQn+1−QnPn+1=−1. /square
+By the way that ( Pnj) and (P1+nj) have no common accumulation point
+ify /∈supp(σ) follows also from [5]. Thus y,y /∈suppσ,is an accumulation
+point of zeros of ( Pnν) ((Qnν)) if and only if yis a common accumulationACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 17
+point of zeros of ( PnνQnν) and (PnνQ1+nν) (of (PnνQnν) and (P1+nνQnν)).
+In the following it will be convenient to use this way of expre ssion.
+Theorem 4.3. Letµbe given by (8)withw∈Sz(E).Then
+lim
+ν/parenleftbigg/integraldisplay
+xP2
+nνdµ,...,/integraldisplay
+xl−1P2
+nνdµ,/radicalbig
+λ2+nν/integraldisplay
+xP1+nνPnνdµ,...,
+/radicalbig
+λ2+nν/integraldisplay
+xl−1P1+nνPnνdµ/parenrightbigg
+=/parenleftbigg/integraldisplay
+ExG(x)dx
+h(x),...,/integraldisplay
+Exl−1G(x)dx
+h(x),/integraldisplay
+ExF(x)
+2dx
+h(x),...,/integraldisplay
+Exl−1F(x)
+2dx
+h(x)/parenrightbigg
+,(54)
+whereGandFare defined in (49),((y1,δ1),...,(yl−1,δl−1))∈Xl−1
+j=1
+(((a2j,a2j+1)\supp(µ))×({±1})),if and only if for j= 1,...,l−1the point
+yj, yj∈(a2j,a2j+1)\supp(µ),is a common accumulation point of zeros of
+(PnνQnν)and(P(1+δj)/2+nνQ(1−δj)/2+nν), δj∈ {±1}.
+Proof.Necessity. In the proof of the sufficiency part of Theorem 3.2a ) we
+have shown that relation (54) implies with the help of Coroll ary 2.5 that the
+relations (50) hold. Moreover by (30) the δj,nν’s from (29) satisfy δj,nν−→
+ν→∞
+δj, j= 1,...,l−1,whereδj∈ {−1,1},since every zero of Glies in the
+interior of the gaps. Taking a look at the three limit relatio ns (21), (36)
+and (37) in conjunction with (29) the assertion is proved usi ng Hurwitz’s
+Theorem [8] about the zeros of uniform convergent sequences of analytic
+functions.
+Sufficiency. First let us note that for any subsequence ( nκ) of (nν) for
+whichthelimitin(21), (36)and(37)existsthelimitfuncti onslim κˆg(nκ)=G
+and lim κˆf(1+nκ)=Fare monic polynomials of degree l−1 andl, respec-
+tively, which satisfy, using the assumption and Hurwitz The orem,G(yj) = 0
+andF(yj) =δj/radicalbig
+H(yj) forj= 1,...,l−1. ThusGandF, recall (30), are
+unique; in other wordsthelimit of all threesequences ( PnνQnν), (PnνQ1+nν)
+and(P1+nνQnν))existsuniformlyonΩandisgivenby GandFwhichproves
+by (18)-(20), taking into consideration the last relation f rom (51) and (52)
+respectively, the sufficiency part. /square
+CombiningTheorem 4.3 andTheorem 3.2 b) weobtain immediate ly (for a
+wider class of measures) a result of the author [17, Theorem 3 .9] concerning
+thedensenessof zerosof ( Pn) inthegaps. For absolutely continuous, smooth
+measures the existence of a sequence ( nν) such that ( Pnν) has no zeros in
+the gaps was shown first in [30].
+5.Consequences for the recurrence coefficients
+First let us demonstrate how to express Theorem 4.3 in terms o f limit
+relations of the recurrence coefficients and of the accumulat ion points of
+zeros of (Pn) and (Qn).It is well known that/integraltext
+xjP2
+nand/radicalbig
+λn+2/integraltext
+xjPn+118 F. PEHERSTORFER
+Pn,j∈N,can be expressed in terms of the recurrence coefficients using the
+recurrence relation of the Pn’s, e.g.
+/integraldisplay
+xP2
+n=αn+1,/integraldisplay
+x2P2
+n=λn+2+α2
+n+1+λn+1,...
+and
+/radicalbig
+λn+2/integraldisplay
+xPn+1Pn=λn+2,/radicalbig
+λn+2/integraldisplay
+x2Pn+1Pn=λn+2(αn+2+αn+1),...
+for a closed formula see [14]. Solving the system of equation s (recall Lemma
+3.1)
+(55)/integraldisplay
+ExjG(x)dx
+h(x)= lim
+ν/integraldisplay
+xjP2
+nνj= 0,...,l−1,
+respectively,
+(56)/integraldisplay
+ExjF(x)dx
+h(x)= lim
+ν2/radicalbig
+λ2+nν/integraldisplay
+xjP1+nνPnνj= 0,...,l−1
+with the help of (44) and (45) we obtain explicit expressions for the coeffi-
+cients ofG(x),respectively, F(x) in terms of the coefficients cjof the series
+expansion of 1 //radicalbig
+H(z),see (44), and of the accumulation points of the re-
+currence coefficients. This enables us to write condition (54 ) of Theorem 4.3
+in terms of accumulation points of recurrence coefficients an d of zeros only.
+Let us demonstrate this for the case of two and three interval s.
+Corollary 5.1. LetE= [a1,a2]∪[a3,a4],dµ=w(x)dxwithw∈Sz(E),
+and letδ∈ {±1}.Theny,y∈(a2,a3),is a common accumulation point of
+(PnνQnν)and(P(1+δ)/2+nνQ(1−δ)/2+nν)if and only if
+(57) lim
+να1+nν=−y+c1and lim
+νλ2+nν=y(c1−y)+c2−c2
+1−δ/radicalbig
+H(y)
+Furthermore, for every (y,δ)∈(a2,a3)× {−1,1}there exists a subse-
+quence(nν)of the natural numbers such that (57)holds, ifEis not the
+inverse image under a polynomial map.
+IfEis the inverse image under a polynomial map of degree Nthen for
+eachb= 0,...,N−1,either(PνN+b)n∈Nor(QνN+b)n∈Nhas an accumulation
+point of zeros at the zero yof theb−th associated polynomial p(b)
+N−1which
+lies in(a2,a3).The limits of (α1+νN+b)and(λ2+νN+b)are given by (57),
+whereδ=∓1ifyis an accumulation point of zeros of (PνN+b),respectively,
+of(QνN+b).
+Proof.Solving the system (55) and (56) of equations we obtain that
+(58)G(x) =x+lim
+να1+nν−c1andF(x) =x2−c1x+2lim
+νλ2+nν+c2
+1−c2
+By (49) and the statements of Theorem 4.3 the assertion follo ws, ifEis
+not the inverse image.
+In the case when Eis the inverse image under a polynomial see [19]. /squareACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 19
+In the case when E= [a1,a2]∪[a3,a4]∪[a5,a6] the solution of the system
+of equations (55) and (56) yields
+G(x) =x2+(˜α1+nν−c1)x+˜λ2+nν+˜λ1+nν+ ˜α2
+1+nν−˜α1+nνc1+c2
+1−c2
+F(x) =x3−c1x2+(2˜λ2+nν+c2
+1−c2)x+2˜λ2+nν(˜α2+nν+ ˜α1+nν−c1)
+−(c3
+1−2c1c2+c3)(59)
+where tilde denotes the limits, that is, ˜ α1+nν:= lim
+να1+nν,....Equating
+coefficients with the polynomials in (49) gives easily the con dition on the
+limits of the recurrence coefficients such that yj, yj∈(a2j,a2j+1),j= 1,2,
+are common accumulation points of zeros of ( PnνQnν) and (P(1+δj)/2+nν
+Q(1+δj)/2+nν),j= 1,2,for givenδj∈ {±1},j= 1,2.
+To obtain a complete picture of the behaviour of the accumula tion points
+of the recurrence coefficients let us first show that there is a u nique corre-
+spondence to that ones of the moments of the Green’s function s.
+Proposition 5.2. a) Letm∈N,(xm,ym) := (x[m
+2]+1,...,x1,..., x−[m−1
+2]+1,
+y[m−1
+2]+2, ...,y2,...,y−[m
+2]+2),wherexj,yj∈R,and put for k,j∈N0, j≥k,
+Ij,k:=Ij,k(xm,ym) =/summationdisplay
+−1≤ki≤1
+i=1,2,...,j,/summationtextj
+i=1ki=kz0,k1zk1,k1+k2...zk1+...+kj−1,k1+...+kj
+(60)
+where
+zk,j=
+
+√yj+1k=j−1
+xj+1k=j
+√yj+2k=j+1
+Then
+Im:Rm×Rm
++→R2m
+(xm,ym)/ma√sto−→(I1,0,√y2I1,1,I2,0,√y2I2,1,...,Im,0,√y2Im,1)
+is a continuous one to one map.
+b) Letσbe a positive measure and suppose that (α1+n(dσ),λ2+n(dσ))
+is bounded and (λ2+n(dσ))n∈Nis bounded away from zero. Let m∈Nbe
+fixed, and put (αm
+1+n(dσ),λm
+2+n(dσ))n∈N:= (α[m
+2]+1+n(dσ),..., α1+n(dσ),...,
+α−[m−1
+2]+1+n(dσ), λ[m−1
+2]+2+n(dσ), ..., λ2+n(dσ), ..., λ−[m
+2]+2+n(dσ))n∈N.
+Then forn>m
+Im((αm
+1+n(dσ),λm
+2+n(dσ)) = (/integraldisplay
+xP2
+ndσ,/radicalbig
+λ2+n(dσ)/integraldisplay
+xPnP1+ndσ,
+...,/integraldisplay
+xmP2
+ndσ,/radicalbig
+λ2+n(dσ)/integraldisplay
+xmPnP1+ndσ)(61)
+andImis a homeomorphism between the set of accumulation points of the
+two sequences.20 F. PEHERSTORFER
+Proof.a) First let us note that by (60) Ij,k, k∈ {0,1}, j∈ {1,...,m},
+depends on the variables ( xj,yj) only and that
+(62)
+Im,0(xm,ym) =
+
+y−m
+2+21/productdisplay
+i=−m
+2+3yi+e1(xm−1,ym−1) if mis even
+x−(m−1
+2)+11/productdisplay
+i=−(m−1
+2)+2yi+e2(xm−1,ym−1) ifmis odd
+since in (60) the lowest possible indices iofxi,yiappear only for the choice
+(k1,...,km) = (−1,...,−1,1,...,1),where∓1 appearsm/2−times, respec-
+tively for ( −1,...,−1,0,1,...,1) where ∓1 appears ( m−1)/2 times. By the
+way, the highest possible indices are obtained by reversing the order, that is,
+for the choice (1 ,...,1,−1,...,−1),respectively, (1 ,...,1,0,−1,...,−1) which
+is needed later.
+Furthermore for m>1
+(63)Im,1(xm,ym)√y2=
+
+xm
+2+1m
+2+1/productdisplay
+i=3yi+e3(xm−1,ym−1) ifmis even
+ym−1
+2+2m−1
+2+1/productdisplay
+i=3yi+e4(xm−1,ym−1) ifmis odd
+since in (60) the highest possible indices iofxi,yiappear only for the choice
+(k1,...,km) = (1,...,1,0,−1,...,−1),where +1 appears m/2 and−1 appears
+(m−2)/2 times, respectively, for (1 ,...,1,−1,...,−1) where ±1 appears (m±
+1)/2 times. Now we are able to prove the assertion, that is,
+(64) Im(xm,ym) =Im(˜xm,˜ym) implies ( xm,ym) = (˜xm,˜ym)
+by induction arguments with respect to m.SinceIj,k,k∈ {0,1},dependsfor
+j= 0,...,m−1 on (˜xm−1,˜ym−1) only it follows by the induction hypothesis
+that
+(65) ( xm−1,ym−1) = (˜xm−1,˜ym−1)
+Thus it remains to be shown that
+(66) x±[m
+2]+1= ˜x±[m
+2]+1andy∓[m
+2]+2= ˜y∓[m
+2]+2
+ifmis even, respectively odd. Since by (64)
+Im,0(˜xm,˜ym) =Im,0(xm,ym) andIm,1(˜xm,˜ym) =Im,1(xm,ym)
+it follows by (62) and (63) in conjunction with (65) that (66) holds and thus
+part a) is proved.
+b) In [14] it has been shown that, n>j,
+(67) Ij,k(αm
+1+n(dσ),λm
+2+n(dσ)) =/integraldisplay
+xjPnPk+ndσACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 21
+which gives (61). By the assumptions on the recurrence coeffic ients it
+follows that the set of accumulation points is a compact set c ontained
+inRm×Rm
++and thus its image under Imis a compact set contained
+in the range of Im,which is by (67) and continuity of Imequal to the
+set of accumulation points of (/integraltext
+xP2
+ndσ,/radicalbig
+λ2+n/integraltext
+xPnP1+n,...,/integraltext
+xmP2
+ndσ,/radicalbig
+λ2+n/integraltext
+xmPnP1+n)n∈N. /square
+We point out that the starting point of Proposition 5.2a) was formula
+(67) due to Nevai [14]. Combining Theorem 3.2 a) and Proposit ion 5.2 b)
+we obtain
+Theorem 5.3. Letµbe given by (8)withw∈Sz(E)and put(αl−1
+1+n(dµ),
+λl−1
+2+n(dµ))n∈N:= (α[l−1
+2]+1+n(dµ),..., α1+n(dµ),..., α−[l−2
+2]+1+n(dµ),
+λ[l−2
+2]+2+n(dµ), ..., λ2+n(dµ), ..., λ−[l−1
+2]+2+n(dµ))n∈N.
+a)(αl−1
+1+nν(dµ),λl−1
+2+nν(dµ))ν∈Nconverges if and only if
+/parenleftbigg
+(nν−(l−1)/2)ωk(∞)+1
+π/integraldisplay
+Elog(w(ξ))∂ωk(ξ)
+∂n+
+ξdξ−m/summationdisplay
+j=1ωk(dj)
+
+ν∈N(68)
+converges modulo 1fork= 1,...,l−1. Furthermore T ◦ A ◦τ−1◦ Il−1is
+a homeomorphism between the sets of accumulation points, wh ereAis the
+Abel map from (27),τis given in Theorem 3.2 and Tis the map between
+JacR/Rand the real torus [0,1]l−1.
+b)τ−1◦Il−1is a homeomorphism from the set of accumulation points of
+(αl−1
+1+n(dµ),λl−1
+2+n(dµ))n∈Ninto the torus Xl−1
+j=1([a2j,a2j+1]+∪[a2j,a2j+1]−),
+if1,ω1(∞),...,ωl−1(∞)are linearly independent over Q.
+Proof.a) By Proposition 5.2b) and Theorem 3.2 a) the sequence
+(αl−1
+1+nν,λl−1
+2+nν)ν∈Nconverges if and only if ( x1,nν,...,xl−1,nν)ν∈Nconverges,
+where (x1,nν,...,xl−1,nν) is given by (26). Next let us recall that, by the form
+of JacR/Rand the bijectivity of the Abel map A, for every ( x1,...,xl−1)∈
+Xl−1
+j=1([a2j,a2j+1]+∪[a2j,a2j+1]−)thereisanunique( t1,...,tl−1)∈[−1/2,1/2]l−1
+such that
+(69) A(x1,...,xl−1) = (Bij)/parenleftbig
+(t1,...,tl−1)t+(m1,...,ml−1)t/parenrightbig
+wheremκ∈Z. Using (23) we obtain
+l−1/summationdisplay
+κ=1
+1
+2l−1/summationdisplay
+j=1δj,nνωκ(xj,nν)
+Bkκ=l−1/summationdisplay
+j=11
+2/integraldisplayxj,nν
+x∗
+j,nνϕk=l−1/summationdisplay
+κ=1tκ,nνBkκ
+withtκ,nν∈[−1/2,1/2] fork= 1,...,l−1,hence
+(70) tκ,nν=1
+2l−1/summationdisplay
+j=1δj,nνωκ(xj,nν) modulo 122 F. PEHERSTORFER
+Since,by(69), ( x1,nν,...,xl−1,nν)ν∈Nconverges ifandonlyif( t1,nν,...,tl−1,nν)ν∈N
+converges modulo 1 the assertion follows by (14) and (70).
+b) Follows immediately by Theorem 3.2 b) and Proposition 5.2 b)./square
+Moreover under the assumptions of Theorem 5.3 b) the number o f ac-
+cumulation points of the recurrence coefficients is infinite, which has been
+proved for the isospectral torus in [12] already, see also [1 1].
+Corollary 5.4. Let(nν)be such that (αl−1
+1+nν(dµ),λl−1
+2+nν(dµ))ν∈Nconverges.
+Then the sequence (nν+1−nν)ν∈Nis unbounded if 1,ω1(∞),...,ωl−1(∞)are
+linearly independent over Q.
+Proof.By (12) and Theorem 5.3 a) we get that the RHS of the system of
+equationsk= 1,...,l−1,
+(nν+1−nν)ωk(∞)−2(mk,nν+1−mk,nν) =
+=1
+2l−1/summationdisplay
+j=1(δj,nν+1ωk(xj,nν+1)−δj,nνωk(xj,nν))(71)
+tends to zero, where mk,nν+1,mk,nν∈Z.Assume that ( nν+1−nν) is
+bounded and thus that the LHS takes on modulo 1 a finite number o f values
+only and is not equal to zero, since ωk(∞) must not be rational, which yields
+the desired contraction. /square
+In particular Theorem 5.3 holds true for measures from the is ospectral
+torus and thus holds true even for measures, whose recurrenc e coefficients
+satisfy (10), whicharedescribedin[20]. Thereforeit look s likewecouldhave
+restricted to the isospectral torus. The problem is that for accumulation
+points of zeros which is the other main point of our studies th ere are no
+corresponding statements like (10) available.
+Thus the sequences ( nν) for which the recurrence coefficients ( αl−1
+1+nν(dµ),
+λl−1
+2+nν(dµ))ν∈Nconverge are determined by the harmonic measures E, only
+the values of the accumulation points depend on µ. We believe that the
+property ”( αl−1
+1+nν(dσ),λl−1
+2+nν(dσ))ν∈Nconverges if and only if ( nνω(∞))ν∈N
+converges modulo 1” holds for a wide class of measures σwhose essential
+support is E; loosely speaking that it is the counterpart of the measures
+whose essential support is a single interval and for which th e recurrence
+coefficients converge, for instance as in Rakhmanov’s [24] an d Denisov’s [4]
+case.
+As an immediate consequence of Thm. 5.3 and Prop. 5.2 in conju nction
+with Cor. 2.5 we obtain
+Corollary 5.5. The Green’s functions (G(z,nν,nν))ν∈Nand(G(z,1 +nν,
+nν))ν∈N, defined in (18)and(19), converge simultaneously uniformly on
+compact subsets of Ωif and only if (nνω(∞))ν∈Nconverges modulo 1.ACCUMULATION POINTS OF RECURRENCE COEFFICIENTS 23
+Theorem 5.6. The following statements hold for the recurrence coefficients
+of any measure µof the form (8)withw∈Sz(E). a) For every k∈Z
+lim
+ν(αl−1
+k+1+nν,λl−1
+k+2+nν)exists iflim
+ν(αl−1
+1+nν,λl−1
+2+nν)exists.
+b)Put(˜αl−1
+k+1+(nν),˜λl−1
+k+2+(nν)) := lim
+ν(αl−1
+k+1+nν,λl−1
+k+2+nν). The limits are
+related to each other by
+(˜αl−1
+k+1+(nν),˜λl−1
+k+2+(nν)) =ψk((˜αl−1
+1+(nν),˜λl−1
+2+(nν)))
+whereψis a continuous map on Rl−1×Rl−1
++which does not depend on µand
+ψkdenotes the k-th composition. (Note that η−1◦ψk◦η,whereη=I−1
+l−1◦τ,
+maps the corresponding points on the torus to each other).
+c) The orbit {ψk((˜αl−1
+1+(nν),˜λl−1
+2+(nν))) :k∈N0}is dense in the set of ac-
+cumulation points of the sequence (αl−1
+1+n,λl−1
+2+n)n∈Nif1,ω1(∞),...,ωl−1(∞)
+are linearly independent over Q.
+Proof.Part a) follows by Proposition 5.2 b) and the fact that by (40) and
+(41) the limits lim
+ν/integraltext
+xjP2
+nνand lim
+ν/radicalbig
+λ2+nν/integraltext
+xjPnνP1+nνexist for every
+j∈Nif they exist for j= 0,...,l−1.
+b)Letusfirstshowtheassertionfor k= 1.Takingalookat( ˜αl−1
+2+(nν),˜λl−1
+3+(nν))
+we see that we have to show only that
+lim
+να[l−1
+2]+2+nν=f(˜αl−1
+1+(nν),˜λl−1
+2+(nν)) and
+lim
+νλ[l−1
+2]+3+nν=g(˜αl−1
+1+(nν),˜λl−1
+2+(nν)),(72)
+wheref,gare continuous functions. By (41) and Proposition 5.2 b)
+lim
+ν/radicalbig
+λ2+nν/integraldisplay
+xlP1+nνPnν=h1(˜αl−1
+1+(nν),˜λl−1
+2+(nν))
+and thus by (63), recall (67),
+lim
+ναl
+2+1+nν=f1(˜αl−1
+1+(nν),˜λl−1
+2+(nν)),respectively ,
+lim
+νλ[l−1
+2]+2+nν=g1(˜αl−1
+1+(nν),˜λl−1
+2+(nν)),(73)
+iflis even, respectively, lis odd.
+Next let us observe that by (60) and (67) for n≥l
+(74)
+/integraldisplay
+xlP2
+1+n=
+
+λl
+2+2+nn+1+l
+2/productdisplay
+i=n+3λi+ ˜e1(αl
+2+1+n,αl−1
+1+n,λl−1
+2+n) if lis even
+α(l−1
+2)+2+nn+2+(l−1
+2)/productdisplay
+i=n+3λi+ ˜e2(λl−1
+2+2+n,αl−1
+1+n,λl−1
+2+n) iflis odd,24 F. PEHERSTORFER
+where ˜e1,˜e2are continuous functions. By (41), (74) applied to l−1,and
+Proposition 5.2 b) it follows that
+lim
+ν/integraldisplay
+xlP2
+1+nν=
+
+h2(˜αl−2
+2+2+(nν),˜αl−1
+1+(nν),˜λl−1
+2+(nν)) iflis even
+h3(˜λl−1
+2+2+(nν),˜αl−1
+1+(nν),˜λl−1
+2+(nν)) iflis odd
+Thus by (74) and (73) we obtain that
+lim
+νλl
+2+2+nν=g2(˜αl−1
+1+(nν),˜λl−1
+2+(nν)),respectively ,
+lim
+ναl−1
+2+2+nν=f2(˜αl−1
+1+(nν),˜λl−1
+2+(nν))
+iflis even, respectively, lis odd, and the relations (72) are proved.
+For arbitrary kthe statement follows immediately by iteration using in-
+duction arguments and the fact that the relations (72) hold f or any (nν) for
+which we have convergence.
+c) Since (nνω(∞)−c)−→
+ν→∞γmodulo 1 implies that ( nν+k)ω(∞)−
+c−→
+ν→∞γ+kω(∞) modulo 1 the assertion follows by part b) and Thm. 5.3
+using the fact that the RHS is dense in [0 ,1]l−1with respect to k,k∈Z./square
+The mapψcan be obtained explicitly by Corollary 2.5 and (67), at leas t
+for smalll. Most likely Theorem 5.6 c) holds true without the assumptio n
+of linear independence of 1 ,ω1(∞),...,ωl−1(∞).Note that Theorem 5.6 is
+of so-called ”oracle type”, see [25]. Finally let us remark t hat we expect
+(investigations are on the way) that Theorem 4.3 and Theorem 5.3 hold
+true for so-called homogeneous sets Eand a possible infinite set of mass
+points lying outside Eand accumulating on E.
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\ No newline at end of file
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+arXiv:1001.0480v1  [cond-mat.mtrl-sci]  4 Jan 2010Rare-earth impurities in Co 2MnSi:
+an opportunity to improve Half-Metallicity at finite temper atures
+E. Burzo and I. Balazs
+Babe¸ s-Bolyai University Cluj-Napoca, RO-800084 Cluj-Na poca, Romania
+L. Chioncel and E. Arrigoni
+Institute of Theoretical and Computational Physics,
+Graz University of Technology, A-8010 Graz, Austria
+F. Beiuseanu
+Faculty of Science, University of Oradea, RO-410087 Oradea , Romania
+Abstract
+We analyse the effects of doping Holmium impurities into the fu ll-Heusler ferromagnetic alloy
+Co2MnSi. Experimental results, as well as theoretical calcula tions within Density Functional The-
+ory in the Local Density Approximation generalized to inclu de Coulomb correlation at the mean
+field level show that the holmium moment is aligned antiparal lely to that of the transition metal
+atoms. According to the electronic structure calculations , substituting Ho on Co sites introduces
+a finite density of states in the minority spin gap, while subs titution on the Mn sites preserves the
+half-metallic character.
+1I. INTRODUCTION
+An important class of materials which are intensively studied in the field of spintronics
+are the so-called half-metallic ferromagnets. These materials are c haracterized by a metallic
+electronic structure for one spin channel, whereas, for the oppo site spin direction, the Fermi
+level issituatedwithinanenergygap[1,2]. Fromamagneticpointofvie w, thesecompounds
+can be either ferro- or ferrimagnets. Quite generally, theoretica l calculations based on the
+Local-Density Approximation (LDA) predict a perfect spin polarizat ion at the Fermi level.
+Antiferromagentic half-metals also exits, although this case is spec ial since no symmetry
+relation connects spin-up and spin-down directions. The existence of half-metallicity in all
+three classes of magnetic orderings was predicted by R. A. de Groot and coworkers some
+years ago [1, 3, 4]. Since then, several possible half-metallic ferrom agnetic (HMF) materials
+have been theoretically investigated by ab initio calculations, and som e of them have been
+synthesized and experimentally studied.
+Devices that exploit spin and charge of the electrons should operat e at not too low
+temperatures. Unfortunately many half-metallic systems exhibit a dramatic suppression of
+spin-polarization well below room temperature [2, 5]. It is therefor e important to study and
+to optimize temperature effects in half-metallic materials. There are several effects poten-
+tially responsible for the suppression of spin polarisation at finite tem perature that can be
+taken into account in a theoretical calculation. The competition bet ween hopping and local
+electron-electron interactions may produce spin disorder, which c an be approximated as ran-
+dom intraatomic exchange interactions [6, 7]. These interactions m odify the local magnetic
+moments and influence the spin polarization[8, 9]. The exchange inter actions described by a
+Heisenberg-type Hamiltonian are also frequently used to discuss fin ite temperature magnetic
+effects. The sign and magnitude of exchange constants give inform ation whether the spin
+structure is collinear or not [7, 10, 11]. Note that non-collinearity ma y be produced also by
+spin-orbit coupling. All the above mentioned mechanisms may be called spin-mixing effects
+[5, 12] and they may significantly influence the half-metallic characte r.
+It is important to point out that in all metallic ferromagnets, the inte raction between
+conduction electrons and spin fluctuations determines their physic al properties. In half-
+metallic ferromagnets the presence of the gap in one of the spin cha nnel determines the
+absence of spin-flip, or one-magnon scattering processes which m odifies considerably the
+2electrons energy spectrum [2, 13, 14]. These electron-magnon int eractions occur both in
+usual ferromagnets and in half-metals, with the difference that fo r the usual ferromagnets
+the states around the Fermi level are quasiparticles for both spin directions, while in half-
+metals an important role is played by incoherent - non-quasiparticle ( NQP) - states [13, 14].
+The origin of these NQP states is connected with spin-polaron proce sses: the spin-down low-
+energyelectronexcitations, whichareforbiddenforHMFintheone -particlepicture, turnout
+to be possible as result of superpositions of spin-up electron excita tions and virtual magnons
+[2, 14–16]. It was demonstrated that these states are important for spin-polarized electron
+spectroscopy [17], nuclear magnetic resonance [18], and subgap tr ansport in ferromagnet-
+superconductor junctions (Andreev reflection) [19]. Recently, t he density of NQP states
+has been calculated using an LDA+dynamical mean field theory (DMFT ) approach [2, 20],
+for several materials such as the semi-Heuslers NiMnSb [21] and (NiF e)MnSb [22], the full
+Heuslers Mn 2VAl [23] and Co 2MnSi [24], the zinc-blende CrAs [25] and VAs [26] and the
+transitionmetaloxide CrO 2[27], demonstrating their importanceastheessential mechanism
+of depolarization in half-metals.
+Among the Heusler half-metals, the full-Heusler Co 2MnSi [28–30] has been intensively
+studied experimentally as well as theoretically. This material is intere sting asit shows a high
+Curie temperature and very little crystallographic disorder [31]. We s howed that Co 2MnSi
+experiences a drastic spin depolarization [24] with increasing temper ature. According to our
+calculations, depolarization is caused by the occurrence of non-qu asiparticle states situated
+justabovetheFermilevel intheminorityspinchannel [2,24]. Inord ertosignificantly reduce
+depolarization a strategy to eliminate the NQP states from the Ferm i level is required. In
+other words, one should try to suppress finite-temperature mag nonic excitations while at
+the same time preserving the minority spin gap. It was suggested th at the substitution
+of a rare-earth (R) element within the Mn sublattice of NiMnSb would in troduce gaps in
+the magnonic spectrum due to the impurity R(4 f) - Mn(3d) coupling, possibly preserving
+the half-metallic gap [2, 32, 33]. In pure NiMnSb the ferromagnetic Cu rie temperature
+TC= 740K is determined by the strength of the Mn-Mn (3 d−3d) coupling [2, 10]. The
+strength of (3 d−4f) couplings in NiMn 1−xRxSb compounds, with R=Nd, Pm, Ho, U was
+discussed previously [2, 33]. A large 3 d−4fcoupling was obtained for the case of Nd
+substitution, and the weakest coupling was realized in the case of Ho . For all rare-earth
+substitutions the electronic-structure calculations showed that the half-metallic gap is not
+3affected, and for temperatures smaller than the values of the 3 d−4fcouplings, interaction
+will lock the Mn(3 d) magnetic moment fluctuation. As a consequence, it was concluded
+that Nd substitution is of practical importance for high-temperat ure applications [2], while
+half-metallicity with Ho substitutions would persist up to temperatur es of around 4K as
+predicted by the calculation [32, 33].
+For the case of Co 2MnSi, preliminary results showed that Ho can enter in the Co 2MnSi
+lattice [34]. Recently, Rajanikanth et. al. [35] showed that substituting Co with 1 .25at%
+Nd results in a single L2 1phase alloy with a thin layer of Nd-enriched phase boundaries. A
+suppression of electron-magnon scattering was experimentally ob served [35].
+As an ongoing investigation of impurity effects on the depolarization o f conduction elec-
+tronsinCo 2MnSibasedalloyweanalyzeinthispaperthepossibilityofsubstituting holmium
+in both Co or Mn lattice sites, as well as changes introduced in band st ructure of the parent
+compound. The first task is to analyze whether holmium can really rep lace Mn or Co in a
+Fm¯3mtype structure. Consequently, we prepared samples in which the s ubstitution at both
+transition metal sites was considered. X-rays and electron micros copy studies were carried
+out. Finally, the magnetic properties and band structures of the d oped compounds were
+analyzed.
+II. PHYSICAL PROPERTIES OF HO-DOPED CO 2MNSI SAMPLES
+The Co 2Mn1−xHoxSi and Co 2−xHoxMnSi alloys with nominal compositions x= 0.05
+andx= 0.1 were prepared by arc melting the constituent elements under pur ified argon
+atmosphere. These were remelted several times in order to ensur e a good homogeneity.
+The alloys were annealed at 900oCfor 12 days. The X-ray analyses were performed by
+using a Brucker V8 diffractometer. Rietveld refinements were made by using the Topas
+code. Electron microscope studies were performed by using a JEOL type equipment. The
+compositions of the samples and distributions of the constituent ele ments were analysed.
+Magnetic measurements were made with an Oxford Instruments ty pe equipment in fields up
+to 5 T and in the temperature range 4.2-1000 K.
+4A. Experimental results
+The refined X-ray diffraction (XRD) spectra of Co 2Mn1−xHoxSi samples with x= 0.05
+andx= 0.1 are plotted in Fig.1. The same type of spectra were obtained when H o sub-
+stitutes Co. The alloys crystallize in a cubic structure having Fm¯3mspace group. As seen
+from Table I, the lattice parameters increase little when increasing h olmium content. The
+linewidths of XRD spectra are a little larger in samples having a composit ionx= 0.1 as
+compared to those having x= 0.05. This can be correlated with a small deformation of the
+lattices as the holmium content increases.
+The better agreement between the computed spectra and those experimentally deter-
+mined, in Co 2Mn1−xHoxSi alloys is obtained when holmium substitutes manganese located
+in 4c sites, according to the starting composition. The goodness of fit (GOF) values are
+indeed worse (larger) if one assumes that Ho occupies cobalt 4a or 4 b sites. On the other
+hand, therefinementsofthecrystalstructuresofCo 2−xHoxMnSisamplesshowedGOFvalues
+rather close when Ho was considered either to replace cobalt or man ganese. In addition, a
+further analysis of our data strongly suggests that holmium occup ies lattice sites in Co 2MnSi
+sample, and does not sit in interstitial positions.
+The compositions of the samples were determined by Scanning Electr on Microscopy
+(SEM), for a surface area of ≈1mm2and a depth from surface up to about 200 ˚A. The
+determined holmium content in samples having initial compositions x= 0.05 and 0 .1
+were 1.30±0.15at% and (2 .7−2.8)±0.30at% respectively in both Co 2Mn1−xHoxSi and
+Co2−xHoxMnSi systems Figs.2a,b. These values are in agreement with the expe cted con-
+tent of 1.25at% and 2.5at%, respectively. The manganese contents in Co 2Mn1−xHoxSi were
+22.8±0.7at%(x= 0.05) and 22 .0±0.6at%(x= 0.1), while the cobalt content was close to
+50at% - Figs.2a,b. The above data suggest that holmium is distributed mainly in manganese
+sites in this compound.
+Ontheotherhand, inthecaseofCo 2−xHoxMnSisamplesthereisadecreaseofbothcobalt
+andmanganese content as comparedto expected compositions, s uggesting thatholmium can
+occupyboththesesites. Thedistributionofholmiumatomsinthesam pleswasalsoanalysed.
+As example, we show in Figs.2c,d the SEM patterns of two manganese s ubstituted samples.
+A random distribution can be seen, with no clustering effects.
+The magnetizations isotherms, at 4 .2K, are shown in Fig. 3. The determined saturation
+5magnetizations, M, per formula unit are given in Table II. The Mvalues are smaller than
+the ones measured in Co 2MnSi, namely 5 µB[36], and decrease when increasing holmium
+content. This suggests that the holmium moments are antiparallely o riented to the cobalt
+and manganese ones. Therefore, the magnetization measuremen ts are in agreement with
+the result of structural and compositional analysis and confirm th at holmium atoms occupy
+lattice sites, and that their moment is antiparallely oriented to cobalt and manganese ones.
+A better agreement is obtained for a composition x= 0.05. For samples having higher
+holmium content a little deviation from the expected stoichiometry or a possible location of
+Ho in both Co and Mn lattice sites cannot be excluded. The analysis of t hermal variations
+of magnetizations in low field, shows the presence of only one magnet ic phase. The Curie
+temperatures of the Ho-doped samples are only minimally reduced wit h respect to that of
+Co2MnSi (Tc= 985 K) as seen in Table I.
+B. Electronic Structure of Ho doped Co 2MnSi
+To compute the electronic structure we used the standard repre sentation of the L2 1struc-
+ture with four inter-penetrating fcc sub-lattices. The atomic pos itions are (0 ,0,0) for Co1,
+(1/2,1/2,1/2) for Co2, and (1 /4,1/4,1/4) and (3 /4,3/4,3/4) for Mn and Si, respectively.
+We note that the Co1 and Co2 sites are equivalent. We used the expe rimental lattice con-
+stanta= 5.65˚Aand the linear muffin-tin orbital method (LMTO)[37] extended to inclu de
+the mean-field Hubbard +U approximation (LDA+U) [38]. It is generally known that the
+LDA+Umethodappliedtometallic compoundscontaining rare-earthe lements, gives aqual-
+itative improvement compared with the LDA not only for excited-sta te properties such as
+energy gaps but also for ground-state properties such as magne tic moments and interatomic
+exchange parameters. Therefore it is expected that it will provide a correct description for
+the narrow Ho(4f) states, the minority spin gap and the 3 d−4fcoupling in the Ho doped
+Co2MnSi.
+The origin of the minority band gap in the full Heusler alloy was discusse d byGalanakis
+et al.[39]. Based on the LDA band-structure calculations for the undope d Co2MnSi, it was
+shown that Co1 and Co2 couple forming bonding/anti-bonding hybrid s. The Co1(t 2g/eg)
+- Co2(t 2g/eg) hybrid bonding orbitals hybridize with the Mn(t 2g/eg) manifold, while the
+Co-Co hybrid anti-bonding orbitals remain uncoupled due to their sym metry. The Fermi
+6energy is situated within the minority spin gapformed by the triply deg enerate anti-bonding
+Co-Co(t 2g) and the double degenerate anti-bonding Co-Co(e g) (Fig. 4). Due to the orbitals
+occupation, the majority spin channel has a metallic character, wit h a small density of states
+at the Fermi level. In agreement with previous [10, 11, 39, 40] calc ulations, an integer total
+spin magnetic moment of 5 .00µB, and a minority spin gap of magnitude 0 .42eVis obtained.
+The behavior of the magnetic moment as a function of temperature in undoped Co 2MnSi
+was discussed in the framework of the LDA+DMFT method [24]. Both magnetisation and
+spin polarisation decrease with increasing temperature, although t he latter decreases much
+faster. Above the Fermi level, non-quasiparticle (NQP) states [2, 14–16] are present, which
+influence significantly the finite temperature properties [24].
+We have performed LDA+U calculations for different values of the av erage Coulomb
+interaction parameter U and different sizes of the unit cell, in order t o investigate the
+sensitivity of the results with respect to the these parameters. T he parameters study was
+performed for the energetically most favorable Ho substitution, n amely the replacement of
+a Mn atom by Ho with an antiparallel coupling of the Ho spin moment with r espect to
+the Mn itinerant electrons spin moment. For all our values of U a half- metallic solution is
+obtained with a minority gap having similar width as in the spin polarized LD A calculation,
+however depending on the strength of U the Fermi level moves tow ards the middle of the
+gap. In Fig. 5 we present the computed value of the Ho spin moments together with the
+difference between the bottom of the conduction band ( EB) and the Fermi level EB−EF
+as a function of the average coulomb parameter U. As one can see there is no significant
+dependence in the studied range of the U parameter 8 −12eV: a slight increase of the
+magnetic moment from 4 .04 up to 4 .15µBis visible, while the maximum EB−EFdifference
+is of order of 100 meV. Therefore in the following calculations we take the values U Ho=
+8eVandJHo= 0.9eVfor the Coulomb and exchange parameters, which agrees with the
+values reported in literature for metallic rare-earth compounds [38 ]. Using these values we
+performed calculations for three supercells containing 16 atoms in t he case of Co 8HoMn 3Si4,
+32 atoms for Co 16Mn7HoSi8and respectively 64 atoms for the Co 32Mn15HoSi16cell. In Fig.
+6 we shows the spin resolved density of states for the two largest s upercells. In the inset the
+concentration dependence of the Ho spin moment is presented for all three supercells. Again
+one can see that the results does no significantly change with the dim ension of the cells.
+Therefore for computational convenience, and in order to obtain the value for the orbital
+7magnetic moment of Ho, we carried out the LDA+U calculation including both spin-orbit
+coupling and non-collinearity effects for the smallest supercell in disc ussion Co 8HoMn 3Si4.
+Our calculation yields a self-consistent collinear antiparallel configur ation of the holmium
+moments with respect to the transition metal moments, with values of spin and orbital
+moments of3 .917µBand5.914µB, respectively. Thetotalmagneticmoment (9 .83µB)isclose
+to the usual value determined by neutron diffraction in compounds h aving cubic structure
+with similar space group, such as HoFe 2[41]. Similarly to the Ho spin moments, which
+were shown not to change significantly form one supercell to the ot her, we do not expect
+significant changes of the Ho orbital moments as well, since the magn itude of Ho spin-orbit
+coupling does not change appreciably with concentration, provided one uses the same type
+of structure and similar lattice parameters. For this reason, it is ju stified to neglect spin-
+orbit coupling for the band structure calculation in the larger super cell presented below. In
+this case, in order to compare the total magnetic moment with the e xperimentally measured
+values, the orbital magnetic moment calculated above for the smalle r supercell is then added
+to the spin moments to obtain the results shown in Table II.
+As discussed above, in the case of Co 2MnSi rare-earth impurities can in principle en-
+ter in both the Co and the Mn- sublattices. For this reason, we pres ent results for the
+electronic-structure calculations carried out on the Co 16Mn7HoSi8supercell to simulate Ho
+substitution at the Mn sites, and for a Co 15HoMn 8Si8cell when Ho replaces Co. The
+self-consistent calculations were performed assuming a collinear, p arallel and respectively
+anti-parallel orientation of the Ho(4 f) magnetic moment with respect to the Mn/Co(3 d)
+ones. The resulting density of states for Co 16Mn7HoSi8with both parallel and antiparallel
+orientation between Ho(4 f)-Mn(3d) moments are plotted in Fig. 7. The half-metallic state
+is stable in both cases, with a gap of similar magnitude. However, the p resence of rare-earth
+impurities induces a redistribution of states such that the Fermi lev el moves slightly towards
+the center of the gap, thus making the half-metallic state more sta ble. It is important to
+mention that in the case of Co 16Mn7HoSi8Ho(4f)-orbitals do not hybridize with the 3 d
+orbitals near the Fermi level, the behaviour of DOS near EFis very similar to the undoped
+case, so the nature of carriers around EFis not changed.
+To compute the values of the Ho(4 f)-Mn(3d) exchange coupling, we use a two-sublattice
+modelthatdescribesthesublatticeofHo(4 f)spinmomentsasbeingantiparalelyorparallely
+oriented to the Mn(3 d) spins sublattice. In this simplified model the exchange coupling, J,
+8correspondstotheintersublatticecouplings. InthismodeltheMn (3d)andtheHo(4 f)-states
+are treated within a mean field LDA+U approach, whereas the Ho(4 f)-Mn(3d) interaction
+was treated as a perturbation. The corresponding mean-field Ham iltonian can be written in
+the form:
+H≈HLDA+U−J/summationdisplay
+i,δσ3d
+iSf
+i+δ (1)
+Here,σ3d
+irepresents the spin operator of the 3d electrons at site ri, andSf
+i+δthe spin of the
+4f shell at the site ri+δ. Within this model the Mn(3 d) local moment fluctuation could be
+quenched by a strong Ho(4 f)-Mn(3d) coupling affecting the magnon excitations of the 3 d
+conduction electron spins of the Mn-sublattice. Given the geometr y of the cell, when Ho is
+substituted into the Mn- sublattice, twelve pairs of Ho(4 f)-Mn(3d) are formed. The corre-
+sponding Ho(4 f)-Mn(3d) coupling constant is calculated as the energy difference between
+the parallel EPand the antiparallel EAPconfiguration. In this way, we obtain a value of
+J≃88K. In analysing the behaviour of exchange constants in Co 2MnSi, it was recently
+pointed out that Co-Mn interactions are responsible for the stabilit y of the ferromagnetism
+[10, 11], in particular the main exchange parameter corresponds to the nearest-neighbour
+Co1-Mninteraction which alreadygives 70%of theof thetotal cont ribution toJ being about
+ten times larger than the Co-Co and Mn-Mn interactions [40, 42]. For the small Ho content
+in the Co 2MnSi, it is shown experimentally , see table Tab.I, that the ferromagn etic Curie
+temperature is not significantly reduced, so one can conclude that the Co1-Mn interaction
+is not considerably affected, and in particular this interaction domina tes over the 3 d−4f
+coupling.
+The resulting total magnetic moments of the supercell, taking into a ccount orbital Ho
+moments as well, were 26 .09µBand 45.91µBfor antiparallel and parallel orientations of Ho
+with respect to the Mn and Co moments, respectively.
+As shown in Ref. [39] and briefly described above (Fig. 4), the minor ity gap is formed by
+the triply degenerate Co-Co anti-bonding t 2gand the double degenerate Co-Co anti-bonding
+eg. Thus, it is expected that a substitution in the Co sublattice would ha ve a stronger effect
+on the electronic structure than the substitution in the Mn sublatt ice, described above.
+Results of the selfconsistent spin polarized LDA+U calculation for th e Co15HoMn 8Si8
+supercell, in which Ho atoms substitutes one of the cobalt sites is sho wn in Fig.8. Identical
+results are obtained when Ho substitutes Co1 or Co2 because coba lt occupies equivalent
+9sites. The magnetic moments per formula unit in the doped material s how a departure from
+the integer values corresponding to the half-metallic case. For Ho im purity on the cobalt
+site, neglecting the orbital contribution, in the anti-parallel 4 f−3dconfiguration we obtain
+µ(AP)
+Ho/Co1= 34.81µBper supercell, while for the parallel case we have µ(P)
+Ho/Co1= 41.87µB
+per supercell. Also in this case, the antiparallel configuration is the m ost stable one, in
+agreement with the magnetic measurements discussed above. An in teresting aspect is that,
+in contrast to the case of substitution at the Mn site, substitution at Co sites fills the
+minority-spin gap, as can be seen in Fig. 8. Notice that Ho(4 f) and (5d) states shown in
+Fig. 8 were multiplied by ten in order to evidence possible hybridization e ffects. As one
+can see (Fig. 8) the Ho(4 f) states show no significant contribution in a large energy range
+(EF±3eV) around the Fermi level. Occupied and empty Ho(4 f) states are present in the
+energy range -8 to -6eV, and 2 to 4eV respectively, not shown in Fig . 8. The anti-bonding
+Co3d(eg)-Ho5d(eg) states appear for both parallel and anti-parallel configurations above the
+Fermi level up to energies EF+0.2eVand these states determine the shift of the Fermi level
+within the conduction band, thus the closure of the half-metallic gap .
+Noticethatasresultofhybridizationeffects, asmallnegativemagn eticmomentisinduced
+on silicon. Values of −0.017µBare obtained when Ho is located on Mn sites and −0.047µB
+when Ho replaces Co. The computed magnetic moments of the atoms when Ho replaces Mn
+or Co in both the antiparallel and parallel alignments are listed in Table I I. The computed
+magnetic moments per formula unit for Co 2Mn7/8Ho1/8Si and Co 15/8MnHo 1/8Si are also
+shown. These compare reasonably with experimental data, when c orrections for different
+compositions are made. This is done by extrapolating the experiment al data obtained with
+x= 0.05 and 0.10 to the theoretically computed case x= 0.125. A rather good agreement
+between computed and experimental values is obtained when the ho lmium replaces cobalt
+atoms in the antiparallel configuration, while a worse agreement is sh own when holmium
+replaces manganese, see Tab.II, although energetically substitut ion in the former way is less
+favorable.
+III. CONCLUSION
+The behavior of spin polarization as a function of temperature is an im portant issue for
+many spintronic materials. In our recent paper [24], we interpreted the measured drastic
+10depolarization in Co 2MnSi as due to the existence of non-quasiparticle states just abo ve the
+Fermi level. Since the origin of these states is related to the electro n-magnon interaction, a
+possibility to reduce this depolarisation effect would be to quench the magnon excitations,
+while keeping the minority spin channel gapped. This possibility was alre ady discussed in
+connection to light-rare earth (Nd) insertion into the host Co 2MnSi [35], supporting the va-
+lidity of the proposed scenario. In the present work, we first demo nstrate experimentally the
+possibility that a heavy rare-earth element (Ho) enters into the tr ansition-metal sublattices
+in Co2MnSi. In addition, we carried out an electronic-structure calculatio n whose results
+suggest that the half-metallicity is maintained when Ho replace Mn ato m in an anti-parallel
+magnetic Ho(4f)-Mn(3d) configuration. Since the calculated exch ange coupling Jturns out
+to be about 88 K, we expect that for temperatures smaller than Jsuch a substitution could
+in principle influence the magnonic excitations, while at the same time lea ving the half-
+metallic gap unchanged. On the other hand, our results show that w hen the substitution
+takes place at Co sites, the minority-spin gap is filled, and half-metallic ity is lost.
+The present calculations does not take into account relaxation effe cts which might appear
+as a result of substitution. This will constitute the subject of futu re investigations. Dynamic
+correlation effects are also expected to be important. In particula r, it would be interesting
+to investigate up to which extent magnonic excitations and electron -magnon interactions
+are affected by Ho substitution. We plan to investigate this effect in t he future.
+M.I. Katsnelson and A.I. Lichtenstein are acknowledged for useful discussions. LC and
+EA acknowledge financial support by the Austrian science fund (FW F project P18505-N16).
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+PhD thesis, University Nijmegen, 2004.
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+(2005).
+13Alloy Lattice constant Saturation magnetization at 4.2K Tc
+(˚A) (µB/fu) (K)
+Co2Mn0.95Ho0.05Si5.651(2) 4.40±0.07 984
+Co2Mn0.90Ho0.10Si5.658(2) 4.05±0.07 982
+Co1.95Ho0.05MnSi 5.655(3) 4.30±0.07 983
+Co1.90Ho0.10MnSi 5.659(2) 3.94±0.07 982
+TABLE I: Lattice parameters, saturation magnetizations an d Curie temperatures for the studied
+compounds.
+SampleMagnetic Computed magnetic moments ( µB)Experimental∗
+coupling MCoMMnMHoMSiMtotM(µB/f.u.)
+Co2MnSi 1.0033.026 --0.035.002 -
+Co2Ho1/8Mn7/8Si P 1.0132.8669.779-0.018 5.738 -
+Co2Ho1/8Mn7/8SiAP 1.0092.866-9.979-0.017 3.261 3.87±0.2
+Co15/8Ho1/8MnSiAP(Co1/Co2) 0.9413.110-9.625-0.047 3.626 3.76±0.1
+TABLE II: Calculated magnetic moments for different Ho-subst ituted compounds. The experi-
+mental data∗are obtained by extrapolating the measured values ( x= 0.05 and 0.10) tox= 0.125
+composition.
+14FIG. 1: (Color online) Refined X-ray spectra for Co 2Mn1−xHoxSi samples with x=0.05 (a) and 0.1
+(b).15FIG.2: (Color online)CompositionsoftheCo 2Mn0.95Ho0.05Si(a)andCo 2Mn0.9Ho0.1Si(b)samples
+as well as the holmium distributions in Co 2Mn0.95Ho0.05Si (c) and Co 2Mn0.9Ho0.1Si (d) samples.
+16FIG. 3: (Color online) Magnetization isotherms at 4.2 K.
+17-1-0.500.51
+Co-Co(eg) Co-Co(t2g)Co(t2g)
+Co(eg)
+-2-1.5-1-0.500.511.52
+E-EF (ev)-1-0.500.51DOS(states/atom)Mn (t2g)
+Mn(eg)Co2MnSi
+Mn(t2g)
+minority spin majority spinminority spin majority spin
+Mn(eg)
+FIG. 4: (Color online) Spin and orbital resolved density of s tates of the (stoichiometric) Co 2MnSi
+full-Heusler alloy obtained by spin-polarized LDA. The hal f-metallic gap in the minority spin
+channel is formed between the hybrid Co-Co(t 2g) and Co-Co (e g) orbitals. No significant Mn-
+orbitals contributions to the edges of the minority spin gap can be seen in the lower panel.
+184.044.084.124.16MHo(  B)
+8 9 10 11 12
+U(eV)95100105110EB -EF (meV)µCo16Mn7HoSi8
+FIG. 5: (Color online) The minority spin half-metallic gap a s function of the average Coulomb
+parameter U. No significant U dependence is present which dem onstrates the lack on 4f-3d hy-
+bridizations around the Fermi level.
+19-6 -5 -4 -3 -2 -1 0 1 2 3 4
+E-EF(eV)-10-5051015DOS(states/eV/cell)x=1/8
+x=1/16
+0 0.1 0.2 0.3
+x(%)4.044.054.064.074.084.09MHo(  B)
+UHo=8eV, JHo=0.9eVCo2Mn1-xHoxSi
+µ
+FIG. 6: (Color online) Spin resolved density of states for di fferent eight respectively 16 times larger
+supercells of Co2MnSi. The inset shows the composition depe ndence Ho magnetic spin moments.
+No essential changes are visible around the Fermi level.
+20-50-2502550
+Total
+-50-2502550DOS(states/eV)Total
+Ho
+-8-6-4 -2 0 2 4
+E-EF (eV)-50-2502550
+Total
+HoCo16Mn8Si8
+Co16Mn7HoSi8
+Co16Mn7HoSi8UHo=8eV, J=0.9eVUHo=8eV, J=0.9eVLSDA+U
+LSDA+U
+(AP)(P)
+FIG. 7: (Color online) Spin resolved density of states of the Co16Mn7HoSi8supercell in the parallel
+(P) and anti-parallel (AP) configuration between Ho(4f)−Mn(3d) moments. For comparison,
+the Co 2MnSi density of states is also plotted. In the presence of Ho, Fermi level shifts towards the
+middle of the minority-spin gap, however the magnitude of th e gap remains unchanged.
+21-60-3003060Co15HoMn8Si8
+Ho - 4f
+Ho - 5d
+Co16Mn8Si8
+-2-1.5 -1-0.5 00.5 1
+E-EF(eV)-60-3003060DOS(states/eV)(AP)(P)majority spin
+minority spin
+majority spin
+minority spin
+FIG. 8: (Color online) Spin resolved density of states aroun d the Fermi level for the Co 15HoMn8Si8
+supercell, in the parallel and anti-parallel configuration s, where Ho is substituted into Co1 lattice
+sites. The Ho-4f and -5d states are magnified by a factor of ten in order to evidence hybridzation
+effects. The minority-spin gap is closed due to the Co3d(eg)-Ho5d(eg) anti-bounding coupling which
+pushes the Fermi level within the conduction band.
+22
\ No newline at end of file
diff --git a/1001.0481.txt b/1001.0481.txt
new file mode 100644
index 0000000000000000000000000000000000000000..bb7e34fbafe71cf928976edca982183a6ac98773
--- /dev/null
+++ b/1001.0481.txt
@@ -0,0 +1,860 @@
+arXiv:1001.0481v2  [nucl-th]  12 Jul 2010Analysis of the Q2dependence of charged-current quasielastic processes
+in neutrino-nucleus interactions
+Artur M. Ankowski,1,2,∗Omar Benhar,2and Nicola Farina2
+1Institute of Theoretical Physics, University of Wroc/suppress law,
+Wroc/suppress law, Poland,
+2INFN and Department of Physics,“Sapienza” Universit` a di R oma,
+I-00185 Roma, Italy
+(Dated: June 21, 2018)
+We discuss the observed disagreement between the Q2distributions of neutrino-nucleus quasielas-
+tic events, measured by a number of recent experiments, and t he predictions of Monte Carlo sim-
+ulations based on the relativistic Fermi gas model. The resu lts of our analysis suggest that these
+discrepancies are likely to be ascribable to both the breakd own of the impulse approximation and
+the limitations of the Fermi gas description. Several issue s related to the extraction of the Q2distri-
+butions from the experimental data are also discussed, and n ew kinematical variables, which would
+allow for an improved analysis, are proposed.
+PACS numbers: 13.15.+g, 25.30.Pt
+I. INTRODUCTION
+A number of recent experiments reported a sizable
+disagreement between the measured Q2distributions
+of neutrino-nucleus scattering events and the results of
+Monte Carlo (MC) simulations [1–3]. The largest dis-
+crepancies occur in the region of low Q2, typically Q2/lessorsimilar
+0.2 GeV2, where the observed number of charged-current
+quasielastic (CCQE) events is significantly lower than
+MCpredictions. Toaccountforthisdifferences,theMini-
+BooNE Collaboration introduced an additional parame-
+ter in the relativistic Fermi gas (RFG) model used in the
+MC code [2]. This procedure leads to a modification of
+the treatment of Pauli blocking, whose physical interpre-
+tation is quite questionable. In Ref. [2] it was also argued
+that a more refined treatment of nuclear effects may be
+needed. However, the results of Ref. [4] suggest that re-
+placing the RFG model with the approach based on the
+use of a realistic spectral function (SF) does not lead to
+a consistent description of low- and high- Q2data. Note
+that, although strictly speaking the spectral function can
+also be defined in the RFG model, in which case it re-
+duces to a collection of δ-function peaks, in the following
+wewillusetheacronymSFtoindicateaspectralfunction
+obtained from more advanced dynamical models.
+The basic assumption underlying MC simulations is
+the validity of the impulse approximation (IA), imply-
+ing that the scattering process involves only one nucleon,
+while the remaining ( A−1) particles act as spectators.
+This scheme is likely to be applicable if the space resolu-
+tion of the incoming neutrino, λ∼ |q|−1, whereqis the
+momentum transfer, is small compared to the average
+separation between nucleons in the target nucleus.
+In this paper we analyze possible sources of the ob-
+serveddiscrepancies,associatedwithboththebreakdown
+∗Artur.Ankowski@roma1.infn.itof the IA and the limitations of the RFG model, and
+point out some issues related to the extraction of the Q2
+distributions from the experimental data.
+In Section II we discuss the bias of the Q2reconstruc-
+tion from the kinematics of the CCQE events, while Sec-
+tion III is devoted to a quantitative investigation of the
+limits of applicability of the IA scheme. In Section IV we
+discuss the main difficulties involved in the comparison
+between theoretical calculations and experimental data,
+and put forth the proposal of new kinematical variables,
+whose use may allow for a more effective data analysis.
+Finally, in Section V, we summarize our results and state
+the conclusions.
+II. UNCERTAINTIES IN THE Q2
+RECONSTRUCTION
+Consider the yield of CCQE events averaged over the
+MiniBooNE neutrino flux. We have compared the dis-
+tributions of events plotted as a function of Q2and its
+reconstructed value, defined as [1, 2]
+Q2
+rec=−m2
+µ+2Erec
+ν(Eµ−|k′|cosθ),(1)
+wheremµ,k′, andθdenote the muon mass, momentum,
+and scattering angle, respectively. In the above equation
+Erec
+ν=2Eµ(Mn−ε)−(ε2−2Mnε+m2
+µ+∆M2)
+2(Mn−ε−Eµ+|k′|cosθ),(2)
+with ∆M2=M2
+n−M2
+p,MnandMpbeing the neutron
+and proton mass, is the reconstructed energy of the in-
+coming neutrino. Note that Eqs. (1) and (2) have to
+be regarded as definitions of two quantities used in data
+analysis and should not be identified with the true Q2
+and neutrino energy.
+The results ofthe calculations, carriedout fora carbon
+targetusingboththeRFGmodel, withFermimomentum2
+SF,Q2
+recSF,Q2RFG,Q2
+recRFG,Q2
+Q2orQ2
+rec(GeV2)105events/GeV2
+1 0.80.60.40.206
+5
+4
+3
+2
+1
+0
+FIG. 1. (color online). Distributions of CCQE events in car-
+bon as a function of Q2andQ2
+rec. The calculations have been
+carried out for the MiniBooNE flux, using the RFG model
+(upper curves) and the SF approach (lower curves).
+pF= 220 MeV and separation energy ε= 34 MeV, and
+the SF approach, with the spectral function of Ref. [5],
+are shown in Fig. 1. In the SF approach Pauli blocking
+hasbeentakenintoaccountusingrealisticnuclearmatter
+momentum distributions and the local density approxi-
+mation, within the scheme discussed in Section III. It
+is apparent that in both cases the definition of Q2
+recin
+terms of the measured quantities |k′|andθallows one to
+reproduce quite well the results obtained using the true
+Q2. However, as the results shown in Fig. 1 involve a
+flux average, one may still ask the question whether the
+two quantities, Q2andQ2
+rec, are totally equivalent.
+To clarify this point, in Fig. 2 we show a compari-
+son between the differential cross sections dσ/dQ2and
+dσ/dQ2
+recobtained from the RFG model at fixedneu-
+trino energies Eν= 1.2 GeV, 0.8 GeV, and 0.4 GeV.
+With the exception of the lowest energy, the maxima
+are again in very good agreement. For this reason the
+cross sections averaged over a not-too-low-energy flux,
+like MiniBooNE’s, are in good agreement.
+On the other hand, the cross sections show quite a rich
+structure,exhibitingbumps, dips, andknees, whichmake
+theQ2andQ2
+recdistributions clearly different. While
+this structure is somewhat emphasized within the RFG
+model, its origin is largely model independent, as it
+can be traced back to kinematics. In this context, the
+most relevant feature is the location of the single parti-
+cle strength, expressed by the average separation energy,
+which turns out to be quite close in the RFG and SF
+models.
+In spite of the fact that the structure is not visible in
+Fig. 1, as it is washed out by the flux average, the results
+shown in Fig. 2 imply that neutrinos of fixed energy con-
+tribute in a slightly different manner to dσ/dQ2and to
+dσ/dQ2
+rec. At neutrino energy 0.8 GeV, corresponding to
+the peak of the MiniBooNE flux, the difference between
+the two cross sections is pronounced.
+Before discussing why the replacement of Q2withQ2
+rec
+leads to a significantchangeofthe crosssection, let us fo-12
+6C(νµ,µ−), RFG,Eν= 1.2 GeV
+reconstructed Q2
+trueQ2
+Q2/parenleftbig
+GeV2/parenrightbig1038dσ/dQ2/parenleftbig
+cm2/GeV2/parenrightbig
+21.61.20.80.4012
+10
+8
+6
+4
+2
+0
+12
+6C(νµ,µ−), RFG,Eν= 0.8 GeV
+reconstructed Q2
+trueQ2
+Q2/parenleftbig
+GeV2/parenrightbig1038dσ/dQ2/parenleftbig
+cm2/GeV2/parenrightbig
+1 0.80.60.40.20.012
+10
+8
+6
+4
+2
+0
+12
+6C(νµ,µ−), RFG,Eν= 0.4 GeV
+reconstructed Q2
+trueQ2
+Q2/parenleftbig
+GeV2/parenrightbig1038dσ/dQ2/parenleftbig
+cm2/GeV2/parenrightbig
+1 0.80.60.40.20.014
+12
+10
+8
+6
+4
+2
+0
+FIG. 2. (color online). Comparison of the differential cross
+sections dσ/dQ2anddσ/dQ2
+recfor neutrino energy 1.2 GeV
+(top panel), 0.8 GeV (middle panel), and 0.4 GeV (bottom
+panel) calculated within the RFG model.
+cus on the mechanism responsible for the observed struc-
+ture in the case of dσ/dQ2. To find the value of Q2cor-
+responding to the maximum we note that, due to Pauli
+blocking, the cross section may increase with momentum
+transfer up to the value |q|∗= 2pF, needed to knock out
+nucleons sitting at the bottom of the Fermi sea. The3
+0.407 GeV20.607 GeV20.807 GeV2Constant Q2:hadronicleptonicConditions:12
+6C(νµ,µ−), RFG,Eν= 0.8 GeV
+ω(GeV)|q|(GeV)
+0.80.70.60.50.40.30.20.101.8
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0
+0.106 GeV20.306 GeV20.506 GeV2Constant Q2
+rec:hadronicleptonicConditions:12
+6C(νµ,µ−), RFG,Eν= 0.8 GeV
+ω(GeV)|q|(GeV)
+0.80.70.60.50.40.30.20.101.8
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0
+FIG. 3. (color online). Upper panel: curves of fixed Q2in
+the (ω,|q|) plane, superimposed to the kinematically allowed
+region (within the dotted boundaries) for 0.8-GeV muon neu-
+trino scattering. Lower panel: same as in upper panel but for
+fixed values of Q2
+rec.
+energy transfer can be written as
+ω=Ep′−Ep+ε
+=/radicalbig
+M2+(p+q)2−/radicalbig
+M2+p2+ε,(3)
+whereM= (Mn+Mp)/2, while Epandp(Ep′and
+p′=p+q) are the neutron (proton) energy and mo-
+mentum. The above equation shows that, for any given
+|p|, knocking out nucleons with momenta parallel to q
+requires the highest energy. As a consequence, the mo-
+mentum and energy transfer |q|∗andω∗, with
+ω∗=/radicalbig
+M2+(3pF)2−/radicalBig
+M2+p2
+F+ε(4)
+correspond to the maximum of the Q2distribution. The
+position of the maximum, Q2= 0.146 GeV2, is indepen-
+dent of neutrino energy, as long as Eνis high enough for
+ω∗and|q|∗tolie within the kinematicallyallowedregion,
+and not too close to its boundary. The bottom panel of
+Fig. 2 shows that for Eν= 0.4 GeV this condition is not
+fulfilled.
+The knee of the cross section, particularly visible for
+Eν= 0.8 GeV, results from the phase space shrinkage
+above a certain Q2. The boundaries of the kinematically
+allowed region in the ( ω,|q|) plane, determined from theconditions
+Eν−|k′| ≤ |q| ≤Eν+|k′|,
+|p′|−|p| ≤ |q| ≤ |p′|+|p|,
+0≤ |p| ≤pF,(5)
+where|k′|=/radicalBig
+E2
+k′−m2µand|p′|=/radicalBig
+E2
+p′−M2, are
+h−pF≤ |q| ≤h+pF,
+Eν−l≤ |q| ≤Eν+l,(6)
+with
+h=/radicalbig
+(EF+/tildewideω)2−M2,
+l=/radicalBig
+(Eν−ω)2−m2µ,
+EF=/radicalBig
+M2+p2
+F,
+/tildewideω=ω−ε.(7)
+AsillustratedintheupperpanelofFig.3, wherethekine-
+matically allowedregion for Eν= 0.8 GeV lies within the
+dotted lines corresponding to the above conditions, the
+availablephasespacestartstodecreaseabove Q2= 0.607
+GeV2. As a consequence, the differential cross section
+also decreases, exhibiting a knee starting at this value
+ofQ2. Comparison between the results corresponding to
+Eν= 0.8GeVand1 .2GeVshowsthatastheneutrinoen-
+ergy gets higher this effect becomes less significant. This
+pattern can be understood consideringthat for higher Eν
+the reduction of the phase space starts at higher |q|and
+Q2, where the cross section is smaller. In addition, the
+shrinkage of the phase space is less pronounced, as the
+allowedQ2range is much broader.
+The relation between Q2andQ2
+recis by no means sim-
+ple. For example, knowing only the energy loss ωand
+the momentum transfer |q|, in MC simulation we cannot
+determine Q2
+rec; oneadditional independent quantity, e.g.
+the neutrino or muon energy, is necessary. This implies
+that using Q2
+recwe are not describing the intrinsic target
+response: some information on the interaction vertex is
+also involved. To calculate Q2
+recwe may use the equality
+Eµ−|k′|cosθ=Q2+m2
+µ
+2Eν, (8)
+following from the identity
+k·k′≡ −1
+2/bracketleftbig
+(k−k′)2−m2
+µ/bracketrightbig
+, (9)
+wherek= (Eν,k) andk′= (Eµ,k′) are the neutrino
+and muon four-momenta, respectively. When we map
+Q2ontoQ2
+recusing Eqs. (1) and (8), the resulting val-
+ues are shifted as listed in Table I, which explains why
+the structures in dσ/dQ2anddσ/Q2
+recappear at differ-
+ent positions. The lower panel of Fig. 3 illustrates the
+reason of the enhancement of the cross sections’ knees,
+which turns them into bumps: where the phase space for4
+TABLE I. Position of the knees or bumps of the distributions
+shown in Fig. 2.
+Eν(GeV) ω(GeV) |q|(GeV) Q2(GeV2)Q2
+rec(GeV2)
+0.4 0.2536 0.5013 0.187 0.107
+0.8 0.5923 0.9788 0.607 0.306
+1.2 0.9576 1.4181 1.094 0.494
+Q2
+rec= 0.306 GeV2Q2= 0.607 GeV2hadronicleptonicConditions:12
+6C(νµ,µ−), RFG,Eν= 0.8 GeV
+ω(GeV)|q|(GeV)
+0.80.70.60.50.40.30.20.101.8
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0
+FIG. 4. (color online). Comparison of curves of fixed Q2
+andQ2
+reccorresponding to the knee (bump) of the differential
+cross section dσ/dQ2(dσ/Q2
+rec).
+dσ/dQ2shrinks, we observe an increase of the allowed
+range of Q2
+rec; the behavior in the two variables is thus
+completely different.
+In Fig. 4 we compare the lines of constant Q2= 0.607
+GeV2andQ2
+rec= 0.306 GeV2corresponding to Eν= 0.8
+GeV. The results clearly show that the two variables, in
+spite ofleadingto verysimilardistributions ofevents, are
+quite different. It clearly appears that the Q2andQ2
+rec
+distributions are determined by the nuclear response in
+different regions of the ( ω,|q|) plane, the latter being
+sensitive to significantly lower values of the momentum
+transfer. This feature implies that the Q2
+recdistribu-
+tion receives contributions from the kinematical region
+in which the validity of the IA may become questionable.
+III. BREAKDOWN OF THE IMPULSE
+APPROXIMATION
+In order to pin down the boundary of the kinemati-
+cal region in which the IA is applicable, we have studied
+the response of nuclear matter (a translationally invari-
+ant system consisting of equal number of protons and
+neutrons subject to strong interactions only) to a scalar
+probe delivering momentum qand energy ω:
+S(q,ω) =/summationdisplay
+n/angbracketleft0|ρ†
+q|n/angbracketright/angbracketleftn|ρq|0/angbracketrightδ(ω+E0−En).(10)
+In the above equation, the operator ρq=/summationtext
+pa†
+p+qap,
+a†
+pandapbeing nucleon creation and annihilation op-
+erators, describes the fluctuations of the target densityinduced by the interaction with the probe. The target
+ground and final states, |0/angbracketrightand|n/angbracketright, are eigenstates of
+the nuclear Hamiltonian Hbelonging to the eigenvalues
+E0andEn, respectively. Note that scattering cross sec-
+tion of a scalar probe (e.g. thermal neutrons scattering
+off liquid helium) can be written in the simple form
+dσ
+dΩ=/parenleftbiggdσ
+dΩ/parenrightbigg
+0S(q,ω), (11)
+where (dσ/dΩ)0is the elementary cross section describ-
+ingscatteringoffindividualtargetconstituents, whilethe
+response defined in Eq. (10) is an intrinsic property, fully
+determined by internal target dynamics.
+In the case of electron- or neutrino-nucleus scattering,
+Eq. (10) can be readily generalized, replacing the density
+fluctuation operator with the operators describing the
+electromagneticandweaknuclearcurrents. Theresulting
+response tensor reads
+Wνµ=/summationdisplay
+n/angbracketleft0|J†
+ν|n/angbracketright/angbracketleftn|Jµ|0/angbracketrightδ4(q+p0−pn).(12)
+In the IA regime, Eq. (10) can be rewritten in the form
+S(q,ω) =/integraldisplay
+d3pdE P h(p,E)Pp(p+q,ω−E),(13)
+wheretheholeandparticlespectralfunctions, PhandPp,
+describe the energy and momentum distributions of the
+struck particle in the initial and final states, respectively.
+The simplest implementation of the IA (usually referred
+to as plane wave impulse approximation, or PWIA) is
+based on the further assumption that final state inter-
+actions (FSI) between the knocked out nucleon and the
+spectator particles can be neglected. As a consequence,
+the nuclear matter particle spectral function reduces to
+Pp(p+q,ω−E) = [1−4
+3πp3
+Fn(p+q)]
+×δ(ω−E+M−Ep+q),(14)
+wheren(p+q) is the occupation probability of the single
+particle state of momentum p+qandEpdenotes the en-
+ergy of a free nucleon carrying momentum p. Note that,
+in the above definition, Pauli blocking of the phase space
+available to the struck nucleon is treated in a consistent
+fashion, using the momentum distribution obtained from
+the hole spectral function through
+n(p) =/integraldisplay
+dE Ph(p,E). (15)
+In most calculations of electron- and neutrino-nucleus
+scattering cross sections, 4 πp3
+Fn(p)/3 is replaced with
+the RFG result θ(pF−|p|), yielding
+Pp(p′,E) =/bracketleftbig
+1−θ(pF−|p′|)/bracketrightbig
+δ(E+M−Ep′)
+=θ(|p′|−pF)δ(E+M−Ep′).(16)5
+It has to be pointed out that the above prescriptions,
+while being justified in the case of uniform nuclear mat-
+ter, are questionable when applied to nuclei. In nuclear
+matter, due to translation invariance, the linear momen-
+tum is a goodquantumnumber, that can be used tolabel
+single particle states. As a consequence, the momentum
+distribution also provides the occupation probability of
+the states. In finite nuclei, on the other hand, single
+particle states must be labeled according to the total an-
+gular momentum J. In this case, for any given p,n(p)
+receives contributions from states of different J, and may
+even exceed unity.
+The available results of accurate nuclear matter calcu-
+lationscanbeusedtomodeltheparticlespectralfunction
+of finite nuclei within the framework of the local density
+approximation [5], i.e. using the definition of Eq. (14)
+with
+4
+3πp3
+Fn(p)→/integraldisplay
+d3r4
+3πp3
+FnNM[ρ(r),p]ρA(r),(17)
+whereρA(r) is the nuclear density distribution, normal-
+ized to unity, and nNM[ρ(r),p] is the momentum distri-
+bution of nuclear matter at uniform density ρ(r). This
+procedure has been used in all calculations of nuclear
+cross sections discussed in this paper.
+Within the nonrelativistic approximation, in which
+both the response and the hole spectral function can be
+evaluated using realistic nuclear Hamiltonians, the valid-
+ity of the IA can be tested comparing S(q,ω) of Eq. (10)
+to
+SPWIA(q,ω) =/integraldisplay
+d3pdE P h(p,E)/bracketleftbigg
+1−4π
+3p3
+Fn(p+q)/bracketrightbigg
+×δ/parenleftbigg
+ω−E−|p+q|2
+2M/parenrightbigg
+, (18)
+for different values of the momentum transfer q.
+The nuclear matter S(q,ω) at equilibrium density,
+ρ0= 0.16 fm−3(corresponding to pF= 262.4 MeV), has
+been recently computed using the correlated basis func-
+tion formalism and an effective interaction derived from
+a state-of-the-art parametrization of the nucleon-nucleon
+potential [6]. To analyze the interplay between short-
+and long-range correlations, the response defined as in
+Eq. (10) has been evaluated in both the Hartree-Fock
+and Tamm-Dancoff approximations.
+Figure 5 shows a comparison between the responses
+of Ref. [6], obtained using Eq. (10) and the correlated
+Hartree-Fock approximation, and those obtained from
+Eq. (18) using the nuclear matter hole spectral function
+of Ref. [7]. The main difference between the two calcu-
+lations lies in the treatment of the target final state. In
+the IA scheme the state describing the struck particle is
+factored out, while in the approach of Ref. [6] the final
+A-nucleon state includes both statistical and dynamical
+correlations between the struck particle and the specta-
+tors.To make the comparison fully consistent, the PWIA
+response has been computed including only the contribu-
+tions of one-hole final states to Ph(p,E). The results ofFIG. 5. Energy dependence of the nuclear matter response
+S(q,ω). Solid lines: correlated Hartree-Fock approxima-
+tion [6]. Dashed lines: PWIA results, obtained from Eq. (18)
+usingthe SFof Ref. [7]. Dot-dashedlines: results ofthe Fer mi
+gas model (with nonrelativistic energies) at kF= 262.4 MeV,
+corresponding to the equilibrium density of nuclear matter .
+The panels are labeled according to the value of |q|.
+the Fermi gas model with nonrelativistic kinetic energy
+spectrum are also displayed.
+The results of Fig. 5 clearly show that at |q|<2pF
+the response obtained from Eq. (18) does not exhibit
+the linear behavior at low ωresulting from the antisym-
+metrization of the final state. On the other hand, as the
+momentum transfer increases, the PWIA response draws
+closer to the one obtained in Ref. [6]. At |q| ∼600 MeV
+the results of the two approaches are within 10% of one
+another in the region of the maximum. Note that inclu-
+sion of dynamical FSI, e.g. according to the approach
+of Ref. [8], would produce a quenching of the PWIA re-
+sponse in the top panel of Fig. 5, thus bringing the solid
+and dashed lines in even better agreement. Theoretical
+studies of electron-nucleus scattering [15] suggest that at
+|q| ∼600 MeV, the FSI effect at the quasifree peak is
+∼10 %.
+The emerging pattern suggests that the assumptions
+underlyingtheIAarelikelytobevalidatmomentalarger
+than∼2pF, while at lower |q|factorization does not ap-
+pear to providean adequate description ofthe final state.6
+|q|>0.45 GeV
+all|q|SF,×1.12:κ-fit
+Q2
+rec(GeV2)105events/GeV2
+1 0.80.60.40.206
+5
+4
+3
+2
+1
+0
+FIG. 6. (color online). Comparison of the MiniBooNE
+parametrization of the data (dotted line), labeled as the κ-
+fit, to the spectral function calculation (dashed line). The
+solid line depicts the contribution to the latter from the re -
+gion where the IA is expected to be valid. The SF results are
+multiplied by a factor 1.12 to make them match the κ-fit.
+In addition, it has to be kept in mind that at |q| ∼pF
+long-rangecorrelations,involvingmorethan onenucleon,
+also play a significant role [6].
+Despite the fact that, being based on the nonrelativis-
+tic approximation, the approach of Ref. [6] should not
+be used in the calculation of the MiniBooNE event dis-
+tribution, it helps to clarify why the Q2
+recdistribution
+of events measured by this experiment can be described
+by the RFG model [2] and the SF approach only for
+Q2
+rec≥0.25 GeV2.
+To improve the description at lower Q2, the Mini-
+BooNE Collaboration introduced the ad hocadditional
+parameter κin the RFG model. While the authors of
+Ref. [2] argue that this procedure allows for a better
+treatment of Pauli blocking, we think that the inclusion
+ofκcannot be justified on physics ground. Therefore, we
+refer to the modified RFG model of Ref. [2] as the κ-fit.
+Figure6showsthatintheregionoflowervaluesof Q2
+rec
+the contribution of |q|’s below 2 pFis dominant. Hence,
+to explain the neutrino-nucleus cross section dσ/dQ2
+recat
+quantitative level in the whole range of Q2
+recone needs a
+consistent description of both short- and long-range cor-
+relations. Unfortunately, due to the difficulties involved
+in the treatment of relativistic particles in the final state,
+such a description has not been fully developed yet.
+IV. EXPERIMENTAL ISSUES
+The average axial mass extracted by past experiments
+turns out to be significantly lower than the values re-
+cently reported by K2K and MiniBooNE [9]. Although
+the early experiments were carried out using a variety of
+targets,about60%ofthetotalnumberofrecordedevents
+were collected on deuteron. The results of deuteron mea-
+surement with highest statistics [10–12] are very consis-
+tent with one another and have small error bars. Com-pared to deuteron experiments, those carried out with
+heavier targets have much poorer statistics, the typical
+number of events being lower by one order of magnitude.
+Moreover, using a deuteron target allows one to mini-
+mize the systematic error associated with the treatment
+of nuclear effects.
+A lower value of the axial mass is also supported by
+the result recently obtained by the NOMAD Collabora-
+tion [13] using a carbon target.
+On the other hand, MiniBooNE collected more events
+than all other experiments combined, and carried out
+an analysis based on the shape of the reconstructed Q2
+distribution. MiniBooNE reported 193709 events sur-
+viving the cuts, of which about 180000 correspond to
+0≤Q2
+rec<1 GeV2. In the analysis not involving the
+additional parameter κ, two cuts were applied: one at
+lowQ2
+rec, to exclude the region where the RFG model
+is expected to break down, and one at high Q2
+rec, to ex-
+clude the region of low statistics. After rejection of the
+events excluded by the cuts, the data sample employed
+to extract the axial mass reduced to ∼112000 events, i.e.
+∼62% of the total. Note, however, that this figure is still
+∼60 times larger than the number of events typically col-
+lected in deuteron experiments.
+A. Identification of CCQE events
+The first problem to be addressed in the comparison
+betweentheoreticalcalculationsand experimentalresults
+is background simulation, which is, to a significant ex-
+tent, detector dependent.
+The extraction of the axial mass requires an accurate
+selection of the CCQE events and a quantitative descrip-
+tion of the irreducible backgrounds which may change
+the topology of the observed event. For example, a
+∆-production process followed by pion absorption may
+be undistinguishable from a CCQE interaction, whereas
+a primary CCQE process may yield pions due to final
+state interactions. Therefore, a more complete theoreti-
+cal analysis should account for intranuclear cascade in a
+consistent way.
+Although cascade calculations including nucleon-
+nucleon correlations have not been developed yet, we can
+gauge the relevance of these effects using the results of a
+comparison of the MC generators employed in the anal-
+ysis of neutrino experiments, reported in Ref. [14]. The
+predicted ratio between trueandobserved CCQE events
+turns out to be in the range 82%–89% for a 1-GeV νµ
+beam and oxygen target. On the other hand, the frac-
+tion of events misidentified as inelastic due to pion pro-
+duced in final state interactions is typically 0.5%–3%.
+The uncertainty in the modeling of pion production and
+absorption may be partly responsible for the disagree-
+ment between the values of the axial mass reported by
+K2K and MiniBooNE and those resulting from different
+experiments.
+Scattering off a correlated pair of nucleons may also7
+be misinterpreted as pion production, as it produces two
+hadron tracks in the final state. However, even neglect-
+ing nucleon absorption this background does not exceed
+3% of the CCQE events in the MiniBooNE kinematical
+range, and is therefore not likely to be significant. Be-
+cause of the weaker Q2dependence, resulting from the
+higher nucleon removal energy, a proper interpretation
+of these events may only marginally increase the value of
+axial mass extracted from the analysis.
+The effect of FSI, negligible in light nuclei and not
+taken into account in this paper, is expected to make the
+Q2
+recdistribution flatter, quenching its maximum and re-
+distributing strength towards higher values of the four-
+momentum transfer [15]. This behavior is due to the
+fact that FSI couple one-hole states to one-particle–two-
+hole states, thus leading to an increase of the average
+removal energy. As a result, the neutrino-nucleon inter-
+action takes place at higher Q2. Hence, part of the dis-
+crepancy between the results of K2K and MiniBooNE,
+on the one hand, and deuteron-based experiments, on
+the other hand, may be ascribable to FSI effects.
+It is also very important to realize that the influence
+of reaction mechanisms beyond the IA cannot be mini-
+mized increasing the beam energy [16]. It turns out that
+low momentum transfers ( |q|/lessorsimilarpF) provide almost the
+same contributions to the CCQE cross section for neu-
+trino energy 0.4 GeV and 100 GeV. In the absence of a
+fully consistent theoretical description of all the relevant
+mechanisms, the most reasonable option for the experi-
+mental analysis appears to be a cutoff, to reject events
+with|q|/lessorsimilar2pF.
+It should be kept in mind that in the region where the
+IA is not valid the cross section may not scale with the
+number of nucleons. As a consequence, the axial mass
+(or any other parametrization of the axial form factor)
+extracted from neutrino-nucleus data at low Q2
+recmay
+turn out to be target dependent.
+B. Proposal of new kinematical variables
+The axial mass is currently extracted from experimen-
+taldatausingthe shapeofthe Q2
+recdistributionofCCQE
+events. However, as pointed out in Section II, Q2
+reccan-
+not be directly measured; its definition [see Eq. (2)] in-
+volves the nucleon separation energy and depends on the
+applied approximations, e.g. the assumption that the
+struck nucleon be at rest.
+It may be convenient replacing Q2
+recwith the new,
+model independent, variable
+β=Eµ−|k′|cosθ, (19)
+whosedefinitiononlyinvolvesmeasuredquantities. From
+β=k·k′
+Eν=Q2+m2
+µ
+2Eν, (20)
+it follows that the βdistribution exhibits the same be-
+havior as the Q2
+recdistribution, and can be comparablySF
+κ-fitFG
+β(GeV)105events/GeV
+0.80.70.60.50.40.30.20.109
+8
+7
+6
+5
+4
+3
+2
+1
+0
+FIG. 7. (color online). Distribution of the MiniBooNE data
+calculated using the “ κfit” (dashed line), the RFG model
+(dotted line), and the SF approach (solid line) as a function
+of the variable β, defined in Eq. (19).
+useful in the extraction of the axial mass or in the anal-
+ysis of nuclear effects. Figure 7 shows the distributions
+obtained for the MiniBooNE flux as a function of β.
+An even better choice appears to be provided by the
+variable
+φ=1
+mµ+β. (21)
+Figure 8, showing the φdistributions of the Mini-
+BooNE data obtained using different approaches, clearly
+illustrates that the main advantage of using φlies in the
+fact that the deviations of the κfit from the RFG model,
+reflecting the breakdown of the IA, show up in the high
+φtail of the distribution. As a consequence, a single cut
+atφ= 3 GeV−1allows one to reject both the region of
+low statistics and the region where effects beyond the IA
+are expected to be important. Hence, the extraction of
+the axial mass from the φdistribution would be based on
+a larger data sample. Moreover, the dependence of the φ
+distribution on the axialmassturns out to be onlyvisible
+atφ >1.6 GeV−1, correspondingto the region of highest
+statistics. Note that in the case of the Q2
+recdistribution
+the situation is reversed.
+V. CONCLUSIONS
+In this paper, we have analyzed possible sources of the
+observed disagreement between the Q2
+recdistributions of
+CCQEeventsreportedby severalrecent experimentsand
+the prediction of Monte Carlo simulations.
+As far as the treatment of nuclear effects is concerned,
+our work suggests that, in addition to the known limita-
+tions of the RFG model, discussed in, e.g., Refs. [15, 17],
+the most critical feature of present analyses is the as-
+sumptionthattheIAschemebeapplicableoverthewhole
+range of Q2. Electron scattering studies have provided
+ample evidence that the IA breaks down in the region
+of low momentum transfer, which turns out to provide a8
+SF
+κ-fitFG
+φ(1/GeV)104events·GeV
+10864209
+8
+7
+6
+5
+4
+3
+2
+1
+0
+FIG. 8. (color online). Same as in Fig. 7, but plotted as a
+function of the variable φ, defined in Eq. (21).
+significant fraction of the observed CCQE events, inde-
+pendent of neutrino energy.
+The failure of the IA is clearly exposed by the com-
+parison between the full nuclear matter response and the
+IA result, discussed in Section III, showing that at mo-
+mentum transfers |q|<2pFthe contributions of more
+complex reactionmechanismsbecome important, oreven
+dominant.
+While our calculations focus on the effects of factor-
+ization of the final state, it must be pointed out that dif-
+ferent mechanisms should also be considered. For exam-
+ple, meson exchange currents, which are long known to
+provide appreciable contributions to the electron-nucleus
+cross section at the quasi elastic peak and beyond, are
+also expected to contribute to the backgroundin the case
+of neutrino-nucleus scattering.
+The development of a consistent treatment of scat-
+tering processes at low and high momentum transfer
+within a formalism easily implementable in MC simu-
+lation, while being feasible, involves severe difficulties,and will require a significant effort in the years to come.
+On the other hand, we believe that introducing ad hoc
+modifications of the available models, lacking a sound
+physical interpretation, will not help to clarify the origin
+of the disagreement between theoretical predictions and
+observations.
+Among the issues related to data analysis, identifica-
+tion of CCQE processes appears to be prominent. The
+discussion of Section IVA, based on the results of MC
+simulations, indicates that it may be at least partly re-
+sponsible for the disagreement between the values of the
+axial mass recently reported by K2K and MiniBooNE
+and those obtained from different measurements.
+Improving event identification will require a more real-
+istic description of FSI, combining the intranuclear cas-
+cade approach with a fully realistic description of the
+target nucleus, and including the relevant inelastic chan-
+nelsleadingtopionproduction. Thekeyelementsneeded
+to pursue this project, i.e. in-medium nucleon and pion
+cross sections and nuclear wave functions including cor-
+relation effects, can be extracted from the available data
+and from theoretical results of accurate many-body cal-
+culations.
+Finally, we suggest that data analysis might be im-
+proved using a new kinematical variable which, unlike
+Q2
+rec, canbe defined in terms ofmeasuredquantitiesonly.
+In addition, using the new variablewould allowone to re-
+ject both the regionoflow statistics and the regionwhere
+effects beyond the IA are expected to be important with
+a single cut. As a result, the extraction of the axial mass
+would be based on higher event statistics.
+ACKNOWLEDGMENTS
+A.M.A. was supported by the Polish Ministry
+of Science and Higher Education under Grant No.
+550/MOB/2009/0. O.B. and N.F. were supported by
+INFN under Grant PI31.
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+[3] K. Hiraide et al.(SciBooNE Collaboration), Phys. Rev.
+D78, 112004 (2008).
+[4] O. Benhar and D. Meloni, Phys. Rev. D 80, 073003
+(2009).
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+Phys.A579, 493 (1994).
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+A505, 267 (1989).
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+Pandharipande, and I. Sick, Phys. Rev. C 44, 2328(1991).
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+Phys. J. C 54, 517 (2008).
+[10] T. Kitagaki et al., Phys. Rev. D 42, 1331 (1990).
+[11] K.L. Miller et al., Phys. Rev. D 26, 537 (1982).
+[12] N.J. Baker et al., Phys. Rev. D 23, 2499 (1981).
+[13] V.V. Lyubushkin et al.(NOMAD Collaboration), Eur.
+Phys. J. C 63, 355 (2009).
+[14] M. Antonello et al., Acta Phys. Pol. B 40, 2519 (2009).
+[15] O. Benhar, N. Farina, H. Nakamura, M. Sakuda, and
+R. Seki, Phys. Rev. D 72, 053005 (2005).
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\ No newline at end of file
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+arXiv:1001.0484v1  [astro-ph.IM]  4 Jan 2010MESO-NH SIMULATIONS OF THE ATMOSPHERIC
+FLOW ABOVE THE INTERNAL ANTARCTIC PLATEAU
+Franck Lascauxa, Elena Masciadria, Susanna Hagelina,band Jeff Stoesza
+aINAF/Osservatorio Astrofisico di Arcetri, 5 L.go E. Fermi, 50125 F lorence, Italy;
+bUppsala Universitet, Department of Earth Sciences, Villav¨ agen 16 , S-752 36 Uppsala, Sweden
+ABSTRACT
+Mesoscale model such as Meso-Nh have proven to be highly reliable in r eproducing 3D maps of optical turbu-
+lence (see Refs. 1, 2, 3, 4) above mid-latitude astronomical sites. These last years ground-based astronomy has
+been looking towards Antarctica. Especially its summits and the Inte rnal Continental Plateau where the optical
+turbulence appears to be confined in a shallow layer close to the icy su rface. Preliminary measurements have
+so far indicated pretty good value for the seeing above 30-35 m: 0.3 6” (see Ref. 5) and 0.27” (see Refs. 6, 7) at
+Dome C. Site testing campaigns are however extremely expensive, in struments provide only local measurements
+and atmospheric modelling might represent a step ahead towards th e search and selection of astronomical sites
+thanks to the possibility to reconstruct 3D C2
+Nmaps over a surface of several kilometers. The Antarctic Plateau
+represents therefore an important benchmark test to evaluate the possibility to discriminate sites on the same
+plateau. Our group8has proven that the analyses from the ECMWF global model do not d escribe with the re-
+quired accuracy the antarctic boundary and surface layer in the p lateau. A better description could be obtained
+with a mesoscale meteorological model. In this contribution we prese nt the progress status report of numerical
+simulations (including the optical turbulence - C2
+N) obtained with Meso-Nh above the internal Antarctic Plateau.
+Among the topic attacked: the influence of different configuration s of the model (low and high horizontal res-
+olution), use of the grid-nesting interactive technique, forecast ing of the optical turbulence during some winter
+nights.
+Keywords: Atmospheric effects, optical turbulence, mesoscale model, site te sting
+1. INTRODUCTION
+The Internal Antarctic Plateau is, at present, a site of potential great interest for astronomical applications. The
+extreme low temperatures, the dryness, the typical high altitude of the Internal Antarctic Plateau (more than
+2500 m), joint to the fact that the optical turbulence seems to be concentrated in a thin surface layer whose
+thickness is of the order of a few tens of meters do of this site a plac e in which, potentially, we could achieve
+astronomical observations otherwise possible only by space.
+In spite of the exciting first results (see Refs. 6, 9, 7) the uncert ainties on the effective gain that astronomers
+might achieve from ground-based astronomical observation from this location still suffers from serious uncer-
+tainties and doubts that have been pointed out in previous work (se e Refs. 10, 11, 8). A better estimate of the
+properties of the optical turbulence above the Internal Antarc tic Plateau can be achieved with both dedicated
+measurements done in simultaneous ways with different instruments and simulations provided by atmospheric
+models. Simulations offer the advantage to provide volumetric maps o f the optical turbulence ( C2
+N) extended on
+the whole Internal Plateau and, ideally, to retrieve comparative es timates in a relative short time and homoge-
+neous way on different places of the Plateau. In a previous paper8our group performed a detailed analysis of the
+meteorological parameters from which the optical turbulence dep ends on, provided by the General Circulation
+Model (GCM) of the European Center for Medium-range Weather F orecasts (ECMWF). In that work we quan-
+tified the accuracy of the ECMWF estimates of all the major meteor ological parameters and, at the same time,
+we pointed out which are the limitations of the General Circulation Mod els. In contexts in which the GCMs fail,
+mesoscale models can supply more information. The latter are indeed conceived to reconstruct phenomena (such
+Send correspondence to Franck Lascaux and Elena Masciadri.
+E-mail: lascaux@arcetri.astro.it - masciadri@arcetri.a stro.it.6000 km
+3000 km 400 km 80 km(b)
+(c)(a)
+(d) (e)
+Figure 1. Orography of Antarctica as seen by the Meson-Nh model (polar stereographic projection) in (a) the monomodel simulation , with
+∆X= 100km; (b) zoom of (a) above the Dome C area (the black square repres ents the same area as in (e)); (c) the dad model of the
+grid-nested simulation with ∆ X= 25km; (d) the second domain of the grid-nested simulation with ∆ X= 5km; (e) the innermost domain
+of the grid-nested simulation with ∆ X= 1km. The dot labelled ’C’ is located at the Concordia Station. Th e dot labelled ’A’ is the Dome
+A.
+as the optical turbulence) that develop at a too small spatial and t emporal scale to be described by a GCM.
+In spite of the fact that mesoscale models can attain higher resolut ion than the GCM, thes parameters such as
+the optical turbulence are not explicitly resolved but are paramete rized, i.e. the fluctuations of the microscopic
+physical quantities are expressed as a function of the correspon ding macroscopic quantities averaged on a larger
+spatial scale (cell of the model). For classical meteorological para meters the use of a mesoscale model should be
+useless if GCMs such as the one of the ECMWF could provide estimate w ith equivalent level of accuracy. For
+this reason the Hagelin et al paper8has been a first step towards the exploitation of the mesoscale Mes o-Nh
+model. We retrieved all what it was possible from the ECMWF analyses a nd we defined their limitations at the
+same time. We concluded that in the first 10-20 m, the ECMWF analyse s show a discrepancy with respect to
+measurements of the order of 2-3 m.s−1for the wind speed and of 4-5 K for the temperature.
+Preliminary tests concerning the optimization of the model configur ation and sensitivity to the horizontal
+and the vertical resolution with the Meso-Nh model have already be en conducted by our team12 for the Internal
+Antarctic Plateau. In this paper we present further progress of that work. More precisely, we intend:
+•To compare the performances of the mesoscale Meso-Nh model an d the ECMWF General Circulation
+Model in reconstructingwind speed and absolute temperature (ma in meteorologicalparametersfrom which
+the optical turbulence depends on) with respect to the measurem ents. This analysis will quantify the
+performances of the Meso-Nh model with respect to the GCM from the ECMWF.
+•To perform simulations of the optical turbulence above Dome C emplo ying different model configurations
+and compare the typical simulated thickness of the surface layer w ith the one measured by Trinquet et al7.
+In this way we aim to establish which configuration is necessary to rec onstruct correctly the C2
+N.2. MESO-NH MESOSCALE MODEL: NUMERICAL SET-UP
+Meso-Nh13is the french non-hydrostatic mesoscale research model develop ed jointly by M´ et´ eo-France and Lab-
+oratoire d’A´ erologie.
+It can simulate the temporal evolution of the three-dimensional at mospheric flow over any part of the globe.
+The prognostic variables forecasted by this model are the three c artesian components of the wind u,v,w, the
+dry potential temperature Θ, the pressure P, the turbulent kinetic energy TKE.
+The system of equation used is based upon the Lipps and Hemler14anelastic formulation allowing for an
+effective filtering of acoustic waves. A Gal-Chen and Sommerville15coordinate on the vertical and a C-grid
+in the formulation of Arakawa and Messinger16for the spatial digitalization is used. The temporal scheme is
+an explicit three-time-level leap-frog scheme with a time filter.17The turbulent scheme is a one-dimensional
+1.5 closure scheme18with the Bougeault and Lacarr` ere19mixing lenght. The surface exchanges are computed
+in an externalized surface scheme (SURFEX) including different phys ical packages, among which ISBA20for
+vegetation.
+Masciadri et al (see Refs. 1, 2) implemented the optical turbulenc e package in order to be able to forecast
+not only the standard meteorological parameters with Meso-Nh, b ut also the optical turbulence ( C2
+N3D maps)
+and all the astroclimatic parameters deduced from the C2
+N. We will refer to the Astro-Meso-Nh code to indicate
+this package. To compare simulations with measurements the integr ated astroclimatic parameters are calculated
+integrating the C2
+Nwith respect to the zenith in the Astro-Meso-Nh-code. The param eterization of the optical
+turbulence and the reliability of the Astro-Meso-Nh model have bee n proved in successive studies in which
+simulationshavebeen comparedto measurementsprovidedby differ entinstruments(See Refs. 4,21). Adedicated
+calibration has been proposed and validated3.
+The atmospheric Meso-Nh model is conceived for research develop ment and for this reason is in constant
+evolution. One of the major advantages of Meso-Nh that was not a valaible at the time of the Masciadri’s studies
+is that it allows now the use of the interactive grid-nesting technique22. This technique consists in using different
+imbricated domains with increasing horizontal resolutions with mesh- sizes that can reach 10 meters.
+We use in this study of the atmosphere above the Antarctica Platea u the same optical turbulence package,
+implemented in the most recent version of the atmospheric Meso-Nh model. We list here down the differences
+in the model configuration that we implemented to do a step ahead in t he research:
+•We use a higher vertical resolution near the ground with respect to previous studies. We still work with a
+logarithimic stretching near the ground up to 3.5 km but we start with a first grid point of 2 m (instead
+of 50 m) with 12 points in the first hundred meters. This configuratio n has been allowed now thanks
+to a better description of the model pressure solver (available in th e new atmospheric Meso-Nh version)
+and it is obviously preferable because it permits to better quantify t he turbulence contribution in the thin
+vertical slabs in the first hundred of meters. Above 3.5 km the vert ical resolution is constant and equal to
+∆H= 600 m as well as in Masciadri’s previous work. The maximum altitude is 22 kilometers.
+•The grid-nesting is implemented with 3 imbricated domains allowing a maxim um horizontal resolution of
+1 km in a region around the telescope ( ∼80 km×80 km).
+•The simulation are forced at synoptic times by analyses from the Eur opean Center for Medium-range
+Weather Forecasts (ECMWF). This permits to perform a real fore cast of the optical turbulence and it
+representsastep aheadwith respect to the Masciadriet al.’sprev iouswork. We highlightindeed, that in all
+the Masciadri’s work the model has been used in a simple domain configu ration permitting a quantification
+of the mean optical turbulence during a night and not really a foreca st of the optical turbulence.
+In spite of the fact that the orographic morphology is almost flat ab ove Antarctica, we know that even a
+weak slope on the ground can be an important factor to induce a cha nge in the wind speed at the surface in
+these regions. The physics of the optical turbulence strongly dep end on a delicate balance between the wind
+speed and temperature gradients. In order to study the sensitiv ity of the model to the horizontal resolution we
+performed two sets of simulations with different model configuratio ns.In the first configuration(that wewill call monomodel ) weuse an horizontalresolution∆ X= 100km covering
+the whole Antarctic continent (Figure 1a,b). This permits to perfor m low time consuming simulations. Such a
+horizontal resolution has been also employed by Swain & Gall´ ee23in a study on the boundary layer seeing done
+with regional atmospheric model MAR.
+In the second configuration we used the grid-nesting technique to test the impact of the high-resolution on
+the optical forecasting. The grid-nested simulations involved thre e domains. The biggest one has a 25 km
+mesh-size and covers all the Antactic Plateau with 120x120 points ( Figure 1c). The second one has a horizontal
+resolution of 5 km, 80x80 points and is centered above the Dome C (F igure 1d). The innermost domain has a 1
+km mesh-size, 80x80 points and is centered above the Concordia St ation area near the Dome C (1e).
+The use of high-resolution has one first major impact: the Dome C ar ea is more fairly reproduced in the
+grid-nested simulation than in the low horizontal resolution simulation (Figure 1b,e). The altitude above mean
+sea level of the Concordia Station with high resolution is around 3230 m, whereas it is around 3200 m with the
+low resolution grid. More over in Fig. 1e we can observe that the Conc ordia Station is not located exactly in
+correspondence of the summit of Dome C but at ∼60 km far away with a slightly lower altitude (∆h ∼15 m).
+Such a morphology can not be distinguished with the mono-model con figuration.
+In section 4 we will compare the typical thickness of the optical tur bulence surface layer reconstructed by the
+model with the two different configurationswith the measurements performed recently7to identify if the model is
+sensitive to this parameter (horizontal resolution) and in case this answer is positive, which configuration better
+matches with observations.
+In the next section we first start to compare results from a Gener al Circulation model (ECMWF) and the
+two configurations from the mesoscale model Meso-Nh.
+All the simulations are initialized and forced every 6 hours with the ana lyses of the European Center of
+Medium-range Weather Forecast (ECMWF).
+Figure 2. Mean temperature profiles at the Concordia Station in the Dom e C area over 47 winter nights (bold lindes) and the correspon ding
+standard deviation (dashed lines). In black, from radiosou ndings; in blue, from the ECMWF analyses; in green, from the M eso-Nh low
+horizontal resolution simulation; in red, from the Meso-Nh grid-nested simulation. On the left, profiles are plotted ov er 20 km. On the right,
+over the first kilometer.
+3. MESO-NH SIMULATIONS IN WINTER ABOVE DOME C - METEOROLOGIC
+PARAMETERS
+The purpose of this section is to verify the performances of the me soscale Meso-Nh model above the Internal
+Antarctic Plateau and to verify if such a mesoscale model can provid e a better estimate of the atmospheric flow
+than a General Circulation model.Radiosondes ECMWF
+Windspeed (m.s−1)4.02 (±2.55) 6.51 (±2.51)
+Temperature (K) 212.90 (±7.64)216.64 (±5.83)
+Table 1. Mean values on 47 winter days at Concordia Station near Dome C of wind speed and temperature at the surface level, for
+radiosoundings, ECMWF analyses. Into brackets, the corres ponding sigma.
+Meso-Nh
+1-MOD Grid-N
+Windspeed (m.s−1)4.23 (±1.77) 3.98 (±1.95)
+Temperature (K) 214.92 (±4.64)214.50 (±4.97)
+TKE (m2.s−2) 0.39 (±0.30) 0.35 (±0.33)
+Table 2. Mean values on 47 days at Concordia Station near Dome C of wind speed and temperature at the surface level, for Meso-Nh
+simulations: 1-MOD is for the low horizontal simulation wit h ∆X=100 km and Grid-N is for the high horizontal grid-nested simulation with
+∆X=1 km for the innermost model centered above the Dome C area . Into brackets, the corresponding sigma.
+An important number of winter nights (47) were simulated with the Me so-Nh mesocale model. We analyze
+here the key meteorologic parameters from which the optical turb ulence depends on: the temperature and
+the wind speed. Both configurations (low horizontal resolution mon omodel, and high horizontal resolution
+grid-nesting) are tested and evaluated. All the simulations start a t 00 UTC and are integrated for 12 hours.
+Simulations outputs at 12 UTC can be compared with measurements ( http://www.climantartide.it ) as well as
+to the analysis from the General Circulation model from the ECMWF. Every 6 hours we force the simulations
+with the ECMWF analyses in order to avoid that the model diverge and /or correct the atmospheric flow as a
+function of the predictions at larger spatial scales.
+In this section a statistical study of the wind and temperature pro files at Concordia Station, Dome C, is
+performed. The 47 nights have been selected in June, July and Augu st 2005 and July 2006. For all the 47 nights
+selected, we respected the following criterion:
+•A radiosounding is available at the end of the simulation (at 12 UTC of th e selected night) to perform
+comparisons between Meson-Nh outputs, ECMWF analysis and obse rvations.
+•For the selected nights, the radiosoundings cover the longest pat h along the z-axis (perpendicolar to the
+ground) before to explode. It was impossible to collect in winter time 4 7 nights in which all the balloons
+reached the 20 km.
+3.1 Temperature profiles at Dome C
+3.1.1 General behaviour of Meso-Nh
+The figure 2 shows the mean temperature profiles at the Concordia Station computed over 47 winter nights from
+the twomodel configurations(low and high horizontalresolution), the ECMWF analysesand the radiosoundings.
+All profiles have been interpolated on a regular 5 m vertical grid, in or der to ease the comparison. The mean
+temperature profiles are very similar over the entire free atmosph ere (Figure 2, left). Some light discrepancies
+appearonthefirstkilometerabovetheground, whereboththeEC MWFanalysesandtheMeso-Nhmodelpresent
+anegativebiasof4-5Kupto200m. Atthis altitude, thegrid-nested simulationandthelowresolutionsimulation
+give similar results and are not necessarily better than the ECMWF. T he temperature gradients provided by
+the two Meso-Nh configurations and the ECMWF analyses are not as pronounced as the one obtained with
+the radiosoundings. In the next section we will analyse the perform ances of ECMWF analyses and Meso-Nh
+model at the very surface level. i.e. the region in which the energetic fluxes budget between the ground and the
+atmosphere takes place. For this reason most of the ability for a mo del in reconstructing the correct atmospheric
+evolution near the ground relies on its ability in reconstructing the te mperature of the surface.
+3.1.2 Surface temperature
+Tables 1, 2, 3 and 4 report the mean and median temperature at the surface measured and simulated by the
+ECMWF analyses and the Meso-Nh model in the two configurations (m ono-model and grid-nesting).Figure 3. Mean wind profiles at the Concordia Station in the Dome C area o ver 47 winter nights (bold lines) and the corresponding stan dard
+deviation (dashed lines). In black, from radiosoundings; i n blue, from the ECMWF analyses; in green, from the Meso-Nh lo w horizontal
+resolution simulation; in red, from the Meso-Nh grid-neste d simulation. Units in m.s−1. On the left, profiles are plotted over 20 km. On the
+right, over the first kilometer.
+One can see that the ECMWF analyses, as already reported in a prev ious paper of our team8, are too warm
+at the surface (almost 4 K for the mean, and 3 K for the median) in win ter with respect to the observations.
+However, the surface temperatures (both mean and median) simu lated by Meso-Nh after 12 hours are closer to
+the observations than the ECMWF analyses.
+The difference between ECMWF analyses and observed mean temper ature is 3.74 K (2.97 K for the median).
+The mean surface temperature in the low-resolution simulation is 2.02 K higher (1.23 K for the median) than in
+the observations. It is only 1.60 K higher for the grid nested simulatio n (0.78 K for the median). One can see
+that the surface temperature are even better retrieved with th e use of the high horizontal resolution (∆X=1 km
+in the innermost domain) than the low horizontal resolution (∆X=100 km).
+3.2 Wind profiles at Concordia Station, Dome C area
+3.2.1 General behaviour of Meso-Nh
+Figure 3 shows the mean wind during the 47 days, with the correspon ding standard deviation. The highest
+mean altitude reached by the balloons is 10 km. From the ground to 10 km, analyses and radiosoundings mean
+wind speed are well correlated. Above 10 km the wind speed reconst ructed by the ECMWF analyses is larger
+Radiosondes ECMWF
+Median 75% 25%Median 75% 25%
+Windspeed (m.s−1)3.10 4.80 2.30 6.08 8.64 4.60
+Temperature (K) 213.10 219.15 205.85 216.03 221.81 211.07
+Table 3. Median values on 47 days and the corresponding quartiles at C oncordia Station of wind speed and temperature at the surfac e
+level, for radiosoundings and ECMWF analyses and relatives percentiles.
+Meso-Nh
+1-MOD Grid-N
+Median 75% 25%Median 75% 25%
+Windspeed (m.s−1)4.08 5.64 2.80 3.42 4.99 2.42
+Temperature (K) 214.33 218.44 210.56 213.88 217.13 210.57
+TKE (m2.s−2) 0.32 0.49 0.14 0.20 0.44 0.10
+Table 4. Median values on 47 days and the corresponding quartiles at C oncordia Station of wind speed and temperature at the surfac e level,
+for Meso-Nh simulations: 1-MOD is for the low horizontal sim ulation with ∆X=100 km and Grid-N is for the high horizontal g rid-nested
+simulation with ∆X=1 km for the innermost model centered at C oncordia Station.than that reproduced by Meso-Nh (monomodel and grid-nesting) (left of figure 3). It is hard to say whether the
+ECMWF analyses or the Meso-Nh simulation is the best since no mean va lue from the observations is available
+at this altitude. On the first kilometer above the ground, and espec ially around 150 m, it is well visible that
+Meso-Nh better reconstructs the strong wind shear than the EC MWF analyses. The wind speed provided by
+the ECMWF analyses is a bit too weak (11 m.s−1instead of 13 m.s−1in the observations). At the same altitude
+the Meso-Nh simulations give better results, with a mean wind profile p erfectly correlated to the one measured
+by the radiosoundings. The improvement is even better in the case o f the high-resolution model. The differences
+between low horizontal and high horizontal simulations is more import ant above 12 km, with an increase in
+intensity of the wind more important in the high-resolution simulation.
+3.2.2 Surface wind
+Tables 1, 2, 3 and 4 show the mean and median values of the wind speed at the first interpolated point of the
+profiles. The mean wind speed in the ECMWF analyses is higher than the observed wind speed (6.51 m.s−1
+against 4.02 m.s−1), thus a difference of 2.49 m.s−1)(∗). The difference in the median is of the same order of
+intensity: 3.02 m.s−1. Both low and high horizontal simulations reproduce better the sur face wind than the
+ECMWF analyses. With a mesh-size of 100 km in Meso-Nh, the differenc e in simulated mean wind speed ans
+observations is of 0.21 m.s−1(0.98 m.s−1for the median). The grid nested simulations give even better result s
+with a difference of 0.04 m.s−1only for the mean and 0.32 m.s−1for the median.
+4. FORECAST OF OPTICAL TURBULENCE: FIRST NIGHTS
+In this section the optical turbulence package developed by our te am1and implemented in our Meso-Nh version
+of the model is used to perform the first simulations of some winter n ights above the Antarctic Plateau.
+All the 11 nights in the winter time (June, July and August) document ed in Trinquet et al7were simulated.
+We present here the first results obtained with Meso-Nh and conce rning the height of the surface layer with the
+observations present in Trinquet et al7.
+As an example, figure 4 displays the temporal evolution of the C2
+Nprofiles for three nights. The temporal
+windows starts at 11 UTC and ends at 17 UTC for each night. All balloo ns have been launched between 14 and
+15 UTC. For each simulation, the average height of the surface laye r is computed for the time period extended
+from 11 UTC to 17 UTC. The optical turbulence is indeed a parameter ized parameter therefore it is a little
+hazardous to consider a precise time forecasted time as in the case of an explicitely resolved parameter such as
+the wind speed or the absolute temperature.
+In order to verify how much simulations match with measurements we compute the typical height of the
+surface layer using the same criterion as in Trinquet et al7. They defined the thickness hslof the surface layer
+as the part containing 90% of the first kilometer boundary layer opt ical turbulence:
+/integraltexthsl
+8mC2
+N(z)dz
+/integraltext1km
+8mC2
+N(z)dz<0.90 (1)
+This criterion is misleading, because it does not take into account the turbulence in the entire free atmosphere.
+We propose to use a different criterion based on the computation of the total integrated C2
+Nover the whole 20
+kilometers:/integraltexthsl
+8mC2
+N(z)dz
+/integraltext20km
+8mC2
+N(z)dz<0.90 (2)
+The observed surface layer thickness for 11 winter nights from Tr inquet et al7are reported on table 5. All mean
+values computed by Meso-Nh are reported in tables 6 (criterion des cribed by Eq.1) and 7 (criterion described by
+Eq.2). Different time intervals were chosen (between 12 UTC and 16 U TC, 11 UTC and 17 UTC, and 12 UTC
+∗Difference from the Hagelin et al paper8values are just due to the fact that in that paper, all the nigh ts of the three
+months (June, July and August) are used while in this paper we simulated just 47 nights selected in two years. The
+statistical sample is not therefore the same.Figure 4. Temporal evolution of the C2
+Nprofile at Concordia Station obtained with Meso-Nh between 1 1 UTC and 17 UTC for (a) 2005 July
+11, low horizontal resolution simulation, (b) 2005 July 11, grid-nested simulation, (c) 2005 July 25, low horizontal re solution simulation, (d)
+2005 July 25, grid-nested simulation, (e) 2005 August 29, lo w horizontal resolution simulation and (f) 2005 August 29, g rid-nested simulation.
+The mean value of the computed mean height of the surface laye r is written on the top-right of each graph (see text for calcu lation details).
+The thick black line represent the evolution of the surface l ayer height.
+and 18 UTC) for the mean calculation. Independently from the time in terval chosen for computing the mean,
+using the criterion of Trinquet et al7the grid-nested simulations give a mean surface thickness f around 48 m
+and the monomodel a thickness of 60 m. Both configurations provid e some higher mean thickness with respect
+to the observed one (36.9 m) but the grid-nested configuration is c loser to observations (∆h ∼10 m). In spite
+of the more expensive simulations, the grid-nested configuration s eems to be necessary to better reconstruct the
+concentration of the turbulence in a thin layer near the surface.
+However, using our criterion (Eq.2), the mean thickness is now 64 m f or the grid-nested simulation (75 m
+for the monomodel). This criterion provides, for both configuratio ns (grid-nested and mono model) a surface
+layer∼15 m thicker than what obtained with the criterion given in Eq.1. This is f ar from being negligible forDate Observed surface
+layer thickness (m)
+13/06/05 23
+04/07/05 30
+07/07/05 21
+11/07/05 98
+18/07/05 26
+21/07/05 47
+25/07/05 22
+01/08/05 40
+08/08/05 30
+12/08/05 22
+29/08/05 47
+Mean 36.9
+Table 5. Surface layer heights for 11 winter nights7. Units in m. The criterion used is the one from Eq.1.
+Date Surface thickness - Meso-Nh - Grid-N Surface thickness - Meso-Nh - 1-MOD
+11-17 UTC 12-16 UTC 12-18 UTC 11-17 UTC 12-16 UTC 12-18 UTC
+13/06/05 23.12 22.37 22.02 27.10 25.99 27.28
+04/07/05 26.18 25.43 25.77 42.14 42.55 41.01
+07/07/05 58.91 58.69 58.55 54.26 54.28 53.42
+11/07/05 81.89 81.36 77.62 110.38 110.01 106.68
+18/07/05 54.15 53.54 51.80 84.87 84.88 81.73
+21/07/05 66.05 67.42 67.19 76.80 75.15 79.26
+25/07/05 17.00 16.56 17.16 25.84 25.62 25.86
+01/08/05 34.62 34.39 32.60 44.51 44.36 42.39
+08/08/05 40.66 41.75 38.52 64.09 69.42 60.36
+12/08/05 28.50 29.76 33.50 37.74 37.82 40.90
+29/08/05 97.09 98.26 95.04 95.15 96.51 93.47
+Mean 48.02 48.14 47.25 60.26 60.60 59.31
+Table 6. Surface layer heights for 11 winter nights deduced from Meso -Nh computations using the criterion in Eq.1. The mean value is also
+reported. Units in m.
+astronomical applications. The solution to overcome the turbulenc e surface layer with a tower might be put in
+serious discussion as a consequence of this result.
+The three nights presented here exhibit different turbulent condit ions. They show the ability of the model to
+react and predict the evolution of different surface layers. During the first night (2005 July 11) the surface layer
+is constant in time, with a mean forecasted thickness of 82 m (with th e grid-nested simulation, fig. 4b). The
+observation for this night7gave a value of 98 m at 14:10 UTC. The second night (2005 July 25) has a forecasted
+mean surface layer height lower (with the grid-nested simulation, fig . 4d): 17 m. The observed value was 22 m
+at 13:53 UTC. The third and last night displayed (2005 August 29) has a forecasted mean surface layer height
+of 97 m (always with the grid-nested simulation, 4f). The observed v alue for this night was 47 m at 14:47 UTC.
+In two of these three nights, the thickness of the surface layer r etrieved by the model is well correlated with
+the observed one. One can notice that for these two nights, the m onomodel simulations give higher values than
+the grid-nested simulations. The third night shows some interesting variability of the surface layer height, which
+are not present in the other nights and that is a signature of an evid ent temporal evolution of the turbulent
+energy distribution even in conditions of a pretty stratified atmosp here. For this night Meso-Nh gives a thickness
+for the surface layer of around 97 m instead of the observed one ( 47 m).Date Grid-N 1-MOD
+11-17 UTC 12-16 UTC 12-18 UTC 11-17 UTC 12-16 UTC 12-18 UTC
+13/06/05 23.70 23.07 22.57 27.82 26.54 27.72
+04/07/05 26.83 26.25 26.38 42.73 43.16 41.88
+07/07/05 59.93 59.47 59.54 55.88 56.13 55.32
+11/07/05 83.48 82.92 79.06 112.07 111.76 108.33
+18/07/05 55.16 54.52 52.70 86.01 85.98 82.82
+21/07/05 173.91 194.67 168.58 156.87 150.07 170.65
+25/07/05 59.30 52.32 74.59 26.16 25.85 26.15
+01/08/05 35.08 34.87 33.03 45.45 45.33 43.31
+08/08/05 55.42 55.66 55.50 142.80 173.55 135.57
+12/08/05 29.40 30.86 34.52 38.69 38.61 41.85
+29/08/05 98.39 99.53 96.20 95.86 97.31 94.26
+Mean 63.70 64.92 63.88 75.49 77.66 75.26
+Table 7. Surface layer heights for 11 winter nights deduced from Meso -Nh computations using the criterion in Eq.2. The mean value is also
+reported. Units in m.
+5. CONCLUSIONS
+In this paper we studied the performances of a mesoscale meteoro logicalmodel, Meso-Nh, in reconstructing wind
+and temperature vertical profiles aboveConcordiaStation, a site in the Internal Antarctic Plateau. Two different
+configurations were tested: monomodel low horizontal resolution , and grid-nesting high horizontal resolution.
+The results were compared to the ECMWF General Circulation Model and radiosoundings.
+Several conclusions can be drawn:
+(1) We showed that near the surface, Meso-Nh retrieved better the wind vertical gradient than the ECMWF
+analyses, thanks to the use of a highest vertical resolution. More over, the analysis of the first vertical
+grid point permits us to conclude that, as is, the Meso-Nh model sur face temperature is closest to the
+observations than the ECMWF General Circulation Model which is too warm.
+(2) The outputs from the grid-nested simulations are closer to the observations than the monomodel simula-
+tions. This study highlighted the necessity of the use of high horizon tal resolution to reconstruct a good
+meteorological field in Antarctica, even if the orography is almost fla t over the Internal Antarctica Plateau.
+The computations estimates from a previous study23are probably affected by the low horizontal resolution
+they used in their simulations.
+(3) For what concerns the optical turbulence, both configuratio ns predict a mean surface layer height higher
+than in the observations. However, it is always inferior at 100 m (whe n we used the criterion 1 of Trinquet
+et al7). The configuration of Meso-Nh giving better estimate (closer to o bservations) is in grid-nesting
+mode.
+(4) The results of Meso-Nh concerning the computation of the mea n thickness of the surface layer are not very
+dependent of the time interval used to average it. This widely simplifie s the analysis of simulations.
+(5) The criterion used in Trinquet et al7appears to be misleading. Indeed, it underestimates the typical
+thickness of the turbulent surface layer. We propose the use of a nother critera (presented in the last
+section of the paper) instead. It gives a mean value higher of aroun d 15 m for the eleven nights. This
+result has an important implication for astronomical application. It is indeed not so realistic to envisage a
+telescope placed above a tower of more than 50 m and the use of the Adaptive Optics remains the unique
+path to follow to envisage astronomical facilities at Dome C.
+This optical turbulence study is based upon a little number of nights ( 11). We need to extend it to a higher
+number of nights in order to have a more reliable statistical analysis o f the results.Inthispaperwedidnotpresentresultsconcerningtheseeing. Wew illfocusourworkaheadonthisparameter,
+discriminating between two different partial contributions: the see ing in the free atmosphere, and the seeing of
+the surface layer. We plan to make comparisons between Meso-Nh o utput and observations.
+ACKNOWLEDGMENTS
+The study has been carried out using radiosoundings from the Prog etto di Ricerca ”Osservatorio Meteo Clima-
+tologico” of the Programma Nazionale di Ricerche in Antartide (PNRA ),http://www.climantartide.it . ECMWF
+analyses are extracted from the MARS catalog, http://www.ecmwf.int . This study has been funded by the Marie
+Curie Excellence Grant (FOROT) - MEXT-CT-2005-023878.
+REFERENCES
+[1] E. Masciadri, J. Vernin and P. Bougeault, ”3Dmapping of optical turbulence using an atmospheric numerical
+model. I: A useful tool for the ground-based astronomy.”, A&AS S, 137, pp. 185-202, 1999.
+[2] E. Masciadri, J. Vernin and P. Bougeault, ”3Dmapping of optical turbulence using an atmospheric numerical
+model. II: First results at Cerro Paranal.”, A&ASS, 137, pp. 203-2 16, 1999.
+[3] E. Masciadri and P. Jabouille, ”Improvementsin the Optical Tur bulence Parameterizationfor 3D simulations
+in a region around a telescope”, A&A, 376, pp. 727-734, 2001.
+[4] E. Masciadri, R. Avila and L.J. Sanchez, ”Statistic reliability of the Meso-Nh atmospherical model for 3D
+C2
+Nsimulations”, RMxAA, 40, pp. 3-14, 2004.
+[5] A. Agabi, E. Aristidi, M. Azaouit, E. Fossat, F. Martin, T. Sadibe kova, J. Vernin and A. Ziad, ”First whole
+atmosphere nighttime seeing measurements at Dome C, Antarctica ”, PASP, 118, pp. 344-348, 2006.
+[6] J. Lawrence, M. Ashley, A. Tokovinin, T. Travouillon, ”Exceptio nal astronomical seeing conditions above
+Dome C in Antarctica”, Nature, 431, pp. 278-281, 2004.
+[7] H. Trinquet, A. Agabi, J. Vernin, M. Azout, E. Aristidi and E. Fo ssat, ”Nighttime optical turbulence vertical
+str ucture above Dome C in Antarctica”, PASP, 120, pp. 203-211, 2008.
+[8] S. Hagelin, E. Masciadri, F. Lascaux and J. Stoesz, ”Compariso n of the atmosphere above the South Pole,
+Dome C and Dome A: first attempt”, MNRAS, accepted.
+[9] E. Aristidi, K. Agabi, M. Azouit, E. Fossat, J. Vernin, T. Travou illon, J.S. Lawrence, C. Meyer, J.W.V.
+Storey, B. Halter, W.L Roth and V. Walden, ”An analysis of temperat ures and wind speeds above Dome C,
+Antarctica”, A&A, 430, pp. 739-746, 2005.
+[10] E. Masciadri, F. Lascaux, J. Stoesz, S. Hagelin and K. Geissler , ”A different ”glance” to the site testing
+above Dome C”, Roscoff Arena Workshop 13-20 October 2006.
+[11] K. Geissler and E. Masciadri, ”Meteorological parameter analy sis above Dome C using data from the
+European Centre for Medium-Range Weather Forecasts”, PASP, 118, 845, pp. 1048-1065, 2006.
+[12] F.Lascaux,E.Masciadri,J.StoeszandS.Hagelin,”Mesosca lesimulationsaboveAntarcticaforastronomical
+applications: first approaches”, Symposium on Seeing, Kona-Hawa ii, 20-22 March 2007. Proceeding available
+athttp://weather.hawaii.edu/symposium/publications .
+[13] J. P. Lafore, J. Stein, N. Asencio, P. Bougeault, V. Ducrocq , J. Duron, C. Fischer, P. Hereil, P. Mascart,
+V. Masson, J.-P. Pinty, J.-L. Redelsperger, E. Richard and J. Vil` a- Guerau de Arellano, ”The Meso-Nh
+atmosphericsimulationsystem.PartI:Adiabaticformulationandco ntrolsimulations”, AnnalesGeophysicae,
+16, pp. 90-109, 1998.
+[14] F. Lipps and R. S. Hemler, ”A scale analysis of deep moist convec tion and some related numerical calcula-
+tions”, J. Atmos. Sci., 39, pp. 2192-2210, 1982.
+[15] T. Gal-Chen and C.J. Sommerville, ”On the use of a coordinate tr ansformation for the solution of the
+Navier-Stokes equations”, J. Comput. Phys., 17, pp. 209-228, 1 975.
+[16] A. Arakawa and F. Messinger, ”Numerical methods used in atm ospheric models”, GARP Tech. Rep., 17,
+WMO/ICSU, Geneva, Switzerland, 1976.
+[17] R. Asselin, ”Frequency filter for time integration”, Mon. Weat her. Rev., 100, pp. 487-490, 1972.
+[18] J.Cuxart, P.Bougeaultand J.-L.Redelsperger, ”Aturbulen ce schemeallowingformesoscaleand large-eddy
+simulations”, Q. J. R. Meteorol. Soc., 126, pp. 1-30, 2000.[19] P. Bougeault and P. Lacarr` ere, ”Parameterization of orog raphic induced turbulence in a mesobeta scale
+model”, Mon. Weather. Rev., 117, pp. 1972-1890, 1989.
+[20] J. Noilhan and S. Planton, ”A simple paramterization of land surf ace processes for meteorological models”,
+Mon. Weather. Rev., 117, pp. 536-549, 1989.
+[21] E. Masciadri and S. Egner, ”First Seasonal Study of Optical Turbulence with an Atmospheric Model”,
+PASP, 118, 849, pp. 1604-1619, 2006.
+[22] J. Stein, E. Richard, J. P. Lafore, J.-P. Pinty, N. Asencio and S. Cosma, ” High-resolution non-hydrostatic
+simulations offlashfloodepisodes with grid nestingand ice phase para meterization”, Meteorol.Atmos. Phys.,
+72, pp. 203-221, 2000.
+[23] M. Swain and H. Gall´ ee, ”Antarctic boundary layer seeing”, P ASP, 118, pp. 1190-1197, 2006.
\ No newline at end of file
diff --git a/1001.0485.txt b/1001.0485.txt
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+arXiv:1001.0485v1  [math.CV]  4 Jan 2010DEGENERATING BEHAVIOR OF GREEN’S FUNCTION
+F. PEHERSTORFER1
+Abstract. Let the unions of real intervals I=∪l
+j=1[a2j−1,a2j], a1<
+... < a 2l,andIn=∪m
+k=1[Bk,n,Ck,n] be such that ∩∞
+k=1[Bk,n,Ck,n] =
+{ck}fork= 1,...,mand dist( E,In)≥const > 0.We show how to
+express asymptotically the Green’s function φ(z,∞,E∪In) ofE∪In
+atz=∞in terms of the Green’s function φ(z,∞,E) andφ(z,ck,E).
+The formula yields immediately asymptotics for φn(z,∞,E∪In) with
+respect to nwhich are important in many problems of approximation
+theory. Another consequence is an asymptotic representati on ofcap(E∪
+In) in terms of cap(E) andφ(z,ck,E) and of the harmonic measure
+ω(∞,Ej,E∪In).
+LetE=/uniontextl
+j=1[a2j−1,a2j],−∞<a1<a2<...<a 2l<∞,be a union of l
+disjoint intervals, put H(x) =/producttext2l
+j=1(x−aj) and letφ(z,z0,E) be a so-called
+complex Green’s function, that is, a mapping which maps ¯C\Eonto the
+exterior of the unit circle, which has a simple pole at z=z0∈¯C\Eand
+satisfies|φ(z,z0)| →1 forz→x∈E; or in other words log |φ|is the Green’s
+function. It is known, see e.g. [7], that
+φ(z,∞,E) :=φ(z,∞) = exp/parenleftBigg/integraldisplayz
+a2lr∞(ξ)dξ/radicalbig
+H(ξ)/parenrightBigg
+wherer∞(ξ) =ξl−1+...is the unique polynomial such that
+(1)/integraldisplaya2j+1
+a2jr∞(ξ)dξ/radicalbig
+H(ξ)= 0 forj= 0,...,l−1
+and that for x∈R\E
+(2) φ(z,x0,E) :=φ(z,x0) = exp/parenleftBigg/integraldisplayz
+a2lrx0(ξ)
+ξ−x0dξ/radicalbig
+H(ξ)/parenrightBigg
+whererx0∈Pl−1is such that
+(3) rx0(x0) =−/radicalbig
+H(x0)
+and
+(4) −/integraldisplaya2j+1
+a2jrx0(ξ)
+ξ−x0dξ/radicalbig
+H(ξ)= 0 forj= 1,...,l−1.
+1The manuscript was prepared by the author in the two months pr eceding his passing
+away in November 2009. The manuscript remained unsubmitted and is not published
+elsewhere (submitted by P. Yuditskii and I. Moale).
+12 F. PEHERSTORFER
+The so-called capacity of Eis given by
+(5) cap(E) = lim
+z→∞|z
+φ(z,∞,E)|
+Byω(z,K,E) we denote the harmonic measure of the set K⊂Ewith
+respect toE.Recall that
+(6) ω(∞,[a2j−1,a2j],E) =/integraldisplaya2j
+a2j−1|r∞(x)|dx/radicalbig
+|H(x)|
+Theorem 0.1. (Asymptotic representation of the Green’s function in the
+degenerate case) Let l,m∈N, E=/uniontextl
+j=1[a2j−1,a2j]and for a strictly
+monotone increasing subsequence (n)ofNletIn=/uniontextm
+k=1[Bk,n,Ck,n]with
+dist(E,In)≥const>0and/intersectiontext∞
+n=1[Bk,n,Ck,n] ={ck}and letωk,n=
+ω(∞,[Bk,n,Ck,n],E∪In)withωk,n−→
+n→∞0fork= 1,...,m.Then uniformly
+on compact subsets of C\(E∪{c1,...,cm})
+(7)φ(z,∞,E∪In) =φ(z,∞,E)
+m/producttext
+k=1(φ(z,ck,E))ωk,n(1+O( max
+1≤k≤mω2
+k,n))
+Proof.For abbreviation we put
+˜En=E∪In, H˜E,n(x) =H(x)˜Hn(x)
+where
+H(x) =2l/productdisplay
+j=1(x−aj),˜Hn(x) =m/productdisplay
+k=1(x−Bk,n)(x−Ck,n)
+By [7] we know that there exists a monic polynomial r˜E,n(x) of degree l+
+m−1 such that
+φ(z,∞,˜En) = exp
+/integraldisplaynr˜E,n(x)
+/radicalBig
+H˜E,n(x)dx
+
+Now let us represent r˜E,nin the form
+r˜E,n(x) = ˜sn(x)m/productdisplay
+k=1(x−ck)+tn(x)
+=rE(x)m/productdisplay
+k=1(x−ck)+sn(x)m/productdisplay
+k=1(x−ck)+tn(x)
+whererE∈Pl−1is the monic polynomial associated with the Green’s func-
+tionφ(z,∞,E),sn(x) = ˜sn(x)−rE(x)∈Pl−2andtn∈Pm−1.Putting
+Ln= max
+1≤k≤n|Ck,n−Bk,n|DEGENERATING BEHAVIOR OF GREEN’S FUNCTION 3
+we have on compact subsets of C\{c1,...,cm}
+1/radicalBig
+H˜E,n(x)=1
+m/producttext
+k=1(x−ck)/radicalbig
+H(x)+O(Ln)
+and thus
+(8)r˜E,n(x)
+/radicalBig
+H˜E,n(x)dx=rE(x)/radicalbig
+H(x)dx+sn(x)/radicalbig
+H(x)dx
++tn(x)
+m/producttext
+k=1(x−ck)dx/radicalbig
+H(x)+O(Ln)
+Partial fraction expansion gives, recall tn∈Pm−1,
+(9)tn(x)
+m/producttext
+k=1(x−ck)=m/summationdisplay
+k=1λk,n
+x−ck
+To determine the λk,n’s we integrate both sides in (8) counterclockwise
+around a circle with center ckand fixed small radius ε.Taking into con-
+sideration the facts, that the limiting values ±/radicalBig
+H˜E,n(x) := lim
+z→x±
+x∈˜E,n1√
+H(z)
+from the upper- and lower half plane satisfy +/radicalBig
+H˜E,n(x) =−/radicalBig
+H˜E,n(x)
+and that
++1/radicalBig
+H˜E,n(x)=sgn r˜E,n(x)
+i/radicalBig
+|H˜E,n(x)|forx∈˜E,n,
+we obtain for the LHS by shrinking the circle to the interval [ Bk,n,Ck,n] that
+(10)/contintegraldisplayr˜E,n(ξ)
+/radicalBig
+H˜E,n(ξ)dξ=2
+i/integraldisplayBk,n
+Ck,n|r˜E,n(x)|
+/radicalBig
+|H˜E,n(x)|dx= 2iπωk,n
+On the other hand the RHS from (8) becomes, note that all the te rms at
+the RHS are analytic on the circle up to 1 /(x−ck),
+/contintegraldisplayr˜E,n(x)
+/radicalBig
+H˜E,n(x)dx=λk,n/contintegraldisplay1
+x−ckdx/radicalbig
+H(x)+O(Ln)
+=λk,n2πi/radicalbig
+H(ck)+O(Ln)
+that is,
+(11) λk,n=ωk,n/radicalbig
+H(ck)+O(Ln)4 F. PEHERSTORFER
+Summarizing, on compact subsets of C\{c1,...,cm}we have
+(12)r˜E,n(x)
+/radicalBig
+H˜E,n(x)dx=rE(x)/radicalbig
+H(x)dx+sn(x)dx/radicalbig
+H(x)
++/summationdisplay
+ωk,n/radicalbig
+H(ck)
+x−ckdx/radicalbig
+H(x)+O(Ln)
+Finally let us determine sn(x) asymptotically. We have by (8), (9) and
+(11) that
+−iπων,n=/integraldisplayCν,n
+Bν,nr˜E,n(x)
+/radicalBig
+H˜E,n(x)dx=/integraldisplaya2ν+1
+a2νr˜E,n(x)
+/radicalBig
+H˜E,n(x)dx
+=/integraldisplaya2ν+1
+a2ν/parenleftBigg
+sn(x)+m/summationdisplay
+k=1ωk,n/radicalbig
+H(ck)
+x−ck/parenrightBigg
+dx/radicalbig
+H(x)+O(Ln)
+Observing that
+−/integraldisplaya2ν+1
+a2νrck(ξ)
+x−ckdξ/radicalbig
+H(ξ)=/integraldisplaya2ν+1
+a2νrck(x)−rck(ck)
+x−ckdx/radicalbig
+H(x)
+−/radicalbig
+H(ck)−/integraldisplaya2ν+1
+a2ν1
+ξ−ckdξ/radicalbig
+H(ξ)
+and that by (3)
+(13) −/integraldisplaya2ν+1
+a2ν1
+ξ−ckdξ/radicalbig
+H(ξ)=iπδkν
+it follows that for ν= 1,...,l−1
+(14) −/integraldisplaya2ν+1
+a2ν/parenleftBigg
+sn(ξ)−m/summationdisplay
+k=1ωk,nqck(ξ)dξ/radicalbig
+H(ξ)/parenrightBigg
+dξ=O(Ln)
+Recall that sn(ξ)−/summationtextm
+k=1ωk,nqck(ξ)∈Pl−2.Since a polynomial q∈Pl−2
+satisfying
+(15)/integraldisplaya2k+1
+a2kq(x)dx/radicalbig
+H(x)= 0 fork= 1,...,l−1
+is identically the zero polynomial it follows with the help o f Cramer’s rule
+that
+sn(x)−m/summationdisplay
+k=1ωk,nqck,n(x) =O(Ln)
+which gives by (12) the assertion, if we are able to show that
+(16) Ln=O( max
+1≤k≤nω2
+k,n)
+Ifck∈(a1,a2l) let us consider
+Ek,n= [a1,Ck,n]∪[a2k+1,a2l],respectively ,Ek,∞= [a1,ck]∪[a2k+1,a2l]DEGENERATING BEHAVIOR OF GREEN’S FUNCTION 5
+andletAk,nandAk,∞bethecriticalpointofln φ(x,∞,Ek,n)andlnφ(x,∞,Ek,∞),
+respectively. Then it follows that
+Ak,n→Ak,∞∈(ck,a2k+1).
+Thus we obtain for n≥n0
+ω(∞,[Bk,n,Ck,n],˜En)≥ω(∞,[Bk,n,Ck,n],Ek,n)
+=/integraldisplayCk,n
+Bk,nx−Ak,n/radicalbig
+(x−a1)(x−a2l)(x−Ck,n)(x−a2k+1)dx
+≥const/integraldisplayCk,n
+Bk,ndx/radicalbigCk,n−x=const/radicalbig
+Ck,n−Bk,n,
+whereconst>0 is independent of n,and this proves (16).
+Ifck∈R\[a1,a2l] then we consider Ek,n:= [a1,Ck,n]⊃˜En,respectively,
+[Bk,n,a2l] and the assertion follows even simpler as above. /square
+Remark 0.2.By conformal transplantation and the invariance property o f
+the harmonic measure it follows that the Theorem holds for mu ch more
+general sets as circular slits in the plane, etc. .
+Corollary 0.3. As above let ωk,n=ω(∞,[Bk,n,Ck,n],E∪/uniontextm
+k=1[Bk,n,Ck,n]).
+Then the following statements hold:
+a)cap(E∪/uniontextm
+k=1[Bk,n,Ck,n]) = (capE)m/producttext
+k=1|φ(∞,ck,E)|ωk,n(capE)(1+
+O(max
+kω2
+k,n))
+b)ω(∞,Ej,E∪/uniontextm
+k=1[Bk,n,Ck,n]) =ω(∞,Ej,E)−m/summationtext
+k=1ωk,nω(ck,Ej,E)+
+O(max
+kω2
+k,n)
+Proof.a) Follows immediately by the fact that for a compact set K
+lim
+z→∞|z
+φ(z,∞,K)|=cap(K)
+b) follows by taking the log in (7) and using (6). /square
+Notation 0.4.a) Letpn(x) =anxn+...; bylc(pn) =anwe denotethe leading
+coefficient of pn.
+b)x1,...,xm∈Ewithx1<x2<...<x mare called alternation points of
+f∈C[a,b] iff(xi) =±(−1)i||f||∞,Efori= 1,...,m.By♯A(f;K) we denote
+the number of alternation points of fonK.
+Theorem 0.5. Letl,m∈N,E=/uniontextl
+j=1[a2j−1,a2j]andIn=/uniontextm
+k=1[Bk,n,Ck,n]
+withdist(E,In)≥const>0for a strictly monotone increasing subsequence
+ofNand/intersectiontext∞
+k=1[Bk,n,Ck,n] ={ck}.
+Suppose that there is a sequence of polynomials (pn)such thatpn∈Pn
+hasnreal zeros, min{|pn(yi)|:p′
+n(yi) = 0} ≥1,||pn||E∪In≤1and6 F. PEHERSTORFER
+♯A(pn;Ik,n) =νk+ 1fork= 1,...,mandn∈N.Then on compact sub-
+sets ofC\(E∪{c1,...,cm})pnhas an asymptotic representation of the form
+(17) 2 pn(x) =/parenleftbigg
+ψn(x)+1
+ψn(x)/parenrightbigg
+(1+o(1))
+where
+ψn(x) =φn(x,∞,E)
+m/producttext
+k=1φνk(x,ck,E)
+Furthermore the norm of the monic polynomial ˆpn(x) =pn(x)/lc(pn)is
+given by
+(18) ||ˆpn||E= 1/lc(pn) = 2(cap E)nm/productdisplay
+j=1φνk(cj;∞,E)(1+o(1))
+where0<r<1,and concerning the alternation points on Eνwe have
+♯A(pn;Eν) =nω(∞,Eν,E)+l−1/summationdisplay
+k=1ω(ck,Eν,E)+O(1
+n) forν= 1,...,l
+Proof.By the properties of pnit follows that p−1
+n([−1,1]) =E∪Inwhich
+implies, see e.g. [3, 4] that
+φ(z,∞,E∪In) =/parenleftBig
+pn(z)−/radicalbig
+p2n(z)−1/parenrightBig1
+n
+= exp/parenleftBigg
+1
+n/integraldisplayz
+a2lp′
+n(x)/radicalbig
+p2n(x)−1dx/parenrightBigg
+Thus
+2pn(z) =φn(z,∞,E∪In)+1
+φn(z,∞,E∪In)
+and the representation (17) follows by Theorem 0.1.
+Concerning relation (18) let us observe that, with the help o f (17) and
+|φ(z,∞,E)|>1 onC\E,
+lc(pn) = lim
+z→∞pn(z)
+zn= lim
+z→∞/parenleftbigg1
+2/parenleftbiggφ(z,∞,E)
+z/parenrightbiggn1/producttextm
+k=1φνk(z,ck,E)/parenrightbigg
+(1+o(1))
+/square
+BytheAlternationTheoremitfollowsimmediatelythatmany L∞-minimal
+polynomials without or with restrictions as minimal polyno mials on several
+intervals satisfy the assumptions of Theorem 0.5 with νk= 1, k= 1,...,m,
+and thus have a representation of the form (17). One of the rem aining chal-
+lenging problems is to describe when ck’s appear between two consecutive
+Eν’s. For a solution of this problem in the case of minimal polyn omials on
+several intervals, see [2].DEGENERATING BEHAVIOR OF GREEN’S FUNCTION 7
+References
+[1] F. Peherstorfer, Deformation of minimal polynomials , J. Approx. Theory 111 (2001),
+180-195.
+[2] F. Peherstorfer, Asymptotic representation of minimal polynomials on sever al inter-
+vals, preprint.
+[3] F. Peherstorfer, R. Steinbauer, Orthogonal and Lq- extremal polynomials on inverse
+images of polynomial mappings , J. Comput. Appl. Math. 127 (2001), 297-315.
+[4] T. Ransford, Potential theory in the complex plane , Cambr. Univ. Press., 1995.
+[5] V. Totik, Polynomial inverse images and polynomial inequalities , Acta Math. (Scan-
+dinavian) 187 (2001), 139-160.
+[6] V. Totik, Chebyshev constants and the inheritance problem , to appear in J. Approx.
+Theory.
+[7] H. Widom, Extremal polynomials associated with a system of curves in t he complex
+plane, Adv. Math. 3 (1969), 127-232.
\ No newline at end of file
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@@ -0,0 +1,251 @@
+arXiv:1001.0486v1  [hep-ph]  4 Jan 2010TU-862
+Constraints on Scalar Phantoms
+Masaki Asano and Ryuichiro Kitano
+Department of Physics, Tohoku University, Sendai 980-8578 , Japan
+Abstract
+We update the constraints on the minimal model of dark matter , where a
+stable real scalar field is added to the standard model Lagran gian with a renor-
+malizable coupling to the Higgs field. Once we fix the dark matt er abundance,
+there are only two relevant model parameters, the mass of the scalar field and
+that of the Higgs boson. The recent data from the CDMS II exper iment have
+excluded a parameter region where the scalar field is light su ch as less than
+about 50 GeV. In a large parameter region, the consistency of the model can be
+tested by the combination of future direct detection experi ments and the LHC
+experiments.Weakly interacting massive particle (WIMP) is one of the most attrac tive scenarios
+to explain the dark matter component of the universe. The simplest and the most
+economical model to realize this scenario is the model of the scalar d ark matter, where
+a new gauge singlet real scalar field Sis introduced to the standard model and it has
+an interaction with the Higgs field, H. The following Lagrangian density is added to
+the standard model:
+LS=1
+2∂µS∂µS−m2
+2S2−k
+2S2|H|2−h
+4!S4. (1)
+This model has been first proposed in Ref. [1] (it is called scalar phant oms) and further
+studied in Refs. [2, 3, 4, 5, 6, 7, 8, 9]. One can obtain the correct siz e of the dark matter
+density via the standard thermal decoupling process in the early un iverse.
+The model can be either thought of as a device to accommodate dar k matter in new
+physics models or as a fundamental theory to describe the physics up to the Planck
+scale. If we take the latter attitude, the parameter region for kandhis subject to the
+constraint from perturbativity and stability of the potential [4, 5, 6, 9].
+Because the model is compact, there are only four unknown param eters: the Higgs
+boson mass m2
+h, the mass of dark matter m2
+S(≡m2+k/angbracketleftH/angbracketright2=m2+kv2), and two
+coupling constants kandh. Herev≃174GeV. Since the coupling constant of
+the dark matter self-interaction his not important for most of interesting physical
+observables, there are only three relevant parameters, one of w hich can be fixed by
+requiring thecorrect darkmatterabundance. Therefore, them odel ishighlypredictive.
+The parameter space is a two-dimensional mh−mSplane.
+In this article, we update the constraints on the model in light of new data from
+the CDMS II experiment [12] as well as the Tevatron exclusion [13] of a Higgs boson
+mass region. We discuss the interplay between the dark matter det ection experiments
+and the Higgs boson searches at the LHC experiments.∗
+The scalar field Sis stable due to a discrete symmetry S↔ −S, and therefore
+is a candidate for a non-baryonic cold dark matter. In the WIMP sce nario, the
+number density of Sis determined by the annihilation cross section, /angbracketleftσann.v/angbracketright, which is
+proportional to k2. One can fix the coupling constant kby requiring the abundance
+to explain the dark matter component of the energy density of the universe [14]. The
+∗Similar studies have been done in Refs. [10, 11] very recently.
+2energy density of dark matter is given by
+ΩS≃1.8×10−10GeV−2
+/angbracketleftσann.v/angbracketright, (2)
+The annihilation cross section is given by
+/angbracketleftσann.v/angbracketright=4k2v2
+(4m2
+S−m2
+h)2+m2
+hΓ2
+H·ΓH|mh=2mS
+2mS(3)
+formS< mh, and
+/angbracketleftσann.v/angbracketright=4k2v2
+(4m2
+S−m2
+h)2+m2
+hΓ2
+H·ΓH|mh=2mS
+2mS+k2/radicalbig
+1−m2
+h/m2
+S
+64πm2
+S/parenleftbigg
+1+3m2
+h
+4m2
+S−m2
+h/parenrightbigg
+,(4)
+formS≥mh. Here Γ Hand Γ H|mh=2mSare the total decay width of the Higgs boson
+and that with a mass 2 mS, respectively. (One should take out h→SSmode in
+ΓH|mh=2mS.) By using the WMAP data [14], Ω S≃0.11, one can fix k.
+Darkmatter Sinteractswithnuclei throughanexchange oftheHiggsboson. Dire ct
+detection experiments such as CDMS II can therefore put constr ains on the model.
+The spin-independent cross section of the S-nucleus elastic scattering normalized to a
+nucleon is given by
+σSI=1
+4π(1 GeV)2
+A2m2
+S(Zfp+(A−Z)fn)2, (5)
+whereAandZare mass and atomic numbers of the target nucleus, respectively. The
+fpandfnfactors are given by
+fN=kmN
+m2
+h/summationdisplay
+qfN
+q,andfN
+q=mq/angbracketleftN|¯qq|N/angbracketright
+mN, (6)
+whereq=u,d,s,c,b,t andN=p,n. In this study, we use
+fp
+u= 0.021, fp
+d= 0.029, fp
+s= 0.0, (7)
+fn
+u= 0.016, fn
+d= 0.037, fn
+s= 0.0, (8)
+fp
+c=fp
+b=fp
+t=fn
+c=fn
+b=fn
+t= 0.070, (9)
+which give a conservative estimate of the scattering cross section [15, 16, 17, 18].
+The Higgs boson can decay into a pair of SifmS< mh/2. This invisible decay
+width is given by:
+Γh→SS=k2v2
+16πmh/radicalBigg
+1−4m2
+S
+m2
+h. (10)
+3 [GeV] [GeV]
+ 0 20 40 60 80 100 120 140
+ 100  120  140  160  180  200mS
+mh
+LEP exclusionTevatron exclusion
+CDMS II
+     exclusionXENON100
+projectionBinv=1%
+50%
+75%
+perturbativity
+30%
+99%disfavored byelectroweak precision fit
+mS=mh/ 2
+60%
+Figure 1: Experimental constraints and contours of the branchin g ratio of the invisible
+Higgs boson decay. The region inside the dashed lines satisfies the st ability and
+triviality bounds for h(mZ) = 0.
+The strategy to search for the Higgs boson in collider experiments w ill be tightly
+connected to dark matter physics in this model.
+There have been studies of the invisible Higgs decay at the LHC. The d iscovery
+potential is a function of
+ξ2=σBSM
+σSM·Binv, (11)
+whereBinvis the branching fraction of the invisible decay mode, and σBSMandσSM
+are the production cross sections of the Higgs boson in new physics models and in the
+standard model, respectively. At the ATLAS detector with 30 fb−1, it is possible to
+discover the invisible decay of the Higgs boson in the 100 GeV < mh<200 GeV region
+ifξ2/greaterorsimilar60 % [19]. In the present model, ξ2=Binvbecause the production cross section
+is same as the one in the standard model.
+Let us discuss the current experimental constraints on this mode l and the size
+ofBinvwhich will be important for the Higgs boson search. As explained befo re,
+one can discuss the constraints/prediction of the model on the mS−mhplane by
+requiring the correct dark matter abundance (Fig. 1)†. The recent data from the
+†In calculating the dark matter abundance and the decay width of th e Higgs boson, we have used
+the program HDECAY [20].
+4CDMS II experiment excludes the region with 10 GeV /lessorsimilarmS/lessorsimilar50 GeV although
+there are uncertainties from the local density of dark matter. We also overlaid the
+projected sensitivity of the XENON100 experiment [21]. The bound f rom the Higgs
+boson searches are also shown. The shaded region in the left side is e xcluded by the
+LEP-II direct Higgs boson searches [22, 23]. The excluded region fr om the Tevatron
+experiments, 163GeV < mh<166 GeV[13], isalso shown where the bounddisappears
+formS/lessorsimilar70 GeV due to large Binv. The region mh>186 GeV is disfavored by the
+electroweak precision measurements (95% CL) [24]. A very light dark matter region,
+mS/lessorsimilar1GeV, is ruled out by a large branching ratio of the B→KSSdecay [25, 26, 27,
+28]. It has been studied that a heavier region can be explored by the invisible decay of
+Υ (or other heavy meson) [29]. Inside the dashed line, the model is pe rturbative and
+the Higgs potential is stable up to the Planck scale. It is interesting t o notice that the
+recent experimental results started to explore a parameter reg ion of the model.
+We also show contours of Binv(solid lines) on the same plane. In the dark matter
+detection and the Higgs search experiments, there are in principle f our physical ob-
+servables, mh,Binv,mS, andσSI, whereas in this model BinvandσSIcan be calculated
+in terms of other two, mhandmS. Therefore, one can check the consistency by using
+data from both experiments. Almost entire region in Fig.1 will be cover ed by next
+generation experiments such as SuperCDMS, XENON, and XMASS [21 , 30, 31] except
+for a region very close to the mS=mh/2 line. Also, the Higgs boson search at the LHC
+will cover most of the region either by the invisible mode or by the ordin ary standard
+model Higgs searches.
+Finally, we comment on the case where two events reported by CDMS II is a signal
+of dark matter. In this case, the parameter region is restricted n ear the boundary of
+the CDMS II exclusion region. Interestingly, the mass region prefe rred by the data,
+mS/lessorsimilar100GeV [32, 11] , is consistent with the prediction in Fig. 1. Since Binvis rather
+large in the region, the LHC experiments should confirm the scenario by observing the
+invisible decay of the Higgs boson when mh/lessorsimilar160 GeV‡.
+The positron excess reported by the PAMELA experiment [33] cann ot be simul-
+taneously explained. One can assume that decays of dark matter p rovide a source of
+high-energy positronsthrougha small breaking ofthe Z2symmetry. Inorder toexplain
+the spectrum, mSis required to be larger than 200 GeV, that is not compatible with
+‡It may be possible to discover the invisible mode even for mh/greaterorsimilar160 GeV, i.e., ξ2∼30 %, once
+systematic uncertainties are better understood [19].
+5the explanation of the CDMS II data in this model.
+Acknowledgments
+This work was supported in part by the Grant-in-Aid for the Global C OE Program
+Weaving Science Web beyond Particle-matter Hierarchy from the Min istry of Educa-
+tion, Culture, Sports, Science and Technology of Japan (M. A.) and the Grant-in-Aid
+for Scientific Research from the Ministry of Education, Science, Sp orts, and Culture of
+Japan, no. 21840006 (R.K.).
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+7
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+arXiv:1001.0487v2  [astro-ph.SR]  5 Jan 2010Astrophysics and Space Science
+DOI 10.1007/s •••••-•••-••••-•
+Numerical simulations of the κ-mechanism with
+convection
+T. Gastine1•B. Dintrans1
+c/circlecopyrtSpringer-Verlag ••••
+Abstract A strong coupling between convection and
+pulsations is known to play a major role in the dis-
+appearance of unstable modes close to the red edge of
+the classical Cepheid instability strip. As mean-field
+models of time-dependent convection rely on weakly-
+constrainedparameters(e.g.Baker1987), wetacklethis
+problem by the means of 2-D Direct Numerical Simu-
+lations (DNS) of κ-mechanism with convection.
+Using a linear stability analysis, we first determine
+the physical conditions favourable to the κ-mechanism
+to occur inside a purely-radiative layer. Both the in-
+stability strips and the nonlinear saturation of unstable
+modes are then confirmed by the corresponding DNS.
+We next present the new simulations with convection,
+where a convective zone and the driving region over-
+lap. The coupling between the convective motions and
+acoustic modes is then addressed by using projections
+onto an acoustic subspace.
+Keywords Hydrodynamics - Instabilities - Stars: os-
+cillations - Convection - Methods: numerical
+1 Introduction
+Eddington (1917) discovered an excitation mechanism
+of stellar oscillations that is related to the opacity be-
+haviour in ionisation zones: the κ-mechanism, where
+κdenotes the opacity. This mechanism occurs when
+the opacity varies during compression phases so as
+to block the emerging radiative flux (Zhevakin 1963;
+Cox and Whitney 1958). Ionisation regions correspond
+to a strong increase in opacity that leads to the opac-
+ity bumps responsible for the local excitation of modes.
+T. Gastine
+B. Dintrans
+1LATT, Universit´ e de Toulouse, CNRS, 14 avenue Edouard Be-
+lin, F-31400 Toulouse, FranceThese driving zones should nevertheless be located at
+a precise radius inside the star, neither too close to
+the surface nor to deep, in order to balance the damp-
+ing that mainly occurs at the surface. It defines the
+so-called transition region that shapes the limit be-
+tweenthe quasi-adiabaticinteriorandthe stronglynon-
+adiabatic surface. For classical Cepheids that pul-
+sate in the fundamental acoustic mode, this transi-
+tion region is located at a temperature T≃4×
+104K corresponding to the second helium ionisation
+(Baker and Kippenhahn 1965). However, the bump lo-
+cation is not solely responsible for the acoustic instabil-
+ity. A careful treatment of the κ-mechanism would in-
+volve dynamical couplings with convection, metallicity
+effects and both realistic equations of state and opac-
+ity tables (Bono, Marconi, and Stellingwerf 1999). The
+purpose of our model is to simplify the hydrodynamic
+approachwhileretainingthe leading-order phenomenon
+–e.g. thelocationofthe opacitybump oritsamplitude–
+such that feasible direct numerical simulations of the κ-
+mechanism with convection can be achieved.
+2 The first step: radiative models in 1-D
+We first focus on radial modes propagating in a purely-
+radiative and partially-ionised layer and then restrict
+our study to the 1-D case. Our model consists of a
+local zoom about an ionisation region and is composed
+byamonatomicandperfect gas(with γ=cp/cv= 5/3)
+underboth aconstant gravity /vector gand kinematic viscosity
+ν. All quantities in our model are dimensionless, e.g.
+temperature is given in unit of the surface temperature
+of the star, length in unit of a fraction of the star radius
+and velocity in unit of the sound speed at the surface.
+In order to easily investigate the influence of the
+opacity bump on the stability, we adopt the following2
+Fig. 1 Influence of the hollow parameters on the conduc-
+tivity profile for Kmax= 10−2andTbump/Tsurf= 3.5.
+conductivity hollow to mimic this bump (as κ∝1/K,
+see Gastine and Dintrans 2008a, hereafter Paper I):
+K0(T0) =Kmax/bracketleftbigg
+1+A−π/2+arctan( σT+T−)
+π/2+arctan( σe2)/bracketrightbigg
+,(1)
+withA= (Kmax−Kmin)/KmaxandT±=T0−Tbump±
+e. HereT0istheequilibriumtemperatureprofile, Tbump
+is the hollow central temperature, and σ, eandAde-
+note its slope, width and relative amplitude, respec-
+tively. Examples of common values of these parameters
+are provided in Fig. 1.
+2.1 Conditions for the instability
+Using the well-known work integral formalism (e.g.
+Unnoet al.1989), it is easy to show that the follow-
+ing condition is necessary to get unstable modes by κ-
+mechanism in variable stars:
+dKT
+dz<0 where KT≡/parenleftbigg∂lnK0
+∂lnT0/parenrightbigg
+ρ0. (2)
+However, this condition is not sufficient as the ionisa-
+tion region should also be located at a given location
+in the star, neither too deep nor too close to its sur-
+face. This area is the so-called transition region that
+separates the quasi-adiabatic interior from the strongly
+non-adiabatic surface and is defined by (e.g. Cox 1980)
+Ψ =∝angb∇acketleftcvT0∝angb∇acket∇ight∆m
+ΠL≃1, (3)
+where ∆ mis the integrated mass between the consid-
+eredpointandthesurface,Πthemodeperiodand Lthe
+Fig. 2 a ) Vertical profile of the Ψ coefficient (Eq. 3).
+The superimposed green zone represents the location where
+Ψ = 1±0.5.b) The real part of the work integral plotted
+for the three different equilibrium models. The dot-dashed
+black line is for the constant radiative conductivity case.
+c) Corresponding radiative conductivity profiles. d) Corre-
+sponding equilibrium fields dKT/dz(Eq. 2).
+luminosity. Ψ is the ratio between the thermal energy
+embedded betweenthe givenradiusandthe surfaceand
+the energy radiated during an oscillation period.
+2.2 The linear stability analysis
+Inordertocheckthesetwoconditions, weperformalin-
+ear stability analysis by using the Linear Solver Builder
+code(Valdettaro et al.2007). Bystartingwith the con-
+ductivity profiles shown in Fig. 1, we first compute the
+equilibrium setup that is solution of both the hydro-
+static and radiative equilibria given by
+/vector∇p0=ρ0/vector gand div[ K0(T0)/vector∇T0] = 0, (4)
+wherep0andρ0denote the equilibrium pressure and
+density. Then, we solve the linear oscillation equations
+for the perturbations by seeking normal modes of the
+form exp( λt) withλ=τ+iω, whereωis the mode
+frequency and τits growth or damping rate (unstable
+modes corresponding to τ >0). By varying the hol-
+low parameters, we then determine the physical con-
+ditions that lead to unstable modes excited by the κ-
+mechanism in our layer (cf. Paper I).
+The two criteria (2) and (3) predict that modes are
+unstable for a “sufficient” hollow (adequate amplitudeNumerical simulations of the κ-mechanism with convection 3
+and width) that is located in the transition region.
+Using the work integral computed from the obtained
+eigenvectors, we indeed recover instability strips for the
+κ-mechanism by considering the three different equilib-
+rium setups (Fig. 2):
+•The first case (dotted blue line, Tbump/Tsurf= 1.7)
+corresponds to a hot star with an ionisation region
+close to the surface. As the surrounding density is
+small, Ψ ≪1 and the conductivity hollow hardly in-
+fluencestheworkintegral: drivingisunabletoprevail
+over damping, leading to τ <0 orstablemodes.
+•In the second case (solid green line, Tbump/Tsurf=
+2.1), the radiative conductivity begins to decrease
+significantly at the location of the transition region
+where Ψ ≃1. As a consequence, driving becomes im-
+portant. Furthermore, the radiative conductivity in-
+crease occurs where non-adiabatic effects are already
+significant, i.e. Ψ <1 there. It means that no damp-
+ing occurs between the hollow position and the sur-
+face as the radiative flux perturbations are frozen-in.
+Driving is overcomingdamping in this case, therefore
+τ >0 and modes are unstable.
+•The third case (dashed red line, Tbump/Tsurf= 2.8)
+corresponds to a cold star of which ionisation re-
+gionislocateddeeperinthestellaratmospherewhere
+Ψ≫1. Ionisation then occurs in a quasi-adiabatic
+location. As a consequence, the excitation provided
+by the conductivity hollow cannot balance the damp-
+ing arising at the top of the layer, thus τ <0 and
+modes are stable.
+In conclusion, our first radiative model of the κ-
+mechanism is well able to reproduce the main physics
+involved in the oscillations of Cepheid stars: (i)one
+should have dKT/dz <0 to drive the oscillations; (ii)
+the thermal engine underlying the conductivity hollow
+should also be located in the transition region where
+Ψ≃1 (Fig. 2).
+2.3 Study of the nonlinear saturation
+To confirm both the results obtained previously in the
+linear stability analysis and to investigate the result-
+ing nonlinear saturation, we perform direct numerical
+simulations of our problem. The idea is to start from
+the most favourable initial conditions found in the sta-
+bility analysis and then advance the general hydro-
+dynamic equations in time. We use a modified ver-
+sion of the public-domain finite-difference Pencil Code2
+that includes an implicit solver of our own for the
+radiative diffusion term in the temperature equation
+(Gastine and Dintrans 2008b, hereafter Paper II).
+2Seehttp://pencil-code.googlecode.com/ .
+Fig. 3 a) Temporal power spectrum for the momentum
+in the (z, ω)-plane. b)The resulting spectrum after in-
+tegrating over depth. c)Comparison between normalised
+momentum profiles for n= (0,2) modes according to the
+DNS power spectrum (solid black line) and to the linear
+stability analysis (dotted blue line).
+To determine which modes are present in the DNS
+in the nonlinear-saturation regime, we first perform
+a temporal Fourier transform of the momentum field
+ρu(z,t) and plot the resulting power spectrum in the
+(z,ω)-plane (Fig. 3a). The mean spectrum follows by
+integrating /hatwiderρu(z,ω) over depth (Fig. 3b). With this
+method, acoustic modes are extracted because they
+emerge as “shark-fin profiles” about definite eigenfre-
+quencies (Dintrans and Brandenburg 2004).
+Several discrete peaks corresponding to normal
+modes well appear but the fundamental mode close
+toω0= 5.439 clearly dominates. Moreover, the lin-
+ear eigenfunctions are compared to the mean profiles
+computed from a zoom taken in the DNS power spec-
+trumabouteigenfrequencies ω0= 5.439andω2= 11.06
+(Fig. 3c). The agreement between the linear eigenfunc-
+tions (dotted blue lines) and the DNS profiles (solid
+black lines) is remarkable. In summary, Fig. 3 shows
+that several overtones are present in the DNS, even for
+long times. These overtones are however linearly stable
+suggesting that some underlying energy transfers occur
+between modes through nonlinear couplings.
+These nonlinear interactions are investigated thanks
+to a tool that measures the sound generation by turbu-
+lent flows (e.g. Bogdan, Cattaneo, and Malagoli 1993).4
+It involves projections of the DNS physical fields onto
+the regular and adjoint eigenvectors that are solutions
+to the linear oscillation equations. The projection co-
+efficients then give the time evolution of each acoustic
+mode that is present in the DNS separately . Further-
+more, as the evolution of the kinetic energy of each
+mode is also accessible with this method, one can fol-
+low the energy transfer between modes.
+Fig. 4 Time evolution of the kinetic energy ratio for the
+n= 0 and n= 2 acoustic modes.
+Figure 4 shows an example of the time evolution
+of the kinetic energy ratio En/Etot
+kinfor the two modes
+n= 0 (i.e. the fundamental one) and n= 2 (i.e. the
+second overtone). Here Endenotes the kinetic energy
+embedded in the acoustic mode oforder n, whileEtot
+kinis
+the total kinetic energy in the simulation. After its lin-
+ear transient growth, a given fraction of the fundamen-
+tal mode energy is progressively transferred to the sec-
+ond overtone until the nonlinear saturation is achieved
+abovetime t≃150, with finally 85% ofthe total kinetic
+energy in the n= 0 mode and the remaining 15% in the
+n= 2 one. We recall that it is at a first glance surpris-
+ing to detect this second overtoneat long times because
+it is linearly stable and should normally be damped by
+diffusive effects.
+The reason for its presence lies in a favored coupling
+in the period ratio between these two modes. Indeed,
+the fundamental period is P0= 2π/ω0≃1.155 in this
+simulation while the n= 2 one is P2≃0.568. As a
+consequence, the corresponding period ratio is close to
+one half ( P2/P0≃0.491) such that the second over-
+tone is involved in the nonlinear saturation through a
+2:1 resonance with the fundamental mode. The lin-
+ear growth of the fundamental mode is then balanced
+by the pumping of energy to the stable second over-
+tone that behaves as an energy sink, leading to the full
+limit-cycle stability.3 Convective models in 2-D
+3.1 From radiative to convective setups
+After the previous results obtained on purely radia-
+tive models in 1-D, we next address the convection-
+pulsation coupling that is suspected to quench the ra-
+dial oscillations of Cepheids close to the red edge. The
+idea is to slightly modify the unstable radiative setup
+to obtain a convective zone superimposed to the ion-
+isation region in 2-D simulations. The convective in-
+stability obeys Schwarzschild’s criterion given by (e.g.
+Chandrasekhar 1961)
+/vextendsingle/vextendsingle/vextendsingle/vextendsingledT0
+dz/vextendsingle/vextendsingle/vextendsingle/vextendsingle>g
+cp=⇒Fbot>gK0(T0)
+cp. (5)
+whereFbot=K0|dT0/dz|is the imposed bottom flux.
+For a given gravity gand conductivity hollow K0(T0),
+convectionwillthereforedevelopforalargeenoughbot-
+tom flux satisfying Eq.(5).
+Fig. 5 Snapshot of the vorticity field /vector∇×/vector vin a 2-D model
+with convection and κ-mechanism.
+Figure 5 displays a snapshot of the vorticity field ob-
+tained in such a convective simulation at time t= 2500.
+In order to ensure that the thermal relaxation is well
+achieved, we ran this simulation for a much longer time
+than in the purely-radiative case and the resolution is
+256×256gridpoints. As expected, convection develops
+in the middle of the box where the conductivity hollow
+is located1. Moreover, because of the density contrast
+across the convection zone, the vorticity is trapped in
+strong convective plume downdrafts that easily over-
+shoot in the stable zone below due to their low P´ eclet
+number (Dintrans 2009).
+1An animation of this simulation is provided at
+http://www.ast.obs-mip.fr/users/tgastine/conv_kappa .avi.Numerical simulations of the κ-mechanism with convection 5
+3.2 Evolution of acoustic modes with convection
+To address the time evolution of the acoustic modes
+that arepresentin the DNS with convection, weproject
+as before the physical fields onto an appropriate acous-
+tic subspace to get the projection coefficient cℓn(t).
+Hereℓdenotes the mode degree while nis its order
+(withkx= (2π/Lx)ℓandLxis the width of the box).
+Fig. 6 Evolution of the acoustic kinetic energy ratio for
+two simulations with a weak convection (solid blue line) and
+a stronger one (dashed green line).
+Fig. 6 shows the total energy embedded in acoustic
+modes for two simulations with convection that differ
+by their Rayleigh number given by
+Ra =−gd4
+conv
+νχcpds
+dz=(∇−∇ adia)gd4
+conv
+νχHp, (6)
+whereHp=−(dlnp0/dz)−1is the pressure scale
+height,dconvthe vertical extent of the convection zone,
+χ=K0/ρ0cpthe radiativediffusivityand sthe entropy.
+This number allows to quantify the strength of the con-
+vective motions, that is, the critical Rayleigh number
+above which convection develops is about 103for poly-
+tropic stratifications (e.g. Gough et al.1976).
+In the first simulation (Ra = 4 ×104, solid blue line),
+the Rayleigh number is slightly super-critical and the
+convection intensity is weak. In this case, we see that
+the acoustic energy ratio remains large (i.e. /greaterorsimilar80%,
+the remaining 20% being in the convection) and the
+nonlinear saturation is reached in the same way than
+in the previous radiative models. In other words, the
+mode propagationis not affected by convectivemotions
+and the convection can be seen as frozen-in .
+On the contrary, increasing the Rayleigh number
+leads to a more vigorous convection and the acoustic
+energyratioismuchmoreaffectedbyconvectiveplumes(dashed green line). Indeed, after a long transient dur-
+ing which it reaches a non-trifling value ( ∼30%), it
+progressively tends to zero at long times while the con-
+vective motions are responsible for the whole kinetic
+energy content. The radial oscillations excited by κ-
+mechanism are thus quenched by convection and this
+situation is relevant to the red edge where the unstable
+acoustic modes are known to be damped by convective
+motions occuring at the star’s surface.
+We thus have shown that the Rayleigh number is
+a key parameter for this problem. But further inves-
+tigations of the convection-pulsation coupling are in
+progress, by especially checking time-dependent theo-
+ries of convection for which several free parameters can
+be constrained by such kind of direct numerical simu-
+lations (e.g. Yecko, Koll´ ath, and Buchler 1998).
+Acknowledgements Calculations were carried out
+on the CALMIP machine of the University of Toulouse
+(http://www.calmip.cict.fr/ ).
+References
+Baker, N., Kippenhahn, R.: Astrophys. J. 142, 868 (1965)
+Baker, N.H.: In: Hillebrandt, W., Meyer-Hofmeister, E.,
+Thomas, H.C. (eds.) Physical Processes in Comets, Stars
+and Active Galaxies, p. 105 (1987)
+Bogdan, T.J., Cattaneo, F., Malagoli, A.: Astrophys. J.
+407, 316 (1993)
+Bono, G., Marconi, M., Stellingwerf, R.F.: Astrophys. J.
+Suppl. Ser. 122, 167 (1999)
+Chandrasekhar, S.: Hydrodynamic and hydromagnetic
+stability. Oxford University Press: Clarendon, Oxford
+(1961)
+Cox, J.P.: Theory of stellar pulsation. Princeton Universi ty
+Press, Princeton (1980)
+Cox, J.P., Whitney, C.: Astrophys. J. 127, 561 (1958)
+Dintrans, B.: Communications in Asteroseismology 158, 45
+(2009)
+Dintrans, B., Brandenburg, A.: A&A 421, 775 (2004)
+Eddington, A.S.: The Observatory 40, 290 (1917)
+Gastine, T., Dintrans, B.: Astron. Astrophys. 484, 29
+(2008a)
+Gastine, T., Dintrans, B.: Astron. Astrophys. 490, 743
+(2008b)
+Gough, D.O., Moore, D.R., Spiegel, E.A., Weiss, N.O.: As-
+trophys. J. 206, 536 (1976)
+Unno, W., Osaki, Y., Ando, H., Saio, H., Shibahashi, H.:
+Nonradial oscillations of stars. University of Tokyo Press ,
+Tokyo (1989)
+Valdettaro, L., Rieutord, M., Braconnier, T., Fraysse, V.:
+JCoAM 205(1), 382 (2007)
+Yecko, P.A., Koll´ ath, Z., Buchler, J.R.: Astron. Astrophy s.
+336, 553 (1998)
+Zhevakin, S.A.: Annu. Rev. Astron. Astrophys. 1, 367
+(1963)
+This manuscript was prepared with the AAS L ATEX macros v5.2.
\ No newline at end of file
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+arXiv:1001.0488v1  [astro-ph.EP]  4 Jan 2010Astronomy & Astrophysics manuscript no. montecoag2 c∝circleco√yrtESO 2018
+November 1, 2018
+The outcome of protoplanetary dust growth:
+pebbles, boulders, or planetesimals?⋆
+II. Introducing the bouncing barrier
+A. Zsom1, C. W. Ormel1, C. G¨ uttler2, J. Blum2, C. P. Dullemond1
+1Max-Planck-Institute f¨ ur Astronomie, K¨ onigstuhl 17, D- 69117 Heidelberg, Germany. Email: zsom@mpia.de
+2Institut f¨ ur Geophysik und extraterrestrische Physik, Te chnische Universit¨ at Braunschweig, Mendelssohnstr. 3, 3 8106
+Braunschweig, Germany. Email: c.guettler@tu-bs.de
+November 1, 2018
+ABSTRACT
+Context. The sticking of micron sized dust particles due to surface fo rces in circumstellar disks is the first stage in the
+production of asteroids and planets. The key ingredients th at drive this process are the relative velocity between the
+dust particles in this environment and the complex physics o f dust aggregate collisions.
+Aims.Here we present the results of a collision model, which is bas ed on laboratory experiments of these aggregates.
+We investigate the maximum aggregate size and mass that can b e reached by coagulation in protoplanetary disks.
+Methods. We use the results of laboratory experiments to establish th e collision model (G¨ uttler et al. (2009)). The
+collision model is based on some necessary assumptions: we m odel the aggregates as spheres having compact and
+porous ‘phases’ and a continuous transition between these t wo. We apply this collision model to the Monte Carlo
+method of Zsom & Dullemond (2008) and include Brownian motio n, radial drift and turbulence as the sources of
+relative velocity between dust particles.
+Results.Wemodel thegrowth ofdustaggregates at 1AUatthemidplanea t threedifferentgas densities. Wefindthatthe
+evolution of the dust does not follow the previously assumed growth-fragmentation cycles. Catastrophic fragmentatio n
+hardly occurs in the three disk models. Furthermore we see lo ng lived, quasi-steady states in the distribution function
+of the aggregates due to bouncing. We explore how the mass and the porosity change upon varying the turbulence
+parameter and by varying the critical mass ratio of dust part icles. Upon varying the turbulence parameter, the system
+behaves in a non-linear way and the critical mass ratio has a s trong effect on the particle sizes and masses. Particles
+reach Stokes numbers of roughly 10−4during the simulations.
+Conclusions. The particle growth is stopped by bouncing rather than fragm entation in these models. The final Stokes
+number of the aggregates is rather insensitive to the variat ions of the gas density and the strength of turbulence. The
+maximum mass of the particles is limited to ≈1 g (chondrule sized particles). Planetesimal formation ca n proceed via
+the turbulent concentration of these aerodynamically size -sorted chondrule-sized particles.
+Key words. planets and satellites - formation, accretion, accretion d isks, methods: numerical
+1. Introduction
+In the coreaccretionparadigmofplanet formation(Mizuno
+(1980); Pollack et al. (1996)) planets are the outcome of
+an accretion process that starts with micron-size dust
+grains and covers 40 magnitudes in mass. It can be divided
+into three stages. The first stage of the formation of rocky
+planets and the rocky cores of gas giant planets starts
+with the coagulation of dust in the protoplanetary disks
+surrounding many pre-main-sequence stars (Safronov
+(1969), Weidenschilling & Cuzzi (1993), Blum & Wurm
+(2008)). The next stage of planet formation is the for-
+mation of protoplanetary cores from the planetesimals.
+The idea is that the kilometer size planetesimals are
+⋆This paper is dedicated tothe memory ofour dear friend and
+colleague Frithjof Brauer (14th March 1980 - 19th September
+2009) who developed powerful models of dust coagulation and
+fragmentation, and thereby studied the formation of planet esi-
+mals beyond the meter size barrier in his PhD thesis. Rest in
+peace, Frithjof.so large, that gravity starts to take over and leads to
+the gravitational agglomeration of these bodies to rocky
+planets. This scenario was studied already by Safronov
+(1969), and has since been modeled using numerical meth-
+ods by Weidenschilling (1980), Nakagawa et al. (1983),
+Mizuno et al. (1988), Schmitt et al. (1997), Wetherill
+(1990), Nomura & Nakagawa (2006), Garaud & Lin
+(2004), Tanaka et al. (2005) and several more authors.
+These models solve for the size distribution of dust ag-
+gregates in the disk as a function of time, and investigate
+if, where and how larger dusty bodies form, and how
+long that takes. Finally, in the third stage, gas accretes
+onto these protoplanets forming giant planets or – in the
+absence of gas – gravitational encounters between these
+protoplanets result in a chaotic, giant impact phase, until
+orbital stability has been achieved (Chambers (2001);
+Kokubo et al. (2006); Thommes et al. (2008)).
+In this study, we focus on the first phase and address
+the fundamental question of how effective dust growth2 A. Zsom et al.: The outcome of protoplanetary dust growth: p ebbles, boulders, or planetesimals? II.
+by surface forces really is; that is, how big do particles
+become by simple sticking processes only. It is known that
+initially, for micron size grains, the growth is driven by
+Brownian motion. This typically leads to slow collisions
+and forms aggregates of fractal structure (Kempf et al.
+(1999); Blum et al. (1996)). In the current picture of
+dust growth, as these aggregates grow, at some point the
+growth will leave the fractal regime, and collisions will
+start to lead to compaction and breaking of the aggregates
+(Blum & Wurm (2000)), embedding of small bodies into
+larger aggregates (leading to ‘filling up’ of these larger
+aggregates and compaction due to the force of the collision
+(Ormel et al. (2007)). As the size of the dust aggregates
+increases, differential vertical settling (Safronov (1969)),
+radial drift (Whipple (1972)) and turbulence (V¨ olk et al.
+(1980); Mizuno et al. (1988); Ormel & Cuzzi (2007))
+will become important new mechanisms driving relative
+velocities between aggregates. The increasing relative
+velocities caused by these mechanisms will at least partly
+compensate the lower collision probability due to lower
+surface-over-mass ratio of large aggregates. When the
+aggregates grow to sizes of millimeter to meter, however,
+the sticking efficiency drops strongly (e.g. Blum & M¨ unch
+(1993)) and the relative velocities become so large that
+aggregates can fragment (Blum & Wurm (2008), so called
+‘fragmentation barrier’). Another hurdle that the particles
+have to circumvent is the ‘drift barrier’ (Weidenschilling
+(1977a)), namely that millimeter, centimeter sized particles
+are lost to the star due to radial drift in a short timescale.
+Recently, Okuzumi (2009) pointed out the existence of a
+‘charge barrier’, which possibly halts the particle growth
+already at an early stage of fractal aggregates. Despite
+manyyearsofefforts,it is not knownif the coagulationpro-
+cess can overcome these barriers. These barriers have been
+and still are the main open question of the initial stages
+of planet formation: the growth from dust to planetesimals.
+Several mechanisms have been proposed to overcome
+this problem, among which are the trapping of dust in vor-
+tices (Barge & Sommeria (1995); Klahr & Henning (1997),
+Lyra et al. (2009)), trapping of decimeter-sized boulders
+in turbulent eddies and the subsequent gravitational col-
+lapse of swarms of these trapped boulders (Johansen et al.
+(2007)), thetrappingofparticlesina pressurebump caused
+by the evaporation front of water (Kretke & Lin (2007);
+Brauer et al. (2008b)) and many more scenarios. However,
+the correct modeling of any of these scenarios requires the
+detailed knowledge of the collisional physics, and these
+models have so far relied either on simplified input phy-
+isics or on simplified initial conditions.
+Becauseoftheircomplexity,collisionalevolutionmodels
+have to make simplifying assumption concerning the out-
+comeofdustaggregatecollisions,forexamplethatcollisions
+always result in sticking, or otherwise use simple recipes
+for the collisional outcome. Ideally, one requires to know
+the detailed outcome of every collision. But modelling this
+microphysics within an evolution model is simply unprac-
+tical. There are computer programs that model such indi-
+vidual collisions in detail (e.g. Dominik & Tielens (1997),
+Suyama et al. (2008); Geretshauser et al, in prep.), but
+each model collision takes anywhere from hours to weeks
+to run on a computer. They are therefore not practical
+to use at run-time in a model that computes the overall
+time-dependent evolution of the dust size distribution in-side protoplanetary disks. Moreover, such collision models
+themselves often depend on poorly known input physics.
+Another approach to obtain the collisional outcome of
+dust aggregatesistomodel thesecollisionin the laboratory.
+From the many experiments that have as of now been per-
+formed a picture emerges of the outcome of dust aggregate
+collision under a variety of conditions in the protoplanetary
+disk (PPD). In a companion paper (G¨ uttler et al. (2009),
+henceforth Paper I), we have collected data from over 19
+experiments, and constructed a set of formulaethat reason-
+ably well describe the outcomes of these collisions in such a
+way that they can be used as input for models that address
+the temporal evolution of the dust size distribution.
+In this paper we will directly rely on the outcome of
+these laboratory experiments for modeling the dust aggre-
+gate size distribution. As described in Paper I, we have
+produced a mapping of all available collision experiments
+regarding silicate-like particles. The velocity range of these
+experiments is also sufficiently wide to cover various disk
+models which roughly correspondto the conditions at 1 AU
+in the PPD. For details regarding the collisional mapping,
+we will refer to paper I, but we will summarizethe elements
+of our new collision model in Sect. 3.1.
+We build this collision kernel into a Monte Carlo code
+for modeling the size- and porosity distribution of dust
+in a protoplanetary disk (Zsom & Dullemond (2008),
+hereafter ZsD08). The outcome of our laboratory-driven
+dust coagulation model is hard to a priori predict since the
+key variables involved depend on a non-trivial interplay
+between the collisionkernel (Paper I) and the velocity field.
+We can, however, anticipate two scenarios. In the first,
+particle growth will proceed beyond the meter-size barrier,
+all the way to planetesimals. In the second scenario, growth
+will terminate at an intermediate size. In this case further
+growth to planetesimal sizes may proceed through con-
+centration and subsequent gravitational collapse of these
+particles (Johansen et al. (2007), Cuzzi et al. (2008)).
+Thus, our model will provide the starting conditions for
+these concentration models. We do emphasize, however,
+that in this work we do not in any way ’optimize’ the
+outcome by laboriously scanning all the parameter space
+or treating environments that may be more conducive for
+growth, like nebula pressure bumps or trapping of dust in
+vortices (Kretke & Lin (2007), Lyra et al. (2009)). These
+are obvious expansions of our work. But by considering
+the sensitivity of a few key parameters (e.g., gas density,
+and turbulence strength) on the outcome of the growth
+process, we do obtain a picture of where the arrow of coag-
+ulation typically points to in protoplanetary environments:
+pebbles, boulders or planetesimals.
+In this paper we describe the three nebulae models
+used in this work and the sources of relative velocity be-
+tween the aggregates (Sect. 2), how we build the coagula-
+tion/fragmentation model of Paper I into the Monte Carlo
+code (Sect. 3), and what these first results look like (Sect.
+4). We also test the sensitivity of the results with respect to
+variations in gas density, the velocity field, and other key
+model parameters. Section 5 reflect the importance of our
+result in the context of planetesimal formation and provide
+suggestions for future experiments. Finally, Sect. 6 lists our
+main conclusions.A. Zsom et al.: The outcome of protoplanetary dust growth: pe bbles, boulders, or planetesimals? II. 3
+2. The nebulae model
+2.1. Disk models
+In this Section we briefly describe the disk models consid-
+ered in this paper.
+The low density model: Resolved millimeter emission maps
+of protoplanetary disks seem to indicate a shallow sur-
+face density profile (Andrews & Williams (2007)): Σ g(r)∝
+r−0.5. Systematic effects of some of their assumptions, such
+as the disk inclinations or the simplified treatment of the
+temperature distribution, may suggest somewhat steeper
+profiles. Therefore, Brauer et al. (2008a) adopted the fol-
+lowing profile:
+Σg(r) = 45g
+cm2/parenleftBigr
+AU/parenrightBig−0.8
+. (1)
+Here we assumed that the central star is of solar mass, the
+disk extends from 0.03 AU until 150 AU and that the total
+mass of the disk is 0.01 M ⊙. Assuming that the pressure
+scale-height is Hp= 0.05×rand the vertical structure is
+gaussian:
+ρg(z,r) =Σg(r)√
+2πHpexp(−z2/2H2
+p), (2)
+the density at 1 AU in the midplane ( z= 0) is 2 .4×10−11
+g cm−3, approximately two orders of magnitude lower than
+the Minimum Mass Solar Nebulae (MMSN) value.
+MMSN model: The Minimum Mass Solar Nebulae model
+(MMSN) was introduced by Weidenschilling (1977b) and
+Hayashi et al. (1985). From the present state of the Solar
+System today, it is possible to obtain a lower limit to
+the mass if the solar nebulae from which the planets were
+formed. The model assumes that the planets were formed
+wheretheyarecurrentlylocated(nomigrationincluded). It
+also assumes that all the solid material presented in the so-
+lar nebula had been incorporated in the planets. The lossof
+solid material due to radial drift is not taken into account.
+Despite these uncertainties, the MMSN model is frequently
+used as a benchmark. The surface density of the MMSN
+disk is given by:
+Σg(r) = 1700g
+cm2/parenleftBigr
+AU/parenrightBig−1.5
+, (3)
+which corresponds to a total disk mass of 0.01 M ⊙con-
+tained between 0.4 and 30 AU (between the orbits of
+Mercury and Neptune). Assuming that the vertical struc-
+ture of the gas follows a gaussian distribution, leads to a
+midplane density at 1 AU of 1 .4×10−9g cm−3.
+The high density model: Desch (2007) introduced a ‘re-
+vised MMSN model’ by adopting the starting positions
+of the planets in the ’Nice’ model of planetary dynamics
+(Tsiganis et al. (2005)) thus taking into account planetary
+migration.Themodelpredictsthat thesolarsystemstarted
+out in a much more compact configuration and its surface
+density profile is given by:
+Σg(r) = 5.1×104g
+cm2/parenleftBigr
+AU/parenrightBig−2.2
+. (4)This model is consistent with a decretion disk which is
+being photoevaporated by the central star. Although the
+model of Desch (2007) was defined for the outer solar
+system, we extrapolate the profile to 1 AU in order to
+cover a broad range of surface density values in our
+calculations. Assuming, as in the MMSN model, a gaussian
+vertical distribution, the density at 1 AU in the midplane
+is 2.7×10−8g cm−3.
+For simplicity, we adopt a midplane temperature of 200
+K (isothermal sound speed of cs= 8.5×104cm s−1) in all
+the three models.
+2.2. Relative velocities
+We consider three sources for relative velocities between
+dust aggregates. These are Brownian motion, radial drift
+and turbulence. In the following, we discuss these sources.
+The average relative velocity of two particles with mass
+m1andm2in a region of a disk with temperature Tdue
+to Brownian motion is
+∆vB(m1,m2) =/radicalBigg
+8kT(m1+m2)
+πm1m2. (5)
+For micron sized particles, the relative velocity is of the or-
+der of0.1cm s−1, but forcm sized particlesthis value drops
+severalorders of magnitude. Therefore,Brownian motion is
+only effective for collisions between small particles during
+the initial stages of growth. Coagulation due to Brownian
+motion results in fluffy aggregates of fractal dimension
+around 2 and 3 (Blum et al. (1996); Kempf et al. (1999);
+Blum et al. (2000); Krause & Blum (2004)). In practice
+there is no growth due to Brownian motion for aggregates
+larger than 100 micron.
+The second source for relative velocity is turbu-
+lence. Relative velocity of aggregates due to the ran-
+dom motion of turbulent eddies were calculated numer-
+ically by V¨ olk et al. (1980), Mizuno et al. (1988) and
+Markiewicz et al. (1991). We use the closed form expres-
+sions presented by Ormel & Cuzzi (2007). We assume that
+turbulence is parameterized by the Shakura & Sunyaev
+(1973)αparameter
+νT=αcsHg, (6)
+whereνTis the turbulent viscosity, csis the isothermal
+sound speed and Hgis the pressure scale height of the disk.
+The value of the αparameter reflects the strength of the
+turbulence in the disk. Typical values of αin this paper
+range between 10−3and 10−5. The turbulent relative ve-
+locityis afunction ofthe stoppingtimes ofthe twocolliding
+particles. The stopping time (or friction time) is the time
+the particle needs to react to the changes in the motion of
+the surrounding gas. As long as the radius of the particle
+is smaller than the mean free path of the gas ( a <9
+4λmfp),
+the particle is in the Epstein regime, where the stopping
+time is (Epstein (1924)):
+ts=tEp=3m
+4vthρgA, (7)
+wheremandAare the mass and the cross section of the
+particle, ρgandvthare the gas density and the thermal
+velocity. At high gas densities, where the mean free path4 A. Zsom et al.: The outcome of protoplanetary dust growth: p ebbles, boulders, or planetesimals? II.
+Fig.1.The combined relative velocities caused by Brownian motion, radial dr ift and turbulence for fluffy particles
+(Ψ = 20) in the three disk models for equal sized particles (a) and for different sized particles with a mass ratio of
+100 (b). The solid line indicates the low density model of Brauer et al. ( 2008a). Physical parameters of the disk: the
+distance from the central star is 1 AU, temperature is 200 K, the d ensity of the gas is 2 .4×10−11g cm−3, and the
+turbulence parameter, α= 10−4. The dotted line represents the MMSN model. The density is 1 .4×10−9g cm−3, the
+other parameters are the same. The dashed line corresponds to t he high density disk. The gas density is 2 .7×10−8g
+cm−3.
+is low or in case of larger particles, the first Stokes regime
+applies and the stopping time is
+ts=tSt=3m
+4vthρgA×4
+9a
+λmfp. (8)
+In the first Stokes regime the stopping time is indepen-
+dent of the particle-gas relative velocity as well as the
+gas density. This regime can be used as long as the par-
+ticle Reynolds number is smaller than unity. The particle
+Reynolds number calculated as (Weidenschilling (1977a)):
+Rep=2a∆vpg
+η, (9)
+where ∆ vpgis the relative velocity between the particle
+and the gas, and ηis the gas viscosity. For particles outside
+the Epstein regime, it can be assumed that the systematic
+velocity (radial drift) dominates over the random veloci-
+ties (turbulence); therefore, ∆ vpg≈vD, wherevDis the
+drift velocity of the particle, defined in the next paragraph.
+The particle Reynolds number never exceeds unity in our
+simulations. Therefore, we do not include further Stokes
+regimes.
+Radial drift also leads to relative velocities between ag-
+gregates. Radial drift ( vD) has two sources: drift of individ-
+ual particles ( vd) and drift due to accretion processes of the
+gas(vda),thusthetotalradialdriftvelocityis vD=vd+vda.
+The radial drift of individual dust aggregates with mass m
+is (Weidenschilling (1977a))
+vd=−2vN
+St+1/St, (10)
+whereStis the Stokes number of the aggregate ( St=tsΩ,
+where Ω is the orbital frequency) and vNis the maximum
+radial drift velocity (Whipple (1972)).
+Thesecondpartoftheradialvelocityisduetotheaccre-
+tion of the gas. This part of the radial velocity is calculatedFig.2.The Stokes number as a function of the particle
+radius in the three models. The parameters of the dust
+for all of the models are the following: monomer radius is
+a0= 0.75µm, material density is ρ0= 2 g cm−3, and
+Ψ = 1.
+as follows (Kornet et al. (2001)):
+vda=vgas
+1+St2, (11)
+wherevgasis the accretion velocity of the gas
+(Takeuchi & Lin (2002)).
+The relative velocity due to radial drift is then simply
+the difference between the radial velocity of particle 1 and
+particle 2. However, as the Stokes number of the aggregates
+is always smaller than 10−3(see Sec. 4), the second term
+of the radial velocity ( vda) can be safely neglected:
+∆vD=|vD1−vD2| ≈ |vd1−vd2|. (12)A. Zsom et al.: The outcome of protoplanetary dust growth: pe bbles, boulders, or planetesimals? II. 5
+This study uses two quantities to describe the porosity
+of the aggregates. The volume filling factor is:
+φ=V∗/Vtot= (A∗/A)3/2, (13)
+whereV∗is the volume occupied by the monomers and
+Vtotis the total volume of the aggregate, including pores,
+andAandA∗are the surface area equivalents of these
+quantities. In this way, the filling factors also enters the
+definition of the friction time (Eqs. 7 and 8). The density
+of aggregates then follows as ρ=ρ0φ, where ρ0= 2 g
+cm−3is the material density of the silicate. In this study
+we will also use the reciprocal parameter of the filling fac-
+tor, which is denoted the enlargement parameter, Ψ = φ−1.
+We illustrate the relative velocity between equal sized
+and different sized aggregates with Ψ = 20 ( φ= 0.05) in
+Fig. 1forthe disk modelsconsideredin this work.Adopting
+a threshold (fragmentation) velocity of 1 m s−1, the maxi-
+mum particle size, which can be reached in the models are:
+0.025 cm in the low density model, 1.4 cm in the MMSN
+model and 1.7 cm in the Desch model. The Stokes num-
+bers of these particles are the same in all the three models,
+4.7×10−3. The constant fragmentation velocity of 1 m s−1
+is the typical velocity at which silicate particles will frag-
+ment (Birnstiel et al. (2009)). In our collision model this is
+not the case for all combinations of mass ratio and porosity
+(Paper I), but the m s−1threshold is still a useful proxy
+for the point where fragmentation processes will become
+important.
+Figure 2 shows the Stokes number as a function of par-
+ticle radii in the three models. Initially, particles are in the
+Epstein regime, where the stopping time, thus the Stokes
+number, depends on the gas density. When the particles
+enter the Stokes regime, the stopping time becomes inde-
+pendent of the gas density (see Eq. 8). One can see that
+particles in the Desch model are in the Stokes regime at
+Stokes number of 4 .7×10−3(when the particles have rel-
+ative velocities of 1 m s−1), while the aggregates in the
+MMSN model are close to it, which explains why the max-
+imum particle size is almost the same in these two models.
+As discussed in Ormel & Cuzzi (2007), particles are ini-
+tially in the ‘tightly coupled particle’ regime, where the ed-
+dies are all of class I type, meaning that the turnover time
+of all eddies is longer than the friction time of the parti-
+cles (V¨ olk et al. (1980)). A particle, upon entering a class I
+eddy, will therefore forget its initial motion and align itself
+tothegasmotionsoftheeddy beforethe eddydecaysorthe
+particle leaves it. This regime is apparent in Fig. 1a and b.
+Different sized particles are in this relative velocity regime
+as long as their masses are less than 10−8g in the low den-
+sity model, 10−3g in the MMSN model and 10−2g in the
+high density model assuming fluffy particles (Ψ = 20). If
+the particles leave this regime and enter the ‘intermediate
+particle’ regime, their relative velocity increases. This tran-
+sition affects the particle evolution, as discussed in e.g. Sect
+4.3.
+3. Collision model and implementation
+Inthisworkweuseastatisticalor‘particleinabox’method
+to compute the collisional evolution. That is, we assumethat all particles are homogeneously distributed within a
+certain volume (the simulation volume). In reality however,
+the particlescould leavethe simulated volumeornew parti-
+cles could enter from outside due to radial drift or random
+motions (turbulence and Brownian motion). Since we do
+not resolve the spatial dependence of the aggregates, we
+will simply assume that local conditions hold during the
+run. The gas and dust densities are kept constant and par-
+ticlescannotleaveorenterthesimulationvolume(hereafter
+‘local approach’).
+3.1. Short overview of the collision model
+Many laboratory experiments on dust aggregate collisions
+have been performed in the past years, see Blum & Wurm
+(2008). The growth begins as fractal growth and we use
+the recipe of Ormel et al. (2007) to describe this initial
+stage. However, once aggregates have restructured into
+non-fractal, macroscopic aggregates (e.g. /greaterorsimilar100µm),
+laboratory experiments show that the collisional outcomes
+become verydiverse. In this regime, many new experiments
+were performed with dust aggregates consisting of 1.5
+µm diameter SiO 2monomers either with high porosity
+φ= 0.15 (Blum & Schr¨ apler (2004)), or intermediate
+porosity ( φ= 0.35). Paper I compiled 19 experiments
+with different aggregate masses, collision velocities, and
+aggregate porosities.
+Fromthese experimentswehaveidentified ninedifferent
+collisional outcomes involving sticking, bouncing, or frag-
+mentation (see Fig. 3). The occurrence of these regimes
+mainly depends on aggregate masses and collision veloci-
+ties. However, it also depends on the porosity of the par-
+ticles and on the critical mass ratio. For example, Paper I
+finds fragmentation in collisions between a porous aggre-
+gate and a solid wall, whereas Langkowski et al. (2008)
+find sticking of a porous projectile by penetrating an also
+porous target. Likewise, Heißelmann et al., in prep. find
+bouncingoftwosimilar-sized,porousdustaggregates,while
+Langkowski et al. (2008) find sticking for the same veloc-
+ity where one collision partner (target) was significantly
+bigger. To address the importance of the mass ratio and
+porosity, we have identified eight different collision regimes
+(look-up tables) based on a binary treatment of porosity
+and mass ratio: i.e., (i) similarly sized or differently sized
+collision partners and (ii) porous or compact collision part-
+ners. The further distinction between target, which we al-
+ways define as the heavier collision partner, and projectile
+then results in eight different collision regimes. We denote
+these regimes as ‘pP’(porous projectile, porous target; tar-
+get significantly bigger than the projectile), ‘pc’(porous
+projectile, compact target; target of similar size than the
+projectile), etc.
+In Paper I we have classified each of these 19 experi-
+ments into one or more of these eight regimes (see Fig. 10
+of Paper I). Based on extrapolation of experimental find-
+ings, we decide in which mass and velocity range collisions
+resultin sticking,bouncing,orfragmentation.Theseresults
+are presented in Fig. 11 in Paper I.
+It should be noted that the critical mass ratio be-
+tween the equal-size ( ‘pp’,‘cc’, etc.) and the different size
+regimes ( ‘pP’,‘cC’, etc.) is ill-constrained by experiments.
+Therefore, we use critical mass ratios of rm= 10, 100 and
+1000 to explore the effect of this parameter.6 A. Zsom et al.: The outcome of protoplanetary dust growth: p ebbles, boulders, or planetesimals? II.
+Fig.3.The collision types considered in this paper. We distinguish between sim ilar sized and different sized particles.
+Some of the collision types only happens for one of the mass ratios. G rey color indicates that during the given collision
+type the particle is compact, or part of the mass will be compacted.
+3.2. Porosity
+PaperI defined a binaryrepresentationofthe porosity, par-
+ticles are either porous or compact. Following the simple
+model of Weidling et al. (2009), we include a continuous
+transition between these two ‘phases’. They showed that
+the compaction of particles due to bouncing can be de-
+scribed by porous and compacted sites on the surface of
+the aggregate. A site of the aggregate is porous if it did not
+encounter any collisions yet (e.g. bouncing), a compacted
+site encountered at least one collision already but any fur-
+ther collision happening at that part of the surface cannot
+change the porosity of this site anymore. We describe the
+probability of hitting a passive site of the aggregate in the
+following way:
+Pp=φc−φ
+φc−φp, (14)
+whereφis the volume filling factor of the aggregate, φcis
+the critical porosity ( φc= 0.4, see Paper I), and φpis the
+volume filling factor of the porous site, which is chosen to
+be 0.15. If φis between 0.15 and 0.4, a random number
+decides whether the particle collided with a porous or a
+compact site. Such a treatment of the porosity ensures
+a continuous transitionfrom porousto compact aggregates.
+During the initial Hit & Stick (S1) phase, particles are
+in the fractalgrowth regime(Ossenkopf(1993), Blum et al.
+(2000),Krause & Blum(2004)).Particlesgrowinitiallydue
+to Brownianmotionand laterdue to turbulence. The struc-
+ture of the aggregate depends on whether the collision
+happened between a cluster and a monomer (PCA) or
+between two clusters (CCA). The latter results in fluffy
+aggregates with fractal dimension of 2, while the formerleads to more compact structures with fractal dimension
+of 3. The hit&stick recipe of Ormel et al. (2007) attempts
+to interpolate between these two fractal models. At one
+point, collisional energies become large enough to invali-
+date this assumption. This occurs when the collision energy
+is five times higher than the rolling energy of monomers
+(Dominik & Tielens (1997), Blum & Wurm (2000)). The
+internal structure then becomes homogenous. In our model
+we assume that once the fractal growth due to S1 is over,
+Bouncingwithcompaction(B1)restructurestheaggregates
+producing compact structures with fractal dimension of 3.
+As our model can not follow the exact shape of the par-
+ticles, we assume that the aggregates are spheres and can
+be described with a single density ( ρ) thus neglecting the
+effects of e.g. the ‘toothing radius’ of Ossenkopf (1993) or
+the craters forming during Penetration (S3).
+3.3. The Monte Carlo method
+Using the expressions for the relative velocity, the col-
+lisional cross section between the dust particles, and
+the collisional outcome, we solve for the temporal evo-
+lution of the dust size distribution. Traditionally, the
+Smoluchowski equation is solved to follow the evolution of
+the mass distribution function (e.g. Dullemond & Dominik
+(2004), Dullemond & Dominik (2005), Tanaka et al.
+(2005), Brauer et al. (2008a)). The continuous form of
+the Smoluchowski equation used in these works lacks
+the stochasticity of the coagulation problem (Safronov
+(1969)). All bodies with mass mwill grow in the same way
+thus the spatial and temporal fluctuations of the particle
+ensemble are averaged out. In reality, however, particles
+with similar masses might follow a different evolutionaryA. Zsom et al.: The outcome of protoplanetary dust growth: pe bbles, boulders, or planetesimals? II. 7
+path depending on what other particles they collide
+with. The collision model typically used in these works
+is, by necessity, rather simple as in the Smoluchowski
+formulation the collision and time evolution steps are
+linked together. These collision models consist of sticking
+and fragmentation and only the mass of the particles
+is followed. The advantage of such a model is that it is
+computationally not too expensive: The entire disk can be
+modeled.
+Ormel et al. (2007) introduced a new Monte Carlo
+method to solve for the mass and the porosity distribu-
+tion function simultaneously. Their collision model consists
+of sticking and compaction; ZsD08 added a simple frag-
+mentation model as well. Although these models are more
+detailed, one can see that they still lack the full complex-
+ity which is observed at “the zoo” of laboratory collision
+experiments.
+The MC-approach used in this study has previously
+been presented by ZsD08. It can be characterized by two
+key properties: (1) the number of MC-particles (also re-
+ferred to as representative particles) is kept constant; (2)
+the method follows the mass of the particle distribution.
+Property (1) is required to preserve good statistics.
+Because of the√
+Nnoise of MC-methods, a large fluctua-
+tion ofNwould severely affect the accuracy of the method
+(Ormel & Spaans (2008)). The second property states that
+our primary interest lies in the particles that contain most
+ofthe massofthe system.Moreover,it has been shownthat
+following the particle’s mass distribution – rather than the
+number distribution – is also a prerequisite to preserve a
+good correspondence with systems that experience strong
+growth (Ormel & Spaans (2008)).
+Property (2) ensures that the MC method samples the
+parameter space only where a signicant portion of the total
+dust mass is. However, this is not always desirable. For
+instance, radiative transfer calculations require the surface
+area distribution of the aggregates, which determines the
+opacity. If most of the particle mass is contained in big
+particles (which are not observable) the amount of small
+particles (which could contain most of the surface area and
+determinestheIRappearanceofthedisk)mightberesolved
+with a bad statistics. But if we are interested in following
+the evolution of the dominant portion of the dust, then MC
+methods naturally focus on these parts of the phase space.
+A required condition for the ZsD08 method to work is
+that the number of the representative particles Nis much
+less than the number of actual aggregates present in the
+system under consideration – a condition that is safely
+met in any of our simulation runs. Then, a representative
+particle will collide only with the non-representative
+particles, whose distribution is assumed to be the same as
+that of the representative particles. We refer to ZsD08 for
+details regarding the precise implementation and accuracy
+of the method; here we further concentrate on how the
+method operates under the new collisional setup.
+The collisionkernelisdefinedastheproductofthe cross
+section of the colliding particles and their relative velocity:
+Ki,k=σi,k∆vi,k, (15)
+where the index icorrespondsto the representativeparticle
+andkis the index of the non-representative particle. The
+kernel is proportional to the probability of a collision. Thevalue of Ki,kis calculated for every possible particle pair,
+and random numbers determine which of the collision will
+occur first and at which time interval.
+The above properties and conditions specify the essence
+of the ZsD08 method: one of the two collision particles is a
+representativeparticle and, by property (1), only one of the
+collisional products becomes the new representative parti-
+cle. By property (2) the choice for the new representative
+particle is weighed by the mass of the collision products.
+A very helpful analogy here is that of the representative
+‘atom’, which is contained within the representative par-
+ticle. The choice for the new representative particle after
+the collision is then proportional to the probability of the
+representative ‘atom’ ending up in the collision products.
+If, for instance, a collision leads to the production of an en-
+tire distribution of debris particles, the probability that a
+particular debris fragment becomes the new representative
+particle is proportional to the likelihood of this fragment to
+contain the representative ‘atom’.
+3.4. Implementation of the collision types
+Wedescribetheimplementationofthecollisionmodelusing
+the representative ‘atom’ concept. We refer to Paper I for
+details of the various collision types described below.
+Hit & Stick (S1), Sticking through surface effects (S2),
+Penetration (S3): All three of these collision types result
+in sticking and increase the mass of the aggregate by that
+of the projectile, but the porosity changes in a different
+manner (see Paper I). The new mass of the representative
+particleiis then the sum of the original particle masses,
+mi,new=mi+mk, wheremiis the mass of the representa-
+tive particle and mkis the mass of the non-representative
+particle.
+Mass transfer (S4): In the case of Mass transfer (S4), a
+certain percentage of the mass of the projectile sticks to
+the target, while the left-over mass of the projectile will
+fragment into a power law distribution (see Paper I).
+There are two situations to consider:
+1. The representative ‘atom’ is part of the target. The
+mass of the new aggregate will be the mass of the orig-
+inal aggregate plus the transferred mass from the non-
+representative particle ( mi,new=mi+mtrans, where
+mtransis the transferred mass calculated according to
+Paper I).
+2. The representative ‘atom’ is part of the projectile.
+Again, we have two situations.
+(a) The representative ‘atom’ will be transferred to the
+non-representative particle. The mass of the new
+representative particle will be the mass of the non-
+representative particle plus the transferred material
+(mi,new=mk+mtrans). The probability of trans-
+ferring (removing) the representative atom from the
+projectile is simply P=mtrans/mi, the ratio be-
+tween the transferred mass and the mass of the pro-
+jectile.
+(b) The representative ‘atom’ remains in one of the
+fragments. The probability of this event is P=
+(mi−mtrans)/mi, the ratio between the fragmented8 A. Zsom et al.: The outcome of protoplanetary dust growth: p ebbles, boulders, or planetesimals? II.
+mass to the original mass of the representative par-
+ticle. As discussed in Paper I, the fragments follow a
+power law mass distribution. The distribution is de-
+fined by the maximum mass of the fragments, which
+is a function of the relative velocity and the total
+mass of the fragments. The total mass of the frag-
+ments is mi−mtrans. We randomly choose from the
+fragment distribution to find the new mass of the
+representative particle (to find which of the frag-
+ments will contain the representative ‘atom’).
+Bouncingwithcompaction(B1): Upon Bouncingwith com-
+paction (B1) particles collide and bounce. Bouncing itself
+does not change the mass of the particles, but it compact-
+ifies them according to Paper I. As observed in laboratory
+experiments (Weidling et al. (2009)), there is a small prob-
+ability (Pfrag= 10−4) that the bouncing particle will break
+apart. If this happens, we break the particle into two equal
+mass pieces.
+Bouncing with mass transfer (B2): Bouncing with mass
+transfer (B2) is, from the implementation point of view,
+similar to Mass transfer (S4). The recipe to define the
+new representative particle is as in Mass transfer (S4). The
+difference is that the projectile does not fragment during
+the collision, and that the porosity changes differently (see
+Paper I).
+Fragmentation (F1): Fragmentation only happens between
+similar sized aggregates in the ‘pp’and‘cc’regimes. The
+fragments follow a power law mass distribution where the
+maximum mass of the fragments is determined by the rel-
+ative velocity of the particles and the total mass that goes
+into the fragments (Paper I). We randomly choose from
+these distribution to determine the new mass of the repre-
+sentative particle.
+Erosion(F2): Erosion(F2)happensbetweendifferent sized
+particles only. During the collision the projectile“kicksout”
+pieces from the target aggregate. These pieces follow a
+power law distribution (see Paper I). We have to consider
+two cases.
+1. The representative ‘atom’ is in the target. Again, we
+have two possibilities.
+(a) The representative ‘atom’ will stay in the target af-
+ter the collision. The mass of the new particle will
+bemi,new=mi−mer, wheremeris the eroded mass.
+The probability of this event is P= (mi−mer)/mi,
+that is the ratio between the left-over mass (which
+does not erode) and the mass ofthe originalparticle.
+(b) The representative ‘atom’ is part of the eroded par-
+ticles. As the erodedparticlesfollow apowerlawdis-
+tribution,werandomlypickfromthisdistribution to
+determine the new mass of the representative parti-
+cle. The likelihood for this event is the ratio between
+the eroded massandthe originalmassofthe particle
+(P=mer/mi).
+2. The representative ‘atom’ is part of the small particle
+which caused the erosion. As the particles do not stick
+and the small particle does not fragment, the represen-
+tative particle remains unaffected.Fragmentation with mass transfer (F3): In Fragmentation
+with mass transfer (F3) the porous particle gets destroyed
+by the compact one and transfers a certain amount of mass
+to the compact particle. Fragmentation with mass transfer
+(F3) only happens in the ‘cp’regime. Again, we have two
+possibilities.
+1. The representative ‘atom’ is part of the compact parti-
+cle. In this case the representative ‘atom’ cannot leave
+the particle. The new mass of the representative parti-
+cle will be mi,new=mi+mtrans, the sum of the original
+mass plus the transferred mass.
+2. The representative ‘atom’ was part of the porous aggre-
+gate.
+(a) The representative ‘atom’ is part of the material
+which is transferred to the compact particle. In this
+case, the new mass of the particle will be that of
+the compact (non-representative) particle plus the
+transferred material ( mi,new=mk+mtrans). The
+probability of this event is P=mtrans/mi.
+(b) The representative ‘atom’ is part of the fragments.
+As before, the mass distribution will follow a power
+law and we randomly pick from this distribution to
+determine the new mass of the representative par-
+ticle. The probability of this event is P= (mi−
+mtrans)/mi.
+3.5. Evolving the particle properties in time
+We summarize how the particle properties are evolved in
+time using the above described kernel. We start with the
+size and porosity distribution of the particles at a given
+time,t. Att= 0, we must give the initial size and porosity
+distribution, see Sect. 4.1. Knowing these:
+–We calculate the cross sections of all possible collision
+partners, as well as their relative velocities using the
+equations described in Sect. 2.2. Both are used to de-
+termine the collision rates between the particle pairs.
+–Byusingrandomnumbers,weidentifyfromthecollision
+rates the representative particle, which is involved in
+the collision, as well as the non-representative particle
+itcollideswithandatwhattimethecollisiontakesplace
+(t+∆t).
+–Knowing the masses (mass ratio) and porosities of the
+collision partners, we identify in which of the eight
+regimes the collision takes place (e.g. ‘pP’, or‘pC’,
+etc.).
+–Next, we identify which of the nine collision types ma-
+terializes (Fig. 3) using the relative velocity of the par-
+ticles and the mass of the projectile (see Paper I).
+–Based on the collision recipe described in Paper I and
+Sect. 3.4, the new mass and new porosity of the rep-
+resentative particle is calculated and the new size and
+porosity distribution of the particles at time t+ ∆tis
+obtained.
+–In the final step, we update the collision rates.
+3.6. Numerical issues
+As mentioned in ZsD08, a sufficiently high number of rep-
+resentative particles is needed to properly reproduce the
+physics of the collision kernel. We performed simulations
+with an increasing number of representative particles andA. Zsom et al.: The outcome of protoplanetary dust growth: pe bbles, boulders, or planetesimals? II. 9
+found that for more than 200-300 particles the results
+of the simulations do not change significantly. For all of
+the simulations described in the following sections 500
+representative particles are used and we average the results
+of 20 simulations to decrease the numerical noise of the
+code.
+The required computational time strongly depends on
+the collision rate of the particles thus determined by the
+dust density and the relative velocity (the αturbulence
+parameter mostly) and the length of the simulation. On a
+2.83GHzCPU,runningthesimulationsonasinglecore,the
+CPUtime variesbetweennine hours(the lowdensitymodel
+withα= 10−5)andthreedays(thehighdensitymodelwith
+α= 10−3). Both simulations covered 106years of particle
+evolution. The high density simulation with α= 10−4, crit-
+ical mass ratio of 100 and t= 107years of evolution takes
+twelve days to simulate.
+4. Results
+4.1. Initial conditions, setup of simulations
+All simulations start with silicate monomers of 1.5 µm di-
+ameter and 2 g cm−3material density (monodisperse size
+distribution). We simulate the dust evolution at the mid-
+plane of our disk models at a distance of 1 AU from the
+centralstar.The gasdensity is obtained fromthe disk mod-
+els described in Sect. 2.1. We assume a typical 1:100 dust
+to gas ratio. We follow the history of each collision: the
+mass and porosity of the colliding particles, their relative
+velocity, the occurred collision type and the new mass and
+porosity of the particles. In this way we can reconstruct the
+history of the dust evolution.
+Theparameterswevaryinthisstudyarethegasdensity
+ρgand the turbulence parameter α. We also treat the crit-
+ical mass ratio rmas a free parameter in order to explore
+its effect on the dust evolution.
+We provide a detailed description of the low density
+model with α= 10−4and critical mass ratio of 100 in Sect.
+4.2. We then compare this with the MMSN model and the
+high density model using the same turbulence parameter
+and the critical mass ratio (Sects. 4.3 and 4.4). In Sects.
+4.5 and 4.6 we discuss the effects of changing the turbu-
+lence parameter and critical mass ratio by comparing those
+results with the two example runs.
+4.2. The low density model
+The gas density in this disk model at 1 AU is 2 .4×10−11
+g cm−3, the turbulence parameter is α= 10−4, the critical
+mass ratio is rm= 100. As shown in Fig. 1, the particles
+reach the fragmentation velocity (1 m s−1) already at sizes
+smaller than millimeter because the particles in low gas
+density environment decouple from the gas alreadyat these
+small radii.
+Figure 4 shows the evolution of the mass distribution
+(a),the porositydistribution (b)andthe collisionfrequency
+of the various collision types (c). The x-axis shows the time
+in a logarithmic scale. The y-axis of Fig. 4a, b shows the
+mass and enlargement distributions, respectively. Here, the
+intensity of the color reflects the number density of repre-
+sentative particles, which, as explained in ZsD08, measures
+the mass density of the distribution. Thus, in Fig. 4a theintensity levels directly reflect the mass density, while in
+Fig. 4b the colours indicate the mass weighted enlargement
+parameter. The black lines show the average of these quan-
+tities over the particle distribution. The y-axis in Fig. 4c
+represents the nine collision types used in this paper. Every
+stripe shows the total collision rate of the collision types at
+a given time.
+Figure 5 represents the collision history in the eight col-
+lision regimes. The x-axis is the velocity, the y-axis shows
+the mass of the projectile. A mass-velocity grid is created
+and for all grid cells we calculate how many collisions hap-
+pened inside that given grid cell during which our ‘local
+approach’ assumption is correct, that is 4 ×105yr. The
+different collision types and their border lines, as well as
+the areas which are covered with laboratory experiments
+(indicated with grey colors) are plotted. For more details
+on the experiments, see Paper I.
+In the‘cc’panel, we indicate two curves with dotted
+lines. These curves are evolution tracks. The left curve is
+obtained by calculating the relative velocity between equal
+sized particles with an enlargement parameter of 2.5 (vol-
+ume filling factor of 40%). The right curve represents the
+relative velocity between particles having a mass ratio of
+100. These two curves serve as a guide to our results, as
+collisions should happen between these two curves in the
+‘cc’panel. The lower part of the left curve, where the rela-
+tive velocity decreases with increasing mass, is a sign that
+relative velocities between equal sized particles are domi-
+nated by Brownian motion. For higher masses, the relative
+velocity is dominated by turbulence. These curves do not
+precisely match the contours because we assumed a con-
+stant enlargement parameter of 40% when calculating the
+evolution tracks, whereas Ψ is a free parameter in the sim-
+ulation.
+4.2.1. Early evolution
+We discuss here the evolution of the distribution functions
+until the ‘local approach’ assumption becomes invalid (4 ×
+105yr). The long term evolution of the dust is discussed in
+Section 4.2.3.
+We distinguish two distinct phases here. During the
+first 300 yr, particles grow by the Hit & Stick (S1) mecha-
+nism. The second phase is Bouncing with compaction (B1)
+dominated; the particles leave the S1 regimes. During this
+phase the mass of the particles is slowly decreasing and the
+enlargement parameter asymptotically reaches a minimum
+value of2.23. As discussed in PaperI, keeping the bouncing
+velocity of a particle constant, the porosity of the aggre-
+gate will asymptotically reach a maximum value, φmax(see
+Paper I). The relative velocity of a particle is a function of
+the friction time (Eq. 7), which depends on the ratio of the
+mass to surface area, m/A. Since particle growth is halted
+at this point in the simulation ( mstays constant), only a
+decrease in Adue to compaction can further increase the
+velocity between particles. The particle radius can decrease
+until either φmaxfor the given relative velocity is reached,
+or until particles reach the maximum compaction possible.
+The latter limit, random close packing (RCP), corresponds
+to an enlargement parameter of 1.6 (volume filling factor
+of∼60%).
+We find that fragmentation does not play a role during
+the evolution of these particles indicated by Fig. 4c. As
+can be seen in Fig. 5, their evolution is halted by bouncing10 A. Zsom et al.: The outcome of protoplanetary dust growth: pebbles, boulders, or planetesimals? II.
+Fig.4.The evolution of the mass distribution (a), enlargement parameter distribution (b) and the collision frequency of
+the nine different collision types (c) in the low density model with α= 10−4and critical mass ratio of 100. The x-axis
+shows the time. The y-axis of the (a) and (b) figures show the logar ithmic mass and the linear enlargement parameter
+respectively. The contours represent the normalized mass densit y and the mass weighted enlargement parameter. The
+black lines represents the average of the mass and enlargement pa rameter at a given time. The y-axis on the (c) figure
+represents the nine collision types. Each stripe shows the total co llision rate of the collision types. Two distinct phases
+can be distinguished. During the initial 300 yr particles grow by Hit & St ick (S1), after that the evolution is governed by
+Bouncing with compaction (B1). The white lines indicate how long our ‘loc al approach’ assumption is valid (discussed
+in Sect. 3).A. Zsom et al.: The outcome of protoplanetary dust growth: pe bbles, boulders, or planetesimals? II. 11
+Fig.5.The collision history of the eight regimes in the low density model for α= 10−4. The x-axis is the relative velocity,
+the y-axis shows the projectile mass. The different collision types, t heir border lines, as well as the areas covered with
+laboratory experiments (grey) are plotted. A relative velocity - ma ss grid is created and in these grid cells we calculate
+how many collisions happened until the ‘local approach’assumption is valid (4×105yr). This is representedby the colors:
+yellow and red indicate a high collision frequency. The two dotted lines o n the‘cc’regime are evolution tracks. Assuming
+a constant (40%) volume filling factor, the relative velocity between equal sized particles (left curve) and particles with
+a mass ratio of 100 (right curve) can be calculated. The collisions in th e simulation should lay between these two lines.
+The small deviations are due to the fact that the volume filling factor is not exactly 40% during the simulation.12 A. Zsom et al.: The outcome of protoplanetary dust growth: pebbles, boulders, or planetesimals? II.
+before the particles could reach the fragmentation barrier.
+The two dominant collision types are Hit & Stick (S1) and
+Bouncing with compaction (B1).
+4.2.2. Termination of growth
+As we can see from Fig. 5, sticking at higher energies than
+the Hit & Stick (S1) border lines is only possible inside the
+‘pP’regime. As soon as we no longer have collisions inside
+this regime or the S1 regimes, the growth is halted. There
+can be two reasons why this is happening: 1.) All parti-
+cles are compact; there are simply no collisions in the ‘pP’
+regime. 2.) The width of the particle mass distribution is
+less than the critical mass ratio ( rm), such that all collision
+take place in the equal-size regimes ( ‘pp’,‘pc’, etc.).
+In the case of the current simulation, the small particles
+have been ‘consumed’. Once the heavy particles grow into
+the Bouncingwith compaction(B1)areaofthe ‘pp’regime,
+their growth in the ‘pp’regime stops. The heavy particles
+collect the small ones via collisions in the ‘pP’regime and
+by doing so, the width of the distribution is reduced to
+a value which is less than rm. Therefore, before particles
+could reach the fragmentation barrier, growth is halted.
+Due to B1, particles get compacted and collisions in the
+‘cc’,‘cp’and‘pc’regimes appear.
+4.2.3. Long term evolution
+Beforediscussingthelongtermevolutionofthedistribution
+functions, we must consider for how long our starting as-
+sumptions (‘local approach’ and constant gas density) hold
+true.
+Using Eq. 11, we calculate that a particle with Stokes
+number 10−4drifts a distance of 1 AU in roughly 4 ×105
+yr. This is the drift timescale beyond which the ‘local ap-
+proach’ assumption (discussed in Sect. 3) is not valid any-
+more: particles become separated from each other on this
+timescale.
+Another process through which particles separate is by
+viscous spreading. We determine the viscous timescale of
+the disk at 1 AU:
+tvis=r2/νT, (16)
+whereris the distance from the central star (1 AU), νTis
+defined in Eq. 6. The viscous timescale in our model, using
+α= 10−4, is of the order of 106yr.
+One has to consider the results of the simulation
+with caution for longer times than the drift or viscous
+timescales. The equilibrium or final state of the particles
+is reached when mass decrease during Bouncing with
+compaction (B1) and Bouncing with mass transfer (B2)
+and mass increaseby Hit & Stick (S1) and Sticking through
+surface effects (S2) are in equilibrium. Or in other words,
+the final state is reached when the evolution of the average
+mass and the enlargement parameter is only determined
+by the stochastic fluctuations of the simulation. We find
+that the equilibrium state of the particles is hardly reached
+within these timescales. Upon neglecting these warnings,
+we find that the equilibrium state of the dust is reached
+att= 4×105yr. The equilibrium is reached between
+the bouncing collisions resulting in breakage and Hit &
+Stick (S1) (see Fig. 4c). The equilibrium average mass andporosity of the particles are ¯ mfin= 2×10−8g,¯Ψfin= 2.77.
+To be able to compare the distribution functions of dif-
+ferent runs, we define some quantities using the mean of
+the distribution functions shown with black lines in Fig. 4a
+and b: max(¯ m), the maximum of the mean mass; max( ¯Ψ),
+maximum of the mean enlargement parameter; Ψ min, the
+minimum mean enlargement parameter when particles do
+not compact anymore; tnoc, the time when Ψ minis reached,
+that is when the time derivative of ¯Ψ is zero ( d¯Ψ/dt= 0);
+andmax( ¯St),themaximumaverageStokesnumberreached
+during the simulation. The values of these quantities are
+listed in Table 1 (model id ‘Lt1d-4m100’). In this table,
+Col. 1 describes the model names. ‘L’ stands for the low
+density model, ‘M’ is the MMSN model, ‘H’ is the high den-
+sity model, the letter ‘t’ and the following number indicates
+the value of the turbulence parameter, and the letter ‘m’
+and the number shows the used critical mass ratio values.
+Columns 2, 3 and 4 show the gas density, turbulence pa-
+rameter and the critical mass ratio respectively. Columns
+5, 6, 7, 8 and 9 list the parameters defined to character-
+ize the distribution functions. These are max(¯ m) in Col. 5,
+max(¯Ψ) in Col. 6, Ψ minin Col. 7, tnocin Col. 8 and finally
+max(¯St) in Col. 9.
+4.3. The MMSN model
+The gas density in the MMSN model at 1 AU at the mid-
+plane is 1 .4×10−9g cm−3,α= 10−4, the criticalmass ratio
+is 100. As shown in Fig. 1, the particles grow to bigger sizes
+than in the low density model, as they are better coupled
+to the gas and the relative velocities are suppressed. As in
+the previous Section, we first discuss the evolution of the
+distribution functions for as long as the ‘local approach’
+assumption holds true (6 ×105yr in this model).
+Figure 6 shows again the time evolution of the mass (a),
+enlargement parameter (b), and the collision frequency (c).
+Figure 7 shows the collision history. These figures show a
+rather different evolution than the previous model.
+4.3.1. Early evolution
+We find that during the fractal growth regime, the collision
+rate of Hit & Stick (S1) is much higher than in the low
+density model (Fig. 6c). This is due to the higher dust
+densities. We can see from Fig. 7, ‘cc’regime, that growth
+starts with Brownian motion because the relative velocity
+decreases with increasing particle mass for particle masses
+less than 10−9g. As a result of these low velocity collisions,
+some particles reach enlargement parameter values higher
+than 30 (volume filling factor less than 3.3%). At 200 yr,
+some particles grow above the border line of Hit & Stick
+(S1) and enter the area of Sticking through surface effects
+(S2) in the ‘pP’plot, and Bouncing with compaction (B1)
+in the‘pp’plot. Growth due to S1 and S2 continues until
+different sized particles enter the transition regime in the
+‘pP’plot. One can see in Fig. 6a, that some particles reach
+1 g in mass. However when particle collisions enter the
+transition regime between Bouncing with mass transfer
+(B2) and Sticking through surface effects (S2) in the ‘pP’
+plot, their masses are equalized due to the mass transfer of
+the B2 collisions and the collisions shift to the similar sized
+regime (B1). We find that after roughly 104yr particlesA. Zsom et al.: The outcome of protoplanetary dust growth: pe bbles, boulders, or planetesimals? II. 13
+Fig.6.Same as Fig. 4 but for the MMSN model. We magnify the spike of the mas s distribution at ∼2.5×103yr in
+Fig. (a). Four phases can be distinguished here. Initially (first 300 y r) particles grow purely by Hit & Stick (S1). After
+this the growth slows down because Bouncing with compation (B1) st arts and all particles leave the Hit & Stick (S1)
+regime. Between 3 ×103and 104yr, particles enter the transition regime between Sticking through surface effects (S2)
+and Bouncing with mass transfer (B2) on the ‘pP’regime. Some particles reach masses of 1 g, but their masses are fa stly
+reduced by B2. The last phase is Bouncing with compaction (B1) domin ated. The solid/dotted white lines indicate how
+long our ‘local approach’ assumptions are valid (discussed in Sect. 3 ).14 A. Zsom et al.: The outcome of protoplanetary dust growth: pebbles, boulders, or planetesimals? II.
+Fig.7.Same as Fig. 5 but for the MMSN model. The particles are better coup led to the gas due to the higher gas
+density. Therefore, they grow to bigger sizes than in the low densit y model.A. Zsom et al.: The outcome of protoplanetary dust growth: pe bbles, boulders, or planetesimals? II. 15
+mostly bounce and compact. The enlargement parameter
+reaches a minimum value of 1.85 (54% volume filling
+factor), the mass distribution function slowly decreases
+due to a small probability of breakage. Collisions at this
+point are mainly happening in the ‘cc’regime.
+A peculiar feature of Fig. 6a is a peak at t= 2.5×103
+yr, which is accompanied by a fast decrease in the enlarge-
+ment parameter in Fig. 6b and an increased collision rate
+of Sticking through surface effects (S2) and Bouncing with
+mass transfer (B2) in Fig. 6c. At this point, the relative
+velocity due to turbulence increases. As discussed in Sect.
+2.2, particles leave the ‘tightly coupled particle’ regime and
+enter the ‘intermediate particle’ regime (see the relative ve-
+locity bump in Fig. 1b). We calculate the growth timescale
+of the heaviest particle with mass Min the simulation as
+follows:
+tgr=/parenleftbigg1
+MdM
+dt/parenrightbigg−1
+. (17)
+This is illustrated with a dotted line in Fig. 8. As a com-
+parison, we also calculate the minimum growth timescale
+that a particle can have (solid line). That is:
+tmax=/parenleftbiggM
+ρd∆vσM/parenrightbigg−1
+, (18)
+whereσMis the cross section of the largest particle. Here,
+we assume that the ‘swept up’ particles have masses of
+M/100, therefore we use the relative velocity curve pre-
+sented in Fig. 1b, dotted line. The effect of the relative
+velocity bump and the increased growth rate is seen at 0.1
+g.
+The relative velocity ‘boost’ happens shortly after the
+particles enter the transition regime of S2 and B2 in the
+‘pP’plot. The heaviest particle, which encounters the
+velocity transition the earliest, experiences higher relative
+velocities leading to an increased collision rate with the
+other particles. As the particles are initially located at the
+lower part of the S2-B2 transition regime (with masses
+of 10−3g, see Fig. 6a), the heaviest particle experiences
+fast growth and reaches masses of 30 g. The simulated
+timescale, however, does not reach the minimum growth
+timescale due to the Bouncing with mass transfer (B2)
+collisions which are reducing the mass of the heaviest
+particle. The rest of the particle population increases in
+mass because of B2 and the growth rate of the heaviest
+particle decreases. Eventually, the fast growth of the
+heaviest particle is halted, the growth timescale at m= 30
+g is infinity. From this point on, the heaviest particle re-
+duced in mass,and B2 equalizesthe masses ofthe particles.
+4.3.2. Long term evolution
+We calculate the drift and viscous timescales to determine
+how long our assumptions of ‘local approach’ and constant
+gas density are valid. Assuming Stokes number 10−4par-
+ticles, we find that the drift timescale is of the order of
+6×105yr, the viscous timescale is 106yr. These timescales
+are indicated with solid and dotted white lines in Fig. 6.
+We find that the final equilibrium is reached at
+t= 2×106yr, which is longer than the drift and the
+viscous timescales. The equilibrium is reached between theFig.8.The dotted line and the ‘+’ signs represent the
+growthtimescale ofthe heaviest particle in the MMSN sim-
+ulation with α= 10−4andrm= 100. As a comparison, we
+show the minimum growth timescale a particle can have in
+this simulation (solid line).
+growth mechanisms of Hit & Stick (S1), Sticking through
+surface effects (S2) and the destruction mechanisms of
+bouncing resulting in breakage and Bouncing with mass
+transfer (B2). The final average mass and porosity of the
+particles are ¯ mfin= 2×10−3g,¯Ψfin= 3.3.
+We conclude that the dust evolution is more complex in
+the MMSN model than in the low density model because
+the complex interaction of the velocity field and the col-
+lision kernel is apparent in this model. As in the previous
+model,Bouncingwithcompaction(B1)isthemostfrequent
+collision type and Hit & Stick (S1) determines the initial
+particle growth, but Sticking through surface effects (S2)
+and Bouncing with mass transfer (B2) are of importance in
+this model. The final equilibrium is not reached within the
+drift and viscous timescales.
+4.4. The high density model
+The gas density in this model is 2 .7×10−8g cm−3at the
+midplane ofthe disk at 1 AUdistance from the centralstar.
+The values of α,rmand the dust to gas ratio are the same
+as in the previous models.
+Figure 1, dashed line, shows the relative velocity field
+of fluffy aggregates in this model. As already discussed in
+Sect. 2.2, the aggregates reach 1 m s−1relative velocities
+at similar masses as the MMSN model due to the Stokes
+drag. Therefore, we expect that the final aggregate sizes
+and masses will be similar to the particles produced in the
+MMSN model.
+Figure 9 shows the time evolution of the mass (a), en-
+largement parameter (b) and the collision frequency (c).
+Figure 10 illustrates the collision history.
+4.4.1. Early evolution
+As seen in Fig. 10, Brownianmotion is the dominant source
+of relative velocity, as long as particles stay below masses
+of 10−8g (that is an order of magnitude higher than in16 A. Zsom et al.: The outcome of protoplanetary dust growth: pebbles, boulders, or planetesimals? II.
+the MMSN model). Therefore, the enlargement parameter
+of the aggregates is also higher than in the MMSN model.
+As the Hit & Stick (S1) collisions are more frequent than
+in the MMSN model due to the higher dust densities, the
+particles reach the Sticking through surface effects (S2) –
+Bouncing with mass transfer (B2) transition regime earlier,
+att= 200 yr. The peak in the mass distribution is not as
+pronounced as in the MMSN model. The relative velocity
+boost happens for heavier aggregates (10−2g, see Fig. 1b)
+due to the higher gas density of the model. When the fast
+growthoftheheaviestparticlestarts,mostoftheprojectiles
+are already in the transition regime. Here, the B2 collisions
+soon reduce the mass of the heaviest particle and narrow
+the mass distribution.
+In contrast to the MMSN model, the mass of the parti-
+cles is not reduced due to the low probability of breakagein
+Bouncingwithcompaction(B1),butiskeptnearlyconstant
+in time. This is the result of the increased collision rate of
+Sticking through surface effects (S2). The S2 collision rate
+increased because of low velocity collisions, which are oc-
+curring when particles are in the tightly couple regime and
+havesimilarstoppingtimes. TheseS2collisionsarehappen-
+ing in the ‘pp’regime as seen in Fig. 10. These collisions
+cancel out the effect of breakage in B1.
+The maximum Stokes number reached in this model is
+3.6×10−5(see Table 1, model id ‘Ht1d-4m100’),lowerthan
+in the MMSN model. The growth in this model is halted by
+the Bouncing with mass transfer(B2) collisions in the tran-
+sition regime of the ‘pP’panel. This shows us that particles
+cannot reach masses much larger than 1 g independently
+from the gas density (or Stokes number), because at this
+point, particles enter the S2-B2 transition regime and the
+growth is halted. Further increasing the gas density would
+result in even lower Stokes numbers.
+4.4.2. Long term evolution
+The drift and the viscous timescales in the high density
+model are both 106yr. As seen in Fig. 9a, the particle
+masses do not change significantly after t= 103yr. The
+porosity is reduced due to Bouncing with compaction (B1)
+and it reaches a final value of 5.41 at t= 3×106yr.
+4.5. Varying the turbulence parameter
+To explore the effects of turbulence, we perform two more
+simulations in each of the disk models. We keep the critical
+mass ratio fixed (100) and vary only the turbulence param-
+eter (α) to have values of 10−3, 10−4and 10−5. The results
+are shown in Table 1, the first nine models and in Fig. 11a.
+TheworkofBrauer et al.(2008a)suggeststhatinsitua-
+tions where fragmentationlimits the growth,a lowerturbu-
+lence strength results in bigger aggregates. This, of course,
+directly reflects the shift of the fragmentation threshold (1
+m/s) to larges sizes when αis lower (Fig. 1). In this study
+it is fragmentation that balances the growth, which results
+in a (quasi) steady-state. For the low density models we do
+see a decrease of the final particle mass, but it is bouncing
+that balances it. In the low density model, particles grow
+only in the Hit & Stick (S1) regimes. When particles leave
+these regimes, the growth stops due to bouncing. The bor-
+der of the S1 regime is determined by the collision energy
+being lower than 5 ×Eroll, whereErollis the rolling en-ergy of monomers (see Paper I). As the collision energy is
+Ecoll= 1/2µ(∆v)2, particles in strong turbulence leave the
+S1 regimes at lower particle masses.
+On the other hand, the MMSN and high density mod-
+els show that the maximum mass of the particles can even
+increase with α. The precise value of the max(¯ m) is deter-
+mined bythe intensity ofthe peak in the mass-densityplots
+(Sect. 4.3.1) and this may vary somewhat between the sim-
+ulations. In the ‘Mt1d-4m100’ model we have argued that
+the spike is exceptionally pronounced due to the high prob-
+ability of Sticking through surface effects (S2) collisions at
+the initial part of the fast growth. However the main point
+is that in the MMSN/high density simulations the maxi-
+mum particle masses all end up around 1 g, independent of
+the turbulent strength.
+The reasonfor this is the nature ofthe S2-B2transition,
+which occurs at projectile masses of 10−4g in the ‘pP’
+plot. As explained before, collisions in the ‘pP’plot are
+the only way by which particles can grow after the Hit
+& Stick (S1) phase is finished. Thus, we require a broad
+distribution for a high growth rate. However, B2 collisions
+worksin the opposite way: it transfers mass from the target
+to the projectile, narrowing the distribution and decreasing
+the overall probability for the ‘pP’process. Thus, once B2
+becomes effective, there is a shift from the ‘pP’panel to
+the‘pp’panel. For the MMSN/high density models this
+behavior is always present and the important quantities
+involved (i.e., relative probability of B2 over S2) scale with
+massandnot with velocity.The resultis that the maximum
+masses particles achieve are ∼1 g and rather insensitive to
+the strength of the turbulence.
+4.6. Varying the critical mass ratio
+We perform simulations in the disk models with α= 10−4
+but with a varying critical mass ratio. We explore how the
+dust distributions change upon using rm= 10, 100 and
+1000. Table 1, lines 10 to 18, shows the parameters de-
+scribing the distribution functions, and Fig. 11b illustrates
+the maximum particle mass as a function of the critical
+mass ratio.
+By examining Table 1 we see that using rm= 10 in
+the low density model (‘Lt1d-4m10’) results in heavier and
+more compact particles. The low critical mass ratio means
+that the biggest particles in the different sized regimes can
+sweepuptheprojectilesandgrowtobiggersizes,eventually
+reaching the fragmentation line, where growth stops. As
+discussed in Sect. 2.2, assuming a fragmentation velocity of
+1 m s−1, the maximum Stokes number of the aggregates is
+4.7×10−3. This value is almost reached in this model.
+We find that there is no significant difference between
+therm= 100 and 1000 simulations in the low density
+model. The explanation for this can be found by exam-
+ining the width of the mass distribution in the Hit & Stick
+(S1) phase. This initial phase is happening in the same way
+independently of the critical mass ratio. If the critical mass
+ratiormis equal to or larger than the width of the distri-
+bution function, collisions between different size particles
+in the‘pP’regime are inhibited. After the S1 phase, the
+width of the distribution in the low density regime is ap-
+proximately 100. Therefore, we do not see any difference
+when the mass threshold is shifted from rm= 100 to rm
+= 1000; in both cases collisions occur between equal-sizeA. Zsom et al.: The outcome of protoplanetary dust growth: pe bbles, boulders, or planetesimals? II. 17
+Fig.9.Same as Fig. 4 but for the high density model. As in Fig. 6a, we zoom in on the peak at the mass distribution.
+The solid white line indicate how long our ‘local approach’ assumptions a re valid at t= 106yr (discussed in Sect. 3).18 A. Zsom et al.: The outcome of protoplanetary dust growth: pebbles, boulders, or planetesimals? II.
+Fig.10. Same as Fig. 5 but for the high gas density model.A. Zsom et al.: The outcome of protoplanetary dust growth: pe bbles, boulders, or planetesimals? II. 19
+Table 1. Overview and results of all the simulations.
+Model ρg α r mmax(¯m) max( ¯Ψ) Ψ mintnoc max(¯St)
+[g cm−3] [g] [yr]
+(1) (2) (3) (4) (5) (6) (7) (8) (9)
+Lt1d-3m100 2 .4×10−1110−3100 8 ×10−87.27 1.77 2 ×1042.5×10−4
+Lt1d-4m100 2 .4×10−1110−4100 9 .7×10−87.12 2.23 8 ×1042.2×10−4
+Lt1d-5m100 2 .4×10−1110−5100 2 .66×10−77.72 3.78 3 ×1052.1×10−4
+Mt1d-3m100 1 .4×10−910−3100 8.13 24.41 3.88 1045.1×10−4
+Mt1d-4m100 1 .4×10−910−4100 4.18 21.9 1.85 2 ×1052.8×10−4
+Mt1d-5m100 1 .4×10−910−5100 7 .7×10−230.0 4.13 7 ×1052.1×10−4
+Ht1d-3m100 2 .7×10−810−3100 3.77 34.1 5.61 1051.4×104
+Ht1d-4m100 2 .7×10−810−4100 0.23 38.0 5.41 3 ×1063.6×10−5
+Ht1d-5m100 2 .7×10−810−5100 0.28 43.9 4.94 4 ×1067.7×10−5
+Lt1d-4m10 2 .4×10−1110−410 9 .2×10−45.88 2.28 1053.8×10−3
+Lt1d-4m100 2 .4×10−1110−4100 9 .7×10−87.12 2.23 8 ×1042.2×10−4
+Lt1d-4m1000 2 .4×10−1110−41000 9 .7×10−87.09 2.29 8 ×1042.2×10−4
+Mt1d-4m10 1 .4×10−910−410 2 .5×10−219.4 2.1 2 ×1052.2×10−4
+Mt1d-4m100 1 .4×10−910−4100 4.18 21.9 1.85 2 ×1052.8×10−4
+Mt1d-4m1000 1 .4×10−910−41000 9 .5×10−323.1 2.9 2 ×1051.3×10−4
+Ht1d-4m10 2 .7×10−810−410 0.15 34.6 2.46 2 ×1064.5×10−5
+Ht1d-4m100 2 .7×10−810−4100 0.23 38.0 5.41 3 ×1063.6×10−5
+Ht1d-4m1000 2 .7×10−810−41000 8 .8×10−240.0 7.1 1053.5×10−5
+In this table, Col. 1 describes the model names. ‘L’ stands fo r the low density model, ‘M’ is the MMSN model, ‘H’ is the high
+density model, the letter ‘t’ and the following number indic ates the value of the turbulence parameter, the letter ‘m’ an d the
+number shows the used critical mass ratio values. Columns 2, 3 and 4 shows the gas density, turbulence parameter and the cr itical
+mass ratio respectively. Columns 5, 6, 7, 8 and 9 list the para meters defined to characterize the distribution functions. These are
+the average maximum mass in Col. 5, the average maximum enlar gement parameter in Col. 6, the minimum enlargement paramet er
+in Col. 7, the end of the compaction phase in Col. 8 and finally t he average maximum Stokes number in Col. 9.
+Fig.11. The maximum mean particle mass as a function of the turbulence para meter (a) and critical mass ratio (b).
+particles only and these are either S1 or (when this stage is
+over) B1.
+For the high density models (MMSN/Desch) we find
+that the outcome is again similar: growth halts at ∼0.1
+g (within a factor of 10) and no clear dependence on rm
+is seen. For the high mass ratios, growth is always in the
+similar-size regime. Here, it is the gas density that deter-
+mines the velocity, i.e., whether we have a sticking (S1) or
+a bouncing (B1) collision. Therefore, if rm= 1000, the highdensity model produces heavier particles than the MMSN
+model (see Fig. 11b). For lower rmit is again the nature of
+the S2-B2transitionregime that limits the maximum mass.
+Thus, the critical mass ratio is an important parame-
+ter since it determines the relative likelihood of collisions
+occurring in the different-size regime, which are in general
+moreconducivetogrowth.Conversely,insimulationswhere
+B2 collisions are important – which have the effect to nar-
+row the distribution – the width of the distribution will20 A. Zsom et al.: The outcome of protoplanetary dust growth: pebbles, boulders, or planetesimals? II.
+Fig.12. The collision frequencies of the 9 collision types in the MMSN model with α= 10−3andrm= 100.
+correspond to the value of the rmparameter, although we
+have also seen that the absolute size/mass is rather insen-
+sitive to it. Overall, these arguments indicate that a good
+knowledge of this parameter is important.
+5. Discussion
+We performed simulations with varying turbulence param-
+eter and critical mass ratio values in three disk models hav-
+ing low, intermediate and high gas densities. We find that
+Hit & Stick (S1) and Bouncing with compaction (B1) are
+the most dominant collision types. All simulations showthe
+presence of long lived, quasi-steady states. Fragmentation
+is rarely present, but even then, only for a limited time pe-
+riod. The absence of fragmentation is due to the bouncing
+collisions.
+5.1. The sensitivity of the results
+As presented in Sect. 4, the outcome of our simulations
+is determined by the collision kernel, the relative velocity
+field. A significant change in one, or both can alter the
+evolution of the aggregates.
+Here we present the results of a test simulation, where
+the Sticking through surface effects (S2) – Bouncing with
+mass transfer (B2) transition regime in the ‘pP’plot is
+neglected and replaced by S2 collisions. This alternative
+transition regime provides a good opportunity to further
+examine the fast growth presented in Sect. 4.3.1, as the
+kernel is now simplified. The new kernel also gives us the
+possibility to see how much the outcome of our simulations
+can be altered by changing critical areas of the parameter
+space. As the transition regime is only constrained by one
+experiment in a rather small area (see e.g. Fig. 5 or Fig. 11
+in Paper I), further experiments may make it necessary to
+change this part of the parameter space. We use the same
+initial conditions as in the ‘Mt1d-4m100’ model described
+in Sect. 4.3.In this case, the heaviest particle experiences increased
+relative velocities, as soon as it reaches m= 0.1 g, and
+the particle undergoes a fast growth period (as in the orig-
+inal MMSN simulation, Sect. 4.3). Figure 13 illustrates the
+growth timescale of the heaviest particle (dotted line) and
+the minimum growth timescale possible (solid line). As
+there are no B2 collisionsto reduce the mass of the heaviest
+particle, the growth timescale reaches the maximum that is
+possible. The heaviest particle increases in mass until the
+rest of the particle population enters the B2 regimes above
+0.1 g in the ‘pP’plot. In this simulation, the maximum av-
+erage mass is 27 g, whereas in the original simulation with
+the transition regime, the value is 4.18 g.
+This work, together with Paper I, is the first attempt to
+calculate dust growth in protoplanetary disks on an empir-
+ical, thus more realistic basis. However, a few more cycles
+of the feedback loop between the laboratory experiments,
+the models of the kind described by PaperI and the models
+described in the paper have to be conducted before we can
+get near a truly reliable model of dust growth in protoplan-
+etary disks.
+5.2. Retention of small grains
+Dullemond & Dominik (2005) showed that without a
+mechanism that reduces the sticking probability of parti-
+cles in the upper layers of the disk or without a continuous
+source of small particles, the observed SEDs of TTauri
+stars would show very weak infrared excess. The SEDs
+of TTauri stars have strong IR excess (e.g. Furlan et al.
+(2005), Kessler-Silacci et al. (2006)); therefore, some kind
+of grain-retention mechanism is needed to explain these
+SEDs. Previous models of grain growth assumed a continu-
+ous cycle of growth and fragmentation, which provides the
+necessary amount of small particles (see e.g. Brauer et al.
+(2008a), Dullemond & Dominik (2005), Birnstiel et al.
+(2009)). Our simulations, however, showed that the mass
+distribution function is narrow. Small, monomer sized
+particles are not present and fragmentation is ineffectiveA. Zsom et al.: The outcome of protoplanetary dust growth: pe bbles, boulders, or planetesimals? II. 21
+Fig.13. Growth timescale in the test simulation where
+the S2-B2 transition regime is replaced by S2 collisions
+only. The dotted line represents the growth timescale of
+the heaviest particles, the solid line is the minimum growth
+timescale.In this scenario,the growthtimescale reachesthe
+maximum possible value.
+in providing small particles, which could be transported to
+disk atmospheres. The question naturally arises: how can
+small grains be produced in our collision model?
+One possible solution might come from bouncing.
+Weidling et al. (2009) performed bouncing experiments
+by putting an aggregate onto an oscillating metal plate
+and measuring the porosity of particles due to collisions
+with the plate. They observed that approximately 10%
+of the projectile mass eroded during the experiment (see
+Table 1 of their paper). This mass loss can happen due to
+the initial collisions; thus the eroded mass sticked to the
+baseplate. It is also possible that small pieces of fragments
+grind off when the aggregates bounce, which cannot be
+observed in the experiment. These ground off particles can
+then diffuse out of the midplane and provide the necessary
+amount of small particles to the upper layers of the disk.
+Future laboratory experiments are needed to quantify the
+level of ground off particles in bouncing collisions.
+The second possible explanation is provided by dust
+growth at the upper layers of the disk. We performed two
+simulations at four pressure scale-heightsin the low density
+model and in the MMSN model using α= 10−4. We find
+that the relative velocity of two monomers in the Brauer
+model is 2 m s−1, thus monomers at these heights do not
+coagulate, only bounce. The particles in the MMSN model
+can form aggregates of maximum of 10 µm in size. Using
+a higher α(as is mostly assumed in the upper layers of
+the disk) can completely halt even this limited growth.
+Therefore, bouncing could be the key ingredient the mech-
+anism that reduces the sticking probability of the parti-
+cles. However, if substantial vertical turbulent mixing takes
+place, this may not help, because these monomers would
+then be “vacuum cleaned” away by the bigger particles at
+the interior of the disk. Further studies of 1D vertical slices
+of disk models are needed to investigate this scenario.5.3. Implications for planetesimal formation models
+One can also see that coagulation only cannot produce
+planetesimals with the conditions presented in this work.
+Even if the turbulence parameter is taken to be zero,
+relative velocity due to radial drift is preventing particles
+to cross the so called ’meter size barrier’. An ideal environ-
+ment for particle growth is a pressure bump in the dead
+zone where both the turbulent and radial relative velocities
+are reduced. Such an environment is located around the
+snow line (Kretke & Lin (2007)). Brauer et al. (2008b)
+showed that in these pressure bumps relative velocities
+stayed below a presumed fragmentation threshold of 10
+m s−1, presenting a window through which particles can
+overcomethe m-sizebarrier,although they assumedperfect
+sticking (no bouncing) below the fragmentation barrier.
+Future studies have to verify whether planetesimals can be
+formed with the collision model presented in this study.
+Another planetesimal forming mechanism is the grav-
+itational collapse of swarms of boulders (Johansen et al.
+(2007)). This scenario assumes that large amount of
+the solid material is presented in dm sized boulders
+(St≥0.1) at the midplane of the disk. These boulders
+then concentrate in long-lived high pressure regions in the
+turbulent gas and these initial over-densities are further
+amplified by the streaming instability. This mechanism
+forms 100 km sized objects on a very short timescale (some
+orbits). However, our simulations produce particles with
+St≈10−4which is due to Bouncing with compaction
+(B1) and the low (1 m s−1) fragmentation velocity of
+silicates. Using a ’stickier’ material such as ices or parti-
+cles with organic mantels may produce bigger particles.
+Molecular dynamic simulations (e.g. Dominik & Tielens
+(1997), Wada et al. (2007), Wada et al. (2008)) showed
+that icy aggregates could have fragmentation velocities
+of about 10 m s−1, although these findings have yet to
+be confirmed by laboratory experiments. Similarly, it is
+conceivable that the enhanced sticking capabilities of ices
+will prevent the bouncing, which is so omnipresent for
+small particles in our simulations, or shifts it to largersizes.
+Cuzzi et al. (2008) outlined an alternative concentra-
+tion mechanism to obtain gravitationally unstable clumps
+of particles, which can then undergo sedimentation and
+form a ‘sandpile’ planetesimal. In this model turbulence
+causes dense concentrations of aerodynamicallysize-sorted,
+chondrule-size particles (Cuzzi et al. (2001))– more pre-
+cisely, particles of Stokes numbers St=Re−1/2≈10−4in
+our simulations. Since growth in our models is typically
+halted at these Stokes numbers, this concentration mech-
+anism is an obvious successor to coagulation – at least
+where it concerns the conditions adopted in this paper
+(1AU, silicates).
+However, it should be emphasized that the formation
+of a gravitationally unstable clump does not imply plan-
+etesimals will form unimpededly. An important question
+to address is how collisions will affect the collapse. In
+the Cuzzi et al. (2008) scenario the collapse occurs on a
+sedimentation timescale and for these high densities col-
+lisions between particles will be frequent. Likewise, in the
+Johansen scenario – where the collapse occurs on an orbital
+timescale and involves St∼0.1 particles – collisions can22 A. Zsom et al.: The outcome of protoplanetary dust growth: pebbles, boulders, or planetesimals? II.
+be rather violent. Collisional fragmentation or erosion may
+change the appearance of the collapse, because the small
+fragments arecarried awayby the gas.The role of collisions
+in these situations is certainly an important question, and
+our new collision model provides a tool to quantitatively
+address this issue in future studies.
+5.4. Consequences for laboratory experiments
+One can see from Fig. 11, in Paper I, that only a small part
+ofthe parameterspaceis coveredbyexperiments.Although
+laboratory experiments cannot be made at every point of
+the parameter space, we suggest future ones based on Figs.
+5, 7 and 10 in order to better understand dust growth in
+the early stages of planet formation.
+–More experiments in the ‘cc’and‘cC’regimes are
+needed as particles get compactified by the end of their
+evolution. Thus, most of the collisions happen in this
+regime, at velocities between 0.1 and 100 cm s−1, at
+masses between 10−7and 10 g.
+–As seen in Figs. 5, 7 and 10, the ‘hot spots’, where most
+of the collisions are happening, are located in the equal
+sized regimes, at the left side of the fragmentation line.
+Therefore, it is important to map these areas of the
+parameter space in detail.
+–We define a sharp border line between the Hit & Stick
+(S1) and Bouncing with compaction (B1) collisions. If
+there is a continuous transition between S1 and B1, the
+growth of particles would not be halted by bouncing at
+suchlowparticlesizes.Asmanycollisionsarehappening
+in the‘pp’and‘cc’regimes, even a small probability of
+growth could increase the particle sizes.
+–As seen in Fig. 6b, particles in high gas density environ-
+ments can have enlargement parameters much higher
+than 6.6 ( φ= 0.15). An interesting question is whether
+the collision types and regimes are also valid for parti-
+cles with such a low volume filling factors, or whether
+these particles have a different collision behavior?
+–The Sticking through surface effects (S2) – Bouncing
+with masstransfer(B2) transitionregimegreatlyaffects
+the outcome of the simulations (see Sect. 5.1). However,
+thetransitionregimeisonlymappedatthehighvelocity
+and low mass regions. Therefore, it is essential to better
+constrain this part of the parameter space.
+–The critical mass ratio affects the particle masses and
+porosities. Experiments are needed to constrain its
+value.
+–The bouncing model, described in Paper I, has impor-
+tant implications for the evolution of dust aggregates in
+protoplanetary disks but it is unfortunately still based
+ontoofew experiments.Furtherexperimentsareneeded
+to refine the model, as Bouncing with compaction (B1)
+is the most frequent collision type in all of the simula-
+tions.
+6. Summary
+We performed simulations of dust growth using the Monte
+Carlo code of ZsD08 and a dust collision model based on
+laboratory experiments (Paper I). We performed simula-
+tions at the midplane of three disk models having low
+(2.4×10−11g cm−3), intermediate (1 .4×10−9g cm−3) and
+high(2.7×10−8gcm−3)gasdensitiesat1AUdistancefromthe central star. We vary the turbulence parameter ( α) and
+the critical mass ratio ( rm) to explore their effects on the
+mass and porosity distribution functions. Our main results
+are:
+–Upon using α= 10−4, the low density / MMSN / high
+density model produces particles with maximum mean
+mass of 9 .7×10−8g / 4.18 g / 0.23 g, the maximum av-
+erage enlargement parameter of these particles are 7.12
+/ 21.9 / 38.0. The maximum average Stokes numbers
+are 2.2×10−4/ 2.8×10−4/ 3.6×10−5.
+–We find that particle evolution does not follow the pre-
+viouslyassumedgrowth-fragmentationcycles.Although
+catastrophic fragmentation is present for a short pe-
+riod of time in some of the models (typically when
+α= 10−3), it has a fringe effect. Particles in most of
+the simulations do not reach the fragmentation barrier
+because their growth is halted by bouncing.
+–Wesee longlived, quasi-steadystatesinthe distribution
+function of the aggregates due to bouncing. The final
+equilibrium state is not reached within the drift or the
+viscous timescales.
+–We performed simulations with varying turbulence
+strength. We find that the system is ‘non-linear’: The
+maximum mass of particles is not a decreasing function
+of the turbulence parameter and is not an increasing
+function of the gas density.
+–We explored the effects of the critical mass ratio. We
+find that different critical mass ratios can affect the
+particle evolution. Small critical mass ratios can pro-
+duce heavier particles, while big values of rmcan halt
+the growth earlier.
+–The maximum Stokes number is rather independent of
+the gas density and the strength of the turbulence.
+–The maximum mass of the aggregates is limited to ≈1
+g due to the S2-B2 transition regime.
+–The Stokes number 10−4particles can be concentrated
+in turbulence by aerodynamical size-sorting, thus plan-
+etesimals can form from these particles.
+Acknowledgements. A. Zs. thanks to Frithjof Brauer, Tilman
+Birnstiel and Felipe Gerhard for useful discussions. C.G. w as funded
+by the Deutsche Forschungsgemeinschaft within the Forsche rgruppe
+759 “The Formation of Planets: The Critical First Growth Pha se” un-
+der grant Bl 298/7-1. A. Zs. acknowledges the support of the I MPRS
+for Astronomy & Cosmic Physics at the University of Heidelbe rg and
+C.W.O. acknowledges the financial support from the Alexande r von
+Humboldt Foundation. We also thank our referee, Sascha Kemp f for
+providing useful comments which helped to improve the paper .
+Appendix A: Time evolution and animations
+We offer six animations to illustrate the time evolution
+of the aggregates. These animations can be viewed at
+www.mpia.de/homes/zsom/Site/animations2.html . The
+massvspsilowd.mpg file shows the time evolution of
+the masses and enlargement parameters of our repre-
+sentative particles. The x-axis is the particle mass in
+gram units, the y-axis is the enlargement parameter. The
+regimes lowd.mpg file illustrates the time evolution of
+Fig. 5. The contour levels are collision frequencies / grid
+cell. The same movies are provided for the MMSN model,
+these are massvspsimmsn.mpg andregimes mmsn.mpg
+respectively. And also for the high density model:
+massvspsihighd.mpg andregimes highd.mpg .A. Zsom et al.: The outcome of protoplanetary dust growth: pe bbles, boulders, or planetesimals? II. 23
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+arXiv:1001.0489v1  [math.KT]  4 Jan 2010Absence of torsion for NK 1(R) over associative rings
+Rabeya Basu
+Department of Mathematical Sciences,
+Indian Institute of Science Education and Research, Kolkat a,
+West Bengal, India
+rbasu@iiserkol.ac.in
+2000 Mathematics Subject Classification:13H99, 15A24, 16R 50, 19B14
+Key words: linear group, K1,NK1,SK1, torsion, Witt vectors.
+Abstract
+WhenRis a commutative ring with identity, and if k∈N, withkR=R, then it
+was shown in [11] that SK 1(R[X]) has no k-torsion. We reprove this result for any
+associative ringRwith identity in which kR=R.
+1 Introduction
+LetRbe an associative ring with identity element 1. Let K 1(R) denote the White-
+head group. In case Ris commutative, let SK 1(R) be the kernel of the determinant
+map from K 1(R) to the group of units of R. Let W( R) be the ring of big Witt vectors.
+We denote NK 1(R) = ker(K 1(R[X])→K1(R));X= 0. In [10], J. Stienstra, using
+ideas of S. Bloch in [2], showed that NK 1(R) is a W( R)-module. Consequently, as
+noted by C. Weibel in ([11], §3), ifkis a unit in R, then SK 1(R[X]) has no k-torsion,
+whenRis a commutative local ring. For a commutative ring with identity NK 1(R)
+coincides with SK 1(R[X]) if we take Rto be local. In this note we generalize Weibel’s
+observation for NK 1(R), where Ris a commutative local ring.
+We prove for an associative ringRwith identity, NK 1(R) has no k-torsion if kis
+a unit in R. In particular, this shows Weibel’s result is a special case. The metho d
+of proof may be considered as a simplified version of J. Stienstra’s ap proach via big
+Witt vectors. This will help the reader to appreciate why big Witt vect ors come into
+the picture in the functorial approach of S. Bloch, et al.
+Theorem 1.1 LetRbe an associative ring with identity. If kis a unit in R, then
+NK1(R)has nok-torsion.
+Corollary 1.2 LetRbe a commutative local ring with identity. If kis a unit in R,
+thenSK1(R[X])has nok-torsion.
+As a consequence of Theorem 1 we prove
+1Theorem 1.3 LetR=R0⊕R1⊕ ···be a graded commutative ring with identity.
+Letkbe a unit in R0. LetN=N0+N1+···+Nr∈Mr(R)be a nilpotent matrix. If
+[(I+N)]k= [I]inSK1(R), then[I+N] = [I+N0]. In particular, if R0is a reduced
+local ring, then SK1(R)has nok-torsion.
+2 Prologue
+LetRbe an associative ring with identity. GL n(R) denotes the group of invertible
+matrices, SL n(R) its subgroupofmatricesof determinant 1 (when Ris a commutative
+ring), E n(R) the subgroup of elementary matrices, i.e.generated by {Eij(λ) :λ∈
+R,i/\e}atio\slash=j}, where E ij(λ) =I+λeijandeijis the matrix with 1 on the ij-th position
+and 0’s elsewhere. For α∈Mr(R),β∈Ms(R) we have α⊥β∈Mr+s(R), where
+α⊥β=/parenleftbiggα0
+0β/parenrightbigg
+.
+There is an infinite counterpart: identifying each matrix α∈GLn(R) with the
+large matrix ( α⊥1) gives an embedding of GL n(R) into GL n+1(R). Let GL( R) =
+∪∞
+n=1GLn(R), SL(R) =∪∞
+n=1SLn(R), and E( R) =∪∞
+n=1En(R) be the corresponding
+infinite linear groups.
+The well known Whitehead’s Lemma asserts that if α∈GLn(R) then we have
+(α⊥α−1)∈E2n(R). Thus we have
+[GL(R),GL(R)] = [E(R),E(R)] = E(R)
+and hence E( R) is a normal subgroup of GL( R). The quotient GL( R)/E(R) is called
+theWhitehead group of the ring Rand is denoted by K 1(R). Forα∈GLn(R)
+let [α] denote its equivalence class in K 1(R). Also, as a consequence of Whitehead’s
+lemma one sees that if α,β∈GLn(R) then [α,β]∈E2n(R); whence K 1(R) is an
+abelian group. For details cf.[1].
+In caseRis commutative the determinant map from GL n(R) toR∗induces a
+map, det : K 1(R)→R∗given by αE(R)/mapsto→detα. The kernel of the map is denoted
+by SK 1(R) and equals to SL( R)/E(R).
+We write NK 1(R) for ker(K 1(R[X])→K1(R));X= 0,i.e.the subgroup con-
+sisting of elements [ α(X)]∈K1(R[X]) such that [ α(0)] = [I]. Note that if Ris a
+commutative local ring then SK 1(R[X]) coincides with NK 1(R); indeed, if Ris a
+local ring then SL n(R) = En(R) for all n >0. Therefore, we may replace α(X) by
+α(X)α(0)−1and assume that [ α(0)] = [I].
+For a commutative ring Rthe group W( R) of bigWitt vectors is defined by:
+W(R) = (1+ XR[[X]])×.
+ForP(X)∈(1+XR[[X]]), letω(P) denote the corresponding element of W( R). The
+group structure in W( R) is given by:
+ω(P)+ω(Q) =ω(P.Q).
+AnyP(X)∈(1+XR[[X]]) can be written uniquely as a product:
+P(X) = Π
+n≥1(1−anXn)−1, an∈R.
+2The elements ( a1,a2,......) are called the Witt co-ordinates ofω(P). Also, there
+exists a unique structure of commutative ring on W( R) such that
+ω((1−aXm)−1).ω((1−bXn)−1) =ω((1−an/rbm/rXmn/r)−r),
+wherer= g.c.d( m,n). The identity element in W( R) is presented by the power
+series (1 −X)−1. For details cf.([2], Prop. I.I), [7].
+For the W( R)-module structure of NK 1(R) see [12] (for previous articles cf.[2],
+[3], [10], [11]).
+3 Higman Linearization
+Two matrices α∈Mr(R) andβ∈Ms(R) are said to be stably equivalent if there
+existsε1,ε2∈Et(R) (for some t≥max{r,s}) such that ε1(α⊥It−r)ε2= (β⊥It−s).
+Lemma 3.1 (Higman Linearization Process) Letα(X)be a matrix over R[X].
+Thenα(X)is stably equivalent to a linear matrix in Ms(R[X])for some s.
+Proof.We may assume that n≥2. Let
+α(X) =a0+a1X+a2X2+···+anXn∈Mr(R), r >1.
+Thenα(X) is stably equivalent to a matrix of degree n−1 overR[X] in the following
+manner:
+/parenleftbiggIr−anX
+0Ir/parenrightbigg/parenleftbigga0+···+anXn0
+0 Ir/parenrightbigg/parenleftbiggIr0
+Xn−1IrIr/parenrightbigg
+=/parenleftbigg
+a0+···+an−1Xn−1−anX
+Xn−1Ir Ir/parenrightbigg
+=α1
+has degree ( n−1). Hence αis stably equivalent to α1. Repeating the above process
+(n−2) times we get the result. ✷
+Corollary 3.2 LetRbe an associative ring with identity. Let α(X)∈GLr(R[X])
+withα(0) =In. Then in K1(R[X])we have [α(X)] = [Is+NX]for some s >0and
+some matrix N∈Ms(R).
+Proof. By Lemma 3.1 there exists ε1,ε2∈Et(R[X]) (for some t > r) such that
+(α(X)⊥It−r) =ε1((Is+NX)⊥It−s)ε2for some s >0 andN∈Ms(R). Now as
+E(R) is a normal subgroup of GL( R), for some integer uthere exists ε′
+1∈Et+u(R[X])
+such that
+(ε1⊥Iu)((Is+NX)⊥It−s+u) = ((Is+NX)⊥It−s+u)ε′
+1.
+Hence in K 1(R[X]) we have
+[α(X)] = [((Is+NX)⊥It−s+u)ε′
+1(ε2⊥Iu)] = [Is+NX].
+✷
+34 Main Theorem
+LetRtdenote the ring R[X]/(Xt+1).
+Lemma 4.1 LetRbe a ring and P(X)∈R[X]be any polynomial. Then the following
+identity holds in the ring Rt:
+(1+XrP(X)) = (1+ XrP(0))(1+ Xr+1Q(X)),
+wherer >0andQ(X)∈R[X], withdeg(Q(X))< t−r.
+Proof.Let us write P(X) =a0+a1X+···+atXt. Then we can write P(X) =
+P(0)+XP′(X) for some P′(X)∈R[X]. Now, in Rt
+(1+XrP(X))(1+XrP(0))−1= (1+XrP(0)+Xr+1P′(X))(1+XrP(0))−1
+= 1+Xr+1P′(X)(1−XrP(0)+X2r(P(0))2−···)
+= 1+Xr+1Q(X)
+whereQ(X)∈R[X] with deg( Q(X))< t−r. Hence the lemma follows. ✷
+Remark. Iterating the above process we can write for any polynomial P(X)∈R[X],
+(1 +XP(X)) = Πt
+i=1(1 +aiXi) inRt, for some ai∈R. By ascending induction it
+will follow that the ai’s are uniquely determined. In fact, if Ris commutative then
+ai’s are the i-th component of the ghost vector corresponding to the big Witt v ector
+of (1+XP(X))∈W(R) = (1+ XR[[X]])×. For details see ([2], §I).
+Lemma 4.2 LetRbe a ring with1
+k∈RandP(X)∈R[X]. Assume P(0)lies in the
+center of R. Then
+(1+XrP(X))kr= 1⇒(1+XrP(X)) = (1+ Xr+1Q(X))
+in the ring Rtfor some r >0andQ(X)∈R[X]withdeg(Q(X))< t−r.
+Proof.By Lemma 4.1
+(1+XrP(X)) = (1+ XrP(0))(1+ Xr+1P1(X)), (1)
+for some P1(X)∈R[X] with deg( P1(X))< t−r. Therefore, in Rt
+(1+XrP(X))kr= 1⇒(1+XrP(0))kr= (1+Xr+1P1(X))−kr.
+As1
+k∈R, we have
+(1+krXrP(0)+Xr+1P2(X)) = (1+ Xr+1P1(X))−kr.
+This implies
+(1+krXrP(0)) = (1+ Xr+1P1(X))−kr(1+(1+ krXrP(0))−1Xr+1P2(X))−1
+= (1+Xr+1P3(X))
+for some P2(X),P3(X)∈R[X] with deg P2(X),P3(X)< t−r. Now, applying
+homomorphism X/mapsto→1
+kXwe get
+(1+XrP(0)) = (1+ Xr+1P4(X))
+4for some P4(X)∈R[X] with deg( P4(X))< t−r. Substituting this in (1) we get
+(1+XrP(X)) = (1+ Xr+1Q(X))
+for some Q(X)∈R[X] with deg( Q(X))< t−r. ✷
+Proof of Theorem 1.1. Letα(X)∈GLn(R[X]) with [α(0)] = [I] be ak-torsion. By
+Corollary 3.2 in K 1(R[X]), [α(X)] = [(Is+NX)] for some s >0 andN∈Ms(R[X]).
+Since (Is+NX) is invertible, Nis nilpotent. Let Nt+1= 0. Since [( Is+NX)]k= [I]
+in K1(R[X]), it follows that
+[Is+kNX+N2X2P1(NX)] = [I]
+for some P1(X)∈R[X]. Hence, as before, as1
+k∈R,
+[(Is+kNX)−1] = [(Is+(Is+kNX)−1N2X2P1(NX)]
+= [Is+(Is−kNX+N2X2P2(NX))N2X2P1(NX)]
+= [Is+N2X2P(NX)]
+for some P(X)∈R[X]. Since [( Is+N2X2P(NX))]k= [I], arguing as in the proof
+of Lemma 4.2 we get in K 1(R[X])
+[Is+N2X2P(NX)] = [Is+N3X3Q(NX)]
+for some Q(X)∈R[X]. Now by repeating the above mentioned argument we get
+[Is+N2X2P(NX)] = [I].
+Finally, applying homomorphism X/mapsto→1
+kXwe get the desired result. ✷
+Now we prove Theorem 1.3 as a consequence of Swan-Weibel homoto py trick. For
+details see ([6], Proof of Prop. 2.22). First we recall the Local-Globa l Principle for a
+graded ring.
+Graded Local-Global Principle: LetR=R0⊕R1⊕···be a graded commutative
+ring with ka unit in R0andα(X)∈GLn(R[X]) withα(0) =In. Ifαm(X)∈
+En(Rm[X]) for all m∈Max(R0), thenα(X)∈En(R[X]).
+This is derived from the usual Local-Global Principle by using the follow ing ho-
+motopy trick due to Swan and Weibel.
+Proof of Theorem 1.3. Consider the ring homomorphism θ:R→R[X], given by
+(a0+a1+···)/mapsto→a0+a1X+···. Then
+[(I+N)]k= [I]⇒θ([(I+N)]k) = [(θ(I+N))]k= [I]
+⇒[(I+N0+N1X+···+NrXr)]k= [I]
+Letmbe a maximal ideal in R0. By Theorem 1
+[(I+N0+N1X+···+NrXr)] = [I]
+in SK1(R0)m.Hence by the Graded Local-Global Principle [( I+N)] = [(I+N0)] in
+SK1(R).
+In particular, if R0is a reduced local ring then units of R0[X] = units of R0. Since
+N0is nilpotent, det( I+N0X)= constant = 1. Hence I+N0is in SL r(R0) = Er(R0).
+This completes the proof. ✷
+55 Examples
+We show by an example that the the condition1
+k∈Ris necessary in Theorem 1.3,
+whence in Theorem 1.1.
+Example. LetRbe a commutative ring with identity and αa 2×2 completion
+of (1−XY,X2) overR[X2,XY,Y2]. In ([5], §8, Example 8.2) it is shown that α∈
+SK1(R[X2,XY,Y2])\SK1(R). Now take R=Z2. Then the square of the Mennicke
+symbol of the vector (1 −XY,X2) is clearly trivial, i.e.α2∈E3(Z2[X2,XY,Y2]).
+(See [1] for definition of Mennicke symbol.)
+Remark. LetPbe a finitely generated projective R-module. A theorem of M.R.
+Gabel in [4] asserts that mP=P⊕···⊕P(m times) is free, for some m. (A result
+of T.Y. Lam in [8] sharpens this bound on m).
+Ravi A. Rao has asked if the converse of the above is true over poly nomial rings
+R[X] withRlocal. More precisely, if R is a local ring with kR = R, does K 0(R[X])
+have non-trivial k-torsion?
+References
+[1] H. Bass; Algebraic K-theory. W. A. Benjamin, Inc., New York-Amsterdam (1968).
+[2] S. Bloch; Algebraic K-Theory and crystalline cohomolog y,Publ. Math. I.H.E.S. 47 (1977),
+187–268.
+[3] S. Bloch; Some formulas pertaining to the K-theory of com mutative group schemes, Journal of
+Algebra 53 (1978), 304–326.
+[4] M.R. Gabel; Generic orthogonal stably free projectives ,Journal of Algebra 29 (1974), 477–488.
+[5] J. Gubeladze; Non triviality of SK 1(R[M]),Journal of Pure and Applied Algebra 104 (1995),
+169–190.
+[6] J.Gubeladze; Classicalalgebraic K-theoryofmonoid al gebras, K -theory and homological algebra
+(Tbilisi, 1987–88), 36–94, Lecture Notes in Math. 1437 (199 0), Springer, Berlin.
+[7] S. Lang; Algebra, Third Edition. Pearson Education Asia.
+[8] T.Y. Lam; Series summation of stably free modules, Quart. J. Math. Oxford Series (2) 27
+(1976), no. 105, 37–46.
+[9] P.M. Cohn; On the Structure of GL 2of a Ring, Publ. Math. 30 (1966), 5–54.
+[10] J. Stienstra; Operation in the linear K-theory of endom orphisms, Current Trends in Algebraic
+Topology , Conf. Proc. Can. Math. Soc. 2 (1982).
+[11] C. Weibel; Mayer-Vietoris Sequence and module structu re on NK 0,Springer Lecture Notes in
+Mathematics 854 (1981), 466–498.
+[12] C. Weibel; Module Structures on the K-theory of Graded R ings,Journal of Algebra 105 (1987),
+465–483.
+Indian Institute of Science Education and Research (IISER- K),
+Mohanpur Campus, P.O. BCKV Campus Main Office ,
+Mohanpur, Nadia - 741252, West Bengal, India.
+Fax: 03473-2334-4107
+email : rabeya.basu@gmail.com, rbasu@iiserkol.ac.in
+6
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+arXiv:1001.0490v1  [hep-th]  4 Jan 2010On Geodesic Motion in Hoˇ rava-Lifshitz Gravity
+Amir Esmaeil Mosaffa1
+Department of Physics, Sharif University of Technology
+P.O. Box 11365-9161, Tehran, Iran
+Abstract
+WeproposeanactionforafreeparticleinHoˇ rava-Lifshitz gravity basedonFoliation
+Preserving Diffeomorphisms. The action reduces to the usual r elativistic action in the
+low energy limit and allows for subluminal and superluminal motions with upper and
+lower bounds on velocity respectively. We find that deviatio n from general relativity is
+governed by a position dependent coupling constant which al so depends on the mass of
+the particle. As a result, light-like geodesics are not affect ed whereas massive particles
+follow geodesics that become mass dependent and hence the eq uivalence principle is
+violated. We make an exact study for geodesics in flat space an d a qualitative analysis
+for those in a spherically symmetric curved background.
+Dedicated to Farhad Ardalan on his 70th birthday
+1E-mail:mosaffa@physics.sharif.edu1 Introduction and Setup
+Thelongquestforunificationremainsanoutstandingchallengefort heoreticalphysicsdespite
+impressive progressinthisregardinthelast fewdecades. The onela st step inthisprogramis
+to unify quantum theory with gravity. Gravitational interactions, as predicted by Einstein’s
+theory, turn out to be irrelevant such that a naive quantization of general relativity breaks
+down at high energies. A possible resolution for this problem is to add c ovariant higher
+derivative terms to Einstein-Hilbert action such that the superficia l degree of divergence
+decreases. A new problem arises though because the higher tempo ral derivatives introduce
+ghost fields in the theory.
+A new idea in this regard has been to add only higher spatial derivative s to the Einstein-
+Hilbert action. By doing this, and at the expense of breaking genera l covariance, no ghost
+fields appear and at the same time the higher derivatives render the theory renormalizable.
+Inspired by similar ideas in Lifshitz’s scalar field theory, and based on s ome earlier works
+[1, 2], Hoˇ rava introduced a renormalizable theory for gravity in [3] w hich has been named
+Hoˇ rava-Lifshitz (HL) gravity. This theory has attracted a lot of interest since then and its
+various aspects have been studied, such as finding solutions, addr essing fundamental issues
+and introducing modifications in the theory and studying its cosmoligic al consequences.
+Some of these works are listed in [4, 5, 6, 10].
+Abasicquestioninthistheoryistounderstandhowmatterfieldsinflu encegeometry, that
+is, in this new theory for gravitation, what is the analogue of the mat ter energy-momentum
+tensor that acts as a gravitational source in Einstein’s field equatio ns in general relativity.
+A first step to answer this question is to study how particles move in a given background or
+in other words what are the geodesics in the HL theory.
+The above question has been addressed in some works recently[7, 8 , 9]. Among these [8]
+makes a comprehensive study of particles in HL gravity as the optica l limit of a scalar field
+theory. They find new features such as superluminal motions and m assive luminal particles
+as well as the dependence of geodesics on the mass of the particle. Similar results are found
+in [9] by studying super Hamiltonian formalism.
+In this note we study geodesic motion for free particles in the same s pirit that HL theory
+is built itself. We write down a kinetic action compatible with the symmetr ies of the theory
+and deform it in a way that it reproduces the expected relativistic ac tion in the low energy
+limit.
+Our results share some of the features with the mentioned works; it predicts sub and
+superluminal motions as two different sectors for motion. In the fo rmer sector, we find an
+upper bound on the velocity, quite similar to relativistic dynamics, whe reas in the latter
+we find a lower bound on the velocity. We also find that even in the sublu minal sector a
+particle’s path in general deviates from its relativistic counterpart . The deviation is in terms
+of a coupling constant which depends on the mass of the particle and is in general position
+dependent. Massless particles, on the other hand, move exactly a s predicted by general
+relativity.
+Intheremainder ofthisintroductionwe briefly list thebasic ingredien ts oftheHL gravity
+which we need in the rest of the work.
+1We consider a D+1 space-time with the metric written in the ADMform as
+ds2=−N2dt2+gijduiduj
+where
+dui=dxi+Nidt, i = 1,2,...,D
+andNandNiare the usual lapse and shift functions and gijis the metric on the spatial
+section. HL gravity is based on the general coordinate transform ations which preserve the
+foliation of space-time into the time direction and spatial sections an d thus the name “foli-
+ation preserving diffeomorphisms” or FPD in short (one way of identif ying these is to make
+a nonrelativistic contraction of diffeomorphisms). FPD in its finite for m turns out to be
+t→t′(t), xi→x′i(t,/vector x)
+under which
+gij→g′
+ij=gkl∂xk
+∂x′i∂xl
+∂x′j
+Ni→N′
+i=gkl∂xk
+∂x′i∂xl
+∂t′+Nk∂xk
+∂x′idt
+dt′
+N→N′=Ndt
+dt′
+FPD allows for more invariants than diffeomorphisms. For example for any two spacetime
+vectorsAµandBνone can form the following invariant products
+N2AtBt, gij(Ai+NiAt)(Bj+NjBt),1
+N2(At−NiAi)(Bt−NjBt), gijAiBj
+Apart from FPD, the HL gravity assumes that the microscopic theo ry enjoys a scaling
+symmetry that acts anisotropically on space and time. This symmetr y is characterized by
+an exponent z >1 and acts as follows
+xi→bxi, t→bzt
+This means that associating a scaling dimension −1 toxi, time will have scaling dimension
+−z. We now move on to study the motion of particles in this theory.
+2 Particles in HL gravity
+We want to write down an action for a particle that exhibits FPD symme try at all regimes of
+energy. In the UV the action must have an anisotropic scaling symme try between time and
+space, specified by a critical exponent z, and in the IR it should roll down to a relativistic
+theory. Themicroscopicactionwhichgoverns theUVregimeconsist sofkinetic andpotential
+terms. As for the kinetic term, the critical exponent only shows up in the scaling dimension
+of the coupling constant, that is, the form of the kinetic term is not sensitive to z. The form
+of the potential terms, however, is severely restricted by the va lue ofz. In the following we
+2study particles that move freely and hence feel no potential in the UV. Such a UV action is
+most naturally written as
+SUV∼/integraldisplay
+dτ gij˙ui˙uj(2.1)
+whereτis an invariant of FPD and ”dot” means derivative with respect to τ. It is natural
+to choose time to parameterize the world line of the particle and henc e setdτ=Ndt. By
+this choice the dynamical fields will be xi(t) which define a valid set of degrees of freedom
+required to specify a curve in spacetime. This choice of the paramet er can be incorporated
+in the action by writing the following
+SUV∼1
+2/integraldisplay
+dτ/bracketleftbigg1
+egij˙ui˙uj+e
+N2˙t2gij˙ui˙uj/bracketrightbigg
+(2.2)
+wheree(τ) is the worldline einbein. Note that we now have xi(τ),t(τ) ande(τ) as the
+degrees of freedom, two more than required. In return we have t he equation of motion for e
+and a new symmetry, reparameterization invariance, which is define d by
+τ→τ′(τ), e(τ)→e′(τ′) =e(τ)dτ
+dτ′,(xi,t)→(xi,t) (2.3)
+One can now use the reparametrization invariance to fix eat a constant value, say 1. The
+equation of motion for e
+e2=N2˙t2(2.4)
+then fixes the parameter as dτ=Ndtand the resulting action will read
+SUV∼/integraldisplaydt
+Ngij(dxi
+dt+Ni)(dxj
+dt+Nj) (2.5)
+As for the IR action, we want it to be of the form
+SIR∼/integraldisplay
+dτ/radicalBig
+c2N2˙t2−gij˙ui˙uj (2.6)
+where now τis a full diffeomorphism invariant parameter and cis the speed of light. One
+can write a quadratic form of this action by introducing the world line e inbein
+SIR∼/integraldisplay
+dτ/bracketleftbigg1
+e(gij˙ui˙uj−c2N2˙t2)−e/bracketrightbigg
+(2.7)
+where again we have the same symmetry of (2.3). The same procedu re stated above is
+applicable to(2.7); use(2.3)tofix e, useequationofmotionfor etofixdτ2∼c2N2˙t2−gij˙ui˙uj
+and the resulting action will be the familiar relativistic action for a free particle.
+Now let’s put everything together, we start with the microscopic ac tion (2.2) and, in
+order to end up with (2.7) in the IR, we add to it a deformation of the f orm
+Sdef∼/integraldisplay
+dτ/bracketleftbigg1
+ec2N2˙t2+e/bracketrightbigg
+(2.8)
+3so that the final form for the action will be SUV+Sdef. One should insert various coupling
+constants to have the correct dimensions for the fields. Given all t his, we propose the
+following action for the motion of a free particle in the HL gravity
+S=1
+2/integraldisplay
+dτ/bracketleftbigg1
+e(gij˙ui˙uj−c2N2˙t2)+eM2
+N2˙t2(gij˙ui˙uj−m2
+M2c2N2˙t2)/bracketrightbigg
+(2.9)
+In the following we will relate the coupling constants mandMto physical quantities. The
+engineering dimensions of the fields and constants are as follows
+[x] =−1,[t] = [τ] =−z ,[m] = [M] = [e−1] = 2−z ,[c] =z−1 (2.10)
+To identify the physical characteristics of the particle in terms of t he coupling constants
+we consider the case of flat space, that is, when N= 1 andNi= 0. The equation of motion
+foregives
+e2M2=˙t2˙xi˙xi−c2˙t2
+˙xi˙xi−m2
+M2c2˙t2,˙xi˙xi=gij˙xi˙xj(2.11)
+One immediately identifies two distinct cases; the numerator and den ominator in the above
+expression are either both positive or both negative. In the forme r case one faces superlu-
+minal and in the latter subluminal motions. Let us first consider the s ubluminal motions.
+Substituting efrom (2.11) in (2.9) and after some algebra we arrive at
+SSUB=−mc2/integraldisplay
+dt/radicalbigg
+1−˙xi˙xi
+c2/radicalBigg
+1−/parenleftbiggM
+m/parenrightbigg2˙xi˙xi
+c2,˙xi=dxi
+dt(2.12)
+If we want a relativistic point particle action in the IR, we need the sec ond square root in the
+above expression to capture corrections to the relativistic action . This requires the second
+square root to be almost equal to one even when the particle’s veloc ity is comparable to the
+speed of light. A minimum requirement for this is to assumem
+M>1. The largerm
+Mis, the
+better we can approximate the IR motion with a relativistic action. A s econd consequence
+of (2.12) is to identify mwith the rest mass of the particle.
+Now let’s move to superluminal motions. This is when both the numerat or and the
+denominator in (2.11) are positive. Again substituting efrom (2.11) in (2.9) gives
+SSUP=M/integraldisplay
+dt˙xi˙xi/radicalBigg
+1−c2
+˙xi˙xi/radicalBigg
+1−/parenleftBigm
+M/parenrightBig2c2
+˙xi˙xi,˙xi=dxi
+dt(2.13)
+For large velocities the above action approximates (2.5) in flat space .Mhas the same
+dimension as mso we identify it with some sort of a “mass” in the superluminal regime.
+Collecting everything together, the action (2.9), in the case of a fla t space, allows sub
+and superluminal motions for a particle. The subluminal motion is gove rned by relativistic
+action plus corrections
+SSUB=−mc2/integraldisplay
+dt/radicalbigg
+1−˙xi˙xi
+c2/bracketleftbigg
+1−1
+2g2˙xi˙xi
+c2+···/bracketrightbigg
+(2.14)
+4whereas the superluminal motion is governed by the non relativistic a ction for the particle
+plus corrections
+SSUP=M/integraldisplay
+dt˙xi˙xi/bracketleftbigg
+1−1
+2c2
+˙xi˙xi+···/bracketrightbigg /bracketleftbigg
+1−1
+21
+g2c2
+˙xi˙xi+···/bracketrightbigg
+(2.15)
+where
+g=M
+m(2.16)
+Some comments are in order
+•Of the three constants appearing in the gauge fixed action, cis the emergent speed of
+light and is provided by the gravitational part of the HL theory wher easmandMare
+physical properties of the particle under study.
+•For a given set of constants, c,mandM, not all velocities are allowed for a particle.
+Either there is an upper bound on the velocities, c, or a lower bound c/g. The initial
+condition of the motion determines whether the motion is subluminal o r superluminal.
+•Given the excellent validity of special relativity even at velocities close to the speed
+of light, the action SSUBmust give a small correction to the relativistic action. We
+should thus assume that g≪1 orm≫Mfor usual particles.
+•Forg≪1, the action SSUP, having an expansion in 1 /gbecomes “strongly coupled”
+even at very high velocities and is approximated by SUVin a very narrow range of
+velocities.
+Now that we have identified the coupling constants and also realized t he necessity of the
+assumption g≪1, we move on to the general case of a curved background. Here a gain the
+action (2.9) predicts sub and superluminal motions. The former is go verned by the usual
+relativistic action from general gravity plus corrections
+SSUB=−mc2/integraldisplay
+dtN/radicalbigg
+1−1
+N2˙ui˙ui
+c2/radicalbigg
+1−g2
+N2˙ui˙ui
+c2
+=/integraldisplay
+dtLGR/bracketleftbigg
+1−1
+2g2
+c2N2(˙xi+Ni)(˙xi+Ni)+···/bracketrightbigg
+(2.17)
+whereas the superluminal motion is governed by the non relativistic a ction plus corrections
+SSUP=M/integraldisplay
+dt1
+N˙ui˙ui/radicalBigg
+1−N2c2
+˙ui˙ui/radicalBigg
+1−N2
+g2c2
+˙ui˙ui
+= =/integraldisplay
+dtLNR/bracketleftbigg
+1−1
+2N2c2
+˙ui˙ui···/bracketrightbigg /bracketleftbigg
+1−1
+2N2
+g2c2
+˙ui˙ui+···/bracketrightbigg
+(2.18)
+where
+LGR=−mc2N/radicalbigg
+1−1
+N2˙ui˙ui
+c2,LNR=M
+N˙ui˙ui,˙ui=dxi
+dt+Ni(2.19)
+In the next section we study the equations of motion following from ( 2.9) to find geodesics
+in some simple backgrounds.
+53 Geodesics in HL Gravity
+In what follows we assume that the constants mandMare finite and that g=M/m≪1.
+To find the geodesics of a particle from the action (2.9), we first writ e down the equation of
+motion for e
+e2m2=N2˙t2˙ui˙ui−c2N2˙t2
+g2˙ui˙ui−c2N2˙t2(3.1)
+Thisequationspecifiestheallowedrangeofvelocitiesandthequalitat ivecharacteristicsofthe
+motion. There are as usual two possible routes to find the geodesic s. One is to solve efrom
+above, replace in the original action (2.9), and solve the equations o f motion following from
+the resulting action (2.17)/(2.18). The second route is to use repa rametrization invariance
+to fixeat a suitable constant which one can choose to be e= 1/m2. Replacing this in (2.9)
+gives
+SGF=m
+2/integraldisplay
+dτ/bracketleftbigg
+˙ui˙ui−c2N2˙t2+g2˙ui˙ui
+N2˙t2/bracketrightbigg
+(3.2)
+There is also a constraint which comes from gauge fixing
+˙ui˙ui−c2N2˙t2=g2˙ui˙ui
+N2˙t2−c2(3.3)
+To find the geodesics we should solve the equations of motion coming f rom (3.2) together
+with the constraint (3.3) that specifies the world line parameter. No te that for g= 0, the
+above system of equations is exactly that in general relativity for g eodesic motion. In the
+following we first solve this system of equations for flat space. In th e passing, we study some
+limiting velocities and masses including the zero mass limit. We then make a n approximate
+analysis for a spherically symmetric background to examine some qua litative features of
+particle motion.
+3.1 Flat Space
+SetN= 1 and Ni= 0.SGFresults in constant ˙ xiand˙t. This gives a straight line in space
+with a constant velocity. Using spatial rotations we project the mo tion on the axis x. The
+constraint equation determines the range of allowed velocities and p ossible lower and upper
+bounds. So we have
+1 =a2v2/c2−1
+v2/c2−1/g2, v=dx
+dt(3.4)
+The following situations can be identified
+•0< a <1 and 1/g2< v2/c2. This is superluminal motion. In the limit a→1−, the
+velocity goes to infinity.
+•1< a <1/g. This is not an allowed initial condition.
+•a≥1/gand 0≤v2/c2<1. This is subluminal motion.
+6Limiting velocities and massless case
+There are some special limits on the masses which allow the particle to a ssume limiting
+velocities, the lower and upper bounds in the super and subluminal se ctors respectively. To
+study these cases we go back to the equation of motion for ebefore fixing the gauge
+e2=a2
+m2v2/c2−1
+g2v2/c2−1(3.5)
+As the first limit we consider g→0 withmkept fixed. In this limit the above relation
+becomes theusual relativistic constraint equation. All that remain s isthe subluminal motion
+and the theory is just special relativity. Obviously one can then tak e them= 0 limit which
+givesv=c. This is of course the upper bound for velocity in the subluminal sect or.
+The other special case is when m→ ∞andgis kept fixed and still smaller than one. In
+this limit the denominator in (3.5) should become zero, i.e., v/c= 1/g, which is the lower
+bound for velocity in the superluminal sector.
+3.2 Spherically Symmetric Background
+We study geodesic motion in the Schwarzschild solution. This is not an e xact solution of HL
+theory but is an approximation at sufficiently large distances from th e origin. Here we are
+only interested in general features of the geodesic motion and pos tpone a detailed analysis to
+further works. The following results should specially be considered w ith care when talking
+about regions near the horizon.
+Start with the following metric
+ds2=−(1−R/r)dt2+dr2
+1−R/r+r2dΩ2
+d−1 (3.6)
+whereRis a positive constant andΩ is the transverse sphere. Because of s pherical symmetry
+we can have purely radial motions and that is what we consider below. Far from the origin
+whereN2= 1−R/r∼1 we effectively have a flat space and the above arguments about
+flat space apply. Choosing the initial value then decides whether we s tart in the sub or
+superluminal sectors. Ay any given value of radius, the upper and lo wer bounds of velocity
+are determined by the position dependent coupling constant, g/N2. In terms of the radial
+velocity, v=dr/dt, the constraint equation is written as
+1 =N2˙t2
+g2v2/c2−N4
+v2/c2−N4/g2(3.7)
+The first thing to note is that as long as N2does not change sign during the course of motion
+(the particle does not cross any horizons), once we start in a sect or we remain in that all
+through the motion. In the subluminal sector the upper bound is no w shifted from ctocN2
+whereas in the superluminal sector the lower bound has shifted fro mc/gtocN2/g. As the
+particle moves towards the origin, both the upper and lower bounds decrease in size. For
+subluminal motion this results in the familiar fact that it takes infinite c oordinate time for
+7the particle to reach the horizon. In the superluminal sector this m eans that the particle
+need not necessarily slow down towards the origin.
+One may try to solve equations for the subluminal sector to compar e with the general
+relativistic geodesics. For g≪1 this can be done perturbatively. One can either use the
+action (2.17) directly or solve (3.7) together with
+c2N2˙t+g2v2
+N4˙t2=a (3.8)
+whereais a constant and v < cN2. Either method shows that the corrections to general
+relativisticequationsofmotiondependonthemassoftheparticleth roughg,thatis, particles
+with different masses follow different paths in space. This violates the equivalence principle
+as expected.
+Interestingly for a massless particle for which ghas to be zero, no superluminal sector
+exists and the geodesics are those predicted by general relativity . If one defines light by such
+an assumption then its motion is the same in both Einstein and HL theor ies for gravity.
+4 Conclusions
+In this note we have proposed an action to describe particles’ motio ns in HL gravity. The
+guiding principle in our construction is the fundamental symmetry of the theory, FPD, and
+the requirement that general relativity is reproduced in low energie s. Starting with a non-
+relativistic action in UV, we add deformations such that the dominant terms in low energies
+sum up to the familiar relativistic action for particle motion. In doing so we encounter
+various coupling constants that are fixed by considering the flat sp ace case.
+The action allows for two distinct sectors for motion, one where velo cities have to be
+smaller than an upper bound and the other which imposes a lower boun d on velocities. We
+call the first sector subluminal and the second superluminal.
+Focusing on subluminal motions in flat space, we can describe the mot ion in terms of the
+particle’s mass, m, and a coupling constant g. Insisting on the validity of special relativity
+even at velocities close to that of light, we have to assume that g≪1. This allows us to
+write the correction to special relativity, imposed by HL theory, in p owers of g.
+For superluminal motions in flat space, the theory is most naturally w ritten in terms of
+gand a constant with mass dimension M. The assumption g≪1 results in the fact that
+the relevant action for this sector becomes strongly coupled for a rbitrarily large velocities.
+Therefore the non-relativistic action that we started with is valid in a n arbitrarily small
+range of velocities.
+We also considered some limiting cases of the constants and velocities . We find that for
+g= 0 the superluminal sector disappears. A particle with m= 0 is then shown to travel
+at the upper bound velocity. The other limiting velocity, the lower bou nd in superluminal
+sector, is reached when Mgoes to infinity.
+One might adopt a different point of view, i.e., require that the deform ations to the
+microscopic action introduce perturbative corrections to the non -relativistic action. This
+requries to assume that g≫1 and results in consequences opposite to the above; the
+8subluminal sector becomes strongly coupled at arbitrarily low energ ies and effectively special
+relativity never becomes a good approximation2.
+For curved backgrounds the above features remain valid. The cou pling constant now
+depends on position and since it also depends on the mass of the part icle, the corrections to
+the relativistic action become mass dependent. As a result particles with different masses
+follow different geodesics and the equivalence principle is violated. Dev iations from general
+relativistic geodesics are measured in terms of g. An interesting result is that light, for which
+m=g= 0, is insensitive to the HL corrections to general relativity and still follows the
+usual light-like geodesics.
+Solvingtheequationsofmotionofouractiontofindtheactualgeod esics caningeneralbe
+difficult. In the subluminal sector, however, one can do this pertur batively. In this note we
+studied the flat space geodesics in both sectors and pointed out so me qualitative features of
+geodesics in a spherically symmetric background in its subluminal sect or. Exact calculations
+in this regard, and more general backgrounds, are postponed to future works.
+Formulating the geodesic motion in the Lagrangianlanguage, our app roach may be taken
+as a first step towards an understanding of how matter behaves a s a source in HL theory.
+This can be achieved by considering a system of the HL gravity plus th e particle actions and
+treating the latter as a source to gravity. This is also postponed to future works.
+Acknowledgments
+It is a pleasure to thank F.Ardalan and B.Lynn for useful discussions and M.Alishahiha for
+useful discussions and also comments on the draft. My sincere tha nks are due to L.Alvarez-
+Gaume for posing the problem, very illuminating discussions and for a r eading of the draft.
+I would also like to thank the theory division of CERN for hospitality whe re part of this
+work was done.
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+11
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+arXiv:1001.0491v1  [math.CA]  4 Jan 2010ASYMPTOTIC REPRESENTATION OF MINIMAL
+POLYNOMIALS ON SEVERAL INTERVALS
+F. PEHERSTORFER
+Abstract. Asymptotic representation of minimal polynomials on sev-
+eral intervals is given. The last modifications and correcti ons of this
+manuscript were done by the author in the two months precedin g his
+passing away in November 2009. The manuscript remained unsu bmitted
+and is not published elsewhere1.
+1.Introduction
+LetE=/uniontextl
+k=1Ek,whereEk= [a2k−1,a2k],a1< a2< ... < a 2l, be a
+system of intervals and let W∈C(E) be a positive weight function on E.It
+is a classical problem to find the minimal polynomial on Ewith respect to
+a given norm and weight function W, that is in case of the maximum norm,
+to find the unique monic polynomial ˆMn(x;W) :=ˆMn(x) =xn+...such
+that
+||ˆMn(.;W)
+W||∞:=max
+x∈E|ˆMn(x;W)
+W(x)|
+=min
+aimax
+x∈E/vextendsingle/vextendsingle/vextendsingle/vextendsinglexn+an−1xn−1+...+a0
+W(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle,
+and respectively, to find the normalized minimal polynomial
+(1) Mn(x;W) :=ˆMn(x;W)/En,∞(W),
+where
+En,∞(w) :=||ˆMn(.;W)/W||∞
+denotes the minimum deviation. In the case of a single interv al, say [−1,1],
+and weight function W(x)≡1, it is well known that the Chebyshev poly-
+nomial
+(2) 2 Tn(z) = (φ(z,∞,[−1,1]))n+(φ(z,∞,[−1,1]))−n
+is the normalized minimal polynomial, where
+(3) φ(z,∞,[−1,1]) =z+/radicalbig
+z2−1
+1Submitted by P. Yuditskii and I. Moale.
+12 F. PEHERSTORFER
+is the complex Green’s function of ¯C\[−1,1] with pole at infinity, while,
+even for the single interval case, asymptotic representati ons with respect to
+a weight function have been proved only recently [15].
+For several intervals not much is known with respect to the L∞-norm
+(in contrast to the L2-norm, that is, for orthogonal polynomials, whose
+asymptotic behavior is well understood nowadays thanks to t he papers
+[4, 5, 28, 35, 45, 47], see also the forthcoming book [38]). In deed in the thir-
+ties of the last century for two intervals Achieser [1] deriv ed an asymptotic
+representation of the minimum deviation En,∞(W) with the help of elliptic
+functions and in the late sixties of the last century Widom [4 7] found an
+asymptotic representation of the minimum deviation for sev eral intervals.
+But (up to the very special case that Eis an inverse image of [ −1,1] un-
+der a polynomial mapping, see [30]) the main points of intere st the explicit
+or asymptotic representations of the minimal polynomials r emained open,
+though these open problems have been pointed out in [47, p. 12 8, 205].
+To state our main results we need some notations. By φ(z,z0) we denote
+a so-called complex Green’s function for C\Eluniquely determined up to a
+multiplication constant of absolute value one (chosen conv eniently below),
+that is,φ(z,z0) is a multiple valued function which is analytic on C\Eup to
+a simple pole at z=z0,has no zeros on C\Eand satisfies |φ(z,z0)| →1 for
+z→x∈Equasi-everywhere; or in other words log |φ(z,z0)|is the Green’s
+function with pole at z=z0∈C\E, as usual denoted by g(z,z0).In the
+case under consideration, as it is known [47, 14], a complex G reen’s function
+may be represented as
+(4) φ(z,∞) = exp/parenleftBigg/integraldisplayz
+a2lr∞(x)dx/radicalbig
+H(x)/parenrightBigg
+where
+(5) H(x) =2l/productdisplay
+j=1(x−aj)
+andr∞(x) =xl−1+...is the unique polynomial such that
+(6)/integraldisplaya2j+1
+a2jr∞(x)dx/radicalbig
+H(x)= 0 forj= 0,...,l−1
+and that for x∈C\E
+(7) φ(z,x0) = exp/parenleftBigg/integraldisplayz
+a2lrx0(x)
+x−x0dx/radicalbig
+H(x)/parenrightBigg
+whererx0∈Pl−1is such that rx0(x0) =−/radicalbig
+H(x0) and
+−/integraldisplaya2j+1
+a2jrx0(x)
+x−x0dx/radicalbig
+H(x)= 0 forj= 1,...,l−1.ASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 3
+Recall that the so-called capacity of Eis given by
+(8) cap(E) = lim
+z→∞|z
+φ(z,∞,E)|.
+Byω(z,B;¯C\E) we denote the harmonic measure of B⊆Ewith respect
+to¯C\Eatz,which is that harmonic and bounded function on C\Elwhich
+satisfies for ξ∈Elthat lim
+z→ξω(z,B,C\El) =iB(ξ),whereiBdenotes the
+characteristic function of B.For abbreviation we put
+ω(z,Ek;¯C\E) =ωk(z).
+Furthermore let us recall that for a given weight function Wwhich is
+Lipschitz continuous on E,there is a unique multi-valued analytic function
+W(z) withW(∞)>0 which has no zeros or poles in ¯C\Eand is such
+that the limiting values W±(x) = limz→x
+±Imz>0W(z) from the upper and lower
+half-plane are Lipschitz-continuous on Ewith the property that
+(9)/radicalbig
+W+(x)W−(x) =W(x) forx∈E.
+Note that
+W(∞) = exp/braceleftBigg
+1
+2π/integraldisplay
+ElogW(x)∂g(ξ;∞)
+∂n+
+ξ|dξ|/bracerightBigg
+.
+Notation 1.1.For givenEand weight function Wput fork= 1,...,l−1
+(10)γk,n(W) :=nωk(∞)+1
+2π/integraldisplay
+ElogW(x)∂ωk(ξ)
+∂n+
+ξ|dξ|+σk,n
+whereσk,n∈ {0,1}is such that γk,n(W)∈[0,1] modulo 2 .Note thatσk,nis
+uniquely determined.
+Forabbreviationset ω(∞) = (ω1(∞),...,ωl−1(∞))andγn(W) =(γ1,n(W),
+...,γl−1,n(W)),i.e.,γn(W)∈[0,1]l−1modulo 2.
+Theorem 1.2. LetW∈C1+α(E), α >0,be positive on E.Letcj,n∈
+[a2j,a2j+1], j= 1,...,l−1,be the unique points such that
+(11)l−1/summationdisplay
+j=1ωk(cj,n) =γk,n(W) modulo 2 for k= 1,...,l−1,
+and suppose that lim
+n∈Mcj,n=cjwithcj∈(a2j,a2j+1)forj= 1,...,l−1,
+whereMis an infinite subset of N.Then the normalized minimal polynomial
+Mn(x;W)has forn∈Mthe following uniform asymptotic representation
+onE:
+(12) 2 Mn(x;W) =ψ+
+n(x)+ψ−
+n(x)+o(1),
+where
+(13) ψn(z) =φ(z;∞)n
+l−1/producttext
+j=1φ(z;cj)W(z)4 F. PEHERSTORFER
+and the constant in the o()term is independent of nandx, x∈E.
+Furthermore on any compact subset Kof¯C\E
+(14)2Mn(z;W)
+φ(z;∞)n=W(z)
+l−1/producttext
+j=1φ(z;cj)+o(1)
+uniformly on K.
+Remark 1.3.We note that condition (11) implies Widom’s condition [47,
+Theorem 5.4], that is, that there exists a unique l′≤l,depending on n,and
+unique points c1,n,...,cl′−1,nfrom the open gaps such that
+(15)l′−1/summationdisplay
+j=1ωk(cj,n) =nωk(∞)+1
+2π/integraldisplay
+ElogW(x)∂ωk(ξ)
+∂n+
+ξ|dξ|modulo 1.
+In fact we will show that (11) and (15) are equivalent. (11) ha s the big
+advantage that it can be written as a Jacobi-inversion probl em from which
+many informations can be extracted.
+The asymptotic representation (14) on compact subsets of ¯C\Ewas con-
+jectured by Widom in [47, p. 205]. In fact he conjectured that (14) holds
+for arbitrary arcs in the complex plane, but this does not hol d in general [ ?].
+Lettingz→ ∞in relation (14) we obtain by (8) immediately the following
+expression for the minimum deviation due to Widom [47, Theor em 11.5],
+derived by him in a completely different way and for a more gener al class of
+weight functions.
+Corollary 1.4. Forn∈M,
+2||Mn(·;W)||
+(capE)n=l−1/producttext
+j=1φ(cj;∞)
+W(∞)+o(1)
+We mention that in the case of the three intervals a formula fo r the capac-
+ity in terms of Theta functions has been given by T. Falliero a nd A. Sebbar
+[10] recently.
+Theorem 1.2 enables us to obtain a precise description of the location of
+the zeros of the minimal polynomial in the gaps ( a2k,a2k+1),k= 1,...,l−1
+and to give the number of zeros in the intervals Ek,denoted by ♯Z(Mn,Ek),
+as in the gaps. By the way it is known that Mnhas at most one zero
+in each gap, which can be proved by Kolmogorov’s criteria for the best
+approximation.
+Corollary 1.5. Under the assumptions of Theorem 1.2 the following state-
+ments about the zeros hold:
+a)Mn(z;W)has exactly one zero in each interval [cj−ε,cj+ε], j=
+1,...,l−1,ε>0forn∈M,n≥n0.ASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 5
+b) For every n∈M,n≥n0andk= 1,...,l−1
+♯Z(Mn(·;Ek)) =−l−1/summationdisplay
+j=1ωk(cj,n)+nωk(∞)+1
+2π/integraldisplay
+ElogW(ξ)∂ωk(ξ)
+∂nξ|dξ|.
+For the study of the limit behavior of the solutions cn= (c1,n,...,cl−1,n)
+of (11) it is crucial that the map into the torus [0 ,1]l−1
+˜A:Xl−1
+j=1(a2j,a2j+1)→/parenleftbigg
+0,1
+2/parenrightbiggl−1
+(x1,...,xl−1)/ma√sto→
+1
+2l−1/summationdisplay
+j=1ω1(xj),...,1
+2l−1/summationdisplay
+j=1ωl−1(xj)
+
+is a homeomorphismus. In fact it turns out that ˜Ais the (linearly trans-
+formed) Abel map restricted to the positive sheet of the Riem ann surface
+y2=H.
+We say that c∈Xl−1
+j=1(a2j,a2j+1) is an accumulation point of zeros of
+(Mnν(x;W))ν∈Nif there exists a sequence ynν= (y1,nν,...,yl−1,nν)∈
+Xl−1
+j=1(a2j,a2j+1)suchthatMnν(yj,nν;W) = 0forj= 1,...,l−1andlim νynν=
+c.
+Theorem 1.6. The following statements are equivalent:
+a) The solutions cnν= (c1,nν,...,cl−1,nν)of(11)satisfy
+lim
+νcnν=cwithc∈Xl−1
+j=1(a2j,a2j+1)
+b)limνγnν(W) =˜A(c)modulo2with˜A(c)∈(0,1)l−1.
+c)c∈Xl−1
+j=1(a2j,a2j+1)is an accumulation point of zeros of (Mnν(·,W))ν∈N.
+Furthermore, the set of accumulation points of the sequence of solutions
+(cn)n∈Nof(11)as well as the set of accumulation points of zeros of
+(Mn(·,W))n∈Nin the gaps is dense in Xl−1
+j=1[a2j,a2j+1],if1,ω1(∞),...,ωl−1(∞)
+are linearly independent over Q.
+If the harmonic measures of the intervals are rational, that is,
+(16) ωk(∞) =mk
+N, mk∈ {1,...,N−1}fork= 1,...l
+then the points cj,nappear periodically with respect to n.More precisely we
+have
+Corollary 1.7. Suppose that (16)holds and let b∈ {0,...,N−1}.
+a) The points cj,nsolving(11)satisfy
+(17) cj,b=cj,b+kNfor allk∈N
+b) Eachcj,blocated in the open gap (a2j,a2j+1)is an accumulation point of
+zeros of (MkN+b(·;W))k∈Nand there are no other accumulation points of
+zeros of(Mn(·;W))n∈Nin the open gaps.6 F. PEHERSTORFER
+We mention that the case when the intervals have rational har monic mea-
+sure is covered by Theorem 1.2 also taking into consideratio n the following
+remark and relation (17).
+Remark 1.8.Theorem 1.2 holds true, if some of the cj,n’s,j∈ {1,...,l−1}
+coincide with a boundarypoint a2jora2j+1ofEfor eachn∈M.In this case
+for the corresponding cjone has to put φ(z,cj)≡1 in (14), respectively, in
+Corollary 1.4. Furthermore Corollary 1.5 a) about the zeros holds only with
+respect to such cj’s which do not coincide with a boundary point of E.
+Finally we conjecture that the asymptotics from Theorem 1.2 (with the
+above modifications) hold true when a boundary point of Eis a limit point
+of thecj,n’s.
+There is also an interesting connection between the L∞-minimum devi-
+ation and minimal polynomials (for that see the Remark at the end of the
+paper) and that ones with respect to suitable weighted L2-norm. First we
+need some notation. For convenience we set
+1/h(x) :=
+
+(−1)l−k
+/radicalbig
+|H(x)|,forx∈int(Ek)
+0,elsewhere
+LetE={R:R(x) =xl−1+...,Rvanishes either at a2jora2j+1forj=
+1,...,l−1}and letE(1−x)={(1−x)R:R∈ E}.Note that for R∈ Ewe
+have that
+R
+h=1/radicalbig
+(x−a1)(a2l−x)l−1/productdisplay
+j=1/parenleftbiggx−a2j+1
+x−a2j/parenrightbiggεj/2
+>0 on int(E)
+and
+(1−x)R
+h=/radicalbigg
+1−x
+1+xl−1/productdisplay
+j=1/parenleftbiggx−a2j+1
+x−a2j/parenrightbiggεj/2
+>0 on int(E)
+whereεj∈ {±1}.The minimum deviation of xnonEwith respect to the
+L2-norm squared is denoted by
+En−1,2(xn;W) = min
+q∈Pn−1/integraldisplay
+E|xn−q(x)|2W(x)dx
+and by♯Z(f;A) we denote the number of zeros of fonA.
+Theorem 1.9. Suppose that the assumptions of Theorem 1.2 ?? hold.
+a)
+E2n−1,∞(x2n,W)∼
+1
+2max
+εj∈{±1}En−1,2
+xn;W(x)/radicalbig
+(x−a1)(x−a2l)l−1/productdisplay
+j=1/parenleftbiggx−a2j+1
+x−a2j/parenrightbiggεj/2
+ASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 7
+whereσnis given by (??)andR(σn)∈ Eis uniquely determined by the
+condition
+♯Z(R(σn),[a2j−1,a2j]) =σj,nmodulo 2 for j= 1,...,l−1
+b)
+E2n,∞(x2n,W)∼1
+2max
+εj∈{±1}En−1,2
+xn;W(x)/radicalbigg
+1−x
+1+xl−1/productdisplay
+j=1/parenleftbiggx−a2j+1
+x−a2j/parenrightbiggεj/2
+
+=1
+2En−1,2(xn;W(x)((1−x)R)(σn)/h)
+whereσnis given by (??)and((1−x)R)(σn)∈ E(1−x)is uniquely deter-
+mined by the condition
+♯Z((1−x)R(σn),[a2j−1,a2j]) =σj,nmodulo 2 for j= 1,...,l−1
+We note that the asymptotic representation (14) and (12) may be given
+in terms of Theta functions also in the following way.
+Corollary 1.10. The function ψn/Wfrom Theorem 1.2 may also be written
+in the form
+cψn(z)
+W(z)= exp{−l−1/summationdisplay
+k=1(1
+2π/integraldisplay
+logW(ξ)∂ωk(ξ)
+∂n+
+ξ|dξ|)/integraldisplayz
+a1ϕk}
+
+ϑ(z;/integraltext∞−
+a1ϕk)
+ϑ(z;/integraltext∞+
+a1ϕk)
+nϑ(z;l−1/summationtext
+j=1/integraltextc−
+j,n
+z0ϕk)
+ϑ(z;l−1/summationtext
+j=1/integraltextc+
+j,n
+z0ϕk)
+ϑdenotes the Riemann Theta function, i.e., for given constan tsbj, j=
+1,...,l−1 and a given point z0on the Riemann surface Rofy2=H
+ϑ(z;bk) :=ϑ(z;b1,...,bl−1) :=ϑ(/integraldisplayz
+z0ϕ1−b1−k1,...,/integraldisplayz
+z0ϕl−1−bl−1−kl−1)
+wherek1,...,kl−1are the so-called Riemann-constants and ϕj,j= 1,...,l−1,
+is a basis of normalized differentials of first kind, see the beg inning of the
+next Section.
+Briefly and roughly speaking we derive the above statements i n the fol-
+lowing way. First we consider the case when the weight functi on is a poly-
+nomial, which includes the particular interesting case W(x)≡1 and, what
+is important, can be treated with the help of rational functi ons on the Rie-
+mann surface y2=H. More precisely, we write the system of equations
+(11) in terms of Abelian integrals of first kind, see (36) belo w. The trans-
+formed system of equations guarantees by Abel’s Theorem the existence of
+a rational function Rnon the Riemann surface which is the keystone in ob-
+taining asymptotics for the minimal polynomials. In fact it turns out that
+Rn:=R+
+n+R−
+n, is a rational function on C, which equioscillates n+1 times8 F. PEHERSTORFER
+onEand thus has zero as best approximation with respect to any li near
+subspace of dimension n. Showing that the rational function is asymptoti-
+cally a polynomial of degree nwe obtain, using the Lipschitz continuity of
+the operator of best approximation and deriving a so-called strong unicity
+constant, that this polynomial is asymptotically the minim al polynomial.
+To get the asymptotic representation for general weight fun ctions we ap-
+proximate the weight function W(x) by polynomials where it is important
+that the approximating sequence of polynomials ( ρν) can be chosen such
+that the boundary value problem I+(x)I−(x) =W(x)/ρν(x) can be solved
+uniquely by a function I(z) which is a nonvanishing, single valued, analytic
+function on C\E.
+2.Asymptotics with respect to rational weights
+In this section we derive an asymptotic formula for the minim al polyno-
+mial with respect to rational weight functions of the form 1 /ρν,whereρνis
+a polynomial of degree νwhich is positive on E, which on the one hand con-
+tains the particular interesting case W(x)≡1 (with geometric convergence
+of the error even) and on the other hand is the basis to obtain a symptotics
+with respect to general weight functions roughly speaking b y approximating
+the weight function by a sequence of 1 /ρν’s. Hence in the following let
+(18) ρν(x) =±ν∗/productdisplay
+j=1(x−wj)νjwithρν>0 onE.
+LetRdenote the hyperelliptic Riemann surface of genus l−1 defined by
+y2=H(z) with branch cuts [ a1,a2],[a2,a3],...,[a2l−1,a2l].The two sheets
+ofRare denoted by R+andR−, where on R+the branch of/radicalbig
+H(z) is
+chosen for which/radicalbig
+H(x)>0 forx > a2l. To indicate that zlies on the
+first resp. second sheet we write z+andz−. Furthermore let the cycles
+{αj,βj}l−1
+j=1be the usual canonical homology basis on R, i.e., the curve αj
+lies on the upper sheet R+ofRand encircles there clockwise the interval
+Ejand the curve βjoriginates at a2jarrives ata2l−1along the upper sheet
+and turns back to a2jalong the lower sheet. R′denotes now the simple
+connected canonical dissected Riemann surface. Let {ϕ1,...,ϕ l−1}, where
+ϕj=/summationtextl−1
+s=1dj,szs√
+H(z)dz,dj,s∈C, be a base of the normalized differential of
+the first kind i.e.
+(19)/integraldisplay
+αjϕk= 2πiδjkand/integraldisplay
+βjϕk=Bjkforj,k= 1,...,l−1
+whereδjkdenotes the Kronecker symbol here. Note that the dj,n’s are real
+since/radicalbig
+H(z) is purely imaginary on Ejand since/radicalbig
+H(z) is real on R\E
+theBjk’s are also real. In the following η(P,Q) denotes the differential of
+the third kind which has simple poles at P and Q with residues 1 and -1,ASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 9
+respectively, and is normalized such that
+(20)/integraldisplay
+αjη(P,Q)dz= 0 for j= 1,...,l−1
+Next we are going to show that the system of equations (11) has a unique
+solution. This is done by writing (11) in terms of Abelian-in tegrals and
+considering the resulting system as a (real) Jacobi inversi on problem, see
+Lemma 2.4 a).
+We need some preliminaries concerning the connection of har monic mea-
+sures and Abelian integrals more or less known.
+Lemma 2.1. a) Letc∈¯C\Eand¯cits conjugate number. Then the follow-
+ing relation holds:
+(21)/integraldisplayc+
+c−ϕj+/integraldisplay¯c+
+¯c−ϕj=l−1/summationdisplay
+k=1(ωk(c)+ωk(¯c))Bjkmod (2B)
+b) Letlog|W|be integrable. Then
+l−1/summationdisplay
+κ=1/parenleftBigg
+1
+2π/integraldisplay
+Elog|W(ξ)|∂ωκ(ξ)
+∂n+
+ξ|dξ|/parenrightBigg
+Bkκ=−2
+πi/integraldisplay
+E+ϕ+
+klog|W|
+c)
+2
+πi/integraldisplay
+E+ϕ+
+klog|ρν|=ν∗/summationdisplay
+j=1νj(/integraldisplay∞+
+∞−ϕk−/integraldisplayw+
+j
+w−
+jϕk)
+Proof.a) Let us represent the Green’s functions G(z;c) = lnφ(z;c),ex-
+tended analytically to the Riemann surface by φ(˜z;c) = 1/φ(z;c) in the
+form
+(22) dG(z;c) =l−1/summationdisplay
+k=1µkϕk+η(z;c+,c−)
+which implies by integrating along the αj-cycles
+(23) −2πiωj(c) =/contintegraldisplay
+αjdG(z;c) =l−1/summationdisplay
+k=1µkδjk2πi
+where the first equality follows by the representation of the harmonic mea-
+sure in terms of Green’s function, see e.g. [21], and the last equality by the
+normalizations of the differentials.
+Integrating along the βj-cycle and using the bilinear relation for abelian
+differentials of the first and third kind, i.e.,
+(24)/contintegraldisplay
+βjη(X,Y) =/integraldisplayY
+Xϕj10 F. PEHERSTORFER
+we obtain
+(25)/contintegraldisplay
+βjdG(z;c) =−l−1/summationdisplay
+k=1ωk(c)Bjk+/integraldisplayc+
+c−ϕj
+Next let us note that the polynomial rc(ξ) from (7) can be represented in
+the form
+rc(ξ) = (ξ−c)(r∞(ξ)−Mc(ξ))−/radicalbig
+H(c)
+whereMc(ξ) =ξl−1+...∈Pl−1is the unique polynomial which satisfies
+−/integraldisplaya2j+1
+a2j/radicalbig
+H(c)
+c−xdx/radicalbig
+H(x)=/integraldisplaya2j+1
+a2jMc(x)dx/radicalbig
+H(x)forj= 1,...,l
+Hence we obtain that
+M¯c(ξ) =Mc(ξ)
+and
+d(G(z,c) +G(z,¯c)) =rc(ξ)
+ξ−c+rc(ξ)
+ξ−¯cdξ/radicalbig
+H(ξ)
+which implies that the integral/integraltext
+βjd(G(z,c) +G(z,¯c)) is purely imaginary,
+since the integrand becomes purely imaginary when ξapproaches Eand
+the integrals over the gaps cancel out by the opposite opposi te direction of
+integration. On the other hand it follows by (25) that the int egral is real
+and thus relation (21) is proved.
+b) Denote by
+wk(z) =ωk(z)+iω∗
+k(z)
+the analytic extension of the harmonic measure on Cand, as usual, let
+(26) wk=−ωk+iω∗
+k
+betheextension to the second sheet. Since wk,k= 1,...,l−1,is just another
+basis of Abelian differentials of first kind we may represent/integraltext
+ϕjas linear
+combinations of the wk’s. Integrating along the βkcycle,k∈ {1,...,l−1},
+and recalling the fact that the integral along a βk-cycle is the difference of
+values along the αkcycle (which is Ek) we obtain with the help of (26) that
+(27)/integraldisplay
+ϕj=−1
+2l−1/summationdisplay
+k=1wkBjk
+Using again the fact that ωk(z) = 1 onEk, k∈ {1,...,l},we obtain that
+1
+iw′
+kdz=1
+idwk(z) =∂ωk
+∂nds
+when we approach from the upper half plane to E,hence by (27)
+(28) −2
+πiϕ+
+j=1
+πl−1/summationdisplay
+k=1/parenleftbigg∂ωk
+∂n+ds/parenrightbigg
+Bkj
+from which part b) follows.ASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 11
+Concerning part c) recall that, consider the periods of the h armonic func-
+tiong(ξ,z)−g(ξ,∞)+log|ξ−z|,
+ωk(z)−ωk(∞) =1
+2π/integraldisplay
+E∂ωk(ξ)
+∂n+
+ξlog|ξ−z||dξ|
+Using (21) and (28) the equality follows. /square
+Notation 2.2.By JacRwe denote the Jacobi variety of R,that is, the
+quotient space Cl−1/(2πi/vector n+/vectorB/vector m),B= (Bjk) the matrix of periods, /vector n,/vector m∈
+Zl−1.JacR/R:=Rl−1/B/vector mdenotes the Jacobi variety restricted to the
+reals. Finally, let [ a2j,a2j+1]±denote the two copies of [ a2j,a2j+1], j=
+1,...,l−1,inR±.Note that [a2j,a2j+1]+∪[a2j,a2j+1]−is a closed loop on
+R.
+The following important statement holds, see e.g., [14, The orem 5.12] and
+[32, Lemma 3.5 (a)]:
+The restricted Abel map
+A:Xl−1
+j=1([a2j,a2j+1]+∪[a2j,a2j+1]−)→JacR/R
+(ζ1,...,ζl−1)/ma√sto→1
+2
+l−1/summationdisplay
+j=1/integraldisplayζj
+ζ∗
+jϕ1,...,l−1/summationdisplay
+j=1/integraldisplayζj
+ζ∗
+jϕl−1
+
+is a holomorphic bijection. Moreover the so called real Jaco bi inversion
+problem has a unique solution, that is, for given ( η1,...,ηl−1)∈Rl−1there
+exists a unique ( ζ1,...,ζl−1)∈Xl−1
+j=1([a2j,a2j+1]+∪[a2j,a2j+1]−) such that
+1
+2l−1/summationdisplay
+j=1/integraldisplayζj
+ζ∗
+jϕk=ηkmod (B).
+Lemma 2.3. a) Lets∈Rl−1andδ∈({−1,1})l−1be given. Then there
+exists a unique (κ1,...,κl−1)∈Xl−1
+j=1Rδjand a unique /vectorσ∈({0,1})l−1such
+that
+1
+2l−1/summationdisplay
+j=1/integraldisplayκj
+κ∗
+jϕk=sk+l−1/summationdisplay
+j=1σj
+2Bkjmodulo (B) fork= 1,...,l−1
+b) Choosing in a) δj= 1forj= 1,...,l−1the representation
+(29)1
+2l−1/summationdisplay
+j=1/integraldisplayκj
+κ∗
+jϕk=sk+l−1/summationdisplay
+j=1σj
+2Bkj=l−1/summationdisplay
+j=1tjBkjmod (B)
+with
+(30) tκ=1
+2l−1/summationdisplay
+j=1ωκ(pr(κj))∈[0,1
+2] forκ= 1,...,l−1
+holds.12 F. PEHERSTORFER
+Choosing in a) δj=−1forj= 1,...,l−1thentjhas a representation of
+the form (30)withtκ∈[−1
+2,0]forκ= 1,...,l−1.
+Proof.The simplest way to prove part a) is to change the model of the
+Riemann surface by choosing now as in [14, Section 5.5.2] the canonical
+homology basis {α′
+j,β′
+j}l−1
+j=1,where the curve α′
+joriginates at a2j,arrives at
+a2j+1along the upper sheet and turns back to a2jalong the lower sheet and
+β′
+jlies in the upper sheet and encircles the interval [ a1,a2j] clockwise. Note
+that theα′andβ′periods can be expressed easily in terms of the αandβ
+periods from (19). More precisely, the period matrix of the n ew model is
+obtained by a linear transformation with entire coefficients of the original
+model. Thus modulo periods it does not matter which model we t ake.
+We knowthat thereexists an uniquesolution /vectorξ∈Xl−1
+j=1R,see[14, Theorem
+5.12] and [32, Lemma 3.5 (a)], such that
+(31)1
+2l−1/summationdisplay
+j=1/integraldisplayξj
+ξ∗
+jϕk=skfork= 1,...,l−1 modB′
+If/vectorξ∈Xl−1
+j=1Rδjwe are done. If ξp∈ R−δpthen we consider the inversion
+problem for ( s1+B′
+p1
+2,...,sl−1+B′
+pl−1
+2) that is, we add the half period
+(B′)p.Then forj/\e}atio\slash=pall pointsξjfrom (31) remain in place and ξpwill be
+displaced by ξ∗
+pifξp/∈ {aj}2l
+j=1.Ifξp∈ {aj}2l
+j=1then this point is displaced
+from one end of ( a2j,a2j+1) to the other. Thus part a) follows by going back
+to our model.
+b) By (??) we know that (29) holds with tj∈[−1/2,1/2].Taking into
+consideration the fact that κj=x+
+jandκ∗
+j=x−
+jand writing the LHS of
+(29) in terms of harmonic measures with the help of (21) it fol lows that
+tk=1
+2l−1/summationdisplay
+j=1ωk(pr(κj))≥0
+which proves part b). /square
+Lemma 2.4. a) The unique solution (ζ1,n,...,ζl−1,n)of the Jacobi inversion
+problem,κ= 1,...,l−1,
+(32)
+l−1/summationdisplay
+j=1/integraldisplayζj,n
+ζ∗
+j,nϕκ=n/integraldisplay∞+
+∞−ϕκ+2
+πi/integraldisplay
+E+ϕ+
+κlog|W|+l−1/summationdisplay
+j=1σj,n(W)Bκjmodulo 2(B)
+whereσj,n(W)is given by (10), has the property that ζj,n∈ R+forj=
+1,...,l−1.Moreover, putting ζj,n=c+
+j,n,henceζ∗
+j,n=c−
+j,nand using Lemma
+2.1 the system of equations (32)becomes (11).ASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 13
+b) The sequence of solutions (c+
+nν)ν∈Nof(32)converges if and only if
+(γnν(W))ν∈Nconverges modulo 2,whereγnν(W)is given in (10). Further-
+more, the transformed Abel map ˜A=B−1◦Ais a real analytic homeomor-
+phism between the sets of accumulation points of (c+
+n)n∈Nand(γn(W))n∈N
+modulo2.
+Proof.Assume that not all ζj,n’s are from R+.Then by Lemma 2.3 there
+is a˜σ∈ {0,1}l−1,˜σ/\e}atio\slash=σn(W),and a˜ζ∈Xl−1
+j=1R+which solves the corre-
+spondingly modified Jacobi inversion problem (32). Applyin g Lemma 2.1 to
+the RHS of (32) and using representation (29) with property ( 30) it follows
+thatγn(W)−σn(W)+˜σ∈[0,1]l−1modulo 2.But this implies by the def-
+inition of σn(W) and its uniqueness, recall (10), that σn(W) =˜σ,which is
+a contradiction.
+With the help of Lemma 2.1 it follows by straightforward calc ulation that
+(32) is equivalent to (11).
+b) Writing relation (32) in the form
+2/parenleftbig
+A((c+)nν)/parenrightbigt=Bγt
+nν
+wheretdenotes the transpose of the vectors, the assertion follows immedi-
+ately by the bijectivity and the other properties of the Abel mapA./square
+Theorem 2.5. Letcj,n∈[a2j,a2j+1], j= 1,...,l−1,be such that
+(33)l−1/summationdisplay
+j=1ωk(cj,n) =γn(1/ρν)
+and suppose that for n∈M⊆Nthere holds cj,n∈[a2j+δ,a2j+1−δ], δ>0,
+j= 1,...,l−1.Then forn∈Mthe following asymptotic representation holds
+uniformly on E
+(34)2Mn(x;ρν)
+ρν(x)=/parenleftbigg
+R+
+n(x;ρν)+1
+R+n(x;ρν)/parenrightbigg
++O(qn),
+where
+(35) R+
+n(x;ρν) =(φ+(x;∞))n−νν∗/producttext
+j=1(φ+(x;wj))νj
+l−1/producttext
+j=1φ+(x;cj,n)
+and whereq∈(−1,1)and the constant in the O-term does not depend on n
+andx, x∈E.
+Proof.For simplicity of writing let us assume that νj= 1 forj= 1,...,ν.
+Firstletustransformcondition (33)intotheequivalent co ndition onAbelian
+differentials of first kind. Multiplying each equation from (3 3) byBκkand14 F. PEHERSTORFER
+summing up we obtain that (33) is equivalent to
+(36)l−1/summationdisplay
+j=1/integraldisplayc+
+j,n
+c−
+j,nϕκ=n/integraldisplay∞+
+∞−ϕκ−ν/summationdisplay
+j=1/parenleftBigg/integraldisplay∞+
+∞−ϕκ−/integraldisplayw+
+j
+w−
+jϕκ/parenrightBigg
++l−1/summationdisplay
+k=1σkn(1/ρν)Bκkforκ= 1,...,l−1.
+Thus by Abel’s Theorem, see e.g. [23, Theorem ], there is a rat ional
+function Rnon the Riemann surface y2=Hsuch that
+(37)x=∞±is a pole (zero) of Rnwith multiplicity n−ν,
+x=w±
+jis a simple pole (zero) of Rn
+x=c±
+j,nis a simple zero (pole) of Rn
+ThusRnis of the form
+(38) Rn=Pn+l−1+√
+HQn−l−1
+g(n)ρν,
+where
+(39) g(n)(z) =l−1/productdisplay
+j=1(x−cj,n),
+Pn+l−1andQn+l−1are polynomials of degree n+l−1 andn−l−1 which
+are such that the numerator in (38) has the properties that
+Pn+l−1+√
+HQn−l−1has a double zero at cj,nforj= 1,...,l−1
+and
+Pn+l−1−√
+HQn−l−1has a double zero at wjforj= 1,...,ν.
+By the way, since the points cj,nandwjare real the rational function Rn(x)
+has the same properties (37) as Rn(¯x),i.e.Rn(¯x) =Rn(x),or in other
+words the coefficients of Pn+l−1(x) andQn−1(x) are real. Taking involution
+(denoted by ˜ x), in (37), which corresponds to multiplication of relation (36)
+by−1,we obtain that
+(40)1
+Rn(z)=Rn(˜z) =Pn+l−1(z)−/radicalbig
+H(z)Qn−l−1(z)
+g(n)(z)ρν(z)
+Moreover, putting
+(41) Rn:=R:=Pn+l−1
+ρνg(n)andSn:=S:=Qn−l−1
+ρνg(n)
+i.e.
+(42) Rn=R+√
+HSand1
+Rn=R−√
+HSASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 15
+it follows that
+(43) R2−HS2= 1
+Note that for x∈E
+(44) R±
+n(x) =R(x)±i/radicalbig
+−H(x)S(x)
+hence forx∈E
+(45) 2 Rn(x) =R+
+n(x)+R−
+n(x) =R+
+n(x)+1
+R+n(x),
+where the last equality follows by (43), and moreover
+(46) |R±
+n(x)|2=R+
+n(x)R−
+n(x) = 1
+Now we claim that
+(47) Rn(z) = (φ(z;∞))n−νν/producttext
+j=1φ(z;wj)
+l−1/producttext
+j=1φ(z;cj,n)
+Indeed, since the function
+f(z) =Rn(z)(φ(z;∞))−(n−ν)l−1/producttext
+j=1φ(z;cj,n)
+ν/producttext
+j=1φ(z;wj)
+has by (37) neither zeros nor poles on ¯C\Eand satisfies by the definition
+ofφand by (33) that |f±|= 1 onE.Thus log |f|is a harmonic bounded
+function on C\Ewhich has a continuous extension to Eand thusf≡1,
+which proves the claim (47).
+Nextletusdemonstratethat |R| ≤1onEandthatRhasn+1alternation
+points onEi.e. there exist n+ 1 points yifromE, y1< y2< ... < y n+1,
+such that
+(48) ( −1)n+1−i=R(yi) =Pn+l−1(yi)
+ρν(yi)g(n)(yi)
+which implies by the Alternation Theorem that
+(49) 0 is a best approximation to Rwith respect to L{xj/ρν}n−1
+j=0
+Since−H >0 on˚E the property that |R| ≤1 onEfollows immediately by
+(43). Furthermore, |R|= 1 onEif and only if HQn−1−l= 0.Thus if we are
+able to show that the zeros of HQn−1−landPn+l−1are simple and strictly
+interlacing on Eand thatQn−l−1has exactly two zeros in each open gap16 F. PEHERSTORFER
+(a2j,a2j+1), j= 1,...,l−1,the alternation property (48) and thus (49) will
+follow. First let us recall that
+(50)
+R=1
+2/parenleftbigg
+Rn+1
+Rn/parenrightbigg
+= coshln Rnand√
+HS=1
+2/parenleftbigg
+Rn−1
+Rn/parenrightbigg
+= sinhln Rn
+Hence
+(51) HQn−l−1(z) = 0 if and only if Rn(z) =±1
+where forz∈Eone has to take the limiting value R+
+n.Note, that by
+representation (47), (4) and (7)
+(52) R+
+n(x) =eiχn(x)
+where
+χn(x) =−(n−ν)π/integraldisplayx
+a1r∞(t)
+h(t)dt
++/integraldisplayx
+a1( bounded function with respect to xandn)dt
+h(t)(53)
+where we have used the fact that
+limz→x
+Imz>0r∞(z)/radicalbig
+H(z)=−iπr∞(x)
+h(x),
+where
+(54)1
+h(x)=
+
+(−1)l−k
+π/radicalbig
+−H(x)forx∈Ek
+0 elsewhere
+Since, by (17), the polynomial r∞(x) has exactly one zero in each open gap
+(a2j,a2j+1), j= 1,...,l−1,it follows that
+(55) |r∞(x)| ≥const>0 onEandr∞(x)/h(x)>0 on˚E.
+Thusχnis strictly monotone on Efor sufficiently large n.Now by (50) and
+(52)
+(56) R(x) = cosχn(x) andi/radicalbig
+H(x)S(x) = sinχn(x),
+hence it follows that on Ethe zeros of Pn+l−1andHQn−l−1are simple and
+strictly interlace.
+Next let us prove that Qnk−l−1has exactly two zeros in ( cj−ε,cj+ε)⊂
+(a2j,a2j+1) forj= 1,...,l−1,if lim
+kcj,nk=cj.
+By (47)Rnhas a simple zero at cj,n.SinceRnis real on R\Eand since
+|φ(z;∞)|>1 onC\E,Rnis unbounded with respect to non compact
+subsets of ( a2j,a2j+1)\{cj}, j= 1,...,l−1,it follows that Rntakes the
+values−1 and +1 on ( cj−ε,cj+ε)⊂(a2j,a2j+1) forj= 1,...,l−1.Hence
+Qnk−l−1has at least two zeros in ( cj−ε,cj+ε).ASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 17
+Toshowthat Qnk−l−1hasexactly twozerosin( cj−ε,cj+ε),j= 1,...,l−1,
+we derive first an explicite formula for the number of zeros of Pn+l−1inEj
+which is of interest by itself. Indeed by (47)
+(57)dlnRn(z) =(n−ν)η(z;∞−,∞+)+ν/summationdisplay
+κ=1η(z;w−
+κ,w+
+κ)
+−l−1/summationdisplay
+κ=1η(z;c−
+κ,n,c+
+κ,n)+l−1/summationdisplay
+j=1ej,nϕj
+where we claim that
+(58)ej,n=♯Z(Pn+l−1;Ej) := number of zeros of Pn+l−1inEj.
+and that
+(59)♯Z(HQn−l−1;Ej)−1 =♯Z(Pn+l−1;Ej)
+=nωj(∞)−l−1/summationdisplay
+κ=1ωj(cκ,n)+1
+π/integraldisplay
+Elog|ρν(ξ)|∂ωk
+∂n+
+ξ|dξ|
+Indeed, by (47) it follows that dlnRnhas a representation of the form (57)
+withej,n∈C.Now by (normalization)
+(60) 2 πiej,n=/integraldisplay
+αjdlnRn(z)
+and on the other hand by shrinking αjtoEjand (52), (41) and (56) we
+obtain
+(61)/integraldisplay
+αjdlnRn(z) =−i∆EjargRn=χn(a2j)−χn(a2j−1)
+=−2πi♯Z(Pn+l−1;Ej)
+wherethelast equality follows by (56), recalling thestric tly interlacing prop-
+erty ofPn+l−1andHQn−l−1.
+Considering the βk-cycles gives with the help of (58) and Riemann’s bi-
+linear relation (24) that
+(62)/integraldisplay
+βkdlnRn(z) = (n−ν)/integraldisplay∞+
+∞−ϕk+ν/summationdisplay
+κ=1/integraldisplayw+
+κ,n
+w−
+κ,nϕk−l−1/summationdisplay
+κ=1/integraldisplayc+
+κ,n
+c−
+κ,nϕk
+−l−1/summationdisplay
+j=1♯Z(Pn+l−1;Ej)
+Let us consider the integral at the LHS. By the direction of in tegration the
+integrals along the gaps cancel out and along the Ek’s the real part of the
+differential becomes zero because of (47). Hence the real part of the integral
+at the LHS (62) is zero and thus zero, since the RHS is real. Wri ting the
+integrals with the help of the formulas from Lemma 2.1 a) and b ) it follows
+that (59) holds.18 F. PEHERSTORFER
+Sincel/summationtext
+k=1ωk(z) = 1 we obtain by summing up (59) that
+n−(l−1) =♯Z(HQn−l−1;E)−l
+i.e.HQn−l−1hasn+1 zeros in E(which implies that Qnk−l−1has at most
+and thus exactly two zeros in each interval ( cj−ε,cj+ε), j= 1,...,l−1)
+and thusRhasn+1 alternation points on Ewhich proves statement (49).
+Nextletusshowthat Pn+l−1/g(n)isasymptoticallyequalonΩ \{l−1/uniontext
+j=1Uε(cj)}
+to a polynomial ˜Mnof degreen.Indeed partial fraction expansion gives
+(63)Pn+l−1(z)
+g(n)(z)=˜Mn(z)+l−1/summationdisplay
+j=1λj,n
+z−cj,n
+Thus
+(64)λj,n= lim
+z→cj,n(z−cj,n)Pn+l−1(z)
+g(n)(z)
+= lim
+z→cj,n(z−cj,n)l−1/producttext
+j=1φ(z;cj,n,E)
+φ(z;∞,E)nρν(z)
+ν/producttext
+j=1φ(z;wj)
+i.e.
+(65) λj,n=O(qn), q<1,
+where we used the facts that |φ(z,∞)| ≥1/qon Ω and that the cj,n’s stay
+away from this set.
+Finally let us show that asymptotically 0 is a best approxima tion to
+Mn(·;ρν)/ρνbydemonstratingthatthebestapproximationof Rn=Pn+l−1/ρνg(n)
+andMn(·;ρν)/ρνdiffer only slightly from each other for nsufficiently large.
+To prove this fact we use the so-called strong unicity consta nt defined for a
+functionf∈C(E) with respect to a linear space Gnby
+(66) γ(f;Gp) :=γ(f) := sup
+p∈Gn||p−p∗(f)||
+||f−p||−||f−p∗(f)||
+IfGnis a Haar system on Eof dimension nandf−g∗has the alternation
+pointsy1,...,yn+1,then it is known that
+(67) γ(f)≤max
+1≤k≤n+1||pk||
+where thepk’s fromGnare uniquely defined by
+(68)pk(yk) =sgn(f−p∗(f))(yk) =:σkandpk(yj) = 0 forj/\e}atio\slash=k
+Furthermore, let us recall that the operator of best approxi mation is Lips-
+chitz continuous that is, if h∈C(E) is another function, then
+||p∗(f)−p∗(h)|| ≤2γ(f)||f−h||.ASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 19
+In the case under consideration we put f=Rn, Gn=L{xj/ρν(x)}n−1
+j=0,
+h=˜Mn/ρνwhich yields by (49) and (64) that
+||p∗(˜Mn)|| ≤2γ(Rn)O(qn)
+Since we will prove in the final step that
+(69) γ(Rn) =O(n)
+it follows by (64), recall that ||Rn||= 1,that
+||˜Mn
+ρν−p∗(˜Mn)||= 1+O(qn)
+and thus
+(70)Mn(x)
+ρν(x)=˜Mn(x)−p∗(x;˜Mn)
+ρν(x)||˜Mn
+ρν−p∗(˜Mn)||=˜Mn(x)
+ρν(x)+O(qn)
+=Rn(x)+O(qn)
+which is by (45) the assertion of the theorem.
+Thus let us prove (69). First we note that in the case under con sideration
+pk(y) =σkρν(yk)lk,n(y)
+ρν(y)
+wherelk,nis thefundamentalLagrange polynomialwith respecttothen odes
+y1,...,yn+1which, as we have proved above, are by (48), (43) and (41) the
+zeros of
+(71) H˜Q,whereQn−l−1=˜Qv2l−2,n
+Recall that we have shown that
+(72)v2l−2,n(x)−→
+n→∞l−1/productdisplay
+j=1(x−cj)2and thatg(n)(x)−→
+n→∞l−1/productdisplay
+j=1(x−cj).
+Thus by (67)
+γ(Rn)≤constmax
+1≤k≤n+1||lk,n(y)||
+and therefore it suffices to show that
+(73) ||lk,n(y)||=||(H˜Q)(y)
+(y−yk)(H˜Q)′(yk)||=O(n)
+With the help of the mean value and Markov’s inequality we get that
+||H˜Q(y)
+y−yk|| ≤n2const||H˜Q|| ≤n2/tildewiderconst||√
+HQn−l−1
+g(n)||=O(n2)
+where we took into consideration (72) and (43). Finally let u s show that at
+the zerosykofH˜Q
+(74) |(H˜Q)′(yk)| ≥const n20 F. PEHERSTORFER
+Using (41), (56) and (71) it follows that at the zeros ykof˜Q
+(HQn−l−1)′=±(√
+−Hρνg(n)χ′
+n)(yk)
+where we used the fact that cos χn(yk) =±1.By (53), (55) and (71) in-
+equality (74) follows at the zeros of ˜Q.
+At a boundary point of E,sayaj,we have
+(75) ( HQn−l−1)′=ρνg(n)H′Qn−l−1
+Now by (53) and (56), recall that sin χn(aj) = 0,
+limx→aj
+x∈EQn−l−1
+ρνg(n)= lim
+x→aj/radicalbig
+−H(x)sinχn(x)
+−H(x)
+which implies, with the help of |r∞(aj)|>0,that (74) holds at the zeros of
+Halso. /square
+Corollary 2.6. For the minimum deviation the following asymptotics hold
+forn∈M
+(76) En−1(xn;ρν) = (capE)n−νl−1/producttext
+j=1φ(cj,n;∞)
+ν∗/producttext
+j=1φ(wj;∞)νj(1+O(qn))
+Proof.Recalling that
+Rn=Pn+l−1
+ρνg(n)=1
+2/parenleftbigg
+Rn+1
+Rn/parenrightbigg
+it follows by (47) and |φ|>1 onC\Ethat
+(77)2lc(Pn+l−1) = 2 lim
+z→∞Pn+l−1
+zn−νρνg(n)= lim
+z→∞Rn
+zn−ν+O(qn)
+= lim
+z→∞/parenleftbiggφ(z;∞)
+z/parenrightbiggn−νν∗/producttext
+j=1φ(wj;∞)νj
+l−1/producttext
+j=1φ(cj,n;∞)
+Since by (48) at the zeros yiofH˜Q,˜Qdefined in (71),
+(−1)n+1−i=Rn(yi) =˜Mn(yi)
+ρν(yi)+O(qn)
+it follows by Vall´ ee-Poussin’s Theorem, see [],
+(78) lc(˜Mn)En−1(xn;ρν) = 1+O(qn)
+Because of (63) we have that
+(79) lc(Pn+l−1) =lc(˜Mn)
+which gives by (77) and (78) the assertion. /squareASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 21
+3.Proof of Theorem 1.2
+The link with the weights of the form 1 /ρis given by the following two
+Lemmatas. For the next lemma compare [15].
+Lemma 3.1. LetW∈C(E)andρνbe positive on E.Then
+(80)||Mn(x;W)
+W(x)−Mn(x;ρν)
+ρν(x)||=O(n)/parenleftBig
+O(||1−ρν
+W||)+O(qn)/parenrightBig
+whereq∈(−1,1).
+Proof.Put
+(81) an= 1/En−1(xn;W) andbn= 1/En−1(xn;ρν)
+thatis,Mn(x;W) =anxn+...andMn(x;ρν) =bnxn+....Usingtheextremal
+property of Mn(·;W) andMn(·;ρν) we obtain that
+1
+an≤ ||Mn(x;ρν)
+W|| ≤1
+bn||ρν
+W||
+and an analogous estimate for Mn(x;W)/ρνyields
+1/||ρν
+W|| ≤an
+bn≤ ||W
+ρν||
+which implies using
+||W
+ρν−1|| ≤ ||W
+ρν||||ρν
+W−1||and 1≤ ||ρν
+W||||W
+ρν||
+that
+(82)/vextendsingle/vextendsingle/vextendsingle/vextendsinglean
+bn−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ ||W
+ρν||||ρν
+W−1||
+Let
+Rn(x;ρν) :=Rn(x) =Pn+l−1
+ρνg(n)
+be given by (41). Since ||Mn(x;W)/W||= 1 and by (70)
+||Mn(x;ρν)
+ρν−Rn(x;ρν)||=O(qn)
+it suffices to show that
+||Mn(x;W)
+ρν−Rn(x;ρν)||=O(n)/parenleftBig
+O(||1−ρν
+W||)+O(qn)/parenrightBig
+We may write
+Mn(x;W)−Pn+l−1(x)
+g(n)(x)= (1−an
+bn)Mn(x;W)+Mn(x;ρν)−Pn+l−1(x)
+g(n)(x)+h(x)
+where
+h(x) =an
+bnMn(x;W)−Mn(x;ρν)∈Pn−122 F. PEHERSTORFER
+Thus by (41)
+||−h
+ρν|| ≤γ(Rn)/parenleftbigg
+||Rn+h
+ρν||−||Rn||/parenrightbigg
+=γ(Rn)/parenleftbigg
+||Rn−Mn(x,ρν)
+ρν−(1−an
+bn)Mn(x;W)
+ρν+Mn(x;W)
+WW
+ρν||−1/parenrightbigg
+=γ(Rn)/parenleftBig
+O(qn)+O(||ρν
+W−1||)/parenrightBig
+which gives by (80) the assertion. /square
+The following lemma is due to Achieser and Tomcuk [4] and show s that
+the RHS in (80) tends to zero, if W∈C1+α, α>0.
+Lemma 3.2. LetW∈Cm(E)be positive on Ewithlim
+n→∞ω2(1
+n)logn= 0
+whereω2denotes the modulus of continuity of second order. Then there is
+a sequence of polynomials ρνof degreeν, ρνpositive on E,such that
+(83) |ρν(x)
+W(x)−1| ≤const
+νmω2(1
+ν)
+and
+(84)/integraldisplay
+Elog|ρν(ξ)|∂ωk(ξ)
+∂n+
+ξ|dξ|=/integraldisplay
+Elog|W(ξ)|∂ωk(ξ)
+∂n+
+ξ|dξ|fork= 1,...,l−1
+Proof.In [4] instead of relation (84) the relation
+(85)/integraldisplay
+Exjlogρν(x)
+W(x)dx
+h(x)= 0j= 0,...,l−1
+is given which obviously is equivalent to
+(86)/integraldisplay
+Eϕ+
+jlogρν(x)
+W(x)= 0j= 0,...,l−1
+Now (85) follows by (28). /square
+Proof of the asymptotic representation (12)of the minimal polynomial
+Mn(x;W)onE.Letρνbe such a sequence of polynomials whose existence
+is guaranteed by Lemma 3.2. Then in conjunction with Lemma 3. 1 and
+Theorem 2.5 it follows that for ν≥ν0,uniformly on E
+(87) 2Mn(x;W)
+W(x)=R+
+n(x;ρν)+R−
+n(x;ρν)+o(1)
+whereo(1) is uniformly bounded with respect to nandνandRn(z;ρν) :=
+Rn(z) is given by (47). First let us demonstrate that for ν≥ν0uniformly
+onE
+(88) R±
+n(x;ρν) =φ±(x;∞)n
+l−1/producttext
+j=1φ±(x,cj,n)/radicalBigg
+W±(x)
+W∓(x)(1+o(1)).ASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 23
+Recall, see [47], that there exists a function Ω ν(z) which has neither ze-
+ros nor poles, is analytic outside E,and is such that Ω+
+νand Ω−
+νextend
+continuously to Eand satisfies for ξ∈E
+(89) |Ων(x)|=/radicalBig
+Ω+ν(x)Ω−ν(x) =ρν(x)>0
+Since
+f(z) =ρν(z)
+ων(z)ν/productdisplay
+j=1φ(z;wj,ν)
+φ(z;∞)
+has neither zeros nor poles on ¯C\Eand satisfies |f±|= 1 onEit follows
+as above that log |f|is a harmonic bounded function on ¯C\Ewhich has a
+continuous extension to Eand therefore f≡1,that is,
+(90)Ων(z)
+ρν(z)=ν/productdisplay
+j=1φ(z;wj,ν)
+φ(z;∞)
+Note that
+(91)/radicalBigg
+Ω±ν(x)
+Ω∓ν(x)=Ω±
+ν(x)
+ρν(x)=ν/productdisplay
+j=1φ±(x;wj,ν)
+φ±(x;∞)
+Next let us consider the function
+(92) Iν(z) = exp{/radicalbig
+H(z)
+2π/integraldisplay
+E1
+z−xlogW(x)
+ρν(x)dx
+h(x)}
+Because of (85) Iν(z) is analytic and nonzero on ¯C\El.SinceW/ρν∈C1+α
+we may apply the Sochozki-Plemelj formula which yields that forx∈E
+(93) I±
+ν(x) = exp{1
+2log|W(x)
+ρν(x)|±iΨ(x)},
+where
+(94) Ψ( x) =/radicalbig
+|H(x)|
+2π−/integraldisplay
+Elog|W(t)/ρν(t)|
+x−tdt/radicalbig
+|H(t)|.
+Moreover
+(95) |Iν(x)|=/radicalBig
+I+ν(x)I−ν(x) =W(x)
+ρν(x)
+Hence, recalling (89), the unique function W(z) with property (9), intro-
+duced in the introduction, is given by
+(96) W(z) =Iν(z)Ων(z).
+Since by (83), (93) and (94)
+(97)Iν(z)−→
+ν→∞1 uniformly on compact subsets of C\Eas well as
+I±
+ν(x)−→
+ν→∞1 uniformly on E24 F. PEHERSTORFER
+it follows that
+W±(x) =I±
+ν(x)Ω±
+ν(x) = Ω±
+ν(x)(1+o(1))
+hence
+(98)/radicalBigg
+W±(x)
+W∓(x)=/radicalBigg
+Ω±ν(x)
+Ω∓ν(x)(1+o(1))
+Thus by (47) and (91) relation (88) is proved.
+Finally let us recall that because of (83)
+(99) cj,n(ρν) =cj,n(W)
+where thecj,n(ρν)’s are the points (33) and the cj,n(W)’s the points sat-
+isfying (11), using the fact that the associated Jacobi-inv ersion problem is
+uniquely solvable. Since
+(100)l−1/productdisplay
+j=1φ±(x;cj,n) =l−1/productdisplay
+j=1φ±(x;cj)(1+o(1))
+weobtain by (88), (87) and(9) theasymptotic representatio n (12) onE./square
+The asymptotic representation (14) on ¯C\Eand the asymptotic value of
+the minimum deviation will be derived after the Lemma.
+Proof of the asymptotic representation (14)outside ofE:PutMn(·;W) =
+Mn,let˜Mnbe the polynomial from (64) and set
+(101) dn=lc(Mn)
+lc(˜Mn)=1
+lc(˜Mn)En(xn;W)= 1+o(1)
+where the last equality follows by the fact, see (79), that
+1
+lc(˜Mn)=En−1(xn;ρν)+O(qn)
+and (81) and (82).
+Now let us consider
+Mn(z)−cn˜Mn(z)
+φn(z)
+Since (Mn−dn˜Mn)/φnis a single valued function which vanishes at z=∞
+we may apply Plemelj-Sochozki’s formula and obtain,
+|(Mn−dn˜Mn)(z)
+φn(z)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+E(Mn−dn˜Mn)(ξ)
+z−ξ/parenleftbigg1
+φ+(ξ)n−1
+φ−(ξ)n/parenrightbigg
+dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=o(1) uniformly on compact subsets of ¯C\E
+using the fact that by (87) and (44), (41) and (63)
+Mn(ξ)−dn˜Mn(ξ) =o(1) uniformly on E.ASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 25
+Since||˜Mn||Eis bounded we know by the so-called Bernstein-Walsh Lemma
+that|˜Mn(z)| ≤const|φn(z)|forz∈¯C\E,hence
+Mn(z)
+φn(z)=˜Mn(z)
+φn(z)+o(1) =Ων(z)
+l−1/producttext
+j=1φ(cj,n;∞)+o(1)
+which, by (96) and (97), proves (14). /square
+Proof of Corollary 1.5: a) Since 1/l−1/producttext
+j=1φ(z;cj) has a simple zero at cj,j=
+1,...,l−1,andisboundedfrombelowoncompactsubsetsof( a2j,a2j+1)\[cj−
+ε,cj+ε] it follows by (14) that Mn(x;1/W) has different sign on ( a2j+ε,cj−
+ε) and (cj+ε,a2j−ε), j= 1,...,l−1,and thus at least one zero in each
+(cj−ε,cj+ε),and therefore exactly one zero.
+b) The assertion follows by (59), (84), (99), (63) and the rel ations
+♯Z(Pn+l−1,Ek) =♯Z(˜Mn,Ek) =♯Z(Mn,Ek)
+recalling the fact that at the boundary points |˜Mn/ρν|tends to one. /square
+Proof of Theorem 1.6. The equivalence of statement a) and b) follows by
+Lemma 2.4 b) and the equivalence of a) and c) by Corollary 1.5. /square
+4.Proof of Theorem 1.9
+Lemma 4.1. LetPnbe orthonormal on Ewith respect to the weight function
+R/hρν=R/√
+−Hρν,R/hρ ν>0onint(E)andρν>0onEand assume for
+simplicity of writing that the zeros wjofρνare simple. Then the following
+relation holds
+(102) RP2
+n−SQ2
+m= 2ρνg(n)
+with
+(RPn)(wj) = (√
+HQm)(wj),
+g(n)(x) =l−1/productdisplay
+j=1(x−xj,n),wherexj,n∈[a2j,a2j+1] forj= 1,...,l−1
+and
+RPn(xj,n) =δj,n√
+HQm(xj,n)
+whereδj,n∈ {±1}.Furthermore putting
+(103) R1=RP2
+n
+ρνg(n)−1
+and
+(104)√
+HR2=√
+HQmPn
+ρνg(n)
+we obtain
+R2
+1−HR2
+2= 126 F. PEHERSTORFER
+R1(xj,n) =δj,n√
+HR2(xj,n)
+and
+R1(wj) =√
+HR2(wj)
+and there holds for n≥n0, k= 1,...,l−1
+(105)
+l−1/summationdisplay
+j=1/integraldisplayκj,n
+κ∗
+j,nϕk=l−1/summationdisplay
+j=1δj,n/integraldisplayx+
+j,n
+x−
+j,nϕk=−(2n+1+∂R−(ν+l))/integraldisplay∞+
+∞−ϕk
+−ν/summationdisplay
+j=1/integraldisplayw+
+j
+w−
+jϕk+l−1/summationdisplay
+j=1(2♯Z(Pn,Ej)+♯Z(R,Ej))Bkj
+wherepr(κj,n) =xj,nandκj,n∈ Rδj,n.
+TheL2-minimum deviation is given by
+(106)/parenleftbigg/integraldisplay
+p2
+n/parenrightbigg
+lc(ρν) = 2(cap E)2n+∂R−(l−1)−ν/producttextl−1
+j=1φ(∞;xj,n)−δj,n
+/producttextν
+j=1φ(∞;wj)+O(qn),
+where0<q<1.
+Proof.The first statements follow by [25] and relation (105) follow s by [32,
+Lemma 3.1]. It has been shown in [32, Lemma 2.3] that
+R1=1
+2/parenleftbigg
+ψn+1
+ψn/parenrightbigg
+and√
+HR2=1
+2/parenleftbigg
+ψn−1
+ψn/parenrightbigg
+where
+ψn(z) =φ(z,∞)2n+1+∂R−(ν+l)ν/productdisplay
+j=1φ(z;wj)l−1/productdisplay
+j=1φ(z,xj,n)δj,n
+Thus we obtain by (105) that
+1/(/integraldisplay
+p2
+n)lc(ρν) = lim
+z→∞1
+z2n+1+∂R−(ν+l)/parenleftbiggRP2
+n
+ρνg(n)−1/parenrightbigg
+= lim
+z→∞1
+z2n+1+∂R−(ν+l)1
+2/parenleftbigg
+ψn+1
+ψn/parenrightbigg
+=1
+2(cap E)−(2n+1+∂R−(ν+l))ν/productdisplay
+j=1φ(z,wj)/productdisplay
+φ(z,xj,n)δj,n+o(qn),
+0< q <1,where we used the fact that lim z→∞1/(ψnz2n+1+∂R−(ν+l)) =
+o(qn). /square
+Lemma 4.2. For every ˜R,R∈ E,respectively, ˜R,R∈ E(1−x)the solutions
+of(105)satisfy
+(107)xj,n(R) =pr(κj,n(R)) =pr(κj,n(˜R)) =xj,n(˜R)j= 1,...,l−1ASYMPTOTIC REPRESENTATION OF MINIMAL POLYNOMIALS 27
+ifκn(R),κn(˜R)∈Xl−1
+j=1(a2j,a2j+1)−∪(a2j,a2j+1)+.In particular
+g(n)(x;R) =g(n)(x;˜R)
+Proof.Since/summationtextl−1
+j=1♯Z(R,Ej)Bkj/2 and/summationtextl−1
+j=1♯Z(˜R,Ej)Bkj/2 differ modulo
+2 by half-periods only statement (107) follows, see the proo f of Lemma 2.3,
+since the solutions of the Jacobi inversion problem (105) wi th respect to R
+and˜Rdiffer with respect to the sheet only. /square
+Proof of Theorem 1.9. a) Forσn:=σn(1/ρν) given by (10) there exists a
+R(σn) such that
+σn=♯Z(R(σn)) mod 2
+Now by the uniqueness of the real Jacobi inversion problem it follows that
+cn=κn(σn)
+that is,
+(108) cj,n=xj,nand−1 =δj,nj= 1,...,l−1
+whereκn(σn) are the solutions from (105) and cnthe solution from (11).
+By Lemma 4.2 and Φ( ∞,x)>1 forx /∈Eit follows that the RHS takes its
+maximum for δj,n=−1, j= 1,...,l−1,hence by (108) for R(σn).Lemma
+4.2 and Corollary ??yield
+1
+2/parenleftbigg/integraldisplay
+p2
+nR(σn)
+ˆρνh/parenrightbigg
+=1
+2En−1,2(xn;R(σn)/ˆρνh) =E2n−1,∞(x2n;1/ˆρν)
+whereˆmeans monic.
+Now by the assumptions on the weight function W,see e.g. [4], there is a
+sequence of ρν’s positive on Esuch that on E
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleρν(x)
+W(x)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle<const
+νω2/parenleftbigg1
+ν/parenrightbigg
+which implies, as we have demonstrated above,
+En−1,∞(xn,1/ρν) =En−1,∞(xn,W)(1+o(1))
+and, see [47] or [4], that
+(109) En−1,2(xn;WR(σn)/h) =En−1,2(xn;R(σn)/hρν)
+which gives the assertion.
+b) Replacing R(σn) from a) by ((1 −x)R)(σn) the assertion follows anal-
+ogously. /square
+It can be shown quite similarly as in the second part of the pro of of
+Theorem 2.5 that the normalized minimal polynomial M2n(·;1/ρν) with28 F. PEHERSTORFER
+||M2n(·;1/ρν)||= 1 is given asymptotically by the orthogonal polynomial
+as follows
+R(x;σn)P2
+n(x;R(σn)/ρνh)−ρν(x)g(n)(x;σn)
+ρνg(n)(x;σn)
+=M2n(x;1/ρν)g(n)(x,σn)+q(x)
+ρν(x)g(n)(x;σn)
+=M2n(x;1/ρν)
+ρν(x)+O(rn),
+where 0< r <1,uniformly on compact subsets of C\{c1,...,cl−1}where
+c1,...,cl−1are the accumulation points of zeros of g(n)(x,σn).For oddn’s
+the assertion holds analogously.
+Proof of Corollary 1.10 . By (22) and (23)
+dlnφ(z;c) =−l−1/summationdisplay
+k=1ωk(c)ϕk+η(z;c+,c−),
+hence by (35)
+ψn(z)
+W(z)= exp{l−1/summationdisplay
+k=1(−nωk(∞)+l−1/summationdisplay
+j=1ωk(cj))/integraldisplayz
+aϕk}
+.(exp/integraltext
+η(z;∞+,∞−))n
+l−1/producttext
+j=1e/integraltextη(z;c+
+j,c−
+j)
+Note that the first factor can be written by (11) in terms of int egrals of
+logW.Now
+l−1/productdisplay
+j=1e/integraltext
+η(z;c+
+j,c−
+j)=ϑ(z;/summationtext
+j/integraltextc+
+j,n
+a1ϕk)
+ϑ(z;/summationtext
+j/integraltextc−
+j,n
+a1ϕk)
+since the functions at the LHS and RHS have the same zeros and p oles
+and the same β−periods and coincide at the point a1.Analogously the
+representation for e/integraltext
+η(z,∞+,∞−)follows, using (11) with W≡1,or by taking
+a look at [10, Proposition 2.1]. The assertion follows by (11 ). /square
+References
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\ No newline at end of file
diff --git a/1001.0492.txt b/1001.0492.txt
new file mode 100644
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@@ -0,0 +1,2473 @@
+arXiv:1001.0492v1  [math.ST]  4 Jan 2010The Annals of Statistics
+2010, Vol. 38, No. 1, 1–50
+DOI:10.1214/08-AOS648
+c/circlecopyrtInstitute of Mathematical Statistics , 2010
+THE SPECTRUM OF KERNEL RANDOM MATRICES
+By Noureddine El Karoui1
+University of California, Berkeley
+We place ourselves in the setting of high-dimensional stati stical
+inference where the number of variables pin a dataset of interest is
+of the same order of magnitude as the number of observations n.
+We consider the spectrum of certain kernel random matrices, in
+particular n×nmatriceswhose( i,j)thentryis f(X′
+iXj/p)orf(/bardblXi−
+Xj/bardbl2/p)wherepisthedimensionofthedata,and Xiareindependent
+data vectors. Here fis assumed to be a locally smooth function.
+The study is motivated by questions arising in statistics and com-
+puter science where these matrices are used to perform, amon g other
+things, nonlinear versions of principal component analysis . Surpris-
+ingly, we show that in high-dimensions, and for the models we ana-
+lyze, the problem becomes essentially linear—which is at odd s with
+heuristics sometimes used to justify the usage of these meth ods. The
+analysis also highlights certain peculiarities of models wi dely stud-
+ied in random matrix theory and raises some questions about th eir
+relevance as tools to model high-dimensional data encounte red in
+practice.
+1. Introduction. Recent years has seen newfound theoretical interest in
+the properties of large-dimensional sample covariance mat rices. With the
+increase in the size and dimensionality of datasets to be ana lyzed, questions
+have been raised about the practical relevance of informati on derived from
+classical asymptotic results concerning spectral propert ies of sample covari-
+ance matrices. To address these concerns, one line of analys is has been the
+consideration of asymptotics where both the sample size nand the num-
+ber of variables pin the dataset go to infinity jointly while assuming, for
+instance, that p/nhad a limit.
+Received December 2007; revised August 2008.
+1Supported by NSF Grant DMS-06-05169.
+AMS 2000 subject classifications. Primary 62H10; secondary 60F99.
+Key words and phrases. Covariance matrices, kernel matrices, eigenvalues of cova riance
+matrices, multivariate statistical analysis, high-dimens ional inference, random matrix the-
+ory, machine learning, Hadamard matrix functions, concent ration of measure.
+This is an electronic reprint of the original article publis hed by the
+Institute of Mathematical Statistics inThe Annals of Statistics ,
+2010, Vol. 38, No. 1, 1–50 . This reprint differs from the original in pagination and
+typographic detail.
+12 N. EL KAROUI
+This type of questions concerning the spectral properties o f large-dimen-
+sional matrices have been and are being addressed in variety of fields, from
+physicstovariousareasofmathematics.Whilethetopicisc lassical,withthe
+seminalcontribution[ 43]datingbackfromthe1950s,therehasbeenrenewed
+and vigorous interest in the study of large-dimensional ran dom matrices in
+the last decade or so. This has led to new insights and the appe arance of
+new “canonical” distributions [ 38], new tools (see [ 41]) and, in Statistics, a
+sensethatoneneedstoexertcautionwithfamiliartechniqu esofmultivariate
+analysis when the dimension of the data gets large and the sam ple size is of
+the same order of magnitude as that dimension.
+So far in Statistics, this line of work has been concerned mos tly with
+the properties of sample covariance matrices. In a seminal p aper, Marˇ cenko
+and Pastur [ 30] showed a result that, from a statistical standpoint, may be
+interpreted as saying, roughly, that asymptotically, the h istogram the eigen-
+values of a sample (i.e., random) covariance matrix is (asym ptotically) a
+deterministic nonlinear deformation of the histogram of the eigenvalues o f
+the population covariance matrix. Remarkably, they manage d to character-
+ize this deformation for fairly general population covaria nces. Their result
+was shown in great generality and introduced new tools to the field including
+one that has become ubiquitous, the Stieltjes transform of a distribution. In
+its best-known form, their result says that when the populat ion covariance
+is identity, and hence all the population eigenvalues are eq ual to 1, in the
+limit the sample eigenvalues are split, and if p≤n, they are spread between
+[(1−/radicalbig
+p/n)2,(1+/radicalbig
+p/n)2]accordingtoafullyexplicitdensityknownnowas
+the density of the Marˇ cenko–Pastur law. Their result was la ter re-discovered
+independently in [ 42] (under slightly weaker conditions) and generalized to
+the case of nondiagonal covariance matrices in [ 37] under some particular
+distributional assumptions which we discuss later in the pa per.
+On the other hand, recent developments have been concerned w ith fine
+propertiesofthelargesteigenvalueofrandommatriceswhi chbecameamenable
+to analysis after mathematical breakthroughs which happen ed in the 1990s
+(see [38,39] and [40]). Classical statistical work on joint distribution of
+eigenvalues of sample covariance matrices (see [ 1] for a good reference) then
+became usable for analysis in high-dimensions. In particul ar, in the case
+of gaussian distributions, with Id covariance, it was shown in [27] and [16]
+that the largest eigenvalue of the sample covariance matrix is Tracy–Widom
+distributed. More recent progress [ 17] managed to carry out the analysis
+for essentially general population covariance. On the othe r hand, models for
+which the population covariance has a few separated eigenva lues have also
+been of interest (see, for instance, [ 8] and [31]). Beside the particulars of the
+different type of fluctuations that can be encountered (Tracy –Widom, Gaus-
+sian or other), researchers have been able to precisely loca lize these largest
+eigenvalues. One interesting aspect of those results is the fact that in theSPECTRUM OF KERNEL RANDOM MATRICES 3
+high-dimensional setting of interest to us, the largest eig envalues are always
+positively biased, with the bias being sometime large. (We a lso note that
+in the case of i.i.d. data—which naturally is less interesti ng in statistics—
+results on the localization of the largest eigenvalue have b een available for
+quite some time now, after the works [ 21] and [45], to cite a few.) This is
+naturally in sharp contrast to classical results of multiva riate analysis which
+show√n-consistency of all sample eigenvalues; though the possibi lity of bias
+is a simple consequence of Jensen’s inequality.
+On the other hand, there has been much less theoretical work o n kernel
+random matrices. By this term, we mean matrices with ( i,j) entry of the
+form,
+Mi,j=k(Xi,Xj),
+whereMis ann×nmatrix,Xiis ap-dimensional data vector and kis a
+function of two variables, often called a kernel, that may de pend onn. Com-
+mon choices of kernels include, for instance, k(Xi,Xj)=f(/bardblXi−Xj/bardbl2/t),
+wherefis a function and tis a scalar, or k(Xi,Xj)=f(X′
+iXj/t). For the
+functionf, common choices include f(x)=exp(−x),f(x)=exp(−xa), for a
+certain scalar a,f(x)=(1+x)a, orf(x)=tanh(a+bx), wherebis a scalar.
+We refer the reader to [ 33], Chapter 4, or [ 44] for more examples.
+Inparticular,wearenotawareofanyworkinthesettingofhi gh-dimensional
+data analysis, where pgrows with n. However, given the practical success
+and flexibility of these methods (we refer to [ 35] for an introduction), it is
+natural to try to investigate theoretically their properti es. Further, as illus-
+trated in the data analytic part of [ 44], ann/pboundedness assumption is
+not unrealistic as far as applications of kernel methods are concerned. One
+aim of the present paper is to shed some theoretical light on t he properties
+of these kernel random matrices and to do so in relatively wid e generality.
+We note that the choice of renormalization that we make below is motivated
+in part by the arguments of [ 44] and their practical choices of kernels for
+data of varying dimensions.
+Existing theory on kernel random matrices (see, for instanc e, the inter-
+esting [28]) for fixed-dimensional input data predicts that the eigenv alues
+of kernel random matrices behave—at least for the largest on es—like the
+eigenvalues of the corresponding operator on L2(dP), if the data is i.i.d.
+with probability distribution P. To be more precise, if Xiis a sequence of
+i.i.d. random variables with distribution P, under regularity conditions on
+the kernelk(x,y), it was shown in [ 28] that, for any index l, thelth largest
+eigenvalue of the kernel matrix Mwith entries,
+Mi,j=1
+nk(Xi,Xj),4 N. EL KAROUI
+converges to the lth largest eigenvalue of the operator Kdefined as
+Kf(x)=/integraldisplay
+k(x,y)f(y)dP(y).
+Theseinsightshavealsobeenderivedthroughmoreheuristi cbutnonetheless
+enlightening arguments in, for instance, [ 44]. Further, more precise fluctua-
+tion results are also given in [ 28]. We also note interesting work on Laplacian
+eigenmaps (see, e.g., [ 9]) where, among other things, results have been ob-
+tained showing convergence of eigenvalues and eigenvector s of certain Lapla-
+cian random matrices (which are quite closely connected to k ernel random
+matrices) computed from data sampled from a manifold, to cor responding
+quantities for the Laplace–Beltrami operator on the manifo ld.
+These results are in turn used in the literature to explain th e behavior
+of nonlinear versions of standard procedures of multivaria te statistics, such
+as Principal Component Analysis (PCA), Canonical Correlat ion Analysis
+(CCA) or Independent Component Analysis (CCA). We refer the reader
+to [36] for an introduction to kernel-PCA, and to [ 2] for an introduction to
+kernel-CCAandkernel-ICA.Attheheartofthesetechniques arethespectral
+properties of kernel random matrices. Because these techni ques are used in
+bioinformatics, a field where large datasets are common and b ecoming the
+norm, it is natural to ask what can be said about these spectra l properties
+for high-dimensional data.
+We show that for the models we analyze (ICA-type models and ge neral-
+izations that go beyond the linear setting of ICA), kernel ra ndom matrices
+essentially behave like sample covariance matrices and hen ce their eigen-
+values suffer from the same bias problems that affect sample co variance
+matrices in high-dimensions. In particular, if one were to t ry to apply the
+heuristics of [ 44] which were developed for low-dimensional problems, to the
+high-dimensional case, the predictions would be quite wild ly wrong. (A sim-
+ple example is provided by the Gaussian kernel with i.i.d. Ga ussian data
+where the computations can be done completely explicitly as explained in
+[44].) We also note that the scaling we use is different from the on e used
+in low dimensions, where the matrices are scaled by 1 /n. This is because
+the high-dimensional problem would be completely degenera te if we used
+this normalization in our setting. However, our results sti ll give information
+about the problem when it is scaled by 1 /n.
+From a random matrix point of view, our study is connected to t he study
+ofEuclideanrandommatricesanddistancematriceswhichis ofsomeinterest
+in, for instance, Physics. We refer to [ 11] and [12] for work in this direction
+in the low (or fixed) dimensional setting. We also note that at the level of
+generality we place ourselves in, the random matrices we stu dy do not seem
+to be amenable to study through the classical tools of random matrix theory.SPECTRUM OF KERNEL RANDOM MATRICES 5
+Hence, beside their obvious statistical interest, they are also interesting on
+purely mathematical grounds.
+Wenowturntothegistofourpaper,whichwillshowthathigh- dimensional
+kernel random matrices behave spectrally essentially like matrices closely
+connected to sample covariance matrices. We will get two typ es of results:
+in Theorems 2.1and2.2, we get a strong approximation result (in operator
+norm) for standard models (ICA-like) studied in random matr ix theory. In
+Theorems 2.3and2.4, we characterize the limiting spectral distribution of
+our kernel random matrices, for a wider class of data distrib utions. In Sec-
+tion2, we also state clearly the consequences of our theorems and r eview
+the relevant theory of high-dimensional sample covariance matrices. From a
+technical standpoint, we adopt a point of view centered on th e concentra-
+tion of measure phenomenon, as exposed for instance in [ 29] as it provides
+a unified way to treat the two types of results we are intereste d in. Finally,
+we discuss in our (self-contained) conclusion (Section 3), the consequences
+of our results and in particular some possible limitations o f “standard” ran-
+dom matrix models as tools to model data encountered in pract ice focusing
+on geometric properties of datasets drawn according to thos e models. As
+explained in more details there, vectors drawn according to these standard
+random matrix models essentially live close to spheres and a re almost or-
+thogonal to one another, a property that may or may not be pres ent in
+datasets to be analyzed and can be seen as a key to many classic al and less
+classical random matrix results (see also [ 19]).
+2. Spectrum of kernel random matrices. Kernel random matrices do not
+seem to be amenable to analysis through the usual tools of ran dom matrix
+theory. In particular, for general f, it seems difficult to carry out either a
+method of moments proof, or a Stieltjes transform proof, or a proof that
+relies on knowing the density of the eigenvalues of the matri x.
+Hence, we take an indirect approach. Our strategy is to find ap proxi-
+mations of the kernel random matrix that have two properties . First, the
+approximation matrix is analyzable or has already been anal yzed in random
+matrix theory. Second, the quality of the approximation is g ood enough that
+spectral properties of the approximating matrix can be show n to carry over
+to the kernel matrix.
+The strategy in the first two theorems is to derive an operator norm “con-
+sistent” approximation of our kernel matrix. In other words , if we call M
+our kernel matrix, we will find Ksuch that /bardbl|M−K/bardbl|2→0, asnandp
+tend to∞. Note that both MandKare real symmetric (and hence Her-
+mitian) here. We explain after the statement of Theorem 2.1why operator
+norm consistency is a desirable property. But let us say that in a nutshell,
+it implies consistency for each individual eigenvalue as we ll as eigenspaces
+corresponding to separated eigenvalues.6 N. EL KAROUI
+For the second set of theorems (Theorems 2.3and2.4), we will relax
+the distributional assumptions made on the data, but, at the expense of
+the precision of the results we will obtain, we will characte rize the limiting
+spectral distribution of our kernel random matrices.
+Our theorems below show that kernel random matrices can be we ll ap-
+proximated by matrices that are closely connected to large- dimensional co-
+variance matrices. The spectral properties of those matric es have been the
+subject of a significant amount of work in recent and less rece nt years, and
+hence this knowledge, or at least part of it, can be transferr ed to kernel
+random matrices. In particular, we refer the reader to [ 4,5,8,17,19,21,
+27,30,31,37,42] and [45] for some of the most statistically relevant results
+in this area. We review some of them now.
+2.1.Some results on large-dimensional sample covariance matri ces.Since
+our main theorems are approximating theorems, we first wish t o state some
+of the properties of the objects we will use to approximate ke rnel random
+matrices. In what follows, we consider an n×pdata matrix, with, say p/n
+having a finite nonzero limit. Most of the results that have be en obtained are
+of two types: either they are so-called “bulk” results and co ncern essentially
+the spectral distribution (or loosely speaking the histogr am of eigenvalues)
+of the random matrices of interest; or they concern the local ization and
+fluctuation behavior of extreme eigenvalues of these random matrices.
+2.1.1.Spectral distribution results. An object of interest in random ma-
+trix theory is the spectral distribution of random matrices . Let us call lithe
+decreasinglyorderedeigenvaluesofourrandommatrix,and letusassumewe
+are working with an n×nmatrix,Mn. The empirical spectral distribution
+of an×nmatrix is the probability measure which puts mass 1 /nat each
+of its eigenvalues. In other words, if we call Fnthis probability measure, we
+have
+dFn(x)=1
+nn/summationdisplay
+i=1δli(x).
+Note that the histogram of eigenvalues represents an integr ated version of
+this measure.
+For random matrices, this measure Fnis naturally a random measure. A
+key result in the area of covariance matrices is that if we obs erve i.i.d. data
+vectorsXi, withXi= Σ1/2
+pYi, where Σ pis a positive semi-definite matrix
+andYiis a vector with i.i.d entries, under weak moment conditions onYi
+and assuming that the spectral distribution of Σ phas a limit (in the sense of
+weak convergence of distributions), Fnconverges to a nonrandom measure
+which we call F.SPECTRUM OF KERNEL RANDOM MATRICES 7
+We call the models Xi=Σ1/2
+pYithe “standard” models of random matrix
+theory because most results have been derived under these as sumptions. In
+particular, various results [ 5,6,21] show, among many other things, that
+when the entries of the vector Yhave 4 (absolute) moments, the largest
+eigenvalues of the sample covariance matrix X′X/n, whereXinow occupies
+theith row of the n×pmatrixX, stay close to the endpoint of the support
+ofF.
+A natural question is therefore to try to characterize F. Except in par-
+ticular situations, it is difficult to do so explicitly. Howev er, it is possible to
+characterize a certain transformation of F. The tool of choice in this context
+is theStieltjes transform of a distribution. It is a function defined on C+by
+the formula, if we call St Fthe Stieltjes transform of F,
+StF(z)=/integraldisplaydF(λ)
+λ−z,Im[z]>0.
+In particular for empirical spectral distributions, we see that, ifFnis the
+spectral distribution of the matrix Mn,
+StFn(z)=1
+nn/summationdisplay
+i=11
+li−z=1
+ntrace((Mn−zId)−1).
+The importance of the Stieltjes transform in the context of r andom matrix
+theory stems from two facts: on the one hand, it is connected f airly explic-
+itly to the matrices that are being analyzed; on the other han d, pointwise
+convergence of Stieltjes transform implies weak convergen ce of distributions,
+if a certain mass preservation condition is satisfied. This i s how a number
+of bulk results are therefore proved. For a clear and self-co ntained introduc-
+tion to the connection between Stieltjes transforms and wea k convergence
+of probability measures, we refer the reader to [ 22].
+The result of [ 30], later generalized by [ 37] for standard random matrix
+models with nondiagonal covariance, and more recently by [ 19], away from
+those standard models, is a functional characterization of the limitF. If
+one callswn(z) the Stieltjes transform of the empirical spectral distrib ution
+ofXX′/n,wn(z) converges pointwise (and almost surely after [ 37]) to a
+nonrandom w(z) which, as a function, is a Stieltjes transform. Moreover, w,
+the Stieltjes transform of F, satisfies the equation, if p/n→ρ,ρ>0,
+−1
+w(z)=z−ρ/integraldisplayλdH(λ)
+1+λw,
+whereHis the limiting spectral distribution of Σ p, assuming that such
+a distribution exists. We note that [ 37] proved the result under a second
+moment condition on the entries of Yi.8 N. EL KAROUI
+From this result, [ 30] derived that in the case where Σ p=Id, and hence
+dH=δ1, the empirical spectral distribution has a limit whose dens ity is, if
+ρ≤1,
+fρ(x)=1
+2πρ/radicalbig
+(b−x)(x−a)
+x,
+wherea=(1−ρ1/2)2andb=(1+ρ1/2)2. The difference between the popu-
+lation spectral distribution (a point mass at 1, of mass 1) an d the limit of
+the empirical spectral distribution is quite striking.
+2.1.2.Largest eigenvalues results. Another line of work has been focused
+onthebehaviorofextremeeigenvaluesofsamplecovariance matrices.Inpar-
+ticular, [ 21] showed, under some moment conditions, that when Σ p=Idp,
+l1(X′X/n)→(1+/radicalbig
+p/n)2almost surely. In other words, the largest eigen-
+value stays close to the endpoint of the limiting spectral di stribution of
+X′X/n. This result was later generalized in [ 45], and was shown to be true
+under the assumption of finite fourth moment only, for data wi th mean 0.
+In recent years, fluctuation results have been obtained for t his largest eigen-
+valuewhichisofpracticalinterestinPrincipalComponent sAnalysis(PCA).
+Under Gaussian assumptions, [ 16] and [27] (see also [ 20] and [26]) showed
+that the fluctuations of the largest eigenvalue are Tracy–Wi dom distributed.
+For the general covariance case, similar results, as well as localization infor-
+mation, were recently obtained in [ 17]. We note that the localization infor-
+mation (i.e., a formula) that was discovered in this latter p aper was shown
+to hold for a wide variety of standard random matrix models th rough appeal
+to [5]. We refer the interested reader to Fact 2 in [ 17] for more information.
+Interesting work has also been done on so-called “spiked” mo dels where a
+few population eigenvalues are separated from the bulk of th em. In partic-
+ular, in the case where all population eigenvalues are equal , except for one
+that is significantly larger (see [ 7] for the discovery of an interesting phase
+transition), [ 31] showed, in the Gaussian case, inconsistency of the largest
+sample eigenvalue, as well as the fact that the angle between the popula-
+tion and sample principal eigenvectors is bounded away from 0. Paul [ 31]
+also obtained fluctuation information about the largest emp irical eigenvalue.
+Finally, we note that the same inconsistency of eigenvalue r esult was also
+obtained in [ 8], beyond the Gaussian case.
+2.1.3.Notation. Let us now define some notation and add some clarifi-
+cations.
+We denote by A′the transpose of A. The matrices we will be working
+with all have real entries. We remind the reader that if AandBare two
+rectangular matrices, ABandBAhave the same eigenvalues, except forSPECTRUM OF KERNEL RANDOM MATRICES 9
+possibly, a certain number of zeros. We will make repeated us e of this fact,
+for example, for matrices like X′XandXX′. In the case where AandB
+are both square, ABandBAhave exactly the same eigenvalues.
+We will also need various norms on matrices. We will use the so -called
+operator norm, which we denote by /bardbl|A/bardbl|2which corresponds to the largest
+singular value of A, that is, max i/radicalbig
+li(A′A). We occasionally denote the
+largest singular value of Abyσ1(A). Clearly, for positive semi-definite ma-
+trices, the largest singular value is equal to the largest ei genvalue. Finally,
+we will sometimes need to use the Frobenius (or Hilbert–Schm idt) norm of
+a matrixA. We denote it by /bardblA/bardblF. By definition, it is simply, because we
+are working with matrices with real entries,
+/bardblA/bardbl2
+F=/summationdisplay
+i,jA2
+i,j.
+Further, we use ◦to denote the Hadamard (i.e., entrywise) product of two
+matrices.Wedenoteby µmthemthmomentofarandomvariable.Notethat
+by a slight abuse of notation, we might also use the same notat ion to refer to
+themth absolute moment (i.e., E|X|m) of a random variable, but if there is
+any ambiguity, we will naturally make precise which definiti on we are using.
+Finally, in the discussion of standard random matrix models that follows,
+there will be arrays of random variables and a.s. convergenc e. We work
+with random variables defined on a common probability space. To each
+ωcorresponds an infinite-dimensional array of numbers. Unle ss otherwise
+noted, the n×pmatrices we will use in what follows are the “upper-left”
+corner of this array.
+We now turn to the study of kernel random matrices. We will sho w that
+we can approximate them by matrices that are closely connect ed to sample
+covariance matrices in high-dimensions and, therefore, th at a number of the
+results we just reviewed also apply to them.
+2.2.Inner-product kernel matrices: f(X′
+iXj/p).
+Theorem 2.1 (Spectrum of inner product kernel random matrices). Let
+us assume that we observe ni.i.d. random vectors, XiinRp. Let us consider
+the kernel matrix Mwith entries
+Mi,j=f/parenleftbiggX′
+iXj
+p/parenrightbigg
+.
+We assume that:
+(a)n≍p, that is,n/pandp/nremain bounded as p→∞.
+(b) Σpis a positive semi-definite p×pmatrix, and /bardbl|Σp/bardbl|2=σ1(Σp)remains
+bounded in p, that is, there exists K>0, such that σ1(Σp)≤K, for all
+p.10 N. EL KAROUI
+(c) Σp/phas a finite limit, that is, there exists l∈Rsuch that limp→∞trace(Σ p)/
+p=l.
+(d)Xi=Σ1/2
+pYi.
+(e)The entries of Yi, ap-dimensional random vector, are i.i.d. Also, denot-
+ing byYi(k)thekth entry ofYi, we assume that E(Yi(k))=0,var(Yi(k))=
+1andE(|Yi(k)|4+ε)<∞for someε>0. (We say that Yihas4+εab-
+solute moments.)
+(f)fis aC1function in a neighborhood of l=limp→∞trace(Σ p)/pand a
+C3function in a neighborhood of 0.
+Under these assumptions, the kernel matrix Mcan (in probability) be
+approximated consistently in operator norm, when pandntend to∞, by
+the matrixK, where
+K=/parenleftbigg
+f(0)+f′′(0)trace(Σ2
+p)
+2p2/parenrightbigg
+11′+f′(0)XX′
+p+υpIdn,
+where
+υp=f/parenleftbiggtrace(Σ p)
+p/parenrightbigg
+−f(0)−f′(0)trace(Σ p)
+p.
+In other words,
+/bardbl|M−K/bardbl|2→0in probability, when p→∞.
+The advantages of obtaining an operator norm consistent est imator are
+many. We list some here:
+•Asymptotically, MandKhave the same j-largest eigenvalue, for any
+j; this is simply because for symmetric matrices, if ljis thejth largest
+eigenvalue of a matrix, Weyl’s inequality (see, e.g., Corol lary III.2.6 in
+[10]) implies that
+|lj(M)−lj(K)|≤/bardbl|M−K/bardbl|2.
+Hence our result implies that |lj(M)−lj(K)|→0 in probability as pand
+ngo to infinity.
+•The limiting spectral distributions of MandK(if they exist) are the
+same. This is a consequence of Lemma 2.1, page31below. So in partic-
+ular, when Khas a limiting spectral distribution (in the sense of weak
+convergence of probability measures), the empirical spect ral distribution
+ofMconverges to that distribution (in the sense of weak converg ence of
+distributions) in probability.
+•We have subspace consistency for eigenspaces correspondin g to separated
+eigenvalues. (For a proof, we refer to [ 18], Corollary 3.) So, when Khas
+eigenvalues that stay separated from the bulk of this matrix ’s eigenvalues,
+thenMhas in probability the same property, and the angle between t he
+corresponding eigenspaces for KandMgo to 0 in probability.SPECTRUM OF KERNEL RANDOM MATRICES 11
+(Note that the statements we just made assume that both MandKare
+symmetric, which is the case here.)
+The strategy for the proof is the following. According to the results of
+LemmaA.3,thematrix X′
+iXj/phas“small”entriesoffthediagonal,whereas
+onthediagonal,theentriesareessentiallyconstantandeq ualtotrace(Σ p)/p.
+Hence, it is natural to try to use the δ-method (i.e., do a Taylor expansion)
+entry by entry. By contrast to standard problems in Statisti cs, the fact that
+we have to perform n2of those Taylor expansions means that the second
+order term is not negligible a priori. The proof shows that th is approach can
+be carried out rigorously, and that, perhaps surprisingly, the second order
+term is not too complicated to approximate in operator norm. It is also
+shown that the third order term plays essentially no role.
+Before we start the proof, we want to mention that we will drop the
+indexpin Σpbelow to avoid cumbersome notation. Let us also note, more
+technically,thatanimportantstepoftheproofistoshowth at,whenthe Yi’s
+have enough moments, they can be treated without much error i n spectral
+results has bounded random variables—the bound depending o n the number
+of moments, nandp. This then enables us to use concentration results
+for convex Lipschitz functions of independent bounded rand om variables at
+various important points of the proof and also in Lemma A.3whose results
+underly much of the approach taken here.
+Proof of Theorem 2.1.First, let us call
+τ/definestrace(Σ)
+p.
+Using Taylor expansions, we can rewrite our kernel matrix as
+f(X′
+iXj/p)=f(0)+f′(0)X′
+iXj/p+f′′(0)
+2(X′
+iXj/p)2
++f(3)(ξi,j)
+6(X′
+iXj/p)3ifi/ne}ationslash=j,
+f(/bardblXi/bardbl2
+2/p)=f(τ)+f′(ξi,i)/parenleftbigg/bardblXi/bardbl2
+2
+p−τ/parenrightbigg
+on the diagonal .
+The proof can be separated in different steps. We will break th e kernel
+matrix into a diagonal term and an off-diagonal term. The resu lts of Lemma
+A.3, after they are shown, will allow us to take care of the diagon al matrix
+at relatively lost cost. So we postpone that part of the analy sis to the end
+of the proof and we first focus on the off-diagonal matrix.
+In what follows, we call “second order term” the matrix Awith entries,
+Ai,j=f′′(0)
+2(X′
+iXj/p)21i/negationslash=j.12 N. EL KAROUI
+We call “third order term” the matrix Bwith entries,
+Bi,j=f(3)(ξi,j)
+6(X′
+iXj/p)31i/negationslash=j.
+The “off-diagonal” matrix is the sum A+B.
+(A)Study of the off-diagonal matrix.
+•Truncation and centralization. Following the arguments of Lemma 2.2
+in [45], we see that because we have assumed that we have 4+ εabsolute
+moments, and n≍p, the array Y=Y1≤i≤n,1≤j≤pis almost surely equal to
+the array/tildewideYof same dimensions with
+/tildewideYi,j=Yi,j1|Yi,j|≤BpwhereBp=p1/2−δandδ>0.
+We will therefore carry out the analysis on this /tildewideYarray. Note that most of
+the results we will rely on require vectors of i.i.d. entries with mean 0. Of
+course,/tildewideYi,jhas in general a mean different from 0. In other words, if we cal l
+µ=E(/tildewideYi,j), we need to show that we do not lose anything in operator norm
+by replacing/tildewideYi’s byUi’s withUi=/tildewideYi−µ1. Note that, as seen in Lemma A.3,
+by plugging in t=1/2−δin the notation of this lemma, which corresponds
+to the 4+εmoment assumption here, we have
+|µ|≤p−3/2−δ.
+Now let us call Sthe matrix XX′/p, except that its diagonal is replaced
+by zeros. From [ 45], and the fact that n/pstays bounded, we know that
+/bardbl|XX′/p/bardbl|2≤σ1(Σ)/bardbl|YY′/bardbl|2/pstays bounded. Using Lemma A.3, we see
+that the diagonal of XX′/pstays bounded a.s. in operator norm. Therefore,
+/bardbl|S/bardbl|2is bounded a.s.
+Now, as in the proof of Lemma A.3, we have
+Si,j=U′
+iΣUj
+p+µ/parenleftbigg1′ΣUj
+p+1′ΣUi
+p/parenrightbigg
++µ21′Σ1
+p/definesU′
+iΣUj
+p+Ri,ja.s.
+Note that this equality is true a.s. only because it involves replacingYby
+/tildewideY. The proof of Lemma A.3shows that
+|Ri,j|≤µ2σ1/2
+1(Σ)(σ1/2
+1(Σ)+p−δ/2)+µ2σ1(Σ) a.s.
+We conclude that, for some constant C,
+/bardblR/bardbl2
+F≤Cn2µ2≤Cn2p−3−2δ→0 a.s.
+Therefore /bardbl|R/bardbl|2→0 a.s. In other words, if we call SUthe matrix with i,j
+entryU′
+iΣUj/poff the diagonal and 0 on the diagonal,
+/bardbl|S−SU/bardbl|2→0 a.s.SPECTRUM OF KERNEL RANDOM MATRICES 13
+Now it is a standard result on Hadamard products (see for inst ance, [10],
+Problem I.6.13, or [ 25], Theorems 5.5.1 and 5.5.15) that for two matrices A
+andB,/bardbl|A◦B/bardbl|2≤/bardbl|A/bardbl|2/bardbl|B/bardbl|2. Since the Hadamard product is commuta-
+tive, we have
+S◦S−SU◦SU=(S+SU)◦(S−SU).
+We conclude that
+/bardbl|S◦S−SU◦SU/bardbl|2≤/bardbl|S−SU/bardbl|2(/bardbl|S/bardbl|2+/bardbl|SU/bardbl|2)→0 a.s.,
+since/bardbl|S−SU/bardbl|2→0 a.s., and /bardbl|S/bardbl|2and hence /bardbl|SU/bardbl|2stay bounded, a.s.
+The conclusion of this study is that to approximate the secon d order term
+in operator norm, it is enough to work with SUand notS, and hence, very
+importantly, with bounded random variables with zero mean. F urther, the
+proof of Lemma A.3makes clear that σ2
+U, the variance of the Ui,j’s goes
+to 1, the variance of the Yi,j’s, very fast. So if we can approximate the
+matrix with ( i,j)-entryU′
+iΣUj/(pσ2
+U) consistently in operator norm by a
+matrix whose operator norm is bounded, this same matrix will constitute
+an operator norm approximation of U′
+iΣUj/p.
+In other words, we can assume that, when working with matrice s of di-
+mensionn×p, the random variables we will be working with have variance
+1 without loss of generality and that they have mean 0 and are b ounded by
+Bp,Bpdepending on pand going to infinity.
+•Control of the second order term. We now focus on approximating in
+operator norm the matrix with ( i,j)th entry,
+f′′(0)
+2(X′
+iXj/p)21i/negationslash=j.
+As we just explained, we assume from now on in all the work conc erning
+the second order term that the vectors Yihave mean 0, and that their entries
+have variance 1 and are bounded by Bp=p1/2−δ. This is because we just
+saw that replacing YibyUi/σUwould not change (a.s. and asymptotically)
+the operator norm of the matrix to be studied. We note that to m ake clear
+that the truncation depends on p, we might have wanted to use the notation
+Y(p)
+i, but since there will be no ambiguity in the proof, we chose to use the
+less cumbersome notation Yi.
+The control of the second order term turns out to be the most de licate
+part of the analysis, and the only place where we need the assu mption that
+Xi=Σ1/2Yi. Let us call Wthe matrix with entries
+Wi,j=
+
+(X′
+iXj)2
+p2,ifi/ne}ationslash=j,
+0, ifi=j.14 N. EL KAROUI
+Note that when i/ne}ationslash=j,
+E(Wi,j)=E(trace(X′
+iXjX′
+jXi))/p2=E(trace(XjX′
+jXiX′
+i))/p2
+=trace(Σ2)/p2.
+Becauseweassumethat trace(Σ) /phasafinitelimit,and n/pstaysbounded
+away from 0, we see that the matrix E(W) has a largest eigenvalue that, in
+general, does not go to 0. Note also that under our assumption s,E(Wi,j)=
+O(1/p). Our aim is to show that Wcan be approximated in operator norm
+by this constant matrix. So let us consider the matrix /tildewiderWwith entries
+/tildewiderWi,j=
+
+(X′
+iXj)2
+p2−trace(Σ2)/p2,ifi/ne}ationslash=j,
+0, ifi=j.
+Simple computations show that the expected Frobenius norm s quared of
+this matrix does not go to 0. Hence more subtle arguments are n eeded to
+control its operator norm. We will show that E(trace(/tildewiderW4)) goes to zero
+which implies that E(/bardbl|/tildewiderW/bardbl|4
+2) goes to zero because /tildewiderWis real symmetric.
+Theelementscontributingtotrace( /tildewiderW4)aregenerallyoftheform /tildewiderWi,j/tildewiderWj,k×
+/tildewiderWk,l/tildewiderWl,i. We are going to study these terms according to how many indic es
+are equal to each other.
+(i)Terms involving 4 different indices: i/ne}ationslash=j/ne}ationslash=k/ne}ationslash=l.We first focus on
+the case where all these indices ( i,j,k,l) are different. Recall that Xi=
+Σ1/2Yi,whereYihasi.i.d.entries.Wewanttocompute E(/tildewiderWi,j/tildewiderWj,k/tildewiderWk,l/tildewiderWl,i),
+so it is natural to focus first on
+E(/tildewiderWi,j/tildewiderWj,k/tildewiderWk,l/tildewiderWl,i|Yi,Yk).
+Now, note that
+/tildewiderWi,j=1
+p2{Y′
+iΣYjY′
+jΣYi−trace(Σ2)}
+=1
+p2{Y′
+iΣ(YjY′
+j−Id)ΣYi+trace(Σ2(YiY′
+i−Id))}.
+Hence, calling
+Mj/definesYjY′
+j−Id,
+we have
+p4/tildewiderWi,j/tildewiderWj,k=(Y′
+iΣMjΣYiY′
+kΣMjΣYk)+(Y′
+iΣMjΣYi)trace(Σ2Mk)
++(Y′
+kΣMjΣYk)trace(Σ2Mi)+trace(Σ2Mi)trace(Σ2Mk).SPECTRUM OF KERNEL RANDOM MATRICES 15
+Now, of course, we have E(Mj)=E(Mj|Yi,Yk)=0. Hence,
+p4E(/tildewiderWi,j/tildewiderWj,k|Yi,Yk)=(Y′
+iΣE(MjΣYiY′
+kΣMj|Yi,Yk)ΣYk)
++trace(Σ2Mi)trace(Σ2Mk).
+IfMis a deterministic matrix, we have, since E(YjY′
+j)=Id,
+E(MjMMj)=E(YjY′
+jMYjY′
+j)−M.
+If we now use Lemma A.1, and, in particular, ( 4), page41, we finally have,
+recalling that here σ2=1,
+E(MjMMj)=(M+M′)+(µ4−3)diag(M)+trace(M)Id−M
+=M′+(µ4−3)diag(M)+trace(M)Id.
+In the case of interest here, we have M=ΣYiY′
+kΣ, and the expectation is
+to be understood conditionally on Yi,Yk, but because we have assumed that
+the indices are different and the Ym’s are independent, we can do the compu-
+tation of the conditional expectation as if Mwere deterministic. Therefore,
+we have
+(Y′
+iΣE(MjΣYiY′
+kΣMj|Yi,Yk)ΣYk)
+=Y′
+iΣ[ΣYkY′
+iΣ+(µ4−3)diag(ΣYiY′
+kΣ)+(Y′
+kΣ2Yi)Id]ΣYk
+=[(Y′
+iΣ2Yk)2+(µ4−3)Y′
+iΣdiag(ΣYiY′
+kΣ)ΣYk+(Y′
+iΣ2Yk)2].
+Naturally,wehave E(/tildewiderWi,j/tildewiderWj,k|Yi,Yk)=E(/tildewiderWk,l/tildewiderWl,i|Yi,Yk),andtherefore,by
+using properties of conditional expectation, since all the indices are different,
+p8E(/tildewiderWi,j/tildewiderWj,k/tildewiderWk,l/tildewiderWl,i)
+=E([2(Y′
+iΣ2Yk)2+(µ4−3)Y′
+iΣdiag(ΣYiY′
+kΣ)ΣYk
++trace(Σ2Mi)trace(Σ2Mk)]2).
+By convexity, we have ( a+b+c)2≤3(a2+b2+c2), so to control the above
+expression, we just need to control the square of each of the t erms appearing
+in it. In other words, we need to understand the terms
+T1=E((Y′
+iΣ2Yk)4),
+T2=E([Y′
+iΣdiag(ΣYiY′
+kΣ)ΣYk]2)
+and
+T3=E([trace(Σ2Mi)trace(Σ2Mk)]2).
+Study ofT1.Let us start by the term T1=E((Y′
+iΣ2Yk)4). A simple re-
+writing shows that
+(Y′
+iΣ2Yk)4=Y′
+iΣ2YkY′
+kΣ2YiY′
+iΣ2YkY′
+kΣ2Yi.16 N. EL KAROUI
+Using (4) in Lemma A.1, we therefore have, using the fact that Σ2YiY′
+iΣ2
+is symmetric,
+E((Y′
+iΣ2Yk)4|Yi)
+=Y′
+iΣ2[2Σ2YiY′
+iΣ2+(µ4−3)diag(Σ2YiY′
+iΣ2)
++trace(Σ2YiY′
+iΣ2)Id]Σ2Yi
+=3(Y′
+iΣ4Yi)2+(µ4−3)Y′
+iΣ2diag(Σ2YiY′
+iΣ2)Σ2Yi.
+Finally, we have, using ( 5) in Lemma A.1,
+E((Y′
+iΣ2Yk)4)=3[2trace(Σ4)+(trace(Σ4))2+(µ4−3)trace(Σ4◦Σ4)]
++(µ4−3)E(Y′
+iΣ2diag(Σ2YiY′
+iΣ2)Σ2Yi).
+Now we have
+Y′
+iΣ2diag(Σ2YiY′
+iΣ2)Σ2Yi=trace(Σ2YiY′
+iΣ2diag(Σ2YiY′
+iΣ2))
+=trace(Σ2YiY′
+iΣ2◦Σ2YiY′
+iΣ2).
+Callingvi=Σ2Yi, we note that the matrix whose trace is taken is ( viv′
+i)◦
+(viv′
+i)=(vi◦vi)(vi◦vi)′(see [24], page 458 or [ 25], page 307). Hence,
+Y′
+iΣ2diag(Σ2YiY′
+iΣ2)Σ2Yi=/bardblvi◦vi/bardbl2
+2.
+Now let us call mkthekth column of the matrix Σ2. Using the fact that
+Σ2is symmetric, we see that the kth entry of the vector viisvi(k)=m′
+kYi.
+Sovi(k)4=Y′
+imkm′
+kYiY′
+imkm′
+kYi. Calling Mk=mkm′
+k, we see, using ( 5) in
+LemmaA.1, that
+E(vi(k)4)=2trace( M2
+k)+[trace( Mk)]2+(µ4−3)trace(Mk◦Mk).
+Using the definition of Mk, we finally get that
+E(vi(k)4)=3/bardblmk/bardbl4
+2+(µ4−3)/bardblmk◦mk/bardbl2
+2.
+Now, note that if Cis a generic matrix and Ckis itskth column, denoting
+byekthekth vector of the canonical basis, we have Ck=Cek, and hence
+/bardblCk/bardbl2
+2=e′
+kC′Cek≤σ2
+1(C) whereσ1(C) is the largest singular value of C.
+So in particular, if we call λ1(D) the largest eigenvalue of a positive semi-
+definite matrix D, we have /bardblmk/bardbl4
+2≤λ1(Σ4)/bardblmk/bardbl2
+2.
+After recalling the definition of mk, and using the fact that/summationtext
+k/bardblmk◦
+mk/bardbl2
+2=/bardblΣ2◦Σ2/bardbl2
+F, we deduce that
+E(/bardblvi◦vi/bardbl2
+2)=3/summationdisplay
+k/bardblmk/bardbl4
+2+(µ4−3)/summationdisplay
+k/bardblmk◦mk/bardbl2
+2
+≤3λ1(Σ4)trace(Σ4)+(µ4−3)trace([Σ2◦Σ2]2).SPECTRUM OF KERNEL RANDOM MATRICES 17
+Therefore, we can conclude that
+E((Y′
+iΣ2Yk)4)≤3λ1(Σ4)trace(Σ4)+(µ4−3)trace([Σ2◦Σ2]2).
+Now recall that, according to Theorem 5.5.19 in [ 25], ifCandDare positive
+semidefinite matrices, λ(C◦D)≺wd(C)◦λ(D) whereλ(D) is the vector
+of decreasingly ordered eigenvalues of D, andd(C) denotes the vector of
+decreasingly ordered diagonal entries of C(because all the matrices are
+positive semidefinite, their eigenvalues are their singula r values). Here ≺w
+denotes weak (sub)majorization. In our case, of course, C=D=Σ2. Using
+the results of Example II.3.5(iii) in [ 10], with the function φ(x)=x2, we see
+that
+trace((Σ2◦Σ2)2)=/summationdisplay
+λ2
+i(Σ2◦Σ2)≤/summationdisplay
+d2
+i(Σ2)λ2
+i(Σ2)≤λ1(Σ4)trace(Σ4).
+Finally, we have
+T1=E((Y′
+iΣ2Yk)4)≤(3+|µ4−3|)λ1(Σ4)trace(Σ4). (1)
+This bounds the first term, T1, in our upper bound.
+Study ofT3.Letusnowturntothethirdterm, T3=E([trace(Σ2Mi)trace×
+(Σ2Mk)]2). We remind the reader that Mi=YiY′
+i−Id. By independence of
+YiandYk, it is enough to understand E([trace(Σ2Mi)]2). Note that
+E([trace(Σ2Mi)]2)=E([Y′
+iΣ2Yi−trace(Σ2)]2)
+=E(Y′
+iΣ2YiY′
+iΣ2Yi)−trace(Σ2)2.
+Using (5) in Lemma A.1, we conclude that
+E([trace(Σ2Mi)]2)=2trace(Σ4)+(µ4−3)trace(Σ2◦Σ2).
+Using the fact that we know the diagonal of Σ2◦Σ2, we conclude that
+T3=E([trace(Σ2Mi)]2[trace(Σ2Mk)]2)
+(2)
+≤{2trace(Σ4)+|µ4−3|λ1(Σ2)trace(Σ2)}2.
+So we have an upper bound on T3.
+Study ofT2.Finally,letusturntothemiddleterm, T2=E([Y′
+iΣdiag(ΣYiY′
+k×
+Σ)ΣYk]2). Before we square it, the argument of the expectation has th e form
+Y′
+iΣdiag(ΣYkY′
+iΣ)ΣYk. Calluk= ΣYk. Making the same computations as
+above, we find that
+Y′
+iΣdiag(ΣYkY′
+iΣ)ΣYk=trace(diag(Σ YkY′
+iΣ)ΣYkY′
+iΣ)
+=trace((ΣYkY′
+iΣ)◦(ΣYkY′
+iΣ))
+=trace((uku′
+i)◦(uku′
+i))=trace(( uk◦uk)(ui◦ui)′)
+=(ui◦ui)′(uk◦uk).18 N. EL KAROUI
+Wededuce,usingindependenceandelementarypropertiesof innerproducts,
+that
+E([Y′
+iΣdiag(ΣYkY′
+iΣ)ΣYk]2)≤E(/bardblui◦ui/bardbl2
+2)E(/bardbluk◦uk/bardbl2
+2).
+Note that to arrive at ( 1), we studied expressions similar to E(/bardblui◦ui/bardbl2
+2).
+So we can similarly conclude that
+T2=E([Y′
+iΣdiag(ΣYkY′
+iΣ)ΣYk]2)≤{(3+|µ4−3|)λ1(Σ2)trace(Σ2)}2.
+(3)
+With our assumptions, the terms ( 1), (2) and (3) are O(p2). Note that in
+the computation of the trace, there are O( n4) such terms. Finally, note
+that the expectation of interest to us corresponds to the sum of the three
+quadratic terms divided by p8. So the total contribution of these terms is
+in expectation O( p−2). This takes care of the contribution of the terms
+involving four different indices, as it shows that
+0≤E/parenleftbigg/summationdisplay
+i/negationslash=j/negationslash=k/negationslash=l/tildewiderWi,j/tildewiderWj,k/tildewiderWk,l/tildewiderWl,i/parenrightbigg
+=O(p−2).
+(ii)Terms involving three different indices: i/ne}ationslash=j/ne}ationslash=k.Note that because
+/tildewiderWi,i=0, terms involving 3 different indices with a nonzero contri bution are
+necessarily of the form ( /tildewiderWi,j)2(/tildewiderWi,k)2, since terms with a cycle of length 3
+all involve a term of the form /tildewiderWi,iand hence contribute 0. Let us now fo-
+cus on those terms, assuming that j/ne}ationslash=k. Note that we have O( n3) such
+terms and that it is enough to focus on the W2
+i,jW2
+i,k, since the contri-
+bution of the other terms is, in expectation, of order 1 /p4[with our as-
+sumptions trace(Σ2)/p2=O(1/p)], and because we have only n3terms in
+the sum, this extra contribution is asymptotically zero. No w, we clearly
+haveE(W2
+i,jW2
+i,k|Yi)=[E(W2
+i,j|Yi)]2, by conditional independence of the two
+terms. The computation of E(W2
+i,j|Yi) is similar to the ones we have made
+above, and we have
+p4E(W2
+i,j|Yi)=2(Y′
+iΣ2Yi)2+(µ4−3)Y′
+iΣdiag(ΣYiY′
+iΣ)ΣYi
++(trace(ΣYiY′
+iΣ))2.
+Using the fact that Ki=ΣYiY′
+iΣ is positive semidefinite, and hence its di-
+agonal entries are nonnegative, we have trace( Ki◦Ki)≤(trace(Ki))2, and
+we conclude that
+p4E(W2
+i,j|Yi)≤(3+|κ4−3|)(Y′
+iΣ2Yi)2≤(3+|κ4−3|)σ1(Σ)4/bardblYi/bardbl4
+2.
+Hence,
+E(W2
+i,jW2
+i,k)≤1
+p8(3+|κ4−3|)2σ1(Σ)8/bardblYi/bardbl8
+2.SPECTRUM OF KERNEL RANDOM MATRICES 19
+Now, the application Fwhich takes a vector and returns its Euclidean norm
+is trivially a convex 1-Lipschitz function, with respect to Euclidean norm.
+Because the entries of Yiare bounded by Bp, we see that, according to
+Corollary 4.10 in [ 29],F(Yi) =/bardblYi/bardbl2satisfies a concentration inequality,
+namely, for r>0,P(|/bardblYi/bardbl2−mF|>r)≤4exp(−r2/16B2
+p) wheremFis a
+median ofF(Yi)=/bardblYi/bardbl2(hencemFis a deterministic quantity). A simple
+integration (see, for instance, the proof of Proposition 1. 9 in [29], and change
+the power from 2 to 8) then shows that
+E(|/bardblYi/bardbl2−mF|8)=O(B8
+p).
+Now we know, according to Proposition 1.9 in [ 29], that ifµFis the mean
+ofF(Yi), that is,µF=E(/bardblYi/bardbl2),µFexists and |mF−µF|=O(Bp). Since
+µ2
+F≤µF2=E(/bardblYi/bardbl2
+2)=p, we conclude that, if Cdenotes a generic constant
+that may change from display to display,
+E(/bardblYi/bardbl8
+2)≤E(|/bardblYi/bardbl2−mF+mF|8)≤27(E(|/bardblYi/bardbl2−mF|8)+m8
+F)
+≤C(E(|/bardblYi/bardbl2−mF|8)+|mF−µF|8+µ8
+F)≤C(B8
+p+p4).
+Now our original assumption about the number of absolute mom ents of the
+random variables of interest imply that Bp=O(p1/2−δ). Consequently,
+E(/bardblYi/bardbl8
+2)=O(p4).
+Therefore,
+E(W2
+i,jW2
+i,k)=O(p−4)
+and
+/summationdisplay
+i/summationdisplay
+j/negationslash=i,k/negationslash=i,j/negationslash=kE(W2
+i,jW2
+i,k)=O(p−1).
+Hence, we also have
+/summationdisplay
+i/summationdisplay
+j/negationslash=i,k/negationslash=i,j/negationslash=kE(/tildewiderW2
+i,j/tildewiderW2
+i,k)=O(p−1).
+(iii)Terms involving two different indices: i/ne}ationslash=j.The last terms we have
+to focus on to control E(trace(/tildewiderW4)) are of the form /tildewiderW4
+i,j. Note that we have
+n2terms like this. Since by convexity, ( a+b)4≤8(a4+b4), we see that it
+is enough to understand the contribution of W4
+i,jto show that/summationtext
+i,jE(/tildewiderW4
+i,j)
+tendstozero.Now,letuscallforamoment v=ΣYiandu=Yj.Thequantity
+of interest to us is basically of the form E((u′v)8). Let us do computations
+conditional on v. We note that since the entries of uare independent and
+have mean 0, in the expansion of ( u′v)8, the only terms that will contribute
+a nonzero quantity to the expectation have entries of uraised to a power20 N. EL KAROUI
+greater than 2. We can decompose the sum representing E((u′v)8|v) into
+subterms, according to what powers of the terms are involved . There are 6
+terms: (2,2,2,2) (i.e., all terms are raised to the power 2), (3 ,3,2) (i.e., two
+terms are raised to the power 3, and one to the power 2), (4 ,2,2), (4,4),
+(5,3), (6,2) and (8). For instance the subterm corresponding to (2 ,2,2,2)
+is, before taking expectations,
+/summationdisplay
+i1/negationslash=i2/negationslash=i3/negationslash=i4u2
+i1u2
+i2u2
+i3u2
+i4(vi1vi2vi3vi4)2.
+After taking expectations conditional on v, we see that it is obviously non-
+negative and contributes
+(σ2)4/summationdisplay
+i1/negationslash=i2/negationslash=i3/negationslash=i4(vi1vi2vi3vi4)2≤/parenleftBig/summationdisplay
+v2
+i/parenrightBig4
+=(Y′
+iΣ2Yi)4≤σ1(Σ)8/bardblYi/bardbl8
+2.
+Note that we just saw that E(/bardblYi/bardbl8
+2)=O(p4) in our context. Similarly, the
+term (3,3,2) will contribute
+µ2
+3σ2/summationdisplay
+i1/negationslash=i2/negationslash=i3v3
+i1v3
+i2v2
+i3.
+In absolute value, this term is less than
+µ2
+3σ2/parenleftBig/summationdisplay
+|vi|3/parenrightBig2/parenleftBig/summationdisplay
+v2
+i/parenrightBig
+.
+Now,notethatif zissuchthat /bardblz/bardbl2=1,wehave,for p≥2,/summationtext|zi|p≤/summationtextz2
+i=
+1.Appliedto z=v//bardblv/bardbl2,weconcludethat/summationtext|vi|p≤/bardblv/bardblp
+2.Consequently,the
+term (3,3,2) contributes in absolute value less than
+µ2
+3σ2/bardblv/bardbl8
+2.
+The same analysis can be repeated for all the other terms whic h are all found
+to be less than /bardblv/bardbl8
+2times the moments of uinvolved. Because we have
+assumed that our original random variables had 4+ εabsolute moments,
+the moments of order less than 4 cause no problem. The moments of order
+higher than 4, say 4+ k, can be bounded by µ4Bk
+p. Consequently, we see
+that
+E(W4
+i,j)=E(E(W4
+i,j|Yi))≤CB4
+pE/parenleftbigg/bardblYi/bardbl8
+p8/parenrightbigg
+=O(B4
+p/p4)=O(p−(2+4δ)).
+Since we have n2such terms, we see that
+/summationdisplay
+i/negationslash=jE(W4
+i,j)→0 asp→∞.
+Using our earlier convexity remark, we finally conclude that
+/summationdisplay
+i/negationslash=jE(/tildewiderW4
+i,j)→0 asp→∞.SPECTRUM OF KERNEL RANDOM MATRICES 21
+(iv)Second order term: combining all the elements. We have therefore
+established control of the second order term and seen that th e largest sin-
+gular value of /tildewiderWgoes to 0 in probability, using Chebyshev’s inequality.
+Note that we have also shown that the operator norm of Wis bounded in
+probability and that
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsingle/vextendsingleW−trace(Σ2)
+p2(11′−Id)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+2→0 in probability.
+•Control of the third order term. We note that the third order term is
+of the form f(3)(ξi,j)X′
+iXj
+pWi,j. According to Lemma A.5, ifMis a real
+symmetric matrix with nonnegative entries, and Eis a symmetric matrix
+such that max i,j|Ei,j|=ζ, then
+σ1(E◦M)≤ζσ1(M).
+Note thatWis real symmetric matrix with nonnegative entries. So all
+we have to show to prove that the third order term goes to zero i n operator
+norm is that max i/negationslash=j|X′
+iXj/p|goes to 0 because we have just established
+that/bardbl|W/bardbl|2remains bounded in probability. We are going to make use of
+LemmaA.3, page45in theAppendix . In our setting, we have Bp=p1/2−δ,
+or 2/m=1/2−δ. The lemma implies, for instance, that
+max
+i/negationslash=j|X′
+iXj/p|≤p−δlog(p) a.s.
+So max i/negationslash=j|X′
+iXj/p|→0 a.s. Note that this implies that max i/negationslash=j|ξi,j|→0 a.s.
+Since we have assumed that f(3)exists and is continuous and hence bounded
+in a neighborhood of 0, we conclude that
+max
+i,j|f(3)(ξi,j)X′
+iXj/p|=o(p−δ/2) a.s.
+If we callEthe matrix with entry Ei,j=f(3)(ξi,j)X′
+iXj/poff-the diagonal
+and 0 on the diagonal, we see that Esatisfies the conditions put forth in
+our discussion earlier in this section and we conclude that
+/bardbl|E◦W/bardbl|2≤max
+i,j|Ei,j|/bardbl|W/bardbl|2=o(p−δ/2) a.s.
+Hence, the operator norm of the third order term goes to 0 almo st surely.
+[To maybe clarify our arguments, let us repeat that we analyz ed the second
+order term by replacing the Yi’s by, in the notation of the truncation and
+centralization discussion, Ui. Let us call WU=SU◦SU, again using notation
+introducedinthetruncation andcentralizationdiscussio n. Aswesaw, /bardbl|W−
+WU/bardbl|2→0 a.s., so showing, as we did, that /bardbl|WU/bardbl|2remains bounded (a.s.)
+implies that /bardbl|W/bardbl|2does too, and this is the only thing we need in our
+argument showing the control of the third order term.]22 N. EL KAROUI
+(B)Control of the diagonal term. The proof here is divided into two
+parts. First, we show that the error term coming from the first order ex-
+pansion of the diagonal is easily controlled. Then we show th at the terms
+added when replacing the off-diagonal matrix by XX′/p+trace(Σ2)/p211′
+can also be controlled. Recall the notation τ=trace(Σ)/p.
+•Errors induced by diagonal approximation. Note that Lemma A.3guar-
+antees that for all i,|ξi,i−τ|≤p−δ/2, a.s. Because we have assumed that f′
+is continuous and hence bounded in a neighborhood of τ, we conclude that
+f′(ξi,i) is uniformly bounded in p. Now Lemma A.3also guarantees that
+max
+i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bardblXi/bardbl2
+2
+p−τ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤p−δa.s.
+Hence, the diagonal matrix with entries f(/bardblXi/bardbl2
+2/p) can be approximated
+consistently in operator norm by f(τ)Id a.s.
+•Errors induced by off-diagonal approximation. When we replace the off-
+diagonal matrix by f′(0)XX′/p+[f(0)+f′′(0)trace(Σ2)/2p2]11′, we add a
+diagonal matrix with ( i,i) entryf(0)+f′(0)/bardblXi/bardbl2
+2/p+f′′(0)trace(Σ2)/2p2
+which we need to subtract eventually. We note that 0 ≤trace(Σ2)/p2≤
+σ2
+1(Σ)/p→0 whenσ1(Σ) remains bounded in p. So this term does not create
+anyproblem.Now,wejustsawthatthediagonalmatrixwithen tries/bardblXi/bardbl2
+2/p
+can be consistently approximated in operator norm by (trace (Σ)/p)Id. So
+thediagonalmatrixwith( i,i)entryf(0)+f′(0)/bardblXi/bardbl2
+2/p+f′′(0)trace(Σ2)/2p2
+can be approximated consistently in operator norm by ( f(0)+f′(0)trace ×
+(Σ)/p)Id a.s.
+This finishes the proof. /square
+2.3.Kernel random matrices of the type f(/bardblXi−Xj/bardbl2
+2/p).As is to be
+expected, the properties of such matrices can be deduced fro m the study of
+inner product kernel matrices, with a little bit of extra wor k. We need to
+slightly modify the distributional assumptions under whic h we work, and
+consider the case where we have 5+ εabsolute moments for the entries of
+Yi. We also need to assume that fis regular is the neighborhood of different
+points. Otherwise, the assumptions are the same as that of Th eorem2.1.
+We have the following theorem:
+Theorem 2.2 (Spectrum of Euclidean distance kernel matrices). Con-
+sider then×nkernel matrix Mwith entries
+Mi,j=f/parenleftbigg/bardblXi−Xj/bardbl2
+2
+p/parenrightbigg
+.SPECTRUM OF KERNEL RANDOM MATRICES 23
+Let us call
+τ=2trace(Σ)
+p.
+Let us callψthe vector with ith entryψi=/bardblXi/bardbl2
+2/p−trace(Σ)/p. Suppose
+that the assumptions of Theorem 2.1hold, but that conditions (e)and(f)
+are replaced by:
+(e′)The entries of Yi, ap-dimensional random vector, are i.i.d. Also, de-
+noting byYi(k)thekth entry of Yi, we assume that E(Yi(k)) = 0,
+var(Yi(k)) = 1andE(|Yi(k)|5+ε)<∞for someε>0. (We say that
+Yihas5+εabsolute moments.)
+(f′)fisC3in a neighborhood of τ.
+ThenMcan be approximated consistently in operator norm (and in pr ob-
+ability) by the matrix K, defined by
+K=f(τ)11′+f′(τ)/bracketleftbigg
+1ψ′+ψ1′−2XX′
+p/bracketrightbigg
++f′′(τ)
+2/bracketleftbigg
+1(ψ◦ψ)′+(ψ◦ψ)1′+2ψψ′+4trace(Σ2)
+p211′/bracketrightbigg
++υpId,
+υp=f(0)+τf′(τ)−f(τ).
+In other words,
+/bardbl|M−K/bardbl|2→0in probability.
+Proof. Note that here the diagonal is just f(0)Id and it will cause no
+trouble. The work, therefore, focuses on the off-diagonal ma trix. In what
+follows, we call τ=2trace(Σ)
+p. Let us define
+Ai,j=/bardblXi/bardbl2
+2
+p+/bardblXj/bardbl2
+2
+p−τ
+and
+Si,j=X′
+iXj
+p.
+With these notation, we have, off the diagonal, that is, when i/ne}ationslash=j, by a
+Taylor expansion,
+Mi,j=f(τ)+[Ai,j−2Si,j]f′(τ)+1
+2[Ai,j−2Si,j]2f′′(τ)
++1
+6f(3)(ξi,j)[Ai,j−2Si,j]3.24 N. EL KAROUI
+Wenotethatthematrix Awithentries Ai,jisarank2matrix.Asamatter
+of fact, it can be written, if ψis the vector with entries ψi=/bardblXi/bardbl2
+2
+p−τ/2,
+A= 1ψ′+ψ1′. Using the well-known identity (see, e.g., [ 23], Chapter 1,
+Theorem 3.2),
+det(I+uv′+vu′)=det/parenleftbigg
+1+u′v/bardblu/bardbl2
+2
+/bardblv/bardbl2
+21+u′v/parenrightbigg
+;
+we see immediately that the nonzero eigenvalues of Aare
+1′ψ±√n/bardblψ/bardbl2.
+After these preliminary remarks, we are ready to start the pr oof per se.
+•Truncation and centralization. Since we assume 5 + εabsolute mo-
+ments, we see, using Lemma 2.2 in [ 45], that we can truncate the Yi’s at
+levelBp=p2/5−δwithδ>0 and a.s. not change the data matrix. We then
+needtocentralizethevectorstruncatedat p2/5−δ.Notethatbecausewework
+withXi−Xj=Σ1/2(Yi−Yj), centralization creates absolutely no problem
+here since it is absorbed in the difference. So in what follows we can assume
+without loss of generality that we are working with vectors Xi= Σ1/2Yi
+where the entries of Yiare bounded by p2/5−δandE(Yi)=0. The issue of
+variance 1 is addressed as before, so we can assume that the en tries ofYi
+have variance 1.
+•Concentration of /bardblXi−Xj/bardbl2
+2/p.By plugging in the results of Corol-
+laryA.2, with 2/m=2/5−δ, we get that
+max
+i/negationslash=j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bardblXi−Xj/bardbl2
+2
+p−2trace(Σ)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤log(p)p−1/10−δ.
+Also, using the result of Lemma A.3, we have
+max
+i|ψi|=max
+i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bardblXi/bardbl2
+2
+p−trace(Σ)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤log(p)p−1/10−δ.
+Note that, as explained in the proof of Lemma A.3, these results are true
+whether we work with Yior their truncated and centralized version.
+•Control of the second order term. The second order term is the matrix
+with (i,j)-entry
+1i/negationslash=j1
+2f′′(τ)(Ai,j−Si,j)2.
+Let us call Tthe matrix with 0 on the diagonal and off-diagonal entries
+Ti,j=(Ai,j−2Si,j)2. In other words,
+Ti,j=1i/negationslash=j/parenleftbigg/bardblXi−Xj/bardbl2
+2−2trace(Σ)
+p/parenrightbigg2
+.SPECTRUM OF KERNEL RANDOM MATRICES 25
+We simply write ( Ai,j−2Si,j)2=A2
+i,j−4Ai,jSi,j+4S2
+i,j. In the notation
+of the proof of Theorem 2.1, the matrix with entries S2
+i,joff the diagonal
+and 0 on the diagonal is what we called W. We have already shown that
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsingle/vextendsingleW−trace(Σ2)
+p2(11′−Id)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+2→0 in probability .
+Now, let us focus on the term Ai,jSi,j. Let us call Hthe matrix with
+Hi,j=(1−δi,j)Ai,jSi,j.
+Let us denote by /tildewideSthe matrix with off-diagonal entries Si,jand 0 on the
+diagonal. If we call S=XX′/p, we have
+/tildewideS=S−diag(S).
+Now note that Ai,j=ψi+ψj. Therefore, we have, if diag( ψ) is the diagonal
+matrix with ( i,i) entryψi,
+H=/tildewideSdiag(ψ)+diag(ψ)/tildewideS.
+We just saw that under our assumptions, max i|ψi|→0 a.s. Because for any
+n×nmatricesL1,L2,/bardbl|L1L2/bardbl|2≤/bardbl|L1/bardbl|2/bardbl|L2/bardbl|2, we see that to show that
+/bardbl|H/bardbl|2goes to 0, we just need to show that /bardbl|/tildewideS/bardbl|2remains bounded.
+Now we clearly have, /bardbl|S/bardbl|2≤/bardbl|Σ/bardbl|2/bardbl|Y′Y/p/bardbl|2. We know from [ 45] that
+/bardbl|Y′Y/p/bardbl|2→σ2(1+/radicalbig
+n/p)2, a.s. Under our assumptions on nandp, this
+is bounded. Now
+diag(S)=diag(ψ)+trace(Σ)
+pId,
+soourconcentrationresultsonceagainimplythat /bardbl|diag(S)/bardbl|2≤trace(Σ)/p+
+ηa.s., for any η>0. Because /bardbl|·/bardbl|2is subadditive, we finally conclude that
+/bardbl|/tildewideS/bardbl|2is bounded a.s.
+Therefore,
+/bardbl|H/bardbl|2→0 a.s.
+Putting together all these results, we see that we have shown that
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsingle/vextendsingleT−(A◦A−diag(A◦A))−4trace(Σ2)
+p2(11′−Id)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+2→0 in probability .
+•Control of the third order term. The third order term is the matrix L
+with 0 on the diagonal and off-diagonal entries
+Li,j=f(3)(ξi,j)
+6/parenleftbigg/bardblXi−Xj/bardbl2
+2−2trace(Σ)
+p/parenrightbigg3
+/definesE◦T,26 N. EL KAROUI
+whereTwas the matrix investigated in the control of the second orde r term.
+On the other hand, Eis the matrix with entries
+Ei,j=(1−δi,j)f(3)(ξi,j)
+6/parenleftbigg/bardblXi−Xj/bardbl2
+2−2trace(Σ)
+p/parenrightbigg
+.
+We have already seen that through concentration, we have
+max
+i/negationslash=j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bardblXi−Xj/bardbl2
+2
+p−2trace(Σ)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤log(p)p−1/10−δa.s.
+This naturally implies that
+max
+i/negationslash=j/vextendsingle/vextendsingle/vextendsingle/vextendsingleξi,j−2trace(Σ)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤log(p)p−1/10−δa.s.
+Soiff(3)isboundedinaneighborhoodof τ,weseethatwithhigh-probability
+so is max i/negationslash=j|f(3)(ξi,j)|. Therefore,
+max
+i/negationslash=j|Ei,j|≤Klog(p)p−1/10−δ.
+We are now in position to apply the Hadamard product argument (see
+LemmaA.5) we used for the control of the third order term in the proof of
+Theorem 2.1. To show that the third order term tends in operator norm to
+0, we hence just need to show that /bardbl|T/bardbl|2remains small compared to the
+bound we just gave on max i,j|Ei,j|. Of course, this is equivalent to showing
+that the matrix that approximates Thas the same property in operator
+norm.
+Clearly, because σ1(Σ) stays bounded, trace(Σ2)/pstays bounded and so
+does/bardbl|trace(Σ2)/p2(11′−Id)/bardbl|2. So we just have to focus on A◦A−diag(A◦
+A). Recall that Ai,i=2(/bardblXi/bardbl2
+2/p−trace(Σ)/p), and soAi,i=2ψi. We have
+already seen that our concentration arguments imply that ma xi|ψi|→0 a.s.
+So/bardbl|diag(A◦A)/bardbl|2=max iψ2
+igoes to 0 a.s. Now,
+A=1ψ′+ψ1′,
+and hence, elementary Hadamard product computations [rely ing onab′◦
+uv′=(a◦u)(b◦v)′] give
+A◦A=1(ψ◦ψ)′+2ψψ′+(ψ◦ψ)1′.
+Therefore,
+/bardbl|A◦A/bardbl|2≤2(√n/bardblψ◦ψ/bardbl2+/bardblψ/bardbl2
+2).
+Using Lemma A.1, and in particular equation ( 5), we see that
+E(ψ2
+i)=2σ4trace(Σ2)
+p2+(µ4−3σ4)trace(Σ◦Σ)
+p2,SPECTRUM OF KERNEL RANDOM MATRICES 27
+and therefore, E(/bardblψ/bardbl2
+2) remains bounded. On the other hand, using Lemma
+2.7 of [5], we see that if we have 5+ εabsolute moments,
+E(ψ4
+i)≤C/parenleftbigg(µ4trace(Σ2))2
+p4+µ5+εB3−ε
+ptrace(Σ4)
+p4/parenrightbigg
+.
+Now recall that we can take Bp=p2/5−δ. Therefore nE(/bardblψ◦ψ/bardbl2
+2) is, at most,
+of orderB3−ε
+p/p. We conclude that
+P(/bardbl|A◦A/bardbl|2>log(p)/radicalBig
+B3−εp/p)→0.
+Note that this implies that
+P(/bardbl|T/bardbl|2>log(p)/radicalBig
+B3−εp/p)→0.
+Now, note that the third order term is of the form E◦T. Because we have
+assumed that we have 5+ εabsolute moments, we have already seen that
+our concentration results imply that
+max
+i/negationslash=j|Ei,j|=O/parenleftBigg
+log(p)/radicalBigg
+B2p
+p/parenrightBigg
+=O(log(p)p−1/10−δ) a.s.
+Using the fact that Thas positive entries and therefore (see Lemma A.5),
+/bardbl|E◦T/bardbl|2≤maxi,j|Ei,j|/bardbl|T/bardbl|2, we conclude that with high-probability,
+/bardbl|E◦T/bardbl|2=O/parenleftBigg
+(log(p))2/radicalBigg
+B5−εp
+p2/parenrightBigg
+=O((log(p))2p−δ′) where δ′>0.
+Hence,
+/bardbl|E◦T/bardbl|2→0 in probability .
+•Adjustment of the diagonal. Toobtainthecompactformoftheapprox-
+imation announced in the theorem, we need to include diagona l terms that
+are not present in the matrices resulting from the Taylor exp ansion. Here,
+we show that the corresponding matrices are easily controll ed in operator
+norm.
+When we replace the zeroth and first order terms by
+f(τ)11′+f′(τ)/bracketleftbigg
+1ψ′+ψ1′−2XX′
+p/bracketrightbigg
+,
+we add to the diagonal the term f(τ) +f′(τ)(2ψi−2/bardblXi/bardbl2
+2/p) =f(τ)−
+2f′(τ)trace(Σ)
+p. In the end, we need to subtract it.28 N. EL KAROUI
+When we replace the second order term by1
+2f′′(τ)[1(ψ◦ψ)′+2ψψ′+(ψ◦
+ψ)1′+4trace(Σ2)
+p211′], we add to the diagonal the diagonal matrix with ( i,i)
+entry,
+2f′′(τ)/bracketleftbigg
+ψ2
+i+trace(Σ2)
+p2/bracketrightbigg
+.
+With our assumptions, max i|ψi| →0 a.s. andtrace(Σ2)
+premains bounded,
+so the added diagonal matrix has operator norm converging to 0 a.s. We
+conclude that we do not need to add it to the correction in the d iagonal of
+the matrix approximating our kernel matrix. /square
+AninterpretationoftheproofsofTheorems 2.1and2.2isthattheyrelyon
+alocal“multiscale”approximationoftheoriginalmatrix( i.e.,thetermsused
+intheentry-wiseapproximationareallofdifferentorderof magnitudes,orat
+different “scales”). However, globally, that is, when looki ng at eigenvalues of
+the matrices and not just at each of their entries there is a bi t of a mixture
+between the scales which creates the difficulties we had to dea l with to
+control the second order term.
+2.3.1.A note on the Gaussian Kernel. The Gaussian kernel corresponds
+tof(x)=exp(−γx) in the notation of Theorem 2.2. We would like to discuss
+it a bit more because of its widespread use in applications.
+The result of Theorem 2.2gives accurate limiting eigenvalue information
+for the case where we renormalize the distances by the dimens ion which
+seems to be implicitly or explicitly what is often done in pra ctice.
+However, it is possible that information about the nonrenor malized case
+might also be of interest in some situations. Let us assume no w that trace(Σ)
+grows to infinity at least as fast as p1/2+2/m+δwhereδ>0 is such that
+1/2 + 2/m+δ<1 which is possible since m≥5 +εhere. We of course
+still assume that its largest singular value, σ1(Σ), remains bounded. Then
+Corollary A.2guarantees that
+min
+i/negationslash=j/bardblXi−Xj/bardbl2
+2
+p>trace(Σ)
+pa.s.
+Hence
+max
+i/negationslash=jexp(−/bardblXi−Xj/bardbl2
+2)≤exp(−trace(Σ)) ≤exp(−p1/2+2/m+δ) a.s.
+Hence, in this case, if Mis our kernel matrix with entries exp( −/bardblXi−
+Xj/bardbl2
+2), we have
+/bardbl|M−Id/bardbl|2≤nexp(−p1/2+2/m+δ) a.s.,
+and the upper bound tends to zero extremely fast.SPECTRUM OF KERNEL RANDOM MATRICES 29
+2.4.More general models. In this subsection, we consider more general
+models than the ones considered above. In particular, we wil l here focus
+on data models for which the vectors Xisatisfy a so-called dimension-free
+concentration inequality. As was shown in [ 19], under these conditions, the
+Marˇ cenko–Pastur equation holds (as well as generalized ve rsions of it). Note
+that these models are more general than the one considered ab ove (the
+proofs in the Appendix illustrate why the standard random matrix models
+can be considered as subcases of this more general class of ma trices) and
+can describe various interesting objects like vectors with certain log-concave
+distributions or vectors sampled in a uniform manner from ce rtain Rieman-
+nian submanifolds of Rpendowed with the canonical Riemannian metric
+inherited from Rp.
+Before we state more precisely the theorem and give examples of distribu-
+tions that satisfy its assumptions, let us give some motivat ion. One potential
+criticism of Theorems 2.1and2.2is that they deal with data models that are
+inherently quite linear. So a natural question is to underst and whether our
+linear approximation result is limited to these linear sett ings. Also, kernel
+methods are often advocated for their handling of nonlinear ities, and though
+the linear case is probably a basic one that needs to be unders tood, a null
+model of sorts, it is important to be able to go beyond it. As we will soon
+see, the next theorems allow us to get results beyond the line ar setting.
+Our generalization of Theorem 2.1to more general distributions is the
+following.
+Theorem 2.3. Consider a triangular “array” of matrices where each
+row of the array consists of a n×pmatrix. We assume these matrices are
+independent. We call Xi,i=1,...,nthe rows of this matrix.
+Suppose the vectors {Xi}n
+i=1∈Rpare i.i.d. mean 0 and have the property
+that for any 1-Lipschitz function F(with respect to Euclidean norm), if mF
+is a median of F(Xi),
+∀t>0P(|F(Xi)−mF|>t)≤Cexp(−ctb)for someb>0,
+whereCis independent of pandcmay depend on pbut is required to satisfy
+c≥p−(1/2−ε)b/2.bis fixed and independent of nandp.
+Consider the n×nkernel random matrix MwithMi,j=f(X′
+iXj/p).
+Assume that p≍n.
+(a)CallΣthe covariance matrix of the Xi’s and assume that σ1(Σ)stays
+bounded and trace(Σ)/phas a limit.
+(b)Suppose that fis a real valued function which is C2around0andC1
+aroundtrace(Σ)/p.30 N. EL KAROUI
+Then the spectrum of this matrix is asymptotically nonrando m and has, a.s.,
+the same limiting spectral distribution as that of
+/tildewiderM=f(0)11′+f′(0)XX′
+p+υpIdn,
+whereυp=f(trace(Σ)
+p)−f(0)−f′(0)trace(Σ)
+p.
+We note that the term f(0)11′does not affect the limiting spectral distri-
+butionof/tildewiderMsincefiniterankperturbationsdonothaveanyeffectonlimit ing
+spectral distributions (see, e.g., [ 3], Lemma 2.2). Therefore, it could be re-
+moved from the approximating matrix, but since it will clear ly be present in
+numerical work and simulations, we chose to leave it in our ap proximation.
+We also note that the limiting distribution of XX′/punder these assump-
+tions has been obtained in [ 19].
+Here are a few examples of models satisfying the distributio nal assump-
+tions stated above. (Unless otherwise noted, b=2 in all these examples.)
+Examples of distributions for which the previous theorem ap plies.
+•Gaussianrandomvariableswith /bardbl|Σ/bardbl|2boundedandtrace(Σ) /pconverges.
+The assumptions of the theorem apply according to [ 29], Theorem 2.7,
+withc(p)=1//bardbl|Σ/bardbl|2.
+•Vectors of the type√prwhereris uniformly distributed on the unit
+(ℓ2-) sphere is dimension p. Theorem 2.3 in [ 29] shows that Theorem 2.3
+applies, with c(p)=(1−1/p)/2, after noticing that a 1-Lipschitz function
+with respect to Euclidean norm is also 1-Lipschitz with resp ect to the
+geodesic distance on the sphere.
+•Vectors Γ√prwithruniformly distributed on the unit ( ℓ2-)sphere in Rp
+and with ΓΓ′=Σ where Σ satisfies the assumptions of the theorem.
+•Vectors with log-concave density of the type e−U(x)with the Hessian of
+Usatisfying, for all x, Hess(U)≥cIdpwherec>0 has the characteristics
+ofc(p) above (see [ 29], Theorem 2.7). Here we also need /bardbl|Σ/bardbl|2to satisfy
+the assumptions of the theorem.
+•Vectorsrdistributed according to a (centered) Gaussian copula, wit h cor-
+respondingcorrelationmatrix Σ having /bardbl|Σ/bardbl|2bounded.HereTheorem 2.3
+applies, since if ˜ rhas a Gaussian copula distribution, then its ith entry
+satisfies ˜ri=Φ(vi) wherevis multivariate normal with covariance matrix
+Σ, Σ being a correlation matrix, that is, its diagonal is 1. He re Φ is the cu-
+mulative distribution function of a standard normal distri bution which is
+trivially Lipschitz. Now taking r= ˜r−1/2 gives a centered Gaussian cop-
+ula. The fact that the covariance matrix of rthen has bounded operator
+norm requires a bit of work and is shown in the Appendix of [ 19].SPECTRUM OF KERNEL RANDOM MATRICES 31
+•Vectors sampled uniformly from certain compact connected s mooth Rie-
+mannian submanifolds, M, ofRp, canonically equipped with the Rieman-
+nian metric gdefined by restricting to each tangent space the ambient
+scalar product in Rp. The curvature properties of these submanifolds need
+to satisfy the assumptions of Theorem 2.4 in [ 29]. Also, the covariance ma-
+trix Σ need to satisfy the assumptions of Theorem 2.3. We note that since
+the length of a curve in Mis equal to its length in Rp, the same remark
+that we made in the case of the sphere of unit radius applies he re, too.
+In particular, a 1-Lipschitz function with respect to Eucli dean norm is
+1-Lipschitz with respect to the geodesic distance on the man ifold.
+•Vectors of the type p1/br, 1≤b≤2 whereris uniformly distributed in
+the 1-ℓbball or sphere in Rp. (See [29], Theorem 4.21, which refers to
+[34] as the source of the theorem.) We also refer the reader to [ 29], pages
+37 and 38 for some of subtleties involved in the definition of t he uniform
+distribution on the sphere. Fact A.1applies to them, with c(p) depending
+only onb. Also, the concentration function is of the form exp( −c(b)tb)
+here.
+We now turn to the proof of Theorem 2.3. The first step in the proof is
+the following lemma.
+Lemma 2.1. SupposeKnis ann×nreal symmetric matrix, whose spec-
+tral distribution converges weakly (i.e., in distribution ) to a limit. Suppose
+Mnis ann×nreal symmetric matrix.
+1.SupposeMnis such that /bardblMn−Kn/bardblF=o(√n). ThenMnandKnhave
+the same limiting spectral distribution.
+2.SupposeMnis such that /bardbl|Mn−Kn/bardbl|2→0. ThenMnandKnhave the
+same limiting spectral distribution.
+Before we prove the lemma, we note that our assumptions imply that the
+limiting spectral distribution of Knis a probability distribution. Therefore,
+to obtain the results of the lemma, we just need to show pointw ise con-
+vergence of Stieltjes transforms and then rely on the result s of [22], and in
+particular Corollary 1 there.
+Proof of Lemma 2.1.We call St Knand St Mnthe Stieltjes transforms
+of the spectral distributions of these two matrices. Suppos ez=u+iv. Let
+us callli(Mn) theith largest eigenvalue of Mn.
+Proof of statement 1. We first focus on the Frobenius norm part of the
+lemma. We have
+|StKn(z)−StMn(z)|=1
+n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay
+i=11
+li(Kn)−z−1
+li(Mn)−z/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1
+nn/summationdisplay
+i=1|li(Mn)−li(Kn)|
+v2.32 N. EL KAROUI
+Now, by Holder’s inequality,
+n/summationdisplay
+i=1|li(Mn)−li(Kn)|≤√n/radicaltp/radicalvertex/radicalvertex/radicalbtn/summationdisplay
+i=1|li(Mn)−li(Kn)|2.
+Using Lidskii’s theorem [i.e., the fact that, since MnandKnare hermitian,
+the vector with entries li(Mn)−li(Kn) is majorized by the vector li(Mn−
+Kn)], with, in the notation of [ 10], Theorem III.4.4 Φ( x)=x2, we have
+n/summationdisplay
+i=1|li(Mn)−li(Kn)|2≤n/summationdisplay
+i=1l2
+i(Mn−Kn)=/bardblMn−Kn/bardbl2
+F.
+We conclude that
+|StKn(z)−StMn(z)|≤/bardblMn−Fn/bardblF√nv2,
+since|li(Kn)−z|≥|Im[li(Kn)−z]|=v, and therefore 1 /|li(Kn)−z|≤1/v.
+Under the assumptions of the lemma, we therefore have
+|StKn(z)−StMn(z)|→0.
+ThereforetheStieltjestransformofthespectraldistribu tionofMnconverges
+pointwise to the Stieltjes transform of the limiting spectr al distribution of
+Kn. Hence, by, e.g., Corollary 1 in [ 22], the spectral distribution of Mn
+converges in distribution to the limiting spectral distrib ution ofKn, which,
+as noted earlier, is a probability distribution by our assum ptions.
+Proof of statement 2. Let us now turn to the operator norm part of
+the lemma. By the same computations as above, we have, using W eyl’s
+inequality,
+|StKn(z)−StMn(z)|=1
+n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay
+i=11
+li(Kn)−z−1
+li(Mn)−z/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤1
+nn/summationdisplay
+i=1|li(Mn)−li(Kn)|
+v2
+≤/bardbl|Mn−Kn/bardbl|2
+v2.
+Hence if /bardbl|Mn−Kn/bardbl|2→0, it is clear that the two Stieljtes transforms are
+asymptotically equal, and the conclusion follows. /square
+We now turn to the proof of the theorem.SPECTRUM OF KERNEL RANDOM MATRICES 33
+Proof of Theorem 2.3.For the weaker statement required for the
+proof of Theorem 2.3, we will show that in the δ-method we need to keep
+only the first term of the expansion as long as fhas a second derivative that
+is bounded in a neighborhood of 0, and a first derivative that i s bounded
+in a neighborhood of trace(Σ) /p. In other words, we will split the problem
+into two parts: off the diagonal, we write
+f/parenleftbiggX′
+iXj
+p/parenrightbigg
+=f(0)+f′(0)X′
+iXj
+p+f′′(ξi,j)
+2/parenleftbiggX′
+iXj
+p/parenrightbigg2
+ifi/ne}ationslash=j;
+on the diagonal, we write
+f/parenleftbiggX′
+iXi
+p/parenrightbigg
+=f/parenleftbiggtrace(Σ)
+p/parenrightbigg
++f′(ξi,i)/parenleftbiggX′
+iXi
+p−trace(Σ)
+p/parenrightbigg
+.
+•Control of the off-diagonal error matrix. Here we focus on the matrix /tildewiderW
+with (i,j) entry
+/tildewiderWi,j=1i/negationslash=jf′′(ξi,j)
+2/parenleftbiggX′
+iXj
+p/parenrightbigg2
+.
+The strategy is going to be to control the Frobenius norm of th e matrix
+Wi,j=
+
+/parenleftbiggX′
+iXj
+p/parenrightbigg2
+,ifi/ne}ationslash=j,
+0, ifi=j.
+According to Lemma 2.1, it is enough for our needs to show that the Frobe-
+nius norm of this matrix is o(√n) a.s. to have the result we wish. Hence,
+the result will be shown, if we can for instance show that
+max
+i,jWi,j≤p−(1/2+ε)(log(p))1+δa.s.,for someδ>0.
+Now Lemma A.4or FactA.1gives, for instance,
+max
+i/negationslash=j/vextendsingle/vextendsingle/vextendsingle/vextendsingleX′
+iXj
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤(pc2/b(p))−1/2[log(p)]2/ba.s.
+Therefore, with our assumption on c(p), we have
+max
+i,jWi,j≤p−(1/2+ε)(log(p))4/ba.s.
+Now,/bardblW/bardblF≤nmaxi,j|Wi,j|, so we conclude that in this situation, with our
+assumptions that n≍p,
+/bardblW/bardblF=o(√n) a.s.
+Now let us focus on
+/tildewiderWi,j=f′′(ξi,j)Wi,j,34 N. EL KAROUI
+whereξi,jis between 0 and X′
+iXj/p. We just saw that with very high-
+probability,thislatterquantitywasless(inabsoluteval ue)thanp−(1/4+ε/2)×
+(log(p))2/b, ifc≥p−(1/2−ε)b/2. Therefore if f′′is bounded by Kin a neigh-
+borhood of 0, we have, with very high probability that
+/bardbl/tildewiderW/bardblF≤K/bardblW/bardblF=o(√n).
+•Control of the diagonal matrix. We first note that when we replace the
+off-diagonal matrix by f(0)11′+f′(0)XX′/p, we add to the diagonal certain
+terms that we need to subtract eventually.
+Hence, our strategy here is to show that we can approximate (i n operator
+norm) the diagonal matrix Dwith entries
+Di,i=f/parenleftbiggtrace(Σ)
+p/parenrightbigg
++f′(ξi,i)/parenleftbiggX′
+iXi
+p−trace(Σ)
+p/parenrightbigg
+−f′(0)X′
+iXi
+p−f(0)
+byυpIdp. To do so, we just have to show that the diagonal error matrix Z,
+with entries
+Zi,i=(f′(ξi,i)−f′(0))/parenleftbiggX′
+iXi
+p−trace(Σ)
+p/parenrightbigg
+goes to zero in operator norm.
+As seen in Lemma A.4or FactA.1, ifc≥p−(1/2−ε)b/2, with very high-
+probability,
+max
+i/vextendsingle/vextendsingle/vextendsingle/vextendsingleX′
+iXi
+p−trace(Σ)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤p−(1/4+ε/2)(log(p))2/b.
+Iff′is continuous and hence bounded aroundtrace(Σ)
+p, we therefore see that
+the operator (or spectral) norm of Zsatisfies with high-probability
+/bardbl|Z/bardbl|2≤Kp−(1/4+ε/2)(log(p))2/b.
+•Final step. We clearly have
+/tildewiderM−M=W+Z.
+It is also clear that /tildewiderMhas a limiting spectral distribution, satisfying, up
+to centering and scaling, the Marˇ cenko–Pastur equation; t his was shown in
+[19]. By Lemma 2.1, we see that/tildewiderMand/tildewiderM−Zhave the same limiting spec-
+tral distribution, since their difference is Zand/bardbl|Z/bardbl|2→0. Using the same
+lemma, we see that Mand/tildewiderM−Zhave (in probability) the same limiting
+spectral distribution, since their difference is Wand we have established
+that the Frobenius norm of this matrix is (in probability) o(√n). Hence,M
+and/tildewiderMhave (in probability) the same limiting spectral distribut ion./square
+We finally treat the case of kernel matrices computed from Euc lidean
+norms, in this more general distributional setting.SPECTRUM OF KERNEL RANDOM MATRICES 35
+Theorem 2.4. Let us call τ=2trace(Σ) /pwhereΣis the covariance
+matrix of the Xi’s. Suppose that fis a real valued function which is C2
+aroundτandC1around0.
+Under the assumptions of Theorem 2.3, the kernel matrix Mwith(i,j)
+entry
+Mi,j=f/parenleftbigg/bardblXi−Xj/bardbl2
+2
+p/parenrightbigg
+has a nonrandom limiting spectral distribution which is the same as that of
+the matrix
+/tildewiderM=f(τ)11′−2f′(τ)XX′
+p+υpIdn,
+whereυp=f(0)+τf′(τ)−f(τ).
+We note once again that the term f(τ)11′does not affect the limiting
+spectral distribution of M. But we keep it for the same reasons as before.
+Proof of Theorem 2.4.Notethatthediagonaltermissimply f(0)Id,
+so this term does not create any problem.
+The rest of proof is similar to that of Theorem 2.3. In particular the
+control of the Frobenius norm of the second order term is done in the same
+way, by controlling the maximum of the off-diagonal term, usi ng Corollary
+A.3and Fact A.1(and hence Lemma A.4).
+Therefore,weonlyneedtounderstandthefirstorderterm,in otherwords,
+the matrix with 0 on the diagonal and off-diagonal entry
+Ri,j=/bardblXi−Xj/bardbl2
+2
+p−τ
+=/bracketleftbigg/bardblXi/bardbl2
+2
+p−trace(Σ)
+p/bracketrightbigg
++/bracketleftbigg/bardblXj/bardbl2
+2
+p−trace(Σ)
+p/bracketrightbigg
+−2X′
+iXj
+p.
+As in the proof of Theorem 2.2, let us call ψthe vector with ith entry
+ψi=/bardblXi/bardbl2
+2
+p−trace(Σ)
+p. Clearly,
+Ri,j=δi,j/parenleftbigg
+1ψ′+ψ1′−2XX′
+p/parenrightbigg
+.
+Simple computations show that
+R−2trace(Σ)
+pId=1ψ′+ψ1′−2XX′
+p.36 N. EL KAROUI
+Now, obviously, 1 ψ′+ψ1′is a matrix of rank at most 2. Hence, Rhas the
+same limiting spectral distribution as
+2trace(Σ)
+pId−2XX′
+p
+since finite rank perturbations do not affect limiting spectr al distributions
+(see, for instance, [ 3], Lemma 2.2). This completes the proof. /square
+2.5.Some consequences of the theorems. In practice, it is often the case
+that slight variant of kernel random matrices are used. In pa rticular, it is
+customary to center the matrices, that is, to transform Mso that its row
+sum, or column sum or both are 0. Note that these operations co rrespond
+to right and/or left multiplication by the matrix H=Idn−11′/n.
+Inthesesituations,ourresultsstillapply;thefollowing factmakesitclear.
+Fact 2.1 (Centered kernel random matrices). LetHbe then×nmatrix
+Idn−11′/n.
+1.If the kernel random matrix Mcan be approximated consistently in op-
+erator norm by K, then, ifa,b∈{0,1},
+HaMHbcan be approximated consistently in operator norm by HaKHb.
+2.If the kernel random matrix Mhas the same limiting spectral distribution
+as the matrix K, then, ifa,b∈{0,1},
+HaMHbhas the same limiting spectral distribution as K.
+A nice consequence of the first point is that the recent hard wo rk on
+localizing the largest eigenvalues of sample covariance ma trices (see [ 8,17]
+and [31]) can be transferred to kernel random matrices and used to gi ve
+some information about the localization of the largest eige nvalues ofHMH,
+for instance. In the case of the results of [ 17], Fact 2 and the arguments of
+[19], Section 2.3.4, show that it gives exact localization info rmation. In other
+words, we can characterize the a.s. limit of the largest eige nvalue ofHMH
+(orHMorMH) fairly explicitly, provided Fact 2 in [ 17] applies. Finally,
+let us mention the obvious fact that since two square matrice sAandB,
+ABandBAhave the same eigenvalues, we see that HMHhas the same
+eigenvalues as MHandHMbecauseH2=H.
+Proof of Fact 2.1.The proofs are simple. First note that His posi-
+tive semi-definite and /bardbl|H/bardbl|2=1. Using the submultiplicativity of /bardbl|·/bardbl|2, we
+see that
+/bardbl|HaMHb−HaKHb/bardbl|2≤/bardbl|M−K/bardbl|2/bardbl|Ha/bardbl|2/bardbl|Hb/bardbl|2=/bardbl|M−K/bardbl|2.SPECTRUM OF KERNEL RANDOM MATRICES 37
+This shows the first point of the fact.
+The second point follows from the fact that HaMHbis a finite rank
+perturbation of M. Hence, using Lemma 2.2 in [ 3], we see that these two
+matrices have the same limiting spectral distribution, and since, by assump-
+tion,Khas the same limiting spectral distribution as M, we have the result
+of the second point. /square
+On Laplacian-like matrices. Finally, we point out a simple consequence
+of our results for Laplacian-like matrices. In light of rece nt results on man-
+ifold learning (see [ 9]) where these matrices play a key role, it is natural
+to ask what happens to them in our context. Suppose Mis ann×nker-
+nel random matrix as defined in the previous theorems, and con sider the
+(Laplacian-like) matrix Ldefined by
+
+
+Li,j=−Mi,j/n, ifi/ne}ationslash=j,
+Li,i=1
+n/summationdisplay
+i/negationslash=jMi,j.
+CallDLthe diagonal matrix made up of the diagonal elements of L. We
+note that our concentration results (Lemmas A.3andA.4, as well as Fact
+A.1) imply that DLcan be approximated in operator norm by f(0)Idnin
+the scalar product kernel matrix case and by f(2trace(Σ p)/p)Idnin the
+Euclidean distance kernel matrix case. [It is so because Li,iis an average of
+almost constant (and equal) quantities, so with high-proba bilityLi,icannot
+deviate from this constant value, for that would require tha t at least one of
+the components of the average deviate from the constant valu e in question.]
+Hence, there exists γpsuch that /bardbl|DL−γpIdn/bardbl|2tends to 0 almost surely.
+We also recall that the diagonal of the matrix Mcan be consistently ap-
+proximated in operator norm by a (finite) multiple of the iden tity matrix,
+so the diagonal of M/ncan be consistently approximated in operator norm
+by 0. Therefore, /bardbl|L+M/n−DL/bardbl|2tends to 0 almost surely, and therefore,
+/bardbl|L+M/n−γpIdn/bardbl|2tends to zero almost surely. In other words, Lcan be
+consistently approximated in operator norm by γpIdn−M/n. Consequently,
+when we can, as in Theorems 2.1and2.2, consistently approximate Min
+operator norm by a linearized version, K, ofM, thenLcan be consistently
+approximated in operator norm by γpIdn−K/n, and we can deduce spectral
+properties of Lfrom that of γpIdn−K/n. When we know only about the
+limiting spectral distribution of M, as in Theorems 2.3and2.4, the operator
+norm consistent approximation of LbyγpIdn−M/ncarries over to give us
+information about the limiting spectral distribution of Lsince the effect of
+γpIdnis just to “shift” the eigenvalues by γp. We note that getting informa-
+tion about the eigenvectors of Lwould require finer work on the properties
+of the matrix DLsince approximating it by a multiple of the identity does
+not give us any information about its eigenvectors.38 N. EL KAROUI
+3. Conclusions. The main result of this paper is that under various tech-
+nical assumptions, in high-dimensions, kernel random matr ices [i.e.,n×n
+matrices with ( i,j)th entryf(X′
+iXj/p) orf(/bardblXi−Xj/bardbl2
+2/p) where{Xi}n
+i=1
+are i.i.d. random vectors in Rpwithp→∞andp≍n] which are often used
+to create nonlinear versions of standard statistical metho ds and essentially
+behave like covariance matrices, that is, linearly, a resul t that is in sharp
+contrast with the low-dimensional situation where pis assumed to be fixed,
+and where it is known that, under some regularity conditions , spectral prop-
+erties of kernel random matrices mimick those of certain int egral operators.
+Under ICA-like assumptions, we were able to get a “strong app roximation”
+result (Theorems 2.1and2.2), that is, an operator norm consistency re-
+sult that carries information about individual eigenvalue s and eigenvectors
+corresponding to separated eigenvalues. Under more genera l and less linear
+assumptions (Theorems 2.3and2.4), we have obtained results concerning
+the limiting spectral distribution of these matrices using a “weak approxi-
+mation” result relying on bounds on Frobenius norms.
+Beside the mathematical results obtained above, this study raises several
+statistical questions, both about the richness—or lack the reof—of models
+that are often studied in random matrix theory and about the e ffect of
+kernel methods in this context.
+3.1.On kernel random matrices. Our study, motivated in part by nu-
+merical experiments we read about in the interesting [ 44], has shown that in
+the asymptotic setting we considered which is generally con sidered relevant
+for high-dimensional data analysis, the kernel random matr ices considered
+here behave essentially like matrices closely connected to sample covariance
+matrices. This is in sharp contrast to the low-dimensional s etting where it
+was explained heuristically in [ 44], and proved rigorously in [ 28], that the
+eigenvalues of kernel random matrices converged (under cer tain assump-
+tions) to those of a canonically related operator.
+This suggests that kernel methods could suffer from the same p roblems
+that affect linear statistical methods, such as Principal Co mponent Anal-
+ysis, in high-dimensions. The practical significance of our result is that in
+high-dimensions, the nonlinear methods that rely on kernel matrices may be
+behaving like their linear counterparts. Our study also per mits the transfer
+of some recent random matrix results concerning large-dime nsional sample
+covariance matrices to kernel random matrices. We now discu ss some possi-
+ble practical settings to highlight when our results are and are not relevant.
+On kernel-PCA. An important motivation for this study was to try
+to understand the properties of kernel-PCA in high-dimensi ons. We refer
+the reader to [ 36] pages 48–50 for a primer on kernel-PCA, but let us saySPECTRUM OF KERNEL RANDOM MATRICES 39
+that kernel-PCA performs a spectral decomposition of a (row and column-
+centralized) kernel matrix to efficiently perform a nonlinea r version of PCA;
+instead of doing standard PCA, the algorithm performs PCA in feature
+space. Our results, and in particular Theorems 2.1and2.2clearly show that
+in high-dimensions, when the assumptions of the theorems ar e satisfied, the
+algorithm may essentially be performing a linear PCA despit e appearances
+to the contrary. By contrast, in low-dimension, results suc h as [28] show
+that the intuition behind kernel-PCA is correct and that the algorithm then
+performs a genuinely nonlinear PCA. Being aware of the differ ence between
+the two settings should be helpful to practitioners in that i t will inform them
+about the possible limitations of kernel-PCA as a nonlinear method. For an
+example of applications, we refer the reader to, for instanc e, [44] and [36],
+Chapter 10. We note also that from a slightly more “numerical analysis”
+standpoint, our results basically say that the Nystr¨ om met hod (see [ 44] and
+references therein) to approximate eigenfunctions of inte gral operators can
+be unreliable in high-dimensions.
+On kernel-ICA. The setting of Theorems 2.1and2.2is naturally well-
+suited for applications to ICA-like problems. Since we are d ealing with vec-
+torsXihere, we would be considering multidimensional ICA problem s (see
+[2], page 33). For instance, in the formulation of [ 2], kernel-ICA is solved by
+solvingakernel-CCAproblem,thatis,ageneralizedeigenv alueproblemwith
+kernel matrices as input. We refer the reader to equations (1 0) and (13) in
+[2] for more details. The results of Theorems 2.1and2.2are directly relevant
+here, since the matrices at stake in these kernel-CCA proble ms can be ap-
+proximated consistently in operator norm by linear counter parts, and hence
+the solution of the kernel-CCA problem can be consistently a pproximated
+by the solution of the problem obtained by linearizing the ke rnel matrices
+at stake, provided the smallest singular value of the linear ized version of
+the kernel matrices in question stay bounded away from 0. We n ote that
+in practice this latter requirement can be checked using our fairly detailed
+knowledge of the properties of extreme eigenvalues of sampl e covariance ma-
+trices. We note that in this setting, our theorems confirm the predictions in
+[2], page 33, that problems might arise with the algorithm they propose in
+high-dimensions due to slow decay of eigenvalues.
+Geostatistics applications. Certain kernel matrices, corresponding to co-
+variance functions of, for instance, Gaussian processes, a lso appear in geo-
+statistics and spatial statistics in techniques such as kri ging (see, e.g., [ 15],
+Chapter 3 and, for instance, pages 106–110). For examples of kernels that
+correspondtocovariancefunctionsofGaussianprocesses, wereferthereader
+to [33], Chapter 4. Naturally, in those sort of applications, the d imension of
+the data vectors is low (at most 3), and therefore “classical results” such as40 N. EL KAROUI
+those of [ 28] apply whereas our results are limited to the high-dimensio nal
+setting more often encountered in some problems of multivar iate statistics,
+machine learning or bioinformatics.
+3.2.Limitations of standard random matrix models. In the study of
+spectral distribution of large-dimensional sample covari ance matrices, it has
+been somewhat forcefully advocated that the study should be done under
+the assumptions that the data are of the form Xi=Σ1/2Yiwhere the entries
+ofYihave, for instance, finite fourth moment. At first sight, this idea is ap-
+pealing,asitseemstoallowagreatvarietyofdistribution sandhenceflexible
+modeling. A possible drawback however, is the assumption th at the data are
+linear combinations of i.i.d. random variables or the neces sary presence of
+independence in the model. This has however been recently ad dressed (see,
+e.g., [19]) and it has been shown that one could go beyond models requir ing
+independence in a lurking random vector which the data linea rly depend on.
+Data analytic consequences. However,aseriouslimitationisstillpresent.
+As the results of Lemmas A.3,A.4and Fact A.1make clear, under the
+models for which the limiting spectral distribution of the s ample covariance
+matrix has been shown to satisfy the Marˇ cenko–Pastur equati on, the norms
+of the data vectors are concentrated, and the corresponding data vectors
+are almost orthogonal to one another. In other words, under t he “standard”
+ICA-like random matrix models (used in Theorems 1 and 2), tha t is, the
+random vectors {Xi}n
+i=1are i.i.d. in Rp,Xi=Σ1/2YiwithYihaving i.i.d.
+entries with mean 0, variance 1 and 4+ εabsolute moments, we have, as-
+suming that {Yi}n
+i=1are i.i.d.,p→∞,p/nand/bardbl|Σ/bardbl|2remain bounded; the
+vectors{Xi}n
+i=1have the property that for a deterministic (and computable)
+sequencecp, we have
+max
+1≤i≤n|/bardblXi/bardbl2
+2/p−cp|→0
+and
+max
+i/negationslash=j|X′
+iXj|/p→0.
+Both these statements hold almost surely. Geometrically, t his means that
+the vectors {Xi/√p}n
+i=1are close to a sphere and almost orthogonal to one
+another. These properties also hold for the more general (an d less linear)
+models we considered in Theorems 2.3and2.4.
+Hence, if one were to plot a histogram of {/bardblXi/bardbl2
+2/p}n
+i=1, this histogram
+would look tightly concentrated around a single value—the s pread of this
+histogram being computable from our concentration results (Lemmas A.3,
+A.4and Fact A.1). Though the models appear to be quite rich, the geometry
+that we can perceive by sampling nsuch vectors, with n≍p, is, arguably,SPECTRUM OF KERNEL RANDOM MATRICES 41
+relativelypoor.Theseremarksshouldnotbetakenasaiming todiscreditthe
+interesting body of work that has emerged out of the study of s uch models.
+Their aim is just to warn possible users that in data analysis , a good first
+step would be to plot the histogram of {/bardblXi/bardbl2
+2/p}n
+i=1and check whether
+it is concentrated around a single value. Similarly, one mig ht want to plot
+the histogram of inner products {X′
+iXj/p}and check that it is concentrated
+around 0. If this is not the case, then insights derived from r andom matrix
+theoretic studies would likely not be helpful in the data ana lysis.
+We note, however, that recent random matrix work (see [ 13,14,19,32])
+has been concerned with distributions which could be loosel y speaking be
+called of “elliptical” type—though they are more general th an what is usu-
+ally called elliptical distributions in Statistics. In tho se settings, the data
+is, for instance, of the form Xi=riΣ1/2Yiwhereriis a real-valued random
+variable, independent of Yi. This allows the data vectors to not approxi-
+mately live on spheres (but does not change anything about an gles between
+different vectors), and is a possible way to address some of th e concerns we
+just raised. The characterization of the limiting spectrum gets quite a bit
+more involved than in the “standard” setting, that is, ri=1, and the results
+show a lack of robustness to the “indirect” assumption that t he data vectors
+live close to a sphere.
+Finally, this geometric discussion applies also to theoret ical studies under-
+taken under the assumptions that the XiareN(0,Σp) and that the problem
+is high dimensional. It should highlight some possibly seve re limitations of
+the normality assumption in high-dimensions.
+APPENDIX
+In this appendix, we collect a few useful results that are nee ded in the
+proof of our theorems, and whose content we thought would be m ore acces-
+sible if they were separated from the main proofs.
+(A) Some useful results. We have the following elementary facts.
+Lemma A.1. SupposeYis a vector with i.i.d. entries and mean 0. Call
+its entriesyi. Suppose E(y2
+i)=σ2andE(y4
+i)=µ4. Then ifMis a deter-
+ministic matrix,
+E(YY′MYY′)=σ4(M+M′)+(µ4−3σ4)diag(M)+σ4trace(M)Id. (4)
+Further, we have (Y′MY)2=trace(MYY′MYY′)and
+E(trace(MYY′MYY′))
+(5)
+=σ4trace(M2+MM′)+σ4(trace(M))2+(µ4−3σ4)trace(M◦M).42 N. EL KAROUI
+Herediag(M)denotes the matrix consisting of the diagonal of the matrix
+Mand0off the diagonal. The symbol ◦denotes Hadamard multiplication
+between matrices.
+Proof. LetuscallR=YY′MYY′.Theproofofthefirstpartiselemen-
+tary and consists merely in writing the ( i,j)th entry of the corresponding
+matrix. As a matter of fact, we have
+Ri,j=yiyj/summationdisplay
+i,jyiyjMi,j=/summationdisplay
+k,lyiyjykylMk,l.
+Using the fact that entries of Yare independent and have mean 0, we see
+that, in the sum, the only terms that will not be 0 in expectati on are those
+for which each index appears at least twice. If i/ne}ationslash=j, only the terms of the
+formy2
+iy2
+jhave this property. So if i/ne}ationslash=j,
+E(Ri,j)=E(y2
+iy2
+j(Mi,j+Mj,i))=σ4(Mi,j+Mj,i).
+Let us now turn to the diagonal terms. Here again, only the ter msy2
+iy2
+k
+matter. So on the diagonal,
+E(Ri,i)=µ4Mi,i+σ4/summationdisplay
+j/negationslash=iMj,j=(µ4−σ4)Mi,i+trace(M).
+We conclude that
+E(R)=σ4(M+M′)+(µ4−3σ4)diag(M)+trace(M)Id.
+The second part of the proof follows from the first result, aft er we remark
+that, ifDis a diagonal and Lis general matrix, trace( LD)=trace(L◦D),
+from which we conclude that trace( Mdiag(M)) = trace(M◦diag(M)) =
+trace(M◦M)./square
+Lemma A.2 (Concentration of quadratic forms). Suppose the vectors Z
+is a vector in Rpwith i.i.d. entries of mean 0 and variance σ2. Suppose that
+their entries are bounded by Bp. LetMbe a symmetric matrix, with largest
+singular value σ1(M). Call
+ζp=128exp(4π)σ1(M)B2
+p
+p,
+νp=/radicalbig
+σ1(Σ).
+Then we have, if r/2>ζp,
+P/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleZ′MZ
+p−σ2trace(M)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle>r/parenrightbigg
+≤8exp(4π)exp(−p(r/2−ζp)2/(32B2
+p(1+2νp)2σ1(M))) (6)
++8exp(4π)exp(−p/(32B2
+p(1+2νp)2σ1(M))).SPECTRUM OF KERNEL RANDOM MATRICES 43
+Proof. We can decompose, using the spectral decomposition of M,
+M=M+−M−whereM+is positive semi-definite and M−is positive defi-
+nite (or 0 if Mis itself positive semi-definite). We can do so by replacing t he
+negative eigenvalues of Mby 0 in the spectral decomposition and get M+in
+that way. Note that then, the largest singular values of M+andM−are also
+bounded by σ1(M) sinceσ1(M) is absolute value of the largest eigenvalue
+ofMin absolute value, and the nonzero eigenvalues of M+are a subset of
+the eigenvalues of Mand so are the eigenvalues of M−whenM−is not 0.
+Now it is clear that the function Fwhich associates to a vector xinRpthe
+scalar/radicalbig
+x′M+x/p=/bardblM1/2
++x/√p/bardbl2is a convex,/radicalbig
+σ1(M)/p-Lipschitz func-
+tion with respect to Euclidean norm. Calling mFthe median of F(Z), we
+have, using Corollary 4.10 in [ 29],
+P(|F(Z)−mF|>r)≤4exp(−pr2/(16B2
+pσ1(M))).
+Let us now call µFthe mean of F(Z) (it exists according to Proposition 1.8
+in [29]). Following the arguments given in the proof of this Propos ition 1.8,
+and spelling out the constants appearing in the last result o f Proposition 1.8
+in [29], we see that
+P(|F(Z)−µF|>r)≤4exp(4π)exp(−pr2/(32B2
+pσ1(M))).
+(Using the notation of Proposition 1.8 in [ 29], we picked κ2= 1/2, and
+C′=exp(πC2/4);showingthatthisisavalidchoicejustrequiresonetoca rry
+out some of the computations mentioned in the proof of that Pr oposition.)
+Let us call A,B,Dthe sets
+A/defines/braceleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleZ′M+Z
+p−µ2
+F/vextendsingle/vextendsingle/vextendsingle/vextendsingle>r/bracerightbigg
+,
+B/defines/braceleftBigg/radicalBigg
+Z′M+Z
+p+µF≤1+2µF/bracerightBigg
+=/braceleftBigg/radicalBigg
+Z′M+Z
+p−µF≤1/bracerightBigg
+and
+D/defines/braceleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalBigg
+Z′M+Z
+p−µF/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>r/(1+2µF)/bracerightBigg
+.
+Of course, we have P(A)≤P(A∩B)+P(Bc). Now note that A∩B⊆D,
+simply because for positive reals, a−b/(√a+√
+b)=√a−√
+b. We conclude
+that
+P(A)≤4exp(4π)[exp(−pr2/(32B2
+p(1+2µF)2σ1(M)))
++exp(−p/(32B2
+pσ1(M)))].44 N. EL KAROUI
+Let us know call σ2the variance of the each of the component of Z. We
+know, according to Proposition 1.9 in [ 29], that
+var(F)=E(Z′M+Z)
+p−µ2
+F=σ2trace(M+)
+p−µ2
+F≤ζp
+=128exp(4π)σ1(M)B2
+p
+p.
+Hence, we conclude that, if r>ζp,
+P/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleZ′M+Z
+p−σ2trace(M+)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle>r/parenrightbigg
+≤4exp(4π)exp(−p(r−ζp)2/(32B2
+p(1+2µF)2σ1(M)))
++4exp(4π)exp(−p/(32B2
+p(1+2µF)2σ1(M))).
+To get the announced result, we note that for the sum of two rea ls to be
+greater than rin absolute value, one needs to be greater than r/2, and that
+ourboundsbecomeconservativewhenwereplace µF(anditscounterpartfor
+M−) byνp. [Note that we get conservative bounds when replacing the µF’s
+by max(E(/radicalbig
+Z′M+Z/p),E(/radicalbig
+Z′M−Z/p)), and that this quantity is clearly
+bounded by σσ1(Σ).] Hence, we have, as announced, if r/2>ζp,
+P/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleZ′MZ
+p−σ2trace(M)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle>r/parenrightbigg
+≤8exp(4π)exp(−p(r/2−ζp)2/(32B2
+p(1+2µF)2σ1(M)))
++8exp(4π)exp(−p/(32B2
+p(1+2µF)2σ1(M))).
+Finally, we note that the proof makes clear that the same resu lt would hold
+for different choices of M+andM−, as long as max( σ1(M+),σ1(M−))≤
+σ1(M)./square
+We therefore have the following useful corollary:
+Corollary A.1. LetYiandYjbe i.i.d. random vectors as in Lemma
+A.2with variance 1. Suppose that Σis a positive semi-definite matrix. We
+have, with
+ζp=128exp(4π)σ1(Σ)B2
+p
+p
+and
+νp=/radicalbig
+σ1(Σ),SPECTRUM OF KERNEL RANDOM MATRICES 45
+that ifr/2>ζpandK=8exp(4π),
+P/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleY′
+iΣYj
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle>r/parenrightbigg
+≤Kexp(−p(r/2−ζp)2/(32B2
+p(1+2νp)2σ1(Σ)))
+(7)
++Kexp(−p/(32B2
+p(1+2νp)2σ1(Σ))).
+Proof. The proof relies on the results of Lemma A.2. Remark that,
+since Σ is symmetric,
+Y′
+iΣYj=1
+2(Y′
+iY′
+j)/parenleftbigg
+0 Σ
+Σ 0/parenrightbigg/parenleftbigg
+Yi
+Yj/parenrightbigg
+.
+Now the entries of the vector made by concatenating YiandYjare i.i.d.
+and so we fall back into the setting of Lemma A.2. Finally, here M+and
+M−are known explicitly. A possible choice is M+=1/2/parenleftBig
+Σ
+ΣΣ
+Σ/parenrightBig
+andM−=
+1/2/parenleftBig
+Σ
+00
+Σ/parenrightBig
+.νpis obtained by upper bounding the expectation of the square
+ofFin the notation of the proof of the previous lemma for these ex plicit
+matrices. Note that their largest singular values are both s maller that σ1(Σ),
+so the results of the previous lemma apply. /square
+Lemma A.3. Let{Yi}n
+i=1be i.i.d. random vectors in Rp, whose entries
+are i.i.d., mean 0, variance 1 and have bounded (in p)m≥4absolute mo-
+ments. Suppose that {Σp}is a sequence of positive semi-definite matrices
+whose operator norms are uniformly bounded in pandn/pis asymptotically
+bounded. We have, for any given ε>0,
+max
+i,j/vextendsingle/vextendsingle/vextendsingle/vextendsingleY′
+iΣpYj
+p−δi,jtrace(Σ p)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤p−1/2+2/m(log(p))(1+ε)/2a.s.
+Proof. Throughout the proof, we assume without loss of generality
+thatm<∞.
+Callt= 2/m. It is clear that with our moment assumptions, t≤1/2.
+According to Lemma 2.2 in [ 45], the maximum of the array of {Yi}n
+i=1is a.s.
+less thanpt. So to control the maximum of the inner products of interest,
+it is enough to control the same quantity when we replace Yiby/tildewideYiwith
+/tildewideYi,l/definesYi,l1|Yi,l|≤pt. Now note that /tildewideYisatisfies the boundedness assumption of
+Corollary A.1, but its mean is not necessarily zero and its variance is not 1 .
+Note however, that all the entries of /tildewideYihave the same mean, /tildewideµ. SinceYihas
+mean 0, we have
+|/tildewideµ|≤E(|Y1,1|1|Y1,1|>pt)≤E(|Y1,1|mp−t(m−1))≤µmp−2+t.
+Similarly, if we call /tildewideσ2the variance of /tildewideY, we have
+/tildewideσ2=E(|Y1,1|21|Y1,1|≤pt)−/tildewideµ2=1−(E(|Y1,1|21|Y1,1|>pt)+/tildewideµ2).46 N. EL KAROUI
+Hence, 0 ≤1−/tildewideσ2, and
+1−/tildewideσ2=E(|Y1,1|21|Y1,1|>pt)+/tildewideµ2
+≤E(|Y1,1|mp−t(m−2))+/tildewideµ2
+≤µmp−2+2t+µ2
+mp−4+2t=O(p−2+2t).
+Let us call Ui=/tildewideYi−/tildewideµ1pand/tildewideUi=Ui//tildewideσwhere/tildewideσ2is also the variance of
+Ui. Corollary A.1applies to the random variables /tildewideUiwithBp=2ptwhenp
+is large enough. So ζp=O(p1−2t). Let us now call, for some ε>0,
+r(p)=pt−1/2(log(p))(1+ε)/2.
+Since, forplarge enough, r(p)/2>ζp, we can apply the conclusions of
+Corollary A.1, and by plugging in the different quantities, we see that
+P(|/tildewideU′
+iΣp/tildewideUj/p|>r(p))≤exp(−K(log(p))1+ε),
+whereKdenotes a generic constant (that may change from display to d is-
+play). In particular, Kis independent of pand is hence trivially bounded
+away from 0 as pgrows. The bound we just obtained on 1 −/tildewideσ2also implies
+that forplarge enough, /tildewideσ2>1/2 from which we conclude that for another
+Kwith the same properties,
+P(|U′
+iΣpUj/p|>r(p))≤exp(−K(log(p))1+ε).
+In other respects, the arguments of Lemma A.2show that, since /tildewideσ2is the
+variance of Ui,
+P(|U′
+iΣpUi/p−/tildewideσ2trace(Σ p)/p|>r(p))≤exp(−K(log(p))1+ε).
+Now
+/tildewideY′
+iΣp/tildewideYj
+p=U′
+iΣpUj
+p+/tildewideµ(1′ΣpUj+U′
+iΣp1)
+p+/tildewideµ21′Σp1
+p.
+Remark that 1′Σp1≤pσ1(Σp), and|1′ΣpUj|≤/radicalbig
+1′Σp1/radicalBig
+U′
+jΣpUj. We con-
+clude, using the results obtained in the proof of Lemma A.2that with proba-
+bility greater than 1 −exp(−K(log(p))1+ε), the middle term is smaller than
+2/radicalbig
+σ1(Σp)(/radicalbig
+σ1(Σp)+r(p))|/tildewideµ|. As a matter of fact,/radicalBig
+U′
+jΣpUj/pis concen-
+trated around its mean which is smaller than /tildewideσ/radicalbig
+trace(Σ p)/pwhich is itself
+smaller than/radicalbig
+σ1(Σp). Now recall that |/tildewideµ|= O(p−2+t) = o(r(p)). We can
+therefore conclude that
+P/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideY′
+iΣp/tildewideYj
+p−δi,j/tildewideσ2trace(Σ p)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle>2r(p)/parenrightbigg
+≤2exp(−K(log(p))1+ε).SPECTRUM OF KERNEL RANDOM MATRICES 47
+Now note that 0 ≤1−/tildewideσ2=O(p−2+2t)=o(r(p)) sincet≤1/2<3/2. With
+our assumptions, trace(Σ p)/premains bounded, so we have finally
+P/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideY′
+iΣp/tildewideYj
+p−δi,jtrace(Σ p)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle>3r(p)/parenrightbigg
+≤2exp(−K(log(p))1+ε).
+And therefore,
+P/parenleftbigg
+max
+i,j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideY′
+iΣp/tildewideYj
+p−δi,jtrace(Σ p)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle>3r(p)/parenrightbigg
+≤2n2exp(−K(log(p))1+ε).
+Using the Borel–Cantelli lemma, we reach the conclusion tha t
+max
+i,j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideY′
+iΣp/tildewideYj
+p−δi,jtrace(Σ p)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤3r(p)=3p2/m−1/2log(p) a.s.
+Because the left-hand side is a.s. equal to |Y′
+iΣpYj
+p−δi,jtrace(Σ p)
+p|, we reach the
+announced conclusion but with r(p) replaced by 3 r(p). Note that, of course,
+any multiple of r(p), where the constant is independent of p, would work in
+the proof. In particular, by taking ˜ r(p)=r(p)/3, we reach the announced
+conclusion. /square
+Corollary A.2. Under the same assumptions as that of Lemma A.3,
+if we callXi=Σ1/2
+pYi, we also have
+max
+i/negationslash=j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bardblXi−Xj/bardbl2
+2
+p−2trace(Σ p)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤p−1/2+2/m(log(p))(1+ε)/2a.s.
+Proof. The proof follows immediately from the results of Lemma A.3,
+after we write
+/bardblXi−Xj/bardbl2
+2−2trace(Σ p)
+=[YiΣpYi−trace(Σ p)]+[YjΣpYj−trace(Σ p)]−2Y′
+iΣpYj.
+Note that as explained in the proof of Lemma A.3, the constants in front of
+theboundingsequencedonotmatter,sowecanreplace3 p−1/2+2/m(log(p))(1+ε)/2
+byp−1/2+2/m(log(p))(1+ε)/2, and the result still holds. [In other words, we
+are really using Lemma A.3with upper bound p−1/2+2/m(log(p))(1+ε)/2/3.]
+/square
+Lemma A.4. Let{Xi}n
+i=1be i.i.d. random vectors in Rpwhose entries
+are i.i.d., mean 0, having the property that for 1-Lipschitz ( with respect to
+Euclidean norm) functions F, if we denote by mFthe median of F(Xi),
+P(|F(Xi)−mF|>r)≤Cexp(−c(p)r2),48 N. EL KAROUI
+whereCis independent of pandcis allowed to vary with p(if it goes to
+zero, we assume it does so like p−α,0≤α<1). CallΣpthe covariance
+matrix ofX1. Assume that σ1(Σp)remains bounded in p. Then, under the
+triangular array construction of Theorem 2.3, we have, for any ε>0,
+max
+i,j/vextendsingle/vextendsingle/vextendsingle/vextendsingleX′
+iXj
+p−δi,jtrace(Σ p)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤(pc(p))−1/2(log(p))(1+ε)/2a.s.
+Proof. The proof once again relies on concentration inequalities. First
+note that Proposition 1.11 combined with Proposition 1.7 in [29] shows that
+ifXiandXjare independent and satisfy concentration inequalities wi th
+concentration function α(r) (with respect to Euclidean norm), then the vec-
+tor/parenleftbigYi
+Yj/parenrightbig
+also satisfies concentration inequalities with concentrat ion function
+2α(r/2) with respect to Euclidean norm in R2p. (We note that Proposition
+1.11 is proved for the metric on R2p/bardbl · /bardbl2+/bardbl · /bardbl2where each Euclidean
+norm is a norm in Rp, but the same proof goes through for Euclidean norm
+onR2p. Another argument would be to say that the metric /bardbl·/bardbl2+/bardbl·/bardbl2is
+equivalent to the norm of the full R2pwith the constants in the inequalities
+being 1 and√
+2 simply because for a,b>0,√
+a2+b2≤a+b≤√
+2√
+a2+b2.)
+Therefore,theargumentsofLemma A.2gothroughwithoutanyproblems
+with Σ p= Id andB2
+p= 4/c(p). So a result similar to Corollary A.1holds
+and we can apply the same ideas as in the proof of Lemma A.3and get the
+announced result. /square
+Corollary A.3. Under the assumptions of Lemma A.4, we have, for
+anyε>0,
+max
+i/negationslash=j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bardblXi−Xj/bardbl2
+2
+p−2trace(Σ p)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤(pc(p))−1/2(log(p))(1+ε)/2a.s.
+Proof. The proof is an immediate consequence of Lemma A.4, along
+the same lines as the proof of Corollary A.2./square
+Finally, allow the same lines of proof; we have the following fact.
+Fact A.1. Let{Xi}n
+i=1be i.i.d. random vectors in Rpwhose entries
+are i.i.d., mean 0, having the property that for 1-Lipschitz ( with respect to
+Euclidean norm) functions F, if we denote by mFthe median of F(Xi),
+P(|F(Xi)−mF|>t)≤Cexp(−c(p)tb)for someb>0,
+whereCis independent of pandcis allowed to vary with p(if it goes to
+zero, we assume it does so like p−α,0≤α<b/2). CallΣpthe covarianceSPECTRUM OF KERNEL RANDOM MATRICES 49
+matrix ofX1. Assume that σ1(Σp)remains bounded in p. Then, we have,
+under the triangular array construction of Theorem 2.3, for anyε>0,
+max
+i,j/vextendsingle/vextendsingle/vextendsingle/vextendsingleX′
+iXj
+p−δi,jtrace(Σ p)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤(pc2/b(p))−1/2(log(p))(1+ε)/ba.s.
+Also, we then have
+max
+i/negationslash=j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bardblXi−Xj/bardbl2
+2
+p−2trace(Σ p)
+p/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤(pc2/b(p))−1/2(log(p))(1+ε)/ba.s.
+The proof of this last fact follows the same step as that of Lem maA.4,
+with a slight adjustment since we need to replace 2 by b. For a related
+question and more details, we refer the reader to [ 19].
+(B) A linear algebraic result. Finally, we finish this appendix with a
+linear algebraic lemma which we need in our approximations a nd is of inde-
+pendent interest.
+Lemma A.5. Suppose M is real symmetric matrix with nonnegative en-
+tries. Suppose that Eis a real symmetric matrix such that maxi,j|Ei,j|≤ζ,
+for someζ≥0. Then, ifσ1(A)is the largest singular value of matrix Aand
+if◦represents the Hadamard product (i.e., entrywise multipli cation of two
+matrices), we have
+σ1(E◦M)≤ζσ1(M).
+Proof. We first note that E◦Mis real symmetric. Therefore,
+σ1(E◦M)= lim
+k→∞[trace((E◦M)2k)]1/(2k).
+Now we claim that
+|trace((E◦M)2k)|≤ζ2ktrace(M2k).
+To see this, recall that for a p×pmatrixA,
+trace(Ak)=/summationdisplay
+1≤i1,i2,...,ik≤pAi1,i2Ai2,i3···Aik,i1.
+Now,
+|Ai1,i2Ai2,i3···Aik,i1|≤|Ai1,i2||Ai2,i3|···|Aik,i1|.
+WhenA=E◦M,Ai,j=Ei,jMi,j. SinceMi,j≥0, we therefore have |Ei,j×
+Mi,j|≤ζMi,j. Hence,
+|trace((E◦M)k)|≤/summationdisplay
+1≤i1,i2,...,ik≤pζkMi1,i2Mi2,i3···Mik,i1=ζktrace(Mk).50 N. EL KAROUI
+So
+[trace((E◦M)2k)]1/(2k)≤ζ[trace(M2k)]1/(2k).
+Taking limits as k→∞concludes the proof. /square
+Acknowledgments. I would like to thank Bin Yu for stimulating my in-
+terestinthequestionsconsideredinthispaperandforinte restingdiscussions
+on the topic. I would like to thank Elizabeth Purdom for discu ssions about
+kernel analysis and Peter Bickel for many stimulating discu ssions about ran-
+dom matrices and their relevance in statistics. I would also like to thank an
+anonymous referee for useful and constructive comments tha t resulted in an
+improved presentation of the paper.
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+Department of Statistics
+University of California, Berkeley
+367 Evans Hall
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+USA
+E-mail: nkaroui@stat.berkeley.edu
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+arXiv:1001.0493v4  [hep-th]  14 Sep 2010arXiv: 1001.0493v4[hep-th]
+A simple strategy for renormalization: QED at one-loop leve l
+Ji-Feng Yang∗,†, Zhao-Ting Pan∗
+∗Department of Physics, East China Normal University, Shang hai 200241, China
+†Kavli Institute for Theoretical Physics China,
+Chinese Academy of Sciences, Beijing 100190, China
+We demonstrate our simple strategy for renormalization wit h QED at one-loop level, basing on
+an elaboration of the effective field theory philosophy. No ar tificial regularization or deformation
+of the original theory is introduced here and hence no manipu lation of infinities, ambiguities arise
+instead of infinities. Ward identities first come to reduce th e number of ambiguities, the residual
+ones could in principle be removed by imposing physical boun dary conditions. Renormalization
+group equations arise as ”decoupling theorems” in the under lying theory perspective. In addition,
+a technical theorem concerning routing of external momenta is also presented and illustrated with
+the self-energy and vertex function as examples.
+PACS numbers: 11.10.Gh;11.30.-j;11.10.Hi;12.20.-m
+Keywords: regularization; renormalization; differential equations; underlying theory
+I. INTRODUCTION
+All the presently known quantum field theories for practical use ar e beset with ultraviolet divergences
+to certain extent in perturbative frameworks. These divergence s imply that such theories are not well
+defined at short distances before renormalization is carried out, a t least within perturbative frameworks.
+The celebrated BPHZ program is the standard algorithm for renorm alization[1–3] that is widely employed
+by particle and field theorists, which can lead to a mathematically well-d efined perturbative formulation
+of a field theory in the end. But the procedures involved are not eas y to be endowed with a natural logic
+that would be readily to accept. In this connection, Epstein and Glas er (EG) had already shown that
+finite perturbation theory of quantum field theories can be obtaine d through constructive methods basing on
+causalityconditions[4, 5] where Feynman rules areonly valid at tree le vel, which providesmoremathematical
+rationales for conventional procedures like BPHZ[6, 7]. Recently, o ther mathematical structures hidden in
+the perturbative formulation (especially in BPHZ) of quantum field th eories has become a rapidly growing
+area of research[8–10], which implies that there are more regularitie s worth exploring under the veil of
+renormalization.
+Frommorephysicalpointofview,accordingtotheeffectivetheory philosophy,theappearanceofultraviolet
+divergences can be understood from the fact that the presently known formulation of a quantum field theory
+is necessarily an effective one with the sophisticated structures or parameters dominant at short distances
+or larger energy scales being ignored. Unfortunately, the sophist icated structures underlying the present
+formulation of quantum field theories are actually unavailable or not d iscovered yet. Nor could us simply
+guess about the exact answers. The divergences or the ill-defined objects are the prices we paid for such
+forced ignorance or simplification. This is the start of the effective fi eld theory philosophy that is now widely
+accepted and applied by physicists. Fortunately, this effective field theory view also implies that the ignored
+short-distance details could at best affect the physical behaviors of effective field theories through local
+interactions or operators that could be parametrized as correct ions to the actions of quantum field theories,
+which, after being absorbed into the normalizationfactors for effe ctive field theory’s couplings and operators,
+leads to the renormalization of quantum (effective) field theories. Otherwise they would be ”cap tured” as
+explicit and active degrees of freedom.
+In practical computations, one usually has to first introduce vario us regularization schemes and interme-
+diate divergences to be removed later, which are supposed to imitat e the roles played by the true underlying
+structures or parameters. Actually, with such operations, the p erturbative expansion only takes rather for-
+mal meaning due to the presence of infinite counter-terms. Moreo ver, for the regularization schemes or
+deformations to be legitimate and useful, they must not harm the bu lk structures of the original theory as
+well as being self-consistent. In particular, important symmetries and global properties like the topologies
+of spacetime and Hilbert space, unitarity and causality, etc., which s erve as foundations for a quantum field
+theory, should be preserved as much as possible, at least no drast ic or uncontrollable changes should be
+introduced. However, the status of the principles just stated ar e extremely hard to precisely or profoundly
+assess within any known regularization. As a matter of fact, the co mprehensive and global structures of2
+a well-defined quantum field theory have never been fully delineated, and we heavily rely on rather formal
+evaluations to deal with this very difficult issue. Sometimes, to pursu e efficiency, even the logical foundations
+of the theory would be severely mutilated[11]. Thus, we only accumula te our faiths in one regularization
+method through intensive tests against experiment data. Up till no w, there exists no regularization that
+could be satisfactorily in service in all contexts, each has certain sh ortcomings that will fail somewhere, or
+alternatively, no one could prove to work everywhere1. For example, the celebrated dimensional regulariza-
+tion works excellently for standard model, especially for the strong interaction, but not for supersymmetric
+theories at all[12–19]. Of course, other schemes also have their own shortcomings, for a comprehensive
+discussions of the advantages and disadvantages of various exist ing regularization methods, please refer to
+Ref.[20]. This is also an indirect reflection of our ignorance of the gran d structures or contents of quantum
+field theories in depth. Therefore, most regularization schemes ar e actually used in a superficial or formal
+sense, only appearingin the intermediate stagesofcalculations. Af ter the artificial and transientexcursionto
+parametrize our ignorance of the underlying short-distance phys ics, we ”return” to our ”original” quantum
+field theories with the deformations removed somehow. This practic e is quite established within perturbative
+regimes for many physically interested quantities or operators. Ne vertheless, the ultimate theoretical values
+for such doings still remain to be seen, at least in nonperturbative r egimes.
+On account of the above reasonings, it is definitely desirable or wort hwhile to explore of any approach or
+procedure that would introduce no or as less as possible deformatio n of the bulk aspects of a quantum field
+theory and the associated infinities or divergences. For physicists , it is also more convenient to work with
+Feynman rules and Feynman diagrams that are physically intuitive and well defined at tree level in such a
+manner that the radiative corrections or loop diagrams could be com puted without introducing too much
+mathematicalauxiliaries. Actually, aswillbedemonstratedandargu edbelow,theremayexistanapproachor
+strategy that seems to be free of the problems mentioned above o r satisfy the requirements just enumerated.
+It is actually based on an elaboration of the effective field theory philo sophy[21–25], introducing no artificial
+deformations, and more pleasantly, no ultraviolet infinities and the a ssociated subtraction of infinities. Only
+local ambiguities for each loop integral may occur as a parametrizat ion of our ignorance, which, as in the
+last step of renormalization, are to be fixed through appropriate b oundary conditions. We feel all these
+virtues should make our strategy a simple and natural approach to start with. It has already been partially
+employed in a number of field theoretical issues[26–30].
+From this report on, we wish to launch serial works to demonstrate , explain and further develop our
+strategy, with our ultimate goals targeting at a consistent and effic ient program. In the course of these
+works, we will illustrate through concrete examples that many tech nical subtleties such as external momenta
+routing (or shift of integration variables) and overlappingdivergen t diagramsdo not cause any trouble. From
+our demonstrations, one could find that our strategy could be well applied beyond the standard perturbative
+framework in terms of Feynman diagrams, see the applications in non perturbative problems[29, 30], unlike
+those principally devised for and limited to relativistic perturbative fo rmulation. Another virtue of our
+approach is that we may examine important issues such as symmetry status of the radiative corrections
+(perturbative or nonperturbative) without being stuck to some s pecific prescription that might lead us to
+wrong conclusions or judgement[26–28]. For perturbative amplitud es, our strategy could reproduce the
+EG approach’s results. In fact, our physically motivated strategy share the following consensus with the
+mathematically motivated EG approach: the perturbative quantum field theories are not unambiguously
+defined in the short distances, some elementary Feynman or loop dia grams contain local ambiguous terms
+that could only be fixed through appropriate boundary conditions. In this report, we employ the simple
+and well-known objects in quantum electrodynamics (QED) for a clea r and pedagogical illustration of the
+”naturalness” and simplicity of the principles and perspective that a re adopted in our approach. In a sense,
+we wish to motivate an elaboration of the effective field theories philos ophy with respect to renormalization
+of all quantum processes.
+The organization of this report is as follows: We describe our strate gy in Sec. II along with some technical
+issues. In Sec. III, we computethe elementaryone-loopamplitude s orFeynmangraphs(that areill defined in
+QED) using natural differential equations following from the existen ce of a complete underlying theory. The
+Ward identities between the vertex and self-energy diagrams will be demonstrated as a valid constraint and
+1The reason is obvious: Were it found, it must at least be equiv alent to the true underlying theory that is well-defined in ev ery
+aspect, including the correct formulation of quantum gravi ty.3
+the four-photon (light by light scattering) diagrams will be shown to be finite as an additional consequence
+of gauge invariance. In Sec. IV, we consider how to impose appropr iate boundary conditions to fix the
+ambiguities, how this step is linked to the conventional procedures, and related issues. We will also answer
+the origin of renormalization group equations in Sec. IV. The whole pr esentation is summarized in Sec. V,
+where some conceptual issues will also be addressed.
+II. DESCRIPTIONS OF OUR STRATEGY
+Our strategy is in fact to determine the ill-defined (or undefined) loo p amplitudes through solving natural
+differential equations together with imposition of physical boundar y conditions. In terms of Feynman dia-
+grams, this is to ”calculate” divergent diagrams using convergent o nes. To proceed, we denote a superficially
+divergent or ill-defined diagram and its underlying theory version with the same symbol Γ which should be
+a function of external momenta [ p], couplings and masses [ g] and the underlying parameters {σ}(which will
+be usually hidden to avoid heavy formulae) that render the diagram fi nite or mathematically meaningful.
+As the underlying parameters {σ}must be very small in ”size”, an ill-defined diagram should be determine d
+through the ”decoupling” limit of the version with the underlying para meters. Not knowing the details of
+the underlying theory, a natural and yet legitimate operation abou t the underlying theory version of an
+ill-defined diagram is to differentiate it with respect to external mome nta [p] for enough times so that the
+resulting loop integration could be computed in terms of the convent ional Feynman rules of quantum field
+theories2:
+∂ωΓ+1
+pΓ([p],[g]) =˜Γ(ωΓ+1)([p],[g]), (1)
+withωΓbeing the superficial divergence degree of diagram Γ and ˜Γ(ωΓ+1)denoting the well-defined diagrams
+generated by the operation ∂ωΓ+1
+p. Then we solve the above differential equation to arrive at a finite
+expression of Γ that contains a polynomial in terms of external mom enta [p] with ambiguous coefficients.
+So, the general solution for the ill-defined diagram Γ reads:
+Γ([p],[g]) =˜Γ([p],[g])+ΓPoly
+[ωΓ]([p],[g],[c]). (2)
+Here, [c] denotes the integration constants collectively. As last step of ou r strategy, the ambiguities are
+removed through imposing reasonable symmetries AND appropriate boundary conditions[31]. It is obvious
+that in our strategy, no artificial deformation is introduced at all .
+As our method rely on operations with respect to external moment a, there may be concerns about the
+issue of external momenta routing. Below we will show that:
+Theorem of Routing II.1 The routing of external momenta in a loop diagram does not mat ter at all in
+our strategy, i.e., different routings will at most yield a re parametrization of the ambiguous polynomials.
+The routing of external momenta has been first addressed in Refs .[24] and [25] more than a decade ago.
+Here, we give a more concrete formulation of the remarks given the re. There may be two kind of routings:
+(a) one keeps the distribution of external momenta at vertices int act; (b) the other amounts to relabeling
+the external momenta at vertices: [ p]→[p′], or a redefinition of the external momenta at vertices. First, we
+prove it for one-loop amplitudes.
+Proof. Case (a) : As the underlying theory version is well-defined, one could perform a integration
+variable transformation so that a different routing is exactly result ed, thus,
+Γ([p],[g];{σ})rt1= Γ([p],[g];{σ})rt2, (3)
+with the underlying parameters {σ}being now explicitly included for illustration. That is, this variable
+transformation should not alter the evaluation. Now apply the differ entiation with respect to the external
+momenta∂ωΓ
+[p]on Eq.(3), we end up with the following equality:
+∂ωΓ
+[p]Γ([p],[g];{σ})rt1=∂ωΓ
+[p]Γ([p],[g];{σ})rt2. (4)
+2In underlying theory perspective, the loop integration and ”decoupling” limit (or low-energy projection) commute wit h each
+other on such differentiated diagrams, or that the ”decoupli ng” limit can be performed before loop integration is done[2 1–23].4
+Taking the ”low-energy” limit so that {σ}become ”decoupled”, we have:
+Limit{σ}/braceleftBig
+∂ωΓ
+[p][Γ([p],[g];{σ})rt1−Γ([p],[g];{σ})rt2]/bracerightBig
+=∂ωΓ
+[p][Γ([p],[g])rt1−Γ([p],[g])rt2] = 0. (5)
+That means different routings differ at most by an polynomial ΓPolythat would be annihilated by ∂ωΓ
+[p]:
+ΓPoly([p],[g])≡Γ([p],[g])rt1−Γ([p],[g])rt2, (6)
+which could well be absorbed into the ambiguous polynomials in Γ (to be fi xed later through physical
+boundaries) as a reparametrization of the expression.
+Case (b) : In this case the operationabout externalmomenta should alsono t alter the amplitude provided
+the integration is done in underlying theory:
+Γ([p],[g];{σ}) = Γ([p′],[g];{σ}). (7)
+Now apply the differentiation with respect to the external momenta [p]0that are in a subset shared by [ p]
+and [p′] to Eq.(7), we have,
+∂ωΓ
+[p]0[Γ([p],[g];{σ})−Γ([p′],[g];{σ})] = 0. (8)
+Then we can take the ”decoupling” limit and perform the loop integrat ions in the two routing, just as in
+case (a). Again, the net difference between the two routings is at m ost a polynomial with respect to [ p]0that
+could again be absorbed into the ambiguous part, after we return t o the original label of external momenta,
+the same expression for Γ should be restored.
+The multi-loop diagrams could be treated in similar fashion[21–25]: First , differentiate the diagram with
+respect to the momenta that areexternal to the overalldiagram to annihilate the overalldivergences; Second,
+consider each individual sub-diagram that is divergent and different iate it with respect to the momenta that
+areexternal to this sub-diagram to annihilate the ”overall” divergences of the sub-diagram; Third, c ontinue
+this operation till the smallest sub-diagrams are thus differentiated ; Fourth, carry out all the resulting loop
+integrations and integrate back indefinitely with respect to the cor responding momenta that are external
+to the corresponding loops; The final outcome will be a finite expres sions in terms of momenta and masses
+external to the whole diagram, containing ambiguities associated ea ch individual divergent or ill-defined
+loops. This operation naturally dissolves the overlapping divergence s that are notoriously difficult to deal
+with in conventionaltreatments, thanks to the workbyCaswellan d Kennedy[32]. Since the treatmentshould
+be done loop by loop, thus, the theorem of routing applies to each loo p integration as well and finally to the
+whole diagram. Q.E.D.
+This theorem would allow us to choose any routing we like or that is conv enient. Obviously, simple routing
+of external momenta would yield simplicity in calculations.
+III. QED AT ONE-LOOP LEVEL
+We will work with the following standard covariant gauge QED in 3+1-dim ensional spacetime:
+LQED=¯ψ(i/ D−m)ψ−1
+4FµνFµν−1
+2ξ(∂µAµ)2, (9)
+withξbeing the dimensionless gauge parameter. Our metric convention re ads
+gµν=gµν= (+,−,−,−). (10)
+At one-loop level, only the following five elementary vertex functions are superficially UV divergent and
+hence have to be renormalized: the self-energy Σ, the electron-p hoton vertex Λ µ, the vacuum polarization
+Πµν, the three- and four-point photon vertices. The Feynman diagra ms for the first three vertices are given
+in Fig.1. The three-point photon vertex is zero due to Furry’s theor em, while, as will be shown below, the
+four point photon vertex is actually definite in our approach due to g auge invariance.5
+We first consider the vertex functions listed in Fig.1 whose Feynman in tegrals read,
+−iΣ(1)(p)≡(−ie)2/integraldisplayd4l
+(2π)4i/bracketleftbig
+(1−ξ)lκlτ
+l2−gκτ/bracketrightbig
+l2+iǫγκi
+/ l+/ p−m+iǫγτ; (11)
+Λ(1)
+µ(p,˜p)≡(−ie)2/integraldisplayd4l
+(2π)4i/bracketleftbig
+(1−ξ)lκlτ
+l2−gκτ/bracketrightbig
+l2+iǫγκi
+/ l+/ p−m+iǫ
+×γµi
+/ l+/˜p−m+iǫγτ; (12)
+iΠ(1)
+µν(q)≡(−ie)2/integraldisplayd4l
+(2π)4(−1)Tr/bracketleftbigg
+γµi
+/ l−m+iǫγνi
+/ l+/ q−m+iǫ/bracketrightbigg
+, (13)
+wherep,˜pandqdenotetheexternalmomentaforelectronandphotonrespectiv ely. Thesuperficialdivergence
+degrees for the three diagrams are well-known: ω(Σ(1)) = 1,ω(Λ(1)
+µ) = 0,ω(Π(1)
+µν) = 2. Note the routing
+of the external momenta are so chosen that they flow through th e fermionic internal lines, which will yield
+convenience for calculations and a clear diagrammatic interpretatio n of the operation associated with the
+differentiation with respect to external momenta, see below. We will consider other routings for the self-
+energy and the vertex diagrams in the Appendix A.
+A. Differential equations for divergent integrals
+Below, we treat these superficially divergent integrals with the stra tegy described above, according to
+which the three differential equations are,
+∂pα∂pβΣ(1)(p) = Σ(1);αβ(p), (14)
+∂pαΛ(1)
+µ(p,˜p) = Λ(1);α
+µ(p,˜p), (15)
+∂qα∂qβ∂qγΠ(1)
+µν(q) = Π(1);αβγ
+µν(q), (16)
+where the right hand sides (RHS) of these differential equations ar e convergent Feynman diagrams, their
+corresponding loop integrals are listed in Appendix A. Actually, in term s of Feynman diagram the RHS of
+Eqs.(14,15,16) are just obtained through insertion of photon prob es with zero momentum, provided that the
+external momentum flows through internal fermion lines, cf. Fig.2 f or an example.
+The loop integration on the RHS of Eqs.(14,15,16) can now be straight forwardly carried out:
+−iΣ(1);αβ(p) =8ie2
+(4π)2/integraldisplay1
+0z3dz/braceleftBigg
+(z/ p−2m)pαpβ
+z(zp2−m2)2−gαβ(/ p−2m
+z)+γ{αpβ}
+2z(zp2−m2)
++(1−ξ)/bracketleftbigg(p2−m2)/ ppαpβ
+(zp2−m2)3−(p2−m2)(gαβ/ p+γ{αpβ})
+4z(zp2−m2)2
+−(6/ p−2m)pαpβ
+4z(zp2−m2)2+gαβ(/ p−m
+2)+γ{αpβ}
+2z2(zp2−m2)/bracketrightBigg/bracerightBigg
+, (17)
+Λ(1);α
+µ(p,˜p) =−e2
+(4π)2/integraldisplay1
+0dy/integraldisplay1−y
+0dz/braceleftbigg
+−4zγµ(p−P)α
+∆+2z(p−P)αGµ
+∆2
++γσ(1−z)γαγµ(/˜p+m−/ P)−z(/ p+m−/ P)γµγα
+∆γσ
++(1−ξ)(1−y−z)/bracketleftbigg4z(p−P)αF1;µ
+∆3
++[zγα(/ p+m−/ P)+(1−z)/ Pγα]γµ(/˜p+m−/ P)/ P
+∆2
++z/ P(/ p+m−/ P)γµ[(/˜p+m−/ P)γα−γα/ P]
+∆2
++2z(p−P)αF2;µ
+∆2+3zγµ/˜pγα+3zγα/ pγµ+3/ Pγαγµ
+∆6
++/˜pγµγα+12z(p−2P)αγµ+(6z−2)mgα
+µ
+∆/bracketrightbigg/bracerightbigg
+, (18)
+iΠ(1);αβγ
+µν(q) =24ie2
+(4π)2/integraldisplay1
+0z4(1−z)4dz
+
+g{α
+µqβgγ}
+ν+q{µg{α
+ν}gβγ}−2gµνg{αβqγ}
+z2(1−z)2(m2−z(1−z)q2)
++q{µg{α
+ν}qβqγ}−12gµνqαqβqγ+(qµqν−p2gµν)g{αβqγ}
+z(1−z)(m2−z(1−z)q2)2
++8(qµqν−q2gµν)qαqβqγ
+(m2−z(1−z)q2)3/bracerightbigg
+, (19)
+P≡zp+y˜p,∆≡(y+z)m2+P2−zp2−y˜p2,
+Gµ≡γσ(/ p+m−/ P)γµ(/˜p+m−/ P)γσ,
+F1;µ≡/ P(/ p+m−/ P)γµ(/˜p+m−/ P)/ P,
+F2;µ≡/˜pγµ/ p+3γµ/˜p/ P+3/ P/ pγµ+(m2−6P2)γµ+2m(3Pµ−pµ−˜pµ). (20)
+Here symmetrization about Greek indices grouped in the bracket {···}) is frequently used for brevity. The
+three definite functions then determine the original loop amplitudes up to certain degree of ambiguity. Next,
+we proceed to solving the differential equations using the above defi nite functions.
+B. Solutions
+Evidently, the direct integrating back with respect to external mo menta would be cumbersome. Instead of
+doing so, we may first extract the differentiation with respect to ex ternal momenta in the following manner,
+−iΣ(1);αβ(p) =∂pα∂pβ/braceleftbiggie2
+(4π)2/integraldisplay1
+0dz/bracketleftbig
+[(1−2z−ξ)/ p+(3+ξ)m]ln(m2−zp2)
+−(1−ξ)z(p2−m2)/ p
+m2−zp2/bracketrightbigg/bracerightbigg
+, (21)
+Λ(1);α
+µ(p,˜p) =∂pα/braceleftbigg
+−e2
+(4π)2/integraldisplay1
+0dy/integraldisplay1−y
+0dz/bracketleftbigg
+2γµln∆+Gµ
+∆+(1−ξ)(1−y−z)
+×/parenleftbigg
+−6γµln∆+F1;µ
+∆2+F2;µ
+∆/parenrightbigg/bracketrightbigg/bracerightbigg
+, (22)
+iΠ(1);αβγ
+µν(q) =∂qα∂qβ∂qγ/braceleftbigg
+−8ie2
+(4π)2(qµqν−gµνq2)/integraldisplay1
+0dzz(1−z)
+×ln(m2−z(1−z)q2)/bracerightbig
+, (23)
+then the solutions could be readily found as:
+−iΣ(1)(p) =ie2
+(4π)2/braceleftbigg/integraldisplay1
+0dz/bracketleftbig
+[(1−2z−ξ)/ p+(3+ξ)m]ln(m2−zp2)
+−(1−ξ)z(p2−m2)/ p
+m2−zp2/bracketrightbigg
++Cψ/ p+Cm/bracerightbigg
+, (24)
+Λ(1)
+µ(p,˜p) =−e2
+(4π)2/braceleftbigg/integraldisplay1
+0dy/integraldisplay1−y
+0dz/bracketleftbigg
+2γµln∆+Gµ
+∆+(1−ξ)(1−y−z)
+×/parenleftbigg
+−6γµln∆+F1;µ
+∆2+F2;µ
+∆/parenrightbigg/bracketrightbigg
++C⊥γµ/bracerightbigg
+, (25)
+iΠ(1)
+µν(q) =−8ie2
+(4π)2/bracketleftbigg
+(qµqν−gµνq2)/integraldisplay1
+0dzz(1−z)ln(m2−z(1−z)q2)
++CAqµqν+˜CAgµνq2+C{µqν}+Cµν+gµνCΠ;0/bracketrightBig
+, (26)7
+with [C···] (Cµνis traceless, Cµ
+µ= 0) being the corresponding integration constants that are ambig uous at
+present stage. They are uniquely defined only in the underlying form ulation that is well defined at short
+distances, but in the effective theory, i.e., QED, they arefree cons tantsto be determined through appropriate
+boundary conditions[31].
+Actually, for photon vacuum polarization, Lorentz invariance could remove the constants Cµ,Cµν. The
+presence of CΠ;0would violate gaugeinvariance, i.e, making the photon massive. To ens ure gauge invariance,
+we must impose CA+˜CA=CΠ;0= 0. For self-energy, both CψandCmpreserve Lorentz invariance, but are
+gauge dependent. In our previous reports, the constants [ C···] are termed as ”agent”constants as they should
+be consequences of taking the ”decoupling” limit of {σ}. To further determine the residual constants, we
+must resort to more constraints from symmetries and finally from p hysical boundary conditions, which will
+be pursued in the following section. Before imposing boundary condit ions, the net outcomes of conventional
+renormalization programs are also ambiguities, the true reflection o f ill-definedness. The same is also true in
+EG approach. However, our strategy leads to the same outcome w ithneithercomplicated (also unnecessary)
+manipulation of infinities normodification of Feynman rules at loop levels, hence, it is simple. So, the real
+problem is ambiguity (rather than divergence) to be fixed through a ppropriate boundary conditions.
+Here, some more technical remarks are in order: (1) In the differe ntial equation approach, any spurious
+violation of symmetries due to local ambiguities in the loop amplitudes co uld be easily removed. Therefore,
+the physical breakdown of symmetries (i.e., anomalies) must come fr omdefiniteproperties of the loop
+diagrams. Actually, it is shown in our previous studies that chiral and trace anomalies[33–41] do originate
+from the existence of definite or nonlocal terms like ( kµkν/k2)ˇΓ in certain divergent integrals3[26, 27, 42, 43].
+After contraction with kµorgµν, they give rise to local operators: (a) kµ·(kµkν/k2)ˇΓ =kνˇΓ (chiral
+anomaly)[26, 27, 42], (b) gµν·(kµkν/k2)ˇΓ =ˇΓ (trace anomaly)[42, 43]. In contrast to spontaneous breaking
+of global symmetries where massless poles appear at tree level as c onsequences, certain massless poles show
+upinloopamplitudesasoriginsofthequantummechanicalviolationofc anonicalsymmetries. (2)Amusingly,
+this perspective of anomalies also imply that there may exist propert ies in divergent integrals that are both
+definite (nonlocal) and unexpected from canonical deductions. Th erefore, such divergent diagrams contain
+more about quantum field theories[44, 45]. These properties might b e viewed as indirect but definite links to
+underlying theory. (3) The differential equations used here also imp ly some structural relations between the
+Feynman diagrams. Actually, in QED, as there is no pure gauge boson loops, all the differentiations with
+respect to external momenta effectively amount to inserting the e lementary QED vertex, or external photon
+with zeromomentum. Thefinal resultamountsto theWardidentity in differential form. Itwill be interesting
+to see what will happen in more complicated theories with pure gauge b oson loops, like non-Abelian theories.
+(4) Technically, one could not obtain a polynomial/local term from con volution of non-polynomial/nonlocal
+functions. A mathematically natural way to yield a polynomial/local te rm is to perform certain limit
+operation. To our interest, this mathematical scenario corrobor ates the underlying theory scenario: The
+computationshouldbefollowedbythe”low-energy”limitduetothewid eseparationofscales,thensomelocal
+terms arise from the ”low-energy” limit, indicating the existence of u nderlying structures. Conventionally,
+a regularization is introduced as a rough substitute of the underlyin g structures and is taken to zero in the
+final stage after removing the ”dusts” (infinities) brought about by the regularization. In our strategy, we
+simply appreciate the existence of the underlying structures and t hen solve the differential equations that
+must be satisfied by the limiting objects.
+C. Comparing with Dimensional Regularization
+Now we list out the results from dimensional scheme for a comparison . Our conclusion is that at one-loop
+level the only difference between the two approacheslies in the local part, which is in fact prescription depen-
+dent. The definite (nonlocal) part must be prescriptionindependen t and hence physical. The twoapproaches
+are equivalent if one takes the local part in dimensional scheme as am biguous. From this equivalence one
+could devise a simple method for computation: do the calculations in dim ensional regularization, then re-
+place the local part with ambiguous polynomials of the corresponding degree. However, this equivalence is
+3Here,kdenotes the external momentum at the corresponding vertex, whileˇΓ the additional polynomial factors in terms of
+external momenta at other vertices.8
+only valid after sub-divergences are subtracted in case of multi-loo p diagrams [46].
+The results from dimensional scheme read,
+−iΣ(1)(p;ǫ) =ie2
+(4π)2/braceleftbigg/integraldisplay1
+0dz/bracketleftbigg
+[(1−2z−ξ)/ p+(3+ξ)m]lnm2−zp2
+4πµ2
+−(1−ξ)z(p2−m2)/ p
+m2−zp2/bracketrightbigg
++/parenleftbigg
+ǫ−1−γE+1
+2/parenrightbigg
+(/ p−4m)
+−(1−ξ)/parenleftbig
+ǫ−1−γE+1/parenrightbig
+(/ p−m)/bracerightbig
++o(ǫ), (27)
+Λ(1)
+µ(p,˜p;ǫ) =−e2
+(4π)2/braceleftbigg/integraldisplay1
+0dy/integraldisplay1−y
+0dz/bracketleftbigg
+2γµln∆+Gµ
+∆+(1−ξ)(1−y−z)
+×/parenleftbigg
+−6γµln∆+F1;µ
+∆2+F2;µ
+∆/parenrightbigg/bracketrightbigg
+−ǫ−1+γE+2
++(1−ξ)/parenleftbigg
+ǫ−1−γE−5
+6/parenrightbigg
+γµ/bracerightbigg
++o(ǫ), (28)
+iΠ(1)
+µν(q;ǫ) =−8ie2
+(4π)2/bracketleftbigg
+(qµqν−gµνq2)/integraldisplay1
+0dzz(1−z)lnm2−z(1−z)q2
+4πµ2/bracketrightbigg
++4ie2
+3(4π)2/parenleftbig
+ǫ−1−γE/parenrightbig
+(qµqν−gµνq2)+o(ǫ). (29)
+Wenotethatintermsoftheaboveparametrization,thefollowingco rrespondencebetweentheagentconstants
+and the local constants in dimensional regularization could be easily e stablished:
+Cψ↔Cψ(ǫ)≡ξ/parenleftbig
+ǫ−1−γE+1+ln/parenleftbig
+4πµ2/parenrightbig/parenrightbig
+−1
+2,
+Cm
+m↔Cm(ǫ)
+m≡ −(3+ξ)/parenleftbig
+ǫ−1−γE+ln/parenleftbig
+4πµ2/parenrightbig/parenrightbig
+−1−ξ,
+C⊥↔C⊥(ǫ)≡ −ξ/parenleftbigg
+ǫ−1−γE−5
+6+ln/parenleftbig
+4πµ2/parenrightbig/parenrightbigg
++7
+6,
+CA↔CA(ǫ)≡ −1
+6/parenleftbig
+ǫ−1−γE+ln/parenleftbig
+4πµ2/parenrightbig/parenrightbig
+. (30)
+Obviously, the dimensional regularized results exactly agree with ou rs obtained from Eqs.(14,15,16) after
+replacing the divergent constants by the agent constants followin g the correspondence (30). In other words,
+the dimensional regularization results (after or before subtract ion) could be seen as a particular solution to
+our differential equations at one-loop level. At multi-loop level, the dim ensional results will also satisfy such
+differential equations loop by loop after all the corresponding sub- divergences are removed[46].
+D. Reducing ambiguities with Ward identities
+Since gauge invariance is encoded in the Ward identities among various vertices, it is natural to ask if these
+identities are satisfied for the loop amplitudes computed in our appro ach. Specifically, we wish to examine
+the status of the following Ward identity for self-energy and verte x function,
+∂pµΣ(1)(p) =−Λ(1)
+µ(p,p), (31)
+in our approach.
+To proceed, we make use of the results in dimensional regularization and look at the Feynman gauge
+components, i.e., those remain when ξ= 1, the rest components will be delegated to Appendix B. Carrying
+out the parametric integrations, we have,
+−iΣ(1)(p;ǫ) =ie2
+(4π)2/braceleftbigg/bracketleftbigg
+ǫ−1−γE+1−lnm2
+4πµ2+δ+(δ2−1)lnδ−1
+δ/bracketrightbigg
+/ p9
+−/bracketleftbigg
+ǫ−1−γE+3
+2−lnm2
+4πµ2+(δ−1)lnδ−1
+δ/bracketrightbigg
+4m/bracerightbigg
+=ie2
+(4π)2/braceleftbigg/bracketleftbigg
+Cψ(ǫ)/bardblξ=1+1
+2−lnm2+δ+(δ2−1)lnδ−1
+δ/bracketrightbigg
+/ p
+−/bracketleftbiggCm(ǫ)
+4m/bardblξ=1+1−lnm2+(δ−1)lnδ−1
+δ/bracketrightbigg
+4m/bracerightbigg
+, (32)
+Λ(1)
+µ(p,p;ǫ) =e2
+(4π)2/braceleftbigg/bracketleftbigg
+ǫ−1−γE+1−lnm2
+4πµ2+δ+(δ2−1)lnδ−1
+δ/bracketrightbigg
+γµ
+−2/bracketleftbigg
+2/parenleftbigg
+δlnδ−1
+δ+1/parenrightbigg
+(δ/ p−2m)+/ p/bracketrightbiggpµ
+p2/bracerightbigg
+=e2
+(4π)2/braceleftbigg/bracketleftbigg
+−C⊥(ǫ)/bardblξ=1+3−lnm2+δ+(δ2−1)lnδ−1
+δ/bracketrightbigg
+γµ
+−2/bracketleftbigg
+2/parenleftbigg
+δlnδ−1
+δ+1/parenrightbigg
+(δ/ p−2m)+/ p/bracketrightbiggpµ
+p2/bracerightbigg
+, (33)
+withδ≡m2
+p2. The constants Cψ(ǫ),C⊥(ǫ) have been defined in the correspondence (30). Now differentiatin g
+Σ(1)(p;ǫ) with respect to pµ, it is trivial to verify the Ward identity (31) in dimensional regularizat ion:
+−i∂pµΣ(1)(p;ǫ) =ie2
+(4π)2/braceleftbigg/bracketleftbigg
+ǫ−1−γE+1−lnm2
+4πµ2+δ+(δ2−1)lnδ−1
+δ/bracketrightbigg
+γµ
+−2/bracketleftbigg
+2/parenleftbigg
+δlnδ−1
+δ+1/parenrightbigg
+(δ/ p−2m)+/ p/bracketrightbiggpµ
+p2/bracerightbigg
+=iΛ(1)
+µ(p,p;ǫ). (34)
+It is easy to see that in dimensional regularization this Ward identity is ensured by the following relation,
+Cψ(ǫ)/bardblξ=1+C⊥(ǫ)/bardblξ=1=5
+2.
+In general gauge, we have (Cf. Appendix B):
+Cψ(ǫ)+C⊥(ǫ) =4+11ξ
+6. (35)
+Evidently, replacing Cψ(ǫ),C⊥(ǫ) withCψ,C⊥, one enters our approach. Thus the Ward identity (31)
+could hold in our approach only when the agent constants CψandC⊥satisfy the same relation as that
+betweenCψ(ǫ) andC⊥(ǫ):
+Cψ+C⊥=4+11ξ
+6. (36)
+That means, gauge invariance constrains the agent constants CψandC⊥according to Eq.(36). Obviously,
+such constraints are just welcome in our approach to reduce the a mbiguities. Of course, at higher orders,
+such constraints must be also consistently imposed. At one-loop lev el, there is in principle no obstacle to
+remove the local violations of any symmetries. As removal of the loc al ambiguities never affects the nonlocal
+and hence definite part, any physical anomaly must be originated fr om definite and hence prescription-
+independent sources. As mentioned above, at least for chiral and trace anomalies, this is indeed the case:
+a type of mass-independent and definite terms is the true source o f quantum mechanical violations of the
+canonical chiral and scale transformations[26, 27, 42, 43].
+Therefore, due to Lorentz and gauge invariance, we are left with t hree ambiguous constants to fix: Cψ,
+CmandCA. This statement is, however, a little bit cursory before the finitene ss of the rest superficially
+divergent diagrams, i.e., the three- and four-photon vertex func tions, is established. Below we turn to
+these diagrams. Since the three-photon vertex does not contrib ute to cross sections or physical observables
+according to Furry’s theorem, we only need to verify the finiteness of the four-photon vertex, that is, there
+is no ultraviolet ambiguity in this vertex function which is again ensured by gauge invariance.10
+E. Finiteness of the four-photon vertex
+To verify that the four-photon vertex is free of ambiguity, it suffic es to show that the most divergent piece
+in each individual diagram cancel out against each other due to gaug e invariance.
+The external momenta and polarizations of the four external pho ton are arranged in the following manner:
+(p1,µ),(p2,ν),(p3,ρ),(p4,σ) withp4=−p1−p2−p3due to conservationof the four dimensional momentum,
+i.e.,p1always enters at vertex µ,p2atν, etc. The four-photon vertex consists of six inequivalent individua l
+Feynman diagrams that differ from each other by the arrangement of the relative positions of the external
+photons along the internal fermion loop. It is convenient to fix the p osition of the first photon of ( p1,µ) and
+let the other photons move to obtain the rest inequivalent loop diagr ams. The notations for the four-photon
+vertex and the six individual diagrams are as follows,
+Γµνρσ(p1,p2,p3,p4) = Γ 1;µνρσ(p1,p2,p3,p4)+Γ2;µρνσ(p1,p3,p2,p4)
++Γ3;µνσρ(p1,p2,p4,p3)+Γ4;µσρν(p1,p4,p3,p2)
++Γ5;µσνρ(p1,p4,p2,p3)+Γ6;µρσν(p1,p3,p4,p2). (37)
+The Feynman integral for Γ 1;µνρσreads,
+iΓ1;µνρσ= (−ie)4/integraldisplay−d4l
+(2π)4Tr/parenleftbiggi
+/ l−m+iǫγµi
+/ l+/ p1−m+iǫγν
+×i
+/ l+/ p1+/ p2−m+iǫγρi
+/ l+/ p1+/ p2+/ p3−m+iǫγσ/parenrightbigg
+. (38)
+By rearranging the external momenta and the elementary vertice s, one could readily obtain the integrals for
+the rest five diagrams. Since the superficial divergence degree of such diagrams is 0, then it suffices to verify
+that the logarithmic divergence is absent in the sum of the six diagram s in order that gauge invariance is
+preserved. Obviously, it suffices to do computation with Γ 1;µνρσas others could be obtained by permutations
+of the vertices ( p1,µ),(p2,ν),···.
+Differentiating the integral in Eq.(38) with respect to p1;α, the resulting integral becomes definite and
+gives,
+∂p1;αiΓ1;µνρσ=8ie4
+(4π)2T1;µνρσ/integraldisplay1
+0dx/integraldisplayx
+0dy/integraldisplayy
+0dz∂p1;αln˜∆1+∂p1;αiˇΓ1;µνρσ, (39)
+T1;µνρσ≡gµνgρσ−2gµρgνσ+gµσgνρ, (40)
+˜∆1≡m2−(x−x2)p2
+1−(y−y2)p2
+2−(z−z2)p2
+3
+−2y(1−x)p1p2−2z(1−x)p1p3−2z(1−y)p2p3, (41)
+where the ˇΓ1;µνρσdenotes the definite parts after integrating back with respect to p1;αand is not concerned
+here. Next, integrating back with respect to p1;α, the diagram Γ 1;µνρσcould be parametrized as follows:
+iΓ1;µνρσ=8ie4
+(4π)2T1;µνρσ/integraldisplay1
+0dx/integraldisplayx
+0dy/integraldisplayy
+0dzln˜∆1
+C1;4γ+ˇΓ1;µνρσ, (42)
+where an integration constant C1;4γis shown explicitly, which is exactly the ambiguity that corresponds to
+the logarithmic divergence of the integral in Eq.(38). The results fo r other diagrams are given in Appendix
+C. For later convenience, we integrate by parts with respect to th e Feynman parametric integration to arrive
+at the following expressions for Γ 1;µνρσ,
+iΓ1;µνρσ=4ie4
+3(4π)2T1;µνρσlnm2
+C1;4γ+˜Γ1;µνρσ, (43)
+where the definite terms arisingfrom this operation have been abso rbed into ˇΓ1;µνρσand collectively denoted
+as˜Γ1;µνρσ, cf. Appendix D.
+Now, in order that the ill-defined piece in the full amplitude vanish
+iΓµνρσ/bardblill-defined =4ie4
+3(4π)2/bracketleftBigg6/summationdisplay
+k=1Tk;µνρσlnm2
+Ck;4γ/bracketrightBigg
+= 0, (44)11
+with/summationtext6
+k=1Tk;µνρσ= 0, the constant Ck;4γmust be identical for each k(say,Ck;4γ=¯C4γ,∀k). Otherwise,
+gauge invariance will be violated. Thus, similar as the case of vacuum p olarization for two-photon vertex,
+gauge invariance also remove the ambiguities in the four-photon ver tex function.
+In fact, after setting all the external momenta to zero, we will en d up with the same ill-defined integral in
+all the six diagrams:
+/integraldisplayd4l
+(2π)4lalblcld
+(l2−m2)4=−i
+24(4π)2[gabgcd+···]lnm2
+¯C4γ. (45)
+Thus it is natural to deem that the constants Ck;4γare identical in any reasonable prescription.
+In dimensional regularization, one would find the following correspon dence in each of the six diagrams:
+lnm2
+Ck;4γ↔ −ǫ−1+γE+lnm2
+4πµ2,
+which means the full four-photon vertex is finite in dimensional regu larization due to the mechanism demon-
+strated above.
+F. Brief summary of differential equation approach to QED at o ne-loop level
+Now, we have treated all the superficially divergent or ill-defined loop diagrams of QED at one-loop level,
+without resorting to any artificial deformation of spacetime and sy mmetries or any form of cutoffs. At one-
+loop level, only three local ambiguities remain to be fixed in QED: Cψ,CmandCAthat appear in self-energy
+Σ, vertex Λ µand vacuum polarization Π µν, respectively. This is a naturalconclusion following from Lorentz
+and gauge invariance and the existence of a complete underlying the ory.
+In particular, from our treatment, one could naturally infer that, any fundamental theory that is well de-
+fined and underlies the presentlyknownQEDmust alsoyield the same o requivalentfunctional expressionsas
+given here after taking the ”decoupling” limit, the only difference is th at the unknown agent constants could
+be unambiguously computed in such underlying theory. Therefore, in the underlying theory perspective, the
+QED processes defined by the elementary Feynman diagrams (in cov ariant gauge) represented by Fig.1 must
+take the following form,
+−iΣ(1)(p) =ie2
+(4π)2/braceleftbigg/integraldisplay1
+0dz/bracketleftbigg
+[(1−2z−ξ)/ p+(3+ξ)m]lnm2−zp2
+µ2
+UD
+−(1−ξ)z(p2−m2)/ p
+m2−zp2/bracketrightbigg
++c0
+ψ/ p+c0
+mm/bracerightbigg
+, (46)
+Λ(1)
+µ(p,˜p) =−e2
+(4π)2/braceleftbigg/integraldisplay1
+0dy/integraldisplay1−y
+0dz/bracketleftbigg
+2γµln∆+Gµ
+∆+(1−ξ)(1−y−z)
+×/parenleftbigg
+−6γµln∆+F1;µ
+∆2+F2;µ
+∆/parenrightbigg/bracketrightbigg
++/parenleftbigg4+11ξ
+6−c0
+ψ/parenrightbigg
+γµ/bracerightbigg
+, (47)
+iΠ(1)
+µν(q) =8ie2(gµνq2−qµqν)
+(4π)2/bracketleftbigg/integraldisplay1
+0dzz(1−z)lnm2−z(1−z)q2
+µ2
+UD+c0
+A/bracketrightbigg
+,
+(48)
+where we have explicitly ”separated out” a dimensional constant µUDfrom all the constants C···to balance
+the dimension in the logarithmic parts, the residual parts are purely dimensionless numbers, each denoted
+asc0
+···. This way of parametrizing the constants will be more precise and us eful for the following discussions.
+We should also note that, in the underlying theory perspective, all t he constants or parameters that
+appear in the Lagrangian should be defined from the ”decoupling” limit of the underlying theory also as
+”agents”. In our view, these ”agents” in lagrangian are only elemen tary from the perspective of (effective)
+field theories, not necessary so in the perspective of underlying th eory. As the agent constants appear in the
+local part of the elementary vertexfunctions within perturbative framework, the Lagrangianconstants would
+mix with these local terms containing the agent constants in the cor responding vertex functions. This is in
+fact the origin of renormalization group equations in our approach, without resorting to the conventional
+renormalization theory, see, Sec. IV for a brief discussion.12
+IV. REMOVING AMBIGUITIES AND ”RENORMALIZATION” SCHEME
+Now we turn to determining the ambiguous agent constants. Obviou sly, the conventional renormalization
+schemes would be reproduced provided that the ambiguous consta nts as well as mass and couplings are fixed
+through the conventional renormalization conditions. An appropr iate choice of the whole set {m,e;µ,c0
+···}
+defines a renormalizationprescription or scheme. So, our simple str ategy is compatible with the conventional
+renormalization programs. In principle, any scheme can be employed provided the same scattering matrix
+elements or the same physical observables could be obtained. This f reedom just leads to the renormalization
+group in the general sense of St¨ uckelberg-Peterman[47]. Fixing t he dimensionless constants in a scheme and
+letting the scale µUDvary, one could obtain the renormalization group equations in the na rrow sense.
+To proceed, the following parametrization would be used useful,
+Σ(1)(p) =A(1)(p2)/ p+B(1)(p2)m, (49)
+¯u(p)Λ(1)
+µu(˜p) = ¯u(p)γµu(˜p)F(1)(k2)+i¯u(p)σµνkνu(˜p)
+2mG(1)(k2), (50)
+Π(1)
+µν(q) =−Π(1)(q2)(gµνq2−qµqν), (51)
+A(1)(p2)≡ −α
+4π/bracketleftbigg
+ξ/parenleftbigg/parenleftbig
+δ2−1/parenrightbig
+lnδ−1
+δ+δ−lnm2
+µ2
+UD/parenrightbigg
++1
+2+c0
+ψ/bracketrightbigg
+, (52)
+B(1)(p2)≡α
+4π/bracketleftbigg
+(3+ξ)/parenleftbigg
+(δ−1)lnδ−1
+δ+1−lnm2
+µ2
+UD/parenrightbigg
+−c0
+m/bracketrightbigg
+, (53)
+F(1)(k2)≡α
+4π/bracketleftbigg
+2arccoth[θk]/bracketleftbigg/parenleftbigg
+1+lnλγ
+m2/parenrightbigg/parenleftbigg
+θk+1
+θk/parenrightbigg
++θk/bracketrightbigg
++2
+−Ξ(k2;1−ξ)−lnm2
+µ2
+UD−c0
+⊥/bracketrightbigg
+, (54)
+G(1)(k2)≡α
+2π/parenleftbigg
+θk−1
+θk/parenrightbigg
+arccoth[θk], (55)
+Π(1)(q2)≡α
+3π/bracketleftbigg
+θq/parenleftbig
+θ2
+q−3/parenrightbig
+arccoth[θq]−θ2
+q−lnm2
+µ2
+UD+8
+3−6c0
+A/bracketrightbigg
+, (56)
+δ=m2
+p2, k=p−˜p, θx≡/radicalbigg
+1−4m2
+x2. (57)
+Note that the vertex function is sandwiched between on-shell sta tes [u(˜p),¯u(p)] of electron. Obviously, the
+scalar factors A(1),B(1),F(1)and Π(1)contain ultra-violet ambiguities (depending on c0
+ψ,c0
+m,c0
+⊥andc0
+A,
+respectively). Moreover, the first three factors are also gauge dependent and infrared singular when the
+external momenta are on mass shell. In F(1), we have explicitly put in a photon mass squared ( λγ) to
+regulate the infrared singularity, with Ξ denoting the terms propor tional to 1 −ξ. In contrast, the scalar
+factorG(1)is both ultra-violet and infrared finite and also gauge independent. I ts value at zero momentum
+transfer (k2= 0) gives the well-known anomalous magnetic moment of electron at o ne-loop level:
+G(1)(0) =α
+2π.
+A. Various boundary conditions for vertex function
+Now we impose various conditions on the elementary vertices (here, Σ(1),Λ(10
+µ,Π(1)
+µν)4. In doing so, both
+the tree level parameters ( m,e) and the agent constants must be understood as scheme-depen dent so that
+the observables computed using these parameters and vertices a re physical and independent of schemes.
+The lagrangian or tree level mass or coupling needs not be just the p hysical one, which may differ by a
+4They could be found in many textbooks for field theory, see, e. g. Ref.[48]13
+”(re)normalization” constant in the conventional terminology. In our strategy, such ”(re)normalization”
+constants could also be introduced but are nevertheless ultra-vio let finite. Actually, as is already pointed out
+in Ref.[49]: ”the renormalization of mass and field has nothing directly t o do with the presence of infinities,
+and would be necessary even in a theory in which all momentum space in tegrals are convergent”.
+1. On-shell conditions
+First, let us consider the so-called on-shell conditions in Feynman ga uge, which read,
+Σ(1)/bardbl/ p=m= 0, ∂/ pΣ(1)/bardbl/ p=m= 0,Λ(1)
+µ/bardbl/ p=/˜p=m= 0,Π(1)/bardblq2=0= 0. (58)
+Using the expressions given above, we could find that:
+Σ(1)/bardbl/ p=m=α
+4π/parenleftbigg
+−3lnm2
+µ2
+UD+5
+2−c0
+ψ−c0
+m/parenrightbigg
+m,
+∂/ pΣ(1)/bardbl/ p=m=α
+4π/parenleftbigg
+lnm2
+µ2
+UD−2lnλγ
+m2−7
+2−c0
+ψ/parenrightbigg
+,
+Λ(1)
+µ/bardbl/ p=/˜p=m=α
+4π/parenleftbigg
+−lnm2
+µ2
+UD+2lnλγ
+m2+7
+2+c0
+ψ/parenrightbigg
+γµ,
+Π(1)/bardblq2=0=α
+3π/parenleftbigg
+−lnm2
+µ2
+UD−6c0
+A/parenrightbigg
+, (59)
+wherec0
+ψ+c0
+⊥= 5/2 has been used. Note that all the infrared singularities have been r egularized with a
+photon mass, and mnow denotes the physical mass for electron. According to conven tional programs, the
+couplingeorα=e2/(4π) appearing in these conditions is understood to be the physical one .
+Then the on-shell conditions could be fulfilled provided the constant s [c0
+···] take the following values:
+c0
+ψ= lnm2
+µ2
+UD−2lnλγ
+m2−7
+2=5
+2−c0
+⊥,
+c0
+m=−4lnm2
+µ2
+UD+2lnλγ
+m2+6, c0
+A=−1
+6lnm2
+µ2
+UD. (60)
+In this manner, all the ambiguities are gone in the vertices, including t he scaleµUD:
+A(1)
+os(p2)≡ −α
+4π/bracketleftbigg/parenleftbig
+δ2−1/parenrightbig
+lnδ−1
+δ+δ−3−2lnλγ
+m2/bracketrightbigg
+, (61)
+B(1)
+os(p2)≡α
+4π/bracketleftbigg
+4(δ−1)lnδ−1
+δ−2−2lnλγ
+m2/bracketrightbigg
+, (62)
+F(1)
+os(k2)≡α
+4π/bracketleftbigg
+2arccoth[θk]/bracketleftbigg/parenleftbigg
+1+lnλγ
+m2/parenrightbigg/parenleftbigg
+θk+1
+θk/parenrightbigg
++θk/bracketrightbigg
+−4−2lnλγ
+m2/bracketrightbigg
+(63)
+Π(1)
+os(q2)≡α
+3π/bracketleftbigg
+θq/parenleftbig
+θ2
+q−3/parenrightbig
+arccoth[θq]−θ2
+q+8
+3/bracketrightbigg
+. (64)
+The on-shell scheme works for field theories without unstable fields as elementary quantum degrees of free-
+dom. The treelevel massis identified asthe physicalpole mass. But f or complicatedtheories likeelectroweak
+with unstable sectors, the on-shell scheme runs into trouble, see Ref.[50–59].
+2. ”Minimal” conditions
+In analogy to the famous ”Minimal Subtraction”, one may define a sc heme where [ c0
+···] are removed so that
+only the ∼lnµ2pieces are left over, which will lead to the well-known mass-independe nt running behavior.
+We will denote this scheme as ”Ms”, which may also be interpreted as ” Minimal specification”.14
+Sincec0
+ψandc0
+⊥can not be zero at the same time due to gauge invariance, we choose the following:
+c0
+ψ+1
+2=c0
+m=c0
+A= 0, c0
+⊥= 3. (65)
+Then the prescription-dependent scalar factors would take the f ollowing neater forms (in Feynman gauge):
+A(1)
+Ms(p2)≡ −α
+4π/bracketleftbigg/parenleftbig
+δ2−1/parenrightbig
+lnδ−1
+δ+δ−lnm2
+µ2
+Ms/bracketrightbigg
+, (66)
+B(1)
+Ms(p2)≡α
+π/bracketleftbigg
+(δ−1)lnδ−1
+δ+1−lnm2
+µ2
+Ms/bracketrightbigg
+, (67)
+F(1)
+Ms(k2)≡α
+4π/bracketleftbigg
+2arccoth[θk]/bracketleftbigg/parenleftbigg
+1+lnλγ
+m2/parenrightbigg/parenleftbigg
+θk+1
+θk/parenrightbigg
++θk/bracketrightbigg
+−1−lnm2
+µ2
+Ms/bracketrightbigg
+, (68)
+Π(1)
+Ms(q2)≡α
+3π/bracketleftbigg
+θq/parenleftbig
+θ2
+q−3/parenrightbig
+arccoth[θq]−θ2
+q+8
+3−lnm2
+µ2
+Ms/bracketrightbigg
+. (69)
+Here the subscript ” Ms” for massmand coupling eare omitted to avoid heavy symbolism and the scale µUD
+is now replaced by µMs. Of course, the concrete values for mandein this scheme need to be determined in
+terms of physical values of mass using appropriate physical obser vables[31]. In this prescription, the scalar
+factorsAandBat general off-shell momenta are not beset with infrared singularit ies in contrast to the
+on-shell prescription.
+3. Boundary conditions at vanishing off-shell momenta
+One could also impose boundary conditions at off-shell momenta. Amo ng these off-shell prescriptions, the
+zero momenta case is much simpler for QED with massive fermion. In Fe ynman gauge, we have:
+Σ(1)/bardbl/ p=0=α
+4π/parenleftbigg
+−4lnm2
+µ2
+UD−c0
+m/parenrightbigg
+m,
+∂/ pΣ(1)/bardbl/ p=0=α
+4π/parenleftbigg
+lnm2
+µ2
+UD−c0
+ψ/parenrightbigg
+,
+Λ(1)
+µ/bardbl/ p=/˜p=0=α
+4π/parenleftbigg
+−lnm2
+µ2
+UD+5
+2−c0
+⊥/parenrightbigg
+γµ,
+Π(1)/bardblq=0=α
+3π/parenleftbigg
+−lnm2
+µ2
+UD−6c0
+A/parenrightbigg
+, (70)
+so they vanish if
+c0
+ψ= lnm2
+µ2
+UD=5
+2−c0
+⊥, c0
+m=−4lnm2
+µ2
+UD, c0
+A=−1
+6lnm2
+µ2
+UD. (71)
+Then the scalar factors read:
+A(1)
+zm(p2)≡ −α
+4π/bracketleftbigg/parenleftbig
+δ2−1/parenrightbig
+lnδ−1
+δ+δ+1
+2/bracketrightbigg
+, (72)
+B(1)
+zm(p2)≡α
+π/bracketleftbigg
+(δ−1)lnδ−1
+δ+1/bracketrightbigg
+, (73)
+F(1)
+zm(k2)≡α
+4π/bracketleftbigg
+2arccoth[θk]/bracketleftbigg/parenleftbigg
+1+lnλγ
+m2/parenrightbigg/parenleftbigg
+θk+1
+θk/parenrightbigg
++θk/bracketrightbigg
+−1
+2/bracketrightbigg
+, (74)
+Π(1)
+zm(q2)≡α
+3π/bracketleftbigg
+θq/parenleftbig
+θ2
+q−3/parenrightbig
+arccoth[θq]−θ2
+q+8
+3/bracketrightbigg
+. (75)15
+The nice point of off-shell conditions at zero momenta (hence the su bscript ”zm”) is that, the infrared singu-
+larities do not show up in the scalar factors at general off-shell mom enta, just like in the ” Ms” prescription.
+Such conditions possess an intuitive interpretation: at zero momen ta, quantum fluctuations are suppressed,
+and an electron would serve as a classical static source of electrom agnetic field, with the tree level coupling
+”e” being pertinent to static or Coulomb charge somehow.
+4. Physical boundary conditions in terms of observables
+In principle, boundary conditions may be imposed upon many other fie ld-theoretical objects that are
+constructed with elementary vertex functions or Green function . It would be better to work with the
+observables that could be readily confronted with physical data an d depend on the elementary parameters
+(masses, couplings, and agent constants) in an as simple as possible manner, sparing the intermediate
+renormalization.
+For example, one may proceed as below in QED: First, physical values are assigned to the Lagrangian
+parameters (masses and couplings), the agent constants are tr eated as unknown or arbitrary; Second, appro-
+priate observables are selected that could be readily expressed in t erms of Feynman diagrams that definitely
+contain the agent constants; Finally, the agent constants could b e fixed in terms of the physical parameters
+and data of these observables. In formulae,
+(1){m,e;[C···]} ⇒ {m(phy),e(phy);[C(phy)
+···]},
+(2)Qn/parenleftBig
+m(phy),e(phy);[C(phy)
+···]/parenrightBig
+=Q(data)
+n, n= 1,2,···
+⇒C(phy)
+···=f···/parenleftBig
+[Q(data)
+n];m(phy),e(phy)/parenrightBig
+. (76)
+The set of observables to be employed for this purpose should fulfill some natural requirements such as
+simplicityandanalyticitywithrespecttotheirdependenceuponmass ,couplingandtheagentconstants,well-
+definedness in the infrared, etc. In principle, even theories like qua ntum chromodynamics (QCD) that are
+elusiveofphysicalpole massesofelementaryquantumfieldscanbe t reatedin thismanner, wherethe physical
+contents of the lagrangian or tree level masses and couplings beco mes a nontrivial issue. Such imposition
+of physical boundary conditions is general and natural from unde rlying theory perspective. Actually, this is
+exactly what is done in literature for the determination of the ”phys ical” coupling of strong interaction, see,
+e.g., Ref.[31], chapter 12.
+5. A remark on boundary conditions
+In order that the same observables are obtained, different choice s of ”intermediate” boundary conditions
+or prescriptions for vertex functions must be related to each oth er somehow, or more specifically, the trans-
+formations across different prescriptions defined by the boundar y conditions should in principle be feasible,
+at least within perturbation theory. In conventional programs, t his is the so-called scheme dependence in
+perturbation theory, where different renormalization conditions o r prescriptions, differ by a set of finite
+renormalization constants, and the observables computed in differ ent prescriptions should agree with each
+other to the perturbation order computed. This is in fact encoded in the renormalization group equation
+in the sense of St¨ uckelberg-Peterman as mentioned in the very be ginning of this section. Furthermore, a
+general prescription would also contain an arbitrary scale as ”reno rmalization point”. The variation of this
+arbitrary scale within the specified prescription should not affect ob servables, which would in effect lead to
+renormalization group equations that are themselves prescription - or scheme-dependent.
+The above statements make perfect sense within perturbative fr amework. However, things might become
+complicated in nonperturbative contexts. As an simple example, we r efer to our previous work[28], where
+it is shown that some scheme in incompatible with dynamical symmetry b reaking of massless λφ4theory
+in nonperturbative context. Strictly speaking, any intermediate b oundary conditions or prescriptions to be
+employed must be in the same ”orbit” as the physical boundaries. Th is in turn requires a complete solution
+of the full theory of quantum fields, which is presently out of our re ach. So, scheme dependence is a real
+issue for quantum field theories beyond perturbative regime.16
+6. Influences of infrared divergences
+It is well-known that QED is beset with infrared divergences. For on- shell scheme, infrared divergences
+show up already in the boundary conditions for the loop amplitudes, a s is evident above. Simply speaking,
+infrared singularity arises because Fock states could not work well in presence of massless particles. Thus
+introducing certain extent of coherence, the infrared divergenc es in QED could be removed by use of Bloch-
+Nordsieck theorem[60], which has been described in detail in many sta ndard field theory textbooks, e.g.,
+[31, 48, 49]. At one loop level, one regularize the infrared singularities in the loop diagrams somehow (here
+in this report, in terms of a fictitious photon mass) and they will canc el out against those from real diagrams
+in physical observables.
+Of course, for more complicated non-Abelian gauge theories, seve re infrared divergences are present and
+their treatments aremore involved. Nevertheless, as our approa chdo not alter anything in the infrared, so all
+the conventional methods for dealing with infrared singularities or s imilar singularities except the ultraviolet
+ones may well apply. As is clear from our above discussions, one may w ork with a set of renormalization
+conditions that are not plagued by infrared singularities to avoid fur ther complications, this could be done
+with conditions or ”boundaries” defined at off-shell momenta where everything is infrared finite[61, 62], e.g.,
+at zero four-momenta as illustrated above.
+B. Lagrangian perspective
+In this subsection, we reexamine the issue in terms of lagrangian or a ction, which allows us to examine
+a number of issues in a more transparent manner. Such effort is also helpful to resolve some conceptual
+difficulties that have been ”afflicting” the conventional practices.
+1. Brief review of conventional programs
+The conventional program of renormalization starts with a bare lag rangian:
+L(ψB,Aµ
+B;mB,eB) =¯ψB(i/ DB−mB)ψB−1
+4FB;µνFµν
+B−1
+2ξB(∂µAµ
+B)2, (77)
+withDµ
+B=∂µ−ieBAµ
+B. Then all the 1PI vertices (Γn
+B([p1,···,pn],mB,eB)) are renormalized through the
+following renormalization rescaling,
+ψB=Z1/2
+ψψ, Aµ
+B=Z1/2
+AAµ, mB=Zm
+Zψm, eB= (ZA)−1/2e, ξB=ZAξ, (78)
+and
+L(ψB,Aµ
+B;mB,eB) =Zψ¯ψi/ Dψ−Zmm¯ψψ−ZA
+4FµνFµν−1
+2ξ(∂µAµ)2
+=L(ψ,Aµ;m,e)+∆L(ψ,Aµ;Zψ−1,Zm−1,ZA−1), (79)
+so that
+Γ(n)
+B([p1,···,pn],mB,eB) =Z−nψ/2
+ψZ−nA/2
+AΓ(n)([p1,···,pn],m,e). (80)
+Here, ∆Lserves to provide all the necessary counter-terms for canceling out any divergences in the loop
+diagrams. As the BPHZ algorithm is equivalent to the above program, below, we will refer to both as
+conventional approachesor programs. One caveat is that the fo rmalism for such programsis only established
+within the realm of perturbation and hence must be so implemented.
+2. Underlying theory perspective
+Now, let us reexamine the same issue from underlying theory perspe ctive. It is convenient to work with
+the path integral formulation. In terms of a underlying theory des cription, the generating functional for the17
+QED processes would formally take the following form:
+Z(Jµ,η,¯η;{σ}) =/integraldisplay
+dµ[φUT]exp/braceleftbigg
+i/integraldisplay
+d4x[LUT([φUT],{σ})
++JµOµ
+A({σ})+O¯ψ({σ})η+ ¯ηOψ({σ})/bracketrightbig/bracerightbig
+, (81)
+where [φUT] is used to label all the necessary degrees of freedom, and Oµ
+A,O¯ψ,Oψdenotes the photon and
+electron field parameters in the UT description which may well be comp osite ones.
+At ”lower” scales where QED become prominent or effective process es, all the typical higher energy modes
+are actually integrated out, resulting in a well-defined functional in t erms of dominant or effective degrees,
+Z(Jµ,η,¯η;{σ}) =/integraldisplay
+dAµd¯ψdψexp/braceleftbigg
+i/integraldisplay
+d4x[Leff(ψ,Aµ;{σ})
++JµAµ+¯ψη+ ¯ηψ/bracketrightbig/bracerightbig
+, (82)
+with/integraltext
+d4xLeff(ψ,Aµ;{σ}) being the effective action generated from the integrating-out. T aking the ”decou-
+pling limit” ( ˘PLE) na¨ ıvely on the effective lagrangian would yield to a classical lagrangia n of QED,
+˘PLE[Leff(ψ,Aµ;{σ})] =LQED(ψ,Aµ), (83)
+which in turn leads to ill-defined path integral as the limit operation and the functional integration does not
+commute. Obviously, the details taken away with the limit operation,
+δL(ψ,Aµ;{σ})≡ L eff(ψ,Aµ;{σ})−˘PLE[Leff(ψ,Aµ;{σ})]
+=Leff(ψ,Aµ;{σ})−LQED(ψ,Aµ),
+is just what we are missing for a well defined path integral. Therefor e, we must include it into the path
+integral,
+Z(Jµ,η,¯η;{σ}) =/integraldisplay
+dAµd¯ψdψexp/braceleftbigg
+i/integraldisplay
+d4x[LQED(ψ,Aµ)+δL(ψ,Aµ;{σ})
++JµAµ+¯ψη+ ¯ηψ/bracketrightbig/bracerightbig
+. (84)
+Evidently, in conventional programs, δLis implemented through regularization and subtraction that are
+encoded in ∆ L. For certain interactions, the job of ∆ Lcould be ”autonomously” done using operators
+appearing in the classical lagrangian, leading in effect to the ”renorm alization” of these operators. Such
+autonomous cases are conventionally termed as renormalizable, wh ile the rest as un-renormalizable. As a
+”realization” or substitute of δL, ∆Lmust also take the responsibility to clear all the side effects like ultra-
+violet infinities associated with a regularization. Only in this sense, the manipulation of infinities (through
+counter terms) may be justifiable, and the bare lagrangian may ser ve as a symbolic and rough representation
+of the underlying theory: Leff(ψ,Aµ;{σ}), which definitely contains the sophisticated underlying quantum
+details that could not be simply described in terms of ”bare” objects that are originally coined in classical
+field theory.
+As we could at best obtain the parametric form of the ”decoupling” e ffects of {σ}, there may be certain
+arbitrariness in our choice of LQED=˘PLE[Leff] (that is actually unknown to us yet) as our ”starting”
+lagrangian for calculations or the in the separation of it out of Leff:
+Leff=˜LQED+˜δL(···;{σ}) =LQED+δL(···;{σ}),
+where˜LQEDandLQEDwould at most differ by a series of local operators (for renormalizab le theories, by a
+finite ”renormalization” of the lagrangian) that could be absorbed in toδL(···;{σ}):
+δL(···;{σ}) =δvarLQED+˜δL(···;{σ}), δvarLQED≡˜LQED−LQED.
+Obviously, any sensible prescription of ˘PLE[Leff] should necessarily include a reference scale that are widely
+separated from the underlying ones. This natural appearance of a reference scale just corresponds to and
+hence remove the mystery around the dimensional transmutation phenomenon in the conventionalprograms.18
+C. Origin of renormalization group equations
+We note that renormalization group (RG) and RG equations (RGE) ap pear in terms of intermediate
+renormalization reparametrization, for which a lucid discussion can b e found in Ref.[63]. If we could proceed
+with all parameters being physical, there seems to be no room for it. However, as discussed above, even
+starting with a physical parametrization, one could still arrive at an arbitrary prescription, provided the
+prescription is related to the physical one via proper transformat ions. In this short subsection, we will not
+repeat the derivation of RGE in a specific scheme defined by the corr esponding boundary conditions as is
+done in conventional programs, instead, we briefly show that the R GE like equations could be generically
+derived as a ”decoupling” theorem in terms of underlying structure s even if the parameters are determined
+by physical boundary conditions[64–66].
+First let us look at the scaling law of QED in terms of underlying theory. For any 1PI n-point vertex
+function Γ(n), the scaling law would read,
+Γ(n)([λp],[λm,e];{λdσσ}) =λdΓ(n)Γ(n)([p],[m,e];{σ}), (85)
+withd···denoting the canonical scaling dimension (equals to the mass dimensio n) of the corresponding
+parameter. The differential form of this scaling law reads,
+/braceleftBigg/summationdisplay
+pp·∂p+m∂m+/summationdisplay
+σdσσ∂σ−dΓ(n)/bracerightBigg
+Γ(n)([p],[m,e];{σ}) = 0. (86)
+Now applying the ”decoupling” limit to this differential equation, we hav e,
+/braceleftBigg/summationdisplay
+pp·∂p+m∂m+/summationdisplay
+idCiCi∂Ci−dΓ(n)/bracerightBigg
+Γ(n)([p],[m,e];[Ci]) = 0, (87)
+with
+/summationdisplay
+idCiCi∂CiΓ(n)=˘PLE/parenleftBigg/summationdisplay
+σdσσ∂σΓ(n)/parenrightBigg
+(88)
+denoting the contributions arising from the ”decoupling” limit operat ion.
+Note that, in whatever prescription for the agent constants [ Ci], the scaling of these constants would just
+elicit insertion of appropriate local operators with respect to each loop where agent constants show up:
+/summationdisplay
+idCiCi∂CiΓ(n)=/summationdisplay
+aδOaˇIOaΓ(n), (89)
+whereˇIOadenotestheinsertionofalocaloperator Oathatcorrespondstoavertexwhose1PIloopcorrections
+contain local ambiguities with δOabeing the associated coefficients. At least at one-loop level in QED, o nly
+logarithmic ambiguities will be really contributing to these ”decoupling” effects that are anomalous in terms
+of the canonical contents of quantum field theories. Thus, δOais the primitive ”anomalous dimension” of
+operatorOa. A closer analysis will tell that among the operators [ Oa], there must be kinetic ones for which
+the above variations would induce an extra finite ”renormalization” o f certain field operators, the rest must
+be characterizinginteractions for which the variations would give ris e to beta functions of their couplings[64–
+66]. So, Eq.(89) is our primitive form of renormalization group equatio ns for a general prescription, which
+could be naturally interpreted as a ”decoupling theorem” of the und erlying structures that regulate the
+ultra-violet regions of our quantum field theories. Obviously, imposin g different boundary conditions would
+lead to different form of/summationtext
+idCiCi∂Ciand/summationtext
+aδOaˇIOa.
+In fact, one could recast the scaling law of Eq.(87) into the following c oncise form
+{p·∂p−dΓ(n)}Γ(n)([p],[m,e];[Ci]) =ˆIi˜ΘΓ(n)([p],[m,e];[Ci])
+= Γ(n)
+i˜Θ([0;p],[m,e];[Ci]), (90)
+withˆIi˜Θ≡ −m∂m−/summationtext
+idCiCi∂Cidenotingthe insertionofthe full traceofthe stresstensor ˜Θµνthat contains
+trace anomalies due to renormalization. This equation is just an alter native form of the Callan-Symanzik
+equation, from which some low-energy theorems of QCD follow as imme diate corollaries[67].19
+V. DISCUSSIONS AND SUMMARY
+Before closing our presentation, we wish to make the following remar ks about our simple strategy for
+renormalization: (1) As we introduce no artificial deformation of all the canonical structures or symmetries,
+hence not affecting spacetime dimension and Dirac algebra, thus our strategy could be applied to any field
+theory, such as supersymmetric and chiral field theories. (2) Sinc e our ignoranceabout the underlying theory
+is parametrized in a general manner, the effects of regularizationu pon field theory and correspondingphysics
+could be examined in a general manner. In particular, the physical o rigin of field theoretical properties like
+anomalies could be unambiguously identified and disentangled with the r egularizationeffects in this strategy,
+surpassing the traditional interpretations that rely on effects of regularization. (3) Various operations such
+as external momenta routing and shift of loop variables are allowed in our simple strategy, sparing many
+subtleties associated these operations in the conventional appro aches. Hence, one could better focus on more
+physical issues in calculations or identify true physical origins of var ious phenomena within field theoretical
+contexts. (4)In theoreticalperspective, ourapproachisalsou seful in exploringwhetherortowhatextent the
+underlying structures are compatible with the canonical symmetrie s or properties of quantum field theories,
+such as Lorentz invariance, various gauge symmetries, unitarity a nd so on. From a more practical viewpoint,
+our approach or similar strategies allows us to examine how our ignora nce about underlying structures could
+be safely treated with the helps of these canonical symmetries. Ob viously, more works need to be done both
+in the construction and applications of efficient programs using the s imple strategy and in the exploration
+of various issues that are intricate with respect to the issue of reg ularization and renormalization.
+In summary, we demonstrated in details with QED at one-loop level ho w a simple strategy for renormal-
+ization could work without introducing any specific form of regulariza tion and manipulations of ultra-violet
+infinities. Some technical operations like loop variable shifting or mome nta routing that are subtle issues
+in conventional programs are shown to be of no problem. In this simp le strategy, ambiguities arise instead
+of infinities, which could be further reduced by imposing appropriate Ward identities. For the Lorentz and
+gauge invariant QED at one-loop level, it is shown in a prescription-inde pendent manner that there are
+only three ambiguities to be fixed using appropriate boundary condit ions. The conventional renormalization
+programs were also analyzed from the underlying theory perspect ive with the rationale of manipulation of
+ultra-violet infinities being explicated. Finally, the renormalization-gr oup-like equations arise generically as
+”decoupling theorems” of the underlying structures.
+Acknowledgement
+The project is supported in part by the National Natural Science F oundation under Grant No.10205004
+and the Ministry of Education of China.
+Appendix A
+In this appendix, we list the convergent integrals that appear in the RHS of Eqs.(14,15,16),
+−iΣ(1);αβ(p) = (−ie)2/integraldisplayd4l
+(2π)4i/bracketleftbig
+(1−ξ)lκlτ
+l2−gκτ/bracketrightbig
+l2+iǫγκi
+/ l+/ p−m+iǫ
+×/bracketleftbigg
+γα−1
+/ l+/ p−m+iǫγβ+(α↔β)/bracketrightbigg−1
+/ l+/ p−m+iǫγτ; (A1)
+Λ(1);α
+µ(p,˜p) = (−ie)2/integraldisplayd4l
+(2π)4i/bracketleftbig
+(1−ξ)lκlτ
+l2−gκτ/bracketrightbig
+l2+iǫγκi
+/ l+/ p−m+iǫγα
+×−1
+/ l+/ p−m+iǫγµi
+/ l+/˜p−m+iǫγτ; (A2)
+iΠ(1);αβγ
+µν(q) = (−ie)2/integraldisplay(−)d4l
+(2π)4Tr/bracketleftbigg
+γµi
+/ l−m+iǫγνi
+/ l+/ q−m+iǫγα
+×−1
+/ l+/ q−m+iǫγβ−1
+/ l+/ q−m+iǫγγ−1
+/ l+/ q−m+iǫ/bracketrightbigg20
++permutations of (α,β,γ). (A3)
+Obviously, these diagrams result in from zero momentum insertion of photon lines into the original diagrams
+that are ill-defined, Cf. Fig. 2 for a diagrammatic representation fo r the case of self-energy diagram.
+As stated in Sec. II and III, other choices of routing only lead to eq uivalent results. For the self-energy
+diagram, we could let the external momentum pof fermion flow through the photon line, so that the RHS
+of Eq.(14) reads,
+−iΣ(1);alt(p) = (−ie)2/integraldisplayd4l
+(2π)4i/bracketleftBig
+(1−ξ)(l−p)κ(l−p)τ
+(l−p)2−gκτ/bracketrightBig
+(l−p)2+iǫγκi
+/ l−m+iǫγτ.
+(A4)
+It is nothing else but the original one with the loop momentum shifted b yp. Then applying our method to
+this integration, we arrived the following,
+−iΣ(1);alt(p) =ie2
+(4π)2/braceleftbigg/integraldisplay1
+0dz[[(z(2−3z)+z(3z−4)ξ)/ p+(4+2z(ξ−1))m]
+×ln(m2−zp2)−(1−ξ)z(1−z)p2(m+z/ p)
+m2−zp2/bracketrightbigg
++˜Cψ/ p+˜Cm/bracerightbigg
+,
+(A5)
+whichisequivalenttotheformoftheself-energydiagramasgiveninE q.(24)aftercarryingouttheparametric
+integrations, and the constants may differ from the those in Eq.(24 ) by a finite amount. Indeed, computing
+the difference between this routing and that given in Eq.(24), we hav e,
+−i(Σ(1);alt−Σ(1)) =ie2
+(4π)2/braceleftbigg/parenleftbigg
+˜Cψ−Cψ+(ξ−1)
+2/parenrightbigg
+/ p+˜Cm−Cm/bracerightbigg
+. (A6)
+Thus the two routings produce identical result after setting
+˜Cψ=Cψ+(1−ξ)
+2,˜Cm=Cm, (A7)
+which is natural from the underlying theory perspective where we s hould have −i(Σ(1);alt−Σ(1)) = 0. Of
+course, one could also verify this point using a consistent regulariza tion method like dimensional scheme.
+This is an illustration of the theorem of routing in case (a) given in Sec. II.
+Alternatively, we could proceed as follows: First, we could find the fo llowing from what we obtained above
+−iΣ(1)/bardblp=0=ie2
+(4π)2/bracketleftbig
+Cm+(3+ξ)mlnm2/bracketrightbig
+, (A8)
+−iΣ(1);alt/bardblp=0=ie2
+(4π)2/bracketleftBig
+˜Cm+(3+ξ)mlnm2/bracketrightBig
+. (A9)
+Next note that,
+−iΣ(1)/bardblp=0= (−ie)2/integraldisplayd4l
+(2π)4i/bracketleftbig
+(1−ξ)lκlτ
+l2−gκτ/bracketrightbig
+l2+iǫγκi
+/ l−m+iǫγτ
+=−iΣ(1);alt/bardblp=0, (A10)
+which implies Cm=˜Cm. In the same fashion, we could find that
+−i∂pαΣ(1)/bardblp=0=ie2
+(4π)2/bracketleftbigg
+Cψ−ξlnm2+1−ξ
+2/bracketrightbigg
+γα, (A11)
+−i∂pαΣ(1);alt/bardblp=0=ie2
+(4π)2/bracketleftBig
+˜Cψ−ξlnm2/bracketrightBig
+γα. (A12)21
+Then from underlying theory perspective the two quantities −i∂pαΣ(1)/bardblp=0and−i∂pαΣ(1);alt/bardblp=0should be
+identical and hence ˜Cψ=Cψ+(1−ξ)
+2. In fact, the integral forms of these two quantities look different from
+each other, but it could be verified in any consistent regularization ( e.g., dimensional regularization) that
+their difference is zero,
+−i∂pα(Σ(1)−Σ(1);alt)/bardblp=0=/integraldisplayd4l
+(2π)4e2(3−ξ)γβ
+l2(l2−m2)/bracketleftbigg
+gβα−2lβlα
+l2−2lβlα
+l2−m2/bracketrightbigg
+= 0, (A13)
+therefore they are identical in any consistent regularization sche me (including the postulated underlying
+theory), and the case (a) of the theorem of routing in Sec. II is ve rified.
+We also note that the routing for the vertex diagram given above ar e so chosen that the calculations are
+less laborious, i.e., the two external fermion lines carry independent momenta so that the differentiation with
+respect toponly operates on one internal line. One could well try the following mor e conventional routing,
+Λ(1);α
+µ(p,p+q) = (−ie)2/integraldisplayd4l
+(2π)4i/bracketleftbig
+(1−ξ)lκlτ
+l2−gκτ/bracketrightbig
+l2+iǫγκi
+/ l+/ p−m+iǫ
+×/bracketleftbigg
+γα−1
+/ l+/ p−m+iǫγµ+γµ−1
+/ l+/ p+/ q−m+iǫγα/bracketrightbigg
+×i
+/ l+/ p+/ q−m+iǫγτ. (A14)
+The final results for Λ µshould be the same after we set q= ˜p−pin the solutions. After quite some algebras,
+we found that the vertex computed using this routing reads,
+Λ(1);α
+µ(p,p+q) =∂pα/braceleftBigg
+−e2
+(4π)2/integraldisplay1
+0dy/integraldisplay1−y
+0dz/bracketleftBigg
+2γµln˜∆+˜Gµ
+˜∆+(1−ξ)(1−y−z)
+×/parenleftBigg
+−6γµln˜∆+˜F1;µ
+˜∆2+˜F2;µ
+˜∆/parenrightBigg/bracketrightBigg/bracerightBigg
+,
+˜P≡(y+z)p+yq,˜∆≡(y+z)m2+˜P2−zp2−y(p+q)2,
+˜Gµ≡γσ(/ p+m−/˜P)γµ(/ p+/ q+m−/˜P)γσ,
+˜F1;µ≡/˜P(/ p+m−/˜P)γµ(/ p+/ q+m−/˜P)/˜P,
+˜F2;µ≡(/ p+/ q)γµ/ p+3γµ(/ p+/ q)/˜P+3/˜P/ pγµ+(m2−6˜P2)γµ
++2m(3˜Pµ−2pµ−qµ). (A15)
+It is immediate to see that this result exactly agrees with that given a bove after setting ˜ p=p+q, as asserted
+by the theorem of routing in case (b) presented in Sec. II.
+Appendix B
+Here, we consider the Ward identity for the components proportio nal to 1−ξthat are listed as below:
+−i∆Σ(1)(p;ǫ)
+1−ξ=−ie2
+(4π)2/braceleftbigg/bracketleftbigg
+ǫ−1−γE+1−lnm2
+4πµ2+δ+(δ2−1)lnδ−1
+δ/bracketrightbigg
+/ p
+−/bracketleftbigg
+ǫ−1−γE+2−lnm2
+4πµ2+(δ−1)lnδ−1
+δ/bracketrightbigg
+m/bracerightbigg
+=−ie2
+(4π)2/braceleftbigg/bracketleftbigg∆Cψ(ǫ)
+ξ−1−lnm2+δ+(δ2−1)lnδ−1
+δ/bracketrightbigg
+/ p
+−/bracketleftbigg∆Cm(ǫ)
+(1−ξ)m+1−lnm2+(δ−1)lnδ−1
+δ/bracketrightbigg
+m/bracerightbigg
+, (B1)22
+∆Λ(1)
+µ(p,p;ǫ)
+1−ξ=−e2
+(4π)2/braceleftbigg/bracketleftbigg
+ǫ−1−γE+1−lnm2
+4πµ2+δ+(δ2−1)lnδ−1
+δ/bracketrightbigg
+γµ
+−2/bracketleftbigg/parenleftbigg
+δlnδ−1
+δ+1/parenrightbigg
+(2δ/ p−m)+/ p/bracketrightbiggpµ
+p2/bracerightbigg
+=−e2
+(4π)2/braceleftbigg/bracketleftbigg∆C⊥(ǫ)
+1−ξ+11
+6−lnm2+δ+(δ2−1)
+×lnδ−1
+δ/bracketrightbigg
+γµ−2/bracketleftbigg/parenleftbigg
+δlnδ−1
+δ+1/parenrightbigg
+(2δ/ p−m)+/ p/bracketrightbiggpµ
+p2/bracerightbigg
+, (B2)
+withδ≡m2/p2. Differentiating ∆Σ(1)(p;ǫ) with respect to pµ, we have,
+−i∂pµ/braceleftbigg∆Σ(1)(p;ǫ)
+1−ξ/bracerightbigg
+=−ie2
+(4π)2/braceleftbigg/bracketleftbigg
+ǫ−1−γE+1−lnm2
+4πµ2+δ+(δ2−1)
+×lnδ−1
+δ/bracketrightbigg
+γµ−2/bracketleftbigg/parenleftbigg
+δlnδ−1
+δ+1/parenrightbigg
+(2δ/ p−m)+/ p/bracketrightbiggpµ
+p2/bracerightbigg
+=i∆Λ(1)
+µ(p,p;ǫ)
+1−ξ. (B3)
+Obviously, the Ward identity for the components proportional to ( 1−ξ) is guaranteed by the following
+relation between ∆ Cψ(ǫ) and ∆C⊥(ǫ):
+∆Cψ(ǫ)+∆C⊥(ǫ) =11(ξ−1)
+6. (B4)
+Now, collecting all the components for Cψ(ǫ) andC⊥(ǫ):
+Cψ(ǫ) =Cψ(ǫ)/bardblξ=1+∆Cψ(ǫ), C⊥(ǫ) =C⊥(ǫ)/bardblξ=1+∆C⊥(ǫ),
+we have:
+Cψ(ǫ)+C⊥(ǫ) =5
+2+11(ξ−1)
+6=4+11ξ
+6. (B5)
+In addition, we note that in Landau gauge,
+−iΣ(1)(p;ǫ)/bardblξ=0=−3ie2
+(4π)2/bracketleftbigg
+ǫ−1−γE+4
+3−lnm2
+4πµ2
++(δ−1)lnδ−1
+δ/bracketrightbigg
+m, (B6)
+Λ(1)
+µ(p,p;ǫ)/bardblξ=0=6e2
+(4π)2/bracketleftbigg
+δlnδ−1
+δ+1/bracketrightbiggmpµ
+p2, (B7)
+as it is well known that the vertex is ultra-violet definite in this gauge.
+Appendix C
+The differentiation of other diagrams with respect to p1;αyields,
+∂pk;αiΓk;µνρσ=/parenleftbigg8ie4
+(4π)2Tk;µνρσ/integraldisplay1
+0dx/integraldisplayx
+0dy/integraldisplayy
+0dz∂p1;αln˜∆k/parenrightbigg
++∂p1;αˇΓk;µνρσ, k= 2,3,4,5,6, (C1)23
+with
+T2;µνρσ=gµρgνσ−2gµνgρσ+gµσgνρ,
+T3;µνρσ=gµνgρσ−2gµσgνρ+gµρgνσ,
+T4;µνρσ=gµσgνρ−2gµρgνσ+gµνgρσ,
+T5;µνρσ=gµσgνρ−2gµνgρσ+gµρgνσ,
+T6;µνρσ=gµρgνσ−2gµσgνρ+gµνgρσ, (C2)
+˜∆2≡m2−(x−x2)p2
+1−(z−z2)p2
+2−(y−y2)p2
+3
+−2z(1−x)p1p2−2y(1−x)p1p3−2z(1−y)p2p3,
+˜∆3≡m2−(x−z−(x−z)2)p2
+1−(y−z−(y−z)2)p2
+2−(z−z2)p2
+3
+−2(y−z)(1−x+z)p1p2−2z(x−z)p1p3−2z(y−z)p2p3,
+˜∆4≡m2−(x−y−(x−y)2)p2
+1−(y−y2)p2
+2−(y−z−(y−z)2)p2
+3
+−2y(x−y)p1p2−2(x−y)(y−z)p1p3−2(y−z)(1−y)p2p3,
+˜∆5≡m2−(x−y−(x−y)2)p2
+1−(y−z−(y−z)2)p2
+2−(y−y2)p2
+3
+−2(x−y)(y−z)p1p2−2y(x−y)p1p3−2(y−z)(1−y)p2p3,
+˜∆6≡m2−(x−z−(x−z)2)p2
+1−(z−z2)p2
+2−(y−z−(y−z)2)p2
+3
+−2z(x−z)p1p2−2(y−z)(1−x+z)p1p3−2z(y−z)p2p3. (C3)
+Thus the solutions:
+iΓk;µνρσ=8ie4
+(4π)2Tk;µνρσ/integraldisplay1
+0dx/integraldisplayx
+0dy/integraldisplayy
+0dzln˜∆k
+Ck;4γ
++ˇΓk;µνρσ, k= 2,3,4,5,6, (C4)
+with the same convention of notations. Here we have deliberately de note the integration constants in the
+way that seems diagram-dependent, i.e., C;4γmay differ among the six diagrams. The gauge invariance will
+require that they must be equal to each other. One could also arriv e at the same conclusion by noting that
+the loop integral for each diagram is identical after setting all the e xternal momenta to zero, for example,
+iΓ1;µνρσ(0,0,0,0) =−e4Tr[γαγµγβγνγδγργηγσ]/integraldisplayd4l
+(2π)4lαlβlδlη
+(l2−m2)4. (C5)
+Appendix D
+Now we demonstrate the integration by parts on the logarithmic ter m ln˜∆1:
+/integraldisplay1
+0dx/integraldisplayx
+0dy/integraldisplayy
+0dzln˜∆1
+C1;4γ
+=/integraldisplay1
+0dx/integraldisplayx
+0dy/bracketleftBigg/parenleftBigg
+zln˜∆1
+C1;4γ/parenrightBigg
+/bardblz=y
+z=0−/integraldisplayy
+0dzz∂z˜∆1
+˜∆1/bracketrightBigg
+=/integraldisplay1
+0dx/integraldisplayx
+0dyyln˜˜∆1
+C1;4γ+˜D1z
+=/integraldisplay1
+0dx/bracketleftBigg/parenleftBigg
+y2
+2ln˜˜∆1
+C1;4γ/parenrightBigg
+/bardbly=x
+y=0−/integraldisplayx
+0dyy2
+2∂y˜˜∆1
+˜˜∆1/bracketrightBigg
++˜D1z
+=/integraldisplay1
+0dxx2
+2lnm2−(x−x2)p2
+4
+C1;4γ+˜D1y+˜D1z
+=1
+6lnm2
+C1;4γ+˜D1x+˜D1y+˜D1z=1
+6lnm2
+C1;4γ+˜D1, (D1)24
+with
+˜˜∆1≡˜∆1/bardblz=y,˜D1z≡ −/integraldisplay1
+0dx/integraldisplayx
+0dy/integraldisplayy
+0dzz∂z˜∆1
+˜∆1,
+˜D1y≡ −/integraldisplay1
+0dx/integraldisplayx
+0dyy2
+2∂y˜˜∆1
+˜˜∆1,˜D1x≡1
+6/integraldisplay1
+0dxx3(1−2x)
+x2−x+m2/p2
+4,
+˜D1≡˜D1x+˜D1y+˜D1z. (D2)
+Here˜D1is definite and hence absorbed into ˇΓ1in Eq.(43), that is, ˜Γ1≡ˇΓ1+˜D1. Then we arrive at the
+form of Γ 1;µνραgiven in Eq.(43). In the same fashion, we have,
+/integraldisplay1
+0dx/integraldisplayx
+0dy/integraldisplayy
+0dzln˜∆k=1
+6lnm2
+Ck;4γ+˜Dk, k= 2,3,4,5,6, (D3)
+with [˜Dk,k= 2,3,4,5,6] being definite functions of external momenta.
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+FIG. 1: The elementary one-loop diagrams in QED
+∂pα∂pβ{ }
+∂pα∂pβΣ(1) =α β 
+=+β α 
+Σ(1); αβ 
+FIG. 2: The graphical representation of Eq.(14).
\ No newline at end of file
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+arXiv:1001.0494v7  [math.NT]  22 Jun 2010CONDITIONAL AND UNCONDITIONAL LARGE GAPS BETWEEN
+THE ZEROS OF THE RIEMANN ZETA-FUNCTION
+S. H. SAKER
+Abstract. In this paper, first by employing inequalities derived from t he Opial
+inequality due to David Boyd with best constant, we will esta blish new unconditional
+lower bounds for the gaps between the zeros of the Riemann zet a function. Second,
+on the hypothesis that the moments of the Hardy Z−function and its derivatives
+are correctly predicted, we establish some explicit formul ae for the lower bounds of
+the gaps between the zeros and use them to establish some new c onditional bounds.
+In particular it is proved that the consecutive nontrivial z eros often differ by at
+least 6.1392 (conditionally) times the average spacing. This value improves the value
+4.71474396 that has been derived in the literature.
+1.Introduction
+The Riemann zeta function ζ(s) is defined on {s∈C: Re(s)>1}by the series
+ζ(s) := 1+1
+2s+1
+3s+1
+4s+1
+5s+...,
+which converges in the region described by the Cauchy integral tes t. It is of funda-
+mental importance because it can also be represented just in term s of the primes. This
+representation is given by
+ζ(s) :=/productdisplay
+p/parenleftbigg
+1−1
+ps/parenrightbigg−1
+,for Re s >1,
+where the product is taken over all prime numbers. Thus its analytic properties are
+related to the distribution of prime numbers. Among the integers th e primes appear to
+be scattered at random. It is known that they are infinite in number , but there is no
+useful formula which generate them. However, on average they o bey simple laws. For
+example, the prime number theorem states that the number of prim es which occur up to
+a given integer X,π(X), is approximately X/log(X), the approximation getting better
+asXincreases. The actual numbers found for different Xwill fluctuate about this value.
+Riemann gave an exact formula for the counting function π(X), in which fluctuations
+about the average are related to the values of sfor which ζ(s) = 0.These are isolated
+points in the complex plan. In [29] the authors presented a numerica l study of Riemann’s
+formula for the oscillating part of the density of the primes and their integer powers.
+The formula consists of an infinite series of oscillatory terms, one fo r each zero of the
+zeta function on the critical line, and was derived by Riemann in his pap er on primes,
+assuming the Riemann hypothesis. They also showed that high-reso lution spectral lines
+can be generated by the truncated series at all integer powers of primes and demonstrate
+1991Mathematics Subject Classification. 11M06, 11M26.
+Key words and phrases. Riemann zeta function, zeros the Riemann zeta function .
+12 S. H. SAKER
+explicitly that the relative line intensities are correct. They then der ived a Gaussian
+sum rule for Riemann’s formula and used to analyze the numerical con vergence of the
+truncated series.
+Riemann conjectured that all nontrivial (non-real) zeros are dist ributed symmetrically
+with respect to the critical line Re s= 1/2 and the real axis. This is the Riemann
+hypothesis. Riemann showed that the zeta-function satisfies a fu nctional equation of the
+form
+(1.1) π−s/2Γ(s
+2)ζ(s) =π−(1−s)/2Γ(1−s
+2)ζ(1−s),
+where Γ is the Euler gamma function. Clearly, there are no zeros in th e half-plane
+of convergence Re( s)>1,and it is also known that ζ(s) does not vanish on the line
+Re(s) = 1.In the negative-half plane ζ(s) and its derivative are oscillatory and from the
+functional equation there exist-so called trivial (real) zeros at s=−2mfor any positive
+integerm(correspondingto the poles of the appearingGamma-factors). I t is conjectured
+that all or at least almost all nontrivial zeros of the zeta-function are simple, see [5] and
+[7].
+Reimann’s connection between the nontrivial zeros and the primes h as particularly
+interesting form: it bears a striking resemblance to the Gutzwiller fo rmula, with the
+zeros behaving like energy levels and the primes labelling the periodic or bits of some
+chaotic classical system. Montgomery [24] studied the distribution of pairs of nontrivial
+zeros 1/2+iγand 1/2+iγ′and conjectured, for fixed α, βsatisfying 0 < α < β, that
+lim
+T→∞1
+N(T)#/braceleftBigg
+0< γ,γ′< T:α≤γ′−γ′
+(2π/logT)≤β/bracerightBigg
+=/integraldisplayβ
+α/parenleftBigg
+1−/parenleftbiggsinπx
+πx/parenrightbigg2/parenrightBigg
+dx.
+This so-called pair correlation conjecture plays a complementary ro le to the Riemann
+hypothesis. This conjecture implies the essential simplicity hypothe sis that almost all
+zeros of the zeta-function are simple. On the other hand the integ ral on the right hand
+side is the same as the one observed in the two point correlation of th e eigenvalues
+which are the energy levels of the corresponding Hamiltonian that ar e usually not known
+with uncertainty. This observation is due to Dyson and it restored s ome hope in an old
+idea of Hilbert and Polya that the Riemann hypothesis follows from the existence of a
+self-adjoint Hermitian operator whose spectrum of eigenvalues co rrespond to the set of
+nontrivial zeros of the zeta function.
+Odlyszko [28] published the results of a remarkable series of comput er calculations of
+the zeroswhich showedthat they werethe sameas thoseoflargeH ermitian matriceswith
+randomly picked entries. These suggests that the zeros might well be the energy levels of
+some as yet unidentified quantum system whose classical motion is ch aotic, and without
+symmetry under time reversal. The connections to quantum chaos and semiclassical
+physics are discussed in [29]. So that the distribution of zeros of the Riemann zeta-
+function is of fundamental importance in number theory as well as in physics.ZEROS OF THE RIEMANN ZETA FUNCTION 3
+The number N(t) of the non-trivial zeros of ζ(s) with ordinate in the interval [0 , T]
+is asymptotically given by the Riemann-von Mangoldt formula (see [12])
+N(T) =T
+2πlog(T
+2πe)+O(logT).
+Consequently there are infinitely many nontrivial zeros, all of them lying in the critical
+strip0<Res <1,andthefrequencyoftheirappearanceisincreasingas T→ ∞.Assume
+that (βn+iγn) are the zeros of ζ(s) in the upper half-plane (arranged in non-decreasing
+order and counted according multiplicity) and γn≤γn+1are consecutive ordinates of
+all zeros. Define
+(1.2) λ:= lim sup
+n→∞(γn+1−γn)
+(2π/logγn),andµ:= lim inf
+n→∞(γn+1−γn)
+(2π/logγn),
+where (2 π/logγn) is the average spacing between zeros. The values of λandµhave
+received a great deal of attention. In fact, important results ha ve been obtained by some
+authors. It generally conjectured that
+(1.3) µ= 0,andλ=∞.
+As mentioned by Montogomery[24] it would be interestingto see how numerical evidence
+compare with the above conjectures. Now, several results has b een obtained, however
+the failure of Gram’s low (see [19]) indicates that the asymptotic beha vior is approached
+very slowly. Thus the numerical evidence may not be particularly illumin ating. So that
+any numerical values of µandλmay be help in proving (1.3), which is one of our aims
+in this paper. Selberg [30] proved that 0 < µ <1< λand the average of rnis 1.Mueller
+[26] obtained λ >1.9 assuming the Riemann hypothesis. Montogomery and Odlyzko [25]
+showed, assuming the Riemann hypothesis, that λ >1.9799 and µ <0.5179.Conrey,
+Ghosh and Gonek [6] showed that, if the Riemann hypothesis is true, thenλ >2.337,
+andµ <0.5172.Bui, Milinovich and Ng [3] obtained λ >2.69,andµ <0.5155 assuming
+the Riemann hypothesis. Conrey, Ghosh and Gonek [8] obtained a ne w lower bound and
+proved that λ >2.68 assuming the generalized Riemann hypothesis for the zeros of th e
+Dirichlet L−functions. Ng in [27] proved that λ >3 assuming the generalized Riemann
+hypothesis for the zeros of the Dirichlet L−functions. Bui[2] proved that λ >3.0155
+assuming the generalized Riemann hypothesis for the zeros of the D irichletL−functions.
+The main results in [2, 3, 8, 27] are based on the idea of Mueller [26].
+Hall [13] supposed that the sequence of distinct positive zeros of t he Riemann zeta-
+function ζ(1
+2+it) which arranged in non-decreasing order and counted according m ulti-
+plicity is given by {tn}and defined
+(1.4) Λ := lim sup
+n→∞tn+1−tn
+(2π/logtn),
+which is the quantity in (1.2) where only zeros1
+2+itnon the critical line with the
+idea that this could be bounded from below unconditionally. Note that the Riemann
+hypothesis implies that the tncorresponded to the positive ordinates of non-trivial zeros
+of the zeta function, i.e., N(T)∼(TlogT)/2π.The averagespacing between consecutive
+zeros with ordinates of order Tis 2π/log(T) which tends to zero as T→ ∞.
+Hall [15] showed that Λ ≥λ, and the lower bound for Λ bear direct comparison with
+such bounds for λdependent on the Riemann hypothesis, since if this were true the
+distinction between Λ and λwould be nugatory. Of course Λ ≥λand the equality holds
+if the Riemann hypothesis is true. So that if the Riemann hypothesis is true, we see4 S. H. SAKER
+that any improvement of Λ (unconditionally) will lead to the improveme nt ofλand vice
+versa. The behavior of ζ(s) on the critical line is reflected by the Hardy Z−function Z(t)
+as a function of a real variable, defined by
+(1.5) Z(t) =eiθ(t)ζ(1
+2+it),whereθ(t) :=π−it/2Γ(1
+4+1
+2it)/vextendsingle/vextendsingleΓ(1
+4+1
+2it)/vextendsingle/vextendsingle.
+Alsoitfollowsthat Z(t)isaninfinitelyoftendifferentiableandrealforreal tandmoreover
+|Z(t)|=|ζ(1/2+it)|. Consequently, the zeros of Z(t) correspond to the zeros of the
+Riemannzeta-functiononthecriticalline. In[14]HallprovedaWirtin ger-typeinequality
+and used the moment
+(1.6)/integraldisplayT
+0Z4(t)dt=1
+2π2Tlog4(T)+O(Tlog3T),
+due to Ingham [21]) and the moments
+(1.7)/integraldisplayT
+0(Z′(t))4dt=1
+1120π2Tlog8(T)+O(Tlog7T),
+(1.8)/integraldisplayT
+0Z2(t)(Z′(t))2dt=1
+120π2Tlog6(T)+O(Tlog5T),
+due to Conrey [9], and obtained unconditionally that Λ ≥2.3452.
+The moments Ik(T) of the Hardy Z−function Z(t) and the moments Mk(T) of its
+derivative are defined by
+Ik(T) :=/integraldisplayT
+0|Z(t)|2kdt,andMk(T) :=/integraldisplayT
+0/vextendsingle/vextendsingle/vextendsingleZ′(t)/vextendsingle/vextendsingle/vextendsingle2k
+dt.
+For positive real numbers k, it is believed that Ik(T)∼C(k)T(logT)k2
+andMk(T)∼
+L(k)T(logT)k2+2kfor positive constants CkandLkwill be defined later. Keating and
+Snaith [22] based on considerations from random matrix theory con jectured that
+(1.9) Ik(T)∼a(k)b(k)T(logT)k2
+,
+wherea(k) is a product over the primes which is defined by
+a(k) :=/producttext
+p(/parenleftbigg
+1−1
+p2/parenrightbigg∞/summationdisplay
+m=0/parenleftbiggΓ(m+k)
+m!Γ(k)/parenrightbigg2
+p−m,
+and
+(1.10) b(k) :=k−1/productdisplay
+j=0j!
+(j+k)!.
+Using the relation (1.10) one can obtain the value of b(k) for any real positive number
+k.Conrey, Rubinstein and Snaith [10] conjectured that
+(1.11) Mk(T)∼a(k)c(k)T(logT)k2+2k,
+where
+c(k) := (−1)k(k+1)
+2k/summationdisplay
+m∈Pk+1
+O(2k)/parenleftbigg
+2k
+m/parenrightbigg/parenleftbigg−1
+2/parenrightbiggm0/parenleftBiggk/productdisplay
+i=11
+(2k−i+mi)!/parenrightBigg
+Mi,j,ZEROS OF THE RIEMANN ZETA FUNCTION 5
+where
+(1.12) Mi,j:=
+k/productdisplay
+1≤i,j≤k(mj−mi+i−j)
+,
+andPk+1
+O(2k) denotes the set ofpartitions m= (m0,...,mk) of2kinto nonnegativeparts.
+In this paper, we will determine the values of b(k)/c(k) fork= 1,2,...,15 that we will
+use derive the new conditional lower bounds for Λ .
+Hall in [15, 18] used the moments of mixed powers of the form
+(1.13)/integraldisplayT
+0Z2k−2h(t)(Z′(t))2hdt∼C(h,k)T(logT)k2+2h,
+whereZ(t) is the Hardy Z−function, and k∈N, 0≤h≤kand a complicated vari-
+ation problem together with a new Wirtinger-type inequality designed exclusively for
+this problem and obtained some conditional lower bounds of Λ. The mo ments in (1.13)
+has been predicted by Random Matrix Theory (RMT) by Hughes [20] w ho stated an
+interesting conjecture on the moments of the zeta function and it s derivatives at its zeros
+subject to the truth of Riemann’s hypothesis when the zeros are s imple. This conjecture
+includes for fixed k >−3/2 the asymptotes formula of the moments of the higher order
+of the Riemann zeta function and its derivative. We suppose furthe r that if kis a fixed
+positive integer and h∈[0, k] is an integer then the formula
+(1.14)/integraldisplayT
+0Z2k−2h(t)(Z′(t))2hdt∼a(k)b(h,k)T(logT)k2+2h,
+holds. Note that this was predicted by Keating and Snaith [22] in the c ase when h= 0,
+with wider range Re( k)>−1/2 and by Hughes [20] in the range min( h,k−h)>−1/2,
+a(k) is a product over the primes and b(h,k) is rational: indeed for integral h, it is
+obtained that
+(1.15) b(h,k) =b(k)/parenleftbigg(2h)!
+8hh!/parenrightbigg
+H(h,k),
+whereH(h,k) is an explicit rational function of kfor each fixed h.The functions H(h,k)
+as introduced by Hughes [20] are given in the following table where K= 2k:
+H(0,k) = 1, H(1,k) =1
+K2−1, H(2,k) =1
+(K2−1)(K2−9)
+H(3,k) =1
+(K2−1)2(K2−25), H(4,k) =K2−33
+(K2−1)2(K2−9)(K2−25)(K2−49)
+H(5,k) =K4−90K2+1497
+(K2−1)2(K2−9)2(K2−25)(K2−49)(K2−81),
+H(6,k) =K6−171K4+6867K2−27177
+(K2−1)3(K2−9)2(K2−25)(K2−49)(K2−81)(K2−121),
+H(7,k) =K8−316K6+30702K4−982572K2+6973305
+(K2−1)3(K2−9)2(K2−25)2(K2−49)(K2−81)(K2−121)(K2−169),
+Table 1. The values of H(h,k), whereK= 2k.
+This sequence continuous, and it is believed that both the nominator and denominator
+are polynomials in k2, moreover that the denominator is actually (see [11])
+(1.16)/productdisplay
+a odd>0/braceleftbigg
+(K2−a2)α(a,h):α(a,h) =4h
+a+√
+a2+8h/bracerightbigg
+.6 S. H. SAKER
+Using the equation (1.15) and the definitions of the functions H(h,k), we can obtain the
+values of b(0,k)/b(k,k) fork= 1,2,...,7.As indicated in [18] Hughes [20] conjectured
+the first four functions and then writes that numerical experimen t suggests the next
+three. Hall [18] shown that in the case when h= 3, (H(3,k)) requires adjustment to fit
+with (1.16) in that extra factor K2−9 should be introduced in both the nominator and
+denominator. To use (1.14) Hall [15] proved a new generalized Wirting er-type inequality
+of the form
+(1.17)/integraldisplayπ
+0H/parenleftBig
+y′(t)/y(t)/parenrightBig
+y2k(t)dt≥(2k−1)L/integraldisplayπ
+0y2k(t)dt,
+wherey(t)∈C2[0, π], y(0) =y(π) = 0,L=L(k,H) is determined from the solution of
+the equation/integraldisplay∞
+0G′(u)
+G(u)+(2k−1)Ldu
+u=kπ, fork∈N,
+whereG(u) :=uH′(u)−H(u),H(u) be an even function, increasing, strictly convex on
+R+and satisfies H(0) =H′(0) = 0,anduH′′(u)→0 asu→0.The inequality (1.17)
+is proved by using the calculus of variation which depends on the minimiz ation of the
+integral on the left hand side subject to the constrains y(0) = 0 and/integraltextπ
+0y2k(t)dt= 1.
+Assuming that (1.14) is correctly predicted, Hall employed the inequ ality (1.17) when
+H(u) :=k/summationdisplay
+h=12k−1
+2h−1/parenleftbiggh
+k/parenrightbigg
+υhu2h,υh≥0,υk= 1,
+and obtained an explicit formula for Λ( k) which is given by
+Λ2≥X,
+whereXis the real positive root of the equation
+k/summationdisplay
+h=12k−1
+2h−1/parenleftbiggk
+h/parenrightbigg
+R(h,k)υhXh−(2k−1)λ(υ1,υ1,...,υk−1)R(0,k) = 0,
+and
+R(h,k) := 4k−hb(h,k)
+b(k,k).
+He then derived a new value of Λ (when k= 3) which is given by
+(1.18) Λ ≥/radicalbig
+7533/901= 2.8915.
+The main challenge in [15] was to maximize X=κ2(which is not an easy task) where
+Xsatisfies the equation
+27X3+385µX2+10395ϑX−121275L= 0,
+andLobtained form the equation
+/integraldisplay∞
+−∞x4+2µx2+υ
+x6+3µx4+3υx2+Ldx=π.
+Hall [16] simplified the calculations in [15] and converted the problem int o one of the
+classical theory of equations involving Jacobi-Schur functions and proved that Λ(4) ≥
+3.392272,Λ(5)≥3.858851,andΛ(6) ≥4.2981467.Hall [18]developed thetheoryset used
+in [16] and proved that Λ(7) ≥4.215007assuming that (1.14) is correctly predicted. The
+improvement of this value as obtained in [18] is given by Λ(7) ≥4.71474396 assuming
+that (1.14) is correctly predicted. The question now is: If it is possib le to employ newZEROS OF THE RIEMANN ZETA FUNCTION 7
+inequalities with best constants to find new explicit formulae for the g aps and use them
+to find new series of the lower bounds?
+The paper gives an affirmative answer to this question. In fact, we w ill derive new
+unconditional and conditional lower bounds for Λ .The main results will be proved by
+employing two inequalities derived from the Opial inequality with a best c onstant due to
+David Boyd [4] who applied a variational technique to reduce the dete rmination of the
+best constant to a nonlinear eigenvalue problem for an integral ope rator.
+The main results will be proved in the next section which is organized as follows:
+First, we derive new unconditional lower bounds for Λ. Second on th e hypothesis that
+the moments of the Hardy Z−function and its derivatives are correctly predicted, we
+establish new explicit formulae of the gaps between the zeros and es tablish some lower
+bounds for Λ. In particular, we will prove that Λ ≥6.1392 which improves the value
+Λ≥4.71474396 .
+2.Main Results
+Before we state and prove the main results, we derive some inequalit ies from the Opial
+inequality due to David Boyd [4] that we will use in this section. The Opia l inequality
+due to David Boyd [4] is presented in the following theorem.
+Theorem A. Ify∈C1[a, b]withy(a) = 0(ory(b) = 0), then
+(2.1)/integraldisplayb
+a|y(t)|p/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingleq
+dt≤K(p,q,r)(b−a)r−q/parenleftBigg/integraldisplayb
+a/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingler
+dt/parenrightBiggp+q
+r
+,
+wherep >0,r >1,0≤q < r,
+K(p,q,r) : =(r−q)pp
+(r−1)(p+q)βp+q−r(I(p,q,r))−p,
+β: =/braceleftbiggp(r−1)+(r−q)
+(r−1)(p+q)/bracerightbigg1
+r
+,
+and
+I(p,q,r) :=/integraldisplay1
+0/braceleftbigg
+1+r(q−1)
+r−qt/bracerightbigg−(p+q+rp)/rp
+[1+(q−1)t]t1/p−1dt.
+First, we will derive a new inequality from the Opial inequality (2.1) of th e from (1.17)
+which allows us to use the moments (1.7) and (1.8) to derive the new un conditional lower
+bound of Λ .For a special case of Theorem A, when r=p+q, we have
+(2.2)/integraldisplayb
+a|y(t)|p/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingleq
+dt≤K(p,q,p+q)(b−a)p/parenleftBigg/integraldisplayb
+a/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsinglep+q
+dt/parenrightBigg
+,
+where
+(2.3) K(p,q,p+q) :=q(p+q)p−1
+(pL(p,q)+q)p, q/ne}ationslash= 0,
+and
+(2.4) L(p,q) :=/integraldisplay1
+0/parenleftbigg1
+1−λsp/parenrightbigg
+ds, where λ=(p+q)(q−1)
+(p+q−1)q.8 S. H. SAKER
+The inequality (2.2) has immediate application to the case where y(a) =y(b) = 0.
+Choosec= (a+b)/2 and apply (2.12) to [ a,c] and [c,b] and then add to obtain
+/integraldisplayb
+a|y(t)|p/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingleq
+dt
+≤K(p,q,p+q)(b−a
+2)p/parenleftBigg/integraldisplayc
+a/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsinglep+q
+dt+/integraldisplayb
+c/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsinglep+q
+dt/parenrightBigg
+≤K(p,q,p+q)(b−a
+2)p/parenleftBigg/integraldisplayb
+a/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsinglep+q
+dt/parenrightBigg
+.
+So that if y(0) =y(π) = 0, we have
+(2.5)/integraldisplayπ
+0|y(t)|p/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingleq
+dt≤K(p,q,p+q)(π
+2)p/parenleftbigg/integraldisplayπ
+0/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsinglep+q
+dt/parenrightbigg
+.
+If we choose p= 2 and q= 2, we get that
+(2.6)/integraldisplayπ
+0y2(t)(y′(t))2dt≤K(2,2,4)(π
+2)2/parenleftbigg/integraldisplayπ
+0/parenleftBig
+y′(t)/parenrightBig4
+dt/parenrightbigg
+.
+Using the definition of K, we see that K(2,2,4) = 0.34613,where we used the value of
+L(2,2) =/integraldisplay1
+0/parenleftbigg1
+1−2
+3s2/parenrightbigg
+ds= 1.4038.
+So that the inequality (2.6) becomes
+(2.7)/parenleftbigg/integraldisplayπ
+0/parenleftBig
+y′(t)/parenrightBig4
+dt/parenrightbigg
+≥4
+(0.34613)π2/integraldisplayπ
+0y2(t)(y′(t))2dt.
+By a suitable linear transformation, we deduce that if y∈C1[a,b] withy(a) = 0 =y(b),
+then we have
+(2.8)/integraldisplayb
+a/parenleftbiggb−a
+π/parenrightbigg2/parenleftBig
+y′(t)/parenrightBig4
+dt≥4
+(0.34613)π2/integraldisplayb
+ay2(t)(y′(t))2dt.
+In the following, assuming the Riemann hypothesis, we will apply the ine quality (2.8)
+and using the moments (1.7) and (1.8) to find a new unconditional valu e of Λ. One
+can see that the value that we will establish does not improve the obt ained values,
+but the technique is a simple one and depends only on the application of an inequality
+derived from the well-known Opial inequality. Note that, as mentione d by Hall, when
+the Riemann hypothesis is true we have λ= Λ.
+Theorem 2.1 .Letε(T)→0in such a way that ε(T)logT→ ∞.Then for sufficiently
+largeT, there exists an interval contained in [T,(1 +ε(T))T]which is free of zeros of
+Z(t)and having length at least
+/radicalBigg
+112
+12(0.34613)π2/braceleftbigg
+1+O/parenleftbigg1
+ε(T)logT/parenrightbigg/bracerightbigg2π
+logT.
+Thus
+(2.9) Λ ≥1
+π/radicalBigg
+112
+12(0.34613)= 1.6529.ZEROS OF THE RIEMANN ZETA FUNCTION 9
+Proof. We follow the arguments in [14] to prove our theorem. Suppose that tlis the
+first zero of Z(t) not less than Tandtmthe last zero not greater than (1 + ε)Twhere
+ε(T)→0 in such a way that ε(T)logT→ ∞.Suppose further that for l≤n < m, we
+have
+(2.10) Ln=tn+1−tn≤2πκ
+logT.
+Applying the inequality (2.8) with y(t) =Z(t), we have
+/integraldisplaytn+1
+tn/bracketleftBigg/parenleftbiggLn
+π/parenrightbigg4/parenleftBig
+Z′(t)/parenrightBig4
+−4
+(0.34613)π2Z2(t)(Z′(t))2/bracketrightBigg
+dt≥0.
+Since the inequality remains true if we replace Ln/πby 2κ/logT, we have
+(2.11)/integraldisplaytn+1
+tn/bracketleftBigg/parenleftbigg2κ
+logT/parenrightbigg4/parenleftBig
+Z′(t)/parenrightBig4
+−4
+(0.34613)π2Z2(t)(Z′(t))2/bracketrightBigg
+dt≥0.
+Summing (2.11) over n,using (1.7) and (1.8), we obtain
+1
+1120π2/parenleftbigg2κ
+logT/parenrightbigg2
+Tlog8(T)+O(Tlog7)
+−4
+(0.34613)π21
+120π2Tlog6(T)+O(Tlog5T)
+=(2κ)2
+1120π2Tlog6(T)+O(Tlog7T)
+−4
+(0.34613)π21
+120π2(Tlog6T)+O(Tlog5T).
+Follows the proof of Theorem 1 in [14], we obtain
+κ2≥112
+12(0.34613)π2+O(1/ε(T)logT).
+Then, we have (noting ( ε(T)logT→ ∞asT→ ∞) that
+Λ≥/radicalBigg
+112
+12(0.34613)π2= 1.6529,
+which the desired value (2.9). The proof is complete.
+Nextinthefollowing,wewillderiveaninequalityfromtheOpialinequality (2.1)which
+allows use to use the moments (1.6) and (1.7) ((1.14)) to derive a new unconditional
+(conditional) lower bound for Λ .As a special case of ((2.1) if q= 0,andp=r= 2k,
+then the inequality (2.1) reduces to
+(2.12)/integraldisplayb
+a|y(t)|2kdt≤A2k(k)(b−a)2k/integraldisplayb
+a/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingle2k
+dt,
+where
+(2.13) A(k) :=/parenleftbigg2(k)
+2(k)−1/parenrightbigg1
+2k
+(2(k))2(k)−1
+2(k)/parenleftbigg
+Γ(1
+2(k))Γ(2(k)−1
+2(k))/parenrightbigg−1
+,10 S. H. SAKER
+and Γ(u) is the Euler gamma function. The inequality (2.12) has immediate applic ation
+to the case where y(a) =y(b) = 0. Choose c= (a+b)/2 and apply (2.12) to [ a, c] and
+[c, b] and then add to obtain
+/integraldisplayb
+a|y(t)|2kdt≤A2k(k)/parenleftbiggb−a
+2/parenrightbigg2k/braceleftBigg/parenleftbigg/integraldisplayc
+a/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingle2k
+dt/parenrightbigg
++/parenleftBigg/integraldisplayb
+c/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingle2k
+dt/parenrightBigg/bracerightBigg
+≤A2k(k)/parenleftbiggb−a
+2/parenrightbigg2k/integraldisplayb
+a/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingle2k
+dt. (2.14)
+From this, we have
+/integraldisplayπ
+0|y(t)|2kdt≤A2k(k)/parenleftBigπ
+2/parenrightBig2k/integraldisplayπ
+0/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingle2k
+dt.
+withy(0) = 0 = y(π).From this inequality, we deduce that if y∈C1[a,b] withy(a) =
+0 =y(b),then we have
+(2.15)/integraldisplayb
+a/parenleftbiggb−a
+π/parenrightbigg2k/parenleftBig
+y′(t)/parenrightBig2k
+dt≥/parenleftbigg2
+π/parenrightbigg2k1
+A2k(k)/integraldisplayb
+a(y(t))2kdt.
+Using the formula (2.13), we have the following values of A(k) fork= 1,2,....15.
+A(1)A(2)A(3)A(4)A(5)
+0.636620.684090.730260.764090.7896
+A(6)A(7)A(8)A(9)A(10)
+0.809550.825620.838880.850030.85956
+A(11)A(12)A(13)A(14)A(15)
+0.86780.875020.881410.887090.89219
+Table 2. The values of A(k) fork= 1,2,...,15.
+In the following theorem, assuming the Riemann hypothesis, we apply the inequality
+(2.15) when k= 2 and using the moments (1.6) and (1.7) to derive an unconditional
+value for Λ .
+Theorem 2.2 .Letε(T)→0in such a way that ε(T)logT→ ∞.Then for sufficiently
+largeT, there exists an interval contained in [T,(1 +ε(T))T]which is free of zeros of
+Z(t)and having length at least
+1
+π(0.68409)4√
+560/braceleftbigg
+1+O/parenleftbigg1
+ε(T)logT/parenrightbigg/bracerightbigg2π
+logT.
+Thus
+(2.16) Λ ≥1
+π(0.68409)4√
+560 = 2.2635.
+Proof. As in the proof of Theorem 2.1, we follow the arguments in [14] to pro ve our
+theorem. Suppose that tlis the first zero of Z(t) not less than Tandtmthe last zero
+not greater than (1+ ε)Twhereε(T)→0 in such a way that ε(T)logT→ ∞.Suppose
+further that for l≤n < m, we have
+(2.17) Ln=tn+1−tn≤2πκ
+logT.ZEROS OF THE RIEMANN ZETA FUNCTION 11
+Applying the inequality (2.15) with y(t) =Z(t) andk= 2, we have
+/integraldisplaytn+1
+tn/bracketleftBigg/parenleftbiggLn
+π/parenrightbigg4/parenleftBig
+Z′(t)/parenrightBig4
+−/parenleftbigg2
+πA(2)/parenrightbigg4
+(Z(t))4/bracketrightBigg
+dt≥0.
+Since the inequality remains true if we replace Ln/πby 2κ/logT, we have
+(2.18)/integraldisplaytn+1
+tn/bracketleftBigg/parenleftbigg2κ
+logT/parenrightbigg4/parenleftBig
+Z′(t)/parenrightBig4
+−/parenleftbigg2
+πA(2)/parenrightbigg4
+(Z(t))4/bracketrightBigg
+dt≥0.
+Summing (2.18) over n,using (1.6) and (1.7), we obtain
+1
+1120π2/parenleftbigg2κ
+logT/parenrightbigg4
+Tlog8(T)+O(Tlog7)
+−/parenleftbigg2
+πA(2)/parenrightbigg41
+2π2(Tlog4T)+O(Tlog3T)
+=(2κ)4
+1120π2Tlog4(T)+O(Tlog7T)
+−/parenleftbigg2
+πA(2)/parenrightbigg41
+2π2(Tlog4T)+O(Tlog3T).
+Follows the proof of Theorem 1 in [14], we obtain
+κ4≥1
+24/parenleftbigg2
+πA(2)/parenrightbigg41120π2
+2π2+O(1/ε(T)logT).
+Now, using the value of A(2) from Table 2, we have (noting ( ε(T)logT→ ∞asT→ ∞)
+that
+Λ≥1
+π1
+(0.68409)4/radicalbigg
+1120
+2= 2.2635,
+which the desired value (2.16). The proof is complete.
+In the following, we will establish some explicit formulae for the gaps be tween the
+zeros of the Riemann zeta function and use them to find new conditio nal series of lower
+bounds. First, we will apply the inequality (2.5). As usual, we assume t hat the Riemann
+hypothesis is true. As a special case of (2.5), if y(0) =y(π) = 0,p= 2k−2handq= 2h,
+we have
+(2.19)/integraldisplayπ
+0|y(t)|2k−2h/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingle2h
+dt≤K(h,k)(π
+2)2k−2h/parenleftbigg/integraldisplayπ
+0/vextendsingle/vextendsingle/vextendsingley′(t)/vextendsingle/vextendsingle/vextendsingle2k
+dt/parenrightbigg
+,
+where
+(2.20) K(h,k) =hk2k−1
+((k−h)L(k,h)+h)2k, h, k/ne}ationslash= 0,
+and
+(2.21) L(h,k) =/integraldisplay1
+0/parenleftbigg1
+1−λs2k−2h/parenrightbigg
+ds,λ=k(2h−1)
+h(2k−1).
+Theorem 2.3. On the hypothesis that the Riemann hypothesis is true and (1. 14) is
+correctly predicted, we have
+(2.22) Λ ≥Λ∗(h,k) :=1
+π/parenleftbigg1
+K(h,k)b(h,k)
+b(k,k)/parenrightbigg1
+2k−2h
+, h/ne}ationslash=k/ne}ationslash= 012 S. H. SAKER
+where
+(2.23) b(h,k) :=2h!
+8hh!H(h,k)
+k−1/productdisplay
+j=0j!
+(j+k)!
+.
+Proof.As in the proof of Theorem 2.2 by applying the inequality (2.19) with y=Z(t),
+we have
+/integraldisplaytn+1
+tn/bracketleftBigg/parenleftbiggLn
+π/parenrightbigg2k/parenleftBig
+Z′(t)/parenrightBig2k
+−22k−2h
+K(h,k)π2k−2h/parenleftbiggLn
+π/parenrightbigg2h
+|Z(t)|2k−2h/vextendsingle/vextendsingle/vextendsingleZ′(t)/vextendsingle/vextendsingle/vextendsingle2h/bracketrightBigg
+dt≥0.
+Since the inequality remains true if we replace Ln/πby 2κ/logT, we have
+/integraldisplaytn+1
+tn/parenleftbigg2κ
+logT/parenrightbigg2k/vextendsingle/vextendsingle/vextendsingleZ′(t)/vextendsingle/vextendsingle/vextendsingle2k
+−/integraldisplaytn+1
+tn22k−2h
+K(h,k)π2k−2h/parenleftbigg2κ
+logT/parenrightbigg2h
+|Z(t)|2k−2h/vextendsingle/vextendsingle/vextendsingleZ′(t)/vextendsingle/vextendsingle/vextendsingle2h
+dt≥0.
+Summing (2.25) over nand using (1.14), we obtain
+/parenleftbigg2κ
+logT/parenrightbigg2k
+a(k)b(k,k)T(logT)k2+2k
+−22k−2h
+K(h,k)π2k−2h/parenleftbigg2κ
+logT/parenrightbigg2h
+a(k)b(h,k)T(logT)k2+2hdt≥0.
+This implies that
+T(logT)k2/braceleftbigg
+(2κ)2ka(k)b(k,k)−22k−2h
+K(h,k)π2k−2h(2κ)2ha(k)b(h,k)/bracerightbigg
+≥o(T(logT)k2),
+whence
+κ2k−2h≥1
+π2k−2hK(h,k)b(h,k)
+b(k,k)+o(1),(asT→ ∞).
+This implies that
+Λ2k−2h(k)≥1
+π2k−2hK(h,k)b(h,k)
+b(k,k), h/ne}ationslash=k/ne}ationslash= 0.
+which is the desired inequality and completes the proof.
+To apply the inequality (2.22), we will need the following values of b(1,k) andb(k,k)
+which are determined from (2.23) where H(h,k) are defined as in Table 1:
+b(1,2) =1
+720, b(2,2) =1
+6720,b(1,3) =1
+1209600, b(3,3) =1
+496742400,
+b(1,4) =1
+219469824000,b(4,4) =31
+271159356948480000,
+b(1,5) =1
+8760533070643200000, b(5,5) =227
+12854317559387145633792000000,
+b(1,6) =1
+127288050516627176816640000000,
+b(6,6) =133933
+25516459094444104187401241999966208000000000,
+b(1,7) =1
+998707926079695101611943783301120000000000,
+b(7,7) =2006509
+895370835179281010419215815294340559070476369920000 000000000.ZEROS OF THE RIEMANN ZETA FUNCTION 13
+Table 3: The values of b(1,k) andb(k,k) fork= 2,...,7.
+Also, we need the following values of K(1,k) which are determined from the formula
+(2.20) by using (2.21) for k= 2,3,...,7 :
+K(1,2) = 0 .23961, K(1,3) = 0.16187, K(1,4) = 0.12227,
+K(1,5) = 9 .8238×10−2, K(1,6) = 8.2128×10−2, K(1,7) = 0.07055.
+Now, we are ready to derive a series of the lower bounds of Λ( k) fork= 2,3,...,7.These
+lower bounds are determined by using the formula (2.22) and presen ted in the following
+table:
+Λ(2) Λ(3) Λ(4) Λ(5) Λ(6) Λ(7)
+1.98662.25912.64073.02083.38003.7124
+Table 3: The lower bounds for Λ( k) fork= 2,3,...,7 by using the formula (2.22).
+In the following, we will apply the inequality (2.15) to establish a new exp licit formula
+for Λ(k). As usual, we assume that the Riemann hypothesis is true and the m oments in
+(1.9) and (1.11) are correctly predicted.
+Theorem 2.4 .Assuming the Riemann hypothesis and the moments in (1.9) and
+(1.11) are correctly predicted, we have
+(2.24) Λ( k)≥1
+πA(k)/parenleftbiggbk
+ck/parenrightbigg1
+2k
+,fork≥3,
+whereA(k)is defined as in (2.13).
+Proof.AsintheproofofTheorem2.2by applyingtheinequality(2.15)with y=Z(t),
+we have/integraldisplaytn+1
+tn/bracketleftBigg/parenleftbiggLn
+π/parenrightbigg2k/parenleftBig
+Z′(t)/parenrightBig2k
+−/parenleftbigg2
+πA(k)/parenrightbigg2k
+(Z(t))2k/bracketrightBigg
+dt≥0.
+Since the inequality remains true if we replace Ln/πby 2κ/logT, we have
+(2.25)/integraldisplaytn+1
+tn/bracketleftBigg/parenleftbigg2κ
+logT/parenrightbigg2k/parenleftBig
+Z′(t)/parenrightBig2k
+−/parenleftbigg2
+πA(k)/parenrightbigg2k
+(Z(t))2k/bracketrightBigg
+dt≥0.
+Summing (2.25) over n,using (1.9) and (1.11), we obtain
+akck/parenleftbigg2κ
+logT/parenrightbigg2k
+T(logT)k2+2k−akbk/parenleftbigg2
+πA(k)/parenrightbigg2k
+T(logT)k2
+=/parenleftBigg
+akckκ2k(22k)−akbk/parenleftbigg2
+πA(k)/parenrightbigg2k/parenrightBigg
+T(logT)k2
+≥O(Tlogk2T),
+whence
+κ2k≥akbk
+22kakck1
+A2k(k)=bk
+22kck/parenleftbigg2
+πA(k)/parenrightbigg2k
+,(asT→ ∞).
+This implies that
+Λ2k(k)≥bk
+22kck/parenleftbigg2
+πA(k)/parenrightbigg2k
+,
+and then we obtain the desired inequality (2.24). The proof is complet e.14 S. H. SAKER
+Conrey, Rubinstein and Snaith [10] gave some explicit values of the pa rameter c(k)
+fork= 1,2,...,15.Using the values of c(k) due to Conrey, Rubinstein and Snaith [10]
+and the relation (1.10), we have the following values of b(k)/c(k) fork= 1,2,...,15 that
+will be used in this paper:
+b1
+c1= 22·3,b2
+c2=26·3·5·7
+223,b3
+c3=212·32·52·72·11
+26335,b4
+c4=220·310·54·72·11·13
+31·21235537,
+b5
+c5=230·312·56·74·11·132·17·19
+227·220395573,b6
+c6=242·319·59·76·113·133·17·19·23
+67·1999·230315577511,
+b7
+c7=256·328·513·78·114·133·172·192·23
+43·46663·242321597711313,b8
+c8=272·334·516·711·116·134·173·192·23·29·31
+46743947 ·25632851279115133,
+b9
+c9=290·342·521·714·118·136·173·193·232·29·31
+19583·16249·27233651671111713517,b10
+c10=2110·355·525·717·1110·138·175·194·233·29·31·37
+3156627824489 ·29034452071311913717319,
+b11
+c11=2132·363·531·718·1112·1310·175·195·234·292·312·37·41·43
+59·11332613 ·33391·21103535247161111139175193,
+b12
+c12=2156·375·537·723·1115·1312·178·197·234·293·312·41·43·47
+241·251799899121593 ·21323635287201113131117719523,
+b13
+c13=2182·390·542·728·1117·1314·1710·198·235·293·313·372·41·43·47
+285533·37408704134429 ·215637353472411151313179197233,
+b14
+c14=2210·3100·550·731·1120·1317·1712·1910·237·294·314·372·41·43·47·53
+197·1462253323 ·6616773091 ·2182386542728111713151711199235,
+b15
+c15=2240·3117·557·737·1122·1319·1714·1911·239·295·315·373·412·432·47·53·59
+1625537582517468726519545837 ·22103102550732111913171713191123729.
+Table 4. The values of b(k)/c(k) fork= 1,2,...,15.
+Using the formula (2.24), the values of b(k)/c(k) in Table 4, and the values of A(k) in
+Table 2, we have the following new lower bounds for Λ( k) fork= 1,2,...,15:
+Λ(1) Λ(2) Λ(3) Λ(4) Λ(5)
+1.73212.26352.70803.12573.5177
+Λ(6) Λ(7) Λ(8) Λ(9) Λ(10)
+3.88134.21504.51964.79855.0560
+Λ(11) Λ(12) Λ(13) Λ(14) Λ(15)
+5.29625.52255.73735.94246.1392
+Table 4. The lower bounds for Λ( k) fork= 1,2,...,15.
+From this table we have the following theorem.
+Theorem 2.5. On the hypothesis that the Riemann hypothesis is true, (1.9) and
+(1.11) are correctly predicted, we have
+(2.26) Λ ≥6.1392
+Remark 1. Using the explicit formulae for the b(k)andc(k)(which would via (2.24)
+help to decide whether the conjecture Λ =∞is true subject to the Riemann hypothesis),ZEROS OF THE RIEMANN ZETA FUNCTION 15
+we have the following formula
+Λ(k)≥1
+πA(k)
+k−1/productdisplay
+j=0j!
+(j+k)!
+1
+2k
+×(−1)k(k+1)
+2k/summationdisplay
+m∈Pk+1
+O(2k)/parenleftbigg
+2k
+m/parenrightbigg/parenleftbigg−1
+2/parenrightbiggm0/parenleftBiggk/productdisplay
+i=11
+(2k−i+mi)!/parenrightBigg
+Mi,j,
+whereA(k)is defined as in (2.13), Mi,jas defined in (1.12) and Pk+1
+O(2k)denotes the
+set of partitions m= (m0,...,mk)of2kinto nonnegative parts.
+Remark 2. The lower bound in (2.26) means that consecutive nontrivial zeros often
+differ by at least 6.1392times the average spacing. This value improves the value of
+Λ≥4.71474396 that has been obtained by Hall.
+References
+[1] P. R. Beesack, Hardy’s inequality and its extensions, Pa cific J. Math. 11 (1961), 39-61.
+[2] H. M. Bui, large gaps between consecutive zeros of the Rie mann zeta function, preprint
+[3] H. M. Bui, M. B. Milinovich and N. Ng, A note on the gaps betw een consecutive zeros of the
+Riemann zeta-function, arXiv:0910.2052v1.
+[4] D. Boyd, Best constants in class of integral inequalitie s, Pac. J. Math. 30 (1969), 367-383.
+[5] A. Y. Cheer and A. D. Goldston, Simple zeros of the Riemann zeta-function, Proc. Amer. Math.
+Soc. 118 (1993), 356-373.
+[6] J. B. Conrey, A. Gosh and S. M. Gonek, A note on gaps between zeros of the zeta function, Bull.
+London. Math. Soc. 16 (1984), 421-424.
+[7] J. B. Conrey, More than two fifth of zeros of the Riemann zet a-function are on the critical line, J.
+Riene Angew. Math. 399 (1989), 1-22.
+[8] J. B. Conrey, A. Gosh and S. M. Gonek, Large gaps between ze ros of the zeta function, Mathematica
+33 (1986), 212-238.
+[9] J. B. Conrey, The fourth moment of derivative of the Riema nn zeta function, Quart. J. Math. 2
+(1988), 21-36.
+[10] J. B. Conrey, M. O. Bubinstein and N. C. Snaith, Moments o f the derivative of the characteristic
+polynomials with an applications to the Riemann zeta functi on, Cummun. Math. Phys. 267 (2006),
+611-629.
+[11] P. -O. Dehaye, Joint moments of derivatives of characte ristic polynomials of the Riemann zeta-
+function, (in press).
+[12] R. Garunkˇ stis and J. Steuding, Simple zeros and discre te moments of the derivative of the Riemann
+zeta-function, J. Number Theor. 115 (2005), 310-321.
+[13] R. R. Hall, The behavior of the Riemann zeta function on t he critical line, Mathematica 46 (1999),
+281-313.
+[14] R. R. Hall, A Wirtinger type inequality and the spacing o f the zeros of the Riemann zeta-function,
+J. Number Theory 93 (2002), 235-245.
+[15] R. R. Hall, Generalized Wirtinger inequalities, rando m matrix theory, and the zeros of the Riemann
+zeta-function, J. Number Theory 97 (2002), 397-409.
+[16] R. R. Hall, Large spaces between the zeros of the Riemann zeta-function and random matrix theory,
+J. Number Theory 109 (2004), 240-256.
+[17] R. R. Hall, A new unconditional result about large space s between zeta zeros, Mathematica 52
+(2005), 101-113.
+[18] R. R. Hall, Large spaces between the zeros of the Riemann zeta-function and random matrix theory
+II, J. Number Theory 128 (2008), 2836-2851.
+[19] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd Ed. Cambridge Univ. Press 1952.
+[20] C. P. Hughes, Thesis, University of Bristol, Bristol, 2 001.
+[21] A. E. Ingham, Mean theorems in the theorem of the Riemann zeta-function, Proc. London Math.
+Soc. 27 (1928), 273-300.16 S. H. SAKER
+[22] J. P. Keating and N. C. Snaith, Random matrix theory and ζ(1/2 +it),Comm. Math. Phys. 214
+(2000), 57-89.
+[23] D. S. Mitinovi´ c, J. E. Peˇ cari´ c and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer
+Academic Publisher , 1993.
+[24] H. L. Montgomery, The pair correlation of zeros of the Ri emann zeta-function on the critical line,
+Proc. Synpos. Pure Math. 24, Amer. Math. Soc., Providence, R I, 1973, 181-193.
+[25] H. L. Montgomery and A. M. Odlyzko, Gaps between zeros of zeta function, in ”Colloq. Math. Soc.
+Janos and Bolyai,” Vol. 34, North-Holand, Amsterdam, 1981.
+[26] J. Mueller, On the difference between consecutive zeros of the Riemann zeta function, J. Number.
+Theo. 14 (1982), 327-331.
+[27] N. Ng, Large gaps between the zeros of the Riemann zeta fu nction, J. Numb. Theor. 128 (2008),
+509-556.
+[28] A. M. Odlyzko, On the distribution of spacing between ze ros of the zeta function, Math. Comp. 48
+(1987), 273-308.
+[29] J. Sakhr, R. K. Bhaduri, and B. P. van Zyl, Zeta function z eros, powers of primes, and quantum
+chaos, Phys. Rev. E 68, 026206 (2003) [7 pages].
+[30] A. Selberg, The zeta-function and the Riemann hypothes is, Skand. Math. 10 (1946), 187-200.
+Department of Mathematics Skills, PYD, King Saud Universit y, Riyadh 11451,
+Saudi Arabia, Department of Math., Faculty of Science, Mans oura University,
+Mansoura 35516, Egypt.
+E-mail address :shsaker@mans.edu.eg, mathcoo@py.ksu.edu.sa
\ No newline at end of file
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+arXiv:1001.0495v1  [math-ph]  4 Jan 2010Ma-Xu quantization rule and exact WKB condition for transla tionally shape invariant
+potentials
+Y. Grandati and A. B´ erard
+Institut de Physique,, ICPMB, IF CNRS 2843, Universit´ e Pau l Verlaine, 1 Bd Arago, 57078 Metz, Cedex 3, France
+For translationally shape invariant potentials, the exact quantization rule proposed by Ma and
+Xu is a direct consequence of exactness of the modified WKB qua ntization condition proved by
+Barclay. We propose here a very direct alternative way to cal culate the appropriate correction for
+the whole class of translationally shape invariant potenti als.
+PACS numbers:
+INTRODUCTION
+In2005,MaandXu[1,2], onthebasisofapreviousworkofCao[3,4]a ndal., proposedanewimprovedquantization
+rule permitting to retrieve the exact spectrum of some exactly solv able quantum systems. Since, numerous papers
+have been published on the possible applications of the Ma-Xu formula to different systems [5–14]. Until now, the
+link between the exactness of the Ma-Xu formula and the solvability o f the considered system seems to be stayed
+unclear. In fact, the translationally shape invariance of the poten tial suffies to ensure the exactness of the Ma-Xu
+formula. This is a direct consequence of a result obtained first by Ba rclay [19] in 1993 and largely overlooked. He
+established that for these specific potentials the higher order ter ms in the WKB series can be resummed, yielding to
+an energy independent correction and he showed that this Maslov in dex can be obtained in a closed analytical form.
+More than ten years later, Bhaduri and al. [20] proposed another interesting derivation of this result which rests on
+periodic orbit theory (POT) [21].
+In the present article, after having established the connection be tween the Ma-Xu formula and Barclay’s result, we
+propose an alternative way to calculate the Maslov index for every t ranslationally shape invariant potential (TSIP).
+Starting from a classification of TSIP which uses new criterions [22], equivalent to the Barclay-Maxwell ones [18], we
+show how to obtain this index using simple complex analysis tools.
+MA-XU QUANTIZATION FORMULA
+Consider the stationary one dimensional Schr¨ odinger equation ( /planckover2pi1= 1, m= 1/2):
+ψ′′(x)+p2(x)ψ(x) = 0, (1)
+wherep(x) =/radicalbig
+E−V(x) is the classical momentum function for an energy E. Ifx1andx2are the classical turning
+points then p2(x)≥0 forx∈[x1,x2].
+Defining:
+w(x) =−ψ′(x)
+ψ(x), (2)
+we have:
+w′(x) =−ψ′′(x)
+ψ(x)+w2(x).
+In other words, if ψ(x) satisfies Eq.(1), w(x) is a solution of the following Riccati equation:
+w′(x) =p2(x)+w2(x) =E−V(x)+w2(x). (3)2
+Every node of ψ(x) (necessarily a simple zero) corresponds to a simple pole of w(x) which decreases in the interval
+[x1,x2]. Consequently,when xrunsthrough[ x1,x2], ateachnodeof ψ(x)thefunction w(x)issubjecttoadiscontinuity
+from−∞to +∞. If we define the phase θ(x) via:
+tanθ(x) =−p(x)
+w(x), (4)
+we can write:
+θ(x) = arctan/parenleftbigg
+−p(x)
+w(x)/parenrightbigg
++nπ, (5)
+where arctan y∈[−π/2,π/2] is the principal determination of the reciprocal of the tangent f unction and where n
+increases of 1 at each node of w(x).
+Then, ifx1andx2are not nodes of ψ(x):
+/integraldisplayx2
+x1θ′(x)dx=Nπ, (6)
+Nbeing the total number of nodes of w(x) on [x1,x2].
+But using Eq.(3) we also have:
+θ′(x) =p(x)−p′(x)w(x)
+w′(x). (7)
+Considering the nthbound state at energy Enfor whichNn=n+1, we obtain the following identity:
+/integraldisplayx2,n
+x1,npn(x)dx=nπ+γ(En), (8)
+where the correction γ(En) to the lowest order WKB condition is given by the following integral:
+γ(En) =π+/integraldisplayx2,n
+x1,nwn(x)p′
+n(x)
+w′n(x)dx. (9)
+Ma and Xu [1, 2] have observed that que for a large class of exactly s olvable potentials, this integral correction is
+in fact independent of the considered energy level, and can be calcu lated from the ground state:
+γ(En) =γ(E0) =π+/integraldisplayx2,0
+x1,0w0(x)p′
+0(x)
+w′
+0(x)dx. (10)
+Inserting Eq.(10) in Eq.(8) gives the Ma-Xu formula:
+/integraldisplayx2,n
+x1,npn(x)dx=nπ+γ(E0). (11)
+The knowledge of the ground state characteristics permits then t o calculate the γ(E0) correction and reaches to
+an implicit formula for the other energy levels En. The explicit calculations have been performed for many exactly
+solvable examples [5–14]. Nevertheless the question of a direct link be tween the exact solvability of the potential and
+the validity of the above formula is still stayed open.
+As much as we know, all the closed form analytically solvable quantum m echanical systems belongs to the set of
+translationallyshapeinvariantpotentials(TSIP) in the senseofSUS Y quantummechanics[15]. In fact forthis set, the
+validity of the Ma-Xu formula follows from a result established in 1993 b y Barclay [19]. In this article, he established3
+that for these potentials, the WKB series can be resummed beyond the lowest order giving an energy independent
+correction which can be absorbed in the Maslov index and written in a c losed analytical form. He showed equally
+that this result is directly correlated to the exactness of the lowes t order SWKB quantization condition [15, 16]. The
+starting point is the definition of two classes of potentials, each cha racterized by a specific change of variable which
+brings the potential into a quadratic form. This two classes are sho wn to coincide to the Barclay-Maxwell classes [18]
+which are based on a functional characterization of superpotent ials and which cover the whole set of TSIP. In both
+cases, the action variable for an energy Enis of the form:
+In=/contintegraldisplay
+Enpn(x)dx= 2/integraldisplayx2,n
+x1,npn(x)dx= 2(nπ+γ), (12)
+whereγis an energy-independent correction characteristic of the consid ered potential. As shown later by Bhaduri
+and al. [20], this results can be retrieved in an elegant way using POT [21 ].
+The translational shape invariance of the potential is then a sufficie nt condition for the validity of the Ma-Xu
+prescription which reaches to Eq.(11).
+In the following, we propose an alternative way to recover Barclay’s result. It lies on a different characterization
+of the Barclay-Maxwell classes that we recently presented [22]. In both cases the action variable can be rewritten
+as a complex integral on an uniformization domain including only one bra nch-cut. Using standard tools of complex
+analysis, it can be readily calculated to recover the explicit form of γfor the whole set of TSIP.
+Note that analogous complex analysis techniques have been employe d by Bhalla and al [24–26] in the frame of
+the quantum Hamilton-Jacobi (QHJ) formalism [28, 29] where, star ting from the exact QHJ quantization condition,
+they have computed the spectrum of numerous potentials for whic h they have also showed the exactness of the
+lowest order SWKB quantization condition. Moreover, Cherqui et a l [27] have recently shown that for algebraic and
+hyperbolic translationally shape invariant potentials, the shape inva riance condition provides sufficient information
+on the singularity structure of the quantum momentum function to determine directly the energy spectrum of the
+system from the exact QHJ quantization condition.
+FIRST CATEGORY POTENTIALS
+Wesaythataonedimensionalpotentialisoffirstcategory[22]ifth ereexistsachangeofvariable x→utransforming
+the potential into an harmonic one V(x)→V(u) =/tildewideλ2u2+/tildewideλ1u+/tildewideλ0, such that u(x) satisfy a constant coefficient
+Riccati equation:
+du(x)
+dx=A0+A1u(x)+A2u2(x), (13)
+du/dxbeing of constant sign in all the range of values of xandu.
+The one dimensional harmonic oscillator correspond to the special c aseA1=A2= 0 and the Morse potential is
+generically associated to the case A1/negationslash= 0, A2= 0.
+Harmonic oscillator
+It is perfectly well knownthat the EBK quantizationcondition is exac t for the harmonicoscillatorimplying then the
+exactness of the Ma-Xu formula with a Maslov index equal to 1 /2. Nevertheless, we will examine this case completely
+as a first example of use of the complex variable integration techniqu e.
+The harmonic oscillator potential with zero ground state energy is:
+V(x,ω) = (ω/2)2x2−ω/2. (14)
+The classical turning points ±x0,nat energyEn=nωsatisfy:
+x2
+0,n=4
+ω/parenleftbigg
+n+1
+2/parenrightbigg
+. (15)4
+The half action variable for a classical periodic orbit of energy Enis then:
+In=/integraldisplayx2,n
+x1,npn(x)dx=ω
+2/integraldisplayx0,n
+−x0,ndx/radicalBig
+x2
+0,n−x2. (16)
+The complex function fH(z) =/radicalBig
+x2
+0,n−z2, defined on C−]−x0,n,x0,n[, admits only one isolated singularity at
+infinity. Then:
+/integraldisplayx0,n
+−x0,ndxfH(x) =πiRes(fH(z),∞). (17)
+The asymptotic behaviour at infinity on the Riemann sheet for which/radicalBig
+u2
+0,n−z2=/radicalBig
+x2
+0,n+y2>0 on the positive
+half imaginary axe ( z=iy,y>0) is given by:
+/radicalBig
+x2
+0,n−z2∼
+z→∞−iz/parenleftBigg
+1−x2
+0,n
+z2/parenrightBigg1
+2
+∼
+z→∞−iz+ix2
+0,n
+21
+z+O(1
+z2). (18)
+Consequently:
+Res(fH(z),∞) =−ix2
+0,n
+2(19)
+and:
+/integraldisplayx2,n
+x1,npn(x)dx=/parenleftbigg
+n+1
+2/parenrightbigg
+π. (20)
+which is the expected result.
+Morse potential
+Asfortheharmonicoscillator,theEBKformulaisexactfortheMors epotential. Thissecondexampleisnevertheless
+very instructive. The Morse potential with zero energy ground st ate is [15]:
+V(x) =A2+B2e−2αx−2B/parenleftBig
+A+α
+2/parenrightBig
+e−αx, α>0. (21)
+Using the change of variable y= exp(−αx), dy=−αydx, it becomes:
+V(y) =B2(y−y0)2+V0 (22)
+withy0= (A+α/2)/Band:
+V0=A2−B2y2
+0=/parenleftBig
+A+α
+2/parenrightBig2
+−A2. (23)
+This is a translationally shape invariant potential with a spectrum give n by [22]:
+En=a2−a2
+n. (24)5
+witha=Aandak=A−kα.
+As for the classical turning points, they are given by:
+/braceleftbiggy2,n=y0+u0,n
+y1,n=y0−u0,n(25)
+with:
+u0,n=/radicalbigg
+y2
+0−a2n
+B2. (26)
+The half action variable for a classical periodic orbit of energy Enis then:
+/integraldisplayx2,n
+x1,npn(x)dx=−1
+α/integraldisplayy2,n
+y1,n/radicalbig
+En−V(y,a)dy
+y(27)
+=B
+α/integraldisplayu0,n
+−u0,nfM(u)du,
+where:
+fM(u) =/radicalBig
+u2
+0,n−u2
+u+y0. (28)
+The complex extended function fM(z) =/radicalBig
+u2
+0,n−z2/(z+y0)T,defined on C−]−u0,n,u0,n[, admits two isolated
+singularities at infinity and in −y0. Then:
+/integraldisplayu0,n
+−u0,ndufM(u) =πi(Res(fM(z),−y0)+Res(fM(z),∞)). (29)
+With the same choice as before for the square root determination w e have:
+fM(z)∼
+z→∞i/parenleftbigg
+−1+y0
+z+O(1
+z2)/parenrightbigg
+(30)
+which implies:
+Res(fM(z),∞) =−ia+α/2
+B. (31)
+The residue in −y0(which is a simple pole) is readily obtained as:
+Res(fM(z),−y0) =ia−nα
+B. (32)
+Then:
+/integraldisplayx2,n
+x1,npn(x)dx=/parenleftbigg
+n+1
+2/parenrightbigg
+π. (33)
+which is the expected result.6
+Effective radial potential for the Kepler-Coulomb problem
+Consider the effective radial potential for the Kepler-Coulomb. It can be written as [22]:
+V(y) =l(l+1)y2−γy+γ2
+4(l+1)2, k>0, (34)
+wherey= 1/xanddy=−y2dx. We can rather use the equivalent form:
+V(y) =l(l+1)(y−y0)2+V0, (35)
+withy0=γ/2l(l+1) and
+V0=γ2
+4(l+1)2−γ2
+4l(l+1). (36)
+.
+This is a translationally shape invariant potential characterized by t he following energy spectrum [22]:
+En(a) =γ2
+4a2−γ2
+4a2n. (37)
+wherea=l+1 andak=l+1+k,.
+The classical turning points are given by:
+/braceleftbigg
+y2,n=y0+u0,n
+y1,n=y0−u0,n(38)
+with:
+u0,n=1/radicalbig
+l(l+1)/radicalBigg
+γ2
+4l(l+1)−γ2
+4a2n. (39)
+The half action variable for a classical periodic orbit of energy Enis then:
+/integraldisplayx2,n
+x1,npn(x)dx=−/integraldisplayy2,n
+y1,n/radicalbig
+En−V(y,a)dy
+y2(40)
+=−/radicalbig
+l(l+1)/integraldisplayu0,n
+−u0,ndufK(u),
+where:
+fK(u) =/radicalBig
+u2
+0,n−u2
+(u+y0)2. (41)
+Again, the complex function fK(z) =/radicalBig
+u2
+0,n−z2/(z+y0)2, defined on C−]−u0,n,u0,n[, admits two isolated
+singularities at infinity and in −y0. Then:
+/integraldisplayu0,n
+−u0,ndufK(u) =πi(Res(fK(z),−y0)+Res(fK(z),∞)). (42)7
+With the same choice as before for the square root determination, we have:
+fK(z)∼
+z→∞−i
+z/parenleftBigg
+1−u2
+0,n
+z2/parenrightBigg1
+2/parenleftBig
+1+y0
+z/parenrightBig−2
+∼
+z→∞−i
+z+O(1
+z2) (43)
+which implies:
+Res(fK(z),∞) =i. (44)
+The residue in −y0which is a double pole is readily obtained as:
+Res(fK(z),−y0) =ian/radicalbig
+l(l+1). (45)
+Then:
+/integraldisplayx2,n
+x1,npn(x)dx=nπ+γ, (46)
+where:
+γ=π/parenleftBig
+l+1+/radicalbig
+l(l+1)/parenrightBig
+which is the expected result.
+Other first category potentials
+If we except the two preceding examples, all the other first categ ory potentials are such that there exists a change
+of variable x→ytransforming the potential into an harmonic one:
+V(x)→V(y) =a(a∓α)y2+λ1y+λ0(a),
+withλ0(a) =λ2
+1/4a2−αa(in order to have a zero energy ground state) and where y(x) satisfies a constant coefficient
+Riccati equation of the form [22]:
+dy
+dx=α±αy2(x)>0. (47)
+The potential can still be written [22]:
+V±(y,a) =a(a∓α)(y−y0)2+V0, (48)
+where:
+y0=−λ1/a(a∓α), V0=λ0(a)−a(a∓α)y2
+0 (49)
+.
+Vis a translationally shape invariant potential and its energy spectru m is then given by [22]:
+En(a) =φ1,±(a)−φ1,±(an), (50)8
+whereak=a±kαand:
+φ1,±(a) =∓a2+λ2
+1
+4a2. (51)
+The classical turning points yi,nfor an energy Enare determined by the condition:
+En=V(yi,n)⇔En=a(a∓α)u2
+0,n+V0, (52)
+where:
+/braceleftbigg
+y2,n=y0+u0,n
+y1,n=y0−u0,n, (53)
+and:
+u0,n=/radicalBigg
+∓1+y2
+0−φ1,±(an)
+a(a∓α). (54)
+Consider first the case where dy/dx=α+αy2(x). The half action variable for a classical periodic orbit of energy
+Enis given by:
+/integraldisplayx2,n
+x1,npn(x)dx=/integraldisplayy2,n
+y1,n/radicalbig
+En−V(y,a)dy
+α(y−i)(y+i)(55)
+=/radicalbig
+a(a−α)
+α/integraldisplayu0,n
+−u0,nf1,+(u)du,
+where:
+f1,+(z) =/radicalBig
+u2
+0,n−z2
+(z−z0)(z−z0)(56)
+withz0=−y0+i.
+The complex extended integrand f1,+(z), defined on C−]−u0,n,u0,n[, admits three isolated singularities at infinity,
+z0andz0. Then:
+/integraldisplayu0,n
+−u0,nduf1,+(u) =πi(Res(f1,+(z),z0)+Res(f1,+(z),z0)+Res(f1,+(z),∞)). (57)
+Whenz→ ∞:
+f1,+(z)∼
+z→∞−i
+z+O(1
+z2), (58)
+which implies:
+Res(f1,+(z),∞) =i. (59)
+As for the residues in z0andz0, using Eq.(52), Eq.(53), Eq.(54) and Eq.(50) we deduce:
+Res(f1,+(z),z0) =−1
+2/radicalbig
+a(a−α)/parenleftbigg
+ian+λ1
+2an/parenrightbigg
+(60)9
+and:
+Res(f1,+(z),z0) =−1
+2/radicalbig
+a(a−α)/parenleftbigg
+−ian+λ1
+2an/parenrightbigg
+. (61)
+Then:
+/integraldisplayx2,n
+x1,npn(x)dx=nπ+γ (62)
+with:
+γ=πa
+α/parenleftbigg
+1−/radicalbigg
+1−α
+a/parenrightbigg
+. (63)
+Consider now the case dy/dx=α−αy2(x). The half action variable for a classical periodic orbit of energy Enis:
+/integraldisplayx2,n
+x1,npn(x)dx=/integraldisplayy2,n
+y1,n/radicalbig
+En−V(y,a)dy
+α(1−y)(1+y)(64)
+=−/radicalbig
+a(a−α)
+α/integraldisplayu0,n
+−u0,nf1,−(u)du,
+where:
+f1,−(u) =/radicalBig
+u2
+0,n−u2
+(u−z1)(u−z2)(65)
+withz1=−1−y0andz2= 1−y0.
+The complex extended integrand f1,+(z), defined on C−]−u0,n,u0,n[, admits three isolated singularities at infinity,
+z1andz2. Then:
+/integraldisplayu0,n
+−u0,nduf1,−(u) =πi(Res(f1,−(z),z1)+Res(f1,−(z),z2)+Res(f1,−(z),∞)). (66)
+Whenz→ ∞:
+f1,+(z)∼
+z→∞−i
+z+O(1
+z2), (67)
+which implies:
+Res(f1,+(z),∞) =i. (68)
+As for the residues in z0andz0, from Eq.(52), Eq.(91), Eq.(92) and Eq.(50) we deduce:
+Res(f1,−(z),z2) =i
+2/radicalbig
+a(a−α)/parenleftbigg
+an+λ1
+2an/parenrightbigg
+(69)
+and:
+Res(f1,−(z),z1) =i
+2/radicalbig
+a(a−α)/parenleftbigg
+an−λ1
+2an/parenrightbigg
+. (70)10
+Note that in this case, we had to change the square root determina tion when we pass from z1toz2, since we have
+either−1<y1,n<y2,n<1(wheny= tanh(αx+ϕ)), that is −1−y0<−u0,n<u0,n<1−y0, ory1,n<−1<1<y2,n
+(wheny= coth(αx+ϕ)), that is −u0,n<−1−y0<1−y0<u0,n.
+Then:
+/integraldisplayx2,n
+x1,npn(x)dx=nπ+γ (71)
+with:
+γ=−πa
+α/parenleftbigg
+1+/radicalbigg
+1−α
+a/parenrightbigg
+. (72)
+We recover the results obtained in [20] for the first class potentials .
+SECOND CATEGORY POTENTIALS
+We say that a one dimensional potential is of second categoryif the re exists a change of variable x→utransforming
+the potential into an isotonic one V(x)→V(u) =/tildewideλ2u2+/tildewideλ0+/tildewideµ2
+u2, such that u(x) satisfies a constant coefficients
+Riccati equation of the form:
+du(x)
+dx=A0+A2u2(x), (73)
+du(x)/dxbeing of constant sign in all the range of values of xandu.
+Isotonic oscillator
+The isotonic potential with a zero energy ground state ( E0= 0) is [22, 30]:
+V−(x) =ω2
+4x2+l(l+1)
+x2−ω/parenleftbigg
+l+3
+2/parenrightbigg
+, l>0. (74)
+Its energy spectrum is given by [22]:
+En(a) = 2nω, (75)
+wherea=/parenleftbigω
+2,l+1/parenrightbig
+andak=/parenleftbigω
+2,l+1+k/parenrightbig
+.
+The classical turning points xi,nfor an energy Enare determined by the condition:
+En=V(xi,n)⇔ω2
+4x4
+i,n−ω/parenleftbigg
+2n+l+3
+2/parenrightbigg
+x2
+i,n+l(l+1) = 0, (76)
+that is:
+/braceleftbiggx2
+2,n=u0,n+δn
+x2
+1,n=u0,n−δn(77)
+with:
+u0,n= 22n+l+3
+2
+ω, δn=2
+ω/radicalBigg/parenleftbigg
+2n+l+3
+2/parenrightbigg2
+−l(l+1). (78)11
+The half action variable for a classical periodic orbit of energy Enis then:
+/integraldisplayx2,n
+x1,npn(x)dx=ω
+4/integraldisplayx2
+2,n
+x2
+1,ndufI(u),
+where we have defined u=x2and:
+fI(u) =/radicalBig/parenleftbig
+u−x2
+1,n/parenrightbig/parenleftbig
+x2
+2,n−u/parenrightbig
+u.
+The complex extended function fI(z) =/radicalBig/parenleftbig
+z−x2
+1,n/parenrightbig/parenleftbig
+x2
+2,n−z/parenrightbig
+/z, defined on C−/bracketrightbig
+x2
+1,n,x2
+2,n/bracketleftbig
+, admits two isolated
+singularities at infinity and in 0. Then:
+/integraldisplayx2
+2,n
+x2
+1,ndufI(u) =πi(Res(fI(z),0)+Res(fI(z),∞)). (79)
+Whenz→ ∞:
+fI(z)∼
+z→∞−i/parenleftBigg
+1−x2
+2,n+x2
+1,n
+2z+O(1
+z2)/parenrightBigg
+, (80)
+which implies:
+Res(fI(z),∞) =−iu0,n=−2i2n+l+3
+2
+ω2. (81)
+As for the residue in 0, from Eq.(52) we have:
+Res(fI(z),0) =/radicalBig
+−x2
+1,nx2
+2,n=i2
+ω/radicalbig
+l(l+1). (82)
+Then:
+/integraldisplayx2,n
+x1,npn(x)dx=nπ+γ, (83)
+with:
+γ=π
+2/parenleftbigg
+l+3
+2−/radicalbig
+l(l+1)/parenrightbigg
+. (84)
+Other second category potentials
+If we except the preceding example, all the other second categor y potentials are such that there exists a change
+of variable x→ytransforming the potential into an isotonic one V(x)→V(y) =λ2y2+µ2
+y2+λ0, wherey(x)>0
+satisfies a constant coefficient Riccati equation of the form [22]:
+dy
+dx=α±αy2(x), (85)12
+dy(x)/dxbeing of constant sign in all the range of values of xandy.
+As shown in [22], the potential can be written:
+V(y,a) =λ(λ∓α)y2+µ(µ−α)
+y2+λ0(a), (86)
+wherea= (λ,µ) and:
+λ0(a) =−α(λ±µ)−2λµ. (87)
+Vis a translationally shape invariant potential and its energy spectru m is given by [22]:
+En(a) =±/parenleftbig
+φ2,±(an)−φ2,±(a)/parenrightbig
+, (88)
+wherea=a0,ak= (λk,µk) = (λ±kα,µ+kα) and:
+φ2,±(a) =φ2,±(λ,µ) = (λ±µ)2. (89)
+The classical turning points yi,nfor an energy Enare determined by the condition:
+En=V(yi,n)⇔λ(λ∓α)y4
+i,n+(λ0(a)−En)y2
+i,n+µ(µ−α) = 0, (90)
+that is:
+/braceleftbiggy2
+2,n=u0,n+δn
+y2
+1,n=u0,n−δn(91)
+with:
+u0,n=En−λ0(a)
+2λ(λ∓α), δn=/radicalBigg/parenleftbiggEn−λ0(a)
+2λ(λ∓α)/parenrightbigg2
+−µ(µ−α)
+λ(λ∓α). (92)
+In the case dy/dx=α+αy2, the half action variable for a classical periodic orbit of energy Enis:
+/integraldisplayx2,n
+x1,npn(x)dx=/integraldisplayy2,n
+y1,n/radicalbig
+En−V(y,a)dy
+α(y2+1)(93)
+=/radicalbig
+λ(λ−α)
+2α/integraldisplayy2
+2,n
+y2
+1,nduf2,+(u),
+where we have defined u=y2and:
+f2,+(z) =/radicalBig/parenleftbig
+z−y2
+1,n/parenrightbig/parenleftbig
+y2
+2,n−z/parenrightbig
+z(z+1). (94)
+This last integral is readily calculated by using a complex variable forma lism. We have indeed:
+/integraldisplayy2
+2,n
+y2
+1,nduf2,+(u) =πi(Res(f2,+(z),0)+Res(f2,+(z),−1)+Res(f2,+(z),∞)). (95)13
+Whenz→ ∞:
+f2,+(z)∼
+z→∞−i
+z+O(1
+z2), (96)
+which implies:
+Res(f2,+(z),∞) =i. (97)
+As for the residues in 0 and −1, from Eq.(90) we have:
+Res(f2,+(z),0) =/radicalBig
+−y2
+1,ny2
+2,n=i/radicalBigg
+µ(µ−α)
+λ(λ−α). (98)
+Using Eq.(90), Eq.(91), Eq.(92) and Eq.(88), we deduce ( −1<0<y2
+1,n<y2
+2,n):
+Res(f2,+(z),−1) =−/radicalBig
+−/parenleftbig
+1+y2
+1,n/parenrightbig/parenleftbig
+y2
+2,n+1/parenrightbig
+=−i(λn+µn)/radicalbig
+λ(λ−α). (99)
+Then:
+/integraldisplayx2,n
+x1,npn(x)dx=nπ+γ (100)
+with:
+γ=πλ
+2α/parenleftbigg
+1−/radicalbigg
+1−α
+λ/parenrightbigg
++πµ
+2α/parenleftbigg
+1−/radicalbigg
+1−α
+µ/parenrightbigg
+. (101)
+A similar analysis can be led in the case dy/dx=α−αy2(x). If we except the specific case of the Scarf II potential
+[22], we can write:
+/integraldisplayx2,n
+x1,npn(x)dx=−/radicalbig
+λ(λ+α)
+2α/integraldisplayy2
+2,n
+y2
+1,nduf2,−(u), (102)
+where we have defined u=y2and:
+f2,−(z) =/radicalBig/parenleftbig
+z−y2
+1,n/parenrightbig/parenleftbig
+y2
+2,n−z/parenrightbig
+z(z−1). (103)
+We have:
+/integraldisplayy2
+2,n
+y2
+1,nduf2,−(u) =πi(Res(f2,−(z),0)+Res(f2,−(z),1)+Res(f2,−(z),∞)). (104)
+Whenz→ ∞:
+f2,−(z)∼
+z→∞−i
+z+O(1
+z2), (105)14
+which implies:
+Res(f2,−(z),∞) =i. (106)
+As for the residues in 0 and 1, using Eq.(90), Eq.(91), Eq.(92) and Eq .(88), we deduce:
+Res(f2,−(z),0) =−/radicalBig
+−y2
+1,ny2
+2,n=−i/radicalBigg
+µ(µ−α)
+λ(λ+α)(107)
+and
+Res(f2,−(z),1) =/radicalBig/parenleftbig
+1−y2
+1,n/parenrightbig/parenleftbig
+y2
+2,n−1/parenrightbig
+=−i(λn−µn)/radicalbig
+λ(λ+α). (108)
+Note that in this case, we had to change the square root determina tion when we pass from 0 to 1, since we have
+0<y2
+1,n<y2
+2,n<1 (y= tanh(αx+ϕ)).
+Then:
+/integraldisplayx2,n
+x1,npn(x)dx=nπ+γ (109)
+with:
+γ=πµ
+2α/parenleftbigg
+1−/radicalbigg
+1−α
+µ/parenrightbigg
+−πλ
+2α/parenleftbigg
+1−/radicalbigg
+1+α
+λ/parenrightbigg
+. (110)
+In the case of the Scarf II potential, the variable y(x) = tanh(αx/2+iπ/4) is a pure phase factor ( |y|= 1) and the
+path of integration in the action variable surrounds the branch cut which is an arc of the unit circle. Nevertheless,
+we can follow the same reasoning as before and we recover the resu lt given in Eq.(110).
+Eq.(101) and Eq.(110) correspond to the results obtained in [20] fo r the second class potentials.
+CONCLUSION
+We have shown how to calculate exactly and in a general way the actio n variable for the whole set of translationally
+shape invariant potentials. The correction to lowest order WKB for mula appears as a constant term redefining the
+Maslov index. This result implies immediately the exactness of the Ma-X u formula for every TSIP. The employed
+techniques of complex analysis are standard but even so instructiv e. The two basics cases are the harmonic and
+isotonic ones (for which we have an equispaced quantum spectrum a nd isochronicity at the classical level). In this two
+cases, the action integral involved is a sum of at most two residues a nd the linear ndependence ( nbeing the energy
+quantum number) comes from the residue at infinity. For all the oth er cases, the integral involved is obtained from
+the two basic cases by deforming the integration measure with a weig ht which is the inverse of an at most quadratic
+polynomial. The residue at infinity becomes energy independent and t he linearndependence takes its origin in the
+finite poles (with an eventual compensation of nonlinear ndependent contributions).
+ACKNOLEDGMENTS
+We would like to thank Y. Kasri for calling our attention to the Ma-Xu f ormula and for stimulating and useful
+discussions.
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\ No newline at end of file
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+arXiv:1001.0496v2  [cond-mat.mes-hall]  27 Sep 2011Bloch-Wannier theory ofpersistent current ina ring madeof the band insulator: Exactresult for
+one-dimensional latticeandestimates forrealisticlatti ces
+A. Mošková, M. Moško,∗and J. Tóbik
+Institute of Electrical Engineering, Slovak Academy of Sci ences, 841 04 Bratislava, Slovakia
+(Dated: September 29, 2018)
+It is known that a mesoscopic normal-metal ringpierced by ma gnetic flux supports persistent current. It was
+proposed recentlythatthe persistent current existsalsoi naringmade of a bandinsulator, where itiscarriedby
+electrons in the fully occupied valence band. In this work, p ersistent currents in rings made of band insulators
+are analyzed theoretically. We first formulate a recipe whic h determines the Bloch states of a one-dimensional
+(1D) ring from the Bloch states of an infinite 1D crystal creat ed by the periodic repetition of the ring. Using
+the recipe,we derive an expression forthe persistent curre nt ina 1D ringmade of aninsulator withanarbitrary
+valenceband ǫ(k). Tofindanexactresultforaspecificinsulator,weconsidera 1Dringrepresentedbyaperiodic
+latticeof Nidentical sites witha single value of the on-site energy. If the Bloch states inthe ring are expanded
+over a complete set of Non-site Wannier functions, the discrete on-site energy spl its into the energy band.
+At full filling, the band emulates the valence band of the band insulator and the ring is insulating. It carries a
+persistentcurrentequaltotheproduct of Nandthederivativeoftheon-siteenergywithrespecttothem agnetic
+flux. This current is not zero if one takes into account that th e on-site Wannier function and consequently the
+on-site energy of each ring site depend on magnetic flux. To de rive the current analytically, we expand all N
+Wannier functions of the ring over the infinite basis of Wanni er functions of the constituting infinite 1D crystal
+andeventuallydeterminethecrystalWannierfunctionsbya methodoflocalizedatomicorbitals. Intermsofthe
+crystal Wannier functions, the current at full fillingarise s because the electroninthe ringwith Nlattice sites is
+allowed to make a single hop from site ito its periodic replicas i±N. In terms of the ring Wannier functions,
+the same current is due to the flux dependence of the Wannier fu nctions basis, and the longest allowed hop is
+fromitoi±Int(N/2)only. Finally,we estimate the persistent current at full fil lingin a few realisticsystems.
+Inparticular,theringsmadeofrealbandinsulators(GaAs, Ge,InAs)withafullyfilledvalencebandandempty
+conductionbandarestudied. Thecurrentdecayswiththerin glengthexponentiallyduetotheexponentialdecay
+of the Wannierfunctions. Inspite of that, itcanbe of measur able size.
+PACS numbers: 73.23.-b, 73.61.Ey
+I. INTRODUCTION
+It is known that a conducting ring pierced by a constant
+magneticflux cansupportpersistentelectroncurrentcircu lat-
+ing around the ring [1]. The persistent current is well known
+to exist in a superconductingring,andalso in a normal-meta l
+ringaslongasitismesoscopic(ofsizecomparableorsmalle r
+than the electron coherence length). Early work dealing wit h
+persistent current and flux quantization in superconductin g
+rings [2] also contains results relevant to the normal metal
+rings. Lateritwasmentioned[3]thatonecanhavecirculati ng
+currentsforfreeelectronsinballistic normal-metalring s,and
+thereference[4]proposedthatpersistentcurrentsshould exist
+in the disordered normal-metal rings. The persistent curre nts
+were indeed observed in the disordered normal-metal rings
+[5–9] as well as in ballistic conductingrings[10, 11]. Deta ils
+arereviewedinvariouspapers[12–14].
+Oneofusrecentlyproposed[15]thatpersistentcurrentex-
+ists alsoin a ringmadeofa bandinsulator,whereit iscarrie d
+exclusively by Bloch electrons in the fully occupied valenc e
+band. The effect is interesting for two reasons [15]. First, it
+seemstobetheonlyeffectmanifestingelectrontransporti na
+fully occupiedvalenceband. In a standardconductancemea-
+surement, the band insulator is biased by two metallic elec-
+trodesandoneobservesthetunnelingofconductionelectro ns
+from metal to metal through the energy gap of the insulator.
+One does not observe the transport of valence band electronsin the insulator. Second, in the fully occupied valence band ,
+Blochelectronscannotbescatteredbyphononsorotherelec -
+trons. Therefore, the electron coherence length should be
+hugeevenatroomtemperatureifthevalencebandisseparate d
+from the conduction band by a large energy gap. The persis-
+tent current in the insulating ring should thus be observabl e
+up to high temperatures as a temperature-independent effec t
+as long as excitations into the (empty) conduction band are
+negligible. By contrast, persistent currents in metallic r ings
+areobservableonlyatlow temperatures. [9].
+Wereportonastudyofpersistentcurrentsinringsmadeof
+bandinsulatorsusingtheformalismoftheBlochandWannier
+functions. It was partly motivated by a pioneering analysis
+[16]ofpersistentcurrentsina1Dring-shapedperiodiclat tice
+composedofthelattice siteswitha singlevalueoftheon-si te
+energy. The authors derived persistent current Ifor spinless
+electrons at zero temperature assuming that they move along
+thelatticebymeansofthenearest-neighbor-sitehopping. The
+resultreads
+I=−4πΓ1
+Nφ0sinπ
+NNe
+sinπ
+Nsin/bracketleftbigg2π
+Nφ
+φ0/bracketrightbigg
+,−φ0
+2≤φ<φ0
+2,
+(1)
+foroddNeand
+I=−4πΓ1
+Nφ0sinπ
+NNe
+sinπ
+Nsin/bracketleftbiggπ
+N/parenleftbigg2φ
+φ0−1/parenrightbigg/bracketrightbigg
+,0≤φ<φ0,
+(2)2
+forevenNe,whereNeistheelectronnumberinthering, Nis
+the numberofthe lattice sites, φis themagneticfluxpiercing
+thering,φ0≡h/eisthefluxquantum,and Γ1isthehopping
+amplitude. Equations (1) and (2) show a nonzero current for
+a conductor ( Ne< N). However, they show a zero current
+for an insulator ( Ne=N), which contradicts our claim that
+persistentcurrentexistsalso inaninsulatingring.
+Therefore,oneofourgoalsistogiveexactproofthataring
+made of a band insulator does support a non-zero persistent
+current carried by the fully filled valence band. We also pro-
+vide a simple tool allowing to estimate persistent currents in
+rings made of realistic band insulators. We finally present a
+derivation of an exact persistent current expression for a s pe-
+cific modelinsulator. Thispaperisstructuredasfollows.
+In Sect. II, we formulate a recipe which determines the
+Bloch states of a one-dimensional (1D) ring from the Bloch
+states of an infinite 1D crystal created by the periodic repet i-
+tion of the ring. In Sect. III we use the recipe to derive an
+expression for the persistent current in a 1D ring made of a
+band insulator with an arbitrary valence band ǫ(k). The ex-
+pressiongenerallyshowsthatthecurrentisnotzero,altho ugh
+it does turn to zero for a cosine-shaped energy band, which
+was used in the analysis with nearest-neighbor-site hoppin g
+[16].
+Toderiveanexactresultforaspecificinsulator,inSect. IV
+we consider a 1D ring represented by a periodic lattice of N
+identical sites with a single value of the on-site energy. If the
+Bloch states in the ring are expanded over a complete set of
+Non-siteWannierfunctions,thediscreteon-siteenergyspl its
+into the energy band. At full filling, the band emulates the
+valence band of the band insulator and the ring is insulating .
+We findthatit carriesapersistentcurrentequaltotheprodu ct
+ofNand the derivative of the on-site energy with respect to
+themagneticflux. AnalternativeexplanationofwhyEqs. (1)
+and (2) givezero currentat full filling is that they do not tak e
+intoaccounttheflux-dependenceofthe on-siteenergy.
+To get closer to an exact formula, in Sect. IV we expand
+allNWannierfunctionsoftheringoverthebasisofWannier
+functions of the constituting infinite 1D crystal. In terms o f
+the crystal Wannier functions, the current at full filling ar ises
+becausetheelectronintheringwith Nlatticesitesisallowed
+tomakeasinglehopfromsite itoitsperiodicreplicas i±N.
+Alternatively,intermsoftheringWannierfunctionsthesa me
+currentisduetothefluxdependenceoftheWannierfunctions
+basis, and the longest allowed hop is from itoi±Int(N/2)
+only.
+In Sect. V we eventually derive the crystal Wannier func-
+tions by a method of localized atomic orbitals (LCAO) and
+expressthepersistentcurrentatfullfillingbymeansofane x-
+act formula. The formula is found to be in good agreement
+with a numerical LCAO calculation. The numerical data are
+provided for a 1D ring made of an artificial band insulator,
+namely for a GaAs ring subjected to a periodic potential em-
+ulatingthe insulatinglattice.
+InSect. VIweestimatepersistentcurrentsintheringsmade
+of real band insulators (GaAs, Ge, InAs) with a fully filledvalence band and empty conductionband. The currentat full
+filling decays with the ring length exponentially due to the
+exponential decay of the Wannier function tails. In spite of
+that,it canbeofmeasurablesize.
+Our results are summarized in Sect. VII. We relate them
+briefly to the paper by Kohn [17], which defines the insulat-
+ing state as a property of the many-body ground state, and
+whichcontainsa remarkimplicitlyadmittingthe existence of
+nonzeropersistentcurrentininsulatingrings.
+We note that some of the results of Sect. V were obtained
+in the conference contribution [18] by means of a formally
+different approach. In another conferencepaper [19] the pe r-
+sistent current at full filling was analyzed for a 1D ring with
+Kronig-Penneypotential.
+II. ELEMENTARYTHEORY AND THEGENERAL RECIPE
+To review basic concepts, we consider the circular 1D ring
+with circumference L, pierced by magnetic flux φ. The elec-
+tronsinsuchringaredescribedbytheSchrödingerequation
+/bracketleftbigg1
+2m(−i/planckover2pi1d
+dx+eφ
+L)2+V(x)/bracketrightbigg
+ψφ(x) =ε(φ)ψφ(x),(3)
+wherexistheelectronpositionalongtheringcircumference,
+mis the electron mass, V(x)is an arbitrary potential applied
+alongthering, ε(φ)istheelectroneigen-energy,and ψφ(x)is
+the electron wave function. Due to the ring geometry, ψφ(x)
+hasto fulfilltheperiodiccondition
+ψφ(x) =ψφ(x+L). (4)
+We definethewavefunction ϕφ(x)bytransformation
+ϕφ(x) = exp/parenleftbigg
+i2π
+Lφ
+φ0x/parenrightbigg
+ψφ(x). (5)
+Settingψφ(x) = exp( −i2π
+Lφ
+φ0x)ϕφ(x)into (3) and (4) we
+obtaintheSchrödingerequation
+/bracketleftbigg
+−/planckover2pi12
+2md2
+dx2+V(x)/bracketrightbigg
+ϕφ(x) =ε(φ)ϕφ(x)(6)
+withtheboundarycondition
+ϕφ(x+L) = exp(i2πφ
+φ0)ϕφ(x). (7)
+Herethemagneticfluxentersonlythe boundarycondition.
+Intheringgeometry, V(x)obeystheperiodiccondition
+V(x) =V(x+L), (8)
+which is the same as for the infinite 1D crystal with lattice
+constantL. Therefore, equation (6) with condition (8) is
+mathematicallyidenticalwiththe Schrödingerequation
+/bracketleftbigg
+−/planckover2pi12
+2md2
+dx2+V(x)/bracketrightbigg
+ϕk(x) =ε(k)ϕk(x),(9)3
+whereV(x)is the infinite periodic potential with period L,
+obtainedbytheperiodicrepetitionoftheringpotential V(x).
+Equation(9)hasthewellknownBlochsolution
+ϕk(x) = exp(ikx)uk(x), (10)
+wherethefunction uk(x)fulfillstheperiodiccondition
+uk(x) =uk(x+L), (11)
+andkisthe electronwavevectorfromtheinterval (−∞,∞).
+Clearly, the wave function (5) and Bloch solution (10) coin-
+cidefork=2π
+Lφ
+φ0. Letusdiscussthiscoincidencein detail.
+Consider first the ring with zero magnetic flux. To obtain
+the wave functions in such ring, it suffices to take the Bloch
+function(10) andto restrictit bytheperiodiccondition
+ϕk(x) =ϕk(x+L), (12)
+which is the condition (7) for φ= 0. Due to the condition
+(12), thewavevector kbecomesdiscrete:
+k=2π
+Ln, n= 0,±1,±2,... (13)
+Thus,intheringwith zeromagneticfluxandspecifiedpoten-
+tialV(x), the eigen-function ϕn(x)and eigen-energy εncan
+be calculated simply by setting k=2π
+Lninto the Bloch so-
+lutionsϕk(x)aε(k), calculated for the same potential V(x)
+repeatedwith period Lfromx=−∞tox=∞. Thisrecipe
+canbegeneralizedtononzeromagneticfluxasfollows.
+Arbitrarymagneticflux φcanbewritteninthe form
+φ=nφ0+φ,, (14)
+whereφ,is the reduced flux from the range <−φ0
+2,φ0
+2)or
+alternatively from <0,φ0), andnis one of the values n=
+0,±1,±2,.... Setting (14) into (5) one can write (5) in the
+form
+ϕn,φ,(x) = exp/parenleftbigg
+i2π
+Lφ,
+φ0x/parenrightbigg
+ψ,
+n,φ,(x),(15)
+wherethefunction ψ,
+n,φ,(x)≡exp/parenleftbig
+i2π
+Lnx/parenrightbig
+ψφ(x)obeysthe
+periodic condition ψ,
+n,φ,(x) =ψ,
+n,φ,(x+L). The boundary
+condition(7)nowreads
+ϕ,
+n,φ,(x+L) = exp(i2πφ,
+φ0)ϕ,
+n,φ,(x).(16)
+Similarly, in the Bloch function theory it is customary to
+expressthe wavevector kbymeansoftherelation
+k=2π
+Ln+k′, (17)
+wherek′is the reduced wave vector from the first Brillouin
+zoneandtheinteger nplaystheroleoftheenergybandnum-
+ber. Using(17)we canwritetheBlochfunction(10) as
+ϕn,k,(x) = exp(ik,x)u,
+n,k,(x), (18)wherethefunction u′
+n,k′(x)≡exp/parenleftbig
+i2π
+Lnx/parenrightbig
+uk(x)fulfillsthe
+periodiccondition u′
+n,k′(x) =u′
+n,k′(x+L). Itistheneasyto
+verifythat
+ϕ,
+n,k,(x+L) = exp(ik,L)ϕ,
+n,k,(x).(19)
+The equations (15) and (16) coincide with the equations
+(18) and(19),respectively,if
+k′=2π
+Lφ,
+φ0, (20)
+oralternatively,if
+k=2π
+L(n+φ,
+φ0). (21)
+One can thus formulate the following general recipe. In
+the ring with a known potential V(x), the eigen-function
+ϕ,
+n,φ,(x)≡ϕφ(x)and eigen-energy εn(φ′)≡ε(φ)can be
+calculated by setting (20) or (21) into the Bloch solutions
+ϕn,k,(x)≡ϕk(x)aεn(k,)≡ε(k), calculated for the same
+potentialV(x)repeated with period Lfromx=−∞to
+x=∞.
+Thegeneralrecipeholdsforanypotential V(x)obeyingthe
+cyclic condition (8). It therefore holds also for any potent ial
+whichadditionallyobeystheperiodiccondition
+V(x) =V(x+a), (22)
+wherea=L/NandNis the (integer) number of periods
+awithin the period L. This type of potential will be con-
+sidered in the rest of the paper. Figure 1 illustrates how an
+infinite 1D crystal is created from a 1D ring with lattice pe-
+riodaand lengthL=Na. Such crystal is still described by
+Schrödingerequation(9), except that now V(x) =V(x+a)
+andconsequently uk(x) =uk(x+a). However,thecondition
+uk(x) =uk(x+L)holdsaswellbecause V(x) =V(x+L).
+We can rederive the general recipe briefly, if we compare
+uk(x) =uk(x+L)withψφ(x) =ψφ(x+L)andϕk(x) =
+exp(ikx)uk(x)withϕφ(x) = exp/parenleftBig
+i2π
+Lφ
+φ0x/parenrightBig
+ψφ(x). We see
+thatϕφ(x)coincideswith ϕk(x)fork=2π
+Lφ
+φ0.
+Itremainstoexpressthepersistentcurrent. TheBlochelec -
+tron in state (n,k′)moves with velocity vn(k′) =1
+/planckover2pi1∂εn(k′)
+∂k′.
+We set into the Bloch electron velocity the formula (20). We
+obtain the expression vn(φ′) =L
+e∂εn(φ′)
+∂φ′, which is the elec-
+tron velocity in state (n,φ′)in the ring. The current carried
+bytheelectroninstate (n,φ′)reads
+In(φ′) =−evn(φ′)
+L=−∂εn(φ′)
+∂φ′. (23)
+Thetotalpersistentcurrentcirculatingintheringis
+I(φ′) =−∂
+∂φ′/summationdisplay
+nεn(φ′), (24)
+where we sum (at zero temperature) over all occupied states
+n= 0,±1,±2,...uptotheFermilevel. Inthefollowingtext
+we usethe formulas(23) and(24) with symbol φ′changedto
+φ, whereφ∈<−φ0
+2,φ0
+2)oralternatively φ∈<0,φ0).4
+-3L/2-L-L/20L/2L3L/2aa
+V(x)
+xV(x)the 1D ring with
+three "atoms"L
+FIG.1. Thetopfigureshowsschematicallythe1Dringcompose dof
+the periodic lattice with Nlattice sites ( N= 3), periodicity a, and
+lengthL=Na. For simplicity, the lattice potential of the ring is
+modeled byasquare-well potential V(x). Thebottom figure depicts
+the infinite 1D periodic potential obtained by the periodic r epetition
+of the ringpotential.
+III.PERSISTENTCURRENTIN 1D RINGMADEOF A
+CRYSTALWITH ARBITRARYVALENCEBAND ε(k)
+Consider first an infinite 1D crystal created froma 1D ring
+withthelatticeperiod aandlengthL=Naasitisillustrated
+in Fig. 1. Let εv(k)be the energy dispersion of the valence
+band of Bloch electronsin that crystal. The wave vector kin
+εv(k)is supposedto be from the Brillouin zone <−π
+a,π
+a>.
+In what follows we skip the valence band index ( v) for sim-
+plicity and we use notation ε(k). We can express ε(k)by
+meansoftheinfiniteFourierexpansion
+ε(k) =a0+a1cos(ka)+
+a2cos(2ka)+...aNcos(Nka)+...,
+(25)
+where
+a0=1
+π/a/integraldisplayπ/a
+0ε(k)dk (26)
+and
+aj=2
+π/a/integraldisplayπ/a
+0ε(k)cos(jka)dk, j= 1,2,...(27)
+Now we apply the general recipe from Sect. II. We set for
+kin Eq. (25) the equation k=2π
+Na(n+φ
+φ0). We obtain the
+expressionfortheringenergies εn(φ):
+εn(φ)≡ε(kn(φ)) =∞/summationdisplay
+j=0ajcos(kn(φ)ja),(28)
+where
+kn(φ)≡2π
+Na(n+φ
+φ0). (29)
+As shown in appendix A, expansion (28) can also be written
+inthe form
+εn(φ) = Ω0+M/summationdisplay
+j=1/braceleftBig
+ΩR
+jcos(kn(φ)ja)+ΩI
+jsin(kn(φ)ja)/bracerightBig
+,
+(30)whereM= Int(N/2)andtheFouriercoefficients Ωaregiven
+by equations (102) and (103). Expressions (28) and (30) are
+mathematicallyequivalent,butexpression(30) revealsan im-
+portantdifferencefrom the Bloch energy(25). Generally,t he
+Bloch energy (25) consists of an infinite number of Fourier
+terms, with the coefficients ajbeing independent on mag-
+netic fluxφ. On the other hand, expression (30) shows that
+the Bloch energy in the ring can be expressed through a fi-
+nite number of Fourier terms, with the Fourier coefficients Ω
+dependingon φ[see Eqs. (102)and(103)].
+IntherestofthissectionweuseEq. (28). Thesameresults
+can be obtained by means of Eq. (30), but derivations are
+morecumbersome.
+Setting Eq. (28) into Eq. (24) we obtainthe persistent cur-
+rent
+I(φ) =2π
+N1
+φ0∞/summationdisplay
+j=1jaj
+×/bracketleftBigg
+cos/parenleftbigg2π
+Nφ
+φ0j/parenrightbigg/summationdisplay
+nsin/parenleftbigg2π
+Nj·n/parenrightbigg
++ sin/parenleftbigg2π
+Nφ
+φ0j/parenrightbigg/summationdisplay
+ncos/parenleftbigg2π
+Nj·n/parenrightbigg/bracketrightBigg
+.(31)
+If the ring contains Nespinless electrons and Neis odd,
+summation over nin Eq. (31) includes the occupied states
+n= 0,±1,±2,...,±(Ne−1)/2. Aftersimplemanipulations
+I(φ) =2π
+N1
+φ0∞/summationdisplay
+j=1jajsin/parenleftbigg2π
+Nφ
+φ0j/parenrightbigg
+×
+2(Ne−1)/2/summationdisplay
+n=0cos/parenleftbigg2π
+Nj·n/parenrightbigg
+−1
+.(32)
+Performingthesummationover nwe arrivetotheformula
+I(φ) =2π
+N1
+φ0/bracketleftBigg∞/summationdisplay
+j=1
+j/negationslash=N,2N,...jajsin/parenleftbigg2π
+Nφ
+φ0j/parenrightbiggsin/parenleftbigπ
+NjNe/parenrightbig
+sin/parenleftbigπ
+Nj/parenrightbig
++Ne∞/summationdisplay
+j=N,2N,...jajsin/parenleftbigg2π
+Nφ
+φ0j/parenrightbigg/bracketrightBigg
+, (33)
+whereφ∈<−φ0
+2,φ0
+2). Note that the first sum in (33) omits
+thetermsj=N,2N,3N,...,includedinthesecondsum.
+IfNeisevenandφ∈<0,φ0),wesuminEq. (31)overthe
+occupied states n= 0,±1,±2,...,±(Ne/2−1),−Ne/2.
+Similarmanipulationsasbeforeleadtothe result
+I(φ) =2π
+N1
+φ0
+×/bracketleftBigg∞/summationdisplay
+j=1
+j/negationslash=N,2N,...jajsin/parenleftbiggπ
+N/parenleftbigg2φ
+φ0−1/parenrightbigg
+j/parenrightbiggsin/parenleftbigπ
+NjNe/parenrightbig
+sin/parenleftbigπ
+Nj/parenrightbig
++Ne∞/summationdisplay
+j=N,2N,...jajsin/parenleftbigg2π
+N/parenleftbiggφ
+φ0/parenrightbigg
+j/parenrightbigg/bracketrightBigg
+, (34)5
+whereφ∈<0,φ0). The results (33) and (34) hold for the
+conducting( Ne<N) aswell asinsulating( Ne=N)rings.
+ForNe=Nthe first sum on the right hand side of Eqs.
+(33)and(34) becomeszeroandbothequationstaketheform
+I(φ) =−(2π/φ0)∞/summationdisplay
+j=N,2N,...jajsin/parenleftbigg2π
+Nφ
+φ0j/parenrightbigg
+,(35)
+whereφ∈<−φ0
+2,φ0
+2)andNcan be even as well as odd.
+Equation (35) expresses the persistent current in a 1D ring
+madeofa bandinsulatorwithanarbitraryvalenceband εk.
+An important property of the formula (35) can be recog-
+nized without specifying the dispersion law εk. The disper-
+sionε(k)isaperiodicfunctionofperiod 2π/a. Aslongasitis
+analytic in the whole Brillouin zone <−π
+a,π
+a>, the Fourier
+coefficientsajinthe expansion(25) obeytheinequality[20]
+|aj| ≤Ce−ξj, (36)
+whereCandξarepositiveconstants. Equation(35)combined
+with inequality (36) shows that the persistent current in th e
+insulating ring is in general not zero albeit it decreases wi th
+the ring length at least exponentially. In the next sections ,
+we will demonstrate for various specific insulating rings th e
+exponential decrease, and we will see that the current can be
+ofmeasurablesize inrealistic systems.
+The pioneer results (1) and (2) can be obtained from Eqs.
+(33) and (34) if we keep only the term j= 1and we put
+a1≡ −2Γ1. Equations(1)and(2)predictforinsulatingrings
+(Ne=N) a zero persistent current, which contradicts equa-
+tion (35). Origin of this contradiction is evident. Keeping
+only the term j= 1means that only the nearest-neighbor-
+site hoppingis operative. Thisapproach,also knownastigh t-
+binding approximation, simplifies the general dispersion l aw
+(25) to the cosine shaped dispersion ε(k)∝cos(ka). How-
+ever, equation (35) shows that nonzero persistent current i n
+the insulating ring of length Nis (mainly) due to the Fourier
+coefficientaNwhich is in general not zero for a realistic dis-
+persionlaw. Wewillseeinthenextsectionthat aNphysically
+represents the hopping amplitude for a single hop from site i
+toits periodicreplicas i±N.
+IV.PERSISTENTCURRENTIN INSULATING1D RINGIN
+THE WANNIERFUNCTIONSFORMALISM
+We consider again the 1D ring with lattice period aand
+lengthL=Na, sketched in Fig. 1. Simultaneously,we con-
+sidertheinfinite1Dcrystalobtainedbytheperiodicrepeti tion
+of the 1D ring, as it is illustrated in Fig. 1. For the sake of
+clarity we recall a few major points. First, the Bloch func-
+tionϕk(x)and Bloch energy ε(k)in the infinite crystal are
+describedbySchrödingerequation
+Hϕk(x) =ε(k)ϕk(x),
+H=−/planckover2pi12
+2md2
+dx2+V(x), V(x) =V(x+a).(37)Second,accordingto the generalrecipe of Sect. II, the Bloc h
+functionandBlochenergyinthering, ϕn,φ(x)andεn(φ),are
+givenbyequations
+ϕn,φ(x) =ϕk=kn(φ)(x), εn(φ) =ε(k=kn(φ)),
+kn(φ)≡2π
+Na(n+φ
+φ0). (38)
+Third,the Bloch vector kis assumed to be fromthe Brillioun
+zone<−π
+a,π
+a>, or alternatively from <0,2π
+a>. This
+meansthatε(k)is the dispersion of a single energyband and
+ϕk(x)are the Bloch functions of that band. As before, the
+functionsϕk(x)andε(k)do not involveanyband index,and
+ε(k)is supposed to represent the valence band of the crystal.
+BelowwedefinetheWannierfunctionsbyrestrictingustothe
+Blochstatesofthatvalenceband.
+A.Wannierfunctionsof the infinitecrystal
+We first define the Wannier functions for the crystal. Ac-
+cording to a standard textbook definition, the Wannier func-
+tionW(x−la)at thelattice site x=laisdefinedas
+W(x−la)≡Wl(x) =a
+2π/integraldisplayπ
+a
+−π
+ae−iklaΦk(x)dk,(39)
+whereΦk(x)is the Bloch function of the infinite crystal. We
+notethatΦk(x)is thesame Bloch functionas ϕk(x)in equa-
+tion(37). Theonlydifferenceisthat Φk(x)isnormalizedas
+/integraldisplaya
+0|Φk(x)|2dx= 1 (40)
+while the normalizationof ϕk(x)in the infinite crystal is un-
+specified. In the following text, Φk(x)represents the nor-
+malized Bloch function of the infinite crystal and the symbol
+ϕk(x)withk=kn(φ)represents the Bloch function of the
+ring. [Inthering ϕk(x)isnormalizedbyequation(48).]
+Since the crystal is infinite, kis a continuous variable and
+theBlochfunctionsfulfilltheorthogonalitycondition
+/integraldisplay∞
+−∞Φk′(x)∗Φk(x)dx=2π
+aδ(k′−k).(41)
+One can then easy verify the orthogonality condition for the
+Wannierfunctions,
+/integraldisplay∞
+−∞W∗
+l′(x)Wl(x)dx=δl,l′. (42)
+Finally,definition(39)canalsobewrittenintheinvertedf orm
+Φk(x) =/summationdisplay
+leiklaWl(x) (43)
+whereonesumsoverall lattice pointsofthe (infinite)cryst al.6
+We definethematrixelements Γjbyequations
+Γl′,l=−/integraldisplay∞
+−∞Wl′(x)ˆHWl(x)dx= Γl−l′,Γj≡Γl−l′,
+(44)
+wherel,l′= 0,±1,±2,···± ∞and it is taken into account
+thatΓl′,ldependsonlyonthedifference j=l−l′. TheBloch
+energyǫ(k)can be written as ǫ(k) =∝angbracketleftΦk|ˆH|Φk∝angbracketright/∝angbracketleftΦk|Φk∝angbracketright.
+Ifwe set for Φkequation(43),we easyfind
+ε(k) =Γ0+2∞/summationdisplay
+j=1Γjcosjka. (45)
+Comparison of Eq. (45) with Eq. (25) shows that aj= 2Γj.
+Obviously,
+Γj=1
+π/2a/integraldisplayπ/a
+−π/aε(k)cos(jka)dk, (46)
+whichisanalternativeformofequation(44).
+Finally,setting aj= 2Γjintotheresult (35) weget
+I=−(4π/φ0)∞/summationdisplay
+j=N,2N,...jΓjsin/parenleftbigg2π
+Nφ
+φ0j/parenrightbigg
+.(47)
+Equation (47) together with Eq. (44) express the persistent
+currentintheinsulatingringthroughtheWannierfunction sof
+the constituting infinite crystal. We defer discussion of th ese
+expressions to subsection C, where equation (47) is derived
+fromtheWannierfunctionsofthering.
+B.Wannierfunctionsofthe ring
+Let us define the Wannier functions of the ring. In the
+ring withNlattice sites and a fixed value of magnetic flux
+only the discrete Bloch vectors kn(φ)are allowed, where
+kn(0)∈<−π
+a,π
+a>or alternatively kn(0)∈<0,2π
+a>. In
+bothcases thereare Nallowedvaluesof n(specifiedbelow),
+whichmeansthattheoriginallycontinuousband ε(k)consists
+ofNdiscrete levels εn(φ). The eigen-functionsof these lev-
+elsrepresentacompletesetof NBlochfunctionsobeyingthe
+orthogonalitycondition
+/integraldisplayNa
+0ϕ∗
+k′(x)ϕk(x)dx=δk,k′=δn,n′,(48)
+wherek=kn(φ). Consequently,inthe1Dringwith Nlattice
+sitesthe Wannierfunctionat site x=lacanbedefinedas
+w(x−la)≡wl(x) =1√
+N/summationdisplay
+ke−iklaϕk(x),(49)
+wherek=kn(φ)and summation over kmeans the summa-
+tionoverthe Nallowedvaluesof n. Definition(49)coincides
+with definition (3.7) in Kohn’s paper [17], except that Kohn
+assumesk=kn(0).In the following calculations we choose the Brillouin zone
+kn(0)∈<0,2π
+a>,forwhichtheallowed naresimply
+n= 0,1,...,N−1 (50)
+withoutregardstotheparityof Nandsignofφ.
+Using Eqs. (48) and (49) we can easy verify the orthogo-
+nalitycondition
+/integraldisplayNa
+0wl′(x)∗wl(x)dx=δl′,l. (51)
+Finally,definition(49)canalsobewrittenintheinvertedf orm
+ϕk(x) =1√
+NN−1/summationdisplay
+l=0eiklawl(x). (52)
+Inthispaperthe Wannierfunctions wl(x)are oftencalled the
+ringWannierfunctionswhiletheWannierfunctions Wl(x)are
+called the crystal Wannier functions. The ring Wannier func -
+tions are magnetic-flux dependent, the crystal Wannier func -
+tionsarenot.
+We note that the results obtained in the rest of this section
+canalsobeobtainedbychoosing kn(0)∈<−π
+a,π
+a>. Inthat
+casethe allowedvaluesof nare
+n= 0,±1,±2,...,±(N−1)/2 (53)
+foroddN,whileforeven Nonehas
+n= 0,±1,±2,...,±(N/2−1),−N/2forφ>0(54)
+and
+n= 0,±1,±2,...,±(N/2−1),+N/2forφ<0.(55)
+Evidently,it ismorestraightforwardtoworkwithEq. (50).
+Nowweexpressthepersistentcurrentatfullfillinginterms
+oftheringWannierfunctions. We write εn(φ)in theform
+εn(φ) =/integraldisplayNa
+0ϕ∗
+kn(φ)(x)Hϕkn(φ)(x)dx. (56)
+We needto rewrite the right handside of Eq. (56) in termsof
+thematrixelements
+Tl′,l=/integraldisplayNa
+0w∗
+l′(x)H wl(x)dx, (57)
+wherel,l′= 0,1,...N−1. WesetintoEq. (57)theequation
+(49). We find
+Tl′,l=1
+N/summationdisplay
+n′/summationdisplay
+neikn′(φ)l′ae−ikn(φ)la
+×/integraldisplayNa
+0ϕ∗
+kn′(φ)(x)Hϕkn(φ)(x)dx
+=1
+Nei2π
+Nφ
+φ0(l′−l)N−1/summationdisplay
+n=0ei2π
+Nn(l′−l)εn(φ).
+(58)7
+SinceTl′,ldepends only on the difference between landl′,
+we canwrite
+Tl′,l=Tl′−l≡Tj=1
+Nei2π
+Nφ
+φ0jN−1/summationdisplay
+n=0ei2π
+Nnjεn(φ).(59)
+InvertingEq. (59)we get
+εn(φ) =N−1/summationdisplay
+j=0e−ikn(φ)jaTj. (60)
+By means of Eq. (60), the ground state energy of the fully
+filledvalencebandisreadilyobtainedintheform
+E(φ) =N−1/summationdisplay
+n=0εn(φ) =N·T0(φ). (61)
+Finally,the persistentcurrentat full-fillingreads
+I=−dE
+dφ=−NdT0
+dφ. (62)
+One can see from Eq. (57) that the matrix element T0rep-
+resents the on-site energy of the Wannier state wl(x). Since
+wl(x)depends on magnetic flux, so does T0. This however
+means that the current (62) is in general not zero. We have
+thus arrived at the same conclusion as in the preceding sec-
+tion, where the persistent current in the insulating ring (E q.
+35)wasderivedfromtheBlochstates andgeneralrecipe.
+C. RingWannierfunctionsinthe basisof crystal Wannier
+functions
+The result (47), or equivalently the result (35), can be ob-
+tained from the result (62), if the ring Wannier functions ar e
+expanded over the basis of the crystal Wannier functions. To
+showthisweproceedasfollows.
+The Bloch functions of the ring are given by Eq. (38) and
+normalized as/integraltextNa
+0ϕ∗
+k(x)ϕk(x)dx= 1, wherek=kn(φ)
+[see Eq. (48)]. The Bloch functions φk(x)coincide with
+ϕk(x)exceptforthenormalizationconstant. Astheyarenor-
+malizedbymeansofEq. (40),intheringwehavetherelation
+ϕk(x) =1√
+Nφk(x), k=kn(φ),(63)
+We set forφk(x)in Eq. (63) the definition (43). Then we set
+Eq. (63) intotherighthandsideofEq. (49). We obtain
+wl(x) =1
+N/summationdisplay
+k∞/summationdisplay
+l′=−∞e−ik(l−l′)aWl′(x),(64)
+wherek=kn(φ)and summation over kmeans the summa-
+tionovern= 0,1,...,N−1. Usingsubstitution l′=Nr+s,wheres= 0,1,...(N−1)andr= 0,±1,±2,···±∞, we
+furtherobtain
+wl(x) =∞/summationdisplay
+r=−∞N−1/summationdisplay
+s=0WNr+s(x)ei2π
+Nφ
+φ0(rN+s−l)
+×1
+NN−1/summationdisplay
+n=0ei2π
+Nn(s−l).
+(65)
+Eventually
+wl(x) =∞/summationdisplay
+r=−∞ei2πφ
+φ0rWNr+l(x).(66)
+Equation(66) expressesthe ring Wannier functionat the rin g
+sitel,wl(x), through the crystal Wannier functions Wl′(x).
+Forφ= 0equation (66) coincides with equation (5.9) in
+Kohn’spaper[17].
+SettingEq. (66) intoEq. (57)we obtain
+Tj=/integraldisplayNa
+0w∗
+j(x)ˆHw0(x)dx
+=∞/summationdisplay
+r=−∞∞/summationdisplay
+r′=−∞ei2πφ
+φ0(r−r′)/integraldisplayNa
+0WNr′+j(x)ˆHWNr(x)dx
+=∞/summationdisplay
+s=−∞e−i2πφ
+φ0s∞/summationdisplay
+r=−∞/integraldisplayNa−rNa
+0−rNaWNs+j(x)ˆHW0(x)dx
+=∞/summationdisplay
+s=−∞e−i2πφ
+φ0s/integraldisplay∞
+−∞WNs+j(x)ˆHW0(x)dx.
+(67)
+By means of the matrix elements Γj, equation (67) can be
+writtenas
+Tj=∞/summationdisplay
+s=−∞e−2πiφ
+φ0sΓsN+j. (68)
+Forj= 0equation(68)gives
+T0= Γ0+2∞/summationdisplay
+s=1ΓsNcos(2πφ
+φ0s).(69)
+We substitute Eq. (69) into the right hand side of Eq. (62).
+We obtainthe persistentcurrentat fullfillingin theform
+I=−4πN
+φ0∞/summationdisplay
+s=1sΓsNsin(2πφ
+φ0s),(70)
+or equivalently in the form (47) which coincides with the re-
+sult (35) for aj= 2Γj.
+Intherestofthissectionwediscussthephysicalmeaningof
+the coefficients ΓN,Γ2N, etc., appearing in expression (47).
+Sinceaj= 2Γj,themeaningofthecoefficients aN,a2N,etc.,
+isthesameanddoesnotneedan extradiscussion.8
+According to the definition (44), in the infinite 1D crystal
+the matrix element ΓNrepresents the hopping amplitude of
+a single hop from the Wannier state at site lto the Wannier
+state at site l+N. In the ring with Nlattice sites the site
+l+Nis a periodic replica of site l, which means the same
+site. Appearance of ΓNin expression (47) therefore implies,
+that the electron in the ring with Nsites can make a single
+hop from the Wannier state at site lto its periodic replica at
+sitel+N. Inotherwords,theelectronhopsfromsite ltothe
+samesitebymakingoneroundaroundthering,and ΓNisthe
+hopping amplitude of the process. Note that we speak about
+hopping in the ring, but in terms of the Wannier states of the
+infinitecrystal(c.f. Eq. 44).
+Similarly, the coefficient Γ2Ndescribes the process in
+which the electron hops from site lto the same site by mak-
+ing two rounds around the ring. It follows from relation (36)
+and equation aj= 2Γj, thatΓ2Nis much smaller than ΓN.
+Its contributionto the right hand side of Eq. (47) is therefo re
+negligible. Themeaningofthecoefficients Γ3N,Γ4N, etc.,is
+analogousandtheircontributionisnegligibleevenmore.
+Wecansummarizetheresult(47)asfollows. Thepersistent
+current in the insulating ring with Nlattice sites, derived in
+terms of the crystal Wannier functions, is due to the fact tha t
+the electron at the ring site lis allowed to make a single hop
+from sitelto its periodic replicas l±N. (The sign minus is
+allowedaswell duetothesymmetryreason.)
+Theprocessinwhichanelectronattheringsite lhopsfrom
+sitelto the same site by making one round around the ring
+may seem meaningless, because such process does not exist
+in the basis of the ring Wannier functions. In that basis the
+energy spectrum εn(φ)is given by equation (60). As shown
+inappendixB, equation(60)canberewrittenforodd Nas
+εn(φ) =N−1
+2/summationdisplay
+j=−N−1
+2e−ikn(φ)jaTj, (71)
+andforeven Nas
+εn(φ) =N
+2/summationdisplay
+j=−N
+2+1e−ikn(φ)jaTj, (72)
+whereT−j=T∗
+j. Accordingtothedefinition(57),thematrix
+elementTjrepresentstheamplitudeofhoppingfromthe ring
+Wannier state wl(x)to the ring Wannier state wl′(x), with
+j=l−l′being the hopping length. The maximum hopping
+length betweenthe ringWannier states, allowed by equation s
+(71) and (72), is j=±Int(N/2)only, and the current in
+the insulating ring (Eq. 62) is due to the fact that the ring
+Wannier states depend on magnetic flux. Hopping from lto
+l±N,l±2N,etc.,emergesaftertheringWannierstatesare
+expressed (see Eq. 64) through the flux-independent crystal
+Wannier states. This provides a different picture of the sam e
+physics.v(x)
+εa(x)ϕa
+x
+2dV(x) a
+2d0
+−V0
+FIG. 2. Left: One-dimensional model of a single isolated ato m, de-
+scribed by equation (73). The atomic potential, v(x), is modeled
+by the potential well of width 2d, embedded between twoinfinitely-
+thick potential barriers of height V0. Right: The potential V(x)in a
+ring-shaped periodic lattice composed of None-dimensional atoms
+(N= 6). The latticeperiod is a,the ringcircumference is L=Na.
+In the numerical calculations of Sect. V, the parameters of t he ring
+are chosen to emulate the GaAs ring with conduction electron s sub-
+jected to the periodic potential of Nquantum dots: the electron ef-
+fective mass is m= 0.067m0, the period of the quantum-dot lattice
+isa= 40nm, the ring length is L=Na, and the size and depth of
+the potential well (quantum dot) are 2d= 15nm andV0= 18meV,
+respectively. For these parameters the potential well supp orts one
+bound state, the ground state with energy εa=−10.4meV. The
+ground-state orbital ϕa(x)issketched schematically.
+V.THE LCAOAPPROXIMATION,APPLICATIONTO THE
+1D LATTICEWITH RECTANGULARPOTENTIALWELLS
+To evaluate the persistent current (Eq. 47) for a specific
+crystallattice,weneedtodeterminethecrystalWannierfu nc-
+tions of the lattice. As an example we consider the ring-
+shaped 1D lattice in Fig. 2. The lattice is composed of the
+artificial 1D atoms with a single atomic level ǫa. The atomic
+orbitalϕa(x)andenergyǫaintheisolatedatompositionedat
+x= 0obeytheSchrödingerequation
+/bracketleftbigg
+−/planckover2pi12
+2md2
+dx2+v(x)/bracketrightbigg
+ϕa(x) =ǫaϕa(x),(73)
+wherev(x)is the atomic potential modeled as a rectangular
+potentialwell centeredat x= 0(figure2, left).
+In accord with the general recipe of Sect. II, assume that
+theringwith Natoms(figure2,right)isperiodicallyrepeated
+as shown in figure 1. This repetition gives rise to the infinite
+1D crystal with potential V(x)which is periodic with period
+aand ranges from x=−∞tox=∞. Evidently, V(x)is a
+sumofall isolatedatomicpotentials,thatmeans
+V(x) =∞/summationdisplay
+l=−∞v(x−la), (74)
+wherelais the position of the l-th lattice point. The Bloch
+functionΦk(x)and Bloch energy ǫ(k)of the electron mov-
+ingin suchperiodicpotentialcanbeexpressedanalyticall yin
+the approximation of localized atomic orbitals (LCAO). The
+BlochfunctionisapproximatedbytheLCAO ansatz[21]
+Φk(x) =C∞/summationdisplay
+l=−∞eiklaϕa(x−la),(75)9
+whereϕ(x−la)is the atomic orbital at the lattice point la
+andCisthenormalizationconstant(givenbelow). TheBloch
+energyǫ(k)canbeexpressedas
+ǫ(k) =∝angbracketleftΦk|ˆH|Φk∝angbracketright/∝angbracketleftΦk|Φk∝angbracketright, (76)
+where
+ˆH=−/planckover2pi12
+2md2
+dx2+V(x), V(x) =∞/summationdisplay
+l=−∞v(x−la).(77)
+IfweusetheLCAOansatz(75),asimpletextbookcalculation
+[21]givestheformula
+ǫ(k) =ǫa−γ0+2/summationtext∞
+j=1γjcos(kja)
+1+2/summationtext∞
+j=1αjcos(kja),(78)
+where
+αj=/integraldisplay∞
+−∞dxϕa(x−ja)ϕa(x), (79)
+γj=−/integraldisplay∞
+−∞dxϕa(x−ja)V′(x)ϕa(x), γ0=γj=0,(80)
+and
+V′(x) =V(x)−v(x), (81)
+withv(x)beingtheatomicpotentialatthelatticesite l= 0. In
+Appendix C we express αjandγjanalytically for the model
+atomiclatticeofFig. 2.
+Using the general recipe of section II, we obtain the spec-
+trum of the ring, ǫn(φ), directly from the Bloch energy (78).
+Itreads
+ǫn(φ) =ǫa−γ0+2/summationtext∞
+j=1γjcos[kn(φ)ja]
+1+2/summationtext∞
+j=1αjcos[kn(φ)ja],(82)
+wherekn(φ)≡2π
+Na(n+φ
+φ0). Onecansubstitutethespectrum
+(82)intothepersistentcurrentformula(24)andonecaneva l-
+uate the sum on the right hand side of Eq. (24) numerically.
+We will refer to this approach as to the numerical LCAO ap-
+proach. The approach provides a straightforward numerical
+evaluation of persistent current in insulating rings ( Ne=N)
+as well as in conductingrings( Ne<N). It does not provide
+understanding of the effect, however, it is useful for verifi ca-
+tionofanalyticalresults.
+Wemainlywanttoverifyourresultforinsulatingrings,the
+formula (47). For a meaningfulcomparison with the numeri-
+calLCAOapproach,thehoppingamplitude Γjintheformula
+(47)hasto bedeterminedin theLCAO approximation.
+We start by specifying the constant Cin the ansatz (75).
+SettingEq. (75)intothenormalizationcondition(40)wefin d
+1
+|C|2= 1+2∞/summationdisplay
+j=1αjcosjka≡N(k).(83)Thesameresultisobtainedifweset Eq. (75)intotheorthog-
+onalitycondition(41).
+ThecrystalWannierfunctionsaredefinedbyequation(39).
+Wesetintoequation(39)theansatz(75)with C= 1//radicalbig
+N(k).
+We obtainthe crystalWannierfunctioninthe form[22,23]
+Wl(x) =∞/summationdisplay
+l′=−∞cl′−lϕa(x−l′a),(84)
+where
+cn=a
+2π/integraldisplayπ
+a
+−π
+acos(kna)/radicalbig
+N(k)dk. (85)
+Expression (84) is the crystal Wannier function in the LCAO
+approximation. It is easy to verify that the expression (84)
+fulfillsthe orthogonalitycondition(41)exactly.
+If we express the crystal Wannier functions in Eq. (44) by
+meansofequation(84),we findthat
+Γj=∞/summationdisplay
+n=−∞∞/summationdisplay
+n′=−∞cn−jcn′(εaαn−n′−γn−n′).(86)
+Equation (86) is the hopping amplitude Γjin the LCAO ap-
+proximation. AnalternativeformofEq. (86)istheequation
+Γj=a
+2π/integraldisplayπ/a
+−π/adkcos(jka)
+×/bracketleftBigg
+ǫa−γ0+2/summationtext∞
+j′=1γj′cos(kj′a)
+1+2/summationtext∞
+j′=1αj′cos(kj′a)/bracketrightBigg
+,
+(87)
+obtained by setting the Bloch energy (78) into the Fourrier
+transformation(46).
+We also note that the Bloch energy (78) is an alternative
+form of the expression (45) with coefficients Γjgiven by
+equation (86), since both forms follow from the same LCAO
+approximation.Proofofcoincidenceofbothformsisstraig ht-
+forward,butlengthyandthereforenotshown.
+The LCAO expression (86) together with expressions (85)
+and(83)are still complicatedforanalyticalcalculations . For-
+tunately,theexpressionscanbe simplified,ifwe take intoa c-
+count that the coefficients αjandγjdecay with increasing j
+exponentially. Exponential decay is due to the fact that the
+atomic orbitals ϕaare exponentially localized at the lattice
+sites, which is the case for any realistic orbital. If the loc al-
+ization is strong enough, then 1≫|α1|≫|α2|≫...and
+1≫|γ1|≫|γ2|≫.... In that limit the expressions (85)
+and(86)are calculatedinappendixD. Theresultsare
+cn≃(−α1)n|An|, n≥0, c−n=cn,(88)
+whereA0= 1and
+An= (−1)n1
+22n/parenleftbigg2n
+n/parenrightbigg
+≃(−1)n1√πn, n≥1,(89)10
+and
+Γj≃ −γ1(−α1)(j−1), j≥1. (90)
+Expressing Γjin equation (47) by means of Eq. (90) and
+keepingonlythetermwith j=Nweobtaintheresult
+I(φ) =−2π
+φ0γ1N(−α1)(N−1)sin/parenleftbigg
+2πφ
+φ0/parenrightbigg
+.(91)
+Thelastequationisthepersistentcurrentintheinsulatin gring
+in the LCAO approximation. The following features of the
+result(91)areworthstressing.
+Forpositive α1equation(91)predictstheexponentiallyde-
+caying current with alternating sign, I(N)∼(−α1)(N−1).
+Just this result holds for the ring in figure 2. Indeed, α1is
+positiveforapositive ϕa(x)suchlikethegroundstate orbital
+infigure2,whichfollowsfromdefinition(79)aswellasfrom
+thefinal expressionfor αj(appendixC).
+For negative α1the dependence I(N)∼(−α1)(N−1)im-
+plies the exponentially decaying current with fixed sign. Fo r
+thesquare-wellmodelinfigure2thecoefficient α1isnegative
+for the first excited bound state (not shown in the figure), be-
+causetheatomicorbitalofthestate isanantisymmetricfun c-
+tion ofxwith respect to the well center. Thus, if the atomic
+levelεaand atomic orbital ϕa(x)belong to the first excited
+boundstate, thesignofthecurrentisthe sameforany N.
+Equations(1)and(2)showthatthepersistentcurrentinthe
+conducting ring ( Ne< N) exhibits the alternating sign in
+dependence on the parity of Ne. This result is independent
+on the specific properties of the conduction band. If the ring
+is insulating ( Ne=N), the result (91) shows that the sign
+of the current with respect to Nis either alternating or fixed,
+depending on the properties of the valence band. More pre-
+cisely, this statement holdsfor the modelring in figure 2. We
+will see in the next section that the sign behaviorcan be even
+morecomplicatedinringsmadeoftherealbandinsulators.
+Comparison of equation (91) with Eqs. (1) and (2) also
+shows another important difference between the insulating
+and conducting rings. In the former case the current decays
+withNexponentially,inthelatter caseit decayslike 1/N.
+Origin of all these differences is easy to see. Equations
+(1) and (2) depend only on the hopping amplitude Γ1, while
+the result (91) is determined by the hopping amplitude ΓN.
+The amplitude ΓNinvolvesoverlap of the Wannier functions
+Wl(x)andWl+N(x). If we express the coefficient cl′−lin
+equation(84) bymeansofequations(88)and(89), weobtain
+Wl(x)≃∞/summationdisplay
+l′=−∞|A|l′−l||(−α1)|l′−l||ϕa(x−l′a).(92)
+Ifwe keepx=naandl= 0forsimplicity,weobtain
+W0(na)≃1/radicalbig
+π|n|(−α1)|n|ϕa(0),|n|≥1.(93)
+It can be seen that the Wannier functiondecayswith distance
+from sitel= 0exponentially [24] and even shows the alter-
+nating sign for positive α1. All these properties are reflected
+bypersistentcurrentatfullfilling.-0.5 -0.25 00.25 0.5
+φ/φ0-0.4-0.200.20.4I [nA]
+-0.5 -0.25 00.25 0.5
+φ/φ0-0.02-0.0100.010.02 I [nA]-0.5 -0.25 00.25 0.5
+φ/φ0-80-4004080I [nA]
+-0.5 -0.25 00.25 0.5
+φ/φ0-6-3036I [nA]Ne=N=1
+Ne=N=3Ne=N=2
+Ne=N=4
+FIG. 3. Persistent current in the insulating 1D ring as a func tion of
+magneticfluxforvariousringlengths N. Theringistheperiodiclat-
+tice with Nsingle-level potential wells (Fig. 2) which contains Ne
+spinless electrons. It is insulating due to full filling ( Ne=N). The
+full lines are the results of the LCAO formula (91) and the das hed
+lines are the results of the numerical LCAO approach. The rin g pa-
+rameters used in these calculations are specified in the capt ion of
+figure 2. The coefficients αjandγjin Eqs. (91) and (82) are given
+byexpressions presented inappendix C.
+Finally,wecalculatethepersistentcurrentquantitative lyfor
+the numerical parameters in figure 2. These parameters em-
+ulate the 1D GaAs ring modulated by a periodic lattice of N
+quantumdots. In such lattice, the discrete energylevel of t he
+quantum dot splits into the energy band. Therefore, the ring
+behaves at full filling ( Ne=N) like a band insulator and at
+partialfilling( Ne<N)asametal.
+The figure 3 shows the persistent current in the insulating
+ringasa functionofmagneticflux forvarious N. Theresults
+evaluated by means of the LCAO formula (91) are compared
+with the results obtained by the numerical LCAO approach.
+ThenumericalLCAOapproachnicelyconfirmsbasicfeatures
+of the formula (91) including the alternating sign of the cur -
+rent. The quantitative difference between the numerical an d
+analytical LCAO data could be suppressed by choosing the
+ringwith amorestronglylocalizedatomicorbitals.
+The dependence on the ring size is presented in figure 4.
+Theleft panelshowsthe persistent currentas a functionof N
+intheinsulatingringandintheconductingringathalffilli ng.
+The conducting ring shows for large Nthe decayI∝1/N,
+intheinsulatingringthecurrentdecayswith Nexponentially.
+Forexample,theinsulatingringwith N= 5supportstheper-
+sistent current as small as ∼0.001nA. Such small persistent
+currentsaredetectablebysensitivetechniques[9, 25,26] .
+The right panel of figure 4 shows the transition from the
+conducting state to the insulating state at Ne=N, achieved
+by varyingNefor fixedN. Experimentally, Necouldbe var-
+ied say by using a suitably designed metallic gate, while the
+quantum-dot lattice might be realized either by means of the
+periodic array of gates or by producing the 1D ring with a11
+0 2 4 68
+ NeN=3
+N=4
+N=5
+N=6
+N=7
+0 4 8 12
+N 10-710-410-1102|I| [nA] Ne=N/2
+Ne=N
+Ne=Nφ = 0.25φ0φ = 0.25φ0
+FIG. 4. Persistent current in the same ringas in the precedin g figure
+for magnetic flux φ= 0.25φ0. The leftpanel shows the dependence
+onNfor the insulating ring ( Ne=N) and conducting ring at half
+filling(Ne=N/2). Therightpanelshowsthedependenceon Nefor
+various values of N. In both panels, the data for Ne=Nshown by
+open symbols were obtained bymeans of theLCAOexpression (9 1)
+and the data for partial filling ( Ne< N) were obtained by means
+of expressions (1) and (2). The full circles in the left panel show
+thedatafor Ne=N,obtainedusingthenumericalLCAOapproach.
+Notethatonlythesizeofthecurrent, |I|,isshown: forallpresented
+data the signof the current varies withvarying parityof Ne.
+periodicallyalternatingcross-sectionsize.
+VI.ESTIMATESOFPERSISTENTCURRENTIN RINGS
+MADEOFREALBANDINSULATORS
+Intheprevioussection,wehavediscussedtheringmadeof
+the artificial band insulator, namely the 1D GaAs ring with a
+few conduction electrons subjected to the periodic quantum -
+dot potential. Now we want to estimate persistent current in
+rings made of the real crystalline band insulators (with ful ly
+occupied valence band and empty conduction band). The
+problem is that such rings are three-dimensional and the 1D
+theoryfromtheprecedingtextis notapplicabledirectly.
+Theproblemisoutlinedinfigure5. Thetoprightsketch in
+thefigureisa2Dsketchofthe3Dring(orahollow3Dcylin-
+der) of width w, created from the cubic 3D lattice of atoms.
+Along such ring there is no periodicitywith the lattice peri od
+a, there is only periodicity with L. However, the theory in
+Sects. III- V wasdevelopedforthe1D ringswith periodicity
+aalongtheringcircumference. Thequestionishowtoextend
+the 1D theory to be applicable to the ring-shaped 3D cluster
+ofatomssketchedin thefigure5.
+In principle, for L≫wit is reasonable to write the 3Dw
+aw
+L / 2π
+w
+L
+FIG. 5. The top figure shows the 3D ring (on the right hand side)
+created from the cubic 3D atomic lattice (on the left hand sid e) with
+thelatticeperiod a. Theringofwidth wisdefinedbytwoconcentric
+circles with the center positioned at random. The resulting ring is a
+3D cluster of atoms which canno longer be viewed as the 1D latt ice
+withperiod d. Thebottomfigureshowstheringartificiallylinearized
+by means of elastic deformation. Repeating the linearized r ing with
+periodLin thexdirection and with period win theydirection, we
+obtaintheinfinitecrystal. Inthelimit L≫wtheeffectoftheelastic
+deformation has to disappear and the Bloch states in the obta ined
+infinite crystal have to coincide with the electronic states in the real
+ring (top right sketch). This allows us to use the equations ( 94) and
+(95). We callthis model the model of the linearized3D cluste r.
+versionofequation(24)in theform
+I(φ) =−∂
+∂φ/summationdisplay
+n,kzεn(φ,ky= 0,kz),(94)
+with the spectrum of the ring, εn(φ,ky= 0,kz), determined
+asfollows:
+εn(φ,ky= 0,kz) =ε(kx=2π
+L(n+φ
+φ0),ky= 0,kz),(95)
+whereε(kx,ky,kz)is theBloch energyof the3D crystalcre-
+atedbyperiodicrepetitionofthelinearized3Dcluster(Fi g.5).
+Note that the unit cell of the crystal is the linearized clust er.
+The Bloch energy ε(kx,ky,kz)should therefore not be con-
+fused with the Bloch energy of the crystal with unit cell a3.
+We callthisapproachthemodelofthelinearized3Dcluster.
+Themodelalso allowsto rewritefor3D ringsall equations
+of Sect. III. First of all, since the period aalong the ring no
+longer exists, it has to be replaced by the period L(replacea
+byLandNby unity). Second, the Bloch energy ε(kx)has
+to be replaced by the Bloch energy ε(kx,ky= 0,kz)just in-
+troduced. Third, the current (35) has to be summed over the
+occupied states kz. Clearly, the model of the linearized 3D
+cluster allows to avoid a full 3D treatment of the ring-shape d
+3D cluster. However,the modelis still cumbersomefor prac-
+ticalusebecausetheBlochspectrum ε(kx,ky,kz)needstobe
+calculated for a nontrivial artificial crystal resulting fr om the
+periodicrepetitionofthe linearizedcluster.
+Amoresimpleapproachistoconsidertheartificial3Dring
+obtainedby elastic bendingof the bulk crystal, as it is show n12wa
+L
+w
+FIG. 6. The 3D ring of length Land width w, created by elastic
+bending of the cubic 3D atomic lattice with lattice period a. In the
+limitw≪Ltheeffectoftheelasticbendinghastodisappearandthe
+electronic states in such ring have to coincide with the Bloc h states
+in the constituting 3D crystal as described in the text (equa tion 97).
+Wecallthismodelthemodeloftheelasticallybent3Dcrysta l. Obvi-
+ously,theresulting3Dringshowsfor w≪Lanartificialperiodicity
+withperiod a,whichis not the case for the realistic3D ring(fig. 5).
+in figure 6. In this case we have instead of the formula (94)
+theformula
+I(φ) =−∂
+∂φ/summationdisplay
+n,ky,kzεn(φ,ky,kz), (96)
+wheretheenergyspectrum εn(φ,ky,kz)isgivenbyequation
+εn(φ,ky,kz) =ε(kx=2π
+Na(n+φ
+φ0),ky,kz),(97)
+withε(kx,ky,kz)being the Bloch energy in the bulk crystal
+beforeit is bent. We call this approachthe model of the elas-
+ticallybent3Dcrystal. Inthismodelthereisanartificialperi-
+odicitywithperiod aalongthering. Inspiteofthis,themodel
+can providea reasonablequantitativeinformationon how th e
+persistentcurrentinthe3Dringdependsontherealistic ba nd
+structureoftheconstitutingbulkcrystal. Inthemodel,th eper-
+sistentcurrentatfullfillingisgivenbyequations(47)and (46)
+modified as follows. The Bloch energy ε(kx)in the Fourrier
+transformation (46) has to be replaced by the Bloch energy
+ε(kx,ky,kz)and thecurrent(47) shouldbe summedoverthe
+occupiedstates kyandkz. Practicalcalculationsarerelatively
+straightforward,becausetheBlochspectrumofthebulkcry s-
+tal,ε(kx,ky,kz), isknownformanyinsulators.
+In what follows we apply the model of the elastically
+bent 3D crystal to the rings with thickness equal to the unit
+cell of the bulk crystal. The energy dispersion of the ring,
+εn(φ,ky= 0,kz= 0), is extracted from the bulk crystal
+dispersionε(kx,ky= 0,kz= 0), calculated by microscopic
+methods [28, 29]. The obtained persistent current should be
+viewed as a minimum value estimate, because the thickness
+ofrealistic ringsismuchlargerthantheunitcell.
+Using standard methods [28, 29], in the figure 7 we nu-
+merically reproduce the well-known band structure of the
+band insulators like the GaAs, Ge, and InAs. Specifically,10-1510-1010-5 1   10-1510-1010-5 1   10-1510-1010-5 1   
+  0
+ -4
+ -8
+-12  0
+ -4
+ -8
+-12  0
+ -4
+ -8
+-12
+_ _ _, , ___(−1)x(−1)x
+(−1)xInAsGaAs
+Ge GeGaAs
+InAs
+25 50 75 0 100 −πa−πa−πaL( ) aπaπaπL(, ,) Γ(0,0,0)
+ΓN
+N(k)[eV]
+ [eV]ε
+FIG. 7. The panels in the right column show the valence bands
+of various band insulators like the GaAs, Ge, and InAs. Speci f-
+ically, the ǫ(k)dispersion curves of the light-hole band (full
+line), double-degenerate heavy-hole band (dashed line), a nd low-
+est band (dotted line) are plotted for the k-values on the line
+(L[−π/a,−π/a,−π/a],Γ[0,0,0],L[π/a,π/a,π/a ]). The panels
+in the left column show the hopping amplitude ΓN, calculated sep-
+arately for each band: the ǫ(k)dependence of the band is Fourier-
+transformed by using the equation (46). The full, dashed, a d otted
+lines show ΓNfor the light-hole, heavy-hole, and lowest bands, re-
+spectively. When compared with ΓNof the light-hole and heavy-
+hole band , ΓNof the lowest band has opposite sign because of the
+oppositecurvatureofthe ǫ(k)curveandthepresenteddatashouldbe
+multipliedby −1as shown inthe figure (butsee the comment [27]).
+the figure summarizes the ǫkcurves of the light-hole band,
+heavy-hole band and lowest band, calculated for the line
+(L[−π/a,−π/a,−π/a],Γ[0,0,0],L[π/a,π/a,π/a ]) in the
+first Brillouin zone. We set each of these ǫ(k)curves numer-
+ically into the Fourier transformation (46) and we calculat e
+for each band the hopping amplitude ΓN. The resulting ΓN
+dependenciesareshownin thefigure7.
+By meansof the figure7 we estimate the persistent current
+asfollows. We formtheringbyelasticbendingofthecrystal .
+We choose the ring orientation for which the kstates on the
+line (L[−π/a,−π/a,−π/a],Γ[0,0,0],L[π/a,π/a,π/a ])
+are directed along the ring circumference. If the ring is
+one-dimensional (with thickness equal to a single unit cell
+of the crystal), the persistent current is given by the 1D
+formula (47) with the hopping amplitude ΓNgiven by
+numerical values in the figure 7. As expected, ΓNdecays
+with increasing Nexponentially, however, for the light-hole
+band the decay is much slower than for other two bands.
+Therefore it is sufficient to calculate the persistent curre nt in
+the light-hole band. The results are shown in the figure 7.
+We recall that these results represent the persistent curre nt
+carried by electrons in the fully occupied valence band. The
+followingfeaturesareworthtopointout.
+The persistent current in the insulating ring decreases wit h13
+10-1510-1010-5
+200150100500GaAs GeInAs
+NI [A]
+FIG. 8. Persistent current versus the ring length Nin the ring made
+of the real crystalline band insulator (GaAs, Ge, InAs). The persis-
+tentcurrentisestimatedbymeansofthe1Dformula(47),whe reΓN
+is the hopping amplitude of the electron in the light-hole ba nd, cal-
+culated in the preceding figure. The lattice constants of the GaAs,
+Ge, and InAs crystals are ∼0.6nm, i.e., the maximum ring lengths
+considered in the figure are L=Na∼120nm. The rings of such
+lengthcould be realizable bymeans of advanced nanotechnol ogy.
+the increasing ring length exponentially, but a careful cho ice
+ofthe insulatingmaterialallowsto achievethe persistent cur-
+rent of measurable value for a technologically realizable r ing
+size. Indeed,the maximumringlengthsconsideredin thefig-
+ure 8 are close to the value ∼120nm, in principle achiev-
+able by modern nanotechnology. Concerning the amplitude
+of the persistent current, the value ∼0.0003nAestimated for
+the InAs ring of length ∼120nm is very small, but in prin-
+ciple detectable say by the experimental technique of Refs.
+[9, 25, 26]. The considered insulators are ordered as GaAs,
+Ge and InAs for the persistent currents ordered increasingl y.
+This suggests a simple rule: the smaller the light-hole effe c-
+tivemassthelargerthepersistentcurrent.
+Weconcludewithafewremarksshowingthattheabove1D
+estimates hold roughly also for the 3D rings. Similar calcu-
+lations as in the figure 7 were performed for the kvalues on
+the line (X[−2π/a,0,0],Γ[0,0,0],X[2π/a,0,0]). We have
+found the energies ǫ(k)and amplitudes ΓN(not shown) very
+similar to those in the figure 7, with the resulting persisten t
+currentsbeingof the same orderof magnitudeas those in the
+figure 8. Moreover, we have observed similar results when
+testingthekvaluesonthelines( K,Γ,K)and(W,Γ,W). The
+modeloftheelasticallybentcrystalthusprovidesroughly the
+same persistent currentindependentlyonthe fact whichcry s-
+talline axis is chosen to be directed along the ring circum-
+ference. This suggests that anisotropy of the valence band
+withrespecttothe /vectorkdirectiondoesaffecttheresultingpersis-
+tent current dramatically. Therefore, the estimated value s of
+the current should hold reasonably also for the realistic ri ng-
+shapedcrystalssimilartothe ringsketchedinfigure5.VII.SUMMARYANDCONCLUDING REMARKS
+We have analyzed theoretically persistent currents in ring s
+made of band insulators. We started by formulating a recipe
+which determinesthe Bloch states of a one-dimensional(1D)
+ringfromtheBloch states ofan infinite 1Dcrystal createdby
+the periodic repetition of the ring. Using the recipe, we hav e
+derived an expression for the persistent current in a 1D ring
+madeofaninsulatorwithanarbitraryvalenceband ǫ(k).
+Tofindanexactresultforaspecificinsulator,wehavecon-
+sidereda1Dringrepresentedbyaperiodiclatticeof Nidenti-
+calsiteswithasinglevalueoftheon-siteenergy. IftheBlo ch
+states in the ring are expanded over a complete set of Non-
+site Wannier functions, the discrete on-site energy splits into
+theenergyband. Atfullfilling,thebandemulatesthevalenc e
+bandofthebandinsulatorandtheringisinsulating.
+We have found that the ring carries a persistent current
+equal to the product of Nand the derivative of the on-site
+energy with respect to the magnetic flux. This current is not
+zero because the on-site Wannier function and consequently
+theon-siteenergyofeachringsite dependonmagneticflux.
+Further, we have expandedall NWannier functions of the
+ring over the basis of Wannier functions of the constituting
+infinite 1D crystal. In terms of the crystal Wannier function s,
+thecurrentatfullfillingarisesbecausetheelectroninthe ring
+withNlattice sites is allowed to make a single hop from site
+ito its periodic replicas i±N. Alternatively, in terms of
+the ring Wannier functionsthe same currentis due to the flux
+dependence of the Wannier functions basis, and the longest
+allowedhopisfrom itoi±Int(N/2)only.
+WehaveeventuallyderivedthecrystalWannierfunctionsin
+the LCAO approximation, and have expressed the persistent
+currentatfullfillingbymeansofanexactformula. Thevalid -
+ityoftheformulawasverifiedbycomparisonwiththenumer-
+ical LCAO approach. We have presented quantitative results
+for a 1D ring made of an artificial band insulator, namely for
+a GaAs ring subjected to a periodic potential emulating the
+insulatinglattice.
+Moreover, we have provided estimates for the rings made
+of real band insulators (GaAs, Ge, InAs) with a fully filled
+valence band and empty conductionband. The currentat full
+filling decays with the ring length exponentially due to the
+exponential decay of the Wannier function tails. In spite of
+that,it canbeofmeasurablesize.
+We have ignored the effect of nonzero temperature. We
+expect the persistent currents in rings made of the band in-
+sulators to be temperature-independent as long the thermal
+energykBTis much smaller than the energy gap separating
+the valencebandfromtheconductionband. Forexample,the
+energygapintheInAscrystalis ∼400meV.Hence,thezero-
+temperaturevalues of the persistent current, predicted fo r the
+insulatingInAsring,couldpersistevenat roomtemperatur e.
+Wehavealsoignoredtheelectron-electroninteraction. Th e
+many-bodystudies[30,31]suggestthatinthe1Dringconsid -
+ered in this paper the repulsive electron-electron interac tion14
+wouldonlyrenormalizetheelectrontransmissionthrought he
+potential barrier separating two potential wells. That mea ns
+that the electron-electron interaction would not essentia lly
+changeourresults.
+Finally, we relate our results to the paper by Kohn [17],
+which defines the insulating state as a property of the many-
+bodygroundstate(seealsothereviewbyResta [32]). Kohn’s
+papercontainsa discussion implicitlyadmittingthe exist ence
+of a negligibly small persistent current in the insulating r ing
+[see the discussion of equations (2.23) - (2.31) in that work ].
+Kohn points out the exponential localization of the electro n
+states in the ring. However, he clearly states by writing his
+equation (2.30), that the current is zero only approximatel y.
+Just this negligibly small current is calculated in our pape r.
+Surprisingly,it canbeofmeasurablesize.
+ACKNOWLEDGEMENTS
+This work was supported by grants APVV-51-003505
+and VVCE-0058-07 from the Grant Agency APVV and
+by grants VEGA-2/0633/09 and VEGA-2/0206/11 from the
+GrantAgencyVEGA.We thankPavol Quittnerforimportant
+helpwiththemathematicalderivationinAppendixD.
+APPENDIXA:DERIVATIONOF EQUATION(30)FROM
+EQUATION (28)
+We start bywritingEq. (28)as
+εn(φ) =∞/summationdisplay
+j′=0aj′cos(kn(φ)j′a), (98)
+Using substitution j′=Ns+j, wheres= 0,1,2,...and
+j= 0,1,...(N−1),we obtain
+εn(φ) =N−1/summationdisplay
+j=0∞/summationdisplay
+s=0asN+jcos[kn(φ)(sN+j)a].(99)
+Assumingthat NisoddwerewriteEq. (99) as
+εn(φ) =∞/summationdisplay
+s=0asNcos[kn(φ)sNa]
++M/summationdisplay
+j=1∞/summationdisplay
+s=0asN+jcos[kn(φ)(sN+j)a]
++N−1/summationdisplay
+j=M+1∞/summationdisplay
+s=0asN+jcos[kn(φ)(sN+j)a],
+(100)whereM= Int(N/2). Equation(100) canberewrittenas
+εn(φ) =∞/summationdisplay
+s=0asNcos[kn(φ)sNa]
++M/summationdisplay
+j=1∞/summationdisplay
+s=0/braceleftBig
+asN+jcos[kn(φ)(sN+j)a]
++a(sN+N−j)cos[kn(φ)(sN+N−j)a]/bracerightBig
+.
+Utilizingdefinition(29) wefindfromEq. (101) theresult
+εn(φ) = Ω0+M/summationdisplay
+j=1/braceleftBig
+ΩR
+jcos(kn(φ)ja)+ΩI
+jsin(kn(φ)ja)/bracerightBig
+,
+(101)
+where
+Ω0(φ) =∞/summationdisplay
+s=0asNcos(2πφ
+φ0s),
+ΩR
+j(φ) =aj+∞/summationdisplay
+s=1cos(2πφ
+φ0s)[asN+j+asN−j],
+ΩI
+j(φ) =∞/summationdisplay
+s=1sin(2πφ
+φ0s)[asN−j−asN+j].
+(102)
+We have sofar assumed odd N. For even N, derivation is
+similar and we find again the results (101) and (102), except
+thatforj=N/2wehaveinsteadofEqs. (102) theequations
+ΩR
+N/2(φ) =∞/summationdisplay
+s=0asN+N/2cos(2πφ
+φ0s),
+ΩI
+N/2(φ) =−∞/summationdisplay
+s=1asN+N/2sin(2πφ
+φ0s).
+(103)
+Equation(101)coincideswith Eq. (30).
+APPENDIXB:VARIOUSFORMSOFEQUATION(60)
+We first assumeodd N. We canrewriteEq. (60) as
+εn(φ) =T0+M/summationdisplay
+j=1e−ikn(φ)jaTj+N−1/summationdisplay
+j=M+1e−ikn(φ)jaTj,
+(104)
+whereM= Int(N/2). Further,
+εn(φ) =T0+M/summationdisplay
+j=1e−ikn(φ)jaTj+M/summationdisplay
+j=1e−ikn(φ)(N−j)aTN−j
+=T0+M/summationdisplay
+j=1e−ikn(φ)jaTj+eikn(φ)jaT∗
+j,
+(105)15
+wherewehaveusedthedefinition(29)andrelations
+T∗
+j=T−j, T N−j=ei2πφ
+φ0T∗
+j,(106)
+which follow from Eq. (59) for real εn(φ). For evenNwe
+obtainin asimilar waytheequation
+εn(φ) =T0+M−1/summationdisplay
+j=1e−ikn(φ)jaTj+eikn(φ)jaT∗
+j
++e−ikn(φ)N
+2aTN
+2.
+(107)
+Equations(105)and(107) canbewrittenintheunifiedform
+εn(φ) = Λ0+M/summationdisplay
+j=1/braceleftBig
+ΛR
+jcos(kn(φ)ja)+ΛI
+jsin(kn(φ)ja)/bracerightBig
+,
+(108)
+where
+Λ0=T0,
+ΛR
+j= (Tj+T∗
+j),
+ΛI
+j=1
+i(Tj−T∗
+j)
+(109)
+for oddNas well as even N, except that for even Nand
+j=N/2we haveinsteadofequations(109) theequations
+ΛR
+N/2=1
+2(Tj+T∗
+j),
+ΛI
+N/2=1
+2i(Tj−T∗
+j).
+(110)
+Equation (108) is formally equivalent with Eq. (101). Coin-
+cidence of both equationsbecomes obviousif we realize that
+equations(109)and(110)coincidewithEqs. (102)and(103) .
+To observe this coincidence, it is sufficient to express Tjin
+equations (109) and (110) by means of expression (68), and
+touse theequations a0= Γ0andaj= 2Γj.
+APPENDIXC:ANALYTICALFORMULASFOR αjANDγj
+Forthe square-wellpotentialinfigure1 theatomicorbitals
+ϕa(x−la)can be foundeasy and the integrals(79) and (80)
+canbecalculatedanalytically. For j= 1,2,...weobtain
+αj≃cos2(Ad)
+1+Bde−B(ja−2d)/bracketleftbigg−4ǫa
+V0+B(ja−2d)/bracketrightbigg
+,(111)
+γj≃V0cos2(Ad)
+1+Bde−B(ja−2d)/bracketleftbigg−2ǫa
+V0+2dB(j−1)/bracketrightbigg
+,
+(112)
+where
+A=/radicalbigg
+2m
+/planckover2pi12(V0+ǫa), B=/radicalbigg
+2m
+/planckover2pi12|ǫa|.(113)Inaddition,for j= 0
+γ0=V0cos2(Ad)
+1+Bde−2B(a−2d)1−e−4Bd
+1−e−2Ba.(114)
+Expressions (111) and (112) are not exact (we have skipped
+some exponentially small terms), but they are in excellent
+agreementwithnumericalintegrationoftheintegrals(79) and
+(80). As expected, the coefficients αjandγjdecay with in-
+creasingjexponentially. Thesimplestformsare
+γj≃γ1e−(j−1)Ba, αj≃α1e−(j−1)Ba, j≥1.(115)
+APPENDIXD: APPROXIMATEFORMULASFOR
+COEFFICIENTS cnANDΓj
+Equation (85) defines cnby means of the integral which can
+be calculated analytically in a certain limit. In what follo ws
+the calculation is presented for positive nbecausecn=c−n.
+We first expressthenormalizationconstant1√
+N(k)as
+1/radicalbig
+N(k)=
+1+2∞/summationdisplay
+j=1αjcosjka
+−1
+2
+=∞/summationdisplay
+s=0As
+2∞/summationdisplay
+j=1αjcosjka
+s
+,(116)
+where
+As= (−1)s1
+22s/parenleftbigg2s
+s/parenrightbigg
+. (117)
+AcloserinspectionoftherighthandsideofEq. (116)shows,
+thattheequation(116) canbeapproximatedas
+1/radicalbig
+N(k)≃∞/summationdisplay
+s=0As[2α1coska]s,(118)
+ifαjfalls withjexponentially as in equation (115), and if
+α2
+1≫α2. We note that the latter condition takes the form
+α1≫exp(−Ba)forαjderived in the preceding appendix.
+In general, both conditions are fulfilled simultaneously if the
+localizationoftheatomicorbitalsisstrongenoughincomp ar-
+isonwiththe latticeconstant a.
+We set the expansion (118) into the formula (85) and we
+performtheintegration. We get
+cn≃∞/summationdisplay
+s=0As[2α1]sIn,s, (119)
+where
+In,s=1
+π/integraldisplayπ
+−πcos(nx)coss(x)dx. (120)
+Itcanbeseen that
+In,s= 0, (121)16
+ifs<norif(s−n)isodd. Itcanalso beseenthat
+In,s=2
+2n+2l/parenleftbiggn+2l
+l/parenrightbigg
+, (122)
+ifs=n+2l,wherel= 0,1,....
+We set equations (122) and (121) into the equation (119).
+We get anapproximateexpressionfor cn,
+cn= (−α1)n∞/summationdisplay
+l=0An+2lα2l
+1/parenleftbiggn+2l
+l/parenrightbigg
+.(123)
+Thelast equationcanbefurtherrewrittenas
+cn= (−α1)n|An|∞/summationdisplay
+l=0An+2l
+Anα2l
+1/parenleftbiggn+2l
+l/parenrightbigg
+<(−α1)n|An|∞/summationdisplay
+l=0α2l
+1/parenleftbiggn+2l
+l/parenrightbigg
+, (124)
+wheretherighthandsideisaproperlychosenupperlimit(be -
+causeAn+2l/An<1).
+Ifwe usetherelation
+∞/summationdisplay
+l=0(α2
+1)l/parenleftbiggn+2l
+l/parenrightbigg
+=1/radicalbig
+1−4α2
+1/parenleftBigg
+2
+1+/radicalbig
+1−4α2
+1/parenrightBiggn
+,
+(125)
+theinequality(124) reads
+cn<(−α1)n|An|1/radicalbig
+1−4α2
+1/parenleftBigg
+2
+1+/radicalbig
+1−4α2
+1/parenrightBiggn
+.
+(126)
+Forα1≪1/2it canbefurtherapproximatedas
+cn<(−α1)n|An|/parenleftbig
+1+α2
+1/parenrightbign(127)
+We returnagaintothe equation(123) andwe replaceits right
+handside bya properlychosenlowerlimit:
+cn>(−α1)n|An|n/summationdisplay
+l=0/parenleftbiggn
+l/parenrightbigg
+(α1)2l
+>(−α1)n|An|/parenleftbig
+1+α2
+1/parenrightbign,
+(128)
+wherewehaveutilizedtheinequality
+An+2l
+An/parenleftbiggn+2l
+l/parenrightbigg
+>/parenleftbiggn
+l/parenrightbigg
+. (129)
+Notice now that the right hand sides of the inequalities (127 )
+and(128)coincide. Thisimpliesthat
+cn= (−α1)n/parenleftbig
+1+α2
+1/parenrightbign|An| ≃(−α1)n|An|.(130)
+Now we attempt to find an approximateexpression for the
+coefficient Γj. Equation (86) can be rewritten by means of
+substitution n′−n=mas
+Γj=∞/summationdisplay
+m=−∞(εaαm−γm)∞/summationdisplay
+n=−∞cncj+m+n.(131)Sincecn=c−n,αn=α−n, andγn=γ−n, weobtain
+Γj= (εaα0−γ0)/bracketleftBig
+c0cj+∞/summationdisplay
+n=1cn(cj+n+cj−n)/bracketrightBig
++∞/summationdisplay
+m=1(εaαm−γm)/bracketleftBig
+c0(cj+n+cj−n)
++∞/summationdisplay
+n=1cn(cj+m+n+cj+m−n+cj−m+n+cj−m−n)/bracketrightBig
+.
+(132)
+Owingto theexponentialdependence(130), the last equatio n
+canbesimplifiedas
+Γj≃j/summationdisplay
+m=0(εaαm−γm)j−m/summationdisplay
+n=0cncj−m−n.(133)
+SettingforcnandAntheequations(130) and(117)we have
+Γj≃j/summationdisplay
+m=0(εaαm−γm)(−α1)j−m.(134)
+Expressingγnandαnbymeansoftheequation(115)wefur-
+therobtain
+Γj≃(εa−γ0)(−α1)j
++j/summationdisplay
+m=1(εaα1−γ1)e−(m−1)Ba(−α1)j−m.
+(135)
+Ifα1≫exp(−Ba), wecanskip alltermswith m≥2. Thus
+Γj≃(−α1)j−1[−γ1+γ0α1)]≃ −(−α1)j−1γ1.(136)
+∗martin.mosko@savba.sk
+[1] Y. Imry, Introduction to Mesoscopic Physics (Oxford Univer-
+sityPress,Oxford, UK, 2002).
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+Phys.Rev. B 37, 6050 (1988).
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+(2008).
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+(AmericanMathematical Society, 2002).
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+[23] W.Andreoni, Phys. Rev. B 14, 4247 (1976).
+[24] Precisely, equation (93) shows the decay |n|−1/2|α1||n|.
+A purely exponential decay was predicted by W. Kohn,
+Phys. Rev. 115, 809 (1959). A very sophisticated estimates by
+L. He and D. Vanderbilt, Phys. Rev. Lett. 86, 5341 (2001), and
+by A. Bruno-Alfonso and D. R. Nacbar, Phys. Rev. B 75,115428 (2007), predict (in our notations) the decay
+|n|−3/4|α1||n|. For our purposes, exact nature of the
+power lawprefactor is ofminor importance.
+[25] A. C. Bleszynski-Jayich, W. E. Shanks, and J. G. E. Harri s,
+Appl. Phys. Lett 92, 013123 (2008).
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+Harris, J.Vac.Sci.Technol 26, 1412 (2008).
+[27] In fact,the detailsof the Γndependence of the lowest band are
+not essential for the resulting persistent current since th e dom-
+inant contribution to the current is due to the light-hole ba nd
+(see the text). We note only for completeness that in the GaAs
+and InAs the ΓNdependence of the lowest band exhibits the
+samesignforall Nexceptforafewisolatedvaluesof Nwhere
+the sign is opposite. We plot the whole ΓNcurve of the lowest
+band witha fixed sign for simplicity. Due to this simplificati on
+theΓNcurve exhibits a few local minima: the ΓNvalues at
+theseminimainfactdonothavethesamesignastherestofthe
+curve.
+[28] D.J. Chadi and M.L. Cohen, Phys. Status Solidi B 68,
+405 (1975).
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+Phys. Rev. B 79, 245323 (2009).
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\ No newline at end of file
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+arXiv:1001.0497v1  [q-fin.ST]  4 Jan 2010January 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+Multiscaled Cross-Correlation Dynamics in Financial Time-S eries
+T. CONLON∗, H.J. RUSKIN and M. CRANE
+School of Computing, Dublin City University, Ireland†
+*tconlon@computing.dcu.ie
+Received 06 October 2008
+The cross-correlation matrix between equities comprises multiple interact ions between
+traders with varying strategies and time horizons. In this paper, we use the Maximum
+Overlap Discrete Wavelet Transform (MODWT) to calculate correlation ma trices over
+different time scales and then explore the eigenvalue spectrum over sliding time windows.
+The dynamics of the eigenvalue spectrum at different times and scales provi des insight
+into the interactions between the numerous constituents involved.
+Eigenvalue dynamics are examined for both medium and high-frequency equity re-
+turns, with the associated correlation structure shown to be dependent on b oth time
+and scale. Additionally, the Eppseffect is established using this multivariate method
+and analysed at longer scales than previously studied. A partition of th e eigenvalue
+time-series demonstrates, at very short scales, the emergence of negative returns when
+the largest eigenvalue is greatest. Finally, a portfolio optimisati on shows the importance
+of time-scale information in the context of risk management.
+1. Introduction
+In recent years, the equal-time cross-correlation matrix h as been studied extensively
+for a variety of multivariate data sets across different disc iplines such as financial
+data [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], electroencephalograph ic (EEG) recordings [12,
+13, 15], magnetoencephalographic (MEG) recordings [16] an d others. In particular,
+Random Matrix Theory (RMT) has been applied to filter the rele vant information
+from the statistical fluctuations inherent in empirical cro ss-correlation matrices,
+constructed for various types of financial, [1, 2, 3, 4, 5, 6, 7 , 8, 9], and MEG,
+[16], time-series. In this paper, we extend previous Comple x Systems research, by
+examining time and scale dependent dynamics of the correlat ion matrix, in order
+to better understand the ever-changing level of synchronis ation between financial
+time-series.
+The dynamics of the largest eigenvalue of a cross-correlati on matrix over small
+windows of time was studied, [17], for the Dow Jones Industri al Average (DJIA) and
+DAX indices. This analysis revealed evidence of time-depen dence between ‘draw-
+downs’/‘drawups’ and an increase/decrease in the largest e igenvalue. The inter-
+actions between stocks of two different markets (DAX and DJIA ) were then in-
+†School of Computing, Dublin City University, Glasnevin, Dublin 9, Ireland
+1January 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+2T.Conlon, H.J.Ruskin, M.Crane
+vestigated, [18], revealing two distinct eigenvalues of th e combined cross-correlation
+matrix,correspondingtoeachmarket.Adjustingfortime-z onedelays,thetwoeigen-
+values coincided, implying that one market leads the dynami cs in the other.
+It has been suggested recently by several authors, [19, 20, 2 1, 22, 23] that there
+may, in fact, be some real correlation information hidden in the RMT defined
+part of the eigenvalue spectrum. A multivariate technique i nvolving the equal-time
+cross-correlation matrix has been shown, [12], to characte rise dynamical changes
+in nonstationary multivariate time-series. Using this tec hnique, the authors also
+demonstrated how correlation information can be observed i n the noise band. As
+the synchronisation of ktime-series within an M−dimensional multivariate time-
+series increases, a repulsion between eigenstates of the co rrelation matrix results, in
+whichklevels participate. Through the use of artificially-create d time-series with
+pre-defined correlation dynamics, it was demonstrated that there exist situations,
+where the relative change in eigenvalues from the lower edge of the spectrum is
+greater than that for the large eigenvalues, implying that i nformation drawn from
+the smaller eigenvalues is highly relevant.
+This technique was subsequently applied to the dynamic anal ysis of the eigen-
+value spectrum of the equal-time cross-correlation matrix of multivariate Epileptic
+Seizure time-series, using sliding windows. The authors de monstrated that informa-
+tion about the correlation dynamics is visible in boththe lower and upper eigen-
+states. A further study, [15], which investigated temporal dynamics of focal onset
+epileptic seizuresa, showed that the zero-lag correlations between multichann el EEG
+signals tend to decrease during the first half of a seizure and increase gradually be-
+fore the seizure ends. Information about cross correlation s was also found in the
+RMT bulk of eigenvalues, [13], with that extracted at the loweredge statistically
+more significant than that from the larger eigenvalues. Application of this t echnique
+to multichannel EEG data showed small eigenvalues to be more sensitive to detec-
+tion of subtle changes in the brain dynamics than the largest . A time and frequency
+based approach, has recently demonstrated the importance o f correlation dynamics
+at high-frequencies during seizure activity, [14].
+Although originally applied to complex EEG seizure time-se ries, [12, 13, 15],
+these techniques were adapted to financial time-series, [11 ], where the correlation
+dynamics for medium-frequency data were calculated using s hort time-windows.
+Changes in the correlation structure were shown to be visibl e in both large and
+small eigenvalue dynamics. Further, ‘drawdowns’ and ‘draw ups’ were shown to be
+market dependent, with corresponding characteristic chan ges in relative eigenvalue
+size found at both ends of the eigenvalue spectrum. This anal ysis was carried out
+using daily data, with the effects of varying data granularit y not examined. In this
+paper, we build upon this analysis by studying the multiscal e effects on correlation
+dynamics using medium and high-frequency data.
+aA focal onset or partial seizure occurs when the discharge starts in one ar ea of the brain and then
+spreads over other areas.January 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+Multiscaled Cross-Correlation Dynamics in Financial Time-Seri es3
+The correlation structure for various markets was studied i n detail, [24], for
+data encompassing a number of time horizons ranging from 5 to 255 minutes. By
+removing the centre of mass, (the market), the authors found that the correlation
+structure is well defined at high frequency, while for the ori ginal data the structure
+emerges at longer time horizons, (using stocks from the New Y ork Stock Exchange).
+Wavelet multiscale analysis has been used by numerous autho rs to decompose
+economic and financial time-series into orthogonal time-sc ale components of varying
+granularities. Gen¸ cayandco-workers have variouslyexam ined thescaling properties
+of foreign exchange volatility, [25], and volatility model s without intraday season-
+alities, [26], using wavelet multiscaling techniques. The systematic risk in a Capital
+Asset Pricing Model was estimated over different granularit ies, [27], where it was
+shown that the return of a portfolio and its Betabbecome stronger as the scale
+increases for the S&P 500.
+Anextensivelystudiedcharacteristicofstockmarketbeha viour,istheincreaseof
+stock return cross-correlations as the sampling time scale increases, a phenomenon
+known as the Eppseffect, [28]. More recently, analysis of time-dependent cor rela-
+tions between high-frequency stocks demonstrated, howeve r, that market reaction
+times have increased due to greater efficiency, [29]. A diminu tion of the Epps effect
+with time is one consequence of increased market efficiency. T rading asynchronicity
+was demonstrated to be not solely responsible for the effect, [30], with the char-
+acteristic time apparently independent of the trading freq uency. Further analysis
+using a toy model of Brownian motion and memoryless renewal p rocess, [31], found
+an exact expression for the Epps frequency dependence, with reasonable fitting also
+for empirical data. In fact, the effect was shown, [32], not to scale with market
+activity but to be due to reaction times, rather than market a ctivity.
+In this paper, we extend the multivariate correlation techn ique, first applied
+to complex EEG seizure data, [12, 13, 15], and later to medium frequency finan-
+cialdata,[11].Thisinterdisciplinaryapproach,allows c haracterisationofcorrelation
+changes intime. Bydecomposing equity returnsintotheir co mponent scalesinshort
+time-windows, we are able to study the correlation and assoc iated eigenspectrum
+at various scales, allowing a time-scale analysis of the eig enspectrum. Using both
+medium and high-frequency stock returns, we examine the dyn amics of the large
+eigenvalues and demonstrate the time-scale dependence of t he correlation structure.
+We show how the Epps effect, [28], can be demonstrated through a multivariate ap-
+proach and expand previous work by examining considerably l onger scales. Finally,
+we look at some applications of the work, with scale dependen t risk characterisa-
+tion and portfolio optimisation examined. Sections (2 -3) describe the techniques
+and data used, Section (4) details the results obtained and c onclusions are given in
+Section (5).
+bThe Beta of an asset is a measure of the volatility, or systematic risk, of a security or a portfolio
+in comparison to the market as a whole.January 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+4T.Conlon, H.J.Ruskin, M.Crane
+2. Methods
+2.1.Wavelet multiscale analysis
+Wavelets provide an efficient means of studying the multireso lution properties of a
+signal, allowing decomposition into different time horizon s or frequency components
+(Discrete Wavelet Transform, DWT), [33, 34]. The definition s of the two basic
+wavelet functions, the father φand mother ψwavelets are given as:
+φj,k(t) = 2j
+2φ/parenleftbig
+2jt−k/parenrightbig
+(1)
+ψj,k(t) = 2j
+2ψ/parenleftbig
+2jt−k/parenrightbig
+(2)
+wherej= 1,...Jin aJ-level decomposition. The father wavelet integrates to 1
+and reconstructs the longest time-scale component of the se ries, while the mother
+wavelet integrates to 0 and is used to describe the deviation s from the trend. The
+wavelet representation of a discrete signal f(t) inL2(R) is:
+f(t) =/summationdisplay
+ksJ,kφJ,k(t)+/summationdisplay
+kdJ,kφJ,k(t)+...+/summationdisplay
+kd1,kφ1,k(t) (3)
+whereJis the number of multiresolution levels (or scales) andkranges from 1 to
+the number of coefficients in the specified level. The coefficien tssJ,kanddJ,kare
+the smooth and detail component coefficients respectively an d given by
+sJ,k=/integraldisplay
+φJ,kf(t)dt (4)
+dj,k=/integraldisplay
+ψj,kf(t)dt(j= 1,...J) (5)
+Each of the coefficient sets SJ,dJ,dJ−1,...d1is called a crystal.
+In this paper, we apply the Maximum Overlap Discrete Wavelet Transform
+(MODWT), [33, 34], a linear filter that transforms a series in to coefficients related
+to variations over a set of scales. Like the DWT it produces a s et of time-dependent
+wavelet and scaling coefficients with basis vectors associat ed with a location tand a
+unitless scale τj= 2j−1for each decomposition level j= 1,...,J 0. The MODWT,
+unlike the DWT, has a high level of redundancy, however, is no northogonal and
+can handle any sample size N. It retains downsampledcvalues at each level of the
+decomposition that would be discarded by the DWT. This reduc es the tendency for
+larger errors at lower frequencies when calculating freque ncy dependent variance
+and correlations, (Section 2.3), as more data is available. Having decomposed the
+equity returns into their component time-scales, we then ca lculate the correlation
+between these components.
+cDownsampling or decimation of the wavelet coefficients retains half of th e number of coefficients
+that were retained at the previous scale and is applied in the DWTJanuary 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+Multiscaled Cross-Correlation Dynamics in Financial Time-Seri es5
+2.2.Correlation dynamics
+The equal-time correlation matrix between time-series of s tock returns is calculated
+using a sliding time window where the number of stocks, N, is smaller than the
+window size T. Given time-series of stock returns Ri(t),i= 1,...,N, we normalise
+the time-series within each window as follows:
+ri(t) =Ri(t)−/hatwiderRi(t)
+σi(6)
+whereσiis the standard deviation of stock i= 1,...,Nand/hatwiderRiis the time average
+ofRiover a time window of size T. Then, the equal time cross-correlation matrix,
+expressed in terms of ri(t), is
+Cij≡ /angbracketleftri(t)rj(t)/angbracketright (7)
+The elements of Cijare limited to the domain −1≤Cij≤1, whereCij= 1 defines
+perfect positive correlation, Cij=−1 corresponds to perfect negative correlation
+andCij= 0 corresponds to no correlation. In matrix notation, the co rrelation is
+expressed as C=1
+TRRt, whereRis anN×Tmatrix with elements rit.
+The eigenvalues λiand eigenvectors ˆ viof the correlation matrix Care found
+usingCˆ vi=λiˆ viand then ordered according to size, such that λ1≤λ2≤...≤
+λN. Given that the sum of the diagonal elements of a matrix (the T race) remains
+constant under a linear transformation,/summationtext
+iλimust always equal the trace of the
+original correlation matrix. Hence, if some eigenvalues in crease then others must
+decrease, to compensate, and vice versa ( Eigenvalue Repulsion ).
+There are two limiting cases for the distribution of the eige nvalues [12, 15],
+with perfect correlation, Ci≈1, when the largest is maximised with value N(all
+others taking value zero). When each time-series consists o f random numbers with
+average correlation Ci≈0, the corresponding eigenvalues are distributed around
+1, (where deviations are due to spurious random correlation s). Between these two
+extremes, the eigenvalues at the lower end of the spectrum ca n be much smaller
+thanλmax. By calculating the eigenspectrum associated with the corr elation matrix
+at different scales in each time window, we get time-series of eigenvalues λmn(τ),
+wheremdenotes the eigenvalue number and nis the scale studied in a particular
+time-window, τ.
+To study the market behaviour when eigenvalues take extreme values, we need
+to partition the later depending on their size. This is achie ved by first normalising
+each eigenvalue in time using
+˜λi(t) =/parenleftbig
+λi−¯λ(τ)/parenrightbig
+σλ(τ)(8)
+where¯λ(τ) andσλ(τ)are the mean and standard deviation of the eigenvalues over
+a particular reference period, τ. By expressing the eigenvalue time-series in terms
+of standard deviation units (SDU), we can then easily partit ion each according to
+it’s relative size.January 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+6T.Conlon, H.J.Ruskin, M.Crane
+Combining the techniques of multivariate correlation anal ysis and multiscaling,
+we are able to provide some novel insight into the dependence of correlation on both
+time and scale. To enable this analysis, we first show how corr elations are calculated
+using wavelet coefficients.
+2.3.Wavelet covariance and correlation
+The wavelet covariance between functions f(t) andg(t) is defined to be the co-
+variance of the wavelet coefficients at a given scale. The unbiased estimator of the
+wavelet covariance at the jthscale is given by
+νfg(τj) =1
+MjN−1/summationdisplay
+t=Lj−1˜Df(t)
+j,t˜Dg(t)
+j,t (9)
+where all the wavelet coefficients affected by the boundary are removed [33] and
+Mj=N−Lj+1. The wavelet variance at a particular scale ν2
+f(τj) is found similarly.
+The MODWT estimate of the wavelet cross correlation between functionsf(t)
+andg(t) may then be calculated using the wavelet covariance and the square root
+of the wavelet variance of the functions at each scale j. The MODWT estimator
+[34] of the wavelet correlation is then given by:
+ρfg(τj) =νfg(τj)
+νf(τj)νg(τj)(10)
+3. Data
+The initial data set comprises the 49 equities of the Dow Jone s (DJ) Euro Stoxx 50
+where full price data is available from May 1999 to August 200 7, resulting in 2183
+daily returns, Fig. 1(a). The Dow Jones Euro Stoxx 50 is a stoc k index of Eurozone
+equities, designed to provide a blue-chip representation o f Eurozone supersector
+leaders. The small number of stocks in the index allows calcu lation of the cross-
+correlation matrices for small time windows, without reduc ing the matrix rank.
+Theseconddatasetstudiedconsists of high-frequencyretu rnsfor theDow Jones
+(DJ) Euro Stoxx 50 from May 2008 to April 2009, resulting in 10 9,540 one-minute
+Returns, Fig. 1(b). High-frequency data allows us to examin e additional features
+and analyse whether the correlation dynamics previously fo und, [11], are inherent
+at all time-scales or if there is a gradual emergence. The tim e-frame studied is
+of particular interest, with sustained market drops and hig h levels of volatility.
+Understanding market interactions in a period such as this m ay assist in forming
+portfolios that are robust to large downwards moves.
+4. Results
+To decompose the data into component time-scales, we select ed the least asymmet-
+ric (LA) wavelet, (known as the Symmlet), which exhibits nea r symmetry aboutJanuary 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+Multiscaled Cross-Correlation Dynamics in Financial Time-Seri es7
+2000 2002 2004 2006 200820002500300035004000450050005500
+YearsDJ Euro Stoxx 50, Daily Prices, May 1999 − June 2008
+(a)
+12345678910
+x 1041600180020002200240026002800300032003400
+MinutesDJ Euro Stoxx 50, 1 Minute Prices, May 2008 − April 2009
+(b)DJ Euro Stoxx 50 DJ Euro Stoxx 50
+Fig. 1. a) DJ Euro Stoxx 50, Daily Prices May 1999 - June 2008 (b) D J Euro Stoxx 50, 1 Minute
+Prices May 2008 - April 2009
+the filter midpoint. LA filters are available in even widths an d the optimal filter
+width is dependent on the characteristics of the signal and t he length of the data.
+The filter width chosen for this study was the LA8, (where the 8 refers to the width
+of the scaling function) since it enabled us to accurately ca lculate wavelet correla-
+tions to relatively long scales, dependent upon the time win dow used. Although the
+MODWT can accommodate any level J0, in practise the largest level is chosen so as
+to prevent decomposition at scales longer than the total len gth of the data series.
+4.1.Medium Frequency Eigenvalue dynamics
+We first focus on medium-frequency equity returns measured a t a one-day hori-
+zon. For each stock, using a sliding time-window of 100 days, we decompose the
+returns into their component scales in each window, using th e wavelet transform.
+The correlation between the wavelet coefficients, correspon ding to each equity, is
+then calculated at each scale, as described (Section 2.3). T ime-series of eigenvalues
+are found at each scale, by calculating the associated eigen spectrum. This approach
+of calculating the eigenspectrum in sliding time-windows, allows analysis of the
+dependence of correlation dynamics on granularity.
+The time-window was chosen such that Q=T
+N= 2.04, thus ensuring that the
+data would be close to stationary in each sliding window, (di fferent values of Q
+were examined, [11], with little variation in the results). Fig. 1(a) shows the valueJanuary 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+8T.Conlon, H.J.Ruskin, M.Crane
+2000 2002 2004 2006 20081015202530Unfiltered Data
+(a)
+2000 2002 2004 2006 20081015202530Wavelet Level 1  (3 days)
+(b)
+2000 2002 2004 2006 2008102030Wavelet Level 2  (6 days)
+(c)
+2000 2002 2004 2006 2008102030Wavelet Level 3  (11 days)
+(d)
+Fig. 2. a) Largest eigenvalue dynamics original data (b) 3 day scale ( c) 6 day scale (d) 11 day scale
+of the DJ Euro Stoxx Index over the period studied. Fig. 2(a) d isplays the largest
+eigenvalue, calculated using the unfiltered (one day) time- series data for sliding time
+windows. As shown, the largest eigenvalue is far from static , rising from a minimum
+of 7.5 to a maximum of 30 .5 from early 2001 to late 2003, (coinciding with the
+bursting of the “tech” bubble). This corresponds to an incre ase in the influence of
+the “Market”, with the behaviour of traders becoming more co rrelated. The next
+major increase occurred in early 2006, followed by a relativ ely marked decline until
+the beginning of the “Credit Crunch” in 2007. Similar to [17] , we note an increase in
+the value of the largest eigenvalue during times of market st ress, with lower values
+during more “normal” periods.
+We next calculate, using the MODWT (Section 2.1), the value o f the largest
+eigenvalue of the cross-correlation matrix over longer tim e horizons of 3, 6 and 11
+days (Fig. 2(b -d)). Certain traders, (such as Hedge Fund managers), may hav e very
+short trading horizons while others, (such as Pension Fund m anagers), have much
+longer horizons. By looking at the value of the largest eigen value at different scales,
+we try to characterise the impact of these different trading h orizons on the cross-
+correlation dynamics between large capitalisation stocks . In Fig. 2(b -d), we see that
+the main features found in the unfiltered data are preserved o ver longer time scales.
+However, certain features, such as the sizeable drop in the l argest eigenvalue at the
+longestscaleinlate2003,arenotseenatshorterscales,bu ttheaggregateimpactfor
+the unfiltered data is a moderate drop. Other features, such a s the increase in 2006,January 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+Multiscaled Cross-Correlation Dynamics in Financial Time-Seri es9
+are not preserved across all scales. The different features, found at various scales,
+suggestthatthecorrelationmatrixismadeupofinteractio nsbetweenstocks,traded
+byinvestorswithdifferenttimehorizons.Thishasimplicat ionsforriskmanagement,
+as the correlation matrix used for input in a portfolio optim isation should depend
+on the investor’s time horizon. We look more closely at this i n Section 4.5.
+4.2.High-Frequency Correlation Analysis
+Using the entire high-frequency data set for the Euro Stoxx 5 0, described earlier, we
+calculated the average correlation and eigenvalue spectru m at a number of scales,
+Table 1. As expected, the largest eigenvalue increases as th e average correlation
+between stocks increases, in keeping with the ‘one-factor’ toy-model of correlations
+described previously, [11]. Analysis of Table 1 gives some i ntuition for the results
+shown, Fig. 2, with increases in the largest eigenvalue in ti me corresponding to an
+increase in the average correlation between stocks, (‘sing le factor’ model, [11]).
+Time Average 1st2nd3rd
+Scale Horizon Correlation Eigenvalue Eigenvalue Eigenvalue
+1 1 min 0.35 20.24 6.66 1.085
+2 5 min 0.39 20.66 5.53 1.26
+3 30 min 0.45 23.33 4.229 1.64
+5 120 min 0.47 24.7 4.18 1.74
+8 480 min 0.52 27.72 3.54 2.42
+10 1100 min 0.51 27.15 3.83 1.97
+12 2000 min 0.41 18.42 2.78 2.39
+Table 1. Correlation and Eigenspectrum Analysis. The 1steigenvalue is the largest eigenvalue of
+the correlation matrix of Euro Stoxx 50 high frequency data from Ma y 2008 to April 2009.
+As mentioned, (Section 1), the Epps effect is the gradual buil d-up of correlation
+as the scale at which stock returns are measured, increases. In previous work, [29,
+30], the Epps effect has been studied using either the bivaria te correlation between
+pairs of stocks or the average of a number of such pairs. Howev er, as seen in Table 1,
+this effect is also visible using a multivariate approach, wi th the largest eigenvalue
+increasing from 20 .24 at the smallest scale to 27 .72 at the 480 min ( ≈1 day) scale,
+with corresponding increase in average system correlation (from 0.35 to 0.52).
+In previous studies, Authors have calculated the correlati on at different scales
+by recalculating the returns, resulting in reduced quantit ies of available data, and
+constraining the maximum possible scale. However, using Wa velet Multiscaling to
+decompose the data into component scales, we can calculate t he correlations and
+eigenvalues at much longer scales. Of particular interest i s the increase in the av-
+erage correlation to a maximum at a scale of 480 minutes and th e subsequent
+decrease at longer scales of 2000 min ( ≈4 days) to 0 .41. This decrease in correla-January 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+10T.Conlon, H.J.Ruskin, M.Crane
+tion corresponds to a decrease in the ‘Market Effect’, with tr aders having longer
+time-horizons acting in a less synchronous fashion. The dec rease may be due to the
+additional discontinuities incorporated in the data at sca les longer than one day as
+additional information is incorporated as stock markets op en for the day, (visible
+in discontinuous jumps in the equity prices).
+4.3.High-Frequency Correlation Dynamics
+Building on the analysis of previous Sections, we now examin e the dynamics of the
+eigenvalues over time using the high-frequency data descri bed. The methodology
+is the same as for daily data, described in Section 4.1, using a time window of
+10,000 minutes. The results for small scales are shown, Fig. 3(a ), with a general
+increase in the largest eigenvalue for longer scales, corre sponding to the increase in
+correlation shown in Section 4.2. However, for small scales (1−5 minutes), some
+distinct dynamics are found, with a sharp increase at around 50,000 minutes and
+a decrease at longer scales (30 −60 minutes). These distinct dynamics emerge even
+more distinctly at longer scales, Fig. 3(b), with eigenvalu es actually having opposite
+behaviour at certain points, (the largest eigenvalue incre ases at the 1 minute scale
+but actually decreases at the longest scales, at approximat ely 50,000 minutes).
+0 1 2 3 4 5 6 7 8 9 10
+x 1045101520253035
+Time (Minutes)Euro Stoxx 50 Correlation Dynamics, Short Scales
+(a)
+0 1 2 3 4 5 6 7 8 9 10
+x 10451015202530354045
+Time (Minutes)Euro Stoxx 50 Correlation Dynamics, Long Scales
+(b)1 minute
+5 minutes
+30 minutes
+60 minutes
+1 minutes
+480 minutes
+1100 minutes
+2000 minutes
+Fig. 3. Eigenvalues dynamics
+Of particular interest is the gradual increase in the larges t eigenvalue moving toJanuary 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+Multiscaled Cross-Correlation Dynamics in Financial Time-Seri es11
+longer scales, in keeping with the Epps effect. As found in the previous Section at
+longer scales, (2 ,000 minutes), using the complete data set, the largest eigen value
+is smaller than for shorter scales at certain points in time. This variation in the
+largesteigenvaluecorrespondstoachangeintheaverageco rrelationbetweenstocks.
+The distinct behaviour at longer time scales may be due to a nu mber of factors,
+with longer scales incorporating additional information a nd smoothing out high
+frequency noise.
+0 1 2 3 4 5 6 7 8 9 10
+x 104024681012
+Time (Minutes)Euro Stoxx 50 Correlation Dynamics, 2nd Largest Eigenvalue
+(a)1 minute
+120 minutes
+480 minutes
+2000 minutes
+0 1 2 3 4 5 6 7 8 9 10
+x 1040.511.522.533.544.555.5
+Time (Minutes)Euro Stoxx 50 Correlation Dynamics, 3rd Largest Eigenvalue
+(b)1 minute
+120 minutes
+480 minutes
+2000 minutes
+Fig. 4. Eigenvalues dynamics, 2ndand 3rdlargest eigenvalues
+The dynamics of the 2ndand 3rdlargest eigenvalue are also shown in Fig. 4.
+Again, the relative size and dynamics of the eigenvalues are found to be scale-
+dependent,withabroadincreaseintheeigenvaluesforlong erscales.Thisisinsharp
+contrast to the results found in Section 4.2, where the 2ndlargest eigenvalue was
+found to strictly decrease as the scale increased. It appear s that for small windows,
+thesecond largest eigenvalue is of greater relative importanc e(at longer scales) with
+a corresponding redistribution of correlation structure a cross the eigenspectrum,
+(eigenvalue repulsion). In the context of previous work don e on Random Matrix
+Theory,[4,9],thismayimplythatdifferentinformationist obefoundinthe2ndand
+subsequent eigenvalues at longer scales. These eigenvalue s are said to correspond to
+differentsectoralgroupings,withpossiblegroupchangesa swemovedownthescales.
+Byremovingthemarketmode,researchers[24],foundthecor relationstructuretobeJanuary 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+12T.Conlon, H.J.Ruskin, M.Crane
+invariant across scales. However, this paper did not examin e different time-windows
+or longer time-scales ( >255 Mins). By combining the method of these authors with
+thoseproposedabove,itmaybepossibletoinvestigate thei nvarianceofcorrelations
+across time as well as scale.
+4.4.Drawdown analysis
+As indicated, (Section 1 and [11, 17]), drawdowns, (or perio ds of large negative
+returns), and drawups, (periods of large positive returns) , tend to be accompanied
+by an increase in different eigenstates of the cross-correla tion matrix. In this sec-
+tion, we attempt to characterise the market according to the relative size of the
+eigenvalues, as well as through the use of eigenvalue ratios .
+The returns, correlation matrix and eigenvalue spectrum ti me-series for over-
+lapping windows of 10 ,000 minutes were calculated and normalised using the mean
+and standard deviation over the entire series, (Eqn. 8). By r epresenting normalised
+eigenvalues in terms of standard deviation units (SDU), we c an partition the eigen-
+values according to their magnitude. The average return of t he index is shown in
+Table2,duringperiodswhenthelargesteigenvalueis ±1SDU).Asthemarkettrend
+was predominantly negative, average returns were negative both for periods when
+the largest eigenvalue was equal to its mimimum and maximum v alues. However,
+distinct behaviour still emerged across all scales.
+Scale No. Std >1<-1
+Original -8.06% -3.39%
+5 Minutes -7.95% -3.06%
+60 Minutes -9.27% -3.64%
+360 Minutes -8.41% -3.52%
+1100 Minutes -10.79% 0.01%
+2000 Minutes -12.06% -6.32%
+Table 2. Drawdown/Drawup analysis. Average Index Returns when var ious eigenvalue partitions
+in SDU are >1 and<−1.
+Looking first at the original or unfiltered data, we see that wh en the largest
+eigenvalue is >1 SDU, the return of the index was −8.06%. In contrast, when the
+largest eigenvalue is <−1 SDU, the return was −3.06%. Moving up the scales,
+similar behaviour was found with large eigenvalues being ch aracterised by more
+pronounced downward moves. The emergence of this character istic at very small
+scales (≈5 minutes), implies that it is not a result of data granularit y.
+4.5.Portfolio Optimisation
+As mentioned previously, the time-scale dependence of corr elation structure has
+implications for risk management. In this Section we demons trate, using high-January 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+Multiscaled Cross-Correlation Dynamics in Financial Time-Seri es13
+frequency 1minute stock returns, that the optimal portfoli o found from classic port-
+folio optimisation is time-scale dependent. The correlati on matrix between stocks
+of the Euro Stoxx 50 was calculated for two time-frames, the fi rst being a short
+time window of 10 ,000 minutes, (corresponding to the first time window studied in
+Section 4.3), the second the complete data set of 109 ,540 minutes, (corresponding
+to the data studied in Section 4.2). The portfolio optimisat ion was unconstrained,
+allowing ‘short-selling’ of stocks, which means that posit ive expected returns for the
+portfolio were possible, even though the majority of expect ed returns were negative.
+The expected returns and risk for the equities were the same f or both optimisations,
+meaning that all deviations in the risk/return profile are du e to changes in the cor-
+relation structure.
+0 10 20 30 40 50 6000.511.522.533.544.5
+Risk (%)Return (%)Efficient Frontier Different Scales Using Time Window of 10,000 Minutes
+(a)
+1 Minute
+60 Minutes
+360 Minutes
+750 Minutes
+2000 Minutes
+0 10 20 30 40 50 60 70 8000.511.522.533.544.5
+Risk (%)Return (%)Efficient Frontier Different Scales Using Time Window of 109,000 Minutes
+(b)
+1 Minute
+60 Minutes
+360 Minutes
+750 Minutes
+2000 Minutes
+Fig. 5. Portfolio Optimisation
+The portfolio optimisation was performed in a standard fash ion, [35, 37, 38],
+and the results, shown in Fig. 5, are highly dependent on the t ime-scale studied.
+Interestingly, for the smaller time-window of 10 ,000 minutes, the portfolios tended
+to be less risky at longer scales. In contrast, using the larg er time-window to cal-
+culate the correlations between stocks, the least risky por tfolio was found using a 1
+minute scale. Portfolios calculated at the 60 −750 minute scales were the riskiest,
+with longer-period scales found to be less risky. This is in k eeping with the find-January 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+14T.Conlon, H.J.Ruskin, M.Crane
+ings of Section 4.2, where the average correlation peaked at a scale of 480 minutes,
+meaning that further diversification benefits (lower correl ation) can be found at the
+shortest and longest scales.
+Theoptimalportfolioisclearlydependentonthetime-scal eusedtocalculatethe
+correlation matrix, which in turn, is dependent on the time w indow used. Obviously,
+this means that care must be taken by investors in the choice o f both time-window
+and granularity used in the calculation of correlation matr ices.
+5. Conclusions
+Using a multivariate technique first applied to complex EEG s eizure data, the corre-
+lation structure between medium and high-frequency financi al time-series was stud-
+ied. Application of the Maximum Overlap Discrete Wavelet Tr ansform extended the
+study of changes in the cross-correlation structure across both scale and time.
+(1) Using the MODWT and a sliding window, the dynamics of the l argest eigen-
+value of the correlation matrix were examined and shown to be time dependent
+at all scales, (using both medium and high-frequency equity returns). Similar
+dynamics were visible across all scales, but with particula r features markedly
+apparent at certain scales. This suggests that the correlat ion matrix between
+equities consists of interactions caused by traders with di fferent time horizons.
+(2) Study of the 2ndand 3rdlargest eigenvalues over both time and scale revealed
+a large increase in sectoral correlations for longer scales , using high-frequency
+data.
+(3) High frequency data were used to calculate the average co rrelation and largest
+eigenvalue for various scales. The Eppseffect was found using this multivariate
+technique and analysis of longer scales revealed a drop in th e average system
+correlation at scales beyond one day.
+(4) A partition of the time-normalised eigenvalues demonst rated quantitatively the
+relationship between the size of the largest eigenvalue and the return of the
+index. This feature emerged at very small scales ( <5 minutes), implying that
+this relationship is not due to the granularity of the data st udied.
+(5) Finally, we looked at the problem of portfolio optimisat ion at various scales.
+Results show that the optimal portfolio depends not only on s cale but on the
+time-window used in the calculation of the correlation matr ix.
+The correlation technique used in this paper to measure the i nteraction between
+agents suffersfrom the drawback of being linear and hence neg lects any higher-order
+relationships. Future work includes the application of non -linear information-theory
+based dependence measures, which will allow the detection o f complex changes in
+synchronisation behaviour around extreme financial events . These non-linear tech-
+niques may uncover further emergent features to those highl ighted above and may
+result in signals that warn of likely future market turmoil. The interaction of various
+agents with competing strategies and operating over differe nt time scales, requiresJanuary 4, 2010 21:55 WSPC/INSTRUCTION FILE MultiscaledCr oss-
+CorrelationDynamics
+Multiscaled Cross-Correlation Dynamics in Financial Time-Seri es15
+further investigation in order to improve understanding of complex phenomema
+such as financial crashes. Additionally, while this financia l work has benefited from
+parallels drawn with EEG system analysis, insights gained h ere may also raise ques-
+tions with regards to the behaviour of other complex systems .
+References
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+Dependent Multivariate Dynamics , To Appear: Comput. Bio. Med. 2009
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+arXiv:1001.0498v1  [math-ph]  4 Jan 2010Particle dynamics inside shocks in Hamilton–Jacobi
+equations
+Kostya Khanin1and Andrei Sobolevski2,3
+1Department of Mathematics, University of Toronto, Toronto, On tario, Canada
+2Institute for Information Transmission Problems of the Russian Ac ademy of
+Sciences, Moscow, Russia
+3Laboratoire J.-V. Poncelet (UMI 2615 CNRS), Moscow, Russia
+E-mail:khanin@math.toronto.edu ,sobolevski@iitp.ru
+Abstract. Characteristic curves of a Hamilton–Jacobi equation can be seen a s action
+minimizing trajectories of fluid particles. For nonsmooth “viscosity” solutions, which
+give rise to discontinuous velocity fields, this description is usually pur sued only up
+to the moment when trajectories hit a shock and cease to minimize th e Lagrangian
+action. In this paper we show that for any convex Hamiltonian there exists a uniquely
+defined canonical global nonsmooth coalescing flow that extends p article trajectories
+and determines dynamics inside the shocks. We also provide a variatio nal description
+of the corresponding effective velocity field inside shocks, and discu ss relation to the
+“dissipative anomaly” in the limit of vanishing viscosity.
+PACS numbers: 47.10.Df, 47.40.Nm, 02.30.Yy, 02.40.Ft, 02.40.Xx
+Submitted to: Philosophical Transactions of the Royal Society AParticle dynamics inside shocks in Hamilton–Jacobi equati ons 2
+1. Introduction
+The Hamilton–Jacobi equation
+∂φ
+∂t(t,x)+H(t,x,∇φ(t,x)) = 0 (1)
+plays an important role in a large variety of mathematical and physica l problems.
+Apart from analytical mechanics, it appears in description of a whole range of extended
+dissipative systems featuring nonequilibrium turbulent processes, from microscales of
+condensed matter and statistical physics through mesoscale set ting of free-boundary
+fluid to macroscale cosmological evolution (see, e.g., a non-exhaust ive collection of
+references in [1]). The central issue in a study of nonlinear evolution for (1) is to
+understand, both from the mathematical and physical points of v iew, the behaviour of
+the system after the inevitable formation of singularities.
+A theory of weak solutions for a general Hamilton–Jacobi equation , employing
+the regularization by infinitesimal diffusion, exists since the 1970s [2, 3, 4]. In the one-
+dimensionalsettingthistheoryisessentiallyequivalenttotheearlier theoryofhyperbolic
+conservation laws in fluid mechanics, developed in the 1950s [5, 6, 7]. I n more than one
+dimension, however, the two theories are no longer parallel.
+The theory of weak solutions for the Hamilton–Jacobi equation is clo sely related
+to calculus of variations, and from this point of view one can say that introduction of
+diffusion is motivated essentially by stochastic control arguments [ 8]. In the present
+paper we adopt a related but somewhat complementary viewpoint, in which the
+Hamilton–Jacobi equation is considered as a fluid dynamics model, and construct the
+flow of “fluid particles” inside the shock singularities of a weak solution .
+A useful example to be borne in mind when thinking about (1) — and arg uably
+the most widely known variant thereof — is the Riemann, or inviscid Bur gers, equation
+∂u
+∂t+u·∇u= 0,∇×u= 0. (2)
+It is obtained for the Hamiltonian H(t,x,p) =|p|2/2 by setting u(t,x) =∇φ(t,x).
+Note that in (2) it is essential that the velocity field uis curl-free, so this model is
+indeed equivalent to the Hamilton–Jacobi equation (1). The Riemann equation may in
+turn be considered as a limit of vanishing viscosity of the Burgers equ ation
+∂uµ
+∂t+uµ·∇uµ=µ∇2uµ,∇×uµ= 0, (3)
+so solutions of (2) can be defined as limits of smooth solutions to (3) a s the positive
+parameterµgoes to zero.
+The Burgers equation is in fact very special: it can be exactly mapped by the Cole–
+Hopf transformation into the linear heat equation and therefore e xplicitly integrated [5].
+Nonetheless the qualitative behaviour of solutions to a parabolic reg ularization of (1)
+for a general convex Hamiltonian
+∂φµ
+∂t+H(t,x,∇φµ) =µ∇2φµ(4)Particle dynamics inside shocks in Hamilton–Jacobi equati ons 3
+in the limit of vanishing viscosity is similar to that for the Burgers equat ion. It turns
+out that as µ→0 there exists a limit φ(t,x) = limφµ(t,x) which is called the viscosity
+solution. Remarkably the viscosity solution can be described by a purely varia tional
+construction which does not use the viscous regularization at all. Be low we briefly
+recall the main ideas of this variational approach.
+Assume that the Hamiltonian function H(t,x,p) is smooth and strictly convex in
+the momentum variable p, i.e., is such that for all ( t,x) the graph of H(t,x,p) as a
+function of plies above any tangent plane and contains no straight segments. Th is
+implies that the velocity v=∇pH(t,x,p) is a one-to-one function of p. In addition,
+the Lagrangian function
+L(t,x,v) = max p[p·v−H(t,x,p)]
+under the above hypotheses is smooth and strictly convex in v, although it may not be
+everywhere finite: e.g., the relativistic Hamiltonian H(t,x,p) =/radicalbig
+1+|p|2corresponds
+to the Lagrangian L(t,x,v) that is defined for |v| ≤1 as−/radicalbig
+1−|v|2and should be
+considered as taking value + ∞elsewhere. This will not happen if in addition one
+assumes that the Hamiltonian Hgrows superlinearly in |p|.
+The relation between the Lagrangian and the Hamiltonian is symmetric : they are
+Legendre conjugate to one another. This relation can also be expr essed in the form of
+the Young inequality:
+L(t,x,v)+H(t,x,p)≥p·v.
+This inequality holds for all vandpand turns into equality whenever v=∇pH(t,x,p)
+orp=∇vL(t,x,v). The two functions ∇pH(t,x,p) and∇vL(t,x,v) are inverse to
+each other; we will call them the Legendre transforms at (t,x) ofpand ofv. (Usually
+the term “Legendre transform” refers to the relation between t he conjugate functions H
+andL; here we follow the usage adopted by A. Fathi in his works on weak KA M theory
+[9], which is more convenient in the present context.)
+Note that if H(t,x,p) =|p|2/2, thenL(t,x,v) =|v|2/2 and the Legendre
+transform reduces to the identity v≡p, blurring the distinction between velocities
+and momenta. This is another very special feature of the Burgers equation.
+Now assume that φ(t,x) is a strong solution of the inviscid equation (1), i.e., a
+smooth function that satisfies the equation in the classical sense. For an arbitrary
+trajectory γ(t) the full time derivative of φalongγis given by
+dφ(t,γ)
+dt=∂φ
+∂t+∇φ·˙γ=∇φ·˙γ−H(t,γ,∇φ)≤L(t,γ,˙γ), (5)
+where at the last step the Young inequality is used. This implies a bound for the
+mechanical action corresponding to the trajectory γ:
+φ(t2,γ(t2))≤φ(t1,γ(t1))+/integraldisplayt2
+t1L(s,γ(s),˙γ(s))ds. (6)
+Equality in (5) is only achieved if ˙γis the Legendre transform of ∇φat (t,γ):
+˙γ(t) =∇pH(t,γ,∇φ(t,γ)). (7)Particle dynamics inside shocks in Hamilton–Jacobi equati ons 4
+This represents one of Hamilton’s canonical equations, with moment um given for the
+trajectory γbypγ(t) :=∇φ(t,γ(t)). The other canonical equation, ˙p=−∇xH, follows
+from (1) and (7) because
+˙pγ(t) =∂∇φ
+∂t+(∇⊗∇φ)·˙γ=−∇xH(t,γ,∇φ)−∇pH·(∇⊗∇φ)+(∇⊗∇φ)·˙γ
+Therefore the bound (6) is achieved for trajectories satisfying H amilton’s canonical
+equations. This is a manifestation of the variational principle of the least action :
+Hamiltonian trajectories ( γ(t),pγ(t)) are (locally) action minimizing. In particular,
+if the initial condition
+φ(t= 0,y) =φ0(y), (8)
+is a fixed smooth function, the identity
+φ(t,x) =φ0(γ(0))+/integraldisplayt
+0L(s,γ(s),˙γ(s))ds
+holds for a minimizer γsuch that γ(t) =x.
+The least action principle can be used to construct the viscosity solution
+corresponding to the initial data (8):
+φ(t,x) = min γ:γ(t)=x/parenleftBig
+φ0(γ(0))+/integraldisplayt
+0L(s,γ(s),˙γ(s))ds/parenrightBig
+. (9)
+This is the celebrated Lax–Oleinik formula (see, e.g., [10] or [9]), which r educes a PDE
+problem (1), (8) to the variational problem (9) where minimization is e xtended to all
+smooth curves γsuch that γ(t) =x.
+The function φdefined by (9) is smooth at those points ( t,x) where the minimizing
+trajectory is unique. In this case, the minimizer can be embedded in a smooth family
+of minimizing trajectories whose endpoints at time 0 and tare continuously distributed
+aboutγ(0)andγ(t) =x. Apieceofinitialdata φ0getscontinuouslydeformedaccording
+to (5) along this bundle of trajectories into a piece of smooth solutio nφto (1) defined
+in a neighbourhood of xat timet. Of course the Hamilton–Jacobi equation is satisfied
+in a strong sense at all points where φis differentiable.
+However, the crucial feature of (9) is that generally there will be p oints (t,x) with
+several minimizers γi, which start at different locations γi(0) but bring the same value
+of action to x=γi(t). Just as above, each of these Hamiltonian trajectories will be
+responsible for a separate smooth “piece” of solution. Thus for loc ationsx′close tox
+the function φwill be represented as a pointwise minimum of smooth pieces φi:
+φ(t,x′) = min iφi(t,x′).
+As allγihave the same terminal value of action, all these pieces intersect a t (t,x):
+φ1(t,x) =φ2(t,x) =...=φ(t,x). Thus the neighbourhood of xat timetis
+partitioned into domains where φcoincides with each of the smooth functions φiand
+satisfies the Hamilton–Jacobi equation (1) strongly. These domain s are separated
+by surfaces of various dimensions where two, or possibly three or m ore, pieces φi
+intersect and hence φis not differentiable. Such surfaces are called shocks. Note thatParticle dynamics inside shocks in Hamilton–Jacobi equati ons 5
+a function φdefined by the Lax–Oleinik formula is continuous everywhere, includin g
+the shocks; it is its gradient that suffers a discontinuity. In genera l, there are infinitely
+many continuous functions that match the initial condition (8) and a re differentiable
+and satisfy the Hamilton–Jacobi equation (1) apart from some sho ck surfaces, just
+asφ. A standard one-dimensional example of such nonuniqueness is pro vided by
+φα(t,x) = min(α|x| −α2t/2,0), which for any α >0 satisfies the initial condition
+φα(0,x) = 0 and the equation ∂φα/∂t+|∂φα/∂x|2/2 = 0 apart from the shock rays
+x=±αt/2,x= 0. Whatdistinguishes thefunction φdefinedby(9)fromallthese“weak
+solutions,” and grants it with important physical meaning, is that φappears in the limit
+of vanishing viscosity for the regularized equation (4) with the initial condition (8) (see,
+e.g., [3]).
+For a smooth Hamiltonian it can be proved that once shocks are crea ted they never
+disappear, although they can merge with one another. Another imp ortant physical
+feature of viscosity solutions is that minimizers can only merge with sh ocks but never
+leave them: all minimizers coming to some ( t,x) in (9) originate at t= 0. It is easy to
+see that this is not so in the above example of φα, where minimizers emerge from x= 0
+at all times t>0.
+Moreover, in a solution φgiven by (9) a minimizer that has come to a shock
+cannot be continued any longer as a minimizing trajectory. Indeed, wherever it comes,
+there will be other trajectories originated at t= 0 that will bring smaller values of
+action to the same location. Hence for the purpose of the least act ion description (9)
+Hamiltonian trajectories are discontinued as soon as they are abso rbed by shocks. The
+set of trajectories which survive as minimizers until time t>0 is decreasing with t, but
+at all times it is sufficiently large to cover the whole space of final posit ions.
+Let us now adopt an alternative viewpoint and consider the Hamilton– Jacobi
+equation as a fluid dynamics model, assuming that Hamiltonian traject ories (7) are
+described by material “particles” transported by the velocity field u(t,x), which is the
+Legendre transform of the momenta field p(t,x) =∇φ(t,x). From this new perspective
+it is no longer natural to accept that particles annihilate once they r each a shock.
+Can therefore something be said about the dynamics of those part icles that got into the
+shock, notwithstanding thefactthattheirtrajectoriesceaset ominimizetheaction? The
+problem here comes from the discontinuous nature of the velocity fi eldu, which makes
+it impossible to construct classical solutions to the transport equa tion˙γ(t) =u(t,γ).
+In dimension 1 the answer to the question above is readily available. Sh ocks at
+each fixedtare isolated points in the xspace and as soon as a trajectory merges with
+one of them, it continues to move with the shock at all later times. Th is description
+is related to C. Dafermos’ theory of generalized characteristics [1 1] which, in fact, can
+be extended to a much more general situation of nonconvex Hamilto nians and systems
+of conservation laws. However, in several space dimensions shock s become extended
+surfaces and already for equation (1) with a strictly convex Hamilto nian dynamics of
+trajectories inside shocks is by no means trivial. The main goal of the present paper is
+to describe a natural and canonical construction of such dynamic s.Particle dynamics inside shocks in Hamilton–Jacobi equati ons 6
+First results in this direction were obtained by I. Bogaevsky [12, 13] for the Burgers
+equation (3). Bogaevsky suggested to consider the transport p roblem for a smooth
+velocity field uµ:
+˙γµ(t) =uµ(t,γµ),γµ(0) =y.
+Sinceuµforµ >0 is a smooth vector field, this equation defines a family of particle
+trajectories forming a smooth flow. The next step is to take the limit of this flow as
+µ↓0. Bogaevsky proved thatthislimitexists asanon-differentiable con tinuous flow, for
+which the forward derivative ˙γ(t+0) = lim τ↓0[γ(t+τ)−γ(t)]/τis defined everywhere.
+Ifγ(t) is located outside shocks, this derivative coincides with u(t,γ(t)). Otherwise
+there are several limit values of velocity ui, and Bogaevsky discovered an interesting
+explicit representation for ˙γ(t+0): it coincides with the center of the smallest ball that
+contains all ui. It should be remarked that uniqueness of a limit flow in the case of
+a quadratic Hamiltonian was earlier observed by P. Cannarsa and C. S inestrari in the
+context of propagationofsingularities fortheeikonal equation an ddifferential inclusions
+[14, Lemma 5.6.2].
+The original approach in [12, 13] is based on the specific properties o f the Burgers
+equation and cannot be applied in the case of general convex Hamilto nians. In
+particular, the method uses the identity of velocities and momenta, which does not hold
+in the general setting. At the same time the common wisdom says tha t all Hamilton–
+Jacobi equations with convex Hamiltonians must have similar propert ies.
+In this work we consider the above strategy, consisting in the para bolic
+regularization of the Hamilton–Jacobi equation and investigation of the vanishing
+viscosity limit for the corresponding regularized flow, in the case of g eneral convex
+Hamiltonians. We show that such a limit exists, and derive an explicit rep resentation
+for the forward velocity of the limit flow that extends the above res ult for the Burgers
+equation. Yet the mechanism powering these results in the general case is completely
+different. It is based on the fundamental uniqueness of a possible lim it behaviour of γµ,
+which we discuss in detail below.
+The paper is organized as follows. In Section 2 we study the local str ucture of a
+viscosity solution near a singularity. We also introduce the notion of a dmissible velocity
+at a singularity and show that it can be determined uniquely. Moreove r, the unique
+admissible velocity provides a solution to a certain convex minimization p roblem, which
+generalizes Bogaevsky’s construction of the center of the smalles t ball. In Section 3 we
+demonstrate that for any nonsmooth viscosity solution there exis ts a unique continuous
+nonsmooth flow of trajectories tangent to the field of admissible ve locities. Section 4
+contains concluding remarks.
+Acknowledgments
+This work is supported by the NSERC Discover Grant and the Russian Foundation for
+Basic Research, project RFBR-CNRS–07–01–92217; the second author is supported inParticle dynamics inside shocks in Hamilton–Jacobi equati ons 7
+part by the French Agence Nationale de la Recherche, project ANR –07–BLAN–0235
+OTARIE. We acknowledge the hospitality of Observatoire de la Cˆ ote d’Azur (Nice,
+France), where part of this work has been performed, as well as s upport of the French
+Ministry for National Education.
+We are sincerely grateful to J´ er´ emie Bec, Patrick Bernard, Ilya Bogaevsky, Yann
+Brenier, Philippe Choquard, and Boris Khesin for numerous valuable d iscussions. It is
+our special pleasure to express gratitude to Uriel Frisch, who mad e crucial contributions
+to the current revival of interest in the “Burgers turbulence” an d related fields, and who
+is turning 70 in 2010.
+2. Local structure of viscosity solutions and admissible mo menta
+Letφbe a viscosity solution to the Hamilton–Jacobi equation (1) with initial data (8).
+If there is a single minimizer coming to ( t,x), thenφis differentiable at this point and
+φ(t+τ,x+ξ) =φ(t,x)+∂φ
+∂tτ+∇φ·ξ+···
+=φ(t,x)−H(t,x,∇φ)τ+∇φ·ξ+···,
+where dots ···stand for higher-orger terms. If φis not differentiable at ( t,x), this
+means that there are several minimizers γisuch that γi(t) =x, each bringing to ( t,x)
+a different piece φiof solution. Then the Lax–Oleinik formula implies that
+φ(t+τ,x+ξ) = min iφi(t+τ,x+ξ)
+=φ(t,x)+min i(−Hiτ+pi·ξ)+···,
+wherepi:=∇φi(t,x) andHi:=H(t,x,pi).
+In the latter case neither of the expressions −Hiτ+pi·ξprovides an adequate
+linear approximation to the difference φ(t+τ,x+ξ)−φ(t,x), but they all majorize
+this difference up to a remainder that is linear or higher-order depen ding onτandξ.
+Evidently, so does too the linear form −Hτ+p·ξfor any convex combination
+p=/summationdisplay
+iλipi, H=/summationdisplay
+iλiHi
+withλi≥0,/summationtext
+iλi= 1. In convex analysis these convex combinations are called
+supergradients of φat (t,x) and the whole collecton of them, which is a convex
+polytope with vertices ( −Hi,pi), is called the superdifferential ofφ[14, 15]. We use
+R. T. Rockafellar’s notation ∂φ(t,x) for the superdifferential [15].
+To avoid a possible misunderstanding it should be noted that, althoug h uniqueness
+of minimizer coming to ( t,x) implies differentiability of φattand earlier times, it does
+notimply its differentiability at any t+τ > t. The following example shows how this
+may happen. The function defined for τ≥0 by
+φ(t+τ,x+ξ) =φ(t,x)−4
+3τ3−2|ξ|τ−4
+3(|ξ|+τ2)3/2
+is a viscosity solution of the equation ∂φ/∂t+|∂φ/∂x|2/2 = 0 that satisfies the smooth
+initial condition φ(t,x+ξ)−φ(t,x) =−4
+3|ξ|3/2. Forτ >0 a shock appears at ξ= 0, butParticle dynamics inside shocks in Hamilton–Jacobi equati ons 8
+differentiability at τ= 0 is recovered because ∂φ(t+τ,x) ={−8τ2}×[−4τ,4τ] shrinks
+to (0,0) asτ↓0. Such points ( t,x) are called preshocks [1].
+The particular case of preshocks is an instance of a general fact: replacing gradients
+with superdifferentials allows to recover continuous differentiability, but in a weaker
+sense. Namely, suppose ( tn,xn) converges to ( t,x) and the sequence ( −Hn,pn)∈
+∂φ(tn,xn) has a limit point ( −H,p). By definition of superdifferential,
+φ(tn+τ,xn+ξ)−φ(tn,xn)≤ −Hnτ+pn·ξ+···; (10)
+passing here to the limit and using continuity of φ, we see that ( −H,p)∈∂φ(t,x).
+Therefore the superdifferential ∂φ(tx) contains all the limit points of superdifferentials
+∂φ(tn,xn) as (tn,xn) converges to ( t,x).
+Suppose (t,x) is a point of shock where ksmooth branches φimeet. It follows from
+the above discussion that for a particle moving from a shock point ( t,x) all possible
+values of the velocity vmust be such that the corresponding momentum p(v) belongs
+to the convex hull of the available momenta pi=∇φi(t,x), 1≤i≤k. However,
+one can say even more. For small positive τnot all the branches φiwill contribute to
+the solution φat a point ( t+τ,x+vτ), but only those of them that correspond to a
+minimum in min i(−Hi+pi·v). Denote the corresponding set of indices
+I(v) :={1≤j≤k:−Hj+pj·v= min
+i(−Hi+pi·v)}. (11)
+The setI(v) has the following physical meaning: if particle moves away from a sho ck
+with a given velocity vthen onlyφjandpjforj∈I(v) are relevant. Geometrically
+one can say that the convex hull of pj,j∈I(v), is the p-projection of the face of
+the superdifferential ∂φ(t,x) that looks toward an infinitesimal observer who has just
+left (t,x) with velocity v.
+This implies that any possible velocity vmust satisfy the following condition.
+Admissibility Condition. A velocity v∗is admissible if and only if the
+corresponding momentum belongs to the convex hull of the mom entapj,j∈I(v∗).
+Namely,
+p∗(v∗) =/summationdisplay
+j∈I(v∗)λjpj, λ j≥0,/summationdisplay
+j∈I(v∗)λj= 1. (12)
+To make the admissibility argument rigorous it is necessary to have so me control
+over the remainder term in (10). A natural function class that con tains viscosity
+solutions of Hamilton–Jacobi equations and in which such control is p ossible is formed
+by semiconcave functions [14]. We refer a reader interested in care ful proofs of this and
+other convex analytic results used in this paper to monographs [14 , 15].
+Remarkably the Admissibility Condition determines the velocity v∗uniquely.
+Lemma 1 (Uniqueness) Letφbe a viscosity solution to the Cauchy problem (1), (8).
+Then at any (t,x)there exists a unique admissible velocity v∗. Moreover this admissible
+velocityv∗is the unique point of the global minimum for the following fu nction
+ˆL(v) :=L(t,x,v)−mini(−Hi+pi·v). (13)Particle dynamics inside shocks in Hamilton–Jacobi equati ons 9
+Proof.Recall that L(t,x,v) is a strictly convex function of vbecause of assumptions
+formulated in the Introduction. Rewriting
+Li(v) :=L(t,x,v)+Hi−pi·v,ˆL(v) = max iLi(v), (14)
+we see that ˆL(v) is a poitwise maximum of a finite number of strictly convex functions
+and therefore is strictly convex itself. Furthermore, because th e Hamiltonian H(t,x,p)
+is assumed to be finite for all p, its conjugate Lagrangian L(t,x,v) grows faster than
+any linear function as |v|increases, and all its level sets are bounded. Thus ˆL(v) attains
+its minimum at a unique value of velocity v∗.
+Let us show that this point of minimum v∗satisfies the admissibility condition.
+Indeed,
+∇vLi(v∗) =∇vL(t,x,v∗)−pi=p∗−pi.
+Suppose that p∗does not belong to the convex hull of pj,j∈I(v∗). Then there exists a
+vectorhsuch that ( p∗−pj)·h<0 for allj∈I(v∗). It follows that Lj(v∗+ǫh)<Lj(v∗)
+for allj∈I(v∗) ifǫ >0 is sufficiently small. Hence, ˆL(v∗+ǫh)<ˆL(v∗) for
+sufficiently small ǫ, which contradicts our assumption that v∗is a point of minimum.
+This contradiction proves that v∗is admissible.
+To prove uniqueness we show that if ˆvis admissible then it is a unique point of
+global minimum for the function ˆL. Using the strict convexity of ˆL, we obtain
+Lj(ˆv+h) =L(t,x,ˆv+h)+Hj−pj·(ˆv+h)
+>Lj(ˆv)+∇vL(t,x,ˆv)·h−pj·h=Lj(ˆv)+(ˆp−pj)·h,
+whereˆpis the Legendre transform of ˆv. Sinceˆvis admissible, ˆp=/summationtext
+jλjpj, where all
+λj≥0and/summationtext
+jλj= 1. Hence,/summationtext
+jλj(ˆp−pj)·h= [(/summationtext
+jλj)ˆp−/summationtext
+jλjpj]·h= [ˆp−ˆp]·h= 0.
+It follows that ( ˆp−pj)·h≥0 for at least one j∈I(ˆv). ThusˆL(ˆv+h)>ˆL(ˆv), which
+implies that ˆvis a unique point of global minimum for ˆL. This observation concludes
+the proof.
+The admissibility property, first formulated above in a somewhat unm anageable
+combinatorial form (12), turns out to be the optimality condition in a convex
+minimization problem given by (13), i.e., a much simpler object. In partic ular, ifφis
+differentiable at ( t,x), thenˆL(v) =L(t,x,v)+H(t,x,∇φ)−∇φ·vand the minimum
+in (13) is achieved at the Legendre transform of ∇φ. We thus recover Hamilton’s
+equation (7).
+The following reformulation will clarify the connection between admiss ibility and
+Bogaevsky’s original construction for the Burgers equation. Let vi=∇pH(t,x,pi)
+be the velocity corresponding to the limit momentum piand observe that pi=
+∇vL(t,x,vi). The Young inequality implies that Hi=H(t,x,pi) =pi·vi−L(t,x,vi)
+and therefore (14) assumes the form
+ˆL(v) = max i[L(t,x,v)−L(t,x,vi)−∇vL(t,x,vi)·(v−vi)].
+The quantity in square brackets is known as the Bregman divergence between vectors
+vandvi, a specific measure of their separation with respect to the convex functionLParticle dynamics inside shocks in Hamilton–Jacobi equati ons 10
+[16]. When L(t,x,v) =|v|2/2, the Bregman divergence reduces to (half) the squared
+distance between the two vectors; hence the admissible velocity v∗exactly conicides
+with the centre of smallest ball containing all vi, and Bogaevsky’s result is recovered.
+Finally, wediscussthephysical meaningofthefunction ˆL. Consideraninfinitesimal
+movement from ( t,x) with velocity v. Obviously φ(t,x)+L(t,x,v)dt−φ(t+dt,x+
+vdt)≥0. It is easy to see that in the linear approximation in d t
+φ(t,x)+L(t,x,v)dt−φ(t+dt,x+vdt) =ˆL(v)dt.
+Hence the unique admissible velocity v∗minimizes the rate of the difference in action
+between the true minimizers and trajectories of particles on shock s. In other words,
+the trajectory on a shock cannot be a minimizer but it does its best t o keep its surplus
+action growing as slowly as possible.
+3. The vanishing viscosity limit
+In the preceding section we constructed a canonical vector field v∗=∇pH(t,x,p∗)
+correspondingtoagivenviscosity solution φoftheCauchy problem(1),(8). Thebasisof
+thisconstruction, theadmissibility condition, appearsasa natural consistency condition
+between velocities and supergradients. This condition together wit h the variational
+principle (13) guarantees uniqueness of the admissible pair ( v∗,p∗).
+The vector field v∗(t,x) can be decomposed into a union of smooth tangent vector
+fields defined on connected pieces of smooth shock surfaces of va rious dimensions as well
+asonthedomainwhere φisdifferentiable, sodynamicsinthelatterdomainorinsideany
+piece of a smooth shock surface is given locally by a smooth flow. But g lobally the field
+v∗is discontinuous, and it is not immediately clear if there exists an overa ll continuous
+flow of trajectories γthat is compatible with v∗in the sense that ˙γ(t+0) =v∗(t,γ).
+Even less obvious is the uniqueness of such a flow.
+In order to answer these questions affirmatively we employ the vanis hing viscosity
+limit for the parabolic regularization
+∂φµ
+∂t+H(t,x,∇φµ) =µ∇2φµ, µ> 0, (15)
+of the Hamilton–Jacobi equation (1). For sufficiently smooth initial d ataφµ(t= 0,y) =
+φ0(y)equation(15)hasagloballydefinedstrongsolution, which islocallyLip schitzwith
+a constant independent of µ. Moreover, φµconverges as µ↓0 to the unique viscosity
+solutionφcorresponding to the same initial data. Proof of these facts may b e found,
+e.g., [3], where they are established for φ0∈C2,α.
+Consider now the differential equation
+˙γµ(t) =∇pH(t,γµ,∇φµ(t,γµ)),γµ(0) =y. (16)
+Forµ >0 this equation has a unique solution γµ
+y, which continuously depends on the
+initial location y. Fix a point ( t,x) witht>0 and pick for all sufficiently small µ>0
+trajectories γµsuch that γµ(t) =x. The uniform Lipschitz property of solutions φµ
+implies that the curves γµare uniformly bounded and equicontinuous on some intervalParticle dynamics inside shocks in Hamilton–Jacobi equati ons 11
+containingt. Hencethereexistsacurve ¯γandasequence µi↓0suchthatlim µi↓0γµi=¯γ
+uniformly. Note that all γµiand¯γare also Lipschitz with a constant independent of µ
+and that ¯γ(t) =x. Let furthermore ¯vbe a limit point of the “forward velocity” of the
+curve¯γat (t,x), i.e., let for some sequence τk↓0
+¯v= limτk↓01
+τk[¯γ(t+τk)−¯γ(t)].
+Of course neither the curve ¯γnor the velocity ¯varea priori defined uniquely.
+Nevertheless it turns out that ¯vmust satisfy the admissibility condition with respect
+to the solution φand therefore it coincides with v∗. Also, trajectories of the flow γµ
+converge as µ↓0 to segments of integral trajectories of the vector field v∗on smooth
+shock surfaces, establishing the uniqueness of the limit flow ¯γ. Moreover, the limit flow
+iscoalescing : if two trajectories intersect at time t, they coincide for all t′>t. All these
+statements follow from the following fact.
+Lemma 2 The flow defined by (16) for sufficiently small µ >0collapses the
+neighbourhood of the shock surface that passes through (t,x)and is tangent to (1,v∗)
+onto this surface.
+Here is a sketch of proof. Let v∗be the admissible velocity corresponding to ( t,x),
+p∗the corresponding momentum, and define
+H∗=p∗·v∗−min
+i(pi·v∗−Hi). (17)
+The full time derivative of the function ψµ(t+τ,x+ξ) =φµ(t+τ,x+ξ)−p∗·ξ+H∗τ
+along an integral trajectory γµof equation (16) which passes through x+ξat timet+τ
+is given by
+dψµ(t+τ,γµ(t+τ))
+dτ=∂φµ
+∂t+H∗+(∇φµ−p∗)·˙γµ.
+Convergence of viscosity solutions implies convergence of their sup erdifferentials (to see
+this, it is enough to replace in (10) the function φwith a sequence of functions φn
+converging pointwise). Therefore limit points of ( ∂φµ/∂t,∇φµ) belong to ∂φ(t,x) for
+all (t,x), and for sufficiently small µ >0,τ >0,ξthere exists ( −Hµ,pµ)∈∂φ(t,x)
+andvµ=∇pH(t,x,pµ) such that, to the linear approximation, vµ=˙γµ+···and
+dψµ
+dτ=−Hµ+H∗+(pµ−p∗)·vµ+··· (18)
+=pµ·v∗−Hµ−min
+i(pi·v∗−Hi) (19)
++(pµ−p∗)·(vµ−v∗)+··· (20)
+Lines (19) and(20) contain nonnegative quantities, which are posit ive ifpµ/ne}ationslash=p∗. To see
+this for (20), observe that by the strict convexity H(t,x,pµ)>H(t,x,p∗)+(pµ−p∗)·v∗
+andH(t,x,p∗)>H(t,x,pµ)+(p∗−pµ)·vµ, and add these two inequalities.
+On the other hand, the superdifferential of the function
+lim
+µ↓0ψµ(t+τ,x+ξ) =φ(t+τ,x+ξ)−p∗·ξ+H∗τParticle dynamics inside shocks in Hamilton–Jacobi equati ons 12
+at (τ= 0,ξ= 0) contains zero because p∗corresponds to an admissible velocity.
+Therefore in the linear approximation the point ξ= 0 and other points of the shock
+surface ofψtangent to (1 ,v∗) are local maxima of ψup to terms of higher order, and
+the full time derivative of ψalong the curve ξ=v∗τvanishes. As d φµ/dtis positive
+for trajectories that start outside the shock, we see that the fl ow (16) collapses them
+asymptotically on the shock surface. This completes the argument .
+The full details of this proof and the rigorous derivation of uniquene ss of the limit
+flow¯γwill begiven ina forthcoming article [17]. Here we just formulate thema in result.
+Theorem 3 Letφbe a viscosity solution to the Cauchy problem for the Hamilto n–
+Jacobi equation (1) with initial data (8). There exists a uni que flow γyof continuous
+trajectories such that γy(0) =y,˙γy(t+ 0)is defined for all t,yand coincides with
+the admissible velocity v∗(t,γy(t))given by solution to the convex minimization problem
+for (13). The trajectory γycontinuously depends on y. After a trajectory γycomes to
+a shock, it stays inside the shock manifold for all later time s. The flow is coalescing: if
+two trajectories γy′andγy′′coincide at time t, thenγy′(t′) =γy′′(t′)for allt′>t.
+4. Concluding remarks
+Starting from a viscosity solution φto the Hamilton–Jacobi equation, we have
+constructed a unique continuous coalescing flow γycompatible with the admissible
+velocity field v∗defined in Section 2 in the sense that ˙γy(t+0) =v∗(t,γy). This flow is
+a natural extension of the smooth flow defined by Hamilton’s equatio n (7). We conclude
+with a few observations concerning this construction.
+1. Recall an important observation made in the original work of Boga evsky: pieces
+of the shock manifold, irrespective of their dimension, are classified intorestraining and
+nonrestraining depending on whether p∗belongs to the interior or the boundary of
+the convex polytope formed by projection of ∂φ(t,x) on the pspace. Particles stay
+on restraining shocks but leave non-restraining shocks along piece s of shock manifold
+of lower codimension corresponding to faces of the boundary cont ainingp∗. Shocks
+of codimension one are always restraining; in particular, so are all sh ocks in the one-
+dimensional case, which makes the construction of the coalescing fl owγytrivial, as
+remarked in the introduction. Interestingly, this classification of s hocks, introduced
+in [12] (“acute” and “obtuse” superdifferentials of φ), seems to have been overlooked by
+physicists before.
+2. Note that the construction of the admissible velocity v∗is purely kinetic: when
+the Lagragian is “natural,” i.e., has the form L(t,x,v) =K(v)−U(t,x), the value v∗
+is the same for all choices of the potential term U(t,x) as long as kinetic energy K(v)
+is fixed. We owe this observation to P. Choquard.
+3. Seen as a family of continuous maps of variational origin from initial coordinates
+yto current coordinates x, the flow γyis clearly relevant for optimal transportation
+problems [18, 19]. An interesting problem suggested by B. Khesin is to study the
+extremal properties of this flow. Indeed it is known from [20] that b efore the firstParticle dynamics inside shocks in Hamilton–Jacobi equati ons 13
+shock formation the flow γyis and action minimizing flow of diffeomorphisms, while the
+first shock formation time t∗marks a conjugate point in the corresponding variational
+problem. According to the suggested view, the flow constructed a bove may be seen
+as a kind of saddle-point, rather than minimum, for a suitable transp ort optimization
+problem.
+4. Another natural context to place our construction in is that of differential
+inclusions (see, e.g., [21]). The flow consructed here may be seen as a solution of
+differential inclusion
+˙γ∈ ∇pH(t,γ,Prp∂φ(t,γ)), (21)
+where Pr pis thepprojection of the superdifferential ∂φ. In comparison with standard
+constructions of the theory of differential inclusions the flow γysolves (21) in a stronger
+sense: the forward derivative ˙γy(t+0) exists everywhere. Also, a simple modification
+of Lemma 2 gives a proof of uniqueness for inclusion (21).
+5. The flow γywas constructed as a limit of a parabolic regularization, and it was
+noticed above that the limit of ( −∂φµ/∂t,∇φµ) belongs to ∂φ(t,x) for any (t,x). This
+statement can be refined if one considers the values of ∇φµalong trajectories of the
+flow (16). Namely, arguments of Section 3 imply that under the succ essive limits µ↓0
+andτ↓0, the gradient ( ∂φµ/∂t,∇φµ) taken at ( t+τ,γµ(t+τ)) converges to ( −H∗,p∗).
+Therefore
+lim
+τ↓0lim
+µ↓0µ∇2φµ(t+τ,γµ(t+τ)) = lim
+τ↓0lim
+µ↓0/bracketleftBig∂φµ
+∂t+H(t,γµ(t+τ),∇φµ)/bracketrightBig
+=H(t,x,p∗)−H∗
+= min
+i(pi·v∗−Hi)−L(t,x,v∗)
+=ˆL(v∗) = min vˆL(v),
+where we took into account formulas (17) and (13). In other word s, minimum of the
+convex minimization problem (13) coincides with the value of the “dissip ative anomaly”
+in the parabolic regularization (4) of the Hamilton–Jacobi equation ( 1).
+6. Observe also that convergence of superdifferentials makes it po ssible to use
+other smoothing procedures for φ(e.g., convoluting it with a standard mollifier),
+giving the same limit γy. However, one can imagine the following completely different
+regularization of the discontinuous velocity field ∇pH(t,x,∇φ(t,x)). Physically
+speaking, this regularization may be characterized by a “zero Pran dtl number” in
+contrast with the previous class of regularizations featuring an “in finite Prandtl
+number.”
+Consider the stochastic equation
+dγǫ=∇pH(t,γǫ,∇φ(t,γǫ))dt+ǫdW(t),
+whereWis the standard Wiener process. The corresponding stochastic flo w is well
+defined in spite of the fact that ∇φdoes not exist everywhere: whenever the trajectory
+γǫhitsshocks, noiseinthesecondtermwillinstantaneouslysteeritina randomdirection
+away from them.Particle dynamics inside shocks in Hamilton–Jacobi equati ons 14
+Assume that as ǫ↓0 the stochastic flow γǫ
+ytends to a limit flow ˜γy, which is also
+forward differentiable. It is easy to see that, due to the averaging , the forward velocity
+v†(t,˜γ) :=˙˜γy(t+0) must belong to the convex hull of vj,j∈I(v†). Namely,
+v†=/summationdisplay
+j∈I(v†)πjvj, π j≥0,/summationdisplay
+j∈I(v†)πj= 1, (22)
+where the velocities vj(t,x) are Legendre transforms of the corresponding momenta
+pj=∇φi(t,x) at a singular point ( t,x). The coefficients πiare equal to asymptotic
+values of shares of time that a trajectory γǫspends in each of the domains where
+φ=φj. Condition (22) above is another compatibility condition, in certain se nse dual
+to the admissibility condition in Section 2. For want of a better term let us call such a
+velocityv†self-consistent .
+The self-consistent velocity is a convex combination of velocities seen by an
+infinitesimal observer leaving ( t,x) with velocity v†. Compare this with the definition
+of admissible momentum p∗, which is a convex combination of momenta see by a similar
+observer moving vith velocity v∗. WhenH(t,x,p) =|p|2/2 andv=p, self-consistent
+velocities and admissible velocities coincide. It is however clear that in t he case of a
+general nonlinear Legendre transform v†/ne}ationslash=v∗=∇pH(t,x,p∗).
+In view of theanalogy between self-consistent velocities and admiss ible momenta, it
+istemptingtoconjecturethattheadmissible velocityisalsounique. G enerallyspeaking,
+this statement is wrong, although it holds in one and two space dimens ions. In higher
+dimensions there exist Hamiltonians and sets of limiting momenta for wh ich there is
+more than one admissible velocity [17]. It is an interesting problem neve rtheless to see
+whether a limiting flow still exists in the limit of weak noise in spite of nonun iqueness
+of admissible velocity. This problem carries a certain similarity with the p roblem of
+limit behaviour for one-dimensional Gibbs measures in the zero-temp erature limit, in
+the case of nonunique ground states.
+References
+[1] Bec J and Khanin K 2007 Burgers turbulence Phys. Rep. 4471–66
+[2] Kruzhkov S 1975 Weak solutions for Hamilton–Jacobi equations o f the eikonal type Mat. Sbornik
+98450–93 (in Russian)
+[3] Lions P-L 1982 Generalized Solutions of Hamilton–Jacobi Equations (Research Notes in
+Mathematics vol 69) (Boston: Pitman)
+[4] Crandall M G, Ishii H and Lions P-L 1992 User’s guide to viscosity so lutions of second order partial
+differential equations Bull. Amer, Math. Soc. 271–67
+[5] Hopf E 1950 The partial differential equation ut+uux=µuxxCommun. Pure Appl. Math. 3201–30
+[6] Lax P 1954 Weak solutions of nonlinear hyperbolic equations and th eir numerical computaton
+Comm. Pure Appl. Math. 7159–93
+[7] Oleinik O 1954 On the Cauchy problem for nonlinear equations in the c lass of discontinuous
+functions Dokl. Akad. Nauk SSSR 95451–5 (in Russian)
+[8] Fleming W H and Soner H M 2005 Controlled Markov Processes and Viscosity Solutions (Stochastic
+Modelling and Applied Probability vol 25) (Berlin etc: Springer)
+[9] Fathi A 2009 Weak KAM Theorem in Lagrangian Dynamics (Cambridge: Cambridge University
+Press)Particle dynamics inside shocks in Hamilton–Jacobi equati ons 15
+[10] E W, Khanin K M, Mazel A E, Sinai Ya G 2000 Invariant measures fo r Burgers equation with
+stochastic forcing Ann. of Math. 151877–960
+[11] Dafermos C M 2005 Hyperbolic Conservation Laws in Continuum Physics (Grundlehren in
+Mathematischen Wissenschaften vol 325) (Berlin etc: Springer)
+[12] Bogaevsky I 2004 Matter evolution in Burgulence Preprint math-ph/0407073
+[13] Bogaevsky I 2006 Discontinuous gradient differential equation s and trajectories in the calculus of
+variations Mat. Sbornik 1971723–51
+[14] Cannarsa P and Sinestrari C 2004 Semiconcave Functions, Hamilton–Jacobi Equations, and
+Optimal Control (Progress in nonlinear differential equations and their appl ications vol 58)
+(Boston: Birkh¨ auser)
+[15] Rockafellar R T 1970 Convex Analysis (Princeton: Princeton University Press)
+[16] Bregman L M 1967 The relaxation method of finding the common po ints of convex sets and its
+application to the solution of problems in convex programming USSRComputational Mathematics
+and Mathematical Physics 7200–17
+[17] Khanin K and Sobolevski A 2009 On dynamics of Lagrangian particles for Hamilton–Jacobi
+equations , in preparation
+[18] Gangbo W and McCann R J 1996 The geometry of optimal transpo rtationActa Math. 177113–61
+[19] Villani C 2009 Optimal Transport: Old and New (Berlin etc: Springer)
+[20] Khesin B A and Misio/suppress lek G 2007 Shock waves for the Burgers equa tion and curvatures
+of diffeomorphism groups Analysis and singularities. Part 2, Collected papers. Dedi cated to
+Academician Vladimir Igorevich Arnold on the occasion of hi s 70th birthday vol. 259 of Tr.
+Mat. Inst. Steklova (Moscow: Nauka) pp 77–85 ( also available as preprint math/0702196)
+[21] Aubin J-P and Cellina A 1984 Differential Inclusions: Set-Valued Maps and Viability The ory
+(Berlin etc: Springer)
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+arXiv:1001.0500v2  [hep-ph]  15 Mar 2010Observation of the Higgs Boson of strong interaction
+via Compton scattering by the nucleon∗
+Martin Schumacher
+mschuma3@gwdg.de
+Zweites Physikalisches Institut der Universit¨ at G¨ ottin gen, Friedrich-Hund-Platz 1
+D-37077 G¨ ottingen, Germany
+Abstract
+It is shown that the Quark-Level Linear σModel (QLL σM) leads to a prediction for the
+diamagnetic term of the polarizabilities of the nucleon which is in excellen t agreement with
+experimental data. The bare mass of the σmeson is predicted to be mσ= 666 MeV and
+the two-photon width Γ( σ→γγ) = (2.6±0.3) keV. It is argued that the mass predicted
+by the QLL σM corresponds to the γγ→σ→N¯Nreaction, i.e. to a t-channel pole of the
+γN→Nγreaction. Large-angle Compton scattering experiments revealing effects of the
+σmeson in the differential cross section are discussed. Arguments a re presented that these
+findings may be understood as an observation of the Higgs boson of strong interaction while
+being a part of the constituent quark.
+1 Introduction
+Theσmeson introduced by Schwinger [1] and Gell-Mann–Levy [2] ha s attracted great interest
+because there are good reasons to consider it as the Higgs bos on of strong interaction, an
+aspect which recently has been emphasized by several author s [3–6]. The mass of the σmeson
+is predicted by the Quark-Level Linear σModel (QLL σM) [7] (see also [8,9] and references
+therein) to be mσ= 666 MeV, whereas π-πscattering analyses led to√sσ=Mσ−iΓσ/2
+withMσ= 441+16
+−8MeV and Γ σ= 544+18
+−25MeV in one recent evaluation (CCL) [10], or√sσ=
+(476−628)−i(226−346) MeVwhentestsofthestabilityoffitstodataaretakenin toaccount [11].
+Itcertainly isextremely important tounderstandhow these twomasses arerelated toeach other.
+Thefindingisthat mσ= 666 MeV isthebaremassof the σmesonwhichis observedinspace-like
+Compton scattering γγ→σ→N¯N, i.e. as at-channel pole of Compton scattering γN→Nγ.
+For the electromagnetic polarizabilities space-like Comp ton scattering is equally important as
+time-like Compton scattering γN→Nγ(s-channel). The σand non-σcomponents of the
+electromagnetic polarizabilities fortheproton( p)andtheneutron( n) areαp,n(σ) =−βp,n= 7.6,
+αp(non−σ) = 4.4,βp(non−σ) = 9.5,αn(non−σ) = 5.8,βn(non−σ) = 9.4 in units of 10−4fm3.
+On the quark level the reaction γγ→σ→N¯Nimplies that two photons with parallel planes of
+linear polarization interact with σmesons, being parts of the constituent quarks. In this sense
+it is justified to consider space-like Compton scattering as an in-situ observation of the Higgs
+boson of strong interaction.
+In addition to the specific aspects of the σmeson as outlined in the preceding paragraph this
+particleisofinterestbecauseithasbeenobservedasanint ermediate stateinmanyreactions [12].
+Therefore, thereis nodoubtthat thisparticle exists andth at it belongstoascalar nonet ( σ(600),
+f0(980),a0(980),κ(800)) below 1 GeV. A problem may appear due to the fact that th ere is also
+a scalar nonet above 1 GeV. This has led to the assumption that the scalar nonet above 1 GeV
+should be understood as q¯qstates whereas the scalar nonet below 1 GeV should be underst ood
+asqq¯q¯qstates (see e.g. [4] and references therein). Other version s consider meson molecules and
+gluonic components (see [3–6,13–16] and references therei n). It is hard to see that strict criteria
+∗R´ esum´ e of an invited talk presented for the A2 collaborati on at MAMI (Mainz)
+1can be found giving proof of the validity of one model and excl uding the validity of an other.
+The best way to proceed is to start with a model for the low-mas s scalar mesons where q¯qis
+a3P0core state which may couple to other hadronic or gluonic confi gurations [15]. Then the
+essential properties as e.g. the two-photon width Γ( σ→γγ) may be understood in terms of
+theq¯qcore whereas other components may show up in hadronic reacti ons where scalar mesons
+appear as intermediate states.
+The present work is a continuation of a systematic series of s tudies [8,17–20] on the elec-
+tromagnetic structure of the nucleon, following experimen tal work on Compton scattering and
+a comprehensive review on this topic [21]. These recent inve stigations have shown [8,17–20]
+that a systematic study of all partial resonant and nonreson ant photo-excitation processes of
+the nucleon and of their relevance for the fundamental struc ture constants of the nucleon as
+there are the electric polarizability ( α), the magnetic polarizability ( β) and the backward spin-
+polarizability ( γπ) is essential for an understanding of the electromagnetic s tructure of the nu-
+cleon. In addition it has been found that the structure of the constituent quarks and their
+coupling to pseudoscalar and scalar mesons is important for the understanding of the electric
+and magnetic polarizabilities and of the backward spin-pol arizability. The main purpose of the
+present work is to prove that the method of calculating the t-channel contribution from the
+reactionγγ→σ→N¯Nwhere the properties of the σmeson are taken from the QLL σM is a
+precise procedure and largely superior to previous approac hes where the combination of the two
+reactionsγγ→σ→ππandγγ→σ→N¯Nis exploited.
+2 The dynamical linear σmodel on the quark level and the mass
+of theσ-meson
+In the following we give a r´ esum´ e of the dynamical linear σmodel on the quark level [7] which
+in short term may be named Quark-Level Linear σModel (QLL σM) [9].
+The Lagrangian of the Nambu–Jona-Lasinio (NJL) model has be en formulated in two equiv-
+alent ways [22–25]
+LNJL=¯ψ(i/ ∂−m0)ψ+G
+2[(¯ψψ)2+(¯ψiγ5τψ)2], (1)
+and
+L′NJL=¯ψi/ ∂ψ−g¯ψ(σ+iγ5τ·π)ψ−1
+2δµ2(σ2+π2)+gm0
+Gσ,(2)
+where
+G=g2/δµ2andδµ2= (mcl
+σ)2. (3)
+Eq. (1) describes the four-fermion version of the NJL model a nd Eq. (2) the bosonized version.
+The quantity m0= (mu+md)/2 denotes the average current quark mass for two flavors, ψthe
+spinor of constituent quarks with two flavors. The quantity Gis the coupling constant of the
+four-fermion version, gthe Yukawa coupling constant and δµa mass parameter entering into
+the mass counter-term of Eq. (2). The coupling constants G,gand the mass parameter δµare
+related to each other and to the σmeson mass in the chiral limit (cl), mcl
+σ, as given in (3). The
+relationδµ2= (mcl
+σ)2can easily be derived by applying spontaneous symmetry brea king toδµ
+in analogy to spontaneous symmetry breaking predicted by th e linearσmodel (LσM) for the
+mass parameter µentering into that model [26].
+2Using diagrammatic techniques the following equations may be found [7,25,27]
+M∗=m0+8iGNc/integraldisplayΛd4p
+(2π)4M∗
+p2−M∗2, M=−8iNcg2
+(mclσ)2/integraldisplayd4p
+(2π)4M
+p2−M2,(4)
+f2
+π=−4iNcM∗2/integraldisplayΛd4p
+(2π)41
+(p2−M∗2)2, fcl
+π=−4iNcgM/integraldisplayd4p
+(2π)41
+(p2−M2)2,(5)
+m2
+π=−m0
+M∗1
+4iGNcI(m2π), (6)
+I(k2) =/integraldisplayΛd4p
+(2π)41
+[(p+1
+2k)2−M∗2][(p−1
+2k)2−M∗2].
+The expression given on the l.h.s. of (4) is the gap equation w ithM∗being the mass of the
+constituent quark with the contribution m0of the current quarks included and Nc= 3 being
+the number of colors. The quantity Mon the r.h.s. of Eq. (4) is the mass of the constituent
+quark in the chiral limit. The l.h.s. of Eq. (5) represents an expression for the pion decay
+constant having the experimental value fπ= (92.42±0.26) MeV [12]. On the r.h.s. of Eq. (5)
+the corresponding expression for fcl
+πis given which is the same quantity in the chiral limit. The
+relation for the pion mass in Eq. (6) has no counterpart in the chiral limit because there the
+pion mass is zero.
+In principle the gap parameter M∗and through this the mass of the σmesonmσcan be
+determined using the l.h.s. relations of Eqs. (4) to (6). Thi s has been carried out in [27] leading
+tomσ≈2M∗≈668 MeV.
+A more elegant way to predict mσon an absolute scale introduced by Delbourgo and
+Scadron [7] is obtained by exploiting the r.h.s. of Eqs. (4) a nd (5). Making use of the identity
+(dimensional regularization [7,28])
+−i(mcl
+σ)2
+16Ncg2=/integraldisplayd4p
+(2π)4/bracketleftbiggM2
+(p2−M2)2−1
+p2−M2/bracketrightbigg
+=−iM2
+(4π)2(7)
+we arrive at
+(mcl
+σ)2=Ncg2M2
+π2, (8)
+and withmcl
+σ= 2Mat the Delbourgo-Scadron relation [7]
+g=gπqq=gσqq= 2π//radicalbig
+Nc= 3.63. (9)
+Theσ-meson mass corresponding to this coupling constant is
+mσ= 666.0 MeV, (10)
+where use has been made of m2
+σ= (mcl
+σ)2+ ˆm2
+π,fcl
+π= 89.8 MeV [31], M= 325.8 MeV and
+ˆmπ= 138.0 MeV. This result may also be compared with information from magnetic moments2.
+It has to be noted that the present version of the QLL σM is incomplete in the sense that
+the coupling of the σmeson to meson loops is not taken into account. These meson lo ops lead
+to contributions to the two-photon decay width Γ( σ→γγ) of theσmeson in addition to the
+dominantq¯qcontribution (see [9] and Table 1).
+2It should be noted that the magnetic moments lead to mσ≈2M∗≈664 MeV when averaged over the results
+for the proton and the neutron (see the second reference in [1 9]).
+33 The two-photon decay width of the σmeson
+In the quark model the σmeson has the structure
+|n¯n/angbracketright=1√
+2|u¯u+d¯d/angbracketright. (11)
+Then the principal contributions to the amplitude M(σ→γγ) comes from the up and down
+quark triangle diagrams [9], yielding (with Nc= 3)
+M(σ→γγ) =5αe
+3πfπVq(ξ), (12)
+Vq(ξ) = 2ξ[2+(1−4ξ)I(ξ)], (13)
+whereαe=e2/4π= 1/137.04,ξ= ˆm2
+q/m2
+σandI(ξ) is the triangle loop integral given by
+I(ξ)
+
+=π2
+2−2log2/bracketleftBig/radicalBig
+1
+4ξ+/radicalBig
+1
+4ξ−1/bracketrightBig
++2πilog/bracketleftBig/radicalBig
+1
+4ξ+/radicalBig
+1
+4ξ−1/bracketrightBig
+(ξ≤0.25),
+= 2arcsin2/bracketleftBig/radicalBig
+1
+4ξ/bracketrightBig
+(ξ≥0.25).(14)
+With the average constituent quark mass ˆ mq= 330 MeV and mσ= 666 MeV we arrive at a
+correction factor of Vq(ξ) = 1.025, whereas in the chiral limit, ˆ mq→Mandmσ→2M, we have
+Vq(ξ) = 1.0. We see that the possible correction due to the factor Vq(ξ) is small and may be
+disregarded in view of possible other uncertainties. Then w ithVq(ξ) = 1 and
+Γ(σ→γγ) =m3
+σ
+64π|M(σ→γγ)|2(15)
+we arrive at
+Γ(σ→γγ) = (2.6±0.3) keV. (16)
+The value for the two-photon decay width given in (16) leads t o an excellent agreement with
+the experimental electromagnetic polarizabilities of the nucleon and, therefore, is experimentally
+confirmed through this agreement. This is especially true fo r the electric polarizability αpof
+the proton which has the smallest experimental error, viz.∆αp/αp≈5%. Using ∆Γ( σ→
+γγ)/Γ(σ→γγ)≈2∆αp/αp≈10% the error given in (16) is obtained (see Table 2 for detail s).
+4 Properties of the reaction γγ→σ→N¯N
+Let us first discuss the kinematics of Compton scattering. Th e conservation of energy and
+momentum in nucleon Compton scattering
+γ(k,λ)+N(p)→γ′(k′,λ′)+N′(p) (17)
+is given by
+k+p=k′+p′(18)
+wherekandk′are the 4-momenta of the incoming and outgoing photon, λandλ′the respective
+helicities and pandp′the 4-momenta of the incoming and outgoing nucleon. Mandels tam
+variables are introduced through the relations
+s= (k+p)2= (k′+p′)2, (19)
+t= (k−k′)2= (p′−p)2, (20)
+u= (k−p′)2= (k′−p)2, (21)
+s+t+u= 2m2, (22)
+4wheremis the mass of the nucleon. At negative tthis quantity can be interpreted in terms of
+the c.m. scattering angle θs:
+sin2θs
+2=−st
+(s−m2)2. (23)
+Compton scattering may be described by invariant amplitude sAi(s,t) (i= 1−6) which are
+analytic functions in the two variables sandtand, therefore, may be treated in terms of
+dispersion relations. The third variable uis not taken into account because of the constraint
+given in (22). The degrees of freedom of the nucleon includin g the structure of the constituent
+quarks enter into the invariant amplitudes via a cut on the re al axis of the complex s-plane
+and in terms of pointlike singularities on the positive real axis of the t-plane. The s-channel
+cut contains the total photoabsorption cross section and th rough a decomposition in terms of
+excitation mechanisms the complete electromagnetic struc ture of the nucleon as seen in one-
+photon processes. The pointlike singularities on the posit ive realt-axis may be related to
+the structure of the constituent quarks which couple to all m esons with a nonzero meson-quark
+coupling constant. Of these the π0andtheσmeson are of special interest. Onthe one handthey
+couple to two photons with perpendicular and parallel plane s of linear polarization, respectively,
+and on the other hand they enter into the QLL σM as described in Section 2. In a formal
+sense the singularities on the positive real t-axis correspond to the fusion of two photons with
+4-momenta k1andk2and helicities λ1andλ2to form at-channel intermediate state |π0/angbracketrightor|σ/angbracketright
+from which – in a second step – a proton-antiproton pair is cre ated. The corresponding reaction
+may be formulated in the form
+γ(k1,λ1)+γ(k2,λ2)→¯N(p1)+N(p2). (24)
+In the present case the N¯Npair creation-process is virtual, i.e. the energy is too low to put the
+proton-antiproton pair on the mass shell. In the c.m. system the 3-momenta of the two photons
+are related by k1+k2= 0, leading to t= (ω1+ω2)2= (Wt)2whereWtis the total energy.
+Theπ0singularity enters into the invariant amplitude A2and theσsingularity into the
+invariant amplitude A1[29]. Let us first discuss the well known case of the π0pole contribution.
+The fixedθ=πdispersion relation is
+˜Aπ0
+2(s,t) =1
+π/integraldisplay∞
+t0Imt˜A2(s′,t′)dt′
+t′−t−i0(25)
+with the solution for a pointlike singularity [21,30]
+˜Aπ0
+2(t) =M(π0→γγ)gπNN
+t−m2
+π0τ3. (26)
+The analogous formula for the σmeson is
+˜Aσ
+1(t) =M(σ→γγ)gσNN
+t−m2σ. (27)
+The relation given in (26) is valid in the whole complex t-plane especially also in the physical
+range of the Compton scattering process at θ=πwhere kinematical constraints are absent.
+For (27) the same is true provided the intermediate state in t he reaction γγ→σ→N¯Nhas a
+definite mass of mσ= 666 MeV as conjectured here. The observation that the predi cted mass
+mσ= 666 MeV together with thepredicted two-photon widthΓ( σ→γγ) = 2.6 keV obtained for
+theσ=1√
+2(u¯u+d¯d) configuration is in perfectagreement with the experimenta l electromagnetic
+polarizabilities is one of the main achievements of the pres ent and our related previous works,
+thus supportingour supposition that this description of th eσmeson pole contribution is correct.
+54.1 Analysis of the γγ→σ→ππreaction
+Table 1 summarizes the results available for the two-photon width Γ(σ→γγ) obtained from
+the most recent analyses of the γγ→σ→ππreactions. For comparison we also give the
+present result obtained from the QLL σM model including the q¯qcomponent only and the full
+calculation of [9] where also meson loops are included. From the large number of very different
+resultsrecently publishedforthe γγ→σ→ππreaction it isdifficulttodecidewhether2 .6 or3.5
+given in lines 3 and 4 of Table 1 are in better agreement with th eγγ→σ→ππdata. However,
+we should keep in mind that the result (2 .6±0.3) keV is supported by the experimental electric
+polarizability αp(seeTable2). Theweighted average oftheresults1–8isΓ( σ→γγ) = (2.3±0.4)
+Table 1: Two-photon width of the σmeson from the γγ→ππreaction compared with the
+QLLσM. In line 3 the q¯qcontribution and information obtained from the electric po larizability
+αpof the proton is taken into account. In line 4 the full calcula tion including meson loops is
+taken into account.
+Γ(σ→γγ) method reference
+[keV] [MeV]
+2.6±0.3 QLLσMq¯qandαp present
+3.5 QLL σMq¯qand meson loops [9]
+4.1±0.3γγ→π0π0[13] (result 1)
+1.8±0.4γγ→π0π0[32] (result 2)
+2.1±0.3γγ→π0π0[32] (result 3)
+3.0±0.3γγ→π0π0[32] (result 4)
+1.68±0.15γγ→π0π0[33] (result 5)
+3.9±0.6γγ→π0π0,π+π−[34] (result 6)
+3.1±0.5γγ→π+π−[35] (result 7)
+2.4±0.4γγ→π+π−[35] (result 8)
+2.3±0.4 average of the results (1 −8)
+keV and thus consistent with the present result and also cons istent with the result of [9]. We
+consider this general agreement as quite satisfactory.
+4.2 Properties of the reaction γγ→σ→N¯Nand the t-channel part of the
+polarizability difference (α−β)t
+The pole at√sσ= 666 MeV has gained importance because it provides an easy ac cess to the
+calculation of the t-channel part of Compton scattering as given by the process γγ→σ→N¯N.
+This is illustrated in Figure 1. In a) and b) the t-channel pole contributions corresponding to
+theπ0andσmeson are compared with each other. These two poles differ by th eir locations at
+mπ0= 135.0 MeV and mσ= 666 MeV, respectively, and their two-photon widths. In add ition
+they differ by the fact that the pseudoscalar meson couples to t wo photons with perpendicular
+planes of linear polarization whereas the scalar meson coup les to two photons with parallel
+planes of linear polarization. Except for this there are no f urther differences in the two t-
+channel contributions. The σmeson pole contribution was originally used in [29] in conne ction
+with the prediction of differential cross sections for Compto n scattering. Its relation to the
+QLLσM was first described in [8]. Prior to this discovery [8] the ca lculation of the scalar t-
+channel contribution to nucleon Compton scattering had to r ely on the available information
+6b) c)cut
+σσσ
+a)π0
+Figure 1:π0andσpole graphs.
+on the two reactions γ→σ→ππandππ→σ→N¯Nwhere theσmeson was taken into
+account through the ππphase relation. This corresponds to the graph c) in Figure 1. Here the
+σpropagator has to be cut into two pieces for the purpose of the calculation. This procedure
+has also been applied by BEFT [36] to make predictions for the t-channel contribution of the
+difference (α−β)tof the electric and magnetic polarizability.
+In more detail the procedures of calculating ( α−β)tfrom the graphs b) and c) in Figure 1
+are given in Eqs. (28) and (29), respectively. Using the γγ→σ→N¯Nreaction the result
+(α−β)t=M(σ→γγ)gσNN
+2πm2σ= 15.2 (28)
+(in units of 10−4fm3) is obtained when inserting M(σ→γγ) = (5αe)/(3πfπ),mσ= 666 MeV
+andgσNN= 13.169±0.057 [37]. In case of graph c) in Figure 1 we restrict ourselves in the
+calculation of the t-channel absorptive part to intermediate states with two pi ons with angular
+momentum J≤2. Then the t-channel part of the BEFT sum rule [36] takes the form:
+(α−β)t=1
+16π2/integraldisplay∞
+4m2πdt
+t216
+4m2−t/parenleftbiggt−4m2
+π
+t/parenrightbigg1/2/bracketleftBig
+f0
++(t)F0∗
+0(t)
+−/parenleftbigg
+m2−t
+4/parenrightbigg/parenleftbiggt
+4−m2
+π/parenrightbigg
+f2
++(t)F2∗
+0(t)/bracketrightBig
+, (29)
+wheref(0,2)
++(t) andF(0,2)
+0(t) are the partial-wave helicity amplitudes of the processes N¯N→ππ
+andππ→γγwith angular momentum J= 0 and 2, respectively, and isospin I= 0. Though
+the quantities entering into Eq. (29) have a clear-cut defini tion the calculation is not easy. This
+is reflected by the long history of different approaches to get r eliable numbers for ( α−β)tbased
+on this equation. A description of these approaches is given in [21].
+Very recently [38] the experimental value of ( α−β)thas been used to make a prediction for
+the two-photon width Γ( σ→γγ) of theσmeson using Eq. (29), with the σmeson being a pole
+at 441-i272 MeV on the second second Riemann sheet. The resul t of this latter calculation is
+Γpole(σ→γγ) = (1.2±0.4) keV which still is compatible with the range of data listed in Table
+1 and, therefore, is in line with the suggestion made by Figur e 1 that the pole representation
+leads to a good approximation of the scalar t-channel contribution.
+5 Compton scattering and polarizabilities of the nucleon
+The differential cross section for Compton scattering may be w ritten in the form [39]
+dσ
+dΩ= Φ2|Tfi|2(30)
+7with Φ =1
+8πmω′
+ωin the laboratory frame and Φ =1
+8π√sin the c.m. frame. The quantity mis the
+mass of the nucleon, ωthe energy of the incoming photon, ω′the energy of the outgoing photon
+in the laboratory frame and√sthe total energy. For the following discussion it is conveni ent to
+use the laboratory frame and to consider special cases for th e amplitude Tfi. These special cases
+are the extreme forward ( θ= 0) and extreme backward ( θ=π) direction where the amplitudes
+for Compton scattering may be written in the form [39]
+1
+8πm[Tfi]θ=0=f0(ω)ǫ′∗·ǫ+g0(ω)iσ·(ǫ′∗×ǫ), (31)
+1
+8πm[Tfi]θ=π=fπ(ω)ǫ′∗·ǫ+gπ(ω)iσ·(ǫ′∗×ǫ). (32)
+In(31)f0(ω) is theforward scattering amplitudewith thetwo photonsin parallel planes of linear
+polarization, g0(ω) the forward scattering amplitude with the two photons in pe rpendicular
+planes of linear polarization, fπ(ω) the backward scattering amplitude with the two photons in
+parallel planes of linear polarization and gπ(ω) the backward scattering amplitude with the two
+photons in perpendicular planes of linear polarization.
+Following Babusciet al. [39]theequations (31) and(32)can beusedtodefinetheelectromag-
+netic polarizabilities and spin-polarizabilities as the l owest-order coefficients in an ω-dependent
+development of the nucleon-structure dependent parts of th e scattering amplitudes:
+f0(ω) =−(e2/4πm)q2+ω2(α+β)+O(ω4), (33)
+g0(ω) =ω/bracketleftbig
+−(e2/8πm2)κ2+ω2γ0+O(ω4)/bracketrightbig
+, (34)
+fπ(ω) =/parenleftbig
+1+(ω′ω/m2)/parenrightbig1/2[−(e2/4πm)q2+ωω′(α−β)+O(ω2ω′2)],(35)
+gπ(ω) =√
+ωω′[(e2/8πm2)(κ2+4qκ+2q2)+ωω′γπ+O(ω2ω′2)], (36)
+whereqeis the electric charge ( e2/4π= 1/137.04),κthe anomalous magnetic moment of the
+nucleon and ω′=ω/(1+2ω
+m).
+In the relations for f0(ω) andfπ(ω) the first nucleon structure dependent coefficients are
+the photon-helicity non-flip ( α+β) and photon-helicity flip ( α−β) linear combinations of the
+electromagnetic polarizabilities αandβ. In the relations for g0(ω) andgπ(ω) the corresponding
+coefficients are the spin polarizabilities γ0andγπ, respectively. The relations between the
+amplitudes fandgand the invariant amplitudes Ai[29,39] are
+f0(ω) =−ω2
+2π[A3(ν,t)+A6(ν,t)], g 0(ω) =ω3
+2πmA4(ν,t), (37)
+fπ(ω) =−ωω′
+2π/parenleftbigg
+1+ωω′
+m2/parenrightbigg1/2/bracketleftbigg
+A1(ν,t)−t
+4m2A5(ν,t)/bracketrightbigg
+, (38)
+gπ(ω) =−ωω′
+2πm√
+ωω′/bracketleftbigg
+A2(ν,t)+/parenleftbigg
+1−t
+4m2/parenrightbigg
+A5(ν,t)/bracketrightbigg
+, (39)
+ω′(θ=π) =ω
+1+2ω
+m, ν=1
+2(ω+ω′), t(θ= 0) = 0, t(θ=π) =−4ωω.′(40)
+For the electric, α, and magnetic, β, polarizabilities and the spin polarizabilities γ0andγπ
+for the forward and backward directions, respectively, we o btain the relations
+α+β=−1
+2π/bracketleftbig
+AnB
+3(0,0)+AnB
+6(0,0)/bracketrightbig
+, α−β=−1
+2π/bracketleftbig
+AnB
+1(0,0)/bracketrightbig
+,
+γ0=1
+2πm/bracketleftbig
+AnB
+4(0,0)/bracketrightbig
+, γ π=−1
+2πm/bracketleftbig
+AnB
+2(0,0)+AnB
+5(0,0)/bracketrightbig
+,(41)
+whereAnB
+iare the non-Born parts of the invariant amplitudes.
+85.1 Compton scattering and electromagnetic fields
+Polarizabilities may be measured by simultaneous interact ion of two photons with the nucleon.
+In case of static fields this may be written in the form
+H(2)=−1
+24παE2−1
+24πβH2, (42)
+where the quantity H(2)is the energy change in the electric and magnetic fields due to the
+polarizabilities. The first part of the r.h.s. of Eq. (42) is r ealized in experiments where slow
+neutrons are scattered in the electric field of a heavy nucleu s, leading to a measurement of the
+electric polarizability. Compton scattering in the forwar d and backward directions lead to more
+EE,
+H
+H,E
+E,
+,H
+H
+EE
+H H ,,
+,E
+E,
+HH(1) (2)
+(3)
+(4)
+Figure 2: Compton scattering viewed as simultaneous intera ction of two electric field vectors
+E,E′and magnetic field vectors H,H′for four different cases. (1) Helicity independent forward
+Compton scattering as given by the amplitude f0(ω). (2) Helicity dependent forward Compton
+scattering as given by the amplitude g0(ω). (3) Helicity independent backward Compton scat-
+tering as given by the amplitude fπ(ω). (4) Helicity dependent backward Compton scattering as
+given by the amplitude gπ(ω). Longitudinal photons can only provide two electric vecto rs with
+parallel planes of linear polarization as shown in the upper part of panel (1). In panels (2) and
+(4) the direction of rotation leading from EtoE′depends on the helicity difference |λp−λγ|.
+general combinations of electric and magnetic fields. This i s depicted in Figure 2. Panel (1)
+corresponds to the amplitude f0(ω), i.e. to Compton scattering in the forward direction with
+parallel electric and magnetic fields. This case can be compa red with Eq. (42). Electromagnetic
+scattering of neutrons corresponds to the upper part of pane l (1) where two electric vectors are
+shown. The cases described in panels (2) – (4) can be realized with real photons only. Panel
+(2) corresponds to helicity dependent Compton scattering i n the forward direction as described
+by the amplitude g0(ω). Panel (3) corresponds to the amplitude fπ(ω) and panel (4) to the
+amplitudegπ(ω). According to the definitions given in Eq. (41), panel (3) co rresponds to the
+case where a σmeson while attached to a constituent quark interacts with t he two photons and
+panel (4) to the case where a π0meson while attached to a constituent quark interacts with t he
+two photons.
+We have argued that electromagnetic scattering of neutrons corresponds to the upper part of
+panel (1). Furthermore, we have argued that the reaction γγ→σ→N¯Nmakes a contribution
+to the case of panel (3). At a first sight there may be a contradi ction to the observation
+that electromagnetic scattering of neutrons and Compton sc attering measure the same electric
+9polarizability. The explanation that there is no contradic tion is as follows. In terms of electric
+and magnetic fields the σmeson always makes a contribution if the two field vectors are either
+parallel or antiparallel. However, with α(σ) =−β(σ) these two parts cancel in the case of
+panel (1) for real photons but do not cancel for virtual photo ns at low neutron velocities where
+magnetic fields are absent.
+5.2 Composition of the Compton scattering amplitudes
+Thetotal Compton differential cross section is represented v ia the graphs3of Figure 3. Compton
+σ  f   a π  η  η0 00π
+π
+Ν ΝΝ ΝΝ ΝΝ Ν∗
+∗a) b) c)
+d) e) f)
+Figure 3: Graphs of processes contributing to Compton scatt ering by the nucleon3. a) Born
+terms, b) Compton scattering via nonresonant 1 πexcitation of the nucleon, c) Compton scat-
+tering via excitation of the resonant states of the nucleon, d) summary of other nonresonant or
+partly resonant excitation processes of the nucleon contri buting to Compton scattering. The
+correspondingcrossed graphshave beenomitted. e) pseudos calart-channel exchange, f) scalar t-
+channel exchange. Regge pole exchanges contributing in the forward direction are not explicitly
+shown.
+scattering takes place via Born terms a) without internal ex citation of the nucleon and non-Born
+terms b) to f) which make contributions to the polarizabilit ies. The graphs corresponding to
+thes-channel, viz.a) to d), have to be supplemented by crossed graphs which are n ot shown
+here. Graph f) is of special interest because it represents t he scalart-channel contribution. In
+addition to the contribution of the σmeson there are small contributions of the f0(980) and
+a0(980) mesons. Figure 4 gives an overview over the spin-indep endent (left panel) and spin-
+dependent (right panel) contributions to graph c) of Figure 3. The cross sections are dominated
+by theP33(1232),D13(1520) and F15(1680) resonances. The main difference between the spin-
+independent cross section (left panel) and the spin-depend ent cross section (right panel) is
+the enhancement of the P33(1232) contribution by a factor 1.26 which can be traced back to
+the E2/M1 ratio of the P33(1232) resonance [20]. Without the multipolarity mixing th e two
+resonance curves would be the same for the P33(1232) resonance. The nonresonant cross section
+corresponding to graph b) in Figure 3 has contributions from the single-pion E0+, (M,E)(1/2)
+1+
+andM(3/2)
+1−CGLN amplitudes with the E0+contribution being the largest. The polarizabilities
+of the nucleon have been measured with high precision and com pared with predictions from
+dispersion theory in several investigations. The most rece nt one is published in [20]. Therefore,
+it is not necessary to give an extensive coverage of the polar izabilities here but it is sufficient
+3Please note that the graphs are shown for illustration only w hereas the calculations are based on dispersion
+theory.
+1025050075010001250150017500100200300400500600
+ω   [MeV]  
+σ
+T   [µ 
+b]
+  
+proton
+25050075010001250150017500100200300400500600
+ω[MeV]σ3/2−σ1/2  [µ b] proton
+Figure 4: Resonant part of the total photo absorption cross s ection forσT=1
+2(σ1/2+σ3/2) and
+σ3/2−σ1/2. Thick line: Sum of all resonances. Thin lines: Contributin g single resonances.
+Table 2: Predicted polarizabilities for the proton compare d with experimental data. The unit
+is 10−4fm3(for details see [20])
+αp βp
+P33(1232) −1.1 +8.3
+other resonances +1.1 +0.2
+E0+nonresonant +3.2 −0.3
+other nonresonant +1.3 +1.2
+σ t-channel +7.6 −7.6
+f0,a0t-channel −0.1 +0.1
+sum +12.0 +1.9
+experiment +12 .0±0.6 +1.9∓0.6
+to present the essential results. The s-channel contributions αsandβsof the electromagnetic
+polarizabilities have been calculated from the multipole c ontent of the photoabsorption cross
+section whereas the t-channel contribution is given by the transition matrix ele mentM(σ→γγ)
+of Eq. (12), the σ-nucleon coupling constant gπNN=gσNN= 13.169±0.057 [37] and the σ-
+meson mass mσ= 666 MeV via αt=−βt=M(σ→γγ)gσNN/4πm2
+σ= 7.6. Predictions for
+partial contributions to the polarizabilities of the proto n are given in Table 2. The purpose
+of Table 2 is to show that the experimental data are well under stood in terms of excitation
+processes of the nucleon and that the σ-mesont-channel makes a dominant contribution.
+From Table 2 the following conclusions may be drawn: (i) the p rediction obtained for the
+two-photon width Γ( σ→γγ) from the QLL σM has been confirmed with a high level of precision
+through the excellent agreement of the experimental electr ic polarizability αpof the proton with
+the corresponding predicted quantity. (ii) an error for the prediction of Γ( σ→γγ) may be
+obtained from the experimental error of αpbeing ∆αp/αp≈5%. Since the σmeson contribution
+to the electric polarizability is by far the largest it is pos sible to relate the errors of αpand
+Γ(σ→γγ) to each other. Using ∆Γ( σ→γγ)/Γ(σ→γγ)≈2∆αp/αp≈10% the error given in
+Eq. (16) is obtained.
+116 Observation of the σ-meson contribution to the differential
+cross section for Compton scattering
+TheBEFT sum rule[36] may bederived from the non-Born (nB) pa rt of the invariant amplitude
+˜A1(s,u,t)≡A1(s,u,t)−t
+4m2A5(s,u,t) (43)
+by applying the fixed- θdispersion relation for θ=π
+Re˜AnB
+1=1
+πP/integraldisplay∞
+s0/parenleftbigg1
+s′−s+1
+s′−u−1
+s′/parenrightbigg
+Ims˜A1(s′,u′,t′)ds′
++1
+πP/integraldisplay∞
+t0Imt˜A1(s′,u′,t′)dt′
+t′−t(44)
+withsu=m4. Then the difference of the electromagnetic polarizabilitie s is given by
+α−β=−1
+2πRe˜AnB
+1(s=m2,u=m2,t= 0) = (α−β)s+(α−β)t. (45)
+A calculation analogous to the derivation of the BEFT sum rul e (see e.g. [21]) but for general
+sandtand for the pole representation of the t-channel part leads to a generalized BEFT sum
+rule in the form
+(α−β)s(ω) =1
+2π2P/integraldisplay∞
+ω0/radicalbigg
+1+2ω′
+mω′3+ω′2(m/2)
+ω′3+ω′2a(ω)+ω′b(ω)+c(ω)σfπ(ω′)dω′
+ω′2(46)
+(α−β)t(ω) =−1
+2πM(σ→γγ)gσNN
+t(ω)−m2σ(47)
+withσfπ(ω) =σ(ω,E1, M2,···)−σ(ω,M1, E2,···) and
+a(ω) =−4ω2+2mω+m2
+2(m+2ω), b(ω) =−2mω2
+m+2ω, c(ω) =−m2ω2
+2(m+2ω), t(ω) =−4ω2
+1+2ω
+m.(48)
+It should be noted that Eq. (47) follows from the reaction γγ→σ→N¯Nwhereσis a genuine
+q¯qstate. This means that the σmeson has a definite mass of mσ= 666 MeV as predicted by
+the QLLσM. A complex mass parameter or a mass distribution are exclud ed as explained in the
+text following Eq. (27).
+The generalized polarizabilities as defined in (46) and (47) are depicted in Figure 5. The
+solid curve starting at ( α−β)(0) =−9.4 represents the contribution of the P33(1232) resonance.
+Thesolid curve starting at ( α−β)(0) = +3.5 represents the contribution of the nonresonant E0+
+amplitude. The curves starting at ( α−β)(0) = +15.2 represent the t-channel contribution of the
+σ-meson calculated for different σ-meson masses. The upper curve (dark grey or red) has been
+calculated for mσ= 800 MeV, the middle curve (grey or green) for mσ= 600 MeV and the lower
+curve (light grey of blue) for mσ= 400 MeV. In principle also a contribution of the D13(1520)
+resonance has to be taken unto account. This contribution is not shown in order not to overload
+the figure. The effect of this contribution is to cancel the E0+contribution in the energy range
+from 400 to 700 MeV which is the most relevant part of the spect rum for the comparison with
+experimental data as discussed later. It is apparent that th e contribution of the σmeson enters
+via a constructive interference with the contribution of th eP33(1232) resonance. This strongly
+enhances the effects of the σmeson in the total differential cross section for Compton scat tering
+at large scattering angles in the range between 400 and 700 Me V.
+120200400600800-20-15-10-5051015
+ω   [MeV]
+(α−β)(ω)  
+[10   fm  ]-43
+Figure 5: Difference of generalized polarizabilities ( α−β)(ω) versus photon energy ω. Solid
+curve starting at ( α−β)(0) =−9.4: contribution of the P33(1232) resonance. Solid curve
+starting at ( α−β)(0) = +3.5: Contribution of the nonresonant E0+amplitude. Curves starting
+at (α−β)(0) = +15.2:t-channel contribution of the σ-meson calculated for different σ-meson
+masses. Uppercurve(darkgreyorred): mσ= 800MeV.Middlecurve(greyorgreen): mσ= 600
+MeV. Lower curve (light grey or blue): mσ= 400 MeV. Not shown is the contribution of the
+D13(1520) resonance which cancels the E0+contribution in the relevant energy range from 400
+to 700 MeV.
+Differential cross sections for Compton scattering by the pro ton have been measured at the
+high duty-factor electron facility MAMI (Mainz) [40,41]. I n this experiment it was possible to
+cover the total energy range from 250 MeV to 800 MeV and angula r range in the laboratory from
+30◦to 150◦in one experimental run. The result of this experiment is sho wn in Figure 6. The
+experimental data collected in angular intervals are given for the c.m. scattering angles 75◦, 90◦
+and 125◦and compared with predictions based on dispersion theory. A s in Figure 5 the mass of
+theσmeson has been varied using 800 MeV, 600 MeV and 400 MeV while k eeping the quantity
+(α−β)(0) constant. From kinematical reasons effects of the σmeson in the differential cross
+section for Compton are only expected in the backward direct ion and are largest at θ= 180◦.
+This is in agreement with the observations made in Figure 6. A tθc.m.= 125◦there are three
+different curves visible in the energy range between 400 MeV an d 700 MeV corresponding to the
+differentσmasses: 800 MeV (upper), 600 MeV (center) and400MeV (lower) . Thisapparently is
+in a complete agreement with our expectation from Figure 5. T his means that we have seen the
+contribution from the σmeson pole in the experimental differential cross section. Fu rthermore,
+the mass we have determined is mσ≈600 MeV in agreement with the prediction made by the
+QLLσM for the reaction γγ→σ→N¯N,viz.mσ= 666 MeV. This mass corresponds to the
+location of the pole in the t-channel of Compton scattering as outlined above.
+7 Discussion
+In the present paper we have shown that the evaluation of the r eactionγγ→σ→N¯Nwith
+properties of the σmeson as given by the QLL σM leads to predictions for the electromagnetic
+polarizabilities of the nucleon being valid with a high leve l of precision. This strongly confirms
+the supposition that the σmeson indeed enters into the polarizabilities via the γγ→σ→N¯N
+process with the σmeson having a mass of mσ= 666 MeV. An even more direct observation of
+theγγ→σ→N¯Nreaction has also been made through its large contribution t o the differential
+cross section for Compton scattering by the proton in the ene rgy range from ω= 400 MeV to
+700 MeV where such a large contribution is expected. On the qu ark level this process may be
+13200 300 400 500 600 700 800
+Eγ [MeV]050100150200250dσ/dΩ [nb/sr] TOKY8050100150200250dσ/dΩ [nb/sr] TOKY64
+ CORN63
+ TOKY78
+ CORN65
+ BONN81
+ MAMI97
+ LEGS9750100150200250300dσ/dΩ [nb/sr] TOKY80
+ BONN81
+ MAMI97
+ LEGS97 θc.m. 
+γ    = 75°
+θc.m. 
+γ    = 90°
+θc.m. 
+γ    = 125°
+Figure 6: Differential cross sections for Compton scattering by the proton versus photon energy.
+The three panels contain data corresponding to the c.m.-ang les of 750, 900and 1250. The three
+curves are calculated for different mass parameters mσ= 800 MeV (upper, dark grey or red),
+600 MeV (center, grey or green) and 400 MeV (lower, light grey or blue).
+understood as Compton scattering by the σmeson while being a part of the constituent quark.
+Prior to the discovery [8,29] of the σmeson pole representation of the scalar-isoscalar t-
+channel of Compton scattering with the σ-meson having a mass of mσ= 666 MeV, it was
+only possible to construct this quantity from a combination of the two reactions γγ→ππ
+andππ→N¯Nwith theσmeson being described via a pole on the second Riemann sheet
+at√sσ= (441−i272) MeV. Both procedures are correct and equivalent to each other. But
+certainly the calculation on the basis of Eq. (28) is easier t o perform than the calculation on
+the basis of Eq. (29). A further result of the present investi gation is that rather firm arguments
+are obtained for the two-photon width of the σ-meson being Γ( σ→γγ) = (2.6±0.3) keV, thus
+removing a large systematic uncertainty left over by differen t recent evaluations [13,32–35,38]
+of theσ-meson pole at√sσ= (441−i272) MeV.
+Theσmeson was originally introduced by Schwinger [1] in his gene ral attempt to explain
+symmetry breaking including also the electro-weak (EW) sec tor. Later on it was shown by
+Gell-Mann–Levy [2] that the σmeson may be used to explain the mass of the nucleon, whereas
+Higgs later on discussed the EW symmetry breaking in a form wh ich nowadays is generally
+accepted [42] and which enters into the standard model (SM). Similarities between the σmeson
+andtheHiggs bosonpossiblymay gobeyondthisinterchange o fnames. Sincethe σmesoniswell
+understoodon a quark level there have been attempts to use it as a guide for models of the Higgs
+boson (see e.g. [43,44] and references therein). At present [45] the SM Higgs boson is excluded
+at 95% C.L. for a mass lower than 114.4 GeV and for a mass in the r ange 160< mH<170
+14GeV. This result is only valid for the SM Higgs boson and does n ot contain information about
+a non-SM Higgs boson [45].
+It is interesting to note that in a new development of ChPT the possible role of the f0(600)
+meson has been discussed [46].
+Acknowledgment
+The author is indebted to J.A. Oller for providing valuable i nformation concerning his work.
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+arXiv:1001.0501v1  [astro-ph.EP]  4 Jan 2010Mon. Not.R. Astron. Soc. 000, 1–10 (2009) Printed 31 October 2018 (MN L ATEX style file v2.2)
+Tightconstraintson theexistenceof additionalplanetsar ound
+HD189733⋆†
+M. Hrudkov ´a1,2,3‡, I.Skillen2, C.R. Benn2,N. P. Gibson4,5, D. Pollacco4, D. Nesvorn ´y6,
+T. Augusteijn7, S. M. Tulloch2and Y. C.Joshi4,8
+1Astronomical Institute ofthe Charles University, VHoleˇ s oviˇ ck´ ach 2,CZ-180 00 Praha8, Czech Republic
+2Isaac Newton Group ofTelescopes, Apartado deCorreos 321, E -38700 Santa Cruz dela Palma,Canary Islands, Spain
+3Th¨ uringer Landessternwarte Tautenburg, Sternwarte 5,D- 07778 Tautenburg, Germany
+4Queen’s University Belfast, University Road, BT71NN, Belf ast, UK
+5School ofPhysics, University ofExeter, EX44QL,Exeter, UK
+6Department ofSpace Studies, Southwest Research Institute , 1050 Walnut Street, Suite 400, Boulder, CO 80302
+7Nordic Optical Telescope, Apartado de Correos474, E-387 00 Santa Cruz dela Palma, Canary Islands, Spain
+8Aryabhatta Research Institute ofObservational Sciences, Manora Peak,Nainital-263129, India
+31 October 2018
+ABSTRACT
+We report a transit timing study of the transiting exoplanet ary system HD 189733. In total
+we observedten transitsin 2006and2008with the2.6-mNordi cOpticalTelescope,and two
+transits in 2007 with the 4.2-m William Herschel Telescope. We used Markov-Chain Monte
+Carlo simulations to derive the system parameters and their uncertainties, and our results
+are in a good agreement with previously published values. We performed two independent
+analysesof transit timingresiduals to place uppermass lim its on putativeperturbingplanets.
+The results show no evidence for the presence of planets down to 1 Earth mass near the 1:2
+and2:1resonanceorbits,andplanetsdownto2.2Earthmasse snearthe3:5and5:3resonance
+orbitswithHD189733b.Thesearethestrongestlimitstodat eonthepresenceofotherplanets
+inthissystem.
+Keywords: planetarysystems–stars: individual:HD189733–techniqu es:photometric.
+1 INTRODUCTION
+Ground-based radial velocity and photometric transit surv eys have
+proved tobe the most successful methods for discovering exo plan-
+ets over the past decade, yielding more than 400 extrasolar p lanets
+discovered to date1. Most of the exoplanets detected are of Jupiter
+mass, but Earth-mass planets remain to be found. An addition al
+planet in a transiting system will perturb the motion of the t ran-
+siting planet, and the interval between the mid-eclipses wi ll not be
+constant.Deviationsfromthepredictedmid-transittimes canthere-
+forerevealthepresence ofotherbodiesinthesystem,orpla celim-
+its on their existence. Short-term variations can uncover t he exis-
+⋆Basedonobservations madewiththeNordicOpticalTelescop e, operated
+on the island of La Palma jointly by Denmark, Finland, Icelan d, Norway,
+and Sweden, in the Spanish Observatorio del Roque de los Much achos of
+the Instituto deAstrof´ ısica deCanarias.
+†Based on observations made with the William Herschel Telesc ope oper-
+ated on the island of La Palma by the Isaac Newton Group in the S panish
+Observatorio delRoquedelosMuchachos oftheInstituto deA strof´ ısica de
+Canarias.
+‡E-mail:marie@tls-tautenburg.de
+1TheExtrasolar Planets Encyclopedia: http://exoplanet.e utence of other planets (Agol et al. 2005; Holman &Murray 2005 ),
+moons (Sartoretti& Schneider 1999; Kipping 2009) and also T ro-
+jans (Ford&Holman 2007), whereas long-term variations res ult
+from orbital decay (Rasioet al. 1996) and from orbital prece ssion
+inducedbyanotherplanet,stellaroblatenessandgeneralr elativistic
+effects (Miralda-Escud´ e 2002; Heyl & Gladman 2007). Disco very
+ofadditional bodies canconstrain theoriesof planetarysy stem for-
+mation and evolution. In this paper we describe a transit tim ing
+studyof the transitingexoplanet system HD189733.
+The HD 189733 transiting system is one of the best studied
+systems from the ground. HD 189733 is a bright star with mag-
+nitudeV=7.67 which is orbited by a transiting Jupiter-mass planet
+in a period of ∼2.22 days(Bouchyet al. 2005), and which also
+has a distant mid-M dwarf binary companion (Bakos et al. 2006 a).
+In 2006 HD 189733 was observed with the MOST(Microvariabil-
+ity and Oscillations of STars) satellite and these data were used
+to search for the existence of other bodies in the system. Fir st,
+Crollet al. (2007) searched for transits from exoplanets ot her than
+the known hot Jupiter, with the result that any additional cl ose-in
+exoplanets on orbital planes near that of HD 189733b with siz es
+rangingfromabout1.7–3.5 R⊕,whereR⊕istheEarthradius,are
+ruled out. Second, an analysis of transit timing variations (TTVs)
+c/circlecop†rt2009 RAS2M. Hrudkov ´aet al.
+Table 1.Observations of the HD 189733 system. The UT date is the date o f the beginning of each night. The cycle number is in periods f rom the ephemeris
+given by Agol etal. (2009).For somenights theexposure time was changed during theobservations; this is indicated by th e second value in parentheses. The
+data rmsisper exposure for the ratio ofintensities ofthe ta rget and the comparison star. Thebarycentric mid-transit t imes of the HD189733 system aregiven
+with uncertainties defined as 68 per cent confidence limits.
+Telescope UT date Cycle no. CCDwindow size Exposure Data rms Mid-transit time O−CComment
+(pixels) (s) (mmag) (BJD - 2450000) (s)
+NOT 2006 July 18 -155 [1040:200] 2.5 2.9 3935.55805±0.00028 38 ±25
+NOT 2006 August 07 -146 [1040:200] 2.5 2.6 3955.52509±0.00014 26 ±12
+NOT 2006 August 27 -137 [1040:200] 2.5 (3.0) 2.7 3975.49194±0.00021 −2±18
+WHT 2007 August 17 +23 [1071,546] 10.0 4.6 4330.46305±0.00042 −79±36
+WHT 2007 September 17 +37 [1071,546] 3.0 (3.5) 4.4 4361.52352±0.00044 −43±38
+NOT 2008 June 07 +156 [1040:200] 3.5 (3.0) 2.6 4625.53404±0.00038 −35±33partial
+NOT 2008 June 18 +161 [1040:200] 3.5 2.3 4636.62768±0.00018 31 ±15
+NOT 2008 July 08 +170 [1040:200] 3.5 (4.0) 2.4 4656.59451±0.00012 1 ±11
+NOT 2008 July 17 +174 [1040:200] 3.5 (3.0) 3.4 4665.46951±0.00029 62 ±25
+NOT 2008 July 28 +179 [1040:200] 3.0 2.3 4676.56188±0.00019 18 ±16
+NOT 2008 August 26 +192 [1655:200] 3.5 2.7 4705.40332±0.00019 15 ±16
+NOT 2008 September 15 +201 [1655:200] 2.5 2.9 4725.37064±0.00053 28 ±46partial
+inthesedata hasbeencarriedout byMiller-Riccietal.(200 8)who
+found that there are no TTVs greater than ±45 s, which rules out
+planets of masses larger than 1 and 4 M⊕, whereM⊕is the Earth
+mass, inthe 2:3and1:2inner resonances, respectively, and planets
+greater than 20M⊕inthe outer 2:1resonance of the known planet
+and greater than 8M⊕inthe 3:2resonance.
+Analyses of transit times similar to Miller-Ricciet al. (20 08)
+have been carried out for other transiting planetary system s.
+Steffen&Agol (2005) found no evidence for a second planet in
+the TrES-1system, excluding planets down to Earthmass near the
+low-order, mean-motion resonances of the transiting plane t. Simi-
+larly,Gibsonet al.(2009a,b)foundnoevidenceforadditio nalplan-
+ets down to sub-Earth masses in the interior and exterior 2:1 reso-
+nances of the TrES-3andHAT-P-3systems.
+To measure times of mid-transits with sufficient accuracy to
+detect terrestrial mass planets requires high quality phot ometry,
+free of systematic effects. HD 189733 is known to have surfac e
+spots; Pont et al. (2007) observed two spot events in HST(Hubble
+Space Telescope) data when the flux during the transit change d by
+1 and0.4 mmag. The presence of surface spots on HD 189733
+complicates any transit timing analysis (Miller-Ricciet a l. 2008).
+The light curve can be distorted if the planet transits in fro nt of a
+spot or due to intrinsic variability of the star. The system p arame-
+ters and the mid-eclipse times derived can then be affected b y an
+inappropriate fittingmodel.
+Instrumental effects during transit ingress or egress can a lso
+influence the accuracy and determination of transit times. F or ex-
+ample, if the transit light curve is not properly normalised so that
+all data points in egress have a flux level that is slightly too high,
+the transittimewillbe determined tooearly.Correct norma lisation
+is especially problematic for partial transit light curves . Both in-
+strumentaleffectsandstellarvariabilitycancausethata lightcurve
+isimproperly normalised.
+It is also important to have a light curve that is well-sample d
+during both ingress and egress, because the transit timingi nforma-
+tion is contained in these parts. When using large telescope s for
+such a bright star, only short exposure times are needed to ge t suf-
+ficient signal-to-noise and to avoid saturation, and so the c adence
+of observation is higher. For a given data accuracy, higher c adence
+leads tomore accurately determined transittimes.
+In section §2 we describe our observations, and in section §3we present our data reduction. In section §4 we explain the tech-
+niquesusedtoestimateuncertaintiesinourdataandtomeas urethe
+system parameters. Finally, in section §5 we describe the 3-body
+simulations used toplace limitson the existence of other bo dies in
+the system, andwe conclude and discuss our results insectio n§6.
+2 OBSERVATIONS
+We observed eight full and two partial transits of HD 189733 w ith
+the2.6-mNordicOpticalTelescope(NOT),LaPalma,Spain,u sing
+ALFOSC (the Andalucia Faint Object Spectrograph and Camera ),
+and two full transits using the AG2 camera on the 4.2-m Willia m
+Herschel Telescope (WHT) of the Isaac Newton Group (ING), La
+Palma,Spain(Table1).
+ALFOSChasa 2048×2048back-illuminatedCCDwithscale
+0.19 arcsec /pixelandfieldof view (FOV) 6.5×6.5 arcmin2.To
+reduce the readout time of each exposure and the duty cycle of
+observation we windowed the CCD with the window sizes sum-
+marized in Table 1. We used a Str¨ omgren y filter to minimize ef -
+fects of colour-dependent atmospheric extinction on the di fferen-
+tialphotometry andtheeffectoflimbdarkening onthetrans itlight
+curves.Wedefocusedthetelescopetypicallyto 3.4arcsec,spread-
+ing the light inside full width at half maximum (FWHM) of the
+Point Spread Function (PSF) over ∼250 pixels, in order to mini-
+mize the impact of pixel-to-pixel sensitivity variations, and to pre-
+vent saturation. Exposure times were chosen to keep counts b elow
+50,000 per pixel to avoid saturation of features such as hot s pots
+and speckles in the defocused stellar images, and to ensure d ata
+linearity. The typical exposure time for the NOT data was 3 s(Ta-
+ble 1).
+AG2 is a frame-transfer CCD mounted at the WHT’s folded
+Cassegrainfocus, based onan ING-designed autoguider head . The
+FOV is3.3×3.3 arcmin2and the scale is 0.4 arcsec/pixel. We
+used a Kitt Peak R filter and defocused the telescope to 10 and
+12 arcsec for the two nights, spreading the FWHM-light over ∼
+490and700pixels,respectively.Thecorrespondingexposu retimes
+were 10and 3s.
+Themid-timeofeachexposure wasconvertedtotheBarycen-
+c/circlecop†rt2009 RAS, MNRAS 000,1–10Limitson additionalplanetsinHD 189733 3
+tric Julian Date (BJD) using the program BARCOR2. We use BJD
+throughout this paper, because for this system the Heliocen tric Ju-
+lianDatewould accumulate anerror of up to4seconds.
+3 DATA REDUCTION
+Bias subtraction, flat-fieldcorrection and aperture photom etry was
+performed usingstandard IRAF3procedures.
+To ensure a signal-to-noise in excess of 1,000 in our
+Str¨ omgreny-filterflat fieldsfor the NOT data we generated a m as-
+ter flat field for each night using individually weighted norm alised
+flat fields from the entire observing season combined withwei ghts
+W= 1−D/S, whereDis the time interval between each night
+and date of observation, and Sis the season length. Applying flat-
+fieldcorrections has onlyaminor effect onthe resultingNOT pho-
+tometry, because of the heavilydefocused PSF.
+For the WHT data we determined master flat fields with a
+signal-to-noise greater than 1,000 for both nights. Howeve r, we
+identified a position-angle dependent scattered light comp onent in
+the flatfields,whichintroduced systematic noise inour WHTp ho-
+tometry. Therefore we didnot apply flat-fieldcorrections.
+We used the star 2MASS 20003818+2242065 as our com-
+parison star for the WHT data. In our NOT data there are
+two available comparison stars, 2MASS 20003818+2242065 an d
+2MASS 20003286+2241118. We found the ratio of their mea-
+sured intensities varies by a few mmag with time, as the tele-
+scope tracks across the meridian. This variation correlate s with
+small drifts in the positions of the stars on the CCD, suggest -
+ing that some light is being lost from the aperture around one of
+the stars due to the wings of the PSF drifting out of that aper-
+ture. A similar variation is seen for the ratio of the intensi ties of
+2MASS20003286+2241118 andout-of-transitHD189733,butn ot
+for2MASS20003818+2242065 andHD189733,suggestingthati t
+islightfrom2MASS20003286+2241118 whichisbeinglost.Th is
+star is the farther of the two from HD 189733, and we conclude
+that the variation in measured intensity is due to a combinat ion of
+the small drifts in stellar position on the CCD, and the varia tion
+of the defocused PSF across the FOV. We therefore used only th e
+comparison star whichis closer toHD189733.
+We used circular, equal diameter, photometric apertures fo r
+bothHD189733andthecomparisonstar.Arangeofaperturesi zes
+was tried and that producing the minimum noise in the out-of-
+transit data was adopted and fixed during each night. The aper ture
+radius for all stars ranged from 18 – 29 pixels for different N OT
+nights and the typical FWHM was around 18 pixels ( 3.4 arcsec).
+For the two WHT nights the aperture radius was 28 and 30 pixels ,
+respectively, and the corresponding FWHM was 25 and 30 pixel s
+(10and12arcsec ).
+We ensured the apertures tracked small drifts in the stellar
+positions on each image by using a large centroiding box of si ze
+4×FWHM. During each night drifts in the stellar positions on
+the CCD were less than 7 (NOT) and 4 pixels (WHT). The sky
+background was subtracted using an estimate of its brightne ss de-
+termined within an annulus centred on each star with a width o f
+10pixels.Foreachnight,differentialphotometrywascomp utedby
+2http://sirrah.troja.mff.cuni.cz/˜mary
+3The Image Reduction and Analysis Facility (IRAF) is distrib uted by the
+National Optical Astronomy Observatories, which are opera ted by the As-
+sociationofUniversitiesforResearchinAstronomy,Inc., undercooperative
+agreement with the National Science Foundation.taking the ratio of counts from HD 189733 to the counts from th e
+comparison star.We normalisedour data usinglinear fitstha t were
+computedtogetherwithothersystemparametersasdescribe din§4.
+The normalised unbinned NOT light curves and binned WHT
+light curves, averaged into 10-second bins to have the simil ar ca-
+dence as the NOT data, are shown in Fig. 1 along with their best -
+fittingmodels, residuals and data error bars, as derived in §4.
+4 LIGHT-CURVEMODELLING
+To estimate the system parameters we used a parametrized mod el
+whereweassumedacircularorbitaroundthecentreofmassto cal-
+culatethenormalisedseparation, z,oftheplanetandstarcentresas
+afunctionoftime.TheanalyticformulasofMandel & Agol(20 02)
+wereusedtocalculatethefractionofthestellarfluxoccult edbythe
+planet using zand the planet-to-star radius ratio, ρ. We assumed a
+quadratic limbdarkening law:
+Iµ
+I0= 1−u1(1−µ)−u2(1−µ)2, (1)
+whereIis the intensity, µis the cosine of the angle between the
+lineofsightandthenormaltothestellarsurface,and u1andu2are
+the linear and quadratic limb darkening coefficients. For th e NOT
+data we allowedthe limbdarkening coefficients tobe freepar ame-
+ters, inorder toinclude possible errors in the limbdarkeni ng coef-
+ficients intoour final system parameters and mid-transit tim es. For
+the WHT data we adopted values u1= 0.4970andu2= 0.2195
+from the tables of Claret (2000) and fixed them in the subseque nt
+analysis.ThesecorrespondtotheJohnsonRfilterwhichhass imilar
+characteristics as the KittPeakRfilterused.
+To compute our model we folded all the NOT light curves of
+full transit except for the night 2008 July 17 which displays obvi-
+oussystematicchangesduringtransit.Inourphotometrywe cannot
+easily distinguish spot effects from systematic instrumen tal errors;
+todosowouldrequiretheinstrumentalsystematicnoisetob emuch
+less than the predicted spot signatures. We fitted simultane ously
+planetary and stellar radius, RpandR⋆, respectively, the orbital
+inclination, i, two limb darkening coefficients, u1andu2, transit
+time,T0,n, and additional two parameters for each night n– the
+out-of-transit flux, foot,n, and a time gradient, tGrad,n. These two
+parameterswereallowedtobefreetoaccountforanynormali sation
+errorsinthedata.Foreachchangeof R⋆,thestellarmass, M⋆,was
+recomputed using the scaling relation R⋆∝M1/3
+⋆. We fixed the
+planetarymass value Mp= 1.15±0.04MJ(Bouchy etal.2005),
+adopted a period P= 2.21857503 ±0.00000037 d (Agol et al.
+2009), and using Kepler’s third law we updated the orbital se mi-
+major axis for eachchoice of M⋆.
+We ran Markov-Chain Monte Carlo (MCMC) simulations
+(Tegmarketal. 2004; Ford 2006; Holmanet al. 2006) with the
+Metropolis–Hastings algorithm (Ford 2005) to estimate the best-
+fitting parameters and their uncertainties. From an initial point, a
+chain is generated by iterating a jump function, which adds a ran-
+domvalueselectedfromaGaussiandistributionwithmean0a nda
+standard deviation 1, scaled by a factor specific for each par am-
+eter so that ∼44per cent of each parameter sets are accepted
+(Gelmanetal. 2003; Ford 2006). In each step of the generated
+chain the χ2fitting statistic for old and new parameter values is
+computed:
+χ2=NDOF/summationdisplay
+i=1/bracketleftbigg
+fi(obs)−fi(theor)
+σi/bracketrightbigg2
++(M⋆−M0)2
+σ2
+M0.(2)
+c/circlecop†rt2009 RAS,MNRAS 000, 1–104M. Hrudkov ´aet al.
+Figure 1. Differential photometry of the HD 189733 system over-plott ed with the best-fitting model (solid line) from the MCMC fit. T he residuals and 1σ
+error bars are also plotted, offset by a constant flux for clar ity. The phase was computed using best-fitting transit times presented in Table 1. The photometry
+for NOTdata is unbinned and for WHTis binned in time with 10 se cond bins to give thesimilar cadence as for NOTdata for clari ty.
+c/circlecop†rt2009 RAS, MNRAS 000,1–10Limitson additionalplanetsinHD 189733 5
+Herefi(obs)is the flux observed at time i,σiis the correspond-
+ing uncertainty, fi(theor)is the flux calculated using formulas of
+Mandel & Agol(2002)and NDOFisthenumber ofmeasurements
+ineach light curve. The new parameter is then accepted if its χ2is
+lowerthanthatforthepreviousparameter,oracceptedwith aprob-
+abilityp= exp/parenleftbig
+−∆χ2/2/parenrightbig
+if itsχ2is higher. The second term in
+Eq. (2) is a Gaussian prior placed on M⋆, whereM0= 0.82 M⊙
+andσM0= 0.03 M⊙is the stellar mass and its uncertainty, es-
+timated from stellar spectra by Bouchy etal. (2005). This en sures
+that errors in the stellar mass, which are the greatest sourc e of un-
+certainty when deriving the system parameters and transit t imes,
+are taken intoaccount.
+The scale factors were chosen so that ∼44per cent of pa-
+rameter sets were accepted (Gelmanet al. 2003; Ford 2006). F or
+each simulation we created 10 independent chains, with leng th at
+least 100,000 points per chain to ensure convergence. Each c hain
+was initiated by a parameter that was within ±5σof a previously
+known best-fittingparameter value using the estimatedunce rtainty
+σ. The first 20 per cent of each chain was discarded to minimize
+theeffectoftheinitialconditions.Wecheckedconvergenc e ofgen-
+eratedchainsusingtheGelman& Rubin(1992)Rstatisticand cre-
+atedchains until R <1.03, a good sign ofconvergence.
+To estimate appropriate error bars in our data accounting
+for any correlated noise, we used a procedure similar to that of
+Gillonetal. (2006) and Naritaet al. (2007). We assigned the same
+error bars to all the data points including only Poisson nois e. An
+initialMCMC analysis of the folded NOTlight curves was used to
+estimatetheparameters Rp,R⋆,i,u1,u2,T0,n,foot,nandtGrad,n.
+The first model light curve was used to find the differences be-
+tween the data and the model for each individual night. Then w e
+rescaled the error bars to satisfy the condition χ2/NDOF= 1.0,
+whereNDOFis the number of measurements in each light curve.
+For the night of 2008 July 17, the nights of the two partial tra n-
+sits (2008 June 07 and 2008 September 15) and for the WHTlight
+curves (2007 August 17 and 2007 September 17) we adopted our
+first model and ran MCMC analysis to find initial parameters T0,
+footandtGradfor each night independently. Then we rescaled the
+error bars similarly as before. We assume that our initial mo del is
+a good description of the light curve. Compared to this model , we
+foundthatfortheNOTdataerrorsarehigherbyfactorsof2.3 –3.4
+than errors including only Poisson noise, and for the WHTdat a by
+factors of4.6and4.4forthe twonights, respectively. Thed atarms
+errorsperexposurearepresentedinTable1.Thepredictedr msdue
+tophotonnoise,whichisdominatedbythefaintercompariso nstar,
+and to atmospheric scintillation, is ∼2.5 mmag for the NOT data
+and∼3mmag for the WHTdata.
+The amplitude of systematic trends inthe photometry was es-
+timated from the standard deviation over one residual point ,σ1,
+andfromthestandarddeviationoftheaverageoftheresidua lsover
+Nsuccessive points, σN. We solved the following system of two
+equations given byGillonetal. (2006):
+σ2
+1=σ2
+w+σ2
+r, (3)
+σ2
+N=σ2
+w
+N+σ2
+r, (4)
+to obtain the amplitude of the white noise, σw, which is uncorre-
+latedandaveragesdownas (1/N)1/2,andtherednoise, σr,which
+is correlated and remains constant for specified N. The error bars
+were then adjusted by multiplying by [1 +N(σr/σw)2]1/2and
+these rescaled uncertainties were used for the subsequent fi tting
+procedure.Toaccountproperlyforthesystematicerrors,t heresult-
+ingmultiplyingfactorwascomputedastheaverageofvalues usingTable 2. System parameters of HD 189733. The uncertainties are 68 per
+cent confidence limits.
+Parameter Symbol Value Units
+Planet radius Rp1.142±0.014 R J
+Star radius R⋆0.755±0.009 R ⊙
+Orbital inclination i 85.70±0.03 deg
+Planet/star radius ratio ρ0.1556±0.0027
+Totaltransit duration Td1.807±0.023 h
+Impact parameter b 0.667±0.009
+differentNin the range 15 – 30 minutes (the typical time-scale of
+ingress and egress).
+Tocreateourfinalmodel,weproceededasbeforebutthistime
+including systematic noise in our data and therefore proper ly esti-
+mating parameter uncertainties. We ran MCMC using the folde d
+NOT light curves and fitting the parameters as described earl ier.
+We created 10 chains, each with length 2,000,000 points in or der
+toachieveconvergence. Ultimately,weusedourfinalmodelt ofind
+individual mid-eclipse times and two normalisation parame ters for
+thenightof2008July17,thenightsofthetwopartialtransi ts(2008
+June 07 and 2008 September 15) and for the WHT light curves
+(2007 August 17 and2007 September 17).
+5 RESULTS
+The final system parameters are presented in Table 2 and are
+consistent within ∼2σerror bars with the previously published
+values (Bakos et al. 2006b; Pontet al. 2007; Winnetal. 2007;
+Miller-Ricciet al.2008).The resultinglimbdarkening coe fficients
+for the NOTdata were u1= 0.46±0.10andu2= 0.35±0.13.
+The final barycentric transit times can be found in Table 1.
+The uncertainties are defined as 68 per cent confidence limits . To
+compute the observed-minus-calculated values ( O−C) we used
+the ephemeris given by Agol etal. (2009):
+Tc(E) = HJD(2454279 .436741±0.000023) + (5)
+(2d.21857503 ±0d.00000037) ×E.
+Theresulting O−Cresidualstogetherwithalltheotherpreviously
+published values (Bakos et al. 2006b; Pont etal. 2007; Winne t al.
+2007; Miller-Riccietal. 2008; Knutson et al. 2009) are plot ted
+in Fig. 2. Our observations did not bring any refinement of the
+ephemeris and we confirm that presented by Agol et al. (2009).
+For the night 2006 August 07 a transit timing measurement of
+HD 189733 was also presented by Miller-Riccietal. (2008) fr om
+theMOSTdata and it is consistent within 2σerror bars with our
+measurement.
+5.1 Transit timingvariations analysis
+Forallourobservations whichspanmorethantwoyears,them ean
+O−C= 5±38 s, where the quoted error is the rms scatter in
+theO−Cvalues and is slightly larger than the average O−Cun-
+certainty∼25s. None of our O−Cmeasurements is a significant
+outlier.Thetwolargest O−Cvaluesforthenights2007August17
+and2008July17coincide withobvious systematicchanges du ring
+thetransit(see Fig.1)andbothhavethe same orlargeruncer tainty
+than the average value. Therefore the rms scatter in the O−Cval-
+ues of 38 s is a good estimate for placing limits on the presenc e of
+other planets inthe system.
+c/circlecop†rt2009 RAS,MNRAS 000, 1–106M. Hrudkov ´aet al.
+-100 0 100 200-100-50050100
+-300 -200 -100 0 100 200-600-400-2000200400
+-150 -140 -130-1000100200
+Cycle number
+Figure2. Top:NOTandWHT O−Cresiduals ofmid-transittimesoftheHD189733systeminclu ding bothpartial(starsymbol)andfulltransits(filled cir cle
+symbol).Middle:Previously published values plotted together withNOTandW HTresults.Filled squares:Bakos etal.(2006b),ground-ba sed; filledtriangles:
+Pontet al.(2007), HST;open squares:Winn etal. (2007),ground-based; open circl es: Miller-Ricci et al.(2008), MOST;open triangles: Knutson et al. (2009),
+Spitzer; filled circles and stars: this work. Bottom:The same as the middle but zoomed for clarity. The cycle numbe r is in periods from the ephemeris given
+by Agolet al. (2009). A horizontal line is plotted in each pan el atO−C= 0to guide the eye. Our timing measurements are the most accura te from known
+ground-based observations.
+c/circlecop†rt2009 RAS, MNRAS 000,1–10Limitson additionalplanetsinHD 189733 7
+We used this conclusion to place mass limits on the exis-
+tence of planets on orbits interior and exterior to HD 189733 b.
+First, we selected the mass, semimajor axis and eccentricit y of
+the putative perturbing planet. The orbital inclination wa s set so
+that HD 189733b and the perturbing planet have coplanar orbi ts.
+The two-planet system was then numerically integrated usin g the
+Bulirsch-Stoer integrator (Presset al. 1992). We determin ed all
+mid-transit times of HD 189733b over a time-span of 500 days,
+an interval long enough to cover at least 14 orbits of all pert urbing
+planets wecanexclude, and usedthese datatoestimateTTVs. The
+mass, initial semimajor axis and initial eccentricity of th e perturb-
+ingplanetwerevariedtodeterminetheTTVamplitudefordif ferent
+planetary configurations.
+Fig. 3 shows the range of the inner and outer planet’s orbits
+thatproduceTTVssmaller/largerthan ±38sandarethuscompati-
+ble/incompatiblewithourTTVobservationsoftheHD189733 sys-
+tem. The shaded area in Fig. 3 excludes a range of possible ecc en-
+tricities and semimajor axes for a putative 1 Earth-mass (to p) and
+2 Earth-masses (bottom) inner (left) and outer (right) plan et in the
+system.Basedonthisanalysis,our observations oftheHD18 9733
+system show no evidence for the presence of planets down to 1
+Earthmassinthe2:1,3:2and5:3exteriorresonance orbits, planets
+down to 1 Earth mass in the 1:2, 1:3, 2:3 and 2:5 interior reso-
+nance orbits,andplanetsdownto2Earthmasses inthe1:4int erior
+resonance orbit with HD 189733b. However, not all of these re s-
+onant orbits are Hill stable. We computed Hill stability acc ording
+to Eq. (21) of Gladman (1993) for both inner and outer perturb ing
+planet and displayed the result in Fig. 3 using thin solid lin e. For
+theinner/outerperturbingplanetalltheorbitsontheleft /righttothe
+thin solid line are Hill-stable, which means that close appr oaches
+between two planets are forbidden. For the rest of the parame ter
+space the Hill stability of the system is unknown; the system still
+maybe Hill-stable.
+Nesvorn´ y (2009) showed that the TTV signal can be signifi-
+cantly amplified for planetary systems with substantial orb ital in-
+clinations of the transiting and perturbing planet and/or i n the case
+of transiting planet in an eccentric orbit with an anti-alig ned orbit
+of the perturbing planetary companion. Therefore for most o rbital
+architectures ofexoplanetary systemswe determinethe per turber’s
+uppermassfromourTTVsundertheassumptionofcoplanarorb its
+of transitingandperturbing planets.
+However, the above mentioned analysis does not take intoac-
+count time sampling of our measured transit times and their u n-
+certainties. It is possible to have a system whose TTV amplit ude
+exceeds 38 s but remains consistent with the available trans it tim-
+ingdata.Toassurethatthelimitsonadditionalplanetspre sentedin
+this paper are not overestimated, inaddition toour previou s analy-
+siswecomparedmodeltimingresidualsagainstthetransitt imesto
+place upper mass limits for a putative perturbing planet. We used
+the same procedure as Gibson etal. (2009a,b), where more det ails
+can be found. To compute model timing residuals we integrate d
+the equations of motion for a three body system using a 4th-or der
+Runge-Kuttamethod,withthefirsttwobodiesrepresentingt hestar
+andplanet of theHD 189733 system, andthethirdbody represe nt-
+ing a putative perturbing planet. The transit times were ext racted
+whenthestarandtransitingplanetwerealignedalongthedi rection
+of observation, and the residuals from a linear fit were taken to be
+the model timing residuals. For each model, TTVs were extrac ted
+for 6 equally spaced directions of observations, and we simu lated
+3 years of TTVs to cover the full range of observations. The re -
+sulting TTVs were then compared to transit times presented i n the
+middle panel of Fig. 2, i. e. all available transit times. Due tocom-putational limitations we assumed that the amplitude of the timing
+residuals scales proportionally to the perturber mass (Ago l et al.
+2005; Holman& Murray 2005), that a perturber has circular or bit
+andthat the orbits of the planets are coplanar.
+Wecreatedmodelswithperiodratiosintherange0.2–5.0,in -
+creasingthesamplingaroundthe interior1:2andexterior2 :1reso-
+nances.Themaximumallowedmassforeachmodelwascalculat ed
+asinGibsonet al.(2009a,b).Wescaledtheperturbermassun tilthe
+χ2of the model fitincreased bya value ∆χ2= 9(Steffen& Agol
+2005) from that of a linear ephemeris. Then we minimized the χ2
+along the epoch, and rescaled the perturber mass again until the
+maximum allowed mass was determined. This was repeated for
+each direction of the observation, and the maximum mass foun d
+wassetasourupper masslimit.Thisprocesswasrepeatedtwi ceto
+verifyourassumptionthatthetimingresidualsscalepropo rtionally
+withthe mass of the perturbing planet.
+Theresultinguppermasslimitsareplottedasafunctionoft he
+periodratioinFig.4.Thesolidlinerepresentstheupperma sslimits
+from our three-body simulations, and the horizontal dashed line
+shows an Earth-mass planet. Based on this analysis, the avai lable
+data were sufficiently sensitive to probe for masses as small as 0.2
+and0.15 M⊕neartheinterior1:2andexterior2:1resonances with
+HD 189733b, respectively. The corresponding upper masses n ear
+the3:5and5:3resonances withHD189733b are2.2and 0.54M⊕.
+In the rest of the space outside the region between the 2:3 and 3:2
+resonanceswithHD189733btheuppermasslimitsareoftheor der
+of a few tens of Earth masses to a few Jupiter masses. However,
+these upper mass limits result from the assumption of a circu lar
+orbitofaperturber.EccentricorbitsmayleadtosmallerTT Vs,and
+hence planets larger than our upper mass limits in eccentric orbits
+couldexistintheseregions.Unfortunately,accountingfo reccentric
+orbits is computationally unfeasible using these models du e to a
+large parameter space. Thus real upper mass limits of a pertu rber
+in a low eccentric orbit can be as much as an order of magnitude
+larger(Gibson etal. 2009b).
+We also consider the possible presence of Trojans in the sys-
+tem.AccordingtoFord& Holman(2007)transittimesarethes ame
+for a system without a Trojan and for a system where the tran-
+siting planet and Trojan have equal eccentricities and the T rojan
+resides exactly at the Lagrange L4/L5 fixed point. TTV analys is
+alone is not suitable for constraining the presence of Troja ns in
+transiting systems. However, a comparison of the photometr ically
+observed transit time and the transit time calculated from t he ra-
+dial velocity data assuming zero Trojan mass can reveal a Tro jan
+or place upper limits on its mass. Such an analysis was done by
+Madhusudhan & Winn(2009)whofound anupper limitof 22M⊕
+for a Trojan in the HD 189733 system. In addition, Crollet al.
+(2007) searched for Trojan transits in MOSTphotometry, assum-
+ingsimilarinclinationsoftheTrojan’sandtransitingpla net’sorbits,
+andconcludedthatTrojanswitharadiusabove 2.7R⊕shouldhave
+beendetectedwith95percentconfidence.Usingameandensit yof
+ρ∼3000 kgm−3, thiscorresponds to 11 M⊕.We usedEq. (1) of
+Ford& Holman (2007) to estimate what Trojan’s mass can be ex-
+cluded inthe system based on 38s rms of our TTVs.However, the
+amplitudeoftheangulardisplacementofaputativeTrojanf romthe
+Langrangepointisnotknown.Iftheselibrationamplitudes aresim-
+ilar as for Trojans orbiting near the Sun-Jupiter Langrange points,
+that is 5 – 30 deg (Murray & Dermott 2000), our TTVs show no
+evidence for Trojans withmasses higher than 5.3 M⊕.
+For an Earth-mass exomoon in a circular orbit about
+HD 189733b Kipping (2009) predicted TTV amplitude of 1.51 s
+and transit duration variation (TDV) amplitude of 2.94 s. In creas-
+c/circlecop†rt2009 RAS,MNRAS 000, 1–108M. Hrudkov ´aet al.
+Figure 3. A numerical survey of the HD 189733 system showing 38-s TTVs c aused by an inner (left plots) and outer (right plots) 1 Earth -mass planet
+(m2= 3×10−6M⊙, top) and 2 Earth-masses planet ( m2= 6×10−6M⊙, bottom). The shaded area excludes a range of possible eccen tricities and
+semimajor axes for a putative 1 and 2 Earth-masses inner/out er planet in the system based on our observational non-detec tion of TTVs greater than ±38 s.
+We do not display plots for Jupiter-mass planets as these wou ld easily be detected in radial velocity searches. The thick solid line shows a boundary where a
+collision between the two planets can occur. It is defined so t hat the apocentre/pericentre of the inner/outer perturbin g planet equals to the semi-major axis of
+thetransiting planet. Thethin solid linerepresents aHill stability computed according toGladman (1993).Onthetopo ftheupper panel weindicate themajor
+resonances of the putative perturbing planet and HD 189733b .
+c/circlecop†rt2009 RAS, MNRAS 000,1–10Limitson additionalplanetsinHD 189733 9Upper Mass / MJ
+Upper Mass / MEarth
+Period Ratio10-310-210-1100101
+0.2 1.0 5.010-1100101102103104
+Figure 4. Upper mass limits on a putative second planet in the HD 189733 system as a function of period ratio based on the comparison o f model timing
+residuals and all available transit times. The solid line re presents the upper mass found using three-body simulations . The horizontal dashed line shows an
+Earth-mass planet. Thegrey area isthe region wherean Earth -mass perturbing planet is not guaranteed to beHill stable.
+ing the eccentricity of the moon’s orbit decreases TTV ampli tude,
+butincreasesTDVamplitude.However,fortheHD189733syst em
+these variations are toosmalltobe detectable inour data.
+6 CONCLUSIONSANDDISCUSSION
+Miller-Riccietal. (2008) found no TTVs greater than ±45 sin
+MOSTdata,andexcluded super-Earthsof masseslargerthan1and
+4M⊕in the 2:3 and 1:2 inner resonance, respectively, and planet s
+greater than 20M⊕inthe outer 2:1resonance of the known planet
+and greater than 8 M⊕in the 3:2 resonance. Miller-Ricciet al.
+(2008)assumedthattheorbitoftheperturbingplanetiscir cularand
+that additional planets in eccentric orbits would produce s tronger
+perturbations. However, Nesvorn´ y(2009)showed thatanec centric
+planet can produce stronger or weaker perturbations depend ing on
+the relative angular position of itsorbital pericentre.
+In this paper we used two different methods to determine the
+uppermasslimitsforaputativeperturbingplanetintheHD1 89733
+system and thus the results of both analyses can be directly c om-
+pared. Our first analysis does not take into account time samp ling
+of the measured transit times and their uncertainties. On th e other
+hand, it was possible to probe for eccentric orbits of a pertu rb-
+ing planet, which is more rigorous than assuming a circular o rbit
+(Nesvorn´ y2009).Furtheranalysiswasperformedtoassure thatthe
+limitsonadditional planetspresentedinthispaperarenot overesti-
+mated. Unfortunately, applying this mothod for eccentric o rbits of
+a perturbing planet would increase the number of parameters enor-
+mously, thus we assumed acircular orbit for the perturber.
+Due to the limitations of our TTV analyses, we adopt the
+least constaining limits to conclude what upper masses of a p u-tativeperturbing planet canbe excluded inthe HD189733 sys tem.
+The results show no evidence for the presence of planets down to
+1 Earth mass near the 1:2 and 2:1 resonance orbits, and planet s
+downto2.2Earthmassesnearthe3:5and5:3resonanceorbits with
+HD189733b. Thesearethestrongest limitstodateonthepres ence
+ofotherplanetsinthissystem,basedonresultsoftwoindep endent
+TTVsanalyses.WealsodiscussthepossiblepresenceofTroj ansin
+the system, and conclude that the highest limit on a Trojan ma ss is
+5.3 M⊕if its libration amplitude is similar as for Trojans orbitin g
+near the Sun-Jupiter Lagrange points.
+ACKNOWLEDGMENTS
+We would like to thank the anonymous referee for useful sug-
+gestions and improvements. We are grateful to P. Harmanec an d
+R. Karjalainen for their careful reading of the manuscript a nd
+comments. The data presented here have been taken using AL-
+FOSC,whichisownedbytheInstitutodeAstrofisicadeAndalu cia
+(IAA) and operated at the Nordic Optical Telescope under agr ee-
+ment between IAA and the NBIfAFG of the Astronomical Obser-
+vatory of Copenhagen. The research was supported by the gran ts
+205/08/H005 and 205/06/0304 of the Czech Science Foundatio n
+and from the Research Program MSM0021620860 of the Ministry
+of Education of the Czech Republic. We acknowledge the use of
+the electronic bibliography maintained byNASA/ADSsystem and
+bythe CDSinStrasbourg.
+c/circlecop†rt2009 RAS,MNRAS 000, 1–1010M. Hrudkov ´aet al.
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+Preprint typeset in JINST style - HYPER VERSION
+Alignment of the ALICE Inner Tracking System with
+cosmic-ray tracks
+ALICE collaboration
+ABSTRACT : ALICE (A Large Ion Collider Experiment) is the LHC (Large Hadron Collider) exper-
+iment devoted to investigating the strongly interacting matter created in nucleus-nucleus collisions
+at the LHC energies. The ALICE ITS, Inner Tracking System, consists of six cylindrical layers
+of silicon detectors with three different technologies; in the outward direction: two layers of pixel
+detectors, two layers each of drift, and strip detectors. The number of parameters to be determined
+in the spatial alignment of the 2198 sensor modules of the ITS is about 13,000. The target align-
+ment precision is well below 10 mm in some cases (pixels). The sources of alignment information
+include survey measurements, and the reconstructed tracks from cosmic rays and from proton–
+proton collisions. The main track-based alignment method uses the Millepede global approach.
+An iterative local method was developed and used as well. We present the results obtained for the
+ITS alignment using about 105charged tracks from cosmic rays that have been collected during
+summer 2008, with the ALICE solenoidal magnet switched off.
+KEYWORDS : Particle tracking detectors (Solid-state detectors); Detector alignment and
+calibration methods.arXiv:1001.0502v4  [physics.ins-det]  28 Sep 2017Contents
+ALICE collaboration 1
+1. Introduction 10
+2. ITS detector layout 11
+2.1 Silicon Pixel Detector (SPD) 12
+2.2 Silicon Drift Detector (SDD) 14
+2.3 Silicon Strip Detector (SSD) 15
+3. Alignment target and strategy 16
+4. Cosmic-ray run 2008: data taking and reconstruction 17
+5. Validation of the survey measurements with cosmic-ray tracks 19
+5.1 Double points in SSD module overlaps 20
+5.2 Track-to-point residuals in SSD 20
+6. ITS alignment with Millepede 22
+6.1 General principles of the Millepede algorithm 22
+6.2 Millepede for the ALICE ITS 22
+6.3 Results on alignment quality 24
+6.4 Prospects for inclusion of SDD in the Millepede procedure 28
+7. SPD alignment with an iterative local method 30
+8. Conclusions 32
+ALICE collaboration
+K. Aamodt78, N. Abel43, U. Abeysekara30, A. Abrahantes Quintana42, D. Adamová86, M.M. Aggarwal25,
+G. Aglieri Rinella40, A.G. Agocs18, S. Aguilar Salazar66, Z. Ahammed55, A. Ahmad2, N. Ahmad2,
+S.U. Ahn50 i, R. Akimoto100, A. Akindinov68, D. Aleksandrov70, B. Alessandro102, R. Alfaro Molina66,
+A. Alici13, E. Almaráz Aviña66, J. Alme8, T. Alt43 ii, V . Altini5, S. Altinpinar32, C. Andrei17,
+A. Andronic32, G. Anelli40, V . Angelov43 ii, C. Anson27, T. Anti ˇci´c113, F. Antinori40 iii, S. Antinori13,
+K. Antipin37, D. Anto ´nczyk37, P. Antonioli14, A. Anzo66, L. Aphecetche73, H. Appelshäuser37,
+S. Arcelli13, R. Arceo66, A. Arend37, N. Armesto92, R. Arnaldi102, T. Aronsson74, I.C. Arsene78 iv,
+A. Asryan98, A. Augustinus40, R. Averbeck32, T.C. Awes76, J. Äystö49, M.D. Azmi2, S. Bablok8,
+– 1 –M. Bach36, A. Badalà24, Y .W. Baek50 i, S. Bagnasco102, R. Bailhache32 v, R. Bala101, A. Baldisseri89,
+A. Baldit26, J. Bán58, R. Barbera23, G.G. Barnaföldi18, L. Barnby12, V . Barret26, J. Bartke29,
+F. Barile5, M. Basile13, V . Basmanov94, N. Bastid26, B. Bathen72, G. Batigne73, B. Batyunya35,
+C. Baumann72 v, I.G. Bearden28, B. Becker20 vi, I. Belikov99, R. Bellwied34, E. Belmont-Moreno66,
+A. Belogianni4, L. Benhabib73, S. Beole101, I. Berceanu17, A. Bercuci32 vii, E. Berdermann32,
+Y . Berdnikov39, L. Betev40, A. Bhasin48, A.K. Bhati25, L. Bianchi101, N. Bianchi38, C. Bianchin79,
+J. Biel ˇcík81, J. Biel ˇcíková86, A. Bilandzic3, L. Bimbot77, E. Biolcati101, A. Blanc26, F. Blanco23 viii,
+F. Blanco63, D. Blau70, C. Blume37, M. Boccioli40, N. Bock27, A. Bogdanov69, H. Bøggild28,
+M. Bogolyubsky83, J. Bohm96, L. Boldizsár18, M. Bombara12 ix, C. Bombonati79 x, M. Bondila49,
+H. Borel89, V . Borshchov51, A. Borisov52, C. Bortolin79 xl„ S. Bose54, L. Bosisio103, F. Bossú101,
+M. Botje3, S. Böttger43, G. Bourdaud73, B. Boyer77, M. Braun98, P. Braun-Munzinger32;33 ii,
+L. Bravina78, M. Bregant103 xi, T. Breitner43, G. Bruckner40, R. Brun40, E. Bruna74, G.E. Bruno5,
+D. Budnikov94, H. Buesching37, P. Buncic40, O. Busch44, Z. Buthelezi22, D. Caffarri79, X. Cai111,
+H. Caines74, E. Camacho64, P. Camerini103, M. Campbell40, V . Canoa Roman40, G.P. Capitani38,
+G. Cara Romeo14, F. Carena40, W. Carena40, F. Carminati40, A. Casanova Díaz38, M. Caselle40,
+J. Castillo Castellanos89, J.F. Castillo Hernandez32, V . Catanescu17, E. Cattaruzza103, C. Cavicchioli40,
+P. Cerello102, V . Chambert77, B. Chang96, S. Chapeland40, A. Charpy77, J.L. Charvet89, S. Chattopadhyay54,
+S. Chattopadhyay55, M. Cherney30, C. Cheshkov40, B. Cheynis62, E. Chiavassa101, V . Chibante Barroso40,
+D.D. Chinellato21, P. Chochula40, K. Choi85, M. Chojnacki106, P. Christakoglou106, C.H. Christensen28,
+P. Christiansen61, T. Chujo105, F. Chuman45, C. Cicalo20, L. Cifarelli13, F. Cindolo14, J. Cleymans22,
+O. Cobanoglu101, J.-P. Coffin99, S. Coli102, A. Colla40, G. Conesa Balbastre38, Z. Conesa del Valle73 xii,
+E.S. Conner110, P. Constantin44, G. Contin103 x, G.J. Contreras64, Y . Corrales Morales101, T.M. Cormier34,
+P. Cortese1, I. Cortés Maldonado84, M.R. Cosentino21, F. Costa40, M.E. Cotallo63, E. Crescio64,
+P. Crochet26, E. Cuautle65, L. Cunqueiro38, J. Cussonneau73, A. Dainese59 iii, H.H. Dalsgaard28,
+A. Danu16, I. Das54, S. Das54, A. Dash11, S. Dash11, G.O.V . de Barros93, A. De Caro90, G. de Cataldo6
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+D. Elia6, D. Emschermann44 xiv, A. Enokizono76, B. Espagnon77, M. Estienne73, D. Evans12,
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+C. Schmidt32, H.R. Schmidt32, K. Schossmaier40, S. Schreiner40, S. Schuchmann37, J. Schukraft40,
+Y . Schutz73, K. Schwarz32, K. Schweda44, G. Scioli13, E. Scomparin102, G. Segato79, D. Semenov98,
+S. Senyukov1, J. Seo50, S. Serci19, L. Serkin65, E. Serradilla63, A. Sevcenco16, I. Sgura5, G. Shabratova35,
+R. Shahoyan40, G. Sharkov68, N. Sharma25, S. Sharma48, K. Shigaki45, M. Shimomura105, K. Shtejer42,
+Y . Sibiriak70, M. Siciliano101, E. Sicking40 xxxiii, E. Siddi20, T. Siemiarczuk107, A. Silenzi13, D. Silvermyr76,
+E. Simili106, G. Simonetti5 x, R. Singaraju55, R. Singh48, V . Singhal55, B.C. Sinha55, T. Sinha54,
+B. Sitar15, M. Sitta1, T.B. Skaali78, K. Skjerdal8, R. Smakal81, N. Smirnov74, R. Snellings3,
+H. Snow12, C. Søgaard28, O. Sokolov65, A. Soloviev83, H.K. Soltveit44, R. Soltz60, W. Sommer37,
+C.W. Son85, H.S. Son95, M. Song96, C. Soos40, F. Soramel79, D. Soyk32, M. Spyropoulou-Stassinaki4,
+B.K. Srivastava109, J. Stachel44, F. Staley89, E. Stan16, G. Stefanek107, G. Stefanini40, T. Steinbeck43 ii,
+E. Stenlund61, G. Steyn22, D. Stocco101 xxxiv, R. Stock37, P. Stolpovsky83, P. Strmen15, A.A.P. Suaide93,
+– 4 –M.A. Subieta Vásquez101, T. Sugitate45, C. Suire77, M. Šumbera86, T. Susa113, D. Swoboda40,
+J. Symons10, A. Szanto de Toledo93, I. Szarka15, A. Szostak20, M. Szuba108, M. Tadel40, C. Tagridis4,
+A. Takahara100, J. Takahashi21, R. Tanabe105, D.J. Tapia Takaki77, H. Taureg40, A. Tauro40, M. Tavlet40,
+G. Tejeda Muñoz84, A. Telesca40, C. Terrevoli5, J. Thäder43 ii, R. Tieulent62, D. Tlusty81, A. Toia40,
+T. Tolyhy18, C. Torcato de Matos40, H. Torii45, G. Torralba43, L. Toscano102, F. Tosello102, A. Tournaire73 xxxv,
+T. Traczyk108, P. Tribedy55, G. Tröger43, D. Truesdale27, W.H. Trzaska49, G. Tsiledakis44, E. Tsilis4,
+T. Tsuji100, A. Tumkin94, R. Turrisi80, A. Turvey30, T.S. Tveter78, H. Tydesjö40, K. Tywoniuk78,
+J. Ulery37, K. Ullaland8, A. Uras19, J. Urbán57, G.M. Urciuoli88, G.L. Usai19, A. Vacchi104,
+M. Vala35 ix, L. Valencia Palomo66, S. Vallero44, A. van den Brink106, N. van der Kolk3, P. Vande Vyvre40,
+M. van Leeuwen106, L. Vannucci59, A. Vargas84, R. Varma71, A. Vasiliev70, I. Vassiliev43 xxxii,
+M. Vassiliou4, V . Vechernin98, M. Venaruzzo103, E. Vercellin101, S. Vergara84, R. Vernet23 xxxvi,
+M. Verweij106, I. Vetlitskiy68, L. Vickovic97, G. Viesti79, O. Vikhlyantsev94, Z. Vilakazi22, O. Vil-
+lalobos Baillie12, A. Vinogradov70, L. Vinogradov98, Y . Vinogradov94, T. Virgili90, Y .P. Viyogi11 xxxvii,
+A. V odopianov35, K. V oloshin68, S. V oloshin34, G. V olpe5, B. von Haller40, D. Vranic32, J. Vrláková57,
+B. Vulpescu26, B. Wagner8, V . Wagner81, L. Wallet40, R. Wan111 xii, D. Wang111, Y . Wang44,
+Y . Wang111, K. Watanabe105, Q. Wen7, J. Wessels72, J. Wiechula44, J. Wikne78, A. Wilk72, G. Wilk107,
+M.C.S. Williams14, N. Willis77, B. Windelband44, C. Xu111, C. Yang111, H. Yang44, A. Yasnopolsky70,
+F. Yermia73, J. Yi85, Z. Yin111, H. Yokoyama105, I-K. Yoo85, X. Yuan111 xxxviii, V . Yurevich35,
+I. Yushmanov70, E. Zabrodin78, B. Zagreev68, A. Zalite39, C. Zampolli40 xxxix, Yu. Zanevsky35,
+S. Zaporozhets35, A. Zarochentsev98, P. Závada82, H. Zbroszczyk108, P. Zelnicek43, A. Zenin83,
+A. Zepeda64, I. Zgura16, M. Zhalov39, X. Zhang111 i, D. Zhou111, S. Zhou7, J. Zhu111, A. Zichichi13 xxii,
+A. Zinchenko35, G. Zinovjev52, Y . Zoccarato62, V . Zychá ˇcek81, and M. Zynovyev52
+Affiliation notes
+iAlso at26
+iiAlso at36
+iiiNow at80
+ivNow at32
+vNow at37
+viNow at22
+viiNow at17
+viiiAlso at46
+ixNow at57
+xNow at40
+xiNow at49
+xiiNow at99
+xiiiNow at6
+xivNow at72
+xvNow at: University of Technology and Austrian Academy of Sciences, Vienna, Austria
+xviAlso at60
+xviiAlso at40
+xviiiNow at115
+xixDeceased
+xxNow at74
+xxiNow at105
+– 5 –xxiiAlso at114
+xxiiiNow at5
+xxivAlso at41
+xxvNow at101
+xxviNow at30
+xxviiNow at89
+xxviiiAlso at78
+xxixNow at44
+xxxNow at33
+xxxiNow at8
+xxxiiNow at4
+xxxiiiAlso at72
+xxxivNow at73
+xxxvNow at62
+xxxviNow at: Centre de Calcul IN2P3, Lyon, France
+xxxviiNow at55
+xxxviiiAlso at79
+xxxixAlso at14
+xlAlso at Dipartimento di Fisica dell´Università, Udine, Italy
+Collaboration institutes
+1Dipartimento di Scienze e Tecnologie Avanzate dell’Università del Piemonte Orientale and Gruppo Collegato INFN,
+Alessandria, Italy
+2Department of Physics Aligarh Muslim University, Aligarh, India
+3National Institute for Nuclear and High Energy Physics (NIKHEF), Amsterdam, Netherlands
+4Physics Department, University of Athens, Athens, Greece
+5Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy
+6Sezione INFN, Bari, Italy
+7China Institute of Atomic Energy, Beijing, China
+8Department of Physics and Technology, University of Bergen, Bergen, Norway
+9Faculty of Engineering, Bergen University College, Bergen, Norway
+10Lawrence Berkeley National Laboratory, Berkeley, California, United States
+11Institute of Physics, Bhubaneswar, India
+12School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom
+13Dipartimento di Fisica dell’Università and Sezione INFN, Bologna, Italy
+14Sezione INFN, Bologna, Italy
+15Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia
+16Institute of Space Sciences (ISS), Bucharest, Romania
+17National Institute for Physics and Nuclear Engineering, Bucharest, Romania
+18KFKI Research Institute for Particle and Nuclear Physics, Hungarian Academy of Sciences, Budapest, Hungary
+19Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy
+20Sezione INFN, Cagliari, Italy
+21Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil
+22Physics Department, University of Cape Town, iThemba Laboratories, Cape Town, South Africa
+23Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Catania, Italy
+– 6 –24Sezione INFN, Catania, Italy
+25Physics Department, Panjab University, Chandigarh, India
+26Laboratoire de Physique Corpusculaire (LPC), Clermont Université, Université Blaise Pascal, CNRS–IN2P3, Clermont-
+Ferrand, France
+27Department of Physics, Ohio State University, Columbus, Ohio, United States
+28Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
+29The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland
+30Physics Department, Creighton University, Omaha, Nebraska, United States
+31Universidad Autónoma de Sinaloa, Culiacán, Mexico
+32ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany
+33Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt, Germany
+34Wayne State University, Detroit, Michigan, United States
+35Joint Institute for Nuclear Research (JINR), Dubna, Russia
+36Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany
+37Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany
+38Laboratori Nazionali di Frascati, INFN, Frascati, Italy
+39Petersburg Nuclear Physics Institute, Gatchina, Russia
+40European Organization for Nuclear Research (CERN), Geneva, Switzerland
+41Laboratoire de Physique Subatomique et de Cosmologie (LPSC), Université Joseph Fourier, CNRS-IN2P3, Institut
+Polytechnique de Grenoble, Grenoble, France
+42Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba
+43Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany
+44Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany
+45Hiroshima University, Hiroshima, Japan
+46University of Houston, Houston, Texas, United States
+47Physics Department, University of Rajasthan, Jaipur, India
+48Physics Department, University of Jammu, Jammu, India
+49Helsinki Institute of Physics (HIP) and University of Jyväskylä, Jyväskylä, Finland
+50Kangnung National University, Kangnung, South Korea
+51Scientific Research Technological Institute of Instrument Engineering, Kharkov, Ukraine
+52Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
+53University of Tennessee, Knoxville, Tennessee, United States
+54Saha Institute of Nuclear Physics, Kolkata, India
+55Variable Energy Cyclotron Centre, Kolkata, India
+56Fachhochschule Köln, Köln, Germany
+57Faculty of Science, P.J. Šafárik University, Košice, Slovakia
+58Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia
+59Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy
+60Lawrence Livermore National Laboratory, Livermore, California, United States
+61Division of Experimental High Energy Physics, University of Lund, Lund, Sweden
+62Université de Lyon 1, CNRS/IN2P3, Institut de Physique Nucléaire de Lyon, Lyon, France
+63Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain
+64Centro de Investigación y de Estudios Avanzados (CINVESTA V), Mexico City and Mérida, Mexico
+– 7 –65Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico
+66Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico
+67Institute for Nuclear Research, Academy of Sciences, Moscow, Russia
+68Institute for Theoretical and Experimental Physics, Moscow, Russia
+69Moscow Engineering Physics Institute, Moscow, Russia
+70Russian Research Centre Kurchatov Institute, Moscow, Russia
+71Indian Institute of Technology, Mumbai, India
+72Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, Münster, Germany
+73SUBATECH, Ecole des Mines de Nantes, Université de Nantes, CNRS-IN2P3, Nantes, France
+74Yale University, New Haven, Connecticut, United States
+75Budker Institute for Nuclear Physics, Novosibirsk, Russia
+76Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States
+77Institut de Physique Nucléaire d’Orsay (IPNO), Université Paris-Sud, CNRS-IN2P3, Orsay, France
+78Department of Physics, University of Oslo, Oslo, Norway
+79Dipartimento di Fisica dell’Università and Sezione INFN, Padova, Italy
+80Sezione INFN, Padova, Italy
+81Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Repub-
+lic
+82Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic
+83Institute for High Energy Physics, Protvino, Russia
+84Benemérita Universidad Autónoma de Puebla, Puebla, Mexico
+85Pusan National University, Pusan, South Korea
+86Nuclear Physics Institute, Academy of Sciences of the Czech Republic, ˇRež u Prahy, Czech Republic
+87Dipartimento di Fisica dell’Università ‘La Sapienza’ and Sezione INFN, Rome, Italy
+88Sezione INFN, Rome, Italy
+89Commissariat à l’Energie Atomique, IRFU, Saclay, France
+90Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Sezione INFN, Salerno, Italy
+91California Polytechnic State University, San Luis Obispo, California, United States
+92Departamento de Física de Partículas and IGFAE, Universidad de Santiago de Compostela, Santiago de Compostela,
+Spain
+93Universidade de São Paulo (USP), São Paulo, Brazil
+94Russian Federal Nuclear Center (VNIIEF), Sarov, Russia
+95Department of Physics, Sejong University, Seoul, South Korea
+96Yonsei University, Seoul, South Korea
+97Technical University of Split FESB, Split, Croatia
+98V . Fock Institute for Physics, St. Petersburg State University, St. Petersburg, Russia
+99Institut Pluridisciplinaire Hubert Curien (IPHC), Université de Strasbourg, CNRS-IN2P3, Strasbourg, France
+100University of Tokyo, Tokyo, Japan
+101Dipartimento di Fisica Sperimentale dell’Università and Sezione INFN, Turin, Italy
+102Sezione INFN, Turin, Italy
+103Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy
+104Sezione INFN, Trieste, Italy
+105University of Tsukuba, Tsukuba, Japan
+– 8 –106Institute for Subatomic Physics, Utrecht University, Utrecht, Netherlands
+107Soltan Institute for Nuclear Studies, Warsaw, Poland
+108Warsaw University of Technology, Warsaw, Poland
+109Purdue University, West Lafayette, Indiana, United States
+110Zentrum für Technologietransfer und Telekommunikation (ZTT), Fachhochschule Worms, Worms, Germany
+111Hua-Zhong Normal University, Wuhan, China
+112Yerevan Physics Institute, Yerevan, Armenia
+113Rudjer Boškovi ´c Institute, Zagreb, Croatia
+114Centro Fermi – Centro Studi e Ricerche e Museo Storico della Fisica “Enrico Fermi”, Rome, Italy
+115Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru
+Corresponding author: Andrea Dainese ( andrea.dainese@pd.infn.it )
+– 9 –1. Introduction
+The ALICE experiment [1] will study nucleus–nucleus, proton–proton and proton–nucleus colli-
+sions at the CERN Large Hadron Collider (LHC). The main physics goal of the experiment is to
+investigate the properties of strongly-interacting matter in the conditions of high energy density
+(>10 GeV =fm3) and high temperature (>0:2 GeV), expected to be reached in central Pb–Pb col-
+lisions atpsNN=5:5 TeV. Under these conditions, according to lattice QCD calculations, quark
+confinement into colourless hadrons should be removed and a deconfined Quark–Gluon Plasma
+should be formed [2]. In the past two decades, experiments at CERN-SPS (psNN=17:3 GeV)
+and BNL-RHIC (psNN=200 GeV) have gathered ample evidence for the formation of this state of
+matter [3].
+The ALICE experimental apparatus, shown in Fig. 1, consists of a central barrel, a forward
+muon spectrometer and a set of small detectors in the forward regions for trigger and other func-
+tions. The coverage of the central barrel detectors allows the tracking of particles emitted within a
+pseudo-rapidity range jhj<0:9 over the full azimuth. The central barrel is surrounded by the large
+L3 magnet that provides a field B =0:5 T.
+The ITS (Inner Tracking System) is a cylindrically-shaped silicon tracker that surrounds the
+interaction region. It consists of six layers, with radii between 3.9 cm and 43.0 cm, covering the
+pseudo-rapidity range jhj<0:9. The two innermost layers are equipped with Silicon Pixel De-
+tectors (SPD), the two intermediate layers contain Silicon Drift Detectors (SDD), while Silicon
+Strip Detectors (SSD) are used on the two outermost layers. The main task of the ITS is to pro-
+vide precise track and vertex reconstruction close to the interaction point. In particular, the ITS
+was designed with the aim to improve the position, angle, and momentum resolution for tracks
+Figure 1. General layout of the ALICE experiment [1].
+– 10 –reconstructed in the Time Projection Chamber (TPC), to identify the secondary vertices from the
+decay of hyperons and heavy flavoured hadrons, to reconstruct the interaction vertex with a reso-
+lution better than 100 mm, and to recover particles that are missed by the TPC due to acceptance
+limitations (very low momentum particles not reaching the TPC and very high momentum ones
+propagating along the 10% inactive area between adjacent TPC chambers).
+The measurement of charm and beauty hadron production in Pb–Pb collisions at the LHC is
+one of the main items of the ALICE physics program, because it will allow to investigate the mech-
+anisms of heavy-quark propagation and hadronization in the large, hot and dense medium formed
+in high-energy heavy-ion collisions and it will serve as a reference for the study of the effects of the
+medium on quarkonia states [4]. To measure the separation, from the interaction vertex, of the de-
+cay vertices of heavy flavoured hadrons, which have mean proper decay lengths ct100–500 mm,
+requires a resolution on the track impact parameter (distance of closest approach to the vertex) well
+below 100 mm. This requirement is met by the ITS. The design position resolution in the plane
+transverse to the beam line for charged-pion tracks reconstructed in the TPC and in the ITS is ex-
+pected to be approximately 10 mm+53mm=(ptp
+sinq), where ptis the transverse momentum in
+GeV=candqis the polar angle with respect to the beam line [4]. The ITS is made of thousands
+of separate modules, whose position is different from the ideal due to the limitations associated
+with the assembly and integration of the different components, and the forces these components
+experience. In order to achieve the required high precision on the track parameters, the relative
+position (location and orientation) of every module needs to be determined precisely. We refer to
+the procedure used to determine the modules relative position as alignment. The ITS alignment
+procedure starts from the positioning survey measurements performed during the assembly, and
+is refined using tracks from cosmic-ray muons and from particles produced in LHC pp collisions.
+Two independent methods, based on tracks-to-measured-points residuals minimization, are con-
+sidered. The first method uses the Millepede approach [5], where a global fit to all residuals is
+performed, extracting all the alignment parameters simultaneously. The second method performs
+a (local) minimization for each single module and accounts for correlations between modules by
+iterating the procedure until convergence is reached.
+In this article, we present the alignment methods for the ITS and the results obtained using the
+cosmic-data sample collected during summer 2008 with B =0 (a small data set with B =0:5 T
+was also collected; we used it for a few specific validation checks). In section 2we describe in
+detail the ITS detector layout and in section 3we discuss the strategy adopted for the alignment.
+In section 4we describe the 2008 sample of cosmic-muon data. These data were used to validate
+the available survey measurements (section 5) and to apply the track-based alignment algorithms:
+the Millepede method (section 6) and a local method that we are developing (section 7). We draw
+conclusions in section 8.
+2. ITS detector layout
+The geometrical layout of the ITS layers is shown in the left-hand panel of Fig. 2, as it is im-
+plemented in the ALICE simulation and reconstruction software framework (AliRoot [6]). The
+ALICE global reference system has the zaxis on the beam line, the xaxis in the LHC (horizontal)
+plane, pointing to the centre of the accelerator, and the yaxis pointing upward. The axis of the ITS
+– 11 –SPD 
+SDD 
+SSD 87.2 cm 
+xy
+z
+loczlocy
+locx
+locφlocθ
+locψFigure 2. Layout of the ITS (left) and orientation of the ALICE global (middle) and ITS-module local (right)
+reference systems. The global reference system has indeed its origin in the middle of the ITS, so that the z
+direction coincides with the beam line.
+barrel coincides with the zaxis. The module local reference system (Fig. 2, right) is defined with
+thexlocandzlocaxes on the sensor plane and with the zlocaxis in the same direction as the global
+zaxis. The local xdirection is approximately equivalent to the global rj. The alignment degrees
+of freedom of the module are translations in xloc,yloc,zloc, and rotations by angles yloc,qloc,jloc,
+about the xloc,yloc,zlocaxes, respectively1.
+The ITS geometry in AliRoot is described in full detail, down to the level of all mechani-
+cal structures and single electronic components, using the ROOT [7] geometrical modeler. This
+detailed geometry is used in Monte Carlo simulations and in the track reconstruction procedures,
+thereby accounting for the exact position of the sensor modules and of all the passive material that
+determine particle scattering and energy loss.
+The geometrical parameters of the layers (radial position, length along beam axis, number of
+modules, spatial resolution, and material budget) are summarized in Table 1. The material budget
+reported in the table takes into account the f-averaged material (including the sensors, electronics,
+cabling, support structures, and cooling) associated with radial paths through each layer. Another
+1.30% of radiation length comes from the thermal shields and supports installed between SPD
+and SDD barrels and between SDD and SSD barrels, thus making the total material budget for
+perpendicular tracks equal to 7.66% of X0.
+In the following, the features of each of the three sub-detectors (SPD, SDD and SSD) that are
+relevant for alignment issues are described (for more details see [1]).
+2.1 Silicon Pixel Detector (SPD)
+The basic building block of the ALICE SPD is a module consisting of a two-dimensional sensor
+matrix of reverse-biased silicon detector diodes bump-bonded to 5 front-end chips. The sensor
+1The alignment transformation can be expressed equivalently in terms of the local or global coordinates.
+– 12 –Table 1. Characteristics of the six ITS layers.
+Number Active Area Material
+Layer Type r[cm]z[cm] of per module Resolution budget
+modules rjz[mm2]rjz[mm2]X=X0[%]
+1 pixel 3.9 14.1 80 12.870.7 12100 1.14
+2 pixel 7.6 14.1 160 12.870.7 12100 1.14
+3 drift 15.0 22.2 84 70.1775.26 3525 1.13
+4 drift 23.9 29.7 176 70.1775.26 3525 1.26
+5 strip 38.0 43.1 748 7340 20830 0.83
+6 strip 43.0 48.9 950 7340 20830 0.86
+matrix consists of 256 160 cells, each measuring 50 mm (rj) by 425 mm (z).
+Two modules are mounted together along the zdirection to form a 141.6 mm long half-stave.
+Two mirrored half-staves are attached, head-to-head along the zdirection, to a carbon-fibre support
+sector, which also provides cooling. Each sector (see Fig. 3, right) supports six staves: two on the
+inner layer and four on the outer layer. The sensors are mounted in such a way that there is a 2%
+overlap between the active regions in rj, but along z there is a gap between each two consecutive
+sensors. Five sectors are then mounted together to form a half-barrel and finally the two (top and
+bottom) half-barrels are mounted around the beam pipe to close the full barrel (shown in the left-
+hand side of Fig. 3), which is actually composed of 10 sectors. In total, the SPD includes 60 staves,
+consisting of 240 modules with 1200 readout chips for a total of 9 :8106cells.
+The spatial precision of the SPD sensor is determined by the pixel cell size, the track inci-
+dence angle on the detector, and by the threshold applied in the readout electronics. The values of
+resolution along rjandzextracted from beam tests are 12 and 100 mm, respectively.
+zy
+x
+Figure 3. SPD drawings. Left: the SPD barrel and the beam pipe (radius in mm). Right: a Carbon Fibre
+Support Sector.
+– 13 –2.2 Silicon Drift Detector (SDD)
+The basic building block of the ALICE SDD [8] is a module divided into two drift regions where
+electrons move in opposite directions under a drift field of 500 V/cm (see Fig. 4, right), with
+hybrids housing the front-end electronics on either side. The SDD modules are mounted on linear
+structures called ladders. There are 14 ladders with six modules each on the inner SDD layer
+(layer 3), and 22 ladders with eight modules each on the outer SDD layer (layer 4). Modules and
+ladders are assembled to have an overlap of the sensitive areas larger than 580 mm in both rjandz
+directions, so as to provide full angular coverage over the pseudo-rapidity range jhj<0:9 (Fig. 4,
+left).
+The modules are attached to the ladder space frame and have their anode rows parallel to the
+ladder axis ( z). The ladders are mounted on a support structure made of two cones and four support
+rings to form the two cylindrical layers [9]. The support rings are mechanically fixed to the cones
+and bear ruby spheres, used as a reference for the ladder positioning as well as for the geometrical
+survey of the module positions in the ladder reference system.
+Thezcoordinate is reconstructed from the centroid of the collected charge along the anodes.
+The position along the drift coordinate ( xlocrj) is reconstructed starting from the measured drift
+time with respect to the trigger time. An unbiased reconstruction of the xloccoordinate requires
+therefore to know with good precision the drift velocity and the time-zero ( t0), which is the mea-
+sured drift time for particles with zero drift distance. The drift velocity depends on temperature (as
+T2:4) and it is therefore sensitive to temperature gradients in the SDD volume and to temperature
+variations with time. Hence, it is important to calibrate this parameter frequently during the data
+taking. Three rows of 33 MOS charge injectors are implanted at known distances from the collec-
+tion anodes in each of the two drift regions of a SDD module [10] for this purpose, as sketched
+in Fig. 4(right). Finally, a correction for non-uniformity of the drift field (due to non-linearities
+in the voltage divider and, for a few modules, also due to significant inhomogeneities in dopant
+concentration) has to be applied. This correction is extracted from measurements of the systematic
+deviations between charge injection position and reconstructed coordinates that was performed on
+MOS injectorsHighest Voltage CathodeDrift Drift
+Figure 4. Left: scheme of the SDD layers. Right: scheme of a SDD module, where the drift direction is
+parallel to the xloccoordinate. Units are millimeters.
+– 14 –all the 260 SDD modules with an infrared laser [11].
+The spatial precision of the SDD detectors, as obtained during beam tests of full-size pro-
+totypes, is on average 35 mm along the drift direction xlocand 25 mm for the anode coordinate
+zloc.
+2.3 Silicon Strip Detector (SSD)
+The basic building block of the ALICE SSD is a module composed of one double-sided strip
+detector connected to two hybrids hosting the front-end electronics. Each sensor has 768 strips on
+each side with a pitch of 95 mm. The stereo angle is 35 mrad, which is a compromise between
+stereo view and reduction of ambiguities resulting from high particle densities. The strips are
+almost parallel to the beam axis ( z-direction), to provide the best resolution in the rjdirection.
+The modules are assembled on ladders of the same design as those supporting the SDD [9].
+A view of the SSD ladder is shown in Fig. 5. The innermost SSD layer (layer 5) is composed of
+34 ladders, each of them being a linear array of 22 modules along the beam direction. Layer 6
+(the outermost ITS layer) consists of 38 ladders, each made of 25 modules. In order to obtain full
+pseudo-rapidity coverage, the modules are mounted on the ladders with small overlaps between
+successive modules, that are staggered by 600 mm in the radial direction. The 72 ladders, carrying
+a total of 1698 modules, are mounted on support cones in two cylinders. Carbon fiber is lightweight
+(to minimize the interactions) and at the same time it is a stiff material allowing to minimize the
+bending due to gravity, which is expected to give shifts of at most 50 mm, for the modules at the
+centre of the lateral ladders of the outer SSD layer.
+For each layer, neighbouring ladders are mounted at one of two slightly different radii ( Dr=
+6 mm) such that full azimuthal coverage is obtained. The acceptance overlaps, present both along
+zandrj, amount to 2% of the SSD sensor surface. The positions of the sensors with respect to
+reference points on the ladder were measured during the detector construction phase, as well as the
+ones of the ladders with respect to the support cones.
+The spatial resolution of the SSD system is determined by the 95 mm pitch of the sensor
+readout strips and the charge-sharing between those strips. Without making use of the analogue
+information the r.m.s spatial resolution is 27 mm. Beam tests [12] have shown that a spatial resolu-
+tion of better than 20 mm in the r jdirection can be obtained by analyzing the charge distribution
+within each cluster. In the direction along the beam, the spatial resolution is of about 830 mm.
+The SSD gain calibration has two components: overall calibration of ADC values to energy
+loss and relative calibration of the P and N sides. This charge matching is a strong point of double
+sided silicon sensors and helps to remove fake clusters. Both the overall and relative calibration
+Figure 5. View of one SSD ladder (from layer 5) as described in the AliRoot geometry.
+– 15 –are obtained from the data. Since the signal-to-noise ratio is larger than 20, the detection efficiency
+does not depend much on the details of the gain calibration.
+3. Alignment target and strategy
+For silicon tracking detectors, the typical target of the alignment procedures is to achieve a level
+of precision and accuracy such that the resolution on the reconstructed track parameters (in par-
+ticular, the impact parameter and the curvature, which measures the transverse momentum) is not
+degraded significantly with respect to the resolution expected in case of the ideal geometry without
+misalignment. For the ALICE ITS, this maximum acceptable degradation has been conventionally
+set to 20% (a similar target is adopted also for the ATLAS Inner Detector [13]). The resolutions
+on the track impact parameter and curvature are both proportional to the space point resolution, in
+the limit of negligible multiple scattering effect (large momentum). If the residual misalignment is
+assumed to be equivalent to random gaussian spreads in the six alignment parameters of the sensor
+modules, on which space points are measured, a 20% degradation in the effective space point res-
+olution (hence 20% degradation of the track parameters in the large momentum limit) is obtained
+when the misalignment spread in a given direction isp
+120%2100%270% of the intrinsic
+sensor resolution along that direction. With reference to the intrinsic precisions listed in Table 1,
+the target residual misalignment spreads in the local coordinates on the sensor plane are: for SPD,
+8mm in xlocand 70 mm in zloc; for SDD, 25 mm in xlocand 18 mm in zloc; for SSD, 14 mm in
+xlocand 500 mm in zloc. Note that these spreads represent effective alignment spreads, including
+the significant effect of the qlocangle (rotation about the axis normal to the sensor plane) on the
+spatial resolution. In any case, these target numbers are only an indication of the precision that is
+required to reach an acceptable alignment quality. We will aim at getting even closer to the design
+performance expected in case of ideal geometry.
+The other alignment parameters ( yloc,yloc,jloc) describe movements of the modules mainly
+in the radial direction. These have a small impact on the effective resolution, for tracks with a
+small angle with respect to the normal to the module plane, a typical case for tracks coming from
+the interaction region. However, they are related to the so-called weak modes : correlated misalign-
+ments of the different modules that do not affect the reconstructed tracks fit quality ( c2), but bias
+systematically the track parameters. A typical example is radial expansion or compression of all
+the layers, which biases the measured track curvature, hence the momentum estimate. Correlated
+misalignments for the parameters on the sensor plane ( xloc,zloc,qloc) can determine weak modes
+as well. These misalignments are, by definition, difficult to determine with tracks from collisions,
+but can be addressed using physical observables [14] (e.g. looking for shift in invariant masses of
+reconstructed decay particles) and cosmic-ray tracks. These offer a unique possibility to correlate
+modules that are never correlated when using tracks from the interaction region, and they offer a
+broad range of track-to-module-plane incidence angles that help to constrain also the yloc,ylocand
+jlocparameters, thus improving the sensitivity to weak modes.
+As already mentioned in the introduction, the sources of alignment information that we use
+are the survey measurements and the reconstructed space points from cosmic-ray and collision
+particles. These points are the input for the software alignment methods, based on global or local
+minimization of the residuals.
+– 16 –The general strategy for the ITS first alignment starts with the validation of the construction
+survey measurements of the SSD detector with cosmic-ray tracks and continues with the software
+alignment of the SPD and the SSD detectors, which also uses cosmic-ray tracks, collected without
+magnetic field. The initial alignment is more robust if performed with straight tracks (no field),
+which help to avoid possible biases that can be introduced when working with curved tracks (e.g.
+radial layer compression/expansion). Then, the already aligned SPD and SSD are used to confirm
+and refine the initial time-zero calibration of SDD, obtained with SDD standalone methods. These
+first steps are described in this report, which presents the status of the ITS alignment before the
+start of the LHC with proton–proton collisions.
+The next step will be, after the validation of the SDD survey measurements with cosmic-ray
+tracks, the alignment of the full detector (SPD, SDD, SSD) with tracks from cosmic rays and,
+mainly, from proton–proton collisions collected with magnetic field B =0 and B =0:5 T. In
+particular, the data with magnetic field switched on will allow us to study the track quality and
+precision as a function of the measured track momentum, thus separating the detector resolution
+and residual misalignment from multiple scattering. The tracks from collisions will provide a
+uniform coverage of the detector modules and will also be used to routinely monitor the quality
+of the alignment during data taking, and refine the corrections if needed. The last step will be
+the relative alignment of the ITS and the TPC with tracks, when both detectors will be internally
+aligned and calibrated. In addition, the relative movement of the ITS with respect to the TPC is
+being monitored, and to some extent measured, using a dedicated system based on lasers, mirrors
+and cameras [15].
+4. Cosmic-ray run 2008: data taking and reconstruction
+During the 2008 cosmic run, extending from June to October, about 105events with reconstructed
+tracks in the ITS were collected. In order to simplify the first alignment round, the solenoidal
+magnetic field was switched off during most of this data taking period. The status of the three ITS
+sub-detectors during the data taking is summarized in the following paragraph (for more details
+on the sub-detectors commissioning, see Refs. [16–18]). The corresponding status during the first
+LHC runs with proton–proton collisions is given in Ref. [19].
+For the SPD, 212 out of 240 modules (88%) were active. Noisy pixels, corresponding to less
+than 0.15% of the total number of pixels, were masked out, and the information was stored in the
+Offline Conditions Database (OCDB) to be used in the offline reconstruction. For the SDD, 246 out
+of 260 modules (95%) participated in the data acquisition. The baseline, gain and noise for each
+of the 133,000 anodes were measured every 24 hours by means of dedicated calibration runs that
+allowed us also to tag noisy ( 0:5%) and dead (1%) channels. The drift velocities were measured
+with dedicated injector runs collected every 6 hours, stored in the OCDB and successively used in
+the reconstruction. For the SSD, 1477 out of 1698 modules (87%) were active. The fraction of bad
+strips was1:5%. The normalized difference in P- and N-charge had a FWHM of 11%. The gains
+proved to be stable during the data taking.
+The events to be used for the ITS alignment were collected with a trigger provided by the pixel
+detectors (SPD). The SPD FastOR trigger [1] is based on a programmable hit pattern recognition
+system (on FPGA) at the level of individual readout chips (1200 in total, each reading a sensor area
+– 17 –of about 1 :41:4 cm2). This trigger system allows for a flexible selection of events of interest,
+for example high-multiplicity proton–proton collisions, foreseen to be studied in the scope of the
+ALICE physics program. For the 2008 cosmic run, the trigger logic consisted of selecting events
+with at least one hit on the upper half of the outer SPD layer ( r7 cm) and at least one on
+the lower half of the same layer. This trigger condition enhances significantly the probability of
+selecting events in which a cosmic muon, coming from above (the dominant component of the
+cosmic-ray particles reaching the ALICE cavern placed below 30 m of molasse), traverses the
+full ITS detector. This FastOR trigger is very efficient (more than 99%) and has purity (fraction of
+events with a reconstructed track having points in both SPD layers) reaching about 30–40%, limited
+mainly by the radius of the inner layer ( 4 cm) because the trigger assures only the passage of
+a particle through the outer layer ( 7 cm). For the FastOR trigger, typically 77% of the chips
+(i.e. about 90% of the active modules) could be configured and used. The trigger rate was about
+0.18 Hz.
+The following procedure, fully integrated in the AliRoot framework [6], is used for track
+reconstruction. After the cluster finding in the ITS (hereafter, we will refer to the clusters as
+“points”), a pseudo primary vertex is created using three aligned points in two consecutive layers
+(starting the search from the SPD). Track reconstruction is then performed using the ITS standalone
+tracker (as described in [4, 20]), which finds tracks in the outward direction, from the innermost
+SPD layer to the outermost SSD layer, using the previously found pseudo primary vertex as its seed;
+all found tracks are then refitted using the standard Kalman-filter fit procedure as implemented in
+the default ITS tracker. During the track refit stage, when the already identified ITS points are used
+in the Kalman-filter fit in the inward direction, in order to obtain the track parameters estimate at
+the (pseudo) vertex, “extra” points are searched for in the ITS module overlaps. For each layer, a
+search road for these overlap points in the neighbouring modules is defined with a size of about
+seven times the current track position error. Currently, the “extra” points are not used to update the
+track parameters, so they can be exploited as a powerful tool to evaluate the ITS alignment quality.
+A clean cosmic event consists of two separate tracks, one “incoming” in the top part of the ITS
+and one “outgoing” in the bottom part. Their matching at the reference median plane ( y=0) can
+be used as another alignment quality check. These two track halves are merged together in a single
+array of track points, which is the single-event input for the track-based alignment algorithms. A
+typical event of this type, as visualized in the ALICE event display, is shown in Fig. 6.
+The uncorrected zenith-azimuth 2D distribution of the (merged) tracks with at least eight
+points in the ITS is shown in Fig. 7, where the azimuth angle is defined in a horizontal plane start-
+ing from the positive side of the zglobal axis. The modulations in the azimuthal dependence of the
+observed flux are due to the presence of inhomogeneities in the molasse above the ALICE cavern,
+mainly the presence of two access shafts. These are seen as the structures at zenith angle 30
+and azimuth180(large shaft) and270(small shaft). On top of these structures, the effect
+of the SPD outer layer geometrical acceptance is visible: the azimuthal directions perpendicular to
+thezaxis (around 90and 270) have larger acceptance in the zenith angle.
+The main limitation of the usage of cosmic-ray tracks for the alignment of a cylindrical de-
+tector like the ITS is that the occupancy of the side modules (zenith angles approaching 90) is
+small, especially for the external layers [21]. In the case of the SSD outer layer, which has the
+smallest fractional coverage, about 75% of the ladders are covered. This is due to the small size
+– 18 –Figure 6. (colour online) A clean cosmic event reconstructed in the ITS (left), as visualized in the ALICE
+event display. The zoom on the SPD (right) shows an “extra” point in one of the rjacceptance overlaps of
+the outer layer.
+azimuth angle [deg]0 50 100 150 200 250 300 350zenith angle [deg]
+0102030405060708090
+020406080100120140160180200220240
+Figure 7. (colour online) Uncorrected distribution of the zenith-azimuth angles of the cosmic tracks recon-
+structed in the ITS.
+of the triggering detector (SPD), the dominance of small zenith angles for cosmic-ray particles and
+the cut on the track-to-module incidence angle ( >30) that we apply to reject large and elongated
+clusters.
+5. Validation of the survey measurements with cosmic-ray tracks
+The SDD and SSD were surveyed during the assembling phase using a measuring machine. The
+survey, very similar for the two detectors, was carried out in two stages: the measurement of the
+positions of the modules on the ladders and the measurement of the positions of the ladder end
+points on the support cone.
+– 19 –In the first stage, for SDD for example [22], the three-dimensional positions of six reference
+markers engraved on the detector surface were measured for each module with respect to ruby
+reference spheres fixed to the support structure. The precision of the measuring machine was 5 mm
+in the coordinates on the ladder plane and about 10 mm in the direction orthogonal to the plane.
+The deviations of the reference marker coordinates on the plane with respect to design positions
+showed an average value of 1 mm and a r.m.s. of 20 mm. In the second stage, the positions of
+the ladder end points with respect to the cone support structure were measured with a precision
+of about 10 mm. However, for the outer SSD layer, the supports were dismounted and remounted
+after the survey; the precision of the remounting procedure is estimated to be around 20 mm in the
+rjdirection [1].
+In the following we describe the results for the validation of the SSD survey measurements
+with cosmic-ray data. The validation of the SDD survey will be performed after completion of the
+detector calibration.
+5.1 Double points in SSD module overlaps
+As already mentioned, the modules are mounted with a small (2 mm) overlap for both the longi-
+tudinal ( z, modules on the same ladder) and transverse directions ( rj, adjacent ladders). These
+overlaps allow us to verify the relative position of neighbouring modules using double points pro-
+duced by the same particle on the two modules. Since the two points are very close in space and
+the amount of material crossed by the particle between the two points is very limited, multiple
+scattering can be neglected.
+We define the distance Dxlocbetween the two points in the local xdirection on the module plane
+(rj) by projecting, along the track direction, the point of one of the two modules on the other
+module plane. Figure 8(left) shows the Dxlocdistribution with and without the survey corrections,
+for both SSD layers (it was verified that the distributions for the two layers are compatible [21,
+23]) . When the survey corrections are applied, the spread of the distributions, obtained from a
+gaussian fit, is s25:5mm. This arises from the combined spread of the two points, thus the
+corresponding effective position resolution for a single point is estimated to be smaller by a factor
+1=p
+2, i.e.18mm, which is compatible with the expected intrinsic spatial resolution of about
+20mm. This indicates that the residual misalignment after applying the survey is negligible with
+respect to the intrinsic spatial resolution. This validation procedure was confirmed using Monte
+Carlo simulations of cosmic muons in the detector without misalignment, which give a spread in
+Dxlocof about 25 mm, in agreement with that obtained from the data.
+5.2 Track-to-point residuals in SSD
+Another test that was performed uses two points in the outer SSD layer to define a straight track
+(no magnetic field) and inspects the residuals between points on the inner layer and the track. The
+residuals are calculated using the position along the track corresponding to the minimum of the
+weighted (dimensionless) distance to the point2. Figure 8(right) shows the distribution of the rj
+2The different expected resolutions in rjandzhave been taken into account in the calculation of the distance of
+closest approach by dividing the deviations by the expected uncertainties, i.e. making use of a dimensionless distance
+measure.
+– 20 –m]µ [locx∆ -200 -150 -100 -50 0 50 100 150 200 entries
+050100150200SSD in-ladder overlaps
+w/o survey
+with surveymµ 0.7 ± Mean -2.1 
+mµ 0.5 ± Sigma 25.4 
+ [cm]φ∆r-0.04 -0.02 0 0.02 0.04number of events
+0200400600
+mµ 0.4 ± Sigma  29.4 mµ 0.4 ± Mean  3.7 w/o survey
+with surveySSD layer5-to-layer6 residualsFigure 8. (colour online) SSD survey validation. Left: distribution of Dxloc, the distance between two points
+in the module overlap regions along zon the same ladder. Right: distribution of the rjresiduals between
+straight-line tracks defined from two points on layer 6 and the corresponding points on layer 5. In both cases,
+gaussian fits to the distributions with survey applied are shown (in right-hand panel the fit range is 2s, i.e.
+[60mm;+60mm]).
+residuals between tracks through layer 6 and points on layer 5. The distribution exhibits significant
+non-gaussian tails, due to multiple scattering of low-momentum particles. The effect of multiple
+scattering on the residuals was analytically estimated to be of about 300 mm for p=0:5 GeV =c
+and to be negligible for p>7 GeV =c. This is roughly compatible with the observed distribution of
+residuals. The width of the central part of the distribution is quantified by performing a gaussian fit
+truncated at 2 s, that gives srj=29mm. The spread contains a contribution from the uncertainty
+in the track trajectory due to the uncertainties in the points on the outer layer. Assuming the same
+resolution on the outer and inner layers and taking into account the geometry of the detector, the
+effective single point resolution spread is 1 =p
+1:902 times the overall spread [23], that is 21 mm.
+This spread is larger than the effective resolution of about 18 mm that is extracted from the double
+points in module overlaps. This difference could be partly due to the multiple scattering, relevant
+for this analysis and negligible for the overlaps analysis, but we can not rule out that additional
+misalignments with a r.m.s. up to about 15 mm are present in the SSD. The mean residual is also
+non-zero, (3:70:4)mm, which suggests that residual shifts at the 5–10 mm level could be present.
+These misalignments would have to be at the ladder level to be compatible with the result from the
+study with sensor module overlaps.
+The same analysis was performed for the residuals in the zdirection [21, 23], not shown here.
+The distributions without and with survey were found to be compatible and the corresponding
+effective single point resolution was found to be compatible with the expected intrinsic resolution
+of about 800 mm. This indicates that the residual misalignment in zis much smaller than the
+intrinsic SSD resolution.
+A third method that was used to verify the SSD survey consisted in performing tracking with
+– 21 –pairs of points (2 points on layer 5 and two points on layer 6 or two sets of points on layer 5 and
+6), and comparing the track parameters of both track segments. The conclusion from this method
+is consistent with the results from the track-to-point method. For details see [23].
+6. ITS alignment with Millepede
+In general, the task of the track-based alignment algorithms is determining the set of geometry
+parameters that minimize the global c2of the track-to-point residuals:
+c2
+global=å
+modules ;tracks~dT
+t;pV1
+t;p~dt;p: (6.1)
+In this expression, the sum runs over all the detector modules and all the tracks in a given data set;
+~dt;p=~rt~rpis the residual between the data point ~rpand the reconstructed track extrapolation ~rt
+to the module plane; Vt;pis the covariance matrix of the residual. Note that, in general, the re-
+constructed tracks themselves depend on the assumed geometry parameters. This section describes
+how this minimization problem is treated by Millepede [5, 24] —the main algorithm used for ITS
+alignment— and presents the first alignment results obtained with cosmic-ray data.
+6.1 General principles of the Millepede algorithm
+Millepede belongs to the global least-squares minimization type of algorithms, which aim at de-
+termining simultaneously all the parameters that minimize the global c2in Eq. ( 6.1). It assumes
+that, for each of the local coordinates, the residual of a given track tto a specific measured point
+pcan be represented in a linearized form as dt;p=~a
¶dt;p=¶~a+~at
¶dt;p=¶~at, where ~ais the
+set of global parameters describing the alignment of the detector (three translations and three ro-
+tations per module) and ~atis the set of local parameters of the track. The corresponding c2
+global
+equation for ntracks with nlocal parameters per track and for mmodules with 6 global parameters
+(N=6mtotal global parameters) leads to a huge set of N+nnnormal equations. The idea behind
+the Millepede method is to consider the local ~aparameters as nuisance parameters that are elimi-
+nated using the Banachiewicz identity [25] for partitioned matrices. This allows to build explicitly
+only the set of Nnormal equations for the global parameters. If needed, linear constraints on the
+global parameters can be added using the Lagrange multipliers. Historically, two versions, Mille-
+pede and Millepede II, were released. The first one was performing the calculation of the residuals,
+the derivatives and the final matrix elements as well as the extraction of the exact solution in one
+single step, keeping all necessary information in computer memory. The large memory and CPU
+time needed to extract the exact solution of a NNmatrix equation effectively limited its use to
+N<10;000 global (alignment) parameters. This limitation was removed in the second version,
+Millepede II, which builds the matrices (optionally) in sparse format, to save memory space, and
+solves them using advanced iterative methods, much faster than the exact methods.
+6.2 Millepede for the ALICE ITS
+Following the development of Millepede, ALICE had its own implementation of both versions,
+hereafter indicated as MP and MPII, within the AliRoot framework [6]. Both consist of a detector
+independent solver class, responsible for building and solving the matrix equations, and a class
+– 22 –m]µ SPD local residuals [-1000 -500 0 500 1000entries
+10210310410not aligned
+alignedlocthick line: x
+locthin line: z
+Sectors Half-staves Modules [deg]locφ∆
+-4-2024Figure 9. (colour online) Left: example of Millepede residuals in the local reference frame of the SPD
+modules before and after the alignment. Right: the corrections to the jlocangle obtained in the hierarchical
+SPD alignment with Millepede.
+interfacing the former to specific detectors. While MP closely follows the original algorithm [5],
+MPII has a number of extensions. In addition to the MinRes matrix equation solution algorithm
+offered by the original Millepede II, the more general FGMRES [26] method was added, as well as
+the powerful ILU(k) matrix preconditioners [27]. All the results shown in this work are obtained
+with MPII.
+The track-to-point residuals, used to construct the global c2, are calculated using a parametric
+straight line ~r(t) =~a+~btor helix ~r(t) =fax+rcos(t+j0);ay+rsin(t+j0);az+bztgtrack
+model, depending on the presence of the magnetic field. The full error matrix of the measured
+points is accounted for in the track fit, while multiple scattering is ignored, since it has no systematic
+effect on the residuals.
+Special attention was paid to the possibility to account for the complex hierarchy of the
+alignable volumes of the ITS, in general leading to better description of the material budget distri-
+bution after alignment. This is achieved by defining explicit parent–daughter relationships between
+the volumes corresponding to mechanical degrees of freedom in the ITS. The alignment is per-
+formed simultaneously for the volumes on all levels of the hierarchy, e.g. for the SPD the correc-
+tions are obtained in a single step for the sectors, the half-staves within the sectors and the modules
+within the half-staves. Obviously, this leads to a degeneracy of the possible solutions, which should
+be removed by an appropriate set of constraints. We implemented the possibility to constrain either
+the mean or the median of the corrections for the daughter volumes of any parent volume. While
+the former can be applied via Lagrange multipliers directly at the minimization stage, the latter,
+being non-analytical, is applied after the Millepede minimization in a special post-processing step.
+The relative movement dof volumes for which the survey data is available (e.g. SDD and SSD
+modules) can be restricted to be within the declared survey precision ssurvey by adding a set of
+gaussian constraints d2=s2
+survey to the global c2.
+We report here two example figures to illustrate the bare output results from Millepede II for
+the alignment of the SPD detector (in this case), while the analysis of the alignment quality will be
+presented in the next section. The left-hand panel of Fig. 9shows an example of the residuals in
+– 23 –Entries  17214
+Constant  13.5 ± 1195.7 
+Mean      0.5 ±  -0.7 
+Sigma     0.5±  48.8 
+m]µ [y=0xy|∆ track-to-track -600 -400 -200 0 200 400 600events
+020040060080010001200Entries  17214
+Constant  13.5 ± 1195.7 
+Mean      0.5 ±  -0.7 
+Sigma     0.5±  48.8 
+not alignedalignedSPD
+Entries  12273
+Mean   0.005348
+RMS     89.45
+m]µ [y=0xy|∆ track-to-track -400 -300 -200 -100 0 100 200 300 400events
+0100200300400500600700 Entries  12273
+Mean   0.005348
+RMS     89.45SPD+SSDFigure 10. Left: distribution of Dxyjy=0for SPD only, before and after alignment. Right: distribution Dxyjy=0
+for track segments reconstructed in the upper and lower parts of SPD+SSD layers; each track segment is
+required to have four assigned points; SSD survey and Millepede alignment corrections are applied. In both
+cases, the distributions are produced from the sample of events used to obtain the alignment corrections.
+SPD (in the local reference frame of the modules) before and after alignment. The right-hand panel
+of Fig. 9shows the obtained corrections for the jlocangle (rotation of the volume with respect to
+itszlocaxis), indicating that the largest misalignments are at the level of the half-staves with respect
+to the carbon fiber support sectors.
+6.3 Results on alignment quality
+The SPD detector was first aligned using 5 104cosmic-ray tracks, with two points in the inner
+layer and two points in the outer layer, collected in 2008 with the magnetic field switched off.
+As described in the previous section, the hierarchical alignment procedure consisted in: aligning
+the ten sectors with respect to each other, the twelve half-staves of each sector with respect to the
+sector, and the two modules of each half-stave with respect to the half-stave.
+The following two observables are mainly used to check the quality of the obtained alignment:
+the top half-track to bottom half-track matching at the plane y=0, and the track-to-point distance
+for the “extra” points in the acceptance overlaps.
+For the first observable, the cosmic-ray track is split into the two track segments that cross
+the upper ( y>0) and lower ( y<0) halves of the ITS barrel, and the parameters of the two seg-
+ments are compared at y=0. The main variable is Dxyjy=0, the track-to-track distance at y=0 in
+the(x;y)plane transverse to beam line. This observable, that is accessible only with cosmic-ray
+tracks, provides a direct measurement of the resolution on the track transverse impact parameter
+d0; namely: sDxyjy=0(pt) =p
+2sd0(pt). Since the data used for the current analysis were collected
+without magnetic field, they do not allow us to directly assess the d0resolution (this will be the
+subject of a future work). However, also without a momentum measurement, Dxyjy=0is a powerful
+indicator of the alignment quality, as we show in the following.
+Figure 10(left) shows the distribution of Dxyjy=0for SPD, without and with the alignment
+– 24 –corrections. The two track segments are required to have a point in each of the SPD layers and
+to pass, in the transverse plane, within 1 cm from the origin (this cut selects tracks with a similar
+topology to those produced in collisions and rejects tracks that have small incidence angles on
+the inner layer modules). A gaussian fit to the distribution in the range [100mm;+100mm]
+gives a centroid compatible with zero and a spread s50mm. For comparison, a spread of
+38mm is obtained from a Monte Carlo simulation, with the ideal geometry of the ITS (without
+misalignment), of cosmic muons generated according to the momentum spectrum measured by
+the ALICE TPC in cosmic runs with magnetic field. When only the SPD detector is used and the
+tracks are straight lines (no magnetic field), the spread of the Dxyjy=0distribution can be related in a
+simple way to the effective spatial resolution sspatial , which includes the intrinsic sensor resolution
+and of the residual misalignment. For tracks passing close to the beam line (as in our case, with the
+cut at 1 cm), we have:
+s2
+Dxyjy=02(r2
+SPD1s2
+spatial ;SPD1+r2
+SPD2s2
+spatial ;SPD2)
+(rSPD1rSPD2)22r2
+SPD1+r2
+SPD2
+(rSPD1rSPD2)2s2
+spatial; (6.2)
+where the inner and outer SPD layers are indicated as SPD1 and SPD2, respectively. This relation
+neglects the effect of multiple scattering in the pixels and in the beam pipe, which is certainly one
+of the reasons why the Dxyjy=0distribution is not gaussian outside the central region, most likely
+populated by the high-momentum component of the cosmic muons. Using the fit result, sDxyjy=0
+50mm, obtained in the central region [100mm;+100mm], we estimate the value sspatial14mm,
+not far from the intrinsic resolution of about 11 mm extracted from the simulation. However, a
+precise estimation of the effective spatial resolution with this method requires the measurement of
+the track momentum, to account properly for the multiple scattering contribution. The statistics
+collected in 2008 with magnetic field did not allow a momentum-differential analysis.
+The next step in the alignment procedure is the inclusion of the SSD detector. As shown
+in section 5, the survey measurements already provide a very precise alignment, with residual
+misalignment levels of less than 5 mm for modules on the ladder and of about 20 mm for ladders.
+Because of the limited available statistics ( 2104tracks with four points in SPD and four
+points in SSD), the expected level of alignment obtained with Millepede on single SSD modules is
+significantly worse than the level reached with the survey measurements. For this reason, Millepede
+was used only to align the whole SPD barrel with respect to the SSD barrel and to optimize the
+positioning of large sets of SSD modules, namely the upper and lower halves of layers 5 and 6.
+For this last step, the improvement on the global positioning of the SSD layers was verified by
+comparing the position and direction of the pairs of SSD-only track segments built using: two
+points in the upper and two in the lower half-barrel (upper–lower configuration) or two points in
+the inner and two in the outer layer (inner–outer configuration). Before the alignment (only the
+survey corrections applied), the mean of Dxyjy=0is(1207)mm and (1:80:6)mm for the
+upper–lower and inner–outer configurations, respectively. After the alignment, it is (56)mm
+and(0:50:6)mm, respectively, that is, compatible with zero for both configurations.
+The right-hand panel of Fig. 10shows the distribution of Dxyjy=0for pairs of track segments,
+each reconstructed with two points in SPD and two in SSD, i.e. the merged cosmic-ray track has
+eight points in SPD+SSD. It can be seen that, when the SSD survey and the Millepede alignment
+are applied, the distribution is centred at zero and very narrow (FWHM 60mm), but it shows
+– 25 –Entries  1784
+Constant  2.3± 70.7 
+Mean      0.505 ± 0.785 
+Sigma     0.5± 18.3 
+m]µ [locx∆ track-to-point -300 -200 -100 0 100 200 300entries
+01020304050607080Entries  1784
+Constant  2.3± 70.7 
+Mean      0.505 ± 0.785 
+Sigma     0.5± 18.3 
+aligned
+not alignedSPD overlaps
+)  [deg]2α+1α sum of incidence angles (10 20 30 40 50 60 70 80 90m]µ2 r.m.s. [±) fit in 
+locx∆(σ
+051015202530
+data
+simulation (no misalignment)
+mµ = 10 σ
+m (3 seeds)µ = 7 σsimulation (random gaussian misal.):SPD overlapsFigure 11. (colour online) SPD double points in acceptance overlaps. Left: track-to-point Dxlocfor “extra”
+points before and after alignment. Right: sof the Dxlocdistributions as a function of the track-to-module
+incidence angle selection; 2008 cosmic-ray data are compared to the simulation with different levels of
+residual misalignment. See text for details.
+non-gaussian tails, most likely due to multiple scattering. A more precise alignment of the SSD
+using high-momentum tracks will be performed with the 2009 cosmic-ray and proton–proton data.
+The second alignment quality observable is the Dxlocdistance between points in the region
+where there is an acceptance overlap between two modules of the same layer. Because of the short
+radial distance between the two overlapping modules (a few mm), the effect of multiple scattering is
+negligible. However, in order to relate the spread of Dxlocto the effective resolution, the dependence
+of the intrinsic sensor resolution on the track-to-module incidence angle has to be accounted for. In
+particular, for SPD, due to the geometrical layout of the detector (Fig. 3, left), the track-to-module
+incidence angles in the transverse plane are in general not equal to 90and they are very different
+for two adjacent overlapping modules crossed by the same track. If Dxlocis defined as described
+in section 5, the error on Dxloccan be related to the effective spatial resolution of the two points,
+sspatial , as:
+s2
+Dxloc=s2
+spatial(a2)+s2
+spatial(a1)cos2(j12) (6.3)
+where the 1 and 2 subscripts indicate the two overlapping points, aiis the incidence angle of the
+track on the module plane, and j12is the relative angle between the two module planes, which is
+18and 9on the inner and outer SPD layer, respectively. Note that, for SSD overlaps on the same
+ladder, we have a1=a2'90andj12=0; therefore, sDxloc=p
+2sspatial , which is the relation we
+used in section 5.
+We start by showing, in Fig. 11(left), the track-to-point distance Dxlocfor the SPD “extra”
+points in the transverse plane, before and after the Millepede alignment. The extra points are not
+used in the alignment procedure. The spread of the distribution is s18mm, to be compared to
+s15mm from a Monte Carlo simulation with ideal geometry. An analysis of the Dxlocdistance
+as a function of the aincidence angle has been performed: five windows on the sum ( a1+a2)
+– 26 –Entries  7448
+Constant  7.1 ± 417.8 
+Mean      0.7 ± -1.0 
+Sigma     0.7± 48.8 
+m]µ [y=0xy|∆ track-to-track -600 -400 -200 0 200 400 600even-number events
+050100150200250300350400450Entries  7448
+Constant  7.1 ± 417.8 
+Mean      0.7 ± -1.0 
+Sigma     0.7± 48.8 SPD
+aligned with
+odd-number events
+ track-to-point residuals [cm]φr-0.02 -0.01 0 0.01 0.02arb. units
+020406080100
+B = 0
+B = +0.5 T
+B = -0.5 Ttrack fit: outer SPD + SSD
+residuals: inner SPDFigure 12. (colour online) Alignment stability tests. Left: for SPD only, distribution of the Dxyjy=0distance
+obtained when aligning with every second track and checking the alignment with the other tracks. Right:
+track-to-point residuals in the inner SPD layer (track fit in outer SPD layer and in the two SSD layers), for
+B=0 and B =0:5 T (the three histograms are normalized to the same integral).
+of the incidence angles on the two overlapping modules have been considered. These windows
+define increasing ranges of incidence angles from 0to 50. Figure 11(right) shows the spread
+of the Dxlocdistribution for the different incidence angle selections: a clear dependence of the
+spread (hence of the spatial resolution) on the incidence angle can be seen. This dependence was
+already observed in SPD test beam measurements [28, 29], which were used to tune the detector
+response simulation in the AliRoot software. In the same figure, Monte Carlo simulation results
+are reported for comparison: simulation with ideal geometry (open circles) and with a misaligned
+geometry obtained using a random gaussian residual misalignment (dashed lines: misalignments
+withs=7mm and three different seeds; dotted line: misalignments with s=10mm). The 2008
+data are well described by the simulation with a random residual misalignment with s7mm.
+However, this conclusion is based on the assumption that the intrinsic resolution is the same in
+the real detector and in the simulation. Since the intrinsic resolution can slightly vary depending
+on the working conditions of the detector (e.g. the settings used for the bias voltage and for the
+threshold), the value of 7 mm for the residual misalignment should be taken only as an indication.
+Furthermore, this is an equivalent random misalignment, while the real misalignments are likely
+non-gaussian and to some extent correlated among different modules.
+The robustness of the obtained results was tested by dividing the data sample in two parts
+and using every second track to align the SPD and the others to check the alignment quality. The
+corresponding Dxyjy=0distribution is presented in the left-hand panel of Fig. 12: the distribution is
+centred at zero and has the same s50mm as in the case of aligning with all tracks.
+Finally, the data with the 0.5 T magnetic field switched on (a few thousand events collected
+at the end of the 2008 cosmic run) were used to perform dedicated checks to evaluate a possible
+effect of the field on the alignment. The alignment corrections extracted from data with B =0 were
+– 27 –applied to data with B =0:5 T and alignment quality was verified using both the extra points
+method and the track-to-point residuals method. The distributions of the track-to-point distance for
+extra points in SPD acceptance overlaps for B =0,+0:5 T and0:5 T data were found to have
+compatible widths ( s[mm]: 18:30:5, 17:82:3, 18:41:8, respectively) [21]. Another check
+was performed using the track-to-point residuals, calculated by fitting the tracks in the SSD layers
+and the outer SPD layer and evaluating the residuals in the inner SPD layer. In Fig. 12(right) the
+comparison between the residuals without magnetic field, with +0:5 T, and with0:5 T is shown.
+Also in this case, the distributions without field and with the two field polarities are compatible.
+6.4 Prospects for inclusion of SDD in the Millepede procedure
+The alignment of the SDD detectors for the xloccoordinate (reconstructed from the drift time) is
+complicated by the interplay between the geometrical misalignment and the calibration of drift ve-
+locity and t0(defined in section 2.2). The t0parameter accounts for the delays between the time
+when a particle crosses the detector and the time when the front-end chips receive the trigger signal.
+Two methods have been developed in order to obtain a first estimate of the t0parameter. The first,
+and simpler, method consists in extracting the t0from the minimum measured drift time on a large
+statistics of reconstructed SDD points. The sharp rising part of the distribution of measured drift
+times is fitted with an error function. The t0value is then calculated from the fit parameters. The
+second method measures the t0from the distributions of residuals along the drift direction ( xloc) be-
+tween tracks fitted in SPD and SSD layers and the corresponding points reconstructed in the SDD.
+These distributions, in case of miscalibrated t0, show two opposite-signed peaks corresponding to
+the two separated drift regions of each SDD module, where electrons move in opposite directions
+(see Fig. 4, right). The t0can be calculated from the distance of the two peaks and the drift ve-
+locity. This second procedure has the advantage of requiring smaller statistics, because it profits
+from all the reconstructed tracks, with the drawback of relying on SDD calibration parameters (the
+drift velocity and possibly the correction maps). Moreover, being based on track reconstruction, it
+might be biased by SPD and/or SSD misalignments.
+Depending on the available statistics, the t0determination with these two methods can be done
+at the level of SDD barrel, SDD ladders or SDD modules. The t0parameter needs actually to be cal-
+ibrated individually for each of the 260 SDD modules, because of differences in the overall length
+of the cables connecting the DAQ cards and the front-end electronics. In particular, a significant
+difference is expected between modules of the A ( z>0) and C sides ( z<0), due to the6 m
+difference in the length of the optical fibres connecting the ITS ladders to the DAQ cards. With the
+first 2000 tracks, it is possible to determine the t0from track-to-point residuals for 4 sub-samples
+of modules, i.e. separating sensors connected to sides A and C of layers 3 and 4. An example of
+residual distributions for the left and right drift sides of the modules of layer 4 side C is shown
+in Fig. 13. The Millepede alignment corrections for SPD and SSD are applied in this case, and it
+has been checked that, if they are not applied, the centroid positions in this figure are not affected
+significantly, while the spread of the distributions increases, as it could be expected. A difference
+of 25 ns between sides A and C of each SDD layer has been observed, in agreement with the 6 m
+difference in fibre lengths (the propagation time of light in optical fibres is 4.89 ns/m). With larger
+statistics (35,000 tracks), it is possible to extract the t0for each half-ladder, which requires pro-
+ducing 36(ladders) 2(A/C sides) pairs of histograms like the ones shown in Fig. 13(left). Further
+– 28 –Drift Region: Left
+Entries  2760
+Constant    119
+Mean      -0.0497
+Sigma     0.0327
+ [cm]loctrack-to-point residuals in x-0.4 -0.2 0 0.2 0.4events
+020406080100120140160Drift Region: Left
+Entries  2760
+Constant    119
+Mean      -0.0497
+Sigma     0.0327Drift Region: Right
+Entries  2527
+Constant    112
+Mean      0.0599
+Sigma     0.0314Drift Region: Right
+Entries  2527
+Constant    112
+Mean      0.0599
+Sigma     0.0314Drift Region: Right
+Entries  2527
+Constant    112
+Mean      0.0599
+Sigma     0.0314
+Drift Region: Left
+Drift Region: RightSDD
+Layer 4 side C
+ [cm]locdrift coordinate x-3 -2 -1 0 1 2 3 residuals [mm]locx
+-0.1-0.0500.050.10.15Geometry only
+Geometry+CalibrationFigure 13. SDD calibration and alignment. Left: distribution of track-to-point residuals in the two drift
+regions for the SDD modules of layer 4 side C ( z<0); tracks are fitted using only their associated points
+in SPD and SSD; the Millepede alignment corrections for SPD and SSD are included, as well as the SSD
+survey. Right: residuals along the drift coordinate for one SDD module as a function of drift coordinate after
+Millepede alignment with only geometrical parameters and with geometrical+calibration parameters.
+cable-length differences, which introduce t0difference, exist at the level of individual half ladders
+and at the level of individual modules. These differences are of the order of 1.5 m (7 ns) and 20 cm
+(<1 ns) respectively, but have not been measured in detail yet, because of the limited size and
+coverage of the cosmic-ray tracks data sample. It should be noted that given the 6:5mm/ns of
+drift velocity, a bias of 1 ns on the t0can lead to a significant effect on the reconstructed position
+along the drift coordinate xloc.
+After a first calibration with these methods, a refinement of the t0determination is obtained by
+running the Millepede minimization with the t0as a free global parameter for each of the 260 SDD
+modules. Similarly, the drift velocity is considered as a free parameter for those SDD modules
+(about 35%) with mal-functioning injectors. For these modules a single value of drift velocity
+is extracted for the full data sample analyzed, thus neglecting the possible dependence of drift
+velocity on time due to temperature instabilities. However, this is not a major concern since the
+drift velocity was observed to be remarkably stable during the data taking [17]. This allows to
+assess at the same time geometrical alignment and calibration parameters of the SDD detectors.
+About 500 tracks are required to align and calibrate a single SDD module. An example is shown
+for a specific SDD module in the right-hand panel of Fig. 13, where the xlocresiduals along the
+drift direction are shown as a function of xloc. The result obtained using only the geometrical
+rotations and translations as free parameters in the Millepede minimization is shown by the circle
+markers. The clear systematic shift between the two drift regions ( xloc<0 and xloc>0) is due to
+both miscalibrated t0and biased drift velocity (this is a module with non-working injectors). These
+systematic effects are no longer present when also the calibration parameters are fitted by Millepede
+(square markers). It should be pointed out that the width of the SDD residual distributions shown
+in Fig. 13does not correspond to the expected resolution on SDD points along drift coordinate
+because of jitter between the time when the muon crosses the detectors and the SPD FastOR trigger,
+which has an integration time of 100 ns. For the about 100 SDD modules with highest occupancy,
+– 29 –the statistics collected in the 2008 cosmic run allowed to check the reliability of the calibration
+parameters ( t0and drift velocity) extracted with Millepede by comparing the values obtained from
+independent analyses of two sub-samples of tracks. From this study, a precision of 0 :025mm/ns
+for the drift velocity and 10 ns for t0was estimated. It should be noted that these precisions are
+limited by the available statistics as well as by the trigger jitter effect mentioned above.
+7. SPD alignment with an iterative local method
+We developed an alignment method that performs a (local) minimization for each single mod-
+ule and accounts for correlations between modules by iterating the procedure until convergence
+is reached. A similar approach is considered by both the CMS and ATLAS experiments [30–32].
+The main difference between this method and the Millepede algorithm is that only in the latter
+the correlations between the alignment parameters of all modules are explicitly taken into account.
+Conversely, the local module-by-module algorithm assumes that the misalignments of the modules
+crossed by a given track are uncorrelated and performs the minimization of the residuals indepen-
+dently for each module. The comparison of the alignment parameters from this method and from
+Millepede would provide a further validation of the results achieved with the Millepede.
+In the local method we minimize, module-by-module, the following local c2function of the
+alignment parameters of a single module:
+c2
+local(~atra;Arot) =å
+tracks~dT
+t;pV1
+t;p~dt;p
+=å
+tracks(~rtArot~rp~atra)T(Vt+Vp)1(~rtArot~rp~atra): (7.1)
+Here, the sum runs over the tracks passing through the module, ~rpis the position of the measured
+point on the module while ~rtis the crossing point on the module plane of the track tfitted with
+all points but ~rp.VtandVpare the covariance matrices of the crossing point and of the measured
+point, respectively. The six alignment parameters enter this formula in the vector ~atra, the alignment
+correction for the position of the centre of the module, and in the rotation matrix Arot, the alignment
+correction for the orientation of the plane of the module. The alignment correction is supposed to
+be small so that the rotation matrix can be approximated as the unity matrix plus a matrix linear
+in the angles. In this way, the c2
+localis a quadratic expression of the alignment parameters and the
+minimization can be performed by simple inversion. The c2
+localfunction in Eq. ( 7.1) can be written
+in the same way also for a set of modules considered as a rigid block. The track parameters are not
+affected by the misalignment of the module under study, because the track point on this module
+is not used in the fit, while the positions of the crossing points are affected, because the tracks are
+propagated to the plane of the module defined in the ideal geometry. This is taken into account by
+adding a large error along the track direction to the covariance matrix of the crossing point.
+Given that this is a local method, it is expected to work best if two conditions are fulfilled:
+the correlation between the misalignments of different modules is small and the tracks used to
+align a given module cross several other modules. In order to limit the bias that can be introduced
+by modules with low statistics, for which the second condition is normally not met, we align the
+modules following a sequence of decreasing number of points. To reduce the residual correlation
+– 30 –Entries  7488
+Constant  5.2 ± 294.8 
+Mean      0.706 ± -1.745 
+Sigma     0.71± 48.21 
+m]µ [
+y=0xy|∆ track-to-track -600 -400 -200 0 200 400 600events
+050100150200250300Entries  7488
+Constant  5.2 ± 294.8 
+Mean      0.706 ± -1.745 
+Sigma     0.71± 48.21 SPD
+Entries  1877
+Constant  7.5 ± 232.5 
+Mean      0.519 ± -1.095 
+Sigma     0.47± 19.97 
+m]µ [locx∆ track-to-point -300 -200 -100 0 100 200 300entries
+050100150200250Entries  1877
+Constant  7.5 ± 232.5 
+Mean      0.519 ± -1.095 
+Sigma     0.47± 19.97 SPD overlapsFigure 14. SPD alignment quality results for the iterative local method. Left: track-to-track Dxyjy=0distri-
+bution defined in section 6.3, using only SPD points. Right: track-to-point Dxlocdistribution for extra points
+in acceptance overlaps.
+between the alignment parameters obtained for the different modules, we iterate the procedure
+until the parameters converge. Simulation studies with misalignments of the order of 100 mm have
+shown that the convergence is reached after about 10 iterations.
+For the ITS alignment using the 2008 cosmic-ray data, we aligned only the SPD modules
+using this method. Like for Millepede, we adopted a hierarchical approach. Given the excellent
+precision of the SSD survey measurements, we used these two layers as a reference. We aligned
+as a first step the whole SPD barrel with respect to the SSD, then the two half-barrels with respect
+to the SSD, then the SPD sectors with respect to the SSD. In the last step, we used SPD and
+SSD points to fit the tracks and we aligned the individual sensor modules of the SPD. Figure 14
+shows the top–bottom track-to-track Dxyjy=0distribution obtained using only the SPD points (left-
+hand panel) and the track-to-point Dxlocfor the double points in acceptance overlaps (right-hand
+panel), after alignment. Both distributions are compatible (mean and sigma from a gaussian fit)
+with the corresponding distributions after Millepede alignment. This is an important independent
+verification of the Millepede results. Since the two methods are in many aspects independent,
+comparing the two sets of alignment parameters could provide a check for the presence of possible
+systematic trends. Figure 15shows the correlation of the inner SPD layer parameter values obtained
+with the iterative method and those obtained with Millepede. A correction was applied to account
+for a possible global roto-translation of the whole ITS, which does not affect the quality of the
+alignment and can be different for the two methods. The closed (open) markers represent the
+modules with more (less) than 500 track points. Most of the modules are clustered along the
+diagonal lines where the parameters from the two methods are exactly the same. There are some
+outlier modules, that are far from the ideal result. However, these outliers mostly correspond to
+modules with low statistics (open markers) at the sides of the SPD barrel.
+– 31 –m]µ [Millepedexδ-2000-1500-1000-500 0 500 1000 1500 2000m]µ [Iterativexδ
+-2000-1500-1000-5000500100015002000
+modules with:
+more than 500 points
+less than 500 points
+m]µ [
+Millepedeyδ-2000-1500-1000-500 0 500 10001500 2000m]µ [Iterativeyδ
+-2000-1500-1000-5000500100015002000
+m]µ [Millepedezδ-800 -600 -400 -200 0 200 400 600 800m]µ [Iterativezδ
+-800-600-400-2000200400600800
+ [mdeg]
+Millepedeψδ-300 -200 -100 0 100 200 300 [mdeg]
+Iterativeψδ
+-300-200-1000100200300
+ [mdeg]Millepedeθδ-300 -200 -100 0 100 200 300 [mdeg]Iterativeθδ
+-300-200-1000100200300
+ [mdeg]
+Millepedeφδ-3000 -2000 -1000 0 1000 2000 3000 [mdeg]
+Iterativeφδ
+-3000-2000-10000100020003000Figure 15. Correlation between the alignment parameters obtained from Millepede (horizontal axis) and the
+iterative method (vertical axis), for the inner SPD layer modules. Modules with more (less) than 500 points
+are represented by the closed (open) markers.
+8. Conclusions
+The results on the first alignment of the ALICE Inner Tracking System with cosmic-ray tracks,
+collected in 2008 in the absence of magnetic field, have been presented.
+The initial step of the alignment procedure consisted of the validation of the survey measure-
+ments for the Silicon Strip Detector (SSD). The three methods applied for this purpose indicate
+that the residual misalignment spread for modules on ladders is less than 5 mm, i.e. negligible with
+respect to the intrinsic resolution of this detector in the most precise direction, while the residual
+misalignment spread for the ladders with respect to the support cones amounts to about 15 mm.
+The procedure continues with track-based software alignment performing residuals minimiza-
+tion. We presented the results obtained with a sample of about 105cosmic-ray tracks, reconstructed
+in events selected by the FastOR trigger of the Silicon Pixel Detector (SPD). We mainly use the
+Millepede algorithm, which minimizes a global c2of residuals for all alignable volumes and a
+large set of tracks.
+We start from the SPD, which is aligned in a hierarchical approach, from the largest mechanical
+structures (10 support sectors) to the 240 single sensor modules. About 90% of the latter were active
+during the 2008 cosmic run, and about 85% had enough space points ( >50) to perform alignment.
+Then, we align the SPD barrel with respect to the SSD barrel. The SSD coverage provided by the
+cosmic-ray tracks is insufficient to align the SSD at the level of ladders, especially for the ladders
+– 32 –close to the horizontal plane y=0. Therefore, for the time being we only align the SSD at the level
+of large sets of ladders.
+The two intermediate ITS layers, the Silicon Drift Detectors (SDD), represent a special case,
+because the reconstruction of one of the two local coordinates requires dedicated calibration pro-
+cedures (drift velocity and drift time zero extraction), which are closely related to the alignment.
+Indeed, one of the approaches that we are developing for the time zero calibration is based on the
+analysis of track residuals in a standalone procedure, initially, and then directly within the Mille-
+pede algorithm. Once these procedures are stable and robust, the SDD will be included in the
+standard alignment chain. For all six layers, the completion of the alignment for all modules will
+require tracks from proton–proton collisions; a few 106events (collected in a few days) should
+allow us to reach a uniform alignment level, close to the target, over the entire detector.
+We use mainly two observables to assess the quality of the obtained alignment: the matching
+of the two half-tracks produced by a cosmic-ray particle in the upper and lower halves of the ITS
+barrel, and the residuals between double points produced in the geometrical overlaps between ad-
+jacent modules. For the SPD, both observables indicate an effective space point resolution of about
+14mm in the most precise direction, only 25% worse than the resolution of about 11 mm extracted
+from the Monte Carlo simulation without misalignments. In addition, the measured incidence angle
+dependence of the spread of the double points residuals is well reproduced by Monte Carlo simu-
+lations that include random residual misalignments with a gaussian sigma of about 7 mm. Further
+confidence on the robustness of the results is provided, to some extent, by the cross-checks we per-
+formed using a small data set with magnetic field switched on and, mainly, by the comparison of the
+Millepede results to those from a second, independent, alignment method. This second method,
+which iteratively minimizes a set of local module-by-module c2functions, yields, compared to
+Millepede, a similar alignment quality and a quite compatible set of alignment corrections.
+Using the present data set with magnetic field off, since the track momenta are not known,
+the multiple scattering effect, which is certainly not negligible, cannot be disentangled from the
+residual misalignment effect. Therefore, a more conclusive statement on the SPD residual mis-
+alignment will be possible only after the analysis of cosmic-ray data collected with magnetic field
+switched on. The same applies for combined tracking with SPD, SDD and SSD points: in this case,
+the momentum-differential analysis of the transverse distance between the two half-tracks (upper
+and lower half-barrels) will allow us to measure the track transverse impact parameter resolution,
+which is a key performance figure in view of the ALICE heavy flavour physics program.
+Acknowledgements
+The ALICE collaboration would like to thank all its engineers and technicians for their invaluable
+contributions to the construction of the experiment, and, in particular, of the Inner Tracking System.
+The ALICE collaboration acknowledges the following funding agencies for their support in
+building and running the ALICE detector:
+Calouste Gulbenkian Foundation from Lisbon and Swiss Fonds Kidagan, Armenia;
+– 33 –Conselho Nacional de Desenvolvimento CientŠfico e Tecnol ˚Ugico (CNPq), Financiadora
+de Estudos e Projeto (FINEP), Fundação de Amparo à Pesquisa do Estado de São Paulo
+(FAPESP);
+National Natural Science Foundation of China (NSFC), the Chinese Ministry of Education
+(CMOE) and the Ministry of Science and Technology of China (MSTC);
+Ministry of Education and Youth of the Czech Rebublic;
+Danish National Science Research Council and the Carlsberg Foundation;
+The European Research Council under the European Community’s Seventh Framework Pro-
+gramme;
+Helsinki Institute of Physics and the Academy of Finland;
+French CNRS-IN2P3, the ‘Region Pays de Loire’, ‘Region Alsace’, ‘Region Auvergne’ and
+CEA, France;
+German BMBF and the Helmholtz Association;
+Hungarian OTKA and National Office for Research and Technology (NKTH);
+Department of Atomic Energy and Department of Science and Technology of the Govern-
+ment of India;
+Istituto Nazionale di Fisica Nucleare (INFN) of Italy;
+MEXT Grant-in-Aid for Specially Promoted Research, Japan;
+Joint Institute for Nuclear Research, Dubna;
+Korea Foundation for International Cooperation of Science and Technology (KICOS);
+CONACYT, DGAPA, México, ALFA-EC and the HELEN Program (High-Energy physics
+Latin-American–European Network);
+Stichting voor Fundamenteel Onderzoek der Materie (FOM) and the Nederlandse Organistie
+voor Wetenschappelijk Onderzoek (NWO), Netherlands;
+Research Council of Norway (NFR);
+Polish Ministry of Science and Higher Education;
+National Authority for Scientific Research - NASR (Autontatea Nationala pentru Cercetare
+Stiintifica - ANCS);
+Federal Agency of Science of the Ministry of Education and Science of Russian Federation,
+International Science and Technology Center, Russian Federal Agency of Atomic Energy,
+Russian Federal Agency for Science and Innovations and CERN-INTAS;
+– 34 –Ministry of Education of Slovakia;
+CIEMAT, EELA, Ministerio de Educación y Ciencia of Spain, Xunta de Galicia (Consellería
+de Educación), CEADEN, Cubaenergía, Cuba, and IAEA (International Atomic Energy
+Agency);
+Swedish Reseach Council (VR) and Knut & Alice Wallenberg Foundation (KAW);
+Ukraine Ministry of Education and Science;
+United Kingdom Science and Technology Facilities Council (STFC);
+The United States Department of Energy, the United States National Science Foundation, the
+State of Texas, and the State of Ohio.
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+arXiv:1001.0503v2  [math-ph]  5 Jul 2010Covariant star product on symplectic and Poisson
+spacetime manifolds
+M. Chaichiana,b, M. Oksanena, A. Tureanua,band G. Zetc
+aDepartment of Physics, University of Helsinki, P.O. Box 64, FI-00014 Helsinki, Finland
+bHelsinki Institute of Physics, P.O. Box 64, FI-00014 Helsin ki, Finland
+cDepartment of Physics, ”Gh. Asachi”Technical University,
+Bd. D. Mangeron 67, 700050 Iasi, Romania
+Abstract
+AcovariantPoissonbracketandanassociated covariant sta rproductinthesense
+of deformation quantization are defined on the algebra of ten sor-valued differential
+forms on a symplectic manifold, as a generalization of simil ar structures that were
+recently defined on the algebra of (scalar-valued) differenti al forms. A covariant
+star product of arbitrary smooth tensor fields is obtained as a special case. Finally,
+we study covariant star products on a more general Poisson ma nifold with a linear
+connection, first for smooth functions and then for smooth te nsor fields of any type.
+Some observations on possible applications of the covarian t star products to gravity
+and gauge theory are made.
+1 Introduction
+Due to several convincing arguments arising from the quantum the ory and the Einstein’s
+theory of gravity, it is generally believed that the manifold structur e of spacetime does
+not exist at distances equal and shorter than the Planck length an d that the correct
+description of spacetime should be somehow noncommutative. Field t heories defined on
+noncommutative spacetimes have been extensively studied during t he last decades (for
+some reviews see [1, 2]). The canonically noncommutative spacetime s tructure, generated
+by the coordinate commutation relations
+[ˆxµ,ˆxν] =iθµν(1.1)
+with a constant antisymmetric θµν, and its Moyal star product have received most atten-
+tion. Also the Lie algebraic structure, the quantum space structu res and the symplectic
+and Poisson manifolds have been considered as possible descriptions of noncommutative
+spacetime. We consider the last two cases where the θµν(ˆx) is a generally ˆ x-dependent
+bivector field.
+The main effects of the noncommutativity of spacetime on the theor ies of particle
+physics, most notably the Standard Model, have been extensively s tudied and by now
+some of their features are well understood. Understanding grav ity on noncommutative
+spacetimes has proven to be a challenging effort. This is due to the diffi culty to accom-
+modate both the gravitational and the noncommutative structur es of spacetime — the
+classical geometrical large-distance structure and the noncomm utativity of coordinates at
+short distances.
+One of the standing issues of noncommutative gravity is the genera l covariance of
+the star product under spacetime diffeomorphisms. The diffeomorp hism-covariance of a
+star product can be achieved in many ways. One way is to construct a star product
+that is by definition covariant under conventional spacetime diffeom orphism. This is the
+1approach we will consider in this work. More specifically we consider sp acetime as a
+symplectic manifold — later as a more general Poisson manifold — and se ek to quantize
+such a spacetime by introducing a (noncommutative) covariant sta r product. This is
+done in the light of two recent approaches [3, 4, 5] to the quantiza tion of a symplectic
+spacetime manifold. We construct a diffeomorphism-covariant Poiss on bracket and an
+associated star product of tensor-valued differential forms on s uch spacetime. A covariant
+star product of tensor fields is obtained as the special case of ten sor-valued zero-forms.
+Possible applications of the obtained covariant star product to gra vity and gauge theory
+are discussed.
+Deformation quantization of more general Poisson manifolds with a t orsion-free linear
+connection has also been studied recently [6] and a universal covar iant star product of
+functions has been constructed. We define a covariant Poisson br acket on a smooth
+manifold with a linear connection and propose an associated covarian t star product of
+tensor fields on the Poisson manifold. The constraints that the con nection is imposed
+to satisfy by these structures are studied. The possibility to relax the torsion-freeness
+condition of [6] in the case of a star product of functions is also cons idered.
+For a recent review of deformation quantization see [7].
+2 On covariant derivative of tensors and differential
+forms
+The intent of this section is to review the concepts of connection an d covariant deriva-
+tive on smooth manifolds, providing some of the definitions and result s that are used in
+the following sections, and to discuss some misunderstandings foun d in recent literature
+regarding these things.
+2.1 Connections and covariant derivatives
+We consider a smooth manifold Mand a linear connection on the tensor bundle T(M) of
+Mand the associated covariant derivative.1The linear connection is given by a covariant
+derivative∇that is a linear map
+∇:Tk,l(M)→Tk,l+1(M), (2.1)
+whereTk,l(M) is the vector space of smooth tensor fields of type ( k,l) onM, i.e. the
+space of smooth sections of the tensor product bundle ⊗kTM⊗lT∗M
+Tk,l(M) = Γ(⊗kTM⊗lT∗M), (2.2)
+whereTMandT∗Mare the tangent bundle of Mand the cotangent bundle of M,
+respectively,⊗kTMdenotes the k-th tensor power of TMand Γ denotes the space of all
+smooth sections of the argument fiber bundle. We shall denote the algebra of tensor fields
+onMby
+T(M) =∞/circleplusdisplay
+k,l=0Tk,l(M). (2.3)
+1Wecouldequallywelltalkaboutanaffineconnectioninsteadofalinearc onnection. See[8], Chapter3,
+Theorem 3.3, for their relation.
+2The covariant derivative ∇Xalong a vector field X∈X(M) = Γ(TM) is a linear deriva-
+tion that preserves the type of tensors
+∇X:Tk,l(M)→Tk,l(M) (2.4)
+and it is related to the connection (2.1) by
+(∇XA)(α1,...,αk,X1,...,Xl) = (∇A)(X;α1,...,αk,X1,...,Xl),(2.5)
+wherethevector field Xinthecovariantderivative ∇XAofA∈Tk,l(M) takestheplace of
+the additional vector argument in ∇A∈Tk,l+1(M) provided by (2.1) (see [8], Chapter 3,
+Section 2).2This together with the requirements that ∇Xcommutes with all contractions
+and acts on functions as the vector X(directional derivative)
+∇Xf=X(f), f∈F(M) = Γ(M×R) (2.6)
+ensures that∇satisfies the properties of a covariant differentation on T(M).3The co-
+variant derivative (2.5) can be written
+(∇XA)(α1,...,αk,X1,...,Xl) =∇X/parenleftbig
+A(α1,...,αk,X1,...,Xl)/parenrightbig
+−k/summationdisplay
+i=1A(α1,...,∇Xαi,...,αk,X1,...,Xl)
+−l/summationdisplay
+i=1A(α1,...,αk,X1,...,∇XXi,...,Xl),(2.7)
+which follows from ∇Xbeing a derivation that commutes with all contractions (see [8],
+Chapter 3, Proposition 2.10). Thus the second covariant derivativ e ofA∈Tk,l(M) is
+(∇2A)(X;Y;) =∇X(∇YA)−∇∇XYA, (2.8)
+where each term is in Tk,l(M) (see [8], Chapter 3, Proposition 2.12). The n-th covariant
+derivative can be obtained inductively.
+Differential forms The vector space of differential forms of degree ponMis the space
+of smooth sections of the p-th exterior power of the cotangent bundle,
+Ωp(M) = Γ(∧pT∗M). (2.9)
+The algebra of differential forms on M— with the exterior product ∧as multiplication
+— is the direct sum of the spaces of p-forms of all degrees pand it shall be denoted by
+Ω(M) =dimM/circleplusdisplay
+p=0Ωp(M). (2.10)
+The covariant derivative of a differential form on Mis defined similarly as for any other
+tensor field on M(see above). However, the algebra Ω( M) is not closed under a covariant
+differentation∇. For example restricting the domain of ∇to Ωp(M) we have
+∇: Γ(∧pT∗M)→Γ(T∗M⊗∧pT∗M), (2.11)
+2The additional argument Xin the (2.5) is the first one, because we want to have the arguments of
+∇Ain the same order as the corresponding tensor indices in the compon ent notation∇ρAµ1···µkν1···νl.
+3The linearity of a tensor ∇Ain its arguments guarantees that ∇fX=f∇Xand∇X+Y=∇X+∇Y,
+for arbitrary f∈F(M) andX,Y∈X(M).
+3where the range is the space of covector-valued p-forms. Thus we have to consider tensor-
+valued differential forms.
+The vector space of ( k,l)-tensor-valued differential forms of degree pshall be denoted
+by
+Ωp(M,Tk,l) = Γ(⊗kTM⊗lT∗M⊗∧pT∗M), (2.12)
+whereTk,labbreviatesthetensorproductbundle ⊗kTM⊗lT∗M.4NotethatΩ0(M,Tk,l) =
+Tk,l(M) and Ωp(M,T0,0) = Ωp(M). The algebra of all tensor-valued differential forms is
+defined as
+Ω(M,T) =dimM/circleplusdisplay
+p=0∞/circleplusdisplay
+k,l=0Ωp(M,Tk,l), (2.13)
+with the multiplication given by the generalized exterior product
+∧: Ωp(M,Tk,l)×Ωq(M,Tm,n)→Ωp+q(M,Tk,l⊗Tm,n) = Ωp+q(M,Tk+m,l+n),(2.14)
+The covariant derivative ∇maps (k,l)-tensor-valued p-forms to (k,l+ 1)-tensor-valued
+p-forms
+∇: Ωp(M,Tk,l)→Ωp(M,Tk,l+1). (2.15)
+We also define an exterior covariant derivative Dthat is the natural extension of the
+exterior derivative d : Ωp(M)→Ωp+1(M) and∇on Ω(M,T). It maps tensorial p-forms
+to tensorial ( p+1)-forms of the same type
+D: Ωp(M,Tk,l)→Ωp+1(M,Tk,l), (2.16)
+which we shall discuss more shortly (see also [8], Chapter 2, Section 5).
+Local smooth frames, the connection one-form, the torsion a nd the curvature
+two-forms and the exterior covariant derivative A connection one-form ωa
+bof∇
+is associated to a local smooth frame {ea}dimM
+a=1of the tangent bundle TMover an open
+setUofMover which TMis trivial. It is defined by
+∇eb=ωa
+b⊗ea. (2.17)
+The connection∇onTM(restricted over U) is given by
+∇φ= (dφa+ωa
+bφb)⊗ea, (2.18)
+whereφ=φaea∈Γ(TM) overUand d is the exterior derivative. On the cotangent
+bundleT∗M, the dual bundle of TM, we can setup a local smooth frame {ea}dimM
+a=1over
+Uthat is dual to the frame of TM,/an}b∇acketle{tea,eb/an}b∇acket∇i}ht=δa
+b. Thus the connection on T∗MoverUis
+given by∇ea=−ωa
+b⊗eband
+∇ψ= (dψb−ωa
+bψa)⊗eb, (2.19)
+whereψ=ψaea∈Γ(T∗M). Extension to the tensor bundle T(M) is straightforward,5
+e.g. forA=Aa1···ak
+b1···blea1⊗···⊗eak⊗eb1⊗···⊗ebl∈Γ(⊗kTM⊗lT∗M) overUwe
+4We shall also refer to elements of Ωp(M,Tk,l) as (k,l)-tensor-valued p-forms.
+5∇has the standard Leibniz rule, ∇(A⊗B) =∇A⊗B+A⊗∇B, and similarly for the exterior
+product,∇(A∧B) =∇A∧B+A∧∇B.
+4have
+∇A=/parenleftBigg
+dAa1···ak
+b1···bl+k/summationdisplay
+i=1ωai
+cAa1···ai−1cai+1···ak
+b1···bl−l/summationdisplay
+i=1ωc
+biAa1···ak
+b1···bi−1cbi+1···bl/parenrightBigg
+⊗ea1⊗···⊗eak⊗eb1⊗···⊗ebl.(2.20)
+All the other local smooth frames of T∗MandTMcan be obtained through local
+linear transformations
+e′a= Λa
+beb, e′
+a=eb(Λ−1)b
+a, (2.21)
+whereinthegeneral case Λ ∈GL(TpM)∼=GL(dimM,R), but additionalstructures on M
+can restrict the local symmetry group to a subgroup of GL(dimM,R). The components
+of tensor fields transform as
+A′a1···ak
+b1···bl= Λa1
+c1···Λak
+ckAc1···ck
+d1···dl(Λ−1)d1
+b1···(Λ−1)dl
+bl(2.22)
+and the connection one-form has the transformation rule
+ω′a
+b= Λa
+cωc
+d(Λ−1)d
+b−dΛa
+c(Λ−1)c
+b. (2.23)
+Fortensor-valueddifferential formswe use notationwhere thete nsor indices arevisible
+and the antisymmetric form components are hidden, e.g. A∈Ωp(M,Tk,l) is written
+Aa1···ak
+b1···bl=1
+p!Aa1···ak
+b1···blc1···cpec1∧···∧ecp. (2.24)
+The torsion two-form Taand the curvature two-form Ra
+bof the connection are defined by
+Ta=Dea= dea+ωa
+b∧eb, (2.25)
+Ra
+b= dωa
+b+ωa
+c∧ωc
+b, (2.26)
+whereDis the exterior covariant derivative (2.16) that is defined for a tens or-valued
+differential form (2.24) as the linear map
+DAa1···ak
+b1···bl= dAa1···ak
+b1···bl+k/summationdisplay
+i=1ωai
+c∧Aa1···ai−1cai+1···ak
+b1···bl
+−l/summationdisplay
+i=1ωc
+bi∧Aa1···ak
+b1···bi−1cbi+1···bl.(2.27)
+Unlike the exterior derivative d Aa1···ak
+b1···blthe exterior covariant derivative (2.27) has the
+correct tensor transformation rule (2.22) under local frame tra nsformations (2.21). The
+second exterior covariant derivative consist of contractions with the curvature two-form
+(2.26)
+D2Aa1···ak
+b1···bl=k/summationdisplay
+i=1Rai
+c∧Aa1···ai−1cai+1···ak
+b1···bl−l/summationdisplay
+i=1Rc
+bi∧Aa1···ak
+b1···bi−1cbi+1···bl.(2.28)
+Taking exterior covariant derivatives of (2.25) and (2.26) yields the Bianchi identities
+DTa=Ra
+b∧eb, (2.29)
+DRa
+b= 0. (2.30)
+5Local coordinates Introducing a local coordinate system {xµ}dimM
+µ=1on the open set U
+ofMenables us to use the full component notation of tensor calculus — t he formalism
+conventionallyused inphysics. Itenables ustolocallywritethecovar iantderivative(2.20)
+of a tensor field A∈Tk,l(M) along the basis vector∂
+∂xµas
+∇µAa1···ak
+b1···bl=∂µAa1···ak
+b1···bl+k/summationdisplay
+i=1ωai
+µ cAa1···ai−1cai+1···ak
+b1···bl
+−l/summationdisplay
+i=1ωc
+µ biAa1···ak
+b1···bi−1cbi+1···bl,(2.31)
+whereωa
+µ bdxµ=ωa
+b,∇µeb=ωa
+µ beaand∇µea=−ωa
+µ beb. This is the local form of (2.7).
+Since the fibers of TMandT∗Mover eachp∈Mare the tangent space TpMand
+the cotangent space T∗
+pMofMatprespectively, the local frames of TMandT∗Mover
+eachp∈Mare smoothly related to the coordinate bases∂
+∂xµand dxµofTpMandT∗
+pM
+respectively through (orientation preserving) linear transforma tions
+ea=eµ
+a∂
+∂xµ, ea=ea
+µdxµ, (2.32)
+whereeµ
+aas a matrix is a GL+(dimM,R)-valued smooth function on Mandea
+µis the
+inverse ofeµ
+a;eµ
+aeb
+µ=δb
+a,eµ
+aea
+ν=δµ
+ν.6The functions eµ
+aandea
+µenable us to transform
+components of tensors between the coordinate and noncoordina te bases.
+A (k,l)-tensor-valued p-form (2.24) behaves as a ( k,l+p)-tensor field under the co-
+variant derivative (2.20)
+∇µAa1···ak
+b1···bl=1
+p!/parenleftBig
+∇µAa1···ak
+b1···blc1···cp/parenrightBig
+ec1∧···∧ecp, (2.33)
+where the expression inside the parenthesis is given by (2.31).
+Using a coordinate basis for T(M) We can even choose the local frames of TM
+andT∗Mto coincide with a coordinate basis of tangent spaces, ea=∂
+∂xa, and cotangent
+spaces,ea= dxa. When this choice is made, we conventionally choose to work with
+one kind of indices, a→µetc., and rename the connection one-form ωa
+b→Γρ
+νand the
+connection coefficients ωa
+µ b→Γρ
+µν. The covariant derivative is now defined by
+∇µ:Tk,l(M)→Tk,l+1(M), (2.34)
+∇ρAµ1···µk
+ν1···νl=∂µAµ1···µk
+ν1···νl+k/summationdisplay
+i=1Γµi
+ρσAµ1···µi−1σµi+1···µk
+ν1···νl
+−l/summationdisplay
+i=1Γσ
+ρνiAµ1···µk
+ν1···νi−1σνi+1···νl.(2.35)
+General coordinate transformations, x→x′=x′(x), are a specific class of frame
+transformations (2.21) with the local transformation matrix
+Λµ
+ν=∂x′µ
+∂xν. (2.36)
+6GL+(dimM,R) ={g∈GL(dimM,R) : detg >0}
+62.2 Criticism
+It is important to understand that the algebra of differential form s Ω(M) is not closed
+under the covariant derivation ∇(equivalently under ∇µin a coordinate basis). The
+covariant derivative ∇ωof ap-formωis a smooth section of the product bundle T∗M⊗
+∧pT∗M. In other words∇µωis a (0,1)-tensor-valued p-form. This is not acknowledged
+in [4], where the covariant derivative ∇µωof ap-formωalong the basis vector eµis
+considered to be a p-form, which leads to some serious problems.
+Differential forms are frame-independent objects that exist inde pendent of anycoor-
+dinate system.∇µωis clearly a frame-dependent object that transforms as a compon ent
+of a covector under general coordinate transformations.
+The convention “∇µacts nontrivially only on the bases eµand dxµ” in [4] is incon-
+sistently executed. The property (2.7) is violated, when some of th e contractions are
+differentiated with ∇µ. As an example we consider the covariant derivative of the con-
+traction of a bivector θµνand two covariant derivatives ∇µαand∇νβof differential forms
+αandβ,
+∇µ(θνρ∇να∇ρβ) = (∇µθνρ)∇να∇ρβ+θνρ(∇µ∇να∇ρβ+∇να∇µ∇ρβ).(2.37)
+Clearly we cannotwrite∇µθνρ=∂µθνρ, as is done in similar calculations of [4] (see,
+[4] Appendices B.5 and C for these calculations), without trivializing th e connection.
+The tensorial nature of ˜Rµνis correctly recognized in these calculations (see also the
+Appendix A of [4]), but the bivector θµνis treated as a function.
+Moreover, in [4] the second covariant derivatives ∇µ∇ναof ap-formαare incorrectly
+calculated, so that the commutator of second covariant derivativ es ofα,
+[∇µ,∇ν]αρ1···ρp=−Tσ
+µν∇σαρ1···ρp−p/summationdisplay
+i=1Rσ
+ρiµναρ1···ρi−1σρi+1···ρp,(2.38)
+contains only the curvature contributions, but not the torsion co ntribution.7This is an
+implication of the failure to fully recognize the additional argument ve ctor provided by
+the covariant derivative.
+Due to these problem in the covariant derivative of [4], the star prod uct proposedin [4]
+is neither truly associative nor covariant. The associativity proper ty of the star product is
+found to be satisfied only because the covariant derivatives in the d ouble Poisson brackets
+like{{α,β},γ}are calculated incorrectly.
+These problems with the covariant derivative found in [4] have been r ecently corrected
+in [3], where the formalism of [4] is reconsidered by using correct defin itions. In [3] the
+covariant derivative ∇µis correctly taken on tensor fields of any type and one does not
+try to extend the algebra of differential forms by the covariant de rivatives.
+7If one wants to use the above mentioned convention for ∇µ, one should calculate the second covariant
+derivative of αas∇µ(dxν⊗(∇να)).
+73 Generalization of the Poisson structure and the
+star product of differential forms to the algebra
+of tensor-valued differential forms on a symplectic
+manifold
+3.1 Poisson algebra of differential forms
+Consider the graded differential Poisson algebra of differential for ms on a symplectic
+manifoldMstudied in [9, 4, 3, 10].
+The Poisson bracket of functions f,g∈F(M) is defined by
+{f,g}=θ(df,dg) =θµν∂µf∂νg. (3.1)
+The Jacobi identity of the Poisson bracket requires that the Poiss on bivector satisfies
+/summationdisplay
+(µ,ν,ρ)θµσ∂σθνρ= 0, (3.2)
+where the sum is over cyclic permutations. The Poisson bivector θis assumed to be
+nondegenerate, so that it has an inverse ωthat satisfies ωµνθνρ=δρ
+µ. It can be shown
+that (3.2) is equivalent to ωbeing a closed form, d ω= 0 [9]. The closed nondegenerate
+two-formωonMis called the symplectic form.
+The Poisson bracket of a function and a differential form α∈Ω(M) (of degree one at
+first and then of any degree)
+{f,α}=∇Xfα=θµν∂µf∇να (3.3)
+is a covariant derivation of αand therefore defines a linear connection on M. By using the
+connection coefficients Γρ
+µνwe can define two connections ∇and˜∇with the connection
+one-forms
+Γρ
+ν= Γρ
+µνdxµand˜Γρ
+µ= Γρ
+µνdxν(3.4)
+respectively, which are different when the torsion (2.25),
+Tρ= Γρ
+ν∧dxν= dxµ∧˜Γρ
+µ, (3.5)
+does not vanish, Tρ
+µν= 2Γρ
+[µν]/ne}ationslash= 0. The Leibniz rule of the Poisson bracket, d {f,g}=
+{df,g}+{f,dg}, implies that the connection ˜∇satisfies
+˜∇µθνρ=∂µθνρ+Γν
+σµθσρ+Γρ
+σµθνσ= 0, (3.6)
+i.e.˜∇is a symplectic connection.8Together (3.2) and (3.6) imply two covariant versions
+of the Jacobi identity
+/summationdisplay
+(µ,ν,ρ)θµσ∇σθνρ= 0 and/summationdisplay
+(µ,ν,ρ)θµσθνλTρ
+σλ= 0. (3.7)
+8We call a connection ∇symplectic if ωµνor equivalently θµνis covariantly constant under the
+covariant derivative.
+8Imposing either∇µθνρ= 0 orTρ
+µν= 0 would lead to a single torsion-free symplectic
+connection∇=˜∇, but this is not necessary. The curvature two-forms (2.26) of ∇and˜∇
+are given by
+Rµ
+ν= dΓµ
+ν+Γµ
+ρ∧Γρ
+νand˜Rµ
+ν= d˜Γµ
+ν+˜Γµ
+ρ∧˜Γρ
+ν (3.8)
+respectively, and we use the Poisson bivector θµνto raise their lower index, e.g.
+˜Rµν=θµρ˜Rν
+ρ. (3.9)
+The curvature two-form of a symplectic connection ˜∇is symmetric ˜Rµν=˜Rνµ.9Note
+that, unlike ˜∇µ, the covariant derivative ∇µdoes not commute with the raising of indices
+withθµν, because∇is not symplectic. Indeed (3.6) implies
+∇µθνρ=Tν
+µσθσρ+Tρ
+µσθνσ. (3.10)
+The unique Poisson bracket of differential forms α,β∈Ω(M) of nonzero degrees that
+is consistent with the graded differential Poisson algebra has been d efined in [4, 3]
+{α,β}=θµν∇µα∧∇νβ+(−1)deg(α)˜Rµν∧iµα∧iνβ, (3.11)
+where deg(α) denotes the degree of αandiµαis the interior product of αwith theµ-th
+basis vector. Covariant derivatives of contractions like (2.37), inc luding multiple covariant
+derivatives, are present when several Poisson brackets (3.11) a re taken, e.g.{α,{β,γ}}.
+InorderforthePoissonbracket (3.11)tosatisfythegradedJac obiidentitytheconnections
+have to satisfy the following additional constraints
+Rµ
+νρσ= 0, (3.12)
+∇λ˜Rµν
+ρσ= 0, (3.13)
+/summationdisplay
+(µ,ν,ρ)˜Rµσ∧iσ˜Rνρ= 0, (3.14)
+where the last constraint (3.14) is, however, implied by the two form er constraints, the
+Leibniz rule and the Jacobi identity (3.2) [10, 3].10
+3.2 Poisson algebra of tensor-valued differential forms
+We want to extend the graded differential Poisson algebra of differe ntial forms by the
+covariant derivation ∇. This is achieved by generalizing the Poisson bracket (3.11) for
+tensor-valued differential forms. In other words the Poisson bra cket should be generalized
+to accept forms whose components have additional tensor indices . Then Poisson brackets
+like{∇µα,β}will be naturally defined. This would enable us to define the related sta r
+product for all tensor-valued differential forms, which enlarges t he applicability of the for-
+malism. The curvature two-forms Rµνand˜Rµνand the torsion two-form Tµare examples
+of such forms. Such star product could indeed be useful for defin ing noncommutative
+deformations of gravitational theories, whose actions involve the curvature two-form(s).
+Next we propose such a formalism that generalizes the approach of [4, 3], and also corrects
+the misunderstandings found in [4].
+9(3.6) implies: 0 = [ ˜∇ρ,˜∇σ]θµν=−Tλ
+ρσ˜∇λθµν+˜Rµ
+λρσθλν+˜Rν
+λρσθµλ=−˜Rνµ
+ρσ+˜Rµν
+ρσ.
+10(3.12) is implied by the Jacobi identity for two functions and one one- form and (3.13) by the Jacobi
+identity for one function and two one.forms [9].
+9The algebra of tensor-valued differential forms
+We choose to work in a local coordinate system {xµ}dimM
+µ=1ofM. This approach can,
+however, be repeated by using any local smooth frame of Ω( M,T), with the frame trans-
+formations (2.21) defined to be compatible with the symplectic struc ture ofM.
+The exterior product (2.14) of two tensor-valued differential for msA∈Ωp(M,Tk,l)
+andB∈Ωq(M,Tm,n),
+Aµ1···µk
+ν1···νl=1
+p!Aµ1···µk
+ν1···νlρ1···ρpdxρ1∧···∧dxρp, (3.15)
+Bµ1···µm
+ν1···νn=1
+q!Bµ1···µm
+ν1···νnρ1···ρqdxρ1∧···∧dxρq, (3.16)
+is a tensor-valued differential form A∧B∈Ωp+q(M,Tk+m,l+n) defined by
+(A∧B)µ1···µk+m
+ν1···νl+n=1
+(p+q)!(A∧B)µ1···µk+m
+ν1···νl+nρ1···ρp+qdxρ1∧···∧dxρp+q
+=1
+p!q!Aµ1···µk
+ν1···νlρ1···ρpBµk+1···µk+m
+νl+1···νl+nρp+1···ρp+qdxρ1∧···∧dxρp+q
+=Aµ1···µk
+ν1···νl∧Bµk+1···µk+m
+νl+1···νl+n.(3.17)
+The exterior product (3.17) satisfies the following properties for a rbitrary tensor-valued
+differential forms A,BandC:
+1.A∧B= 0 if deg(A)+deg(B)>dim(M).
+2. Degree:
+deg(A∧B) = deg(A)+deg(B). (3.18)
+3. Symmetry:
+Aµ1···µk
+ν1···νl∧Bρ1···ρm
+σ1···σn= (−1)deg(A)deg(B)Bρ1···ρm
+σ1···σn∧Aµ1···µk
+ν1···νl.(3.19)
+4. Associativity: ( A∧B)∧C=A∧(B∧C).
+It is necessary to write Aµ1···µkν1···νlinstead of just Ain the exterior product (3.17) when
+the order of the factors is changeable as in (3.19), because the te nsor product is generally
+noncommutative, A⊗B/ne}ationslash=B⊗A.11
+The interior product of tensor-valued differential forms can be de fined so that it rec-
+ognizes only the form part of tensor-valued differential forms. Th e interior product of
+A∈Ωp(M,Tk,l) with the coordinate basis vector∂
+∂xµis the map
+iµ: Ωp(M,Tk,l)→Ωp−1(M,Tk,l+1),
+iρAµ1···µkν1···νl=1
+(p−1)!Aµ1···µkν1···νlρσ2···σpdxσ2∧···∧dxσp.(3.20)
+It satisfies
+iρ/parenleftbig
+Aµ1···µkν1···νl∧Bρ1···ρm
+σ1···σn/parenrightbig
+=iρAµ1···µkν1···νl∧Bρ1···ρm
+σ1···σn
++(−1)deg(A)Aµ1···µk
+ν1···νl∧iρBρ1···ρm
+σ1···σn(3.21)
+11One can write (3.19) equivalently as A∧B= (−1)deg(A)deg(B)(B∧A)◦σ(k,l), where the map σ(k,l)
+movesthefirst kcovectorargumentsandthefirst lvectorargumentsovertherestofthe argumentsofeach
+type,σ(k,l)(α1,...,α k+m,X1,...,X l+n) = (αk+1,...,α k+m,α1,...,α k,Xl+1,...,X l+n,X1,...,X l).
+We, however, prefer to keep track of the order of the argument s with the tensorial indices.
+10andiµiνA=−iνiµA. Zero-forms vanish under the interior product iµ.
+The exterior covariant derivative D(2.16) is used instead of the exterior derivative,
+because the latter maps tensorial differential forms to nontenso rial ones. The exterior
+covariant derivative (2.27) is now written
+DAµ1···µk
+ν1···νl= dAµ1···µk
+ν1···νl+k/summationdisplay
+i=1Γµi
+ρ∧Aµ1···µi−1ρµi+1···µk
+ν1···νl
+−l/summationdisplay
+i=1Γρ
+νi∧Aµ1···µk
+ν1···νi−1ρνi+1···νl,(3.22)
+where the exterior derivative d is given by
+dAµ1···µk
+ν1···νl=1
+p!∂σAµ1···µk
+ν1···νlρ1···ρpdxσ∧dxρ1∧···∧dxρp. (3.23)
+Dsatisfies the same Leibniz rule as d,
+D/parenleftbig
+Aµ1···µk
+ν1···νl∧Bρ1···ρm
+σ1···σn/parenrightbig
+=DAµ1···µk
+ν1···νl∧Bρ1···ρm
+σ1···σn
++(−1)deg(A)Aµ1···µkν1···νl∧DBρ1···ρm
+σ1···σn.(3.24)
+Theexterior covariant derivative ˜Doftheotherconnectionisdefined analogouslyby using
+the connection one-form ˜Γµ
+νinstead of Γµ
+ν.
+A connection also provides the covariant derivative on Ω( M,T)
+∇µ: Ωp(M,Tk,l)→Ωp(M,Tk,l+1) (3.25)
+that is defined in (2.31) and (2.33) (see also (2.34)–(2.35) for the pr esent case of a coor-
+dinate basis). The covariant derivative of a tensor-valued differen tial form can be written
+in a compact form as
+∇ρAµ1···µkν1···νl=∂ρAµ1···µkν1···νl+k/summationdisplay
+i=1Γµi
+ρσAµ1···µi−1σµi+1···µkν1···νl
+−l/summationdisplay
+i=1Γσ
+ρνiAµ1···µk
+ν1···νi−1σνi+1···νl−˜Γσ
+ρ∧iσAµ1···µk
+ν1···νl,(3.26)
+where we denote
+∂σAµ1···µk
+ν1···νl=1
+p!∂σAµ1···µk
+ν1···νlρ1···ρpdxρ1∧···∧dxρp. (3.27)
+Defintion for the other covariant derivative ˜∇µis analogous (replace Γρ
+µνwith Γρ
+νµand˜Γρ
+µ
+with Γρ
+µ). When the second covariant derivative ∇ρ∇σAµ1···µkν1···νlis taken, the subscript
+σis treated as a covariant tensor index. The commutator of second covariant derivatives
+reads
+[∇ρ,∇σ]Aµ1···µk
+ν1···νl=−Tλ
+ρσ∇λAµ1···µk
+ν1···νl+k/summationdisplay
+i=1Rµi
+λρσAµ1···µi−1λµi+1···µk
+ν1···νl
+−l/summationdisplay
+i=1Rλ
+νiρσAµ1···µk
+ν1···νi−1λνi+1···νl−Rλ
+τρσdxτ∧iλAµ1···µkν1···νl.
+(3.28)
+11The covariant derivative has the Leibniz rule
+∇λ/parenleftbig
+Aµ1···µk
+ν1···νl∧Bρ1···ρm
+σ1···σn/parenrightbig
+=∇λAµ1···µk
+ν1···νl∧Bρ1···ρm
+σ1···σn
++Aµ1···µk
+ν1···νl∧∇λBρ1···ρm
+σ1···σn.(3.29)
+We can find a useful relation for Dand∇ρby multiplying (3.26) with d xρ∧from left
+dxρ∧∇ρAµ1···µkν1···νl=DAµ1···µkν1···νl−Tρ∧iρAµ1···µkν1···νl. (3.30)
+We can even write it as a local operator identity
+D= dxµ∧∇µ+Tµ∧iµ. (3.31)
+Once again a similar relation holds for the other connection
+˜D= dxµ∧˜∇µ−Tµ∧iµ. (3.32)
+We shall occasionally refer to both ∇andDas the connection — similarly for ˜∇and
+˜D.
+The Poisson bracket
+Now we can extend the Poisson bracket (3.11) for tensor-valued d ifferential forms
+/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
+=θλτ∇λAµ1···µk
+ν1···νl∧∇τBρ1···ρm
+σ1···σn
++(−1)deg(A)˜Rλτ∧iλAµ1···µk
+ν1···νl∧iτBρ1···ρm
+σ1···σn.(3.33)
+If eitherAµ1···µkν1···νlorBρ1···ρmσ1···σn(or both) is a tensor field of zero form degree, the
+Poisson bracket is defined by
+/braceleftbig
+Aµ1···µkν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
+=θλτ∇λAµ1···µkν1···νl∇τBρ1···ρm
+σ1···σn,(3.34)
+which is also consistent with (3.1) and (3.3).12
+The Poisson bracket (3.33) of tensor-valued differential forms sa tisfies the following
+properties of the graded differential Poisson algebra. For A∈Ωp(M,Tk,l) andB∈
+Ωq(M,Tm,n) andC∈Ωr(M,Ti,j) we have:
+1. Bracket degree
+deg/parenleftBig/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig/parenrightBig
+= deg(A)+deg(B), (3.35)
+is implied by the following properties. The covariant derivative (3.25) d oes not
+change the degree of tensor-valued differential forms, deg( ∇µA) = deg(A). The
+interior product (3.20) reduces the degree by one, deg( iµA) = deg(A)−1. The
+exterior product (3.17) has the degree (3.18).
+2. Graded symmetry
+/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
+= (−1)deg(A)deg(B)+1/braceleftbig
+Bρ1···ρm
+σ1···σn,Aµ1···µk
+ν1···νl/bracerightbig
+,
+(3.36)
+follows from the symmetry property of the exterior product (3.19 ) and from the
+antisymmetry of θµνand the symmetry of ˜Rµνunderµ↔ν.
+12We essentially consider that the interior product of a zero-form is z ero.
+123. Graded product
+/braceleftbig
+Aµ1···µk
+ν1···νl∧Bρ1···ρm
+σ1···σn,Cλ1···λi
+τ1···τj/bracerightbig
+=Aµ1···µk
+ν1···νl∧/braceleftbig
+Bρ1···ρm
+σ1···σn,Cλ1···λi
+τ1···τj/bracerightbig
++(−1)deg(B)deg(C)/braceleftbig
+Aµ1···µk
+ν1···νl,Cλ1···λi
+τ1···τj/bracerightbig
+∧Bρ1···ρm
+σ1···σn,(3.37)
+follows from the Leibniz rule for ∇µ(3.29) and the similar property for iµ(3.21)
+and the symmetry property of the exterior product (3.19).
+4. Leibniz rule
+D/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
+=/braceleftbig
+DAµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
++(−1)deg(A)/braceleftbig
+Aµ1···µk
+ν1···νl,DBρ1···ρm
+σ1···σn/bracerightbig
+,(3.38)
+By applying the Leibniz rule (3.24) to the left-hand side of (3.38) we ob tain
+D/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
+=Dθλτ∧∇λAµ1···µk
+ν1···νl∧∇τBρ1···ρm
+σ1···σn
++θλτ/parenleftBig
+D∇λAµ1···µkν1···νl∧∇τBρ1···ρm
+σ1···σn
++(−1)deg(A)∇λAµ1···µk
+ν1···νl∧D∇τBρ1···ρm
+σ1···σn/parenrightBig
++(−1)deg(A)D˜Rλτ∧iλAµ1···µk
+ν1···νl∧iτBρ1···ρm
+σ1···σn
++(−1)deg(A)˜Rλτ∧/parenleftBig
+DiλAµ1···µkν1···νl∧iτBρ1···ρm
+σ1···σn
++(−1)deg(A)−1iλAµ1···µk
+ν1···νl∧DiτBρ1···ρm
+σ1···σn/parenrightBig
+.(3.39)
+Thenweuse(3.31)tocalculatetherelationof D∇λAµ1···µkν1···νland∇λDAµ1···µkν1···νl.
+First we calculate
+D∇µ= dxν∧∇ν∇µ+Tν∧iν∇µ (3.40)
+and
+∇µD= dxν∧∇µ∇ν+∇µTν∧iν+Tν∧∇µiν (3.41)
+and find out that ∇µandiνcommute (follows from (3.20) and (3.26) by direct
+calculation, recalling iνiµ=−iµiν)
+iν∇µAµ1···µk
+ν1···νl=∇µiνAµ1···µk
+ν1···νl, (3.42)
+which then together imply
+D∇µ=∇µD+dxν∧[∇ν,∇µ]−∇µTν∧iν. (3.43)
+Thus we obtain the relation
+D∇λAµ1···µk
+ν1···νl=∇λDAµ1···µk
+ν1···νl+dxρ∧[∇ρ,∇λ]Aµ1···µk
+ν1···νl
+−∇λTρ∧iρAµ1···µk
+ν1···νl.(3.44)
+This result can as well be derived directly from the definitions (3.22) a nd (3.26),
+but it is a lengthy calculation. By using the definitions (3.22) and (3.20) we obtain
+the relation of DiλAµ1···µkν1···νlandiλDAµ1···µkν1···νl,
+(Diλ+iλD)Aµ1···µkν1···νl=∇λAµ1···µkν1···νl+iλTρ∧iρAµ1···µkν1···νl,(3.45)
+13where we have also used (3.23), (3.26) and iλTρ=˜Γρ
+λ−Γρ
+λ. Introducing the results
+(3.44) and (3.45) into (3.39) yields
+D/braceleftbig
+Aµ1···µkν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
+=/braceleftbig
+DAµ1···µkν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
++(−1)deg(A)/braceleftbig
+Aµ1···µk
+ν1···νl,DBρ1···ρm
+σ1···σn/bracerightbig
++Dθλτ∧∇λAµ1···µk
+ν1···νl∧∇τBρ1···ρm
+σ1···σn
++(−1)deg(A)/parenleftBig
+D˜Rλτ+˜Rφτ∧iφTλ+˜Rλφ∧iφTτ/parenrightBig
+∧iλAµ1···µkν1···νl∧iτBρ1···ρm
+σ1···σn
++θφτ/parenleftBig
+˜Rλ
+φ−∇φTλ/parenrightBig
+∧iλAµ1···µk
+ν1···νl∧∇τBρ1···ρm
+σ1···σn
++(−1)deg(A)θλφ/parenleftBig
+˜Rτ
+φ−∇φTτ/parenrightBig
+∧∇λAµ1···µkν1···νl∧iτBρ1···ρm
+σ1···σn
++θλτdxφ∧/parenleftBig
+[∇φ,∇λ]Aµ1···µk
+ν1···νl∧∇τBρ1···ρm
+σ1···σn
++∇λAµ1···µk
+ν1···νl∧[∇φ,∇τ]Bρ1···ρm
+σ1···σn/parenrightBig
+,(3.46)
+where some regrouping and simplifications have been done. For furt her simplifica-
+tion we calculate
+D˜Rµν+˜Rρν∧iρTµ+˜Rµρ∧iρTν= d˜Rµν+˜Γµ
+ρ∧˜Rρν+˜Γν
+ρ∧˜Rµρ
+=˜D˜Rµν=˜D/parenleftBig
+θµρ˜Rν
+ρ/parenrightBig
+=˜Dθµρ∧˜Rν
+ρ,(3.47)
+where (2.30) has been used in the last equality. As a final step we intr oduce (3.28)
+into the right-hand side of (3.46) and combine the contributions of t he first and the
+last term of (3.28) to the third, fifth and sixth term of (3.46). In th e third term of
+the resulting expression we calculate
+Dθµν+θρνiρTµ+θµρiρTν=˜Dθµν. (3.48)
+Thus we obtain the result
+D/braceleftbig
+Aµ1···µkν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
+=/braceleftbig
+DAµ1···µkν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
++(−1)deg(A)/braceleftbig
+Aµ1···µk
+ν1···νl,DBρ1···ρm
+σ1···σn/bracerightbig
++˜Dθλτ∧∇λAµ1···µk
+ν1···νl∧∇τBρ1···ρm
+σ1···σn
++(−1)deg(A)˜Dθλφ∧˜Rτ
+φ∧iλAµ1···µk
+ν1···νl∧iτBρ1···ρm
+σ1···σn
++θφτ/parenleftBig
+˜Rλ
+φ−∇φTλ+iφRλ
+χ∧dxχ/parenrightBig
+∧iλAµ1···µk
+ν1···νl∧∇τBρ1···ρm
+σ1···σn
++(−1)deg(A)θλφ/parenleftBig
+˜Rτ
+φ−∇φTτ+iφRτ
+χ∧dxχ/parenrightBig
+∧∇λAµ1···µk
+ν1···νl∧iτBρ1···ρm
+σ1···σn
+−θλτ/bracketleftBigg/parenleftBiggk/summationdisplay
+i=1iλRµi
+φ∧Aµ1···µi−1φµi+1···µk
+ν1···νl−l/summationdisplay
+i=1iλRφ
+νi∧Aµ1···µk
+ν1···νi−1φνi+1···νl/parenrightBigg
+∧
+∧∇τBρ1···ρm
+σ1···σn+∇λAµ1···µkν1···νl∧
+∧/parenleftBiggm/summationdisplay
+i=1iτRµi
+φ∧Bρ1···ρi−1φρi+1···ρm
+σ1···σn−n/summationdisplay
+i=1iτRφ
+νi∧Bρ1···ρm
+σ1···σi−1φσi+1···σn/parenrightBigg/bracketrightBigg
+.
+(3.49)
+Hence for the Leibniz rule (3.38) to hold for arbitrary tensor-value d differential
+forms, we have to introduce the following constraints:
+141. The connection ˜Dis symplectic
+˜Dθµν=/parenleftBig
+˜∇ρθµν/parenrightBig
+dxρ= 0. (3.50)
+2. The interior product of the curvature of ∇vanishes
+iµRν
+ρ= 0. (3.51)
+This implies that the curvature of ∇has to vanish, Rν
+ρ=1
+2dxµ∧iµRν
+ρ= 0.
+3. The curvature two-forms and the torsion two-form satisfy
+˜Rµ
+ν−∇νTµ+iνRµ
+ρ∧dxρ= 0. (3.52)
+Taking (3.51) into account we obtain
+˜Rµ
+ν=∇νTµ. (3.53)
+It would be quite tempting to require that the other connection ˜Dsatisfies a similar
+Leibniz rule as (3.38). Such property would impose additional constr aints on the
+connections and further restrict the geometry. However, we do not require such
+property for ˜D, because the Poisson bracket (3.33) has been defined with ∇, not
+with˜∇, which makes Dthe natural choice for the Leibniz rule (3.38).
+5. Graded Jacobi identity
+/braceleftBig
+Aµ1···µk
+ν1···νl,/braceleftbig
+Bρ1···ρm
+σ1···σn,Cλ1···λi
+τ1···τj/bracerightbig/bracerightBig
++(−1)deg(A)[deg(B)+deg(C)]/braceleftBig
+Bρ1···ρm
+σ1···σn,/braceleftbig
+Cλ1···λi
+τ1···τj,Aµ1···µk
+ν1···νl/bracerightbig/bracerightBig
++(−1)[deg(A)+deg(B)]deg(C)/braceleftBig
+Cλ1···λi
+τ1···τj,/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig/bracerightBig
+= 0,(3.54)
+First we calculate the Poisson bracket
+/braceleftBig
+Aµ1···µk
+ν1···νl,/braceleftbig
+Bρ1···ρm
+σ1···σn,Cλ1···λi
+τ1···τj/bracerightbig/bracerightBig
+=θφ1χ1∇χ1θφ2χ2∇φ1Aµ1···µk
+ν1···νl∧∇φ2Bρ1···ρm
+σ1···σn∧∇χ2Cλ1···λi
+τ1···τj
++θφ1χ1θφ2χ2/parenleftBig
+∇φ1Aµ1···µkν1···νl∧∇χ1∇φ2Bρ1···ρm
+σ1···σn∧∇χ2Cλ1···λi
+τ1···τj
++∇φ1Aµ1···µk
+ν1···νl∧∇φ2Bρ1···ρm
+σ1···σn∧∇χ1∇χ2Cλ1···λi
+τ1···τj/parenrightBig
++(−1)deg(B)θφ1χ1∇χ1˜Rφ2χ2∧∇φ1Aµ1···µk
+ν1···νl∧iφ2Bρ1···ρm
+σ1···σn∧iχ2Cλ1···λi
+τ1···τj
++θφ1χ1˜Rφ2χ2∧/parenleftBig
+(−1)deg(B)∇φ1Aµ1···µk
+ν1···νl∧iφ2∇χ1Bρ1···ρm
+σ1···σn∧iχ2Cλ1···λi
+τ1···τj
++(−1)deg(B)∇φ1Aµ1···µkν1···νl∧iφ2Bρ1···ρm
+σ1···σn∧iχ2∇χ1Cλ1···λi
+τ1···τj
++(−1)deg(A)iφ2Aµ1···µk
+ν1···νl∧iχ2∇φ1Bρ1···ρm
+σ1···σn∧∇χ1Cλ1···λi
+τ1···τj
++(−1)deg(A)+deg(B)iφ2Aµ1···µk
+ν1···νl∧∇φ1Bρ1···ρm
+σ1···σn∧iχ2∇χ1Cλ1···λi
+τ1···τj/parenrightBig
++(−1)deg(B)−1˜Rφ1χ1∧iχ1˜Rφ2χ2∧iφ1Aµ1···µk
+ν1···νl∧iφ2Bρ1···ρm
+σ1···σn∧iχ2Cλ1···λi
+τ1···τj
++(−1)deg(A)+deg(B)˜Rφ1χ1∧˜Rφ2χ2∧/parenleftBig
+iφ1Aµ1···µk
+ν1···νl∧iχ1iφ2Bρ1···ρm
+σ1···σn∧iχ2Cλ1···λi
+τ1···τj
++(−1)deg(B)+1iφ1Aµ1···µk
+ν1···νl∧iφ2Bρ1···ρm
+σ1···σn∧iχ1iχ2Cλ1···λi
+τ1···τj/parenrightBig
+,(3.55)
+15wherewehaveused(3.42). Cyclingthrough Aµ1···µkν1···νl,Bρ1···ρmσ1···σnandCλ1···λi
+τ1···τj,
+using the symmetry property (3.19) of the exterior product and in troducing the ex-
+pression (3.28) for the commutators of covariant derivatives, giv es the left-hand side
+of the graded Jacobi identity (3.54) as
+/braceleftBig
+Aµ1···µkν1···νl,/braceleftbig
+Bρ1···ρm
+σ1···σn,Cλ1···λi
+τ1···τj/bracerightbig/bracerightBig
++(−1)deg(A)[deg(B)+deg(C)]/braceleftBig
+Bρ1···ρm
+σ1···σn,/braceleftbig
+Cλ1···λi
+τ1···τj,Aµ1···µk
+ν1···νl/bracerightbig/bracerightBig
++(−1)[deg(A)+deg(B)]deg(C)/braceleftBig
+Cλ1···λi
+τ1···τj,/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig/bracerightBig
+=/bracketleftbigg
+θφ1χ1/parenleftbig
+∇χ1θφ2χ2−θφ2ψTχ2
+χ1ψ/parenrightbig
++θφ2χ1/parenleftBig
+∇χ1θχ2φ1−θχ2ψTφ1
+χ1ψ/parenrightBig
++θχ2χ1/parenleftBig
+∇χ1θφ1φ2−θφ1ψTψ2
+χ1ψ/parenrightBig/bracketrightbigg
+∇φ1Aµ1···µk
+ν1···νl∧∇φ2Bρ1···ρm
+σ1···σn∧∇φ3Cλ1···λi
+τ1···τj
++θφ1χ1θφ2χ2/bracketleftBigg/parenleftBiggk/summationdisplay
+i=1Rµi
+ψφ1φ2Aµ1···µi−1ψµi+1···µk
+ν1···νl−l/summationdisplay
+i=1Rψ
+νiφ1φ2Aµ1···µk
+ν1···νi−1ψνi+1···νl/parenrightBigg
+∧
+∧∇χ1Bρ1···ρm
+σ1···σn∧∇χ2Cλ1···λi
+τ1···τj+∇φ1Aµ1···µk
+ν1···νl∧
+∧/parenleftBiggk/summationdisplay
+i=1Rρi
+ψχ1φ2Bρ1···ρi−1ψρi+1···ρm
+σ1···σn−l/summationdisplay
+i=1Rψ
+σiχ1φ2Bρ1···ρm
+σ1···σi−1ψσi+1···σn/parenrightBigg
+∧
+∧∇χ2Cλ1···λi
+τ1···τj+∇φ1Aµ1···µk
+ν1···νl∧∇φ2Bρ1···ρm
+σ1···σn∧
+∧/parenleftBiggk/summationdisplay
+i=1Rλi
+ψχ1χ2Cλ1···λi−1ψλi+1···λi
+τ1···τj−l/summationdisplay
+i=1Rψ
+τiχ1χ2Cλ1···λi
+τ1···τi−1ψτi+1···τj/parenrightBigg
+−Rψ
+ωφ1φ2dxω∧/parenleftBig
+iψAµ1···µk
+ν1···νl∧∇χ1Bρ1···ρm
+σ1···σn∧∇χ2Cλ1···λi
+τ1···τj
++(−1)deg(A)+1∇χ1Aµ1···µk
+ν1···νl∧iψBρ1···ρm
+σ1···σn∧∇χ2Cλ1···λi
+τ1···τj
++(−1)deg(A)+deg(B)∇χ1Aµ1···µk
+ν1···νl∧∇χ2Bρ1···ρm
+σ1···σn∧iψCλ1···λi
+τ1···τj/parenrightBig/bracketrightBigg
++θφ1χ1∇χ1˜Rφ2χ2∧/parenleftBig
+(−1)deg(B)∇φ1Aµ1···µk
+ν1···νl∧iφ2Bρ1···ρm
+σ1···σn∧iχ2Cλ1···λi
+τ1···τj
++(−1)deg(A)+deg(B)+1iχ2Aµ1···µk
+ν1···νl∧∇φ1Bρ1···ρm
+σ1···σn∧iφ2Cλ1···λi
+τ1···τj
++(−1)deg(A)iφ2Aµ1···µk
+ν1···νl∧iχ2Bρ1···ρm
+σ1···σn∧∇φ2Cλ1···λi
+τ1···τj/parenrightBig
++(−1)deg(B)−1/parenleftBig
+˜Rφ1χ1∧iχ1˜Rφ2χ2+˜Rφ2χ1∧iχ1˜Rχ2φ1+˜Rχ2χ1∧iχ1˜Rφ1φ2/parenrightBig
+∧
+∧iφ1Aµ1···µkν1···νl∧iφ2Bρ1···ρm
+σ1···σn∧iχ2Cλ1···λi
+τ1···τj.(3.56)
+Since the graded Jacobi identity (3.54) requires that the right-ha nd side of (3.56)
+vanishes, we have to introduce the following constraints:
+1. A covariant version of the Jacobi identity for the Poisson bivect or
+/summationdisplay
+(µ,ν,ρ)θµσ/parenleftbig
+∇σθνρ−θνλTρ
+σλ/parenrightbig
+=/summationdisplay
+(µ,ν,ρ)θµσθνλTρ
+σλ= 0,(3.57)
+where (3.10) has been used in the first equality. This constraint is alr eady
+satisfied (3.7).
+162. The curvature tensor of the connection ∇vanishes (3.12).
+3. The curvature two-form of the connection ˜∇is covariantly constant under ∇,
+∇µ˜Rνρ= 0. (3.58)
+Thisisequivalent tothecurvaturetensor of ˜∇havingthesameproperty (3.13).
+4. The curvature ˜Rµνsatisfies (3.14).
+Comparing the constraints needed to satisfy the graded different ial Poisson algebra of
+tensor-valued differential forms to the constraints (3.6) and (3.1 2)–(3.14) for differential
+forms obtained in the literature, we find that there is no need for ne w constraints. There
+are new conditions (3.51), (3.53) and (3.57) on the connections, bu t they are all satisfied
+due to the vanishing of the curvature of the connection ∇(3.12), the definition of the
+two connections (3.4) in terms of the same set of connection coeffic ients and the covariant
+Jacobi identities (3.7).13Thus this generalization to tensor-valued differential forms does
+not require any additional constraints on the connections.
+It has been shown [9] that due to the constraints (3.12) and (3.6) t here exists a lo-
+cal coordinate system {Φα}where the connection coefficients are given in terms of the
+invertible Poisson bivector θαβ={Φα,Φβ}as
+Γα
+βγ=θαδ∂βωδγ. (3.59)
+Here we refer to these coordinates by the first part of the alphab etα,β,γ,... . The form
+(3.59) of the connection coefficients Γα
+βγis covariant under the group of affine transforma-
+tions of the coordinates Φα,
+Φα→Nα
+βΦβ+Vα, (3.60)
+whereNα
+βandVαare constants, since both sides of (3.59) transform like tensors u nder
+such affine transformations. The torsion tensor and the (nonvan ishing) curvature tensor
+are, of course, also given by the Poisson structure in these coord inates, e.g. Tα
+βγ=
+θαδ∂δωβγ. Another special basis is provided by the one-forms PαβdΦβ, with respect to
+which the connection ∇is trivial, that simplifies many calculations. Most importantly
+one finds that the Poisson bivector is quadratic in the coordinates Φαby solving the
+identity ˜Rγδ
+αβ=∂βTγδ
+αfor the torsion and then the torsion Tβγ
+α=∂αθβγforθαβ,14
+θαβ={Φα,Φβ}=1
+2˜Rαβ
+γδΦγΦδ+fαβ
+γΦγ+gαβ, (3.61)
+where˜Rγδ
+αβ,fαβ
+γandgαβare constants (all antisymmetric under α↔β). This is some-
+what analogous to Darboux’s theorem for symplectic geometry.
+We provide some further analysis on the constraints imposed on the connections. First
+wecalculatethevanishingcovariantderivative ∇µof˜Rνρ(3.58)byusingtheformula(3.10)
+that is implied by the symplecticity of ˜∇:
+∇µ˜Rνρ=∇µ/parenleftBig
+θνσ˜Rρ
+σ/parenrightBig
+=/parenleftbig
+Tν
+µλθλσ+Tσ
+µλθνλ/parenrightbig˜Rρ
+σ+θνσ∇µ˜Rρ
+σ= 0.(3.62)
+13See [9, 3] for how the condition (3.53) is implied by the definition of the t wo connections (3.4), the
+vanishing of the curvature of the connection ∇(3.12) and the so called first Bianchi identity (2.29) in its
+tensorial form.
+14Here we have used θαβandωαβto raise and lower indices respectively. See [9] for details.
+17Multiplying by the symplectic form ωτν(sum overν), introducing the constraint (3.53)
+and renaming some of the indices yields
+/parenleftbig
+Tλ
+µτωνλθτσ+Tσ
+µν/parenrightbig
+∇σTρ+∇µ∇νTρ= 0. (3.63)
+Thus the second covariant derivatives of the torsion can be writte n in terms of first
+covariant derivatives of the torsion multiplied by the torsion, the Po isson bivector and the
+symplectic form.
+Let us consider the antisymmetric and the symmetric parts of (3.63 ) with respect
+to the indices µandν. According to (3.28) and the vanishing of the curvature of the
+connection∇(3.12) we have [∇µ,∇ν] =−Tρ
+µν∇ρ. Hence we can decompose
+∇µ∇ν=∇(µ∇ν)+∇[µ∇ν]=∇(µ∇ν)−1
+2Tρ
+µν∇ρ. (3.64)
+Thus the antisymmetric part of (3.63) is
+1
+2/parenleftBig/parenleftbig
+Tλ
+µτωνλ−Tλ
+ντωµλ/parenrightbig
+θτσ+Tσ
+µν/parenrightBig
+∇σTρ= 0. (3.65)
+Assuming (3.53) does not vanish, (3.65) implies
+/parenleftbig
+Tλ
+µτωνλ−Tλ
+ντωµλ/parenrightbig
+θτσ+Tσ
+µν= 0 (3.66)
+or equivalently/summationdisplay
+(µ,ν,ρ)Tσ
+µνωσρ= 0. (3.67)
+Together (3.7) and (3.67) impose a fairly strict set of conditions on t he torsion — though
+not enough to fix it completely.
+The symmetric part of (3.63), which can be written
+∇(µ∇ν)Tρ=1
+2/parenleftbig
+Tσ
+µλωνσ+Tσ
+νλωµσ/parenrightbig
+θλτ∇τTρ, (3.68)
+does not provide such an interesting result.
+3.3 Star product
+The star product for tensor-valued differential forms can be defi ned similarly as in [3]
+Aµ1···µk
+ν1···νl⋆Bρ1···ρm
+σ1···σn=Aµ1···µk
+ν1···νl∧Bρ1···ρm
+σ1···σn
++∞/summationdisplay
+n=1/planckover2pi1nCn/parenleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/parenrightbig
+,(3.69)
+whereCnare bilinear covariant differential operators of at most order nin each argument,
+which are constructed from the covariant derivatives ∇µ, the Poisson bivector θ, the
+torsion tensor and the curvature tensor(s). Further the oper atorsCnare chosen so that
+the star product (3.69) satisfies the following properties:
+1. The star product is associative
+Aµ1···µk
+ν1···νl⋆/parenleftbig
+Bρ1···ρm
+σ1···σn⋆Cλ1···λi
+τ1···τj/parenrightbig
+=/parenleftbig
+Aµ1···µk
+ν1···νl⋆Bρ1···ρm
+σ1···σn/parenrightbig
+⋆Cλ1···λi
+τ1···τj.(3.70)
+182. The first order deformation is given by the Poisson bracket (3.33 )
+C1/parenleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/parenrightbig
+=/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
+.(3.71)
+3. The constant function, M∋x/mapsto→1, is the identity: 1 ⋆A=A⋆1 =A.
+4. EveryCnis of ordernin the Poisson bivector θ(including its covariant derivatives
+(3.10) and the curvature (3.9)) and it has the degree
+deg/parenleftBig
+Cn/parenleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/parenrightbig/parenrightBig
+= deg(A)+deg(B).(3.72)
+5. The operators Cnhave the generalized Moyal symmetry
+Cn/parenleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/parenrightbig
+= (−1)deg(A)deg(B)+nCn/parenleftbig
+Bρ1···ρm
+σ1···σn,Aµ1···µk
+ν1···νl/parenrightbig
+.
+(3.73)
+To the second order in the deformation parameter /planckover2pi1the star product is given by
+C2/parenleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/parenrightbig
+=1
+2θλ1τ1θλ2τ2∇λ1∇λ2Aµ1···µk
+ν1···νl∧∇τ1∇τ2Bρ1···ρm
+σ1···σn
++1
+3/parenleftbigg
+θλ1τ1∇τ1θλ2τ2+1
+2θλ2φθτ2χTλ1
+φχ/parenrightbigg/parenleftBig
+∇λ1∇λ2Aµ1···µk
+ν1···νl∧∇τ2Bρ1···ρm
+σ1···σn
++∇τ2Aµ1···µk
+ν1···νl∧∇λ1∇λ2Bρ1···ρm
+σ1···σn/parenrightBig
++(−1)deg(A)θλ1τ1˜Rλ2τ2∧∇λ1iλ2Aµ1···µk
+ν1···νl∧∇τ1iτ2Bρ1···ρm
+σ1···σn
+−1
+2˜Rλ1τ1∧˜Rλ2τ2∧iλ1iλ2Aµ1···µkν1···νl∧iτ1iτ2Bρ1···ρm
+σ1···σn
+−1
+3˜Rλ1τ1∧iτ1˜Rλ2τ2∧/parenleftBig
+(−1)deg(A)iλ1iλ2Aµ1···µkν1···νl∧iτ2Bρ1···ρm
+σ1···σn
++iλ2Aµ1···µk
+ν1···νl∧iλ1iτ2Bρ1···ρm
+σ1···σn/parenrightBig
+.(3.74)
+The second term of (3.74) can be simplified by using (3.10) and (3.7),
+θµσ∇σθνρ+1
+2θνσθρλTµ
+σλ=−1
+2θνσθρλTµ
+σλ, (3.75)
+but we choose to keep the similarity with the star product of [3].15Proof of the asso-
+ciativity of the star product (3.69) to O(/planckover2pi12) is completely analogous with [3].16At the
+classical levelO(1) the associativity is trivially implied by the associativity of the exter ior
+product. AtO(/planckover2pi1) the associativity is implied by the graded symmetry rule (3.37). At
+15There is a sign difference in the second factor of the second term of (3.74) compared to [3] that is
+enabled by the antisymmetry of the first factor under λ2↔τ2. The motivation for this cosmetic change
+is to emphasize the symmetry property (3.73) of C2.
+16Due to the vanishing of the curvature of the connection ∇the tensorial indices can mostly be ignored
+in the calculation verifying the associativity (3.70) to O(/planckover2pi12).
+19O(/planckover2pi12) the associativity condition
+Aµ1···µkν1···νl∧C2/parenleftBig
+Bρ1···ρm
+σ1···σn,Cλ1···λi
+τ1···τj/parenrightBig
+−C2/parenleftBig
+Aµ1···µk
+ν1···νl∧Bρ1···ρm
+σ1···σn,Cλ1···λi
+τ1···τj/parenrightBig
++C2/parenleftBig
+Aµ1···µkν1···νl,Bρ1···ρm
+σ1···σn∧Cλ1···λi
+τ1···τj/parenrightBig
+−C2/parenleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/parenrightbig
+∧Cλ1···λi
+τ1···τj
+=C1/parenleftBig
+C1/parenleftbig
+Aµ1···µkν1···νl,Bρ1···ρm
+σ1···σn/parenrightbig
+,Cλ1···λi
+τ1···τj/parenrightBig
+−C1/parenleftBig
+Aµ1···µk
+ν1···νl,C1/parenleftBig
+Bρ1···ρm
+σ1···σn,Cλ1···λi
+τ1···τj/parenrightBig/parenrightBig(3.76)
+can be shown to hold by using the properties of the Poisson bracket , the constraints these
+properties imply — namely (3.6), (3.7), (3.12), (3.13) and (3.14) — and the properties of
+the covariant derivative and the interior product — including the com mutativity of the
+two (3.42), iµiν=−iνiµand the decomposition (3.64).
+As discussed in [3] the next order /planckover2pi13deformation could be derived with a considerable
+amount of calculation by finding an ansatz that satisfies the require d conditions.
+If the torsion vanishes, we have a flat symplectic connection ∇. Then the star product
+(3.69) can be defined by
+Aµ1···µk
+ν1···νl⋆Bρ1···ρm
+σ1···σn/vextendsingle/vextendsingle
+T=0=Aµ1···µk
+ν1···νl∧Bρ1···ρm
+σ1···σn
++∞/summationdisplay
+n=1/planckover2pi1n
+n!θλ1τ1···θλnτn∇λ1···∇λnAµ1···µk
+ν1···νl∧∇τ1···∇τnBρ1···ρm
+σ1···σn,(3.77)
+since now the covariant derivatives commute both with each other a nd with the Poisson
+bivectorθµν.
+3.4 On the algebra of tensors
+Thus by starting from the graded differential Poisson structure o n the algebra of forms
+Ω(M), we have generalized it to the algebra of tensor-valued differentia l forms (2.13) and
+consequently to the subalgebra of all tensor fields on M,
+T(M) =∞/circleplusdisplay
+k,l=0Ω0(M,Tk,l)⊂Ω(M,T). (3.78)
+For such tensor-valued zero-forms the Poisson bracket (3.33) is reduced to (3.34) and in
+the star product,
+Aµ1···µk
+ν1···νl⋆Bρ1···ρm
+σ1···σn=Aµ1···µk
+ν1···νlBρ1···ρm
+σ1···σn
++∞/summationdisplay
+n=1/planckover2pi1nCn/parenleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/parenrightbig
+,(3.79)
+the deformation of order /planckover2pi12is written
+C2/parenleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/parenrightbig
+=1
+2θλ1τ1θλ2τ2∇λ1∇λ2Aµ1···µk
+ν1···νl∇τ1∇τ2Bρ1···ρm
+σ1···σn
++1
+3/parenleftbigg
+θλ1τ1∇τ1θλ2τ2+1
+2θλ2φθτ2χTλ1
+φχ/parenrightbigg/parenleftBig
+∇λ1∇λ2Aµ1···µk
+ν1···νl∇τ2Bρ1···ρm
+σ1···σn
++∇τ2Aµ1···µk
+ν1···νl∇λ1∇λ2Bρ1···ρm
+σ1···σn/parenrightBig
+.(3.80)
+20In the case of vanishing torsion we obtain the simple star product of tensor fields
+Aµ1···µk
+ν1···νl⋆Bρ1···ρm
+σ1···σn/vextendsingle/vextendsingle
+T=0=Aµ1···µk
+ν1···νlexp/parenleftBig
+/planckover2pi1← −∇λθλτ− →∇τ/parenrightBig
+Bρ1···ρm
+σ1···σn.(3.81)
+In the recent work [5] a covariant star product of functions was d efined on a symplectic
+manifold with vanishing torsion and curvature ( T=R= 0). It was also proposed that
+this star product could be straightforwardly extented for tenso r fields. We recognize that
+the reduced ( T=R=˜R= 0) case (3.81) of our more general star product of tensor fields
+(3.79) is exactly what the extension of the star product of [5] to te nsor fields would be.
+3.5 Discussion
+When we consider possible applicationsof these star products (3.69 ) and(3.79) in physics,
+particularly gravity and gauge theory, the problem (perhaps also a possibility) is that the
+structure of the graded differential Poisson algebra of (tensor- valued) differential forms
+requires strict constraints on the underlying symplectic manifold. D ue to the required
+constraints the torsion and the curvature are rather restricte d, which is likely to cause
+some problems particularly for theories of gravity. Still the connec tion∇can have a
+nonvanishing torsion and in this case the symplectic connection ˜∇also has curvature.
+This should open up the possibility for some nontrivial gravitational d ynamics.
+In the extremely restricted ( T=R=˜R= 0) case (3.81) that was also recently
+studied in [5] there is virtually impossible to have a nontrivial theory of gravity, because
+neither the energy-momentum tensor nor the spin density tensor are supported due to
+the vanishing of both the curvature and the torsion. Setting up th e equivalence principle
+would clearly be impossible. Thus this star product (3.81) can be used only in cases where
+the curvature and the torsion vanish in the corresponding commut ative theory. Then in
+the noncommutative extension of the theory we would find correct ions to the geometrical
+objects in the higher orders of the deformation parameter /planckover2pi1due to the star product. In
+the gravitational field equations these corrections would require c ompensating corrections
+to the energy-momentum tensor and possibly to the spin density te nsor depending on the
+chosen action. This is problematic since, as we noted, matter fields a re not supported in
+this case.17An example of such theory is the two-dimensional noncommutative d ilaton
+gravity studied in [5].
+In the case of gauge theory these restrictions are not quite as se vere as in the case
+of gravity. Noncommutative gauge theory with Yang-Mills actions ha s been studied [11,
+12, 13] in this setting. The former work employed the popular Seiber g-Witten map [14].
+In [12, 13] the star product of differential forms was generalized t o Lie algebra-valued
+differential forms in order to be able to apply it to the connection one -form of the gauge
+theory, as well as to the gauge transformation parameter and th e field strength, which are
+allLiealgebra-valued. Thisgeneralizationisfairlysimpletoachievesinc ethegeneratorsof
+theinternalgaugesymmetrycommutewiththecovariantderivatio n∇. Thegeneralization
+to tensor-valued differential forms we have presented can be fur ther generalized to Lie
+algebra-valued objects,
+Aµ1···µk
+ν1···νl=Aµ1···µka
+ν1···νlTa, (3.82)
+17Such corrections to the right hand (energy-momentum) side of fie ld equations frequently appear in
+noncommutativetheories of gravitywhen a star product is introdu ced. Particularlyin the case of vacuum
+field equations such corrections cannot be associated to matter fi elds, because presumably there is no
+matter in empty space. So the correctionswould have to be physica lly interpreted as some kind of energy-
+momentum inherent to the noncommutative spacetime. However, a t this point such interpretations are
+mere speculations.
+21whereTaare the generators of the Lie algebra, along the lines of [12] with rela tive ease.18
+The star product is defined by
+Aµ1···µk
+ν1···νl⋆Bρ1···ρm
+σ1···σn=Aµ1···µka
+ν1···νl∧Bρ1···ρmb
+σ1···σnTaTb
++∞/summationdisplay
+n=1/planckover2pi1nCn/parenleftbig
+Aµ1···µka
+ν1···νl,Bρ1···ρmb
+σ1···σn/parenrightbig
+TaTb,(3.83)
+where the operators Cnare defined as before. In order to obtain a star commutator that
+is consistent with [12, 13] we have required the symmetry propert y (3.73) for Cn, though
+it is not required in [3].
+/bracketleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracketrightbig
+⋆≡Aµ1···µk
+ν1···νl⋆Bρ1···ρm
+σ1···σn
+−(−1)deg(A)deg(B)Bρ1···ρm
+σ1···σn⋆Aµ1···µk
+ν1···νl
+=Aµ1···µka
+ν1···νl∧Bρ1···ρmb
+σ1···σn[Ta,Tb]
++∞/summationdisplay
+n=1/planckover2pi1nCn/parenleftbig
+Aµ1···µka
+ν1···νl,Bρ1···ρmb
+σ1···σn/parenrightbig
+[Ta,Tb](n),(3.84)
+where [Ta,Tb](n)=TaTb−(−1)nTbTais the anticommutator, {Ta,Tb}, for every odd nand
+the commutator, [ Ta,Tb], for every even n.
+4 Covariant star product on a Poisson manifold
+In this section we discuss a covariant star product on a regular Pois son manifold M, first
+for functions and then for tensor fields. Since Mis regular, we can require that a linear
+connection exists on M. On a nonregular Poisson manifold we would generally define a
+different connection on each symplectic leaf of M, or define a contravariant connection on
+Mand use the associated contravariant derivative instead of a cova riant one [15, 16, 17].
+It was shown by Kontsevich [18] that a star product can be constr ucted for smooth
+functions on Rdwith any Poisson structure θin the sense of deformation quantization, so
+that at first order in the deformation parameter the star produc t is given by the Poisson
+bracket of functions. A path integral formulation of the Kontsev ich quantization has been
+developed [19]. The Kontsevich formula is not well-suited for calculatin g the star product
+beyond/planckover2pi12, because it contains integrals that cannot be solved by any standa rd method.
+The star product of functions has been calculated up to /planckover2pi14by using a simpler iterative
+approach [20].
+The existence of a covariant star product of functions on any Pois son manifold ( M,θ)
+with a torsion-free linear connection has been shown in [6] and given explicitly to /planckover2pi13as
+an example.
+4.1 Star product of functions
+The Poisson structure on the algebra of smooth functions f,g∈F(M) is defined by
+{f,g}=θ(df,dg) =θµν∂µf∂νg, (4.1)
+18Pleasenotethatoneofthemisunderstandingsof[4]hasbeeninhe ritedto[12]. Namely, [ ∇µ,∇ν]α= 0
+is not required for any α∈Ω(M), since it would also imply that the torsion vanishes, which is not
+necessary.
+22whereθis a Poisson bivector field, i.e. a smooth section of ∧2TM. The Poisson bracket
+(4.1) satisfies the required properties
+1. Antisymmetry: {f,g}=−{g,f}
+2. Jacobi identity:
+{f,{g,h}}+{g,{h,f}}+{h,{f,g}}= 0 (4.2)
+3. Derivation in the second argument:
+{f,gh}={f,g}h+g{f,h} (4.3)
+when the bivector θµνsatisfies the Jacobi identity (3.2).
+The star product of functions f,g∈F(M) is defined by
+f ⋆g=fg+∞/summationdisplay
+n=1/planckover2pi1nCn(f,g), (4.4)
+where the bidifferential operators Cn:F(M)×F(M)→F(M) are constructed from the
+torsion-free linear connection ∇, the Poisson bivector and the curvature tensor. At order
+/planckover2pi1one has
+C1(f,g) ={f,g}. (4.5)
+The star product (4.4) is required to be associative to all orders in /planckover2pi1,
+f ⋆(g⋆h) = (f ⋆g)⋆h. (4.6)
+Such star product of functions is given to order /planckover2pi13by [6]
+C2(f,g) =1
+2θµνθρσ∇µ∇ρf∇ν∇σg+1
+3θµσ∇σθνρ(∇µ∇νf∇ρg+∇ρf∇µ∇νg) (4.7)
++1
+6∇ρθµν∇µθρσ∇νf∇σg,
+C3(f,g) =−1
+6θρσ/parenleftbig
+LXf∇/parenrightbigµ
+νρ/parenleftbig
+LXg∇/parenrightbigν
+µσ, (4.8)
+whereLXf∇is the tensor defined by the Lie derivative of the connection ∇along the
+Hamiltonian vector field Xf=i(df)θ:
+/parenleftbig
+LXf∇/parenrightbigµ
+νρ=θσµ∇ν∇ρ∇σf+∇νθσµ∇ρ∇σf+∇ρθσµ∇ν∇σf
++∇ν∇ρθσµ∇σf+Rµ
+σνρθσλ∇λf.(4.9)
+Note that since the torsion vanishes, Tρ
+µν= 0, the covariant derivatives commute
+[∇µ,∇ν]f=−Tρ
+µν∇ρf= 0, (4.10)
+for everyf∈F(M). This star product exists for any Poisson manifold ( M,θ) and any
+torsion-free connection ∇.
+A covariant star product of functions can alternatively be defined directly according
+to the Kontsevich universal formula [18] by replacing the partial de rivatives∂µwith the
+covariant derivatives ∇µin allCn,n>1. By using the results of [20] one can write this
+23star product up to order /planckover2pi14. At orders higher than /planckover2pi12, where second and higher covariant
+derivatives of the bivector θappear, we have to introduce another condition
+[∇µ,∇ν]θρσ= 0, (4.11)
+in addition to the vanishing of the torsion (4.10), in order to ensure t he associativity of
+the star product. Thus the curvature tensor of the connection satisfies
+[∇µ,∇ν]θρσ=−Tλ
+µν∇λθρσ+Rρ
+λµνθλσ+Rσ
+λµνθρλ
+=−θσλRρ
+λµν+θρλRσ
+λµν= 0(4.12)
+or equivalently
+Rµν
+ρσ=Rνµ
+ρσ. (4.13)
+It is sufficient for θto be covariantly constant, ∇µθνρ= 0, but it is not necessary.19
+Without the above conditionfor the curvature we would have to add terms with curvature
+contributions to the star product in order to satisfy the associat ivity requirement. Indeed
+this approach is nothing more than a special case of the universal s tar product studied in
+[6].
+Relaxing the condition of [6] that the connection is torsion-free app ears to be very
+difficult without imposing some constraints on both the curvature an d the torsion. We
+shall discuss this briefly while considering a star product of tensor fi elds.
+4.2 Star product of tensor fields
+Although we have found a covariant star product of tensor fields o n a symplectic manifold
+as a special case of a star product of tensor-valued differential f orms in the Section 3, we
+would like a find a construction with less constraints on the connectio n. Since it is the
+definitionofthe Poisson bracket that primarilyimposes theconstra ints ontheconnections
+in the case of tensor-valued differential forms, we attempt to defi ne a Poisson bracket of
+tensor fields with a minimal set of properties.
+The Poisson structure (4.1) can be extended on the algebra of smo oth tensor fields
+A,B∈T(M) by
+/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
+=θλτ∇λAµ1···µk
+ν1···νl∇τBρ1···ρm
+σ1···σn.(4.14)
+For a function f∈F(M) the bracket{f,·}is a covariant derivation with respect to the
+second argument/braceleftbig
+f,Aµ1···µk
+ν1···νl/bracerightbig
+=∇XfAµ1···µk
+ν1···νl, (4.15)
+with the Hamiltonian vector field Xµ
+f=θνµ∇νf. We postulate the following properties
+for the Poisson bracket as a straightforward generalization of th e usual case of functions.
+1. Antisymmetry:
+/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
+=−/braceleftbig
+Bρ1···ρm
+σ1···σn,Aµ1···µk
+ν1···νl/bracerightbig
+(4.16)
+19In the case of a symplectic manifold the condition ∇µθνρ= 0 would be equivalent to the symplectic
+formωto be covariantly constant, i.e. having a symplectic torsion-free co nnection.
+242. Jacobi identity:
+/braceleftBig
+Aµ1···µk
+ν1···νl,/braceleftbig
+Bρ1···ρm
+σ1···σn,Cλ1···λi
+τ1···τj/bracerightbig/bracerightBig
++/braceleftBig
+Bρ1···ρm
+σ1···σn,/braceleftbig
+Cλ1···λi
+τ1···τj,Aµ1···µk
+ν1···νl/bracerightbig/bracerightBig
++/braceleftBig
+Cλ1···λi
+τ1···τj,/braceleftbig
+Aµ1···µkν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig/bracerightBig
+= 0(4.17)
+3. “Derivation in the second argument”:
+/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σnCλ1···λi
+τ1···τj/bracerightbig
+=/braceleftbig
+Aµ1···µk
+ν1···νl,Bρ1···ρm
+σ1···σn/bracerightbig
+Cλ1···λi
+τ1···τj
++Bρ1···ρm
+σ1···σn/braceleftbig
+Aµ1···µk
+ν1···νl,Cλ1···λi
+τ1···τj/bracerightbig
+(4.18)
+The Jacobi identity (4.17) imposes the following two constraints on t he connection∇:
+/summationdisplay
+(µ,ν,ρ)θµσ/parenleftbig
+∇σθνρ−θνλTρ
+σλ/parenrightbig
+= 0, (4.19)
+θµλθντRρ
+σλτ= 0. (4.20)
+Note that according to (4.20) the curvature tensor does not nee d to vanish everywhere
+since the Poisson bivector θµνis not necessarily invertible.
+Then we quantize the Poisson manifold by defining a covariant star pr oduct of tensor
+fields as in (3.79). The order /planckover2pi1deformation, C1, is again defined to be the Poisson
+bracket (4.14). The operators Cnare chosen to satisfy the same properties as in the
+Section (3.3).20A propriate ansatz for C2can be found by calculating the side of the
+associativity condition at order /planckover2pi12that depends on C1and choosing a C2that produces
+a similar expression on the other side of the condition. We choose C2to be of the same
+form as in (3.80). The associativity property of the star product im poses the additional
+constraint/summationdisplay
+(µ,ν,ρ)θµσ/parenleftbigg
+∇σθνρ+1
+2θνλTρ
+σλ/parenrightbigg
+= 0. (4.21)
+In order to satisfy both (4.19) and (4.21) we require that the conn ection satisfies the
+covariant Jacobi identities (3.7), so that the cyclic sum over each o f the terms of (4.19)
+and (4.21) is zero. Thus the constraints (3.7) and (4.20) are all tha t is needed for a
+covariant star product of tensor fields on a Poisson manifold up to o rder/planckover2pi12.
+In the case of a star product of functions there is no need for the constraint (4.20).
+However, the other constraints (3.7) are required, and they con strain the connection so
+that both the torsion and the curvature are affected. Thus relax ing the torsion-freeness
+constraint has lead to having some constraints for both the torsio n and the curvature.
+At present it is unclear whether additional constraints need to be in troduced for the
+connection at higher orders in /planckover2pi1.
+5 Conclusion
+We have generalized the recently defined covariant star product o f differential forms on a
+symplecticmanifold[3]totensor-valueddifferentialformsandcons equently totensorfields
+20The sign factor in the symmetry property (3.73) is obviously replace d with (−1)n.
+25of any type. This generalization does not require any new constrain ts on the connections.
+Possible applications of the star product to gravity and gauge theo ry have been discussed,
+considering the rather strict constraints the connections have t o satisfy. Further study of
+both of these applications is required.
+Then we proposed a covariant star product of tensor fields on a Po isson manifold with
+a linear connection that has less constraints than in the first case. Thus this star product
+could be a more viable option for theories of gravity.
+We also discussed the possibility to relax the torsion-freeness cond ition of the linear
+connection of the universal covariant star product of functions defined on a Poisson man-
+ifold in [6]. It was found that this requires one to impose some constra ints on both the
+torsion and the curvature, namely (3.7) in our case.
+Finally, a remark about the Poisson algebra of tensor fields is in order . A Poisson
+algebra consists of a commutative associative algebra endowed with a Poisson bracket. A
+graded-commutative associative algebra — like the algebra of differe ntial forms — can be
+turned into a graded Poisson algebra by introducing a graded Poisso n bracket. However,
+the algebra of tensor fields is neither commutative nor graded-com mutative. This is
+the reason why the Poisson structure of tensor fields (4.14)—(4.1 8) was defined for the
+components of the tensors, which are of course commutative fun ctions. This raises the
+question could a Poisson structure for tensor fields be defined som e other way compared
+to the definition given above?
+Acknowledgments
+The support of the Academy of Finland under the Projects No. 121 720 and 127626 is
+greatly acknowledged. The work of M. O. was fully supported by the Jenny and Antti
+WihuriFoundation. G.Z.acknowledges thesupportofCNCSIS-UE FISCSU GrantID-620
+of the Ministry of Education and Research of Romania.
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+ory using a covariant star product defined between Lie-valued diffe rential forms,
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+27
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+arXiv:1001.0504v1  [math.RT]  4 Jan 2010Intersection theory on punctual Hilbert schemes
+and graded Hilbert schemes
+Laurent Evain (laurent.evain@univ-angers.fr)
+November 8, 2018
+Abstract
+The rational Chow ring A∗(S[n],Q) of the Hilbert scheme S[n]parametrising
+the length nzero-dimensional subschemes of a toric surface Scan be described
+with the help of equivariant techniques. In this paper, we explain the general
+method and we illustrate it through many examples. In the last sectio n, we
+present results on the intersection theory of graded Hilbert sche mes.
+R´ esum´ e
+Les techniques ´ equivariantes permettent de d´ ecrire l’anneau de Chow rationnel
+A∗(S[n],Q) du sch´ ema de Hilbert S[n]param´ etrant les sous-sch´ emas ponctuels
+de longueur nd’une surface torique S. Dans cet article, nous pr´ esentons la
+d´ emarche g´ en´ erale et nous l’illustrons au travers de nombreux e xemples. La
+derni` ere section expose des r´ esultats de th´ eorie d’intersect ion sur des sch´ emas
+de Hilbert gradu´ es.
+Introduction
+LetSbe a smooth projective surface and S[n]the Hilbert scheme parametrising
+thelength nzero-dimensionalsubschemesof S. Howtodescribethecohomology
+ringH∗(S[n],Q) and the Chow ring A∗(S[n],Q) ?
+A first approach is based on the work of Nakajima, Grojnowski and Lehn
+among others [12], [16], [13], [14], [2]. The direct sum ⊕n∈NH∗(S[n],Q) is an
+(infinite dimensional) irreducible representation and carries a Fock s pace struc-
+ture [15]. Lehn settles a connection between the Fock space struc ture and the
+intersection theory of the Hilbert scheme via the action of the Cher n classes of
+tautological bundles [11].
+An other method, independent of the Fock space formalism introdu ced by
+Nakajima, has been developed in [5] when Sis a toric surface. The point is
+that the extra structure coming from the torus action brings into the scene an
+1equivariant Chow ring which is easier to compute than the classical Ch ow ring.
+The classical Chow ring is a quotient of the equivariant Chow ring.
+The computations of this equivariant approach are explicit. They re ly on
+the standard description of the cohomology of the Grassmannians and on a
+description of the tangent space to the Hilbert scheme at fixed poin ts.
+The main goal of this paper is to present this equivariant approach. We
+follow the general theory and we illustrate it with the case S=P2andn= 3
+as the main example.
+In the last section, we bring our attention to graded Hilbert scheme s, which
+playedanimportant rolein the equivariantcomputations. We presen tresultson
+the set theoretic intersection of Schubert cells, which suggest th at intersection
+theory on graded Hilbert schemes could be described in terms of com binatorics
+of plane partitions.
+Throughout the paper, we use the formalism of Chow rings and work over
+any algebraically closed field k. When k=C, the Chow ring co¨ ıncides with
+usual cohomology since the action of the two-dimensional torus TonSinduces
+an action of TonS[d]with a finite number of fixed points.
+1 Equivariant intersection theory
+1.1 General results
+In this section, we recall the facts about equivariant Chow rings th at we need.
+To simplify the presentation, we workwith rationalcoefficients and t he notation
+A∗(X) :=A∗(X,Q) denotes the rational Chow ring.
+The construction of an equivariant Chow ring associated with an alge braic
+space endowed with an action of a linear algebraic group has been set tled by
+Edidin and Graham [3]. Their construction is modeled after the Borel c onstruc-
+tion in equivariant cohomology.
+Proposition 1. LetGbe an algebraic group, Xan equidimensional quasi-
+projective scheme with a linearized G-action and i,j∈Z,i≤dim(X),j≥0.
+There exists a representation VofGsuch that
+•Vcontains an open set Uon which Gacts freely,
+•U→U/Gexists as a scheme and is a principal Gbundle,
+•codim VV\U >dim(X)−i.
+The quotient XG= (X×U)/Gunder the diagonal action exists as a scheme.
+The groups AG
+i(X) :=Ai+dim(V)−dim(G)(XG)andAj
+G(X) :=AG
+dim(X)−j(X)are
+independent of the choice of the couple (U,V).
+Definition 2. The group Ai
+G(X)is by definition the equivariant Chow group
+ofXof degree i.
+Example 3. IfG=T= (k∗)nis a torus, then a possible choice for the couple
+(U,V)isV= (kl)nwithl >>0, andU= (kl−{0})nwithTacting on Vby
+(t1,...,tn)(x1,...,x n) = (t1x1,...,tnxn). The quotient U/Tis isomorphic to
+(Pl−1)n.
+2Example 4. Letpbe a point and T= (k∗)nthe torus acting trivially on p.
+ThenA∗
+T(p)≃Q[h1,...,h n]wherehihas degree 1for alli.
+Proof. By the above example, A∗
+T(p) = lim l→∞A∗((Pl−1)n) =
+liml→∞Q[h1,...,h n]/(hl
+1,...,hl
+n) =Q[h1,...,h n], where hihas degree 1 ( ac-
+cording to the definition of the equivariant Chow group, the limit cons idered is
+a degreewise stabilisation thus the limit is the polynomial ring and not a p ower
+series ring).
+IfXis smooth, then ( X×U)/Gis smooth too and A∗
+T(X) is a ring : the
+intersection of two classes u,v∈A∗
+T(X) takes place in the Chow ring A∗((X×
+U)/G).
+Example 5. The isomorphism A∗
+T(p)≃Q[h1,...,h n]of the last example is an
+isomorphism of rings.
+Definition 6. LetEbe aG-equivariant vector bundle on XandEG→XGthe
+vector bundle with total space EG= (E×U)/G. The equivariant Chern class
+cG
+j(E)is defined by cG
+j(E) =cj(EG)∈Aj(XG) =Aj
+G(X).
+The identification of A∗
+T(p) with a ring of polynomials Rcan be made in-
+trinsic using equivariant Chern classes.
+Proposition 7. LetˆTbe the character group of a torus T≃(k∗)n. Any
+character χ∈ˆTdefines a one-dimensional representation of Tbyt.k=χ(t)k,
+hence an equivariant bundle over the point and an equivarian t Chern class cT
+1(χ).
+The map χ→cT
+1(χ)∈A1
+T(p)extends to an isomorphism R=SymQ(ˆT)→
+A∗
+T(p), whereSymQ(ˆT)is the symmetric algebra over Qof the group ˆT.
+Example 8. LetT=k∗be the one dimensional torus acting on the projective
+spacePr=Proj k[x0,...,x r]byt.(x0:···:xr) = (tn0x0:···:tnrxr). Then
+A∗
+T(Pr) =Q[t,h]/p(h,t)wherep(h,t) =/summationtextr
+i=0hr−iei(n0t,...,n rt),eibeing the
+i-th elementary symmetric polynomial.
+Proof. XTis thePrbundleP(O(n0)⊕ ··· ⊕O (nr)) overPl−1. The rational
+Chow ring of this projective bundle is Q[h,t]/(p(h,t),tl). We have the result
+whenltends to ∞.
+Example 9. LetVbe a representation of GandG(k,V)the corresponding
+Grassmannian. Then A∗
+G(G(k,V))is generated as an Rmodule by the equiv-
+ariant Chern classes of the universal quotient bundle.
+Proof.The quotient ( G(k,V)×U)/Gis a Grassmann bundle over U/Gwith
+fiber isomorphic to G(k,V). Since the Chow rings of Grassmann bundles are
+generated over the Chow ring of the base by the Chern classes of t he universal
+quotient bundle, the result follows.
+31.2 Results specific to the action of tori
+Brion [1] pushed further the theory of equivariant Chow rings when the group
+is a torus Tacting on a variety X.
+Theorem 10. [1] LetXbe a smooth projective T-variety. The restriction
+morphism i∗
+T:A∗
+TX→A∗
+TXTis injective.
+Example 11. LetT=k∗act onP1byt.(x:y) = (tx:y). The inclusion
+i∗
+T:A∗
+T(P1)→A∗
+T({0,∞}) =R2identifies A∗
+T(P1)with the couples (P,Q)of
+polynomials ∈R=Q[t]such that P(0) =Q(0).
+Proof.LetV=Vect(x,y) be the 2 dimensional vector space with P(V) =P1.
+By the above A∗
+T(P1) is generated by the Chern classes of the universal quotient
+bundle as an R-module. On the point ∞=ky∈P(V), the quotient bundle Q
+is isomorphic to kxandTacts with character t. Thusc1(Q)∞=t. Similarly,
+the restriction of Qto the point 0 = kxis a trivial equivariant bundle and
+c1(Q)0= 0. Thus c1(Q) restricted to {0,∞}is (0,t). Obviously, c0(Q) = (1,1).
+ThusA∗
+T(P1) =Q[t](0,t)+Q[t](1,1) as expected.
+IfT′⊂Tis a one codimensional torus, the localisation morphism i∗
+Tfactor-
+izes:A∗
+T(X)→A∗
+T(XT′)i∗
+T′→A∗
+T(XT) =RXT. Brion has shown
+Theorem 12. [1] LetXbe a smooth projective variety with an action of T.
+The image Im(i∗
+T)satisfies Im(i∗
+T) =∩T′Im(i∗
+T′)where the intersection runs
+over all subtori T′of codimension one in T.
+An important point is that the equivariantChow groups determine th e usual
+Chowgroups. Thefibersof XT→U/Tareisomorphicto X. Letj:X→XTbe
+the inclusion of a fiber and j∗:A∗
+T(X)→A∗(X) the corresponding restriction.
+Theorem 13. [1] LetR+=ˆTR⊂Rbe the set of polynomials with positive
+valuation. The morphism j∗is surjective with kernel R+A∗
+T(X).
+Example 14. A∗(P1) = (Q[t](t,0) +Q[t](1,1))/(Q[t]+(t,0) +Q[t]+(1,1))≃
+Q[t]/(t2). The isomorphism sends (P=/summationtextpiti,Q=/summationtextqiti)withp0=q0to
+(p0,p1−q1).
+Finally, we have an equivariant Kunneth formula for the restriction t o fixed
+points, proved in [5].
+Theorem 15. LetXandYbe smooth projective T-varieties with finite set of
+fixed points XTandYT. LetA∗
+T(X)⊂RXT,A∗
+T(Y)⊂RYT, andA∗
+T(X×
+Y)⊂RXT×YTthe realisation of their equivariant Chow rings via localis ation
+to fixed points. The canonical isomorphism RXT⊗RRYT≃RXT×YTsends
+A∗
+T(X)⊗A∗
+T(Y)toA∗
+T(X×Y).
+Example 16. LetTbe the one dimensional torus acting on P1×P1by
+t.([x1,y1],[x2:y2]) = ([tx1,y1],[tx2:y2]). For each copy of P1,A∗
+T(P1)is
+4generated as an R-module, by the elements e= 1.{0}+ 1.{∞}= (1,1)and
+f= 0.{0}+t.{∞}= (0,t). By the Kunneth formula, A∗
+T(P1×P1)⊂Q[t]4is
+generated by the elements (1,1,1,1) = (1,1)⊗(1,1),(0,t,0,t) = (1,1)⊗(0,t),
+(0,0,t,t) = (0,t)⊗(1,1),(0,0,0,t2) = (0,t)⊗(0,t)where the coordinates are
+the coefficients with respect to the four points (a,b)∈P1×P1witha,b∈ {0,∞}.
+The strategy
+Let’s sum up the situation. The equivariant Chow ring A∗
+T(X) satisfies the
+usual functorial properties of a Chow ring: there is an induced pus hforward
+for a proper morphism , an induced pull-back for a flat morphism, equ ivari-
+ant vector bundles have equivariant Chern classes... When the fixed point set
+XTis finite, the computations are identified with calculations in products of
+polynomial rings.
+Since it is possible to recoverthe usual Chowring from the equivarian tChow
+ring, the point is to compute the equivariant Chow ring and its restric tion to
+fixedpoints. ThestrategythatwillbefollowedinthecaseoftheHilbe rtschemes
+is to use theorem 12 above. It is not obvious a priori that the geome try and
+the equivariant Chow rings of ( S[n])T′and their restriction to fixed points are
+easier to describe than the equivariant Chow ring of the original var ietyS[n].
+This is precisely the work to be done.
+2 Iarrobino varieties and graded Hilbert scheme
+In our computations of the Chow ring of the Hilbert scheme, a centr al role will
+be played bythe Iarrobinovarieties orgradedHilbert schemesthat weintroduce
+now.
+Fix a set of dimensions H= (Hd)d∈Nsuch that Hd= 0 ford >>0.
+Definition-Proposition 17. The Iarrobino variety Hhom,His the set
+of homogeneous ideals I=⊕Id⊂k[x,y]of colength/summationtext
+dHdsuch
+thatcodim(Id,k[x,y]d) = Hd. This is a subvariety of G=/producttext
+d s.t. H d/ne}ationslash=0Grass(Hd,k[x,y]d). Moreover, Hhom,His empty or irreducible.
+Proof. Each vector space Idcorresponds to a point in the Grassman-
+nianGrass(Hd,k[x,y]d) andIcorresponds to a point in the product G=/producttextGrass(Hd,k[x,y]d). Accordingly, Hhom,His a subvariety of G. The irre-
+ducibility of Hhom,His shown in [9].
+The Chow ring A∗(Hhom,H) is related to the Chow ring of G, as shown by
+King and Walter [10].
+Theorem 18. Leti:Hhom,H֒→Gdenote the inclusion. The pull back i∗:
+A∗(G)→A∗(Hhom,H)is surjective.
+There are natural generalisations of the Iarrobino varieties, intr oduced by
+Haiman and Sturmfels [8] and called graded Hilbert schemes. In our ca se, the
+5graded Hilbert schemes we are interested in are the quasi-homogen eous Hilbert
+schemes.
+Definition-Proposition 19. Letweight(x) =a∈N,weight(y) =b∈N
+with(a,b)/\e}atio\slash= (0,0). Consider the set Hab,Hof quasi-homogeneous ideals
+I=⊕d∈NId⊂k[x,y]withcodim(Id,k[x,y]d) =Hd. There is a closed em-
+beddingi:Hab,H֒→G= Πd∈N,Hd/ne}ationslash=0Grass(H(d),k[x,y]d).
+Remark 20. We could also consider the case a∈Zand/orb∈Z. However
+whenab <0,Hab,Hwould be empty or a point. Moreover, changing (a,b)with
+(−a,−b)gives an isomorphic graduation. Consequently, any non triv ial variety
+Hab,Hcan be realized with a≥0andb≥0. We thus consider a∈Nandb∈N
+without loss of generality.
+One wants to extend in this context the results by Iarrobino and Kin g-
+Walter.
+Extending Iarrobino’s irreducibility result is possible, but not immediat e, as
+Iarrobino’s argument does not extend.
+Theorem 21. [7] The graded Hilbert scheme Hab,His empty or irreducible.
+Idea of the proof. SinceHab,Hissmoothasthefixedlocusof( A2)[/summationtextHd]underthe
+action of a one dimensional torus, irreducibility is equivalent to conne ctedness.
+To prove connectedness, Hab,Hadmits a stratification where the cells are the
+inverse images of the product of Schubert cells on Gby the immersion Hab,H→
+G. Each cell is an affine space. The cells being connected spaces, it suffi ces
+to connect together the different cells to prove the connectedne ss ofHab,H. To
+this aim, one writes down explicit flat families over P1. These flat families
+correspond to curves drawn on Hab,Hthat give the link between the different
+cells.
+The arguments of King and Walter generalise easily to the quasi-
+homogeneous case. Moreover, the affine plane A2is a toric variety. The action
+ofT=k∗×k∗onA2induces an action of TonHab,H. It is possible to generalise
+the results of King-Walter to the equivariant setting. With minor mod ifications
+of their method, one gets the following theorem.
+Theorem 22. The natural restriction morphisms i∗:A∗(G)→A∗(Hab,H)and
+i∗
+T:A∗
+T(G)→A∗
+T(Hab,H)are surjective.
+Corollary 23. The above theorem induces a description of A∗
+T(Hab,H). Indeed,
+ifj: (Hab,H)T→Hab,His the inclusion of the T-fixed points, then A∗(Hab,H) =
+im(j∗
+T) =im(ij)∗
+Tand the computation of (ij)∗
+T:A∗
+T(G)→A∗
+T(HT
+ab,H)follows
+easily from the description of A∗
+T(G)using equivariant Chern classes.
+Example 24. LetHhom,Hbe the Iarrobino variety parametrising the ho-
+mogeneous ideals in k[x,y]with Hilbert function H= (1,1,0,0,...). The
+torusT=k∗acts byt(x,y) = (tx,y). The two fixed points are the ideals
+6(x2,y)and(x,y2). The Iarrobino variety Hhom,Hembeds in the Grassman-
+nianGrass(1,k[x,y]1)of one dimensional subspace of linear forms. The uni-
+versal quotient Qover the Grassmannian is a T-bundle. Its restriction over
+the points (x2,y)and(x,y2)is aT-representation corresponding to the charac-
+terst/mapsto→tandt/mapsto→1. Thus the first Chern class of the universal quotient is
+(t,0)∈RHT
+hom,H=R2. Finally A∗
+THhom,H= (cT
+1(Q),cT
+0(Q)) =R(t,0)+R(1,1).
+3 Geometry of the fixed locus
+3.1 Geometry of (S[d])T
+LetSbe a smooth projective toric surface. The 2-dimensional torus Twhich
+acts onSacts naturally on S[d]. According to theorems 10 and 12, one main
+ingredient to describe the equivariant Chow ring A∗
+T(S[d]) is to describe the
+geometry of the fixed loci ( S[d])Tand (S[d])T′of the Hilbert scheme S[d]under
+the action of the two dimensional torus Tand under the action of any one-
+dimensional torus T′⊂T.
+Consider the example S=P2=Proj k[X,Y,Z] and (P2)[3]the associ-
+ated Hilbert scheme. The action of T=k∗×k∗onP2is (t1,t2).XaYbZc=
+(t1X)a(t2Y)bZc. First, we describe the finite set (( P2)[3])T. Obviously, a
+subscheme Z∈((P2)[3])Thas a support included in {p1,p2,p3}where the
+pi’s are the toric points of P2fixed under T. Through each toric point,
+there are two toric lines with local equations x= 0 and y= 0. Since
+ZisT-invariant, the ideal I(Z) is locally generated by monomials xαyβ.
+1 xline x=0
+line y=0Monomials
+1 xxyy x^2y
+x^2Ideal (x^2,y)
+Using the two lines, we can represent graphically the monomials xαyβas
+shown. An ideal I⊂k[x,y] generated by monomials is represented by the set
+of monomials which are not in the ideal. For instance, the ideal ( x2,y) which
+does not contain the monomials 1 ,xis drawn in the above figure.
+A B C D E
+p_1p_3
+p_2
+Proposition 25. In(P2)[3], there are a finite number of subschemes invari-
+ant under the action of the two-dimensional torus. Up to perm utation of the
+7projective variables X,Y,ZofP2, there are five such subschemes A,B,C,D,E .
+Proof.By the above, the invariant subschemes are represented by mono mials
+around each toric point of P2. The number of monomials is the degree of the
+subscheme, ie. 3in oursituation. Up to permutation ofthe axes, all the possible
+casesA,B,C,D,E are given in the picture.
+In general, we have:
+Proposition 26. The points of (S[d])Tare in one-to-one correspondence with
+the tuples of staircases (Ei)i∈STsuch that/summationtext
+i∈STcardinal (Ei) =d.
+3.2 Tangent space at p∈(S[d])T
+We recall the description of the tangent space at a point p∈(S[d])T([7], but
+see also [4] for an other description) .
+The six cleft couples for the 6−dimensional tangent space 
+For simplicity, we suppose that the subscheme pis supported by a single
+point. Recall that we have associated to pthe staircase Fof monomials xayb
+not inI(p) wherex,yare the toric coordinates around the support of p. A cleft
+forFis a monomial m=xayb/∈Fwith (a= 0 orxa−1yb∈F) and (b= 0
+orxayb−1∈F). We order the clefts of Faccording to their x-coordinates:
+c1=yb1,c2=xa2yb2,...,c p=xapwitha1= 0< a2<···< ap. Anx-cleft
+couple for Fis a couple C= (ck,m), where ckis a cleft ( k/\e}atio\slash=p),m∈F, and
+mxak+1−ak/∈F. By symmetry, there is a notion of y-cleft couple for F. The
+set of cleft couples is by definition the union of the ( xory)-cleft couples.
+Theorem 27. The vector space TpS[d]is in bijection with the formal sums/summationtextλiCi, whereCiis a cleft couple for p.
+Example 28. (P2)[3]is a 6 dimensional variety. A basis for the tangent space
+at a point with local equation (x2,xy,y2)is the set of cleft couples shown in the
+figure.
+With equivariant techniques, it is desirable to describe the tangent s pace as
+a representation. The torus Tacts on the monomials ckandmwith characters
+χkandχm. We let χC=χm−χk.
+Proposition 29. Under the correspondence of the above theorem, the cleft cou -
+pleCis an eigenvector for the action of Twith character χC.
+3.3 Geometry of (S[d])T′
+We come now to the description of ( S[d])T′whereT′is a one-dimensional
+subtorus of T. We start with an example.
+8Example 30. IfS=P2andT′≃k∗acts onA2⊂P2viat.(x,y) = (tx,ty)
+the irreducible components of ((P2)[3])T′through the points A,B,C,D,E are
+isomorphic to an isolated point, P1,P1×P1,P1×P1,P2.
+Proof.The tangent space to (( P2)[3])T′at a point p∈((P2)[3])Tis the tangent
+spaceto ( P2)[3]invariantunder T′. Using the description ofthe tangent spaceto
+(P2)[3]as a representation in the previous section, one computes the dime nsion
+of the tangent space of (( P2)[3])T′at each of the points A,B,C,D,E . The
+corresponding dimensions are 0 ,1,2,2,2.
+In particular, an irreducible variety through A(resp.B,C,D,E ) invariant
+underT′with dimension 0 (resp. 1 ,2,2,2) is the irreducible component of
+((P2)[3])T′through A(resp. through B,C,D,E ). It thus suffices to exhibit
+such irreducible varieties.
+Ais an isolated point and there is nothing to do.
+The component P1passing through Bcan be described geometrically. The
+set of lines through the origin of A2is aP1. To each such line D, we consider
+the subscheme Zof degree 3 supported by the origin and included in D. The set
+of such subschemes Zmoves in a P1. It is the component of (( P2)[3])T′through
+B. This component can be identified with the Iarrobino variety with Hilbe rt
+function H= (1,1,1,0,0,...).
+With the same set of lines through the origin, one can consider the su b-
+schemes Zof degree 2 supported by the origin and included in a line D. Since
+Zmoves in a P1,Z∪pwherepis a point on the line at infinity moves in a
+P1×P1. It is a component through CandD.
+Finally, a subscheme Zof degree 2 included in the line at infinity moves in
+aP2. Thus the union of Zand the origin moves in a P2which is the component
+of (S[d])T′through E.
+The followingproposition saysthat all but a finite number ofreprese ntations
+ofT′give a trivial result.
+Proposition 31. Letaandbbe coprime with |a| ≥3or|b| ≥3. Suppose
+thatT′=k∗acts onA2⊂P2viat.(x,y) = (tax,tby). Then the irreducible
+components of ((P2)[3])T′through the points A,B,C,D,E are isolated points.
+Proof.The tangent space at these points is trivial.
+We see in the examples that the irreducible components of ( P2)T′are the
+three toric points of P2for a general T′. For some special T′, there are two
+components, namely a toric point and the line joining the remaining two toric
+points.
+For a general toric surface S, the situation is similar.
+Proposition 32. For anyT′one codimensional subtorus of T,ST′is made up
+of isolated toric points (wi)and of toric lines (yj)joining pairs of the remaining
+toric points.
+9Definition 33. We denote by PFix(T′) ={wi}the set of isolated toric points
+inST′and byLFix(T′) ={yj}the set of lines in ST′.
+In the (P2)[3]example, we identified the irreduciblecomponents of(( P2)[3])T′
+with products B1×...×Brof projective spaces. Some of the projective spaces
+were identified with a graded Hilbert scheme. For instance, the comp onent of
+((P2)[3])T′throughBhas been identified with the Iarrobinovariety with Hilbert
+function H= (1,1,1,0,0,...).
+In general, the irreducible components of ( S[d])T′are products B1×...×Br
+where the components Biare projective spaces or graded Hilbert schemes.
+Let’s look at the situation more closely to describe these component s. If
+Z∈(S[d])T′, the support of Zis invariant under T′. The invariant locus in Sis
+a union of isolated points ( wi) and lines ( yi). We denote by Wi(Z) andYi(Z)
+the subscheme of Zsupported respectively by the point wiand by the line yi.
+By construction, we have:
+Proposition 34. A subscheme Z∈(S[d])T′admits a decomposition Z=
+∪wi∈PFix(T′)Wi(Z)∪yi∈LFix(T′)Yi(Z).
+Obviously, ( S[d])T′is not irreducible : when Zmoves in in a connected
+component, the degree of Wi(Z) andYi(Z) should be constant. But fixing
+the degree of Wi(Z) andYi(Z) is not sufficient to characterize the irreducible
+components of ( S[d])T′as shown by the components identified in example 30.
+Example 35. The components of ((P2)[3])T′throughAandBare 2 distinct
+Iarrobino varieties corresponding to the same degrees 3on the point (0,0)inA2
+and 0 on the line at infinity.
+The finer invariant which distinguishes the irreducible components is s imilar
+to the one used for the Iarrobino varieties. It is a Hilbert function t aking into
+account the graduation provided by the action.
+Example 36. LetOA=k[x,y]/(x2,xy,y2)andOB=k[x,y]/(y,x3)be the
+ring functions corresponding to the points AandBin example 30. The action
+ofT′onOAis diagonalizable with characters 1,t,twhereas the action of T′on
+OBacts with characters 1,t,t2.
+LetZ=Spec O Z∈(S[d])T′. The torus T′acts onOZwith a diagonalizable
+action. In symbols, OZ=⊕Vχ, whereVχ⊂OZis the locus where T′acts
+through the character χ∈ˆT′.
+Definition 37. IfZ∈(S[d])T′, we define HZ:ˆT′→N,χ→dimVχand we
+letZ=∪Wi(Z)∪Yi(Z)the decomposition of Zintroduced above. The tuple of
+functions HZ= (HWi(Z),HYi(Z))indexed by the connected components {wi,yj}
+ofST′is by definition the Hilbert function associated to Z.
+Theorem 38. [7] The set of Hilbert functions HZ= (HWi(Z),HYi(Z))corre-
+sponding to the subschemes Z∈(S[d])T′is a finite set. Moreover, the irreducible
+components of (S[d])T′are in one-to-one correspondence with this set of Hilbert
+functions HZ.
+10Thedecompositionof( S[d])T′asaproductfollowseasilyfromthedescription
+of the Hilbert functions. Let Bwi(H) be the set of subschemes Wi⊂S,T′fixed,
+supported by the fixed point wiwithHWi=H. Similarly, let Byi(H) be the set
+of subschemes Yi⊂S,T′fixed, with support in the fixed line yiandHYi=H.
+A reformulation of the above is thus:
+Theorem 39. S[d],T′=∪Bw1(Hw1)×...Bwr(Hwr)×By1(Hy1)...×Bys(Hys)
+where{w1,...,w r}=PFix(T′)are the isolated fixed points, {y1,...,y s}=
+LFix(T′)are the fixed lines, and the union runs through all the possibl e tuples
+of Hilbert functions (Hw1,...,H wr,Hy1,...,H ys).
+The next two propositions identify geometrically the factors Bwi(Hwi) and
+Byj(Hyj) of the above product.
+By the very definition, we have:
+Proposition 40. For every isolated fixed point wi∈PFix(T′)and every func-
+tionHwi:ˆT′→N, the variety Bwi(Hwi)is a graded Hilbert scheme.
+Example 41. In example 30, take w= (0,0)∈A2and Hilbert function H(χ) =
+1for the three characters χ= 1,t,t2andH(χ) = 0otherwise. Then Bw(H)is
+the component of ((P2)[3])T′passing through B, which has been identified with
+the homogeneous Hilbert scheme Hhom,H.
+As for the other components, we have:
+Proposition 42. For every fixed line yi∈LFix(T′)and every function Hyi:
+ˆT′→N,Byi(Hyi)is a product of projective spaces.
+Illustration of the last proposition on an example. Consider T′=k∗acting on
+A2⊂P2viat.(x,y) = (tx,y). The line x= 0 isT′-fixed. We say that a scheme
+Zis horizontal of length nif it is in the affine plane and I(Z) = (y−a,xn),
+or if it is a limit of such schemes when the support (0 ,a) moves to infinity.
+Two horizontal schemes Z1andZ2of respective length n1/\e}atio\slash=n2move in a
+P1×P1. When the supports of Z1andZ2are distinct, Z1∪Z2is a scheme of
+lengthn1+n2. We thus obtain a rational function P1×P1···>(P2)[n1+n2].
+The schemes being horizontal, the limit of Z1∪Z2is completely determined by
+the support when the schemes Z1andZ2collide. More formally, the rational
+function extends to a well defined morphism ϕ:P1×P1→(P2)[n1+n2]. This
+morphism is an embedding and gives an isomorphism between P1×P1and one
+of the components Byi(Hyi) introduced above.
+Ifn1=n2in the above paragraph, ϕis not an embedding any more because
+oftheactionofthesymmetricgroupwhichexchangestherolesof Z1andZ2. But
+taking the quotient, we obtain an embedding P2= (P1×P1)/σ2→(P2)[n1+n2]
+which is an isomorphism on a component.
+114 Description of the Chow ring
+4.1 Description using generators
+LetT′⊂Tbe a one dimensional subtorus. Recall that each irreducible com-
+ponentCof (S[d])T′is a product Bw1(Hw1)×...Bwr(Hwr)×By1(Hy1)...×
+Bys(Hys) where{w1,...,w r}=PFix(T′) and{y1,...,y s}=LFix(T′).
+The factors Bwi(Hwi) are graded Hilbert schemes. We have seen in example
+24 the computation of generators for their equivariant Chow ring. We denote
+byMwi,T′,Hwithis equivariant Chow ring.
+The factors Byi(Hyi) are product of projective spaces. We have seen in
+examples 8, 9 or 16 the computation of generators for their equiva riant Chow
+ring. We denote by Nyi,T′,Hyithis equivariant Chow ring.
+Then the equivariantChow ringofthe component Cis givenbythe Kunneth
+formula of theorem 15:
+A∗
+T(C) =/circlemultiplydisplay
+wi∈PFix(T′)Mwi,T′,Hwi/circlemultiplydisplay
+yi∈Lfix(T′)Nyi,T′,Hyi
+WhenH= (Hw1,...,H wr,Hy1,...,H ys) runs through the possible Hilbert
+functions to describe all the irreducible components Cof (S[d])T′and using
+theorem 12, we finally get:
+Theorem 43. [5]
+A∗
+T(S[d]) =/intersectiondisplay
+T′⊂T/circleplusdisplay
+#H=d(/circlemultiplydisplay
+wi∈PFix(T′)Mwi,T′,Hwi/circlemultiplydisplay
+yi∈LFix(T′)Nyi,T′,Hyi)
+4.2 Second description of the Chow ring: From generators
+to relations
+In the last formula, the modules Mwi,T′,HwiandNyi,T′,Hyiwere described with
+explicit generators. It is possibleto adoptthe relationspoint ofview ratherthan
+the generators point of view. This is an algebraic trick which relies on B ott’s
+integration formula, proved by Edidin and Graham in an algebraic cont ext.
+The equivariant Chow ring is then described as a set of tuples of polyn omials
+satisfying congruence relations.
+The propositionbelow that makesthe transitionfrom generatorst o relations
+involvesequivariantChernclassesoftherestrictions TS[d],pofthetangentbundle
+TS[d]at fixed points p∈(S[d])T. Since we havedescribed the fiber of the tangent
+bundle at these points as a T-representation, computing the equivariant Chern
+classes is straightforward and the set of relations can be compute d.
+Proposition 44. [5] Letβi= (βip)p∈(S[d])Tbe a set of generators of the
+Q[t1,t2]-module i∗
+TA∗
+T(S[d])⊂Q[t1,t2](S[d])T. Letα= (αp)∈Q[t1,t2](S[d])T.
+Then the following conditions are equivalent.
+•α∈i∗
+TA∗
+T(S[d])
+12• ∀i, the congruence
+/summationdisplay
+p∈(S[d])T(αpβip/productdisplay
+q/ne}ationslash=pcT
+dimS[d](TS[d],q))≡0 (mod/productdisplay
+p∈(S[d])TcT
+dimS[d](TS[d],p))
+holds.
+We apply the method to ( P2)[3]. There are 22 fixed points thus the equiv-
+ariant Chow ring is a subring of Q[t1,t2]22. Five of the fixed points A,...,E
+have been introduced in the examples. The other fixed points are ob tained
+from these five by a symmetry. For instance, A12=σ(A) whereσis the toric
+automorphism of P2exchanging the points p1andp2.
+Theorem 45. [5] The equivariant Chow ring A∗
+T((P2)[3])⊂Q[t1,t2]{A,A12,...,E}
+is the set of linear combinations aA+a12A12+···+eEsatisfying the relations
+•a+a13−d−d13≡0 (mod t2
+2)
+•d−d13≡0 (mod t2)
+•a−a13≡0 (mod t2)
+•a−b≡0 (mod2t1−t2)
+•b−b13≡0 (mod t2)
+• −b+3c−3c12+b12≡0 (mod t3
+1)
+• −b+c+c12−b12≡0 (mod t2
+1)
+•3b−c+c12+−3b12≡0 (mod t1)
+•b−b23≡0 (mod t2−t1)
+•c−d+c23−d23≡0 (mod(t1−t2)2)
+•c+d−c23−d23≡0 (mod(t1−t2))
+•c23−d23≡0 (mod(t1−t2))
+•c−c13≡0 (mod t2)
+•d−2e+d12≡0 (mod t2
+1)
+•d−d12≡0 (mod t1)
+•all relations deduced from the above by the action of the symm etric group
+S3.
+The Chow ring A∗((P2)[3])is the quotient of A∗
+T((P2)[3])by the ideal generated
+by the elements fA+···+fE,f∈Q[t1,t2]+.
+5 Graded Hilbert schemes revisited
+The quasi-homogeneous Hilbert schemes played a central role in the computa-
+tionofA∗
+T(S[d])andtheirChowringwascomputedusingequivarianttechniques.
+In this section, we present a result suggesting that their Chow ring could admit
+an alternative description in terms of combinatorics of partitions.
+Tosimplifythe notations, werestrictfromnowonourattentiontot hehomo-
+geneous case of Iarrobino varieties, but the statements below ca n be formulated
+in the quasi-homogeneous case [6].
+13Recall that the gradedHilbert scheme Hhom,Hembeds in aproduct ofGrass-
+mannians: Hhom,H֒→G=/producttext
+d∈N,Hd/ne}ationslash=0Grass(Hd,k[x,y]d). The Grassmanni-
+ans are stratified by their Schubert cells, constructed with respe ct to the flag
+F1=Vect(xd)⊂F2=Vect(xd,xd−1y)⊂...Fd+1=k[x,y]d. The product G
+is stratified by the product of Schubert cells, and Hhom,His stratified by the
+restrictions of these products of Schubert cells. We still call thes e restrictions
+Schubert cells on Hhom,H.
+Example 46. A Schubert cell of a Iarrobino variety Hhom,Hcontains a unique
+monomial ideal I⊂k[x,y]that we represent as usual by the set of monomials
+E(I) ={xayb/∈I}.
+The Grassmannians in the product Gare trivial when Hd= dimk[x,y]d.
+When the numbers of non trivial Grassmannians in Gis one, the inclusion
+Hhom,H⊂Gis an isomorphism. In this Grassmannian case, the closures of the
+Schubert cells form a base of A∗(Hhom,H) and the intersection is classically
+described in terms of combinatorics involving the partitions associat ed to the
+cells.
+Question :Is it possible to describe the intersection theory in terms o f
+partitions when Hhom,His not a Grassmannian ?
+The intersection theory when Hhom,His not a Grassmannian is more com-
+plicated than in the Grassmannian case. On the set theoretical leve l, the inter-
+sectionC∩Dbetween the closures of two cells CandDis difficult to determine.
+The closure Cof a cellCis not a union of cells any more.
+However, a necessary condition for the incidence C∩D/\e}atio\slash=∅is known and
+expressed in terms of reverse plane partitions with shape E(J), where Jis the
+unique monomial ideal in D.
+5
+2
+10
+0 0
+0 0 0
+0 00 0
+0 0 0005
+2
+10
+0 0
+0 0 0
+0 00 0
+0 0 000
+E(I)E(J) and the plane partition
+Definition 47. A reverse plane partition with shape E⊂N2is a two-
+dimensional array of integers nij,(i,j)∈Esuch that ni,j≤ni,j+1,ni,j≤
+ni+1,j
+Definition 48. A monomial ideal Iis linked to a monomial ideal Jby a reverse
+plane partition nijwith support E(J)ifE(I) ={xa+na,byb−nab,(a,b)∈E(J)}
+Example 49. In the above figure, E(I)is linked to E(J).
+Definition 50. IfI⊂k[x,y]is a monomial ideal of colength n, the complement
+ofIis the ideal C(I)such that E(C(I))contains the monomials xayb,a < n,
+b < nwithxn−1−ayn−1−b∈I(see the figure below).
+14E(I) E(C(I))
+Theorem 51. [6] LetCandC′⊂Hhom,Hbe two cells containing the monomial
+idealsIandI′. IfC∩C′/\e}atio\slash=∅, then
+•Iis linked to I′by a reverse plane partition.
+•C(I)is linked to C(I′)by a reverse plane partition.
+By analogy with the Grassmannian case, we are led to the following ope n
+question: Can we describe the intersection theory on the Graded H ilbert
+schemes in terms of combinatorics of the plane partitions ?
+References
+[1] M Brion. Equivariant chow groups for torus actions. Transformations
+Groups, (2):225–267, 1997.
+[2] K Costello and I Grojnowski. Hilbert schemes, hecke algebras and the
+Calogero-Sutherland system. math.AG/0310189 .
+[3] Dan Edidin and William Graham. Equivariant intersection theory. Invent.
+Math., 131(3):595–634, 1998.
+[4] Geir Ellingsrud and Stein Arild Strømme. On the homology of the Hilber t
+scheme of points in the plane. Invent. math. , (87):343–352, 1987.
+[5] L. Evain. The Chow ring of punctual Hilbert schemes on toric surf aces.
+Transform. Groups , 12(2):227–249, 2007.
+[6] Laurent Evain. Incidence relations among the Schubert cells of e quivariant
+punctual Hilbert schemes. Math. Z. , 242(4):743–759, 2002.
+[7] Laurent Evain. Irreducible components of the equivariant punc tual Hilbert
+schemes. Adv. Math. , 185(2):328–346, 2004.
+[8] Mark Haiman and Bernd Sturmfels. Multigraded Hilbert schemes. J. Al-
+gebraic Geom. , 13(4):725–769, 2004.
+[9] Anthony A. Iarrobino. Punctual Hilbert schemes. Mem. Amer. Math. Soc. ,
+10(188):viii+112, 1977.
+[10] A King and C Walter. On chow rings of fine moduli spaces of modules .J.
+reine angew. Math. , 461:179–187, 1995.
+[11] M Lehn. Chern classes of tautological sheaves on hilbert schem es of points
+on surfaces. Invent. math. , (136):157–207, 1999.
+15[12] Manfred Lehn and Christoph Sorger. Symmetric groups and th e cup prod-
+uct on the cohomology of Hilbert schemes. Duke Math. J. , 110(2):345–357,
+2001.
+[13] Manfred Lehn and Christoph Sorger. The cup product of Hilber t schemes
+forK3 surfaces. Invent. Math. , 152(2):305–329, 2003.
+[14] Wei-Ping Li, Zhenbo Qin, and Weiqiang Wang. The cohomology rings o f
+Hilbert schemes via Jack polynomials. In Algebraic structures and moduli
+spaces, volume38of CRM Proc. LectureNotes ,pages249–258.Amer.Math.
+Soc., Providence, RI, 2004.
+[15] H Nakajima. Heisenberg algebra and Hilbert schemes of points on projec-
+tive surfaces. Ann. of Math. , 2(145):379–388, 1997.
+[16] Eric Vasserot. Sur l’anneau de cohomologie du sch´ ema de Hilbert deC2.
+C. R. Acad. Sci. Paris S´ er. I Math. , 332(1):7–12, 2001.
+16
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+arXiv:1001.0505v1  [nlin.CG]  4 Jan 2010CONSTRUCTION OF AN ISOTROPIC CELLULAR
+AUTOMATON FOR A REACTION-DIFFUSION EQUATION
+BY MEANS OF A RANDOM WALK
+A. NISHIYAMA AND T. TOKIHIRO
+Abstract. We propose a new method to construct an isotropic cel-
+lular automaton corresponding to a reaction-diffusion equa tion. The
+method consists of replacing the diffusion term and the react ion term
+of the reaction-diffusion equation with a random walk of micr oscopic
+particles and a discrete vector field which defines the time ev olution of
+the particles. The cellular automaton thus obtained can ret ain isotropy
+and therefore reproduces the patterns found in the numerica l solutions
+of the reaction-diffusion equation. As a specific example, we apply the
+method to the Belousov-Zhabotinsky reaction in excitable m edia.
+1.introduction
+Reaction-diffusion equations are used extensively for model ing pattern
+formation observed in natural and social phenomena. The equ ations are
+deduced from the simple idea that concentration changes of a material in a
+system are caused by reactions between the materials contai ned in the sys-
+tem and by diffusion of each material. Numerical solutions of t he equations
+show a variety of patterns, and they are used widely in variou s fields such
+as chemistry, biology, medical science etc. [1].
+An alternative approach to modeling pattern formation is to usea cellular
+automaton (hereafter abbreviated as CA)[2]. CAs are mathem atical models
+with discrete time, space and state variables, and are define d by simple time
+evolution rules. They are able to reproduce complex pattern s, and therefore
+have a lot of applications. For example, the lattice gas CAs r epresent very
+well the various features of fluid dynamics and reaction-diffu sion phenomena
+[3]. Ingeneral, CAshave theadvantage of beingableto simul ate phenomena
+at lower computational cost than when using partial different ial equations.
+A lot of models for cooperative phenomena of excitable media , derived by
+using partial differential equations or cellular automata ha ve been proposed.
+The so-called ”oregonator” model [5, 6, 7] expressed in term s of differential
+equations can explain the Belousov-Zhabotinsky(BZ) react ion [4] known as
+a typical example of a reaction-diffusion system. On the other hand, some
+approaches by means of cellular automata for excitable medi a have been
+proposed in [8, 9, 10]. However, these early CA models featur ed update
+rules based on nearest neighbor connections, and hence, fac ed several seri-
+ous shortcomings [11]. The most serious of these is the lack o f curvature
+12 A. NISHIYAMA AND T. TOKIHIRO
+and dispersion effects and unwanted anisotropy of the front mo tion [12, 13].
+To overcome these problems, several automata models have be en proposed
+[12, 14, 15, 13, 17, 18]. Gerhardt, Shuster and Tyson introdu ced bigger
+neighborhoods to model the curvature effects and make the thre shold a lin-
+ear function to take dispersion into account [12, 14, 15]. We imar, Tyson and
+Watson improved themodels by introducingamask, i.e. a weig hted summa-
+tion of automaton values over large neighborhoods [17, 18], and Henze and
+Tyson extended it to three spatial dimensions [19]. These mo dels recover
+curvatureanddispersioneffects well, buttheanisotropy of w ave propagation
+is not completely eliminated.
+Incellularautomata, torecovertheisotropyofthetime-ev olution patterns
+is in fact a difficult problem. If the adopted lattice used in th e modeling
+has periodicity, such as a square lattice or a hexagonal latt ice, then, due to
+thisperiodicity thetime-evolution patternsobtained fro mthemodelbecome
+anisotropic. In [16] Markus and Hess have proposed an isotro pic model for
+excitable media. In their model, they use a square lattice, b ut instead of
+placing each grid point at the center of a unit cell, they assi gn each grid
+point to a random location within its unit cell. The isotropy is recovered by
+taking a large number of neighboring cells within a circular area with radius
+R. As other CA models for excitable mediawhich seem to recove r anisotropy
+to some extent, one can cite the ”Moving Average CA” method by Weimar
+[21], the lattice gas method CAs by Raymond et al. [22] or Chen [23] et al.,
+the isotropic CA model for the growth process of a bacterial c olony by using
+a Voronoi lattice [24] proposed in [25]. In the previous pape r [20], Tanaka
+and the authors also considered an isotropic CA model for the BZ reaction.
+This simple square lattice model adopts a Moore neighborhoo d with radius
+1 as for the neighborhood cells and recovers the isotropy by i ntroducing
+spatial randomness in relation to a threshold controlling t he excitation of
+reaction.
+In the present paper, we propose a new method to construct a CA model
+corresponding to a reaction-diffusion system. The method is b ased on the
+random walk and on the phase diagram of reaction equations. W e require
+the method to satisfy the following two conditions: (i)The t ime evolution
+pattern of the CA preserves the isotropy of the reaction-diffu sion equation.
+(ii)The CA model explicitly contains the control parameter s corresponding
+to the reaction terms and diffusion terms in the reaction-diffus ion equation.
+In the next section, we explain the general method to constru ct the
+CA corresponding to a given reaction-diffusion equation. In s ection 3, the
+method introduced in section 2 is applied to the BZ reaction, and the ob-
+tained evolution patterns are examined. A summary or the res ults is given
+in section 4.CONSTRUCTION OF AN ISOTROPIC CELLULAR AUTOMATON FOR A REACT ION-DIFFUSION EQUATION BY MEANS OF A RANDOM WALK 3
+2.methodology
+In this section, a method to construct the CA corresponding t o a given
+reaction-diffusion equation is introduced. The idea is very s imple: to replace
+the diffusion term by a random walk process and the reaction ter ms by time
+evolution along a discrete vector field obtained from the pha se diagram.
+First, we explain the reaction-diffusion equation in brief, t hen we introduce
+our CA model.
+2.1.Reaction-Diffusion Equation. Suppose that there are Ndifferent
+reactive materials U1,U2,···,UNin some spatial region. Let the densities
+of these materials at position rand time tbeu1(r,t),u2(r,t),···,uN(r,t)
+respectively, and put u:=(u1,u2,···,uN)T∈RN. The reaction-diffusion
+equation is given as
+(1)∂u
+∂t=f(u)+D∇2u,
+whereD= diag(d1,d2,···,dN) i.e.Dis aN×Ndiagonal matrix which has
+the diffusion coefficient djfor each material as a diagonal element. The vec-
+torf(u) = (f1(u),f2(u),···,fN(u))Texpresses the interactions between
+the materials and ∇:=∂
+∂ris the nabla symbol, in particular ∇2=∂2
+∂x2+∂2
+∂y2
+in two spatial dimension: r= (x,y).
+2.2.The CA Model. We now introduce our cellular automaton model
+for the reaction-diffusion equation (1). For simplicity, we a ssume that the
+equation is defined in two spatial dimensions and that the CA m odel is
+defined on a two dimensional square lattice. Generalization to higher di-
+mensional systems and different types of lattices will be stra ightforward.
+In the reaction-diffusion equation (1), the variable urepresents the density
+of reactive material. In our CA model, we replace the density of material
+u(r,t) by the number of microscopic particles ut
+mn∈Z+Nat the corre-
+sponding lattice point ( m,n)∈Z2at time step t∈Z/2. Then the time
+evolution of our CA model is determined by
+ut+1/2
+mn=ut
+mn+R(ˆut
+mn),
+ut+1
+mn=ut+1/2
+mn+F(ut+1/2
+mn),(2)
+whereˆut
+mndenotes a set of concentration variables around ( m,n) at time
+stept,R(ˆut
+mn) andF(ut
+mn) denote discrete vector fields which are obtained
+by replacing the diffusion and reaction terms of the reaction- diffusion equa-
+tion respectively with the following processes:
+(i)The diffusion term D∇2ucorresponds to a random walk of particles.
+Let us consider a random walk of particles defined in the Neuma nn neigh-
+borhood ˆut
+mn={ut
+m,n,ut
+m−1,n,ut
+m+1,n,ut
+m,n−1,ut
+m,n+1}. One can equally
+adopt the Moor neighborhood or other neighborhoods. We deno te byU→t
+mn
+the stochastic variable which defines the number of particle s moving from
+the (m,n) site to the ( m+1,n) site due to the random walk of the particles,4 A. NISHIYAMA AND T. TOKIHIRO
+byU→t
+m−1,nthat from from ( m−1,n) to (m,n), and so on. Then R(ˆut
+mn)
+equals the difference between the number of outgoing particle s and that of
+the incoming particles due to the random walk at position ( m,n) and time
+t:
+R(ˆut
+mn) =U→t
+m−1,n+U←t
+m+1,n+U↑t
+m,n−1+U↓t
+m,n+1
+−(U→t
+mn+U←t
+mn+U↑t
+mn+U↓t
+mn). (3)
+If we define pjas the transition probability of particles to one neighbor-
+ing cell in the random walk, 1 −4pjis the probability of particles to stay
+on site and the expectation of U→t
+mn,U→t
+m−1,netc. are given respectively
+by/an}b∇acketle{tU→t
+m,n/an}b∇acket∇i}ht=Put
+m−1,n,/an}b∇acketle{tU→t
+m−1,n/an}b∇acket∇i}ht=Put
+m−1,nwith a diagonal matrix P=
+diag(p1,p2,···,pN). We can see that Pcorresponds to the diffusion coeffi-
+cientDof the reaction-diffusion equation.
+(ii)The reaction term f(u) is replaced with an appropriate discrete func-
+tionF(ut
+mn) = (F1(ut
+mn),F2(ut
+mn),···,FN(ut
+mn))T∈ZN. In equation (1),
+the reaction term f(u) is the vector field that defines the velocity vector∂u
+∂t.
+HenceF(x) should be so chosen such that the time evolution of xis consis-
+tent with the typical orbits in the phase diagram of the ordin ary differential
+equation
+du
+dt=f(u).
+SinceF(ut
+mn) sometimes returns a negative number, it is practically con -
+venient to use the discrete vector field, G(ut
+mn) :=ut
+mn+F(ut
+mn)≥0.
+Then our CA model is rewritten as
+ut+1/2
+mn=ut
+mn+R(ˆut
+mn),
+ut+1
+mn=G(ut+1/2
+mn).(4)
+We can obtain the time evolution pattern by successive subst itution of
+R(ˆut
+mn) andG(ut
+mn) for appropriate initial conditions.
+It is well known that the continuous limit of the random walk i s equiva-
+lent to a diffusion equation, and that in the large scale limit, isotropy of the
+distribution function of the particles is guaranteed by a ra ndom walk. Since
+the discrete vector field F(u) is chosen such that it is essentially equivalent
+to the vector field f(u) in the continuous limit, we expect that the patterns
+obtained from our CA model become almost isotropic in certai n large sys-
+tems. It should be noted that our model naturally contains th e parameters
+{pi}which correspond to the diffusion coefficients and all other con trol pa-
+rameters for reactions, which are necessarily contained in the discrete vector
+field.
+3.CA model for the Belousov-Zhabotinsky reaction
+In this section, we apply the method introduced in the previo us section
+to the Belousov-Zhabotinsky (BZ) reaction as a specific exam ple. First,CONSTRUCTION OF AN ISOTROPIC CELLULAR AUTOMATON FOR A REACT ION-DIFFUSION EQUATION BY MEANS OF A RANDOM WALK 5
+we briefly explain the BZ reaction and oregonator known as a ma thematical
+modelforthisreaction. Next, weintroduceourCAmodelofth eBZreaction.
+3.1.BZ Reaction. The BZ reaction is known as an oscillating oxidation-
+reduction reaction which occurs by mixing some chemical com pounds (such
+as Ce4+, BrO3−, CH2(COOH) 2, H2SO4). If the BZ reaction is spread spa-
+tially, then it forms trigger waves, spiral waves or target p atterns [4]. The
+BZ reaction is often modeled by using partial differential equ ations. Among
+them, the oregonator, which is a system of simultaneous ordi nary differen-
+tial equations with two variables, is widely considered to b e the simplest
+possible model [6]. The time evolution of the spatial patter ns in the BZ
+reaction is described by the following equation which adds d iffusion terms
+to the oregonator:
+∂u
+∂t=1
+ǫ/bracketleftbigg
+u(1−u)−bv(u−a)
+u+a/bracketrightbigg
++du∇2u,
+∂v
+∂t=u−v+dv∇2v,(5)
+wherebandǫ(or 1/ǫ) are, respectively, a threshold which gives the excita-
+tion and a parameter which defines the excitability of reacti on respectively.
+Depending on the parameters a,bandǫ, Eq. (5) shows two typical states:
+an excitable state with one stable equilibrium point and an o scillatory state
+with one unstable equilibrium point. Here we consider only t he excitable
+state (or excitable media).
+Letf(u,v) :=1
+ǫ[u(1−u)−bv(u−a)
+u+a] andg(u,v) :=u−v. Figure 1 shows
+the phase diagram for the excitable state of the BZ reaction. The null
+clines that are obtained from f(u,v) =g(u,v) = 0 are shown by solid lines
+and a typical orbit for the excitable state is shown by a dashe d line. The
+intersecting point of f= 0 andg= 0 is a stable point if there is no diffusion,
+however it becomes unstable when the strength of the perturb ation (mainly
+due to the diffusion effects) exceeds a certain threshold δ. Then the state
+of the medium becomes unstable and changes along the dashed l ine shown
+in the phase diagram, until it returns to the equilibrium poi nt once again.
+The repetition of this process induces spacial patterns suc h as spiral waves.
+3.2.The CA Model. Our CA model for Eq. (5) is described in the form
+of Eq. (4), introduced in section 2.2. According to Eq. (5), w e put
+ut
+mn:= (ut
+mn,vt
+mn)T,G(ut
+mn) := (Gu(ut
+mn),Gv(ut
+mn))TandR(ˆut
+mn) :=
+(Ru(ˆut
+mn),Rv(ˆvt
+mn))T.
+First, we define the discrete vector field G(ut
+mn) by imitating the solution
+orbit of the phase diagram shown in Figure 1. The discrete vec tor field we
+configured is shown in Table 1 and illustrated in Figure 2. We c an see that
+Figure 2 is a simplification of the phase diagram of Figure 1. L etut
+mn∈Z+,
+vt
+mn∈ {0,1}in this example of an excitable BZ reaction. Parameters α,
+βandγcontrol the rate of reaction, and ∆ is the threshold which det er-
+mines whether the excitation occurs or not. Parameter N∈Z+denotes the6 A. NISHIYAMA AND T. TOKIHIRO
+Figure 1. The phase diagram for the excitable BZ reaction.
+expected maximum value of variable ut
+mn. All of these five parameters are
+positive integers.
+The value of the discrete vector field G(ut
+mn,vt
+mn) is determined in func-
+tion of the values of ut
+mnandvt
+mn, as shown in Table 1. For 0 ≤ut
+mn<
+∆,vt
+mn= 0, the state of the medium returns to the equilibrium point w ith
+velocityα, because the state cannot exceed the threshold ∆ due to inade -
+quate diffusion effects. For ∆ ≤ut
+mn< N−1−β,vt
+mn= 0, the variable ut
+mn
+increases at therate βuntilut
+mn≥N−1−β. Intherange γ < ut
+mn,vt
+mn= 1,
+the variable ut
+mndecreases at the rate of γ. In the two remaining ranges,
+the variable vt
+mnchanges between 0 and 1. Here we assumed vt
+mn∈ {0,1},
+because two states are sufficient to separate the phases, acco rding to the
+medium, corresponding to the variable vt
+mn.
+Table 1. Example of discrete vector field G(ut
+mn) for an
+excitable medium.
+(ut
+mn,vt
+mn) =ut
+mn (Gu(ut
+mn),Gv(ut
+mn)) =G(ut
+mn)
+(0≤ut
+mn<∆,vt
+mn= 0) (max[ut
+mn−α,0],0)
+(∆≤ut
+mn< N−1−β,vt
+mn= 0) (ut
+mn+β,0)
+(N−1−β≤ut
+mn,vt
+mn= 0) (N−1,1)
+(γ < ut
+mn,vt
+mn= 1) (ut
+mn−γ,1)
+(0≤ut
+mn≤γ,vt
+mn= 1) (0,0)
+InEq. (5), theexcitation ofexcitable mediais determinedb ythediffusion
+ofu, i.e.,du≫dv. Hence, in this example, we only consider the random
+walk of the variable ut
+mnand we simulate the time evolution by putting
+P= diag(pu,0).CONSTRUCTION OF AN ISOTROPIC CELLULAR AUTOMATON FOR A REACT ION-DIFFUSION EQUATION BY MEANS OF A RANDOM WALK 7
+Figure 2. Outline of the discrete vector field G(ut
+mn) in Table1.
+Consequently, our CA model for the excitable BZ reaction is g iven as:
+ut+1/2
+mn=ut
+mn+Ru(ˆut
+mn),
+vt+1/2
+mn=vt
+mn,
+ut+1
+mn=Gu(ut+1/2
+mn),
+vt+1
+mn=Gv(ut+1/2
+mn).(6)
+3.3.Numerical Results. In this section, we show several time evolution
+patterns obtained from Eq. (6) and discuss the results.
+3.3.1.Patterns. The first example is a single trigger wave (Figure 3). It
+is produced by the initial condition u0
+mn= [h·exp(−((m−L/2)2+ (n−
+L/2)2)/w2)] andv0
+mn= 0 on a 2-dimensional square area with L×Lcells.
+Here [ ] is Gauss’ symbol, i.e.: [ x] is the largest integer that is less than or
+equal to xandhandware positive real numbers. We observed that from
+a pulse triggered at the center, a ring-shaped wave spreads o utwards in an
+almost isotropic fashion.
+In order to generate a spiral wave, appropriate initial cond itions are nec-
+essary. Firstly, wegenerate asingletrigger wave like theo neshownin Figure
+3. Then we cut off one part of the ring pattern as shown in Figure 4 and
+use the remainder as the initial state. The spiral wave obtai ned from our
+model is shown in Figure 5.
+The third example is a target pattern (Figure 6). The initial condition is
+the same as in the case of the trigger wave, but with different pa rameters.
+From the central pulse, ring-shaped waves are produced repe atedly.
+3.3.2.Anisotropy. We evaluate the anisotropy of the trigger wave in Figure
+3 by measuring the residual error when compared with the aver age radius of
+the ring, and plot it in Figure 7 as a function of the propagati on direction.
+We find that the wave fronts of the trigger wave propagate in ea ch direction
+with an anisotropy in the range of ±2.5 percent. In Figure 8 we plot the
+variation from the circle as a function of the radius of the tr igger wave. It
+shows that the trigger wave indeed grows closer to a complete circle as the
+radius (and therefore time) increases.8 A. NISHIYAMA AND T. TOKIHIRO
+Figure 3. Single trigger wave. 500 ×500 cells, t= 5850,
+N= 100,pu= 0.2, ∆ = 21, α= 1,β= 1,γ= 1.
+Figure 4. A cut trigger wave. 200 ×200 cells, t= 250,
+N= 30,pu= 0.2, ∆ = 6, α= 1,β= 2,γ= 1.
+Next, we evaluate the parameter dependency of the anisotrop y for the
+patterns observed from our model. In Figures 9 and 10, the var iations of
+the wavefronts of trigger waves propagated to radius 450, ar e plotted as a
+function of the transition probability puof particles by changing parameters
+β(Fig. 9) and∆ (Fig. 10). We findthat thepatterns becomemorei sotropic
+when diffusion becomes stronger, and that they tend to be isotr opic for
+smallerβ, but depend less on ∆.
+3.3.3.Parameters. Our CA model has six parameters N,pu, ∆,α,βand
+γ. The parameter ∆ corresponds to bin Eq. (5) and defines the threshold
+of excitation; βcorresponds to 1 /ǫwhich defines the excitability of reaction.
+Figure 11 is the phasediagram of our model obtained by changi ng the values
+of ∆ and β. The remaining parameters are chosen as N= 50,pu= 0.1,α=
+γ= 1. We confirm that spiral waves can be generated in a wide regi on of
+the parameters. In an even larger parameter range, we find the re exist threeCONSTRUCTION OF AN ISOTROPIC CELLULAR AUTOMATON FOR A REACT ION-DIFFUSION EQUATION BY MEANS OF A RANDOM WALK 9
+Figure 5. Spiral wave. 200 ×200 cells, t= 963,N= 30,
+pu= 0.2, ∆ = 6, α= 1,β= 2,γ= 1.
+Figure 6. Target pattern. 300 ×300 cells, t= 626,N= 100,
+pu= 0.04, ∆ = 2, α= 1,β= 10,γ= 1.
+different regions; one without propagating waves, one allowi ng for trigger
+waves (or broken trigger waves), and one chaotic pattern reg ion.
+The parameter range for the spiral waves obtained from Eq. (5 ) has been
+examined in detail by Jahnke and Winfree [7]. The general ten dency is that,
+whenbis sufficiently large, the spiral wave does not appear and for 1 /ǫ≫1
+the spiral wave destabilizes and chaotic behaviour is obser ved. As shown in
+Figure 11, the phase diagram of our model has the same feature as that for
+Eq. (5), and we may conclude that the parameters ∆ and βplay the same
+role as the control parameters of the reaction terms in the re action-diffusion
+equation.
+4.summary
+We have proposed a method to construct a CA model correspondi ng to
+a reaction-diffusion equation, in which the diffusion effect is re placed by10 A. NISHIYAMA AND T. TOKIHIRO
+0 π/2 π 3π/2 2π
+Angle (rad)-5-2.502.55Residual Error (%)
+Figure 7. Plot of the anisotropy of the model, measured as
+the residual error of the trigger pattern in Figure3.
+0 100 200 300 400
+Radius1525Variation (%)
+Figure 8. The relationship between the radius of the ring
+and the variation of the radius.
+a random walk with transition probability matrix Pand the reaction by
+discrete vector fields. The model can include control parame ters for both
+diffusion and reaction, as is the case in the reaction-diffusion equation. AsCONSTRUCTION OF AN ISOTROPIC CELLULAR AUTOMATON FOR A REACT ION-DIFFUSION EQUATION BY MEANS OF A RANDOM WALK 11
+0.1 0.15 0.2
+pu11.522.53Variatiaon (%)β=1
+β=2
+β=3
+β=4
+β=5
+Figure 9. The parametric dependency of the anisotropy of
+the model for the parameter β.N= 100, ∆ = 15, α= 1,
+γ= 1.
+an example, we have shown that our model can successfully rep roduce the
+patterns of BZ reaction. Applications to other reaction-di ffusion systems
+such as FitzHugh-Nagumo equation [26] are interesting futu re problems. In
+the present method, however, there are still various candid ates for the time
+evolution rules, depending on the choice of the discrete vec tor fields that are
+supposed to have similar features to those of the continuous vector fields,
+given by the reaction-diffusion equation. Of course we can als o adopt dis-
+crete vector fields by investigating the reaction process fr om a microscopic
+point of view. The determination a suitable time evolution r ule will de-
+pend on the individual phenomenon and we will investigate th is problem
+extensively in the future.
+Acknowledgments
+We would like to thank professors Hiroshi Tanaka and Ralph Wi llox for
+useful discussions and comments.12 A. NISHIYAMA AND T. TOKIHIRO
+0.1 0.15 0.2
+pu0.60.811.21.41.61.82Variation (%)∆=10
+∆=15
+∆=20
+Figure 10. The parameter dependence of the anisotropy of
+the model on the parameter ∆. N= 100,α= 1,β= 1,
+γ= 1.
+References
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+Figure 11. The phase diagram of the patterns obtained
+from our CA. N= 50,pu= 0.1,α=γ= 1.
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\ No newline at end of file
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+arXiv:1001.0506v2  [astro-ph.SR]  18 Jan 2010The asteroseismic potential of Kepler: first results for solar-type
+stars
+W. J. Chaplin1, T. Appourchaux2, Y. Elsworth1, R. A. Garc´ ıa3, G. Houdek4, C. Karoff1,
+T. S. Metcalfe5, J. Molenda- ˙Zakowicz6, M. J. P. F. G. Monteiro7, M. J. Thompson8,
+T. M. Brown9, J. Christensen-Dalsgaard10, R. L. Gilliland11, H. Kjeldsen10,
+W. J. Borucki12, D. Koch12, J. M. Jenkins13, J. Ballot14, S. Basu15, M. Bazot7,
+T. R. Bedding16, O. Benomar2, A. Bonanno17, I. M. Brand˜ ao7, H. Bruntt18,
+T. L. Campante7,10, O. L. Creevey19,20, M. P. Di Mauro21, G. Do˘ gan10, S. Dreizler22,
+P. Eggenberger23, L. Esch15, S. T. Fletcher24, S. Frandsen10, N. Gai15,25, P. Gaulme2,
+R. Handberg10, S. Hekker1, R. Howe26, D. Huber16, S. G. Korzennik27, J. C. Lebrun28,
+S. Leccia29, M. Martic28, S. Mathur30, B. Mosser31, R. New24, P.-O. Quirion10,32,
+C. R´ egulo19,20, I. W. Roxburgh33, D. Salabert19,20, J. Schou34, S. G. Sousa7, D. Stello16,
+G. A. Verner33, T. Arentoft10, C. Barban31, K. Belkacem35, S. Benatti36, K. Biazzo37,
+P. Boumier2, P. A. Bradley38, A.-M. Broomhall1, D. L. Buzasi39, R. U. Claudi40,
+M. S. Cunha7, F. D’Antona41, S. Deheuvels31, A. Derekas42,16, A. Garc´ ıa Hern´ andez43,
+M. S. Giampapa26, M. J. Goupil18, M. Gruberbauer44, J. A. Guzik38, S. J. Hale1,
+M. J. Ireland16, L. L. Kiss42,16, I. N. Kitiashvili45, K. Kolenberg4, H. Korhonen46,
+A. G. Kosovichev34, F. Kupka47, Y. Lebreton48, B. Leroy31, H.-G. Ludwig48, S. Mathis3,
+E. Michel31, A. Miglio35, J. Montalb´ an35, A. Moya49, A. Noels35, R. W. Noyes27,
+P. L. Pall´ e20, L. Piau3, H. L. Preston39,50, T. Roca Cort´ es19,20, M. Roth51, K. H. Sato3,
+J. Schmitt52, A. M. Serenelli53, V. Silva Aguirre53, I. R. Stevens1, J. C. Su´ arez43,
+M. D. Suran54, R. Trampedach55, S. Turck-Chi` eze3, K. Uytterhoeven3, R. Ventura17– 2 –
+1School of Physics and Astronomy, University of Birmingham, Edgba ston, Birmingham, B15 2TT, UK
+2Institut d’Astrophysique Spatiale, Universit´ e Paris XI – CNRS (UMR 8617), Batiment 121, 91405 Orsay
+Cedex, France
+3Laboratoire AIM, CEA/DSM – CNRS – Universit´ e Paris Diderot – IRFU /SAp, 91191 Gif-sur-Yvette
+Cedex, France
+4Institute of Astronomy, University of Vienna, A-1180 Vienna, Aus tria
+5High Altitude Observatory and, Scientific Computing Division, Nationa l Center for Atmospheric Re-
+search, Boulder, Colorado 80307, USA
+6Astronomical Institute, University of Wroc/suppress law, ul. Kopernika, 11 , 51-622 Wroc/suppress law, Poland
+7Centro de Astrof´ ısica, Universidade do Porto, Rua das Estrelas, 4150-762, Portugal
+8School of Mathematics and Statistics, University of Sheffield, Houn sfield Road, Sheffield S3 7RH, UK
+9Las Cumbres Observatory Global Telescope, Goleta, CA 93117,USA
+10Department of Physics and Astronomy, Aarhus University, DK-80 00 Aarhus C, Denmark
+11Space Telescope Science Institute, Baltimore, MD 21218, USA
+12NASA Ames Research Center, MS 244-30, Moffett Field, CA 94035, U SA
+13SETI Institute/NASA Ames Research Center, MS 244-30, Moffett Field, CA 94035, USA
+14Laboratoire d’Astrophysique de Toulouse-Tarbes, Universit´ e de Toulouse, CNRS, 14 av E. Belin, 31400
+Toulouse, France
+15Department of Astronomy, Yale University, P.O. Box 208101, New H aven, CT 06520-8101, USA
+16Sydney Institute for Astronomy (SIfA), School of Physics, Univ ersity of Sydney, NSW 2006, Australia
+17INAF Osservatorio Astrofisico di Catania, Via S.Sofia 78, 95123, Ca tania, Italy
+18Observatoire de Paris, 5 place Jules Janssen, 92190 Meudon Princip al Cedex, France
+19Departamento de Astrof´ ısica, Universidad de La Laguna, E-2820 7 La Laguna, Tenerife, Spain
+20Instituto de Astrof´ ısica de Canarias, E-38200 La Laguna, Tener ife, Spain
+21INAF-IASF Roma, Istituto di Astrofisica Spaziale e Fisica Cosmica, v ia del Fosso del Cavaliere 100,
+00133 Roma, Italy
+22Georg-August Universit¨ at, Institut f¨ ur Astrophysik, Friedric h-Hund-Platz 1, D-37077 G¨ ottingen
+23Geneva Observatory, University of Geneva, Maillettes 51, 1290, S auverny, Switzerland
+24Materials Engineering Research Institute, Faculty of Arts, Compu ting, Engineering and Sciences,
+Sheffield Hallam University, Sheffield, S1 1WB, UK
+25Beijing Normal University, Beijing 100875, P.R. China
+26National Solar Observatory, 950 N. Cherry Ave., POB 26732, Tucs on, AZ 85726-6732, USA– 3 –
+27Harvard-Smithsonian Center for Astrophysics, 60 Garden Stree t, Cambridge, MA 02138, USA
+28LATMOS, University of Versailles St Quentin, CNRS, BP 3, 91371 Verr ieres le Buisson Cedex, France
+29INAF-OAC, Salita Moiariello, 16 80131 Napoli, Italy
+30Indian Institute of Astrophysics, Koramangala, Bangalore, 5600 34, India
+31LESIA, CNRS, Universit´ e Pierre et Marie Curie, Universit´ e, Denis D iderot, Observatoire de Paris, 92195
+Meudon cedex, France
+32Canadian Space Agency, 6767 Boulevard de l’A´ eroport, Saint-Hub ert, QC, J3Y 8Y9, Canada
+33Astronomy Unit, Queen Mary, University of London, Mile End Road, L ondon, E1 4NS, UK
+34HEPL, Stanford University, Stanford, CA 94305-4085, USA
+35D´ epartement d’Astrophysique, G´ eophysique et Oc´ eanograph ie (AGO), Universit´ e de Li` ege, All´ ee du 6
+Aoˆ ut 17 4000 Li` ege 1, Belgique
+36CISAS (Centre of Studies and Activities for Space), University of P adova, Via Venezia 15, 35131, Padova,
+Italy
+37Arcetri Astrophysical Observatory, Largo Enrico Fermi 5, 501 25 Firenze, Italy
+38Los Alamos National Laboratory, Los Alamos, NM 87545-2345 USA
+39Eureka Scientific, 2452 Delmer Street Suite 100, Oakland, CA 94602 -3017
+40INAF Astronomical Observatory of Padova, vicolo Osservatorio 5 35122 Padova, Italy
+41INAF Osservatorio di Roma, via di Frascati 33, I-00040 Monte Po rzio, Italy
+42Konkoly Observatory of the Hungarian Academy of Sciences, Buda pest, Hungary
+43Instituto de Astrof´ ısica de Andaluc´ ıa (CSIC), CP3004, Granada , Spain
+44Department of Astronomy and Physics, Saint Mary’s University, Ha lifax, NS B3H 3C3, Canada
+45Center for Turbulence Research, Stanford University, 488 Esco ndido Mall, Stanford, CA 94305 USA
+46European Southern Observatory, Karl-Schwarzschild-Str. 2, D -85748 Garching bei M¨ unchen, Germany
+47Faculty of Mathematics, University of Vienna, Nordbergstraße 15 , A-1090 Wien, Austria
+48GEPI, Observatoire de Paris, CNRS, Universit´ e Paris Diderot, 5 Pla ce Jules Janssen, 92195 Meudon,
+France
+49Laboratorio de Astrof´ ısica Estelar y Exoplanetas, LAEX-CAB (IN TA-CSIC), Villanueva de la Ca˜ nada,
+Madrid, PO BOX 78, 28691, Spain
+50Department of Mathematical Sciences, University of South Africa , Box 392, UNISA 0003, South Africa
+51Kiepenheuer-Institut f¨ ur Sonnenphysik, Sch¨ oneckstr. 6, 79 104 Freiburg, Germany
+52Observatoire de Haute-Provence, F-04870, St.Michel l’Observat oire, France
+53Max Planck Institute for Astrophysics, Karl Schwarzschild Str. 1 , Garching, D-85741, Germany– 4 –
+ABSTRACT
+We present preliminary asteroseismic results from Kepleron three G-type
+stars. The observations, made at one-minute cadence during the first 33.5d of
+science operations, reveal high signal-to-noise solar-like oscillation spectra in all
+three stars: About 20 modes of oscillation may be clearly distinguishe d in each
+star. We discuss the appearance of the oscillation spectra, use th e frequencies
+and frequency separations to provide first results on the radii, ma sses and ages
+of the stars, and comment in the light of these results on prospect s for inference
+on other solar-type stars that Keplerwill observe.
+Subject headings: stars: oscillations — stars: interiors — stars: late-type
+1. Introduction
+TheKeplermission (Koch et al. 2010) will realize significant advances in our under -
+standing of stars, thanks to its asteroseismology program, part icularly for cool (solar-type)
+main-sequence andsubgiant starsthat show solar-likeoscillations, i.e., small-amplitude oscil-
+lations intrinsically damped and stochastically excited by the near-su rface convection (e.g.,
+Christensen-Dalsgaard 2004). Solar-like oscillation spectra have m any modes excited to ob-
+servable amplitudes. The rich information content of these seismic s ignatures means that
+the fundamental stellar properties (e.g., mass, radius, and age) m ay be measured and the
+internal structures constrained tolevels thatwouldnototherwis e bepossible (e.g., see Gough
+1987; Cunha et al. 2007).
+High-precision results are presently available on a selection of bright solar-type stars
+(e.g., see Aerts et al. 2010, and references therein). Recent exa mples include studies of
+CoRoTsatellite data (e.g., Michel et al. 2008; Appourchaux et al. 2008), an d data collected
+by episodic ground-based campaigns (e.g., Arentoft et al. 2008). H owever, the Kepler
+spacecraft will increase by more than two orders of magnitude the number of stars for which
+high-quality observations will be available, and will also provide unprec edented multi-year
+observations of a selection of these stars.
+During the initial ten months of science operations, Keplerwill survey photometrically
+more than 1500 solar-type targets selected for study by the Kepler Asteroseismic Science
+54Astronomical Institute of the Romanian Academy, Str. Cutitul de Argint, 5, RO 40557,Bucharest,RO
+55JILA, University of Colorado, 440 UCB, Boulder, CO 80309-0440, U .S.A.– 5 –
+Consortium (KASC) (Gilliland et al. 2010a). Observations will be one month long for
+each star. In order to aid preparations for analyses of these sta rs,Keplerdata on three
+bright solar-type targets – KIC 6603624, KIC 3656476 and KIC 11 026764 – have been made
+available in a preliminary release to KASC (see Table 1 for the 2MASS ID o f each star). The
+stars are at the bright end of the Keplertarget range, having apparent magnitudes of 9.1,
+9.5 and 9.3mag, respectively. The Kepler Input Catalog (KIC; Latham et al. 2005), from
+which all KASC targets were selected, categorizes them as G-type stars. In this Letterwe
+present initial results on these stars.
+2. The Frequency Power Spectra of the Stars
+Thestarswereobservedforthefirst 33.5daysofscience operat ions(2009May12toJune
+14). Detection of oscillations of cool main-sequence and subgiant s tars demands use of the
+58.85-sec, high-cadence observations since the dominant periods in some solar-type stars can
+be as short as two minutes. Timeseries data were then prepared fr om the raw observations in
+the manner described by Gilliland et al. (2010b). Fig. 1 shows the freq uency-power spectra
+on a log-log scale (grey). Heavily smoothed spectra (Gaussian filter ) are over-plotted as
+continuous black lines. The quality of these data for asteroseismolo gy is excellent, with each
+spectrum showing a clear excess of power due to solar-like oscillation s. The excess is in each
+case imposed upon a background that rises slowly toward lower freq uencies.
+2.1. Power Spectral Density of the Background
+The average power spectral density at high frequencies provides a good measure of the
+power due to photon shot noise. The high-frequency power is in all c ases close to pre-launch
+estimates, given the apparent magnitudes of the targets, sugge sting that the data areclose to
+being shot-noise limited (e.g., see Gilliland et al. 2010b). This result lends further confidence
+to our expectations – from hare-and-hounds exercises with simula ted data – that we will
+be able to conduct asteroseismology of solar-like KASC survey targ ets down to apparent
+magnitudes of 11 and fainter (e.g., see Stello et al. 2009).
+There are also background components of stellar origin, and we des cribe these com-
+ponents by power laws in frequency. Kinks in the observed backgro unds of KIC 6603624
+and KIC 3656476 (kinks in solid black curves, indicated by arrows on F ig. 1) suggest the
+presence of two stellar components in the plotted frequency rang e. One component (dot-
+ted lines) is possibly the signature of bright faculae. A similar signatur e, due to faculae, is– 6 –
+seen clearly in frequency-power spectra of Sun-as-a-star data collected by the VIRGO/SPM
+photometers on SOHO(e.g., Aigrain et al. 2004). We do not see a kink in the background
+of KIC 11026764, and it may be that the characteristic “knee” of t he facular component
+coincides in frequency with the oscillation envelope, making it hard to d iscriminate from the
+other components. The other stellar component, which is shown by all three stars (dashed
+lines), carries the signature of stellar granulation.
+2.2. The Oscillation Spectra
+Fig. 2 shows in more detail the oscillation spectra of the three stars . The high signal-
+to-noise ratios observed in the mode peaks allows about 20 individual modes to be identified
+very clearly in each star.
+Stars KIC 6603624 and KIC 3656476 present patterns of peaks t hat all show nearly
+regular spacings in frequency. These peaks are due to acoustic (p ressure, or p) modes of high
+radial order, n, with frequencies approximately proportional to√¯ρ, ¯ρ∝M/R3being the
+mean density of the star, with mass Mand surface radius R. The most obvious spacings are
+the “large frequency separations”, ∆ ν, between consecutive overtones of the same spherical
+angular degree, l. These large separations are related to the acoustic radii of the s tars.
+The “small frequency separations” are the spacings between mod es adjacent in frequency
+that have the same parity angular degree. Here, we see clearly the small separations δν02
+between modes of l= 0 and l= 2 (photometric observations have low sensitivity to l= 3
+modes, and so they cannot be seen clearly in these frequency-pow er spectra). The small
+separations are very sensitive to the gradient of the sound speed in the stellar cores and
+hence the evolutionary state of the stars.
+The near regularity of the frequency separations of KIC 6603624 and KIC 3656476 is
+displayed in the ´ echelle diagrams in Fig. 3. Here, we have plotted estim ates of the mode
+frequencies of the stars (see Section 3 below) against those freq uencies modulo the average
+large frequency separations. In a simple asymptotic description of high-order p modes (e.g.,
+Tassoul 1980) the various separations do not change with freque ncy. Stars that obeyed this
+description would show vertical, straight ridges in the ´ echelle diagra m (assuming use of the
+correct ∆ ν). Solar-type stars do in practice show departures of varying deg rees from this
+simple description. These variations with frequency carry signatur es of, for example, regions
+of abrupt structural change in the stellar interiors, e.g., the near -surface ionization zones and
+the bases of the convective envelopes (Houdek & Gough 2007).
+WhilethevariationsareclearlymodestinKIC6603624andKIC365647 6,KIC11026764– 7 –
+presents a somewhat different picture. Its l= 1 ridge is noticeably disrupted, and shows
+clear evidence of so-called avoided crossings (Osaki 1975; Aizenma n et al. 1977), where
+modes resonating in different cavities exist at virtually the same freq uency.
+These avoided crossings are a tell-tale indicator that the star has e volved significantly.
+In young solar-type stars there is a clear distinction between the f requency ranges that
+will support p modes and buoyancy (gravity, or g) modes. As stars evolve, the maximum
+buoyancy (Brunt-V¨ ais¨ al¨ a) frequency increases. After exha ustion of the central hydrogen,
+the buoyancy frequency in the deep stellar interior may increase to such an extent that it
+extends into the frequency range of the high-order acoustic mod es. Interactions between
+acoustic modes and buoyancy modes may then lead to a series of avo ided crossings, which
+affect (or “bump”) the frequencies and also change the intrinsic pr operties of the modes,
+with some taking on mixed p and g characteristics. The jagged appea rance of the l= 1
+ridge of KIC 11026764 illustrates these effects, which are strikingly similar to those seen
+in ground-based asteroseismic data on the bright stars ηBoo (Christensen-Dalsgaard et al.
+1995; Kjeldsen et al. 1995) and βHyi (Bedding et al. 2007). Deheuvels & Michel (2009)
+and Deheuvels et al. (2009) have also reported evidence for avoide d crossings in CoRoT
+observations of the oscillation spectrum of the star HD49385.
+Next, we consider the peaks of individual modes. The observed osc illation modes in
+solar-typestarsareintrinsically stable(e.g. Balmforth1992a; Hou deket al. 1999)butdriven
+stochastically by the vigorous turbulence in the superficial stellar la yers (e.g. Goldreich &
+Keeley 1977; Balmforth 1992b; Samadi & Goupil 2001). Solar-like mo de peaks have an
+underlying form that follows, to a reasonable approximation, a Lore ntzian. The widths of
+the Lorentzians provide a measure of the linear damping rates, while the amplitudes are
+determined by the delicate balance between the excitation and damp ing. Measurement of
+these parameters therefore provides the means to infer various important properties of the
+still poorly understood near-surface convection.
+The observed maximum mode amplitudes are all higher than solar. This is in line with
+predictions from simple scaling relations (Kjeldsen & Bedding 1995; Sa madi et al. 2007),
+which use the inferred fundamental stellar properties (see Sectio n 3 below) as input. Data
+froma larger selection of survey stars are required beforewe can say anything more definitive
+about the relations.
+It is clear even from simple visual inspection of the spectra that the intrinsic linewidths
+of the most prominent modes are comparable in size in all three stars to the linewidths
+shown by the most prominent solar p modes ( ≈1µHz). It would not otherwise be possible
+to distinguish the l= 0 and l= 2 modes so easily. The intrinsic frequency resolution of
+these short survey spectra ( ∼0.35µHz) makes it difficult to provide more definitive widths.– 8 –
+However, the appearance of the modes in these stars is consisten t with the suggestion of
+Chaplin et al. (2009) that the linewidths are a strong function of effe ctive temperature.
+The three stars here all have effective temperatures that are sim ilar to, or slightly cooler
+than, solar; while the F-type main-sequence stars observed by CoRoT– which have effective
+temperatures a few hundred degrees hotter than the Sun – exhib it linewidths that areseveral
+times larger than solar (e.g., see Appourchaux et al. 2008; Barban e t al. 2009; Garc´ ıa et al.
+2009). (We add that CoRoT sees linewidths in the G-type star HD493 85 (Deheuvels et al.
+2009) that are also narrower than those observed in F-type star s.)
+Given the inference on the linewidths of the stars, we are able to sta te with renewed
+confidence that it should be possible to extract accurate and prec ise frequencies of l= 0, 1
+and2 modes in many of the brighter solar-type KASC targets observed byKepler, because
+the peaks will be clearly distinguishable. The prospects on cool star s selected for long-
+term observations are particularly good. Here, the combination of improved resolution in
+frequency and modest (i.e., solar-like) linewidths will in principle allow for robust estimation
+of frequency splittings of modes. These splittings have contributio ns from the internal stellar
+rotation and magnetic fields. We add that longer-term observation s of the three survey stars
+reported in this Letterwill be needed to show indisputable evidence of frequency splitting.
+3. Inference on the Stellar Properties
+An estimate of the average separations ∆ νandδν02provides a complementary set
+of seismic data well suited to constraining the stellar properties (Ch ristensen-Dalsgaard
+1993). In faint KASC survey targets – where lower signal-to-noise ratios will make it difficult
+to extract robust estimates of individual frequencies – the avera ge separations will be the
+primary seismic input data. The signatures of these separations ar e quite amenable to
+extraction, owing to their near-regularity.
+Different teams extracted estimates of the average separations of the three stars, with
+analysis methods based largely on use of the autocorrelation of eith er the timeseries or
+the power spectrum (e.g., see Huber et al. 2009; Mosser & Appourc haux 2009; Roxburgh
+2009; Hekker et al. 2010; Mathur et al. 2010). We found good agre ement between the
+different estimates (i.e., at the level of precision of the quoted para meter uncertainties).
+Representative estimates of the separations are given in Table 1. T he teams also used
+peak-fitting techniques (like those applied to CoRoT data; see, for example, Appourchaux
+et al. 2008) to make available to the modeling teams initial estimates of the individual
+mode frequencies. The ´ echelle diagrams plotted in Fig. 3 show repre sentative sets of these
+frequencies.– 9 –
+Several modeling teams then applied codes to estimate the stellar pr operties using the
+frequency separations, and other non-seismic data (see below), as input; the results from
+these analyses were then used as the starting points for further modeling, involving com-
+parisons of the observed frequencies with frequencies calculated from evolutionary models.
+Use of individual frequencies increases the information content pr ovided by the seismic data
+(e.g., see Monteiro et al. 2000; Roxburgh & Vorontsov 2003; Mazum dar et al. 2006; Cunha
+& Metcalfe 2007). (For a general discussion of the modeling method s, see Cunha et al. 2007;
+Stello et al. 2009; Aerts et al. 2010; and references therein. Furt her detailed presentations
+of the modeling techniques applied here will appear in future papers.)
+The frequencies and the frequency separations depend to some e xtent on the detailed
+physics assumed in the stellar models (Monteiro et al. 2002). Conseq uently, a more secure
+determination of the stellar properties is possible when other comple mentary information
+– such as effective temperature Teff, luminosity (or log g) and metallicity – is known to
+sufficiently high accuracy and precision. The potential to test the in put physics of models of
+field stars (e.g., convective energy transport, diffusion, opacities , etc.) requires non-seismic
+data for the seismic diagnostics to be effective (e.g. see Creevey et al. 2007). The modeling
+analyses therefore also incorporated non-seismic constraints, u singTeff, metallicity ([Fe/H])
+and loggfrom complementary ground-based spectroscopic observations (see Table 1). The
+uncertainties in these spectroscopically estimated parameters ar e significantly lower than
+those given by the KIC parameters. Preliminary results given by diffe rent groups on the
+same ground-based spectra of these stars do however suggest that the true, external errors
+are higher than the quoted errors. Follow-up spectroscopic and p hotometric observations,
+andfurthercomparativeanalyses, arenowbeingcarriedoutforo therbrightsolar-typeKASC
+targets by more than twenty members of KASC (e.g., Molenda- ˙Zakowicz et al. 2008).
+Our initial estimates of the stellar radii and masses are presented in Table 1. Given
+the close relation between the global properties of the stars and t heir oscillation frequencies,
+these seismically inferred properties are more precise, and more ac curate, than properties
+inferred without the seismic inputs. The radii of KIC 6603624 and KI C 3656476 have been
+determined to better than 2%, and the masses to better than 6%. The modeling suggests
+both stars may be near the end of their main-sequence lifetimes.
+The precision achieved for KIC 11026764 is not quite as good (about 5% in the radius,
+and about 10% in the mass). This star has evolved off the main sequen ce, and is harder
+to model. KIC 11026764 demonstrates that when mixed modes are o bserved individual
+frequenciescanprovidemorestringenttestsofthemodelingthan cantheaverageseparations.
+Initial results from modeling individual frequencies show it is possible t o reproduce the
+disrupted l= 1 frequency ridge (Fig. 3), indicating an age in the range 6 and 7Gyr . The– 10 –
+modeling also suggests that some of the l= 2 modes may have mixed character.
+Initial modeling results may also help to interpret the observed seism ic spectra, allowing
+further mode frequencies to be identified securely. A possible exam ple for KIC 11026764
+concerns the prominent mode at ∼723µHz. The peak lies on the l= 0 ridge, yet its
+appearance – a narrow peak, indicating a lightly damped mode having m ixed characteristics
+– suggests a possible alternative explanation: the modeling points st rongly to the observed
+power being predominantly from an l= 1 mode that has been shifted so far in frequency
+that it lies on top of an l= 0 mode.
+In summary, all three stars are clearly excellent candidates for lon g-term observations
+byKepler. With up to 1yr of data we would, for example, expect to measure th e depths
+of the near-surface convection zones, and the signatures of ne ar-surface ionization of He. It
+should also be possible to constrain the rotational frequency splitt ings. With 2yr of data
+and more, we would also hope to begin to constrain any long-term cha nges to the frequencies
+and other mode parameters due to stellar-cycle effects (Karoff et al. 2009). More detailed
+modeling will also allow us to characterize the functional form of any r equired near-surface
+corrections to the model frequencies (see Kjeldsen et al. 2008).
+Funding for this Discovery mission is provided by NASA’s Science Mission Directorate.
+The authors wish to thank the entire Keplerteam, without whom these results would not be
+possible. We also thank all funding councils and agencies that have su pported the activities
+of KASC Working Group1, and the International Space Science Ins titute (ISSI). The anal-
+yses reported in this Letter also used observations made with FIES at the Nordic Optical
+Telescope, and with SOPHIE at Observatoire de Haute-Provence.
+Facilities: The Kepler Mission
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+Fig. 1.— Frequency-power spectra of the three stars, smoothed over 1µHz, plotted on a log-
+log scale (grey). The continuous black lines show heavily smoothed sp ectra (Gaussian filter).
+The other lines show best-fitting estimates of different component s: the power envelope
+due to oscillations, added to the offset from shot noise (dot-dashe d); faculae (dotted) and
+granulation (dashed). The arrows indicate kinks in the rising backgr ound (see text).– 15 –
+Fig. 2.— Frequency-power spectra of the three stars, plotted on a linear scale over the
+frequency ranges where the mode amplitudes are most prominent. Examples of the charac-
+teristic large (∆ ν) and small ( δν02) frequency separations are also marked on the spectrum
+of KIC 3656476.
+Table 1. Non-seismic and seismic parameters, and preliminary stellar p ropertiesa
+Star 2MASS Teff logg [Fe/H] ∆ ν δν 02 R M
+ID [K] [dex] [dex] [ µHz] [ µHz] [R ⊙] [M ⊙]
+KIC 6603624b19241119+4203097 5790 ±100 4.56 ±0.10 0.38 ±0.09 110.2 ±0.6 4.7 ±0.2 1.18 ±0.02 1.05 ±0.06
+KIC 3656476c19364879+3842568 5666 ±100 4.32 ±0.06 0.22 ±0.04 94.1 ±0.6 4.4 ±0.2 1.31 ±0.02 1.04 ±0.06
+KIC 11026764b19212465+4830532 5640 ±80 3.84 ±0.10 0.02 ±0.06 50.8 ±0.3 4.3 ±0.5 2.10 ±0.10 1.10 ±0.12
+aNon-seismic parameters are Teff, loggand [Fe/H]; seismic parameters are ∆ νandδν02; and the stellar properties inferred
+from the non-seismic and seismic data are MandR.
+bNon-seismic parameters for KIC 6603624 and KIC 11026764 fro m observations made with FIES at the Nordic Optical
+Telescope (NOT); data reduced in the manner of Bruntt et al. ( 2004, 2008).
+cNon-seismic parameters for KIC 3656476 from observations m ade with SOPHIE at Observatoire de Haute-Provence; data
+reduced in the manner of Santos et al. (2004).– 16 –
+Fig. 3.— ´Echelle diagrams of the observed frequencies in each star, showing thel= 0 (filled
+red symbols), l= 1 (open blue symbols) and l= 2 (small black symbols) ridges.
\ No newline at end of file
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+arXiv:1001.0507v2  [astro-ph.SR]  1 Feb 2010Automated classification of variable stars in the asterosei smology
+program of the Kepler space mission
+J. Blomme1, J. Debosscher1, J. De Ridder1, C. Aerts1,2, R.L. Gilliland3, J.
+Christensen-Dalsgaard4, H. Kjeldsen4, T.M. Brown5, W.J. Borucki6, D. Koch6, J.M.
+Jenkins7, D.W. Kurtz8, D. Stello9, I.R. Stevens10, M.D. Suran11, A. Derekas9,12
+ABSTRACT
+Wepresentthefirstresultsoftheapplicationofsupervisedclassifi cationmethodstothe Kepler
+Q1 long-cadence light curves of a subsample of 2288 stars measure d in the asteroseismology
+program of the mission. The methods, originally developed in the fram ework of the CoRoT
+and Gaia space missions, are capable of identifying the most common t ypes of stellar variability
+in a reliable way. Many new variables have been discovered, among whic h a large fraction
+are eclipsing/ellipsoidal binaries unknown prior to launch. A compariso n is made between our
+classification from the Keplerdata and the pre-launch class based on data from the ground,
+showing that the latter needs significant improvement. The noise pr operties of the Keplerdata
+are compared to those of the exoplanet program of the CoRoT sat ellite. We find that Kepler
+improves on CoRoT by a factor 2 to 2.3 in point-to-point scatter.
+Subject headings: methods: data analysis — methods: statistical — (stars:) bi naries: eclipsing — stars:
+variables: other — techniques: photometric
+1Instituut voor Sterrenkunde, Katholieke Universiteit
+Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium
+2IMAPP, Department of Astrophysics, Radbout Uni-
+versity Nijmegen, PO Box 9010, 6500 GL Nijmegen, the
+Netherlands
+3Space Telescope Science Institute, 3700 San Martin
+Drive, Baltimore, MD 21218, USA
+4Department of Physics and Astronomy, Aarhus Uni-
+versity, DK-8000 Aarhus C, Denmark
+5Las Cumbres Observatory Global Telescope, Goleta,
+CA 93117, USA
+6NASA Ames Research Center, MS 244-30, Moffett
+Field, CA 94035, USA
+7SETI Institute/NASA Ames Research Center, MS 244-
+30, Moffett Field, CA 94035, USA
+8Jeremiah Horrocks Institute of Astrophysics, Univer-
+sity of Central Lancashire, PR1 2HE, UK
+9Sydney Institute for Astronomy (SIfA), School of
+Physics, University of Sydney, NSW 2006, Australia
+10School of Physics and Astronomy, University of Birm-
+ingham, Edgbaston B15 2TT, United Kingdom
+11Astronomical Institute of the Romanian Academy, Str.
+Cutitul de Argint 5, RO 40557, Bucharest, RO
+12Konkoly Observatory, Hungarian Academy of Sciences,
+H-1525 Budapest, P.O. Box 67, Hungary1. Introduction
+TheKeplersatellite, launched in March 2009,
+is NASA’s first mission capable of finding Earth-
+size and smaller planets (Borucki et al. 2010;
+Koch et al. 2010). It has an 0.95-m aperture
+Schmidt telescope with a photometer comprised
+of 42 CCDs having a fixed field of view of 105
+square degrees in the constellations Cygnus and
+Lyrae. It is designed to monitor continuously the
+brightness of 160000 stars during the first year,
+reduced to 100000 stars later in the mission. This
+results in high-quality light curves, not only in-
+teresting for the detection of planets, but also of
+great importance for asteroseismology.
+In this Letter, we present data from the astero-
+seismology program of the NASA Kepler Mission
+(Gilliland et al. 2010). We present a search for
+variable stars and the application of supervised
+classification methods to the Keplerlong-cadence
+Q1 data of its asteroseismology program, cover-
+ing 33.5 days in total. All the light curves have
+29.4-min time sampling. Both the total time span
+1and sampling are very well suited to study short
+period eclipsing and ellipsoidal binaries, classical
+pulsators such as RRLyr stars and Cepheids, and
+nonradial pulsators such as βCep stars, Slowly
+pulsating B stars (SPBs), δSct stars and γDor
+stars (see Aerts et al. (2010) for a definition of all
+these classes). We used the stellar fluxes as they
+were delivered to us after preliminary data pro-
+cessing(Jenkins et al.2010). Intotal, weanalyzed
+2288Keplerlight curves.
+We compare our results to a pre-launch classi-
+fication based on data in the Kepler Input Catalog
+(KIChereafter) and prepared by the Kepler As-
+teroseismic Science Consortium (KASC, Gilliland
+et al. 2010). Finally, we compare the point-to-
+point scatter of the Keplerdata to the one of
+CoRoT’s exoplanet data of the first long run (5
+months) of that mission.
+2. Adopted Methodology
+Todetectandextractthevariables,wereliedon
+theautomatedvariabilitycharacterizationmethod
+as described in detail in Debosscher et al. (2007,
+2009). This method searches for three indepen-
+dent frequencies for every star, which are used to
+make a harmonic best-fit to the trend-subtracted
+time series. In this way we obtain a homogeneous
+set of parameters for each star, irrespective of its
+variability nature. The goal is notto achieve a
+good light curve model, but rather to deduce a
+set of light curve parameters which is sufficient to
+classify the variability.
+Using this set of parameters, we classified
+the stars using a modified version of the clas-
+sifier based on Gaussian mixtures, described in
+Debosscher et al. (2007), with the definition stars
+from Debosscher et al. (2009). We improved
+the performance of the algorithm presented in
+Debosscher et al. (2009) by classifying the objects
+using a multi-stage tree. In each node, we decide
+which groups of stars we want to distinguish. The
+best parameters are then selected for that node
+and for each group the parameter distribution is
+approximatedwith amixtureofmulti-dimensional
+Gaussians. To each variable stellar target we as-
+sign a probability that it belongs to a particular
+group.
+In order to obtain the final probability for
+each variability class we multiply the probabili-ties alongthe correspondingroot-to-leafpath (see,
+e.g., Ripley et al. (1996) for a general introduc-
+tion and definition of tree-structured classifiers).
+In practice, this works as follows for our applica-
+tion: the probability of a stellar target being an
+RRLyr star of type ab is, e.g., the probability of
+not being an eclipsing binary (first stage) times
+the probability of belonging to the RRLyr group
+(second stage) times the probability of being an
+RRLyr star of type ab.
+This procedure was followed to compute class
+probabilities for each target. In order to ob-
+tain the best candidates, we additionally used
+the Mahalanobis distance, which is a multi-
+dimensional generalization of the one-dimensional
+statistical or standard distance as described in
+Debosscher et al. (2009). A visual check has been
+performed as well.
+3. Classification results
+The variability classes we currently take into
+account, and the number of good candidate class
+members, are listed in Table1. While we classified
+more than 200 stars securely, the majority of tar-
+gets still has a too ambiguous class assignment,
+mainly due to the limited time base and to the
+large fraction of red giants among the sample (see
+below). Nevertheless, we managed to identify nu-
+merous new pulsators and binaries from the short
+time series and from early data reduction. All the
+illustrations presented in this paper contain stars
+that wereeither unknownasvariablesorweremis-
+classified prior to launch.
+We evaluated ourclassification resultsby visual
+inspection and manual analysis of the best can-
+didates based on the Mahalanobis distance, and
+by comparing them with the pre-launch classifica-
+tion which is often insecure due to limited ground
+data. The results are summarized in Table1. In
+the second and third column, a Mahalanobis dis-
+tance smaller than 3 and 2 is taken, respectively,
+without visual inspection. In the fourth column
+the final numbers of candidates are given, taking a
+Mahalanobis distance less than 3 and performing
+an additional visual inspection. In the last col-
+umn, the pre-launch classification done by KASC
+members is given. Note that several targets occur
+in more than one pre-launch class. For the ma-
+jority of the classes in Table1, our results appre-
+2ciably improve the pre-launch results, which were
+necessarily based on ground-based data and could
+not take into account Kepler’s high quality light
+curves. Good agreement is obtained for binaries
+and classical pulsators such as RRLyr stars and
+Cepheids, but even there, we are able to improve
+the pre-launch results as some RR Lyr candidates
+were reclassified by us as eclipsing binaries. One
+example is given in the third panel of Fig.1.
+We could also identify many new (eclipsing) bi-
+naries, some of which are shown in Fig.1. An ex-
+ample of a red giant pulsator with solar-like oscil-
+lations in an eclipsing binary is discussed in detail
+in Hekker et al. (2010).
+For main-sequence nonradial pulsators, such as
+βCep,δSct, SPB and γDor stars, there is much
+discrepancy between the pre-launch and our clas-
+sification. Few of the pre-launch candidates turn
+out to be actual class members. Indeed, we do
+not find high-probability candidates among these
+stars with a pre-launch class assignment. On the
+other hand, we identified new nonradial pulsators
+not present in the pre-launchlists. Some examples
+are shown in Fig.2. Similar light curves, albeit for
+fainter stars, were found in the CoRoT exoplanet
+database (Degroote et al. 2009, e.g.).
+We have not yet been able to compare our re-
+sults for the long period variables along or past
+the Asymptotic Giant Branch, such as Miras or
+RVTau stars, since the current total time span of
+the light curves is only a fraction of the typical
+pulsation periods of those objects. Moreover, we
+did not yet search for short-period pulsators, such
+as solar-like pulsators along the main sequence,
+rapidly oscillating Ap stars, subdwarf OB vari-
+ables and white dwarf pulsators (see Aerts et al.
+(2010) for class definitions), given that we do not
+yet have short-cadence data.
+Another point of attention is the classification
+of solar-like pulsators. Stochastic pulsations are
+more easily recognized from a broad power excess
+than from the methodology adopted here. Indeed,
+the automated selection of the three highest fre-
+quency peaks will almost always be peaks due to
+granulation and/or background noise. Thus, to
+find such pulsators, one better uses an extractor-
+type approach. This involves fitting and subtrac-
+tion of the granulation and background signal to
+characterize the type of star, after which the oscil-
+lations can be sought. We refer to Chaplin et al.(2010) , Bedding et al. (2010) and Stello et al.
+(2010) for a study of solar-like pulsators among
+Keplertargets.
+Finally, we stress that our classifiers solely
+use the information contained in the Keplerlight
+curves. This implies that we cannot discriminate
+well between the class pairs of B-type βCep and
+A-typeδSct stars with periodicities of the order
+of hours, and B-type SPB versus F-type γDor
+stars with oscillation periods of days. Both pairs
+of classes contain light curves with very similar
+characteristics. To discriminate between them, at
+leastsomespectralinformation, suchasaproperly
+dereddened B-V color index or a stellar spectrum,
+is needed. According to the Initial Mass Func-
+tion, most of these candidates should be AF-type
+pulsators.
+4. Noise properties of the Kepler data in
+the asteroseismology program
+Figure3revealsthepoint-to-pointscatterinthe
+Keplerdataofits asteroseismologyprogramasde-
+scribed in Gilliland et al. (2010), as well as a com-
+parison with that for the CoRoT space mission’s
+exoplanet data for its first long run of 5 months
+(Auvergne et al. 2009; Aigrain et al. 2009). We
+rescaled the CoRoT data to the same integration
+time of 29.4min as for the long-cadence mode of
+Keplerin order to ensure an appropriate com-
+parison. Duty cycles of these CoRoT and Kepler
+data are some 90% and 99%, respectively. It can
+be seen that the Keplerdata with still prelimi-
+nary processing (Jenkins et al. 2010) already out-
+performs the CoRoT exoplanet data by a factor
+∼2 for objects with a visual magnitude of around
+14 to a factor of ∼2.3 for objects of magnitude
+around 16. This is more or less as expected based
+on the difference in aperture size of the two in-
+struments. Importantly for follow-up studies of
+the most interesting variable stars, the Kepleras-
+teroseismologysample focuses on brighterobjects.
+Typical noise levels of the least variable stars in
+theKeplerasteroseismology program, estimated
+as the average amplitude in the Fourier transform
+avoiding the low-frequency regime below 1 cycle
+per day, range from 1.3 µmag for an 8th magni-
+tude star to 34 µmag for a 16th magnitude star.
+More than 70% of the KASC stars are variable
+in one way or another, even taking into account
+3Fig. 1.— Some newly discovered eclipsing binaries, which were not identifi ed as such prior to launch. The
+pre-launch KASC class is given on the right.
+4Fig. 2.— Some examples of multiperiodic pulsators, not present in the p re-launch class lists of pulsators.
+The pre-launch KASC class is given on the right.
+5Table 1
+Stellar variability classes considered in this work. See Ch apter 2 in Aerts et al. (2010)
+for a definition of the classes. MD stands for the Mahalanobi s distance as defined in
+Debosscher et al. (2009). A hyphen ‘-’ indicate variability types not taken into account
+in our classification scheme.
+Stellar variability classes MD <3 MD <2 MD <3 KASC
++ visual inspection pre-launch class
+Mira variables (MIRA) 0 0 0 305
+RV-Tauri stars (RVTAU) 1 1 0 2
+Cepheids (CEP) 7 3 6 35
+Compact pulsators (Compact) - - - 4
+RR-Lyr stars (RRLYR) 28 28 28 51
+βCep orδSct stars (BCEP/DSCUT) 65 15 28 57
+SPB orγDor stars (SPB/GDOR) 28 13 28 20
+Ellipsoidal variables (ELL) 31 8 23 0
+Eclipsing binaries (ECL) 101 92 101 100
+Red giants (RG) - - - 1513
+Rapidly oscillating Ap stars (ROAP) - - - 4
+Oscillations in clusters (OS-clus) - - - 178
+Solar-like oscillations (SOL) - - - 73
+residual instrumental effects. Note that this is
+a much higher fraction than for the CoRoT ex-
+oplanet programme, because the KASC target se-
+lection was aimed at focusing on variable stars
+while the CoRoT exoplanet sample is unbiased
+with respect to variability.
+5. Conclusions
+We presented the first results of the applica-
+tion of automated supervised classification meth-
+ods to 2288 Keplerlight curves. Comparison with
+existing pre-launch classification and manual clas-
+sification of the light curves shows the capabili-
+ties of our methodology: we are able to improve
+the pre-existingclassificationresultsseriously, and
+to identify new class members, unknown prior to
+launch.
+We will repeat the classifications as more and
+longer time-span light curves become available.
+This way, we will also be able to identify vari-
+ables with longer periodicities and have much bet-
+ter capacity to unravel beat periods in multiperi-
+odic pulsators. More classes will be included in
+the classification scheme, and, even more impor-
+tantly, the class definition stars will be updated
+using the high quality Keplerlight curves them-
+selves. Access to 1-min cadence data will allow us
+toclassifyshorterperiodvariables,inadditionalto8 9 10 1112 13 14 1516
+V mag00.00050.0010.00150.0020.00250.0030.00350.004point-to-point scatter (mag)
+Fig. 3.— Point-to-point scatter of the long-
+cadence light curves in Kepler’s asteroseismology
+program(Gilliland et al.2010, black)comparedto
+theoneofthelightcurvesintheCoRoTLRc01ex-
+oplanet programme (Auvergne et al. 2009, gray)
+6those presented here. With each new set of data,
+our results will be updated and made available.
+We gratefully thank the entire Keplerteam,
+whose excellent efforts have made these results
+possible. Funding for this Discovery mission is
+provided by NASA’s Science Mission Directorate.
+The research leading to these results has received
+funding from the European Research Council un-
+der the European Community’s Seventh Frame-
+work Programme (FP7/2007–2013)/ERC grant
+agreement n◦227224 (PROSPERITY), as well as
+from the Research Council of K.U.Leuven grant
+agreement GOA/2008/04 and from the Belgian
+Federal Science Office. DWK acknowledges sup-
+port by the UK Science and Technology Facilities
+Council.
+Facilities: The Kepler Mission
+REFERENCES
+Aerts C., Christensen-Dalsgaard, J., & Kurtz,
+D.W. 2010, Asteroseismology, Springer-Verlag
+(ISBN 978-1-4020-5178-4)
+Aigrain, S., et al. 2009, A&A, 425, 429
+Auvergne, M., et al. 2009, A&A, 506, 411
+Bedding, T.R., et al. 2010, ApJ, submitted
+Borucki, W.J., et al. 2010, PASP, in press
+Chaplin, B.W., et al. 2010, ApJ, submitted
+Debosscher, J., Sarro, L.M., Aerts, C., Cuypers,
+J., Vandenbussche, B., Garrido, R., & Solano,
+E. 2007, A&A, 475, 1159
+Debosscher, J., et al. 2009, A&A, 506, 519
+Degroote, P., et al. 2009, A&A, 506, 471
+Gilliland, R.L., et al. 2010, PASP, in press
+Hekker, S., et al. 2010, ApJ, submitted
+Jenkins, J.M., et al. 2010, ApJ, submitted
+Koch, D., et al. 2010, PASP, in press
+Ripley, B.D. 1996, Pattern Recognition and Neu-
+ral Networks, Cambridge University Press
+Stello, D., et al. 2010, ApJ, submitted
+This 2-column preprint was prepared with the AAS L ATEX macros v5.2.
+7
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+arXiv:1001.0508v3  [hep-th]  27 Apr 2010Noncommutative gauge theory using covariant star product
+defined between Lie-valued differential forms
+M. Chaichiana,b, M. Oksanena, A. Tureanua,band G. Zetc
+aDepartment of Physics, University of Helsinki, P.O. Box 64,
+FI-00014 Helsinki, Finland
+bHelsinki Institute of Physics, P.O. Box 64, FI-00014 Helsin ki, Finland
+cDepartment of Physics, The“Gheorghe Asachi”Technical Uni versity of Iasi,
+Bd. D. Mangeron 67, 700050 Iasi, Romania
+Abstract
+We develop an internal gauge theory using a covariant star pr oduct. The
+space-time is a symplectic manifold endowed only with torsi on but no curvature.
+It is shown that, in order to assure the restrictions imposed by the associativity
+property of the star product, the torsion of the space-time h as to be covariant
+constant. Anillustrativeexampleisgivenanditisconclud edthatinthiscasethe
+conditions necessary to define a covariant star product on a s ymplectic manifold
+completely determine its connection.
+1 Introduction
+Noncommutative gravity has been intensively studied in the last year s. One important
+motivation is the hope that such a theory could offer the possibility to develop a
+quantum theory of gravity, or at least to give an idea of how this cou ld be achieved
+[1, 2, 3, 4, 5, 6]. There are two major candidates to quantum gravit y: string theory
+[7] and loop quantum gravity [8]. Noncommutative geometry and, in pa rticular, gauge
+theoryofgravityareintimatelyconnected withboththeseapproa ches andtheoverlaps
+are considerable [2]. String theory is one of the strongest motivatio ns for considering
+noncommutative space-time geometries and noncommutative grav itation. It has been
+shown, for example, that in the case when the end points of strings in a theory of open
+strings are constrained to move on Dbranes in a constant B-field background and
+one considers the low-energy limit, then the full dynamics of the the ory is described
+by a gauge theory on a noncommutative space-time [9]. Recently, it h as been argued
+that the dynamics of the noncommutative gravity arising from strin g theory [10] is
+much richer than some versions proposed for noncommutative gra vity. It is suspected
+that the reason for this is the noncovariance of the Moyal star pr oduct under space-
+time diffeomorphisms. A geometrical approach to noncommutative g ravity, leading to
+a general theory of noncommutative Riemann surfaces in which the problem of the
+frame dependence of the star product is also recognized, has bee n proposed in [11] (for
+further developments, see [12, 13]).
+Now, one important problem is to develop a theory of gravity conside ring curved
+noncommutative space-times. The main difficulty is that the noncomm utativity pa-
+rameterθµνis usually taken to be constant, which breaks the Lorentz invarianc e of
+the commutation relations between coordinates [see (2.1) below], an d implicitly of any
+noncommutative field theory. One possible way to solve this problem is to consider θµν
+depending on coordinates and use a covariant star product. In [14 ] such a product has
+been defined between differential forms and the property of asso ciativity was verified
+up to the second order in θµν.
+1In this paper we will adopt the covariant star product defined in [14 ] and extend
+the result to the case of Lie-algebra-valued differential forms. We will follow the same
+procedure as in our previous paper [15]. But, in order to simplify the expression of the
+covariant product and to give an illustrative example, we will consider the case when
+the noncommutative space-time is a symplectic manifold Mendowed only with torsion
+(and no curvature). The restrictions imposed by such a covariant star product requires
+alsothatthetorsioniscovariantconstant. Themotivationforado ptingsuchamanifold
+is thatit allowstheconstructionofnoncommutative teleparallel gra vity (fortheidea of
+teleparallelism in gravity see [16]). It has been shown that a very diffic ult problem like
+the definition of a tensorial expression for the gravitational ener gy-momentum density
+can be solved in teleparallel gravity [17, 18]. This density is conserved in covariant
+sense. It has also been argued that the quantization problem is muc h more convenient
+to handle in teleparallel gravity [19] than in general relativity, due to the possibility of
+decomposing torsion into irreducible pieces under the global Lorent z group.
+Teleparallel gravity is also a very natural candidate for an effective noncommuta-
+tive fieldtheory of gravitation[20]. In additionit possesses many fe atureswhich makes
+it particularly well-suited for certain analyses. For instance it enable s a pure tenso-
+rial proof of the positivity of the energy in general relativity [21], it y ields a natural
+introduction of Ashtekar variables [22], and makes it possible to stud y the torsion at
+quantum level [23], for example, in the gravitational coupling to spino r fields.
+On the other hand, we can try to apply the covariant star product to the case
+when the space-timeis asymplectic manifoldwhich hasonly curvature , but the torsion
+vanishes. Then, the restriction imposed by the Jacobi identity for the Poisson bracket
+requires also the vanishing curvature. The corresponding connec tion is flat symplectic
+and this reduces drastically the applicability area of the covariant st ar product. Of
+course, it is possible to have a manifold with both curvature and tors ion.
+In Section 2, considering a manifold Mendowed only with torsion, we give the defi-
+nition of the covariant star product between two arbitrary Lie-alg ebra-valued differen-
+tial forms and some of its properties. Then, the star bracket bet ween such differential
+forms is introduced and some examples are given.
+Section 3 is devoted to the noncommutative internal gauge theory formulated with
+the new covariant star product. The noncommutative Lie-algebra -valued gauge po-
+tential and the field strength 2-form are defined and their gauge t ransformation laws
+are established. It is shown that the field strength is gauge covaria nt and satisfies a
+deformed Bianchi identity.
+An illustrative example is presented in Section 4. It is shown that in our simple
+example, the conditions necessary to define a covariant star prod uct on a symplectic
+manifold Mcompletely determine its connection.
+Section 5 is devoted to the discussion of the results and to the inter pretation of
+noncommutative gauge theory formulated by using the covariant s tar product between
+Lie-algebra-valued differential forms on symplectic manifolds. Some other possible
+applications of this covariant star product are also analyzed.
+The Appendix A contains a detailed verification of the associativity pr operty of
+the covariant star product including only the torsion in its definition.
+22 Covariant star product
+We consider a noncommutative space-time Mendowed with the coordinates xµ, µ=
+0,1,2,3, satisfying the commutation relation
+[xµ,xν]⋆=iθµν(x), (2.1)
+whereθµν(x) =−θνµ(x) is a Poisson bivector [14]. The space-time is organized as a
+Poisson manifold by introducing the Poisson bracket between two fu nctionsf(x) and
+g(x) by
+{f,g}=θµν∂µf∂νg. (2.2)
+In order that the Poisson bracket satisfies the Jacobi identity, t he bivector θµν(x) has
+to obey the condition [24, 25]
+θµσ∂σθνρ+θνσ∂σθρµ+θρσ∂σθµν= 0. (2.3)
+If a Poisson bracket is defined on M, thenMis called a Poisson manifold (see [24] for
+mathematical details).
+Suppose now that the bivector θµν(x) has an inverse ωµν(x), i.e.
+θµρωρν=δµ
+ν. (2.4)
+Ifω=1
+2ωµνdxµ∧dxνis nondegenerate (det ωµν/ne}ationslash= 0) and closed (d ω= 0), then it is
+called a symplectic 2-form and Mis a symplectic manifold. It can be verified that the
+condition d ω= 0 is equivalent with the equation (2.3) [14, 24, 26]. In this paper we
+will consider only the case when Mis symplectic.
+Because the gauge theories involve Lie-valued differential forms su ch as
+A=Aa
+µ(x)Tadxµ=Aµdxµ, Aµ=Aa
+µ(x)Ta, whereTaare the infinitesimal generators
+of a symmetry group G, we need to generalize the definition of the Poisson bracket
+to differential forms and define then an associative star product f or such cases. These
+problems were solved in [14, 24, 26]. In [15] we generalized these res ults to the case of
+Lie-algebra-valued differential forms. This generalization has the e ffect that the com-
+mutator of differential forms can be a commutator or an anticommu tator, depending
+on their degrees.
+Assuming that θµν(x) is invertible, we can always write the Poisson bracket {x,dx}
+in the form [14, 24, 26]
+{xµ,dxν}=−θµρΓν
+ρσdxσ, (2.5)
+where Γν
+ρσare some functions of xtransforming like a connection under general coor-
+dinate transformations. As Γν
+ρσis generally not symmetric, one can use the connection
+1-forms
+˜Γµ
+ν= Γµ
+νρdxρ,Γµ
+ν= dxρΓµ
+ρν (2.6)
+to define two kinds of covariant derivatives ˜∇and∇, respectively. The curvatures for
+these two connections are
+˜Rν
+λρσ=∂ρΓν
+λσ−∂σΓν
+λρ+Γν
+τρΓτ
+λσ−Γν
+τσΓτ
+λρ, (2.7)
+Rν
+λρσ=∂ρΓν
+σλ−∂σΓν
+ρλ+Γν
+ρτΓτ
+σλ−Γν
+στΓτ
+ρλ. (2.8)
+Because the connection coefficients Γρ
+µνare not symmetric/parenleftbig
+Γρ
+µν/ne}ationslash= Γρ
+νµ/parenrightbig
+the symplectic
+manifold Mhas also a torsion defined as usual [14]
+Tρ
+µν= Γρ
+µν−Γρ
+νµ. (2.9)
+3The connection ∇satisfies the identity [14]
+[∇µ,∇ν]α=−Rσ
+ρµνdxρ∧iσα−Tρ
+µν∇ρα, (2.10)
+and an analogous formula applies for ˜∇. Here,αis an arbitrary differential k-form
+α=1
+k!αµ1···µkdxµ1∧···∧dxµk (2.11)
+andiσαdenotes the interior product which maps the k-formαinto a (k−1)-form
+iσα=1
+(k−1)!ασµ2···µkdxµ2∧···∧dxµk. (2.12)
+It has been proven that in order for the Poisson bracket to satisf y the Leibniz rule
+d{f,g}={df,g}+{f,dg} (2.13)
+the bivector θµν(x) has to obey the property [14]
+˜∇ρθµν=∂ρθµν+Γµ
+σρθσν+Γν
+σρθµσ= 0. (2.14)
+Thusθµνis covariant constant under ˜∇, and˜∇is an almost symplectic connection.
+One can use the Leibniz condition (2.14) together with the Jacobi ide ntity for the
+Poisson bivector θµνto obtain the cyclic relation for torsion
+/summationdisplay
+(µ,ν,ρ)θµσθνλTρ
+σλ= 0. (2.15)
+Note that while this relation shows that a torsion-free connection id entically satisfies
+the property (2.15), the Jacobi identity does not require the con nection to be torsion-
+less. Also note that (2.14) and the Jacobi identity for the Poisson b ivector can be
+combined to obtain the following cyclic relation:
+/summationdisplay
+(µ,ν,ρ)θµσ∇σθνρ= 0. (2.16)
+If in addition to ˜∇ρθµν= 0, one imposes ∇ρθµν= 0, the torsion vanishes, Tρ
+µν= 0,
+and there is only one covariant derivative ∇=˜∇. In this paper, we do not require
+that∇ρθµν= 0.
+Using the graded product rule, one arrives at the following general expression of
+the Poisson bracket between differential forms [14, 26]
+{α,β}=θµν∇µα∧∇νβ+(−1)|α|˜Rµν∧(iµα)∧(iνβ), (2.17)
+where|α|is the degree of the differential form α, and
+˜Rµν=1
+2˜Rµν
+ρσdxρ∧dxσ,˜Rµν
+ρσ=θµλ˜Rν
+λρσ (2.18)
+In order that (2.17) satisfies the graded Jacobi identity
+{α,{β,γ}}+(−1)|α|(|β|+|γ|){β,{γ,α}}+(−1)|γ|(|α|+|β|){γ,{α,β}}= 0,(2.19)
+4the connection Γρ
+µνmust obey the following additional conditions [14]
+Rν
+λρσ= 0, (2.20)
+∇λ˜Rµν
+ρσ= 0. (2.21)
+A covariant star product between arbitrary differential forms ha s been defined
+recently in [14] having the general form
+α⋆β=α∧β+∞/summationdisplay
+n=1/parenleftbiggi/planckover2pi1
+2/parenrightbiggn
+Cn(α,β), (2.22)
+whereCn(α,β)are bilinear differential operatorssatisfying the generalized Moya l sym-
+metry [26]
+Cn(α,β) = (−1)|α||β|+nCn(β,α). (2.23)
+The operator C1coincides with the Poisson bracket, i.e. C1(α,β) ={α,β}. An
+expression for C2(α,β) has been obtained also in [14] so that the star product (2.22)
+satisfies the property of associativity
+(α⋆β)⋆γ=α⋆(β ⋆γ). (2.24)
+Inthispaperweconsiderthecasewhenthesymplecticmanifold Mhasonlytorsion,
+i.e. in addition to the necessary constraints (2.14), (2.20) and (2.21 ) we require
+˜Rσ
+µνρ= 0. (2.25)
+Since the curvature Rσ
+ρµνvanishes (2.20), one obtains the following relation between
+the curvature ˜Rand the torsion T[14]
+˜Rσ
+µνρ=∇µTσ
+νρ. (2.26)
+This relation shows that the condition (2.25) requires that the tors ionTσ
+νρis covariant
+constant, i.e.
+∇µTσ
+νρ= 0. (2.27)
+Therefore, if the torsion is covariant constant, the symplectic ma nifoldMhas only
+torsion but not curvature.
+For such a symplectic manifold, the bilinear differential operators C1(α,β) and
+C2(α,β) in the star product (2.22) proposed in [14] reduce to the simpler fo rms
+C1(α,β) ={α,β}=θµν∇µα∧∇νβ, (2.28)
+C2(α,β) =1
+2θµνθρσ∇µ∇ρα∧∇ν∇σβ+1
+3/parenleftbigg
+θµσ∇σθνρ+1
+2θνσθρλTµ
+σλ/parenrightbigg
+(2.29)
+×(∇µ∇να∧∇ρβ−∇να∧∇µ∇ρβ).
+We can verify that the covariant star product with torsion defined in (2.28)–(2.29)
+is associative [see the Appendix A]. In the next section we apply this co variant star
+product in order to develop a noncommutative internal gauge theo ry.
+53 Noncommutative gauge theory
+Let us consider the internal symmetry group Gand develop a noncommutative gauge
+theory on the symplectic manifold Mendowed with the covariant star product (with
+torsion) defined above. We proceed as in [15], but considering the st ar product defined
+in (2.28)–(2.29). This product differs from that used in [15]:
+•The curvature ˜Rσ
+µνρis supposed here to vanish as well as Rσ
+µνρ;
+•The ordinary derivative ∂σθνρ(see (2.17) in [15]) is replaced with the covariant
+derivative ∇σθνρ;
+•In (2.29) it appears an additional term1
+2θνσθρλTµ
+σλcompared with previous ver-
+sion (see [14] for details, and also [26, 25] for other aspects).
+The results given in [15] apply with the corresponding changes mentio ned above. Be-
+fore presenting them we make some observations on other possible applications of the
+covariant star product.
+1. It will be interesting to see if the Seiberg-Witten map can be gener alized to the
+case when the ordinary derivatives are replaced with the covariant derivatives
+and the Moyal star product is replaced by the covariant one.
+2. We can consider that the symplectic manifold Mis associated to a gauge theory
+of gravitation with Poincar´ e group Pas local symmetry (see [27, 28, 29] for
+notations and definitions) and using the covariant star product as in [15]. In
+this case we introduce the Poincar´ e gauge fields ea
+µ(tetrads) and ωab
+µ(spin
+connection) and then we define the covariant derivative as
+∇µ=∂µ−1
+2ωab
+µΣab. (3.1)
+It can be shown that by imposing the tetrad postulate [27]
+∇µea
+ν−Γρ
+µνea
+ρ= 0 (3.2)
+one introduces the connection Γρ
+µνin the Poincar´ e gauge theory and the strength
+tensorsFab
+µν, Fa
+µνdetermine the curvature and torsion of M
+Rρσ
+µν=Fab
+µν¯eρ
+a¯eσ
+b, Tρ
+µν=Fa
+µν¯eρ
+a, (3.3)
+where ¯eρ
+adenote the inverse of ea
+µ, i.e.
+¯eρ
+aea
+σ=δρ
+σ,¯eρ
+aeb
+ρ=δb
+a. (3.4)
+Now, let us suppose that we develop an internal gauge theory with t he symmetry
+groupGon the symplectic manifold M. It is very important to remark that
+making theminimal prescription ∂µ→ ∇µthe strengthtensor Fµνofthe internal
+gauge fields Aµ=Aa
+µTamust be written as
+Fµν=∇µAν−∇νAµ−i[Aµ,Aν]+AρTρ
+µν, (3.5)
+in order to assure its covariance both under Poincar´ e group Pand internal group
+G. The gauge invariance under the gauge transformations of Gbecomes quite
+6clear if we observe that the expression (3.5) is identical with the usu al one.
+Indeed, using the definitions of the covariant derivative ∇µand torsion Tρ
+µν, we
+obtain
+Fµν=∂µAν−AρΓρ
+µν−∂νAµ+AρΓρ
+νµ−i[Aµ,Aν]+Aρ/parenleftbig
+Γρ
+µν−Γρ
+νµ/parenrightbig
+=∂µAν−∂νAµ−i[Aµ,Aν].(3.6)
+Also, because the components of the gauge parameter λ=λaTaare considered
+as functions, we can write the gauge transformations as
+δAµ=∇µλ−i[Aµ,λ],∇µλ=∂µλ. (3.7)
+In what follows, we will use these expressions (3.5) and (3.7) in order to show
+their invariances explicitly.
+Supposenowthatwehaveaninternalgaugegroup Gwhoseinfinitesimalgenerators
+Tasatisfy the algebra
+[Ta,Tb] =ifc
+abTc, a,b,c = 1,2,...,m, (3.8)
+with the structure constants fc
+ab=−fc
+baand that the Lie-algebra-valued infinitesimal
+parameter is
+ˆλ=ˆλaTa. (3.9)
+We use the hat symbol “ˆ” to denote the noncommutative quantitie s of our gauge
+theory. The parameter ˆλis a zero-form, i.e. ˆλaare functions of the coordinates xµon
+the symplectic manifold M.
+Now, we define the gauge transformation of the noncommutative L ie-valued gauge
+potential
+ˆA=ˆAa
+µ(x)Tadxµ=ˆAµdxµ,ˆAµ=ˆAa
+µ(x)Ta, (3.10)
+by
+ˆδˆA= dˆλ−i/bracketleftig
+ˆA,ˆλ/bracketrightig
+⋆. (3.11)
+Here we consider the following formula for the commutator [ α,β]⋆of two arbitrary
+differential forms αandβ
+[α,β]⋆=α⋆β−(−1)|α||β|β ⋆α. (3.12)
+Then, using the definition (2.22) of the star product, we can write ( 3.11) as
+ˆδˆAa= dˆλa+fa
+bcˆAbˆλc+/planckover2pi1
+2da
+bcC1/parenleftig
+ˆAb,ˆλc/parenrightig
+−/planckover2pi12
+4fa
+bcC2/parenleftig
+ˆAb,ˆλc/parenrightig
++O/parenleftbig
+/planckover2pi13/parenrightbig
+,(3.13)
+where we noted {Ta,Tb}=dc
+abTc. In fact, this notation is valid if the Lie algebra closes
+also for the anticommutator, as it happens, for example, in the cas e of unitary groups.
+In general, the commutators like/bracketleftig
+ˆA,ˆλ/bracketrightig
+⋆take values in the enveloping algebra [30].
+Therefore, the gauge field ˆAand the parameter ˆλtake values in this algebra. Let us
+write for instance ˆA=ˆAITIandˆλ=ˆλITI, then
+/bracketleftig
+ˆA,ˆλ/bracketrightig
+⋆=1
+2/braceleftig
+ˆAI,ˆλJ/bracerightig
+⋆[TI,TJ]+1
+2/bracketleftig
+ˆAI,ˆλJ/bracketrightig
+⋆{TI,TJ}.
+7Thus, allproductsofthegenerators TIwillbenecessaryinordertoclosetheenveloping
+algebra. Its structure can beobtained by successively computing the commutatorsand
+anticommutators starting from the generators of Lie algebra, un til it closes [30],
+[TI,TJ] =ifK
+IJTK,{TI,TJ}=dK
+IJTK.
+Therefore, in our above notations and in what follows we understan d this structure in
+general.
+The operators Cnof the star product are defined similarly for noncommutative
+differential forms like ˆAaas for commutative ones. In particular C1/parenleftig
+ˆAb,ˆλc/parenrightig
+and
+C2/parenleftig
+ˆAb,ˆλc/parenrightig
+are given by (2.28)–(2.29). Here the covariant derivative concern s the
+space-time manifold M, not the gauge group G, so we use the definition
+∇µˆAa=/parenleftig
+∂µˆAa
+ν−Γρ
+µνˆAa
+ρ/parenrightig
+dxν≡/parenleftig
+∇µˆAa
+ν/parenrightig
+dxν. (3.14)
+We define also the curvature 2-form ˆFof the gauge potentials by
+ˆF=1
+2dxµ∧dxνˆFµν= dˆA−i
+2/bracketleftig
+ˆA,ˆA/bracketrightig
+. (3.15)
+Then, using the definition (2.22) of the star product and the prope rty (2.23) of the
+operators Cn/parenleftbig
+αa,βb/parenrightbig
+, we obtain from (3.15)
+ˆFa= dˆAa+1
+2fa
+bcˆAb∧ˆAc+1
+2/planckover2pi1
+2da
+bcC1/parenleftig
+ˆAb,ˆAc/parenrightig
+−1
+2/planckover2pi12
+4fa
+bcC2/parenleftig
+ˆAb,ˆAc/parenrightig
++O/parenleftbig
+/planckover2pi13/parenrightbig
+(3.16)
+More explicitly, in terms of components we have
+ˆFa
+µν=∇µˆAa
+ν−∇νˆAa
+µ+fa
+bcˆAb
+µˆAc
+ν+ˆAa
+ρTρ
+µν+/planckover2pi1
+2da
+bcC1/parenleftig
+ˆAb
+µ,ˆAc
+ν/parenrightig
+−/planckover2pi12
+4fa
+bcC2/parenleftig
+ˆAb
+µ,ˆAc
+ν/parenrightig
++O/parenleftbig
+/planckover2pi13/parenrightbig
+,
+(3.17)
+where we used the definition Cn/parenleftig
+ˆAb,ˆAc/parenrightig
+=Cn/parenleftig
+ˆAb
+µ,ˆAc
+ν/parenrightig
+dxµ∧dxνwith
+C1/parenleftig
+ˆAb
+µ,ˆAc
+ν/parenrightig
+=/braceleftig
+ˆAb
+µ,ˆAc
+ν/bracerightig
+=θρσ∇ρˆAb
+µ∇σˆAc
+ν, (3.18)
+C2/parenleftig
+ˆAb
+µ,ˆAc
+ν/parenrightig
+=1
+2θρσθλτ∇ρ∇λˆAb
+µ∇σ∇τˆAc
+ν+1
+3/parenleftbigg
+θρτ∇τθσλ+1
+2θστθλφTρ
+τφ/parenrightbigg
+(3.19)
+×/parenleftig
+∇ρ∇σˆAb
+µ∇λˆAc
+ν−∇σˆAb
+µ∇ρ∇λˆAc
+ν/parenrightig
+.
+Under the gauge transformation (3.11) the curvature 2-form ˆFtransforms as
+ˆδˆF=i/bracketleftig
+ˆλ,ˆF/bracketrightig
+⋆, (3.20)
+where we used the Leibniz rule
+d/parenleftig
+ˆα⋆ˆβ/parenrightig
+= dˆα⋆ˆβ+(−1)|α|ˆα⋆dˆβ (3.21)
+which we admit to be valid to all orders in /planckover2pi1. The Leibniz rule was verified only in
+the first order. To second order the proof is very cumbersome. W e believe, however,
+motivated by the associativity of the proposed star product, tha t the Leibniz rule
+8is valid to all orders. This issue remains a challenge to be proven. In te rms of the
+components (3.20) gives
+ˆδˆFa=fa
+bcˆFbˆλc+/planckover2pi1
+2da
+bcC1/parenleftig
+ˆFb,ˆλc/parenrightig
+−/planckover2pi12
+4fa
+bcC2/parenleftig
+ˆFb,ˆλc/parenrightig
++O/parenleftbig
+/planckover2pi13/parenrightbig
+. (3.22)
+In the zeroth order, the formula (3.22) reproduces therefore t he result of the commu-
+tative gauge theory
+δFa
+µν=fa
+bcFb
+µνλc⇔δF=i[λ,F]. (3.23)
+Using again the Leibniz rule, we obtain the deformed Bianchi identity
+dˆF−i/bracketleftig
+ˆA,ˆF/bracketrightig
+⋆= 0. (3.24)
+If we apply the definition (3.12) of the star commutator, we obtain
+dˆF+i/bracketleftig
+ˆF,ˆA/bracketrightig
+=/bracketleftbigg/planckover2pi1
+2da
+bcC1/parenleftig
+ˆFb,ˆAc/parenrightig
+−/planckover2pi12
+4fa
+bcC2/parenleftig
+ˆFb,ˆAc/parenrightig/bracketrightbigg
+Ta+O/parenleftbig
+/planckover2pi13/parenrightbig
+,(3.25)
+or in terms of components
+dˆFa−fa
+bcˆFb∧ˆAc=/planckover2pi1
+2da
+bcC1/parenleftig
+ˆFb,ˆAc/parenrightig
+−/planckover2pi12
+4fa
+bcC2/parenleftig
+ˆFb,ˆAc/parenrightig
++O/parenleftbig
+/planckover2pi13/parenrightbig
+.(3.26)
+We remark that in zeroth order we obtain from (3.25) the usual Bian chi identity
+dF+i[F,A] = 0. (3.27)
+In addition, if the gauge group is U(1), the Bianchi identity (3.24) becomes
+dˆF=/planckover2pi1C1/parenleftig
+ˆA,ˆF/parenrightig
++O/parenleftbig
+/planckover2pi13/parenrightbig
+. (3.28)
+This result is also in accord with that of [24].
+Having established the previous results, we can construct a nonco mmutative Yang-
+Mills (NCYM) action. We will consider the case when the gauge group is U(N). Let
+Gµνbe a metric on the noncommutative space-time M[15]. We suppose that the
+metricGµνbelongs to the adjoint representation of U(1)⊂U(N) in the sense that
+Gµν=GµνI, whereIis the unity matrix of U(N) in this representation. Therefore,
+we consider the components of Gµνas Lie-algebra-valued zero-forms. The covariant
+derivative of the metric Gµνis
+∇µGνρ=∂µGνρ+GνσΓρ
+µσ+Γν
+µσGσρ. (3.29)
+IfGµνis not constant, we have to modify it to be a covariant metric ˆGµνfor the
+(NCYM) action [31], so that it transforms like ˆF[see (3.20)]
+ˆδˆGµν=i/bracketleftig
+ˆλ,ˆGµν/bracketrightig
+⋆, (3.30)
+Then, using the definition (3.12) for the star commutator, we obta in from (3.30)
+ˆδˆGµν=θρσ∇ρˆGµν∂σˆλ+O/parenleftbig
+/planckover2pi13/parenrightbig
+. (3.31)
+9We can use the Seiberg-Witten map with the covariant star product for a field, which
+is in the adjoint representation (as we consider Gµνto be), in order to obtain [32]
+ˆGµν=Gµν−A0
+ρθρσ∇σGµν+O/parenleftbig
+/planckover2pi13/parenrightbig
+, (3.32)
+whereA0
+µis the gauge field in the U(1) sector of U(N).
+Inorder to construct the NCYM actionfor thegaugefields Aa
+µ(x),µ= 1,2,3,0,a=
+0,1,2,...,N2−1, we use the definition for the integration /an}b∇acketle{tf/an}b∇acket∇i}htof a function f(or of
+any other quantity) over the noncommutative space Mas (see [31] for details)
+/an}b∇acketle{t·/an}b∇acket∇i}ht=/integraldisplay
+dx4|Pf(B)|(·), (3.33)
+whereB=θ−1andPf(B) denotes the Pfaffian of B, i.e.|Pf(B)|=/radicalbig
+det(B). The
+notation has a connection with the very important result that for a D-brane in a B
+field background (with Bconstant or nonconstant), its low-energy effective theory
+lives on a noncommutative space-time with the Poisson structure [31 , 33, 34]. More
+exactly, it has been shown that the metric Gintroduced on the Poisson manifold M
+is related to the metric gappearing in the fundamental string (open or closed) action
+byG=−B−1gB−1[9, 31, 33].
+Now, we define the NCYM action by (see [15, 31])
+ˆSNCYM=−1
+2g2c/angbracketleftig
+tr/parenleftig
+ˆG⋆ˆF ⋆ˆG⋆ˆF/parenrightig/angbracketrightig
+=−1
+4g2c/angbracketleftig
+tr/parenleftig
+ˆGµν⋆ˆFνρ⋆ˆGρσ⋆ˆFσµ/parenrightig/angbracketrightig
+,(3.34)
+wheregcistheYang-Millsgaugecouplingconstant,andwehaveusedthenorm alization
+property
+tr(TaTb) =1
+2δabI. (3.35)
+Usingthepropertiesofgaugecovariance(3.20)and(3.30)for ˆFandˆG, respectively,
+we obtain
+ˆδˆSNCYM=−/planckover2pi1
+4g2c/angbracketleftig
+C1/parenleftig
+tr/parenleftig
+ˆGˆFˆGˆF/parenrightig
+,ˆλ/parenrightig/angbracketrightig
++O/parenleftbig
+/planckover2pi13/parenrightbig
+. (3.36)
+Since the integral (3.33) is cyclic in the Poisson limit [31], the integral of the Poisson
+bracket vanishes, i.e. /an}b∇acketle{tC1(f,h)/an}b∇acket∇i}ht= 0 for any integrable functions fandh, and thus
+(3.36) becomes
+ˆδˆSNCYM= 0+O/parenleftbig
+/planckover2pi13/parenrightbig
+. (3.37)
+Therefore, the action ˆSNCYMis invariant up to the second order in /planckover2pi1. The expression
+(3.34) of the action can be further simplified as [15, 31]
+ˆSNCYM=−1
+2g2
+c/angbracketleftig
+tr/parenleftig
+ˆGˆFˆGˆF/parenrightig/angbracketrightig
++O/parenleftbig
+/planckover2pi13/parenrightbig
+=−1
+4g2
+c/angbracketleftig
+tr/parenleftig
+ˆGµνˆFνρˆGρσˆFσµ/parenrightig/angbracketrightig
++O/parenleftbig
+/planckover2pi13/parenrightbig
+,
+(3.38)
+Using the previous results we can obtain solutions for the noncommu tative gauge
+field equations. An example is given in Section 4, using the symplectic ma nifoldM
+endowed with a covariantly constant torsion.
+We can add fields into our noncommutative gauge model in the usual w ay. As
+an example, we mention the case when the noncommutative U(N) gauge theory is
+10coupled to a Higgs multiplet ˆΦ(x) =ˆΦaTain the adjoint representation. The action
+integral for ˆΦ(x) is [35]
+ˆSHIGGS=−1
+4g2/angbracketleftig
+tr/parenleftig
+ˆDµˆΦ⋆ˆGµν⋆ˆDνˆΦ/parenrightig/angbracketrightig
+, (3.39)
+where
+ˆDµˆΦ =∂µˆΦ−ig/bracketleftig
+ˆΦ,ˆAµ/bracketrightig
+⋆(3.40)
+is the noncommutative gauge covariant derivative. Because this de rivative is gauge
+covariant, in the sense
+ˆδ/parenleftig
+ˆDµˆΦ/parenrightig
+=i/bracketleftig
+ˆλ,ˆDµˆΦ/bracketrightig
+⋆, (3.41)
+the action ˆSHIGGSis invariant as well as ˆSNCYMup to the second order in /planckover2pi1. The action
+of the noncommutative U(N) coupled to the Higgs multiplet ˆΦ(x) reads
+ˆSNC=−1
+4g2/angbracketleftig
+tr/parenleftig
+ˆGµν⋆ˆFνρ⋆ˆGρσ⋆ˆFσµ+ˆDµˆΦ⋆ˆGµν⋆ˆDνˆΦ/parenrightig/angbracketrightig
+. (3.42)
+This action can be used for obtaining solutions of the noncommutativ e version of the
+Yang-MillsHiggsmodelbyusing thecovariantstarproductonthesy mplectic manifold
+M, as an extension of the results of [35], where the usual Moyal star product is used.
+4 An illustrative example
+As a very simple example we consider the Poincar´ e gauge theory to c onstruct the
+manifold M. Then, suppose that we have the gauge fields ea
+µand fix the gauge
+ωab
+µ= 0 [36]. We define the connection coefficients
+Γρ
+µν= ¯eρ
+a∂µea
+ν, (4.1)
+where ¯eρ
+adenotes the inverse of ea
+µ. Obviously, the connection Γ defined by these
+coefficients is not symmetric, i.e. Γρ
+µν/ne}ationslash= Γρ
+νµ. Define then the torsion by formula
+Tρ
+µν= Γρ
+µν−Γρ
+νµ. (4.2)
+In order to simplify the calculation, we consider the case of spherica l symmetry
+and choose the gauge fields ea
+µas
+ea
+µ= diag/parenleftbigg
+A,1,1,1
+A/parenrightbigg
+,¯eµ
+a= diag/parenleftbigg1
+A,1,1,A/parenrightbigg
+, (4.3)
+whereA=A(r) is a function depending only on the radial coordinate r. Then,
+denoting the spherical coordinates on Mby (xµ) = (r,ϑ,ϕ,t), µ= 1,2,3,0, the non-
+null components of the connection coefficients are
+Γ0
+10=−A′
+A,Γ1
+11=A′
+A. (4.4)
+It is easy to see that the only non-null components of the torsion a re
+T0
+01=−T0
+10=A′
+A. (4.5)
+11Also, using the definitions (2.7) and (2.8) of the curvatures, we obt ain
+˜R0
+101=−˜R0
+110=AA′′−2A′2
+A2, Rλ
+µνρ= 0 (4.6)
+and all other components of ˜Rλ
+µνρvanish. In these expressions, we denote the first and
+second derivatives of A(r) byA′andA′′, respectively. The vanishing curvature Rλ
+µνρ
+agrees with the constraint (2.20).
+Introduce then the noncommutativity parameters θµνand suppose that we choose
+them to be
+(θµν) =
+0 0 01
+A(r)
+0 0 b0
+0−b0 0
+−1
+A(r)0 0 0
+, (4.7)
+wherebis a nonvanishing constant. Then we have
+˜∇1θ01=−˜∇1θ10= 0,∇1θ01=−∇1θ10=A′
+A2. (4.8)
+This agrees with the constraint (2.14) that θµνis covariant constant under ˜∇.
+Finally, if we impose also the condition that the curvature ˜Rµν
+ρσ=θµλ˜Rν
+λρσvanishes
+(equivalent with (2.25) due to (2.4)), that implies ∇λ˜Rµν
+ρσvanishes too (2.21), then
+from (4.6) we obtain the following differential equation of the second order for the
+unknown function A(r):
+AA′′−2A′2= 0. (4.9)
+The solutions of this equation is
+A(r) =−1
+c1r+c2, (4.10)
+wherec1andc2are two arbitrary constants of integration. Therefore, in our sim ple
+example, the conditions necessary to define a covariant star prod uct on a symplectic
+manifold Mcompletely determine its connection. In addition, it is very interestin g to
+see that the covariant derivative of the torsion, defined as
+∇µTν
+ρσ=∂µTν
+ρσ+Γν
+µλTλ
+ρσ−Γλ
+µρTν
+λσ−Γλ
+µσTν
+ρλ, (4.11)
+has the following non-null components
+∇1T0
+01=−∇1T0
+10=AA′′−2A′2
+A2. (4.12)
+Then, taking into account the equation (4.9), we conclude that the torsion is covariant
+constant, ∇µTν
+ρσ= 0, a result which is in concordance with the condition (2.27).
+Nowweconstruct anoncommutative U(2)gaugetheoryonthespace-timemanifold
+Mpresented in the previous example. Denote the generators of the U(2) group by
+Ta,a=k,0, withk= 1,2,3; hereTk=σk(Pauli matrices) generates the U(2) sector
+andT0=I(the unit matrix) — the U(1) sector of the gauge group . These generators
+satisfy the algebra (3.8), where only the structure constant fi
+jk= 2ǫijk(ǫijkare the
+totally antisymmetric Levi-Civita symbols) of the SU(2) sector are nonvanishing, the
+other components of fa
+bcbeing equal to zero. The anticommutator {Ta,Tb}=dc
+abTc
+12also belongs to the algebra of U(2), where d0
+bc= 2δbc,da
+b0= 2 are the only nonvanishing
+components.
+We choose the 1-form gauge potential of the form [37, 38]
+A=uT3dt+w(T2dϑ−sinϑT1dϕ)+cosϑT3dϕ+vT0dt, (4.13)
+whereu,w,vare functions depending only on the radial coordinate r. We consider
+the metric Gµνand its inverse Gµνof the form
+Gµν= diag/parenleftbigg1
+N,r2,r2sin2ϑ,−N/parenrightbigg
+(4.14)
+and
+Gµν= diag/parenleftbigg
+N,1
+r2,1
+r2sin2ϑ,−1
+N/parenrightbigg
+, (4.15)
+respectively, where Nisalsoafunctiondependingonlyon r. Forexample, thefollowing
+set of functions
+u=u0+Q
+r, w= 0, v= 0, N= 1−2M
+r+Q2+1
+r2(4.16)
+describes a colored black hole in the SU(2) sector [38]. The metric Gµνis of Reissner-
+Nordstr¨omtype withelectric charge Qandunitmagneticcharge [38]. Itis thesimplest
+solution of the Einstein-Yang-Mills field equations with a nontrivial gau ge field.
+We can obtain the noncommutative Yang-Mills field equations and their solutions
+by imposing the variational principle ˆδˆSNCYM= 0. However, it is simpler and equiv-
+alent to use the Seiberg-Witten map and determine the noncommuta tive gauge fields
+ˆAµ, the field strength ˆFµνand the metric ˆGµνorder by order in the deformation pa-
+rameter.
+To this end, we denote the noncommutative quantities of our model byˆλ=ˆλaTa
+(the gauge parameter), ˆA=ˆAµdxµ=ˆAa
+µTadxµ(the 1-form gauge potential) and
+ˆGµν=ˆGµνI(the metric), and expand them as formal power series in θ
+ˆλ=λ+λ(1)+λ(2)+···,
+ˆAµ=Aµ+A(1)
+µ+A(2)
+µ+···, (4.17)
+ˆGµν=Gµν+Gµν(1)+Gµν(2)+···,
+where the zeroth order terms λ,AµandGµνare the ordinary counterparts of ˆλ,ˆAµ
+andˆGµν, respectively. Using the Seiberg-Witten map for the noncommutat ive gauge
+theory with the covariant star product [32] we obtain the following expressions for the
+first order deformations:
+λ(1)=1
+4θρσ{∂ρλ,Aσ}, (4.18)
+A(1)
+µ=−1
+4θρσ{Aρ,∇σAµ+Fσµ}, (4.19)
+Gµν(1)=−θρσA0
+ρ∇σGµν. (4.20)
+Note that due to the particular formof the parameter θµν(4.7) and the solution(4.10),
+there are in fact three noncommutativity parameters in our model: c1,c2andb. From
+13now on we denote them by c1=θ1(of dimension T),c2=θ2(of dimension LT) and
+b=θ3(dimensionless).
+Thefirstorderdeformationsofthefieldstrengthcanbeobtained fromthedefinition
+(3.15) by using (4.19):
+F(1)
+µν=−1
+4θρσ({Aρ,∇σFµν+DσFµν}−{Fµρ,Fνσ}), (4.21)
+where
+∇σFµν=∂σFµν−Γρ
+σµFρν−Γρ
+σνFµρ (4.22)
+is the covariant derivative (it concerns the space-time manifold M) and
+DσFµν=∇σFµν−i[Aσ,Fµν] (4.23)
+is the gauge covariant derivative (it concerns the gauge group).
+In particular, for the colored black hole solution (4.16) we obtain
+Aa(1)
+0=/parenleftbigg
+0,0,1
+2sin(2ϑ)θ3,−1
+2/parenleftbigg
+u0+Q
+r/parenrightbigg/bracketleftbigg/parenleftbigg
+u0+3Q
+r/parenrightbigg
+θ1+2Q
+r2θ2/bracketrightbigg/parenrightbigg
+,(4.24)
+Gµν(1)= 0, (4.25)
+F0(1)
+µν=
+0 0 0 B(r)(rθ1+θ2)
+0 0 cos(2 ϑ)θ3 0
+0 −cos(2ϑ)θ30 0
+−B(r)(rθ1+θ2) 0 0 0
+,(4.26)
+wherea= 1,2,3,0 and
+B(r) =Q
+r3/parenleftbigg
+2u0+3Q
+r/parenrightbigg
+, (4.27)
+and other components of Fa(1)
+µνare equal to zero. When the U(1) sector is not empty
+v(r)/ne}ationslash= 0, we obtain a nonvanishing first order deformation of the metric Gµν(1)/ne}ationslash= 0,
+but for simplicity we prefer to consider only the colored black hole solu tion here.
+Allthesefirstorderdeformationsvanishifthenoncommutativityp arametersvanish
+θ1,θ2,θ3→0. However, this limit cannot be achieved because the symplectic str ucture
+of the space-time Mimposes the condition det( θµν)/ne}ationslash= 0.
+Finally, we mention that colored black holes and their generalizations w ith angular
+momentum and cosmological term, as well as solutions with cylindrical and plane
+symmetries have also been obtained [38]. It would be of interest to e xtend these
+results to the noncommutative theory.
+5 Conclusions and discussions
+We developed a noncommutative gauge theory by using a covariant s tar product be-
+tween differential forms on symplectic manifolds defined as in [14]. We f ollowed the
+same way as in our recent paper [15], extending the results of [14] to the case of
+Lie-valued differential forms.
+To simplify thecalculations, we considered a space-time endowed only with torsion.
+It has been shown that, in order to satisfy the restrictions impose d by the associativity
+propertyofthecovariantstarproduct, thetorsionofthespac e-timehastobecovariant
+14constant, ∇µTν
+ρσ= 0. On the other hand, it has been argued that a covariant star
+product defined in the case when the space-time is a symplectic manif old endowed
+only with curvature is not possible. This is due to the restrictions impo sed by the
+associativity property of the covariant star product which requir es also the vanishing
+curvature. The corresponding connection is therefore flat symp lectic and this reduces
+the applicability area of the covariant star product.
+AnillustrativeexamplehasbeenpresentedstartingfromthePoinca r´ egaugetheory.
+Using the gaugefields ea
+µandfixing the gauge ωab
+µ= 0 [36] we defined the nonsymmet-
+ric connection Γρ
+µν= ¯eρ
+a∂µea
+ν. We deduced that, in this case, the conditions necessary
+to define a covariant star product on a symplectic manifold Mcompletely determine
+its connection.
+Some other possible applications of this covariant star product hav e been also
+analyzed. First, it will be very important to generalize the Seiberg-W itten map to
+the case when the ordinary derivatives are replaced with covariant derivatives and the
+Moyal star product is replaced by the covariant one. Second, we c an try to develop a
+noncommutative gaugetheory of gravity considering the symplect ic manifold Mas the
+background space-time. For such a purpose, we have to verify if t he noncommutative
+field equations do not impose too many restrictive conditions on the c onnection Γρ
+µν,
+in addition to those required by the existence of the covariant prod uct. However, the
+problem of which gauge group we can choose remains unsolved. The P oincar´ e group
+cannotbeusedbecauseitdoesnotclosewithrespect tothestarp roduct. Apossibility
+will be to choose the group GL(2,C), but in this case we obtain a complex theory of
+gravitation [39, 40]. Another possibility is to consider the universal e nveloping of the
+Poincar´ e group, but this is infinite dimensional and we must find crite ria to reduce
+the number of the degrees of freedom to a finite one. Some possible ideas are given
+for the case of SU(N) or GUT theories in [41], where it is argued that the infinite
+number of parameters can in fact all be expressed in terms of right number of classical
+parameters and fields via the Seiberg-Witten maps.
+Two ways to generalize the Poisson bracket and the covariant star product to the
+algebra of tensor fields on a symplectic manifold have been proposed recently [42, 43].
+In the latter work one studied covariant star products on spaces of tensor fields defined
+over a Fedosov manifold with a given symplectic structure and a given flat torsionless
+symplectic connection. In the former approach one considers suc h generalizations on
+symplectic manifolds and Poisson manifolds that impose as few constr aints as possible
+on the connections. Both of these approaches are in part motivat ed by their possible
+application to noncommutative gravity.
+Acknowledgements. The support of the Academy of Finland under the Projects
+No. 121720 and No. 127626 is gratefully acknowledged. The work of M. O. was fully
+supportedby theJenny andAntti Wihuri Foundation. G.Z.ackno wledges thesupport
+of CNCSIS-UEFISCSU Grant ID-620 of the Ministry of Education an d Research of
+Romania.
+A Appendix
+We verify the associativity property up to the second order in /planckover2pi1. For simplicity we
+denote the exterior product by α∧β=αβ.
+15Introducing (2.22) into (2.24) we obtain successively
+/bracketleftigg
+αβ+i/planckover2pi1
+2C1(α,β)+/parenleftbiggi/planckover2pi1
+2/parenrightbigg2
+C2(α,β)+···/bracketrightigg
+⋆γ
+=α⋆/bracketleftigg
+βγ+i/planckover2pi1
+2C1(β,γ)+/parenleftbiggi/planckover2pi1
+2/parenrightbigg2
+C2(β,γ)+···/bracketrightigg
+(A.1)
+or
+(αβ)γ+i/planckover2pi1
+2[C1(αβ,γ)+C1(α,β)γ]
++/parenleftbiggi/planckover2pi1
+2/parenrightbigg2
+[C2(αβ,γ)+C1(C1(α,β),γ)+C2(α,β)γ]+···
+=α(βγ)+i/planckover2pi1
+2[C1(α,βγ)+αC1(β,γ)]
++/parenleftbiggi/planckover2pi1
+2/parenrightbigg2
+[C2(α,βγ)+C1(α,C1(β,γ))+αC2(β,γ)]+···.
+Identifying the terms of different orders in /planckover2pi1we obtain
+(αβ)γ=α(βγ), (A.2)
+C1(αβ,γ)+C1(α,β)γ=C1(α,βγ)+αC1(β,γ), (A.3)
+C2(αβ,γ)+C1(C1(α,β),γ)+C2(α,β)γ=C2(α,βγ)+C1(α,C1(β,γ))+αC2(β,γ).
+(A.4)
+(A.2) is verified because the exterior product has this property.
+Using (2.28), the (A.3) becomes
+θµν[∇µ(αβ)(∇νγ)+(∇µα)(∇νβ)γ−(∇µα)∇ν(βγ)−α(∇µβ)(∇νγ)] = 0
+that is satisfied due to the Leibniz rule ∇µ(αβ) = (∇µα)β+α(∇µβ).
+In order to verify (A.4) we write it as
+δC2(α,β,γ)≡C2(α,βγ)−C2(α,β)γ−C2(αβ,γ)+αC2(β,γ)
+={{α,β},γ}−{α,{β,γ}}(A.5)
+We calculate the right-hand side of (A.5) first
+{{α,β},γ}−{α,{β,γ}}= (−1)|α|(|β|+|γ|){β,{γ,α}}
+= (−1)|α|(|β|+|γ|)θµν(∇µβ)∇ν(θρσ(∇ργ)(∇σα)) =−θµν(∇νθρσ)(∇ρα)(∇µβ)(∇σγ)
++θµνθρσ[(∇µ∇ρα)(∇νβ)(∇σγ)−(∇µα)(∇ρβ)(∇σ∇νγ)],(A.6)
+where the graded symmetry property (2.23) (for n= 1) and the graded Jacobi identity
+(2.19) of the Poisson bracket are used in the first equality, the exp ression (2.28) for the
+Poissonbracket inthesecondstep, andthesymmetry properties ofθµνandtheexterior
+product, αβ= (−1)|α||β|βα, in the last equality. Then we introduce the decomposition
+of the second covariant derivative of an arbitrary differential for m as
+∇µ∇ρα=1
+2{∇µ,∇ρ}α−1
+2Tλ
+µρ∇λ, (A.7)
+16implied by (2.10) and (2.20), into (A.6). Finally, using the cyclic propert y (2.15) for
+the torsion, we obtain
+{{α,β},γ}−{α,{β,γ}}=−/parenleftbigg
+θµσ∇σθνρ+1
+2θνσθρλTµ
+σλ/parenrightbigg
+(∇να)(∇µβ)(∇ργ)
++1
+2θµνθρσ[{∇µ,∇ρ}α(∇νβ)(∇σγ)−(∇µα)(∇ρβ){∇ν,∇σ}γ],(A.8)
+Next, wecalculatetheleft-handsideof (A.5), theHochschildcobou ndaryofC2(see
+[14] for details). First we calculate C2(α,βγ)−C2(α,β)γandC2(αβ,γ)−αC2(β,γ),
+then substracting them yields
+δC2(α,β,γ) =1
+2θµνθρσ/bracketleftbig
+(∇µ∇ρα)2(∇(νβ∇σ)γ)−2(∇(µα∇ρ)β)(∇ν∇σγ)/bracketrightbig
++1
+3/parenleftbigg
+θµσ∇σθνρ+1
+2θνσθρλTµ
+σλ/parenrightbigg/parenleftbig
+(∇ρα)2(∇(µβ∇ν)γ)−2(∇(µα∇ν)β)(∇ργ)/parenrightbig
+,(A.9)
+where we denote ∇(µα∇ρ)β=1
+2[(∇µα)(∇ρβ)+(∇ρα)(∇µβ)]. By using the symme-
+tries of the factor θµνθρσand the cyclic relation implied by (2.16) and (2.15),
+/summationdisplay
+(µ,ν,ρ)/parenleftbigg
+θµσ∇σθνρ+1
+2θνσθρλTµ
+σλ/parenrightbigg
+= 0, (A.10)
+we find
+δC2(α,β,γ) =1
+2θµνθρσ[{∇µ,∇ρ}α(∇νβ)(∇σγ)−(∇µα)(∇ρβ){∇ν,∇σ}γ]
+−/parenleftbigg
+θµσ∇σθνρ+1
+2θνσθρλTµ
+σλ/parenrightbigg
+(∇να)(∇µβ)(∇ργ),(A.11)
+This is the same result as in (A.8) and therefore we have verified the a ssociativity of
+our star product to the second order in /planckover2pi1.
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+arXiv:1001.0509v3  [math.NA]  5 Oct 2010Approximation error of the Lagrange reconstructing polyno mial
+G.A. Gerolymos
+Universit´ e Pierre-et-Marie-Curie ( UPMC ), Case 161, 4 place Jussieu, 75005 Paris, France
+Abstract
+The reconstruction approach [Shu C.W.: SIAM Rev. 51(2009) 82–126] for the numerical approxi-
+mation of f′(x)is based on the construction of a dual function h(x)whose sliding averages over the
+interval [x−1
+2∆x,x+1
+2∆x]are equal to f(x)(assuming an homogeneous grid of cell-size ∆x). We study
+the deconvolution problem [Harten A., Engquist B., Osher S. , Chakravarthy S.R.: J. Comp. Phys. 71
+(1987) 231–303] which relates the Taylor polynomials of h(x)andf(x), and obtain its explicit solution,
+by introducing rational numbers τndefined by a recurrence relation, or determined by their gene r-
+ating function, gτ(x), related with the reconstruction pair of ex. We then apply these results to the
+specific case of Lagrange-interpolation-based polynomial reconstruction, and determine explicitly the
+approximation error of the Lagrange reconstructing polyno mial (whose sliding averages are equal to
+the Lagrange interpolating polynomial) on an arbitrary ste ncil defined on a homogeneous grid.
+Keywords: reconstruction, (Lagrangian) interpolation and reconstr uction, hyperbolic PDEs, finite
+differences, finite volumes
+2000 MSC: 65D99, 65D05, 65D25, 65M06, 65M08
+1. Introduction
+The Godunov approach [1] to hyperbolic conservation laws
+∂tu+∂xF(u)=0 (1)
+is based on space-time averaging of the PDE (1). Assuming an homogeneous time-independent grid
+(∆x=const ), space-averaging of (1), over the interval [x−1
+2∆x,x+1
+2∆x], leads to the exact relation [1]
+∂
+∂t¯u(x,t)+1
+∆x/bracketleftig
+F/parenleftig
+u(x+1
+2∆x,t)/parenrightig
+−F/parenleftig
+u(x−1
+2∆x,t)/parenrightig/bracketrightig
+=0 (2)
+where
+¯u(x,t) :=/integraldisplay+1
+2
+−1
+2u(x+ξ∆x,t)dξ (3)
+are the sliding cell-averages of the solution. Defining the s liding cell-averages F(u), by applying the
+operator (3) on F(u), we have immediately by differentiation, provided that ∆x=const ,
+∂F(u(x,t))
+∂x=F/parenleftig
+u(x+1
+2∆x,t)/parenrightig
+−F/parenleftig
+u(x−1
+2∆x,t)/parenrightig
+∆x(4)
+exactly, so that, combining (2) and (4)
+∂t¯u+∂xF(u)=0 (5)
+Email address: georges.gerolymos@upmc.fr (G.A. Gerolymos)
+Preprint submitted to J. Approx. Theory November 6, 2018iethe equation for the sliding cell-averages, for ∆x=const , has the same form as the original equa-
+tion [2]. For this reason, it is assumed that what is computed (and stored at the nodes of the computa-
+tional grid [3, 4]) are the cell-averages of the solution.
+In the discretization of (2) we are led to consider the comput ation of the derivative of a function f(x)
+(corresponding to ¯u) sampled on the computational grid, by differences at x±1
+2∆xof the values of an
+unknown function h(x)(corresponding to u), which has to be reconstructed [2, 5, 6, 3, 4] from the values
+of its cell-averages sampled on the grid. In the following, w e concentrate on the spatial discretization
+problem, vizcompute f′(x)via reconstruction of h(x±1
+2∆x)[2, 5, 6, 3, 4].
+Reconstruction (Definition 2.1) is the basis of ENO [7, 2, 8, 9] and WENO [5, 6, 3, 10, 11, 12, 13,
+14, 4, 15] schemes. Although the term polynomial reconstruc tion is quite general, including Hermite-
+interpolation [16], spectral methods [17], and spectral el ement techniques [18], we concentrate in
+the present work on reconstruction approaches based on Lagr angian interpolation of the function
+averages. There exist several algorithms for Lagrangian-i nterpolation-based polynomial reconstruc-
+tion [2, 3, 4], and these have been successfully used for the c onstruction of progressively higher-order
+schemes [6, 11, 15], using symbolic calculation [13, 14]. Th ereconstruction via primitive approach [2]
+is probably the most widely used in order-of-accuracy proof s [3, 4], while the reconstruction via decon-
+volution approach [2] has been formulated with respect to the solutio n of linear systems. Most of these
+schemes and associated order-of-accuracy relations [6, 11 , 13, 14, 15] were developed for particular
+values of the order-parameter r(determining the discretization stencil), using symbolic computation.
+On the other hand, analytical relations for the order-of-ac curacy of the approximation of h(x), for arbi-
+trary reconstruction-order-parameter rare not available. To obtain such relations it seems necessa ry
+to study in detail the relations between a function h(x)(which is reconstructed) and its cell-averages
+f(x). This is the reconstruction via deconvolution approach defined by Harten et al. [2]. Up to now,
+these relations were obtained by solving, using symbolic ca lculation, the associated linear system [2,
+(3.13b), p. 244], up to a certain order. Although the solutio n by symbolic computation of the linear
+system [2, (3.13b), p. 244] is not difficult, it is only valid u p to a certain O(∆xq), and the non availability
+of an explicit solution hinders the development of general e xpressions of the approximation error of
+the reconstruction.
+Along with the numerous successful developments of practic alWENO schemes based on the re-
+construction via primitive approach [3, 4], the unknown fun ctionh(x)which is reconstructed by its
+cell-averages f(x)appears explicitly in recent analyses [13, 14] of the trunca tion error. Analyzing
+the reconstruction error in terms of the unknown function h(x)and its derivatives ( reconstruction via
+deconvolution [2]) is a more intuitive approach, especially when consider ing the discretization error
+of the WENO approximation to f′(x)and potential improvements in the formulation of the nonlin ear
+weights [13, 14]. Analyses based on the reconstruction via primitive [2] approach are somehow less
+straightforward as they involve the primitive of the recons tructed function/integraltextx
+x1h(ζ)dζ.1The main mo-
+tivation of the present work is to contribute to the developm ent of analytical tools, applicable to the
+study of the truncation error [13, 14], determination of the loss-of-accuracy at smooth extrema [13]
+and research for the improvement of WENO schemes [14], maintaining the in-built scalability (with
+the stencil width) towards higher order-of-accuracy of WENO schemes [6, 11, 15]. For these reasons, in
+the present work we are not interested in the development of a new algorithm for the solution of the
+reconstruction problem. Instead, we focus on reconstructi on relations of general validity, iestencil-
+independent, which are necessary for the study of the approx imate reconstruction order-of-accuracy.
+In§2 we study the general relations underlying the reconstruct ion approach for the numerical ap-
+proximation of the 1-derivative f′(x)of a function f(x). Initially we study the relations between the
+derivatives of a function f(x)and those of a dual function h(x), whose sliding averages, over a con-
+stant length∆x, are equal to f(x). We will call the functions, f(x)andh(x), satisfying this relation
+a reconstruction pair for the discretization of f′(x)(Definition 2.1). We introduce the rational num-
+bersτn∈Q, defined either by a recurrence relation (Lemma 2.5) or throu gh a generating function
+1The lower bound of the integral being of no particular conseq uence [3, 4, 19] we can chose the coordinate of grid-point x1
+instead of the usual (but more abstract) lower bound at −∞[3, 4, 19].
+2(Theorem 2.9), which are used to develop explicit series rep resentations of h(x)(and of its derivatives)
+with respect to powers of ∆xand the derivatives of f(x). The principal new result in §2 is that we
+are able to give explicit solutions to the fundamental relat ions of the reconstruction via deconvolution
+approach [2, (3.13), pp. 244–246], which (Lemma 2.5) are wid ely used throughout the paper. The gen-
+erating function of the rational numbers τn∈Qappears in the expression of the reconstruction pair of
+ex(Theorem 2.9).
+In§3 we study the particular case of polynomial reconstruction . We show (Lemma 3.1) that for
+every polynomial pf(x)of degree Minxwe can define, using the numbers τn(Lemma 2.5), a polynomial
+ph(x), also of degree Minx, so that pfandphare a unique reconstruction pair (Definition 2.1). Initiall y
+(§3.2) the numbers τn(Lemma 2.5) were introduced, using a matrix algebra approac h to study the
+relation between pf(x)andph(x). This part of the paper ( §3.2) gives the explicit inversion of the matrix
+appearing in the reconstruction via deconvolution theory [ 2, (3.13b), p. 244].
+In practice f(x)is usually approximated by its Lagrange interpolating poly nomialpf(x;Si,M−,M+,∆x)
+on a given stencil Si,M−,M+(Definition 4.1), and h(x)is approximated by the reconstruction pair of
+pf(x;Si,M−,M+,∆x),ph(x;Si,M−,M+,∆x), which we will call the Lagrange reconstructing polynomial [4]. In
+§4 we study the approximation error of the Lagrange reconstru cting polynomial, Eh(x;Si,M−,M+,∆x) :=
+ph(x;Si,M−,M+,∆x)−h(x), and obtain an explicit relation for the expansion of this er ror in powers of ∆x
+(Proposition 4.7). This is only possible through the explic it solution of the deconvolution problem
+(Lemma 2.5). In §5 we briefly summarize the existence and uniqueness results c oncerning the re-
+constructing polynomial. Finally, in §6 we briefly describe some applications of the present result s to
+practical WENO schemes, and discretization methods in general, highlight ing the merits of the recon-
+struction via deconvolution approach, as developed in the p resent work.
+Standard results referring to the Lagrange interpolating p olynomial [20, 21] are included only
+when they are necessary for the proof of the new results conce rning the reconstructing polynomial.
+Useful relations for summation indices in multiple sums [22 , 23], and other identities, used throughout
+the paper, are summarized in Appendix A.
+2. Reconstruction pairs and exact reconstruction relation s
+Before proceeding to a detailed examination of the reconstr uction of polynomials we examine the
+general relations underlying the reconstruction approach for the evaluation of the derivative f′(x)of a
+function f(x), via the construction of a function h(x)(reconstruction pair of f(x); Definition 2.1), whose
+sliding (with x) averages [3, 4] on the interval [x−1
+2∆x,x+1
+2∆x]are equal to f(x), over an appropriate
+interval x∈I⊂R. We express in particular the derivatives of h(x)as series of the derivatives of
+f(x), with coefficients determined by the derivatives at ∆x=0of the function gτ(∆x)appearing in the
+reconstruction pair of the exponential function (Theorem 2 .9).
+2.1. Reconstruction pairs
+The basic idea underlying reconstruction procedures to com pute the derivative f′(x)of a function
+f(x)follows directly from the Leibniz rule [24, pp. 411–412] giv ing the derivative of a definite integral
+with respect to its (variable) bounds. To this end we need to c onstruct a function h(x)whose sliding
+(withx) average over an interval [x−1
+2∆x,x+1
+2∆x]of constant width∆xis equal to f(x).
+Definition 2.1 (Reconstruction pair). Assume that∆x∈R>0is a constant length, and that the
+functions f:I−→Randh:I−→Rare defined on the interval I=[a−1
+2∆x,b+1
+2∆x]⊂R, satisfying
+everywhere
+f(x)=1
+∆x/integraldisplayx+1
+2∆x
+x−1
+2∆xh(ζ)dζ∀x∈[a,b] (6a)
+assuming the existence of the integral in (6a). We will note t he functions f(x)andh(x)related by (6a)
+h=R(1;∆x)(f) (6b)
+f=R−1
+(1;∆x)(h) (6c)
+3and will call fandha reconstruction pair on [a,b], in view of the computation of the 1-derivative. /square
+By definition (6a), R−1
+(1;∆x)(6c) is defined by
+[R−1
+(1;∆x)(h)](x) :=1
+∆x/integraldisplayx+1
+2∆x
+x−1
+2∆xh(ζ)dζ∀x∈[a,b]
+h: [a−1
+2∆x,b+1
+2∆x]−→R/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayx+1
+2∆x
+x−1
+2∆xh(ζ)dζ∈R∀x∈[a,b](7)
+ieR−1
+(1;∆x)is a mapping applicable to all real functions defined on [a−1
+2∆x,b+1
+2∆x]⊂Rfor which the
+integral (6a) exists. Then, R(1;∆x)is defined as the inverse mapping of R−1
+(1;∆x)(7), assuming that the
+inverse mapping exists ( cfRemark 2.8).
+Lemma 2.2 (Reconstruction). Consider the functions f(x)andh(x)constituting a reconstruction pair
+on[a,b]⊂R(Definition 2.1) . Assume that f(x)andh(x)are of class CN(N∈N)on the interval I=
+[a−1
+2∆x,b+1
+2∆x]⊂R. Then
+f(n)(x)=h(n−1)(x+1
+2∆x)−h(n−1)(x−1
+2∆x)
+∆x∀x∈[a,b]∀n∈{1,···,N} (8)
+PROOF . Direct differentiation of (6a), yields
+f′(x)=h(x+1
+2∆x)−h(x−1
+2∆x)
+∆x∀x∈[a,b] (9)
+by application of the Leibniz rule [24, pp. 411–412], and tak ing into account that ∆xis constant∀x.
+Successive differentiation of (9) yields (8). /square
+All reconstruction-based approaches [7, 2, 5, 6, 11, 25, 13, 26, 14] for the numerical approximation
+ofPDEs are based on, or can be shown to be related to, Lemma 2.2. Thes e relations (8) are exact rela-
+tions concerning the continuous functions fandh. When f(x)andh(x)are numerically approximated
+consistently, iein a way satisfying (6) up to a given order ∆xM+1, then (8) are satisfied up to some order
+≤M+1.
+Definition 2.3 (Lagrange reconstructing polynomial). Letpfbe the Lagrange interpolating poly-
+nomial of the function fon the arbitrary stencil {i−M−,···,i+M+}ofM+1equidistant points ( M:=
+M−+M+) around point i. Its reconstruction pair (Definition 2.1) will be called the Lagrange recon-
+structing polynomial on the stencil {i−M−,···,i+M+}. /square
+Remark 2.4 (Homogeneous grid). The basic relations underlying reconstruction, which are g iven
+in Lemma 2.2, hold iff ∆x=const ,ie, when used as basis for the numerical approximation of f′(x), these
+relations are only applicable on a homogeneous grid. In the c ase of an inhomogeneous grid, where the
+spacing∆x(x)is a function of position ( ∆x:R−→R>0) these relations should be modified to include ∆x′
+and(∂∆xh)∆x′. The general case of an inhomogeneous grid requires specific study. /square
+2.2. Deconvolution
+Obviously, the relations between fandh(Lemma 2.2) imply that the Taylor-polynomials of f(x)
+can be expressed with respect to the derivatives h(n)(x±1
+2∆x), which can themselves be replaced by
+Taylor-polynomials of h(x). We have
+Lemma 2.5 (Deconvolution of h=R(1;∆x)(f)).Letf(x)andh(x)=[R(1;∆x)(f)](x)be a reconstruction
+pair (Definition 2.1) , satisfying the conditions of Lemma 2.2 . Then∀NTJ∈N:NTJ<N−1
+f(n)(x)=⌊NTJ
+2⌋/summationdisplay
+ℓ=0∆x2ℓ
+22ℓ(2ℓ+1)!h(n+2ℓ)(x)+O(∆x2⌊NTJ
+2⌋+2)∀x∈[a,b]
+∀n∈N0:n<N−2⌊NTJ
+2⌋(10a)
+4Inversely,
+h(n)(x)=⌊NTJ
+2⌋/summationdisplay
+ℓ=0τ2ℓ∆x2ℓf(n+2ℓ)(x)+O(∆x2⌊NTJ
+2⌋+2)∀x∈[a,b]
+∀n∈N0:n<N−2⌊NTJ
+2⌋(10b)
+where the numbers τ2ℓ(Tab. 1) are defined by the recurrence relations
+τ0=1 ;τ2k=k−1/summationdisplay
+s=0−τ2s
+22k−2s(2k−2s+1)!=k/summationdisplay
+s=1−τ2k−2s
+22s(2s+1)!k>0 (10c)
+PROOF . Approximating h(ζ)(which was assumed to be of class CNin Lemma 2.2) in (6) by the corre-
+sponding Taylor-polynomial (Taylor-jet) of order NTJ[27, pp. 219–232] around ζ=xyields,∀NTJ∈N:
+NTJ<N−1,
+f(x)=1
+∆x/integraldisplayx+1
+2∆x
+x−1
+2∆xNTJ/summationdisplay
+ℓ=0(ζ−x)ℓ
+ℓ!h(ℓ)(x)+O/parenleftig
+(ζ−x)NTJ+1/parenrightigdζ
+=1
+∆x/integraldisplayx+1
+2∆x
+x−1
+2∆xNTJ/summationdisplay
+ℓ=0(ζ−x)ℓ
+ℓ!h(ℓ)(x)dζ+O(∆xNTJ+1)
+=1
+∆xNTJ/summationdisplay
+ℓ=0/integraldisplay1
+2∆x
+−1
+2∆xηℓ
+ℓ!dηh(ℓ)(x)+O(∆xNTJ+1)
+=1
+∆xNTJ/summationdisplay
+ℓ=0/parenleftigg∆xℓ+1
+2ℓ(ℓ+1)!1−(−1)ℓ+1
+2/parenrightigg
+h(ℓ)(x)+O(∆xNTJ+1)∀x∈[a,b] (11)
+and since∀k∈N0
+ℓ+1=2k+1 (k∈N0)=⇒1−(−1)ℓ+1=2 (12a)
+ℓ+1=2k(k∈N0)=⇒1−(−1)ℓ+1=0 (12b)
+we obtain
+f(x)=⌊NTJ
+2⌋/summationdisplay
+ℓ=0∆x2ℓ
+22ℓ(2ℓ+1)!h(2ℓ)(x)+O(∆x2⌊NTJ
+2⌋+2) (13)
+which is (10a) for n=0. Successive differentiation of (13) by xyields (10a).
+To invert (10a) we search for numbers τ2s(s∈N0)satisfying∀MTJ∈N:MTJ<N−1and∀n∈N0:n<
+N−2⌊MTJ
+2⌋
+h(n)(x)=MTJ/summationdisplay
+s=0τ2s∆x2sf(n+2s)(x)+O(∆x2MTJ+2)
+=MTJ/summationdisplay
+s=0MTJ/summationdisplay
+ℓ=0∆x2ℓ
+22ℓ(2ℓ+1)!h(n+2s+2ℓ)(x)+O(∆x2MTJ+2)τ2s∆x2s+O(∆x2MTJ+2)
+=MTJ/summationdisplay
+s=0MTJ/summationdisplay
+ℓ=0/parenleftiggτ2s∆x2s+2ℓ
+22ℓ(2ℓ+1)!h(n+2s+2ℓ)(x)/parenrightigg
++O(∆x2MTJ+2)
+=2MTJ/summationdisplay
+k=0min(k,MTJ)/summationdisplay
+s=max(0,k−MTJ)τ2s
+22k−2s(2k−2s+1)!∆x2kh(n+2k)(x)+O(∆x2MTJ+2)
+=MTJ/summationdisplay
+k=0k/summationdisplay
+s=0τ2s
+22k−2s(2k−2s+1)!∆x2kh(n+2k)(x)+O(∆xMTJ+2) (14)
+5because of (A.3). (14) holds, provided that ( δk0is the Kronecker δ)
+k/summationdisplay
+s=0τ2s
+22k−2s(2k−2s+1)!=δk0∀k∈N0 (15)
+which is satisfied if the numbers τ2kare defined by (10c). Truncating (13) to O(∆x2⌊NTJ
+2⌋)yields (10b).
+The remainder in (10a) and (10b) is O(∆x2⌊NTJ
+2⌋+2), because only even powers of ∆xappear in these
+expressions. /square
+Table 1: Numbers τn(19c) satisfying recurrence (10c), for 0≤n≤21.
+τ0= 1
+τ1= 0
+τ2=−1
+24
+τ3= 0
+τ4=7
+5,760
+τ5= 0
+τ6=−31
+967,680
+τ7= 0
+τ8=127
+154,828,800
+τ9= 0
+τ10=−73
+3,503,554,560
+τ11= 0
+τ12=1,414,477
+2,678,117,105,664,000
+τ13= 0
+τ14=−8,191
+612,141,052,723,200
+τ15= 0
+τ16=16,931,177
+49,950,709,902,213,120,000
+τ17= 0
+τ18=−5,749,691,557
+669,659,197,233,029,971,968,000
+τ19= 0
+τ20=91,546,277,357
+420,928,638,260,761,696,665,600,000
+τ21= 0
+Remark 2.6 (Relation to previous work [2, 28]). The results in Lemma 2.5 expressing the deriva-
+tives of the sliding cell-averages f(x)with respect to the derivatives of the function h(x)=[R(1;∆x)(f)](x),
+are straightforward. In particular (10a) corresponds to [2 8, (2.15), p. 299]. The new results of
+Lemma 2.5 are the inversion relations (10b), which are based on the introduction of the numbers
+τn(10c). These results are the general explicit solution of th e linear system written in Harten et al. [2,
+(3.13b), p. 244], and provide the exact deconvolution relat ion between f(x)and[R(1;∆x)(f)](x)(Defini-
+tion 2.1), in the case of a homogeneous ( ∆x=const ) grid. The general case of an inhomogeneous grid
+requires specific study. The inversion relations (10b) are t he main building block of the present work,
+as far as error analysis of the reconstruction is concerned. We will show that the numbers τn(10c) can
+also be defined by a generating function (Theorem 2.9). /square
+6Corollary 2.7 (Taylor-polynomial of h(x+ξ∆x)).Assume the conditions of Lemma 2.5 . Then
+h(x+ξ∆x)=NTJ/summationdisplay
+s=0⌊s
+2⌋/summationdisplay
+ℓ=0τ2ℓξs−2ℓ
+(s−2ℓ)!∆xsf(s)(x)+O(∆xNTJ+1) (16)
+PROOF . Since
+2/floorleftbiggn
+2/floorrightbigg
++2=/braceleftiggn+1∀n=2k−1k∈N
+n+2∀n=2k k∈N(17a)
+(10b) can be rewritten as
+∆xmh(m)(x)
+m!=⌊NTJ−m
+2⌋/summationdisplay
+ℓ=0τ2ℓ(m+2ℓ)!
+m!∆xm+2ℓf(m+2ℓ)(x)
+(m+2ℓ)!+O(∆x2⌊NTJ
+2⌋+2) (17b)
+In that form (17b) we have a relation between the coefficients of the Taylor-polynomials of f(x+ξ∆x)
+and ofh(x+ξ∆x), expressed in powers of ξ. In particular, using (17b), we have
+h(x+ξ∆x)=NTJ/summationdisplay
+m=0ξm∆xmh(m)(x)
+m!+O(∆xNTJ+1)
+=NTJ/summationdisplay
+m=0⌊NTJ−m
+2⌋/summationdisplay
+ℓ=0τ2ℓ∆xm+2ℓf(m+2ℓ)(x)
+m!ξm+O(∆xNTJ+1) (18a)
+=NTJ/summationdisplay
+s=0⌊s
+2⌋/summationdisplay
+ℓ=0τ2ℓ∆xsf(s)(x)
+(s−2ℓ)!ξs−2ℓ+O(∆xNTJ+1) (18b)
+where we used (A.3) and (A.2), and the fact that NTJ+1≤2⌊NTJ
+2⌋+2. This completes the proof. /square
+This expression (16) is useful in computing the error of nume rical approximations to h(x)(Proposi-
+tion 4.6).
+Remark 2.8 (Existence and uniqueness). From Definition 2.1 it follows immediately (proof by con-
+tradiction) that every reconstruction pair h=R(1;∆x)(f), withh(x)continuous, if it exists, is unique. For
+everyh(x)analytic in Iwith radius of convergence rCh(x), the series (10a) with n=0converges, as
+NTJ−→∞ ,∀∆x∈/parenleftig
+0,2rCh(x)/parenrightig
+, so that (because of uniqueness), for every analytic functi onh(x)there
+exists a unique function f=R−1
+(1;∆x)(h). Whether the converse is always true, is an open question.
+Assuming f(x)analytic in Iwith radius of convergence rCf(x), does not automatically imply the con-
+vergence of (10b) with n=0asNTJ−→∞ , because limn→∞(τ2n(2n)!)=∞. The necessary conditions of
+existence require further study. Nonetheless, since limn→∞τ2n=0(Tab. 1) andτ2nτ2n+2<0∀n∈N0
+(Tab. 1), the class of functions f(x)for which (10b) with n=0is convergent as NTJ−→∞ is not empty.
+It is easy to verify that most of the basic functions f(x)have reconstruction pairs h=R(1;∆x)(f), as
+do all polynomials of finite degree ( §3.1). Whenever any of the series (10) converges as NTJ−→∞ ,
+the upper limit of the sums can be readily replaced by ∞, to yield complete converging expansions
+(power-series). The Godunov approach [1] to hyperbolic con servation laws∂tu+∂xF=0(1), is based
+on space-time averaging of the PDE (1), to obtain the corresponding PDE,∂t¯u+∂x¯F=0(5), for the
+cell-averages ¯u(3). Therefore, with respect to the notation used in Definiti on 2.1,¯ucorresponds to f
+anducorresponds to h. In the context of reconstruction procedures [2, 5, 6, 3, 4] f or the discretization
+of hyperbolic conservation laws, the existence of the solut ion (integrable function) u(ieh) is assumed,
+so that the existence of the sliding-averages ¯u(ief) follows (Remark 2.8). Hence, the results obtained
+in§2 (where the existence of his assumed) are directly applicable to the Godunov approach for the
+numerical computation of hyperbolic conservation laws. /square
+72.3. Generating function of τnand the reconstruction pair of exp(x) :=ex
+As mentioned above (Remark 2.8) most of the basic functions h ave reconstruction pairs. The re-
+construction pair of the exponential function plays an impo rtant role in the reconstruction relations
+(Lemma 2.5), because it defines the generating function of th e numbersτn(Tab. 1).
+Theorem 2.9 ( R(1;∆x)(exp)).The reconstruction pair of exp(x) :=exis
+[R(1;∆x)(exp)](x)=1
+2∆x
+sinh1
+2∆xex=gτ(∆x)ex(19a)
+where the function
+gτ(x) :=1
+2x
+sinh1
+2x(19b)
+is the generating function of the numbers τn(Tab. 1) satisfying (10c)
+τn:=1
+n!g(n)
+τ(0)∀n∈N0 (19c)
+Furthermore
+τ2n+1:=1
+(2n+1)!g(2n+1)
+τ(0)=0∀n∈N0 (19d)
+PROOF . From (10b), since exis of class C∞, we have∀NTJ∈N
+[R(1;∆x)(exp)](x)=NTJ/summationdisplay
+n=0τ2n∆x2nd2n
+dx2nex+O(∆x2NTJ+2)=NTJ/summationdisplay
+n=0τ2n∆x2nex+O(∆x2NTJ+2) (20)
+Sincelimn→∞τ2n=0andτ2nτ2n+2<0∀n∈N0, the alternating ( ∆x2n>0∀n∈N0) series in (20) converges
+asNTJ−→∞ , at least∀∆x∈(0,1). Defining the function gτ(x)
+gτ(x) :=∞/summationdisplay
+n=0τ2nx2n(21)
+suggests that∃gτ:R−→Rsuch that
+[R(1;∆x)(exp)](x)=gτ(∆x)ex(22)
+Using (22) in (6a)
+ex=1
+∆x/integraldisplayx+1
+2∆x
+x−1
+2∆xgτ(∆x)eζdζ=1
+∆xgτ(∆x)/parenleftig
+ex+1
+2∆x−ex−1
+2∆x/parenrightig
+(23)
+gives
+gτ(∆x)=∆x
+e1
+2∆x−e−1
+2∆x=1
+2∆x
+sinh1
+2∆x(24)
+proving (19a). It is a simple exercise to show that the functi ongτ(x)(19b) is continuous at x=0, and
+has continuous derivatives of arbitrary order at x=0, satisfying
+gτ(0)=1 (25a)
+g(2n+1)
+τ(0)=0∀n∈N0 (25b)
+Comparing the Taylor-series of gτ(x)(19b) with the series definition of gτ(x)(21), and taking into ac-
+count (25) proves (19c). (25b) yields (19d). /square
+83. Reconstruction of polynomials
+Reconstruction of polynomials (Definition 2.1) is the basis ofENO [7, 2] and WENO [5, 6, 11, 13,
+14, 15] reconstructions. We investigate in detail the coeffi cients of polynomial ( §3.1) reconstruction
+pairs (Definition 2.1).
+3.1. Polynomial reconstruction pair
+In this section we consider the case where either f(x)orh(x)in Definition 2.1 is a polynomial.
+Lemma 3.1 ( Polynomial reconstruction pair). Letph(x,xi,∆x)be a polynomial of degree M
+ph(x;xi,∆x) :=M/summationdisplay
+m=0chm/parenleftbiggx−xi
+∆x/parenrightbiggm
+(26a)
+Thenpf(x;xi,∆x)defined by (Definition 2.1)
+pf(x;xi,∆x) :=1
+∆x/integraldisplayx+1
+2∆x
+x−1
+2∆xph(ζ;xi,∆x)dζ (26b)
+is a polynomial also of degree M, with coefficients cfmwhich can be computed from the coefficients chmof
+ph(x;xi,∆x)
+pf(x;xi,∆x)=M/summationdisplay
+m=0cfm/parenleftbiggx−xi
+∆x/parenrightbiggm
+(26c)
+cfm=⌊M−m
+2⌋/summationdisplay
+k=0chm+2k
+22k(2k+1)/parenleftiggm+2k
+2k/parenrightigg
+∀m∈{0,···,M} (26d)
+m!cfm=⌊M−m
+2⌋/summationdisplay
+k=0(m+2k)!
+22k(2k+1)!chm+2k∀m∈{0,···,M} (26e)
+Inversely, the coefficients chmofph(x;xi,∆x)can be computed from the coefficients cfmofpf(x;xi,∆x)
+chm=1
+m!⌊M−m
+2⌋/summationdisplay
+k=0τ2kcfm+2k(m+2k)!∀m∈{0,···,M} (26f)
+where the numbers τ2k(Tab. 1) are defined by (19c) and satisfy the recurrence (10c) .
+PROOF . Computing the integral in (26b) gives
+pf(x;xi,∆x)=/integraldisplayx
+∆x+1
+2
+x
+∆x−1
+2M/summationdisplay
+m=0chm/parenleftbigg
+ζ−xi
+∆x/parenrightbiggmdζ=M/summationdisplay
+m=0chm
+m+1/parenleftbiggx−xi
+∆x+1
+2/parenrightbiggm+1
+−M/summationdisplay
+m=0chm
+m+1/parenleftbiggx−xi
+∆x−1
+2/parenrightbiggm+1
+=M/summationdisplay
+m=0chm
+m+1m+1/summationdisplay
+n=0/parenleftiggm+1
+n/parenrightigg/parenleftbiggx−xi
+∆x/parenrightbiggn1
+2m−n1−(−1)m+1−n
+2
+=M/summationdisplay
+m=0chm
+m+1m/summationdisplay
+n=0/parenleftiggm+1
+n/parenrightigg/parenleftbiggx−xi
+∆x/parenrightbiggn1
+2m−n1−(−1)m+1−n
+2 (27)
+9where in the last line of (27)/summationtextm+1
+n=0was changed to/summationtextm
+n=0because n=m+1=⇒1−(−1)m+1−n=1−(−1)0=0.
+This proves that both ph(x;xi,∆x)andpf(x;xi,∆x)are of degree M. Since
+m+1−n=2k+1k∈N0=⇒1−(−1)m+1−n=2 (28a)
+m+1−n=2k k∈N0=⇒1−(−1)m+1−n=0 (28b)
+0≤n=m−2k≤m k∈N0=⇒0≤2k≤m⇐⇒0≤k≤⌊m
+2⌋ (28c)
+upon substituting 2k:=m−n, (27) becomes
+pf(x;xi,∆x)=M/summationdisplay
+m=0chm
+m+1⌊m
+2⌋/summationdisplay
+k=01
+22k/parenleftiggm+1
+m−2k/parenrightigg/parenleftbiggx−xi
+∆x/parenrightbiggm−2k=M/summationdisplay
+m=0⌊m
+2⌋/summationdisplay
+k=0chm
+22k(m+1)/parenleftiggm+1
+m−2k/parenrightigg/parenleftbiggx−xi
+∆x/parenrightbiggm−2k
+(29)
+and, using (A.3), (29) reads
+pf(x;xi,∆x)=M/summationdisplay
+ℓ=0⌊M−ℓ
+2⌋/summationdisplay
+k=0chℓ+2k
+22k(ℓ+2k+1)/parenleftiggℓ+2k+1
+ℓ/parenrightigg/parenleftbiggx−xi
+∆x/parenrightbiggℓ
+(30)
+Using the identity (A.4) and changing the summation index ℓtomgives
+pf(x;xi,∆x)=M/summationdisplay
+m=0⌊M−m
+2⌋/summationdisplay
+k=0chm+2k
+22k(2k+1)/parenleftiggm+2k
+2k/parenrightigg/parenleftbiggx−xi
+∆x/parenrightbiggm
+(31)
+which proves (26d). In practice, the coefficients cfmare computed by solving a Vandermonde sys-
+tem [29], and the linear system (26d) must be solved to comput e the coefficients chm[2]. The general
+solution can be obtained using backward substitution witho ut making reference to the basic recon-
+struction relations ( §2). This alternative, matrix-algebra-oriented, proof of L emma 3.1 is given in §3.2.
+However, the solution can be obtained immediately, by obser ving (26e) that the relation between
+cfmm!andchm+2k(m+2k)!in (26d) is identical to the relation between f(m)(x)and∆x2kh(m+2k)(x)in (10a), with
+the only difference that the upper limit of the sum is finite. T he inverse relation is exactly analogous
+to (10b), because, using (26e) in the right-hand-side of (26 f)
+1
+m!⌊M−m
+2⌋/summationdisplay
+s=0τ2scfm+2s(m+2s)!=1
+m!⌊M−m
+2⌋/summationdisplay
+s=0τ2s⌊M−m
+2⌋−s/summationdisplay
+ℓ=0(m+2s+2ℓ)!chm+2s+2ℓ
+22ℓ(2ℓ+1)!=⌊M−m
+2⌋/summationdisplay
+s=0⌊M−m
+2⌋−s/summationdisplay
+ℓ=0/parenleftigg
+τ2s(m+2s+2ℓ)!chm+2s+2ℓ
+22ℓ(2ℓ+1)!m!/parenrightigg
+=⌊M−m
+2⌋/summationdisplay
+k=0k/summationdisplay
+s=0τ2s
+22k−2s(2k−2s+1)!(m+2k)!chm+2k
+m!=⌊M−m
+2⌋/summationdisplay
+k=0δk0(m+2k)!chm+2k
+m!=chm (32)
+where we used (15), and (A.3) and (A.2). This completes the pr oof. /square
+The extension of the above results (Lemma 3.1) to infinite pow er-series (assuming that they are
+convergent) is straightforward.
+3.2. Matrix inversion proof of Lemma 3.1
+In this section we summarize the matrix inversion relations which can be used for an alternative,
+matrix-algebra-oriented, proof (Lemma 3.4) of Lemma 3.1. B y (26d) the coefficients cfnofpfare ex-
+pressed as linear combinations of the coefficients chnofph. This system (26d), whose solution expresses
+chnas linear combinations of cfnis the deconvolution linear system [2, (3.13b), p. 244].2Since the sum-
+mation relations (26d) involve increments with step 2, we ca n split (26d) into 2 independent linear
+2more precisely, the system in [2, (3.13b), p. 244] relates n!cfnwithm!chm.
+10systems
+cfM−2ℓ=ℓ/summationdisplay
+k=0chM−2ℓ+2k
+(2k+1) 22k/parenleftiggM−2ℓ+2k
+2k/parenrightigg
+ℓ=0,···,⌊M
+2⌋ (33a)
+cfM−1−2ℓ=ℓ/summationdisplay
+k=0chM−1−2ℓ+2k
+(2k+1) 22k/parenleftiggM−1−2ℓ+2k
+2k/parenrightigg
+ℓ=0,···,⌊M−1
+2⌋ (33b)
+for[chM−2⌊M
+2⌋,···,chM]T(33a) and for [chM−1−2⌊M−1
+2⌋,···,chM−1]T(33b), respectively. In matrix-form, we have
+1···
+............
+0 0 11
+(2+1) 22/parenleftiggM−2
+2/parenrightigg1
+(4+1) 24/parenleftiggM
+4/parenrightigg
+0 0··· 11
+(2+1) 22/parenleftiggM
+2/parenrightigg
+0 0··· 0 1
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipupright
+U(⌊M
+2⌋,M)chM−2⌊M
+2⌋
+...
+chM−4
+chM−2
+chM=cfM−2⌊M
+2⌋
+...
+cfM−4
+cfM−2
+cfM(34a)
+1···
+............
+0 0 1...1
+(4+1) 24/parenleftiggM−1
+4/parenrightigg
+0 0···11
+(2+1) 22/parenleftiggM−1
+2/parenrightigg
+0 0···0 1
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipupright
+U(⌊M−1
+2⌋,M−1)chM−1−2⌊M−1
+2⌋
+...
+chM−5
+chM−3
+chM−1=cfM−1−2⌊M−1
+2⌋
+...
+cfM−5
+cfM−3
+cfM−1(34b)
+where the matrices U(⌊M
+2⌋,M)(34a) and U(⌊M−1
+2⌋,M−1)(34b) are upper unitriangular [30]. The corresponding
+linear systems (34) can be solved using backward-substitut ion [30]. To obtain the general solution, we
+initially remind, without going into the details of a formal proof, a standard result of matrix calcu-
+lus [30], concerning the inverse of an upper unitriangular m atrix.
+Lemma 3.2 (Inverse of an upper unitriangular matrix). LetU∈Rn×nbe an upper unitriangular
+matrix
+ui,i=1 1≤i≤n (35a)
+ui,j=0j<i
+1<i≤n(35b)
+U=1u1,2···u1,n−1u1,n
+0 1···u2,n−1u2,n
+............
+0 0···1un−1,n
+0 0···0 1(35c)
+11Its inverse U−1exists and is also an upper unitriangular matrix
+ˇui,i=1 1≤i≤n (36a)
+ˇui,j=0j<i
+1<i≤n(36b)
+U−1=1 ˇu1,2···ˇu1,n−1ˇu1,n
+0 1···ˇu2,n−1ˇu2,n
+............
+0 0···1 ˇun−1,n
+0 0···0 1(36c)
+whose nonzero elements ˇui,j(j≥i) satisfy the recurrence relations
+ˇun,n=1 (36d)
+ˇun−k,n−k+s=−s/summationdisplay
+ℓ=1un−k,n−k+ℓˇun−k+ℓ,n−k+s1≤k<n
+1≤s≤k(36e)
+PROOF . It is straightforward to show, by induction, that detU=1. The proof by induction of (36) is
+a simple exercise of matrix calculus, directly obtained fro m the backward-substitution algorithm for
+solvingUx=b[30]. /square
+This recurrence is applied to compute the inverse of the uppe r unitriangular matrices (34) of the
+linear system (26d) of Lemma 3.1.
+Lemma 3.3 (Inverse of the matrices in Lemma 3.1). Assume N≤⌊M
+2⌋+1. LetU(N,M)∈RN×Nbe an
+upper unitriangular matrix whose elements are given by
+(U(N,M))N−ℓ,N−ℓ−k=0 0 ≤k≤N−1−ℓ
+(U(N,M))N−ℓ,N−ℓ=1
+(U(N,M))N−ℓ,N−ℓ+k=1
+(2k+1)22k/parenleftiggM−2ℓ+2k
+2k/parenrightigg
+0≤k≤ℓ; 0≤ℓ<N−1 ;N≤⌊M
+2⌋+1
+(37a)
+Its inverse U−1
+(N,M)is also an upper unitriangular matrix whose elements are giv en by
+(U−1
+(N,M))N−ℓ,N−ℓ−k=0 0 ≤k≤N−1−ℓ
+(U−1
+(N,M))N−ℓ,N−ℓ=1
+(U−1
+(N,M))N−ℓ,N−ℓ+k=τ2k(M−2ℓ+2k)!
+(M−2ℓ)!0≤k≤ℓ; 0≤ℓ<N−1 ;N≤⌊M
+2⌋+1(37b)
+where the numbers τ2k(Tab. 1) are defined by the recurrence (10c) .
+PROOF . To simplify notation let (U(N,M))ij=uijand(U−1
+(N,M))ij=ˇuijBy Lemma 3.2 U−1
+(N,M)is also an upper
+unitriangular matrix. It is easy to verify, by straightforw ard computation, using (36), that (37b) holds
+for0≤ℓ≤3. To prove that (37b) is valid for 0≤ℓ≤N−1, by induction, suppose that (37b) is valid for
+1≤ℓ≤m. Then, from (36e)
+ˇuN−(m+1),N−(m+1)+k=−k/summationdisplay
+s=1uN−(m+1),N−(m+1)+sˇuN−(m+1)+s,N−(m+1)+k
+=−k/summationdisplay
+s=1uN−(m+1),N−(m+1)+sˇuN−(m+1−s),N−(m+1−s)+(k−s) (38a)
+12and since s≥1=⇒m+1−s≤m, we may replace ˇuN−(m+1−s),N−(m+1−s)+(k−s)in (38a) by (37b), so that
+ˇuN−(m+1),N−(m+1)+k=k/summationdisplay
+s=1−1
+22s(2s+1)/parenleftiggM−2(m+1)+2s
+2s/parenrightigg
+τ2k−2s(M−2(m+1−s)+2(k−s))!
+(M−2(m+1−s))!
+=k/summationdisplay
+s=1−τ2k−2s
+22s(2s+1)!(M−2(m+1)+2k)!
+(M−2(m+1))!
+=k/summationdisplay
+s=1−τ2k−2s
+22s(2s+1)!(M−2(m+1)+2k)!
+(M−2(m+1))!=τ2k(M−2(m+1)+2k)!
+(M−2(m+1))!(38b)
+because, setting ℓ:=k−s
+k/summationdisplay
+s=1−τ2k−2s
+22s(2s+1)!=k−1/summationdisplay
+ℓ=0−τ2ℓ
+22k−2ℓ(2k−2ℓ+1)!=τ2k (38c)
+by (10c). This completes the proof of (37b) by induction. /square
+Lemma 3.4 (Solution of the linear system (26d) ).The solution of the linear system (26d) is given
+by(26f) .
+PROOF . The unitriangular matrices U(⌊M
+2⌋,M)(34a) and U(⌊M−1
+2⌋,M−1)(34b) are of the type defined in
+Lemma 3.3. Using the result (37b) of Lemma 3.3 for the inverse matrices U−1
+(⌊M
+2⌋,M)andU−1
+(⌊M−1
+2⌋,M−1),
+the solution of the linear systems (34) is
+chM−2ℓ=ℓ/summationdisplay
+k=0τ2kcfM−2ℓ+2k(M−2ℓ+2k)!
+(M−2ℓ)!ℓ=0,···,⌊M
+2⌋ (39a)
+chM−1−2ℓ=ℓ/summationdisplay
+k=0τ2kcfM−1−2ℓ+2k(M−1−2ℓ+2k)!
+(M−1−2ℓ)!ℓ=0,···,⌊M−1
+2⌋ (39b)
+where the numbers τ2k(Tab. 1) are defined by the recurrence (10c). Since
+m=M−2ℓ=⇒2ℓ=M−m=⇒ℓ=⌊M−m
+2⌋ (40a)
+m=M−2ℓ−1=⇒2ℓ+1=M−m=⇒ℓ=⌊M−m
+2⌋ (40b)
+the 2 solutions (39) can be grouped into (26f), which complet es the proof. /square
+4. Error of polynomial reconstruction
+We consider in this paper reconstruction on a homogeneous gr id (recall that (8) hold iff ∆x=const ).
+The reconstruction polynomials are computed by interpolat ingf(x)sampled on an appropriately cho-
+sen stencil (Definition 4.1). We examine the relations and or der-of-accuracy of polynomial reconstruc-
+tion (Definition 2.3) on an arbitrary stencil Si,M−,M+(Definition 4.1) defined on a homogeneous grid. The
+WENO [5, 6, 11, 13, 14, 15] schemes are based on the convex combinat ion of polynomial reconstructions
+on a family of substencils. For the development of the order- of-accuracy relations, it is necessary to
+develop results on the approximation-error of polynomial r econstruction for the general stencil Si,M−,M+,
+around point i(not necessarily contained in the stencil), with M−neighbours on the left, and M+neigh-
+bours on the right (Definition 4.1).
+134.1. Polynomial reconstruction
+The part concerning the approximation of f(x)by a polynomial pf(x;Si,M−,M+,∆x)is found in most
+textbooks of numerical analysis [20, 21]. It is only briefly i ncluded here for use in deriving the results
+concerning the approximation of h(x)by the polynomial ph(x;Si,M−,M+,∆x)which forms a reconstruction
+pair with pf(Definition 2.1). To obtain the relations concerning ph(x;Si,M−,M+,∆x)it is not very practical
+to work with the Newton divided-differences form of pf[20, 21], which are widely used in WENO
+theory [7, 2, 5, 3, 4]. It is, instead, preferable to work with the standard form of pfexpanded in powers
+of(x−xi), whose coefficients can be readily expressed (Proposition 4 .5) from the coefficients of the
+inverse of the Vandermonde matrix [31, 32] corresponding to the stencil Si,M−,M+(Definition 4.1). This
+representation of pfallows direct use of the formulas relating the coefficients o fphandpf(Lemma 3.1).
+Definition 4.1 (Stencil). Consider a 1-D homogeneous computational mesh
+xi=x1+(i−1)∆x∆x=const∈R>0 (41a)
+Assume
+M:=M−+M+≥0 (41b)
+The set of contiguous points
+Si,M−,M+:={i−M−,···,i+M+} (41c)
+is defined as the discretization-stencil in the neighbourho od ofi, withM−neighbours to the left and
+M+neighbours to the right. The stencil Si,M−,M+(41c) contains M+1>0points and has a length of M
+intervals. If M±≥0then the stencil contains the pivot-point i. IfM−M+<0then the stencil does not
+contain the pivot-point i. We will note
+[Si,M−,M+] :=[xi−M−,xi+M+]⊂R (41d)
+the interval defined by the extreme points of the stencil. /square
+Remark 4.2 (Stencils and notation). In our notation the stencil is defined by a reference (pivot)
+pointi, and by the number of neighbours M±on each side of point i(Definition 4.1). The position of
+the pivot point iin the stencil is arbitrary. This is necessary for obtaining relations for all of the WENO
+stencils with reference to the same point i. In the following developments, there appear quantities
+depending both on M±and oni(and eventually on the values of fsampled at the points of the stencil).
+We will systematically note these quantities as functions o f the stencil Si,M−,M+. On the other hand,
+there appear quantities, which depend on M±but not on the pivot point i(neither on the values of f
+sampled at the points of the stencil). We will systematicall y note these quantities as functions of M−
+andM+, and not of Si,M−,M+. This difference is important when considering order-of-a ccuracy relations
+(egCorollary 4.9). /square
+Definition 4.3 (Vandermonde matrix on Si,M−,M+).LetM:=M−+M+and assume M≥0. The matrix
+M+
+M−V∈R(M+1)×(M+1)with elements (M+
+M−V)ij
+M+
+M−V:=(−M−)0(−M−)1···(−M−)M
+...
+(+M+)0(+M+)1···(+M+)MM:=M−+M+≥0 (42)
+is the Vandermonde matrix [31, 32] defined on the stencil Si,M−,M+(Definition 4.1). SinceM+
+M−Vis a
+Vandermonde matrix, its inverseM+
+M−V−1exists [29, 33]. The elements ofM+
+M−V−1∈R(M+1)×(M+1)will be
+noted(M+
+M−V−1)ij. /square
+14Lemma 4.4 (Inverse Vandermonde matrix on Si,M−,M+).Assume the conditions of Definition 4.3 .
+Then the entries of the inverse of the Vandermonde matrixM+
+M−V(42)onSi,M−,M+are given by
+(M+
+M−V−1)ij=M+1−i/summationdisplay
+n=0(M−)n/parenleftiggn+i−1
+n/parenrightigg
+(M
+0V−1)i+n,j∀i,j∈{1,···,M+1}
+M:=M−+M+(43a)
+whereM
+0V−1is the inverse of the Vandermonde matrixM
+0VonSi,0,M={i,···,i+M}(Definition 4.3) , whose
+entries are given by3
+(M
+0V−1)ij=(−1)i+jM+1/summationdisplay
+k=11
+(k−1)!/parenleftiggk−1
+j−1/parenrightigg/bracketleftiggk−1
+i−1/bracketrightigg
+∀i,j∈{1,···,M+1} (43b)
+Define
+νM−,M+,m,k:=M+/summationdisplay
+ℓ=−M−(M+
+M−V−1)m+1,ℓ+M−+1ℓk(43c)
+Then the following identities hold
+νM−,M+,m,k=M+/summationdisplay
+ℓ=−M−(M+
+M−V−1)m+1,ℓ+M−+1ℓk=δmk0≤k≤M
+0≤m≤M(43d)
+M/summationdisplay
+m=0νM−,M+,m,kℓm=ℓk∀k∈N0
+∀ℓ∈{−M−,···,M+}(43e)
+PROOF .4SinceM+
+M−V(42) is an (M+1)×(M+1)Vandermonde matrix on M+1distinct nodes its inverse
+M+
+M−V−1exists [29, 33]. Macon and Spitzbart [34, 29] have given expl icit expressions for the inverse of the
+Vandermonde matrix on integer nodes. To prove (43b) we start from [33, Theorem 1, p. 973], giving the
+inverse of the Vandermonde matrix on nequidistant nodes on [0,1],ieon(n−1)xi=(i−1)∀i∈{1,···,n},
+as
+/parenleftiggi−1
+n−1/parenrightiggj−1
+,i,j∈{1,···,n}−1
+ij=(−1)i+j(n−1)i−1n/summationdisplay
+k=11
+(k−1)!/parenleftiggk−1
+j−1/parenrightigg/bracketleftiggk−1
+i−1/bracketrightigg
+(44a)
+3/bracketleftiggn
+k/bracketrightigg
+are the unsigned Stirling numbers of the first kind [22, 23, 33 ] satisfying
+/bracketleftiggn
+0/bracketrightigg
+=δn0
+/bracketleftiggn+1
+k/bracketrightigg
+=n/bracketleftiggn
+k/bracketrightigg
++/bracketleftiggn
+k−1/bracketrightigg
+m/bracketleftiggn
+n−m/bracketrightigg
+=m−1/summationdisplay
+k=0/parenleftiggn−k
+m+1−k/parenrightigg/bracketleftiggn
+n−k/bracketrightigg
+n/summationdisplay
+k=1(−1)k(m−1)k−1/bracketleftiggn−1
+k−1/bracketrightigg
+=(−1)n(n−1)!/parenleftiggm−1
+n−1/parenrightigg
+4Proof of (43d) is most easily obtained using Proposition 4.5 , and proof of (43e) is most easily obtained using Propositio n 4.6,
+which are proved below. Notice that (43d) is not used in the pr oof of Proposition 4.5, nor is (43e) in the proof of Propositi on 4.6.
+15which directly implies, setting n=M+1,
+/parenleftiggi−1
+M/parenrightiggj−1
+,i,j∈{1,···,M+1}−1
+ij=(−1)i+jMi−1M+1/summationdisplay
+k=11
+(k−1)!/parenleftiggk−1
+j−1/parenrightigg/bracketleftiggk−1
+i−1/bracketrightigg
+(44b)
+Obviously, Mi−1andMj−1in (44b) are scaling factors (for M+1equidistant nodes on [0,1]we have
+M∆x=1). This is clearly seen by writing the Vandermonde matrix on Si,0,M(42) as
+M
+0V:=/bracketleftig
+(i−1)j−1,i,j∈{1,···,M+1}/bracketrightig
+=/parenleftiggi−1
+M/parenrightiggℓ−1
+,i,ℓ∈{1,···,M+1}/bracketleftig
+Mℓ−1δℓj,ℓ,j∈{1,···,M+1}/bracketrightig
+(44c)
+and since/bracketleftig
+Mℓ−1δℓj,ℓ,j∈{1,···,M+1}/bracketrightig
+is a diagonal matrix
+M
+0V−1=/bracketleftbiggδiℓ
+Mi−1,i,ℓ∈{1,···,M+1}/bracketrightbigg/parenleftiggℓ−1
+M/parenrightiggj−1
+,ℓ,j∈{1,···,M+1}−1
+(44d)
+which, by (44b), proves (43b).
+To obtain the final expression (43a), we observe that, for M:=M−+M+, the stencils Si,M−,M+(corre-
+sponding Vandermonde matrixM+
+M−V; Definition 4.3) and Si−M−,0,M(corresponding Vandermonde matrix
+M
+0V; Definition 4.3) correspond by Definition 4.1 to the same set o f points{i−M−,···,i+M+}. Therefore,
+∀f∈C[xi−M−,xi+M+], by the uniqueness of the Lagrange interpolating polynomia l [20], we have (using
+the notation of Proposition 4.5)
+pf(x;Si,M−,M+,∆x)=pf(x;Si−M−,0,M,∆x)∀x∈R
+∀f∈C[xi−M−,xi+M+](44e)
+the only difference being in the choice of the pivot point ( xiforSi,M−,M+andxi−M−=xi−M−∆xforSi−M−,0,M)
+used for the representation (45b) of the interpolating poly nomial of f(x)on the nodes{i−M−,···,i+M+}.
+By (45b), (44e) reads
+M/summationdisplay
+m=0cf,Si,M−,M+,m/parenleftbiggx−xi
+∆x/parenrightbiggm
+=M/summationdisplay
+s=0cf,Si−M−,0,M,s/parenleftbiggx−xi−M−
+∆x/parenrightbiggs
+=M/summationdisplay
+s=0cf,Si−M−,0,M,s/parenleftbiggx−xi
+∆x+M−/parenrightbiggs
+=M/summationdisplay
+s=0s/summationdisplay
+n=0cf,Si−M−,0,M,s/parenleftiggs
+n/parenrightigg
+(M−)n/parenleftbiggx−xi
+∆x/parenrightbiggs−n
+m:=s−n=M/summationdisplay
+m=0M−m/summationdisplay
+n=0cf,Si−M−,0,M,m+n/parenleftiggm+n
+n/parenrightigg
+(M−)n/parenleftbiggx−xi
+∆x/parenrightbiggm∀x∈R
+∀f∈C[xi−M−,xi+M+](44f)
+implying
+cf,Si,M−,M+,m=M−m/summationdisplay
+n=0cf,Si−M−,0,M,m+n/parenleftiggm+n
+n/parenrightigg
+(M−)n∀m∈{0,···,M} (44g)
+which by (49a) gives, ∀f∈C[xi−M−,xi+M+]
+M+/summationdisplay
+ℓ=−M−(M+
+M−V−1)m+1,ℓ+M−+1fi+ℓ=M−m/summationdisplay
+n=0M/summationdisplay
+s=0(M
+0V−1)m+n+1,s+1fi−M−+s/parenleftiggm+n
+n/parenrightigg
+(M−)n
+=M/summationdisplay
+s=0M−m/summationdisplay
+n=0(M
+0V−1)m+n+1,s+1fi−M−+s/parenleftiggm+n
+n/parenrightigg
+(M−)n
+ℓ:=s−M−=M+/summationdisplay
+ℓ=−M−M−m/summationdisplay
+n=0/parenleftiggm+n
+n/parenrightigg
+(M−)n(M
+0V−1)m+n+1,ℓ+M−+1fi+ℓ (44h)
+16and since fi+ℓ(ℓ∈{−M−,···,M+}) are linearly independent we have
+(M+
+M−V−1)m+1,ℓ+M−+1=M−m/summationdisplay
+n=0/parenleftiggm+n
+n/parenrightigg
+(M−)n(M
+0V−1)m+n+1,ℓ+M−+1∀m∈{0,···,M}
+∀ℓ∈{−M−,···,M+}(44i)
+which proves (43a).
+To prove the identities containing νM−,M+,m,k(43c), notice that the elements ofM+
+M−V(42) read
+(M+
+M−V)ij=(i−1−M−)j−11≤i≤M+1
+1≤j≤M+1(44j)
+Explicit expression of the elements of the product (M+
+M−V−1)·(M+
+M−V)=IM+1(whereIM+1∈R(M+1)×(M+1)is the
+identity matrix) yields
+δm+1,k+1=/parenleftbigg
+(M+
+M−V−1)·(M+
+M−V)/parenrightbigg
+m+1,k+1=νM−,M+,m,k0≤k≤M
+0≤m≤M(44k)
+and as a consequence (43d). To prove (43e), consider the erro r (51b) of the polynomial interpolation
+pf(xi+ξ∆x;Si,M−,M+,∆x)on the stencil Si,M−,M+(Proposition 4.6). By construction, we have
+pf(xi+ℓ∆x;Si,M−,M+,∆x)=fi+ℓ(51b)=⇒Ef(xi+ℓ∆x;Si,M−,M+,∆x)=0∀ℓ∈{−M−,···,M+} (44l)
+which, using (51e) and (51g) in (51b), proves (43e). /square
+Proposition 4.5 (Lagrange polynomial reconstruction on Si,M−,M+).Let
+ph(x;Si,M−,M+,∆x) :=M/summationdisplay
+m=0ch,Si,M−,M+,m/parenleftbiggx−xi
+∆x/parenrightbiggm
+(45a)
+pf(x;Si,M−,M+,∆x) :=M/summationdisplay
+m=0cf,Si,M−,M+,m/parenleftbiggx−xi
+∆x/parenrightbiggm
+(45b)
+be 2 polynomials of degree
+M:=M−+M+ (45c)
+constituting a polynomial (Lemma 3.1) reconstruction pair (Definition 2.1) ph=R(1;∆x)(pf). Assume that
+the polynomial pf(x;Si,M−,M+,∆x)is obtained by interpolation of the values of f(x)on the points of the
+stencil Si,M−,M+(Definition 4.1) . Then
+ph(xi+ξ∆x;Si,M−,M+,∆x)=M+/summationdisplay
+ℓ=−M−αh,M−,M+,ℓ(ξ)fi+ℓ (45d)
+pf(xi+ξ∆x;Si,M−,M+,∆x)=M+/summationdisplay
+ℓ=−M−αf,M−,M+,ℓ(ξ)fi+ℓ (45e)
+whereαh,M−,M+,ℓ(ξ)andαf,M−,M+,ℓ(ξ)are polynomials of degree Min
+ξ:=x−xi
+∆x(45f)
+with coefficients depending only on the 3 indices ( M−,M+,ℓ)
+αh,M−,M+,ℓ(ξ) :=M/summationdisplay
+m=0⌊M−m
+2⌋/summationdisplay
+k=0τ2k(m+2k)!
+m!(M+
+M−V−1)m+2k+1,ℓ+M−+1ξm(45g)
+αf,M−,M+,ℓ(ξ) :=M/summationdisplay
+m=0(M+
+M−V−1)m+1,ℓ+M−+1ξm(45h)
+17where(M+
+M−V−1)ijare the elements of the inverse Vandermonde matrix on Si,M−,M+(Lemma 4.4) , and the
+numbersτ2k(Tab. 1) are defined by (19c) and satisfy the recurrence (10c) .
+PROOF . Define
+xi+ℓ:=xi+ℓ∆x−M−≤ℓ≤M+ (46a)
+fi+ℓ:=f(xi+ℓ)−M−≤ℓ≤M+ (46b)
+TheM+1coefficients cf,(Si,M−,M+),m(m=0,···,M)are computed by equating the polynomial pf(xi+ℓ;Si,M−,M+,∆x)(45b)
+to known values fi+ℓ
+fi−M−=pf(xi−M−;Si,M−,M+,∆x)
+...
+fi+M+=pf(xi−M+;Si,M−,M+,∆x)(47)
+Expanding (47) results in an (M+1)×(M+1)Vandermonde (Definition 4.3) linear system
+(−M−)0(−M−)1···(−M−)M
+...
+(+M+)0(+M+)1···(+M+)M
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipupright
+M+
+M−Vcf,Si,M−,M+,0
+...
+cf,Si,M−,M+,M=fi−M−
+...
+fi+M+(48)
+Hence (Definition 4.3)
+cf,Si,M−,M+,m=M+/summationdisplay
+ℓ=−M−(M+
+M−V−1)m+1,ℓ+M−+1fi+ℓ∀m∈{0,···,M} (49a)
+ch,Si,M−,M+,m=1
+m!⌊M−m
+2⌋/summationdisplay
+k=0τ2kcf,Si,M−,M+,m+2k(m+2k)!∀m∈{0,···,M} (49b)
+where we used the deconvolution formula (26f) for computing ch,Si,M−,M+,m. Injecting (49a) into (45b) we
+have
+pf(xi+ξ∆x;Si,M−,M+,∆x)=M/summationdisplay
+m=0M+/summationdisplay
+ℓ=−M−(M+
+M−V−1)m+1,ℓ+M−+1fi+ℓξm=M+/summationdisplay
+ℓ=−M−M/summationdisplay
+m=0(M+
+M−V−1)m+1,ℓ+M−+1ξmfi+ℓ(50a)
+proving (45e) and (45h). Injecting (49b) into (45a) we have
+ph(xi+ξ∆x;Si,M−,M+,∆x)=M/summationdisplay
+m=0⌊M−m
+2⌋/summationdisplay
+k=0τ2k(m+2k)!
+m!cf,Si,M−,M+,m+2kξm
+=M/summationdisplay
+m=0⌊M−m
+2⌋/summationdisplay
+k=0τ2k(m+2k)!
+m!M+/summationdisplay
+ℓ=−M−(M+
+M−V−1)m+2k+1,ℓ+M−+1fi+ℓξm
+=M+/summationdisplay
+ℓ=−M−M/summationdisplay
+m=0⌊M−m
+2⌋/summationdisplay
+k=0τ2k(m+2k)!
+m!(M+
+M−V−1)m+2k+1,ℓ+M−+1ξmfi+ℓ (50b)
+proving (45d) and (45g). /square
+184.2. Approximation error of Lagrange polynomial reconstru ction
+Of course the accuracy relations for the approximation of f(x)are well-known [20], but this section
+(§4.2) is concerned with the accuracy of the approximation of h(x), using Lagrange polynomial recon-
+struction based on the knowledge of the values of f(x)on an arbitrary stencil defined on a homogeneous
+grid (§4.1).
+Proposition 4.6 (Error of Lagrange polynomial reconstruct ion on Si,M−,M+).Letpf(x;Si,M−,M+,∆x)
+andph(x;Si,M−,M+,∆x)be a polynomial (Lemma 3.1) reconstruction pair (Definition 2.1) ph=R(1;∆x)(pf),
+satisfying the conditions of Proposition 4.5 . Then,pf(x;Si,M−,M+,∆x)approximates f(x)toO(∆xM+1), and
+ph(x;Si,M−,M+,∆x)approximates h(x)toO(∆xM+1)
+ph(x;Si,M−,M+,∆x)=h(x)+Eh(x;Si,M−,M+,∆x)=h(x)+O(∆xM+1) (51a)
+pf(x;Si,M−,M+,∆x)=f(x)+Ef(x;Si,M−,M+,∆x)=f(x)+O(∆xM+1) (51b)
+where the approximation errors constitute a reconstructio n pairEh=R(1;∆x)(Ef)(Definition 2.1) and,
+∀NTJ≥M+1, are given by (assuming fandhare of class CNTJ+1)
+Eh(xi+ξ∆x;Si,M−,M+,∆x)=NTJ/summationdisplay
+s=M+1µh,M−,M+,s(ξ)∆xsf(s)
+i+O(∆xNTJ+1) (51c)
+=NTJ/summationdisplay
+s=M+1⌊s−M−1
+2⌋/summationdisplay
+ℓ=0µh,M−,M+,s−2ℓ(ξ)
+22ℓ(2ℓ+1)!∆xsh(s)
+i+O(∆xNTJ+1) (51d)
+Ef(xi+ξ∆x;Si,M−,M+,∆x)=NTJ/summationdisplay
+s=M+1µf,M−,M+,s(ξ)∆xsf(s)
+i+O(∆xNTJ+1) (51e)
+whereµh,M−,M+,s(ξ)andµf,M−,M+,s(ξ)are polynomials of degree sinξ(45f)
+µh,M−,M+,s(ξ) :=⌊s
+2⌋/summationdisplay
+k=0−τ2k
+(s−2k)!ξs−2k+M/summationdisplay
+m=0⌊M−m
+2⌋/summationdisplay
+k=0τ2kνM−,M+,m+2k,s(m+2k)!
+s!m!ξm(51f)
+µf,M−,M+,s(ξ) :=1
+s!−ξs+M/summationdisplay
+m=0νM−,M+,m,sξm (51g)
+whereνM−,M+,m,sare defined by (43c) , and the numbers τ2k(Tab. 1) are defined by (19c) and satisfy the
+recurrence (10c) .
+PROOF . To prove (51b) we start by Taylor-expanding fi+ℓin (49a), and using (43d)
+cf,Si,M−,M+,m=M+/summationdisplay
+ℓ=−M−(M+
+M−V−1)m+1,ℓ+M−+1NTJ/summationdisplay
+s=0ℓs∆xsf(s)
+i
+s!+O(∆xNTJ+1)
+=NTJ/summationdisplay
+s=0M+/summationdisplay
+ℓ=−M−/parenleftbigg
+(M+
+M−V−1)m+1,ℓ+M−+1ℓs/parenrightbigg∆xsf(s)
+i
+s!+O(∆xNTJ+1)
+=NTJ/summationdisplay
+s=0νM−,M+,m,s∆xsf(s)
+i
+s!+O(∆xNTJ+1)=M/summationdisplay
+s=0δms∆xsf(s)
+i
+s!+NTJ/summationdisplay
+s=M+1νM−,M+,m,s∆xsf(s)
+i
+s!+O(∆xNTJ+1)
+=∆xmf(m)
+i
+m!+NTJ/summationdisplay
+s=M+1νM−,M+,m,s∆xsf(s)
+i
+s!+O(∆xNTJ+1) (52)
+19Injecting (52) into (45b), and replacing f(xi+ξ∆x)by its Taylor-polynomial, we have
+Ef(xi+ξ∆x;Si,M−,M+,∆x)=pf(xi+ξ∆x;Si,M−,M+,∆x)−f(xi+ξ∆x)
+=M/summationdisplay
+m=0∆xmf(m)
+i
+m!+NTJ/summationdisplay
+s=M+1νM−,M+,m,s∆xsf(s)
+i
+s!+O(∆xNTJ+1)ξm−f(xi+ξ∆x)
+=M/summationdisplay
+m=0∆xmf(m)
+i
+m!ξm−f(xi+ξ∆x)+M/summationdisplay
+m=0NTJ/summationdisplay
+s=M+1νM−,M+,m,s∆xsf(s)
+i
+s!ξm+O(∆xNTJ+1)
+=NTJ/summationdisplay
+m=M+1−∆xmf(m)
+i
+m!ξm+M/summationdisplay
+m=0NTJ/summationdisplay
+s=M+1νM−,M+,m,s∆xsf(s)
+i
+s!ξm+O(∆xNTJ+1)
+=NTJ/summationdisplay
+s=M+1−∆xsf(s)
+i
+s!ξs+NTJ/summationdisplay
+s=M+1M/summationdisplay
+m=0νM−,M+,m,sξm∆xsf(s)
+i
+s!+O(∆xNTJ+1) (53)
+proving (51b), (51e) and (51g).
+To prove (51a) we use the expression (52) for cf,Si,M−,M+,min (49b) to obtain
+ch,Si,M−,M+,m=1
+m!⌊M−m
+2⌋/summationdisplay
+k=0τ2k(m+2k)!∆xm+2kf(m+2k)
+i
+(m+2k)!+NTJ/summationdisplay
+s=M+1νM−,M+,m+2k,s∆xsf(s)
+i
+s!)+O(∆xNTJ+1)
+=⌊M−m
+2⌋/summationdisplay
+k=0τ2k∆xm+2kf(m+2k)
+i
+m!+⌊M−m
+2⌋/summationdisplay
+k=0τ2k(m+2k)!
+m!NTJ/summationdisplay
+s=M+1νM−,M+,m+2k,s∆xsf(s)
+i
+s!+O(∆xNTJ+1) (54)
+Injecting (54) into (45a), and replacing h(xi+ξ∆x)by its Taylor-polynomial (16), we have5
+Eh(xi+ξ∆x;Si,M−,M+,∆x)=ph(xi+ξ∆x;Si,M−,M+,∆x)−h(xi+ξ∆x)=M/summationdisplay
+m=0⌊M−m
+2⌋/summationdisplay
+k=0τ2k∆xm+2kf(m+2k)
+i
+m!ξm
++M/summationdisplay
+m=0⌊M−m
+2⌋/summationdisplay
+k=0NTJ/summationdisplay
+s=M+1τ2k(m+2k)!
+m!νM−,M+,m+2k,s∆xsf(s)
+i
+s!ξm−NTJ/summationdisplay
+m=0⌊NTJ−m
+2⌋/summationdisplay
+k=0τ2k∆xm+2kf(m+2k)
+i
+m!ξm+O(∆xNTJ+1) (55a)
+which simplifies to
+Eh(xi+ξ∆x;Si,M−,M+,∆x)=M/summationdisplay
+m=0⌊M−m
+2⌋/summationdisplay
+k=0NTJ/summationdisplay
+s=M+1τ2k(m+2k)!
+m!νM−,M+,m+2k,s∆xsf(s)
+i
+s!ξm
++M/summationdisplay
+m=0⌊NTJ−m
+2⌋/summationdisplay
+k=⌊M−m
+2⌋+1−τ2k
+m!∆xm+2kf(m+2k)
+iξm+NTJ/summationdisplay
+m=M+1⌊NTJ−m
+2⌋/summationdisplay
+k=0−τ2k
+m!∆xm+2kf(m+2k)
+iξm+O(∆xNTJ+1) (55b)
+Using (A.3) and (A.2), (55b) reads (the summation indices on line 1 remaining unchanged)
+Eh(xi+ξ∆x;Si,M−,M+,∆x)=NTJ/summationdisplay
+s=M+1M/summationdisplay
+m=0⌊M−m
+2⌋/summationdisplay
+k=0τ2k(m+2k)!
+s!m!νM−,M+,m+2k,sξm∆xsf(s)
+i
++NTJ/summationdisplay
+s=M+1⌊s
+2⌋/summationdisplay
+k=⌈s−M
+2⌉−τ2k
+(s−2k)!ξs−2k∆xsf(s)
+i+NTJ/summationdisplay
+s=M+1⌈s−M
+2⌉−1/summationdisplay
+k=0−τ2k
+(s−2k)!ξs−2k∆xsf(s)
+i+O(∆xNTJ+1) (55c)
+5h(x+ξ∆x)(16)=NTJ/summationdisplay
+s=0⌊s
+2⌋/summationdisplay
+ℓ=0τ2ℓξs−2ℓ
+(s−2ℓ)!∆xsf(s)(x)+O(∆xNTJ+1)m:=s−2ℓ=NTJ/summationdisplay
+m=0⌊NTJ−m
+2⌋/summationdisplay
+ℓ=0τ2ℓ∆xm+2ℓf(m+2ℓ)(x)
+m!ξm+O(∆xNTJ+1)
+20and definingµh,M−,M+,s(ξ)by (51f) we obtain (51a) and (51c).
+Finally, using (10a) in (51c)
+Eh(xi+ξ∆x;Si,M−,M+,∆x)=NTJ/summationdisplay
+s=M+1µh,M−,M+,s(ξ)⌊NTJ−s
+2⌋/summationdisplay
+ℓ=0∆xs+2ℓh(s+2ℓ)
+i
+22ℓ(2ℓ+1)!+O(∆xNTJ+1)
+=NTJ/summationdisplay
+s=M+1⌊NTJ−s
+2⌋/summationdisplay
+ℓ=0µh,M−,M+,s(ξ)
+22ℓ(2ℓ+1)!∆xs+2ℓh(s+2ℓ)
+i+O(∆xNTJ+1) (55d)
+which, by (A.3) and (A.2), proves (51d). /square
+Proposition 4.7 (Approximation error of Lagrange polynomi al reconstruction on Si,M−,M+).Asssume
+the conditions and definitions of Proposition 4.6 . Then
+Eh(xi+ξ∆x;Si,M−,M+,∆x)=NTJ/summationdisplay
+n=M+1λh,M−,M+,n(ξ)∆xnh(n)(xi+ξ∆x)+O(∆xNTJ+1) (56a)
+Ef(xi+ξ∆x;Si,M−,M+,∆x)=NTJ/summationdisplay
+n=M+1λf,M−,M+,n(ξ)∆xnf(n)(xi+ξ∆x)+O(∆xNTJ+1) (56b)
+whereλh,M−,M+,n(ξ)andλf,M−,M+,n(ξ)are polynomials of degree ninξ(45f)
+λh,M−,M+,n(ξ) :=n−M−1/summationdisplay
+ℓ=0µh,M−,M+,n−ℓ(ξ)(−1)ℓ+1
+(ℓ+1)!/parenleftig
+(ξ−1
+2)ℓ+1−(ξ+1
+2)ℓ+1/parenrightig
+(56c)
+λf,M−,M+,n(ξ) :=n−M−1/summationdisplay
+ℓ=0(−ξ)ℓ
+ℓ!µf,M−,M+,n−ℓ(ξ) (56d)
+whereµh,M−,M+,n(ξ)is defined by (51f) andµf,M−,M+,n(ξ)is defined by (51g) .
+PROOF . Taylor-expanding fiin (51e), around the point xi+ξ∆x,6we have
+Ef(xi+ξ∆x;Si,M−,M+,∆x)=NTJ/summationdisplay
+s=M+1µf,M−,M+,s(ξ)∆xsNTJ−s/summationdisplay
+ℓ=0(−ξ)ℓ
+ℓ!∆xℓf(s+ℓ)(xi+ξ∆x)+O(∆xNTJ+1)
+=NTJ/summationdisplay
+s=M+1NTJ−s/summationdisplay
+ℓ=0µf,M−,M+,s(ξ)(−ξ)ℓ
+ℓ!∆xs+ℓf(s+ℓ)(xi+ξ∆x)+O(∆xNTJ+1)
+=NTJ/summationdisplay
+n=M+1n−M−1/summationdisplay
+ℓ=0µf,M−,M+,n−ℓ(ξ)(−ξ)ℓ
+ℓ!∆xnf(n)(xi+ξ∆x)+O(∆xNTJ+1) (57a)
+which proves (56b).
+6f(n)(x)=NTJ/summationdisplay
+ℓ=0(−ξ)ℓ
+ℓ!∆xℓf(n+ℓ)(x+ξ∆x)+O(∆xNTJ+1)
+21Replacing f(s)
+iin (51c) by its expansion7in terms of the derivatives ∆xℓh(s+ℓ)(xi+ξ∆x)we have
+Eh(xi+ξ∆x;Si,M−,M+,∆x)=NTJ/summationdisplay
+s=M+1µh,M−,M+,s(ξ)∆xsNTJ−s/summationdisplay
+ℓ=0(−1)ℓ+1
+(ℓ+1)!/parenleftig
+(ξ−1
+2)ℓ+1−(ξ+1
+2)ℓ+1/parenrightig
+∆xℓh(s+ℓ)(xi+ξ∆x)
++O(∆xNTJ+1)
+=NTJ/summationdisplay
+s=M+1NTJ−s/summationdisplay
+ℓ=0µh,M−,M+,s(ξ)(−1)ℓ+1
+(ℓ+1)!/parenleftig
+(ξ−1
+2)ℓ+1−(ξ+1
+2)ℓ+1/parenrightig
+∆xs+ℓh(s+ℓ)(xi+ξ∆x)
++O(∆xNTJ+1)
+=NTJ/summationdisplay
+n=M+1n−M−1/summationdisplay
+ℓ=0µh,M−,M+,s(ξ)(−1)ℓ+1
+(ℓ+1)!/parenleftig
+(ξ−1
+2)ℓ+1−(ξ+1
+2)ℓ+1/parenrightig
+∆xnh(n)(xi+ξ∆x)
++O(∆xNTJ+1) (57b)
+which proves (56a). Obviously, by (56d), deg(λf,M−,M+,n(ξ))=n. It is easy8to verify that, by (56c),
+deg(λh,M−,M+,n(ξ))=n, which completes the proof. /square
+4.3. Approximation error of hi±1
+2and off′
+i
+One of the principal uses of the reconstructing polynomial b eing the numerical approximation
+off′
+i:=f′(xi)via (9), we give in this section the relations concerning the approximation error of
+hi±1
+2:=h(xi±1
+2∆x)(Corollary 4.8) and of f′
+i(Corollary 4.9), which are readily obtained by application of
+Proposition 4.7.
+Corollary 4.8 (Accuracy at i+1
+2of Lagrange polynomial reconstruction on Si,M−,M+).Letpf(x;Si,M−,M+,∆x)
+andph(x;Si,M−,M+,∆x)be a polynomial (Lemma 3.1) reconstruction pair (Definition 2.1) ph=R(1;∆x)(pf),
+satisfying the conditions of Proposition 4.5 . Then, the reconstructed value at xi+1
+2:=xi+1
+2∆x, which will
+be noted ˆhSi,M−,M+,i+1
+2, approximates hi+1
+2:=h(xi+1
+2)toO(∆xM+1)withM:=M−+M+≥0. The error of the ap-
+proximation can be expanded in powers of ∆xwith coefficients involving the derivatives h(m)
+i+1
+2:=h(m)(xi+1
+2)
+ˆhSi,M−,M+,i+1
+2:=ph(xi+1
+2;Si,M−,M+,∆x) (58a)
+=M+/summationdisplay
+ℓ=−M−aM−,M+,ℓfi+ℓ (58b)
+=hi+1
+2+NTJ/summationdisplay
+s=M+1ΛM−,M+,s∆xsh(s)
+i+1
+2+O(∆xNTJ+1)
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipupright
+O(∆xM+1)(58c)
+7Approximating h(ζ)(which was assumed to be of class CNin Lemma 2.2) in (6a) by the corresponding Taylor-polynomia l
+(Taylor-jet) of order NTJ[27, pp. 219–232] around ζ=x+ξ∆xyields,∀NTJ∈N:NTJ<N,
+f(x)=1
+∆x/integraldisplayx+1
+2∆x
+x−1
+2∆xNTJ/summationdisplay
+ℓ=0(ζ−x−ξ∆x)ℓ
+ℓ!h(ℓ)(x+ξ∆x)+O/parenleftig
+(ζ−x−ξ∆x)NTJ+1/parenrightigdζ
+=1
+∆x/integraldisplayx+1
+2∆x
+x−1
+2∆xNTJ/summationdisplay
+ℓ=0(ζ−x−ξ∆x)ℓ
+ℓ!h(ℓ)(x+ξ∆x)dζ+O(∆xNTJ+1)=1
+∆xNTJ/summationdisplay
+ℓ=0/integraldisplay(+1
+2−ξ)∆x
+(−1
+2−ξ)∆xηℓ
+ℓ!dηh(ℓ)(x+ξ∆x)+O(∆xNTJ+1)
+=NTJ/summationdisplay
+ℓ=0/parenleftigg(−1)ℓ+1
+(ℓ+1)!/parenleftig
+(ξ−1
+2)ℓ+1−(ξ+1
+2)ℓ+1/parenrightig
+∆xℓh(ℓ)(x+ξ∆x)/parenrightigg
++O(∆xNTJ+1)
+8/parenleftig
+(ξ−1
+2)ℓ+1−(ξ+1
+2)ℓ+1/parenrightig
+=ℓ+1/summationdisplay
+k=0/parenleftigg/parenleftiggℓ+1
+k/parenrightigg
+ξℓ+1−k/parenleftig
+−1
+2/parenrightigk/parenrightigg
+−ℓ+1/summationdisplay
+k=0/parenleftigg/parenleftiggℓ+1
+k/parenrightigg
+ξℓ+1−k/parenleftig
++1
+2/parenrightigk/parenrightigg
+=ξℓ+1/parenleftbigg/parenleftig
+−1
+2/parenrightig0−/parenleftig
++1
+2/parenrightig0/parenrightbigg
++ℓ+1/summationdisplay
+k=1/parenleftigg/parenleftiggℓ+1
+k/parenrightigg
+ξℓ+1−k/parenleftbigg/parenleftig
+−1
+2/parenrightigk−/parenleftig
++1
+2/parenrightigk/parenrightbigg/parenrightigg
+22where the constants ΛM−,M+,sare given by
+ΛM−,M+,s:=λh,M−,M+,s(1
+2)=s−M−1/summationdisplay
+ℓ=0(−1)ℓ
+(ℓ+1)!µh,M−,M+,s−ℓ(1
+2) (58d)
+withλh,M−,M+,s(ξ)being the degree sinξpolynomial defined by (56c) ,µh,M−,M+,s(ξ)being the degree sinξ
+polynomial defined by (51f) , and9
+aM−,M+,ℓ:=αh,M−,M+,ℓ(1
+2) (58e)
+withαh,M−,M+,ℓ(ξ)being the degree Minξpolynomial defined by (45g) .
+PROOF . Using (45d) and (56a), in the definition of ˆhSi,M−,M+,i+1
+2(58a), we have immediately
+ˆhSi,M−,M+,i+1
+2:=ph(xi+1
+2;Si,M−,M+,∆x)=M+/summationdisplay
+ℓ=−M−αh,M−,M+,ℓ(1
+2)fi+ℓ
+=hi+1
+2+NTJ/summationdisplay
+s=M+1λh,M−,M+,s(1
+2)∆xsh(s)
+i+1
+2+O(∆xNTJ+1)
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipupright
+Eh(xi+1
+2∆x;Si,M−,M+,∆x)(59)
+and using the definition (56c) to compute λh,M−,M+,s(1
+2)completes the proof. /square
+Corollary 4.9 (Order-of-accuracy of Lagrange polynomial r econstruction). Assume the condi-
+tions of Proposition 4.5 . Then
+ˆhSi,M−,M+,i+1
+2−ˆhSi−1,M−,M+,i−1
+2
+∆x=f′
+i+NTJ/summationdisplay
+n=M+1ΛM−,M+,n∆xnf(n+1)
+i+O(∆xNTJ+1)=f′
+i+O(∆xM+1) (60)
+whereˆhSi−1,M−,M+,i−1
+2=ˆhSi−1,M−,M+,i−1+1
+2(58a) , and the constants ΛM−,M+,nare defined by (58d) .
+PROOF . The constants ΛM−,M+,n(58d) depend only on the 3 indices ( M−,M+,n), and not on the point
+indexi(Remark 4.2), because the polynomials µh,M−,M+,s−ℓ(ξ)(51f) are also independent of the point
+indexi. Hence, we have, by (58c),
+ˆhSi−1,M−,M+,i−1
+2=ˆhSi−1,M−,M+,i−1+1
+2=hi−1
+2+NTJ/summationdisplay
+s=M+1ΛM−,M+,s∆xsh(s)
+i−1
+2+O(∆xNTJ+1) (61)
+Subtracting (61) from (58c) yields
+ˆhSi,M−,M+,i+1
+2−ˆhSi−1,M−,M+,i−1
+2
+∆x=hi+1
+2−hi−1
+2
+∆x+NTJ/summationdisplay
+s=M+1ΛM−,M+,s∆xsh(s)
+i+1
+2−h(s)
+i−1
+2
+∆x+O(∆xNTJ+1) (62)
+and using the exact relations (8) we obtain (60). /square
+9Notice that Shu [3], following a different route, has shown t hat
+aM−,M+,ℓ=M+1/summationdisplay
+m=ℓ+M−+1M+1/summationdisplay
+p=0
+p/nequalmM+1/productdisplay
+q=0
+q/nequalm
+q/nequalp(M−−q+1)
+M+1/productdisplay
+p=0
+p/nequalm(m−p)
+is an equivalent expression for the coefficients aM−,M+,ℓ(58e).
+23Remark 4.10 (Order-of-accuracy). The previous result (Corollary 4.9) illustrates that the O(∆xM+1)
+accuracy in approximating f′is achieved, using O(∆xM+1)interpolates for f, because of the exact recon-
+struction relations (Lemma 2.2). Liu et al. [5] note this as a nO(∆xM)accuracy increased to O(∆xM+1)
+at one chosen point, vizxi. /square
+5. Interpolating and reconstructing polynomial
+We briefly summarize how the existence and uniqueness proper ties of the interpolating polynomial
+carry on to the reconstructing polynomial. Consider first th e general case of a polynomial reconstruc-
+tion pair ( §3.1). Combining the existence (Lemma 3.1) and uniqueness (R emark 2.8) of polynomial
+reconstruction pairs, we can formulate
+Theorem 5.1 (Vector spaces of polynomial reconstruction pa irs). Consider the (M+1)-dimensional
+vector space of polynomials with real coefficients of degree ≤Minx,RM[x]. Then the reconstruction
+mapping R(1;∆x)(Definition 2.1) is a bijection of RM[x]onto itself.
+PROOF . By construction (Lemma 3.1) ∀p(x)∈RM[x]∃q(x)=[R(1;∆x)(p)](x)∈RM[x], and inversely
+∀q(x)∈RM[x]∃p(x)=[R−1
+(1;∆x)(q)](x)∈RM[x]. Furthermore, since the elements of RM[x]are continu-
+ous functions, the reconstruction pair q(x)=[R(1;∆x)(p)](x)is unique (Remark 2.8), which completes the
+proof. /square
+In his recent review of WENO schemes, Shu [4] stresses the difference between WENO interpola-
+tion and WENO reconstruction. In this sense, pf(x;Si,M−,M+,∆x)in Proposition 4.5 is the interpolating
+polynomial of f(x)onSi,M−,M+, andph(x;Si,M−,M+,∆x)is the reconstructing polynomial (Definition 2.3). Of
+course
+Proposition 5.2 (Lagrange reconstructing polynomial). Assume the conditions of Proposition 4.6 .
+The Lagrange reconstructing polynomial ph(x;Si,M−,M+,∆x)approximates h(x)toO(∆xM+1)but, unless f(x)
+is a polynomial of degree ≤M, it does not interpolate h(x)onSi,M−,M+,ie, iff(x)is not a polynomial of
+degree≤M, we have in general
+ph(xi+ℓ∆x;Si,M−,M+,∆x)/nequalh(xi+ℓ∆x)∀ℓ∈{−M−,···,M+} (63)
+PROOF . Proof is obtained by contradiction. It suffices to give an ex ample where the inequalities (63)
+hold. Consider the reconstruction pair (Theorem 2.9)
+f(x) :=ex−xi;h(x)=[R(1;∆x)(f)](x)=gτ(∆x)ex−xi(64a)
+withgτdefined by (19b). Consider the polynomial reconstruction of f(x)(Proposition 4.5) on Si,1,1. By
+(45d) and (45g)
+ph(xi+ξ∆x;Si,1,1,∆x)=fi−1/parenleftig1
+2ξ2−1
+2ξ−1
+24/parenrightig
++fi/parenleftig13
+12−ξ2/parenrightig
++fi−1/parenleftig1
+2ξ2+1
+2ξ−1
+24/parenrightig
+(64b)
+We have fi=1andfi±1=e±∆x, and evaluating ph(xi+ℓ∆x;Si,1,1,∆x)−h(xi+ℓ∆x), using (64b) and (64a),
+forℓ=−1,0,1, and for different values of ∆x(eg∆x=1
+100), we verify (63). /square
+Most of the results of existence and uniqueness properties o f the interpolating polynomial hold,
+with appropriate adjustments, for the reconstructing poly nomial, because of Theorem 5.1. We briefly
+summarize in the following those necessary to prove WENO reconstruction relations [3, 4].
+Theorem 5.3 (Existence and uniqueness of the Lagrange recon structing polynomial). Assume
+the conditions of Proposition 4.6 . There exists a unique Lagrange reconstructing polynomial ph(x;Si,M−,M+,∆x)
+of the form (45d) which approximates h(x)toO(∆xM+1).
+PROOF . Existence, with αh,M−,M+,ℓ(ξ)given by (45g), is proved in Proposition 4.5 by construction . We
+know from approximation theory [20, 21] that there is a uniqu e Lagrange interpolating polynomial
+pf(x;Si,M−,M+,∆x)onSi,M−,M+, and that the reconstruction pair ph(x;Si,M−,M+,∆x)=[R(1;∆x)(pf)](x;Si,M−,M+,∆x)
+is unique (Remark 2.8), which completes the proof. /square
+246. Examples of applications
+The analytical relations developed in the present work can p rove quite useful in the analysis of
+practical WENO schemes, and more generally in the development of discretiz ation schemes. Providing
+detailed analysis of such applications is beyond the scope o f the present paper. We sketch, nonetheless,
+in the following, 3 applications (the complete proofs will b e given elsewhere), to illustrate the useful-
+ness of the reconstruction pair concept, of the associated a pplication of the deconvolution Lemma 2.5,
+and of the explicit expressions for the Lagrange reconstruc ting polynomial.
+6.1. Representation of the Lagrange reconstructing polyno mial by combination of substencils
+All WENO [3, 4] schemes for reconstruction on the general homogeneou s stencil Si,M−,M+(Defini-
+tion 4.1) are based on the weighted combination of the recons tructions on Ks+1≤M:=M−+M+
+substencils10
+Si,M−−ks,M+−Ks+ks={i−M−+ks,···,M+−Ks+ks}M±∈Z:M=M−+M+≥2
+1≤Ks≤M−1
+ks∈{0,···,Ks}(65)
+with appropriate weights, which are nonlinear in the cell-a verages f(x), to ensure monotonicity at
+discontinuities [35], and such that the weighted combinati on of the Lagrange reconstructing polyno-
+mials on the substencils, at regions where h(x)is smooth, approximates to O(∆xM+1)[6] or higher [13]
+the Lagrange reconstructing polynomial on the big stencil Si,M−,M+, which (Proposition 4.6) is O(∆xM+1)-
+accurate. The starting point for scheme design is the determ ination of the underlying linear scheme,
+iethe determination of weight-functions σh,M−,M+,Ks,ks(ξ)(ks∈{0,···,Ks}), independent of f(x), which
+combine the Lagrange reconstructing polynomials on the sub stencils exactly into the Lagrange recon-
+structing polynomial on the big stencil Si,M−,M+
+ph(xi+ξ∆x;Si,M−,M+,∆x)=Ks/summationdisplay
+ks=0σh,M−,M+,Ks,ks(ξ)ph(xi+ξ∆x;Si,M−−ks,M+−Ks+ks,∆x) (66a)
+Obviously, using (56a) in (66a), the weight functions σh,M−,M+,Ks,ks(ξ)must satisfy the consistency condi-
+tion
+Ks/summationdisplay
+ks=0σh,M−,M+,Ks,ks(ξ)=1 (66b)
+Shu [3] indicated examples of instances where it was impossi ble to find such weights, as well as in-
+stances where convexity of the combination was lost (presen ce of negative weights). The corresponding
+problem for the Lagrange interpolating polynomial
+pf(xi+ξ∆x;Si,M−,M+,∆x)=Ks/summationdisplay
+ks=0σf,M−,M+,Ks,ks(ξ)pf(xi+ξ∆x;Si,M−−ks,M+−Ks+ks,∆x) (67a)
+Ks/summationdisplay
+ks=0σf,M−,M+,Ks,ks(ξ)=1 (67b)
+is directly related to the Neville-Aitken algorithm [21, pp . 11–13], which constructs the interpolat-
+ing polynomial on {i−M−,···,i+M+}by recursive combination of the interpolating polynomials on
+10Notice that the family of subdivisions (65) includes (when v aryingKs) all possible subdivisions to substencils of equal length
+(M−Ksintervals) whose union is the entire stencil Si,M−,M+.
+25substencils, with weight-functions which are also polynom ials ofx[21, pp. 11–13]. Carlini et al. [36]
+have given the explicit representation of the polynomial we ights for the construction of the Lagrange
+interpolating polynomial on Si,M−,M+, by combination of the Lagrange interpolating polynomials on the
+Ks+1substencils, for a certain family of stencils/subdivision s. Liu et al. [19] have extended the family
+of stencils/subdivisions studied. Since for the Lagrange i nterpolating polynomial case the weight-
+functions are polynomials, (67) are valid ∀ξ∈R.
+The corresponding problem for the Lagrange reconstructing polynomial (66) was studied, only very
+recently, by Liu et al. [19]. It turns out that, for the recons truction case, the weight-functions which
+satisfy (66) are rational functions of ξ, implying that (66) is valid ∀ξ∈R\Sσh,M−,M+,Ks,ieeverywhere
+except at the union Sσh,M−,M+,Ksof the poles of the Ks+1rational functions σh,M−,M+,Ks,ks(ξ)(ks∈{0,···,Ks}).
+Liu et al. [19] studied the family of stencils Si,⌊M
+2⌋,M−⌊M
+2⌋, for the Ks=/ceilingleftigM
+2/ceilingrightig
+-level subdivision, in the range
+M∈{2···,11}, and used symbolic computation to give explicit expression s of the weight-functions, and
+to study their poles and regions of convexity.
+We highlight in the following how the identification of recon struction pairs (Definition 2.1), and the
+analytical expressions for the Lagrange reconstructing po lynomial (Proposition 4.5) and its approxi-
+mation error (Proposition 4.7), can be used to develop gener al analytical expressions (valid ∀M±∈Z:
+M:=M−+M+≥2and∀Ks∈{1,···,M−1}) for the weight-functions, prove that there can be no poles
+at cell-interfaces ( n+1
+2∀n∈Z), and extend the important results obtained in Liu et al. [19 ]. It is
+quite straightforward, using the definition of the Lagrange reconstructing polynomial (Definition 2.3),
+to show by (6a) that the polynomial αh,M−,M+,ℓ(ξ)(45g) appearing in the representation (45d) of the La-
+grange reconstructing polynomial on the stencil Si,M−,M+is the reconstruction pair, on a unit-spacing
+grid, of the corresponding polynomial αf,M−,M+,ℓ(ξ)(45h) appearing in the representation (45e) of the
+Lagrange interpolating polynomial on the same stencil
+αh,M−,M+,ℓ(ξ)=/bracketleftig
+R(1;1)(αf,M−,M+,ℓ)/bracketrightig
+(ξ)⇐⇒αf,M−,M+,ℓ(ξ)=/integraldisplayξ+1
+2
+ξ−1
+2αh,M−,M+,ℓ(η)dη/braceleftigg∀ℓ∈{−M−,···,M+}
+∀ξ∈R(68)
+It is easy to prove by (68), using the mean value theorem for th e definite integral [27, pp. 350–359],
+and the knowledge of the Mroots ofαf,M−,M+,ℓ(ξ)(αf,M−,M+,ℓ(n)=0∀n∈{M−,···,M+}\{ℓ}[20, 21]) that
+αh,M−,M+,ℓ(n+1
+2)/nequal0/braceleftigg∀ℓ∈{−M−,···,M+}
+∀n∈Z(69)
+and that all of the Mroots ofαh,M−,M+,ℓ(ξ)are real. It can be shown that both/braceleftbigαh,M−,M+,ℓ(ξ),ℓ∈{−M−,···,M+}/bracerightbig
+and/braceleftig
+αf,M−,M+,ℓ(ξ),ℓ∈{−M−,···,M+}/bracerightig
+form a basis of the (M+1)-dimensional vector space of polynomials
+of degree≤Minξ,RM[ξ], and therefore, none of these polynomials is identically 0. We can work out
+several identities for the polynomials αh,M−,M+,ℓ(ξ), with corresponding identities for αf,M−,M+,ℓ(ξ)because
+of (68), and show that an analytical expression for the ratio nal weight-functions σh,M−,M+,Ks,ks(ξ)is given
+by the recurrence11
+σh,M−,M+,Ks,ks(ξ)=αh,M−,M+,−M−+ksM(ξ)
+αh,M−−ks,M+−1+ks,−M−+ksM(ξ)Ks=1
+min(Ks−1,ks)/summationdisplay
+ℓs=max(0,ks−1)σh,M−,M+,Ks−1,ℓs(ξ)σh,M−−ℓs,M+−(Ks−1)+ℓs,1,ks−ℓs(ξ)Ks≥2
+∀ks=0,···,Ks≤M−1 (70)
+This analytical formulation, which only requires (45g) as i nput, is easily programmed in any symbolic
+computation package, and can generate the rational weight- functions∀M±∈Z:M=M−+M+≥2. Since
+11The recurrence relation (70) for Ks≥2holds also forσf,M−,M+,Ks,ks(ξ).
+26all of the Mroots of any of the polynomials αh,M−,M+,ℓ(ξ)are real, by (70), we can show by induction that
+all of the poles of the rational weight-functions σh,M−,M+,Ks,ks(ξ)are real. These analytical results can
+then be used to extend the results of Liu et al. [19] ∀M∈N≥2, and for arbitrarily biased stencils in
+a homogeneous grid, providing at the same time simple symbol ic computation routines for roots and
+poles.
+Going into further details and results is beyond the scope of the present work. We include however
+the following result. For the Ks=/ceilingleftigM
+2/ceilingrightig
+-level subdivision of the usual WENO stencils [19] Si,⌊M
+2⌋,M−⌊M
+2⌋,
+we know from direct computation [19] that the weight-functi onsσh,⌊M
+2⌋,M−⌊M
+2⌋,⌈M
+2⌉,ks(1
+2)≥0. Using the
+analytical expression (70) we have obtained computational ly the following result
+Result 6.1 (Positivity of linear weights at i+1
+2).Assume that|M±|≤9, satisfying (65). Then if for
+the subdivision level KsofSi,M−,M+(65)all substencils contain either point ior point i+1
+Si,M−−ks,M+−Ks+ks∩{i,i+1}/nequal∅∀ks∈{0,···,Ks} ⇐⇒/braceleftigg−M−≤0<M+
+Ks≤min(M−+1,M+)>0(71a)
+then the rational weight-functions (70)satisfy
+σh,M−,M+,Ks,ks(1
+2)>0∀ks∈{0,···,Ks} (71b)
+/square
+6.2. Truncation error of WENO approximations to f′(x)
+Nonlinearity, ensuring monotonicity [35], in WENO schemes is introduced by nonlinear weighting-
+out of stencils for which the reconstructing polynomial is n onsmooth. Smoothness is almost invari-
+ably [6, 3, 10, 11, 12, 16, 13, 14, 4, 15] measured using the Jia ng-Shu smoothness indicators [6]. Let
+u:R−→R,M∈N≥1and∆x∈R>0, and define
+βM(x;∆x;u) :=M/summationdisplay
+k=1/integraldisplayx+1
+2
+x−1
+2∆x∆x2k−1/parenleftiggdk
+dxku(ζ)/parenrightigg2
+dζ (72a)
+=M/summationdisplay
+k=1/integraldisplay1
+2
+−1
+2/parenleftiggdk
+dξku(x+ξ∆x)/parenrightigg2
+dξ (72b)
+By (72b) it is seen that, upon normalization by ∆x,βM(x;∆x;u)is [37] the usual norm of/parenleftig
+dξu/parenrightig
+in the
+Sobolev space HM−1/parenleftig
+(−1
+2,+1
+2)/parenrightig
+:=WM−1,2/parenleftig
+(−1
+2,+1
+2)/parenrightig
+. The Jiang-Shu smoothness indicator, when consider-
+ing reconstruction at xi+1
+2onSi,M−,M+(Corollary 4.8), is defined12[6, 3, 11, 4] by
+βph,Si,M−,M+,i+1
+2:=βM/parenleftig
+xi;∆x;ph(·;Si,M−,M+,∆x)/parenrightig
+(72c)
+with, as usual, M:=M−+M+. The realization of the optimal order-of-accuracy by the WENO schemes
+hinges upon the fact [6, 13, 14, 4] that βph,Si,M−,M+,i+1
+2for different stencils of equal length M:=M−+
+M+but different biasing around the pivot-point xionly differ to O(∆xM+2), the lower-order part being
+common to all stencils of equal length M. Expansions in powers of ∆xsfn
+ifs−n
+i(n≤ ⌊s
+2⌋) have been
+12The interval of integration was defined by Jiang and Shu [6] as the cell[xi−1
+2,xi+1
+2], with center the pivot point xiwhere we
+wish to approximate the derivative f′
+i:=f′(xi). This introduces some sort of upwinding, since the interval [xi,xi+1]could have
+been used (this would correspond to using βM(xi+1
+2;∆x;ph(·;Si,M−,M+,∆x))instead in (72c). This is the choice made in the context
+of central WENO interpolation by Carlini et al. [36]. The choice of [xi,xi+1]as the integration interval for upwind-biased WENO
+schemes has not been studied. Nonetheless, in the context of cell-centered finite-volume schemes [38] the choice of the c ell as
+the volume of integration seems natural.
+27given in the literature, up to the WENO17(composed by 9 substencils of length M=8cells [15]), using
+symbolic calculation [6, 13, 14, 15]. Using the analytical e xpressions for the error of the reconstructing
+polynomial with respect to the unknown function h(x)which is reconstructed, eg(56c), (51c) or (51d),
+it is quite straightforward to explain the existence of the c ommon part. Using (56a) in (72c) yields, by
+(72b)
+βph,Si,M−,M+,i+1
+2=M/summationdisplay
+k=1/integraldisplay1
+2
+−1
+2/parenleftiggdk
+dξk/parenleftbigg
+h(xi+ξ∆x)+Eh(xi+ξ∆x;Si,M−,M+,∆x)/parenrightbigg/parenrightigg2
+dξ
+=M/summationdisplay
+k=1/integraldisplay1
+2
+−1
+2/parenleftiggdk
+dξk/parenleftbigg
+h(xi+ξ∆x)/parenrightbigg/parenrightigg2
+dξ
++M/summationdisplay
+k=1/integraldisplay1
+2
+−1
+22/parenleftiggdk
+dξkh(xi+ξ∆x)/parenrightigg/parenleftiggdk
+dξkEh(xi+ξ∆x;Si,M−,M+,∆x)/parenrightigg
++/parenleftiggdk
+dξkEh(xi+ξ∆x;Si,M−,M+,∆x)/parenrightigg2dξ
+(73)
+iethe common (stencil-independent) part is indeed the Sobole v norm of dξh(xi+ξ∆x)and the non-
+common part involves the approximation error of the Lagrang e reconstructing polynomial (Proposi-
+tion 4.7), and is therefore stencil-dependent. The deconvo lution Lemma 2.5 is necessary to compute
+analytically the expansion of the common part in terms of pow ers∆xsfn
+ifs−n
+i(n≤⌊s
+2⌋), and (51c) com-
+bined with the deconvolution Lemma 2.5 is used to evaluate an alytically the expansion of the stencil-
+dependent part of (73), whose knowledge is essential for the evaluation and improvement [6, 13, 14] of
+the design of nonlinear weights, especially when intereste d in developing weights maintaining one of
+the great advantages of the Jiang-Shu weights, vizthe straightforward extension to arbitrarily high-
+order accuracy [11, 15]. Furthermore these expressions wer e used to compute analytically the leading
+2 terms of the asymptotic expansions of the Jiang-Shu nonlin ear weights [6] and from these the lead-
+ing term of the truncation error of WENO and WENOM [13] schemes. These developments are quite
+lengthy, and will be reported elsewhere.
+6.3. Extension to higher derivatives by multiple reconstru ction
+The reconstruction approach can also be used to approximate f′′(x)(and in general f(n)(x)), and in
+particular to compute interface fluxes, for high-order cons ervative discretization of diffusive terms in
+finite-volume methods, egin the direct numerical simulation ( DNS) of turbulence [39].
+Assume the conditions of Definition 2.1, for f(x)andh(x), but defined on I=[a−∆x,b+∆x]⊂R, and
+assume that∃h:I−→Rwhich satisfies
+h=R(1;∆x)(h)(6a)=⇒h(x)=1
+∆x/integraldisplayx+1
+2∆x
+x−1
+2∆xh(ζ)dζ ∀x∈[a−1
+2∆x,b+1
+2∆x] (74a)
+(8)=⇒h(n)(x)=h(n−1)(x+1
+2∆x)−h(n−1)(x−1
+2∆x)
+∆x∀x∈[a−1
+2∆x,b+1
+2∆x]
+∀n∈{1,···,N}(74b)
+assuming f,h,h∈CN[a−∆x,b+∆x], withN∈N≥2. Recall (Remark 2.8) that reconstruction pairs of
+continuous functions, if they exist, are unique. Combining (6a) with (74a), we have
+f(x)=[R−1
+(1;∆x)(h)](x)(74a)=[R−1
+(1;∆x)◦R−1
+(1;∆x)/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipupright
+R−1
+(2;∆x)(h)](x)=1
+∆x2/integraldisplayx+1
+2∆x
+x−1
+2∆x/integraldisplayη+1
+2∆x
+η−1
+2∆xh(ζ)dζdη (75)
+and we may write h=R(2;∆x)(f)the reconstruction pair of f(x)for the computation of the 2-derivative.
+28Differentiating (75) twice with respect to x, we readily obtain the exact relation
+f′′(x)=h(x+∆x)−2h(x)+h(x−∆x)
+∆x2(76a)
+=1
+∆x/bracketleftiggh(x+∆x)−h(x)
+∆x/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipupright
+(74b)=h′(x+1
+2∆x)−h(x)−h(x−∆x)
+∆x/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehtipupright
+(74b)=h′(x−1
+2∆x)/bracketrightigg
+(76b)
+By successive application of the deconvolution Lemma 2.5, a ssuming the conditions of Lemma 2.5, we
+can obtain the deconvolution relations for h(x)
+f(n)(x)=⌊NTJ
+2⌋/summationdisplay
+ℓ=0ℓ/summationdisplay
+s=0/parenleftigg2ℓ+2
+2s+1/parenrightigg∆x2ℓ
+22ℓ(2ℓ+2)!h(n+2ℓ)(x)+O(∆x2⌊NTJ
+2⌋+2)∀x∈[a,b]
+∀n∈N0:n<N−2⌊NTJ
+2⌋(77a)
+h(n)(x)=⌊NTJ
+2⌋/summationdisplay
+ℓ=0ℓ/summationdisplay
+s=0τ2sτ2ℓ−2s∆x2ℓf(n+2ℓ)(x)+O(∆x2⌊NTJ
+2⌋+2)∀x∈[a,b]
+∀n∈N0:n<N−2⌊NTJ
+2⌋(77b)
+and redevelop all the results concerning R(1;∆x)forR(2;∆x), and in general for reconstruction procedures
+determining the interface fluxes for the computation of f′′(x). In an analogous manner we can tackle
+the important practical problem of very-high-order conser vative discretizations of (fA(x)f′
+B(x))′[40].
+7. Conclusions
+The results in this paper concern both the general relations between two functions constituting a
+reconstruction pair (Definition 2.1), and the analysis of th e approximation error of the reconstructing
+polynomial (Definition 2.3).
+We call a function h(x)whose sliding averages over a constant length ∆xare equal to f(x)the
+reconstruction pair of f(x),h=R(1;∆x)(f)(Definition 2.1). The exact relations ∆xf(n)(x)=h(n−1)(x+1
+2∆x)−
+h(n−1)(x−1
+2∆x)(8) are the basis of the numerical approximation of f′(x)by reconstruction [7, 2, 8, 9].
+The reconstruction pair of the exponential function is [R(1;∆x)(exp)](x)=gτ(∆x)ex(Theorem 2.9). The
+function gτ(x)(19b) is the generating function of the numbers τn(Tab. 1) satisfying recurrence (10c).
+The numbersτn(19c) define the coefficients of the analytical solution (Lem ma 2.5) of the deconvolution
+problem for Taylor polynomials [2, (3.13), pp. 244–246]. Th is analytical solution (Lemma 2.5) is one of
+the main results of this work. It was also obtained by an alter native matrix-algebra oriented approach
+(§3.2).
+The reconstruction pair of a polynomial of degree M∈Nis also a polynomial of degree M(Lemma 3.1),
+whose coefficients can be explicitly determined by (26f) usi ng the numbers τn(Tab. 1), R(1;∆x)being a
+bijection of the vector space of polynomials of degree ≤M∈Nonto itself (Theorem 5.1).
+The Lagrange reconstructing polynomial (Definition 2.3) on an arbitrary stencil Si,M−,M+:={i−M−,···,i+M+},
+on a homogeneous grid, in the neighbourhood of point i(Definition 4.1), is the reconstruction pair of
+the Lagrange interpolating polynomial on Si,M−,M+:={i−M−,···,i+M+}. The Lagrange reconstructing
+polynomial on Si,M−,M+is of degree M:=M−+M+(Proposition 4.5) and approximates h(x)toO(∆xM+1)
+(Proposition 4.7). The complete expansion (56a) of the appr oximation error of the Lagrange recon-
+structing polynomial in terms of powers of ∆xcan be expressed using the polynomials λh,M−,M+,n(ξ)
+defined by (56c). Most of the standard results of existence an d uniqueness of the interpolating polyno-
+mial apply to the reconstructing polynomial (Theorem 5.3).
+Typical applications include the analytical expression an d results on the roots and poles of the ra-
+tional weight-functions combining the Lagrange reconstru cting polynomials on substencils of Si,M−,M+:=
+{i−M−,···,i+M+}into the Lagrange reconstructing polynomial on the entire s tencil, the analytical
+29expression of the Taylor expansions of the Jiang-Shu [6] smo othness indicators and of the truncation
+error of WENO schemes, and the analysis of the discretization of f(n)(x)byn-reconstruction. It is hoped
+that the theoretical relations on reconstruction pairs and the analytical expressions of the approxima-
+tion error of the reconstructing polynomial will be useful i n the analysis and improvement of practical
+discretization schemes.
+Appendix A. Useful relations for summation indices
+We summarize here several relations [22, 23] used in the text to manipulate the limits of summa-
+tion indices, and some other useful formulas.
+α≤n⇐⇒ ⌈α⌉≤n
+α<n⇐⇒ ⌊α⌋<n
+n<β⇐⇒n<⌈β⌉
+n≤β⇐⇒n≤⌊β⌋∀α,β∈R
+∀n∈Z(A.1)
+s≤2k⇐⇒ ⌈s
+2⌉≤k
+s<2k⇐⇒ ⌊s
+2⌋<k
+2k<s⇐⇒k<⌈s
+2⌉
+2k≤s⇐⇒k≤⌊s
+2⌋∀s,k∈Z (A.2)
+Nmax/summationdisplay
+n=NminMmax/summationdisplay
+m=Mminanm=Nmax+Mmax/summationdisplay
+s=Nmin+Mminmin(Nmax,s−Mmin) /summationdisplay
+n=max(Nmin,s−Mmax)an,s−n=Nmax+Mmax/summationdisplay
+s=Nmin+Mminmin(Mmax,s−Nmin) /summationdisplay
+m=max(Mmin,s−Nmax)as−m,m (A.3)
+1
+ℓ+2k+1/parenleftiggℓ+2k+1
+ℓ/parenrightigg
+=1
+2k+1/parenleftiggℓ+2k
+2k/parenrightigg
+(A.4)
+Acknowledgments
+The present work was partly supported by the EU-funded research project ProBand, ( STREP –
+FP6AST4–CT–2005–012222 ). Computations were performed using HPC resources from GENCI –IDRIS
+(Grants 2010–066327 and 2010–022139).
+Symbolic calculations were performed using maxima (http://sourceforge.net/projects/maxima ).
+The corresponding package reconstruction.mac is available at http://www.aerodynamics.fr .
+References
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+31
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diff --git a/1001.0510.txt b/1001.0510.txt
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+arXiv:1001.0510v1  [cond-mat.stat-mech]  4 Jan 2010Interplay among helical order, surface effects and range of i nteracting layers in
+ultrathin films.
+F. Cinti(1,2,3), A. Rettori(2,3), and A. Cuccoli(2)
+(1)Department of Physics, University of Alberta, Edmonton, Al berta, Canada T6G 2J1
+(2)CNISM and Department of Physics, University of Florence, 50 019 Sesto Fiorentino (FI), Italy. and
+(3)CNR-INFM S3National Research Center, I-41100 Modena, Italy
+(Dated: November 10, 2018)
+The properties of helical thin films have been thoroughly inv estigated by classical Monte Carlo
+simulations. The employed model assumes classical planar s pins in a body-centered tetragonal
+lattice, where the helical arrangement along the film growth direction has been modeled by nearest
+neighbor and next-nearest neighbor competing interaction s, the minimal requirement to get helical
+order. We obtain that, while the in-plane transition temper atures remain essentially unchangedwith
+respect to the bulk ones, the helical/fan arrangement is sta bilized at more and more low temperature
+when the film thickness, n, decreases; in the ordered phase, increasing the temperatu re, a softening
+of the helix pitch wave-vector is also observed. Moreover, w e show also that the simulation data
+around both transition temperatures lead us to exclude the p resence of a first order transition for all
+analyzed sizes. Finally, by comparing the results of the pre sent work with those obtained for other
+models previously adopted in literature, we can get a deeper insight about the entwined role played
+by the number (range) of interlayer interactions and surfac e effects in non-collinear thin films.
+PACS numbers: 64.60.an,64.60.De,75.10.Hk,75.40.Cx,75. 70.Ak.
+I. INTRODUCTION
+The study of low dimensional frustrated magnetic
+systems1still raises great interest, both in consequence
+of theoretical aspects, related to their peculiar criti-
+cal properties2, and in view of possible technological
+applications3. Indeed, beside conventional ferromagnetic
+or antiferromagnetic phase transitions, in many new ma-
+terials other nontrivial and unconventional forms of or-
+dering have been observed4,5. A quantity of particular
+interest in this context is the spin chirality, an order pa-
+rameter which turned out to be extremely relevant in,
+e.g., magnetoelectric materials6, itinerant MnSi7, binary
+compounds as FeGe8, glass transition of spins9, andXY
+helimagnets, as Holmium, Terbium or Dysprosium10. In
+the latter case, a new universality class was predicted be-
+cause aZ2×SO(2) symmetry is spontaneously broken
+in the ordered phase2: In fact, when dealing with such
+systems, in addition to the SO(2) symmetry of the spin
+degrees of freedom /vectorSi, one has to consider also the Z2
+symmetry of the spin chirality κij∝/bracketleftBig
+/vectorSi×/vectorSj/bracketrightBigz
+.
+For these rare-earth elements, the development of new
+and sophisticated experimental methods11has allowed to
+obtain ultra-thin films where the non-collinear modula-
+tion is comparable with the film thickness. Under such
+conditions the lack of translational invariance due to the
+presence of surfaces results decisive in order to observe
+a drastic change of the magnetic structures12. Recent
+experimental data on ultra-thin Holmium films13have
+been lately interpreted and discussed14,15on the basis
+of detailed classical Monte Carlo (MC) simulations of a
+spin Hamiltonian, which is believed to give a realistic
+modeling of bulk Holmium. Such Hamiltonian, proposed
+by Bohr et al.16, allows for competitive middle-range in-teractions by including six different exchange constants
+along the ccrystallographic axis, and gives a helix pitch
+wave-vector Qzsuch that Qzc′≃30◦, wherec′=c/2 is
+the distance between nearest neighboring spin layerspar-
+alleltothe abcrystallographicplanes,henceforthdenoted
+also asx−yplanes, while zwill be taken parallel to c.
+Forn >16,nbeing the number ofspin layersin the film,
+a correct bulk limit is reached, while for lower nthe film
+properties are clearly affected by the strong competition
+among the helical pitch and the surface effects, which in-
+volve the majority of the spin layers. In the thickness
+rangen= 9−16, i.e. right for thickness values com-
+parable with the helical pitch, three different magnetic
+phases emerged, with the high-temperature, disordered,
+paramagneticphaseandthe low-temperature, long-range
+ordered one separated by an intriguing, intermediate-
+temperature block phase, where outer ordered layers co-
+exist with some inner disordered ones, the phase tran-
+sition of the latter eventually displaying the signatures
+of a Kosterlitz-Thouless one. Finally, for n≤7 the film
+collapses once and for all to a quasi-collinear order.
+The complex phase diagram unveiled by such MC sim-
+ulations awaken however a further intriguing question:
+to what extent the observed behavior may be considered
+a simple consequence of the competition between helical
+order and surface effects? I.e., is it just a matter of hav-
+ing such a competition or does the range of interactions
+also play a relevant role? Indeed, when the range of the
+interactions is large enough we have a greater number of
+planes which can be thought of as ”surface planes”, i.e.
+for which the number of interacting neighbors are sig-
+nificantly reduced with respect to the bulk layers; there-
+fore, we expect that the larger the interaction range, the
+stronger should be the surface effects. But, at the same
+time, the same modulation of the magnetic order can2
+xz
+yJJ
+J1 
+02 
+FIG. 1: (colors online) (a): body-centered tetragonal (BCT)
+lattice with J0in-plane coupling constant, and out-of-plane
+J1, andJ2competing interactions.
+be achieved with different number of interacting layers:
+notably, nearest and next-nearest layers competitive in-
+teractions are enough to get a helical structure with a
+whatever pitch wavevector. Such observation gives us a
+possible wayto solvethe conundrum previouslyemerged,
+as we have the possibility of varying the range of inter-
+actions without modifying the helical pitch, thus decou-
+pling the two relevant length scales along the film growth
+direction, and making accessible a range of nof the or-
+der of, or smaller than, the helical pitch, but still large
+enough that a substantial number of layers can behave
+as “bulk” layers. Therefore, while in the previous papers
+wehavestudiedthepropertiesofultrathinmagneticfilms
+of Ho assuming a model with six interlayer exchange in-
+teractions, here we investigate by MC simulations the
+properties of the same system by making use of the sim-
+plest model Hamiltonian able to describe the onset of a
+helical magnetic order in Holmium, i.e. we consider only
+two inter-layer coupling constants, as previously done in
+Ref. 11.
+The paper is organized as follows: In Sec. II the model
+Hamiltonian will be defined, and the MC techniques, and
+all the thermodynamic quantities relevant for this study,
+will be introduced. In Sec. III the results obtained for
+differentthicknesseswill bepresented, bothin the matter
+ofthe criticalpropertiesofthe modelandofthe magnetic
+ordered structures observed. Finally, in Sec. IV we shall
+discuss such results, drawing also some conclusions.
+II. MODEL HAMILTONIAN AND MONTE
+CARLO OBSERVABLES
+ThemodelHamiltonianweuseinoursimulationsisthe
+minimal one able to describe helimagnetic structures:
+H=−
+J0/summationdisplay
+/angbracketleftij/angbracketright/vectorSi·/vectorSj+J1/summationdisplay
+/angbracketleftik/angbracketright/vectorSi·/vectorSk+J2/summationdisplay
+/angbracketleftil/angbracketright/vectorSi·/vectorSl
+.
+(1)/vectorSiare classical planar unit vectors representing the di-
+rection of the total angular momentum of the magnetic
+ions, whose magnitude/radicalbig
+j(j+1) (j= 8 for Holmium
+ions) is already encompassed within the definition of the
+interaction constants J0,1,2. As sketched in Fig. 1, the
+magnetic ions are located on the sites of a body-centered
+tetragonal (BCT) lattice; the first sum appearing in the
+Hamiltonian describes the in-plane ( xy) nearest neigh-
+bor(NN) interaction,whichistakenferromagnetic(FM),
+with exchange strength J0>0; the second sum rep-
+resents the coupling, of exchange strength J1, between
+spins belonging to nearest neighbor (NN) planes along
+thez-direction (which we will assume to coincide with
+the film growth direction); finally, the third sum takes
+into account the interaction, of exchange strength J2, be-
+tween spins lying on next-nearestneighbor (NNN) planes
+alongz. In order to have frustration, giving rise to non-
+collinear order along zin the bulk, NN interaction J1
+can be taken both ferro- or antiferromagnetic, but NNN
+coupling J2has necessarily to be antiferromagnetic, and
+the condition |J2|>|J1|/4 must be fulfilled. Such simpli-
+fied Hamiltonian was already employed to simulate he-
+lical ordering in bulk systems by Diep1,17and Loison18.
+In the bulk limit, the state of minimal energy of a sys-
+tem described by Eq.(1) correspondsto a helical arrange-
+ment of spins. The ground state energy per spin is equal
+toeg(Qz) = [−4J0−2J1(4cos(Qzc′)+δcos(2Qzc′))]
+wherec′is the distance between NN layers, δ=J2
+J1,
+andQzc′= arccos/parenleftbig
+−1
+δ/parenrightbig
+is the angle between spins ly-
+ing on adjacent planes along the z-direction. The ob-
+served helical arrangement in bulk holmium corresponds
+toQzc′≃30.5◦10: such value can be obtained from
+the formula above with the set of coupling constants
+J0=67.2K, J1=20.9K, and J2=−24.2K, that we have
+employedin our simulations. The given values for the ex-
+change constants are the same already used by Weschke
+et al.in Ref. 13 to interpret experimental data on
+Holmium films on the basis of a J1−J2model, after
+a proper scaling by the numbers of NN and NNN on
+neighboring layers of a BCT lattice.
+In the following we will denote with nthe film thick-
+ness, i.e. the number of spin layers along the zdirection,
+and with L×Lthe number of spins in each layer (i.e., L
+is the lattice size along both the xandydirections). In
+our simulations thickness values from 1 to 24 were con-
+sidered, while the range of lateral size Lwas from 8 to
+64. Periodic boundary conditions were applied along x
+andy, while free boundaries were obviously taken along
+the film growth direction z.
+Thermal equilibrium was attained by the usual
+Metropolis algorithm19, supplemented by the over-
+relaxed technique20in order to speed-up the sampling
+of the spin configuration space: a typical “Monte Carlo
+step” was composed by four Metropolis and four-five
+over-relaxed moves per particle. Such judicious mix of
+moves is able both to get faster the thermal equilibrium
+and to minimize the correlation “time” between succes-
+sive samples, i.e. the undesired effects due to lack of in-3
+dependence of different samples during the measurement
+stage. For each temperature we have usually performed
+three independent simulations, each one containing at
+least 2×105measurements, taken after discarding up to
+5×104Monte Carlosteps in orderto assurethermal equi-
+libration.
+In the proximity of the critical region the multiple his-
+togram (MH) technique was also employed21, as it allows
+us to estimate the physical observables of interest over a
+whole temperature range in a substantially continuous
+way by interpolating results obtained from sets of simu-
+lations performed at some different temperatures.
+For all the quantities of interest, the averagevalue and
+the error estimate were obtained by the bootstrap re-
+sampling method22given that, as pointed out in Ref. 23,
+for a large enough number of measurements, this method
+turns out to be more accurate than the usual blocking
+technique. In our implementation, we pick out randomly
+a sizable number of measurements (typically, between 1
+and 1×103for the single simulation, and between 1 and
+5×104fortheMHtechnique), anditeratethere-sampling
+at least one hundred times.
+The thermodynamic observables we have investigated
+include the FM order parameter for each plane l:
+ml=/radicalBig
+(mx
+l)2+(my
+l)2, (2)
+which is related to the SO(2) symmetry breaking. At the
+same time, it turns out to be significant also the average
+order parameter of the film, defined as
+M=1
+nn/summationdisplay
+l=1ml. (3)
+Turning to the helical order, which is the relevant
+quantity for the Z2×SO(2) symmetry, we can explore
+it along two different directions. The first one is by the
+introduction of the chirality order parameter1,2
+κ=1
+4(n−1)L2sinQz/summationdisplay
+/angbracketleftij/angbracketright/bracketleftbig
+Sx
+iSy
+j−Sy
+iSx
+j/bracketrightbig
+,(4)
+where the sum refers to spins belonging to NN layers
+iandj, respectively, while Qzis the bulk helical pitch
+vector along the zdirection. The second possibility is
+that of looking at the integral of the structure factor:
+MHM=1
+K/integraldisplayπ
+0dqzS(/vector q) (5)
+whereS(/vector q), with/vector q= (0,0,qz), is the structure factor24
+(i.e. the Fourier transform of the spin correlation func-
+tion) along the z-direction of the film, while the normal-
+ization factor Kis the structure factor integral at T= 0.
+Although the use of the last observable can be seen as a
+suitable and elegant way to overcome the intrinsic diffi-
+culties met in defining a correct helical order parameter,
+free of any undue external bias (as the wave-vector Qz0 20 40 60 80 100 120 140
+T (K)00.511.522.5cv / kB
+L = 24
+L = 32
+L = 48
+L = 6420 30 40 50 60 702.12.22.32.42.52.6
+cv,max
+L
+FIG. 2: (color online) Specific heat cvper spin vs. temper-
+ature for thickness n= 16 (for lateral dimension, see the
+legend inside the figure). Inset: Maximum of cvvs.Lob-
+tained through MH technique. The continuum red line is a
+power law fit.
+entering the definition of κin Eq. (4)), we remind that
+such quantity has generally to be managed with particu-
+lar care, as discussed in details in Refs.14,15, where it was
+shown that the presence of block structures prevents us
+to unambiguously relate the evolution of S(/vector q) with the
+onset of helical order. However, for the specific case of
+the model under investigation such integrated quantity
+can still be considered a fairly significant order parame-
+ter, as no block structures emerge from the simulations
+(see below).
+In order to get a clear picture of the critical region and
+to give an accurate estimate of the critical temperature,
+we look also at the following quantities
+cv=nL2β2/parenleftbig
+∝angbracketlefte2∝angbracketright−∝angbracketlefte∝angbracketright2/parenrightbig
+, (6)
+χo=nL2β/parenleftbig
+∝angbracketlefto2∝angbracketright−∝angbracketlefto∝angbracketright2/parenrightbig
+, (7)
+∂βo=nL2(∝angbracketleftoe∝angbracketright−∝angbracketlefto∝angbracketright∝angbracketlefte∝angbracketright), (8)
+u4(o) = 1−∝angbracketlefto4∝angbracketright
+3∝angbracketlefto2∝angbracketright2, (9)
+whereβ= 1/kBT, andois one of the relevant observ-
+ables, i.e. ml,M,κ,M HM. In this paper, we shall mainly
+locate the critical temperature by lookingat the intersec-
+tion of the graphs of the Binder cumulant25, Eq. (9), as a
+function of Tobtained at different L. For clarity reasons,
+we introduce also the following symbols: by TN(n) we
+will denote the helical/fan phase transition temperature
+for thickness n,TC(n) will instead indicate the order-
+ing temperature of the sample as deduced by looking at
+the behaviour of the average order parameter (3), while
+Tl
+C(n) will be the l-th plane transition temperature re-
+lated to the order parameter defined in Eq. (2).4u4 (M)
+0.620.640.66
+130 131 132 133 134 135136 137 138
+T (K)0.50.550.60.65u4 (MHM)
+0 0.2 0.4 0.60.8qzS(qz ) (a.u.)(a)
+(b)0.66
+0.64
+0.62
+FIG. 3: (color online) Binder cumulants at thickness n=
+16, colors as in Fig. 2. (a): Binder cumulant for the order
+parameter definedin Eq.(3). (b): Bindercumulant extracted
+from the integral of the structure factor (see Sec. II). Inse t:
+structure factor for L= 64 between T= 131 K (upper curve)
+andT= 140 K (lower), with 1 K temperature step.
+III. RESULTS
+The results obtained by MC simulations of the model
+introduced in Sec. II will be presented starting from
+n= 16, i.e. the highest investigated film thickness which
+still displays a bulk-like behaviour. In Fig. 2 the spe-
+cific heat for samples with n= 16 and lateral dimension
+L= 24,32,48,64 is shown. The location of the specific
+heat maximum shows a quite definite evolution toward
+the bulk transition temperature, THo
+N≃132K10(it is
+worthwhiletonote thatforthis XYmodel the meanfield
+theory predicts a critical temperature THo
+N,MF≃198K).
+The intensity of the maximum of cvhas been analyzed
+by the MH technique for the same lateral dimensions (see
+inset of Fig. 2): it clearly appears as it increases with L
+in a smooth way.
+The Binder cumulant for the average order parameter
+defined inEq.(3)wasobtainedclosetothe cvpeakand is
+reportedinFig.3a; itsanalysisleadstoanestimateofthe
+critical temperature of the sample (given by the location
+of the common crossing point of the different curves re-
+ported in the figure) of TC(16) = 133 .2(5) This value can
+be considered in a rather good agreementwith the exper-
+imental ordering temperature of Holmium THo
+N, the rel-
+ative difference being about 1%. Even such a mismatch
+between THo
+NandTC(16) could be completely eliminated
+by slightly adjusting the in-plane coupling constant J0,
+but, as discussed in Sec. II, we shall preserve the value
+reportedin Refs. 13, and 12in orderto allowfora correct
+comparison with the results reported in those papers.
+The development of the helical arrangement of magne-
+tization along the film growth direction was investigated
+by looking at the integral of the structure factor S(/vector q)
+along the z-direction, i.e. by taking /vector q= (0,0,qz), and
+making again use of the cumulant analysis in order to
+locate the helical transition temperature at TN(16) =20 40 6080 100 120 140
+T (K)χκ(a.u.)00.20.40.6κ
+00.511.52cv / kB
+20 40 6080 100 120 140
+T (K)0.10.20.30.40.50.6u4(κ)132 134 136
+T (K)∂βκ  (a.u.)(a)
+(c)(b)
+(d)
+FIG.4: (color online)Thermodynamic quantitiesobtainedf or
+thickness n= 8 in the temperature range 0-150K. Colors and
+symbols as in Fig. 2. (a): specific heat; (b): chirality order
+parameter. (c): susceptibility χκ.(d): Binder cumulant for
+κ.
+133.1(3)K (see Fig. 3b). The crossing points of the
+Binder’s cumulants of the helical order parameter imme-
+diately appear to be located, within the errorbars, at the
+same temperature of those for the averagemagnetization
+previously discussed. In addition, it is worthwhile to ob-
+serve that the peak evolution of S(0,0,qz), in particular
+close to TN(16) (inset of Fig. 3b), displays the typical
+behaviour expected for an helical structure. We can thus
+conclude that for n= 16, as it is commonly observed
+in bulk samples, the establishment of the in-plane order
+coincides with onset of the perpendicular helical arrange-
+ment at TN(16). However, due to helix distortion in the
+surface regions, the maximum of S(0,0,qz) stabilizes at
+values of qzsensibly smaller (e.g. Qz(TN(16))≈16◦,
+andQz(T= 10K)≈28◦) with respect to the bulk one
+(QHo
+z= 30.5◦).
+The MC simulations outcomes for n= 16 we just pre-
+sented appear quite different with respect to those ob-
+tained at the same thickness for the model with six cou-
+pling constants along the zdirection14,15. Indeed, for
+theJ1-J2model here investigated, we observe that all
+layers order at the same temperature, and we do not find
+any hint of the block-phase, with inner disordered planes
+intercalated to antiparallel quasi-FM four-layer blocks,
+previously observed; sample MC runs we made using the
+samehcplattice employed in Refs. 14,15 shows that the
+presence or absence of the block phase is not related to
+the lattice geometry, but it is a consequence of the inter-
+action range only.
+We now move to describe and discuss MC simulation
+data for thinner samples. A graphical synthesis of the
+results obtained for n= 8 in reported in Fig. 4a-d. The
+specific heat cv, shown in Figs. 4a, reveals very small
+finite-size effects, which, however, cannot be unambigu-
+ously detected for the largest lattice size ( L= 64), as
+they fall comfortably within the error range. Surpris-
+ingly, the specific heat maximum is located close to the
+bulk transition temperature as found for n= 16, and5
+0 2 4 6 8 10 12 14 16 18 20
+ n020406080100120140TN (n) , TC (n)  (K)
+TN (n)
+TC (n)
+TNbulk
+FIG. 5: Transition temperatures TN(n) andTC(n) vs. film
+thickness n.
+the same is true for the crossing point of the Binder cu-
+mulant of the average magnetization M(not reported in
+figure), which is located at TC(8) = 133 .3(3)K. These
+data give a first rough indication that also for n= 8 all
+the planes of the sample are still ordering almost at the
+same temperature; such property has been observed for
+all the investigated thicknesses nbelow 16, so that TC(n)
+results quite n-independent (see also Fig. 5) .
+Although the layer subtraction does not seem to mod-
+ifyTC(n), the onset of helical arrangement is observed to
+shift at lower temperatures as ndecreases. The chirality
+κdefinedinEq.(4)isreportedinFig4bfor n= 8. Asthe
+temperature decreases, around T∼80K we can identify
+a finite-size behaviour of κwhich, at variance with the
+previous one, can be easily recognized as typical of an
+effective phase transition. Such conclusion is confirmed
+by the analysis of the chiral susceptibility χκ(Fig. 4c),
+which for the largest Lhas a maximum at T= 85K. As-
+suming that the order parameter (4) is the relevant one
+to single out the onset of the fan arrangement, we can
+get a more accurate estimate of TN(8) by looking at the
+Binder cumulant u4(κ), reported in Fig. 4d. By making
+use of the MH technique, we locate the crossing point at
+TN(8) = 92(2)K. Finally, it is worthwhile to observe as
+the specific heat does not show any anomaly at TN(8),
+being the entropy substantially removed at TC(8).
+The scenario just outlined for n= 8 results to be cor-
+rect in the thickness range 6 ≤n/lessorsimilar15, where a clear
+separation between TN(n) andTC(n) can be easily fig-
+ured out. In such temperature window, the strong sur-
+face effects produce a quasi-FM set-up of the magnetic
+film structure along the z-direction. While leaving to the
+next Section amore detailed discussionof this regime, we
+report in Fig. 5 a plot of TN(n) andTC(n) vs.nfor all
+the simulated thicknesses. The separation between the
+two critical temperatures is maximum for n= 6, where
+TN(6) = 38(4), that is TN(6)∼1
+3TC(6). For films with
+less than six layersno fan orderis observed, i.e. for n= 5
+and below the chirality does not display any typical fea-
+ture of fan ordering at any temperature below TC(n). As
+a representative quantity we finally look at the rotation0 1 2 3 4 5 605101520 ∆ϕl(deg.)
+T=10K
+T=20K
+T=30K
+T=40K
+T=50K
+0 1 2 3 4 5 l012345(a)   n = 6
+(b)   n = 5
+FIG. 6: Rotation angle ∆ ϕlbetween magnetic moments on
+NN layers ( l+ 1,l) at some low temperatures, for thickness
+n= 5 and n= 6, and lateral dimension L= 64.
+angle of the magnetization between nearest planes:
+∆ϕl=ϕl+1−ϕl= arccos/bracketleftbig
+Mx
+lMx
+l+1+My
+lMy
+l+1/bracketrightbig
+(10)
+where (Mx
+l,My
+l) is the magnetic vector profile for each
+planel. ∆ϕlis displayed in Fig. 6a and Fig. 6b, for
+n= 6 and n= 5, respectively. In Fig. 6a, a quite clear
+fan stabilization is observed when the temperature de-
+creases, while in Fig. 6b, i.e. for n= 5, ∆ϕlkeeps an
+almosttemperatureindependent verysmallvalue; what’s
+more, ∆ϕlseems to loose any temperature dependence
+asT= 0 is approached. We attribute the absence of fan
+arrangement for n≤5 as simply due to the lack of “bulk
+planes” inside the film, so that we are left with only a 2d
+trend at TC(n), i.e. at the temperature where the order
+parameters defined in Eqs. (2) and (3) show a critical
+behaviour.
+IV. DISCUSSION AND CONCLUSION
+A possible framework to analyze the results presented
+in the previous Section is suggested by Fig. 5, where we
+can easily distinguish three significant regions: i) high
+thickness, n/greaterorequalslant16, where the films substantially display a
+bulk behaviour, with the single planes ordering tempera-
+ture coinciding with the helical phase transition one; ii)
+intermediate thickness, 6 ≤n/lessorsimilar15, where the tempera-
+ture corresponding to the onset of in-plane order, TC(n),
+isstill≃THo
+N, butwherethe helical/fanarrangementsta-
+bilizes only below a finite temperature TN(n)< TC(n);
+iii) low thickness,1 ≤n≤5, where TC(n)/lessorsimilarTHo
+Nbut no
+fan phase is present at any temperature.
+The observed behaviour in region iii) can be reason-
+ably attributed to the decreasing relevance of the con-
+tribution to the total energy of the system coming from
+the competitive interactions among NNN planes as the
+film thickness decreases; moreover, the thinness of the6
+0 20 40 60 80 100 120 140
+T (K)0102030 ∆ϕl,l+1(T)   (deg.)TN(16) TN(8)
+FIG. 7: (color online) ∆ ϕl(T) vs. temperature for the surface
+planes,l= 1 (triangles), l= 2 (squares), l= 3 (diamonds),
+l= 4 (circles). Straight lines and full symbols: n= 8. Dashed
+lines and open symbols: n= 16.
+film leads to an effective 2d-like trend. Region ii) looks
+however more intriguing, and requires a more accurate
+discussion, which can benefit from a careful comparison
+of the behaviour of a given quantity in regions i) andii).
+For this purpose, we look at the temperature depen-
+dence of the rotation angle of the magnetization between
+NN planes. In Fig. 7, ∆ ϕl(T) forn= 8 and n= 16
+(continuous and dashed lines, respectively), is plotted for
+the outermost planes, l= 1...4. For both thicknesses, a
+monotonictrendisobservedforall l, but at variancewith
+what happens for the highest thickness, for n= 8 we see,
+starting from a temperature T/lessorsimilarTN(8), an abrupt drop
+of ∆ϕ3and ∆ϕ4, which rapidly reach an almost con-
+stant value, only slightly larger than ∆ ϕ1. In the tem-
+perature range TN(8)/lessorsimilarT < T C(8) we thus substantially
+observe the same small magnetic phase shifts between all
+NN layers, testifying an energetically stable quasi-FM
+configuration giving no contribution to the helical order
+parameters. The latter point can be made clearer by
+looking at the the peak position Qz,maxof the structure
+factorS(0,0,qz). In Fig. 8 the average of Qz,maxvs Tis
+reported, again for n= 8 and for different lateral dimen-
+sionsL26. As expected from the previous argument, we
+see that Qz,max= 0 forTN(8)< T < T C(8), while it be-
+gins to shift to higher values as soon as the temperature
+decreases below TN(8), making apparent a progressive
+fan stabilization with Qz,max∝negationslash= 0 and reaching a value
+of about 21◦forT= 10K.
+In a previous study, where the magnetic properties of
+Ho thin films were investigated by MC simulations of a
+Heisenberg model with easy-plane single-ion anisotropy
+and six out-of-plane coupling constants (as obtained by
+experimental neutron scattering measurements16) on a
+HCPlattice14,15, itwasfoundthatforthicknessescompa-
+rable with the helical pitch the phase diagram landscape
+is quite different from what we find here. Indeed, for
+n= 9−16, three different magnetic phases could be sin-0 20 40 60 80 100
+T (K)051015202530Qz, max(deg.)
+FIG. 8: (color online) Qz, position of the maximum of S(/vector q),
+vs. temperature for thickness n= 8. Inset: magnetic vector
+(mx
+l,my
+l) profile for some temperatures for L= 64. Colors
+and symbols as in Fig. 2.
+0 1 2 3 4 5678 9 10 11 12
+l020406080∆ϕl(deg)T=100K
+     130K
+     135K
+     140K
+     145K
+FIG. 9: ∆ ϕlfor a BCT lattice and n= 12, when the six
+coupling constants set employed in Ref. 14,15 (see text) is
+used. The temperature range has been chosen around TC(n)
+(error bars lye within point size).
+gled out, with the high-temperature, paramagneticphase
+separated from the low-temperature, long-range ordered
+one, by an intermediate-temperature block phase where
+outer ordered 4-layersblocks coexist with some inner dis-
+ordered ones. Moreover, it was observed that the phase
+transition of such inner layers turns out to have the sig-
+natures of a Kosterlitz-Thouless one.
+The absence of the block phase in the J1−J2model
+here investigated has to be attributed to the different
+range of interactions, rather than to the different lattice
+structure. We came to this conclusion by doing some
+simulations using the same set of interaction constants
+employed in Refs. 14,15, but using a BCT lattice: the
+results we obtained for ∆ ϕlwithn= 12 are reported in
+Fig. 9. The latter is absolutely similar to Fig.7 of Ref. 15
+and clearlydisplaysthe footmarksofthe block phase(see
+down-triangle),withtwoexternalblocksoforderedlayers
+(l=1...5 and 8...12 ), where ∆ ϕlis roughly 10◦, sep-
+arated by a block of disordered layers, and with almost7
+-140 -139 -138 -137 -136 -135-134 -133 -132 -131
+e00.51 Pe90K
+91K
+92K
+93K
+94K
+95K
+-94 -92 -90 -88 -86 -84 -82 -80 -78
+e00.20.4Pe129K
+130K
+131K
+132K
+133K
+134KTC (8) = 133.3(3)KTN (8) = 92(2)K(a)
+(b)
+FIG. 10: (colors online) Equilibrium probability distribu tion
+of the energy for the thickness n= 8 for some temperatures
+aroundTN(8),(a), andTC(8),(b), respectively.
+opposite magnetization. We can thus confidently assert
+that, regardless of the underlying lattice structure, by
+decreasing the number of the out-of-plane interactions,
+for thicknesses close to the helical bulk pitch, the blockphase is replaced by a quasi-FM configuration in the in-
+termediate temperature range TN(n)< T < T C(n) .
+As a final issue we address the problem of the order
+of the transitions observed at TN(n) andTC(n), respec-
+tively. In particular, we focus our attention to the thick-
+ness ranges where the chiral order parameter is relevant,
+i.e. regions i) andii) as defined at the beginning of
+this Section. In Fig. 10 the equilibrium probability dis-
+tribution of the energy for temperatures around TN(8)
+(Fig. 10a) and TC(8) (Fig. 10b) is plotted: for both
+temperatures, no double peak structure is observed, so
+that we have no direct indication for a first order tran-
+sition even if, according to precedent studies of Loison
+and Diep17,18, the presence of a first-order transition at
+TN(n), cannot be completely excluded, as it could reveal
+itself only when the lateral dimension Lare much larger
+than the largest correlation length. The same conclusion
+about the order of transition is reached for any other in-
+vestigatedfilm thickness, asthe energyprobabilitydistri-
+butionshapedoesnot qualitativelychange. Thisfindings
+agree with the results we got in previous MC simulations
+discussed in Ref. 15, so that we may conclude that the
+order of the observed transitions is not affected by the
+range of interactions.
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+20F. R. Brown and T. J. Woch, Phys. Rev. Lett. 58, 2394
+(1987).
+21D. P. Landau, and K. Binder, A Guide to Monte Carlo
+Simulation in Statistical Physics , Cambridge University
+Press, Cambridge (2000).
+22M. E.J. Newman, and G. T. Barkema, Monte Carlo Meth-
+ods in Statistical Physics , Clarendon Press, Oxford (1999).
+23B. Efron, The Annals of Statistics 7, 1 (1979).
+24P. M. Chaikin, T. C. Lubensky Principles of condensed
+matter physics , Cambridge University Press, New York
+(1995).
+25K. Binder, Z. Phys. B 43, 119 (1981). K. Binder, Phys.
+Rev. Lett. 47, 693 (1981).
+26Such observable has been obtained from instantaneous
+evaluation of the structure factor during the stochastic
+process, and subsequently statistically analyzed as all th e
+other macroscopic quantities.
\ No newline at end of file
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+ 1 Mapping between the order of thermal denaturation a nd the shape of the critical line of 
+mechanical unzipping in 1-dimensional DNA models 
+ 
+ 
+Sahin Buyukdagli and Marc Joyeux (#)  
+Laboratoire de Spectrométrie Physique (CNRS UMR 558 8), 
+Université Joseph Fourier - Grenoble 1, 
+BP 87, 38402 St Martin d'Hères, FRANCE  
+ 
+ 
+PACS classification : 87.14.Gg, 87.15.Aa, 64.70.-p 
+ 
+Short title : mechanical unzipping versus thermal denaturation of DNA 
+ 
+Abstract : In this Letter, we investigate the link between t hermal denaturation and 
+mechanical unzipping for two models of DNA, namely the Dauxois-Peyrard-Bishop model 
+and a variant thereof we proposed recently. We show  that the critical line that separates 
+zipped from unzipped DNA sequences in mechanical un zipping experiments is a power-law 
+in the temperature-force plane. We also prove that for the investigated models the 
+corresponding critical exponent is proportional to the critical exponent α, which characterizes 
+the behaviour of the specific heat in the neighbour hood of the critical temperature for thermal 
+denaturation. 
+ 
+ 
+(#)  email : Marc.Joyeux@ujf-grenoble.fr  2 1 - INTRODUCTION 
+ 
+ Several fundamental biological processes, like tra nscription and replication, involve 
+the separation and recombination of the two strands  that compose double-stranded (zipped) 
+DNA molecules. Since transcription and replication are quite complex mechanisms, many 
+early studies focused on thermal denaturation, that  is the separation of the two strands upon 
+heating [1-7]. In cells, unzipping is however not m ediated by thermal activation but rather by 
+proteins, which apply forces to separate and stretc h complementary DNA strands. Compared 
+to thermal denaturation studies, the recent mechani cal unzipping experiments, where forces 
+are applied to adjacent 5’ and 3’ strands of indivi dual DNA molecules [8-16], therefore 
+represent a great step towards the understanding of  real biological processes. These 
+experiments triggered in turn much theoretical effo rt (see [17-27] and references therein). 
+ The purpose of the present paper is to point out t hat there actually exists a tight link 
+between thermal denaturation and mechanical unzippi ng. Indeed, we will show that for the 
+two 1-dimensional mesoscopic DNA models we investig ated it is possible to establish a 
+mapping between the order of the thermal denaturati on phase transition and the shape of the 
+critical line of mechanical unzipping in the temper ature-force plane. More precisely, we will 
+show that the critical line that separates zipped f rom unzipped sequences in mechanical 
+unzipping experiments is a power-law, whose exponen t is proportional to the critical 
+exponent α that characterizes the behaviour of the specific h eat in the neighbourhood of the 
+critical temperature for thermal denaturation. This  work therefore complements that of Singh 
+and Singh, who showed that the critical force varie s as the square root of the temperature gap 
+to denaturation for the Dauxois-Peyrard-Bishop mode l with a large anharmonic stacking 
+parameter ρ [26].  3  The remainder of this paper is organized as follow s. Sect. 2 provides a brief 
+description of the two models used in this work, as  well as the bases of the Transfer-Integral 
+formalism. Calculation of the critical exponent α is next discussed in Sect. 3. In Sect. 4, we 
+establish that the critical unzipping force evolves  as a power of the temperature gap to 
+denaturation, and that the corresponding critical e xponent is related to α by a linear scaling 
+law. Finally, we discuss the relevance of this stud y for real mechanical unzipping experiments 
+and conclude in Sect. 5. 
+ 
+2 – MODELS AND TRANSFER-INTEGRAL CALCULATIONS 
+ 
+ The potential energy of the two DNA models whose u nzipping behaviour is studied in 
+this paper is of the form 
+∑
+=− + =N
+nnn nM yyWyVV
+11)) ,()((  ,        (2.1) 
+where ny is the deviation from equilibrium of the distance between the bases of the nth  pair. 
+The one-particle Morse potential 
+2)1 ()(nya
+nM eDyV−−=          (2.2) 
+models the binding energy of the hydrogen bonds tha t connect paired bases. ) ,(1−nnyyW  is a 
+nearest-neighbor potential, which describes the stac king interaction between successive bases 
+belonging to the same strand. The choice for ) ,(1−nnyyW  is crucial, since the shape of the 
+thermal denaturation transition, which is a collect ive effect, depends primarily on its form. 
+The Dauxois-Peyrard-Bishop (DPB) model [28] assumes t hat the stacking interaction is of the 
+form 
+) 1 ()(2),() ( 2
+1 11−+−
+− − + −=nnyy
+nn nn e yyKyyWαρ  .      (2.3)  4 The coupling constant of this interaction drops fro m )1 (ρ+K to K as the paired bases 
+separate, which decreases the rigidity of DNA seque nces close to dissociation and results in a 
+sharp first-order transition. Numerical values of t he parameters used in this work are those of 
+Ref. [26], that is D=0.063 eV, a=4.2 Å-1, K=0.025 eV Å-2, α=0.35 Å-1, except that we 
+performed calculations with several values of ρ ranging from 0 to 5. 
+ The Joyeux-Buyukdagli (JB) model [29] instead assu mes that 
+2
+1) (
+1 ) () 1 (),(2
+1
+−−−
+− −+ −Δ=−
+nnbyyb
+nn yyK eCHyyWnn ,      (2.4) 
+where the first term describes the finite stacking interaction and the second one the stiffness 
+of the phosphate-sugar backbone. The sharpness of t he melting transition predicted by the JB 
+model is precisely due to the finite depth CH/Δ of the stacking interaction. In this work, we 
+performed calculations with two sets of parameters,  namely that of Refs. [29,30], that is 
+D=0.04 eV, a=4.45 Å-1, ΔH=0.44 eV, C=2, b=0.10 Å-2 and Kb=10 -5 eV Å-2 (set JB-1), and 
+that of Ref. [7], that is D=0.048 eV, a=6.0 Å-1, ΔH=0.409 eV, b=0.80 Å-2 and Kb=4 10 -4 eV Å-
+2 (set JB-2). 
+ Transfer-Integral (TI) calculations are made possi ble by the fact that only nearest-
+neighbor interactions are considered in the potenti al energy of Eq. (2.1). One can therefore 
+define a kernel 
+)]) ,()(21)(21[1exp( ),(1 1 1 − − − + + −=nn nM nM
+Bnn yyWyVyVTkyyK  ,    (2.5) 
+and write the partition function of a homogeneous D NA sequence of length N, whose first 
+base pair is kept at a given distance yy=1 from equilibrium and whose end base pair is 
+anchored (i.e. 0 =Ny), in the form 
+N N NM M
+BNN
+dy dy dy yyyyVyVTkyyKyyKyyKyZ
+... )()()]) ()([21exp( ),()... ,(),()(
+21 1 11 23 12
+δ δ− + −× =∫ −
+  (2.6)  5 The basic trick of the TI method consists in expand ing the kernel in an orthogonal basis 
+)()( ),(1 1 − − ΦΦ=∑ nk
+knkk nn yy yyK λ  ,       (2.7) 
+where the kΦ and kλ are the eigenvectors and eigenvalues of the integr al operator and satisfy 
+)( )(),( y xyxKdx kk k Φ=Φ∫λ  .        (2.8) 
+By substituting the kernel expansion of Eq. (2.7) i n Eq. (2.6), one gets 
+∑Φ=−
+kkkN
+k yc yZ )() 0 ( )(1λ  ,         (2.9) 
+where  
+]2)(exp[ )()(TkyVyyc
+BM
+k k − Φ=  .                 (2.10) 
+Similar considerations show that the partition func tion free Z of the sequence without external 
+constraint (except for the last base pair that rema ins anchored) can be written in the form 
+∑Φ=−
+kk kN
+kC Z ) 0 (1
+free λ  ,                   (2.11) 
+where  
+∫=dy ycCk k )( .                   (2.12) 
+The work done in stretching the first base pair at distance y, )(yW, is the free energy 
+difference between the stretched and the free chain s, that is 
+free )(ln )(ZyZTkyWB−=  ,                   (2.13) 
+The derivative of )(yW with respect to y provides the average force )(yF, which is needed 
+to keep the first base pair separated by y 
+dy ydZ 
+yZTk
+dy ydW yFB)(
+)()()( −= =  .                  (2.14)  6 In the thermodynamic limit of infinitely long chain s ( ∞→N), the sums in Eqs. (2.9) and 
+(2.11) are dominated by the ground state of the TI operator with eigenvector 0Φ and 
+eigenvalue 0λ. In this limit, the free energy per base pair, f, may therefore be obtained from 
+0ln λTkfB−=                    (2.15) 
+for both the stretched and free chains, and Eqs. (2 .13) and (2.14) simplify to 
+.)(
+)()()(ln )(
+0
+000
+dy ydc 
+ycTkyFCycTkyW
+BB
+−=−=
+                  (2.16) 
+The critical force cF is defined as the average force, which is needed t o keep the bases of the 
+first pair at an infinite distance from one another , that is 
+)(∞→=yFFc  .                   (2.17) 
+At that point, it must be emphasized that cF is usually substantially smaller than the force, 
+which is actually needed to separate the two strand s. This later force indeed corresponds to 
+the maximum of )(yF when y is increased from zero to infinity. It turns out t hat there usually 
+exists a very large force barrier at short y separations (see for examples Figs. 5 and 6 of [26 ]). 
+cF does not correspond to the value of )(yF at the maximum of the barrier, but rather to the 
+asymptotic value of )(yF at large values of y. 
+ At last, one might wish to write the free energy p er base pair of the unstretched 
+sequence, f, as the sum of a non-singular part, ns f, and a singular part, sing f,  
+sing ns fff+=   .                   (2.18) 
+The non-singular part of the free energy, ns f, should behave smoothly as temperature is 
+raised up to the melting temperature cT, while the singular part, sing f, is expected to vary 
+more sharply and remain constant once the two DNA s trands are separated. Singularities at cT  7 in the temperature evolution of the entropy and/or specific heat of the system should arise 
+from sing f and not ns f. The decomposition of Eq. (2.18) is not unique, bu t the requirement 
+that sing f remains constant above cT, that is when the two strands are widely separated , 
+indicates that the most natural choice consists in considering that ns f is the free energy of 
+non-interacting DNA single-strands, that is when th e pairing potential ) (nMyV is omitted in 
+Eq. (2.1). One consequently obtains 
+ss BzTkf ln ns −=  ,                    (2.19) 
+where 
+du Tkuhz
+Bss 
+ss ∫−= ])(exp[  .                  (2.20) 
+The function ) (uhss  in Eq. (2.20) is equal to 
+2
+2)( uKDuhss +=                    (2.21) 
+for the DPB model and to 
+)1 (2)(22 ub
+b ss eHuKDuh−−Δ++=                  (2.22) 
+for the JB one. The integral in Eq. (2.20) can be e valuated analytically for the DPB model, 
+leading to 
+)2ln( 22
+ns KaTkTkDfB Bπ−=  .                 (2.23) 
+For the JB model, ns f can either be estimated numerically or, at the cos t of a slight 
+approximation, be computed from 
+))] 0 () 0 ((ln[ 2 1 ns IIaTkDfB + −=  ,                  (2.24)  8 where ) 0 (1I and ) 0 (2I are obtained by setting 0=F in the expressions for 1I and 2I in Eqs. 
+(2.8) and (2.9) of Ref. [7]. Note that a factor a was omitted in Eq. (2.7) of Ref. [7], which 
+should actually read β/)] (ln[ ),(21IIaDFTgu +−= . 
+ 
+3 – DETERMINATION OF THE CRITICAL EXPONENT α 
+ 
+ In this section, we calculate the critical exponen t α of unconstrained DNA sequences 
+described by the DPB and JB models. α is the exponent, which describes the behaviour of the 
+specific heat per base pair, Vc, in the neighborhood of the critical temperature cT, that is 
+α−∝
+∂∂−= t
+TfTcV 22
+ ,         (3.1) 
+where t is the reduced temperature 1 /−=cTTt . Strictly speaking, α should be computed 
+from the singular part of Vc. Experimentalists, however, usually have no means to separate 
+the singular from the non-singular part of the meas ured specific heat. Since the singular part is 
+expected to vary much more sharply than the non-sin gular part in the neighbourhood of cT, 
+the usual approximation dating back to the pioneeri ng work of Kadanoff et al [31] consists in 
+estimating α from log-log fits of the evolution of Vc close to cT. This amounts to consider 
+that “transients” arising from the non-singular par t do not alter the estimation of α. This 
+method is still used today with only minor modifica tions (see for example Refs. [32-34]). 
+Determination of α with this method is illustrated in Fig. 2 of Ref. [30] and Fig. 10 of Ref. [7] 
+for the JB model and sets of parameters JB-1 and JB -2, respectively. The obtained values of α 
+(α=1.13 and α=1.33, respectively) are reported in the first colu mn of Table 1. We performed 
+TI calculations with the same grid of y values as in Refs. [7,30] for the DPB model and fo ur 
+values of ρ ranging from 0.0 to 5.0. Results are illustrated i n Fig. 1 and reported in the first 
+column of Table 1. Note that in all these calculati ons, as well as in those discussed below, the  9 critical temperature cT is obtained as the temperature for which the corre lation length ξ, 
+computed as in [7,29,30], is maximum. 
+ We also determined α from the temperature evolution of the singular par t of the free 
+energy, sing f, which is expected to vary according to 
+α−∝2
+sing tf  ,           (3.2) 
+where sing f is obtained from Eqs. (2.15), (2.18) and (2.19). F ig. 2 shows the evolution of 
+)/(log sing 10 Df−  as a function of )(log 10 t for the DPB model and four values of ρ ranging 
+from 0.0 to 5.0, obtained with the same grid of y values as in Refs. [7,30]. The values of α 
+deduced from these plots are reported in column (2)  of Table 1. Also reported in the same 
+column are the values of α obtained from similar calculations dealing with th e JB model and 
+sets of parameters JB-1 and JB-2. 
+ Comparison of columns (1) and (2) of Table I indic ates that both methods agree in 
+predicting that, for the DPB model, α increases with ρ. This is due to the fact that, for the 
+DPB model, the order of the melting phase transitio n decreases from 2 to 1 as ρ increases 
+[26,35,36]. However, for both the DPB and the JB mo del, the values of α obtained from the 
+temperature evolution of Vc are systematically larger than those obtained from  the evolution 
+of sing f. Moreover, the temperature evolution of sing f leads to “well-behaved” values of α, 
+that is, to values that are systematically comprise d between 0 and 1, while the values of α 
+obtained from the evolution of Vc are larger than 1 for the two sets of parameters o f the JB 
+model and for the DPB model with ρ=5.0. The reason for this is that the switching fro m 
+second to first order phase transition causes the t emperature evolution of the specific heat to 
+depart from a power-law and approach a Dirac peak w ith infinite slope. This can be clearly 
+seen in Fig. 3, which shows the temperature evoluti on of the entropy per base pair, 
+Tfs∂−∂ =/, for increasing values of ρ. At last, one may note that additional calculation s (not  10 discussed here) show that the values of α obtained from the temperature evolution of sing f all 
+satisfy the hyperscaling law 2=+να  (where ν is the critical exponent of the correlation 
+length ξ), while values obtained from the temperature evolu tion of Vc do not [7,30]. 
+ 
+4 – THE LINEAR SCALING LAW RELATING σ TO α 
+ 
+ In this section, we first show that for the models  introduced in Sect. 2 the critical force 
+)(TFc behaves according to a power-law in the neighbourh ood of cT, and then that the 
+corresponding critical exponent σ is linked to α by a linear scaling law. 
+ Fig. 4 shows the critical force ) (TFc for the DPB model and four values of ρ ranging 
+from 0.0 to 5.0, obtained from Eqs. (2.16) and (2.1 7) and the same grid of y values as in Refs. 
+[7,30]. It is seen that the shape of the critical l ine ) (TFFcc= between zipped and unzipped 
+sequences varies markedly with ρ : it indeed transforms from a straight line into a  curve as the 
+phase transition evolves from second order (small ρ) to first order (large ρ). Similar plots for 
+the JB model and sets of parameters JB-1 and JB-2 c an be found in Fig. 1 of Ref. [7]. At that 
+point, it should be emphasized that the only models , which reproduce reasonably the 
+experimentally observed thermal evolution of the cr itical force, are the JB model with set of 
+parameters JB-2 (these parameters were precisely ad justed in order to reproduce the ) (TFc 
+curve [7]) and, to a somewhat lesser extent, the DP B model with 5=ρ. Nonetheless, the 
+other models are used here for the purpose of compa rison and to check the validity of the 
+scaling law derived below. 
+ Log-log plots, such as those shown in Fig. 5, furt her indicate that cF actually evolves 
+as a power of t : 
+σtFc∝ ,           (4.1)  11 with a critical exponent σ that decreases from 1 (for ρ=0) to 1/2 (for large values of ρ). Note 
+that the value 2 / 1=σ was already reported by Singh and Singh for the DP B model with 
+5=ρ [26]. The corresponding values of σ, as well as those obtained for the JB model and th e 
+JB-1 and JB-2 sets of parameters, are reported in c olumn (3) of Table 1. Most interestingly, it 
+is possible to relate the two critical exponents α and σ through a linear scaling law. This can 
+be achieved by noticing that the free energy of the  stretched unzipped sequence, ug, writes, 
+for the DPB model, 
+KFfgu22
+ns −=  ,          (4.2) 
+where ns f is given in Eq. (2.23). Since the critical force cF is such that fgu=, comparison 
+of Eqs. (2.18) and (4.2) shows that cF satisfies 
+KFfc
+22
+sing −= .           (4.3) 
+Similarly, for the JB model ug writes 
+)] (ln[ 21IIaTkDgB u + −=  ,          (4.4) 
+where the expressions for 1I and 2I can be found in Eqs. (2.8)-(2.9) of Ref. [7]. A li ttle bit of 
+algebra then shows that for sets of parameters JB-1  and JB-2 one has, to a good 
+approximation 
+bc
+KF
+IIIf4) 0 () 0 () 0 (2
+2 11
+sing +−≈  .         (4.5) 
+By replacing Eqs. (3.2) and (4.1) in Eqs. (4.3) and  (4.5), one immediately sees that for both 
+models α and σ should consequently satisfy 
+σα22=−  .           (4.6) 
+Comparison of columns (2) and (4) of Table 1 shows that this scaling law is, indeed, very 
+well satisfied for both models.  12  At that point, it should be mentioned that other c ritical exponents for cF and/or other 
+scaling laws involving σ have been derived for models that differ markedly from the ones 
+studied here. For example, Bhattacharjee has shown that if stretched DNA is treated as two 
+flexible interacting elastic strings tied together at one end, then the critical force cF evolves 
+according to [18] 
+d
+c cF−−∝2/ 1vv  ,          (4.7) 
+where v is the depth of the pairing potential (take n as a Dirac function) and cv  the critical 
+value of v at which mechanical unzipping occurs at a given temperature T. It is difficult to 
+check the validity of Eq. (4.7) for the more comple x DPB and JB models studied here, 
+because for these models the order of the transitio n depends on geometrical factors not taken 
+into account in [18], like for example the ratio of  the widths of the pairing and stacking 
+interactions [35]. 
+ On the other hand, it was shown that for the Polan d-Scheraga model where self-
+avoiding interactions are accounted for (both withi n loops and between loops and the rest of 
+the chain), the critical force scales like [37] 
+νtFc∝ .           (4.8) 
+In Eq. (4.8), ν does not stand for the critical exponent of the co rrelation length ξ but rather the 
+correlation length exponent of a self-avoiding rand om walk (the radius of gyration GR of a 
+random walk of length L scales as νL). Numerically, ν is equal to 3/4 in dimension d=2 and to 
+approximately 0.588 in dimension d=3. It therefore appears that the two models lead t o 
+different predictions. 
+ At last, when stretched DNA is described as a self -avoiding walk on the three-
+dimensional Sierpinski gasket, one gets [38]  13 θνσα=−2  ,           (4.9) 
+where θν is the critical exponent of the end-to-end distanc e. Eq. (4.10) coincides with Eq. 
+(4.6) when θν is assigned its mean-field value 2 / 1 =θν . 
+ 
+5 – CONCLUSION 
+ 
+ We have shown that for both the DPB and JB models the critical force cF evolves as a 
+power of TTc− and that the corresponding critical exponent, σ, is related to the critical 
+exponent of the specific heat, α, through the linear relation σα22=− . Quite interestingly, 
+numerical calculations indicate that the values of α derived from σ according to this relation 
+are in excellent agreement with the statistically r elevant values of α determined from the 
+evolution of the singular part of the free energy, sing f. Mechanical unzipping experiments 
+may therefore provide an accurate method to estimat e the order of the thermal denaturation 
+phase transition. One must however be conscious of three limitations. 
+ First, real DNA is a heteropolymer and sequence in homogeneity has marked effects on 
+both thermal [39-46] and mechanical [8,9,13,15,16,2 3,47,48] unzipping. We showed in Ref. 
+[45] that the essential effect of heterogeneity on thermal denaturation is to let different 
+portions of the investigated sequences open at slig htly different temperatures. We also pointed 
+out that, besides this macroscopic effect, the loca l aperture of each portion is however very 
+similar to that of a homogeneous sequence of the sa me length. Nonetheless, the precise effect 
+of heterogeneity on the validity of the scaling rel ation of Eq. (4.6) still has to be investigated. 
+ Moreover, mechanical unzipping experiments are not  straightforward and the range of 
+values of t spanned by today experiments [14,47] is not suffic iently broad to confirm the 
+power-law evolution of the critical force.  14  At last, the theoretical result itself may be ques tioned. As discussed at the end of Sect. 
+4, different models for DNA unzipping may indeed le ad to different critical behaviors and 
+scaling laws. Moreover, even when focusing on Hamil tonian models expressed in terms of 
+continuous dynamical variables, the obtained result  depends essentially on the facts that (i) 
+the free energy of stretched unzipped sequences, ug, varies as the square of the applied force 
+(see Eqs. (4.2) and (4.4)), and (ii) the singular p art of the free energy of unstretched zipped 
+sequences, sing f, evolves as a power of t in the neighbourhood of the critical temperature cT. 
+To the best of our knowledge, point (i) is relative ly robust in the sense that, for all the models 
+we are aware of, ug displays the same 2F dependence. This includes of course the DPB (Eq. 
+(4.2)) and JB (Eq. (4.4)) models, as well as the fr eely jointed chain model (Eq. (3) of Ref. 
+[14]), but we checked that the same dependence also  holds for the more involved helicoidal 
+DNA model of Barbi et al [49]. In contrast, point ( ii) seems to be more model-dependent. 
+Indeed, while sing f evolves as a power of t in the neighbourhood of cT for the DPB and JB 
+models, this is no longer true for the helicoidal D NA model of Barbi et al [49]. Figs. 5 and 6 
+of Ref. [49] show that for this model sing f rather behaves as )/exp( ta− in the neighbourhood 
+of cT. This indicates that the helicoidal DNA model does  not describe thermal denaturation as 
+a phase transition but as a Kosterlitz-Thouless sin gularity. While there still exists, for this 
+model, a mapping between the shape of the critical line of mechanical unzipping in the 
+temperature-force plane and the sharpness of the th ermal denaturation transition, this mapping 
+no longer assumes the very simple form of Eq. (4.6) . Further work, both experimental and 
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+061909  18 TABLE CAPTION 
+ 
+Table 1  : Values of the critical exponents α and σ obtained according to different methods 
+described in the text and for different sets of par ameters for the DPB and JB models. α is the 
+critical exponent of the specific heat and σ that of the critical force cF. Note that, according to 
+Eq. (4.6), α and σ are related through )1 ( 2σα−= . 
+  19 FIGURE CAPTIONS 
+ 
+Figure 1  (color online) : Plot of ) /(log 10 BVkc as a function of )(log 10 t for the DPB model 
+and four values of ρ ranging from 0.0 to 5.0. The dash-dotted lines sho w the slopes from 
+which the critical exponents α are estimated. These values of α are reported in column (1) of 
+Table 1. 
+ 
+Figure 2 (color online) : Plot of ) /(log sing 10 Df−  as a function of )(log 10 t for the DPB model 
+and four values of ρ ranging from 0.0 to 5.0. The dash-dotted lines sho w the slopes from 
+which the critical exponents α−2 are estimated. The corresponding values for α are reported 
+in column (2) of Table 1. 
+ 
+Figure 3  (color online) : Plot of the entropy per base pair  s (in units of the Boltzmann 
+constant kB) as a function of the reduced temperature t for the DPB model and four values of ρ 
+ranging from 0.0 to 5.0. 
+ 
+Figure 4  (color online) : Plot of the critical force cF (expressed in pN) as a function of 
+temperature T for the DPB model and four values of ρ ranging from 0.0 to 5.0. 
+ 
+Figure 5 (color online) : Plot of ) (log 10 cF as a function of )(log 10 t for the DPB model and 
+four values of ρ ranging from 0.0 to 5.0. cF is expressed in pN. The dash-dotted lines show 
+the slopes from which the critical exponents σ are estimated. These values of σ are reported in 
+column (3) of Table 1.  20 TABLE 1 
+ 
+ 
+ 
+ α 
+(1) α 
+(2) σ 
+(3) 2(1-σ) 
+(4) 
+DPB, ρ=0.0  0.00 0.02 0.99 0.02 
+DPB, ρ=0.5 0.58 0.35 0.78 0.44 
+DPB, ρ=1.0 0.92 0.66 0.67 0.66 
+DPB, ρ=5.0 1.53 0.99 0.50 1.00 
+JB-1 1.13 0.82 0.59 0.82 
+JB-2 1.33 0.57 0.72 0.56 
+ 
+(1) obtained from the plot of ) /(log 10 BVkc as a function of )(log 10 t (see Fig. 1). 
+(2) obtained from the plot of ) /(log sing 10 Df−  as a function of )(log 10 t (see Fig. 2). 
+(3) obtained from the plot of ) (log 10 cF as a function of )(log 10 t (see Fig. 5). 
+(4) compare with the values of α in column (2). 
+  21 FIGURE 1 
+ 
+ 
+ 
+ 
+  22 FIGURE 2 
+ 
+ 
+ 
+ 
+  23 FIGURE 3 
+ 
+ 
+ 
+ 
+  24 FIGURE 4 
+ 
+ 
+ 
+ 
+  25 FIGURE 5 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
\ No newline at end of file
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new file mode 100644
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+++ b/1001.0512.txt
@@ -0,0 +1,772 @@
+arXiv:1001.0512v1  [astro-ph.SR]  4 Jan 2010Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 1 November 2018 (MN L ATEX style file v2.2)
+AOVel: The role of multiplicity in the development of
+chemical peculiarities in late B-type stars⋆
+J. F. Gonz´ alez1†, S. Hubrig2, and F. Castelli3
+1Instituto de Ciencias Astron´ omicas, de la Tierra y del Espa cio, Casilla 467, 5400 San Juan, Argentina
+2Astrophysical Institute Postdam, An der Sternwarte 16, 144 82 Potsdam, Germany
+3Instituto Nazionale di Astrofisica - Osservatorio Astronom ico di Trieste, Via Tiepolo 11, I-34131 Trieste, Italy
+1 November 2018
+ABSTRACT
+We present high-resolution, high signal-to-noise UVES spectra of A OVel, a quadruple
+system containing an eclipsing BpSi star. From these observations we reconstruct
+the spectra of the individual components and perform an abundan ce analysis of all
+four stellar members. We found that all components are chemically p eculiar with
+different abundances patters. In particular, the two less massive stars show typical
+characteristicsof HgMn stars. The two most massive stars in the s ystem show variable
+line profiles indicating the presence of chemical spots. Given the you th of the system
+and the notable chemical peculiarities of their components, this sys tem could give
+important insights in the origin of chemical anomalies.
+Key words: stars: binaries – stars: chemically peculiar – stars:individual: HD 6882 6
+1 INTRODUCTION
+AOVel (=HD68826) was classified as BpSi by
+Bidelman & MacConnell (1973) and was reported as
+an eclipsing binary by Clausen, Gim´ enez, & van Houten
+(1995). In fact it is one of the only two double-lined
+eclipsing binaries with a BpSi component known to date
+(Hensberge et al. 2004). From light-time effect on the times
+of minima, Clausen et al. (1995) deduced the presence of a
+third body.
+In our previous paper (Gonz´ alez et al. 2006), using
+FEROS spectroscopic observations, we discovered that this
+system is actually a spectroscopic quadruple system with
+components close to the ZAMS. The four stars form two
+close spectroscopic pairs (periods of 1.58 and 4.15 days)
+bound gravitationally to each other in a wide eccentric orbi t
+with a period of 41 yr. In that paper we combined our radial
+velocity ( RV) measurements with the available photomet-
+ric datatoderive orbital parameters for both binary system s
+andtocalculate the absolute parameters ofthe eclipsing sy s-
+tem. For the first time, direct determination of the radius
+and the mass was obtained for a BpSi star.
+In this work we present high-resolution, high signal-to-
+noise UVES spectra, which were used to perform an abun-
+dance analysis of all four components of this multiple sys-
+⋆Based on observations obtained at the European Southern Ob-
+servatory, Paranal, Chile, ESO programmes 076.D-0169(A) a nd
+072.D-0235(B).
+†E-mail: fgonzalez@icate-conicet.gob.artem. In Sec.2 we present the observations and describe the
+reconstruction of the component spectra. In Sec.3 we an-
+alyze the spectral characteristics of each component and
+present the results of the abundance analysis. In the last
+Section we discuss the main results and the occurrence of
+chemical peculiarities in binary and multiple systems.
+2 OBSERVATIONS AND SPECTRA
+RECONSTRUCTION
+Three spectra were obtained in service mode with UVES
+at VLT-UT2 telescope in October 2005. The spectra have
+been taken with the 0.4arcsec slit for the blue arm and the
+0.3arcsec slit for the red arm on three consecutive nights
+with two different dichroics to achieve the highest UVES
+resolution of 110,000 in the red spectral region and 80,000
+in the blue spectral region. We used exposure times of 20–30
+min, obtaining a S/N ratio above 200 in the spectral range
+4000–8000 ˚A.
+These spectra are analyzed here along with five FEROS
+spectra described in the previous paper by Gonz´ alez et al.
+(2006). To calculate separate spectra for the four compo-
+nents of the system and to measure their RVs, the iter-
+ative method described by Gonz´ alez & Levato (2006) was
+adapted for the multiple system AO Vel. This algorithm
+computes the spectra of the individual components and the
+RVs iteratively. In each step the computed spectra are used
+to remove the spectral features of all but one component
+from the observed spectra. The resulting single-lined spec -
+c/circlecop†rt0000 RAS2J. F. Gonz´ alez et al.
+Table 1. Modified Julian dates of UVES observations, orbital phases, measured RVs, and signal-to-noise ratio at three different w ave-
+length.
+MJD Phase AB Phase CD RV A RVB RVC RVD S/Nλ4200S/Nλ5600S/Nλ8500
+kms−1kms−1kms−1kms−1
+53669.3357 0.7133 0.8722 172 ±4 -180 ±4 88.2 ±2.2 -49.7 ±0.9 250 300 175
+53670.3346 0.3436 0.1129 -149 ±3 172 ±5 -38.6 ±2.2 88.2 ±0.9 310 375 190
+53671.3523 0.9859 0.3582 52 ±8 -9 ±8 -50.2 ±4.7 102.2 ±1.6 235 320 180
+tra are used to measure the RV of that component and to
+compute its spectrum by combining them appropriately.
+In these calculations we combined the three new UVES
+spectra with our four FEROS observations taken out of
+eclipse. Since the resolution and S/N-ratio of the UVES
+spectra were much higher than for FEROS spectra, we as-
+signed higher weight to these spectra in the last iterations .
+Table1 lists the measured RVs along with the phases
+(zero at conjunction) corresponding to the orbits AB and
+CD. For the calculation of orbital phases we have applied
+a time correction of +0.0236d and -0.0337d for the system
+AB and CD respectively, in order to take into account the
+light time effect. The last spectrum in this table was taken
+at a phase of partial eclipse of the primary star A. We have
+taken into account this fact during the calculations of the
+component spectra, considering that the light contributio n
+of A is smaller in that spectrum and therefore the line in-
+tensities are expected to differ from those in the remaining
+spectra. If, for out-of-eclipse phases, star A contributes a
+fraction lato the total light of the system, and during a
+phase of partial eclipse the light of this star is reduced by a
+factorf, then the apparent intensity of the spectral lines of
+the star Ais reduced bya factor f/(f·la+1−la), and that of
+the remaining components is increased by 1 /(f·la+1−la).
+These scaling factors were estimated to be, in the case of
+our third UVES spectrum, 0.85 and 1.10, respectively.
+We note that the amplitude of the RV curves of the sys-
+tem AB might be affected by the asymmetry of the spectral
+line profiles, which present variations for several element s
+(see next Section). This error source has not been considere d
+in the formal error quoted in Table1, since it is difficult to
+be estimated without determining the surface distribution
+of the various atomic species giving rise for spectral lines
+used for RV measurements.
+The obtained new RV measurements were in general
+agreement with the orbits published in our previous pa-
+per (Gonz´ alez et al. 2006), especially for the eclipsing bi -
+nary AB, for which some orbital parameters are fixed by the
+photometric data. In the case of the pair CD, the new data
+allowed a significant improvement of its orbital parameters .
+The orbital parameters recalculated combining all availab le
+RV measurements are listed in Table2. In Fig.1 we present
+the RV curves for the eclipsing binary system AB and for
+the less massive system CD. In the case of the binary AB,
+which presents apsidal motion, the solid line corresponds t o
+the orbit calculated for the epoch of FEROS observations
+and the dotted line to the epoch of UVES observations.
+Once the spectra of the stellar components have been
+reconstructed, they must be scaled to recover the intrinsic
+intensities of the spectral lines in those stars. To this aim ,
+as in our previous paper, we used the photometric relativeTable 2. Orbital parameters for the spectroscopic binaries AB
+and CD.
+Parameter Unit Value
+PCD days 4.14933 ±0.00004
+TCD(conj) MJD 53300.539 ±0.007
+VγCD kms−121.5±0.4
+KC kms−194.6±1.2
+KD kms−1103.9 ±0.7
+eCD R⊙ 0.017 ±0.006
+ωCD deg 261 ±22
+aCDsiniCD R⊙ 16.48 ±0.12
+qCD 0.917 ±0.016
+MCsin3iCD M⊙ 1.82±0.03
+MDsin3iCD M⊙ 1.67±0.05
+VγAB kms−18.1±1.8
+KA kms−1167.8 ±3.4
+KB kms−1184.1 ±3.4
+aAB R⊙ 10.99a±0.15
+qAB 0.911 ±0.026
+MA M⊙ 3.68a±0.14
+MB M⊙ 3.35a±0.14
+aCalculated using the photometric orbital inclination iAB= 88.5
+deg.
+fluxes given by Clausen et al. (1995). The relative contribu-
+tion to the observed continuum near λ5000 was estimated to
+be0.40, 0.28, 0.16, 0.16 for componentsA,B, C, andD.Gen-
+erally, when applyinga spectral disentangling technique, the
+resulting spectra of the individual components have a very
+high quality, since the S/N-ratio increases roughly as the
+square root of the number of observed spectra involved in
+the calculations (usually several tens), and diminishes pr o-
+portionally to the relative flux of the component under con-
+sideration (usually close to 0.5). In the present work a smal l
+number of spectra are available for a system of 4 spectral
+components. For that reason, the reconstructed spectra, af -
+ter their continuum has been normalized to one, have a S/N-
+ratio lower than the composite observed spectra. In partic-
+ular, for the less luminous stars C and D the effective S/N
+resulted between 50 and 80.
+3 SPECTRAL CHARACTERISTICS AND LINE
+PROFILE VARIABILITY
+The primary component of this subsystem is a typical vari-
+able BpSi star with weak He lines. The spectrum of the com-
+ponent B appears rather normal with strong He and Si lines.
+However, a careful inspection of the high-S/N UVES spectra
+revealed thepresence ofsome asymmetrical and variable lin e
+c/circlecop†rt0000 RAS, MNRAS 000, 000–000The quadruple system AOVel 3
+Figure 1. Radial velocity curves forthe system AB (upper panel)
+CD (lower panel). Squares, triangles, and small circles are the
+UVES, FEROS, CASLEO observations, respectively. Filled sy m-
+bols correspond to the more massive component in each subsys -
+tem (A and C).
+profiles in the spectrum of both stars A and B, which could
+be related to the presence of He, Si, and Mg spots. Since
+a basic assumption for the reconstruction of the component
+spectra is the constancy with phase of the spectral morphol-
+ogy of each star, the disentangled spectra may have small
+artificial features around spectral lines presenting varia bil-
+ity. For that reason we preferred, for the analysis of profile
+variability, touseonly those spectral lines andorbital ph ases
+for which the line under consideration is clearly free of con -
+tamination from any other line belonging to the remaining
+components. The first two UVES spectra are particularly
+useful for this purpose since in them both binary systems
+are near opposite quadratures, allowing us toidentify clea rly
+the four spectral components.
+In Fig.2 we show the profile of several Si ii, Mgii, and
+Heilines, the atomic species that present the most conspic-
+uous lines in the spectra of stars A and B. Both stars A and
+B exhibit line profile variability. Figure3 shows the spectr al
+region around the Mg iidoublet at λ4481 and the He iline
+λ4471. The upper spectra are the two UVES spectra taken
+at quadratures. In the lower spectra, synthetic spectra of C
+and D have been subtracted from the former. The Mg iiline
+atλ4481 shows the same behavior as silicon lines. In this
+figure we have plotted the observed spectra to demonstrateFigure 2. Line profiles variations for stars A (panel a) and B
+(panel b) near two opposite quadratures: phase 0.34 (left-h and
+part of each panel) and 0.71 (right-hand part of each panel). The
+spectral line profiles shown are, from top to bottom: He iλλ4471,
+4026, Si iiλλ6371, 6347, 5056, 5041, and Mg iiλ4481.
+that the four components are well resolved, especially star
+B. Consequently, the removal of components C and D does
+not affect significantly the shape of the line profiles. As a
+matter of fact, the profile difference between the two phases
+is evident even in the original composite spectra. In the cas e
+of star A, the lines are blended with those of component
+C in one of the two UVES spectra, making somewhat less
+confident the analysis of profile variations.
+We note that, even though the reflexion effect due to
+the mutual irradiation of the components would distort the
+line profiles qualitatively in the same way as it is observed
+for Mgiior Siiilines, its expected contribution to the line
+profile variability is much smaller than the observed varia-
+tions. In fact, from the temperature-ratio and the relative
+separation of the components, we estimate that the profile
+variations caused by reflexion effect would be about 5 times
+smaller than the variations observed in Mg iiλ4481. There-
+fore, the cause of these profile variations is not related to
+proximity effects but to asymmetries in the surface chemi-
+cal distribution.
+Helium lines are weak, especially in star A, so it is not
+possible to detect clear variations in the line profiles from
+the available spectroscopic data. However, we detect that
+the strength of He lines at λ4026 and λ4471 is variable.
+Other strong He ilines (λλ4121, 4922, 5016, 5876) appear
+blended with Fe or Si lines belonging to the companions,
+making them less useful for the analysis of the line profile
+variability.
+For star B the spectral line profiles of Mg iiand
+Siiiappear strongly variable, as presented in Figs.2
+and 3. Usually, magnesium does not present large
+equivalent width variations in magnetic peculiar stars
+(Leone, Catalano, & Malaroda 1997). However, line profile
+c/circlecop†rt0000 RAS, MNRAS 000, 000–0004J. F. Gonz´ alez et al.
+Figure 3. Line profiles variations in components A and B.
+The two upper spectra are the observed UVES spectra for
+MJD53669.3 and 53670.3, respectively. Labels A, B, C, and D
+mark the position of the Mg iilineλ4481 for the four compo-
+nents. In the two lower spectra C and D components have been
+removed. The lines Mg iiλ4481 and He iλ4471 are marked for
+components A and B.
+variability of the magnesium doublet at λ4481 has been re-
+portedbyKuschnig et al. (1999),whopublishedthefirstMg
+map for an Ap star (CUVir). They found that, even with-
+out showing important equivalent width variations, Mg ii
+λ4481 exhibits notorious profile variations, similar to thos e
+observed in AOVel for the components A and B. The pro-
+file shape suggests that Mg is more abundant in the region
+facing the companion star. A more complete study of line
+profiles, that would allow to determine the surface distribu -
+tion of chemical elements, will require a larger number of
+spectra obtained at different phases.
+We can assume that the stellar rotation in the short-
+period eccentric binary AB is synchronized with the angular
+velocity at periastron, so that the rotational period differ s
+from the orbital period by a factor 1.163 ( Porb= 1.5846 d,
+Prot= 1.3622 d). This fact implies that the stellar surface
+visible at a given orbital phase varies with time, completin g
+one cycle in 9.7 days, which corresponds to about 6 orbital
+cycles. A multi-epoch study of AOVel would be of great
+interest to prove whether the location of stellar spots is fix ed
+on the star surface or is related to the position of the stella r
+companion.
+In our spectra of the C and D components the most
+interesting, and rather unexpected fact is the presence of
+spectral lines typical of HgMn stars. The Hg iilineλ3984
+is present in both stars C and D, while strong Y iiandFigure 4. Mass-radius diagram for the components of the more
+massive binary. The parameters of stars A and B, and their err or
+(1-sigma) are represented by the position and size of the eli pses.
+Continuous lines are the ZAMS and the isochrone log(age)=7. 6.
+Dashed lines, from left to right, are isotherms in the range
+T=12000–14500K, with a step of 500K.
+Ptiilines are present in the spectrum of component D. Al-
+though comparatively weak, some Mn iilines have been also
+identified in both components. In the reconstructed spec-
+trum of star D we identified Pt iilines at 4046 and 4514
+and more than two dozens Y iilines. No noticeable lines of
+Zrii, Scii, Baii, Pii, Xeii, Gaiiwere detected in stars C or
+D. The measured central wavelength of Hg iiline at 3984,
+3983.975 ˚A and 3984.08 ˚A for stars C and D, respectively,
+indicates that heavy Hg isotopes are predominant in these
+components. We will return to this point in the next section.
+The weakness of spectral lines for the two less massive
+stars makes impossible to study line profile variations in
+their spectra.
+4 CHEMICAL ABUNDANCES
+The abundances were determined with the synthetic spec-
+trum method using ATLAS9 model atmospheres and the
+SYNTHE code (Kurucz 1993). Stellar temperatures of
+companions C and D were estimated from the correla-
+tion of excitation potentials of Fe lines with abundances
+(Gonz´ alez, Nesvacil, & Hubrig 2008). Microturbulent velo c-
+ities were derived from the correlation of equivalent width s
+of Fe lines with abundances.
+In case of stars A and B, however, it was not possible
+to apply the same procedure since Fe lines are weaker and
+they appear broader because of the higher rotation. Esti-
+matedtemperaturesforthesestarswerederivedfromtheob-
+served masses and radii, interpolating in the grid of Geneva
+theoretical stellar models (Lejeune & Schaerer 2001) for so -
+lar composition. Fig.4 shows the position of stars A and B
+in the mass-ratio diagram. Both stars are located close the
+ZAMS.Asis shown inthisfigure, themasses andradii deter-
+mined from light and RV curves correspond to temperatures
+of 13900±500K and 13200 ±450K for components A and B,
+respectively. Surface gravity were calculated directly fr om
+masses and radii. The adopted atmospheric parameters are
+listed in Table3.
+c/circlecop†rt0000 RAS, MNRAS 000, 000–000The quadruple system AOVel 5
+Table 3. Atmospheric parameters
+Star T eff logg ζ v sini
+K kms−1kms−1
+A 13900 ±500 4.26 ±0.03 2 65
+B 13200 ±450 4.31 ±0.03 2 60
+C 12000 ±300 4.3 ±0.2 2 40
+D 11500 ±300 4.2 ±0.2 1 18
+Table 4. Abundances (log[ N/N(H)]). Less precise values (error
+= 0.2–0.4) are marked with a colon.
+Element Star A Star B Star C Star D Sun
+He -2.41 -1.16 -2.0: ≤-3.0: -1.07
+C -4.62 -3.87 -3.5: -3.5: -3.48
+O -3.71 -3.56 -3.7: -3.2: -3.17
+Mg -5.50 -4.76 -4.76 -4.96 -4.42
+Si -3.81 -4.74 -4.28 -4.84 -4.45
+P -5.7: -6.55
+S -4.67 -4.67 -4.67 -4.67
+Ca -5.69
+Ti ≤-7.5≤-8.0 -7.0: -6.37 -6.98
+Cr ≤solar -6.3: -5.87 -6.33
+Mn -5.65 -6.61
+Fe -5.09 -4.99 -4.25 -4.56 -4.50
+Ni ≤-6.8 -5.77
+Sr -8.27 -9.03
+Y -7.15 -9.76
+Pt -5.94 -10.2
+Hg -5.05 -5.70 -10.9
+The results of the abundance analysis are presented in
+Table4. For comparison we list in the last column the solar
+abundances adopted from the work by Grevesse & Sauval
+(1998). The abundances of elements not listed in this table
+were assumed to be solar. The complete list of the spec-
+tral lines used for abundaces calculation is included in the
+AppendixA.
+Star A exhibits an overabundance of Si by 0.64dex and
+underabundance of He by -1.3 ±0.3dex. This explains why
+He lines are stronger in the star B, even when A is hot-
+ter than B according to the depths of photometric minima.
+Star B appears similar to a normal late B-type star, having
+the abundances of helium and silicon close to solar values.
+However, Fe seems to be slightly underabundantin its atmo-
+sphere. No line of Ti has been observed either in star A or
+B, indicating that Ti is underabundant in both stars. Mg is
+underabundantin star A by −1.1 dex. Similar values are fre-
+quently observed in magnetic Ap and Bp stars (Leone et al.
+1997).
+Wenotethatthepresentedabundancesinstars AandB
+should be taken with caution, since the detected line profile
+variations suggest a non-uniform chemical distribution of at
+least a few elements.
+The subsystem CD shows abundances which are typi-
+cal for HgMn stars. Star C shows a strong overabundance
+of Hg by 5.8dex. Other metals like Fe, Ti, Mg, and Si show
+normal solar abundances in star C. Abundances of C, O,Ga, Sc, and Sr seem also to be solar. Helium lines are only
+marginally detected in star C and the corresponding abun-
+dance is consequently rather uncertain (about 0.4 dex). It i s
+clear, however, that He is underabundant in star C.
+Star D exhibits overabundances of Hg by 5.2dex, Y by
+2.5dex, Pt by 5.2dex, and Sr by 0.8dex, relative to the solar
+values. Ga and Sc lines are not observed.
+The uncertainties of the abundance determination, typ-
+ically 0.1–0.2, are somewhat larger than in usual analyses
+even though the observed spectra have high S/N-ratio and
+high resolution. Three different reasons contribute to the
+degradation of the accuracy. First, the effective S/N-ratio
+of the normalized spectra of the individual components is
+comparatively low, since the line strength is diluted by the
+continua of the 4 components. In the case of stars C and D,
+for example, the S/N is diminished by a factor 6. Second,
+the disentangling process using a small number of observed
+spectra might generate small fake features, especially clo se
+to strong variable lines. Finally, the uncertainty of the li ne
+equivalent widths involves the uncertainty in the relative
+light contribution of each component to the total light of
+the system, i.e. the normalization scale factors adopted fr om
+the photometric data. We note however that, the chemical
+peculiarities mentioned above are much larger than the un-
+certainties, and the classification of stars C and D as HgMn
+stars is absolutely out of doubt.
+In Fig.5 we present the synthetic spectrum of star
+D in the spectral region containing Pt ii4046, Pt ii4061,
+Srii4077, and several Fe and Cr lines. As it is visible in this
+figure,Niis underabundantinthis star. Fig.6shows thesyn-
+thetic spectra for the regions of the HgII line at λ3984. Even
+when the rotational broadening is significant, some informa -
+tion about the isotopic composition can be derived from the
+reconstructed spectra of components C and D. With this
+aim the line profile was fitted with a synthetic spectrum in
+which Hg is assumed to be a mixture of the lightest (196)
+and the heaviest (204) isotopes. As it can be seen in the fig-
+ure, the best fit suggests that the heaviest isotope is by far
+the most abundant in both components of the less massive
+subsystem.
+5 DISCUSSION
+Using high-S/N high-resolution spectra obtained with
+UVES, we have determined chemical abundance for all four
+components of the multiple system AOVel. Further, line
+profile variations were detected in the spectra of the two
+most massive stars belonging to the AB system.
+Since the multiple system AOVel is formed by four
+gravitationally bound stars, we can assume that they have
+the same original chemical composition and the same age.
+All components are close to the ZAMS and have already de-
+veloped different chemical anomalies in their atmospheres,
+depending on their temperature and probably due to the
+membership in binary systems. The observed masses and
+radii for the components of the eclipsing pair suggest that
+the system age is less than about 40 million years.
+The connection between HgMn peculiarity and mem-
+bership in binary and multiple systems seems to be sup-
+ported by two observational facts. On the one hand there is
+a high frequency of binaries and multiples among HgMn
+c/circlecop†rt0000 RAS, MNRAS 000, 000–0006J. F. Gonz´ alez et al.
+Figure 5. Comparison of the reconstructed spectrum of the star D and th e synthetic spectrum (gray line, red in online version) in th e
+spectral region around Pt and Sr lines. In the calculation of this synthetic spectrum, Ni abundance was assumed to be sola r so that the
+large Ni underabundance is better appreciated.
+Figure 6. Observed spectra (thin line) and synthetic spectra
+(heavy line) for stars C and D in the region of the Hg iiline
+λ3984 and the Y iilineλ3982. The isotopic composition for Hg
+was assumed to be 196=0.9%, 204=99.1% for starC and 196=4%,
+204=96% for star D.
+stars (Hubrig, Ageorges, & Sch¨ oller 2008). On the other
+hand, in some binaries the location of chemical spots seems
+to be related with the position of the binary companion
+(Hubrig et al. 2006a, see also Savanov et al. 2009). These
+studies indicates that the presence of a companion can play
+a significant role in the development and distribution of
+chemical anomalies on the surface of HgMn stars, proba-
+bly due to proximity effects on the atmospheric temper-
+ature or due to the magnetic field induced in a close bi-
+nary system. The role that magnetic fields possibly play in
+the development of anomalies in HgMn stars, which mostlyappear in binary systems, has never been critically tested
+by astrophysical dynamos. Recent magnetohydrodynamical
+simulations revealed a distinct structure for the magnetic
+field topology similar to the fractured elemental rings ob-
+served on the surface of HgMn stars (see e.g. Fig. 1 in
+Hubrig, Gonz´ alez, & Arlt 2008). The combination of differ-
+ential rotation and a poloidal magnetic field was studied nu-
+merically by the spherical MHD code of Hollerbach (2000).
+The presented typical patterns of the velocity and the mag-
+netic field on the surface of the star may as well be an indi-
+cation for element redistribution on (or in) the star.
+Generally He and Si variable Bp stars possess large-
+scale organized magnetic fields which in many cases appear
+to occur essentially under the form of a single large dipole
+located close to the centre of the star. The magnetic field is
+usually diagnosed through observations of circular polari za-
+tion in spectral lines. Hubrig et al. (2006b) used the multi-
+mode instrument FORS1 mounted on the 8 m Kueyen tele-
+scopeoftheVLTtomeasurethemeanlongitudinalmagnetic
+field in the system AO Vel. No detection was achieved with
+the single low-resolution measurement (R=2000) resulting
+in/angbracketleftBz/angbracketright= 80±64G. We should, however, keep in mind that
+the observed spectropolarimetric spectrum is a composite
+spectrum consisting of four spectropolarimetric spectra b e-
+longing to A, B, C and D components. Each of the com-
+panions can possess an individual magnetic field of different
+geometry, strength and polarity. Since the inhomogeneous
+elemental distribution of Si and Mg is also observed on the
+surface of the component B, the presence of a large-scale
+organized magnetic field is quite possible in this component
+too. The components C and D show the peculiarity char-
+acter typical of HgMn stars. Hubrig et al. (2006b) showed
+that longitudinal magnetic fields in HgMn stars are rather
+weak, of the order of a few hundred Gauss and less, and
+the structure of their fields must be sufficiently tangled so
+c/circlecop†rt0000 RAS, MNRAS 000, 000–000that it does not produce a strong net observable circular
+polarization signature.
+Thus, the non-detection of a magnetic field in the sys-
+tem AO Vel can be possibly explained by the dilution of the
+magnetic signal due to the superposition of four differently
+polarized spectra. On the other hand, the magnetic field of
+each component should be detectable with high resolution
+spectropolarimeters making use of the Zeeman effect in in-
+dividual metal lines.
+The time scale for the peculiarities to be developed re-
+main unclear since the results of studies of evolutionary st a-
+tus of chemically peculiar stars of different type show some-
+what contradictory results. The early work by Abt (1979)
+suggested that the frequency of peculiar stars of Si and
+HgMn groups increases with age and particularly there ex-
+ist no peculiar star on the ZAMS. In a more recent pho-
+tometric search for peculiar stars in five young clusters,
+Paunzen, Pintado, & Maitzen (2002) also concluded that
+theCPphenomenonneedsatleastseveralMyrtostartbeing
+effective. On the other hand, some chemically peculiar stars
+has been detected within very young associations or even
+in star forming regions like Ori OB1 (Abt & Levato 1977;
+Woolf & Lambert 1999), Lupus 3 (Castelli & Hubrig 2007),
+and L988 (Herbig & Dahm 2006). The results of the present
+paper show the existence of coeval stars with chemical pecu-
+liarities of Si and HgMn types, close to the ZAMS indicat-
+ing that the age threshold for these peculiarities is simila r
+for both subgroups of chemically peculiar stars. Our study
+supports the idea that these chemical peculiarities origin ate
+quite soon after the star formation.
+It is noteworthy, that the subsystem CD, the less mas-
+sive binary of the AOVel, shows several similarities with
+the eclipsing binary ARAur belonging to the Aur OB1
+association. Both systems are formed by stellar compo-
+nents very close to the ZAMS or even in the pre-MS stage
+(Nordstrom & Johansen 1994; Gonz´ alez et al. 2006), and
+they have almost the same orbital period, 4d.13 and 4d.15
+for ARAur and AOVelCD, respectively. In both systems
+the primary is aHgMn star, but while in the AOVel CD sys-
+tem the secondary companion shares the same peculiarity,
+in ARAur the secondary is a normal star (Khokhlova et al.
+1995). In fact, the primary component of ARAur should
+be compared with star D of the system AOVel, since they
+have a similar effective temperature. This similarity in the
+physical and orbital properties seems to have been trans-
+lated into the development of similar pattern of chemical
+abundances: notorious overabundance of Hg and Pt, over-
+abundance of Sr, Y, and Mn at the level of 1–3 dex, slight
+overabundance of Cr and Ti, and the abundance of Si and
+Mg close to the solar abundance, or slightly subsolar. We se-
+lected a few additional HgMn spectroscopic binaries with ef -
+fective temperatures close to 11,000K: HD32964, HD89822,
+HD173524, and HD191110. In Fig.7 the abundances of var-
+ious elements in ARAur and selected binaries are compared
+with those of the AOVel D companion. The abundances are
+presented for HgMn stars member of double-lined spectro-
+scopic binaries with periods between 4 and 12 days and hav-
+ing effective temperatures in the range 10700–11700K. For
+HD173524 and HD191110 we plotted the abundances de-
+termined in both components. The abundances values have
+been taken from Adelman (1994), Catanzaro & Leto (2004),
+and Catanzaro, Leone, & Leto (2003). It is remarkable, howFigure 7. Abundances of the AOVel D companion
+(filled squares) compared with published abundances for an-
+other seven binary components with similar effective tem-
+peratures: HD32964B (circles), HD34364A (open squares),
+HD89822A (pentagons), HD173524A (stars), HD173524B (tri-
+angles), HD191110A (plusses), and HD191110B (crosses). Th e
+dashed line represents the solar abundance.
+much similar the chemical composition of these stars is. We
+note, however, that the abundance distribution on the stel-
+lar surface of HgMn stars is probably inhomogeneous, and
+consequently, the chemical composition derived from spec-
+troscopic observations that do not cover the rotational cy-
+cle, should be taken just as indicative values until accurat e
+abundancevalues obtained usingDoppler imaging technique
+become available. The spotted character of magnetic Bp-Ap
+stars is well known, but also in the case of HgMn stars the
+presence of chemical spots or belts on the surface is not
+uncommon (Hubrig, Gonz´ alez, & Arlt 2008; Savanov et al.
+2009).
+As already mentioned above, the atmospheric chemical
+composition of ARAur exhibits a non-uniform surface
+distribution with a very interesting pattern related to the
+position of the companion (Hubrig et al. 2006a). A similar
+study in AOVel would be worthwhile to verify whether
+the elemental distribution on the stellar surfaces of C and
+D components show a similar behavior. Furthermore, an
+intensive spectroscopic campaign that would allow the
+reconstruction of chemical maps for all four components
+should provide important information for the proper
+understanding of the origin of chemically peculiar stars.
+APPENDIX A: LIST OF SPECTRAL LINES
+TableA lists all the spectral lines used for abundance deter -
+mination. The lines marked with an asterisk are blends. The
+abundance was determined using the fitting by a synthetic
+spectrum.
+We thank to Charles R. Cowley and Rainer Arlt for
+helpful discussions.[]
+Table A1. Spectral lines used for abundace determination.
+ion λ(˚A) loggf Source log N/H
+A B C D
+HeI 4026.1844∗−2.625 NIST3 −2.3−1.2−2.0≤−3.0
+4026.1859∗−1.448 NIST3
+4026.1860∗−0.701 NIST3
+4026.1968∗−1.449 NIST3
+4026.1983∗−0.972 NIST3
+4026.3570∗−1.324 NIST3
+4387.9291 −0.883 NIST3 −− − 1.2−− −−
+4471.4704∗−2.203 NIST3 −3.0−1.25−2.0≤−3.0
+4471.4741∗−1.026 NIST3
+4471.4743∗−0.278 NIST3
+4471.4856∗−1.025 NIST3
+4471.4893∗−0.550 NIST3
+4471.6832∗−0.903 NIST3
+4921.9310 −0.435 NIST3 −1.8−1.2−− −−
+5015.6776 −0.820 NIST3 −− − 1.0−− −−
+5875.5987∗−1.516 NIST3 −2.7−0.95≤−2.4−2.0
+5875.6139∗−0.341 NIST3
+5875.6148∗+0.408 NIST3
+5875.6251∗−0.340 NIST3
+5875.6403∗+0.137 NIST3
+5875.9663∗−0.215 NIST3
+6678.1517 +0.329 NIST3 −2.2−0.9≤−3.0−1.8
+CII 4267.001∗+0.563 NIST3 −4.62−3.87−3.52−3.52
+4267.261∗+0.716 NIST3
+4267.261∗−0.584 NIST3
+OI 6155.961∗−1.363 NIST3 −3.71−3.56−3.71−3.21
+6155.971∗−1.011 NIST3
+6155.989∗−1.120 NIST3
+6156.737∗−1.487 NIST3 −3.71−3.56−3.71−3.21
+6156.755∗−0.898 NIST3
+6156.778∗−0.694 NIST3
+6158.149∗−1.841 NIST3 −3.71−3.56−3.71−3.21
+6158.172∗−0.995 NIST3
+6158.187∗−0.409 NIST3
+MgII 4481.126∗+0.749 NIST3 −5.5−4.76−4.76−4.96
+4481.150∗−0.553 NIST3
+4481.325∗+0.594 NIST3
+SiII 3853.665 −1.341 NIST3 −4.4−4.3−5.6−5.0
+3856.018 −0.406 NIST3 −4.4−4.9−5.6−5.0
+3862.595 −0.757 NIST3 −4.0−4.9−5.6−5.0
+3954.300∗−1.040 KP −3.6−− −− −−
+3954.504∗−0.880 KP
+4128.054 +0.359 NIST3 −4.1−4.8−4.3−5.2
+4130.872∗−0.783 NIST3 −4.1−4.8−4.3−5.0
+4130.894∗+0.552 NIST3
+4187.128∗−1.050 KP −3.7−− −− −−
+4187.128∗−2.590 KP
+4187.151∗−1.160 KP
+4190.724 −0.351 LA −3.7−− −− −−
+4198.133 −0.611 LA −3.7−− −− −−
+4200.658∗−0.889 NIST3 −3.7−− −− −−
+4200.887∗−2.034 NIST3
+4200.898∗−0.733 NIST3
+5041.024 +0.029 NIST3 −3.8−4.6−3.9−4.5[]
+Table A1 –continued
+ion λ(˚A) loggf Source log N/NH
+A B C D
+5055.984∗+0.523 NIST3 −4.0−4.7−4.3−4.8
+5056.317∗−0.492 NIST3
+5185.232∗+0.472 LA −3.8−− −− −−
+5185.520∗−1.603 NIST3
+5185.520∗−0.302 NIST3
+5185.555∗−0.456 NIST3
+5688.817 +0.126 NIST3 −3.6−− −− −−
+5701.370 −0.057 NIST3 −3.6−− −− −−
+5957.559 −0.225 NIST3 −4.0−4.9−4.6−4.8
+5978.930 +0.084 NIST3 −4.0−5.4−4.6−4.8
+6347.103 +0.149 NIST3 −3.6−4.4−4.0−4.6
+6371.371 −0.082 NIST3 −3.7−4.5−4.3−4.6
+PII 4178.463 −0.409 HI −6.59−− −− −−
+6024.178 +0.137 NIST3 −5.39−− −− −−
+6034.039 −0.220 NIST3 −5.39−− −− −−
+6043.084 +0.416 NIST3 −5.39−− −− −−
+SII 4153.068 +0.617 NIST3 −4.71−4.71−4.71−4.71
+5027.203 −0.705 NIST3 −4.71−4.71−4.71−4.71
+TiII 4399.765 −1.190 PTP ≤−7.5≤−8.0−7.02−−
+4411.072 −0.670 PTP ≤−7.5≤−8.0−7.02−6.37
+4418.714 −1.970 PTP ≤−7.5≤−8.0−7.02−6.37
+4468.492 −0.620 NIST3 ≤−7.5≤−8.0−7.02−6.37
+CrII 4261.913 −1.531 K03Cr −− −− − 6.37−5.87
+5237.329 −1.160 NIST3 −− −− − 6.37−5.87
+5313.590 −1.650 NIST3 −− −− − 6.37−5.87
+MnII 4206.367 −1.590 K03Mn −− −− − 5.65−5.65
+4478.635 −0.942 K03Mn −− −− − 5.65−5.65
+FeII 4178.862 −2.440 FW06 −5.75−5.0−− −−
+4233.172 −1.809 FW06 −− −− − 4.24−4.54
+4303.176 −2.610 FW06 −5.75−5.0−− −−
+4385.387 −2.580 FW06 −5.75−5.0:−− −−
+4416.830 −2.600 FW06 −5.75−5.0−3.50−4.14
+4923.927 −1.210 FW06 −5.75−5.3−4.24−4.85
+5018.440 −1.350 FW06 −4.90−4.8−4.00−4.54
+5100.607∗+0.144 K09 −− −− − 4.24−4.54
+5100.734∗+0.671 J07
+5169.033 −0.870 FW06 −5.20−4.8−4.24−4.54
+5197.577 −2.054 FW06 −− −− − 4.34−4.70
+5227.483 +0.831 J07 −− −− − 4.24−4.54
+5234.625 −2.210 FW06 −− −− − 4.34−4.54
+5276.002 −1.900 FW06 −4.8−5.8−4.54−4.74
+5316.615 −1.780 FW06 −4.8−5.0−4.34−4.74
+5362.741∗−0.708 K09 −− −− − 4.00−4.34
+5362.869∗−2.855 K09
+5362.967∗+0.008 K09
+NiII 4015.474 −2.410 K03Ni −− −− −− ≤− 6.79
+4067.031 −1.834 K03Ni −− −− −− ≤− 6.79
+SrII 4077.709 +0.151 NIST3 −− −− −− − 8.27[]
+Table A1 –continued
+ion λ(˚A) loggf Source log N/NH
+A B C D
+YII 3982.592 −0.493 NIST3 −− −− −− − 7.45
+4235.727 −1.509 NIST3 −− −− −− − 6.60
+4309.620 −0.745 NIST3 −− −− −− − 6.95
+4883.684 +0.071 NIST3 −− −− −− − 7.25
+4900.120 −0.090 NIST3 −− −− −− − 7.25
+5200.406 −0.579 NIST3 −− −− −− − 7.45
+5205.722 −0.342 NIST3 −− −− −− − 7.15
+PtII 4046.443 −1.190 ENG −− −− −− − 5.94
+4061.644 −1.890 ENG −− −− −− − 5.94
+4288.371 −1.570 ENG −− −− −− − 5.94:
+HgII 3983.890 −1.510 NIST3 −− −− −− − 6.55
+NIST3: NIST atomic spectra Database, version 3 at http://ph ysics.nist.gov;
+FW06 : Fuhr & Wiese (2006)
+PTP: Pickering, Thorne, & Perez (2002)
+LA : Lanz & Artru (1985)
+KP : Kurucz & Peytremann (1975)
+HI : Hibbert (1988)
+J07: Johannson (2007), private communication
+K03Cr:Kurucz (2003), http://kurucz.harvard.edu/atoms/ 2401/gf2401.pos
+K03Mn:Kurucz (2003), http://kurucz.harvard.edu/atoms/ 2501/gf2501.pos
+K03Ni:Kurucz (2003), http://kurucz.harvard.edu/atoms/ 2801/gf2801.pos
+K09: Kurucz (2009), http://kurucz.harvard.edu/atoms/26 01/gf2601.pos
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\ No newline at end of file
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+arXiv:1001.0513v1  [math.PR]  4 Jan 2010Intersection local times of independent
+fractional Brownian motions as generalized
+white noise functionals
+Maria Jo˜ ao Oliveira
+Universidade Aberta, P 1269-001 Lisbon, Portugal
+CMAF, University of Lisbon, P 1649-003 Lisbon, Portugal
+oliveira@cii.fc.ul.pt
+Jos´ e Lu´ ıs da Silva
+DME, University of Madeira, P 9000-390 Funchal, Portugal
+CCM, University of Madeira, P 9000-390 Funchal, Portugal
+luis@uma.pt
+Ludwig Streit
+Forschungszentrum BiBoS, Universit¨ at Bielefeld, D 33501 Bielefeld, Germany
+CCM, University of Madeira, P 9000-390 Funchal, Portugal
+streit@physik.uni-bielefeld.de
+Abstract
+In this work we present expansions of intersection local tim es of
+fractional Brownian motions in Rd, for any dimension d≥1, with
+arbitrary Hurst coefficients in (0 ,1)d. The expansions are in terms
+of Wick powers of white noises (corresponding to multiple Wi ener
+integrals), being well-defined in the sense of generalized w hite noise
+functionals. As an application of our approach, a sufficient c ondition
+ondfor the existence of intersection local times in L2is derived,
+extending the results in [NOL07] to different and more general Hurst
+coefficients.
+Keywords: Fractional Brownian motion; White noise analysis; Local time
+2000 AMS Classification: 60H40, 60G15, 60J55, 28C20, 46F25, 82D60
+11 Introduction
+In the recent years the fractional Brownian motion has become an object
+of intense study, namely, due to its special properties, such as sh ort/long
+range dependence and self-similarity, yielding its proper and natura l uses in
+several applications in different fields (e.g. mathematical finances [M O08],
+telecommunications engineering [NRT03]).
+Besidesitsownspecificproperties, theintersectionpropertiesof fractional
+Brownian motion paths have been studied by many authors as well, se e
+e.g. the works done by Gradinaru et al. [GRV03], Nualart et al. [HN07],
+[HN05], Rosen [Ros87], and the references therein.
+One may consider intersections of sample paths with themselves, as in
+[DOS08] and references therein, or with other independent fract ional Brow-
+nian motions, as in [NOL07].
+This work concerns the latter standpoint. Within the white noise ana ly-
+sis framework (Section 2), a first purpose of this work is an extens ion of the
+results presented in [AOS01] to two d-dimensional independent fractional
+Brownian motions BH1andBH2with different Hurst coefficients, H1and
+H2. Technically, this approach has the advantage that the underlying prob-
+ability space does not depend on any Hurst coefficient under conside ration.
+As a consequence, one may analyze the intersection local time of an y two
+independent fractional Brownian motions, without any restriction on the
+corresponding Hurst coefficients.
+From the viewpoint of applications to physics, this absence of restr ictions
+on the Hurst coefficients under consideration is meaningful to widen the
+modelling of polymers towards polymers molecules handling different ty pes
+of polymers.
+For low dimensions, that is, either for d= 1 or for d= 2, the white noise
+analysis framework allows the definition of the intersection local time of any
+two independent fractional Brownian motions BHiin terms of an integral
+over a Donsker’s δ-function
+L≡/integraldisplay
+d2tδ(BH1(t1)−BH2(t2)),
+intended to sum up the contributions from each pair of moments of t imet1,
+t2for which the fractional Brownian motions BHiarrive at the same point.
+A rigorous definition, such as, e.g., through a sequence of Gaussian s ap-
+2proximating the δ-function,
+(2πε)−d/2exp/parenleftbigg
+−|x|2
+2ε/parenrightbigg
+, ε >0,
+will make Lincreasingly singular, and various “renormalizations” have to be
+done as the dimension dincreases. Of course, besides the dimension of the
+space, the type of “renormalizations” needed depends as well on t he Hurst
+coefficients Hi∈(0,1)dbeing considered. For d >2 with 1 /maxjH1,j+
+1/maxjH2,j≤d, the expectation diverges in the limit and must be sub-
+tracted. Depending on the values of max jHi,j, further kernel terms must be
+also subtracted (Theorem 9).
+In this work we are particularly interested in the chaos decompositio n
+ofL. We expand Lin terms of Wick powers [HKPS93] of white noise, an
+expansion which corresponds to that in terms of multiple Wiener integ rals
+when one considers the Wiener process as the fundamental rando m variable.
+This allows us to derive the kernels for L. Due to the local structure of the
+Wickpowers, thekernel functionsarerelatively simpleandexhibitcle arlythe
+dimension dependence singularities of L(Proposition 8). For comparison, we
+also calculate the regularized kernel functions corresponding to t he Gaussian
+δ-sequence mentioned above (Theorem 9).
+As an application of this approach, in Theorem 11 we derive a sufficient
+condition for the existence of the intersection local times in L2, extending the
+results obtained in [NOL07] to different and more general Hurst coe fficients.
+2 Gaussian white noise calculus
+In this section we briefly recall the concepts and results of white no ise anal-
+ysis used throughout this work (for a detailed explanation see e.g. [B K88],
+[HKPS93], [HØUZ96], [Kuo96], [Oba94]).
+2.1 Fractional Brownian motion
+Thestartingpointofwhitenoiseanalysisfortheconstructionoftw o indepen-
+dentd-dimensional, d≥1, fractional Brownian motions is the real Gelfand
+triple
+S2d(R)⊂L2
+2d(R)⊂S′
+2d(R),
+3whereL2
+2d(R) :=L2(R,R2d) is the real Hilbert space of all vector valued
+square integrable functions with respect to the Lebesgue measur e onR, and
+S2d(R),S′
+2d(R) are the Schwartz spaces of the vector valued test functions
+and tempered distributions, respectively. We shall denote the L2
+2d(R)-norm
+by|·|2d(or if there is no risk of confusion simply by |·|) and the dual pairing
+between S′
+2d(R) andS2d(R) by/an}bracketle{t·,·/an}bracketri}ht2d, or simply by /an}bracketle{t·,·/an}bracketri}ht, which is defined as
+the bilinear extension of the inner product on L2
+2d(R), i.e.,
+/an}bracketle{tg,f/an}bracketri}ht2d=2d/summationdisplay
+i=1/integraldisplay
+Rdxgi(x)fi(x),
+for allg= (g1,...,g2d)∈L2
+2d(R) and all f= (f1,...,f2d)∈S2d(R). By the
+Minlos theorem, there is a unique probability measure µon theσ-algebraB
+generated by the cylinder sets on S′
+2d(R) with characteristic function given
+by
+C(f) :=/integraldisplay
+S′
+2d(R)dµ(/vector ω)ei/an}bracketle{t/vector ω,f/an}bracketri}ht=e−1
+2|f|2,f∈S2d(R).
+In this way we have defined the white noise measure space ( S′
+2d(R),B,µ).
+To construct two independent d-dimensional fractional Brownian motions
+we shall consider a 2 d-tuple of independent Gaussian white noises
+/vector ω:= (/vector ω1,/vector ω2), /vector ωi= (ωi,1,...,ωi,d),i= 1,2.
+Within this formalism, a version of a d-dimensional Wiener Brownian motion
+is given by
+B(t) :=/parenleftbig
+/an}bracketle{tω1,1 1[0,t]/an}bracketri}ht,...,/an}bracketle{tωd,1 1[0,t]/an}bracketri}ht/parenrightbig
+,(ω1,...,ωd)∈S′
+d(R),
+where 1 1 Adenotes the indicator function of a set Aand/an}bracketle{t·,·/an}bracketri}ht=/an}bracketle{t·,·/an}bracketri}ht1. For
+an arbitrary d-dimensional Hurst parameter H= (H1,...,H d)∈(0,1)d, a
+version of a d-dimensional fractional Brownian motion is given by
+BH(t) :=/parenleftbig
+/an}bracketle{tω1,MH11 1[0,t]/an}bracketri}ht,...,/an}bracketle{tωd,MHd1 1[0,t]/an}bracketri}ht/parenrightbig
+,(ω1,...,ωd)∈S′
+d(R),
+where, for a 1-dimensional Hurst parameter H∈(0,1) and for a generic real
+4valued function f,
+(MHf)(x) :=
+
+(1
+2−H)KH
+Γ/parenleftbig
+H+1
+2/parenrightbiglim
+ε→0+/integraldisplay∞
+εdyf(x)−f(x+y)
+y3
+2−H, H∈(0,1/2) (∗)
+f(x), H=1
+2
+KH
+Γ/parenleftbig
+H−1
+2/parenrightbig/integraldisplay∞
+xdyf(y)(y−x)H−3
+2, H∈(1/2,1) (∗∗),
+provided the limit in ( ∗) exists for almost all x∈Rand the integral in
+(∗∗) exists for all x∈R(for more details see e.g. [Ben03] and [PT00] and
+the references therein). Independently of the case under cons ideration, the
+normalizing constant KHis given by
+KH= Γ/parenleftbigg
+H+1
+2/parenrightbigg/parenleftbigg1
+2H+/integraldisplay∞
+0ds/parenleftBig
+(1+s)H−1
+2−sH−1
+2/parenrightBig/parenrightbigg−1
+2
+.
+There are several examples of functions ffor which MHfexists for any
+H∈(0,1),namely, for f= 1 1[0,t]witht >0orforf∈S1(R). Formoredetails
+and proofs see e.g. [Ben03], [BHØZ08], [Mis08], [PT03], and the referen ces
+therein.
+2.2 Hida distributions and characterization results
+Let us now consider the complex Hilbert space ( L2) :=L2(S′
+2d(R),B,µ). For
+simplicity one introduces the notation
+n= (n1,···,nd)∈Nd, n=d/summationdisplay
+i=1ni,n! =d/productdisplay
+i=1ni!.
+The space ( L2) is canonically isomorphic to the symmetric Fock space of
+symmetric square integrable functions,
+(L2)≃/parenleftBig∞/circleplusdisplay
+k=0SymL2(Rk,k!dkx)/parenrightBig⊗2d
+,
+5which leads to the chaos expansion of the elements in ( L2),
+F(/vector ω1,/vector ω2) =/summationdisplay
+m/summationdisplay
+k/an}bracketle{t:/vector ω⊗m
+1:⊗:/vector ω⊗k
+2:,fm,k/an}bracketri}ht
+=/summationdisplay
+m/summationdisplay
+k/angbracketleftBiggd/circlemultiplydisplay
+i=1:ω⊗mi
+1,i:⊗d/circlemultiplydisplay
+j=1:ω⊗mj
+2,j:,fm,k/angbracketrightBigg
+,
+with kernel functions fm,kin the Fock space, that is, square integrable func-
+tions of the m+karguments and symmetric in each mi-,kj-tuple.
+To proceed further we have to consider a Gelfand triple around the space
+(L2). We will use the space ( S)∗of Hida distributions (or generalized Brow-
+nian functionals) and the corresponding Gelfand triple ( S)⊂(L2)⊂(S)∗.
+Here (S) is the space of white noise test functions such that its dual space
+(with respect to ( L2)) is the space ( S)∗. Instead of reproducing the ex-
+plicit construction of ( S)∗(see e.g. [HKPS93]), in Theorem 2 below we
+characterize this space through its S-transform. We recall that given a
+f= (f1,f2)∈S2d(R), and the Wick exponential
+: exp(/an}bracketle{t/vector ω,f/an}bracketri}ht) ::=/summationdisplay
+m/summationdisplay
+k1
+m!k!/an}bracketle{t:/vector ω⊗m
+1:⊗:/vector ω⊗k
+2:,f⊗m
+1⊗f⊗k
+2/an}bracketri}ht=C(f)e/an}bracketle{t/vector ω,f/an}bracketri}ht2d,
+we define the S-transform of a Φ ∈(S)∗by
+SΦ(f) :=/an}bracketle{t/an}bracketle{tΦ,: exp(/an}bracketle{t·,f/an}bracketri}ht) :/an}bracketri}ht/an}bracketri}ht,∀f∈S2d(R). (1)
+Here/an}bracketle{t/an}bracketle{t·,·/an}bracketri}ht/an}bracketri}htdenotes the dual pairing between ( S)∗and (S) which is defined as
+the bilinear extension of the sesquilinear inner product on ( L2). We observe
+that the multilinear expansion of (1),
+SΦ(f) :=/summationdisplay
+m/summationdisplay
+k/an}bracketle{tFm,k,f⊗m
+1⊗f⊗k
+2/an}bracketri}ht,
+extends the chaos expansion to Φ ∈(S)∗with distribution valued kernels
+Fm,ksuch that
+/an}bracketle{t/an}bracketle{tΦ,ϕ/an}bracketri}ht/an}bracketri}ht=/summationdisplay
+m/summationdisplay
+km!k!/an}bracketle{tFm,k,ϕm,k/an}bracketri}ht, (2)
+for every generalized test function ϕ∈(S) with kernel functions ϕm,k.
+In order to characterize the space ( S)∗through its S-transform we need
+the following definition.
+6Definition 1 A function F:S2d(R)→Cis called a U-functional whenever
+1. for every f1,f2∈S2d(R)the mapping R∋λ/ma√sto−→F(λf1+f2)has an entire
+extension to λ∈C,
+2. there are constants K1,K2>0such that
+|F(zf)| ≤K1eK2|z|2/bardblf/bardbl2,∀z∈C,f∈S2d(R)
+for some continuous norm /bardbl·/bardblonS2d(R).
+We are now ready to state the aforementioned characterization r esult.
+Theorem 2 ([KLPSW96], [PS91]) TheS-transform defines a bijection be-
+tween the space (S)∗and the space of U-functionals.
+As a consequence of Theorem 2 one may derive the next two statem ents.
+The first one concerns the convergence of sequences of Hida dist ributions
+and the second one the Bochner integration of families of distributio ns of
+the same type (for more details and proofs see e.g. [HKPS93], [KLPSW 96],
+[PS91]).
+Corollary 3 Let(Φn)n∈Nbe a sequence in (S)∗such that
+(i) for all f∈S2d(R),((SΦn)(f))n∈Nis a Cauchy sequence in C,
+(ii) there are constants K1,K2>0such that for some continuous norm /bardbl·/bardbl
+onS2d(R)one has
+|(SΦn)(zf)| ≤K1eK2|z|2/bardblf/bardbl2,∀z∈C,f∈S2d(R),n∈N.
+Then(Φn)n∈Nconverges strongly in (S)∗to a unique Hida distribution.
+Corollary 4 Let(Ω,B,m)be a measure space and λ/ma√sto→Φλbe a mapping
+fromΩto(S)∗. We assume that the S-transform of Φλfulfills the following
+two properties:
+(i) the mapping λ/ma√sto→(SΦλ)(f)is measurable for every f∈S2d(R),
+(ii) theSΦλobeys aU-estimate
+|(SΦλ)(zf)| ≤C1(λ)eC2(λ)|z|2/bardblf/bardbl2, z∈C,f∈S2d(R),
+for some continuous norm /bardbl·/bardblonS2d(R)and for some C1∈L1(Ω,m),
+C2∈L∞(Ω,m).
+7Then /integraldisplay
+Ωdm(λ)Φλ∈(S)∗
+and
+S/parenleftbigg/integraldisplay
+Ωdm(λ)Φλ/parenrightbigg
+(f) =/integraldisplay
+Ωdm(λ) (SΦλ)(f).
+3 Chaos expansions
+Let us now consider two independent d-dimensional fractional Brownian mo-
+tionsBH1(t) andBH2(t) with Hurst multiparameters H1= (H1,1,...,H 1,d)
+andH2= (H2,1,...,H 2,d), respectively. That is, given a 2 d-tuple of indepen-
+dent white noises ( ω1,1,...,ω1,d,ω2,1,...,ω2,d),
+BHi(t) :=/parenleftbig
+/an}bracketle{tωi,1,MHi,11 1[0,t]/an}bracketri}ht,...,/an}bracketle{tωi,d,MHi,d1 1[0,t]/an}bracketri}ht/parenrightbig
+, i= 1,2.
+Proposition 5 For each tandsstrictly positive real numbers the Bochner
+integral
+δ(BH1(t)−BH2(s)) :=/parenleftbigg1
+2π/parenrightbiggd/integraldisplay
+Rddλeiλ(BH1(t)−BH2(s))
+is a Hida distribution with S-transform given by
+Sδ(BH1(t)−BH2(s))(f)
+=/parenleftbigg1√
+2π/parenrightbiggdd/productdisplay
+j=11√
+t2H1,j+s2H2,j· (3)
+·e−1
+2/summationtextd
+j=11
+t2H1,j+s2H2,j(/integraltext
+Rdx(f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x)))2
+,
+for allf= (f1,1,...,f1,d,f2,1,...,f2,d)∈S2d(R).
+Proof.The proof of this result follows from an application of Corollary 4 to
+theS-transform of the integrand function
+Φ(/vector ω1,/vector ω2) :=eiλ(BH1(t)−BH2(s)), /vector ωi= (ωi,1,...,ωi,d),i= 1,2,
+with respect to the Lebesgue measure on Rd. For this purpose we begin by
+observing that since the fractional Brownian motions are independ ent one
+has
+SΦ(f) =SeiλBH1(t)(f1)·Se−iλBH2(s)(f2)
+8for every f= (f1,f2)∈S2d(R),f1:= (f1,1,...,f1,d),f2:= (f2,1,...,f2,d). Hence,
+according e.g. to [HKPS93], for all λ= (λ1,...,λd)∈Rdwe obtain
+SΦ(f) =d/productdisplay
+j=1eiλj/integraltext
+Rdx(f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x))e−1
+2λ2
+j(t2H1,j+s2H2,j),
+(4)
+which clearly fulfills the measurability condition. Moreover, for all z∈Cwe
+find
+|SΦ(zf)|
+=d/productdisplay
+j=1e−1
+4λ2
+j(t2H1,j+s2H2,j)·
+·d/productdisplay
+j=1/vextendsingle/vextendsingle/vextendsinglee−1
+4λ2
+j(t2H1,j+s2H2,j)+izλj/integraltext
+Rdx(f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x))/vextendsingle/vextendsingle/vextendsingle
+≤d/productdisplay
+j=1e−1
+4λ2
+j(t2H1,j+s2H2,j)·
+·d/productdisplay
+j=1e−1
+4λ2
+j(t2H1,j+s2H2,j)+|z||λj||/integraltext
+Rdx(f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x))|,
+where, for each j= 1,...,d, the corresponding term in the second product is
+bounded by
+exp/parenleftBigg
+|z|2
+t2H1,j+s2H2,j/parenleftbigg/integraldisplay
+Rdx/parenleftbig
+f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x)/parenrightbig/parenrightbigg2/parenrightBigg
+,
+9because
+−1
+4λ2
+j(t2H1,j+s2H2,j)
++|z||λj|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Rdx/parenleftbig
+f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=−/parenleftbigg|z|√
+t2H1,j+s2H2,j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Rdx/parenleftbig
+f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+−|λj|
+2/radicalbig
+t2H1,j+s2H2,j/parenrightbigg2
++|z|2
+t2H1,j+s2H2,j/parenleftbigg/integraldisplay
+Rdx/parenleftbig
+f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x)/parenrightbig/parenrightbigg2
+.
+As a result,
+|SΦ(zf)| ≤e−1
+4/summationtextd
+j=1λ2
+j(t2H1,j+s2H2,j)·
+·e|z|2/summationtextd
+j=11
+t2H1,j+s2H2,j(/integraltext
+Rdx(f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x)))2
+,
+where, as a function of λ, the first exponential is integrable on Rdand the
+second exponential is constant.
+An application of the result mentioned above completes the proof. I n
+particular, it yields (3) by integrating (4) over λ. /squaresolid
+In order to proceed further the next result shows to be very use ful. It
+improves the estimate obtained in [Ben03, Theorem 2.3] towards the charac-
+terization results stated in Corollaries 3 and 4.
+Lemma 6 ([DOS08]) Let H∈(0,1)andf∈S1(R)be given. There is a
+non-negative constant CHindependent of fsuch that
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Rdxf(x)(MH1 1[0,t])(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤CHt/parenleftbigg
+sup
+x∈R|f(x)|+sup
+x∈R|f′(x)|+|f|/parenrightbigg
+for allt >0.
+In particular, the use of Lemma 6 allows to state the next result on
+intersection local times LH1,H2as well as on their subtracted counterparts
+L(N)
+H1,H2. There, and throughout the rest of this work as well, given a H=
+(H1,...,H d)∈(0,1)dwe shall use the notation
+¯H:= max
+j=1,...,dHj.
+10Theorem 7 LetT >0be given. For any pair of integer numbers d≥1,
+N≥0and for any pair of Hurst multiparameters H1,H2∈(0,1)dsuch that
+max{¯H1,¯H2}/parenleftbigg
+N+d
+2−1
+2min{¯H1,¯H2}/parenrightbigg
+< N+1
+2,
+the Bochner integral
+L(N)
+H1,H2:=/integraldisplayT
+0dt/integraldisplayT
+0dsδ(N)(BH1(t)−BH2(s))
+is a Hida distribution.
+Proof.To prove this result we shall again use Corollary 4 with respect to
+theLebesgue measure on[0 ,T]2. Forthis purpose let us denotethe truncated
+exponential series by
+expN(x) :=∞/summationdisplay
+n=Nxn
+n!.
+It follows from (3) that for every t,s >0 theS-transform of δ(N)(BH1(t)−
+BH2(s)) is given by
+Sδ(N)(BH1(t)−BH2(s))(f) (5)
+=/parenleftbigg1√
+2π/parenrightbiggdd/productdisplay
+j=11√
+t2H1,j+s2H2,jexpN/parenleftBigg
+−1
+2d/summationdisplay
+j=11
+t2H1,j+s2H2,j·
+·/parenleftbigg/integraldisplay
+Rdx/parenleftbig
+f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x)/parenrightbig/parenrightbigg2/parenrightBigg
+,
+which is a measurable function.
+In order to check the boundedness condition, on S2d(R) let us consider
+the norm /bardbl·/bardbldefined for all f= (f1,...,f2d)∈S2d(R) by
+/bardblf/bardbl:=/parenleftBigg2d/summationdisplay
+i=1/parenleftbigg
+sup
+x∈R|fi(x)|+sup
+x∈R|f′
+i(x)|+|fi|/parenrightbigg2/parenrightBigg1
+2
+. (6)
+We observe that on S1(R) this norm reduces to the continuous norm
+/bardblf/bardbl= sup
+x∈R|f(x)|+sup
+x∈R|f′(x)|+|f|, f∈S1(R),
+11which implies the continuity of the norm (6) for higher dimensions.
+By Lemma 6, for each j= 1,...,d, we obtain
+/parenleftbigg/integraldisplay
+Rdx/parenleftbig
+f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x)/parenrightbig/parenrightbigg2
+≤2/parenleftbigg/integraldisplay
+Rdxf1,j(x)(MH1,j1 1[0,t])(x)/parenrightbigg2
++2/parenleftbigg/integraldisplay
+Rdxf2,j(x)(MH2,j1 1[0,s])(x)/parenrightbigg2
+≤2t2C2
+H1,j/bardblf1,j/bardbl2+2s2C2
+H2,j/bardblf2,j/bardbl2,
+and thus, for all z∈Cand allf∈S2d(R),
+/vextendsingle/vextendsingleS(δ(N)(BH1(t)−BH2(s)))(zf)/vextendsingle/vextendsingle
+≤/parenleftbigg1√
+2π/parenrightbiggdd/productdisplay
+j=11√
+t2H1,j+s2H2,jexpN/parenleftbigg
+|z|2C2
+H1,H2t2+s2
+t2H(1)+s2H(2)/bardblf/bardbl2/parenrightbigg
+withCH1,H2:= max{CH1,j,CH2,j:j= 1,...,d}and
+H(1):=
+
+¯H1= max
+j=1,...,dH1,j,0< t≤1
+min
+j=1,...,dH1,j, t > 1, H(2):=
+
+¯H2= max
+j=1,...,dH2,j,0< s≤1
+min
+j=1,...,dH2,j, s > 1.
+Therefore, for 0 < t,s≤1 one has
+t2+s2
+t2H(1)+s2H(2)=t2+s2
+t2¯H1+s2¯H2≤1,
+and thus
+expN/parenleftbigg
+|z|2C2
+H1,H2t2+s2
+t2¯H1+s2¯H2/bardblf/bardbl2/parenrightbigg
+≤/parenleftbiggt2+s2
+t2¯H1+s2¯H2/parenrightbiggN
+e|z|2C2
+H1,H2/bardblf/bardbl2;
+while either for t >1 or fors >1 one finds
+expN/parenleftbigg
+|z|2C2
+H1,H2t2+s2
+t2H(1)+s2H(2)/bardblf/bardbl2/parenrightbigg
+≤/parenleftbiggt2+s2
+t2H(1)+s2H(2)/parenrightbiggN
+e|z|2C2
+H1,H2/parenleftbigg
+t2+s2
+t2H(1)+s2H(2)+N/parenrightbigg
+/bardblf/bardbl2
+.
+12As a consequence, independently of Tbeing smaller or greater than 1 there
+is always a function C=C(t,s)>0 bounded on [0 ,T]2such that
+/vextendsingle/vextendsingleS(δ(N)(BH1(t)−BH2(s)))(zf)/vextendsingle/vextendsingle
+≤/parenleftbigg1√
+2π/parenrightbiggdd/productdisplay
+j=11√
+t2H1,j+s2H2,j/parenleftbiggt2+s2
+t2H(1)+s2H(2)/parenrightbiggN
+e|z|2C2
+H1,H2C(t,s)/bardblf/bardbl2.(7)
+The proof then amounts to prove the integrability on [0 ,T]2of the expression
+d/productdisplay
+j=11√
+t2H1,j+s2H2,j/parenleftbiggt2+s2
+t2H(1)+s2H(2)/parenrightbiggN
+appearing in (7). For this purpose one observes that due to the sin gular
+point at the origin this expression is integrable on [0 ,T]2if and only if it
+is integrable on [0 ,1]2. As shown in the Appendix (Lemma 14), this occurs
+whenever
+2max{¯H1,¯H2}/parenleftbigg
+N+d
+2−1
+2min{¯H1,¯H2}/parenrightbigg
+−2N <1.
+The proof is then completed by an application of Corollary 4. /squaresolid
+As a consequence, one may derive thechaos expansion for the (tr uncated)
+local times L(N)
+H1,H2.
+Proposition 8 Under the conditions of Theorem 7, L(N)
+H1,H2has the chaos
+expansion
+L(N)
+H1,H2(/vector ω1,/vector ω2) =/summationdisplay
+m/summationdisplay
+k/an}bracketle{t:/vector ω⊗m
+1:⊗:/vector ω⊗k
+2:,FH1,H2,m,k/an}bracketri}ht
+where the kernel functions FH1,H2,m,kare given by
+FH1,H2,m,k=
+/parenleftbigg1
+π/parenrightbiggd
+2(−1)m+3k
+2/parenleftbigm+k
+2/parenrightbig
+!/parenleftbigg1
+2/parenrightbiggm+k+d
+2/parenleftbiggm+k
+m/parenrightbigg/integraldisplayT
+0dt/integraldisplayT
+0dsd/productdisplay
+j=1/parenleftbigg1
+t2H1,j+s2H2,j/parenrightbiggmj+kj+1
+2
+·
+·d/circlemultiplydisplay
+j=1/parenleftbig
+(MH1,j1 1[0,t])⊗mj⊗(MH2,j1 1[0,s])⊗kj/parenrightbig
+13for each m= (m1,...,m d)and each k= (k1,...,k d)such that m+k≥
+2Nand all sums mj+kj,j= 1,...,d, are even numbers. All other kernel
+functions FH1,H2,m,kare identically equal to zero.
+Proof.According to Corollary 4, the S-transform of the (truncated) local
+timeL(N)
+H1,H2is obtained by integrating (5) over [0 ,T]2. Hence, given a f=
+(f1,1,...,f1,d,f2,1,...,f2,d)∈S2d(R) one has
+SL(N)
+H1,H2(f)
+=/parenleftbigg1√
+2π/parenrightbiggd/integraldisplayT
+0dt/integraldisplayT
+0dsd/productdisplay
+j=11√
+t2H1,j+s2H2,j·
+·∞/summationdisplay
+n=N(−1)n
+2nn!/summationdisplay
+n1,···,nd
+n1+···+nd=nn!
+n1!···nd!d/productdisplay
+j=1/parenleftbigg1
+t2H1,j+s2H2,j/parenrightbiggnj
+·
+·/parenleftbigg/integraldisplay
+Rdx/parenleftbig
+f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x)/parenrightbig/parenrightbigg2nj
+(8)
+with (8) being equal to
+/summationdisplay
+mj,kj
+mj+kj=2nj(−1)kj/parenleftbigg2nj
+mj/parenrightbigg/parenleftbigg/integraldisplay
+Rdxf1,j(x)(MH1,j1 1[0,t])(x)/parenrightbiggmj
+·
+·/parenleftbigg/integraldisplay
+Rdxf2,j(x)(MH2,j1 1[0,s])(x)/parenrightbiggkj
+.
+From these calculations follow the equality
+SL(N)
+H1,H2(f) =/parenleftbigg1
+π/parenrightbiggd
+2/integraldisplayT
+0dt/integraldisplayT
+0ds∞/summationdisplay
+n=N/summationdisplay
+n1,···,nd
+n1+···+nd=n/summationdisplay
+m1,...,md,k1,...,kd
+mj+kj=2nj,j=1,...,d
+
+d/productdisplay
+j=1(−1)mj+3kj
+2/parenleftBig
+mj+kj
+2/parenrightBig
+!/parenleftbigg1
+2(t2H1,j+s2H2,j)/parenrightbiggmj+kj+1
+2
+
+·
+·/braceleftBiggd/productdisplay
+j=1/parenleftbiggmj+kj
+mj/parenrightbigg/parenleftbigg/integraldisplay
+Rdxf1,j(x)(MH1,j1 1[0,t])(x)/parenrightbiggmj
+·
+·/parenleftbigg/integraldisplay
+Rdxf2,j(x)(MH2,j1 1[0,s])(x)/parenrightbiggkj/bracerightBigg
+,
+14which is equivalent to
+/parenleftbigg1
+π/parenrightbiggd
+2/integraldisplayT
+0dt/integraldisplayT
+0ds/summationdisplay
+m,k
+m+k≥2N
+mj+kjeven,j=1,...,d(−1)m+3k
+2/parenleftbigm+k
+2/parenrightbig
+!/parenleftbigg1
+2/parenrightbiggm+k+d
+2/parenleftbiggm+k
+m/parenrightbigg
+·
+·d/productdisplay
+j=1/parenleftbigg1
+t2H1,j+s2H2,j/parenrightbiggmj+kj+1
+2/parenleftbigg/integraldisplay
+Rdxf1,j(x)(MH1,j1 1[0,t])(x)/parenrightbiggmj
+/parenleftbigg/integraldisplay
+Rdxf2,j(x)(MH2,j1 1[0,s])(x)/parenrightbiggkj
+.
+Comparing with the general form of the chaos expansion
+/summationdisplay
+m/summationdisplay
+k/an}bracketle{t:/vector ω⊗m
+1:⊗:/vector ω⊗k
+2:,FH1,H2,m,k/an}bracketri}ht,
+one concludes that the kernels FH1,H2,m,kvanish whenever either there is a
+j= 1,...,dsuch that mj+kjis an odd number or m+k <2N, while for all
+other cases
+FH1,H2,m,k=
+/parenleftbigg1
+π/parenrightbiggd
+2(−1)m+3k
+2/parenleftbigm+k
+2/parenrightbig
+!/parenleftbigg1
+2/parenrightbiggm+k+d
+2/parenleftbiggm+k
+m/parenrightbigg/integraldisplayT
+0dt/integraldisplayT
+0dsd/productdisplay
+j=1/parenleftbigg1
+t2H1,j+s2H2,j/parenrightbiggmj+kj+1
+2
+·
+·d/circlemultiplydisplay
+j=1/parenleftbig
+(MH1,j1 1[0,t])⊗mj⊗(MH2,j1 1[0,s])⊗kj/parenrightbig
+.
+/squaresolid
+Theorem7showsthatfor d= 1ord= 2allintersectionlocaltimes LH1,H2
+are well-defined for all possible Hurst multiparameters H1,H2in (0,1)d. For
+d >2, intersection local times are well-defined only for 1 /¯H1+ 1/¯H2> d.
+Under these conditions, Proposition 8 in addition yields
+Eµ(LH1,H2) =FH1,H2,0,0=/parenleftbigg1√
+2π/parenrightbiggd/integraldisplayT
+0dt/integraldisplayT
+0dsd/productdisplay
+j=11√
+t2H1,j+s2H2,j.
+Informally speaking, for 1 /¯H1+ 1/¯H2≤dwithd >2, the local times
+only become well-defined once subtracted the divergent terms. Th is “renor-
+15malization” procedure motivates the study of a regularization. As a compu-
+tationally simple regularization we discuss
+LH1,H2,ε:=/integraldisplayT
+0dt/integraldisplayT
+0dsδε(BH1(t)−BH2(s)), ε >0,
+where
+δε(BH1(t)−BH2(s)) :=/parenleftbigg1√
+2πε/parenrightbiggd
+e−(BH1(t)−BH2(s))2
+2ε.
+Theorem 9 Letε >0be given. For all H1,H2∈(0,1)dand all dimensions
+d≥1the intersection local time LH1,H2,εis a Hida distribution with kernel
+functions given by
+FH1,H2,ε,m,k=/parenleftbigg1
+π/parenrightbiggd
+2(−1)m+3k
+2/parenleftbigm+k
+2/parenrightbig
+!/parenleftbigg1
+2/parenrightbiggm+k+d
+2/parenleftbiggm+k
+m/parenrightbigg
+/integraldisplayT
+0dt/integraldisplayT
+0dsd/productdisplay
+j=1/parenleftbigg1
+ε+t2H1,j+s2H2,j/parenrightbiggmj+kj+1
+2d/circlemultiplydisplay
+j=1/parenleftbig
+(MH1,j1 1[0,t])⊗mj⊗(MH2,j1 1[0,s])⊗kj/parenrightbig
+for allm= (m1,...,m d),k= (k1,...,kd)∈Nd
+0such that all sums mi+kj,j=
+1,...,d, are even numbers, and FH1,H2,ε,m,k≡0if at leastone of the sums mi+
+kiis an odd number. Moreover, if max{¯H1,¯H2}/parenleftBig
+N+d
+2−1
+2min{¯H1,¯H2}/parenrightBig
+<
+N+1
+2, then when εtends to zero the (truncated) intersection local time
+L(N)
+H1,H2,εconverges strongly in (S)∗to the (truncated) local time L(N)
+H1,H2.
+Proof.As before, the first part of the proof follows from the Corollary 4
+with respect to the Lebesgue measure on [0 ,T]2. By the definition of the
+S-transform, for all f= (f1,1,...,f1,d,f2,1,...,f2,d)∈S2d(R) one finds
+Sδε(BH1(t)−BH2(s))(f)
+=d/productdisplay
+j=11/radicalbig
+2π(ε+t2H1,j+s2H2,j)·
+·e−1
+2/summationtextd
+j=11
+ε+t2H1,j+s2H2,j(/integraltext
+Rdx(f1,j(x)(MH1,j1 1[0,t])(x)−f2,j(x)(MH2,j1 1[0,s])(x)))2
+,
+16which is measurable. Hence, similarly to the proof of Theorem 7, an ap pli-
+cation of Lemma 6 yields for all z∈Cand allf∈S2d(R)
+|S(δε(BH1(t)−BH2(s)))(zf)|
+≤d/productdisplay
+j=11/radicalbig
+2π(ε+t2H1,j+s2H2,j)e|z|2C2
+H1,H2t2+s2
+ε+t2H(1)+s2H(2)/bardblf/bardbl2
+witht2+s2
+ε+t2H(1)+s2H(2)bounded on [0 ,T]2and/producttextd
+j=11√
+2π(ε+t2H1,j+s2H2,j)integrable
+on [0,T]2. By Corollary 4, one may then conclude that LH1,H2,ε∈(S)∗and,
+moreover, for every f= (f1,1,...,f1,d,f2,1,...,f2,d)∈S2d(R),
+SLH1,H2,ε(f) =/integraldisplayT
+0dt/integraldisplayT
+0dsSδε(BH1(t)−BH2(s))(f)
+=/parenleftbigg1
+π/parenrightbiggd
+2/integraldisplayT
+0dt/integraldisplayT
+0ds/summationdisplay
+m,k
+mj+kjeven,j=1,...,d(−1)m+3k
+2/parenleftbigm+k
+2/parenrightbig
+!/parenleftbigg1
+2/parenrightbiggm+k+d
+2/parenleftbiggm+k
+m/parenrightbigg
+·
+·d/productdisplay
+j=1/parenleftbigg1
+ε+t2H1,j+s2H2,j/parenrightbiggmj+kj+1
+2/parenleftbigg/integraldisplay
+Rdxf1,j(x)(MH1,j1 1[0,t])(x)/parenrightbiggmj
+/parenleftbigg/integraldisplay
+Rdxf2,j(x)(MH2,j1 1[0,s])(x)/parenrightbiggkj
+.
+As in the proof of Proposition 8, it follows from the latter expression that
+the kernels FH1,H2,ε,m,kappearing in the chaos expansion of LH1,H2,εvanish
+if at least one of the mi+kiinm+k= (m1+k1,...,m d+kd) is an odd
+number, otherwise they are given by
+FH1,H2,ε,m,k=/parenleftbigg1
+π/parenrightbiggd
+2(−1)m+3k
+2/parenleftbigm+k
+2/parenrightbig
+!/parenleftbigg1
+2/parenrightbiggm+k+d
+2/parenleftbiggm+k
+m/parenrightbigg
+/integraldisplayT
+0dt/integraldisplayT
+0dsd/productdisplay
+j=1/parenleftbigg1
+ε+t2H1,j+s2H2,j/parenrightbiggmj+kj+1
+2d/circlemultiplydisplay
+j=1/parenleftbig
+(MH1,j1 1[0,t])⊗mj⊗(MH2,j1 1[0,s])⊗kj/parenrightbig
+.
+To complete the proof amounts to check the convergence. For th is purpose
+we shall use Corollary 3. Since
+SL(N)
+H1,H2,ε(f) =/integraldisplayT
+0dt/integraldisplayT
+0dsSδ(N)
+ε(BH1(t)−BH2(s))(f),
+17for every z∈Cand every f∈S2d(R), a similar procedure used to prove
+Theorem 7 yields
+/vextendsingle/vextendsingle/vextendsingleSL(N)
+H1,H2,ε(zf)/vextendsingle/vextendsingle/vextendsingle≤/integraldisplayT
+0dt/integraldisplayT
+0ds/vextendsingle/vextendsingleSδ(N)
+ε(BH1(t)−BH2(s))(zf)/vextendsingle/vextendsingle
+≤/parenleftbigg1√
+2π/parenrightbiggd
+e|z|2/bardblf/bardbl2C2
+H1,H2supt,s∈[0,T]C(t,s)·
+·/integraldisplayT
+0dt/integraldisplayT
+0dsd/productdisplay
+j=11√
+t2H1,j+s2H2,j/parenleftbiggt2+s2
+t2H(1)+s2H(2)/parenrightbiggN
+,
+showing the boundedness condition. Furthermore, we have
+/vextendsingle/vextendsingleSδ(N)
+ε(BH1(t)−BH2(s))(f)/vextendsingle/vextendsingle≤
+/parenleftbigg1√
+2π/parenrightbiggdd/productdisplay
+j=11√
+t2H1,j+s2H2,j/parenleftbiggt2+s2
+t2H(1)+s2H(2)/parenrightbiggN
+eC2
+H1,H2/bardblf/bardbl2supt,s∈[0,T]C(t,s),
+which allows the use of theLebesgue dominated convergence theor emto infer
+the other condition needed for the application of Corollary 3. /squaresolid
+Given any pair of Hurst multiparameters H1,H2∈(0,1)d,d≥1, such
+thatd <1/¯H1+1/¯H2, according to the convergence result stated inTheorem
+9, for any f∈S2d(R) fixed, the SLH1,H2,ε(f) converges to SLH1,H2(f). This
+fact combined with the characterization result of the convergenc e in (L2)
+in terms of the S-transform, recalled in Proposition 10, allows to improve
+the previous statements concerning the intersection local times ( Theorem 11
+below). In particular, this theorem extends the results obtained in [NOL07]
+to different and more general Hurst multiparameters.
+Proposition 10 Let(Φn)n∈Nbe a sequence in (L2)andΦ∈(L2). The
+following two assertions are equivalent:
+(i)(Φn)n∈Nconverges in (L2)toΦ;
+(ii) the sequence (/bardblΦn/bardbl)n∈Nconvergesto /bardblΦ/bardbland, for all f∈S2d(R),(SΦn(f))n∈N
+converges to SΦ(f).
+Here/bardbl·/bardbldenotes the norm defined on (L2).
+Theorem 11 Forany pair of Hurst multiparameters H1,H2∈(0,1)d,d≥1,
+such that d <1/¯H1+1/¯H2, the intersection local times LH1,H2as well as all
+LH1,H2,ε,ε >0, exist in (L2), and the sequence of LH1,H2,εconverges in (L2)
+toLH1,H2asεtends to zero.
+18Proof.According to the previous considerations, the proof amounts to s how
+thatLH1,H2,LH1,H2,ε∈(L2), for all ε >0, and that the convergence (in ε) of
+their (L2)-norms holds. For this purpose we begin by showing that the sums
+/summationdisplay
+m/summationdisplay
+km!k!|FH1,H2,ε,m,k|2
+(L2
+2d(R))⊗(m+k),/summationdisplay
+m/summationdisplay
+km!k!|FH1,H2,m,k|2
+(L2
+2d(R))⊗(m+k),
+(9)
+converge, where FH1,H2,ε,m,kandFH1,H2,m,kare the kernels given by Theorem
+9andProposition8,respectively. By(2),thiswillprovethat LH1,H2,ε,LH1,H2∈
+(L2) with/bardblLH1,H2,ε/bardbl2given by the first sum appearing in (9) and /bardblLH1,H2/bardbl2
+by the second one.
+Similar calculations done to prove Theorem 9 yield
+/summationdisplay
+m/summationdisplay
+km!k!|FH1,H2,ε,m,k|2
+(L2
+2d(R))⊗(m+k)
+=/summationdisplay
+m/summationdisplay
+km!k!/parenleftbigg1
+2π/parenrightbiggd(−1)m+3k
+/parenleftbig/parenleftbigm+k
+2/parenrightbig
+!/parenrightbig2/parenleftbigg1
+2/parenrightbiggm+k/parenleftbiggm+k
+m/parenrightbigg2
+/integraldisplayT
+0dt/integraldisplayT
+0ds/integraldisplayT
+0dt′/integraldisplayT
+0ds′d/productdisplay
+j=1/parenleftBigg
+1/radicalbig
+(ε+t2H1,j+s2H2,j)(ε+t′2H1,j+s′2H2,j)/parenrightBiggmj+kj+1
+·/an}bracketle{tMH1,j1 1[0,t],MH1,j1 1[0,t′]/an}bracketri}htmj/an}bracketle{tMH2,j1 1[0,s],MH2,j1 1[0,s′]/an}bracketri}htkj,
+with the inner products being equal to
+/an}bracketle{tMH1,j1 1[0,t],MH1,j1 1[0,t′]/an}bracketri}ht=1
+2/parenleftBig
+t2H1,j+t′2H1,j−|t−t′|2H1,j/parenrightBig
+,
+/an}bracketle{tMH2,j1 1[0,s],MH2,j1 1[0,s′]/an}bracketri}ht=1
+2/parenleftBig
+s2H2,j+s′2H2,j−|s−s′|2H2,j/parenrightBig
+,
+for eachj= 1,...,d,
+=/parenleftbigg1
+2π/parenrightbiggd/integraldisplayT
+0dt/integraldisplayT
+0ds/integraldisplayT
+0dt′/integraldisplayT
+0ds′d/productdisplay
+j=11/radicalbig
+(ε+t2H1,j+s2H2,j)(ε+t′2H1,j+s′2H2,j)
+∞/summationdisplay
+n=01
+4nn!/summationdisplay
+n1,···,nd
+n1+···+nd=nn!
+n1!...nd!d/productdisplay
+j=1(2nj)!
+nj!/parenleftbigg1
+(ε+t2H1,j+s2H2,j)(ε+t′2H1,j+s′2H2,j)/parenrightbiggnj
+·/parenleftbigg1
+2/parenleftBig
+t2H1,j+t′2H1,j−|t−t′|2H1,j+s2H2,j+s′2H2,j−|s−s′|2H2,j/parenrightBig/parenrightbigg2nj
+.
+19Concerningtheintegrandfunction, observethatusingthefollowin gequalities
+for the Gamma function,
+(2n!)
+n!=22n
+√πΓ/parenleftbigg
+n+1
+2/parenrightbigg
+,
+Γ/parenleftbigg
+n+1
+2/parenrightbigg
+= Γ/parenleftbigg1
+2/parenrightbiggn−1/productdisplay
+i=0/parenleftbigg1
+2+i/parenrightbigg
+=√πn−1/productdisplay
+i=0/parenleftbigg1
+2+i/parenrightbigg
+,
+one may rewrite it as
+d/productdisplay
+j=11/radicalbig
+(ε+t2H1,j+s2H2,j)(ε+t′2H1,j+s′2H2,j)·
+·∞/summationdisplay
+n=0/summationdisplay
+n1,···,nd
+n1+···+nd=nd/productdisplay
+j=1Γ/parenleftbig
+nj+1
+2/parenrightbig
+√πnj!/parenleftbigg1
+(ε+t2H1,j+s2H2,j)(ε+t′2H1,j+s′2H2,j)/parenrightbiggnj
+·/parenleftbigg1
+2/parenleftBig
+t2H1,j+t′2H1,j−|t−t′|2H1,j+s2H2,j+s′2H2,j−|s−s′|2H2,j/parenrightBig/parenrightbigg2nj
+=d/productdisplay
+j=11/radicalbig
+(ε+t2H1,j+s2H2,j)(ε+t′2H1,j+s′2H2,j)∞/summationdisplay
+n=0/summationdisplay
+n1,···,nd
+n1+···+nd=nd/productdisplay
+j=11
+nj!/parenleftBiggnj−1/productdisplay
+i=0/parenleftbigg1
+2+i/parenrightbigg/parenrightBigg
+·
+·d/productdisplay
+j=1/parenleftBig
+t2H1,j+t′2H1,j−|t−t′|2H1,j+s2H2,j+s′2H2,j−|s−s′|2H2,j/parenrightBig2nj
+4nj(ε+t2H1,j+s2H2,j)nj(ε+t′2H1,j+s′2H2,j)nj. (10)
+Hence, taking into account that for any 0 ≤t,t′,s,s′≤Tand for any
+j= 1,...,d
+0≤/parenleftBig
+t2H1,j+t′2H1,j−|t−t′|2H1,j+s2H2,j+s′2H2,j−|s−s′|2H2,j/parenrightBig2
+4(ε+t2H1,j+s2H2,j)(ε+t′2H1,j+s′2H2,j)<1,
+one recognizes that the sum in (10) is indeed the Taylor expansion of the
+function
+d/productdisplay
+j=1
+/radicalBigg
+1−/parenleftbig
+t2H1,j+t′2H1,j−|t−t′|2H1,j+s2H2,j+s′2H2,j−|s−s′|2H2,j/parenrightbig2
+4(ε+t2H1,j+s2H2,j)(ε+t′2H1,j+s′2H2,j)
+−1
+,
+20and thus
+/summationdisplay
+m/summationdisplay
+km!k!|FH1,H2,ε,m,k|2
+(L2
+2d(R))⊗(m+k)
+=/parenleftbigg1
+2π/parenrightbiggd/integraldisplayT
+0dt/integraldisplayT
+0ds/integraldisplayT
+0dt′/integraldisplayT
+0ds′d/productdisplay
+j=1/parenleftBig
+(ε+t2H1,j+s2H2,j)(ε+t′2H1,j+s′2H2,j)
+−1
+4/parenleftBig
+t2H1,j+t′2H1,j−|t−t′|2H1,j+s2H2,j+s′2H2,j−|s−s′|2H2,j/parenrightBig2/parenrightbigg−1
+2
+.
+Independently of the dimension dand the Hurst multiparameters under con-
+sideration, clearly such a multiple integral is always finite.
+Concerning the second sum in (9), first we note that the expressio n of the
+kernelsFH1,H2,m,kcoincides with FH1,H2,ε,m,kforε= 0. Thus, one may apply
+the previous scheme, just replacing εby zero, with the slight difference that
+in this case one has
+0</parenleftBig
+t2H1,j+t′2H1,j−|t−t′|2H1,j+s2H2,j+s′2H2,j−|s−s′|2H2,j/parenrightBig2
+4(t2H1,j+s2H2,j)(t′2H1,j+s′2H2,j)<1,
+only for 0 < t,t′,s,s′≤Tsuch that t/ne}ationslash=t′ands/ne}ationslash=s′. Thus, only for
+0< t,t′,s,s′≤Tsuch that t/ne}ationslash=t′ands/ne}ationslash=s′the sum corresponding to the
+sum in (10) converges. As a result, in this case we obtain
+/summationdisplay
+m/summationdisplay
+km!k!|FH1,H2,m,k|2
+(L2
+2d(R))⊗(m+k)
+= 4/parenleftbigg1
+2π/parenrightbiggd/integraldisplayT
+0dt/integraldisplayt
+0dt′/integraldisplayT
+0ds/integraldisplays
+0ds′d/productdisplay
+j=1/parenleftBig
+(t2H1,j+s2H2,j)(t′2H1,j+s′2H2,j)
+−1
+4/parenleftBig
+t2H1,j+t′2H1,j−(t−t′)2H1,j+s2H2,j+s′2H2,j−(s−s′)2H2,j/parenrightBig2/parenrightbigg−1
+2
+.
+Due to the existence of singular points, an additional analysis is now n eeded
+in order to show the convergence of this multiple integral. As before , it is
+enough to consider the case T= 1.
+21As a first step we use the fact that for each j= 1,...,dfixed one has
+(t2H1,j+s2H2,j)(t′2H1,j+s′2H2,j) (11)
+−1
+4/parenleftBig
+t2H1,j+t′2H1,j−|t−t′|2H1,j+s2H2,j+s′2H2,j−|s−s′|2H2,j/parenrightBig2
+≥t2H1,jt′2H1,j−1
+4/parenleftBig
+t2H1,j+t′2H1,j−|t−t′|2H1,j/parenrightBig2
+(12)
++s2H2,js′2H2,j−1
+4/parenleftBig
+s2H2,j+s′2H2,j−|s−s′|2H2,j/parenrightBig2
+, (13)
+with the advantage that, in contrast to (11), (12) as well as (13) only depend
+of a unique Hurst parameter. Moreover, (12) and (13) are both o f the type
+ϕH(u,v) :=u2Hv2H−1
+4/parenleftbig
+u2H+v2H−|u−v|2H/parenrightbig2,
+which, as a function of uandv, is an homogeneous function of order 4 H.
+Therefore, for every 0 < v < u < 1 one has
+u2Hv2H−1
+4/parenleftbig
+u2H+v2H−(u−v)2H/parenrightbig2
+=u4H/bracketleftBigg/parenleftBigv
+u/parenrightBig2H
+−1
+4/parenleftbigg
+1+/parenleftBigv
+u/parenrightBig2H
+−/parenleftBig
+1−v
+u/parenrightBig2H/parenrightbigg2/bracketrightBigg
+,
+where, for v/u∈(0,1) fixed, the expression between the square brackets is a
+decreasing function of H∈(0,1).
+As a consequence,
+/summationdisplay
+m/summationdisplay
+km!k!|FH1,H2,m,k|2
+(L2
+2d(R))⊗(m+k)
+≤4/parenleftbigg1
+2π/parenrightbiggd/integraldisplay1
+0dt/integraldisplayt
+0dt′/integraldisplay1
+0ds/integraldisplays
+0ds′/parenleftbigg
+t2¯H1t′2¯H1−1
+4/parenleftBig
+t2¯H1+t′2¯H1−(t−t′)2¯H1/parenrightBig2
++s2¯H2s′2¯H2−1
+4/parenleftBig
+s2¯H2+s′2¯H2−(s−s′)2¯H2/parenrightBig2/parenrightbigg−d
+2
+. (14)
+Now the proof follows closely the one in [NOL07, Proof of Lemma 4],
+based on the fact that
+λ−d
+2=1
+Γ(d
+2)/integraldisplay+∞
+0dze−λzzd
+2−1,
+22which allows to rewrite the multiple integral in (14) as
+1
+Γ(d
+2)/integraldisplay+∞
+0dzzd
+2−1/parenleftbigg/integraldisplay1
+0dt/integraldisplayt
+0dse−zϕ¯H1(t,s)/parenrightbigg/parenleftbigg/integraldisplay1
+0dt/integraldisplayt
+0dse−zϕ¯H2(t,s)/parenrightbigg
+.
+(15)
+Since
+∀z∈[0,1],/integraldisplay1
+0dt/integraldisplayt
+0dse−zϕ¯Hi(t,s)<+∞, i= 1,2,
+the convergence of the integral (15) then will follow from the conv ergence of
+the integral
+/integraldisplay+∞
+1dzzd
+2−1/parenleftbigg/integraldisplay1
+0dt/integraldisplayt
+0dse−zϕ¯H1(t,s)/parenrightbigg/parenleftbigg/integraldisplay1
+0dt/integraldisplayt
+0dse−zϕ¯H2(t,s)/parenrightbigg
+.
+As in [NOL07, Proof of Lemma 4], the homogeneity property of ϕ¯H1and
+ϕ¯H2yields
+/integraldisplay+∞
+1dzzd
+2−1/parenleftbigg/integraldisplay1
+0dt/integraldisplayt
+0dse−zϕ¯H1(t,s)/parenrightbigg/parenleftbigg/integraldisplay1
+0dt/integraldisplayt
+0dse−zϕ¯H2(t,s)/parenrightbigg
+=/integraldisplay+∞
+1dzzd
+2−1−1
+2¯H1−1
+2¯H2
+/integraldisplayz1
+4¯H1
+0dx/integraldisplayx
+0dye−ϕ¯H1(x,y)
+
+/integraldisplayz1
+4¯H2
+0dx/integraldisplayx
+0dye−ϕ¯H2(x,y)
+,
+where a double change of coordinates leads for each i= 1,2 to
+/integraldisplayz1
+4¯Hi
+0dx/integraldisplayx
+0dye−ϕ¯Hi(x,y)
+≤1
+4¯Hi/integraldisplayπ/4
+0dθ(ϕ¯Hi(cosθ,sinθ))−1
+2¯Hiγ/parenleftbigg1
+2¯Hi,22¯Hizϕ¯Hi(cosθ,sinθ)/parenrightbigg
+.
+Hereγis the lower incomplete gamma function, that is,
+γ(α,x) :=/integraldisplayx
+0dye−yyα−1, α >0,
+which, as shown in [NOL07, Lemma 2], is bounded by
+γ(α,x)≤K(α)xǫ, K(α) := max/braceleftbigg1
+α,Γ(α)/bracerightbigg
+,
+23for allx >0 and for every 0 < ǫ < α.
+Hence, for all 0 < ǫ <1
+2max{¯H1,¯H2}one finally obtains
+/integraldisplay+∞
+1dzzd
+2−1/parenleftbigg/integraldisplay1
+0dt/integraldisplayt
+0dse−zϕ¯H1(t,s)/parenrightbigg/parenleftbigg/integraldisplay1
+0dt/integraldisplayt
+0dse−zϕ¯H2(t,s)/parenrightbigg
+≤22ǫ(¯H1+¯H2)
+16¯H1¯H2K/parenleftbigg1
+2¯H1/parenrightbigg
+K/parenleftbigg1
+2¯H2/parenrightbigg/integraldisplay+∞
+1dzzd
+2−1−1
+2¯H1−1
+2¯H2+2ǫ· (16)
+·/parenleftBigg/integraldisplayπ/4
+0dθ(ϕ¯H1(cosθ,sinθ))ǫ−1
+2¯H1/parenrightBigg/parenleftBigg/integraldisplayπ/4
+0dθ(ϕ¯H2(cosθ,sinθ))ǫ−1
+2¯H2/parenrightBigg
+.
+Concerning the integral in z, clearly it converges provided ǫ <¯H1+¯H2−d¯H1¯H2
+4¯H1¯H2,
+being¯H1+¯H2−d¯H1¯H2
+4¯H1¯H2always a positive number, because d <1/¯H1+ 1/¯H2.
+These facts combined mean that in (16) one shall fix a
+0< ǫ <min/braceleftbigg1
+2max{¯H1,¯H2},¯H1+¯H2−d¯H1¯H2
+4¯H1¯H2/bracerightbigg
+.
+For such a ǫfixed, also both integrals in θconverge, cf. [NOL07, Proof of
+Lemma 4], and thus (16) converges.
+In this way we have shown that, on the one hand, LH1,H2∈(L2), and,
+on the other hand, that one may apply a Lebesgue dominated conve rgence
+argument to infer the convergence in εof/bardblLH1,H2,ε/bardbl2to/bardblLH1,H2/bardbl2./squaresolid
+Remark 12 Under the conditions of Theorem 11, one has
+min/braceleftbigg1
+2max{¯H1,¯H2},¯H1+¯H2−d¯H1¯H2
+4¯H1¯H2/bracerightbigg
+=¯H1+¯H2−d¯H1¯H2
+4¯H1¯H2
+whenever d≥1
+min{¯H1,¯H2}−1
+max{¯H1,¯H2}, while for d <1
+min{¯H1,¯H2}−1
+max{¯H1,¯H2},
+min/braceleftbigg1
+2max{¯H1,¯H2},¯H1+¯H2−d¯H1¯H2
+4¯H1¯H2/bracerightbigg
+=1
+2max{¯H1,¯H2}.
+Appendix
+Lemma 13 Let0< H1< H2<1be given. The integral
+/integraldisplay1
+0dt/integraldisplay1
+0ds(t2+s2)N
+(t2H1+s2H2)N+d
+2
+is finite if and only if 2H2/parenleftbig
+N+d
+2/parenrightbig
+<1+2N+H2
+H1.
+24Proof.Through the change of variables t=uH2
+H1one obtains
+/integraldisplay1
+0dt/integraldisplay1
+0ds(t2+s2)N
+(t2H1+s2H2)N+d
+2
+=H2
+H1N/summationdisplay
+n=0/parenleftbiggN
+n/parenrightbigg/integraldisplay1
+0du/integraldisplay1
+0dsu2nH2
+H1s2(N−n)
+(u2H2+s2H2)N+d
+2uH2
+H1−1,(17)
+where each double integral appearing in (17) is finite if and only if the
+integrand function is integrable on the unit ball B1(0)⊂R2. For each
+n= 0,1,...,Na polar change of coordinates yields
+/integraldisplay
+B1(0)dudsu2nH2
+H1s2(N−n)
+(u2H2+s2H2)N+d
+2uH2
+H1−1
+=/integraldisplay2π
+0dθcos(2n+1)H2
+H1−1θ·sin2(N−n)θ
+/parenleftbig
+cos2H2θ+sin2H2θ/parenrightbigN+d
+2/integraldisplay1
+0dr1
+r2H2(N+d
+2)−(2n+1)H2
+H1−2(N−n),
+which is finite if and only if the integral in rconverges, that is, if and only
+if 2H2(N+d
+2)−(2n+ 1)H2
+H1−2(N−n)<1. This shows that a necessary
+and sufficient condition for the convergence of the sum in (17) is give n by
+2H2(N+d
+2)−2N−H2
+H1<1. /squaresolid
+Lemma 14 GivenHi= (Hi,1,...,H i,d)∈(0,1)d,i= 1,2, assume that ¯H1<
+¯H2. Then
+/integraldisplay1
+0dt/integraldisplay1
+0dsd/productdisplay
+j=11√
+t2H1,j+s2H2,j/parenleftbiggt2+s2
+t2¯H1+s2¯H2/parenrightbiggN
+<∞
+whenever 2¯H2/parenleftbig
+N+d
+2/parenrightbig
+<1+2N+¯H2¯H1.
+Proof.Since in [0 ,1]2the following inequality holds
+d/productdisplay
+j=11√
+t2H1,j+s2H2,j/parenleftbiggt2+s2
+t2¯H1+s2¯H2/parenrightbiggN
+≤(t2+s2)N
+/parenleftbig
+t2¯H1+s2¯H2/parenrightbigN+d
+2,
+the proof reduces to an application of the previous lemma. /squaresolid
+25Acknowledgments
+We truly thank F. P. da Costa for the helpful discussions. Financial support
+of projects PTDC/MAT/67965/2006, PTDC/MAT/100983/2008 a nd FCT,
+POCTI-219, ISFL-1-209 is gratefully acknowledged.
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+W.Westerkamp. GeneralizedfunctionalsinGaussianspaces: The
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+1996.
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+Raton, New York, London, and Tokyo, 1996.
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+and Related Processes , volume 1929 of LNM. Springer Verlag,
+Berlin and Heidelberg, 2008.
+[MO08] R. Vilela Mendes and M. J. Oliveira. A data-reconstructed fra c-
+tional volatility model. Economics: The Open-Access, Open-
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\ No newline at end of file
diff --git a/1001.0514.txt b/1001.0514.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d1757615a741a8f010f5395e8540d438cd72f1e7
--- /dev/null
+++ b/1001.0514.txt
@@ -0,0 +1,1841 @@
+arXiv:1001.0514v1  [math.CO]  4 Jan 2010Freie Universit¨ at Berlin
+Fachbereich Mathematik
+Diplomarbeit
+Classification of smooth lattice polytopes
+with few lattice points
+Supervisor: Dr. Christian Haase
+Abstract: Aftergivingashort introductiononsmoothlatticepolytopes, Iwill p resent a
+proofforthefiniteness ofsmoothlatticepolytopeswithfewlattice points. Theargument
+is then turned into an algorithm for the classification of smooth lattic e polytopes in fixed
+dimension with an upper bound on the number of lattice points. Additio nally I have
+implemented this algorithm for dimension two and three and used it, to gether with
+a classification of smooth minimal fans by Tadao Oda, to create lists o f all smooth
+2-polytopes and 3-polytopes with at most 12 lattice points.
+by: Benjamin Lorenz
+lorenz.benjamin@fu-berlin.deContents
+1. Introduction 3
+1.1. Polytopes and Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
+1.2. Cones and Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
+1.3. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
+2. Finiteness 7
+2.1. Finiteness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
+2.2. General Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
+2.2.1. Generating Normal Fans . . . . . . . . . . . . . . . . . . . . . . . 10
+2.2.2. Generating Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . 10
+3. Classification 14
+3.1. Blowing-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
+3.2. Smooth Polytopes in Dimension Two . . . . . . . . . . . . . . . . . . . . 15
+3.2.1. Classifying the Normal Fans . . . . . . . . . . . . . . . . . . . . . 15
+3.2.2. Creating the Polytopes . . . . . . . . . . . . . . . . . . . . . . . . 17
+3.2.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
+3.3. Smooth Polytopes in Dimension Three . . . . . . . . . . . . . . . . . . . 18
+3.3.1. Classifying the Normal Fans . . . . . . . . . . . . . . . . . . . . . 18
+3.3.2. Creating the Polytopes . . . . . . . . . . . . . . . . . . . . . . . . 24
+3.3.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
+4. Final Comments 25
+A. List of Smooth Polygons 27
+B. List of Smooth Polytopes 28
+C.polymake extension 29
+C.1. Parameterized Fans ( pfan.rules ) . . . . . . . . . . . . . . . . . . . . . . 29
+C.2. Fans ( fan.rules ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
+C.3. Additional Script ( checkiso ) . . . . . . . . . . . . . . . . . . . . . . . . 39
+C.4. Minimal Fans as polymake files . . . . . . . . . . . . . . . . . . . . . . . 40
+C.4.1. Dimension Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
+C.4.2. Dimension Three . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
+21. Introduction
+1.1. Polytopes and Lattices
+Our main goal is the classification of smooth lattice polytopes with few lattice points,
+so we will start by introducing polytopes and lattice polytopes.
+Definition. A setP⊂Rdis called a polytope if it is the convex hull of a finite set of
+pointsV={v1,...,v n} ⊂Rd. IfVis minimal up to inclusion, then the points v1,...,v n
+are the vertices of P.
+P= conv(V) =/braceleftBigg/summationdisplay
+v∈Vλvv∈Rd|/summationdisplay
+v∈Vλv= 1,λv≥0∀v∈V/bracerightBigg
+Remark. In the following we will always assume Vto be the vertex set of the polytope.
+For the definition of smooth lattice polytopes, and to make precise w hat we mean by
+polytopes with few lattice points, we first need to define lattices in Rd.
+Definition. For some finite set B={b1,...,b n} ⊂Rd, we define the latticeΛ(B):
+Λ(B) =/braceleftBigg/summationdisplay
+b∈Bλbb∈Rd|λb∈Z∀b∈B/bracerightBigg
+⊂Rd
+If the vectors in Bare linearly independent then Bis called a basisof Λ(B).
+Remark. We will now restrict to the lattice Λ = Zd⊂Rd, so any set B={b1,...,b d} ⊂
+Zdis a basis for Zdif det(B) =±1.
+Now, for a lattice Λ = Λ( B)⊂Rd, thedual lattice is
+Λ∗={γ∈(Rd)∗|γ(x)∈Z∀x∈Λ}
+and one can easily see that for Λ( B) =Zdwe have Λ∗= Λ((B−1)T)∼=Zd. So for some
+lattice basis Bdefine the dual lattice basis B∗= (B−1)T.
+We can now define smooth lattice polytopes:
+Definition. LetPbe a polytope. Then Pis called a latticepolytope if all vertices
+have integer coordinates, i.e. V⊆Zd.
+When we talk about the lattice points L(P) of a polytope P⊂Rdwe mean the elements
+ofZdthat lie in the polytope, namely L(P) =P∩Zd. So we want to limit the number
+|L(P)|for our polytopes.
+For a lattice polytope Pto be smooth it has to be simple. So for v∈Vdenote by nb( v)
+the set of adjacent vertices, i.e. for every w∈nb(v) there is an edge containing vand
+w. Then a polytope P⊂Rdissimpleif|nb(v)|=dfor all vertices v∈V.
+3Definition. A simple lattice polytope Pis calledsmooth if, at each vertex, the lattice
+minimal edge-directions form a lattice basis, or more formally
+det/parenleftbigg/braceleftbiggw−v
+gcd(w−v)|w∈nb(v)/bracerightbigg/parenrightbigg
+=±1∀v∈V
+1.2. Cones and Fans
+In our main algorithm we want to approach the polytopes from the du al and therefore
+we now introduce cones and fans and define some important proper ties.
+Definition. ForS⊂Rdfinite, the polyhedral cone C= cone(S) is
+cone(S) =/braceleftBigg/summationdisplay
+r∈Sarr∈Rd|ar∈R≥0∀r∈S/bracerightBigg
+A coneC⊂Rdis called pointed if there is α∈(Rd)∗such that
+C∩{x∈Rd|α(x)≤0}={0}
+IfC= cone(S) is pointed and Sis minimal up to inclusion, then the vectors r∈Sare
+called the raysofC.
+Remark. Some restrictions on the cones we will consider later on.
+•All our cones are pointed.
+•All our cones are rational, which means that there is S⊂Qdfinite, such that
+C= cone(S).
+•And we assume all rays r∈Sto be primitive vectors, i.e. gcd( r) = 1 for all r∈S.
+Thedimension of a cone C= cone(S) is the dimension of its linear hull dim( C) =
+dim(span( S)). Ak-dimensional cone Cis called simplicial if there is S={r1,...,r k}
+such that C= cone(S). One may already see the corresondence to simple polytopes.
+We will continue in this direction by defining smooth cones. Later we will see that both
+definitions of smoothness coincide if we choose the right construct ion to define a fan
+from a polytope.
+Definition. LetC⊂Rdbe ak-dimensional simplicial cone, then Cissmooth if there
+arez1,...,z d∈Zdsuch that
+•C= cone({z1,...,z k})
+•det(z1,...,z d) =±1
+4Definition. AfanFis a finite set of cones with the following properties.
+(1)C1,C2∈F⇒C1∩C2∈F
+(2)C∈F, C′/ne}ationslash=∅, C′≤C(i.e.C′is a non-empty face of C)⇒C′∈F
+For a fan F, denote by F(k)the set of cones of dimension k. A fanFinRdis called
+complete if it covers the whole Rd, which means/uniontext
+C∈FC=Rd.
+The properties of cones defined above can of course be extended to fans, a fan is sim-
+plicialorsmooth if all its cones are.
+1.3. Duality
+In the third section of the introduction we will now state a duality the orem of polyhedral
+theory and establish a connection between polytopes and fans. Th is will turn out useful
+later in the classification.
+Definition. For a matrix A∈Rm×dand a vector b∈Rmthepolyhedron P(A,b)⊆Rd
+is the intersection of finitely many halfspaces
+P(A,b) =/intersectiondisplay
+airow ofA{x∈Rd| /an}bracketle{tai,x/an}bracketri}ht ≤bi}={x∈Rd|Ax≤b}
+For the duality theorem we need the Minkowski sum P+Qof two subsets P,Q⊂Rd:
+P+Q={x+y∈Rd|x∈P, y∈Q}
+Theorem. A setP⊆Rdis a polyhedron if and only if it is the Minkowski sum of a
+polytope and a cone.
+Proof.The proof of this duality can be found in most textbooks containing p olyhedral
+theory, e.g. Lectures on Polytopes [8].
+Remark. In particular we have the following results for polytopes:
+•A polytope is a bounded polyhedron.
+•ForP=P(A,b) bounded, there is V⊂Rdfinite, such that P= conv(V).
+•And forP= conv(V) there are A∈Rm×dandb∈Rmsuch that P=P(A,b).
+This is very useful as it allows us to switch between both representa tions as we need it.
+Now let us define the normal fan of a polytope, the main constructio n we will use for
+the classification of the polytopes.
+5Definition. LetP=P(A,b)⊂Rdbe a polytope, the normal cone of a face F⊂Pis
+NCP(F) = cone( {airow ofA| /an}bracketle{tai,x/an}bracketri}ht=bi∀x∈F})
+Thenormal fan ofPis defined by the normal cones of all faces of the polytope.
+NFP={NCP(F)|F⊂Pface}
+To see that this does indeed produce a fan, observe that for ever y faceFof the polytope
+the faces containing Fare defined by subsets of the rays defining F. Thus, the faces
+ofPcontaining Fdefine the faces of NC P(F). Additionally this construction always
+produces a complete fan.
+This yields aconstruction to buildafanfromagiven polytope. Wecan
+also use this in the other direction. For a given matrix Acontaining
+the rays of the fan we can define a polytope P(A,b) by choosing a
+right-hand side vector b. We just need to make sure that the normal
+fan of the polytope is again the fan we started with, since there can be
+different polytopes with the same set of facet normals.
+However, we only want to classify smooth polytopes, so the obvious question is whether
+there is a property of the normal fan that coincides with the smoot hness of the polytope.
+And the obvious answer is also correct:
+Proposition. LetPbe lattice d-polytope and NFPits normal fan. Then Pis smooth if
+and only if NFPis smooth.
+Proof.First we will show that Pis simple if and only if NF Pis simplicial.
+•Let NF Pbe a simplicial fan, then the normal cone of a vertex vofPis defined by
+exactlydrays. Each ( d−1) subset of the facets defined by these rays can define an
+edge and since there are dsuch subsets, there can be at most dedges containing v.
+Because Pis ad-dimensional polytope, there must be at least dedges containing
+vand thus Pis simple.
+•LetPbe a simple polytope and vbe a vertex of P. Now, every facet containing
+vis itself a ( d−1)-dimensional polytope and thus there must be at least ( d−1)
+edges containing vin each facet. Since there are only d(d−1)-subsets of the d
+edges at vthere can be at most ddifferent facets and thus NF Pis simplicial.
+We choose again one vertex vinP. To prove the smoothness we use the matrizes
+E, Adefined by the edge-directions e1,...,e datvand the normal cone of the vertex
+NP(v) = cone( {a1,...,a d}).
+For given smooth edge-directions EchooseA=−(E−1)T, obviously Adefines the
+normal cone. By Cramer’s rule and since det( E) =±1 we know that Ais integral and
+det(A) =∓1, thus the cone is smooth. And for given smooth normal fan with ra ys in
+AchooseE=−(A−1)T. By the same argument E∈Zd×d, det(E) =∓1 and thus E
+contains the lattice minimal edge-directions.
+62. Finiteness
+2.1. Finiteness Theorem
+Now we can state the main theorem [5].
+Theorem. For integers dandN, there are only finitely many smooth d-polytopes with
+at mostNlattice points.
+Since the following proof inspired the technique for the classification algorithm following
+subsequently, we will repeat its details.
+Proof.We will proof this in three steps. First we want to limit the number of co mbina-
+torial types of smooth fans, then the number of smooth fans with each of those types
+and finally the number of smooth polytopes having one of those as no rmal fan, all using
+the limit on the number of lattice points.
+In the following we will again assume all rays to be primitive.
+•Assume we have a combinatorial type of a complete simplicial fan Fwithnrays and
+at mostNfull-dimensional cones. A full-dimensional cone of this fan can be defi ned by
+d-subsetsC⊂ {1,...,n},|C|=d. The fan can then be defined by the set of all such
+full-dimensional cones F(d)⊂/parenleftbig{1,...,n}
+d/parenrightbig
+.Note that we use a different notation here since
+we consider only the combinatorial type of the fan and do not c are about the rays. We
+know that the fan has at most Nfull-dimensional cones, so |F(d)| ≤N. Additionally
+every ray needs to appear in at least dfull-dimensional cones:
+di=|{C∈F(d)|i∈C}| ≥d∀i∈ {1,...,n}
+By counting all appearances of all rays in F(d)in two different ways we have:
+d·n≤n/summationdisplay
+i=1di≤d·N
+And thus n≤N, which means that we have at most Nrays in the simplicial fan F.
+Hence our fan is fully defined by choosing
+F(d)⊂/parenleftbigg{1,...,n}
+d/parenrightbigg
+⊂/parenleftbigg{1,...,N}
+d/parenrightbigg
+SinceNanddare fixed there are only finitely many such subsets, and thus only fin itely
+many possible combinatorial types of simplicial fans.
+•Given the combinatorial type we now want to define the fan Fup to isomorphism:
+LetC= cone(n1,n2,...,n d−1)∈Fbe a codimension 1 cone which is contained in two
+full-dimensional cones C1= cone(C,r1) andC2= cone(C,r2). Since all the cones are
+7smoothr1andr2have lattice distance one from C, but lie on different sides of C. So
+r1+r2lies in the span of Cand thus
+r1+r2=d−1/summationdisplay
+i=1aC,ini
+The coefficients aC,iare the edge-paremeters of a fan F. We know aC,i∈ZbecauseC
+is smooth and thus {n1,...,n d−1}is a Hilbert basis of C.
+We can also use the edge-parameters aC,ifor 1≤i≤d−1, C∈F(d−1)together with the
+combinatorial type to reconstruct fan Fusing the following technique: Every smooth
+fan can, while preserving the lattice, be transformed to contain th e cone defined by all
+dunit vectors. This means we can assume the fan to contain this cone . Then there
+are codimension 1 cones for every ( d−1)-subset of the unit vectors, so, given the edge-
+parameters for each one of those cones, one can calculate the ra y on the other side of
+this codimension 1 cone with the formula above. Iterate until all ray s are calculated.
+Since the edge-parameters are in Zthere may be infinitely many such smooth fans, but
+in the next step we will show a way to bound those parameters.
+•In the last step we have to bound the number of smooth polytopes Pwith at most
+Nlattice points having a given normal fan Fand the number of smooth fans the above
+construction yields.
+We therefore consider the edge eof the polytope dual to a codimension 1 cone C∈F.
+Letl(e) be the lattice length of the edge eandx1,x2be the vertices defining this edge.
+Denote by u1,...,u dthe lattice-minimal edge-directions at x1which are dual to the
+inner normals −n1,...,−nd−1,−r1aroundx1. So we have x2=x1+l(e)ud. Using the
+edge-directions at x1and the edge-parameters we can calculate the edge-directions at
+x2.
+Lemma. Forx1,x2as above, the edge-directions at x2are
+u′
+i=ui+aC,iudfori∈ {1,...,d−1}
+u′
+d=−ud
+Proof.The inner normals around x2are−n1,...,−nd−1,−r2and
+−r2=r1−d−1/summationdisplay
+i=1aC,ini
+so we have
+/an}bracketle{t−ni,u′
+j/an}bracketri}ht=/an}bracketle{t−ni,uj+aC,jud/an}bracketri}ht=/an}bracketle{t−ni,uj/an}bracketri}ht−aC,j/an}bracketle{tni,ud/an}bracketri}htfori,j < d
+=/an}bracketle{t−ni,uj/an}bracketri}ht=δi,j fori,j < d
+/an}bracketle{t−ni,u′
+d/an}bracketri}ht=/an}bracketle{t−ni,−ud/an}bracketri}ht= 0 for i < d
+8/an}bracketle{t−r2,u′
+j/an}bracketri}ht=/an}bracketle{tr1,uj+aC,jud/an}bracketri}ht−d−1/summationdisplay
+i=1aC,i/an}bracketle{tni,uj+aC,jud/an}bracketri}ht forj < d
+=aC,j/an}bracketle{tr1,ud/an}bracketri}ht−aC,j/an}bracketle{tnj,uj/an}bracketri}ht= 0 for j < d
+/an}bracketle{t−r2,u′
+d/an}bracketri}ht=/an}bracketle{tr1,−ud/an}bracketri}ht−d−1/summationdisplay
+i=1aC,i/an}bracketle{tni,−ud/an}bracketri}ht=/an}bracketle{t−r1,ud/an}bracketri}ht= 1
+which proves that the matrix containing these edge-directions is inv erse to the given
+inner normals, finishing the proof of the lemma.
+Since in our smooth lattice polytope every edge must have length at le ast 1 the lattice
+points defined by the vertex x2plus one of the edge-directions must of course lie in the
+polytope and thus
+/an}bracketle{tr1,x2+aC,iud/an}bracketri}ht ≥ /an}bracketle{tr1,x1/an}bracketri}ht
+/an}bracketle{tr1,x1+l(e)ud+aC,iud/an}bracketri}ht ≥ /an}bracketle{tr1,x1/an}bracketri}ht
+(l(e)+aC,i)/an}bracketle{tr1,ud/an}bracketri}ht ≥0
+aC,i≥ −l(e)≥ −N
+This gives a lower bound for the edge-parameters. For an upper bo und let us first
+introduce the thickened edge.
+x2+u1+aC,1u3x2+u2+aC,2u3
+x1+u1x1+u2x1 x2=x1+l(e)u3
+Figure 1: A thickened edge
+Definition. Using the notation from above we define the thickened edge in a smoo th
+polytope as the convex hull of all those points we just discussed.
+c(e) = conv( {x1,x1+u1,...,x 1+ud−1,
+x2,x2+u1+aC,1ud,...,x 2+ud−1+aC,d−1ud})
+We obviously have c(e)⊆Pand as shown in Figure 1 c(e) contains d−1 lines from
+x1+uitox2+ui+aC,iudeach containing l(e)+1+aC,ilattice points and the line from
+x1tox2containing l(e)+1 lattice points, so
+|c(e)∩Zd|=d(l(e)+1)+d−1/summationdisplay
+i=1aC,i
+9Since the edge length must be positive and we want the number of latt ice points of P
+and thus of c(e) to be less than N, this gives us an upper bound for the sum of all
+edge-parameters. Together with the lower bound this leaves us wit h only finitely many
+choices for the edge-parameters. This means for an upper bound Non the number of
+lattice points in the polytope there are only finitely many possible smoo th normal fans.
+Now for a given normal fan we can define the polytope up to isomorph ism by fixing all
+edge lengths. Just assume one vertex at the origin and the edge-d irections to be the
+unit vectors, then the edge lengths define the next vertices and t he normal fan defines
+the next edge-directions. Iterate until all vertices are fixed. An d we want of course the
+edge length l(e) to be positive and less than N, so for every edge there are only finitely
+many possible edge lengths. And thus only finitely many polytopes for this fan.
+So altogether we have finitely many combinatorial types of fans, fin itely many choices
+for the parameters and finitely many choices for the edge lengths a nd thus only finitely
+many smooth lattice d-polytopes with at most Nlattice points. This completes the
+proof of the finiteness theorem.
+2.2. General Algorithm
+The proof gives rise to the following approach for the classification o f such polytopes.
+2.2.1. Generating Normal Fans
+The first step is to find all combinatorial types of complete simplicial f ans in dimension
+dwith at most Nfull-dimensional cones. One can then proceed by finding parameter s
+aC,iwhich produce smooth fans, for all codimension 1 cones.
+In dimension three we can use that the combinatorial types of simplic ial fans are equiv-
+alent to the combinatorial types of triangulations of S2with at most N2-simplices,
+which have been classified by R. Bowen and S. Fisk in Generation of triangulations of
+the sphere [2]. And Tadao Oda has done the classification of smooth two-dimensio nal
+fans which are minimal up to equivariant blowing-ups and the same in dim ension three
+for at most eight rays in Convex Bodies and Algebraic Geometry [7]. We will explain
+and use those results later in chapter 3.
+2.2.2. Generating Polytopes
+The next step now is to find the polytopes. In the proof we used the edge lengths
+together with the normal fan to define the polytope. Here we will us e the right-hand
+side of its inequality description P(A,b) ={x|Ax≤b}instead of the edge lengths, but
+after expressing the edge length in terms of the b-vector we can use the edge lengths to
+find all those b-vectors defining smooth lattice polytopes with the correct norma l fan.
+10Right-Hand Side
+For a given fan Fwe now need to find all possible b-vectors which yield smooth lattice
+polytopes with at most Nlattice points and Fas normal fan. Therefore define the set
+of all such vectors
+RHS(F,N) ={b∈Zm|NFP(A,b)=F,|P(A,b)∩Zd| ≤N}
+where the rows of Aare the rays of F. In the following we want to define a polytope
+containing this set and enumerate all its lattice points.
+Calculating the edge lengths
+Lemma. LetC= cone(n1,n2,...,n d−1) be a codimension 1 cone and r1,r2∈Zdsuch
+thatC1= cone(C,r1), C2= cone(C,r2)∈Fare the full-dimensional cones containing
+C. Then the length l(e) of the edge ein the polytope P(A,b) dual to Ccan be expressed
+in terms of the right-hand side bas follows:
+l(e) =br1+br2−d−1/summationdisplay
+i=1aC,ibni
+whereaC,i(1≤i≤d−1) such that r1+r2=/summationtextd−1
+i=1aC,ini.
+Proof.The coefficients a∗are the same as in the proof of the theorem. Now let x1andx2
+be the vertices dual to C1andC2respectively. From the proof of the finiteness theorem
+we know /an}bracketle{tud,−r1/an}bracketri}ht= 1 and thus l(e) =/an}bracketle{tx2−x1,−r1/an}bracketri}ht. Sincer1+r2=/summationtextd−1
+i=1aC,ini, this
+yields
+l(e) =/an}bracketle{tx1,r1/an}bracketri}ht−/an}bracketle{tx2,r1/an}bracketri}ht
+=br1−/an}bracketle{tx2,d−1/summationdisplay
+i=1aC,ini−r2/an}bracketri}ht
+=br1−d−1/summationdisplay
+i=1aC,i/an}bracketle{tx2,ni/an}bracketri}ht+/an}bracketle{tx2,r2/an}bracketri}ht
+=br1−d−1/summationdisplay
+i=1aC,ibni+br2
+Approximating the Right-Hand Sides
+Since we can express l(e) linearly in terms of the b-vector we can now use bounds on the
+edge length to approximate RHS( F,N). For a codimension 1 cone C, denote by eCthe
+edge in the polytope dual to C.
+11Lemma. Foragiven normalfan Findimension dandamaximal number oflatticepoints
+N, every polytope defined by Fand a vector from RHS( F,N) satisfies the following
+inequalities
+l(eC)≤N−/summationtextd−1
+i=1aC,i
+d−1 ∀C∈F(d−1)
+l(eC)≥1 ∀C∈F(d−1)
+bi= 0 ∀1≤i≤d/summationdisplay
+C∈F(d−1)(l(eC)−1)≤N−|F(d)|
+Usingl(e) as defined in the previous lemma, these inequalities define a polytope
+B(F,N)⊃RHS(F,N)
+Proof.We first show that RHS( F,N) satisfies the inequalities:
+(i) We can now use an idea from the proof of the finiteness theorem. The convex hull
+of the thickened edge c(e) contains d(l(e)+1)+/summationtextd−1
+i=1aC,ilattice points for an edge
+ewith the parameters aC,i, so we have the following bound for l(e).
+l(e)≤N−/summationtextd−1
+i=1aC,i
+d−1
+(ii) The fan completely defines the combinatorial type of the polytop e, so every codi-
+mension 1 cone must define a non-degenerate (i.e. l(e)>0) edge in the polytope
+and since all vertices have integer coordinates the edge length mus t be at least
+one. This gives the second set of inequalities.
+(iii) We now assume Fto be lattice-transformed such that the first drays are the
+unit vectors, this is possible because Fis smooth. Since we do not care about
+translations, we can fix the first dright-hand sides to be zero.
+(iv) The total number of lattice points on all edges of the polytope is the number of
+relative interior lattice points of each edge ( l(e)−1) plus the number of vertices,
+so this sum must be less than the given bound Non the number of lattice points.
+So all inequalities hold for any b∈RHS(F,N), they are all linear in terms of band thus
+B(F,N) is a polyhedron.
+Now to prove that this polyhedron is bounded we use a similar approac h as we used to
+construct the fan from the edge-parameters. For every codime nsion 1 cone we have the
+following inequalities
+1≤l(e) =br1+br2−d−1/summationdisplay
+i=1aC,ibni≤N
+12Starting with the full-dimensional cone defined by the unit-vectors , where all bi(1≤
+i≤d) are zero, we can always find codimension 1 cones where only one of t heb∗in the
+inequalities above is not yet bounded and thus bound it. We can iterat e this until all b∗
+are bounded and thus B( F,N) is a polytope.
+Enumeration
+Now, given an inequality description of B( F,N) we can calculate all lattice vectors
+b∈B(F,N)∩Zm. Togetherwiththematrix A,eachonedefinesasmoothlatticepolytope
+P(A,b) with possibly too many lattice points. Thus, after removing all polyt opes with
+more than Nlattice points, we obtain a list of smooth lattice polytopes with normal fan
+Fand at most Nlattice points. The last step is to remove all polytopes that are lattic e-
+isomorphic to another one. All these computations can be done with polymake [6] using
+interfaces to 4ti2[1],Latte macchiato [4] andnormaliz2 [3].
+133. Classification
+After a short definition of blowing-ups, the sections 3.2 and 3.3 cont ain a detailed de-
+scription of how to classify smooth two- and three-dimensional latt ice polytopes with at
+most 12 lattice points based on the technique described above.
+3.1. Blowing-ups
+We want to use the results of Tadao Oda who classified smooth fans in dimension two
+and three which are minimal up to equivariant blowing-ups. This means that all normal
+fans we need can be generated by successive equivariant blowing-u ps of the fans given in
+Convex Bodies and Algebraic Geometry by Tadao Oda [7]. But first we need to clarify
+what such ablowing-upis. Inshort, itmeans subdividing aconeandall cones containing
+it.
+Definition. LetˆCbe a cone in the fan Fandr∈ˆC. The blowing-up b(C,r) of a cone
+Catris
+b(C,r) ={cone({r,C′})|C′∈F,C′≤C,r /∈C′} ifr∈C
+b(C,r) ={C} ifr /∈C
+whereC′≤Cmeans that C′is a face of C.
+Now the blowing-up b(F,r) ofFatris
+b(F,r) =/uniondisplay
+C∈Fb(C,r)
+One can easily see that this is indeed again a fan.r
+A blowing-up is equivariant ifr=/summationtext
+s∈Ssfor some cone C= cone(S)∈F, whereSis
+minimal up to inclusion and all rays are primitive. We denote by b(F,C) the blowing-up
+in coneC.
+For a smooth fan, an equivariant blowing-up is again smooth. Since we want to classify
+smooth polytopes and thus smooth normal fans, we only have to co nsider equivariant
+blowing-ups.
+Remark. Since we have a connection between the fan and the polytope one co uld ask
+what such a blowing-up of the normal fan corresponds to on the po lytope side.
+We first choose a cone in the fan, which is equivalent to choosing a fac e in the polytope,
+and then add the sum of all rays of the cone as new ray. This corres ponds to adding a
+new facet by cutting-off that face of the polytope.
+143.2. Smooth Polytopes in Dimension Two
+3.2.1. Classifying the Normal Fans
+Asalreadymentioned wedo notstartwithcombinatorial types offa ns, but withminimal
+smooth fans, and in dimension two there are only two such fans as giv en by the following
+theorem.
+Minimal Fans
+Theorem. (Tadao Oda [7]) Every smooth fan in dimension two can, up to isomor-
+phism, be created by finitely many successive equivariant bl owing-ups of one of the fol-
+lowing fans.
+(i) The fan Fprepresenting the complex projective space P2(C).
+F(1)
+p={a0,a1,a2}={(1,0),(0,1),(−1,−1)}
+F(d)
+p={cone({a0,a1}),cone({a1,a2}),cone({a2,a0})}
+(ii) The fans Farepresenting the Hirzebruch surfaces for a∈Z.
+F(1)
+a={a0,a1,a2,a3}={(1,0),(0,1),(−1,−a),(0,−1)}
+F(d)
+a={cone({a0,a1}),cone({a1,a2}),cone({a2,a3}),cone({a3,a0})}
+(1,0)(0,1)
+(−1,−1)Fp (1,0)(0,1)
+(0,−1)(−1,−a)Fa
+Tree of Blowing-ups
+Instead of generating combinatorial types or finding parameters we have a different task
+now, namely equivariant blowing-ups. In dimension two this means cho osing one full-
+dimensional cone C= cone(r1,r2), inserting the ray r=r1+r2and replacing the cone
+CwithC1={r1,r}andC2={r,r2}.
+So beginning with a fan Fwe can generate all normal fans with the following recursion:
+1. For every full-dimensional cone C∈F(d):
+15a) Create the blowing-up b(F,C),
+b) Ifb(F,C) has less than 12 full-dimensional cones: Start recursion.
+Which is a depth-first search in a rather large rooted tree, having 1 + (3! + 4! + ···+
+11!)/2! = 21977356 nodes for up to 12 full-dimensional cones for the fan Fp, but lots
+or even most of those fans are isomorphic. So we need a way to redu ce the number of
+nodes to consider:
+We first assign an order to all full-dimensional cones. Now, since blow ing upb(F,Ci) in
+Cjandb(F,Cj) inCiwill give the same fan, we can ignore all cones Cjwith index j < i
+in the subtree of the blowing-up b(F,Ci). (When blowing up in cone Ciin a fan with k
+full-dimensional cones C1,...,C kwe will index the two new cones with iandk+1.)
+This reduces the amount of blowing-ups to 58785 for Fpand 35072 for Fa.
+Parameters
+The fanFaand all further blowing-ups of it represent classes of infintely many fans with
+a∈Z. But, from the proof of our main theorem, we know that for some u pper bound
+Non the number of lattice points, there can be only finitely many of tho se. The proof
+even gives us a direct way to bound the parameter a.
+Lemma. For the fan Faand a maximal number of lattice points Nfor the polytope, we
+can restrict ato 0≤a≤Nanda/ne}ationslash= 1.
+Proof.For a pair of two-dimensional cones C1={r1,r}andC2={r2,r}sharing a
+common ray rwe can write r1+r2in terms of r, i.e.r1+r2=arr. Additionally we
+knowar≥ −l(e), where l(e) is the length of the edge edual to the ray r. Obviously
+l(e)≤N, so we have ar≥ −N.
+•For the cones C1={(1,0),(0,1)},C2={(−1,−a),(0,1)}this yields
+r1+r2= (1,0)+(−1,−a) = (0,−a) =−a·(0,1)
+and thus −a=a(0,1),1≥ −N.
+•And forC1={(1,0),(0,−1)},C2={(−1,−a),(0,−1)}we have
+r1+r2= (1,0)+(−1,−a) = (0,−a) =a·(0,−1)
+anda=a(0,−1),1≥ −N.
+Because of the symmetry of the fan we can restrict to positive aand skip a= 1 since
+Fathen equals Fpafter an equivariant blowing-up in the cone C= cone({a0,a2}).
+16Parameters in the Blowing-up
+We could find bounds on the parameter aof the Hirzebruch surface using an edge of
+the corresponding polytope, but when blowing up the fan, which mea ns cutting off a
+vertex of the polytope, this edge may have different parameters a nd the bounds may
+be different too. We could of course do the same calculation for all blo wing-ups, but a
+faster approach is to find a way to use the bounds we found for som e fanFon all its
+blowing-ups.
+Assume we have a pair of cones C1={r1,r}, C2={r2,r} ∈Faround a ray rwith
+r1+r2=arr, this means our bound is ar≥ −N. Now, in the blowing-up b(F,C2), we
+have a new cone C′
+2={r′
+2,r}withr′
+2=r+r2. This yields
+r1+r′
+2=arr−r2+r+r2= (ar+1)r
+So we now have ar+1≥ −Nand thus ar≥ −N−1. This means we have to loosen the
+bounds on a parameter by one if it appears as edge-parameter at o ne of the rays of the
+current cone.
+Thus, after creating a new fan for every valid value of afor each class of fans, we now
+have all possible smooth normal fans for the smooth polytopes.
+3.2.2. Creating the Polytopes
+Now we can continue as in the general case, for every fan Fwe define the polytope
+B(F,N) approximating RHS( F,N) and enumerate all its lattice points b. For some fan
+with rays Aand a right-hand side bcheck if|P(A,b)∩Z2| ≤N. And finally remove
+polytopes that are lattice-isomorphic to another one.
+3.2.3. Results
+We can now state the result of the computation:
+Proposition. There are 41 smooth lattice polygons with at most 12 lattice p oints.
+Vertices 3 4 5 6 7 8 ≥9
+Polygons 3 30 3 4 0 1 0
+The complete list of polytopes can be found in the appendix.
+173.3. Smooth Polytopes in Dimension Three
+3.3.1. Classifying the Normal Fans
+Minimal Fans
+As in the two-dimensional case, we start with a list of smooth minimal f ans from Tadao
+Oda [7].
+Theorem. (Tadao Oda [7]) Every smooth fan in dimension three, with at most eight
+rays, can, up to isomorphism, be created by finitely many succ essive equivariant blowing-
+ups of one of the 19fans with labels 34,(3243)′,(3243)′′,46,324362,314353,4552,4662,
+324472,33415163,32425262,31445162,32415461,(31435361)′,(31435361)′′,(3256)′,(3256)′′,
+(4454)′and(4454)′′and weights as given in Convex Bodies and Algebraic Geometry .
+Remark. i
+•All drawings of such fans in the following chapter are stereographic projections of
+the corresponding triangulation of S2.
+•The labels give the occurence ofvertices with different degrees inth e triangulation,
+forexample 324362means2vertices ofdegree3, 3vertices ofdegree4 and2vertices
+of degree 6. See the pictures later in this chapter.
+Remark. With at most eight rays in the fan, from Eulers formula and the fact t he the
+fan is simplicial we know that there are at most twelve full-dimensional cones in the fan
+and thus at most twelve vertices in the polytope. This is the main reas on we chose 12
+as upper bound for the number of lattice points in the polytope.
+Even in the two-dimensional case, the blowing-up tree was huge so it comes in handy
+that we can use the two-dimensional classification now to reduce th e number of fans
+we have to consider. From the 19 fans of the theorem there will be o nly five left after
+applying thenowfollowingcriterionanditwill helpusalotlaterduring the computation
+of all blowing-ups.
+Smooth Polygons
+We use the smooth polygons we generated earlier to remove fans wh ose polytopes would
+have too many lattice points on the boundary. Let PN(k) be the set of all smooth lattice
+k-gons with at most Nlattice points. Define the following numbers:
+lN(k) = min{|P∩Z2|forP∈ PN(k)}
+iN(k) = min{|int(P)∩Z2|forP∈ PN(k)}
+bN(k) = min{|P∩Z2|−|int(P)∩Z2|forP∈ PN(k)}
+18From the results of the previous section we can determine the value s shown in the table
+below.
+k3 4 5 6 7 8 >9
+l12(k)3 4 8 7 >12 12 >12
+i12(k)0 0 1 1 4
+b12(k)3 4 7 6 8
+Denote by krthe number of full-dimensional cones in Fa rayris contained in and let
+kFbe the maximal krover all rays rinF.
+All vertices of the polytope defined by the full-dimensional cones ar ound a ray rlie on
+the boundary of the facet defined by the ray r, so for every ray rthere is a kr-gon on
+the boundary of the polytope. Since the polytope is smooth, all the se polygons have to
+be smooth, too. This yields the following lower bound for the number o f lattice points,
+which depends solely on the combinatorial type of the fan.
+Lemma. (smooth lattice polygon criterion) For a smooth lattice polytope Pwith
+the normal fan F.
+|P∩Z3| ≥ |F(d)|+(bN(kF)−kF)+/summationdisplay
+r∈F(1)iN(kr)
+Remark. This bound can be improved by considering the combinatorial type of the fan
+more precisely. However, for our computation it suffices.
+Remaining Minimal Fans
+Now we can check the fans from the theorem for the above criterio n, the labels of the
+fans are exactly the numbers k∗
+rwe are looking for.
+•First, consider all fans with eight rays and twelve full-dimensional co nes. That are the
+fans labeled 4662, 324472, 33415163, 32425262, 31445162, 32415461, (31435361)′, (31435361)′′,
+(3256)′, (3256)′′, (4454)′and (4454)′′, which cannot be blown up as this would add two
+additional full-dimensional cones. From the labels one can see that t hey all have at least
+one ray contained in five, six or seven full-dimensional cones (5∗,6∗or 7∗), this means
+the corresponding polytope would have a five-, six- or seven-gon o n the boundary. But
+since any smooth five- or six-gon has at least one interior lattice poin t and there is no
+smooth seven-gon with less than 13 lattice points at all, the polytope would have at
+least 13 lattice points on the boundary. So none of these fans admit s a smooth lattice
+polytope with at most 12 lattice points.
+•Now consider the fans labeled 4552and 314353with ten full-dimensional cones.
+19(4552) Here we have two non-incident rays each contained in five full-
+dimensional cones, which means we have two edge-disjoint five-
+gons in the polytope. But we know that there is no smooth
+five-gon with less than eight lattice points and thus any smooth
+polytope with this fan as normal fan would have at least 16 ver-
+tices.4552
+A smooth polytope corresponding to any blowing-up of this fan would have 12
+vertices and either two six-gons or a five-gon and a six-gon and thu s have at least
+14 lattice points.
+(314353) Thisfancontainsthreerayswithfiveneighbors, soanypolytope
+with this normal fan would have at least three lattice points in
+the relative interior of the facets and thus at least 13 lattice
+points in total. Any blowing-up would leave at least one of the
+five-gons unchanged and thus there would still be at least 13
+lattice points, since we have now two additional vertices from
+the blowing-up and one in the interior lattice points in the five-
+gon.314353
+•By the above criterion, the fans numbered 34,(3243)′,(3243)′′,46and 324362could
+appear as normal fans for smooth lattice polytopes with at most 12 lattice points and
+will be the input to the algorithm.
+This leaves us with the following five fans or classes thereof with rays and edge-para-
+meters as drawn [7].
+(0,1,0)
+(1,0,0) (−1,−1,−1)34 ∞= (0,0,1)
+−1 −1−1−1
+−1−1−1−1
+−1−1
+−1−1(−1,−1,−a) (0,0,1)(1,0,0) (0,0,−1)(3243)′∞= (0,1,0)
+000000 1
+a
+−1aa−1
+−1−a−1−a−a−1
+x
+x
+20(0,−1,−1) (1,0,0)(0,1,0) (−1,b,c)(3243)′′∞= (0,0,1)
+c−b−b−cb−cbc −1
+0
+−100−1
+−10−100−1
+(0,−1,b) (1,0,0)(0,1,0) (−1,−a,c)
+(0,0,−1)46∞= (0,0,1)
+c−abaaab−c−c−a−ac
+0
+−b
+−b
+0b
+0
+0000
+b00000
+x
+x
+(0,1,−1) (0,1,0)(0,0,1) (0,0,−1)324362 ∞= (1,0,0)
+−120−a−a0 0
+a+1
+a+1
+0
+−122−1
+110000
+2−1
+2−1 −122a+ 1−2 −2
+−2a−1
+(−1,2,−1)(0,−1,−a)
+Tree of Blowing-ups
+Again, the first step of the algorithm is to generate all possible norm al fans by successive
+blowing-ups. There are now two possible cases, blowing up in a 3-dimen sional cone or
+a 2-dimensional cone, which corresponds in the polytope to cutting off a vertex or an
+edge respectively, as shown in figure 2.
+Figure 2: Blowing-up of a three-dimensional fan and the cutting-off in the corresponding
+polytope
+Generating all blowing-ups successively gives again a huge rooted tr ee, but we now have
+more ways to reduce the number of nodes we have to check.
+21(1) We can again limit the depth of the tree since we want at most 12 fu ll-dimensional
+cones and each blowing-up increases the number of such cones by t wo.
+(2) Blowing-ups in full-dimensional cones can again be done in any orde r. LetF
+containkfull-dimensional cones. When blowing up in the cone Cithe three new
+cones get the numbers i,k+ 1,k+ 2, so we can ignore all full-dimensional cones
+with index strictly less than iat all nodes in the subtree of this blowing-up.
+(3) For codimension 1 cones there are two cases. Let Cbe a codimension 1 cone
+contained in two full-dimensional cones C1andC2.
+(3a) Blowing up b(F,C) in another codimension 1 cone C′, which is contained in
+C1orC2, will yield a different result than blowing up b(F,C′) in the cone C.
+All cones have to be considered in this case.
+(3b) Blowing up b(F,C) in a codimension 1 cone C′, which is not contained in C1
+orC2, will yield the same result as blowing up b(F,C′) inC. So we can use
+an ordering on the codimension 1 cones to skip cones with lower index if they
+do not fall into case (3 a).
+(4) Blowing up in a full-dimensional cone and later in one of its codimensio n 1 faces
+yields the same rays and the same combinatorial type as blowing up in t he codi-
+mension 1 cone first and then in on new codimension 1 cone as shown be low.
+x′
+1
+r1r2nd−1
+n1x1
+r1r2nd−1
+n1r1r2nd−1
+n1x1x2
+r1r2nd−1
+n1x′
+2x′
+1r1r2nd−1
+n1
+x1=x′
+2x2=x′
+1
+And we have the following new rays:
+x2=d−1/summationdisplay
+i=1ni x′
+1=d−1/summationdisplay
+i=1ni=x2
+x1=r1+d−1/summationdisplay
+i=1ni x′
+2=x′
+1+r1=r1+d−1/summationdisplay
+i=1ni=x1
+So only one of these cases has to be considered.
+(5) As already mentioned, we can use the smooth lattice polygon crit erion described
+earlier. Fans that violate this criterion will be considered for furthe r blowing-ups,
+22but will not be kept for the later steps of the algorithm. We cannot c ut off the
+tree here because the criterion is not monotone:
+Assume we have a polytope with one vertex that is contained in three five-gons,
+then cutting-off this vertex will replace all with six-gons. So in total this new
+polytope will have two more vertices but may have less lattice points t han the
+original polytope, since the smallest smooth five-gon has 8 lattice po ints in total
+and the smallest smooth six-gon is the hexagon with only 7 total lattic e points.
+This has been implemented with one variable for each full-dimensional a nd each codi-
+mension 1 cone that can have the values ‘always ignore’, ‘ignore’, ‘cons ider’, ‘always
+consider’. We will create the blowing-up and continue the recursion o nly for cones with
+value ‘consider’ or ‘always consider’. The specific cases are:
+(2) Visited full-dimensional cones are marked ‘ignore’ for the subtr ee.
+(3a) Visited codimension 1 cones are marked ‘ignore’ for the subtre e if they haven’t
+been marked ‘always consider’ or ‘always ignore’.
+(3b) Codimension 1 cones are marked ‘always consider’ if they were o n ‘ignore’ before
+and are contained in one of the full-dimensional cones containing the codimension
+1 cone of the current blowing-up.
+(4) Codimension 1 cones are marked ‘always ignore’ if they were cont ained in the
+full-dimensional cone of the current blowing-up.
+(x) Newly created cones are always set to ‘consider’.
+Parameters for the Minimal Fans
+Most of the remaining minimal fans are actually classes of fans with pa rameters a,b,c∈
+Z. Hence we have to find bounds for their parameters and find ways t o use those bounds
+for their blowing-ups.
+Wecanfindboundsfortheirparametersanalogouslytothetwo-dim ensional case: When-
+ever a parameter xappears as aC,i=xandaC′,j=−xwe know −N≤x≤Nand
+get:
+[34] has no parameters,
+[(3243)′] 0≤a≤Nusing the symmetry of the fan,
+[(3243)′′]−N≤b,c≤N,
+[46]−N≤a,b,c≤N,
+[324362] here we use aC,i= 2a+1 andaC′,j=−2a−1, so−N−1
+2≤a≤N−1
+2.
+23Parameters in the Blowing-up
+In three dimensions we have to consider the following two different ca ses of blowing-ups:
+•When blowing-up in a full-dimensional cone we can do the same as in the t wo-
+dimensional case.
+Denote by C1={r1,n1,n2}, C2={r2,n1,n2}two full-dimensional cones contain-
+ing a codimension 1 cone C={n1,n2}withr1+r2=aC,1n1+aC,2n2. We blow
+up in the cone C2, so the new full-dimensional cones containing CareC1and
+C′
+2={r′
+2,n1,n2}withr′
+2=r2+n1+n2. We obtain
+r1+r′
+2=aC,1n1+aC,2n2+n1+n2= (aC,1+1)n1+(aC,2+1)n2
+andthisgives aC,i+1≥ −Nandthus aC,i≥ −N−1. Thatmeanswehavetoloosen
+the bounds on a parameter by one if it appears at one of the codimen sion 1 cones
+contained in the cone of the current blowing-up, just as in the two- dimensional
+case.
+•For the blowing-up in a codimension 1 cone choose C1, C2andCas in the full-
+dimensional case. When blowing up in the cone Catn=n1+n2we get six
+new cones C′
+1={r1,n1,n}, C′
+2={r2,n1,n}containing C′={n1,n}andC′′
+1=
+{r1,n2,n}, C′′
+2={r2,n2,n}containing C′′={n,n2}. Now check the parameters
+at the new cone C′:
+r1+r2=aC,1n1+aC,2n2=aC,1n1+aC,2(n−n1) = (aC,1−aC,2)n1+aC,2n
+and atC′′we get equivalently:
+r1+r2=aC,1n1+aC,2n2=aC,1(n−n2)+aC,2n2=aC,1n+(aC,2−aC,1)n2
+So, in the blowing-up, we have found two codimension 1 cones each ha ving one of
+the edge-parameters of the original fan, aC′,2=aC,2andaC′′,1=aC,1. These edge-
+parameters of the two new cones C′, C′′yield the same bounds for the parameters
+of the fan as the codimension 1 cone Cin the original fan. Which means we can
+just pass on the bounds to the new blowing-up.
+So we have again all neccessary bounds for the parameters of all f ans and can create a
+list of completely defined smooth fans that serve as normal fans fo r our polytopes.
+3.3.2. Creating the Polytopes
+We can now continue as in the general and the two-dimensional case usingpolymake
+in combination with 4ti2,normaliz2 andLatte macchiato to compute all smooth
+polytopes with at most 12 lattice points.
+243.3.3. Results
+Running this algorithm for the remaining five minimal smooth fans yields the following
+result.
+Proposition. There are 33 smooth three-dimensional lattice polytopes wi th at most 12
+lattice points.
+Vertices 4 6 8 ≥10
+Polytopes 2 25 6 0
+Again the list of polytopes can be found in the appendix.
+4. Final Comments
+We now have a list of smooth lattice polytopes in dimension two and thre e with at most
+12 lattice points. For dimension three the bound 12 may seem rather low and in fact
+none of the 33 lattice polytopes from this classification has an interio r lattice point. So
+these polytopes may not bring great insight, but this is a first step a nd there are several
+points in the algorithm where it could be improved.
+By implementing a different way to directly generate all smooth norma l fans one could
+skip the big recursion calculating all blowing-ups, which is one of the mo st time-consu-
+ming steps in the classification algorithm. And one could then overcom e the limits of at
+most 12 vertices and thus lattice points and dimension at most three .
+The second point where one could work on is the calculation of lattice p oints of the
+polytope containing all right-hand sides. The dimension of this polyto pe is equal to
+the number of rays of the fan minus the dimension. So when we want t o reach more
+complex smooth polytopes we will have rather high-dimensional polyt opes containing
+the right-hand sides, so we might need better bounds for the right -hand sides and/or
+faster algorithms for computing the lattice points.
+25References
+[1] 4ti2 team, 4ti2—a software package for algebraic, geometric and combi natorial prob-
+lems on linear spaces , Available at http://www.4ti2.de .
+[2] Robert Bowen and Stephen Fisk, Generation of triangulations of the sphere , Math-
+ematics of Computation 21(1967), no. 98, 250–252.
+[3] W. Bruns and B. Ichim, NORMALIZ . computing normalizations of
+affine semigroups. , With contributions by C. S¨ oger. Available from
+http://www.math.uos.de/normaliz .
+[4] Jes´ usA.DeLoera,RaymondHemmecke, RurikoYoshida, andJe remyTauzer, lattE,
+2005,http://www.math.ucdavis.edu/ ~latte/.
+[5] Alicia Dickenstein, Sandra Di Rocco, Christian Haase, Milena Hering , Diane Macla-
+gan, BenjaminNill, PacoSantos, HalSchenck, andGregSmith, There are few smooth
+d-polytopes with n lattice points , 2009.
+[6] Ewgenij Gawrilow and Michael Joswig, polymake: a framework for analyzing convex
+polytopes , Polytopes — Combinatorics and Computation (Gil Kalai and G¨ unter M.
+Ziegler, eds.), Birkh¨ auser, 2000, pp. 43–74.
+[7] Tadao Oda, Convex bodies and algebraic geometry, an introduction to th e theory of
+toric varieties , Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Ve rlag,
+1987.
+[8] G.M. Ziegler, Lectures on polytopes , Graduate Texts in Mathematics, vol. 152,
+Springer-Verlag, 1995.
+26A. List of Smooth Polygons
+(1,0) (0,0)
+(0,1) (1,1)(0,0) (0,1)
+(2,0) (2,1)(0,0) (0,1)
+(3,0) (3,1)(0,0) (0,1)
+(4,0) (4,1)(0,0) (0,1)
+(5,0) (5,1)
+(2,0) (0,0)
+(0,2) (2,2)(0,2) (0,0)
+(3,0) (3,2)(1,0) (0,0)
+(0,1) (3,1)(0,0) (0,1)
+(2,0) (4,1)(0,0) (0,1)
+(3,0) (5,1)
+(0,0) (0,1)
+(4,0) (6,1) (0,0) (1,0)
+(5,2) (0,2)(1,0) (0,0)
+(0,1) (4,1)
+(0,0) (0,1)
+(2,0) (5,1)(0,0) (0,1)
+(3,0) (6,1)(1,0) (0,0)
+(0,1) (5,1)
+(0,0) (0,1)
+(2,0) (6,1)(0,0) (0,1)
+(3,0) (7,1)(1,0) (0,0)
+(0,1) (6,1)
+(0,0) (0,1)
+(2,0) (7,1)(1,0) (0,0)
+(0,1) (7,1)(0,0) (0,1)
+(2,0) (8,1)
+(1,0) (0,0)
+(0,1) (8,1)(1,0) (0,0)
+(0,1) (9,1)(0,1) (1,0)
+(0,0) (0,0) (2,0)
+(0,2)
+(0,0) (3,0)
+(0,3)(1,0) (0,1)
+(0,2) (2,0) (2,0) (0,2)
+(0,3) (3,0)(1,0) (0,1)
+(0,3) (3,0)(1,1) (3,0)
+(0,2) (0,4)
+(4,0)(4,1) (0,3)
+(0,2) (3,0)
+(1,1) (4,0)
+(1,3) (3,2)
+(0,3) (0,2)
+(3,0) (1,1)
+(4,0) (1,3)(2,0) (0,1)
+(1,0) (0,3)
+(2,1)(2,0) (0,1)
+(1,0) (0,4)
+(2,2)(0,2) (1,0)
+(0,1) (2,0)
+(2,1) (1,2)(0,3) (2,0)
+(0,2) (3,0)
+(3,1) (1,3)(0,3) (1,0)
+(0,1) (3,0)
+(3,1) (1,3)
+(0,0) (1,0)
+(0,4) (1,3)(2,0) (0,0)
+(0,4) (2,2) (0,0) (1,0)
+(0,5) (1,4)
+27B. List of Smooth Polytopes
+(0,0,0) (0,1,0)
+(0,0,1) (7,0,0)
+(1,1,0) (1,0,1)(0,0,0) (0,1,0)
+(0,0,1) (6,0,0)
+(1,1,0) (1,0,1)(0,0,0) (0,1,0)
+(0,0,1) (6,0,0)
+(2,1,0) (1,0,1)
+(0,0,0) (0,1,0)
+(0,0,1) (5,0,0)
+(1,1,0) (1,0,1)(0,0,0) (0,1,0)
+(0,0,1) (5,0,0)
+(2,1,0) (1,0,1)(0,0,0) (0,1,0)
+(0,0,1) (5,0,0)
+(3,1,0) (1,0,1)
+(0,0,0) (0,1,0)
+(0,0,1) (4,0,0)
+(1,1,0) (1,0,1)(0,0,0) (0,1,0)
+(0,0,1) (5,0,0)
+(2,1,0) (2,0,1)(0,0,0) (0,1,0)
+(0,0,1) (4,0,0)
+(2,1,0) (1,0,1)
+(0,0,0) (0,1,0)
+(0,0,1) (4,0,0)
+(3,1,0) (1,0,1)(0,0,0) (0,1,0)
+(0,0,1) (4,0,0)
+(4,1,0) (1,0,1)(0,0,0) (0,1,0)
+(0,0,1) (3,0,0)
+(1,1,0) (1,0,1)
+(0,0,0) (0,1,0)
+(0,0,1) (4,0,0)
+(2,1,0) (2,0,1)(0,0,0) (0,1,0)
+(0,0,1) (3,0,0)
+(2,1,0) (1,0,1)(0,0,0) (0,1,0)
+(0,0,1) (4,0,0)
+(3,1,0) (2,0,1)(0,0,0) (0,1,0)
+(0,0,1) (3,0,0)
+(3,1,0) (1,0,1)
+(0,0,0) (0,1,0)
+(0,0,1) (2,0,0)
+(2,1,0) (1,0,1)(0,0,0) (0,1,0)
+(0,0,1) (3,0,0)
+(3,1,0) (2,0,1)(0,0,1) (0,1,0)
+(1,0,0) (0,0,0) (0,0,0) (2,0,0)
+(0,2,0) (0,0,2)(1,0,0) (0,1,0)
+(1,0,1) (0,1,1)
+(0,2,0) (2,0,0)(2,0,0) (0,2,0)
+(2,0,1) (0,2,1)
+(0,3,0) (3,0,0)
+(0,1,0) (1,0,0)
+(0,0,0) (0,0,1)
+(1,0,1) (0,1,1)(0,0,0) (0,0,1)
+(2,0,0) (0,2,0)
+(2,0,1) (0,2,1)(0,0,0) (1,0,0)
+(0,1,0) (0,0,2)
+(1,0,2) (0,1,2)(0,0,0) (1,0,0)
+(0,1,0) (0,0,3)
+(1,0,3) (0,1,3)(0,0,0) (0,0,1)
+(2,0,0) (0,2,0)
+(1,0,1) (0,1,1)(0,1,0) (0,0,0)
+(0,0,1) (0,1,1)
+(3,0,0) (3,1,0)
+(1,0,1) (1,1,1)
+(0,1,0) (0,0,0)
+(0,0,1) (0,1,1)
+(2,0,0) (3,1,0)
+(1,0,1) (2,1,1)(0,1,0) (0,0,0)
+(0,0,1) (0,1,1)
+(2,0,0) (2,1,0)
+(1,0,1) (1,1,1)(0,1,0) (0,0,0)
+(0,0,1) (0,2,1)
+(2,0,0) (2,1,0)
+(1,0,1) (1,2,1)(1,1,0) (0,1,0)
+(0,0,0) (1,0,0)
+(0,0,1) (1,0,1)
+(0,1,1) (1,1,1)(0,1,0) (0,0,0)
+(0,0,1) (0,1,1)
+(2,0,0) (2,1,0)
+(2,0,1) (2,1,1)
+28C.polymake extension
+C.1. Parameterized Fans ( pfan.rules )
+# a parameterized class of fans
+declare object ParameterFan <Rational> {
+# maximal number of lattice points
+property POINT_LIMIT : Int;
+# dimension of the fan
+property DIM : Int;
+# incidence sets (of the rays) defining the full-dimensiona l cones or
+# vertices of the polytope
+property VERTICES : Array<Set>;
+# number of vertices
+property N_VERTICES : Int;
+# incidence sets (of the vertices) defining the codimension 1 cones or
+# edges between vertices of the polytopes
+property EDGES : Array<Set>;
+# number of edges
+property N_EDGES : Int;
+# rays of the fan
+property RAYS : Matrix<Rational>;
+# number of rays
+property N_RAYS : Int;
+# number of different parameters
+property N_PARAMETER : Int;
+# rows with positions in RAYS matrix, parameter number and co efficient
+property PARAMETER_POS : Matrix<Int>;
+# lower and upper bound for each parameter (rowwise)
+property PARAMETER_BOUNDS : Matrix<Int>;
+# searches full-dimensional cones sharing a common codimen sion 1 cone
+rule EDGES : VERTICES , N_VERTICES , DIM {
+my $vert = $this->VERTICES;
+my @edges = ();
+local $_;
+# check all possible pairs
+foreach (all_subsets_of_k(2,0..($this->N_VERTICES -1) )) {
+my $r = $vert->[$_->[0]] * $vert->[$_->[1]];
+if ($#$r == ($this->DIM - 2)) {
+push @edges, new Set($_);
+}
+}
+$this->EDGES = new Array<Set>(@edges);
+}
+rule DIM : RAYS {
+$this->DIM = $this->RAYS->cols;
+}
+rule N_EDGES : EDGES {
+$this->N_EDGES = $#{$this->EDGES} + 1;
+29}
+rule N_VERTICES : VERTICES {
+$this->N_VERTICES = $#{$this->VERTICES} + 1;
+}
+rule N_RAYS : RAYS {
+$this->N_RAYS = $this->RAYS->rows;
+}
+# main method creating the list of polytopes for this fan
+# calls all necessary other functions
+# returns: the polytopes as perl array of Polytope objects
+# input: integer value determining the print- or debug-leve l
+user_method POLYTOPES {
+my ($this,$print) = @_;
+my @poly = ();
+print "Creating blowing -ups ...\n" if($print);
+my @blowups = $this->create_blowups($print);
+print "Done creating blowing -ups.\n" if($print);
+my $i=1;
+print "Generating polytopes ...\n" if($print);
+foreach my $pfan (@blowups) {
+print "blowup ",$i," of ",scalar(@blowups),"\n" if ($prin t);
+my @fans = $pfan->getfans();
+my $j = 1;
+foreach my $fan (@fans) {
+print "\tfan ",$i,":",$j," of ",scalar(@blowups),":",
+scalar(@fans),"\n" if ($print);
+my $num = $fan->N_POLYTOPES;
+if ($num > 0) {
+push @poly, $fan->POLYTOPE_LIST;
+}
+$j++;
+}
+$i++;
+}
+print "Done generating polytopes.\n" if($print);
+print "Found ",scalar(@poly), " polytopes including dupli cates .\n" if($print);
+print "Removing duplicates ...\n" if($print);
+my @final_poly = ();
+outer: for (my $i = scalar(@poly)-1; $i >= 0; $i--) {
+my $p = @poly[$i];
+for (my $j = $i-1; $j >= 0; $j--) {
+my $q = @poly[$j];
+# reduce neccessary isomorphy checks by checking number of v ertices first
+if ($p->N_VERTICES == $q->N_VERTICES) {
+if (lattice_isomorphic_smooth_polytopes($p,$q)) {
+next outer;
+}
+}
+}
+push @final_poly , $p;
+}
+print "Found ",scalar(@final_poly), " non-isomorphic pol ytopes with this normal fan.\n";
+return @final_poly;
+}
+# starts the recursion generating all blowing-ups returnin g them in an perl array
+# printlimit specifies up to which depth to print messages
+user_method create_blowups {
+my ($fan,$printlimit) = @_;
+# incidence vectors storing which vertices and edges should be blown-up in the tree,
+# zero vectors at the beginning
+my $vertexlist = new Vector <Int>($fan->POINT_LIMIT );
+30my $edgelist = undef;
+$edgelist = new Vector<Int>(($fan->DIM * $fan->POINT_LIM IT )/2) if ($fan->DIM > 2);
+my @blowups = recursion($fan,$vertexlist ,$edgelist,0,$ printlimit);
+return @blowups;
+}
+# creates a new fan for every combination of parameter values and
+# returns the list of fans as perl array
+user_method getfans {
+my $bf = shift;
+my $rays = $bf->RAYS;
+my @fans = ();
+if ($bf->N_PARAMETER > 0) {
+my $npar = $bf->N_PARAMETER;
+my $pb = $bf->PARAMETER_BOUNDS;
+# vector storing the current parameter values
+my $par = new Vector<Int>($pb->col(0));
+my $cont = 1;
+do {
+my $r = new Matrix<Rational>($rays);
+# add the parameter values to every position it occurs
+foreach my $pos (@{$bf->PARAMETER_POS}) {
+$r->($pos->[0],$pos->[1]) += $par->[$pos->[2]] * $pos-> [3];
+}
+push @fans,new Fan<Rational>(POINT_LIMIT=>($bf->POINT _LIMIT),DIM=>($bf->DIM),
+RAYS=>$r,VERTICES=>($bf->VERTICES),EDGES=>($bf->EDG ES));
+# generate next vector of parameter values
+# increase first parameter and check bounds,...
+$par->[0] ++;
+for (my $i = 0; $i < $npar; $i++) {
+if($par->[$i] > $pb->[$i]->[1]) {
+if($i == $npar-1) {
+$cont = 0;
+} else {
+$par->[$i] = $pb->[$i]->[0];
+$par->[$i+1]++;
+}
+}
+}
+} while($cont == 1);
+} else {
+push @fans,new Fan<Rational>(POINT_LIMIT=>($bf->POINT _LIMIT),DIM=>($bf->DIM),
+RAYS=>$rays,VERTICES=>($bf->VERTICES),EDGES=>($bf-> EDGES));
+}
+return @fans;
+}
+# main recursion creating the blowing-ups
+sub recursion {
+my ($fan,$vertexlist ,$edgelist,$depth,$printlimit) = @ _;
+my @b = ();
+if ($fan->DIM > 2) {
+# in dimension > 2 check smooth polygon criterion
+if (checkpolygons($fan) <= $fan->POINT_LIMIT) {
+push @b, $fan;
+}
+} else {
+push @b, $fan;
+}
+# if blowing-up does not exceed maximal number of vertices
+if ($fan->N_VERTICES <= $fan->POINT_LIMIT - ($fan->DIM-1 )) {
+# blowing-up of full-dimensional cones
+31for(my $i = 0; $i < $fan->N_VERTICES; $i++) {
+if ($vertexlist ->[$i] < 1) {
+print "do vertexblowup:",$depth ,"-",$i,"\n"if ($depth < $printlimit);
+my ($newfan ,$elist) = do_vertex_blowup($fan,$i,$edgeli st);
+push @b, recursion($newfan ,(new Vector<Int>($vertexlis t)),
+$elist ,$depth+1,$printlimit);
+$vertexlist ->[$i]=1;
+}
+}
+# if dim > 2 blowing-up of codim 1 cones
+if($fan->DIM > 2) {
+for (my $i = 0; $i < $fan->N_EDGES; $i++) {
+if ($edgelist->[$i] < 1) {
+print "do edgeblowup:",$depth ,"-",$i,"\n" if ($depth < $p rintlimit );
+my ($newfan ,$vlist,$elist) = do_edge_blowup($fan,$i,$v ertexlist ,$edgelist);
+push @b, recursion($newfan ,$vlist,$elist ,$depth+1,$pr intlimit);
+if($edgelist->[$i] == 0) {
+$edgelist->[$i] = 1;
+}
+}
+}
+}
+}
+return @b;
+}
+# create the new fan after the blowing-up of full-dim cone $i
+sub do_vertex_blowup {
+my ($fan,$i,$edgelist) = @_;
+# set of rays defining the vertex
+my $list = $fan->VERTICES->[$i];
+my $nrays = $fan->N_RAYS;
+my $elist = undef;
+$elist = new Vector<Int>($edgelist) if (defined($edgelis t));
+# create and add the new ray
+my $ray = new Vector<Rational>($fan->DIM);
+local $_;
+foreach (@$list) {
+$ray += $fan->RAYS->[$_];
+}
+my $newrays = $fan->RAYS / $ray;
+# prepare edges array, vertices array and array with positio ns for new vertices
+my @newedges = @{$fan->EDGES};
+my @pos = ($i);
+my @vertices = @{$fan->VERTICES};
+for (my $k = 0; $k < $fan->DIM - 1; $k++) {
+push @pos, ($fan->N_VERTICES+$k);
+push @vertices, new Set();
+}
+local $_;
+foreach (all_subsets_of_k(2,@pos)) {
+push @newedges, (new Set($_));
+}
+my $k = 0;
+my $newvertices = new Array<Set>(@vertices);
+my $newedges = new Array<Set>(@newedges);
+# find neighbors of vertex we are cutting off, correct the edg es
+# and create new vertices
+for (my $j = 0; $j < $fan->N_EDGES; $j++) {
+my $edge = $fan->EDGES->[$j];
+if ($edge->contains($i)) {
+my $v = ($edge - $i)->[0];
+# new vertex
+32$newvertices ->[@pos[$k]] = ($fan->VERTICES->[$v] * $lis t) + $nrays;
+# set new edge endpoints
+$newedges->[$j] = new Set($v,@pos[$k]);
+# set neighboring edges to always ignore,
+# we do vertex -> neighboring edge via edge -> edge
+$elist ->[$j] = 2 if (defined($edgelist));
+$k++;
+}
+}
+my $newfan;
+# add parameters for the new ray and loosen bounds if neccessa ry
+my $par = new Matrix<Int>();
+if ($fan->N_PARAMETER > 0) {
+my @param = @{$fan->PARAMETER_POS};
+local $_;
+my $p_indices = new Vector<Int>($fan->N_PARAMETER );
+# find parameters appearing in the rays defining the current ly blown-up cone
+foreach (@{$fan->PARAMETER_POS}) {
+if ($list->contains($_->[0])) {
+push @param, (new Vector <Int>($nrays ,$_->[1],$_->[2],$ _->[3]));
+$p_indices->[$_->[2]]=1;
+}
+}
+$par = new Matrix<Int>(@param);
+# loosen bounds on all parameters found earlier
+my $pbounds = new Matrix<Int>($fan->PARAMETER_BOUNDS);
+for (my $j = 0; $j < $fan->N_PARAMETER; $j++) {
+if ($p_indices ->[$j] > 0) {
+$pbounds->[$j]->[0] --;
+$pbounds->[$j]->[1] ++;
+}
+}
+# create new fan with all modified data
+$newfan = new ParameterFan <Rational>(POINT_LIMIT=>$fan ->POINT_LIMIT ,
+DIM=>$fan->DIM,VERTICES=>$newvertices ,EDGES=>$newed ges,RAYS=>$newrays,
+PARAMETER_POS=>$par,PARAMETER_BOUNDS=>$pbounds,N_PA RAMETER=>$fan->N_PARAMETER );
+} else {
+$newfan = new ParameterFan <Rational>(POINT_LIMIT=>$fan ->POINT_LIMIT ,
+DIM=>$fan->DIM,VERTICES=>$newvertices ,EDGES=>$newed ges,RAYS=>$newrays,
+PARAMETER_POS=>$par,N_PARAMETER=>$fan->N_PARAMETER ) ;
+}
+return ($newfan ,$elist);
+}
+# create the new fan after the blowing-up of full-dim cone $i
+sub do_edge_blowup {
+my ($fan,$i,$vertexlist ,$edgelist) = @_;
+my $vlist = new Vector<Int>($vertexlist );
+my $elist = new Vector<Int>($edgelist);
+# indices of the two vertices of the edge
+my $edge = $fan->EDGES->[$i];
+my $e1 = $edge->[0];
+my $e2 = $edge->[1];
+# sets of the rays defining these vertices
+my $v1 = $fan->VERTICES->[$e1];
+my $v2 = $fan->VERTICES->[$e2];
+# two rays defining this edge (as set and separate)
+my $er = $v1*$v2;
+my $er1 = $er->[0];
+my $er2 = $er->[1];
+# total number of vertices and rays
+my $nv = $fan->N_VERTICES;
+my $nrays = $fan->N_RAYS;
+# add new ray
+33my $newrays = $fan->RAYS / ($fan->RAYS->[$er->[0]] + $fan- >RAYS->[$er->[1]]);
+# new edgearray
+my @newedges = @{$fan->EDGES};
+# create edges to and between the two vertices that will be put at the end
+push @newedges, new Set($nv,$nv+1);
+push @newedges, new Set($e1,$nv);
+push @newedges, new Set($e2,$nv+1);
+my $k = 0;
+# create new vertices
+my $newvertices = new Array<Set>(@{$fan->VERTICES},new S et(), new Set());
+$newvertices ->[$e1] = $v1 - $er2 + $nrays;
+$newvertices ->[$e2] = $v2 - $er2 + $nrays;
+$newvertices ->[$nv] = $v1 - $er1 + $nrays;
+$newvertices ->[$nv+1] = $v2 - $er1 + $nrays;
+# set the new vertices to ’consider’ for later blowing-ups
+$vlist ->[$e1] = 0;
+$vlist ->[$e2] = 0;
+# create the new edges and update all old edges
+# we skip all just added edges and the cut-off edge
+my $newedges = new Array<Set>(@newedges);
+for (my $j = 0; $j < $fan->N_EDGES; $j++) {
+next if ($i==$j);
+my $e = $fan->EDGES->[$j];
+if ($e->contains($e1) || $e->contains($e2)) {
+my $v = ($e - $edge)->[0];
+if (!($fan->VERTICES->[$v]->contains($er1))) {
+$newedges->[$j] = new Set($v,$nv) if ($e->contains($e1)) ;
+$newedges->[$j] = new Set($v,$nv+1) if ($e->contains($e2 ));
+}
+$elist ->[$j] = -1 if ($elist ->[$j]==1);
+}
+}
+my $newfan;
+# check if the new ray has parameters
+my $par = new Matrix<Int>();
+if ($fan->N_PARAMETER > 0) {
+my @param = @{$fan->PARAMETER_POS};
+local $_;
+# search for parameters for the rays defining the new one
+foreach (@{$fan->PARAMETER_POS}) {
+if ($er->contains($_->[0])) {
+push @param, (new Vector <Int>($nrays ,$_->[1],$_->[2],$ _->[3]));
+}
+}
+$par = new Matrix<Int>(@param);
+$newfan = new ParameterFan <Rational>(POINT_LIMIT=>$fan ->POINT_LIMIT ,
+DIM=>$fan->DIM,VERTICES=>$newvertices ,EDGES=>$newed ges, RAYS=>$newrays,
+PARAMETER_POS=>$par,PARAMETER_BOUNDS=>$fan->PARAMET ER_BOUNDS ,
+N_PARAMETER=>$fan->N_PARAMETER);
+} else {
+$newfan = new ParameterFan <Rational>(POINT_LIMIT=>$fan ->POINT_LIMIT ,
+DIM=>$fan->DIM,VERTICES=>$newvertices ,EDGES=>$newed ges, RAYS=>$newrays,
+N_PARAMETER=>$fan->N_PARAMETER);
+}
+return ($newfan ,$vlist,$elist);
+}
+# check the smooth polygon lower bound for the number of latti ce points
+sub checkpolygons{
+my ($fan) = @_;
+my $rays = $fan->N_RAYS;
+34# results from the 2-dimensional classification,
+# "13" means not possible for <= 12 lattice points
+# FIXME for more than POINT_LIMIT > 12
+my $innercounts = new Vector<Rational>(’0 0 1 1 13 4 13 13’);
+my $bordercounts = new Vector<Rational>(’3 4 7 6 13 8 13 13’);
+# collect degrees of all rays (number of vertices they appear in)
+my $sizes = new Vector<Int>( ’0 ’ x $rays );
+local $_;
+foreach (@{$fan->VERTICES}) {
+foreach my $x (@$_) {
+$sizes ->[$x] ++;
+}
+}
+# find maximal degree, and sum up all interior lattice points
+my $count = 0;
+my $max = 0;
+local $_;
+foreach (@$sizes) {
+if ($_ > $max) {
+$max = $_;
+}
+next if ($_ > 8);
+$count += $innercounts ->[$_-3];
+}
+if ($max > 8) {
+return 13;
+}
+# add boundary lattice points for the largest polygon
+$count += $bordercounts ->[$max-3];
+$count += $fan->N_VERTICES - $max;
+return $count;
+}
+}
+C.2. Fans ( fan.rules )
+# a completely defined fan
+declare object Fan<Rational> {
+# maximal number of lattice points
+property POINT_LIMIT : Rational;
+# dimension of the fan
+property DIM : Int;
+# incidence sets (of the rays) defining the full-dimensiona l cones or
+# vertices of the polytope
+property VERTICES : Array<Set>;
+# number of vertices
+property N_VERTICES : Int;
+# incidence sets (of the vertices) defining the codimension 1 cones or
+# edges between vertices of the polytopes
+property EDGES : Array<Set>;
+# number of edges
+property N_EDGES : Int;
+# rays of the fan
+35property RAYS : Matrix<Rational>;
+# number of rays
+property N_RAYS : Int;
+# polytope approximating the set of valid right-hand sides
+property RHS_POLYTOPE : Polytope<Rational>;
+# set of right-hand side vectors defining polytopes
+# with at most POINT_LIMIT lattice points
+property RHS_VECTORS : Set<Vector<Integer >>;
+# set of right-hand side vectors defining polytopes
+# with at most POINT_LIMIT lattice points
+property N_POLYTOPES : Integer;
+# returns a perl array of all polytopes with this normal fan an d
+# at most POINT_LIMIT lattice points
+user_method POLYTOPE_LIST {
+my $fan = shift;
+my @list = ();
+# create polytope for every rhs vector
+foreach my $rhs (@{$fan->RHS_VECTORS }) {
+my $ineq = (new Vector <Rational>($rhs)) | ($fan->RAYS);
+my $p = new Polytope<Rational>(INEQUALITIES=>$ineq);
+#my$v =$p->N_VERTICES;
+#my$lp =$p->N_LATTICE_POINTS;
+push @list, $p;
+}
+return @list;
+}
+# searches full-dimensional cones sharing a common codimen sion 1 cone
+rule EDGES : VERTICES , N_VERTICES , DIM {
+my $vert = $this->VERTICES;
+my @edges = ();
+local $_;
+# check all possible pairs
+foreach (all_subsets_of_k(2,0..($this->N_VERTICES -1) )) {
+my $r = $vert->[$_->[0]] * $vert->[$_->[1]];
+if ($#$r == ($this->DIM - 2)) {
+push @edges, new Set($_);
+}
+}
+$this->EDGES = new Array<Set>(@edges);
+}
+rule DIM : RAYS {
+$this->DIM = $this->RAYS->cols;
+}
+rule N_EDGES : EDGES {
+$this->N_EDGES = $#{$this->EDGES} + 1;
+}
+rule N_VERTICES : VERTICES {
+$this->N_VERTICES = $#{$this->VERTICES} + 1;
+}
+rule N_RAYS : RAYS {
+$this->N_RAYS = $this->RAYS->rows;
+}
+rule N_POLYTOPES : RHS_VECTORS {
+$this->N_POLYTOPES = $#{$this->RHS_VECTORS }+1;
+}
+36# searches in RHS_POLYTOPE for RHS-vectors defining polyto pes
+# with at most POINT_LIMIT lattice points
+rule RHS_VECTORS : RHS_POLYTOPE , RAYS , POINT_LIMIT , DIM {
+my $valid = $this->RHS_POLYTOPE;
+my $limit = $this->POINT_LIMIT;
+my $dim = $this->DIM;
+my $rays = $this->RAYS;
+# check the polytope for feasibility and number of lattice po ints first
+if ($valid ->FEASIBLE == 0) {
+$this->RHS_VECTORS=new Set<Vector<Integer >>();
+return;
+}
+if ($valid ->N_LATTICE_POINTS == 0) {
+$this->RHS_VECTORS=new Set<Vector<Integer >>();
+return;
+}
+# remove first coordinate (homogenized)
+my $rhs = $valid ->LATTICE_POINTS ->minor(All,range(1,$r ays->rows()-$dim));
+my @rhs = @$rhs;
+@rhs = sort @rhs;
+my @good_rhs = ();
+my $i=0;
+my $sched2;
+while (@rhs > 0) {
+# create new RHS vector and inequality matrix
+my $x = ((zero_vector <Int>($dim)) | shift(@rhs));
+my $ineq = (new Vector <Rational>($x)) | $rays;
+# define polytope and check number of lattice points
+my $p = new Polytope<Rational>(INEQUALITIES=>$ineq);
+if ($p->N_LATTICE_POINTS <= $limit) {
+# found a new polytope
+push @good_rhs, $x;
+$i++;
+} else {
+# we only keep RHS-vectors with at least one entry with smalle r
+# absolute value than the current vector
+my @newrhs = ();
+point: for my $v (@rhs) {
+for(my $j = 0; $j < $rays->rows()-$dim; $j++) {
+if (abs($v->[$j]) < abs($x->[$j+$dim])) {
+push @newrhs ,$v;
+next point;
+}
+}
+}
+@rhs = @newrhs;
+}
+}
+$this->RHS_VECTORS=new Set<Vector <Integer >>(@good_rh s);
+}
+# creates the polytope approximating the set of right-hand s ides
+rule RHS_POLYTOPE : RAYS , VERTICES , EDGES , POINT_LIMIT , N_ EDGES , N_VERTICES {
+my $vert = $this->VERTICES;
+my $edges = $this->EDGES;
+my $rays = $this->RAYS;
+my $dim = $rays->cols();
+my $r = $rays->rows();
+my $limit = $this->POINT_LIMIT;
+37# stores the sum of all edge-lengths
+my $totallength = new Vector<Rational>($r+1);
+# first stores l(e) >= 1 inequalities , second one inequaliti es for l(e)<=...
+my @ineq = ();
+my @l_ineq = ();
+# calculate all edge lengths and create inequalities
+foreach (@$edges) {
+# vertices defining the edge
+my $fromvert = $vert->[$_->[0]];
+my $tovert = $vert->[$_->[1]];
+# rays defining the edge (n_1,...,n_{d-1})
+my $neighbors = $fromvert * $tovert;
+# rays limiting this edge (r_1,r_2)
+my $from = $fromvert - $neighbors;
+my $to = $tovert - $neighbors;
+$from = $from->[0];
+$to = $to->[0];
+# calculate inequalities
+my ($edgelimit,$pointlimit) = edgelimits($rays,$from,$ to,$limit ,$neighbors);
+$totallength += $pointlimit;
+push @ineq, $edgelimit;
+push @l_ineq , $pointlimit;
+}
+# right-hand side for total length inequality
+$totallength ->[0] = $limit + $this->N_EDGES - $this->N_VE RTICES;
+push @l_ineq , $totallength;
+my $ineq = new Matrix<Rational>(\@ineq);
+my $l_ineq = new Matrix<Rational>(\@l_ineq);
+# for all rays with all entries positive we know that the right -hand side must
+# be postive and vice versa (we want the negative RHS-vectors )
+my $signs = new Matrix<Rational>($r,$r+1);
+my @signs = ();
+row: for (my $i = 0; $i < $r; $i++) {
+my $sign = 0;
+for (my $j = 0; $j < $dim; $j++) {
+if ($rays->($i,$j) < 0) {
+next row if ($sign > 0);
+$sign = -1;
+} elsif ($rays->($i,$j) > 0) {
+next row if ($sign < 0);
+$sign = 1;
+}
+}
+$signs ->($i,$i+1) = -$sign;
+}
+# put all inequalities together and remove the first d column s after the rhs entry
+# because those variables are always zero (we do not want tran slations)
+$ineq = $ineq / $signs;
+$ineq = $ineq->minor(All,range(0,0)+range($dim+1,$r)) ;
+$l_ineq = $l_ineq ->minor(All,range(0,0)+range($dim+1, $r));
+my $limited = (new Matrix<Rational>($ineq)) / (new Matrix< Rational>($l_ineq));
+$this->RHS_POLYTOPE = new Polytope<Rational>( INEQUALIT IES=>$limited );
+}
+38# calculates the inequalities in terms of the rhs vector give n by 1<=l(e)<=...
+sub edgelimits {
+my ($rays, $from, $to, $limit, $nb) = @_;
+my $dim = $rays->cols();
+# limiting rays
+my $r1 = $rays->row($to);
+my $r2 = $rays->row($from);
+# rays defining the edge
+my $N = $rays->minor($nb,All);
+# determine edge-parameters a_{C,i}
+$N = new Matrix<Rational>(transpose($N));
+my $v = new Vector<Rational>($r1+$r2);
+my $a = lin_solve($N,$v);
+my $sum = 0;
+# and sum them up for the right-hand side
+$sum += $a->[$_] for (0 .. ($dim-2));
+# vectors storing the edgelength >= 1 and <=... inequalities
+my $edgelimit = new Vector<Rational>($rays->rows() + 1);
+my $pointlimit = new Vector <Rational>($rays->rows() + 1);
+# rhs of the inequalities
+$edgelimit ->[0] = (-1);
+$pointlimit ->[0] = (($limit - $sum)/$dim-1);
+# set coefficients
+for (my $i = 0; $i < $dim-1; $i++) {
+my $k = $nb->[$i] + 1;
+$edgelimit ->[$k] = (-1) * $a->[$i];
+$pointlimit ->[$k] = $a->[$i];
+}
+$edgelimit->[$to+1] = 1;
+$pointlimit ->[$to+1] = -1;
+$edgelimit->[$from+1] = 1;
+$pointlimit ->[$from+1] = -1;
+return ($edgelimit,$pointlimit );
+}
+}
+C.3. Additional Script ( checkiso )
+# removes all lattice isomorphic polytopes from a list
+# only the first one of an isomorphy class is kept
+sub checkiso {
+my (@list) = @_;
+my @res = ();
+outer: for (my $i = scalar(@list)-1; $i >= 0; $i--) {
+my $p = @list[$i];
+for (my $j = $i-1; $j >= 0; $j--) {
+my $q = @list[$j];
+if ($p->N_VERTICES == $q->N_VERTICES) {
+if ($p->N_LATTICE_POINTS == $q->N_LATTICE_POINTS) {
+next outer if (lattice_isomorphic_smooth_polytopes($p, $q));
+}
+}
+}
+push @res, $p;
+}
+return @res;
+}
+39C.4. Minimal Fans as polymake files
+C.4.1. Dimension Two
+Projective Space
+<?xml version="1.0" encoding="utf-8"?>
+<object name="P" type="polytope::ParameterFan&lt;Rati onal&gt;" version="2.9.7"
+tm="4b0d8c7b" xmlns="http://www.math.tu-berlin.de/po lymake/#3">
+<property name="POINT_LIMIT" value="12" />
+<property name="DIM" value="2" />
+<property name="VERTICES">
+<m>
+<v>0 1</v>
+<v>1 2</v>
+<v>0 2</v>
+</m>
+</property>
+<property name="RAYS">
+<m>
+<v>1 0</v>
+<v>0 1</v>
+<v>-1 -1</v>
+</m>
+</property>
+<property name="N_RAYS" value="3" />
+<property name="N_VERTICES" value="3" />
+<property name="N_PARAMETER" value="0" />
+</object>
+Hirzebruch surfaces
+<?xml version="1.0" encoding="utf-8"?>
+<object name="H" type="polytope::ParameterFan&lt;Rati onal&gt;" version="2.9.7"
+tm="4af96f7b" xmlns="http://www.math.tu-berlin.de/po lymake/#3">
+<property name="POINT_LIMIT" value="12" />
+<property name="DIM" value="2" />
+<property name="VERTICES">
+<m>
+<v>0 1</v>
+<v>0 2</v>
+<v>1 3</v>
+<v>2 3</v>
+</m>
+</property>
+<property name="RAYS">
+<m>
+<v>1 0</v>
+<v>0 1</v>
+<v>0 -1</v>
+<v>-1 0</v>
+</m>
+</property>
+<property name="N_RAYS" value="4" />
+<property name="N_VERTICES" value="4" />
+<property name="PARAMETER_POS">
+<m>
+<v>3 1 0 1</v>
+</m>
+40</property>
+<property name="PARAMETER_BOUNDS">
+<m>
+<v>0 12</v>
+</m>
+</property>
+<property name="N_PARAMETER" value="1" />
+</object>
+C.4.2. Dimension Three
+Fan 34
+<?xml version="1.0" encoding="utf-8"?>
+<object name="f" type="polytope::ParameterFan&lt;Rati onal&gt;" version="2.9.7"
+tm="4b0d8c1d" xmlns="http://www.math.tu-berlin.de/po lymake/#3">
+<property name="POINT_LIMIT" value="12" />
+<property name="DIM" value="3" />
+<property name="VERTICES">
+<m>
+<v>0 1 2</v>
+<v>0 1 3</v>
+<v>0 2 3</v>
+<v>1 2 3</v>
+</m>
+</property>
+<property name="RAYS">
+<m>
+<v>1 0 0</v>
+<v>0 1 0</v>
+<v>0 0 1</v>
+<v>-1 -1 -1</v>
+</m>
+</property>
+<property name="N_RAYS" value="4" />
+<property name="N_VERTICES" value="4" />
+<property name="N_PARAMETER" value="0" />
+</object>
+Fan(3242)′
+<?xml version="1.0" encoding="utf-8"?>
+<object name="f" type="polytope::ParameterFan&lt;Rati onal&gt;" version="2.9.7"
+tm="4b0d571c" xmlns="http://www.math.tu-berlin.de/po lymake/#3">
+<property name="POINT_LIMIT" value="12" />
+<property name="DIM" value="3" />
+<property name="VERTICES">
+<m>
+<v>0 1 2</v>
+<v>0 1 3</v>
+<v>0 2 4</v>
+<v>0 3 4</v>
+<v>1 2 4</v>
+<v>1 3 4</v>
+</m>
+</property>
+<property name="RAYS">
+<m>
+<v>1 0 0</v>
+<v>0 1 0</v>
+41<v>0 0 1</v>
+<v>0 0 -1</v>
+<v>-1 -1 0</v>
+</m>
+</property>
+<property name="N_RAYS" value="5" />
+<property name="N_VERTICES" value="6" />
+<property name="PARAMETER_POS">
+<m>
+<v>4 2 0 -1</v>
+</m>
+</property>
+<property name="PARAMETER_BOUNDS">
+<m>
+<v>0 12</v>
+</m>
+</property>
+<property name="N_PARAMETER" value="1" />
+</object>
+Fan(3242)′′
+<?xml version="1.0" encoding="utf-8"?>
+<object name="f" type="polytope::ParameterFan&lt;Rati onal&gt;" version="2.9.7"
+tm="4b0d8bd2" xmlns="http://www.math.tu-berlin.de/po lymake/#3">
+<property name="POINT_LIMIT" value="12" />
+<property name="DIM" value="3" />
+<property name="VERTICES">
+<m>
+<v>0 1 2</v>
+<v>0 1 3</v>
+<v>0 2 3</v>
+<v>1 2 4</v>
+<v>1 3 4</v>
+<v>2 3 4</v>
+</m>
+</property>
+<property name="RAYS">
+<m>
+<v>1 0 0</v>
+<v>0 1 0</v>
+<v>0 0 1</v>
+<v>0 -1 -1</v>
+<v>-1 0 0</v>
+</m>
+</property>
+<property name="N_RAYS" value="5" />
+<property name="N_VERTICES" value="6" />
+<property name="PARAMETER_POS">
+<m>
+<v>4 1 0 1</v>
+<v>4 2 1 1</v>
+</m>
+</property>
+<property name="PARAMETER_BOUNDS">
+<m>
+<v>-12 12</v>
+<v>-12 12</v>
+</m>
+</property>
+<property name="N_PARAMETER" value="2" />
+</object>
+42Fan 46
+<?xml version="1.0" encoding="utf-8"?>
+<object name="f" type="polytope::ParameterFan&lt;Rati onal&gt;" version="2.9.7"
+tm="4b0d8a3c" xmlns="http://www.math.tu-berlin.de/po lymake/#3">
+<property name="POINT_LIMIT" value="12" />
+<property name="DIM" value="3" />
+<property name="VERTICES">
+<m>
+<v>0 1 2</v>
+<v>0 1 3</v>
+<v>0 2 4</v>
+<v>0 3 4</v>
+<v>1 2 5</v>
+<v>1 3 5</v>
+<v>2 4 5</v>
+<v>3 4 5</v>
+</m>
+</property>
+<property name="RAYS">
+<m>
+<v>1 0 0</v>
+<v>0 1 0</v>
+<v>0 0 1</v>
+<v>0 0 -1</v>
+<v>0 -1 0</v>
+<v>-1 0 0</v>
+</m>
+</property>
+<property name="N_RAYS" value="6" />
+<property name="N_VERTICES" value="8" />
+<property name="PARAMETER_POS">
+<m>
+<v>5 1 0 -1</v>
+<v>4 2 1 1</v>
+<v>5 2 2 1</v>
+</m>
+</property>
+<property name="PARAMETER_BOUNDS">
+<m>
+<v>-12 12</v>
+<v>-12 12</v>
+<v>-12 12</v>
+</m>
+</property>
+<property name="N_PARAMETER" value="3" />
+</object>
+Fan 324362
+<?xml version="1.0" encoding="utf-8"?>
+<object name="f" type="polytope::ParameterFan&lt;Rati onal&gt;" version="2.9.7"
+tm="4b0d42cd" xmlns="http://www.math.tu-berlin.de/po lymake/#3">
+<property name="POINT_LIMIT" value="12" />
+<property name="DIM" value="3" />
+<property name="VERTICES">
+<m>
+<v>0 1 2</v>
+<v>0 1 6</v>
+<v>0 2 5</v>
+<v>0 3 4</v>
+<v>0 3 5</v>
+43<v>0 4 6</v>
+<v>1 2 6</v>
+<v>2 5 6</v>
+<v>3 4 6</v>
+<v>3 5 6</v>
+</m>
+</property>
+<property name="RAYS">
+<m>
+<v>1 0 0</v>
+<v>0 1 0</v>
+<v>0 0 1</v>
+<v>0 0 -1</v>
+<v>0 1 -1</v>
+<v>0 -1 0</v>
+<v>-1 2 -1</v>
+</m>
+</property>
+<property name="N_RAYS" value="7" />
+<property name="N_VERTICES" value="10" />
+<property name="PARAMETER_POS">
+<m>
+<v>5 2 0 -1</v>
+</m>
+</property>
+<property name="PARAMETER_BOUNDS">
+<m>
+<v>-6 5</v>
+</m>
+</property>
+<property name="N_PARAMETER" value="1" />
+</object>
+44
\ No newline at end of file
diff --git a/1001.0515.txt b/1001.0515.txt
new file mode 100644
index 0000000000000000000000000000000000000000..be58ebf7e6057830f96738f4fcac6c124b42158a
--- /dev/null
+++ b/1001.0515.txt
@@ -0,0 +1,332 @@
+arXiv:1001.0515v1  [math.LO]  4 Jan 2010Stable embeddedness and NIP
+Anand Pillay∗
+University of Leeds
+January 4, 2010
+Abstract
+We give some sufficient conditions for a predicate Pin a complete
+theoryTto be “stably embedded”. Let PbePwith its “induced ∅-
+definable structure”. The conditions are that P(or rather its theory)
+hasfinitethornrank, PhasNIPinTandthat Pisstably1-embedded
+inT. This generalizes a recent result of Hasson and Onshuus[4] w hich
+deals with the case where Piso-minimal in T. Our proofs make use of
+the theory of strict nonforking and weight in NIPtheories ([2], [8]).
+1 Introduction and preliminaries
+The notion of “stable embeddedness” of a predicate (or definable s et) in a
+theory (or structure) is rather important in model theory, and s ays roughly
+that no new structure is added by external parameters. The cur rent paper is
+somewhattechnical, andheavilyinfluencedby[4]onthestableembed dability
+ofo-minimal structures, which we wanted to cast in a more general con text.
+Stable embeddedness usually refers to a structure Mwhich is inter-
+pretable (without parameters) in another structure N, and says that any
+subset ofMnwhich is definable, with parameters, in N, is definable, with
+parameters in M. I prefer to think of the universe of Mas simply an ∅-
+definable set PinN, and to say that Pis stably embedded in Nif every
+subset ofPnwhich is definable (in N) with parameters, is definable (in N)
+with parameters from P. See below.
+∗Supported by EPSRC grant EP/F009712/1
+1Soourgeneralframeworkisthatofacompletetheory Tinlanguage Land
+a distinguished predicate or formula or sort P(x). We work in a saturated
+modelNofT, and letPalso denote the interpretation of PinN. Unless
+we say otherwise, “definability” refers to definability with paramete rs in the
+ambient structure N. We will also discuss the notion “ Pis stable in N” just
+to clarify current notation and relationships.
+Definition 1.1. (i)Pis stable in T(or inN) if there do NOT exist a
+formulaφ(¯x,y)(where¯xis a tuple of variables each of which is of sort P,
+andyis an arbitrary tuple of variables) and ¯ai⊂Pandbi∈Neqfori<ω
+such thatN|=φ(¯ai,bj)iffi≤j.
+(ii)PisNIPinT(orN) if there do NOT exist φ(¯x,y)(with same proviso as
+before) and ¯ai⊂Pfori<ωandbs∈Neqfors⊆ωsuch thatN|=φ(¯ai,bs)
+iffi∈s.
+(iii)Pis stably embedded in T(orN) if for allnevery subset of Pnwhich
+is definable in Nwith parameters, is definable in Nwith parameters from P.
+(iv)Pis1-stably embedded, if (iii) holds for n= 1.
+We letPdenote the structure whose universe is Pand whose basic re-
+lations are those which are ∅-definable in N. Note that if PisNIPinN
+andPis the structure with universe Pand relations all subsets of various
+Pnwhich are ∅-definable in N, thenTh(P) hasNIPtoo. Of course if T
+(=Th(N)) hasNIPthenPhasNIPinN, and even in this case there are
+situations where Pis not stably embedded in N. For example when Tis the
+theory of dense pairs of real closed fields and where Pis the bottom model.
+Remark 1.2. (i) ForPto be stable in Nit is enough that Definition 1.1(i)
+holds in the case where ¯xis a single variable xranging over P. Likewise for
+PbeingNIPinNand Definition 1.1(ii). Also if Pis stable in Nthen it is
+NIPinN.
+(ii) Suppose that <is a distinguished ∅-definable total ordering on P. Define
+Pto beo-minimal in Nif every definable (in N, with parameters) subset
+ofPis a finite union of intervals (with endpoints in Ptogether with plus
+or minus ∞) and points. Then IF Piso-minimal in Nthen it is 1-stably
+embedded in NAND hasNIPinN.
+(iii) IfPis stable in NthenPis stably embedded in N.
+Comments. (i) (forPbeingNIPinT) and (ii) were already mentioned in
+an earlier draft of [4]. For (i) the point is that well-known results redu cing
+2stability and NIPto the case of formulas φ(x,y) wherexranges over ele-
+ments rather than tuples, adapt to the current “relative” conte xt. (iii) is also
+well-known.
+We now discuss the finite thorn rank condition. There is an extensive theory
+in place around rosy theories and U-thorn-rank (e.g. [6]). From this point
+of view, a theory T′has “finite rank” if T′is rosy and there is a finite bound
+on theU-thorn-ranks of types in any given imaginary sort. If T′is 1-sorted
+it suffices to have a bound on the U-thorn-ranks of types of elements in the
+home sort. See [3] for more details on this point of view. We will now give an
+equivalent abstract definition which is the one we will work with and whic h
+does not need any acquaintance with rosy theories and/or thorn f orking. In
+the following T′is a complete theory, a,b,c,..range over possibly imaginary
+elements of a saturated model of T′andA,B,C,.. range over small sets of
+imaginaries from such a model.
+Definition 1.3. We say that T′hasfinite rank if there is an assignement
+to everya,Bof a nonnegative integer rk(a/B), depending only on tp(a,B)
+with the following properties:
+(i)rk(a,b/C) =rk(b,a/C) =rk(a/b,C)+rk(b/C),
+(ii)rk(a/B) = 0iffa∈acl(B),
+(iii) for any aandB⊆C, there isa′such thattp(a′/B) =tp(a/B)and
+rk(a′/C) =rk(a/B),
+(iv) for any (imaginary) sort Zthere is a finite bound on rk(a)fora∈Z.
+IfT′is a finite rank theory then we obtain a good notion of independence
+by defining ato be independent from BoverCifrk(a/BC) =rk(a/C). Of
+course ano-minimal theory is an example.
+We can now state our main result, reverting to our earlier notation ( T,P,P,
+etc.).
+Theorem 1.4. Suppose that Th(P)is finite rank, PhasNIPinTandP
+is1-stably embedded in N. ThenPis stably embedded in N.
+Our previous discussion shows that if Piso-minimal in Tthen all the hy-
+potheses ofTheorem 1.4 aresatisfied. Let us note immediately that Theorem
+1.4 needs the “ NIPinN” hypothesis on P. For example let Nbe the struc-
+ture with two disjoint unary predicates P,Q, and a random bi-partite graph
+3relationRbetweenP(2)(unordered pairs from P) andQ. Then one checks
+thatPis 1-stably embedded in Nbut not stably embedded.
+Hrushovski suggested the following example showing that the finite rank
+hypothesis is needed. Let Tbe some completion of the theory of proper
+dense elementary pairs of algebraically closed valued fields F1< F2, which
+have the same residue field and value group. Then F1is 1-stably embedded
+inF2. But ifa∈F2\F1, then the function taking x∈F1tov(x−a) cannot
+be definable with parameters from F1.
+Thanks to Ehud Hrushovski and Alf Onshuus for helpful comments , sugges-
+tions, and discussions.
+2 Forking
+In this section we fix a complete theory Tand work in a saturated model ¯M.
+As usualA,B,..denote small subsets of ¯M. Likewise for small elementary
+substructures M,M0etc.a,b,.. usually denote elements of ¯Mequnless we
+say otherwise. Likewise for variables. Dividing and forking are meant in
+the sense of Shelah. Namely a formula φ(x,b) divides over Aif someA-
+indiscernible sequence ( φ(x,bi) :i < ω) (withb0=b) is inconsistent. And
+a partial type forks over Aif it implies a finite disjunction of formulas each
+of which divides over A. By a global type we mean a complete type over ¯M
+(or over a sufficiently saturated M). Note that for a global type p(x),pdoes
+not fork over Aiffpdoes not divide over A(i.e. every formula in pdoes not
+divide over A). Also any partial type does not fork over Aif and only if it
+has an extension to a global type which does not divide (fork) over A. Let
+M0denote a small model (elementary substructure of ¯M).
+Fact 2.1. (i) Suppose that ThasNIP. Letp(x)be a global type. Then p
+does not fork over M0if and only if pisAut(¯M/M0)-invariant.
+(ii) If the global type p(x)isAut(¯M/M0)-invariant, and (ai:i<ω)are such
+thatan+1realizesp|(M0a0..an)for alln, then(ai:i < ω)is indiscerinible
+overM0and its type over M0depends only on p.(ai:i < ω)is called a
+Morley sequence in poverM0.
+Comment. (i) appears explicitly in [1], and also in [5] but is implicit in [8].
+(ii) is well-known, but see Chapter 12 of [7] for a nice account.
+4Definition 2.2. (i) Letp(x)be a global type (or complete type over a satu-
+rated model), realized by c. We say that pstrictly does not fork over Aifp
+does not fork over Aandtp(¯M/Ac)does not fork over A(namely for each
+smallBcontainingAand realization c′ofp|B,tp(B/Ac)does not fork over
+A).
+(ii) LetA⊆B, andp(x)∈S(B). Thenpstrictly does not fork over A(or
+pis a strict nonforking extension of p|A) ifphas an extension to a global
+complete type q(x)which strictly does not fork over A.
+Fact 2.3. Assume that ThasNIPand letM0be a model.
+(i) For any formula φ(x,b),φ(x,b)divides over M0iffφ(x,b)forks overM0.
+(ii) Anyp(x)∈S(M0)has a global strict nonforking extension.
+(iii) Suppose p(x)is a global complete type which strictly does not fork over
+M0. Let(ci:i < ω)be a Morley sequence in poverM0. Supposeφ(x,c0)
+divides over M0(whereφ(x,y)is overM0). Then {φ(x,ci) :i < ω}is
+inconsistent.
+Proof.This is all contained in [2], where essentially only the property NTP2
+(implied by NIP) is used. (i) is Theorem 1.2 there (as any model is an
+“extension base” for nonforking). (ii) is Theorem 3.29 there. And ( iii) is
+Claim 3.14 there.
+Corollary 2.4. Suppose that ThasNIPandM0is a model of cardinality
+κ≥ |T|. Letq(y)be a global complete type which strictly does not fork over
+M0and let(cα:α<κ+)be a Morley sequence in qoverM0. Then for any
+(finite) tuple a, there isα<κ+such thattp(a/M0cα)does not fork over M0.
+Proof.Suppose not. So, using Fact 2.3(i), for each αthere is some formula
+φα(x,y)withparametersfrom M0suchthat |=φα(a,cα)andφα(x,cα)divides
+overM0. AsM0has cardinality κ≥ |T|there isφ(x,y) overM0andα0<
+α1< α2< ... < κ+such thatφ(x,y) =φαi(x,y) for alli < ω. By Fact 2.3
+(iii){φ(x,cαi) :i<ω}is inconsistent, which is a contradiction as this set of
+formulas is supposed to be realized by a.
+Some remarks are in order concerning the notions introduced abov e and
+the last corollary. Strict nonforking in the NIPcontext was introduced by
+Shelah [8] (where the study of forking in NIPtheories was also initiated)
+and all the results above are closely connected in one way or anothe r with
+Shelah’s work. A version of Corollary 2.4, which on the face of it is incor rect
+5due possibly to typographical errors, appears in [8] as Claim 5.19. A b etter
+version of Corollary 2.4 appears in [9].
+3 Proof of Theorem 1.4
+We revert to the context of section 1, and Theorem 1.4. Namely Tis an
+arbitrary theory, Na saturated model, and Pa∅-definable set in N. As
+therePis the structure with universe Pand relations subsets of various Pn
+which are ∅-definable in N. So if for example aandbare tuples from P
+(or even from Peq) which have the same type in Pthen they have the same
+type inN. We will assume that Th(P) is finite rank, as in Definition 1.3.
+We assume that PhasNIPinN(from which it follows that any sort in
+PeqhasNIPinNand also the structure PhasNIPin its own right).
+We also assume that Pis 1-stably embedded in N. Note that any sort (or
+definable set) in Peqis also a sort (or definable set) in Neq. IfSis a sort
+ofPeq, andXis a subset of Swhich is definable (with parameters) in N,
+we will say, hopefully without ambiguity, that Xiscoded inPifXcan be
+defined inNwith parameters from P, which is equivalent to saying that
+a canonical parameter for Xcan be chosen in Peq, and is also of course
+equivalent to saying that Xis definable with parameters in the structure P.
+Our aim is to prove that for any n, any subset of Pndefinable in Nis coded
+inP. The 1-stable embeddedness of Pstates that any subset of Pdefinable
+with parameters in Nis coded in P. We first aim towards the following key
+proposition, which extends this to definable functions on P.
+Proposition 3.1. LetZbe a sort in Peq. Letf:P→Zbe a function
+definable in N. Thenfis coded in P.
+Proof of Proposition 3.1.
+This will go through a couple of steps. The first is an adaptation of ide as
+from [4], using just the finite rank hypothesis on P:
+Lemma 3.2. (In the situation of Proposition 3.1) There is a relation R(x,z)
+which is definable in Nwith parameters from P, and some k<ω, such that
+N|= (∀x∈P)(R(x,f(x))∧(∃≤kz∈Z)R(x,z))
+Proof.Note that we are free to add parameters from Pto the language (but
+not, of course, from outside P). We try to construct ai∈P,bi∈Zfori<ω
+6such that
+(i)an/∈acl((ai,bi)i<n),
+(ii)bn=f(an), and
+(iii)bn/∈acl((aibi)i<n,an).
+If at stagenwe cannot continue the construction it means that for all a∈P
+such thata /∈acl((aibi)i<n,f(a)∈acl((aibi)i<n,a). Compactness will yield
+the required relation R(x,z) (defined with parameters from P).
+So we assume that the construction can be carried out and aim for a contra-
+diction. Wenowworkinthefiniterankstructure P. Notethat rk(anbn/(aibi)i<n)>
+1 for allnand moreover there is a finite bound to rk(anbn) asnvaries.
+Choose minimal s >1 such that for some m, for infinitely many n > m,
+rk(anbn/(aibi)i≤m) =s. It is then easy to find m<i 0<ii<..such that for
+allj,
+rk(aij,bij/a0,b0,..,am,bm,ai0,bi0,..,aij−1,bij−1) =rk(aij,bij/a0,b0,..,am,bm) =
+s.
+So after adding constants (in P) fora0,b0,..,am,bm, thinning the sequence,
+and relabelling, we have
+(iv)rk(an,bn) =rk(anbn/(aibi)i<n) =s>1 for alln, which means that
+(v){(a0b0),(a1b1),(a2b2),....}is an “independent” set of tuples.
+Of course we still have that ai/∈acl(∅),bi/∈acl(ai) for alli, as well as
+bi=f(ai) for alli. Nowfis definable in Nwith some parameter eso let us
+writefasfeto exhibit the dependence on e.
+Claim.For eachS⊆ωthere arebS
+i∈Zfori<ωsuch that
+(a) Fori∈ω,i∈SiffbS
+i=bi.
+(b)tp((aibi)i<ω) =tp((aibS
+i)i<ω).
+Proof of claim. Fixn /∈S. By(v) above, rk(bn/an,(aibi)i/negationslash=n)>0, so letbS
+nbe
+such thattp(bS
+n/an,(aibi)i/negationslash=n) =tp(bn/an,(aibi)i/negationslash=n) andrk(bS
+n/(aibi)i<ω)>0.
+So we see that tp((anbn),(aibi)i/negationslash=n) =tp(anbS
+n),(aibi)i/negationslash=n) andbS
+n/\e}atio\slash=bn. Iterate
+this to obtain the claim.
+By (b) of the claim, and automorphism, for each S⊆ω, there iseSsuch that
+feS(an) =bS
+nfor alln∈ω. By (a) of the claim, we have that feS(an) =bniff
+n∈S. This contradicts PhavingNIPinT, and proves Lemma 3.2
+Note that if Th(P) had Skolem functions, or even Skolem functions for “al-
+gebraic” formulas, we could quickly deduce Proposition 3.1 from Lemm a 3.2.
+(And this is how it works in the o-minimal case.) Likewise if we can choose
+7Rin Lemma 3.2 such that k= 1, we would be finished. So let us choose
+Rin Lemma 3.2 such that kis minimized, and work towards showing that
+k= 1. Here we use nonforking in the NIPtheoryTh(P). Let us fix a small
+elementary substructure M0ofPwhich contains the parameters from R. So
+for anya∈P,f(a)∈acl(M0,a), and this will be used all the time.
+Lemma 3.3. There is a finite tuple cfromPeqsuch that whenever a∈P
+andb=f(a)andtp(a/M0c)does not fork over M0(in the structure P) then
+b∈dcl(M0,c,a)(inPor equivalently in N).
+Proof.Let us suppose the lemma fails (and aim for a contradiction). We will
+first find inductively an,bn,dnforn < ωsuch that writing cn= (an,bn,dn)
+we have
+(i)an∈P,bn/\e}atio\slash=dnare inZandbn=f(an).
+(ii)tp(anbn/M0c0..cn−1) does not fork over M0, and
+(iii)tp(anbn/M0c0...cn−1) =tp(andn/M0c0...cn−1).
+Suppose have found ai,bi,difori < n. Putc=c0...cn−1(so ifn= 0,c
+is the empty tuple.) As the claim fails for c, we finda∈P, such that
+f(a) =b,tp(a/M0c) does not fork over M0, andb /∈dcl(M0ca). Noting that
+b∈acl(M0,a) we also have that tp(ab/M0c) does not fork over M0. Letd
+realize the same type as boverM0cawithd/\e}atio\slash=b. Putan=a,bn=b,dn=d.
+So the construction can be accomplished.
+Claim.LetS⊆ω. Definegn=bnifn∈Sandgn=dnifn /∈S. Then
+tp((aigi)i<ω/M0) =tp((aibi)i<ω/M0(inPor equivalently in N).
+Proof of claim. Assume, by induction, that
+(*)tp((aigi)i<n/M0) =tp((aibi)i<n/M0),
+and we want the same thing with n+1 in place of n. By (iii) above it suffices
+to prove that tp((aigi)i<n(anbn)/M0) =tp((aibi)i<n(anbn)/M0). This is true
+by (*) and Fact 2.1(i). (Namely as tp(anbn/M0c) does not fork over M0it has
+a global complete extension qsay which does not fork over M0so by 2.1(i)
+qis fixed by automorphisms which fix M0pointwise. So whether or not a
+formulaφ(x,z,d) say is inqor not depends just on tp(d/M0), so the same is
+true fortp(anbn/M0c).) The claim is proved.
+As earlier we write fasfewhereeis a parameter in Nover which fis
+defined. By the claim, and automorphism, for each S⊆ωwe can find eS
+inNsuch thatfeS(ai) =gifor alli < ω. In particular, as di/\e}atio\slash=bifor all
+i,feS(ai) =biif and only if i∈S, showing that Phas the independence
+property in N, a contradiction. Lemma 3.3 is proved.
+8We now complete the proof of Proposition 3.1 by showing that k= 1. Let
+us assume k >1 and get a contradiction. Let c∈ Peqbe as given by the
+Lemma 3.3. Then, (by the previous two lemmas), whenever a∈Pand
+tp(a/M0c) does not fork over M0, THEN there is d∈Zsuch that |=R(a,d)
+andd∈dcl(M0,a,c). Note that fis not mentioned here, so this statement
+is purely about P. Hence:
+(*) whenever c′∈ Phas the same type over M0asc, then whenever a∈P
+andtp(a/M0c′) does not fork over M0thend∈dcl(M0,a,c) for somed∈Z
+such that |=R(a,d).
+Wework in P. ByFact 2.3(ii) let q(y)bea globalstrict nonforking extension
+oftp(c/M0), and let ( cα:i <|T|+) be a Morley sequence in qoverM0.
+By Corollary 2.4, for any a∈P, there isαsuch thattp(a/M0cα) does
+not fork over M0, and hence by (*), there is dsuch that |=R(a,d) and
+d∈dcl(M0,a,cα). By compactness there is a partial function g(−) defined
+overM0∪{cα:α<|T|+}such that
+(**) for all a∈P, there isd∈Zsuch that |=R(a,d) andg(a) =d.
+Now by our assumption that Pis 1-stable embedded, {a∈P:g(a) =f(a)}
+is definable over parameters from Pby some formula ψ(x) say. LetR′(x,z)
+be (ψ(x)∧z=g(x))∨(¬ψ(x)∧R(x,z)∧z/\e}atio\slash=g(x)). Then clearly R′satisfies
+Lemma 3.2 with k−1. This contradiction proves Proposition 3.1.
+Proof of Theorem 1.4.
+There is no harm in adding a few constants for elements of P(so that we
+can do definition by cases). We will prove by induction on nthat any subset
+ofPnwhich is definable in Nwith parameters, is coded in Peq. Forn= 1
+this is the 1-stable embeddednes hypothesis. Assume true for nand we’ll
+prove forn+1. Letφ(x1,..,xn,xn+1,e) be a formula, with parameter e∈N
+defining the set X⊆Pn+1. For each a∈P, letXabe the subset of Pn
+defined by φ(x1,x2,..,xn,a,e). By induction hypothesis Xais coded in P,
+namely has a canonical parameter casay inPeq. By compactness, we assume
+that there is a formula ψ(x1,..,xn,z) (ofL), wherezranges over a sort Z
+ofPeq, such that for each a∈P,Xais defined by ψ(x1,..,xn,ca) (andca
+is still a canonical parameter for Xa). The map taking atocais clearly an
+e-definable function fsay fromPtoZ. By Proposition 3.1, fis coded in P
+(i.e. can be defined with parameters from P). As the original set X⊆Pn+1
+is defined by the “formula” ψ(x1,..,xn,f(xn+1)), it follows that Xis also
+coded inP. The proof of Theorem 1.4 is complete.
+9References
+[1] H. Adler, Introduction to theories without the independence pr operty,
+to appear in Archive Math. Logic.
+[2] A. Chernikov and I. Kaplan, Forking and dividing in NTP2theories, to
+appear in Journal of Symbolic Logic.
+(number 147 on the MODNET preprint server)
+[3] C.Ealy, K.Krupinski, andA.Pillay, Superrosydependentgroups having
+finitelysatisfiablegenerics, AnnalsofPureandAppliedLogic151(200 8),
+1 - 21.
+[4] A. Hasson and A. Onshuus, Embedded o-minimal structures, to appear
+in Journal of the London Math. Soc.
+[5] E. Hrushovski and A. Pillay, On NIPand invariant measures, preprint
+2009 (revised version).
+http://arxiv.org/abs/0710.2330
+[6] A. Onshuus, Properties and consequences of thorn independe nce, Jour-
+nal of Symbolic Logic, 71 (2006), 1-21.
+[7] B. Poizat, A course in Model theory; an introduction to contemp orary
+mathematical logic, Springer 2000.
+[8] S. Shelah, Dependent first order theories, continued, Israel J. Math. 173
+(2009), 1-60.
+http://arxiv.org/abs/0406440
+[9] A. Usvyatsov, Morley sequences in dependent theories, prepr int.
+http://arxiv.org/abs/0810.0733
+10
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+arXiv:1001.0516v1  [astro-ph.EP]  4 Jan 2010Astronomy& Astrophysics manuscriptno.13065 c/circlecopyrtESO 2018
+November15,2018
+Lettertothe Editor
+The lunar phases of dustgrains orbiting Fomalhaut
+M.Min1, M.Kama2, C.Dominik2,3, and L.B.F.M.Waters2,4
+1Astronomical InstituteUtrecht, Universityof Utrecht,P. O.Box80000, NL-3508 TAUtrecht,The Netherlands
+2Astronomical InstituteAnton Pannekoek, Universityof Ams terdam, Kruislaan 403, 1098 SJAmsterdam, The Netherlands
+3Afdeling Sterrenkunde, Radboud UniversiteitNijmegen, Po stbus 9010, 6500 GLNijmegen, The Netherlands
+4Insituut voor Sterrenkunde, K.U.Leuven, Celestijnenlaan 200 D,B-3001 Leuven, Belgium
+Received August 5, 2009; accepted December 18, 2009
+Abstract
+Opticalimages of thenearby starFomalhaut show a ringof dus t orbitingthecentral star.Thisdust isinmany respects exp ectedtobe
+similartothezodiacal dustinthe solarsystem.Theringdis plays aclearbrightness asymmetry, attributedtoasymmetr ic scatteringof
+the centralstarlightby thecircumstellar dust grains.Rec ent measurements show thatthe bright side ofthe Fomalhaut r ingis oriented
+away from us. This implies that the grains in this system scat ter most of the light in the backward direction, in sharp cont rast to the
+forward-scatteringnatureofthegrainsinthesolarsystem .Inthisletter,weshowthatgrainsconsiderablylargertha nthosedominating
+the solar system zodiacal dust cloud provide a natural expla nation for the apparent backward scattering behavior. In fa ct, we see the
+phases of the dust grains inthe same way as we can observe the p hases of the Moon and other large solar system bodies. We outl ine
+how the theory of the scattering behavior of planetesimals c an be used to explain the Fomalhaut dust properties. This ind icates that
+the Fomalhaut dust ring is dominated by very large grains. Th e material orbiting Fomalhaut, which is at the transition be tween dust
+andplanetesimals, can, withrespect totheir optical behav ior, best be described as micro-asteroids.
+Key words. Keywords should be given
+1. Introduction
+Optical images of some nearby stars show the presence of
+orbiting solid material through the starlight it scatters ( e.g
+Kalasetal., 2005). This has been interpreted as evidence of an
+evolvingplanetarysystem.Collisionsbetweenlargebodie ssuch
+as asteroids are believed to be responsible for the producti on
+of these small solid particles. This is similar to our own so-
+lar system, in which interplanetary dust grains are found th at
+are thought to originate from asteroids and comets. The size
+and chemical composition of these grains determine the way i n
+which they reflect starlight. Laboratorymeasurementsand l ight
+scattering theoryshow that small solid particlesscatter m ost in-
+cident light in the forward direction, in agreement with ast ro-
+nomical observations. However, recent observations of the dust
+orbitingthe nearbystar Fomalhauthaveshown forthe first ti me
+that the grains in this system scatter most of the light in the
+backward direction (LeBouquinet al., 2009). This is in shar p
+contrast with the current understanding of light scatterin g and
+challengesthe analogybetween interplanetarydust and the dust
+observedin Fomalhaut.In this paper, we show that an excelle nt
+matchtotheobservedbackscatteringbehaviorofthemateri alor-
+bitingFomalhautcanbeobtainedbyusingthetheoryofregol ith
+coveredsurfaces.
+2. Apparentbackward scattering
+The circumstellar ring of Fomalhaut as observedby the Hubbl e
+space telescope clearly shows a dark side and a bright side
+(Kalaset al., 2005). Regarding the orientation of the ring, the
+general forward scattering behavior of dust grains has led t o
+Send offprint requests to : M. Min, e-mail:M.Min@uu.nlthe conclusion that the bright side of the ring should be the
+onetilted towardsthe observer.Using theVeryLargeTelesc ope
+Interferometer (VLTI), the rotation axis of the central sta r and
+subsequently the inclination axis of the system was inferre d
+(LeBouquinetal., 2009). This inclination is opposite to pr e-
+vious assumption in that the direction of scattering is reve rsed
+fromforward-dominatedtobackward-dominated.Thiscontr asts
+withthe currentconsensusinthefield oflightscattering.
+Arelativelysimpleexplanationforthiscanbefoundbycon-
+sidering bodies much larger than interplanetary dust grain s. As
+an example let us take the Moon, which for all practical pur-
+posescan beconsidereda giganticdust grain.However,it is ev-
+ident that the Moon is bright when viewed in a backward scat-
+tering situation (full Moon) and it is dark in the forward sca t-
+tering case (new Moon). The question then arises: is the Moon
+truly backscattering?The answer is “no”. The apparentsurp rise
+that the Moon is forward scattering is caused by the sharp and
+in most cases invisible di ffraction peak it causes. For the Moon
+this diffraction spike, caused by interference e ffects due to the
+shadow of the Moon, is so narrowly forward peaked that it can
+onlybeseenintheforward-mostfractionofadegree1,i.e.when
+the Sunis rightbehindthe Moon.Forall practicalpurposes, the
+Moonappearsto bebackscattering,anddi ffractioncanbeomit-
+ted.
+For smaller grains, the di ffraction spike often dominatesthe
+overallscatteringbehavioroverabroaderrangeofscatter ingan-
+gles. It appears that the disk around Fomalhaut contains dus t
+grains that behave backward-reflecting like asteroidal bod ies.
+This must be because the grains are very large, making the
+diffraction spike so narrowly forward-peaked that it is outside
+1For objects the size of the Moon, the di ffracted energy at visible
+wavelengths isconfined towithinthe forward-most µ-arcsecond.2 M. Min etal.:The lunar phases of dust grains orbitingFomal haut
+Figure1. Angularscattering functionsforthe regolithparticles,
+showing the relative intensity scattered at a scattering an gleθ.
+The pure reflectance function (solid, black curve) is given, and
+diffractioneffectsare addedforparticleswith a narrowsize dis-
+tribution around 3µm grains (dashed curve). The exact scatter-
+ing function for this narrow size distribution of spheres as ob-
+tainedfromtheMietheoryisalso shown(solid,greycurve).
+theobservablerangeofangles.FortheFomalhautsystem,wh ich
+has an inclination of 25◦, the range of observablescattering an-
+gles is 25◦<θ<155◦. Kalasetal. (2005) already noted that
+the apparent single scattering albedo of the dust grains is i n the
+rangeof5−10%,whichislowforanykindofdustgrains.Such
+values are more typical for the geometric albedo of asteroid s.
+The geometric albedo of asteroids does not include the contr i-
+bution from diffraction. For large grains, it is easily shown that
+when diffraction is included, the single scattering albedo is al-
+ways larger than 50% (vande Hulst, 1957, Chapter 12). Thus,
+the true single scattering albedo of the grains, averaged ov er all
+angles and including di ffraction, must be much higher than the
+observedvalue, indicating that most of the scattered light is not
+detected.
+3. Modelingthescatteringfunctionofasteroidal
+bodies
+Below we treat the dust grains in the Fomalhaut system as
+micro-asteroids, meaning that we use the theory of reflectan ce
+by regolith-coveredasteroidal surfacesto computetheir s catter-
+ingbehavior.Forthereflectanceofasteroidalsurfaces,we usean
+analyticalmodelforthebidirectionalreflectance(Hapke, 1981).
+Forsimplicityweusethetheoryomittinge ffectsofmacroscopic
+roughness as discussed in Hapke (1984). This basically mean s
+that we set the macroscopic roughness parameter as discusse d
+in that paper to zero. In addition to this, we ignore the oppo-
+sition effect. This mainly plays a role near backward-scattering
+and thus largely falls outside the range of scattering angle s that
+wehaveaccesstointheFomalhautsystem.Tointegrateovert he
+surfaceofthe body,weuse MonteCarlo raytracing.
+3.1. The regolithparticles
+Asinputinthesecomputations,weneedtoknowthealbedoand
+angular scattering function for the regolith particles tha t cover
+thesurfaceofthegrain.Notethatsincetheregolithpartic lesare
+closely packed, we need to know these parameters omitting th e
+contributionfromdi ffraction(Hapke,1981).Thecontributionof
+diffractioncan be separated when we computethe angular scat-
+tering function and single scattering albedo using geometr ical
+optics. More exact computational methods like the Mie theor y
+do not allow the separation of the contributionfrom di ffraction.
+Wethusassumetheregolithparticlesarelargeenoughsotha tge-ometricalopticscanbeapplied,andinadditionweonlycons ider
+surface reflections, i.e. transmitted rays are ignored. We r ealize
+that using geometrical optics is a rough approximation for t he
+grainsizes we areconsidering.Nevertheless,since we arei nter-
+estedonlyintheoverallshapeoftheangularscatteringfun ction
+mostly at intermediate scattering angles, the approximati on is
+not so bad for grain sizes a few times the wavelength of radia-
+tion, especially for particles with a considerable imagina ry part
+of the refractive index (Wielaardet al., 1997). In this appr oxi-
+mation the scattering propertiesof the particlesare indep endent
+oftheirsize andshapeaslongastheyhaveaconvexsurface.T o
+compute the reflections we need to specify the refractive ind ex
+of the regolith particles which we take to be m=1.6+0.1i,
+thought to be typical for cosmic materials. However, we find
+that the final result is hardly influenced by modest variation s
+in the refractive index (with the real part ∼1.5−1.8 and the
+imaginary part/lessorsimilar0.3). The angular scattering function for the
+regolithparticleswhichresultsfromthisisdominantlyfo rward-
+scattering, while their single scattering albedo is 11% (om itting
+diffraction). In Fig. 1 we show three di fferent curves: the angu-
+lar scattering function omitting di ffraction (solid, black curve),
+the angular scattering function with added contribution fr om
+diffraction by a narrow size distribution of spheres with diam-
+etersaround3µm(dashedcurve),andthefullangularscattering
+functioncomputedusingtheMietheoryforthisnarrowsized is-
+tribution(solid,greycurve).All curvesare plottedasa fu nction
+ofthescatteringangle,i.e.0◦isforward-scatteringwhile180◦is
+backward-scattering. Note that the curve obtained using re flec-
+tionplusdiffractionisquitesimilartothatobtainedusingthefull
+Mie theory.It can be argued that 3 µm is rather large for the re-
+golith particles coveringthe grains we discuss below. Howe ver,
+itdoesallowthegrainstobecoveredbyorcomposedoftheser e-
+golith particles. The parametersderivedaboveare used as i nput
+for the optical properties of the regolith particles in the H apke
+theory.
+3.2. The grainsasa whole
+Theresultingsinglescatteringpropertiesoftheregolith particles
+obtained in the previous paragraph are inserted into the Hap ke
+theory. Note that we use the theory in its most simple form, i. e.
+we ignore effects of macroscopic roughness and the opposition
+effect.In thatcase, theonlyremainingparametersare thesing le
+scattering albedo and the angular scattering function of th e re-
+golith particles obtained above.As we will show below, we ca n
+findanalmostperfectfitwithoutconsideringadditionalpar ame-
+tersonmacroscopicroughnessandoppositione ffect.Thismeans
+that using only the measurements available so far, we cannot
+constrainthemanddecidedtousethemostsimpleformthatca n
+still reproduce the observations. Much more accurate obser va-
+tions of the angular scattering function of the Fomalhaut gr ains
+areneededtocompareallHapkeparameterstothosederivedf or,
+forexample,theMoonorsolarsystem asteroids.
+In Fig. 2 we show the resulting angular scattering function
+of the model Fomalhaut dust grains. The curves show the rel-
+ative intensity scattered at an angle θ. The grey area indicates
+the observablerangeofscatteringanglesforthe Fomalhaut sys-
+tem.NotethatthegrainswesimulateusingtheHapketheorya p-
+pear to be predominantlybackward-scattering(we see the cr es-
+cents of the grains), while the small regolith particles cov ering
+the surface of these larger grains are predominantly forwar d-
+scattering, as discussed above. The angular scattering fun ction
+resultingfromonlyusingtheHapkereflectancetheory,i.e. with-
+out including diffraction, is shown by the long dashed line. ToM. Min etal.:The lunar phases of dust grains orbitingFomalh aut 3
+Figure2. Angularscatteringfunctionsforlargedustgrains,showin gtherelativeintensityscatteredatascatteringangle θ.Thepure
+reflectance function (long-dashed curve) is given, and di ffraction effects are added for 10, 30 and 100 µm grains (dotted, dashed
+and solid curves). The empirical scattering function of the Fomalhaut disk grains from Kalasetal. (2005) is also shown ( thick
+grey curve, albedo chosen to match the computations). The gr ey area indicates the observable range of scattering angles for the
+Fomalhautsystem.
+simulate the finite size of the dust grains, we added the con-
+tribution from diffraction using a diameter of the grains of 10,
+30 and 100µm in dotted, dashed and solid lines, respectively.
+For the diffraction, we took simple Fraunhofer di ffraction by
+a spherical aperture. To get rid of the resonance structures as-
+sociated with Fraunhofer di ffraction, averaging was performed
+over a narrow size distribution (a flat distribution from 0.8 to
+1.2 times the average size). Grains of 100 µm diameter clearly
+fit the best to the empirical angular scattering function (sh own
+by the thickgrayline) in agreementwith Dentetal. (2000). F or
+theobservedscatteringfunctionofthegrainsorbitingFom alhaut
+we used the Henyey-Greenstein parameterization obtained b y
+Kalasetal. (2005), where we simply switched the forward and
+backwardscatteringdirections,i.e.switchingtheasymme trypa-
+rametergfrom+0.2to−0.2.Theangularscatteringfunctionob-
+tainedinthiswaywasmultiplicativelyscaledtomatchthec om-
+putations. This scaling factor gives the single scattering albedo
+ofthegrains.Wefoundthatwhenweintegratedtheangularsc at-
+tering function resulting from the Hapke theory over the ang les
+accessible in the Fomalhautsystem (the area indicated in gr ay),
+the albedo of the grains averaged over the observable range o f
+angles is 5%. This is consistent with the low albedo found by
+Kalasetal. (2005).
+If we consider the assumptions on the regolith particles and
+the Hapke theory it is clear that the model we present is quite
+rough.However,afirmoutcomeofthemodelisthatfortherang e
+ofscatteringanglesweobservethescatteringisdominated byre-
+flectionratherthanbydi ffraction.Thisimplies,asshownabove,
+thatthegrainsarelargerthan100 µmindiameter.Thisoutcome
+doesnotdependontheassumptionsinthemodelandistherefo re
+a solidconclusion.
+4. Conclusion
+We conclude that the scattering surface in the disk around
+Formalhautisdominatedbygrainsofatleast100 µminsize.Our
+findingsagreewiththefactthattheinfraredspectrumisfea ture-
+less (Stapelfeldtetal., 2004),indicatingthat grainssma llerthanafewmicronareheavilydepletedinthesystem.Also,Dentet al.
+(2000)findthatlargegrainsdominatethethermalemission. Itis
+remarkable that such large grains are observed directly in o pti-
+cal light in an astronomical object. The reason for this is th at
+while the dust mass in many systems is dominated by large
+grains (see e.g. Wilneretal., 2005; Testi etal., 2003), the y are
+normallyover-shonebyacomponentofsmallgrains.Thiscom -
+ponent,thoughlessmassive,dominatestheopticalcrossse ction,
+and therefore appearance, of the disk. The fact that very lar ge
+grains are so visible on a global scale in the Formalhaut disk
+meansthatsmallgrainshavebeencleanedoutfromthisdiske x-
+tremely efficiently, and that all there is left are these very large
+grains,directlyprobedbyreflectedstellar light.
+We concludethat we are observingthe transition part of pa-
+rameterspace,goingfromdustscatteringtoplanetesimalr eflec-
+tion. In fact, one might conclude that, almost 400 years afte r
+Galileo observed the crescent of Venus, we are seeing the cre s-
+centsofthelargedustgrainsin thediskaroundFomalhaut.
+Acknowledgements. We are indebted to the referee, Ludmilla Kolokolova, for
+constructive comments leading to asignificant improvement of the paper.
+References
+Dent, W. R. F., Walker, H. J., Holland, W. S., & Greaves, J. S. 2 000, MNRAS,
+314, 702
+Hapke, B.1981, J.Geophys. Res., 86, 3039
+Hapke, B.1984, Icarus, 59, 41
+Kalas, P.,Graham, J.R.,&Clampin, M. 2005, Nature, 435,106 7
+LeBouquin, J.-B.,Absil, O.,Benisty, M.,et al. 2009, A&A,4 98, L41
+Stapelfeldt, K.R.,Holmes, E.K.,Chen, C.,et al. 2004, ApJS ,154, 458
+Testi, L.,Natta, A.,Shepherd, D.S.,&Wilner, D.J.2003, A& A,403, 323
+vandeHulst,H.C.1957,LightScattering bySmallParticles (NewYork:Wiley)
+Wielaard, D. J., Mishchenko, M. I., Macke, A., & Carlson, B. E . 1997,
+Appl. Opt.,36, 4305
+Wilner, D. J., D’Alessio, P., Calvet, N., Claussen, M. J., & H artmann, L. 2005,
+ApJ,626, L109
\ No newline at end of file
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+arXiv:1001.0517v2  [nucl-th]  11 Aug 2011Reduced neutron widths in the nuclear data ensemble:
+Experiment and theory do not agree
+P. E. Koehler∗
+Physics Division, Oak Ridge National Laboratory,Oak Ridge , TN 37831
+(Dated: September 24, 2018)
+Abstract
+I have analyzed reduced neutron widths (Γ0
+n) for the subset of 1245 resonances in the nuclear
+data ensemble (NDE) for which they have been reported. Rando m matrix theory (RMT) predicts
+for the Gaussian orthogonal ensemble (GOE) that these width s should follow a χ2distribution
+having one degree of freedom ( ν= 1) - the Porter Thomas distribution (PTD). Careful analysi s of
+the Γ0
+nvalues in the NDE rejects the validity of the PTD with a statis tical significance of at least
+99.97% ( ν= 0.801±0.052). This striking disagreement with the RMT prediction is most likely due
+to the inclusion of significant p-wave contamination to the supposedly pure s-wave NDE. When an
+energy dependent threshold is used to remove the p-wave contamination, the PTD is still rejected
+with a statistical significance of at least 98.17% ( ν= 1.217±0.092). Furthermore, examination of
+the primary references for the NDE reveals that many resonan ces in most of the individual data
+sets were selected using methods derived from RMT. Therefor e, using the full NDE data set to
+test RMT predictions seems highly questionable. These resu lts cast very serious doubt on claims
+that the NDE represents a striking confirmation of RMT.
+∗Electronic address: koehlerpe@ornl.gov
+1I. INTRODUCTION
+The nuclear data ensemble (NDE) [1, 2] is a set of 1407 resonance en ergies consisting
+of 30 sequences in 27 different nuclides. The ensemble was assembled to test predictions of
+random matrix theory (RMT) [3]. Fluctuation properties of resonan ce energies in the NDE
+were found to be in remarkably close agreement with RMT predictions for the Gaussian
+orthogonalensemble (GOE).Althoughtherehave beenseveral o ther successful tests ofRMT
+using nuclear resonances (e.g., [4, 5]),the NDE is perhaps the most imp ortant because, as
+stated in Ref. [3], ”As a result of these analyses, it became generally accepted that proton
+and neutron resonances in medium weight and heavy nuclei agree wit h GOE predictions.”
+Hence, the NDE routinely is cited as providing striking confirmation of RMT.
+Reduced neutron widths (Γ0
+n) have been reported for a subset of 1245 resonances in the
+NDE of Ref. [2], consisting of 14 to 178 measurements for 24 nuclide s. If the GOE correctly
+describes the data, RMT predicts these widths should follow a χ2distribution having one
+degree of freedom ( ν= 1) - the Porter Thomas distribution (PTD). It has be argued that
+the PTD is more generally valid [6] and more robust [7] than other RMT predictions. Also,
+it has been demonstrated [8] that a much more reliable analysis of sp ectra fluctuations can
+be performed using neutron widths than using resonance energies . In addition, it has been
+shown[9]thatresonancewidthfluctuationsaremoresensitivetha nenergyfluctuationstothe
+degree of chaos in model quantum systems. In addition, it is straigh tforward [10] to account
+for experimental effects such as missed or spurious resonances, and to use the statistically
+efficient maximum-likelihood (ML) method while using width data, but the same cannot be
+said for energy data. For these reasons, a test of the PTD using t he NDE neutron widths
+could be very valuable.
+In Ref. [2] such a test on a subset of the NDE data was described fr om which it was
+concluded that there was satisfactory accord between theory a nd experiment. However,
+there are several problems with the analysis of Ref. [2], which I will d escribe below. I find
+that when the data are analyzed more carefully, they do not agree with the PTD.
+I will show below that the NDE suffers from significant p-wave contamination. For such
+resonances, reported reduced neutron widths Γ0
+nare actually only ”effective” reduced widths
+gΓn/√En, whereg=2J+1
+(2I+1)(2j+1)(withJ,I, andjbeing spins of the resonance, target, and
+neutron, respectively) is the spin statistical factor. All s-wave resonances for NDE nuclides
+2included in my analysis have J=1
+2and hence g= 1 andgΓ0
+n= Γ0
+n. However, there are two
+resonance spins possible for p-wave resonances for these nuclides; J=1
+2or3
+2, and hence
+g= 1 or 2. As a reminder of these facts, I will use gΓ0
+nwhen referring to reduced neutrons
+widths in the remainder of this paper.
+II. RECONSTRUCTING THE NDE
+To my knowledge, the resonance energies and neutron widths in the NDE have never
+been published as a set. It should be possible, however, to reconst ruct this information from
+the nuclides and corresponding number of resonances given in the N DE papers [1, 2] and the
+primary references cited therein. Unfortunately, several of th e references listed in the NDE
+papers are ”private communication”, there are a few errors in the number of resonances
+reported in Refs. [2], and this reference does not explain why some o f the nuclides in their
+NDE were excluded from their analysis of the neutron widths. For ex ample, in Ref. [2] it
+is stated that 1182 neutron widths in 21 nuclides were included in the a nalysis. However,
+the sum of the number of resonances in these 21 nuclides reported in this same reference is
+actually 1194. One resonance in182W was reported without a neutron width in the primary
+reference [11], so only 40 of the 41 resonances for this nuclide can b e included in the width
+analysis. Therefore, the total number of widths is actually 1193, n ot 1182. Also, 19, 54, 47,
+and 21 resonances have been reported [8, 12] for154,156,158,160Gd, respectively, not 19, 47, 21,
+and 54, respectively as stated in Ref. [2]. In addition, there are th ree nuclides (160Dy,164Dy,
+and186W, with 18, 20, and 14 resonances, respectively) in the NDE of Ref. [2] that were
+not included in their width analysis, even though neutron widths were available [11, 13].
+In total then, the NDE of Ref. [2] should contain 1245 neutron res onances in 24 nuclides
+for which widths have been reported. I obtained resonance energ ies and neutron widths for
+these nuclides from the primary references given in Table I, and cro ss checked these data
+with information in the EXFOR/CSISRS [14] database. Given the prob lems noted above,
+I cannot be certain that I have analyzed the same data as those in t he NDE of Ref. [2].
+However, using data from the primary references should minimize an y differences with Ref.
+[2] as well as make it easier for others to reconstruct the data set I have used.
+3TABLE I: NDE nuclides.
+Nuclide Ref. E max(keV) ML Results
+Minimum Threshold p-free Threshold
+TmaxNresνmin TmaxNres νpf
+64Zn [15, 16] 367.55 0.024 103 1.35+0.24
+−0.220.05 99 1.54+0.29
+−0.26
+66Zn [15, 16] 297.63 0.025 65 0.68+0.23
+−0.210.05 61 0.74+0.27
+−0.25
+68Zn [15, 16] 247.20 0.019 45 0.75+0.27
+−0.250.05 41 0.95+0.36
+−0.32
+114Cd [17] 3.3336 0.137 17 0.35+0.54
+−0.340.45 11 2.0+1.5
+−1.2
+152Sm [18] 3.665 0.025 70 1.14+0.27
+−0.250.1 62 1.55+0.40
+−0.38
+154Sm [18] 3.0468 0.036 27 0.76+0.38
+−0.360.1 22 1.32+0.65
+−0.55
+154Gd [12] 0.2692 0.15 19 0.44+0.58
+−0.430.2 18 0.49+0.64
+−0.48
+156Gd [8] 1.9908 0.009 54 1.22+0.27
+−0.260.2 46 1.44+0.51
+−0.49
+158Gd [12] 3.9827 0.012 47 0.75+0.25
+−0.220.2 35 1.17+0.54
+−0.47
+160Gd [12] 3.9316 0.013 21 0.55+0.34
+−0.330.2 16 0.83+0.75
+−0.65
+160Dy [13] 0.4301 0.07 18 0.83+0.57
+−0.510.2 14 1.41+1.0
+−0.83
+162Dy [13] 2.9572 0.046 46 1.02+0.33
+−0.320.2 40 0.99+0.47
+−0.43
+164Dy [13] 2.9687 0.04 20 0.82+0.51
+−0.440.2 16 2.3+1.2
+−1.0
+166Er [19] 4.1693 0.02 109 0.85+0.18
+−0.170.3 78 1.85+0.49
+−0.45
+168Er [19] 4.6711 0.04 48 0.80+0.30
+−0.270.3 37 1.32+0.62
+−0.55
+170Er [19] 4.7151 0.05 31 0.36+0.34
+−0.31.03 17 3.6+1.6
+−1.3
+172Yb [20] 3.9000 0.05 55 0.71+0.27
+−0.260.06 54 0.70+0.30
+−0.26
+174Yb [20] 3.2877 0.02 19 0.80+0.44
+−0.390.06 16 1.29+0.68
+−0.58
+176Yb [20] 3.9723 0.015 23 0.04+0.29
+−0.030.06 15 1.05+0.65
+−0.55
+182W [11] 2.6071 0.069 40 0.76+0.37
+−0.350.15 34 1.50+0.62
+−0.55
+184W [11] 2.6208 0.04 30 0.62+0.34
+−0.310.15 26 0.99+0.54
+−0.48
+186W [11] 1.1871 0.07 14 1.23+0.78
+−0.620.15 13 1.32+0.93
+−0.75
+232Th [21] 2.988 0.016 178 0.76+0.13
+−0.120.26 123 1.78+0.36
+−0.34
+238U [21] 3.0151 0.0045 146 0.79 ±0.12 0.47 84 1.02+0.39
+−0.34
+W.A. - - 1245 0.801 ±0.052 - 978 1.217 ±0.092
+4III. IMPORTANCE OF THRESHOLD IN ML ANALYSIS OF NEUTRON
+WIDTHS
+Because the PTD is a special case (degrees-of-freedom ν= 1) of the family of χ2dis-
+tributions, it is assumed that the data are distributed accordingly a nd the ML method is
+used to estimate the most likely value of ν. Ideally, the data should be complete (no missing
+resonances) and pure (all resonances have the same parity). Un fortunately, all experiments
+from which neutron widths have been determined have finite thresh olds for detecting reso-
+nances and for separating small s- from large p-wave resonances. Not properly accounting
+for these effects can result in substantial systematic errors in ML estimates of ν. Given
+the shape of the χ2distribution as a function of ν, neglecting the effect of missed s-wave
+resonances below threshold will, in general, lead to a falsely large value ofνfrom the ML
+analysis. Conversely, including even just a few p-wave resonances in an s-wave set will, in
+general, lead to a falsely small value of ν. This potential problem is especially important
+for many of the NDE nuclides because they are near the peaks of th ep- and minimum of
+thes-wave neutron strength functions, and therefore neutron widt h is a much less reliable
+indicator of resonance parity.
+Below I will show that the NDE is seriously contaminated by p-wave resonances. It is
+also almost certainly incomplete, as illustrated in Figs. 1 and 2. Reduce d neutron widths for
+all 1245 resonances in the NDE are shown as a function of resonanc e energy in Fig. 1, from
+which it can be seen that their are fewer small widths as the energy in creases. This is just
+what is expected from well-known [22] instrumental effects that de crease sensitivity as the
+energy increases; hence, more small resonances are missed at hig her energies. This is further
+illustrated in Fig. 2 in which distributions of reduced neutron widths ar e shown for the 100
+lowest- and highest-energy NDE resonances. As can be seen in this figure, the distribution
+of the highest-energy set is substantially narrower (and hence is in better agreement with
+aχ2distribution having a larger ν) than the lowest energy one. Again, this is just what is
+expected if more resonances are missed as energy increases.
+Typically, these difficulties have been surmounted by using a energy- independent thresh-
+old as an integral part of the ML analysis, implicitly assuming that all s-wave resonances
+above threshold have been observed. We recently have shown [23 ] that an energy-dependent
+threshold (on gΓ0
+n) of the form T=aEn, whereais a constant factor, offers three ad-
+5vantages compared to using a energy-independent threshold. Fir st,p-wave contamination
+is eliminated equally effectively at all energies. This is because the pene trability factor for
+p-waves differs from s-waves by (to good approximation) a factor of En. Second, experiment
+thresholds have approximately this same energy dependence; thu s, possible diffusiveness of
+the instrumental threshold can be surmounted equally effectively a t all energies. Third,
+statistical precision of the analysis is maximized by allowing the largest p-wave-free set of
+s-wave resonances to be included.
+Analyzing a data set comprised of many different nuclides such as the NDE involves
+at least two additional potential pitfalls. First, as is evident from Fig . 3, the apparent
+sensitivities of the various experiments from which the NDE was deriv ed differ by several
+orders of magnitude. Therefore, if the entire NDE were analyzed a s a single set (as was
+done in Ref. [2]), the threshold must be at least as high as the highest apparent individual
+threshold. But doing this will substantially reduce the statistical pr ecision of the result
+and at least partially negate the reason for assembling the NDE in the first place. Second,
+for a given set of data and threshold, the ML-estimated average r educed neutron width
+might be substantially different from the one estimated from the dat a (e.g., from a simple
+average or by assuming ν= 1). Therefore, it is important to include the average reduced
+width as a parameter in the ML analysis. Furthermore, the differenc e between these two
+estimated values will likely vary from one NDE nuclide to the next. For t hese two reasons,
+it is important that separate ML analyses be made for each NDE nuclid e.
+To minimize the effects of the above problems, I have done separate ML analyses for
+each nuclide in the NDE using (a range of) separate energy-depend ent thresholds, and then
+combined these individual results for comparison to theory.
+IV. AN IMPROVED ML ANALYSIS OF THE NDE NEUTRON WIDTHS
+The analysis technique was briefly described in Ref. [23]. Each resona nceλhas an energy
+Eλand reduced neutron width gΓ0
+λn. The probability density function (PDF) f(x|ν) for a
+χ2distribution is given by,
+f(x|ν)dx=ν
+2G(ν
+2)(νx
+2)ν
+2−1exp(−νx
+2)dx, (1)
+60.0 0.2 0.4 0.6 0.8 1.0
+E/Emax10-310-210-1100101gΓn0/<gΓn0>
+NDE
+p wave or uncertain parity
+FIG. 1: (Color online) All 1245 reduced neutron widths in the NDE (red X’s). Data for each
+nuclide have been normalized to their respective average re duced neutron widths and maximum
+energies. Blue circles depict those resonances which have b een identified as being pwave or of
+uncertain parity.
+10-210-1100
+(gΓn0/<gΓn0>)1/20.00.20.40.60.81.0Fraction Above ( gΓn0/<gΓn0>)1/2
+First 100 NDE res.
+Last 100 NDE res.
+GOE (ν = 1)
+ν = 1.5
+FIG. 2: (Color online) Reduced neutron width distributions for the 100 lowest- (blue cirles) and
+highest- (red X’s) energy resonances in the NDE. The blue sol id curve depicts the RMT prediction
+for the GOE (the PTD) whereas the red dashed curve is for a χ2distribution having ν=1.5. See
+text for details.
+750 100 150 200 250
+Mass Number10-310-210-1gΓ0
+n, min/<gΓn0>NDE nuclides
+FIG. 3: (Color online) Blue circles depict minimum reduced n eutron widths, normalized to their
+respective averages, for each of the nuclides in the NDE, ver sus mass number.
+whereG(ν
+2) is the gamma function forν
+2, andx→gΓ0
+λn/E[gΓ0
+λn], where E[ gΓ0
+λn] is the
+expectation(average)valueofthereducedneutronwidth, withE [•]denotingtheexpectation
+value operator. The ML method is used to estimate most likely values o fνand E[Γ0
+λn] as
+well as their uncertainties.
+To facilitate comparison of the various nuclides in the NDE (e.g., Figs.1, 2, 3, and 5, and
+Table I), I have normalized data for each nuclide to their respective average reduced widths
+< gΓ0
+n>(as reported in the primary references or in Ref. [24]) and maximum energiesEmax.
+Hence, thresholds (on gΓ0
+n/ < gΓ0
+n>) can be expressed as T′=TmaxEn/Emaxand thus,
+Tmaxis the maximum value of the threshold relative to the average reduce d neutron width.
+Values of Tmaxused in the analyses are given in Table I.
+The joint PDF for statistical variables gΓ0
+λnandEλis defined in a 2D region Igiven by
+inequalities Eλ< EmaxandgΓ0
+λn> T(Eλ). The expression for this PDF reads
+h0/parenleftbig
+Eλ,gΓ0
+λn|ν,E[gΓ0
+λn]/parenrightbig
+=Cf/parenleftbigggΓ0
+λn
+E[gΓ0
+λn]/vextendsingle/vextendsingle/vextendsingle/vextendsingleν/parenrightbigg
+. (2)
+The factor C, ensuring a unit norm of h0, isν- and E[gΓ0
+λn]-dependent. The ML function
+was calculated from all n0pairs/bracketleftbig
+Eexp
+λi,gΓexp
+λin/bracketrightbig
+, obtained from the experiment, which fall into
+8the region I. Specifically,
+L/parenleftbig
+ν,E[gΓ0
+λn]/parenrightbig
+=n0/productdisplay
+i=1h0/parenleftbig
+Eexp
+λi,gΓ0exp
+λin|ν,E[gΓ0
+λn]/parenrightbig
+. (3)
+For the initial analyses, thresholds just below the smallest observe d resonance for each
+nuclide were used. ML results with these thresholds are given in colum nνminof Table I.
+Because experiment thresholds might not beprecisely sharp, it is ex pected that the resulting
+νminvalues would be systematically a bit large. However, almost all νminvalues are less than
+the PTD value of 1.0 .
+Contour plots of ML functions for232Th, calculated at two different thresholds, in the
+form
+z/parenleftbig
+ν,E[gΓ0
+λn]/parenrightbig
+= 21
+2/bracketleftbig
+lnLmax−lnL/parenleftbig
+ν,E[gΓ0
+λn]/parenrightbig/bracketrightbig1
+2(4)
+areshown in Fig. 4. Here, Lmaxis the maximum of the ML function. Contours at fixed z=k
+encircle approximatelythe kσconfidence region, andwereusedtoderivethe1 σuncertainties
+given in Table I. Careful statistical analysis in Ref. [23] verified that these contours are
+reliable. The weighted average of results at minimum threshold for th e 24 nuclides in the
+NDE isν= 0.801±0.052, which is 3.8 standard deviations smaller than the predicted resu lt
+ofν= 1. Hence, these data reject the PTD with a statistical significanc e of 99.98%. To
+check this result, the combined probability for the 24 NDE nuclides wa s calculated using
+Fisher’s and Stouffer’s (both weighted and unweighted) techniques [25]. These methods
+yielded combined confidence levels of 99.97%, 99.98%, and 99.99%, res pectively, in good
+agreement with the weighted-average result. Hence, at minimum th resholds the NDE data
+reject the PTD with high confidence and indicate that νis significantly less than 1.0. A ν
+value significantly less than the PTD could be a sign of interesting phys ics [9, 23, 26–29].
+However, a more likely explanation is that the NDE contains sizeable p-wave contamination.
+V. CLEANSING THE NDE OF P-WAVES
+Results of the maximum-likelihood analyses for three NDE nuclides are shown in Fig.
+5. On the left of this figure, normalized reduced neutron widths are plotted as functions of
+normalized resonance energy. On the right side of this figure, νvalues from the ML analyses
+are plotted versus threshold coefficients Tmax.
+90.8 1.0 1.2 1.4 1.6 1.8
+E [gΓn0] (meV)0.40.50.60.70.80.91.01.1ν
+ 1
+ 2
+ 3 4 4
+ 5 5 6 7232Th, Tmax = 0.016
+1.2 1.4 1.6 1.8 2.0 2.2 2.41.01.52.02.5
+ 1
+ 2 3
+ 4 4
+ 5 5 6232Th, Tmax = 0.26
+3  
+FIG. 4: Plots of z(ν,E[gΓ0
+λn]) constructed from ML analyses, at minimum (left) and p-wave-free
+(right) thresholds, of the NDE232Th data. On each plot, a filled circle indicates the location o f
+Lmax, and a dashed horizontal line is drawn at ν= 1, the PTD value.
+That the NDE is contaminated by p-wave resonances is evident from this figure in two
+ways and from Figs. 1 and 6. First, resonances in the NDE that have been identified (in
+Refs. [24, 30] and references contained therein) as p-wave or of uncertain parity are shown
+as open circles in Figs. 1 and 5. For example, in Ref. [31], 58 p-wave resonances in232Th
+were assigned on the basis of γ-cascade information, 13 of which are in the NDE. One of
+these 13 resonances also is known [32, 33] to be pwave by its observed parity-violating
+asymmetry. Percentages of p-wave or unknown-parity resonances for NDE nuclides, as a
+function of mass number, are shown in Fig. 6. For over half (14/24) of the NDE nuclides
+analyzed herein, these resonances account for 5% or more of the total, for 10 of the 24 they
+are at least 10%, and in the three worst cases about 35%.
+1010-310-210-1100101
+114Cd
+01234
+114Cd
+10-310-210-1100101gΓn0/<gΓn0>
+166Er
+0123 Degrees of Freedom166Er
+E/Emax0.0 0.5 1.010-310-210-1100101
+232Th
+0.0 0.2 0.4 0.6
+Tmax0123
+232Th
+FIG. 5: (Color online) Left: normalized reduced neutron wid ths versus normalized resonance
+energies for114Cd,166Er, and232Th resonances in the NDE. Red X’s depict all resonances in the
+NDE whereas blue circles show resonances previously identi fied as being pwave or of uncertain
+parity. Right: Red X’s depict νvalues from ML analyses versus thresholds used, for the same
+three nuclides. Error bars represent 1 σconfidence levels. Black dashed vertical lines correspond
+to thresholds depicted by black dashed curves in the left par t of this figure. See text for details.
+Second, that many of the NDE resonances are in fact p-wave is reinforced by the behavior
+of theνvalues from the ML analyses as functions of threshold, as shown in t he right side of
+Fig. 5. In all three cases shown, νsystematically increases with threshold before gradually
+stabilizing. This is just the behavior expected for a populationof s-wave resonances contam-
+inated by p-wave resonances. Similar fractions of previously identified p-wave resonances
+and trends in νwith threshold are seen for several of the other NDE nuclides.
+1150 100 150 200 250
+Mass Number010203040% p wave or unknown parityNDE
+FIG. 6: (Color online) Blue circles depict percentages of ND E resonances for each nuclide which
+are known to be pwave or are of uncertain parity, as a function of mass number.
+Removingeffectsofthese p-waveresonancesfromtheNDEMLanalysisisasimplematter
+of raising thresholds until they are above the largest previously-id entifiedp-wave resonance
+and/orνstabilizes as a function of threshold. Typical ” p-wave free” thresholds for three
+NDE nuclides are shown as dashed curves in the left-hand part of Fig . 5, and corresponding
+values of Tmaxare depicted by dashed vertical lines in the right-hand part of this fi gure.
+Degrees-of-freedom values for each of the NDE nuclides at these ”p-wave free” thresholds
+are given in column five ( νpf) of Table I. The resulting weighted average for the NDE is still
+in conflict with the RMT prediction for the GOE, albeit in the opposite dir ection from the
+result using the lowest thresholds: ν= 1.217±0.092, corresponding to a confidence level of
+98.17% for excluding the PTD. Fisher’s and unweigthed and weighted S touffer’s confidence
+levels are somewhat higher; 99.15%, 99.81%, and 99.37%, respective ly. Hence, when the
+NDE is cleansed of p-wave resonances, the data still reject the PTD with high confiden ce.
+VI. DISCUSSION
+I have shown that when neutron widths in the NDE are analyzed care fully and in such a
+way as to eliminate p-wave resonances, the data exclude the PTD with fairly highconfide nce
+(ν= 1.217±0.092). At the same time, it has been shown [1, 2] that the NDE as a wh ole
+12agrees remarkably well with energy-level fluctuations predicted b y RMT for the GOE. Given
+the greater sensitivity of widths over energies to the degree of ch aos in the system [9], and
+to effects related to the ”openess” of the system [27–29, 34, 35], perhaps this dichotomy can
+be resolved. However, I am not aware of any published model which d oes so while at the
+same time yielding ν >1.
+Forexample, inthecalculationsofRef. [9], collective effects sometim es resulted inregions
+of model space where transition-strength distributions deviated strongly from the PTD but,
+at the same time, energy-level fluctuations were consistent with G OE predictions. How-
+ever, transition-strength distributions in this model deviated fro m the PTD in the direction
+opposite ( ν <1) of the NDE.
+Reaction effects also should be considered to explain the dichotomy. For example, it
+recently was proposed [35] that the standard transformation of measured to reduced s-
+wave neutron widths, Γ0
+n= Γn/√
+E, should be modified for nuclides near the peaks of
+thes-wave neutron strength function. This newly proposed transfor mation, Γ0
+n= Γn/√
+E×
+π(h2
+2m)1/2(E+|E0|), whereE0istheenergy(relativetothreshold) ofthe s-wavesingle-particle
+state responsible for the peak in the neutron strength function, could affect (relative to the
+standard transformation) the shape of the Γ0
+ndistribution for values of E0near the resolved-
+resonance energy range. However, as most of the NDE nuclides ar e not near the peaks of the
+s-wave neutron strength function, this proposal should not affec t the NDE, and therefore
+cannot explain the dichotomy.
+Several explanations related to the ”openess” of the system [27, 28, 34] and to correla-
+tions between incoming and outgoing channels [29] have been propos ed for the observation
+[23] that Γ0
+ndistributions for192,194Pt resonances are significantly broader ( ν≈0.5) than
+the PTD. However, width distributions [28, 29] predicted by these t heories are better char-
+acterized by ν <1 rather than ν >1 as observed for the NDE.
+Considering how data in the NDE were selected suggests another so lution. It often has
+been stated that ∆ 3is very sensitive to missing or misassigned resonances (see e.g., [36–
+38] for recent work on this subject). Therefore, if the NDE was p ure but incomplete, or
+vice versa, it would not be expected to agree well with the spacing st atistics used in Refs.
+[1, 2]. However, it seems plausible that because the NDE is both impure and incomplete,
+it can be made to agree with these statistics, especially considering t hat there are expected
+to be many more p- thans-wave resonances at the small widths where it was not possible
+13(using means independent of RMT) to differentiate the two parities, and hence an extremely
+large number (see below) of possible ” s-wave” sets can be constructed from the observed
+resonances.
+Many different selections were applied to obtain the NDE. For example , in Ref. [2] it is
+stated that ”The criterion for inclusion in the NDE is that the individua l sequences be in
+general agreement with the GOE.” Furthermore, data from many o f the included nuclides
+were selected, at least in part, using measures derived from RMT fo r the GOE.
+Data for all but three of the 24 nuclides considered herein were obt ained by the group
+at Columbia University. According to their publications (e.g., Ref.[19]), they had ”...no
+specific tests for svsplevels, so there may be errors in these assignments.” Therefore,
+they relied on theoretical guidance, specifically measures derived f rom the GOE, to perform
+these separations. For example, for six of the 24 nuclides consider ed herein, including the
+two having the largest number of resonances, separation of p- froms-wave resonances was
+accomplished [39] by i) assuming all resonances having neutron width s larger than a certain
+(unspecified) sizewere swave, ii)calculating thenumber of s-wave resonancesbelowthissize
+byassumingthePTDiscorrect, andiii)decidingwhichoftheresonanc esbelowthethreshold
+defined in the first step and needed to achieve the total number ca lculated in the second
+step, were s-wave by requiring good agreement with four spacing statistics (th e Wigner
+nearest-neighboor spacing distribution, ρ(Sj,Sj+1), ∆3, and the Dyson Ftest) derived from
+the GOE. Separation of s- fromp-wave resonances for several of the other NDE nuclides
+followed the first two steps described above followed by a Bayesian a nalysis to decide which
+of the resonances below the threshold defined in the first step to a ssign to the s-wave set.
+The Bayesian analysis again assumes that the PTD is correct (for bo ths-andp-wave
+resonances) and furthermore requires the average widths for s- andp-wave resonances. The
+latter quantity usually is known only approximately, if at all. Such Baye sian analyses are
+known to be unreliable. For example, several neutron resonances in64Zn [15] are known to
+be definitely pwave by their symmetrical shape in transmission (total cross sect ion) data,
+but nevertheless have a Bayesian probability of >99 % of being swave.
+Giventheseselectionproceduresthen, itisperhapsunderstanda blethattheNDEappears
+to agree well with the spacing statistics examined in Refs. [1, 2] desp ite that fact that the
+data are neither complete nor pure. Consider, for example, the ca se of232Th, the NDE
+nuclide with the largest number of resonances. The authors of the primary reference from
+14which these data were taken [21, 39] state that 80% of the s-wave set (i.e. the set included
+in the NDE) were chosen to be swave because they were too large to be pwave. Hence,
+36 of the 178232Th resonances fall below this size and had to be selected as swave using
+measures derived from the GOE, as noted above. In this same refe rence, 62 resonances are
+assigned as pwave (and hence by definition are below the s-wave threshold). Therefore,
+there are 98 resonances from which to choose the 36 needed to co mplete the s-wave set.
+Therefore, the number of possible s-wave sets is astronomically large (˜1027), and hence it
+should not be surprising that at least one set can be found to agree with the various RMT
+measures used in Ref. [39]. Similar numbers apply to the other data s ets from Columbia.
+Finally, given that a significant fraction of the NDE data were selecte d using measures
+derived from RMT for the GOE, it seems highly questionable to use the full NDE for any
+test of this same theory. In contrast, the ” p-free” results presented herein circumvent this
+problem by eliminating these problematical resonances from the tes t.
+VII. CONCLUSIONS
+Ihave shownthatneutronwidthsintheNDE[2], whenanalyzedcaref ully, reject thePTD
+with high confidence. Given that reported deviations from the PTD f or individual nuclides
+[21, 23, 40–44] have occurred on both sides of ν= 1, the statistical tests used herein likely
+underestimate the confidence with which the PTD can be rejected f or the combined set of
+nuclides in the NDE. Furthermore, I have shown that the NDE is not p ure and very likely
+incomplete. Also, measures derived from RMT for the GOE often wer e used in deciding
+which resonances to include in the NDE. These facts cast very serio us doubt on repeated
+claims that the NDE constitutes a striking confirmation of RMT.
+Measurement techniques have improved considerably since the dat a in the NDE were
+acquired. In particular, new methods for determining resonance s pins [44, 45] hold the
+promise of surmounting the difficulty of separating small s- from large p-wave resonances,
+which hasbeenoneofthemost troublesome barrierstoobtaining be tterdata. Unfortunately
+most experimentalists continue to use RMT to correct their data fo r instrumental effects
+rather than use their data to test RMT. Exceptions to this practic e are recent tests [23, 44]
+which have revealed significant disagreements between new neutro n data and the PTD.
+These new data, together with previously reported disagreement s [21, 40–43], which have
+15largely been ignored, are potentially very interesting. Therefore, I urge experimentalists
+obtaining new and improved data to use them, when possible, to test RMT.
+Acknowledgments
+This work was supported by the Office of Nuclear Physics of the U.S. D epartment of
+Energy under Contract No. DE-AC05-00OR22725 with UT-Battelle , LLC.
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+18
\ No newline at end of file
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+++ b/1001.0518.txt
@@ -0,0 +1,242 @@
+arXiv:1001.0518v2  [hep-ph]  7 Jan 2010b→sℓ+ℓ−inthehigh q2regionattwo-loops
+VolkerPilipp∗, ChristofSchüpbach
+Albert EinsteinCenterforFundamentalPhysics
+InstituteforTheoreticalPhysics,Universityof Bern,Sid lerstrasse5,3012Bern,Switzerland
+E-mail:volker.pilipp@itp.unibe.ch ,christof.schuepbach@itp.unibe.ch
+We report on the first analytic NNLL calculation for the matri x elements of the operators O1
+andO2for the inlusive process b→Xsl+l−in the kinematical region q2>4m2
+c, whereq2is the
+invariantmasssquaredofthelepton-pair.
+RADCOR 2009- 9thInternationalSymposiumonRadiativeCorr ections(ApplicationsofQuantumField
+Theoryto Phenomenology)
+October25-302009
+Ascona,Switzerland
+∗Speaker.
+c/circlecopyrtCopyrightownedbytheauthor(s)underthetermsoftheCreat ive CommonsAttribution-NonCommercial-ShareAlikeLicen ce. http://pos.sissa.it/b→sℓ+ℓ−inthehighq2regionattwo-loops VolkerPilipp
+1. Introduction
+Inthe Standard Model, the flavor-changing neutral current p rocessb→Xsl+l−only occurs at
+the one-loop level and is therefore sensitive to new physics . In the kinematical region where the
+lepton invariant mass squared q2is far away from the c¯c-resonances, the dilepton invariant mass
+spectrum and the forward-backward asymmetry can be precise ly predicted using large mbexpan-
+sion, where the leading term isgiven by the partonic matrix e lement of the effective Hamiltonian
+Hef f=−4GF√
+2V∗
+tsVtb10
+∑
+i=1Ci(µ)Oi(µ). (1.1)
+We neglect the CKM combination V∗
+usVuband the operator basis is defined as in [1]. In [2] we
+published the first analytic NNLL calculation of the high q2region of the matrix elements of the
+operators
+O1=(¯sLγµTacL)(¯cLγµTabL),O2=(¯sLγµcL)(¯cLγµbL), (1.2)
+whichdominate theNNLLamplitudenumerically. Earlierthe seresultswereonlyavailable analyt-
+ically in the region of low q2[3, 4]. Using equations of motion the NNLL matrix elements of the
+effective operators take theform
+/angbracketleftsℓ+ℓ−|Oi|b/angbracketright2-loops=−/parenleftBigαs
+4π/parenrightBig2/bracketleftBig
+F(7)
+i/angbracketleftO7/angbracketrighttree+F(9)
+i/angbracketleftO9/angbracketrighttree/bracketrightBig
+, (1.3)
+whereO7=e/g2
+smb(¯sLσµνbR)FµνandO9=e2/g2
+s(¯sLγµbL)∑l(¯lγµl).
+2. Calculations
+b s
+c
+a)O1,2
+b s
+c
+b)O1,2b s
+c
+c)O1,2
+b s
+c
+d)O1,2b s
+c
+e)O1,2
+b s
+c
+f)O1,2
+Figure 1: Diagrams that have to be taken into account at order αs. The circle-crosses denote the possible
+locationswherethevirtualphotonisemitted(seetext).
+Thediagrams contributing at order αsare shown inFigure 1. Weset ms=0and define
+ˆs=q2
+m2
+bandz=m2
+c
+m2
+b, (2.1)
+2b→sℓ+ℓ−inthehighq2regionattwo-loops VolkerPilipp
+whereqis the momentum of the virtual photon. After reducing occurr ing tensor-like Feynman
+integrals [5]theremaining scalar integrals canbefurther reduced tomaster integrals using integra-
+tion by parts (IBP) identities [6]. Considering the region ˆ s>4z, we expanded the master integrals
+inzand kept the full analytic dependence in ˆ s.
+For power expanding Feynman integrals we use a combination o fmethod of regions [7] and
+differential equation techniques [8,9]: Consider aset of Feynmanintegrals I1,...,Independing on
+the expansion parameter zand related by a system of differential equations obtained b y differenti-
+atingIαwith respect to zand applying IBPidentities:
+d
+dzIα=∑
+βhαβIβ+gα, (2.2)
+wheregαcontainssimplerintegralswhichposenoseriousproblems. Expandingbothsidesof(2.2)
+inε,zand lnz
+Iα=∑
+i,j,kI(j,k)
+α,iεizj(lnz)k,hαβ=∑
+i,jh(j)
+αβ,iεizj,gα=∑
+i,j,kg(j,k)
+α,iεizj(lnz)k,(2.3)
+and inserting (2.3) into (2.2) weobtain algebraic equation s for the coefficients I(j,k)
+α,i
+0=(j+1)I(j+1,k)
+α,i+(k+1)I(j+1,k+1)
+α,i−∑
+β∑
+i′∑
+j′h(j′)
+αβ,i′I(j−j′,k)
+β,i−i′−g(j,k)
+α,i. (2.4)
+This enables us to recursively calculate higher powers of zonce the leading powers are known. In
+practice thismeansthat weneed the I(0,0)
+α,iandsometimes alsothe I(1,0)
+α,iasinitial condition to(2.4).
+These initial conditions can becomputed using method of reg ions. Anon trivial check isprovided
+by the fact that the leading terms containing logarithms of zcan be calculated by both method of
+regions and the recurrence relation (2.4).
+The summation index jin (2.3) can take integer or half-integer values, depending on the
+specific set of integrals Iα. In order to determine the possible powers of zand ln(z)we used the
+algorithm described in[9]. Agiven D-dimensional L-loopFeynmanintegral I(z)readsinFeynman
+parameterization
+I(z)=(−1)N/parenleftbiggi
+(4π)D/2/parenrightbiggL
+Γ(N−LD/2)/integraldisplay
+dNxδ(1−N
+∑
+n=1xn)UN−(L+1)D/2
+(zF1+F2)N−LD/2,(2.5)
+whereU,F1andF2are polynomials in xn. Using Mellin-Barnes representation (2.5) can be cast
+into thefollowing form
+I(z)=(−1)N/parenleftbiggi
+(4π)D/2/parenrightbiggL1
+2πi/integraldisplayi∞
+−i∞dszsΓ(−s)Γ(s+N−LD/2)
+×/integraldisplay
+dNxδ(1−N
+∑
+n=1xn)UN−(L+1)D/2Fs
+1F−s−N+LD/2
+2. (2.6)
+By closing the integration contour over sto the right hand side the poles on the positive real axis
+turn into powers of z. If we apply the technique of sector decomposition [10] to (2.6) we end up
+with terms of the following form
+N
+∑
+l=1∑
+k/integraldisplay1
+0dN−1t/parenleftBigg
+N−1
+∏
+j=1tAj−Bjε−Cjs
+j/parenrightBigg
+UN−(L+1)D/2
+lkFs
+1,lkF−s−N+LD/2
+2,lk, (2.7)
+3b→sℓ+ℓ−inthehighq2regionattwo-loops VolkerPilipp
+whereUlk,F1,lkandF2,lkcontain terms that are constant in /vectort. From (2.7) we can read off that the
+poles insarelocated at:
+sjn=1+n+Aj−Bjε
+Cj, (2.8)
+wheren∈N0.
+Additionally, theproceduredescribedaboveallowsustoev aluatethecoefficientsoftheexpan-
+sioninznumerically whichweusedtoagaintesttheinitial conditio ns ofthedifferential equations.
+3. Results
+In order to get accurate results we keep terms up to z10. Our results agree with the previous
+numerical calculation [11] within less than 1% difference. To demonstrate the convergence of the
+power expansions, we show in Figure 2 the form factors defined in (1.3) as functions of ˆ s, where
+weinclude allordersupto z6,z8andz10. Weuseasdefault value z=0.1suchthat the c¯c-threshold
+islocated at ˆ s=0.4. Oneseesfromthefiguresthat far awayfrom the c¯c-threshold, i.e.for ˆ s>0.6,
+the expansions for all form factors arewell behaved.
+Theimpact of our results onthe perturbative part of the high q2-spectrum [3]
+R(ˆs)=1
+Γ(¯B→Xce−¯νe)dΓ(¯B→Xsℓ+ℓ−)
+dˆs(3.1)
+isshowninFigure3(left), whereweusedthesameparameters asin[2]. Thefinitebremsstrahlung
+corrections calculated in [4] are neglected. From Figure 3 ( left) we conclude that for µ=mbthe
+contributions of our results lead to corrections of the orde r 10%−15%. Integrating R(ˆs)over the
+high ˆsregion, wedefine
+Rhigh=/integraldisplay1
+0.6dˆsR(ˆs). (3.2)
+Figure3(right)showsthedependence oftheperturbativepa rtofRhighontherenormalization scale.
+Weobtain
+Rhigh,pert=(0.43±0.01(µ))×10−5, (3.3)
+where we determined the error by varying µbetween 2 GeV and 10 GeV. The corrections due to
+our results lead toadecrease of the scale dependence to 2%.
+Acknowledgments
+ThisworkispartiallysupportedbytheSwissNationalFound ation,byEC-ContractMRTNCT-
+2006-035482 (FLAVIAnet) and by the Helmholtz Association t hrough funds provided to the vir-
+tual institute ”Spin and strong QCD” (VH-VI-231). The Alber t Einstein Center for Fundamental
+Physics is supported by the ”Innovations- und Kooperations projekt C-13 of the Schweizerische
+Universitätskonferenz SUK/CRUS”.
+4b→sℓ+ℓ−inthehighq2regionattwo-loops VolkerPilipp
+-0.88-0.86-0.84-0.82-0.8-0.78-0.76
+0.4 0.5 0.6 0.7 0.8 0.9 1ℜF(7)
+1
+ˆs-0.5-0.4-0.3-0.2-0.100.1
+0.4 0.5 0.6 0.7 0.8 0.9 1ℑF(7)
+1
+ˆs
+-50-40-30-20-1001020
+0.4 0.5 0.6 0.7 0.8 0.9 1ℜF(9)
+1
+ˆs-70-60-50-40-30-20
+0.4 0.5 0.6 0.7 0.8 0.9 1ℑF(9)
+1
+ˆs
+4.54.64.74.84.955.15.25.3
+0.4 0.5 0.6 0.7 0.8 0.9 1ℜF(7)
+2
+ˆs-0.500.511.522.5
+0.4 0.5 0.6 0.7 0.8 0.9 1ℑF(7)
+2
+ˆs
+810121416182022
+0.4 0.5 0.6 0.7 0.8 0.9 1ℜF(9)
+2
+ˆs-30-20-10010
+0.4 0.5 0.6 0.7 0.8 0.9 1ℑF(9)
+2
+ˆs
+Figure 2: Real and imaginaryparts of the formfactors F(7,9)
+1,2as functionsof ˆ s. To demonstratethe conver-
+gence of the expansion in zwe included all orders up to z6,z8andz10in the dotted, dashed and solid lines
+respectively. We put µ=mbandusedthedefaultvalue z=0.1.
+5b→sℓ+ℓ−inthehighq2regionattwo-loops VolkerPilipp
+00.511.522.533.5
+0.6 0.65 0.7 0.75 0.8 0.85 0.9R(ˆs)[10−5]
+ˆs0.350.40.450.50.55
+1 2 3 4 5 6 7 8 9 10Rhigh[10−5]
+µ[GeV]
+Figure 3: Perturbative part of R(ˆs)(left) and Rhigh(right) at NNLL. The solid lines represents the NNLL
+result, whereas in the dotted lines the order αscorrections to the matrix elements associated with O1,2are
+switchedoff. Intheleftfigureweuse µ=mb. See textfordetails.
+References
+[1] C. Bobeth,M. MisiakandJ.Urban, Nucl.Phys. B574,291(2000).
+[2] C. Greub,V. PilippandC. Schüpbach,JHEP 0812,040(2008).
+[3] H.H. Asatryan,H. M.Asatrian,C. GreubandM. Walker, Phy s.Rev.D65,074004(2002).
+[4] H.H. Asatryan,H. M.Asatrian,C. GreubandM. Walker, Phy s.Rev.D66,034009(2002).
+[5] G.PassarinoandM.J. G.Veltman, Nucl.Phys. B160,151(1979).
+[6] K.G. ChetyrkinandF. V. Tkachov, Nucl.Phys. B192,159(1981);F. V. Tkachov, Phys.Lett. B100,
+65(1981).
+[7] M.BenekeandV.A. Smirnov, Nucl.Phys. B522,321(1998);S.G. Gorishnii, Nucl.Phys. B319,633
+(1989);V. A.Smirnov, Commun.Math.Phys. 134,109(1990);V. A.Smirnov, SpringerTractsMod.
+Phys.177,1(2002).
+[8] R.Boughezal,M.CzakonandT.Schutzmeier, JHEP 09,072(2007);A.V.Kotikov, Phys.Lett. B254,
+158(1991);V. Pilipp, Nucl.Phys. B794,154(2008);E.Remiddi, NuovoCim. A110,1435(1997).
+[9] V. Pilipp, JHEP 09,135(2008).
+[10] T.BinothandG.Heinrich,Nucl.Phys.B 585,741(2000).
+[11] A.Ghinculov,T.Hurth,G. IsidoriandY.P. Yao, Nucl.Ph ys.B685,351(2004).
+6
\ No newline at end of file
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@@ -0,0 +1,442 @@
+arXiv:1001.0519v1  [cond-mat.quant-gas]  4 Jan 2010Strongly correlated gases of Rydberg-dressed atoms: quant um and classical dynamics
+G. Pupillo,1A. Micheli,1M. Boninsegni,2,1I. Lesanovsky,3and P. Zoller1
+1Institute for Theoretical Physics, University of Innsbruc k,
+and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck, Austria
+2Department of Physics, University of Alberta, Edmonton, Al berta, Canada T6G 2J1
+3School of Physics and Astronomy, University of Nottingham, Nottingham, UK
+We discuss techniques to generate long-range interactions in a gas of groundstate alkali atoms, by
+weakly admixing excited Rydberg states with laser light. Th is provides a tool to engineer strongly
+correlatedphaseswithreduceddecoherencefrominelastic collisions andspontaneousemission. Asan
+illustration, we discuss the quantumphases of dressed atom s with dipole-dipole interactions confined
+in a harmonic potential, as relevant to experiments. We show that residual spontaneous emission
+from the Rydberg state acts as a heating mechanism, leading t o a quantum-classical crossover.
+PACS numbers: 32.80.Ee, 34.20.Cf, 03.75.-b
+There is currently significant interest in the physics
+of dipolar quantum degenerate gases [1]. Strong long-
+range dipole-dipole interactions promise the realization
+of novel many-body phases in neutral gases, such as self-
+assembled crystals [2], topological superfluids and quan-
+tum phaseswithhidden topologicalorder[3]. Theregime
+of strong dipolar couplings is easily accessible with inter-
+actingelectricdipole moments, as realized in particu-
+lar in quantum gases with polar molecules prepared in
+their rovibrational ground state [4]. In contrast, quan-
+tum gasesofgroundstateatomstypicallyinteractviathe
+much smaller magnetic interactions [5]. The question,
+therefore, is to what extent this regime of strong dipo-
+lar interactions can also be realized with present atomic
+gases experiments with alkali atoms. Here we propose
+and investigate a setup where the huge electric dipole
+moments d∼n2[6–8] of atomic Rydberg states with
+principal quantum number nare weakly admixed to the
+atomic ground state, thus providing an atomic gas of in-
+teracting effective electric dipoles comparable to the case
+of polar molecules. A central question, and the main dif-
+ference to the molecular case, is decoherence and heating
+mechanisms associated with spontaneous emission from
+Rydberg states, and possible inelastic collisions. Below
+we show that (i) dipolar crystals can be realized with
+Rydberg-dressed atoms confined to two dimensions, with
+negligible collisionallosses, and (ii) spontaneousemission
+γrprovides an intrinsic heating mechanism, allowing the
+study of thermalization phenomena with full tunability
+of system parameters. We study the associated crystal
+melting with molecular dynamics calculations.
+The setup we have in mind is illustrated in Fig. 1: we
+propose to weakly couple with laser light the ground-
+state|g∝angbracketrightof each atom to a Stark-split Rydberg state
+|r∝angbracketrightwith large dipole moment d0, in the kDebye. For
+large enough detuning ∆ from resonance and interpar-
+ticle distances, interactions are of the dipole-dipole type
+V3D
+int∝(Ω/∆)4d2
+0/r3, with Ω the Rabi frequency. By
+confining the particles to a 2D plane using an optical
+field, the effective in-plane interactions V2D
+intare then(a) (b) 
+eF 
+∆E n
+∆E n|r >
+|r >|r >
+|r' >|r >+ |r >|r' >|g >|g >
+|g >|r >+ |r >|g >|r' >
+|r'' >
+|g >n=19 
+/endash.cap10 
+0 1 22 4 6
+3r/R010 /endash.cap4 
+/endash.cap5 0
+Ω
+γr, kR∆dr
+(c) 
+Rn/R0E/∆ 
+0 3 5 710
+γeff, kR
+FIG. 1: (color online)(a) Sketch of the energy levels consti tut-
+ingtheStarkfan ofanalkali metalatom exposedtoanelectri c
+field.|g/angbracketrightand|r/angbracketrightare coupled by a laser with Rabi frequency Ω
+and (blue) detuning ∆. γris the spontaneous emission from
+|r/angbracketright, with momentum kRof the photon recoil. (b) BO poten-
+tials in the ( x−y)-plane in the dressed picture. R0is the
+resonant Condon point. The effective interaction potential
+V3D
+int(r) is the higher-energy curve (red). Level crossings oc-
+cur forRn< R0. Inset: blow-up of V3D
+int(r) (continuous line)
+compared to 1 /r3(dashed line). (c) Sketch of experimental
+setup: Rydberg-dressed atoms are confined to 2D by a strong
+confining laser beam, with dipoles polarized perpendicular to
+the plane. An effective spontaneous emission with rate γeff
+provides for an intrinsic heating mechanism (see text).
+purely repulsive, with negligible collisional losses. This
+opens the way to the study of the many-body phases
+of 2D dipoles in these systems. As an illustration, we
+show the existence ofmesoscopic supersolids and crystals
+with Rydberg-dressed atoms under realistic conditions
+of in-plane harmonic confinement, using exact quantum
+Monte-Carlo simulations. Residual spontaneous emis-
+sionγeff∼(Ω/∆)2γrfrom the Rydberg state introduces
+an intrinsic heating mechanism, driving the quantum
+phases into the classical regime. We study the quan-
+tum/classical crossover by means of molecular dynam-2
+ics simulations, and show the emergence of a dynamical
+thermalization timescale in these systems.
+Effective interaction potentials with suppressed deco-
+herence from atomic collisions and spontaneous emission
+are obtained as follows. In the presence of a homoge-
+neous electric field (field strength F, oriented along ez)
+the energy levels of an alkali atom show the well-known
+Stark structure [6]. We are interested here in high an-
+gular momentum states which show a linear Stark ef-
+fect. The energy of these states is well approximated by
+ǫnn1n2= (3/2)ea0n(n1−n2)F, withn1andn2parabolic
+quantum numbers, a0the Bohr radius and ethe elec-
+tron charge. The highest-energy state |r∝angbracketrightof a given
+manifold n(n1=n−1,n2= 0) is energetically sep-
+arated from the next (lower lying) state in the mani-
+fold by ∆ En=e(3/2)a0nF, while adjacent manifolds
+remain well separated by an energy ∆ EIT≫∆Enfor
+field strengths smaller than FIT≈1
+3n5m2e5
+128π3ǫ3
+0/planckover2pi14(Inglis-
+Teller limit). In our scheme, Fig. 1, each atom is ef-
+fectively reduced to a two-level system. Within the
+rotating-wave approximation and with a product ba-
+sis/braceleftig
+|r∝angbracketrighti|r∝angbracketrightj,|r∝angbracketrighti|g∝angbracketrightj,|g∝angbracketrighti|r∝angbracketrightj,|g∝angbracketrighti|g∝angbracketrightj/bracerightig
+the Hamiltonian
+governing the internal dynamics of two atoms is given by
+Hij=
+−2∆+vijΩ Ω 0
+Ω−∆ 0 Ω
+Ω 0 −∆ Ω
+0 Ω Ω 0
+, (1)
+withvij=D(1−3cosθij)|Ri−Rj|−3≡DνijR−3
+ij, the
+dipole-dipole interaction between two atoms at position
+Ri(Rj), andD=d2
+0/(4πǫ0).θijis the angle between
+the vector Ri−Rjand the dipole moment aligned par-
+allel to the z-axis. The interaction potential which is
+experienced by the dressed ground state atoms is given
+by the Born-Oppenheimer (BO) energy surface that adi-
+abatically connects to the energy of the product state
+|g∝angbracketrighti|g∝angbracketrightjas ∆→ ∞. Here, we focus on the weak driv-
+ing limit Ω <∆. For two atoms approaching each other
+and blue detuning ∆ >0, the groundstate BO potential
+will be approximately given by V3D
+int(Rij,θij)≃Dνij/R3
+ij
+forRij< R0, withR0≃(D//planckover2pi1∆)1/3a resonant Condon
+pointwithtypicalvaluesinthehundredsofnm, Fig. 1(b).
+Diabatic crossings with different potential surfaces of the
+same and different n-manifolds leading to collisional two-
+bodylosseswilloccurinthisregion Rij< R0, atdistances
+of order Rn≃(D/∆En)1/3andRIT≃(D/∆EIT)1/3,
+respectively, with R0> Rn≫RIT, reminiscent of blue-
+shielding techniques [2, 9].
+ForRij/greaterorsimilarR0, two-body diabatic losses are absent and
+V3D
+int(Rij,θij)≃(Ω/∆)4Dνij/R3
+ij. In this work we will
+focus on this parameter regime. This has two additional
+advantages: (i) spontaneous emission rates are strongly
+reduced to values γeffand (ii) effective interactions are
+reducedtovaluescompatiblewithtrappingofatomswith
+FIG. 2: (color online) (a-d)Monte Carlo snapshots of theden -
+sity of particles in all mesoscopic phases for N= 13 dipoles,
+as a function of the effective mass τ. (a) superfluid; (b) su-
+persolid; (c) ring-like crystals; (d) classical crystal.
+optical fields (e.g., optical lattice), and confinement to
+low-dimensional geometries.
+Collisional losses for Rij> R0are linked to popula-
+tion of the attractive part of the dipole-dipole interac-
+tion. Sampling of this attractive part can be suppressed
+by confining atoms to 2D using a tight (optical) trap-
+ping along z, with harmonicfrequency /planckover2pi1ω⊥=/planckover2pi12/ma2
+⊥>
+V3D
+int(R), withmthe atomic mass, and a⊥the transverse
+harmonic oscillator length. The BO potential is then
+purely repulsivein 3D: there is an energy barrierbetween
+the long-distance repulsion and the attractive short-
+distance regime. Residual losses are linked to tunneling
+below this energybarrier, and can be computed semiclas-
+sicaly as Γ coll=ωattexp[−c(Ω4Dm//planckover2pi12∆4a⊥)2/5], withc
+aconstantoforderunity, and ωattthe attempt frequency,
+of order of the average kinetic energy in the gas. For
+Rij> R0the effective dynamics is purely 2D, with inter-
+actionsV2D
+int= (Ω/∆)4D/ρ3≡˜D/ρ3, withρ=|ρ|, and
+ρa vector in the ( x−y)-plane.
+Mesoscopic crystals with Rydberg-dressed atoms: We
+consider a setup where Nbosonic dressed atoms are con-
+fined to a 2D plane by applying a strong transverse trap-
+ping field [2], e.g a 1D optical lattice, and are aligned
+perpendicular to the plane [Fig. 1(c)]. We assume an
+additional in-plane parabolic trap with frequency ω, as
+realized in experiments by a magnetic dipole trap, or
+a single site of a large spacing optical lattice. Defin-
+ing length and energy scales r0= (˜D/mω2)1/5and
+˜ǫ=mω2r2
+0=˜D/r3
+0= (m3ω6˜D2)1/5, respectively, the
+2D Hamiltonian in dimensionless form reads
+H
+˜ǫ=N/summationdisplay
+i=1/bracketleftbigg
+−1
+2τ2∂2
+∂ρ2
+i+1
+2ρ2
+i/bracketrightbigg
++/summationdisplay
+i>j1
+|ρi−ρj|3,(2)3
+g(r)
+0 1 2 3 400.5 11.5 22.5 
+τ=1 τ=2.5 τ=5 τ=20 
+00.1 0.2 0.3 f
+n’ 1 7 13 τ=1 
+τ=2.5 
+r/r0
+FIG. 3: (color online) Radial densityprofiles g(r)for the cases
+of Fig. 3. Inset: statistics of computed particle exchanges f
+as a function of the number of particles n′participating to the
+exchange for the cases of finite superfluid fraction τ= 1 and
+2.5. In a superfluid, fis finite for all n′≤N. The supersolid
+phase with τ= 2.5 has finite density modulation (figure) and
+the same f-distribution of τ= 1 (inset).
+whereτ≡˜ǫ//planckover2pi1ω= (r0/ℓ)2=/parenleftig
+m˜D//planckover2pi12ℓ/parenrightig2/5
+characterizes
+the strength of the dipole-dipole interactions in the trap
+withℓ=/radicalbig
+/planckover2pi1/mωthe harmonic oscillator length. Equa-
+tion (2) shows that τplays the role of an effective mass
+representing a control parameter, which can be increased
+by increasing the strength of dipole-dipole interactions,
+or by compressing the trap.
+For small τ/lessorsimilar1, we expect the kinetic energy to dom-
+inate, and the cluster to be in a weakly interacting su-
+perfluid (SF) phase, while for τ≫1 the kinetic energy
+becomes negligible, and the system ground state should
+resemble the classical crystalline (CC) configuration ob-
+tained by minimizing the last two terms of Eq. ( 2). We
+obtainan estimate ofthe critical τcforthe crossoverfrom
+the superfluid to the crystal by noting that for a homo-
+geneous system the superfluid-crystal transition occurs
+atrQM=Dm//planckover2pi12a= 18±4 [2, 10], where rQMrepre-
+sents the ratio of the dipolar interactions D/a3to the
+kinetic energy /planckover2pi12/ma2withathe mean interparticle dis-
+tance. By rewriting τc= (rQMa/ℓ)2/5, and approximat-
+inga∼ℓ, we obtain the prediction τc≃3, which is
+essentially N-independent .
+We determine the generic zero-temperature phase di-
+agram for trapped dipolar atoms by means of Quantum
+Monte Carlo simulations based on the continuous-space
+Worm Algorithm [11]. We find that for mesoscopic clus-
+ters with N/lessorsimilar40 four different quantum phases exist: (i)
+a SF for τ≪τc, (ii) a mesoscopic supersolid (MS) for
+τ/lessorsimilarτc, (iii) a ring-shaped crystal (RC) for τ/greaterorsimilarτc, and
+(iv) a CC for τ≫τc. These phases are distinguished by
+measuring the superfluid fraction ρsand the radial den-
+sity profile g(r): the SF and MS phases are superfluid
+withρs= 1, while the RC and CC phases have ρs= 0;
+the SF has a flat, featureless g(r), while the MS, RC and
+CC phases have significant density modulations. In thecrystalline RC phase, particles are arrangedin concentric
+rings with a fixednumber of particles per ring. For small
+enough temperatures, these rings are free to rotate in-
+dependently of each other. In the crystalline CC phase,
+atoms are arranged in the classical crystal configuration,
+at fixed relative positions. Mesoscopic crystals have been
+also found in excitonic materials in Refs.[12].
+In the following we provide example results for N= 13
+atoms, which we found to display all general features of
+mesoscopic clusters with N/lessorsimilar40. We find that for N≫
+40the system resemblesthe homogeneoussituation, with
+a sharp crossover between SF and CC phases around τc.
+Panels (a-d) in Fig. 3are snapshots of the particle
+density for N= 13 and 1 ≤τ≤20 in the four meso-
+scopic phases described above. These results correspond
+to a low enough temperature T≪˜ǫ, that they can be
+regarded as groundstate estimates. Panel (a) shows a
+SF forτ= 1 with a flat density profile (see also g(r)
+in Fig.3), where the various particle probability clouds
+overlap. This overlap is directly connected to superfluid-
+ity [13], and consistently we here measure ρs= 1. The
+emergence of a MS phase for τ= 2.5/lessorsimilarτcis signaled
+by a distinguishable density modulation in Fig. 1(b) (see
+also Fig. 3), combined to a measured ρs= 1. We find
+that the superfluid properties are completely unaltered
+by the increased strength of interactions with respect to
+the caseτ= 1. This is quantified in the inset of Fig. 3by
+measuring the statistics of exchange cycles fin a many-
+particle path [13]: in a superfluid, the probability fthat
+n′particles exchange is finite for all n′≤N, while in a
+crystal is approximately zero. In the inset of Fig. 3it
+is shown that fis finite and equal for τ= 1 and 2.5,
+computed at corresponding low temperatures (i.e., same
+fractions of ˜ ǫ).
+Forτ/greaterorsimilarτc[panels (c-d)] we observe crystallization of
+the atomic cloud, with ρs= 0 and finite density modu-
+lations, consistently with the estimates above. In partic-
+ular, for τ= 5 we obtain a RC phase, with concentric,
+independent rings [14], while for τ= 20 particles are ar-
+ranged in the CC configuration for N= 13, with 4 par-
+ticles at the center, and 9 outside. This classical crystal,
+characterized by large peaks in g(r) in Fig. 3, is found to
+be the groundstate configuration for all τ/greaterorsimilar20.
+Effective heating : The residual effective single-particle
+spontaneous emission rate is γeff∼(Ω/∆)2γr. In this
+work, we focus on the case where after the spontaneous
+emission the atom is found in |g∝angbracketright, and then instanta-
+neously re-dressed, which is the relevant situation for
+n/lessorsimilar30 [15]. Spontaneous emission results in the release
+of a photon, with momentum /planckover2pi1kand recoil energy ER,
+in the tens of kHz. For dipoles parallel to F, emission
+will be preferentiallyalong z, andERwill be absorbedby
+the confining lattice potential with /planckover2pi1ω⊥> ER. However,
+any finite component of the momentum /planckover2pi1kxyof the emit-
+ted photon in the ( x−y)-plane will result in a random
+”kick” of the atoms, which will ultimately translate into4
+an effective intrinsic heating rate for the many-body sys-
+tem. We conservatively estimate the crystalline phases
+(τ≫1) tobe observableup toan average“melting”time
+tm∝TM/(γeffER), withTMthe (classical) melting tem-
+perature in the trap. We computed TMfor allN <30
+by means of molecular dynamics simulations for the 2D
+system in the classical limit ( τ→ ∞) [16]. By monitor-
+ing the behavior of the specific heat and density-density
+correlations, we determine the value TM= 0.4(2)˜ǫ. This
+can result in characteristic lifetimes of order of several
+milliseconds. For example, for87Rb atoms coupled to
+the Rydberg-state ( n= 20,n1= 19,n2= 0) with
+d0≈1.45kDebye, γr∼100kHz, ER≃25kHz, via a laser
+with Ω/2π= 100MHz, ∆ /2π= 1GHz and a DC field
+F= 25kV/m, for ω/2π= 5kHz we obtain r0≃680nm,
+τ≃20, ˜ǫ≃4ER, andtm/greaterorsimilar5ms.
+We further investigated the melting dynamics. Spon-
+taneous emission was simulated by applying random in-
+plane kicks to each particle with momentum /planckover2pi1kxy=
+(2mERxy)1/2, with fixed ERxy/lessorsimilarER, at an average rate
+γeff. Panels (a) and (b) in Fig. 4show our results for
+ERxy= 0.5˜ǫand 2.5 ˜ǫas a function of time t, respec-
+tively, for N= 13 particles [see Fig. 3]. Thin lines are
+single classical-dynamics trajectories, while thick dashed
+lines are averages over many trajectories. We find that
+the average energy Etotincreases linearly with time t, as
+expected. For the smaller ERxyof panel (a), we find that
+the crystal melts roughly at the expected time tm(cor-
+responding to several spontaneous emission processes),
+while in panel (b) the observed melting time teqis signif-
+icantly longer than tm(roughly the time of one sponta-
+neous emission). This prolonged stability is a dynamical
+process: classicalmelting can occur only after the system
+has reached thermal equilibrium. The latter corresponds
+tothetime teqatwhichthe averagepotentialenergy Epot
+and kinetic energy Ekinbecome equal and equipartition
+of energy applies. teq, and thus the crystal lifetime, can
+be (much) longer than tm.
+Since interparticle distances can be of the order of a
+µm ormore, the spatial structure ofthe crystallinephases
+00.2 0.6 (a) N=13 
+γeff  =0.32ω 
+ER=0.5ε N=13 
+γeff  =0.64ω 
+Er=2.5ε 
+Ekin 
+Epot Etot 
+Ekin 
+Epot Etot 
+ωt/2π E/N ε
+0 0.1 0.15 0.25 0.2 0 0.2 0.1 0.4 0.5 013(b) 
+ωt/2π E/N ε
+tm teq tm teq 
+FIG. 4: (color online) Molecular dynamics (MD) simulation
+of crystal heating. Black, red and blue lines are total en-
+ergyEtot, kinetic Ekinand potential Epotenergies vs. time
+t, respectively. Thin continuous lines: single MD trajector ies.
+Thick dashed lines: averages over many MD trajectories. teq
+andtmare the equilibration and melting time, respectively.can be imaged using, e.g., tightly focused beams.
+The authors thank H.P. B¨ uchler and F. Ferlaino for
+discussions. This work was supported by IQOQI, the
+Austrian FWF, the US through MURI and EOARD, the
+EU through STREP FP7-ICT-2007-C project NAME-
+QUAM, the Canadian NSERC through G121210893.
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+Lukinet al.,ibid.87, 037901 (2001); M. Saffman and K.
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+M. D. Lukin, arXiv:0911.1427 .
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+Rev. Mod. Phys. 71, 1 (1999).
+[10]G. E. Astrakharchik et al., Phys. Rev. Lett. 98, 060405
+(2007).
+[11]M. Boninsegni, N. V. Prokof’ev and B. V. Svistunov,
+Phys. Rev. Lett. 96, 070601 (2006) and Phys. Rev. E
+74, 036701 (2006).
+[12]A.I. Belousov and Yu.E. Lozovik, Eur. Phys. J. D 8, 251
+(2000); Yu.E. Lozovik, S. Yu. Volkov, and M. Willander,
+JETP Lett. 79, 473, (2004); P. Ludwig et al., New J.
+Phys.10, 083031 (2008). For Coulomb clusters: V.M.
+BedanovandF.M. Peeters, Phys.Rev.B 49, 2667(1994);
+A. Filinov et al.,ibid.77, 214527 (2008).
+[13]P. Sindzingre, M. L. Klein and D. M. Ceperley, Phys.
+Rev. Lett. 63, 1601 (1989).
+[14]Finite clusters can rotate at a temperature T/lessorsimilar/planckover2pi12/I,I
+being the classical moment of inertia. We have observed
+this forτ= 5:fshows two sharp peaks for n= 4 and 9
+(inner and outer rings, respectively), other cycles being
+absent. However, this is not a “superfluid”.
+[15]Choosing larger ncan make lifetimes longer, since γr∼
+1/n3. However, then itis notpossible toneglect processes
+wheretheatomisfoundinanexcitedstate |r′/angbracketright /negationslash=|r/angbracketrightafter
+the spontaneous emission; S. Thwaite et al., unpublished.
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\ No newline at end of file
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+arXiv:1001.0520v1  [math.OA]  4 Jan 2010EXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS
+PIOTR M. SO/suppress LTAN
+Abstract. We present two examples of actions of non-regular locally co mpact quantum groups
+on their homogeneous spaces. The homogeneous spaces are defi ned in a way specific to these
+examples, but the definitions we use have the advantage of bei ng expressed in purely C∗-algebraic
+language. We also discuss continuity of the obtained action s. Finally we describe in detail a
+general construction of quantum homogeneous spaces obtain ed as quotients by compact quantum
+subgroups.
+1.Introduction
+The main aim of this paper is to present two examples of quantum homo geneous spaces for
+non-regular quantum groups. We will construct quotients of the q uantum “ az+b” and quantum
+E(2) groups (from [27, 18, 17] and [21] respectively) by their class ical subgroups. In both cases our
+definition of the quantum homogeneous space will be deeply rooted in the particular form of the
+C∗-algebras related to these quantum groups. On the other hand, o ur definitions of quantum ho-
+mogeneous spaces will be given in purely C∗-algebraic language, without any use of von Neumann
+algebras. Let us also remark that non-regular quantum groups do not fit into the elaborate and
+powerful framework of [20]. We will prove that in both cases the ac tion of the quantum group on
+its homogeneous space is continuous in an appropriate sense. The s econd example (presented in
+Section 4) can also be used as basis of a simple definition of a quantum h omogeneous space of the
+form /BZ/ /C3, where /BZis a quantum group and /C3is acompact quantum subgroup of /BZ. We develop
+this idea in the last section. The notion of a quantum subgroup we will u se is more restrictive
+than that of [20]. On the other hand, our construction works for g eneral bisimplifiable Hopf C∗-
+algebras, not only for locally compact quantum groups. In particula r we can use universal versions
+of quantum groups ([8], [19, Section 5]).
+We will precede our examples of quantum homogeneousspaces with a brief discussion of actions
+of classical locally compact groups on C∗-algebras. We shall provide a description of such actions
+in language appropriate for generalization to quantum groups. Con tinuity of actions is discussed
+and a definition of a continuousaction ofa quantum groupis propose dand compared with existing
+approaches.
+The tools used to prove our results will range from the notion of G-product and Lanstad
+algebra to C∗-algebras generated by unbounded affiliated elements. The materia l on crossed
+products by of C∗-algebras by actions of abelian groups (including the notion of a G-product
+and Lanstad algebra) can be found in [11, 13] and [7]. For the notions of multiplier algebra of a
+C∗-algebra, strict topology, morphisms of C∗-algebras and elements affiliated to C∗-algebras we
+refer to [24, 10, 25]. Let us only mention here that by a morphism form a C∗-algebra Ato another
+C∗-algebra Bwe shall mean a ∗-homomorphism ϕformAto the multiplier algebra M( B) ofB
+which is nondegenerate, i.e. one whose image contains an approximate unit for B. The set of all
+morphisms from AtoBwill be denoted by Mor( A,B). In [25] the reader will also find a detailed
+exposition of the concept of a C∗-algebra generated by unbounded elements affiliated with it. We
+will also use some earlier work on one of our our examples from [16] as w ell as a wide range of
+results related to the considered quantum groups ([22, 21, 23, 27 , 18, 17]).
+Date : November 2, 2018.
+2000Mathematics Subject Classification. 46L89, 46L55, 46L85.
+Key words and phrases. quantum group, quantum group action, C∗-algebra, quantum homogeneous space.
+Research partially supported by Polish government grant no . N201 1770 33.
+12 PIOTR M. SO/suppress LTAN
+The definition of a locally compact quantum group is due to Kusterman s and Vaes ([9]), but
+we will not be making any use of the Haar weights of the quantum grou ps under consideration.
+On the other hand our quantum groups will have a continuous counit . For the theory of compact
+quantum groups we refer to [26].
+Let us briefly describe the contents of the paper. We begin with a se ction which carefully
+rephrases the theory of actions of locally compact groups on C∗-algebras int the language suitable
+for quantum groups(much like variousaspectsof locallycompact gr oupactionson locally compact
+spacesweretreated in [5]). There arenonew resultsin that section , but wefeel that some concepts
+present in the literature require clarification. At the end of this sec tion we define continuous
+actions of quantum groups on C∗-algebras (quantum spaces) and provide a short explanation of
+the standard problems encountered in construction of quantum h omogeneous spaces. In Section
+3 we introduce the quantum “ az+b” groups for various values of the deformation parameter and
+define the homogeneous space obtained as the quotient by a classic al subgroup. This construction
+was already performed in [16]. Then we analyze the action of the quan tum “az+b” group on its
+homogeneous space to show that it is continuous. The example from Section 4 is very similar in
+spirit, but the methods of analyzing this example are significantly diffe rent.
+Motivated by the Example from Section 4, in the last section we gener alize known construction
+ofaquotientquantumhomogeneousspaceforcompactquantumg roup(cf.[14,15])tothesituation
+where the groupis no longercompact, but the subgroup is. We show that the action ofthe original
+quantum group on the resulting quantum homogeneous space is con tinuous.
+2.Continuity of classical actions
+Let (B,G,α) be a C∗-dynamical system. Then for each x∈Bwe have the function
+α(x) :G∋t/mapsto−→αt(x)∈B. (2.1)
+This function has constant norm and is norm-continuous by assump tion. In particular it does not
+vanish at infinity on Gunlessx= 0. It follows that α(x)/\e}atio\slash∈C0(G)⊗Bfor non-zero x. However,
+it is easy to see that αis a∗-homomorphism from Bto M/parenleftbig
+C0(G)⊗B/parenrightbig
+, where the last algebra
+is naturally identified with the space of all norm-bounded functions G→M(B) continuous in the
+strict topology.
+This raises the question how to describe the special property that for each x∈Bthe function
+α(x) has its values in B(not merely in M( B)) and is norm-continuous (not only strictly contin-
+uous). The well known answer is that these properties are equivale nt to the fact that for each
+f∈C0(G) the element
+(f⊗ /BD)α(x)∈C0(G)⊗B
+(consider a function fconstant and non-zero on a neighborhood of a given point t∈G).
+Let us note here the fact which is well known, but difficult to find in the literature:
+Proposition 2.1. Let(B,G,α)be aC∗-dynamical system and for each x∈Bletα(x)be defined
+by(2.1). Then the linear span of the set
+/braceleftbig
+(f⊗ /BD)α(x)f∈C0(G), x∈B/bracerightbig
+is dense in C0(G)⊗B.
+We will give a very short and easy proof of this fact once appropriat e structure on C 0(G) has
+been defined (in the proof of Proposition 2.3). Nevertheless we wan t to include an elementary
+proof of Proposition 2.1 which does not use the additional structur e of C 0(G).
+Proof of Proposition 2.1. We need to approximate any element of C 0(G)⊗Bby functions of a
+specific form. It is enough to approximateelements from a dense su bspace C c(G,B)⊂C0(G,B)∼=
+C0(G)⊗B. Therefore let Fbe a fixed continuous function G→Bwith compact support. For
+t∈suppFlet
+Ut=/braceleftbig
+s∈G/vextenddouble/vextenddoubleF(s)−αst−1/parenleftbig
+F(t)/parenrightbig/vextenddouble/vextenddouble< ε/bracerightbig
+.EXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS 3
+By compactness of supp Fthere exist t1,...,tnsuch that supp F⊂n/uniontext
+i=1Uti. Let (χi)i=1,...,nbe a
+partition of unity subordinated to the covering ( Uti)i=1,...,nof suppFand define
+X(t) =n/summationdisplay
+j=1χj(t)αt/parenleftBig
+αt−1
+j/parenleftbig
+F(tj)/parenrightbig/parenrightBig
+.
+In other words
+X=n/summationdisplay
+j=1(χj⊗ /BD)α/parenleftBig
+αt−1
+j/parenleftbig
+F(tj)/parenrightbig/parenrightBig
+.
+Take now t∈suppF. We have
+/vextenddouble/vextenddoubleX(t)−F(t)/vextenddouble/vextenddouble=/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay
+j=1χj(t)αtt−1
+j/parenleftbig
+F(tj)/parenrightbig
+−F(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+=/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay
+j=1χj(t)αtt−1
+j/parenleftbig
+F(tj)/parenrightbig
+−n/summationdisplay
+j=1χj(t)F(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+≤n/summationdisplay
+j=1χj(t)/vextenddouble/vextenddoubleαtt−1
+j/parenleftbig
+F(tj)/parenrightbig
+−F(t)/vextenddouble/vextenddouble.
+Now the j-th term of the last sum is non-zero only if t∈Utj. It follows that each term of this
+sum is either zero or less than χj(t)ε.
+We have thus shown that for each iwe have sup
+t∈Uti/vextenddouble/vextenddoubleX(t)−F(t)/vextenddouble/vextenddouble≤ε. On the other hand, outside
+n/uniontext
+i=1UtibothXandFare zero. /square
+An immediate corollary of Proposition 2.1 is that the linear span of
+/braceleftbig
+(f⊗y)α(x)f∈C0(G), x,y∈B/bracerightbig
+is dense in C 0(G)⊗B. Indeed, if f⊗yis a simple tensor in C 0(G)⊗Bthen we can approximate
+it by finite sums of the form/summationdisplay
+(fi⊗ /BD)α(xi).
+Now if ( eλ) is a (bounded) approximate unit for Bthen, by the diagonal argument, we can
+approximate f⊗yby sums of the form/summationtext(fi⊗eλ)α(xi) because
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+(fi⊗eλ)α(xi)−g⊗y/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble( /BD⊗eλ)/parenleftbigg/summationdisplay
+(fi⊗ /BD)α(xi)−g⊗y/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble+/vextenddouble/vextenddoubleg⊗(eλy−y)/vextenddouble/vextenddouble.
+Since the approximation works for simple tensors, it also works for g eneral elements of C 0(G)⊗B.
+Corollary 2.2. Let(B,G,α)be aC∗-dynamical system and for each x∈Bletα(x)be defined
+by(2.1). Thenα∈Mor/parenleftbig
+B,C0(G)⊗B/parenrightbig
+.
+Let us note here one other density condition appearing in the literat ure. Quite clearly the linear
+span of/braceleftbigg/integraldisplay
+ϕ(t)αt(x)dtϕ∈L1(G), x∈B/bracerightbigg
+is dense in B(anyx∈Bis the limit of such integrals with δ-like net of integrable functions). In
+order to rewrite this condition in a more convenient way let us introdu ce for each ϕ∈L1(G) the
+continuous functional
+ωϕ: C0(G)∋f/mapsto−→/integraldisplay
+ϕ(t)f(t)dt.
+Then we see that the linear span of
+/braceleftbig
+(ωϕ⊗id)α(x)ϕ∈L1(G), x∈B/bracerightbig
+is dense in B.4 PIOTR M. SO/suppress LTAN
+Let us introduce the standard comultiplication ∆ ∈Mor/parenleftbig
+C0(G),C0(G)⊗C0(G)/parenrightbig
+:
+∆(f)(s,t) =f(st). (2.2)
+Then one easily checks that (∆ ⊗id)◦α= (id⊗α)◦α.
+Clearly one can also see that ( ǫ⊗id)◦α= id, where ǫis the evaluation functional
+C0(G)∋f/mapsto−→f(e)∈ /BV.
+We will now state and prove a simple proposition dealing with various con ditions defining
+continuity of a group action in the language adaptable to the more ge neral context of quantum
+groups.
+Proposition 2.3. LetGbe a locally compact group and let Bbe aC∗-algebra. Let
+α∈Mor/parenleftbig
+B,C 0(G)⊗B/parenrightbig
+be such that
+•with∆defined by (2.2)we have
+(∆⊗id)◦α= (id⊗α)◦α (2.3)
+•for anyf∈C0(G)andx∈Bwe have
+(f⊗ /BD)α(x)∈C0(G)⊗B,
+Then the following conditions are equivalent
+(1) (ǫ⊗id)◦α= id, whereǫis the evaluation at the neutral element of G,
+(2) kerα={0},
+(3)the linear span of/braceleftbig
+(f⊗ /BD)α(x)f∈C0(G), x∈B/bracerightbig
+(2.4)
+is dense in C0(G)⊗B,
+(4)the linear span of/braceleftbig
+(ωϕ⊗id)α(x)ϕ∈L1(G), x∈B/bracerightbig
+(2.5)
+is dense in B.
+Moreover, if the equivalent conditions (1)–(4)are satisfied then then there exists a continuous
+actionαofGonBsuch that αis defined by (2.1).
+Proof.Let us begin with defining a family ( αt)t∈Gof maps B→Bby
+αt(x) =α(x)(t) = (δt⊗id)α(x),
+whereδtis the evaluation functional C 0(G)∋f/mapsto→f(t). One immediately find that each αtis
+an endomorphism of Band that for each x∈Bthe mapping G∋t/mapsto→αt(x) is norm continuous.
+Moreover, it follows from (2.3) that
+αt◦αs=αts
+for allt,s∈G. In particular αeis an idempotent mapping B→Bwhich commutes with all αt’s.
+The range of all αt’s is equal to the range of αeand their kernels are all equal to ker αe.
+It follows that (1) ⇔(2). Indeed, (1) clearly implies (2), but (2) means that for any non -zero
+x∈Bthe function t/mapsto→αt(x) is non zero. Thus if x∈kerαethenαt(x) = 0 for all tand sox= 0.
+In other words, αeis an idempotent with zero kernel, so αe= id. This is exactly (1).
+In particular if (1) is satisfied then all αt’s are automorphisms and αbecomes a continuous
+action of GonB. To see that this implies (3) without using Proposition2.1 note that th e mapping
+C0(G)⊗algB∋(f⊗x)/mapsto−→(f⊗ /BD)α(x)∈C0(G)⊗B (2.6)
+extends to an automorphism of C 0(G)⊗B. Indeed, the map (2.6) is bounded and multiplicative
+(because C 0(G) is commutative). The inverse mapping is given by
+(f⊗x)/mapsto−→(f⊗ /BD)(id⊗κ)/parenleftbig
+α(x)/parenrightbig
+,
+whereκis the automorphism of C 0(G) given by κ(f)(t) =f(t−1). Therefore the linear span of
+(2.4) is dense in C 0(G)⊗Bas the image of C 0(G)⊗algBunder an automorphism of C 0(G)⊗B.EXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS 5
+The fact that (3) ⇒(4) is standard: approximate a simple tensor f⊗y∈C0(G)⊗Bby sums of
+the form/summationdisplay
+(fi⊗ /BD)α(xi).
+Then take ϕ∈L1(G) such that ωϕ(f) = 1. Then the elements
+/summationdisplay
+(ωϕfi⊗id)α(xi)
+approximate yand consequently the linear span of (2.5) is dense in B.
+Finally let us see that (4) implies (1). If (1) is not satisfied then we kno w that the ranges of
+allαt’s are equal to the range of αewhich is strictly contained in Band closed (as the image of a
+C∗-algebra under a ∗-homomorphism). Then it follows that for any ϕ∈L1(G) the element
+(ωϕ⊗id)α(x) =/integraldisplay
+ϕ(t)αt(x)dt
+also lies in the range of αeand we see that (4) is not satisfied. /square
+Inwhatfollowswewilldealwith quantumgroupsoftheform /BZ= (A,∆) with continuouscounit
+ǫ. Wewillsaythat /BZactscontinuouslyonaC∗-algebraBifthereisamorphism α∈Mor(B,A⊗B)
+such that
+•(∆⊗id)◦α= (id⊗α)◦α,
+•for anyc∈Aandx∈Bwe have ( c⊗ /BD)α(x)∈A⊗B,
+•(ǫ⊗id)◦α= id.
+In the literature the last condition is often replaced by either requir ement that the linear span
+of/braceleftbig
+(c⊗ /BD)α(x)c∈A, x∈B/bracerightbig
+be dense in A⊗Bor that/braceleftbig
+(ω⊗id)α(x)ω∈A∗, x∈B/bracerightbig
+be dense in B(whereA∗is the space of normal linear functionals on A, cf.[19]). In [3, Proposition
+5.8] it is shown that these conditions are equivalent for regularlocally compact quantum groups
+and that the second condition is strictly weaker for semi-regular, but non-regular locally compact
+quantum groups.
+Ifα∈Mor(B,A⊗B) is such that (∆ ⊗id)◦α= (id⊗α)◦αand kerα={0}then the
+established terminology is that αis areducedaction of /BZonB(see e.g. [20, 6]). As Proposition
+2.3 shows, in case when /BZis a classical locally compact group, all these conditions are equivalen t.
+Let /BZ= (A,∆) and /C3= (C,∆C) be quantum groups. We say that a morphism π∈Mor(A,C)
+identifies /C3as aclosed quantum subgroup of /BZifπis a surjective ∗-homomorphism A→Cand
+(π⊗π)◦∆ = ∆ C◦π. As we mentioned in Section 1, such a definition of a quantum subgrou p
+is more restrictive than that of [20]. In the following sections we will giv e examples of situations
+where this notion of quantum subgroup applies.
+Let us explain the difficulty which we come acrossas we try to generaliz ethe standard construc-
+tion of a quotient quantum homogeneous space from [15]. Let us ass ume that the C∗-algebras
+AandBare commutative, so that A= C 0(G) andC= C 0(K), where Gis a locally com-
+pact group and Kis a closed subgroup of G. In this situation πis the restriction morphism
+A∋f/mapsto→f/vextendsingle/vextendsingle
+K∈M(C). The range rarely coincides with C. In the language of noncommutative
+topology the homogeneous space G/Kis described by the C∗-algebra B= C 0(G/K). It is a
+nontrivial question how to identify this C∗-algebra as a subalgebra of M( A) = Cb(G). Clearly any
+elementx∈Bshould satisfy
+(id⊗π)∆(x) =x⊗ /BD, (2.7)
+but this is certainly not enough. In fact all elements of M( B) also satisfy this condition. In the
+examples of Sections 3 and 4 /BZ= (A,∆) will be a non-regular locally compact quantum group
+with a subgroup /C3= (C,∆C) (which in both cases will be a classical group) and we will produce
+a candidate for B. It will be a C∗-subalgebra of M( B) consisting of elements xsatisfying (2.7)
+and some additional conditions tailored to these specific examples. T he next question which arises
+naturally after the definition of a homogeneous space is whether th e restriction of ∆ to Bactually6 PIOTR M. SO/suppress LTAN
+defines an action of /BZon /BZ/ /C3. We will have to check that α= ∆/vextendsingle/vextendsingle
+Bis a morphism from Bto
+A⊗Band that algebraic conditions of a (co)action are satisfied. More imp ortantly we would like
+to show that the action αis continuous. None of these facts is totally obvious and we prove th em
+by application of the theory of C∗-algebras generated by unbounded elements ([25]).
+3.The action of the quantum “ az+b” group on its homogeneous space
+The construction of the quantum “ az+b” group starts with choosing the value of the deforma-
+tion parameter q. There are at least three possibilities to choose q. They are described in detail
+in [17]. One of these possibilities is to take q∈]0,1[ (cf. [27, Appendix A]). It is not important for
+our purposes to dwell on this choice since the aspect of quantum “ az+b” groups we shall need
+are the same regardless of the value of the deformation paramete r. All details can be found in
+[27, 18, 17].
+We let Γ be the subgroup of the multiplicative group /BV\{0}generated by qand{qitt∈ /CA}
+with an appropriate fixed choice of logarithm of q. The group Γ is then self-dual and we denote
+byχthe nondegenerate bicharacter Γ ×Γ→ /CCsuch that
+χ(γ,γ′) =χ(γ′,γ),
+χ(γ,qit) =|γ|it,
+χ(γ,q) = Phase γ
+for allγ,γ′∈Γ andt∈ /CA.
+Then we let Γ be the closure of Γ in /BVand letβbe the action of Γ on Γ by multiplication. We
+defineA= C 0(Γ)⋊βΓ. An affiliated element bηAis introduced as the image of the canonical
+generator of C 0(Γ) inA, whileaηAas the unique affiliated element such that Sp a⊂Γ,ais
+invertible (with a−1ηA) and/parenleftbig
+χ(a,γ)/parenrightbig
+γ∈Γis the canonical family of unitary elements of M( A)
+implementing the action βon C 0(Γ).
+By the results of [27, 18] (the actual reference depends on the v alue ofq, cf. [17]) there exists
+a unique ∆ ∈Mor(A,A⊗A) such that
+∆(a) =a⊗a,∆(b) =a⊗b˙+b⊗ /BD
+and /BZ= (A,∆) is a quantum group (in fact a non-regular locally compact quantum group cf. also
+[28]).
+Let us first see that Γ is a subgroup of /BZin the sense described at the end of Section 2, i.e. there
+exists a morphism π∈Mor/parenleftbig
+A,C0(Γ)/parenrightbig
+such that
+∆◦π= (π⊗π)◦∆
+andπis a surjection onto C 0(Γ). Indeed, since the C∗-algebra Ais generated by unbounded
+elements a,a−1andbaffiliated with it (cf. [27, 18, 17]), it is enough to define πon the generators
+by putting π(b) = 0 and letting π(a) be the canonical generator of C 0(Γ) (the identity function
+Γ∋γ/mapsto→γ∈ /BV).
+Let us remark here that in [16] it is the dual group of Γ which is identifie d with a subgroup of/BZwhich we are describing. This is correct since Γ ∼=/hatwideΓ, but the distinction has no bearing on our
+construction of quantum homogeneous space, so we will stick with t he choice of Γ as a subgroup
+of /BZ.
+We shall now define a quantum homogeneous space /BZ/Γ. By definition the C∗-algebra Bof
+continuous functions vanishing at infinity on /BZ/Γ is the set of those x∈M(A) for which
+(1) (id⊗π)∆(x) =x⊗ /BD,
+(2) for any y∈C∗(Γ)⊂M(A) the element yx∈A,
+(3) the mapping Γ ∋γ/mapsto→χ(a,γ)∗xχ(a,γ) is continuous.
+Condition (1) selects elements of M( A) which are “constant along the fibers” of the fibration/BZ→ /BZ/Γ. However, they cannot correspond to functions vanishing at infi nity on /BZ/Γ (e.g. the
+unit /BDsatisfies this condition). Therefore condition (2) is introduced to s elect those “continuous
+bounded functions on /BZ” which not only are constant on cosets of Γ, but in addition vanish
+at infinity “in the direction transversal to the cosets.” Condition (3 ) does not have a classicalEXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS 7
+analogue. This definition of the algebra of “continuous functions va nishing at infinity on /BZ/Γ”
+was first introduced in [16].
+It is not difficult to check that B=/braceleftbig
+f(b)f∈C0(Γ)/bracerightbig
+. This is because conditions (1)–(3)
+are really the Lanstad conditions for the Γ-product structure on A= C 0(Γ)⋊βΓ (cf. [11, 13]).
+Therefore the homogeneous space /BZ/Γ is a classical space, i.e. one described by a commutative
+C∗-algebra.
+It was pointed out already in [16] that the restriction of ∆ to B⊂M(A) defines a morphism
+α∈Mor(B,A⊗B) and that
+(∆⊗id)◦α= (id⊗α)◦α.
+Moreover one can easily see that ( ǫ⊗id)◦α= id, where ǫis the counit of /BZdefined by ǫ(a) = 1,
+ǫ(b) = 0. Therefore, in order to say that the action of /BZon the homogeneous space /BZ/Γ is
+continuous we only need to prove the following:
+Theorem 3.1. Let /BZ= (A,∆)be the quantum “ az+b” group and let B⊂M(A)be theC∗-algebra
+of functions vanishing at infinity on the quantum homogeneou s space /BZ/Γ. Letα∈Mor(B,A⊗B)
+be the morphism defined above. Then for any c∈Aand any x∈Bthe element
+(c⊗ /BD)α(x)∈A⊗B. (3.1)
+Proof.We shall identify Bwith C 0(Γ) andA⊗BwithA⊗C0(Γ)∼=C0(Γ,A). The element bηB
+corresponds to the function Γ∋z/mapsto→z∈ /BVandα(b) to
+Γ∋z/mapsto−→(az˙+b)ηA.
+(cf. [25, Formula 2.6]).
+It follows that for any f∈C0(Γ) the image of f(b) underαis identified with the element
+Γ∋z/mapsto−→f(az˙+b)
+of Cstrict
+b/parenleftbig
+Γ,M(A)/parenrightbig
+. Therefore with x=f(b) the element (3.1) corresponds to the function
+Γ∋z/mapsto−→cf(az˙+b)
+which is an element of C b(Γ,A). We want to show that this is in fact an element of C 0(Γ,A) =
+C0(Γ)⊗A.
+According to results [22, 27, 18] (again the reference depends on the choice of q) there is a
+continuous function Fq:Γ→ /CCsuch that az˙+b=Fq(z−1ba−1)∗azFq(z−1ba−1), so that
+f(az˙+b) =Fq(z−1ba−1)∗f(az)Fq(z−1ba−1). (3.2)
+Our aim is to show that for any fixed c∈Athe norm of
+cFq(z−1ba−1)∗f(az)Fq(z−1ba−1)
+tends to 0 as zgoes to∞ofΓ. Clearly this norm is equal to the norm of
+cFq(z−1ba−1)∗f(az).
+Sinceba−1ηA, it is known (see e.g. [27, Theorem 5.1]) that Fq(z−1ba−1) tends to /BDstrictly in
+M(A) azz→ ∞(note that Fq(0) = 1). Therefore for any ε >0 there exists a constant Rεsuch
+that/vextenddouble/vextenddoublecFq(z−1ba−1)∗−c/vextenddouble/vextenddouble< ε.
+for allz∈Γ with|z|> Rε. Thus the norm
+/vextenddouble/vextenddoublecFq(z−1ba−1)∗f(az)/vextenddouble/vextenddouble≤/vextenddouble/vextenddoublecf(az)/vextenddouble/vextenddouble+/vextenddouble/vextenddouble/parenleftbig
+cFq(z−1ba−1)∗−c/parenrightbig
+f(az)/vextenddouble/vextenddouble</vextenddouble/vextenddoublecf(az)/vextenddouble/vextenddouble+ε/bardblf/bardbl.
+To see that the norm of cf(az) goes to zero as z→ ∞we can assume that fhas compact support
+and that cis of the form
+c=˜f(b)g(a)
+for some ˜f∈C0(Γ) andg∈C0(Γ) (the set of such elements is linearly dense in A). Then
+cf(za) =˜f(b)Gz(a),8 PIOTR M. SO/suppress LTAN
+whereGz(z′) =g(z′)f(zz′). Since gvanishes at 0, it is easy to see that /bardblGz/bardbl −−−→
+z→∞0 and it
+follows that/vextenddouble/vextenddoublecf(za)/vextenddouble/vextenddouble< εfor sufficiently large |z|. /square
+4.The action of the quantum E(2)group on its homogeneous space
+ForthedescriptionofthequantumE(2)groupwewillfixaHilbert spa ceHwithanorthonormal
+basis (ei,j)i,j∈ /CI. We will change our notation slightly in order to be in agreement with co nventions
+established in the literature [21, 23]. Let q∈]0,1[ be a parameter and let /BVqbe the subgroup of/BV\{0}generated by qand{qitt∈ /CA}— this is Γ from Section 3 for the case of real q. As before
+we let /BVqbe the closure of /BVqin /BV.
+We will now define two operators on H. We let vbe the unitary operator defined uniquely by
+vei,j=ei−1,jfor ai,j∈ /CIand letnbe the closed linear operator on Hsuch that the linear span
+of the orthonormal basis ( ei,j)i,j∈ /CIis a core of nandnei,j=qiei,j+1for alli,j∈ /CI. Thennis a
+normal operatorwith Sp n⊂ /BVq. We define Aas the closure of the set of finite linear combinations
+/summationdisplay
+fl(n)vl, (4.1)
+wherefl∈C0( /BVq) for all l∈ /CI. With this definition Ais a non-degenerate C∗-subalgebra of
+B(H) isomorphic to C 0( /BVq)⋊β
+/CI, whereβis the natural action by multiplication by q(cf. [21]).
+Moreover v∈M(A) andnηAandAis generated by nandv([25, Example 4]).
+It is shown in [21] that there exists a unique ∆ ∈Mor(A,A⊗A) such that
+∆(v) =v⊗v,∆(n) =v⊗n˙+n⊗v∗
+and /BZ= (A,∆) is a non-regular locally compact quantum group ([1]) called the quantum E(2)
+group.
+The classical group /CCis a subgroup of /BZ, i.e. we have a quantum group morphism π∈
+Mor/parenleftbig
+A,C( /CC)/parenrightbig
+defined uniquely by putting π(n) = 0 and letting π(v) be the canonical unitary
+generator of C( /CC). We wish to describe the quantum homogeneous space /BZ/ /CC.
+To that end let us define structure of a /CI-product on Adifferent from the standard one (arising
+from the fact that Ais the crossed product of C 0( /BVq) by /CI). We define a representation
+V: /CI∋n/mapsto−→Vn=vn∈M(A)
+and action of /CConAby automorphisms ( ˜βµ)µ∈ /CCsuch that
+˜βµ(v) =µv,˜βµ(n) =µ−1n
+for allµ∈ /CC. The triple ( A,V,˜β) is a /CI-product. This /CI-product structure on Ais not a “twist”
+of the standard /CI-product structure on Aas defined in [7], but let us note that if we denote by ˆβ
+the action of /CConAdual toβthen for each µ∈ /CC.
+˜βµ(y) =Uµˆβµ(y)U∗
+µ,
+whereUµis the unitary operator such that Uµei,j=µ−jei,j.
+Following the example from Section 3 we define the C∗-algebra of Bof continuous functions
+vanishing at infinity on /BZ/ /CCas the set of those x∈M(A)
+(1) (id⊗π)∆(x) =x⊗id,
+(2) for any y∈C∗( /CI)⊂M(A) the element yx∈A.
+The above conditions are, as in Section 3, precisely the Lanstad con ditions for the /CI-product
+structure we have defined (in particular Breally is a C∗-algebra). Condition 1 is exactly the
+condition of being ˜β-invariant, while condition 2 says that the “function” xvanishes at infinity
+in the direction transversal to that of the subgroup /CC. In fact condition (2) is equivalent to
+demanding that x∈A(because C∗( /CI) is unital). The third Lanstad condition is empty because/CIis a discrete group.
+Before proceeding let us define a certain C∗-algebra which will turn out to be the algebra
+of continuous unctions vanishing at infinity on /BZ/ /CC. Let C 0( /CI∪ {+∞}) be the C∗-algebra of
+sequences ( xn)n∈ /CIfor which lim
+n→−∞xn= 0 and lim
+n→+∞xnexists and is finite. We let Cbe theEXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS 9
+crossed product C= C 0( /CI∪{+∞})⋊ /CI, where the action of /CIis by translation. The C∗-algebra
+Cis an extension of Kby C( /CC). More precisely we have
+C∼=/braceleftbigg/bracketleftbiggx y
+z u/bracketrightbigg
+x∈ T, y,z,u∈ K/bracerightbigg
+(whereTis the Toeplitz algebra) and the extension is
+0/d47/d47K∼=M2(K) /d47/d47Cρ/d47/d47C( /CC) /d47/d470
+withρsending a matrix/bracketleftbigx y
+z u/bracketrightbig
+to the symbol of x.
+Proposition 4.1. LetBbe the algebra of continuous functions vanishing at infinity on /BZ/ /CC.
+Then
+(1)TheC∗-algebraBis the closure of the set of finite sums of the form
+/summationdisplay
+fk(n)vk(4.2)
+where for each k∈ /CIthe function fk∈C0(Γ)is such that
+fk(µz) =µkfk(z)
+for allz∈Γandµ∈ /CC.
+(2)The operator vnis affiliated with B,
+(3)Bis generated by vn.
+(4)Bis isomorphic to C.
+Proof.We shall first prove point (1). Let B0be the closure of the set of finite sums described in
+Statement (1). Since each term fk(n)vnis invariant under ˜βand is already contained in A(as
+opposed to M( A)) we have B0⊂B. Moreover B0is invariant under conjugation with v. Finally
+let us note that
+C∗( /CI)B0C∗( /CI) (4.3)
+is dense in A(as before, we identify C∗( /CI) with the C∗-subalgebra of M( A) generated by v). This
+is because the functions fkappearing in (4.2) are trigonometric monomials along the individual
+circles of /BVq. Therefore we can uniformly approximate on each circle sufficiently r egular functions
+(e.g. twice differentiable) and those are dense in all continuous func tions on the circle. This can
+be done on each circle separately. As a result all elements of the for m (4.1) lie in the closure of
+(4.3). By [7, Lemma 2.6] B0=B.
+Now we shall deal with points (2)–(4). The operator vnis affiliated with B. This is because
+thez-transform of vnclearly belongs to B0and the element /BD−z∗
+vnzvn= ( /BD+n∗n)−1is a strictly
+positive element of B(cf. [25, Section 1]). Let us denote by Xthe operator vnand letX=U|X|
+be the polar decomposition of X. We have Uei,j=ei−1,j+1and|X|ei,j=qiei,j. One easily finds
+that forfk∈C0( /BVq) such that fk(µz) =µkfk(z) we have
+fk(n)vk=fk/parenleftbig
+|X|/parenrightbig
+Uk. (4.4)
+A moment of reflection shows now that B∼=Cand point (4) follows.
+Letρbe a representation of Bon a Hilbert space L. Then ρ(vn) is a closed operator on
+L. Moreover by (4.4) the image under ρof any element x∈B0is determined by the polar
+decomposition of ρ(vn). This means that vnseparates representations of Band it follows that B
+is generated by vnby [25, Theorem 3.3], since/parenleftbig/BD+(vn)∗(vn)/parenrightbig−1∈B. /square
+Before proceeding let us emphasize the fact that B⊂A. Moreover since B⊂Awe have that
+B=/braceleftbig
+x∈A˜βµ(x) =xfor allµ∈ /CC/bracerightbig
+=/braceleftbig
+x∈A(id⊗π)∆(x) =x⊗ /BD/bracerightbig
+.
+We also have
+Lemma 4.2. A⊗B=/braceleftbig
+X∈A⊗A(id⊗˜βµ)(X) =Xfor allµ∈ /CC/bracerightbig
+.10 PIOTR M. SO/suppress LTAN
+Proof.The inclusion “ ⊂” is obvious. For the converse one let us take X∈A⊗Asuch that
+(id⊗˜βµ)(X) =Xfor allµ∈ /CC.Xis a limit of finite sums of simple tensors:
+X= lim
+k→∞Nk/summationdisplay
+i=1a(k)
+i⊗b(k)
+i.
+Letdµbe the normalized Haar measure on /CC. We have
+X=/integraldisplay
+(id⊗˜βµ)(X)dµ= lim
+k→∞Nk/summationdisplay
+i=1a(k)
+i⊗/integraldisplay
+˜βµ(b(k)
+i)dµ
+Clearly the elements c(k)
+i=/integraltext˜βµ(b(k)
+i)dµare˜β-invariant, so that they belong to B. It follows that
+X∈A⊗B. /square
+Now let us describe the action of /BZon its homogeneous space /BZ/ /CC. The image of vnηAunder
+∆ is
+v2⊗vn˙+vn⊗ /BD.
+By[29, Theorem6.1]and[22, Theorem5.1]thiselementisaffiliatedwith A⊗B. Sincevngenerates
+B, we see that the map α= ∆/vextendsingle/vextendsingle
+Bis a morphism form BtoA⊗B. Clearly we have
+(∆⊗id)◦α= (id⊗α)◦α
+and (ǫ⊗id)◦α= id, where ǫis the counit of /BZ(defined by ǫ(v) = 1 and ǫ(n) = 0).
+SinceB⊂Awe have α(B)⊂A⊗Aby [23, Formula (57)]. Note also that
+(id⊗˜βµ)◦∆ = ∆◦˜βµ. (4.5)
+In view of Lemma 4.2, this implies that in fact α(x)∈A⊗Bfor anyx∈B. In particular for any
+c∈Aandx∈Bwe have ( c⊗ /BD)α(x)∈A⊗B. Thus we obtain
+Corollary 4.3. α∈Mor(B,A⊗B)is a continuous action of /BZon /BZ/ /CC.
+Remark 4.4.Let us also note that the set of all elements ( c⊗ /BD)α(x) withc∈Aandx∈Bis not
+only contained (as shown above), but also linearly dense in A⊗B. Indeed if we denote by Ethe
+conditional expectation A∋c/mapsto→/integraltext˜βµdµ∈Bused in the proof of Lemma 4.2 then it follows from
+(4.5) that (id ⊗E)◦∆ = ∆◦E. Using the fact that the span of/braceleftbig
+(a⊗ /BD)∆(b)a,b∈A/bracerightbig
+is dense
+inA⊗Awe see that
+span/braceleftbig
+(c⊗ /BD)α(x)c∈A, x∈B/bracerightbig
+= span/braceleftbig
+(c⊗ /BD)∆(x)c∈A, x∈B/bracerightbig
+= span/braceleftbig
+(c⊗ /BD)∆/parenleftbig
+E(d)/parenrightbig
+c,d∈A/bracerightbig
+= span/braceleftbig
+(c⊗ /BD)(id⊗E)/parenleftbig
+∆(d)/parenrightbig
+c,d∈A/bracerightbig
+= span/braceleftbig
+(id⊗E)/parenleftbig
+(c⊗ /BD)∆(d)/parenrightbig
+c,d∈A/bracerightbig
+is dense in A⊗B(cf. Lemma 4.2).
+5.Quotients by compact subgroups
+A pair /BZ= (A,∆) consisting of a C∗-algebra Aand a coassociative ∆ ∈Mor(A,A⊗A) such
+that
+span/braceleftbig
+∆(a)( /BD⊗a′)a,a′∈A/bracerightbig
+=A⊗A, (5.1a)
+span/braceleftbig
+(a⊗ /BD)∆(a′)a,a′∈A/bracerightbig
+=A⊗A. (5.1b)
+is usually called a bisimplifiable Hopf C∗-algebra (see e.g. [2]), while in [12] the such objects are
+called “proper C∗-bialgebras with cancellation property.” We will stick to the former te rminology.
+This section is devoted to the proof of the next theorem which is a dir ect generalization of
+the construction in [14][Section 6], [15, Section 1] (cf. also [4]) to the situation where the original
+quantumgroup(orHopfC∗-algebra) /BZisnotcompact. Weincludetheassumptionthat /BZpossessesEXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS 11
+a continuous counit in order to fit the scheme of earlier sections. In case /BZdoes not have a counit
+the all statements of Theorem 5.1 (except (3)) are still true.
+Note also that many of the results of Section 4 are in fact conseque nces of Theorem 5.1, but
+the proofs given in Section 4 are different and independent of the ne xt result.
+Theorem 5.1. Let /BZ= (A,∆)be a bisimplifiable Hopf C∗-algebra with a continuous counit ǫ
+and let /C3= (C,∆C)be a compact quantum group. Let πbe a∗-homomorphism A→Cwhich is
+surjective and satisfies
+(π⊗π)◦∆ = ∆ C◦π.
+LetB=/braceleftbig
+x∈A(id⊗π)∆(x) =x⊗ /BD/bracerightbig
+. Then
+(1)Bis a nondegenerate C∗-subalgebra of A,
+(2)the map α= ∆|Bhas values in M(A⊗B)and belongs to Mor(A,A⊗B),
+(3) (ǫ⊗id)◦α= id,
+(4)for anyb∈Banda∈Awe have (a⊗ /BD)α(b)∈A⊗B.
+Proof.Denoteγ= (id⊗π)∆∈Mor(A,A⊗C). We note the identities
+(γ⊗id)◦γ= (id⊗∆C)◦γ, (5.2a)
+(∆⊗id)◦γ= (id⊗γ)◦∆ (5.2b)
+which follow directly form the coassociativity of ∆. Next we note that
+span/braceleftbig
+γ(a)( /BD⊗c)a∈A, c∈C/bracerightbig
+=A⊗C, (5.3a)
+span/braceleftbig
+(a⊗ /BD)γ(a′)a,a′∈A/bracerightbig
+=A⊗C (5.3b)
+(in particular both sets on the left hand side are contained in A⊗C). Both (5.3a) and (5.3b)
+follow from (5.1) by the surjectivity of π. If /C3has a continuous counit then it can be shown that
+ǫ=ǫC◦πand then (id ⊗ǫC)◦γ= id. This means that γdefines a continuous (right) action of /C3
+onA.
+Lethbe the Haar measure of /C3and let
+E:A∋a/mapsto−→(id⊗h)γ(a).
+Here we use the strictly continuous extension of (id ⊗h) to M(A⊗C). By (5.3a) Ehas values in
+A. Indeed we have that h=vh′for some h′∈C∗andv∈C, so that
+(id⊗h)γ(a) = (id⊗h′)/parenleftbig
+γ(a)( /BD⊗v)/parenrightbig
+is an element of A. Moreover E2=Ebecause
+E2(a) = (id⊗h⊗h)/parenleftbig
+(γ⊗id)γ(a)/parenrightbig
+= (id⊗h⊗h)/parenleftbig
+(id⊗∆C)γ(a)/parenrightbig
+= (id⊗h)γ(a) =E(a).
+In the above calculation we must remember that we are dealing with ex tensions of strictly con-
+tinuous maps to multiplier algebras. For example the identity ( h⊗h)◦∆C=hfollows from the
+definition of the Haar measure, but (id ⊗h⊗h)◦(id⊗∆C) = (id⊗h) is an identity between maps
+on M(A⊗C). We obtain it by extending both sides from A⊗Cto M(A⊗C) by strict continuity.
+Similarly we show that E(A)⊂B. Indeed, denoting by Hthe map C∋c/mapsto→h(c) /BDwe have
+γ/parenleftbig
+E(a)/parenrightbig
+=γ/parenleftbig
+(id⊗h)γ(a)/parenrightbig
+= (id⊗id⊗h)/parenleftbig
+(γ⊗id)γ(a)/parenrightbig
+= (id⊗id⊗h)/parenleftbig
+(id⊗∆C)γ(a)/parenrightbig
+=/parenleftbig
+id⊗[(id⊗h)◦∆C]/parenrightbig
+γ(a)
+= (id⊗H)γ(a) =E(a)⊗ /BD.(5.4)
+As before the identities satisfied by the maps involved in the above ca lculation are extended by
+strict continuity to multiplier algebras.12 PIOTR M. SO/suppress LTAN
+An immediate consequence of (5.4) is that B=E(A). To prove point (1) of Theorem 5.1 we
+note that it follows from (5.3b) that
+span/braceleftbig
+aba∈A, b∈B/bracerightbig
+= span/braceleftbig
+aE(a′)a,a′∈A/bracerightbig
+= span/braceleftbig
+a(id⊗h)γ(a′)a,a′∈A/bracerightbig
+= span/braceleftbig
+(id⊗h)/parenleftbig
+(a⊗ /BD)γ(a′)/parenrightbig
+a,a′∈A/bracerightbig
+is a dense subset of A. In other words the inclusion of BintoAbelongs to Mor( B,A).
+In particular M( B) is the idealizer of Bin M(A) and M( A⊗B) is the idealizer of A⊗Bin
+M(A⊗B) ([10, Proposition 2.3]), i.e.
+M(A⊗B) =/braceleftbig
+X∈M(A⊗A)XY,YX ∈A⊗Bfor allY∈A⊗B/bracerightbig
+. (5.5)
+Let us also note that we have
+A⊗B=/braceleftbig
+Y∈A⊗A(id⊗γ)(Y) =Y⊗ /BD/bracerightbig
+. (5.6)
+Indeed, the inclusion “ ⊂” is clear. For “ ⊃” we use the conditional expectation Ein the same
+way as in the proof of Lemma 4.2. Any Ybelonging to the right hand side of (5.6) satisfies
+(id⊗E)(Y) =Yand it is not difficult to see that A⊗Bis the image of (id ⊗E). Using (5.6) we
+can show that
+M(A⊗B) =/braceleftbig
+X∈M(A⊗A)(id⊗γ)(X) =X⊗ /BD/bracerightbig
+. (5.7)
+The inclusion “ ⊃” follows because if X∈M(A⊗A) satisfies (id ⊗γ)(X) =X⊗ /BDthen for any
+Y∈A⊗Bwe haveXY,YX ∈A⊗B, so that X∈M(A⊗B). Conversely, if X∈M(A⊗B) then
+for anyY∈A⊗Bwe have XY∈A⊗B, so by (5.6) we have
+/parenleftbig
+(id⊗γ)(X)/parenrightbig
+(Y⊗ /BD) =/parenleftbig
+(id⊗γ)(X)/parenrightbig/parenleftbig
+(id⊗γ)(Y)/parenrightbig
+= (id⊗γ)(XY) = (XY⊗ /BD) = (X⊗ /BD)(Y⊗ /BD).
+This means that T= (id⊗γ)(X)−X⊗ /BD∈M(A⊗A⊗B) satisfies
+T(Y⊗ /BD) = 0
+for allY∈A⊗B. It follows that T(a⊗a′⊗b) is zero for all a,a′∈A,b∈B, so that T= 0 and
+we obtain “ ⊂” in (5.7).
+Let us now address point (2) of Theorem 5.1. The map α= ∆/vextendsingle/vextendsingle
+Bis a composition of the
+inclusion of BintoAwith ∆. Therefore it is an element of Mor( B,A⊗A). We want to show
+that it belongs to Mor( B,A⊗B). For this it is enough to demonstrate that the range of αlies in
+M(A⊗B).
+By application of (id ⊗id⊗h) to both sides of (5.2b) we obtain
+∆◦E= (id⊗E)◦∆.
+This shows that the image of αis in the image of (id ⊗E) extended to M( A⊗A). Clearly for any
+X∈A⊗AandY∈A⊗Bwe have
+(id⊗γ)/bracketleftbig/parenleftbig
+(id⊗E)(X)/parenrightbig
+Y/bracketrightbig
+=XY⊗ /BD. (5.8)
+Both sides of this formula are strictly continuous with respect to X, so (5.8) holds also for X∈
+M(A⊗A). In view of (5.7), formula (5.8) (together with the analogous one ( id⊗γ)/bracketleftbig
+Y/parenleftbig
+(id⊗
+E)(X)/parenrightbig/bracketrightbig
+=YX⊗ /BDand its extension) for proves that the range of αis contained in M( A⊗B).
+Point (3) of Theorem 5.1 is a straightforwardconsequence of the d efinition of α. To prove point
+(4) we note that
+/braceleftbig
+(a⊗ /BD)α(b)a∈A, b∈B/bracerightbig
+=/braceleftbig
+(a⊗ /BD)∆/parenleftbig
+E(a′)/parenrightbig
+a,a′∈A/bracerightbig
+=/braceleftbig
+(a⊗ /BD)(id⊗E)∆(a′)a,a′∈A/bracerightbig
+=/braceleftbig
+(id⊗E)/parenleftbig
+(a⊗ /BD)∆(a′)/parenrightbig
+a,a′∈A/bracerightbig
+so that ( a⊗ /BD)α(b)∈A⊗Bfor alla∈A,b∈B. /squareEXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS 13
+Remark 5.2.Note that it follows easily from the last lines of the proof of Theorem 5 .1 that the
+linear span of the set/braceleftbig
+(a⊗ /BD)α(b)a∈A, b∈B/bracerightbig
+is dense in A⊗B(cf. Remark 4.4).
+References
+[1]S. Baaj: R´ epresentation r´ eguli` ere du groupe quantique Eµ(2) de Woronowicz. C. R. Acad. Paris, t.314, S´ erie
+I, (1992), 1021–1026.
+[2]S. Baaj & G. Skandalis: Unitaries muliplicatifs et dualit´ e pour les poiduits croi s´ es de C∗-alg´ ebres. Ann. Sci-
+ent.´Ec. Norm. Sup. 4` e s´ erie, t. 26(1993), 425–488.
+[3]S. Baaj, G. Skandalis & S. Vaes: Non-semi-regular quantum groups coming from number theory .Com-
+mun. Math. Phys. 235no. 1 (2003), 139–167.
+[4]F. Boca: Ergodic actions of compact matrix pseudogroups on C∗-algebras. In Recent Advances in Operator
+algebras. Ast´ erisque 232(1995), pp. 93–109.
+[5]D.A. Ellwood: A new characterization of principal actions. J. Funct. Anal. 173(2000), 49–60.
+[6]R. Fischer: Volle verschr¨ ankte Produkte f¨ ur Quantengruppen und ¨ aqu ivariante KK-Theorie. Ph.D. thesis,
+University of M¨ unster, 2003.
+[7]P. Kasprzak: Rieffel deformation via crossed products. J. Funct. Anal. 257no. 5 (2009), 1288–1332.
+[8]J. Kustermans: Locally compact quantum groups in the universal setting. Int. J. Math. 12(2001), 289–338.
+[9]J. Kustermans & S. Vaes: Locally compact quantum groups. Ann. Scient. ´Ec. Norm. Sup. 4` e s´ erie, t. 33
+(2000), 837–934.
+[10]E.C. Lance: Hilbert C∗-modules, a toolkit for operator algebraists. London Mathematical Society Lecture
+Note Series 210, Cambridge University press 1995.
+[11]M.B. Landstad: Duality theory for covariant systems. Trans. Amer. Math. Soc. 248, no. 2 (1979), 223–267.
+[12]T. Masuda, Y. Nakagami & S.L. Woronowicz: A C∗-algebraic framework for quantum groups. Int. J. Math.
+14no. 9 (2003) 903–1001.
+[13]G.K. Pedersen: C∗-algebras and their automorphism groups. Academic Press 1979.
+[14]P. Podle ´s:Quantum spheres. Lett. Math. Phys. 14(1987), 193–202.
+[15]P. Podle ´s:Symmetries of quantum spaces. Subgroups and quotient space s of quantum SU(2) and SO(3)
+groups.Commun. Math. Phys. 170(1995), 1–20.
+[16]W. Pusz & P.M. So/suppress ltan: Analysis on a homogeneous space of a quantum group. Preprint , University of
+Warsaw. arxiv:math.OA/0509610v3.
+[17]W. Pusz & P.M. So/suppress ltan: On some low dimensional quantum groups. Preprint, Universi ty of Warsaw, to
+appear in Quantum symmetries in noncommutative geometry P.M. Hajac editor. arXiv:0712.3369v1.
+[18]P.M. So/suppress ltan: New quantum “ az+b” groups. Rev. Math. Phys. 17no. 3 (2005), 313–364.
+[19]P.M. So/suppress ltan & S.L. Woronowicz: From multiplicative unitaries to quantum groups II. J. Funct. Anal. 252
+no. 1 (2007), 42–67.
+[20]S. Vaes: A new approach to induction and imprimitivity results. J. Func. Anal. 229(2005), 317–374.
+[21]S.L. Woronowicz: Quantum E(2)-group and its Pontryagin dual. Lett. Math. Phys. 23(1991), 251–263.
+[22]S.L. Woronowicz: Operator equalities related to quantum E(2) group. Commun. Math. Phys. 144 (1992),
+417–428.
+[23]S.L. Woronowicz: Quantum SU(2) and E(2) Groups. Contraction Procedure. Commun. Math. Phys. 149
+(1992), 637–652.
+[24]S.L. Woronowicz: Unbounded elements affiliated with C∗-algebras and non-compact quantum groups. Com-
+mun. Math. Phys. 136(1991), 399–432.
+[25]S.L. Woronowicz: C∗-algebras generated by unbounded elements. Rev. Math. Phys. 7, no. 3 (1995), 481–521.
+[26]S.L. Woronowicz: Compact quantum groups. In Sym´ etries Quantiques, les Houches, Session LXIV 1995,
+Elsevier 1998, pp. 845–884.
+[27]S.L. Woronowicz: Quantum ‘ az+b’ group on complex plane. Int. J. Math. 12, no. 4 (2001), 461–503.
+[28]S.L. Woronowicz: Haar weight on some quantum groups. Group 24: Physical and mathematical aspects
+of symmetries, Proceedings of the 24th International Collo quium on Group Theoretical Methods in Physics
+Paris, 15–20 July 2002, Institute of Physics, Conference Series Number 173, pp. 763–772.
+[29]S.L. Woronowicz & K. Napi ´orkowski: Operator theory in the C∗-algebra framework. Rep. Math. Phys. 31
+(1992), 353–371.
+Institute of Mathematics, Polish Academy of Sciences
+and
+Department of Mathematical Methods in Physics, Faculty of P hysics, University of Warsaw
+E-mail address :piotr.soltan@fuw.edu.pl
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+arXiv:1001.0521v1  [math.OA]  4 Jan 2010WHEN IS A QUANTUM SPACE NOT A GROUP?
+PIOTR M. SOŁTAN
+Abstract. We give a survey of techniques from quantum group theory whic h can be used to
+show that some quantum spaces (objects of the category dual t o the category of C∗-algebras)
+do not admit any quantum group structure. We also provide a nu mber of examples which
+include some very well known quantum spaces. Our tools inclu de several purely quantum group
+theoretical results as well as study of existence of charact ers and traces on C∗-algebras describing
+the considered quantum spaces as well as properties such as n uclearity.
+1.Introduction
+LetXbe a topological space. One can easily turn Xinto an associative topological semigroup.
+A possible definition of multiplication is x·y=yfor allx,y∈X. This is clearly not a group
+(unlessXconsists of a single point). One is therefore lead to a more re fined question whether X
+can be turned into a topological group.
+It is not difficult to find topological spaces which cannot be gi ven a structure of a topological
+group at all. As an example of this phenomenon consider the in terval[0,1]. Clearly this topological
+space cannot be a topological group because there is no homeo morphism of [0,1]onto itself carrying
+an end point onto an interior point (the endpoints have neigh borhoods which are connected after
+the endpoint is removed and interior points do not have such n eighborhoods) and there would
+have been one if [0,1]were a topological group. The same argument shows that no man ifold with
+boundary can be endowed with a structure of a topological gro up.
+One is forced apply much more sophisticated tools to prove, f or example, that the two-sphere/CB2is not a topological group with its usual topology. Of course , the arguments produced above no
+longer work, as /CB2is a homogeneous space. Still, using some results about vect or bundles or by
+noticing that the cohomology ring of /CB2does not admit a Hopf algebra structure, the conclusion
+that /CB2is not a group can be reached ([8, Section 3.C]).
+In this paper we want to address similar questions, but inste ad of topological spaces we want to
+consider objects of noncommutative topology orquantum spaces, i.e. objects of the category dual
+to the category of C∗-algebras ([4, 24]). Perhaps not unexpectedly the tools we w ill use to show
+that some well known quantum spaces do not admit a group struc ture (they are not quantum
+groups as defined in Section 2) are of completely different nature tha n those known from classical
+topology.
+Let us briefly describe the contents of the paper. In Section 2 we give definitions of objects of
+our study such as quantum spaces, quantum semigroups and com pact quantum groups. We also
+provide some basic examples and introduce standard termino logy. Section 3 is devoted to a survey
+of results from the theory of compact quantum groups which we need in following sections. In
+particular we describe the constructions of the reduced anduniversal versions of a given compact
+quantum group and define Woronowicz characters . The next section introduces the tools used to
+show that some well known quantum spaces do not admit a compac t quantum group structure.
+The results are tailored to suit applications and proofs of s ome known facts (e.g. from [1]) are
+considerably simplified. Finally, in Section 5 we describe i n detail examples of quantum spaces
+not admitting any compact quantum group structure. At the en d of this section we discuss some
+partial results about the quantum disk .
+Research partially supported by Polish government grant no . N201 1770 33.
+12 PIOTR M. SOŁTAN
+2.Compact quantum semigroups and compact quantum groups
+Let us first define compact quantum spaces. The category of compact quantum spaces is by
+definition the category dual to the category of unital C∗-algebras with unital ∗-homomorphisms
+as morphisms (the compactness of our quantum spaces is encod ed in the fact that all considered
+C∗-algebras will have a unit). The choice to restrict attentio n to compact quantum spaces is
+motivated mainly by the amazingly rich theory of compact quantum groups as defined below.
+Definition 2.1.
+(1) A compact quantum semigroup is a pair /BZ= (A,∆), whereAis a unital C∗-algebra and
+∆is a morphism A→A⊗A(minimal tensor product of C∗-algebras) such that
+(id⊗∆)◦∆ = (∆⊗id)◦∆.
+The morphism ∆is called the comultiplication of /BZ.
+(2) A compact quantum group is a compact quantum semigroup /BZ= (A,∆)such that the
+linear spans of the sets
+/braceleftbig
+∆(a)( /BD⊗b)a,b∈A/bracerightbig
+and/braceleftbig
+(a⊗ /BD)∆(b)a,b∈A/bracerightbig
+are dense in A⊗A.
+Example 2.2. LetGbe a compact associative semigroup and let A= C(G). We define ∆ :A→
+A⊗A= C(G×G)by
+(∆f)(s,t) =f(st).
+Then /BZ= (A,∆)is a compact quantum semigroup as in Definition 2.1(1). The de nsity conditions
+from Definition 2.1(2) are equivalent to cancellation laws:
+/parenleftbig
+s·t=s′·t/parenrightbig
+=⇒/parenleftbig
+s=s′/parenrightbig
+,
+/parenleftbig
+s·t=s·t′/parenrightbig
+=⇒/parenleftbig
+t=t′/parenrightbig
+,(1)
+so /BZ= (A,∆)is a compact quantum group if and only id the implications (1) hold for any
+s,s′,t,t′∈G, i.e. precisely when Gis a compact group.
+It is a fact that any compact quantum group /BZ= (A,∆)withAcommutative is necessarily of
+the form described in Example 2.2.
+Example 2.3. LetΓbe a discrete group and let A= C∗(Γ). SinceAis the completion of ℓ1(Γ)in
+the maximal C∗-norm, there is a copy of Γinside the unitary group of A. Moreover Ais generated
+by these elements. One can easily see that there exists a uniq ue∆ :A→A⊗Asuch that
+∆(γ) =γ⊗γ
+for allγ∈Γ. It is clear that /BZ= (A,∆)is a compact quantum group.
+The comultiplication of the compact quantum group /BZ= (A,∆)described in Example 2.3 is
+cocommutative, i.e.∆ =σ◦∆, (whereσ:A⊗A→A⊗Ais the flip). One is tempted to write that
+all cocommutative compact quantum groups are of the form des cribed in Example 2.3. However
+this statement is not true. It is true for universal compact quantum groups which we will define
+in Subsection 3.4.
+There are numerous other examples of compact quantum groups in the literature for which we
+refer the reader to e.g. [23, 22, 7].
+3.Additional structure
+Throughout this section we let /BZ= (A,∆)be a compact quantum group. The C∗-algebra
+Acarries very rich additional structure. As we will see in Sec tion 5, this fact may be used to
+decide whether a given compact quantum space can be endowed w ith a compact quantum group
+structure.
+The results about compact quantum groups formulated in this section are all covered in [25]
+except for the material of Subsection 3.4 which can be found i n [1, Section 3].WHEN IS A QUANTUM SPACE NOT A GROUP? 3
+3.1.Hopf algebra. The first result we want to state relates compact quantum grou ps to Hopf ∗-
+algebras. For the theory of Hopf algebras we refer to [21]. Ho pf∗-algebras and their generalizations
+are discussed e.g. in [15].
+Definition 3.1. Ann-dimensional unitary representation of /BZis a unitary matrix
+u=
+u1,1···u1,n
+.........
+un,1···un,n
+∈Mn( /BV)⊗A=Mn(A)
+such that
+(2) ∆(ui,j) =n/summationdisplay
+k=1ui,k⊗uk,j.
+Elements {ui,j}i,j=1,...,nare called the matrix elements of u.
+The concept of a finite dimensional unitary representation o f a compact quantum group is a
+straightforward generalization of the notion of a finite dim ensional unitary representation of a
+compact group. It is easy to see that in the case presented in E xample 2.2 representations of the
+compact quantum group are the same objects as ordinary repre sentations of the group.
+Theorem 3.2 (S.L. Woronowicz) .Let /BZ= (A,∆)be a compact quantum group and let Abe the
+linear span of matrix elements of all finite dimensional unita ry representations of /BZ. Then
+(1)∆/vextendsingle/vextendsingle
+A(A)⊂A⊗algA,
+(2)Ais dense unital ∗-subalgebra in A,
+(3)if∆Ais the restriction of ∆toAthen(A,∆A)is a Hopf ∗-algebra. In particular there
+exist antipode κ:A→Aand counit e:A→ /BV.
+(4)Ais the unique Hopf ∗-algebra which can be embedded in (A,∆)as a dense ∗-subalgebra
+with the same comultiplication.1
+Let /BZ= (A,∆)be a compact quantum group. The dense Hopf ∗-algebra AofAis called the
+Hopf∗-algebra associated to /BZ.In case of a classical group G(as in Example 2.2) the associated
+Hopf∗-algebra is the algebra of regular functions on G. In the case described in Example 2.3 the
+associated Hopf ∗-algebra is the group algebra of Γ.
+The antipode and counit of Amay fail to have continuous extensions to A. We will return to
+the case when eextends to a character of Ain Subsection 4.1.
+A quantum group /BZ= (A,∆)with the property that the antipode of Aextends to a continuous
+linear map A→Ais called a quantum group of Kac type. The situation when κhas a continuous
+extension to Acan be characterized in many different ways. We will give some equivalent conditions
+for this phenomenon to hold in Subsection 3.2.
+3.2.Haar measure.
+Theorem 3.3 (S.L. Woronowicz) .Let /BZ= (A,∆)be a compact quantum group. Then there exists
+a unique state honAsuch that
+(id⊗h)∆(a) = (h⊗id)∆(a) =h(a) /BD
+for alla∈A.
+The statehintroduced in Theorem 3.3 is called the Haar measure of /BZ. Clearly, in the case
+from Example 2.2, the state hcorresponds to integration with respect to the normalized H aar
+measure on G. In Example 2.3 the Haar measure is the well known von Neumann trace. A more
+detailed analysis of the latter example shows that the Haar m easure of a compact quantum group
+need not be a faithful state. Note that in the two examples of H aar measures we just discussed
+the statehis a trace, i.e. h(ab) =h(ba)for alla,b∈A. One can ask if the Haar measure hof a
+compact quantum group /BZ= (A,∆)is always a trace. This is not the case. The following theorem
+describes this situation.
+1The reference for the last statement of Theorem 3.2 is [1, The orem 5.1].4 PIOTR M. SOŁTAN
+Theorem 3.4 (S.L. Woronowicz) .Let /BZ= (A,∆). Lethbe the Haar measure of /BZand letκbe
+the antipode of the Hopf ∗-algebra A⊂A. Then the following conditions are equivalent:
+(1) /BZis of Kac type,
+(2)his a trace,
+(3)κhas a bounded extension to A,
+(4)κ2= id,
+(5)κis a∗-antiautomorphism of A.
+3.3.Reduced quantum group. As we noted in Subsection 3.2 the Haar measure of a compact
+quantum group /BZ= (A,∆)may not be faithful. However the following theorem (essenti ally due
+to S.L. Woronowicz) shows that we can always pass to a “new ver sion” of /BZwhich has a faithful
+Haar measure.
+Theorem 3.5. Let /BZ= (A,∆)be a compact quantum group with Haar measure h. Then the left
+kernel ofh
+J=/braceleftbig
+a∈Ah(a∗a) = 0/bracerightbig
+is a two-sided ideal in A. LetArbe the quotient A/Jand letρr:A→Arbe the quotient map.
+Then
+(1)there exists a unique ∆r:Ar→Ar⊗Arsuch that the diagram
+A
+ρr
+/d15/d15∆/d47/d47A⊗A
+ρr⊗ρr
+/d15/d15
+Ar∆r/d47/d47Ar⊗Ar
+is commutative,
+(2) /BZr= (Ar,∆r)is a compact quantum group,
+(3)ρris injective on A⊂Aandρr(A)is the Hopf ∗-algebra associated to /BZr.
+Let /BZ= (A,∆)be a compact quantum group with associated Hopf ∗-algebra A. The compact
+quantum group /BZris called the reduced version of /BZ. In the situation from Example 2.2 we have
+Ar=Abecause the Haar measure is faithful. On the other hand in cas e from Example 2.3 where
+A= C∗(Γ)for a discrete group Γ, we haveAr= C∗
+r(Γ)— the reduced group C∗-algebra of Γ.
+One can also adopt a different point of view and treat AandAras completions of Awith
+respect to different C∗-norms for which ∆Ais continuous.
+3.4.Universal quantum group.
+Theorem 3.6. Let /BZ= (A,∆)be a compact quantum group with associated Hopf ∗-algebra A.
+There exists the enveloping C∗-algebraAuofA. Letρu:Au→Abe the canonical epimorphism.
+Then
+(1)there exists a unique ∆u:Au→Au⊗Ausuch that the diagram
+Au
+ρu
+/d15/d15∆u/d47/d47A⊗A
+ρu⊗ρu
+/d15/d15
+A∆/d47/d47A⊗A
+is commutative,
+(2) /BZu= (Au,∆u)is a compact quantum group,
+The compact quantum group /BZudescribed in Theorem 3.6 is called the universal version of /BZ.
+SinceAuis defined as the completion of Awith respect to a maximal possible C∗-norm, it is clear
+that the Hopf ∗-algebra associated to /BZuisA.
+IfAis commutative (i.e. in the situation of Example 2.2) we alwa ys haveAu=A=Ar.
+Moreover if Aris commutative then so is Aand consequently Au=A=Ar.WHEN IS A QUANTUM SPACE NOT A GROUP? 5
+3.5.Woronowicz characters. Let /BZ= (A,∆)be a compact quantum group and let φandψbe
+continuous functionals on A. Then the functional (φ⊗ψ)◦∆is called the convolution ofφandψ
+and is denoted by φ∗ψ. Similarly if a∈Athen we can define leftandright convolutions φ∗a
+anda∗ψofawithφandψrespectively by
+φ∗a= (id⊗φ)∆(a)anda∗ψ= (ψ⊗id)∆(a).
+These definitions are straightforward generalizations of t he notion of convolution of measures and
+measures and continuous functions on a compact group (cf. Ex ample 2.2).
+IfAis the Hopf ∗-algebra associated to /BZandφandψare linear functionals on Athen the
+we can use the same formulas to define φ∗ψandφ∗a,a∗ψfora∈A.
+Theorem 3.7 (S.L. Woronowicz) .Let /BZ= (A,∆)be a compact quantum group with associated
+Hopf∗-algebra A. Then there exists a unique family (fz)z∈ /BVof non-zero multiplicative functionals
+onAsuch that
+(1)for anya∈Athe function z/mapsto→fz(a)is entire holomorphic,
+(2)f0=eandfz1∗fz2=fz1+z2for allz1,z2∈ /BV,
+(3)for anyz∈ /BVanda∈A
+fz/parenleftbig
+κ(a)/parenrightbig
+=f−z(a)andfz(a∗) =f−z(a)
+(4)for anya∈Awe haveκ2(a) =f−1∗a∗f1.
+It is clear from Theorem 3.7(3) that (fit)t∈ /CAis a one-parameter group (under the operation of
+convolution) of ∗-characters of A. They extend to characters of Auwhich we call Woronowicz
+characters. The whole family (fz)z∈ /BVis the family of modular functionals of /BZ.
+It is important to note that the family (fz)z∈ /BVmay be trivial in the sense that fz=efor allz.
+Indeed it is always the case for Kac algebras:
+Theorem 3.8. Let /BZ= (A,∆)be a compact quantum group and (fz)z∈ /BVbe the family of modular
+functionals of /BZ. Then the following are equivalent:
+(1)fz=efor allz,
+(2) /BZis of Kac type.
+Theorem 3.8 implies, in particular, that commutative and co commutative examples (Examples
+2.2 and 2.3) have trivial families of modular functionals.
+3.6.The modular group. Let /BZ= (A,∆)be a compact quantum group with associated Hopf
+∗-algebra Aand modular functionals (fz)z∈ /BV. The formula
+σt(a) =fit∗a∗fit
+fort∈ /CAanda∈Adefines a one parameter group of automorphisms of A. This group is called
+themodular group of /BZ. Let us note that the modular group has a continuous extensio n toAu,
+because all Woronowicz characters extend to characters of Auand it is easy to see that the family
+(fit)t∈ /CAis continuous on Au. A theorem of S.L. Woronowicz ([25, Theorem 2.6]) asserts th at the
+one parameter group (σt)t∈ /CAis also continuous on Arand is intimately connected with the failure
+of the Haar measure hof /BZto be a trace ( his aσ-KMS state).
+4.Some tools
+4.1.Continuity of Woronowicz characters. Let /BZ= (A,∆)be a compact quantum group and
+letGbe the set of non-zero multiplicative functionals on A. ClearlyGis a weak∗compact subset
+of the unit sphere in A∗and convolution of functionals defines on Ga structure of a compact
+associative semigroup. There are many ways to see that Gis in fact a compact group. Indeed, let
+γ:A→C(G)be the Gelfand transform. Clearly γis a surjective ∗-homomorphism and if ∆Gis
+the map C(G)→C(G×G)given by
+(∆G(f))(φ,ψ) =f(φ∗ψ)6 PIOTR M. SOŁTAN
+then(γ⊗γ)◦∆ = ∆ G◦γby the very definition of convolution product (in other words γis a
+quantum group morphism ). Therefore we have the density of linear spans of
+/braceleftbig
+∆G(f)( /BD⊗g)f,g∈C(G)/bracerightbig
+and/braceleftbig
+(f⊗ /BD)∆(g)f,g∈C(G)/bracerightbig
+inC(G)⊗C(G)which is equivalent to cancellation laws in G(cf. Example 2.2). It follows that G
+is a compact group. In particular it has the unit element φ0.
+The next thing we want to check is that φ0is the extension to Aof the counit edefined on
+the Hopf ∗-algebra Aassociated with /BZ. This follows by considering a finite dimensional unitary
+representation u∈Mn( /BV)⊗Aand the unitary matrix U= (id⊗φ0)u. It follows from (2) that
+U=U2, which for a unitary matrix means U= /BD. This means that φ0(ui,j) =δi,jfor any finite
+dimensional unitary representation u= (ui,j)of /BZ. The counit eofAhas the same values on
+matrix elements of representations, and Ais spanned by these matrix elements. Therefore φ0=e
+onA.
+Theorem 4.1. Let /BZ= (A,∆)be a compact quantum group such that the C∗-algebraArpossesses
+a character. Then all Woronowicz characters are continuous on A.
+Proof. From the discussion preceding statement of the theorem we kn ow that the nonempty set
+of characters of Aris a compact group whose unit is the extension to Arof the counit eof the
+Hopf∗-algebra Aassociated with /BZr. We will still write eto denote this extension.
+Now for any a∈Aandt∈ /CAwe have
+e/parenleftbig
+σt(a)/parenrightbig
+= (fit⊗e⊗fit)(∆A⊗id)∆ A(a)
+= (fit⊗fit)∆A(a) = (fit∗fit)(a) =f2it(a)
+and we know that both eand(σt)t∈ /CAare continuous on Ar. This means that Woronowicz charac-
+ters(fit)t∈ /CAalso have continuous extensions to Ar. SinceAris the image of Aunderρr(cf. Sub-
+section 3.3) which is the identity on A, all Woronowicz characters also extend to characters of
+A. /square
+It has to be noted that in fact a statement much stronger than T heorem 4.1 is true. Namely,
+existence of a character on Arimplies that ρuandρrare isomorphisms, so in particular, Ar=
+A=Au([1]) and Woronowicz characters must extend to A. However, we will make use solely of
+the weaker statement presented above.
+4.2.Other results. We will need two more results that will help us disprove exist ence of compact
+quantum group structure on some compact quantum spaces. The first one follows from a very
+deep theorem of Bédos, Murphy and Tuset ([2, Theorem 1.1]).
+Theorem 4.2. Let /BZ= (A,∆)be a compact quantum group of Kac type with associated Hopf
+∗-algebra A. Then the following are equivalent:
+(1)theC∗-algebraAris nuclear,
+(2)the counit of Ais continuous on Ar.
+Recall that a C∗-algebraAisnuclear if for any C∗-algebraBthe algebraic tensor product
+A⊗algBadmits a unique C∗-norm. This property can be also characterized in many differ ent
+ways, but all we will need is the fact that a crossed product of a commutative C∗-algebra by an
+action of a commutative group is nuclear ([18, Proposition 2 .1.2]).
+The next fact we will need is the following:
+Theorem 4.3 ([19, Remark A.2]) .Let /BZ= (A,∆)be a compact quantum group such that A
+admits a faithful family of tracial states. Then /BZis of Kac type.
+5.Examples
+In this section we will give examples of compact quantum spac es which do not admit any
+compact quantum group structure. In each case we will use the results collected in Section 4 to
+prove non-existence of such a structure. The quantum spaces under consideration are all discussed
+in the survey article [5], where further references can be fo und.WHEN IS A QUANTUM SPACE NOT A GROUP? 7
+Example 5.1 (The quantum torus) .Let us fixθ∈]0,1[. Thequantum two-torus /CC2
+θis the quantum
+space corresponding to the rotation C∗-algebraAθ, i.e. the universal C∗-algebra generated by two
+unitary elements uandvsatisfying the relation uv= e2πiθvu([16]). This C∗-algebra is nuclear
+(indeed,Aθ= C( /CC)⋊ /CI, where the action is by rotation by 2πθ). Moreover it possesses a faithful
+trace (unique if θis irrational).
+Let us assume that there exists ∆ :Aθ→Aθ⊗Aθsuch that /BZ= (Aθ,∆)were a compact
+quantum group. We know from theorem 4.3 that the Haar measure of /BZmust then be a trace and/BZis of Kac type. Moreover, since Aθis nuclear, we know from Theorem 4.2 that (Aθ)rmust have
+a character (because then (Aθ)ris also nuclear). But if this were the case then Aθwould admit
+a character, since (Aθ)ris a quotient of Aθ(in fact, for irrational θthe algebra Aθis simple, so
+there are no proper quotients of Aθ) and we know that Aθdoes not admit any characters.
+The same reasoning can be applied to show that higher dimensi onal quantum tori ([17]) do
+not admit a compact quantum group structure for nontrivial d eformation parameters. If the
+deformation parameters are trivial (that means θ= 0for the two-torus) the resulting C∗-algebra
+Aθis just the algebra of continuous function on a torus and, as s uch, carries a compact quantum
+group structure.
+It should be noted that the quantum two-torus can be made into a “part” of a compact quantum
+group. This was done by P.M. Hajac and T. Masuda in [7].
+There is one more interesting remark. The theory of quantum g roups on operator algebra
+level has its version in the language of von Neumann algebras (see e.g. [10]). The passage to von
+Neumann algebras is achieved by taking weak closure in the GN S representation defined by the
+Haar measure. If /BZ= (Aθ,∆)were a compact quantum group for some irrational θthen the
+unique tracial state of Aθwould be its Haar measure (as we said earlier this follows fro m Theorem
+4.3 and Theorem 3.4). Therefore the resulting von Neumann al gebra would be the hyperfinite
+factor of type II1. We already know that the quantum torus is not a quantum group . However
+the “algebra of measurable essentially bounded functions” on the quantum torus is the same as
+that of “measurable functions” on many compact quantum grou ps. Indeed we get the same von
+Neumann algebra when we start with C∗(Γ)for any amenable i.c.c. discrete group Γ.
+Example 5.2 (Batteli-Elliott-Evans-Kishimoto quantum two-spheres) .The algebra of continuous
+functions on a Bratteli-Elliott-Evans-Kishimoto quantum two-sphere is by definition the fixed point
+subalgebra CθofAθconsidered in Example 5.1 under the action of /CI2sendinguandvto their
+adjoints ([3, 5]). For θ= 0we haveCθ= C( /CB2)and this C∗-algebra carries no compact quantum
+group structure because /CB2is not a topological group. For θ∈]0,1[the argument used to show
+that /CC2
+θis not a compact quantum group works perfectly well for the BE EK quantum two-spheres.
+All we need is to know that Cθis a nuclear C∗-algebra with a faithful trace which admits no
+characters.
+Example 5.3 (Standard Podleś quantum sphere) .In [13] Piotr Podleś defined and studied a
+class of quantum spaces which later came to be known as Podleś (quantum) spheres. These are
+compact quantum spaces endowed with an action of the quantum SU(2) group ([23]) mimicking
+the standard action of SU(2) on /CB2. Podleś found all such objects and showed that they form
+a family ( /CB2
+q,c)labeled by a parameter c∈[0,∞](the parameter qis related to the particular
+quantum SU(2) group). Only for c= 0can /CB2
+q,cbe considered as a quotient homogeneous space
+the quantum SU(2) and this quantum sphere is referred to as the standard Podleś quantum sphere
+(cf. [14]).
+The algebra of continuous functions on /CB2
+q,0is isomorphic to K+, i.e. the minimal unitization
+of the algebra of compact operators on a separable Hilbert sp ace. This C∗-algebra has a unique
+proper ideal which is maximal. This makes it easy to see that i f there existed ∆ :K+→K+⊗K+
+such that /BZ= (K+,∆)were a compact quantum group then the Haar measure hof /BZwould have
+to be faithful. Otherwise the left kernel Jofhwould be either {0}or so big, that the quotient
+Ar=K+/Jwould be commutative (equal to /BV). But we already said in Subsection 3.4 that
+compact quantum groups described by commutative C∗-algebras are automatically universal. It
+follows that /BZmust be equal to its reduced version. Note that on K+there exists a nontrivial
+character.8 PIOTR M. SOŁTAN
+There are no faithful traces on K+, so that the family of modular functionals must be nontrivia l.
+But since K+has a character, the Woronowicz characters must be continuo us and nontrivial on
+K+(Theorem 4.1). However there is only one character on K+which shows that existence of
+∆is impossible.
+Example 5.4 (Natsume-Olsen quantum two-spheres) .In [11] a family of quantum spaces was
+introduced which we call the family of Natsume-Olsen quantu m two-spheres. The family is
+parametrized by a parameter t∈/bracketleftbig
+0,1
+2/bracketleftbig
+. TheC∗-algebraBtcorresponding to tis the univer-
+salC∗-algebra generated by two elements zandζsatisfying the following relations:
+ζ∗ζ+z2= /BD=ζζ∗+(tζζ∗+z)2,
+ζz−zζ=tζ( /BD−z2).
+Fort= 0we haveBt= C( /CB2)and /CB2is not a compact group.
+In caset >0let us suppose that /BZ= (Bt,∆)is a compact quantum group for some comul-
+tiplication ∆ :Bt→Bt⊗Bt. Then we note that Bthas an ideal Iisomorphic to C( /CC)⊗K
+such thatBt/I= /BV2([11]). This can be used to show that the Haar measure hof /BZcannot
+be a trace. Indeed, if it were then a simple argument shows tha t the left kernel Jofhwould
+contain I. Then the quotient (Bt)r=Bt/Jwould be commutative and we would arrive at a
+contradiction as e.g. in Example 5.3. Then one must do a littl e work to show that (Bt)radmits
+a character ([20]). By Theorem 4.1 all Woronowicz character s must be continuous on Btand by
+Theorems 3.8 and 3.4 and the fact that his not a trace we see that Btmust have a nontrivial
+one-parameter continuous family of characters. This contr adicts a simple fact that the character
+space ofBtconsists of two points. It follows that Natsume-Olsen quant um spheres do not admit
+a compact quantum group structure.
+6.On the quantum disk
+Let us now describe some partial results on the question whet her the quantum disk could have
+a quantum group structure. The quantum disk is the compact qu antum space described by the
+Toeplitz algebra T([9]). We do not have a proof that the quantum disk does not adm it a compact
+quantum group structure, but we can go quite far along the lin es of our previous examples.
+If we assume that /BZ= (T,∆)is a compact quantum group then the Haar measure hof /BZmust
+be faithful. This is because Kis an essential ideal of T, and the left kernel Jofhwould have
+a non zero intersection with Kwhich by simplicity of Kwould be all of K. In particular J
+would contain Kand the quotient would be commutative. As we stated in Subsec tion 3.4 this
+would imply that the C∗-algebra Tis commutative. In particular hcannot be a trace since there
+is no faithful trace on T.
+Ashis faithful, we see that /BZ= /BZrand since Tadmits a character, all Woronowicz characters
+must be continuous on T. The character space of the C∗-algebra Tis homeomorphic to /CCand
+carries a structure of a compact group. Let us first see that th is is exactly the set of Woronowicz
+characters with the group structure of quotient of /CA(recall that fit∗fis=fi(t+s)). Indeed, let G
+be the compact group of characters of Tand letGWbe the subgroup consisting of Woronowicz
+characters. GWis nontrivial (because his not tracial) and it is a connected subset of G. Therefore
+GWcontains the unit of Gas an interior point (cf. Section 1) and thus a neighborhood o f the unit
+ofG. ButGis connected, so it is algebraically generated by any open ne ighborhood of its unit.
+It follows that GW=G.
+The Gelfand transform γ:T →C( /CC)is a morphism of compact quantum groups, where
+the comultiplication on C( /CC)comes from group structure of /CC=G=GW. Using some known
+results about Toeplitz algebras ([6]) one can show that ther e is an isometry xwhich generates
+Tand is mapped to a group like generator zofC( /CC)under the Gelfand transform (this means
+that the comultiplication on C( /CC)sendsztoz⊗z). It follows that ∆(x) =x⊗x+X, where
+X∈kerγ⊗γ=T ⊗K+K⊗ T. Now, ifXcould be shown to be zero we would be done,
+since in that case xwould be group like and by [25, Theorem 2.6(2)] would belong t o the Hopf
+∗-algebra associated to /BZ. However group like elements of Hopf algebras must be invert ible and x
+is not. Unfortunately, although we can show a number of prope rties ofX, as for now the propertyWHEN IS A QUANTUM SPACE NOT A GROUP? 9
+thatX= 0is not within our reach. In fact, using techniques from the th eory of Hopf-Galois
+extensions, one could show that the weaker property that (id⊗γ)(X) = 0 would be enough to
+disprove existence of a compact quantum group structure on t he quantum disk.
+Acknowledgments
+Research partially supported by Polish government grant no . N201 1770 33. This paper is
+based on a lecture delivered at the 19thInternational Conference on Banach Algebras held at
+Będlewo, July 17–27, 2009. The support for the meeting by the Polish Academy of Sciences,
+the European Science Foundation, and the Faculty of Mathema tics and Computer Science of the
+Adam Mickiewicz University at Poznań is gratefully acknowl edged.
+References
+[1] E. Bédos, G.J. Murphy & L. Tuset: Co-amenability of compa ct quantum groups, J. Geom. Phys. 40 (2001),
+no. 2, 130–153.
+[2] E. Bédos, G.J. Murphy & L. Tuset: Amenability and co-amen ability of algebraic quantum groups. II.
+J. Funct. Anal. 201 (2003), no. 2, 303–340.
+[3] O. Bratteli, G.A. Elliott, D.E. Evans & A. Kishimoto: Non -commutative spheres I, Int. J. Math. 3 (1991),
+139–166.
+[4] A. Connes: Noncommutative geometry, Academic Press 199 4.
+[5] L. Dąbrowski: The garden of quantum spheres, in: Noncommutative geometry and quantum groups (Warsaw,
+2001), Banach Center Publ. 61, Polish Acad. Sci., Warsaw, 2003, 37 –48,
+[6] R.G. Douglas: Banach algebra techniques in operator the ory, Academic Press 1972.
+[7] P.M. Hajac & T. Masuda: Quantum double-torus, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 6,
+553–558.
+[8] A. Hatcher: Algebraic Topology, Cambridge University P ress 2002.
+[9] S. Klimek & A. Leśniewski: Quantum Riemann surfaces I. Th e unit disk, Commun. Math. Phys. 146 (1992),
+103–122.
+[10] J. Kustermans & S. Vaes: Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand.
+92 (2003), no. 1, 68–92.
+[11] T. Natsume & C.L. Olsen: A new family of noncommutative 2-spheres, J. Func. Anal. 202 (2003), 363–391.
+[12] G.K. Pedersen: C∗-algebras and their automorphism groups, Academic Press 19 79.
+[13] P. Podleś: Quantum spheres, Lett. Math. Phys. 14 (1987), no. 3, 193–202.
+[14] P. Podleś: Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) andSO(3) groups,
+Commun. Math. Phys. 170 (1995), no. 1, 1–20.
+[15] P. Podleś & E. Müller: Introduction to quantum groups, Rev. Math. Phys. 10 (1998), no. 4, 511–551.
+[16] M.A. Rieffel: C∗-algebras associated to irrational rotations, Pac. J. Math. 93 (1981) 415–429.
+[17] M.A. Rieffel: Non-commutative tori — a case study of non- commutative differentiable manifolds, Contempo-
+rary Mathematics 105 (1990), 191–211.
+[18] M. Rørdam: Classification of Nuclear C∗-Algebras, Encyclopedia of Mathematical Sciences Vol. 126, editors
+J. Cuntz & V.F.R. Jones, Springer Verlag 2002.
+[19] P.M. Sołtan: Quantum Bohr compactification, Il. J. Math. 49 (2005), no. 4, 1245–1270.
+[20] P.M. Sołtan: Quantum spaces without group structure, t o appear in Proc. Amer. Math. Soc. Archive:
+arXiv:0904.3019v2 [math.OA].
+[21] M.E. Sweedler: Hopf algebras, W.A. Benjamin Inc. 1969.
+[22] A. Van Daele & S. Wang: Universal quantum groups, Int. J. Math. 7 (1996), no. 2, 255–263.
+[23] S.L. Woronowicz: Twisted SU(2) group. An example of noncommutative differential calculus, Publ. RIMS,
+Kyoto University 23 (1987), 117–181.
+[24] S.L. Woronowicz: Unbounded elements associated with C∗-algebras and non-compact quantum groups, Com-
+mun. Math. Phys. 136 (1991), 399–432.
+[25] S.L. Woronowicz: Compact quantum groups, in: Symétries Quantiques, les Houches, Session LXIV 1995 ,
+Elsevier 1998, 845–884.
+Institute of Mathematics of the Polish Academy of Sciences, and, Department of Mathematical
+Methods in Physics, Faculty of Physics, University of Warsa w
+E-mail address :piotr.soltan@fuw.edu.pl
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+arXiv:1001.0522v2  [nucl-th]  5 Jan 2010QCD sum rules for D mesons in dense and hot
+nuclear matter
+T. Hilger, R. Schulze and B. K¨ ampfer
+Forschungszentrum Dresden-Rossendorf, PF 510119, D-0131 4 Dresden, Germany
+TU Dresden, Institut f¨ ur Theoretische Physik, D-01062 Dresde n, Germany
+E-mail:t.hilger@fzd.de
+Abstract. Open charm mesons (pseudo-scalar and scalar as well as axial-vect or and
+vector) propagating or resting in nuclear matter display an enhanc ed sensitivity to the
+chiral condensate. This offers new prospects to seek for signals o f chiral restoration,
+in particular in pAand ¯pAreactions as envisaged in first-round experiments by the
+CBM and PANDA collaborations at FAIR. Weinberg type sum rules for c harming
+chiral partners are presented, and the distinct in-medium modifica tions of open-charm
+mesons are discussed. We also address the gluon condensates nea rTcand their impact
+on QCD sum rules.
+PACS numbers: 21.65.Jk, 12.40.Yx, 14.40.Lb, 24.85.+p
+Submitted to: J. Phys. G: Nucl. Phys.QCD sum rules for Dmesons in dense and hot nuclear matter 2
+1. Introduction
+The chiral condensate seems to be of utmost importance as candid ate of an order
+parameter of chiral symmetry which is spontaneously and explicitly b roken in nature
+(vacuum). The approach to restored chiral symmetry in a dense a nd/or hot medium
+may be signaled by a modified hadron spectrum. In particular, the sp ectral strengths
+of hadrons are expected to become modified when embedded in stro ngly interacting
+matter. While for vector mesons and strange mesons fairly exhaus tive studies have been
+performed in this context, charm degrees of freedom require fur ther investigations. The
+charmed hadron spectroscopy in pAand ¯pAcollisions will be addressed by two major
+experimental set-up’s of FAIR, CBM and PANDA. With this motivation we report in
+section 2 our results of a QCD sum rule analysis of spectral changes of pseudo-scalar
+open charm Dmesons in cold nuclear matter.
+In section 3 we consider differences of spectral strengths of chir al partners thus
+exposing the role of the chiral condensate also for charm. Since, w ithin the QCD sum
+rule approach which we employ here, the temperature and density d ependence of QCD
+condensates is important, we present in section 4 a first explorativ e study of two gluon
+condensates near the deconfinement temperature but at finite n et baryon density.
+2. QCD sum rules for D and ¯D mesons
+QCDsumrulesoffertheuniquepossibility torelate(non-perturbativ e) QCDparameters
+and hadronic observables. To be specific, the current-current c orrelation function
+Π(q) = i/integraldisplay
+d4xeiqx/angbracketleft/angbracketleftT/bracketleftbig
+j(x)j†(0)/bracketrightbig
+/angbracketright/angbracketright, (1)
+withcurrents j(x) (withthequarkstructurei ¯d(x)γ5c(x)forthepseudo-scalar Dmesons,
+for instance) is related to the integrated spectral density
+Π(q) =1
+π/integraldisplay+∞
+−∞ds∆Π(s,|/vector q|)
+s−q0, (2)
+where ∆Π in turn is related to hadronic observables. We are here inte rested in an
+exploration of spectral changes. Quite general, one may consider the moments
+Sn(M)≡/integraldisplays+
+0
+s−
+0dssn∆Π(s)e−s2/M2(3)
+which allow to define the quantities
+∆m≡1
+2S1S2−S0S3
+S2
+1−S0S2, (4)
+m+m−≡ −S2
+2−S1S3
+S2
+1−S0S2, (5)
+m2≡∆m2+m+m−, (6)
+which may be interpreted, e.g. by exploiting the pole + continuum ansa tz, as mass
+splitting (actually, ∆ mis half of the splitting) and mass shift of Dand¯D. In line withQCD sum rules for Dmesons in dense and hot nuclear matter 3
+a similar pattern in the strange meson sector, one expects shifts o f spectral strengths by
+an ambient nuclear medium as exhibited schematically in figure 1. An eva luation of the
+Figure 1. Expected spectral changes of Dmesons in a cold nuclear medium.
+above equations leads to the results displayed in figure 2 (for details and a discussion
+of different parameter sets cf. [1, 2]). These results point to a neg ative mass splitting
+/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53/s45/s48/s46/s48/s52/s45/s48/s46/s48/s51/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48
+/s32/s32/s109/s32/s91/s32/s71/s101/s86/s32/s93
+/s110/s32/s91/s32/s102/s109/s45/s51
+/s32/s93/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53/s49/s46/s56/s52/s49/s46/s56/s54/s49/s46/s56/s56/s49/s46/s57/s48/s49/s46/s57/s50/s49/s46/s57/s52
+/s32/s32/s109/s32/s91/s32/s71/s101/s86/s32/s93
+/s110/s32/s91/s32/s102/s109/s45/s51
+/s32/s93
+Figure 2. (color online) The D-¯Dmass splitting ∆ mfrom Eq. (4) and the center of
+gravitymfrom Eq. (6) as a function of nuclear matter density n. The colored region
+indicates the confidence range.
+D+−D−(left panel), while a safe average shift of D++D−cannot be extracted from
+the present approach due to uncertainties of input parameters ( see right panel of figure
+2). Similar considerations apply for BandDsmesons [1].
+A discussion of finite width effects always meets the problem of additio nal
+parameters which have to be fixed by the same set of equations. He nce, one only is
+able to give the mass of the particle (i.e. the center of gravity of the spectral function)
+as a function of its width. In the case of Dmesons the widths for particle (+) and
+anti-particle ( −) enter, and the mass–width correlation of the particle is locked with the
+width of the antiparticle as well. Employing the following Breit-Wigner an satz for the
+spectral function
+∆Π(ω) =F+
+πωΓ+
+(ω2−m2
++)2+ω2Γ2
++Θ(ω)+F−
+πωΓ−
+(ω2−m2
+−)2+ω2Γ2
+−Θ(−ω)(7)
+and determining m±, Γ±such that they fulfill Eqs. (4) and (6). For ∆ m=−30 MeV
+andm= 1.915 GeV one obtains the results depicted in figure 3.QCD sum rules for Dmesons in dense and hot nuclear matter 4
+/s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56/s49/s46/s56/s53/s49/s46/s57/s48/s49/s46/s57/s53/s50/s46/s48/s48/s50/s46/s48/s53/s50/s46/s49/s48
+/s32/s32/s109
+/s49/s32/s91/s32/s71/s101/s86/s32/s93
+/s49/s32/s91/s32/s71/s101/s86/s32/s93/s32
+/s50/s32/s61/s32/s48/s46/s48/s49/s32/s71/s101/s86
+/s32
+/s50/s32/s61/s32/s48/s46/s48/s51/s32/s71/s101/s86
+/s32
+/s50/s32/s61/s32/s48/s46/s48/s53/s32/s71/s101/s86
+/s32
+/s50/s32/s61/s32/s48/s46/s48/s55/s32/s71/s101/s86
+/s32
+/s50/s32/s61/s32/s48/s46/s48/s57/s32/s71/s101/s86
+/s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56/s49/s46/s57/s48/s49/s46/s57/s53/s50/s46/s48/s48/s50/s46/s48/s53/s50/s46/s49/s48/s50/s46/s49/s53/s50/s46/s50/s48/s32
+/s49/s32/s61/s32/s48/s46/s48/s49/s32/s71/s101/s86
+/s32
+/s49/s32/s61/s32/s48/s46/s48/s51/s32/s71/s101/s86
+/s32
+/s49/s32/s61/s32/s48/s46/s48/s53/s32/s71/s101/s86
+/s32
+/s49/s32/s61/s32/s48/s46/s48/s55/s32/s71/s101/s86
+/s32
+/s49/s32/s61/s32/s48/s46/s48/s57/s32/s71/s101/s86
+/s32/s32/s109
+/s50/s32/s91/s32/s71/s101/s86/s93
+/s50/s32/s91/s32/s71/s101/s86/s32/s93
+Figure 3. Mass–width correlations for m+(left panel) and m−(right panel) for
+different assumptions of widths of the respective other particle. ∆ m=−30 MeV,
+m= 1.915 GeV and F+=F−are assumed. In a first step, the integration limits in
+Eq. (3) have been extended to ±∞without including the continuum contribution, to
+avoid the influence of possible threshold effects.
+lV−A, light-light V−A, heavy-light P−S, heavy-light
+−1F2
+π
+0 0 8 mQ/angbracketleft¯qq/angbracketright 2mQ/angbracketleft¯qq/angbracketright
+1 O6 8m3
+Q/angbracketleftqq/angbracketright+4mQ/angbracketleft∆/angbracketright −2m3
+Q/angbracketleftqq/angbracketright+
+mQ/angbracketleftqgσGq/angbracketright−mQ/angbracketleft∆/angbracketright
+28m5
+Q/angbracketleftqq/angbracketright−
+4m3
+Q/angbracketleftqgσGq/angbracketright−
+12/angbracketleft∆/angbracketright+O7−2m5
+Q/angbracketleftqq/angbracketright+
+3m3
+Q/angbracketleftqgσGq/angbracketright−
+3/angbracketleft∆/angbracketright+O7
+Table 1. A few selected moments of the spectral differences/integraltext
+dssl/parenleftbig
+∆ΠV(P)−∆ΠA(S)/parenrightbig
+for chiral partners. Oidenotes a condensate or a
+combination of condensates of mass dimension iand/angbracketleft∆/angbracketrightstands for a medium-specific
+combination of condensates, i.e. it vanishes at zero density and tem perature.
+3. Weinberg type sum rules
+Exploiting the Operator Product Expansion one can evaluate in the s pirit of (1)
+the differences of spectral moments for vector and axial-vector currents, as has been
+pioneered by Weinberg [3], and applied to a hot medium by Shuryak and Kapusta [4].
+Our results for the differences of vector (V) – axial-vector (A) an d the scalar (S) –
+pseudo-scalar (P) correlators in case of mesons with one light quar k,m1≪ΛQCD, and
+one heavy quark, mQ=m2≫ΛQCD, are summarized in table 1. The employed currents
+arejV(A)
+µ=q1(γ5)γµq2,jS(P)= (i)q1(γ5)q2. In table 1, also the first spectral difference
+moments for mesons consisting of two light quarks, m1,2≪ΛQCDare listed for V−A.
+In particular, one recognizes for heavy-light mesons that the diffe rences betweenQCD sum rules for Dmesons in dense and hot nuclear matter 5
+chiral partners are driven by the chiral condensate /angbracketleft¯qq/angbracketrightin conjunction with the heavy
+quark mass mQ. Furthermore, in case of light-light vector mesons the first conde nsate
+which qualifies as anorder parameter is a certain combination of four -quark condensates
+for the 1st moment in next-to-leading order of the strong coupling αs, whereas the chiral
+condensate can only occur in lowest order of αsand in the 0th moment of the spectral
+differences if one of the quarks is heavy. In this respect, one may id entify such heavy-
+light quark bound states (hadrons) as useful probes for chiral r estoration to be linked
+with the density dependence of the chiral condensate /angbracketleft¯qq/angbracketrightn.
+4. Condensates near T c
+According to the consideration in [5] one may relate the gluon conden sate at finite
+temperature to the QCD trace-anomaly for Nf= 3 and the equation of state:
+/angbracketleftBigαs
+πG2/angbracketrightBig
+T=/angbracketleftBigαs
+πG2/angbracketrightBig
+0−8
+9(e−3p), (8)
+/angbracketleftbiggαs
+π/parenleftbigg
+(uG)2−G2
+4/parenrightbigg/angbracketrightbigg
+T=−3
+4αs
+π(e+p), (9)
+whereas contributions from light quarks to (8) have been omitted in a first step, as
+we focus on the continuation to finite densities. The quantities e−3pande+pare
+accessible by QCD lattice evaluations [6] and may be extrapolated to fi nite baryon
+densities by employing the Rossendorf quasiparticle model [7] which is here adjusted
+to [6]. Both condensates (8) and (9) are depicted in figure 4 in the re gion nearTc
+at quark chemical potential µq= 0 (left panel) and µq= 270 MeV (right panel);
+energy density eand pressure phave been employed from the quasiparticle model which
+allows for a thermodynamic consistent extrapolation to finite densit ies. The curves
+/s49/s46/s48/s48 /s49/s46/s48/s50 /s49/s46/s48/s52 /s49/s46/s48/s54 /s49/s46/s48/s56 /s49/s46/s49/s48/s45/s48/s46/s48/s49/s48/s45/s48/s46/s48/s48/s53/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s48/s46/s48/s49/s53
+/s32/s97/s115/s113/s116/s97/s100/s32 /s61/s54
+/s32/s97/s115/s113/s116/s97/s100/s32 /s61/s56
+/s32/s32/s32/s32/s32 /s32/s112/s52/s32 /s61/s54
+/s32/s32/s32/s32/s32 /s32/s112/s52/s32 /s61/s56/s103/s108/s117/s111/s110/s32/s99/s111/s110/s100/s101/s110/s115/s97/s116/s101/s115/s32/s91/s71 /s86/s52
+/s93
+/s47
+/s99/s49/s46/s48/s48 /s49/s46/s48/s50 /s49/s46/s48/s52 /s49/s46/s48/s54 /s49/s46/s48/s56 /s49/s46/s49/s48/s45/s48/s46/s48/s49/s48/s45/s48/s46/s48/s48/s53/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s103/s108/s117/s111/s110/s32/s99/s111/s110/s100/s101/s110/s115/s97/s116/s101/s115/s32/s91/s71 /s86/s52
+/s93
+/s47
+/s99
+Figure 4. Temperature dependence of the gluon condensates (8) (upper c urves) and
+(9) (lower curves) at µq= 0 (left panel) and µq= 270 MeV (right panel, same line and
+color code as in left panel).
+for the condensate (8) are flattened with increasing lattice tempo ral extend Nτ; [5]
+is reproduced by the p4 action for Nτ= 6. On the other hand, the condensate (9)
+seems not to be affected by such a choice. At finite densities, the glu on condensate forQCD sum rules for Dmesons in dense and hot nuclear matter 6
+/angbracketleftαs
+πG2/angbracketrightTdrops significantly due to the non-vanishing chemical potential. The symmetric
+and traceless gluon condensate/angbracketleftBig
+αs
+π/parenleftBig
+(uG)2−G2
+4/parenrightBig/angbracketrightBig
+Tis much less influenced by density
+effects. These effects have to be studied in detail for the J/ψ, which depends essentially
+on the considered gluon condensates [5].
+As mentioned above, finite current quark masses modify the trace of the energy
+momentum tensor to [8] ( Nf= 3)
+Tµ
+µ=−9αs
+8πGa
+µνGµν
+a+mu¯uu+md¯dd+ms¯ss+... , (10)
+where higher order αsterms have been neglected. (The contributions of heavy quarks
+canbeabsorbedinto thefirst termbyusing theheavy-quark mass -expansion.) Aprecise
+analysis would hence also have to include the density and temperatur e dependence of
+the quark condensates. However, as the chiral condensate is su pposed to vanish above
+Tcdue to the restoration of chiral symmetry, it is justified to neglect their influence on
+the gluon condensates in (8) and (9). Furthermore, if light quark c ontributions can be
+neglected in specific QCD sum rule evaluations, it is consistent with neg lecting such
+terms in (8) and (9), too. This offers the avenue towards the evalu ation of QCD sum
+rules near/above the deconfinement transition at non-zero bary on densities.
+5. Summary
+In summary we analyzed in-medium modifications of Dmesons in cold nuclear matter
+and find good prospects for an experimental verification, as the s pectral strengths of D
+and¯Dmay be displaced by about 60 MeV at saturation density. The key her e is the
+amplification of the chiral condensate by the heavy charm-quark m ass. We also had a
+glimpse on an extension of the celebrated Weinberg and Kapusta-Sh uryak sum rules for
+chiral partners and considered, furthermore, the determinatio n of the gluon condensate
+via the QCD trace anomaly in the deconfinement region at non-zero b aryon density.
+Acknowledgments
+The work is supported by BMBF Grant 06DR9059D and GSI-FE.
+References
+[1] T. Hilger, R. Thomas, B. K¨ ampfer, Phys. Rev. C 79, 025202 (2009).
+[2] T. Hilger, B. K¨ ampfer, arXiv:0904.3491.
+[3] S. Weinberg, Phys. Rev. Lett. 18, 507 (1967).
+[4] J. I. Kapusta, E. V. Shuryak Phys. Rev. D 49, 9 (1994).
+[5] K. Morita, S. H. Lee, Phys. Rev. Lett. 100, 022301 (2008).
+[6] A. Bazavov et al., Phys. Rev. D 80, 014504 (2009).
+[7] R. Schulze, M. Bluhm, and B. K¨ ampfer, Eur. Phys. J. ST 155, 17 7 (2008), arXiv:0709.2262; R.
+Schulze and B. K¨ ampfer, Prog. Part. Nucl. Phys. 62, 386 (2009) , arXiv:0811.0274; R. Schulze,
+B. K¨ ampfer, arXiv:0912.2827.
+[8] T. D. Cohen, R. J. Furnstahl, D. K. Griegel, Phys. Rev. C 45, 1881 (1992).
\ No newline at end of file
diff --git a/1001.0523.txt b/1001.0523.txt
new file mode 100644
index 0000000000000000000000000000000000000000..b6e365aaf7cd0a05458c677f6346c4cf154d4e7c
--- /dev/null
+++ b/1001.0523.txt
@@ -0,0 +1,261 @@
+IEEE ANTENNAS AND WIRELESS PROP AGA TION LETTERS, VOL. 8, 2009 1341
+On the Bandwidth of High-Impedance Frequency
+Selective Surfaces
+Filippo Costa , Student Member , IEEE , Simone Genovesi , Member , IEEE , and
+Agostino Monorchio , Senior Member , IEEE
+Abstract— In this letter , the bandwidth of high-impedance sur-
+faces (HISs) is discussed by an equivalent circuit approach. Even ifthese surfaces have been employed for almost 10 years, it is some-times unclear how to choose the shape of the frequency selective
+surface (FSS) on the top of the grounded slab in order to achieve
+the largest possible bandwidth. Here, we will show that the con-ventional approach describing the HIS as a parallel connection be-tween the inductance given by the grounded dielectric substrateand the capacitance of the FSS may induce inaccurate results inthe determination of the operating bandwidth of the structure. In-deed, in order to derive a more complete model and to provide amore accurate estimate of the operating bandwidth, it is also neces-sary to introduce the series inductance of the FSS. We will presentthe explicit expression for defining the bandwidth of a HIS, and wewill show that the reduction of the FSS inductance results in thebest choice for achieving wide operating bandwidth in correspon-dence with a given frequency.
+Index T erms— Artificial magnetic conductor (AMC), bandwidth
+of high-impedance surfaces (HISs).
+I. I NTRODUCTION
+HIGH-IMPEDANCE surfaces (HISs) [1] have been fre-
+quently employed in electromagnetic devices, as for
+instance in the design of low-profile antennas [2], [3] or im-proved electromagnetic absorbers [4], [5]. As is well known,
+these structures mimic the perfect magnetic conductor (PMC)
+condition within a small frequency range, and for this reason,they are often referred to as artificial magnetic conductors(AMCs). It is therefore evident that the bandwidth of this meta-
+surface is one of the key features of the structure. Some papers
+available in literature often present novel artificial surfacesclaiming a large operational bandwidth. However, in order tomake a fair comparison, structures with equal substrate should
+be considered since the substrate parameters are essential in
+defining the HIS bandwidth. In this work, we show that theconventional representation of the HIS as a parallel circuit
+between the inductance of the grounded dielectric and the
+capacitance given by the capacitive frequency selective surface(FSS) is not fully accurate for the determination of the devicebandwidth. In particular, we demonstrate that it is necessary
+to include the series inductance of the FSS in the equivalent
+circuit for obtaining the correct expression of the bandwidth.
+Manuscript received November 09, 2009; revised November 20, 2009. First
+published December 11, 2009; current version published December 29, 2009.
+The authors are with the Department of Information Engineering, Univer-
+sity of Pisa, 56122 Pisa, Italy (e-mail: filippo.costa@iet.unipi.it; simone.gen-
+ovesi@iet.unipi.it; a.monorchio@iet.unipi.it).
+Color versions of one or more of the figures in this letter are available online
+at http://ieeexplore.ieee.org.
+Digital Object Identifier 10.1109/LA WP.2009.2038346Afterward, the influence of the cell shape on the HIS bandwidth
+is analyzed. Simple considerations on the equivalent circuit
+allow us to define the shape of the periodic cell ensuringthe widest frequency band. In particular, we are interestedin deriving the most wideband FSS shape in correspondence
+with a fixed frequency since this issue has many interesting
+practical implications. Some examples of AMC structures,whose substrate thickness and permittivity are kept constant,are presented and discussed.
+II. F
+ORMULA TION
+The equivalent input impedance of the reactive surface
+can be interpreted as the parallel connection of the FSS
+impedance
+ and the equivalent input impedance of
+the grounded dielectric slab,
+ . For normal incidence, the
+impedance of the (thin) grounded dielectric slab behaves asan inductor (
+,
+ and
+ being the permeability and
+thickness, respectively, of the dielectric slab). The FSS element
+can be modeled as a single capacitance in the quasi-static
+region or by a RLC series circuit for a larger frequency range.
+Considering the FSS element as a pure capacitive element
+, the bandwidth
+ of the HIS structure, defined as the
+range where the phase of reflection coefficient is comprised
+between
+ and
+ , can be written as the bandwidth of
+a parallel LC circuit
+(1)
+where
+ represents the inductance of the substrate,
+ is the an-
+gular resonance frequency, and
+ is the free-space impedance.
+From the relation (1), it results that an increase of the substrate
+thickness enlarges the bandwidth of the structure due to the
+growth of the inductance. We also mention that the use of amagnetic substrate would be a valuable strategy for obtaininga higher inductance value and, as a consequence, a bandwidth
+increment [6].
+However, from (1), one might conclude that, while keeping
+the substrate thickness and the resonance frequency constant,a highly capacitive element such as a patch with a very small
+gap would be characterized by a smaller bandwidth than a less
+coupled element as a cross. Moreover, if we model the FSS byusing a capacitor only, we could speculate that different shapes
+of the patch, with different gaps and different periodicities but
+with the same value of averaged capacitance [7], would have thesame bandwidth. Contrarily, these considerations are incorrectbecause the simple LC parallel circuit, even if it is an excellent
+tool for studying these structures [5], [7], does not allow to cor-
+rectly predict the operating bandwidth.
+1536-1225/$26.00 © 2009 IEEE1342 IEEE ANTENNAS AND WIRELESS PROP AGA TION LETTERS, VOL. 8, 2009
+Fig. 1. Equivalent transmission line of a (a) high-impedance surface and (b) its
+corresponding lumped circuit model allowing to correctly determine the band-
+width of the structure.
+Fig. 2. Qualitative behavior of the reactive impedance in (2) and graphical rep-resentation of the HIS bandwidth.
+In order to properly evaluate the bandwidth of the HIS struc-
+tures, it is necessary to consider a more complete circuit wherethe series inductance
+of the FSS has been introduced (as
+shown in Fig. 1).
+The impedance of the circuit reported in Fig. 1(b) reads
+(2)
+As is well known, the high-impedance condition is verified
+when the denominator of (2) is null, namely at the angular res-onance frequency
+. The bandwidth of a
+high-impedance surface is defined as the frequency range where
+the absolute value of the impedance is larger than the free-space
+impedance (see Fig. 2).
+This condition corresponds to a phase of reflection coefficient
+comprised within the range [
+ ,
+ ] [1]. In order to define
+the bandwidth of the new equivalent circuit, it is necessary to
+find the two positive frequencies for which the impedance in (2)is equal to
+and
+ (see Fig. 2). In the former case, the
+equation has two positive solutions for
+ , and we must select
+the first one; indeed, the second one would be obtained when the
+HIS impedance is equal to
+ at higher frequency, well after the
+AMC band (see Fig. 2). In the latter case, the equation has onlya positive solution for
+. The two equations can be solved by
+employing the theory of the cubic equations [8]. By dividing the
+Fig. 3. Fractional bandwidth of a high-impedance surface as a function of the
+FSS series inductance /76(by fixing the grounded substrate inductance /76
+ to 7
+nH) and as a function of /76
+ (by fixing /76to 4 nH) . The capacitance is varied
+in order to preserve the same resonance frequency that is fixed to 6 GHz in all
+cases.
+equations by the coefficient of
+ , we can write the coefficients
+in the other three terms as follows:
+(3)
+By defining the three following intermediate coefficients:
+(4)
+we can get the two right solutions and, after some algebra, we
+can also derive the final expression of the fractional bandwidthof the new circuit (
+)a s
+(5)
+In order to highlight how the FSS series inductance influences
+the fractional bandwidth of the high-impedance surface, the re-lation (5) is plotted in Fig. 3 as a function of
+and
+ . When
+the values of the inductances are varied, the capacitance value
+is changed as well in order to maintain the resonance frequencyfixed. In this case, the resonance frequency of the resonator ischosen to be equal to 6 GHz. The values for the inductances and
+the capacitance are realistic values in a practical design. As ex-
+pected, increasing the grounded substrate inductance leads to anenlargement of the fractional bandwidth. Conversely, it is worthunderlining that an increase of the series inductance implies a
+reduction of the fractional bandwidth.COST A et al. : ON THE BANDWIDTH OF HIGH-IMPEDANCE FREQUENCY SELECTIVE SURFACES 1343
+Fig. 4. FSS element shapes under investigation.
+The fractional bandwidth of this resonant circuit can be max-
+imized both by decreasing the inductance and the capacitance
+of the FSS (considering that the inductance of the substrate isfixed). Anyway, in order to preserve the same resonance fre-quency, an increase of the capacitance value must correspond
+to a decrease of the series inductance and vice versa. As it is
+apparent from Fig. 3, the minimization of the inductance of theelement and the consequent increase of the capacitance resultsin the best choice for maximizing the fractional bandwidth. The
+inductance of an FSS can be lowered by using large elements
+with respect to the unit cell (as, for instance, the solid interiorelements defined in [4]). This argument can be intuitively ex-plained by considering the inductance of a straight wire above
+the ground plane. According to the expression reported in [9], it
+follows that the inductance of a printed structure can be loweredby using large element for a constant value of substrate thick-ness. For these reasons, we expect that a cross element will be
+the most narrowband, whereas the patch-type element will be
+the most wideband even if its capacitance is higher.
+III. N
+UMERICAL RESUL TS
+To demonstrate these intuitive considerations, we analyze six
+different FSS shapes on the same grounded dielectric substrate(Fig. 4). The substrate parameters are
+and
+thickness equal to 1.6 mm (typical of the commercial FR4).
+Patch elements characterized by a different ratio
+ between the
+patch size
+ and the periodicity
+ are analyzed
+together with a cross-shaped FSS, an interdigitated patch, anda meandered shape [10], [11]. Another candidate offering large
+operational bandwidths is the hexagon element, but it does not
+guarantee better performance in terms of bandwidth with re-spect to the square patch.
+To obtain a resonance in correspondence with the same fre-
+quency, the different elements must have different periodicities.Fig. 5 shows the phase of reflection coefficient of different HISconfigurations with the same substrate and a periodicity de-
+signed so that all the analyzed structures resonate at the same
+frequency. To confirm the validity of our model [and thereforethe bandwidth relation presented in (5)], the impedances of theFSSs under analysis were determined by the inversion method
+proposed in [12]. Once computed, each FSS impedance, the
+values of the capacitance, and the inductance can be obtained byan iterative procedure. The impedances for each FSS elementemployed in the HIS configurations are reported in Fig. 6 to-
+gether with its approximation by the series LC circuit. In Table I,
+a summary of the capacitances and inductances for each ana-lyzed shape is reported together with the fractional bandwidthcomputed in accordance to the relation (5). From these values,
+it is apparent that an increase of the FSS capacitance does not
+Fig. 5. Phase of reflection coefficient of different HIS on the same substrate
+and a periodicity ( /68) designed so that all the analyzed structures resonate at the
+same frequency.
+Fig. 6. Reactive impedance of the FSS of the different cell shapes.
+T ABLE I
+INDUCT ANCE AND CAP ACIT ANCE VALUES OF THE FSS SUNDER ANAL YSIS
+AND THE PERCENT AGE FRACTIONAL BANDWIDTH /70/66/87
+ ,C OMPUTED BY
+THE RELA TION (5)
+correspond to a decrease of the bandwidth as one could infer by
+observing the relation (1). This happens because, as for instance1344 IEEE ANTENNAS AND WIRELESS PROP AGA TION LETTERS, VOL. 8, 2009
+in the patch shape, the reduction of the distance between adja-
+cent patches implies an increase of the capacitance but, at thesame time, a strong reduction of the inductance.
+Nevertheless, the simplified relation (1) correctly predicts the
+enlargement of the fractional bandwidth in the patch-type FSSwhen the distance between adjacent patches is increased andthe periodicity is kept constant [13] (however, this case is not of
+practical interest since we do not obtain the resonance in cor-
+respondence with the same frequency). We also conclude thatthe bandwidth of the cross element is the smallest, whereas,among the patches, a higher ratio
+leads to a larger bandwidth
+of the FSS. The meandered structure is characterized both by
+a high capacitance, due to the strong coupling between adja-cent lines, and a high inductance, due to the narrowness and thelength of the lines. Indeed, these high values are instrumental
+for a substantial miniaturization of the cell in an AMC configu-
+ration. This small periodicity determines a total capacitance anda total inductance (depending on the periodicity also [7]) similarto other elements (see Table I).
+Similar results for the bandwidth of crosses and patches were
+qualitatively explained in [14] by exploiting the theory of smallantennas. The authors argued that the AMC bandwidth is re-duced by the degree to which the fields are localized within the
+surface texture and do not uniformly fill its volume. Finally, it is
+worth mentioning that the patch-type FSS with very small gapis the more effective solution for blocking the TE surface waves.Indeed, the effectiveness of the FSS in blocking the normal mag-
+netic field determines the TE mode cutoff frequency, and hence
+the TE bandwidth [15].
+IV . C
+ONCLUSION
+In this letter, a novel circuit approach for studying the band-
+width of HISs has been presented. As discussed in the bodyof this letter, the conventional representation of the HIS as a
+parallel LC circuit leads to an inaccurate estimate of the de-
+vice bandwidth. For such a reason, the expression of the frac-tional bandwidth by using a more complete equivalent circuit,including also the series FSS inductance, has been derived. We
+have shown that the increase of the series inductance implies
+a reduction of the fractional bandwidth. It has therefore beendemonstrated that the patch FSS element with a small gap be-tween adjacent patches, even if strongly capacitive, is the bestchoice for designing a wideband AMC in correspondence with
+a fixed frequency. We have also shown that an extremely minia-turized HIS can be designed by employing a meandered shape
+element while maintaining wideband properties.
+R
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+Italy, Oct. 22–24, 2007, pp. 811–814.
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\ No newline at end of file
diff --git a/1001.0524.txt b/1001.0524.txt
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@@ -0,0 +1,432 @@
+arXiv:1001.0524v1  [cond-mat.mes-hall]  4 Jan 2010Large Magnetic Susceptibility Anisotropy of Metallic Carb on Nanotubes
+T. A. Searles,1Y. Imanaka,2T. Takamasu,2H. Ajiki,3J. A. Fagan,4E. K. Hobbie,4and J. Kono1,∗
+1Department of Electrical and Computer Engineering, Rice Un iversity, Houston, Texas 77005, USA
+2National Institute for Materials Science, 3-13 Sakura, Tsu kuba, Ibaraki 305-0003, Japan
+3Photon Pioneers Center, Osaka University, Suita 565-0871, Japan
+4National Institute of Standards and Technology, Gaithersb urg, Maryland 20899, USA
+(Dated: October 29, 2018)
+Throughmagneticlineardichroismspectroscopy, themagne ticsusceptibilityanisotropyofmetallic
+single-walled carbon nanotubes has been extracted and foun d to be 2-4 times greater than values
+for semiconducting single-walled carbon nanotubes. This l arge anisotropy is consistent with our
+calculations and can be understood in terms of large orbital paramagnetism of electrons in metallic
+nanotubes arising from the Aharonov-Bohm-phase-induced g ap opening in a parallel field. We also
+compare our values with previous work for semiconducting na notubes, which confirm a break from
+the prediction that the magnetic susceptibility anisotrop y increases linearly with the diameter.
+PACS numbers: 78.67.-n, 78.67.Ch
+Unusual magnetic properties of single-walled carbon
+nanotubes (SWNTs) havegenerated much theoreticalin-
+terestduringthe lasttwodecades[1–8]. Theorbitalmag-
+netic susceptibility χof SWNTs is predicted to be large,
+two orders of magnitude larger than their spin magnetic
+susceptibility and comparable in magnitude to the well-
+known large diamagnetic susceptibility of graphite. In
+addition, the value of χis predicted to be strongly de-
+pendent onthe strengthofthe appliedmagneticfield, the
+Fermi energy, and the field orientation. Furthermore, the
+sign ofχcan be either positive (paramagnetic) or nega-
+tive (diamagnetic), depending on the nanotube chirality
+as well as the orientation of the magnetic field relative
+to the nanotube axis. At the core of these properties is
+the Aharonov-Bohm effect [1, 9], which changes the elec-
+tronicband structureofthe SWNTs via atube-threading
+magnetic flux in a truly non-intuitive manner.
+The magnetic susceptibility anisotropy, ∆ χ=χ/bardbl−χ⊥,
+whereχ/bardbl(χ⊥) is the parallel (perpendicular) susceptibil-
+ity, of SWNTs has been calculated using different meth-
+ods [2, 4, 5, 8]. Metallic SWNTs are predicted to be
+paramagnetic along the tube axis and diamagnetic in the
+perpendicular direction ( χM
+/bardbl>0> χM
+⊥), whereas semi-
+conductingSWNTsarepredictedtobediamagneticinall
+directions but most strongly diamagnetic in the perpen-
+dicular direction ( χS
+⊥< χS
+/bardbl<0) [2, 4, 5]. See, e.g., Fig. 1
+for ourk·pcalculations for (6,6) and (6,5) nanotubes at
+300 K. Thus, all SWNTs are expected to have positive
+∆χ, while metallic SWNTs are expected to have greater
+values of ∆ χthan semiconducting SWNTs. A finite
+∆χresults in the alignment of SWNTs in the magnetic
+field direction, which, combined with the anisotropic op-
+tical properties of SWNTs, allows for an estimation of
+∆χvalues for SWNTs through magneto-optical spec-
+troscopy. Previous magneto-optical studies found that
+∆χ∼1.5×10−5emu/mol for semiconducting SWNTs
+with∼1 nm diameters [9–14], while no information on
+∆χis currently available for metallic SWNTs. ! "" 
+paramagnetic diamagnetic 
+B
+#
+-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5  " $ 
+90 75 60 45 30 15 0
+#%&'( (6,6) 
+(6,5) 300 K 
+FIG.1: Predictedmagneticsusceptibilityanisotropyofsi ngle-
+walled carbon nanotubes. The magnetic susceptibility χwas
+calculated through a k·pmodel and is expressed in units of
+χ∗= (2πγ/a)(πa2/φ0)2a−2= 1.46×10−4emu/mol, where
+γ= 6.46 eV ·˚A,a= 2.5˚A, andφ0=e/his the magnetic
+flux quantum. The (6,6) nanotube (metallic) is paramagnetic
+along the tube axis and diamagnetic in the perpendicular di-
+rection ( χ/bardbl>0> χ⊥), whereas the (6,5) nanotube (semicon-
+ducting) is diamagnetic in all directions but most strongly
+diamagnetic in the perpendicular direction ( χ⊥< χ/bardbl<0).
+In both cases, ∆ χ=χ/bardbl−χ⊥>0, leading to alignment in an
+external magnetic field B.
+Here, we present the first experimental estimation of
+the ∆χof metallic SWNTs through high-field magnetic
+linear dichroism spectroscopy of CoMoCAT nanotubes
+individually suspended in aqueous solution at room tem-
+perature. The sample had a much smaller diameter dis-
+tribution than the HiPco samples used in the previous
+studies [10, 11], which allowed us to clearly identify, and
+closely examine the magnetic field dependence of, ab-
+sorption peaks for metallic nanotubes. We made a de-
+tailedcomparisonofthe∆ χofmetallicand semiconduct-
+ing nanotubes with similar diameters and lengths within2
+0.8
+0.6
+0.4Absorbance(8,3)(7,5)(6,4)(6,5)
+(6,5)
+phonon(7,4)(6,6)
+(5,5)(a)
+A// @ 35 T
+A/s94 @ 35 T 0 T & A0 Data
+ A0 = (A// + 2A⊥)/3
+0.6
+0.5
+0.4LDr
+3.43.23.02.82.62.42.22.01.8
+Energy (eV)35 T (b)
+(6,4)(6,5)(7,4)(6,6)
+(5,5)
+FIG. 2: (a) Absorption spectra (solid black) for 0 T and 35 T
+with all peaks assigned to specific chiralities. No trace is
+intentionally offset, indicating greater (smaller) absorp tion
+for parallel (perpendicular) polarization. The unpolariz ed
+isotropic absorbance (dashed red), calculated from the 35 T
+spectra via Eq. (1), agrees well with the 0 T data. (b) Re-
+duced linear dichroism versus energy from measured data.
+The largest peak is from metallic nanotubes (6,6) and (7,4).
+the same sample and found that the values of ∆ χare
+2-4 times greater in metallic tubes. We also compared
+our values with previous work for semiconducting nan-
+otubes [13], which confirm a break from the prediction
+that ∆χshould increase linearly with the tube diameter.
+Polarized magneto-optical absorption measurements
+on length-sorted, (6,5)-enriched CoMoCAT SWNTs were
+performed using the 35 T hybrid magnet at the National
+Institute for Materials Science in Tsukuba, Japan. The
+SWNTs were suspended in 1% sodium deoxycholate and
+length-sorted by dense liquid ultracentrifugation to have
+an average length of ∼500 nm. A Xe lamp, fiber-coupled
+to a custom optical probe, allowed for broadband white-
+light excitation of the E11metallic, E22semiconductor,
+andE33semiconductor interband transitions of SWNTs.
+The sample was held in a cuvette with a film polarizer
+directly on the front face to ensure that the incident
+light was linearly polarized. The transmitted light was
+collimated by a lens, collected with another fiber, and
+dispersed through a monochromator equipped with a Si
+CCD. All measurements were done at ∼300 K.
+Figure 2(a) shows polarization-dependent optical ab-
+sorption spectra at 0 and 35 T. None of the spectra are
+intentionally offset, indicating an increase (decrease) in
+absorbance for light polarized parallel (perpendicular) to
+the field. This is a direct result of magnetic alignment,togetherwith the fact that only the light-fieldcomponent
+parallel to the tube axis is strongly absorbed in SWNTs.
+Namely, at 35 T there is a finite degree of alignment in
+the magnetic field direction within the nanotube ensem-
+ble, resulting in stronger (weaker) absorption for light
+polarized parallel (perpendicular) to the field. From the
+theory of linear dichroism for an ensemble of anisotropic
+molecules [15], the following quantity can be shown to be
+constant, independent of the degree of alignment:
+A0≡A/bardbl+2A⊥
+3, (1)
+whereA/bardbl(A⊥) isthe absorptionfor lightpolarizedparal-
+lel (perpendicular) to the orientation axis, i.e., the mag-
+netic field direction. We calculated A0at 35 T using
+Eq. (1) and plotted it in Fig. 2(a) as a red dashed line.
+The agreement between the calculated A0and the zero-
+field absorbance spectrum confirms that A0is indepen-
+dent of alignment (or B), because at 0 T A/bardbl=A⊥=
+A0. We further confirmed that at any fields between 0
+and 35 T the increase in A/bardbland decrease in A⊥are such
+thatA0is preserved through Eq. (1).
+Linear dichroism, LD=A/bardbl−A⊥, is a measure of the
+degree of alignment. However, since it directly depends
+on the absorbance, LDalone cannot be used for compar-
+ing different spectral features. To adjust for differences
+in relative absorbances due to the fact that our sample
+is enriched for (6,5), the LDwas divided by A0, yield-
+ing thereducedlinear dichroism, LDr=LD/A 0[15, 16],
+which is plotted in Fig. 2(b) for 35 T. It is immediately
+evident from this plot that the spectral region where
+metallicpeaks[(6,6)and(7,4)]existhasmuchlarger LDr
+than the region where the most prominent semiconduct-
+ingpeaks [(6,5)and (6,4)] exist. This is evidence that the
+(6,6) and (7,4) tubes are aligning more strongly than the
+(6,5) and (6,4) tubes. In order to quantitatively deter-
+mine the magnetic alignment properties of each chirality
+present in our sample, we performed detailed spectral
+analysis, as described below.
+Eachabsorptionspectrum wasfit by a superpositionof
+multiple Lorentzian peaks plus a polynomial offset. The
+zero-field spectrum was fit first, as shown in Fig. 3(a),
+with each Lorentzian assigned to a specific chirality. The
+offset was included to account for the contributions from
+theπplasmon peaks, bundles, and light scattering [17–
+19]. For each magnetic field, the parallel and perpendic-
+ular spectra were fit simultaneously with the constraint
+that each Lorentzian independently satisfy Eq. (1). Fig-
+ure 3(b) shows representative results for the (7,5), (6,5),
+(6,6), and (5,5) nanotubes at 0 and 35 T. Note that we
+required the width and position of each Lorentzian to be
+the same as the values obtained at 0 T, which ensured
+that the calculated LDrfor each chirality is constant as
+a function of photon energy, as shown in Fig. 3(c) for
+35 T. Here, each LDrline is a direct result of calculation
+from each Lorentzian of different chirality, while LDrin3
+0.4
+0.3
+0.2
+0.1
+0.0Absorbance(b)
+(6,6)
+(5,5)(6,5)
+(7,5) 35 T, parallel
+ 0 T
+ 35 T, perpendicular
+1.5
+1.0
+0.5LDr @ 35 T
+3.0 2.5 2.0
+Energy (eV) Metal   Semiconductor (c)
+(6,6)
+(5,5)
+(7,4)
+(8,3)
+(7,5)
+(6,5)(6,4)0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0Absorbance(a)
+(8,3)(7,5)
+(6,4)(6,5)
+(6,5)
+phonon(7,4)(6,6)(5,5)(7,5) E33 0 T, data
+ 0 T, fit (total)
+ individual fits
+FIG. 3: (a) Fitting results for 0 T (b) Lorentzians for differe nt
+chirality nanotubes at 0 and 35 T. (c) LDrvs. Energy (eV)
+derived from fitting for each individual chirality nanotube in
+our sample. The metallic tubes (red) are higher than the
+semiconducting nanotubes (blue dotted).
+Fig. 2(b) was calculated from the entire spectra. As a re-
+sult, one can extract chirality-specific information on the
+alignment degree from Fig. 3(c). It is evident that the
+values are much larger for the metallic (red lines) than
+the semiconducting (blue lines) nanotubes.
+Foranisotropicmoleculeswith the dipolemoment fully
+oriented along the long axis, the reduced linear dichro-
+ism can be expressed as LDr= 3S[15, 16]. Here, the
+nematicorderparameter S= (3/angbracketleftcos2θ/angbracketright−1)/2isadimen-
+sionless measure of the degree of alignment, being equal
+to zero for completely randomly oriented nanotubes and
+one for completely aligned nanotubes, and θis the angle
+between the nanotube axis and the orientation axis (i.e.,
+the magnetic field direction in the present case). Values
+forSwere calculated for representative nanotubes from
+theLDrvalues in Fig. 3(c) and plotted as a function
+of magnetic field in Fig. 4(a). Here, we can clearly see
+that the ( n,n)-chirality, or armchair, metallic nanotubes
+[(6,6) and (5,5)] show much stronger alignment than the
+semiconducting nanotubes [(6,5) and (6,4)].1.0
+0.8
+0.6
+0.4
+0.2
+0.0S
+0.124
+124
+1024
+u(b) 0.5
+0.4
+0.3
+0.2
+0.1
+0.0S
+403020100
+Magnetic Field (T) (6,6)
+ (5,5)
+ (6,5)
+ (6,4)(a)
+0.6
+0.4
+0.2
+0.0S
+56789
+10234
+Magnetic Field (T) (6,6)
+ fit for (6,6)
+ (6,5)
+ fit for (6,5)(c)3.0
+2.5
+2.0
+1.5χ// /χ∗
+403020100
+Magnetic Field (T)(d)
+30 K
+300 K(6,6)
+FIG. 4: (a) The nematic order parameter Sas a function
+of magnetic field for armchair nanotubes (5,5) and (6,6) and
+semiconducting nanotubes (6,5) and (6,4), calculated from
+the measured reduced linear dichroism values. (b) Sversusu
+[Eq. (3)]. (c) Comparison of Sversus magnetic field for (6,5)
+and (6,6) nanotubes. (d) Calculated magnetic field depen-
+dence of the parallel magnetic susceptibility of (6,6) nano tube
+forT=30Kand300Kinunitsof χ∗=1.46×10−4emu/mol.
+In order to model the observed magnetic field depen-
+dence of S, we approximate the angular distribution of
+nanotubes at temperature Tin a magnetic field Bby the
+Maxwell-Boltzmann distribution function [10–13, 20]
+Pu(θ) =exp(−u2sin2θ)sinθ
+/integraltextπ/2
+0exp(−u2sin2θ)sinθdθ,(2)
+whereu=/radicalBig
+B2N∆χ
+2kBTis a dimensionless measure of the
+relative importance of the magnetic alignment energy
+(B2N∆χ/2) and the thermal energy ( kBT),Nis the
+number of carbon atoms in each nanotube with a length
+of500nm(in moles), and kBrepresentsBoltzmann’scon-
+stant. Using this distribution function and the definition
+ofS, one can derive
+S(u) =/integraldisplayπ/2
+0Pu(θ)/parenleftbigg3cos2θ−1
+2/parenrightbigg
+dθ
+=−1
+2+/parenleftbigg3
+4u2/parenrightbigg/parenleftBigg
+u
+√π
+2e−u2erfi(u)−1/parenrightBigg
+,(3)
+where erfi is the imaginary error function [erfi( u) =
+2
+i√π/integraltextiu
+0e−t2dt]. Equation (3) is plotted in Fig. 4(b).
+We were able to fit the SversusBcurves with Eq. (3),
+asshownfor(6,6)and(6,5)inFig.4(c), toextract∆ χfor
+each nanotube chirality. However, care must be taken in4
+TABLE I: Comparison of calculated and measured values of
+magnetic susceptibility anisotropy for seven types of SWNT s.
+For each chirality ( n,m), the diameter d, the chiral index ν
+= mod 3 ( n−m), and the chiral angle αare given, followed
+by theoretical (30 K and 300 K) and experimental (300 K)
+values of ∆ χ. All values for ∆ χare×10−5emu/mol.
+(n,m)d(nm) ν α(◦) ∆χ30K
+th∆χ300K
+th∆χ300K
+exp
+(6,6) 0.83 0 30 5.78 3.92 3.63
+(5,5) 0.69 0 30 4.95 3.39 3.35
+(7,4) 0.77 0 21 5.42 3.70 2.44
+(8,3) 0.78 -1 15 1.49 1.46 2.13
+(7,5) 0.83 -1 24 1.58 1.55 1.66
+(6,5) 0.76 1 27 1.45 1.42 1.01
+(6,4) 0.69 -1 23 1.33 1.29 1.24
+using this procedure, since the values of ∆ χare expected
+to depend on BandT, especially for metallic nanotubes.
+Thus, we calculated ∆ χas a function of BandTfor
+all the seven SWNTs studied in this work. Figure 4(d)
+shows the Bdependence of χ/bardblfor the (6,6) nanotube for
+30 K and 300K. Although χ/bardbldecreasesby ∼27% at 30 K
+as the field increases from 0 T to 40 T, it stays constant
+within∼0.7% at 300 K in the same field range, ensuring
+that the magnetic field dependence can be neglected for
+our room temperature measurements.
+Theoretical and experimental values of ∆ χobtained
+in the present work are summarized in Table I. The val-
+ues for the three metallic nanotubes, (7,4), (5,5), and
+(6,6), are all higher than those for the semiconducting
+nanotubes. In particular, the ∆ χof armchair, or ( n,n),
+carbonnanotubesare2-4times largerthan thosein semi-
+conducting nanotubes, depending on the diameter. This
+large difference in magnetic susceptibility anisotropy is a
+direct consequence of the Aharonov-Bohm physics caus-
+ing the band structure to change in the form of bandgap
+opening in metallic nanotubes and bandgap shrinking for
+semiconducting nanotubes. Specifically for metallic nan-
+otubes, this causes a large paramagnetism in the direc-
+tion along the tube axis.
+Finally, the experimentalvalues for metallic nanotubes
+in Table I do not follow a strict diameter dependence,
+something that is predicted for zigzag semiconducting
+tubes [8] and shown experimentally in semiconducting
+tubes [13]. It is also important to note that the (7,4)
+nanotube has a larger value of ∆ χthan the semicon-
+ducting tubes, but it is not as large as those of the arm-
+chair tubes. Unfortunately, it is the only non-armchair
+mod-3 nanotube in our sample, but a detailed study on
+a metallic-enriched sample [21] should yield many more
+metallic nanotubes to investigate in the future.
+In conclusion, we have successfully measured ∆ χfor
+metallic single-walled carbon nanotubes for the first timeand confirmed that they are much larger than those of
+semiconducting nanotubes. We also calculated the mag-
+netic field and temperature dependence of magnetic sus-
+ceptibilities of the SWNTs studied experimentally, which
+support the experimental findings. Lastly, we were able
+to confirm previous experimental results for the chiral-
+ity dependence of the magnetic susceptibility anisotropy
+in semiconducting nanotubes and found that this is also
+true for metallic nanotubes.
+This work was supported by DOE-BES (through
+Grant No. DEFG02-06ER46308), NSF (through Grant
+No. OISE-0530220), and the Robert A. Welch Founda-
+tion (through Grant No. C-1509). We thank Noe Alvarez
+for help with AFM length measurements and Ajit Srivas-
+tava for assistance with data analysis. Official contribu-
+tion of the National Institute of Standards and Technol-
+ogy; not subject to copyright in the United States.
+∗kono@rice.edu; corresponding author.
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+(2006).
+[8] M. A. L. Marques, M. d’Avezac, and F. Mauri, Phys.
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+Boca Raton, 2006), chap. 5, pp. 119–151.
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+[15] A.RodgerandB.Norden, Circular Dichroism and Linear
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\ No newline at end of file
diff --git a/1001.0525.txt b/1001.0525.txt
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index 0000000000000000000000000000000000000000..d8a7fdd179ddf66fd5663dc204b059a39a4b854b
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+++ b/1001.0525.txt
@@ -0,0 +1,450 @@
+ 
+Capillarity Theory for the Fly-Casting Mechanism 
+Emm anuel Trizaca, Yaakov Levyb, and Peter G. Wolynesc*
+ 
+aLaboratoire de Physique Théorique et Modé les Statistiques, CNRS UM R 8626, Université 
+Paris-Sud, F -91405 Orsay Cedex, France; bDepartm ent of Struct ural Biology, W eizmann 
+Institute of Science, Rehovot 76100, Israel ; and cDepartm ent of Che mistry and Biochem istry, 
+University of California, San Diego, 9500 Gilm an Drive, La Jolla, CA 92093-0371 
+ 
+* Corresponding author:  pwolynes@ucsd.edu 
+ 
+Running title: “Biophysical criteria for fly-casting binding”  
+ 
+Keyw ords: protein folding, binding m echanis m, disordered proteins, fast folding  
+ 
+ 1Abstract 
+Biom olecular folding and function are often coup led. During m olecular recognition events, one 
+of the binding partners m ay tran siently or pa rtially unf old, allowing more rapid access to a  
+binding site. W e describe a sim ple model for this  flycas ting mechanism based on  the capillarity 
+approxim ation and polym er chain statistics. The m odel shows that f lycasting  is m ost effective  
+when the protein unfolding barrier is s mall and th e part of the chain which extends towards the 
+target is relatively rigid. These features are often seen in known exam ples of flycasting in 
+protein-DNA binding. Sim ulations  of protein-DNA binding base d on well-funneled native-
+topology m odels with electrostatic forces conf irm the trends of the analytical theory. 
+ 
+ 2Introductio n 
+ It is becom ing clear that protein foldi ng and protein functioning are of ten overlapping 
+processes. For cooperatively fo lding proteins, a free e nergy ba rrier separates the folded 
+configurations from  the unfolded ones. If this  barrier is sufficientl y high, we expect the 
+traditional "function f ollows f olding " paradigm  to be va lid. Many pro teins however  are unf olded  
+before they function(1-5). Other proteins have native ensem bles separated by only a sm all barrier 
+from  their disordered  states. It h as recently been  argued that in som e circum stances there m ay be 
+no folding barrier at all(6-8). Cl early in all these cases  folding and function m ust be coupled. In 
+addition, partially surmounting the folding barrier , even if it is high, m ay short circuit the 
+otherwise la rge barrie rs that would accom pany allosteric ch anges, if the protein were required to 
+always rem ain intact. These two ways of co upling the f olding landscape and the functional 
+landscap e have been term ed "fly-casting" (9-11 ) (for the binding process)  and "cracking" (for  
+allos teric ch ange)(12 -14). In the fly-casting scen ario, a protein m ay bind fr om a relatively la rge 
+distan ce, th ereby enhan cing its cap ture r adius: the tr ade-of f is between the entrop y cost f rom 
+extending a subdom ain and the energy gained upon binding to a target. W hile the disordered 
+proteins m ay have slower translational diffusion comparing to globular pr oteins, their intrinsic 
+flexibility imposes fewer constraints on binding and therefore they m ay have faster binding to 
+their bindin g partners (10, 15). It is possible th at the biologic al function of  downhill a nd ultra-f ast 
+folding(16-20) is to achieve fast  binding via fly-casting. In this paper, we explore the interplay 
+between the fly-casting and the fo lding barrier using th e capillarity m odel fo r protein folding, 
+which in its sim plest form ignores many of th e structural features of the native protein. 
+ 
+ 3 Despite the intrinsic complexity of the folding of a ra ndom heteropolym er, most proteins  
+have evolved to fold via a funneled energy la ndscape(21, 22). This evolved feature greatly 
+simplifies the physics of  the folding process beca use the primary free en ergy scales involved are 
+then the entropy of organizing th e chain into a specific topology and the stabilization energy. 
+These free energies m ust balance n ear th e phy siological f olding tem perature, giving a single 
+param eter for that determ ines the specific topo logy free energy profile. The rem aining sm aller 
+energy scales com e from  the ruggedness of the la ndscape (which is refl ected in the rate of 
+confor mational diffusion) across the lands cape and a coop erativity  energy that comes from  the 
+nonadditivity of inter-residue forces which ha s a large component from  solvation. The latte r 
+effect is very analogous to the surface tension of  a liquid condensing from  a gas. The capillarity 
+pictu re of folding takes over th e pic ture of  nucle ation at a  first orde r tran sition  more or les s intact 
+to describe the change from  folded to unfolde d states(23). This picture m ight be called the 
+"spherical cow" approxim ation to folding since it i gnores, to zeroth order, th e specifics of protein 
+topology an d connectiv ity. This great oversim plification is, however, a positiv e featu re when we 
+wish to understand trends, such as the scal ing of folding rate with length etc.  
+ 
+ In this paper, we use the cap illarity picture to look at fly-casting. Th e model gives crisp 
+criteria for the conditions under w hich flycasting can occur. The analysis shows the key role 
+played by the ratio of the binding affinity to the folding barrier in determ ining whether fly-
+casting will occur. In ge neral f ly-casting is enco uraged by high af finities and low barriers. Fly -
+casting and cracking are thought to play a special ro le in how transcription factors navigate along 
+DNA and brachiate between DNA ch ains(24). On the ba sis of the present theory it turns out then 
+to be quite natural that DNA binding proteins  have been found to be am ong the fastest 
+ 4folders(10, 17). Rigidity of the pr otein in the denatured state al so encourages fly-casting thus  
+suggesting an increased possible role for re sidual structure in th e denatured state. 
+ 
+In addition to outlin ing the capilla rity m odel of  fly-cas ting in  this pap er, we com pare the theo ry 
+with results fro m a topology based simulation m odel of the protein-DNA binding process. This 
+analys is con firms the tre nds expecte d from the capillar ity model. 
+ 
+Methods 
+The capilla rity mode l of fly -casting associat ion 
+For our purposes, the protein m ade up of N residues is split into thre e reg ions. The f irst part with 
+N1 residues is structured (i.e., folded) and in cont act with the  targe t (assumed to be a rela tively 
+simple geom etry, such as an interface, m embrane , or DNA). At the other term inus of the protein,  
+we have another structured region of  N2 residues. Finally, the fly-casti ng effect trans lates into an 
+intermediate  region with  N3 = N-N 1-N2 unstructured residues, with end-to-end extension L, and 
+perpendicular to the binding substrate. This inte rmediate s ection will be trea ted a s a standard  
+homo-polym er coil under stretching. The N 1 section  being  in d irect co ntact with  the ta rget, the 
+protein center-of-m ass distan ce to the substrate is R=L(N 2+N 3/2)/N. We implicitly neglec t the  
+size of ordered regions (1 and 2) compared to th e "arm " extension L, and we e mphasize that the 
+confor mations under study here im pose a contact between the protei n and the binding substrate.  
+Under those assum ptions, the tota l free energy is written as 
+F=Fcap(N1)+Fcap(N2)+Fbind(N1)+Fent(N3)                                                            (1) 
+ Both N 1 and N 2 term ini contribute to the free ener gy through a capillarity term (23, 25) 
+Fcap(Ni)=γ(Ni2/3−N−1/3Ni)+τNi                                                                         (2) 
+ 5where τ∝T−Tf denotes the deviation from  folding tem perature T f. The quantity γ is analogous t o 
+a surface tension. At T f it is the single param eter that governs the free energy barrier to 
+completely f old the p rotein: 
+Fcapbarrier=4γN2/3
+27(1−τN1/3/γ)2                                                                                   (3) 
+In what follows, we will refe r to the free energy barrier for folding as the above quantity, 
+evaluated at folding temperatu re: 
+B=4γN2/3
+27                                                                                                      (4) 
+If ν denotes the Flory exponent( 26) of the interm ediate N 3 part, the corresponding entropy cost 
+of stretching reads Fent(N3), up to an irrelevant prefactor  
+Fent∝kTεL
+aN3ν⎛ 
+⎝ ⎜ ⎞ 
+⎠ ⎟ 1/(1−ν)
+                                                                                            (5) 
+where a denotes the residue size (a pproxim ately 0.3 nm ), and kT is the therm al energy. W hether  
+we consider a Gauss ian chain  (ν=1/2) or a swollen  coil with  excluded  volume (ν≈0.6) is 
+immaterial, and we will assum e for sim plicity tha t ν=1/2 in the following. More im portant is the 
+prefactor ε that accoun ts for the rig idity of  the chain: it can be viewed as the ratio a/lp where lp  
+is the chain pers istence length: ε=1 for a fully flex ible chain, whereas we will consid er ε=0.2 for 
+a typical protein. F inally, as far as the binding  term  is concerned, several m odels can be 
+considered; in the sim plest approxim ation, we shall take Fbind(N1)=−δN1kT. Several constraints 
+should be enforced, such as L≤N3a, or others resulting from  working at fixed R. 
+ 
+Simulation model of biomolecu lar association  
+We study the effect of the folding barrier and binding affinity on the fly-casting 
+charac teristics using  a coarse-g rained simulation  model for a system  that has been reported to 
+ 6follow the fly-casting mechanism . The association of  the Ets protein to it s specific D NA binding 
+site was studied using a native topology based model (G o model) supplem ented by electrostatic 
+interactions. The native topology based m odel takes into account only intera ctions that exist in 
+the native structure and, therefore, includes mainly topological frustration. Adding nonspecific 
+electrostatic interactions cont ributes energetic frustration betw een the protein and the DNA as  
+well as within the protein. Purely  structure based m odels have already been used to study the 
+folding of m any m onom eric protei ns that fold in a two-state fashion(27-30) or that fold via a 
+more com plicated folding schem e. They have al so been  used to  study higher molecularity 
+reactions such as dim erization and tetram erization(11, 31-33).  
+In our study, the protein and the DNA are m odeled using a re duced representation. Each 
+residue is represented by a single bead centered on its α-carbon (C α) position. Adjacent beads are 
+strung together into a polym er chain by m eans of a potential encoding bond length and angle 
+constraints. The secondary structure is enc oded in the dihedral angle potential and the non-
+bonded (native contact) potential.  In the framework of the m odel, all native contacts are 
+represented by a 10-12 Lennard Jones form  without any discrim ination between the various 
+chem ical types of interaction. Th e residues do not have any chem ical identity (the inform ation 
+for folding is encoded in the structure) but so me have a point charge. Accordingly, positively 
+charged residues (Arg and Lysine) have positive point charge and the negatively charged residue 
+(Asp and Glu) have a negative poi nt charge.  The other beads  are n eutral.  Details of the coarse-
+grained m odeling of the protein-DNA system  can be found in previous publications(10, 34). 
+To enhance the sam pling of binding events, a constraint is applie d on the protein-DNA 
+system  so that they are confined to a sphere of  radius 40 Å centered at the center of m ass of the 
+DNA that is  kept frozen. For each electros tatic strength, several cons tant tem perature molecular 
+dynam ics simulations w ere perform ed starting from an unbound (folded or unfolded) protein 
+confor mation. At a give n tem perature several trajec tories were collected starting from  different 
+initial conf ormation and velocitie s. The length of  the sim ulations was determ ined by the dem and 
+of having several tran sition s of folding/unf olding or binding/unbin ding at the transitio n 
+temperature s. The m ultiple tra jectories were co mbined using the W eighted Histogra m Analysis 
+Method (WHAM(35)) to calculate therm odyna mic properties of the system s. 
+To investigate the effect of the folding barrier of Ets and its affinity to DNA, the 
+association of Ets with its DNA cogna te was studied for different valu es of dielectric constant. In 
+ 7addition, the folding barrier wa s tuned by increasing the cooper ativity by incorporating a non-
+additive te rm to the Ham ilitonian(3 6).  
+    
+Results 
+The persistent limit 
+The case of a persistent fly-casting "arm " is strai ghtforward to work out a nd yield s an interesting 
+benchm ark. In this lim it indeed, the ratio ε is sm all, and the entropy cost of the arm irrelevant. 
+We therefore assum e here that th e middle part  of the protein is fully stretched, so that L=N3a, 
+which therefore provides the m ost favorable cond itions for the occurrence of fly-casting. It can 
+be shown th at whenever the to tal free energy F becom es negative, the only folded region of the 
+protein is that part in c ontact with the m embrane ( N2=0) and N1 takes its maximum value  
+allowed (for a given value of R): a NR N N / 21 −= . Starting from a large value of R, F first 
+increases upon decreasing R, which corresponds to the energy penalty of folding of N 1-like units,  
+then decreas es and vanis hes at R=R* with 
+R*=1
+21+δN1/3
+γ⎛ 
+⎝ ⎜ ⎞ 
+⎠ ⎟ −3
+−1⎡ 
+⎣ ⎢ 
+⎢ ⎤ 
+⎦ ⎥ 
+⎥ 2
+                                                                                          (6) 
+For R<R*, F<0; of course, the s tates with pos itive free en ergy a t R>R* are m etastable, and are 
+obtained enf orcing a con tact between  the protei n and the binding substrate for all adm issible R. 
+As a consequence, in the range R>R*, it is  more favorable  to a pproach the  substrate  in the  
+collap sed state (with N1=N2=0), with a vanishing free energy a nd no contact to the substrate. 
+The free energy profile is shown in Fig 2.   
+ 
+Increasing the ratio δ/γ increas es R*, but pres erves the sh ape of the curves sho wn in Fig 2. 
+Although it is therm odynam ically favorable to fly-cast when R<R*, kinetic effects m ay preempt 
+ 8the phenom enon, and it is of intere st to com pute the associated fr ee energy barrier. To this end, 
+we notice th at as far as the folding o f N1 residues is  concern ed, a fold ing temperature m ismatch τ 
+without binding is equivalent to  a (negative) binding term  at the folding tem perature; taking 
+advantage of this δ↔−τ correspondence, E quation (3) yields an energy barrier  
+Fε=0barrier=4γN2/3
+27(1+δN1/3/γ)2<B                                                                           (7) 
+If the quan tity on th e left hand side above exceed s som e threshold Fthres, fly-casting will not take 
+place. Mo re explicitly, such a s ituation of kinetic h indrance occurs for Fthres<B under the 
+condition that for  
+δNkT <27
+4BB
+Fthres−1⎛ 
+⎝ ⎜ ⎜ ⎞ 
+⎠ ⎟ ⎟                                                                                   (8) 
+In the rem ainder, we will refer to δNkTas the affinity. 
+ 
+The general case ( ε≠0) 
+For a given value of R, the free energy in (1) d epends on 2 variables (e.g., N3 and L), while N2 
+follows from   and N/)/N N(LR 23 2− =3 2 1 N NN N − −= . At var iance with the  persistent 
+limit where the "arm " extension L can take its maxim um value ( N3a) at no entropy cost, L is here 
+a variab le that follows from  minimizing the total free energy.  In the relev ant param eter range, at 
+fixed R and L, F(N3) is a concave function, which ther efore reaches its minimum at the 
+boundaries of the accessible N3 dom ain. This sim plifies th e analysis when it com es to finding the  
+optim al L for a given center-of-m ass distance  R. Once the resulting function L(R) is known, we  
+can com pute the fly cas ting distan ceR* for which F becom es negative. 
+  
+ 9 In Fig 3, the critical fly casting distan ce corresponding to the onset of negative free 
+energies is while the corresponding value of   is . Increasing the  
+barrier for fol ding B from  5kT to 10 kT significantly reduces Na R 042.0*≈1N 60.0*
+1≈N
+R* to 0.022 (see also F igure 5), 
+with . On the other hand, decrea sing the rigidity param eter ε from  0.2 to 0.1 facilitates 
+fly casting (  with never theless , alm ost unaffected), keeping γ and δ as in 
+Fig 3. 75.0*
+1≈N
+Na R 06.0*≈ 60.0*
+1≈N
+It turns out that the sta tes that m inimize th e free energy for *RR<  are such that  (num ber of  
+residues in the “arm ”, see Fig. 1) takes it s maximum  value, and that in addition,  vanishes. 
+This la tter f act a llows o ne to easily rela te the temperature dependence in  our m odel (all figures 
+shown above are at the folding temperature 3N
+2N
+fTT= ) to its δ –dependence. Decreasing T is 
+equivalent to increasing δ; more precise ly, a giv en affinity p arameter δ at  (i.e fTT≠ 0≠τ ) is 
+equivalent to an affinity )/(kTτδ−  at fTT= . 
+Of course, several variants of the present m odel can be put forward, to refine th e analysis.  Our 
+goal here has been to devise a m inimal framework. W e have nevertheless checked that the 
+present phenom enology is unaffected by a m odifica tion of the binding energy function, so that  
+its  dependence saturates beyond a prescribed threshold.   1N
+ 
+Compariso n to simula tions 
+We exam ine the effect of the folding kinetics  and of the protein affinity to the target on the 
+existence of fly-casting  mechanism using coarse -grained modeling of a ssocia tion of the Ets 
+protein to its specific DNA binding site (Fig. 6a). We have previously show n that the Ets protein 
+binds its cognate DNA sequence vi a a fly-casting m echanism (10). It  was shown that the protein 
+ 10flexibility significantly enhances binding to the DNA com paring to a scenario when  the protein  
+binds the DNA as a  rigid m olecule. The fly-casti ng can be probed by plotting the free energy as a 
+function of the separation distance between the pr otein and DNA (Fig. 6b ). These plots indicated 
+a coupling between folding and protein-DNA asse mbly, even for we ak electrostatic forces, 
+resulting in a change of the capture radius of  the target. A sharper decrease of the free energy 
+curve is ob served when flexibility  and electrost atics are both inclu ded in contrast to when 
+electros tatic forces alone  guide a rigi d protein. 
+ To exam ine the effect of the folding barrier  and affinity to the DNA on th e mechanism  of 
+protein-DNA association and in pa rticularly the onset of fly-cas ting, we study the assem bly of 
+the Ets protein with its cognate DNA at va rious dielectric constants and various folding 
+cooperativity(36). These two parameters affect the height of the free energy folding barrier and 
+the free energy of the bound com plex. Increasing the folding barrier by incorporating a non-
+additive ter m in the protein Ham iltonian re sults in a m ilder f ly-cas ting ef fect as r eflected by the 
+milder slope of the free energy when plotted al ong the separation distan ce of protein-DNA (Fig . 
+6B). Accordingly, as the folding barrier incr eases, fly -casting  takes place only  with closer 
+proxim ity of the protein to the DNA. W e probe the fly-casting by com paring the distance R** of 
+the free energy curves of protein-DNA assem bly at a given energy value (Fig. 6B). We point out  
+that R* and R** are defined differe ntly, yet bo th distances in dicate stron ger fly-casting effect as 
+they are larg er.  
+ Figure 6C illustrates that R ** gets s maller as th e folding barrier incre ases for different 
+values of dielectric constants. This observa tion is in harm ony with the capilla rity theo ry 
+prediction shown in Figures 4 and 5a. The coarse-g rained m odel ag rees with  the analytical m odel 
+with respect to the effect of the protein-DNA a ffinity on the fly-casting. R** is larger for high 
+ 11affinity protein-DNA interactions (Fig. 6D), sim ilarly to the trend shown in Figure 4. W e note  
+that it is  not prac tical to make a one to on e comparison between the analytical an d sim ulation  
+models due to differences in the tw o models a nd the ir parameterization  (e.g., in th e analytical 
+model the substrate attraction to the target is short range, while it is longer range in the 
+simulations due to e lectrostatic inte raction ). Non etheless, de spite the d ifferences  betw een the two 
+models, they concur in the unraveling of the key biophysical param eters that can affect the 
+binding via the fly-casting mechanis m.      
+ 
+Conclusions 
+The conceptual separation often m ade between folding processes and functional dynam ics relies 
+on the idea that there is clear distinction in thermodyna mic term s between the two parts of the 
+energy landscape--the denatured and native ensem bles. In t his paper we have put m athematical 
+flesh on this notion by using a sim ple model. The capillar ity m odel of  folding was generalized to  
+study the folding-binding proces s. Criteria describing when tr ansient unfolding dominates the  
+binding m echanism  can be explicitly  found in this m odel. The exte nt of  flycasting is shown to 
+depend on the ratio of the binding affi nity to unfolding barrier height.  
+ 
+Flycasting is also shown to be encouraged by lo w chain entropy for the unfolded region reducing 
+the free en ergy cost of reaching fa r towards the ta rget. Thes e trends a re confirmed in sim ulations 
+of protein-D NA binding using a well funneled, stru cture based m odel with structural detail. The 
+predicted criteria for fly casting shou ld be sus ceptible to tes t by appropriate protein engineering 
+studies of folding/bi nding kinetics(37). 
+ 12Figure legends 
+FIGURE 1:  A schem atic view of a protein that inter acts with a targ et (ill ustrated by the flat 
+surface) via the fly-cas ting m echanism . The prot ein is com posed of 3 reg ions: two folded  
+regions, L1 and L2, and a fl exible lin ker of  length  L3. 
+ 
+FIGURE 2:  Persis tent lim it ε =0 
+a) F/B as a function of center of m ass dis tance R to th e subs trate.  Here, B denotes the barrier for 
+folding defined in (4), i.e., in the ab sence of binding; W e took B=5 kT, N=100, δ=0.1 (which  
+corresponds to a binding en ergy of -10 kT). The value of R wh ere F vanishes defines R* (see the  
+arrow) and discrim inates the states  with positiv e free energ ies at R>R* (shown with a dashed 
+line) from the stab le one s with R<R*. Consequently, the m ost stable configuration when R>R* is 
+for a protein  "detached" from  the substrate, which can be either a diso rdered state or the folded 
+state (as suming we are at the fold ing temperatur e). Hence, th e optim al free energy p rofile is the 
+one shown by the thick continuous line. Fo r the param eters chosen, Eq. (6) gives R* = 0.15 Na, 
+which is indeed observed in the figure. Taking for concreteness a residue size of 0.3 nm , this 
+corresponds to R* = 4.5 nm . b) Plot of the number of resi dues in the re gion in contact with th e 
+substrate, as a function of R, for the sam e param eters as in  a). M etastable s tates are again  shown 
+by the dashed lines while the stable states  correspond to the thick continuous curve.  
+ 
+Figure 3.  Free energy profile versus R for the sam e parameters as in Fig 2,  but with ε = 0.2 
+instead of 0. The inset sh ows the corresponding value of N1. Metas table s tates are no t display ed. 
+ 
+Figure 4.  iso R* contour lines in (barrier  B versus Affinity) plot. 
+ 
+Figure 5.  Cuts through the contour plots of Fig 4. a) . Shows the critical fly-casting distance R* 
+as a function of barrier for folding B in the m ain graph (at constant binding energy δN kT = 10 
+kT). Inset: sam e, as a function of binding energy, at constant barrier B = 5 kT). Here, the rigidity 
+param eter is  ε = 0.2. b). Shows N 1* as a function of barrier for folding B. 
+ 13 
+ 
+Figure 6. Simulation of fly-casting bindi ng in protein-DNA interactions . The binding of the Ets  
+protein to its cognate DNA was sh own by nativ e topology based m odel to follow fly-casting 
+mechanism . A). The cry stal stru cture of the specif ic com plex of  the Ets pr otein with its cognate 
+DNA (pdb code 1BC8). B). Free energy curves as  a function of the separation distance between 
+protein and DNA for si mulation with different  degree of non-additivity. Larger non-additivity 
+results with higher folding barrier. The stars co rrespond to the R** distances that com pare the 
+fly-casting strength. C). The R** as  a function o f the free en ergy hei ght of the folding barrier for 
+different values of d ielectric constant of the Coulom b intera ction s. The bar rier heigh t was  
+modulated b y the non-ad ditivity term (36). d).  The R ** as a function of binding affinity of the Ets  
+to its cognate DNA.     
+ 
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+within the capillar ity approximation P Natl Acad Sci USA  94:6170-6175. 
+24. Vuzman, D. , Azia, A., Levy, Y. (In p ress) Searching DNA via a "mon key bar" 
+mechanism: the sign ificance of dis ordered tails J Mol Biol . 
+25. Finkelstein, A., Badretdinov, A. (1997) Ra te of protein folding near the point of 
+thermodyn amic equilibrium betw een the co il and the most stable ch ain fold Fold 
+Des 2:115-121. 
+26. Gennes, P.G. de ed. (1979) Scalin g concepts in polymer p hysics,  (Cor nell Univer sity 
+Press, Ithaca, NY). 
+ 1527. Clementi, C ., Nymeyer, H., Onuchic, J. N.  (2000) Topological and energetic factors: 
+what determines the structural details o f the transition state ensemb le and "en-
+route" intermediates for protein foldin g? An investigation for small globular 
+proteins J Mol Biol  298:937-953. 
+28. Chave z, L. L., Onuchic, J. N., Clementi, C. (2004) Quantifying the roughness on the 
+free energy landscape: entropic bo ttlenecks and protein fo lding rates J Am Chem 
+Soc 126:8426-8432. 
+29. Clementi, C ., Garcia, A . E., Onuchic, J. N.  (2003) Interplay among tertiary contacts, 
+secondary s tructure formation and side-c hain packing in t he protein f olding 
+mechanism all-a tom representa tion study of protein L. J Mol Biol  326:879-890. 
+30. Finke, J. M., Onuchic, J. N. (2005) Equilib rium and kinetic fo lding pathw ays of a 
+TIM barrel w ith a funneled energ y landscape Biophysical Journal  89:488-505. 
+31. Levy, Y., Cho, S. S., Onuchic, J. N., Wolyn es, P. G. (2005) A survey of flexible 
+protein bin ding mechanisms and their transi tion states using native topology based 
+energy landscapes J Mol Biol  346:1121-1145. 
+32. Levy, Y., Onuchic, J. (2006) Mech anisms of p rotein assembly: lesso ns from 
+minima list models Acc Chem Res  39:284-290. 
+33. Levy, Y., Caflisch, A., Onuchic, J. N., Wolynes, P. G. (2004) The folding and 
+dimeriz ation of HIV-1 protease: evidence for a stable mo nomer fro m simu latio ns J 
+Mol Biol  340:67-79. 
+34. Toth-Petroc zy, A., Simon, I., Fuxreiter,  M., Levy, Y. (2009) Disordered Tails of 
+Homeodomains Facilitate DNA Re cognitio n by Providing a Trade-Off between 
+Folding and Specific B inding Journal of the A merican Chemical Society  131:15084-+. 
+35. Kumar, S., Bouz ida, D., Sw endsen, R. H ., Kollman, P. A., Rosenberg, J. M. (1992) 
+The Weighted Histog ram Analysis Met hod for Free-Energ y Calculations on 
+Biomolecu les .1. The Method Journal of Computational Chemistry  13:1011-1021. 
+36. Ejtehadi, M. R., Avall, S. P., Plotkin, S. S. (2004) Three-bo dy Interactions Improv e 
+the Prediction of Rate and Mechanism in Pro tein Foldin g Models Proc Natl Aca d Sci 
+USA  101:15088-15093. 
+37. Ferreiro, D.  U., Sanche z, I. E., de Prat Gay, G. (2008) Transiti on state for protein-
+DNA recog nition Proc Natl Acad Sci U S A  105:10797-10802. 
+ 
+ 
+ 16N1
+N3N2
+L
+Figure 1A B
+Figure 2Figure 3B (kT)
+Binding affinity (kT)R* (nm)
+Figure 4A
+B
+Figure 5Figure 62468123456
+  R** (A)
+Folding barrier height ( kT) ε =  75
+ ε = 100
+ ε = 125
+ ε = 150
+-6 -5 -4 -3 -2 -10123456 ε =  75
+ ε = 100
+ ε = 125
+ ε = 150
+  R** (A)
+Affinity  (kT)10 15 20-4-3-2-1
+  
+Higher f olding barri erFree Energy ( kT)
+Distance protein-DNA (A)
+A B
+C D
\ No newline at end of file
diff --git a/1001.0526.txt b/1001.0526.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1ac2f597e2b0f7d245055f5bc418ebf7de775f3c
--- /dev/null
+++ b/1001.0526.txt
@@ -0,0 +1,1312 @@
+arXiv:1001.0526v2  [cond-mat.mes-hall]  19 Mar 2010Effective continuous model for surface states and thin films o f three-dimensional
+topological insulators
+Wen-Yu Shan, Hai-Zhou Lu, and Shun-Qing Shen∗1
+1Centre of Theoretical and Computational Physics,
+The University of Hong Kong, Pokfulam Road, Hong Kong, China∗
+Two-dimensional effective continuous models are derived fo r the surface states and thin films of
+the three-dimensional topological insulator (3DTI). Star ting from an effective model for 3DTI based
+on the first principles calculation [Zhang et al, Nat. Phys. 5, 438 (2009)], we present solutions for
+both the surface states in a semi-infinite boundary conditio n and in thin film with finite thickness.
+The coupling between opposite topological surfaces and str ucture inversion asymmetry (SIA) give
+rise to gapped Dirac hyperbolas with Rashba-like splitting s in energy spectrum. Besides, the SIA
+leads to asymmetric distributions of wavefunctions for the surface states along the film growth
+direction, making some branches in the energy spectra much h arder than others to be probed by
+light. These features agree well with the recent angle-reso lved photoemission spectra of Bi 2Se3films
+grown on SiC substrate [Zhang et al, arXiv: 0911.3706]. More importantly, using the parameters
+fitted by experimental data, the result indicates that the th in film Bi 2Se3lies in quantum spin Hall
+region based on the calculation of the Chern number and the Z2invariant. In addition, strong SIA
+always intends to destroy the quantum spin Hall state.
+PACS numbers:
+I. INTRODUCTION
+Topological insulators (TIs), which are band insula-
+tors with topologically protected edge or surface states,
+have attracted increasing attention recently1. A well-
+known paradigm of topological insulator is the quantum
+Hall effect, in which the cyclotron motion of electrons
+in a strong magnetic field gives rise to insulating bulk
+states but one-way conducting states propagating along
+edges ofsystem2. The idea was generalizedto a graphene
+model with spin-orbit coupling, which exhibits the quan-
+tum spin Hall (QSH) state3,4. Later, the realization
+of an existing QSH matter was predicted theoretically5
+and soon confirmed experimentally6,7in two-dimensional
+(2D) HgTe/CdTe quantum wells. Furthermore it was
+found that the QSHstate can be induced even by the dis-
+ordersorimpurities8–10. Meanwhile,theconceptwasalso
+generalized for three-dimensional (3D) TIs, which are
+3D band insulators surrounded by 2D conducting surface
+states with quantum spin texture11–14. BixSb1−x, an al-
+loywithcomplexstructureofsurfacestates,wasfirstcon-
+firmed as a 3DTI15,16. Soon after that it was verified by
+both experiments17,18and first-principles calculations19
+that stoichiometric crystals Bi 2X3(X=Se, Te) are TIs
+with well-defined single Dirac cone of surface states and
+extra large band gaps comparable with room tempera-
+ture. The Dirac fermions in the surface states of 3DTI
+obey the 2+1 Dirac equations and reveal a lot of uncon-
+ventional properties and possible applications, such as
+the topological magneto-electric effect20and Majorana
+fermions for fault-tolerant quantum computing21–26.
+Thanks to the state-of-art semiconductor technologies,
+low-dimensionalstructuresofBi 2X3canberoutinelyfab-
+ricatedinto ultra-thin films27,28and nanoribbons29. This
+stimulates several theoretical works on the thin films of
+3DTIs30–32. For further studies of the transport and op-tical properties of 3DTI films and their potential appli-
+cations in spintronics and quantum information, it is de-
+sirable to establish an effective continuous model for thin
+films of TIs.
+In this paper, we present an effective continuous model
+for the surface states and ultra-thin film of TIs. Start-
+ing with a 3D effective low-energy model based on the
+first principles calculations19, we first present the solu-
+tions for the surface states and the correspondingspectra
+for a semi-infinite boundary condition of gapless Dirac
+Fermions and for the thin film of TIs. The finite size
+effect of spatial confinement in a thin film leads to a
+massive Dirac model which may exhibit the QSH ef-
+fect. Within the same theoretical framework, a structure
+inversion asymmetry (SIA) term is further introduced
+in this work to account for the influence of substrate,
+providing a description of the Rashba-like energy spec-
+tra observed in the angle resolved photoemission spec-
+tra (ARPES) in the recent experiment on Bi 2Se3films28.
+We derived the parameter conditions for the formation
+of QSH effect in a thin film in the absence and presence
+of the SIA. By analyzing the fitting parameters with the
+help of the Chern number and the Z2invariant, we iden-
+tified the ultrathin films of Bi 2Se3in the QSH phase in
+the experiment.
+The paper is organized as follows. In Sec. II we intro-
+duce an anisotropic 3D Hamiltonian for 3DTI, which is
+a starting point of the present work. With this Hamilto-
+nian, we present detailed solutions to the thin film in two
+different boundary conditions. In Sec. III, effective con-
+tinuous models are established for the surface states and
+thin film of 3DTI. Within the framework of this effective
+continuous model, the structure inversion asymmetry is
+taken into account and an effective Hamiltonian for SIA
+is derived in Sec. IV. In Sec. V, we apply the model to
+newly fabricated thin film Bi 2Se3and demonstrate that2
+thin films of Bi 2Se3are in the QSH regime. Finally, a
+conclusion is presented in Sec. VI.
+II. MODEL AND GENERAL SOLUTIONS FOR
+3DTI
+A. Model for 3DTI
+FIG. 1: Schematic of a topological insulator film grown on
+substrate. The grown direction is defined as zaxis. The
+thickness of the film is L.
+As shown in Fig. 1, we will consider a thin film grown
+alongzdirection. The thickness of the film is L. We
+assume translational symmetry in x-yplane so that the
+wavenumbers kxandkyaregoodquantum numbers. We
+start with the effective model proposed to describe the
+bulk states near the Γ point for the bulk Bi 2Se319. The
+states are mainly contributed by four hybridized states
+of Se and Bi pzorbitals, denoted as {|p1+
+z,↑/an}b∇acket∇i}ht,|p2−
+z,↑/an}b∇acket∇i}ht,
+|p1+
+z,↓/an}b∇acket∇i}ht,|p2−
+z,↓/an}b∇acket∇i}ht}, where + ( −) stands for the even (odd)
+parity. The Hamiltonian is given by
+H(k) =ǫ0(k)I4×4+
+M(k)−iA1∂z0A2k−
+−iA1∂z−M(k)A2k−0
+0A2k+M(k)iA1∂z
+A2k+0iA1∂z−M(k)
+,
+(1)
+wherek±=kx±iky,ǫ0(k) =C−D1∂2
+z+D2k2,M(k) =
+M+B1∂2
+z−B2k2, andk2=k2
+x+k2
+y, withA1,A2,B1,B2,
+C,D1,D2, andMthemodelparameters. Thismodelhas
+the time reversal symmetry and the inversion symmetry.
+Though we start with a concrete model, the conclusion
+in this paper should be applicable to other topological
+insulator films. We shall demonstrate that this model
+for the bulk states can produce the surface states with
+appropriate boundary condition.
+B. General solutions of the surface states
+Following the method by Zhou et al.33, the general
+solution for either the bulk states or the surface states
+canbederivedanalytically. Despitetheexistenceoftime-
+reversalsymmetry, the A2k±term couples opposite spins
+in Hamiltonian (1), and one has to solve a 4 ×4 matrix,
+instead of the simplified 2 ×2 one in the 2D case33. Byputting a four-component trial solution
+ψ=ψλeλz(2)
+into the Schr¨ odinger equation ( Eis the eigenvalue of en-
+ergy)
+H(k,−i∂z)ψ=Eψ, (3)
+the secular equation
+det|H(k,−iλ)−E|= 0 (4)
+gives four solutions of λ(E), denoted as βλα(E), with
+α∈ {1,2},β∈ {+,−}, and
+λα(E) =/bracketleftBigg
+−F
+2D+D−+(−1)α−1√
+R
+2D+D−/bracketrightBigg1
+2
+,(5)
+where for convenience we have defined
+F=A2
+1+D+(E−L1)+D−(E−L2),
+R=F2−4D+D−[(E−L1)(E−L2)−A2
+2k+k−],
+D±=D1±B1,
+L1=C+M+(D2−B2)k2,
+L2=C−M+(D2+B2)k2. (6)
+Because of double degeneracy, each of the four βλα(E)
+corresponds to two linearly independent four-component
+vectors, found as
+ψαβ1=
+D+λ2
+α−L2+E
+−iA1(βλα)
+0
+A2k+
+, (7)
+ψαβ2=
+A2k−
+0
+iA1(βλα)
+D−λ2
+α−L1+E
+. (8)
+The general solution should be a linear combination of
+these eight functions
+Ψ(E,k,z) =/summationdisplay
+α=1,2/summationdisplay
+β=±/summationdisplay
+γ=1,2Cαβγψαβγeβλαz,(9)
+withthesuperpositioncoefficients Cαβγtobedetermined
+by boundary conditions. In the following, we will con-
+sider two different boundary conditions: one is semi-
+infinite focusing on only one surface at z= 0; the other
+includes two opposite surfaces at z=±L/2. In both
+cases we assume open boundary conditions (Ψ = 0) for
+the surface states at the surfaces.
+C. Solutions for the surface states with
+semi-infinite boundary conditions
+The surface states have a finite distribution near the
+boundary. For a film thick enough that the states at op-
+posite surfaces barely couple to each other, we can focus3
+on just one surface. Without loss of generality, we study
+a system from z= 0 to + ∞. The boundary condition is
+given as
+Ψ(z= 0) = 0 and Ψ( z→+∞) = 0.(10)
+The condition of Ψ( z→+∞) = 0 requires that Ψ con-
+tains only the four terms in which β=−and the real
+part ofλαis positive.
+Applying the boundary conditions of Eq. (10) to the
+general solution of Eq. (9), the secular equation of the
+nontrivial solution to the coefficients Cαβγleads to
+(λ1+λ2)2=−A2
+1
+D+D−, (11)
+which along with Eq. (5) gives the dispersion of the sur-
+face states
+E±=C+D1M
+B1±A2/radicalbigg
+1−(D1
+B1)2k+(D2−B2D1
+B1)k2.
+(12)
+Near the Γ point, the dispersion shows a massless
+Dirac cone in kspace, with the Fermi velocity vF=
+(A2//planckover2pi1)/radicalBig
+1−(D1
+B1)2, instead of plain A2//planckover2pi1as in Ref.19.
+The wave functions for E±are found as
+Ψ+=C0
++
+i
+2/radicalBig
+D+
+B1
+−1
+2/radicalBig
+−D−
+B1
+−1
+2/radicalBig
+D+
+B1eiϕ
+i
+2/radicalBig
+−D−
+B1eiϕ
+(e−λ+
+2z−e−λ+
+1z),
+Ψ−=C0
+−
+−i
+2/radicalBig
+D+
+B1
+1
+2/radicalBig
+−D−
+B1
+−1
+2/radicalBig
+D+
+B1eiϕ
+i
+2/radicalBig
+−D−
+B1eiϕ
+(e−λ−
+2z−e−λ−
+1z),(13)
+whereλ±
+αare short for λα(E=E±) according to Eq.
+(5), tanϕ≡ky/kx, andC0
+±arethenormalizationfactors.
+Thepropertiesofthesolutionto λαdeterminethe spatial
+distribution of the wave functions. Generally speaking,
+the edge states exist if λ1andλ2are both real or com-
+plex conjugate partners. For either case, there should be
+inequality relations
+M
+B1>0,
+D+D−<0. (14)
+The edge states distribute mostly near the surface ( z=
+0), with the scale of the decay length about λ−1
+1,2for real
+λ1,2or [Re(λ1,2)]−1for complex λ1,2. In the former case,
+thewavefunctionsdecayexponentiallyandmonotonously
+away from the surface (not from z= 0); while in lattercase the decaying is accompanied by a periodical oscilla-
+tion, which can be easily seen from the wavefunctions in
+Eq. (13). In addition, there exist complex solutions to
+λαwhen
+A2
+1
+−D+D−<4M
+B1. (15)
+D. Solutions for finite-thickness boundary
+conditions
+When the thickness of the film is comparable with the
+characteristic length 1 /λ1,2of the surface states, there is
+coupling between the states on opposite surfaces. One
+has to consider the boundary conditions at both surfaces
+simultaneously. Without loss of generality, we will con-
+siderthetopsurfaceislocatedat z=L/2andthebottom
+surface at −L/2. The boundary conditions are given as
+Ψ(z=±L
+2) = 0. (16)
+In this case, the general solution consists of all eight
+linearly independent functions. Applying the boundary
+conditions in Eq. (16) to the general solution of Eq. (9),
+the secular equation of the nontrivial solution to the su-
+perposition coefficients Cαβγleads to a transcendental
+equation
+D+D−
+A2
+1(λ2
+1−λ2
+2)2+(λ2
+1+λ2
+2)
+λ1λ2=tanhλ2L
+2
+tanhλ1L
+2+tanhλ1L
+2
+tanhλ2L
+2.
+(17)
+In a largeLlimit, tanhλαL
+2reduces to 1 , then Eq. (17)
+can recover the result in Eq. (11). With the help of Eq.
+(5), Eq. (17) can be used to identify the energy spectra
+and the values of λαnumerically.
+Due to the finite size effect33, the coupling between
+the states at the top and bottom surfaces will open an
+energy gap30–32. We define the gap as ∆ = E+−E−at
+the Γ point, where E+andE−are two solutions of Eq.
+(17). ForλαL≫1 andλ2≫λ1(Lcan be finite), the
+approximate expression for ∆ can be found. If λαis real,
+the gap can be approximated by
+∆≃4|A1D+D−M|/radicalbig
+B3
+1(A2
+1B1+4D+D−M)e−λ1L,(18)
+which decays exponentially as a function of L. Fig. 2(a)
+shows the gap as a function of thickness, in which a set of
+model parameters used to fit the ARPES of 4QL Bi 2Se3
+thin film is employed, as listed in the first row of Tab. I.
+Forsomeothermaterialstheremayexistcomplex λ1=
+λ∗
+2and we can define λ1=a−ibandλ2=a+ib, where
+a>0,b>0 according to Eq. (5). In this case, the gap
+is found as
+∆≃8|A1D+D−M|/radicalbig
+−B3
+1(A2
+1B1+4D+D−M)e−aLsin(bL),(19)4
+/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48
+/s32/s32/s32/s32/s32
+/s76/s32/s91/s81/s76/s93/s32/s91/s101/s86/s93/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48
+/s40/s98/s41/s40/s97/s41
+/s32/s110/s117/s109/s101/s114/s105/s99/s97/s108
+/s32/s50/s66
+/s49/s50
+/s47/s76/s50
+/s32
+/s32/s37/s32/s40/s63/s89/s41
+/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s49/s48/s50/s48/s51/s48
+/s40/s100/s41/s40/s99/s41/s66/s32/s91/s101/s86 /s197/s50
+/s93
+/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s45/s54/s48/s45/s53/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48
+FIG. 2: (Color online) [(a)(b)] The energy gap ∆ ≡E+−E−
+and [(c)(d)] the model parameter B[defined in Eq. (42)] as
+functions of the film thickness L. Circles correspond to the
+numerical results of the transcendental equations (30) and
+(31). Solid and dash lines correspond to the approximate
+formulas to ∆ when Lis finite [Eqs. (18) and (19)] or very
+small [Eq. (50)], respectively. All the parameters are adop ted
+(a) by fitting experimental results of 4QL Bi 2Se3, and (b)
+from the numerical fitting for the first principles calculati on
+of Bi2Se319, as listed in Tab. I.
+TABLE I: Two sets of parameters for the 3D Dirac model.
+The first row is extracted from our effective model parameters
+for 4QL Bi 2Se3film in table II, and the second row is adopted
+from the first principles calculation19.
+M A 1A2B1B2C D 1D2
+(eV) (eV ˚A) (eV˚A) (eV˚A2) (eV˚A2) (eV) (eV ˚A2) (eV˚A2)
+0.28 3.3 4.1 1.5 -54.1 -0.0068 1.2 -30.1
+0.28 2.2 4.1 10 56.6 -0.0068 1.3 19.6
+with
+a⋍A1
+2/radicalbig
+−D+D−, (20)
+b⋍/radicalBigg
+M
+B1+A2
+1
+4D+D−. (21)
+According to this result, the oscillation period of the gap
+π/bbecomesπ/radicalbig
+B1/MwhenA1= 0, in accordance with
+the result obtained by Liu et al32. Fig. 2(b) shows the
+gap oscillation by using the model parameters listed in
+the second entry of Tab. I. Besides, the sine function
+implies that ∆ may be negative. Later we will see that
+the sign of ∆ can be found by solving E0
++andE0
+−from
+Eqs. (30) and (31), respectively.III. EFFECTIVE CONTINUOUS MODELS
+The solutions of the surface states and thin film of
+3DTI can be applied to calculate physical properties ex-
+plicitly. For instance, we can see whether the ground
+state of a thin film exhibits QSHE or not by calculating
+the Chern number or Z 2invariant. It is also desirable
+to establish an effective continuous model to explore the
+properties of these surface states especially when other
+interactions have to be taken into account. For this pur-
+pose, in this section we derivean effective low-energyand
+continuous models for the surface states and thin film of
+3DTI.
+Due to the low-energy long-wavelength nature of the
+Dirac cone of the surface electrons, we can use the solu-
+tions of the surface states at the Γ point as a basis to ex-
+pand the Hamiltonian H(k) in Eq.(1), which will be valid
+when the energy is limited within the band gap between
+the conduction and valence bands. This is equivalent to
+a truncation approximation as we exclude the solutions
+for the bulk states in the basis. In this approach, the
+Hamiltonian in Eq. (1) can be expressed as
+H(/vectork) =H0(k= 0)+∆H, (22)
+where
+H0=/bracketleftBigg
+h(A1) 0
+0h(−A1)/bracketrightBigg
+, (23)
+with
+h(A1) =/bracketleftBigg
+−D−∂2
+z+C+M −iA1∂z
+−iA1∂z−D+∂2
+z+C−M/bracketrightBigg
+,
+(24)
+and
+∆H=/bracketleftBigg
+D2k2−B2k2σzA2k−σx
+A2k+σxD2k2−B2k2σz/bracketrightBigg
+.(25)
+The first term can be solved exactly, and the last term
+describes the behaviors of electrons near the Γ point.
+A. Basis states at Γpoint
+H0in Eq. (23) is block-diagonal. Its solution can be
+found by solving each block separately, i.e., h(A1)Ψ↑=
+EΨ↑andh(−A1)Ψ↓=EΨ↓. Because the lower block
+is the ”time” reversal of the upper block, the solutions
+satisfy Ψ ↓(z) = ΘΨ ↑(z), where Θ = −iσyKis the
+time-reversal operator, with σytheycomponent of the
+Paulimatrices and Kthe complex conjugationoperation.
+Equivalently, we can replace A1by−A1in all the results
+for the upper block, to obtain those for the lower block.
+Therefore, we only need to solve h(A1). Following the
+same approach in Sec. II, we put a two-component trial
+solution
+ψ↑=ψ↑
+λeλz(26)5
+into
+h(A1,−i∂z)ψ↑=Eψ↑, (27)
+the secular equation for a nontrivial solution yields four
+roots ofλ(E), denoted as βλα, withβ∈ {+,−}and
+a∈ {1,2}. Note that here λαis short for λα(k= 0)
+in Eq. (5 ). Each βλαcorresponds to a two-component
+vector
+ψ↑
+αβ=/bracketleftBigg
+D+λ2
+α−l2+E
+−iA1(βλα)/bracketrightBigg
+. (28)
+The general solution is a linear combination of the four
+linearly independent two-component vectors
+Ψ↑=/summationdisplay
+α=1,2/summationdisplay
+β=+,−Cαβψ↑
+αβeβλαz. (29)
+Applying the boundary conditions Eq. (16) to this gen-
+eral solution, we obtain two transcendental equations,
+(C−M−E−D+λ2
+1)λ2
+(C−M−E−D+λ2
+2)λ1=tanh(λ2L
+2)
+tanh(λ1L
+2),(30)
+and
+(C−M−E−D+λ2
+1)λ2
+(C−M−E−D+λ2
+2)λ1=tanh(λ1L
+2)
+tanh(λ2L
+2).(31)
+The solutions to Eqs. (30) and (31) give two energies at
+the Γ point, designated as E0
++≡E+(k= 0) andE0
+−≡
+E−(k= 0), respectively. The eigen wavefunctions for E0
++
+andE0
+−are, respectively,
+ϕ(A1)≡Ψ+
+↑=C+/bracketleftBigg
+−D+η+
+1f+
+−
+iA1f+
++/bracketrightBigg
+,(32)
+χ(A1)≡Ψ−
+↑=C−/bracketleftBigg
+−D+η−
+2f−
++
+iA1f−
+−/bracketrightBigg
+,(33)
+whereC±arethenormalizationfactors. Thesuperscripts
+off±
+±andη±
+1,2stand forE0
+±, and the subscripts of f±
+±for
+parity, respectively. The expressions for f±
+±andη±
+1,2are
+given by
+f±
++(z) =cosh(λ1z)
+cosh(λ1L
+2)−cosh(λ2z)
+cosh(λ2L
+2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+E=E0
+±,(34)
+f±
+−(z) =sinh(λ1z)
+sinh(λ1L
+2)−sinh(λ2z)
+sinh(λ2L
+2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+E=E0
+±,(35)
+η±
+1=(λ1)2−(λ2)2
+λ1coth(λ1L
+2)−λ2coth(λ2L
+2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+E=E0
+±,(36)
+η±
+2=(λ1)2−(λ2)2
+λ1tanh(λ1L
+2)−λ2tanh(λ2L
+2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+E=E0
+±.(37)The energy spectra and wavefunctions of the lower block
+h(−A1) ofH0can be obtained directly by replacing A1
+by−A1. Based on the above discussions, the four eigen-
+states ofH0can be given by
+Φ1=/bracketleftBigg
+ϕ(A1)
+0/bracketrightBigg
+,Φ2=/bracketleftBigg
+χ(A1)
+0/bracketrightBigg
+,
+Φ3=/bracketleftBigg
+0
+ϕ(−A1)/bracketrightBigg
+,Φ4=/bracketleftBigg
+0
+χ(−A1)/bracketrightBigg
+,(38)
+with Φ 1→Φ3and Φ 2→Φ4under the time-reversal op-
+eration. We should emphasize that these four solutions
+are for the surface states, and the solutions for the bulk
+states are not presented here. We use the four states as
+the basis states, and other states are discarded (except
+that in Fig. 3 where four extra bulk states are also in-
+cluded by the same approach), because of a large gap
+between the valence and conduction bands.
+B. Effective model for 3DTI films
+With the help of the four states Eq. (38) at the Γ
+point, we can expand Hamiltonian Eq. (1) to obtain a
+new effective Hamiltonian
+Heff≡/integraldisplayL/2
+−L/2dz[Φ1,Φ4,Φ2,Φ3]†H[Φ1,Φ4,Φ2,Φ3],(39)
+where for convenience, we organize the sequence of the
+basis states following {Φ1, Φ4, Φ2, Φ3}. Under the reor-
+ganized basis, the effective Hamiltonian is found as
+Heff=/bracketleftBigg
+h+0
+0h−/bracketrightBigg
+, (40)
+with
+h+=E0−Dk2+/bracketleftBigg
+∆
+2−Bk2˜A2k−
+˜A∗
+2k+−∆
+2+Bk2/bracketrightBigg
+,
+h−=E0−Dk2+/bracketleftBigg
+−∆
+2+Bk2−˜A∗
+2k−
+−˜A2k+∆
+2−Bk2/bracketrightBigg
+,(41)
+and
+B=˜B1−˜B2
+2, D=˜B1+˜B2
+2−D2,
+E0= (E++E−)/2,∆ =E+−E−,
+˜B1=B2/an}b∇acketle{tϕ(A1)|σz|ϕ(A1)/an}b∇acket∇i}ht,
+˜B2=B2/an}b∇acketle{tχ(A1)|σz|χ(A1)/an}b∇acket∇i}ht,
+˜A2=A2/an}b∇acketle{tϕ(A1)|σx|χ(−A1)/an}b∇acket∇i}ht. (42)
+We find that ˜A2here can be either real or purely imagi-
+nary(seeAppendix VII fordetails), classifyingthe model
+into two cases:6
+Case I is for a real ˜A2≡/planckover2pi1vF, and the effective Hamil-
+tonian is further written as
+hτz=E0−Dk2+/planckover2pi1vFτz/vector σ·/vectork+τzσz(∆
+2−Bk2),(case I);
+(43)
+and case II for a purely imaginary ˜A2≡i/planckover2pi1vF,
+hτz=E0−Dk2+/planckover2pi1vF(/vector σ×/vectork)z+τzσz(∆
+2−Bk2),(case II)
+(44)
+withτz=±1 corresponding to the upper (lower) 2 ×2
+block in Eq. (40), vFthe defined Fermi velocity and /vector σ
+and/vectorkhere only refer to the components in x-yplane. In
+fact, these two effective Hamiltonian can consist of the
+invariantsof the irreducible representation D1/2of SU(2)
+group34.
+Eq. (41) can also be expressed in terms of the Pauli
+matrices
+hτz=E0−Dk2+d·σ, (45)
+with
+d=
+
+τz(/planckover2pi1vFkx,/planckover2pi1vFky,∆
+2−Bk2),(case I)
+(/planckover2pi1vFky,−/planckover2pi1vFkx, τz(∆
+2−Bk2)).(case II)
+(46)
+d(k) vectors in case I and case II, respectively, corre-
+spond to Dresselhaus- and Rashba-like textures. Note
+that case I is essentially the effective 4 ×4 model for the
+CdTe/HgTe quantum wells5. However, we find that case
+I only occurs for quite a small range of thickness. For
+mostthicknessesofinterest, ˜A2ispureimaginary. There-
+fore, we only focus on case II in the following discussions.
+By far, we have reduced the anisotropic 3D Dirac model
+into a generalizedeffective model for 2Dthin films, under
+the freestanding open boundary conditions.
+C. Effective continuous model for surface states
+Despite the simple explicit form, the parameters in
+Hamiltonian(40)needtobedetermined numerically. Be-
+fore that, we can take two limits to see their behaviors.
+The first limit is λαL≫1, forα= 1,2. In this case,
+tanh(λαL
+2)⋍1, and both Eqs. (30) and (31) reduce to
+(C−M−E−D+λ2
+1)λ2= (C−M−E−D+λ2
+2)λ1.(47)
+Solving this equation, we have an effective continuous
+model for the surface states (ss) of 3D topological insu-
+lator as
+Hss=C+D1M
+B1+(D2−B2D1
+B1)k2
++A2/radicalbigg
+1−(D1
+B1)2(σxky−σykx),(48)which has the same dispersion as Eq. (12) and the
+same Fermi velocity vF=A2
+/planckover2pi1/radicalBig
+1−(D1
+B1)2as for the
+semi-infinite boundary conditions. In an isotropic case,
+D1=D2andB1=B2, the quadratic term disappears
+and we have a linear dispersion for the Dirac cone. Fi-
+nallyitisnoticedthat themodelsforthesurfacestatesat
+the top and bottom surface have the same form assumed
+λαL≫1.We will see that these results work well even
+for films down to five quintuple layers (QL) of atoms in
+thickness (1 QL is about 1 nm).
+D. The ultra-thin limit
+Another opposite limit is L→0, which is a little bit
+complicated since λαLdoes not approach to zero when
+Lis very small. In Eq. (30), the left side has an order
+ofL2whenL→0, so tanh(λ1L
+2) must have the order of
+L−2, which means
+tanh(λ1L
+2) = 0⇒λ1=iπ
+L. (49)
+Combining this result with Eq. (5), the model becomes
+hτz=D1π2
+L2+D2k2+A2(/vector σ×/vectork)z+τz(B1π2
+L2+B2k2)σz.
+(50)
+It is found that a finite energy gap opens at k= 0, i.e.,
+∆ = 2B1π2/L2as shown in Fig. 2. Note that this result
+in theL→0 limit even provides a rough estimate of the
+gap for most thicknesses. Besides, the continuum limit
+generally assumed in this work also works well even for
+only several quintuple layers.
+IV. STRUCTURE INVERSION ASYMMETRY
+A. Structure Inversion Asymmetry
+A recent experiment28revealed that the substrate on
+which the film is growninfluences dramaticallyelectronic
+structure inside the film. Because the top surface of the
+film is usually exposed to the vacuum and the bottom
+surfaceisattachedtoasubstrate,theinversionsymmetry
+does not hold along zdirection, leading to the Rashba-
+like energy spectra for the gapped surface states. In this
+case, an extra term that describes the structure inversion
+asymmetry (SIA) needs to be taken into account in the
+effective model.
+We use the same method as that in Sec. III to in-
+clude the SIA term. Without loss of generality, we add
+a potential energy V(z) into the Hamiltonian. Generally
+speaking,V(z) can be expressed as V(z) =Vs(z)+Va(z),
+in whichVs(z) =Vs(−z) andVa(z) =−Va(−z). The
+symmetric term Vscould contribute to the mass term ∆
+in the effective model, which may lead to an energy split-
+ting of the Dirac cone at the Γ point. We do not discuss7
+it in details in this paper. Here we focus on the case
+of the antisymmetric term, V(z) =Va(z), which breaks
+the top-bottom inversion symmetry in the Hamiltonian.
+A detailed analysis demonstrates that Va(z) couples Φ 1
+(Φ3) to Φ 2(Φ4), which can be readily seen according
+to their spin and parity natures. The modified effective
+Hamiltonian in the presence of V(z) becomes
+HSIA
+eff=Heff+
+0 0˜V0
+0 0 0 ˜V∗
+˜V∗0 0 0
+0˜V0 0
+,(51)
+where
+˜V=/integraldisplayL/2
+−L/2dz/an}b∇acketle{tϕ(A1)|Va(z)|χ(A1)/an}b∇acket∇i}ht.(52)
+Comparingthis definition with that of ˜A2in Eq. (42), we
+find that ˜Valsocanbe eitherrealorpurelyimaginary. In
+the case of a purely imaginary (case II) ˜A2,˜Vmust be
+real (see Appendix VII), and the effective Hamiltonian
+with SIA can be written as
+HSIA
+eff=/bracketleftBigg
+h+(k)˜Vσ0
+˜Vσ0h−(k)/bracketrightBigg
+. (53)
+In the case of a real ˜A2,˜Vmust be purely imaginary, and
+the effective Hamiltonian with SIA then has the form
+HSIA
+eff=/bracketleftBigg
+h+(k)˜Vσz
+−˜Vσzh−(k)/bracketrightBigg
+. (54)
+Without the SIA term, the effective Hamiltonian (44)
+gives the energy spectra of the gapped surface states as
+E±=E0−Dk2±/radicalbigg
+(∆
+2−Bk2)2+(/planckover2pi1vF)2k2,(55)
+where + ( −) sign stands for the conduction (valence)
+band, each of which has double spin degeneracy due to
+time-reversal symmetry. When the SIA term is included,
+the Hamiltonian (51) gives
+E1,±=E0−Dk2±/radicalbigg
+(∆
+2−Bk2)2+(|˜V|+/planckover2pi1vFk)2,
+E2,±=E0−Dk2±/radicalbigg
+(∆
+2−Bk2)2+(|˜V|−/planckover2pi1vFk)2,
+(56)
+where the extra index 1 (2) stands for the inner (outer)
+branches of the conduction or valence bands. The en-
+ergy spectra in the presence of ˜Vis shown in Fig. 3.
+Each spin-degenerate dispersion in Eq. (55) shifts away
+from each other along kaxis. Both the conduction and
+valence bands show Rashba-like splitting. An intuitive
+understanding of the energy spectra in Fig. 3 can begiven with the help of Fig. 4. On the left is for a thicker
+freestanding symmetric TI film, and it has a single gap-
+less Dirac cone on each of its two surfaces, with the solid
+and dash lines for the top and bottom surface, respec-
+tively. The two Dirac cones are degenerate. The top of
+Fig. 4 indicates that the inter-surface coupling across
+an ultrathin film will turn the Dirac cones into gapped
+Dirac hyperbolas. On the bottom of Fig. 4, SIA lifts
+the Dirac cone at the top surface while lowers the Dirac
+cone at the bottom surface. The potential difference at
+the top and bottom surfaces removes the degeneracy of
+the Dirac cones. On the right of Fig. 4, the coexistence
+of both the inter-surface coupling and SIA leads to two
+gapped Dirac hyperbolas which also split in k-direction,
+as shown in Fig. 3.
+−0.1 −0.05 0 0.05 0.1−0.8−0.6−0.4−0.200.20.40.60.8
+kx(˚A−1)E (eV)
+  
+4 QLTop surface
+Bottom
+surface
+FIG. 3: (Color online) Energy spectra of surface states (fou r
+branches in the middle) and several branches of bulk states
+(those at the top andthe bottom) for afilm with thethickness
+of 4 QL in the presence of the structure inversion asymmetry.
+The color of lines corresponds to the spatial distribution o f
+the wavefunctions in zdirection. Dark blue (light green) rep-
+resents thatthewavefunctions mainly distributeon thesid e of
+the top (substrate) surface. The model parameters are liste d
+in the first row of Table I.
+B. Location of the surface states
+Location of the surface states can be revealed by eval-
+uating the expectation of position z of these states. The
+spatial distributions along the z direction of a state ψα
+can be evaluated by the expectation of position in zdi-
+rection/an}b∇acketle{tz/an}b∇acket∇i}ht,
+/an}b∇acketle{tz/an}b∇acket∇i}htα=/integraldisplayL
+2
+−L
+2z|ψα|2dz. (57)
+By this definition, /an}b∇acketle{tz/an}b∇acket∇i}htα∈[−L
+2,L
+2] and/an}b∇acketle{tz/an}b∇acket∇i}htαbecomes 0
+for a symmetric spatial distribution.8
+FIG. 4: (Color online) Theevolution ofthedoubly-degenera te
+gapless Dirac cones for the 2D surface states, in the pres-
+ence of both the inter-surface coupling and structure inver -
+sion asymmetry, into the gapped hyperbolas that also split
+ink-direction. The blue solid and green dashed lines corre-
+spond to the states residing near the top and bottom surfaces ,
+respectively.
+With the SIA, the eigen wavefunctions are found as
+ψ1±=1
+2/radicalBig
+Ein
+±(Ein
+±+t)k
+i(t+Ein
+±)k
+(|˜V|+/planckover2pi1vFk)k+
+i(|˜V|+/planckover2pi1vFk)k
+(t+Ein
+±)k+
+,(58)
+withEin
+±=E1±−E0+Dk2,t=∆
+2−Bk2, and,
+ψ2±=1
+2/radicalbig
+Eout
+±(Eout
+±+t)k
+−i(t+Eout
+±)k
+(|˜V|−/planckover2pi1vFk)k+
+i(−|˜V|+/planckover2pi1vFk)k
+(t+Eout
+±)k+
+(59)
+withEout
+±=E2±−E0+Dk2. Fig. 3 demonstrates /an}b∇acketle{tz/an}b∇acket∇i}htby
+the brightness of lines, with dark blue for /an}b∇acketle{tz/an}b∇acket∇i}ht=L
+2(the
+top surface), and light green for /an}b∇acketle{tz/an}b∇acket∇i}ht=−L
+2(the substrate
+or bottom surface).
+For a thin film of 4 quintuple layers (QL), L= 3.8nm,
+it is found that the two surface states are well separated
+and dominantly distributed near the two surfaces. The
+averaged /an}b∇acketle{tz/an}b∇acket∇i}ht ≃ ±L
+3, which is about 2/3 of a QL ( ≈L/6)
+deviating from the surface. In this case the top and bot-
+tom surface states are well defined even without the SIA
+(˜V= 0). The average value remains almost unchanged
+in a large range of k. However, the crossing point of the
+spectra of the top and bottom surface states, the aver-
+aged/an}b∇acketle{tz/an}b∇acket∇i}htchanges from + L/3 to 0,and then goes to the
+value of −L/3. This demonstrates that the finite thick-
+ness makes the two states couple with each other as their
+wavefunctions alongthe z direction have a finite overlap.As a result the two states open an energy gap as in the
+case of edge states in QSH system33. The value of the
+gap is a function of Las shown in Fig. 2(a) and (b).
+Near this region, /an}b∇acketle{tz/an}b∇acket∇i}htvaries from /an}b∇acketle{tz/an}b∇acket∇i}ht ≃L/3 to−L/3, and
+becomes zero exactly when two states are mixed com-
+pletely. For a large L, we find that the averaged distance
+of the surface states deviating from the surface remains
+about 1 QL.
+Simply speaking, the states close to the top surface
+are easier to be probed by light than those close to the
+bottom surface. This provides a hint to understand why
+there are branches in energy spectra with much more
+faint ARPES signals28.
+V. THIN FILM B i2Se3AND QSH STATES
+In this section, we will investigate the realization of
+QSH effect in thin films and apply the effective model to
+thethinfilmBi 2Se3. Whenthesystemdoesnotbreakthe
+inversion symmetry, the effective Hamiltonian is block-
+diagonalized by τz=±1. This is in a good agreement
+with the theory by Murakami et al35. In this case we
+can define a τz-dependent Chern number (Hall conduc-
+tance) for each block like the spin Chern number36, from
+which the nontrivial QSH phase can be identified. Af-
+ter introducing the SIA term, the τz-dependent Chern
+number loses its meaning as the two blocks are mixed to-
+gether. However, we can still employ the Z 2topological
+classification4, which requires no inversion symmetry, to
+identify possible QSH thin films in experiment.
+A. QSH effect without SIA
+Considering the block-diagonal form of the effective
+model without SIA (40), we can derive the Hall conduc-
+tance for each block, separately. For the 2 ×2 Hamilto-
+nian in terms of the d(k) vectors and Pauli matrices in
+Eq. (45), the Kubo formula for the Hall conductance can
+be generally expressed as37,38
+σxy=−e2
+2Ω/planckover2pi1/summationdisplay
+k(fk,−−fk,+)
+d3ǫαβγ∂da
+∂kx∂dβ
+∂kydγ(60)
+where Ω is the volume of the system, dthe norm of
+(dx,dy,dz),fk,±= 1/{exp[(E±(k)−µ)/kBT] + 1}the
+Fermi distribution function of electron (+) and hole ( −)
+bands, with µthe chemical potential, kBthe Boltzmann
+constant, and Tthe temperature.
+At zero temperature and when the chemical potential
+µlies between the band gap ( −|∆|
+2,|∆|
+2), the Fermi func-
+tions reduce to fk,+= 0 andfk,−= 1. In this case we
+have31
+στz
+xy=−τze2
+2h[sgn(∆)+sgn( B)]. (61)
+This result intuitively shows that only when Band ∆
+have the same sign, the Chern number is equal to +1 or9
+−1, which is topologically nontrivial, and the Hall con-
+ductance is quantized to be ±e2/h. In other words, the
+QSH depends not only on the sign of ∆ at the Γ point
+but also on that of Bforklarge enough. Experimen-
+tally, theτz-dependent Hall conductance can be probed
+by the nonlocal measurement, just like that for the 2D
+CdTe/HgTe quantum wells7.
+B. QSH effect with SIA: Z 2invariant
+0.01 0.1 1-1.0-0.50.00.51.0~(a) V = 0 P(k)
+k [Å-1]~
+0.01 0.1 1-1.0-0.50.00.51.0  ∆>0, B>0
+ ∆<0, B>0
+ ∆>0, B<0
+ ∆<0, B<0(b) V = 0.053 eV 
+FIG. 5: (Color online) [(a) and (b)] P(k) for four combina-
+tions of ∆ and B, in absence (a) and the presence (b) of a
+small SIA ˜V.|∆|=0.07 eV and |B|=10 eV ˚A2. (c) The con-
+tourCis used to count the number of the pairs of the zeros
+ofP(k), which form a circle ring when ∆ B >0.
+In the presence of SIA, ˜Vcouples the blocks h+and
+h−, so theτz-dependent Hall conductance becomes non-
+sense. Following Kane and Mele4, we can employ the Z 2
+topological classification to give a criterion of the QSH
+phase, because it does not require inversion symmetry as
+a necessary condition. The Z 2index can be obtained
+by counting the number of pairs of complex zeros of
+P(k)≡Pf[A(k)], where the Pfaffian is defined as
+Pf[A(k)] =1
+2nn!/summationdisplay
+Permutationsof {i1,...,i2n}(−1)NAi1i2...Ai2n−1i2n,
+(62)
+in whichNcounts the number of times of permutations,
+andA(k) is a 2norder anti-symmetric matrix defined by
+the overlaps of time reversal
+Aij(k) =/an}b∇acketle{tui(k)|Θ|uj(k)/an}b∇acket∇i}ht (63)withi,jrun over all the bands below the Fermi surface,
+i.e.,ψ1−andψ2−in the present case according to Eqs.
+(58) and (59). Based on the spin nature of the basis
+states{Φ1,Φ4,Φ2,Φ3}in our effective model, the time-
+reversaloperatorhere is defined as Θ ≡iσx⊗σyK, where
+σxandσyare thex- andy-component of Pauli matrices,
+respectively, and Kthe complex conjugate operator. The
+number of pairs of zeros can be counted by evaluating
+the winding of the phase of P(k) around a contour C
+enclosing half of the complex plane of k=kx+iky,
+I=1
+2πi/contintegraldisplay
+Cdk·∇klog[P(k)+iδ].(64)
+Becausethemodelisisotropic,wecanchoose Ctoenclose
+the upper half plane, the integral then reduces to only
+the path along kx-axis while the part of the half-circle
+integral vanishes for δ>0 and|k| →+∞.
+In the absence of the SIA term, P(k) is found for the
+Hamiltonian (44) as
+P(k) =−∆
+2+Bk2
+/radicalBig
+(∆
+2−Bk2)2+(/planckover2pi1vF)2k2,(65)
+in which one can check that the zero points exist only
+whenk2= ∆/2B >0, and form a circle ring. Along
+kx-axis only one of a pair of zeros in the ring is enclosed
+in the contour C, which gives a Z2indexI= 1. This
+defines the nontrivial QSH phase, and is in consistence
+with the conclusion by the Hall conductance in Eq. (61).
+In the presence ofa small SIA term ˜V </planckover2pi1vF/radicalbig
+|∆/2B|,
+with the help of the eigen wavefunctions (58) and (59),
+realP(k) can be found (after a U(1) rotation) as
+P(k) =(t+Ein
+−)(t+Eout
+−)+|˜V|2−(/planckover2pi1vFk)2
+2/radicalBig
+Ein
+−Eout
+−(t+Ein
+−)(t+Eout
+−)
+×/braceleftBigg
+sgn(/planckover2pi1vFk−|˜V|),∆>0
+1, ∆≤0,(66)
+where the sgn is to secure the continuity of P(k). One
+can check that P(0) =−sgn(∆) and P(∞) = sgn(B)
+. Besides, for a small ˜V, the behavior of P(k) between
+P(0) andP(∞) will not change qualitatively (see Fig.
+5). Therefore, for ∆ B >0,P(kx,0) should still have odd
+pairs of zeros. For a large ˜V≥/planckover2pi1vF/radicalbig
+|∆/2B|,
+P(k) =(t+Ein
+−)(t+Eout
+−)+|˜V|2−(/planckover2pi1vFk)2
+2/radicalBig
+Ein
+−Eout
+−(t+Ein
+−)(t+Eout
+−)
+×/braceleftBigg
+sgn(/planckover2pi1vFk−|˜V|), B <0
+1, B ≥0.(67)
+One can check for this case P(0)P(∞) is always positive
+thusP(k) has even pairs of zeros, regardless of the signs
+and values of ∆ and B. In other words, a large SIA will
+always destroy the quantum spin Hall phase.10
+C. Thin film Bi 2Se3and QSH effect
+TABLE II: Fitting parameters to the Bi 2Se3thin films, using
+the energy spectra Eq. (56) from our effective model.[adopte d
+from Ref.28]
+Layers E0D ∆ B v F|˜V|
+(QL) (eV) (eV ˚A2) (eV) (eV ˚A2) (105m/s) (eV)
+2 -0.470 -14.4 0.252 21.8 4.47 0
+3 -0.407 -9.7 0.138 18.0 4.58 0.038
+4 -0.363 -8.0 0.070 10.0 4.25 0.053
+5 -0.345 -15.3 0.041 5.0 4.30 0.057
+6 -0.324 -13.0 0 0 4.28 0.068
+Recently, thickness-dependent band structure of
+molecular beam epitaxy grown ultrathin films Bi 2Se3
+was investigated by in-situ angle-resolved photoemission
+spectroscopy28. An energy gap was first observed exper-
+imentally in the surface states of Bi 2Se3below the thick-
+ness of six quintuple layers, which confirms theoretical
+prediction as a finite size effect30–33.
+Table II shows the fitting parameters to the ARPES
+data of Bi 2Se3thin films28using the energy spectra for-
+mula [Eq. (56)]. For the films with thickness ranging
+from 2 QL to 5 QL, all of them satisfy sgn(∆ B)>0 and
+˜V </planckover2pi1vF/radicalbig
+|∆/2B|, hence the films are possibly in the
+QSH regime. We identify that only 2 QL, 3 QL, and 4
+QL belongto the nontrivialcasefor potential QSHeffect.
+5 QL is an exceptional case that the fitted parameters B
+andDdo not satisfy the existence condition of an edge
+states solution33. The condition of B2<D2will lead to
+the band gap closing for a large k. However, it is under-
+stood that the model is only valid near the Γ point, and
+the fitting parameters are limited to the case of small k.
+And the band gap was measured clearly for the film of
+5QL.
+It was previously predicted, using the parameters from
+the first-principles calculation19, that the gap ∆ should
+oscillateasafunctionofthefilmthickness30–32. However,
+this oscillation is not reflected in the measured results.
+D. QSH effect of SIA and the edge states
+In the quantum Hall effect the Chern number of the
+bulk states has an explicit correspondence to the num-
+ber of edge states in an open boundary condition39. In
+topological insulator or QSH system, the Z 2topological
+invariant has also a relation to the number of helical edge
+states40. As a supplementary support to the above con-
+clusion, we demonstrated the presence of edge states in
+a periodic boundary condition along the x-direction and
+an open boundary condition (say along y-direction) im-
+posed in a geometry of strip of the thin film by means
+of numerical calculation. Using the parameters in TableI, we have concluded that a stripe of 2 - 4 QL will ex-
+hibit helical edgestates. Morespecifically, we presentthe
+energy dispersion for 4QL in Fig. 6. There is a doubly-
+degenerate Dirac point inside the gap of the 2D surface
+states for 4 QL in consistence with the results obtained
+in the above sections.
+−0.1−0.05 0 0.05 0.1
+−0.1 −0.05  0  0.05  0.1E (eV)
+kx(Å−1)4QL
+FIG. 6: (Color online) The energy spectra by the tight bind-
+ing calculation for 4 QL of Bi 2Se3thin films with the width
+alongy-direction 200 nm. The parameters are given in table
+II.
+VI. CONCLUSIONS
+We derived two-dimensional effective continuous mod-
+els for the surface states and thin films of three-
+dimensional topological insulators (3DTI). A gapless
+Diracconewasconfirmedforthesurfacestatesofa3DTI.
+For a thin film, the coupling between opposite topologi-
+cal surface states in space opens an energy gap, and the
+Dirac cone evolves into a gapped Dirac hyperbola. The
+thin film may break the top-bottom symmetry. For ex-
+ample, the thin film grows on a substrate, and possesses
+thestructuralinversionasymmetry(SIA). ThisSIAleads
+to a Rashba-like coupling and energy splitting in the mo-
+mentum space. It also leads to asymmetric distributions
+of states along film growth direction.
+The ARPES measurements on Bi 2Se3films have
+demonstrated that the surface spectra opens a visible en-
+ergygap when the thickness is below 6QLs.28The energy
+gap was observed to be a function of the thickness of thin
+film, andinagoodconsistencewiththeoreticalprediction
+as a finite size effect of the thickness of thin film. The
+Rashba-like splitting was measured clearly in the thin
+film of 2 to 6 QLs. This can be explained very well from
+the inclusion of the SIA. Since the thin film was grown
+on a SiC substrate and the other surface is exposed to
+the vacuum this fact results in the SIA in the thin film.
+Another direct evidence to support the SIA is the sig-
+nal intensity pattern of the energy spectra of ARPES.11
+Usually the surface states are located dominantly near
+the top and bottom surfaces. The signal intensity for
+these two branches of energy spectra of ARPES are dif-
+ferent. The SIA will cause the coupling between two
+surface states near their crossing point. That is why the
+Rashba-like splitting of the ARPES spectra has a bright
+crossing point near the Γ point, with one branch bright
+and the other almostinvisible. Thus the SIA term can be
+used to describe the ARPES measurements on the thin
+film Bi 2Se3very well.
+Our effective model demonstrates that the 3DTI can
+be reducedto an two-dimensionalquantum spin Hall sys-
+tem due to the spatial confinement. Strictly speaking,
+the system is no longer a 3DTI in the original sense once
+the energy gap opens in the surface bands, since the Z2
+invariant for the bulk states becomes zero. However the
+surface bands themselves may contribute a non-trivial
+one in the Z2 invariant even when the SIA term is in-
+cluded. Our calculation demonstrates that a strong SIA
+always intends to destroy the quantum spin Hall effect.
+A critical value for SIA exists, at the point there is a
+transition from a topologicaltrivial to non-trivial phases.
+Based on the model parameters fitted from the experi-
+mental data of ARPES, we conclude that the thin film
+Bi2Se3should exhibit quantum spin Hall effect once the
+energygap opens in the surfacespectra due to the spatial
+confinement of the thin film.
+Acknowledgments
+We thank Ke He and Qi-Kun Xue for providing ex-
+perimental data prior to publication, and Qian Niu for
+helpful discussions. This work was supported by the Re-
+search Grant Council of Hong Kong under Grant No.
+HKU 7037/08P and HKU 10/CRF/08.
+VII. APPENDIX: MODEL PARAMETERS ˜A2
+AND˜V
+TABLE III: Four possible combinations of λ1andλ2, accord-
+ing to Eq. (5), and resulting f±andη1,2according to Eq.
+(38). According to Eq. (5), λ2
+1< λ2
+2, so there does not exist
+a case when λ1is real and λ2is pure imaginary.
+λ1 λ2 f± η1,2
+case A λ∗
+2 λ∗
+1 imaginary real
+real real real real
+case B imaginary real real real
+imaginary imaginary real realIn this appendix, we demonstrate that both the pa-
+rameters ˜A2and˜Vin the effective model (40) can be
+either real or purely imaginary, and the product of ˜A2˜V
+must be pure imaginary. By putting the wavefunctions
+TABLE IV: Four possible groups of E+andE−, and resulting
+values of ˜A2and˜V.
+E+ E−/tildewideA2/tildewideV
+case A case A imaginary real
+case A case B real imaginary
+case B case A real imaginary
+case B case B imaginary real
+Eqs. (32) and (33) into the definitions in Eqs. (42) and
+(52), we have
+˜A2=iA1A2D+C+C−/integraldisplayL
+2
+−L
+2dz[η−
+2f+∗
++f−
+++η+∗
+1f+∗
+−f−
+−]
+˜V=C+C−/integraldisplayL
+2
+−L
+2dzV(z)[D2
++η+∗
+1η−
+2f+∗
+−f−
+++A2
+1f+∗
++f−
+−].
+(68)
+For arbitrary energy, Eq. (5) requires that the values of
+λ1andλ2can only be one of the combinations shown in
+Tab. III. By putting these combinations into Eq. (34)-
+(37), one can show that for the first entry, η1andη2are
+real whilef+andf−are pure imaginary, referred as the
+case A; while for entries 2 - 4, all of η1,η2,f+, andf−
+are real, referred as the case B. For an arbitrary group of
+E+andE−, each of them belongs to either the case A or
+B, leading to four possibilities, as shown in Tab. IV. In
+particular, according to Eq. (15), when A2
+1/(−D+D−)>
+4M/B1, there is no complex λ1,2, corresponding to the
+last row of Tab. IV, i.e., ˜A2is pure imaginary while ˜Vis
+real.
+References
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+045125
\ No newline at end of file
diff --git a/1001.0527.txt b/1001.0527.txt
new file mode 100644
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+arXiv:1001.0527v2  [nucl-th]  1 Apr 2013Squeezed correlations of strange
+particle-antiparticles
+Sandra S Padula1, Danuce M Dudek1and O Socolowski Jr2
+1Inst. F´ ısica Te´ orica - UNESP, C. P. 70532-2, 01156-970 S˜ ao Pa ulo, SP, Brazil
+2Departamento de F´ ısica, FURG, C. P. 474, 96201-900 Rio Grande, RS, Brazil
+E-mail:padula@ift.unesp.br
+Abstract. Squeezed correlations of hadron-antihadron pairs are predicted to appear
+if their masses are modified in the hot and dense medium formed in high e nergy heavy
+ion collisions. If discovered experimentally, they would be an unequivo cal evidence
+of in-medium mass shift found by means of hadronic probes. We discu ss a method
+proposed to search for this novel type of correlation, illustrating it by means of Ds-
+mesons with in-medium shifted masses. These particles are expecte d to be more easily
+detected and identified in future upgrades at RHIC.
+PACS numbers: 25.75.Gz, 25.75.-q, 21.65.Jk
+1. Introduction: The hadronic squeezed states
+About ten years ago, M. Asakawa, T. Cs¨ org˝ o and M. Gyulassy[1] finalized a
+model description for the effects of in-medium hadronic mass modific ation leading to
+correlations of boson-antiboson pairs, also known as Back-to-Ba ck Correlations (BBC).
+This type of correlation between a particle and its own antiparticle wa s first noted by
+R. Weiner et al.[2]. Within a short period after the final proposition in Re f.[1], P. K.
+Panda et al.[3] showed that similar correlations between a fermion and a antifermion
+should appear, if their masses were shifted in the hot and dense med ia formed in high
+energy heavy ion collisions.
+Inboththebosonicandthefermioniccases, thein-mediumquasi-pa rticlesproduced
+in those collisions are related to the asymptotic, observed particles , by means of a
+Bogoliubov-Valatin (BV) transformation. This transformation links the creation and
+annihilation operators in both environments, i.e., the asymptotic ope ratorsaanda†, to
+their in-medium counterparts, bandb†. The corresponding Hamiltonians are given by
+H0andHm=H0+H′, whereH′contains the parameter expressing the mass-shift.
+The BV transformation relating the operators a(a†) tob(b†) are given by
+ak=ckbk+s∗
+−kb†
+−k;a†
+k=c∗
+kb†
+k+s−kb−k, whereck= cosh(fk) andsk= sinh(fk). The
+argument is the squeezing parameter , named in this way because the BV transformation
+isequivalenttoasqueezingoperation. Itiswrittenas fk=1
+2log/parenleftBig
+ωk
+Ωk/parenrightBig
+,withω2
+k=k2+m2,
+mbeing the asymptotic mass, Ω2
+k=k2+m2
+∗,m∗being the in-medium modified mass,Squeezed correlations of strange particle-antiparticles 2
+andkis the momentum. A constant mass-shift is considered here, homog eneously
+distributed over all the system, and related to the asymptotic mas s bym∗=m±δM.
+More generally, however, it could be a function of the coordinates in side the system
+and the momenta, δM=δM(|r|,|k|). Both the bosonic and the fermionic squeezed
+correlations are positive, have unlimited intensity, and are also desc ribed by similar
+formalisms[1,3]. ThisisillustratedinRef. [3]forthecaseof φφand ¯pppairs, evidencing
+the resemblance of the correlations for bosons and for fermions o f similar asymptotic
+masses. In the remainder of this paper, we will focus on the bosonic case.
+The effects of the shifted mass on the squeezed particle-antipart icle correlations
+can be understood by analyzing the the joint probability for observ ing two particles,
+N2(k1,k2) =ωk1ωk2/bracketleftBig
+/angbracketlefta†
+k1ak1/angbracketright/angbracketlefta†
+k2ak2/angbracketright+/angbracketlefta†
+k1ak2/angbracketright/angbracketlefta†
+k2ak1/angbracketright+/angbracketlefta†
+k1a†
+k2/angbracketright/angbracketleftak2ak1/angbracketright/bracketrightBig
+, which
+results fromtheapplicationofageneralization ofWick’s theoremfor locallyequilibrated
+systems[4, 6]; /angbracketleft.../angbracketrightmeans thermal averages. In this expression, all three contribut ions
+could survive when the boson is its own antiparticle, as is the case of φ-mesons or π0’s.
+The last term is in general identically zero. However, if the particle’s m ass is shifted
+in-medium, it can contribute significantly. It is identified with the squa re modulus of
+the squeezed amplitude, Gs(k1,k2) =√ωk1ωk2/angbracketleftak1ak2/angbracketright. The first term corresponds to the
+product of the spectra of the two identical bosons, N1(ki)=ωkid3N
+dki=ωki/angbracketlefta†
+kiaki/angbracketright, and the
+second term, to the identical particle contribution, identified with t he square modulus
+of the chaotic amplitude, Gc(k1,k2) =√ωk1ωk2/angbracketlefta†
+k1ak2/angbracketright.
+The full two-particle correlation function for φφorπ0π0can be written as
+C2(k1,k2) = 1+|Gc(k1,k2)|2
+Gc(k1,k1)Gc(k2,k2)+|Gs(k1,k2)|2
+Gc(k1,k1)Gc(k2,k2). (1)
+In case of charged mesons, such as D±
+s, the terms in Eq.(1) would act independently,
+i.e., either the first and the second terms together would contribut e to the HBT effect
+(D±
+sD±
+s), or the first and the last terms, to the BBC effect ( D+
+sD−
+s).
+In Refs.[1, 3], an infinite system was considered. Later, a finite expa nding
+system was studied, within a non-relativistic approach, assuming flo w-independent
+squeezing parameter, which allowed for obtaining analytical solution s to the problem[7].
+Considering a hydrodynamical ensemble[1, 5, 7] the squeezed amplit ude results in
+Gs(k1,k2) =E1,2
+(2π)3
+2|c0||s0|/braceleftBig
+R3exp/bracketleftBig
+−R2
+2(k1+k2)2/bracketrightBig
++2n∗
+0R3
+∗exp/bracketleftBig
+−(k1−k2)2
+8m∗T/bracketrightBig
+×exp/bracketleftBig/parenleftBig
+−im/angbracketleftu/angbracketrightR
+2m∗T∗−1
+8m∗T∗−R2
+∗
+2/parenrightBig
+(k1+k2)2/bracketrightBig/bracerightBig
+, (2)
+and the spectrum in
+Gc(ki,ki) =Ei,i
+(2π)3
+2/braceleftBig
+|s0|2R3+n∗
+0R3
+∗(|c0|2+|s0|2)exp/parenleftBig
+−k2
+i
+2m∗T∗/parenrightBig/bracerightBig
+,(3)
+whereR∗=R/radicalbig
+T/T∗andT∗=T+m2/angbracketleftu/angbracketright2
+m∗[7]. For the sake of simplicity, the system
+was supposed to be Gaussian in shape, with a cross-sectional area of radius R, andT
+is the freeze-out temperature; R∗andT∗are, respectively, the flow-modified radius and
+temperature. The flow velocity, introduced before estimating the results in Eqs. (2)Squeezed correlations of strange particle-antiparticles 3
+and (3), was written as v=/angbracketleftu/angbracketrightr/R, where the values /angbracketleftu/angbracketright= 0,0.23 or 0.5 were used in
+the present work. For finite particle emissions, we considered a Lor entzian distribution,
+F(∆t) = [1 + ( ω1+ω2)2∆t2]−1, multiplying the third term in Eq. (1). We adopt here
+¯h=c= 1. The results in Eq.(2) and (3) are then introduced, respectively , in the third
+and first terms of Eq.(1), leading to the squeezed correlation func tion.
+2. Illustrative results
+We previously applied the analytical results shown above to φ-meson squeezed
+correlations, and later, to K+K−pairs. We investigated the behavior of the squeezed
+correlation function for precise back-to-back pairs, i.e., for part icle-antiparticle pairs
+emitted with exactly opposite momenta, studying Cs(/vectork,−/vectork,m∗) as a function of |/vectork|and
+m∗. Preliminary results for those particles where shown in previous mee tings[7]-[13].
+Recent results considering φφpairs from simulation are in Ref. [14].
+Figure 1. Squeezed correlation function versus the shifted mass and the
+momenta of the particles for back-to-back D+
+sD−
+spairs.
+The analytical results can be applied to any other particles subject ed to in-medium
+mass-shift and compatible with the non-relativistic limit considered in t he formulation.
+Recently, STAR reconstructed D0+¯D0’s through their decay into K∓π±, by measuring
+the invariant mass distribution of those decay products[15]. The ide ntification of D-
+mesons could be improved after the detector’s upgrade. Motivate d by this possibility,
+andconsideringthatchargedmesonsmaybeeasiertoobserve, we applythatformulation
+to the case of D+
+sD−
+spairs. Similar to what was previously done for φφandK+K−,
+we analyze the behavior of Cs(/vectork,−/vectork,m∗) forD+
+sD−
+spairs in the ( |/vectork|,m∗)-plane. Fig. 1
+shows that the intensity of Cs(k,−k,m∗) vs.|k|vs.m∗increases for decreasing freeze-
+out temperatures, which can be seen by comparing the two left plot s withT= 220
+MeV, with the bottom right plot with T= 140 MeV, in all of which ∆ t= 2 fm/c. Fig.
+1 also illustrates another striking feature, i.e., finite emission interva ls can dramaticallySqueezed correlations of strange particle-antiparticles 4
+reduce the strength of the squeezed correlation function. This c an be seen by comparing
+the two plots in the right panel: a Lorentzian emission distribution with ∆t= 2 fm/c
+has the effect of reducing the signal by about three orders of mag nitude, as compared
+to the instantaneous emission (∆ t= 0). Finally, the left panel shows how the behavior
+ofCs(k,−k,m∗) is affected by the presence of flow. We see that the growth of the
+signal’s intensity for increasing values of |k|is faster in the static case than when
+< u >/negationslash= 0. Nevertheless, flow seems to enhance the overall intensity of Cs(k,−k,m∗) in
+thewhole regionof |k|investigated. Naturally, at m∗=mDs∼1969MeV, thesqueezing
+disappears, i.e., Cs(k,−k,m∗)≡1.
+A panoramic view of the variation of Cs(k,−k,m∗) in the ( |k|,m∗)-plane can be
+better appreciated by contour plots of the previous results, as s hown in Fig. 2.
+0.20.40.60.81
+1.96 1.98
+0.20.40.60.81
+1.96 1.980.20.40.60.81
+1.8 2 2.2
+0.20.40.60.81
+1.8 2 2.2
+Figure 2. Contour plots of the results of Fig. 1, correspondi ng to the
+squeezed correlation functions vs. |k|andm∗.
+We can see that the location of the maxima of Cs(k,−k,m∗) forD+
+sD−
+spairs
+is shifted about ≈2−2.5% from the value of the particle’s asymptotic mass, either
+above or below it. A similar behavior could be observed in the case of φφpairs [8]-
+[14]. However, for K+K−pairs, the maximum was shifted about 30% from the K’s
+asymptotic mass [12, 13], perhaps signaling to the limit of applicability of the non-
+relativistic approximations considered in the model, in this case.
+The outcome properties shown above are important for understa nding the expected
+behavior ofthesqueezed correlationfunction, withintheapproxim ationsoftheproposed
+model. However, for practical purposes of searching for it exper imentally, the approach
+analyzed so far is not very helpful, since the modified mass of particle s is not
+a measurable quantity, existing only inside the hot and dense medium. Besides,
+the measurement of particle-antiparticle pairs with exactly back-t o-back momenta is
+unrealistic. Thus, a promising form to empirically search for the hadr onic squeezed
+correlation was proposed, in analogy with HBT [11]-[14], i.e., to measure the squeezedSqueezed correlations of strange particle-antiparticles 5
+correlation function in terms of the momenta of the particles combin ed asK12=
+1
+2(k1+k2), andq12= (k1−k2). Ina relativistic treatment, as proposed by M. Nagy[11],
+we should construct the momentum variable as Qµ
+back= (ω1−ω2,k1+k2) = (q0,2K).
+In fact, it is preferable to redefine this variable as Q2
+bbc=−(Qback)2= 4(ω1ω2−KµKµ),
+whose non-relativistic limit is Q2
+bbc→(2K12)2, correctly recovering that variable.
+The results for Cs(2K12,q12) are shown in Fig. 3. In the plots on the left panel,
+the radius of the system was fixed to be R= 4 fm, and on the right, to R= 7 fm. In
+both, it was assumed that the mass was shifted by the amount corr esponding to the
+lower maximum in Figs. 1 and 2, a relative reduction in the mass of about 2%.
+Figure 3. Squeezed correlation functions for R= 4fm (left panel) and
+R= 7fm (right panel), considering a reduction of the in-medium m ass to
+m∗= 1930MeV.
+From Fig. 3 we can see that radial flow ( /angbracketleftu/angbracketright= 0.23) does have an effect on
+Cs(2K12,q12), making it more intense if compared to /angbracketleftu/angbracketright= 0, in all the investigated
+region of the ( 2K12,q12)-plane, from about 40%, at low |q12|, to roughly 15%, at high
+|q12|. Fig. 3 also shows that the inverse width of Cs(2K12,q12) reflects the size of the
+squeezing region, being narrower for larger systems ( R= 7 fm) than for smaller ones
+(R= 4 fm). We also note that the intensity of the squeezed correlation function is high
+enough for stimulating its experimental search, even after applyin g the time reduction
+factor corresponding to an emission during a finite interval of ∆ t= 2 fm/c.
+3. Summary and conclusions
+The results of Fig. 1 and 2 showed that the squeezed correlation fu nction survives both
+finite system sizes and flow, with measurable intensity. The finite emis sion process has
+a strong reduction effect in its strength, even if it happens in a sudd en manner, as
+considered. The plots in the right panel of Fig. 1 show that the time m ultiplicative
+factor reduces Cs(k,−k,m∗) about three orders of magnitude, for D+
+sD−
+spairs. TheSqueezed correlations of strange particle-antiparticles 6
+left panel in Fig. 1 shows that, if the system is subjected to flow, th is could enhance
+the squeezed correlation signal, facilitating its potential discovery in experiments. The
+freeze-out temperature is an essential ingredient, the lower the temperature, the higher
+the intensity of Cs(k,−k,m∗). The expectations are that Ds’s would decouple early.
+From Fig. 1 we can see that, even in a high temperature limit and subje cted to the
+time-reducing factor, the squeezed correlation function still has measurable intensity.
+We should remember that, if the shift in the mass is away from the valu e considered
+in the calculation leading to Fig. 3, Cs(2K12,q12) would attain smaller values than the
+ones shown, but the signal could still be high enough to be searched for. Another
+important point to emphasize is that it is crucial to accumulate high st atistics and look
+for the effect in the 3 −Dconfiguration shown in the plots, since projecting it into
+theK12-axis can drastically reduce the signal, making it more difficult to discov er the
+hadronic squeezing effect. Finally, comparing with previous results f orφφ[8]-[14] and
+K+K−[11, 12, 13]-[16, 17], we can conclude that, within this model, the stre ngth of
+squeezed correlation function seems to grow with the asymptotic m ass of the particles
+involved, making it even more promising to look for BBC’s for heavier pa rticles.
+Acknowledgments
+We are grateful to FAPESP and CAPES for the support to participa te in the SQM ’09.
+OSJ acknowledges funding from CNPQ.
+References
+[1] Asakawa M, Cs¨ org˝ o T and Gyulassy M 1999 Phys. Rev. Lett. 834013
+[2] Andreev I V, Pl¨ umer M and Weiner R M 1991 Phys. Rev. Lett. 673475
+[3] Panda P K, Cs¨ org˝ o T, Hama Y, Krein G and Padula Sandra S 2001 Phys. Lett B51249
+[4] Gyulassy M, Kauffmann S K and Wilson L W 1979 Phys. Rev. C 202267
+[5] Makhlin A and Sinyukov Yu 1987 Sov. J. Nucl. Phys. 46354
+[6] Sinyukov Yu 1994 Nucl. Phys. A566589c
+[7] Padula Sandra S, Hama Y, Krein G, Panda P K and Cs¨ org˝ o T 2006 Phys. Rev. C73044906
+[8] Padula Sandra S, Hama Y, Krein G, Panda P K and Cs¨ org˝ o T 2006 Proc. Quark Matter ’05
+(Budapest) Nucl. Phys. A 774, 615
+[9] Padula Sandra S, Hama Y, Krein G, Panda P K and Cs¨ org˝ o T 2006 Proc. Workshop on Particle
+Correlations and Femtoscopy (Kromeriz) AIP Conf. Proc. 828, 645
+[10] Cs¨ org˝ o T and Padula Sandra S 2007 Proc. WPCF 2006 (S˜ ao Paulo) Braz. J. Phys. 37949
+[11] Padula Sandra S, Socolowski Jr O, Cs¨ org˝ o T and Nagy M 2008 Proc. Quark Matter 2008 (Jaipur)
+J. Phys. G: Nucl. Part. Phys. 35, 104141
+[12] Padula Sandra S, Dudek Danuce M and Socolowski Jr O, 2009 Pro c. WPCF 2008 (Krakow) A.
+Phys. Pol. 40, N. 4, 1225
+[13] Padula Sandra S, Socolowski Jr O and Dudek Danuce M 2009 Proc. ISMD 2008 (Hamburg)
+DESYPROC170917 271 (arXiv:0812.1784v1 (nucl-th) and arXiv:0902 .0377 (hep-ph) p271)
+[14] Padula Sandra S, Socolowski Jr O 2009 Searching for squeezed particle-antiparticle correlations in
+high energy heavy ion collisions arXiv:1001.0126-v1 (nucl-th).
+[15] Abelev B I et al., STAR Collaboration arXiv:0805.0364 (nucl-ex).
+[16] Dudek Danuce M 2009 Master DissertationSqueezed correlations of strange particle-antiparticles 7
+[17] Padula Sandra S and Dudek Danuce M, in preparation.
\ No newline at end of file
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+arXiv:1001.0528v1  [cond-mat.stat-mech]  4 Jan 2010Can one identify non-equilibrium in a three-state system by analyzing two-state
+trajectories?
+Christian P. Amann, Tim Schmiedl, and Udo Seifert
+II. Institut f¨ ur Theoretische Physik, Universit¨ at Stutt gart, 70550 Stuttgart, Germany
+For a three-state Markov system in a stationary state, we dis cuss whether, on the basis of data
+obtained from effective two-state (or on-off) trajectories, it is possible to discriminate between an
+equilibrium state and a non-equilibrium steady state. By ca lculating the full phase diagram we
+identify a large region where such data will be consistent on ly with non-equilibrium conditions.
+This regime is considerably larger than the region with osci llatory relaxation, which has previously
+been identified as a sufficient criterion for non-equilibrium .
+PACS numbers: 05.40.-a, 05.70.Ln, 82.37.-j
+For the emerging field of thermodynamics of small
+systems, or stochastic thermodynamics1–3, enzymes and
+proteins like molecular motors constitute paradigmatic
+systems since their conformational changes can be ob-
+served on a single molecule level using a variety of exper-
+imental techniques4. The analysis of trajectories record-
+ing the transition between microstates of the enzyme can
+reveal insights into both the underlying kinetics and the
+thermodynamic implications of such changes5–12. If the
+surrounding conditions like temperature or concentra-
+tions of other reactants are constant, the enzyme reaches
+a stationary state for which two classes must be distin-
+guished. First, in a genuine equilibrium state, no net
+flux between any two microstates of the enzyme occurs.
+Second, in a non-equilibrium steady state (NESS), the
+probability to find the enzyme in any microstate is still
+time-independent. However, non-zero net fluxes between
+microstates indicate that the system is externally driven
+with concomitant entropy production. If all microstates
+are experimentally accessible, in principle, distinguish-
+ing between these two alternatives is trivial, given long
+enough trajectories. It suffices to infer both the station-
+ary probabilities ps
+ifor the system to be in state iand
+the rates kijfor a transition between iandj, and then
+to check for genuine equilibrium in which the detailed
+balance condition ps
+ikij=ps
+jkjifor any two states must
+hold.
+In practice, rarely all relevant microstates are ex-
+perimentally accessible or distinguishable. The result-
+ing only coarse-grained information affects and aggra-
+vates a thermodynamic consistent analysis of individual
+trajectories13,14. In standard applications, often only a
+two-state trajectory is obtained in which each observ-
+able state may in fact contain several microstates. In
+an ambitious program, Flomenbom and co-workers15–19
+have investigated which information about the underly-
+ing topology of the state space, or ‘network’, can be in-
+ferredfromthe observationofsuchtwo-statetrajectories.
+But even if the underlying topology is known, two-state
+trajectories cannot reveal the full information like the
+value of all rates between microstates.
+For time-independent external conditions, arguably,
+the most relevant question is, whether or not such anenzyme acts in an equilibrium state or in a NESS. The
+simplest system for which this problem can be posed
+is a three-state system like the stochastic Michaelis-
+Menten scheme for an enzyme turning substrate into
+a product7,20,21. Suppose that two of the three states
+are experimentally not distinguishable, can we decide in
+which of the two classes of stationary states the enzyme
+is in, if we are given a sufficiently long two-state trajec-
+tory? It is well-known that oscillations in the correla-
+tion function of such a two-state trajectory imply that
+the underlying three-state system is in a NESS rather
+than in equilibrium22. Here, we show that the range of
+ascertained non-equilibrium can be extended much be-
+yond this oscillatory regime, i.e., even for decaying cor-
+relations it can be possible to infer an underlying NESS
+state. However, there will remain parameters for which
+the question posed in the title cannot be answered af-
+firmatively. Only if all three states are visible, like in
+dual-color flourescence correlation spectroscopy, one can
+discriminate between equilibrium and non-equilibrium
+steady states23,24.
+We consider a three-state Markov system with states
+i= 1,2,3 and transition rates kij. The dynamics of the
+probability pi(t) tobe instate iattimetthen isgoverned
+by the master equation
+∂tpi=3/summationdisplay
+j=1Kijpj (1)
+with rate matrix
+K=
+−k12−k13k21 k31
+k12−k21−k23k32
+k13 k23−k31−k32
+.(2)
+We assume that state 1 (the on-state) is fluorescently
+labelled and can be observed, but that it cannot be de-
+termined experimentally, whether the system is in state
+2 or state 3 (together called the off-state), as sketched in
+fig. 1. The experiment then yields an effective two-state
+trajectory which no longer follows a Markovian dynam-
+ics.
+Four quantities can be extracted from such two-state
+trajectories and identified with their theoretical corre-
+spondences as follows:2
+Fig. 1: Three-state Markov model with states 2 and 3 lumped
+into a compound state.
+(i) The waiting time distribution in state 1 is exponen-
+tial with a decay rate
+L≡(k12+k13). (3)
+(ii) The waiting time distribution woff(t) in the off-
+state can be calculated from a modified Markov model
+where state 1 is treated as an absorbing state which
+amountsto setting the rate constants k12andk13to zero.
+This yields the effective rate matrix
+/tildewideK=/parenleftbigg
+−k21−k23k32
+k23−k31−k32/parenrightbigg
+, (4)
+whichgovernsthetimeevolutionoftheprobabilities /tildewidep2(t)
+and/tildewidep3(t) to be in state 2 and 3, respectively, if we stop
+the dynamics when the system reaches state 1. The wait-
+ing time distribution then is the negative time derivative
+of the compound probability to be either in state 2 or
+state 3,
+woff(t) =−d
+dt(/tildewidep2(t)+/tildewidep3(t)). (5)
+This waiting time distribution is a sum of exponentials
+with exponents given by the eigenvalues ˜λiof the modi-
+fied rate matrix /tildewideKwhich can be calculated as
+˜λi=1
+2/parenleftBig
+−S±/radicalbig
+S2−T/parenrightBig
+(6)
+with
+S≡k21+k23+k31+k32and (7)
+T≡4(k21k31+k21k32+k23k31). (8)
+Ifboth decaytimes can be determined from experimental
+time traces, both quantities SandTcan be inferred.
+(iii) Last, one can measure the time-dependent proba-
+bilityp1(t) to be in state 1 with initial condition p1(0) =
+1. This quantity can also be calculated from the master
+equation (1), yielding a sum of exponentials
+p1(t) =ps
+1+c1eλ1t+c2eλ2t. (9)
+The exponents λiare eigenvalues of the rate matrix K
+(2) and can be written as
+λi=1
+2/parenleftBig
+−L−S±/radicalbig
+(S+L)2−T−M/parenrightBig
+(10)with
+M≡4k12(k23+k31+k32)
++4k13(k21+k23+k32).(11)
+The prefactors c1,2can be obtained from the linear com-
+bination of eigenvectors of the matrix Kwhich yield the
+initial state p1(0) = 1.
+In summary, the four quantities L, S, TandMcan be
+determined from experimental time traces, provided the
+statistics is good enough to allow fits parametrizing the
+given exponential behavior. In principle, one could de-
+rive some other set of four independent quantities rather
+thanL, S, T andM. Our choice is convenient due to
+two reasons. First, all four quantities are non-negative
+(even positive if no rate constant kijvanishes). Second,
+for the distinction between equilibrium and a NESS the
+absolute time-scale is irrelevant. This fact can be used to
+scale all times by 1 /Lleading to the three dimensionless
+quantities
+/hatwideS≡S/L,/hatwideT≡T/L2and/hatwiderM≡M/L2.(12)
+A rescaling of time in the original model would lead to
+five independent rate constants /hatwidekij≡kij/L(with/hatwidek12+
+/hatwidek13= 1). Hence given measured reduced rates /hatwideS,/hatwideTand
+/hatwiderMthere still remains a two-parameter manifold in the
+three-state system dynamics that is consistent with the
+observed two-state trajectory.
+The three-state system is in equilibrium if and only if
+the original rates obey the condition
+k12k23k31=k32k21k13. (13)
+An analysis of the functional dependence of the rates
+kijon/hatwideS,/hatwideT ,/hatwiderMand the equilibrium condition
+(13) assisted by Wolfram’s algebraic software package
+MATHEMATICA 7 leads to the identification of three
+regimesinthephasediagramshowninfig.2. Itischecked
+whether or not for given reduced rates /hatwideSand/hatwiderMpositive
+rate constants kijcompatible with (13) can be found.
+This is the case only if
+0</hatwideT/lessorequalslant/hatwideT1≡/hatwiderM/hatwideS−/hatwiderM2
+4, (14)
+which defines the regime denoted EQ/NESS in fig. 2. In
+this regime, the measured data is principally insufficient
+for discriminating a NESS from an equilibrium state.
+Hence for any measured /hatwideTwith/hatwideT >/hatwideT1the system is
+definitely in a NESS. This criterion constitutes our main
+result. We now show that this parameter space is much
+larger than the set of parameters for which oscillations in
+the correlation function occur. The latter property has
+previously been identified as a sufficient criterion for a
+NESS. Oscillations in the autocorrelation function p1(t)
+(9) occur if the decay rates λiin eq. (10) acquire an
+imaginary part, i.e., if
+/hatwideT >/hatwideT2≡(/hatwideS+1)2−/hatwiderM. (15)3
+/LParen1b/RParen1
+EQ/Slash1NESSOSC
+NESSNESS
+01234560.00.51.01.52.0
+ /LParen1a/RParen1
+EQ/Slash1NESSNESS
+0.0 0.1 0.2 0.3 0.40.0000.0020.0040.0060.0080.010
+/LParen1d/RParen1
+EQ/Slash1NESSOSC
+NESS
+0.00.51.01.52.02.50.00.51.01.52.02.5
+k23k31/LParen1c/RParen1
+EQ/Slash1NESSOSC
+NESS
+0 10 20 30 40020406080100
+Fig. 2: (color online) (a)-(c): Phase diagram in the ( /hatwiderM,/hatwideT) plane for different values of /hatwideSwith (a) /hatwideS= 0.1, (b)/hatwideS= 1.5 and (c)
+/hatwideS= 10. The white area EQ/NESS allows both equilibrium states and NESSs, the purple area NESScan contain only NESSs
+and the brown area OSCcorresponds to NESSs with oscillatory correlation functio n. TheOSCregime touches the EQ/NESS
+regime at points/parenleftBig
+2+2/hatwideS,−1+/hatwideS2/parenrightBig
+. The checkered area is excluded since no choice of non-negat ive rate constants will lead to
+such an ( /hatwiderM,/hatwideT) value. The green line represents equilibrium states with s ome of the rate constants equal to zero. The black
+lines are the projection of the subdomain of the phase diagra m shown in (d) that corresponds to the given /hatwideSvalue.
+(d): Phase diagram in the ( k23, k31) plane for fixed rate constants k13= 1,k21= 1,k32= 1 and k12= 4. For each point in
+the diagram, the parameters L, S, T andMare calculated and the three regimes accordingly identified . The blue hyperbola
+comprises the genuine equilibrium state according to eq. (1 3).
+This regime is denoted by OSCin fig. 2. It is easy
+to check that /hatwideT2(/hatwiderM)/greaterorequalslant/hatwideT1(/hatwiderM) (with equality only for
+/hatwiderM= 2(/hatwideS+1)). For all measured /hatwideTwith/hatwideT2/greaterorequalslant/hatwideT >/hatwideT1,
+denoted by NESSin fig. 2, our new criterion establishes
+a NESS on the basis of two-state trajectories which do
+not exhibit oscillatory behaviour. While for /hatwideS≃1 the
+newly established regionof NESSs beyond the oscillatory
+regime is rather small, see fig. 2(b), both for /hatwideS≪1 andfor/hatwideS≫1 the present analysis shows a large regime of
+newly ascertained NESSs. In the latter two cases, the
+oscillatory regime is, for /hatwideS/lessorequalslant0.5, totally absent or rather
+small, see figs. 2(a) and (c), respectively.
+For fixed /hatwideSthe phase diagrams of physically admissi-
+ble (/hatwiderM,/hatwideT) values are restricted to /hatwiderM/lessorequalslant4/hatwideSand/hatwideT/lessorequalslant/hatwideS2,
+since values outside this range would require negative kij
+values. Moreover, for /hatwiderM/lessorequalslant2/hatwideS, onlyT/lessorequalslant/hatwideT1can be re-4
+alized with non-negative kij. The region /hatwideS2/greaterorequalslantT >/hatwideT1,
+which is excluded by the latter condition, is shown check-
+ered in figs. 2(a)-(c). The equalities in these constraints
+of physically admissible values hold only if some of the
+rates are zero.
+Finally, an analysis of the boundary /hatwideT1(/hatwiderM), which re-
+quires some rate constants kijto be zero, shows that for
+0/lessorequalslant/hatwiderM/lessorequalslant2/hatwideSthe system is definitely in equilibrium on
+this boundary. For /hatwideT= 0 the system is also definitely in
+equilibrium.
+Forrepresentativeexamplesoftheseresultsinthe orig-
+inal space of the six rates kij, consider fig. 2(d). There
+we show the three regimes EQ/NESS ,NESSandOSCin
+the (k23,k31) plane holding the other four rate constants
+fixed. The line of genuine equilibrium (13) is embedded
+into region EQ/NESS where information from the state
+trajectory is not sufficient to uniquely identify equilib-
+rium. However for ( k23,k31) parameter values in the
+regionNESSour analysis predicts correctly a NESS be-
+yond the previously established OSCregion.
+Experimental measurements of the waiting time distri-
+butionwoff(t) (5) and autocorrelation function p1(t) (9)
+will cause some uncertainty due to finite data. For a
+rough estimate, we assume a relative error of ±εin all
+eigenvalues ˜λi(6) andλi(10), while it should be possible
+to determine the decay time L(3) with better precision.
+A linear error propagation then yields a constant relative
+error for /hatwideSand/hatwideTof±εand±2εrespectively. The ef-
+fect on/hatwiderMis a relative error of ±(2ε+ 4ε/hatwideT//hatwiderM). Thus
+error bars of experimentally determined pairs of ( /hatwiderM,/hatwideT)values would increase linearly on straight lines through
+the origin in fig. 2(a)-(c). Since the maximum value of /hatwideT
+increases quadratically with the maximum value of /hatwiderM,
+the uncertainty on the phase boundary between NESS
+andEQ/NESS in fig. 2(a)-(c) increases with increasing
+/hatwideS. Hence, a clear discrimination between the two regions
+wouldbeaffectedbyerrorslessinthesmall /hatwideSregimethan
+for large /hatwideS. For the data shown in fig. 2(a) and ε= 0.05,
+the phase boundary effectively broadens to about 11%
+on the top and 10% at the bottom.
+In summary, we have derived a criterion sufficient to
+identify a non-equilibrium steady state in a three-state
+system based on information extracted from sufficently
+long two-state trajectories. While the new criterion ex-
+tends the region of ascertained NESSs significantly be-
+yond the previously identified region based on oscillation
+in the autocorrelation function, there remain parameter
+valuesforwhichtwo-statetrajectoriescontainprincipally
+not enough information to discriminate NESS from an
+equilibrium state. It will be interesting to perform a
+similar analysis for four-state systems or even more com-
+plex networks, where one would expect that the region
+of ascertained NESSs on the basis of two-state trajec-
+tory information becomes relatively smaller than in the
+three-state case. Generalization of our approach to cases
+where three-state trajectories, obtained, e.g., via dual-
+colour fluorescence, are available on four or more-state
+networks is feasible in principle, but may, in practice,
+become algorithmically challenging.
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+arXiv:1001.0529v2  [cs.GT]  3 Feb 2010Symposium on TheoreticalAspects of Computer Science 2010( Nancy, France), pp. 107-118
+www.stacs-conf.org
+ON ITERATED DOMINANCE, MATRIX ELIMINATION, AND
+MATCHED PATHS
+FELIX BRANDT1AND FELIX FISCHER1AND MARKUS HOLZER2
+1Institut f¨ ur Informatik, Ludwig-Maximilians-Universit ¨ at M¨ unchen, 80538 M¨ unchen, Germany
+E-mail address :{brandtf,fischerf}@tcs.ifi.lmu.de
+2Institut f¨ ur Informatik, Universit¨ at Gießen, 35392 Gieß en, Germany
+E-mail address :holzer@informatik.uni-giessen.de
+Abstract. We study computational problems arising from the iterated r emoval of weakly
+dominated actions in anonymous games. Our main result shows that it is NP-complete
+to decide whether an anonymous game with three actions can be solved via iterated weak
+dominance. The two-action case can be reformulated as a natu ral elimination problem on
+a matrix, the complexity of which turns out to be surprisingl y difficult to characterize and
+ultimately remains open. We however establish connections to a matching problem along
+paths in a directed graph, which is computationally hard in g eneral but can also be used
+to identify tractable cases of matrix elimination. We finall y identify different classes of
+anonymous games where iterated dominance is in P and NP-comp lete, respectively.
+1. Introduction
+Ananonymous game ischaracterized bythefactthatplayersdonotdistinguish between
+other players in the game, i.e., their payoff only depends on t he number of other players
+playing the different actions, but not on their identities. An onymous games constitute a
+very natural class of multi-player games which is also highl y relevant in practice (cf. [7]).
+Symmetric games additionally have identical payoff functions for all p layers. A strategyof
+a player is a probability distribution over his actions, and we say that an action is weakly
+dominated if there exists a strategy of the same player guaranteeing hi m at least the same
+payoff for any combination of strategies of the other players , and strictly more payoff for
+some such combination.1Dominated actions may be discarded for the simple reason tha t
+the player will never face a situation where he would benefit f rom using these actions.
+The solution concept of iterated dominance works by removing a dominated action and
+applying the same reasoning to the reduced game (e.g., [15]) . A game is then called solvable
+by iterated dominance if there is a sequence of eliminations that leaves only one action for
+1998 ACM Subject Classification: F.2.2, J.4.
+Key words and phrases: Algorithmic Game Theory, Computational Complexity, Itera ted Dominance,
+Matching.
+1Some authors (e.g., [10, 13]) use the terms weak dominance or dominance to refer to a weaker notion
+that does notrequire the dominating strategy to sometimes yield a strict ly higher payoff. This notion is
+called very weak dominance by other authors (e.g., [2, 4]).
+c/circlecopyrtF.Brandt,F. Fischer,andM. Holzer
+CC/circlecopyrtCreativeCommonsAttribution-NoDerivsLicense108 F. BRANDT, F. FISCHER, AND M. HOLZER
+each player. Interestingly, anonymous games often arise in the context of voting, where
+dominance solvability was originally introduced [14].
+Unlike iterated strictdominance, which requires the dominating action in each ste p to
+be strictly better for everycombination of strategies of the other players, proper epis temic
+foundations for iterated weakdominance are fairly hard to come by (e.g., [3, 19]). Never-
+theless, iterated weak dominance is an established and well -studied solution concept that
+occurs in virtually every textbook on game theory. Its compu tational properties, however,
+are not well understood, particularly in restricted classe s like anonymous games. Potential
+computational hardness of iterated weak dominance stems fr om the fact that the result of
+the elimination process generally depends on the order in wh ich actions are eliminated.
+Related Work. Deciding whether a game in normal form can be solved by iterat ed weak
+dominance is NP-complete already for games with two players and two different payoffs and
+when restricted to dominance by pure strategies [10, 6]. In t wo-player constant-sum games,
+both solvability and eliminability of a given action become tractable, while reachability of
+a subgame remains NP-complete [4]. The corresponding probl ems forstrictdominance can
+generally be solved in polynomial time [6].
+All of the above results concern games with few players and an unbounded number
+of actions. Unlike general normal-form games, anonymous an d symmetric games allow for
+a succinct representation even when the number of players is unbounded. Computational
+aspects of these games, particularly with respect to Nash eq uilibrium, have recently come
+underincreasedscrutinyduetotheirimportanceinmodelin glargeanonymousenvironments
+like the Internet. A Nash equilibrium of a symmetric game can be found in polynomial time
+if the number of actions is not too large compared to the numbe r of players [16]. In the
+larger class of anonymous games, Nash equilibria admit a pol ynomial-time approximation
+scheme when there is only a constant number of actions [7]. Th e pure equilibrium problem
+is tractable in anonymous games with a constant number of act ions, and NP-complete if
+the number of actions grows in the number of players [5].
+Results and Paper Structure. We begin by introducing the relevant game-theoretic
+concepts. InSection 3weshowthatiterated dominancesolva bility isNP-hardforsymmetric
+games with an unbounded number of actions, and tractable for symmetric games with a
+constant numberof actions. Therest of thepaperis then conc erned withthe only remaining
+class, anonymous games with a constant number of actions. In Section 4, we show how
+the two-action case can be reformulated as a natural elimina tion problem on a matrix.
+The complexity of this problem remains open, but in Section 5 we draw connections to
+a matching problem on paths of a directed graph. The latter pr oblem, which may be of
+independent interest, is intractable in general but allows us to obtain efficient algorithms
+for restricted versions of matrix elimination. In Section 6 we finally use the matching
+formulation to show NP-hardness of iterated dominance in an onymous games with three
+actions. Proofs are omitted due to space constraints, and wi ll be given in the full version
+of the paper.
+2. Preliminaries
+An accepted way to model situations of strategic interactio n is by means of a normal-
+form game (e.g., [15]).ON ITERATED DOMINANCE, MATRIX ELIMINATION, AND MATCHED PAT HS 109
+Definition 2.1 (normal-form game) .Agame in normal-form is a tuple Γ=
+(N,(Ai)i∈N,(pi)i∈N), where Nis a finite set of playersand for each player i∈N,Ai
+is a finite set of actionsavailable to player iandpi: (/producttext
+i∈NAi)→Ris a function mapping
+each action profile, i.e., each combination of actions, to a r eal-valued payofffor player i.
+We write Si=∆(Ai) for the set of (mixed) strategies of player i∈N, i.e., the set of
+probability distributions over his actions, and call a stra tegypureif it selects some action
+with probability one. A vector s∈/producttext
+i∈NSiwill be called a strategy profile . Payoff functions
+naturally extend to strategy profiles, and we write pi(s) for the expected payoff of player i
+in strategy profile s. We further write n=|N|for the number of players in a game, sifor
+theith element of strategy profile s, ands−ifor the vector of all elements of sbutsi.
+We will henceforth concentrate on games where Ai=Afor alli∈Nand some set A.
+Such a game is anonymous if the payoff of player iis invariant under any automorphism π′:
+AN→ANof the set of actions profiles induced by a permutation π:N→Nof the set of
+players that satisfies π(i) =i(e.g., [5]). An intuitive way to describe anonymous games is
+in terms of equivalence classes of the automorphism group of π′, using a notion introduced
+by Parikh [18] in the context of context-free languages. Giv en a set Aof actions, the
+commutative image of an action profile aN∈ANis given by #( aN) = (#(a,aN))a∈Awhere
+#(a,aN) =|{i∈N:ai=a}|. In other words, #( a,aN) denotes the number of players
+playing action ain action profile aN, and #( aN) is the vector of these numbers for all
+the different actions. This definition naturally extends to ac tion profiles for subsets of the
+players. We consider four types of anonymity (cf. [5]).
+Definition 2.2 (anonymity) .LetΓ= (N,(Ai)i∈N,(pi)i∈N) be a normal-form game, Aa
+set of actions such that Ai=Afor alli∈N.Γis called
+•anonymous ifpi(aN) =pi(a′
+N) for alli∈Nand allaN,a′
+N∈ANwithai=a′
+iand
+#(a−i) = #(a′
+−i),
+•symmetric ifpi(aN) =pj(a′
+N) for alli,j∈Nand allaN,a′
+N∈ANwithai=a′
+jand
+#(a−i) = #(a′
+−j),
+•self-anonymous ifpi(aN) =pi(a′
+N) for alli∈Nand allaN,a′
+N∈ANwith #(aN) =
+#(a′
+N), and
+•self-symmetric ifpi(aN) =pj(a′
+N)foralli,j∈NandallaN,a′
+N∈ANwith#(aN) =
+#(a′
+N).
+When talking about anonymous games, we write pi(ai,x−i) for the payoff of player i
+under any action profile aNwith #(a−i) =x−i. For self-anonymous games, pi(x) is used to
+denote the payoff of player iunder any profile aNwith #(aN) =x. Unless noted otherwise,
+weassumethat anonymousgames aregivenexplicitly, i.e., a salist ofpayoffs forthedifferent
+commutative images.
+A well-known method for simplifying strategic games is the r emoval of actions that
+are weakly dominated by some strategy of the same player, in t he sense that playing
+the latter is never worse than playing the former and sometim es strictly better. The re-
+moval of one or more dominated actions may render additional actions dominated, which
+may then iteratively be removed. To make these notions preci se, we need some notation.
+Given a game Γ= (N,(Ai)i∈N,(pi)i∈N), call an elimination sequence of Γa finite se-
+quence (D1,D2,...,D k) of subsets of the disjoint union of the sets Ai, i.e.,Dj⊆ ∪i∈NA∗
+i
+for alljwith 1≤j≤k, where A∗
+i=Ai× {i}. For a set D⊆ ∪i∈NA∗
+i, denote110 F. BRANDT, F. FISCHER, AND M. HOLZER
+byΓ(D) the induced subgame of Γwhere the actions in Dhave been removed, i.e.,
+Γ(D) = (N,(A′
+i)i∈N,(pi|/producttext
+i∈NA′
+i)i∈N) whereA′
+i={a: (a,i)∈A∗
+i\D}.
+Definition 2.3 (iterated dominance) .LetΓ= (N,(Ai)i∈N,(pi)i∈N) be a game. An action
+di∈Aiis said to be (weakly) dominated by strategy si∈Siif for all b∈/producttext
+j∈NAj,
+pi(b−i,di)≤/summationtext
+ai∈Aisi(ai)pi(b−i,ai) and for at least one b∈/producttext
+j∈NAj,pi(b−i,di)</summationtext
+ai∈Aisi(ai)pi(b−i,ai). An elimination sequence ( D1,D2,...,D m) ofΓis called valid
+if either it is the empty sequence, or if ( D1,D2,...,D m−1) is valid in Γand every
+dm∈Dmis dominated in Γ(∪1≤j≤m−1Dj). An action a∈ ∪i∈NAiis called eliminable
+if there exists a valid elimination sequence ( D1,D2,...,D m) such that ais weakly dom-
+inated in Γ(∪1≤j≤mDj). Game Γis called solvable if it is possible to obtain a game
+where only one action remains for each player, i.e., if there exists a valid elimination se-
+quence (D1,D2,...,D m) such that Γ(∪1≤j≤mDj) = (N,(A′
+i)i∈N,(p′
+i)i∈N) with|A′
+i|= 1 for
+alli∈N.
+We call iterated dominance solvability ( IDS) and eliminability ( IDE) the computational
+problems that ask for solvability of a game and eliminabilit y of a particular action. In
+contrast to iterated strictdominance, which requires the inequality to be strict for ev ery
+action profile of the other players, the result of iterated we ak dominance depends on the
+order in which actions are removed, since the elimination of an action may render actions
+of another player undominated (e.g., [2]).
+Restricted types of iterated dominance can be obtained by re quiring that the dominat-
+ing strategy siis pure, or that the elements of an elimination sequence are s ingletons and
+actions thus have to be eliminated one at a time (e.g., [2]). A s far as dominance by pure
+and mixed strategies is concerned, we will frequently explo it that the two versions coincide
+in games with two actions, and also in games with only two differ ent payoffs [6]. All results
+hold for dominance by pure strategies andfor dominance by mixed strategies. Valid elim-
+ination sequences consisting of singletons possess a somew hat less complicated structure.
+We therefore in some cases restrict our attention to this spe cialization, and refer to the
+corresponding computational problems as stepwiseIDSandIDE. The results ultimately
+obtained for the two variants will be very similar. A different notion of solvability merely
+requires the remaining action profiles to yield a unique payo ff to each of the players (e.g.,
+[14]). We note, but do not show here, that all hardness and tra ctability results extend to
+this notion as well.
+3. Complexity of Iterated Dominance
+Intuitively, a large number of actions neutralizes the comp utational advantage obtained
+from anonymity, by allowing for a distinction of the players by means of the actions they
+play. The search for pure Nash equilibria, for example, is tracta ble for anonymous games
+with a constant number of actions, but becomes NP-hard as soo n as the number of actions
+grows in the number of players [5]. In the latter case, the siz e of the explicit representation
+growsexponentiallyinthenumberofplayers, andonewoulde xpectnaturalinstancesofsuch
+games to be described succinctly (cf. [16]). While as a matte r of fact the results of Brandt
+et al. [5] are established via a specific encoding of the payoff functions, namely Boolean
+circuits, they nevertheless provide interesting insights into the influence of restricted classes
+of payoff functions on the complexity of solving a game. We giv e a similar result for iterated
+dominance in self-symmetric games, hardness for the other c lasses follows by inclusion.ON ITERATED DOMINANCE, MATRIX ELIMINATION, AND MATCHED PAT HS 111
+Theorem 3.1. IDSandIDEare NP-hard for all four classes of anonymous games, even if
+the number of actions grows only logarithmically in the numb er of players, if only dominance
+by pure strategies is considered, and if there are only two di fferent payoffs.
+Inthecaseofsymmetricgames, iterateddominancebecomest ractablewhenthenumber
+of actions is bounded by a constant.
+Theorem 3.2. For symmetric and self-symmetric games with a constant numb er of actions,
+IDSandIDEcan be decided in polynomial time.
+In light of these two results, only one interesting class rem ains, namely anonymous
+games with a constant number of actions. To gain a better unde rstanding of the problem,
+we restrict ourselves even further to games with two actions . It turns out that in this case
+iterated dominance can be reformulated in a natural way as an elimination problem on
+matrices. The latter is the topic of the following section.
+4. A Matrix Elimination Problem
+LetΓ= ([n],({0,1})i∈N,(pi)i∈N) be a self-anonymous game with two actions for each
+player, and observe that the payoffs of Γcan be represented by a matrix XΓ= (xi,j)(n+1)×n
+theith row of which contains the payoff profile when exactly i−1 players play action 1,
+i.e.,xij=pj(i−1). It will be instructive to view iterated dominance elimin ation in Γin
+terms of the corresponding operations on the matrix XΓ. For now, we restrict our attention
+to the case where actions are eliminated one by one, and more g enerally consider matrices
+with an arbitrary number of rows and columns. It suffices to loo k at matrices whose entries
+are natural numbers.
+LetXbe anm×nmatrix with entries from the natural numbers. Call a column cofX
+increasing for an interval Iover therows of Xif theentries in caremonotonically increasing
+inI, with a strict increase somewhere in this interval. Analogo usly, call cdecreasing forI
+if its entries are monotonically decreasing in I, with a strict decrease somewhere in this
+interval. Say that cisactiveforIif it is either increasing or decreasing for this interval.
+Now consider a process that starts with Xand successively eliminates pairs of a row and a
+column. Rows will only beeliminated from the top or bottom, s uch that the remaining rows
+always form an interval over the rows of X. A column will only be eliminated if it is active
+for the remaining rows. Elimination of an increasing column is accompanied by elimination
+of the top row. Analogously, a decreasing column and the bott om row are eliminated at
+the same time. The process ends when no active columns remain .
+Let us define the problem more formally. For a set A,v∈An, anda∈A, denote
+by #(a,v) =|{ℓ≤n:vℓ=a}|thecommutative image ofaandv, and write v...k=
+(c1,c2,...,ck) for the prefix of vof length k≤n. Further denote [ n] ={1,2,...,n}and
+[n]0={0,1,...,n}.
+Definition 4.1 (matrix elimination) .LetX∈Nm×nbe a matrix. Call a column k∈[n]
+ofXincreasing in an interval [ i,j]⊆[m] if the sequence xik,xi+1,k,...,x jkis monotonically
+increasing and xik< xjk,decreasing in [i,j]⊆[m] ifxik,xi+1,k,...,x jkis monotonically
+decreasing and xik> xjk, andactiveif it is either increasing or decreasing. Then, an
+elimination sequence of length kforXis a pair ( c,r) such that c∈[m]k,r∈ {0,1}k, and
+for alli,jwith 1≤i < j≤k,ci/\e}atio\slash=cjand either ri= 0 and column ciis increasing in112 F. BRANDT, F. FISCHER, AND M. HOLZER
+a b c d
+1321
+0221
+0230
+0230
+3230a b c d
+121
+021
+030
+030a b c d
+21
+21
+30a b c d
+1
+0
+Figure 1: A matrix and a sequence of eliminations
+[#(0,r...i−1)+1,m−#(1,r...i−1)], orri= 1 and column ciis decreasing in [#(0 ,r...i−1)+
+1,m−#(1,r...i−1)].
+Consider for example the sequence of matrices shown in Figur e 1, obtained by starting
+with the 5 ×4 matrix on the left and successively eliminating columns b,a,c, andd. In this
+particular example, the process ends when all rows and colum ns of the matrix have been
+eliminated. If instead we eliminated columns canda, no further eliminations would be
+possible. In fact, it would be obvious after the first elimina tion step that we cannot obtain
+a sequence of length 4: one of the columns not eliminated so fa r, column b, contains the
+same value in every row; this column cannot become active any more, and, as a consequence,
+will never be eliminated.
+What matters are not the actual matrix entries, but rather th e difference between
+successive entries in a column. A more intuitive way to look a t the problem may thus be in
+terms of a matrix with the number of rows reduced by one, and ar rows pointing downward
+or upward if the value increases or decreases between two adj acent entries. A column can
+be deleted if it contains at least one arrow, and if all arrows in this column point in the
+same direction. The corresponding row to be deleted is the on e at the base of the arrows.
+We will be interested in two computational problems. Matrix elimination ( ME) asks
+whether there exists an elimination sequence that deletes t he whole matrix, i.e., one of
+length min( m−1,n). Eliminability of a column ( CE) is given k∈[n] and asks whether
+there exists an elimination sequence ( c,r) such that for some i,ci=k. Without restrictions
+onmandn,MEandCEturn out to be equivalent. Indeed, both of them are equivalen t
+to the problem of deciding whether there exists an eliminati on sequence eliminating certain
+numbers of rows from the top and bottom of the matrix. Several other questions, like the
+one of an elimination sequence of a certain length, are equiv alent as well.
+Lemma 4.2. CEandMEare equivalent under disjunctive truth-table reductions.
+When restricted to the case m > n,CEis at least as hard as MEin the sense that the
+latter canbereducedtotheformerwhilethereisnoobvious r eductionintheotherdirection.
+The problem MEitself might be harder when the number of columns significant ly exceeds
+the number of rows, because then the set of columns effectively needs to be partitioned into
+two sets of sizes mandn−mof columns that have to be deleted and columns that can be
+discarded right away.
+It is not hard to see that elimination of a matrix Xis closely related to iterated dom-
+inance in the self-anonymous game described at the beginnin g of this section, where each
+player has two actions 0 and 1, and the payoff of player jwhen exactly i−1 players play
+action 1 is given by matrix entry xij. Given actions for the other players, player jcan
+choose between two adjacent entries of column j, so one of his two actions is dominated
+by the other one if the column is increasing or decreasing. El iminating one of two actionsON ITERATED DOMINANCE, MATRIX ELIMINATION, AND MATCHED PAT HS 113
+effectively removes a player from the game, and elimination of the top or bottom row of
+the matrix mirrors the fact that the number of players who can still choose between both
+of their actions is reduced by one. Let us formally establish this relationship.
+Lemma 4.3. StepwiseIDSandIDEin anonymous games with two actions are equivalent
+under disjunctive truth-table reductions to MEandCE, respectively, restricted to instances
+withm=n+1.
+We could have well allowed the simultaneous elimination of c olumns, and it is fairly
+obvious that the resulting computational problems would be equivalent to IDSandIDE.
+So why do we require columns to be eliminated one at a time? For one, solving MEand
+CEas defined above turns out to be intricate enough to begin with , and we will ultimately
+not be able to characterize their complexity. On the other ha nd, the additional structure
+afforded by stepwise elimination will help us to gain addition al insights, which we will then
+use to prove the main result of this paper: NP-hardness of IDSandIDEin games with three
+actions, both for stepwise and simultaneous eliminations. Finally, much of the complexity
+of matrix elimination already appears to be present in the st epwise version, and any result
+for that version can probably be extended to simultaneous el iminations as well.
+SolvingMEin general turns out to be surprisingly complicated. A natur al restriction
+can beobtained by requiring that all columns are increasing or decreasing in [1 ,m]. It is not
+too hard to show that this makes the problem tractable irresp ective of the dimensions of
+the matrix, and we do so in the next section as a corollary of a s lightly more general result.
+Unfortunately, tractability of this restricted case does n ot tell us a lot about the complexity
+ofMEin general. The latter obviously becomes almost trivial if t he order of elimination for
+the columns is known, i.e., if we are given c∈[n]kand ask whether there exists r∈ {0,1}k
+such that ( c,r) is an elimination sequence. This observation directly imp lies membership in
+NP. More interestingly, deciding whether there exists c∈[n]kfor a given r∈ {0,1}ksuch
+that (c,r) is an elimination sequence is also tractable. The reason is the specific “life cycle”
+of a column. Consider a matrix X, two intervals I,J⊆[m] over the rows of Xsuch that
+J⊆I, and a column c∈[n] that is active in both IandJ. Then,cmust also be active for
+any interval Ksuch that J⊆K⊆I, andcmust either be increasing for all three intervals,
+or decreasing for all three intervals. Thus, rdetermines for every i∈[k] a set of possible
+values for ci, and leaves us with a matching problem in a bipartite graph wi th edges in
+[n]×[k]. The latter can be solved in polynomial time. Closer inspec tion reveals that it can
+in fact be decomposed into two independent matching problem s on convex bipartite graphs,
+for which the best known upper bound is NC2[11].
+But whatif nothingabout candrisknown? Despitethefact that wecan onlyeliminate
+thetoporbottomrowofthematrixineach step, thisstillamo untstoanexponentialnumber
+of possible sequences. The best upper boundfor matching in c onvex bipartite graphs means
+that there currently is not much hope for constructing an alg orithm that determines r
+nondeterministically and computes a matching on the fly. We c an nevertheless use the
+above reasoning to recast the problem in the more general fra mework of matching on paths .
+For this, we will respectively identify intervals and pairs of intervals over the rows of X
+with vertices and edges of a directed graph G, and will then label each edge ( I,J) by the
+identifiers of the columns of Xthat take ItoJ. An elimination sequence of length kforX
+then corresponds to a path of length kinGwhich starts at the vertex corresponding to the
+interval [1 ,m], such that there exists a matching of size kbetween the edges on this path
+and the columns of X. In particular, by fixing a particular path, we obtain the bip artite114 F. BRANDT, F. FISCHER, AND M. HOLZER
+matching problem described above. A more detailed discussi on of this problem is the topic
+of the following section. We first study the problem itself, a nd return to matrix elimination
+toward the end of the section.
+5. Matched Paths
+The matching problem described in the previous section gene ralizes the well-studied
+class of matching problems between two disjoint sets, or bip artite matching problems, by
+requiring that the elements of one of the two sets form a certa in sub-structure of a combina-
+torial structure. Most interesting from a computational pe rspective are variants where the
+underlying combinatorial structure can be identified in pol ynomial time, as it is the case
+for paths or for spanning trees.
+Definition 5.1 (matching, matched path) .LetXbe a set, Σan alphabet, and σ:X→2Σ
+a labeling function assigning sets of labels to elements of X. Then, a matching ofσis a
+total function f:X→Σsuch that for all x,y∈X,f(x)∈σ(x) andf(y)/\e}atio\slash=f(x) ify/\e}atio\slash=x.
+LetG= (V,E) be a directed graph, Σan alphabet, and σ:E→2Σa labeling function
+for edges of G. Then, a matched path of length kinGis a sequence e1,e2,...,eksuch that
+for alliwith 1≤i < k, there exist u,v,w∈Vsuch that ei= (u,v) andei+1= (v,w), and
+the restriction of σto{ei: 1≤i≤k}has a matching.
+We call matched path( MP) thecomputational problemthat asks, foran explicitly giv en
+directed graph Gwith corresponding labeling function σand an integer k, whether there
+exists a matched path of length kinG. Variants of this problem can be obtained by asking
+for a matching that contains a certain set of labels, or a matc hed path between a particular
+pair of vertices. These variants have an interesting interp retation in terms of sequencing
+with resources and multi-dimensional constraints on the ut ilization of these resources: every
+resource can be used in certain states corresponding to vert ices of a directed graph, and
+their use causes transitions between states. The goal then i s to find a sequence that uses a
+specific set or a certain number of resources, or one that reac hes a certain state.
+In the context of this paper, we are particularly interested in instances of MPcorre-
+sponding to instances of ME. We will see later that the graphs of such instances are layer ed
+grid graphs (e.g., [1]), and that the labeling function sati sfies a certain convexity property.
+But let us look at the general problem for a bit longer. Greenl aw et al. [12] consider the
+relatedlabeled graph accessibility problem , which, given a directed graph Gwith a single
+label attached to each edge, asks whether there exists a path such that the concatenation
+of the labels along the path is a member of a context free langu ageLgiven as part of the
+input. This problem is P-complete in general and LOGCFL-com plete ifGis acyclic. A
+matching, however, corresponds to a partial permutation of the members of the alphabet,
+and the number of nonterminal symbols of any context-free gr ammar in Chomsky normal
+form for the permutation language over Σgrows super-polynomially in the size of Σ[8].
+It thus should not come as a surprise that the problem becomes harder when we ask for
+a matching. Indeed, MPbears some resemblance to the NP-complete problem forbidden
+pairsof finding a path in a directed or undirected graph if certain p airs of nodes or edges
+may not be used together [9]. Instead of reducing forbidden p airs toMP, however, we show
+NP-hardness of a restricted version of MPusing a more complicated construction, on which
+we will be able to build in Section 6. To formally state the res ult we need some terminology.ON ITERATED DOMINANCE, MATRIX ELIMINATION, AND MATCHED PAT HS 115
+LetG= (V,E) be a directed graph with vertex set V= [m]0×[n]0. Call (u,v)∈Ea
+south edge if for some iandj,u= (i,j) andv= (i+1,j), and an east edge if for some i
+andj,u= (i,j) andv= (i,j+ 1). Then, Gis called an m×nlayered grid graph if it
+contains only south and east edges. In labeled graphs, nonex istent edges and edges that are
+mapped to the empty set by the labeling function are equivale nt. We therefore concentrate
+oncomplete layered grid graphs, i.e., those containing all south and al l east edges.
+Theorem 5.2. MPis NP-complete. Hardness holds even if Gis a complete layered grid
+graph,|σ(e)| ≤1for every e∈E, and|{e∈E:λ∈σ(e)}| ≤2for every λ∈Σ.
+The proof of this theorem starts by looking at a complete m×ngrid graph Gfor
+appropriate values of mandn, and at a labeling function σ: [m]0∪[n]0→Σ. The
+latter can be interpreted as a labeling function for edges of Gwhere a label either appears
+on all the edges in a given row or column or on none of them. Labe ls inΣcorrespond
+to variable occurrences in an instance of the NP-complete pr oblembalanced one-in-three
+3SAT[17], and σis defined in such a way that a path through the graph correspon ds to
+an assignment of truth values to variable occurrences. The o verall structure of the graph
+consist of two parts. In the first part, consistency of the ove rall assignment is ensured by
+placing labels corresponding to different occurrences of the same variable on the same path.
+In the second part, the same labels are used again to verify th at all clauses are satisfied
+by the assignment. To obtain Theorem 5.2 and get a better unde rstanding of the minimal
+requirements for hardness, the graph is then modified furthe r. An important property of
+the labeling function in this context seems to be that the sam e label can appear at least
+twice in different parts of the graph.
+The labeling function σcan also more generally be interpreted as belonging to a more
+general graph where transitions can take place from any vert ex to any other vertex to the
+south and east of it, as long as the distance in columns betwee n the two vertices is at most
+the number of unused labels that appear on the row associated with the former vertex, and
+the same condition holds for the distance in rows and the numb er of labels on the column.
+Intuitively, this type of transition occurs when several do minated actions of a game are
+eliminated simultaneously. It will play an important role i n the proof of Theorem 6.1.
+Let us now return to matrix elimination. In light of Theorem 5 .2, an efficient algorithm
+forMEwould have to exploit additional structure of MPinstances induced by instances
+ofME. This structure is indeed quite restricted in that edges car rying a particular label λ
+satisfy a “directed” convexity condition: if λappears on two edges e= (u,v) ande′=
+(u′,v′), thenλmust appear on allsouth edges or on all east edges that lie on a path from u
+tov′, but not both. In particular, if there is such a path, it canno t be that one of eande′is
+a south edge and the other is an east edge. This fact is illustr ated in Figure 2, which shows
+the labeled graph for the MEinstance of Figure 1, as well as a matched path corresponding
+to an elimination sequence of maximum length.
+Definition 5.3 (directed convexity) .LetG= (V,E) be a complete layered grid graph. A
+labeling function σ:E→2ΣforGis called directed convex if for every label λ∈Σand for
+every set of three edges e1= (u1,v1),e2= (u2,v2),e3= (u3,v3), such that u2is reachable
+fromu1,u3is reachable from u2, andλ∈σ(e1)∩σ(e3), it holds that e1ande3have the
+same direction and λ∈σ(e2) if and only if e2has the same direction as well.
+It is not too hard to see that instances corresponding to MEhave a directed convex
+labeling function.116 F. BRANDT, F. FISCHER, AND M. HOLZER
+(0,0) (0,4)
+(1,3)
+(2,2)
+(3,1)
+(4,0){b,d}{a,b,d}{a,b,d}{a}
+{c}{c}{c}∅
+{b,d}{b,d}{b,d}
+{a,c}{c}{c}
+∅ ∅
+{a}∅
+∅
+{a}
+Figure 2: Labeled graph for the matrix elimination instance of Figure 1. A matched path
+and its matching are shown in bold.
+Lemma 5.4. MEis polynomial time many-one reducible to MPrestricted to layered grid
+graphs and directed convex labeling functions.
+Directed convexity of the labeling function means that we ca nnot show NP-hardness of
+MEby a construction similar to the one used in the proof of Theor em 5.2. On the other
+hand, it is not quite clear how the additional structure prov ided by directed convexity can
+be exploited to obtain a polynomial-time algorithm for ME. The case m≤nwill probably
+add additional complications. We therefore leave the compl exity ofMEas an open problem,
+albeit quite an elegant one.
+Here we consider a more special case of MP, which provides additional insights. In
+the corresponding instances of ME, all columns are active at the beginning of the matrix
+elimination process, or all columns are active in the interv al of length one at the end of the
+elimination process.
+Definition 5.5 (backward and forward closure) .LetG= (V,E) be a complete layered
+grid graph. Let sbe the unique vertex of Gwith in-degree zero, tthe unique vertex with
+outdegree zero. Then, a labeling function σ:E→2ΣforGis called backward closed if
+{λ∈σ(s,v) : (s,v)∈E}=Σ. Similarly, σis called forward closed if{λ∈σ(s,v) : (v,t)∈
+E}=Σ.
+It may not have gone unnoticed that these properties are clos ely related to closure prop-
+erties found respectively in matroids and antimatroids. To gether with directed convexity,
+each of the closure properties further implies that each lab el appears only on east edges or
+only on south edges. This allows us to consider two distinct m atching problems, one for
+east and one for south edges, and obtain a tractability resul t.
+Theorem 5.6. LetG= (V,E)be a complete layered grid graph, σa labeling function for G
+that is directed convex and either backward or forward close d. Then, MPforGandσcan
+be solved in nondeterministic logarithmic space.
+A generalization of both backward and forward closure can be obtained by considering
+labeling functions that are connected in the sense that the edges carrying a particularON ITERATED DOMINANCE, MATRIX ELIMINATION, AND MATCHED PAT HS 117
+0
+21 0 1
+30010 011
+0
+21 0 1
+30010 01
+0
+21 00010 01
+3 1010 01
+3
+2
+Figure 3: Payoffs of a particular player in a self-anonymous ga me with n= 3 and k=
+3. Initially all actions are pairwise undominated. If one of the other players
+eliminates action 1, action 3 weakly dominates action 1. Act ion 1 then becomes
+undominated if some player deletes action 3, and dominated b y action 2 if one
+more player deletes action 3, and some player deletes action 2.
+label, together with all edges in the respective other direc tion, form a weakly connected
+graph. This property introduces a dependence between the ma tching problems for the two
+directions, and a very interesting question is whether Theo rem 5.6 can be generalized to
+this setting.
+6. Self-Anonymous Games with a Constant Number of Actions
+It is natural to ask whether iterated dominance for games wit h more than two actions
+canstillbeinterpretedintermsofeliminationsinamatrix ormatrix-likestructure. Consider
+a self-anonymous game with kactions. As before, the payoff of a particular player ionly
+depends on the number of players, including the player itsel f, that play each of the different
+actions. They can thus be written down as entries in a discret e simplex of dimension k−1.
+The elimination of the ℓth action by some player can then be interpreted as a cut along
+theℓth 0-face of the simplex of every player.
+The left hand side of Figure 3 shows the payoffs of a particular p layer in a self-
+anonymous game with n= 3 and k= 3. Compared to matrix elimination as introduced
+in Definition 4.1 and illustrated in Figure 1, we notice an int eresting shift, which curiously
+has nothing to do with the added possibility of dominance by m ixed strategies. Rather,
+a particular action a∈Amay now be eliminated by either one of several other actions
+inA\{a}, and the situations where acan be eliminated no longer form a convex set.
+This already indicates that it might be possible to construc t a layered grid graph with
+corresponding labeling function for which the existence of a matched path is NP-hard to
+decide, and which is induced by a self-anonymous game with th ree actions for each player.
+To obtain our main result we however have to overcome one addi tional obstacle: when
+dropping the assumption that actions are eliminated one at a time, the equivalence between
+elimination sequences and labeled paths in a layered grid gr aph breaks down. We therefore
+start from the construction used in the proof of Theorem 5.2, and use additional vertices
+and labels to make it work for the more general type of transit ions corresponding to the
+simultaneous elimination of actions.
+Theorem 6.1. IDSandIDEare NP-complete. Hardness holds even for self-anonymous
+games with three actions and two different payoffs, and also ap plies to stepwise IDSand
+IDE.118 F. BRANDT, F. FISCHER, AND M. HOLZER
+Acknowledgements
+This material is based upon work supported by the Deutsche Fo rschungsgemeinschaft
+undergrants BR 2312/3-1 andBR 2312/3-2. We thankHermann Gr uber, Paul Harrenstein,
+Tim Roughgarden, Inbal Talgam, and Michael Tautschnig for v aluable discussions, and
+apologize to Edith Hemaspaandra for spoiling the sunset at W hite Sands.
+References
+[1] E. Allender, D. A. Mix Barrington, T. Chakraborty, S. Dat ta, and S. Roy. Grid graph reachability
+problems. In Proceedings of the 21st Annual IEEE Conference on Computati onal Complexity (CCC) ,
+pages 299–313, 2006.
+[2] K. R. Apt. Uniform proofs of order independence for vario us strategy elimination procedures. Contri-
+butions to Theoretical Economics , 4(1), 2004.
+[3] A. Brandenburger, A. Friedenberg, and H. J. Keisler. Adm issibility in games. Econometrica , 76(2):
+307–352, 2008.
+[4] F. Brandt, M. Brill, F. Fischer, and P. Harrenstein. On th e complexity of iterated weak dominance
+in constant-sum games. In M. Mavronicolas and V. G. Papadopo ulou, editors, Proceedings of the
+2nd International Symposium on Algorithmic Game Theory (SA GT), volume 5814 of Lecture Notes in
+Computer Science (LNCS) , pages 287–298. Springer-Verlag, 2009.
+[5] F. Brandt, F. Fischer, and M. Holzer. Symmetries and the c omplexity of pureNash equilibrium. Journal
+of Computer and System Sciences , 75(3):163–177, 2009.
+[6] V. Conitzer and T. Sandholm. Complexity of (iterated) do minance. In Proceedings of the 6th ACM
+Conference on Electronic Commerce (ACM-EC) , pages 88–97. ACM Press, 2005.
+[7] C. Daskalakis and C. H. Papadimitriou. Discretized mult inomial distributions and Nash equilibria in
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+IEEE Computer Society Press, 2008.
+[8] K. Ellul, B. Krawetz, J. Shallit, and M.-W. Wang. Regular expressions: New results and open problems.
+Journal of Automata, Languages and Combinatorics , 9(2–3):233–256, 2004.
+[9] H. N. Gabow, S. N. Maheshwari, and L. Osterweil. On two pro blems in the generation of program test
+paths.IEEE Transactions on Software Engineering , 2(3):227–231, 1976.
+[10] I. Gilboa, E. Kalai, and E. Zemel. The complexity of elim inating dominated strategies. Mathematics
+of Operations Research , 18(3):553–565, 1993.
+[11] F. Glover. Maximum matching in convex bipartite graphs .Naval Research Logistics Quarterly , 14:
+313–316, 1967.
+[12] R. Greenlaw, H. J. Hoover, and W. L. Ruzzo. Limits to Parallel Computation . Oxford University Press,
+1995.
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+games.Operations Research Letters , 7:103–107, 1988.
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+[15] R. B. Myerson. Game Theory: Analysis of Conflict . Harvard University Press, 1991.
+[16] C. H. Papadimitriou and T. Roughgarden. Computing equi libria in multi-player games. In Proceedings
+of the 16th Annual ACM-SIAM Symposium on Discrete Algorithm s (SODA) , pages 82–91. SIAM, 2005.
+[17] I. Parberry. On the computational complexity of optima l sorting network verification. In Proceedings of
+the Conference on Parallel Architectures and Languages Eur ope (PARLE) , volume 505 of Lecture Notes
+in Computer Science (LNCS) , pages 252–269. Springer-Verlag, 1991.
+[18] R. Parikh. On context-free languages. Journal of the ACM , 13(4):570–581, 1966.
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+313, 1992.
+This work is licensed under the Creative Commons Attributio n-NoDerivsLicense. To view a
+copyofthislicense,visit http://creativecommons.org/licenses/by-nd/3.0/ .
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+arXiv:1001.0530v1  [math.CV]  4 Jan 2010LOCALLY CONFORMALLY K ¨AHLER METRICS
+ON KATO SURFACES
+MARCO BRUNELLA
+Abstract. We show that every Kato surface admits a locally confor-
+mally K¨ ahler metric.
+In this note we shall prove the following theorem, which impr oves and
+completes our previous result [Bru].
+Theorem 1. Every Kato surface admits a locally conformally K¨ ahler met -
+ric.
+Wereferto[Bel]and[Bru], andreferencestherein, forsome backgroundon
+locally conformally K¨ ahler metrics on complex surfaces. W e just recall the
+followingfact, whichwillbeusedbelow: acomplexsurface Sadmitsalocally
+conformally K¨ ahler metric if and only if its universal cove ring/tildewideSadmits a
+K¨ ahler metric ωwhich transforms conformally under the action of the deck
+transformations ( g∗(ω) =cgω,cg∈R+, for every deck transformation g).
+In particular, we observe the following consequence of Theo rem 1.
+Corollary 2. The universal covering of a Kato surface is K¨ ahlerian.
+It would be interesting to know if this last statement holds f orevery
+compact complex surface. Note, however, that there are comp act complex
+surfaces which do not admit locally conformally K¨ ahler met rics [Bel], hence
+ageneralization of Corollary 2 to any compact complex surfa ceshouldfollow
+a different path.
+The proof of Theorem 1 is based on a point of view already explo ited in
+[Bru,§2], but we shall not need the stability result proved in [Bru, §3].
+Kato surfaces. Let us briefly recall the definition and the construction of
+Kato surfaces [Kat] [Dlo]. Let Brdenotes the (open) ball of radius rinC2,
+and setB=B1. Take a sequence of blow-ups
+π:/hatwideB−→B
+over the origin 0 ∈B, subject to the following constraint: at each step, we
+blow-up a point belonging to the exceptional divisor create d at the previous
+step. Take also a holomorphic (up to the boundary) embedding
+σ:B−→/hatwideB
+which sends the origin to a point belonging to the last except ional divisor
+created by π. Set
+W=/hatwideB\σ(B).
+1METRICS ON KATO SURFACES 2
+Wecangluetogetherthetwoboundarycomponentsof ∂W= (∂/hatwideB)∪(σ(∂B))
+using the real analytic CR-diffeomorphism
+σ◦π:∂/hatwideB−→σ(∂B).
+The result is a minimal compact complex surface S=Sπ,σ, with first Betti
+numberequal to 1and second Betti numberequal to thenumbero f blow-ups
+inπ. It is called a Kato surface , or asurface with a global spherical shell .
+Of course, such a surface cannot be K¨ ahlerian, for its first B etti number is
+odd.
+The following remark is very simple, yet of capital importan ce for our
+construction. For every r∈(0,1], the embedding σsendsBrinto/hatwideBr=
+π−1(Br): indeed, the composite map π◦σ:B→Bfixes the origin, and
+hence by Schwarz Lemma it maps BrinsideBr, for every r≤1, which
+precisely means that σ(Br)⊂π−1(Br). If we set
+Wr=/hatwideBr\σ(Br),
+then we can glue, as before, the two components of ∂Wrusingσ◦π. The
+result is the samecompact complex surface S. Indeed, the open subsets
+Wr,r∈(0,1], may be understood as different fundamental domains in the
+same universal covering /tildewideSfor the same action of the deck transformations.
+This phenomenon is also at the origin of the fact that the clas sification of
+Kato surfaces can be reduced to the classification of germsof contracting
+mappings of the type π◦σ[Dlo]. In fact, we may replace the balls Brwith
+any domain U⊂Bcontaining 0 and such that σ(U)⊂π−1(U).
+A K¨ ahler metric on /hatwideB.On/hatwideB, there obviously exists a K¨ ahler form ω0,
+smooth up to the boundary. In order to prove Theorem 1, we first ly modify
+ω0on a neighbourhood of σ(0).
+Considerσ∗(ω0). It is a K¨ ahler form on B, which admits a potential:
+σ∗(ω0) = ddcϕ
+withϕsmooth and strictly plurisubharmonic on B. Up to adding an affine
+(hencepluriharmonic)function,wemayassumethat ϕ(0) = 0and(d ϕ)(0) =
+0. The Taylor expansion of ϕat 0 has the form
+ϕ(z) =L(z,z)+ReM(z,z)+O(/bardblz/bardbl3)
+whereLis the Levi form at 0 and Mis a homogeneous polynomial of degree
+2. Because Re Mis pluriharmonic, we may replace ϕwithϕ−ReM, and
+we obtain in this way a new potential for σ∗(ω0), still denoted by ϕ, whose
+Taylorexpansionat0is L(z,z)+O(/bardblz/bardbl3). Thismeansthat ϕhasaminimum
+point at 0, of Morse type.
+We can choose λ>0 and 0<r<r′<1 such that the function
+ρ(z) =λ(1+/bardblz/bardbl2)
+is greater than ϕonBrand smaller than ϕon∂Br′. The function /tildewideϕonB
+defined by /tildewideϕ=ϕonB\Br′and/tildewideϕ= regularized maximum between ϕandMETRICS ON KATO SURFACES 3
+ρonBr′is therefore smooth, strictly plurisubharmonic, and equal toρon
+some neighbourhood of Br.
+TheK¨ ahler form σ∗(ddc/tildewideϕ) onσ(B) glues toω0outsideσ(B), giving a new
+K¨ ahler form ω1on/hatwideB. By construction, this K¨ ahler form has the property:
+σ∗(ω1) = ddcρonBr.
+Another K¨ ahler metric on /hatwideBr.We now modify the previous ω1on a
+neighbourhood of ∂/hatwideBr. Chooses<rsufficiently close to r, so that /hatwideBsstill
+containsσ(Br), i.e. (π◦σ)(Br)⊂Bs.
+divisorB
+BsBrBrπ
+σB
+exceptional
+Take a potential ψfor the direct image π∗(ω1):
+π∗(ω1) = ddcψonB.
+Note thatψis necessarily singular at 0, but it doesn’t matter. We can
+choosec1∈R+andc2∈Rsuch that the function
+c1ρ+c2
+is greater than ψon∂Brand smaller than ψon∂Bs. The function /tildewideψ
+onBrdefined by /tildewideψ=ψonBsand/tildewideψ= regularized maximum between
+ψandc1ρ+c2onBr\Bsis therefore smooth, strictly plurisubharmonic,
+equal toc1ρ+c2on some neighbourhood of ∂Brand equal to ψon some
+neighbourhood of Bs.
+By pulling-back /tildewideψwithπand taking ddc, we get on /hatwideBra new K¨ ahler
+formω2with the following two properties:
+(1)ω2=ω1on/hatwideBs⊃σ(Br), so thatσ∗(ω2) = ddcρonBr;
+(2)π∗(ω2) = ddc(c1ρ+c2) =c1ddcρon some neighbourhood of ∂Br.METRICS ON KATO SURFACES 4
+Conformal glueing. We can now complete the proof of Theorem 1. The
+glueing map σ◦π:/hatwideBr→σ(Br) satisfies the conformal property
+(σ◦π)∗(ω2|U) =1
+c1ω2|V
+whereU⊂σ(Br) andV⊂/hatwideBrare suitable neighbourhoods of σ(∂Br) and
+∂/hatwideBr. Because the Kato surface Sis obtained by identifying these two neigh-
+bourhoods via σ◦π, we get on Sa locally conformally K¨ ahler metric.
+Remark. Becausec1can be chosen arbitrarily large, we see also that, in
+the terminology of [Bru, Rem. 9], the open subset I(S)⊂(0,1) contains
+an interval (0 ,ε) (withε, in principle, depending on S). We do not know if
+I(S)isan interval.
+Remark. Theprevious construction can bedone parametrically: if {St}t∈D
+is a family of Kato surfaces, then, up to restricting the base D, we can find
+a family of locally conformally K¨ ahler metrics {ωt}t∈D, smoothly depending
+ont. This is, however, slightly weaker than the stability resul t of [Bru],
+which states that everylocally conformally K¨ ahler metric ω0onS0extends
+to a family {ωt}t∈D.
+References
+[Bel] F. A. Belgun, On the metric structure of non-K¨ ahler complex surfaces , Math. Ann.
+317 (2000), 1–40
+[Bru] M. Brunella, Locally conformally K¨ ahler metrics on certain non-K¨ ahle rian surfaces ,
+Math. Ann. 346 (2010), 629–639
+[Dlo] G. Dloussky, Structure des surfaces de Kato , M´ em. Soc. Math. France (N.S.) No.
+14 (1984)
+[Kat] M. Kato, Compact complex manifolds containing “global” spherical s hells, in Proc.
+Int. Symp. on Algebraic Geometry (Kyoto, 1977), Kinokuniya (1978), 45–84
+Marco Brunella, Institut de Math ´ematiques de Bourgogne – UMR 5584 –
+9 Avenue Savary, 21078 Dijon, France
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+arXiv:1001.0531v2  [math.QA]  5 Jan 2010The paper has been withdrawn.
+1
\ No newline at end of file
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+arXiv:1001.0532v1  [hep-ex]  4 Jan 2010DESY 09-185 ISSN 0418-9833
+October 2009
+MeasurementofLeadingNeutronProductionin
+Deep-InelasticScatteringat HERA
+H1Collaboration
+Abstract
+Theproduction ofleading neutrons, wheretheneutron carri es alarge fraction xLofthein-
+coming proton’s longitudinal momentum, is studied in deep- inelastic positron-proton scat-
+tering at HERA. The data were taken with the H1 detector in the years 2006 and 2007
+and correspond to an integrated luminosity of 122 pb−1. The semi-inclusive cross section
+is measured in the phase space defined by the photon virtualit y6< Q2<100 GeV2,
+Bjorken scaling variable 1.5·10−4< x <3·10−2, longitudinal momentum fraction
+0.32< xL<0.95andneutron transverse momentum pT<0.2 GeV. Theleading neutron
+structure function, FLN(3)
+2(Q2,x,xL), and the fraction of deep-inelastic scattering events
+containing a leading neutron are studied as a function of Q2,xandxL. Assuming that
+thepionexchange mechanismdominates leadingneutronprod uction, thedataprovidecon-
+straints onthe shape of the pion structure function.
+Submittedto Eur.Phys.J.CF.D. Aaron5,49, C. Alexa5, K. Alimujiang11,51, V. Andreev25, B. Antunovic11, S. Backovic30,
+A. Baghdasaryan38, E. Barrelet29, W. Bartel11, K. Begzsuren35, A. Belousov25, J.C. Bizot27,
+V.Boudry28,I.Bozovic-Jelisavcic2,J.Bracinik3,G.Brandt11,M.Brinkmann12,51,V.Brisson27,
+D.Bruncko16,A.Bunyatyan13,38,G.Buschhorn26,L.Bystritskaya24,A.J.Campbell11,K.B.Can-
+tunAvila22,K.Cerny32,V.Cerny16,47,V.Chekelian26,A.Cholewa11,J.G.Contreras22,J.A.Coughlan6,
+G. Cozzika10, J. Cvach31, J.B. Dainton18, K. Daum37,43, M. De´ ak11, B. Delcourt27, J. Delvax4,
+E.A. De Wolf4, C. Diaconu21, V. Dodonov13, A. Dossanov26, A. Dubak30,46, G. Eckerlin11,
+V. Efremenko24, S. Egli36, A. Eliseev25, E. Elsen11, A. Falkiewicz7, L. Favart4, A. Fedotov24,
+R.Felst11,J.Feltesse10,48,J.Ferencei16,D.-J.Fischer11,M.Fleischer11,A.Fomenko25,E.Gabathuler18,
+J.Gayler11,S.Ghazaryan11,A.Glazov11,I.Glushkov39,L.Goerlich7,N.Gogitidze25,M.Gouzevitch11,
+C.Grab40,T.Greenshaw18,B.R.Grell11,G.Grindhammer26,S.Habib12,D.Haidt11,C.Helebrant11,
+R.C.W.Henderson17,E.Hennekemper15,H.Henschel39,M.Herbst15,G.Herrera23,M.Hildebrandt36,
+K.H.Hiller39,D.Hoffmann21,R.Horisberger36,T.Hreus4,44,M.Jacquet27,X.Janssen4,L.J¨ onsson20,
+A.W. Jung15, H. Jung11, M. Kapichine9, J. Katzy11, I.R. Kenyon3, C. Kiesling26, M. Klein18,
+C.Kleinwort11,T.Kluge18,A.Knutsson11,R.Kogler26,P.Kostka39,M.Kraemer11,K.Krastev11,
+J.Kretzschmar18,A. Kropivnitskaya24,K. Kr¨ uger15, K.Kutak11, M.P.J.Landon19, W.Lange39,
+G. Laˇ stoviˇ cka-Medin30, P. Laycock18, A. Lebedev25, V. Lendermann15, S. Levonian11, G. Li27,
+K.Lipka11,51,A.Liptaj26,B.List12,J.List11,N.Loktionova25,R.Lopez-Fernandez23,V.Lubimov24,
+L.Lytkin9,A.Makankine9,E.Malinovski25,P.Marage4,Ll.Marti11,H.-U.Martyn1,S.J.Maxfield18,
+A. Mehta18, A.B. Meyer11, H. Meyer11, H. Meyer37, J. Meyer11, S. Mikocki7, I. Milcewicz-
+Mika7, F. Moreau28, A. Morozov9, J.V. Morris6, M.U. Mozer4, M. Mudrinic2, K. M¨ uller41,
+P.Mur´ ın16,44,Th.Naumann39,P.R.Newman3,C.Niebuhr11,A.Nikiforov11,D.Nikitin9,G.Nowak7,
+K.Nowak41,J.E.Olsson11,S.Osman20,D.Ozerov24,P.Pahl11,V.Palichik9,I.Panagouliasl,11,42,
+M.Pandurovic2,Th.Papadopouloul,11,42,C.Pascaud27,G.D.Patel18,O.Pejchal32,E.Perez10,45,
+A. Petrukhin24, I. Picuric30, S. Piec39, D. Pitzl11, R. Plaˇ cakyt˙ e11, B. Pokorny32, R. Polifka32,
+B. Povh13, V. Radescu14, A.J. Rahmat18, N. Raicevic30, A. Raspiareza26, T. Ravdandorj35,
+P. Reimer31, E. Rizvi19, P. Robmann41, B. Roland4, R. Roosen4, A. Rostovtsev24, M. Rotaru5,
+J.E. Ruiz Tabasco22, S. Rusakov25, D.ˇS´ alek32, D.P.C. Sankey6, M. Sauter14, E. Sauvan21,
+S. Schmitt11, L. Schoeffel10, A. Sch¨ oning14, H.-C. Schultz-Coulon15, F. Sefkow11, R.N. Shaw-
+West3, L.N. Shtarkov25, S. Shushkevich26, T. Sloan17, I. Smiljanic2, Y. Soloviev25, P. Sopicki7,
+D. South8, V. Spaskov9, A. Specka28, Z. Staykova11, M. Steder11, B. Stella33, G. Stoicea5,
+U. Straumann41, D. Sunar11, T. Sykora4, V. Tchoulakov9, G. Thompson19, P.D. Thompson3,
+T.Toll12,F.Tomasz16,T.H.Tran27,D.Traynor19,T.N.Trinh21,P.Tru¨ ol41,I.Tsakov34,B.Tseepeldorj35,50,
+J. Turnau7, K. Urban15, A. Valk´ arov´ a32, C. Vall´ ee21, P. Van Mechelen4, A. Vargas Trevino11,
+Y.Vazdik25,S.Vinokurova11,V.Volchinski38,M.vondenDriesch11,D.Wegener8,Ch.Wissing11,
+E.W¨ unsch11,J.ˇZ´ aˇ cek32,J.Z´ aleˇ s´ ak31,Z.Zhang27,A.Zhokin24,T.Zimmermann40,H.Zohrabyan38,
+andF. Zomer27
+1I. PhysikalischesInstitutder RWTH, Aachen,Germany
+2VincaInstituteof Nuclear Sciences,Belgrade,Serbia
+3Schoolof Physicsand Astronomy,UniversityofBirmingham, Birmingham,UKb
+4Inter-UniversityInstitutefor High Energies ULB-VUB, Bru ssels and UniversiteitAntwerpen,
+Antwerpen,Belgiumc
+5NationalInstituteforPhysicsand Nuclear Engineering(NI PNE) , Bucharest,Romania
+6RutherfordAppletonLaboratory,Chilton,Didcot,UKb
+7Institutefor NuclearPhysics, Cracow,Polandd
+18Institutf¨urPhysik, TU Dortmund,Dortmund,Germanya
+9JointInstituteforNuclear Research,Dubna,Russia
+10CEA, DSM/Irfu,CE-Saclay,Gif-sur-Yvette, France
+11DESY, Hamburg,Germany
+12Institutf¨ur Experimentalphysik,Universit ¨atHamburg,Hamburg,Germanya
+13Max-Planck-Institutf ¨ur Kernphysik,Heidelberg,Germany
+14PhysikalischesInstitut,Universit ¨atHeidelberg,Heidelberg,Germanya
+15Kirchhoff-Institutf ¨urPhysik,Universit ¨atHeidelberg, Heidelberg,Germanya
+16InstituteofExperimentalPhysics,SlovakAcademyof Scien ces, Koˇ sice, SlovakRepublicf
+17DepartmentofPhysics, UniversityofLancaster,Lancaster ,UKb
+18DepartmentofPhysics, UniversityofLiverpool,Liverpool ,UKb
+19Queen MaryandWestfieldCollege, London,UKb
+20PhysicsDepartment,Universityof Lund,Lund, Swedeng
+21CPPM, CNRS/IN2P3 - Univ. Mediterranee,Marseille,France
+22DepartamentodeFisicaAplicada,CINVESTAV, M ´erida,Yucat ´an,Mexicoj
+23DepartamentodeFisica,CINVESTAV IPN, M ´exico City,Mexicoj
+24InstituteforTheoreticaland ExperimentalPhysics, Mosco w,Russiak
+25Lebedev PhysicalInstitute,Moscow,Russiae
+26Max-Planck-Institutf ¨ur Physik,M ¨unchen,Germany
+27LAL, UniversityParis-Sud,CNRS/IN2P3, Orsay,France
+28LLR, EcolePolytechnique,CNRS/IN2P3, Palaiseau,France
+29LPNHE, Universit ´esParisVIandVII, CNRS/IN2P3, Paris,France
+30FacultyofScience, Universityof Montenegro,Podgorica,M ontenegroe
+31InstituteofPhysics,Academyof Sciences oftheCzech Repub lic,Praha,Czech Republich
+32FacultyofMathematicsandPhysics,Charles University,Pr aha,Czech Republich
+33DipartimentodiFisicaUniversit `adi RomaTreand INFNRoma 3,Roma, Italy
+34InstituteforNuclear Researchand NuclearEnergy, Sofia,Bu lgariae
+35Institute of Physics and Technology of the Mongolian Academ y of Sciences, Ulaanbaatar,
+Mongolia
+36PaulScherrer Institut,Villigen,Switzerland
+37FachbereichC, Universit ¨atWuppertal,Wuppertal,Germany
+38Yerevan PhysicsInstitute,Yerevan, Armenia
+39DESY, Zeuthen, Germany
+40Institutf¨ur Teilchenphysik,ETH, Z ¨urich,Switzerlandi
+41Physik-Institutder Universit ¨atZ¨urich,Z¨urich,Switzerlandi
+42Also at Physics Department, National Technical University , Zografou Campus, GR-15773
+Athens,Greece
+43Alsoat Rechenzentrum,Universit ¨atWuppertal,Wuppertal,Germany
+44Alsoat UniversityofP.J. ˇSaf´arik,Koˇ sice, SlovakRepublic
+45Alsoat CERN, Geneva, Switzerland
+46Alsoat Max-Planck-Institutf ¨urPhysik, M ¨unchen,Germany
+47Alsoat ComeniusUniversity,Bratislava,SlovakRepublic
+48Alsoat DESYandUniversityHamburg,HelmholtzHumboldtRes earchAward
+49Alsoat Facultyof Physics,UniversityofBucharest,Buchar est,Romania
+250Alsoat UlaanbaatarUniversity,Ulaanbaatar,Mongolia
+51SupportedbytheInitiativeandNetworking FundoftheHelmh oltzAssociation(HGF)under
+thecontractVH-NG-401.
+aSupported by the Bundesministerium f ¨ur Bildung und Forschung, FRG, under contract num-
+bers05H09GUF,05H09VHC, 05H09VHF,05H16PEA
+bSupportedbytheUKScienceandTechnologyFacilitiesCounc il,andformerlybytheUKPar-
+ticlePhysicsand AstronomyResearchCouncil
+cSupportedby FNRS-FWO-Vlaanderen,IISN-IIKW and IWTand by InteruniversityAttraction
+PolesProgramme,BelgianScience Policy
+dPartiallySupportedbyPolishMinistryofScienceandHighe rEducation,grantPBS/DESY/70/2006
+eSupportedbytheDeutscheForschungsgemeinschaft
+fSupportedbyVEGA SRgrantno. 2/7062/27
+gSupportedbytheSwedishNaturalScienceResearchCouncil
+hSupported by the Ministry of Education of the Czech Republic under the projects LC527,
+INGO-1P05LA259andMSM0021620859
+iSupportedbytheSwissNationalScienceFoundation
+jSupportedbyCONACYT, M ´exico, grant48778-F
+kRussianFoundationforBasicResearch (RFBR),grantno1329 .2008.2
+lThis project is co-funded by the European Social Fund (75%) a nd National Resources (25%)
+-(EPEAEKII)-PYTHAGORASII
+31 Introduction
+The production of leading baryons in deep-inelastic scatte ring (DIS) provides a testing ground
+for the theory of strong interactions in the soft regime. Eve nts containing a neutron, which
+carries a large fraction xLof the longitudinal momentum of the incoming proton, have be en
+observed in electron-proton collisions at HERA [1–7]. The p rocess leading to such events,
+ep→e′nX, is illustrated in Fig. 1a. The pion exchange mechanism, ill ustrated in Fig. 1b,
+is expected to dominate leading neutron production at large xLand low transverse momentum
+of the neutron [8–14]. In this picture of leading neutron pro duction, the proton fluctuates into
+a state consisting of a positively charged pion and a neutron p→nπ+. The virtual photon
+subsequently interacts with a parton from the pion. Consequ ently, the cross section factorises
+into two parts (proton vertex factorisation): one factor de scribes the proton fluctuation into
+anπ+state, the other describes the photon-pion scattering [8,9 ]. The production of leading
+neutrons in DIS at HERA may therefore provideconstraints on the structure of the pion at low
+to mediumBjorken- x, whiletheknowledgeof thepion structurefrom fixed target e xperiments
+[15] is limited to higher xvalues. Previous H1 and ZEUS studies of semi-inclusive lead ing
+neutron DIS cross sections [1,2] demonstrate that these mea surements are indeed sensitive to
+the structure of the pion and can distinguish between differ ent parameterisations of the pion
+structurefunction.
+Energetic neutrons can also be produced in the fragmentatio n of the proton remnant. The
+comparison of leading neutron production with inclusive DI S provides tests of fragmentation
+mechanisms. The hypothesis of limiting fragmentation [16, 17] states that, in the high-energy
+limit, the cross section for the inclusive production of par ticles in the target fragmentation re-
+gion becomes independent of the incident projectile energy . This hypothesis implies that, in
+DIS, leading neutron productionis insensitiveto Bjorken- xand thevirtualityofthe exchanged
+photonQ2.
+Inthispaperameasurementofthesemi-inclusivecrosssect ionforleadingneutronproduc-
+tioninDISispresented. Thisanalysisisbasedonadatasamp lecorrespondingtoanintegrated
+luminositywhichis 36timeslargerthanthatofthepreviousH1publication[1]. An ewneutron
+calorimeterwith improvedperformance is used. The larger d ata set together with better exper-
+imental capabilities allow the extension of the kinematic r ange of the measurement to higher
+valuesof Q2andxand thereductionofthetotaluncertaintyoftheresults.
+2 EventKinematicsandReconstruction
+The kinematics of semi-inclusive leading neutron producti on are shown in Fig. 1a, where the
+four-vectorsoftheincomingand outgoingparticles and oft heexchanged virtualphoton γ∗are
+indicated. Thekinematicvariables Q2,xandyareusedtodescribetheinclusiveDISscattering
+process. Theyare defined as
+Q2=−q2, x =Q2
+2p·q, y =p·q
+p·k, (1)
+4e(k)
+p(p)
+te(k )
+n(p  )nX(p  )X(q)*e
+pe
+nX*
++
+Figure 1: (a) Generic diagram for leading neutron productio nep→e′nXin deep-inelastic
+scattering. (b)Diagramofthesameprocess assumingthat it proceeds viapionexchange.(a) (b)
+wherep,kandqare the four-momenta of the incident proton, the incident po sitron and the
+exchanged virtual photon, respectively. These variables a re reconstructed using the technique
+introducedin[18], whichoptimisestheresolutionthrough outthemeasured yrangebyexploit-
+inginformationfrom boththescattered positronand thehad ronicfinal state.
+Thekinematicvariablesusedtodescribethefinalstateneut ronarethelongitudinalmomen-
+tumfraction xLandthesquaredfour-momentumtransfer tbetweentheincidentprotonandthe
+final stateneutron
+xL= 1−q·(p−pn)
+q·p≃En/Ep, t= (p−pn)2≃ −p2
+T
+xL−(1−xL)/parenleftBigg
+m2
+n
+xL−m2
+p/parenrightBigg
+,(2)
+wherempis the proton mass, pnis the four-momentum of the final state neutron, mnis the
+neutronmassand EnandpTare theneutronenergy and transversemomentum,respective ly.
+Thefour-folddifferentialcrosssectionforleadingneutr onproductioncanbeparameterised
+byasemi-inclusivestructurefunction, FLN(4)
+2, defined by
+d4σ(ep→enX)
+dxdQ2dxLdt=4πα2
+xQ4/parenleftbigg
+1−y+y2
+2/parenrightbigg
+FLN(4)
+2(Q2,x,xL,t). (3)
+The contribution from longitudinally polarised photons ca n be neglected in the phase space
+studied in this analysis. Integrating Eq. 3 over tyields the semi-inclusive structure function
+FLN(3)
+2measuredin thisanalysis
+d3σ(ep→enX)
+dxdQ2dxL=/integraldisplaytmin
+t0d4σ(ep→enX)
+dxdQ2dxLdtdt
+=4πα2
+xQ4/parenleftbigg
+1−y+y2
+2/parenrightbigg
+FLN(3)
+2(Q2,x,xL), (4)
+wheretheintegrationlimitsare
+tmin=−(1−xL)/parenleftBigg
+m2
+n
+xL−m2
+p/parenrightBigg
+andt0=−(pmax
+T)2
+xL+tmin. (5)
+5Herepmax
+Tis the upper limit of the neutron transverse momentum used fo r theFLN(3)
+2mea-
+surement. Smaller values of pmax
+Tare expected to enhance the relative contribution of pion
+exchange[10,11]. Inthisanalysis pmax
+Tissetto0.2GeV,asinthepreviousH1publication[1].
+3 ExperimentalProcedureandDataAnalysis
+The data used in this analysis were collected with the H1 dete ctor at HERA in the years 2006
+and 2007 and correspond to an integrated luminosity of 122 pb−1. During this period HERA
+collided positrons and protons with energies of Ee= 27.6 GeVandEp= 920 GeV , respec-
+tively.
+3.1 H1detector
+AdetaileddescriptionoftheH1detectorcanbefoundelsewh ere[19–23]. Here,abriefaccount
+isgivenofthecomponentsmostrelevanttothepresentanaly sis. Theoriginoftheright-handed
+H1 coordinate system is the nominal epinteraction point. The direction of the proton beam
+definesthepositive z-axis(forwarddirection);thepolarangle θismeasuredwithrespecttothis
+axis. Transversemomentaaremeasured inthe x−yplane.
+Theepinteractionregionissurroundedbyatwo-layeredsilicons tripdetectorandtwolarge
+concentricdriftchambers. Usingthesedetectorscharged p articlemomentaaremeasured inthe
+angular range 25◦< θ <155◦with a resolution of σ(pT)/pT= 0.005pT/GeV⊕0.015[24].
+The tracking system is surrounded by a finely segmented Liqui d Argon (LAr) calorimeter,
+which covers a range in polar angle of 4◦< θ <154◦with full azimuthal acceptance. The
+LAr calorimeter consists of an electromagnetic section wit h lead absorber and a hadronic sec-
+tion with steel absorber. The total depth of the LAr calorime ter ranges from 4.5to8hadronic
+interactionlengths. Itsenergyresolution,determinedin testbeammeasurements,is σ(E)/E≈
+12%//radicalbig
+E[GeV]⊕1%for electrons and σ(E)/E≈50%//radicalbig
+E[GeV]⊕2%for charged pi-
+ons [25]. The backward region ( 153◦< θ <177.8◦) is covered by a lead/scintillating-fibre
+calorimeter, the SpaCal. Its main purpose is the detection o f scattered positrons. The energy
+resolution for positrons is σ(E)/E≈7.1%//radicalbig
+E[GeV]⊕1%. The LAr and SpaCal calorime-
+ters are surrounded by a superconducting solenoid which pro vides a uniform magnetic field of
+1.16T alongthebeamdirection.
+TheluminosityismeasuredviatheBethe-Heitlerprocess ep→e′pγ. Thefinalstatephoton
+isdetected inadedicated calorimetersituatednear thebea m pipeat z=−103 m.
+3.2 Detectionof leading neutrons
+Leading neutrons are detected in the forward neutron calori meter (FNC). The FNC is situated
+at a polar angle of 0◦beyond the magnets used to deflect the proton beam, at z= 106 m. A
+schematic view of the FNC is shown in Fig. 2a. It consists of th e Main Calorimeter and the
+6PreshowerCalorimeter. In addition,two layers of vetocoun ters situatedat a distanceof 2 min
+frontofthePreshowerCalorimeterare usedto vetocharged p articles.
+The Preshower Calorimeter is a 40 cmlong lead-scintillator sandwich calorimeter, corre-
+sponding to about 60radiation lengths or 1.6hadronic interaction lengths. It is composed of
+24planes: the first 12planes each consist of a lead plate of 7.5 mmthickness and a scintil-
+lator plate of 2.6 mmthickness, the second 12planes each consist of a lead plate of 14 mm
+thickness and a scintillator plate of 5.2 mmthickness. The transverse size of the scintillating
+platesis26×26 cm2. Eachscintillatingplatehas 45parallelgroovesholding 1.2mmdiameter
+wavelength shifter (WLS) fibres. In order to obtain good spat ial resolution, the orientation of
+fibres alternates from horizontal to vertical in consecutiv e planes. At each plane the fibres are
+bundled into nine strips of five fibres. Longitudinally, the s trips are combined into 9vertical
+and9horizontaltowerswhichare finallyconnected to 18photomultipliers.
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+/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
+/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1
+/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
+/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
+/0/0/0/0/0/1/1/1/1/1/0/0/0/0/0/1/1/1/1/1
+1 2
+5
+8     FNC 
+     PRESHOWER BEAM PIPEMAIN CALORIMETER
+      600
+260
+600     2060     
+XY
+Z/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
+600
+6001 2
+3 4 5
+6 7 8Beam
+q=0pipe
+o
+Figure 2: (a) Schematic view of the FNC. (b) Layout of tiles on an active board of the Main
+Calorimeter; the position of readout fibres is indicated by d otted lines; the hatched area shows
+the geometrical acceptance window defined by the beam line el ements. The position corre-
+spondingto θ= 0◦isalsoindicated. All dimensionsaregivenin mm.
+TheMainCalorimeteroftheFNCisasandwich-typecalorimet erconsistingoffouridentical
+sections. Each sectionis 51.5cmlongwithtransversedimensionsof 60×60 cm2and consists
+of25leadabsorberplatesof 14mmthicknessand 25activeboardswith 3mmthickscintillators.
+Each active board is made of 6scintillating tiles with a transverse size of 20×20 cm2and2
+tiles of transverse size 20×26 cm2, as shown in Fig. 2b. Each scintillating tile has a circular
+grooveholdinga 1mmdiameterWLSfibrewhichisattachedvialightconnectorstot wo1mm
+diameter transparent fibres. Longitudinally, for each sect ion the fibres from 25tiles with the
+sametransverse positionare bundled into a towerand connec ted to one photomultiplier. There
+are then8towers in each section, making a total of 32towers in the Main Calorimeter. The
+proton beam pipe is located in a rectangular space along the t op of the calorimeter, as can
+be seen in Fig. 2. The total length of the Main Calorimeter is 206 cm, corresponding to 8.9
+hadronicinteractionlengths.
+7All modules of the FNC were initially calibrated at CERN usin g electron beams with en-
+ergies between 120and230 GeV and hadron beams with energies between 120and350 GeV.
+The FNC was positioned on a movable platform which allowed th e response of each module
+and tower to be measured separately. After this initial cali bration, the FNC had an approxi-
+mately uniform response independent of impact position. Th e linearity of the energy response
+of the FNC was measured at HERA from beam-gas interactions du ring dedicated runs, when
+theprotonbeamwas accelerated tofiveintermediateenergie s between 150GeV and920GeV.
+The energy response of the FNC is linear to a precision of 3%and the hadronic energy resolu-
+tion isσ(E)/E≈63%//radicalbig
+E[GeV]⊕3%. The energy resolution for electromagnetic showers,
+whicharealwaysfullycontainedinthePreshowerCalorimet er,isσ(E)/E≈20%//radicalbig
+E[GeV].
+For hadronic showers starting in the Main Calorimeter, the s patial resolution is σ(x,y)≈
+10 cm//radicalbig
+E[GeV]⊕0.6 cm. A better spatial resolution of about 2 mmis achieved for the
+electromagneticshowersandforthosehadronicshowerssta rtinginthePreshowerCalorimeter.
+The acceptance of the FNC is defined by the aperture of the HERA beam line magnets and
+is limited to neutron scattering angles of θn∼<0.8 mradwith an approximately 30%azimuthal
+coverage. Thegeometricalacceptance windowoftheFNCis in dicatedinFig. 2b.
+AfterthecalorimeterwasinstalledintheH1beamline,thes tabilityofcalibrationconstants
+was monitored using interactions between the proton beam an d residual gas molecules in the
+beam pipe. The neutron energy spectrum was compared with the results of Monte Carlo sim-
+ulation based on pion exchange. From this monitoring, the ti me dependent variations of the
+calibration constants of order of a few percent were determi ned. Short term variations of pho-
+tomultiplier gain were monitored using LED signals. The LED s were operated during empty
+bunches at a frequency of 0.8 Hz. The averaged LED signal responses were used to provide
+offlineenergy corrections appliedduringthereconstructi on.
+The longitudinalsegmentation of the FNC allows efficient di scriminationof neutrons from
+charged particles or electromagnetic energy deposits. Cha rged particles are rejected using sig-
+nals from the veto counters. Events with an energy deposit on ly in the Preshower Calorimeter
+are identified as electromagnetic showers (photons, π0mesons). All other patterns of energy
+depositionare identified as hadronicshowers, which are sub sequentlyclassified as either those
+startinginthePreshowerCalorimeterorthosestartingint heMainCalorimeter,thelatterhaving
+worsespatialresolution. In thisanalysisbothhadronicsh owertypesare used.
+3.3 Eventselection
+The data sample was collected using a trigger which requires the scattered positron to be mea-
+suredintheSpaCal andatleast 100GeV energytobedepositedintheFNC. Thetriggersignal
+fromtheFNCisformedusingananaloguesumofsignalsfromth ePreshowerCalorimeterand
+the central towers of the Main Calorimeter. The trigger effic iency is above 97%in most of the
+analysisphasespace, decreasing to 94%forxL<0.4.
+TheselectionofDISeventsisbasedontheidentificationoft hescatteredpositronasthemost
+energeticcompact calorimetricdepositin theSpaCal with a n energyE′
+e>11 GeVand apolar
+angle156◦< θ′
+e<175◦. The energy weighted cluster radius is required to be less th an4 cm,
+as expected for an electromagnetic shower [26]. The z-coordinate of the primary event vertex
+8isrequiredtobewithin ±35cmofthenominalpositionoftheinteractionpoint. Theremain ing
+clustersinthecalorimetersandthechargedtracksarecomb inedtoreconstructthehadronicfinal
+state. To suppress events with initial state hard photon rad iation, as well as events originating
+fromnon- epinteractions,thequantity E−pz,summedoverallreconstructedparticlesincluding
+the positron, is required to lie between 35 GeVand70 GeV. This quantity is expected to be
+twicetheelectron beamenergy forDISeventswithoutQEDrad iation. Furthermore,eventsare
+selectedwithinthekinematicrange 6< Q2<100GeV2,0.02< y <0.6and1.5·10−4< x <
+3·10−2.
+Eventscontainingaleadingneutronareselected byrequiri ngahadronicclusterintheFNC
+with an energy above 275 GeV and a polar angle below 0.75 mrad. The cut on polar angle,
+definedbythegeometricalacceptanceoftheFNC, restrictst heneutrontransversemomenta pT
+totherange pT< xL·0.69GeV.
+The final data samplecontains 315960events which satisfy these selection criteria. For the
+measurementof FLN(3)
+2, an additionalrequirement onthetransversemomentumofth eneutron
+pT<0.2 GeVis applied to enhance the relative contribution of pion exch ange and to avoid
+havingan xLdependent pTcut. Thenumberofeventsselectedwiththisadditionalrequ irement
+is209150.
+3.4 MonteCarlo simulationand corrections tothe data
+Monte Carlo simulations are used to correct the data for the e ffects of detector acceptance,
+inefficiencies and migrations between measurement interva ls due to finite resolution and QED
+radiation. All generated eventsare passed through a GEANT3 [27]based simulationof theH1
+apparatus and are processed using the same reconstruction a nd analysis framework as is used
+forthedata.
+The DJANGO [28] program generates inclusive DIS events. It i s based on leading order
+electroweakcrosssectionsandtakesQCDeffectsintoaccou ntuptoorder αs. Thehadronicfinal
+state is simulated using ARIADNE [29], based on the Colour Di pole Model, with subsequent
+hadronisation effects modelled using the Lund string fragm entation model as implemented in
+JETSET[30]. DJANGO is also used inthis analysisto simulate eventswhereleading neutrons
+originatefrom proton remnant fragmentation. RAPGAP [31] i s a general purposeevent gener-
+atorfor inclusiveand diffractive epinteractions. HigherorderQCD effects are simulatedusing
+parton showers and the final state hadrons are obtained via Lu nd string fragmentation. Higher
+order electroweak processes in the DJANGO and RAPGAP genera tors are simulated using an
+interfacetoHERACLES [32].
+IntheversiondenotedbelowasRAPGAP- π, theprogramsimulatesexclusivelythescatter-
+ing of virtual or real photons off an exchanged pion. Here, th e cross section for photon-proton
+scatteringtothefinal state nXtakes theform
+dσ(ep→e′nX) =fπ+/p(xL,t)·dσ(eπ+→e′X), (6)
+wherefπ+/p(xL,t)represents the pion flux associated with the splittingof a pr oton into a π+n
+system and dσ(eπ+→e′X)is the cross section of the positron-pion interaction. Ther e are
+9several parameterisations of the pion flux [9–13]. In this an alysis, the pion flux factor is taken
+fromthelight-conerepresentation[10]as
+fπ+/p(xL,t) =1
+2πg2
+pπn
+4π(1−xL)−t
+(m2π−t)2exp/parenleftbigg
+−R2
+πnm2
+π−t
+1−xL/parenrightbigg
+, (7)
+wheremπis the pion mass, g2
+pπn/4π= 13.6is thepπncoupling constant deduced from phe-
+nomenological analyses of low-energy data [33] and Rπn= 0.93 GeV−1is the radius of the
+pion-protonFock state[10].
+TheDJANGO and RAPGAP- πMonteCarlo simulationsare calculated usingGRV leading
+orderpartondistributionsfortheproton[34]and thepion[ 35], respectively.
+Figure 3 shows the observed energy and pTdistribution of the neutron for selected data
+sample together with Monte Carlo simulations. The differen ces between the RAPGAP- πand
+DJANGO generators are particularly visible in the neutron e nergy distributions (Fig. 3a). The
+RAPGAP- πsimulation peaks at En∼650 GeV and describes the shape of the distribution at
+highenergies. At lowerenergies ( En∼<600 GeV)theRAPGAP- πsimulationdoes not describe
+the data. In this region additional physics processes which are expected to contribute signifi-
+cantly are not simulated. In contrast, the DJANGO simulatio n predicts a large contribution to
+the cross section in this region. The best description of the data is achieved if the predictions
+of the RAPGAP- πand DJANGO Monte Carlo programs are added, using weighting f actors
+of0.65and1.2for RAPGAP- πand DJANGO, respectively. This Monte Carlo combination is
+labelledas“ 0.65×RAPGAP- π+ 1.2×DJANGO”inthefiguresandisusedtocorrectthedata.
+Cross sections at hadron level are determined from selected events by applying bin depen-
+dent correction factors. These factors are determined from the combination of DJANGO and
+RAPGAP- πMonteCarlo simulationsas theratiosofthecrosssectionso btainedfrom particles
+athadronlevelwithoutQEDradiationtothecrosssectionca lculated usingreconstructedparti-
+cles and including QED radiation effects. The typical value of these factors is about 1.2at the
+highestxLincreasing to 4at thelowest xLdue to thenon-uniformazimuthalacceptance ofthe
+FNC.
+The binning in Q2,xandxLused to measure FLN(3)
+2is given in Table 1. The bin purities,
+definedasthefractionofeventsreconstructedinaparticul arbinthatoriginatedfromthatbinon
+hadronlevel,andthebinstabilities,definedasthefractio nofeventsoriginatingfromaparticular
+bin on hadron level that are reconstructed in that bin, are hi gher than 50%for(Q2,x)-bins and
+higherthan 60%forxLbins.
+The measured distributions may contain background arising from different sources. The
+backgroundfrom photoproductionprocesses, wheretheposi tronisscattered intothebackward
+beampipeandaparticlefromthehadronicfinalstatefakesth epositronsignatureintheSpaCal,
+is estimated using the PHOJET MonteCarlo generator [36]. Th is background is negligibleex-
+cept at the highest yvalues where it can reach 6%. Background also arises from the random
+coincidenceof DIS events, causing activity in the central d etector, with proton beam-gas inter-
+actions, which give a neutron signal in the FNC. This contrib ution is estimated by combining
+DIS events with neutrons originating from beam-gas interac tions in the bunch-crossings adja-
+cent to the bunch-crossing of the DIS event. It is found to be s maller than 1%. The estimated
+backgroundcontributionsarenotsubtracted fromthemeasu rements.
+10The contribution from proton dissociation, where the leadi ng neutron originates from the
+decayofahighermassstate,isestimatedusinganimplement ationinRAPGAPofthedissocia-
+tionmodeloriginallydevelopedfortheDIFFVM[37]MonteCa rlogenerator. Thiscontribution
+can be up to 30%at lowxLvalues, but is less than 5%forxL>0.7. It is included in the cross
+sectiondefinition.
+3.5 Systematicuncertainties
+The effects of various systematic uncertainties on the cros s section measurements are deter-
+mined using Monte Carlo simulations, by propagating the cor responding estimated measure-
+mentuncertaintythroughthefull analysischain.
+The acceptance of the FNC is defined by the interaction point a nd the geometry of the
+HERA magnets and is determined using Monte Carlo simulation s. The angular distribution of
+the neutrons studied in this analysis is sharply peaked in th e forward direction. Therefore the
+acceptancedependscriticallyonsmallinclinationsofthe incomingprotonbeamwithrespectto
+its nominal direction. The uncertainty on the position of th e neutron impact point is estimated
+tobe5 mm, which resultsin a 4.4%uncertainty in theFNCacceptance. Theuncertainty inthe
+neutrondetectionefficiency and theuncertaintyof 2%on theabsoluteenergy scaleoftheFNC
+lead to a systematicerror on the cross section of 2%and6%, respectively. An additional 0.5%
+uncertainty is attributed to the trigger efficiency. These e ffects are strongly correlated between
+measurementintervalsand mainlycontributetotheoverall normalisationuncertainty.
+The uncertainties on the measurements of the scattered posi tron energy ( 1%) and angle
+(1 mrad) in the SpaCal lead to a combined systematic uncertainty of t ypically1.9%on the
+cross section. The uncertainty of the energy measurement of the hadronic final state in the
+centraldetectorsaffectthereconstructionofthekinemat icvariables y,Q2andx. Thishadronic
+energy scale uncertainty is estimated to be 4%for this measurement, leading to an uncertainty
+onthecrosssection of 0.4%on average.
+Thesystematicerrorontheefficiencytoreconstructtheeve ntvertexisdeterminedbycom-
+paringthereconstructionefficienciesforthedataandtheM onteCarlosimulation. Thediscrep-
+ancyisless than 1%.
+Thesystematicuncertaintyarisingfromtheradiativecorr ectionsandthemodeldependence
+of the data correction are estimated by varying the DJANGO an d RAPGAP- πscaling factors
+describedinSect.3.4withinvaluespermittedbythedata. T heresultinguncertaintyonthecross
+sectionisbelow 5%inmostofthebinsandtypically 2%.
+The luminosity measurement uncertainty for the selected ru n period leads to an overall
+normalisationuncertainty of 5%.
+Thetotalsystematicerrorineach biniscalculatedasthequ adraticsumofallcontributions.
+Systematicerrors aretypically 10%forFLN(3)
+2and14%fordσ/dxLmeasurements.
+114 Results
+The single differential leading neutron DIS cross sections as a function of xLare presented in
+Fig. 4 and Table 2. In Fig. 4b the differential cross section i nxLis shown for pT<0.2 GeV.
+The difference between the shapes of distributions in Fig. 4 a and Fig. 4b, in particular a steep
+fall at low xLin Fig. 4a, is due to the geometrical acceptance of the FNC whi ch restricts the
+accessible pTrangetopT< xL·0.69GeV.
+The measured cross sections are compared with the Monte Carl o simulations. For large
+values of xL∼>0.7, the RAPGAP- πsimulationdescribes the shape of the xLdistributions well,
+in agreement with theassumptionthat at high xLthe dominantmechanismfor leading neutron
+productionispionexchange. Thefull xLrangeiswelldescribedbythesumoftheRAPGAP- π
+and DJANGO Monte Carlo generators, using the scaling factor s discussed in Sect. 3.4. This
+indicatesthatthe πexchangemechanismdominatesathigh xL,whileprotonremnantfragmen-
+tationgivesasignificantcontributionat low xL.
+The measured cross sections are also compared with the predi ctions of the Soft Colour In-
+teraction model (SCI) [38], implemented in the LEPTO Monte C arlo generator program [39].
+In the SCI model, the production of leading baryons and diffr action-like configurations is en-
+hanced via non-perturbative colour rearrangements betwee n the outgoing partons. A refined
+versionofthemodelusesageneralisedarealaw(GAL)[40]fo rthecolourrearrangementprob-
+ability. Compared to DJANGO MC predictions, SCI-GAL improv es the description at higher
+xL, as can be seen in Fig. 4. However, for lower pTvalues the predicted cross section for
+xL>0.5isstilltoolow.
+Figure 5 and Tables 3-6 present the measurement of the semi-i nclusive structure function
+FLN(3)
+2(Q2,x,xL)in the range 6< Q2<100 GeV2,1.5·10−4< x <3·10−2,0.32<
+xL<0.95andpT<0.2 GeV. In all(Q2,x)bins, the shape of the FLN(3)
+2distribution as a
+functionof xLissimilartotheshapeofthesingledifferentialcrosssect ioninxL(Fig.4b). The
+distributions are reasonably well described by the combina tion of RAPGAP- πand DJANGO
+simulations.
+The measurement of FLN(3)
+2allows the validityof the hypothesisof limitingfragmenta tion
+tobetested, accordingtowhichtheproductionofleadingne utronsintheprotonfragmentation
+regionisindependentof Q2andx. Toinvestigatethisprediction,theratioofthesemi-incl usive
+structure function FLN(3)
+2to the proton structure function F2is studied as a function of Q2in
+bins ofxandxL(Fig. 6). The values of F2are obtained from the H1PDF2009 parameterisa-
+tion [26]. The horizontal lines indicatethe average value o f the ratio for a given xLbin. These
+average values decrease from 7%to2%with increasing xL, reflecting the general behaviour
+observed in Figs. 4 and 5. The ratios are almost independent o fxandQ2in eachxLbin, im-
+plyingthat FLN(3)
+2andF2haveasimilar (Q2,x)behaviour,as expectedfrom thehypothesisof
+limitingfragmentation.
+Assuming that leading neutrons are produced via the exchang e of a colour singlet particle,
+e.g. theπ+, thestructurefunction FLN(3)
+2factorises into aflux factor which is a function of xL
+and a structure function FLN(2)
+2which depends on Q2andβ=x/(1−xL). The quantity β
+can be interpreted as the fraction of the exchanged particle ’s momentum carried by the parton
+12interacting with the virtual photon. The value of FLN(3)
+2(Q2,β,xL)is obtained by replacing
+the variable xwithβin Eq. 4. The distribution of FLN(3)
+2, shown in Fig. 7, has a similar
+dependence on βin all(Q2,xL)bins. This behaviour can be approximated by a power law
+functionFLN(3)
+2∝β−λ. In each (Q2,xL)bin a fit of the parameter λis performed. Within
+uncertainties the value of the fitted parameter λis independent of xLwhich is consistent with
+protonvertexfactorisation.
+However, as a function of Q2, the values of λincrease from 0.23at lowest Q2to0.3at
+highestQ2. Thisslow Q2dependenceissimilartotherisetowardslow xoftheprotonstructure
+functionF2measured in inclusive DIS [41]. It is further investigated b y fitting the measured
+FLN(3)
+2(Q2,β,xL)assumingthefollowingfunctionalform
+FLN(3)
+2(Q2,β,xL) =c(xL)·β−λ(Q2)withλ=a·ln(Q2/Λ2) (8)
+wherea,Λand thenormalisations c(xL)are the free parameters of thefit. Thisnine parameter
+fitoverthewhole xLrangehasagood χ2,supportingthevalidityoftheemployedansatz(Eq.8).
+Withintheexperimentaluncertaintiestheobtainedvalues oftheparameters a= 0.052±0.003
+andΛ = 0.416±0.052GeV areinreasonableagreementwiththoseobtainedintheanaly sisof
+theprotonstructurefunction F2[41]. Thisdemonstratesthesimilaritybetweenthe Q2evolution
+ofFLN(3)
+2and theQ2evolutionoftheprotonstructurefunction F2.
+Sincepionexchangedominatesleadingneutronproductiona thighxLandlowpT,themea-
+surement of FLN(3)
+2in the range 0.68< xL<0.77can be used to estimate the pion structure
+function at low Bjorken- x, following the procedure introduced in [1]. Assuming proto n vertex
+factorisation,whichissupportedbythepresentdataasexp lainedabove,thequantity FLN(3)
+2/Γπ
+can beinterpretedas beingequal tothestructurefunctiono fthepion,where
+Γπ(xL) =/integraldisplaytmin
+t0fπ/p(xL,t)dt (9)
+is the integral of the pion flux over the measured t-range, where t0andtminare given by
+Eqs. 5. The pion flux from Eq. 7 used for the RAPGAP- πsimulation yields Γπ= 0.13for
+pmax
+T= 0.2 GeVatxL= 0.73, which is the central value of the chosen xLrange. Using other
+parameterisationsofthepionflux,e.g. from[9,11–13], lea dstovaluesofthepionfluxintegral
+which may differ by up to 30%. In this evaluation of the pion structure function contribu tions
+from background processes like the exchange of ρanda2-mesons, proton diffractive dissocia-
+tion and∆production are not taken into account. Within the narrow xLrange considered here
+they are only expected to affect the absolute normalisation of the results. The contribution of
+neutrons from fragmentation is of order 25−35%, as can be estimated using the DJANGO
+prediction. The relative size of this contribution is large ly independent of Q2andβand thus
+haslittleimpactontheshapeofthedistribution.
+Figure 8 shows FLN(3)
+2/Γπas a function of Q2in bins of βwhile Fig. 9 shows FLN(3)
+2/Γπ
+as a function of βin bins of Q2. In Fig. 9 the contribution of neutrons from fragmentation,
+as predicted by DJANGO and scaled by a weighting factor 1.2, a s described in Sect. 3.4, is
+indicated. The data are compared to predictions of the param eterisations of the pion struc-
+ture function GRSc- π[42] and ABFKW- π[43]. The measurements are also compared to the
+H1PDF2009 parameterisation of the proton structure functi onF2(Q2,x)[26] which is scaled
+13by a factor of 2/3 in order to naively account for the differen t number of valence quarks in the
+pion and proton, respectively. The values of F2(Q2,x)are calculated at Bjorken- xequal toβ.
+TheQ2distribution exhibits a rise with increasing Q2(i.e. scaling violation) for all βvalues
+in the measured range, which is similar in size and shape to th at seen in the parameterisations
+of the inclusive structure functions of both the pion and pro ton (Fig. 8). The βdistributions
+show a steep rise with decreasing βfor allQ2values (Fig. 9). This behaviour is in reasonable
+agreementwiththepionandprotonstructurefunctionparam eterisations. Inabsolutevaluesthe
+parameterisations lie above the measurements. Other param eterisations of the pion structure
+function[44,45] were ruled out by previousH1 and ZEUS measu rements[1,2] as theyshow a
+muchflatterbehaviouras a functionof β.
+Thecomparisonofthemeasured FLN(3)
+2/Γπandtheparameterisationsofthepionstructure
+functionis affected by uncertaintieson thepion flux normal isation,as explainedabove. It may
+alsodependonabsorptivecorrections[46–49],whichareno ttakenintoaccountinthisanalysis.
+Neutronabsorptionmayoccurthroughrescatteringwhichtr ansformstheneutronintoacharged
+baryonorshiftstheneutron tolowerenergy orhigher pT.
+Theresultspresentedhereareconsistentwiththeprevious measurementbytheH1Collabo-
+ration[1]. AsimilaranalysishasbeenpublishedbytheZEUS Collaboration[2]. Thereisgood
+agreementbetweenthetwocrosssectionmeasurements. Fort heextractionofthepionstructure
+function, the flux factor normalisations used in the ZEUS ana lysis are different from the one
+usedhere. Withintheuncertaintiesofthenormalisationth eH1and ZEUSresultsagree.
+5 Summary
+The cross section for leading neutron production in deep-in elastic positron-proton scattering
+dσ/dx Land the semi-inclusivestructure function FLN(3)
+2(Q2,x,xL)are measured in the kine-
+matic region 6< Q2<100 GeV2,1.5·10−4< x <3·10−2,0.32< xL<0.95and
+pT<0.2GeV. Thepresent measurementshaveexperimentaluncertaintie sof10to15%.
+The measurements are well described by a Monte Carlo simulat ion including neutron pro-
+ductioninfragmentationandneutronsproducedfrom π+exchange,aspredictedbytheDJANGO
+andRAPGAP programsrespectively. At large xL∼>0.7theπ+-exchangeprocess dominates.
+Within the measured kinematic range, the semi-inclusivest ructure function FLN(3)
+2and the
+inclusive structure function F2have similar (Q2,x)behaviour, which is consistent with the
+hypothesis of limiting fragmentation. The dependence of FLN(3)
+2on the variable βis similar
+forallxLbins,inaccordancewiththeexpectationfromprotonvertex factorisation. Thescaling
+violationsobservedin FLN(3)
+2aresimilarinsizeandshapetothoseseenintheparameteris ations
+oftheinclusivestructurefunctionsofthepionandtheprot on. Thedataareusedtoestimatethe
+structurefunctionofthepion,uptouncertaintiesontheba ckgroundcontributionandtheoverall
+normalisation,intheframeworkoftheonepionexchangemod elfortheneutronkinematicrange
+0.68< xL<0.77andpT<0.2 GeV.
+14Acknowledgements
+We are grateful to the HERA machine group whose outstanding e fforts have made this ex-
+periment possible. We thank the engineers and technicians f or their work in constructing and
+maintaining the H1 detector, our funding agencies for financ ial support, the DESY technical
+staffforcontinualassistanceandtheDESYdirectoratefor supportandforthehospitalitywhich
+theyextendtothenon-DESYmembersofthecollaboration.
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+17Q2-binQ2range[GeV2]x-bin xrange xL-binxLrange
+16.00÷9.00 11.50·10−4÷3.20·10−410.32÷0.41
+29.00÷13.523.20·10−4÷6.82·10−420.41÷0.50
+313.5÷20.036.82·10−4÷1.45·10−330.50÷0.59
+420.0÷30.041.45·10−3÷3.10·10−340.59÷0.68
+530.0÷45.053.10·10−3÷6.60·10−350.68÷0.77
+645.0÷67.066.60·10−3÷1.41·10−260.77÷0.86
+7 67.0÷100 71.41·10−2÷3.00·10−270.86÷0.95
+Table 1: Bins in Q2,xandxLas used for the measurement of the semi-inclusive structure
+functionFLN(3)
+2.
+xLrange dσ/dx L[nb] dσ/dx L[nb]
+pT< xL·0.69GeV pT<0.2 GeV
+0.30÷0.3710.6±0.11±1.38.21±0.09±0.99
+0.37÷0.4412.0±0.11±1.16.75±0.07±0.53
+0.44÷0.5114.0±0.11±1.36.26±0.06±0.45
+0.51÷0.5815.6±0.11±1.45.92±0.05±0.46
+0.58÷0.6516.5±0.11±1.45.81±0.05±0.42
+0.65÷0.7216.7±0.10±1.45.71±0.05±0.40
+0.72÷0.7916.0±0.10±1.35.57±0.05±0.31
+0.79÷0.8612.4±0.08±1.24.54±0.04±0.25
+0.86÷0.937.3±0.06±1.22.57±0.03±0.38
+0.93÷1.002.3±0.03±0.90.63±0.01±0.26
+Table 2: Differential cross section dσ/dxLof leading neutron production in deep-inelastic
+scattering in the kinematic range 6< Q2<100 GeV2,1.5·10−4< x <3·10−2and
+0.32< xL<0.95. The first uncertainty is statistical and the second systema tic. Normalisation
+uncertaintiesof 5%are notincluded.
+18Q2[GeV2] x x L β FLN(3)
+2
+7.3 2 .24·10−40.365 3 .53·10−40.0724±0.0035±0.0058
+7.3 2 .24·10−40.455 4 .12·10−40.0573±0.0024±0.0044
+7.3 2 .24·10−40.545 4 .93·10−40.0582±0.0022±0.0049
+7.3 2 .24·10−40.635 6 .14·10−40.0589±0.0020±0.0042
+7.3 2 .24·10−40.725 8 .16·10−40.0557±0.0018±0.0036
+7.3 2 .24·10−40.815 1 .21·10−30.0489±0.0017±0.0023
+7.3 2 .24·10−40.905 2 .36·10−30.0236±0.0012±0.0043
+7.3 4 .78·10−40.365 7 .53·10−40.0627±0.0030±0.0052
+7.3 4 .78·10−40.455 8 .77·10−40.0547±0.0022±0.0037
+7.3 4 .78·10−40.545 1 .05·10−30.0490±0.0018±0.0046
+7.3 4 .78·10−40.635 1 .31·10−30.0500±0.0017±0.0034
+7.3 4 .78·10−40.725 1 .74·10−30.0503±0.0016±0.0034
+7.3 4 .78·10−40.815 2 .58·10−30.0425±0.0015±0.0021
+7.3 4 .78·10−40.905 5 .03·10−30.0205±0.0010±0.0030
+7.3 1 .02·10−30.365 1 .60·10−30.0531±0.0026±0.0045
+7.3 1 .02·10−30.455 1 .87·10−30.0471±0.0020±0.0036
+7.3 1 .02·10−30.545 2 .24·10−30.0421±0.0016±0.0031
+7.3 1 .02·10−30.635 2 .79·10−30.0445±0.0015±0.0030
+7.3 1 .02·10−30.725 3 .71·10−30.0413±0.0014±0.0028
+7.3 1 .02·10−30.815 5 .51·10−30.0352±0.0013±0.0019
+7.3 1 .02·10−30.905 1 .07·10−20.0156±0.0008±0.0023
+7.3 2 .17·10−30.365 3 .42·10−30.0461±0.0025±0.0037
+7.3 2 .17·10−30.455 3 .99·10−30.0385±0.0018±0.0028
+7.3 2 .17·10−30.545 4 .77·10−30.0383±0.0016±0.0030
+7.3 2 .17·10−30.635 5 .95·10−30.0338±0.0013±0.0021
+7.3 2 .17·10−30.725 7 .90·10−30.0346±0.0013±0.0026
+7.3 2 .17·10−30.815 1 .17·10−20.0293±0.0011±0.0015
+7.3 2 .17·10−30.905 2 .29·10−20.0147±0.0008±0.0024
+11 2 .24·10−40.365 3 .53·10−40.0866±0.0058±0.0069
+11 2 .24·10−40.455 4 .12·10−40.0740±0.0042±0.0060
+11 2 .24·10−40.545 4 .93·10−40.0692±0.0036±0.0049
+11 2 .24·10−40.635 6 .14·10−40.0701±0.0033±0.0052
+11 2 .24·10−40.725 8 .16·10−40.0702±0.0031±0.0049
+11 2 .24·10−40.815 1 .21·10−30.0517±0.0025±0.0028
+11 2 .24·10−40.905 2 .36·10−30.0273±0.0019±0.0046
+11 4 .78·10−40.365 7 .53·10−40.0771±0.0044±0.0061
+11 4 .78·10−40.455 8 .77·10−40.0652±0.0032±0.0047
+11 4 .78·10−40.545 1 .05·10−30.0580±0.0026±0.0051
+11 4 .78·10−40.635 1 .31·10−30.0635±0.0025±0.0046
+11 4 .78·10−40.725 1 .74·10−30.0570±0.0022±0.0034
+11 4 .78·10−40.815 2 .58·10−30.0483±0.0020±0.0023
+11 4 .78·10−40.905 5 .03·10−30.0226±0.0014±0.0039
+11 1 .02·10−30.365 1 .60·10−30.0623±0.0037±0.0052
+11 1 .02·10−30.455 1 .87·10−30.0554±0.0028±0.0041
+11 1 .02·10−30.545 2 .24·10−30.0513±0.0023±0.0042
+11 1 .02·10−30.635 2 .79·10−30.0483±0.0020±0.0030
+11 1 .02·10−30.725 3 .71·10−30.0493±0.0020±0.0035
+11 1 .02·10−30.815 5 .51·10−30.0419±0.0018±0.0021
+11 1 .02·10−30.905 1 .07·10−20.0182±0.0011±0.0033
+Table 3: The semi-inclusive structure function FLN(3)
+2(Q2,x,xL), for neutrons with
+pT<0.2GeV. The bin centre values of Q2,x,xLandβin the corresponding bins are also
+given. Thefirstuncertaintyisstatisticalandthesecondsy stematic. Normalisationuncertainties
+of5%are notincluded.
+19Q2[GeV2] x x L β FLN(3)
+2
+11 2 .17·10−30.365 3 .42·10−30.0505±0.0032±0.0041
+11 2 .17·10−30.455 3 .99·10−30.0447±0.0024±0.0035
+11 2 .17·10−30.545 4 .77·10−30.0428±0.0021±0.0032
+11 2 .17·10−30.635 5 .95·10−30.0417±0.0018±0.0034
+11 2 .17·10−30.725 7 .90·10−30.0420±0.0018±0.0026
+11 2 .17·10−30.815 1 .17·10−20.0339±0.0016±0.0016
+11 2 .17·10−30.905 2 .29·10−20.0167±0.0011±0.0031
+11 4 .63·10−30.365 7 .29·10−30.0460±0.0035±0.0041
+11 4 .63·10−30.455 8 .50·10−30.0412±0.0027±0.0032
+11 4 .63·10−30.545 1 .02·10−20.0348±0.0021±0.0027
+11 4 .63·10−30.635 1 .27·10−20.0344±0.0019±0.0025
+11 4 .63·10−30.725 1 .68·10−20.0307±0.0016±0.0018
+11 4 .63·10−30.815 2 .50·10−20.0269±0.0015±0.0012
+11 4 .63·10−30.905 4 .87·10−20.0157±0.0012±0.0023
+16 4 .78·10−40.365 7 .53·10−40.0829±0.0059±0.0069
+16 4 .78·10−40.455 8 .77·10−40.0652±0.0039±0.0049
+16 4 .78·10−40.545 1 .05·10−30.0679±0.0037±0.0049
+16 4 .78·10−40.635 1 .31·10−30.0666±0.0033±0.0052
+16 4 .78·10−40.725 1 .74·10−30.0708±0.0034±0.0049
+16 4 .78·10−40.815 2 .58·10−30.0578±0.0030±0.0030
+16 4 .78·10−40.905 5 .03·10−30.0300±0.0022±0.0051
+16 1 .02·10−30.365 1 .60·10−30.0787±0.0054±0.0068
+16 1 .02·10−30.455 1 .87·10−30.0706±0.0043±0.0050
+16 1 .02·10−30.545 2 .24·10−30.0529±0.0029±0.0040
+16 1 .02·10−30.635 2 .79·10−30.0580±0.0029±0.0043
+16 1 .02·10−30.725 3 .71·10−30.0568±0.0028±0.0038
+16 1 .02·10−30.815 5 .51·10−30.0469±0.0024±0.0023
+16 1 .02·10−30.905 1 .07·10−20.0233±0.0017±0.0041
+16 2 .17·10−30.365 3 .42·10−30.0674±0.0050±0.0055
+16 2 .17·10−30.455 3 .99·10−30.0546±0.0035±0.0036
+16 2 .17·10−30.545 4 .77·10−30.0512±0.0030±0.0039
+16 2 .17·10−30.635 5 .95·10−30.0513±0.0027±0.0042
+16 2 .17·10−30.725 7 .90·10−30.0485±0.0025±0.0032
+16 2 .17·10−30.815 1 .17·10−20.0416±0.0023±0.0021
+16 2 .17·10−30.905 2 .29·10−20.0195±0.0015±0.0036
+16 4 .63·10−30.365 7 .29·10−30.0544±0.0045±0.0039
+16 4 .63·10−30.455 8 .50·10−30.0425±0.0029±0.0030
+16 4 .63·10−30.545 1 .02·10−20.0417±0.0026±0.0032
+16 4 .63·10−30.635 1 .27·10−20.0421±0.0024±0.0037
+16 4 .63·10−30.725 1 .68·10−20.0377±0.0021±0.0021
+16 4 .63·10−30.815 2 .50·10−20.0307±0.0018±0.0017
+16 4 .63·10−30.905 4 .87·10−20.0151±0.0012±0.0023
+24 4 .78·10−40.365 7 .53·10−40.0945±0.0091±0.0076
+24 4 .78·10−40.455 8 .77·10−40.0903±0.0076±0.0075
+24 4 .78·10−40.545 1 .05·10−30.0716±0.0054±0.0047
+24 4 .78·10−40.635 1 .31·10−30.0664±0.0047±0.0049
+24 4 .78·10−40.725 1 .74·10−30.0784±0.0052±0.0059
+24 4 .78·10−40.815 2 .58·10−30.0640±0.0047±0.0032
+24 4 .78·10−40.905 5 .03·10−30.0281±0.0029±0.0062
+Table 4: The semi-inclusive structure function FLN(3)
+2(Q2,x,xL), for neutrons with
+pT<0.2GeV, continuedfrom Table3.
+20Q2[GeV2] x x L β FLN(3)
+2
+24 1 .02·10−30.365 1 .60·10−30.0871±0.0073±0.0070
+24 1 .02·10−30.455 1 .87·10−30.0782±0.0056±0.0064
+24 1 .02·10−30.545 2 .24·10−30.0679±0.0044±0.0040
+24 1 .02·10−30.635 2 .79·10−30.0678±0.0040±0.0054
+24 1 .02·10−30.725 3 .71·10−30.0617±0.0035±0.0044
+24 1 .02·10−30.815 5 .51·10−30.0494±0.0031±0.0028
+24 1 .02·10−30.905 1 .07·10−20.0231±0.0021±0.0036
+24 2 .17·10−30.365 3 .42·10−30.0684±0.0060±0.0058
+24 2 .17·10−30.455 3 .99·10−30.0602±0.0045±0.0046
+24 2 .17·10−30.545 4 .77·10−30.0601±0.0041±0.0057
+24 2 .17·10−30.635 5 .95·10−30.0545±0.0034±0.0028
+24 2 .17·10−30.725 7 .90·10−30.0471±0.0029±0.0031
+24 2 .17·10−30.815 1 .17·10−20.0429±0.0028±0.0020
+24 2 .17·10−30.905 2 .29·10−20.0206±0.0019±0.0039
+24 4 .63·10−30.365 7 .29·10−30.0531±0.0050±0.0041
+24 4 .63·10−30.455 8 .50·10−30.0503±0.0040±0.0031
+24 4 .63·10−30.545 1 .02·10−20.0467±0.0034±0.0038
+24 4 .63·10−30.635 1 .27·10−20.0426±0.0028±0.0033
+24 4 .63·10−30.725 1 .68·10−20.0421±0.0027±0.0030
+24 4 .63·10−30.815 2 .50·10−20.0328±0.0022±0.0019
+24 4 .63·10−30.905 4 .87·10−20.0174±0.0017±0.0028
+24 9 .87·10−30.365 1 .55·10−20.0452±0.0052±0.0035
+24 9 .87·10−30.455 1 .81·10−20.0421±0.0042±0.0034
+24 9 .87·10−30.545 2 .17·10−20.0398±0.0035±0.0038
+24 9 .87·10−30.635 2 .70·10−20.0345±0.0028±0.0018
+24 9 .87·10−30.725 3 .59·10−20.0341±0.0027±0.0028
+24 9 .87·10−30.815 5 .34·10−20.0266±0.0022±0.0017
+24 9 .87·10−30.905 1 .04·10−10.0143±0.0015±0.0029
+37 1 .02·10−30.365 1 .60·10−30.1051±0.0109±0.0088
+37 1 .02·10−30.455 1 .87·10−30.0708±0.0063±0.0053
+37 1 .02·10−30.545 2 .24·10−30.0726±0.0058±0.0054
+37 1 .02·10−30.635 2 .79·10−30.0720±0.0052±0.0056
+37 1 .02·10−30.725 3 .71·10−30.0621±0.0045±0.0036
+37 1 .02·10−30.815 5 .51·10−30.0497±0.0039±0.0025
+37 1 .02·10−30.905 1 .07·10−20.0236±0.0026±0.0044
+37 2 .17·10−30.365 3 .42·10−30.0757±0.0081±0.0065
+37 2 .17·10−30.455 3 .99·10−30.0667±0.0060±0.0041
+37 2 .17·10−30.545 4 .77·10−30.0571±0.0046±0.0045
+37 2 .17·10−30.635 5 .95·10−30.0593±0.0044±0.0042
+37 2 .17·10−30.725 7 .90·10−30.0546±0.0039±0.0041
+37 2 .17·10−30.815 1 .17·10−20.0440±0.0035±0.0038
+37 2 .17·10−30.905 2 .29·10−20.0213±0.0024±0.0037
+37 4 .63·10−30.365 7 .29·10−30.0647±0.0075±0.0053
+37 4 .63·10−30.455 8 .50·10−30.0550±0.0053±0.0039
+37 4 .63·10−30.545 1 .02·10−20.0493±0.0043±0.0036
+37 4 .63·10−30.635 1 .27·10−20.0439±0.0035±0.0029
+37 4 .63·10−30.725 1 .68·10−20.0458±0.0035±0.0027
+37 4 .63·10−30.815 2 .50·10−20.0362±0.0030±0.0017
+37 4 .63·10−30.905 4 .87·10−20.0177±0.0021±0.0038
+Table 5: The semi-inclusive structure function FLN(3)
+2(Q2,x,xL), for neutrons with
+pT<0.2GeV, continuedfrom Table4.
+21Q2[GeV2] x x L β FLN(3)
+2
+37 9 .87·10−30.365 1 .55·10−20.0469±0.0057±0.0046
+37 9 .87·10−30.455 1 .81·10−20.0446±0.0046±0.0030
+37 9 .87·10−30.545 2 .17·10−20.0390±0.0037±0.0038
+37 9 .87·10−30.635 2 .70·10−20.0372±0.0032±0.0023
+37 9 .87·10−30.725 3 .59·10−20.0335±0.0028±0.0019
+37 9 .87·10−30.815 5 .34·10−20.0326±0.0028±0.0018
+37 9 .87·10−30.905 1 .04·10−10.0138±0.0017±0.0023
+55 2 .17·10−30.365 3 .42·10−30.0962±0.0125±0.0074
+55 2 .17·10−30.455 3 .99·10−30.0621±0.0072±0.0074
+55 2 .17·10−30.545 4 .77·10−30.0727±0.0072±0.0045
+55 2 .17·10−30.635 5 .95·10−30.0622±0.0057±0.0034
+55 2 .17·10−30.725 7 .90·10−30.0540±0.0048±0.0041
+55 2 .17·10−30.815 1 .17·10−20.0486±0.0048±0.0027
+55 2 .17·10−30.905 2 .29·10−20.0204±0.0029±0.0039
+55 4 .63·10−30.365 7 .29·10−30.0695±0.0093±0.0052
+55 4 .63·10−30.455 8 .50·10−30.0614±0.0069±0.0052
+55 4 .63·10−30.545 1 .02·10−20.0448±0.0048±0.0032
+55 4 .63·10−30.635 1 .27·10−20.0482±0.0048±0.0035
+55 4 .63·10−30.725 1 .68·10−20.0498±0.0046±0.0035
+55 4 .63·10−30.815 2 .50·10−20.0415±0.0042±0.0034
+55 4 .63·10−30.905 4 .87·10−20.0164±0.0023±0.0022
+55 9 .87·10−30.365 1 .55·10−20.0592±0.0084±0.0048
+55 9 .87·10−30.455 1 .81·10−20.0429±0.0053±0.0027
+55 9 .87·10−30.545 2 .17·10−20.0409±0.0046±0.0030
+55 9 .87·10−30.635 2 .70·10−20.0372±0.0038±0.0027
+55 9 .87·10−30.725 3 .59·10−20.0338±0.0034±0.0022
+55 9 .87·10−30.815 5 .34·10−20.0304±0.0032±0.0018
+55 9 .87·10−30.905 1 .04·10−10.0148±0.0022±0.0037
+55 2 .10·10−20.365 3 .31·10−20.0515±0.0089±0.0046
+55 2 .10·10−20.455 3 .86·10−20.0347±0.0051±0.0021
+55 2 .10·10−20.545 4 .62·10−20.0286±0.0038±0.0038
+55 2 .10·10−20.635 5 .76·10−20.0334±0.0040±0.0018
+55 2 .10·10−20.725 7 .65·10−20.0303±0.0036±0.0020
+55 2 .10·10−20.815 1 .14·10−10.0266±0.0033±0.0023
+55 2 .10·10−20.905 2 .21·10−10.0098±0.0018±0.0018
+82 4 .63·10−30.365 7 .29·10−30.0588±0.0099±0.0045
+82 4 .63·10−30.455 8 .50·10−30.0537±0.0080±0.0036
+82 4 .63·10−30.545 1 .02·10−20.0465±0.0061±0.0036
+82 4 .63·10−30.635 1 .27·10−20.0520±0.0063±0.0052
+82 4 .63·10−30.725 1 .68·10−20.0480±0.0056±0.0033
+82 4 .63·10−30.815 2 .50·10−20.0417±0.0053±0.0040
+82 4 .63·10−30.905 4 .87·10−20.0180±0.0032±0.0028
+82 9 .87·10−30.365 1 .55·10−20.0484±0.0085±0.0041
+82 9 .87·10−30.455 1 .81·10−20.0514±0.0077±0.0036
+82 9 .87·10−30.545 2 .17·10−20.0366±0.0052±0.0025
+82 9 .87·10−30.635 2 .70·10−20.0507±0.0064±0.0038
+82 9 .87·10−30.725 3 .59·10−20.0361±0.0044±0.0027
+82 9 .87·10−30.815 5 .34·10−20.0311±0.0041±0.0029
+82 9 .87·10−30.905 1 .04·10−10.0170±0.0030±0.0027
+82 2 .10·10−20.365 3 .31·10−20.0477±0.0093±0.0040
+82 2 .10·10−20.455 3 .86·10−20.0319±0.0052±0.0025
+82 2 .10·10−20.545 4 .62·10−20.0366±0.0053±0.0024
+82 2 .10·10−20.635 5 .76·10−20.0372±0.0050±0.0030
+82 2 .10·10−20.725 7 .65·10−20.0443±0.0057±0.0028
+82 2 .10·10−20.815 1 .14·10−10.0241±0.0035±0.0020
+82 2 .10·10−20.905 2 .21·10−10.0132±0.0026±0.0027
+Table 6: The semi-inclusive structure function FLN(3)
+2(Q2,x,xL), for neutrons with
+pT<0.2GeV, continuedfrom Table5.
+220500010000150002000025000
+300 400 500 600 700 800 900
+En [GeV]EventsH1 Data
+RAPGAP- p
+DJANGO
+0.65×RAPGAP- p + 1.2×DJANGO
+H1
+0500010000150002000025000
+0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
+pT [GeV]EventsH1 Data
+RAPGAP- p
+DJANGO
+0.65×RAPGAP- p + 1.2×DJANGO
+H1(a) (b)
+Figure 3: The observed neutron energy (a) and transverse mom entum (b) distributions in the
+kinematicrange 6< Q2<100GeV2and1.5·10−4< x <3·10−2. Thedataarecomparedtothe
+predictions of RAPGAP- π(dashed line) and DJANGO (dotted line) Monte Carlo simulati ons.
+Also shown is a weighted combination of those two simulation s (full line), as described in
+Sect. 3.4.
+230510152025
+0.4 0.6 0.8 1
+xLds/dxL [nb]
+pT < xL× 0.69 GeVH1H1 Data
+RAPGAP- p
+DJANGO
+LEPTO-SCI-GAL
+0.65×RAPGAP- p + 1.2×DJANGO
+0246810
+0.4 0.6 0.8 1
+xLds/dxL [nb]
+pT < 0.2 GeVH1H1 Data
+RAPGAP- p
+DJANGO
+LEPTO-SCI-GAL
+0.65×RAPGAP- p + 1.2×DJANGO(a) (b)
+Figure 4: The cross section as a function of the fractional en ergy of the neutron xLin the
+kinematicrange 6< Q2<100GeV2and1.5·10−4< x <3·10−2. Thetransversemomentum
+of the neutron is restricted to pT< xL·0.69 GeV(a) andpT<0.2 GeV(b). The data are
+compared to the predictions of RAPGAP- π(dashed line), DJANGO (dotted line) and LEPTO-
+SCI-GAL (dash-dotted line) Monte Carlo simulations. Also s hown is a weighted combination
+ofRAPGAP- πand DJANGO simulations(fullline), as describedin Sect. 3. 4.
+240.1
+0.1
+0.1
+  xL0.4               1
+00.1
+  xL0.4               1
+00.1
+  xL0.4               1
+00.1
+  xL0.4               1  xL0.4               1
+00.1
+  xL0.4               1  xL0.4               1Q2 = 7.3 GeV2Q2 = 11 GeV2Q2 = 16 GeV2Q2 = 24 GeV2Q2 = 37 GeV2Q2 = 55 GeV2Q2 = 82 GeV2
+2.2×10-4x
+4.8×10-4x
+1.0×10-3x
+2.2×10-3x
+4.6×10-3x
+9.9×10-3x
+2.1×10-2x
+H1 Data
+RAPGAP- p
+DJANGO
+0.65×RAPGAP- p + 1.2×DJANGOFLN(3)
+2       (Q2
+ ,x,xL)
+H1
+Figure 5: The semi-inclusive structure function FLN(3)
+2(Q2,x,xL), for neutrons with
+pT<0.2GeV,comparedtothepredictionsofRAPGAP- π(dashedline)andDJANGO(dotted
+line)MonteCarlosimulations. Alsoshownisaweightedcomb inationofthosetwosimulations
+(fullline),as described inSect. 3.4.
+2500.050.1 H1 Data
+Average
+00.050.1
+00.050.1
+00.050.1
+00.050.1
+00.050.1
+10 10210 10210 10210 10210 10210 10210 102
+Q2 [GeV2]xL = 0.37 xL = 0.46 xL = 0.55 xL = 0.64 xL = 0.73 xL = 0.82 xL = 0.91
+2.2×10-4x
+4.8×10-4x
+1.0×10-3x
+2.2×10-3x
+4.6×10-3x
+9.9×10-3xFLN(3)
+2       (Q2
+ ,x,xL)/F2(Q2
+ ,x) H1
+Figure6: Theratioofthesemi-inclusivestructurefunctio nFLN(3)
+2(Q2,x,xL),forneutronswith
+pT<0.2GeV, to the proton structure function F2(Q2,x)obtained from the H1PDF2009 fit to
+inclusiveDIS data[26]. Thelinesshowtheaveragevaluefor agivenxLbin.
+2600.1Q2 = 7.3 GeV2Q2 = 11 GeV2Q2 = 16 GeV2Q2 = 24 GeV2Q2 = 37 GeV2Q2 = 55 GeV2Q2 = 82 GeV2
+00.1
+00.1
+00.1
+00.1
+00.1
+00.1
+10-110-310-110-310-110-310-110-310-110-310-110-3
+b10-110-3H1 Data
+FitxL = 0.37
+xL = 0.46
+xL = 0.55
+xL = 0.64
+xL = 0.73
+xL = 0.82
+xL = 0.91FLN(3)
+2       (Q2
+ ,b,xL) H1
+Figure 7: The semi-inclusive structure function FLN(3)
+2(Q2,β,xL), for neutrons with
+pT<0.2GeV, shown as a function of βin bins of Q2andxL. The lines are the results of
+thefit withafunction c(xL)·β−λ(Q2)as described inSect. 4.
+2700.511.522.533.544.55
+10 102j = 0 , b = 0.077j = 1 , b = 0.036j = 2 , b = 0.017j = 3 , b = 0.0079j = 4 , b = 0.0037j = 5 , b = 0.0017j = 6 , b = 0.00082
+Q2 [GeV2]FLN(3)
+2       (xL= 0.73)/Gp + 0.5×j
+H1 Data
+GRSc-p LO
+ABFKW- p Set 1 NLO
+2/3 F2  H1PDF2009FLN(3)
+2       (xL= 0.73)/Gp , Gp = 0.13
+H1
+Figure 8: The semi-inclusive structure function FLN(3)
+2, for neutrons with pT<0.2 GeV, di-
+videdbythepionflux Γπintegratedover tatthecentralvalue xL= 0.73,shownasafunctionof
+Q2inbinsof β. ThepionfluxisdefinedinEq.7. Thedataarecomparedtotwodi fferentparam-
+eterisations of the pion structure function Fπ
+2[42,43] and to the H1PDF2009 parameterisation
+oftheprotonstructurefunction[26], which hasbeen scaled by 2/3.
+2800.20.40.6
+10-310-2
+bQ2 = 7.3 GeV2
+00.20.40.6
+10-310-2
+bQ2 = 11 GeV2
+00.20.40.6
+10-2
+bQ2 = 16 GeV2
+00.20.40.6
+10-2
+bQ2 = 24 GeV2
+00.20.40.6
+10-2
+bQ2 = 37 GeV2
+00.20.40.6
+10-210-1
+bQ2 = 55 GeV2
+00.20.40.6
+10-1
+bQ2 = 82 GeV2H1 Data
+GRSc-p LO
+ABFKW- p Set 1 NLO
+2/3 F2  H1PDF2009
+DJANGO ×1.2/GpFLN(3)
+2       (xL= 0.73)/Gp , Gp = 0.13 H1
+Figure 9: The semi-inclusive structure function FLN(3)
+2, for neutrons with pT<0.2 GeV, di-
+vided by the pion flux Γπintegrated over tat the central value xL= 0.73, shown as a function
+ofβin bins of Q2. The pion flux is defined in Eq. 7. The data are compared to two di fferent
+parameterisationsof the pion structurefunction Fπ
+2[42,43] and to the H1PDF2009 parameter-
+isationoftheproton structurefunction [26], which has bee n scaled by 2/3. The contributionof
+neutronsfromfragmentation,aspredictedbyDJANGOandsca ledbyafactor1.2,asdescribed
+inSect. 3.4, isindicated.
+29
\ No newline at end of file
diff --git a/1001.0533.txt b/1001.0533.txt
new file mode 100644
index 0000000000000000000000000000000000000000..565906986d9cddd60450d6c777384d3d215245b8
--- /dev/null
+++ b/1001.0533.txt
@@ -0,0 +1,341 @@
+arXiv:1001.0533v1  [astro-ph.HE]  4 Jan 2010PROCEEDINGS OF THE 31stICRC, Ł´OD´Z 2009 1
+Observation of the Galactic Cosmic Ray Moon shadowing effect
+with the ARGO-YBJ experiment
+Roberto Iuppa∗ †, Daniele Martello‡§, Bo Wang¶and Giovanni Zizzi‡§
+on behalf of the ARGO-YBJ Collaboration
+∗INFN Sezione Roma Tor Vergata, via della Ricerca Scientifica 1, Rome - Italy
+†Dipartimento di Fisica, Universit ´a Roma Tor Vergata, via della Ricerca Scientifica 1, Rome - Ita ly
+‡INFN Sezione di Lecce, via per Arnesano, 73100 Lecce - Italy
+§Dipartimento di Fisica dell’Universit `a del Salento, via per Arnesano, 73100 Lecce - Italy
+¶Key Laboratory of Particle Astrophysics, IHEP - Chinese Aca demy of Science, P.O. Box 918, 100049 Beijing, P.R. China
+Abstract. Cosmic rays are hampered by the Moon
+and a deficit in its direction is expected (the so-called
+Moon shadow ). The Moon shadow is an important
+method to determine the performance of an air
+shower array. In fact, the westward displacement of
+the shadow center, due to the propagation of cosmic
+rays in the geomagnetic field, allows to calibrate the
+energy scale of the primary particles observed by
+the detector. In addition, the shape of the shadow
+allows a measurement of the angular resolution and
+the position of the deficit at high energy allows the
+evaluation of the pointing accuracy of the detector.
+In this paper we present the observation of the
+galactic cosmic rays Moon shadowing effect per-
+formed by the ARGO-YBJ experiment in the multi-
+TeV energy region. The measured angular resolution
+as a function of the shower size is compared with the
+expectations from a MC simulation.
+Keywords : Moon Shadow observation, Cosmic
+Rays, ARGO-YBJ experiment
+I. INTRODUCTION
+TheangularresolutionisacriticalfeatureofanExten-
+sive Air Shower (EAS) array in gamma-ray astronomy.
+In fact, the rejection of the nearly isotropic background
+of charged cosmic rays is mainly performed by im-
+proving the angular resolution, thus reducing the source
+region extension. Hence the tuning of a firm calibration
+technique of the angular resolution is mandatory.
+The CYGNUS experiment in 1991 reported the first
+determination of the angular resolution of an EAS de-
+tector by exploiting the analysis of the shadow of the
+Moon [1]. In fact, as pointed out in 1957 by Clark [2],
+the cosmic rays are hampered in their propagationto the
+Earth due to the Moon’s presence and a deficit of events
+in its direction is expected: the so-called Moon shadow .
+At high energy, the Moon shadow would be observed
+by an ideal detector as a 0.26◦wide circular deficit of
+events, centered on the Moon position. The deviation
+from such an ideal case gives us information about the
+Point Spread Function (PSF) of the detector. The shape
+of the deficit allows the measurement of the angularresolution and its position allows the evaluation of the
+absolute pointing accuracy of the detector. In addition,
+positively charged particles are eastward deflected, due
+to the geomagnetic field, with an energy dependence
+∆θ∼1.6◦Z/ETeV. As a consequence, the observation
+of the displacement of the Moon provides a direct check
+of the relation between shower size and primary energy.
+Therefore, the analysis of the Moon shadow allows the
+calibration of the performance of an EAS array.
+The same shadowing effect can be observed in the
+direction of the Sun but the interpretation of the shad-
+owing phenomenology is more complex. In fact, the
+displacement of the shadow from the apparent position
+of the Sun could be explained by the joint effects of
+the geomagnetic field and of the solar and Interplane-
+tary Magnetic Fields, whose configuration considerably
+changes with the phases of the solar activity cycle [3].
+Results about the Sun shadow observation with the
+ARGO-YBJ experiment are discussed in [4].
+Inthispaperwepresenttheobservationofthegalactic
+cosmic rays Moon shadowing effect carried out by
+the ARGO-YBJ experiment. We report on the angular
+resolutionofthedetectorinthemulti-TeVenergyregion.
+The pointing error is also investigated.
+II. THEARGO-YBJ EXPERIMENT
+The ARGO-YBJ detector, located at the YangBaJing
+Cosmic Ray Laboratory (Tibet, P.R. China, 4300 m
+a.s.l.),istheonlyexperimentexploitingthe fullcoverage
+approach at very high altitude. The detector is consti-
+tuted by a central carpet ∼74×78 m2, made of a single
+layer of Resistive Plate Chambers(RPCs) with ∼92%of
+active area, enclosed by a partially instrumented guard
+ring that extends the detector surface up to ∼100×110
+m2. The apparatus has a modular structure, the basic
+data acquisitionelement beinga cluster (5.72 ×7.64m2),
+namely a group of 12 RPCs (2.80 ×1.25 m2each).
+Each chamber is read by 80 strips of 7 ×62 cm2(the
+spatial pixel), logically organized in 10 independent
+pads of 56 ×62 cm2representing the time pixel of the
+detector. The RPCs are operated in streamer mode with
+a standard gas mixture (Argon 15%, Isobutane 10%,2 R. IUPPA et al.MOON SHADOW OBSERVATION
+TetraFluoroEthane 75%), the High Voltage settled at 7.2
+kV ensures an overall efficiency of about 96% [5]. The
+central carpet contains 130 clusters (hereafter ARGO-
+130) and the full detector is composed of 153 clusters
+for a total active surface of ∼6700 m2.
+AlleventsgivinganumberoffiredpadsN pad≥Ntrig
+in the central carpet within a time window of 420 ns are
+recorded. The spatial coordinates and the time of any
+fired pad are then used to reconstruct the position of the
+shower core and the arrival direction of the primary.
+The ARGO-YBJ experiment started recording data
+with the whole central carpet in June 2006. Since 2007
+November the full detector is in stable data taking at
+the multiplicity trigger threshold N trig≥20 and a duty
+cycle∼90%: the trigger rate is about 3.6 kHz.
+The reconstruction of shower parameters is split into
+the following steps. First the shower core position is
+derived with the Maximum Likelihood method from the
+lateral density distribution of the secondary particles.
+In the second step, given the shower core position, the
+shower axis is reconstructed by means of an iterative
+weighted planar fit being able to reject the time values
+belonging to the non-gaussian tails of the arrival time
+distributions. A conical correction with a slope fixed to
+α= 0.03 rad is applied to the surviving hits in order to
+improve the angular resolution [6].
+III. MONTECARLOSIMULATION
+The air showers development in the atmosphere has
+been generated with the CORSIKA v. 6.500 code in-
+cluding the QGSJET-II.03 hadronic interaction model
+for primary energy above 80 GeV and the FLUKA code
+for lower energies [7]. Cosmic ray spectra of p, He and
+CNO have been simulated in the energy range from
+30 GeV to 1 PeV following [8]. The relative fractions
+(in % of the total) after triggering by the ARGO-
+YBJ detector for events with N strip≥30 are: p ∼88%,
+He∼10%, CNO ∼2%. About 3 ·1011showers have been
+distributed in the zenith angle interval 0-60 degrees. The
+secondary particles have been propagateddown to a cut-
+off energy of 1 MeV. The experimental conditions have
+beenreproducedviaaGEANT3-basedcode.Theshower
+core positions have been randomly distributed sampling
+in energy-dependent area up to 103×103m2, centered
+on the detector.
+IV. DATA ANALYSIS
+For the analysis of the shadowing effect a 10◦×10◦
+sky map in celestial coordinates (right ascension and
+declination) with 0.1◦×0.1◦bin size, centered on the
+Moon location, is filled with the detected events. The
+background is evaluated with both the time swapping
+[10] and the equi-zenith angle [11] methods.
+With the time swapping method, N ”fake”events
+are generated for each detected one, by replacing the
+measured arrival time with new ones. These events
+are randomly selected within a 3 hours wide buffer of
+recorded data. Swapping the time is swapping the rightWest                                                       East     -4 -3 -2 -1 0 1 2 3 4South                                                     North 
+-4-3-2-101234
+-40-35-30-25-20-15-10-50
+Fig. 1: Moon shadow significance map observed by the
+ARGO-YBJ detector in 2063 hours on-source for events
+with N strip≥30 and zenith angle θ <50◦. The color
+scale gives the statistical significance.
+ascension, keeping unchanged the declination. A new
+sky map (background map) is built by using 10 such
+fake events for each real one, so that the statistical error
+on the background can be kept small enough.
+With the equi-zenith angle method 6 off-source bins
+are symmetricallyalignedon bothsides of the on-source
+field, at the same zenith angle. The off-source bins are
+each set at an azimuth distance 5◦/sinθfrom the on-
+source bin, where θis the zenith angle of the Moon
+position. Other off-source bins are located every 5◦/sinθ
+from the nearest off-source bins. The average of the
+event densities inside these bins was taken to be the
+background.
+Tomaximizethesignaltonoiseratio,thebinsarethen
+groupedover a circular area of radius ψ, i.e. every bin is
+filled with the content of all the surrounding bins whose
+center is closer than ψfrom its center. The value of ψ
+is related to the angular resolution of the detector, and
+corresponds to the radius of the observational window
+that maximizes the signal to noise ratio, which in turn
+depends on the number of fired pads of the event: when
+the PSF is a Gaussian with rms σ,ψ=σ·1.58and
+contains ∼72%of the events. Finally, the integrated
+background map is subtracted from the corresponding
+integrated event map, thus obtaining the ”source map”.
+For each bin of such a map, the deficit significance with
+respect to the background is calculated according to the
+Li and Ma formula [12]. Notice that in the integrated
+maps neighboring bins are correlated.
+The analysis reported in this paper refers to events
+collected after the following event selection: (1) each
+event should fire more than 30 strips on the ARGO-
+130 central carpet to avoid any threshold effect; (2) the
+zenith angle of the shower arrival direction should be
+less than 50◦; (3) the reconstructed core position should
+be inside an area 250 ×250 m2centered on the detector;PROCEEDINGS OF THE 31stICRC, Ł´OD´Z 2009 3
+)°) (mδ) cos(mα - α(-6 -5 -4 -3 -2 -1 0 1 2 3 4deficit count
+-12000-10000-8000-6000-4000-200002000 30-60
+)°) (mδ) cos(mα - α(-3 -2 -1 0 1 2 3deficit count
+-8000-6000-4000-20000100-300
+)°) (mδ) cos(mα - α(-6 -5 -4 -3 -2 -1 0 1 2 3 4deficit count
+-6000-5000-4000-3000-2000-10000100060-100
+)°) (mδ) cos(mα - α(-3 -2 -1 0 1 2 3deficit count
+-2000-1500-1000-5000500300-500
+Fig. 2: Deficit counts measured around the Moon pro-
+jected along the East-West axis for different multiplicity
+bins.
+(4) the reduced χ2of the final temporal fit should be
+less than 100 ns2. According to our simulation studies,
+the median energy of the selected protons firing 30 ÷
+60 strips is E 50≈1.4 TeV (mode energy ∼0.30 TeV).
+In Fig.1 the Moon shadow observed with all data
+recorded since June 2006 (2063 hours on-source) for
+events with N strip≥30 and zenith angle θ <50◦is
+shown. The statistical significance of the observation is
+about 43 standard deviations.
+V. RESULTS
+The deficit counts observed around the Moon pro-
+jected to the East-West axis are shown in Fig. 2 for 4
+multiplicity bins. We use the events contained in an an-
+gular slice parallel to the East-West axis and centered to
+the observed Moon position. The widths of these bands
+are function of the N strip-dependent angular resolution:
+±3.3◦in 30≤Nstrip<60,±2.6◦in 60≤Nstrip<100,
+±2.0◦in 100≤Nstrip<300,±1.5◦in 300≤Nstrip<
+500. As an expected effect of the geomagnetic effect,
+the profile of the shadow is broadened and the peak
+positions shifted westward as the multiplicity (i.e., the
+cosmic ray primary energy) decreases. We note that in
+thelowestmultiplicitybin(N strip=30-60)theMoonis
+shifted by about 1◦, as expected for a primaryof rigidity
+1.6TeV/Z: thisis the first time that an EAS-arrayis able
+to detect showers with such a low primary energy.
+The best procedure to evaluate the pointing accuracy
+is to observe the position of the Moon shadow pro-
+duced by high-energy cosmic rays which are negligibly
+affectedby the geomagneticfield. For protonsof 30 TeV
+we expect a deflectionof about0.05◦. For heaviernuclei
+this deflection will increase but as the composition of
+cosmic rays in this energy range is dominated by the
+light component (nuclei heavier than CNO contribute to
+theratelessthan3%inthewholestripmultiplicityrange
+[13]) we expect only a small contribution from heavy
+ions to the blurringof the Moon shadow.As can be seen
+from the Fig. 3, the observed high energy (N pad≥1000,West                                                               East    -4 -3 -2 -1 0 1 2 3 4South                                              North    
+-4-3-2-101234
+-20-15-10-50
+Fig. 3: Moon shadow significance map observed by the
+ARGO-YBJ detector in 2063 hours on-source for events
+with N strip≥1000 and zenith angle θ<50◦. The color
+scale gives the statistical significance.
+E50
+p∼30 TeV) Moon shadow position is centered in
+the East-West direction but we observe a residual shift
+towards the North. Since the displacement along the
+North-South axis is not affected by the geomagnetic
+field at the Yangbajing latitude [9], we are able to
+investigate this pointing error without the Moon shadow
+simulation as a function of the multiplicity. The analysis
+has been performed both with the time-swapping and
+the equi-zenith angle methods: the results are in good
+agreement and suggest that there is a residual systematic
+shift towards North of (0.20 ±0.05)◦, independent of
+multiplicity. As a conservative estimate we assume our
+systematic errors to coincide with this displacement
+in both North-.South and East-West projection. These
+upper limits for the systematic errors are however much
+smaller than our angular resolution, at least for the bulk
+of data, and can therefore be neglected in the point
+source searches.
+In the Fig. 4 the displacement of the Moon shadow in
+the East-West direction is shown. Two different methods
+agree with each other quite well. A comparison between
+the measurement and the simulations allows to attribute
+this displacement to an absolute energy calibration of
+the detector.
+The PSF of the detector, studied in the North-South
+projection, not affected by the geomagnetic field, is
+Gaussian for N strip≥100, while for lower multiplicities
+it can be described with an additional Gaussian, which
+contributes for about 20%. For these events the angular
+resolution is calculated as the weighted sum of the σ2of
+each gaussian. In Fig. 5 the measured angular resolution
+is compared to expectations from a MC simulation as
+a function of the multiplicity. As can be seen, the
+values are in fair agreement: the angular resolution σ
+of the ARGO-YBJ experiment for cosmic ray-induced
+air showers is less than 0.6◦for Nstrip≥300. The effect
+of the finite angular width of the Moon on the angular4 R. IUPPA et al.MOON SHADOW OBSERVATION
+number of fired strips210310410)   °(α ∆
+-1.5-1-0.500.5[TeV]/Z50E
+1 10210[TeV]/Z50E
+1 10210
+EW displacement
+equi-zenith
+time-swapping
+Fig. 4: Observed displacement of the Moon shadow in
+the East-West direction as a function of multiplicity.
+The upper scale refers to the median energy of rigidity
+(TeV/Z) in each multiplicity bin (shown by the horizon-
+tal errors). The experimental data calculated with two
+different methods (see text).
+number of fired strips210310(deg.)σ
+00.20.40.60.811.21.41.61.822.2[TeV]/Z50E
+1 10
+Angular resolution
+MC expectation
+Data
+Fig. 5: Measured angular resolution of the ARGO-YBJ
+detector compared to expectations from MC simulation
+as a function of the multiplicity. The upper scale refers
+to the median energy of rigidity (TeV/Z) in each multi-
+plicity bin (shown by the horizontal errors).
+resolution (less than 5% if σ >0.4◦and only 1.7% if
+σ >0.7◦) is discussed in [9]. From MC simulations we
+draw that the angular resolution for γ-induced showers
+at the threshold is at least 30% lower due to their better
+defined time profile [6]. A new reconstruction algorithm
+based on a weighted fit of the time profile in understudy
+in order to improve the angular resolution.
+A new reconstruction algorithm based on a weighted
+fit of the time profile is under study in order to improve
+the angular resolution.
+VI. CONCLUSIONS
+The galactic cosmic ray Moon shadowing effect has
+been observed by the ARGO-YBJ experiment in themulti-TeV energy region with high statistical signifi-
+cance. The analysis has been performed both with the
+time-swappingmethodandthe equi-zenithanglemethod
+in order to investigate possible biases in the background
+calculation. The measured angular resolution is in good
+agreement with MC simulations, making us confident
+in the reconstruction algorithms, we can further find an
+absolute energy calibration of the detector.
+REFERENCES
+[1] D.E. Alexandreas et al., Phys. Rev. D431735, 1991.
+[2] G.W. Clark, Phys. Rev. 108450, 1957.
+[3] M. Amenomori et al., ApJ5411051, 2000.
+[4] F.R. Zhu et al., these proceedings
+[5] G. Aielli et al., NIMA56292, 2006.
+[6] G. Di Sciascio et al., Proc. of 30th ICRC 2009, (preprint:
+arXiv:0710.1945).
+[7] D. Heck et al., Report FZKA 6019 1998.
+[8] B. Wiebel-Sooth et al., A&A330389, 1998.
+[9] G. Di Sciascio and R. Iuppa, these proceedings
+[10] D.E. Alexandreas et al., NIMA311350, 1992.
+[11] M. Amenomori et al., Phys. Rev. D472675, 1993.
+[12] T. Li and Y. Ma, ApJ,272317, 1993.
+[13] S. Catalanotti er al., these proceedings.
\ No newline at end of file
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+arXiv:1001.0534v2  [math.DG]  21 Dec 2011MULTIPLICATIVE FORMS AT THE INFINITESIMAL LEVEL
+HENRIQUE BURSZTYN AND ALEJANDRO CABRERA
+Abstract. Wedescribearbitrarymultiplicativedifferentialforms on Liegroupoids
+infinitesimally, i.e., in terms of Lie algebroid data. This d escription is based on the
+study of linear differential forms on Lie algebroids and enco mpasses many known
+integration results related to Poisson geometry. We also re visit multiplicative mul-
+tivector fields and their infinitesimal counterparts, drawi ng a parallel between the
+two theories.
+Contents
+1. Introduction 1
+Acknowledgments 4
+Notation, conventions and identities 4
+2. Linear forms on vector bundles 5
+2.1. Tangent and cotangent bundles of vector bundles 5
+2.2. The structure of linear forms on vector bundles 6
+2.3. Tangent lifts 8
+3. Linear forms on Lie algebroids 9
+3.1. Core and linear sections 9
+3.2. Tangent Lie algebroids 10
+3.3. IM-forms 11
+4. Infinitesimal description of multiplicative forms 15
+4.1. From multiplicative to IM forms 16
+4.2. Integration of IM forms 18
+5. Relation with the Weil algebra and Van Est isomorphism 22
+6. The dual picture: multiplicative multivector fields 25
+6.1. Linear multivector fields and derivations 26
+6.2. Infinitesimal description of multiplicative multivec tor fields 29
+6.3. The universal lifiting theorem revisited 30
+References 35
+1.Introduction
+This paper is devoted to the study of multiplicative different ial forms on Lie
+groupoids, with focus on their infinitesimal counterparts. Given a Lie groupoid
+Gover a manifold M, recall that a k-formω∈Ωk(G) is called multiplicative if
+m∗ω= pr∗
+1ω+pr∗
+2ω, wherem:G(2)=G ×MG → Gis the groupoid multiplication,
+and pri:G(2)→ G,i= 1,2, are the natural projections. Our goal is to characterize
+multiplicative forms on Gsolely in terms of information from its Lie algebroid. We
+will also discuss the analogous problem for multiplicative multivector fields.
+12
+Multiplicative differential forms and multivector fields on L ie groupoids have been
+studied for over 20 years in a variety of contexts. On Lie grou ps, multiplicative
+bivector fields came to notice in the late 1980s when the conce pt of Poisson Lie group
+(see e.g. [20] and references therein) was systematized. Mu ltiplicative 2-forms on
+Lie groupoids appeared around the same time as the symplecti c forms on symplectic
+groupoids [9, 29], originally introduced as part of a quanti zation scheme for Poisson
+manifolds. Poisson Lie groups and symplectic groupoids nat urally led to Poisson
+groupoids [30], which gave further impetus to the study of mu ltiplicative structures.
+In particular, multiplicative vector fields on Lie groupoid s turn out to encompass
+classical lifting processes of general interest in different ial geometry, see [23]. There
+are further connections between multiplicative 2-forms an d bivector fields and the
+theory of moment maps, found e.g. in [4, 5, 18, 25, 31].
+A central issue when considering multiplicative geometric al structures on Lie
+groupoids concerns their infinitesimal description, i.e., their description in terms
+of Lie-algebroid data. As usual in Lie theory, relating glob al and infinitesimal ob-
+jects involves suitable differentiation and integration pro cedures. Important classes
+of examples of multiplicative structures and their infinite simal counterparts include
+Poisson Lie groups and Lie bialgebras (see e.g. [20]), sympl ectic groupoids and Pois-
+son manifolds (see [7, 9]), and, more generally, the corresp ondence between Poisson
+groupoids and Lie bialgebroids [22, 24]. Dirac structures [ 10, 28] fit into a similar
+picture, being the infinitesimal versions of certain multip licative 2-forms, see [5]. As
+observed in [11], generalized complex structures [16, 17] p rovide another class of
+geometrical structures encoding infinitesimal informatio n that can be integrated to
+multiplicative structures on Lie groupoids. In establishi ng these infinitesimal-global
+correspondences, different methods have been employed, some based on the inte-
+grability of Lie-algebroid morphisms, e.g. [3, 22, 24], oth ers on infinite-dimensional
+arguments, e.g. [5, 7, 8, 18]. Although identifying the infin itesimal versions of mul-
+tiplicative forms is key in some of the aforementioned works , only special cases of
+this problem have been considered. In this paper, we treated it in full generality.
+Fromyetanotherperspective, multiplicativedifferentialf ormsariseasconstituents
+of the Bott-Schulman double complex of Lie groupoids [2] (se e also [1] and references
+therein), which computes the cohomology of their classifyi ng spaces. So the problem
+of understanding multiplicative forms infinitesimally may be seen as part of the
+problem of finding infinitesimal models for the cohomology of classifying spaces.
+This broader viewpoint is explored in the recent work [1], le ading to results closely
+related to ours; a comparison between them is also discussed in this paper.
+Ourapproach to describemultiplicative forms infinitesima lly starts with thestudy
+oflineardifferential forms on vector bundles A→M. We observe (Theorem 2.5)
+that any linear k-form onAis equivalent to a pair ( µ,ν) of vector-bundle maps
+µ:A→ ∧k−1T∗M,ν:A→ ∧kT∗M, covering the identity on M. IfAcarries a
+Lie algebroid structure, with bracket [ ·,·] and anchor ρ, we say that the pair ( µ,ν)
+is anIMk-form(IMstanding for infinitesimally multiplicative ) if the following
+compatibility conditions are satisfied: for all u,v∈Γ(A),
+(1)iρ(u)µ(v) =−iρ(v)µ(u),
+(2)µ([u,v]) =Lρ(u)µ(v)−iρ(v)dµ(u)−iρ(v)ν(u),
+(3)ν([u,v]) =Lρ(u)ν(v)−iρ(v)dν(u).3
+We proveinTheorem4.6that multiplicative k-formsonasource-simply-connected
+Lie groupoid GoverMare in one-to-one correspondence with IM k-forms on its Lie
+algebroidA→M. Concretely, the IM k-form (µ,ν) associated with a multiplicative
+k-formω∈Ωk(G) is defined by
+/an}bracketle{tµ(u),X1∧...∧Xk−1/an}bracketri}ht=ω(u,X1,...,Xk−1),
+/an}bracketle{tν(u),X1∧...∧Xk/an}bracketri}ht= dω(u,X1,...,Xk),
+whereXi∈TM,i= 1,...,k, and we view M⊆ GandA⊆TG|M.
+A special class of IM-forms is obtained as follows. Any close d formφ∈Ωk+1(M)
+determines a map ν:A→ ∧kT∗M,ν(u) =−iρ(u)φ, satisfying condition (3) above.
+The IMk-forms (µ,ν) withνof this type are referred to as IMk-forms relative to
+φ; they are the infinitesimal versions of multiplicative k-forms satisfying
+dω=s∗φ−t∗φ,
+wheresandtdenote the groupoid source and target maps. For k= 2, IM forms
+relativetoφincludeφ-twistedPoissonandDiracstructures[28], andourTheorem 4.6
+recovers their known integrations [5, 8]. For arbitrary k, IM forms relative to φwere
+studied in [1] in connection with the Weil algebra of a Lie alg ebroid. These and
+other examples are discussed in this paper.
+Themethod we useto integrate IM forms on Lie algebroids to mu ltiplicative forms
+on Lie groupoids relies entirely on the known correspondenc e between Lie-algebroid
+and Lie-groupoid morphisms (Lie’s second theorem for Lie al gebroids); in particular,
+we do not resort to the path spaces of [7, 12, 27], hence avoidi ng infinite dimensional
+constructions. Although our method is inspired by [3, 22, 24 ], it brings a technical
+difference in that we represent differential k-forms on a manifold Nbyfunctions
+⊕kTN→R(as opposed to maps ⊕k−1TN→T∗N); this small variation greatly
+simplifies computations, so even when restricted to known si tuations, our general
+proof seems more direct than existing ones. The integration of IM forms is carried
+out in two steps: first, we show that an IM k-form on a Lie algebroid A→M
+defines an element in Ωk(A) whose associated function ⊕kTA→Ris a Lie-algebroid
+morphism; second, upon integration, one obtains a groupoid morphism ⊕kTG →R
+which defines a multiplicative k-form1.
+In the last part of the paper, we revisit multiplicative mult ivector fields on Lie
+groupoids, as in [18]. We show how the very same techniques us ed to study mul-
+tiplicative forms apply to the dual situation of multivecto r fields, leading to an
+alternative proof of the universal lifting theorem of [18] ( not involving path spaces)
+and drawing a clear parallel between the two theories.
+As a final remark, we note that the results in this paper admit a natural formu-
+lation in terms of graded geometry. Multiplicative forms an d multivector fields on
+a given Lie groupoid Gmay be seen as multiplicative functions on the associated
+gradedLie groupoids T[1]GandT∗[1]G, respectively. On ordinary Lie groupoids,
+the infinitesimal counterpart of a multiplicative function is a Lie-algebroid cocycle.
+The same holds at the graded level and, from this perspective , our results consist
+1Forms on Gare commonly viewed as sections G → ∧kT∗G. The multiplicative ones, for k= 1,
+are such that G →T∗Gis a groupoid morphism. However, this viewpoint does not ext end to
+k≥2, as in this case ∧kT∗Ginherits no canonical groupoid structure in general; in con trast,⊕kTG
+is always a groupoid and multiplicative forms are convenien tly described by groupoids morphisms
+⊕kTG →R. An analogous discussion holds for multivector fields.4
+in using the geometry of T[1]GandT∗[1]Gto obtain concrete descriptions of their
+Lie-algebroid cocycles. For example, using the the natural multiplicative vector field
+onT[1]G(the de Rham differential on G), one identifies its Lie-algebroid cocycles
+with IM forms (see Theorem 3.1); for an analogous descriptio n of the Lie-algebroid
+cocycles of the graded groupoid T∗[1]G(see Theorem 6.1), ones uses its canonical
+multiplicative symplectic structure (defined by the Schout en bracket on G). We will
+not elaborate on the supergeometric viewpoint in this paper , though it makes our
+results more intuitive.
+Thepaperisorganizedasfollows. InSection2, weconsiderl ineardifferentialforms
+on vector bundles A→Mand establish their correspondence with pairs of vector-
+bundle maps ( µ,ν), whereµ:A→ ∧k−1T∗Mandν:A→ ∧kT∗M. In Section 3, we
+defineIMk-forms on Liealgebroids andprove acompatibility resultwi th tangent Lie
+algebroid structures (Theorem 3.1). Section 4 is devoted to Theorem 4.6, which is
+the correspondence between IM forms on Lie algebroids and mu ltiplicative forms on
+Lie groupoids; we also discuss several special cases of this result. Section 5 explains
+therelationship between Theorem4.6 andtheVan Estisomorp hismof [1]. InSection
+6, we revisit the theory of multivector fields from [18].
+Acknowledgments. We thank D. Iglesias Ponte and C. Ortiz for helpful discus-
+sions related to this project. Cabrera thanks CNPq for financ ial support and IMPA
+for its hospitality and stimulating environment during the development of this work.
+Bursztyn’s research has been supported by CNPq and Faperj. T he authors acknowl-
+edge the anonymous referees for their many useful suggestio ns, especially regarding
+improvements in the introduction.
+Notation, conventions and identities. For vector bundles A→MandB→M
+over the same baseM, a vector-bundle map Ψ : A→Bis always assumed to cover
+the identity map on M, unless stated otherwise. We denote its transpose, or dual, by
+Ψt:B∗→A∗. We denote the k-fold direct sum of a vector bundle qA:A→Mby
+⊕k
+MA, or simply ⊕kAif there is no risk of confusion. We may also use the notation/producttextk
+qAAif we want to be explicit about the projection map qA(this is relevant when
+dealing with double vector bundles).
+For a Lie groupoid GoverM, we usually denote its source and target maps by
+sandt. The set G(2)⊂ G × G of composable pairs is defined by the condition
+s(g) =t(h), and the multiplication is denoted by m:G(2)→ G,m(g,h) =gh. The
+unit mapǫ:M→ Gis often used to identify Mwith its image in G. The Lie
+algebroid of GisAG= ker(Ts)|M, with anchor Tt|A:A→Mand bracket induced
+by right-invariant vector fields. For a Lie algebroid A→M, we denote its anchor
+byρAand bracket by [ ·,·]A(or simplyρand [·,·], if there is no risk of confusion).
+We introduce some notation and collect some identities that will beuseful for later
+computations. If U1,...,Umare vector fields on a manifold M, we set
+(1.1) IU
+m,r:=iUm...iUr, r≤m,
+whereiUis the usual contraction. An inductive application of Carta n’s formula gives
+(1.2) IU
+m,1d =m/summationdisplay
+l=1(−1)l+1IU
+m,l+1LUlIU
+l−1,1+(−1)mdIU
+m,1,5
+whereddenotes thedeRhamdifferential and LUis theLiederivative. Given another
+vector field Xand recalling the commutator formula i[X,Ul]=LXiUl−iUlLX, we
+obtain
+(1.3) LXIU
+m,1=m/summationdisplay
+l=1IU
+m,l+1i[X,Ul]IU
+l−1,1+IU
+m,1LX.
+Given a differential form α, we also have
+(1.4)IU
+m,1(df∧α) =m/summationdisplay
+l=1(−1)l+1df(Ul)IU
+m,l+1IU
+l−1,1α+(−1)mdf∧IU
+m,1α.
+We often use Einstein’s summation convention when there is n o risk of confusion.
+2.Linear forms on vector bundles
+In order to define linear forms, we recall a few facts about tan gent and cotangent
+bundles of vector bundles.
+2.1.Tangent and cotangent bundles of vector bundles. LetqA:A−→Mbe
+a vector bundle, and let TAbe the tangent bundle of the total space A. Besides its
+natural vector bundlestructureover A, with projection map denoted by pA:TA−→
+A, it is also a vector bundle over TM, with respect to the map TqA:TA−→TM.
+It is useful to consider a coordinate description of these bu ndles. Let ( xj) be
+coordinates on M,j= 1,...,dim(M), and let {ed}be a basis of local sections
+ofA,d= 1,...,rank(A). The corresponding coordinates on Aare denoted by
+(xj,ud), and tangent coordinates on TAby (xj,ud,˙xj,˙ud). In this notation, given
+x= (xj), the coordinates ( ud) specify a point in Ax, (˙xj) a point in TxM, whereas
+(˙ud) determines a point on a second copy of Ax, tangent to the fibres of A−→M.
+Note thatpA(xj,ud,˙xj,˙ud) = (xj,ud), andTqA(xj,ud,˙xj,˙ud) = (xj,˙xj).
+Similarly, considerthecotangent bundle T∗A, withlocalcoordinates( xj,ud,pj,ξd),
+where (pj) determines a point in T∗
+xM, and (ξd) a point in A∗
+x, dual to the direction
+tangent to the fibres of A−→M. In this case, besides the natural vector bundle
+structurecA:T∗A−→A,cA(xj,ud,pj,ξd) = (xj,ud),T∗Ais also a vector bundle
+overA∗[23], with respect to the projection map given in coordinate s by
+(2.1) r:T∗A−→A∗, r(xj,ud,pj,ξd) = (xj,ξd).
+The total spaces TAandT∗Aare examples of double vector bundles , see [21, 26].
+They fit into the following commutative diagrams:
+TATqA/d47/d47
+pA
+/d15/d15TM
+pM
+/d15/d15
+AqA/d47/d47MT∗Ar/d47/d47
+cA
+/d15/d15A∗
+qA∗
+/d15/d15
+AqA/d47/d47M
+where
+(2.2)pM:TM−→M, pM(xj,˙xj) = (xj), qA∗:A∗−→M, qA∗(xj,ξd) = (xj),
+are the natural projections. Recall, see e.g. [21], that the intersection of the kernels
+of the top and left arrows on each diagram defines a vector bund le overM, known6
+as thecore. In the case of TA, the core is identified with A−→M, with coordinates
+(xj,˙ud); forT∗A, the core is T∗M, with coordinates ( xj,pj).
+2.2.The structure of linear forms on vector bundles. LetA−→Mbe a
+vector bundle, with local coordinates ( xj,ud), and let us consider the k-fold direct
+sum ofTAoverA,
+⊕k
+ATA:=TA×A...×ATA,
+locally described by coordinates ( xj,ud,˙xj
+1,...,˙xj
+k,˙ud
+1,...,˙ud
+k). It is a vector bundle
+overA, with projection map
+(xj,ud,˙xj
+1,...,˙xj
+k,˙ud
+1,...,˙ud
+k)/mapsto→(xj,ud),
+and also a vector bundle over ⊕kTM=TM×M...×MTM, with projection map
+(xj,ud,˙xj
+1,...,˙xj
+k,˙ud
+1,...,˙ud
+k)/mapsto→(xj,˙xj
+1,...,˙xj
+k).
+Given ak-form Λ ∈Ωk(A) on the total space of A−→M, let us consider the
+induced maps
+Λ♯:⊕k−1
+ATA−→T∗A,Λ♯(U1,...,Uk−1) =iUk−1...iU1Λ, (2.3)
+Λ :⊕k
+ATA−→R, Λ(U1,...,Uk) =iUk...iU1Λ, (2.4)
+which are alternating and linear in each of their entries2.
+Definition 2.1. Ak-formΛis called linearif the induced map Λ♯(2.3)is a mor-
+phism of vector bundles with respect to the vector bundle str uctures⊕k−1
+ATA−→
+⊕k−1TMandT∗A−→A∗. The space of linear k-forms onAis denoted by Ωk
+lin(A).
+In particular, Λ♯covers a base map λ:⊕k−1TM−→A∗,
+(2.5) ⊕k−1
+ATAΛ♯/d47/d47
+/d15/d15T∗A
+r
+/d15/d15
+⊕k−1TMλ/d47/d47A∗.
+The mapλis skew symmetric on its entries, so it can be viewed as a vecto r-bundle
+map∧k−1TM−→A∗. Its transpose is the vector-bundle map
+(2.6) λt:A−→ ∧k−1T∗M.
+A simple computation in coordinates shows the following.
+Lemma 2.2. Given ak-formΛ∈Ωk(A), the following are equivalent:
+(1) Λis linear.
+(2)In local coordinates (xj,ud)onA,Λhas the form
+Λ =1
+k!Λi1...ik,d(x)uddxi1∧...∧dxik+ (2.7)
+1
+(k−1)!λi1...ik−1d(x)dxi1∧...∧dxik−1∧dud,
+whereλi1...ik−1d=/an}bracketle{tλ(∂xi1,...,∂xik−1),ed/an}bracketri}ht, and∂xj=∂
+∂xj.
+2Notice that, since Λ♯(resp.Λ) ismultilinear in its entries, it is nota vector-bundle morphism
+from the direct sum ⊕k−1TA→A(resp.⊕kTA→A) toT∗A→A∗, unlessk= 2 (resp. k= 1).7
+(3)The map Λ :⊕k
+ATA−→Rdefines a vector-bundle map
+(2.8) ⊕k
+ATAΛ/d47/d47
+/d15/d15R
+/d15/d15
+⊕kTM /d47/d47{∗}.
+Given a vector-bundle map µ:A−→ ∧kT∗M, let us consider the linear k-form
+ΛµonAgiven at a point u∈Aby
+(2.9) (Λ µ)u:=TqA|t
+uµ(u).
+In local coordinates ( xi,ud) onA, Λµis written as
+(2.10) (Λ µ)u=1
+k!µi1...ik,d(x)uddxi1∧...∧dxik,
+whereµi1...ik,dis defined by
+µi1...ik,d=/angbracketleftbigg
+µ(ed),∂
+∂xi1∧...∧∂
+∂xik/angbracketrightbigg
+.
+Example 2.3. Whenk= 1, a direct computation in coordinates shows that the
+linear 1-form Λµ, defined by the vector-bundle map µ:A−→T∗M, satisfies
+(2.11) Λ µ=µ∗θcan,
+whereθcan=pidxiis the canonical 1-form on T∗M. (WhenA=M−→Mis the
+vector bundle with zero fibres, (2.11)recovers the well-known “tautological” property
+µ∗θcan=µ.)
+Lemma 2.4. A lineark-formΛcovers the fibrewise zero map in (2.5)if and only
+if it is of the form Λµ(as in(2.9)) for a vector bundle map µ:A−→ ∧kT∗M.
+Proof.We can use Lemma 2.2, or argue more globally as follows. Consi der the
+projectionr:T∗A−→A∗, as in (2.1). One can directly check that
+ker(r)u= (ker(TqA|u))◦= im(TqA|t
+u),
+where◦stands for the annihilator. It follows from (2.9) that r◦Λ♯
+µ= 0, which means
+that Λµcovers the fibrewise zero map in (2.5). Conversely, if Λ cover s the fibrewise
+zero map, then r◦Λ♯= 0; so, given U1,...,Uk∈TuA, Λ♯(U1,...,Uk−1) =TqA|∗
+uα
+for someα∈T∗
+qA(u)M. Since Λ is skew symmetric, we conclude that Λ( U1,...,Uk)
+only depends on TqA|u(Uj),j= 1,...,k. Hence, for each u∈A, there exists
+µ(u)∈ ∧lT∗
+qA(u)Msuch that (Λ) u=TqA|∗
+uµ(u). The linear dependence of µ(u) on
+ufollows from the linear dependence of (Λ) uonu, see (2.7); the resulting vector
+bundle map µ:A−→ ∧kT∗Mis smooth by the local expression (2.10). /square
+Proposition 2.5. There is a one-to-one correspondence between linear k-formsΛon
+A, covering a map λ:⊕k−1TM−→A∗, and pairs (µ,ν), whereµ:A−→ ∧k−1T∗M
+andν:A−→ ∧kT∗Mare vector bundle morphisms. The correspondence is given by
+(2.12) Λ = dΛ µ+Λν,
+whereµ= (−1)k−1λt.8
+Proof.Let Λ be a linear k-form onA, and setµ= (−1)k−1λt. A direct computation
+using the local expression (2.10) and Lemma 2.2 shows that th ek-formdΛµis linear
+and covers the same map λ, hence the linear k-form Λ−dΛµcovers the fibrewise zero
+map. ByLemma2.4, thereisaunique ν:A−→ ∧kT∗MsuchthatΛ −dΛµ= Λν./square
+A direct consequence of (2.12) is that if Λ is a linear form, th en so is dΛ.
+Example 2.6. LetΛ∈Ω2(A)be a linear 2-form with dΛ = 0. According to the
+previous proposition, we can write it as Λ = dΛµ+Λν, andΛbeing closed amounts
+todΛν= 0; this condition immediately implies that ν= 0, soΛ =dΛµ. Using
+(2.11), it follows that
+Λ = (λt)∗ωcan,
+whereωcan=−dθcan= dxi∧dpithe canonical symplectic form on T∗M(see[19,
+Sec. 7.3], and also [3, Prop. 4.3] ).
+2.3.Tangent lifts. We now briefly discuss linear forms obtained via the tangent
+liftoperation [13, 32] (see also [3] and [23]), that assigns to an yk-form on a manifold
+Ma lineark-form on the total space of its tangent bundle pM:TM→M.
+Let us consider the operation
+(2.13) τ: Ωl(M)−→Ωl−1(TM), τ(β)|X:= (TpM|X)t(iXβ),
+whereX∈TMandl≥1; i.e., forU1,...,Ul−1∈TX(TM),
+iUl−1...iU1τ(β)|X=β(X,TpM(U1),...,TpM(Ul−1)).
+In the notation of Section 2.2, τ(β) is a linear ( l−1)-form on the vector bundle
+A=TMof type Λ ν, where
+ν:TM−→ ∧l−1T∗M, ν(X) =iXβ.
+It directly follows from Example 2.3 that, if ω∈Ω2(M), thenτ(ω) = (ω♯)∗θcan,
+whereθcan=pidxiis the canonical 1-form on T∗M.
+Thetangent lift operation,
+(2.14) Ωk(M)−→Ωk(TM), α/mapsto→αT,
+assigns toα∈Ωk(M) the formαT∈Ωk(TM) defined by the Cartan-like formula
+(2.15) αT= dτ(α)+τ(dα).
+It follows directly from (2.15) that αTis linear and that the operation (2.14) is
+compatible with exterior derivatives, in the sense that (d α)T= dαT.
+We will also need an equivalent characterization of the tang ent lift, see e.g. [13].
+Givenα∈Ωk(M), consider the associated map
+α:⊕kTM−→R,(X1,...,Xk)/mapsto→α(X1,...,Xk).
+Let/producttextk
+TpMT(TM) denote the fibred product with respect to the vector bundle
+TpM:T(TM)−→TM, Tp M(xj,˙xj,δxj,δ˙xj) = (xj,δxj),
+where (xj,˙xj,δxj,δ˙xj) are the local coordinates on T(TM) induced by the tan-
+gent coordinates ( xj,˙xj) onTM. We have a natural identification T(⊕kTM)∼=9
+/producttextk
+TpMT(TM), so we can view the differential of the function αinC∞(⊕kTM) as a
+map
+dα:k/productdisplay
+TpMT(TM)−→R.
+Note that the canonical involution
+(2.16) JM:T(TM)−→T(TM), JM(xj,˙xj,δxj,δ˙xj) = (xj,δxj,˙xj,δ˙xj),
+induces an identification
+J(k)
+M:k/productdisplay
+pTMT(TM)−→k/productdisplay
+TpMT(TM).
+One can prove (see e.g. [13]) that, given α∈Ωk(M), its tangent lift αT∈Ωk(TM)
+is uniquely determined by the condition
+(2.17) αT= dα◦J(k)
+M:k/productdisplay
+pTMT(TM)−→R.
+3.Linear forms on Lie algebroids
+3.1.Core and linear sections. Any vector bundle that fits into a double vector
+bundleadmits two distinguishedtypes of sections, known as linearandcoresections;
+a detailed discussion can be found e.g. in [15, Sec. 2.3] and [ 21]. For our purposes,
+we are mostly interested in the particular (double) vector b undles⊕k
+ATA(and, later
+in Section 4, also in ⊕k
+AT∗A), so we restrict ourselves to these cases.
+Each section uof a vector bundle A→Mdefines a core section /hatwideuand a linear
+sectionTuofTA→TMas follows. The tangent bundle of Aalong its zero section
+M ֒→Anaturally splits as TA|M=TM⊕A, and we can define, for each X∈TxM,
+/hatwideu(X) :=X(x)+u(x), where the sum is with respect to the previous decompositio n
+ofTA|M. The linear section Tu:TM→TAis obtained by applying the tangent
+functor tou:M→A.
+Let us consider local coordinates ( xj) onM, a basis of local sections {ed}ofA,
+and dual basis {ed}ofA∗. As in Section 2, we denote the corresponding coordinates
+onAby (xj,ud), and onA∗by (xj,ξd), while coordinates on TAare denoted by
+(xj,ud,˙xj,˙ud), and onT∗Aby (xj,ud,pj,ξd).
+The core and linear sections of TA→TMdefined by a local section eaofAare
+explicitly given by
+(3.1) /hatwideea(xj,˙xj) = (xj,0,˙xj,δd
+a), Tea(xj,˙xj) = (xj,δd
+a,˙xj,0),
+whereδd
+ais thed-th component of ea, i.e., 1 ifd=aor zero otherwise.
+More generally, ealocally defines two types of local sections of ⊕k
+ATA−→ ⊕kTM
+as follows: the first type is given, for each n∈ {1,...,k}, by
+(3.2)/hatwideea,n(˙x1⊕...⊕˙xk) :=/hatwide0(˙x1)⊕...⊕/hatwide0(˙xn−1)⊕/hatwideea(˙xn)⊕/hatwide0(˙xn+1)⊕...⊕/hatwide0(˙xk),
+where ˙xl= (xj,˙xj
+l) belongstothe l-thcomponentof ⊕kTMand/hatwide0(˙xl) = (xj,0,˙xj
+l,0);
+the second type is
+(3.3) ( Tea)k(˙x1⊕...⊕˙xk) :=Tea(˙x1)⊕...⊕Tea(˙xk).10
+The sections /hatwideea,nand (Tea)kare the core and linear sections on ⊕k
+ATA, re-
+spectively. A key property is that they generate the module o f local sections of
+⊕k
+ATA−→ ⊕kTM. Note also that, under the natural projection ⊕k
+ATA−→A, core
+sections/hatwideea,nare sent to the zero section of A−→M, while linear sections ( Tea)k
+map to the section ea.
+3.2.Tangent Lie algebroids. Supposethat A−→Mcarries aLie algebroid struc-
+ture(seee.g. [6,21]), withLiebracket [ ·,·]AonΓ(A)andanchormap ρA:A−→TM.
+Then the vector bundle TA−→TMinherits a natural Lie algebroid structure,
+known as the tangent Lie algebroid , see e.g. [22]. We will need local expressions for
+the tangent Lie algebroid in terms of the coordinates introd uced in Section 3.1.
+The Lie algebroid A−→Mis locally determined by structure functions ρj
+aand
+Cc
+abdefined by
+(3.4) ρA(ea) =ρj
+a∂
+∂xj,[ea,eb]A=Cc
+abec.
+The tangent Lie algebroid structure on TA−→TMis defined in terms of core and
+linear sections (3.1) by
+[/hatwideea,/hatwideeb]TA= 0,[Tea,/hatwideeb]TA=Cc
+ab/hatwideec,[Tea,Teb]TA=Cc
+abTec+ ˙xi∂Cc
+ab
+∂xi/hatwideec, (3.5)
+ρTA(Tea) =ρj
+a∂
+∂xj+ ˙xi∂ρj
+a
+∂xi∂
+∂˙xj, ρTA(/hatwideea) =ρj
+a∂
+∂˙xj. (3.6)
+In(3.6), wehaveidentifiedpointsin T(TM), writtenincoordinatesas( xj,˙xj,δxj,δ˙xj),
+with tangent vectors
+δxj∂
+∂xj+δ˙xj∂
+∂˙xj/vextendsingle/vextendsingle/vextendsingle
+(xj,˙xj).
+We notice that the tangent Lie algebroid induces a Lie algebr oid structure on the
+direct sum ⊕k
+ATA−→ ⊕kTM. This is a general property of VB-algebroids [15, Sec.
+2.1], which we directly verify in this example. A simple cons equence of (3.5) and
+(3.6) is that if UandVare local sections of TA−→TM, each of type /hatwideeaorTea,
+then
+(3.7)TpM(ρTA(U)) =ρA(pA(U)), pA([U,V]TA) = [pA(U),pA(V)]A,
+wherepM:TM−→MandpA:TA−→Aare the natural projections. It follows
+from the first equation in (3.7) that if U1⊕...⊕Uk∈ ⊕k
+ATAis of type (3.2) or (3.3),
+then
+TpM(ρTA(Ul)) =TpM(ρTA(Um)),∀l,m∈ {1,...,k}.
+As a result, ( ρTA(U1),...,ρ TA(Uk)) defines an element in/producttextk
+TpMT(TM). Using the
+natural identification/producttextk
+TpMT(TM) =T(⊕kTM), we obtain a vector bundle map
+ρk:⊕k
+ATA−→T(⊕kTM),
+(3.8) ρk(U1⊕...⊕Uk) :=ρTA(U1)⊕...⊕ρTA(Uk).11
+Writing⊕kTMin local coordinates ( xj,˙xj
+1,...,˙xj
+k), we have the following explicit
+formulas:
+ρk(/hatwideea,n) =ρj
+a∂
+∂˙xj
+n, (3.9)
+ρk((Tea)k) =ρj
+a∂
+∂xj+k/summationdisplay
+n=1Wj
+a,n∂
+∂˙xj
+n, (3.10)
+whereWj
+a,n= ˙xi
+n∂ρj
+a
+∂xi∈C∞(⊕kTM).
+The second equation in (3.7) implies that if U1⊕...⊕UkandV1⊕...⊕Vkare
+local sections of ⊕k
+ATA−→ ⊕kTMof type (3.2) or (3.3), then
+(3.11) [ U1⊕...⊕Uk,V1⊕...⊕Vk]k:= [U1,V1]TA⊕...⊕[Uk,Vk]TA
+is a well-defined local section of ⊕k
+ATA−→ ⊕kTM. Explicitly, we have:
+[/hatwideea,n,/hatwideeb,m]k= 0, (3.12)
+[(Tea)k,/hatwideeb,m]k=Cd
+ab/hatwideed,m (3.13)
+[(Tea)k,(Teb)k]k=Cd
+ab(Ted)k+k/summationdisplay
+n=1˙xi
+n∂Cd
+ab
+∂xi/hatwideed,n. (3.14)
+The induced Lie algebroid structure on ⊕k
+ATA−→ ⊕kTMis defined by ρkand
+the extension of [ ·,·]kto all sections via the Leibniz rule3.
+3.3.IM-forms. Let Λ∈Ωk(A) be a linear k-form on a Lie algebroid A−→M,
+k≥1. Following Prop. 2.5, let µ:A−→ ∧k−1T∗Mandν:A−→ ∧kT∗Mbe the
+vector-bundle maps such that Λ = dΛ µ+Λν. Let us consider the bundle map
+(3.15) ⊕k
+ATAΛ/d47/d47
+/d15/d15R
+/d15/d15
+⊕kTM /d47/d47{∗}.
+The following is the main result of this section.
+Theorem 3.1. The map (3.15)is a Lie algebroid morphism if and only if the fol-
+lowing holds for all u,v∈Γ(A):
+iρ(u)µ(v) =−iρ(v)µ(u) (3.16)
+µ([u,v]) =Lρ(u)µ(v)−iρ(v)dµ(u)−iρ(v)ν(u) (3.17)
+ν([u,v]) =Lρ(u)ν(v)−iρ(v)dν(u). (3.18)
+For a Lie algebroid A→Mand vector-bundle maps
+µ:A−→ ∧k−1T∗M, ν:A−→ ∧kT∗M, k≥1,
+we say that the pair ( µ,ν) is anIMk-formonAif conditions (3.16), (3.17) and
+(3.18) aresatisfied. Theterminology IM stands for infinitesimally multiplicative , and
+it will be clarified in Section 4. The space of IM k-forms onAis denoted by Ω IM(A).
+3We adopt the simplified notation ρk, [·,·]k, instead of ρ⊕k
+ATAand [·,·]⊕k
+ATA; in particular, ρ1=
+ρTAand [·,·]1= [·,·]TA.12
+We note that Theorem 3.1 can be alternatively phrased in term s of the map Λ♯
+(2.5), as this map is a Lie algebroid morphism if and only if so isΛ.
+Remark 3.2. Given an IM-form (µ,ν), it follows from (3.17), using the skew-
+symmetry and Jacobi identity for the Lie algebroid bracket [·,·]onΓ(A), thatν
+automatically satisfies
+iρ(u)ν(v) =−iρ(v)ν(u), (3.19)
+iρ(w)(Lρ(v)ν(u)−Lρ(u)ν(v))+c.p.= 0 (3.20)
+for allu,v,w∈Γ(A), wherec.p.stands for cyclic permutations in u,v,w.
+Example 3.3. Consider a Lie algebroid A→Mand ak-formη∈Ωk(M). Then
+the pair(µ,ν)of vector-bundle maps
+µ:A→ ∧k−1T∗M, µ(u) =−iρ(u)η,andν:A→ ∧kT∗M, ν(u) =−iρ(u)dη,
+defines an IM k-form onA.
+Example 3.4. LetA→Mbe a Lie algebroid, and let φ∈Ωk+1(M)be such that
+iρ(u)dφ= 0,∀u∈Γ(A). One directly checks that the vector-bundle map ν:A−→
+∧kT∗Mgiven by
+(3.21) ν(u) :=−iρ(u)φ
+verifies(3.18). The particular IM k-forms(µ,ν)onAfor whichνis given as in
+(3.21)for a closed form φ∈Ωk+1(M)are called IMk-forms relative to φ. These
+special types of IM forms have first appeared in [5](fork= 2), and more recently in
+[1](for arbitrary k), in the study of multiplicative forms (see Section 4).
+Remark 3.5. LetιC:C֒→Mbe an orbit of the Lie algebroid A→M, i.e., an
+integral leaf of the distribution ρ(A)⊂TM. If(µ,ν)is an IMk-form onA, then
+we have induced forms µC∈Ωk(C)andνC∈Ωk+1(C)defined by
+iρ(u)µC=ι∗
+Cµ(u), iρ(u)νC=ι∗
+Cν(u).
+It follows from (3.16)and(3.19)that the formulas above do define differential forms
+onC; moreover, (3.17)implies that dµC=νC. In particular, we see that any IM
+k-form on a transitive Lie algebroid is like the one in Example 3 .3.
+In order to prove Thm. 3.1, we need some lemmas. We work in loca l coordinates
+(xj,ud) onA, induced by coordinates ( xj) onMand the choice of a basis of local
+sections{ed}ofA(see Section 3.1).
+Lemma 3.6. Let˙x= (˙x1,...,˙xk)∈ ⊕kTM, where˙xl= (xj,˙xj
+l)belongs to the l-th
+copy ofTM. Then:
+Λ(/hatwideea,n(˙x1,...,˙xk)) = (−1)n−1I˙x
+k,n+1I˙x
+n−1,1µ(ea), (3.22)
+Λ((Tea)k(˙x1,...,˙xk)) =I˙x
+k,1(dµ(ea)+ν(ea)), (3.23)
+seen as functions in C∞(⊕kTM)(see(1.1)for notation).13
+Proof.Writing Λ = dΛ µ+Λνand recalling the local expressions of Λ µand Λν(see
+(2.10)), we have
+Λ|(xj,ud)=1
+(k−1)!uddµi1...ik−1,d(x)∧dxi1∧...∧dxik−1+ (3.24)
+1
+(k−1)!µi1...ik−1,d(x)dud∧dxi1∧...∧dxik−1+
+1
+kνi1...ik,d(x)uddxi1∧...∧dxik.
+We write points in TAwith coordinates ( xj,ud,˙xj,˙ud) in terms of horizontal tangent
+vectors∂
+∂xjand vertical tangent vectors∂
+∂udas
+˙xj∂
+∂xj+ ˙ud∂
+∂ud/vextendsingle/vextendsingle/vextendsingle
+(xj,ud).
+In particular, recalling the local sections /hatwideea,/hatwide0 andTeaofTA→TMfrom Section
+3.1, we have
+/hatwide0(˙x) = ˙xj∂
+∂xj/vextendsingle/vextendsingle/vextendsingle
+(xj,0),/hatwideea(˙x) = ˙xj∂
+∂xj+∂
+∂ua/vextendsingle/vextendsingle/vextendsingle
+(xj,0), Tea(˙x) = ˙xj∂
+∂xj/vextendsingle/vextendsingle/vextendsingle
+(xj,δda),
+where ˙x= (xj,˙xj)∈TM. Using (3.2) and (3.3), formulas (3.22) and (3.23) follow
+from a direct calculation. /square
+Let (xj,˙xj
+1,...,˙xj
+k) be local coordinates on ⊕kTM, and fixn∈ {1,...,k}.
+Lemma 3.7. Letα∈Ωl(⊕kTM)be such that L∂
+∂˙xj
+nα= 0∀j, and consider on
+⊕kTMthe local vector fields ˙xn= ˙xj
+n∂
+∂xj,Vv=vj(x)∂
+∂˙xj
+n, andVh=vj(x)∂
+∂xj.
+ThenLVvi˙xnα=iVhα.
+Proof.The proof follows from the identity i[X,Y]=LXiY−iYLXand the fact that
+[vi(x)∂
+∂˙xin,˙xj
+n∂
+∂xj] =vj∂
+∂xj−˙xj
+n∂vi
+∂xj∂
+∂˙xin. /square
+We now proceed to the proof of the main result.
+Proof.(of Theorem 3.1)
+To show that the map Λ in (3.15) is a Lie algebroid morphism (see e.g. [21]), the
+only condition to be verified is
+(3.25) Λ([U,V]k) =Lρk(U)Λ(V)−Lρk(V)Λ(U)
+for allU,Vsections of ⊕k
+ATA−→ ⊕kTM. Since sections of type /hatwideea,n(core) and
+(Teb)k(linear) locally generate the space of sections of ⊕k
+ATA−→ ⊕kTM, it suffices
+to verify (3.25) taking UandVto be of these types.
+Core-Core: Let us consider two core sections /hatwideea,nand/hatwideeb,m. Since [/hatwideea,n,/hatwideeb,m]k= 0
+(3.12), condition (3.25) in this case becomes
+(3.26) Lρk(/hatwideea,n)Λ(/hatwideeb,m)−Lρk(/hatwideeb,m)Λ(/hatwideea,n) = 0.
+Using (3.9) and (3.22), we see that
+Lρk(/hatwideea,n)Λ(/hatwideeb,m) = (−1)n−1Lρia∂
+∂˙xinI˙x
+k,m+1I˙x
+m−1,1µ(eb).14
+This condition is trivially satisfied when n=m, so we may assume that n>m(the
+casen < mleads to the same). Using Lemma 3.7, we see that the right-han d side
+of the last equation agrees with
+(−1)n−1I˙x
+k,n+1iρ(ea)I˙x
+n−1,m+1I˙x
+m−1,1µ(eb) =
+(−1)n−1(−1)n−2I˙x
+k,n+1I˙x
+n−1,m+1I˙x
+m−1,1iρ(ea)µ(eb).
+Hence we obtain
+Lρk(/hatwideea,n)Λ(/hatwideeb,m) =−I˙x
+k,n+1I˙x
+n−1,m+1I˙x
+m−1,1iρ(ea)µ(eb).
+An analogous computation leads to
+Lρk(/hatwideeb,m)Λ(/hatwideea,n) =I˙x
+k,n+1I˙x
+n−1,m+1I˙x
+m−1,1iρ(eb)µ(ea).
+It follows that (3.26) is equivalent to
+iρ(ea)µ(eb) =−iρ(eb)µ(ea).
+Core-Linear: We now consider sections /hatwideeb,mand (Tea)k, so that (3.25) reads
+(3.27) Λ([(Tea)k,/hatwideeb,m]k) =Lρk((Tea)k)Λ(/hatwideeb,m)−Lρk(/hatwideeb,m)Λ((Tea)k).
+Using the linearity of Λ, (3.13) and (3.22), we have
+Λ([(Tea)k,/hatwideeb,m]k) =Λ(Cd
+ab/hatwideed,m) =Cd
+ab(−1)m−1I˙x
+k,m+1I˙x
+m−1,1µ(ed) (3.28)
+= (−1)m−1I˙x
+k,m+1I˙x
+m−1,1µ([ea,eb]).
+For each fixed n, consider the functions Wj
+a,n=∂ρj
+a
+∂xi˙xi
+ndefined in (3.10), noticing
+the following identity (of local vector fields on ⊕kTM):
+(3.29) Wj
+a,n∂
+∂xj=−[ρ(ea),˙xn],
+where ˙xn= ˙xi
+n∂
+∂xi. Using (3.29) and Lemma 3.7, we see that
+Lρk((Tea)k)Λ(/hatwideeb,m) =/parenleftBigg
+Lρ(ea)+k/summationdisplay
+l=1LWi
+a,l∂
+∂˙xi
+l/parenrightBigg
+(−1)m−1I˙x
+k,m+1I˙x
+m−1,1µ(eb)
+= (−1)m−1/parenleftBigg
+Lρ(ea)IU
+k,1µ(eb)−k/summationdisplay
+l=1IU
+k,l+1i[ρ(ea),Ul]IU
+l−1,1µ(eb)/parenrightBigg
+whereU= (U1,...,Uk−1) = (˙x1,...,˙xm−1,˙xm+1,...,˙xk). It follows from (1.3) that
+(3.30) Lρk((Tea)k)Λ(/hatwideeb,m) = (−1)m−1I˙x
+k,m+1I˙x
+m−1,1Lρ(ea)µ(eb).
+Using (3.23) and Lemma 3.7, we obtain
+Lρk(/hatwideeb,m)Λ((Tea)k) =Lρia∂
+∂˙ximI˙x
+k,1(dµ(ea)+ν(ea)) =I˙x
+k,m+1iρ(ea)I˙x
+m−1,1(dµ(ea)+ν(ea))
+= (−1)m−1I˙x
+k,m+1I˙x
+m−1,1iρ(ea)(dµ(ea)+ν(ea)).
+Combining this last equation with (3.28) and (3.30), we see t hat (3.27) is equivalent
+to
+(3.31) µ([ea,eb]) =Lρ(ea)µ(eb)−iρ(eb)dµ(ea)−iρ(eb)ν(ea).15
+Linear-Linear: We finally consider condition (3.25) for two linear sections :
+(3.32) Λ([(Tea)k,(Teb)k]k) =Lρk((Tea)k)Λ((Teb)k)−Lρk((Teb)k)Λ((Tea)k).
+Using (3.14) and the linearity of Λ, we have
+Λ([(Tea)k,(Teb)k]k) =Cd
+abΛ((Ted)k)+k/summationdisplay
+n=1dCd
+ab(˙xn)Λ(/hatwideed,n)
+=Cd
+abI˙x
+k,1(dµ(ed)+ν(ed))+k/summationdisplay
+n=1(−1)n−1dCd
+ab(˙xn)I˙x
+k,n+1I˙x
+n−1,1µ(ed). (3.33)
+It follows from (1.4) (also using that I˙x
+k,1µ(ed) = 0, since µ(ed) is a (k−1)-form)
+that
+I˙x
+k,1Cd
+abdµ(ed) =I˙x
+k,1d(Cd
+abµ(ed))−I˙x
+k,1(dCd
+ab∧µ(ed))
+=I˙x
+k,1d(Cd
+abµ(ed))−k/summationdisplay
+n=1(−1)n+1dCd
+ab(˙xn)I˙x
+k,n+1I˙x
+n−1,1µ(ed).
+Comparing with (3.33), we conclude that
+Λ([(Tea)k,(Teb)k]k) =I˙x
+k,1(dµ(Cd
+abed)+ν(Cd
+abed)) (3.34)
+=I˙x
+k,1(dµ([ea,eb])+ν([ea,eb])).
+Using Lemma 3.7, (3.29) and (1.3), we directly obtain
+Lρ((Tea)k)Λ((Teb)k) =/parenleftBigg
+Lρ(ea)+k/summationdisplay
+n=1LWia,n∂
+∂˙xin/parenrightBigg
+I˙x
+k,1(dµ(eb)+ν(eb)) (3.35)
+=I˙x
+k,1Lρ(ea)(dµ(eb)+ν(eb)).
+Similarly
+(3.36) Lρ((Teb)k)Λ((Tea)k) =I˙x
+k,1Lρ(eb)(dµ(ea)+ν(ea)).
+Combining (3.34), (3.35) and (3.36), we see that (3.32) is eq uivalent to
+dµ([ea,eb])+ν([ea,eb]) =Lρ(ea)(dµ(eb)+ν(eb))−Lρ(eb)(dµ(ea)+ν(ea)).
+We may assume that (3.31) holds, in which case one can directl y check that the last
+equation is equivalent to
+ν([ea,eb]) =Lρ(ea)ν(eb)−iρ(eb)dν(ea).
+/square
+4.Infinitesimal description of multiplicative forms
+In this section, we relate IM-forms on Lie algebroids with mu ltiplicative forms
+on Lie groupoids. Let Gbe a Lie groupoid over M, with source and target maps
+denoted by s,t:G −→M, respectively, multiplication m:G(2)−→ G, and unit map
+ǫ:M−→ G(that we often use to view Mas a submanifold of G). The Lie algebroid
+ofGis denoted by A(G), or simply Aif there is no risk of confusion; see Section 1.
+Ak-formα∈Ωk(G) is called multiplicative if
+(4.1) m∗α=pr∗
+1α+pr∗
+2α,16
+wherepr1,pr2:G(2)−→ Gare the natural projections. Alternatively, one may define
+multiplicative forms in terms of a natural groupoidstructu re onTGoverTM, known
+as thetangent groupoid , see e.g. [21]; it has source (resp. target) map Ts:TG −→
+TM(resp.Tt:TG −→TM), multiplication Tm: (TG)(2)=TG(2)−→TG, and
+unit mapTǫ:TM−→TG. This groupoid structure can be naturally extended to
+the direct sum ⊕k
+GTG,k≥1, making it a Lie groupoid over ⊕kTM, with source
+(resp. target) map ⊕kTs(resp.⊕kTt), multiplication map ⊕kTm, etc.
+Letα∈Ωk(G), and let us consider the associated map
+(4.2) α:⊕k
+GTG −→R,α(U1,...,Uk) =iUk...iU1α.
+The following observation is immediate from (4.1).
+Lemma 4.1. αis multiplicative if and only if αis a groupoid morphism. (Here R
+is viewed as an additive group.)
+We denote the space of multiplicative k-forms on Gby Ωk
+mult(G) .
+4.1.From multiplicative to IM forms. LetGbe a Lie groupoid over M, and
+consider the tangent lift operation Ωk(G)→Ωk(TG),α/mapsto→αT, recalled in Section
+2.3. Using the natural inclusion ιA:A=AG֒→TG, we define a map
+(4.3) Lie : Ωk(G)−→Ωk(A), α/mapsto→Lie(α) =ι∗
+AαT.
+Givenα∈Ωk(G), let us consider the associated bundle maps µ:A−→ ∧k−1T∗M
+andν:A−→ ∧kT∗M,
+/an}bracketle{tµ(u),X1∧...∧Xk−1/an}bracketri}ht=α(u,X1,...,Xk−1), (4.4)
+/an}bracketle{tν(u),X1∧...∧Xk/an}bracketri}ht= dα(u,X1,...,Xk), (4.5)
+forX1,...,Xk∈TMandu∈A(here we use the natural inclusions TM ֒→TG|M
+andA֒→TG|M).
+Lemma 4.2. Thek-formLie(α)∈Ωk(A)is linear and satisfies
+Lie(α) = dΛµ+Λν.
+Proof.Letβ∈Ωl(G) be anyl-form on G, and let us consider the l−1-form onA
+given byι∗
+Aτ(β) (see (2.13)), i.e.,
+ι∗
+Aτ(β)|u= (TιA|u)tτ(β)|ιA(u)= (TιA|u)t(TpG|ιA(u))tiιA(u)β
+= (T(pG◦ιA)|u)tiιA(u)β.
+From the commutative diagram
+AιA/d47/d47
+qA
+/d15/d15TG
+pG
+/d15/d15
+Mǫ/d47/d47G,
+we see that
+ι∗
+Aτ(β)|u= (TqA|u)t(Tǫ|qA(u))tiιA(u)β.
+It immediately follows (see (2.9)) that
+ι∗
+Aτ(α) = Λµ, ι∗
+Aτ(dα) = Λν.17
+Using (2.15), we see that
+Λ =ι∗
+A(dτ(α)+τ(dα)) = dι∗
+Aτ(α)+ι∗
+Aτ(dα) = dΛµ+Λν.
+/square
+Recall thatanygroupoidmorphism ψ:G1−→ G2definesaLiealgebroidmorphism
+Lie(ψ) :AG1−→AG2that fits into the diagram
+(4.6) TG1Tψ/d47/d47TG2
+AG1Lie(ψ)/d47/d47ιA1/d79/d79
+AG2ιA2/d79/d79
+Whenα∈Ωk(G) is multiplicative, we saw in Lemma 4.1 that α:⊕k
+GTG −→Ris a
+groupoid morphism; we consider its infinitesimal counterpa rt,
+Lie(α) :A(⊕k
+GTG)−→R,
+where now Ris viewed as the trivial Lie algebroid over a point. The natur al projec-
+tionpG:TG −→ G is a groupoid morphism, and there is a canonical identificati on
+of Lie algebroids4
+A(⊕k
+GTG) =k/productdisplay
+Lie(pG)A(TG).
+Our next goal is to compare the following two maps:
+Lie(α) :⊕k
+AT(AG)−→Rand Lie(α) :k/productdisplay
+Lie(pG)A(TG)−→R.
+The involution JG:T(TG)−→T(TG) (see (2.16)) defines an identification of Lie
+algebroidsjG:T(AG)−→A(TG) via the diagram
+(4.7) T(AG)jG/d47/d47
+TιAG
+/d15/d15A(TG)
+ιA(TG)
+/d15/d15
+T(TG)JG/d47/d47T(TG)
+Note that the property TpG◦JG=pTGimplies that
+Lie(pG)◦jG=pA.
+As a result, we have a natural identification of Lie algebroid s,
+(4.8) j(k)
+G:⊕k
+ATA=k/productdisplay
+pAT(AG)∼−→k/productdisplay
+Lie(pG)A(TG),
+4The Lie algebroid A(TG)→TMis aVB-algebroid [15, Sec. 2.1] with respect to the vector
+bundle structure Lie( pG) :A(TG)→A; the algebroid structure on A(TG) can be extended to/producttextk
+Lie(pG)A(TG) in terms of core and linear sections, just as described in Se ction 3.2.18
+fitting into the diagram
+(4.9)/producttextk
+pAT(AG)j(k)
+G/d47/d47
+(TιAG)k
+/d15/d15/producttextk
+Lie(pG)A(TG)
+(ιA(TG))k
+/d15/d15/producttextk
+pTGT(TG)
+J(k)
+G/d47/d47/producttextk
+TpGT(TG).
+Lemma 4.3. Letα∈Ωk(G)be multiplicative. Then
+(4.10) Lie( α)◦j(k)
+G=Lie(α).
+In particular, Lie(α) :⊕k
+AT(AG)−→Ris a Lie algebroid morphism.
+Proof.By definition, Lie( α) = dα◦(ιA(TG))k, and using (4.9) and (2.17) we obtain
+Lie(α)◦j(k)
+G= dα◦(ιA(TG))k◦j(k)
+G= dα◦J(k)
+G◦(TιAG)k
+=αT◦(TιAG)k=ι∗
+AαT.
+/square
+Proposition 4.4. Letα∈Ωk(G)be multiplicative, and let µandνbe defined as in
+(4.4)and(4.5). Then(µ,ν)is an IMk-form onAG.
+Proof.The result is a direct consequence of Lemmas 4.2, 4.3, and The orem 3.1. /square
+4.2.Integration of IM forms. LetGbea Lie groupoidover M, with Liealgebroid
+A=AG. Assume that Gissource-simply-connected (i.e., the s-fibres are connected
+with trivial fundamental group), so that ⊕k
+GTGis also a source-simply-connected
+groupoid5. Let Λ ∈Ωk(A) be ak-form onAfor which Λ :⊕k
+ATA−→Ris a Lie
+algebroid morphism.
+Lemma 4.5. There is a unique multiplicative k-formα∈Ωk(G)such that Lie(α) =
+Λ(see(4.3)).
+Proof.SinceΛ is a morphism of Lie algebroids, the identification (4.8) al so leads to
+a Lie algebroid morphism
+(4.11) Λ◦(j(k)
+G)−1=k/productdisplay
+Lie(pG)A(TG)∼=A(⊕kTG)−→R.
+As⊕k
+GTGis a source-simply-connected groupoid, we can use Lie’s sec ond theorem
+(see e.g. [21]) to obtain a unique groupoid morphism
+(4.12) IΛ:⊕k
+GTG −→R
+5Given any X= (X1,...,X k)∈ ⊕kTxM, the projection ( pG)k:⊕k
+GTG −→ G makes the source
+fibre ((Ts)k)−1(X)⊆TGinto an affine bundle over the source fibre s−1(x)⊆ G19
+integrating the morphism (4.11), i.e., such that Lie( IΛ) =Λ◦(j(k)
+G)−1. To check that
+IΛ=α, forα∈Ωk(G), it suffices to verify that the following conditions hold:
+IΛ(U1,...,Ui,...,Uj,...,Uk) =−IΛ(U1,...,Uj,...,Ui,...,Uk), (4.13)
+IΛ(U1,...,Ui−1,cUi,Ui+1,...,Uk) =cIΛ(U1,...,Uk), (4.14)
+IΛ(U1,...,Ui−1,Ui+U′
+i,Ui+1,...,Uk) =IΛ(U1,...,Ui,...,Uk)+ (4.15)
+IΛ(U1,...,U′
+i,...,Uk),
+for allUi,U′
+i∈TgG,g∈ G,c∈R, where 1 ≤i < j≤k. As we now show, all
+conditions can be verified with the same type arguments (cf. [ 23]).
+To prove that (4.13) holds, one directly checks that the map I(ij)
+Λ:⊕k
+GTG −→R,
+I(ij)
+Λ(U1,...,Ui,...,Uj,...,Uk) :=−IΛ(U1,...,Uj,...,Ui,...,Uk),
+is a groupoid morphism, and Lie( I(ij)
+Λ) :/producttextk
+Lie(pG)A(TG)−→Rsatisfies
+Lie(I(ij)
+Λ)(V1,...,Vi,...,Vj,...,Vk) =−Lie(IΛ)(V1,...,Vj,...,Vi,...,Vk)
+=−Λ◦(j(k)
+G)−1(V1,...,Vj,...,Vi,...,Vk)
+= Lie(IΛ)(V1,...,Vi,...,Vj,...,Vk),
+since Λ is skew-symmetric. So Lie( I(ij)
+Λ) = Lie(IΛ), and the uniqueness of integration
+in Lie’s second theorem implies that I(ij)
+Λ=IΛ, which is (4.13).
+Similarly, for a fixed c∈R, one can directly show that both the left and right-
+hand sides of (4.14) define groupoid morphisms ⊕k
+GTG −→R, whose infinitesimal
+counterparts agree at the level of Lie algebroids due to the m ultilinearity of Λ. Then
+(4.14) follows again by the uniqueness part of Lie’s second t heorem.
+The last condition (4.15) can be treated in a completely anal ogous way, by first
+noticing that both sides of (4.15) define groupoid morphisms ⊕k+1
+GTG −→R, where
+now we need an extra copy of TGforU′
+i. Again, these morphisms agree at the
+infinitesimal level due to the multilinearity of Λ, and hence agree globally.
+The fact that αis multiplicative follows from Lemma 4.1, and the equality Λ =
+Lie(α) is a consequence of Lemma 4.3. /square
+A direct consequence of Lemmas 4.3 and 4.5 is that the map
+Ωk
+mult(G)→Ωk(A), α/mapsto→Lie(α),
+is a bijection onto the subspace of k-forms Λ ∈Ωk(A) such that Λ :⊕k
+ATA→Ris
+a morphism of Lie algebroids. By the correspondence in Theor em 3.1, this bijection
+can be alternatively phrased in terms of IM-forms on A:
+Theorem 4.6. LetGbe a source-simply-connected Lie groupoid over Mwith Lie
+algebroidA−→M. For each positive integer k, there is a 1-1 correspondence
+(4.16) Ωk
+mult(G)−→Ωk
+IM(A), α/mapsto→(µ,ν),
+whereµ,νare given by
+/an}bracketle{tµ(u),X1∧...∧Xk−1/an}bracketri}ht=α(u,X1,...,Xk−1), (4.17)
+/an}bracketle{tν(u),X1∧...∧Xk/an}bracketri}ht= dα(u,X1,...,Xk). (4.18)
+Proof.The result follows from Lemma 4.2 and Theorem 3.1. /square20
+The following is a simple example of correspondence in Theor em 4.6.
+Example 4.7. Let us equip A=T∗M→Mwith the trivial Lie algebroid structure
+(both anchor and bracket are identically zero), so we may ide ntifyG=T∗M(with
+groupoid multiplication given by fibrewise addition). Fixi ngµ= Id :T∗M→T∗M,
+then any vector-bundle map ν:T∗M→ ∧2T∗Mdefines an IM 2-form (µ,ν). When
+ν= 0, then(µ,ν)corresponds under (4.16)to the canonical symplectic form ωcan
+onG=T∗M; for an arbitrary ν, the corresponding multiplicative 2-form is given,
+at eachg= (qj,pj)∈T∗M, by
+ω|g=ωcan|g+c∗
+Mν(g)
+wherecM:T∗M→Mis the natural projection.
+Let us list some immediate consequences of Theorem 4.6, illu strating how the
+correspondence (4.16) restricts to subclasses of multipli cative and IM forms:
+(a) Letη∈Ωk(M). Following Example 3.3, we know that ( µ,ν), whereµ(u) =
+−iρ(u)ηandν(u) =−iρ(u)dη, defines an IM k-form. One directly verifies
+that the corresponding multiplicative k-form isα=s∗η−t∗η.
+(b) Letφ∈Ωk+1(M) be a closed k+1-form. Then Theorem 4.6 gives a bijective
+correspondence between IM k-forms onArelative toφ(i.e.,ν(u) =−iρ(u)φ,
+see Example 3.4) and multiplicative k-formsαsatisfying d α=s∗φ−t∗φ.
+To verify this fact, just notice that d αis a multiplicative ( k+1)-form cor-
+responding to an IM ( k+ 1)-form of the type discussed in item (a). This
+recovers [5, Thm. 2.5] when k= 2 (cf. [3]), as well as [1, Thm. 2] when kis
+an arbitrary positive integer.
+(c) Letα∈Ωk
+mult(G) be a given closed multiplicative k-form, with associated IM
+k-form (µα,να) (note that να= 0, necessarily). It follows from Theorem 4.6
+that there is a 1-1 correspondence between multiplicative ( k−1)-formsθ
+with dθ=αand vector-bundle maps µ:A→ ∧k−2T∗Msatisfying, for all
+u,v∈Γ(A),
+iρ(u)µ(v) =−iρ(v)µ(u), µ([u,v]) =Lρ(u)µ(v)−iρ(v)dµ(u)−iρ(v)µα(u).
+The reason is that ( µ,µα) is the IM ( k−1)-form associated with θ(note that
+µαsatisfies (3.18) as a result of ( µα,να) being an IM k-form forαandνα
+being zero). This correspondence is the content of [1, Thm. 3 ].
+Fork= 2, one has further refinements of Theorem 4.6 based on the gen eral study
+of multiplicative 2-forms carried out in [5, Section 4], lea ding to natural generaliza-
+tions of twisted Poisson and Dirac structures (in the sense o f [28]). On the vector
+bundleTM⊕T∗M, consider the pairing /an}bracketle{t(X,α),(Y,β)/an}bracketri}ht=β(X) +α(Y), and the
+natural projections prT:TM⊕T∗M→TMandprT∗:TM⊕T∗M→T∗M. As
+in Theorem 4.6, we denote by Gthe source-simply-connected Lie groupoid with Lie
+algebroidA→M.
+(d) Given an IM 2-form ( µ,ν) onA, we consider the vector-bundle map ( ρ,µ) :
+A→TM⊕T∗Mand its image
+(4.19) L={(ρ(u),µ(u))|u∈A} ⊂TM⊕T∗M,
+which is a subbundle whenever it has constant rank. By (3.16) , over each
+point ofM,Lis isotropic with respect to the pairing /an}bracketle{t·,·/an}bracketri}ht. It follows from21
+[5, Cor. 4.8] that, under the assumption that dim( G) = 2dim(M), the cor-
+respondence (4.16) restricts to a bijection between multip licative 2-forms ω
+such that
+(4.20) ker( Ts)x∩ker(ω)x∩ker(Tt)x={0},∀x∈M,
+and IM 2-forms ( µ,ν) for which L=L⊥(i.e.,Lislagrangian with respect
+to/an}bracketle{t·,·/an}bracketri}ht; in particular, it is a subbundle with rank( L) = dim(M)) and
+(4.21) ( ρ,µ) :A−→L⊂TM⊕T∗M
+is an isomorphism of vector bundles. Moreover, t:G →MrelatesωandLas
+aforward Dirac map (see, e.g., [5, Sec. 2.1]). Ifwedefine νL:L→ ∧2T∗Mby
+νL((ρ(u),µ(u))) =ν(u), then the identification (4.21) induces a Lie algebroid
+structure on Lwith anchor prT|Land bracket on Γ( L) given by
+(4.22) [( X,α),(Y,β)]L:= ([X,Y],LXβ−iYdα−iYνL(X,α)).
+Conversely, if a lagrangian subbundle L⊂TM⊕T∗Mis equipped with
+νL:L→ ∧2T∗Mfor which (4.22) is a Lie bracket on Γ( L), thenA=Lis a
+Lie algebroid with anchor prT|L; ifνLalso satisfies (3.18), then ( prT∗|L,νL)
+is an IM 2-form. Then, under the bijection (4.16), ( prT∗|L,νL) corresponds
+to a multiplicative 2-form ωonGsatisfying (4.20). Taking νLto be of type
+νL(X,α) =−iXφfor a closed 3-form φ∈Ω3(M), we recover the integration
+of twisted Dirac structures by twisted presymplectic group oids of [5, Sec. 2].
+(e) When a multiplicative 2-form is nondegenerate, then dim (G) = 2dim(M)
+automatically (see, e.g., [5, Lem. 3.3]), and (4.20) trivia lly holds. Under
+(4.16), this situation corresponds to the case where the IM 2 -form (µ,ν) is
+such thatµ:A→T∗Mis an isomorphism; in other words, Lis the graph
+of a bivector field πonM:L={(iαπ,α)|α∈T∗M}. Following (d) above,
+the fact that Γ( L) is closed under the bracket (4.22) is expressed by the
+compatibility condition
+1
+2[π,π](α,β,γ) =νL((iαπ,α))(iβπ,iγπ)
+for allα,β,γ∈Ω1(M). WhenνLis of typeνL(X,α) =−iXφfor a closed
+3-formφ∈Ω3(M), one recovers twisted Poisson structures, and (4.16) give s
+their integration to twisted symplectic groupoids (cf. [8] ).
+Remark 4.8 (Higher Dirac structures) .Just as Dirac structures are special cases of
+IM 2-forms, the higherDirac structures of [33]are particular cases of higher-degree
+IMk-forms; so Theorem 4.6 can be used to integrate higher Dirac st ructures as well
+(c.f.[33, Prop. 3.7] ).
+Remark 4.9 (Path-space construction) .We now relate Theorem 4.6 to the path-
+spaceapproach to integration found in [5, 7, 8]. For an integrable Lie algebroid A,
+there is an explicit model for its integrating source-simpl y-connected Lie groupoid
+G(A) [12, 27] ; namely, G(A)is the quotient PA/∼, wherePAis the subspace
+ofA-pathsin the Banach manifold of all C1paths from the interval I= [0,1]
+intoAwithC2projection to M, and∼is the equivalence relation defined by A-
+homotopies , see[12]. IfΛ∈Ωk(A)is ak-form onA, we use the evaluation map22
+ev:PA×I→A,(a(·),t)/mapsto→a(t)to define a k-form/tildewideΛ∈Ωk(PA)by the formula
+(4.23) /tildewideΛ =/integraldisplay
+Idt ev∗Λ.
+One verifies that, if Λ∈Ωk(A)is a linear k-form, then the k-form/tildewideΛ (4.23) is
+basicwith respect to the quotient projection q:PA→ G(A), i.e.,/tildewideΛ =q∗αfor
+α∈Ωk(G(A)), if and only if the induced map ¯Λ (2.4)is a Lie algebroid morphism
+(i.e., ifΛis an IMk-form). In this case, the multiplicative k-form obtained by
+integration of the morphism ¯ΛtoG(A)(see Lemma 4.5) agrees with α, showing
+how the correspondence in Thm. 4.6 is viewed in the light of the path-space method
+(generalizing the approach in [5, Sec. 5] ).
+5.Relation with the Weil algebra and Van Est isomorphism
+This section clarifies how linear and IM-forms on Lie algebro ids fit into the Weil
+algebra of [1, Sec. 3], and how the infinitesimal description of multiplicative forms
+relates to the general Van Est isomorphism of [1, Sec. 4].
+LetAbe a Lie algebroid over M. We consider the associated Weil algebra W(A)
+as in [1, Sec. 3], which is a bi-graded differential algebra. Th e space of elements
+of degree (p,k) is denoted by Wp,k(A), and the differential on W(A) is a sum of
+differentials dh+dv, where
+dh:Wp,k(A)−→Wp+1,k(A), dv:Wp,k(A)−→Wp,k+1(A).
+We will be mostly concerned with Wp,k(A) forp= 0,1,2.
+Forp= 0, we have W0,k(A) = Ωk(M). An element Λ W∈W1,k(A) is given by a
+pair ((ΛW)0,(ΛW)1), where
+(5.1) (Λ W)0: Γ(A)−→Ωk(M),(ΛW)1∈Ωk−1(M,A∗),
+subject to the compatibility condition
+(5.2) (Λ W)0(fu) =f(ΛW)0(u)−df∧(ΛW)1(u),
+forf∈C∞(M),u∈Γ(A), and (Λ W)1viewed as a C∞(M)-linear map
+(5.3) (Λ W)1: Γ(A)−→Ωk−1(M).
+An element cW∈W2,k(A) is a triple (( cW)0,(cW)1,(cW)2), where
+(cW)0: Γ(A)×Γ(A)−→Ωk(M),(cW)1: Γ(A)−→Ωk−1(M,A∗), (5.4)
+(cW)2∈Ωk−1(M,S2(A∗)),
+and such that ( cW)0is skewsymmetric and R-bilinear, subject to suitable compati-
+bility conditions (extending (5.2)) that we will not need ex plicitly.
+We need to recall the expression for dhrestricted to W1,k(A). By definition (see
+[1, Sec. 3.1]), for Λ W= ((ΛW)0,(ΛW)1)∈W1,k(A),dhΛW∈W2,k(A) is given by
+(cf. (5.4))
+(dhΛW)0(u,v) =−(ΛW)0([u,v])+Lρ(u)((ΛW)0(v))−Lρ(v)((ΛW)0(u)), (5.5)
+(dhΛW)1(u)(v) =Lρ(u)((ΛW)1(v))−(ΛW)1([u,v])+iρ(v)((ΛW)0(u)), (5.6)
+(dhΛW)2(u) =−iρ(u)((ΛW)1(u)), (5.7)
+foru,v∈Γ(A).23
+We also need the expression for dvin the following particular situation. Any
+bundlemap µ:A→ ∧kT∗M(equivalently seenasa C∞(M)-linear map µ: Γ(A)−→
+Ωk(M)) defines an element µW∈W1,k(A) by
+(5.8) ( µW)0=µ,(µW)1= 0.
+In this case, dv(µW)∈W1,k+1(A) is defined by (see [1, Sec. 3.1])
+(5.9) ( dvµW)0(u) =−dµ(u),(dvµW)1=µ.
+Proposition 5.1. Consider the map ψ: Ωk
+lin(A)−→W1,k(A),k= 1,2,...,
+Λ = dΛµ+Λν/mapsto→ΛW:=−dvµW+νW.
+The following holds:
+(1)ψinduces aC∞(M)-linear isomorphism Ω•
+lin(A)∼−→W1,•(A).
+(2)ψ◦d =−dv◦ψ.
+(3)ψrestricts to a linear isomorphism Ωk
+IM(A)∼−→ker(dh|W1,k(A)).
+Proof.It is clear from (5.8) and (5.9) that the map ψis injective. Let us check
+that any Λ W∈W1,k(A) can be written in the form −dvµW+νWforC∞(M)-
+linear maps µ: Γ(A)−→Ωk−1(M),ν: Γ(A)−→Ωk(M). Let us write Λ W=
+((ΛW)0,(ΛW)1), and setµ=−(ΛW)1. Then (dvµW)1=−(ΛW)1, so the element
+c=dvµW+ ΛW∈W1,k(A) is such that c1= 0, which implies that c=νWfor a
+bundle map ν:A−→ ∧kT∗M. TheC∞(M)-linearity of ψresults from the following
+properties: fdΛµ= dΛfµ−Λdf∧µanddv(fµ)W=fdvµW−(df∧µ)W. Hence (1)
+is proven.
+To prove (2), writing Λ = dΛ µ+ Λν, we have dΛ = dΛ ν. By definition of ψ, it
+followsthat ψ(dΛ) =−dvνW. Ontheotherhand, −dv(ψ(Λ)) =−dv(−dvµW+νW) =
+−dvνW, hence (2) holds.
+For (3), we must consider the condition dhΛW= 0. Written in terms of its
+components (5.5), (5.6), and (5.7), we obtain three equatio ns involving (Λ W)0and
+(ΛW)1, which must be shown to agree with conditions (3.16), (3.17) and (3.18) in
+Thm. 3.1. Using (5.8), (5.9), we see that
+(5.10) (Λ W)0(u) = dµ(u)+ν(u),(ΛW)1(u) =−µ(u)
+for allu∈Γ(A), and it is clear that (5.7) and (5.6) coincide with conditio ns (3.16)
+and (3.17), respectively.
+For the degree-0 condition (5.5), using (5.10) and (3.17), w e find
+ν([u,v]) = diρ(v)dµ(u)+diρ(v)ν(u)+Lρ(u)ν(v)−Lρ(v)dµ(u)−Lρ(v)ν(u).
+Using Cartan’s formula LX=iXd+diX, one directly verifies that the last equation
+agrees with (3.18). /square
+LetGbeasource-simply-connected Liegroupoidover M, withLiealgebroid A−→
+M. There is a double complex Ωk(G(p)) associated to G, known as the Bott-Shulman
+complex, see[2]. Itis equippedwitha differential ∂: Ωk(G(p))−→Ωk(G(p+1)), as well
+as the de Rham differential d : Ωk(G(p))−→Ωk+1(G(p)). TheVan Est isomorphism
+constructed in [1] relates the cohomologies of Ωk(G(p)) andWp,k(A). We will only
+need a few results of the theory, for p= 0,1.
+Forp= 0, Ωk(G(0)) = Ωk(M) =W0,k(A), and
+∂: Ωk(M)−→Ωk(G), ∂(η) =t∗η−s∗η.24
+Forp= 1, the differential ∂: Ωk(G)−→Ωk(G(2)) is
+∂(α) =pr∗
+1α−m∗α+pr∗
+2α,
+and the Van Est map of [1, Sec. 4] restricts to a map
+(5.11) V: Ωk
+mult(G) = ker(∂|Ωk(G))−→ker(dh|W1,k(A))⊂W1,k(A),
+given by
+(5.12) V(α)0(u) =ǫ∗(diuα+iudα),V(α)1(u) =−ǫ∗(iuα)
+whereα∈Ωk
+mult(G),u∈Γ(A) (we view Aas a subbundle of TG|M) andǫ:M→ G
+is the unit map of G. The map Vsatisfies
+(5.13) V◦d =−dv◦V,V(∂(η)) =dhη,
+forη∈Ωk(M). The general Van Est isomorphism of [1] implies that the ind uced
+map
+(5.14)Ωk
+mult(G)
+Im(∂|Ωk(M))−→ker(dh|W1,k(A))
+Im(dh|Ωk(M))
+is a bijection. In this specific situation, a stronger fact ho lds.
+Proposition 5.2. The map V: Ωk
+mult(G)−→ker(dh|W1,k(A))is a bijection.
+The proof of the proposition uses the following observation (cf. [1, Sec. 6]).
+Lemma 5.3. Letσ∈W1,k(A),ω∈Ωk+1
+mult(G)be such that dhσ= 0andV(ω) =
+−dvσ. Then there exists a unique β∈Ωk
+mult(G)such that
+V(β) =σ,dβ=ω.
+Proof.The key fact to prove the lemma is shown in [1, Lem. 6.3]: for a c losedk-form
+α∈Ωk
+mult(G),V(α) = 0 if and only if α= 0. As an application, we see that ωis
+necessarily closed, since V(dω) =dv(dvσ) = 0.
+Sincedhσ= 0, the isomorphism (5.14) implies that there exists /tildewideβ∈Ωk
+mult(G) such
+thatV(/tildewideβ) =σ+dhη. Ifβ=/tildewideβ−∂η, then by (5.13) we have
+V(β) =V(/tildewideβ)−V(∂η) =σ.
+To conclude that d β=ω, note that d β−ωis multiplicative, closed, and V(dβ−ω) =
+−dvσ+dvσ= 0. /square
+We can now prove the proposition.
+Proof.(of Prop. 5.2) Let us fix ξ∈W1,k(A),dhξ= 0. Letσ=−dvξ. Thendvσ= 0,
+dhσ= 0, and Lemma 5.3 implies that there exists a unique β∈Ωk+1
+mult(G) such that
+(5.15) V(β) =σ,dβ= 0.
+SinceV(β) =−dvξ, and, by assumption, dhξ= 0, we can apply Lemma 5.3 to
+conclude that there exists a unique θ∈Ωk
+mult(G) such that V(θ) =ξand dθ=β.
+But notice that the condition d θ=βis automatically satisfied if V(θ) =ξ, since
+V(dθ) =σand the conditions in (5.15) determine βuniquely. /square25
+Composing the bijection (5.11) with the identification Ωk
+IM(A)∼=ker(dh|W1,k(A))
+of (3) in Prop. 5.1, we obtain a bijection
+Ωk
+mult(G)∼−→Ωk
+IM(A).
+Using (5.10) and (5.12), we see that this bijection is explic itly given by α/mapsto→(µ,ν),
+whereµ,νare defined as in (4.4), (4.5), hence agreeing with Theorem 4. 6.
+6.The dual picture: multiplicative multivector fields
+In this section, we illustrate how the techniques used in the paper to study in-
+finitesimal versions of multiplicative forms can be equally applied to multiplicative
+multivector fields.
+We keep the notation introduced in Section 2.1. We focus on th e cotangent
+bundlecA:T∗A→Aof a vector bundle A→M, described in local coordi-
+nates (xj,ud,pj,ξd), where (xj,ud) are relative to a basis of local sections {ed}of
+A. The local coordinates on A∗relative to the dual basis {ed}are denoted by
+(xj,ξd); recall from (2.1) that we have a vector-bundle structure r:T∗A→A∗,
+(xj,ud,pj,ξd)/mapsto→(xj,ξd). As in Section 2.2, we also consider the k-fold direct sum
+⊕k
+AT∗A, describedbylocal coordinates( xj,ud,p1
+j,...,pk
+j,ξ1
+d,...,ξk
+d), asavector bun-
+dle over ⊕kA∗, with projection map ( xj,ud,p1
+j,...,pk
+j,ξ1
+d,...,ξk
+d)/mapsto→(xj,ξ1
+d,...,ξk
+d).
+As in Section 3.1, we will need special sections of the bundle ⊕k
+AT∗A→ ⊕kA∗.
+For the bundle T∗A−→A∗, we consider local sections
+(6.1) d /hatwidexi(xj,ξd) = (xj,0,δi
+j,ξd), eL
+a(xj,ξd) = (xj,δd
+a,0,ξd),
+which are core and linear sections, respectively; these sec tions generate the module
+of local sections of T∗A−→A∗, and the projection T∗A−→Amaps core sections to
+the zero section of A−→Mand linear sections eL
+ato the section ea. More generally,
+local sections of ⊕k
+AT∗A→ ⊕kA∗are generated by sections of types
+d/hatwidexi,n(ξ1⊕...⊕ξk) = 0(ξ1)⊕...0(ξn−1)⊕d/hatwidexi(ξn)⊕0(ξn+1)...⊕0(ξk), (6.2)
+(eL
+a)k(ξ1⊕...⊕ξk) =eL
+a(ξ1)⊕...⊕eL
+a(ξk), (6.3)
+whereξ1⊕...⊕ξk∈ ⊕kA∗and 0 :A∗→T∗A, 0(xj,ξd) = (xj,0,0,ξd), is the zero
+section. For each k, we will use these sections to express the natural Lie algebr oid
+structure on ⊕k
+AT∗A→ ⊕kA∗, similarly to Section 3.2.
+Using the notation in (3.4), the defining relations for the co tangent Lie algebroid
+structure on T∗A→A∗are
+[d/hatwidexi,d/hatwidexj]T∗A= 0, (6.4)
+[eL
+a,d/hatwidexj]T∗A=∂ρj
+a
+∂xid/hatwidexi,[eL
+a,eL
+b]T∗A=−∂Cc
+ab
+∂xiξcd/hatwidexi+Cc
+abeL
+c, (6.5)
+ρT∗A(d/hatwidexi) =ρi
+d∂
+∂ξd, ρT∗A(eL
+a) =ρj
+a∂
+∂xj+Cc
+abξc∂
+∂ξb. (6.6)
+This Lie algebroid structure is extended to direct sums ⊕k
+AT∗A→ ⊕kA∗in total
+analogy to what was done for the tangent Lie algebroid in Sect ion 3.2; we adopt the
+simplified notation ρk=ρ⊕k
+AT∗Aand [·,·]k= [·,·]⊕k
+AT∗Afor the resulting anchor and26
+bracket6. Explicitly, the anchor is given by
+(6.7) ρk(d/hatwidexi,n) =ρi
+d∂
+∂ξn
+d, ρk((eL
+a)k) =ρj
+a∂
+∂xj+Cc
+abξn
+c∂
+∂ξn
+b,
+whereas for the bracket we have
+[d/hatwidexi,n,d/hatwidexj,m]k= 0,[(eL
+a)k,d/hatwidexj,m]k=∂ρj
+a
+∂xid/hatwidexi,m(6.8)
+[(eL
+a)k,(eL
+b)k]k=Cd
+ab(eL
+d)k−∂Cc
+ab
+∂xiξn
+cd/hatwidexi,n. (6.9)
+6.1.Linear multivector fields and derivations. Letπ∈ Xk(A) = Γ(∧kTA) be
+ak-vector field on the total space of a vector bundle qA:A→M. Let us consider
+the function (cf. (2.4))
+π:⊕k
+AT∗A→R,π(Υ1,...,Υk) =iΥk...iΥ1π.
+We say that π∈ Xk(A) islinearifπdefines a vector-bundle map
+(6.10) ⊕k
+AT∗Aπ/d47/d47
+/d15/d15R
+/d15/d15
+⊕kA∗ /d47/d47{∗},
+similarly to (2.8). One can directly verify that the notion o f linear multivector field
+agrees with the one considered in [18, Section 3.2]. The spac e of linear k-vector
+fields is denoted by Xk
+lin(A). As in Lemma 2.2 (cf. (2.7)), πis expressed in local
+coordinates ( xj,ud) ofAas
+π=1
+k!πb1...bk
+d(x)ud∂
+∂ub1∧...∧∂
+∂ubk+ (6.11)
+1
+(k−1)!πb1...bk−1j(x)∂
+∂ub1∧...∧∂
+∂ubk−1∧∂
+∂xj.
+WehavethefollowinganalogofProposition2.5forlinearmu ltivector fields, proven
+in [18, Prop. 3.7]: there is a 1-1 correspondence between ele ments in Xk
+lin(M) and
+pairs (δ0,δ1), whereδ0:C∞(M)→Γ(∧k−1A) andδ1: Γ(A)→Γ(∧kA) are linear
+maps satisfying
+(6.12) δ0(fg) =gδ0f+fδ0g, δ1(fu) = (δ0f)∧u+fδ1u,
+for allf,g∈C∞(M) andu∈Γ(A). Equivalently, one may view such pairs ( δ0,δ1)
+as restrictions of linear maps δ: Γ(∧•A)→Γ(∧•+k−1A) satisfying the property
+(6.13) δ(u∧v) = (δu)∧v+(−1)p(k−1)u∧(δv)
+foru∈Γ(∧pA) andv∈Γ(∧qA); i.e.,δis a degree-( k−1) derivation of the exterior
+algebra Γ( ∧•A). For this reason, we denote both maps δ0andδ1byδ. The explicit
+6Since tangent Lie algebroids are not used in this section, th is notation should not cause any
+confusion with the one in Section 3.2.27
+correspondence between πandδis given by7(see [18, Section 3.2])
+π(dlξ1,...,dlξk−1,dq∗
+Af) =q∗
+A/angbracketleftBig
+δf,ξ1∧...∧ξk−1/angbracketrightBig
+, (6.14)
+π(dlξ1,...,dlξk)(u) =k/summationdisplay
+i=1(−1)i+kπ(dlξ1,...,/hatwidestdlξi,...,dlξk,dq∗
+A/angbracketleftbig
+ξi,u/angbracketrightbig
+) (6.15)
+−/angbracketleftBig
+δu,ξ1∧...∧ξk/angbracketrightBig
+,
+wheref∈C∞(M),u∈Γ(A),ξ1,...,ξk∈Γ(A∗), andlξi∈C∞(A) is the linear
+functionlξi(u) =/angbracketleftbig
+ξi,u/angbracketrightbig
+. In coordinates, we have
+(6.16)δxi=1
+(k−1)!πb1...bk−1i(x)eb1∧...∧ebk−1, δea=−1
+k!πb1...bka(x)eb1∧...∧ebk,
+where{ed}is a basis of local sections of A.
+LetA→Mbe a Lie algebroid. The Lie bracket [ ·,·] on Γ(A) has a natural
+extension (still denoted by [ ·,·]) to the exterior algebra Γ( ∧•A),
+[·,·] : Γ(∧pA)×Γ(∧qA)→Γ(∧p+q−1A),
+making it into a Gerstenhaber algebra (see e.g. [6]): for u∈Γ(∧pA),v∈Γ(∧qA),
+andw∈Γ(∧rA), we have
+[u,v] =−(−1)(p−1)(q−1)[v,u], (6.17)
+[u,v∧w] = [u,v]∧w+(−1)(p−1)qv∧[u,w]. (6.18)
+The next result is the analog of Theorem 3.1 for linear multiv ector fields.
+Theorem 6.1. Letπ∈ Xk
+lin(A)be a linear k-vector field on a Lie algebroid A, and
+letδ: Γ(∧•A)→Γ(∧•+k−1A)be the associated derivation (as in (6.14)and(6.15)).
+Then the map π(6.10)is a Lie algebroid morphism if and only if
+(6.19) δ[u,v] = [δu,v] +(−1)(p−1)(k−1)[u,δv],
+for allu∈Γ(∧pA),v∈Γ(∧qA)(i.e.,δis a(k−1)-derivation of the Gerstenhaber
+bracket).
+To draw a clear parallel with Theorem 3.1, we denote by Xk
+IM(A) the space of
+degree (k−1) derivations δ: Γ(∧•A)→Γ(∧•+k−1A) of the Gerstenhaber structure
+(i.e., (6.13) and (6.19) hold), in analogy with IM k-forms.
+Proof.We work locally, so the condition that πis a Lie algebroid morphism is
+(6.20) π([Υ1,Υ2]k) =Lρk(Υ1)π(Υ2)−Lρk(Υ2)π(Υ1),
+whereΥ 1,Υ2are local sections of ⊕k
+AT∗A→ ⊕kA∗of types (6.2) or (6.3) (cf. (3.25));
+hence, just as in the proof of Theorem 3.1, there are 3 cases to be analyzed. The
+assertion of Theorem 6.1 is a direct consequence of the follo wing claims:
+(c1) Let Υ 1= d/hatwidexi,land Υ 2= d/hatwidexj,m. Ifl=m, then (6.20) is automatically
+satisfied; if l/ne}ationslash=m, then (6.20) is equivalent to
+(6.21) δ[xi,xj] = [δxi,xj]+(−1)k−1[xi,δxj] = 0.
+7To see that (6.14) and (6.15) determine the linear k-vectorπ, note that fibres of T∗A→Aare
+generated by elements of types d lξand dq∗
+Af, and by linearity idq∗
+Af2idq∗
+Af1π= 0.28
+(c2) Let Υ 1= d/hatwidexi,land Υ2= (eL
+b)k. Then (6.20) is equivalent to
+(6.22) δ[xi,eb] = [δxi,eb]+(−1)k−1[xi,δeb].
+(c3) Let Υ 1= (eL
+a)kand Υ2= (eL
+b)k. Then (6.20) is equivalent to
+(6.23) δ[ea,eb] = [δea,eb]+[ea,δeb].
+In order to prove claims (c1), (c2) and (c3), we need some gene ral observations.
+For any function F:⊕kA∗→Rwhich isk-linear over C∞(M) and skew symmetric,
+let ΦF∈Γ(∧kA) be the unique element such that
+F(ξ1,...,ξk) =/angbracketleftBig
+ΦF,ξ1∧...∧ξk/angbracketrightBig
+.
+E.g., forF(ξ1,...,ξk) =Fb1...bkξ1
+b1...ξk
+bkwithFb1...bktotally antisymetric in its
+indices, we have Φ F=1
+k!Fb1...bkeb1∧...∧ebk. We will consider the cases where
+F=π(d/hatwidexj,m) andF=π((eL
+a)k). Using the local expressions (6.2), (6.3) as well as
+(6.11) and (6.16), one may directly verify the following ide ntities:
+π(d/hatwidexj,m(ξ1,...,ξk)) = (−1)k−m/angbracketleftBig
+δxj,ξ1∧...∧/hatwiderξm∧...∧ξk/angbracketrightBig
+, (6.24)
+π((eL
+a)k(ξ1,...,ξk)) =−/angbracketleftBig
+δea,ξ1∧...∧ξk/angbracketrightBig
+, (6.25)
+where the notation ξ1∧...∧/hatwiderξm∧...∧ξkmeans that ξmis omitted.
+Let us now consider F(ξ1,...,ξk) =Fb1...bkξ1
+b1...ξk
+bkandthevector fields ρk(d/hatwidexi,l)
+andρk((eL
+a)k) on⊕kA∗, see (6.7). Then a direct computation shows the following
+identities:
+Lρk(d/hatwidexi,l)(F(ξ1,...,ξk)) = (−1)l/angbracketleftBig
+[xi,ΦF],ξ1∧...∧/hatwideξl∧...∧ξk/angbracketrightBig
+, (6.26)
+Lρk((eLa)k)(F(ξ1,...,ξk)) =/angbracketleftBig
+[ea,ΦF],ξ1∧...∧...∧ξk/angbracketrightBig
+. (6.27)
+From (6.24) and (6.26), we directly see, assuming that l<m, that
+Lρk(d/hatwidexi,l)(π(d/hatwidexj,m)(ξ1,...,ξk)) =
+(−1)l+k−m/angbracketleftBig
+[xi,δxj],ξ1∧...∧/hatwideξl∧...∧/hatwiderξm∧...,ξk/angbracketrightBig
+,
+Lρk(d/hatwidexj,m)(π(d/hatwidexi,l)(ξ1,...,ξk)) =
+(−1)m−1+k−l/angbracketleftBig
+[xj,δxi],ξ1∧...∧/hatwideξl∧...∧/hatwiderξm∧...∧ξk/angbracketrightBig
+.
+Combining these two equations with (6.17), we conclude that claim (c1) holds.
+To prove the other two claims, note first that the derivation p roperty for functions
+in (6.12) implies that
+(6.28) δf=∂f
+∂xjδxj, f∈C∞(M).
+As a result, since [ eb,xi] =Lρ(eb)xi=ρi
+b, we have that
+(6.29) δ[xi,eb] =−δ[eb,xi] =−∂ρi
+b
+∂xjδxj.29
+Using the second formula in (6.8) together with (6.24) and (6 .29), we obtain
+π([d/hatwidexi,l,(eL
+b)k]k(ξ1,...,ξk)) =−(−1)k−l/angbracketleftbigg∂ρi
+b
+∂xjδxj,ξ1∧...∧/hatwideξl∧...∧ξk/angbracketrightbigg
+(6.30)
+= (−1)k−l/angbracketleftBig
+δ[xi,eb],ξ1∧...∧/hatwideξl∧...∧ξk/angbracketrightBig
+.
+Combining (6.24) and (6.27), as well as (6.25) and (6.26), we immediately get
+Lρk((eL
+b)k)(π(d/hatwidexi,l(ξ1,...,ξk))) = (−1)k−l/angbracketleftBig
+[eb,δxi],ξ1∧...∧/hatwideξl∧...∧ξk/angbracketrightBig
+, (6.31)
+Lρk(d/hatwidexi,l)(π((eL
+b)k(ξ1,...,ξk))) =−(−1)l/angbracketleftBig
+[xi,δeb],ξ1∧...∧/hatwideξl∧...∧ξk/angbracketrightBig
+. (6.32)
+Now claim (c2) is a direct consequence of (6.30), (6.31) and ( 6.32).
+Finally, to prove (c3), we observe a few facts. From (6.28), w e see that
+(6.33) δ[ea,eb] =δ(Cc
+abec) =∂Cc
+ab
+∂xjδxj∧ec+Cc
+abδec.
+The usual formula for the wedge product gives us the identity
+∂Cc
+ab
+∂xj/angbracketleftBig
+δxj∧ec,ξ1∧...∧ξk/angbracketrightBig
+=k/summationdisplay
+n=1(−1)k−n∂Cc
+ab
+∂xjξn
+c/angbracketleftBig
+δxj,ξ1∧...∧/hatwiderξn∧...∧ξk/angbracketrightBig
+;
+using it, we immediately obtain from (6.9), (6.24) and (6.25 ) that
+(6.34) π([(eL
+a)k,(eL
+b)k]k(ξ1,...,ξk)) =−/angbracketleftBig
+δ[ea,eb],ξ1∧...∧ξk/angbracketrightBig
+.
+On the other hand, from (6.25) and (6.27) we have that
+(6.35) Lρk((eLa)k)(π((eL
+b)k(ξ1,...,ξk))) =−/angbracketleftBig
+[ea,δeb],ξ1∧...∧ξk/angbracketrightBig
+.
+Using (6.34) and (6.35), we can immediately verify that clai m (c3) holds. /square
+6.2.Infinitesimal description of multiplicative multivector fields. We now
+discuss the analogs of the results in Section 4 for multiplic ative multivector fields.
+LetGbe a Lie groupoid over M. Its cotangent bundle T∗Ghas a natural Lie
+groupoid structure over A∗, known as the cotangent groupoid ofG, see [9] and
+[21] for a full description. For us, it will suffice to recall th at the unit map /tildewideǫ:A∗→
+T∗G|MidentifiesA∗with the annihilator of TM⊂TG, and that the source map
+/tildewides:T∗G →A∗is defined by
+(6.36) /an}bracketle{t/tildewides(αg),u/an}bracketri}ht=/an}bracketle{tαg,Tlg(u−Tt(u))/an}bracketri}ht, αg∈T∗
+gG, u∈As(g),
+wherelgdenotes left translation in G. Note that/tildewidesis a vector-bundle map covering
+s:G →M; using coordinates ( zl) onG, it has the form
+(6.37) /tildewides(zl,αl) = (s(z)j,Cl
+d(z)αl)∈A∗|s(z).
+We will not need the explicit expression for Cl
+d(z), just to note that /tildewides(dt∗f) = 0 for
+allf∈C∞(M) (by (6.36)), which implies that
+(6.38) Cl
+d(z)∂(t∗f)
+∂zl= 0,∀f∈C∞(M).30
+Similarly to what happensfor the tangent groupoid, the cota ngent groupoidstruc-
+ture extends to direct sums ⊕k
+GT∗Gover⊕kA∗. A multivector field Π ∈ Xk(G) is
+calledmultiplicative if the associated map
+(6.39) Π :⊕k
+GT∗G →R,Π(ζ1,...,ζk) =iζk...iζ1Π
+is a groupoid morphism (cf. Lemma 4.1). We denote the space of multiplicative
+k-vector fields on GbyXk
+mult(G).
+Remark 6.2. We may equivalently consider the map
+(6.40) Π♯:⊕k−1
+GT∗G →TG,Π♯(ζ1,...,ζk−1) =iζk−1...iζ1Π,
+and verify that Πis multiplicative if and only if Π♯is a groupoid morphism.
+Let us recall, see e.g. [23], the identification of Lie algebr oids
+(6.41) θG:A(T∗G)∼−→T∗(AG),
+which extends to an identification θk
+G:A(⊕k
+GT∗G) =/producttext
+Lie(cG)A(T∗G)→ ⊕k
+AT∗A
+(cG:T∗G → Gis a groupoid morphism). Given Π ∈ Xk
+mult(G), we consider the
+infinitesimal map Lie( Π) :A(⊕k
+GT∗G)→R(see (4.6)), as well as the composition
+(6.42) Lie( Π)◦(θk
+G)−1:⊕k
+AT∗A→R.
+The exact same arguments as in Lemma 4.5 directly show that th ere is a unique
+k-vector field Lie(Π) ∈ Xk(A) satisfying
+(6.43) Lie(Π) = Lie( Π)◦(θk
+G)−1;
+moreover, the map
+Xk
+mult(G)−→ Xk(A),Π/mapsto→Lie(Π),
+is a bijection onto the subspaceof k-vector fields π∈ Xk
+lin(A) for which π:⊕k
+AT∗A→
+Ris a morphism of Lie algebroids. An immediate consequence of Theorem 6.1 is
+Corollary 6.3. There is a bijective correspondence
+(6.44) Xk
+mult(G)−→ Xk
+IM(A),Π/mapsto→δ,
+whereδis the derivation associated with π= Lie(Π) ∈ Xk
+lin(A)(via(6.14)and
+(6.15)).
+This result is parallel to Theorem 4.6, except that it provid es no explicit way of
+computing δdirectly out of Π (analogous to (4.17) and (4.18)). This miss ing aspect
+will be clarified in the next section.
+6.3.The universal lifiting theorem revisited. Foru∈Γ(∧pA), let us denote
+byurthe corresponding right-invariant p-vector field on G. As observed in [18,
+Section 2], given Π ∈ Xk
+mult(G), then [Π,ur] is again right invariant, which means
+that there exists δΠu∈Γ(∧p+k−1A) such that ( δΠu)r= [Π,ur]. One can check that
+the mapδΠ: Γ(∧pA)→Γ(∧p+k−1A) is a derivation of the Gerstenhaber structure,
+i.e.,δΠ∈ Xk
+IM(M).
+Proposition 6.4. The map Xk
+mult(G)−→ Xk
+IM(A),Π/mapsto→δΠ, whereδΠis defined by
+(6.45) ( δΠf)r= [Π,t∗f],(δΠu)r= [Π,ur],
+forf∈C∞(M)andu∈Γ(A), coincides with the map (6.44); in particular, it is a
+bijection.31
+The fact that the correspondence in Proposition 6.4 is a bije ction is the universal
+lifting theorem of [18] (see Theorem 2.34 therein), which we recover here as a con-
+sequence of Corollary 6.3. We need to collect some observati ons before getting into
+the proof of Proposition 6.4.
+Let us consider the isomorphism
+(6.46) Θ : T(T∗G)−→T∗(TG),(zj,αj,˙zj,˙αj)/mapsto→(zj,˙zj,˙αj,αj),
+which is related to the identification θGin (6.41) via
+(6.47) θG= (TιA)t◦Θ◦ιA(T∗G),
+where (TιA)tis the fibrewise dual to the vector-bundle map TιA:TA→ι∗
+AT(TG)
+(the composition in (6.47) is well defined since Θ ◦ιA(T∗G)(A(T∗G))⊂ι∗
+AT∗(TG); this
+can be derived directly from (6.37)). For a k-vector field Π ∈ Xk(G), itstangent
+liftis thek-vector field Π T∈ Xk(TG) defined by the condition (cf. (2.17))
+(6.48) ΠT= dΠ◦(Θ−1)k,
+where dΠ :T(⊕k
+GT∗G) =/producttextk
+TcGT(T∗G)→Ris the differential of the function Π in
+C∞(⊕k
+GT∗G) defined by (6.39).
+Remark 6.5. As observed in [13], one may alternatively define the tangent lift ΠT
+in terms of Π♯(6.40):
+(6.49) Π♯
+T:⊕k−1
+TGT∗(TG)→T(TG),Π♯
+T= (JG)−1◦TΠ♯◦(Θ−1)(k−1),
+whereJG:T(TG)→T(TG)is the involution (2.16).
+When Π is multiplicative, it follows from (6.43) that
+(6.50) π=Lie(Π) = ΠT◦(Θ◦ιAT∗G◦θ−1
+G)k.
+We will need this characterization of πin the proof of Proposition 6.4.
+For local computations, it will beconvenient to consider ad apted local coordinates
+(6.51) ( xj,yd) onGaroundM⊂ G,
+whereydare coordinates along the s-fibres. We will also use the induced coordinates
+((xj,yd),(˙xj,˙yd)) onTG, and similarly for T∗G,T(T∗G) andT∗(TG). In these co-
+ordinates,ιA:A→TG|M,ιA(xj,ud) = ((xj,0),(0,ud)), andTιA:TA→ι∗
+AT(TG)
+is given by
+TιA/parenleftbigg
+˙xj∂
+∂xj+ ˙ud∂
+∂ud/parenrightbigg/vextendsingle/vextendsingle/vextendsingle
+u= ˙xj∂
+∂xj+ ˙ud∂
+∂˙yd/vextendsingle/vextendsingle/vextendsingle
+ιA(u), u∈A,
+whereas for ( TιA)t:ι∗
+AT∗(TG)→T∗Awe have
+(TιA)t(pjdxj+γadya+pjd˙xj+γad˙ya)|ιA(u)= (pjdxj+γadua)|u, u∈A.
+Since the unit map /tildewideǫ:A∗→T∗G|MidentifiesA∗with the annihilator of TM⊂TG,
+givenξ∈Γ(A∗), locally written as ( xj,ξd), the local 1-form on Ggiven by
+(6.52) /tildewideξ(xj,yd) =ξd(x)dyd
+extends/tildewideǫ(ξ(x)) to a neighborhood of MinG. We denote by l/tildewideξ∈C∞(TG) the
+linear function determined by /tildewideξ. The following lemma is key to compare the map in
+Proposition 6.4 with the map (6.44).32
+Lemma 6.6. LetJ= Θ◦ιAT∗G◦θ−1
+G:T∗A→ι∗
+AT∗(TG), and letu0∈A. Then,
+for anyf∈C∞(M)andξ∈Γ(A∗), we have
+J(dq∗
+Af|u0) = d(t∗f)∨|ιA(u0), (6.53)
+J(dlξ|u0) = (dl/tildewideξ+dh∨)|ιA(u0), (6.54)
+where∨means the pull-back of functions on GbypG:TG → G, andh∈C∞(G)is a
+function that vanishes on M⊂ G.
+Proof.Theprooffollows fromsomeobservations, all ofwhichcan be checked through
+computations in adapted local coordinates ( xj,yd) as in (6.51).
+The first observation one can directly verify is that
+(6.55) ( TιA)t(d(t∗f)∨|ιA(u)) = dq∗
+Af|u, u∈A.
+Using the local expression (6.37) for the source map /tildewides, the property (6.38), and
+the definition of Θ, a direct computation shows that
+(6.56) T/tildewides(Θ−1(d(t∗f)∨|ιA(u))) = 0∈TA∗|qA(u), qA(u)∈M⊂A∗.
+It follows that Θ−1(d(t∗f)∨|ιA(u)) is in the image of ιA(T∗G), hence there is a unique
+Υ∈T∗A|qA(u)such that
+Θ−1(d(t∗f)∨|ιA(u)) =ιA(T∗G)(θ−1
+G(Υ)),i.e.,J(Υ) = d( t∗f)∨|ιA(u).
+Using (6.55) and (6.47). we conclude that Υ = d q∗
+Af|u, which proves (6.53).
+With respect to the coordinates (( xj,yd),(˙xj,˙yd)) onTG, one can write
+(6.57) d l/tildewideξ|ιA(u)=∂ξa
+∂xiuadxi+ξad˙ya∈T∗(TG)|ιA(u),
+from where we conclude that
+(6.58) ( TιA)t(dl/tildewideξ|ιA(u)) =/parenleftbigg∂ξa
+∂xjuadxj+ξadua/parenrightbigg/vextendsingle/vextendsingle/vextendsingle
+u=q∗
+Adlξ|u∈T∗A|u.
+Let us now consider J(q∗
+Adlξ)∈ι∗
+AT∗(TG). Since Θ−1(J(q∗
+Adlξ)) lies inT(T∗G)|A∗,
+one can directly verify that J(q∗
+Adlξ) can be written as
+pjdxj+γadya+γad˙ya,
+i.e., its components relative to d˙ xjvanish. By (6.47), ( TιA)t(J(q∗
+Adlξ)|u) =q∗
+Adlξ|u,
+so from the second equality in (6.58) we conclude that pj=∂ξa
+∂xjuaandγa=ξa, i.e.,
+J(q∗
+Adlξ|u) =/parenleftbigg∂ξa
+∂xjuadxj+γadya+ξad˙ya/parenrightbigg/vextendsingle/vextendsingle/vextendsingle
+ιA(u).
+For each given u0∈A, one can find h∈C∞(G) vanishing on M⊂ Gand such
+that dh∨|ιA(u0)=γa(ιA(u0))dya, and (6.54) follows by a direct comparison with
+(6.57). /square
+Wewillneedthefollowingimmediateobservationsaboutlin earfunctionsonvector
+bundles.
+Lemma 6.7. LetqB:B→Nbe a vector bundle, with coordinates (xj,bd)relative
+to a basis of local sections {ed}, and consider b=bded∈Γ(B),b∨=bd∂
+∂bd∈ X(B),33
+andβ=βded∈Γ(B∗). Letlβ∈C∞(B)be the linear function defined by β, and fix
+b0=b(x0)∈B, for a given x0∈N. ThenLb∨lβ=q∗
+B/an}bracketle{tβ,b/an}bracketri}htand
+(6.59) lβ(b0) = (Lb∨lβ)(b0).
+We now prove Proposition 6.4.
+Proof.(of Proposition 6.4)
+Letπ= Lie(Π), and consider ξ1,...,ξk−1∈Γ(A∗) andf∈C∞(M). Let us fix
+u0∈A,x0=qA(u0)∈M. By (6.50) and Lemma 6.6, we have
+π(dlξ1,...,dlξk−1,dq∗
+Af)|u0= ΠT(Jdlξ1,...,Jdlξk−1,Jdq∗
+Af)|ιA(u0)
+= ΠT(dl/tildewideξ1+dh∨
+1,...,dl/tildewideξk−1+dh∨
+k−1,d(t∗f)∨)|ιA(u0), (6.60)
+withhi∈C∞(G),hi|M= 0, and/tildewideξias in (6.52) . We directly check from the
+definition of Π Tthat it is a linear multivector, Π T∈ Xk
+lin(TG), so (see footnote 7)
+(6.61) idf∨
+1idf∨
+2ΠT= 0,∀f1,f2∈C∞(G).
+Hence the expression in (6.60) agrees with
+(6.62) Π T(dl/tildewideξ1,...,dl/tildewideξk−1,d(t∗f)∨)|ιA(u0)= [ΠT,(t∗f)∨](dl/tildewideξ1,...,dl/tildewideξk−1)|ιA(u0),
+where [·,·] is the Schouten bracket on X•(TG).
+Let us consider the vertical lift operation X•(G)→ X•(TG), Π/mapsto→Π∨: in co-
+ordinates (zl) onG, it sends the vector field Y=Yl∂
+∂zltoY∨=Yl∂
+∂˙zl, and this
+is extended to a graded algebra homomorphism of multivector fields. From the
+Schouten bracket relations for vertical and tangent lifts, see e.g. [14], we obtain
+[ΠT,(t∗f)∨] = [Π,t∗f]∨= ((δΠf)r)∨.
+Lettingx0=qA(u0)∈M, a direct computation in coordinates (6.51) shows that
+((δΠf)r)∨(dl/tildewideξ1,...,dl/tildewideξk−1)|ιA(u0)=/angbracketleftBig
+(δΠf)r,/tildewideξ1∧...∧/tildewideξk−1/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle
+ǫ(x0),
+=/angbracketleftBig
+δΠf,ξ1∧...∧ξk−1/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle
+x0,
+from where it follows that
+π(dlξ1,...,dlξk−1,dq∗
+Af)|u0=/angbracketleftBig
+δΠf,ξ1∧...∧ξk−1/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle
+x0=q∗
+A/angbracketleftBig
+δΠf,ξ1∧...∧ξk−1/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle
+u0.
+Comparing with (6.14), we conclude that δ(see (6.44)) and δΠagree onC∞(M). It
+remains to check that they agree on Γ( A).
+We now consider ξ1,...,ξk∈Γ(A∗) and describe π(dlξ1,...,dlξk)|u0in terms of
+δΠ. By (6.50) and (6.54), we have (keeping the notation of Lemma 6.6)
+π(dlξ1,...,dlξk)|u0= ΠT(Jdlξ1,...,Jdlξk)|u0
+= ΠT(dl/tildewideξ1+dh∨
+1,...,dl/tildewideξk+dh∨
+k)|ιA(u0). (6.63)
+From (6.61), we see that the expression Π T(dl/tildewideξ1+ dh∨
+1,...,dl/tildewideξk+ dh∨
+k) can be re-
+written as
+(6.64) Π T(dl/tildewideξ1,...,dl/tildewideξk)+k/summationdisplay
+j=1ΠT(Jdlξ1,...,Jdlξj−1,dh∨
+j,Jdlξj+1,...,Jdlξk).34
+We claim that, for all j= 1,...,k, we have
+(6.65) Π T(Jdlξ1,...,Jdlξj−1,dh∨
+j,Jdlξj+1,...,Jdlξk) = 0.
+To see that, recall from Remark 6.5 that Π Tsatisfies Π♯
+T◦Θ(k−1)= (JG)−1◦TΠ♯,
+and, since Π♯:⊕k−1
+GT∗G →TGis a groupoid morphism (see Remark 6.2),
+TΠ♯◦(ιA(T∗G))(k−1)⊆A(TG).
+Itfollows from(4.7) andthedefinition of Jthat Π♯
+T◦J(k−1)⊆TιA(TA)⊂ι∗
+AT(TG).
+Relative to the adapted coordinates ( xj,yd) in (6.51), elements in TιA(TA)⊂
+ι∗
+AT(TG) are combinations of∂
+∂xjand∂
+∂˙yd, whereas d h∨
+j|ιA(u)is in the span of d yd.
+So (6.65) follows, and we conclude that
+(6.66) π(dlξ1,...,dlξk)|u0= ΠT(dl/tildewideξ1,...,dl/tildewideξk)|ιA(u0).
+To proceed, we observe that Π T(dl/tildewideξ1,...,dl/tildewideξk) defines a linear function on TG,
+and, using (6.59) in Lemma 6.7 (with B=TG), we write
+(6.67) Π T(dl/tildewideξ1,...,dl/tildewideξk)|ιA(u0)=L(ur)∨(ΠT(dl/tildewideξ1,...,dl/tildewideξk))|ιA(u0),
+whereu∈Γ(A) is such that u(x0) =u0. But
+L(ur)∨(ΠT(dl/tildewideξ1,...,dl/tildewideξk)) =(L(ur)∨ΠT)(dl/tildewideξ1,...,dl/tildewideξk)
++k/summationdisplay
+j=1ΠT(dl/tildewideξ1,dl/tildewideξj−1,L(ur)∨(dl/tildewideξj),dl/tildewideξj+1,...,dl/tildewideξk)),
+and note that
+L(ur)∨(dl/tildewideξl) = d(/tildewideξl(ur))∨,L(ur)∨ΠT= [ur,Π]∨=−((δΠu)r)∨,
+where we used the Schouten-bracket relations for tangent an d vertical lifts in the
+second equation. One can directly check that
+((δΠu)r)∨(dl/tildewideξ1,...,dl/tildewideξk)|ιA(u0)=/angbracketleftBig
+(δΠu)r,/tildewideξ1∧...∧/tildewideξk/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle
+ǫ(x0)=/angbracketleftBig
+δΠu,ξ1∧...∧ξk/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle
+x0.
+ThusL(ur)∨(ΠT(dl/tildewideξ1,...,dl/tildewideξk))|ιA(u0)equals
+−/angbracketleftBig
+δΠu,ξ1∧...∧ξk/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle
+x0+k/summationdisplay
+j=1ΠT(dl/tildewideξ1,...,dl/tildewideξj−1,d(/tildewideξj(ur))∨,dl/tildewideξj+1,...,dl/tildewideξk)|ιA(u0).
+Using local coordinates ( xj,yd) as in (6.51), one can check the identity
+d(/tildewideξj(ur))∨|ιA(u0)= d(t∗/angbracketleftbig
+ξj,u/angbracketrightbig
+)∨|ιA(u0)+dh∨
+j|ιA(u0),
+wherehj∈C∞(G) vanishes on M⊂ G. It follows that
+ΠT(dl/tildewideξ1,...,dl/tildewideξj−1,d(/tildewideξj(ur))∨,dl/tildewideξj+1,...,dl/tildewideξk)|ιA(u0)= (6.68)
+ΠT(dl/tildewideξ1,...,dl/tildewideξj−1,d(t∗/angbracketleftbig
+ξj,u/angbracketrightbig
+)∨,dl/tildewideξj+1,...,dl/tildewideξk)|ιA(u0)+
+ΠT(dl/tildewideξ1,...,dl/tildewideξj−1,dh∨
+j,dl/tildewideξj+1,...,dl/tildewideξk)|ιA(u0).35
+Using the linearity of Π T(see footnote 7) and (6.65), we see that
+ΠT(dl/tildewideξ1,...,dl/tildewideξj−1,dh∨
+j,dl/tildewideξj+1,...,dl/tildewideξk)
+= ΠT(Jdl/tildewideξ1,...,Jdl/tildewideξj−1,dh∨
+j,Jdl/tildewideξj+1,...,Jdl/tildewideξk) = 0.
+A direct comparison with (6.60), (6.62) gives that
+ΠT(dl/tildewideξ1,...,dl/tildewideξj−1,d(t∗/angbracketleftbig
+ξj,u/angbracketrightbig
+)∨,dl/tildewideξj+1,...,dl/tildewideξk)|ιA(u0)=
+(−1)k−jπ(dlξ1,...,/hatwidestdlξj,...,dlξk,dq∗
+A/angbracketleftbig
+ξj,u/angbracketrightbig
+)|u0
+Going back to (6.66), we finally conclude that
+π(dlξ1,...,dlξk)|u0=−/angbracketleftBig
+δΠu,ξ1∧...∧ξk/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle
+x0
++k/summationdisplay
+j=1(−1)k−jπ(dlξ1,...,/hatwidestdlξj,...,dlξk,dq∗
+A/angbracketleftbig
+ξj,u/angbracketrightbig
+)|u0.
+Comparing with (6.15), we conclude that δ=δΠ. /square
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+broids. Arxiv: 1003.1004, to appear in J. Symplectic Geom.
+Instituto de Matem ´atica Pura e Aplicada, Estrada Dona Castorina 110, Rio de
+Janeiro, 22460-320, Brasil
+E-mail address :henrique@impa.br
+Department of Mathematics, Universityof Toronto, 40 St. Ge orge Street, Toronto,
+Ontario, M5S 2E4 Canada
+E-mail address :ale.cabrera@gmail.com8
+8A.C.’s current address: Departamento de Matem´ atica Aplic ada, Instituto de Matematica, Uni-
+versidade Federal do Rio de Janeiro, CEP 21941-909, Rio de Ja neiro - RJ, Brazil
\ No newline at end of file
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+arXiv:1001.0535v4  [math.FA]  23 Apr 2011On refined Young inequalities and reverse inequalities
+Shigeru Furuichi∗
+Department of Computer Science and System Analysis,
+College of Humanities and Sciences, Nihon University,
+3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Jap an
+Abstract. In this paper, we show refined Young inequalities for two posi tive operators. Our
+results refine the ordering relations among the arithmetic m ean, the geometric mean and the
+harmonic mean for two positive operators. In addition, we gi ve two different reverse inequalities
+for the refined Young inequality for two positive operators.
+Keywords : Young inequality, reverse inequality, positive operator, operator mean,
+Specht’s ratio and operator inequality
+2000 Mathematics Subject Classification : 15A45
+1 Introduction
+We start from the famous arithmetic-geometric mean inequal ity which is often called Young
+inequality:
+(1−ν)a+νb≥a1−νbν(1)
+for nonnegative real numbers a,bandν∈[0,1].
+Recently, theinequality (1) was refinedby F.Kittaneh andY. Manasrah inthefollowing form,
+for the purpose of the study on matrix norm inequalities.
+Proposition 1.1 ([1, 2, 3]) Fora,b≥0andν∈[0,1], we have
+(1−ν)a+νb≥a1−νbν+r(√a−√
+b)2, (2)
+wherer≡min{ν,1−ν}.
+It is notable that the inequality (2) was first proved in (6.32 ) on page 46 of the reference [1].
+In the section 2 of this paper, we give refined Young inequalit ies for two positive operators
+based on the scalar inequality (2).
+As for the reverse inequalities of the Young inequality, M.T ominaga gave the following in-
+teresting operator inequalities. He called them converse inequalities, however we use the term
+reversefor such inequalities, throughout this paper.
+Proposition 1.2 ([4]) Letν∈[0,1], positive operators AandBsuch that 0< mI≤A,B≤
+MIwithh≡M
+m>1. Then we have the following inequalities for every ν∈[0,1].
+∗E-mail:furuichi@chs.nihon-u.ac.jp
+1(i) (Reverse ratio inequality)
+S(h)A♯νB≥(1−ν)A+νB,
+where the constant S(h)is called Specht’s ratio [5, 6] and defined by
+S(h)≡h1
+h−1
+elogh1
+h−1,(h/ne}ationslash= 1)
+for positive real number h.
+(ii) (Reverse difference inequality)
+hL(m,M)logS(h)+A♯νB≥(1−ν)A+νB, (3)
+where the logarithmic mean Lis defined by
+L(x,y)≡y−x
+logy−logx,(x/ne}ationslash=y)L(x,x)≡x
+for two positive real numbers xandy.
+In the section 3 of this paper, we give reverse ratio type ineq ualities of the refined Young
+inequality for positive operators. In the section 4 of this p aper, we also give reverse difference
+type inequalities of the refined Young inequality for positi ve operators.
+2 Refined Young inequalities for positive operators
+LetHbe a complex Hilbert space. We also represent the set of all bo unded operators on Hby
+B(H). IfA∈B(H) satisfies A∗=A, thenAis called a self-adjoint operator. A self-adjoint
+operator Asatisfies/an}bracketle{tx|A|x/an}bracketri}ht ≥0 for any |x/an}bracketri}ht ∈ H, thenAis called a positive operator. For two
+self-adjoint operators AandB,A≥BmeansA−B≥0.
+Itiswell-known thatwehavethefollowing Younginequaliti esforinvertiblepositiveoperators
+AandB:
+(1−ν)A+νB≥A♯νB≥/braceleftbig
+(1−ν)A−1+νB−1/bracerightbig−1, (4)
+whereA♯νB≡A1/2(A−1/2BA−1/2)νA1/2defined for ν∈[0,1]. The power mean was originally
+introduced in the paper [7]. The simplified and elegant proof for the inequalities (4) was given
+in [8]. See also [9] for the reader having interests in operat or inequalities.
+As a refinement of the inequalities (4), we have the following refined Young inequality for
+positive operators.
+Theorem 2.1 Forν∈[0,1]and positive operators AandB, we have
+(1−ν)A+νB≥A♯νB+2r/parenleftbiggA+B
+2−A♯1/2B/parenrightbigg
+(5)
+≥A♯νB (6)
+≥/braceleftbigg
+A−1♯νB−1+2r/parenleftbiggA−1+B−1
+2−A−1♯1/2B−1/parenrightbigg/bracerightbigg−1
+(7)
+≥/braceleftbig
+(1−ν)A−1+νB−1/bracerightbig−1(8)
+wherer≡min{ν,1−ν}andA♯νB≡A1/2(A−1/2BA−1/2)νA1/2defined for ν∈[0,1].
+To prove Theorem 2.1, we use the following lemma.
+2Lemma 2.2 For invertible positive operators XandY, we have
+(X+Y)−1=X−1−X−1(X−1+Y−1)−1X−1.
+Proof: Since (X+Y)(X+Y)−1X=X, we have X(X+Y)−1X+Y(X+Y)−1X=X. Thus we
+haveX(X+Y)−1X=X−Y(X+Y)−1X=X−(X−1(X+Y)Y−1)−1=X−(X−1+Y−1)−1.
+Multiplying X−1from both sides, we obtain the lemma.
+Proof of Theorem 2.1 : Thesecond inequality (6) is clear, sincewe have 2 r/parenleftbigA+B
+2−A♯1/2B/parenrightbig
+≥
+0. We prove the first inequality. From the inequality (2), we h ave forν∈[0,1] andx≥0
+νx+1−ν−xν−r(√x−1)2≥0.
+By the standard operational calculus, we have
+νT+1−ν≥Tν+r(T1/2−1)2
+=Tν+r(T−2T1/2+1) (9)
+for a positive operator Tandν∈[0,1]. From here, we suppose that Ais an invertible. (For a
+general case, we consider the invertible positive operator Aǫ≡A+ǫIfor positive real number
+ǫ. If we take a limit ǫ→0, the following result also holds. Throughout this paper, w e apply
+this continuity argument, however, from now on, we omit such descriptions for simplicity.)
+Substituting T=A−1/2BA−1/2into the inequality (9), we have
+νA−1/2BA−1/2+1−ν≥/parenleftBig
+A−1/2BA−1/2/parenrightBigν
++r/braceleftbigg
+A−1/2BA−1/2−2/parenleftBig
+A−1/2BA−1/2/parenrightBig1/2
++1/bracerightbigg
+Multiplying A1/2to the above inequality from both sides, we have
+(1−ν)A+νB≥A♯νB+r/parenleftbig
+A+B−2A♯1/2B/parenrightbig
+,
+which proves the inequality (5).
+Replacing AandBbyA−1andB−1in the inequality (5), respectively and taking the inverse
+of both sides, then we have the last inequality (8).
+By Lemma 2.2, the right hand side of the inequality (7) can be c alculated as
+R.H.S.=A♯νB−(A♯νB)/bracketleftBigg
+A♯νB+/braceleftbigg
+2r/parenleftbiggA−1+B−1
+2−A−1♯1/2B−1/parenrightbigg/bracerightbigg−1/bracketrightBigg−1
+(A♯νB).
+Since,/parenleftbig
+A−1♯νB−1/parenrightbig−1=A♯νB≥0 and 2r/parenleftBig
+A−1+B−1
+2−A−1♯1/2B−1/parenrightBig
+≥0, we have the third
+inequality (7), which completes the proof.
+In the paper [10], the equivalent relation between the Young inequality and the H¨ older-
+McCarthy inequality [11] was shown by a simplified elegent pr oof. Here we show a kind of the
+refinement of the H¨ older-McCarthy inequality applying The orem 2.1.
+Corollary 2.3 Forν∈[0,1]and any positive operator Aon the Hilbert space Hand any unit
+vector|x/an}bracketri}ht ∈ H, if/an}bracketle{tx|A|x/an}bracketri}ht /ne}ationslash= 0, then we have
+1−/an}bracketle{tx|A|x/an}bracketri}ht−ν/an}bracketle{tx|Aν|x/an}bracketri}ht ≥r/parenleftBig
+1−/an}bracketle{tx|A|x/an}bracketri}ht−1/2/an}bracketle{tx|A1/2|x/an}bracketri}ht/parenrightBig2
+, (10)
+wherer≡min{ν,1−ν}.
+3Proof: Ifν= 0, then the inequality (10) is trivial. It is sufficient that w e prove it for the case
+ofν∈(0,1]. In the inequality (9) , we put T=k1
+νA, for any positive real number kand by the
+unit vector |x/an}bracketri}ht ∈ H, we have
+νk1
+ν/an}bracketle{tx|A|x/an}bracketri}ht+1−ν≥k/an}bracketle{tx|Aν|x/an}bracketri}ht+r/parenleftBig
+k1
+2ν/an}bracketle{tx|A1/2|x/an}bracketri}ht−1/parenrightBig2
+(11)
+In the inequality (11), if we put k=/an}bracketle{tx|A|x/an}bracketri}ht−ν, then we obtain the inequality (10).
+Remark 2.4 From H¨ older-McCarthy inequality [11]:
+/an}bracketle{tx|A|x/an}bracketri}htν≥ /an}bracketle{tx|Aν|x/an}bracketri}ht (12)
+for any unit vector |x/an}bracketri}ht ∈ H, if/an}bracketle{tx|A|x/an}bracketri}ht /ne}ationslash= 0, then we have
+1−/an}bracketle{tx|A|x/an}bracketri}ht−ν/an}bracketle{tx|Aν|x/an}bracketri}ht ≥0.
+The inequality (10) gives a refined one for the above inequalit y which is equivalent to the in-
+equality (12) in the case of /an}bracketle{tx|A|x/an}bracketri}ht /ne}ationslash= 0.
+3 A reverse ratio inequality for a refined Young inequality
+For positive real numbers a,bandν∈[0,1], M.Tominaga showed the following inequality [4]:
+S/parenleftBiga
+b/parenrightBig
+a1−νbν≥(1−ν)a+νb, (13)
+which is called the converse ratio inequality for the Young i nequality in [4]. In this section, we
+show the reverse ratio inequality of the refined Young inequa lity (2).
+Lemma 3.1 For positive real numbers a,bandν∈[0,1], we have
+S/parenleftbigg/radicalbigga
+b/parenrightbigg
+a1−νbν≥(1−ν)a+νb−r(√a−√
+b)2, (14)
+wherer≡min{ν,1−ν}.
+Proof:
+(i) For the case of ν≤1/2,r=ν. We consider the following function.
+gb(ν)≡νb+(1−ν)−ν(√
+b−1)2
+bν,/parenleftbigg
+0≤ν≤1
+2/parenrightbigg
+.
+Then we have
+g′
+b(ν) =2(√
+b−1)−/braceleftBig
+2(√
+b−1)ν+1/bracerightBig
+logb
+bν
+so that the equation g′
+b(ν) = 0 implies
+ν=1
+logb−1
+2(√
+b−1)≡νb.
+4From the Klein inequality:
+1−1√
+b≤log√
+b≤√
+b−1,(b >0)
+we have νb∈[0,1
+2]. We also find that g′
+b(ν)>0 forν < νbandg′
+b(ν)<0 forν > νb. Thus
+the function gb(ν) takes a maximum value when ν=νb,(b/ne}ationslash= 1) and it is calculated as
+follows.
+max
+0≤ν≤1
+2gb(ν) =gb(νb) =2(√
+b−1)/parenleftBig
+1
+logb−1
+2(√
+b−1)/parenrightBig
++1
+b1
+logbb−1
+2(√
+b−1)
+=2(√
+b−1)
+logb
+eb−1
+2(√
+b−1)=/parenleftBig√
+b/parenrightBig1√
+b−1
+elog/parenleftBig√
+b/parenrightBig1√
+b−1=S(√
+b).
+Thus we have the following inequality.
+S(√
+b)bν≥νb+(1−ν)−ν(√
+b−1)2. (15)
+In the case of b= 1, we have the equality in the above inequality, since we hav eS(1) = 1.
+Replacing bbyb
+aand then multiplying ato both sides, we have
+S/parenleftbigg/radicalbigga
+b/parenrightbigg
+a1−νbν≥(1−ν)a+νb−ν(√a−√
+b)2, (16)
+since we have S(x) =S(1/x) forx >0.
+(ii) For the case of ν≥1/2,r= 1−ν. We consider the following function.
+ha(ν)≡ν+(1−ν)a−(1−ν)(1−√a)2
+a1−ν,/parenleftbigg1
+2≤ν≤1/parenrightbigg
+.
+By the similar way to (i), we have
+h′
+a(ν) = 0⇔ν= 1−/parenleftbigg1
+loga−1
+2(√a−1)/parenrightbigg
+≡νa.
+We also find νa∈[1
+2,1] andh′
+a(ν)>0 forν < νaandh′
+a(ν)<0 forν > νa. Thus the
+function ha(ν) takes a maximum value when ν=νa,(a/ne}ationslash= 1) and it is calculated by
+max
+1
+2≤ν≤1ha(ν) =ha(νa) =S(√a).
+Therefore we have the following inequality.
+S(√a)a1−ν≥ν+(1−ν)a−(1−ν)(1−√a)2. (17)
+In the case of a= 1, we have the equality in the above inequality, since we hav eS(1) = 1.
+Replacing abya
+band then multiplying bto both sides, we have
+S/parenleftbigg/radicalbigga
+b/parenrightbigg
+a1−νbν≥(1−ν)a+νb−(1−ν)(√a−√
+b)2. (18)
+5From (i) and (ii), the proof is completed.
+Remark 3.2 We easily find that both sides in the inequality (14) is less th an or equal to thoes
+in the inequality (13) so that neither the inequality (14) no r the inequality (13) is uniformly
+better than the other.
+In addition, our next interest is the ordering between S/parenleftbig/radicalbiga
+b/parenrightbig
+a1−νbνand(1−ν)a+νb. How-
+ever we have no ordering between them, because we have the fol lowing examples. For example,
+leta= 1andb= 10. Ifν= 0.9, then(1−ν)a+νb−S/parenleftbig/radicalbiga
+b/parenrightbig
+a1−νbν≃ −0.246929. And if
+ν= 0.6, then(1−ν)a+νb−S/parenleftbig/radicalbiga
+b/parenrightbig
+a1−νbν≃1.71544.
+Applying Lemma 3.1, we have the reverse ratio inequality of t he refined Young inequality
+for positive operators.
+Theorem 3.3 We suppose two invertible positive operators AandBsatisfy0< mI≤A,B≤
+MI, whereIrepresents an identity operator and m,M∈R. For any ν∈[0,1], we then have
+S(√
+h)A♯νB≥(1−ν)A+νB−2r/parenleftbiggA+B
+2−A♯1/2B/parenrightbigg
+, (19)
+whereh≡M
+m>1andr≡min{ν,1−ν}.
+Proof: In Lemma 3.1, we put a= 1, then we have for all b >0,
+S(√
+b)bν≥νb+(1−ν)−r(√
+b−1)2
+We consider the invertible positive operator Tsuch that 0 < mI≤T≤MI. Then we have the
+following inequality
+max
+m≤t≤MS(√
+t)Tν≥νT+(1−ν)−r(T−2T1/2+1), (20)
+for anyν∈[0,1]. We put T=A−1/2BA−1/2.Since we then have1
+h=m
+M≤A−1/2BA−1/2≤
+M
+m=h, we have
+max
+1
+h≤t≤hS(√
+t)/parenleftBig
+A−1/2BA−1/2/parenrightBigν
+≥νA−1/2BA−1/2+(1−ν)−r/braceleftbigg
+A−1/2BA−1/2−2/parenleftBig
+A−1/2BA−1/2/parenrightBig1/2
++1/bracerightbigg
+.
+Note that h >1 andS(x) is monotone decreasing for 0 < x <1 and monotone increasing for
+x >1 [4]. Thus we have
+S(√
+h)/parenleftBig
+A−1/2BA−1/2/parenrightBigν
+≥νA−1/2BA−1/2+(1−ν)−r/braceleftbigg
+A−1/2BA−1/2−2/parenleftBig
+A−1/2BA−1/2/parenrightBig1/2
++1/bracerightbigg
+.
+Multiplying A1/2to the above inequality from both sides, we have the present t heorem.
+64 A reverse difference inequality for a refined Young inequali ty
+For the classical Young inequality, the following reverse i nequality is known. For positive real
+numbers a,bandν∈[0,1], M.Tominaga showed the following inequality [4]:
+L(a,b)logS/parenleftBiga
+b/parenrightBig
+≥(1−ν)a+νb−a1−νbν, (21)
+which is called the converse difference inequality for the You ng inequality in [4]
+In this section, we show the reverse difference inequality of t he refined Young inequality (2).
+Lemma 4.1 For positive real numbers a,bandν∈[0,1], we have
+ωL(√a,√
+b)logS/parenleftbigg/radicalbigga
+b/parenrightbigg
+≥(1−ν)a+νb−a1−νbν−r/parenleftBig√a−√
+b/parenrightBig2
+, (22)
+whereω≡max/braceleftBig√a,√
+b/bracerightBig
+.
+Proof:
+(i) For the case of ν≤1/2,r=ν. We consider the following function.
+gb(ν)≡νb+(1−ν)−bν−ν(√
+b−1)2,/parenleftbigg
+0≤ν≤1
+2/parenrightbigg
+.
+Fromg′
+b(ν) = 2(√
+b−1)−bνlogb, we have
+g′
+b(ν) = 0⇔ν=log√
+b−1
+log√a
+logb≡νb.
+We also find that νb∈[0,1
+2] by elementaly calculations with the following inequaliti es:
+1−1√
+b≤log√
+b≤√
+b−1,(b >0).
+In addition, we have g′′
+b(ν) =−bν(logb)2<0. Therefore gbtakes a maximum value
+whenν=νb, and it is calculated as gb(νb) =L(1,√
+b)logS(√
+b) by simple but slightly
+complicated calculations. Thus we have
+L(1,√
+b)logS/parenleftBig√
+b/parenrightBig
+≥νb+(1−ν)−bν−ν(√
+b−1)2.
+We putb
+ainstead of bin the above inequality, and then multiplying ato both sides, we
+have
+√aL(√a,√
+b)logS/parenleftbigg/radicalbigga
+b/parenrightbigg
+≥(1−ν)a+νb−a1−νbν−ν(√a−√
+b)2,(23)
+sinceL(x,y) =L(y,x) andS(x) =S(1/x) forx >0.
+(ii) For the case of ν≥1/2,r= 1−ν. We consider the following function.
+ha(ν)≡ν+(1−ν)a−a1−ν−(1−ν)(1−√a)2,/parenleftbigg1
+2≤ν≤1/parenrightbigg
+.
+7By the similar way to (i), we have
+h′
+a(ν) = 0⇔ν= 1−log√a−1
+log√a
+loga≡νa.
+By the similar way to (i), we have νa∈[1
+2,1] andh′′
+a(ν) =−a1−ν(loga)2<0 so that ha
+takes a maximum value when ν=νa, and it is calculated as ha(νa) =L(1,√a)logS(√a).
+Thus we have
+L(1,√a)logS(√a)≥ν+(1−ν)a−a1−ν−(1−ν)(1−√a)2,
+which implies
+√
+bL(√
+b,√a)logS/parenleftbigg/radicalbigga
+b/parenrightbigg
+≥(1−ν)a+νb−a1−νbν−(1−ν)(√a−√
+b)2,(24)
+by replacing abya
+band then multiplying bto both sides.
+From the inequalities (23) and (24), we have the present theo rem, since L(x,y) =L(y,x) and
+S(x) =S(1/x) forx >0.
+Remark 4.2 We easily find that the right hand side of the inequality (21) i s greater than that
+of the inequality (22). Therefore, if the left hand side of the inequality (22) is greater than that
+of the inequality (21), then Theorem 4.1 is trivial one. Howev er, we have not yet found any
+counter-example such that
+L(a,b)logS/parenleftBiga
+b/parenrightBig
+≥ωL(√a,√
+b)logS/parenleftbigg/radicalbigga
+b/parenrightbigg
+, (25)
+whereω= max/braceleftBig√a,√
+b/bracerightBig
+for anya,b >0by the computer calculations. Here we give a remark
+that we have the following inequalities:
+L(a,b)≤ωL(√a,√
+b), and logS/parenleftBiga
+b/parenrightBig
+≥logS/parenleftbigg/radicalbigga
+b/parenrightbigg
+(26)
+for anya,b >0. At least, we actually have many examples satisfying the ine quality (25) so that
+we claim that Theorem 4.1 is nontrivial as a refinement of the in equality (21).
+In addition, it is remarkable that we have no ordering between
+L(a,b)logS/parenleftBiga
+b/parenrightBig
+and
+ωL(√a,√
+b)logS/parenleftbigg/radicalbigga
+b/parenrightbigg
++r/parenleftBig√a−√
+b/parenrightBig2
+for anya,b >0andν∈[0,1]. Therefore we may claim that Theorem 4.1 is also nontrivial fro m
+the sense of finding a tighter upper bound of (1−ν)a+νb−a1−νbν.
+Finally we prove the following theorem. It can be proven by th e similar method in [4].
+8Theorem 4.3 We suppose two invertible positive operators satisfy 0< mI≤A,B≤MI,
+whereIrepresents an identity operator and m,M∈R. For any ν∈[0,1], we then have
+h√
+ML(√
+M,√m)logS(√
+h)≥(1−ν)A+νB−A♯νB−2r/parenleftbiggA+B
+2−A♯1/2B/parenrightbigg
+,(27)
+whereh≡M
+m>1andr≡min{ν,1−ν}.
+Proof: From the inequality (22), we have
+ωL(√
+b,1)logS(√
+b)≥νb+(1−ν)−bν−r(b−2√
+b+1), (28)
+for allν∈[0,1], putting b= 1. We consider the invertible positive operator Tsuch that
+0< mI≤T≤MI. Then we have the following inequality
+max
+m≤t≤Mmax/braceleftBig√
+t,1/bracerightBig
+L(√
+t,1)logS(√
+t)≥νT+(1−ν)−Tν−r(T−2T1/2+1),(29)
+for anyν∈[0,1]. We put T=A−1/2BA−1/2.Since we then have1
+h=m
+M≤A−1/2BA−1/2≤
+M
+m=h, we have
+max
+1
+h≤t≤hmax/braceleftBig√
+t,1/bracerightBig
+L(√
+t,1)logS(√
+t)
+≥νA−1/2BA−1/2+(1−ν)−/parenleftBig
+A−1/2BA−1/2/parenrightBigν
+−r/braceleftbigg
+A−1/2BA−1/2−2/parenleftBig
+A−1/2BA−1/2/parenrightBig1/2
++1/bracerightbigg
+.
+Note that h >1 andL(u,1) is monotone increasing function for u >0. In addition, we note
+thatS(x) is monotone decreasing for 0 < x <1 and monotone increasing for x >1 [4]. Thus
+we have
+√
+hL(√
+h,1)logS(√
+h)
+≥νA−1/2BA−1/2+(1−ν)−/parenleftBig
+A−1/2BA−1/2/parenrightBigν
+−r/braceleftbigg
+A−1/2BA−1/2−2/parenleftBig
+A−1/2BA−1/2/parenrightBig1/2
++1/bracerightbigg
+.
+Multiplying A1/2to the above inequality from both sides, we have
+√
+hL(√
+h,1)logS(√
+h)A≥(1−ν)A+νB−A♯νB−2r/parenleftbiggA+B
+2−A♯1/2B/parenrightbigg
+.
+Since the left hand side in the above inequality is less than
+√
+hL(√
+h,1)logS(√
+h)M=h√
+ML(√
+M,√m)logS(√
+h),
+the proof is completed.
+Remark 4.4 As mentioned in Remark 4.2, we have not yet found the ordering between the right
+hand side of the inequality (27) and that of the inequality (3 ). Therefore Theorem 4.3 is not a
+trivial result.
+95 Concluding remarks
+As we have seen, we gave refined Young inequalities for two pos itive operators. In addition, we
+gave reverse ratio type inequalities and reverse difference t ype inequalities for the refined Young
+inequality for positive operators. Closing this paper, we s hall give a refinement of the weighted
+arithmetic-geometric mean inequality for nreal numbers by a simple proof.
+Proposition 5.1 Leta1,···,an≥0andp1,···,pn>0with/summationtextn
+j=1pj= 1andλ≡min{p1,···,pn}.
+If we assume that the multiplicity attaining λis1, then we have
+n/summationdisplay
+i=1piai−n/productdisplay
+i=1api
+i≥nλ/parenleftBigg
+1
+nn/summationdisplay
+i=1ai−n/productdisplay
+i=1a1/n
+i/parenrightBigg
+, (30)
+with equality if and only if a1=···=an.
+Proof: We suppose λ=pj. For any j= 1,···,n, we then have
+n/summationdisplay
+i=1piai−pj/parenleftBiggn/summationdisplay
+i=1ai−nn/productdisplay
+i=1a1/n
+i/parenrightBigg
+=npj/parenleftBiggn/productdisplay
+i=1a1/n
+i/parenrightBigg
++n/summationdisplay
+i=1,i/negationslash=j(pi−pj)ai
+≥n/productdisplay
+i=1,i/negationslash=j/parenleftBig
+a1/n
+1···a1/n
+n/parenrightBignpjapi−pj
+i
+=ap1
+1···apnn.
+In the above process, the classical weighted arithmetic-ge ometric mean inequality [12, 13] for
+a1,···,an≥0 andp1,···,pn>0 with/summationtextn
+j=1pj= 1;
+n/summationdisplay
+j=1pjaj≥n/productdisplay
+j=1apj
+j, (31)
+with equality if and only if a1=···=an, was used. We note that pi−pj>0 from the
+assumption of the proposition. The equality in the inequali ty (30) holds if and only if
+(a1a2···an)1
+n=a1=a2=···=aj−1=aj+1=···=an
+by the equality condition of the classical weighted arithme tic-geometric mean inequality (31).
+Therefore a1=a2=···=aj−1=aj+1=···=an≡a, then we have a1
+n
+jan−1
+n=afrom the first
+equality. Thus we have aj=a, which completes the proof.
+The inequality (30) gives a refinement of the classical weigh ted arithmetic-geometric mean
+inequality (31). At the sametime, it gives anatural general ization of the inequality (2) proved in
+[2]. It is also notable that Proposition 5.1 can be proven by u sing the bounds for the normalized
+Jensen functional, which were obtained by S.S.Dragomir in [ 14], except for the equality cond-
+tions. Note that the inequality (30) itself holds without th e assumption that the multiplicity
+attaining λis 1. In addition, when we do not impose on this assumption, th e equality in the
+inequality (30) holds if pi=1
+nfor alli= 1,···,n.
+Ackowledgement
+The author thanks Professor S. M. Sitnik for letting me know t he valuable infromation on the
+reference [1] and Dr.F.C.Mitroi for giving me the valuable c omment on the equality condition of
+Propisition 5.1. Theauthoralsothankstheanonymousrefer eeforvaluablecomments toimprove
+the manuscript. The author was supported in part by the Japan ese Ministry of Education,
+Science, SportsandCulture, Grant-in-Aid forEncourageme nt of YoungScientists (B), 20740067.
+10References
+[1] N.A.Bobylev and M. A. Krasnoselsky, Extremum Analysis ( degenerate cases), Moscow,
+preprint, 1981, 52 pages, (in Russian).
+[2] F.Kittaneh and Y.Manasrah, Improved Young and Heinz ine qualities for matrices,
+J.Math.Anal.Appl.,Vol.36(2010), pp.262-269.
+[3] S.Izuminoand N.Nakamura, Weighted Geometric Means of P ositive Operators, Kyungpook
+Math. J. 50(2010), pp.213-228.
+[4] M.Tominaga, Specht’s ratio in the Young inequality, Sci .Math.Japon.,Vol.55(2002),pp.583-
+588.
+[5] W.Specht, Zer Theorie der elementaren Mittel, Math.Z., Vol.74(1960),pp.91-98.
+[6] J.I.Fujii, S.Izumino and Y.Seo, Determinant for positi ve operators and Specht’s theorem,
+Sci.Math.Japon.,Vol.1(1998),pp.307-310.
+[7] F.Kubo and T.Ando, Means of positive operators, Math. An n.,Vol.264(1980),pp.205-224.
+[8] T.Furuta and M.Yanagida, Generalized means and convexi ty of inversion for positive oper-
+ators, Amer.Math.Monthly, Vol.105 (1998),pp.258-259.
+[9] T.Furuta, Invitation to linear operators: From matrix t o bounded linear operators on a
+Hilbert space, Taylor and Francis, 2002.
+[10] T.Furuta, TheH¨ older-McCarthy and theYounginequali ties areequivalent for Hilbert space
+operators, Amer.Math.Monthly, Vol.108(2001),pp.68-69.
+[11] C.A.McCarthy, Cp, Israel J. Math.,Vol.5(1967),pp.249-271.
+[12] G.H.Hardy, J.E.Littlewood and G.P´ olya, Inequalitie s, Cambridge University Press, 1952.
+[13] P.S.Bullen, Handbook of Means and Their Inequalities, Kluwer Acad. Publ., Dordrecht,
+2003.
+[14] S.S.Dragomir, Bounds for the normalised Jensen functi onal, Bull. Australian Math. Soc.,
+Vol.74(2006), pp.471-478.
+11
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+arXiv:1001.0536v2  [cond-mat.supr-con]  22 Feb 2010Itinerant Nature of Magnetism in Iron Pnictides: A first prin ciples study
+Yu-Zhong Zhang,1Ingo Opahle,1Harald O. Jeschke,1and Roser Valent´ ı1
+1Institut f¨ ur Theoretische Physik, Goethe-Universit¨ at F rankfurt,
+Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany
+(Dated: October 24, 2018)
+Within the framework of density functional theory we invest igate the nature of magnetism in
+various families of Fe-based superconductors. (i) We show t hat magnetization of stripe-type anti-
+ferromagnetic order always becomes stronger when As is subs tituted by Sb in LaOFeAs, BaFe 2As2
+and LiFeAs. By calculating Pauli susceptibilities, we attr ibute the magnetization increase obtained
+after replacing As by Sb to the enhancement of an instability at (π,π). This points to a strong con-
+nection between Fermi surface nesting and magnetism, which supports the theory of the itinerant
+nature of magnetism in various families of Fe-based superco nductors. (ii) We find that within the
+family LaOFe Pn(Pn=P, As, Sb, Bi) the absence of an antiferromagnetic phase in L aOFeP and its
+presence in LaOFeAs can be attributed to the competition of i nstabilities in the Pauli susceptibility
+at (π,π) and (0,0), which further strengthens the close relation between Fe rmi surface nesting and
+experimentally observed magnetization. (iii) Finally, ba sed on our relaxed structures and Pauli
+susceptibility results, we predict that LaOFeSb upon dopin g or application of pressure should be
+a candidate for a superconductor with the highest transitio n temperature among the hypothetical
+compounds LaOFeSb, LaOFeBi, ScOFeP and ScOFeAs while the pa rent compounds LaOFeSb and
+LaOFeBi should show at ambient pressure a stripe-type antif erromagnetic metallic state.
+PACS numbers: 74.70.-b 74.25.Ha 74.25.Jb,71.15.Mb,71.15 .Pd
+I. INTRODUCTION
+After the discovery of the first high- Tciron-based su-
+perconductor La[O 1−xFx]FeAs1(denoted as 1111 com-
+pound), the superconducting transition temperature was
+rapidly raised up to 55 K with substitution of La by
+Sm2. While various other families of Fe-based super-
+conductors were reported afterwards, like the 122 com-
+pounds AEFe2As2(AE=Ca,Sr,Ba)3–9, the 111 com-
+poundsAFeAs (A=Li,Na)10–15and the 11 compounds
+FeCh(Ch=Se,Te)16–18, the transition temperature has
+always been lower than the highest one observed in the
+1111 systems. Besides the continuous experimental at-
+tempts to pursue higher superconducting transition tem-
+peratures in the Fe-based compounds and deeper under-
+standing of high- Tcsuperconductivity19, a great effort to
+understand the origin of their phase diagram has also
+been made on the theoretical side.
+Although it is widely believed that magnetically medi-
+ated rather than phonon-mediated pairing dominates the
+superconducting state due to its proximity to a stripe-
+type antiferromagnetic phase20–37, the origin of the mag-
+netism is still highly under debate. Some of the ex-
+perimental work and theoretical studies based on den-
+sity functional theory (DFT) support an itinerant sce-
+nario of magnetism due to the fact that the electron and
+hole sheets of the Fermi surface are nearly nested23,38–44
+and correlation effects are not very strong, resulting in
+a metallic state of the parent compounds45–48. In con-
+trast, some authors favor a localized picture21,34,35,49–53
+since DFT calculations fail to reproduce the experimen-
+tally observed band splitting in the stripe-type antifer-
+romagnetic phase54. In these studies the observed small
+magnetic moment is attributed to highly frustrated su-perexchange interactions which explain the observed low
+energy spin excitations well55. Apart from the oppos-
+ing viewpoints above, various other interpretations co-
+exist, such as those that propose that the magnetism
+could come from the local Hund’s rule coupling56or from
+the coexistence of localized and itinerant electrons57–60.
+A recent LDA+U calculation explains the small mag-
+netic moment by formation of magnetic multipoles62
+while LDA+DMFT (dynamical mean field theory) cal-
+culations in conjunction with angle resolved photoemis-
+sion (ARPES) experiments suggest that involvement of
+non-localfluctuationsmaybe crucial61. Further, arecent
+DMFT calculation stressed the importance of the inter-
+play between frustrated and unfrustrated bands within a
+two-band Hubbard model at half-filling63. Among these
+theories, it is presently hard to decide which is the most
+promisingfor the observedmagnetism in the parent com-
+pounds of the iron-based superconductors.
+In fact, one of the most popular theories mentioned
+above, the itinerant scenario of magnetism, was recently
+substantially challenged both by experiment64and den-
+sity functional theory calculations65. On the one hand,
+an ARPES study on the 11 compound Fe 1+xTe shows no
+evidence of a Fermi surface nesting at ( π,0)64while mag-
+neticorderwithsuchawavevectorisdetectedbyneutron
+scattering66. On the other hand, a DFT calculation on
+LaOFeAs, BaFe 2As2and LiFeAs based on a pseudopo-
+tential method65reveals that the magnetic moments are
+enhancedin all compoundsbyreplacingAs with Sb while
+the Fermi surface nesting with a nesting vector of ( π,π)
+is found with this substitution to be enhanced only in
+the 1111 compound but suppressed in the 122 and 111
+compounds. These studies question altogether the appli-
+cability of the theory of itinerant magnetism to the 11,2
+122 and 111 systems.
+The former discrepancy in the 11 compounds was soon
+resolved by a new DFT calculation based on the full
+potential linear muffin tin orbital (FPLMTO) method
+which reconciles the theory of itinerant magnetism with
+the existing experiments on Fe 1+xTe67. It shows that,
+while the Fermi surface is nested at ( π,π) in the undoped
+FeTeasinotheriron-basedsuperconductors,dopingwith
+0.5 electrons due to the excess of Fe in Fe 1+xTe leads to
+a strong ( π,0) nesting of the Fermi surface which cor-
+responds to the observed magnetic ordering. However,
+up to now, the second question of enhanced Fermi sur-
+face nesting in 1111 versus suppression in 122 and 111
+compounds when As is substituted by Sb still remains.
+In this work, by applying Car-Parrinellomolecular dy-
+namics68based on a projector augmented wave (PAW)
+basis69, we will show that, in contrast to the results from
+a pseudopotential method65, the magnetic moment and
+the Pauli susceptibility at ( π,π) are simultaneously en-
+hancedwhen AsisreplacedbySb inLaOFeAs, BaFe 2As2
+and LiFeAs, which strongly suggests that Fermi sur-
+face nesting is closely related to the magnetic moment
+strength, and the itinerant scenario of nesting-driven
+magnetism is still valid in the 111, 122 and 1111 com-
+pounds. By further comparing the Pauli susceptibilities
+of LaOFe PnwithPn=P, As, Sb, and Bi, we argue that
+the absence and the presence of magnetism at ambient
+pressure in LaOFeP and LaOFeAs, respectively, origi-
+nate from the competition between the instabilities of
+the susceptibility at (0 ,0) and ( π,π), which again in-
+dicates the importance of Fermi surface nesting for the
+description of magnetism. We predict that a stripe-
+type antiferromagnetic metallic state should be present
+in the hypothetical compounds LaOFeSb and LaOFeBi.
+Finally, we study the structural and magnetic proper-
+ties of 1111 compounds including REOFeAs ( RE=Ce,
+Nd, Sm), LaOFe Pn(Pn=P, As, Sb, Bi) and ScOFe Pn
+(Pn=P, As) and predict that LaOFeSb could be a su-
+perconductor with the highest transition temperature
+among these compounds.
+II. METHOD
+Throughoutthis paper, the Car-Parrinello68projector-
+augmented wave69method is employed to optimize the
+lattice parameters and internal atomic positions. These
+optimized structures are then used for all subsequent
+electronic structure calculations unless stated otherwise.
+4×4×4k-points and doubled (√
+2×√
+2×1) unit cells
+with stripe-type antiferromagnetic order are used when
+relaxation of all lattice and electronic degrees of free-
+dom is performed. We use time steps of 0.12 fs and fric-
+tion to cool the systems to zero temperature. Note that
+the structure optimization is performed in the magnetic
+phase. As we shall show below, whenever experimental
+structures are available, our optimized structures com-
+parewell withthe experimentalones. Thisisnot thecaseif structure optimizations are performed within non-spin
+polarized calculations as has been frequently pointed out
+in the literature.39,42,70
+We used high energy cutoffs of 408 eV and 1632 eV
+for the wave functions and charge density expansion,
+respectively. The total energy was converged to less
+than 0.01 meV/atom and the cell parameters to less
+than 0.0005 ˚A. Part of our results are double-checked
+by the full potential linearized augmented plane wave
+(FPLAPW) method as implemented in the WIEN2k
+code71and full potential local orbital (FPLO) method72.
+Results are consistent among these methods. Through-
+out the paper, the Perdew-Burke-Ernzerhof generalized
+gradient approximation (GGA) to DFT has been used if
+not specified otherwise, and comparisons with the results
+from local density approximation (LDA) were also per-
+formed. InordertodetermineiftheFermisurfacenesting
+is the driving force for the low-temperature stripe-type
+antiferromagneticordering,wecalculatethe q-dependent
+Pauli susceptibility at ω=0 and Fermi surface cuts at
+different kzplanes without magnetization. These cal-
+culations were performed with the WIEN2k code using
+RKmax= 7. While 40000 kpoints in each kzplane are
+used in calculating Fermi surface cuts, a three dimen-
+sional grid of 128 ×128×128kandqpoints and the
+constant matrix element approximation are employed for
+the susceptibility. For the calculations including Ce, Nd,
+and Sm atoms in the nonmagnetic phases, we apply the
+opencoreapproximationforthelocalized felectrons. All
+calculations were performed in the scalar relativistic ap-
+proximation,whichusuallyprovidesagooddescriptionof
+structural properties even for heavier elements73. Thus,
+spin-orbit coupling, which could be potentially relevant
+especially for Bi compounds74,75, is neglected in the cal-
+culations for the valence electrons. However, since most
+of the weight of the Bi 6p states is well below the Fermi
+energy and irrelevant to magnetic ordering, we would ex-
+pectonlyminormodificationsfortheresultingFermisur-
+faces and susceptibilities.
+III. PAULI SUSCEPTIBILITIES AND
+MAGNETISM IN 1111, 122 AND 111
+COMPOUNDS
+As is pointed out in Section I, a DFT calculation based
+on a pseudopotential method within the SIESTA code65
+reveals a disconnection between magnetism and Fermi
+surface nesting in 122 and 111 compounds, ( i.e., while
+magnetization is enhanced, the Pauli susceptibility at
+q= (π,π) which is responsible for stripe-type antifer-
+romagnetic ordering is suppressed when As is replaced
+by Sb in BaFe 2As2and LiFeAs), and therefore ques-
+tions the scenario of an itinerant nature of magnetism.
+From our spin-polarizedGGA calculationsfor LaOFe Pn,
+BaFe2Pn2and LiFe Pn(Pn= As, Sb), the same trends
+in magnetism are detected as observed in Ref. 65: the
+ground states are all found to be stripe-type antiferro-3
+FIG. 1: (Color online) Comparison of normalized static q-
+dependent Pauli susceptibilities at fixed qz=πbetween ar-
+senide and antimonide of (a) 111 compounds, (b) 1111 com-
+pounds and (c) 122 compounds. The normalization factors
+are the susceptibilities of the corresponding arsenide sys tems
+for each type of compound at q0= (0,0,π). Please note
+that the peak position is not exactly at qπ= (π,π,π) since
+the electron and hole Fermi surfaces are nearly nested rathe r
+than perfectly nested. Here, GGA is used for the DFT calcu-
+lations.
+magnetic metallic states and magnetic moments increase
+with the substitution of As by Sb in LaOFeAs, BaFe 2As2
+and LiFeAs.
+However, the magnetic moments we obtained
+are 1.6 (2.3) µBin LiFeAs (LiFeSb), 2.0 (2.5) µB
+in BaFe 2As2(BaFe2Sb2) and 1.8 (2.2) µBin
+LaOFeAs (LaOFeSb)76, which are notably smaller than
+those obtained from the pseudopotential method65,87.
+Further comparing the optimized lattice structures, we
+find that, while our results are in good agreement withTABLE I: Comparison between different DFT codes of the
+structures of LiFeAs optimized within GGA. zLi=0.3385 and
+zAs=0.7688 are obtained from our CP-PAW calculations.
+a(˚A)b(˚A)c(˚A)m(µB)dFe-As(˚A)
+SIESTA655.482 5.285 6.190 2.54 2.434
+VASP855.408 5.294 6.237 1.5 2.359
+WIEN2k86- - - 1.58 2.382
+CP-PAW 5.422 5.307 6.255 1.56 2.385
+previous GGA calculations, such as LiFeAs calculated
+with VASP85and WIEN2k86, there are large differences
+between our results and those of Refs. 65,87 as shown
+in Table I. Furthermore, in Table II, we show the com-
+parison between experimental and optimized structural
+data for BaFe 2As2, where we find that our optimized
+structure agrees with the experimental one better than
+that from Ref. 65. Since the electronic band structure
+close to the Fermi level is sensitive to the lattice
+structure20,88, the conclusion of Ref. 65 based on their
+optimized structures that there is no connection between
+Fermi surface nesting and magnetism is questionable.
+Therefore, we reinvestigate the nesting property of the
+Fermi surface.
+TABLE II: Comparison between the experimental structure
+of BaFe 2As2and the optimized structures from different DFT
+codes within GGA. The magnetic moment on each Fe is also
+shown. zAs=0.6495 is obtained from our CP-PAW calcula-
+tions.
+a(˚A)b(˚A)c(˚A)m(µB)dFe-As(˚A)
+Exp.785.616 5.571 12.943 0.87 2.392
+SIESTA655.756 5.590 13.04 2.78 2.436
+CP-PAW 5.693 5.666 13.008 1.98 2.396
+Fig. 1 presents the comparison of normalized q-
+dependent Pauli susceptibilities at fixed qz=πbetween
+arsenides and antimonides in 111, 1111 and 122 com-
+pounds. The normalization factors are the susceptibili-
+ties of the corresponding arsenide for each type of com-
+pounds at q0= (0,0,π). We fix qz=πbecause of the
+factthat spins onironarearrangedantiferromagnetically
+along the zdirection as observed in experiments77–81.
+However, we checked that the conclusion drawn below
+will not be changed if qz= 0 is fixed.
+We find that in all Fe-based families, the situation
+is similar. Two peaks around q0= (0,0,π) andqπ=
+(π,π,π) are detected in both arsenides and antimonides,
+and the peaks around ( π,π,π) are always stronger than
+those around (0 ,0,π), indicating that the instability to-
+wards stripe-type antiferromagnetic ordering dominates,
+which is consistent with our spin-polarized GGA calcu-
+lations. Most importantly, we find that, in contrast to
+Ref. 65, the Pauli susceptibilities are also enhanced to-
+gether with the magnetizations when As is substituted
+by Sb in LaOFeAs, LiFeAs and BaFe 2As2, which demon-4
+FIG. 2: (Color online) Fermi surface cuts for (a) LaOFeP
+and (b) LaOFeAs along different kz-planes, where kz=n/4,
+n∈ {0,1,2}in units of 2 π. The cyan (light gray) curves
+are the electron Fermi surfaces around ( π,π,k z) and the red
+(darkgray)curvestheholeFermisurfaces around(0 ,0,kz). In
+order to show the nesting properties, the hole Fermi surface s
+at (0,0,kz) are shown again, shifted by ( π,π). Here we use
+GGA for the DFT calculations.
+strates a connection between Fermi surface nesting and
+magnetism and consequently strongly suggests that the
+theory of Fermi-surface-nesting-driven magnetism is still
+valid.
+IV. COMPETITION OF INSTABILITIES IN
+PAULI SUSCEPTIBILITIES IN 1111
+COMPOUNDS
+TABLE III: Comparison of representative distances dFe−Pn
+in (˚A) where Pn=P, As, Sb and Bi in LaOFeP, LaOFeAs,
+LaOFeSb, and LaOFeBi between different GGA optimized
+structures and experimental structures, if available.
+Exp.92,93CP-PAW VASP89SIESTA65
+LaOFeP 2.289 2.264 2.232 -
+LaOFeAs 2.408 2.372 2.357 2.446
+LaOFeSb - 2.547 2.50 2.660
+LaOFeBi - 2.639 - -
+FIG. 3: (Color online) Static q-dependent Pauli suscepti-
+bilities at fixed qz=πfor (a) LaOFeP and (b) LaOFeAs.
+The corresponding values of the Pauli susceptibilities at
+q0= (0,0,π) in LaOFeP and LaOFeAs, respectively, are sub-
+tracted. On top, two-dimensional contour maps are shown.
+Here, GGA is used for the DFT calculations.
+TABLE IV: Comparison of representative distances dLa-Oin
+(˚A) in LaOFeP, LaOFeAs, LaOFeSb, and LaOFeBi between
+different GGA optimized structures and experimental struc-
+tures, if available.
+Exp. CP-PAW (ours) VASP (ref.) SIESTA65
+LaOFeP 2.350 2.344 2.349 -
+LaOFeAs 2.363 2.356 2.369 2.374
+LaOFeSb - 2.375 2.394 2.398
+LaOFeBi - 2.382 - -
+In what followswe concentrateon the 1111compounds
+and perform a comparative study among LaOFe Pn
+(Pn=P, As, Sb, Bi). In Table III and IV, we first present
+the comparisons of two representative atomic distances
+among different structures optimized within GGA and
+experimental structures, if available. Our results agree5
+well with the experimental ones.
+Fig. 2 shows the calculated Fermi surface cuts for
+LaOFeP and LaOFeAs on different kz-planes based on
+the experimental lattice structures. The figure is almost
+unchanged if we consider the optimized lattice struc-
+ture. The holeFermisurfacesaround(0 ,0,kz)areshifted
+by (π,π,0) to show the nesting properties. Shifting of
+(π,π,π) was also investigated, and we find that the nest-
+ing properties are nearly unchanged. From the figure,
+it is apparent that the Fermi surface nesting is even
+more perfect in LaOFeP than in LaOFeAs, indicating a
+strongertendencytostripe-typeantiferromagneticorder-
+ing in LaOFeP compared to LaOFeAs. However, experi-
+mentally, while a small magnetization is observed in un-
+doped LaOFeAs77, superconductivity rather than mag-
+netic order is detected in undoped LaOFeP90. These
+observations would indicate that Fermi surface nesting
+might not be connected to magnetization.
+In orderto quantify the Fermi surfacenesting, weshow
+in Fig. 3 the q-dependent Pauli susceptibilities at fixed
+qz=πfor LaOFeP and LaOFeAs with subtraction of the
+corresponding values at q0= (0,0,π). While a peak in
+LaOFePappearsrightat qπ= (π,π,π) indicatingalmost
+perfect nesting properties of the Fermi surfaces, peaks
+are situated close to qπ= (π,π,π) in LaOFeAs sug-
+gesting nearly nested Fermi surfaces, which is consistent
+with the Fermi surface cuts shown in Fig. 2. The most
+interesting finding in Fig. 3 is that the relative values
+χ(π,π,π)-χ(0,0,π) increase from LaOFeP to LaOFeAs
+irrespective of whether the Fermi surface nesting is per-
+fect or not. While the peak at ( π,π,π) favors stripe-
+type antiferromagnetic ordering, the one at (0 ,0,π) rep-
+resents a possible instability towards checkerboard-type
+antiferromagnetic ordering or A-type antiferromagnetic
+ordering where ferromagnetic layers are stacked antifer-
+romagnetically. The heights of these two peaks become
+closer in LaOFeP than in LaOFeAs, implying that com-
+petition between the above-mentioned two types of anti-
+ferromagneticstates becomesstrongerin LaOFePif ther-
+malorquantumfluctuationsaretakenintoaccount. Also
+spin-fluctuation mediated pairing of the superconducting
+state23,26,30coming from inter-band scattering around
+qπ= (π,π,π) takes part in the competition. Eventually,
+as the two types of antiferromagnetism strongly compete
+with each other, the additional superconducting state or-
+der emerges and opens a gap, removing the high instabil-
+ity at the Fermi level and lowering the total energy. This
+could be the scenario to explain why undoped LaOFeP
+is always nonmagnetic but shows superconductivity be-
+low 3.2 K at ambient pressure. As χ(π,π,π)−χ(0,0,π)
+increases beyond a critical value, the stripe-type antifer-
+romagnetic ordering prevails over the checkerboard-type
+one and the pairing. This scenario may apply to the
+low-temperature magnetic phase of LaOFeAs. Further-
+more, we have calculated the total energies of checker-
+board and stripe-type antiferromagnetic phases for both
+LaOFeP and LaOFeAs. We found that the stripe-type
+antiferromagnetic phases are the ground state in both
+FIG. 4: (Color online) Static q-dependent Pauli suscepti-
+bilities at fixed qz=πfor (a) LaOFeSb and (b) LaOFeBi.
+The corresponding values of the Pauli susceptibilities at
+q0= (0,0,π) in LaOFeSb and LaOFeBi, respectively, are sub-
+tracted. On top, two-dimensional contour maps are shown.
+In the DFT calculations GGA is used.
+cases and the energy difference between the two phases
+is smaller in LaOFeP than in LaOFeAs, which is consis-
+tent with the trend of χ(π,π,π)-χ(0,0,π).
+In Fig. 4, we display the q-dependent Pauli suscepti-
+bilities at fixed qz=πfor the hypothetical compounds
+LaOFeSb and LaOFeBi. The corresponding values of
+the Pauli susceptibilities at q0= (0,0,π) in LaOFeSb
+and LaOFeBi, respectively, are again subtracted. Note
+thatχ(π,π,π)−χ(0,0,π) in LaOFeSb and LaOFeBi is
+even larger than that in LaOFeAs (see Fig. 3), indicat-
+ing that the instability towardsstripe-type antiferromag-
+netic ordering could win the competition between dif-
+ferent instabilities in these two compounds. It is also
+interesting to note that the peak at q0= (0,0,π) be-
+comes flatter when we go from LaOFeP to LaOFeBi.
+While the flatness of the peak can be associated with
+a larger number of different magnetic structures lying6
+FIG. 5: (Color online) Static Pauli susceptibilities at q0=
+(0,0,π) andqπ= (π,π,π) for LaOFe PnwithPn=P, As, Sb,
+Bi. Here we use GGA for the DFT calculations.
+within a small energywindow, the strongerpeaks around
+qπ= (π,π,π) compared to q0= (0,0,π) in all the 1111
+compounds we studied makes other magnetic orderings
+besides the stripe-type antiferromagnetic one less favor-
+able. Combining the results from spin-polarized GGA
+(LSDA) calculations where magnetic moments on each
+iron are given as 2.2 (1.3) and 2.4 (1.8) µBin LaOFeSb
+and LaOFeBi, respectively, we predict that the ground
+states of these two compounds at ambient pressure with-
+out doping should show stripe-type antiferromagneticor-
+der although spin-polarized GGA (LSDA) calculations
+overestimate the magnetic moments.
+The increase of the magnetic moment from As to Sb to
+Bi can be understood from Fig. 5 where Pauli suscepti-
+bilities at q0= (0,0,π) andqπ= (π,π,π) for LaOFe Pn
+withPn=P, As, Sb, Bi are explicitly shown. While the
+increasing absolute value of χPn(π,π,π) asPnchanges
+from P to Bi is probably responsible for the increasing
+magnetic moment, the difference between χPn(π,π,π)
+andχPn(0,0,π) dominates the possible competition be-
+tween different ordered states, which again implies a
+strong relation between Fermi surface nesting and mag-
+netism. Due to the fact that LaOFeAs is not a super-
+conductor without doping or application of pressure, we
+argue that possible superconducting states in LaOFeSb
+and LaOFeBi can only occur under doping or application
+of pressure.
+In Fig. 6, we display the DOS for LaOFeAs, LaOFeSb
+and LaOFeBi calculated within spin-polarized GGA cal-
+culations. It is shown that in all three cases the DOS at
+the Fermi level remains finite, suggesting that undoped
+LaOFeSb and LaOFeBi are stripe-type antiferromagnetic
+metals at ambient pressure as is the case of LaOFeAs
+or other iron-based superconductors. Our results from
+spin-polarized GGA calculations cannot corroborate the
+arguments introduced in Ref. 91 where it is argued that
+LaOFeSb and LaOFeBi could be antiferromagnetic Mott
+insulators, though the tendency towards larger Fe mag--1 0 1015
+ LaOFeAs
+ LaOFeSb
+ LaOFeBi
+  DOS (states/eV/unit cell/spin)
+energy (eV) LaOFeP
+FIG. 6: (Color online) Total density of states (DOS) for
+LaOFeAs, LaOFeSb and LaOFeBi in the low-temperature or-
+thorhombic phase with stripe-type antiferromagnetic orde r
+calculated by spin-polarized GGA. Also shown for compari-
+son is the DOS for nonmagnetic LaOFeP calculated by GGA.
+Relatively high DOS at the Fermi level is present in LaOFeP,
+indicating the possible instability with respect to superc on-
+ductivity in the absence of magnetic order.
+netic moments from the As to Bi compounds is observed.
+Such a conclusion is also confirmed by our LSDA calcula-
+tions. InFig.6, theDOSforLaOFePisalsoshown. Since
+LaOFeP is nonmagnetic at low temperature, only the
+non-spinpolarized GGA result is presented. It is found
+that the DOS at the Fermi level remains relatively high,
+which indicates a possible instability with respect to su-
+perconductivity in the absence of magnetic order with
+the opening of a superconducting gap and the lifting of
+the degeneracy at the Fermi level.
+V. NEW SUPERCONDUCTOR CANDIDATES
+Experimentally, the highest recorded superconducting
+transition temperature has been observed in the 1111
+compounds. It is therefore tempting to find ways to pre-
+dictTcfor hypothetical 1111 compounds. One promis-
+ing route is to consider a phenomenological relation be-
+tweenTcand theχ(π,π,π)−χ(0,0,π) of the parent com-
+pound. As we know, with doping or application of pres-
+sure, the χ(π,π,π) instability is suppressed, resulting in
+the disappearanceof stripe-type antiferromagneticorder.
+However, strong inter-band scattering with a wave vec-
+tor around ( π,π,π) compared to intra-band scattering
+with a wave vector around (0 ,0,π) remains, which leads
+to a superconducting state. Therefore we argue that the
+largerthe relative value of χ(π,π,π)-χ(0,0,π) in the par-
+ent compound is, the stronger the antiferromagnetic spin
+fluctuation after suppression of magnetic order, and thus
+the higher the superconducting transition temperature
+will be.7
+FIG. 7: (Color online) Prediction of superconducting tran-
+sition temperatures Tcfrom a phenomenological relation be-
+tweenTcandχ(π,π,π)−χ(0,0,π) for the parent compounds
+of the hypothetical 1111 compounds LaOFeSb, LaOFeBi,
+ScOFeP and ScOFeAs. The phenomenological relation was
+determined by calculating χ(π,π,π)−χ(0,0,π) for several
+1111 compounds LaOFeP, LaOFeAs, CeOFeAs, NdOFeAs,
+and SmOFeAs by GGA where Tc’s and lattice structures are
+given experimentally. The χ(π,π,π)−χ(0,0,π) for LaOFeP
+and LaOFeAs calculated by GGA from DFT optimized struc-
+tures are also shown for comparison. It is found that the re-
+sultingχ(π,π,π)−χ(0,0,π) for LaOFeP and LaOFeAs based
+on our optimized structures is only slightly underestimate d
+compared to that calculated from experimental structures,
+which shows that the result dependsonly weakly on our struc-
+ture optimization.
+In Fig. 7, we plot χ(π,π,π)−χ(0,0,π) versus Tcfor
+the hypothetical 1111 compounds LaOFeSb, LaOFeBi,
+ScOFeP and ScOFeAs. The phenomenological relation
+between Tcandχ(π,π,π)−χ(0,0,π) is determined by
+first calculating χ(π,π,π)−χ(0,0,π) for several typ-
+ical 1111 compounds LaOFeP90,92, LaOFeAs1,93, Ce-
+OFeAs94,95, NdOFeAs96,97, and SmOFeAs98,99, whereTc
+and lattice structures are given experimentally, and then
+fitting the data by an exponential growth function of
+χ(π,π,π)−χ(0,0,π) = 2.86+0.28∗exp(Tc/18.14). With
+this relation, Tc for the compounds which have not yet
+been experimentally reported is predicted by optimizing
+the lattice structureand calculating χ(π,π,π)−χ(0,0,π)
+from DFT calculations. From Fig. 7, we find that the re-
+sultingχ(π,π,π)−χ(0,0,π) for LaOFeP and LaOFeAs
+based on our optimized structures is only slightly under-
+estimated compared to that calculated from experimen-
+tal structures, which shows that the result depends only
+weakly on our structure optimization. Among the four
+1111 compounds, this procedure shows that LaOFeSb
+can give the highest Tcaround 60 K which is above the
+highest recorded Tcof 55 K in SmOFeAs.
+An alternative procedure to predict Tcphenomenolog-
+ically is based on the fact that the physical properties
+FIG. 8: (Color online) Prediction of superconducting tran-
+sition temperatures Tcfrom a phenomenological relation
+between Tcand the ratio of atomic distances dFe-Feand
+dFe-Asfor the hypothetical 1111 compounds LaOFeSb,
+LaOFeBi, ScOFeP and ScOFeAs. A constant ratio of
+r0=d0,Fe-Fe/d0,Fe-Asfor distances d0,Fe-Fe,d0,Fe-Asin a per-
+fect FeAs 4tetrahedron is subtracted. The phenomenologi-
+cal relation is determined by taking into account the same
+compounds LaOFeP, LaOFeAs, CeOFeAs, NdOFeAs, and
+SmOFeAs as in Fig. 7 where Tcand lattice structures are
+given experimentally and doing a linear fit as in Ref. 95.
+The|dFe-Fe/dFe-As−r0|for LaOFeP, LaOFeAs, LaOFeSb,
+and LaOFeBi based on optimized structures within GGA and
+LDA are also shown for comparisons. Comparing the results
+for LaOFeP and LaOFeAs from GGA and LDA optimiza-
+tions with those from experiments, we find that the optimized
+structuresfrom GGA aremore consistent with theexperimen-
+tal one.
+of Fe-based superconductors strongly depend on the As
+position. Tiny shiftings of the As position away from
+or closer to the iron plane will significantly change the
+band structure around the Fermi level20,88. Therefore,
+similar to Refs. 95 and 100, we plot in Fig. 8 Tcversus
+the absolute value of the ratio between atomic distances
+dFe-FeanddFe-Aswhile subtracting r0=d0,Fe-Fe/d0,Fe-As
+where d 0,Fe-Feand d 0,Fe-Asdenote the distances in a
+perfect tetrahedron formed by four nearest neighbor As
+atoms surrounding one Fe atom. Similar to the first
+scheme, we determine the phenomenological relation be-
+tween|dFe-Fe/dFe-As−r0|andTcby taking into account
+the same compounds LaOFeP, LaOFeAs, CeOFeAs, Nd-
+OFeAs, and SmOFeAs as in the first scheme where Tc
+and lattice structures are given experimentally and do-
+ing a linear fit as in Ref. 95. With this relation, the Tcfor
+LaOFeSb, LaOFeBi, ScOFeP and ScOFeAs is predicted
+based on the optimized structure.
+Fig. 8 presents the results based on the structures op-
+timized within both GGA and LDA. Comparing the re-
+sults for LaOFeP and LaOFeAs from GGA and LDA op-
+timizations with those from experiments, we find that
+the optimized structures from GGA are more consis-8
+tent with the experimental one. According to the re-
+lation we fitted, LaOFeSb always gives the highest Tcof
+57.5(50.6)KforLDA(GGA)optimizations,respectively,
+among the four 1111 compounds we studied. Combining
+the two presented phenomenological prediction schemes,
+we clearly obtain that LaOFeSb would be upon doping
+or under pressurea good candidate for superconductivity
+with highest Tcand it would be very interesting to see it
+synthesized.
+VI. CONCLUSIONS
+In the present work we studied the physical proper-
+ties of LaOFe Pn, BaFe 2Pn2and LiFe PnwithPn=As
+and Sb. Our results support the validity of the itin-
+erant nature of magnetism in these compounds where
+magnetization is closely related to Fermi surface nest-
+ing. Furthermore, we concentrated on the 1111 com-
+pounds LaOFe PnwithPn=P, As, Sb, and Bi. We
+found that the increase of the magnetic moment in the
+undoped compounds with Pnvarying from P to Bi is
+due to the increasing instability of the Pauli suscep-
+tibility at qπ= (π,π,π) and the decreasing competi-
+tion to the instability at q0= (0,0,π). The supercon-
+ducting state appearing in undoped LaOFeP at ambi-
+ent pressure is ascribed to the strong competition be-
+tween the instability at qπ= (π,π,π) andq0= (0,0,π).
+Thus, together with the investigation of the DOS in thelow temperature phase, we argue that the hypotheti-
+cal compounds LaOFeSb and LaOFeBi are antiferromag-
+netic metals at ambient pressure without doping. The
+results for LaOFe Pnagain strongly imply that Fermi
+surface nesting plays a dominating role in the physical
+properties of the 1111 compounds. Finally we consider
+two phenomenological relations to predict the supercon-
+ducting transition temperature Tcfor the hypothetical
+1111 compounds and predict that LaOFeSb would be a
+possible candidate for a superconductor with a higher
+Tcthan presently recorded for the known Fe-based su-
+perconductors. Combining the fact that Fermi surface
+nesting dominates the physics in the 122 compounds,
+1111 compounds, 11 compounds, and 111 compounds,
+we argue that magnetism in iron-based superconductors
+is strongly influenced by their itinerant nature. However,
+fromourstudy, the localizedscenarioisnotruledoutand
+may also play an important role in the physics of iron-
+based superconductors. Furthermore, while in our study
+we emphasize the role of the states at the Fermi level and
+accordingly the nesting property of the Fermi surface on
+the itinerant nature of magnetism, the significant contri-
+butions to the finite momentofitinerant magnetism from
+the states in the vicinity of the Fermi level should not be
+ignored.
+Acknowledgments.- We would like to thank the
+Deutsche Forschungsgemeinschaft for financial support
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\ No newline at end of file
diff --git a/1001.0537.txt b/1001.0537.txt
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+arXiv:1001.0537v1  [astro-ph.SR]  4 Jan 2010Mon. Not. R. Astron. Soc. 000, 1–??(2009) Printed 29 October 2018 (MN L ATEX style file v2.2)
+Observations of the quiescent X-ray transients
+GRS1124–684 (=GUMus) and CenX–4 (=V822Cen)
+taken with ultracam on the VLT
+T.Shahbaz,1⋆V.S.Dhillon2, T.R.Marsh3, J.Casares1, C.Zurita1, P.A.Charles4
+1Instituto de Astrof´ ısica de Canarias, 38200 La Laguna, Ten erife, Spain
+2Department of Physics and Astronomy, University of Sheffield , Sheffield, S3 7RH, UK
+3Department of Physics, University of Warwick, Coventry CV4 7AL, England
+4South African Astronomical Observatory, P.O. Box 9, Observ atory, 7935, South Africa
+29 October 2018
+ABSTRACT
+We present high time-resolution multicolour optical observations of the quiescent X-
+ray transients GRS1124–684 (=GUMus) and CenX–4 (=V822Cen) o btained with
+ultracam . Superimposed on the secondary stars’ ellipsoidal modulation in bot h ob-
+jects are large flares on time-scales of 30–60min, as well as severa l distinct rapid flares
+on time-scales of a few minutes, most of which show further variabilit y and unresolved
+structure. Not significant quasi-periodic oscillations are observed and the power den-
+sity spectra of GRS1124–684and CenX–4 can be described by a pow er-law. From the
+colour-colour diagrams of the flare events, for GRS1124–684 we fi nd that the flares
+can be described by hydrogen gas with a density of NH∼1024nucleons cm−2, a tem-
+perature of ∼8000K and arising from a radius of ∼0.3R⊙. Finally we compile the
+values for the transition radius (the radius of the hot advection-d ominated accretion
+flow) estimated from quasi-periodic oscillations and/or breaks in the power density
+spectrum for a variety of X-ray transients in different X-ray stat es. As expected, we
+find a strong correlation between the bolometric luminosity and the t ransition radius.
+Key words: accretion, accretion disc – binaries: close – stars: individual: GRS11 24–
+684 (=GU,Mus) stars: individual: CenX–4 (=V822Cen)
+1 INTRODUCTION
+X-ray transients (XRTs) are a subset of low-mass X-ray
+binaries that display episodic, dramatic X-ray and optical
+outbursts, usually lasting for several months. More than
+70 percent of XRTs are thought to contain black holes
+(Charles & Coe 2006). The black hole X-ray transients are
+known to exhibit five distinct X-ray spectral states, dis-
+tinguished by the presence or absence of a soft black-
+body component at 1keV and the luminosity and spectral
+slope of emission at harder energies; these are known as
+the quiescent, low, intermediate, high and very high states
+(Tanaka & Shibazaki 1996). The quiescent and low states
+can mostly be explained with the advection dominated ac-
+cretion flow (ADAF) model (Narayan, McClintock & Yi
+1996; Esin, McClintock & Narayan 1997). In the context of
+⋆E-mail: tsh@ll.iac.es
+1Based on observations made at the European Southern Obser-
+vatory, Paranal, Chile (ESO program 075.D-0193)the ADAF model, properties similar to the low/hard state
+are expected for the quiescent state, as there is thought to
+be no distinction between the two except that the mass ac-
+cretion rate is much higher and the size of the ADAF region
+is smaller for the low/hard state.
+Similar to the transition between the low/hard and
+high/soft (thermal-dominant) state, where there is a recon -
+figuration of the accretion flow (Esin et al. 1997), there is
+alsoobservational evidenceforastatetransition between the
+low/hard and quiescent states, in that the quiescent state
+power-law appears softer thanin the low/hard state. In both
+thesestates, theADAFmodel predictsthattheinneredgeof
+the disc is truncated at some large radius, with the interior
+region filled by an ADAF. Strong evidence for such a trun-
+cated disc is provided by observations of XTEJ1118+480
+in the low/hard state during outburst (Hynes et al. 2000;
+McClintock et al. 2001; Esin et al. 2001; Chaty et al. 2003),
+wherethedischasaninnerradiusof >55Schwarzschildradii
+(Rsch) and a hot optically-thin plasma in the inner regions.
+c∝circlecopyrt2009 RAS2T. Shahbaz et al.
+Table 1. Log of the VLT+ULTRACAM observations.
+Object UT Date UT Start exp. time (s) No.of images Median seei ng (range)
+starting u′/g′/i′u′/g′/i′(′′)
+GRS1124–684 09/05/05 23:13 5.0/5.0/5.0 4005 1.5 (1.2–2.0)
+GRS1124–684 10/05/05 00:17 5.0/5.0/5.0 3469 1.7 (1.3–3.0)
+CenX–4 18/06/07 01:41 6.0/3.0/3.0 3296/6579/6579/ 0.6 (0. 45 –1.2)
+In quiescence, the ADAF model predicts that the inner disc
+edge will move outward to larger radii (Esin et al. 1997).
+Variability is one of the ubiquitous characteristics of
+accreting black holes extending across all wavelengths. Hi gh
+time resolution optical photometry of XRTs in quiescence
+haveshownvariability withapowerdensityspectrum(PDS)
+very similar to those of low/hard-state XRTs, i.e. a power-
+law band-limited spectrum (Hynes et al. 2003). The simi-
+larity of the PDS suggests that the optical variability coul d
+have a similar origin and might be associated with the cen-
+tral X-raysource. Furthermorethepresenceofoptical/X-r ay
+correlations (Hynes et al. 2004) requires some association
+between the optical and the source of the X-rays. The ori-
+gin of the break in the band-limited spectrum is not known,
+but it is plausible to expect that it approximately scales
+with the size of the inner region. A low break frequency
+in quiescence would then be expected, since the advective
+region is expected to be larger. Here we report on our high-
+timeresolution multi-colouroptical observations oftheb lack
+hole X-ray transient GRS1124–684 and the neutron star X-
+ray transient CenX–4 in quiescence. These observations are
+part of a ongoing campaign with ultracam to obtain high-
+timeresolutionphotometryofX-raybinaries (Shahbaz et al .
+2003; Shahbaz et al. 2005).
+2 OBSERVATIONS AND DATA REDUCTION
+Multi-colour photometric observations of GRS1124–684 and
+CenX–4 were obtained with ultracam on the 8.2-m MELI-
+PAL unit of the Very Large Telescope (VLT) at Paranal,
+Chile. The GRS1124–684 data were taken during the nights
+starting 2005 May 9 to 10, whereas the CenX–4 data were
+taken on the nights starting 2007 June 18 (see Table 1 for a
+log of the observations). ultracam is an ultra-fast, triple-
+beam CCD camera, where the light is split into three broad-
+band colours (blue, green and red) by two dichroics. The de-
+tectors are back-illuminated, thinned, E2V frame-transfe r
+1024×1024 CCDs with a pixel scale of 0.15arcsecs/pixel.
+Due to the architecture of the CCDs the dead-time is essen-
+tially zero (for further details see Dhillon et al. 2007). Ou r
+observations were taken using the Sloan u′,g′andi′filters
+with effective wavelengths of 3550 ˚A, 4750 ˚A and 7650 ˚A re-
+spectively.
+Theultracam pipeline reduction procedureswere used
+to debias and flatfield the data. The same pipeline was
+also used to obtain lightcurves for GRS1124–684, CenX–4
+and several comparison stars by extracting the counts us-
+ing aperture photometry. The most reliable results were ob-
+tained using a variable aperture which scaled with the see-
+ing. The count ratio of the target with respect to a bright
+local standard (which has similar colour to our target) wasthen determined. The magnitude of the target was then ob-
+tained using the calibrated magnitude of the local standard .
+As a check of the photometry and systematics in the re-
+duction procedure, we also extracted lightcurves of a faint
+comparison star similar in brightness to the target.
+The mean magnitude of GRS1124–684 was u′=21.49,
+g′=20.65 and i′=19.92 and we estimate the photometric ac-
+curacyperexposuretobe16.4, 2.8and2.0percentfor the u′,
+g′andi′band respectively. For CenX–4, the mean magni-
+tude was u′=18.69,g′=18.07 and i′=17.27 and we estimate
+the photometric accuracy per exposure to be 5.9, 0.9 and 0.6
+percent for the u′,g′andi′band respectively. The uncer-
+tainties only reflect the 1- σstatistical errors in the relative
+photometry.
+3 SHORT-TERM VARIABILITY
+The optical lightcurves of GRS1124–684 and CenX–4 (Fig-
+ure 1) show short-term variability/flares superimposed on
+the secondary star’s weak ellipsoidal modulation. So if we
+want to determine the flux of the flares, these “steady” con-
+tributions must first be removed from the lightcurves. In or-
+der to isolate the short-term variability in each band we firs t
+de-reddened the observed magnitudes using a colour excess
+of E(B-V)=0.29 (Cheng et al. 1992) for GRS1124–684 and
+E(B-V)=0.10 for CenX–4 (Blair et al. 1984) and adopting
+the ratio AV/E(B−V)=3.1 (Cardelli, Clayton, & Mathis
+1989), and then converted the Sloan AB magnitudes to flux
+density (Fukugita et al. 1996). We then fitted a double si-
+nusoid to the lower-envelope of the lightcurve with periods
+equal to the orbital period [ Porb=0.4320604d for GRS1124–
+684; Casares et al. (1997) and Porb=0.6290496d for CenX–
+4; Torres et al. (2002)] and its first harmonic, where the
+phasing was fixed and the normalizations were allowed to
+float free. We rejected points more than 3- σabove the fit,
+then re-fitted, repeating the procedure until no new points
+were rejected. For GRS1124–684, contrary to Hynes et al.
+(2003) we do not find evidence of any phase shift in the
+data, because one can see from Figure 1 that orbital phase
+0.0 coincides with the minimum light.
+Figures 2 and 3 show the resulting lightcurves (af-
+ter subtracting the secondary star’s ellipsoidal modulati on),
+where large flare events lasting 30-60min and 10–30min
+respectively are clearly seen. For GRS1124–684 what is
+also noticeable are numerous rapid flare events (Figure 5),
+that typically last 5min or less and which are not resolved.
+For both targets there are numerous short-term flares with
+rise/decay times of ∼3–5min, superimposed on larger flares.
+For GRS1124–684 the large flares have time-scales of 30–
+60min, whereas for CenX–4 they typically have time-scales
+c∝circlecopyrt2009 RAS, MNRAS 000, 1–??Observationsofthe quiescentX-raytransients GRS1124–68 4(=GUMus) andCenX–4 (=V822Cen) 3
+of 10–30min. The parameters of the flares as defined by
+Zurita et al. (2002) are given in Table 2.
+A look at the lightcurves of GRS1124–684 shows short-
+term flare events which seem to occur regularly. In Figure 4
+we show tick marks with representative intervals of 6.3min.
+As one can see, some of the flare events seem to recur on a
+regular basis and the period appears to be stable on short
+time-scales. It is clear, however, that these events are not
+strictly periodic, and not strong enough to show up as QPOs
+in a log-log plot of the PDS (see section 4).
+4 THE POWER DENSITY SPECTRUM
+The flare lightcurves (i.e. the after subtracting the sec-
+ondary star’s ellipsoidal modulation from the de-reddened
+lightcurves) of GRS1124–684 and CenX–4 show features
+which seem to be periodic. To see if this is true we com-
+puted the power density spectrum (PDS) of the data. To
+compute the PDS we detrended the data using the dou-
+ble sinusoid fit described in the previous section and then
+added the mean flux level of the data. Although the ul-
+tracam sampling is perfectly uniform over short periods of
+time (tens of minutes), we use the Lomb-Scargle method to
+compute the periodograms (Press et al. 1992) with the same
+normalization method as is commonly used in X-ray astron-
+omy, where the power is normalized to the fractional root
+mean amplitude squared per hertz (van der Klis 1994). We
+used the constraints imposed by the Nyquist frequency and
+the typical duration of each observation to limit the range
+of different frequencies searched.
+The lightcurves of GRS1124–684 and CenX–4 shows
+features that are present in all three bands (see Figure 6). I n
+order to determine the significance of possible peaks above
+the red-noise level, we used a Monte Carlo simulation simi-
+lar to Shahbaz et al. (2005). We generated lightcurves with
+exactly the same sampling and integration times as the real
+data. We started with a model noise lightcurve generated
+usingthe method of Timmer & Koenig (1995) with apower-
+law index as determined from the PDS of the observed data,
+and then addedGaussian noise using the errors derived from
+the photometry. We computed 5000 simulated lightcurves
+and then calculated the 95% confidence level at each fre-
+quency taking into account a realistic number of indepen-
+dent trial (Vaughan 2005). For GRS1124–684 and CenX–4
+the 95% confidence level contour rules out any QPOs and
+indeed it would seem that the PDS is dominated by red-
+noise. ThusthePDSoftheflarelightcurvefor GRS1124–684
+and CenX–4 in each band can be described by a power-law
+model (see Figure 6).
+In Figure 6 the errors in each frequency bin are deter-
+mined from the standard deviation of the points within each
+bin and the white noise level was subtracted by fitting the
+highest frequencies with a white-noise (constant) plus red -
+noise (power-law) model. The power-law index of the fit is
+also given.
+5 THE ORIGIN OF THE FLARES
+In an attempt to interpret the broad-band spectral proper-
+ties of the flare, we compared the observed colours with theprediction for three different emission mechanisms, namely
+a blackbody, an optically-thin layer of hydrogen and syn-
+chrotron emission. We computed the given emission spec-
+trum and then calculate the expected flux density ratios
+fu′/fg′andfg′/fi′using the synthetic photometry package
+SYNPHOT (iraf/stsdas ). Given the intrinsic model flux
+we can then determine the corresponding radius of the re-
+gion that produces the observed de-reddened flux at a given
+distance.
+Figure 7 show the data for the flare events and the
+expected results for different emission models. For stellar
+flares the power-law index of the electron energy distribu-
+tion∼–2 (Crosby, Aschwanden & Schmitt 1993), which cor-
+respondstoaspectral energydistributionwithindex α=–0.5
+(Fν∝να). However, the spectral energy distribution index
+observed in V404Cyg ( α∼–2.0) implies a much steeper
+index for the electron energy distribution (Shahbaz et al.
+2003), which may not be completely implausible given the
+extreme conditions around a black hole. Therefore in Figure
+7 we show the power-law model for αranging from -2 to 2.
+We also show the very unlikely blackbody case, where the
+flares are due to blackbody radiation from a heated region of
+the disc’s photosphere. The most likely model for a thermal
+flare is emission from an optically thin layer of recombin-
+ing hydrogen, which is essentially the mechanism generally
+accepted for solar flares. We therefore determined the con-
+tinuum emission spectrum of an LTE slab of hydrogen for
+different baryon densities, NH= 1021−1024nucleons cm−2,
+and calculated the expected flux ratios.
+For CenX–4 the amplitudes of the flares are small and
+difficult to define. All we can say is that they arise from
+an optically thin region with NH∼1021−25nucleons cm−2,
+a temperature of ∼8,300K and a radius of ∼0.06R ⊙(as-
+suming a distance of 1.2kpc; Chevalier et al. 1989). For
+GRS1124–684 we can comment more because the flares
+are well defined; Figure 7 shows the data for the best re-
+solved small and large flares for GRS1124–684. The flare
+events can be described by hydrogen gas with a density of
+NH∼1024nucleons cm−2and a temperature of ∼8000K,
+which corresponds to a radius of ∼0.3R⊙(assuming a dis-
+tance of 5.1kpc; Gelino, Harrison, & McNamara 2001).
+6 DISCUSSION
+6.1 The origin of the flares
+We can compare the flares in GRS1124–684 to the flare
+properties determined in other quiescent black hole XRTs.
+The large flares in V404Cyg are consistent with optically
+thin gas with a temperature of ∼8000K arising from a
+region with an equivalent blackbody radius of at least
+2R⊙(Shahbaz et al. 2003). One should regard the equiv-
+alent radius estimate as a lower limit, because the emis-
+sion mechanism is unlikely to be blackbody. Similarly
+for XTEJ1118+480, the short-term flares have a black-
+body temperature of ∼3500K and an equivalent radius of
+∼0.10R ⊙. By isolating the spectrum of the flare event in
+A0620–00, we found that it could be represented by opti-
+cally thin gas of hydrogen with radius 0.04R ⊙and tem-
+perature 10,000–14,000K (Shahbaz et al. 2005), which are
+consistent with the bright spot area and temperature. The
+c∝circlecopyrt2009 RAS, MNRAS 000, 1–??4T. Shahbaz et al.
+Table 2. Properties of the GRS1124–684 and CenX–4 flares. vobsis the spectroscopic veiling, v′
+dis the contribution to the non-variable
+disc light, ¯ zfandσzare the mean flare flux and its standard deviation, respective ly.σ∗
+z=σz/v′
+dandηis the fraction of the average
+veiling due to the flares (Zurita et al. 2002).
+Target Band vobsv′
+d¯zfσzσ∗
+zη
+GRS1124–684 u′94% 0.92 0.387 0.25 0.27 0.27
+g′64% 0.57 0.194 0.16 0.28 0.25
+i′15% 0.04 0.133 0.11 2.8 0.78
+CenX–4 u′60% 0.48 0.312 0.13 0.27 0.39
+g′20% 0.78 0.182 0.12 0.15 0.19
+i′5% 0.70 0.042 0.005 0.007 0.06
+Balmer line flux and variations in A0620–00 suggests that
+there are two emitting regions, the accretion disc and the ac -
+cretion stream/disc impact region. The persistent emissio n
+is optically thin and during the flare events there is either
+an increase in temperature or the emission is more optically
+thick than the continuum. For GRS1124–684 we find that
+the flare events reach a maximum radius of ∼0.3R⊙, much
+larger than what is observed in A0620–00 (Shahbaz et al.
+2005), XTEJ1118+480 (Shahbaz et al. 2004) and CenX–4
+(section 5).
+The flares GRS1124–684 and CenX–4 arise from vari-
+ous regions in the accretion disc that in total occupy 3.4%
+and 0.3% of the disc’s area respectively. Thus could be
+from the hotspot, but the H αDoppler maps of GRS1124–
+684 and CenX–4 do not show any evidence of a hotspot
+(Casares et al. 1997; Torres et al. 2002). However, it should
+be noted that the optical state of quiescent transients
+are known to vary significantly from epoch to epoch
+(Cantrell et al. 2008).
+6.2 The transition radius
+Narayan et al. (1997) have shown that the X-ray obser-
+vations of quiescent XRTs can be explained by a two-
+component accretion flow model. The geometry of the flow
+consists of an hot inner ADAF that extends from the black
+hole horizon to a transition radius rtrand a thin accretion
+disc that extends from rtrto the outer edge of the accre-
+tion disc. An ADAF has turbulent gas at all radii, with a
+variety of time-scales, ranging from a slow time-scale at th e
+transition radius down to nearly the free-fall time close to
+the compact object. In principle, interactions between the
+hot inner ADAF and the cool, outer thin disc, at or near
+the transition radius, can be a source of optical variabil-
+ity, due to synchrotron emission by the hot electrons in the
+ADAF(Esin et al. 1997). For an ADAFthe variability could
+be quasi-periodic and would have a characteristic time-sca le
+given by a multiple of the Keplerian rotation period at rtr.
+The ADAF model also predicts that rtrshould get larger as
+the inner disc evaporates during the decline from outburst.
+To see if there is any observational evidence for this,
+we have compiled the values for rtrestimated from spec-
+tral energy distribution (SED) fits, QPOs and/or breaks in
+the PDS for a variety of XRTs in different X-ray states
+(see Table 3). For J1118+480 during outburst there are
+several estimates for rtr. The low/hard state X-ray PDSof J1118+480 (April 2000) shows a low-frequency break
+fbreakat 23mHz and a QPO at ∼80mHz (Hynes et al.
+2003), which is approximately consistent with the relation
+observed in other sources (Wijnands & van der Klis 1999),
+most likely related toan instability inthe accretion flowth at
+modulates the accretion rate. The QPO frequency is most
+likely related to rtr, which can be a source of quasi-periodic
+variability and rtrcan be estimated assuming the QPO fre-
+quency is the Keplerian rotation period at rtr. The observed
+QPO frequency corresponds to rtr=670 (tK= 2πR/vK=
+2π(GM/R3)−1/2∼(M/10M⊙)(r/100)3/2s, where Ris the
+absolute radius and ris in units of R sch).
+For a cellular-automaton ADAF disc
+(Takeuchi & Mineshige 1997) , the break-frequency is
+determined by the maximum peak intensity of the X-ray
+shots which is on the order of the size of the advection-
+dominated region, and corresponds to the inverse of the
+free-fall time-scale of the largest avalanches (Takeuchi,
+Mineshige & Negoro 1995) at rtr. However, it should be
+noted that the break frequency depends not only on the
+size of the ADAF region but also on the propagation speed
+of the perturbation (Mineshige priv. comm.). Since the per-
+turbation velocity should be less than the free-fall veloci ty,
+the free-fall velocity gives an upper limit to the size of the
+ADAF region. Thus equation 13 of Takeuchi et al. (1995)
+gives an upper limit to rtr(rtr<103.2(fbreakMX)−2/3).
+Thus the break frequency observed in J1118+80 during
+outburst corresponds to rtr<5250. The SED fits give
+rtr∼350, which is similar to value determined from the
+QPO to within a factor of 2; note that the variability/QPO
+could have a characteristic time-scale given by a multiple o f
+the Keplerian rotation period at rtr.
+Figure 8 shows rtrversus the bolometric X-ray luminos-
+ity (Campana & Stella 2000) for the sources listed in Table
+3. One can see that there does exist a correlation (the corre-
+lation coefficient is -0.96) with a slope index of ∼-3.0. How-
+ever, more observations of rtrduring different X-ray states
+are required.
+ACKNOWLEDGMENTS
+TS, JC and CZ acknowledge support from the Span-
+ish Ministry of Science and Technology under the
+grant AYA2007–66887. Partially funded by the Span-
+ish MEC under the Consolider-Ingenio 2010 Program
+c∝circlecopyrt2009 RAS, MNRAS 000, 1–??Observationsofthe quiescentX-raytransients GRS1124–68 4(=GUMus) andCenX–4 (=V822Cen) 5
+Table 3. The transition radius for various systems
+Object State Method log[ Rtr/RD] log[LX/LEDD] Reference
+GUMus ultrasoft SED -2.92 -0.16 Misra (1999)
+J1118+480 low/hard SED -1.91 -2.79 Chaty et al. (2003)
+J1118+480 low/hard QPO -1.72 -2.79 Hynes et al. (2003)
+J1118+480 low/hard PDS break -0.77 -2.79 Hynes et al. (2003)
+J1655–40 quiescence Delay -1.51 -5.25 Hameury et al. (1997)
+V404Cyg quiescence QPO -1.29 -4.89 Shahbaz et al. (2003)
+A0620–00 quiescence PDS break -0.05 -7.16 Hynes et al. (2003 )
+J1118+480 quiescence PDS break -0.07 -8.43 Shahbaz et al. (2 005)
+grant CSD2006-00070: “First Science with the GTC”
+(http://www.iac.es/consolider-ingenio-gtc/). ULTRACA M
+is supported by STFC grant PP/D002370/1.
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+Figure 1. The de-reddened u′,g′andi′-band lightcurves of GRS1124–684 and CenX–4 phase-folded u sing the ephemeris given in
+Casares et al. (1997) and Torres et al. (2002) respectively; orbital phase 0.0 is defined as inferior conjunction of the se condary star. The
+solid line is the fitted ellipsoidal modulation. At the botto m of each panel we also show the lightcurve of a comparison sta r of similar
+magnitude to the target, offset vertically for clarity.
+c∝circlecopyrt2009 RAS, MNRAS 000, 1–??Observationsofthe quiescentX-raytransients GRS1124–68 4(=GUMus) andCenX–4 (=V822Cen) 7
+Figure 2. Top: The GRS1124–684 flare de-reddened flux density u′(top panel), g′(middle panel)and i′-band (bottom panel) lightcurves
+obtained by subtracting a fit to the lower-envelope of the lig htcurves shown in Figure 1. The uncertainties in the u′,g′andi′lightcurves
+are 4.3×10−3mJy, 1.2 ×10−4mJy and 2.0 ×10−3mJy respectively. Bottom: the flux ratio lightcurves binned to a time resolution of 150s
+for clarity.
+c∝circlecopyrt2009 RAS, MNRAS 000, 1–??8T. Shahbaz et al.
+−1 0 1 2 3 4 5fu x 0.01 (mJy)
+−0.5 0 0.5 1 1.5fg x 0.01 (mJy)
+0.6 0.65 0.7 0.75−1 0 1 2fi x 0.01 (mJy)
+Figure 3. The top three panels show the CenX–4 flare de-reddened flux den sityu′g′andi′-band lightcurves obtained by subtracting
+a fit to the lower-envelope of the lightcurves shown in Figure 1. The uncertainties in the u′,g′andi′lightcurves are 1.2 ×10−3mJy,
+2.7×10−3mJy and 7.0 ×10−3mJy respectively. The bottom two panels show the flux ratio li ghtcurves binned to a time resolution of
+135s for clarity.
+c∝circlecopyrt2009 RAS, MNRAS 000, 1–??Observationsofthe quiescentX-raytransients GRS1124–68 4(=GUMus) andCenX–4 (=V822Cen) 9
+Figure 4. A close-up of the g′-band lightcurve of GRS1124–684. Short-term flare events se em to be periodic, but only over a few cycles.
+Vertical tick marks are shown with a representative interva l of 6.3min.
+0 5 10 15 200 2 4 6fi x 0.01 (mJy)
+40 45 50 55 609 May 2005
+125 130 135 140 145
+170 175 180 185 1900 2 4 6fi x 0.01 (mJy)
+190 195 200 205 210 235 240 245 250 255
+270 275 280 285 2900 2 4 6
+Time (mins)fi x 0.01 (mJy)
+290 295 300 305 310
+Time (mins)310 315 320 325 330
+Time (mins)40 45 50 55 600 2 4 6
+60 65 70 75 8010 May 2005
+140 145 150 155 1600 2 4 6
+170 175 180 185 190
+190 195 200 205 2100 2 4 6
+Time (mins)230 235 240 245 250
+Time (mins)
+Figure 5. Detailed plot of some of the individual flares in the i′-band lightcurve of GRS1124–684. Note that numerous flare ev ents
+which last for a few minutes are not resolved. The solid point in the top left panels marks the typical uncertainty in the da ta.
+c∝circlecopyrt2009 RAS, MNRAS 000, 1–??10T. Shahbaz et al.
+Figure 6. From top to bottom: the u′,g′andi′PDS of the flare lightcurves of GRS1124–684 taken on 9 and 10 Ma y 2005 (left) and
+CenX–4 taken on 18 June 2007 (right). The dashed line is a powe r–law fit. The slope of the power-law fit to the PDS is indicated in
+each panel.
+c∝circlecopyrt2009 RAS, MNRAS 000, 1–??Observationsofthe quiescentX-raytransients GRS1124–68 4(=GUMus) andCenX–4 (=V822Cen) 11
+Figure 7. Colour-Colour diagram for the clearest small and large flare events in GRS1124–684 and the large flare events in CenX–4.
+The panels labelled ”ALL” show all the data for that particul ar night, where the error bars have been removed for clarity. The dashed
+lines show optically-thin hydrogen slab models for differen t column densities and the open squares shows the temperatur e in 1000K
+units. The solid line shows a power-law model ( Fν∝να) with indices α=-0.5 and -1.5 marked as open circles. The dot-dashed line is a
+blackbody model where the asterisk shows the colour for a tem perature of 6500K.
+c∝circlecopyrt2009 RAS, MNRAS 000, 1–??12T. Shahbaz et al.
+Figure 8. The transition radius (accretion disc units RD) versus bolometric luminosity (Eddington units LEDD) for a variety of X-ray
+transients in different X-ray states. The filled circles are f or systems in quiescence and the crosses are for systems in th e low/hard
+state or in outburst. The rtrvalues have been determined from QPOs (crosses), breaks in t he PDS (upper limit), fits to the spec-
+tral energy distribution (squares) or delays in the optical /X-ray outburst (stars): J1118+480 (Chaty et al. 2003; Shah baz et al. 2005);
+A0620–00 (Hynes et al. 2003); J1655–40 (Hameury et al. 1997) ; V404Cyg (Shahbaz et al. 2003); GRS1124–684 (Misra 1999). T he X-ray
+luminosities are taken from Campana & Stella (2000) and the o rbital parameters from Charles & Coe (2006).
+c∝circlecopyrt2009 RAS, MNRAS 000, 1–??
\ No newline at end of file
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+arXiv:1001.0538v1  [astro-ph.HE]  4 Jan 2010Short Gamma Ray Bursts: marking
+the birth of black holes from coalescing
+compact binaries
+Davide Lazzati1and Rosalba Perna1
+1 Introduction
+As soon as the catalog of gamma-ray bursts (GRBs) detected by t he BATSE
+(Burst And Transient Source Experiment) had enough events to a llow a sta-
+tistical study, it was discovered that GRB light curves could be sepa rated in
+two families [44]. Long GRBs are characterized by a duration of more t han 2
+seconds and a somewhat soft spectrum. Short GRBs, on the othe r hand, are
+characterized by a duration of less than 2 seconds and a harder sp ectrum.
+Not much more could be said in the BATSE era, due to the lack of precis e
+localizationsandimpossibilityoflongwavelengthfollow-upthatplagued both
+the long and short GRB populations. With the launch of the Italian-Du tch
+satelliteBeppoSAX,thesituationforlongGRBschangeddramatically .X-ray,
+optical and radio afterglows were discovered [16, 87, 82] to follow t he prompt
+phase. Spectroscopyrevealedthat the GRBs lie at cosmologicaldis tances and
+that involve explosion energies similar to core collapse supernovae [54 , 45].
+Evidenceofbeamingandassociationtomassivestarsemerged[72,7 7]leading
+to the now widely accepted scenario of long GRBs as collimated relativis tic
+outflowsassociatedtoTypeIb/csupernovaexplosions[79,37]. Un fortunately
+BeppoSAX wasnon optimally designed to detect shortGRBs and none of the
+above information was available for the short bursts that remained elusive
+and mysterious. The only advance came from the discovery that sh ort GRBs
+also have longer wavelength emission on longer timescale [46]. Such disc overy
+was however made on a stacked light curve from past events, and d id not
+allow for any follow-up observation. A general consensus wasreac hed in those
+years that short GRBs could be associated to the merger of compa ct binary
+systems [21], based on a theoretical desire more than on any robus t evidence.
+JILA, University of Colorado, 440 UCB, Boulder, CO 80309-04 40, USA
+lazzati,rosalba@colorado.edu
+12 Davide Lazzati and Rosalba Perna
+More recently, thanks to the HETE-2 satellite and to Swift, afterg low ob-
+servations have been performed also for the class of short GRBs, measuring
+their redshift and energetics, and finally giving some observational corrob-
+oration to the idea that they originate from binary mergers [88, 28]. Not
+everything has been clarified, though, the main problem being now th e one
+of defining what is a long and what is a short bursts, since the two clas ses
+seem to have a gray area between them with bursts sharing a comple x set
+of properties. In this review, we critically present the new discover ies made
+with HETE-2 and Swift, the theoretical advances that were made p ossible
+by those discoveries and the still debated issues and future persp ectives. This
+paper is divided in two main sections, the first observational and the second
+theoretical.
+2 Observations
+2.1 Pre-Swift era
+Observations performed in the pre-Swift (and HETE-2) era are ma inly those
+performed with BATSE and Konus. Figure 1 shows a histogram of the T90
+distribution of 2041 GRBs detected by BATSE. The T90is the time interval
+during which the GRB emits 90 per cent of the total fluence. The solid and
+dashed lines show the Gaussian fit to the distribution for the short a nd long
+bursts, respectively. Even though the bimodality is clear, it is also cle ar that
+the two populations are not entirely separated, since a considerab le tail of
+short burst population (about 25 per cent) extends beyond T90>2 s, the
+traditional dividing line.
+Additional separation between the two classes is provided by the sp ectral
+analysis of the light curves. Figure 2 shows in greytones the two dime nsional
+distribution of BATSE GRBs in the hardness-duration plane. The har dness
+is defined as the ratio of counts in the high energy channels (3 and 4) over
+the counts in the low energy channels (1 and 2). Even though short bursts
+appear to have a systematically harder spectrum, the two populat ion have a
+sizable overlap.
+The origin of the spectral difference has been analyzed in detail by
+Ghirlanda et al. [29]. They performed spectroscopy of a sample of 36 bright
+short bursts comparing the results of several spectral models. They conclude
+that the spectra of short bursts are successfully fit by a single po wer-law
+with an exponential high energy cut-off. They also find that short G RBs
+have harder spectra due to steeper low energy slopes of the powe r-law rather
+than due to a larger perk frequency. It is worth reminding that long GRBs
+are usually fit wit a smoothly broken power-law model or Band functio n [3].Title Suppressed Due to Excessive Length 3
+Fig. 1 Distribution of the durations of 2041 BATSE GRBs. Gaussian fi ts for the short
+(solid line) and long (dashed line) populations are overlai d.
+Nakar & Piran [56] analyzed the light curves of a sample of short GRBs
+from BATSE. They find that, when high resolution data are available, short
+GRB light curves can be resolved into the superposition of many pulse s, with
+statistical properties analogous to those of the long GRBs. This ma y indicate
+that the same dynamical and dissipation processes are powering th e light
+curves, even though such processes are still far from being unde rstood.
+The quest for short GRB afterglows went on for all the BATSE and B ep-
+poSAX era with very limited results. No afterglow of an individual shor t
+GRB was ever found. Lazzati et al. [46] stacked the background s ubtracted
+light curve of the 76 brightest short BATSE GRBs looking for an evide nce of
+afterglow in the hard X-rays. They found (see Fig. 3) that there is indeed an
+excess soft component following the prompt emission of short GRBs lasting
+approximately 100 seconds. They showed that this component is co nsistent
+with being of afterglow origin. As we will see below, it later emerged fro m4 Davide Lazzati and Rosalba Perna
+Fig. 2Hardness-duration distribution of BATSE GRBs.
+Swift observations that this component is more likely residual activit y from
+the central engine.
+2.2 The Swift era
+Many of the riddles of short GRB astrophysics have been solved in th e Swift
+era, with a noticeable contribution from the HETE-2 satellite. Due to the
+characteristics of the BAT detector, Swift observations did not in crease our
+understanding of the prompt emission.
+In the spring and summer of 2005, after years of struggle, after glow of
+short GRBs were finally detected in X-rays, optical and radio wavele ngths
+[28, 88, 14, 23, 38, 17, 6], bringing the short bursts in the afterglo w era.
+2.2.1 Afterglows
+At first sight, afterglows of short GRBs are similar to the afterglow s of long
+GRBs.Theyarefainter,butqualitativelyanalogous.Figure4shows theX-ray
+afterglow of GRB 050724 as observed by the Swift XRT [13]. The afte rglowTitle Suppressed Due to Excessive Length 5
+Fig. 3Light curve of the excess emission found by Lazzati et al, ??in the composite light
+curve of 76 bright BATSE short GRBs. The dashed line is a best fi t afterglow model.
+shows an initially very bright phase, followed by a sharp decline, a poss ible
+t−1power-law decay and a flare at late times. The afterglow is overall ve ry
+faint. This is supposed to be due to a combination of an isotropic equiv alent
+energy smaller than that of long bursts and of a low density interste llar
+medium, down to n∼10−5cm−3[58]. Such low densities are thought to
+be an indication of the binary merger origin of short bursts, since ar e of the
+order of magnitude of what expected in the intergalactic medium.
+The initial bright phase is likely the one detected on BATSE data [46]
+and initially interpreted as the regular afterglow. It is now believed to be
+a sign of continued activity of the central engine (see below). X-ra y flares
+are also common. Differently from the X-ray flares detected on top of long
+GRB afterglows, the flares so far detected on long short GRBs hav e long
+time scale ( δt/t∼1) [49]. Their origin is not clear, a possible detection of
+rapid variation was reported for GRB 050709 [23], but is inconclusive. Since
+rapid variations are associate to late time activity of the engine [49], t heir6 Davide Lazzati and Rosalba Perna
+confirmation would be of great relevance for our understanding of the short
+GRB engine physics.
+Fig. 4Swift X-ray afterglow of GRB 050724 [13]. The dashed line sho ws at−1power-law
+for comparison purposes.
+It is still debated whether short GRB afterglows show conclusively t he
+presence of beaming of the short GRB fireballs. The question is relev ant since
+the total energyof the explosion depends on the beaming factor. In few cases,
+a jet break has been claimed. A good example is that of GRB 051221 [78 ]
+whereasimultaneousbreakinthe opticalandX-raylightcurveswas detected
+approximately 5 days after the explosion. This is however an isolated case
+and in most case only an lower limit to the beaming can be obtained [35]. It
+seems fairto conclude that, evenif beamed, shortGRBs areless be amed than
+long GRBs, for which opening angles of few degrees are routinely mea sured
+[30]. This observations supports the idea that short GRBs are asso ciated to
+the merging of compact objects, for which hydrodynamical collimat ion has a
+much lesser role than in massive stars [47].Title Suppressed Due to Excessive Length 7
+Intensive searches for a supernova component have been perfo rmed in sev-
+eral short GRBs. Due to their moderate redshift, the searches a re very sensi-
+tive, being able to detect the presence of a supernova bump even if the super-
+nova was 10 times fainter than the faintest observed supernova [2 3, 9, 17, 78].
+The lack of any detection strongly favors a different progenitor fo r short and
+long GRBs.
+2.2.2 Host galaxies
+A debated and important aspect of short GRB research has been t hat of
+the identification of the host galaxy. It is important for at least two reasons.
+First, the identification of a host galaxy is usually the only way to obta in a
+redshift for short GRBs. Second, host galaxies can provide import ant clues
+to the nature of the progenitor of the short bursts and on the ph ysics of their
+afterglow emission.
+UnlikelongGRBs,shortburstsprovidedaharderchallengetoastro nomers
+for the identification of the host galaxy. While long bursts explode on top
+of the brightest region of galaxies, making the identification unques tionable
+[22], short GRBs explode in anonymous regions in the outskirt of galax ies,
+sometimes having more than one galaxy within their error circles.
+The first well-localized short GRB offers a good example of the situatio n
+[39, 65, 28]. The X-ray error box was large enough to contain a low-r edshift
+elliptical galaxy, member of a cluster, and several high redshift fain t objects,
+analogous to the host galaxies of long GRBs. The implication of the cho ice of
+host wererelevant.If the nearbyelliptical wasthe host, short GR BS wouldbe
+associated to non star forming objects, they would explode in the o utskirts
+of the host - if not outside of them - and would be low redshift events ,
+involving several orders of magnitude less energy than long events . On the
+other hand, if the progenitorwas one ofthe high redshift objects , short GRBs
+could be analogous to the long ones. Circumstantial evidence favor ing a low
+redshift origin of short bursts was emerging in the meantime, showin g that
+the location of BATSE short GRBs correlates with local bright galaxie s [81]
+or with clusters [31].
+With more short GRB localizations and improvement in the error boxes
+there is now very little doubt that the population of short GRB host g alaxies
+is markedly different from those of long bursts. Short GRB hosts se em to be
+a far less homogeneous sample than those of long ones. With the cav eat that
+there may be some misidentifications, given that short GRBs do not e xplode
+in the brightest parts of their hosts but rather in their outskirts. Long GRB
+hosts are usually dwarf starbursting galaxies at moderate to high r edshift,
+always found in the field. Morphologically, short GRB hosts can be of a ny
+kind, from early type ellipticals to late type spirals, and are found bot h in
+the field and inside clusters. In most cases, short GRB hosts have lo w star
+formation, <1M⊙/y−1(L⋆/L), or about one hundredth of the star formation8 Davide Lazzati and Rosalba Perna
+rate in long GRB hosts [15]. The two populations are different at a very high
+confidence level [34] and at least in some cases, evidence of an old po pulation
+of star was found in the host [17, 78]. Approximately 20 per cent of S wift
+short bursts are associated to clusters of galaxies [7], in agreemen t with the
+fractionofstellarmasscontainedin suchsystems.ShortGRBs see mtherefore
+to be better unbiased samples of the stellar population in the low-mod erate
+redshift universe than long ones.
+2.2.3 Redshift distribution
+Fig. 5Redshift distribution of short (thick line) and long (thin l ine) GRBs.
+The redshift of short GRBs is always measured as the redshift of th e pu-
+tative host galaxy and is therefore less robust than the long GRB re dshifts,
+that are often measured both in absorption on the afterglow spec trum and
+in emission in the host galaxy. Figure 5 shows the short GRB redshift d istri-Title Suppressed Due to Excessive Length 9
+bution with a thick line histogram. Even though some of the values can be
+debated, it is clear that the redshift distribution of short GRBs is ce ntered at
+much smaller redshifts than that of long GRBs. The long GRB redshift dis-
+tribution is shown with a thin histogram for comparison and extends b eyond
+redshift 6 [80, 36, 42].
+2.2.4 Short or not short?
+We have discusses so far a consistent body of observational evide nce coher-
+ently supporting the idea that short and long GRBs have different ph ysical
+origin and are two well separated populations. Long GRBs are long in γ-
+rays and soft, have bright afterglows with jet breaks and supern ova bumps,
+they explode in star forming regions inside high redshift dwarf irregu lar star
+forming galaxy. Short burst, on the other hand, are short and ha rd inγ-rays,
+have dim afterglows and lack any evidence of an associated superno va, ex-
+plode in low-intermediate redshift galaxies of all kind, with little sign of s tar
+formation.
+Such an idyllic interpretation framework was suddenly shaken last su m-
+mer, when two puzzling bursts were detected: GRB 060505 and GRB 060614
+[19, 26, 27]. They appeared to be long in the γ-ray properties, but had faint
+afterglowsandnosignofanassociatedsupernova,downtoverys tringentlim-
+its. Possibleinterpretationsrangefrom shortburstswith apartic ularlybright
+X-ray early afterglow [89] to long bursts with a non-massive star pr ogenitor
+[27], or associated to a SN explosion that did not produce large quant ities of
+56Ni [85]. The puzzle becomes even more intricate if secondary indicato rs are
+used. GRB 060614 is found to be consistent with the Amati correlat ion [1],
+like a long duration GRB, but to share the short GRB properties in ter ms of
+spectral lags (no significant lag was detected) [61].
+Even though it is definitely premature to give up a classification succe ssful
+for decades based on only two events, such observations should c learly ring
+a warning bell. It is obvious that if short GRBs are followed by relatively
+bright early X-ray afterglows [46, 13], the distinction between shor t and long
+events based only on their duration is band dependent and doomed t o fail.
+However, a new robust scheme has not appeared, so far. Spectr al lag are
+a promising tool, but they are still theoretically unexplained and may w ell
+not be associated to the physics of the progenitor, something tha t seem to
+be clearly different between the two classes. A classification based o n the
+host galaxy properties is also a possibility that could be considered, w ith
+the problem that not all GRBs have host galaxy detections and so ma ny
+events would become unclassified. It is clear that more events of diffi cult
+classification have to be detected before we can establish a new, su ccessful,
+classification scheme.10 Davide Lazzati and Rosalba Perna
+2.3 SGRs
+A considerable excitement was caused by the detection of a giant fla re from
+thesoftgammarepeater(SGR)SGR1806-20[64,83,12,52].Thein itialphase
+of the flare is a bright spike lasting less than one second, with a blackb ody
+spectrum and enough energy to be detected by BATSE out to a dist ance
+of approximately 50 Mpc. It is followed by a pulsed X-ray tail that wou ld
+be undetectable with any present instrumentation for an extraga lactic flare.
+These properties suggested that a significant fraction of BATSE s hort GRBs
+could indeed be giant flares from SGRs in the local universe, out to th e Virgo
+cluster of galaxies.A largeeffort was attempted to constrainthis f raction and
+possibly identify the extragalactic SGRs in the BATSE sample. The sea rch
+was based on positional coincidence with nearby galaxies [57, 62, 69], on the
+spectral properties of the prompt emission [48], and on the presen ce of an
+oscillating tail. None of these searchesfound any suitable candidate , implying
+a fraction of at most 15 per cent of BATSE short GRBs being SGR flar es in
+the local universe. This limit is only marginally consistent with the Galact ic
+SGR rate, and requires the SGR1806-20 flare to be an exceptional event. Of
+course, caution should always be considered when statistics is base d on a
+handful of events.
+3 Theory
+4 Binary mergers as progenitors of short GRBs
+The leading candidates as progenitors of short GRBs are mergers o f NS-NS
+binaries [63, 32, 21, 60] and BH-NS binaries [60, 55]. Two compact obj ects in
+a binary are bound to eventually merge due to the emission of gravita tional
+wave radiation that causes a loss of energy resulting in a gradual sh rinking
+of the orbit. For typical binary parameters, binaries are expecte d to coalesce
+within a Hubble time, as discussed in more detail in the following. From an
+energetic point of view, there is enough gravitational binding energ y that is
+liberated during the merger to power the GRB and its subsequent af terglow.
+The engine that helps channeling this energy into a relativistic flow is be -
+lieved to be, like in the case of long GRBs, an hyper-accreting accret ion disk.
+However, while in the case of long bursts (which are thought to be as sociated
+with the collapse of massive stars), the event duration is set by the collapse
+time of the star envelope (on the order of several tens of second s) that keeps
+on feeding the disk, in the case of a binary merger the activity phase is set
+by the viscous timescale of the disk, which is a fraction of a second, t he right
+order of magnitude needed for the power source of short bursts .Title Suppressed Due to Excessive Length 11
+From an observationalstandpoint, double neutron stars arekno wn to exist
+from direct observations in the Galaxy. Well-known examples are the Hulse-
+Taylor binary and the binary pulsar [11]. On the other hand, compact -object
+binarieswhereoneofthecomponentsisablackholehavenotbeenob servedso
+far. However, theoretical modeling of binary evolution predicts th at binaries
+with black holes are expected (e.g. [10, 70, 24, 4]). It should be note d that,
+even if BH-BH binaries are also predicted as a possible end state of bin ary
+evolution, however they are not expected to give rise to a GRB when they
+merge. This is because, in this scenario, there would be no fuel to po wer the
+accretion disk that could then provide the source of energy for th e GRB (see
+§??).
+In order for mergers of compact objects to be progenitors of sh ort bursts,
+a fundamental condition is that the merger event rate be compara ble to
+that of the bursts. With the growing number of short bursts with d etected
+redshift, a comparison can be made not only for the overall (i.e. red shift-
+integrated)eventrate,butalsoforitsredshiftevolution.Asthe samplegrows,
+the calibration ofthe cosmic rate evolution,in combination with obser vations
+of the burst rate as a function of the galaxy type, can allow one to f urther
+constrain various types of binary models.
+The main source of guidance for the expected characteristics of a popu-
+lation of bursts associated with the coalescence of two compact ob jects is
+provided by population synthesis calculations. In the following, we will de-
+scribe the main assumptions, calculation methods, and results from these
+calculations ( §??), with a special emphasis on their specific application to
+short bursts (S??). Finally ( §??), we will discuss the highlights of numerical
+calculations that simulate the final moments of the binary, and the f ormation
+of the accretion disk that eventually leads to the GRB.
+4.1 Binary evolution - theoretical modeling
+In the last few years,severalgroups[70, 4] havedeveloped popu lation synthe-
+siscalculations.Thesesimulations,startingfromsomeassumptions regarding
+the binary star population, track the evolution of stars both individ ually and
+in relation to the companion in the binary system; in output, they are able
+to predict the fraction of systems that end up with a certain type o f compact
+objects, the merger time of each binary, and hence the cosmic eve nt rate.
+In the following, we describe in more detail the highlights of the popula tion
+synthesis calculations.
+The evolution of each star in the binary system is computed given its z ero-
+age main-sequence mass and its metallicity. The code tracks all the s tages
+from the main sequence to the red giant branch to the core helium bu rning
+and the asymptotic giant branch. At each stage of evolution, the b asic stel-
+lar parameters (radius, luminosity, stellar mass, core mass) are de termined.12 Davide Lazzati and Rosalba Perna
+While each star is evolved depending on its initial parameters, a numbe r
+of effects that influence the binary orbit (e.g. mass and angular mom entum
+lossesdue to stellarwinds) aretakeninto account.At everyevolut ionarytime
+step, the codes check for possible binary interactions. If any of t he compo-
+nents fills its Roche lobe, then the resulting mass transfer is comput ed, and
+so the eventual resulting mass and angular momentum losses. If th e binary
+survives the mass transfer event (i.e. both stellar components fit within their
+Roche lobe), then the evolution of the binary keeps on being followed . The
+calculation for each star ends at the formation of a stellar remnant : a white
+dwarf, a neutron star or a black hole. If the remnant is born with a s upernova
+explosion, the effects of the supernova kicks and mass loss on the b inary or-
+bit are computed. Finally, once a binary consists of two remnants, it s merger
+lifetime is calculated, i.e. the time until which the components merge du e to
+emission of gravitational radiation and the consequent orbital dec ay.
+Early population synthesis studies [70, 25] found that, for the NS- NS bi-
+nary systems, merger times are generally long, tmerg>0.1−1 Gyr. Smaller
+timescales were found [86] as a result of the assumption that the se condary
+star, once it becomes a low-mass helium-rich star, can initiate an ext ra mass
+transfer phase. More recently, simulations [4] identified a new popu lation
+of coalescing NS-NS binaries, which merge on a timescale which is much
+shorter than that of the ’classical” population. These very short t imescales,
+tmerg0.001−1 Myr, are the result of allowing both the primary and the
+secondary star to initiate an extra mass transfer phase. Furthe rmore, the in-
+clusion of natal kicks in the simulations [4] further contributes to de crease
+the merger timescales. Accounting for natal kicks, in fact, result s in the dis-
+ruption of the widest binaries and in an eccentricity gain for the rema ining
+ones, which further reduces their lifetimes. This subpopulation of r elatively
+short-lived NS-NS binaries was found to be the dominant channel, ma king
+up about 80% of the total population.
+On the other hand, all population synthesis calculations agree on th e
+typical distribution of merger times of the NS-BH systems, which is f ound
+to be similar to that of the “classical” population of NS-NS binaries, i.e.
+tmerg>0.1−1 Gyr.
+4.2 Theoretical predictions for the observational
+properties of short bursts due to binary mergers
+a) Rates The distribution of merger times obtained from population synthe-
+sis calculations, combined with a prescription for the cosmic star for mation
+rate, allows one to estimate the rate of binary mergers throughou t the life-
+time of the Universe. Since the simulations by Belczynski et al. [5] inclu de
+the population of tight NS-NS binaries (i.e. the short-lived population ), the
+predicted cosmic redshift rates are different for the population of NS-NS andTitle Suppressed Due to Excessive Length 13
+of NS-BH binaries. An example is shown in Figure 1, with data from the s im-
+ulations by Belczynski et al. [5]. Those calculations used as star form ation
+rate which peaks at a redshift of about 3 and has only a mild decline up t o
+redshifts of about 5. Due to the merger time delays, however, the predicted
+binary merger rate peaks at a remarkably lower redshift. The large r the time
+delay, the larger the shift to lower redshifts of the binary merger e vent rate
+compared to that of the underlying star forming rate. This is espec ially em-
+phasized in Figure 1, where the rates due to the fraction of (both N S-NS
+and NS-BH) long-lived binaries ( t >100 Myr) are shown together with the
+overall rates. Whereas the assumed star-formation rate remain s constant up
+to a redshift of ∼5, the rate of the long-lived component drops substantially.
+For the overall binary merger rate, if the assumed SFR peaks at a r edshift of
+∼3, the peak is found in the range ∼1−2.
+Fig. 6Thick lines: the inferred merger rate as a function of redshi ft for the NS-NS and
+BH-NS mergers. Thin lines : the contribution from the long-lived ( t >100 Myr) population.
+Rates are in arbitrary units. Data from the simulations of Be lczynski et al. [5].
+Another method of estimating the binary merger rates starts fro m a count
+of the observed number of NS-NS binaries in the Milky Way. This numbe r is
+then correctedto account for the completeness of the survey a nd transformed
+into a local rate by weighing in the estimated lifetimes of the observed sys-
+tems. Up to date, the latest calculations of the Galactic merger rat es [41]14 Davide Lazzati and Rosalba Perna
+have yielded a value in the range 1 .7×10−5to 2.9×10−4yr−1at the 95%
+level.
+em b) Offsets from host galaxies, densities of the circumburst mediu m,
+afterglow brightnesses and galaxy types
+Since the first studies on GRBs, it was clear that the location of a bur st
+within a galaxy, and hence its environment, would provide important c lues
+to the properties of the progenitor itself. In the past decade, a n umber of
+groups [25, 8, 66, 5] have performed statistical studies of the dis tribution of
+the binary mergersites in a galaxy.Generallyspeaking, progenitors with very
+short lifetimes are typically expected to produce GRBs close to their place
+of birth. However, if the progenitor is endowed with a kick velocity (s uch as
+in the case of binaries), then the merger sites could have a substan tial offset
+from the place of birth. The amount of the offset will generally depen d on a
+combination of the merger time and the velocity of the binary system , as well
+as the potential well in which the systems move. Kick velocities are es timated
+to be in the rangeofseveraltens to severalhundredskm/s [40]. A binarywith
+a velocity in the upper range of the distribution, and a long lifetime ∼a few
+Gpc, can travel a substantial distance, on the order of the Mpc s cale, before
+merging. On the other hand, for binaries with a short lifetime, on the order
+of a million years or less, the place of merger is always going to be close t o
+the place of birth even for the fastest pulsars of the distribution. Therefore,
+it is clear that the expectations for the distribution of the merger s ites are
+going to heavily depend on the expected distribution of merger times of the
+binary population under consideration.
+Another element of consideration is the fact that different types o f galax-
+ies are expected to host different types of stellar populations, with , at one
+extreme, starburst galaxies which are dominated by very young st ars, and at
+the other end, elliptical galaxies which have almost no star formation rate
+at all and hence are dominated by an old stellar population. These diffe rent
+types of galaxies, therefore, will select out different components of the binary
+mergers. Starburst galaxies are expected to be dominated by the short-lived
+binary population, while elliptical galaxieswill mostly host the merger ev ents
+of the long-lived population. Spiral galaxies are in between, since the y host
+both young and hold star populations. As a result of these differenc es, merger
+events taking place in elliptical galaxies will generally occur at larger off sets
+(from the galaxy centers) with respect to merger events that oc cur in star-
+burst and spiral galaxies. Furthermore, within a given morphologica l galaxy
+type, mergers in small galaxies will have substantially larger offsets t han
+merger in large galaxies. This is due to the larger gravitational poten tial of
+the larger galaxies, which prevents the binaries from traveling too f ar.
+Thesitesofthe binarymergersareespeciallyimportant indeterminin g the
+observability of the GRB afterglows. Since the gas density declines w ith the
+distancefromthegalaxycenter,mergersthatoccurintheoutsk irtsofgalaxies
+are expected to give rise to very dim afterglows. Based on the cons iderations
+above,shortburstsoccurringinlargestarburstgalaxiesareexp ectedtobetheTitle Suppressed Due to Excessive Length 15
+brightest, while short bursts occurring in small ellipticals will be gener ally
+the dimmest. A fraction of these bursts are expected to be “nake d”, i.e.
+completely lacking any afterglow emission, especially at wavelengths lo nger
+thanthe X-rayband,wheretheeffect ofthe densitybecomesmor eimportant.
+As more data on short bursts gather, the information on the frac tion of
+bursts as a function of the galaxy type will become an important elem ent of
+study in order to establish whether there is a dominant channel of b inaries
+(i.e. short-lived versus long-lived) that give rise to short bursts. H owever, if
+one wants to extract from these types of statistical studies mea ningful phys-
+ical information, one needs to keep in mind that there is a huge poten tial for
+selection biases. In fact, since bursts are localized through after glow observa-
+tions, and bursts occurring in elliptical galaxies are expected to be g enerally
+dimmer, there is going to be a selection effect toward the observatio n of a rel-
+atively larger fraction of the bursts occurring in starburst and sp iral galaxies,
+i.e. the short-lived component of the binary population. Furthermo re, since
+the mean redshift of merger events for the short-lived population is higher
+than that for the long-lived population, the selection effect describ ed above
+will also result in a bias toward higher redshifts.
+4.3 The final moments: from a merging binary to a
+hyperaccreting disk around a black hole
+Although there is no “direct” evidence for an accretion disk in GRBs, the
+GRB phenomenology provides strong hints in that direction. Firstly, accre-
+tion disks are a powerful way to tap gravitational energy and chan nel it
+into other sources. Second, the overall (short) burst duration s, as discussed
+above, are naturally accounted by the viscous timescale of the disk , while the
+millisecond timescale variability (observed both in long and short burst s) is
+on the same order of the dynamical timescale of a compact disk accr eting
+around a stellar-mass black hole. From an observational point of vie w, on
+the other hand, it is well known that other systems in nature believe d to be
+associated with black holes accreting from a disk (i.e. active galactic n uclei,
+micro-quasars) are able to power relativistic jets, for which we hav e strong
+evidence also in GRBs.
+In the last several years, a number of groups have devoted a sub stan-
+tial effort into simulations of NS-NS and NS-BH mergers, and the res ult-
+ing structure of the hyperaccreting accretion disk. The early simu lations [?]
+were Newtonian, used a polytropic equation of state, and did not inc lude
+the effects of neutrinos. These, on the other hand, were shown t o represent
+a substantial source of cooling in a number of semianalytical, 1D calcu la-
+tions of hyperaccreting disks around black holes [59, 20] . Neutrino effects,
+together with more realistic equations of state, were taken into ac count into
+later simulations [73, 74, 75, 76]. These simulations, independently of the nu-16 Davide Lazzati and Rosalba Perna
+merical method used, found results in general agreement. Althou gh details
+differ depending on whether the initial progenitor is a NS-NS or a NS-B H,
+some common features can be identified. As the binary members spir al in,
+within a few orbital periods the outcome is the formation of a dense a nd
+thick, hot torus, of mass on the order of ∼0.01−0.3M⊙that accretes onto a
+stellar mass black hole. In the case where the initial binary members a re both
+neutron stars, the black hole will be formed as a result of the accre tion of
+mass onto one of the NSs. Depending on the total initial mass of the binary
+system, the collapse of the hypermassive NS into a BH can occur pro mptly,
+or it can be delayed for an initial time during which the star is supporte d by
+differential rotation. The accreting material, on the other hand, is provided
+by the tidally disrupted debris of the NS.
+The duration of the prompt GRB phase phase is set by the time during
+which efficient accretion occurs. Given the observed γ-ray luminosities ( ∼
+1049−1050erg/s assuming a beaming correction of a factor ∼0.1), and
+taking an efficiency of conversion of accretion energy into γ-rays of a fraction
+of percent to a percent, the accretion rates of the disk must the n be in the
+range∼0.01−10M⊙. The resulting hyperaccreting disk is very dense and
+hot, optically thick to photons, and cools mainly by neutrinos. In the upper
+end of the range of accretion rates, the density of the disk can ho wever be
+so high that the innermost regions would become optically thick even t o the
+neutrinos themselves.
+An important component of the studies of GRB accretion disks deals with
+the processes by which the disk black-hole system is able to collimate a nd
+launch the relativistic jets known to power the GRBs. Two mechanism s have
+been suggested as being involved: neutrino-anti neutrino annihilat ion, and
+magnetic fields. In the former process, suggested by a number of authors
+[33, 21, 60, 53, 55], the source of energy is provided by the neutrin os and
+anti-neutrinos emitted in the cooling disk which annihilate in a funnel ab ove
+a disk (which has a lower density). Calculations [ ?] have estimated that the
+maximum efficiency by which the rest mass energy of the accreting ma terial
+is converted into neutrino luminosity does not exceed a value of ∼10−4. For
+a disk mass of ∼0.1M⊙, this would yield an energy output of about 1049
+erg, therefore making this jet-production mechanisms viable only if there is
+a substantial degree of collimation in short bursts.
+The second method of jet collimation and launch relies on the help of ma g-
+netic fields. A number of authors [ ?, 84, 50] have suggesting this possibility
+by noticing how, even if the magnetic field is initially low, it is likely to be
+amplified by the magneto-rotational instability within the disk [2]. Nume ri-
+cal simulations [51] find that magnetically driven jets, whose energy output
+increases with the spin of the BH, are generally more efficient than ne utrino-
+powered jets. Magnetic fields are therefore considered to play an important
+role in collimating and driving the jets from the accretion disk.
+Whereas a substantial progress has been made in this area of rese arch
+of GRB hyperaccreting accretion disks, recent, new observation s withSwiftTitle Suppressed Due to Excessive Length 17
+have shown that the current picture is far from being complete. In particular,
+the detection of energetic X-ray flares superimposed on the smoo th afterglow
+decay, with arrival times and durations from tens to tens of thous ands of
+seconds, requires the presence of an engine with duration much lon ger than
+the fraction of a second that is sufficient to power the prompt phas e of a
+short burst [49]. These observations have prompted a number of s uggestions
+on how to make the lifetime of the accretion disk much longer than its v is-
+cous timescale. Perna, Armitage & Zhang ([67]; see also Piro & Phfal [6 8])
+suggested that fragmentation of the outer parts of the accret ion disk (which
+would then accrete at later times) could be responsible for creating a long-
+lived engine. Proga & Zhang [71], on the other hand, envisaged a scen ario
+in which the accumulation of magnetic flux in the innermost parts of th e
+accretion disk creates a barrier that can then produce intermitte nt accretion.
+Other suggestions, which do not involve accretion disks, include tha t of Dai
+et al. [18]. They showed that, if the NS-NS merger leads to a different ially ro-
+tating, millisecond pulsar, then the differential rotation can cause t he interior
+magnetic field to wind up and break through the stellar surface, hen ce result-
+ing in magnetic reconnection-driven explosive events. These event s would be
+observed as X-ray flares.
+5 Gravitational waves from short GRBs
+Mergers of two compact objects have traditionally been of great in terest as
+sources of gravitational waves. With the likely association of binary mergers
+with short GRBs, the interest of the gravitational wave community has been
+extended to that of short GRBs. The local rate of these sources , however, is
+not large enough to make a detection likely with a blind search with LIGO -
+I. However, the detection probability can be increased if the obser vations
+are made shortly after the γ-ray detection from the burst is detected. Esti-
+mates [43] suggest that in this case a positive detection in coincidenc e with a
+short GRB could be already made with current gravitational wave de tectors.
+Clearly, such a signal, besides giving information on the last moments o f the
+binary merger, would also provide the still needed conclusive evidenc e of the
+association of short bursts with mergers of compact objects.
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+arXiv:1001.0539v2  [quant-ph]  9 Jun 2010Quantum noise and self-sustained radiation of PT-symmetric systems
+Henning Schomerus
+Department of Physics, Lancaster University, Lancaster, L A1 4YB, United Kingdom
+(Dated: September 7, 2018)
+The observation that PT-symmetric Hamiltonians can have real-valued energy level s even if
+they are non-Hermitian has triggered intense activities, w ith experiments in particular focusing on
+optical systems, where Hermiticity can be broken by absorpt ion and amplification. For classical
+waves, absorption and amplification are related by time-rev ersal symmetry. This work shows that
+microreversibility-breaking quantum noise turns PT-symmetric systems into self-sustained sources
+of radiation, which distinguishes them from ordinary, Herm itian quantum systems.
+PACS numbers: 42.50.Lc, 03.65.-w, 03.65.Nk
+A common factor in quantum systems with a non-
+Hermitian Hamiltonian is the non-conservation of par-
+ticle number, either because the system is leaky, or be-
+cause there is loss or gain in an absorbing or amplifying
+medium. Ignoring nonlinear effects such as the feedback
+in a laser, such systems ordinarily do not possess sta-
+tionary states; instead, they only support decaying qua-
+sibound states with complex energy, where the imagi-
+nary part Im E=−1/2τ(setting ¯h≡1) accounts for
+particle loss with decay rate 1 /τ(particle gain is asso-
+ciated to a negative decay rate). A notable exception
+are non-Hermitian systems with loss and gain combined
+such that they are invariant under joint parity ( P) and
+time-reversal ( T) symmetry [1]. If there is no leakage,
+thesePT-symmetric systems generically possess a set of
+real-valued energy levels, as well as complex-conjugate
+pairs of complex energy levels. Systems with entirely
+real spectrum define a consistent unitary extension of
+quantum mechanics [2, 3]. This observation has led to
+intense research efforts delivering a new theoretical per-
+spective on systems as varied as quantum field theories
+and complex crystals [4], while experimental realizations
+in particular focus on optical systems where Hermiticity
+can be violated by absorption and amplification [5].
+For classical waves, amplification and absorption are
+strictly related by time reversal. The existence of sta-
+tionary states with real energy can therefore be seen as a
+consequence of the balance of amplification and absorp-
+tion in parity-related regions of a PT-symmetric system.
+At the heart of absorption and amplification, however,
+are noisy microscopic quantum processes (such as spon-
+taneous and stimulated emission events, and stimulated
+absorption events) which effectively break time-reversal
+symmetry [6]. The objective of this Letter is to show
+that the effects of this quantum noise distinguish PT-
+symmetric systems from Hermitian quantum systems,
+and indeed suggest an alternative interpretation of the
+physics behind non-Hermitian PTsymmetry: (i) Ac-
+counting forquantum noise, PT-symmetricsystems with
+stationary states are self-sustained sources of radiation,
+fed bythe pumping inthe amplifying partsofthe system.
+(ii) That the energy of these states is real means that theâLin â0right 
+âLout â1in 
+â1out â0le/f_t âRin âRout 
+â2in â2out Γ Γ
+bL^bR^
+S1S2=/ScriptCapP/ScriptCapT S1
+}}}}
+SLSR
+FIG. 1: Illustration of the scattering input-output approa ch
+to non-Hermitian PT-symmetric systems, defining the scat-
+tering input-output operators ˆ ain,out
+L,Rand internal bosonic
+noise operators ˆbL,Rin different parts of the system. The op-
+erators ˆaright,left
+0and ˆain,out
+1,2feature in the intermediate steps
+of the calculation (for details see the text). Semitranspar -
+ent mirrors with transmission probability Γ are introduced to
+study the limit Γ →0 of a closed (non-leaky) system. (In
+the context of present experiments with optical fibers [5], t his
+sketch relates to the transverse confinement.)
+system is stabilized at the lasing threshold. (iii) When
+the system is leaky, the emitted radiation breaks parity
+symmetry (i.e., the emission pattern is asymmetric). (iv)
+In the limit of a non-leaking system, the emitted radia-
+tion intensity approaches a constant value, and provides
+a direct measure of the non-Hermiticity of the system.
+The internal energy density of radiation then diverges,
+which entails a practical limitation for the implementa-
+tion ofPTsymmetry in closed systems.
+These conclusions are obtained by employing the
+quantum-optical input-output formalism [7] in its scat-
+tering formulation [8–10]. The scattering approach also
+provides insight into PTsymmetry for classical waves
+[11, 12], which defines the starting point of this Letter.
+Scattering approach to non-Hermitian PT-symmetric
+systems.—Probing the internal dynamics of an optical
+system by external radiation naturally leads to the scat-
+tering scenario depicted in Fig. 1. The relation aout=
+Sainbetween incoming and outgoing wave amplitudes is2
+provided by the scattering matrix
+S(E) =/parenleftbiggr t′
+t r′/parenrightbigg
+, (1)
+which contains blocks describing reflection ( r,r′) and
+transmission( t,t′) whenprobedfromthe left orright, re-
+spectively. Each block consists of an N×N-dimensional
+matrix, where Nis the number of modes at each en-
+trance. The scattering matrix is generally energy depen-
+dent (which arises from the wavenumber dependence of
+constructive and destructive interference), and its poles
+determinetheenergiesofquasiboundstatesinthesystem
+[14].
+In general, the scattering matrix fulfills the following
+two reciprocity relations: The Onsager relation
+S(γ,−B,E) =ST(γ,B,E), (2)
+and the relation
+S(−γ,B,E) = [S†(γ,B,E∗)]−1(3)
+of classicalmicroreversibility. Here, γandBcharacterize
+two possible sources of broken time-reversal symmetry:
+absorption or amplification ( γ >0 orγ <0), which con-
+tribute an imaginary symmetric (non-Hermitian) term
+to the Hamiltonian H, and magneto-optical effects ( B),
+which contribute an imaginary antisymmetric (but still
+Hermitian) term.
+Conventional time-reversal TH=H∗transforms so-
+lutions according to Tψ=ψ∗, which interchanges in-
+comingand outgoingstates, andthereforetransformsthe
+scattering matrix according to
+TS(γ,B,E) = [S∗(γ,B,E)]−1=S(−γ,−B,E∗).(4)
+Assuming that energy is real, a system has Tsymme-
+try,TS=ShenceS∗=S−1, ifS(γ,B) =S(−γ,−B),
+which requires γ=B= 0 [13]. Parity PH(x) =H(−x)
+transformssolutionsaccordingto Pψ(x) =ψ(−x), which
+exchanges the left and right leads and yields
+PS(γ,B,E) =σxS(γ,B,E)σx, (5)
+whereσxis a Pauli matrix operating on the blocks in
+Eq. (1). The PToperation on the scattering matrix is
+therefore given by
+PTS(γ,B,E) =σx[S∗(γ,B,E)]−1σx(6)
+=σxS(−γ,−B,E∗)σx.(7)
+For Hermitian systems, PTsymmetry implies S=
+σxSTσx[15]. For non-Hermitian systems, PTsymme-
+try implies the additional condition Pγ=−γ[γ(x) =
+−γ(−x)], i.e., there is a balance of absorption and am-
+plification in parity-related regions.Letusnowexplorefromthe scatteringperspectivehow
+real-energy bound states appear in PT-symmetric sys-
+tems. As shown in Fig. 1, such systems can be con-
+structed by joining two regions, where the left region,
+withscatteringmatrix S1=/parenleftbigg
+r1t′
+1
+t1r′
+1/parenrightbigg
+, isPT-symmetric
+to the right region, S2=PTS1, which using standard
+block-inversion formulas can be written as
+S2=
+1
+(r′
+1−t1r−1
+1t′
+1)∗(r′−1
+1t1)∗ 1
+(t′
+1r′−1
+1t1−r1)∗
+(r−1
+1t′
+1)∗1
+(t1r−1
+1t′
+1−r′
+1)∗1
+(r1−t′
+1r′
+1−1t1)∗
+.
+(8)
+Bound states can be studied by closing the system off
+by mirrors with small transmission probability Γ ≪1,
+described by a scattering matrix
+SΓ=−
+√1−Γi√
+Γ
+i√
+Γ√
+1−Γ
+. (9)
+The mirrors can be included by wave matching at their
+inward-facing faces, which amounts to an algebraic elim-
+ination of the amplitudes ain,out
+1,2in Fig. 1.
+The scattering matrix of the left half of the system
+then reads
+SL=−
+r1+√
+1−Γ
+1+r1√1−Γit′
+1√
+Γ
+1+r1√1−Γ
+it1√
+Γ
+1+r1√
+1−Γt1t′
+1√
+1−Γ
+1+r1√
+1−Γ−r′
+1
+,
+(10)
+while the scattering matrix SR=PTSLof the right half
+again follows from symmetry. These scattering matrices
+relateamplitudes ofingoingand outgoingmodes (defined
+in Fig. 1) according to
+/parenleftbiggaout
+L
+aright
+0/parenrightbigg
+=SL/parenleftbiggain
+L
+aleft
+0/parenrightbigg
+,/parenleftbigg
+aleft
+0
+aout
+R/parenrightbigg
+=SR/parenleftbigg
+aright
+0
+ain
+R/parenrightbigg
+.
+(11)
+The scattering matrix of the composed system is ob-
+tained by algebraically eliminating the amplitudes aleft
+0
+andaright
+0at the interface between both regions. For
+Γ→0, these amplitudes become singular when
+detIm(r′
+L) = det/bracketleftbigg
+Im/parenleftbigg
+r′
+1−t1t′
+1
+1+r1/parenrightbigg/bracketrightbigg
+= 0,(12)
+which (due to the general connection of scattering poles
+and quasibound states) is the quantization condition of
+the closed system. Determinantal quantization condi-
+tions of this kind have been introduced for general sys-
+tems in Ref. [14]; they also form the basis of exact nu-
+merical techniques as reviewed in [16]. The PT-specific3
+version (12) of the quantization condition requires that
+theNreal column vectors of Im( r′
+L) be linearly depen-
+dent, which generically can be achieved by varying a
+single real parameter such as E(identifying this as a
+problem of codimension one). Therefore, we recover that
+closedPT-symmetric systems typically possess a num-
+ber of bound states with real energy, even though the
+Hamiltonian is not Hermitian. Condition (12) also ad-
+mits complexeigenvalues, which then appear in complex-
+conjugated pairs.
+Quantum noise .—The scattering approach can be ex-
+tended to include quantum noise by passing from wave
+amplitudes ain,aoutto bosonic annihilation operators
+ˆain, ˆaout, respectively. This defines the scattering vari-
+ant of the input-output formalism [7–10], which has been
+used to describe systems that are exclusively absorbing
+or amplifying. To adapt the approach to PT-symmetric
+systems, where both effects are combined, we formally
+separate the absorbing regions from the amplifying re-
+gions, and then join them together similar to the de-
+scription of classical waves, given above.
+For definiteness let us assume that the left half of the
+system is purely absorbing. For this part, the input-
+output scattering relations then take the form
+/parenleftbiggˆaout
+L
+ˆaright
+0/parenrightbigg
+=SL/parenleftbiggˆain
+L
+ˆaleft
+0/parenrightbigg
++QLˆbL,(13)
+which connects the ingoing and outgoing modes to
+bosonic operators ˆbLandˆbRrepresenting quantum fluc-
+tuations in the left and right part of the medium. These
+operators can be defined microscopically in the frame-
+work of system-and-bath approaches, or phenomenologi-
+cally asoperator-valuedLangevinforces[7, 8]. The prop-
+erties of these operators can be determined from the con-
+dition that both ˆ ainand ˆaouthave to satisfy standard
+canonical commutation relations. This dictates that the
+coupling matrix QLsatisfies the fluctuation-dissipation
+theorem[9]
+QLQ†
+L= 1−SLS†
+L. (14)
+In the right half of the system, where the medium is
+amplifying, we have
+/parenleftbiggˆaleft
+0
+ˆaout
+R/parenrightbigg
+=SR/parenleftbigg
+ˆaright
+0
+ˆain
+R/parenrightbigg
++QRˆb†
+R,(15)
+where the commutation relationsnow dictate coupling to
+creation operators, with
+QRQ†
+R=SRS†
+R−1. (16)
+By assumption, the operators ˆb†
+LandˆbRcommute with
+ˆain; however, accordingto Eqs. (13) and (15) they do not
+commute with ˆ aout, which is a manifestation of broken
+micro-reversibility in quantum optics.We can now describe the full PT-symmetric system by
+algebraically eliminating the interface operators ˆ aleft
+0and
+ˆaright
+0. In the absence of incoming radiation, the intensity
+emitted to the left and right is then given by
+IL(E) =1
+2π∝angbracketleftˆaout†
+Lˆaout
+L∝angbracketright, IR(E) =1
+2π∝angbracketleftˆaout†
+Rˆaout
+R∝angbracketright,(17)
+which can be evaluated assuming
+∝angbracketleftˆb†
+LˆbL∝angbracketright= 0,∝angbracketleftˆb†
+RˆbR∝angbracketright= 0 (18)
+(ground-statepopulationintheabsorbingregionsandto-
+tal population inversion in the amplifying regions; these
+conditions minimize the quantum noise).
+Letusfirst considerthecaseofasingle-moderesonator
+(N= 1) with purely ballistic internal dynamics and ab-
+sorption in the left region [17], with scattering matrices
+S1=/parenleftbigg
+0t1
+t10/parenrightbigg
+, S2=/parenleftbigg
+0 1/t∗
+1
+1/t∗
+10/parenrightbigg
+,(19)
+where|t1|<1. Includingthe mirrors,thetotalscattering
+matrix is [18]
+S=
+√
+1−Γ/parenleftbig
+t∗
+12−t2
+1/parenrightbig
+t2
+1(1−Γ)−t∗
+12|t1|2Γ
+t2
+1(1−Γ)−t∗
+12
+|t1|2Γ
+t2
+1(1−Γ)−t∗
+12√
+1−Γ/parenleftbig
+t∗
+12−t2
+1/parenrightbig
+t2
+1(1−Γ)−t∗
+12
+,(20)
+and the quantization condition (12) for the closed res-
+onator takes the form Im t2
+1= 0 (indeed, the scatter-
+ing matrix elements diverge under this condition and
+Γ→0). Following the quantum-optical procedure de-
+scribed above [and using, in particular, Eqs. (14), (16),
+and (18)], we find that this resonator emits radiation of
+intensity
+IL(E) =Γ/parenleftbig
+|t1|−2−1/parenrightbig/parenleftbig
+1−Γ+|t1|2/parenrightbig
+2π|(t1/t∗
+1)2−1+Γ|2,(21)
+IR(E) =Γ/parenleftbig
+1−|t1|2/parenrightbig/parenleftbig
+1−Γ+|t1|−2/parenrightbig
+2π|(t1/t∗
+1)2−1+Γ|2.(22)
+Since|t1|<1 this gives IR>IL, the difference being
+∆I(E) =IR(E)−IL(E) =Γ2/parenleftbig
+|t1|−1−|t1|/parenrightbig2
+2π|(t1/t∗
+1)2−1+Γ|2.(23)
+Therefore, the emission from the right exit, close to the
+amplifying region of the medium, is larger than the emis-
+sion from the left exit, close to the absorbing region of
+the medium. The overall output intensity to both sides
+can be written as
+I(E) =IL(E)+IR(E) =Γ(2−Γ)/parenleftbig
+|t1|−2−|t1|2/parenrightbig
+2π|(t1/t∗
+1)2−1+Γ|2.(24)4
+Close to quantization in the closed system [Γ ≪1,
+E≈E0, whereE0fulfills the quantization condition
+Imt2
+1(E0) = 0], the emission pattern becomes symmetric
+(as a consequence of PTinvariance) and approaches a
+Lorentzian of the form
+IL(E) =IR(E) =Γ/parenleftbig
+|t0|−2−|t0|2/parenrightbig
+2π|2iτ(E−E0)+Γ|2.(25)
+Heret0=t1(E0), while
+τ= 2Im1
+t1dt1
+dE/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+E=E0≈2L/c (26)
+is the transmission delay time of propagation between
+the two mirrors (with Lthe length of each region and c
+the speed of light) . The full width at half maximum is
+given by ∆E= Γ/τ. While this width shrinks to zero as
+the system is closed off, remarkably the total intensity
+Itot=/integraldisplay
+I(E)dE=|t0|−2−|t0|2
+2τ(27)
+remains finite, and can be interpreted as a direct mea-
+sure of the degree of non-Hermiticity of the system (for
+ballistic transport, Hermiticity implies |t0|= 1).
+In the case of a single-mode resonator with backscat-
+tering (where r1andr′
+1are finite), compact expressions
+can still be obtained as long as the leakage remains
+small (Γ ≪1), implying according to Eq. (10) that
+|rL+1|,|tL|,|t′
+L| ≪1. The emission pattern is then still
+symmetric, with intensity
+IL(E) =IR(E) =1
+2π(1−|r′
+L|2)|t′
+L|2
+|2(Imr′
+L)−itLt′
+L|2.(28)
+Linearizationaroundthequantizationconditionagainre-
+veals a Lorentzian line shape, with line width
+∆E= Re/parenleftbiggd[(Imr′
+L)/tLt′
+L]
+dE/parenrightbigg−1
+.(29)
+Accounting for the scaling (10) of scattering coefficients
+with Γ, the totalintensity Itot∝(1−|r′
+L|2) againremains
+finite as Γ →0. In the Hermitian case, this limit would
+imply|r′
+L|= 1, so that the intensity vanishes. There-
+fore, the emitted radiation is still a direct measure of the
+degree of non-Hermiticity.
+Following the general formalism described above, the
+observations for one-dimensional scattering can be ex-
+tendedtothegeneralcaseof PT-symmetricsystemswith
+manymodes, forwhichcompactexpressionsarenolonger
+available. The emitted intensity generally remains finite
+evenin the limit ofaclosedsystem. Becausethe expecta-
+tion values ∝angbracketleftˆaleft†
+0ˆaleft
+0∝angbracketright ∝Γ−1,∝angbracketleftˆaright†
+0ˆaright
+0∝angbracketright ∝Γ−1of the
+internaloperatorsformallydivergeinthislimit, thisisac-
+companied by a diverging internal energy density, which
+can be interpreted as the source of this radiation. De-
+viations from perfect PTsymmetry can be incorporatedinto the scattering formalism by taking SR∝negationslash=PTSL,
+which results in an asymmetric emission pattern even in
+the limit of non-leaky mirrors.
+In standard laser theory [19], an active optical medium
+is below threshold (and stable) as long as all fundamen-
+talmodes aredamped, which ischaracterizedby complex
+energies with a negative imaginarypart. With increasing
+pumping, the loss of the modes within the amplification
+window is counteracted by gain, and as soon as one of
+the energies acquires a positive imaginary part the sys-
+tem becomes instable: the intensity increases exponen-
+tially until saturation sets in. Therefore, the radiation
+features described above are characteristic of a laser sta-
+bilized at threshold, the threshold condition being that
+the energy of the lasing mode is real. The system then
+is marginally stable, and the internal radiation energy
+accumulates due to the quantum fluctuations. In prac-
+tice, this leads to saturation in the amplifying parts of
+the system and therefore identifies an obstacle for the
+implementation of strict PTsymmetry in closed optical
+systems. When the system is slightly opened up, the
+threshold condition is no longer met and the internal en-
+ergy density is finite, while the emitted intensity remains
+a direct measure of the non-Hermiticity of the system.
+Concluding remarks. —In summary, the present Letter
+demonstrates that accounting for quantum noise, opti-
+cal realizations of non-Hermitian PT-symmetric systems
+emit radiation of an intensity that provides a direct mea-
+sure of non-Hermiticity.
+This has both practical as well as fundamental impli-
+cations. The fundamental consequences arise because
+non-Hermitian PT-symmetric systems with an entirely
+real spectrum define a consistent unitary theory of quan-
+tum mechanics [2, 3], formalized by the concept of quasi-
+Hermiticity, which introduces anew scalarproduct based
+on a generalized conjugation operation Cthat satisfies
+C2= 1, [C,H] = [C,PT] = 0. The self-sustained radia-
+tion identified here shows that accounting for quantum
+noise, non-Hermitian PT-symmetric systems are physi-
+cally distinct from ordinary Hermitian quantum systems:
+the canonical commutation relations for the input and
+output operators are only invariant under unitary trans-
+formations, which constraintsthe possibility to introduce
+alternative scalar products. This enforces classical dis-
+tinctions based on the transparency of such systems [12].
+From a practical perspective, the self-sustained radi-
+ation can be used as an indicator of successfully imple-
+mented non-Hermitian PTsymmetry in leaky systems,
+while the accompanying marginal instability and diverg-
+ing internal energy density signifies a practical obstacle
+for its implementation in the limit of no leakage. Fur-
+thermore, these findings identify a hitherto unexplored
+arena to study quantum fluctuations in active systems
+close to the lasing threshold.
+Non-optical realizations of PT-symmetric systems of-
+fer a wide range of connections to quantum field theories5
+[4], and while the results in the present paper are not
+directly transferable in detail, they suggest that quan-
+tum noise should provide fundamental insight into such
+systems, as well.
+[1] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80,
+5243 (1998).
+[2] C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev.
+Lett.89, 270401 (2002); Am. J. Phys. 71, 1095 (2003).
+[3] F. G. Scholtz, H. B. Geyer, and F. J. W. Hahne,
+Ann. Phys. (NY) 213, 74 (1992); A. Mostafazadeh, J.
+Math. Phys. (N.Y.) 43, 205 (2002); ibid.2814 (2002);
+ibid.3944 (2002); J. Phys. A 36, 7081 (2003); preprint
+arXiv:0810.5643.
+[4] C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
+[5] C. E. R¨ uter et al., Nature Phys. 6, 192 (2010); Z. H.
+Musslimani, K. G. Makris, R. El-Ganainy, and D. N.
+Christodoulides, Phys. Rev. Lett. 100, 030402 (2008); K.
+G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z.
+H. Musslimani, ibid.100, 103904 (2008); A. Guo et al.,
+ibid.103, 093902 (2009).
+[6] R. Loudon, The Quantum Theory of Light , 3rd ed. (Ox-
+ford University Press, New York, 2001).
+[7] M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386
+(1984); C. W. Gardiner and M. J. Collett, Phys. Rev. A
+31, 3761 (1985).
+[8] T. Gruner and D.-G. Welsch, Phys. Rev. A 54, 1661
+(1996).
+[9] C. W. J. Beenakker, Phys. Rev. Lett. 81, 1829 (1998).
+[10] H. Schomerus, K. Frahm, M. Patra, and C. W. J.
+Beenakker, Physica A 278, 469 (2000).
+[11] For PT-symmetric scattering theory in one dimension
+see F. Cannata, J. P. Dedonder and A. Ventura, Ann.
+Phys.322, 397 (2007).
+[12] M. V. Berry, J. Phys A 41, 244007 (2008); H. F. Jones,
+Phys. Rev. D 76, 125003 (2007); ibid.78, 065032 (2008).
+[13] In Hermitian systems ( γ= 0),Tis equivalent tothe alternative time-reversal T′H=HT, for which
+T′S(γ,B,E) =ST(γ,B,E) =S(γ,−B,E). ForT′sym-
+metry,S=ST, hence S(γ,B) =S(γ,−B), which re-
+quiresB= 0, while γcan be finite. While absorption or
+amplification generally breaks Tsymmetry, T′symmetry
+is therefore still observed in optical systems which show
+no magneto-optical effects, even when they are absorbing
+or amplifying (in that case H=HTis non-Hermitian
+but symmetric). A system with T T′symmetry obeys
+S−1=S†, henceS(−γ,B) =S(γ,B), requiring γ= 0 so
+that the Hamiltonian is Hermitian.
+[14] E. Doron and U. Smilansky, Phys. Rev. Lett. 68, 1255
+(1992).
+[15] H.U.Baranger andP.A.Mello, Phys.Rev.B 54, R14297
+(1996); V. A. Gopar, S. Rotter, and H. Schomerus, Phys.
+Rev. B73, 165308 (2006); M. Kopp, H. Schomerus, and
+S. Rotter, Phys. Rev. B 78, 075312 (2008); R. S. Whit-
+ney, H. Schomerus, and M. Kopp, Phys. Rev. E 80,
+056209 (2009); ibid.80, 056210 (2009).
+[16] A. B¨ acker, in The Mathematical Aspects of Quantum
+Maps, edited by M. D. Esposito and S. Graffi, Lecture
+Notes in Physics Vol. 618 (Springer, Berlin, 2003), p. 91.
+[17] M. Znojil, Phys. Lett. A 285, 7 (2001).
+[18] The left and right halves of the resonator are described
+by
+SL=−
+√
+1−Γit1√
+Γ
+it1√
+Γt2
+1√
+1−Γ
+,
+SR=−
+√
+1−Γ
+t∗
+12i√
+Γ
+t∗
+1
+i√
+Γ
+t∗
+1√
+1−Γ
+.
+[19] A. E. Siegmann, Lasers(University Science Books, Mill
+Valley, CA, 1986).
\ No newline at end of file
diff --git a/1001.0540.txt b/1001.0540.txt
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+arXiv:1001.0540v1  [nucl-th]  4 Jan 2010Electromagnetic interactions in HaloEffectiveField Theor y
+R. Higa1,2,a
+1Helmholtz-Institutf¨ urStrahlen-undKernphysik,Univer sit¨ atBonn,Nußallee14-16,53115Bonn,Germany
+2KernfysischVersnellerInstituut,RijksuniversiteitGro ningen,Zernikelaan25,9747AA Groningen,TheNetherlands
+Abstract. After a brief discussion of e ffective field theory applied to nuclear clusters, I concentra te on the
+inclusionoftwoparticularaspects,namely, narrowresona nces andelectromagnetic interactions.Asexamples of
+applications, Ipresent the details of our studies onalpha- alpha andproton-alpha scattering.
+1 Introduction
+Two-particle scattering at low enough energies are insen-
+sitive to the details of the interparticle interaction, pro -
+vided that the latter has a limited range of action. The in-
+tuitive argument behind this general property is that mo-
+menta whose wavelengths are much larger than the inter-
+actionrangecannotresolveitsfinerdetails.Theelasticam -
+plitude incorporates unitarity constraints and an univers al
+low-energyexpansion, a.k.a.theeffectiverangeexpansion
+(ERE),whosecoefficientsencodethelow-energybehavior
+of the forces for a particular system. This simple observa-
+tion, made by Bethe and others [1] more than fifty years
+ago,becamea benchmarkfortheoriststryingtomodelthe
+dynamicsoftheinteraction.
+The same line of reasoning is shared by e ffective field
+theories (EFT), where only the relevant low-energy de-
+grees of freedom are explicitly taken into account. Modes
+that are active only at high energies are “frozen” into the
+low-energy constants. In the particular case of EFT with
+short-rangeinteractions,thetwo-bodyscatteringamplit ude
+is equivalent to a low-energy expansion of the ERE am-
+plitude [2]. However,the EFT approachhas extra features
+overtheeffectiverangetheory.First,thesystematicexpan-
+sion of the amplitude allows for a reliable and unbiased
+estimate of the theoretical uncertainties. Second, symme-
+triesof theproblemor the waytheyare brokencanbe im-
+plemented in a straightforward way (for instance, electro-
+magneticandweakcurrents,isospin,parity).Andthird,th e
+ability to extend the formalism to systems with more than
+twoparticles.
+One example of the second feature illustrates a di ffer-
+ence between the EFT and ERE approaches. In the pro-
+cessnp→dγ, the M1 vtransitioncontainsat NLO a four-
+nucleon-photon operator that cannot be generated by the
+ERE approach [3]. The associated coupling can be fixed
+fromcoldneutroncapturerateatincidentneutronspeedof
+2200m/s.RupakimprovedthecalculationofM1 vandE1 v
+transitions to N2LO and N4LO, respectively, lowering the
+ae-mail:R.Higa@rug.nltheoretical uncertainty in the np→dγreaction to about
+1% forcenter-of-massenergiesupto 1MeV[4].
+When extended to three or more particles with large
+scattering length(s), EFT provides new insights and ex-
+plains certain universal properties that are independent o f
+the interaction details. For instance, the related Thomas
+and Efimov effects [5] are intimately connected to the be-
+havior of the leading three-body counterterm under varia-
+tions of the momentumcuto ffin the dynamical equations.
+Suchcountertermis necessaryto properlyrenormalizethe
+theory, as well as to guarantee a unique solution in the
+limit when the cuto ffis taken to infinity [6,7]. It exhibits
+a limit cycle behavior, that is, a log-periodic dependence
+with the cutoff, that explains the geometrically-separated
+bound states in the Efimov spectrum. For nuclear systems
+with three and four nucleons, the theory provides a con-
+vincingexplanationto certaincorrelations,like the almo st
+lineardependenceofthespin-doubletneutron-deuteronsc at-
+tering length with the triton binding energy [8,9] or simi-
+lardependenceofthelatterwiththealphaparticlebinding
+energy [10], known respectively as the Phillips and Tjon
+lines.
+Due to the universal character, e ffective field theories
+with short-range interactions have been successfully ap-
+plied to distinct areas of physics —atomic, particle, and
+few hadron systems. For a more complete overview, see
+Refs. [6,11]. Here I will concentrate on applications to
+nuclei that behave as systems of loosely bound clusters,
+which also include the nowadays popular halo nuclei. In
+Section 2 the generalfeaturesof the so-called halo /cluster
+EFT are presented in certain detail, with emphasis on as-
+pectsquiteoftenfacedwhendealingwithnuclearclusters.
+Sections 3 and 4 present applications of halo /cluster EFT
+to alpha-alpha ( αα) and proton-alpha ( pα) scattering, and
+in Section5aretheconcludingremarks.
+2 EFTfornuclearclusters
+The relevant degrees of freedom in halo /cluster EFT are
+the structureless, weakly boundobjects representedby re-EPJWeb ofConferences
+spective field operators. In the case of halo nuclei, these
+are the corenucleusandthe valencenucleons.E ffectslike
+nucleon excitations inside the core, pion or nucleon ex-
+changes,take place at energieswell abovethe bindingen-
+ergy that holds the clusters together. The former has an
+associatedmomentumscale Mhioftheorderoftheinverse
+of the interaction range, while in the latter case the mo-
+mentum scale Mlo≪Mhiis inversely proportional to the
+sizeofthehalosystem.Asexplainedinthefollowing,this
+separation of scales is, from the EFT point of view, the
+outcomeofafine-tuneinthecouplingconstants,similarto
+the one responsible for generating a shallow bound state
+in the spin-triplet nucleon-nucleon channel. In fact, due
+to its large extension compared to the typical range of the
+proton-neutroninteraction,thedeuteroncanbeseenasthe
+simplest halo nucleus, with a loosely bound neutron sur-
+roundingaprotoncore.
+In orderto understandthe formalism,let us consider a
+system of identical bosons, with mass mα, represented by
+afieldφandinteractingviaapairwise S-waveshort-range
+force.ThemostgeneralLagrangianrespectingtherelevant
+symmetriesofthesystem(parity,totalangularmomentum,
+approximate isospin and non-relativistic Galilean invari -
+ance)is givenby
+L=φ†i∂0+∇2
+2mαφ−d†i∂0+∇2
+4mα−∆d
++g/bracketleftbigg
+d†φφ+(φφ)†d/bracketrightbigg
++···, (1)
+where we introduce an auxiliary (dimeron) field d, with
+“residual mass” ∆, carrying the quantum numbers of two
+bosons in S-wave and coupling with their fields through
+the coupling constant g. The dots stand for higher order
+terms in a derivative expansion. The building blocks are
+the boson and dimeron propagators, the LO coupling g
+between the dimeron and two boson fields, and higher-
+order couplings that can be rewritten, via field redefini-
+tions, as powers of the dimeron kinetic energy.The boson
+anddimeronpropagatorsread
+iSα(q0;q)=i
+q0−q2/2mα+iǫ, (2)
+iD(q0;q)=−i
+q0−q2/4mα−∆+iǫ,(3)
+wherethepreciseformofthelatterdependsonthemagni-
+tudeof eachterm in thedenominator.Thepowercounting
+specifies the amountof fine-tuningin the low-energycou-
+plings,responsibleforassigningawell-definedordertoev -
+erypossibleFeynmandiagramthatcontributestothescat-
+tering amplitude. In a natural scenario, where fine-tuning
+is absent, the effective couplingsin Eq.(1) dependonlyon
+the high-energyscale Mhiof the theory. That corresponds
+to∆∼M2
+hi/mαandg2∼Mhi/m2
+α. The bare dimeronprop-
+agator(3)is static atleadingorder(LO),
+iD(p0;p)≃−i
+−∆+iǫ∼O(mα/M2
+hi).
+The LO tree diagram is proportionalto a0=mαg2/4π∆∼
+O(1/Mhi), wherea0is the boson-boson scattering length.The one-loop graph receives two additional factors from
+thecouplingconstant g,onedimeronpropagator,one-loop
+integration measure dq0d3q∼k5/mα, and two interme-
+diate boson propagators S2
+α∼(mα/k2)2. The net result is
+a suppression of k/Mhicompared to the tree graph, with
+kthe typical low-energy momentum under consideration.
+The next order, k2/M2
+hismaller than the tree level, com-
+prises two-loop contributions proportional to ( a0k)2and
+one kinetic term insertion to the dimeron propagator pro-
+portional to a0r0k2, wherer0=4π/m2
+αg2∼1/Mhiis the
+boson-bosoneffective range. The amplitude then amounts
+to a simple Taylor expansion in powers of the expansion
+parameter k/Mhi.
+However, a natural perturbative treatment is unable to
+accountforaplethoraofnuclearphenomena,wherebound
+states and resonances are the rule rather than exceptions.
+That demands a certain amount of fine-tuning in the pa-
+rameters of the effective Lagrangian.The physics of shal-
+lowS-wave bound states can be described by fine-tuning
+the parameter ∆in Eq.(1), responsible for driving the size
+of the scattering length a0. Setting∆∼MhiMlo/mαleads
+to a large a0∼1/Mlo, withMlothe new momentum scale
+associated to the fine-tuning. This situation is analogous
+tothenucleon-nucleon( NN)case[6,7].Aconsequenceof
+this new scale is that higher-loopgraphsbuilt up from the
+LO boson-boson-dimeron coupling gare no longer sup-
+pressed when k∼Mlo, but contribute at the same order as
+the lower-loop and tree graphs. The resummation of this
+class of diagrams leads to a LO amplitude in the form of
+a truncated effective range expansion up to its first coef-
+ficient. The scattering length then becomes the only rele-
+vant scale and (for positive values) predicts the existence
+of a shallow bound state EB≈1/(mαa2
+0). The renormal-
+ization group (RG) analysis of the Schr¨ odinger equation
+demonstrates that two-body systems with large scattering
+lengthconsist ofstronglyinteractingsystemswithan RG-
+flow towards a non-trivial fixed point [12], in contrast to
+the natural case of weakly interacting particles, where the
+RG flowstowardsastable, trivialfixedpoint.
+When extended to three-body systems, this EFT pro-
+vides new insights on some universal features, as men-
+tioned in the introduction. An investigation searching for
+theseuniversaleffectsintwo-neutronhaloswasperformed
+in Ref. [13], where it was foundthe20C nucleias possible
+candidateto haveanEfimov-likespectrum(seealso [14]).
+Non-universal (effective range) corrections have recently
+beingincorporatedtothisstudy[15].
+Thereisanadditionalphenomenumquiteoftenpresent
+in nuclear clusters, namely, the existence of low-energy
+resonances in the continuum. One could then expect the
+need for extra amount of fine-tuning. This is the case for
+neutron-alpha ( nα) scattering [16,17], where the presence
+of a narrow resonance in the P3/2channel around En≈1
+MeV requires adjustment of the coupling constants of the
+theorytoenhancethe P3/2amplitudeafewordersofmag-
+nitude away from the natural assumption. Another exam-
+ple istheααsystem[18], whichcontainsan S-wavereso-
+nance about 0.1 MeV above the elastic threshold. The αα
+scattering amplitude can be described by the Lagrangian19thInternationalIUPAP ConferenceonFew-BodyProblemsinPhy sics
+(1), but assuming ∆∼M2
+lo/mα. This is necessary to gen-
+erate an even larger ααscattering length, a0∼Mhi/M2
+lo.
+These scalings will be discussed in detail in the next sec-
+tion. With this power-counting the dimeron propagator is
+nolongerstatic, as thekinetic andresidualmasstermsare
+now of comparable order. In momentum space it has the
+form of Eq.(3). The presence of the kinetic term in the
+dimeron propagator in practice resums e ffective range r0
+correctionstoall orders.Thisresummationisnecessaryto
+reproduceanarrowresonanceatlow energy[18,16,17].
++...
+... ++
+Fig. 1.Graphic representation of TCS, as sum of graphs with
+multiple insertions of the bare dimeron propagator (double line)
+andthe “Coulombbubble loop”. The lattercontains intermed iate
+Coulomb photons resummed toallorders (shaded ellipse).
+For the case of chargedbosons, electromagneticinter-
+actions can be incorporatedinto Eq.(1) via the usual min-
+imal substitution. Among them, Coulomb photons are the
+dominantones at low energies[18]. Coulombinteractions
+wereformulatedintheEFTframeworkbyKongandRavn-
+dal[19]forthetwo-protonsystem,andcanbeextendedin
+astraightforwardwaytoincluderesonances[18].Theidea
+reliesonthetwo-potentialformalism,wheretheamplitude
+is separated in a term that contains only the two-particle
+pure Coulomb scattering TC, plus the Coulomb-modified
+strong amplitude TCS. The latter is diagramatically illus-
+tratedbyFig.1(seecaption)andhastheformofageomet-
+ric series. Like in the proton-proton case [19], the power
+countingfornarrowresonancesdemandsitsresummation.
+Up to next-to-leading order (NLO) one gets for our two-
+bosonexample,withreducedmass µandcharge Zα,
+TCS=−2π
+µC2
+ηe2iσ01
+−1
+a0+k2r0
+2−2kCH
++P0
+4k4
+(−1
+a0+k2r0
+2−2kCH)2, (4)
+where
+kC=µZ2
+αe2
+4π=µZ2
+ααem (5)
+is the inverse of the Bohr radius, and P0, the shape pa-
+rameter.ThestrengthofCoulombphotonsisdrivenbythe
+Sommerfeldparameter
+η(k)=kC
+k, (6)
+and the Coulomb-modifiedstrong amplitude is modulated
+bytheSommerfeldfactor,
+C2
+η=2πη
+e2πη−1, (7)which quantifies the probability of finding the two bosons
+at the same point in space. The phase of pure Coulomb
+relativetofreescattering,foragivenpartialwave L, reads
+σL=argΓ(L+1+iη)=1
+2iln/bracketleftiggΓ(L+1+iη)
+Γ(L+1−iη)/bracketrightigg
+.(8)
+Thelast CoulombingredientinEq.(4)is thefunction
+H(η)=ψ(iη)+(2iη)−1−ln(iη), (9)
+and explicitly shows the complicated analytic structure of
+Coulomb scattering at low energies. Nevertheless, apart
+fromtheSommerfeldfactorandthe S-waveCoulombphase,
+onenoticesthatthee ffectofdressingthestrongamplitude
+with non-perturbativeCoulomb photonsamountsto, apart
+fromamultiplicativefactorinthenumerator,replacingth e
+unitarity term−ikby−2kCH(η) in the denominatorof the
+amplitude. Similar expression can be derived for P-wave
+interactionsandwill bepresentedinSection4.
+3ααscattering
+3.1 power counting: pinning the scalesdown
+In order to better understand the scalings of the EFT pa-
+rameters in the presence of a resonance, it is helpful to
+providesomenumbersfortherelevantphysicalquantities.
+It is well-known that the ααsystem is dominated by S-
+waveatlowenergies[20],havingaverynarrowresonance
+atER≃92 keV and width ΓR≈6 MeV. To this reso-
+nanceiscommonlyassignedthe8Begroundstate.The αα
+scattering length a0∼2000 fm is about three orders of
+magnitude larger than the αmatter radius, suggesting the
+largeamountoffine-tuningdiscussedinthelastsection.In
+contrast,theeffectiverange r0≈1fmandshapeparameter
+P0≈−1.7fm3obeytheexpectednaturalscalingsof1 /Mhi
+and 1/M3
+hi, respectively. It seems therefore reasonable to
+start with the scalings ∆∼M2
+lo/µandg2/2π∼Mhi/µ2.
+WiththeevaluationofFeynmandiagramsandafterthere-
+quired resummation described in the previous section one
+finds
+TCS=−2π
+µC2
+ηe2iσ0/bracketleftigg
+−2π∆(R)
+µg2+π
+µg2k2−2kCH(η)/bracketrightigg−1
++···
+(10)
+where∆(R)is the only parameter in the Lagrangian, up to
+NLO,that isrenormalizedbythe Coulombloops,
+∆(R)=∆(κ)+µg2
+2πκ
+D−3
++2kC/bracketleftigg1
+D−4−ln/parenleftigg√πκ
+2kC/parenrightigg
+−1+3
+2CE/bracketrightigg,(11)
+withDthe numberof space-time dimensions, κthe renor-
+malization scale parameter, and CE=0.577...the Euler-
+Mascheroni constant. The amplitude matches with the ex-
+pansion of the ERE amplitude (4), from where one iden-
+tifies the ERE parameters in terms of the EFT couplings.EPJWeb ofConferences
+One thenfinds a0∼Mhi/M2
+lo. The low-energyscale Mlo∼√mαER≈20 MeV is nearly seven times smaller than a
+high momentum scale associated to either the pion mass
+or the excitation energy of the alpha particle, Mhi∼mπ∼/radicalbigmαE∗α≈140 MeV. Based on these numbers, one could
+expectconvergentresultsforobservablesat laboratoryen -
+ergiesupto 3MeV.
+TheBohrmomentum kCprovidesthescaleofCoulomb
+interactions. Due to the large ααreduced mass in its def-
+inition (5), kCturns out to be numerically of the order of
+Mhi. One can therefore expects large and important elec-
+tromagneticcontributions.Nevertheless,itisinteresti ngto
+discuss the other limit kC→0, when Coulomb is turned
+off. In this case one has 2 kCH(η)→ikand the denomina-
+toroftheLO amplitudereads
+T−1
+LO∝−1/a0+r0k2/2−ik. (12)
+For momenta k∼Mlothe first two terms are suppressed
+byMlo/Mhicompared to the last one. Therefore, all that
+is left at LO is the unitary term 1 /(−ik). In this limit, the
+8Be system exhibits non-relativistic conformal invariance
+[21] and the corresponding three-body system,12C, ac-
+quires an exact Efimov spectrum [6,8]. If it was possi-
+ble to somehow shield the charges of the αparticles, the
+above limit strongly suggests that two and three charg-
+lessαparticleswouldprobablybethebestnuclearsystems
+to observe few-bodyuniversality. The above scenario cer-
+tainlychangesforaphysicalvalueof kC,duetothebreak-
+ing of conformal invariance by the 1 /rCoulomb force.
+Nevertheless, the fact that the groundstate of8Be and the
+Hoyle state in12C remain very close to the threshold into
+α-particles suggests that this conformal picture is not far
+fromthereal case.
+The breaking of scale invariance by the Coulomb in-
+teractionintroducesthe scale kCin the propagationof two
+chargedparticles.AswehaveseeninEq.(4),theunitarity
+term is modified. The balance between strong-interaction
+termsandCoulomb-modifiedpropagationnowdependson
+boththestrong-interactionscale MloandkC.Whilethefor-
+meriskR=√mαER∼20MeV,thelatteris kC=αemZ2
+αµ∼
+60 MeV, and therefore a relative strength of kC/kR∼3.
+For momenta k∼kR, we are clearly in the deep non-
+perturbative Coulomb region. The Sommerfeld parameter
+ηreaches large values and the function 2 kCH(η) is very
+different from the usual unitarity term ik. Instead of ham-
+pering any simplification, it actually allows a low-energy
+expansionoftheCoulombfunction H.UsingStirling’sse-
+ries,
+lnΓ(1+z)=1
+2ln2π+/parenleftigg
+z+1
+2/parenrightigg
+lnz−z+1
+12z−1
+360z3+···,
+(13)
+andψ(z)≡(d/dz)lnΓ(z)in Eq.(9) gives
+H(η)=1
+12η2+1
+120η4+···+iπ
+e2πη−1.(14)
+Theunitaritytermisthusreplacedby2 kCH(η)∼k2/6kCat
+LO. This is now a factor k/6kCsmaller in magnitudethanthe unitarity term in the absence of Coulomb,and compa-
+rableto the effective-rangetermcomingfromthe dimeron
+kineticterm. Onecangraspit automaticallyifoneconsid-
+ers3kC∼Mhi, asitappearstobethecase numerically.
+3.2 resonancepole expansion
+Before incorporating the simplifications on the function
+H(η), one shouldtake a carefullookat the EFT amplitude
+uptoNLO.First,anisolatedremark.Sinceweassumethat
+P0is a parametrically small correction, one is allowed to
+reshuffletheperturbativeseriesandresum P0toallorders,
+thusrecoveringtheERE formula,
+T(ERE)
+CS=−2π
+µC2
+ηe2iσ0
+−1/a0+k2r0/2−k4P0/4−2kCH.(15)
+Second, one notices that equation (4) is valid for generic
+momenta k∼Mlo.However,itfailsintheimmediateprox-
+imity ofkR. This situation is familiar from the neutron-
+alpha case [17]. The power counting works for k∼Mlo
+except in the narrow region |k−kR|=O(M2
+lo/Mhi) where
+the LO denominator approaches zero and a resummation
+oftheNLOterm,hereassociatedwiththeshapeparameter,
+isrequired.Asonegetsclosertotheresonancemomentum
+kR, higher-order terms in the ERE are kinematically fine-
+tuned as well. This happensbecause the imaginarypart of
+the denominator is exponentially suppressed by a factor
+exp(−2πηR)∼10−8and the real part is allowed to be ar-
+bitrarily small. Nevertheless, this kinematical fine-tuni ng
+is not a conceptual problem. From the EFT point of view,
+each newfine-tuningcanbe accommodatedbyreshu ffling
+the series and redefining the pole position. Such a proce-
+dure works fine with a small number of kinematical fine-
+tunings, but is not practical in the ααsystem. A better al-
+ternative is to performan expansionaround the resonance
+pole position, starting from the resummed (ERE) ampli-
+tude. The situation here is nearly identical to the NNsys-
+tem,whereonecanchoosetoexpandtheamplitudearound
+the bound-statepole[22] ratherthanaroundzeroenergy.
+Agreatsimplificationisachievedfromthefactthatthe
+resonanceliesinthedeepCoulombregime,whereEq.(14)
+provides an accurate representation of Hup to the preci-
+sion we are considering.The real terms shownin Eq. (14)
+then become an expansion in powers of ∼(k/3kC)2=
+O(k2/M2
+hi). Given the asymptotic feature of the Stirling’s
+series, at somepointthe remainingtermscannolongerbe
+expanded;atthatpointtheremaindershouldbetreatedex-
+actly.Inlowestorders,however,wecanusethesuccessive
+terms shown in Eq. (14): numerically, the terms up to η−4
+worktobetterthan3%for ELAB=3MeV.
+The expansion (14) not only simplifies a complicated
+function of kC/k, but also makes the physics around the
+resonance more transparent. Since the “size” of the res-
+onance, 1/kR, is much larger than the Bohr radius 1 /kC,
+the Coulomb interaction is e ffectively short ranged, and
+the real part of Hresembles the ERE expansion. In the
+amplitude TCSthe different strong and Coulomb coe ffi-
+cientsproportionaltoacommonpowerof kcanbegrouped19thInternationalIUPAP ConferenceonFew-BodyProblemsinPhy sics
+together, where one observes that they have comparable
+sizes. We thereforedefine
+˜r0=r0−1
+3kC,˜P0=P0+1
+15k3
+C,(16)
+andsoon.UptoNLO werewrite TCSas
+TCS=−2π
+µC2
+ηe2iσ0
+−1/a0+˜r0k2/2−˜P0k4/4−ikC2η
+=−2π
+µC2
+ηe2iσ0
+˜r0(k2−k2
+R)/2−˜P0(k4−k4
+R)/4−ikC2η
+=−2π
+µC2
+ηe2iσ01
+˜r0(k2−k2
+R)/2−ikC2η/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipupright
+LO term
++˜P0
+4(k4−k4
+R)
+(˜r0(k2−k2
+R)/2−ikC2η)2
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipupright
+NLO correction+...,(17)
+where
+k2
+R=2
+a0˜r01
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+LO term−˜P0
+a0˜r2
+0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+NLO correction+....(18)
+From this expressionone sees directly that kR∼Mlo, with
+corrections ofO(M2
+lo/M2
+hi). Note that we keep the exact
+form of the imaginary term in Eq. (17): even though it is
+negligible at k∼kR, it has an important exponential de-
+pendence on the energyresponsible for keeping the phase
+shiftsrealintheelastic regime.
+Whena0<0 andr0<1/3kC, we have k2
+R>0 and
+the two poles of Eq. (17) are located in the lower half of
+the complex-momentum plane very near the real axis, as
+expected for a narrow resonance. The amplitude TCScan
+be written in terms of the resonance energy ER=k2
+R/2µ
+andtheresonancewidth Γ(E)as1
+TCS=2πe2iσ0
+µ/radicalbig
+2µEΓ(E)/2
+E−ER+iΓ(E)/2.(19)
+Onefindsthat
+Γ(E)=Γ(ER)e2πkC/kR−1
+e2πkC/k−11
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+LO term
+−µ2˜P0
+2πkC/parenleftig
+e2πkC/kR−1/parenrightigΓ(ER)
+2(E−ER)
+/bracehtipupleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext/bracehext /bracehtipupright
+NLO correction+...,(20)
+where
+Γ(ER)=−4πkC
+µ˜r01
+e2πkC/kR−11
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+LO term+˜P0k2
+R
+˜r0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+NLO correction+....
+(21)
+1SeealsoRef.[23].Thewidthisverysmallbecauseofthelargevalueof2 πkC/kR
+in theexponential.
+In the form of Eqs. (19) and (20) we can keep ERand
+Γ(ER)fixedateachorderintheexpansion.Notethatthese
+equations do not change to this order if one makes a dif-
+ferentchoice— e.g.,(k2−k2
+R)2insteadof k4−k4
+R—forthe
+formofthe ˜P0terminEq.(17).Thebehaviorofthephase
+shiftaround ERisguaranteedtobeofaresonanttype,asit
+automaticallysatisfies theconstraints
+δc
+0(ER)=π
+2(resonanceenergy) ,(22)
+and
+/parenleftiggdδc
+0(E)
+dE/parenrightigg
+ER=2
+Γ(ER)(resonancewidth) .(23)
+3.3 confronting the data
+Unfortunately ααscattering data at low energies are not
+abundant. Nevertheless, all the existing measurements at
+ELABupto5MeVshowthatitisdominatedbythe Swave,
+thankstothepresenceofthe( Jπ,I)=(0+,0)resonanceim-
+mediately abovethreshold.Early determinationsof the 0+
+resonance energy were performed in reactions like11B+
+p→2α+α(see [20] and references therein). Later mea-
+surementsofthescatteringof4Heatomson4He+ions,es-
+peciallyprojectedtoscantheresonanceenergyregionand
+supplemented with a detailed analysis [24,25] improved
+the determination of the resonance energy and width to
+theircurrentlyacceptedvalues, ER
+LAB=184.15±0.07keV
+andΓR
+LAB=11.14±0.50eV [25]. TheresonanceCM mo-
+mentumisthus kR=/radicalig
+µER
+LAB≈18.5MeV.
+The obvious way to extract the ERE parameters from
+theS-wave phase shift is to fit (the cotangent of) the lat-
+tertotheEREformula.However,fitsto ααscatteringdata
+alone are not able to constrain well those parameters and,
+apartfromthelargeuncertainties,don’tseemtopredictth e
+existenceofthe0+resonance.Imposingthatthephaseshift
+crossesthevalue π/2attheresonanceenergyimprovesthe
+situation, though not dramatically. Following a previous
+suggestion [26], Ref. [27] used not only the available res-
+onance energy, but also the width [24] to constrain these
+parameters.Theobtainedvaluesfor a0=−1.76×103fm,
+r0=1.096fm,andP0=−1.654fm3madeuseofthereso-
+nanceenergyandwidth fromRef. [24]. Laterwe compare
+these numberswith theonesfromourEFTfits.
+At energiesbelow ELAB=3 MeV data were obtained,
+and a phase-shift analysisperformed,by Ref. [28]. Values
+of the latter can be found in Table II of the review [20].
+Since it is well-determined experimentally, and due to its
+relevance to the triple-alpha process, we use the 0+reso-
+nance parametersfrom Ref. [25] as important constraints.
+ThisisinlinewiththeEFTapproach,wherelower-energy
+observableshave preferenceoverhigher-energyones. The
+relationshipamongthe EFT parametersand the resonance
+energyandwidthallowsonetoreducethenumberofvari-
+ables to be adjusted at each order in the power counting.EPJWeb ofConferences
+Below, we also show the ERE from Ref. [27] for orien-
+tation, and comment on the extremely large value of the
+scatteringlength a0,whichsuggestsalargeamountoffine-
+tuning in the parameters of the underlying theory away
+fromthenaturalnessassumption.
+Inthepowercountingthatwediscussedforthe ααsys-
+tem,theamplitude TCSforgenericmomentaisgivenupto
+NLObyEq.(17).Asdemonstratedintheprevioussubsec-
+tion, this expression combines the deep-non-perturbative
+Coulomb approximation (14) for the function Hwith the
+expansion around the resonance pole, thus preventing the
+need formultiple kinematicalfine-tunings.In LO, the two
+parameters a0and ˜r0can be obtained from the constraints
+(18) and(21).At NLO, a fit to scatteringdata isneededto
+determine ˜P0.
+ 90 120 150 180
+ 0  1  2  3δ0 [degrees]
+ELAB [MeV]LO
+NLO
+ERE fit
+Afzal et.al.
+ 0 0.05 0.1 0.15 0.2
+ 0  1  2  3K
+ELAB [MeV]LO
+NLO
+ERE fit
+Afzal et.al.
+Fig. 2.EFT results for ααscattering at LO (dotted) and NLO
+(solid),comparedagainstthedata.Toppanel:phaseshift δc
+0.Bot-
+tom panel: K(η)≡C2
+η(cotδc
+0−i)/2η+H(η).
+Figure2 showsour resultswith the resonanceposition
+and width constraints, compared to the available S-wave
+phase shiftsbelow ELAB=3 MeV. In the regionabovethe
+resonance, where scattering data are shown, the LO curve
+isaprediction,whichisconsistentwiththefirstfewpoints
+butthenmovesawayfromthedata.TheNLOcurvehas ˜P0
+as an extra parameter, which is determined from a globalχ2-fittoscatteringdatashown.Asexpectedfromaconver-
+gentexpansion,the descriptionof thelow-energydataim-
+proves with increasing order. At about 3 MeV and above,
+higher-order contributions are expected to be important,
+as suggested by the discussion on the relevant scales and
+manifestinthegrowingdi fferencebetweentheNLOcurve
+andbothLOcurveanddatapoints.Alsoshownareresults
+from a fit using the conventional ERE formula, Eq. (15),
+in ordertostressthedi fferencesbetweenthisandourEFT
+approach.
+Table 1 shows the values of ERE parameters used to
+produce the curves of Fig. 2, and compare them with the
+values [27] obtained from e ffective range theory. At LO,
+our values for a0andr0are consistent with the ones from
+Ref. [27], remarkably r0. The NLO corrections, however,
+spoil this initial LO agreement. The reason for this devia-
+tion could be due to the way the width constraint was ob-
+tainedinRef. [27]. ItsEq.(4)reads
+dh
+dk2(ηR)−1
+µΓ(ER)π
+e2πkC/kR−1=1
+4kC/parenleftig
+r0−P0k2
+R/parenrightig
+,(24)
+whereh(η)≡Re[H(η)]. Following this reference’s pro-
+cedure, we reproduced the quoted value of the width (6.4
+eV) only when dh(ηR)/dk2was approximated by 1 /12k2
+C
+[see Eq. (14)]. However, that is equivalent to neglect the
+electromagnetic term of ˜P0in Eq. (21), which is incon-
+sistentsincethestrongpiececontributesatthesameorder .
+NeglectingthistermexplainstheagreementatLOanddis-
+agreementat NLO betweenourandRef.[27] numbersfor
+r0. With a smaller r0, and therefore larger (negative) ˜ r0,
+one can also understand why Ref. [27] obtains a smaller
+a0, as the product ˜ r0a0is inverselyproportionalto the res-
+onance momentum squared. By repeating the Ref. [27]’s
+procedure including the width constraint consistently, we
+obtained essentially the same values as in our EFT fits.
+ThisupdatedERE fit isalso showninTable 1.
+a0(103fm) r0(fm)P0(fm3)
+LO−1.80 1.083 —
+NLO−1.92±0.091.098±0.005−1.46±0.08
+ERE(our fit)−1.92±0.091.099±0.005−1.62±0.08
+ERE[27]−1.65±0.171.084±0.011−1.76±0.22
+Table 1.ERE parameters extracted from EFTfits in the firsttwo
+orders, compared with values from two ERE fits, our own and
+Ref. [27]’s.
+Our fits reveal both e ffective range r0∼1/(180MeV)
+and shape parameter P0∼1/(170MeV)3scaling with
+powers of Mhi, in agreement with our a prioriestimate.
+The relative errors in a0and ˜r0at LO are estimated to be
+of the order of the EFT parameter expansion, Mlo/Mhi∼
+1/7≈15%.AtNLO,themainsourceofuncertaintycomes
+from the precision of the most recent measurement of the
+resonance width [25], lying between 4–5%. The uncer-
+tainty in ˜P0, givenbythe χ2-fit, is of the same order.Note
+that the small relative error in r0compared to the one in19thInternationalIUPAP ConferenceonFew-BodyProblemsinPhy sics
+˜r0is due to the former being an order of magnitudelarger
+than the latter, as we discuss in the next subsection. The
+NLOerrorsfoundhereareafactoroftwosmallerthanthe
+onesobtainedbyRef. [27].
+One should stress that the accurate value of the reso-
+nancewidth, ΓR=5.57±0.25eV,imposestightconstraints
+onourfits,through a0andr0.Asignificantimprovementin
+our NLO fit and overall agreement with data is observed.
+But lookingin detail, one sees that the theoreticalcurveis
+not able to cross the errorbandsof many scatteringpoints
+below3MeV.Thiscanbeinferredfromthe χ2/datum≃4.
+In principle, a better agreement should be achieved by an
+N2LO calculation. However, that would introduce an ex-
+tra parameter that is mostly determined by the scattering
+data, and an expected agreement could mask any possi-
+ble inconsistencies between the phase shifts and the reso-
+nance parameters. The high NLO χ2/datum suggests that
+theyare not compatiblewith each otheror,at least, oneof
+them has overestimatedprecision. As a test we fitted both
+our NLO EFT and ERE expressions to the scattering data
+without the constraintsfromthe resonancewidth. In these
+twocases, descriptionof S-wavephaseshiftsismuchbet-
+ter at the expense of an underpredicted resonance width,
+ΓR=4.9±0.6eVwithEREand ΓR=2.87±0.23eVwith
+EFT. The ERE result is still consistent with the measured
+ΓRthankstoitslargeerrorbar.InEFT,wherelower-energy
+datahavehigherpriority,thediscrepancyisamplified.The
+problem is even more pronounced if the fit is performed
+using data up to 2.5 MeV instead of 3 MeV: the results
+ΓR=4.2±0.6eVwithEREand ΓR=2.93±0.34eVwith
+EFTfallbeyondthequotedexperimentalerrorbars.Oddly,
+thistendencycontinuesasonelowerstheupperlimitinthe
+fit.Reanalysesoftheexistinglow-energydataorevennew
+measurementsseemnecessarytoresolvethisdiscrepancy.
+3.4 fine-tuning puzzle
+Asurprisingfeatureofthe ααsystemistheverylargemag-
+nitudeofa0,evenifcomparedtothelargescatteringlength
+observedin the nucleon-nucleonsystem.Thelatter case is
+thoughtto bethe outcomeof a fine-tuningin the QCD pa-
+rameters, giving rise to an anomalously low momentum
+scale. It is plausible to expect that this fine-tuning propa-
+gates to heavier systems. However,the enormousvalue of
+a0in theααis suggestive of a more delicate tuning, with
+electromagneticinteractionsplayinga crucialrole.
+To better understand the puzzle it is worth looking in
+details at Eq. (11). The Coulomb loop contributions, pro-
+portional to the curly brackets, scale as M2
+hi/µ, given that
+g2/2π∼Mhi/µ2and 2kC∼Mhi. For the scaling ∆(R)∼
+M2
+lo/µto holdthe scale-dependentparameter ∆(κ)must be
+of the same size of the Coulomb loops, and strongly can-
+cel each other to produce a result ( Mlo/Mhi)2∼100 times
+smaller. This amount of fine-tuning is necessary to obtain
+a resonance at the right position. Similar cancellation is
+observed for the P-wave resonance in nα[17]. However,
+in the present case the fine-tuning is caused by a delicate
+balancebetweenthestrongandelectromagneticforces.Thisfine-tuningscenariobecomesevenmoreenigmatic
+by considering the width. The tiny value of ΓR≈6 eV
+is intimately related to the resonance momentum, as can
+be immediately seen in Eq. (21). That is simply an e ffect
+of the Coulomb repulsion—the resonance trapped inside
+the envelopeformedbythe strongplusCoulombpotential
+has to tunnel through a larger potential barrier as its en-
+ergy gets smaller. The probability of tunneling is propor-
+tional to the resonance width, from where the smallness
+ofΓRfollows. Despite the tiny value, the width has an as-
+sociated scale of 4 πkC/µ˜r0which is quite large. A natural
+estimative for this quantity would be ∼M2
+hi/µ, implying
+that ˜r0∼1/Mhi. However, from the width constraint one
+finds ˜r0∼Mlo/M2
+hi, leading to an r0that roughly cancels
+against 1/3kCwith a remainder about 10% smaller. From
+Eq. (18) onesees that the extrafine-tuningin ˜ r0is respon-
+sibleforanextraincreaseinthescatteringlengthbyafac-
+tor ofMhi/Mlo, becoming effectively|a0|∼M2
+hi/M3
+lo. It is
+remarkable, that if the strong forces generated an r011%
+larger the8Be ground state would be bound, with drastic
+consequencesin the formationof elementsin the universe
+(see alsoRef.[29]).
+4pαscattering
+Thepαscattering at low energiesis mostly dominated by
+theP3/2partialwave,duetothepresenceofaresonanceat
+protonenergies Ep≈2MeV.S-wavealsogivesanimpor-
+tantcontributionfor Ep≤5MeV,whichisthelow-energy
+region we are interested in (see below). In the P1/2wave
+the phase shift varies smoothly up to Ep=18 MeV, sug-
+gesting the existence of a very broadresonancefor proton
+energies between 8 and 15 MeV [30]. However,to the or-
+derinthepowercountingandenergiesthatweconsider,it
+behavesasatypicalhigher-ordere ffect.Higherwavephase
+shifts areless thana few fractionsof degreebelow Ep≈6
+MeV, where D-wavesstart beingrelevant.Therefore,only
+twopartialwavescontributetothelow-energyEFTforthe
+pαsystem,uptoandincludingNLO.
+The Lagrangian for nucleon and alpha particles inter-
+actingvia S1/2andP3/2partialwavesiswrittenas[31,16,17]
+LNα=φ†iD0+D2
+2mαφ+N†iD0+D2
+2mNN
++ς0+s†/bracketleftbigg
+−∆0+/bracketrightbigg
+s+ς1+t†iD0+D2
+2(mα+mN)−∆1+t
+−1
+4FµνFµν+g1+
+2/braceleftbigg
+t†S†·/bracketleftbigg
+NDφ−(DN)φ/bracketrightbigg
++H.c.
+−r/bracketleftbigg
+t†S†·D(Nφ)+H.c./bracketrightbigg/bracerightbigg
++g0+/bracketleftbigg
+s†Nφ+φ†N†s/bracketrightbigg
++ς0+s†/bracketleftigg
+iD0+D2
+2(mα+mN)/bracketrightigg
+s
++g′
+1+t†iD0+D2
+2(mα+mN)2
+t, (25)EPJWeb ofConferences
+Table 2. P3/2pαresonance parameters from the extended R-
+matrixanalysis of Ref.[32].
+kr(MeV) ki(MeV) ER(MeV)ΓR/2(MeV)
+51.1 9.0 1.69 0.61
+whereφandNarethealphaandnucleonfieldswithmasses
+mαandmN,respectively,and
+r=(mα−mN)/(mα+mN). (26)
+The auxiliary dimeron fields tandscouple to NαinP3/2
+andS1/2waves,withleading-ordercouplingconstants g1+
+andg0+,respectively.The Si’sare2×4spin-transitionma-
+trices between J=1/2andJ=3/2 total angularmomen-
+tum states. The sign variables ς0+,ς1+=±1 in front of
+the kinetic term of the dimeron fields are adjusted to re-
+produce the signs of the respective e ffective ranges. The
+gauge-covariant derivative is defined in the usual way in
+termsofthephotonfield Aµ,
+Dµ=∂µ+ieZ1+τ3
+2Aµ, (27)
+whereZis the chargeof the “particle” whose field the co-
+variantderivativeactson, τ3isthez-directionPaulimatrix
+actinginisospinspace,and Aµis thephotonfield.
+Thepowercountinginessenceconsistsofestablishing
+howthesizeofthe(renormalized)EFTcouplingconstants
+dependonthemomentumscalespresentinthesystemone
+wishestodescribe.Theformerhaveadirectrelationtoob-
+servables, namely, the ERE parameters. In the pαsystem
+oneisinterestedindescribingthescatteringregionaroun d
+theP3/2resonancelocatedat k=kp∼Mlo,where
+kp=kr−iki (28)
+is the (complex) resonance momentum in terms of real
+quantities krandki. Table 2 lists the resonance parame-
+tersextractedfromthe so-calledextended R-matrixanaly-
+sis used in Ref. [32], which indicates that kr∼Mlo∼50
+MeV and ki∼M2
+lo/Mhi∼10 MeV. The suppression of
+kirelative to krmatches with narrow character of the P3/2
+resonance. As in the ααcase [18], Mhiis set as the mo-
+mentum required to excite the αcore or the pion mass,
+both around 140 MeV. Our expansion parameter is of the
+order of 1/3, allowing us to expect convergent results for
+protonenergiesupto 5MeV.
+Table 3 shows the numerical values of the S1/2and
+P3/2ERE parameterstaken fromRef. [33]. As we can no-
+tice, whileP1+/4∼r0+/2∼1/Mhiare in agreement with
+the natural assumption, the quantities a0+∼1/Mlo,a1+∼
+1/M3
+lo, andr1+/2∼Mloare fine-tuned. To be consistent
+withthesescalingstheEFTcouplingsinEq.(25)mustbe-
+haveas
+∆(R)
+1+∼M2
+lo
+2µ,g(R)2
+1+
+3π∼1
+µ2Mlo,andg′
+1+
+4∼µ
+MloMhi
+(29)Table 3. S1/2andP3/2pαparameters extractedfrom the EREfit
+of Ref. [33].
+a0+(fm) r0+(fm)
+4.97±0.121.295±0.082
+a1+(fm3) r1+(fm−1)P1+(fm)
+−44.83±0.51−0.365±0.013−2.39±0.15
+in theP3/2channel,and
+∆(R)
+0+∼MhiMlo
+2µandg2
+0+
+π∼Mhi
+µ2(30)
+inS1/2, whereµnow stands for the pαreduced mass. In
+thelattercasethesituationisnearlyidenticaltotheprot on-
+protoncase [19]: upto NLOthe S1/2amplitudereads
+T0+=−2π
+µC2
+ηe2iσ0
+−1/a0+−2kCH(η)/bracketleftigg
+1−r0+k2/2
+−1/a0+−2kCH(η)/bracketrightigg
+.
+(31)
+ForP-wave interactions, Coulomb can be introduced
+in a straightforwardway [31]. The evaluation of Feynman
+diagramsandthenecessaryresummationresultsin
+T1+=−2π
+µC(1)2
+ηe2iσ1k2P1+(θ)
+−1/a1++r1+k2/2−2kCH(1)(η)
+×/bracketleftigg
+1+P1+k4/4
+−1/a1++r1+k2/2−2kCH(1)(η)/bracketrightigg
+,(32)
+where
+C(1)2
+η=(1+η2)C2
+η, (33)
+H(1)(η)=k2(1+η2)H(η), (34)
+and thevariable kC=ZαZpµαemisadaptedto the pαcase.
+TheP3/2projectorisgivenby
+P1+(θ)=2cosθ+σ·ˆnsinθ, (35)
+whereθistheanglebetween kandk′—theinitialandfinal
+momenta in the center-of-mass frame, respectively— and
+ˆn=k×k′/|k×k′|istheunitvectornormaltothescattering
+plane. In this form, the T1+amplitude requires the same
+kinematical fine-tuning discussed in Sect. 3.2. For prag-
+matical reasons, we therefore adopt the expansion around
+the resonancepoleinthischannel.
+4.1P3/2resonance pole expansion
+For practical applications, the more e fficient way of ex-
+pressing the EFT amplitudein the presence of low-energy
+resonances is to perform the resonance pole expansion.
+Onenicefeatureisthat,ateveryorderinthepower-countin g,19thInternationalIUPAP ConferenceonFew-BodyProblemsinPhy sics
+the amplitude maintains its resonant behavior at the ex-
+pected position with the correct magnitude. It is straight-
+forward to extend the ideas outlined in Sect. 3.2 to the
+present case, with a resonance pole in the fourth quadrant
+ofthecomplexmomentumplane[31].Withthepolegiven
+byEq.(28) onerewritesEq.(32)as
+T1+=−2π
+µC(1)2
+ηe2iσ1k2P1+(θ)
+˜r1+
+2(k2−k2p)−2kC/bracketleftbigg
+H(1)/parenleftigkC
+k/parenrightig
+−H(1)/parenleftbigg
+kC
+kp/parenrightbigg/bracketrightbigg
+×1+P1+(k2−k2
+p)2/4
+˜r1+
+2(k2−k2p)−2kC/bracketleftbigg
+H(1)/parenleftigkC
+k/parenrightig
+−H(1)/parenleftbigg
+kC
+kp/parenrightbigg/bracketrightbigg,
+(36)
+where
+˜r1+=−kr2Li/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+LO−2ikiP1+/bracehtipupleft/bracehext/bracehext/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehext/bracehext/bracehtipupright
+NLO(37)
+andLi,Lraredefinedat thepolepositionfrom
+2kCH(1)(kC/kp)=k3
+rLr+2ik2
+rkiLi.(38)
+Thefactorsinfrontof Lr,Li∼O(1)reflectthebehaviorof
+the Coulomb function H(1)(η) around the resonance. No-
+tice that, as in the ααamplitude, T1+is parametrizedonly
+in terms of krandkiat LO, with the extra parameter P1+
+appearingat NLO.
+TheP3/2scatteringlengthande ffectiverangearegiven
+by
+r1+
+2=−krLi−krP1+
+21−k2
+i
+k2r
+≃−krLi/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+LO−krP1+
+2/bracehtipupleft/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehtipupright
+NLO, (39)
+1
+a1+=−k3
+rLr+Li1−k2
+i
+k2r−krP1+
+41+k2
+i
+k2r2
+≃−k3
+rLr+Li/bracehtipupleft/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehtipupright
+LO−krP1+
+4/bracehtipupleft/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehtipupright
+NLO, (40)
+andnicelyshowthescalings1 /a1+∼M3
+loandr1+/2∼Mlo
+thatwereexpectedfromthenumbersofTable3.
+4.2 results
+Inordertotestourpower-countingassumptions,wefitthe
+expressions for the S1/2andP3/2amplitudes to the corre-
+sponding phase shifts and compare the obtained ERE pa-
+rameterswith the ones in the literature. We used the num-
+bers from Table 5 of Ref. [33] in our fit. The results are
+shown in Table 4. In our fits we have assumed that all
+thepointshavethesamestatisticalweight,sinceerrorbar sTable4. S1/2andP3/2pαEREparametersextractedinourfitsat
+LO andNLO.
+order a0+(fm)r0+(fm)
+LO 7.4+8.0
+−2.2—
+NLO 4.81+0.05
+−0.211.7+1.3
+−0.8
+order a1+(fm3)r1+(fm−1)P1+(fm)
+LO−58.0+11.0
+−29.0−0.15+0.14
+−0.09—
+NLO−44.5+1.6
+−0.1−0.40+0.04
+−0.10−2.8+1.0
+−1.8
+Table 5.P3/2pαresonance parameters extracted from our fits at
+LO andNLO.
+P3/2kr(MeV) ki(MeV) ER(MeV)ΓR/2(MeV)
+LO 50.6+1.2
+−2.510.3+1.4
+−0.81.64+0.09
+−0.180.70+0.12
+−0.09
+NLO 50.7+0.5
+−0.69.40+0.01
+−0.101.66+0.04
+−0.040.63+0.01
+−0.02
+werenotprovided.However,weincorporatedtheEFTsys-
+tematicerrorestimatesbyimposingthattheamplitudecan-
+not be reproduced to a precision better than ( k/2mπ)n+1,
+wherekis the CM momentum of the p αsystem,mπour
+high-energy scale (the pion mass), and nthe order in the
+EFT expansion. The maximum and minimum values of
+each ERE parameter were determined by multiplying the
+amplitudebyafactorof1 +x(k/2mπ)n+1,wherexisacon-
+stant that varies between −1 and 1. For the central value,
+such constant is zero. As one can see, our results are con-
+sistent with the ones shown in Table 3 at NLO. At LO the
+central valuesare slightly o ff,neverthelessconsistentwith
+an expansionparameteroftheorderof1 /3.
+In Table 5 are shown our fit results for the P3/2res-
+onance parameters. One should stress that krandkiare
+the variables determined from the fit, from which a1+and
+r1+can be extracted via Eqs. (40) and (39). The results
+are consistent with the values in Table 2, except for ki
+which is slightly o ff. That is probably due to the fact that
+it is a subleading e ffect compared to kr,ki∼M2
+lo/Mhi,
+and to improve the agreement one needs to go one or-
+der higher. Nevertheless, the resonance energy and width
+are consistent with Ref. [32], which used the extended R-
+matrix analysis. The latter is based on the same concept
+that the resonance is a manifestation of a pole of the am-
+plitudeinthelowerhalfofthecomplexenergyplane.Both
+our and Ref. [32] results disagree with the traditional R-
+matrix analysis, which are nowadays considered less reli-
+ablein extractingresonanceproperties[30].
+In Fig. 3 we show the EFT results for S1/2andP3/2
+phase shifts at LO and NLO. A very good convergenceis
+observed,especiallyforthe P3/2channel.Thecurvesusing
+theEREexpressionsareomitted,sincetheyessentiallyfal lEPJWeb ofConferences
+on top of the NLO curve and cannot be distinguished in a
+plot.
+-60-40-20020406080100120140
+0.0 1.0 2.0 3.0 4.0 5.0δ [degrees]
+Ep [MeV]P3/2
+S1/2LO
+NLO
+Arndt et al.
+Fig. 3.EFT results for S1/2andP3/2scattering phase shifts at
+LO (dotted) and NLO (solid), compared against the partial wa ve
+analysis results from Arndt et al.(diamonds).
+Since we obtained a very goodfit of the S1/2andP3/2
+EFTamplitudestothecorrespondingphaseshifts,itisuse-
+ful to compare directly to observables in order to test our
+assumptions about the neglected higher-order terms. This
+is shown in Fig. 4 for the elastic di fferential cross-section
+measured at the laboratoryscattering angle θ=140◦. The
+sharpenhancementatlowenergiesreflectsthecharacteris-
+tic dominance of the Mott cross-section. One clearly sees
+thatalreadyat LO(dottedline)onehasa gooddescription
+of data, especially around the resonance peak. There is a
+slight improvement at NLO (thick solid line). The curves
+alsoagreewellwiththeEREcurve(thinsolidline),which
+uses the ERE formula for the S1/2,P3/2, andP1/2partial
+wavesandthenumericalparametersfromRef. [33].How-
+ever, small deviations start to show up at Ep≈3.5 MeV,
+indicating an increasing importance of higher order terms
+as one goes higher in energy. The deviations, which seem
+to come mainly from the P1/2channel, can be eliminated
+bygoingtohigherordersinthe powercounting.
+5 summary
+I reviewed the programme of halo /cluster EFT suitable to
+study several interesting nuclearprocessesthat play a rol e
+in nuclear astrophysics. I concentrated on two important
+aspects commonly present in those systems, namely, nar-
+rowresonancesandCoulombinteractions.The ααandpα
+interactions were given as examplesof how setting up the
+formalism,thenusedaspracticalapplications.
+Thestudyofthe ααinteractionrevealedaveryinterest-
+ing picture,wherethe scales of the stronginteractioncon-
+spire to produce a system with an almost non-relativistic
+conformal symmetry. One also observed extra amount of
+fine-tuning once electromagnetic interactions between the
+particlesare turnedon, upto the point ofgeneratingan S-
+wave scattering length that is three orders of magnitude 0 0.05 0.1 0.15 0.2 0.25
+ 0 1 2 3 4 5 6dσ/dΩ [barn/sr]
+Ep[MeV]θ=140 degLO
+NLO
+ERE
+Nurmela et.al.
+Fig. 4.EFT results at LO (dotted) and NLO (thick solid) for pα
+elasticcross-sectionat θ=140◦laboratoryscatteringangle,com-
+pared against the partial wave analysis results from Arndt et al.
+(thinsolid) and measured data point from Ref.[34](diamond s).
+larger than the typical range of the interaction. Despite
+the struggle in understanding the subtle cancellations be-
+tweenthestrongandelectromagneticforces,theformalism
+is quite successful phenomenologically. In fact, thanks to
+those cancellations we were able to pin down the S-wave
+effective range parameterswith an improvedaccuracyrel-
+ativeto previousstudies.
+We extended the formalism to include P-waves with
+resonance and Coulomb interactions when dealing with
+thepαsystem. We performed the expansion of the P3/2
+amplitude around the resonance pole, which allowed us
+to extract the resonance properties directly from a fit to
+the phase shift. We derived expressions for the scattering
+lengthandeffectiverangeintermsoftheresonanceparam-
+eters, that explainsthe scalingsof the formerwith the low
+andhighmomentumscales MloandMhiofthetheory.Our
+resultsatLOandNLOexhibitgoodconvergenceataratio
+of about1/3,and the resonanceenergyand width are con-
+sistentwiththeonesusingtheextended R-matrixanalysis.
+Comparisonwiththedi fferentialcross-sectionat140◦lab-
+oratoryanglereassurestheconsistencyofthepowercount-
+ing,withP1/2contributionstartingtoshowuponlyforpro-
+tonenergiesbeyond3.5MeV.
+Theααand nucleon- αinteractions are the basic ones
+beforeconsideringmorecomplicatedclustersof αandnu-
+cleons. An interesting example is the Hoyle state in12C,
+which plays a key role in the triple-alpha reaction respon-
+sible for the formation of heavy elements. A model-study
+with this state in mind was developed in Ref.[35], where
+a perturbative treatment of the Coulomb interaction was
+proposed. This idea might be useful to handle the techni-
+cal difficultiesinvolvingthreechargedparticles.
+Acknowledgments
+IwouldliketothankHans-WernerHammer,BiravanKolck,
+andCarlosBertulaniforstimulatingcollaboration,andth e19thInternationalIUPAP ConferenceonFew-BodyProblemsinPhy sics
+organizersofthe19thInternationalIUPAPFew-BodyCon-
+ference 2009 for the opportunity to present this talk. This
+work was supported by the BMBF under contract number
+06BN411.
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\ No newline at end of file
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+arXiv:1001.0541v2  [gr-qc]  25 Oct 2010Late–time Kerr tails: generic and non–generic initial data sets, “up” modes, and
+superposition
+Lior M. Burko1,2and Gaurav Khanna3
+1Department of Physics, University of Alabama in Huntsville , Huntsville, Alabama 35899, USA
+2Center for Space Plasma and Aeronomic Research,
+University of Alabama in Huntsville, Huntsville, Alabama 3 5899, USA
+3Physics Department, University of Massachusetts at Dartmo uth, N. Dartmouth, Massachusetts 02747, USA
+(Dated: October 18, 2010)
+Three interrelated questions concerning Kerr spacetime la te–time scalar–field tails are considered
+numerically, specifically the evolutions of generic and non –generic initial data sets, the excitation of
+“up” modes, and the resolution of an apparent paradox relate d to the superposition principle. We
+propose to generalize the Barack–Ori formula for the decay r ate of any tail multipole given a generic
+initial data set, to the contribution of any initial multipo le mode. Our proposal leads to a much
+simpler expression for the late–time power law index. Speci fically, we propose that the late–time
+decay rate of the Yℓmspherical harmonic multipole moment because of an initial Yℓ′mmultipole is
+independent of the azimuthal number m, and is given by t−n, wheren=ℓ′+ℓ+1 forℓ< ℓ′and
+n=ℓ′+ℓ+3 forℓ≥ℓ′. We also show explicitly that the angular symmetry group of a multipole
+does not determine its late–time decay rate.
+PACS numbers: 04.70.Bw, 04.25.Nx, 04.30.Nk
+I. INTRODUCTION AND SUMMARY
+Perturbations of black holes decay first with complex frequencies, known as the quasinormal modes of of the black
+hole, taken over at late times by the tails of the perturbation field— d ecaying as power–laws of time. The decay
+rate of the late–time tails in the spacetime of spinning black holes has b een the subject of much debate and some
+confusion. Much of the debate in the literature was focused on the late–time decay rate of an initially pure azimuthal
+hexadecapole ( ℓ′= 4,m= 0) scalar field perturbation. All authors agree that at late times t he field is dominated
+by its monopole moment, and that the decay rate would be according to an inverse power of time, but different
+claims were made as to the value of the power–law exponent. Specific ally, there were claims for decay rate along fixed
+Boyer–Lindquist raccording to t−3[1],t−5[2, 3], and even t−5.5[4].
+In the last couple of years the behavior of the power–law tails has be en clarified in a number of papers [5–7] (and
+the reader is referred to the detail therein), yet there are some remaining interesting detail waiting to be unveiled.
+In particular, it was shown in [7] that the disagreement between th et−3andt−5behaviors can be attributed to the
+use of different initial data sets (see also [6]), but not to the use of d ifferent slicing conditions as was suggested in
+[5]. Most importantly, Boyer–Lindquist slicing is not unique in terms of t he resulting tails as had been argued, but
+rather belongs to an equivalency class of slicing conditions (including a lso ingoing Kerr slicing), all having similar tail
+structure properties.
+For the evolution of scalar fields on a fixed background of a Kerr blac k hole, the value of the power-law index,
+namely the exponent nin the late-time decay rate t−n, was given by Hod [2] as n=ℓ′+m+1 for even ℓ′−m>2,
+n=ℓ′+m+ 2 for odd ℓ′−m >2, andn= 2ℓ′+ 3 forℓ′−m <2, whereℓ′is the multipole moment of the initial
+perturbation and mis its azimuthal number. This expression for nis rather complicated and notably depends on the
+azimuthal number m. We argue that the added tacit requirement that it is the decay rat e of the slowest decaying
+mode that is of usual interest masks the inherent simplicity of nin the Hod formula. Specifically, we propose that
+the decay rate of the ℓmode because of an initial ℓ′mode is independent of m, and is given by n=ℓ′+ℓ+ 1 for
+ℓ < ℓ′andn=ℓ′+ℓ+ 3 forℓ≥ℓ′. Notice that this proposal is consistent with Hod’s formula and with [7 ] when
+they are applicable, and generalizes them also to all other cases. It is therefore a proposal for the general formula for
+the decay rate of any multipole moment of the tail because of any mu ltipole moment of the initial data. It is striking
+that this more general formula is simpler than the formula for the de cay rate of the least damped mode, that was
+the question that [2] was addressing. We believe that this simple, gen eral formula will provide insight into a more
+complete understanding of the tail phenomenon in spinning black hole spacetimes.
+Our general formula allows up to address some questions left open a lso in the analysis of Barackand Ori [8]. Barack
+and Ori studied analytically the late–time tails for generic families of init ial data (only assuming compactly supported
+outgoing initial pulses). Instead of carrying the analysis in the freq uency domain —which turns out to be complicated
+on a Kerr background because the frequency dependence of the spheroidal harmonics implies that separation of the
+two angular variables depends on the frequency— Barack and Ori a nalyzed the evolution of perturbations in the time
+domain. Barackand Ori found the decay rate of the ℓmultipole given generic initial data, which they considered to be2
+data in which all multipoles are present. In fact, because the smalles t multipole contribution to the initial data turns
+out as we show below to dominate the decay rate of the ℓmultipole at late times, Barack and Ori found the decay
+rate of an even (odd) mode ℓexcites by the scalar field monopole (dipole) mode of the initial data. T he Barack–Ori
+formula for the decay rate of the ℓmultipole has not been confirmed by numerical simulations, which we do below for
+the first time. Our proposed formula also generalizes it to any initial m ultipole, not just the monopole (or dipole) as
+in [8].
+Consider next the question of the decay rate of a certain multipole m omentℓ. In the Schwarzschild spacetime
+the decay rate depends only the the value of ℓ, specifically it is t−nwheren= 2ℓ+ 3. Most notably, the decay
+rate is independent of the history of the mode. This is no longer the c ase in the Kerr spacetime: the decay rate
+of a multipole ℓdoes depend on how that mode came into existence, specifically, it de pends on both ℓ,ℓ′. In the
+Schwarzschild case the dependence is only on ℓbecause of the degeneracy ℓ′=ℓand the absence of mode couplings
+(“spherical harmonics are eigenfunctions of the wave operator” ) . Therefore, in the Schwarzschild case one may say
+that “a multipole is a multipole,” and consequently its decay rate is intima tely linked to its angular distribution and
+the associated symmetry group. Specifically, in Schwarzschild ther e is a one–to–one relation of the symmetry group
+of the mode in question and the mode’s late–time decay rate, such th at the symmetry determines the time evolution
+of the mode. Previous studies of tail behavior in Kerr have suggest ed that it is not correct to say in Kerr that “a
+multipole is a multipole,” although the authors are not aware of explicit s tatements in this regard. We bring here
+new evidence that that is indeed the case, and the tail decay rate h as nothing to do with the symmetry group of the
+mode in question. This insight may be helpful in the understanding of c ases where more than one channel contributes
+to a mode of a certain ℓvalue, although each channel contributes so that the decay rate is different, in what might
+be construed as an apparent violation of the superposition principle . The realization that in Kerr one may no longer
+correctly argue that “a multipole is a multipole” completely resolves th is apparent violation.
+The organizationof this Paper is as follows: In Section II we describe the numerical code and the type of initial data
+sets that we use. Specifically, we describe the code development th at was necessary in order to support our proposal
+for the tail decay-rate formula. In this Paper we discuss three re lated questions: First, in Section III we address the
+difference in the time evolution of late–time tails between generic and n on–generic initial data sets. Generic initial
+data sets were first considered by Barack and Ori [8], but their res ults have not been confronted with numerical
+simulations. In addition to verifying the results of [8], we also discuss the meaning of genericity of initial data sets.
+As we demonstrate, not any mixture of all multipole modes is in fact ge neric, and one needs to exclude mixtures
+that are the outcome of the evolution of non-generic initial data, a lthough such mixtures exhibit similar properties to
+mixtures that represent bona fide generic data sets. Then, in Section IV, we consider an aspect of lat e–time tails that
+has not been studied before, specifically the decay rate of excited higher multipole modes, which we call “up”–excited
+modes, namely the case ℓ > ℓ′. The decay rate of “up” excited modes generalized the Barack–Or i result in [8] to
+some interesting cases: the case of non–generic initial data in which (only) the lowest dynamically allowed mode is
+not present, and the case in which only one multipole mode is present in the initial data.
+II. NUMERICAL CODE
+A. Initial data sets
+LinearizedperturbationsofKerrblackholesaredescribedbyTeuk olsky’smasterequation, giveninBoyer–Lindquist
+coordinates for a scalar field ( s= 0)ψ, known as the Teukolsky function, by
+−/bracketleftbigg(r2+a2)2
+∆−a2sin2θ/bracketrightbigg
+∂ttψ−4Mar
+∆∂tϕψ
++∂r(∆∂rψ)+1
+sinθ∂θ(sinθ∂θψ)
++/parenleftbigg1
+sin2θ−a2
+∆/parenrightbigg
+∂ϕϕψ= 0, (1)
+whereM,aarethe blackhole’smassand spin angularmomentum per unit mass, re spectively, and the horizonfunction
+∆ =r2−2Mr+a2.
+To solve Eq. (1) numerically, we define, following [9], b(r,θ) := (r2+a2)/Σ where Σ2= (r2+a2)2−a2∆ sin2θ.
+We next define the ‘momentum’ Π of the field φdefined byψ=eimϕφ, according to
+Π :=∂φ
+∂t+b(r,θ)∂φ
+∂r∗3
+wherer∗is the regular Kerr spacetime ‘tortoise’ coordinate defined by dr∗/dr= (r2+a2)/∆.
+Our code is a 2+1D first–order code for the time evolution of modes mofφ,Π given two independent functions of
+two variables at t= 0 as initial data, based on [9]. The convergence order is taken to be second–order in the radial
+and temporal directions, and sixth order in the angular direction (a lthough some of the higher– ℓresults were obtained
+with tenth–order angular operators). The standard grid resolut ion we use is 64 ,000 grid points in the radial direction,
+and 64 angular grid points (which we denote as 64 K×64), unless stated otherwise, taking the temporal step to be
+the largest step consistent with the Courant condition.
+In practice, we choose as initial data φ(r,θ)|t=0= 0 and Π( r,θ)|t=0=f(r)Pℓ(cosθ), where the radial function
+f(r) is chosen to be a gaussian f(r) = (1/√
+2πσ2) exp[−(r∗−r∗0)2/(2σ2)] withr∗0= 25Mandσ= 6M. The outer
+and inner boundaries are placed at rboundary
+∗=±800M, which allows us to integrate to t= 1,500Matr∗0without
+seeing boundary reflection effects. We choose the Kerr paramete ra= 0.995M. We use quadrupole or octal precision
+floating–point arithmetic throughout according to need.
+We emphasize that our choice of initial data —based on the field and its conjugate momentum— is in accord with
+other common choices. Specifically, when we choose a single value of ℓ, our choice of initial data is a pure multipole
+moment, equivalent to choices made based on the field and its time der ivative. There is no ambiguity in the definition
+of a pure multipole [7].
+B. Code Improvements
+Since the goal of this work is to investigate the late-time decay rate of higher multipole modes of Kerr black holes
+a number of improvements to our time-domain Teukolsky equation co de were necessary. This is because generating
+accurate numerical data in the context of higher multipole modes inv olves a number of challenges. Firstly, these
+simulations need to be rather long — this is because typically the obser ved field exhibits an exponentially decaying
+oscillatory behavior (“quasi- normal ringing”) in the initial part of th e evolution and only much later this transitions
+over to a clean power-law decay. One needs to wait for these oscillat ions to dissipate away. For higher multipoles
+this phase lasts much longer, thus requiring significantly longer evolu tions. Secondly, because each multipole has a
+different decay rate (which increases with an increase in ℓ) at late times one ends up with numerical data in which
+different multipoles have widely different amplitudes (often 30 – 40 ord ers of magnitude apart!). For this reason, not
+only does the numerical solution scheme have to be high-order (to r educe the discretization errors to the required
+levels) but it also requires high-precision floating-point numerical co mputation (due to the large range of amplitudes
+involved).
+Below we list the major improvements made since the publication of [7] to our Teukolsky equation evolution code:
+1.Higher-Order Algorithm: As mentioned above, we require a higher-order numerical evolution scheme to keep
+the discretization errors to sufficiently low levels, otherwise these c an overwhelm the numerical data associated
+to the weakly excited higher multipole modes of interest. It turns ou t that it is sufficient that only the angular
+differentiation (i.e. θ-derivatives of the field) be implemented using a higher-order numer ical stencil. The
+temporal and the radial direction related operations can simply sta y 2nd-orderand such a mixed approachyields
+sufficiently good results [7]. For this reason, in this work we choose th e finite-difference angular differentiation
+operator to be 10th-order accurate and leave the rest of the nu merical scheme as a standard 2nd-order Lax-
+Wendroff algorithm.
+2.Higher-Order Projection Integrals: We also require a matching 10th-order accuracy in mode projection integrals
+to prevent numerical integration errors from overwhelming the hig her multipole numerical data.
+3.Higher-Order Numerical Precision: As pointed out above, we also require high-numerical precision in our
+computations. In particular, double, quadruple and octal precisio n may be required depending upon the details
+of Kerr tails simulation being attempted. This is simply to prevent roun doff error from overwhelming the
+weakly excited modes of interest – especially important for “up” exc ited higher multiple modes! We made use of
+quadruple precision (in some cases, octal precision) floating-point accuracy in this work. Now, very few current
+computer processors support full quadruple-precision (128–bit ) datatype and operations. And to the best of our
+knowledge, no compute hardware natively supports octal-precisio n (256–bit) arithmetic. Thus, we implemented
+such high-precision computations in software using publicly available s oftware libraries. This made our work
+particularly challenging computationally, because with each such incr ease in numerical precision, our code’s
+overall performance drops by an order-of-magnitude.
+4.Code Parallelism and HPC Hardware: In order to perform the numerical simulations in a reasonable time-
+frame, we made use of message-passing (MPI) based parallelism in ou r code and also used sophisticated HPC4
+hardware. In particular, domain-decomposition based parallelism wa s implemented in the code and IBM Cell
+BE and Power7 systems with specialized optimizations [10] were used t o perform these long evolutions in a
+reasonable time-frame.
+5.Other Enhancements: We also enhanced the code further to be able to evolve non-axisymm etric mode compu-
+tations (both odd and even m-modes). The even m-mode case was straightforward to implement, since these
+simply require some additional terms in the evolution equation. For th e oddm-mode case we used the follow-
+ing procedure to simplify our implementation: Instead of evolving the scalar field ψitself, we transformed the
+Teukolsky equation to evolve the field ψ/sinθ. This simple transformation enabled us to reuse several of our
+evenm-mode code algorithms (such as the higher-order angular boundar y condition imposition) and also avoid
+the introduction of non-polynomial functions of the code’s angular grid variable, i.e., cos θ.
+III. NUMERICAL EVOLUTION OF GENERIC INITIAL DATA
+We first show for the first time numerical tests of the Barack–Ori formula [8]. Then, in Section IV we show how
+the Barack–Ori formula is a particular case for “up” excitation of m odes. Barack and Ori noted that in the generic
+case of black hole perturbations, all dynamically allowed modes are pr esent. Specifically, in the case of scalar field
+perturbations, all modes, starting with the monopole ( ℓ= 0) evolve. The decay rate of a multipole ℓwith azimuthal
+numbermis given by t−nwhere
+nℓ=ℓ+|m|+3+q, (2)
+whereq= 0,1 ifℓ+mis even or odd, respectively. Notably, the Barack–Ori formula dep ends explicitly on the
+azimuthal number m, and like the Hod formula includes a certain complexity that has to do w ith whether ℓ+mis
+even or odd.
+The meaning of “generic” initial data is that the relative amplitudes of the various multipoles and their radial
+profiles are not fine tuned. Consider, e.g., the following initial data se t,φ(r,θ)|t=0= 0 and Π( r,θ)t=0=Asin2θf(r),
+wheref(r) is any localized radial function, say a gaussian. These initial data inc lude non-vanishing monopole and
+quadrapole moments. We evolved these data numerically, and found that as expected the late–time tail includes all
+even multipoles, and is dominated by the monopole field that decays ac cording tot−3at late times. The quadrapole
+projection of the late–time field decays according to t−5, in accordance with the Barack–Oriformula. Other choices of
+initial data sets and multipoles are also found to be in agreement with t he Barack–Ori formula. As far as the present
+authors are aware, this is the first direct numerical confirmation o f the Barack–Ori formula.
+Let us now consider the following scenario: one starts with initial dat a as above, and evolves the field to a certain
+value of time, say to t=T. The field at t=Tis a complicated mixture of modes, each with its own radial profile.
+Now consider a different initial value problem, in which at t= 0 only a pure multipole, say the quadrapole moment,
+is non-vanishing, and evolve these initial data to t=T. These pure mode initial data also evolve to a complicated
+mixture of multipole moments at t=T, and at first look one does not notice much qualitative difference bet ween
+the fields at t=T. However, treating the fields at t=Tas new initial value problems, further time evolution leads
+to different decay rates at late times. Specifically, the generic data lead to quadrapole projection that drops off like
+t−5, and the pure data lead to quadrapole projection that drops off like t−7. Considering only the values at t=Tas
+the initial data sets, what is the fundamental difference between t he two sets, that leads to very different late–time
+evolutions? Even though either set appears at t=Tto be a set of amplitudes with weighted ratios for the various
+multipole moments (as functions of the radial coordinate), the set describing the pure mode evolution is made at
+t=Tof a very carefully chosen set of ( rdependent) amplitudes. In fact, it is the outcome of fine–tuned ev olution
+of a pure mode of the original initial value problem. Examination of the data set at t=Tdoes not reveal anything
+qualitative different about it: it is the specific amplitude ratios and rad ial profiles that make it non-generic.
+One may test these ideas by injecting at t=Tanother field, say that of a pure mode. Start at t= 0 with a pure
+quadrapolar field. At t=Tthe projection of the field is a mixture of all even multipoles. Then, at t=T, inject either
+a pure monopole or a pure quadrapole with some radial profile. The mu ltipolar content of the field after the injection
+is changed in that the relative multipolar amplitudes changed (as func tions of the radius). When the injected field is
+quadrapolar, the late–time decay rate of the quadrapole moment is t−7, as should be expected from the fact that we
+have a superposition of two linear pure mode evolutions (which a cert ain time delay between them) (Fig. 1). When
+the injected field is a monopole, the late–time decay rate of the quad rapole moment is t−5(Fig. 2). The addition of
+a monopole field (without fine tuning its amplitude and radial profile) ma kes the multipolar content of the total field
+generic. One may pose the following question: When the injected field is a quadrapole, its time evolution will excite
+a monopole. Why in the presence of this excited monopole the quadra pole projection of the field decays like t−7,
+whereas in the presence of an injected monopole the decay rate is t he slowert−5? There is a fundamental difference5
+0 0.5 1 1.5 2 2.5 3 3.5 4−12−10−8−6−4−20246
+log10 t / Mlog10 | ψ L=2 |
+t−7
+FIG. 1: The quadrapole projection of the full field as a functi on of time, for initial data of a pure quadrapole field, then in jected
+atT= 250Mwith a pure quadrapole field. The late–time field behavior is c omputationally found to be given by ψ∼t−7.05.
+The power law index, and all the indices in this Paper, were ca lculated by extrapolating the local power-law index to infin ite
+time.
+0 0.5 1 1.5 2 2.5 3 3.5 4−10−8−6−4−202
+log10 t / Mlog10 | ψ L=2 |
+t−7t−5
+FIG. 2: The quadrapole projection of the full field as a functi on of time, for initial data of a pure quadrapole field, then in jected
+atT= 250Mwith a pure monopole field. The late–time field behavior is com putationally found to be given by ψ∼t−5.03.
+between the two cases: the excited monopole is not an arbitrary fie ld: it is carefully chosen by the dynamics of the
+problem, and may be viewed as a fine tuned field; the injected monopo le is arbitrary, and is not fine tuned to lead to
+a different late–time decay rate.
+IV. EXCITATION OF “UP” MODES
+Most interest has been devoted to finding the decay rate of the slo west decaying mode, for obvious reasons. Specif-
+ically, the slowest decaying mode determines the late–time behavior o f the full field. As lower multipole modes
+typically have a slower decay rate, most interest has naturally focu sed on the excitation of “down” modes, specifically
+the excitation of lower multipole modes starting with higher multipole mo des.
+Here, we are interested in the converse, i.e., in the excitation and de cay rate of “up” modes, i.e., the excitation and
+decay rate of a higher multipole mode than the one initially excited. Suc h modes are typically sub-dominant, and do
+not dominate the late–time behavior of the full field. Our interest in t he “up” modes is therefore only tangentially6
+0 1 2 3 4 5 6
+x 10−333.13.23.3
+M / t2n0
+0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
+x 10−377.17.27.3
+M / t2n2
+0 0.5 1 1.5 2 2.5 3
+x 10−399.19.29.3
+M / t2n4
+FIG. 3: The ℓ= 0,2,4 multipole projections, for initial data of a pure quadrupo le field as functions of time. The late–time
+field behavior is computationally found to be given by 2ψ0∼t−3.005,2ψ2∼t−7.009, and 2ψ4∼t−9.01, respectively.
+related to the question of the decay rate of the full field. Instead , it is the gaining of more insight towards a more
+complete understanding of the dynamics and mode coupling propert ies thereof we are interested in here. Denoting
+the initially excited puremultipole by ℓ′, and the multipole moment of the field whose excitation and decay are is of
+interest by ℓ(so that for “up”-modes ℓ>ℓ′and for “down”-modes ℓ<ℓ′), we propose that the late–time decay rate
+for excited ℓ,mmodes along a r= const worldline, is given by t−nwhere
+ℓ′nℓ=/braceleftbigg
+ℓ′+ℓ+1 forℓ<ℓ′
+ℓ′+ℓ+3 forℓ≥ℓ′. (3)
+Notably, this formula is independent of m. Specifically, for all (allowed) values of mthe decay rate of the ℓexcited
+multipole because of an initial ℓ′is the same. This formula is simpler than either the Hod or the Barack– Ori
+formulas, but is fully consistent with them in their domain of applicability . Specifically, when ℓ′= 0,1, Eq. (3)
+coincides with the Barack–Ori formula (2). In the Schwarzschild ca se there is no mode coupling, ℓ=ℓ′, and therefore
+n=ℓ′+ℓ+3 = 2ℓ+3. The numerical evidence described below supports Eq. (3) in all c ases we have checked.
+Azimuthal modes
+To test our proposal —and in particular the m-independence of the decay rate, which may be construed as a
+statement on the generally m-dependent Green’s function— we focus now on pure multipole initial d ata sets.
+Letting the initial ℓ′= 2, we project the multipoles ℓ= 0,2,4, and the local power indices are shown in Fig. 3. Of
+particular interest here is the excitation of the azimuthal, hexadec apole mode, ℓ= 4. In this case, ℓ′= 2,ℓ= 4, so
+that our proposal is that n= 2+4 +3 = 9. In our numerical simulations, we have found the value of 2n4= 9.01,
+with uncertainly in the last figure.
+Starting with initial data for a pure ℓ′= 0 monopole mode, we project the ℓ= 0,2 and 4 modes. The local power
+indices 0n0and0n2are shown in Fig. 4. The hexadecapole mode ℓ= 4,0ψ4, is shown in Fig. 5. Figure 5 suggests
+that our grid density is not sufficiently high to obtain accurately the v ery late time hexadecapole projection to the
+required accuracy. Indeed, as Fig. 5 shows, the tail of the field “c urves up” at very late times. However, increasing
+the grid density appears to straighten up the field, from which we inf er that this curving is a numerical artifact.
+We attribute the problem to our calculation of a mode that is a second –order excited “up” mode, i.e., the ℓ= 4
+excited mode from an initial ℓ′= 0 mode. The practical problems in determining the decay rate accu rately are
+accentuated by Fig. 6, which displays the local power indices for thr ee grid densities. Figure 6 suggests that indeed
+a higher resolution simulation would be successful at determining the late–time decay rate with sufficient accuracy.
+Our results do suggest that 0n4= 7 (notice, this is a numerical result with one significant figure, not a n exact value!),
+as indicated by the reference curve in Fig. 5, but any determination of the decay rate at higher accuracy than that
+would require higher resolution simulations.
+We have also tested our proposal with odd modes. Specifically, Let ℓ′= 3 and study the late time tail for the modes
+ℓ= 1,3,5. Figure 7 shows the projections ℓ= 1,3,5 as functions of time, and figure 8 displays the local power indices7
+0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
+x 10−333.053.13.153.2
+log10 M / t0n0
+0 0.5 1 1.5 2 2.5 3 3.5
+x 10−355.055.15.155.2
+log10 M / t0n2
+FIG. 4: The local power indices 0n0(upper panel) and 0n2(lower panel) as functions of the time t.
+0 0.5 1 1.5 2 2.5 3−18−16−14−12−10−8−6−4
+log10 t / Mlog10 |  0 ψ 4  |
+  
+100Kx100
+80Kx80
+64Kx64
+2.62.65 2.72.75 2.82.85 2.9−16−15−14
+t−7 
+3 3.05 3.1 3.15−17.2−17−16.8−16.6
+FIG. 5: The ℓ= 4 multipole projections, for initial data of a pure monopol e field as functions of time. Shown are the fields
+with three different grid densities, 64 K×64 (dash–dotted curve), 80 K×80 (dashed curve), and 100 K×100 (solid curve). The
+inserts show the same for segments of the full data, in additi on to a reference curve ∼t−7.
+TABLE I: Scattering matrix for azimuthal modes. Late–time p ower law indices ℓ′nℓfor theℓ–projections of the fields for pure
+azimuthal ( m= 0) mode initial data sets, ℓ′= 0,1,2,3,4. This table includes both “down” and “up” excitations. Ast erisks
+relate to kinematically disallowed states. All figures are s ignificant.
+Initial Projected Projected Projected Projected Projected Projected
+ℓ′modeℓ= 0ℓ= 1ℓ= 2ℓ= 3ℓ= 4ℓ= 5
+03.007 * 5.01 * 7 *
+1 * 5.002 * 7.003 * 9
+23.005 * 7.009 * 9.01 *
+3 * 5.002 * 9.009 *11.008
+4 5.01 * 7.002 *11.008 *8
+0 0.5 1 1.5 2 2.5
+x 10−355.566.57
+log10 M / t0n4
+  
+1.61.8 22.22.4
+x 10−377.057.17.157.27.25100Kx100
+80Kx80
+64Kx64
+FIG. 6: The local power index 0n4as a function of the time tfor three grid resolutions, 64 K×64 (dash–dotted curve), 80 K×80
+(dashed curve), and 100 K×100 (solid curve).
+0.5 1 1.5 2 2.5 3 3.5−25−20−15−10−505
+log10 t / Mlog10 |  3 ψ L  |L = 3L = 1
+L = 5
+FIG. 7: The ℓ= 1,3,5 multipole projections, for initial data of a pure octupole field as functions of time. The late–time field
+behavior is computationally found to be given by ψ1∼t−5.002,ψ3∼t−9.009, andψ5∼t−11.008, respectively.
+for these modes as functions of time.
+Starting with ℓ′= 1, we project the dipole and octupole modes and their respective lo cal power indices in Fig. 9.
+We encounter the same problem finding the decay rate for ℓ= 5 — i.e., find 1ψ5and1n5— as we did when we
+we calculated 0ψ4and0n4(Figs. 10 and 11). Again, the second–order “up”–excited mode re quires high resolution
+simulations.
+Our proposal (3) naturally extends the Barack–Ori formula (2) t o cases where the initial data do not include the
+monopole ( ℓ′= 0) mode. Recall that the Barack–Ori formula considers the case of generic initial data, which means
+a mixture of all multipole modes with arbitrary relative amplitudes and r adial profiles. In particular, the monopole
+mode is generically present, so that the slowest decaying contribut ion of any “up”–excited ℓmode is generated by
+the initial monopole. Our proposal degenerates to the Barack–Or i formula when ℓ′= 0. When the monopole is not
+present in the generic initial data set, our proposal will produce th e decay rate of a mixture of modes when ℓ′is taken
+to be the smallest multipole present in the generic data set, and as a p articular case will also produce the decay rate
+of theℓmultipole when the initial data is a pure ℓ′(<ℓ) mode, the non–generic case, in the language of Barack and
+Ori. In summary, the Barack–Ori formula Eq. (2) is simply the partic ular case 0nℓof Eq. (3).
+Table I summarizes our results for the azimuthal, m= 0, case. It shows the power law indices for the late9
+0 0.5 1 1.5 2 2.5
+x 10−31111.111.211.3
+M / t3n50 0.5 1 1.5 2 2.5 3 3.5
+x 10−399.19.29.3
+M / t3n30 0.5 1 1.5 2 2.5 3 3.5
+x 10−355.15.25.3
+M / t3n1
+FIG. 8: The local power indices of the ℓ= 1,3,5 multipole projections as functions of time, for initial da ta of a pure octupole
+field (same data as in Fig. 7).
+0 0.5 1 1.5 2 2.5 3−15−10−50
+log10 t / Mlog10 |1ψ L |
+ L=3 
+0 1 2 3
+x 10−377.17.27.37.4
+ M / t1n3
+0 1 2 3 4
+x 10−355.055.15.155.25.255.3
+ M / t1n1 L=1 
+FIG. 9: The dipole ( ℓ= 1) and octupole ( ℓ= 3) projections, 1ψ3and1ψ3, correspondingly (upper panel), and their local power
+indices, 1n1and1n3, respectively (left and right lower panels), as functions o f time, for initial data of a pure dipole ( ℓ′= 1)
+field.
+time tails for both “up”- and “down”-modes. It is in some loose sense a “partial scattering matrix,” that includes
+(partial) information on the excitation amplitude for each of the exc itation channels corresponding with a preparation
+state of the initial data. The scattering matrix is only partial, becau se it includes no information on the pre-factor
+ψ0=ψ0(r0,profile) of the late-time tail ψ(t)∼ψ0t−n, wherer0is the evaluation point, and where ψ0also possibly
+depends on the radial profile of the initial data. The (partial) scatt ering matrix includes only the information on n.
+The scattering matrix has a clear shape of the sum of two triangular matrices: letting rows be associated with ℓ′and
+column with ℓ, the upper triangular matrix elements satisfy n=ℓ′+ℓ+3 and the strictly lower triangular matrix
+elements satisfy n=ℓ′+ℓ+1.
+Non–azimuthal modes
+Our proposal (3) is not limited to azimuthal modes, and predicts tha t the late-time decay rate of the ℓ,mmodes
+that is excited by the initial ℓ′,mmode is independent of m, and depends only on ℓ,ℓ′. We have tested this hypothesis
+in numerous cases, that are summarized in Tables I, II, and III.10
+0 0.5 1 1.5 2 2.5 3−20−15−10−5
+log10 t / Mlog10 |1ψ 5 |
+  
+3.063.083.13.123.143.163.18−21−20.8−20.6−20.4−20.2−20−19.8
+t−9  100Kx100
+ 80Kx80
+ 64Kx64
+FIG. 10: The ℓ= 5 multipole projections, for initial data of a pure monopol e field as functions of time. Shown are the fields
+with three different grid densities, 64 K×64 (dash–dotted curve), 80 K×80 (dashed curve), and 100 K×100 (solid curve). The
+inserts show the same for segments of the full data, in additi on to a reference curve ∼t−9.
+0 0.5 1 1.5 2
+x 10−388.28.48.68.899.2
+ M / t1n5
+  
+ 100Kx100
+ 80Kx80
+ 64Kx64
+FIG. 11: The local power index 1n5as a function of the time tfor three grid resolutions, 64 K×64 (dash–dotted curve),
+80K×80 (dashed curve), and 100 K×100 (solid curve).
+In Fig. 12 we show the fields for m= 1 andm= 2 modes that are created by an initial octupole and excites all
+oddℓmodes. Specifically, we show the ℓ= 3,5 projections for both mvalues. We also show the full fields for both
+mvalues. Notably, for the case m= 1 the dipole mode is also excited, and eventually dominates the full fie ld at late
+times. In the m= 2 case the smallest ℓ-mode that is excited is the octupole, such that the full field in the m= 2 case
+decays more rapidly than in the m= 1 case. We also show the local power indices in Fig. 13. The late time va lues
+of the local power indices for each projected ℓmode approach the same values independently of the value of m, in
+accordance with Eq. (3).
+In Table IV we re-organize the data appearing in Tables I,II, and III , so that the m-independence of the power-law
+indices is brought out in a sharper way. Specifically, the late–time dec ay rates for any allowed excited ℓ′mode from
+an initialℓmode is independent of the value of min all the cases we have examined.11
+22.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3−20−18−16−14−12−10−8−6−4
+L = 5L = 3Full  field
+ log10 ( t / M ) log10 | 3, mψ L, m |
+  
+FIG. 12: The full fields and their ℓ= 3,5 projections as functions of the time tform= 1 (solid) and m= 2 (dashed). To help
+identify the modes, we denote the fields with ℓ′,mψℓ,m. The initial data are for an initial pure octupole moment.
+0 0.5 1 1.5 2 2.5 3
+x 10−38.88.999.19.29.39.4
+ M / t  3,m n 3,m 
+  
+0 0.5 1 1.5 2
+x 10−310.8510.910.951111.0511.111.1511.2
+ M / t  3,m n 5,m 
+  
+ m = 1
+ m = 2 m = 1
+ m = 2
+FIG. 13: The local power indices 3n3and3n5as functions of the time tform= 1,2. To help identify the modes, we denote
+the local power indices with ℓ′,mnℓ,m. The case m= 1 is described by a solid curve, and the case m= 2 is described by the
+dashed curve.
+TABLE II: Scattering matrix for non-azimuthal modes with m= 1. Late–time power law indices ℓ′nℓfor theℓ–projections of
+the fields for pure m= 1 mode initial data sets, ℓ′= 2,3,4,5. This table includes both “down” and “up” excitations. Ast erisks
+relate to kinematically disallowed states. All figures are s ignificant.
+Initial Projected Projected Projected Projected Projected
+ℓ′modeℓ= 1ℓ= 2ℓ= 3ℓ= 4ℓ= 5
+2 * 6.997 * 8.995 *
+35.0001 * 9.009 *11.005
+4 *7.0001 *11.006 *
+57.0006 *9.0001 *13.00812
+TABLE III: Scattering matrix for non-azimuthal modes with m= 2. Late–time power law indices ℓ′nℓfor theℓ–projections of
+the fields for pure m= 2 mode initial data sets, ℓ′= 2,3,4. This table includes both “down” and “up” excitations. Ast erisks
+relate to kinematically disallowed states. All figures are s ignificant.
+Initial Projected Projected Projected Projected
+ℓ′modeℓ= 2ℓ= 3ℓ= 4ℓ= 5
+27.0003 * 8.995 *
+3 *9.0006 *11.001
+47.0001 *11.003 *
+TABLE IV: Presentation of the data in Tables I,II, and III in a form that emphasizes the independence of the decay rate on m.
+Late–time power law indices ℓ′,mnℓ,mfor theℓ–projections of the fields for pure m= 0,1,2 mode initial data sets, ℓ= 2,3,4.
+This table includes both “down” and “up” excitations. Aster isks relate to kinematically disallowed states. All figures are
+significant.
+Initial Projected Projected Projected Projected
+ℓ′,mmodeℓ= 2ℓ= 3ℓ= 4ℓ= 5
+2,0 7.009 * 9.01 *
+2,1 6.997 * 8.995 *
+2,2 7.0003 * 8.995 *
+3,0 * 9.009 *11.008
+3,1 * 9.009 *11.005
+3,2 *9.0006 *11.001
+4,0 7.002 *11.008 *
+4,1 7.0001 *11.006 *
+4,2 7.0001 *11.003 *
+Acknowledgments
+We are indebted to Richard Price and Jorge Pullin for discussions. Mos t of the numerical simulations were executed
+on Alabama Supercomputer Center (ASC), Louisiana Optical Netwo rk Initiative (LONI), and Playstation 3 clusters.
+LMB was supported in part by a Theodore Dunham, Jr. Grant of the F.A.R., by a NASA EPSCoR RID grant,
+and by NSF grants PHY–0757344 and DUE–0941327. GK was suppor ted in part by NSF grants PHY–0831631,
+PHY–0902026, and PHY–1016906.
+[1] L.M. Burko and G. Khanna, Phys. Rev. D 67, 081502(R) (2003).
+[2] S. Hod, Phys. Rev. D 61, 024033 (1999); 61, 064018 (2000).
+[3] E. Poisson, Phys. Rev. D 66, 044088 (2002).
+[4] W. Krivan, Phys. Rev. D 60, 101501(R) (1999).
+[5] R.J. Gleiser, R.H. Price, and J. Pullin, Class. Quantum G rav.25, 072001 (2008).
+[6] M. Tiglio, L. Kidder, and S. Teukolsky, Class. Quantum Gr av.25, 105022 (2008).
+[7] L.M. Burko and G. Khanna, Class. Quantum Grav. 26, 015014 (2009).
+[8] L. Barack and A. Ori, Phys. Rev. Lett. 82, 4388 (1999).
+[9] W. Krivan, P. Laguna, P. Papadopoulos, and N. Andersson, Phys. Rev. D 56, 3395 (1997).
+[10] R. Ginjupalli and G. Khanna, High-Precision Numerical Simulations of Rotating Black Ho les Accelerated by CUDA , Pro-
+ceedings of the International Conference on High Performan ce Computing Systems (HPCS), Orlando, FL (2010).
\ No newline at end of file
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new file mode 100644
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--- /dev/null
+++ b/1001.0542.txt
@@ -0,0 +1,383 @@
+arXiv:1001.0542v2  [cond-mat.str-el]  30 Apr 2010Valence bond glass on an fcc lattice in the double perovskite Ba2YMoO 6.
+M. A. de Vries,1,2,∗A. C. Mclaughlin,3and J.-W. G. Bos4,5
+1School of Physics and Astronomy, E. C. Stoner Laboratory, Un iversity of Leeds, Leeds. LS2 9JT, UK
+2School of Physics & Astronomy, University of St-Andrews, th e North Haugh, KY16 9SS, UK
+3Department of Chemistry, University of Aberdeen, Meston Wa lk, Aberdeen AB24 3UE, UK
+4School of Chemistry, University of Edinburgh, King’s Build ings, Mayfield Road, Edinburgh EH9 3JZ, UK
+5Department of Chemistry - EPS, Heriot-Watt University, Edi nburgh, EH14 4AS, UK
+(Dated: October 29, 2018)
+We report on the unconventional magnetism in the cubic B-sit e ordered double perovskite
+Ba2YMoO 6, using AC and DC magnetic susceptibility, heat capacity and µSR. No magnetic order
+is observed down to 2 K while the Weiss temperature is ∼ −160 K. This is ascribed to the geo-
+metric frustration in the lattice of edge-sharing tetrahed ra with orbitally degenerate Mo5+s= 1/2
+spins. Our experimental results point to a gradual freezing of the spins into a disordered pattern of
+spin-singlets, quenching the orbital degeneracy while lea ving the global cubic symmetry unaffected,
+and providing a rare example of a valence bond glass.
+PACS numbers: 75.10.Jm, 76.75.+i, 75.40.Cx
+Magnetic insulators with lattices in which antiferro-
+magnetic (AF) bonds are geometrically frustrated have
+been studied widely in the pursuit of exotic quantum
+groundstatessuchasspinliquid[1,2]. Suchnon-classical
+ground states have mainly been sought in low dimen-
+sional structures such as the triangular lattice system
+κ-(BEDT-TTF) 2Cu2(CN)3[3] and the kagome antifer-
+romagnet herbertsmithite [4]. Materials with a geomet-
+rically frustrated face centred cubic (fcc) lattice have in
+this respect received much less attention. The 12 near-
+neighbourmagneticbonds J1betweenthe[000]and[1
+21
+20]
+spins on the fcc lattice form a network of edge-sharing
+tetrahedra (Fig. 1). When these bonds are AF ( J1>0)
+the magnetism is geometrically frustrated, giving rise
+to a large (but not macroscopically large [5] as for the
+kagome lattice) ground-state manifold of spin configura-
+tions unrelated by symmetry. Further neighbour inter-
+actions (J2) along the 6 [100] vectors lift this degeneracy
+only partially; J2<0 (along the 6 [100] vectors) leads to
+type I order, weak AF exchange (0 < J2<2J1) to type
+III order and stronger AF exchange J2>2J1to type II
+order. Thermal/quantum fluctuations and quenched dis-
+order have been shown to result in a bias for respectively
+collinearandanti collinearstateswithin these degenerate
+ground state manifolds [6–8], an entropic selection effect
+termed “order from disorder” [9]. This is in agreement
+with experiments on well-known compounds of rock-salt
+structure such as MnO [10, 11], Cd 1−xMnxTe [12], NiO,
+MnSe [10]. Classical type I, II or III order has also been
+confirmed for s= 1/2 [13–15] although less is known
+about the physics at the boundaries between the classi-
+cal phases. In this Letter we describe the unconventional
+magnetism in the compound Ba 2YMoO 6, providing ex-
+perimental evidence that an exotic valence bond glass
+(VBG) [16, 17] state can stabilize at the boundary be-
+tween the known classical phases on the fcc lattice. Such
+a disordered state has been predicted to be possible evenin the absence of structural disorder, as an example of a
+non-equilibrium quantum ground state [16].
+FIG.1: FourMoO 6octahedra(shadedgrey)andYions (large
+spheres) in the cubic unit cell of Ba 2YMoO 6(left). The Mo5+
+ions form a lattice of edge-sharing tetrahedra (right). The
+cubic lattice constant is 8.389 ˚A.
+The B-site ordered double perovskites are of general
+stoichiometry A2BB′O6where the Asite typically hosts
+alkaline-earths and lanthanides and the Bsites can host
+3d, 4dand 5dtransition metal (TM) ions. Depending on
+the combination of BandB′site ions, electronic phases
+from strongly correlated metals via half-metals [18] and
+semiconductors [19, 20] to Mott insulating can be real-
+ized. Mott insulating 4 dand 5dTM compounds are rare.
+Theoccurrenceofthis insulatingphasein thedouble per-
+ovskitesisduetothelargedistancebetweentheTM ions,
+of the order of 5 to 6 ˚A. Examples of Mott insulators
+are Ba 2LaRuO 6and Ca 2LaRuO 6[21], respectively type
+III and type I antiferromagnets. Sr 2CaReO 6[22] and
+Sr2MgReO [23] (the Re6+hass= 1/2) have spin-glass
+ground states, consistent with a negligible J2along the
+pathway Re-O- B′-O-Re. There is a large group of Mo5+
+compoundsBa 2LnMoO6withLn=Nd, Sm, Eu,Gd, Dy,
+Er,YbandY[24]. TheMo5+hasasinglyoccupied4 dt2g
+level with s= 1/2. Due to the strong spin-orbit coupling
+in 4dTM ions in a cubic crystal field this is expected to
+lead to a j= 3/2 triplet [25, 26]. Only the larger lan-
+thanide compounds ( Ln= Nd, Sm and Eu) have N´ eel
+order (Type I, implying J2<0) coinciding with a weak2
+Jahn-Teller distortion [24, 27, 28], while the exchange
+interaction is of the order of 100 K [24]. The other com-
+pounds were found to be paramagnetic and with cubic
+symmetry at all temperatures. Ba 2YMoO 6is the sim-
+plest of these compounds because the Y3+ion does not
+carry a magnetic moment. The magnetic exchange is
+mainly via the 90◦B’-O-O-B’ π−πbonds [21, 24], giv-
+ing rise to 12 near-neighbour AF J1bonds for each spin,
+across the edges of the tetrahedra (Fig. 1).
+Polycrystalline Ba 2YMoO 6was prepared by the solid
+state reaction of stoichiometric oxides of Y 2O3, MoO 3
+and BaCO 3powders of at least 99.99% purity. These
+were ground, die-pressed into a pellet and heated under
+flowing 5% H 2/N2. The final synthesis temperature was
+1200-1250◦C with three intermediate regrinding steps to
+ensure phase homogeneity. It was found that a first heat-
+ing step of ∼2 hr at 900◦C in air and thorough homoge-
+nization helps to prevent the formation of BaMoO 4and
+Y2O3impurities. Phase purity was confirmed by labo-
+ratory X-ray powder diffraction. In a related paper [29]
+neutron powder diffraction results are discussed, which
+show that the Y/Mo site disorder is less than 1%. The
+diamagneticanalog,Ba 2YNbO 6, waspreparedat1200◦C
+in air from YNbO 4and BaCO 3. The sample magnetisa-
+tion was measured on a Quantum Design magnetic prop-
+erty measurement system (MPMS) in fields up to 5 T.
+The heat capacity was measured on a Quantum Design
+physical property measurement system (PPMS), using
+7.0 mg of a sintered pellet. The µSR experiment was
+carried out at MUSR at ISIS, UK.
+0 50 100 150 200 250 300
+T (K)036912χ 103 (emu/mol)Curie-Weiss fits
+"bulk" magnetism
+χ (T) in 1 T
+0 2 4 6 8
+H (T)0123
+M 10-2(emu/mol) M(H), 5 K
+M(H), 2.3 K
+fit to 5 K data
+fit to 2.3 K data
+00.40.81.21.6
+χ-1 10-3 (mol/emu)
+FIG. 2: (color online) The DC magnetic susceptibility χ(x,
+left axis) and χ−1(x, right axis) of Ba 2YMoO 6measured in 1 T.
+Curie-Weiss fits in the two linear regimes in χ−1are indicated
+in grey (red) and the black line gives the difference between t he
+total susceptibility and the low-temperature Curie term. T he
+inset shows M(H)at 2.3 (+) and 5 K ( ◦) and fits to the data
+with Brillouin functions, accounting for 7% of the Mo s= 1/2
+spins at 5K but only for 2% at 2.3 K.
+TheDC magneticsusceptibilitymeasuredina1Tfieldis shown in Fig. 2. A Curie-Weiss fit to the high temper-
+ature susceptibility yields a Weiss temperature of -160 K
+and a Curie constant of 0.25 emu mol−1K−1, small com-
+pared to the 0.38 emu mol−1K−1expected for s= 1/2.
+This difference is attributed to strong quantum fluctu-
+ations common in low-spin antiferromagnets and previ-
+ously observed in double perovskites [26]. Below 25 K a
+second linear regime is observed in χ−1, corresponding
+(for a 1 T field) to a ∼10% fraction of all the s= 1/2
+moments (or ∼5% if they have the full j= 3/2 where
+gJ= 4/3) and a Weiss temperature of −2.3 K indicat-
+ing weak AF exchange. This fraction is too large to be
+ascribed directly to either structural disorder or an im-
+purity phase in the sample. Furthermore, fits to M(H)
+measured at 2.3 and 5 K (inset of Fig. 2) with Brillouin
+functions lead to estimates of respectively 2 and 7% of
+all spins, compared to 10% for fits to the M(T) curve.
+This suggests that the apparently quasi-free spins are an
+emergent property of the (disorder free) system.
+0 50 100 150
+T (K)-0.4-0.200.20.40.60.8
+χ'' 103(emu/mol)1 kHz
+3 kHz
+5 kHz
+9 kHz
+0123456χ' 103 (emu/mol)dc
+FIG. 3: (Color online) The temperature dependence of the
+dispersive (left axis, open symbols) anddissipative (righ taxis,
+filled symbols) components of the ACsusceptibility. The sol id
+lines are guides to the eye.
+The AC susceptibility measured with a field amplitude
+of 5 Oe and zero DC offset field is shown in Fig. 3. The
+dispersive part of the AC susceptibility ( χ′) is almost
+frequency independent and is comparable to the diverg-
+ing low temperature DC susceptibility. The dissipative
+part (χ′′) shows a frequency dependent maximum be-
+tween 26 and 70 K. Remarkably, the maximum gradually
+gets sharperas the frequency increases, instead ofweaker
+as expected for a spin-glass transition. The agreement
+between the DC susceptibility and χ′below 20 K is a
+strong indication that the Curie-term can be ascribed to
+the weakly coupled spins.
+The heat capacity associated with the single Mo5+
+4delectron in Ba 2YMoO 6(as shown in Fig. 4) was ob-
+tainedfromcomparisonwith the heatcapacityofthe dia-
+magnetic analogue Ba 2YNbO 6. The heat capacity from
+phonons of Ba 2YMoO 6is expected to be lower than for
+Ba2YNbO 6by a factor 0.991 due to the mass difference3
+between the Mo and Nb nuclei. However this is small
+compared to the experimental error in the sample mass
+which is known with 10% accuracy. For this reason the
+heat capacity was measured well into the paramagnetic
+regime and matched to the heat capacity of Ba 2YNbO 6
+above 200 K (150 K) for the zero-field (9 T) measure-
+ments [30]. The magnetic entropy is gradually released
+over a wide range of temperatures with a broad maxi-
+mum around 50 K. No anomalies corresponding to phase
+transitions are observed, only a gradualfreezing, quench-
+ing all degrees of freedom associated with the orbitally
+degenerate t2gs= 1/2 4delectrons. As shown in the
+inset of Fig. 4, the total entropy recovered Stot= 12±2
+JK−1mol−1, close to the Rln4 = 11 .5 expected for a
+j= 3/2 quadruplet (the j=l−s= 1/2 doublet lies at
+much higher energies [25, 26]). Below 25 K only ∼5%
+of the entropy is released, in agreement with the Curie
+fit to the low temperature susceptibility which was found
+to correspond to ∼5% of the Mo5+if these remaining
+spins have j= 3/2. In a 9 T magnetic field most of the
+magnetic entropy shifts to lower temperatures.
+0 50 100 150 200
+T (K)00.050.10.150.2C/T (JK-2mol-1)zero field
+9 Tesla
+0 50 100 150 200
+T (K)051015Stot (JK-1mol-1)
+FIG. 4: (Color online) The magnetic heat capacity obtained
+by subtracting the heat capacity of the diamagnetic analogu e
+Ba2YNbO 2in zero field (black dots) and in 9 Tesla (open
+squares). The experimental error for the 9 T data is com-
+parable to that indicated for the zero field data (grey area).
+The inset shows the total entropy release as a function of
+temperature in zero field (black line) and in 9 T (red line).
+To gain a better understanding of the gradual freezing
+and the appearance of apparently weakly-coupled spins
+aµSR experiment was carried out. The zero-field muon
+spin relaxation spectra at 120 K, 5 K and 1.4 K are
+shown in Fig. 5. There is no evidence of muon relax-
+ation due to nuclear spins which confirms that the main
+muon stopping site is near the O2−ions. At 120 K there
+is no muon relaxation, as expected for a paramagnetic
+state. Remarkably, at 5 K a muon relaxation is still
+only just detectable. If the maximum in the AC sus-
+ceptibility is due to a conventional spin-glass transition0 2 4 6 8 10
+t (µs)00.20.40.60.81P
+120 K
+5 K
+1.4 K
+1 10 100
+T (K)00.020.040.06 λ
+FIG. 5: (Color online) The muon spin relaxation at 120, 5and
+1.4 K. The relaxation follows an exponential decay ( P(t) =
+exp[−t/λ]β, solid lines) with β= 1 for all but the 1.4 K data,
+whereβ= 0.7. Thetemperature dependenceoftherelaxation
+rateλis shown in the inset.
+a Lorentzian Kubo-Toyabe muon relaxation is expected
+below the spin glass transition, as observed in the re-
+lated system Sr 2MgReO 6[23]. The very slow muon re-
+laxation observed at 5 K in Ba 2YMoO 6indicates there
+are no static moments. At the same time the heat capac-
+ity data shows that at 5 K most of the magnetic entropy
+associated with j= 3/2 is quenched, implying static or-
+der. The majority of spins must therefore have bound
+into (non-magnetic) static spin-singlet “valence bonds”
+inwhichalsothe orbitaldegreesoffreedomarequenched.
+The moderate increasein the muon relaxationrate below
+5 K is then due to slowing-downof a small fraction of the
+spins which are left isolated as domain walls or defects
+in a (disordered) valence bond crystal (VBC). The best
+characterization of this state is probably a valence bond
+glass (VBG) as described in Ref. 17.
+The magnetic properties of Ba 2YMoO 6are very differ-
+ent to those of the related compound Sr 2MgReO 6[23],
+where a first order transition to a conventional spin glass
+state is observed. That the crossover in Ba 2YMoO 6is
+not a conventional spin-glass transition is also clear from
+the unusual frequency dependence of the AC susceptibil-
+ity. The gradual freezing and cross-over region around
+50 K are consistent with a pseudogap predicted for the
+VBG [17]. This gap, which corresponds to a spin-singlet
+dimerization energy scale, is filled by levels correspond-
+ing to emergent weakly-coupledspins which give rise to a
+diverging susceptibility as the temperature is decreased.
+In close agreementwith Ref. 17 the observedlow temper-
+ature susceptibility follows a power law χ∝(T−Ts)−γ
+withγ≈1. Asnotedearlier,thiscontributionfromeffec-
+tively weakly-coupledspins can not be related one-to-one
+to any structural disorder but arises as a cooperative ef-
+fect, due to the amorphous arrangement of spin-singlets.
+The heat capacity does not become zero at the lowest
+temperature measured which is consistent with a small4
+residual entropy and an ungapped spectrum as expected
+for the VBG.
+The classical ground-stateenergy is highest when J2≈
+2J1, at the cross-over between type III ( J2<2J1) and
+type II magnetism (the energy per spin is −J1/2+J2/4
+forJ2≤2J1). One possibility is that around this cross-
+over a spin-singlet state is energetically favored. A com-
+plete explanation of why spin-singlets stabilize will in the
+present case also involve the orbital degrees of freedom;
+It is the valence-bond formation rather than a Jahn-
+Teller effect that fixes the orbital orientation. This could
+be accompanied by local structural distortions which do
+not lead to experimentally observable [24, 29] structural
+changes because the valence bonds do not form a regular
+pattern within the crystal. As first observed by Goode-
+nough and Battle [21], the band width is an important
+factor in understanding the magnetism in double per-
+ovskites with 4 dand 5dtransition metal ions. Because
+Ba2YMoO 6is a relatively wide-band insulator it could
+also be that the stability of a spin-singet state can be
+understood in terms of the t−Jmodel [31].
+In conclusion, the B-site ordered double perovskite
+Ba2YMoO 6has Mo5+ions with a singly-occupied de-
+generate t2gorbital in a cubic crystal field and with
+s= 1/2. These moments are located on an fcc lattice
+and coupled antiferromagnetically, with a Weiss temper-
+ature of −160 K. At high temperature the single-ionic
+j= 3/2 moments are strongly reduced by quantum fluc-
+tuations, consistent with the formation of stable spin-
+singlet dimers at low temperatures. Remarkably, the
+dimerizationpatternappearstobedisordered,givingrise
+to emergent effectively unpaired spins with a diverging
+susceptibility as the temperature is reduced, in agree-
+ment with theoretical predictions of a VBG [17]. It is
+proposed that the stabilisation of spin-singlets is a result
+of the fine-tuning of the ratio J2/J1to the boundary be-
+tween the classical type II and type III phases. Clearly
+the present results provide a strong motivation for fur-
+ther theoretical explorations of the phase diagram of the
+fcc antiferromagnet and of the interplay between spin
+and orbital degrees of freedom in 4 dand 5dtransition
+metal compounds. This is the first observation of a VBG
+in a quantum magnet. Further experimental studies are
+needed to understand the relationship between the struc-
+tural and magnetic disorder in this class of materials.
+The Royal Society of Edinburgh (JWGB) and Lever-
+hulme Trust (ACM) are acknowledged for financial sup-
+port. C. L. Henley and H. M. Rønnow are gratefully
+acknowledged for fruitful discussions. S. J. Ray, P. King
+and P. Baker are gratefully acknowledged for assistance
+with the µSR experiments.
+∗Electronic address: m.a.devries@physics.org[1] J. E. Greedan, J. of Mat. Chem. 11, 37 (2001).
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+(2005).
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+ner, Phys. Rev. B 65, 144413 (2002).
+[23] C. R. Wiebe et al., Phys. Rev. B 68, 134410 (2003).
+[24] E. J. Cussen, D. R. Lynham, and J. Rogers, Chem. Mat.
+18, 2855 (2006).
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+[27] A. C. Mclaughlin, Solid State Commun. 137, 354 (2006).
+[28] A. C. Mclaughlin, Phys. Rev. B 78, 132404 (2008).
+[29] A. C. McLaughlin, M. A. de Vries, and J.-W. G. Bos, To
+be puslished, 2010.
+[30] The data thus obtained gives a lower limit to the heat
+capacity associated with the singly occupied 4 d t2glevel,
+which is sufficient for the present purpose.
+[31] E. Dagotto, Thet−Jand frustrated Heisenberg Models:
+a status report on numerical studies (World Scientific,
+Singapore 1991).
+[32] After submission of this paper a preprint on Ba 2YMoO 6
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+peared. Their conclusions are broadly in agreement with
+our observations.
\ No newline at end of file
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+arXiv:1001.0543v2  [quant-ph]  10 Oct 2010Optimalreconstruction ofthestates inqutrits system
+Fei Yan,1Ming Yang∗,1and Zhuo-Liang Cao2
+1Key Laboratory of Opto-electronic Information Acquisitio n and Manipulation,
+Ministry of Education, School of Physics and Material Scien ce,
+Anhui University, Hefei 230039, People's Republic of China
+2Department of Physics & Electronic Engineering, Hefei Norm al University, Hefei 230061, People's Republic of China
+Based on mutually unbiased measurements, an optimal tomogr aphic scheme for the multiqutrit states is pre-
+sented explicitly. Because the reconstruction process of s tates based on mutually unbiased states is free of
+information waste, we refer to our scheme as the optimal sche me. By optimal we mean that the number of the
+requiredconditionaloperationsreachestheminimuminthi stomographicschemeforthestatesofqutritsystems.
+Special attention will be paid tohow those different mutual ly unbiased measurements are realized; that is, how
+todecompose each transformation that connects each mutual ly unbiased basis withthe standard computational
+basis. It is found that all those transformations can be deco mposed into several basic implementable single-
+andtwo-qutrit unitaryoperations. Forthe three-qutritsy stem, there exist fivedifferent mutuallyunbiased-bases
+structures with different entanglement properties, so we i ntroduce the concept of physical complexity to min-
+imize the number of nonlocal operations needed over the five d ifferent structures. This scheme is helpful for
+experimental scientists torealize the most economical rec onstruction of quantum states inqutrit systems.
+PACS numbers: 03.65.Wj, 03.65.Ta,03.65.Ud,03.67.Mn
+I. INTRODUCTION
+The quantumstate of a system is a fundamentalconcept in
+quantummechanics,andaquantumstatecanbedescribedby
+a density matrix, which contains all the information one can
+obtainaboutthatsystem. Amaintaskforimplementingquan-
+tumcomputationistoreconstructthedensitymatrixofanun -
+known state, which is called quantum state reconstruction o r
+quantum state tomography[1, 2]. The technique was first de-
+velopedbyStokestodeterminethepolarizationstateofali ght
+beam [3]. Recently, Minimal qubit tomography process has
+been proposed by ˇReh´ aˇ cek et al, where only four measure-
+ment probabilities are needed for fully determining a singl e
+qubitstate,ratherthanthesixprobabilitiesinthestanda rdpro-
+cedure [4]. But the implementation of this tomography pro-
+cess requires measurementsof N-particle correlations[5] .The
+statistical reconstruction of biphotons states based on mu -
+tually complementary measurements has been proposed by
+Bogdanov et al[6, 7]. Ivanov et alproposed a method to
+determine an unknown mixed qutrit state from nine indepen-
+dent fluorescence signals[8]. Moreva et alpaid attention to
+experimentalproblem of the realization of the optimal prot o-
+col for polarization ququarts state tomography [9]. In 2009 ,
+Taguchietaldevelopedthesinglescantomographyofspatial
+three-dimensional(qutrits) state based on the effect of re alis-
+ticmeasurementoperators[10]. Allevi etalstudiedtheimple-
+mentationofthereconstructionoftheWignerfunctionandt he
+densitymatrixforcoherentandthermalstatesbybyswitchi ng
+on/offsinglephotonavalanchephotodetectors[11].
+In orderto obtain the full informationaboutthe system we
+need to perform a series of measurements on a large num-
+ber of identically prepared copies of the system. These mea-
+surement results are not independent of each other, so there
+∗mingyang@ahu.edu.cnisredundancyin these resultsin thepreviouslyused quantu m
+tomography processes [12], which causes a resources waste.
+If we remove this redundancy completely, the reconstructio n
+process will become an optimal one. So, to design an opti-
+mal set of measurements for removing the redundancy is of
+fundamentalsignificanceinquantuminformationprocessin g.
+Mutually unbiased bases (MUBs) have been used in a va-
+riety of topicsin quantummechanics[13–36]. MUBs are de-
+finedbythepropertythatthesquaredoverlapbetweenavecto r
+in one basis and all basis vectorsin the other bases are equal .
+That is to say the detection over a particular basis state doe s
+notgiveanyinformationaboutthestateifitismeasuredina n-
+otherbasis. Ivanovi´ cfirst introducedthe conceptof MUBs t o
+the problem of quantumstate determination[13], and proved
+theexistenceofsuchbasesintheprime-dimensionsystemby
+an explicit construction. Thenit has been shownby Wootters
+andFieldsthatmeasurementsinthisspecialclassofbases, i.e.
+mutuallyunbiasedmeasurements(MUMs)provideaminimal
+as well as optimal way of complete specification of an un-
+known density matrix [14]. They proved that the maximal
+numbersof MUBs is d+1in prime-dimensionsystem. This
+resultalso appliesto theprime-power-dimensionsystem.
+MUBs play a special role in determining quantum states,
+suchasit formsaminimalsetofmeasurementbasesandpro-
+videsanoptimalwayfordeterminingaquantumstate[13–16]
+etc. Recently an optimal tomographicreconstruction schem e
+was proposed by Klimov et alfor the case of determining
+a state of multiqubit quantum system based on MUMs in
+trapped ions system [37]. However, the use of three-level
+systems instead of two-level systems has been proven to be
+securer against a symmetric attack on a quantum key distri-
+bution protocol with MUMs than the currently existing mea-
+surementprotocol[38, 39]. Quantumtomographyin high di-
+mensional (qudit) systems has been proposed and the num-
+ber of requiredmeasurementsis d2n−1withdbeing the di-
+mension of the qudit system and nbeing the number of the
+qudits [12]. This tomography process is not an optimal one,2
+andthereisabigredundancyamongthemeasurementresults
+there. Toremovethisredundancy,wewillproposeanoptimal
+tomography process for qutrits states. This optimal quantu m
+tomography process is the MUBs-based qutrit states tomog-
+raphy, and the number of required measurements is greatly
+reduced. A d-dimensional quantum system is represented by
+a positive semidefinite Hermitian matrix ρwith unit trace in
+d-dimensional Hilbert space, which is specified by d2−1
+realparameters. Anondegeneratemeasurementperformedon
+suchasystemprovides d−1independentprobabilities. So,in
+general, one requiresat least d+1differentorthogonalmea-
+surements for fully determining an unknown ρ. Forn-qutrit
+tomography it only needs 3n+ 1measurements. Through
+analysis, one can find that the tomography process for qubit
+systemcannotbegeneralizedtoquditsystemcaseinatrivia l
+way. The MUBs-based tomographyprocess for qubit system
+proposedbyKlimov et alcannot be directlyappliedto qutrit
+system [37]. This is because the entanglement feature of the
+MUBsofqubitssystemistotallydifferentfromthatofqutri ts
+system. So, we will study the physical implementation of an
+optimal tomographic scheme for the case of determining the
+statesofmulti-qutritsystembasedontheMUMs.
+Fromtheexperimentalpointofview,thephysicalcomplex-
+ity is a key point for the implementation of a scheme. Here,
+for multi-qutrit quantum tomography, the physical complex -
+ity mainly comes from the entanglement bases, i.e. the num-
+ber of the two-qutrit conditional operation needed for the d e-
+composition of these entangled bases. In addition, there ex -
+ists many different MUBs with different entanglement prop-
+erties in multi-qutrit system. So the physical complexity o f
+the quantum tomographyprocess here depends on the entan-
+glementstructureoftheMUBsused,anditbecomesveryim-
+portant to optimize the quantumtomographyprocess overall
+thepossibleMUBsentanglementstructuresofthesystem.
+This paper is arranged as follows. In the next section we
+introduce the MUBs in a d-dimensional system ( d=pn
+(p∝ne}ationslash= 2)) and show how to reconstruct an unknown state by
+MUMs. Here pis a prime. Section III briefly reviews the
+generalmethodtoreconstructaqutritstate,wherethenumb er
+of measurements is 8. However the MUBs-based qutrit to-
+mography proposed here only needs 4measurements, which
+meansthenumberofmeasurementsisreduced. Herethemea-
+surement reductionfor single qutrit case is not obvious, so in
+sectionIVwewillextendtheone-qutritcasetotwo-qutrits ys-
+tem. For the two-qutrit system the number of measurements
+is only10for determining all the elements of the density op-
+erator rather than 34−1 = 80measurements in the scheme
+proposed in Ref. [12]. It means a great reduction of the ex-
+perimentalcomplexity. Sowe concludethattheoptimalmea-
+surementsontheunknownqutritstatesaretheMUMs. Insec-
+tion V we discuss the physical complexity for implementing
+the MUMs in three-qutritsystem, and give the optimal MUB
+forthequtritsystem quantumtomographyprocesswithmini-
+mizedphysicalcomplexity. Thelastsectionistheconclusi on.II. MUTUALLYUNBIASEDBASESANDMUTUALLY
+UNBIASEDMEASUREMENTS
+As shown by Wootters, Fields [14] and Klappenecker,
+R¨ otteler [19], in finite field language, the first MUB in a
+d=pn(p∝ne}ationslash= 2)-dimensionalquantumsystem is the standard
+basisB0givenbythe vector (a(0)
+k)l=δkl,k,l∈̥pn, where
+thesuperscriptdenotesthebasis, kthevectorinthebasis, lthe
+componentand ̥pnisthefieldwith pnelements. Theother d
+MUBs are denoted by Brwhich consists of vectors (a(r)
+k)l
+defined by[14]: (a(r)
+k)l= (1/√
+d)ωTr(r·l2+k·l),r,k,l∈
+̥pn,r∝ne}ationslash= 0.Hereω=exp(2πi/p)andTrθ=θ+θp+
+θp2+· · ·+θpn−1.The set of mutually unbiased projec-
+tors can be given by P(r)
+k=|a(r)
+k∝an}bracketri}ht∝an}bracketle{ta(r)
+k|.It is worth notic-
+ing that|a(r)
+k∝an}bracketri}htcontains the computational basis B0. Here
+Tr(P(s)
+jP(r)
+k) = (1/d)(1−δsr+dδsrδjk).Thenthemeasure-
+ment probabilities given by ω(r)
+k=Tr(P(r)
+kρ)completely
+determine the unknown density operator of a d-dimensional
+system[13]: ρ= Σd
+r=0Σd−1
+k=0ω(r)
+kP(r)
+k−I.
+For instance, in a qutrit system, there are three MUBs be-
+side the computational basis B0={|0∝an}bracketri}ht,|1∝an}bracketri}ht,|2∝an}bracketri}ht}, in the fol-
+lowingformwith ω=exp(2πi/3):
+B1:{|a(1)
+0∝an}bracketri}ht= (1/√
+3)(|0∝an}bracketri}ht+|1∝an}bracketri}ht+|2∝an}bracketri}ht),
+|a(1)
+1∝an}bracketri}ht= (1/√
+3)(|0∝an}bracketri}ht+ω|1∝an}bracketri}ht+ω∗|2∝an}bracketri}ht),
+|a(1)
+2∝an}bracketri}ht= (1/√
+3)(|0∝an}bracketri}ht+ω∗|1∝an}bracketri}ht+ω|2∝an}bracketri}ht)};(1a)
+B2:{|a(2)
+0∝an}bracketri}ht= (1/√
+3)(ω|0∝an}bracketri}ht+|1∝an}bracketri}ht+|2∝an}bracketri}ht),
+|a(2)
+1∝an}bracketri}ht= (1/√
+3)(|0∝an}bracketri}ht+ω|1∝an}bracketri}ht+|2∝an}bracketri}ht),
+|a(2)
+2∝an}bracketri}ht= (1/√
+3)(|0∝an}bracketri}ht+|1∝an}bracketri}ht+ω|2∝an}bracketri}ht)};(1b)
+B3:{|a(3)
+0∝an}bracketri}ht= (1/√
+3)(ω∗|0∝an}bracketri}ht+|1∝an}bracketri}ht+|2∝an}bracketri}ht),
+|a(3)
+1∝an}bracketri}ht= (1/√
+3)(|0∝an}bracketri}ht+ω∗|1∝an}bracketri}ht+|2∝an}bracketri}ht),
+|a(3)
+2∝an}bracketri}ht= (1/√
+3)(|0∝an}bracketri}ht+|1∝an}bracketri}ht+ω∗|2∝an}bracketri}ht)}.(1c)
+III. RECONSTRUCTIONPROCESSFORAN ARBITRARY
+SINGLE QUTRITSTATE
+Anunknownsinglequtritstatecanbeexpressedas[12,40]:
+ρ= (1/3)/summationtext8
+j=0rjλj,whereλ0is an identity operator and
+the other λjare the SU(3) generators [41]. The general
+method to reconstruct the qutrit state is to measure the ex-
+pectation values of the λoperators [12], where rj=∝an}bracketle{tλj∝an}bracketri}ht=
+Tr[ρλj]. Thusonewill findthatthenumberofrequiredmea-
+surements is 8. However, if we choose the MUMs to deter-
+mine the qutrit state, the number of needed MUMs is only 4
+ratherthan 8ofRef. [12]. ThefouroptimalsetofMUBshave
+been presented by Eqs.(1a,1b,1c) plus the standard computa -
+tionalbasisintheprecedingsection. EachofthethreeMUBs
+in Eqs.(1a,1b,1c) is related with the standard computation al3
+TABLE I: The transformations connecting the MUBs with the st an-
+dard computational basis for a qutrit system based on Fourie r trans-
+forms and the phase operations.
+Basis Transformation
+2 F−1
+3 F−1R
+4 F−1R−1
+basisbyaunitarytransformation. Thesetransformationsh ave
+been listed in Table.I. Here, Fdenotes the Fourier transfor-
+mation:
+F|j∝an}bracketri}ht= (1/√
+3)2/summationdisplay
+l=0exp(2πilj/3)|l∝an}bracketri}ht,j= 0,1,2,(2)
+Rdenotesaphaseoperation:
+R=|0∝an}bracketri}ht∝an}bracketle{t0|+ω|1∝an}bracketri}ht∝an}bracketle{t1|+ω|2∝an}bracketri}ht∝an}bracketle{t2|, (3)
+andtheControlledgateis
+X|i∝an}bracketri}ht|j∝an}bracketri}ht=|i∝an}bracketri}ht|j⊖i∝an}bracketri}ht. (4)
+Where⊖denotesthe difference j−imodulo3.Ifthereare n
+qutrits, the number of MUM is 3n+1, which is far less than
+32n−1in Ref. [12]. That is to say the use of MUMs can
+represent a considerable reduction in the operationsand ti me
+requiredforperformingthefullstate determination[37].
+IV. RECONSTRUCTIONPROCESSFOR ANARBITRARY
+TWO-QUTRITSTATE
+Now if we further extend one-qutrit case to two-qutrit
+case, the density matrix can be expressed as: ρ12=
+(1/9)/summationtext8
+j,k=0rjkλj⊗λk,whererjk=∝an}bracketle{tλj⊗λk∝an}bracketri}ht. If we
+use the general method in Ref. [12] to fully determine the
+state,d2n−1 = 34−1 = 80measurementswill be needed.
+So much measurements will inevitably introduce redundant
+informationof the state, which is obviouslya resourcewast e.
+So here we will take advantage of the MUMs to reconstruct
+thetwo-qutritstate. Itis easytofind thatthenineelements of
+̥9(finitefield)are {0,α,2α,1,1+α,1+2α,2,2+α,2+2α}
+byusingtheirreduciblepolynomials θ2+θ+2= 0[14]. Here
+we use the representation {|0∝an}bracketri}ht,|α∝an}bracketri}ht,|2α∝an}bracketri}ht···|2+2α∝an}bracketri}ht}as the
+standardbasis.
+One can find that therewill be only d2+1 = 32+1 = 10
+MUMstobedone,whichismuchlessthan 80ofRef. [12]. It
+meansthattheoperationsandtimeneededforthewholestate
+determinationisgreatreduced. Thedecompositionsforall the
+MUBsofthetwo-qutritsystem havebeenlistedin Table. II.
+V. THEPHYSICALCOMPLEXITYFOR IMPLEMENTING
+THEMUMSINTHE THREE-QUTRITSYSTEM
+In general, the fidelity of single logic gates can be greater
+than99%, but nonlocal gates have a relatively lower fidelity.TABLE II: The decompositions of MUBs for the two-qutrit syst em
+based on Fourier transformations, phase operations and con trolled-
+NOTgates( X12)[42]withthefirstparticleassourceandthesecond
+one as target. The subscript denotes the ithparticle, i= 1,2.
+Basis Decompositions
+2 F−1
+1F−1
+2
+3 F−1
+1R1F−1
+2R2
+4 F−1
+1X12F−1
+2R−1
+2
+5 F−1
+1X−1
+12R1F−1
+2R−1
+2
+6 F−1
+1X−1
+12F−1
+2R−1
+1
+7 F−1
+1R−1
+1F−1
+2R−1
+2
+8 F−1
+1X−1
+12F−1
+2R2
+9 F−1
+1R−1
+1X12F−1
+2R2
+10 F−1
+1R1X12F−1
+2
+The fidelity of a practical CNOT gate can reach a value up to
+0.926for trappedions system in Lab [43]. Klimov et alhave
+introducedthe conceptof the physicalcomplexityof each se t
+ofMUBsasafunctionofthenumberofnonlocalgatesneeded
+for implementing the MUMs [37]. Here the fidelity value of
+the CNOT gates for qubits systems also can be used to eval-
+uate the physicalcomplexityof the MUBs of qutritssystems.
+Why we can say so is because of the following point. Al-
+though the systems involved here are three-state ones, all t he
+operations used in our reconstruction process can be decom-
+posed into effective two-state operations. So the complexi ty
+ofthecurrenttomographyschemeisproportionaltothenum-
+ber of the nonlocalgatesused( C∝6) fortwo qutritssystem.
+AsshowninRef[44],theonlyMUBstructureforatwo-qutrit
+system is (4,6),where4is thenumberofthe separablebases
+and6isthenumberofthe bipartiteentangledbases.
+However in three-qutrit case, there are five
+sets of MUBs with different structures, namely
+{(0,12,16),(1,9,18),(2,6,20),(3,3,22),(4,0,24)}[21, 44].
+It is easy to see that the (0,12,16)set of MUBs has the
+minimumphysicalcomplexity. We say thatthe optimalset of
+the MUBs is (0,12,16). The decompositionsof the MUBs in
+the three-qutrit case for (0,12,16) are listed in Table.III . So
+the set of MUBs (0,12,16) has a complexity C∝44, which
+isa veryimportantvalueinexperimentalrealizationofit.
+For the multi-qutrit system( n >3), it is not easy to get all
+the sets of MUBs, and it is even more difficult to get the ex-
+plicit decompositionsof the optimal set of MUBs. Neverthe-
+less, the results for the two-qutrit and three-qutrit cases have
+providedthe experimentalistsvaluablereferences.
+VI. CONCLUSION
+We have explicitly presented an optimal tomographic
+schemeforthesingle-qutritstates,two-qutritstatesand three-
+qutrit states based on the MUMs. Because the MUBs based
+state reconstruction process is free of information waste,
+the minimal number of required conditional operations are
+needed. So we call our qutrits tomographic scheme the op-4
+TABLEIII:The decompositions of MUBs (0,12,16) for three-q utrit system
+Basis Decompositions
+1 F−1
+2X23F−1
+1F−1
+3
+2 F−1
+1R−1
+1F−1
+2R−1
+2X23F−1
+3R−1
+3
+3 F−1
+1X13R3
+4 F−1
+2R2X−1
+23F−1
+1X12R2F−1
+3
+5 F−1
+1X−1
+12F−1
+3
+6F−1
+2R2X23F−1
+1R1X−1
+13R3F−1
+3R3
+7F−1
+2X−1
+23R−1
+3F−1
+1R−1
+1X−1
+12R−1
+3
+8 F−1
+1R−1
+1X−1
+13F−1
+2R2F−1
+3R−1
+3
+9 F−1
+1X−1
+13F−1
+2R2X−1
+23F−1
+3R3
+10 F−1
+1R2X12X−1
+13F−1
+2
+11 F−1
+1R1X−1
+12F−1
+2R2X−1
+23F−1
+3
+12 F−1
+1R−1
+1X13X−1
+12F−1
+2F−1
+3R−1
+3
+13 F−1
+1X12F−1
+2R−1
+2X−1
+23F−1
+3R−1
+3
+14 F−1
+1R1X12F−1
+2X23F−1
+3R3Basis Decompositions
+15 F−1
+2R2X23
+16 F−1
+1R1X13F−1
+2R2X23F−1
+3R−1
+3
+17 F−1
+1R−1
+1X13F−1
+2F−1
+3
+18F−1
+1R−1
+1X−1
+13F−1
+2R−1
+2X−1
+23R3F−1
+3R−1
+3
+19 F−1
+1R1X−1
+13F−1
+2R−1
+2F−1
+3R3
+20 F−1
+1X12F−1
+3R−1
+3
+21 F−1
+1R−1
+1X−1
+12F−1
+2R2F−1
+3R3
+22 F−1
+1X−1
+12F−1
+2X−1
+23F−1
+3R3
+23 F−1
+1R−1
+1X13F−1
+2R−1
+2X−1
+23
+24 F−1
+1R−1
+1X−1
+13F−1
+2X23R3F−1
+3
+25 F−1
+1R1X−1
+12X−1
+23F−1
+2R−1
+2
+26 F−1
+1R1X12X−1
+23F−1
+2R−1
+2F−1
+3
+27 F−1
+1R−1
+1X12F−1
+2R2
+28 F−1
+1R1F−1
+2X−1
+23
+timal one. Here, we explicitly decompose each measure-
+mentintoseveralbasicsingle-andtwo-qutritoperations. Fur-
+thermore, all these basic operations have been proven imple -
+mentable[42]. ThephysicalcomplexityofasetofMUBsalso
+hasbeencalculated,whichisanimportantthresholdinexpe ri-
+ment. Wehopethesedecompositionscanhelptheexperimen-
+tal scientists to realize the most economicalreconstructi onof
+quantumstates inqutritssystemin Lab.
+Acknowledgments. This work is supported by NationalNatural Science Foundation of China (NSFC) under Grants
+No. 10704001andNo. 61073048,theKeyProjectofChinese
+Ministry of Education.(No.210092), the Key Program of the
+Education Department of Anhui Province under Grants No.
+KJ2008A28ZC, No. KJ2009A048Z,No. 2010SQRL153ZD,
+and No. KJ2010A287, the Talent Foundation of Anhui Uni-
+versity,the personneldepartmentofAnhuiprovince,andAn -
+hui Key Laboratory of Information Materials and Devices
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\ No newline at end of file
diff --git a/1001.0544.txt b/1001.0544.txt
new file mode 100644
index 0000000000000000000000000000000000000000..084bd90c3c38b0eef0958bc2e07fa49ec678e7ff
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@@ -0,0 +1,2107 @@
+arXiv:1001.0544v2  [math.AG]  30 Jul 2010Good formal structures for flat meromorphic
+connections, II: Excellent schemes
+Kiran S. Kedlaya
+July 30, 2010
+Abstract
+Given a flat meromorphic connection on an excellent scheme ov er a field of charac-
+teristic zero, weproveexistence of good formalstructures after blowingup; thisextends
+a theorem of Mochizuki for algebraic varieties. The argumen t combines a numerical
+criterion for good formal structures from a previous paper, with an analysis based on
+the geometry of an associated valuation space (Riemann-Zar iski space). We obtain a
+similar result over the formal completion of an excellent sc heme along a closed sub-
+scheme. If we replace the excellent scheme by a complex analy tic variety, we obtain
+a similar but weaker result in which the blowup can only be con structed in a suitably
+small neighborhood of a prescribed point.
+Introduction
+The Hukuhara-Levelt-Turrittin decomposition theorem gives a clas sification of differential
+modules over the field C((z)) of formal Laurent series resembling the decomposition of
+a finite-dimensional vector space equipped with a linear endomorphis m into generalized
+eigenspaces. It implies that after adjoining a suitable root of z, one can express any differen-
+tial module as a successive extension of one-dimensional modules. T his classification serves
+as the basis for the asymptotic analysis of meromorphic connection s around a (not neces-
+sarily regular) singular point. In particular, it leads to a coherent de scription of the Stokes
+phenomenon , i.e., the fact that the asymptotic growth of horizontal sections n ear a singu-
+larity must be described using different asymptotic series depending on the direction along
+which one approaches the singularity. (See [45] for a beautiful ex position of this material.)
+In our previous paper [25], we gave an analogue of the Hukuhara-Le velt-Turrittin decom-
+position for irregular flat formal meromorphic connections on complex analytic or algebra ic
+surfaces. (The regular case is already well understood in all dimens ions, by work of Deligne
+[12].) The result [25, Theorem 6.4.1] states that given a connection, o ne can find a blowup
+of its underlying space and a cover of that blowup ramified along the p ole locus of the
+connection, such that after passing to the formal completion at a ny point of the cover, the
+connection admits a good decomposition in the sense of Malgrange [31, §3.2]. This implies
+1that one gets (formally at each point) a successive extension of co nnections of rank 1; one
+also has some control over the pole loci of these connections. The precise statement had
+been conjectured by Sabbah [38, Conjecture 2.5.1], and was prove d in the algebraic case by
+Mochizuki [34, Theorem 1.1]. The methods of [34] and [25] are quite diff erent; Mochizuki
+uses reduction to positive characteristic and some study of p-curvatures, whereas we use
+properties of differential modules over one-dimensional nonarchim edean analytic spaces.
+The purpose of this paper is to extend our previous theorem from s urfaces to complex
+analytic or algebraic varieties of arbitrary dimension. Most of the ha rd work concerning
+differential modules over nonarchimedean analytic spaces was alrea dy carried out in [25];
+consequently, thispaperconsists largelyofargumentsofamoret raditionalalgebro-geometric
+nature. As in [25], we do not discuss asymptotic analysis or the Stoke s phenomenon; these
+have been treated in the two-dimensional case by Sabbah [38] (build ing on work of Majima
+[30]), and one expects the higher-dimensional case to behave similar ly.
+The paper divides roughly into three parts. In the remainder of this introduction, we
+describe the contents of these parts in more detail, then conclude with some remarks about
+what remains for a subsequent paper.
+0.1 Birational geometry
+In the first part of the paper ( §1–2), we gather some standard tools from the birational
+geometry of schemes. One of these is Grothendieck’s notion of an excellent ring , which
+encompasses rings of finite type over a field, local rings of complex a nalytic varieties, and
+their formal completions. Using excellent rings and schemes, we can give a unified treatment
+of differential modules in both the algebraic and analytic categories, without having to keep
+track of formal completions.
+Another key tool we introduce is the theory of Krull valuations and Riemann-Zariski
+spaces. The compactness of the latter will be the key to translatin g a local decomposition
+theorem for flat meromorphic connections into a global result.
+0.2 Local structure theory
+In the second part of the paper ( §3–5), we continue the local study of differential modules
+from [25]. (Note that this part of the paper can be read almost entir ely independently
+from the first part, except for one reference to the definition of an excellent ring.) We first
+define the notion of a nondegenerate differential ring , which includes global coordinate rings
+of smooth algebraic varieties, local rings of smooth complex analytic varieties, and formal
+completions of these. We prove an equivalence between different no tions of good formal
+structures, which is needed to ensure that our results really do ad dress a generalization
+of Sabbah’s conjecture. We then collect some descent arguments to transfer good formal
+structures between a power series ring over a domain and the corr esponding series ring over
+the fraction field of that domain. We finally translate the local algebr aic calculations into
+geometric consequences for differential modules on nondegenera te differential schemes and
+complex analytic varieties.
+2It simplifies matters greatly that the numerical criterion for goodf ormal structures estab-
+lished in the first part of [25] is not limited to surfaces, but rather ap plies in any dimension.
+We incorporate that result [25, Theorem 4.4.2] in a more geometric f ormulation (see Theo-
+rem3.5.4andProposition5.2.3): aconnection ona nondegenerate diff erential scheme admits
+a good formal structure precisely at points where the irregularity is measured by a suitable
+Cartier divisor. In other words, there exist a closed subscheme (t heturning locus ) and a
+Cartier divisor defined away from the turning locus (the irregularity divisor ) such that for
+any divisorial valuation not supported entirely in the turning locus, t he irregularity of the
+connection along that valuation equals the multiplicity of the irregular ity divisor along that
+valuation.
+0.3 Valuation-theoretic analysis and global results
+In the third part of the paper ( §6–8), we attack the higher-dimensional analogue of the
+aforementionedconjectureofSabbah[38,Conjecture2.5.1]con cerninggoodformalstructures
+for connections on surfaces. Before discussing the techniques u sed, let us recall briefly how
+Sabbah’s original conjecture was resolved in [25], and why the metho d used there is not
+suitable for the higher-dimensional case.
+As noted earlier, the first part of [25] provides a numerical criterio n for the existence
+of good formal structures. In the second part of [25], it is verified that the numerical
+criterion can be satisfied on surfaces after suitable blowing up. This verification involves
+a combinatorial analysis of the variation of irregularity on a certain s pace of valuations;
+that space is essentially an infinitely ramified tree. (More precisely, it is a one-dimensional
+nonarchimedean analytic space in the sense of Berkovich [5].) Copying this analysis directly
+inahigher-dimensional setting involves replacingthetreebyahigher -dimensional polyhedral
+complex whose geometry is extremely difficult to describe; it seems diffi cult to simulate on
+such spaces the elementary arguments concerning convex funct ions which appear in [25, §5].
+We instead take an approach more in the spirit of birational geometr y (after Zariski).
+Given a connection on a nondegenerate differential scheme, we see k to construct a blowup
+on which the turning locus is empty. To do this, it suffices to check tha t for each centered
+valuation on the scheme, there is a blowup on which the turning locus m isses the center of
+the valuation. The same blowup then satisfies the same condition for all valuations in some
+neighborhood of the given valuation in the Riemann-Zariski space of the base scheme. Since
+the Riemann-Zariski space is quasicompact, there are finitely many blowups which together
+eliminate the turning locus; taking a single blowup which dominates them all achieves the
+desired result.
+The obstruction in executing this approach is a standard bugbear in birational geome-
+try: it is very difficult to classify valuations on schemes of dimension gr eater than 2. We
+overcome this difficulty using a new idea, drawn from our work on semis table reduction for
+overconvergent F-isocrystals [22, 23, 24, 26], and from Temkin’s proof of inseparable local
+uniformization for function fields in positive characteristic [41]. The id ea is to quantify the
+difficulty of describing a valuation in local coordinates using a numerica l invariant called the
+transcendence defect . A valuation of transcendence defect zero (i.e., an Abhyankar valua-
+3tion) can be described completely in local coordinates. A valuationof pos itive transcendence
+defect cannot be so described, but it can be given a good relativedescription in terms of the
+Berkovich openunit discover acompletefieldoflower transcendenc e defect. Thisconstitutes
+a valuation-theoretic formulation of the standard algebro-geome tric technique of fibering a
+variety in curves.
+By returning the argument to the study of Berkovich discs, we for ge a much closer
+link with the combinatorial analysis in [25, §5] than may have been evident at the start
+of the discussion. One apparent difference from [25] is that we are n ow forced to consider
+discs over complete fields which are not discretely valued, so we need some more detailed
+analysis of differential modules on Berkovich discs than was used in [2 5]. However, this
+difference is ultimately illusory: the analysis in question (from our book onp-adic differential
+equations [27]) is already used heavily in our joint paper with Xiao on diffe rential modules
+on nonarchimedean polyannuli [28], on which the first part of [25] is he avily dependent.
+In any case, using this fibration technique, we obtain an analogue of the Hukuhara-
+Levelt-Turrittin decomposition for a flat meromorphic connection o n an integral nondegen-
+erate differential scheme, after blowing up in a manner dictated by a n initial choice of a
+valuation on the scheme (Theorem 7.1.7). As noted above, thanks t o the quasicompactness
+of Riemann-Zariski spaces, this resolves a form of Sabbah’s conje cture applicable to flat
+meromorphic connections on any nondegenerate differential sche me (Theorem 8.1.3). When
+restricted to the case of an algebraic variety, this result reprodu ces a theorem of Mochizuki
+[35, Theorem 19.5], which was proved using a sophisticated combinatio n of algebraic and
+analytic methods.
+0.4 Further remarks
+Using the aforementioned theorem, we also resolve the higher-dime nsional analogue of Sab-
+bah’sconjecturefor formalflatmeromorphicconnectionsonexcellentschemes(Theorem8.2.1) ;
+this case is not covered by Mochizuki’s results even in the case of a fo rmal completion of an
+algebraic variety. We obtain a similar result for complex analytic variet ies (Theorem 8.2.2),
+but it is somewhat weaker: in the analytic case, we only obtain blowups producing good
+formal structures which are locallydefined. That is, the local blowups need not patch to-
+gether to give a global blowup. To eliminate this defect, one needs a m ore quantitative form
+of Theorem 8.1.3, in which one produces a blowup which is in some sense functorial . This
+functoriality is meant in the sense of functorial resolution of singula rities for algebraic va-
+rieties, where the functoriality is defined with respect to smooth mo rphisms; when working
+with excellent schemes, one should instead allow morphisms which are regular(flat with
+geometrically regular fibres). We plan to address this point in a subse quent paper.
+We mention in passing that while the valuation-theoretic fibration arg ument described
+above is not original to this paper, its prior use has been somewhat lim ited. We suspect that
+there are additional problems susceptible to this technique, e.g., in t he valuation-theoretic
+study of plurisubharmonic singularities [9].
+4Acknowledgments
+Thanks to Bernard Teissier, Michael Temkin, and Liang Xiao for helpf ul discussions. Finan-
+cial support was provided by NSF CAREER grant DMS-0545904, DAR PA grant HR0011-
+09-1-0048, MIT (NEC Fund, Cecil and Ida Green Career Developme nt Professorship), and
+the Institute for Advanced Study (NSF grant DMS-0635607, Jam es D. Wolfensohn Fund).
+1 Preliminaries from birational geometry
+Webeginbyintroducingsomenotionsfrombirationalgeometry, not ablyincludingGrothendieck’s
+definition of excellent schemes.
+Notation 1.0.1. ForXan integral separated scheme, let K(X) denote the function field of
+X.
+1.1 Flatification
+Definition 1.1.1. Letf:Y→Xbe a morphism of integral separated schemes. We say fis
+dominant if the image of fis dense inX; it is equivalent to require that the generic point of
+Ymust map onto the generic point of X. We sayfisbirational if there exists an open dense
+subscheme UofXsuch that the base change of ftoUis an isomorphism Y×XU→U; this
+implies that fis dominant. We say fis amodification (ofX) if it is proper and birational.
+Definition 1.1.2. Letf:Y→Xbe a modification of an integral separated scheme X,
+and letg:Z→Xbe a dominant morphism. Let Ube an open dense subscheme of Xover
+whichfis an isomorphism. The proper transform ofgunderfis defined as the morphism
+W→Y, whereWis the Zariski closure of Z×XUinZ×XY; this does not depend on the
+choice ofU.
+WewillusethefollowingspecialcaseofRaynaud-Grusonflatification [36, premi` erepartie,
+§5.2].
+Theorem 1.1.3. Letg:X→Sbe a dominant morphism of finite presentation of finite-
+dimensional noetherian integral separated schemes. Then t here exists a modification f:T→
+Ssuch that the proper transform of gunderfis a flat morphism.
+1.2 Excellent rings and schemes
+The class of excellent schemes was introduced by Grothendieck [18, §7.8] in order to capture
+the sort of algebro-geometric objects that occur most commonly in practice, while excluding
+some pathological examples that appear in the category of locally no etherian schemes. The
+exact definition is less important than the stability of excellence unde r some natural opera-
+tions; the impatient reader may wish to skip immediately to Proposition 1.2.5. On the other
+hand, the reader interested in more details may consult either [18, §7.8] or [32, §34].
+5Definition 1.2.1. A morphism of schemes is regularif it is flat with geometrically regular
+fibres. A ring Ais aG-ringif for any prime ideal pofA, the morphism Spec( /hatwideAp)→Spec(Ap)
+is regular. (Here /hatwideApdenotes the completion of the local ring Apwith respect to its maximal
+idealpAp.)
+Definition 1.2.2. Theregular locus of a locally noetherian scheme X, denoted Reg( X), is
+the set of points x∈Xfor which the local ring OX,xofXatxis regular. A noetherian ring
+AisJ-1if Reg(Spec( A)) is open in A. We sayAisJ-2if every finitely generated A-algebra
+is J-1; it suffices to check this condition for finite A-algebras [32, Theorem 73].
+Definition 1.2.3. A ringAiscatenary if for any prime ideals p⊆qinA, all maximal
+chains of prime ideals from ptoqhave the same finite length. (The finiteness of the length
+of each maximal chain is automatic if Ais noetherian.) A ring Aisuniversally catenary if
+any finitely generated A-algebra is catenary.
+Definition 1.2.4. A ringAisquasi-excellent if it is noetherian, a G-ring, and J-2. A quasi-
+excellent ringis excellent ifitisalsouniversally catenary. Ascheme is (quasi-)excellent ifitis
+locally noetherian and covered by open subsets isomorphic to the sp ectra of (quasi-)excellent
+rings. Note that an affine scheme is (quasi-)excellent if and only if its c oordinate ring is.
+As suggested earlier, the class of excellent rings is broad enough to cover most typical
+cases of interest in algebraic geometry.
+Proposition 1.2.5. The class of (quasi-)excellent rings is stable under format ions of local-
+izations and finitely generated algebras (including quotie nts). Moreover, a noetherian ring is
+(quasi-)excellent if and only if its maximal reduced quotie nt is.
+Proof.See [32, Definition 34.A].
+Corollary 1.2.6. Any scheme locally of finite type over a field is excellent.
+Proof.A fieldkis evidently noetherian, a G-ring, and J-2. It is also universally caten ary
+because for any finitely generated integral k-algebraA, the dimension of Aequals the tran-
+scendence degree of Frac( A) overk, by Noether normalization (see [13, §8.2.1]). Hence k
+is excellent. By Proposition 1.2.5, any finitely generated k-algebra is also excellent. This
+proves the claim.
+Corollary 1.2.7. Any modification of a (quasi-)excellent scheme is again (qua si-)excellent.
+Remark 1.2.8. The classes of excellent and quasi-excellent rings enjoy many additio nal
+properties which we will not be using. For completeness, we mention a few of these. See [32,
+Definition 34.A] for omitted references.
+•If a local ring is noetherian and a G-ring, then it is J-2 and hence quas i-excellent.
+•Any Dedekind domain of characteristic 0, such as Z, is excellent. However, this fails
+in positive characteristic [32, 34.B].
+6•Any quasi-excellent ring is a Nagata ring , i.e., a noetherian ring which is universally
+Japanese. (A ring Aisuniversally Japanese if for any finitely generated integral A-
+algebraBand any finite extension Lof Frac(B), the integral closure of BinLis a
+finiteB-module.)
+Remark 1.2.9. It has been recently shown by Gabber using a weak form of local uni-
+formization (unpublished) that excellence is preserved under comp letion with respect to an
+ideal. This answers an old question of Grothendieck [18, Remarque 7.4 .8]; the special case
+for excellent Q-algebrasof finite dimension had been established previously by Rott haus[37].
+However, we will not need Gabber’s result because we will establish ex cellence of the rings
+we consider using derivations; see Lemma 3.2.5.
+1.3 Resolution of singularities for quasi-excellent schem es
+Upon introducing the class of quasi-excellent schemes, Grothendie ck showed that it is in
+some sense the maximal class of schemes for which resolution of sing ularities is possible.
+Proposition 1.3.1 (Grothendieck) .LetXbe a locally noetherian scheme. Suppose that for
+any integral separated scheme Yfinite overX, there exists a modification f:Z→Ywith
+Zregular. Then Xis quasi-excellent.
+Proof.See [18, Proposition 7.9.5].
+Grothendieck then suggested that Hironaka’s proof of resolution of singularities for va-
+rieties over a field of characteristic zero could be adapted to check that any quasi-excellent
+scheme over a field of characteristic zero admits a resolution of sing ularities. To the best
+of our knowledge, this claim was never verified. However, an analogo us statement has been
+established more recently by Temkin, using an alternative proof of H ironaka’s theorem due
+to Bierstone and Milman.
+Definition 1.3.2. Aregular pair is a pair (X,Z), in which Xis a regular scheme, and Zis
+a closed subscheme of Xwhich is a normal crossings divisor . The latter means that ´ etale
+locally,Zis the zero locus on Xof a regular function of the form te1
+1···tenn, fort1,...,tna
+regular sequence of parameters and e1,...,ensome nonnegative integers.
+Theorem 1.3.3. For every noetherian quasi-excellent integral scheme XoverSpec(Q), and
+every closed proper subscheme ZofX, there exists a modification f:Y→Xsuch that
+(Y,f−1(Z))is a regular pair.
+Proof.See [39, Theorem 1.1].
+Remark 1.3.4. One can further ask for a desingularization procedure which is functorial
+for regular morphisms. This question has been addressed by Bierst one, Milman, and Temkin
+[7, 42, 43]. We will need this in a subsequent paper; see Remark 8.2.5.
+71.4 Alterations
+It will be convenient to use a slightly larger class of morphisms than ju st modifications.
+Definition 1.4.1. Analteration of an integral separated scheme Xis a proper, dominant,
+generically finite morphism f:Y→XwithYintegral. If Xis a scheme over Spec( Q), this
+implies that there is an open dense subscheme UofXsuch thatY×XU→Uis finite ´ etale.
+IfXis excellent, then so is Yby Proposition 1.2.5.
+Remark 1.4.2. Alterations were introduced by de Jong to give a weak form of resolu tion
+of singularities in positive characteristic and for arithmetic schemes ; see [11, Theorem 4.1].
+Since here we onlyconsider schemes over a field of characteristic 0, this benefit is not relevant
+for us; the reason we consider alterations is because the valuation -theoretic arguments of §7
+are easier to state in terms of alterations than modifications. Othe rwise, one must work
+not just with the Berkovich unit disc, but also with more general one -dimensional analytic
+spaces, as in Temkin’s proof of inseparable local uniformization [41].
+Lemma 1.4.3. Letg:Z→Xbe an alteration of a finite-dimensional noetherian integra l
+separated scheme X. Then there exists a modification f:Y→Xsuch that the proper
+transform of gunderfis finite flat.
+Proof.By Theorem 1.1.3, we can choose fso that the proper transform g′ofgunderfis
+flat. Sinceg′is flat, it has equidimensional fibres by [32, Theorem 19]. Hence g′is locally of
+finite type with finite fibres, i.e., g′is quasifinite. Since any proper quasifinite morphism is
+finite by Zariski’s main theorem [19, Th´ eor` eme 8.11.1], we conclude th atg′is finite.
+1.5 Complex analytic spaces
+We formally introduce the category of complex analytic spaces, and the notion of a modifi-
+cation in that category. One can also define alterations of complex a nalytic spaces, but we
+will not need them here.
+Definition 1.5.1. ForXa locally ringed space, a closed subspace ofXis a subset of the
+form Supp( OX/I) for some ideal sheaf IonX, equipped with the restriction of the sheaf
+OX/I.
+Definition 1.5.2. Acomplex analytic space is a locally ringed space Xwhich is locally
+isomorphictoaclosedsubspaceofanaffinespacecarryingtheshea fofholomorphicfunctions.
+We define morphisms of complex analytic spaces, and closed subspac es of a complex analytic
+space, using the corresponding definitions of the underlying locally r inged spaces.
+Definition 1.5.3. Amodification of irreducible reduced separated complex analytic spaces
+isamorphism f:Y→Xwhichisproper(asamapoftopologicalspaces) andsurjective, an d
+whichrestrictstoanisomorphismonthecomplement ofaclosedsubs paceofX. Forinstance,
+the analytification of a modification of complex algebraic varieties is ag ain a modification;
+the hard part of this statement is the fact that algebraic propern ess implies topological
+properness, for which see [20, Expos´ e XII, Proposition 3.2].
+8Definition 1.5.4. In the category of complex analytic spaces, a regular pair will denote a
+pair (X,Z) in which Xis a complex analytic space, Zis a closed subspace of X, and for
+eachx∈X,OX,xis regular (i.e., Xis smooth at x) and the ideal sheaf defining Zdefines a
+normal crossings divisor on Spec( OX,x).
+The relevant form of resolution of singularities for complex analytic s paces is due to
+Aroca, Hironaka, and Vicente [1, 2].
+Theorem 1.5.5. For every irreducible reduced separated complex analytic s paceXand every
+closed proper subspace ZofX, there exists a modification f:Y→Xsuch that (Y,f−1(Z))
+is a regular pair.
+2 Valuation theory
+We need some basic notions from the classical theory of Krull valuat ions. Our blanket
+reference for valuation theory is [44].
+2.1 Krull valuations
+Definition 2.1.1. Avaluation (orKrull valuation ) on a field Fwith values in a totally
+ordered group Γ is a function v:F→Γ∪{+∞}satisfying the following conditions.
+(a) Forx,y∈F,v(xy) =v(x)+v(y).
+(b) Forx,y∈F,v(x+y)≥min{v(x),v(y)}.
+(c) We have v(1) = 0 and v(0) = +∞.
+We sayvistrivialifv(x) = 0 for all x∈F×. Areal valuation is a Krull valuation with
+Γ =R.
+We say the valuations v1,v2areequivalent if for allx,y∈F,
+v1(x)≥v1(y)⇐⇒v2(x)≥v2(y).
+The isomorphism classes of the following objects associated to vare equivalence invariants:
+value group: Γ v=v(F×)
+valuation ring: ov={x∈F:v(x)≥0}
+maximal ideal: mv={x∈F:v(x)>0}
+residue field: κv=ov/mv.
+Note that the equivalence classes of valuations on Fare in bijection with the valuation rings
+of the field F, i.e., the subrings oofFsuch that for any x∈F×, at least one of xorx−1
+belongs to o. (Given a valuation ring o, the natural map v:F→(F×/o×)∪{+∞}sending
+0 to +∞is a valuation with ov=o.)
+9Definition 2.1.2. LetRbe an integral domain, and let vbe a valuation on Frac( R). We
+sayviscentered on Rifvtakes nonnegative values on R, i.e., ifR⊆ov. In this case, R∩mv
+is a prime ideal of R, called the centerofvonR.
+Similarly, let Xbe an integral separated scheme with function field K(X), and letvbe
+a valuation on K(X). ThecenterofvonXis the set of x∈XwithoX,x⊆ov; it is either
+empty or an irreducible closed subset of X(see [44, Proposition 6.2], keeping in mind that
+the hypothesis of separatedness is needed to reduce to the affine case). In the latter case, we
+sayviscentered on X, and we refer to the generic point of the center as the generic center
+ofv. In fact, we will refer so often to valuations on K(X) centered on Xthat we will simply
+call them centered valuations on X.
+IfX= SpecR, thenvis centered on Xif and only if vis centered on R, in which case
+the center of vonRis the generic center of vonX.
+Lemma 2.1.3. (a) LetFbe a field and let vbe a valuation on F. Then for any field E
+containing F, there exists an extension of vto a valuation on E.
+(b) Letf:Y→Xbe a proper dominant morphism of integral separated schemes . For any
+centered valuation vonX, any extension of vtoK(Y)is centered on Y. (Such an
+extension exists by (a).)
+Proof.We may deduce (a) by applying to ovthe fact that every local subring of Eis domi-
+nated by a valuation ring [44, §1]. Part (b) is the valuative criterion for properness; see [16,
+Th´ eor` eme 7.3.8].
+2.2 Numerical invariants
+Here are a few basic numerical invariants attached to valuations.
+Definition 2.2.1. Letvbe a valuation on a field F. Anisolated subgroup (orconvex
+subgroup) of Γvis a subgroup Γ′such that for all α∈Γ′,β∈Γvwith−α≤β≤α, we
+haveβ∈Γ′. In this case, the quotient group Γ v/Γ′inherits a total ordering from Γ v, and we
+obtain a valuation v′onFwith value group Γ v/Γ′by projection. We also obtain a valuation
+vonκv′with value group Γ′. The valuation vis said to be a composite ofv′andv.
+Let ratrank( v) = dim Q(Γv⊗ZQ) denote the rational rank ofv. Forka subfield of κv,
+let trdeg(κv/k) denote the transcendence degree ofκvoverk.
+Theheight(orreal rank ) ofv, denoted height( v), is the maximum length of a chain of
+proper isolated subgroups of Γ v; note that height( v)≤ratrank(v) [44, Proposition 3.5]. By
+definition, height( v)>1ifandonlyifΓ vadmitsanonzeroproperisolatedsubgroup, inwhich
+casevcan be described as a composite valuation as above. On the other ha nd, height(v)≤1
+if and only if vis equivalent to a real valuation [44, Proposition 3.3, Exemple 3].
+There is a fundamental inequality due to Abhyankar (generalizing a r esult of Zariski),
+which gives rise to an additional numerical invariant for valuations ce ntered on noetherian
+schemes. This invariant quantifies the difficulty of describing the valu ation in local coordi-
+nates.
+10Definition 2.2.2. LetXbe a noetherian integral separated scheme. Let vbe a centered
+valuation on X, with center Z. Define the transcendence defect ofvas
+trdefect(v) = codim(Z,X)−ratrank(v)−trdeg(κv/K(Z)).
+(The term defect rank is used in [41, Remark 2.1.1] for the same quantity.) We say vis an
+Abhyankar valuation if trdefect(v) = 0.
+Theorem 2.2.3 (Zariski-Abhyankar inequality) .LetXbe a noetherian integral separated
+scheme. Then for any centered valuation vonXwith center Z,trdefect(v)≥0. Moreover,
+if equality occurs, then Γvis a finitely generated abelian group, and κvis a finitely generated
+extension of K(Z).
+Proof.We may reduce immediately to the case where X= Spec(R) forRa local ring, and
+Zis the closed point of X. In this case, see [47, Appendix 2, Corollary, p. 334] or [44,
+Th´ eor` eme 9.2].
+These invariants behave nicely under alterations, in the following sen se.
+Lemma 2.2.4. Letf:Y→Xbe an alteration of a noetherian integral separated scheme
+X. Letvbe a centered valuation on X.
+(a) There exists at least one extension wofvto a centered valuation on Y.
+(b) For any was in (a), we have height(w) = height( v),ratrank(w) = ratrank( v), and
+trdefect(w)≤trdefect(v)with equality if Xis excellent.
+Proof.For (a), apply Lemma 2.1.3. For (b), we may check the equality of heig hts and
+rational ranks at the level of the function fields. To check the ineq uality of transcendence
+defects, let Zbe the center of vonX, and letWbe the center of wonY; then
+trdefect(v)−trdefect(w) = codim(Z,X)−codim(W,Y)+trdeg(κw/K(W))−trdeg(κv/K(Z))
+= codim(Z,X)−codim(W,Y)+trdeg(κw/κv)−trdeg(K(W)/K(Z)).
+We may check that trdeg( κw/κv) = 0 at the level of function fields; see [44, §5] for this verifi-
+cation. After replacing Xby the spectrum of a local ring, we may apply [18, Th´ eor` eme 5.5.8]
+or [32, Theorem 23] to obtain the inequality
+codim(Z,X)+trdeg(K(Y)/K(X))≥codim(W,Y)+trdeg(K(W)/K(Z)),
+with equality in case Xis excellent. Since fis an alteration, K(Y) is algebraic over K(X)
+and so trdeg( K(Y)/K(X)) = 0. This yields the desired comparison.
+Here is another useful property of Abhyankar valuations.
+11Remark 2.2.5. LetRbe a noetherian local ring with completion /hatwideR. Letvbe a centered
+real valuation on R. Thenvextends by continuity to a function ˆ v:/hatwideR→Γv∪{+∞}. This
+function is in general a semivaluation , in that it satisfies all of the conditions defining a
+valuation except that p= ˆv−1(+∞) is only a prime ideal, not necessarily the zero ideal.
+For instance, choose a=a1x+a2x2+··· ∈k/llbracketx/rrbrackettranscendental over k(x). PutR=
+k[x,y](x,y), andletvbetherestrictionofthe x-adicvaluationof k/llbracketx/rrbracketalongthemap R→k/llbracketx/rrbracket
+defined byx/mapsto→x,y/mapsto→a. Thenvis a valuation on R, butp= (y−a)/\e}atio\slash= 0.
+On the other hand, suppose vis a real Abhyankar valuation. Then ˆ vinduces a valuation
+on/hatwideR/pwith the same value group and residue field as v. By the Zariski-Abhyankar inequal-
+ity, this forces dim( /hatwideR/p)≥dimR; since dim R= dim/hatwideR[13, Corollary 10.12], this is only
+possible for p= 0. That is, if vis a real Abhyankar valuation, it extends to a valuation on
+/hatwideR.
+2.3 Riemann-Zariski spaces
+We need a mild generalization of Zariski’s original compactness theore m for spaces of valua-
+tions, which we prefer to state in scheme-theoretic language. See [40,§2] for a much broader
+generalization.
+Definition 2.3.1. ForRa ring, the patch topology on Spec(R) is the topology generated
+by the sets D(f) ={p∈Spec(R) :f /∈p}and their complements. Note that for any f∈R,
+the open sets for the patch topology on D(f) are also open for the patch topology on R. It
+follows that we can define the patch topology on a scheme Xto be the topology generated
+by the open subsets for the patch topologies on all open affine subs chemes ofX, and this
+will agree with the previous definition for affine schemes. The resultin g topology is evidently
+finer than the Zariski topology.
+Lemma 2.3.2. Any noetherian scheme is compact for the patch topology.
+Proof.Bynoetherianinduction, itsuffices tocheck thatif Xisanoetherianscheme suchthat
+each closed proper subset of Xis compact for the patch topology, then Xis also compact for
+the patch topology. Since a noetherian scheme is covered by finitely many affine noetherian
+schemes, each of which has finitely many irreducible components, we may reduce to the case
+of an irreducible affine noetherian scheme Spec( R).
+Givenanopencover ofSpec( R)forthepatchtopology, theremustbeanopensetcovering
+the generic point. This open set must contain a basic open set of the formD(f)\D(g) for
+somef,g∈R, but this only covers the generic point if D(g) is empty. Hence our open cover
+includes an open set containing D(f) for somef∈Rwhich is not nilpotent. By hypothesis,
+the closed set X\D(f) is covered by finitely many opens from the cover, as then is X.
+Definition 2.3.3. LetXbe a noetherian integral separated scheme. The Riemann-Zariski
+spaceRZ(X)consists oftheequivalence classes ofcentered valuationson X. IfX= Spec(R),
+we also write RZ( R) instead of RZ( X).
+We may identify RZ( X) with the inverse limit over modifications f:Y→X, as fol-
+lows. Given v∈RZ(X), we take the element of the inverse limit whose component on a
+12modification f:Y→Xis the generic center of vonY(which exists by Lemma 2.1.3(a)).
+Conversely, given an element of the inverse limit with value xYonY, form the direct limit
+of the local rings OY,xY; this gives a valuation ring because any g∈K(X) defines a rational
+mapX/axisshort/axisshort/arrowaxisrightP1
+Z, and the Zariski closure of the graph of this rational map is a modific ation
+WofXsuch that one of gorg−1belongs to OW,xW.
+We equip RZ( X) with the Zariski topology , defined as the inverse limit of the Zariski
+topologies on the modifications of X. For any dominant morphism X→Wof noetherian
+integral schemes, we obtainaninduced continuous morphism RZ( X)→RZ(W)using proper
+transforms.
+Theorem 2.3.4. For any noetherian integral separated scheme X, the space RZ(X)is quasi-
+compact.
+Proof.Foreachmodification f:Y→X,YiscompactforthepatchtopologybyLemma2.3.2.
+If we topologize RZ( X) with the inverse limit of the patch topologies, the result is compact
+by Tikhonov’s theorem. The Zariski topology is coarser than this, s o RZ(X) is quasicompact
+for the Zariski topology. (Note that we cannot check this directly because an inverse limit
+of quasicompact topological spaces need not be quasicompact.)
+Proposition 2.3.5. Letf:Y→Xbe a morphism of finite-dimensional noetherian integral
+separated schemes, which is dominant and of finite presentat ion. Then the map RZ(f) :
+RZ(Y)→RZ(X)is open.
+Proof.By Theorem 1.1.3, there exists a modification g:Z→Xsuch that the proper
+transformh:W→Zoffundergis flat. The claim now follows from the fact that a
+morphism which is flat and locally of finite presentation is open [18, Th´ eor` eme 2.4.6].
+3 Nondegenerate differential schemes
+We now explain how the notion of an excellent scheme interacts with de rivations, and with
+our discussion of good formal structures in [25].
+Definition 3.0.1. LetR֒→Sbe an inclusion of domains, and let Mbe a finiteS-module.
+By anR-latticeinM, we mean a finite R-submodule LofMsuch that the induced map
+L⊗RS→Mis surjective.
+3.1 Nondegenerate differential local rings
+We first introduce a special class of differential local rings.
+Definition 3.1.1. Adifferential (local) ring is a (local) ring Requipped with an R-module
+∆Racting onRvia derivations, together with a Lie algebra structure on ∆ Rcompatible
+with the Lie bracket on derivations.
+13Definition 3.1.2. Let (R,∆R) be a differential local ring with maximal ideal mand residue
+fieldκ=R/m. We sayRisnondegenerate if it satisfies the following conditions.
+(a) The ring Ris a regular (hence noetherian) local Q-algebra.
+(b) TheR-module ∆ Ris coherent.
+(c) For some regular sequence of parameters x1,...,xnofR, there exists a sequence
+∂1,...,∂n∈∆Rof derivations of rational type with respect to x1,...,xn. That is,
+∂1,...,∂nmust commute pairwise, and must satisfy
+∂i(xj) =/braceleftBigg
+1 (i=j)
+0 (i/\e}atio\slash=j).(3.1.2.1)
+The existence of such derivations for a single regular sequence of p arameters implies
+the same for any other regular sequence of parameters; see Cor ollary 3.1.9.
+Remark 3.1.3. Note that (3.1.2.1) by itself does not force ∂1,...,∂nto commute pairwise.
+For instance, if R=C(t)/llbracketx1,x2/rrbracket, we can satisfy (3.1.2.1) by taking
+∂1=∂
+∂x1+∂
+∂t, ∂ 2=∂
+∂x2+t∂
+∂t.
+Example 3.1.4. The following are all examples of nondegenerate local rings.
+(a) Any local ring of a smooth scheme over a field of characteristic 0 .
+(b) Any local ring of a smooth complex analytic space.
+(c) Any completion of a nondegenerate local ring with respect to a p rime ideal. (By
+Lemma 3.1.6 below, we may reduce to the case of completion with respe ct to the
+maximal ideal, for which the claim is trivial.)
+We insert some frequently invoked remarks concerning the regular ity condition.
+Remark 3.1.5. LetRbe a regular local ring. Let /hatwideRbe the completion of Rwith respect
+to its maximal ideal; then the morphism R→/hatwideRis faithfully flat [33, Theorem 8.14]. More-
+over, since ´ etaleness descends down faithfully flat quasicompact morphisms of schemes [20,
+Expos´ e IX, Proposition 4.1], any element of /hatwideRwhich is algebraic over Rgenerates a finite
+´ etale extension of Rwithin/hatwideRwith the same residue field.
+Lemma 3.1.6. Let(R,∆R)be a nondegenerate differential local ring with maximal idea l
+m. For any prime ideal qcontained in m, the differential ring (Rq,∆R⊗RRq)is again
+nondegenerate.
+Proof.By [32, Theorem 45, Corollary], Rqis regular, and any regular sequence of parameters
+ofRcontains a regular sequence of parameters for Rq. From these assertions, the claim is
+evident.
+14The following partly generalizes [25, Lemma 2.1.3]. As the proof is identic al, we omit
+details.
+Lemma 3.1.7. Let(R,∆R)be a nondegenerate differential local ring. Suppose further that
+for somei∈ {1,...,n},Risxi-adically complete.
+(a) Fore∈Z,xi∂iacts onxe
+iR/xe+1
+iRvia multiplication by e.
+(b) The action of xi∂ionxiRis bijective.
+(c) The kernel Riof∂ionR[x−1
+i]is contained in Rand projects bijectively onto R/xiR.
+There thus exists an isomorphism R∼=Ri/llbracketxi/rrbracketunder which ∂icorresponds to∂
+∂xi.
+Corollary 3.1.8. Let(R,∆R)be a nondegenerate differential local ring which is complete
+with respect to its maximal ideal m. Then there exists an isomorphism R∼=k/llbracketx1,...,xn/rrbracketfor
+k=R/m, under which ∂icorresponds to∂
+∂xifori= 1,...,n.
+Corollary 3.1.9. Let(R,∆R)be a (not necessarilycomplete) nondegeneratedifferential local
+ring. Then condition (c) of Definition 3.1.2 holds for any reg ular sequence of parameters.
+Proof.Lety1,...,ynbe a second regular sequence of parameters. Define the matrix Aby
+puttingAij=∂i(yj); then det( A) is not in the maximal ideal mofR, and so is a unit in R.
+We may then define the derivations
+∂′
+j=/summationdisplay
+i(A−1)ij∂i(j= 1,...,n),
+and these will satisfy
+∂′
+i(yj) =/braceleftBigg
+1 (i=j)
+0 (i/\e}atio\slash=j).
+It remains to check that the ∂′
+icommute pairwise; for this, it is harmless to pass to the
+case where Ris complete with respect to m. In this case, by Corollary 3.1.8, we have
+an isomorphism R∼=k/llbracketx1,...,xn/rrbracketunder which each ∂icorresponds to the formal partial
+derivative in xi. That isomorphism induces an embedding of kintoRwhose image is
+killed by∂1,...,∂nand hence also by ∂′
+1,...,∂′
+n. We thus obtain a second isomorphism
+R∼=k/llbrackety1,...,yn/rrbracketunder which each ∂′
+icorresponds to the formal partial derivative in yi. In
+particular, these commute pairwise, as desired.
+3.2 Nondegenerate differential schemes
+Wenowconsider moregeneraldifferential ringsandschemes, follow ing[25,§1]. Weintroduce
+the nondegeneracy condition for these and show that it implies exce llence.
+Definition 3.2.1. Adifferential scheme is a scheme Xequipped with a quasicoherent OX-
+moduleDXacting on OXvia derivations, together with a Lie algebra structure on DX
+compatible with the Lie bracket on derivations. Note that the categ ory of differential affine
+schemes is equivalent to the category of differential rings in the obv ious fashion.
+15Definition 3.2.2. We say a differential scheme ( X,DX) isnondegenerate ifXis separated
+and noetherian of finite Krull dimension, DXis coherent over OX, and each local ring of X
+is nondegenerate. We say a differential ring Ris nondegenerate if Spec( R) is nondegenerate;
+this agrees with the previous definition in the local case.
+Remark 3.2.3. Let (R,∆R) be a differential domain. For several types of ring homomor-
+phismsf:R→SwithSalso a domain, there is a canonical way to extend the differential
+structure on Rto a differential structure on S, provided we insist that the differential struc-
+ture besaturated . That is, we equip Swith the subset of ∆ Frac(R)⊗RSconsisting of elements
+which act as derivations on Frac( S) preserving S.
+To be specific, we may perform such a canonical extension for fof the following types.
+(a) A generically finite morphism of finite type.
+(b) A localization.
+(c) A morphism from Rto its completion with respect to some ideal.
+IfRis nondegenerate, it is clear in cases (b) and (c) that Sis also nondegenerate. In case
+(a), one can only expect this if Sis regular, in which case it is true but not immediate; this
+is the content of the following lemma.
+Lemma 3.2.4. LetXbe a nondegenerate differential scheme, and let f:Y→Xbe an
+alteration with Yregular. Then the canonical differential scheme structure o nY(see Re-
+mark 3.2.3) is again nondegenerate.
+Proof.What is needed is to check that every local ring of Yis nondegenerate, so we may
+fixy∈Yandx=f(y)∈X. Since the nondegenerate locus is stable under generization by
+Lemma 3.1.6, it suffices to consider cases where yandxhave the same codimension, as such
+yare dense in each fibre of f.
+Choosearegularsystemofparameters x1,...,xnforXatx. SinceOX,xisnondegenerate,
+wecanchoosederivations ∂1,...,∂nactingonsomeneighborhood Uofxwhichareofrational
+type with respect to x1,...,xn. By Lemma 3.1.7, we can write /hatwideOX,x∼=k/llbracketx1,...,xn/rrbracketin such
+a way that ∂1,...,∂ncorrespond to∂
+∂x1,...,∂
+∂xn.
+Choose a regular system of parameters y1,...,ynforYaty. By our choice of y, the
+residue field ℓofyis finite over k. We may thus identify /hatwidestOY,ywithℓ/llbrackety1,...,yn/rrbracketfor the
+samenin such a way that the morphism /hatwideOX,x→/hatwidestOY,ycarrieskintoℓ. Using such an
+identification, we see (as in the proof of Corollary 3.1.9) that if we defi ne the matrix Aover
+OY,ybyAij=∂i(yj), then put ∂′
+j=/summationtext
+i(A−1)ij∂i, we obtain derivations of rational type with
+respect toy1,...,yn. HenceOY,yis also nondegenerate, as desired.
+Lemma 3.2.5. Let(X,DX)be a nondegenerate differential scheme.
+(a) The scheme Xis excellent.
+(b) For each x∈X, the differential local ring OX,xis simple.
+16(c) IfXis integral, then the subring of Γ(X,OX)killed by the action of DXis a field.
+Proof.For (a), see [33, Theorem 101]. For (b), note that for mX,xthe maximal ideal of OX,x
+andκxthe residue field, the nondegeneracy condition forces the pairing
+∆OX,x×mX,x/m2
+X,x→κx (3.2.5.1)
+to be nondegenerate on the right. The differential ring OX,xis then simple by [25, Proposi-
+tion 1.2.3]. For (c), note that the nondegeneracy condition preven ts anyr∈Γ(X,OX) killed
+by the action of DXfrom belonging to the maximal ideal of any local ring of Xunless it
+vanishes in the local ring.
+Corollary 3.2.6. Every local ring of an algebraic variety over a field of charac teristic0, or
+of a complex analytic variety, is excellent. Moreover, the c ompletion of any such ring with
+respect to any ideal is again excellent.
+Proof.By Proposition 1.2.5, it is enough to check both claims when the variety in question
+is the (algebraic or analytic) affine space of some dimension. In partic ular, such a space has
+regular local rings, so Lemma 3.2.5(a) applies to yield the conclusion.
+Corollary 3.2.7. LetXbe a smooth complex analytic space which is Stein. Let Kbe a
+compact subset of X. Then the localization of Γ(X,OX)atK(i.e., the localization by the
+multiplicative set of functions which do not vanish on K) is excellent, as is any completion
+thereof.
+Proof.The localization is noetherian by [21, Theorem 1.1]; the claim then follows from
+Lemma 3.2.5.
+Remark 3.2.8. The terminology nondegenerate for differential rings arises from the fact we
+had originally intended condition (c) of Definition 3.1.2 to state that th e pairing (3.2.5.1)
+must be nondegenerate on the right. However, it is unclear whethe r this suffices to ensure
+the existence of derivations of rational type with respect to a reg ular sequence of parameters.
+Without such derivations, it is more difficult to work in local coordinate s. To handle this
+situation, one would need to rework significant sections of both [25 ] and [28]; we opted
+against this approach because it is not necessary in the applications of greatest interest, to
+algebraic and analytic varieties.
+3.3∇-modules
+We next consider differential modules.
+Definition 3.3.1. A∇-module over a differential scheme Xis a coherent OX-module F
+equipped with an action of DXcompatible with the action on OX. Over an affine differential
+scheme with noetherian underlying scheme, this is the same as a finite differential module
+over the coordinate ring.
+17Definition 3.3.2. For (X,DX) a differential scheme and φ∈Γ(X,OX), letE(φ) be the
+∇-module free on one generator vsatisfying∂(v) =∂(φ)vfor any open subscheme UofX
+and any∂∈Γ(U,DX).
+Lemma 3.3.3. LetXbe a nondegenerate differential scheme. Then every ∇-module over
+Xis locally free over OX(and hence projective).
+Proof.This follows from Lemma 3.2.5(b) via [25, Proposition 1.2.6].
+Remark 3.3.4. LetRbe a nondegenerate differential domain, and let Mbe a finite dif-
+ferential module over R. By Lemma 3.2.5, Mis projective over R, and so is a direct
+summand of a free R-module. Hence for any (not necessarily differential) domains S,T,U
+withR⊆S,T⊆U, withinM⊗RUwe have
+(M⊗RS)∩(M⊗RT) =M⊗R(S∩T).
+3.4 Admissible and good decompositions
+We next reintroduce the notions of good decompositions and good f ormal structures from
+[25], in the language of nondegenerate differential rings. Remember that these notions do
+not quite match the ones used by Mochizuki; see [25, Remark 4.3.3, Re mark 6.4.3].
+Hypothesis 3.4.1. Throughout §3.4–3.5, let Rbe a nondegenerate differential local ring.
+Letx1,...,xnbe a regular sequence of parameters for R, and putS=R[x−1
+1,...,x−1
+m] for
+somem∈ {0,...,n}. Let/hatwideRbe the completion of Rwith respect to its maximal ideal, and
+put/hatwideS=/hatwideR[x−1
+1,...,x−1
+m]. LetMbe a finite differential module over S.
+Definition 3.4.2. Let ∆log
+Rbe the subset of ∆ Rconsisting of derivations under which the
+ideals (x1),...,(xm) are stable. We say Misregularif there exists a free R-latticeM0in
+Mstable under the action of ∆log
+R. We sayMistwist-regular if End(M) =M∨⊗RMis
+regular.
+Example 3.4.3. For anyφ∈R,E(φ) is regular. For any φ∈S,E(φ) is twist-regular.
+Remark 3.4.4. In caseR=/hatwideR, when checking regularity of M, we may choose an isomor-
+phismR∼=k/llbracketx1,...,xn/rrbracketas in Corollary 3.1.8 and check stability of the lattice just under
+x1∂1,...,xm∂m,∂m+1,...,∂n, as then [25, Proposition 2.2.8] implies stability under all of
+∆log
+R. This means that in case R=/hatwideR, our definitions of regularity and twist-regularity
+match the definitions from [25], so we may invoke results from [25] re ferring to these notions
+without having to worry about our extra level of generality.
+Proposition 3.4.5. SupposeR=/hatwideR. Then the differential module Mis twist-regular if and
+only ifM=E(φ)⊗SNfor someφ∈Sand some regular differential module NoverS.
+Proof.This is [25, Theorem 4.2.3] (plus Remark 3.4.4).
+18Definition 3.4.6. Anadmissible decomposition ofMis an isomorphism
+M∼=/circleplusdisplay
+α∈IE(φα)⊗SRα (3.4.6.1)
+for someφα∈S(indexed by an arbitrary set I) and some regular differential modules
+RαoverS. An admissible decomposition is goodif it satisfies the following two additional
+conditions.
+(a) Forα∈I, ifφα/∈R, thenφαhas the form ux−i1
+1···x−immfor some unit uinRand
+some nonnegative integers i1,...,im.
+(b) Forα,β∈I, ifφα−φβ/∈R, thenφα−φβhas the form ux−i1
+1···x−immfor some unit u
+inRand some nonnegative integers i1,...,im.
+Aramified good decomposition ofMis a good decomposition of M⊗RR′for some connected
+finite integral extension R′ofRsuch thatR′⊗RSis ´ etale over S. By Abhyankar’s lemma
+[20, Expos´ e XIII, Proposition 5.2], any such extension is contained inR′′[x1/h
+1,...,x1/h
+m] for
+some connected finite ´ etale extension R′′ofRand some positive integer h. Agood formal
+structure ofMis a ramified good decomposition of M⊗RR′forR′the completion of Rwith
+respect to ( x1,...,xm). This is not the same as a ramified good decomposition of M⊗R/hatwideR
+(since/hatwideRis the completion with respect to the larger ideal m), but any such decomposition
+does in fact induce a good formal structure (see Proposition 4.4.1) .
+Remark 3.4.7. In Definition 3.4.6, an admissible decomposition need not be unique if it
+exists. However, thereisa unique minimal admissible decomposition, obtainedby combining
+the terms indexed by αandβwheneverφα−φβ∈R. The resulting minimal admissible
+decomposition is good if and only if the original admissible decomposition is good.
+The following limited descent argument will crop up several times.
+Proposition 3.4.8. Suppose that Ris henselian, and that M⊗S/hatwideSadmits a filtration 0 =
+M0⊂ ··· ⊂Mℓ=M⊗S/hatwideSby differential submodules, in which each successive quotie nt
+Mj+1/Mjadmits an admissible decomposition ⊕α∈IjE(φα)⊗/hatwideSRα. Then the φαalways belong
+toS+/hatwideR; in particular, they can be chosen in Sif desired.
+Proof.Fori= 0,...,m, put/hatwideSi=/hatwideR[x−1
+1,...,x−1
+i]. We show that
+φα∈S+/hatwideSi(i=m,...,0;α∈/uniondisplay
+jIj) (3.4.8.1)
+by descending induction on i, the casei=mbeing evident and the case i= 0 yielding the
+desired result. Given (3.4.8.1) for some i >0, we can choose a nonnegative integer hsuch
+that
+xh
+iφα∈S+/hatwideSi−1(α∈/uniondisplay
+jIj). (3.4.8.2)
+19We wish to achieve this for h= 0, which we accomplish using a second descending induction
+onh. Ifh>0, write each xh
+iφαasfα+gαwithfα∈Sandgα∈/hatwideSi−1.
+Chooseα∈Ijfor somej. LetTandUdenote the xi-adic completions of Frac( S) and
+Frac(/hatwideS), respectively. Put N=M⊗SE(−x−h
+ifα). We may then apply [25, Theorem 2.3.3]
+to obtain a Hukuhara-Levelt-Turrittin decomposition of N⊗ST′for some finite extension
+T′ofT.
+We claim that U′=T′⊗TUis a field extension of U, from which it follows that T′is
+the integral closure of TinU′. It suffices to check this after adjoining x1/m
+ifor some positive
+integerm, so we may assume T′is unramified over T. In that case, we must show that
+the residue fields of T′andUare linearly disjoint over the residue field of T, i.e., that for
+any finite extension ℓof Frac(R/xiR),ℓand Frac( /hatwideR/xi/hatwideR) have no common subfield strictly
+larger than Frac( R/xiR). This holds because by Remark 3.1.5, such a subfield would induce
+a finite ´ etale extension of R/xiRwith the same residue field; however, any such extension
+must equal R/xiRbecause the latter ring is henselian (because Ris). This proves the claim.
+We may extend scalars to obtain a Hukuhara-Levelt-Turrittin deco mposition of N⊗SU′
+in which the factors E(r) all haver∈T′. However, since α∈Ij, (N⊗SU′)⊗U′E(−x−h
+igα)
+has a nonzero regular subquotient. This is only possible if one of the f actorsE(r) in the
+decomposition of N⊗SU′satisfiesr≡x−h
+igα(modoU′).
+In particular, if we choose e1,...,ei−1so thatg′
+α=xe1
+1...xei−1
+i−1gαbelongs to /hatwideR, then the
+image ofg′
+αin/hatwideR/xi/hatwideRis algebraic over Frac( R/xiR). We again use the henselian property of
+R/xiRto deduce that the image of g′
+αin/hatwideR/xi/hatwideRmust in fact belong to R/xiR. This allows
+us to replace hbyh−1 in (3.4.8.2), completing both inductions.
+Remark 3.4.9. Proposition 3.4.8 implies that if Ris henselian and M⊗S/hatwideSadmits a
+good decomposition, then the terms φαappearing in (3.4.6.1) can be defined over S. Using
+Theorem 3.5.3 below, we can also realize the regular modules RαoverSprovided that
+we can identify /hatwideRwithk/llbracketx1,...,xn/rrbracketin such a way that kembeds into R. In such cases,
+M⊗S/hatwideSadmits a good decomposition in the sense of Sabbah [38, I.2.1.5]. This o bservation
+generalizes an argument of Sabbah for surfaces [38, Proposition I.2.4.1] and fulfills a promise
+made in [25, Remark 6.2.5].
+On the other hand, Proposition 3.4.8 does not imply that any good decomposition of
+M⊗S/hatwideSdescends to a good decomposition of Mitself. This requires two additional steps
+which cannot always be carried out. One must descend the project ors cutting out the
+summands E(φα)⊗/hatwideSRαof the minimal good decomposition. If this can be achieved, then
+by virtue of Proposition 3.4.8, each summand can be twisted to give a d ifferential module
+NαoverSsuch thatNα⊗S/hatwideSis regular. One must then check that each Nαitself is regular.
+For a typical situation where both steps can be executed, see The orem 4.3.4.
+3.5 Good decompositions over complete rings
+We now examine more closely the case of a nondegenerate differentia lcomplete local ring,
+recalling some of the key results from [25]. Throughout §3.5, continue to retain Hypothe-
+sis 3.4.1.
+20Definition 3.5.1. Use Corollary 3.1.8 to identify /hatwideRwithk/llbracketx1,...,xn/rrbracketforkthe residue
+field ofR, using some derivations ∂1,...,∂nof rational type with respect to x1,...,xn. For
+r= (r1,...,rn)∈[0,+∞)n, let| · |rbe the (e−r1,...,e−rn)-Gauss norm on /hatwideR; note that
+this does not depend on the choice of the isomorphism /hatwideR∼=k/llbracketx1,...,xn/rrbracket. LetFrbe the
+completion of Frac( /hatwideR) with respect to |·|r. LetF(M,r) be the irregularity of M⊗SFr, as
+defined in [25, Definition 1.4.8]. We say Misnumerical ifF(M,r) is a linear function of r.
+The following is a consequence of [25, Theorem 3.2.2].
+Theorem 3.5.2. The function F(M,r)is continuous, convex, and piecewise linear. More-
+over, forj∈ {m+ 1,...,n}, if we fix rifori/\e}atio\slash=j, thenF(M,r)is nonincreasing as a
+function of rjalone.
+We have the following numerical criterion for regularity by [25, Theor em 4.1.4] (plus
+Remark 3.4.4).
+Theorem 3.5.3. Assume that R=/hatwideR. Then the following conditions are equivalent.
+(a)Mis regular.
+(b) There exists a basis of Mon whichx1∂1,...,xm∂mact via commuting matrices over
+kwith prepared eigenvalues (i.e., no eigenvalue or differenc e between two eigenvalues
+equals a nonzero integer), and ∂m+1,...,∂nact via the zero matrix.
+(c) We have F(M,r) = 0for allr.
+We also have the following numerical criterion for existence of a ramifi ed good decompo-
+sition in the complete case.
+Theorem 3.5.4. Assume that R=/hatwideR. The following conditions are equivalent.
+(a) The module Madmits a ramified good decomposition.
+(b) BothMandEnd(M)are numerical.
+Proof.This holds by [25, Theorem 4.4.2] modulo one minor point: since [25, Theo rem 4.4.2]
+does not allow for derivations on the residue field of R, it only gives a good decomposition
+with respect to the action of ∂1,...,∂n. However, if we form the minimal good decomposi-
+tion as in Remark 3.4.7, this decomposition must be preserved by the a ctions of the other
+derivations.
+We will find useful the following consequence of the existence of a go od decomposition.
+Proposition 3.5.5. Assume that R=/hatwideRand thatMadmits a good decomposition. Then
+for some finite ´ etale extension R′ofR,M⊗RR′admits a filtration 0 =M0⊂ ··· ⊂Md=
+M⊗RR′by differential submodules, with the following properties.
+(a) We have rank(Mi) =ifori= 0,...,d.
+21(b) Fori= 1,...,d−1, there exists an endomorphism of ∧iMas a differential module
+with image ∧iMi.
+Proof.We reduce first to the case where Mis twist-regular, then to the case where Mis
+regular. By Theorem 3.5.3, there exists a basis of Mon whichx1∂1,...,xm∂macts via
+commuting matrices over kwith prepared eigenvalues, and ∂m+1,...,∂nact via the zero
+matrix. Let Vbe thek-span of this basis. Choose a finite extension k′ofkcontaining all of
+the eigenvalues of these matrices, and put R′=k′/llbracketx1,...,xn/rrbracket.
+We can now split V⊗kk′as a direct sum such that on each summand, each xi∂iacts
+with a single eigenvalue; this splitting induces a splitting of Mitself. After replacing kwith
+k′, we may now reduce to the case where each xi∂iacts onVwith a single eigenvalue. By
+twisting, we can force that eigenvalue to be zero.
+By Engel’s theorem [14, Theorem 9.9], x1∂1,...,xm∂mact on some complete flag 0 =
+V0⊂ ··· ⊂Vd=VinV. The corresponding submodules 0 = M0⊂ ··· ⊂Md=MofM
+have the desired property: for instance, M1occurs as the image of the composition of the
+projectionM→M/Md−1, an isomorphism M/Md−1→M1, and the inclusion M1→M.
+4 Descent arguments
+In this section, we make some crucial descent arguments for differ ential modules over a
+localized power series ring with coefficients in a base ring.
+4.1 Hensel’s lemma in noncommutative rings
+We need to recall a technical tool in the study of differential module s over nonarchimedean
+rings, a form of Hensel’s lemma for noncommutative rings introduced by Robba. Our pre-
+sentation follows Christol [10].
+Theorem 4.1.1 (Robba, Christol) .LetRbe a nonarchimedean, not necessarily commutative
+ring. Suppose the nonzero elements a,b,c∈Rand the additive subgroups U,V,W⊆Rsatisfy
+the following conditions.
+(a) The spaces U,Vare complete under the norm, and UV⊆W.
+(b) The map f(u,v) =av+ubis a surjection of U×VontoW.
+(c) There exists λ>0such that
+|f(u,v)| ≥λmax{|a||v|,|b||u|}(u∈U,v∈V).
+(Note that this forces λ≤1.)
+(d) We have ab−c∈Wand
+|ab−c|<λ2|c|.
+22Then there exists a unique pair (x,y)∈U×Vsuch that
+c= (a+x)(b+y),|x|<λ|a|,|y|<λ|b|.
+For thisx,y, we also have
+|x| ≤λ−1|ab−c||b|−1,|y| ≤λ−1|ab−c||a|−1.
+Proof.See [10, Proposition 1.5.1] or [27, Theorem 2.2.2].
+4.2 Descent for iterated power series rings
+Using Christol’s factorization theorem, we make a decompletion argu ment analogous to [25,
+§2.6]. We use the language of Newton polygons and slopes for twisted p olynomials, as
+presented in [25, Definition 1.6.1].
+Hypothesis 4.2.1. Throughout §4.2, lethbe a positive integer. Let Abe a differential
+domain of characteristic 0, such that the module of derivations on K= Frac(A) is finite-
+dimensional over K, and the constant subring kofAis also the constant subring of K.
+(In particular, kmust be a field.) Define the ring Rh(K) asK((x1))···((xh)). Define
+the ringR†
+h(A) as the union of A((x1/f))···((xh/f))[f−1] over all nonzero f∈A. Equip
+Rh(K) andR†
+h(A) with the componentwise derivations coming from A, plus the derivations
+∂1,...,∂h=∂
+∂x1,...,∂
+∂xh.
+Lemma 4.2.2. The ringsRh(K)andR†
+h(A)are fields.
+Proof.This is clear for Rh(K), so we concentrate on R†
+h(A). We proceed by induction on h,
+the caseh= 0 being clear because R†
+0(A) =K.
+Givenr∈R†
+h(A), writer=/summationtext
+irixi
+hwithri∈R†
+h−1(A). Letmbe the smallest index
+such thatrm/\e}atio\slash= 0. By the induction hypothesis, rmis a unit in R†
+h−1(A), so we may reduce
+to the case where m= 0 andrm= 1. In this case, for some nonzero f∈Awe have
+1−r∈(xh/f)A((x1/f))···((xh−1/f))/llbracketxh/f/rrbracket[f−1];
+by replacing fby a large power, we may force
+1−r∈(xh/f)A((x1/f))···((xh−1/f))/llbracketxh/f/rrbracket.
+In that case, the formula r−1=/summationtext∞
+j=0(1−r)jshows that r−1∈R†
+h(A).
+Using Christol’s factorization theorem, we obtain the following.
+Proposition 4.2.3. EquipR†
+h(A)with thexh-adic norm (of arbitrary normalization) and the
+derivationxh∂h. Then any twisted polynomial P∈R†
+h(A){T}factors uniquely as a product
+Q1···Qmwith eachQihaving only one slope riin its Newton polygon, and r1<···<rm.
+23Proof.OverRh(K), such a factorization exists and is unique by [22, Lemma 1.6.2]. It thu s
+suffices to check existence over R†
+h(A). For this, we induct on the number of slopes in
+the Newton polygon of P(T) =/summationtext
+iPiTi. Suppose there is more than one slope; we can
+then choose an index icorresponding to an internal vertex of the Newton polygon. By
+Lemma 4.2.2, Piis a unit in R†
+h(A), so we may reduce to the case Pi= 1.
+Sinceicorresponds to an internal vertex of the Newton polygon, there exists a positive
+rational number r/ssuch that −log|Pi−j|−j(r/s)log|xh|>0 forj/\e}atio\slash= 0. That is, the norm
+ofxjr/s
+hPi−jis less than 1 for each j/\e}atio\slash= 0. Since Pj∈R†
+h(A) for allj/\e}atio\slash=i, we can choose
+f∈Anonzero so that
+Pj∈A((x1/f))···((xh/f))[f−1] (j/\e}atio\slash=i).
+We then have
+x(i−j)r/s
+hPj∈(xh/f)1/sA((x1/f))···((xh−1/f))/llbracket(xh/f)1/s/rrbracket[f−1] (j/\e}atio\slash=i).
+As in the proof of Lemma 4.2.2, by replacing fby a large power, we may force
+x(i−j)r/s
+hPj∈(xh/f)1/sA((x1/f))···((xh−1/f))/llbracket(xh/f)1/s/rrbracket(j/\e}atio\slash=i).(4.2.3.1)
+LetUbe the set of twisted polynomials Q(T) =/summationtext
+jQjTj∈R†
+h(A){T}of degree at most
+deg(P)−isuch that
+x−jr/s
+hQj∈A((x1/f))···((xh−1/f))/llbracket(xh/f)1/s/rrbracket(j= 0,...,deg(P)−i).
+LetVbe the set of twisted polynomials Q(T) =/summationtext
+jQjTj∈R†
+h(A){T}of degree at most
+i−1 such that
+x(i−j)r/s
+hQj∈A((x1/f))···((xh−1/f))/llbracket(xh/f)1/s/rrbracket(j= 0,...,i−1).
+LetWbe the set of twisted polynomials Q(T) =/summationtext
+jQjTj∈R†
+h(A){T}of degree at most
+deg(P) such that
+x(i−j)r/s
+hQj∈A((x1/f))···((xh−1/f))/llbracket(xh/f)1/s/rrbracket(j= 0,...,deg(P)).
+ThenU,V,Ware complete for the |xh|−r/s-Gaussnorm and UV⊆W. Puta= 1,b=Ti,c=
+P, so that (u,v)/mapsto→av+buis a surjection of U×VontoW, and|ab−c|<|c|.
+We now invoke Theorem 4.1.1 to obtain a nontrivial factorization Q1Q2ofPin which all
+slopes ofQ1are less than −r/swhile all slopes of Q2are greater than −r/s. (Condition (c)
+of Theorem 4.1.1 may be verified exactly as in [27, Theorem 2.2.1].) We may then invoke
+the induction hypothesis to conclude.
+Proposition 4.2.4. EquipR†
+h(A)with thexh-adic norm (of arbitrary normalization). Let
+Mbe a finite differential module over R†
+h(A). ThenMadmits a unique decomposition
+M=⊕s≥1Msas a direct sum of differential submodules, such that for each s≥1, the
+scale multiset of ∂honMs⊗R†
+h(A)Rh(K)consists entirely of s.
+24Proof.Asin[25,Proposition1.6.3],exceptusingProposition4.2.3inplaceof[25, Lemma1.6.2].
+The following argument is reminiscent of [25, Lemma 2.6.3].
+Lemma 4.2.5. EquipR†
+h(A)with thexh-adic norm (of arbitrary normalization). Let Mbe
+a finite differential module over R†
+h(A)such that the scale of ∂honM⊗R†
+h(A)Rh(K)is equal
+to1. ThenMadmits anR†
+h−1(A)-lattice stable under all of the given derivations on R†
+h(A).
+Proof.PutR=R†
+h(A) andR′=R†
+h−1(A)((xh)). By [25, Proposition 2.2.10], we can find a
+regulating lattice WinM⊗RR′. By [25, Proposition 2.2.11], the characteristic polynomial
+ofxh∂honW/xhWhas coefficients in the constant subring of R, which isk. (Note that in
+order to satisfy the running hypothesis [25, Hypothesis 2.1.1], we ne ed that the module of
+derivations on Kis finite-dimensional over K.)
+Choose a basis of W/xhWon whichxh∂hacts via a matrix over k. SinceRis a dense
+subfield ofR′, we can lift this basis to a basis e1,...,edofWconsisting of elements of M.
+LetNbe the matrix of action of xh∂hon this basis. Then Nhas entries in
+R†
+h(A)∩(k+xhR†
+h−1(A)/llbracketxh/rrbracket).
+We can thus choose f∈Anonzero so that Nhas entries in
+k+(xh/f)A((x1/f))···((xh−1/f))/llbracketxh/f/rrbracket.
+WriteN=/summationtext∞
+i=0Ni(xh/f)iwithNihaving entries in R†
+h−1(A). As in [25, Lemma 2.2.12],
+there exists a unique matrix U=/summationtext∞
+i=0Ui(xh/f)ioverR†
+h−1(A)/llbracketxh/f/rrbracketwithU0equal to the
+identity matrix, such that
+NU+xh∂h(U) =UN0.
+Namely, given U0,...,Ui−1, there is a unique choice of Uisatisfying
+iUi=UiN0−N0Ui−i/summationdisplay
+j=1NjUi−j
+becauseN0has prepared eigenvalues. More explicitly, each entry of Uiis a certain k-linear
+combination of entries of/summationtexti
+j=1NjUi−j. By induction on i, it follows that
+Ui∈A((x1/f))···((xh−1/f)) (i= 1,2,...).
+That is,Uhas entries in A((x1/f))···((xh/f)) and thus in R†
+h(A). The desired result
+follows.
+Proposition 4.2.6. For any finite differential module MoverR†
+h(A), we have
+H0(M) =H0(M⊗R†
+h(A)Rh(K)).
+25Proof.We induct on hwith trivial base case h= 0. Suppose that h >0. PutR=R†
+h(A).
+Pick any v∈M⊗RRh(K). EquipRwith thexh-adic norm (of arbitrary normalization). By
+Proposition 4.2.4, we can split Mas a direct sum M1⊕M2, in which the scale multiset of ∂h
+onM1⊗RRh(K) has all elements equal to 1, while the scale multiset of ∂honM2⊗RRh(K)
+has all elements greater than 1. We must then have v∈M1⊗RRh(K).
+ByLemma4.2.5, M1admitsanR†
+h−1(A)-latticeNstableunderallofthegivenderivations
+onR. Writevformally as/summationtext
+i∈Zvixi
+hwithvi∈N⊗R†
+h−1(A)Rh−1(K). Since the action of
+xh∂honNhas prepared eigenvalues, the equality xh∂h(v) = 0 implies that vi= 0 fori/\e}atio\slash= 0.
+Hencev∈N⊗R†
+h−1(A)Rh−1(K), so the induction hypothesis implies v∈N, and the desired
+result follows.
+4.3 Descent for localized power series rings
+Using the iterated power series rings we have just considered, we o btain some descent results
+for localized power series rings.
+Hypothesis 4.3.1. Throughout §4.3, letAbe a differential domain of characteristic 0,
+such that the module of derivations on K= Frac(A) is finite-dimensional over K, and the
+constant subring kofAis also the constant subring of K. For integers n≥m≥0, let
+R†
+n,m(A) be the union of
+A/llbracketx1/f,...,x n/f/rrbracket[(x1/f)−1,...,(xm/f)−1][f−1]
+over all nonzero f∈A. PutRn,m(K) =K/llbracketx1,...,xn/rrbracket[x−1
+1,...,x−1
+m]. EquipR†
+n,m(A)
+andRn,m(K) with the componentwise derivations on Aplus the derivations ∂1,...,∂n=
+∂
+∂x1,...,∂
+∂xn. Note that R†
+n,0(A) is a henselian local ring which is nondegenerate as a differ-
+ential ring.
+Proposition 4.3.2. For any finite differential module MoverR†
+n,m(A),
+H0(M) =H0(M⊗R†
+n,m(A)Rn,m(K)).
+Proof.EmbedR†
+n,m(A) into thering R†
+n(A) defined inHypothesis 4.2.1. Then within Rn(K),
+R†
+n,m(A) is the intersection of Rn,m(K) andR†
+n(A). By Remark 3.3.4, within M⊗R†
+n,m(A)
+Rn(K) we have
+(M⊗R†
+n,m(A)Rn,m(K))∩(M⊗R†
+n,m(A)R†
+n(A)) =M. (4.3.2.1)
+Givenany v∈H0(M⊗R†
+n,m(A)Rn,m(K)),wehave v∈H0(M⊗R†
+n,m(A)R†
+n(A))byLemma4.2.5.
+By (4.3.2.1), this implies v∈H0(M), proving the claim.
+We also need the following related argument in the regular case.
+Proposition 4.3.3. LetMbe a finite differential module over R†
+n,m(A), such thatM⊗R†
+n,m(A)
+Rn,m(K)is regular. Then Mis regular, in the sense that there exists a basis of Mon which
+x1∂1,...,xn∂nact via matrices over K.
+26Proof.Again, embed R†
+n,m(A) intoR†
+n(A). We can construct bases of M⊗R†
+n,m(A)Rn(K)
+having the desired property in two different fashions. One is to apply [25, Theorem 4.1.4]
+to construct a suitable basis of M⊗R†
+n,m(A)Rn,m(K), and then extend scalars to Rn(K).
+The other is to construct a suitable basis of M⊗R†
+n,m(A)R†
+n(A) by repeated application of
+Lemma 4.2.5, and then extend scalars to Rn(K).
+The resulting bases must have the same K[x1,...,xn][x−1
+1,...,x−1
+m]-span (as in the proof
+of [25, Proposition 2.2.13]). They thus both consist of elements of th e intersection of
+M⊗R†
+n,m(A)Rn,m(K) withM⊗R†
+n,m(A)R†
+n(A). By (4.3.2.1), this intersection equals M; this
+yields the desired result.
+Putting these arguments together yields the following.
+Theorem 4.3.4. LetMbe a finite differential module over R†
+n,m(A). Then any admissible
+(resp. good) decomposition of M⊗R†
+n,m(A)Rn,m(K)descends to an admissible (resp. good)
+decomposition of M.
+Proof.Given an admissible decomposition of M⊗R†
+n,m(A)Rn,m(K), the projectors onto the
+summands are horizontal sections of End( M)⊗R†
+n,m(A)Rn,m(K). These descend to End( M)
+by Proposition 4.3.2. With notation as in (3.4.6.1), the φαcan be chosen in R†
+n,m(A) by
+Proposition 3.4.8. Hence the Rαcan be defined over R†
+n,m(A); they are regular by Proposi-
+tion 4.3.3.
+Corollary 4.3.5. LetMbe a finite differential module over A/llbracketx1,...,xn/rrbracket[x−1
+1,...,x−1
+m].
+Then any admissible (resp. good) decomposition of K/llbracketx1,...,xn/rrbracket[x−1
+1,...,x−1
+m]descends to
+an admissible (resp. good) decomposition of Af/llbracketx1,...,xn/rrbracket[x−1
+1,...,x−1
+m]for some nonzero
+f∈A.
+4.4 Good formal structures
+One application of Theorem 4.3.4 is to relate ramified good decompositio ns over complete
+rings to good formal structures over noncomplete rings.
+Proposition 4.4.1. LetRbe a nondegenerate differential local ring with completion /hatwideR. Let
+x1,...,xnbe a regular sequence of parameters for R, and putS=R[x−1
+1,...,x−1
+m]for some
+m. LetMbe a finite differential module over S. Then any ramified good decomposition of
+M⊗R/hatwideRinduces a good formal structure of M.
+Proof.Wemayassumefromtheoutsetthat Riscompletewithrespecttotheideal( x1,...,xm).
+In addition, by replacing Rwith a finite integral extension R′such thatR′⊗RSis ´ etale over
+S, we may reduce to the case where M⊗R/hatwideRadmits a good decomposition.
+Choose derivations ∂1,...,∂n∈∆Rof rational type with respect to x1,...,xn, then
+identify/hatwideRwithk/llbracketx1,...,xn/rrbracketasinCorollary3.1.8. Let Rmbethejointkernel of ∂1,...,∂mon
+R; then by Lemma 3.1.7, we have an isomorphism R∼=Rm/llbracketx1,...,xm/rrbracket. PutK= Frac(Rm).
+By Theorem 3.5.4, for some finite extension K′ofKand some positive integer h,
+MK′=M⊗SK′/llbracketx1/h
+1,...,x1/h
+m/rrbracket[x−1/h
+1,...,x−1/h
+m]
+27admits a ramified good decomposition; by enlarging R, we may reduce to the case h= 1.
+PutL= Frac(k/llbracketxm+1,...,xn/rrbracket), and letL′be a component of L⊗KK′. By Remark 3.3.4,
+combiningtheminimalgooddecompositionsof M⊗R/hatwideRandMK′yieldsagooddecomposition
+of
+M⊗ST/llbracketx1,...,xm/rrbracket[x−1
+1,...,x−1
+m]
+forTequal to the intersection k/llbracketxm+1,...,xn/rrbracket∩K′withinL′. By Remark 3.1.5, Tis finite
+´ etale overR, yielding the desired result.
+5 Good formal structures
+We collect some basic facts about good formal structures on nond egenerate differential
+schemes, complex analytic varieties, and formal completions there of.
+Hypothesis 5.0.1. Throughout §5, letXbeeithera nondegenerate differential scheme,
+or a smooth (separated) complex analytic space (see §1.5). For short, we distinguish these
+two options as the algebraic case and theanalytic case . LetZbe a closed subspace of X
+containing no irreducible component of X. Let/hatwidestX|Zbe the formal completion of XalongZ
+(in the category of locally ringed spaces). Let Ebe a∇-module over O/hatwidestX|Z(∗Z).
+5.1 Good formal structures
+Definition 5.1.1. Letx∈Zbe a point in a neighborhood of which ( X,Z) is a regular
+pair. We say that Eadmits an admissible decomposition (resp. a good decomposition , a
+ramified good decomposition ) atxif the restriction of Eto/hatwideOX,x(∗Z) admits an admissible
+decomposition (resp. a good decomposition, a ramified good decomp osition). Let Ybe the
+intersection of the components of Zpassing through x; by Proposition 4.4.1, the restriction
+ofEto/hatwideOX,x(∗Z) admits a ramified good decomposition if and only if the restriction of E
+toO/hatwidestX|Y,x(∗Z) does so. We describe this condition by saying that Eadmits a good formal
+structure atx.
+Suppose(X,Z)isaregularpair. Wedefinethe turning locus ofEtobetheset of x∈Zat
+whichEfails to admit a good formal structure; this set may be equipped with the structure
+of a reduced closed subspace of Z, by Proposition 5.1.4 below.
+Remark 5.1.2. One might consider the possibility that the restriction of EtoO/hatwidestX|Z,x(∗Z)
+itself admits a ramified good decomposition, or in Sabbah’s language, t hatEadmits a very
+good formal structure atx. However, an argument of Sabbah [38, Lemme I.2.2.3] shows that
+one cannot in general achieve very good formal structures even after blowing up. For this
+reason, we make no further study of very good formal structur es.
+Remark 5.1.3. IfEis defined over OX(∗Z) itself, one can also speak about good formal
+structures at points outside of Z, but they trivially always exist.
+28Proposition 5.1.4. Suppose (X,Z)is a regular pair. Then the turning locus of Eis the
+underlying set of a unique reduced closed subspace of Z, containing no irreducible component
+ofZ.
+Proof.We first treat the algebraic case. Suppose Wis an irreducible closed subset of Z
+not contained in the turning locus. By the numerical criterion from T heorem 3.5.4, the
+generic point of Walso lies outside the turning locus. Corollary 4.3.5 then implies that the
+intersection of Wwith the turning locus is contained in some closed proper subset of W.
+By noetherian induction, it follows that the turning locus is closed in Z; it thus carries a
+unique reduced subscheme structure. Moreover, the turning loc us cannot contain the generic
+point of any component of Z, because at any such point we may apply the usual Turrittin-
+Levelt-Hukuhara decomposition theorem (or equivalently, becaus e the numerical criterion of
+Theorem 3.5.4 is always satisfied when the base ring is one-dimensional) . Hence the turning
+locus cannot contain any whole irreducible component of Z.
+We next reduce the analytic case to the algebraic case. Recall that Xadmits a neighbor-
+hoodbasisconsisting ofcompactsubsets ofSteinsubspaces of X. LetKbeanelement ofthis
+basis, letUbe a Stein subspace of XcontainingK, and letVbe an open set contained in K.
+By Corollary 3.2.7, the localization Rof Γ(U,OU) atKis noetherian, hence a nondegenerate
+differential ring. Let Ibe the ideal of Rdefined byZ, and putM= Γ(U,E)⊗Γ(U,OU)Ras a
+differential moduleover R. Bythepreviousparagraph, theturninglocusof Mmaybeviewed
+as a reduced closed subscheme of Spec( R/I) not containing any irreducible component. Its
+inverse image under the map V→Spec(R) is then a reduced closed subspace of V∩Znot
+containing any irreducible component. Since we can choose Vto cover a neighborhood of
+any given point of X, we deduce the desired result.
+5.2 Irregularity and turning loci
+It will be helpful to rephrase the numerical criterion for good form al structures (Theo-
+rem 3.5.4) in geometric language. For this, we must first formalize the notion of irregularity.
+Definition 5.2.1. LetEbe an irreducible component of Z. We define the irregularity
+IrrE(E) ofEalongEas follows.
+Suppose first that we are in the algebraic case. Let ηbe the generic point of E. LetLbe
+the completion of Frac( OX,η), equipped with its discrete valuation normalized to have value
+groupZ. We define Irr E(E) as the irregularity of the differential module over Linduced by
+E, in the sense of [25, Definition 1.4.9].
+Suppose next that we are in the analytic case; in this case, we use a “ cut by curves”
+definition. We may assume that ( X,Z) is a regular pair by discarding its irregular locus
+(whichhascodimensionatleast2in X). LetTbetheturninglocusof E; byProposition5.1.4,
+T∩Eis a proper closed subspace of E, so in particular its complement is dense in E. We
+claim that there exists a nonnegative integer mwith the following property: for any curve C
+inXand any isolated point zofC∩Enot belonging to T, the irregularity of the restriction
+ofEtoCat the point zis equal to mtimes the intersection multiplicity of CandEatz.
+29Namely, it suffices to check this assertion on each element of a basis f or the topology of X,
+which may be achieved as in the proof of Proposition 5.1.4. We define Ir rE(E) =m.
+Definition 5.2.2. Anirregularity divisor forEis a Cartier divisor DonXsuch that for
+any normal modification f:Y→Xand any prime divisor EonY, the irregularity of f∗E
+alongEis equal to the multiplicity of f∗DalongE. Such a divisor is unique if it exists.
+Moreover, any Q-Cartier divisor satisfying the definition must have all integer multiplic ities,
+and so must be an integral Cartier divisor.
+Proposition 5.2.3. Suppose that (X,Z)is a regular pair. Then the following conditions
+are equivalent.
+(a) The turning locus of Eis empty.
+(b) Both EandEnd(E)admit irregularity divisors.
+Proof.Given (b), (a) follows by Theorem 3.5.4. Given (a), we may check (b) lo cally around
+a pointx∈Z. We make a sequence of reductions to successively more restrictiv e situations,
+culminating in one where we can read off the claim. Namely, we reduce so as to enforce the
+following hypotheses.
+(a) There exist a regularsequence ofparameters x1,...,xn∈ OX,xforXatx(by shrinking
+X).
+(b) We have Z=V(x1···xm) (by shrinking X).
+(c) The module Eadmits a good decomposition at x(by shrinking X, then replacing X
+by a finite cover ramified along Z).
+(d) With notation as in (3.4.6.1), the φαbelong to OX,x[x−1
+1,...,x−1
+m] (by applying Propo-
+sition 3.4.8).
+In this case, we claim that in some neighborhood of x, the irregularity divisor of Eis the sum
+of the principal divisors −rank(Rα)div(φα) over allα∈Iwithφα/∈ OX,x, while the irregu-
+laritydivisor ofEnd( E)isthesum oftheprincipal divisors −rank(Rα)rank(Rβ)div(φα−φβ)
+over allα,β∈Iwithφα−φβ/∈ OX,x. This may be checked as in the proof of Proposi-
+tion 5.1.4.
+5.3 Deligne-Malgrange lattices
+In the work of Mochizuki [35], the approach to constructing goodf ormal structures is via the
+analysis of Deligne-Malgrange lattices. Since we use a different techn ique to construct good
+formal structures, it is worthindicating how to recover informatio nabout Deligne-Malgrange
+lattices.
+30Definition 5.3.1. LetFbe a coherent sheaf over OX(∗Z) (resp. over O/hatwidestX|Z(∗Z)). Alattice
+ofFis a coherent OX-submodule (resp. O/hatwidestX|Z-submodule) F0ofFsuch that the induced
+mapF0⊗OXOX(∗Z)→ F(resp.F0⊗O/hatwidestX|ZO/hatwidestX|Z(∗Z)→ F) is surjective. We make the
+following observations.
+(a) LetFbe a coherent sheaf over OX(∗Z), and put /hatwideF=F ⊗OX(∗Z)O/hatwidestX|Z(∗Z). Then the
+map
+F0/mapsto→/hatwideF0=F ⊗OXO/hatwidestX|Z
+gives a bijection between lattices of Fand lattices of /hatwideF, as in [31, Proposition 1.2].
+Moreover, F0is locally free if and only if /hatwideFis, because the completion of a noetherian
+local ring is faithfully flat (see Remark 3.1.5).
+(b) In the analytic case, a coherent sheaf over OX(∗Z) orO/hatwidestX|Z(∗Z) need not admit any
+lattices at all. See [31, Exemples 1.5, 1.6].
+By contrast, even inthe analytic case, a ∇-modulealways admits a latticewhich is nearly
+canonical. It only depends on a certain splitting of the reduction mod uloZmap.
+Definition 5.3.2. In the algebraic case, let K0be a field containing each connected com-
+ponent of the subring of Γ( X,OX) killed by the action of all derivations. (Each of those
+components is a field by Lemma 3.2.5(d).) In the analytic case, put K0=C. LetK0be an
+algebraic closure of K0. Letτ:K0/Z→K0be a section of the quotient K0→K0/Z. We
+sayτisadmissible ifτ(0) = 0,τis equivariant for the action of the absolute Galois group
+ofK0, and for any λ∈K0and any positive integer a, we have
+τ(λ)−λ=/ceilingleftbiggτ(aλ)−aλ
+a/ceilingrightbigg
+. (5.3.2.1)
+Such a section always exists by [25, Lemma 2.4.3]. For instance, if K0=C, one may take τ
+to have image {s∈C: Re(s)∈[0,1)}.
+For the remainder of §5.3, fix a choice of an admissible section τ.
+Definition 5.3.3. Suppose that ( X,Z) is a regular pair. A Deligne-Malgrange lattice of the
+∇-moduleEoverO/hatwidestX|Z(∗Z) is a lattice E0ofEsuch that for each point x∈Z, the restriction
+ofE0to/hatwideOX,xis the Deligne-Malgrange lattice of the restriction of Eto/hatwideOX,x(∗Z), in the sense
+of [25, Definition 4.5.2] (for the admissible section τ). Such a lattice is evidently unique if it
+exists.
+Theorem 5.3.4. Suppose that (X,Z)is a regular pair and that Ehas empty turning locus.
+Then the Deligne-Malgrange lattice E0ofEexists and is locally free over O/hatwidestX|Z.
+31Proof.It suffices to check both assertions in case Xis the spectrum of a local ring R,Zis
+the zero locus of x1···xmfor some regular sequence of parameters x1,...,xnofR, andR
+is complete with respect to the ( x1···xm)-adic topology (but not necessarily with respect
+to the (x1,...,xm)-adic topology). For i= 1,...,m, letFibe thexi-adic completion of
+Frac(R), putEi=E ⊗iFi, and let E0,ibe the Deligne-Malgrange lattice of Ei. We define
+E0to be theR-submodule of Econsisting of elements whose image in Eibelongs to E0,ifor
+i= 1,...,m. As in [25, Lemma 4.1.2], we see that E0is a lattice in Eand that E0⊗R/hatwideR
+is the Deligne-Malgrange lattice of E ⊗R[x−1
+1,...,x−1
+m]/hatwideR[x−1
+1,...,x−1
+m]. In particular, E0⊗R/hatwideRis
+a finite free /hatwideR-module by [25, Proposition 4.5.4], so E0is a finite free R-module by faithful
+flatness of completion (Remark 3.1.5 again).
+Remark 5.3.5. LetUbe the open (by Proposition 5.1.4) subspace of Xon which (X,Z)
+is a regular pair and Ehas no turning locus. Malgrange [31, Th´ eor` eme 3.2.1] constructe d
+a Deligne-Malgrange lattice over U; this construction is reproduced by our Theorem 5.3.4.
+Malgrange then went on to establish the much deeper fact that this lattice extends over all
+ofX[31, Th´ eor` eme 3.2.2]. We will only reproduce this result after estab lishing existence of
+good formal structures after blowing up; see Theorem 8.2.3.
+The following property of Deligne-Malgrange lattices follows immediate ly from Proposi-
+tion 4.4.1; we formulate it to make a link with Mochizuki’s work. See Remar k 5.3.7.
+Proposition 5.3.6. Suppose that (X,Z)is a regular pair and that Ehas empty turning
+locus. Choose any x∈Z. ForUan open neighborhood of xinX,f:U′→Ua finite cover
+ramified over Z, andy∈f−1(x), putZ′=f−1(Z)and letYdenote the intersection of the
+irreducible components of Z′passing through y. Then we can choose Uandfso that for any
+y∈f−1(x), any admissible decomposition of the restriction of f∗Eto/hatwideOU′,y(∗Z′)induces a
+corresponding decomposition of the restriction to O/hatwideU′|Y,yof the Deligne-Malgrange lattice E′
+0
+off∗E.
+Remark 5.3.7. Suppose that ( X,Z) is a regular pair, and that both Eand End( E) have
+empty turning locus. (The restriction on End( E) is needed to overcome the discrepancy
+between our notion of a good decomposition and Mochizuki’s definition of a good set of
+irregular values; see [25, Remark 4.3.3, Remark 6.4.3].) The conclusion o f Proposition 5.3.6
+assertsthat E′
+0isanunramifiedly good Deligne-Malgrangelattice inthelanguageofMochizuki
+[35, Definition 5.1.1].
+By virtue of the definition of Deligne-Malgrange lattices in the one-dim ensional case [25,
+Definition 2.4.4], it is built into the definition of Deligne-Malgrange lattices in general that
+f∗E′
+0=E0. HenceE0is agood Deligne-Malgrange lattice in the language of Mochizuki; that
+is, Proposition 5.3.6 fulfills a promise made in [25, Remark 4.5.5].
+6 The Berkovich unit discs
+In [25,§5], we introduced the Berkovich closed and open unit discs over a com plete discretely
+valued field of equal characteristic 0, and used their geometry to m ake a fundamental finite-
+32ness argument as part of the proof of Sabbah’s conjecture. Her e, we need the analogous
+construction over an arbitrary complete nonarchimedean field of c haracteristic 0. To achieve
+this level of generality, we must recall some results from [26, §2], and make some arguments
+as in [26, §4].
+Hypothesis 6.0.1. Throughout §6, letFbe a field complete for a nonarchimedean norm
+|·|F, of residual characteristic 0. Define the real valuation vFbyvF(·) =−log|·|F. LetCF
+denote a completed algebraic closure of F, equipped with the unique extensions of |·|Fand
+vF.
+Notation 6.0.2. ForAa subring of Fandρ >0, let| · |ρdenote the ρ-Gauss norm on
+A[x,x−1] with respect to |·|F. Forα≤β∈(0,+∞), define the following rings.
+•LetA/a\}b∇acketle{tα/x/a\}b∇acket∇i}htdenote the completion of A[x−1] under|·|α.
+•LetA/a\}b∇acketle{tx/β/a\}b∇acket∇i}htdenote the completion of A[x] under|·|β.
+•LetA/a\}b∇acketle{tα/x,x/β /a\}b∇acket∇i}htdenote the Fr´ echet completion of A[x,x−1] under|·|αand|·|β(equiv-
+alently, under |·|ρfor allρ∈[α,β]).
+Forβ= 1, we abbreviate A/a\}b∇acketle{tx/β/a\}b∇acket∇i}ht,A/a\}b∇acketle{tα/x,x/β /a\}b∇acket∇i}httoA/a\}b∇acketle{tx/a\}b∇acket∇i}ht,A/a\}b∇acketle{tα/x,x/a\}b∇acket∇i}ht. Note that none of these
+rings changes if we replace Aby its completion under |·|F.
+6.1 The Berkovich closed unit disc
+We first recall a few facts about the Berkovich closed unit disc over the fieldF. The case
+F=C((x)) was treated in [25, §5], but we need to reference the more general treatment in
+[26,§2.2]. (Note that the treatment there allows positive residual chara cteristic, which we
+exclude here.)
+Definition 6.1.1. The Berkovich closed unit disc D=DFconsists of the multiplicative
+seminorms αonF[x] which are compatible with the given norm on Fand bounded above by
+the1-Gaussnorm. Forinstance, for z∈oCFandr∈[0,1], thefunction αz,r:F[x]→[0,+∞)
+takingP(x) to ther-Gauss norm of P(x+z) is a seminorm; it is in fact the supremum
+seminorm on the disc Dz,r={z′∈CF:|z−z′| ≤r}.
+Lemma 6.1.2. For any complete extension F′ofF, the restriction map DF′→DFis
+surjective.
+Proof.See [5, Corollary 1.3.6].
+Definition 6.1.3. Forα,β∈D, we say that αdominatesβ, notatedα≥β, ifα(P)≥β(P)
+for allP∈F[x]. Define the radiusofα∈D, denotedr(α), to be the infimum of r∈[0,1]
+for which there exists z∈oCFwithαz,r≥α.
+As in [25, Proposition 5.2.2], we use the following classification of points o fD. See [5,
+1.4.4] for the case where Fis algebraically closed, or [26, Proposition 2.2.7] for the general
+case.
+33Proposition 6.1.4. Each element of Dis of exactly one of the following four types.
+(i) A point of the form αz,0for somez∈oCF.
+(ii) A point of the form αz,rfor somez∈oCFandr∈(0,1]∩|C×
+F|F.
+(iii) A point of the form αz,rfor somez∈oCFandr∈(0,1]\|C×
+F|F.
+(iv) The infimum of a sequence αzi,riin which the discs Dzi,riform a decreasing sequence
+with empty intersection and positive limiting radius.
+Moreover, the points which are minimal under domination are precisely those of type (i) and
+(iv).
+By [26, Lemma 2.2.12], we have the following.
+Lemma 6.1.5. For eachα∈Dand eachr∈[r(α),1], there is a unique point αr∈Dwith
+r(αr) =randαr≥α. (By Proposition 6.1.4, if r/\e}atio\slash=r(α), we can always write αr=αz,rfor
+somez∈oC.)
+Corollary 6.1.6. Ifα∈Dis of type (iv) and is the infimum of the sequence αzi,ri, then
+for anyr∈(0,1)withα0,r≥αand anyP∈F/a\}b∇acketle{tx/r/a\}b∇acket∇i}ht, there exists an index i0such that
+αzi,ri(P) =α(P)fori≥i0.
+Proof.The caseP∈F[x] follows from the proof of [26, Proposition 2.2.7]. The general case
+follows by choosing Q∈F[x] such that α0,r(P−Q)<α(P) and applying the previous case
+toQ.
+Except for some points of type (i), every point of Dinduces a valuation on F(x). These
+valuations have the following numerical behavior [26, Lemma 2.2.16].
+Lemma 6.1.7. Letαbe a point of Dof type (ii) or (iii). Let v(·) =−logα(·)be the
+corresponding real valuation on F(x).
+(a) Ifαis of type (ii), then
+trdeg(κv/κvF) = 1,dimQ((Γv/ΓvF)⊗ZQ) = 0.
+(b) Ifαis of type (iii), then
+trdeg(κv/κvF) = 0,dimQ((Γv/ΓvF)⊗ZQ) = 1.
+346.2 More on irrational radius
+Let us take a closer look at the case of Proposition 6.1.4 of type (iii), i.e ., a disc of an
+irrational radius.
+Hypothesis 6.2.1. Throughout §6.2, in addition to Hypothesis 6.0.1, choose r∈(0,1)\
+|C×
+F|F, so thatα0,r∈Dis a point of type (iii). (The case where α0,ris of type (ii) is a bit
+more complicated, and we will not need it here.)
+Lemma 6.2.2. Supposeg∈F/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}htis such that α0,r(g−1)<1. Thengcan be factored
+uniquely as g1g2withg1∈1+xF/a\}b∇acketle{tx/r/a\}b∇acket∇i}ht×,g2∈F/a\}b∇acketle{tr/x/a\}b∇acket∇i}ht×,α0,r(g1−1)<1, andα0,r(g2−1)<1.
+In particular, gis a unit in F/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht.
+Proof.Apply Theorem 4.1.1 with U=xF/a\}b∇acketle{tx/r/a\}b∇acket∇i}ht,V=F/a\}b∇acketle{tr/x/a\}b∇acket∇i}ht,W=F/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht,a=b= 1,
+andc=g. (Compare the proof of Proposition 4.2.3 or [27, Theorem 2.2.1].)
+Lemma 6.2.3. Any nonzero g∈F/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}htcan be factored (not uniquely) as g=xig1g2
+for somei∈Z,g1∈F/a\}b∇acketle{tx/r/a\}b∇acket∇i}ht×, andg2∈F/a\}b∇acketle{tr/x/a\}b∇acket∇i}ht×.
+Proof.(Compare [26, Lemma 2.2.14].) Write g=/summationtext
+i∈Zgixiwith|gi|Fri→0 asi→ ±∞.
+Sincer /∈ |C×
+F|F, there exists a unique index jwhich maximizes |gj|Frj. Lemma 6.2.2 implies
+thatg−1
+jx−jgfactors as a unit in F/a\}b∇acketle{tx/r/a\}b∇acket∇i}httimes a unit in F/a\}b∇acketle{tr/x/a\}b∇acket∇i}ht, yielding the claim.
+Corollary 6.2.4. The completion of F(x)underα0,ris equal toF/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht.
+Proof.The complete ring F/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}htcontainsF[x,x−1] as a dense subring, and is a field
+by Lemma 6.2.3. This proves the claim.
+Lemma 6.2.5. SupposeFis integrallyclosedin the completeextension F′. ThenF/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht
+is integrally closed in F′/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht.
+Proof.We may reduce to the case where both FandF′are algebraically closed. Let f=/summationtext
+i∈Zfixi∈F′/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}htbe an element which is integral over F/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht. LetP(T) be the
+minimal polynomial of foverF/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht. Then for each τ∈Aut(F′/F),/summationtext
+i∈Zτ(fi)xiis
+also a root of P. Hence each fimust have finite orbit under τ, and so must belong to F.
+Proposition 6.2.6. LetAbe a subring of F. LetSbe a complete subring of F/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht
+which is topologically finitely generated over A/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht, such that Frac(S)is finite over
+Frac(A/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht). Then there exists a subring A′ofFwhich is finitely generated over A,
+such that Frac(A′)is finite over Frac(A)andS⊆A′/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht.
+Proof.It suffices to check the claim in case Sis the completion of A/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht[g] for some
+g∈F/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}htwhich is integral over Frac( A/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht), with minimal polynomial P(T).
+By Lemma 6.2.5, the coefficients of gmust belong to the completion of the integral closure
+of Frac(A) withinF. Consequently, for any ǫ >0, we can choose a finitely generated A-
+subalgebra A′ofFwith Frac(A′) finite over Frac( A), so that there exists h∈A′/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}ht
+withα0,r(g−h)< ǫ. Forǫsuitably small, we may then perform a Newton iteration to
+compute a root of P(T) inA′/a\}b∇acketle{tr/x,x/r/a\}b∇acket∇i}htclose toh, which will be forced to equal g.
+356.3 Differential modules on the open unit disc
+We now collect some facts about differential modules on Berkovich dis cs, particularly con-
+cerning their behavior in a neighborhood of a minimal point. The hypot hesis of residual
+characteristic 0 will simplify matters greatly; an analogous but more involved treatment in
+the case of positive residual characteristic is [26, §4].
+Hypothesis 6.3.1. Throughout §6.3–6.4, let Mbe a∇-module of rank dover the open
+Berkovich unit disc
+D0={α∈D:α0,r≥αfor somer∈[0,1)}.
+That is, for each r∈[0,1), we must specify a differential module Mrof rankdoverF/a\}b∇acketle{tx/r/a\}b∇acket∇i}ht,
+plusisomorphisms Mr⊗F/angbracketleftx/r/angbracketrightF/a\}b∇acketle{tx/s/a\}b∇acket∇i}ht∼=Msfor0<s<r< 1satisfyingthecocyclecondition.
+(Note that α0,0/∈D0, contrary to the convention adopted in [25, Definition 5.3.1].)
+Definition 6.3.2. Forα∈D0, putIα= (0,+∞) ifαis of type (i) and Iα= (0,−logr(α)]
+otherwise. For s∈Iα, letαsbe the unique point of D0withαs≥αandr(αs) =e−s(given
+by Lemma 6.1.5). Let Fα,sbe the completion of F(x) underαs. Definef1(M,α,s)≥ ··· ≥
+fd(M,α,s)≥ssothatthescalemultisetof∂
+∂xonM⊗Fα,sconsistsofef1(M,α,s)−s,...,efd(M,α,s)−s.
+Beware that this is a different normalization than in the definition of irr egularity; the new
+normalization is such that /vextendsingle/vextendsingle/vextendsingle/vextendsingle∂
+∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+sp,M⊗Fα,s=ef1(M,α,s).
+PutFi(M,α,s) =f1(M,α,s)+···+fi(M,α,s).
+Proposition 6.3.3. The function Fi(M,α,s)is continuous, convex, and piecewise affine in
+s, with slopes in1
+d!Z. Furthermore, the slopes of Fi(M,α,s)are nonpositive in a neighborhood
+of anysfor whichfi(M,α,s)>s.
+Proof.As in [26, Proposition 4.6.4], this reduces to [27, Theorem 11.3.2].
+The following argument makes critical use of the hypothesis that Fhas residual char-
+acteristic 0. The situation of positive residual characteristic is muc h subtler; compare [26,
+Proposition 4.7.5].
+Proposition 6.3.4. (a) Suppose α∈D0is of type (i). Then in a neighborhoodof s= +∞,
+for eachi∈ {1,...,d},fi(M,α,s) =sidentically.
+(b) Suppose α∈D0is of type (iv). Then in a neighborhood of s=−logr(α), for each
+i∈ {1,...,d}, eitherfi(M,α,s)is constant, or fi(M,α,s) =sidentically.
+Proof.Suppose first that αis of type (i). By Proposition 6.3.3, f1(M,α,s) =F1(M,α,s)
+is convex, and it cannot have a positive slope except in a stretch whe refi(M,α,s) =s
+identically for all i. In particular, we cannot have fi(M,α,s)−s >0 for alls: otherwise,
+the left side would be a convex function on (0 ,+∞) with all slopes less than or equal to −1,
+and so could not be positive on an infinite interval. We thus have fi(M,α,s 0) =s0for some
+36s0; the right slope of fi(M,α,s) ats=s0must be at least 1 because fi(M,α,s)≥sfor
+alls. By convexity, the right slope of fi(M,α,s) at anys≥s0must be at least 1. If we
+ever encounter a slope strictly greater than 1, then that slope is a chieved at some point with
+fi(M,α,s)> s, contradicting Proposition 6.3.3. It follows that fi(M,α,s) =sidentically
+fors≥s0.
+Suppose next that αis of type (iv). Pick some r∈(r(α),1) such that α0,r≥α, and
+putE= Frac(F/a\}b∇acketle{tx/r/a\}b∇acket∇i}ht). By the cyclic vector theorem [25, Lemma 1.3.3], there exists an
+isomorphism M⊗E∼=E{T}/E{T}P(T) for some twisted polynomial P(T) =/summationtext
+iPiTi∈
+E{T}with respect to the derivation∂
+∂x. By Corollary 6.1.6, for each i,αs(Pi) is constant
+forsin a neighborhood of −logr(α). Hence the Newton polygon of Pmeasured using αsis
+also constant for sin a neighborhood of −logr(α). By [25, Proposition 1.6.3], we can read
+offfi(M,α,s) as the greater of sand the negation of the i-th smallest slope of the Newton
+polygonofP. This implies thatinaneighborhoodof −logr(α),fi(M,α,s)iseither constant
+or identically equal to s, as desired.
+6.4 Extending horizontal sections
+We need some additional arguments that allow us, in certain cases, t o extend horizontal
+sections of ∇-modules over D0. Throughout §6.4, retain Hypothesis 6.3.1.
+Proposition 6.4.1. Forα∈D0, the following conditions are equivalent.
+(a) Fori= 1,...,d,fi(M,α,s)is constant until it becomes equal to sand then stays equal
+tosthereafter (see Figure 1).
+(b) There exists s1∈(0,−logr(α))such that for i= 1,...,d, on the range s∈(0,s1],
+fi(M,α,s)is either constant or identically equal to s.
+(c) There exists a direct sum decomposition M=⊕tMtof∇-modules over D0, such that
+fi(Mt,α,s)is equal to a constant value depending only on t(not oni) until it becomes
+equal tosand then stays equal to sthereafter.
+Proof.It is clear that (a) implies (b) and that (c) implies (a). Given (b), (c) h olds by [27,
+Theorem 12.4.1].
+Definition 6.4.2. Wesaythat Misterminal iftheequivalent conditionsofProposition6.4.1
+are satisfied. Note that condition (b) does not depend on α. Fors0∈[0,−logr(α)), if
+condition (a) only holds for s > s0, we say that Mbecomes terminal at s0; this condition
+depends on α, but only via the point αs0.
+ForIa closed subinterval of Iα∪ {0}, we sayMbecomes strongly terminal on Iif for
+i= 1,...,d, over the interior of I,fi(M,α,s) is either everywhere constant, or everywhere
+equal tos. By Proposition 6.4.1, this implies that Mbecomes terminal at the left endpoint
+ofI.
+We have the following criterion for becoming strongly terminal.
+37y
+sy=sy=fi(M,α,s)
+Figure 1: Graph of a function satisfying condition (a) of Proposition 6.4.1.
+Proposition 6.4.3. Suppose that for some j∈ {0,...,d+ 1}and some values 0< s1<
+s2<s3<−logr(α), we have
+fi(M,α,s 1) =fi(M,α,s 2) =fi(M,α,s 3) (i= 1,...,j)
+fi(M,α,s 1)−s1=fi(M,α,s 2)−s2=fi(M,α,s 3)−s3= 0 (i=j+1,...,d).
+ThenMbecomes strongly terminal on [s1,s3]. In particular, Mbecomes terminal at s1.
+Proof.The given conditions imply that for i= 1,...,d,Fi(M,α,s) agrees with a certain
+affine function at s=s1,s2,s3. Since it is convex by Proposition 6.3.3, it must be affine over
+the ranges∈[s1,s3]. This proves the claim.
+Lemma 6.4.4. Suppose that f1(M,α,s) =sidentically. Then Madmits a basis of horizon-
+tal sections.
+Proof.This follows from Dwork’s transfer theorem [27, Theorem 9.6.1].
+Proposition 6.4.5. Suppose that for some s0∈(0,−logr(α)),Mbecomes strongly terminal
+on[0,s0]. Then for any s∈(0,s0),H0(M) =H0(M⊗Fα,s).
+Proof.The hypothesis implies that Misterminal. ByProposition6.4.1, thereisadirect sum
+decomposition M=M0⊕M1such thatfi(M0,α,s)>sforsnear 0 fori= 1,...,rank(M0),
+andfi(M1,α,s) =sidentically for i= 1,...,rank(M1). SinceMbecomes strongly terminal
+on [0,s0], we must have fi(M0,α,s)>sfors∈(0,s0) andi= 1,...,rank(M0).
+Picks∈(0,s0) andv∈H0(M⊗Fα,s). Sincefi(M0,α,s)> sfori= 1,...,rank(M0),
+the projection of vontoM0⊗Fα,smust be zero; that is, v∈H0(M1⊗Fα,s). However, by
+Lemma 6.4.4, M1admits a basis of horizontal sections. If we write vin terms of this basis,
+the coefficients must be horizontal elements of Fα,s, but the only such elements belong to F.
+Hencev∈H0(M1)⊆H0(M), as desired.
+38Remark 6.4.6. The restriction that fi(M,α,s)≥sis in a certain sense a bit artificial.
+Recent work of Baldassarri (in progress, but see [3, 4]) seems to p rovide a better defini-
+tion offi(M,α,s) that eliminates this restriction, which would lead to some simplification
+above. (Rathermoresimplification couldbeexpected intheanalogou sbut morecomplicated
+arguments in [26, §5].)
+7 Valuation-local analysis
+We now make the core technical calculations of the paper. We give a h igher-dimensional
+analogue of the Hukuhara-Levelt-Turrittin decomposition theore m, in terms of a valuation
+on a nondegenerate differential scheme. It will be convenient to st ate both the result and all
+of the intermediate calculations in terms of the following running hypo thesis.
+Hypothesis 7.0.1. Throughout §7, letXbe a nondegenerate differential integral scheme,
+letZbe a reduced closed proper subscheme of X, and let Ebe a∇-module over X\Z,
+identified with a finite differential module over OX(∗Z). (Note that we do not pass to the
+formal completion.) Let vbe a centered valuation on X, with generic center z.
+7.1 Potential good formal structures
+We introduce the notion of a potential good formal structure ass ociated to the valuation v,
+and state the theorem we will be proving over the course of this sec tion. In order to lighten
+notation, we phrase everything in terms of “replacing the input dat a”.
+Definition 7.1.1. GivenaninstanceofHypothesis7.0.1, theoperationof modifying/altering
+the input data will consist of the following.
+•Letf:Y→Xbe a modification/alteration of X. Replacevwith an extension v′of
+vto a centered valuation on Y; such an extension exists by Lemma 2.1.3. Since the
+schemeXisexcellent byLemma3.2.5(a),wehaveheight( v′) = height(v),ratrank(v′) =
+ratrank(v), and trdefect( v′) = trdefect( v) by Lemma 2.2.4.
+•ReplaceXwith an open subscheme X′ofYon whichv′is centered. Replace Zwith
+Z′=X′∩f−1(Z) (viewed as a reduced closed subscheme of X′).
+•ReplaceEwith the restriction of f∗EtoOX′(∗Z′).
+Note that composing operations of one of these forms gives anoth er operation of the same
+form.
+Lemma 7.1.2. Given any instance of Hypothesis 7.0.1, after modifying the input data, we
+can enforce the following conditions.
+(a) The field κvis algebraic over the residue field kofXatz.
+39(b) The pair (X,Z)is regular.
+(c) The scheme X= SpecRis affine.
+(d) There exists a regular system of parameters x1,...,xnofRatz, such that Z=
+V(x1···xm)for some nonnegative integer m.
+(e) There exists an isomorphism /hatwidestOX,z(∗Z)∼=k/llbracketx1,...,xn/rrbracket[x−1
+1,...,x−1
+m]forkas in (a),
+and derivations ∂1,...,∂n∈∆Racting as∂
+∂x1,...,∂
+∂xn.
+Proof.To enforce (a), we may decrease trdeg( κv/k) by picking g∈ovwhose image in
+κvis transcendental over k, then blowing up to force one of gorg−1intoOX,z. This
+condition persists under all further modifications, so we may additio nally enforce (b) using
+Theorem 1.3.3. We can enforce the other conditions by shrinking X, and in the case of (e)
+invoking Corollary 3.1.8.
+Proposition 7.1.3. The following conditions are equivalent.
+(a) After modifying the input data, Eadmits a good formal structure at z(or equivalently
+a ramified good decomposition, by Proposition 4.4.1).
+(b) After altering the input data, Eadmits a good decomposition at z.
+(c) After altering the input data, Eadmits an admissible decomposition at z.
+(d) After altering the input data, the restriction of f∗Eto/hatwidestOX,z(∗Z)admits a filtration
+with successive quotients of rank 1.
+Proof.Given (a), (b) is evident. Given (b), let f:Y→Xbe an alteration such that f∗E
+admits a good decomposition at the generic center of some extensio n ofv. By Lemma 1.4.3,
+we can find a modification g:X′→Xsuch that the proper transform hoffundergis
+finite flat. By Theorem 1.3.3, we can choose X′so that (X′,Z′) is a regular pair, for Z′the
+union ofg−1(Z) with the branch locus of h. Theng∗Eadmits a ramified good decomposition
+at the generic center of vonX′, yielding (a).
+Given (b), (c) is evident. Given (b), (d) holds by Proposition 3.5.5. It remains to show
+that each of (c) and (d) implies (b); we give the argument for (d), a s the argument for (c) is
+similar but simpler.
+Given (d), set notation as in Lemma 7.1.2. By Proposition 3.4.5, each qu otient of the
+filtration has the form E(φα)⊗ Rαfor someφα∈/hatwideOX,z(∗Z) and some regular differential
+moduleRαover/hatwideOX,z(∗Z). By Proposition 3.4.8, we can choose the φαinOX,z(∗Z).
+After altering the input data, we can ensure that the φαobey conditions (a) and (b) of
+Definition 3.4.6. This does not give a good decomposition directly, beca use we do not have
+a splitting of the filtration. On the other hand, if Mis the restriction of Eto/hatwideOX,z(∗Z), and
+Mssdenotes the semisimplification of M, thenF(M,r) =F(Mss,r) andF(End(M),r) =
+F(End(Mss),r) for allr, andMssdoes admit a good decomposition. Using Theorem 3.5.4,
+we deduce that Madmits a good formal structure, yielding (b).
+40Definition 7.1.4. Under Hypothesis 7.0.1, we say that Eadmits a potential good formal
+structure atvif any of the equivalent conditions of Proposition 7.1.3 are satisfied.
+Remark 7.1.5. Note that Eadmits a potential good formal structure (as a meromorphic
+differential module over X) if and only if Ezdoes so (as a meromorphic differential module
+over Spec( OX,z)). Thus for the purposes of checking the existence of potential good formal
+structures, we may always reduce to the case where Xis the spectrum of a local ring and z
+is the closed point.
+On the other hand, we may not replace Spec( OX,z) by its completion, because not every
+alteration of the completion corresponds to an alteration of Spec( OX,z). For instance, if
+X= Spec(k[x1,x2,x3]) andzis the origin, then blowing up the completion of Xatzat the
+ideal (x1−f(x2),x3) fails to descend if f∈k/llbracketx2/rrbracketis transcendental over k(x2).
+Remark 7.1.6. Note that if Z′is another closed proper subscheme of XcontainingZ, and
+the restriction of EtoOX(∗Z′) admits a potential good formal structure at v, thenEalso
+admits a potential good formal structure at v. This is true because the numerical criterion
+for good formal structures (Theorem 3.5.4, or Proposition 5.2.3) is insensitive to adding
+extra singularities.
+The rest of this section is devoted to proving the following theorem.
+Theorem 7.1.7. For any instance of Hypothesis 7.0.1, Eadmits a potential good formal
+structure at v.
+Outline of proof. We prove the theorem by induction on the height and transcendenc e defect
+ofv, as follows.
+•We first note that Theorem 7.1.7 holds trivially for vtrivial. This is the only case for
+which height( v) = 0.
+•WenextprovethatTheorem7.1.7holdsinallcaseswhereheight( v) = 1andtrdefect( v) =
+0. See Lemma 7.2.2.
+•We next prove that for any positive integer e, if Theorem 7.1.7 holds in all cases where
+height(v) = 1 and trdefect( v)< e, then it also holds in all cases where height( v) = 1
+and trdefect( v) =e. See Lemma 7.3.3.
+•We finally prove that for any integer h>1, if Theorem 7.1.7 holds in all cases where
+height(v)<h, then it also holds in all cases where height( v) =h. See Lemma 7.4.1.
+7.2 Abyhankar valuations
+We begin the proof of Theorem 7.1.7 by analyzing valuations of height 1 and transcendence
+defect 0, i.e., all real Abhyankar valuations. This analysis relies on th e simple description of
+such valuations in local coordinates.
+41Lemma 7.2.1. Suppose that height(v) = 1andtrdefect(v) = 0. After modifying the in-
+put data, in addition to the conditions of Lemma 7.1.2, we can ensure that the following
+conditions hold.
+(f) The value group of vis freely generated by v(x1),...,v(xn).
+(g) The valuation vis induced by the (v(x1),...,v(xn))-Gauss valuation on /hatwidestOX,z.
+Proof.This is a consequence of the equality case of Theorem 2.2.3. See [29] f or details. (See
+Lemma 7.3.2 for a similar argument.)
+Lemma 7.2.2. Foranyinstanceof Hypothesis 7.0.1in which height(v) = 1andtrdefect(v) =
+0,Eadmits a potential good formal structure at v.
+Proof.Set notation as in Lemma 7.2.1. Normalize the embedding of the value gr oup ofv
+intoRso thatv(x1···xn) = 1, and put α= (v(x1),...,v(xn)), so that the components of α
+are linearly independent over Q(viewing Ras a vector space over Q).
+By Theorem 3.5.2, F(E,r) andF(End(E),r) are piecewise integral linear in r. Sinceα
+lies on no rational hyperplane, it lies in the interior of a simultaneous do main of linearity
+forF(E,r) andF(End(E),r). Write this domain as the intersection of finitely many closed
+rational halfspaces. We modify the input data as follows: for each o f these halfspaces, choose
+a defining inequality m1r1+···mnrn≥0 withm1,...,mn∈Z, then ensure that xm1
+1···xmnn
+becomes regular. (This amounts to making a toric blowup in x1,...,xn.)
+After this modification of the input data, F(E,r) andF(End(E),r) become linear func-
+tions ofr. By Theorem 3.5.4, Ehas a good formal structure at z, as desired.
+7.3 Increasing the transcendence defect
+We now take the decisive step from real Abhyankar valuations to re al valuations of higher
+transcendence defect. For this, we need to invoke the analysis of ∇-modules on Berkovich
+discs made in §6. The overall structure of the argument is inspired directly by [26, §5],
+and somewhat less directly by [41]; see Remark 7.3.6 for a summary in te rms of the relevant
+notations. (Note that this is the only step where we make essential use of alterations rather
+than modifications.)
+Lemma 7.3.1. Letr≤sbe positive integers. Let c1,...,csbe positive real numbers such that
+c1,...,crform a basis for the Q-span ofc1,...,cs. Then there exists a matrix A∈GLs(Z)
+such thatA−1has nonnegative entries, and
+s/summationdisplay
+j=1Aijcj>0 (i= 1,...,r),s/summationdisplay
+j=1Aijcj= 0 (i=r+1,...,s).
+Proof.The general case follows from the case r=s−1, which is due to Perron. See [46,
+Theorem 1].
+42The following statement and proof, a weak analogue of Lemma 7.2.1, a re close to those
+of [26, Lemma 2.3.5]. (Compare also [26, Lemma 5.1.2].)
+Lemma 7.3.2. Suppose that height(v) = 1. After modifying the input data and enlarging
+Z, in addition to the conditions of Lemma 7.1.2, we can ensure t hat the following condition
+holds.
+(f) We have m= ratrank(v), andv(x1),...,v(xm)are linearly independent over Q.
+Proof.Putr= ratrank(v). We first choose a1,...,ar∈ovwhose valuations are linearly
+independent over Q. We shrink Xto ensure that a1,...,ar∈Γ(X,OX), then enlarge Zto
+ensure that a1,...,ar∈Γ(X\Z,O×
+X).
+Modify the input data as in Lemma 7.1.2, then change notation by repla cing the labels
+x1,...,xnwithx′
+1,...,x′
+nand the label mwiths. In this notation, each aigenerates the
+same ideal as some monomial in x′
+1,...,x′
+s. This implies that v(x′
+1),...,v(x′
+s) must also be
+linearly independent over Q. Since it is harmless to reorder the indices on x′
+1,...,x′
+s, we can
+ensure that in fact v(x′
+1),...,v(x′
+r) are linearly independent over Q.
+Fixanembedding ofΓ vintoR. Apply Lemma 7.3.1 with( c1,...,cs) = (v(x′
+1),...,v(x′
+s)),
+then put
+yi=s/productdisplay
+j=1xAij
+j(i= 1,...,s).
+By modifying the input data (again with a toric blowup), we end up with a new ringRwith
+local coordinates y1,...,ys,x′
+s+1,...,x′
+nat the center of v. Note that for i=r+1,...,s, we
+havev(yi) = 0. Since trdeg( κv/k) = 0,yimust generate an element of κvwhich is algebraic
+overk. Henceyi∈ O×
+X,z, soZmust now be the zero locus of y1···yr. We may now achieve
+the desired result by taking m=rand using any regular sequence of parameters starting
+withy1,...,ym.
+We now state the desired result of this subsection, giving the induct ion on transcendence
+defect.
+Lemma 7.3.3. Lete >0be an integer. Suppose that for any instance of Hypothesis 7. 0.1
+withheight(v) = 1andtrdefect(v)< e,Eadmits a potential good formal structure at v.
+Then for any instance of Hypothesis 7.0.1 with height(v) = 1andtrdefect(v) =e,Eadmits
+a potential good formal structure at v.
+We will break up the proof of Lemma 7.3.3 into several individual lemmat a. These will
+all be stated in terms of the following running hypothesis.
+Hypothesis 7.3.4. During the course of proving Lemma 7.3.3, we will carry hypotheses
+as follows. Let ebe a positive integer such that for any instance of Hypothesis 7.0.1 w ith
+height(v) = 1 and trdefect( v)<e,Eadmits a good formal structure at v.
+Choose an instance of Hypothesis 7.0.1 in which height( v) = 1 and trdefect( v) =e. Fix
+an embedding of Γ vintoR. Putd= rank(E). Set notation as in Lemma 7.3.2 (after possibly
+enlargingZ, which is harmless thanks to Remark 7.1.6).
+43In terms of this hypothesis, we may set some more notation.
+Definition 7.3.5. PutRn=R/xnRand let/hatwideRbe thexn-adic completion of R; note that by
+Lemma 3.1.7, wemayidentify /hatwideRwithRn/llbracketxn/rrbracket. Extendvby continuity toareal semivaluation
+on/hatwideR(see Remark 2.2.5), then let vnbe the restriction to Rn.
+For any real semivaluation wonRn, putpw=w−1(+∞), so thatwinduces a true
+valuation on Rn/pw. Letℓ(w) denote the completion of Frac( Rn/pw) underw, carrying the
+norme−w(·). By extending scalars to ℓ(w)/llbracketxn/rrbracket, form the restriction NwofEto the Berkovich
+open unit disc D0,woverℓ(w).
+Letznbe the center of vnon Spec(Rn/pvn). Letαv∈D0,vnbe the seminorm e−v(·). Define
+αsfors∈(0,−logr(αv)) as in Definition 6.3.2.
+Remark 7.3.6. In terms of the notation from Definition 7.3.5, we can now give a possib ly
+helpful summary of the rest of the proof of Lemma 7.3.3. The basic id ea is to view the
+formal spectrum of Rn/llbracketxn/rrbracketas a family of formal discs over Spec( Rn); however, one can
+give a more useful description of the situation in Berkovich’s languag e of nonarchimedean
+analytic spaces.
+In Berkovich’s theory, one associates to a commutative nonarchim edean Banach algebra
+itsGel’fand transform , which consists of all multiplicative seminorms bounded above by
+the Banach norm. For instance, for Fa complete nonarchimedean field, this construction
+applied toF/a\}b∇acketle{tx/a\}b∇acket∇i}ht(the completion of F[x] for the Gauss norm) produces the closed unit disc
+DFas in Definition 6.1.1.
+EquipRn/llbracketxn/rrbracketwith theρ-Gauss norm for some ρ∈(0,1). The Gel’fand transform of
+Rn/llbracketxn/rrbracketthen fibres over the Gel’fand transform of Rnfor the trivial norm. The fibre over
+a semivaluation w(or rather, over the corresponding multiplicative seminorm e−w(·)) is a
+closed disc over ℓ(w); taking the union over all ρgives the open unit disc over ℓ(w).
+Imagine a two-dimensional picture in which the Gel’fand transform of Rnis oriented
+horizontally, while the fibres over it are oriented vertically. Using the analysis of ∇-modules
+on Berkovich discs from §6, we can control the spectral behavior of ∂non a single fibre. In
+particular, within the fibre over vn, we obtain good control in a neighborhood of v. To make
+more progress, however, we must combine this vertical informatio n with some horizontal
+information. We do this by picking another point in the fibre over vnat which we have access
+to the induction hypothesis. This gives good horizontal control no t just of the irregularity,
+but of the variation of the individual components of the scale multise t of∂n. (This is needed
+because we may not have enough continuous derivations on ℓ(vn) to control the irregularity
+along the fibre over vn.) We ultimately combine the horizontal and vertical information to
+control the behavior of Eover a neighborhood of vnin the Gel’fand transform of Rn/llbracketxn/rrbracket.
+This control leads to a proof of Lemma 7.3.3.
+We now set about the program outlined in Remark 7.3.6. We first give a r efinement of
+Definition 7.1.1 which respects the notation of Definition 7.3.5.
+Definition 7.3.7. Given an instance of Hypothesis 7.3.4, by modifying/altering the input
+data onRn, we will mean performing a sequence of operations of the following fo rm.
+44•LetRn→R′
+nbe a morphism of finite type to another nondegenerate differential
+domain such that Frac( R′
+n) is finite over Frac( Rn),pvnhas a positive but finite number
+of preimages in Spec( R′
+n), and the conclusion of Lemma 7.3.2 holds for some extension
+v′
+nofvn. That is, there must exist a regular sequence of parameters x′
+1,...,x′
+n−1inR′
+n
+at the center of v′
+n, such that for m= ratrank(v), the inverse image of Zin Spec(R′
+n)
+is the zero locus of x′
+1···x′
+m, andv(x′
+1),...,v(x′
+m) are linearly independent over Q.
+•Choose a finite list of generators y1,...,yhofR′
+noverRn. Foreach generator yj, choose
+a lift ˜yjofyjinR′
+n/llbracketxn/rrbracketwhich is integral over Frac( R). LetR′be the ring obtained by
+adjoining ˜y1,...,˜yhtoR; we may identify the xn-adic completion of R′withR′
+n/llbracketxn/rrbracket.
+Apply Lemma 6.1.2 to obtain an extension v′of the semivaluation vtoR′
+n/llbracketxn/rrbracket. By
+restriction, we obtain a true valuation on R′.
+•ReplaceRwithR′andvwithv′. Replacex1,...,xn−1with any lifts of x′
+1,...,x′
+n−1
+toR′.
+We will distinguish this operation from modifying/altering the input data on R, as the latter
+must be done carefully in order to have any predictable effect on Rnand the other structures
+introduced in Definition 7.3.5.
+We next collect some horizontal information, by extracting conseq uences from the induc-
+tion hypothesis on transcendence defect.
+Lemma 7.3.8. Assume Hypothesis 7.3.4. Choose s∈(0,+∞)\(Γv⊗ZQ). Letvsbe the
+valuation on Rinduced from the s-Gauss semivaluation on Rn/llbracketxn/rrbracket(relative to vn).
+(a) Thereexists a finitelygeneratedintegral R-algebraRswithFrac(Rs)finite over Frac(R),
+such thatvsadmits a centered extension to Rswith generic center zs, andEadmits a
+good decomposition at zs.
+(b) For a suitable choice of Rs, there exist ψ1,...,ψd2∈Frac(Rs)such that for any real
+valuationwonRadmitting a centered extension to Rswith generic center zs, the scale
+multiset of ∂nonEnd(E)computed with respect to e−w(·)equalse−w(ψ1),...,e−w(ψd2).
+(c) For a suitable choice of Rs, the completion /hatwideEzsofEatzsadmits a filtration whose
+successive quotients are of rank 1, such that for i= 1,...,rank(E)−1, the step of the
+filtration having rank ihas top exterior power equal to the image of some endomorphis m
+of∧i/hatwideEzs.
+Proof.Todeduce(a),notethatheight( vn) = 1andratrank( vn) = ratrank( v)byLemma7.3.2,
+so trdefect( vn) =e−1. Then note that height( vs) = 1 and ratrank( vs) = ratrank( vn) +1
+by Lemma 6.1.7, so trdefect( vs) =e−1. Hence the induction hypothesis on transcendence
+defects may be invoked, yielding (a). To deduce (b), use Propositio n 3.4.8. To deduce (c),
+use Proposition 3.5.5.
+We next collect some vertical information from the analysis of ∇-modules on Berkovich
+discs.
+45Lemma 7.3.9. Assume Hypothesis 7.3.4. After altering the input data, End(∧jNvn)is
+terminal for j= 1,...,rank(E)−1.
+Proof.Note that trdefect( vn)<trdefect(v), so by Lemma 6.1.7, αvis a point of D0,vnof
+type (i) or (iv). By Proposition 6.3.4, for some s0∈[0,−logr(αv)), End(∧jNvn) becomes
+terminal at s0forj= 1,...,rank(E)−1.
+Chooses1∈(s0,−logr(αv)). Writeαs1=αz,rforr=e−s1and somezin a finite
+extension of Frac( Rn) withvn(z)>0. By altering the input data on Rn, we can force
+z∈Rn. Note that R∩Rnis the kernel of ∂nonR, which is dense in Rnwith respect to vn
+because it contains x1,...,xn−1. Thus we can in fact choose z∈R∩Rn, then modify the
+input data on Rby replacing xnwithxn−z. At this point, we may now take z= 0.
+After altering the input data on Rn, we can produce h∈R∩Rnwithv(h) =s0. We may
+then modify the input data on R, by replacing xnbyxn/h, to achieve the desired result.
+We now begin to mix the horizontal and vertical information. We first use vertical
+information to refine our last horizontal statement (Lemma 7.3.8), as follows.
+Lemma 7.3.10. Assume Hypothesis 7.3.4, and assume that End(Nvn)is terminal. Choose
+s∈(0,+∞)\(Γv⊗ZQ)and an open neighborhood Iofsin(0,+∞). After altering the
+input data on Rn, we may choose Rsandψ1,...,ψd2as in the conclusions of Lemma 7.3.8,
+satisfying the following additional conditions.
+(a) We have ψ1,...,ψd2∈(R∩Rn)∪{xn}.
+(b) For any centered real semivaluation wnonRnwith generic center zn, normalized so
+thatwn(x1···xn−1) =vn(x1···xn−1), there exists s′∈Isuch that
+fi(Nwn,s′) =/braceleftBigg
+wn(ψi) (ψi∈R∩Rn)
+s′(ψi=xn)(i= 1,...,d2).
+Proof.Set notationas in Lemma 7.3.8. After altering theinput data on Rn, fori= 1,...,d2,
+iffi(End(Nvn),αv,s) is constant, we can find an element φiofR∩Rnsuch thatvn(φi) =
+fi(End(Nvn),αv,0). Forifor whichfi(End(Nvn),αv,s) =sidentically, we instead put
+φi=xn. By permuting the ψiappropriately, we may ensure that vs(φi) =vs(ψi) for
+i= 1,...,d2. By replacing Rsby a suitable modification, we can ensure that w(φi) =w(ψi)
+for allw∈RZ(Rs). We may thus replace the ψiwith theφihereafter; this yields (a).
+By modifying Rs, we can ensure that for any centered real valuation wnonRnnormalized
+such thatwn(x1···xn−1) = 1, any centered real semivaluation wonRsextendingwnsatisfies
+w(xn)∈I. On the other hand, by Proposition 2.3.5, the image of RZ( Rs) in RZ(Rn) is open.
+Hence by modifying the input data on Rn, we may thus ensure that RZ( Rs) surjects onto
+RZ(Rn). These two assertions together yield (b).
+We now turn around and use this improved horizontal information to refine our last
+vertical assertion (Lemma 7.3.9), so that it applies not just at the s emivaluation vnbut also
+in a neighborhood thereof.
+46Lemma 7.3.11. Assume Hypothesis 7.3.4. After altering the input data, for any cen-
+tered real valuation wnonRnwith generic center zn,End(∧jNwn)is terminal for j=
+1,...,rank(E)−1.
+Proof.ByLemma7.3.9, wemayassumethatEnd( ∧jNvn)isterminalfor j= 1,...,rank(E)−
+1. Chooses0∈(0,−logr(αv))∩(Γv⊗ZQ) such that α0,s0≥αv. Choose an open subinterval
+Iof [0,s0] on whichfi(End(Nvn),s) is affine for i= 1,...,d2. Choose three nonempty open
+subintervals I1,I2,I3ofIsuch that for any sj∈Ij, we haves1<s2<s3.
+Forj= 1,2,3, choosesj∈Ij\(Γv⊗ZQ); this is possible because Γ vhas finite rational
+rank. By Lemma 7.3.10, after altering the input data on Rn, for any centered real valuation
+wnonRnwithgenericcenter zn,normalizedsuchthat wn(x1···xn−1) =vn(x1···xn−1),there
+existss′
+j∈Ijsuch thatfi(Nwn,s′
+j) =fi(Nvn,s′
+j) fori= 1,...,d2. By Proposition 6.4.3, Nwn
+becomes terminal at s′
+1, and hence also at s0. By a similar argument, ∧jNwnalso becomes
+terminal at s0forj= 2,...,rank(E)−1.
+After altering the input data on Rn, we can produce h∈R∩Rnwithv(h) =s0. We may
+then modify the input data on R, by replacing xnbyxn/h, to achieve the desired result.
+We now combine horizontal and vertical information once more to ob tain potential good
+formal structures.
+Lemma 7.3.12. Under Hypothesis 7.3.4, suppose that for any centered real v aluationwn
+onRnwith generic center zn,End(∧jNwn)is terminal for j= 1,...,rank(E)−1. ThenE
+admits a potential good formal structure at v.
+Proof.Pick any centered height 1 Abhyankar valuation wonRswith generic center zs, such
+that the restriction wnofwtoRnhas generic center zn. Normalize the embedding of Γ winto
+Rso thatwn(x1···xn−1) =vn(x1···xn−1), then letαw∈D0,wnbe the point corresponding
+tow. Note that by Lemma 7.3.10, on some closed interval containing −logr(αw) in its
+interior, End( ∧jNwn) becomes strongly terminal for j= 1,...,rank(E)−1. Let/hatwiderRnbe the
+completion of Rnatzn, and let /hatwiderRsbe the completion of Rsatzs. By Remark 2.2.5, w
+extends to a centered real valuation on /hatwiderRs.
+Forj= 1,...,rank(E)−1, letvj∈End(∧j/hatwideEzs) be the horizontal element corresponding
+to the endomorphism of ∧j/hatwideEzsdescribed in Lemma 7.3.8(c). By Proposition 6.4.5, vjbe-
+longs to End( ∧jE)⊗ℓ(wn)/a\}b∇acketle{txn/r/a\}b∇acket∇i}ht. On the other hand, it also belongs to End( ∧jE)⊗/hatwiderRs; by
+Remark 3.3.4, we thus find it in End( ∧jE)⊗SforSa complete subring of ℓ(wn)/a\}b∇acketle{tr/xn,xn/r/a\}b∇acket∇i}ht
+which is topologically finitely generated over Rn/a\}b∇acketle{tr/xn,xn/r/a\}b∇acket∇i}ht, such that Frac( S) is finite over
+Frac(Rn/a\}b∇acketle{tr/xn,xn/r/a\}b∇acket∇i}ht) (which may be chosen independently of j). By Proposition 6.2.6, vj
+belongs to End( ∧jE)⊗R′
+n/a\}b∇acketle{tr/xn,xn/r/a\}b∇acket∇i}htfor some topologically finitely generated /hatwiderRn-algebra
+R′
+nsuch that Frac( R′
+n) is finite over Frac( /hatwiderRn) (again chosen independently of j). By Re-
+mark 3.3.4, vjbelongs to End( ∧jE)⊗R′
+n/a\}b∇acketle{txn/r/a\}b∇acket∇i}ht, and in particular to End( ∧jE)⊗R′
+n/llbracketxn/rrbracket.
+This last conclusion is stable under altering the input data on Rn. By applying Proposi-
+tion4.3.2,wemayaltertheinputdataon RnsothatvjbelongstoEnd( ∧jE)⊗/hatwiderRn/llbracketxn/rrbracket[x−1
+1,...,x−1
+n−1]
+forj= 1,...,rank(E)−1. We obtain a filtrationof E⊗/hatwiderRn/llbracketxn/rrbracket[x−1
+1,...,x−1
+n−1] with successive
+47quotients of rank 1 by taking the step of rank jto contain elements which wedge to 0 with
+the image of vj. By Proposition 7.1.3, Eadmits a potential good formal structure at v, as
+desired.
+By combining Lemma 7.3.11 with Lemma 7.3.12, we deduce Lemma 7.3.3.
+Remark 7.3.13. Note that in the proof of Lemma 7.3.3, we arrive easily at the situation
+where End( Nvn) is terminal, but it takes more work to reach the situation where End (Nw)
+is terminal for any centered real valuation wonRnwith generic center zn. This extra work
+is not needed in case vnitself extends to a real valuation on the completion of Rnatzn, but
+this does not always occur; see Remark 2.2.5.
+7.4 Increasing the height
+We finally construct good potential formal structures for valuat ions of height greater than
+1. This argument is loosely modeled on [23, Theorem 4.3.4].
+Lemma 7.4.1. Leth >1be an integer. Suppose that for any instance of Hypothesis 7. 0.1
+withheight(v)<h,Eadmits a potential good formal structure at v. Then for any instance
+of Hypothesis 7.0.1 with height(v) =h,Eadmits a potential good formal structure at v.
+Proof.Choose a nonzero proper isolated subgroup of Γ v, then define v′,vas in Defini-
+tion 2.2.1. Note that height( v′) and height( v) are both positive and their sum is height( v) =
+h, so both are at most h−1. By the induction hypothesis, Eadmits a potential good formal
+structure at v′; in particular, after altering the input data, a good decomposition e xists at
+the generic center of v′.
+Afteralteringtheinputdataandpossiblyenlarging Z(whichisharmlessbyRemark7.1.6),
+we may set notation as in Lemma 7.1.2 in such a way that for some r, the center of v′onX
+is the zero locus of xr+1,...,xn. LetR′be the completion of Rfor the ideal ( x1,...,xr). By
+Corollary 3.1.8, we may write R′∼=R1/llbracketx1,...,xr/rrbracket, whereR1is the joint kernel of ∂1,...,∂r
+onR′. PutK= Frac(R1), so that by construction
+E ⊗K/llbracketx1,...,xr/rrbracket[x−1
+1,...,x−1
+r]
+admits a minimal good decomposition. By Theorem 4.3.4, this decompos ition descends to a
+minimal good decomposition of
+E ⊗R†
+r,r(R1).
+With notation as in (3.4.6.1), by the last assertion of Proposition 4.3.3, eachRαadmits
+aK-lattice stable under the action of x1∂1,...,xr∂r. This gives a collection of instances
+of Hypothesis 7.0.1 with Rreplaced by R1andvreplaced by v. Again by the induction
+hypothesis, after altering the input data (and lifting from R1toR, as in Definition 7.3.7),
+we obtain good decompositions of each of these lattices.
+Putting this all together, we obtain an admissible but possibly not goo d decomposition
+ofEatz. By Proposition 7.1.3, this suffices to imply that Eadmits a potential good formal
+structure at v.
+488 Good formal structures after modification
+To conclude, we extract fromTheorem 7.1.7 aglobal theorem onthe existence of goodformal
+structures for formal flat meromorphic connections on nondege nerate differential schemes,
+after suitable blowing up. We also give partial results in the cases of f ormal completions of
+nondegenerate differential schemes and complex analytic varieties .
+8.1 Local-to-global construction of good formal structure s
+Using the compactness of Riemann-Zariski spaces, we are able to p ass from the valuation-
+local Theorem 7.1.7 to a more global theorem on construction of goo dformal structures after
+a blowup, in the case of an algebraic connection.
+Hypothesis 8.1.1. Throughout §8.1, letXbe a nondegenerate integral differential scheme,
+and letZbe a closed proper subscheme of X. LetEbe a∇-module over OX(∗Z).
+Lemma 8.1.2. Letvbe a centered valuation on X. Then there exist a modification fv:
+Xv→X, a centered extension wofvtoXv, and an open subset UvofXvon whichwis
+centered, such that f∗
+vEadmits a good formal structure at each point of Uv.
+Proof.By Theorem 7.1.7, we can choose data as in the statement of the lemm a so that
+f∗
+vEadmits a good formal structure at the generic center of vonUv. (Note that Proposi-
+tion 7.1.3 ensures that we can choose fto be a modification, not just an alteration.) By
+Proposition 5.1.4, this implies that we can rechoose Uvso thatEadmits a good formal
+structure at each point of Uv.
+Theorem 8.1.3. There exists a modification f:Y→Xsuch that (Y,W)is a regular pair
+forW=f−1(Z), andf∗Eadmits a good formal structure at each point of Y.
+Proof.For each valuation v∈RZ(X), set notation as in Lemma 8.1.2. Since v∈RZ(Uv) by
+construction, the sets RZ( Uv) cover RZ( X). Since RZ( X) is quasicompact by Theorem 2.3.4,
+we can choose finitely many valuations v1,...,vn∈RZ(X) such that the sets Ti= RZ(Uvi)
+fori= 1,...,ncover RZ(X). Putfi=fviandXi=Xvi. By applying Theorem 1.3.3 to
+the unique component of X1×X···×XXnwhich dominates X, we construct a modification
+f:Y→Xfactoring through each Xi, such that ( Y,f−1(Z)) is a regular pair.
+We now check that this choice of fhas the desired property. For any z∈Y, we may
+choose a valuation v∈RZ(X) with generic center zonY. For some i, we havev∈Ti, so
+f∗
+iEadmits a good formal structure at the generic center of vonXi. That point is the image
+ofzinXi, sof∗Eadmits a good formal structure at z, as desired.
+Remark 8.1.4. ForXan algebraic variety over an algebraically closed field of characteris-
+tic 0, Theorem 8.1.3 reproduces a result of Mochizuki [35, Theorem 19.5]. (More precisely,
+one must apply Theorem 8.1.3 to both Eand End( E), due to the discrepancy between our
+notion of good formal structures and Mochizuki’s definition. See ag ain [25, Remark 4.3.3,
+Remark 6.4.3].) Mochizuki’s argument is completely different from ours: he uses analytic
+methods to reduce to the case of meromorphic connections on sur faces, which he had previ-
+ously treated [34, Theorem 1.1] using positive-characteristic argu ments.
+498.2 Formal schemes and analytic spaces
+From Theorem 8.1.3, we obtain a corresponding result for formal co mpletions of nondegen-
+erate schemes. We also obtain a somewhat weaker result for forma l completions of complex
+analytic spaces. We do not obtain the best possible result in the analy tic case; see Re-
+mark 8.2.5.
+Theorem 8.2.1. LetXbe a nondegenerate integral differential scheme, let Zbe a closed
+proper subscheme of X, and let/hatwidestX|Zbe the formal completion of XalongZ. LetEbe a
+∇-module over O/hatwidestX|Z(∗Z). Then there exists a modification f:Y→Xsuch that (Y,W)is
+a regular pair for W=f−1(Z), andf∗Eadmits a good formal structure at each point of W.
+Proof.We first consider the case where Xis affine. Put X= Spec(R) andZ= Spec(R/I).
+Let/hatwideRbetheI-adiccompletionof R, andput /hatwideI=I/hatwideR. Put/hatwideX= Spec(/hatwideR)and/hatwideZ= Spec(/hatwideR//hatwideI).
+We can then view Eas a∇-module on O/hatwideX(∗/hatwideZ), and apply Theorem 8.1.3 to deduce the
+claim.
+We now turn to the general case. Since Xis integral and noetherian, it is covered by
+finitely many dense open affine subschemes U1,...,Un. Fori= 1,...,n, we may apply
+the previous paragraph to construct a modification fi:Yi→Uisuch that ( Yi,Wi) is a
+regular pair for Wi=f−1
+i(Ui∩Z), andf∗
+iEadmits a good formal structure at each point
+ofWi. By taking the Zariski closure of the graph of fiwithinYi×SpecZX, we may extend
+fito a modification f′
+i:Y′
+i→X. By Theorem 1.3.3, we may construct a modification
+f:Y→Xfactoring through the fibred product of the f′
+i, such that ( Y,W) is a regular pair
+forW=f−1(Z). This modification has the desired effect.
+Theorem 8.2.2. LetXbe a smooth (separated) complex analytic space. Let Zbe a closed
+subspace of Xcontaining no irreducible component of X. Let/hatwidestX|Zbe the formal completion
+ofXalongZ. LetEbe a∇-module over O/hatwidestX|Z(∗Z). For each z∈Z, there exist an open
+neighborhood UofzinXand a modification f:Y→Usuch that (Y,W)is a regular pair
+forW=f−1(U∩Z), andf∗Eadmits a good formal structure at each point of W.
+Proof.We may reduce to Theorem 8.1.3 as in the proof of Proposition 5.1.4.
+Inboththealgebraicandanalyticcases, we recover Malgrange’sco nstruction ofcanonical
+lattices [31, Th´ eor` eme 3.2.2].
+Theorem 8.2.3. LetXbe either a nondegenerate integral differential scheme or a s mooth
+irreducible complex analytic space. Let Zbe a closed proper subspace of X, and let/hatwidestX|Zbe
+the formal completion of XalongZ. LetEbe a∇-module over O/hatwidestX|Z(∗Z). LetUbe the open
+(by Proposition 5.1.4) subspace of Xover which (X,Z)is a regular pair and Ehas empty
+turning locus. Then the Deligne-Malgrange lattice of E|Uextends uniquely to a lattice of E.
+Proof.Letj:/hatwidestU|Z→/hatwidestX|Zbe the inclusion. Let E0be the Deligne-Malgrange lattice of
+EU; it is sufficient to check that j∗E0is coherent over O/hatwidestX|Z. This may be checked locally
+50around a point z∈Z. After replacing Xby a suitable neighborhood of z, we may apply
+Theorem8.2.1orTheorem8.2.2toconstructamodification f:Y→Xsuchthat(Y,f−1(Z))
+is a regular pair and f∗Ehas empty turning locus. By Theorem 5.3.4, f∗Eadmits a Deligne-
+Malgrange lattice E1. We then have j∗E0=f∗E1, which is coherent because fis proper (by
+[17, Th´ eor` eme 3.2.1] in the algebraic case, and [15, §6, Hauptsatz I] in the analytic case),
+This proves the claim.
+Remark 8.2.4. As noted in [31, Remarque 3.3.2], the lattice constructed in Theorem 8 .2.3
+is reflexive, and hence locally free in codimension 2.
+Remark 8.2.5. In Theorem 8.2.2, we would prefer to give a globalmodification rather
+than a local modification around each point of Z. The obstruction to doing so is that while
+Theorem8.1.3givesaprocedureforconstructing asuitablemodifica tion, theprocedureisnot
+functorialforopenimmersions. Suchfunctorialityisnecessaryto gluethelocalmodifications;
+we cannot instead imitate the proof of Theorem 8.2.1 because the co mplex analytic topology
+is too fine to admit Zariski closures.
+In a subsequent paper, we plan to describe a modification procedur e which is functorial
+for allregularmorphisms of nondegenerate differential schemes. For this, one n eeds a form
+of embedded resolution of singularities for excellent schemes which is functorial for regular
+morphisms. Fortunately, such a result has recently been given by T emkin [43], based on
+earlier work of Bierstone, Milman, and Temkin [7, 42].
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+arXiv:1001.0545v1  [astro-ph.HE]  4 Jan 2010Astron.Nachr./ AN 999,No.88,789–795(2006)/ DOIpleaseset DOI!
+New photometry and astrometry of the isolated neutron star
+RXJ0720.4-3125using recent VLT/FORS observations⋆
+Thomas Eisenbeiss1,⋆⋆, Christian Ginski1, Markus M. Hohle1,2, Valeri V. Hambaryan1, Ralph
+Neuh¨auser1,andTobiasO.B. Schmidt1
+1AstrophysikalischesInstitutundUniversit¨ ats-Sternwa rteJena,Friedrich-Schiller-Universit¨ atJena,Schille rg¨ asschen2-3,
+07745 Jena
+2MaxPlanckInstitut f¨ ur extraterrestrische PhysikGarchi ng, Giessenbachstrasse, 85738 Garching
+The dates of receipt andacceptance should be insertedlater
+Keywords Stars:neutron - Dense matter -Stars:imaging -Techniques: image processing - Stars:individual
+Since the firstoptical detection of RXJ0720.4-3125 various observations have been performed to determine astrometric
+andphotometric data. We present the firstdetection of the is olatedneutron star inthe VBessel filtertostudy the spectra l
+energy distribution and derive a new astrometric position. At ESO Paranal we obtained very deep images with FORS
+1 (three hours exposure time) of RXJ0720.4-3125 in V Bessel fi lter in January 2008. We derive the visual magnitude
+by standard star aperture photometry.Using sophisticated resampling software we correct the images for field distorti ons.
+Then we derive an updated position and proper motion value by comparing its position with FORS 1 observations of
+December 2000. We calculate a visual magnitude of V= 26.81±0.09mag, which is seven times in excess of what
+is expected from X-ray data, but consistent with the extant U , B and R data. Over about a seven year epoch difference
+we measured a proper motion of µ= 105.1±7.4masyr−1towardsθ= 296.951◦±0.0063◦(NW) , consistent with
+previous data.
+c/circlecopyrt2006WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim
+1 Introduction
+Indeepopticalfollow-upobservationsofbrightX-raysour -
+cesfromtheROSATmission(Vogeset al.1999)seventher-
+malemitting isolatedneutronstars(INS), exhibitingnora -
+dioemission, havebeendiscoveredsofar(seeHaberl2004
+2007; vanKerkwijk&Kaplan2007forrecentreviews).
+After the first opticaldetectionof the brightestone,RX
+J1856.4-3754 (Walter, Wolk, & Neuh¨ auser 1996),
+Haberlet al. (1997) found the second brightest INS
+RXJ0720.4-3125. It was detected in the optical B and R
+band by Kulkarni&vanKerkwijk (1998). Using deep ob-
+servationsMotch,Zavlin,&Haberl(2003)derivedthepro-
+per motion of RXJ0720.4-3125 and measured the B mag-
+nitude with VLT/FORS 1. Later Kaplan, van Kerkwijk,
+& Anderson (2007) measured the distance of RXJ0720.4-
+3125(∼360pc)andupdatedthepropermotion usingHub-
+ble SpaceTelescope(HST)observations.
+ComparingblackbodyfitsderivedfromX-Rayobserva-
+tions with the flux measured in the optical/UV wavelength
+(Motchetal. 2003) shows that RXJ0720.4-3125 exceeds
+itsexpectedoptical/UVfluxbyabout anorderofmagnitude
+(Hambaryanet al. 2009). Furthermore Haberletal. (2006)
+detected variations in the X-ray flux. The origin of these
+⋆Based on observations made with ESO Telescopes at the Parana l Ob-
+servatory under programme ID 080.D-0135(A). Based on obser vations
+with XMM-Newton, an ESA Science Mission with instruments an d con-
+tributions directly funded by ESAMember states and the USA ( NASA).
+⋆⋆Corresponding author: e-mail: eisen@astro.uni-jena.devariations is still controversial (Hohle et al. 2009; Kapla n
+et al.2007).
+In this paper we measure the V magnitude for the first
+time andupdatethepropermotionandabsoluteposition.
+In Section 2 we discuss the observations and data re-
+duction.Sections3and4addressesthephotometry.InSec-
+tion 5 we discuss how our newly derived V magnitude fits
+with the investigations of earlier work. Section 6 is dedi-
+cated to our astrometric investigations and in Section 7 we
+summarizeourresults.
+2 Observationsand Data reduction
+InJanuaryof2008weobservedRXJ0720.4-3125forabout
+three hoursexposuretime at ESO Paranalobservatorywith
+the FOcal Reducer/low dispersion Spectrograph (FORS 1,
+Appenzelleretal. 1998), installed in Unit Telescope Kue-
+yen of the Very LargeTelescope(VLT). Observationswere
+carried out in the V Bessel filter. We used a pixel scale of
+0.125′′/pixelwhichleadstoafieldofviewof 4.2′×4.2′.13
+singleexposuresof900sweretakenover1.5nights.
+For astrometric calibration we observed the globular
+cluster 47-Tuc with a pointing at α= 00h22m29.3sand
+δ=−71d59′54.3′′. The observations are summarized in
+Table1. The imageoftheRXJ0720.4-3125 fieldisshown
+in Fig.1.
+All images were flat fielded and bias subtracted. Since
+FORS1 got a new detector in 2007 we were restricted to
+absolute world coordinates for astrometric comparison to
+c/circlecopyrt2006WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim790 T.Eisenbeiss etal.:Vmag andposition of RXJ0720.4-3125
+Table 1 Observationlog
+Date Instrument Filter Exp Pixel Seeing Target
+telescope s (′′) (′′)
+12 Jan. 2008 FORS1/VLT-UT1 VBessel 8.1·1030.125 1.1 RXJ0720.4-3125
+13 Jan. 2008 FORS1/VLT-UT1 V Bessel 3.6·1030.125 0.8 RXJ0720.4-3125
+12 Jan. 2008 00:42 (UT) FORS1/VLT-UT1 VBessel 4·100.125 1.6 47-Tuc
+12 Jan. 2008 00:47 (UT) FORS1/VLT-UT1 VBessel 12 0.125 1.6 La ndolt fieldSA98
+12 Jan. 2008 04:48 (UT) FORS1/VLT-UT1 VBessel 5 0.125 1.1 Lan dolt fieldSA98
+Fig.1 Co-added V band images obtained with FORS1 on Kueyen in Janua ry 2008. The circle shows the position of
+RXJ0720.4-3125as measured in the FORS1 image of December 20 00 (Motchet al. 2003). The arrow labeled 01/2008
+pointsto ourdetectedpositionofRXJ0720.4-3125.
+earlier observations. Furthermore the new chip is divided
+in two parts which limits the number of reference stars on
+eachpart.Inordertoderiveanaccurateastrometricsoluti on
+we observed the globular cluster 47-Tuc at four dither po-
+sitions. Based on the ESO fits header we derived image, as
+well asworldcoordinatesofeachobjectin the imageswith
+theSource Extractor (SE, Bertin &Arnouts 1996). These
+object catalogs are fitted with reference to the the two mi-
+cronallskysurveycatalog(2MASS,Cutri etal.2003),pro-
+vided by the ViZiR database (Ochsenbein, Bauer, & Mar-
+cout2000).AftertheWCS frameiscalibratedinthatwaya
+fifthorderpolynomialisfittedtothedata.This χ2-algorithm
+corrects the remaining field distortions. We used the soft-
+wareSCAMP1(Bertin2006)forthesecalculations. SCAMP
+stores the calibration information (WCS transformation as
+wellasfielddistortioncoefficients)inanexternalheaderf or
+each image. We extracted the field distortion coefficients,
+1Software for Calibrating AstroMetry and Photometrywe derived calibrating the 47-Tuc images and used them
+as input for the calibration procedure of the RXJ0720.4-
+3125images,whichis doneby SCAMPas well inthe same
+way. Similar, but directly without using a calibration clus -
+ter the RXJ0720.4-3125images of the year 2000 observed
+by Motchetal. (2003) were calibrated. The field distorion
+mapofthiscalibrationisshownasanexampleinFig.2.
+Anothersoftwarecalled SWarp(Bertinet al.2002)uses
+the external headers produced by SCAMPto align, resam-
+ple,andco-addtheimages.A meshbasedbackgroundsub-
+traction is applied during co-addition and the two parts of
+thechiparestitchedtogether.ThisisillustratedinFig.3 .
+Typically the semi major axis of the error ellipse of
+2MASSdoesnotexceed0.3arcsec.Thestatistical errorsof
+the source detection procedure are much smaller. SCAMP
+preselects unambiguous, well exposed sources for the fit-
+ting and treats the uncertaintiesin a χ2sense. Nevertheless
+itisunnecessarytotreateacherrorsourceindividually.T he
+c/circlecopyrt2006WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim www.an-journal.orgAstron. Nachr. /AN (2006) 791
+Instrument A1: distortion map
+-00°02 20  -00°02 20  
+-00°01 40  -00°01 40  
+-00°01  0  -00°01  0  
+-00°00 20  -00°00 20  
++00°00 20  +00°00 20  
++00°01  0  +00°01  0  
++00°01 40  +00°01 40  
++00°02 20  +00°02 20  
+-00°02 20  -00°02 20  -00°01 40  -00°01 40  -00°01  0  -00°01  0  -00°00 20  -00°00 20  +00°00 20  +00°00 20  +00°01  0  +00°01  0  +00°01 40  +00°01 40  +00°02 20  +00°02 20  
+0.970.980.991.00(x10-1)arcsec
+pixel scale-101
+%
+Fig.2FielddistortionmapofFORS1detectorinDecem-
+ber 2000.Color codinggoesfromred (middlein the image
+andtopofthescale)throughyellowandgreentoblue(cor-
+nersofthe imageandbottomofthescale.)
+resultingmeasurementpluscalibrationerroriscalculate din
+a statistical way asdescribedinSection6.
+3 Photometry forJanuary 12th, 2008
+To flux-calibrate the image we observed the Landolt stan-
+dardstarfieldSA98(Landolt1992)inthefirstnight. Since
+the standardswere taken only in the first night we used the
+nine imagesof RXJ0720.4-3125taken in the first night for
+standard star photometry. We co-added them in the same
+way as describedabove,but only the upperpart of the chip
+was used this time to avoid possible systematic photometry
+errorsastheymayhappenwhileequalizingthebackground
+ofthetwopartsoftheFORS1detector.Intheupperhalfof
+thechiponly thetwostandardstars98556and98557were
+observed at airmasses 1.656and1.112, respectively, (Fig.
+4).Duringdataanalysisitturnedout,thattherearefainto b-
+jectsnearby.UsingtheLandoltmagnitudesforthestarsand
+the given errors we first calculated the difference between
+thetwomagnitudesofthestarsasmeasuredbyLandoltand
+byus assumingthat
+V556,Land−V557,Land≈V556,inst−V557,inst,
+whereVinstareourinstrumentalmagnitudes.Ourmeasure-
+ments show that the stars marked as c1 and c2 in Fig. 4
+were includedin the apertureof Landolt(1992) so we took
+themintoaccountaccordingly.Aperturephotometryisdone
+usingtheDAOPHOTpackage(Stetson1987).
+We measuredtheinstrumentalmagnitudesofbothstan-
+dardstarsineachfieldandcalculatedthezeropointcorrec-
+tioncandthefirstorderextinctioncoefficient k.Takinginto
+accountall sourcesof error,which isthe 3σerrorofthe in-
+strumental magnitude, the error of the Landolt magnitudes
+andthedifferencebetweenthevaluesbyusingthedifferentGroup #1: detections
+07h20m14s07h20m14s
+07h20m18s07h20m18s
+07h20m22s07h20m22s
+07h20m26s07h20m26s
+07h20m30s07h20m30s
+07h20m34s07h20m34s
+-31°29 -31°29 -31°28 -31°28 -31°27 -31°27 -31°26 -31°26 -31°25 -31°25 -31°24 -31°24 
+Fig.3Illustration how SCAMP co-added the 13 FORS
+1 images (two parts of the chip). We used a dither pat-
+tern of four positions to avoid bad pixels. Diamonds and
+pluses symbolize objects foundin each, or at least some of
+the imagesandusedforcalculationof theastrometricsolu-
+tion while squares symbolize objects not used for calibra-
+tion.Therectanglesillustratethechipshapeandtherelat ive
+position.
+Fig.4VLT/FORS1 image of the Landolt standard stars
+98556 and 98557. There are two objects close to the stan-
+dard stars, marked as c1 and c2, which contribute flux to
+the standard magnitude measured by Landolt (1992) so we
+measured the flux of these objects as well and took their
+contributionintoaccount.
+standardstarswe derive
+c= (−21.0832±0.0017)magand
+k= (0.1562±0.0058)mag .
+Sincethese quantitiesarederivedusingtwo differentstar s,
+their uncertaintiesinclude an estimation of the error,whi ch
+ismadeneglectingthecolorterm.
+Themostpronouncederrorsourcecomesfromthe faint
+INS magnitudemeasurementitself. Fig. 1 showsthe rim of
+www.an-journal.org c/circlecopyrt2006WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim792 T.Eisenbeiss etal.:Vmag andposition of RXJ0720.4-3125
+5 10 15 20−20−1001020304050
+pixelrelative fluxnoise levelRXJ 0720
+S/N 6 σ
+Fig.5Projection of the detection of RXJ0720.4-3125 .
+Theneutronstarisdetectedwith ∼6σ.
+an association of bright O-stars to the left of RXJ0720.4-
+3125.Inourdeepexposuresthesestarsinfluencetheback-
+ground, so we had to choose an aperture smaller than that
+in the standard star image to deal with this problem. The
+INS is detected at ∼6σ. Because of the faintness of the
+source only the top of the PSF has a higher count rate than
+the background hence, for the INS a smaller aperture than
+for the reference stars is possible (Fig. 5). We investigate d
+themagnitude-aperturedependenceofthisfaintsourceand
+concluded from our measurements and Fig. 5 that an aper-
+tureradiusofsevenpixelsissufficient.
+Using the uncertainties of candkand taking the vari-
+ation of the airmass ( YNS= 1.028±0.019) during obser-
+vation into account we finally derive the V magnitude of
+RXJ0720.4-3125tobe
+V= (26.88±0.15)mag,forJan12th ,2008.
+4 Photometry forJanuary 13th, 2008
+Fourimagesweretakeninthesecondnightundermuchbet-
+terseeingconditionsthanthefirstnight.Sincethemostpro -
+nouncedsourceoferrorinthe absolutephotometrywasthe
+measurementofthefaintINSsmagnitudeitself,wedecided
+to use these images too, even thoughno official photomet-
+ric standards were taken that night. We co-added the im-
+ages as before and explained in Section 2. In addition we
+were using the SEto apply a background subtraction. The
+SEuses a mesh based background algorithm with a mesh
+size of 64 pixels. In the resulting image we identified three
+reference stars near RXJ0720.4-3125with known V-band
+magnitudes(Motch& Haberl1998andFig.6).
+Weusedanaperture of15pixelradiusforthereference
+stars, given by the full width at half maximum (FWHM)
+size of≈4.5pixel. For the INS we determined the ob-
+ject’s size. It turned out that the PSF of the INS vanishes
+belowthebackgroundat aradiusoflessthanfivepixels,so
+we chose the aperture radius accordingly. Furthermore we
+[KVA2007] 102
+[KVA2007] 103 [HMB97] CRX J0720.4-3125                              
+Fig.6Co-added and background subtracted image used
+forrelativephotometry.Thereferencestarsusedarenamed
+and marked by arrows. Motch&Haberl (1998) measured
+their V magnitude with an accuracy of 0.05 mag. The INS
+ismarkedandtheapertureradius(5pixel,circle)is shown.
+chosesmallerinnerandouterskyradiicomparedtotheref-
+erence stars to prevent the neighboring stars from falling
+intothebackgroundannulus.
+The difference of the instrumental and the apparent
+magnitudes for each reference star is calculated. The mean
+ofthisdifferencegivesagainthedetectorzeropoint, soth is
+themagnitudeofRXJ0720.4-3125isdeterminedto be
+V= (26.81±0.09)mag,forJan13th ,2008
+whichisfullyconsistentwith ourothermeasurement.
+Note, that if the small aperture would have caused any
+loss of flux the magnitude of the INS would be larger in-
+stead.Sincetheerrorbarissmallerweusetherelativevalu e
+in thefollowingdiscussion.
+5 The Spectral Energy Distributionof
+RXJ0720.4-3125
+The visual magnitude of RXJ0720.4-3125 is consistent
+with an optical excess about an order of magnitude larger
+than the expectedflux extrapolatedfrom the X-ray spectra
+first reported in Kulkarni& vanKerkwijk (1998) and stud-
+ied in detail by Kaplanet al. (2003). However, the origin
+of this enigmatic property as well as its relation to the X-
+ray variations is still unknown and a matter of debate, see
+Haberl (2007) and Haberlet al. (2006) for a review about
+the INSs and RXJ0720.4-3125 in particular. It might be
+possible that the emitting area of the soft X-ray radiationi s
+not strictly connected to the source of optical/UV photons
+(Hambaryanet al.2009). Tr¨ umper(2005);Turolla,Zane&
+Drake (2004); Zane, Turolla & Page (2007); Zane, Turolla
+& Drake (2004) and references therein discuss alternative
+explanationsof the optical excess. The radiusof the X-Ray
+emitting area is ≈4.5km (Haberlet al. 2006), normalized
+to adistanceof300pc.
+c/circlecopyrt2006WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim www.an-journal.orgAstron. Nachr. /AN (2006) 793
+Table 2 Optical,groundbasedobservationsofRXJ0720.4-3125.Onl ymeasurementsofgroundbasedobservationsare
+shown,incontrasttoFig.7,whereHSTobservationsfromKap lanet al.(2003)areincluded.Otherreferencesaregivenin
+the table.
+Filter Mag Fluxdensity λcent Flux Refernence
+[W/m2/Hz]·10−34[˚A] [ergs−1cm−2]·1016
+U 25.68 ±0.17 9.75+1.65
+−1.42 3600 8. 13+1.38
+−1.18 Motch et al.2003
+B 26.58 ±0.25 9.64+2.46
+−1.98 4300 6. 73+1.72
+−1.38 Motch et al.2003
+26.79±0.20 7.94+1.61
+−1.33 4300 5. 54+1.12
+−0.93 Motch et al.2003
+26.44±0.15 11.00+1.60
+−1.45 4300 7. 67+1.12
+−1.01 Motch et al.2003
+26.787±0.040 7.96+0.30
+−0.28 4300 5. 55+0.21
+−0.20 Motch et al.2003
+26.620±0.050 9.29+0.44
+−0.42 4300 6. 48+0.31
+−0.29 Motch et al.2003
+V 26.81 ±0.09 6.69+0.99
+−0.86 5500 3. 65+0.54
+−0.47 this paper
+R 26.9 ±0.3 5.11+1.63
+−1.23 7000 2. 19+0.70
+−0.53 Kulkarni &van Kerkwijk1998
+H>23.07±0.11<7.75+0.80
+−0.72 16500 <12.29+1.32
+−1.12 Posselt etal. 2009 (upper limit)
+For comparison to other optical/UV magnitudes we
+show in Fig. 7 the flux expected from a black body model
+(XMM-Newton EPIC-pn). The effective temperature of
+RXJ0720.4-3125ischanging ontimescalesof years, see
+Hohleet al.(2009)andHaberletal. (2006).
+We plot in Fig. 7 the spectra obtained with
+XMM-NewtonEPIC-pnwith lowest (orbit0078)andhigh-
+est (orbit 0815) temperatures ( 86.5±0.4eV and94.6±
+0.5eV, respectively)measuredyet.Thevalueswereobtain-
+ed by fitting the spectra with standard software xspec12
+(Dorman, Arnaud, & Gordon 2003) in the energy range of
+0.16-1.5keV by applying a black body model with addi-
+tive gaussian absorption line ( line energy = 301 ±3eV,
+σ= 77±2eV) with absorption due to the ISM ( NH=
+1.04±0.02·1020cm−2),seeHohleet al.(2009)fordetails.
+6 Astrometry
+For the longest possible epoch difference we used the B-
+band image of Motchet al. (2003) from the year 2000 and
+compared the position of all objects with the positions in
+ourimagefrom2008.Theobjectsinthetwoimagesareas-
+signedtoeachotherandthechangeofpositioniscalculated
+(Fig.8).
+Afterwards the change of position undergoesa Kolmo-
+gorov-Smirnov2-sample test comparing the absolute posi-
+tion changes of the objects to a simulated Rayleigh distri-
+bution.Thetestisperformedcomparingthecumulativedis-
+tribution functions (CDF) of the test sample and the data.
+If the test fails objects lying outside 2 σfrom the mean of
+thedistributionareexcluded.Themeanandstandarddevia-
+tionofthedistributionarerecalculatedandthemotionofa ll
+objects is shifted by that mean to reduce the last remaining
+systematical errors. This is repeated until the test succee ds.
+What isleftisasampleofRayleighdistributedbackground1014101510161017101810−1610−1510−1410−1310−1210−1110−10
+frequency [Hz]νFν [Hz erg/s/cm2/Hz]
+  
+Hup
+R
+V
+B
+U
+Fig.7Optical/UV flux from R to UV magnitudes (our
+new V-banddata point is marked as circle) comparedto X-
+Ray spectra obtained from XMM-Newton EPIC-pn (right)
+performedinfullframemodewiththinfilter.Thetwospec-
+tra of RXJ0720.4-3125 are taken from orbit 0078 (grey
+dots) and 0815 (black dots) with effective temperatures of
+86.5±0.4eV and 94.6 ±0.5eV, respectively. The solid line
+marks the best fit model for rev. 0078 (including interstel-
+lar extinction), while the dashed lines correspondingto th e
+black bodies for both spectra without any absorption. The
+broadabsorptionfeaturenear 300eV( /hatwide=7.25·1016Hz) can
+be seen in the spectrum of revolution 0815 (black dots).
+The Optical magnitudes are sumarized in table 2, the flux
+in theHbandisanupperlimit.
+stars, without any systematic effects. The standard devia-
+tionσBackofthesestarscouldbeseenastotalcalibrationer-
+ror.SincetheNSismuchfainterthanthebackgroundstars,
+the positional error of its photo-center is relatively larg e.
+We added the intrinsic proper motion error ∆µintr(which
+is again calculated by the position error in both epochsand
+www.an-journal.org c/circlecopyrt2006WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim794 T.Eisenbeiss etal.:Vmag andposition of RXJ0720.4-3125
+0246810121400.10.20.30.40.50.60.70.80.91
+xF(x)
+Fig.9 Cumulative distribution function (CDF) of the
+Rayleigh distributed background stars in the RXJ0720.4-
+3125field, comparedwith the CDF of a synthetic Rayleigh
+distribution. The comparisonshows good agreement,prov-
+ing that there are no systematical errors left in the calcula -
+tion.
+theepochdifference)oftheNS inbothepochstothaterror,
+representingthefullerroroftheneutronstarsmotionas
+∆µNS=/radicalbig
+(∆µintr)2+(σback)2.
+The mean intrinsic error is already included in σBackbut
+is larger for the barely visible NS than for the background
+objects.
+We derivea propermotionvalue
+µα= (−93.2±5.4)mas/yr
+µδ= (48.6±5.1)mas/yr
+µ= (105.1±7.4)mas/yr.
+θ= (296.951±0.0063)◦
+This proper motion is consistent with previously published
+values (Kaplanetal. 2007; Motchet al. 2003).
+RXJ0720.4-3125changeditspositionfromDecember2000
+(MJD = 51902 days) to January 2008 ( MJD = 54477
+days)from
+MJD = 51902days:
+7h20m24.976s−31d25′50.105′′to
+±14.15mas ±12.82mas
+MJD = 54477days:
+7h20m24.926s−31d25′49.776′′
+±19.78mas ±19.30mas,
+Since the total exposure time in the year 2000 image was
+longer and the observation wavelength was shorter, our
+newly reduced position measurement of the year 2000 is
+moreprecise andthesignalto noiseratioisbetter.
+7 Summary
+Taking a deep optical image of RXJ0720.4-3125 we were
+abletodeterminetheVmagnitudeandupdatedtheposition.Ourpropermotion( µ= 105.1±7.4mas)confirmsthemea-
+surement obtained by Kaplanet al. (2007) with the HST
+(107.8±1.2mas)with adifferenceoflessthan3mas/yr.
+Our V magnitude ( V = 26.81±0.09mag) is compat-
+ible with previous observations from Motchet al. (2003)
+(B = 26.620±0.050mag)andfrom(Kulkarni&vanKerk-
+wijk (1998)) ( R = 26.9±0.3mag). These data can be fit-
+ted by the emergent spectrum of the photoionized plasma,
+surroundingtheNS (Hambaryanet al. 2009). Thisscenario
+would also explain the power-law component of the ”opti-
+cal excess” (Kaplanet al. 2003),which isconfirmedby our
+data.
+For further investigations correlated observations in X-
+rayandtheoptical/UVareneededtocollectenoughdatato
+investigate the nature of the flux variations of RXJ0720.4-
+3125andtounderstandthisisolatedneutronstar.
+Wealsonote,thattherearenoadditionalfaint,fastmov-
+ing objects detected in this field (by comparing the year
+2000observationsandourimage),hencenoadditionalsim-
+ilar NSin thisfield.
+Acknowledgements. The authors would like to thank Frank
+Haberl, Fred Walter, Andreas Seifahrt and Tristan R¨ oll for fruit-
+ful discussions and improving comments. Furthermore we tha nk
+the Staff members of ESO Paranal observatory for their help a nd
+assistance carrying out the observations. Based on observa tions
+made with ESO Telescopes at the La Silla or Paranal Observa-
+tories under programme ID 66.D-0286(A). Moreover we thank
+Emmanuel Bertin and his team for providing new data reductio n
+software. This research has made use of the VizieR catalogue ac-
+cess tool, CDS, Strasbourg, France. This work was supported in
+part byDFGgrant SFB/Transregio7”GravitationalWaveAstr on-
+omy”. CG acknowledges support byDFGgrand under project NE
+515/30-1. The work of MMH has been supported by CompStar, a
+researchnetworkingprogrammeoftheEuropeanScienceFoun da-
+tion(ESF).TOBSacknowledges support from Evangelisches S tu-
+dienwerk e.V.Villigst.
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+arXiv:1001.0546v1  [q-bio.BM]  4 Jan 2010Femtonewton Forces Can Control Protein-Meditated DNA
+Looping
+Yih-Fan Chen,1J. N. Milstein,2and Jens-Christian Meiners2,3
+1Department of Biomedical Engineering,
+University of Michigan, Ann Arbor, Michigan 48109
+2Department of Physics, University of Michigan, Ann Arbor, M ichigan 48109
+3LSA Biophysics, University of Michigan, Ann Arbor, Michiga n 48109
+(Dated: November 14, 2018)
+Abstract
+We show that minuscule entropic forces, on the order of 100 fN , can prevent the formation
+of DNA loops–a ubiquitous means of regulating the expressio n of genes. We observe a tenfold
+decrease in the rate of LacI-mediated DNA loop formation whe n a tension of 200 fN is applied
+to the substrate DNA, biasing the thermal fluctuations that d rive loop formation and breakdown
+events. Conversely, once looped, the DNA-protein complex i s insensitive to applied force. Our
+measurements are in excellent agreement with a simple polym er model of loop formation in DNA,
+and show that an anti-parallel topology is the preferred Lac I-DNA loop conformation for a generic
+loop-forming construct.
+1Since Jacob and Monod’s groundbreaking work on gene regulation, t helacoperon has
+become a canonical example of how prokaryotic cells regulate the ex pression of genes in
+response to changes in environmental conditions [1]. The lacoperon is responsible for the
+efficient metabolism of lactose in Escherichia coli bacteria ensuring that enzymes capable
+of digesting lactose are produced only when needed. The lacrepressor-mediated DNA loop,
+which is formed when tetrameric lacrepressor protein binds to two lacoperator sites si-
+multaneously, is an important part of this gene regulatory network and is crucial for the
+repression of lacgenes[2]. Long range genetic regulation by DNA looping, however, is n ot
+unique to the lacoperon, but appears in a variety of contexts within prokaryotes, such as
+thearaorgaloperons and is ubiquitous within eukaryotes [3]. While the biochemistry of
+these processes is generally well understood, the mechanics of th e assembly and breakdown
+of protein-mediated DNAloops has only recently garnered much att ention [4]. In this paper,
+we investigate the role that tension in the substrate DNA plays in the formation and break-
+down of protein-mediated DNA loops, and conclude that loop format ion is acutely sensitive
+to entropic forces on the hundred-femtonewton scale.
+Protein-mediated DNA loop formation is driven by thermal fluctuatio ns in the DNA
+which bring distant operators close enough for loop closure by a pro tein. However, it is quite
+surprising that the magnitude of these fluctuations, which one can estimate as kBT/lp≈80
+fN where lp= 50 nm is the persistence length, is much smaller than the typical pico newton
+forcesthatariseintheintracellularenvironment, frommolecularmo torsorDNA-cytoskeletal
+attachments for example. This observation has led to predictions t hat forces as small as a
+few hundred femtonewtons are sufficient to reduce the loop forma tion rate by more than
+two orders of magnitude [5–7]. Given that the cellular environment is t hought to regularly
+subject DNA-protein complexes to large static or fluctuating forc es, the cell must either use
+mechanical pathways to regulate genetic function, or compensat e for the effects of tension
+to ensure the stable control of gene expression.
+To experimentally study the effects of tension on the kinetics of DNA looping, we used
+optical tweezers inconjunctionwith tethered-particle motion(TP M) measurements to inves-
+tigatetheformationandbreakdownofLacI-mediatedDNAloopsun deraconstantstretching
+force. We report three main results: First, the rate of loop forma tion is extremely sensitive
+to applied tension resulting in a tenfold decrease in loop formation whe n increasing the
+tension from 60 to 183 fN. Second, the lifetime of the looped state a ppears to be completely
+2FIG. 1: (a) Tethered DNA is trapped in the linear region of an o ptical potential (dashed line
+indicates laser focus). The figure also represents a kinetic model of DNA looping. (b) Raw experi-
+mental recording of LacI-mediated DNA looping. The looped/ unlooped threshold is chosen at the
+minimum between the two state distributions displayed in a h istogram of the binned data.
+unaffected by forces as large as 183 fN. Third, our measurements strongly suggest that
+the anti-parallel conformation is the dominant topology of a generic LacI-mediated DNA
+loop [8]. We mechanically attenuate the thermal fluctuations that dr ive loop formation and
+breakdown, and measure the associated changes to the looping an d unlooping rates, by em-
+ploying a variety of optical tweezers that differs from the more con ventional tweezers setup.
+Constant-force axial optical tweezers stretch the molecule awa y from the surface and trap
+the attached microsphere slightly below the laser focus in an approx imately linear region
+of the optical potential. This provides effectively a constant force in the axial direction
+that does not change when the protein binds to or dissociates from the DNA (see Fig. 1a).
+Details of this set-up are described in Ref.[9]. Because of this novel o ptical tweezers setup,
+we have been able to measure the formation and breakdown rates o f LacI-mediated loops as
+3a function of applied tensile force in the femtonewton range.
+The DNA samples used in this study were prepared in a similar way to tha t of other
+TPM experiments [9]. We surface-tethered a 1316-bp ds-DNA molec ule with two symmetric
+lacoperators spaced 305 bp apart and then attached an 800 nm polys tyrene microsphere
+to the other end, which was trapped within the linear regime of the op tical potential. The
+total tension in the DNA was carefully calibrated to account for the applied optical force [9]
+and volume exclusion effects arising from entropic interactions betw een the microsphere and
+the coverslip [10]. The looping and unlooping lifetimes were measured un der four different
+forces: 60 ( ±5), 78 (±6), 121 ( ±9), and 183 ( ±15) fN in the presence of 100 pM of LacI
+protein. In each measurement, the surface-tethered ds-DNA m olecule was stretched by a
+contant force while the CCD camera captured defocused images of the tethered microsphere
+at a frame rate of 100 fps. The looped and unlooped states of the D NA molecule, which
+correspond to different axial positions of the microsphere, can be measured by analyzing the
+resulting images, as shown in Fig. 1b. By directly observing changes in the axial position
+of the microsphere, the temporal resolution for detecting loop fo rmation and breakdown
+events in our experiment is as low as 300 ms, an order of magnitude be tter than conven-
+tional TPM. However, as we decrease the applied tension, it become s increasingly difficult
+to resolve changes in the size of the microsphere, and at zero optic al force, we must resort
+to conventional TPM. Moreover, even in the absence of an optical force, a residual entropic
+force from excluded volume effects remains. For this reason, we we re not able to obtain a
+direct measure of the force free loop formation and breakdown ra tes.
+The data was analyzed by first extracting the elapsed times betwee n loop formation and
+breakdownevents fromtimetracesliketheoneinFig. 1b. Then, for eachforcecondition, the
+lifetime of each state was determined from a fit to the cumulative pro bability distribution,
+as shown in Fig. 2. The resulting distribution displayed by the looped st ate is well fit by a
+single exponential function
+P(t;τ) = 1−e−t/τ, (1)
+with time constant τ. However, the data of the unlooped state is poorly fit to a single
+exponential function, but is well fit to a biexponential function
+P2(t;τ1,τ2) =cP(t;τ1)+(1−c)P(t;τ2), (2)
+with time constants τ1andτ2, and dimensionless fitting parameter c. Results of the fits are
+4FIG. 2: Cumulative probability distributions of the observ ed durations that the DNA molecule
+remains in the looped and unlooped state under increasing fo rce conditions (left to right). The
+data of (a) the looped state and (b) the unlooped state are fit t o the single exponential function
+and a biexponential function respectively.
+shown in Table 1.
+One of the most striking features of the data in Fig. 2 is that the diss ociation time
+constant of DNA loops is unaffected by increasing the force from 60 to 183 fN. This result
+is in contrast to the force dependence of the time necessary to fo rm a loop, which increases
+significantly with only a modest increase in applied tension. To interpre t these observations
+quantitatively, we begin by applying a kinetic model for the underlying processes of protein
+binding, unbinding, loop formation, and breakdown, as illustrated in F ig. 1a. This is the
+simplest model of the kinetics that is both consistent with our data a nd what is currently
+known about LacI mediated looping.
+If we collect all time intervals that start at a loop formation event ( L) and end at a loop
+breakdown event ( S1), then, within this ensemble, simple first-order kinetics are given by
+5the process
+LkU−→S1, (3)
+whereLis the looped state, S1is the state of the DNA with only one operator bound
+to a protein, and kUis the unlooping rate. Therefore, the time dependent probability of
+unlooping is
+S1(t) = 1−e−kUt, (4)
+whichcorrespondstothefitfunction(Eq.1)forthelifetimesofth eloopedstate. Thekinetics
+of loop formation, however, are more complicated because there a re different unlooped sub-
+states that cannot be distinguished within our experiment. We star t by collecting all time
+intervals thatbeginatanunlooping event andendwiththeformation ofa loop. Thekinetics
+may be represented as
+S2k+⇋
+k−S1kL−→L, (5)
+whereS1is the state of one vacant and one occupied operator and may direc tly convert to
+the looped state Lat a rate kL, or remain unlooped and convert to state S2at a rate k−.
+StateS2, however, is an alternate configuration with both or neither opera tor occupied by
+a protein, which is not able to directly form a loop, but may convert to stateS1at a rate
+k+. With the initial condition S1(0) = 1, the first-order kinetics above may be solved for
+the time-dependent probability of forming a loop
+L(t) = 1−1
+2α/bracketleftbig
+(κ−kL+α)e−t/τ1−(κ−kL−α)e−t/τ2/bracketrightbig
+, (6)
+whereκ=k++k−,α= [(κ+kL)2−4k+kL]1/2, and the time constants are defined as
+τ1= 2/(κ+kL−α) andτ2= 2/(κ+kL+α). Equation 6 is again a biexponential distribution
+and corresponds to the fit function of Eq. 2. Therefore, we can u nambiguously extract the
+four rate constants in our kinetic model. The results are shown in Ta ble 2 and plotted in
+Fig. 3. Our main observation is that, within the uncertainties of our m easurements, k+,
+k−, andkUare independent of force whereas kLis acutely force-sensitive on the hundred-
+femtonewton scale. This is consistent with the conventional expec tation that the rate of
+conversion between the unlooped states S1andS2does not vary significantly as a function
+of the stretching force on this scale. The insensitivity to applied for ce of the unlooping rate
+kUcanbeexplained byconsidering thebindingenergyoftheLacIprote intotheDNA,whose
+disassociation from the lacbinding site is necessary to break a loop. With a binding energy
+6TABLE I: Fits to the cumulative probability distributions.
+Force (fN) 60±5 78±6 121±9 183±15
+Looped
+τ(s) 20.8±0.3 22.5±0.2 24.1±0.3 24.8±0.7
+Unlooped
+τ1(s) 3.0±0.2 4.0±0.42 9.7±0.4 14.5±1
+τ2(s) 31±8 54±10 91±10 101±6
+c 0.77±0.03 0.77±0.02 0.74±0.02 0.38±0.03
+of 10−19J [11] and an operator region that spans ∼20 bp, the minimum force needed to
+removetheproteinfromtheoperatoris ∼10pN,whichisseveral ordersofmagnitudegreater
+than the tension we applied. It is clear then why the looped state is re latively insensitive
+to mechanical tension. On the other hand, the sensitivity of the loo ping rate to such small
+forces is quite striking and potentially rich in implications. Since the cha racteristic force
+that results from thermal fluctuations of ds-DNA is approximately 80 fN, and since DNA
+looping is a result of thermal fluctuations, femtonewton forces ca n clearly impact the loop
+formation process.
+Quantitatively useful models of loop formation must explicitly conside r the orientation of
+the operators along the DNA in the looped state, as the exact geom etry of the loop matters
+significantly. Such theories were developed by Blumberg et al.[6] and, independently, by Yan
+et al.[7]. In this paper, we use the model developed by Blumberg et al.so begin by finding
+the difference in the force dependent contributions to the free en ergy between a looped and
+a stretched length of DNA: ∆ F=FL(f,θ)−FS(f). The kink angle θis defined as the angle
+between the tangent vectors of the DNA at the operator sites of the protein-DNA complex.
+A relation for the excess contribution to the free energy as a func tion of kink angle, imposed
+on the DNA by the loop, is given by:
+FL=4f1/2(1−cos(θ/4))
+1+12f−3/2(1−cos(θ/4))/(1+cos( π−θ)), (7)
+where the free energy is in units of kBTand the force fis in units of the characteristic force
+for thermal fluctuations, fc=kBT/lp≈80 fN.
+An analytic relation for the free energy of a stretched segment of DNA is given by the
+7difference between the potential energy of a worm-like chain (WLC) and the work done by
+tension:
+FS=−llx2
+4/parenleftbigg1
+(1−x)2+2/parenrightbigg
+, (8)
+wherellis the loop length of the DNA and xis the relative extension of the DNA in units of
+lp. We can now calculate the characteristic time necessary to form a lo op under an applied
+force using the principle of detailed balance
+τf
+τ0=e−∆F, (9)
+whereτ0is the characteristic time at zero force.
+X-ray studies of the co-crystals of LacI protein bound to short o perator fragments have
+revealed the structure of the DNA protein complex [12]. These resu lts impose constraints
+upon, but do not fully determine the topology of the DNA loop. The pr eferred direction in
+which the DNAenters and leaves the loopedcomplex remains unsettle d, but the correspond-
+ingtopologiesareeitheranti-parallelconformations, withakinkang leofapproximately150◦,
+or parallel conformations, with a kink angle of 30◦.
+To fit the data, we calculate τf= 1/kLfrom Eq. 9 as a function of force using the WLC
+model to provide the relative extension xin Eq. 8. Since we cannot directly measure the
+force free lifetime τ0, we use this as a single adjustable parameter to fit the curves. The
+value for τ0is given by a least squares fit to the data. We then generate a curve for both the
+anti-parallel and parallel conformations and see, from Fig. 3, that the anti-parallel topology
+is more force-sensitive than its parallel counterpart. Our data su ggest that the anti-parallel
+conformation is the dominant topology of a generic LacI-mediated D NA loop. In conclusion,
+our results establish that very small forces, on the order of a hun dred femtonewtons, can
+control the assembly of the regulatory protein-DNA complex nece ssary for expression of
+thelacgene. On the other hand, once formed, the looped complexes are q uite stable
+and cannot easily be disrupted by tension in the substrate DNA, givin g the system much-
+needed robustness. Thus, it appears more than likely that mechan ical pathways can control
+transcription through the application of tiny forces that are gene rated by other intracellular
+processes. We hope that the development of force measurement techniques inside living cells
+will lead to the identification of such pathways. We also conclude that such regulatory forces
+would likely act on the assembly process of these complexes, but not their breakdown.
+8FIG. 3: Measured values for k−(△),k+(⋄),kL(/squaresolid), andkU(◦) (see Table 2). The looping rate
+kLis fit by the theoretical predictions for the anti-parallel ( solid line) and parallel (dashed line)
+topologies illustrated in the insert.
+TABLE II: Rates extracted from kinetic rate equations.
+Force (fN) 60±5 78±6 121±9 183±15
+Kinetic
+Rates
+(10−3/s)kU 48.1±0.6 44.5±0.4 41.5±0.6 40±1
+kL 262±7 196±6 79±1 32±1
+k− 61±9 48±5 21±2 25±4
+k+ 40±10 24±5 14±2 21±3
+WethankJasonKahnforprovidinguswiththeLacIprotein. Thiswor kwassupportedby
+theNationalInstitutesofHealth, GrantNo. GM65934,andtheNa tionalScienceFoundation
+FOCUS, Grant No. 0114336.
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+J. Wilson, H. Zhan, L. Swint-Kruse, K. S. Matthews, Cell. Mol . Life Sci. 64, 3 (2007).
+[2] S. Oehler, E. R. Eismann, H. Kr¨ amer, and B. M¨ uller-Hill , EMBO J. 9, 973 (1990); J. M. G.
+Vilar and S. Leibler, J. Mol. Biol. 331, 981 (2003).
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+[4] L. Finzi and J. Gelles, Science 267, 378 (1995); F. Vanzi, C. Broggio, L. Sacconi, and F. S.
+9Pavone, Nuc. Acids. Res. 34, 3409 (2006); L. Han et al.PLoS ONE 4, e5621 (2009).
+[5] J. F. Marko and E. D. Siggia, Biophys. J. 73, 2173 (1997).
+[6] S. Blumberg, A. V. Tkachenko, and J.-C. Meiners, Biophys . J.88, 1692 (2005); S. Blumberg,
+M. W. Pennington, and J.-C. Meiners, J. Biol. Phys. 32, 73 (2006).
+[7] J. Yan, R. Kawamura, and J. F. Marko, Phys. Rev. E 71, 061905 (2005).
+[8] K. B. Towles, J. F. Beausang, H. G. Garcia, R. Phillips and P. C. Nelson, Phys. Bio. 6, 025001
+(2009).
+[9] Y. F. Chen, G. A. Blab, and J.-C. Meiners, Biophys. J. 96, 4701 (2009).
+[10] Y. F. Chen, D. P. Wilson, K. Raghunathan, and J.-C. Meine rs, Phys. Rev. E 80, 020903(R)
+(2009).
+[11] D. E. Frank, R. M. Saecker, J. P. Bond, M. W. Capp, O. V. Tso dikov, S. E. Melcher, M. M.
+Levandoski, and M. T. Record, J. Mol. Biol. 267, 1186 (1997).
+[12] M. Lewis, G. Chang, N. C. Horton, M. A. Kercher, H. C. Pace , M. A. Schumacher, R. G.
+Brennan, and P. Lu, Science 271, 1247 (1996).
+10
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+ 1 Dispersion supported BB84 quantum key distribution using phase modulated light   J. Mora, A Ruiz, W. Amaya and J. Capmany   iTEAM Research Institute, Universidad Politécnica de Valencia Camino de Vera s/n, 46022 – Valencia (Spain)    ABSTRACT We propose and experimentally demonstrate that, contrary to what it was thought up to now, BB84 operation is feasible using the double phase modulator (PM-PM) configuration in frequency coded systems. This is achieved by exploiting the phase to intensity conversion due to the chromatic dispersion provided by the fiber linking Alice and Bob.  Thus, we refer to this system as dispersion supported or DS BB84 PM-PM configuration.       PACS: 03.67.Dd, 42.50.Ar, 42.79.Hp     2  Quantum cryptography features the unique way of sharing a random sequence of bits between users with a certifiably security not attainable with either public or secret-key classical cryptographic systems. This is achieved by means of quantum key distribution (QKD) techniques, which rely on exploiting the laws of quantum mechanics [1].  QKD deals with the need to distribute a key between a transmitter (Alice) and a receiver (Bob) with complete confidentiality.  If Alice and Bob encode their bits in states of a quantum system, a third party, the eavesdropper (Eve) interested in gaining access to the information they share will modify the quantum state and therefore will destroy its information.  At the same time, the eavesdropper will neither be able to keep a perfect copy of the sequence nor to send it again in order to avoid being detected, as this is not allowed by the non-cloning theorem [2].  Photonics is the principal enabling technology for long distance QKD using optical fiber links. A particularly interesting approach is the so called frequency coding technique proposed by Merolla and co-workers [3] which relies on encoding the information bits on the sidebands of either phase [4] or amplitude [5] radiofrequency (RF) modulated light.  In essence, Alice randomly changes the phase of the electrical signal used to drive a light modulator among four phase values (0, π) and (π/2, 3π/2), which form a pair of conjugated basis.  When arriving to Bob, he modulates the signal again using the same microwave signal frequency and thus his new sidebands will interfere with those created by Alice [3].  The frequency coded approach has the additional added value that its capacity can be upgraded by adding more microwave  3 subcarriers, an approach which is known as Subcarrier Multiplexed Quantum Key Distribution (SCM-QK) [6]-[8]  The original proposal by Merolla and co-workers [5] based on the use of a pair of simple phase modulators (PM-PM configuration) was, in principle, suited only for the implementation of the B92 protocol [9]. In fact, to demonstrate the implementation of the BB84 protocol it had to be modified by replacing the phase modulators by amplitude modulators (AM-AM configuration) [5,10], which, in addition, need to be properly biased for successful operation.   In this letter we demonstrate, contrary to what it was thought up to now, that BB84 operation is feasible using the PM-PM configuration. This is achieved by exploiting the phase to intensity conversion due to the chromatic dispersion provided by the fiber linking Alice and Bob.  Thus, we refer to this system as dispersion supported or DS PM-PM configuration. This result is important for two reasons: first of all because it allows to implement the BB84 protocol rather than the insecure B92 protocol using the simplest frequency coded configuration (PM do not require bias voltage) and secondly because the conditions under which this is possible are compatible with those to be found in practice, since Alice and Bob will be, in general separated by a dispersive optical fiber link.  The system under consideration is shown schematically in Fig. 1. We briefly recall its operation [3]. Assuming that Alice’s transmitter contains a monochromatic optical source that emits photons with an angular frequency ωo. The optical source is externally modulated by a phase modulator (PM1) which is fed by means of a local oscillator OL1.  4 The local oscillator gives an RF signal of frequency Ω in which Alice can introduce a random phase shift ΦA to encode the binary secret key (0 for bit “0” and p for bit “1”).  Therefore, the output signal of Alice’s transmitter can be described when low modulation index mA is considered as:      (1)   Where Eo is the amplitude of the electrical signal related with the average number of photons per bit and tA represents the optical losses of the PM1. After propagation through a fiber link of length L, the light is again externally modulated by another PM2 at Bob’s receiver. PM2 is driven also with a frequency Ω but with variable phase ΦB coming from an OL2. Then, the output optical signal after PM2 is given by:   EBOBt()=Eoe−jωot⋅ejβoLe−αLtAtB×1+jmA2ej12β2LΩ2⋅cosΩt+Φo+ΦA()⎧⎨⎩⎫⎬⎭1+jmB⋅cosΩt+ΦB()⎧⎨⎩⎫⎬⎭ (2)  where α is the optical fiber losses, tB is the optical losses of the PM2 and mB is the modulation index of the PM2. From equation (2), we can observe that fiber chromatic dispersion β2 introduces a phase factor Φo=β2L after propagation between the sidebands and the carrier of the optical signal coming from Alice’s transmitter.  The final stage in the system is composed of two optical filters centered at ωo+Ω and ωo-Ω in order to measure the intensity of each optical sideband carrying the secret key.  5 The output of each filter is then sent to a photon counter. Therefore, the optical power normalized for each sideband is given by: Pωo+Ω()=121+VcosΦB−ΦA()⎡⎣⎤⎦ Pωo−Ω()=121+VcosΦB−ΦA+β2LΩ2()⎡⎣⎤⎦ (3)  Where the parameter V corresponds with visibility of each band, which can be written as:  (4)  From equation (3), we observe that both sidebands have the same value when the fiber dispersion is negligible or compensated, i.e., when the term  can be considered close to zero or negligible, as assumed, for instance in [3]. Therefore, the BB84 protocol cannot be implemented since the amplitudes of both sidebands are always equal regardless of which particular base is chosen. However, we can see that the BB84 protocol can be implemented when the visibility is unity () and the following condition is fulfilled:   (5)  Since then, equation (3) can be written as [5]:   (6)  6   Equation (5) provides a design criterium for the system. If the overall link dispersion β2L is fixed, then (5) gives the required value of the subcarrier frequency. On the other hand, if Ω is fixed, then from (5) we get the value of the minimum required link dispersion.   Implementing the QKD system with a pair of phase modulators brings up several advantages in terms of cost and system complexity. For instance phase modulators are cheaper than amplitude modulators and there is no need to fulfil the counterphase bias condition required when using two amplitude modulators. This point is certainly important since the modulator bias voltage tends to drift with time and keeping this condition would require in practice an additional simultaneous bias tracking circuitry. In addition, we can consider this configuration as a security upgrade since only the BB84 protocol is implemented between Alice and Bob when (5) is fulfilled. For example if  Ω and β2 are fixed and Alice and Bob are separated by a distance L=πβ2Ω2 then any attempt of attack made by Eve at a point 0<z<L between Alice and Bob will result in her retrieving information with an increased QBER(z) as compared to that of a standard BB84 protocol. In fact, a simple computation yields the value:         (7) Where  dB is the detector dark count probability and the contrast C is defined as:    7        (8)  Note that the contrast takes its maximum value which corresponds with the visibility V for  z=L. As an example, figure 2 shows the evolution of the QBER for the DS PM-PM configuration as a function of the link distance z normalized to the total link length L for the case where , dB=8.10-6 and V=98%. For the sake of comparison a broken-trace curve is included showing the QBER evolution for the BB84 system based on the AM-AM configuration [merolla] and negligible dispersion. It becomes clear that the dispersion supported scheme converges to the standard BB84 system performance for z=L.  In order to validate the proposal, we have  tested  the QKD system under classical operation regime (similar results are to be obtained when attenuating the signals as long as they can be represented by coherent states). The experimental demonstration was implemented with an optical laser delivering an output power of 5 dBm at 1550 nm. Both modulators, PM1 and PM2, were electrooptic phase modulators with a 20 GHz electrical bandwidth and a 2.5 dB optical insertion losses. The half-wave voltage was 7.4 V.  The modulation index was 0.35 with a RF signal of 15 GHz. The fiber link had a length around 15 km to comply with equation (5) with n=1,  and the spectra were recorded using an optical spectrum analyzer (OSA).  Figure 3 shows the power spectrum density measured at the OSA for three different fiber lengths when Bob chooses the correct base for the same state that Alice (left) or  8 the orthogonal state (right) which corresponds with a phase difference of 0 or π, respectively. In Fig.3 (a), we can observe that BB84 protocol can be implemented when the condition  is experimentally satisfied. According to expression (4), the optical sidebands  and appear or not depending to the constructive o destructive interference imposed by the phases chosen between Alice and Bob. However, the contrast of the interference is reduced when the length of the fiber link is far from that required by Equation (5) as shown in Figs. 3(b) and 3(c) which plot similar results for a fiber lengths of 7.3 km or a few of meters respectively. Due to the optical fiber losses, the optical power of the optical carrier  is -0.7, -2.6 and -4.0 dBm, respectively.  The theoretical and experimental values of the amplitudes of the detected sidebands are plotted in figures 4.a (upper sideband) and 4.b (lower sideband) respectively as a function of the phase mismatch between Alice and Bob’s RF modulating signals. Results are included for the three different fiber link lengths previously considered. An excellent agreement can be observed between theoretical and experimental results. Furthermore, only for the case where the link length is that designed to fulfill with the condition imposed by Eq (5) one can appreciate the complementary characteristic of the sideband amplitudes, which is a distinctive feature of the BB84 operation.   As an additional supporting experimental evidence, figure 5 plots the theoretical and experimental evolution of the contrast function given by (8). Notice again the excellent agreement between both results.    9 In summary, we have proposed and demonstrated, that by exploiting the phase to intensity conversion, which takes place in an optical fiber link due to chromatic dispersion, BB84 operation is feasible using the PM-PM configuration, which was thought to be valid only up to now for the implementation of the B92 protocol. Experimental results have been provided to support our proposal showing an excellent agreement with the theoretical predictions.    ACKNOWLEDGMENTS  The authors wish to thank  the Spanish Government support through Quantum Optical Information Technology, a CONSOLIDER-INGENIO 2010 Project, and the Generalitat Valenciana through the PROMETEO 2008/092 Research Excellency Award  10 REFERENCES  [1] N. Gisin, G. Ribordy, W. Tittel and H. Zbiden, Rev. Mod. Phys., vol 74, Nº1, p. 145 (2002) [2] W.K. Wootters and W.H. Zurek, Nature, vol 299, p. 802 (1982) [3] J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, and W. T.Rhodes, Phys.Rev. Lett. 82, p. 1656 (1999). [4] J-M. Mérolla, Y. Mazurenko, J. P. Goedgebuer, H. Porte and W. T.Rhodes, Opt. Lett. 24, p. 104 (1999). [5] O. Guerreau, J-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F.J. Malassenet, J. of Selected Topycs in Quantum Electron., 9, p. 1533 (2003). [6] A. Ortigosa-Blanch and J. Capmany, ‘ Phys. Rev. A, (024305), 2006 [7] J. Capmany, Optics Express, vol. 17, no. 8, pp. 6457–6464, April 2009. [8] J. Capmany, A. Ortigosa-Balnch, J. Mora, A. Ruiz, W. Amaya and A. Martínez, J. of Selected Topycs in Quantum Electron., 15, p. 1607 (2009).  [9] C.H. Bennett., Phys. Rev. Lett. 68, p. 3121 (1992). [10] J-M. Mérolla, L. Duraffourg, J. P. Goedgebuer, A. Soujaeff, F. Patois and W. T.Rhodes, Eur. Phys. J. D 18, p. 141 (2002).  11 FIGURE CAPTIONS  Figure 1. Scheme of the frequency coded system using a pair of phase modulators (PM-PM configuration)). Alice an Bob are separated by a dispersive link of length L.  Figure 2. (Solid trace) Evolution of the QBER for the DS PM-PM configuration as a function of the link distance z normalized to the total link length L for  and V=98%. (Broken-trace) QBER evolution for the BB84 system based on the AM-AM configuration [merolla] and negligible dispersion  Figure 3. Power spectra for a fiber link with a length (a) 15 km, (b) 7.3 km and (c) 0 km . Left column results are for ΦA−ΦB=0. Right column results are for ΦA−ΦB=π  Figure 4. Amplitude of sidebands for different phase shifts between Alice and Bob after 15 km (■), 7.3 km (●) y 0 km (▲): (a) left optical sideband and (b) right optical sideband. (Solid traces represent theoretical results) .  Figure 5. Theoretical and experimental evolution of the contrast function given by (8)   12        Figure 1. Scheme of the frequency coded system using a pair of phase modulators (PM-PM configuration)). Alice an Bob are separated by a dispersive link of length L                13   
+ Figure 2. (Solid trace) Evolution of the QBER for the DS PM-PM configuration as a function of the link distance z normalized to the total link length L for  and V=98%. (Broken-trace) QBER evolution for the BB84 system based on the AM-AM configuration [5] and negligible dispersion         14  
+  Figure 3. Power spectra for a fiber link with a length (a) 15 km, (b) 7.3 km and (c) 0 km . Left column results are for ΦA−ΦB=0. Right column results are for ΦA−ΦB=π  15  
+ 
+    Figure 4. Amplitude of sidebands for different phase shifts between Alice and Bob after 15 km (■), 7.3 km (●) y 0 km (▲): (a) left optical sideband and (b) right optical sideband. (Solid traces represent theoretical results)    16  
+ Figure 5. Theoretical and experimental evolution of the contrast function given by (8)  
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+arXiv:1001.0548v1  [math.AC]  4 Jan 2010PROOF OF THE COMBINATORIAL NULLSTELLENSATZ OVER
+INTEGRAL DOMAINS IN THE SPIRIT OF KOUBA
+PETER HEINIG
+Abstract. It is shown that by eliminating duality theory of vector spac es from
+a recent proof of Kouba (O. Kouba, A duality based proof of the Combinatorial
+Nullstellensatz. Electron. J. Combin. 16 (2009), #N9) one o btains a direct
+proof of the nonvanishing-version of Alon’s Combinatorial Nullstellensatz for
+polynomials over an arbitrary integral domain. The proof re lies on Cramer’s
+rule and Vandermonde’s determinant to explicitly describe a map used by
+Kouba in terms of cofactors of a certain matrix.
+That the Combinatorial Nullstellensatz is true over integr al domains is a
+well-known fact which is already contained in Alon’s work an d emphasized in
+recent articles of Micha/suppress lek and Schauz; the sole purpose of the present note is
+to point out that not only is it not necessary to invoke dualit y of vector spaces,
+but by not doing so one easily obtains a more general result.
+Mathematics Subject Classification 2010: 13G05, 15A06
+1.Introduction
+The Combinatorial Nullstellensatz is a very useful theorem (see [ 1]) about mul-
+tivariate polynomials over an integral domain which bears some resem blance to the
+classical Nullstellensatz of Hilbert.
+Theorem1 (Alon, Combinatorial Nullstellensatz (ideal-containment-version), The-
+orem 1.1 in [ 1]).LetKbe a field, R⊆Ka subring, f∈R[x1,...,x n],S1,...,S n
+arbitrary nonempty subsets of K, andgi:=/producttext
+s∈Si(xi−s)for every 1≤i≤n. If
+f(s1,...,s n) = 0for every (s1,...,s n)∈S1×···×Sn, then there exist polynomi-
+alshi∈R[x1,...,x n]with the property that deg(hi)≤deg(f)−deg(gi)for every
+1≤i≤n, andf=/summationtextn
+i=1higi.
+Theorem 2 (Alon, Combinatorial Nullstellensatz (nonvanishing-version), Theo rem
+1.2 in [ 1]).LetKbe a field, R⊆Ka subring, and f∈R[x1,...,x n]. Letc·
+xd1
+1···xdnnbe a term in fwithc/\e}atio\slash= 0whose degree d1+···+dnis maximum among
+all degrees of terms in f. Then every product S1× ··· ×Sn, where each Siis an
+arbitrary finite subset of Rsatisfying |Si|=di+ 1, contains at least one point
+(s1,...,s n)withf(s1,...,s n)/\e}atio\slash= 0.
+Three comments are in order. First, talking about subrings of a field is equivalent
+to talking about integral domains: every subring of a field clearly is an integral
+domain, and, conversely, every integral domain Ris (isomorphic to) a subring of
+its field of fractions Quot( R). Second, strictly speaking, rings are mentioned in [ 1]
+only in Theorem 1, but Alon’s proof in [ 1] of Theorem 2is valid for polynomials
+The author was supported by a scholarship from the Max Weber- Programm Bayern and by
+the ENB graduate program TopMath.
+12 PETER HEINIG
+over integral domains as well. Third, it is intended that the Siare allowed to be
+subsets of Kin Theorem 1but required to be subsets of Rin Theorem 2, since this
+is the slightly stronger formulation: if Theorem 2is true as it is formulated here,
+then by invoking it with R=Kand by viewing an f∈R[x1,...,x n],Rbeing a
+subring of K, as a polynomial in K[x1,...,x n], it is true as well with the Sibeing
+allowed to be arbitrary subsets of K.
+In [1], Theorem 2was deduced from Theorem 1. In [ 3], Kouba gave a beautifully
+simple and direct proof of the nonvanishing-version of the Combinat orial Nullstel-
+lensatz, bypassing the use of the ideal-containment-version. Kou ba’s argument was
+restricted to the case of polynomials over a field and at one step app lied a suitably
+chosen linear form on the vector space K[x1,...,x n] to the given polynomial fin
+Theorem 2.
+However, for Kouba’s idea to work, it is not necessary to have reco urse to duality
+theory of vector spaces and in the following section it will be shown ho w to make
+Kouba’s idea work without it, thus obtaining a direct proof of the full Theorem 2.
+Finally, two relevant recent articles ought to be mentioned. A very s hort di-
+rect proof of Theorem 2was given by Micha/suppress lek in [ 5] who explicitly remarks that
+the proof works for integral domains as well. Moreover, the differe nces/braceleftbig
+s−s′:
+{s,s′} ∈/parenleftbigSk
+2/parenrightbig/bracerightbig
+in the proof below play a similar role in Micha/suppress lek’s proof. In [ 6],
+Schauz obtained far-reaching generalizations and sharpenings of Theorem 2, ex-
+pressly working with integral domains and generalizations thereof t hroughout the
+paper.
+2.Proof of Theorem 2
+The proof of the Theorem 2will be based on the following simple lemma.
+Lemma 3. LetRbe an integral domain. Let S={s1,...,s m} ⊆Rbe an arbitrary
+finite subset. Then there exist elements λ(S)
+1,...,λ(S)
+mofRsuch that
+λ(S)
+1·(1,s1,s2
+1,...,sm−1
+1) +···+λ(S)
+m·(1,sm,s2
+m,...,sm−1
+m)
+= (0,0,0,..., 0,/productdisplay
+1≤i<j≤m(si−sj)). (1)
+Proof. Let [m] :={1,...,m}. Define bto be the right-hand side of the claimed
+equation, taken as a column vector, and let A= (aij)(i,j)∈[m]2be the Vandermonde
+matrix defined by aij:=si−1
+j. Then the statement of the lemma is equivalent to
+the existence of a solution λ(S)∈Rmof the system of linear equations Aλ(S)=b.
+By the well-known formula for the determinant of a Vandermonde ma trix (see [ 4],
+Ch. XIII, §4, example after Prop. 4.10), det( A) =/producttext
+1≤i<j≤m(si−sj).
+SinceSis a set, all factors of this product are nonzero, and since Rhas no zero
+divisors, the determinant is therefore nonzero as well. Now let αijbe the cofactors
+ofA, i.e.αij:= (−1)i+jdet(A(ij)), where A(ij)is the (m−1)×(m−1) matrix
+obtained from Aby deleting the i-th row and the j-th column (see [ 2], Ch. IX,
+§3, before Lemma 1). By Cramer’s rule (see Ch. IX, §3, Corollary 2 of Theorem 6
+in [2] or Theorem 4.4 in [ 4]), for every j∈[m],
+det(A)·λ(S)
+j=m/summationdisplay
+i=1αijbi.NULLSTELLENSATZ OVER INTEGRAL DOMAINS 3
+Usingbm= det(A),bi= 0 for every 1 ≤i < m , and the commutativity of an
+integral domain, this reduces to
+det(A)·/parenleftbig
+λ(S)
+j−αmj/parenrightbig
+= 0.
+Hence, since det( A)/\e}atio\slash= 0 and Rhas no zero divisors, if follows that the cofactors
+λ(S)
+j=αmj∈Rprovide explicit elements with the desired property. /square
+Using this lemma, Kouba’s argument may now be carried out without ch ange in
+the setting of integral domains.
+Proof of Theorem 2.LetRbe an arbitrary integral domain and f∈R[x1,...,x n]
+be an arbitrary polynomial. Let d1,...,d n∈N≥0be the exponents of a term
+c·xd1
+1···xdnnwithc/\e}atio\slash= 0 which has maximum degree in f. For each k∈[n],
+choose an arbitrary finite subset Sk⊆Rand apply Lemma 3withS=Skand
+m=|S|=dk+ 1 to obtain a family of elements ( λ(Sk)
+sk)sk∈SkofR(where in order
+to avoid double indices the coefficients λare now being indexed by the elements of
+Skdirectly, not by an enumeration of each Sk) with the property that
+/summationdisplay
+sk∈Skλ(Sk)
+sk·sℓ
+k= 0 for every ℓ∈ {0,...,d k−1}, (2)
+/summationdisplay
+sk∈Skλ(Sk)
+sk·sdk
+k=/productdisplay
+{s,s′}∈(Sk
+2)(s−s′) =:rk∈R\{0}. (3)
+Using the coefficient families ( λ(Sk)
+sk)sk∈Sk, define, ` a la Kouba, the map
+Φ :R[x1,...,x n]−→R
+g/mapsto−→/summationdisplay
+(s1,...,sn)∈S1×···×Snλ(S1)
+s1···λ(Sn)
+sn·g(s1,...,s n). (4)
+Due to the commutativity of an integral domain, Φ is an R-linear form on the
+R-module R[x1,...,x n], hence its value Φ( f) on a polynomial fcan be evaluated
+termwise as
+Φ(f) =/summationdisplay
+c·ta term in fc·Φ(t). (5)
+Ift=c·xd′
+1
+1···xd′
+nnis an arbitrary term in R[x1,...,x n], then
+Φ(t) =c·Φ(xd′
+1
+1···xd′
+nn) =c·/summationdisplay
+(s1,...,sn)∈S1×···×Snλ(S1)
+s1···λ(Sn)
+sn·sd′
+1
+1···sd′
+nn
+=c·/summationdisplay
+s1∈S1···/summationdisplay
+sn∈Snλ(S1)
+s1···λ(Sn)
+sn·sd′
+1
+1···sd′
+nn
+=c·n/productdisplay
+k=1/parenleftbigg/summationdisplay
+sk∈Skλ(Sk)
+sksd′
+k
+k,/parenrightbigg
+(6)
+where in the last step again use has been made of the commutativity o f an integral
+domain. By ( 6) and ( 2) it follows that for every term t, if there is at least one
+exponent d′
+iwithd′
+i< di, then Φ( t) = 0. Moreover, by the choice of the term
+c·xd1
+1···xdnn, every term c′·xd′
+1
+1···xd′
+nnoffwhich is different from the term c·4 PETER HEINIG
+xd1
+1···xdnnmust, even if it has itself maximum degree in f, contain at least one
+exponent d′
+iwithd′
+i< di. Therefore
+/summationdisplay
+(s1,...,sn)∈S1×···×Snλ(S1)
+s1···λ(Sn)
+sn·f(s1,...,s n)(4)= Φ(f)(2),(6)=c·Φ(xd1
+1···xdn
+n) =
+(3),(6)=c·n/productdisplay
+k=1/productdisplay
+{s,s′}∈(Sk
+2)(s−s′) =c·n/productdisplay
+k=1rk/\e}atio\slash= 0, (7)
+sinceRhas no zero divisors. Obviously this implies that there exists at least o ne
+point (s1,...,s n)∈S1×···×Snwherefdoes not vanish. /square
+3.Concluding question
+Is there any interesting use for the fact that even in the case of in tegral domains
+the coefficients of Kouba’s map can be explicitly expressed in terms of cofactors of
+the matrices ( si−1
+j)?
+Acknowledgement
+The author is very grateful to the department M9 of Technische U niversit¨ at
+M¨ unchen for excellent working conditions.
+References
+1. N. Alon, Combinatorial Nullstellensatz , Combin. Probab. Comput. 8(1999), no. 1, 7–29.
+2. G. D. Birkhoff and S. Mac Lane, Algebra , 3. ed., American Mathematical Society, 1987.
+3. O. Kouba, A duality based proof of the Combinatorial Nullstellensatz , Electron. J. Combin. 19
+(2009), #N9.
+4. S. Lang, Algebra , 3. ed., Graduate Texts in Mathematics, vol. 211, Springer, 2002.
+5. M. Micha/suppress lek, A short proof of Combinatorial Nullstellensatz , arXiv:0904.4573v1 [math.CO]
+(2009).
+6. U. Schauz, Algebraically Solvable Problems: Describing Polynomials as Equivalent to Explicit
+Solutions , Electron. J. Combin. 15(2008), #R10.
+Zentrum Mathematik, Lehr- und Forschungseinheit M9 f ¨ur Angewandte Geometrie
+und Diskrete Mathematik, Technische Universit ¨at M¨unchen, Boltzmannstraße 3, D-
+85747 Garching bei M ¨unchen, Germany
+E-mail address :heinig@ma.tum.de
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+arXiv:1001.0549v1  [math.GN]  4 Jan 2010REMARKS ON NONMEASURABLE UNIONS OF BIG POINT
+FAMILIES
+ROBERT RA/suppress LOWSKI
+Abstract. We show that under some conditions on a family A ⊂Ithere
+exists a subfamily A0⊂ A such that/uniontextA0is nonmeasurable with respect to
+a fixed ideal Iwith Borel base of a fixed uncountable Polish space. Our resul t
+applies to the classical ideal of null subsets of the real lin e and to the ideal of
+first category subsets of the real line.
+1.Introduction
+We use standard set theoretical notations. Among others, we de note byP(X)
+the family of all subsets of the set X. We denote by |X|the cardinality of the
+setX. We denote by [ X]<ωthe set of all finite subsets of X, by [X]ωthe set of
+all countable subsets of Xand finally by [ X]≤ωthe set of all at most countable
+subsets of X. We denote by Rthe real line. If Xis a topological space then we
+denote by B( X) the family of all Borel subsets of X. Suppose that Iis aσ-ideal of
+subsets of X. We denote by B[ I] the least σ-field containing B( X)∪I. Notice that
+B[I] ={B△A:B∈B(X)∧A∈I}. We say that the set A⊂Xis measurable
+with respect to the ideal IiffA∈B[I].
+LetIbe an ideal of subsets of a topological space X. We say that Ihas aBorel
+base if for any A∈Ithere exists a Borel subset BofXsuch that B∈Iand
+A⊆B. The ideal Kof the first category subsets of the real line Rhas a Borel
+base, since every set from Kcan be covered by an Fσsubset from K. The ideal
+Lof Lebesgue measure subsets of the real line has a Borel base too , since every
+measure zero subset of the real line can be covered by a Gδsubset of measure zero.
+The family B[ L] is the family of Lebesgue measurable subsets of the real line and
+B[K] coincides with the family of all subsets of the real line which has the B aire
+property.
+Definition 1.1. A pair (X,I)is aPolish ideal space ifXis an uncountable
+Polish space, I⊆P(X)is a proper σ-ideal with a Borel base and [X]≤ω⊆I. A
+triple (G,I, +)isPolish ideal group if(G,I)is a Polish ideal space, (G,+)is
+abelian topological group and the ideal is translation inva riant.
+Let (X,I) be a Polish ideal space. We denote by B +(I) the family B( X)\I.
+Notice that if A∈B+(I) then|A|= 2ω.
+Definition 1.2. LetX= (X,I)be a Polish ideal space. We say that a set A⊆X
+is completely X-nonmeasurable if
+(∀B∈B+(I))(A∩B/\e}atio\slash=∅∧Ac∩B/\e}atio\slash=∅).
+1991Mathematics Subject Classification. Primary 03E35, 03E75; Secondary 28A99.
+Key words and phrases. Lebesgue measure, Baire property, Cantor set, packing dime nsion,
+algebraic sum.
+12 ROBERT RA/suppress LOWSKI
+Motivation of the above definition is as follows: let A⊂Rbe any Lebesgue
+measurable set then we can find a two Borel sets let say B1,B2∈B(R) such that
+B1⊂A⊂B2withB2\B1∈L. Then the inner Lebesgue measure is equal to outer
+Lebesgue measure. But in a case when C⊂Ris completely ( R,L)-nonmeasurable
+set then the set Chas the inner Lebesgue measure equal to 0 and the set Chas
+full outer Lebesgue measure so the set Cis not Lebesgue measurable.
+LetXbe an uncountable Polish space and let X= (X,[X]≤ω). Then Xis a
+Polish ideal space. A set A⊆Xis completely X-nonmeasurable set if and only if
+Ais a classical Bernstein set. Therefore the above definition is a gene ralisation of
+a classical property.
+An ideal Iis c.c.c. if every family of pairwise disjoint non-empty I-positive Borel
+sets is countable. Now let ( X,I) be a Polish ideal space with Ic.c.c. and A⊆X.
+LetAbe a maximal family of pairwise disjoint I-positive Borel sets contained in
+Ac.SetB= (/uniontextA)c.ThenBis Borel, A⊆Band for every Borel set C⊇A,
+B\C∈I.Any such set Bis called a Borel envelope ofAand will be denoted by
+[A]I.Note that a Borel envelope of Ais unique modulo Iand it is minimal (modulo
+I) Borel set containing A.
+Definition 1.3. LetAbe a family of subsets of a Polish ideal space (X,I). Then
+(1)add(I) = min{|C|:C ⊂I∧/uniontextC/∈I}.
+(2)cov(A) = min{|B|:B ⊂ A∧/uniontextB=X},
+(3)covh(A) = min{|B|:B ⊂ A∧ (∃B∈B+(X))(B⊂/uniontextB)}.
+Notice that if Iis an ideal and A ⊆Ithenadd(I)≤covh(A)≤cov(A). It is
+easy to see that covh(K) =cov(K) andcovh(L) =cov(L).
+Suppose that ( G,+) is and abelian group A,B⊆Gandg∈G. Then we put
+A+g={a+g:a∈A}andA+B={a+b∈G:a∈A∧b∈B}. We call the
+setA+Bthe algebraic sum of the sets AandB.
+2.Nonmeasurable unions of null sets
+Suppose that Ais a family of subsets of a set Xand letx∈X. We put
+Ax={A∈ A:x∈A}. We say that a family A ⊆P(X) ispoint-finite if|Ax|< ω
+for each x∈X. The classical Four Poles Theorem (see [1]) says that if ( X,I) is a
+Polish ideal space and A ⊆Iis a point-finite family such that/uniontextA=Xthen there
+exists a subfamily B ⊆ A that/uniontextBis (X,I)-nonmeasurable. The problem of the
+existence of subfamilies with a completely nonmeasurable unions was d iscussed in
+[10], where a general theorem was proved for ccc σ-ideals with Borel base under
+the assumption of the non existence of quasi-measurable cardinal less or equal
+than continuum. In the ZFC theory the above problem was solved un der some
+assumption on regularity of the family A(see [3]). The complete nonmeasurability
+of the union of ,,small-point” family of subset was investigated in [2]. In t his section
+we consider the unions of ,,big-point” families of the subsets of Polish id eal spaces.
+Theorem 2.1. Let(X,I)be a Polish ideal space. Suppose that a family A ⊆Ihas
+the following properties:
+(1) (∀x∈X)(|Ax|= 2ω),
+(2) (∀x,y∈X) (x/\e}atio\slash=y→ |Ax∩Ay| ≤ω),
+(3)covh(A) = 2ω.
+Then there exists a subfamily A0⊆ Asuch that/uniontextA0is a completely (X,I)-
+nonmeasurable set.REMARKS ON NONMEASURABLE UNIONS OF BIG POINT FAMILIES 3
+Proof. Let B +(I) ={Bα:α < 2ω}. We will build a sequence (( Aξ,dξ))ξ<2ωsuch
+that for all ξ <2ωwe have Aξ∈ A,dξ∈Bξand
+(1) (∀ξ <2ω)(Aξ∩Bξ/\e}atio\slash=∅),
+(2){dξ:ξ <2ω}∩/uniontext
+ξ<2ωAξ=∅.
+Suppose that α < 2ω, letDα={dξ:ξ < α}and that we have built a sequence
+((Aξ,dξ))ξ<αsuch that for all ξ < α we haveAξ∈ A,dξ∈Bξ, (∀ξ < α )(Aξ∩Bξ/\e}atio\slash=
+∅) and{dξ:ξ < α} ∩/uniontext
+ξ<αAξ=∅. Letdα∈Bα\/uniontext
+ξ<αAξwhat is possible by
+assumption (3) and x0∈Bα\(Dα∪{dα}). From assumption (2) we get
+(∀ξ≤α)(|Ax0∩Adξ| ≤ω),
+so, from assumption (1), we get
+Ax0\/uniondisplay
+ξ≤α(Ax0∩Adξ)/\e}atio\slash=∅.
+Let us fix any Aα∈ Ax0\/uniontext
+ξ≤α(Ax0∩Adξ). Then {dξ:ξ≤α} ∩Aα=∅. This
+show that there exists a sequence satisfying properties (1) and ( 2).
+Finally, let us put A0={Aξ∈ A :ξ <2ω}. LetBbe any set from B +(X).
+Then there exists ξ <2ωsuch that B=Bξ. From (1) we deduce that Bξ∩/uniontextA0/\e}atio\slash=∅
+and from (2) we deduce that Bξ\/uniontextA0/\e}atio\slash=∅. Therefore/uniontextA0is a completely ( X,I)-
+nonmeasurable set. /square
+Theorem 2.2. Let(X,I)be a Polish ideal space. Suppose that a family A ⊂I
+satisfies the following conditions:
+(1)/uniontextA=X,
+(2) (∀x,y∈X)(x/\e}atio\slash=y→ |Ax∩Ay| ≤ω),
+(3)covh(A) = 2ω.
+Then there exists a subfamily A0⊂ Asuch that/uniontextA0is(X,I)-nonmeasurable.
+Proof. LetB+(I) ={Bξ:ξ <2ω}. ForZ⊂Xwe putAZ={A∈ A:A∩Z/\e}atio\slash=∅}.
+Let us consider the two alternative possibilities:
+P1: (∀D⊂X)(|D|<2ω→/uniontextAD∈I),
+P2: (∃D⊂X)(|D|<2ω∧/uniontextAD/∈I).
+Case P1 . We will construct by the transfinite induction a sequence (( Aα,dα))α<2ω
+with the following properties:
+(1) (∀α <2ω)(Aα,dα)∈(A×Bα),
+(2) (∀α <2ω)(Aα∩Bα/\e}atio\slash=∅),
+(3) (∀α <2ω)({dξ:ξ < α}∩/uniontext
+ξ<αAξ=∅).
+Suppose that α <2ωand that we have constructed a partial sequence (( Aξ,dξ))ξ<α
+with satisfies the above conditions (restricted to α). LetDα={dξ:ξ < α}. Then
+|Dα| ≤ |α|<2ω. Therefore by the condition P1we have/uniontextADα∈I. Let us fix
+x0∈Bα\/uniontextADα/\e}atio\slash=∅. ThenAx0∩ ADα=∅andAx0/\e}atio\slash=∅. Let us choose any
+Aα∈ Ax0. Notice that Dα∩Aα=∅. Finally, let us choose dα∈Bα\/uniontext
+ξ<α+1Aξ.
+Then the sequence (( Aξ,dξ))ξ<α+1also satisfies the the above three conditions.
+This show that a required sequence exists. It is easy to check that the union of the
+family{Aα:α <2ω}is (X,I)-nonmeasurable.
+Case P2 . LetD⊆Xbe a subset of Xsuch that |D|<2ωand/uniontextAD/∈I. If/uniontextADis (X,I)-nonmeasurable then proof is finished. Suppose hence that the s et/uniontextADis measurable. The there are two possibilities: D/\e}atio\slash∈IorD∈I.4 ROBERT RA/suppress LOWSKI
+Suppose first that D /∈I. ThenDis a nonmeasurable set, since otherwise D
+would contain some perfect set, which is impossible ( |D|<2ω). For each d∈D
+we choose some Ad∈ A such that d∈Adand we put A0={Ad:d∈D}. Then
+D⊆/uniontextA0, so/uniontextA0/∈I. Moreover |A0| ≤ |D|<2ω=covh(A), so for every
+B∈B+(I) we have B\/uniontextA0/\e}atio\slash=∅. This implies that the set/uniontextA0is (X,I)-
+nonmeasurable.
+Suppose now that D∈I. Without loss of generality we may assume that
+A=AD. LetZ=/uniontextADand let{Bξ:ξ <2ω}= (P(Z)∩B(X))\I. We shall
+build a sequence (( Aα,dα))α<2ωsuch that:
+(1) (∀ξ <2ω)(Aξ∈ A∧dξ∈Bξ\D),
+(2) (∀ξ <2ω)(Aξ∩Bξ/\e}atio\slash=∅),
+(3) (∀ξ <2ω)({dβ:β < ξ}∩/uniontext
+β<ξAβ=∅).
+Suppose that α < 2ωand that a sequence (( Aξ,dξ))ξ<αsatisfies the above three
+conditions. Let us observe that for any ξ < α we have |Adξ| ≤ |/uniontext
+d∈DAd∩Adξ| ≤
+|D|, so|A{dξ:ξ<α}| ≤ |α| · |D|<2ω. Therefore Bα\(D∪/uniontext/uniontext
+ξ<αAdξ)/\e}atio\slash=∅. Let
+us choose any x0∈Bα\(D∪/uniontext/uniontext
+ξ<αAdξ) andAα∈ Ax0. Then Aα∩ {dξ:
+ξ < α}=∅and{dξ:ξ < α} ∩/uniontext
+ξ<α+1Aξ=∅. Finally, let us choose any
+dα∈Bα\(D∪/uniontext
+ξ<α+1Aξ). Then the sequence (( Aξ,dξ))ξ<α+1satisfies the three
+inductive assumptions. Therefore the sequence (( Aα,dα))α<2ωwith the above three
+properties exists. It is easy to check that the union of the family {Aα:α <2ω}is
+(X,I)-nonmeasurable. /square
+3.Nonmeasurable algebraic sums of null sets on Polish groups
+In this section we will consider nonmeasurability in abelian Polish groups . Namely,
+the union of the translations of some subset from some fixed σ- ideal with Borel
+base which contains all singletons.
+It is well known that so called Four Poles Theorem see [1] is in some sens e the best
+we can get in ZFC. In case of families that are not point-finite the com plete (X,I)-
+nonmeasurability is independent of ZFC theory including countable po int families
+also see [7]. Then in general case we need additional set theoretic as sumptions as
+in the paper [2] for example:
+Theorem 3.1 (ZFC +cov(I) = 2ω).Let(X,I)be a Polish ideal space and let
+A ⊂Ibe such that:
+(1)/uniontextA=X,
+(2){x∈X:/uniontextAx/∈I} ∈I.
+Then there exists subfamily A0⊂ Asuch that union/uniontextA0is completely (X,I)-
+nonmeasurable.
+But in special cases if we adopt some regularity conditions on our fam ilies then
+the results on nonmeasurability can be proved in the theory ZFC (se e [3]). In this
+section we will introduce the notion of tiny perfect set with respect to a family of
+subsets of the Polish space. This gives some regularity of the family o f subsets A
+and finally covh(A) = 2ω.
+Here we are would like present the probably well known lemma on trans lation.
+The proof of this lemma was inspired by Ryll-Nardzewski.REMARKS ON NONMEASURABLE UNIONS OF BIG POINT FAMILIES 5
+Lemma 3.1. Let(G,L)be a Polish ideal group with a Haar measure λand let
+Perf (G)be a family of all perfect sets in G. Then
+(∀P∈Perf (G))(∀B∈B+(G))(∃x0∈G)(|(P+x0)∩B|= 2ω).
+Moreover, if P∈Perf (G)andB∈B+(G)then
+λ({x∈G:|(P+x0)∩B|= 2ω})>0.
+Proof. Letµbe a regular measure defined on perfect set Psuch that µ(P) = 1 and
+let
+D= (G×P)∩(/uniondisplay
+t∈R{t}×(t+B))⊂G2,
+whereBis a Borel set such that λ(B)>0. Let us observe that
+(t,s)∈D≡(s∈P)∧(∃b∈B)(t+b=s)≡(s∈P)∧(∃b∈B)(t=s−b),
+therefore
+D=/uniondisplay
+s∈P(/uniondisplay
+b∈Bs−b)×{s}=/uniondisplay
+s∈P(s−B)×{s}.
+Let us consider the product measure λ×µ. Then we have:
+(λ×µ)(D) =/integraldisplay
+D1d(λ×µ)(t,s) =/integraldisplay
+Pd(µ)(s)/integraldisplay
+s+Bdλ(t) =/integraldisplay
+Pd(µ)λ(s+B) (1)
+=/integraldisplay
+Pd(µ)λ(B) =λ(B)/integraldisplay
+Pd(µ) =λ(B)µ(P) =λ(B)>0. (2)
+Then, by Egglestone theorem (see [9] and [11]) there are two perfe ct setsP1,P2
+such that λ(P1)>0 andP1×P2⊂D. Note that
+t∈P1→P2⊂P∩(t+B),
+soλ({t∈G:|P∩(t+B)|= 2ω})>0. /square
+Remark 3.1. Let us note that there are other proofs of the above lemma. Fir st
+one was given by by Cicho´ n and uses the Shoenfield absolutene ss argument. The
+second one was due to Ryll-Nardzewski and is based on convolu tion measures and
+was an inspiration for the proof presented above. Another on e was due to Morayne,
+where density point of measure was used.
+Definition 3.1. Let(G,I)be a Polish ideal group, and let A ⊂ P (G). A perfect
+setP⊂Gis atiny perfect set with respect to Aif
+(∀t∈G)(∀A∈ A)(|({t}+P)∩A| ≤ω).
+Lemma 3.2. Let(G,I)be a Polish ideal group and let A ⊂I. Suppose that there
+exists a tiny set with respect to the family A. Thencovh(A) = 2ω.
+Proof. Let us consider any Ipositive Borel set B∈B+(I) and any tiny perfect set
+Pwith respect to our family A ⊂I. Then there exists a translation x0+PofP
+for some x0∈Gsuch that |(P+x0)∩B|= 2ωby Lemma 3.1. Let us choose any
+subfamily A0⊂ A withκ=|A0|<2ω. But for any A∈ A 0|A∩(x0+P)| ≤ω
+then|/uniontextA0∩(x0+P)| ≤κ·ω <2ωand we have ( B∩(x0+P)\/uniontextA0/\e}atio\slash=∅./square
+Proposition 3.1. Let(G,I)be a Polish abelian group and let A ⊂Ibe such that
+(1) (∀x∈G)(|Ax|= 2ω),
+(2) (∀x,y∈G)(x/\e}atio\slash=y→ |Ax∩Ay| ≤ω),6 ROBERT RA/suppress LOWSKI
+(3)there exists tiny perfect set Pwith respect to the family A.
+Then there exists subfamily A0⊂ Asuch that/uniontextA0is completely (G,I)-nonmeasurable
+set inG.
+Proof. The result follows from Theorem 2.1 and Lemma 3.2. /square
+Proposition 3.2. Let(G,I)be a Polish abelian group and let A ⊂Ibe such that
+(1)/uniontextA=G,
+(2) (∀x,y∈G)(x/\e}atio\slash=y→ |Ax∩Ay| ≤ω),
+(3)there exists tiny perfect set with respect to the family A.
+Then there exists subfamily A0⊂ Asuch that/uniontextA0is(G,I)-nonmeasurable.
+Proof. The result follows from Theorem 2.2 and Lemma 3.2. /square
+We present some applications of the above Propositions.
+Theorem 3.2. Let2≤n∈ωbe positive integer. Then there exists a family Lof
+lines inRnsuch that/uniontextLis a completely (Rn,L)-nonmeasurable set.
+Proof. LetAbe family of all lines in Rn. Then for all x,y∈Rnifx/\e}atio\slash=ythen
+|{l∈ A:x∈l}∩{l∈ A:y∈l}|= 1≤ωand|{l∈ A:x∈l}|= 2ω.
+Let us observe that the unit Euclidean sphere Sis a tiny perfect set with respect
+to the family A. So, we get the theorem from Proposition 3.1. /square
+Remark 3.2 (Given by referee) .Even the a stronger statement (with parallel lines)
+is simple: if A⊂Rn−1is completely nonmeasurable, then so is A×R.
+Theorem 3.3. LetAbe a family of circles with radius 1 on the plane R2such
+that/uniontextA=R2. Then there exists A0⊂ Asuch that/uniontextA0is a Lebesgue non-
+measurable (do not have the Baire property). Moreover, if ad ditionally every point
+is covered 2ωmany times then there exists A0⊂ Asuch that/uniontextA0completely
+(R2,I)-nonmeasurable set, where I=LorI=K.
+Proof. The proof is analogous to the previous one, but here we use Propos ition 3.2
+for the first statement. /square
+In the above example we have the situation where our family is a set of one di-
+mensional circles. But under some additional assumptions we can ge t the assertion
+for arbitrary finite dimension of the spheres. Unfortunately here we cannot use our
+Propositions so proof will be a bit longer.
+Theorem 3.4. Letn∈ωbe a fixed positive integer and let us us consider a family
+ofn−1dimensional Euclidean spheres A ⊂ {S(x,r) :x∈Rn∧r >0}with the
+following property:
+∀x∈Rn{y∈Rn:∃r >0x∈S(y,r)∈ A} ∈ B[L]\L,
+then there exists A0⊂ Asuch that/uniontextA0is completely (Rn,L)-nonmeasurable.
+Proof. Let us enumerate the set of all positive Borel sets B+(L) ={Bα:α <2ω}
+and for any αlet us assume that we have defined a transfinite sequence: /a\}bracketle{t(Aξ,dξ)∈
+A×Bξ:ξ < α/a\}bracketri}htwith the following conditions:
+(1) (∀ξ < α )(Aξ∩Bξ/\e}atio\slash=∅),
+(2){dξ:ξ < α}∩/uniontext{Aξ:ξ < α}=∅.REMARKS ON NONMEASURABLE UNIONS OF BIG POINT FAMILIES 7
+Now let us choose any x∈Bα\ {dξ:ξ < α}. For every ξ < α letHξbe the
+perpendicular bisector hyperplane of the segment connecting xanddξ. Let us
+consider the set
+H={Hξ⊂Rn:ξ < α}.
+By Lemma 3.2 a Lebesgue positive set cannot be covered by/uniontextH. Then there exists
+a positive real r >0 andy∈Rnsuch that x∈S(y,r)∈ A andS(y,r)∩{dξ:ξ <
+α}=∅.
+Since our family Ais a tiny with respect to any line in Rn, by Lemma 3.2
+there exists d∈Bα\/uniontext
+ξ<α+1Aξand letdα=d. So we have built a sequence
+/a\}bracketle{t(Aξ,dξ)∈ A ×Bξ:ξ < α + 1/a\}bracketri}htof length α+ 1 which satisfies the following
+conditions:
+(1) (∀ξ < α + 1)(Aξ∩Bξ/\e}atio\slash=∅),
+(2){dξ:ξ < α + 1}∩/uniontextAξ:ξ < α =∅.
+Therefore, by the transfinite induction, we can have the analogou s sequence with
+the length 2ωand then putting A0={Aξ:ξ <2ω}we get the assertion. /square
+From now we will consider nonmeasurability of Cantor-like sets where crucial
+role is played by packing dimension which is closely related to coverings o f the real
+line see [4],[5],[8].
+It was proved in [2] that there exists subset Aof the classical Cantor set Csuch
+thatA+Cis completely ([0 ,2],L)-nonmeasurable. This proof was done by the
+ultrafilter method. Here we give a new proof of this theorem under s ome additional
+set theoretical assumption.
+Theorem 3.5 (ZFC +cov(L) = 2ω).LetR={0,1,2}ωbe the additive group with
+coordinatewise addition mod 3and letC3={0,2}ωbe the Cantor-like set. Then
+there exists a set A⊂ C3such that A+C3is completely (R,L)-nonmeasurable set.
+Proof. Notice that C3+C3=R. Moreover C3and their translations are null sets.
+Therefore we have to check that the second condition in Theorem 3 .1 holds. Let us
+observe that H={x∈ R :|{n∈ω:x(n) = 0}|< ω} ∈ L . LetA={t+C3:t∈
+C3}andD(x) ={n∈ω:x(n) = 0}. Letx∈ R\Hand let us suppose that t∈ C3
+is such that x∈t+C3. Then for every n∈D(x) we have t(n) = 0 (2 + 0 = 2 and
+2 + 2 = 1) and for every c∈ C3we have t(n) +c(n)∈ {0,2}. But|D(x)|=ωfor
+everyx∈ R\H, so{y∈ R:∀n∈D(x)y(n)∈ {0,2}} ∈ L , hence
+/uniondisplay
+Ax⊂ {y∈ R:∀n∈D(x)y(n)∈ {0,2}} ∈ L
+for every x∈ R \H. Let us observe that R \H⊂/uniontext
+x∈R\HAxand that the set/uniontext
+x∈R\HAxhas a full measure. So the second condition of the Theorem 3.1 is
+satisfied, which finishes the proof. /square
+Using the same method we can prove an analogous theorem for the c lassical
+Cantor subset of the real line in the theory ZFC +cov(L) = 2ω.
+Theorem 3.6 (Darji, Keleti [8]) .If a setC⊂Rhas packing dimension less than
+1then for any positive measure Borel set Band any T⊂Rsuch that |T|<2ωwe
+haveB\(T+C)/\e}atio\slash=∅.
+The above theorem is a generalisation of Gruenhage’s theorem abou t the clas-
+sical Cantor set. It is important to note that assumption on packin g dimension is
+necessary (see [4], [5]).8 ROBERT RA/suppress LOWSKI
+LetC4⊂[0,1] be a Cantor-like set constructed as follows:
+(1) remove from the interval [0 ,1] the open segment (1
+4,3
+4),
+(2) do the same with the remaining segments infinitely many times.
+It is easy to check that dim pack(C4) =1
+2, thus by Darji-Keleti Theorem (see
+[8]) we need continuum many translates of the set C4to cover any set of positive
+Lebesgue measure.
+Proposition 3.3 (ZFC+cov(L) = 2ω).There exists a set A⊂1
+2C4such that A+C4
+is a([0,3
+2],L)-completely nonmeasurable subset of the interval [0,3
+2].
+Proof. (Sketch ) First of all one can prove that C4+1
+2C4= [0,3
+2] and that for any
+x∈[0,3
+2] the set{(a,b)∈ C4×1
+2C4:x=a+b}is finite (see [6] Lemma 7 for details).
+LetA={t+C4:t∈1
+2C4} ⊂LthenAis point-finite family. Therefore, by Theorem
+3.1, there exists A0⊂ A such that/uniontextA0is completely ([0 ,3
+2],L)-nonmeasurable in
+the interval [0 ,3
+2]. /square
+Acknowledgement Author is very indebted to Professor Ryll-Nardzewski for an
+idea of the proof of Lemma 3.1, Professors Jacek Cicho´ n and Micha /suppress l Morayne for
+critical remarks. Author would like to thank the referee for finding serious mistakes
+in an earlier version of this work and for giving valuable suggestions.
+References
+[1] J. Brzuchowski, J. Cicho´ n, E. Grzegorek, Cz. Ryll-Nard zewski, On existence of nonmeasur-
+able unions, Bull. Pol. Acad. Sci. Math. 27(1979), pp. 447–448.
+[2] J. Cicho´ n, M. Morayne, R. Ra/suppress lowski, Cz. Ryll-Nardzews ki and Sz. ˙Zeberski, On nonmea-
+surable unions, Topology and its Applications, 154(2007), pp. 884–893.
+[3] R. Ra/suppress lowski and Sz. ˙Zeberski, Complete nonmeasurability in regular families, accepted to
+Houston Journ. of Math.
+[4] M. Elekes and J. Steprans, Less than 2ωmany translates of a compact null set may cover
+the real line, Fund. Math. 181, (2004), no 1, pp. 89–96.
+[5] M. Elekes and A. Toth, Covering locally compact groups by less than 2ωmany translates
+of a compact nullset, Fund. Math. 193, (2007), pp. 243–257.
+[6] K. Ciesielski, H. Fejzi´ c and C. Freiling, Measure zero s ets with non-measurable sum, Real
+Anal. Exchange 27 (2001/02), no. 2, pp. 783–793.
+[7] D. H. Fremlin, Measure additive coverings and measurabl e selectors, Dissertationes Math.
+260(1987)
+[8] U. Darji, T. Keleti, Covering Rwith translates of a compact set, Proc. Amer. Math. Soc.
+131 (2003), no. 8, 2596–2598.
+[9] H. G. Egglestone, Two measure properties of Cartesian pr oduct sets, Quart. J. Math.,
+Oxford Ser. (2) 5(1954), pp. 108–115.
+[10] Sz. ˙Zeberski, On completely nonmeasurable unions, Math. Log. Q uart.53(1) (2007), pp.
+38–42.
+[11] Sz. ˙Zeberski, Nonstandard proofs of Eggelstone like theorems, Proceedings of the Ninth
+Prague Topological Symposium (Prague 2001) pp. 353–357, To pology Atlas, Toronto 2002.
+E-mail address :ralowski@im.pwr.wroc.pl
+Institute of Mathematics and Computer Sciences, Wroc/suppress law Un iversity of Technol-
+ogy, Wybrze ˙ze Wyspia ´nskiego 27, 50-370 Wroc/suppress law, Poland.
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+arXiv:1001.0550v2  [gr-qc]  25 Jan 2010Astrophysics and Space Science
+DOI 10.1007/s •••••-•••-••••-•
+de Sitter expansion with anisotropic fluid in Bianchi
+type-I space-time
+¨Ozg¨ ur Akarsu1•Can Battal Kılın¸ c2
+c/circlecopyrtSpringer-Verlag ••••
+Abstract Some features of the Bianchi type-I uni-
+versesinthepresenceofafluidthatwieldsananisotropic
+equation of state (EoS) parameter are discussed in the
+context of general relativity. The models that exhibit
+de Sitter volumetric expansion due to the constant ef-
+fective energy density (the sum of the energy density
+of the fluid and the anisotropy energy density) are of
+particular interest. We also introduce two locally ro-
+tationally symmetric models, which exhibit de Sitter
+volumetric expansion in the presence of a hypothetical
+fluid that has been obtained by minimally altering the
+conventional vacuum energy. In the first model, the
+directional EoS parameter on the xaxis is assumed to
+be -1, while the ones on the other axes and the energy
+density of the fluid are allowed to be functions of time.
+In the second model, the energy density of the fluid
+is assumed to be constant, while the directional EoS
+parameters are allowed to be functions of time.
+Keywords Bianchitype-I;deSitterexpansion; cosmo-
+logical constant; anisotropic fluid
+1 Introduction
+Inhomogeneous and anisotropic universes have particu-
+larlybeen ofinterestinmathematicalcosmology,rather
+than observational cosmology. This situation can be
+related to two main ingredients of modern cosmology:
+(i) The present universe had very well been described
+with Friedmann-Lemaitre models that are based on the
+¨Ozg¨ ur Akarsu
+Can Battal Kılın¸ c
+Ege University, Faculty of Science, Dept. of Astronomy and
+Space Sciences, 35100 Bornova, ˙Izmir/Turkey.
+1e-mail: ozgur.akarsu@mail.ege.edu.tr
+2e-mail: can.kilinc@ege.edu.trspatially homogeneous and isotropic Robertson-Walker
+space-time. (ii) According to the inflationary paradigm
+the universe should have achieved an almost isotropic
+and homogeneous geometry at the end of the infla-
+tionary era (see Linde 2008 for a review of inflation-
+ary cosmology). However, for a realistic cosmologi-
+cal model one should consider spatially inhomogeneous
+and anisotropic space-times and then show whether
+they can evolve to the observed amount of homogene-
+ity and isotropy. Because of the analytical difficulties
+in studying the inhomogeneous models, as a first step
+towardamorerealisticmodel onemayconsiderBianchi
+type cosmological models, which form a large and al-
+most complete class of relativistic cosmological models,
+which are homogeneous but not necessarily isotropic.
+The only spatially homogeneous but anisotropic mod-
+els other than Bianchi type models are the Kantowski-
+Sachs locally symmetric family. See Ellis & van Elst
+(1999) for generalized, particularly anisotropic, cosmo-
+logical models and Ellis (2006) for a concise review on
+Bianchi type models.
+Inprinciple, oncethe metricisgeneralizedtoBianchi
+types, theequationofstate(EoS)parameterofthefluid
+can also be generalized in a way conveniently to wield
+anisotropy with the considered metric. In such models,
+where both the metric and EoS parameter of the fluid
+are allowed to exhibit an anisotropic character, the uni-
+versecanexhibitnon-trivialisotropizationhistoriesand
+it can be examined whether the metric and/or the EoS
+parameter of the fluid evolve toward isotropy. Thus,
+one gets the opportunity of constructing even more re-
+alistic models than the Bianchi type models where the
+fluid that fills the universe is assumed to be isotropic
+from the beginning.
+Moreover, in recent years Bianchi universes have
+been gaining an increasing interest of observational
+cosmology, since the WMAP data (Hinshaw et al.
+2003, 2007, 2009) seem to require an addition to the2
+standard cosmological model with positive cosmolog-
+ical constant that resembles the Bianchi morphology
+(see Jaffe et al. 2005, 2006a,b; Campanelli et al. 2006,
+2007; Hoftuft et al. 2009). According to this, the uni-
+verse should have achieved a slightly anisotropic spa-
+tial geometry in spite of the inflation, contrary to
+generic inflationary models (Guth 1981; Sato 1981;
+Albrecht & Steinhardt 1982; Linde 1982, 1983, 1991,
+1994) and that might be indicating a non-trivial
+isotropization history of the universe due to the pres-
+ence of an anisotropic energy source. We may talk
+about two main classes of such models; according to
+whether this anisotropization occurs at an early time
+or at late times of the universe. The former class can
+be related with the inflaton field, which drives the infla-
+tion, while the latter one can be related with the dark
+energy (DE), which drives the late time acceleration of
+the universe.
+In the context of the former class, the generic in-
+flationary models can be modified in a way to end
+inflation with a slightly anisotropic spatial geometry;
+see, e.g., Campanelli et al. (2006, 2007). A key in-
+gredient of the inflationary models is an inflaton field,
+which generates an expansion like the one generated
+by the cosmological constant. In these models, space,
+and thus the inflaton field, is assumed to be homo-
+geneous and isotropic from the beginning. The pos-
+sibility of inflation in anisotropic space-time has first
+been investigated by Barrow & Turner (1981). Some
+authors have considered Bianchi type space-times in
+the presence of scalar fields, e.g., Feinstein & Ibanez
+(1993); Aguirregabiria et al. (1993). Ford (1989) dis-
+cussed the possibility of inflationary models in which
+inflation is driven by a vector field, which gives rise
+to an anisotropic EoS parameter, rather than a scalar
+field, for the first time. However, it was suffering from
+the fine-tuning problem. Recently, Koivisto & Mota
+(2008c) have considered several new classes of viable
+vector field as alternatives to the inflaton within in the
+Bianchi type-I framework. Golovnev et al. (2008) have
+constructed a successful model, which either could give
+a completely isotropic universe or slightly anisotropic
+universe at the end of the inflation.
+In the context of the latter class, the isotropy of
+space that has been achieved in the inflationary era
+can be distorted during the contemporary accelera-
+tion of the universe by modifying the DE in a way to
+wield anisotropic pressure, see, e.g., Koivisto & Mota
+(2008a,b,c); Rodrigues (2008). It is well known that
+the inclusion of a positive cosmological constant, which
+ismathematicallyequivalenttothevacuumenergywith
+p=−ρand which is the simplest candidate for the DE,
+may lead to the explanation of this observed accelera-
+tion of the universe.Note that the vacuum energy (the cosmological con-
+stant) is coming into question in both early and late
+time acceleration of the universe. The solution of Ein-
+stein’s field equations in the presence of a positive cos-
+mological constant for a homogeneous and isotropic
+model results in an exponentially expanding universe,
+which is known as the de Sitter model (Gron & Hervik
+2007). Thus, de Sitter and de Sitter like expansions,
+hence the cosmological constant and its alterations, are
+of particular interest for contemporary cosmology.
+Wald (1983) investigated the asymptotic behavior
+of initially expanding homogeneous cosmological mod-
+els with a positive cosmological constant and showed
+that such models, of all Bianchi types (only type-
+IX under some conditions), exponentially evolve to-
+ward the de Sitter solution. Gron (1985) investi-
+gated the expansion anisotropy during the inflationary
+era for a vacuum Bianchi type-I universe with a non-
+vanishing positive cosmological constant, and discussed
+the anisotropic generalization of the de Sitter solution.
+Beesham (1994) showed that Bianchi type-I models in
+the presence of a positive cosmological constant cannot
+be of the pure de Sitter type, unless the gravitational
+”constant” takes negative values. Kalligas et al. (1994)
+showedthat de Sitter inflation is allowedat least at late
+times of the universe within the Bianchi type-I frame-
+work. Arbab (1997) constructed a Bianchi type-I cos-
+mological model, in the presence of variable Gand Λ
+and bulk viscosity, that exhibits de Sitter expansion.
+Kumar & Singh (2007) gave an exponentially expand-
+ing Bianchi type-I model in the presence of a hypo-
+theticalperfectfluid. Akarsu & Kilinc(2010)presented
+a locally rotationally symmetric (LRS) Bianchi type-I
+model that exhibits de Sitter volumetric expansion in
+a mixture of a perfect fluid and a fluid that wields a
+special form of a dynamical and anisotropic EoS pa-
+rameter.
+Barrow (1997) presented an analysis of the cosmo-
+logical evolution of matter sources that possess small
+anisotropicpressures(electricandmagneticfields, colli-
+sionless relativistic particles, gravitons, anti-symmetric
+axion fields in low-energy string cosmologies, spatial
+curvature anisotropies and stresses arising from sim-
+ple topological defects), and he discussed the effects of
+inflation on the anisotropy of the pressures.
+Consideringtheabovediscussion,onemayfirstthink
+to distort the isotropy of the vacuum energy so as to
+obtain anisotropy in the geometry of the universe in
+general relativity. Thus, in this study, Bianchi type-I
+models that exhibit de Sitter volumetric expansion in
+the presence of a vacuum energy that has minimally
+been altered in a way to wield anisotropic EoS param-
+eter will be of our particular interest. We first, in Sect.de Sitter expansion with anisotropic fluid in Bianchi type-I space-time 3
+2, discuss some features of the Bianchi type-I universe
+filled with afluid that wieldsananisotropicEoSparam-
+eter. We define the effective energy density as the sum
+of the energy density of the fluid and the anisotropy en-
+ergydensity, andspecializethediscussiontothe models
+that exhibit de Sitter volumetric expansion due to the
+constant effective energy density. We then, in Sect. 3,
+examine the solutions for a LRS Bianchi type-I space-
+time, where the universe exhibits de Sitter volumetric
+expansion in the presence of a hypothetical fluid, that
+has been obtained by minimally altering the conven-
+tional vacuum energy, and we present two exact mod-
+els. In the first model, in Sect. 3.1, the directional EoS
+parameter on the xaxis is assumed to be -1, while the
+ones on the other axes and the energy density of the
+fluid are allowed to be functions of time. In the sec-
+ond model, in Sect. 3.2, the energy density of the fluid
+is assumed to be constant, while the directional EoS
+parameters are allowed to be functions of time.
+2 A general discussion on the dynamics of the
+models in the presence of an anisotropic fluid
+2.1 The field equations in the presence of anisotropic
+fluid
+Weconsiderthediagonalmetricofthespatiallyflat, ho-
+mogeneous but anisotropic Bianchi type-I space-time,
+ds2=dt2−A(t)2dx2−B(t)2dy2−C(t)2dz2,(1)
+whereA(t),B(t) andC(t) are the directional scale fac-
+tors, functions of the cosmic time, t.
+Within the framework of the metric given by (1),
+the energy-momentum tensor of a fluid can be written,
+most generally, in anisotropic diagonal form as follows:
+Tνµ= diag[T00,T11,T22,T33]. (2)
+TheEinsteinfieldequations,innaturalunits(8 πG=
+1 andc= 1), are
+Rµν−1
+2Rgµν=−Tµν, (3)
+wheregµνis the metric tensor; gµνuµuν= 1 (uµ=
+(1,0,0,0) is the four-velocity vector); Rµνis the Ricci
+tensor; Ris the Ricci scalar, Tµνis the energy-
+momentum tensor.
+In a comoving coordinate system, the Einstein field
+equations (3), for the Bianchi type-I space-time (1) in
+case of the anisotropic energy-momentum tensor given
+by (2), read
+˙A
+A˙B
+B+˙A
+A˙C
+C+˙B
+B˙C
+C=T00, (4)¨B
+B+¨C
+C+˙B
+B˙C
+C=T11, (5)
+¨A
+A+¨C
+C+˙A
+A˙C
+C=T22, (6)
+¨A
+A+¨B
+B+˙A
+A˙B
+B=T33, (7)
+where an overdot denotes d/dt.
+ThedirectionalHubble parameters,whichdetermine
+the expansion rates of the universe in the directions of
+x,yandz, for the metric given in (1) can be defined as
+Hx≡˙A
+A,Hy≡˙B
+BandHz≡˙C
+C,(8)
+respectively, and then the mean Hubble parameter,
+which determines the volumetric expansion rate of the
+universe, can be given as
+H=1
+3˙V
+V=Hx+Hy+Hz
+3, (9)
+whereV=ABCis the volume of the universe.
+2.2 The anisotropy of the expansion
+The anisotropy of the expansion can be parametrized
+by using the directional Hubble parameters (8) and the
+mean Hubble parameter (9) of the expansion,
+∆≡1
+33/summationdisplay
+i=1/parenleftbiggHi−H
+H/parenrightbigg2
+, (10)
+whereHi(i=1,2,3) represents the directional Hubble
+parameters in the directions of x,yandz, respec-
+tively. ∆ = 0 corresponds to isotropic expansion. The
+Bianchi type-I universe approaches spatial isotropy if
+∆→0,V→+∞andT00>0 ast→+∞(see
+Collins & Hawking 1973 for details).
+The anisotropy parameter of the expansion can also
+be given in terms of the components of the energy-
+momentum tensor (2) and the mean Hubble parameter
+(9) by using the evolution equations (5)-(7);
+∆ =1
+9H23/summationdisplay
+i,j=1/bracketleftbigg
+λj+/integraldisplay
+(Tii−Tjj)Vdt/bracketrightbigg2
+V−2andi > j,
+(11)
+where the λjare real constants. One can immediately
+see that the above equation is reduced to the one given4
+by Gron (1985):
+∆ =λ12+λ22+λ32
+9H2V−2(12)
+for an isotropic fluid, i.e., where T11=T22=T33.
+2.3 Anisotropy energy density in the presence of an
+anisotropic fluid
+The energy density of the fluid ρ, which corresponds to
+T00, can immediately be written in terms of the mean
+Hubble parameter Hand the anisotropy parameter of
+the expansion ∆ by using the constraint equation (4),
+ρ=T00= 3H2/parenleftbigg
+1−∆
+2/parenrightbigg
+. (13)
+This is equivalent to the generalized Friedmann equa-
+tionforaBianchitype-Ispace-time(seeEllis & van Elst
+1999 and Barrow 1995 for the generalized Friedmann
+equation). According to this, for a given value of the
+mean Hubble parameter Hin Bianchi type-I space-
+time, the anisotropy of the expansion lowers down the
+energy density; the highest energy density is achieved
+in case of isotropic expansion (i.e., ∆ = 0) and the
+anisotropy of the expansion cannot be arbitrary be-
+cause the condition to observe ρ >0 for a comoving
+observer is ∆ <2.
+By moving from (13), the anisotropy energy density
+(namely, the energy density that associated with the
+anisotropy of the expansion) may be defined as follows:
+ρβ≡3
+2H2∆. (14)
+Then, using (11) in this definition, we obtain
+ρβ=1
+63/summationdisplay
+i,j=1/bracketleftbigg
+λj+/integraldisplay
+(Tii−Tjj)Vdt/bracketrightbigg2
+V−2andi > j,
+(15)
+for the anisotropyenergy density. One can observethat
+the anisotropy of the fluid also contributes to the de-
+termination of the ρβvia the integral term. In case of
+an isotropic fluid the integral term vanishes and ρβis
+reduced to the following form:
+ρβ=1
+6/parenleftbig
+λ12+λ22+λ32/parenrightbig
+V−2, (16)
+whichisequivalenttotheonedefinedbyBarrow & Turner
+(1981) in case of an isotropic fluid. According to this
+equation, (16), ρβdecreases monotonically as the vol-
+ume of the universe increases and converges to null
+asV→+∞, while it diverges as V→0. On theother hand, as can be seen from (15), ρβcan ex-
+hibit non-trivial behaviors if the fluid is allowed to
+be anisotropic. In other words, in the presence of
+an anisotropic fluid, we may have cosmological models
+with non-trivial isotropization histories, e.g., models in
+whichρβdoes not diverge as V→0 and/or V→+∞.
+2.4 Conservation laws for the anisotropic fluid and
+vacuum energy
+Allowing for anisotropy in the pressure of the fluid, and
+thus in its EoS parameter, gives rise to new possibilities
+for the evolution of the energy source. To see this, we
+first parametrize the energy-momentum tensor given in
+(2) as follows:
+Tνµ= diag[ρ,−px,−py,−pz] (17)
+= diag[1,−wx,−wy,−wz]ρ
+= diag[1,−w,−(w+γ),−(w+δ)]ρ,
+wherepx,pyandpzare the pressures and wx,wyand
+wzare the directional EoS parameters on the x,yand
+zaxes, respectively; wis the deviation-free EoS pa-
+rameter of the fluid. The deviation from isotropy is
+parametrized by setting wx=wand then introducing
+skewness parameters δandγ, which are the deviations
+fromw, respectively, on the yandzaxes.w,δandγ
+are not necessarily constants and might be functions of
+the cosmic time, t.
+The conservation of the energy-momentum tensor of
+the fluid parametrized in (17), i.e., Tµν;ν= 0, leads to
+the following equation:
+˙ρ+3(1+w)ρH+δρHy+γρHz= 0, (18)
+where the last two terms, the terms with δandγ, arise
+due to the anisotropy of the fluid. Now we can briefly
+examine what possibilities arise due to these two terms
+when the conventional vacuum energy (which is math-
+ematically equivalent to the cosmological constant Λ
+and which can be represented with an EoS parameter
+in the form of p=−ρ) is minimally altered in a way to
+wield an anisotropic EoS parameter. Obviously, when
+we consider the conventional vacuum energy, which is
+isotropic (i.e., γ=δ= 0), ifw=−1, then the energy
+density ( ρ) is necessarily constant and vice versa. How-
+ever, this is not the case when the energy-momentum
+tensor of the fluid is generalized to the form given in
+(17), namely, when an anisotropic EoS parameter is al-
+lowed for the fluid. In that case, if the energy density
+is assumed to be constant, ρ=const., (18) is reduced
+to the following equation:
+3(1+w)H+δHy+γHz= 0, (19)de Sitter expansion with anisotropic fluid in Bianchi type-I space-time 5
+thus,wis not necessarily constant. Similarly, if w=
+−1, (18) is reduced to the following equation:
+˙ρ+δρHy+γρHz= 0, (20)
+thus, the energy density is not necessarily constant ei-
+ther.
+2.5 Generalized Friedmann equation and de Sitter
+expansion
+Using the energy-momentum tensor parametrized in
+(17), the anisotropy energy density can be written in
+terms of the directional EoS parameters as follows:
+ρβ=1
+63/summationdisplay
+i,j=1/bracketleftbigg
+λj+/integraldisplay
+(wj−wi)ρVdt/bracketrightbigg2
+V−2andi > j.
+(21)
+Note that the behavior of the anisotropy energy den-
+sity is not only determined by the λjconstants and the
+volume of the universe as in (16), but also by the direc-
+tional EoS parameters ( wx,wyandwz) and the energy
+density of the fluid ( ρ) via the integral term.
+WemaywritedownthegeneralizedFriedmannequa-
+tion by considering (13) and (14) and define the effec-
+tiveenergydensity ρef,whichdeterminesthevolumetric
+expansion rate of the universe,
+3H2=ρ+ρβ≡ρef. (22)
+From the definition of the mean Hubble parameter
+given in (9) the volume of the universe can be obtained
+as follows:
+V=c1e3/integraltext
+Hdt, (23)
+wherec1is a positive constant of integration. Thus,
+using (22) in (23) the volume of the universe can be
+obtained in terms of the effective energy density,
+V=c1e√
+3/integraltext√ρefdt. (24)
+According to this, the condition for the de Sitter vol-
+umetric expansion does not correspond to a constant
+energy density of the fluid but to a constant effective
+energy density, i.e., ρef=ρ+ρβ=const., which leads
+to
+V=c1e√3ρeft. (25)
+Obviously, the universe exhibits de Sitter expansion
+whenρβis null in the presence of a positive cosmo-
+logical constant, i.e., in the presence of conventional
+vacuum energy that wields a constant energy density.However, if ρβis not null, then ρefis not constant, but
+a decreasing function of the cosmic tas long as the uni-
+verse expands, since ρβ∝V−2. Thus, as also shown by
+Beesham (1994), the behavior of Bianchi type-I infla-
+tionary solutions cannot be of the pure de Sitter type,
+even when p=−ρ, but are of power-law type.
+What if we consider the anisotropy energy density
+together with a mixture of the conventional vacuum
+energyand perfect fluids? The conventionalperfect flu-
+ids, i.e., radiation, pressureless matter etc. can be de-
+scribed by an EoS parameter in the form of p=wρand
+their energy densities change as V−(1+w). Thus, the
+cosmological constant will be dominant as V→+∞,
+provided that w >−1, and according to the cosmic no-
+hair theorem for Einstein gravity introduced by Wald
+(1983), the universe will exponentially evolve toward
+the de Sitter universe. In other words, Bianchi type-I
+models in the presence of a positive cosmological con-
+stant isotropize and their volumetric expansion rates
+approach de Sitter expansion as V→+∞. On the
+other hand, according to (16), no matter how small the
+anisotropy energy density is, compared with the other
+sources, ρβwill eventually dominate any perfect fluid
+and govern the dynamics of the expansion in the very
+early evolution of the universe as V→0, provided that
+w <1, i.e., the universe will approximate the Kasner
+vacuum solution. However, these cases are not implied
+once the implicitly assumed isotropy of the vacuum en-
+ergy is relaxed. This is because, according to (21), ρβ
+does not have to be proportional with V−2in case of
+an anisotropic fluid, and it may not diverge as V→0
+and/orV→+∞.
+Our final remark in this section concerns the con-
+dition for isotropization of a Bianchi type-I universe
+that exhibits de Sitter volumetric expansion. The con-
+dition for isotropization mentioned in Sect. 2.3 can be
+rewritten as follows: ρβ→0 ast→+∞sinceHis
+constant, and this leads to ˙ ρβ<0 ast→+∞, since
+ρβ>0 by definition. This, in addition, leads to ˙ ρ >0
+ast→+∞from ˙ρβ+ ˙ρ= 0. Thus, the condition for
+isotropization in the Bianchi type-I models that exhibit
+de Sitter volumetric expansion imposes on the fluid the
+constraint to behave like a phantom energy, i.e., to ex-
+hibit an increasing energy density as Vincreases. On
+the other hand, if the energy density of the fluid is con-
+stant (ρ=const.), then the anisotropy energy density
+is also constant ( ρβ=const.). If ˙ρ <0, then ˙ρβ>0;
+thus for models in which the energy density of the fluid
+is decreasing, the anisotropy energy density (thus, the
+anisotropy of the expansion) increases as tincreases.6
+3 Exponentially expanding LRS models in the
+presence of anisotropic fluid
+In the following, we examine the cosmological models
+that exhibit de Sitter volumetric expansion within the
+LRS Bianchi type-I framework and present two exact
+models.
+The LRS Bianchi type-I metric reads
+ds2=dt2−A(t)2dx2−B(t)2(dy2+dz2). (26)
+Thus,Hy=Hz, and in the following they are repre-
+sented by Hy,z. The energy-momentum tensor given in
+(17) can be customized for the LRS Bianchi-I metric by
+choosing δ=γ,
+Tνµ= diag[1,−w,−(w+γ),−(w+γ)]ρ. (27)
+Considering (26) and (27), the Einstein field equations
+(4)-(7) can be reduced to the following system of equa-
+tions:
+˙B2
+B2+2˙A
+A˙B
+B=ρ, (28)
+˙B2
+B2+2¨B
+B=−wρ, (29)
+¨B
+B+˙B
+B˙A
+A+¨A
+A=−(w+γ)ρ. (30)
+Then we have initially five variables ( A,B,ρ,w,γ)
+and three linearly independent equations, namely the
+three Einstein field equations (28)-(30). Thus we will
+need two additional constraints to close the system of
+equations. However, before introducing the constraints
+to close the system of equations, we can give ρ,wandγ
+in terms of the mean Hubble parameter and the direc-
+tional Hubble parameter on the xaxis, manipulating
+the field equations (28)-(30),
+ρ= 3H2−3
+4(Hx−H)2, (31)
+w=d
+dt(Hx−3H)−3
+4(Hx−3H)2
+3H2−3
+4(Hx−H)2, (32)
+γ=3
+2d
+dt(H−Hx)+9
+2(H−Hx)H
+3H2−3
+4(Hx−H)2. (33)
+The anisotropy energy density can also be given in
+terms of HandHxby using (13), (14) and (28),
+ρβ=3
+4(Hx−H)2. (34)One may check that the summation of (31) and (34)
+leads toρef=ρ+ρβ= 3H2.
+As the first constraint to close the system of equa-
+tions, the effective energy density is assumed to be con-
+stant throughout the history of the universe:
+ρef= 3k2, (35)
+wherekis a positive constant and thus from (22)
+H=k, (36)
+which correspondsto the well-knownde Sitter volumet-
+ric expansion, i.e.,
+V=AB2=c1e3kt. (37)
+The solution of (33) by considering (31) and (36) gives
+the following equation for the evolution of the direc-
+tional scale factor on the xaxis:
+A=κekt+λ
+3ke−3kt+2
+9k(e−3kt/integraltext
+e3ktΓ(t)dt−/integraltext
+Γ(t)dt),(38)
+whereκ >0 andλare real constants and Γ( t) =γρ
+is the skewness of the pressure. Using (38) in (37) the
+directional scale factor on the yandzaxes is obtained
+as follows:
+B=/parenleftBigc1
+κ/parenrightBig1/2
+ekt−λ
+6ke−3kt−1
+9k(e−3kt/integraltext
+e3ktΓ(t)dt−/integraltext
+Γ(t)dt).
+(39)
+Using the scale factors (38) and (39), the directional
+Hubble parameters are obtained as follows:
+Hx=k−λe−3kt−2
+3e−3kt/integraldisplay
+Γ(t)e3ktdt, (40)
+Hy,z=k+λ
+2e−3kt+1
+3e−3kt/integraldisplay
+Γ(t)e3ktdt. (41)
+Using (36), (40) and (41) in (10) the anisotropy of the
+expansion, ∆, and thus the anisotropy energy density,
+ρβ, are obtained as
+∆ =2
+3ρβ
+k2=1
+18e−6kt/parenleftbig
+3λ+2/integraltext
+e3ktΓ(t)dt/parenrightbig2
+k2.(42)
+It can be seen that the skewness of the pressure Γ( t)
+also contributes to the evolution of the cosmological
+parameters.
+One can immediately obtain the cosmological pa-
+rameters once a function for Γ( t) is chosen. Alterna-
+tively, one can obtain the parameters by introducing
+one more constraint on one of the cosmological param-
+eters; for instance, on the energy density of the fluid ρde Sitter expansion with anisotropic fluid in Bianchi type-I space-time 7
+or on the EoS parameter. Trivially, one may also as-
+sume that the fluid is a perfect fluid (i.e., γ= 0; thus
+Γ is null) and would then obtain the solutions given by
+Kumar & Singh (2007) for an exponentially expanding
+Bianchi type-I universe. One can observe that for any
+perfect fluid, i.e., in case of Γ = 0, the initial anisotropy
+ofthe expansiondies awaymonotonicallyas tincreases,
+while it may exhibit various non-trivial behaviors ac-
+cording to the function chosen for Γ( t).
+3.1 Model for ρef=const.andw=−1
+Note that we need one more constraint in addition to
+the one given in (35) to fully determine the system.
+Since the first assumptiongivenby(35) causesde Sitter
+volumetricexpansion, it seemsreasonableto chooseour
+last assumption as
+w=−1. (43)
+If we solve the system of equations which consists of
+the three Einstein field equations (28)-(30) and the two
+constraints given by (35) and (43), we obtain the fol-
+lowing exact expressions for the scale factors:
+A=c1ekt/parenleftbig
+κk−1+3λe−3kt/parenrightbig−2/3, (44)
+B=ekt/parenleftbig
+κk−1+3λe−3kt/parenrightbig1/3. (45)
+The directional Hubble parameters are obtained as fol-
+lows:
+Hx=k+6λk2
+κe3kt+3λk, (46)
+Hy,z=k−3λk2
+κe3kt+3λk. (47)
+Theanisotropyparameteroftheexpansionandthusthe
+anisotropy energy density are found to be dynamical,
+∆ =2
+3ρβ
+k2=18λ2k2
+(κe3kt+3λk)2. (48)
+The energy density of the fluid is also found to be dy-
+namical in a way so as to secure ρ+ρβ= 3k2,
+ρ= 3k2/parenleftBigg
+1−9λ2k2
+(κe3kt+3λk)2/parenrightBigg
+(49)
+and the skewness parameter of the EoS parameter is
+obtained as follows:
+γ=−27λ2k2
+κe3kt(κe3kt+6λk). (50)One can check that this solution yields (20). The
+positiveness condition on the energy density of the fluid
+(ρ >0) imposes the restriction that the anisotropy of
+the expansion is smaller than 2 (∆ <2) and that im-
+posesλto be positive. Thus, only those models with
+positiveλvalues are viable, and hence in the following
+we will consider only the positive λvalues.
+∆ decreases monotonically as tincreases and ∆ →0
+ast→+∞. Thus, the space approaches isotropy as
+t→+∞in this model. On the other hand, ∆ →
+2[1−κ/(κ+3λk)]2ast→0.
+ρ→3k2andγ→0ast→+∞. Thatis, the EoSpa-
+rameter of the fluid isotropizes as tincreases and mim-
+ics the cosmologicalconstant as t→+∞. On the other
+hand,ρ→3k2/parenleftBig
+1−/parenleftbig
+1+κ
+λk/parenrightbig−2/parenrightBig
+andγ→ −27λ2k2
+κ2+6λκk,
+whichisalwaysanegativenumber,as t→0. Therefore,
+as expected from the isotropization condition discussed
+in Sect. 2.5., ρbehaves like a phantom energy and in-
+creases as Vincreases. This result is consistent with
+the following situation: γis always negative and thus
+EoS parameters on the yandzaxes are passing the
+phantom divide line, i.e., wy,z<−1.
+3.2 Model for ρef=const.andρ=const.
+Whilewis assumed to be −1 so as to close the system
+in the preceding section, it is allowed to be a function
+of the cosmic time tin this section, but, this time, the
+energy density of the fluid is assumed to be constant:
+ρ(t) =const.=ρ (51)
+If we solve the system of equations which consists of
+the three Einstein field equations (28)-(30) and the two
+constraints given by (35) and (51), we obtain the fol-
+lowing exact expressions for the scale factors:
+A=κ−2c1ekt±2
+3√
+9k2−3ρt, (52)
+B=κekt∓1
+3√
+9k2−3ρt. (53)
+The directional Hubble parameters are obtained as fol-
+lows:
+Hx=k±2
+3/radicalbig
+9k2−3ρ, (54)
+Hy,z=k∓1
+3/radicalbig
+9k2−3ρ. (55)
+The anisotropy parameter of the expansion and thus
+the anisotropy energy density are found to be non-
+dynamical,
+∆ =2
+3ρβ
+k2= 2−2
+3ρ
+k2. (56)8
+The deviation-free EoS parameter of the fluid is ob-
+tained as follows:
+w=ρ−6k2
+ρ±2k
+ρ/radicalbig
+9k2−3ρ=−1−∆∓√
+2∆
+1−∆
+2.(57)
+The skewness parameter of the EoS parameter is ob-
+tained as follows:
+γ=∓3k
+ρ/radicalbig
+9k2−3ρ=∓3/radicalBig
+∆
+2
+1−∆
+2. (58)
+One can check that this solution yields (19). The di-
+rectional Hubble parameters, the anisotropy of the ex-
+pansion (thus, the anisotropy energy density) and the
+directional EoS parameters are all constants through-
+out the history of the universe.
+It is obvious that ρ≤3k2, thusρβ≤3k2and
+∆≤2. Werecovertheconventionalvacuumenergyand
+isotropic expansion when ρ= 3k2, because w=−1,
+γ= 0 and ∆ = 0 in that case. However, when ρ <3k2,
+both the expansion and the fluid deviate from isotropy,
+becausew >−1,γ <0 orw <−1,γ >0 and ∆ >0
+in this case. It can be observed that, while the EoS
+parameter on the xaxis is in the quintessence region
+(w >−1), the ones on the yandzaxes are in the
+phantom region ( w+γ <−1), or vice versa. According
+to this, while the expansion of the xaxis acts so as to
+decrease the energy density of the fluid, the expansion
+of theyzplane acts so as to increase the energy density
+of the fluid or vice versa. However, in total, the decre-
+ments and increments compensate for each other and
+the energy density of the fluid does not change.
+4 Conclusion
+Onmotivatingfromtheincreasingevidencefortheneed
+of a geometry that resembles Bianchi morphology to
+explain the observed anisotropy in the WMAP data,
+we have discussed some features of the Bianchi type-
+I universes in the presence of a fluid that wields an
+anisotropic equation of state (EoS) parameter in gen-
+eral relativity. We have focused on those models that
+exhibit de Sitter volumetric expansion in the presence
+of a hypothetical fluid obtained by distorting the EoS
+parameter of the conventional vacuum energy in a way
+so as to wield anisotropy.
+We have also given two exact solutions within the lo-
+cally rotationally symmetric Bianchi type-I framework.
+In both models the effective energy density (the sum
+of the energy density of the fluid and the anisotropy
+energy density) has been assumed to be constant so as
+to secure the de Sitter volumetric expansion.In the first model, the directional EoS parameter on
+thexaxishasbeen assumedto be -1. The anisotropyof
+the expansion, the energy density and the anisotropy of
+the fluid have been found to be dynamical. While the
+anisotropy of the expansion and the anisotropy of the
+fluid decrease and tend to null as the universe expands,
+the energy density of the fluid increasesand approaches
+to its maximum value. The fluid approximates the con-
+ventional vacuum energy as the universe evolves.
+In the second model, the energy density of the fluid
+has been assumed to be constant. The anisotropy of
+the expansion and the anisotropy of the fluid have been
+found to be non-dynamical. When the maximum value
+of the energy density of the fluid (3 k2, where kis a
+positive constant) is considered, the universe expands
+isotropically and the fluid mimics the conventional vac-
+uum energy. Lower values of the energy density of the
+fluid give rise to an anisotropy both in the expansion
+and EoS parameter of the fluid.
+Acknowledgements ¨Ozg¨ ur Akarsu was supported
+in part by The Scientific and Technological Research
+Council of Turkey (T ¨UB˙ITAK). Some of this work was
+carried out while ¨O. Akarsu was visiting the Depart-
+ment of Applied Mathematics and Theoretical Physics
+(DAMTP), University of Cambridge. ¨O. Akarsu would
+also like to thank Jonathan Middleton for the discus-
+sions he had with him.de Sitter expansion with anisotropic fluid in Bianchi type-I space-time 9
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\ No newline at end of file
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+arXiv:1001.0551v2  [astro-ph.HE]  7 Jul 2010Astronomy & Astrophysics manuscript no. mcmcII c∝circleco√yrtESO 2018
+August 26, 2018
+A Markov Chain Monte Carlo technique to sample transport and
+source parameters of Galactic cosmic rays
+II. Results for the diffusion model combining B/C and radioac tive nuclei
+A. Putze1,2, L. Derome2, and D. Maurin3,4,5
+1The Oskar Klein Centre for Cosmoparticle Physics, Departme nt of Physics, Stockholm University, AlbaNova, SE-
+10691 Stockholm, Sweden
+2Laboratoire de Physique Subatomique et de Cosmologie ( lpsc), Universit´ e Joseph Fourier Grenoble 1, CNRS/IN2P3,
+Institut Polytechnique de Grenoble, 53 avenue des Martyrs, Grenoble, 38026, France
+3Laboratoire de Physique Nucl´ eaire et des Hautes Energies ( lpnhe), Universit´ es Paris VI et Paris VII, CNRS/IN2P3,
+Tour 33, Jussieu, Paris, 75005, France
+4Dept. of Physics and Astronomy, University of Leicester, Le icester, LE17RH, UK
+5Institut d’Astrophysique de Paris ( iap), UMR7095 CNRS, Universit´ e Pierre et Marie Curie, 98 bis bd Arago, 75014
+Paris, France
+Received / Accepted
+ABSTRACT
+Context. Ongoing measurements of the cosmic radiation (nuclear, ele ctronic, and γ-ray) are providing additional insight
+into cosmic-ray physics. A comprehensive picture of these d ata relies on an accurate determination of the transport
+and source parameters of propagation models.
+Aims.A Markov Chain Monte Carlo method is used to obtain these para meters in a diffusion model. By measuring the
+B/C ratio and radioactive cosmic-ray clocks, we calculate t heir probability density functions, placing special empha sis
+on the halo size Lof the Galaxy and the local underdense bubble of size rh. We also derive the mean, best-fit model
+parameters and 68% confidence level for the various paramete rs, and the envelopes of other quantities.
+Methods. The analysis relies on the USINE code for propagation and on a Markov Chain Monte Carlo technique pre-
+viously developed by ourselves for the parameter determina tion.
+Results.The B/C analysis leads to a most probable diffusion slope δ=0.86+0.04
+−0.04for diffusion, convection, and reacceler-
+ation, or δ= 0.234+0.006
+−0.005for diffusion and reacceleration. As found in previous studi es, the B/C best-fit model favours
+the first configuration, hence pointing to a high value for δ. These results do not depend on L, and we provide simple
+functions to rescale the value of the transport parameters t o anyL. A combined fit on B/C and the isotopic ratios
+(10Be/9Be,26Al/27Al,36Cl/Cl) leads to L= 8+8
+−7kpc and rh= 120+20
+−20pc for the best-fit model. This value for rhis
+consistent with direct measurements of the local interstal lar medium. For the model with diffusion and reacceleration,
+L= 4+1
+−1kpc and rh= 3+70
+−3pc (consistent with zero). We vary δ, because its value is still disputed. For the model
+with Galactic winds, we find that between δ= 0.2 and 0.9, Lvaries from O(0) toO(2) ifrhis forced to be 0, but it
+otherwise varies from O(0) toO(1) (with rh∼100 pc for all δ/greaterorsimilar0.3). The results from the elemental ratios Be/B,
+Al/Mg, and Cl/Ar do not allow independent checks of this pict ure because these data are not precise enough.
+Conclusions. We showed the potential and usefulness of the Markov Chain Mo nte Carlo technique in the analysis of
+cosmic-ray measurements in diffusion models. The size of the diffusive halo depends crucially on the value of the dif-
+fusion slope δ, and also on the presence/absence of the local underdensity damping effect on radioactive nuclei. More
+precise data from ongoing experiments are expected to clari fy this issue.
+Key words. Methods: statistical – ISM: cosmic rays
+1. Introduction
+Almost a century after the discovery of cosmic ra-
+diation, the number of precision instruments devoted
+to Galactic cosmic ray (GCR) measurements in the
+GeV-TeV energy range is unprecedented. The GeV γ-
+ray diffuse emission is being measured by the Fermi
+satellite (The Fermi-LAT Collaboration 2009), while the
+TeV diffuse emission is within reach of ground ar-
+rays of Cerenkov Telescopes (e.g., Hess, Aharonian et al.
+2006;Milagro , Abdo et al. 2008). The high-energy spec-
+Send offprint requests to : Antje Putze, antje@fysik.su.setrum of electrons and positrons uncovered some sur-
+prising and still debated features ( Atic, Chang et al.
+2008;Fermi, Abdo et al. 2009; Hess, Aharonian et al.
+2008, 2009; Pamela, Adriani et al. 2009a; Ppp-bets ,
+Torii et al. 2008). For nuclei, many experiments (satellites
+and balloon-borne) have acquired data, that remain to be
+published ( Cream, Ahn et al. 2008; Tracer, Ave et al.
+2008;Atic, Panov et al. 2008; Pamela). Anti-protons are
+also being measured ( Pamela, Adriani et al. 2009b) and
+are targets for future satellite and balloon experiments
+(Ams-02,Bess-Polar). Anti-deuteron detection should be
+achieved in a few years ( Ams-02, Choutko & Giovacchini
+1A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+2008;Gaps, Fuke et al. 2008). A complementary view
+of cosmic-ray propagation is given by anisotropy mea-
+surements from ground experiments of high energy (e.g.,
+the Tibet Air Shower Arrays, Amenomori et al. 2006;
+Super-Kamiokande-I detector, Guillian et al. 2007; Eas-
+top, Aglietta et al.2009). This multi-messengerand multi-
+energy picture will soon be completed: neutrino detectors
+arestillindevelopment(e.g., Icecube ,Km3net),butiden-
+tifying the sources of the GCRs should be within reach a
+few years after data collection (Halzen et al. 2008).
+All these measurements are probes to understand-
+ing and uncovering the sources of cosmic rays, the
+mechanisms of propagation, and the interaction of CRs
+with the gas and the radiation field of the Galaxy
+(Strong et al. 2007). It is important to determine the
+propagation parameters, because their value can be
+compared to theoretical predictions for the transport
+in turbulent magnetic fields (e.g., Casse et al. 2002;
+Ptuskin et al. 2006; Minnie et al. 2007; Tautz et al. 2008,
+Yan & Lazarian 2008 and references therein), or related to
+the source spectra predicted in acceleration models (e.g.,
+Marcowith et al. 2006; Uchiyama et al. 2007; Plaga 2008;
+Reville et al. 2008, Reynolds 2008 and references therein).
+The transport and source parameters are also related to
+Galactic astrophysics (e.g., nuclear abundances and stellar
+nucleosynthesis—Silberberg & Tsao 1990; Webber 1997),
+and to dark matter indirect detection (e.g., Donato et al.
+2004; Delahaye et al. 2008b).
+In the first paper of this series (Putze et al. 2009, here-
+after Paper I), we implemented a Markov Chain Monte
+Carlo (MCMC) to estimate the probability density func-
+tion (PDF) of the transport and source parameters. This
+allowed us to constrain these parameters with a sound sta-
+tistical method, to assess the goodness of fit of the models,
+and as a by-product, to provide 68% and 95% confidence
+level (CL) envelopes for any quantity we are interested in
+(e.g., B/C ratio, anti-proton flux). In Paper I, the analysis
+was performed for the simple Leaky Box Model (LBM) to
+validate the approach. We extend the analysis for the more
+realistic diffusion model, by considering constraints set by
+radioactive nuclei. The model is the minimal reaccelera-
+tion one, with a constant Galactic wind perpendicular to
+the disc plane (e.g., Jones et al. 2001; Maurin et al. 2001),
+allowing for a central underdensity of gas (of a few hun-
+dreds of pc) aroundthe solarneighbourhood (Donato et al.
+2002).
+The paper is organised as follows. In Sect. 2, we recall
+the main ingredients of the diffusion model, in particular
+the so-called local bubble feature. We briefly describe the
+MCMC technique in Sect. 3 (the full description was given
+in Paper I). We then estimate the transport parameters in
+the 1D and 2D geometry. In Sect. 4, this is performed at
+fixedL(halo size of the Galaxy), using the B/C ratio only.
+The analysis is extended in Sect. 5 by taking advantage
+of the radioactive nuclei to break the well-known degener-
+acy between the parameters K0(normalisation of the dif-
+fusion coefficient) and L. We then present our conclusions
+in Sect. 6.
+2. Propagation model
+The set of j= 1...nequations governing the prop-
+agation of nCR nuclei in the Galaxy is described in
+Berezinskii et al.(1990). Itisagenericdiffusion/convectionequationwithenergygainsandlosses.Dependingontheas-
+sumptions made about the spatial and energy dependence
+of the transport coefficients, semi-analytical or fully nu-
+merical procedures are necessary to solve this set of equa-
+tions. The solution also depends on the boundary condi-
+tions, hence on the geometry of the model for the Galaxy.
+Several diffusion models are considered in the lit-
+erature (Webber et al. 1992; Bloemen et al. 1993;
+Strong & Moskalenko1998;Jones et al.2001;Maurin et al.
+2001; Berezhko et al. 2003; Shibata et al. 2006; Evoli et al.
+2008; Farahat et al. 2008). We use a popular two-zone
+diffusion model with minimal reacceleration, where the
+Galactic wind is constant and perpendicular to the
+Galactic plane. The 1D and 2D version of this model are
+discussed, e.g., in Jones et al. (2001) and Maurin et al.
+(2001). For the sake of legibility, the solutions are given in
+Appendix A.
+Below, we reiterate the assumptions of the model, and
+describe the free parametersthat we constrainin this study
+(Sect. 2.4).
+2.1. Transport equation
+The differential density Njof the nucleus jis a function
+of the total energy Eand the position rin the Galaxy.
+Assuming a steady state, the transport equation can be
+written in a compact form as
+LjNj+∂
+∂E/parenleftbigg
+bjNj−cj∂Nj
+∂E/parenrightbigg
+=Sj.(1)
+The operatorL(we omit the superscript j) describes the
+diffusion K(r,E) and the convection V(r) in the Galaxy,
+but also the decay rate Γ rad(E) = 1/(γτ0) if the nu-
+cleus is radioactive, and the destruction rate Γ inel(r,E) =/summationtext
+ISMnISM(r)vσinel(E) for collisions with the interstellar
+matter (ISM), in the form
+L(r,E) =−∇·(K∇)+∇·V+Γrad+Γinel.(2)
+The coefficients bandcin Eq. (1) are respectively first
+and second order gains/losses in energy, with
+b(r,E) =/angbracketleftbigdE
+dt/angbracketrightbig
+ion,coul.−∇.V
+3Ek/parenleftbigg2m+Ek
+m+Ek/parenrightbigg
+(3)
++(1+β2)
+E×Kpp,
+c(r,E) =β2×Kpp. (4)
+In Eq. (3), the ionisation and Coulomb energy losses
+are taken from Mannheim & Schlickeiser (1994) and
+Strong & Moskalenko (1998). The divergence of the
+Galactic wind Vgives rise to an energy loss term re-
+lated to the adiabatic expansion of cosmic rays. The last
+term is a first order contribution in energy from reacceler-
+ation. Equation (4) corresponds to a diffusion in momen-
+tum space, leading to an energy gain. The associated dif-
+fusion coefficient Kpp(in momentum space) is taken from
+the model of minimal reacceleration by the interstellar tur-
+bulence (Osborne & Ptuskin 1988; Seo & Ptuskin 1994). It
+is related to the spatial diffusion coefficient Kby
+Kpp×K=4
+3V2
+ap2
+δ(4−δ2)(4−δ), (5)
+2A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+cV
+cVcVconstant windConvectiveK(E)
+Diffusive HaloL
+2hrhz
+R(z=r=0)
+Fig.1.Sketch of the model: sources and interactions (in-
+cluding energy losses and gains) are restricted to the thin
+disc∝2hδ(r). Diffusion Kand convection Vctransport nu-
+cleiinboththedisc(half-height h)andthehalo(half-height
+L). The Galaxy radial extension is R. The local bubble is
+featured to be a cavity of radius rhin the disc devoid of
+gas.
+whereVais the Alfv´ enic speed in the medium.
+The source term Sjis a combination of i)
+primary sources qj(r,E) of CRs (e.g., super-
+novae), ii) secondary fragmentation-induced sources/summationtextmk>mj
+knISM(r)vσk→j
+frag(E)Nk(r,E), and iii) secondary
+decay-induced sources/summationtext
+kNk(r,E)/(γτk→j
+0). In partic-
+ular, the secondary contributions link one species to all
+heavier nuclei, coupling together the nequations. However,
+the matrix is triangular and one possible approach is to
+solve the equation starting from the heavier nucleus (which
+is always assumed to be a primary).
+2.2. Geometry of the Galaxy and simplifying assumptions
+TheGalaxyis modelledtobe athin discofhalf-thickness h,
+which contains the gas and the sources of CRs. This disc is
+embedded in a cylindrical diffusive halo of half-thickness L,
+where the gas density is assumed to be 0. CRs diffuse into
+both the disc and the halo independently of their position.
+A constant wind Vcperpendicular to the Galactic plane
+is also considered. This is summarised in Fig. 1 (see next
+section for the definition of rh).
+We use the δ(z) approximation introduced in Jones
+(1979), Ptuskin & Soutoul (1990), and Webber et al.
+(1992). Considering the radial extension Rof the Galaxy
+to be either infinite or finite leads to the 1D version or 2D
+version of the model, respectively. The corresponding sets
+of equations (and their solutions) obtained after these sim-
+plifications are presented in Appendix A. These assump-
+tions allow for semi-analytical solutions of the problem,
+as the interactions (destruction, spallations, energy gain
+and losses) are restricted to the thin disc. The gain is in
+the computing time, which is a prerequisite for the use
+of the MCMC technique, where several tens of thousands
+of models are calculated. These semi-analytical models re-
+produce all salient features of full numerical approaches
+(e.g., Strong & Moskalenko 1998), and they are useful for
+systematically studying the dependence on key parame-
+ters, or some systematics of the parameter determination
+(Maurin et al. 2010).
+We note that most of the results of the paper are
+based on the 1D geometry (solutions only depend on z),
+which is less time-consuming than the 2D one in terms ofcomputing time1. The parameter degeneracy is also more
+easily extracted and understood in this case (Jones et al.
+2001; Maurin et al. 2006). Nevertheless, the results for the
+2D geometry are also reported, as it has been used in
+a series of studies inspecting stable nuclei (Maurin et al.
+2001, 2002), β-radioactive nuclei (Donato et al. 2002),
+standard anti-nuclei (Donato et al. 2001, 2008, 2009) and
+positrons (Delahaye et al. 2008a). It has also been used
+to set constraints on dark matter annihilations in anti-
+nuclei (Donato et al. 2004), and positrons (Delahaye et al.
+2008b). The reader is referred to these papers, and espe-
+cially Maurin et al. (2001) for more details and references
+about the 2D case.
+2.3. Radioactive species and the local bubble
+Our model does not take into account all the observed ir-
+regularities of the gas distribution, such as holes, chimneys,
+shell-like structures, and disc flaring. The main reason is
+that as far as stable nuclei are concerned, only the average
+grammage crossed is relevant when predicting their flux
+(which motivates LBM). As such, the thin-disc approxima-
+tion is a good trade-off between having a realistic descrip-
+tion of the structure of the Galaxy and simplicity.
+However, the local distribution of gas affects the flux
+calculation of radioactive species (Ptuskin et al. 1997;
+Ptuskin & Soutoul 1998; Donato et al. 2002). We consider
+a radioactive nucleus that diffuses in an unbound volume
+and decays with a rate 1 /(γτ0). In spherical coordinates,
+appropriate to describe this situation, the diffusion equa-
+tion reads
+−K△rG+G
+γτ0=δ(r). (6)
+The solution for the propagator G(the flux is measured at
+r=0for simplicity) is
+G(r′)∝e−r′/√Kγτ0
+r′. (7)
+Secondary radioactive species, such as10Be, originate from
+the spallations of the CR protons (and He) with the ISM.
+We model the source term to be a thin gaseous disc, except
+in a circular region of radius rhat the origin. In the δ(z)
+approximation (see Fig. 1) and in cylindrical coordinates,
+Q(r,z)∝Θ(r−rh)δ(z), (8)
+where Θ is the Heaviside function. The flux of a radioac-
+tive species is thus given by (we rewrite the propagator in
+cylindrical coordinates)
+N(r=z=0)∝/integraldisplay∞
+0/integraldisplay+∞
+−∞G(/radicalbig
+r′2+z′2)Q(r′,z′)r′dr′dz′.(9)
+The ratio of the flux calculated for a cavity/hole rhto that
+of the flux without hole ( rh= 0) is
+Nrh
+Nrh=0= exp/parenleftbigg−rh√Kγτ0/parenrightbigg
+= exp/parenleftbigg−rh
+lrad/parenrightbigg
+.(10)
+1The 2D solution is based on a Bessel expan-
+sion/resummation [see Eq. (A.14)]. For each Bessel order,
+an equation similar to that for the 1D geometry needs to be
+solved. Nine Bessel orders are in many cases enough to ensure
+convergence (Maurin et al. 2001), but at least 100 orders are
+required in the general case, which multiply the computing
+time by roughly the same amount.
+3A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+The quantity lrad=√Kγτ0is the typical distance on
+which a radioactive nucleus diffuses before decaying. Using
+K≈1028cm2s−1andτ≈1 Myr, the diffusion length is
+lrad≈200 pc. Hence, in principal, any underdensity on a
+scalerh∼100 pc about the Sun leads to an exponential
+attenuation of the flux of radioactive nuclei. This attenu-
+ation is both energy-dependent and species-dependent. It
+is energy-dependent because it decreases with the energy
+as the time-of-flight of a radioactive nucleus is boosted by
+both its Lorentz factor and the increase in the diffusion
+coefficient. It is species-dependent because nuclei half-lives
+for the standard Z <30 cosmic-rayclocks range from 0.307
+Myr for36Cl to 1.51 Myr for10Be.
+In this paper, we model the local bubble to be this sim-
+ple hole in the gaseous disc, as shown in Fig. 1. The expo-
+nential decrease in the flux of this modified DM, as given
+by Eq. (10), is directly plugged into the solutions for the
+standard DM ( rh= 0). In principle, i) the hole has also
+an impact on stable species as it decreases the amount of
+matter available for spallations, and ii) in the 2D geometry,
+a hole at R⊙= 8 kpc breaks down the cylindrical geome-
+try. However, in practice, Donato et al. (2002) found that
+the first effect is minor, and that the hole can always be
+taken to be the origin of the Galaxy (the impact of the R
+boundary being negligible for radioactive species).
+Other subleties exist, which were not considered in
+Donato et al. (2002). Indeed, the damping in the solar
+neighbourhood—combined with the production of the ra-
+dioactive species matching the data at low energy—means
+that at intermediate GeV/n energies, the flux of this ra-
+dioactive species is higher in the modified model (with
+rh∝ne}ationslash= 0) than in the standard one (with rh= 0). It also
+means that everywhere else in the Galactic disc, at all en-
+ergies, the radioactive fluxes are higher in the modified
+model (with damping). There are two consequences: i) all
+spallative products from these radioactive nuclei originate
+in an effective diffusion regionin the disc (Taillet & Maurin
+2003), the size of which may be much largerthan the size of
+the underdense bubble. In this case, these products ought
+to be calculated from the undamped fluxes; ii) the decay
+products of these radioactive nuclei (e.g.,10B, which orig-
+inates from the β-decay of10Be) are stable species that
+originate in an effective diffusion sphere (decay can occur
+not only in the disc, but in the halo). Both these effects
+must be considered because their contributions potentially
+affect the calculation, e.g., of the B/C and Be/B ratios (by
+means of the B flux), which are used to fit the models. We
+confirm that taking spallative products from the damped
+or undamped radioactivefluxes left these ratiosunchanged.
+On the other hand, for the decay products, the effect is of
+the order of 1−10%, which is in general enough to change
+the values of the best-fit parameters. However, the average
+flux (over the effective diffusion zone) from which the decay
+products originate lies between the damped and undamped
+values: the lower the effective diffusive sphere, the closer
+the flux is to the damped one. In particular, at low energy,
+when convection is allowed, the diffusion zone can be small
+(Taillet & Maurin 2003).
+To keep the approach simple, we use here the damped
+flux of radioactive species for all spallative and decay prod-
+ucts(aswasimplicitly assumedinDonato et al.2002).This
+approach is expected to provide the maximal possible size
+forrh(if a non-null value is preferred by the fit).2.4. Input ingredients and free parameters of the study
+2.4.1. Gas density
+The gas density scale height strongly varies with rde-
+pending on the form considered—neutral, molecular, or
+ionised (see, e.g., Ferri` ere 2001). We use the surface den-
+sity measured in the solar neighbourhood as a good es-
+timate of the average gas in the Galactic disc. We set
+nISM= 1 cm−3, which corresponds to a surface density
+ΣISM= 2hnISM∼6×1020cm−2(Ferri` ere 2001). The
+number fraction of H and He is taken to be 90% and
+10%, respectively. The ionised-hydrogen space-averaged
+density may be identified with the free-electron space-
+averaged density, which is the sum of the contributions
+of HIIregions and the diffuse component (G´ omez et al.
+2001; Ferri` ere 2001). The intensity of the latter is well
+measured 0 .018±0.002 cm−3(Berkhuijsen et al. 2006;
+Berkhuijsen & M¨ uller 2008), whereas the former depends
+stronglyon the Galactocentricradius r(Anderson & Bania
+2009). For the total electron density, we choose to set
+∝an}bracketle{tne−∝an}bracketri}ht= 0.033 and Te∼104K (Nordgren et al. 1992).
+The disc half-height is set to be h= 100 pc. It is not
+a physical parameter per sein theδ(z) approximation, al-
+though it is related to the phenomena occurring in the thin
+disc. Physical parameters are related to the surface den-
+sity, which is easily rescaled from that calculated setting
+h= 100 pc (should we use a different hvalue). In the 2D
+geometry, the boundary is set to be R= 20 kpc and the
+sun is located at R⊙= 8.0 kpc.
+2.4.2. Fragmentation cross-sections
+In Paper I, the sets of fragmentation cross-sections
+were taken from the semi-empirical formulation of
+Webber et al. (1990) updated in Webber et al. (1998) (see
+also Maurin et al. 2001 and references therein). In this pa-
+per, they are replaced by the 2003 version, as given in
+Webber et al. (2003). Spallations on He are calculated with
+the parameterisation of Ferrando et al. (1988).
+2.4.3. Source spectrum
+We assume that a universal source spectrum for all nu-
+clei exists, and that it has a simple power-law description.
+As in Paper I, we assume that Q(E)∝βηR−α. The pa-
+rameter αis the spectral index of the sources and ηen-
+codes the behaviour of the spectrum at low energy. The
+normalisations of the spectra are given by the source abun-
+dancesqj, which are renormalised during the propagation
+step to match the data at a specified kinetic energy per nu-
+cleon (usually∼10 GeV/n). The correlations between the
+source and the transport parameters and their impact on
+the transport parameter determination were discussed in
+Paper I. In this study, we set η=−1 andγ=α+δ= 2.65
+(Ave et al. 2008). Constraints on the source spectra from
+the study of the measured primary fluxes are left to a sub-
+sequent paper (Donato et al., in preparation).
+2.4.4. Free parameters
+We have two geometrical free parameters
+–L, the halo size of the Galaxy (kpc);
+4A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+–rh, the size of the local bubble (kpc), which is most of
+the time set to be 0 (to compare with models in the
+literature that do not consider any local underdensity);
+and four transport ones
+–K0,the normalisationofthediffusioncoefficient(in unit
+of kpc2Myr−1);
+–δ, the slope of the diffusion coefficient;
+–Vc, the constant convective wind perpendicular to the
+disc (km s−1);
+–Va, the Alfv´ enic speed (km s−1) regulating the reaccel-
+eration strength [see Eq. (5)].
+The diffusion coefficient is taken to be
+K(E) =βK0Rδ. (11)
+3. MCMC
+The MCMC method, based on the Bayesian statistics, is
+used here to estimate the full distribution—the so-called
+conditional probability-density function (PDF)—given
+some experimental data (and some prior density for these
+parameters). We summarise below the salient features of
+the MCMC technique.Adetailed descriptionofthe method
+can be found in Paper I. The issue of the efficiency, which
+was not raised in Paper I, is discussed in Appendix C.
+The Bayesian approach aims to assess the extent to
+which an experimental dataset improves our knowledge of
+a given theoretical model. Considering a model depending
+onmparameters
+θ≡{θ(1), θ(2), ..., θ(m)}, (12)
+we wish to determine the PDF of the parameters given the
+data,P(θ|data). This so-called posterior probability quan-
+tifies the change in the degree of belief one can have in the
+mparameters of the model in the light of the data. Applied
+to the parameter inference, Bayes theorem reads
+P(θ|data) =P(data|θ)·P(θ)
+P(data), (13)
+whereP(data) is the data probability (the latter does not
+depend on the parameters and hence, can be treated as
+a normalisation factor). This theorem links the posterior
+probability to the likelihood of the data L(θ)≡P(data|θ)
+and the so-called priorprobability, P(θ), which indicates
+the degree of belief one has beforeobserving the data.
+The technically difficult point of Bayesian parameter es-
+timates lies in the determination of the individual poste-
+rior PDF, which requires an (high-dimensional) integration
+of the overall posterior density. Thus an efficient sampling
+method for the posterior PDF is mandatory.
+In general, MCMC methods attempt to studying
+anytargetdistribution of a vector of parameters, here
+P(θ|data), by generating a sequence of npoints/steps
+(hereafter a chain)
+{θi}i=1,...,n≡{θ1,θ2, ...,θn}. (14)
+Eachθiis a vector of mcomponents, e.g., as defined in
+Eq. (12). In addition, the chain is Markovian in the sense
+that the distribution of θn+1is influenced entirely by the
+value of θn. MCMC algorithms are developed to ensureTable 1. Classes of models tested in the paper.
+Model Transport parameters Description
+I {K0, δ, Vc} Diffusion + convection
+II {K0, δ, Va}Diffusion + reacceleration
+III {K0, δ, Vc, Va} Diff. + conv. + reac.
+Table 2. Most probable values for B/C data only ( L=
+4 kpc).
+Model K0×102δ V cVa
+Data (kpc2Myr−1) (kms−1) (kms−1)
+I-F 0 .42+0.03
+−0.040.93+0.02
+−0.0313.5+0.3
+−0.3···
+II-F 9 .7+0.3
+−0.20.234+0.006
+−0.005··· 73+2
+−2
+III-F 0 .46+0.08
+−0.060.86+0.04
+−0.0418.9+0.3
+−0.438+2
+−2
+that the time spent by the Markov chain in a region of the
+parameter space is proportional to the target PDF value
+in this region. Here, the prescription used to generate the
+Markov chains is the so-called Metropolis-Hastings algo-
+rithm, which ensures that the stationary distribution of the
+chain asymptotically tends to the target PDF.
+The chain analysis is based on the selection of a subset
+of points from the chains (to obtain a reliable estimate of
+the PDF). Some steps at the beginning of the chain are
+discarded (burn-in length). By construction, each step of
+the chain is correlated with the previous steps: sets of inde-
+pendent samples are obtained by thinning the chain (over
+the correlation length). The fraction of independent sam-
+ples measuring the efficiency of the MCMC is defined to be
+the fraction of steps remaining after discarding the burn-
+in steps and thinning the chain. The final results of the
+MCMC analysis are the target PDF and all marginalised
+PDFs. They are obtained by merely counting the number
+of samples within the related region of parameter space.
+4. Results for stable species (fixed halo size L)
+For stable species, the degeneracy between the normalisa-
+tion of the diffusion coefficient K0and the halo size of the
+GalaxyLprevents us from being able to constrain both pa-
+rameters at the same time. We choose to set L= 4 kpc (we
+also setrh= 0, i.e., standard DM). The free transport pa-
+rameters are{K0, δ, Vc, Va}. The classes of models consid-
+ered are summarised in Table 1. The reference B/C dataset
+(denoted dataset F) used for the analysis is described in
+Appendix D.1.
+4.1. PDF for the transport parameters
+We begin with the PDFs of the parameters based on the
+B/C constraint(dataset F) for the variousclassesof models
+(I, II, or III). The PDFs are shown in Fig. 2.
+The first important feature is that the marginal distri-
+butions of the transport parameters (diagonals) are mostly
+Gaussian. From the off-diagonal distributions, we remark
+thatK0andδare negatively correlated. This originates in
+the low-energy relation K(E)∝K0Rδ, which should re-
+main approximately constant to reproduce the bulk of the
+data at GeV/n energy. The diffusion slope δis negatively
+5A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+13 1400.51 [km/s]cV
+0.85 0.9 0.95 113140.85 0.9 0.95 1051015δ
+0.003 0.004 0.00513140.85 0.9 0.95 10.0030.0040.005
+0.003 0.004 0.00505001000/Myr]2 [kpc0K
+Model I-F
+/d.o.f = 11.2)
+min2χ(
+0.22 0.240204060δ
+0.22 0.240.090.0950.10.105
+0.09 0.095 0.1 0.105050100150/Myr]2 [kpc0K
+0.22 0.247075
+0.09 0.095 0.1 0.1057075
+70 7500.10.2Va [km/s]Model II-F
+/d.o.f = 4.68)
+min2χ(
+18 19 2000.51 [km/s]cV
+0.8 0.91819200.8 0.90510 δ
+0.004 0.006 0.0081819200.8 0.90.0040.0060.008
+0.004 0.006 0.0080200400600 /Myr]2 [kpc0K
+35 40 451819200.8 0.9354045
+0.004 0.006 0.008354045
+35 40 4500.10.2 Va [km/s]Model III-F
+/d.o.f = 1.47)min2χ(
+Fig.2.From top to bottom: posteriorPDFs ofmodels I, II,
+andIII usingthe B/Cconstraint(datasetF). Thediagonals
+show the 1D marginalised PDFs of the indicated parame-
+ters. Off-diagonal plots show the 2D marginalised posterior
+PDFs for the parameters in the same column and same line
+respectively. The colour code corresponds to the regions of
+increasingprobability(from palerto darkershade), and the
+twocontours(smoothed)delimitregionscontaining,respec-
+tively, 68% and 95% (inner and outer contour) of the PDF.Table 3. Best-fitmodelparametersforB/Cdataonly( L=
+4 kpc).
+Model Kbest
+0×102δbestVbest
+c Vbest
+aχ2/d.o.f
+Data (kpc2Myr−1) (km s−1) (km s−1)
+I-F 0.42 0.93 13.5 ... 11.2
+II-F 9.74 0.23 ... 73.1 4.68
+III-F 0.48 0.86 18.8 38.0 1.47
+correlated with Va, which is related to a smaller δbeing
+obtained if more reacceleration is included. On the other
+hand, the positive correlation between δandVcindicates
+that larger δare expected for larger wind velocities.
+We show in Table 2 the most probable values of the
+transport parameters, as well as their uncertainties, corre-
+sponding to 68% confidence levels (CL) of the marginalised
+PDFs. The precision to which the parameters are obtained
+is excellent, ranging from a few % to 10% at most (for
+the slope of the diffusion coefficient δin III). This corre-
+sponds to statistical uncertainties only. These uncertainties
+are of the order of, or smaller than systematics generated
+from uncertainties in the input ingredients (see details in
+Maurin et al. 2010).
+As found in previous studies (e.g., Lionetto et al. 2005),
+for pure diffusion/reacceleration models (II), the value of
+the diffusion slope δfound is low (≈0.23 here). When con-
+vection is included (I and III), δis large (≈0.8−0.9). This
+scatterin δwasalreadyobservedinJones et al.(2001),who
+also studied different classes of models. The origin of this
+scatter is consistent with the aforementioned correlations
+in the parameters (see also Maurin et al. 2010).
+The best-fit model parameters (which are not always
+the most probable ones) are given in Table 3, along with
+the minimal χ2value per degree of freedom, χ2
+min/d.o.f
+(last column). As found in previous analyses (Maurin et al.
+2001, 2002), the DM with both reacceleration and convec-
+tion reproduces the B/C data more accurately than with-
+out:χ2/d.o.f= 1 .47 for III, 4.90 for II, and 11.6 for I. The
+B/C ratio associated with these optimal χ2values are dis-
+played with the data in Fig. 3. We note that the poorfit for
+II (compared to III) is explained by the departure of the
+model prediction from high-energy HEAO-3 data.
+4.2. Sensitivity to the choice of the B/C dataset
+For comparison purposes, we now focus on several datasets
+for the B/C data. Low-energy data points include ACE
+data, taken during the solar minimum period 1997-1998
+(de Nolfo et al. 2006). Close to submission of this pa-
+per, another ACE analysis was published (George et al.
+2009). The 1997-1998 data points were reanalysed and
+complemented with data taken during the solar maxi-
+mum period 2001-2003. The AMS-01 also provided B/C
+data covering almost the same range as the HEAO-3
+data (Tomassetti & AMS-01 Collaboration 2009). Hence,
+for this section only, we attempt to analyse other B/C
+datasets that include these components:
+–A: HEAO-3 [0 .8−40 GeV/n], 14 data points;
+–C:HEAO-3+lowenergy[0 .3−0.5GeV/n],22datapoints;
+–F: HEAO-3+low+high energy [0 .2−2 TeV/n], 31 data
+points;
+6A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+Ek/n (GeV/n)-110 1 10210310B/C ratio
+00.050.10.150.20.250.30.35
+||
+=225 MVΦ=250 MVΦ=250 MV)ΦIMP7-8 (
+=200 MV)Φ Spacelab-2 (
+=250 MV)Φ HEAO-3 (
+=225 MV)Φ Voyager1&2 (
+=225 MV)Φ ACE 97-98 (
+=425 MV)Φ CREAM (
+/dof=11.2)
+min2χ Model I-F   (
+/dof=4.68)
+min2χ Model II-F  (
+/dof=1.47)
+min2χ Model III-F (
+Fig.3.Best-fit ratio for model I (blue-dotted line), II
+(red-dashed line), and model III (black-solid line) using
+dataset F: IMP7-8, Voyager1&2, ACE-CRIS, HEAO-3,
+Spacelab, and CREAM. The curves are modulated with
+Φ = 250GV (and Φ = 225GV at low energy). The corre-
+sponding best-fit parameters are gathered in Table 3.
+Ek/n (GeV/n) 1 10210B/C ratio
+00.050.10.150.20.250.30.35
+IS
+IMP7-8 (Garcia-Munoz et al. 1987)
+Voyager1&2 (Lukasiak 1999)
+ACE 97-98 (De nolfo et al. 2006)
+ACE 97-98 (George et al. 2009)
+ACE 01-03 (George et al. 2009)AMS-01 (Tomassetti et al. 2009)
+HEAO-3 (Engelmann et al. 1990)
+Spacelab-2 (Swordy et al. 1990)
+CREAM 04 (Ahn et al. 2008)
+Fig.4.B/C data used in this section. Shown are the IS
+data (rescaled from TOA data using EIS
+k=ETOA
+k+Φ, see
+Paper I). For several experiments, in addition to the error
+bars in the ratio, we display the energy interval from which
+the central energy point is obtained.
+–G1: as F, but with new ACE 1997-1998 data, 31 data
+points;
+–G2: as F, but with new ACE 2001-2003 data only, 31
+data points;
+–G1/2: using both 1997-1998 and 2001-2003 ACE data,
+37 data points;
+–H: as F, but HEAO-3replacedby AMS-01 data, 27data
+points.
+The data are shown in Fig. 4. Thanks to the high level
+of modulation for the 2001-2003ACE data, the IS (demod-
+ulated) B/C ratio covers nicely the gap between HEAO-
+3 and lower energy data. HEAO-3 and AMS-01 data also
+show consistency across their whole energy range .
+The best-fit model parameters for these data are shown
+in Table 4. The low-energy data play an important part in
+the fitting procedure: δdecreases by 0.1 when going from
+III-A to III-C, and the diffusion normalisation is decreased.
+When the CREAM data at higher energy are taken into ac-
+count (III-F), the best-fit diffusion slope δagain becomesTable 4. Best-fit model parametersbased on different B/C
+datasets.
+Model Kbest
+0×102δbestVbest
+c Vbest
+aχ2/d.o.f
+Data (kpc2Myr−1) (km s−1) (km s−1)
+III-A 2.51 1.00 21.7 35.4 2.11
+III-C 0.43 0.89 18.9 36.7 1.72
+III-F 0.48 0.86 18.8 38.0 1.47
+III-G1 0.53 0.84 18.0 37.4 1.80
+III-G2 0.46 0.85 20.0 39.6 2.73
+III-G1/2 0.53 0.83 19.0 39.1 2.94
+III-H 1.85 0.51 18.1 54.1 0.25
+slightly lower (from 0.89 to 0.86), but CREAM data un-
+certainty is still too important to be conclusive. The im-
+pact of the low-energy ACE reanalysed data points is seen
+when comparing III-F with III-G1: the scatter between the
+derived best-fit parameters is already of the order of the
+statistical uncertainty (see Table 2). The data taken either
+during the solar minimum period (G1) or the solar maxi-
+mumperiod(G2)coveradifferentenergyrange(seeFig.4).
+Theχ2
+minfor G2 is greater, which is not surprising, given
+the abnormaltrend followed by these data (empty circles in
+Fig. 4). Nevertheless, it is reassuring to see that they lead
+to consistent values of the transport parameters.
+If we now replace the HEAO-3 data with the AMS-
+01 data, the impact on the fit is striking: the best-fit dif-
+fusion slope δgoes from 0.86 to 0.51. As discussed in
+Maurin et al. (2010), HEAO-3 data strongly constrain the
+slope towards δ≈0.8, even if there is a systematic energy
+bias in the HEAO-3 data themselves. From the AMS-01
+data, we see that there could be a way of reconciling the
+presence of a Galactic wind and reasonable values of δ.
+However, the large error bars in AMS-01 data, reflected by
+the lowχ2/d.o.f value, does not allow to draw firm conclu-
+sions. Data in the same energy range from PAMELA would
+be helpful in that respect. Moreover,high energy data from
+subsequent CREAM flights or from the TRACER experi-
+ments will be a crucial test of the diffusion slope: at TeV
+energies, diffusion alone is expected to shape the observed
+spectra, so that the ambiguity with the effect of convection
+or reacceleration is lifted (Castellina & Donato 2005).
+4.3. Comparison of trends for the DM and for the LBM
+For completeness, we briefly comment on the similarities
+and differences between the results found here and in
+Paper I. To follow the organisation of the previous sec-
+tions, the comparison with the LBM is discussed for differ-
+ent classes of models (I, II, and III), and then for different
+datasets (A, B, and C). We note that the best-fit values
+presented below differ slightly for those given in Paper I, as
+an updated set of production cross-section is used.
+We recall that in the LBM (see Paper I), the free pa-
+rameters are the normalisation of the escape length λ0,
+δ, a cut-off rigidity R0, and a pseudo-Alfv´ enic speed Va.
+The latter is linked to a true speed by means of Va=
+Va×(hL)1/2, i.e.,Va= 0.41/2Vaforh= 0.1 kpc and
+L= 4 kpc. The diffusion coefficient at 1 GV is related
+to the escape length by means of K0≈0.5cׯµL/λ0,
+where we use µ= 2hn¯m= 1.34×10−3g cm−2, leading
+7A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+Table 5. Best-fit parameters on B/C data for the LBM.
+Model Kbest
+0×102δbestR0Vbest
+aχ2/d.o.f
+Data (kpc2Myr−1) (GV) (km s−1)
+I-F 2.36 0.56 5.70 ... 5.52
+II-F 5.26 0.38 ... 65.1 1.78
+III-F 4.19 0.43 2.94 53.9 1.56
+III-A 2.32 0.57 4.40 11.3 2.71
+III-C 4.13 0.44 3.10 53.6 2.26
+III-F 4.19 0.43 2.94 53.9 1.56
+toK0(kpc2Myr−1)≈0.82/λ0(g cm−2). The LBM pa-
+rameters gathered in Table 5 are obtained from the above
+conversions, to ease the comparison with the DM results.
+For the different classes of models (I, II and III), a com-
+parison of Table 3 with the first three rows of Table 5 in-
+dicates that the same trend is found. For instance, model I
+(without reacceleration)has a larger δthan those with, and
+model II (without convection/rigidity-cutoff) has a smaller
+δthan those with. The slope for model III (with both
+convection and reacceleration) is in-between. This effect is
+more marked for the DM than for the LBM. We note that
+model II (with reaccelerationbut without convection) is al-
+mostconsistentwithaKolmogorovspectrumofturbulence,
+but is inconsistent with the data. Concerning the different
+datasets (A, C, and F), again, the same trend as for the
+LBM is found (compare Table 4 and the last three rows of
+Table 5).
+The most striking difference between the two models
+(LBM and DM) concerns their δvalues. This difference can
+be explained in terms of non-equivalent parameterisation
+of the low-energy transport coefficient (see Maurin et al.
+2010 for more details). Apart from this, both the value of
+the Alfv´ enic speed and the normalisation of the diffusion
+coefficient K0in the two cases are fairly consistent when
+similar values of δare considered.
+4.4. Dependence of the parameters with L
+Allthepreviousconclusionswerederivedfor L= 4kpc,but
+hold for any other halo size. The evolution of the transport
+parameters with Lis shown in Fig. 5 (the best-fit values
+are consistent with those found in Maurin et al. 2002). In
+the three upper figures, we havesuperimposed the observed
+dependence a parametric formula.
+ForK0(top panel), the formulacan be understood if we
+consider the grammage of the DM. In the purely diffusive
+regime, we have λesc∝L/K. This means that when we
+varyL, to keep the same grammage in the equivalent LBM,
+we need to vary K0accordingly. We find that K0= 1.08×
+10−3(L/1 kpc)1.06kpc2Myr−1instead of K0∝L. The
+origin of the residual L1.06dependence is unclear. It may
+come from the energy loss and gain terms.
+For the reacceleration, the interpretation is also simple.
+From Eq. (5), Vashould scale as√K0, so that Va∝√
+L.
+We find Va= 18.21(L/1 kpc)0.53km s−1. This is ex-
+actlyVa∝√K0, with the dependence√
+L0.06as above.
+The quantities Vcandδare roughly constant with L. The
+χ2
+minsurface is rather flat, although a minimum is observed
+aroundL∼15 kpc (the presence of a minimum may be
+related to the presence of the decayed10Be into10B in
+]-1  Myr2  [kpc0K-310-210-1101.06 L-3=1.08 100K]-1    [km saV2100.53=18.21 LaV]-1    [km scV18.51919.50.016=18.38 LcVδ
+0.840.860.88
+L  [kpc] 1 10/d.o.f2χ
+1.451.51.55
+Fig.5.Best-fit parameters (III-F) as a function of the halo
+size of the Galaxy (blue circles). From top to bottom: K0,
+Va,Vc,δ, and the associated χ2
+min. In the first three figures,
+a parametric function matching the observed dependence
+is shown (dashed-red line).
+the B/C ratio). This flatness is a consequence of the de-
+generacy of K0/Lwhen only stable species are considered.
+Consequently, an MCMC with Las an additional free pa-
+rameter does not converge to the stationary distribution.
+A sampling of the Galactic halo size is possible if radioac-
+tive nuclei are considered to lift the above degeneracy (see
+Sect. 5).
+4.5. Summary of stable species and generalisation to the 2D
+geometry
+The transport parameters for both LBM (Paper I) and 1D
+DM, when fitted to existing B/C data, are consistent with
+both convection and reacceleration. The correlations be-
+tween the various transport parameters, as calculated from
+the MCMC technique, are consistent with what is expected
+8A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+Table 6. Best-fitmodelparametersonB/Cdata:1Dversus
+2D DM ( L= 4 kpc).
+Model Kbest
+0×102δbestVbest
+c Vbest
+aχ2/d.o.f
+Data (kpc2Myr−1) (km s−1) (km s−1)
+1D II-F 9.74 0.23 ... 73.1 4.68
+2D II-F 8.56 0.24 ... 68.6 4.67
+1D III-F 0.48 0.86 18.9 38.0 1.47
+2D III-F 0.42 0.86 18.7 35.5 1.46
+from the relationships between DMs and the LBM (e.g.,
+Maurin et al. 2006). From the B/C analysis point of view,
+it implies that even if we are unable to reach conclusions
+about the value of δ(see Maurin et al. 2010), once this
+value is known, all other transport parametersare well con-
+strained.
+The conclusions obtained for the 1D DM naturally hold
+for the 2D DM. We recall that the main difference between
+the 1D and 2D geometry is that i) the spatial distribution
+of sources, which was constant in 1D, is now q(r); and ii)
+the Galaxy has a side-boundary at a radius taken to be
+R= 20 kpc. As a check, we first used the 2D solution
+(presented in Appendix A.2) with R= 20 kpc, but set q(r)
+to be constant. The best-fit parameters were in agreement
+with those obtained from the 1D solution. We present in
+Table 6 the best-fit parameters for models II and III for
+L= 4 kpc in the 2D solution where q(r) follows the SN
+remnant distribution of Case & Bhattacharya (1998). The
+values for the 1D solution are also reported for the sake
+of comparison. The main difference is in the value of K0,
+which varies by∼10% and also affects Va(by means of the
+ratioVa/√K0, which is left unaffected). This is consistent
+with the variations found by Maurin et al. (2002).
+5. Results for radioactive species (free halo size L)
+We now attempt to lift the degeneracy between the halo
+size and the normalisation of the diffusion coefficient, using
+radioactive nuclei. The questions that we wish to address
+are the following: i) With existing data, how large are the
+uncertainties in Lfor a given model? ii) Do radioactive nu-
+clei provide different answers for models with different δ?
+iii) Is the mean value (and uncertainty) for Lobtained from
+a given isotopic/elemental ratio consistent with or stronger
+constrained than that obtained from another measured iso-
+topic/elemental ratio? iv) How does the presence of a local
+underdense bubble (modelled as a hole of radius rh, see
+Sect. 2.3) affect the conclusions?
+Until now, almost all studies have focused on the
+isotopic ratios of10Be/9Be,26Al/27Al,36Cl/Cl, and
+54Mn/Mn. An alternative, discussed in Webber & Soutoul
+(1998), is to consider the Be/B, Al/Mg, Cl/Ar, and Mn/Fe
+ratios. The advantage of considering these elemental ratios
+is that they are easier to measure than isotopic ratios, and
+thus provide a wider energy range to which we can fit the
+data. Taking ratios such as Be/B maximises the effect of
+radioactive decay, since the numerator represents the de-
+caying nucleus and the denominator the decayed nucleus.
+However, the radioactive contribution is only a fraction of
+the elemental flux, and HEAO-3 data were found to be lessTable 7. Most probable values for models II and III for
+the free parameters of the local bubble radius rhand/or
+the Galactic halo size L(constrained by B/C and10Be/9Be
+data).
+K0×102δ V cVaL r h
+(kpc2Myr−1) (kms−1) (kms−1) (kpc) (pc)
+II 8.6+0.2
+−0.20.239+0.005
+−0.007··· 69+2
+−2[4]···
+II 13+2
+−20.234+0.006
+−0.005··· 84+5
+−65.2+0.7
+−0.6···
+II 11+2
+−30.235+0.008
+−0.004···77+8
+−114+1
+−13+70
+−3
+III 0.41+0.04
+−0.070.86+0.04
+−0.0318.7+0.4
+−0.335+2
+−2[4]···
+III 6.1+0.8
+−0.80.86+0.03
+−0.0519.4+0.4
+−0.3135+11
+−1046+9
+−8···
+III 0.8+1
+−0.70.86+0.03
+−0.0418.7+0.5
+−0.455+31
+−218+8
+−7120+20
+−20
+constraining that the isotopic ratios in Webber & Soutoul
+(1998).
+Below, we consider and compare the constraints from
+both the isotopic ratios and the elemental ratios. The
+data used are described in Appendix D.2. We discard
+54Mn because it suffers more uncertainties than the oth-
+ers in the calculation (and also experimentally) due to the
+electron capture decay channel. The free parameters for
+which we seek the PDF are the four transport parameters
+{K0, δ, Vc, Va}, plus one{L}or two geometrical param-
+eters{L, rh}, depending on the configuration considered.
+The main results of this section are thus in identifying the
+PDF ofLfor the standard DM, and the PDFs of both L
+andrhfor the modified DM.
+5.1. PDFs of Landrhusing isotopic measurements
+We start with a simultaneous fit to B/C and10Be/9Be, for
+both model III (diffusion/convection/reacceleration), and
+model II (diffusion/reacceleration), the latter being fre-
+quently used in the literature.
+5.1.1. Simultaneous fit to B/C and10Be/9Be
+The marginalised posterior PDFs of Landrhand the cor-
+relations between these new free parameters and the prop-
+agation parameters of models II and III are given in the
+Figs. 6 and 7, respectively. The most probable values of the
+parameters are gathered in Table 7.
+For all configurations, the diffusion slope δand the
+Galactic wind Vcare unaffected by the addition of the free
+parameters Landrh. The B/C fit is degenerate in K0/L
+andVa/√
+K, so that the values of K0andVavary asL
+varies. For model III, their evolution follows the relations
+given in Fig. 5. This implies that there is a positive cor-
+relation between K0andVa, andK0andL, as seen from
+Figs. 6 and 7. The uncertainty in the diffusive halo size L
+is smaller for II than for III. This is a consequence of the
+inclusion of the constant wind, which decreases the resolu-
+tion onK0from 2% (Model II) to 10% (Model III)—see
+e.g., Table 2 or 7—hence broadening the distribution of L.
+Below, the results for the standard DM—for which rh
+is set to be 0—and those for the modified DM—for which
+rhis left as an additional free parameter—are discussed
+separately. This allows us to emphasise the impact of rh
+9A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+4 6 800.20.40.6 L [kpc]
+ = 0hr
+4 6 80.220.24δ
+L [kpc]
+4 6 80.10.150.2/Myr]2 [kpc0K
+L [kpc]
+4 6 880100 [km/s]aV
+L [kpc]
+2 4 600.20.4L [kpc]
+ 0≠ hrModel II
+2 4 60.220.24δ
+L [kpc]
+2 4 60.050.10.15/Myr]2 [kpc0K
+L [kpc]
+2 4 6406080100
+ [km/s]aV
+L [kpc]
+0 0.05 0.1 0.15246
+ [kpc]hrL [kpc]
+0 0.05 0.1 0.150.220.24
+ [kpc]hrδ
+0 0.05 0.1 0.150.050.10.15
+ [kpc]hr/Myr]2 [kpc0K
+0 0.05 0.1 0.15406080100
+ [kpc]hr [km/s]aV
+0 0.05 0.1 0.150510 [kpc]hr
+Fig.6.ModelII (diffusion/reacceleration):marginalisedposteriorPDFof the diffusivehalosize L(rightpanelsofthe first
+and second row) and the local bubble radius rh(right panel of the last row) for the standard ( rh= 0, first row) and the
+modified DM ( rh∝ne}ationslash= 0, second and last row), as constrained by using B/C and10Be/9Be data. The correlations between
+the geometrical parameters Landrhand the transport parameters δ,K0, andVa, are shown in the 2D histograms. The
+colour code corresponds to the regions of increasing probability (f rom paler to darker shades), and the two contours
+(smoothed) delimit regions containing respectively 68% and 95% (inne r and outer contour) of the PDF.
+40 60 8000.020.04L [kpc]
+ = 0hr
+40 60 801920 [km/s]cV
+L [kpc]
+40 60 800.80.9δ
+L [kpc]
+40 60 800.040.060.08/Myr]2 [kpc0K
+L [kpc]
+40 60 80120140160 [km/s]aV
+L [kpc]
+0 20 40 6000.020.040.06L [kpc]
+ 0≠ hrModel III
+0 20 40 60181920 [km/s]cV
+L [kpc]
+0 20 40 600.80.9δ
+L [kpc]
+0 20 40 6000.020.040.06/Myr]2 [kpc0K
+L [kpc]
+0 20 40 60050100150 [km/s]aV
+L [kpc]
+0 0.05 0.1 0.150204060
+ [kpc]hrL [kpc]
+0 0.05 0.1 0.15181920
+ [kpc]hr [km/s]cV
+0 0.05 0.1 0.150.80.9
+ [kpc]hrδ
+0 0.05 0.1 0.1500.020.040.06
+ [kpc]hr/Myr]2 [kpc0K
+0 0.05 0.1 0.15050100150
+ [kpc]hr [km/s]aV
+0 0.05 0.1 0.150102030  [kpc]hr
+Fig.7.Model III (diffusion/convection/reacceleration): same as in Fig. 6 . The transport parameters are now δ,K0,Va,
+andVc, with the geometrical parameters Landrh.
+on the other parameters, which is different for models II
+and III.Standard DM ( rh= 0):the parameter Lis constrained to
+be between 4 .6 and 5.9kpc for model II, having a most
+probable value at 5.2kpc—a result compatible with other
+10A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+studies (Moskalenko et al. 2001)—the posterior PDF of L
+extends from 25 to 85kpc for model III (most probable
+value at 46kpc). In terms of statistics, the best-fit model is
+stil model III, for which the χ2/d.o.f is 1.41.
+Modified DM ( rh∝ne}ationslash= 0):the presence of a local bubble re-
+sults in an exponential attenuation of the local radioactive
+flux, see Sect. 2.3 and Eq. (10). We thus expect to have
+a different best-fit parameter for Lin that case. The re-
+sulting posterior PDFs of Landrhand the correlations
+to the propagation parameters for this modified DM are
+given in Figs. 6 and 7 for models II and III respectively.
+The most probable values are gathered in Table 7 (third
+and last lines).
+As expected, the local bubble radius rhis negatively
+correlated with the Galactic halo size L. The effect is more
+striking for model III, where the favoured range for Lex-
+tends from 1 to 50kpc. The most probable value is 8+8
+−7kpc
+for a local bubble radius rh= 120+20
+−20pc. The χ2/d.o.f of
+this configuration is 1.28, instead of 1.41 for the standard
+DM. The improvement to the fit is statistically significant
+according to the Fisher criterion.
+The situation for model II is different. The halo size
+Lis already small for the standard configuration rh= 0.
+Adding the local bubble radius rhto the fit decreases the
+most probable value of Lonly slightly to 4+1
+−1kpc and the
+measured value of rhis compatible with 0pc. In addition,
+theχ2/d.o.f is 3.69 and hence poorer than for the configu-
+ration without the local bubble feature. In this model (dif-
+fusion/reacceleration, no convection), a local underdensity
+is not supported.
+5.1.2. Results and comparison with fits to26Al/27Al and
+36Cl/Cl
+We repeat the analysis for the remaining isotopic ratios.
+The resulting marginalised posterior PDFs of the Galactic
+halosize Landthelocalunderdensity rharegiveninFigs.8
+and 9 for models II and III, respectively. The correlation
+plots with the transport parameters are similar to those of
+Figs. 6 and 7 and are not repeated.
+Standard DM ( rh= 0):as for the10Be/9Be ratio (red-
+dotted line), Lis well constrained in model II at small val-
+ues for the26Al/27Al (green-long dashed-dotted line), and
+36Cl/Cl (blue dashed-dotted line) ratios, covering slightly
+different but consistent ranges from 4 to 14kpc. The width
+of the estimated PDFs increases when moving from the
+10Be/9Be ratio to the36Cl/Cl ratio, due to the decreasing
+accuracy of the data. In the same way, the adjustment to
+the data becomes poorer, as expressed by the increase in
+χ2/d.o.f from 3.59to 4.09. Used alone, the radioactiveratio
+10Be/9Be constrains the most precisely the halo size L, but
+the constraints obtained with the other radioactive ratios
+are completely compatible within the 2 σrange. The most
+likely value of Lis ascertained when the three radioactive
+ratios are fitted simultaneously (black solid line).
+The best-fit model is model III, where the overall cov-
+ered halo size range extends from 20 to 140kpc. The most
+probable value found for Lwith 68% confidence level (CL)
+errors is 62+7
+−10kpc.4 6 8 10 12 14 16 18 2000.10.20.30.40.50.60.70.80.9
+4 6 8 10 12 14 16 18 2000.10.20.30.40.50.60.70.80.9L [kpc]
+Be9Be/10
+Al27Al/26
+Cl/Cl36
+Cl/Cl36Al + 27Al/26Be + 9Be/10
+Model II
+ = 0hr
+2 4 6 8 10 12 14 16 1800.050.10.150.20.250.30.350.4
+2 4 6 8 10 12 14 16 1800.050.10.150.20.250.30.350.4L [kpc]
+Model II
+ 0≠ hr
+0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20246810121416
+0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20246810121416  [kpc]hr
+Be9Be/10
+Al27Al/26
+Cl/Cl36
+Cl/Cl36Al + 27Al/26Be + 9Be/10
+Fig.8.Model II: marginalised posterior PDFs of the
+Galactic geometry parameters for the standard DM ( rh=
+0, top panel) and for the modified DM ( rh∝ne}ationslash= 0, bot-
+tom panels). The four curves result from the combined
+analysis of B/C plus isotopic ratios of radioactive species:
+B/C+10Be/9Be (red dotted line), B/C+26Al/27Al (green
+long dashed-dotted line), and B/C+36Cl/Cl (blue dashed-
+dotted line). The black solid curve represents the extracted
+PDF resulting from a simultaneous fit of B/C plus all three
+isotopic ratios. All PDFs are smoothed.
+40 60 80 100 120 140 16000.010.020.030.040.05
+40 60 80 100 120 140 16000.010.020.030.040.05L [kpc]
+Be9Be/10
+Al27Al/26
+Cl/Cl36
+Cl/Cl36Al + 27Al/26Be + 9Be/10
+Model III
+ = 0hr
+0 20 40 60 80 100 12000.010.020.030.040.050.06
+0 20 40 60 80 100 12000.010.020.030.040.050.06L [kpc]
+Be9Be/10
+Al27Al/26
+Cl/Cl36
+Cl/Cl36Al + 27Al/26Be + 9Be/10
+Model III
+ 0≠ hr
+0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180510152025
+0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180510152025 [kpc]hr
+Fig.9.Same as in Fig. 8, but for model III.
+Modified DM ( rh∝ne}ationslash= 0):the resulting marginalised poste-
+rior PDFs of Lare shown in Figs. 8 and 9 (lower left) for
+models II and III, respectively. Again, the extracted PDFs
+for all radioactiveratiosarecompletely compatible for both
+models. As described above, the decrease in Lis more pro-
+nounced for model III than for model II. This decrease can
+11A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+be observed for all radioactive ratios, independently of the
+model chosen.
+The resulting marginalised posterior PDFs of rhare
+given in Figs. 8 and 9 (lower right) for models II and III,
+respectively. The addition of an underdensity in the local
+interstellar medium is preferred by the data in the best-fit
+model III, but it is disfavoured in model II. The most prob-
+able values for rhrange from 90pc for the36Cl/Cl ratio to
+140pc for the the26Al/27Al ratio, and the overall fit points
+to a most probable radius of 130+10
+−20pc.
+These results confirm and extend the slightly differ-
+ent analysis of Donato et al. (2002), who found that for
+model III, the best-fit values for rhwas∼80 pc (see also
+Appendix B).
+5.1.3. Envelopes of 68% CL
+Confidence contours(for any combination ofthe CR fluxes)
+corresponding to given confidence levels (CL) in the χ2dis-
+tribution can be drawn, as detailed in Appendix A and
+Sect. 5.1.4 of Paper I. From the MCMC calculation based
+on the B/C +10Be/9Be +26Al/27Al +36Cl/Cl constraint,
+we select all sets of parameters for which the χ2meets the
+68% confidence level criterion. For each set of these pa-
+rameters, we calculate the B/C and the three isotopic ra-
+tios. We store for each energy the minimum and maximum
+value of the ratio. The corresponding contours (along with
+the best-fit ratio) for models II (standard DM, red) and III
+(standard and modified DM, blue) are drawn in Fig. 10. To
+ease the comparison with the data, all results correspond
+to IS quantities (the approximation made in the demodu-
+lation procedure, see Paper I, is negligible with respect to
+the experimental error bars).
+We see that the presentdata alreadyconstrainvery well
+the various ratios for the standard DM. The difference be-
+tween the results of models II and III are more pronounced
+at high energy (effect of δ), as seen from the B/C ratio be-
+yond10GeV/n.Allcontoursarepinchedaround10GeV/n,
+which is a consequence of the energy chosen to renormalise
+the flux to the data in the propagation code. In princi-
+ple, the source abundance of each species may be set as
+an additional free parameter in the fit (Paper I), but at
+the cost of the computing time. The three isotopic ratios
+(10Be/9Be26, Al/27Al, and36Cl/Cl) provide a fair match to
+the data for all models, considering the large scatter and
+possible inconsistencies between the results quoted by vari-
+ous experiments. In particular, for10Be/9Be, new data are
+necessary to confirm the high value of the ratio measured
+at∼GeV/n energy.
+The envelope for the modified DM is quite large at
+high energy, because the uncertainty in rhis responsi-
+ble for a larger scatter in the other parameters. The two
+standard DM contain non-overlapping envelopes beyond
+GeV/n energies. This means that to disentangle the mod-
+els, having measurements of the above isotopic ratios in
+the 1−10 GeV/n may be more important than just having
+more and higher quality data at low energy.
+5.1.4. General dependence of Lwithδ(forrh= 0)
+To investigate the difference in the results obtained from
+models II and III, we fit B/C and the three isotopic ratios
+for different values of δ(a similar trend with Lis obtainedif just one isotopic ratio is selected). The analysis relies on
+the Minuit minimisation routine to quickly find the best-fit
+values, as described in Maurin et al. (2010). The evolution
+of the parameterswith δis shown on the left side of Fig. 11.
+The bottom panel shows the evolution of χ2
+min/d.o.f, where
+we recover that the best-fit δfor model II (dashed-blue
+line) lies around δ≈0.2, whereas that for model III (solid-
+black line) lies around δ≈0.8. As already underlined, the
+contribution to the χ2value is dominated by the B/C con-
+tribution because as discussed in Appendix C, the values
+of transport parameters that reproduce the B/C ratio are
+expected to remain within a narrow range. This explains
+what is observed in the various panels showing these com-
+binations. For model II, we emphasise that for δ/greaterorsimilar0.5,
+the best-fit value for Vais zero (Model II becomes a pure
+diffusion model).
+The most important result, given in the top panel, is for
+Las a function of δ, where any uncertainty in the determi-
+nation of δtranslates into an uncertainty in the determina-
+tion ofL. When a Galactic wind is considered (Model III,
+black-solidline),thecorrelationbetween Landδisstronger
+than for model II (no wind). There is no straightforward
+explanation of this dependence. The flux of the radioactive
+isotope can be shown to be Nrad(0)≈hq/√Kγτ0(e.g.,
+Maurin et al. 2006). Since secondary fluxes should match
+the data regardless of the value for δ, this implies that the
+ratio10Be/9Be depends only on√Kγτ0. At the same time,
+to ensure that the secondary-to-primary ratio is constant,
+wemustmaintainaconstant L/K.Thedifficultyisthatthe
+former quantity is a constant at low rigidity where the iso-
+topic ratio is measured, whereas the latter quantity should
+remain as close a possible to the B/C data over the whole
+energy range. Hence, all we can say is that the variation
+inLwithδis related to the variation in K0/Lwithδ, as
+shown in the second figure (left panel) of Fig. 11.
+We note that all the calculations in the paper are based
+on the W03 (Webber et al. 2003) fragmentation cross-
+sections. The impact of using the W03 set or the GAL09
+set (provided in the widely used GALPROP package2) on
+the determination ofthe halo size Lis shown as thin-dotted
+lines (left panel, same figure)3. Any difference existing be-
+tween these two sets of production cross-sections has no
+impact on the best-fit value for L: thin-dashed curves (ob-
+tained with GAL09 cross-sections) almost match the thick-
+solid curves (obtained with W03 cross-sections). For other
+ratios, the effect of the GAL09 cross-sections is always the
+same, so it is not discussed further.
+5.1.5. General dependence of Lwithδ(forrh∝ne}ationslash= 0)
+We repeat the analysis with the underdensity rhas an ad-
+ditional free parameter. The dependence of Landrhon
+the diffusion slope δis shown in the right panel of Fig. 11.
+A comparison between the left and the right panel shows
+that the combinations of parameters K0/L,Va/√K0, and
+Vcare almost unaffected by the presence of a local bubble;
+theχ2/d.o.f is also only slightly affected.
+Forδ/lessorsimilar0.2,rhis consistent with 0 for both
+model II (diffusion/reacceleration) and model III (diffu-
+2http://galprop.stanford.edu/web galprop/galprop home.html
+3Theimpacton thetransport parameters is detailed inSect.7
+of Maurin et al. (2010): the region of the best-fit values is
+slightly displaced, as seen in the figure.
+12A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+Ek/n (GeV/n)-110 1 10210310B/C
+00.050.10.150.20.250.30.35
+ISIMP7-8
+Voyager1&2
+ACE 97-98
+ACE 01-03
+AMS-01
+HEAO-3
+Spacelab-2
+CREAM 04
+Standard DM II
+Standard DM III
+Modified DM III
+Ek/n (GeV/n)-110 1 10210310Be9Be/10
+00.10.20.30.40.50.6
+ISIMP7-8
+Balloon #1
+Balloon #2
+Balloon #3
+ISEE-3
+Ulysses
+Voyager1&2
+ACE 97-98
+ISOMAX
+Standard DM II
+Standard DM III
+Modified DM III
+Ek/n (GeV/n)-110 1 10210310Al27Al/26
+00.020.040.060.080.10.120.14
+ISBalloon
+ISEE-3
+Voyager1&2
+Ulysses
+ACE 97-98Standard DM II
+Standard DM III
+Modified DM III
+Ek/n (GeV/n)-110 1 10210310Cl/Cl36
+00.050.10.150.20.25
+ISUlysses
+ACE 97-98Standard DM II
+Standard DM III
+Modified DM III
+Fig.10. Shown are the envelopes of 68% CL (shaded areas) and best-fit (t hick lines) ratios for the standard DM II
+(rh= 0, red) and for model III (standard and modified DM, blue) in the 1 D geometry (based on the B/C +10Be/9Be
++26Al/27Al +36Cl/Cl constraint). All quantities are IS. The data are demodulated using the approximate procedure
+EIS
+k=ETOA
+k+Φ.
+sion/convection/reacceleration). For model II, the size of
+rhsuddenly jumps to ∼100 pc. But for δ/greaterorsimilar0.3, it returns
+to the pure diffusion regime, rhdecreasing abruptly (to
+a non-vanishing value) and Lbecoming vanishingly small.
+In this regime, the thin-disc approximation is no longer
+valid and nothing can be said about it. For model III, the
+plateaurh∼100 pc is stable for all δ/greaterorsimilar0.2. The under-
+dense bubble also stabilises the value of the halo size L.
+The way of understanding this trend is as for the stan-
+dard DM, but now the flux of the radioactive species reads
+Nrad
+rh(0)∝exp(−rh/√Kγτ0)/√Kγτ0. The weaker depen-
+dence of Lwithδmust be represented by this formula.
+We underline that for all best-fit configurations leading to
+rh∝ne}ationslash= 0, the improvement is statistically meaningful com-
+pared to the case rh= 0.
+5.2. Isotopic versus elemental measurements
+A similar analysis can be carried out using elemental ra-
+tios instead of isotopic ones. As before, the best-fit values
+of well-chosen combinations of the transport parameters
+{K0, δ, Vc, Va}areleft unchangedwhen radioactivespecies
+are added to the fit (same values as in Fig. 11).δ0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8L (kpc)
+10210
+   Model III
+Be/B
+Al/Mg
+Cl/Ar   Model II
+Be/B
+Al/Mg
+Cl/Ar
+   Combined fit         
+B/C + elemental rad. ratio
+Fig.12. Best-fit value of the halo size Las a function of δ
+in standard DM, based on a fit on B/C plus a ratio where
+a radioactive species is present: B/C+Be/B (black small
+symbols), B/C+Al/Mg (blue medium-size symbols), and
+B/C+Cl/Ar(pink largesymbols). The dashed lines(square
+symbols) refer to model II, and the solid lines (circles) refer
+to model III.
+5.2.1. General dependence of Lwithδ
+For the standard DM ( rh= 0), the dependence of the diffu-
+sive halo size Lon the diffusion slope δis shown in Fig. 12,
+13A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+                    Combined fit                   
+Cl/Cl36Al + 27Al/26Be + 9Be/10B/C + 
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1L [kpc]
+110210
+III (diff.+conv.+reac.)
+II (diff.+reac.)= 0]h[r
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ]-1  [ kpc Myrδ/L.500K-110
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1/2)]δ)(4-2δ(4-δ30/[KaV050100
+]1/2 Myr-1 kpc-1[km s
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ] -1 [ km scV
+05101520
+δ    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1/d.o.f2χ
+1100.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1  [kpc]hr
+0.050.10.15
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1L [kpc]
+110210
+III (diff.+conv.+reac.)
+II (diff.+reac.) 0]≠h[r
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ]-1  [ kpc Myrδ/L.500K-110
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1/2)]δ)(4-2δ(4-δ30/[KaV050100
+]1/2 Myr-1 kpc-1[km s
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ] -1 [ km scV
+05101520
+δ    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1/d.o.f2χ
+110
+Fig.11. Left panel: standard DM model ( rh= 0)—thin-dotted lines are derived using the GAL09 instead of the W0 3
+fragmentation cross-sections. Right panel: modified DM model ( rh∝ne}ationslash= 0). For both panels, shown are the best-fit parame-
+ters on B/C +10Be/9Be +26Al/27Al +36Cl/Cl data, as a function of the diffusion slope δ. The latter is varied between
+0.1 and 1.0 for model II (blue lines, open and filled squares) and model III (black lines, open and filled circles). From
+top to bottom, L,K0/L,Va/√K0, andVcas a function of δare shown. The bottom panel shows the best χ2/d.o.f for
+eachδ.
+for the three combinations B/C + Be/B, B/C + Al/Mg,
+and B/C + Cl/Ar. The trend is similar to that for isotopic
+ratios:Lincreases with increasing δ. The main difference is
+that the increase is sharper for both models II and III. For
+the former, only a small region around δ≈0.2 corresponds
+to small halo sizes. For the latter, the halo size increases
+sharply above δ/greaterorsimilar0.6.For completeness, similar fits were carried out for the
+modified DM ( rh∝ne}ationslash= 0). However, adding an additional de-
+gree of freedom only worsens the situation, and the models
+converge to arbitrarily small or high values of Landrh.
+Finally, if we fit the combined B/C data, the three isotopic
+ratiosandthe threeelementalratios,wedonotobtainmore
+constraintsthanwhenfittingB/Candthethreeisotopicra-
+tios. This may indicate that the models have difficulties in
+14A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+Ek/n (GeV/n)-110 1 10210310Be/B
+0.20.250.30.350.40.450.5
+ISIMP7-8
+HEAO-3
+Ulysses
+Voyager1&2
+ACE 97-98
+Standard DM II
+Standard DM III
+Modified DM III
+Ek/n (GeV/n)-110 1 10210310Al/Mg
+0.10.120.140.160.18
+IS
+HEAO-3
+UlyssesStandard DM II
+Standard DM III
+Modified DM III
+Ek/n (GeV/n)-110 1 10210310Cl/Ar
+0.30.350.40.450.50.550.60.650.7
+IS
+HEAO-3
+UlyssesStandard DM II
+Standard DM III
+Modified DM III
+Fig.13. Same asin Fig. 10but forthe ratiosBe/B,Al/Mg,
+and Cl/Ar.
+fitting all these data together: either the model is incom-
+plete or the data themselves may show some inconsisten-
+cies. This is more clearly seen from the comparison of the
+model calculation and the data for these elemental ratios
+(see below).
+5.2.2. Envelopes of 68% CL
+From the same set of constraint as in Sect. 5.1.3 (i.e., B/C
+and the isotopic ratios of radioactivespecies only), we draw
+the CL for the elemental ratios in Fig. 13.
+Given their large error bars, the elemental ratios are in
+overall agreement with the data, except at low energy and
+especially for the Be/B ratio. The main difference betweenthe Be/B ratio and the two other ratios is that Be and B
+are pure secondary species, whereas all other elements may
+contain some primary contribution which can be adjusted
+to more closely match the data. This also explains why the
+Be/B ratio reaches an asymptotical value at high energy
+(related to the respective production cross-sections of Be
+and B), whereas the two others exhibit more complicated
+patterns. The low-energyBe/B ratio is related to either the
+model or the energy biases in the production cross-sections
+for these elements (which is still possible, e.g. Webber et al.
+2003), or to systematics in the data. To solve this issue,
+better data over the whole energy range are required.
+5.3. Summary and generalisation to the 2D geometry
+Using radioactive nuclei in the 1D geometry, we found
+that in model II (diffusion/reacceleration), L∼4 kpc
+andrh∼0, and for the best-fit model III (diffu-
+sion/convection/reacceleration), L∼8 kpc and rh∼
+120 pc. The halo size is an increasing function of the dif-
+fusion slope δ, but in model III the best-fit value for rh
+remains∼100 pc for any δ/greaterorsimilar0.3. This value agrees
+with direct observation of the LISM (see Appendix B).
+Measurement of elemental ratios of radioactive species are
+not yet precise enough to provide valuable constraints.
+For now, there are too large uncertainties and too many
+inconsistencies between the data themselves to enable us to
+point unambiguously toward a given model. Moreover, one
+has to keep in mind that any best-fit model is relative to a
+given set of data chosen for the fit (see Sect. 4.2). We note
+that there may be ways out of reconciling the low-energy
+calculation of the Be/B ratio with present data, e.g., by
+changing the low-energy form of the diffusion coefficient
+(Maurin et al. 2010), but this goes beyond the goal of this
+paper.
+All these trends are found for the models with 2D ge-
+ometry. We calculate in Table 8 the best-fit parameters for
+the standard model II ( rh= 0) and the modified model III
+(rh∝ne}ationslash= 0). The values for the 1D geometry are also reported
+forthe sakeofcomparison.Apartfromafew tensofpercent
+difference in some parameters, as emphasised in Sect. 4.5,
+somedifferences areexpected ifthe size ofthe diffusive halo
+Lis larger than the distance to the side boundary R, which
+isdR= 12 kpc in the 2D geometry. It is a well-known
+result that the closest boundary limits the effective diffu-
+sion region from where CR can originate (Taillet & Maurin
+2003). For model II, Lis smaller than dR. We obtain a
+smaller than 10% difference for K0, and a∼30% difference
+forL. For the modified model III, the halo size has a larger
+scatter(seeprevioussections),with Lbest
+1D= 13.6> dR.The
+geometry is thus expected to affect the determination of L.
+We find that Lbest
+2D= 4.3 and that the value of K0is thus
+Lbest
+1D/Lbest
+2D∼3 times larger, and Vais∼√
+3 times larger
+than in 1D.
+6. Conclusions
+We have used a Markov Chain Monte Carlo technique to
+extract the posterior distribution functions of the free pa-
+rameters of a propagation model. Taking advantage of its
+sound statistical properties, we have derived the confidence
+intervals (as well as confidence contours) of the models for
+15A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+Table 8. Best-fit parameters on
+B/C+10Be/9Be26+Al/27Al +36Cl/Cl: 1D vs 2D DM.
+Config. K0×102δ V cVaL r hχ2/d.o.f
+(kpc2Myr−1) (km s−1) (kpc)
+1D-II-L15.4 0.23 ···92.5 6.2 ···3.09
+2D-II-L14.9 0.24 ···90.8 8.8 ···3.04
+1D-III-Lrh1.90 0.83 18.9 73.5 13.6 0 .13 1.43
+2D-III-Lrh5.24 0.85 18.3 123. 4.3 0 .16 1.48
+fluxes and other quantities derived from the propagation
+parameters.
+In the first paper of this series (Paper I), we focused on
+the phenomenologically well-understood LBM to ease the
+implementation of the MCMC. In contrast, here we have
+analysed a more realistic DM. In agreement with previous
+studies, when B/C only is considered, we have confirmed
+that a model with diffusion/convection/reacceleration is
+more likely than the diffusion/reacceleration case. The for-
+mer would imply that δ∼0.8, whereas the latter would
+imply that δ∼0.2. This result does not depend on the
+halo size: we provided simple parameterisations to obtain
+the value of the transport parameters for any halo size L.
+If mere eye inspection of the published AMS-01 data shows
+consistency with the HEAO-3 data (covering the same en-
+ergyregion),aB/CanalysisbasedonAMS-01data(instead
+of HEAO-3) also indicates that convection and reaccelera-
+tionisrequired,but nowprovidingadiffusion slope δ∼0.5,
+closer to theoretical expectations. Data from PAMELA or
+high-energydata from CREAM and TRACER are required
+to help solving the long-standing uncertainty in the value
+forδ.
+A second important topic of this paper has been the
+halo size Lof the Galaxy and the impact of the underdense
+medium in the solar neighbourhood. The determination of
+Lis for instance crucial to predictions of antimatter fluxes
+from dark matter annihilations. The size of the local un-
+derdense medium is as important, as it can bias the de-
+termination of L. We provided a step-by-step study of the
+various radioactive clocks at our disposal. Our detailed ap-
+proach can serve as a guideline as how to take advantage of
+future high-precision measurements that will soon become
+available(e.g., from AMS). The main conclusionsabout the
+constraintsprovidedby the radioactivespecies are,in diffu-
+sion/reacceleration models, L∼4 kpc and no underdense
+local bubble is necessaryto match the data. For the best-fit
+model, which requires diffusion/convection/reacceleration,
+L∼8 kpc with rh∼120 pc. For both models, the halo
+size found is an increasing function of the diffusion slope
+δ. A striking feature is that in models with convection, the
+best-fit value for rhremains∼100 pc for any δ/greaterorsimilar0.3.
+For instance, the B/C AMS-01 data (which implies that
+δ∼0.5) and the radioactive ratios are consistent with a
+wind and a local underdense bubble. This very value of
+rh∼100 pc is also supported by direct observation of the
+LISM (see Appendix B).
+As emphasised in this study, the determination of the
+value of Landrhstrongly depends on the value of δ.
+For all these parameters, high-energy data of secondary-to-
+primary ratios, data in the ∼1 GeV/n−10 GeV/n range
+for isotopic ratios (of radioactive species), and/or data forthe radioactive elemental ratios in the 1 −100 GeV/n en-
+ergy range are necessary. This is within reach of several fly-
+ing and forthcoming balloon-borne projects and satellites
+(PAMELA, AMS).
+Acknowledgements. We thank C. Combet for a careful reading of the
+paper. A.Pis gratefulforfinancial supportfromthe Swedish Research
+Council (VR) through the Oskar Klein Centre. We acknowledge the
+support of the French ANR (grant ANR-06-CREAM).
+Appendix A: Solutions of the diffusion equation
+We provide below the solutions for the diffusion equation
+with a constant wind Vcand a single diffusion coefficient
+K(E) in the whole Galaxy. In the 1D version of the model
+(e.g., Jones et al. 2001), the source distribution and the gas
+density do not depend on r, so that the propagated fluxes
+depend only on z.
+The derivation of these solutions is very similar
+and has no additional difficulties to those experienced
+by Maurin et al. (2001), to which we refer the reader for
+more details. As both frameworks (1D and 2D) exhibit
+similar forms, formulae are written for the 1D model only.
+Formulae for the 2D case are obtained by replacing some
+1D quantities by their 2D counterparts, as specified below.
+A.1. 1D–model
+The starting point is the transport equation
+(Berezinskii et al. 1990). We assume that the diffusion
+coefficient Kdoes not depend on spatial coordinates. A
+constant wind Vcblows the particle awayfrom the Galactic
+disc, along the zdirection. In the thin-disc approximation
+(e.g., Webber et al. 1992), the diffusion/convection for the
+1D–model (discarding energy redistributions) is
+/braceleftbigg
+−Kd2
+dz2+Vc∂
+∂z+Γrad+2hΓtotδ(z)/bracerightbigg
+N≡LN=Q(z).
+(A.1)
+In this equation, Nis the differential density of a given
+CR species, Γ rad= 1/(γτ) is its decay rate, and Γ tot=/summationtext
+ISMnISMvσISMis its destruction rate in the thin gaseous
+disc (nISM=H, He). The right-handside (r.h.s)ofthe equa-
+tion is a generic source term, that contains one of the fol-
+lowing three contributions, i.e., Q(z) =P(z)+S(z)+R(z):
+i)P(z) = 2hδ(z)×qs
+0Q(E) is the standard primary source
+term for sources located in the thin disc. The quantity
+qs
+0is the source abundance of nucleus jwhose source
+spectrum is Q(E)∝βηR−α.
+ii)S(z) = 2hδ(z)×Γp→s
+totNp(z= 0,E) is the standard sec-
+ondary source term (also in the disc), where Γp→s
+tot=
+nvσp→sis the production cross-section of nucleus pinto
+s. This simple form originates from the straight-ahead
+approximation used when dealing with nuclei (see, e.g.,
+Maurin et al. 2001 for more details).
+iii)R(z) = Γr→s′
+decayNr(z,E) described a contribution from a
+radioactive nucleus r, decaying into s′in both the disc
+and the halo.
+The equation is even in zso that it is enough to solve it
+in the upper-half plane. The use of the standard boundary
+condition N(z=L) = 0 and continuity of the density and
+the currentat the disccrossingcompletely characterisesthe
+solution.
+16A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+A.1.1. Stable species
+For a mixed species, primary and secondary standard
+sources add up, so that, for a nucleus kwith no radioactive
+contribution, the source term is rewritten as
+Qm
+disc(E) =qm
+0Q(E)+/summationdisplay
+k>mΓk→m
+totNp(z= 0,E),(A.2)
+and the corresponding equation to solve is then
+LmNm= 2hδ(z)·Qm
+disc(E).
+We find the solution in the halo, apply the boundary con-
+ditionN(z=L) = 0, and then ensure continuity between
+the disc and the halo, so that
+Nm(z) =Nm(0)·exp(Vcz/2K)sinh(Sm(L−z)/2)
+sinh(SmL/2)(A.3)
+and
+Nm(0) =2hQm
+disc(E)
+Am. (A.4)
+The quantities SmandAmare defined as
+Sm≡/radicalbigg
+V2c
+K2+4Γm
+rad
+K2; (A.5)
+Am≡Vc+2hΓm
+tot+KSmcoth/parenleftbiggSmL
+2/parenrightbigg
+. (A.6)
+A.1.2. Adding a β-decay source term: general solution
+It is emphasisedin Maurin et al. (2001) that the10Be→10B
+channel contributes up to 10% in the secondary boron flux
+at low energy and cannot be neglected. Although the spa-
+tial distribution of a radioactive nucleus decreases expo-
+nentially with z, we have to consider that the source term
+is emitted from the halo, complicating the solution. The
+equation to solve for the nucleus j, which is β-fed by its
+radioactive parent ris
+LjNj= Γr→j
+decayNr(z,E),
+whereNr(z,E) is given by Eq. (A.3). The solution is found
+following the same steps as above, although it has a more
+complicated form (due to a non-vanishing source term in
+the halo).
+If we take into account both the standard source term
+Qj
+disc(E) and the radioactive contribution of the nucleus
+Nr, we obtain:
+Nj(z) =/braceleftbigg
+̟·sinh(Sj(L−z)/2)
+sinh(SjL/2)
+−νΘΛ·cosh(Sjz/2)
+cosh(SjL/2)/bracerightbigg
+×exp(Vcz/2Kj)
++ Θ/braceleftbigg
+λsinh/parenleftbiggSr
+2(L−z)/parenrightbigg
++Λcosh/parenleftbiggSr
+2(L−z)/parenrightbigg/bracerightbigg
+×exp(Vcz/2Kr),(A.7)
+where
+Θ≡−Γk→j
+rad
+Kj(λ2−Λ2)Nr(0)
+sinh(SrL/2)(A.8)and
+̟≡2hQj
+disc
+Aj+Θ
+Aj× (A.9)
+/braceleftbiggνai
+cosh(SjL/2)/bracketleftbig
+Vc+2hΓj/bracketrightbig
+−sinh(SrL/2)/bracketleftbigg
+a/parenleftbigg
+Vc(2−Kj
+Kr)+2hΓj/parenrightbigg
++aiKjSr/bracketrightbigg
+−cosh(SrL/2)/bracketleftbigg
+ai/parenleftbigg
+Vc(2−Kj
+Kr)+2hΓj/parenrightbigg
++aKjSr/bracketrightbigg/bracerightbigg
+,
+where
+κ≡1/Kr−1/Kj
+ν≡eκVcL/2Λ≡κSrVc
+2λ≡κV2
+c
+2Kr+Γr
+rad
+Kr−Γj
+rad
+Kj.
+Thesuperscripton Kindicatesthatthe diffusion coefficient
+is to be evaluated at a rigidity calculated for the nucleus m.
+The latter can differ from one nucleus to another because,
+the calculation is performed at the same kinetic energy per
+nucleon for all the nuclei (hence at slightly different rigidi-
+ties for different nuclei). To compare with the data, the flux
+is calculated at z= 0
+Nj(0)=̟+Θ/bracketleftBigg
+λsinh(SrL
+2)+Λcosh(SrL
+2)−νΛ
+coshSjL
+2/bracketrightBigg
+.
+(A.10)
+A.1.3. Solution including energy redistributions
+When energy redistributions are included, the solution
+Nh(z) in the halo remains the same because our model
+assumes no energy redistributions in that region. Only the
+last step of the calculation changes (ensuring continuity
+duringthedisccrossing).Thenewsolutionisdenoted N(0).
+For the case of a mixed species mwithout radioactive
+contribution, the result is straightforward: the solution for
+the halo is still given by Eq. (A.3), but Nm(0) is now given
+by
+Nm(0)=Nm(0)−2h
+Am/parenleftbigg
+b(E)dNm(0)
+dE+c(E)d2Nm(0)
+dE2/parenrightbigg
+,
+which is solvednumerically, Nm(0) being the solutionwhen
+energy terms are discarded, i.e., Eq. (A.4). The terms a(E)
+andb(E) describing energy losses and gains are discussed
+in Sect. 2.1.
+Whenaradioactivecontributionexists,theconstantleft
+to determine is ̟from Eq. (A.7), which we denote now ̟∗
+̟∗=̟−2h
+Aj/parenleftbigg
+b(E)dNj(0)
+dE+c(E)d2Nj(0)
+dE2/parenrightbigg
+.(A.11)
+As above, ̟denotes the quantity evaluated without energy
+redistribution, whereas Nj(0) denotes the equilibrium flux
+atz= 0. To ensureNj(0) also appears in the l.h.s. of the
+equation, we form the quantity
+Ξ≡Θ/bracketleftBigg
+λsinh(SrL
+2)+Λcosh(SrL
+2)−νΛ
+coshSjL
+2/bracketrightBigg
+.(A.12)
+17A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+HenceNm(0) =̟+ Ξ, and we can add to both sides of
+Eq. (A.11) the quantity Ξ, so that we recover the standard
+form
+Nj(0) =Nj(0)−2h
+Aj/parenleftbigg
+b(E)dNj(0)
+dE+c(E)d2Nj(0)
+dE2/parenrightbigg
+,
+which we solve numerically.
+This is the solution in the disc ( z= 0). The solution for
+anyzis obtained from Eq. (A.7), making the substitution
+̟→̟∗=Nj(0)−Ξ. (A.13)
+We note that for Θ = 0 (i.e., no radioactive contribution)
+the result for standard sources in the disc is recovered.
+A.2. 2D geometry
+Cylindrical symmetry is now assumed, both the CR den-
+sityNand the source terms depending on r. Compared to
+Eq. (A.1), the operator △rnow acts on N(r,z).
+An expansion along the first order Bessel function is
+performed
+N(r,z) =∞/summationdisplay
+i=1Ni(z)J0/parenleftBig
+ζir
+R/parenrightBig
+.(A.14)
+The quantity ζiis thei-th zero of J0, and this form auto-
+matically ensures the boundary condition N(r=R,z) = 0.
+We have
+−△rJ0/parenleftBig
+ζir
+R/parenrightBig
+=ζi
+R2J0/parenleftBig
+ζir
+R/parenrightBig
+,
+so that each Bessel coefficient Ni(z) follows an equation
+very similar to Eq. (A.1), where
+Γrad⇒Γrad+ζi
+R2,
+and where each source term must also be expanded on the
+Bessel basis. More details can be found in Maurin et al.
+(2001).
+The full solutions for mixed species, with stable or ra-
+dioactive parents, is straightforwardly obtained from 1D
+ones, after making the substitutions
+Nj(z)2D model=⇒Nj
+i(z), (A.15)
+Sj2D model=⇒Sj
+i≡/radicalBigg
+V2c
+K2+4ζ2
+i
+R2+4Γj
+rad
+K2,(A.16)
+Aj2D model=⇒Aj
+i≡2hΓj+Vc+KSj
+icoth/parenleftBigg
+Sj
+iL
+2/parenrightBigg
+,(A.17)
+and
+Θj(Sr,Nr(0))=⇒Θj
+i(Sr
+i,Nr
+i(0)), (A.18)
+̟j(Sj,Aj)=⇒̟j
+i(Sj
+i,Aj
+i), (A.19)
+λr(Sr)=⇒λr
+i(Sr
+i). (A.20)
+The above formulae, for the radioactive source, dif-
+fer slightly from those presented in Maurin et al. (2001).
+However, the only difference is in the flux for z∝ne}ationslash= 0, which
+was not considered in this paper.Appendix B: The local bubble
+The underdensity in the local interstellar matter (LISM)
+is coined the local bubble4. The LISM is a region of ex-
+tremely hot gas ( ∼105−106K) and low density ( n/lessorsimilar
+0.005 cm−3) within an asymmetric bubble of radius /lessorsimilar
+65-250 pc surrounded by dense neutral hydrogen walls
+(Sfeir et al. 1999; Linsky et al. 2000; Redfield & Linsky
+2000). This picture has been refined by subsequent stud-
+ies, e.g., Lallement et al. (2003). The Sun is located inside
+a local interstellar cloud (LIC) of typical extension ∼50
+pc whose density NHI∼0.1 cm−3(Gloeckler et al. 2004;
+Redfield & Falcon 2008). Despite these successes, a com-
+plete mapping and understanding of the position and prop-
+erties of the gas/cloudlets filling the LISM, as well as the
+issue of interfaces with other bubbles remains challenging
+(e.g., Redfield & Linsky 2008; Reis & Corradi2008). Based
+on existing data, numerical simulations of the local bubble
+infer that it is the result of 14 −19 SNe occurring in a
+moving group, which passed through the present day local
+HIcavity 13 .5−14.5 Myr ago (Breitschwerdt & de Avillez
+2006). The same study suggests that the local bubble ex-
+panded into the Milky Way halo roughly 5 Myr ago.
+A last important point, is that of the existence of tur-
+bulence in the LISM to scatter off CRs. The impact of the
+underdense local bubble on the production of radioactive
+nuclei as modelled in Eq. (10) depends whether the trans-
+port of the radioactive nuclei in this region is diffusive or
+not. In a study based on a measurement of the radio scintil-
+lation of a pulsar located within the local bubble, Spangler
+(2008) infers that values for the line of sight component of
+the magnetic field are only slightly less, or completely con-
+sistent with, lines of sight through the general interstellar
+medium; the turbulence is unexpectedly high in this region.
+These pieces of observational evidences support the
+model used in Sect. 2.3, leading to an enhanced decrease
+in the flux of radioactive species at low energy. A detailed
+study should take into account the exact morphologyof the
+ISM (asymmetry, cloudlets). However, there are so many
+uncertainties in this distribution and the associated level of
+turbulence, that a crude description is enough to capture a
+possible effect in the CR data.
+Appendix C: MCMC optimisation
+The efficiency of the MCMC increases when the PDFs of
+the parameters are close to resembling Gaussians. Large
+tails in PDFs require more steps to be sampled correctly.
+A usual task in the MCMC machinery is to find some com-
+binations of parameters that ensure that these tails disap-
+pear. This was not discussed in the case of the LBM as the
+efficiency of the PDF calculation wassatisfactory.In 1D(or
+2D) DMs, the computing time is longer and the efficiency is
+foundtobelower.Tooptimiseandspeedupthecalculation,
+we provide combinations of parameters that correspond to
+a Gaussian distribution.
+A typical PDF determination with four free parame-
+ters{Vc, δ, K0, Va}(see next section) is shown in Fig. C.1.
+The diagonal of the left panel shows the PDF of these pa-
+rameters (black histogram), on which a Gaussian fit is su-
+perimposed (red line). We see a sizeable tail for the K0
+4For a state-of-the-art view on the subject, the reader is re-
+ferred to the proceedings of a conference held in 2008: The Local
+Bubble and Beyond II — http://lbb.gsfc.nasa.gov/
+18A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+18 19 2000.51 [km/s]cV
+0.8 0.91819200.8 0.90510 δ
+0.004 0.006 0.0081819200.8 0.90.0040.0060.008
+0.004 0.006 0.0080200400600 /Myr]2 [kpc0K
+35 40 451819200.8 0.9354045
+0.004 0.006 0.008354045
+35 40 4500.10.2 Va [km/s]Burn­In length: 2.9
+Correlation Length: 5.8
+: 135156/400000 = 33.789%indf
+/d.o.f: 1.47min2χChain analysisBinary space partitioning step
+Model III-F
+18 19 2000.51 [km/s]cV
+0.8 0.91819200.8 0.90510 δ
+0.032 0.034 0.0361819200.8 0.90.0320.0340.036
+0.032 0.034 0.0360200400 [kpc/Myr]δ/L*500K
+100 1101819200.8 0.9100110
+0.032 0.034 0.036100110
+100 11000.050.10.15 /kpc]Myr [km/s )δ)(4 - 2δ(4 - Burn­In length: 3.0
+Correlation Length: 1.9
+: 209947/400000 = 52.487%indf
+/d.o.f: 1.47min2χChain analysisBinary space partitioning step
+Model III-F
+Fig.C.1. Posterior PDFs of the model parameters (using the Binary Space P artitioning step—see Paper I, and the B/C
+constraint). The diagonals show the 1D marginalised PDF of the indica ted parameters, and the red line results from a
+Gaussian fit to the histogram. Off-diagonal plots show the 2D margin alised posterior PDFs for the parameters in the
+same column and same line, respectively. The colour code correspon ds to the regions of increasing probability (from paler
+to darker shades), and the two contours (smoothed) delimit regio ns containing respectively 68% and 95% (inner and
+outer contour) of the PDF. Left panel: PDFs for {Vc, δ, K0, Va}. Right panel: the same PDFs but shown for a different
+combination of the parameters {Vc, δ, K0/L×50δ, Va//radicalbig
+K0×3δ(4−δ2)(4−δ)}.
+parameter, and a small asymmetry for the Vaparameter.
+The right panel shows the same PDFs, but for the following
+combinations of the transport parameters:
+K0←→K0
+L×50δ(C.1)
+Va←→Va/radicalbig
+K0×3δ(4−δ2)(4−δ)(C.2)
+These forms are inspired by the known degeneracies be-
+tween parameters. For instance, in diffusion models, the
+secondary to primary ratio is expected to remain un-
+changed as long as the effective grammage∝an}bracketle{tx∝an}bracketri}htof the
+model is left unchanged. For pure diffusion, we obtain
+(Jones et al. 2001; Maurin et al. 2006) for the grammage
+∝an}bracketle{tx∝an}bracketri}ht= ΣvL/(2K) = ΣL/(2cK0(R/1GV)δ), where Σ is the
+surface density. Apart from the K0−Ldegeneracy, the
+parameters K0andδare correlated. We find that the com-
+bination K0×50δisappropriateforremovingthe K0PDF’s
+tail (see Fig. C.1). The origin of the value 50 is unclear. It
+may be related to the energy range coveredby B/C HEAO-
+3 data on which the fits are based. The combination used
+forVa[Eq. (C.2)] comes directly from the form of the reac-
+celeration term Eq. (5). Reacceleration only plays a role at
+low energy, so we can take Rδ≈1 and end up with the
+combination presented in Eq. (C.2). The independent ac-
+ceptance find, defined in Paper I as the ratio of the number
+of independent samples to the total step number, increases
+from 1/3 to 1/2 by using the above described parameter
+combinations for the four parameter model presented in
+Fig. C.1.A last combination is for the local bubble parameter rh:
+rh←→rh√K0. (C.3)
+This comes from the form ofEq. (10), wherethe flux damp-
+ing for radioactive species (due to the local bubble) is ef-
+fective only at low energy ( γ≈1,R≈1).
+Appendix D: Datasets for CR measurements
+D.1. B/C ratio
+Unless specified otherwise, the reference B/C dataset used
+throughout the paper is denoted dataset F: it consists of i)
+low-energy data taken by the IMP7-8 (Garcia-Munoz et al.
+1987), the Voyager 1&2 (Lukasiak et al. 1999), and
+the ACE-CRIS (de Nolfo et al. 2006) spacecrafts;
+ii) intermediate energies acquired by HEA0-3 data
+(Engelmann et al. 1990); and iii) higher energy data from
+Spacelab (Swordy et al. 1990) and the published CREAM
+data (Ahn et al. 2008). Other existing data are discarded
+either because of their too large error bars, or because of
+their inconsistency with the above data (see Paper I).
+D.2. Isotopic and elemental ratios of radioactive species
+For10Be/9Be, the data are taken from balloon
+flights (Hagen et al. 1977; Buffington et al. 1978;
+Webber & Kish 1979), including the ISOMAX balloon-
+borne instrument (Hams et al. 2004), and from
+the IMP-7/8 (Garcia-Munoz et al. 1977), ISEE-3
+(Wiedenbeck & Greiner 1980), Ulysses (Connell 1998),
+19A. Putze et al.: An MCMC technique to sample transport and sou rce parameters of Galactic cosmic rays. II.
+Voyager (Lukasiak et al. 1999), and ACE spacecrafts
+(Yanasak et al. 2001). For26Al/27Al, the data consist of
+a series of balloon flights (Webber 1982), and the ISEE-
+3 (Wiedenbeck 1983), Voyager (Lukasiak et al. 1994),
+Ulysses (Simpson & Connell 1998), and ACE spacecrafts
+(Yanasak et al. 2001). For36Cl/Cl, the data are from
+the CRISIS balloon (Young et al. 1981), and from the
+Ulysses (Connell et al. 1998) and ACE (Yanasak et al.
+2001) spacecrafts.
+The data for the elemental ratioscome from the HEAO-
+3 (Engelmann et al. 1990), Ulysses (Duvernois & Thayer
+1996), and the ACE spacecrafts de Nolfo et al. (2006). The
+published ACE data on Al/Mg and Cl/Ar (George et al.
+2009) were not used as Be/B is not provided.
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+arXiv:1001.0552v1  [math-ph]  4 Jan 2010On Bers generating functions for first order
+systems of mathematical physics
+Vladislav V. Kravchenko1, Marco P. Ramirez T.2
+1Departamento de Matem´ aticas, CINVESTAV del IPN, Unidad Qu er´ etaro,
+Libramiento Norponiente No. 2000, Fracc. Real de Juriquill a,
+Quer´ etaro, Qro. C.P. 76230 MEXICO e-mail: vkravchenko@qr o.cinvestav.mx
+2Escuela de Ingenier´ ıa de la Universidad La Salle,
+Benjam´ ın Franklin No. 47, Col. Condesa, C.P. 06140, M´ exic o D.F.
+e-mail: mramirez@lci.ulsa.mx
+December 28, 2018
+Abstract
+Considering one of the fundamental notions of Bers’ theory o f pseu-
+doanalytic functions the generating pair via an intertwini ng relation we
+introduce its generalization for biquaternionic equation s corresponding to
+different first-order systems of mathematical physics with v ariable coef-
+ficients. We show that the knowledge of a generating set of sol utions of
+a system allows one to obtain its different form analogous to t he com-
+plex equation describing pseudoanalytic functions of the s econd kind and
+opens the way for new results and applications of pseudoanal ytic function
+theory. As one of the examples the Maxwell system for an inhom ogeneous
+medium is considered, and as one of the consequences of the in troduced
+approach we find a relation between the time-dependent one-d imensional
+Maxwell system and hyperbolic pseudoanalytic functions an d obtain an
+infinite system of solutions of the Maxwell system. Other con sidered ex-
+amples are the system describing force-free magnetic fields and the Dirac
+system from relativistic quantum mechanics.
+1 Introduction
+Bers’ theory of pseudoanalytic functions mainly created in fifties o f the last
+century [2] offers interesting and still not fully explored tools for st udying and
+solving linear elliptic equations in the plane. Recent advances in the the ory and
+itsapplications[11] showthat someabstractconstructionspropo sedbyBerscan
+be made completely explicit and applicable to important equations of ma the-
+matical physics. For example, pseudoanalytic formal powers intro duced and
+studied by Bers and later on by many other mathematicians had been obtained
+1only in some very special situations which represented a considerab le obstacle
+for a further development of pseudoanalytic function theory. Th is obstacle has
+been substantially diminished in a recent work reported in [11] due to the fact
+that there has been found a method for constructing formal pow ers explicitly in
+a much more general situation. Moreover, it was shown that this co nstruction
+offers a tool for calculating explicitly complete systems of solutions o f linear
+elliptic second-order equations in the plane. This progress togethe r with some
+otherrecentdevelopmentsposedmoreopenquestionsconcernin gpseudoanalytic
+function theory, its generalizations and applications. For example, a possibil-
+ity to develop a hyperbolic pseudoanalytic function theory with applic ations to
+hyperbolic equations of mathematical physics was explored in [12] ( see also [11]
+and [5]). In the present work we use some of the results of [12] for obtaining
+an infinite system of solutions of the one-dimensional time-depende nt Maxwell
+system. Another related important open question is the developme nt of pseu-
+doanalytic function theory in higher dimensions. This was analized in a n umber
+of papers (see [14], [10], [1] and [11]), and it is clear that in this direction the
+development is barely starting.
+The aim of the present paper is to show that some fundamental idea s of
+pseudoanalytic function theory are valid in a general situation and a pplicable
+to linear systems of mathematical physics both elliptic and hyperbolic . The
+starting point of Bers’ theory is the concept of a generating pair w hich in a
+sense means a substitution of a pair of “rectilinear elements” of the plane 1 and
+ibyapairofquite arbitrary“curvilinearelements”–apairofcomplexf unctions
+FandGwhich only should enjoy the property of independence in the sense t hat
+any complex function wcan be represented in the form w=ϕF+ψGwhere
+ϕandψare real-valued functions. Beginning with generalizations of the firs t
+definitions from analytic function theory like the derivative, Bers sh ows that
+behind a generating pair there is always a corresponding generalized Cauchy-
+Riemann system which is usually called the Vekua orCarleman-Vekuaeq uation.
+The knowledge of a generating pair for a Vekua equation allows one to represent
+it in another form which is the equation for pseudoanalytic functions of the
+second kind. This form is very convenient for introducing a simple for mula for
+calculation of the ( F,G)-derivative in the sense of Bers and of the corresponding
+antiderivative and in fact represents a cornerstone of all furthe r constructions
+of pseudoanalytic function theory including formal powers.
+In this work we show that the concept of a generating set of funct ions in
+the sense of Bers is much more universal and can be introduced in re lation
+with first-order systems of mathematical physics with the aid of hy percomplex
+algebraic tools. This allows one to obtain another form of a correspo nding
+systemanalogoustothatdescribingpseudoanalyticfunctionsoft hesecondkind.
+In order to generalize the concept of a generating pair it resulted t o be fruitful
+to develop a slightly different approachto this concept via a certain in tertwining
+relation. We consider it in the next section. Another tool implemente d in this
+paper is the algebra of biquaternions. The related notations are int roduced also
+in section 2.
+We consider only few examples in this paper chosen in such a way that f rom
+2one side it becomes clear that our approachis generaland is applicab le to a wide
+variety of systems and from the other it can be seen that for each particular
+physically meaningful system some special interesting phenomena m ay occur.
+Thus,weconsiderthetime-dependentMaxwellsystemforinhomog eneousmedia
+(section 3), the system describing force-free magnetic fields (se ction 4) and the
+Dirac equation (section 5). In the case of the Maxwell system we int roduce a
+generating set of solutions, with its aid we obtain the Maxwell system in the
+form of an equation for pseudoanalytic functions of the second kin d. This allows
+us to find a relation of the Maxwell system in the one-dimensional cas e with
+the hyperbolic pseudoanalytic function theory developed in [12]. As a direct
+consequence of this relation we obtain an infinite system of solutions of the
+Maxwell equations.
+In the case of force-free magnetic fields we show that in fact one e xact so-
+lution is sufficient to obtain a corresponding generating quartet and to write
+down the second-kind equation. This interesting phenomenon leads to an ob-
+servation regarding the quotients of solutions. Namely, we obtain a differential
+equation satisfied by the quotients of solutions. This result seems t o be new
+even in the case of monogenic functions (a special case when the pr oportional-
+ity factor in the considered system vanishes identically). Finally, in se ction 5
+we show that the concept of a generating set is applicable to the Dira c system
+with electromagnetic and scalar potentials, and in this case as well th e system
+can be written in another form correspondingto pseudoanalytic fu nctions of the
+second kind.
+2 Preliminaries
+2.1 Biquaternions
+We will denote the algebra of biquaternions or complex quaternions b yH(C)
+with the standardbasicquaternionicunits denoted by e0= 1,e1,e2ande3. The
+complex imaginary unit is denoted by ias usual. The set of purely vectorial
+quaternions q=qis identified with the set of three-dimensional vectors.
+The quaternionic conjugation of a biquaternion q=q0+qwill be denoted
+as
+q=q0−q,
+and byq∗we denote the complex conjugation of q,
+q∗= Req−iImq.
+Sometimes the following notation for the operator of multiplication fr om the
+right-hand side will be used
+Mpq=q·p.
+The main quaternionic differential operator introduced by Hamilton h imself
+andsometimescalledtheMoisil-Theodorescooperatorisdefinedonc ontinuously
+3differentiable biquaternion-valued functions of the real variables x1,x2andx3
+according to the rule
+Dq=3/summationdisplay
+k=1ek∂kq,
+where∂k=∂
+∂xk.
+2.2 Pseudoanalytic functions
+In this subsection we introduce some basic concepts from Bers’ th eory of pseu-
+doanalytic functions [2] and give a slightly varied interpretation of th e notion
+of a generating pair. Precisely this different interpretation allows us to intro-
+duce the generating sets for the considered in the subsequent se ctions systems
+of mathematical physics.
+According to [2] a pair of arbitrary continuously differentiable with re spect
+to the real variables xandycomplex-valued functions FandGsatisfying the
+inequality
+Im(FG)>0 (1)
+in a domain Ω ⊂Cis called a generating pair. This inequality means that F
+andGare independent in the sense that any complex function Wdefined in Ω
+can be expressed in the form
+W=ϕF+ψG
+whereϕandψare real-valued functions.
+For a fixed generating pair and in the case when ϕandψare continuously
+differentiable one can define the ( F,G)-derivative of the function Win the fol-
+lowing way
+˚W=F ∂zϕ+G ∂zψ,
+where∂z=1
+2(∂x−i∂y). The derivative ˚Wexists iff the equality
+F ∂zϕ+G ∂zψ= 0 (2)
+holds. Here ∂z=1
+2(∂x+i∂y).
+Introducing the notation
+a=−F∂zG−G∂zF
+FG−FG, b=F∂zG−G∂zF
+FG−FG,
+equation (2) can be written in the form of a Vekua equation
+∂zW−aW−bW= 0. (3)
+Functionsaandbare known as characteristic coefficients of the generating
+pair (F,G) and solutions of (3) are known as pseudoanalytic functions, or mo re
+exactly (F,G)-pseudoanalytic functions of the first kind. Solutions of the cor-
+responding equation (2) regarded as complex-valued functions w=ϕ+iψare
+4called (F,G)-pseudoanalytic functions of the second kind. Note that by con-
+struction both generating functions FandGare (F,G)-pseudoanalytic of the
+first kind.
+Now let us consider the operator from Vekua equation (3), ∂z−a−bC
+where byCwe denote the operator of complex conjugation. Take an arbitrar y
+real-valued function ϕand consider the equality
+(∂z−a−bC)(ϕf) =f∂zϕ (4)
+wherefis some complex function. It is easy to see that this equality holds for
+any real-valued ϕifffis a particular solution of (3). In order to be able to
+consider a general solution of (3) another particular solution, say ,gis needed.
+In this case we can look for solutions of (3) in the form W=ϕf+ψgwhere
+ϕandψare real-valued functions and fandgare particular solutions of (3)
+if onlyfandgare independent in the sense explained above. Thus we arrive
+at the concept of a generating pair for a Vekua equation via the inte rtwining
+relation (4). We have then
+(∂z−a−bC)(ϕf+ψg) =f∂zϕ+g∂zψ
+which in particular gives us a relation between equations (3) and (2).
+3 Generating sets of solutions for the Maxwell
+system in inhomogeneous media
+Let us consider the Maxwell equations for inhomogeneous media
+rotH=ε∂tE+j, (5)
+rotE=−µ∂tH, (6)
+div(εE) =ρ, (7)
+div(µH) = 0. (8)
+Hereεandµare real-valued functions of coordinates, EandHare real-valued
+vectorfields depending on tand spatial variables, the real-valued scalarfunction
+ρand the real vector function jcharacterize the distribution of sources of the
+electromagnetic field.
+The wave propagation velocity will be denoted by c=1√εµ, the refraction
+index byn=√εµand the intrinsic impedance of the medium by Z=/radicalbigµ
+ε. As
+was shown in [7], [8], introducing the notations
+c=grad√c√c,Z=grad√
+Z√
+ZandV=√εE+i√µH
+one can rewrite system (5)-(8) in the form of a single biquaternionic equation
+(1
+c∂t+iD)V−MicV−MiZV∗=−(√µj+iρ√ε)
+5which in a sourceless situation becomes
+(1
+c∂t+iD)V−MicV−MiZV∗= 0. (9)
+Letϕbe a real-valued function. Then it is easy to see that the equality
+(1
+c∂t+iD−Mic−MiZC)[ϕV] = (1
+c∂t+iD)[ϕ]·V (10)
+holds iff Vis a solution of (9).
+Assume that {V1,...,V6}are solutions of (9) independent in the sense
+that for any complex vector function Vthere exist real valued functions ϕk,
+k= 1,2,...,6 such that V=/summationtext6
+k=1ϕkVk. This can be easily written as a
+condition on a corresponding determinant of a matrix formed by com ponents of
+Vk. Then due to (10) we have that V=/summationtext6
+k=1ϕkVkis a solution of (9) if and
+only if
+6/summationdisplay
+k=1(1
+c∂t+iD)[ϕk]·Vk= 0. (11)
+This equation for real-valued functions ϕk,k= 1,2,...,6 is a Bers’ equation
+for Maxwell pseudoanalytic functions of the second kind.
+Notice that in a frequently encountered in practice case µ= Const we have
+thatc=Zand (9) turns into the equation
+(1
+c∂t+iD)V−i(V+V∗)c= 0, (12)
+for which it is easy to propose a triplet of independent solutions in the form
+V4=ie1,V5=ie2,V6=ie3
+corresponding to a constant magnetic field. Thus, in order to rewr ite Maxwell’s
+system in the form (11) it is sufficient to find another triplet of solutio ns. In
+some cases this can be done relatively easy. Let us consider a strat ified medium,
+that isε=ε(x) and hence c=c1(x)e1=c′(x)
+2c(x)e1. Then the vectors
+V1=ce1,V2=e2
+c,andV3=e3
+c
+aresolutionsof(12). Consequently, theMaxwellsysteminthiscas eisequivalent
+to equation (11) (for Maxwell pseudoanalytic functions of the sec ond kind):
+(1
+c∂t+iD)ϕ1·ce1+3/summationdisplay
+k=2(1
+c∂t+iD)ϕk·ek
+c+6/summationdisplay
+k=4(1
+c∂t+iD)ϕk·iek−3= 0.
+For a better understanding of this equation as well as of equation ( 12) let us
+consider solutions depending on tandxonly:
+(1
+c(x)∂t+ie1∂x)V(t,x)−i(V(t,x)+V∗(t,x))c1(x)e1= 0.
+6One can observe that equations for V1and those for V2,V3are not coupled. We
+have
+(1
+c∂t+ie1∂x)V1−i(V1+V∗
+1)c1e1= 0 (13)
+and
+(1
+c∂t+ie1∂x)(V2e2+V3e3)−i((V2+V∗
+2)e2+(V3+V∗
+3)e3)c1e1= 0.(14)
+The first of these equations can be easily solved. Note that ∂tV1≡0 and
+∂xV1−2c1ReV1= 0. ThatisIm V1≡Constand∂xReV1−c′(x)
+c(x)ReV1= 0which
+gives us a general form of the component V1in the case under consideration,
+V1=a1c(x)+ia2wherea1anda2are arbitrary real constants.
+Now let us consider equation (14). It can be written as the following b icom-
+plex equation
+(1
+c∂t+ie1∂x)Φ+ic1e1(Φ+Φ∗) = 0
+for the bicomplex function Φ = V2+V3e1.
+Denote by Nan antiderivative of the refraction index nand consider the
+following change of the variable x/ma√sto→ξ=N(x). Then the function Ψ( t,ξ(x)) =
+Φ(t,x) as a function of the variables tandξsatisfies the following equation
+(∂t+ie1∂ξ)Ψ(t,ξ)+ie1C′(ξ)
+2C(ξ)(Ψ(t,ξ)+Ψ∗(t,ξ)) = 0
+whereC(ξ(x)) =c(x). Note that introducing a new unity j=ie1which ob-
+viously satisfies the equality j2= 1 and considering the bicomplex function in
+the form Ψ = Ψ 1+Ψ2jwhere Ψ 1,2=u1,2+v1,2e1withu1,2andv1,2being real
+valued functions we can rewrite the last equation as follows
+(∂t+j∂ξ)Ψ(t,ξ)+jC′(ξ)
+2C(ξ)(Ψ(t,ξ)+Ψ∗(t,ξ)) = 0
+where Ψ∗= Ψ1−Ψ2j. Finally, applying the conjugation operator and intro-
+ducing the new bicomplex function W=√
+CΨ∗we arrive at the equation
+1
+2(∂ξ−j∂t)W−f′(ξ)
+2f(ξ)W∗= 0
+wheref=√
+C, which can be written in the form of a hypebolic Vekua equation
+∂zW−fz
+fW= 0 (15)
+where∂z=1
+2(∂ξ−j∂t) and instead of an asterisk we used bar for denoting
+the same conjugation with respect to j. This equation in the case when Whas
+values in the algebra of hyperbolic numbers which we denote as H, that is when
+W=W1+W2jwithW1andW2being real valued, was introduced and studied
+in [12] and [11] in relation to the Klein-Gordon equation and in [5] in relatio n to
+7the Zakharov-Shabat system. Notice that due to the fact that fis real valued,
+equation (15) in fact consists of two separate equations for H-valued functions,
+that is it reduces to a pair of equations which were considered in the p revious
+publications [5], [11] and [12]. In order to observe this one needs to wr ite the
+functionWin the form W=w1+w2e1wherew1andw2areH-valued. Then
+equation (15) is equivalent to the following pair of separate equation s
+∂zw1−fz
+fw1= 0 and ∂zw2−fz
+fw2= 0. (16)
+Now gathering all the introduced transformations we have that
+w1(t,ξ) =/radicalbig
+C(ξ)/parenleftBig/radicalbig
+/tildewideε(ξ)/tildewideE2(t,ξ)−√µ/tildewideH3(t,ξ)j/parenrightBig
+and
+w2(t,ξ) =/radicalbig
+C(ξ)/parenleftBig/radicalbig
+/tildewideε(ξ)/tildewideE3(t,ξ)+√µ/tildewideH2(t,ξ)j/parenrightBig
+where tilde means that the corresponding original function was writ ten as a
+function of ξ, e.g.,/tildewideε(ξ(x)) =ε(x).
+Thus, all theory developed in [12] and [11] is applicable in this case to t he
+hyperbolic pseudoanalytic functions w1andw2. As an interesting application
+let us mention a possibility to construct an infinite system of exact so lutions
+of (15) using the results from [12] and [11] combined with the elegan t formulas
+obtained by Bers and Gelbart in the elliptic case [2]. Namely, we notice t hat
+the pair of functions
+(F,G) = (f,j/f) (17)
+is a generating pair for both equations (16) and moreover it has a fo rm con-
+venient for applying the results of Bers and Gelbart. Following [2] (s ee [11,
+Sect. 4.2] for the slightly corrected formulas), we give explicit form ulas for the
+formal powers corresponding to the generating pair (17) with fdepending on
+one Cartesian variable ξ. For simplicity we assume that z0= 0 andF(0) = 1.
+In this case the formal powers are constructed in an elegant mann er as follows.
+First, denote
+X(0)(ξ) =/tildewideX(0)(ξ) = 1
+and forn= 1,2,...denote
+X(n)(ξ) =
+
+nξ/integraldisplay
+0X(n−1)(x)dx
+f2(x)for an odd n
+nξ/integraldisplay
+0X(n−1)(x)f2(x)dxfor an even n
+8/tildewideX(n)(ξ) =
+
+nξ/integraldisplay
+0/tildewideX(n−1)(x)f2(x)dxfor an odd n
+nξ/integraldisplay
+0/tildewideX(n−1)(x)dx
+f2(x)for an even n
+Then fora=a′+ja′′,a′,a′′∈Randz=ξ+tjwe have
+Z(n)(a,0,z) =f(ξ)Re∗Z(n)(a,0,z)+j
+f(ξ)Im∗Z(n)(a,0,z)
+where
+∗Z(n)(a,0,z) =a′n/summationdisplay
+m=0/parenleftbiggn
+m/parenrightbigg
+X(n−m)jmtm(18)
++ja′′n/summationdisplay
+m=0/parenleftbiggn
+m/parenrightbigg
+/tildewideX(n−m)jmtmfor an odd n
+and
+∗Z(n)(a,0,z) =a′n/summationdisplay
+m=0/parenleftbiggn
+m/parenrightbigg
+/tildewideX(n−m)jmtm(19)
++ja′′n/summationdisplay
+j=0/parenleftbiggn
+m/parenrightbigg
+X(n−m)jmtmfor an even n.
+For anya∈ Handn∈Nthe formal power Z(n)(a,0,z) is a solution of (16).
+Thus, the system of constructed formal powers gives an infinite s ystem of so-
+lutions of the Maxwell equations in the case under consideration. Up to now
+it is an open question what part of the kernel of (16) and conseque ntly of the
+Maxwell system (14) can be approximated by the obtained exact so lutions.
+4 Generating solution for force-freemagnetic fields
+The system describing force-free magnetic fields has the form
+rotB+αB= 0, (20)
+divB= 0,
+whereBis a vector describing the magnetic field and the commonly known
+as proportionality factor αis a real-valued function of spatial variables. This
+systemisofgreatimportanceinsuchfieldsashightemperaturesup erconductors
+9and dynamics of magnetofluids (see e.g. [4], [15], [16]). It has been ana lized
+by different methods including those of quaternionic analysis (see [3 ], [13] for
+the theory in the case when αis a constant and [8], [9] and [17] for some
+developments in a nonconstant case).
+Obviously system (20) can be written in the form of a quaternionic eq uation
+(D+α)B= 0
+which is completely equivalent to (20). In what follows we do not restr ict
+ourselves by purely vectorial null solutions of the operator D+αand consider
+the equation
+(D+α)B= 0 (21)
+on the class of continuously differentiable H(C)-valued functions Bwith the
+proportionality factor αbeing a complex-valued function. Applying the same
+idea as before we find out that for any scalar (complex-valued) fun ctionϕ∈
+C1(Ω) where Ω is some domain in R3the equality
+(D+α)(ϕb) = (Dϕ)b (22)
+holds with bbeing an H(C)-valued function iff bis a solution of (21). Thus,
+chosing four independent solutions of (21) bk,k=0,3 we can look for a general
+solution of (21) in the form B=/summationtext3
+k=0ϕkbkwhere the scalar functions ϕk
+should satisfy the equation
+3/summationdisplay
+k=0Dϕk·bk= 0
+which is an analogue of the Bers equation for pseudoanalytic functio ns of the
+second kind.
+Nevertheless one can notice that in the case of equation (21) in fac t it is
+sufficient to have only one particular solution b. Asbλwithλbeing a constant
+biquaternion is also a solution of (21) the generating quartet bk,k=0,3 can be
+proposed in the form bk=bekif onlybis invertible in any point of the domain
+of interest Ω.
+Thus, we obtain the following statement.
+Proposition 1 Letbbe a particular solution of (21), invertible in any point of
+Ω⊂R3. Then the general solution of (21) can be represented as the p roduct
+B=bΦwhere the components of the H(C)-valued function Φsatisfy the equation
+3/summationdisplay
+k=0DΦk·bek= 0. (23)
+Proof. Under the conditions of the proposition let us consider the function
+B=bΦ =/summationtext3
+k=0bΦkek. Now application of the operator D+αtoBgives
+(D+α)B=/summationtext3
+k=0DΦk·bek+/summationtext3
+k=0Φk·((D+α)b)ekfrom where it is seen
+thatBis a solution of (21) iff (23) is valid.
+10Remark 2 It should be noticed that the last proposition is also valid w henαis
+a full biquaternionic function.
+Fromthe abovepropositionthe followinginterestingfactabout the quotients
+of solutions of (21) follows.
+Proposition 3 Letfandgbe solutions of (21) and fbe invertible in the
+domain of interest. Then the function Φ =f−1gis a solution of the equation
+3/summationdisplay
+k=0DΦk·fek= 0 (24)
+and ifgis also invertible the inverse Ψ = Φ−1=g−1fis a solution of the
+equation
+3/summationdisplay
+k=0DΨk·gek= 0. (25)
+Proof.According to the previous proposition if gis a solution of (21) it can
+be represented in the form g=fΦ where the components of the biquaternionic
+function Φ = f−1gare solutions of (24). Now noticing that if gis invertible as
+well the function Φ is also invertible and hence consideration of f=gΦ−1leads
+to equation (25).
+Remark 4 This proposition is of course also valid in the special case α≡0
+corresponding to quaternionic monogenic or hyperholomorp hic functions. It is
+well known that in general a quotient of two monogenic functi ons must not be
+monogenic. Proposition 3 gives us a precise equation satisfi ed by the quotient.
+5 The Dirac equation
+We considerherethe Diracequationforonespin 1/2particleundert he influence
+of an electromagnetic potential, but in fact the procedure is applica ble to other
+kinds of physical potentials (scalar, pseudoscalar, etc.). The Dira c equation has
+the form /parenleftBigg
+γ0∂t−3/summationdisplay
+k=1γk∂k+im+iφγ0+i3/summationdisplay
+k=1Akγk/parenrightBigg
+Ψ = 0, (26)
+whereγ0,γ1,γ2andγ3are the Dirac γ-matrices (see, e.g., [19]), mis the mass
+of the particle, φis the electric potential and A1,A2andA3are components
+of the magnetic potential− →A. The wave function Ψ is a C4-vector function
+Ψ = (Ψ 0,Ψ1,Ψ2,Ψ3). Equation (26) is considered in some domain G ⊂R4.
+As was shown in [6] (see also [13] and [8]) the Dirac equation (26) can be
+written in the following quaternionic form
+(D−∂tMe1+a−Miφe1+me2)Φ = 0
+11where the purely vectorial quaternion ais obtained from the magnetic potential− →A, and the H(C)-valued function Φ is related to Ψ by an invertible matrix
+transformation (see [13], [8] and [11]). It is worth mentioning that th is form of
+the Dirac equation was recently rediscovered in [18].
+Consideration of solutions of (26) with fixed energy Ψ( t,x) = Ψ ω(x)eiωt
+leads to the biquaternionic equation
+(D+a+Mb)W= 0 (27)
+whereWis anH(C)-valued function of three spatial variables and b=−i(φ+
+ω)e1−me2.
+Proposition 5 Let the biquaternionic functions F0,F1,F2andF3be indepen-
+dent1solutions of (27). Then W=/summationtext3
+k=0ϕkFkwhereϕ0,ϕ1,ϕ2andϕ3are
+scalar functions is a solution of (27) if and only if the follo wing equation is
+satisfied
+3/summationdisplay
+k=0(Dϕk)Fk= 0. (28)
+Obviously, equation (28) is an analogue of equation (2) describing ps eudo-
+analytic functions of the second kind.
+6 Conclusions
+In the present paper it was shown that the concept of a generatin g pair is not
+limited to the classical pseudoanalytic function theory and can be int roduced in
+relation with a variety of first-order systems of mathematical phy sics. Here we
+considered the Maxwell system for inhomogeneous media, the syst em describ-
+ing force-free magnetic fields and the Dirac system from relativistic quantum
+mechanics. Nevertheless the approach presented in this paper an d based on
+the consideration of a generating solution as an intertwining operat or is clearly
+more general. The knowledge of a generating set of solutions makes it possible
+to rewrite the original system in a form analogous to the Vekua equa tion for
+pseudoanalytic functions of the second kind which opens the way to construc-
+tion of classes of solutions as was shown in section 3 for the Maxwell s ystem and
+perhaps more important to the development of Bers’ theory for t he correspond-
+ing system of mathematical physics including such concepts as the d erivative,
+antiderivative, formal powers, Taylor and Laurent series, etc.
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+Theor. 2008, v. 41, issue 6, 065205.
+[13] V. V. Kravchenko and M. V. Shapiro Integral representation s for spatial
+models of mathematical physics. Harlow: Addison Wesley Longman Lt d.,
+Pitman Res. Notes in Math. Series, v. 351, 1996.
+[14] H. Malonek, Generalizing the (F,G)-derivative in the sense of Ber s. In Clif-
+ford Algebras and Their Application in Mathematical Physics, Dordre cht:
+Kluwer Academic Publishers, 247-257, 1996.
+[15] G. E. Marsh, Force free magnetic fields: solutions, topology an d applica-
+tions.World Scientific, 1995.
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+with non-constant proportionality factor and their applications to High
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+arXiv:1001.0553v2  [astro-ph.HE]  7 Jul 2010Astronomy & Astrophysics manuscript no. systematics c∝circlecopyrtESO 2018
+October 17, 2018
+Systematic uncertainties on the cosmic-ray transport para meters
+Is it possible to reconcile B/C data with δ= 1/3orδ= 1/2?
+D. Maurin1,2,3, A. Putze4,5, and L. Derome4
+1Laboratoire de Physique Nucl´ eaire et des Hautes Energies ( lpnhe), Universit´ es Paris VI et Paris VII, CNRS/IN2P3,
+Tour 33, Jussieu, Paris, 75005, France
+2Dept. of Physics and Astronomy, University of Leicester, Le icester, LE17RH, UK
+3Institut d’Astrophysique de Paris ( iap), UMR7095 CNRS, Universit´ e Pierre et Marie Curie, 98 bis bd Arago, 75014
+Paris, France
+4Laboratoire de Physique Subatomique et de Cosmologie ( lpsc), Universit´ e Joseph Fourier Grenoble 1, CNRS/IN2P3,
+Institut Polytechnique de Grenoble, 53 avenue des Martyrs, Grenoble, 38026, France
+5The Oskar Klein Centre for Cosmoparticle Physics, Departme nt of Physics, Stockholm University, AlbaNova, SE-
+10691 Stockholm, Sweden
+Received / Accepted
+ABSTRACT
+Context. The B/C ratio is used in cosmic-ray physics to constrain the t ransport parameters. However, from the same
+set of data, the various published values show a puzzling lar ge scatter of these parameters.
+Aims.We investigate the impact of using different inputs (gas dens ity and hydrogen fraction in the Galactic disc,
+source spectral shape, low-energy dependence of the diffusi on coefficient, and nuclear fragmentation cross-sections) o n
+the best-fit values of the transport parameters. We quantify the systematics produced when varying these inputs, and
+compare them to statistical uncertainties. We discuss the c onsequences for the slope of the diffusion coefficient δ.
+Methods. The analysis relies on the propagation code USINE interface d with the Minuit minimisation routines.
+Results. We find the typical systematic uncertainties to be greater th an the statistical ones. The several published
+values of δ(from 0.3 to 0.8) can be recovered when varying the low-energ y shape of the diffusion coefficient and the
+convective wind strength. Models including a convective wi nd are characterised by δ/greaterorsimilar0.6, which cannot be reconciled
+with the expected theoretical values (1 /3 and 1/2). However, from a statistical point of view ( χ2analysis), models
+with both reacceleration and convection—hence large δ—are favoured. The next favoured models in line yield δ, which
+can be accommodated with 1 /3 and 1/2, but require a strong upturn of the diffusion coefficient at lo w energy (and no
+convection).
+Conclusions. To date, using the best statistical tools, the transport par ameter determination is still plagued by many
+unknowns at low energy ( ∼GeV/n). To disentangle all these configurations, measureme nts of the B/C ratio at TeV/n
+energies and/or combination with other secondary-to-prim ary ratios is necessary.
+Key words. Methods: statistical – ISM: cosmic rays
+1. Introduction
+A central question in Galactic cosmic ray (GCR) physics
+lies in the determination of the transport parameters.
+The standard procedure consists in fitting a secondary-to-
+primary ratio, e.g. B/C. In general, the parameters derived
+in different studies provide inconsistent values, especially
+for the slope δof the diffusion coefficient. It is important
+to understand the origin of these differences.
+The first attempts to get the best-fit value of the trans-
+port parameters, as well as their statistical uncertainties,
+were carried in Maurin et al. (2001) and Lionetto et al.
+(2005). Recently, we implemented a Markov chain Monte
+Carlo(MCMC)toestimatetheprobability-densityfunction
+(PDF) of the transport and source parameters for Galactic
+cosmic rays(Putze et al. 2009, 2010, hereafterPapersI and
+II). This sound statistical technique applied to current B/C
+data allowed us to estimate the statistical uncertainties
+on various parameters. We found typical uncertainties of
+Send offprint requests to : D. Maurin, dmaurin@lpnhe.in2p3.fr∼10−20%. Although such precision is expected given the
+accuracy of the data, one may wonder whether these re-
+sults are not dominated by systematic uncertainties from
+the input ingredients and assumptions made to do the cal-
+culation.
+This is an important issue since in our pre-
+vious studies, the best-fit slope observed in a
+diffusion/constant-convection/reacceleration propagation
+model (Maurin et al. 2001, 2002)—confirmed by our recent
+MCMC analysis (Paper II)—points to δ≈0.75−0.85.
+This is at variance with the result of the GALPROP
+code (Strong & Moskalenko 1998), where the best-fit
+models correspond to smaller δ≈0.3−0.4 (Lionetto et al.
+2005) in either a diffusion/linear-convection or a diffu-
+sion/reacceleration model. Moreover, none of these results
+agree with the best-fit slopes from leaky-box models (with
+rigidity cut-off), where δ≈0.5−0.6 (e.g., Webber et al.
+2003—no reacceleration—, or Paper I—with reaccelera-
+tion). Such a scatter in δwas also observed in Jones et al.
+(2001).
+1D. Maurin et al.: Systematic uncertainties on the cosmic-ra y transport parameters
+The propagation code USINE (Maurin et al., in prepa-
+ration) provides the GCR fluxes in both the framework
+of the leaky-box model (LBM) and the diffusion model
+(DM). Associated with an efficient minimisation tool for
+finding the best-fit parameters of a model (w/wo convec-
+tion, w/wo diffusive reacceleration), it allows these differ-
+ences to be addressed thoroughly. We also investigate how
+sensitive these best-fit parameters are to various input in-
+gredients/parameters.
+The paper is organised as follows. In Sect. 2, the DM
+used is briefly recalled. In Sect. 3, the free parameters of
+the study, our methodology, and the input ingredients—the
+systematic effects of which are studied in this paper—are
+described.InSect.4,wediscusstheconsequencesofvarying
+the gas characteristicson the transport parameter determi-
+nation. In Sect. 5, we focus on the effect of the source spec-
+trum on the best-fit values for δin models with or without
+convection/reacceleration. In Sect. 6, a similar analysis is
+carried out for the low-energy shape of the diffusion coeffi-
+cient. We repeat again the analysis in Sect. 7, using various
+production cross-section sets. In a final step, we explore in
+Sect. 8 the effect of biasing B/C HEAO-3 data at high en-
+ergy. We summarise, discuss our results, and conclude in
+Sect. 9.
+2. Description of the diffusion model
+The models and the equations are described in Paper II, to
+which we refer the reader for a complete discussion.
+2.1. Diffusion equation
+The differential density Njof the nucleus jis a function
+of the total energy Eand the position rin the Galaxy.
+Assuming steady-state, the transport equation can be writ-
+ten in a compact form as
+LjNj+∂
+∂E/parenleftbigg
+bjNj−cj∂Nj
+∂E/parenrightbigg
+=Sj.(1)
+The operator L(we omit the superscript j) describes the
+diffusion K(r,E) and convection V(r) in the Galaxy, the
+decay rate Γ rad(E) = 1/(γτ0) for radioactive species, and
+the destruction rate Γ inel(r,E) =/summationtext
+ISMnISM(r)vσinel(E)
+on the interstellar matter (ISM):
+L(r,E) =−∇·(K∇)+∇·V+Γrad+Γinel.(2)
+The coefficients bandcare respectively first and second
+order gains/losses in energy, with
+b(r,E) =/angbracketleftbigdE
+dt/angbracketrightbig
+ion,coul.−∇.V
+3Ek/parenleftbigg2m+Ek
+m+Ek/parenrightbigg
+(3)
++(1+β2)
+E×Kpp,
+c(r,E) =β2×Kpp. (4)
+Thecoefficient Kppisthediffusion coefficientinmomentum
+space, and it can take several forms (see later).
+2.2. Geometry of the Galaxy
+The Galaxy is modelled to be a thin disc of half-thickness
+h, which contains the gas and the sources of CRs. This discis embedded in a cylindrical diffusive halo of half-thickness
+L(where the gas density is assumed to be equal to 0). A
+constantwind V(r) = sign( z)·Vc×ez, perpendicularto the
+Galactic plane, is assumed. In this framework, CRs diffuse
+in the disc and in the halo independently of their position.
+Theseassumptionsallowforsemi-analyticalsolutionsofthe
+transport equation, as the interactions (destruction, spalla-
+tions, energy gain and losses) are restricted to the thin disc.
+Such semi-analytical models reproduce all salient features
+of full numerical approaches (e.g., Strong & Moskalenko
+1998).
+In this study, the disc half-height is set to h= 100 pc. It
+is not a physical parameter per sein the thin-disc approx-
+imation, but the phenomena occurring in the thin disc are
+related to it. The physical parameter is the surface density
+Σ: should a different hvalue be used, a rescaling always
+allows obtaining the same Σ.
+Considering the radial extension Rof the Galaxy to be
+infinite leads to the 1D version of the DM. This geometry,
+used in Jones et al. (2001) and in our Paper II, is also used
+in this analysis. The corresponding sets of equations and
+their solutions are presented in the Appendix of Paper II.
+They are not repeated here.
+3. Methodology
+As in Papers I and II, three different classes of diffusion
+models are considered. In addition, for completeness, we
+also treat the pure diffusion case:
+–Model 0 = {K0, δ}, i.e. pure diffusion ( Va=Vc= 0);
+–Model I = {K0, δ Vc}, i.e. no reacceleration ( Va= 0);
+–Model II = {K0, δ, Va}, i.e. no convection ( Vc= 0);
+–Model III = {K0, δ, Vc, Va}.
+The parameters K0andδcome from the standard form
+assumed for the diffusion coefficient, namely,
+K(E) =βK0Rδ. (5)
+3.1. Constrained parameters
+At most, for Model III, four transport parameters need to
+be determined from CR data:
+–K0,the normalisationofthediffusioncoefficient(in unit
+of kpc2Myr−1);
+–δ, the slope of the diffusion coefficient;
+–Vc, the constant convective wind perpendicular to the
+disc (km s−1);
+–Va, the Alfv´ enic speed (km s−1) regulating the reaccel-
+eration strength [see Eq. (6)].
+As in other studies, we use the B/C ratio to constrain these
+parameters. In DMs, the halo size of the Galaxy Lis an ex-
+tra free parameter. It cannot be determined solely from the
+B/C ratio because of the well-known degeneracy between
+K0andL. In this study, we choose to fix L. The value of
+the transport parameters for any other values of Lcan be
+obtained from simple scaling laws, as presented in Fig. 5 of
+Paper II (note that δdoes not depend on L). To keep the
+discussion as simple as possible, this paper is based on B/C
+data alone, with four free parameters {K0, δ, Vc, Va}and
+the halo size set to L= 4 kpc.
+2D. Maurin et al.: Systematic uncertainties on the cosmic-ra y transport parameters
+Table 1. Reference values for the inputs.
+Input name Default value/dependence/set
+Gas Σ = 6 .17·1020cm2, fH= 90%
+Source spectrum ηS=−1, α+δ= 2.65
+K(E) andKpp(E) Slab Alfv´ en (SA): Eqs. (5) & (6)
+Cross-sections W03 (Webber et al. 2003)
+Data B/C, dataset F†
+†31 data points from IMP7-8, Voyager 1&2, ACE, HEA0-3,
+Spacelab, and CREAM04
+Note 1. In subsequent tables of the paper, all results obtained with
+the default inputs are in italics.
+3.2. Inputs and default configuration
+The inputs that we examine and vary are the following
+(details and references are given in Paper II, Sect. 2.4):
+–gas surface density Σ ISM= 2hnISMand hydrogen frac-
+tionfH(in number),
+–source spectrum Qj(E) =qjβηSR−α(qjis scaled to
+match the measured elemental fluxes at ∼10 GeV/n),
+–spatial/momentum diffusion coefficients using different
+turbulence models (Schlickeiser 2002),
+–production cross-sections.
+The reference (default) values for these inputs are gathered
+in Table 1. The default diffusion coefficient Kpp(in mo-
+mentum space) is taken from the model of minimal reaccel-
+eration by the interstellar turbulence (Osborne & Ptuskin
+1988; Seo & Ptuskin 1994):
+Kpp×K=4
+3V2
+ap2
+δ(4−δ2)(4−δ), (6)
+whereVais the Alfv´ enic speed in the medium.
+Concerning the experimental data used to fit the B/C
+ratio, as in Paper II, we use the following dataset (de-
+noted F in Paper II), which consists of 31 B/C data points
+(see Fig. 5 of this paper): IMP7-8 (Garcia-Munoz et al.
+1987), Voyager 1&2 (Lukasiak et al. 1999), ACE-CRIS
+(de Nolfo et al. 2006), HEA0-3 (Engelmann et al. 1990),
+Spacelab (Swordy et al. 1990), and CREAM (Ahn et al.
+2008).
+3.3. Minimisation routine
+In Papers I and II, we adapted the MCMC technique for a
+propagation code. This allows the PDF of the parameters
+to be obtained along with their statistical uncertainties.
+However, this technique relies on thousands of calculations
+for a single setting of the inputs, which is not optimal for
+speed.Here,weareonlyinterestedinthebest-fitvalues,not
+in the PDF of the parameters.Therefore, we use the Minuit
+library(aCERNlibrary),whichprovidesminimisationrou-
+tines. Instead of a few hours of distributed calculations for
+the MCMC technique, a few minutes on a workstation are
+enough to obtain the best-fit parameters.
+A few configurations have been cross-checked with the
+MCMC technique, to ensure that the typical widths of the
+PDFsremainthesame,whatevertheinputingredientsused
+in the calculation. This allows us, for a given propagationmodel calculated from different input ingredients, to com-
+pare the resulting scatter in the transport parameter val-
+ues—systematic uncertainties (hereafter SystUnc)—to the
+typical statistical uncertainties (hereafter StatUnc) found
+with the MCMC technique.
+4. Influence of the gas description
+We start by varying the gas parameters. This is discussed
+forthemostgeneralclassoftheDM,i.e.ModelIII(allowing
+forbothconvectionandreacceleration).Forobviousreasons
+(see below), our conclusions also hold for Models 0, I, or II.
+In 1D models, the surface density of the gas in the
+model corresponds to the average of the truegas surface
+density (which depends on the position in the Galaxy) over
+theeffectivediffusionvolume(Taillet & Maurin2003).This
+is a reasonable assumption to make for stable nuclei (see
+Paper II). However, the details of the volume over which
+to calculate this averageare not straightforward.Moreover,
+even if a more realistic distribution of gas were to be used,
+the latter is not free of uncertainties. We thus allow for
+some uncertainty in this input. We also vary the fraction of
+hydrogen relative to helium.
+4.1. Influence of the hydrogen fraction
+In Table 2, the second and third lines (compare with the
+first line) show the effect of changing the hydrogen frac-
+tion in the ISM: the transport parameters are changed at
+most by ∼5%. There is a systematic trend for δto de-
+crease with smaller fractions of helium in the gas. However,
+the uncertainty on the hydrogen and helium fraction is not
+more than a few %. We therefore conclude that this has no
+strong impact on the derived transport parameters.
+4.2. Influence of the surface density
+The fourth and fifth lines show the effect of changing the
+surface gas density Σ ISM= 2hnISM. Whereas it has a small
+impact on the diffusion slope δ, the other transport param-
+eters are strongly affected. The change in the parameters
+can be understood if we look at the grammage of the DM
+(Maurin et al.2002).Inthepurelydiffusiveregime,wehave
+(e.g., Maurin et al. 2006)
+∝an}bracketle{tx∝an}bracketri}htpure−DM=ΣISM¯mvL
+2K, (7)
+where ¯mis the mean mass of the ISM. Let Kref
+0,δref,Vref
+c,
+andVref
+abe the best-fit parameters obtained for a surface
+density of gas Σref(first line of the Table). We remind that
+Lis fixed here. If the surface gas density is rescaled, i.e.
+Σnew=x×Σref, in order to keep the same grammage in
+Eq. (7), we need to have Knew
+0=x×Kref
+0. This is what we
+get in Table 2.
+The same reasoning holds for the convective wind. In
+presence of a constant wind, the full expression for the
+grammage reads (e.g., Maurin et al. 2006)
+∝an}bracketle{tx∝an}bracketri}htVc≡ΣISM¯mv
+2Vc/bracketleftBig
+1−e−VcL
+K/bracketrightBig
+. (8)
+WiththeaboverescalingforΣnewandKnew
+0,wegetVnew
+c=
+x×Vref
+c. Again, this is recovered in Table 2.
+3D. Maurin et al.: Systematic uncertainties on the cosmic-ra y transport parameters
+Table 2. Best-fit transport parameters for different ISM.
+Gas Kbest
+0×102δbestVbest
+cVbest
+aχ2/d.o.f
+ΣfH(kpc2Myr−1) (km s−1) (km s−1)
+Σreffref
+H 0.48 0.86 18.8 38.0 1.47
+Σref95% 0.53 0.83 18.6 38.1 1.24
+Σref80% 0.41 0.90 19.3 37.7 2.14
+1
+2Σreffref
+H 0.25 0.85 9.5 19.4 1.45
+2Σreffref
+H 0.92 0.86 37.3 74.6 1.51
+Note 2. Model III for L= 4 kpc. For all settings, we keep fixed
+∝angbracketleftne−∝angbracketright= 0.033 and Te∼104K. The reference parameters for the sur-
+face density Σrefand the hydrogen fraction fref
+Hare given in Table 1.
+The last parameter is the Alfv´ enic speed, for which we
+need to consider the whole transport equation. The speed
+Vais used to define Kpp[see Eq. (6)], which appears in the
+termsbandc[Eqs. (3) and (4)] of Eq. (1). If we consider
+the latter equation and the transport operator Lin Eq. (2),
+the rescaling Σnew=x×Σrefleads toLnew=x×Lref. This
+impliesbnew=x×bref,cnew=x×cref, andSnew=x×Sref.
+Let us consider the three terms each in turn. For b, if we
+look at Eq. (3), this xfactor is automatically ensured for
+the Coulomb and ionisation losses (they depend on the gas
+surface density), and also for adiabatic losses (as Vnew
+c=
+x×Vref
+c). This implies Knew
+pp=x×Kref
+pp. From Eq. (6), we
+haveKpp∝V2
+a/K0, such that the previous equality yields
+(Vnew
+a)2/Knew
+0=x×(Vref
+a)2/Kref
+0.
+This gives Vnew
+a=x×Vref
+a(the same information is given
+by thecterm), in agreement with the results of Table 2.
+4.3. Summary for the gas surface density
+An uncertainty of x% on the gas surface density leads to
+SystUnc ofx% onK0,VcandVa, but it leaves the diffusion
+slopeδunchanged. This is true, whatever the value for the
+halo size L. For the sake of completeness, we should note
+thatthisalsoaffectsthesourceterm:thesourceabundances
+are nowqnew=x×qref.
+5. Influence of the source spectrum
+In Paper I, we showed that some correlations exist between
+the sourcespectraand the transportparameters.Below,we
+investigatein moredetail how the transport parametersare
+changed if we assume different input for α(spectral index
+of the sources) and ηS(related to the spectral shape at low
+energy:
+Q(E)∝βηSR−α. (9)
+5.1. Influence of the source slope α
+It is usually said that the source spectrum energy depen-
+denceQ(E) factors out of the secondary-to-primary ratio.
+This agrees with the results gathered in Table 3, which
+shows the effect of varying the source slope α(or equiva-
+lently the slope of the measured fluxes γ=α+δ). This
+is especially true for Model I (no reacceleration), whereas
+if reacceleration is included (Models II and III), a few %Table 3. Best-fit transport parameters for different slopes
+γ.
+γ Kbest
+0×102δbestVbest
+c Vbest
+aχ2/d.o.f
+(kpc2Myr−1) (km s−1) (km s−1)
+0: 2.55 4.13 0.39 ... ... 29.7
+0: 2.65 4.08 0.40 ... ... 28.8
+0: 2.75 4.02 0.40 ... ... 27.9
+0: 2.85 3.97 0.40 ... ... 27.0
+I: 2.55 0.42 0.93 14.0 ... 11.4
+I: 2.65 0.42 0.93 13.5 ... 11.2
+I: 2.75 0.42 0.93 13.1 ... 11.0
+I: 2.85 0.42 0.93 12.7 ... 10.9
+II: 2.55 9.41 0.24 ... 74.2 4.52
+II: 2.65 9.76 0.23 ... 73.1 4.73
+II: 2.75 10.1 0.23 ... 72.1 4.94
+II: 2.85 10.4 0.22 ... 71.0 5.15
+III: 2.55 0.52 0.84 18.9 39.7 1.25
+III: 2.65 0.48 0.86 18.8 38.0 1.47
+III: 2.75 0.45 0.87 18.8 36.4 1.75
+III: 2.85 0.42 0.88 18.6 35.0 2.10
+Note 3. Models 0, I, II, and III for L= 4 kpc. The parameter γis
+the slope of the propagated fluxes ( γ=α+δ). The source spectrum
+isQ(E)∝βηSR−α, andηS=−1.
+change on the parameter γleads to a few % change on Va,
+K0, andδ.
+From the last column of the same table, we see that
+on a statistical basis, the best model is Model III, then
+Model II, Model I, and finally Model 0. The relative merit
+(goodness-of-fit) of each class of models (III, II, I, and 0)
+remains unchanged when αis changed.
+5.2. Influence of the low-energy source spectrum through ηS
+The secondparameterthat canbe variedin the sourceterm
+is the low-energy shape βηS[see Eq. (9)], which is a priori
+not very well constrained, both from observationaland the-
+oretical points of view. The evolution of the best-fit trans-
+port parameters as a function of ηSis shown in Fig. 1, for
+Model 0 (dash-dotted lines/stars), I (dotted lines/plusses),
+II (dashed lines/crosses), and III (solid lines/open circles).
+The three configurations have marked minima in their
+χ2
+min(bottom panel), but not at the same ηS. Models 0
+and I have a similar χ2dependence (weak impact of the
+presence of the wind), as have Models II and III (both af-
+fected by reacceleration). Understanding the exact depen-
+dence of the transport parameters with ηSis not the goal
+of the analysis. The important point to underline is that
+changing ηSdoes not allow reconciling convection Models I
+andIII with smaller δ. Asδvariesonlyin the range0 .3−0.5
+for Model 0 (pure diffusion), we conclude that for present
+B/C data, large δare associated to the constant Galactic
+wind. The largest variation is observed for Model II (reac-
+celeration only), for which δis allowed to be either smaller
+or larger than δbest
+II= 0.33 (associated with ηbest
+S≈1) for
+a smaller or a larger ηS. For any ηS, Model III always per-
+forms better than the other two models.
+4D. Maurin et al.: Systematic uncertainties on the cosmic-ra y transport parameters
+η -4 -2 0 2 4]-1  Myr2   [kpc0  K-210-110 (source)Sη Influence of 
+η -4 -2 0 2 4 δ0.51
+Model III
+Model II (no conv.)
+Model I (no reac.)
+Model 0 (pure diff.)Model III
+Model II (no conv.)
+Model I (no reac.)
+Model 0 (pure diff.)Model III
+Model II (no conv.)
+Model I (no reac.)
+Model 0 (pure diff.)Model III
+Model II (no conv.)
+Model I (no reac.)
+Model 0 (pure diff.)
+η -4 -2 0 2 4]-1   [km saV20406080
+η -4 -2 0 2 4]-1   [km scV5101520
+Sη -4 -2 0 2 4/d.o.f2χ
+110
+Fig.1.From top to bottom: best-fit values of K0,δ,Va,
+Vc, and the associated χ2/d.o.f.as a function of ηS[see
+Eq. (9)]. Here, γ=α+δ= 2.65. The vertical grey-dotted
+line isa guidelineforthe resultsat the default configuration
+ηS=−1.
+Table 4. Best-fit values from a simultaneous fit on B/C
+and O.
+Model η(B/C+O)
+S α(B/C+O)δ(B/C+O)χ2
+min/d.o.f.
+0 −0.82 2.22 0.41 25.3
+I −1.23 2.23 0.98 9.23
+II 1 .24 2.24 0.35 3.62
+III 1 .56 2.30 0.95 2.31
+Note 4. The data used are IMP7-8, Voyager 1&2, ACE, HEA0-3,
+Spacelab, and CREAM04 for the B/C ratio, and HEAO-3 for the
+oxygen flux.
+5.3. Fitting αandηSusing a primary flux
+The source parameters αandηSare not completely free.
+They can be fitted directly on primaryelemental fluxes (see
+Paper I). If we determine simultaneously the transport and
+sourceparametersfrom a fit on B/C + O HEAO-3data, we
+obtain the best-fit valuesgatheredin Table 4. ForModels II
+and III,ηB/C+O
+Sis not far from ηbest
+Sfound from the B/C
+fit (see Fig. 1). For Model II, using the above constraint
+(ηB/C+O
+S= 1.24)selects the diffusion slope δB/C+O= 0.35,
+close to the Kolmogorov spectrum for the turbulence.The surprising result is that the value for αB/C+Ois
+quite resilient to any slope obtained for δ. This is odd,
+since it leaves the propagated slope γ=α+δquite uncon-
+strained, leading to either 2.6 (Model II) or 3.2 (Model III).
+This indicates that in Model III, pure diffusive transport
+(i.e.γ=α+δ) has not yet been reached at TeV/n ener-
+gies.
+5.4. Summary for the source effect
+The dependence of the transport parameters with the
+source parameters αandηSare very different. We checked
+that the transport parameters do not significantly depend
+on the source parameter α(in particular, δis unaffected).
+Still, correlations exist between the source and the other
+transport parameters (see Paper I), and from Table 3, we
+can estimate that they affect their best-fit values at most
+by a few ten percent.
+On the other hand, Fig. 1 shows that the transport pa-
+rameters (including δ) are quite sensitive to the low-energy
+sourcespectrumparameterisedby ηS. Inanycase,this does
+not allow reconciliation of Model III with small δ. Taking
+a sizeable range for ηSalways lead to δ/greaterorsimilar0.6. This is even
+worseforModel I (no reacceleration).At variance,Model II
+(no convection) is very sensitive to ηS. However, ηSis con-
+strained by primary fluxes measurements, and by no means
+could it span the range shown in Fig. 1. When this con-
+straint is considered, the best-fit δfor Model II turns out
+to be 0.35. Given the sensitivity of the latter to ηSand the
+possible bias in low-energy data (because of the solar mod-
+ulation), this is consistent with a Kolmogorov spectrum. If
+the constraint of the primary flux is taken into account, the
+uncertainty on the transport parameters (associated to the
+source parameter ηS) can be estimated as ∼10−20%.
+The puzzling point is that, although Model III is pre-
+ferred over Model II on a statistical basis, the latter seems
+to agree more with the expected theoretical value of δ. This
+is more of further if we consider the quantity γ=α+δ,
+the asymptotic slope reached in the purely diffusive regime.
+The TRACER experiment finds γ≈2.65 for all nu-
+clei (Ave et al. 2008), whereas the best-fit slope from high-
+energy H and He data is γ≈2.85 (Donato et al. 2009).
+From the propagated slope’s point of view, Model II is
+again favoured over Model III, since the former leads to
+γ= 2.59,whereas the latter leads to 3 .25(read offTable 4).
+If Model III is confirmed, as alreadyemphasised above,this
+would mean that even high-energy data have not reached
+the purely diffusive transport yet.
+6. Influence of the low-energy diffusion coefficient
+We now turn to the effect of the low-energy shape of the
+diffusion coefficient. Assuming different diffusion schemes
+lead to different forms of the spatial and momentum diffu-
+sion coefficient (Schlickeiser 2002). Recently, Ptuskin et al.
+(2006) also argue that the form of the spatial diffusion coef-
+ficient can change at low energy, due to the possibility that
+the nonlinear MHD cascade sets the power-lawspectrum of
+turbulence.
+We first consider several theoretically-motivated diffu-
+sion forms given in the literature. They mostly differ at low
+energy, so that it is useful to rewrite Eq. (5) as
+K(E) =βηT·K0Rδ. (10)
+5D. Maurin et al.: Systematic uncertainties on the cosmic-ra y transport parameters
+Table 5. K(E) andKppfor different schemes.
+Type of turbulence ηTKppKxx
+4/3p2V2a
+LBI Leaky Box Inspired 01
+δ(4−δ2)(4−δ)
+SA Slab Alfv´ en 11
+δ(4−δ2)(4−δ)
+IFM Isotropic fast magnetosonic 2 −δ β1−δln(v
+Va)
+Mix Mixture SA and IFM 1 −δ β1−δln(v
+Va)
+Note 5. The spatial diffusion coefficient is Kxx=βηT·K0·Rδ.
+Table 6. Best-fit transport parameters based on different
+low-energy dependence of the diffusion coefficient.
+Type Kbest
+0×102δbestVbest
+c Vbest
+aχ2/d.o.f
+(kpc2Myr−1) (km s−1) (km s−1)
+0: LBI 3.48 0.45 ... ... 17.5
+0: SA 4.08 0.40 ... ... 28.8
+0: IFM 4.30 0.38 ... ... 36.7
+0: Mix 3.71 0.43 ... ... 23.7
+I: LBI 0.40 0.94 13.6 ... 12.0
+I: SA 0.42 0.93 13.5 ... 11.2
+I: IFM 0.42 0.93 13.5 ... 11.6
+I: Mix 0.41 0.94 13.5 ... 12.0
+II: LBI 5.50 0.38 ... 65.0 1.61
+II: SA 9.76 0.23 ... 73.1 4.73
+II: IFM 14.0 0.16 ... 18.9 6.86
+II: Mix 7.13 0.32 ... 12.8 2.03
+III: LBI 0.70 0.78 18.0 47.1 0.87
+III: SA 0.48 0.86 18.8 38.0 1.47
+III: IFM 0.49 0.85 18.9 45.6 1.25
+III: Mix 0.73 0.77 17.8 57.4 0.93
+Note 6. Models 0, I, II, and III for L= 4 kpc. SA corresponds to the
+reference DM used throughout the paper.
+In the second step, we let ηTvary over a wide range in
+order to draw more general conclusions.
+6.1. Influence of the turbulence scheme
+A few turbulence schemes are gathered in Table 5. The
+associated best-fit values of the transport coefficients are
+presented in Table 6. When only convection is present
+(Model I), the low-energy form of K(E) is irrelevant, as
+seen in Table 6. For models with reacceleration (Models II
+and III), it significantly affects almost all parameters. If
+there is no wind (Model II), the effect is maximum on δ.
+The model with pure diffusion (Model 0), or with both con-
+vectionandreacceleration(Model III), falls in-between.For
+Model III, the casesSA andIFM, on the one hand, and LBI
+and Mix, on the other, give very similar results. This is eas-
+ily understood as the quantities ηSA
+T= 2−δandηIMF
+T= 1
+(respectively ηLBI
+T= 1−δandηMix
+T= 0) are roughly equal
+forδbest
+III∼1.
+TheimportantresultisthatModelIIIismildlysensitive
+to the diffusion scheme, with an uncertainty of a few ten
+percent scatter on all the transport parameters. However,
+Model IIparametersareextremelysensitivetothe diffusion
+scheme (more than a factor of 2 scatter). Depending onη -4 -3 -2 -1 0 1]-1  Myr2   [kpc0  K-210-110 (transport)Tη Influence of 
+η -4 -3 -2 -1 0 1 δ0.51
+Model III
+Model II (no conv.)
+Model I (no reac.)
+Model 0 (pure diff.)Model III
+Model II (no conv.)
+Model I (no reac.)
+Model 0 (pure diff.)Model III
+Model II (no conv.)
+Model I (no reac.)
+Model 0 (pure diff.)Model III
+Model II (no conv.)
+Model I (no reac.)
+Model 0 (pure diff.)
+η -4 -3 -2 -1 0 1]-1   [km saV20406080
+η -4 -3 -2 -1 0 1]-1   [km scV5101520
+Tη -4 -3 -2 -1 0 1/d.o.f2χ
+110
+Fig.2.Same as Fig. 1, but now as a function of ηT[see
+Eq. (10)]. The vertical grey-dotted line is for the default
+configuration ηT= 1 (SA).
+the case considered, δis found in the range 0 .16−0.38.
+The hierarchy of χ2
+minamong the various models is always
+conserved.
+6.2. Generalisation to any ηT
+We generalise the analysis by allowing for any value of the
+parameter ηTin the diffusion coefficient K(E). In doing so,
+we do not seek to provide sound physical motivations for
+the range tested. In this section, whatever the value of ηT,
+the diffusion coefficient in momentum space is assumed to
+followKppKxx= (4/3)p2V2
+a/(δ(4−δ2)(4−δ)).
+The best-fit transport parametersand χ2
+minevolution as
+a function of ηTare plotted in Fig. 2. For ηT/lessorsimilar−2, all the
+models convergeslowly towardspurely diffusive models (no
+convection, no reacceleration). But this is at the cost of a
+badχ2(see bottom panel). Based on the χ2criterium, high
+values of ηT(/greaterorsimilar2) are also disfavoured. The four configu-
+rations have marked minima in their χ2
+min, corresponding
+toηbest
+T≈ −2.75,−2.5,−0.25,+0.25 for Models 0, I, II,
+and III respectively, for which δbest≈0.6,0.6,0.4,0.8. For
+ηT∼ −2.5,allmodelspointto δ∼0.5,closetoaKraichnan
+spectrum for turbulence.
+ForModelIII,forawell-chosenvalueof ηT,thediffusion
+slope could be decreased at most down to δ= 0.5, but such
+a configuration does not correspond to the minimal χ2for
+6D. Maurin et al.: Systematic uncertainties on the cosmic-ra y transport parameters
+this model so is excluded. For Model II, almost any value
+ofδcan be reached.
+6.3. Summary for the diffusion coefficient effect
+Similars to the effect of the source parameter ηS, varying
+the parameter ηTon a wide range i) does not to allow
+δ/lessorsimilar1/2 to reached for Model III, and ii) strongly affects δ
+for Model II. For the latter, the best-fit ηTleads toδ≈0.4,
+slightly more than the Kolmogorovspectrum ofturbulence.
+From a statistical point of view, Model III is again pre-
+ferred,except in the region −3/lessorsimilarηT/lessorsimilar−1.5whereVcdrops
+to zero,sothat it is equivalentto Model II. For ηT/lessorsimilar−2.75,
+Vaalso drops to zero, so that all models are close to the
+purely diffusive case (Model 0), which favours δ∼1/2. As
+before, one of the interesting features of Model II (reac-
+celeration only) is its versatility. As the range of allowed
+ηTremains unspecified to some extent, Model II does not
+point to any specific value of δ. It can accommodate values
+aslowas1 /3,goingthrough1 /2andevenhighervalues,ifa
+sharp turn-off (negative values of ηT) exists in the diffusion
+coefficient (e.g., Ptuskin et al. 2006).
+7. Influence of the production cross-sections
+Many reaction channels are required to calculate the pro-
+duction of secondary species. Semi-analytical formulae,
+semi-empirical approaches, or even fit to the data are cur-
+rently used in the literature to obtain the full set of pro-
+duction cross-section in CR physics. We show first the dif-
+ferent best-fit values obtained for different sets available in
+the literature. We then inspect what would be the effect
+of energy-biased cross-sections on the transport parameter
+derivation.
+7.1. Using different sets of fragmentation cross-sections
+We use four different sets of cross-sections. Among them,
+the most up-to-date are W03 (Webber et al. 2003) and
+GAL09 (from the GALPROP code1). The former is based
+on the semi-empirical approach of Webber and coworkers,
+initiated in the 90’s (Webber et al. 1990). The latter set
+takesadvantageoftheformerapproach,renormalisingsome
+cross-sections to selected data. The two others, WKS98
+(Webber et al. 1998) and S01 (private communication of
+Aim´ e Soutoul, 2001), are a bit outdated, but they serve as
+an illustration here.
+The effect of these different sets on the parameter de-
+termination is shown in Table 7. The scatter in the values
+of the different parameters is not the same depending on
+the model considered.WKS98 requiresless convectionthan
+any other set, and GAL09 needs more reacceleration than
+the others. Otherwise, it is difficult to find clear trends. For
+instance, the W03 and GAL09 sets give very similar results
+for Model I, but not as similar as for Models II and III.
+If we discard the S01 set (which is based on an un-
+published preliminary analysis), we can conclude that the
+SystUnc on the transport parameters related to the choice
+of the production cross-sectionis /lessorsimilar20%. This figure is sim-
+ilar to the uncertainty quoted for the parameterisation of
+the cross-section themselves ( ∼10−20%).
+1http://galprop.stanford.edu/web galprop/galprop home.htmlTable 7. Best-fit transport parameters for various cross-
+section sets.
+X-files Kbest
+0×102δbestVbest
+c Vbest
+aχ2/d.o.f
+(kpc2Myr−1) (km s−1) (km s−1)
+0: WKS98 2.66 0.53 ... ... 23.5
+0: S01 3.40 0.40 ... ... 34.5
+0: W03 4.08 0.40 ... ... 28.8
+0: GAL09 3.83 0.46 ... ... 28.4
+I: WKS98 0.45 0.95 10.4 ... 12.0
+I: S01 0.25 1.01 11.9 ... 13.5
+I: W03 0.42 0.93 13.5 ... 11.2
+I: GAL09 0.49 0.95 13.6 ... 12.3
+II: WKS98 7.19 0.31 ... 71.5 3.40
+II: S01 9.27 0.22 ... 68.8 6.56
+II: W03 9.76 0.23 ... 73.1 4.73
+II: GAL09 10.0 0.26 ... 85.0 4.03
+III: WKS98 0.69 0.80 15.8 43.5 1.11
+III: S01 0.25 0.98 16.2 27.9 3.01
+III: W03 0.48 0.86 18.8 38.0 1.47
+III: GAL09 0.65 0.82 21.7 49.4 1.53
+Note 7. Models 0, I, II, and III for L= 4 kpc. The four base sets of
+cross-sections are WKS98 from Webber et al. (1998), S01 from a pri-
+vate communication of Aim´ e Soutoul (2001), W03 from Webber et al.
+(2003), and GAL09 from the GALPROP code v50.1p (2009).
+7.2. Influence of a systematic energy bias in W03
+cross-sections
+A striking result of the updated cross-section formula-
+tion of W03 (Webber et al. 2003) compared to WKS98
+(Webber et al. 1998) was a systematic energy bias (smaller
+cross-sections at higher energy). To address the effect of a
+possibleresidualbias, we allow for a systematic energy bias
+in all production cross-sections, either at low energy (LE)
+or at high energy (HE). Biased sets are obtained from the
+reference set W03 by applying a factor ( Ek/n/10)−xbelow
+10 GeV/n (LE bias), and ( Ek/n)xabove 1 GeV/n (HE bias,
+but left constant above 10 GeV/n). This is illustrated for
+the12C→(10B+11B) cross-section, as shown in Fig. 3. As
+seen from this figure and the experimental data, biases x
+as large as 0 .05 are marginally consistent with the data at
+low energy, but are acceptable at high energy.
+In Table 8, we only report the results of the high-energy
+biasesonthebest-fitvaluesofthetransportparameters.We
+checkedthattheresultofalow-energyorahigh-energybias
+gives the same trends. Quite naturally, there is a significant
+correlation of the bias with the transport parameters. We
+note that if δincreases, all the other transport parameters
+K0,Va, andVcdecrease. In principle, we would expect that
+δincreases if the bias is positive: larger cross-sections at
+high-energy produce more secondaries at high-energy, re-
+quiring a larger δto match the same B/C data. This is
+observed for Models 0 and II. However, the reverse effect is
+observed for Models I and III, i.e., for models with convec-
+tive wind.
+Such an exercise has limitations since it is not realistic
+to expect such systematic energy biases for all production
+channels.However,it givessometrend and furtherconfirms
+that Model III cannot be reconcile with small δ.
+7D. Maurin et al.: Systematic uncertainties on the cosmic-ra y transport parameters
+0.2 0.25 0.30204060δ
+0.2 0.25 0.30.080.090.1
+0.08 0.09 0.1050100150 /Myr]2 [kpc0K
+0.2 0.25 0.370758085
+0.08 0.09 0.170758085
+70 75 80 8500.10.2Va [km/s]Model IIBest-fit values
+W03
+WKS98
+S01
+GAL09
+HE-Bias:x=+0.05
+HE-Bias:x=+0.02
+He-Bias:x=-0.02
+He-Bias:x=-0.0516 18 20 2200.51 [km/s]cV
+16 18 20 220.70.80.91
+0.7 0.8 0.9 10510 δ
+16 18 20 220.0050.01
+0.7 0.8 0.9 10.0050.01
+0.005 0.010200400600/Myr]2 [kpc0K
+16 18 20 22304050
+0.7 0.8 0.9 1304050
+0.005 0.01304050
+30 40 5000.10.2 Va [km/s]Model IIIBest-fit parameters
+W03
+WKS98
+S01
+GAL09
+HE-Bias:x=+0.05
+HE-Bias:x=+0.02
+He-Bias:x=-0.02
+He-Bias:x=-0.05
+Fig.4.Left panel: Model II (pure reacceleration). Right panel: Model II I (reacceleration and convection). PDF of the
+transport parameters (for the reference inputs) as obtained in Paper II, along with the best-fit values for different
+production cross-section sets.
+ (GeV/n)k/nE-110 1 10  (mb) B→12C σ
+3035404550
+LE bias (W03)
+x = +0.02
+x = -0.02
+HE bias (W03)
+x = +0.05
+x = +0.02
+x = -0.02
+x = -0.05WKS98
+GAL09
+W03
+Fig.3. Production cross-section for12C+H→10,11B
+(adapted from Webber et al. 2003). The standard sets are
+shown as solid lines (WKS98: red dots; GAL09: red down
+triangles; W03: black stars), and the biased sets in dotted
+(|x|= 0.02) and dashed ( |x|= 0.05) lines.
+7.3. Summary for the influence of fragmentation
+cross-sections
+The results when using different production cross-section
+sets are difficult to interpret. To check that the observed
+trends do not come from an error in the code, we compared
+our results with the LBM analysis of Webber et al. (2003).
+These authors find a significantly smaller escape length de-
+pendence on rigidity (from P0.6toP0.5), going from the
+WKS98 to the W03 set. From both an LBM analysis with
+ourpropagationcode and a DM analysisusing the LBI case
+(see Table 5) with the addition of a rigidity cut-off (not dis-
+cussedin thispaper), wealsofind this decreaseof∆ δ= 0.1.Table 8. High-energy biases on the W03 cross-section set.
+W03 Kbest
+0×102δbestVbest
+c Vbest
+aχ2/d.o.f
+HE bias x(kpc2Myr−1) (km s−1) (km s−1)
+0: +0.05 4.05 0.45 ... ... 26.5
+0: +0.02 4.06 0.42 ... ... 27.7
+0:+0.00 4.08 0.40 ... ... 28.8
+0:−0.02 4.09 0.38 ... ... 30.1
+0:−0.05 4.13 0.34 ... ... 32.5
+I: +0.05 0.61 0.89 13.7 ... 12.1
+I: +0.02 0.49 0.91 13.6 ... 11.4
+I:+0.00 0.42 0.93 13.5 ... 11.2
+I:−0.02 0.35 0.95 13.4 ... 11.2
+I:−0.05 0.27 0.98 13.1 ... 11.8
+II: +0.05 9.20 0.28 ... 78.0 3.26
+II: +0.02 9.51 0.25 ... 75.0 4.13
+II:+0.00 9.76 0.23 ... 73.1 4.73
+II:−0.02 10.0 0.21 ... 71.2 5.35
+II:−0.05 10.4 0.19 ... 68.1 6.27
+III: +0.05 0.98 0.74 19.0 50.0 1.01
+III: +0.02 0.75 0.81 19.0 42.7 1.22
+III:+0.00 0.48 0.86 18.8 38.0 1.47
+III:−0.02 0.35 0.91 18.5 33.4 1.88
+III:−0.05 0.21 1.01 17.8 26.6 2.94
+Note 8. Models 0, I, II, and III for L= 4 kpc. The vari-
+ous sets are obtained after multiplying the W03 set by ( Ek/n)x
+above 1 GeV/n (and kept constant above 10 GeV/n), with x=
++0.05,+0.02,0,−0.02,−0.05. (corresponding to bottom to top
+curves on the right-hand side of Fig. 3).
+This is also confirmed for the more standard Model 0 (pure
+diffusion) where we go from 0.53 to 0.40.
+We thus have to conclude, as seen from Table 7, that
+the effect of these cross-section sets is not the same for dif-
+8D. Maurin et al.: Systematic uncertainties on the cosmic-ra y transport parameters
+Ek/n (GeV/n)1 10210310B/C ratio
+00.050.10.150.20.250.30.35
+IS=250 MV) - Garcia-Munoz et al. 1987ΦIMP7-8 (
+=225 MV) - Lukasiak 1999Φ Voyager1&2 (
+=225 MV) - De Nolfo et al. 2006Φ ACE 97-98 (
+=250 MV) - Engelmann et al. 1990Φ HEAO-3 (
+=200 MV) - Swordy et al. 1990Φ Spacelab (
+=425 MV) - Ahn et al. 2008Φ CREAM 04 (
+Biased HEAO-3 data
+x = +0.10
+x = +0.05
+x = -0.05
+Fig.5.Demodulated B/C data used in this paper (de-
+noted dataset F). Biased HEAO-3 data are shown as solid,
+dashed, and dotted lines.
+ferentclassesofmodels:it stronglydepends onthe presence
+or absence of convection and, to a lesser extent, on reac-
+celeration. Nevertheless, for all models, the typical scatter
+in the best-fit values is a factor of 2 for K0,∼50% forVa,
+∼10% forδ, and∼5% forVc. Figure 4 illustrates that un-
+certainties in some input parameters (here the production
+cross-sections) provide larger SystUnc on the transport pa-
+rameters than their StatUnc calculated using the MCMC
+technique (taken from Paper II).
+As for the other effects inspected in previous sections,
+Model III always provide δlarger than ∼0.6. Model II
+provides values of δin the correct range for a Kolmogorov
+spectrum, but it never does better than Model III in terms
+of the goodness-of-fit.
+8. Influence of an energy bias in HEAO-3 B/C data
+It is obvious that most of the results derived in this pa-
+per rely on the confidence we place in the HEAO-3 data,
+since they are the most constraining data to date. Their
+very small error bars at high energy could over-constrain
+δ. Several other studies use large error bars for high en-
+ergy B/C HEAO-3 points, contradicting the prescription
+given in the original paper (Engelmann et al. 1990), and
+this could bias their results. As an exercise, we allowed for
+an energybias in these data. This approachis disputable so
+we do not wish to defend it strongly. We inspected whether
+a small bias in the data strongly affects the best-fit values
+of the transport parameters.
+The HEAO-3 data are biased above Ec, using
+(Ek/n/Ec)xforEk/n> Ec, withEc= 2.9 GeV/n, as shown
+in Fig. 5. As in previous sections, we fit Models 0, I, II, and
+III on dataset F (as shown in Fig. 5), where the HEAO-
+3 data have been replaced by the biased ones. The results
+have been gathered in Table 9. Intuitively, in a similar fash-
+ion as for the bias in the cross-sections, we expect that a
+higher value of the B/C data at high energy (positive bias)
+leads to a smaller δ. This is what is observed for all models.
+The effect of the bias on δis at its maximum for the pure
+diffusion model (Model 0), where ∆ δ∼0.2. For Models II
+and III, the effect is less pronounced owing to the pres-Table 9. Influence of biasing HEAO-3 data.
+HEAO-3 Kbest
+0×102δbestVbest
+c Vbest
+aχ2/d.o.f
+HE bias x(kpc2Myr−1) (km s−1) (km s−1)
+0: +0.10 4.97 0.28 ... ... 22.5
+0: +0.05 4.50 0.34 ... ... 25.2
+0:+0.00 4.08 0.40 ... ... 28.8
+0:−0.05 3.69 0.46 ... ... 33.0
+I: +0.10 0.45 0.81 13.4 ... 9.77
+I: +0.05 0.44 0.87 13.5 ... 10.2
+I:+0.00 0.42 0.93 13.5 ... 11.2
+I:−0.05 0.40 0.99 13.6 ... 12.5
+II: +0.10 10.1 0.17 ... 56.6 4.54
+II: +0.05 9.95 0.20 ... 64.8 4.27
+II:+0.00 9.76 0.23 ... 73.1 4.73
+II:−0.05 9.56 0.27 ... 81.5 5.79
+III: +0.10 0.31 0.85 19.1 30.0 1.32
+III: +0.05 0.42 0.83 19.0 34.8 1.16
+III:+0.00 0.48 0.86 18.8 38.0 1.47
+III:−0.05 0.50 0.90 18.7 40.0 2.18
+Note 9. Models 0, I, II, and III for L= 4 kpc, where HEAO-3
+data are biased using the formula ( Ek/n/Ec)xforEk/n> Ec, with
+Ec= 2.9 GeV/n and x= +0.1,+0.05,0.,−0.05.
+ence of reacceleration, with a change of only ∆ δ∼0.1.
+However, Model III (convection and reacceleration), which
+is statistically preferred over Model II, still points to un-
+comfortably high values for δ. The only way out would be
+to have a non-systematic energy effect in the data. For in-
+stance, the result on the fit to the recently published AMS-
+01 data (Tomassetti & AMS-01 Collaboration 2009) leads
+toδ≈0.5 (see Paper II).
+Finally, for completeness, we checked that the effect of
+the solar modulation parameters (we took ∆ φ±50 MV for
+the HEAO-3 data) was negligible on all the derived trans-
+port parameters.
+9. Summary and discussion
+The guiding questions for this paper were the following.
+First, do uncertainties in the input ingredients lead to
+systematic uncertainties ( SystUnc) on the derived trans-
+port parameters greater than their statistical uncertainties
+(StatUnc)?Second,canwereproducethevariousvaluesof δ
+given in the literature, and which one should be preferred?
+We elaborate on the answers below before concluding.
+9.1. SystUnc and StatUnc: which ones dominate?
+From the analysis of Paper II, the StatUnc found for the
+various transport parameters typically fall in the range
+5%−10% (see Fig. 4). The SystUnc generated by each
+input we varied are summarised below.
+–A change of x% in the gas surface density Σ ISMtrans-
+lates into a change of x% for all transport parameters
+butδ, which is unaffected. Such a simple scaling can be
+used for comparing models in the literature, as different
+ΣISMare generally used. Depending on how confident
+we are in the measurement of Σ ISM, we may conclude
+9D. Maurin et al.: Systematic uncertainties on the cosmic-ra y transport parameters
+that, for this input, the generated SystUnc is similar to
+or larger than the StatUnc.
+–The transport parameters are very sensitive to the low-
+energy spectral shape of the source parameter ηS, but
+not to the source slope α. Both parameters can be de-
+termined by including data of a primary flux in the fit.
+When doing so, the SystUnc andStatUnc are of the
+same order.
+–If we let ηT—which parameterises the low-energy
+shape of the diffusion coefficient— free in the range
+[−2.,1.], theSystUnc completely dominate the StatUnc.
+Diffusion coefficients with asharpturn-offat low-energy
+are possible, as shown in Fig. 1 of Ptuskin et al. 2006
+(their dashed line corresponds to our ηT=−2).
+–For the cross sections, as seen in Fig. 4, the SystUnc
+can also be larger than the StatUnc for some of the
+transport coefficients. The use of other secondary-to-
+primary ratios, such as Li/C and Be/C, could help to
+cross-checktheconsistencyofeachcross-sectionset,and
+thus decrease the SystUnc.
+These conclusions hold for all classes of models (0, I, II,
+and III).
+9.2. Which value of δshould we trust?
+The two key ingredients for determinaning δare i) the low-
+energy spectral shape of the diffusion coefficient as parame-
+terised by ηT[see Eq. (10)], and ii) the presence or absence
+ofa wind. Thesetwoeffectsallow δto reachvaluesaslowas
+0.2 up to 0.9.Such a wide range is alsofound in Jones et al.
+(2001), depending on the model they consider (convection,
+turbulent diffusion, stochastic reacceleration).
+We recover δ≈0.3 in the pure reacceleration
+case (no convection), as in other propagation codes
+(Strong & Moskalenko 1998; Jones et al. 2001). As soon as
+a convective wind is included, the diffusion slope is large
+(δ≈0.8), as obtained in other studies (Jones et al. 2001;
+Maurin et al. 2001). The effect of the low-energy shape of
+the diffusion coefficient is illustrated by the fact that stan-
+dard LBM and standard diffusion models lead to different δ
+(more details are given in Putze, Derome & Maurin, ICRC
+2009). For LBM, which are equivalent to diffusion models
+ifK(E) =K0Rδ, we recover δ≈0.5−0.6 (Webber et al.
+2003, Paper I.
+It is reassuring to see that the results of the various
+propagation codes used in the literature are consistent.
+However, this does not settle the question of which value of
+δwe should trust. In principle, one advantage of our anal-
+ysis is that it should clearly point to a preferred value for
+δ, by a mere comparison of the best χ2values. Indeed, in
+almost all settings considered, Model III (convection and
+reacceleration) performs better than any other, the second
+best being Model II (no convection). This behaviour is il-
+lustrated by the χ2
+mindependence as a function of δ, as
+shown in Fig. 6. For a standard diffusion coefficient (black-
+solid curve), there is a clear minimum at high δ, and a
+less marked localminimum at low δ. This second minimum
+matches the χ2
+minof Model II (black-dashed curve). The
+value of the χ2
+minslightly decreases when ηTdecreases (red
+curves), then increases for ηT/lessorsimilar0 (grey and violet curves).
+In the process,the best-fit valuefor δis displacedtosmaller
+values, the localminimum at low δovertaking and merging
+with the high δone.δ0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1/d.o.f2χ
+110=+1 (SA)Tη
+=0 (LBI)Tη
+=-1Tη
+=-2TηModels II
+Models IIIModels III
+Fig.6.Evolution of χ2
+min/d.o.f. as a function of δfor III
+(thick lineswith symbols)andII(thin lines).The fourcases
+correspond to different ηT(low-energy shape of the diffu-
+sion coefficient).
+Table 10. Best-fit parameters for a few selected configura-
+tions.
+Model ηTK0×102δ V cVaχ2/d.o.f
+(kpc2Myr−1) (km s−1) (km s−1)
+III†SA 0.481 0.856 18.84 37.98 1.47
+III/II⋆-1.3 3.161 0.512 0. 45.35 2.26
+I/0‡-2.61 2.054 0.613 0. ... 3.29
+II†SA 9.753 0.234 ... 73.14 4.73
+†Best-fit transport parameters for standard Model II and III.
+‡Best-fit parameters with ηTfree (no reacceleration).
+⋆Best-fit parameters for fixed ηT.
+Note 10. Standard models refer to SA diffusion coefficients (see
+Table 5). Alternative models show their different values for ηT,
+as parameterised from Eq. (10).
+For standard diffusion schemes (SA), the best class of
+models is Model III. The main problem is that whatever
+ingredients are changed in this model, we are always left
+withδ/greaterorsimilar0.6, and most of the time with even higher values,
+∼0.9. Such high values, always associated to the presence
+of the constant convective wind, are difficult to explain.
+Should we choose to discard this Model III, we would be
+left with the task of explaining why the statistical analysis
+fails in the context of CR physics. A way out is provided,
+as seen in Fig. 6, if we let free ηT(low-energy shape of the
+diffusion coefficient). Indeed, the only case where Model II
+(reacceleration only) performs as Model III is for ηTin the
+range−3/lessorsimilarηT/lessorsimilar−1.5. In such a configuration, the best-
+fit value for the convective velocity of Model III drops to
+zero, so that the best model only has reacceleration (i.e.
+equivalent to Model II): we term this class of models III/II.
+The next best-fit class of models is obtained if we discard
+reacceleration (Model I-like) and again let ηTgo free. As
+for the previous case, the best-fit value for the convective
+velocity drops to zero: we denote this class I/0. In terms
+of the best χ2
+min, the standard diffusion Model II has only
+the fourth-best χ2value. The best-fit values for these four
+classes of models are gathered in Table 10, sorted according
+to their figure of merit.
+10D. Maurin et al.: Systematic uncertainties on the cosmic-ra y transport parameters
+ (GeV/n)k/nE-110 1 10210310B/C ratio
+00.050.10.150.20.250.30.35
+=250 MV) - Garcia-Munoz et al. 1987ΦIMP7-8 (
+=225 MV) - Lukasiak 1999Φ Voyager1&2 (
+=225 MV) - De nolfo et al. 2006Φ ACE 97-98 (
+=250 MV) - Engelmann et al. 1990Φ HEAO-3 (
+=200 MV) - Swordy et al. 1990Φ Spacelab (
+=425 MV) - Ahn et al. 2008Φ CREAM 04 (
+Model III (SA)
+/d.o.f=1.47)2χ=0.86, δ(Model II (SA)
+/d.o.f=4.73)2χ=0.23, δ(=-1.30)
+Tη Model III/II (
+/d.o.f=2.26)2χ=0.51, δ(
+=-2.61)
+Tη Model I/0 (
+/d.o.f=3.29)2χ=0.61, δ(TOA
+=225 MVΦ=250 MVΦ||
+Fig.7.Best-fit B/C ratio for standard Model II (thick red)
+and III (thick black) using the reference setting (solid and
+thin dashed lines are for the cross-section sets W03 and
+GAL09). The special best-fit Models O/I and III/II are
+also plotted in thin grey lines (light and dark shade respec-
+tively).
+The best-fit curves for these four relevant configura-
+tions are shown in Fig. 7. From this graphic view, we can
+see that Model II matches the data best at very-low en-
+ergy, Model III at intermediate energies, and Models O/I
+and III/II at high energy. While we found that a system-
+atic bias in HEAO-3 data would not change the ordering
+of which is the best model (see Sect. 8), we see from Fig. 7
+that a more complicated pattern in the data may be more
+effective in changing which model is best.
+To summarise, when a standard scheme for the diffu-
+sion coefficient is assumed, Model III is always preferred
+over Model II. However, the latter seems to agree more
+with theoretical expectations for δ. Model II is also more
+consistent with the measured propagated slope γ=α+δ.
+Because of this oddity, we may not be able to give a defi-
+niteansweraboutthevalueof δ.Thisisfurthercomplicated
+whenηTis left free; in that case, best models are always
+without convection, and even the pure diffusive model is
+redeemed (although both have difficulty reproducing the
+B/C peak at GeV/n energies). We see that many uncer-
+tainties show up at GeV/n energies. As also illustrated by
+the various predictions for various δin Fig 7, the higher the
+energy, the closer we can expect to reach the purely diffu-
+sive regime. As a result, high-energy B/C data are desired
+to unambiguously pinpoint the value of δ.
+9.3. Conclusion
+In the past years, we have promoted and used a
+model favouring both convection and reacceleration (e.g.,
+Maurin et al. 2001, 2002, and subsequent studies) from sta-
+tistical criteria.The main andknown problemofthis model
+lies in its uncomfortably high value for δ(δ∼0.8). Such a
+model is also preferred in the present study. In addition, we
+found that the high value for δis extremely resilient to any
+change in the setting, which leaves us with several alterna-
+tives: assume that there are complicated biases in the data
+that conspire to give high δin such models, that this highvalue ofδis real (in that case it needs to be explained theo-
+retically), or that any model with convection should be ex-
+cluded (which contradicts the fact that winds are observed
+in many galaxies). Even if we adopt the last alternative, no
+firm conclusions can be drawn on the value of δ. Indeed, if
+the statistical analysis is relaxed, a large category of mod-
+els are redeemed, attaining any value for δbetween 0.3 and
+0.9. These models may be purely diffusive, with convection
+and/orreacceleration,and areverysensitivetothe shape of
+the low-energy diffusion coefficient (which is not prescribed
+theoretically for the moment). Data at higher energy are
+needed to solve this question. More constraints can also be
+obtained by combining several secondary-to-primary ratios
+(e.g., Webber 1997a,b). This is left for a later study.
+This study has limitations. For instance, we only varied
+the sourceand diffusion parametersaccordingto simple pa-
+rameterisations. More complicated dependences could have
+beeninspected.However,itisworthrecallingthatitmaybe
+dangerous to introduce too many ad hocprescriptions, be-
+cause the statistical meaning—already unclear when com-
+paringthedifferentclassesofmodels—becomeslessandless
+obvious as the number of parameters and models tested
+increase. In the framework of homogeneous and isotropic
+diffusion coefficients, a maybe more important issue is the
+question of the Galactic wind. A constant wind was cho-
+sen because of the simplicity of the solutions (of the cor-
+responding diffusion equation). On the one hand, Galactic
+winds are ubiquitous. On the other, as shown in this study,
+constant wind cannot accommodate a realistic slope of the
+diffusion coefficient. A linear wind may providedifferent re-
+sults. We are implementing a numerical solver in our prop-
+agation code to inspect this issue. Another possibility that
+cannot be ruled out is that the HEAO-3 data suffer from
+non-trivial systematics.
+Finally,thestartingpointofthepaperwasacomparison
+between systematic uncertainties (generated by uncertain-
+ties in the input ingredients) and statistical uncertainties
+for the values of the transport parameters. That the for-
+mer can be larger than the latter shows that many efforts
+are still needed in CR physics, especially for the production
+cross-sections,beforeonecantakefulladvantageofanysta-
+tistical analysis, as performed in Papers I and II with an
+MCMC technique. Some of these issues may be resolved as
+new data on cosmic-ray nuclei are being released ( cream,
+pamela,tracer).
+Acknowledgements. We thank W. R. Webber for providing us with
+the nuclear production cross-sectionsdiscussed inWebber etal.(2003)
+and for useful suggestions. We also thank C. Combet for a care ful
+reading of the paper. A. P. is grateful for financial support f rom the
+Swedish Research Council (VR) through the Oskar Klein Centr e.
+We acknowledge the support of the French ANR (grant ANR-06-
+CREAM).
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+arXiv:1001.0554v1  [math.CV]  4 Jan 2010NIKISHIN SYSTEMS ARE PERFECT
+U. FIDALGO PRIETO AND G. L ´OPEZ LAGOMASINO
+Abstract. K. Mahler introduced the concept of perfect systems in the ge neral theory he de-
+veloped for the simultaneous Hermite-Pad´ e approximation of analytic functions. We prove that
+Nikishin systems are perfect providing, by far, the largest class of systems of functions for which
+this important property holds. As consequences, in the cont ext of Nikishin systems, we ob-
+tain: an extension of Markov’s theorem to simultaneous Herm ite-Pad´ e approximation, a general
+result on the convergence of simultaneous quadrature rules of Gauss-Jacobi type, the logarith-
+mic asymptotics of general sequences of multiple orthogona l polynomials, and an extension of
+the Denisov-Rakhmanov theorem for the ratio asymptotics of mixed type multiple orthogonal
+polynomials.
+Keywords and phrases. Perfect systems, Nikishin systems, multiple orthogonal polynomials ,
+mixed type approximation, simultaneous quadratures, rate of con vergence, logarithmic asymp-
+totics, potential theory, ratio asymptotics.
+A.M.S. Subject Classification. Primary: 30E10, 42C05; Secondary: 41A20.
+1.Introduction
+1.1.Some historical remarks. In 1873, Charles Hermite publishes in [28] his proof of the tran-
+scendence of emaking use ofsimultaneousrationalapproximationof systemsof ex ponentials. That
+paper marked the beginning of the modern analytic theory of numbe rs.
+The formal theory of simultaneous rational approximation for gen eral systems of analytic func-
+tions was initiated by K. Mahler in lectures delivered at the University o f Groningen in 1934-35.
+These lectures were published years later in [32]. Important contrib utions in this respect are also
+due to his students J. Coates and H. Jager, see [12] and [29]. K. Mah ler’s approach to the simulta-
+neous approximation of finite systems of analytic functions may be r eformulated in the following
+terms.
+Letf= (f0,...,f m)beafamilyofanalyticfunctionsinsomedomain Doftheextendedcomplex
+plane containing ∞. Fix a non-zero multi-index n= (n0,... ,n m)∈Zm+1
++,|n|=n0+... ,nm.
+There exist polynomials an,0,...,a n,m, not all identically equal to zero, such that
+i) degan,j≤nj−1,j= 0,...,m(degan,j≤ −1 means that an,j≡0),
+ii)/summationtextm
+j=0an,j(z)fj(z)−dn(z) =O(1/z|n|),z→ ∞,
+for some polynomial dn. Analogously, there exists Qn, not identically equal to zero, such that
+i) degQn≤ |n|,
+ii)Qn(z)fj(z)−Pn,j(z) =O(1/znj+1),z→ ∞,j= 0,...,m,
+for some polynomials Pn,j,j= 0,...,m.
+Initially, the polynomials an,0,... ,a n,mwere called latin and Qngerman polynomials, due to
+the letters employed in denoting them (see the papers of Mahler, Co ates and Jager cited above).
+Thepolynomials dnandPn,j,j= 0,...,m, areuniquelydeterminedfromii)oncetheirpartnersare
+found. Later, thetwoconstructionshavebeencalledtypeIandt ypeII polynomials(approximants)
+of the system ( f0,... ,f m). Algebraically, they are closely related. This is clearly exposed in
+Dedicated to the memory of the outstanding russian mathemat ician E.M. Nikishin who died on December 17,
+1987, at the early age of 42. See [18] for a brief account of his results and list of publications.
+The work of both authors was supported by Ministerio de Cienc ia y Tecnolog´ ıa under grants MTM2006-13000-
+C03-02 and MTM2009-12740-C03-01.
+12 FIDALGO AND L ´OPEZ
+[12],[29],and [32]. When m= 0 both definitions coincide with that of the well-known Pad´ e
+approximation in its linear presentation.
+Apart from Hermite’s result, type I, type II, and a combination of t he two (called mixed type),
+havebeenemployedintheproofoftheirrationalityofothernumber s. Forexample,in[7]F.Beukers
+shows that Apery’s proof (see [1]) of the irrationality of ζ(3) can be placed in the context of mixed
+type Hermite-Pad´ e approximation. See [47] for a brief introductio n and survey on the subject.
+More recently, mixed type approximation has appeared in random ma trix and non-intersecting
+brownian motions theories (see, for example, [6] and [13]).
+In applications in the areas of number theory, convergence of simu ltaneous rational approxima-
+tion, and asymptotic properties of type I and type II polynomials, a central question is that of
+uniqueness, to within a constant factor.
+Definition 1.1.A multi-index nis said to be normal for the system ffor type I approximation
+(respectively, for type II,) if deg an,j=nj−1,j= 0,...,m(respectively, deg Qn=|n|). A system
+of functions fis said to be perfect if all multi-indices are normal.
+It is easy to see that under normality ( an,0,...,a n,m) andQnare uniquely determined to within
+a constant factor.
+Considering the construction at the origin (instead of z=∞which we chose for convenience),
+the system of exponentials consideredby Hermite, ( ew0z,... ,ewmz),wi/\e}atio\slash=wj,i/\e}atio\slash=j,i,j= 0,...,m,
+is known to be perfect for type I and type II. A second example ofa perfect system for both types is
+that given by the binomial functions (1 −z)w0,... ,(1−z)wm,wi−wj/\e}atio\slash∈Z. All multi-indices nsuch
+thatn0≥ ··· ≥nmare known to be type I and type II normal for (logm(1−x),... ,log(1−x),1).
+When normality occurs for multi-indices with decreasing components the system is said to be
+weakly perfect . Except for systems formed by Cauchy transforms of measures , basically these
+are the only examples known of perfect or weakly perfect systems .
+1.2.Markov systems and orthogonality. Letsbe a finite Borel measure with constant sign
+whose compact support consists of infinitely many points and is cont ained in the real line. In the
+sequel, we only consider such measures. By ∆ we denote the smallest interval which contains the
+support, supp s,ofs. We denote this class of measures by M(∆). Let
+/hatwides(z) =/integraldisplayds(x)
+z−x
+denote the Cauchy transform of s. Obviously,/hatwides∈ H(C\∆); that is, it is analytic in C\∆.If we
+apply the construction above to the system formed by /hatwides(m= 0), it is easy to verify that Qnturns
+out to be orthogonal to all polynomials of degree less than n∈Z+. Consequently, deg Qn=n,all
+its zeros are simple and lie in the open convex hull Co(supp s) of supps. Therefore, such systems
+are perfect. Let us see two other examples much more illustrating.
+1.2.1.Angelesco systems. In [2], A. Angelesco considered the following systems of functions. L et
+∆j,j= 0,...,m, be pairwise disjoint bounded intervals contained in the real line and sj,j=
+0,...,m, a system of measures such that Co(supp sj) = ∆j.
+Fixn∈Zm+1
++and consider the type II approximant of the so called Angelesco sys tem of
+functions (/hatwides0,... ,/hatwidesm) relative to n. It turns out that/integraldisplay
+xνQn(x)dsj(x) = 0, ν= 0,...,n j−1, j= 0,...,m.
+Therefore, Qnhasnjsimple zeros in the interior (with respect to the euclidean topology of R)
+of ∆j. In consequence, since the intervals ∆ jare pairwise disjoint, deg Qn=|n|and Angelesco
+systems are type II perfect. Type I perfectness for Angelesco systems has not been studied.
+Unfortunately, Angelesco’spaper receivedlittle attention and suc h systemsreappearmanyyears
+later in [34] where E.M. Nikishin deduces some of their formal propert ies.
+Though type II normality for Angelesco systems is so easy to deduc e, the multiple orthogonal
+polynomials and the rational approximations associated with them do not have good asymptotic
+behavior. In [25] and [3], their logarithmic and strong asymptotic for mulas, respectively, are given.
+In this respect, a different system of Markov functions turns out to be much more interesting and
+foundational from the geometric and analytic points of view.NIKISHIN SYSTEMS ARE PERFECT 3
+1.2.2.Nikishin systems. Inanattempt toconstructgeneralclassesoffunctionsforwhic h normality
+takes place, in [35] E.M. Nikishin introduced the concept of MT-syste m. Let ∆ α,∆βbe two non
+intersecting bounded intervals contained in the real line and σα∈ M(∆α),σβ∈ M(∆β). With
+these two measures we define a third one as follows (using the differe ntial notation)
+d/a\}bracketle{tσα,σβ/a\}bracketri}ht(x) =/hatwideσβ(x)dσα(x);
+that is, one multiplies the first measure by a weight formed by the Cau chy transform of the second
+measure. Certainly, this product of measures is non commutative.
+Above,/hatwideσβdenotes the Cauchy transform of the measure σβ. The reader may argue, and we
+agree, that the appropriate notation is /hatwiderσβ. However, throughout the paper, we will need Cauchy
+transforms of measures with several sub-indices and supra-indic es; for example/hatwidests2
+1,j(and much
+more extended). The correct notation causes space consumptio n and aesthetic inconveniences. So,
+be cautious,/hatwides2
+1,jis not the Cauchy transform of ssub-indexed with 1 ,jand then squared, but
+precisely the Cauchy transform of a measure denoted s2
+1,j. The good news is that powers rarely
+appear in the paper and they are clear from the context (for exam ple, (−1)jorz2).
+Definition 1.2.Take a collection ∆ j,j= 0,...,m, of intervals such that
+∆j∩∆j+1=∅, j= 0,...,m−1.
+Let (σ0,...,σ m) be a system of measures such that Co(supp σj) = ∆j,σj∈ M(∆j),j= 0,...,m.
+We say that ( s0,...,s m) =N(σ0,... ,σ m), where
+s0=σ0, s1=/a\}bracketle{tσ0,σ1/a\}bracketri}ht,... ,s m=/a\}bracketle{tσ0,/a\}bracketle{tσ1,... ,σ m/a\}bracketri}ht/a\}bracketri}ht
+is the Nikishin system of measures generated by ( σ0,... ,σ m).
+Fixn∈Zm+1
++and consider the type II approximant of the Nikishin system of func tions
+(/hatwides0,...,/hatwidesm) relative to n. It is easy to prove that
+/integraldisplay
+xνQn(x)dsj(x) = 0, ν= 0,...,n j−1, j= 0,...,m.
+Allthe measures sjhavethesamesupport; therefore,itisnotimmediatetoconcludet hatdegQn=
+|n|. Nevertheless, if we denote
+sj,k=/a\}bracketle{tσj,σj+1,...,σ k/a\}bracketri}ht, j <k, s j,j=/a\}bracketle{tσj/a\}bracketri}ht=σj,
+the previous orthogonality relations may be rewritten as follows
+/integraldisplay
+(p0(x)+m/summationdisplay
+k=1pj(x)/hatwides1,k(x))Qn(x)dσ0(x) = 0, (1)
+wherep0,... ,p mare arbitrary polynomials such that deg pk≤nk−1,k= 0,...,m.
+Definition 1.3.A system of real continuous functions u0,...,u mdefined on an interval ∆ is
+called an AT-system on ∆ for the multi-index n∈Zm+1
++if for any choice of real polynomials (that
+is, with real coefficients) p0,...,p m,degpk≤nk−1,the function
+m/summationdisplay
+k=0pk(x)uk(x)
+has at most |n|−1 zeros on ∆. If this is true for all n∈Zm+1
++we have an AT system on ∆.
+In other words, u0,...,u mforms an AT-system for non ∆ when the system of functions
+(u0,...,xn0−1u0,u1,...,xnm−1um)
+is a Tchebyshev system on ∆ of order |n|−1. From the properties of Tchebyshev systems (see [30,
+Theorem 1.1]), it follows that given x1,...,x N,N <|n|,points in the interior of ∆ one can find
+polynomials h0,... ,h m,conveniently, with deg hk≤nk−1,such that/summationtextm
+k=0hk(x)uk(x) changes
+sign atx1,... ,x N,and has no other points where it changes sign on ∆ .
+In [35], Nikishin stated without proof that the system of functions ( 1,/hatwides1,1,...,/hatwides1,m) forms
+an AT-system for all multi-indices nsuch thatn0≥ ··· ≥nm(he proved it when additionally
+n0−nm≤1). Due to (1) this implies that Nikishin systems are type II weakly per fect.4 FIDALGO AND L ´OPEZ
+The proof of Nikishin’s assertion is a consequence of [15, Theorem 4.1 ]. Actually, Driver and
+Stahl proved normality for the wider class of multi-indices
+Zm+1
++(⊛) ={n∈Zm+1
++: 0≤j <k≤m⇒nk≤nj+1}
+(see also [26]). In [15, Theorem 4.2], for the same class of multi-indices , the authors also proved
+type I weak perfectness for Nikishin systems. In the same paper ( see remark on page 171), it is
+shown that when m= 1 Nikishin systems are type II perfect. Improvements are also co ntained in
+[9, Theorem 1] (see also [22, Theorem 2]) where normality is proved fo r all multi-indices in
+Zm+1
++(∗) ={n∈Zm+1
++:/\e}atio\slash ∃0≤i<j <k ≤msuch thatni<nj<nk}
+and [19, Theorem 1] containing the proof that for m= 2 Nikishin systems are type II perfect.
+Ever since the appearance of [35], a subject of major interest for those involved in simultaneous
+approximation was to determine whether or not Nikishin systems are perfect. The main result of
+this paper gives a positive answer to this question. Moreover, we will prove perfectness for mixed
+type Nikishin systems, containing type I and type II as particular ca ses. The proof is based on
+the reduction of the problem to the case of multi-indices with decrea sing components (that is, to
+weak perfectness). In the sequel,
+Zm+1
++(•) ={n∈Zm+1
++:n0≥ ··· ≥nm}.
+Notice that
+Zm+1
++(•)⊂Zm+1
++(⊛)⊂Zm+1
++(∗)⊂Zm+1
++.
+Whenm= 0 these sets are equal. For m= 1 the last two coincide. If m≥2 they are all distinct.
+The proof of the main result relies on interesting reduction formulas concerning products and
+ratios of Cauchy transforms of measures. We will see numerous co nsequences of the perfectness of
+Nikishinsystemsin: convergenceofsimultaneousPad´ eapproximat ion,convergenceofsimultaneous
+quadrature rules, and asymptotic properties of multiple orthogon al polynomials.
+1.2.3.Mixed type Nikishin systems. In [42], Sorokin introduced the following construction. Let
+F= (fj.k),
+be an (m2+ 1)×(m1+ 1) dimensional matrix of analytic functions in some domain Dof the
+extended complex plane containing ∞. Fix a multi-index n= (n1;n2)∈Zm1+1
++×Zm2+1
++, such
+that|n1|=|n2|+1. We denote ni= (ni,0,...,n i,mi),i= 1,2.There exists a vector polynomial
+An= (an,0,...,a n,m1), such that
+a)An/\e}atio\slash≡0,degan,k≤n1,k−1,k= 0,...,m 1,
+b) (FAt
+n−Dt
+n)(z) = (O(1/zn2,0+1),... ,O(1/zn2,m2+1)t=:O(1/zn2+1),z→ ∞.
+for somem2+1 dimensional vector polynomial Dn(the super-index tmeans taking transpose and
+0denotes the zero vector). Finding Anreduces to solving a linear homogeneous system of |n2|
+equations determined by the conditions b) on |n1|unknowns (the total number of coefficients of
+the polynomials an,k,k= 0,...,m 1). Since|n2|+1 =|n1|a non trivial solution exists.
+Definition 1.4.A non zero vector Ansatisfying a)-b) is called mixed type vector polynomial
+relative to Fandn∈Zm1+1
++×Zm2+1
++,|n1|=|n2|+ 1. If deg an,k=n1,k−1,k= 0,...,m 1,
+the multi-index nis called mixed type normal. Fis mixed type perfect when all multi-indices in
+Zm1+1
++×Zm2+1
++such that |n1|=|n2|+1 are normal.
+This construction has as particular cases type I ( m2= 0) and type II ( m1= 0) polynomials.
+LetS1= (s1
+0,0,...,s1
+0,m1) =N(σ1
+0,...,σ1
+m1),S2= (s2
+0,0,... ,s2
+0,m2) =N(σ2
+0,... ,σ2
+m2),σ1
+0=
+σ2
+0,be two given Nikishin systems generated by m1+ 1 andm2+ 1 measures, respectively. We
+underline the fact that both Nikishin systems stem from the same ba sis measure σ1
+0=σ2
+0, but
+there is no other restriction on them. Let us introduce the row vec tors
+U= (1,/hatwides2
+1,1,... ,/hatwides2
+1,m2),V= (1,/hatwides1
+1,1,...,/hatwides1
+1,m1)
+and the (m2+1)×(m1+1) dimensional matrix function
+W=UtV.NIKISHIN SYSTEMS ARE PERFECT 5
+Define the matrix Markov type function
+/hatwideS(z) =/integraldisplayW(x)dσ2
+0(x)
+z−x
+understanding that integration is carried out entry by entry on th e matrix W. We say that /hatwideSis a
+mixed type Nikishin system of functions.
+In the rest of the paper, we will study mixed type Nikishin systems an d their mixed type
+polynomials. Occasionally, we reduce the study to type II ( m1= 0). In such cases, for simplicity,
+we reduce the notation. Namely, n= (n0,...,n m), the vector function will be f= (/hatwides0,...,/hatwidesm),
+wherem=m2, and (s0,...,s m) =N(σ0,... ,σ m). The mixed type polynomials Anwill then be
+denotedQn.
+1.3.Statement of the main results. Mixed type Nikishin systems and their associated mixed
+type polynomials satisfy many interesting properties. Let us begin w ith
+Theorem 1.1.Let(s1,1,... ,s 1,m) =N(σ1,...,σ m)be given. Then, the system (1,/hatwides1,1,...,/hatwides1,m)
+forms an AT-system on any interval ∆disjoint from ∆1= Co(supp σ1). Moreover, for each
+n∈Zm+1
++,and arbitrary polynomials with real coefficients pk,degpk≤nk−1,k= 0,...,m, the
+linear form p0+/summationtextm
+k=1pk/hatwides1,k,has at most |n|−1zeros inC\∆1.
+From here, we can prove
+Theorem 1.2.The matrix/hatwideSis mixed type perfect. For each n∈Zm1+1
++×Zm2+1
++,|n1|=|n2|+1,
+the vector polynomial Anis uniquely determined up to a constant factor.
+In particular, this means that Nikishin systems are type I and type I I perfect.
+An easy consequence of Theorem 1.2 (more precisely, of Lemma 2.3 b elow) is that the linear
+form
+An:=an,0+m1/summationdisplay
+k=1an,k/hatwides1
+1,k,
+has at least |n2|sign changes in the interior (with respect to the euclidean topology o fR) of the
+interval ∆ 0. In particular, for any n∈Zm+1
++, the type II polynomial Qn(=an,0) has all its zeros
+located inside Co(supp σ0). This has a striking consequence in terms of the convergence of t ype II
+rational approximation of Nikishin systems.
+In [35], it was proved that for ( s0,s1) =N(σ0,σ1)
+lim
+n→∞Pn,k
+Qn=/hatwidesk, k= 0,1,
+uniformly on compact subsets of C\Co(suppσ0),where the limit is taken along the sequence
+n= (n,n),n∈Z+. Whenm2= 0 the corresponding result is Markov’s classical theorem on the
+convergence of Pad´ e approximants, see [33]. The extension to dia gonal sequences for arbitrary
+mandn∈Λ⊂Zm+1
++(•) such that |n| → ∞and max {n0−nm:n∈Λ}<∞is contained in
+[11, Corollary 1] (which also includes the case when the measures hav e unbounded support and a
+Carleman type condition is satisfied). Combining Theorem 1.2 and [21, T heorem 1] we obtain
+Corollary 1.1.Let(s0,0,...,s 0,m) =N(σ0,... ,σ m)andΛ⊂Zm+1
++be given. Assume that
+there exist constants c>0,κ<1,such that
+nj≥|n|
+m+1−c|n|κ, j= 0,...,m.
+Then,
+lim
+n∈ΛPn,k
+Qn=/hatwides0,kK ⊂C\Co(suppσ0), k= 0,...,m.
+Moreover,
+limsup
+n∈Λ/bardbl/hatwides0,k−Pn,k
+Qn/bardbl1/2|n|
+K≤δK<1, k= 0,...,m,
+where/bardbl·/bardblKdenotes the uniform norm on K,
+δK= max{|ϕt(z)|:z∈ K,t∈Co(suppσ1)∪{∞}},6 FIDALGO AND L ´OPEZ
+andϕtrepresents conformally C\Co(suppσ0)onto the unit circle with ϕt(t) = 0,ϕ′
+t(t)>0.
+Throughout the paper, the notation
+lim
+n∈Λgn(z) =g(z),K ⊂Ω,
+stands for uniform convergence of the sequence of functions {gn},n∈Λ,to the function gon each
+compact subset Kcontained in the indicated region (in this case Ω).
+This result gives a very general extension of Markov’s theorem. No tice that the sequences are
+not required to be close to diagonal (equal components in the multi- indices).
+Since the zeros xn,j,j= 1,...,|n|,ofQnare simple, we can decompose Pn,k/Qnas follows
+Pn,k(z)
+Qn(z)=|n|/summationdisplay
+i=1λn,k,i
+z−xn,i, λ n,k,i= lim
+z→xn,i(z−xn,i)Pn,k(z)
+Qn(z)=Pn,k(xn,i)
+Q′n(xn,i).
+The following analogue of the Gauss-Jacobi quadrature formula ta kes place.
+Corollary 1.2.Let(s0,0,... ,s 0,m) =N(σ0,...,σ m)andn=Zm+1
++be given. Then, for each
+k= 0,...,mand every polynomial p,degp≤ |n|+nk−1
+/integraldisplay
+p(x)ds0,k(x) =|n|/summationdisplay
+i=1λn,k,ip(xn,i).
+Ifn= (n,n+1,...,n+1), then
+sign(λn,k,i) = sign(s0,k), i= 1,...,|n|.
+Consequently, for the sequence of multi-indices {(n,n+1,...,n+1)}n∈Z+⊂Zm+1
++, for any bounded
+Riemann-Stieltjes integrable function fonCo(suppσ0)and eachk= 0,...,m
+/integraldisplay
+f(x)ds0,k(x) = lim
+n→∞|n|/summationdisplay
+i=1λn,k,if(xn,i).
+This result provides convergence of the quadrature formulas simu ltaneously for all the measures
+in the Nikishin system taking the same nodes in all the quadrature for mulas. Simultaneous quad-
+rature formulas were studied in [8] in connection with certain applicat ion to computer graphics
+illuminating bodies. Whenever feasible, simultaneous quadrature for mulas are more efficient, from
+the computational point of view, compared with the use of Gauss-J acobi quadrature indepen-
+dently on each measure. In [22], a more detailed study of simultaneou s quadrature formulas for
+Nikishin systems ofmeasuresmaybe found. We wish to point out that for the classofmulti-indices
+considered in Corollary 1.2 all the statements in [22, Corollary 2] hold t rue for allk= 0,...,m.
+Theorem 1.1 follows easily from
+Theorem 1.3.Let(s1,1,...,s 1,m) =N(σ1,... ,σ m)andn= (n0,... ,n m)∈Zm+1
++be given.
+Then, there exists a permutation λof(0,...,m)which reorders the components of ndecreasingly,
+nλ(0)≥ ··· ≥nλ(m),and an associated Nikishin system S(λ) = (r1,1,... ,r 1,m) =N(ρ1,...,ρ m)
+such that for any real polynomials pk,degpk≤nk−1, there exist real polynomials qksuch that
+p0+m/summationdisplay
+k=1pk/hatwides1,k= (q0+m/summationdisplay
+k=1qk/hatwider1,k)/hatwides1,λ(0),degqk≤nλ(k)−1, k= 0,...,m.
+We wish to point out that here /hatwides1,0denotes the function identically equal to 1; this is relevant
+whenλ(0) = 0. There may be several permutations λfor which the statement holds, each one with
+an associated S(λ). We do not know if there is an S(λ) for eachλwhich reorders the components
+ofndecreasingly. As reference, we can say that there exists S(λ) (but not exclusively) for that λ
+which additionally satisfies that for all 0 ≤j <k≤nwithnj=nkthen alsoλ(j)<λ(k).
+Theorem 1.4.Suppose that S1= (s1
+0,0,...,s1
+0,m1) =N(σ1
+0,... ,σ1
+m1),S2= (s2
+0,0,...,s2
+0,m2) =
+N(σ2
+0,... ,σ2
+m2),σ1
+0=σ2
+0,andn= (n1;n2)∈Zm1+1
++×Zm2+1
++,|n1|=|n2|+1,be given. Let λ2and
+S(λ2) =N(ρ2
+1,...,ρ2
+m2)be a permutation and a Nikishin system associated with N(σ2
+1,...,σ2
+m2)NIKISHIN SYSTEMS ARE PERFECT 7
+andn2by Theorem 1.3. Construct (r2
+0,0,...,r2
+0,m2) =N(ρ2
+0,...,ρ2
+m2), whereρ2
+0=/hatwides2
+1,λ2(0)σ2
+0.
+Then/integraldisplay
+xνAn(x)dr2
+0,k(x) = 0, ν= 0,...,n 2,λ2(k)−1, k= 0,...,m 2.
+Using Theorem 1.4 and results from [23], we can obtain logarithmic and r atio asymptotics for
+sequences {An}n∈Λ,Λ⊂Zm1+1
++×Zm2+1
++,|n1|=|n2|+1,under appropriate assumptions on the
+measures generating S1,S2,and Λ.
+A positive measure σis said to be regular if
+lim
+n→∞κ1/n
+n= 1/cap(suppσ),
+where cap( ·) denotes the logarithmic capacity of the Borel set ( ·) andκndenotes the leading
+coefficient of the n-th orthonormalpolynomial with respect to σ. A negative measure σis regularif
+−σis regular. In either cases we write σ∈Reg. For equivalent forms of defining regular measures,
+see sections 3.1 to 3.3 in [45] (in particular Theorem 3.1.1). For short, we write (S1,S2)∈Regto
+mean that all the measures which generate both Nikishin systems ( S1,S2) are regular.
+A region Ω of the extended complex plane which has a compact complem entEis said to
+be regular if the Dirichlet problem has a solution on Ω for any continuou s function defined on
+∂U=∂E. This is equivalent to proving that the Green’s function on Ω with singu larity at ∞can
+be extended continuously to all C(for details on Green’s function and regular domains see [27,
+Theorem 10.12]). In this case it is usual to say also that Eis a regular compact set.
+Definition 1.5.We say that a compact set Eis quasi-regular when E=/tildewideE∪e, where/tildewideEis a
+regular compact set and eis at most a denumerable set whose accumulation points lie in /tildewideE.
+Let Λ = Λ( p1,0,... ,p 1,m1;p2,0,... ,p 2,m2)⊂Zm1+1
++×Zm2+1
++be an infinite sequence of distinct
+multi-indices such that for each n= (n1,n2)∈Λ,|n1|=|n2|+1, and
+lim
+n∈Λn1,j
+|n1|=p1,j∈(0,1), j= 0,...,m 1,lim
+n∈Λn2,j
+|n2|=p2,j∈(0,1), j= 0,...,m 2.
+The following two results require some normalization on the sequence of linear forms under
+consideration. By Theorem 1.2, for each n∈Zm1+1
++×Zm2+1
++,Anis uniquely determined except
+for a constant factor.
+Definition 1.6.Let min{n1,0,... ,n 1,m1} ≥1. We say that Anis monic if the leading co-
+efficient of an,jis 1, where jis eitherm1whenn1,m1= min{n1,0,... ,n 1,m1}, or ifn1,m1>
+min{n1,0,...,n 1,m1}it is such that n1,j= min{n1,0,... ,n 1,m1}andn1,j<n1,k,j <k≤m1.
+We do not need to normalize Anwhenn1has components equal to zero. We have
+Theorem 1.5.LetΛ = Λ(p1,0,... ,p 1,m1;p2,0,... ,p 2,m2)⊂Zm1+1
++×Zm2+1
++,(S1,S2)∈Reg,
+S1=N(σ1
+0,... ,σ1
+m1),andS2=N(σ2
+0,... ,σ2
+m2)be given. Assume that the supports of the
+measures which generate S1,S2are quasi-regular. Then, the associated sequence of monic m ixed
+type multiple orthogonal linear forms {An},n∈Λ, satisfies
+lim
+n∈Λ|An(z)|1/|n1|=G(z),K ⊂C\(∆1
+0∪∆1
+1),
+where∆1
+i= Co(supp σ1
+i),i= 0,1.
+A formula for Gis given in (49) during the proof of Theorem 1.5. It is expressed in ter ms of the
+solutionofavectorequilibriumproblemforthe logarithmicpotential. T he matrixgoverningthe in-
+teractionbetweenthedifferentpotentialsinthesystemdependso n(p1,0,...,p 1,m1;p2,0,...,p 2,m2).
+By allowing quasi-regularity of the supports, this theorem is already novel for standard orthog-
+onal polynomials ( m1=m2= 0) (see Lemma 5.3 below). Theorem 1 .5 unifies the study of the
+logarithmic asymptotics of type I and type II multiple orthogonal po lynomials under the most
+general conditions on the measures, their supports, and the beh avior of the sequence of multi-
+indices on which the limit is taken. It is motivated by the results of [10], [2 6] and [36] where type
+I and type II were considered separately, and the generating mea sures are supported on intervals
+on which their Radon Nikodym derivative is positive almost everywhere . In [23, Theorem 1.3] an
+analogue was obtained assuming that the components of n1,n2are decreasing and the supports of8 FIDALGO AND L ´OPEZ
+the generating measures are regular sets. The first study of the logarithmic asymptotics of mixed
+type multiple orthogonal polynomials of Nikishin systems was carried o ut in [43].
+For the next result, we assume that supp( σi
+j) =/tildewide∆i
+j∪ei
+j,j= 0,...,m i,i= 1,2, where/tildewide∆i
+jis
+a bounded interval of the real line, |(σi
+j)′|>0 a.e. on/tildewide∆i
+j, andei
+jis at most a denumerable set
+without accumulation points in R\/tildewide∆i
+j. We denote this writing
+S1=N′(σ1
+0,... ,σ1
+m1), S2=N′(σ2
+0,...,σ2
+m2).
+Notice the “prime” on N. In the context of this paper, this condition is the analogue of the o ne
+imposed by S. A. Denisov (see [14]) in his extension of E. A. Rakhmanov ’s celebrated theorem
+on ratio asymptotics of orthogonal polynomials. The original proof of Rakhmanov’s theorem is in
+[39]-[40]. An improved and reduced version of the proof by the autho r may be found in [41].
+Fix a vector l:= (l1;l2) where 0 ≤l1≤m1and 0≤l2≤m2. We define the multi-index
+nl:= (n1+el1;n2+el2) = (nl1
+1;nl2
+2), where elidenotes the unit vector of length mi+1 with all
+components equal to zero except the component ( li+1) which equals 1.
+Fix two permutations λ1,λ2,of(0,...,m 1) and (0,...,m 2), respectively, and a positive number
+C. By Λ(λ1,λ2,C) we denote the set of all multi-indices n= (n1;n2)∈Zm1+1
++×Zm2+1
++such that
+a)|n1|=|n2|+1,
+b)λ1,S(λ1),andλ2,S(λ2),are solutions given by Theorem 1.3 to n1,N(σ1
+1,... ,σ1
+m1),and
+n2,N(σ2
+1,... ,σ2
+m2),respectively.
+c)ni,λi(0)−ni,λi(mi)≤C,i= 1,2,
+Any sequence Λ ⊂Zm1+1
++×Zm2+1
++of distinct multi-indices satisfying a) and
+sup{max{ni,0,...,n i,mi}−min{ni,0,...,n i,mi}:n∈Λ,i= 1,2}<∞
+is contained in ∪λ1,λ2Λ(λ1,λ2,C) for some sufficiently large C, where the union is taken over all
+possible pairs of permutations. Thus, any such Λ can be partitioned in a finite number ofsequences
+of indices satisfying a)-c) for the same pair λ1,λ2of permutations, plus a set containing a finite
+number of multi-indices.
+Theorem 1.6.LetS1=N′(σ1
+0,... ,σ1
+m1),S2=N′(σ2
+0,... ,σ2
+m2)andλ1,λ2,be given. Fix l=
+(l1;l2),0≤l1≤m1,0≤l2≤m2.LetΛ⊂Zm1+1
++×Zm2+1
++be an infinite sequence of distinct
+multi-indices such that for all n∈Λ,n,nl∈Λ(λ1,λ2,C)for some sufficiently large C. Then, the
+associated sequence of monic mixed type multiple orthogona l linear forms {An},n∈Λ, verifies
+lim
+n∈ΛAnl(z)
+An(z)=A(l)(z),K ⊂C\(suppσ1
+0∪Co(suppσ1
+1)),
+whereA(l)is a one to one analytic function in C\(/tildewide∆1
+0∪/tildewide∆1
+1).
+An expression for A(l)will be given in (50) at the end of the proof of this result. The answer is
+given in terms of a conformal representation of an associated Riem ann surface with m1+m2+2
+sheets and genus zero onto the extended complex plane. It also de pends onl,λ1,andλ2. We wish
+to point out that if Λ ⊂Λ(λ1,λ2,C), takingl1=λ1(0) andl2=λ2(0) then nl∈Λ(λ1,λ2,C+1),
+for alln∈Λ. Therefore, for any sequence Λ ⊂Λ(λ1,λ2,C) we can always find at least one pair
+(l1;l2),for which ratio asymptotics can be proved. In general, ( l1;l2) is admissible if λ1andλ2
+applied to nl1
+1andnl2
+2, respectively, are also decreasing for all n∈Λ except at most a finite number
+of multi-indices.
+For type II multiple orthogonal polynomials, with generating measur es supported on intervals,
+and multi-indices in Zm+1
++(•), ratio asymptotics was proved in [5]. Different extensions followed in
+[31] and [23]. Theorem 1.6 is an immediate consequence of Theorem 1.4 a nd [23, Theorem 6.4].
+LetMbe the least common multiple of m1+ 1 andm2+ 1. Denote /tildewiden= (/tildewiden1;/tildewiden2) which is
+obtained adding M/(m1+1) to each component of n1andM/(m2+1) to each component of n2.
+We have
+Corollary 1.3.LetS1=N′(σ1
+0,... ,σ1
+m1),S2=N′(σ2
+0,... ,σ2
+m2)be given. Let Λ⊂Zm1+1
++×
+Zm2+1
++be an infinite sequence of distinct multi-indices such that |n1|=|n2|+1for alln∈Λand
+sup{max{ni,0,... ,n i,mi}−min{ni,0,... ,n i,mi}:n∈Λ}<∞, i= 1,2.NIKISHIN SYSTEMS ARE PERFECT 9
+Then
+lim
+n∈ΛA/tildewiden(z)
+An(z)=A(z),K ⊂C\(suppσ1
+0∪Co(suppσ1
+1)).
+An expression for Aappears in (51).
+The strong asymptotics of type II multiple orthogonal polynomials f or Nikishin systems was
+given by A. I. Aptekarev in [4] for diagonal sequences of multi-indice s and systems of generating
+measures formed by weights satisfying Szeg˝ o’s condition. It rema ins the best result in this respect.
+To conclude the Introduction, we call the reader’s attention to th e excellent survey by J. Nuttall
+[38] on Hermite–Pad´ e polynomials. Here, in the form of a general co njecture, the author draws
+the general picture of the asymptotic behavior of Hermite–Pad´ e polynomials in terms of functions
+which are solutions of boundary value problems on associated Rieman n surfaces. Our asymptotic
+results once more confirm his (at that time somewhat bold) predictio ns.
+2.Proof of Theorems 1.1, 1.2 and Corollary 1.1
+We begin with some auxiliary Lemmas.
+Lemma2.1.Letsk,k= 1,...,m, be finite signed Borel measures with compact support such tha t
+Co(suppsk) = ∆⊂R. LetF(z) =f0(z) +/summationtextm
+k=1fk(z)/hatwidesk(z)∈ H(C\∆),wherefk∈ H(V),k=
+0,...,m, andVis a neighborhood of ∆. IfF(z) =O(1/z2),z→ ∞,then
+m/summationdisplay
+k=1/integraldisplay
+fk(x)dsk(x) = 0, (2)
+whereasF(z) =O(1/z),z→ ∞,implies that
+F(z) =m/summationdisplay
+k=1/integraldisplayfk(x)dsk(x)
+z−x. (3)
+Proof.Let Γ⊂Vbe a positively oriented closed smooth Jordan curve that surround s ∆. If
+F(z) =O(1/z2),z→ ∞,from Cauchy’s theorem, Fubini’s theorem and Cauchy’s integral for mula,
+it follows that
+0 =/integraldisplay
+ΓF(z)dz=m/summationdisplay
+k=1/integraldisplay
+Γfk(z)/hatwidesk(z)dz=m/summationdisplay
+k=1/integraldisplay /integraldisplay
+Γfk(z)dz
+z−xdsk(x) = 2πim/summationdisplay
+k=1/integraldisplay
+fk(x)dsk(x),
+and we obtain (2). On the other hand, if F(z) =O(1/z),z→ ∞,and we assume that zis in the
+unbounded connected component of the complement of Γ, Cauchy ’s integral formula and Fubini’s
+theorem render
+F(z) =1
+2πi/integraldisplay
+ΓF(ζ)dζ
+z−ζ=1
+2πim/summationdisplay
+k=1/integraldisplay
+Γfk(ζ)/hatwidesk(ζ)dζ
+z−ζ=
+m/summationdisplay
+k=1/integraldisplay1
+2πi/integraldisplay
+Γfk(ζ)dζ
+(z−ζ)(ζ−x)dsk(x) =m/summationdisplay
+k=1/integraldisplayfk(x)dsk(x)
+z−x
+which is (3). ✷
+Lemma 2.2.Let(s1,1,...,s 1,m) =N(σ1,...,σ m)andn∈Zm+1
++be given. Consider the linear
+form
+Ln=p0+m/summationdisplay
+k=1pk/hatwides1,k,degpk≤nk−1, k= 0,...,m,
+where the polynomials pkhave real coefficients. Assume that n0= max{n0,n1−1,...,n m−1}. If
+Lnhad at least |n|zeros inC\∆1the reduced form p1+/summationtextm
+k=2pk/hatwides2,kwould have at least |n|−n0
+zeros inC\∆2.
+Proof.The function Lnis symmetric with respect to the real line Ln(z) =Ln(z); therefore, its
+zeros come in conjugate pairs. Thus, if Lnhas at least |n|zeros inC\∆1, there exists a polynomial10 FIDALGO AND L ´OPEZ
+wn,degwn≥ |n|,with real coefficients and zeroscontained in C\∆1such that Ln/wn∈ H(C\∆1).
+This function has a zero of order ≥ |n|−n0+1 at∞. Consequently, for all ν= 0,...,|n|−n0−1,
+zνLn
+wn=O(1/z2)∈ H(C\∆1), z→ ∞,
+and
+zνLn
+wn=zνp0
+wn+m/summationdisplay
+k=1zνpk
+wn/hatwides1,k.
+From (2), it follows that
+0 =/integraldisplay
+xν(p1+m/summationdisplay
+k=2pk/hatwides2,k)(x)dσ1(x)
+wn(x), ν= 0,...,|n|−n0−1,
+taking into consideration that s1,1=σ1andds1,k(x) =/hatwides2,k(x)dσ1(x),k= 2,...,m.
+These orthogonality relations imply that p1+/summationtextm
+k=2pk/hatwides2,khas at least |n|−n0sign changes in
+the interior of ∆ 1. In fact, if there were at most |n|−n0−1 sign changes one can easily construct
+a polynomial pof degree ≤ |n|−n0−1 such that p(p1+/summationtextm
+k=2pk/hatwides2,k) does not change sign on ∆ 1
+which contradicts the orthogonality relations. Therefore, alread y in the interior of ∆ 1⊂C\∆2,
+the reduced form would have the number of zeros claimed. ✷
+Using induction, this lemma already allows to prove the AT property fo r multi-indices in
+Zm+1
++(⊛). That result is due to Driver and Stahl (see [16, Theorem 2.4.1]).
+We reduce the general case to the one with n0= max{n0,n1−1,...,n m−1}with
+Lemma 2.3.Let(s1,1,...,s 1,m) =N(σ1,...,σ m),m≥1,andn∈Zm+1
++be given. Consider the
+linear form Lndefined in Lemma 2.2. Assume that nj= max{n0+1,n1,...,n m}. Then, there
+exist a Nikishin system (s∗
+1,1,...,s∗
+1,m) =N(σ∗
+1,... ,σ∗
+m), a multi-index n∗= (n∗
+0,...,n∗
+m)∈Zm+1
++
+which is a permutation of nwithn∗
+0=nj, and polynomials with real coefficients p∗
+k,degp∗
+k≤
+n∗
+k−1,k= 0,...,m, such that
+Ln=p0+m/summationdisplay
+k=1pk/hatwides1,k= (p∗
+0+m/summationdisplay
+k=1p∗
+k/hatwides∗
+1,k)/hatwides1,j=L∗
+n/hatwides1,j.
+The proof is quite intricate and we leave it to the next section. Inste ad let us prove Theorem
+1.1 assuming that the Lemma 2.3 is true.
+Proof of Theorem 1. Obviously, the first statement of the theorem follows from the sec ond.
+We prove the second one using induction on m. Form= 0 the linear form reduces to a polynomial
+of degree ≤n0−1 and thus has at most n0−1 zeros in the complex plane as claimed.
+Assume that the result is true for any Nikishin system with m−1(≥0) measures and let us
+show that it is also valid for Nikishin systems with mmeasures. To the contrary, let us suppose
+thatLnhas at least |n|zeros on C\∆1.
+Shouldn0= max{n0,n1,... ,n m}, by Lemma 2.2 the linear form p1+/summationtextm
+k=2pk/hatwides2,kwould have
+at least|n|−n0zeros inC\∆2. Now,|n|−n0is the norm of the multi-index ( n1,... ,n m) which
+together with the Nikishin system N(σ2,... ,σ m) define the reduced form. This contradicts the
+induction hypothesis.
+Suppose that nj= max{n0+1,n1,... ,n m}. According to Lemma 2.3, the linear form L∗
+nhas
+the same zeros as LninC\∆1, sinces1,jis never zero on that region. The multi-index n∗which
+determines L∗
+nhas the same norm as nand its first component satisfies the assumptions of Lemma
+2.2. Following the same arguments as before we arrive to a contradict ion. The proof is complete.
+✷
+For the proof of Theorem 1.2 and Corollary 1.1, we also use
+Lemma2.4.Let/hatwideSandn∈Zm1+1
++×Zm2+1
++,|n1|=|n2|+1,be given. Then Ansatisfies
+/integraldisplay
+Ln2(x)An(x)dσ2
+0(x) = 0, (4)NIKISHIN SYSTEMS ARE PERFECT 11
+for any linear form
+Ln2(x) =p0(x)+m2/summationdisplay
+j=1pj(x)/hatwides2
+1,j(x),
+where thepj,j= 0,...,m 2,denote arbitrary polynomials such that degpj≤n2,j−1.Anhas
+exactly|n2|zeros inC\Co(suppσ1
+1), they are simple, and lie in the interior of Co(suppσ1
+0).
+Proof. In fact, from the condition b) of Definition 1.4 it follows that there e xists a polynomial
+dn,jsuch that for any polynomial pj,degpj≤n2,j−1,j∈ {0,...,m 2},
+pj(z)/parenleftiggm1/summationdisplay
+k=0an,k(z)/integraldisplay/hatwides2
+1,j(x)/hatwides1
+1,k(x)dσ2
+0(x)
+z−x−dn,j(z)/parenrightigg
+=O/parenleftbig
+1/z2/parenrightbig
+, z→ ∞,
+(here/hatwides2
+1,0≡1) and the function on the left hand side is holomorphic in C\Co(suppσ2
+0). Using
+Lemma 2.1, it follows that
+/integraldisplay
+pj(x)/hatwides2
+1,j(x)m1/summationdisplay
+k=0an,k(x)/hatwides1
+1,k(x)dσ2
+0(x) = 0.
+Adding these relations for j= 0,...,m 2, we obtain (4).
+From Theorem 1.1, we know that Anhas at most |n1| −1 =|n2|zeros on C\Co(suppσ1
+1).
+From (4) it follows that this form has at least |n2|sign changes in the interior of Co(supp σ2
+0) =
+Co(suppσ1
+0). Therefore, the last statement is obtained. ✷
+Let us prove Theorem 1.2 assuming that Lemma 2.3 (Theorem 1.1) is tr ue.
+Proof of Theorem 1.2. Suppose that for some n,Anis not normal. That is, some component
+an,kofAnhas degan,k≤n1,k−2. According to Theorem 1.1, Ancan have on the interval
+Co(suppσ2
+0) at most |n1|−2 =|n2|−1 zeros. Consequently, this function can have in the interior
+ofCo(supp σ2
+0) atmostN≤ |n2|−1signchanges. Supposethis isthecaseandlet x1,...,x Nbethe
+points where it changes sign. According to Theorem 1.1, (1 ,/hatwides2
+1,1,...,/hatwides2
+1,m2) is also an AT system.
+Using the properties of Tchebyshev systems, we can find polynomia lsp0,... ,p m2,with degpj≤
+n2,j−1,such that Ln(x) =p0(x)+/summationtextm2
+j=1pj(x)/hatwides2
+1,j(x) changes sign at x1,... ,x N,and has no other
+points where it changes sign in the interior of Co(supp σ2
+0). Therefore, the function Ln(x)An(x)
+has constant sign on Co(supp σ2
+0) but this contradicts (4) since σ2
+0is a measure with constant sign
+whose support contains infinitely many points. Thus, deg an,k=n1,k−1,k= 0,...,m 1, and
+perfectness has been established.
+Let us assume that there are two non collinear solutions An,A∗
+n,to a)-b). Then, there exists a
+real constant C/\e}atio\slash= 0 such that An−CA∗
+n/\e}atio\slash≡0 and at least one of the components of An−CA∗
+n
+satisfies deg( an,k−Ca∗
+n,k)≤n1,k−2.This is not possible since An−CA∗
+nalso solves a)-b) and
+according to what was proved above all its components must have m aximum possible degree. ✷
+Definition 2.1.LetEbe a subset of the complex plane and Uthe class of all coverings of Eby
+disksUn. The radius of Unis denoted |Un|. The (one dimensional) Hausdorff content of Eis
+h(E) = inf{/summationdisplay
+|Un|:{Un} ∈ U}.
+Let{fn}n∈Λbe a sequence of functions defined on a region D⊂C. We say that {fn}n∈Λ
+converges to fin Hausdorff content on Dif for every compact set K ⊂Dand anyε>0
+lim
+n∈Λh({z∈ K:|fn(z)−f(z)|>ε}) = 0.
+We denote this by
+H−lim
+n→∞fn=f,K ⊂D.
+In [24, Lemma 1], A.A. Gonchar proved that if the functions fnare holomorphic in Dand
+they converge in Hausdoff content to finD, thenfis in fact holomorphic in D(more precisely,
+differs from a holomorphic function on a set of zero Hausdorff conte nt) and the convergence (to
+the equivalent holomorphic function) is uniform on each compact sub set ofD.12 FIDALGO AND L ´OPEZ
+Proof of Corollary 1.1. In [21, Theorem 1] it was proved that under the assumptions of the
+corollary, for each k= 0,...,m,
+H−lim
+n∈ΛRn,k=/hatwidesk,K ⊂C\Co(suppσ0).
+Due to Gonchar’slemma andthe lastassertionofLemma 2.4, it followst hat convergenceis uniform
+on each compact subset of C\Co(suppσ0). Regarding the proof of the rate of convergence, we
+refer to the last sentence on page 104 of [21] (see also [21, Corollar y 1]). ✷
+3.Proof of Lemma 2.3 and Corollary 1.2.
+It is well known (see appendix in [30] and [45, Theorem 6.3.5]) that for e achs∈ M(∆),there
+exists a measure τ∈ M(∆) andℓ(z) =az+b,a= 1/|s|,b∈R,such that
+1//hatwides(z) =ℓ(z)+/hatwideτ(z), (5)
+where|s|is the total variation of the measure s.For convenience, we call τthe inverse measure of
+s.Such measures will appear frequently in our reasonings, so we will fix a notation to distinguish
+them. They will alwaysrefer to inversesof measures denoted with sand will carryoverto them the
+corresponding sub-indices. The same goes for the polynomials ℓ. For instance, if sα,β=/a\}bracketle{tσα,σβ/a\}bracketri}ht
+1//hatwidesα,β(z) =ℓα,β(z)+/hatwideτα,β(z).
+Forconvenience, sometimeswewrite /a\}bracketle{tσα,σβ/hatwide/a\}bracketri}htinplaceof/hatwidesα,β. Thisisspeciallyusefullateronwhere
+we need the Cauchy transforms of complicated expressions of pro ducts of measures for which we
+do not have a short hand notation. Since sα,α=σα, we also write
+1//hatwideσα(z) =ℓα,α(z)+/hatwideτα,α(z).
+Lemma3.1.Letσα∈ M(∆α),σβ∈ M(∆β),and∆α∩∆β=∅.Then:
+/hatwideσα(z)/hatwideσβ(z) =/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)+/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z), z∈C\(∆α∪∆β), (6)
+/hatwideσα(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)=|σα|
+|/a\}bracketle{tσασβ/a\}bracketri}ht|+/integraldisplay/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(xα)
+/hatwideσβ(xα)dτα,β(xα)
+z−xα=|σα|
+|/a\}bracketle{tσα,σβ/a\}bracketri}ht|+/a\}bracketle{tτα,β
+/hatwideσβ,σβ,σα/hatwide/a\}bracketri}ht(z),(7)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)
+/hatwideσα(z)=|/a\}bracketle{tσα,σβ/a\}bracketri}ht|
+|σα|−/integraldisplay/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(xα)dτα,α(xα)
+z−xα=|/a\}bracketle{tσα,σβ/a\}bracketri}ht|
+|σα|−/a\}bracketle{tτα,α,σβ,σα/hatwide/a\}bracketri}ht(z).(8)
+Proof.In fact, (6) follows from the chain of equalities
+/hatwideσα(z)/hatwideσβ(z) =/integraldisplay /integraldisplaydσα(xα)dσβ(xβ)
+(z−xα)(z−xβ)=/integraldisplay /integraldisplay/parenleftbigg1
+z−xα−1
+z−xβ/parenrightbiggdσα(xα)dσβ(xβ)
+xα−xβ
+=/integraldisplay
+/hatwideσα(xβ)dσβ(xβ)
+z−xβ+/integraldisplay
+/hatwideσβ(xα)dσα(xα)
+z−xα=/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)+/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z).
+Notice that
+/hatwideσα(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)−|σα|
+|/a\}bracketle{tσασβ/a\}bracketri}ht|=O/parenleftbigg1
+z/parenrightbigg
+∈ H/parenleftbig
+C\∆α/parenrightbig
+, z→ ∞.
+From (5) and (6), it follows that
+/hatwideσα(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)=/hatwideσβ(z)/hatwideσα(z)
+/hatwideσβ(z)/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)=/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)+/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)=
+1
+/hatwideσβ(z)+/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)ℓα,β+/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)/hatwideτα,β(z).
+Since−|σα|
+|/angbracketleftσασβ/angbracketright|+1
+/hatwideσβ+/angbracketleftσβ,σα/hatwide/angbracketright
+/hatwideσβℓα,βand/angbracketleftσβ,σα/hatwide/angbracketright
+/hatwideσβare analytic on a neighborhood of the interval ∆ α,
+which contains the support of τα,β, relation (3) implies (7).NIKISHIN SYSTEMS ARE PERFECT 13
+The proof of (8) is similar but somewhat more direct. Again, we have t hat
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)
+/hatwideσα(z)−|/a\}bracketle{tσα,σβ/a\}bracketri}ht|
+|σα|=O/parenleftbigg1
+z/parenrightbigg
+∈ H/parenleftbig
+C\∆α/parenrightbig
+, z→ ∞.
+From (5) and (6), we get
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)
+/hatwideσα(z)=/hatwideσα(z)/hatwideσβ(z)−/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσα(z)=/hatwideσβ(z)−/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)ℓα,α(z)−/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)/hatwideτα,α(z).
+But−|/angbracketleftσα,σβ/angbracketright|
+|σα|+/hatwideσβ−/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}htℓα,αand/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}htare analytic in a neighborhood of ∆ α; therefore,
+(3) implies (8). ✷
+Formulas (5)-(6) are the building blocks for (7)-(8) and many more interesting relations. Let us
+further extend Lemma 3.1. The new formulas may be grouped in two s ince (9) may be regarded
+a special case of (10) and (11)-(12) as special cases of (13). Pu tting each group in one formula
+causes some notational incongruence which we prefer to avoid for the benefit of the reader.
+Lemma3.2.Let(s1,1,... ,s 1,m) =N(σ1,...,σ m)be given. Then:
+/hatwides1,k
+/hatwides1,1=|s1,k|
+|s1,1|−/a\}bracketle{tτ1,1,/a\}bracketle{ts2,k,σ1/a\}bracketri}ht/hatwide/a\}bracketri}ht,1 =j <k≤m, (9)
+/hatwides1,k
+/hatwides1,j=|s1,k|
+|s1,j|+(−1)j/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,... ,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht,2≤j <k≤m,
+(10)
+/hatwides1,1
+/hatwides1,j=|s1,1|
+|s1,j|+/a\}bracketle{tτ1,j
+/hatwides2,j,/a\}bracketle{ts2,j,σ1/a\}bracketri}ht/hatwide/a\}bracketri}ht= (11)
+|s1,1|
+|s1,j|+|/a\}bracketle{ts2,j,σ1/a\}bracketri}ht|
+|s2,j|/hatwideτ1,j−/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht/hatwide/a\}bracketri}ht,1 =k<j≤m,
+/hatwides1,2
+/hatwides1,j=|s1,2|
+|s1,j|−/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht
+/hatwides3,j,/a\}bracketle{ts3,j,σ2/a\}bracketri}ht/hatwide/a\}bracketri}ht= (12)
+|s1,2|
+|s1,j|−|/a\}bracketle{ts3,j,σ2/a\}bracketri}ht|
+|s3,j|/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht/hatwide/a\}bracketri}ht+/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,/a\}bracketle{tτ3,j,s2,j/a\}bracketri}ht/hatwide/a\}bracketri}ht,2 =k<j≤m,
+/hatwides1,k
+/hatwides1,j=|s1,k|
+|s1,j|+(−1)k−1/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,...,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht=
+(13)
+|s1,k|
+|s1,j|+(−1)k−1|/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht|
+|sk+1,j|/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,...,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht/hatwide/a\}bracketri}ht+
+(−1)k/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,...,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht,/a\}bracketle{tτk+1,j,sk,j/a\}bracketri}ht/hatwide/a\}bracketri}ht.3 =k<j≤m.
+Proof.Cauchy transformsequal zeroat infinity; therefore, the const ants appearingon the right
+hand sides in each of the first equalities of (9)-(13) must be as indica ted, if in fact the other term
+is a Cauchy transform. Consequently, we will not pay attention to t he constants coming out of the
+consecutive transformations we make in our deduction and simply de note them with consecutive
+constantsCj.
+Obviously, (9) is deduced from (8) taking σα=σ1=s1,1andσβ=/a\}bracketle{tσ2,···,σk/a\}bracketri}ht=s2,k. Formula
+(10) is obtained applying (8) inside out several times as we will indicate .
+Let 2≤j <k≤m. Using (8) on /hatwidesj,k//hatwidesj,j, we have that
+/a\}bracketle{tσj−1,σj,... ,σ k/hatwide/a\}bracketri}ht=/a\}bracketle{tsj−1,j
+/hatwidesj,j,sj,k/hatwide/a\}bracketri}ht=/a\}bracketle{t/hatwidesj,k
+/hatwidesj,jsj−1,j/hatwide/a\}bracketri}ht=C1/hatwidesj−1,j−/a\}bracketle{tsj−1,j,τj,j,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht.
+(14)
+In particular, if j= 2 we get
+/a\}bracketle{tσ1,σ2,...,σ k/hatwide/a\}bracketri}ht=/a\}bracketle{ts1,2
+/hatwides2,2,s2,k/hatwide/a\}bracketri}ht=/a\}bracketle{t/hatwides2,k
+/hatwides2,2s1,2/hatwide/a\}bracketri}ht=C1/hatwides1,2−/a\}bracketle{ts1,2,τ2,2,/a\}bracketle{ts3,k,σ2/a\}bracketri}ht/hatwide/a\}bracketri}ht,14 FIDALGO AND L ´OPEZ
+and applying (8) on /a\}bracketle{ts1,2,τ2,2,/a\}bracketle{ts3,k,σ2/a\}bracketri}ht/hatwide/a\}bracketri}ht//hatwides1,2, it follows that
+/hatwides1,k
+/hatwides1,2=C1−1
+/hatwides1,2/a\}bracketle{ts1,2,τ2,2,/a\}bracketle{ts3,k,σ2/a\}bracketri}ht/hatwide/a\}bracketri}ht=|s1,k|
+|s1,2|+/a\}bracketle{tτ1,2,/a\}bracketle{tτ2,2,s1,2/a\}bracketri}ht,/a\}bracketle{ts3,k,σ2/a\}bracketri}ht/hatwide/a\}bracketri}ht
+which is (10) for j= 2.
+Assume that j≥3. We write then
+/hatwides1,k=/a\}bracketle{tσ1,... ,σ j−2,/hatwidesj,k
+/hatwidesj,jsj−1,j/hatwide/a\}bracketri}ht
+and on account of (14), we obtain
+/hatwides1,k
+/hatwides1,j=C1−1
+/hatwides1,j/a\}bracketle{tσ1,... ,σ j−2,sj−1,j,τj,j,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht.
+This means that
+/hatwides1,k
+/hatwides1,3=C1−1
+/hatwides1,3/a\}bracketle{ts1,3
+/hatwides2,3,s2,3,τ3,3,/a\}bracketle{ts4,k,σ3/a\}bracketri}ht/hatwide/a\}bracketri}ht, j= 3,
+or
+/hatwides1,k
+/hatwides1,j=C1−1
+/hatwides1,j/a\}bracketle{tσ1,... ,σ j−3,sj−2,j
+/hatwidesj−1,j,sj−1,j,τj,j,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht, j≥4.
+Using (8) again, it follows that
+/a\}bracketle{tsj−1,j,τj,j,sj+1,k,σj/hatwide/a\}bracketri}ht
+/hatwidesj−1,j=C2−/a\}bracketle{tτj−1,j,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht.
+Substituting above, we have
+/hatwides1,k
+/hatwides1,3=C3+(−1)21
+/hatwides1,3/a\}bracketle{ts1,3,τ2,3,/a\}bracketle{tτ3,3,s2,3/a\}bracketri}ht,/a\}bracketle{ts4,k,σ3/a\}bracketri}ht/hatwide/a\}bracketri}ht, j= 3,
+or
+/hatwides1,k
+/hatwides1,j=C3+(−1)21
+/hatwides1,j/a\}bracketle{tσ1,...,σ j−3,sj−2,j,τj−1,j,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht, j≥4.
+Ifj= 3 one more use of (8) brings us to (10). If j≥4 we keep on applying (8) inside out until we
+arrive at
+/hatwides1,k
+/hatwides1,j=C4+(−1)j−11
+/hatwides1,j/a\}bracketle{ts1,j,τ2,j,/a\}bracketle{tτ3,j,s2,j/a\}bracketri}ht,... ,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht,
+which is just one step away from (10) through (8) taking
+σα=s1,j, σ β=/a\}bracketle{tτ2,j,/a\}bracketle{tτ3,j,s2,j/a\}bracketri}ht,... ,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/a\}bracketri}ht.
+Now, let us prove formulas (11)-(13). The second equality in each o ne of these relations is an
+immediate consequence of (8) since from it we get
+/a\}bracketle{tsk+1,j,σk/hatwide/a\}bracketri}ht
+/hatwidesk+1,j=|/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht|
+|sk+1,j|−/a\}bracketle{tτk+1,j,sk,j/hatwide/a\}bracketri}ht. (15)
+The proof of the first equality is obtained, generally speaking, as in p roving (10) except that we
+begin using once relation (7). In fact, when k= 1 formula (11) follows directly from (7) taking
+σα=σ1=s1,1andσβ=s2,j.
+Assume that 2 ≤k<j≤m. Using (7), it follows that
+/a\}bracketle{tσk−1,σk/hatwide/a\}bracketri}ht=/a\}bracketle{tsk−1,j
+/hatwidesk,j,sk,k/hatwide/a\}bracketri}ht=/a\}bracketle{t/hatwidesk,k
+/hatwidesk,jsk−1,j/hatwide/a\}bracketri}ht=C5/hatwidesk−1,j+/a\}bracketle{tsk−1,j,τk,j
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht.
+Consequently,
+/hatwides1,2
+/hatwides1,j=C5+1
+/hatwides1,j/a\}bracketle{ts1,j,τ2,j
+/hatwides3,j,/a\}bracketle{ts3,j,σ2/a\}bracketri}ht/hatwide/a\}bracketri}ht, k= 2,
+or
+/hatwides1,k
+/hatwides1,j=C5+1
+/hatwides1,j/a\}bracketle{tσ1,...,σ k−2,sk−1,j,τk,j
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht, k≥3.NIKISHIN SYSTEMS ARE PERFECT 15
+From this point on we use (8). From this formula, we obtain
+/a\}bracketle{tsk−1,j,τk,j
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht
+/hatwidesk−1,j=C6−/a\}bracketle{tτk−1,j,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht,
+and (12) readily follows if k= 2. Fork≥3 this implies
+/hatwides1,3
+/hatwides1,j=C7−1
+/hatwides1,j/a\}bracketle{ts1,j,τ2,j,/a\}bracketle{tτ3,j,s2,j/a\}bracketri}ht
+/hatwides4,j,/a\}bracketle{ts4,j,σ3/a\}bracketri}ht/hatwide/a\}bracketri}ht, k= 3,
+or
+/hatwides1,k
+/hatwides1,j=C7−1
+/hatwides1,j/a\}bracketle{tσ1,... ,σ k−3,sk−2,j,τk−1,j,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht, k≥4.
+Continuing down, using (8) on each step, we obtain
+/hatwides1,k
+/hatwides1,j=C8+(−1)k−21
+/hatwides1,j/a\}bracketle{ts1,j,τ2,j,/a\}bracketle{tτ3,j,s2,j/a\}bracketri}ht,...,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht.
+One more use of (8) with
+σα=s1,j, σ β=/a\}bracketle{tτ2,j,/a\}bracketle{tτ3,j,s2,j/a\}bracketri}ht,... ,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/a\}bracketri}ht
+gives the first equality of (13). With this we conclude the proof. ✷
+Remark 3.1.We wish to point out that formulas (9)-(10) and the second equalitie s in (11)-(13)
+are contained in [17, Theorem 3.1.3], where they were deduced using t he Stieltjes-Plemelj inversion
+formula. Our explicit expressions of the right hand sides are necess ary for the arguments to follow.
+Additionally, the first equalities in (11)-(13) are of great value for t he proof of the general case.
+Proof of Lemma 2.3 when j= 1.From (5) and (9), we have
+Ln
+/hatwides1,1=p0
+/hatwides1,1+p1+m/summationdisplay
+k=2pk/hatwides1,k
+/hatwides1,1=
+(ℓ1,1p0+p1+m/summationdisplay
+k=2|s1,k|
+|s1,1|pk)+p0/hatwideτ1,1−m/summationdisplay
+k=2pk/a\}bracketle{tτ1,1,s2,k,σ1/hatwide/a\}bracketri}ht=L∗
+n.
+We are done taking n∗= (n1,n0,n2,... ,n m) and
+N(σ∗
+1,...,σ∗
+m) =N(τ1,1,/a\}bracketle{tσ2,σ1/a\}bracketri}ht,σ3,...,σ m)
+since/a\}bracketle{ts2,k,σ1/a\}bracketri}ht=/a\}bracketle{t/a\}bracketle{tσ2,σ1/a\}bracketri}ht,σ3,...,σ k/a\}bracketri}htwhenk≥3. ✷
+In the sequel 2 ≤j≤m. From (5), (10), and the first equalities in (11)-(13), one has
+Ln
+/hatwides1,j=p0
+/hatwides1,j+pj+m/summationdisplay
+k/negationslash=j,k=1pk/hatwides1,k
+/hatwides1,j= (ℓ1,jp0+pj+m/summationdisplay
+k/negationslash=j,k=1|s1,k|
+|s1,j|pk)+
+p0/hatwideτ1,j+p1/a\}bracketle{tτ1,j
+/hatwides2,j,/a\}bracketle{ts2,j,σ1/a\}bracketri}ht/hatwide/a\}bracketri}ht+
+j−1/summationdisplay
+k=2(−1)k−1pk/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,...,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht+
+(−1)jm/summationdisplay
+k=j+1pk/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,... ,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht. (16)
+Now, it is not so clear who the auxiliary Nikishin system should be becaus e some annoying ratios
+of Cauchy transforms have appeared. We shall see that already f orj= 2 there are two candidates,
+and for general jthe number of candidates equals 2j−1.
+We can use (15) (see the second inequalities in (11)-(13)) to obtain
+Ln
+/hatwides1,j= (ℓ1,jp0+pj+m/summationdisplay
+k/negationslash=j,k=1|s1,k|
+|s1,j|pk)+(p0+|/a\}bracketle{ts2,j,σ1/a\}bracketri}ht|
+|s2,j|p1)τ1,j+16 FIDALGO AND L ´OPEZ
+j−1/summationdisplay
+k=2(−1)k−1(pk−1+|/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht|
+|sk+1,j|pk)/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,... ,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht/hatwide/a\}bracketri}ht+
+(−1)j−1pj−1/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,... ,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht/hatwide/a\}bracketri}ht+
+(−1)jm/summationdisplay
+k=j+1pk/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,... ,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht. (17)
+(The sum/summationtextj−1
+k=2is empty if j= 2.)
+If we are in the class Zm+1
++(∗) of multi-indices, and we take jto be the first component for
+whichnj= max{n0+1,n1,... ,n m}, thenn0≥ ··· ≥nj−1. It follows that
+deg(ℓ1,jp0+pj+m/summationdisplay
+k/negationslash=j,k=1|s1,k|
+|s1,j|pk)≤nj−1
+and
+deg(pk−1+|/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht|
+|sk+1,j|pk)≤nk−1−1, k= 1,...,j−1.
+ThusL∗
+nis the right hand side of (17), which is a linear form generated by the m ulti-index
+n∗= (nj,n0...,nj−1,nj+1,...,n m)∈Zm+1
++and the Nikishin system
+N(σ∗
+1,... ,σ∗
+m) =N(τ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,... ,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tσj+1,σj/a\}bracketri}ht,σj+2,... ,σ m).
+This would be sufficient to prove the AT property within the class Zm+1
++(∗) because it is easy
+to observe that then ( n0,... ,n j−1,nj+1,...,n m)∈Zm
++(∗) (see the proof of Theorem 1.2). This
+result was first obtained in [22, Theorem 2].
+Of course, (17) is still valid in the general case but, if it is not true th atn0≥...≥nj−1, some
+of the degrees of the polynomials in the linear form on the right hand b low up with respect to the
+bounds established by the components of n∗. We must proceed with caution. For this, we need
+two more reduction formulas which are contained in the next lemma.
+Letτα,β;γ,γdenote the inverse measure of /a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/a\}bracketri}ht. That is,
+1//a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z) =ℓα,β;γ,γ(z)+/hatwideτα,β;γ,γ(z)
+whereℓα,β;γ,γdenotes a first degree polynomial. This notation seems unnecessar ily complicated.
+It is consistent with the one used later for more general inverse me asures which will be needed.
+Lemma 3.3.Let∆γ,∆αand∆βbe three intervals such that ∆γ∩∆α=∅= ∆β∩∆α.Let
+σγ∈ M(∆γ), σα∈ M(∆α)andσβ∈ M(∆β).Then for any f∈L1(σγ)
+/hatwideσα(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)/a\}bracketle{t/a\}bracketle{tτα,α,σβ,σα/a\}bracketri}ht,fσγ,σα/hatwide/a\}bracketri}ht(z) =/a\}bracketle{t/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht
+/hatwideσβτα,β,fσγ,σα,σβ/hatwide/a\}bracketri}ht(z), (18)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)
+/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z)/a\}bracketle{t/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht
+/hatwideσβτα,β,σγ,σα,σβ/hatwide/a\}bracketri}ht(z) =/a\}bracketle{t/a\}bracketle{tσβ,σα,σγ/hatwide/a\}bracketri}ht
+/hatwideσβ/a\}bracketle{tσγ,σα,σβ/hatwide/a\}bracketri}ht
+/hatwideσγτα,β;γ,γ/hatwide/a\}bracketri}ht(z).
+(19)
+Proof.Let us prove (18). Taking into account (6) and (8), we have that
+/a\}bracketle{t/a\}bracketle{tτα,α,σβ,σα/a\}bracketri}ht,/a\}bracketle{tfσγ,σα/a\}bracketri}ht/hatwide/a\}bracketri}ht(z) =/a\}bracketle{tfσγ,σα/hatwide/a\}bracketri}ht(z)/a\}bracketle{tτα,α,σβ,σα/hatwide/a\}bracketri}ht(z)−/a\}bracketle{t/a\}bracketle{tfσγ,σα/a\}bracketri}ht,τα,α,σβ,σα/hatwide/a\}bracketri}ht(z) =
+/a\}bracketle{tfσγ,σα/hatwide/a\}bracketri}ht(z)/parenleftigg
+|/a\}bracketle{tσα,σβ/a\}bracketri}ht|
+|σα|−/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)
+/hatwideσα(z)/parenrightigg
+−/integraldisplay/parenleftigg
+|/a\}bracketle{tσα,σβ/a\}bracketri}ht|
+|σα|−/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(xγ)
+/hatwideσα(xγ)/parenrightigg
+f(xγ)d/a\}bracketle{tσγ,σα/a\}bracketri}ht(xγ)
+z−xγ=
+/integraldisplay
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(xγ)f(xγ)dσγ(xγ)
+z−xγ−/a\}bracketle{tfσγ,σα/hatwide/a\}bracketri}ht(z)/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)
+/hatwideσα(z).
+This and (7), render
+/hatwideσα(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)/a\}bracketle{t/a\}bracketle{tτα,α,σβ,σα/a\}bracketri}ht,fσγ,σα/hatwide/a\}bracketri}ht(z) =NIKISHIN SYSTEMS ARE PERFECT 17
+/hatwideσα(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)/integraldisplay
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(xγ)f(xγ)dσγ(xγ)
+z−xγ−/a\}bracketle{tfσγ,σα/hatwide/a\}bracketri}ht(z) =
+−/a\}bracketle{tfσγ,σα/hatwide/a\}bracketri}ht(z)+|σα|
+|/a\}bracketle{tσα,σβ/a\}bracketri}ht|/a\}bracketle{tfσγ,σα,σβ/hatwide/a\}bracketri}ht(z)+/a\}bracketle{tfσγ,σα,σβ/hatwide/a\}bracketri}ht(z)/a\}bracketle{tτα,β
+/hatwideσβ,/a\}bracketle{tσβ,σα/a\}bracketri}ht/hatwide/a\}bracketri}ht(z)
+Since
+/hatwideσα(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)/a\}bracketle{t/a\}bracketle{tτα,α,σβ,σα/a\}bracketri}ht,fσγ,σα/hatwide/a\}bracketri}ht(z) =O/parenleftbigg1
+z/parenrightbigg
+∈ H(C\∆α), z→ ∞,
+that−/a\}bracketle{tfσγ,σα/hatwide/a\}bracketri}ht+|σα|
+|/angbracketleftσα,σβ/angbracketright|/a\}bracketle{tfσγ,σα,σβ/hatwide/a\}bracketri}htand/a\}bracketle{tfσγ,σα,σβ/hatwide/a\}bracketri}htare analytic on a neighborhood of ∆ α,
+on account of (3), we obtain (18).
+Now, we prove (19). From (8) and (6)
+/a\}bracketle{t/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht
+/hatwideσβτα,β,σγ,σα,σβ/hatwide/a\}bracketri}ht(z) =
+|/a\}bracketle{tσβ,σα/a\}bracketri}ht|
+|/a\}bracketle{tσβ/a\}bracketri}ht|/a\}bracketle{tτα,β,σγ,σα,σβ/hatwide/a\}bracketri}ht(z)−/a\}bracketle{t/a\}bracketle{tτα,β,σγ,σα,σβ/a\}bracketri}ht,τβ,β,σα,σβ/hatwide/a\}bracketri}ht(z) =
+|/a\}bracketle{tσβ,σα/a\}bracketri}ht|
+|/a\}bracketle{tσβ/a\}bracketri}ht|/a\}bracketle{tτα,β,σγ,σα,σβ/hatwide/a\}bracketri}ht(z)−/a\}bracketle{tτα,β,σγ,σα,σβ/hatwide/a\}bracketri}ht(z)/a\}bracketle{tτβ,β,σα,σβ/hatwide/a\}bracketri}ht(z)+
+/a\}bracketle{t/a\}bracketle{tτβ,β,σα,σβ/a\}bracketri}ht,/a\}bracketle{tτα,β,σγ,σα,σβ/a\}bracketri}ht/hatwide/a\}bracketri}ht(z) =
+/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)/a\}bracketle{tτα,β,σγ,σα,σβ/hatwide/a\}bracketri}ht(z)+/a\}bracketle{t/a\}bracketle{tτβ,β,σα,σβ/a\}bracketri}ht,/a\}bracketle{tτα,β,σγ,σα,σβ/a\}bracketri}ht/hatwide/a\}bracketri}ht(z) =
+/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)/parenleftigg
+|/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/a\}bracketri}ht|
+|/a\}bracketle{tσα,σβ/a\}bracketri}ht|−/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)/parenrightigg
++
+/integraldisplay/parenleftigg
+|/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/a\}bracketri}ht|
+|/a\}bracketle{tσα,σβ/a\}bracketri}ht|−/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(xβ)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(xβ)/parenrightigg
+d/a\}bracketle{tτβ,β,σα,σβ/a\}bracketri}ht(xβ)
+z−xβ=
+−/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)+
+/parenleftigg
+/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)+/a\}bracketle{tτβ,β,σα,σβ/hatwide/a\}bracketri}ht(z)/parenrightigg
+|/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/a\}bracketri}ht|
+|/a\}bracketle{tσα,σβ/a\}bracketri}ht|−/a\}bracketle{tτβ,β,/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z) = (20)
+−/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)+|/a\}bracketle{tσβ,σα/a\}bracketri}ht|
+|/a\}bracketle{tσβ/a\}bracketri}ht||/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/a\}bracketri}ht|
+|/a\}bracketle{tσα,σβ/a\}bracketri}ht|−/a\}bracketle{tτβ,β,/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z) =
+−/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)−|/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/a\}bracketri}ht|
+|σβ|−/a\}bracketle{tτβ,β,/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z) =
+−/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z)
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)+/a\}bracketle{tσβ,σα,σγ/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z).
+In the second last equality above, we employed that
+0 = lim
+z→∞z/hatwideσα(z)/hatwideσβ(z) = lim
+z→∞z/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)+ lim
+z→∞z/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z) =|/a\}bracketle{tσα,σβ/a\}bracketri}ht|+|/a\}bracketle{tσβ,σα/a\}bracketri}ht|,
+which implies that |/a\}bracketle{tσα,σβ/a\}bracketri}ht|=−|/a\}bracketle{tσβ,σα/a\}bracketri}ht|. Analogously, −|/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/a\}bracketri}ht|=−|/a\}bracketle{t/a\}bracketle{tσα,σγ/a\}bracketri}ht,σβ/a\}bracketri}ht|=
+|/a\}bracketle{tσβ,/a\}bracketle{tσα,σγ/a\}bracketri}ht/a\}bracketri}ht|and by (8)
+/a\}bracketle{tσβ,/a\}bracketle{tσα,σγ/a\}bracketri}ht/hatwide/a\}bracketri}ht
+/hatwideσβ=|/a\}bracketle{tσβ,/a\}bracketle{tσα,σγ/a\}bracketri}ht/a\}bracketri}ht|
+|σβ|−/a\}bracketle{tτβ,β,/a\}bracketle{t/a\}bracketle{tσα,σγ/a\}bracketri}ht,σβ/a\}bracketri}ht/hatwide/a\}bracketri}ht=−|/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/a\}bracketri}ht|
+|σβ|−/a\}bracketle{tτβ,β,/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht,
+which is used in the last equality. Therefore, from the chain of equat ions and (7), we derive that
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)
+/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z)/a\}bracketle{t/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht
+/hatwideσβτα,β,σγ,σα,σβ/hatwide/a\}bracketri}ht(z) =−/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)+/a\}bracketle{tσβ,σα,σγ/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)
+/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z)=18 FIDALGO AND L ´OPEZ
+−/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)+/a\}bracketle{tσβ,σα,σγ/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)|/a\}bracketle{tσα,σβ/a\}bracketri}ht|
+|/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/a\}bracketri}ht|+/a\}bracketle{tσβ,σα,σγ/hatwide/a\}bracketri}ht(z)
+/hatwideσβ(z)/a\}bracketle{tτα,β;γ,γ
+/hatwideσγ,σγ,σα,σβ/hatwide/a\}bracketri}ht(z).
+Taking into consideration that
+/a\}bracketle{tσα,σβ/hatwide/a\}bracketri}ht(z)
+/a\}bracketle{t/a\}bracketle{tσα,σβ/a\}bracketri}ht,σγ/hatwide/a\}bracketri}ht(z)/a\}bracketle{t/a\}bracketle{tσβ,σα/hatwide/a\}bracketri}ht
+/hatwideσβτα,β,σγ,σα,σβ/hatwide/a\}bracketri}ht(z) =O/parenleftbigg1
+z/parenrightbigg
+∈ H(C\∆α), z→ ∞,
+whereas −/angbracketleftσβ,σα/hatwide/angbracketright
+/hatwideσβ+/angbracketleftσβ,σα,σγ/hatwide/angbracketright
+/hatwideσβ|/angbracketleftσα,σβ/angbracketright|
+|/angbracketleft/angbracketleftσα,σβ/angbracketright,σγ/angbracketright|and/angbracketleftσβ,σα,σγ/hatwide/angbracketright
+/hatwideσβare analytic on a neighborhood of ∆ α,
+using (3) we obtain (19). ✷
+Proof of Lemma 2.3 when 2≤j≤m.Set
+L∗
+n=p∗
+0+p0/hatwideτ1,j+p1/a\}bracketle{tτ1,j
+/hatwides2,j,/a\}bracketle{ts2,j,σ1/a\}bracketri}ht/hatwide/a\}bracketri}ht+
+j−1/summationdisplay
+k=2(−1)k−1pk/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,...,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht+
+(−1)jm/summationdisplay
+k=j+1pk/a\}bracketle{tτ1,j,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,...,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht, (21)
+where
+p∗
+0=ℓ1,jp0+pj+m/summationdisplay
+k/negationslash=j,k=1|s1,k|
+|s1,j|pk,degp∗
+0≤nj−1 =n∗
+0−1.
+We tookthis function fromthe righthand sideof(16). We must show that thereexist amulti-index
+n∗∈Zm+1
++, which is a permutation of n, and a Nikishin system N(σ∗
+0,... ,σ∗
+m) which allow to
+expressL∗
+nas a linear form generated by them with polynomials with real coefficien ts. So far,n∗
+0
+defined above serves the purpose of being the first component of n∗andp∗
+0of being the polynomial
+part of the linear form.
+First step. Isn0≥n1orn0≤n1? (Whenn0=n1we can proceed either ways.)
+A1)Ifn0≥n1, taken∗
+1=n0andσ∗
+1=τ1,j. Decompose/angbracketlefts2,j,σ1/hatwide/angbracketright
+/hatwides2,jusing (8). Then, the first three
+terms of L∗
+nare
+p∗
+0+p0/hatwideσ∗
+1+p1/a\}bracketle{tσ∗
+1
+/hatwides2,j,/a\}bracketle{ts2,j,σ1/a\}bracketri}ht/hatwide/a\}bracketri}ht=p∗
+0+(p0+|/a\}bracketle{ts2,j,σ1/a\}bracketri}ht|
+|s2,j|p1)/hatwideσ∗
+1−p1/a\}bracketle{tσ∗
+1,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht/hatwide/a\}bracketri}ht.
+Consequently, taking p∗
+1=p0+|/angbracketlefts2,j,σ1/angbracketright|
+|s2,j|p1, we have that deg p∗
+1≤n∗
+1−1(=n0−1).
+In case that j= 2, we obtain
+L∗
+n=p∗
+0+p∗
+1/hatwideτ1,2−p1/a\}bracketle{tτ1,2,/a\}bracketle{tτ2,2,s1,2/a\}bracketri}ht/hatwide/a\}bracketri}ht+m/summationdisplay
+k=3pk/a\}bracketle{tτ1,2,/a\}bracketle{tτ2,2,s1,2/a\}bracketri}ht,/a\}bracketle{ts3,k,σ2/a\}bracketri}ht/hatwide/a\}bracketri}ht
+(compare with (17)). Then, the proof would be complete taking n∗= (n2,n0,n1,n3...,nm) and
+the Nikishin system
+N(σ∗
+1,... ,σ∗
+m) =N(τ1,2,/a\}bracketle{tτ2,2,s1,2/a\}bracketri}ht,/a\}bracketle{tσ3,σ2/a\}bracketri}ht,σ4,... ,σ m).
+(Ifm= 2, then n∗= (n2,n0,n1) and the Nikishin system is N(τ1,2,/a\}bracketle{tτ2,2,s1,2/a\}bracketri}ht).)
+Ifj≥3, we obtain
+L∗
+n=p∗
+0+p∗
+1/hatwideσ∗
+1−p1/a\}bracketle{tσ∗
+1,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht/hatwide/a\}bracketri}ht−p2/a\}bracketle{tσ∗
+1,/a\}bracketle{ts3,j,σ2/hatwide/a\}bracketri}ht
+/hatwides3,j/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht/hatwide/a\}bracketri}ht+
+j−1/summationdisplay
+k=3(−1)k−1pk/a\}bracketle{tσ∗
+1,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,... ,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht+
+(−1)jm/summationdisplay
+k=j+1pk/a\}bracketle{tσ∗
+1,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,...,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht.NIKISHIN SYSTEMS ARE PERFECT 19
+B1)Ifn0≤n1, taken∗
+1=n1andσ∗
+1=/angbracketlefts2,j,σ1/hatwide/angbracketright
+/hatwides2,jτ1,j. We can rewrite (21) as follows
+L∗
+n=p∗
+0+p1/hatwideσ∗
+1+p0/a\}bracketle{t/hatwides2,j
+/a\}bracketle{ts2,j,σ1/hatwide/a\}bracketri}htσ∗
+1/hatwide/a\}bracketri}ht+
+j−1/summationdisplay
+k=2(−1)k−1pk/a\}bracketle{t/hatwides2,j
+/a\}bracketle{ts2,j,σ1/hatwide/a\}bracketri}htσ∗
+1,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,... ,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht+
+(−1)jm/summationdisplay
+k=j+1pk/a\}bracketle{t/hatwides2,j
+/a\}bracketle{ts2,j,σ1/hatwide/a\}bracketri}htσ∗
+1,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,... ,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht, (22)
+Decompose/hatwides2,j
+/angbracketlefts2,j,σ1/hatwide/angbracketrightusing (7). Then, the first three terms of L∗
+nin (22) can be expressed as
+p∗
+0+p1/hatwideσ∗
+1+p0/a\}bracketle{t/hatwides2,j
+/a\}bracketle{ts2,j,σ1/hatwide/a\}bracketri}htσ∗
+1/hatwide/a\}bracketri}ht=p∗
+0+(p1+|s2,j|
+|/a\}bracketle{ts2,j,σ1/a\}bracketri}ht|p0)/hatwideσ∗
+1+p0/a\}bracketle{tσ∗
+1,/hatwides1,j
+/hatwideσ1τ2,j;1,1/hatwide/a\}bracketri}ht.
+Takingp∗
+1=p1+|s2,j|
+|/angbracketlefts2,j,σ1/angbracketright|p0, we have that deg p∗
+1≤n∗
+1−1(=n1−1).
+Ifj= 2, due to (18) (in the next formula /hatwides4,k≡1 ifk= 3)
+/hatwides2,2
+/a\}bracketle{ts2,2,σ1/hatwide/a\}bracketri}ht/a\}bracketle{t/a\}bracketle{tτ2,2,s1,2/a\}bracketri}ht,/a\}bracketle{ts3,k,σ2/a\}bracketri}ht/hatwide/a\}bracketri}ht=/hatwideσ2
+/a\}bracketle{tσ2,σ1/hatwide/a\}bracketri}ht/a\}bracketle{t/a\}bracketle{tτ2,2,s1,2/a\}bracketri}ht,/hatwides4,kσ3,σ2/hatwide/a\}bracketri}ht=
+/a\}bracketle{t/a\}bracketle{tσ1,σ2/hatwide/a\}bracketri}ht
+/hatwideσ1τ2,2;1,1,/hatwides4,kσ3,σ2,σ1/hatwide/a\}bracketri}ht=/a\}bracketle{t/a\}bracketle{tσ1,σ2/hatwide/a\}bracketri}ht
+/hatwideσ1τ2,2;1,1,/a\}bracketle{tσ3,σ2,σ1/a\}bracketri}ht,s4,k/hatwide/a\}bracketri}ht.
+Consequently,
+L∗
+n=p∗
+0+p∗
+1/hatwideσ∗
+1+p0/a\}bracketle{tσ∗
+1,/hatwides1,2
+/hatwideσ1τ2,2;1,1/hatwide/a\}bracketri}ht+m/summationdisplay
+k=3pk/a\}bracketle{tσ∗
+1,/hatwides1,2
+/hatwideσ1τ2,2;1,1,/a\}bracketle{tσ3,σ2,σ1/a\}bracketri}ht,s4,k/hatwide/a\}bracketri}ht.
+In this situation, we would be done considering n∗= (n2,n1,n0,n3,... ,n m) and the system
+N(σ∗
+1,...,σ∗
+m) =N(/a\}bracketle{tσ2,σ1/hatwide/a\}bracketri}ht
+/hatwideσ2τ1,2,/a\}bracketle{tσ1,σ2/hatwide/a\}bracketri}ht
+/hatwideσ1τ2,2;1,1,/a\}bracketle{tσ3,σ2,σ1/a\}bracketri}ht,σ4,...,σ m).
+(Shouldm= 2, then n∗= (n2,n1,n0) and the Nikishin system is N(/angbracketleftσ2,σ1/hatwide/angbracketright
+/hatwideσ2τ1,j,/angbracketleftσ1,σ2/hatwide/angbracketright
+/hatwideσ1τ2,2;1,1).)
+This result already includes new multi-indices for which it is possible to pr ove normality. The only
+sub-case studied previously (see [19]) was for m= 2. Therefore, if j= 2 we are done.
+Let us assume that j≥3. (The algorithm ends after j−1 steps.) So far, we have used the
+notationsk,lonly withk≤l. We will extend its meaning to k>lin which case
+sk,l=/a\}bracketle{tσk,σk−1,...,σ l/a\}bracketri}ht, k>l.
+Notice that if l<k<j , then
+/a\}bracketle{tsk,j,sk−1,l/a\}bracketri}ht=/a\}bracketle{tsk,l,sk+1,j/a\}bracketri}ht.
+The inverse measure of /a\}bracketle{tsk,j,sk−1,l/a\}bracketri}htwe denote by τk,j;k−1,l; that is,
+1//a\}bracketle{tsk,j,sk−1,l/hatwide/a\}bracketri}ht=ℓk,j;k−1,l+/hatwideτk,j;k−1,l
+In particular, τ2,j;1,1denotes the inverse measure of /a\}bracketle{ts2,j,σ1/a\}bracketri}ht.
+Let us transform the measures in/summationtextj−1
+k=2of (22). Regarding the term with p2, using (19) with
+σα=σ2,σβ=s3,jandσγ=σ1, we obtain
+/hatwides2,j
+/a\}bracketle{ts2,j,σ1/hatwide/a\}bracketri}ht/a\}bracketle{t/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht
+/hatwides3,j,/a\}bracketle{ts3,j,σ2/a\}bracketri}ht/hatwide/a\}bracketri}ht=/a\}bracketle{t/a\}bracketle{ts3,j,σ2,σ1/hatwide/a\}bracketri}ht
+/hatwides3,j/hatwides1,j
+/hatwideσ1τ2,j;1,1/hatwide/a\}bracketri}ht.20 FIDALGO AND L ´OPEZ
+Forj= 3,/summationtextj−1
+k=3is empty, so here the formulas make sense when j≥4. Using (18), with σα=s2,j,
+σβ=σ1,σγ=τ3,j, and
+f=f1,j,k=
+
+/a\}bracketle{ts4,j,σ3/hatwide/a\}bracketri}ht
+/hatwides4,j, 3 =k<j≤m,
+/a\}bracketle{t/a\}bracketle{tτ4,j,s3,j/a\}bracketri}ht
+/hatwides5,j,/a\}bracketle{ts5,j,σ4/a\}bracketri}ht/hatwide/a\}bracketri}ht, 4 =k<j≤m,
+/a\}bracketle{t/a\}bracketle{tτ4,j,s3,j/a\}bracketri}ht,... ,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht,5≤k<j≤m,
+we obtain
+/hatwides2,j
+/a\}bracketle{ts2,j,σ1/hatwide/a\}bracketri}ht/a\}bracketle{t/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,... ,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht=
+/a\}bracketle{t/hatwides1,j
+/hatwideσ1τ2,j;1,1,f1,j,kτ3,j,s2,j,σ1/hatwide/a\}bracketri}ht=
+
+
+/a\}bracketle{t/hatwides1,j
+/hatwideσ1τ2,j;1,1,/a\}bracketle{tτ3,j,s2,j,σ1/a\}bracketri}ht
+/hatwides4,j,/a\}bracketle{ts4,j,σ3/a\}bracketri}ht/hatwide/a\}bracketri}ht, 3 =k<j≤m,
+/a\}bracketle{t/hatwides1,j
+/hatwideσ1τ2,j;1,1,/a\}bracketle{tτ3,j,s2,j,σ1/a\}bracketri}ht,/a\}bracketle{tτ4,j,s3,j/a\}bracketri}ht
+/hatwides5,j,/a\}bracketle{ts5,j,σ4/a\}bracketri}ht/hatwide/a\}bracketri}ht, 4 =k<j≤m,
+/a\}bracketle{t/hatwides1,j
+/hatwideσ1τ2,j;1,1,/a\}bracketle{t/hatwideσ1s2,j/hatwide/a\}bracketri}htτ3,j,/hatwides3,jτ4,j,...,/hatwidesk−2,jτk−1,j,/angbracketleft/hatwideσksk+1,j/hatwide/angbracketright/hatwidesk−1,j
+/hatwidesk+1,jτk,j/hatwide/a\}bracketri}ht,5≤k<j≤m.
+In the last row, we used a more compact notation to fit the line. Notic e that it is the same as
+/a\}bracketle{t/hatwides1,j
+/hatwideσ1τ2,j;1,1,/a\}bracketle{tτ3,j,s2,j,σ1/a\}bracketri}ht,/a\}bracketle{tτ4,j,s3,j/a\}bracketri}ht,...,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht.
+As for the terms in/summationtextm
+k=j+1, applying (18) with σα=s2,j, σβ=σ1,σγ=τ3,j, and
+f=f1,j,k=/braceleftigg
+/a\}bracketle{ts4,k,σ3/hatwide/a\}bracketri}ht, 3 =j <k≤m,
+/a\}bracketle{t/a\}bracketle{tτ4,j,s3,j/a\}bracketri}ht,...,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht,4≤j <k≤m,
+it follows that
+/hatwides2,j
+/a\}bracketle{ts2,j,σ1/hatwide/a\}bracketri}ht/a\}bracketle{t/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,... ,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht=
+=/a\}bracketle{t/hatwides1,j
+/hatwideσ1τ2,j;1,1,f1,j,kτ3,j,s2,j,σ1/hatwide/a\}bracketri}ht=
+
+
+/a\}bracketle{t/hatwides1,3
+/hatwideσ1τ2,3;1,1,/a\}bracketle{tτ3,3,s2,3,σ1/a\}bracketri}ht,/a\}bracketle{ts4,k,σ3/a\}bracketri}ht/hatwide/a\}bracketri}ht, 3 =j <k≤m,
+/a\}bracketle{t/hatwides1,j
+/hatwideσ1τ2,j;1,1,/a\}bracketle{tτ3,j,s2,j,σ1/a\}bracketri}ht,/a\}bracketle{tτ4,j,s3,j/a\}bracketri}ht,...,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht,4≤j <k≤m.
+Whenj=mno such terms exist.
+Therefore,
+L∗
+n=p∗
+0+p∗
+1/hatwideσ∗
+1+p0/a\}bracketle{tσ∗
+1,/hatwides1,j
+/hatwideσ1τ2,j;1,1/hatwide/a\}bracketri}ht−p2/a\}bracketle{tσ∗
+1,/a\}bracketle{ts3,j,σ2,σ1/hatwide/a\}bracketri}ht
+/hatwides3,j/hatwides1,j
+/hatwideσ1τ2,j;1,1/hatwide/a\}bracketri}ht+
+j−1/summationdisplay
+k=3(−1)k−1pk/a\}bracketle{tσ∗
+1,/hatwides1,j
+/hatwideσ1τ2,j;1,1,f1,j,kτ3,j,s2,j,σ1/hatwide/a\}bracketri}ht+(−1)jm/summationdisplay
+k=j+1pk/a\}bracketle{tσ∗
+1,/hatwides1,j
+/hatwideσ1τ2,j;1,1,f1,j,kτ3,j,s2,j,σ1/hatwide/a\}bracketri}ht.
+Using the notation for f1,j,kdefined previously, in A1)we ended up with
+L∗
+n=p∗
+0+p∗
+1/hatwideσ∗
+1−p1/a\}bracketle{tσ∗
+1,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht/hatwide/a\}bracketri}ht−p2/a\}bracketle{tσ∗
+1,/a\}bracketle{ts3,j,σ2/hatwide/a\}bracketri}ht
+/hatwides3,j/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht/hatwide/a\}bracketri}ht+
+j−1/summationdisplay
+k=3(−1)k−1pk/a\}bracketle{tσ∗
+1,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,f1,j,kτ3,j,s2,j/hatwide/a\}bracketri}ht+(−1)jm/summationdisplay
+k=j+1pk/a\}bracketle{tσ∗
+1,/a\}bracketle{tτ2,j,s1,j/a\}bracketri}ht,f1,j,kτ3,j,s2,j/hatwide/a\}bracketri}ht.
+Denote
+σ(l1)
+2=/braceleftbigg/hatwides1,jτ2,j, l 1= 1,
+/hatwides1,j
+/hatwideσ1τ2,j;1,1, l1= 0.NIKISHIN SYSTEMS ARE PERFECT 21
+The two formulas for L∗
+nmay be put in one writing
+L∗
+n=p∗
+0+p∗
+1/hatwideσ∗
+1+(−1)l1pl1/a\}bracketle{tσ∗
+1,σ(l1)
+2/hatwide/a\}bracketri}ht−p2/a\}bracketle{tσ∗
+1,/a\}bracketle{ts3,j,s2,l1+1/hatwide/a\}bracketri}ht
+/hatwides3,jσ(l1)
+2/hatwide/a\}bracketri}ht+ (23)
+m/summationdisplay
+k=3,k/negationslash=jδk,jpk/a\}bracketle{tσ∗
+1,σ(l1)
+2,f1,j,kτ3,j,s2,l1+1,s3,j/hatwide/a\}bracketri}ht, n l1= min{n0,n1},
+where
+δk,j=/braceleftbigg
+(−1)k−1, k<j,
+(−1)j, k>j.
+We also have
+degp∗
+0≤nj−1 =n∗
+0−1,degp∗
+1≤max{n0,n1}−1 =n∗
+1−1.
+Forj= 2 we found the following solutions
+max{n0,n1}(n∗
+0,... ,n∗
+m) N(σ∗
+1,...,σ∗
+m)
+n0 (n2,n0,n1,n3,... ,n m)N(τ1,2,/a\}bracketle{tτ2,2,s1,2/a\}bracketri}ht,s3,2,σ4,...,σ m)
+n1 (n2,n1,n0,n3,... ,n m)N(/hatwides2,1
+/hatwides2,2τ1,2,/hatwides1,2
+/hatwides1,1τ2,2;1,1,s3,1,σ4,...,σ m)
+(Ifm= 2, then n∗has only the first three components and the Nikishin system has only the first
+two measures indicated.)
+We are ready for the induction hypothesis but, for the sake of clar ity, let us take one more step.
+Second step. The casej= 2 has been solved; therefore, j≥3. We ask whether min {n0,n1} ≥
+n2or min{n0,n1} ≤n2? (When min {n0,n1}=n2we can proceed either ways.)
+A2)If min{n0,n1} ≥n2, taken∗
+2= min{n0,n1}andσ∗
+2=σ(l1)
+2,l1∈ {0,1}. Decompose
+/angbracketlefts3,j,s2,l1+1/hatwide/angbracketright
+/hatwides3,jusing (8). Then, the first four terms of L∗
+nreduce to
+p∗
+0+p∗
+1/hatwideσ∗
+1+(−1)l1pl1/a\}bracketle{tσ∗
+1,σ∗
+2/hatwide/a\}bracketri}ht−p2/a\}bracketle{tσ∗
+1,/a\}bracketle{ts3,j,s2,l1+1/hatwide/a\}bracketri}ht
+/hatwides3,jσ∗
+2/hatwide/a\}bracketri}ht=
+p∗
+0+p∗
+1/hatwideσ∗
+1+p∗
+2/a\}bracketle{tσ∗
+1,σ∗
+2/hatwide/a\}bracketri}ht+p2/a\}bracketle{tσ∗
+1,σ∗
+2,τ3,j,s2,l1+1,s3,j/hatwide/a\}bracketri}ht, n l1= min{n0,n1},
+withp∗
+2= (−1)l1pl1−|/angbracketlefts3,j,s2,l1+1/angbracketright|
+|s3,j|p2, degp∗
+2≤n∗
+2−1.
+In case that j= 3, we have
+L∗
+n=p∗
+0+p∗
+1/hatwideσ∗
+1+p∗
+2/a\}bracketle{tσ∗
+1,σ∗
+2/hatwide/a\}bracketri}ht+p2/a\}bracketle{tσ∗
+1,σ∗
+2,τ3,3,s2,l1+1,s3,3/hatwide/a\}bracketri}ht−m/summationdisplay
+k=4pk/a\}bracketle{tσ∗
+1,σ∗
+2,f1,3,kτ3,3,s2,l1+1,s3,3/hatwide/a\}bracketri}ht,
+This subcase produces the two solutions (in both min {n0,n1} ≥n2)
+max{n0,n1}(n∗
+0,...,n∗
+m) N(σ∗
+1,...,σ∗
+m)
+n0 (n3,n0,n1,n2,...)N(τ1,3,/hatwides1,3τ2,3,/hatwides2,3τ3,3,s4,3,...)
+n1 (n3,n1,n0,n2,...)N(/angbracketlefts2,3,σ1/hatwide/angbracketright
+/hatwides2,3τ1,3,/hatwides1,3
+/hatwides1,1τ2,3;1,1,/a\}bracketle{tτ3,3,s2,3,σ1/a\}bracketri}ht,s4,3,...)
+Ifm= 3, then n∗has only the first four components and the Nikishin system has only t he first
+three measures indicated. When m≥4
+(n∗
+4,...,n∗
+m) = (n4,... ,n m),(σ∗
+5,...,σ∗
+m) = (σ5,... ,σ m),(ifm≥5).
+Letj≥4. Define
+f=f2,j,k=
+
+/a\}bracketle{ts5,j,σ4/hatwide/a\}bracketri}ht
+/hatwides5,j, 4 =k<j≤m,
+/a\}bracketle{t/a\}bracketle{tτ5,j,s4,j/a\}bracketri}ht
+/hatwides6,j,/a\}bracketle{ts6,j,σ5/a\}bracketri}ht/hatwide/a\}bracketri}ht, 5 =k<j≤m,
+/a\}bracketle{t/a\}bracketle{tτ5,j,s4,j/a\}bracketri}ht,... ,/a\}bracketle{tτk−1,j,sk−2,j/a\}bracketri}ht,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht,6≤k<j≤m,
+/a\}bracketle{ts5,k,σ4/hatwide/a\}bracketri}ht, 4 =j <k≤m,
+/a\}bracketle{t/a\}bracketle{tτ5,j,s4,j/a\}bracketri}ht,... ,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht, 5≤j <k≤m,22 FIDALGO AND L ´OPEZ
+From (23), we obtain
+L∗
+n=p∗
+0+p∗
+1/hatwideσ∗
+1+p∗
+2/a\}bracketle{tσ∗
+1,σ∗
+2/hatwide/a\}bracketri}ht+p2/a\}bracketle{tσ∗
+1,σ∗
+2,τ3,j,s2,l1+1,s3,j/hatwide/a\}bracketri}ht+p3/a\}bracketle{tσ∗
+1,σ∗
+2,/a\}bracketle{ts4,j,σ3/hatwide/a\}bracketri}ht
+/hatwides4,jτ3,j,s2,l1+1,s3,j/hatwide/a\}bracketri}ht+
+m/summationdisplay
+k=4,k/negationslash=jδk,jpk/a\}bracketle{tσ∗
+1,σ∗
+2,/a\}bracketle{tτ3,j,s2,l1+1,s3,j/a\}bracketri}ht,f2,j,kτ4,j,s3,j/hatwide/a\}bracketri}ht, n l1= min{n0,n1}.(24)
+B2)If min{n0,n1} ≤n2, taken∗
+2=n2andσ∗
+2=/angbracketlefts3,j,s2,l1+1/hatwide/angbracketright
+/hatwides3,jσ(l1)
+2. Using (7) to decompose
+/hatwides3,j
+/angbracketlefts3,j,s2,l1+1/hatwide/angbracketright, the first four terms of L∗
+nin (23) become
+p∗
+0+p∗
+1/hatwideσ∗
+1−p2/a\}bracketle{tσ∗
+1,σ∗
+2/hatwide/a\}bracketri}ht+(−1)l1pl1/a\}bracketle{tσ∗
+1,/hatwides3,j
+/a\}bracketle{ts3,j,s2,l1+1/hatwide/a\}bracketri}htσ∗
+2/hatwide/a\}bracketri}ht=
+p∗
+0+p∗
+1/hatwideσ∗
+1+p∗
+2/a\}bracketle{tσ∗
+1,σ∗
+2/hatwide/a\}bracketri}ht+(−1)l1pl1/a\}bracketle{tσ∗
+1,σ∗
+2,/a\}bracketle{ts2,l1+1,s3,j/hatwide/a\}bracketri}ht
+/hatwides2,l1+1τ3,j;2,l1+1/hatwide/a\}bracketri}ht, n l1= min{n0,n1}
+wherep∗
+2=−p2+(−1)l1|/hatwides3,j|
+|/angbracketlefts3,j,s2,l1+1/hatwide/angbracketright|pl1,degp∗
+2≤n∗
+2−1.
+Ifj=m= 3 we are done. Should m≥4 andj= 3, using (18) with σα=σ3=s3,3,σβ=
+s2,l1+1,σγ=s4,k, andf≡1,for the terms in/summationtextm
+k=4, we obtain (see definition of f1,3,k)
+/a\}bracketle{tσ(l1)
+2,f1,3,kτ3,3,s2,l1+1,s3,3/hatwide/a\}bracketri}ht=/a\}bracketle{t/hatwides3,3
+/a\}bracketle{ts3,3,s2,l1+1/hatwide/a\}bracketri}htσ∗
+2,/a\}bracketle{tτ3,3,s2,l1+1,s3,3/a\}bracketri}ht,s4,k,s3,3/hatwide/a\}bracketri}ht=
+/a\}bracketle{tσ∗
+2,/a\}bracketle{ts2,l1+1,s3,3/hatwide/a\}bracketri}ht
+/hatwides2,l1+1τ3,3;2,l1+1,s4,k,s3,l1+1/hatwide/a\}bracketri}ht.
+Therefore,
+L∗
+n=p∗
+0+p∗
+1/hatwideσ∗
+1+p∗
+2/a\}bracketle{tσ∗
+1,σ∗
+2/hatwide/a\}bracketri}ht+(−1)l1pl1/a\}bracketle{tσ∗
+1,σ∗
+2,/a\}bracketle{ts2,l1+1,s3,j/hatwide/a\}bracketri}ht
+/hatwides2,l1+1τ3,j;2,l1+1/hatwide/a\}bracketri}ht−
+m/summationdisplay
+k=4pk/a\}bracketle{tσ∗
+1,σ∗
+2,/a\}bracketle{ts2,l1+1,s3,3/hatwide/a\}bracketri}ht
+/hatwides2,l1+1τ3,3;2,l1+1,s4,k,s3,l1+1/hatwide/a\}bracketri}ht
+The casej= 3 is finished with two additional solutions (in both min {n0,n1} ≤n2)
+max{n0,n1}(n∗
+0,... ,n∗
+m) N(σ∗
+1,... ,σ∗
+m)
+n0 (n3,n0,n2,n1,...)N(τ1,3,/hatwides3,2
+/hatwides3,3/hatwides1,3τ2,3,/hatwides2,3
+/hatwides2,2τ3,3;2,2,s4,2,...)
+n1 (n3,n1,n2,n0,...)N(/angbracketlefts2,3,σ1/hatwide/angbracketright
+/hatwides2,3τ1,3,/hatwides3,1
+/hatwides3,3/hatwides1,3
+/hatwides1,1τ2,3;1,1,/angbracketlefts2,1,s3,3/hatwide/angbracketright
+/hatwides2,1τ3,3;2,1,s4,1,...)
+Ifm= 3, then n∗has only the first four components and the Nikishin system has only t he first
+three measures indicated. When m≥4
+(n∗
+4,...,n∗
+m) = (n4,... ,n m),(σ∗
+5,...,σ∗
+m) = (σ5,... ,σ m),(ifm≥5).
+Let us transform the terms in/summationtextm
+k=3,k/negationslash=jof (23). Assume that j≥4. Consider the function
+multiplying p3; that is, when 3 = k<j≤m. Using (19) with σα=σ3,σβ=s4,j,andσγ=s2,l1+1,
+we obtain
+/a\}bracketle{tσ(l1)
+2,f1,j,3τ3,j,s2,l1+1,s3,j/hatwide/a\}bracketri}ht=/a\}bracketle{t/hatwides3,j
+/a\}bracketle{ts3,j,s2,l1+1/hatwide/a\}bracketri}htσ∗
+2,/a\}bracketle{ts4,j,σ3/hatwide/a\}bracketri}ht
+/hatwides4,jτ3,j,s2,l1+1,s3,j/hatwide/a\}bracketri}ht=
+/a\}bracketle{tσ∗
+2,/a\}bracketle{ts4,j,s3,l1+1/hatwide/a\}bracketri}ht
+/hatwides4,j/a\}bracketle{ts2,l1+1,s3,j/hatwide/a\}bracketri}ht
+/hatwides2,l1+1τ3,j;2,l1+1/hatwide/a\}bracketri}ht, l 1∈ {0,1}.
+To reduce the terms in/summationtextj−1
+k=4when 5≤j≤m,and/summationtextm
+k=j+1for 4≤j < m, apply (18) with
+σα=s3,j, σβ=σ2orσβ=/a\}bracketle{tσ2,σ1/a\}bracketri}ht, σγ=τ4,j, andf=f2,j,k. It follows that
+/a\}bracketle{tσ(l1)
+2,f1,j,kτ3,j,s2,l1+1,s3,j/hatwide/a\}bracketri}ht=/a\}bracketle{t/hatwides3,j
+/a\}bracketle{ts3,j,s2,l1+1/hatwide/a\}bracketri}htσ∗
+2,/a\}bracketle{tτ3,j,s2,l1+1,s3,j/a\}bracketri}ht,f2,j,kτ4,j,s3,j/hatwide/a\}bracketri}ht=NIKISHIN SYSTEMS ARE PERFECT 23
+/a\}bracketle{tσ∗
+2,/a\}bracketle{ts2,l1+1,s3,j/hatwide/a\}bracketri}ht
+/hatwides2,l1+1τ3,j;2,l1+1,f2,j,kτ4,j,s3,j,s2,l1+1/hatwide/a\}bracketri}ht, l 1∈ {0,1}.
+Therefore,
+L∗
+n=p∗
+0+p∗
+1/hatwideσ∗
+1+p∗
+2/a\}bracketle{tσ∗
+1,σ∗
+2/hatwide/a\}bracketri}ht+(−1)l1pl1/a\}bracketle{tσ∗
+1,σ∗
+2,/a\}bracketle{ts2,l1+1,s3,j/hatwide/a\}bracketri}ht
+/hatwides2,l1+1τ3,j;2,l1+1/hatwide/a\}bracketri}ht+ (25)
+p3/a\}bracketle{tσ∗
+1,σ∗
+2,/a\}bracketle{ts4,j,s3,l1+1/hatwide/a\}bracketri}ht
+/hatwides4,j/a\}bracketle{ts2,l1+1,s3,j/hatwide/a\}bracketri}ht
+/hatwides2,l1+1τ3,j;2,l1+1/hatwide/a\}bracketri}ht,
+/summationdisplay
+k=4,k/negationslash=jδk,jpk/a\}bracketle{tσ∗
+1,σ∗
+2,/a\}bracketle{ts2,l1+1,s3,j/hatwide/a\}bracketri}ht
+/hatwides2,l1+1τ3,j;2,l1+1,f2,j,kτ4,j,s3,j,s2,l1+1/hatwide/a\}bracketri}ht, l 1∈ {0,1}.
+Let us write down (24)-(25) in one expression. Recall that j≥4. Take
+σ(l2)
+3=
+
+/hatwides2,jτ3,j, l 1= 1 in (24),
+/a\}bracketle{ts2,j,σ1/hatwide/a\}bracketri}htτ3,j, l 1= 0 in (24),
+/hatwides2,j
+/hatwides2,2τ3,j;2,2, l 1= 1 in (25),
+/angbracketlefts2,1,s3,j/hatwide/angbracketright
+/hatwides2,1τ3,j;2,1, l1= 0 in (25).
+Then
+L∗
+n=p∗
+0+p∗
+1/hatwideσ∗
+1+p∗
+2/a\}bracketle{tσ∗
+1,σ∗
+2/hatwide/a\}bracketri}ht+(−1)l2pl2/a\}bracketle{tσ∗
+1,σ∗
+2,σ(l2)
+3/hatwide/a\}bracketri}ht+p3/a\}bracketle{tσ∗
+1,σ∗
+2,/a\}bracketle{ts4,j,s3,l2+1/hatwide/a\}bracketri}ht
+/hatwides4,jσ(l2)
+3/hatwide/a\}bracketri}ht+
+m/summationdisplay
+k=4,k/negationslash=jδk,jpk/a\}bracketle{tσ∗
+1,σ∗
+2,σ(l2)
+3,f2,j,kτ4,j,s3,l2+1,s4,j/hatwide/a\}bracketri}ht, n l2= min{n0,n1,n2}.(26)
+We also have that deg p∗
+k≤n∗
+k−1,k= 0,1,2,where
+n∗
+k=
+
+nj= max{n0+1,n1,... ,n m}, k= 0,
+max{n0,n1}, k = 1,
+max{min{n0,n1},n2}, k = 2,(27)
+and the measures σ∗
+1,σ∗
+2have also been determined. The polynomials p∗
+0,p∗
+1,p∗
+2, the indices
+n∗
+0,n∗
+1,n∗
+2, and the measures σ∗
+1,σ∗
+2will not change in subsequent reductions. The structure of
+a Nikishin systems brakes down in the Cauchy transform multiplying p3due to an annoying ratio
+of Cauchy transforms modifying the third measure in the correspo nding product.
+With this unified formula, you can repeat the line of reasonings employ ed in step 2 (or 1) and
+so on. Let us write down the eight solutions corresponding to j= 4. In the table, besides (27), we
+have thatn∗
+3= max{min{n0,n1,n2},n3},n∗
+4= min{n0,... ,n 3}, and of course n∗
+0=n4.
+(n∗
+1,...,n∗
+4)N(σ∗
+1,... ,σ∗
+m)
+(n0,n1,n2,n3)N(τ1,4,/hatwides1,4τ2,4,/hatwides2,4τ3,4,/hatwides3,4τ4,4,s5,4,...)
+(n1,n0,n2,n3)N(/angbracketlefts2,4,σ1/hatwide/angbracketright
+/hatwides2,4τ1,4,/hatwides1,4
+/hatwideσ1τ2,4;1,1,/a\}bracketle{tτ3,4,s2,4,σ1/a\}bracketri}ht,/hatwides3,4τ4,4,s5,4,...)
+(n0,n2,n1,n3)N(τ1,4,/angbracketlefts3,4,σ2/hatwide/angbracketright
+/hatwides3,4/hatwides1,4τ2,4,/hatwides2,4
+/hatwideσ2τ3,4;2,2,/a\}bracketle{tτ4,4,s3,2,σ4/a\}bracketri}ht,s5,4,...)
+(n1,n2,n0,n3)N(/angbracketlefts2,4,σ1/hatwide/angbracketright
+/hatwides2,4τ1,4,/angbracketlefts3,4,s2,1/hatwide/angbracketright
+/hatwides3,4/hatwides1,4
+/hatwideσ1τ2,4;1,1,/angbracketlefts2,1,s3,4/hatwide/angbracketright
+/hatwides2,1τ3,4;2,1,/a\}bracketle{tτ4,4,s3,1,σ4/a\}bracketri}ht,s5,4,...)
+(n0,n1,n3,n2)N(τ1,4,/hatwides1,4τ2,4,/hatwides4,3
+/hatwideσ4/hatwides2,4τ3,4,/hatwides3,4
+/hatwideσ3τ4,4;3,3,s5,3,...)
+(n1,n0,n3,n2)N(/angbracketlefts2,4,σ1/hatwide/angbracketright
+/hatwides2,4τ1,4,/hatwides1,4
+/hatwideσ1τ2,4;1,1,/hatwides4,3
+/hatwideσ4/a\}bracketle{tτ3,4,s2,4,σ1/a\}bracketri}ht,/hatwides3,4
+/hatwideσ3τ4,4;3,3,s5,3,...)
+(n0,n2,n3,n1)N(τ1,4,/angbracketlefts3,4,σ2/hatwide/angbracketright
+/hatwides3,4/hatwides1,4τ2,4,/hatwides4,2
+/hatwideσ4/hatwides2,4
+/hatwideσ2τ3,4;2,2,/angbracketlefts3,2,σ4/hatwide/angbracketright
+/hatwides3,2τ4,4;3,2,s5,2,...)
+(n1,n2,n3,n0)N(/angbracketlefts2,4,σ1/hatwide/angbracketright
+/hatwides2,4τ1,4,/angbracketlefts3,4,s2,1/hatwide/angbracketright
+/hatwides3,4/hatwides1,4
+/hatwideσ1τ2,4;1,1,/hatwides4,1
+/hatwideσ4/angbracketlefts2,1,s3,4/hatwide/angbracketright
+/hatwides2,1τ3,4;2,1,/angbracketlefts3,1,σ4/hatwide/angbracketright
+/hatwides3,1τ4,4;3,1,s5,1,...)
+Ifm= 4, then n∗has only the first five components and the Nikishin system has only th e first
+four measures indicated. When m≥5
+(n∗
+5,...,n∗
+m) = (n5,... ,n m),(σ∗
+6,...,σ∗
+m) = (σ6,... ,σ m),(ifm≥6).
+Summarizing, in step 1 we proved the statement of Lemma 2.3 when j= 2, showing that there
+are two solutions, and for j≥3 we obtained formula (23) which allowed us to prove in step 2 that24 FIDALGO AND L ´OPEZ
+the lemma is true when j= 3, with 4 solutions, and for j≥4 to obtain formula (26), similar to
+(23), which provides the instruments to carry out step 3 and so on . The case j= 1 was treated
+separately. The induction will be on the number of successful step s we have been able to carry
+out. The counter of the steps will be denoted j∗. Fixj,2≤j≤m. Assume that we have been
+able to carry out j∗steps. Let us describe what we have obtained in
+Stepj∗. Induction hypothesis. We have proved that the statement of Lemma 2 .3 holds
+whenj=j∗+1, with 2j−1= 2j∗solutions, and when j≥j∗+2,we have defined:
+(1) indices
+n∗
+k=/braceleftbiggnj= max{n0+1,n1,...,n m}, k= 0,
+max{min{n0,... ,n k−1},nk}, k= 1,...,j∗.
+(2) integers lk,k= 0,...,j∗, inductively as follows: l0= 0,l1is the subindex between those
+ofn0,n1not employed in defining n∗
+1, and so forth until lj∗which is the subindex between
+those ofn0,...,n j∗which was not employed in defining n∗
+1,... ,n∗
+j∗. In particular,
+nlk= min{n0,... ,n k}.
+(3) polynomials
+p∗
+k=
+
+ℓ1,jp0+pj+/summationtextm
+i=1,i/negationslash=j|s1,i|
+|s1,j|pi, k= 0,
+(−1)lk−1plk−1+C1,j,kpk, k = 1,...,j∗,min{n0,...,n k−1} ≥nk,
+(−1)k−1pk+C2,j,kplk−1, k = 1,...,j∗,min{n0,...,n k−1} ≤nk.
+whereC1,j,k,C2,j,k,are real constants different from zero.
+(4) functions
+fj∗,j,k=
+
+/a\}bracketle{tsj∗+3,k,σj∗+2/hatwide/a\}bracketri}ht, j∗+2 =j <k≤m,
+/a\}bracketle{t/a\}bracketle{tτj∗+3,j,sj∗+2,j/a\}bracketri}ht,... ,/a\}bracketle{tτj,j,sj−1,j/a\}bracketri}ht,/a\}bracketle{tsj+1,k,σj/a\}bracketri}ht/hatwide/a\}bracketri}ht, j∗+3≤j <k≤m,
+/a\}bracketle{tsj∗+3,j,σj∗+2/hatwide/a\}bracketri}ht
+/hatwidesj∗+3,j, j∗+2 =k<j≤m,
+/a\}bracketle{t/a\}bracketle{tτj∗+3,j,sj∗+2,j/a\}bracketri}ht
+/hatwidesj∗+4,j,/a\}bracketle{tsj∗+4,j,σj∗+3/a\}bracketri}ht/hatwide/a\}bracketri}ht, j∗+3 =k<j≤m,
+/a\}bracketle{t/hatwidesj∗+2,jτj∗+3,j,...,/hatwidesk−2,jτk−1,j,/a\}bracketle{tτk,j,sk−1,j/a\}bracketri}ht
+/hatwidesk+1,j,/a\}bracketle{tsk+1,j,σk/a\}bracketri}ht/hatwide/a\}bracketri}ht, j∗+4≤k<j≤m.
+(5) measures σ∗
+1,...,σ∗
+j∗andσ(lj∗)
+j∗+1whosesupportsarecontainedinthe sameintervals∆ 1,...,
+∆j∗+1asσ1,...,σ j∗+1, respectively, where
+σ(lj∗)
+j∗+1=
+
+τj∗+1,j,sj∗,lj∗−1+1,sj∗+1,j, if max{min{n0,...,n j∗−1},nj∗}=nlj∗−1,
+/a\}bracketle{tsj∗,lj∗−1+1,sj∗+1,j/hatwide/a\}bracketri}ht
+/hatwidesj∗,lj∗−1+1τj∗+1,j;j∗,lj∗−1+1,if max{min{n0,...,n j∗−1},nj∗}=nj∗.
+With these elements, we have proved the formula (analogous to tho se obtained in steps 1 and 2)
+L∗
+n=p∗
+0+j∗/summationdisplay
+k=1p∗
+k/a\}bracketle{tσ∗
+1,...,σ∗
+k/hatwide/a\}bracketri}ht+(−1)lj∗plj∗/a\}bracketle{tσ∗
+1,...,σ∗
+j∗,σ(lj∗)
+j∗+1/hatwide/a\}bracketri}ht+ (28)
+(−1)j∗pj∗+1/a\}bracketle{tσ∗
+1,...,σ∗
+j∗,/a\}bracketle{tsj∗+2,j,sj∗+1,lj∗+1/hatwide/a\}bracketri}ht
+/hatwidesj∗+2,jσ(lj∗)
+j∗+1/hatwide/a\}bracketri}ht+
+m/summationdisplay
+k=j∗+2,k/negationslash=jδk,jpk/a\}bracketle{tσ∗
+1,... ,σ∗
+j∗,σ(lj∗)
+j∗+1,fj∗,j,kτj∗+2,j,sj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht.
+Stepj∗+ 1. Induction proof. To complete the induction we must prove in this step that
+Lemma 2.3 is satisfied when j=j∗+2, with 2j∗+1solutions, and when j≥j∗+3 to produce a
+formula which extends (28) one more step.
+Case A) Suppose that max {min{n0,... ,n j∗},nj∗+1}= min{n0,... ,n j∗}=nlj∗. Take
+n∗
+j∗+1=nlj∗andσ∗
+j∗+1=σ(lj∗)
+j∗+1.NIKISHIN SYSTEMS ARE PERFECT 25
+Using (8) on/angbracketleftsj∗+2,j,sj∗+1,lj∗+1/hatwide/angbracketright
+/hatwidesj∗+2,j, from (28) it follows that
+L∗
+n=p∗
+0+j∗+1/summationdisplay
+k=1p∗
+k/a\}bracketle{tσ∗
+1,... ,σ∗
+k/hatwide/a\}bracketri}ht+(−1)j∗+1pj∗+1/a\}bracketle{tσ∗
+1,...,σ∗
+j∗,σ∗
+j∗+1,τj∗+2,j,sj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht+
+m/summationdisplay
+k=j∗+2,k/negationslash=jδk,jpk/a\}bracketle{tσ∗
+1,... ,σ∗
+j∗,σ(lj∗)
+j∗+1,fj∗,j,kτj∗+2,j,sj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht,
+where
+p∗
+j∗+1= (−1)lj∗plj∗+(−1)j∗|/a\}bracketle{tsj∗+2,j,sj∗+1,lj∗+1/a\}bracketri}ht|
+/hatwidesj∗+2,jpj∗+1,degp∗
+j∗+1≤nj∗+1−1.
+Sincefj∗,j∗+2,k=/a\}bracketle{tsj∗+3,k,σj∗+2/hatwide/a\}bracketri}ht, ifj=j∗+2 we have that L∗
+nis a linear form generated by
+n∗= (n∗
+0,...,n∗
+j∗+1,nj∗+1,nj∗+3,...,n m)
+and the Nikishin system
+N(σ∗
+1,... ,σ∗
+j∗+1,/a\}bracketle{tτj∗+2,j∗+2,sj∗+1,lj∗+1,sj∗+2,j∗+2/a\}bracketri}ht,sj∗+3,j∗+2,σj∗+4,... ,σ m).
+Whenm=j(=j∗+2), the system ends with the measure /a\}bracketle{tτj∗+2,j∗+2,sj∗+1,l+1,sj∗+2,j∗+2/a\}bracketri}ht.
+Ifj≥j∗+3, (28) may be expressed as
+L∗
+n=p∗
+0+j∗+1/summationdisplay
+k=1p∗
+k/a\}bracketle{tσ∗
+1,... ,σ∗
+k/hatwide/a\}bracketri}ht+(−1)j∗+1pj∗+1/a\}bracketle{tσ∗
+1,...,σ∗
+j∗+1,τj∗+2,j,sj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht+
+(−1)j∗+1pj∗+2/a\}bracketle{tσ∗
+1,...,σ∗
+j∗+1,/a\}bracketle{tsj∗+3,j,σj∗+2/hatwide/a\}bracketri}ht
+/hatwidesj∗+3,jτj∗+2,j,sj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht+ (29)
+m/summationdisplay
+k=j∗+3,k/negationslash=jδk,jpk/a\}bracketle{tσ∗
+1,...,σ∗
+j∗+1,/a\}bracketle{tτj∗+2,j,sj∗+1,lj∗+1,sj∗+2,j/a\}bracketri}ht,fj∗+1,j,kτj∗+3,j,sj∗+2,j/hatwide/a\}bracketri}ht,
+wherefj∗+1,j,kis defined as fj∗,j,ksubstituting j∗byj∗+1.
+Case B) If max{min{n0,... ,n j∗},nj∗+1}=nj∗+1, take
+n∗
+j∗+1=nj∗+1andσ∗
+j∗+1=/a\}bracketle{tsj∗+2,j,sj∗+1,lj∗+1/hatwide/a\}bracketri}ht
+/hatwidesj∗+2,jσ(lj∗)
+j∗+1.
+Rewrite (28) as follows
+L∗
+n=p∗
+0+j∗/summationdisplay
+k=1p∗
+k/a\}bracketle{tσ∗
+1,...,σ∗
+k/hatwide/a\}bracketri}ht+(−1)j∗pj∗+1/a\}bracketle{tσ∗
+1,...,σ∗
+j∗,σ∗
+j∗+1/hatwide/a\}bracketri}ht+
+(−1)lj∗plj∗/a\}bracketle{tσ∗
+1,... ,σ∗
+j∗,/hatwidesj∗+2,j
+/a\}bracketle{tsj∗+2,j,sj∗+1,lj∗+1/hatwide/a\}bracketri}htσ∗
+j∗+1/hatwide/a\}bracketri}ht+
+m/summationdisplay
+k=j∗+2,k/negationslash=jδk,jpk/a\}bracketle{tσ∗
+1,... ,σ∗
+j∗,/hatwidesj∗+2,j
+/a\}bracketle{tsj∗+2,j,sj∗+1,lj∗+1/hatwide/a\}bracketri}htσ∗
+j∗+1,fj∗,j,kτj∗+2,j,sj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht.
+Using (7), reduce/hatwidesj∗+2,j
+/angbracketleftsj∗+2,j,sj∗+1,lj∗+1/hatwide/angbracketrightin the term with plj∗. This formula transforms into
+L∗
+n=p∗
+0+j∗+1/summationdisplay
+k=1p∗
+k/a\}bracketle{tσ∗
+1,...,σ∗
+k/hatwide/a\}bracketri}ht+ (30)
+(−1)lj∗plj∗/a\}bracketle{tσ∗
+1,...,σ∗
+j∗+1,/a\}bracketle{tsj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht
+/hatwidesj∗+1,lj∗+1τj∗+2,j;j∗+1,lj∗+1/hatwide/a\}bracketri}ht+
+m/summationdisplay
+k=j∗+2,k/negationslash=jδk,jpk/a\}bracketle{tσ∗
+1,... ,σ∗
+j∗,/hatwidesj∗+2,j
+/a\}bracketle{tsj∗+2,j,sj∗+1,lj∗+1/hatwide/a\}bracketri}htσ∗
+j∗+1,fj∗,j,kτj∗+2,j,sj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht.26 FIDALGO AND L ´OPEZ
+where
+p∗
+j∗+1= (−1)j∗pj∗+1+(−1)lj∗|sj∗+2,j|
+|/a\}bracketle{tsj∗+2,j,sj∗+1,lj∗+1/a\}bracketri}ht|plj∗,degp∗
+j∗+1≤nj∗+1−1.
+Ifm=j∗+2 we are done, since the sum/summationtextm
+k=j∗+2,k/negationslash=jis empty (recall that j∗+2≤j≤m).
+Suppose that j∗+2 =j <m. Using (18) with σα=σj∗+2,σβ=sj∗+1,lj∗+1,σγ=sj∗+3,k, and
+f≡1,for the terms in/summationtextm
+k=j∗+3(see definition of fj∗,j∗+2,k), (30) becomes
+L∗
+n=p∗
+0+j∗+1/summationdisplay
+k=1p∗
+k/a\}bracketle{tσ∗
+1,...,σ∗
+k/hatwide/a\}bracketri}ht+
+(−1)lj∗plj∗/a\}bracketle{tσ∗
+1,...,σ∗
+j∗+1,/a\}bracketle{tsj∗+1,lj∗+1,σj∗+2/hatwide/a\}bracketri}ht
+/hatwidesj∗+1,lj∗+1τj∗+2,j∗+2;j∗+1,lj∗+1/hatwide/a\}bracketri}ht+
+m/summationdisplay
+k=j∗+3δk,jpk/a\}bracketle{tσ∗
+1,... ,σ∗
+j∗+1,/a\}bracketle{tsj∗+1,lj∗+1,σj∗+2/hatwide/a\}bracketri}ht
+/hatwidesj∗+1,lj∗+1τj∗+2,j∗+2;j∗+1,lj∗+1,sj∗+3,k,sj∗+2,lj∗+1/hatwide/a\}bracketri}ht.
+Thus, we conclude with j=j∗+2 taking
+n∗= (n∗
+0,... ,n∗
+j∗+1,nlj∗,nj∗+3,... ,n m)
+and the Nikishin system
+N(σ∗
+1,...,σ∗
+j∗+1,/a\}bracketle{tsj∗+1,lj∗+1,σj∗+2/hatwide/a\}bracketri}ht
+/hatwidesj∗+1,lj∗+1τj∗+2,j∗+2;j∗+1,lj∗+1,sj∗+3,lj∗+1,σj∗+4,...,σ m).
+Ifm=j(=j∗+2), the system ends with the measure/angbracketleftsj∗+1,lj∗+1,σj∗+2/hatwide/angbracketright
+/hatwidesj∗+1,lJ∗+1τj∗+2,j∗+2;j∗+1,lj∗+1.
+Letj∗+3≤j≤m. In (30), the measure multiplying pj∗+2is transformed by means of (19),
+withσα=σj∗+2,σβ=sj∗+3,jandσγ=sj∗+1,lj∗+1. All other terms of/summationtextm
+k=j∗+2,k/negationslash=jare reduced
+employing (18) taking σα=sj∗+2,j,σβ=sj∗+1,lj∗+1,σγ=τj∗+2,jandf=fj∗+1,j,k(remember
+thatfj∗+1,j,kis defined substituting j∗byj∗+1 in the definition of fj∗,j,kas was already used at
+the end of case A)). It is easy to verify that (30) adopts the expr ession
+L∗
+n=p∗
+0+j∗+1/summationdisplay
+k=1p∗
+k/a\}bracketle{tσ∗
+1,...,σ∗
+k/hatwide/a\}bracketri}ht+ (31)
+(−1)lj∗plj∗/a\}bracketle{tσ∗
+1,...,σ∗
+j∗+1,/a\}bracketle{tsj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht
+/hatwidesj∗+1,lj∗+1τj∗+2,j;j∗+1,lj∗+1/hatwide/a\}bracketri}ht+
+(−1)j∗+1pj∗+2/a\}bracketle{tσ∗
+1,...,σ∗
+j∗+1,/a\}bracketle{tsj∗+3,j,sj∗+2,lj∗+1/hatwide/a\}bracketri}ht
+/hatwidesj∗+3,j/a\}bracketle{tsj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht
+/hatwidesj∗+1,lj∗+1τj∗+2,j;j∗+1,lj∗+1/hatwide/a\}bracketri}ht+
+′/summationdisplay
+δk,jpk/a\}bracketle{tσ∗
+1,...,σ∗
+j∗+1,/a\}bracketle{tsj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht
+/hatwidesj∗+1,lj∗+1τj∗+2,j;j∗+1,lj∗+1,fj∗+1,j,kτj∗+3,j,sj∗+2,j,sj∗+1,lj∗+1/hatwide/a\}bracketri}ht
+(/summationtext′=/summationtextm
+k=j∗+3,k/negationslash=j), which has the same structure as (29).
+Letlj∗+1denotethesubindexbetweenthoseof n0,... ,n j∗+1whichwasnotemployedindefining
+n∗
+1,... ,n∗
+j∗+1, then
+nlj∗+1= min{n0,... ,n j∗+1}.
+Notice that in the situation of case A), lj∗+1=lj∗, and in case B), lj∗+1=j∗+1.Define
+σ(lj∗+1)
+j∗+2=
+
+τj∗+2,j,sj∗+1,lj∗+1,sj∗+2,j, if max{min{n0,...,n j∗},nj∗+1}=nlj∗,
+/a\}bracketle{tsj∗+1,lj∗+1,sj∗+2,j/hatwide/a\}bracketri}ht
+/hatwidesj∗+1,lj∗+1τj∗+2,j;j∗+1,lj∗+1,if max{min{n0,...,n j∗},nj∗+1}=nj∗+1.
+With these notations, formulas (29) and (31) may be unified in
+L∗
+n=p∗
+0+j∗+1/summationdisplay
+k=1p∗
+k/a\}bracketle{tσ∗
+1,... ,σ∗
+k/hatwide/a\}bracketri}ht+(−1)lj∗+1plj∗+1/a\}bracketle{tσ∗
+1,...,σ∗
+j∗+1,σ(lj∗+1)
+j∗+2/hatwide/a\}bracketri}ht+NIKISHIN SYSTEMS ARE PERFECT 27
+(−1)j∗+1pj∗+2/a\}bracketle{tσ∗
+1,... ,σ∗
+j∗+1,/a\}bracketle{tsj∗+3,j,sj∗+2,lj∗+1+1/hatwide/a\}bracketri}ht
+/hatwidesj∗+3,jσ(lj∗+1)
+j∗+2/hatwide/a\}bracketri}ht+
+m/summationdisplay
+k=j∗+3,k/negationslash=jδk,jpk/a\}bracketle{tσ∗
+1,... ,σ∗
+j∗+1,σ(lj∗+1)
+j∗+2,fj∗+1,j,kτj∗+3,j,sj∗+2,lj∗+1+1,sj∗+3,j/hatwide/a\}bracketri}ht.
+With this we conclude the induction and Lemma 2.3 has been proved. ✷
+Before moving on, let us write down the expressions of the p∗
+kafter carrying out j−1 steps. We
+will need their structure for further developments. We have
+L∗
+n=p∗
+0+m/summationdisplay
+k=1p∗
+k/a\}bracketle{tσ∗
+1,...,σ∗
+k/hatwide/a\}bracketri}ht
+where
+p∗
+k=
+
+ℓ1,jp0+pj+/summationtextm
+i=1,i/negationslash=j|s1,i|
+|s1,j|pi, k= 0,
+(−1)lk−1plk−1+C1,j,kpk, k = 1,...,j−1,max{nlk−1,nk}=nlk−1,
+(−1)k−1pk+C2,j,kplk−1, k = 1,...,j−1,max{nlk−1,nk}=nk,
+(−1)j−1pj−1 k=j, min{nlj−2,nj−1}=nj−1,
+(−1)lj−1plj−2, k =j, min{nlj−2,nj−1}=nlj−2,
+(−1)jpk, k =j+1,...,m.(32)
+Lemma 2.3 has an immediate consequence in terms of the orthogonalit y conditions satisfied by
+the linear form An(see (4)). We state it as a lemma which is useful to prove Corollary 1.2 and
+the results on the asymptotic behavior of sequences of these linea r forms.
+Lemma 3.4.Let/hatwideSandn= (n1;n2)∈Zm1+1
++×Zm2+1
++,|n1|=|n2|+ 1,be given. Suppose that
+n2,j= max{n2,0+1,n2,1,...,n 2,m2}. Letn∗
+2= (n∗
+2,0,...,n∗
+2,m2)andN(σ2∗
+1,... ,σ2∗
+m2)be a multi-
+index and a Nikishin system associated with n2andN(σ2
+1,...,σ2
+m2)through Lemma 2.3. Set
+dσ2∗
+0=/hatwides2
+1,j(x)dσ2
+0and(s2∗
+0,0,s2∗
+0,1,... ,s2∗
+0,m2) =N(σ2∗
+0,σ2∗
+1,...,σ2∗
+m2). Then,
+/integraldisplay
+xνAn(x)ds2∗
+0,k(x) = 0, ν= 0,...,n∗
+2,k−1, k= 0,...,m 2. (33)
+Proof. Fork= 0, since ds2∗
+0,0=dσ2∗
+0=/hatwides2
+1,j(x)dσ2
+0, (4) reduces to (33) when Ln2(x) =
+xν/hatwides2
+1,j(x),ν= 0,...,n∗
+2,0−1, taking into consideration that n∗
+2,0=n2,j.
+Letj+1≤k≤m2. On account of Lemma 2.3 and (32)
+Ln2(x) =xν/hatwides2
+1,k(x) = (|s2
+1,k|
+|s2
+1,j|xν+(−1)jxν/hatwides2∗
+1,k)/hatwides2
+1,j.
+Consequently, from (4) and the orthogonality for k= 0, we obtain
+0 =/integraldisplay
+xν/hatwides2
+1,k(x)An(x)dσ2
+0(x) =|s2
+1,k|
+|s2
+1,j|/integraldisplay
+xνAn(x)ds2∗
+0,0(x)+(−1)j/integraldisplay
+xνAn(x)ds2∗
+0,k(x) =
+(−1)j/integraldisplay
+xνAn(x)ds2∗
+0,k(x), ν= 0,...,n∗
+2,k−1,
+sincen∗
+2,k=n2,k≤n2,j=n∗
+2,0,k=j+1,...,m 2.Thus, for these values of kthe assertion also
+holds.
+Ifk∈ {1,...,j−1}and max {n2,lk−1,n2,k}=n2,k,from Lemma 2.3 and (32), it follows that
+Ln2(x) =xν/hatwides2
+1,k(x) = (|s2
+1,k|
+|s2
+1,j|xν+(−1)k−1xν/hatwides2∗
+1,k)/hatwides2
+1,j.
+Using the same arguments as in the previous case, we obtain what is n eeded. Similarly, when k=j
+and min{n2,lj−2,n2,j−1}=n2,j−1,then
+Ln2(x) =xν/hatwides2
+1,j−1(x) = (|s2
+1,j−1|
+|s2
+1,j|xν+(−1)j−1xν/hatwides2∗
+1,j)/hatwides2
+1,j
+and integrating the statement follows.28 FIDALGO AND L ´OPEZ
+Assume that (33) holds for all values of the parameter less than k,1≤k≤j,and let us show
+that it is also true for k. When max {n2,lk−1,n2,k}=n2,k,1≤k≤j−1,or min{n2,lj−2,n2,j−1}=
+n2,j−1,k=j,we just proved that (32) is satisfied, so we must only consider the c ases when
+max{n2,lk−1,n2,k}=n2,lk−1,1≤k≤j−1,and min{n2,lj−2,n2,j−1}=n2,lj−2,k=j.In this
+situation, if 1 ≤k≤j−1 we have that
+Ln2(x) =xν/hatwides2
+1,lk−1(x) = (C0xν+k−1/summationdisplay
+i=1C1xν/hatwides2∗
+1,i+(−1)lk−1xν/hatwides2∗
+1,k)/hatwides2
+1,j,(/hatwides2
+1,0≡0).
+whereCi,i= 1,...,k−1,are constants, C0is also a constant if lk−1/\e}atio\slash= 0 and it is a first degree
+polynomial when lk−1= 0. From (4) and the induction hypothesis, it follows that
+0 =/integraldisplay
+xν/hatwides2
+1,lk−1(x)An(x)dσ2
+0(x) =k−1/summationdisplay
+i=0/integraldisplay
+CixνAn(x)ds2∗
+0,i(x)+
+(−1)lk−1/integraldisplay
+xνAn(x)ds2∗
+0,k(x) = (−1)lk−1/integraldisplay
+xνAn(x)ds2∗
+0,k(x), ν= 0,...,n∗
+2,k−1,
+sincen∗
+2,k=n2,lk−1= min{n2,0,...,n 2,k−1} ≤n∗
+2,i,i= 0,...,k−1, and when lk−1= 0 then
+n2,0< n2,j=n∗
+2,0. Notice that we have already proved (33) for all values of the para meter
+up toj−1. Whenk=jand min{n2,lj−2,n2,j−1}=n2,lj−2, we proceed analogously taking
+Ln2=xν/hatwides2
+1,lj−2. Again,n∗
+2,j≤n∗
+2,i,i= 0,...,n∗
+2,j−1and we can complete the induction. ✷
+Remark 3.2.Fixk∈ {1,...,m 2}. Takingpk≡1,pi≡0,i/\e}atio\slash=k,i= 0,...,m 2, one obtains the
+formula that links s2
+0,kwith the measures s2∗
+0,0,...,s2∗
+0,k(not all have to appear).
+Proof of Corollary 1.2. From Lemma 2.4, we have that/integraldisplay
+xνQn(x)ds0,k(x) = 0, ν= 0,...,n k−1, k= 0,...,m. (34)
+Using the definition of type II Pad´ e approximation, we have that fo r any polynomial Q,degQ≤nk
+Q(z)(Qn(z)/hatwides0,k(z)−Pn,k(z)) =O(1/z), z→ ∞.
+On account of (3), it follows that
+Q(z)(Qn(z)/hatwides0,k(z)−Pn,k(z)) =/integraldisplayQ(x)Qn(x)ds0,k(x)
+z−x(35)
+TakingQ≡1, we obtain that
+Pn,k(z) =/integraldisplayQn(z)−Qn(x)
+z−xds0,k(x),Pn,k(z)
+Qn(z)=1
+Qn(z)/integraldisplayQn(z)−Qn(x)
+z−xds0,k(x).
+Consequently, since the zeros of Qn(z) =/producttext|n|
+i=1(z−xn,i) are simple and lie in the interior of
+Co(suppσ0)
+Pn,k(z)
+Qn(z)=|n|/summationdisplay
+i=1λn,k,i
+z−xn,i, λn,k,i= lim
+z→xn,i(z−xn,i)Pn,k(z)
+Qn(z)=/integraldisplayQn(x)
+Q′n(xn,i)ds0,k(x)
+x−xn,i.(36)
+Letpbe an arbitrary polynomials of degree ≤ |n|+nk−1 andℓn(z) =/summationtext|n|
+i=0Qn(z)p(xn,i)
+Q′n(xn,i)(z−xn,i)be
+the Lagrange polynomial of degree |n|−1 which interpolates pat the zeros of Qn. Then
+(p−ℓn)(z) =q(z)Qn(z),degq≤nk−1.
+From (34),/integraldisplay
+(p−ℓn)(x)ds0,k(x) = 0.
+Consequently, using (36)
+/integraldisplay
+p(x)ds0,k(x) =/integraldisplay
+ℓn(x)ds0,k(x) =|n|/summationdisplay
+i=1p(xn,i)/integraldisplayQn(x)ds0,k(x)
+Qn(xn,i)(x−xn,i)=|n|/summationdisplay
+i=1λn,k,ip(xn,i),
+which gives the first statement of the corollary.NIKISHIN SYSTEMS ARE PERFECT 29
+Let us consider the case when n0= max{n0,n1−1,...,n m−1}. According to (35) with Q≡1,
+Fubini’s theorem, and (34)
+/integraldisplay
+tν(Qn(t)/hatwides0,0(t)−Pn,0(t))ds1,k(t) =/integraldisplay
+tν/integraldisplayQn(x)ds0,0(x)
+t−xds1,k(t) =
+/integraldisplay
+Qn(x)/integraldisplaytν∓xν
+t−xds1,k(t)ds0,0(x) =/integraldisplay
+qν(x)Qn(x)ds0,0(x)−/integraldisplay
+xνQn(x)ds0,k(x) = 0,
+for allν= 1,...,n k−1 andk= 1,...,m, sinceqν(x) =/integraltexttν−xν
+t−xds1,k(t) is a polynomial such that
+degqν≤nk−2≤n0−1. This implies that
+/integraldisplay
+(p1(t)+m/summationdisplay
+k=2pk(t)/hatwides2,k(t))(Qn(t)/hatwides0,0(t)−Pn,0(t))ds1,1(t) = 0,
+forarbitrarypolynomials pk,degpk≤nk−1,k= 1,...,m. Since,byTheorem1.1,(1 ,/hatwides2,2,... ,/hatwides2,m)
+is an AT-system, it follows that Qn/hatwides0,0−Pn,0, has at least |n|−n0sign changes in the interior of
+Co(suppσ1).
+LetQn,1be a monic polynomial of degree |n|−n0whose simple zeros are points where Qn/hatwides0,0−
+Pn,0changes sign on Co(supp σ1). It is easy to verify that
+zν(Qn/hatwides0,0−Pn,0)(z)
+Qn,1(z)=O(1/z2), z→ ∞, ν= 0,...,|n|−1.
+Using (2), we obtain
+/integraldisplay
+xνQn(x)ds0,0(x)
+Qn,1(x)= 0, ν= 0,...,|n|−1.
+Then, for any /tildewideQ,deg/tildewideQ≤ |n|,
+/integraldisplay
+Qn(x)/tildewideQ(z)−/tildewideQ(x)
+z−xds0,0(x)
+Qn,1(x)= 0
+which implies that
+/tildewideQ(z)/integraldisplayQn(x)
+z−xds0,0(x)
+Qn,1(x)=/integraldisplay/tildewideQ(z)Qn(x)
+z−xds0,0(x)
+Qn,1(x).
+In particular, with z=xn,i, taking/tildewideQ(x) =Qn,1(x) and then/tildewideQ(x) =Qn(x)
+Q′n(xn,i)(x−xn,i), we have
+λn,0,i=/integraldisplayQn(x)
+Q′n(xn,i)ds0,0(x)
+x−xn,i=/integraldisplayQn,1(x)Qn(x)
+Q′n(xn,i)(x−xn,i)ds0,0(x)
+Qn,1(x)=
+Q′
+n(xn,i)
+Q′n(xn,i)Qn,1(xn,i)/integraldisplayQn(x)
+Q′n(xn,i)(x−xn,i)ds0,0(x)
+Qn,1(x)=
+/integraldisplay/parenleftbiggQn(x)
+Q′n(xn,i)(x−xn,i)/parenrightbigg2Qn,1(xn,i)
+Qn,1(x)ds0,0(x), i= 1,...,|n|.
+Since/parenleftig
+Qn(x)
+Q′n(xn,i)(x−xn,i)/parenrightig2Qn,1(xn,i)
+Qn,1(x)is positive for all x∈Co(supps0,0), the second statement of
+Corollary 1.2 follows for k= 0. Using standard arguments of one-sided polynomial approximat ion
+of Riemann-Stieltjes integrable functions (see e.g. [46, Theorem 15 .2.2] and [22, Lemma 2]), the
+third statement is a consequence of the first two for any sequenc e of multi-indices Λ ⊂Zm+1
++such
+that for all n∈Λ,n0= max{n0,n1−1,...,n m−1}. In particular, the second and third statements
+are valid when n= (n,n+1,...,n+1).
+In the rest of the proof, we restrict our attention to multi-indices of the form n= (n,n+
+1,...,n+1). Fixk∈ {1,...,m}.Since we have that nk=n+1 = max {n0+1,n1,...,n m},we
+can apply Lemma 3.4 with j=kand we obtain that Qnis multiple orthogonal with respect to n∗,
+which hasn+1 in the first component, and a Nikishin system N(σ∗
+0,...,σ∗
+m) whose first measure
+iss∗
+0,0=s0,k. Consequently, the coefficients
+λn,k,i=/integraldisplayQn(x)
+Q′n(xn,i)ds0,k(x)
+x−xn,i=/integraldisplayQn(x)
+Q′n(xn,i)ds∗
+0,0(x)
+x−xn,i
+must all have the same sign as s0,k. The convergence of the quadratures is obtained as before. ✷30 FIDALGO AND L ´OPEZ
+4.Proof of Theorems 1.3-1.4
+Proof of Theorem 1.3. Form= 0 the result is trivially true since Ln=p0. Whenm= 1 it
+is easy to deduce. Indeed, if n0≥n1, takeλthe identity and S(λ) =N(σ1); otherwise, n0<n1
+and by Lemma 2.3
+p0+p1/hatwides1,1= (p∗
+0+p∗
+1/hatwideτ1,1)/hatwides1,1,degp∗
+0≤n1−1,degp∗
+1≤n0−1.
+Hence, the solution is λsuch thatλ(0) = 1,λ(1) = 0,andS(λ) =N(τ1,1). In the following m≥2.
+Next, letusconsiderthecasewhen n0= max{n0,...,n m}. Ifn0≥ ··· ≥nm, theresultistrivial
+takingλthe identity and S(λ) =N(σ1,... ,σ m). Otherwise, there exists m,0≤m≤m−2,such
+thatn0≥ ··· ≥nm,nm= max{nm,...,n m}, andnm+1<max{nm+2,... ,n m}. (Consequently,
+nm+1<nm.)
+We have
+Ln=p0+m/summationdisplay
+k=1pk/hatwides1,k=p0+m/summationdisplay
+k=0pk/hatwides1,k+m/summationdisplay
+k=m+1pk/a\}bracketle{tσ1,... ,σ m,sm+1,k/hatwide/a\}bracketri}ht.
+It is easy to check (see, for example, Lemma 2.1 in [31]) that for each k∈ {m+1,...,m},
+pk/a\}bracketle{tσ1,...,σ m,sm+1,k/hatwide/a\}bracketri}ht=ℓk,0+m/summationdisplay
+i=1ℓk,i/hatwides1,i+/a\}bracketle{tσ1,... ,σ m,pksm+1,k/hatwide/a\}bracketri}ht, (37)
+degℓk,i≤degpk−1,and these polynomials have real coefficients. Since nk≤nm≤nm−1≤n0
+wheneverk∈ {m+1,...,n m}, the polynomials ℓk,i,i= 0,...,m,k=m+1,...,m, are absorbed
+by the polynomials pk,k= 0...,m,without altering the bound on the degrees of the second.
+Therefore, there exist polynomials with real coefficients /tildewidepk,deg/tildewidepk≤nk−1,k= 0...,m,such that
+p0+m/summationdisplay
+k=1pk/hatwides1,k=/tildewidep0+m/summationdisplay
+k=0/tildewidepk/hatwides1,k+/a\}bracketle{tσ1,...,σ m,m/summationdisplay
+k=m+1pksm+1,k/hatwide/a\}bracketri}ht=
+/tildewidep0+m/summationdisplay
+k=0/tildewidepk/hatwides1,k+/a\}bracketle{tσ1,...,σ m,(pm+1+m/summationdisplay
+k=m+2pk/hatwidesm+2,k)σm+1/hatwide/a\}bracketri}ht. (38)
+By assumption nm+1<max{nm+2,...,n m}.So, we can apply Lemma 2.3 on the linear form
+pm+1+/summationtextm
+k=m+2pk/hatwidesm+2,k. Thus, there exist a Nikishin system N(σ∗
+m+2,... ,σ∗
+m), a multi-index
+(n∗
+m+1,... ,n∗
+m)∈Zm−m
++, which is a permutation of ( nm+1,...,n m), and polynomials with real
+coefficients p∗
+k,degp∗
+k≤n∗
+k−1,k=m+1,...,m, such that
+pm+1+m/summationdisplay
+k=m+2pk/hatwidesm+2,k= (p∗
+m+1+m/summationdisplay
+k=m+2p∗
+k/hatwides∗
+m+2,k)/hatwidesm+2,j,
+wherejis such that nj= max{nm+1,... ,n m}andn∗
+m+1=nj. Substitute this formula in (38)
+and reverse the application of (37) to pull out the polynomials /tildewidepkof the product of measures. The
+new polynomials lk,i,k∈ {m+ 1,...,n m}which arise from this second application of (37) are
+also absorbed by the polynomials /tildewidepk,k= 0,...,min (38) without changing the bound on their
+degrees. Therefore, we get that for certain polynomials with real coefficients p∗
+k,k= 0,...,m
+Ln=p∗
+0+m/summationdisplay
+k=1p∗
+k/hatwides1,k+p∗
+m+1/hatwides1,j+m/summationdisplay
+k=m+2p∗
+k/a\}bracketle{tσ1,...,σ m,sm+1,j,σ∗
+m+2,... ,σ∗
+m/hatwide/a\}bracketri}ht,(39)
+which is a linear form generated by the multi-index n∗= (n0,...,n m,nj,n∗
+m+2,...,n∗
+m),and the
+Nikishin system N(σ1,...,σ m,sm+1,j,σ∗
+m+2,...,σ∗
+m).
+Now, we have that n0≥ ··· ≥nm≥njandnj= max{nj,n∗
+m+2,...,n∗
+m}. Ifnm+2≥ ··· ≥nm,
+we are done taking λsuch thatnλ(k)=n∗
+k,/hatwides1,λ(0)≡1,and
+S(λ) =N(σ1,... ,σ m,sm+1,j,σ∗
+m+2,...,σ∗
+m).
+Otherwise, we repeat the process with the linear form on the right h and of (39). The new mwill
+certainly be larger that the previous one and in a finite number of iter ations we reorganize the
+entries of nin decreasing order obtaining with it λandS(λ).NIKISHIN SYSTEMS ARE PERFECT 31
+Ifn0<max{n0,...,n m}, we apply first Lemma 2.3 and then proceed as have done before but
+with the form L∗
+n=p∗
+0+/summationtextm
+k=1p∗
+k/hatwides∗
+1,kand the multi-index n∗arising from that lemma. Thus,
+we find a permutation λof (0,...,m) such that n∗
+λ(0)≥ ··· ≥n∗
+λ(m),S(λ) =N(ρ1,... ,ρ m),and
+polynomials with real coefficients qk,k= 0,...,m, such that
+L∗
+n=q0+m/summationdisplay
+k=1qk/hatwider1,k,degqk≤n∗
+λ(k)−1, k= 0,...,m.
+Ifλ∗is the permutation due to Lemma 2.3 which transports ninton∗, takingλ=λoλ∗and
+S(λ) =S(λ), applying the formula of Lemma 2.3 the assertion of Theorem 1.3 aga in follows. ✷
+Using induction we could have reduced a bit the proof of Theorem 1.3. Nevertheless, for the
+proof of Theorem 1.4 it was convenient to underline the fact that Th eorem 1.3 is a consequence of
+iterating the use of Lemma 2.3 and the trick exhibited in (37)-(39).
+Proof of Theorem 1.4. Ifm2= 0 orm2= 1 andn2,0≥n2,1the result is trivial. For m2= 1
+andn2,0<n2,1the statement is contained in Lemma 3.4. So, we restrict our attent ion tom2≥2.
+First, we consider the case when n2,0= max{n2,0,...,n 2,m2}. In this situation, the result is
+trivial again if n2,0≥ ··· ≥n2,m2. If this is not the case, there exists m,0≤m≤m2−2,such
+thatn2,0≥ ··· ≥n2,m,n2,m= max{n2,m,...,n 2,m2}, andn2,m+1<max{n2,m+2,... ,n 2,m2}.
+According to (39), for any polynomials pk,degpk≤n2,k−1,(39) takes place; that is,
+p0+m2/summationdisplay
+k=1pk/hatwides2
+1,k=p∗
+0+m/summationdisplay
+k=1p∗
+k/hatwides2
+1,k+p∗
+m+1/hatwides2
+1,j+m/summationdisplay
+k=m+2p∗
+k/a\}bracketle{tσ2
+1,... ,σ2
+m,s2
+m+1,j,σ2∗
+m+2,...,σ2∗
+m/hatwide/a\}bracketri}ht,
+wherejis such that n2,j= max{n2,m+1,...,n 2,m2}.This linear form is generated
+n∗
+2= (n∗
+2,0,... ,n∗
+2,m2) = (n2,0,... ,n 2,m,n2,j,n∗
+2,m+2,... ,n∗
+2,m2),
+and the Nikishin system
+N(σ2∗
+1,... ,σ2∗
+m2) =N(σ2
+1,...,σ2
+m,s2
+m+1,j,σ2∗
+m+2,...,σ2∗
+m2).
+Consider the extended Nikishin system N(σ2∗
+0,σ2∗
+1,...,σ2∗
+m2) =N(σ2
+0,σ2∗
+1,...,σ2∗
+m2).The form An
+is of multiple orthogonality with respect to n∗
+2and the extended Nikishin system.
+In fact, by definition, Ansatisfies
+/integraldisplay
+xνAn(x)ds2∗
+0,k(x) = 0, ν= 0,...,n∗
+2,k−1, k= 0,...,m+1,
+sinces2∗
+0,k=s2
+0,k,n∗
+2,k=n2,k,k= 0,...,m,s2∗
+0,m+1=s2
+0,j, andn∗
+2,m+1=n2,j. To prove
+/integraldisplay
+xνAn(x)ds2∗
+0,k(x) = 0, ν= 0,...,n∗
+2,k−1, k=m+2,...,m 2,
+one follows arguments similar to those employed in proving Lemma 3.4, c hoosing particular ex-
+pressions for Ln2of the form xν/hatwides2
+1,k(kis not always equal to k), and taking into consideration
+(37)-(39) as well as (32). The details are left to the reader.
+Once we have proved that Anis of multiple orthogonality with respect to n∗
+2and the extended
+Nikishin system, one repeats the process finding a new m, which is obviously larger than the
+previous one, and in a finite number of iterations the statement follo ws.
+Ifn2,0<max{n2,0,... ,n 2,m2}, the proof is reduced to the previous case by Lemma 3.4. ✷
+5.Proof of Theorems 1.5-1.6 and Corollary 1.3
+If we apply Theorem 1.3 to the form An, we obtain that there exists a permutation λ1of
+(0,...,m 1) and an associated Nikishin system S(λ1) = (r1
+1,1,... ,r1
+1,m1) =N(ρ1
+1,...,ρ1
+m1) such
+that
+An=an,0+m1/summationdisplay
+k=1an,k/hatwides1
+1,k= (bn,0+m1/summationdisplay
+k=1bn,k/hatwider1,k)/hatwides1,λ1(0)=Bn/hatwides1,λ1(0),32 FIDALGO AND L ´OPEZ
+where/hatwides1,λ1(0)≡1 ifλ1(0) = 0, and deg bn,k≤n1,λ1(k)−1,k= 0,...,m 1.On the other hand, from
+Theorem 1.4, we know that there exists a permutation λ2of (0,... ,m 2) and a Nikishin system
+N(ρ2
+0,...,ρ2
+m2), whereρ2
+0=/hatwides2
+1,λ2(0)σ2
+0, such that for each k= 0,...,m 2,
+/integraldisplay
+xνAn(x)dr2
+0,k(x) =/integraldisplay
+xνBn(x)/hatwides1,λ1(0)(x)dr2
+0,k(x) = 0, ν= 0,...,n 2,λ2(k)−1.
+Therefore, Bnis a linear form, generated by the multi-index ( n1,λ1(0),... ,n 1,λ1(m1)) andS(λ1),
+which is of multiple orthogonality with respect to the multi-index ( n2,λ2(0),... ,n 2,λ2(m2)) and the
+Nikishin system ( /hatwides1,λ1(0)r2
+0,r2
+1,... ,r2
+m2) =N(/hatwides1,λ1(0)ρ2
+0,ρ2
+1,...,ρ2
+m2). In other words,
+Bn= (bn,0,... ,b n,m1)
+is the mixed type multiple orthogonal polynomial relative to the pair of Nikishin systems ( /tildewideS1,/tildewideS2)
+and the multi-index /tildewiden= (/tildewiden1;/tildewiden2)∈Zm1+1
++×Zm2+1
++, where
+(/tildewideS1,/tildewideS2) = (N(/hatwides1,λ1(0)ρ2
+0,ρ1
+1,...,ρ1
+m1),N(/hatwides1,λ1(0)ρ2
+0,ρ2
+1,... ,ρ2
+m2)),
+and
+/tildewideni= (ni,λi(0),...,n i,λi(m1)), i= 1,2.
+Both/tildewiden1and/tildewiden2have decreasing components. Therefore, to derive Theorems 1.5 and 1.6 we can
+apply the results of [23].
+Lemma5.1.If(S1,S2)satisfies the hypotheses of Theorem 5 (respectively 6 )the same is true for
+(/tildewideS1,/tildewideS2).
+Proof.The systems /tildewideS1and/tildewideS2are obtained transforming the generating measures of S1and
+S2through inversion of measures and multiplication by Cauchy transfo rms of measures supported
+on disjoint intervals. We have to check that these operations pres erve the quasi-regularity of
+supports, the regularity of measures, and the property concer ning the Radon-Nikodym derivative
+of the measure.
+Letσ∈ M(∆),∆ = Co(supp σ),τis the inverse measure of σ, andgis a continuous function
+on ∆ with constant sign and different from zero on ∆.
+It is trivial that the supports of σandgσcoincide and that σ′>0 if and only if gσ′>0. It
+is well known and easy to verify (using, for example, the minimality pro perty of monic orthogonal
+polynomials) that σ∈Regif and only if gσ∈Regas well.
+Regarding the inversion of measures, the Stieltjes-Plemelj inversio n formula implies that the
+continuous parts of the supports of σandτcoincide. From the formula relating /hatwideσand/hatwideτit is
+obvious that isolated mass points of σoutside its continuous support become zeros of /hatwideτ(thus are
+no longer in the support of τ). On the other hand, in each connected component of ∆ \suppσ,/hatwideσ
+may have at most one zero (counting multiplicity), because /hatwideσis strictly monotonic when restricted
+to any one of those components. Such zeros of /hatwideσbecome mass points of τ. They are isolated,
+so they can only accumulate on supp σ. Therefore, if supp σ=/tildewideE∪e, where/tildewideEis regular with
+respect to the Dirichlet problem and eis at most a denumerable set of points which may only
+accumulate on /tildewideE, then supp τ=/tildewideE∪/tildewideewhere/tildewideeis at most a denumerable set of points which may
+only accumulate on /tildewideE. In particular, the same holds when /tildewideEis an interval (case of Theorem 1.6).
+Ifσ∈Regthenτ∈Reg. Indeed, the denominator Qnof then-th diagonal Pad´ e approximant
+of/hatwideσ, taken with leading coefficient equal to 1, is the n-th monic orthogonalpolynomial with respect
+toσ. The numerator Pn−1is an (n−1)-th orthogonal polynomial with respect to τ. By Markov’s
+theorem
+lim
+nPn−1(z)
+Qn(z)=/hatwideσ(z),K ⊂C\∆,
+In particular, the leading coefficient cn−1ofPn−1satisfies
+lim
+ncn−1= lim
+nlim
+z→∞zPn−1(z)
+Qn(z)= lim
+z→∞z/hatwideσ(z) =σ(∆)/\e}atio\slash= 0.
+Therefore,
+lim
+n|Qn(z)|1/n= cap(supp σ)egΩ(z;∞)⇔lim
+n/vextendsingle/vextendsingle/vextendsingle/vextendsinglePn(z)
+cn/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/n
+= cap(supp σ)egΩ(z;∞),NIKISHIN SYSTEMS ARE PERFECT 33
+uniformly on compact subsets of C\∆, wheregΩ(z;∞) denotes Green’s function of the region
+Ω =C\suppσwith singularity at ∞. But supp σand suppτdiffer on a set of capacity zero so
+their capacities coincide as well as the Green’s function of the comple ment of their supports. The
+limits above are equivalent to regularity (see [45, Theorem 3.1.1]).
+Thatσ′>0 a.e. on an interval is equivalent to τ′>0 a.e. on the same interval follows from
+the Stieltjes–Plemelj inversion formula. ✷
+In order to prove Theorem 5 there is still one thing to be considered . The corresponding result
+[23, Theorem 1.3] for the case of decreasing components in n1,n2,was proved assuming that the
+supports of the measures were regular. We have to extend its app licability to the case of quasi-
+regular supports because, as follows from the proof of the previo us lemma, the regularity of the
+supports of the measures generating ( S1,S2) does not guarantee regularity of the supports of the
+measures which generate ( /tildewideS1,/tildewideS2) since isolated mass points may arise.
+We need some notation. M1(E) denotes the class of probability measures supported on E, and
+Vµ(z) =/integraldisplay
+log1
+|z−ζ|dµ(z)
+the logarithmic potential of the measures µ. Ifqlis a polynomials of degree l,
+µql=1
+l/summationdisplay
+ql(x)=0δx
+is the associated normalized zero counting measure, where δxis the Dirac measure with mass 1 at
+x. In [44, Theorem I.1.3] the authors prove
+Lemma5.2.LetE⊂Cbe a compact subset of the complex plane and φa continuous function on
+E. Then, there exists a unique µ∈ M1(E)and a constant wsuch that
+Vµ(z)+φ(z)/braceleftbigg
+≤w, z∈suppµ,
+≥w, z∈E\A,cap(A) = 0.
+µandware called the equilibrium measure and the equilibrium constant, respe ctively, in pres-
+ence of the external field φon the compact E.
+We are especially grateful to H. Stahl who gave us the clue for the f ollowing improvement of
+[23, Lemma 4.2].
+Lemma5.3.Letσ∈Reg,suppσ⊂R, wheresuppσis quasi-regular. Let {φl},l∈Λ⊂Z+,be a
+sequence of positive continuous functions on suppσsuch that
+lim
+l∈Λ1
+2llog1
+|φl(x)|=φ(x)>−∞,
+uniformly on suppσ. By{ql},l∈Λ,denote a sequence of monic polynomials, degql=l,and
+/integraldisplay
+xkql(x)φl(x)dσ(x) = 0, k= 0,...,l−1.
+Then
+∗lim
+l∈Λµql=µ,
+in the weak star topology of measures, and
+lim
+l∈Λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+|ql(x)|2φl(x)dσ(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/2l
+= exp(−w),
+whereµandware the equilibrium measure and equilibrium constant in the presence of the external
+fieldφonsuppσ=:E. We also have that
+lim
+l∈Λ/parenleftigg
+|ql(z)|
+/bardblqlφ1/2
+l/bardblE/parenrightigg1/l
+= exp(w−Vµ(z)),K ⊂C\Co(supp(σ)),
+where/bardbl·/bardblEdenotes the sup norm on E.34 FIDALGO AND L ´OPEZ
+Proof.Proceeding as in the proof of [23, Lemma 4.2] one shows that for any sequence of monic
+polynomials {pl},l∈Λ,such that deg pl=l,
+limsup
+l∈Λ/parenleftigg
+|pl(z)|
+/bardblplφ1/2
+l/bardblE/parenrightigg1/l
+≤exp(w−Vµ(z)),K ⊂C, (40)
+and
+liminf
+l∈Λ/bardblplφ1/2
+l/bardbl1/l
+E≥exp(−w). (41)
+In particular, these relations hold for {ql},l∈Λ. In [23, Lemma 4.2], it is also proved that
+limsup
+l∈Λ/bardblqlφ1/2
+l/bardbl1/l
+2≤exp(−w), (42)
+where/bardblqlφ1/2
+l/bardbl2is theL2norm ofqlφ1/2
+lwith respect to σ. We may assume, without loss of
+generality, that σis positive. In deducing (40)-(42), the regularity of supp σis not required.
+Combining (41)-(42), it follows that
+liminf
+l∈Λ/parenleftigg
+/bardblqlφ1/2
+l/bardblE
+/bardblqlφ1/2
+l/bardbl2/parenrightigg1/l
+≥1.
+Should
+limsup
+l∈Λ/parenleftigg
+/bardblqlφ1/2
+l/bardblE
+/bardblqlφ1/2
+l/bardbl2/parenrightigg1/l
+≤1, (43)
+then
+lim
+l∈Λ/parenleftigg
+/bardblqlφ1/2
+l/bardblE
+/bardblqlφ1/2
+l/bardbl2/parenrightigg1/l
+= 1.
+and due to (41)-(42), we would have
+limsup
+l∈Λ/bardblqlφ1/2
+l/bardbl1/l
+E= limsup
+l∈Λ/bardblqlφ1/2
+l/bardbl1/l
+2= exp(−w). (44)
+Once (44) is attained, with the help of (40), one can conclude the pr oof as in [23, Lemma 7]. So,
+it remains to show that (43) takes place when we relax the regularity of suppσto quasi-regularity.
+In[45,Theorem3.2.3]itisproved(see(v) ⇒(vi))that(43)holdsforanysequenceofpolynomials
+{pl},l∈Λ,such that deg pl=l, if the same property is satisfied when φl≡1,l∈Λ. Though the
+hypothesis of that theorem also contains the assumption that sup pσbe regular, the proof of this
+assertion is independent of the regularity condition. Therefore, le t us show that (43) holds true
+whenφl≡1,l∈Λ,and suppσis quasi-regular.
+In fact, according to [45, Theorem 3.2.1 ii)] , we have that
+limsup
+l∈Λ/parenleftbigg|pl(z)|
+/bardblpl/bardbl2/parenrightbigg1/l
+≤exp(gΩ(z;∞)),K ⊂C, (45)
+wheregΩ(z;∞) is the Green’s function of the region Ω = C\Ewith singularity at ∞. Since
+E=/tildewideE∪e, where/tildewideEis regular with respect to the Dirichlet problem and cap( e) = 0, we have that
+gΩ(z;∞) =g/tildewideΩ(z;∞),/tildewideΩ =C\/tildewideE, andg/tildewideΩ(z;∞) extends continuously to all C.
+Fixε >0 and letUε={z∈C:g/tildewideΩ(z;∞)< ε}. This is an open set which contains /tildewideEwhere
+g/tildewideΩ(z;∞) = 0 identically. Since the set eis at most denumerable and all its accumulation points
+are contained in /tildewideE, it follows that e\Uεhas at most a finite number of points (or may be empty).
+Let{z1,... ,z N}be the set of such points (should there be any). For each fixed k= 1,...,N,
+|pl(zk)|2
+/bardblpl/bardbl2
+2σ(zk)≤/integraldisplay|pl(x)|2
+/bardblpl/bardbl2
+2dσ(x) = 1.
+Consequently,
+limsup
+l∈Λ/parenleftbigg|pl(zk)|
+/bardblpl/bardbl2/parenrightbigg1/l
+≤1, k= 1,...,N.NIKISHIN SYSTEMS ARE PERFECT 35
+sinceσ(zk)>0. Because of (45)
+limsup
+l∈Λ/parenleftbigg/bardblpl/bardblE∩Uε
+/bardblpl/bardbl2/parenrightbigg1/l
+≤exp(ε)
+This, together with the previous inequality for zk,k= 1,...,N, immediately imply that
+limsup
+l∈Λ/parenleftbigg/bardblpl/bardblE
+/bardblpl/bardbl2/parenrightbigg1/l
+≤exp(ε).
+The arbitrariness of ε>0 renders what we set out to prove. ✷
+The assumption that the points in eonly accumulate on /tildewideEis essential. If this was not the case
+one can construct examples where (43) does not hold.
+Proof of Theorem 1.5. We will prove |n1|-th root asymptotics for the sequence {Bn},n∈Λ.
+Since/hatwides1,λ1(0)(z)/\e}atio\slash= 0,z∈C\∆1
+1,the statement of the theorem readily follows with the same limit.
+For definiteness, in reordering the components of a given n, let us take that unique pair of
+permutations ( λ1,λ2) such that for each i= 1,2,wheneverni,λi(j)=ni,λi(k)for some 0 ≤j <k≤
+mi,thenλi(j)<λi(k).By Λ(λ1,λ2), we denote the set of all multi-indices in Λ whose components
+n1,n2,are reordered decreasingly with λ1andλ2respectively. We are only interested in those
+Λ(λ1,λ2) containing an infinite number of elements of Λ .Fix (λ1,λ2) and let
+(/tildewideS1,/tildewideS2) = (N(/hatwides1,λ1(0)ρ2
+0,ρ1
+1,... ,ρ1
+m1),N(/hatwides1,λ1(0)ρ2
+0,ρ2
+1,...,ρ2
+m2))
+be the pair of Nikishin systems associated with Anby Theorems 1.3-1.4 with respect to which Bn
+is a multiple orthogonal linear form.
+Set
+Pj=m1/summationdisplay
+k=jp1,λ1(k), j= 0,...,m 1, P −j=m2/summationdisplay
+k=jp2,λ2(k), j= 0,...,m 2.
+Define the tri-diagonal matrix
+C=
+P2
+−m2−P−m2P−m2+1
+20 ···0
+−P−m2P−m2+1
+2P2
+−m2+1 −P−m2+1P−m2+2
+2···0
+0 −P−m2+1P−m2+2
+2P2
+−m2+2 ···0
+...............
+0 0 0 ···P2
+m1
+. (46)
+The sub-indices of the entries cj,kofCrun from −m2−1 tom1+1.
+LetM1(Ek) be the subclass of probability measures of M(Ek),
+Ek=/braceleftbiggsuppρ1
+k, k= 1,...,m 1,
+suppρ2
+−k, k=−m2,...,0.
+Denote
+M1=M1(E−m2)×···×M 1(Em1).
+Given a vector measure µ= (µ−m2,... ,µ m1)∈ M1andj∈ {−m2,...,m 1},we define the
+combined potential
+Wµ
+j(x) =m1/summationdisplay
+k=−m2cj,kVµk(x), Vµk(x) =/integraldisplay
+log1
+|x−y|dµk(y).
+Set
+J(µ) =m1/summationdisplay
+k,j=−m2cj,k/integraldisplay /integraldisplay
+log1
+|x−y|dµj(x)dµk(y) =/integraldisplay
+Wµ
+j(x)dµj(x).
+From Propositions 4.1-4.5 in [37, Chapter 5] it follows that there exist s a unique vector measure
+µ= (µ−m2,... ,µm1)∈ M1such that
+J(µ) = inf{J(µ) :µ∈ M1}, (47)36 FIDALGO AND L ´OPEZ
+and that exist constants wµ
+j,j=−m2,...,m 1,for which
+Wµ
+j(x)/braceleftigg
+≤wµ
+j, x∈suppµj,
+≥wµ
+j, x∈Ej\Aj,cap(Aj) = 0.(48)
+for certain Borel sets Aj. For any two vector measures µ1,µ2∈ M1such thatJ(µ1)<∞,J(µ2)<
+∞,straightforward calculations yield
+J(µ2)−J(µ1) =J(µ2−µ1)+2m1/summationdisplay
+j=−m2/integraldisplay
+Wµ1
+j(x)d(µ2
+j−µ1
+j)(x).
+SinceJ(µ2−µ1)≥0 for allµ1,µ2∈ M1(see [37, Proposition 4.2]), and sets of capacity zero are
+negligible for measures with finite energy, if µ1satisfies (48) it also satisfies (47). Thus, (47)-(48)
+are equivalent and a measure verifying any one of the two is unique (a nd so are the constants
+in (48)).µis called the equilibrium vector measure, and wµ= (wµ
+−m2,... ,wµ
+m1) the equilibrium
+vector constant, for the logarithmic potential governed by the in teraction matrix Con the system
+of compact sets Ej,j=−m2,...,m 1.
+From Lemma 5.1 we have that ( /tildewideS1,/tildewideS2)∈Regand the supports of the generating measures are
+quasi-regular. If Anis monic (see Definition 1.6), due to the way in which Bnis constructed (in
+particular, see (32) in the proof of Lemma 2 .3 and the proof of Theorem 1.4) it follows that bn,m1
+is either plus or minus an,λ−1
+1(m1). Thus, its leading coefficient is either 1 or −1; that is, except
+for a sign change, Bnis monic with the normalization imposed in [23, Theorem 5.1]. Following the
+proof of [23, Theorem 5.1], but using Lemma 5.3 instead of [23, Lemma 5 .1], one finds that
+lim
+n∈Λ(λ1,λ2)|Bn(z)|1/|n1= exp/parenleftigg
+P1Vµ1(z)−P0Vµ0(z)−2m1/summationdisplay
+k=1ωµ
+k
+Pk/parenrightigg
+,K ⊂C\(∆1
+0∪∆1
+1),
+whereµ=µ(C) = (µ−m2,... ,µm1) is the equilibrium vector measure and ( ωµ
+−m2,...,ωµ
+m1) is the
+system of equilibrium constants for the vector potential problem d etermined by the interaction
+matrixCdefined in (46) on the system of compact sets Ej,j=−m2,...,m 1.
+It is easy to see that the interaction matrix Cdoes not depend on ( λ1,λ2) and that the compact
+setsEj,j=−m2,... ,m 1,for each fixed j, may differ only on a (denumerable) set of capacity
+zero depending on λ1,λ2(see proof of Lemma 5.1). Therefore, the equilibrium measure and t he
+equilibrium constant are uniquely determined for any Λ( λ1,λ2) containing infinitely many terms
+of Λ. Consequently,
+lim
+n∈Λ|An(z)|1/|n1= exp/parenleftigg
+P1Vµ1(z)−P0Vµ0(z)−2m1/summationdisplay
+k=1ωµ
+k
+Pk/parenrightigg
+,K ⊂C\(∆1
+0∪∆1
+1).
+(49)
+With this we conclude the proof. ✷
+Remark 5.1.If we denote by Qn,0the monic polynomial whose zeros are those of An, under the
+assumptions of Theorem 1.5, we have (see [23, Theorem 4.2])
+∗lim
+n∈ΛµQn,0=µ0.
+There are other linear forms related with Anwhose asymptotic zero distribution and logarithmic
+asymptotic is described in terms of the other components of µand the vector equilibrium constant.
+This allows to give the logarithmic asymptotics of the polynomials an,kas well. For a view of what
+can be expected, see [23, Section 5]. These results can be used to g ive the exact rate of convergence
+of mixed type Hermite Pad´ e approximants, see [23, Section 7] and [2 0, Theorem 7]. For example,
+in case of type II approximation, under regularity of the generatin g measures and quasi–regularity
+of their supports, the following limit exists
+lim
+n∈Λ/bardbl/hatwides0,k−Pn,k
+Qn/bardbl1/2|n|
+K,K ⊂C\(Co(suppσ0)∪Co(suppσ1)), k= 0,...,m.
+Proof of Theorem 1.6. The existence of the limit claimed in Theorem 1.6 follows directly
+from [23, Theorem 1.4], but to give an expression of the limit function, we must introduce some
+notions.NIKISHIN SYSTEMS ARE PERFECT 37
+Letλ1,λ2,andl= (l1;l2) be as given in Theorem 1.6. Consider the ( m1+m2+ 2)-sheeted
+Riemann surface
+R=m1/uniondisplay
+k=−m2−1Rk,
+formed by the consecutively “glued” sheets
+R−m2−1:=C\/tildewide∆−m2,Rk:=C\(/tildewide∆k∪/tildewide∆k+1), k=−m2,...,m 1−1,Rm1:=C\/tildewide∆m1,
+where the upper and lower banks of the slits of two neighboring shee ts are identified. Define
+(/tildewidel1;/tildewidel2) := (λ−1
+1(l1);λ−1
+2(l2)).
+Letψ(/tildewidel)be a singled valued function defined on Ronto the extended complex plane satisfying
+ψ(/tildewidel)(z) =C1
+z+O(1
+z2), z→ ∞(−/tildewidel2−1),
+ψ(/tildewidel)(z) =C2z+O(1), z→ ∞(/tildewidel1),
+whereC1andC2are nonzero constants. Since the genus of Ris zero,ψ(/tildewidel)exists and is uniquely
+determined up to a multiplicative constant. Consider the branches o fψ(/tildewidel), corresponding to the
+different sheets k=−m2−1,...,m 1ofR
+ψ(/tildewidel):={ψ(/tildewidel)
+k}m1
+k=−m2−1.
+Given an arbitrary function F(z) which has in a neighborhood of infinity a Laurent expansion of
+the formF(z) =Czk+O(zk−1),C/\e}atio\slash= 0,andk∈Z,we denote
+/tildewideF:=F/C.
+Because of Theorem 1.4, Lemma 5.1, and the normalization adopted, the sequence {Bn},n∈Λ,
+satisfies all the assumptions of [23, Theorem 6.8]. Consequently,
+lim
+n∈ΛBnl(z)
+Bn(z)=C(/tildewidel)/tildewideψ(/tildewidel)
+0(z),K ⊂C\(suppρ1
+0∪suppρ1
+1),
+whereC(/tildewidel) is a constant, which only depends on /tildewideland can be determined exactly (see (69), (73),
+and (83) in [23]) in terms of the values of the branches of ψ(/tildewidel)at∞. Due to the relation between
+BnandAn, we obtain
+lim
+n∈ΛAnl(z)
+An(z)=C(/tildewidel)/tildewideψ(/tildewidel)
+0(z),K ⊂C\(suppσ1
+0∪Co(suppσ1
+1)), (50)
+since suppσ1
+0= suppρ1
+0and Co(supp ρ1
+1)⊂Co(suppσ1
+1). ✷
+Proof of Corollary 1.3. As in the proof of Theorem 1.5, for definiteness, in reordering the
+components of a given n, let us take that unique pair of permutations ( λ1,λ2) such that for each
+i= 1,2,wheneverni,λi(j)=ni,λi(k)for some 0 ≤j <k≤mi,thenλi(j)<λi(k).By Λ(λ1,λ2), we
+denote the set of all multi-indices in Λ whose components n1,n2,are reordered decreasingly with
+λ1andλ2respectively. We are only interested in those Λ( λ1,λ2) containing an infinite number of
+elements of Λ .Fix (λ1,λ2) and let
+(/tildewideS1,/tildewideS2) = (N(/hatwides1,λ1(0)ρ2
+0,ρ1
+1,... ,ρ1
+m1),N(/hatwides1,λ1(0)ρ2
+0,ρ2
+1,...,ρ2
+m2))
+be the pair of Nikishin systems associated with Anby Theorems 1.3-1.4 with respect to which Bn
+is a multiple orthogonal linear form.
+LetMbe the least common multiple of m1+ 1 andm2+ 1, and define d1:=M/(m1+ 1),
+d2:=M/(m2+ 1). Within the class of pairs l= (l1;l2) with 0 ≤l1≤m1, 0≤l2≤m2, we
+distinguish the subclass
+L:={(l1;l2) :l1≡rmod(m1+1), l2≡rmod(m2+1) for some 0 ≤r≤M−1}.
+It is easy to check that for different r,0≤r≤M−1,the pairs (l1,l2) inLare distinct. Let
+p:= (p1;p2), where p1= (d1,... ,d 1) andp2= (d2,... ,d 2) havem1+1 andm2+1 components,
+respectively. By n+pwe denote the multi-index ( n1+p1;n2+p2); that is, n+p=/tildewiden.38 FIDALGO AND L ´OPEZ
+Givenn∈Λ(λ1,λ2) and 0≤r≤M, letn(r) :=n+q(r) whereq(r) = (q1(r);q2(r)) is the
+multi-index satisfying ( qi(r) = (qi,0(r),... ,q i,mi(r)),i= 1,2)
+qi,λi(j)(r) =/braceleftbigg
+ki+1, j= 0,...,s i−1,
+ki, j=si,... ,m i,r=ki(mi+1)+si,0≤si≤mi.
+Hence,n(0) =n,n(M) =n+p=/tildewiden. It is easy to see that for all r∈ {0,...,M−1}, the same
+pair (λ1,λ2) reorders the components of n(r) giving rise to the same systems ( /tildewideS1,/tildewideS2).
+We have
+An+p(z)
+An(z)=M−1/productdisplay
+r=0An(r+1)(z)
+An(r)(z).
+Due to (50)
+lim
+n∈Λ(λ1,λ2)A/tildewiden(z)
+An(z)=/productdisplay
+(l1,l2)∈LC(/tildewidel)/tildewideψ(/tildewidel)
+0(z),K ⊂C\(suppσ1
+0∪Co(suppσ1
+1)),(51)
+wherel= (l1;l2) is precisely the multi-index satisfying l1≡rmod(m1+1),l2≡rmod(m2+1),
+and/tildewidel= (/tildewidel1;/tildewidel2) = (λ−1
+1(l1);λ−1
+2(l2)). The limit does not depend on ( λ1,λ2) because the set
+/tildewideL={(/tildewidel1;/tildewidel2) : (l1;l2)∈L}is the same for all ( λ1,λ2). The proof is complete. ✷
+Remark 5.2.The linear forms associated with Anmentioned in the previous remark also satisfy
+ratio asymptotics in the spirit of the results contained in [23, Section 6].
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+reprinted in his Oeuvres, Tome III, Gauthier-Villars, Pari s, 1912, 150-181.
+[29]H. Jager. A simultaneous generalization of the Pad´ e table. I-VI. Ind ag. Math. 26(1964), 193-249.
+[30]M. G. Krein and A. A. Nudel’man , “The Markov Moment Problem and Extremal Problems”. Transl . Math.
+Monogr., Vol. 50, Amer. Math. Soc., Providence, R. I., 1977.
+[31]A. L´opez Garc ´ıa and G. L ´opez Lagomasino. Relative asymptotics of multiple orthogonal polynomials f or
+Nikishin systems. J. of Approx. Theory 158(2009), 214–241.
+[32]K. Mahler . Perfect systems. Compos. Math. 19(1968), 95-166.
+[33]A.A. Markov. Deux demonstrations de la convergence de certains fraction s continues. Acta Math. 19(1895),
+93–104.
+[34]E. M. Nikishin . A system of Markov functions. Vestnik Moskov. Univ. Ser. I M at. Mekh (1979):4, 60–63
+(Russian); English translation in Moscow Univ. Math. Bull. 34(1979), 63–66.
+[35]E. M. Nikishin . On simultaneous Pad´ e approximants. Matem. Sb. 113(1980), 499–519 (Russian); English
+translation in Math. USSR Sb. 41(1982), 409–425.
+[36]E. M. Nikishin . Asymptotics of linear forms for simultaneous Pad´ e approx imants. Izv. Vyssh. Uchebn. Zaved.
+Mat. (1986):2, 33–41 (Russian); English translation in Sov iet Math. (Iz. VUZ) 30(1986), 43–52.
+[37]E.M. Nikishin and V.N. Sorokin. “Rational Approximations and Orthogonality”. Transl. Mat h. Monogr.,
+Vol. 92, Amer. Math. Soc., Providence, R. I., 1991.
+[38]J. Nuttall. Asymptotics of diagonal Hermite-Pad´ e polynomials. J. App rox. Theory 42(1984), 299–386.
+[39]E. A. Rakhmanov . On the asymptotic of the ratio of orthogonal polynomials. M at. Sb.103(1977), 237–252
+(Russian); English translation in Math. USSR Sb. 32(1977), 199-213.
+[40]E. A. Rakhmanov . On the asymptotic of the ratio of orthogonal polynomials II . Mat. Sb. 118(1982), 104–117
+(Russian); English translation in Math. USSR Sb. 46(1983), 105-117.
+[41]E. A. Rakhmanov . On asymptotic properties of orthogonal polynomials on the unit circle with weights not
+satisfying Szeg˝ o’s condition. Mat. Sb. 130(1986), 151–169 (Russian); English translation in Math. US SR Sb.
+58(1987), 149-167.
+[42]V.N. Sorokin. On simultaneous approximation of several linear forms. Ves tnik Moskov. Univ. Ser. I Mat. Mekh
+(1983):1, 44–47 (Russian); English translation in Moscow U niv. Math. Bull. 38(1983), 53–56.
+[43]V.N. Sorokin. Hermite-Pad´ e approximants for polylogarithms. Izv. Vyss h. Uchebn. Zaved. Mat. (1994):2,
+49–59 (Russian); English translation in Russian Math. (Iz. VUZ)38(1994), 47-57.
+[44]E.B. Saff and V. Totik. “Logarithmic Potentials with External Fields.” Series of C omprehensive Studies in
+Mathematics, Vol. 316, Springer, New York, 1997.
+[45]H. Stahl and V. Totik “General Orthogonal Polynomials.” Cambridge University P ress, Cambridge, 1992.
+[46]G. Szeg ˝o.“Orthogonal Polynomials.” Coll. Pub. Amer. Math. Soc., Vol . XXIII, 4-th. Ed., Providence, R.I.,
+1975.
+[47]W. Van Assche. Analytic number theory and rational approximation. “Coimb ra Lecture Notes on Orthogonal
+Polynomials”, A. Branquinho and A. Foulqui´ e Eds.. Nova Sci ence Pub., New York, 2008.
+(Fidalgo) Departamento de Matem ´aticas, Universidad Carlos III de Madrid, c/ Universidad 30, 2 8911
+Legan´es, Spain.
+E-mail address , Fidalgo: ufidalgo@math.uc3m.es
+(L´ opez) Departamento de Matem ´aticas, Universidad Carlos III de Madrid, c/ Universidad 30, 2 8911
+Legan´es, Spain.
+E-mail address , L´ opez: lago@math.uc3m.es
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+arXiv:1001.0555v1  [cs.CG]  4 Jan 2010On aTree and aPathwith no Geometric Simultaneous
+Embedding
+P.Angelini†, M.Geyer‡,M. Kaufmann‡andD.Neuwirth‡
+†Dipartimento diInformatica eAutomazione –Università Rom a Tre,Italy
+angelini@dia.uniroma3.it
+‡Wilhelm-Schickard-Institut für Informatik–Universität Tübingen, Germany
+geyer/mk/neuwirth@informatik.uni-tuebingen.de
+Abstract. Two graphs G1= (V,E1)andG2= (V,E2)admit ageomet-
+ric simultaneous embedding if there exists a set of points Pand a bijection
+M:P→Vthat induce planar straight-line embeddings both for G1and for
+G2. While it is known that two caterpillars always admit a geome tric simultane-
+ous embeddingandthattwotreesnot alwaysadmitone,theque stionabout atree
+andapathisstillopenandisoftenregardedasthemostpromi nent openproblem
+in this area. We answer this question in the negative by provi ding a counterex-
+ample. Additionally,since the counterexample uses disjoi ntedge sets forthetwo
+graphs, we also negatively answer another open question, th at is, whether it is
+possible to simultaneously embed two edge-disjoint trees. As a final result, we
+studythe same problem whensome constraints onthetree arei mposed. Namely,
+weshow thata treeofdepth 2andapathalways admita geometricsimultaneous
+embedding.Infact,suchastrongconstraintisnotsofarfro mclosingthegapwith
+the instances not admitting any solution, as the tree used in our counterexample
+has depth 4.
+1 Introduction
+Embeddingplanargraphsisawell-establishedfieldingraph theoryandalgorithmswith
+agreatvarietyofapplications.Keystonesinthisfieldaret heworksofThomassen[15],
+ofTutte[16],andofPachandWenger[14],dealingwithplana randconvexrepresenta-
+tionsofgraphsin theplane.
+Since recently, motivated by the need of contemporarily rep resent several differ-
+ent relationships among the same set of elements, a major foc us in the research lies
+onsimultaneous graph embedding . In this setting, given a set of graphs with the same
+vertex-set, the goal is to find a set of points in the plane and a mapping between these
+points and the vertices of the graphs such that placing each v ertex on the point it is
+mappedto yieldsa planarembeddingfor each of the graphs,if they are displayed sep-
+arately. Problems of this kind frequently arise when dealin g with the visualization of
+evolving networks and with the visualization of huge and com plex relationships, as in
+thecase ofthegraphoftheWeb.
+Among the many variants of this problem, the most important a nd natural one is
+thegeometric simultaneous embedding . Given two graphs G1= (V,E′)andG2=(V,E′′), the task is to find a set of points Pand a bijection M:P→Vthat induce
+planarstraight-lineembeddingsforboth G1andG2.
+In the seminal paper on this topic [2], Brass et al.proved that geometric simulta-
+neous embeddings of pairs of paths, pairs of cycles, and pair s of caterpillars always
+exist. Acaterpillar is a tree such that deleting all its leaves yields a path. On th e other
+hand,manynegativeresultshavebeenshown.Brass etal.[2]presentedapairofouter-
+planargraphsnotadmittinganysimultaneousembeddingand providednegativeresults
+for three paths, as well. Erten and Kobourov [4] found a plana r graph and a path not
+allowing any simultaneous embedding. Geyer et al.[12] proved that there exist two
+treesthatdonotadmitanygeometricsimultaneousembeddin g.However,thetwotrees
+used in the counterexample have common edges, and so the prob lem is still open for
+edge-disjointtrees.
+The most important open problem in this area is the question w hether a tree and a
+pathalwaysadmitageometricsimultaneousembeddingornot .Inthispaperweanswer
+thisquestioninthenegative.
+Manyvariantsofthe problem,wheresome constraintsarerel axed,havebeenstud-
+ied in the literature. If the edges do not need to be straight- line segments, a famous
+result ofPachandWenger[14]showsthatanynumberofplanar graphsadmita simul-
+taneous embedding, since it states that any planar graph can be planarly embedded on
+anygivensetofpointsintheplane.However,thesameresult doesnotholdiftheedges
+that are shared by two graphshave to be represented by the sam e Jordan curve.In this
+settingtheproblemiscalled simultaneousembeddingwith fixededges [9,11,6].
+The research on this problem opened a new exciting field of pro blems and tech-
+niques, like ULP trees and graphs [5,7,8], colored simultan eous embedding [1], near-
+simultaneousembedding[10], andmatcheddrawings[3],dee plyrelated to the general
+fundamentalquestionofpoint-setembeddability.
+In this paper we study the geometric simultaneous embedding problem of a tree
+andapath.Weanswerthequestioninthenegativebyprovidin gacounterexample,that
+is, a tree and a path not admitting any geometric simultaneou s embedding. Moreover,
+since the tree and the path used in our counterexampledo not s hare any edge, we also
+negativelyanswerthequestionontwoedge-disjointtrees.
+Themainideabehindourcounterexampleistouse thepathtoe nforcea partofthe
+tree to be in a certain configuration which cannot be drawn pla nar. Namely, we make
+use of level nonplanar trees [5,8], that is, trees not admitt ing any planar embedding if
+theirverticeshavetobeplacedinsidecertainregionsacco rdingtoaparticularleveling.
+The tree of the counterexample contains many copies of such t rees, while the path is
+used to create the regions. To prove that at least one copy has to be in the particular
+levelingthatdeterminesacrossing,weneedaquitehugenum berofvertices.However,
+sucha hugenumberisoftenneededjust to ensurethe existenc eofparticularstructures
+playingaroleinourproof.Amuchsmallercounterexampleco uldlikelybeconstructed
+withthesametechniques,butwedecidedtopreferthesimpli cityoftheargumentations
+ratherthanthesearchfortheminimumsize.
+The paper is organized as follows. In Sect. 2 we give prelimin ary definitions and
+weintroducetheconceptoflevelnonplanartrees.InSect.3 wedescribethetree Tand
+the pathPused in the counterexample. In Sect. 4 we give an overview of t he proof
+21
+2 43
+5 8 6 9 7 10 91
+25 6
+3
+47
+108l1
+l2
+l3
+l4a1
+l2
+l3 a4a3l
+2a1
+25
+83
+97
+4
+1016
+(a) (b) (c)
+Fig.1.(a) A tree Tu. (b) A level nonplanar tree Twhose underlying tree is Tu. (c) A
+region-levelnonplanartree Twhoseunderlyingtreeis Tu.
+thatTandPdonotadmitanygeometricsimultaneousembedding,whilein Sect.5we
+givethedetailsofsuchaproof.InSect.6wepresentanalgor ithmforthesimultaneous
+embeddingofatreeofdepth 2andapath,andin Sect.7we makesomefinalremarks.
+2 Preliminaries
+A (undirected) k-level tree T= (V,E,φ)onnvertices is a tree T′= (V,E), called
+theunderlying tree ofT, together with a leveling of its vertices given by a function
+φ:V/mapsto→ {1,...,k}, such that for every edge (u,v)∈E, it holds φ(u)/\e}atio\slash=φ(v)
+(See [5,8]). A drawing of T= (V,E,φ)is alevel drawing if each vertex v∈Vsuch
+thatφ(v) =iis placed on a horizontal line li={(x,i)|x∈R}. A level drawing
+ofTisplanarif notwo edgesintersectexcept,possibly,atcommonend-po ints.Atree
+T= (V,E,φ)islevelnonplanar if it doesnotadmitanyplanarleveldrawing.
+We extendthisconcepttotheoneof region-leveldrawing byenforcingthevertices
+ofeachleveltolie insideacertainregionratherthanonaho rizontalline.Let l1,...,lk
+bekpairwise non-crossing straight lines and let r1,...,r k+1be the regions of the
+planesuchthatanystraight-linesegmentconnectingapoin tinriandapointin rh,with
+1≤i < h≤k+1,cutsallandonlythelines li,li+1,...,lh−1,inthisorder.Adrawing
+ofak-leveltree T= (V,E,φ)iscalledregion-leveldrawing ifeachvertex v∈Vsuch
+thatφ(v) =iis placed inside region ri. A region-level drawing of Tisplanarif no
+two edges intersect except, possibly, at common end-points . A treeT= (V,E,φ)is
+region-levelnonplanar ifit doesnotadmitanyplanarregion-leveldrawing.
+The4-leveltree TwhoseunderlyingtreeisshowninFig.1(a)hasbeenshowntob e
+levelnonplanar[8](seeFig.1(b)).Inthenextlemmaweshow thatTisalsoregion-level
+nonplanar(seeFig.1(c)).
+Lemma 1. The4-level tree Twhose underlying tree is shown in Fig. 1(a) is region-
+levelnonplanar.
+Proof:Refer to Fig. 1(c). First observe that, in any possible regio n-level planar
+drawing of T, the paths p1=v5,v2,v8andp2=v6,v3,v9define a polygon Q2(a
+3jhBi
+r333
+23
+23
+3
+1233
+(a) (b)
+Fig.2.(a) A schematization of the complete tree T. Joints and stabilizers are small
+circles, branches are solid triangles, while complete subt rees connected to a joint are
+dashedtriangles.(b)A schematizationofabranch Bi.
+polygonQ3) inside region r2(regionr3). We have that v1is insideQ2, as otherwise
+one of edges (v1,v2)or(v1,v3)would cross one of p1orp2. Hence, vertex v4has to
+be inside Q3, as otherwise edge (v1,v4)would cross one of p1orp2. However,in this
+case, there is no placement for vertices v7andv10that avoids a crossing between one
+ofedges(v4,v7)or(v4,v10)andoneofthe alreadydrawnedges. /square
+Lemma 1 will be vital for proving that there exist a tree Tand a path Pnot ad-
+mittinganygeometricsimultaneousembedding.Infact, Tcontainsmanycopiesofthe
+underlyingtreeof T,whilePconnectsverticesof Tinsuchawaytocreatetheregions
+satisfying the above conditions and to enforce at least one o f such copies to lie inside
+theseregionsaccordingto thelevelingmakingit nonplanar .
+3 The Counterexample
+In this section we describe a tree Tand a path Pnot admitting any geometric simul-
+taneousembedding.
+3.1 Tree T
+ThetreeTcontainsaroot randqverticesj1,...,j qatdistance 1fromr,calledjoints.
+Each joint jh, withh= 1,...,q, is connected to xcopiesB1,...,B xof a subtree,
+calledbranch,and tol:= (s−1)4·32·xvertices of degree 1, called stabilizers . See
+Fig. 2(a). Each branch Biconsists of a root ri,(s−1)·3vertices of degree (s−1)
+adjacentto ri,and(s−2)·(s−1)·3leavesat distance 2fromri. Verticesbelonging
+to a branch Biare called B-verticesand denotedby 1−,2−, or3−vertices,according
+totheirdistancefromtheirjoint.Fig.2(b)displays 1−,2−,and3−verticesofabranch
+Bi.
+Because of the huge number of vertices, in the rest of the pape r, for the sake of
+readability, we use variables n,s, andxas parameters describing the size of certain
+4configurations. Such parameters will be given a value when th e technical details of
+the argumentations are described. At this stage we just clai m that a total number n≤/parenleftbig27·3·x+2
+3/parenrightbig
+ofvertices(see Lemmata5and4) sufficesforthecounterexam ple.
+As a first observation we note that, despite the oversized num ber of vertices, tree
+Thas limited depth, that is, every vertex is at distance from the root at most 4. This
+leadsto thefollowingproperty.
+Property1. Anypathoftreeedgesstartingat theroothasatmost 3bends.
+3.2 Path P
+PathPis given by describing some basic and recurring subpaths on t he vertices of
+Tand how such subpaths are connected to each other. The idea is to partition the set
+of branches Biadjacent to each joint jhinto subsets of sbranches each and to con-
+nect their vertices with path edges, accordingto some featu res of the tree structure, so
+definingthefirstbuildingblock,called cell.Then,cellsbelongingtodifferentbranches
+are connectedto each other,hencecreatingstructures,cal ledformations ,for whichwe
+can ensure certain properties regarding the intersection b etween tree and path edges.
+Further,differentformationsareconnectedtoeachotherb ypathedgesinsuchawayto
+createbiggerstructures,called extendedformations ,whichare,intheirturn,connected
+tocreatea sequenceofextendedformations .
+All of these structures are constructed in such a way that the re exists a set of cells
+such that any four of its cells, connected to the same joint an d being part of the same
+formationorextendedformation,containaregion-levelno nplanartreeforanypossible
+leveling,wherethelevelscorrespondtocells.Hence,prov ingthatfourofsuchcellslie
+indifferentregionssatisfyingthepropertiesofseparati ondescribedaboveisequivalent
+to proving the existence of a crossing in the tree. This allow s us to consider only the
+bigger structures instead of dealing with single copies of t he region-level nonplanar
+tree.
+Inthe followingwe definesuchstructuresmoreformallyands tate theirproperties.
+Cell:Themostbasicstructuredefinedby Pisdefinedbylookingathowitconnects
+verticesof some branches Biconnectedto the same joint jhofT. Consider a set of s
+branches Bi,i= 1,...,s, connected to jh. Assume the vertices of a level inside each
+treetobearbitrarilyordered.Foreach r= 1,...,s,definea cellcr(h)tobecomposed
+ofitshead,itstail, anda number tof stabilizersof jh.
+Theheadofcr(h)consistsoftheunique1-vertexof Br,thefirstthree2-verticesof
+each branch Bk, with1≤k≤sandk/\e}atio\slash=r, that are not already used in a cell ca(h),
+with1≤a < r, and, for each 2-vertex not in cr(h)and not in Br, the first 3-vertices
+notalreadyusedina cell ca(h), with1≤a < r.
+Thetailofcr(h)consistsofasetof 3·s·(s−1)2branches Bkadjacentto jh.This
+setispartitionedinto 3·(s−1)2subsetsof ssubtreeseach.Theverticesofeachofthe
+subsetsaredistributedbetweenthecellsinthesamewayasf ortheverticesofthehead.
+Thisimpliesthateachcellcontainsone1-vertex, 3·(s−1)2-vertices,and 3·(s−
+2)·(s−1)3-vertices of the head, an additional 3·(s−1)21-vertices, 32·(s−1)3
+2-vertices,and 32·(s−2)·(s−1)33-verticesofthetail,plus 32·(s−1)4stabilizers.
+51
+j332233222
+1 1 1 1 1
+h
+Fig.3.A cell.B-verticesof the head are depicted by large white circles, B-vertices of
+thetailarelargegreycircles, B-verticesnotpartofthecell(showingthetreestructure)
+are small grey circles and stabilizers are small white cirlc es. Tree edges are grey and
+pathedgesareblack.
+PathPinside cell cr(h)visits the vertices in the following order: It starts at the
+unique 1-vertex of the head, then it reaches all the 2-vertic es of the head, then all the
+3-verticesofthe head,then all the 2-verticesof the tail, a nd finally all the 3-verticesof
+the tail, visitingeachset in arbitraryorder.After eachoc currenceof a 2-or3-vertexof
+the head, Pvisits a 1-vertexof the tail, and after each occurrenceof a 2 - or a 3-vertex
+ofthetail, it visitsastabilizer ofjoint jh(see Fig.3).
+Note that, by this construction, for each joint there exists a set of cells such that
+eachsubsetofsizefourcontainsregion-levelnonplanartr eeswithallpossiblelevelings,
+wherethe levelscorrespondto the membershipof the vertice sto a cell. We nowdefine
+two bigger structures describing how cells of this set are co nnected to cells of sets
+connectedto otherjoints.
+Formation: In the definition of a cell we described how the path traverses through
+one set of branchesconnectedto the same joint. Now we descri behow cells from four
+differentsetsareconnected.
+Aformation F(H),H= (h1,h2,h3,h4)consistsof592cells, namelyof148cells
+cr(hi)from the set of cells constructed above for each 1≤i≤4. PathPconnects
+these cells in the order ((h1h2h3)37h37
+4)4, that is,Prepeats four times the following
+sequence: It connects c1(h1)toc1(h2), then toc1(h3), then toc2(h1), and so on till
+c37(h3), from which it then connects to c1(h4), toc2(h4), and so on till c37(h4)(see
+Fig.4(a)).Aconnectionbetweentwoconsecutivecells cr(a)andcr(b)isdonewith an
+edgeconnectingtheendverticesoftheparts P(cr(a))andP(cr(b))ofPrestrictedto
+theverticesof cr(a)andcr(b),respectively.Namely,theuniquevertexin cr(a)having
+degree 1 both in P(cr(a))and inTis connected to the unique vertex in cr(b)having
+degree1in P(cr(b))butnotin T. Thefollowingpropertyholds:
+Property2. Foranyformation F(H)andanyjoint jh,withh∈H,iffourcells cr(h)∈
+F(H)arepairwiseseparatedbystraightlines,thenthereexists a crossingin T.
+ExtendedFormation: Formationsareconnectedbythepathinaspecialsequence,
+definedas extendedformation anddenotedby EF(H),whereH= (H1= (h1,...,h 4),
+H2= (h5,...,h 8),..., H x= (h4x−3,...h4x))is a tuple of 4−tuples of disjoint in-
+dexes of joints (see Fig. 4(b)). Let F1(Hi),...,F y−y
+x(Hi)bey−y
+xformations not
+62 1433
+hjhjhjh1c(j  )4c37(j  )4c37(j  )
+rc(j  )37 1
+c2(j  )1
+c(j  )1 1j
+(a) (b)
+Fig.4.(a) A formation.Tree edgesare depictedby greyand path edge sby blacklines.
+Please note in this figure also the bundle of tree edges connec ting the different cells
+belonging to the same branch. (b) A subsequence (H1,...,H x)2of an extended for-
+mation. Formations are inside a table to represent the 4-tuple they belong to and to
+emphasizethat in each repetition(a row ofthe table) a forma tionat a certain 4-tupleis
+missing.
+belonging to any other extended formation and composed of ce lls of the same set S.
+These formations are connected in the order (H1,H2,...,H x)y, but in each of these
+yrepetitions one Hiis missing. Namely, in the k-th repetition the path does not reach
+anyformationat Hm,withm=kmodx.Wesaythatthe k-threpetitionhasa defect
+atm. We call a subsequence (H1,H2,...,H x)xafull repetition insideEF(J). A full
+repetitionhasexactlyonedefectat eachtuple.
+Note that the size of scan now be fixed as the number of formationscreating rep-
+etitions inside one extended formation times the number of c ells inside each of these
+formations,thatis s:= (y−y
+x)·37·4.Weclaimthat x≤7·32·223andy≤72·33·226
+is sufficient throughout the proofs. However, for readabili ty reasons, we will keep on
+usingvariables xandyin theremainderofthepaper.
+SequenceofExtendedFormation: Extendedformationsareconnectedbythepath
+inaspecialsequence,called sequenceofextendedformations anddenotedby SEF(H),
+whereH= (H∗
+1,...,H∗
+12)is a12−tuple of tuples of 4−tuples. For each tuple H∗
+i,
+wherei= 1,...,12, consider 110extended formations (EFi(H∗
+1),...,EF i(H∗
+12)),
+withi= 1,...,110, not already belonging to any other sequence of extended for ma-
+tions.Theseextendedformationsareconnectedinside SEF(H)intheorder (H∗
+1,...,
+H∗
+12)(120). There exist two types of sequences of extended formations. Namely, in the
+first typethereisoneextendedformationmissingineachsub sequence (H∗
+1,...,H∗
+12),
+that we call defect,as forthe extendedformations.In thesecondtype,two cons ecutive
+extended formations are missing. Namely, in the k-th repetition the path skips the ex-
+tended formations connecting at H∗
+mand atH∗
+m+1, withm=kmod12. In this case,
+we saythatthe repetitionhasa doubledefect .
+Since, for each set of 48xjoints,(48x)!different disjoint sequences of extended
+formationsexist,wejustconsiderthesequenceswheretheo rderdefinedbythetupleis
+theorderofthejointsaroundtheroot.
+7rc’(h’) c’(h’)
+jjh’he2e1
+c (h)2c (h)1
+j
+2hjh12h’j
+1h’
+rj
+(a) (b)
+Fig.5.(a) A passage between cells c1(h),c2(h), andc′(h′). (b) Two interconnected
+passages.
+4 Overview
+In this section we present the main argumentationsleading t o the final conclusionthat
+thetreeTandthepath PdescribedinSect.3donotadmitanygeometricsimultaneous
+embedding.Themainideainthisproofschemeistousethestr ucturesgivenbythepath
+tofixapartofthetreeinaspecificshapecreatingspecificres trictionsfortheplacement
+ofthefurthersubstructuresof TandofPattachedtoit.
+We first give some furtherdefinitionsand basic topologicalp ropertieson the inter-
+action among cells that are enforced by the preliminary argu ments about region-level
+planardrawingsandbytheorderinwhichthesubtreesarecon nectedinsideoneforma-
+tion.
+Passage: Considertwocells c1(h),c2(h)thatcannotbeseparatedbyastraightline
+andacell c′(h′),withh′/\e}atio\slash=h.We saythatthereexistsa passagePbetweenc1,c2,and
+c′ifthepolylinegivenbythepathof c′separatesverticesof c1fromverticesof c2(see
+Fig. 5(a)). Since the polylinecan not be straight, there is a vertexof c′lying inside the
+convexhulloftheverticesof c1∪c2, whichimpliesthe following.
+Property3. Inapassagebetweencells c1,c2,andc′thereexist atleast twopath-edges
+e1,e2ofc′suchthatboth e1ande2areintersectedbytree-edgesconnectingverticesof
+c1toverticesof c2.
+For two passages P1betweenc1(h1),c2(h1), andc′(h′
+1), andP2betweenc3(h2),
+c4(h2), andc′(h′
+2)(w.l.o.g., we assume h1< h′
+1,h2< h′
+2, andh1< h2), we distin-
+guish three different configurations: (i) If h′
+1< h2,P1andP2areindependent ; (ii) if
+h′
+2< h′
+1,P2isnestedintoP1; and(iii) if h2< h′
+1< h′
+2,P1andP2areinterconnected
+(seeFig. 5(b)).
+Doors:Letc1(h),c2(h), andc′(h′)be three cells creatinga passage. Consider any
+triangle given by a vertex v′ofc′inside the convex hull of c1∪c2and by any two
+vertices of c1∪c2. This triangle is a doorif it encloses neither any other vertex of
+c1,c2nor any vertex of c′that is closer than v′tojh′inT. A door is openif no tree
+edgeincidentto v′crossestheoppositesideofthetriangle,thatis,thesideb etweenthe
+verticesof c1andc2(see Fig.6(a)),otherwiseit is closed(see Fig.6(b)).
+8Considertwojoints jaandjb,withh,a,h′,bappearinginthiscircularorderaround
+theroot.Anypolylineconnectingtherootto ja,thentojb,andagaintotheroot,without
+crossing tree edges, must traverse each door by crossing bot h the sides adjacent to v′.
+If a door is closed, such a polyline has to bend after crossing one side adjacent to v′
+and beforecrossingthe otherone. Also, if two passages P1andP2are interconnected,
+either all the closed doors of P1are traversed by a path of tree-edges belonging to P2
+or all the closed doorsof P2are traversedby a path of tree-edgesbelongingto P1(see
+Fig.5(b)).
+In the rest of the argumentation we will exploit the fact that the closed door of a
+passage requests a bend in the tree to obtain the claimed prop erty that a large part of
+Thastofollowthesameshape.Inviewofthis,westatethefoll owinglemmatarelating
+theconceptsofdoors,passages,andformations.
+Lemma 2. For each formation F(H), withH= (h1,...,h 4), there exists a passage
+betweensomecells c1(ha),c2(ha),c′(hb)∈F(H), with1≤a,b≤4.
+Lemma 3. Eachpassagecontainsat leastonecloseddoor.
+From the previous lemmata we conclude that each formation co ntains at least one
+closed door.To provethat the effectsofclosed doorsbelong ingto differentformations
+can be combined to obtain more restrictions on the way in whic h the tree has to bend,
+we exploita combinatorialargumentbasedonthe RamseyTheo rem[13] andstate that
+thereexistsa set ofjointspairwisecreatingpassages.
+Lemma 4. Given a set of joints J={j1,...,j y}, with|J|=y:=/parenleftbig27·3·x+2
+3/parenrightbig
+, there
+exists a subset J′={j′
+1,...,j′
+r}, with|J′|=r≥27·3·x, such that foreach pairof
+jointsj′
+i,j′
+h∈J′there exist two cells c1(i),c2(i)creatingapassagewith acell c′(h).
+Now we formally define the claimed property that part of the tr ee has to follow a
+fixed shape by considering how the drawing of the subtrees att ached to two different
+jointsforcethedrawingofthesubtreesattachedtothejoin tsbetweenthemintheorder
+aroundthe root.
+Enclosingbendpoints: Considertwopaths p1={u1,v1,w1}andp2={u2,v2,w2}.
+The bendpoint v1ofp1encloses the bendpoint v2ofp2ifv2is internal to triangle
+△(u1,v1,w1). SeeFig. 7(a).
+xbjb j’byb
+rjb j’b
+xbyb
+r
+(a) (b)
+Fig.6.(a)Anopendoor.(B) Acloseddoor.
+91v
+u2v
+1w
+w2 2u1
+1cs2cs43 cscs
+(a) (b)
+Fig.7.(a)Anenclosingbendpoint.(b)A 3-channelanditschannelsegments.
+Channels: Consider a set of joints J={j1,...,j k}in clockwise order around
+the root. The channelciof a joint ji, withi= 2,...,k−1, is the region given by
+the pair of paths, one path of ji−1and one path of ji+1, with the maximumnumberof
+enclosing bendpoints with each other. We say that ciis anx-channel if the number of
+enclosingbendpointsis x. Observethat, byProperty1, x≤3. A3-channelis depicted
+inFig.7(b).Notethat,givenan x-channelciofji,all theverticesofthesubtreerooted
+atjithat areatdistanceat most xfromtherootlie inside ci.
+Channel segments: Anx-channel ciis composed of x+ 1parts called channel
+segments (see Fig. 7(b)). The first channel segment cs1is the part of cithat is visible
+fromtheroot.The h-thchannelsegment cshistheregionof cidisjointfrom csh−1that
+isboundedbytheelongationsofthepathsof ji−1andji+1aftertheh-thbend.
+Observe that, since the channels are created by tree-edges, any tree-edge connect-
+ing vertices in the channel has to be drawn inside the channel , while path-edges can
+cross other channels. In the following we study the relation ships between path-edges
+and channels. The following property descends from the fact that every second vertex
+reachedby Pina cell iseithera1-vertexora stabilizer.
+Property4. Foranypathedge e= (a,b), atleast oneof aandblieinsideeither cs1or
+cs2.
+Blocking cuts: Ablocking cut is a path edge connecting two consecutive channel
+segmentsbycuttingsomeoftheotherchannelstwice. SeeFig . 8.
+Property5. Letcbe a channel that is cut twice by a blocking cut. If chas vertices in
+boththe channelsegmentscut bythe path edge,thenit hassom everticesin a different
+channelsegment.
+Proof:Considertheverticeslyinginthetwochannelsegmentsof c.Inordertoconnect
+them inT, a vertex vis needed in the bendpoint area of c. However, in order to have
+pathconnectivitybetween vandtheverticesinthetwochannelsegments,somevertices
+ina differentchannelsegmentareneeded. /square
+In the followinglemma we show that in a set of joints as in Lemm a 4 it is possible
+to find a suitable subset such that each pair of paths of tree-e dges starting from the
+root and containing such joints has at least two common enclo sing bendpoints, which
+impliesthat mostofthemcreate 2-channels.
+10Lemma 5. Consider a set of joints J={j1,...,j k}such that there exists a passage
+betweeneachpair (ji,jh),with1≤i,h≤k.LetP1={P|Pconnects ciandc3k
+4+1−i,
+fori= 1,...,k
+4}andP2={P|Pconnectsck
+4+iandck+1−i, fori= 1,...,k
+4}be
+two sets of passages between pairs of joints in J(see Fig. 18). Then, for at leastk
+4of
+thejointsofonesetofpassages,say P1,thereexistpathsin T,startingattherootand
+containingthesejoints,whichtraverse all thedoorsof P2with atleast 2andat most3
+bends.Also,at leasthalfof thesejointscreatean x-channel,with 2≤x≤3.
+By Lemma 5, any formation attached to a certain subset of join ts must use at least
+three different channel segments. In the remainder of the ar gumentation we focus on
+this subset of joints and give some properties holding for it , in terms of interaction
+betweendifferentformationswithrespecttochannels.Sin ceweneedafullsequenceof
+extendedformationsattachedtothesejoints, khastobeatleasteighttimesthenumber
+ofchannelsinsidea sequenceofextendedformations,that i s,k≥8·48x= 27·3x.
+First,we givesomefurtherdefinitions.
+Nested formations A formation Fisnestedin a formation F′if there exist two
+edgese1,e2∈Fandtwoedges e′
+1,e′
+2∈F′cuttingaborder cbofachannel csuchthat
+all the verticesof the pathin Fbetweene1ande2lie inside the regiondelimitedby cb
+andbythe pathin F′betweene′
+1ande′
+2(seeFig. 10(a)).
+A series of pairwise nested formations F1,...,F kisr-nestedif there exist rfor-
+mationsFq1,...,F qr, with1≤q1,...,q r≤k, belonging to the same channel and
+such that, for each pair Fqp,Fqp+1, there exists at least one formation Fz,1≤z≤k,
+belongingtoanotherchannelandsuchthat Fqpisnestedin FzandFzisnestedin Fqp+1
+(seeFig. 10(b)).
+Independent sets of formations LetS1,...,S kbe sets of formations of one ex-
+tendedformationsuchthateachset Sicontainsformations Fi(H1),...,F i(Hr)onthe
+set of4-tuplesH={H1,...,H r}, where the joints of Hiare between the joints of
+Hi−1and ofHi+1in the order around the root. Further, let Fa(Hc)be not nested in
+Fb(Hd), foreach 1≤a,b≤k, a/\e}atio\slash=b,and1≤c,d≤r. Ifforeachpairofsets Sa,Sb
+there exist two lines l1,l2separating the vertices of SaandSbinside channel segment
+cs1andcs2,respectively,thesets are independent (seeFig.11).
+In the following lemmata we prove that in any extended format ion there exists a
+nestingofacertaindepth(Lemma8).Thisimportantpropert ywillbethestartingpoint
+forthefinalargumentationandwillbedeeplyexploitedinth erestofthepaper.Wegetto
+thisconclusionbyfirstprovingthatinanextendedformatio nthenumberofindependent
+Fig.8.Ablockingcut.
+11j j3k/421
+kk/2k/41
+rjj j
+Fig.9.Two setsofpassages P1andP2asdescribedin Lemma5.
+1e e2 1e’ e’2 cbFF’
+c
+(a) (b)
+Fig.10.(a)Aformation Fnestedinaformation F′.(b)Aseriesof r-nestedformations.
+sets of formationsis limited (Lemma6) and thenby showingth at, althoughthereexist
+formations that are neither nested nor independent, in any e xtended formation there
+existsacertainnumberofpairsofformationthathavetobee itherindependentornested
+(Lemma7).
+Lemma 6. Thereexistno n≥222·14independentsetsofformations S1,...,S ninside
+anyextendedformation,whereeach Sicontainsformationsofafixedsetofchannelsof
+sizer≥22.
+1J2J3J121F (J )22F (J )23F (J )S1
+13F (J )11F (J )
+12F (J )
+S2
+J
+Fig.11.Twoindependentsets S1andS2.
+12Lemma 7. Consider four subsequences Q1,...,Q 4, whereQi= (H1,H2,...,H x),
+ofanextendedformation EF,eachconsistingofawholerepetitionof EF.Then,there
+exists eitherapairof nestedsubsequencesorapairof indep endentsubsequences.
+Lemma 8. Consider an extended formation EF(H1,H2,...,H x). Then, there exists
+ak-nesting,where k≥6,amongtheformationsof EF.
+Oncetheexistenceof 2-channelsandofanestingofacertaindepthineachextended
+formationhas been shown,we turn our attention to study how s uch a deep nesting can
+beperformedinsidethechannels.
+Letcsaandcsb, with1≤a,b≤4, be two channel segments. If the elongation
+ofcsaintersects csb, then it is possible to connect from csbtocsaby cutting both the
+sides ofcsa. In this case, csaandcsbhave a2−side connection (see Fig. 12(b)). On
+thecontrary,if theelongationof csadoesnotintersect csb, onlyoneside of csacan be
+used.Inthiscase, csaandcsbhavea1−sideconnection (seeFig. 12(a)).
+(a) (b)
+Fig.12.(a)A1−sideconnection.(b)A 2−sideconnection.
+Based on these different ways of connecting distinct channe l segments, we split
+our proof into three parts, the first one dealing with the sett ing in which only 1-side
+connectionsareallowed,thesecondoneallowingonesingle 2-sideconnection,andthe
+last onetacklingthegeneralcase.
+Proposition1. If there exist only 1−side connections,then TandPdo not admit any
+geometricsimultaneousembedding.
+We provethis propositionby showing that, in this configurat ion,the existence of a
+deep nesting in a single extended formation, proved in Lemma 8, results in a crossing
+ineitherTorP.
+Lemma 9. If an extended formation lies in a part of the channel that con tains only
+1−side connections, then TandPdo not admit any geometric simultaneous embed-
+ding.
+Next,westudythecaseinwhichthereexist 2-sideconnections.Wedistinguishtwo
+types of2-side connections, based on the fact that the elongation of c hannel segment
+csaintersectingchannelsegment csbstartsatthebendpointthatisclosertotheroot,or
+13not. In the first case we have a low Intersection (see Fig. 13(a)), denotedby Il
+(a,b), and
+inthesecondcasewehavea highIntersection (seeFig.13(b)),denotedby Ih
+(a,b),where
+a,b∈ {1,...,4}. We use the notation I(a,b)to describe both Ih
+(a,b)andIl
+(a,b). We say
+thattwointersections I(a,b)andI(c,d)aredisjointifa,d∈ {1,2}andb,c∈ {3,4}.For
+example, I(1,3)andI(4,2)aredisjoint,while I(1,3)andI(2,4)arenot.
+r r
+(a) (b)
+Fig.13.(a)Alow Intersection.(b)AhighIntersection.
+Since consecutive channel segmentscan not create any 2-side connection,in order
+to explore all the possible shapes we consider all the combin ations of low and high
+intersections created by channel segments cs1andcs2with channel segments cs3and
+cs4. With theintentofprovingthat intersectionsofdifferent channelshavetomaintain
+certainconsistencies,we state thefollowinglemma.
+Lemma 10. Consider two channels chp,chqwith the same intersections. Then, none
+of channels chi, wherep < i < q , have an intersection that is disjoint with the inter-
+sectionsof chpandofchq.
+As for Proposition 1, in order to prove that 2-side connectionsare not sufficient to
+obtain a simultaneous embedding of TandP, we exploit the existence of the deep
+nesting shown in Lemma 8. First, we analyze some propertiesr elating such nesting to
+channelsegmentsandbendingareas.A bendingarea b(a,a+1)is theregionbetween
+csaandcsa+1where bendpoints can be placed. We first observe that all the e xtended
+formationshavetoplaceverticesinsidethebendingareaof thechannelsegmentwhere
+the nestingtakes place, and then provethat not manyof the fo rmationsinvolvedin the
+nesting can use the part of the path that creates the nesting t o place vertices in such
+a bending area, which implies that the extended formations h ave to reach the bending
+areaina differentway.
+Lemma 11. Consider an x-nesting of a sequence of extended formations on an inter-
+sectionI(a,b), witha≤2. Then, there exists a triangle tin the nesting that separates
+someof thetrianglesnestingwith tfrom thebendingarea b(a,a+1)(orb(a−1,a)).
+Then,westudysomeofthecasesinvolving 2-sideconnectionsandweshowthatthe
+connectionsbetweenthebendingareaandthe"endpoints"of thenestingcreateafurther
+nestingofdepthgreaterthan 6.Hence,ifnofurther 2−sideconnectionisavailable,this
+secondnestingisnotdrawable.
+14Proposition2. Lettbe a triangle open on a side splitting a channel segment csinto
+two partssuchthateveryextendedformation EFhasverticesinbothparts.Iftheonly
+possibility to connectvertices in different parts of csis with a1-side connectionand if
+anysuchconnectioncreatesa triangleopenona side that isn ested with t, thenTand
+Pdonotadmitanygeometricsimultaneousembedding.
+wcsv
+u
+Fig.14.A situation as in Proposition 2. The chosen turning vertex is represented by a
+bigblackcircleandisinconfiguration β. Theinnerandthe outerareasarerepresented
+byalightgreyanda darkgreyregion,respectively.
+Refer to Fig. 14. Consider the two path-edges e1= (u,v),e2= (v,w)creatingt
+such that the common point vis in the channel segment csthat is split into two parts,
+thatwecall innerarea andouterarea ,respectively.Weassumethat e1,e2donotcutany
+channel segment cs′completely, since such a cut would create more restrictions than
+placinguorwinsidecs′. Consider the path in an extended formation EFconnecting
+the inner and the outer area through a 1-side connection at cs′. As a generalization,
+consider for such a path of EFonly a vertex, called turning vertex , which is placed
+incs′and for which no other path in EFexists that connects the inner and the outer
+area by using a channel segment cs′′such that the subpath to cs′′intersects either cs′′
+or its elongation. If there exist more than one of such vertic es, then arbitrarily choose
+one of them. Observe that the path connecting from the inner a rea to the outer area
+through the turning vertex encloses exactly one of uandw. If it encloses u, it is in
+configuration α, otherwise it is in configuration β. If there exist both paths in αand
+paths inβconfiguration, then we arbitrarily consider one of them. Fin ally, consider
+the connections between different extended formations ins ide a sequence of extended
+formations. Consider a turning vertex vin a channel segment csof a channel chsuch
+that the edges incident to vcut a channel ch′. Then, any connection of an extended
+formation of ch′from the inner to the outer area in the same configuration as chand
+withitsturningvertex v′incsissuchthat v′liesinsidetheconvexhullofthetwoedges
+incidentto v.
+Inthefollowingtwolemmataweshowthatinthesettingdescr ibedinProposition2
+thereexistsa crossingeitherin TorinP.
+15Lemma 12. In a situation as described in Proposition 2, not all the exte nded forma-
+tionsinasequenceofextendedformationscanplaceturning verticesinthesamechan-
+nelsegment.
+Lemma 13. In a situation as described in Proposition 2, TandPdo not admit any
+geometricsimultaneousembedding.
+Based on the property given by Proposition 2, we present the s econd part of the
+proof,in whichwe show that havingtwo intersections I(a,b)andI(c,d)doesnot helpif
+I(a,b)andI(c,d)arenotdisjoint.
+Proposition3. If there exists no pair of disjoint 2-side connections, then TandPdo
+notadmitanygeometricsimultaneousembedding.
+Observethat,inthissetting,itissufficienttorestrictth eanalysistocases I(1,3)and
+I(3,1), sincethe casesinvolving 2and4canbereducedto them.
+Lemma 14. If a shape contains an intersection I(1,3)and does not contain any other
+intersectionthatisdisjointwith I(1,3),thenTandPdonotadmitanygeometricsimul-
+taneousembedding.
+Lemma 15. Ifthereexistsasequenceofextendedformationinanyshape containingan
+intersection I(3,1),thenTandPdonotadmitanygeometricsimultaneousembedding.
+Observe that, in the latter lemma, we proved a property that i s stronger than the
+one stated in Proposition 3. In fact, we proved that a simulta neous embedding cannot
+beobtainedinanyshapecontaininganintersection I(3,1), evenifa secondintersection
+thatisdisjointwith I(3,1)ispresent.
+Finally,in the third partof the proof,we tackle the general case wheretwo disjoint
+intersectionsexist.
+Proposition4. If there exists two disjoint 2-side connections, then TandPdo not
+admitanygeometricsimultaneousembedding.
+Since the cases involving intersection I3,1were already considered in Lemma 15,
+we only have to consider the eight different configurations w here one intersection is
+I(1,3)and the other is one of I(4,{1,2}). In the next three lemmata we cover the cases
+involving Ih
+(1,3)andinLemma19the onesinvolving Il
+(1,3).
+Consider two consecutive channel segments csiandcsi+1of a channel cand let
+ebe a path-edge crossing the border of one of csiandcsi+1, saycsi. We say that e
+createsadoublecut atcif theelongationof ecutscincsi+1. A doublecut is simpleif
+edoesnotcross csi+1(seeFig.15(a))and non-simple otherwise(seeFig.15(b)).Also,
+a double cut of an extended formation EFisextremal with respect to a bending area
+b(x,x+1)ifthereexistsnodoublecutof EFthatiscloserthanit to b(x,x+1).
+Property6. Any edge ekcreating a double cut at a channel kin channel segment csi
+blocksvisibilitytothebendingarea b(i,i+1)forapartof csiineachchannel chhwith
+h > k(withh < k).
+16ci+1csiecs
+cicsi+1 e
+cs
+(a) (b)
+Fig.15.(a)A simpledoublecut.(b)A non-simpledoublecut.
+In the followinglemma we show that a particular orderingof e xtremal doublecuts
+intwoconsecutivechannelsegmentsleadstoanon-planarit yinTorP.Notethat,any
+orderofextremaldoublecutscorrespondsto an orderof thec onnectionsof a subset of
+extendedformationstothebendingarea.
+Lemma 16. Letcsiandcsi+1be two consecutive channelsegments. If there exists an
+orderedset S:= (1,2,...,5)3of extremal doublecutscutting csiandcsi+1such that
+the order of the intersections of the double cuts with csi(withcsi+1) is coherent with
+theorderof S,thenTandPdonotadmitanygeometricsimultaneousembedding.
+Then,weshowthatshape Ih
+(1,3)I(4,2)inducesthisorder.Toprovethis,wefirststate
+theexistenceofdoublecutsinshape Ih
+(1,3)Ih
+(4,2).Theexistenceofdoublecutsinshape
+Ih
+(1,3)Il
+(4,2)canbe easilyseen.
+Lemma 17. Each extended formation in shape Ih
+(1,3)Ih
+(4,2)creates double cuts in at
+least onebendingarea.
+Lemma 18. Every sequenceof extendingformationsin shape Ih
+(1,3)Ih,l
+(4,2)containsan
+ordered set (1,2,...,5)3of extremal double cuts with respect to bending area either
+b(2,3)orb(3,4).
+Finally,weconsidertheconfigurationswhereoneintersect ionisIl
+(1,3)andtheother
+one is one of Ih,l
+(4,2). Observe that, in both cases, channel segment cs2is on the convex
+hull.
+Lemma 19. If channel segment cs2is part of the convex hull, then TandPdo not
+admitanygeometricsimultaneousembedding.
+Basedontheabovediscussion,we state thefollowingtheore m.
+Theorem1. Thereexistatreeandapaththatdonotadmitanygeometricsi multaneous
+embedding.
+17Proof:LetTandPbe the tree andthe pathdescribedin Sect. 3. Then,by Lemma5,
+Lemma10, andProperty1, a partof Thastobe drawninside channelshavingat most
+four channel segments. Also, by Lemma 8, there exists a nesti ng of depth at least 6
+insideeachextendedformation.
+ByProposition1,ifthereexistonly 1-sideconnections,then TandPdonotadmit
+anysimultaneousembedding.ByProposition3,ifthereexis tseitherone 2-sideconnec-
+tions or a pair of non-disjointintersections, then TandPdo not admit any simultane-
+ous embedding. By Proposition 4, even if there exist two disj oint2-side intersections,
+thenTandPdonotadmitanysimultaneousembedding.Sinceitisnotposs ibletohave
+morethantwo disjoint 2-sideintersections,the statementfollows. /square
+5 Detailed Proofs
+Lemma 2 .For each formation F(H), withH= (h1,...,h 4), there exists a passage
+betweensomecells c1(ha),c2(ha),c′(hb)∈F(H), with1≤a,b≤4.
+Proof:Suppose,foracontradiction,thatthereexistsnopassagei nsideF(H). First
+observe that, if two cells c1(ha),c2(ha)∈F(H)are separated by a polyline given by
+thepathpassingthrough F(H),theneithertheyareseparablebyastraightlineorsuch
+apolylineiscomposedofedgesbelongingtoacell c3(ha)ofthesamejoint jha.Since,
+by Property 2, there exists no set of four cells of a given join t insideF(H)that are
+separableby a straight line, it followsthat all the cells of F(H)of a givenjoint can be
+groupedinto at most 3different sets S1,S2, andS3such that cells from different sets
+can be separated by straight lines, but cells from the same se t can not. Therefore, the
+cellsinsideoneofthesesetscanonlybeseparatedbyotherc ellsofthesame set.
+2
+h1S1
+h1S2
+h3
+S3
+h3S1
+h2S2
+h2S3
+S2
+jh2jh3jh1S1
+h3
+h 3
+h1S
+3
+eee
+e
+re1
+2
+4
+5
+Fig.16.Thefivepathedges e1,...,e 5connectingfivecellsofset Sa
+h1withfivecellsof
+setSb
+h2.
+Considertheconnectionsofthepaththrough F(H)withregardtothisnotionofsets
+ofcells.Let Sy
+hx,withx= 1,...,4andy= 1,...,3,bethesetofcellsbelongingtoset
+18Syand attached to joint jhx. Hence, for any two cells c1(hx),c2(hx+1)there are nine
+possible ways to connect between some Sy
+hxandSy′
+hx+1. Since the part of Pthrough
+F(H)visits 37 times cells from jh1,jh2,jh3, in this order, there exist five path edges
+e1,...,e 5connectingfivecellsofset Sa
+h1withfivecellsofset Sb
+h2,where1≤a,b≤3
+(see Fig.16). Withoutlossofgenerality,weassume that edg ese1,...,e 5appearin this
+order in the part of PthroughF(H). Observe that e1,...,e 5, together with the five
+cellsofSa
+h1andthefivecellsof Sb
+h2theyconnect,subdividetheplaneintofiveregions.
+Sincethepathiscontinuousin F(H),itconnectsfromtheendof e1(acellofjoint jh2)
+tothebeginningof e2(acellofjoint jh1),fromtheendof e2tothebeginningof e3,and
+soon.Ifintheregionbetweenedges esandes+1,with1≤s≤4,thereexistsnocellof
+jointjh3, then the path through F(H)will not traversethe regionbetween these edges
+in the opposite direction, since the path contains no edges g oing from a cell of jh2to
+a cell ofjh1and since the start- (and end-) cells of these edges cannot be separated by
+straight lines. Furthermore,note that, in this case, the pa th-connectionfrom estoes+1
+doesnottraversethe regionbetweenthe edges,thereforefo rminga spiral shape, inthe
+sensethatthepartofthepathfollowing es+1isseparatedfromthepartofthepathprior
+toes. Since we have five edges between Sa
+h1andSb
+h2but only 3 possible sets of cells
+on jointjh3, at least one pair of edges exists creating an empty region an d therefore a
+spiralseparatingthepath.
+Bythisargument,itfollowsthatcellsattachedtojoint jh4indifferentrepetitionsof
+the subsequence ((h1h2h3)37h37
+4)inF(H)are separated by path edges of the spirals
+formed by the repeated subsequence of visited cells of the jo intsjh1,jh2,jh3. Since
+four repetitions create four of such separated cells on jh4, by Property 2 there exists a
+pairofcellsthatarenotseparablebyastraightlinebutare separatedbythepath.Since
+the path of the spiral separating them consists only of cells on different joints, any
+possible separating polyline leads to a contradiction to th e non-existence of a passage
+insideF(H). /square
+Lemma 3 .Eachpassagecontainsat leastonecloseddoor.
+h’
+vv’j
+Fig.17.Thereexistsacloseddoorin eachpassage.
+Proof:Refer to Fig. 17. Let P1be a passage between c1(h),c2(h), andc′(h′).
+Consider any vertex vofc′inside the convex hull of C:=c1∪c2. Further, consider
+19all the triangles △(v,v1,v2)created by vwith any two vertices v1,v2∈Csuch that
+△(v,v1,v2)doesnotencloseanyothervertexof C. Thepath oftreeedgesconnecting
+vtojh′enters one of the triangles. Then, either it leaves the trian gle on the opposite
+side,therebycreatingacloseddoor,oritencountersavert exv′ofc′.Sinceatleastone
+vertexof c′liesoutsidetheconvexhullof C,otherwisetheywouldnotbeseparatedby
+c′, it is possible to repeat the argument on triangle △(v′,v1,v2)until a closed door is
+found. /square
+Lemma 4 .Given a set of joints J={j1,...,j y}, with|J|=y:=/parenleftbig27·3·x+2
+3/parenrightbig
+, there
+exists a subset J′={j′
+1,...,j′
+r}, with|J′|=r≥27·3·x, such that foreach pairof
+jointsj′
+i,j′
+h∈J′there exist two cells c1(i),c2(i)creatingapassagewith acell c′(h).
+Proof:By construction of the tree, for each set of four joints, ther e are formations
+that visit only cells of these joints. By Lemma 2, there exist s a passage inside each of
+these formations, which implies that for each set of four joi nts there exists a subset of
+twojointscreatingapassage.Theactualnumberofjointsne ededtoensuretheexistence
+of a subset of joints of size rsuch that passages exist between each pair of joints is
+given by the Ramsey Number R(r,4). This number is defined as the minimal number
+of vertices of a graph Gsuch that Geither has a complete subgraph of size ror an
+independent set of size 4. Since in our case we can never have an independent set of
+size4,weconcludethatasubsetofsize rexistswiththeclaimedproperty.TheRamsey
+numberR(r,4)isnotexactlyknown,butwecanusetheupperbounddirectlye xtracted
+fromtheproofoftheRamseytheoremtoarriveat theboundsta tedabove.[13] /square
+Lemma5 .Considerasetofjoints J={j1,...,j k}suchthatthereexistsapassagebe-
+tweeneachpair (ji,jh),with1≤i,h≤k.LetP1={P|Pconnects ciandc3k
+4+1−i,
+fori= 1,...,k
+4}andP2={P|Pconnectsck
+4+iandck+1−i, fori= 1,...,k
+4}be
+two sets of passages between pairs of joints in J(see Fig. 18). Then, for at leastk
+4of
+thejointsofonesetofpassages,say P1,thereexistpathsin T,startingattherootand
+containingthesejoints,whichtraverse all thedoorsof P2with atleast 2andat most3
+bends.Also,at leasthalfof thesejointscreatean x-channel,with 2≤x≤3.
+j j3k/421
+kk/2k/41
+rjj j
+Fig.18.Thetwosetsofpassages P1andP2describedin Lemma5.
+20Proof:Observefirstthateachpassageof P1isinterconnectedwitheachpassageof
+P2andthat allthe passagesof P1andall thepassagesof P2arenested.
+By Lemma 3 and Property 1, for one of P1andP2, sayP1, either for every joint
+ofP1betweenthe jointsof P2in the orderaroundthe root orfor everyjoint of P1not
+betweenthejointsof P2,thereexistsapath piinT,startingat therootandcontaining
+these joints, which has to traverse all the doorsof P2by making at least 1and at most
+3bends. Also, paths p1,...,p k
+4can be ordered in such a way that a bendpoint of pi
+encloses a bendpoint of phfor eachh > i. It follows that there exist x-channels with
+1≤x≤3foreach joint.Considernow the set of joints J′⊂Jvisitedby these paths.
+We assume thejointsof J′={j′
+1,...j′
+r}tobein thisorderaroundtheroot.
+rp1p
+1e
+2e
+1prp
+e1
+2e
+/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1
+/0/0/0/0/0/0
+/1/1/1/1/1/1
+/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1
+/0/0/0/0/0/0
+/1/1/1/1/1/1
+/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1
+(a) (b)
+Fig.19.(a)Theseparatingcell c′isintheoutermostchannel.(b)Theseparatingcell c′
+isin theinnermostchannel.
+Consider the path p1whose bendpoint encloses the bendpoint of each of all the
+other paths and the path prwhose bendpoint encloses the bendpoint of none of the
+otherpaths(see Figs. 19(a) and 19(b)).Please note that eit herp1visitsj′
+1andprvisits
+j′
+ror vice versa, say p1visitsj′
+1. By construction,there exists a passage between cells
+fromj′
+1and cells from j′
+r. In this passage there exist either two path-edges e1,e2of a
+cellc′(1)separating two cells c1(r),c2(r), thereby crossing the channel of j′
+r, or two
+edges of a cell c′(r)separating two cells c1(1),c2(1), thereby crossing the channel of
+j′
+1. We showthat 1-channelsarenotsufficienttodrawthese pass ages.
+In the first case (see Fig. 19(a)), both separating edges e1,e2cross the path prbe-
+fore and after the bend, thereby creating blocking cuts sepa rating vertices of the same
+cell, sayc1. Since they are connected by the path, by Property 5, an addit ional bend
+is needed. In the other case (see Fig. 19(b)), any edge connec ting vertices of c′(j′
+r)is
+not even crossing any edge of p1and therefore at least another bend is needed in the
+channel. So at least one of the joints needs an additional ben d. Since there are pas-
+sagesbetweeneachpairof jointsin J′, all butonejoint jqhavea paththat hasto bend
+an additional time. We note that the additional bendpoint of each path pkaside from
+p1,pr, andpqhas to enclose all the additional bendpointseither of p1,...,p k−1or of
+21pk+1,...,p r.Itfollowsthat,foratleast halfofthejoints,thereexist x-channelswhere
+2≤x≤3. /square
+Lemma6 .Thereexistno n≥222·14independentsetsofformations S1,...,S ninside
+anyextendedformation,whereeach Sicontainsformationsofafixedsetofchannelsof
+sizer≥22.
+Proof:Assume for a contradiction, that such independent sets S1,...,S nexist.
+By Lemma 2, each formation in each set will contain a passage a nd thereby an edge
+cuttingthechannelborder.ByProperty4eachformationine achset will placeanedge
+to either channel segment cs1orcs2. As can be easily seen, there exists a set S1of
+sizen
+2of sets of formationsthat will have at least one common conne ctionfor a fixed
+formation Fiin each set Sa⊂S1, where1≤n. By repeating the argument we can
+find a subset S2⊂S1of sizen
+4such that these sets will have at least two common
+connections for formations Fi,Fhin each set Sa⊂S2. By continuing this procedure
+we arriveat a subset Srofsizen
+2rthat will haveat least rcommonconnections.Since
+allthesecommonconnectionshavetoconnecttoeither cs1orcs2,wehaveidentifieda
+setS={S′
+1,...,S′n
+2r}ofsizen
+2rofsetsofformationsofsizeatleastr
+2thathasallthe
+connectionsto onespecific channelsegment CS.
+We now consider the cutting edges for each of the formations o fSinCS. Since
+any of those can intersect the channelborderon two differen tsides, at least half of the
+connections for a fixed formation Fr
+4in all the sets will intersect with one side of the
+channelborder,therebycrossingeitherallthechannels 1,...,r
+4−1orallthechannels
+r
+4+ 1,...,r
+2, assume the first. Consider now the formations Fr
+8in each of the sets.
+Theseformationsofthesets S′
+2,S′
+4,...,S′n
+2r+1willbeseparatedon CSbytheedgesof
+theformations Fr
+4ofthesets S′
+3,S′
+5,...,S′n
+2r−1. Toavoidamonotonicorderingofthe
+separated formationsand thereby the existence of an region -levelnonplanartree these
+formations Fr
+8havetoplaceverticesinanadjacentchannelsegment CS′.Thiswillcre-
+ateblockingcutsforeitherallthechannels 1,...,r
+8−1orallthechannelsr
+8+1,...,r
+4,
+assume the first. Considernow the formations F1in eachof the sets. These formations
+of the sets S′
+3,S′
+5,...,S′n
+2r−2will be separated on CSby the edges of the formations
+Fr
+8of the sets S′
+4,S′
+6,...,S′n
+2r−3. By the same argument as above also these forma-
+tions have to place vertices in an adjacent channel segment t hat are visible from some
+of the separated areas of CS. Since the connection from the formations Fr
+8are block-
+ing for the connection to CS′, the formations F1have to use the remaining adjacent
+channel segment CS′′, thereby blocking all the channels 1,...r2. We finally consider
+the formations F2of the sets S′
+4,S′
+6,...,S′
+10. These formations are now separated in
+CSby the blocking edges to CS′of the formations Fr
+8and by the blocking edges to
+CS′′of the formations F1. Therefore,these formationscannot use part of any channel
+segment (tree-)visible to the separated areas in CS. So, by Property 2, we identified a
+region-levelnonplanartree,incontradictionto theassum ption. /square
+Lemma 7 .Consider four subsequences Q1,...,Q 4, whereQi= (H1,H2,...,H x),
+ofanextendedformation EF,eachconsistingofawholerepetitionof EF.Then,there
+exists eitherapairof nestedsubsequencesorapairof indep endentsubsequences.
+Proof:Assume that no pair of nested subsequences exists. We show th at a pair of
+independentsubsequencesexists.
+22First, we considerhow Q1,...,Q 4use the first two channelsegments cs1andcs2.
+Each of these subsequences uses either only cs1, onlycs1, or both to place its forma-
+tions.Observethat,ifa subsequenceusesonly cs1andanotheroneusesonly cs2, then
+such subsequences are clearly independent. So we can assume that all of Q1,...,Q 4
+usea commonchannelsegment,say cs2.
+a
+Fig.20.If three subsequences use the same channel segment cs, then at least two of
+themareeithernestingorseparatedin cs.
+Then we show that, if three subsequences use the same channel segmentcs, then
+at least two of them are separated in cs. In fact, if two subsequences using csare not
+independent, then they contain formations on the same chann elathat intersect with
+different channel borders of a. However, a third subsequence containing a formation
+that intersects a channel border of ais such that there exists either a nesting or a clear
+separation between this subsequence and the other subseque nce intersecting the same
+channelborderof a(see Fig. 20). Thisfact impliesthat if three subsequencesu se only
+cs2, then at least two of them are independent. From this and from the fact that there
+are four subsequences using cs2, we derive that two subsequences, say Q1,Q2, are
+separated in cs2and are not separated in cs1. Then, the third subsequence Q3can be
+placed in such a way that it is not separated from Q1andQ2incs2. However, this
+implies that Q4is separated in cs1from two of Q1,Q2,Q3and incs2from two of
+Q1,Q2,Q3, which implies that Q4is separated in both channel segments from one of
+Q1,Q2,Q3. /square
+Lemma 8 .Consider an extended formation EF(H1,H2,...,H x). Then, there exists
+ak-nesting,where k≥6,amongtheformationsof EF.
+Proof:Assume, for a contradiction,that there is no k-nesting among the sequence
+of formations in EF. We claim that, under this assumption, there exist more than n
+sequences of independent formations in EFfrom the same set of channels C, where
+n≥222·14and|C| ≥22.By Lemma6,sucha claimclearlyimpliesthestatement.
+Consider sequences that use some common channels in channel segments cs1and
+cs2.Then,theirseparationin cs1hastheoppositeorderingwithrespecttotheirsepara-
+tionincs2.
+23Observe that, by Lemma 7, there exist at most (n−1)·3different nestings of
+subsequences such that there are less than nindependent sets of subsequences. Also
+notethat,if someformationsbelongingtotwodifferentsub sequencesarenesting,then
+alltheformationsofthesesubsequenceshavetobepartofso menesting.However,this
+does not necessarily mean for all the formationsto nest with each other and to build a
+singlenesting.
+Since the numberof channelsused inside EFis greaterthan (n−1)·3·3, where
+n≥222·14, we have a nesting consisting of subsequences with at least 3different
+defects.
+Let the nesting consist of subsequences Q1
+1,...,Qr
+1,Q1
+2,...,Qr
+2,...,Q1
+k...,Qr
+k,
+whereQh
+idenotesthe h-thoccurrenceofasubsequenceof EFwithadefectatchannel
+i.Further,letthepathconnectthemintheorder Q1
+1,Q1
+2,...,Q1
+k,Q2
+1,...,Q2
+k,...,Qr
+k.
+We showthat thereexistsa pairofindependentsubsequences withinthisnesting.
+4 3 2 1 1 2 3 4 4 3 2 1 1 2 3 4 2 3 1 2 4 4 3 4 3 2 1 1
+(a) (b) (c)
+Fig.21.(a) and (b) Possible configurations for Q1
+1,Q1
+2, andQ1
+3. (c) The repetitions
+followthe outwardorientation.
+1 2 3 1 2 3 4 4 4 3 21 2 3 1 2 3 4 4 4 3 21 1
+(a) (b)
+Fig.22.The connection between channels 2and4blocks visibility for the following
+repetitions to the part of the channel segment where vertice s of channel 3were placed
+till thatrepetition.
+Consider now the first two nesting repetitions of sequence (H1,H2,...,H x), that
+is,Q1
+1andQ1
+2. Let the nesting consist of a formation F(k)fromQ1
+1nesting in a for-
+241 2 3 1 2 3 4 4 4 3 21
+(a)
+Fig.23.All the channels c,...,xare shifted and the next repetition starts in a com-
+pletelydifferentregion.
+mationF′(s)fromQ1
+2. Consider the edges e1,e2∈F(k)ande′
+1,e′
+2∈F′(s)that are
+responsible for the nesting. Without loss of generality we a ssume the path pthat con-
+nectse′
+2ande1not to contain edges e′
+1,e2. Consider the two parts a,bof the channel
+borderof s,whereaisbetween e1ande′
+1andbisbetween e2ande′
+2.Considernowthe
+closed region delimited by the path through F′(s), the path p, the path through F(k),
+andb.Sucharegionissplit intotwoclosedregions RinandRnestbya(see Fig.24).
+1e’ e’2RinRnest
+e21eF’
+F
+a b
+Fig.24.RegionsRinandRnest.
+Observe that, in order to reach from Rinto the outer region, any path has to cross
+bothaandb.We notethat thepartof Pstartingat e′
+2andnotcontaining F(k)iseither
+completelycontainedintheouterregionorhastocrossover betweenRinandtheouter
+regionbytraversing Rnest.Similarly,thethepartof Pstartingat e1andnotcontaining
+F′(s)either does not reach the outer region or has to cross over bet weenRinand the
+outerregionbytraversing Rnest.Furthermore,anyformation F′′onsuchapathisalso
+eithercrossingoverandtherebycuttingboth aandb,ornot.Inthefirstcase Fisnested
+inF′′andF′′isnestedin F′.
+Consider now the third nesting repetition Q1
+3of sequence (H1,H2,...,H x)(see
+Figs. 21(a) and 21(b)). It is easy to see that if Q1
+3is nested between Q1
+1andQ1
+2, then
+25thereexistsanestingofdepth 1becauseQ1
+3containsa defectata differentchannel.So
+we have to consider the cases when the repetitions create the nesting by strictly going
+either outward or inward. By this we mean that the i-th repetition Q1
+ihas to be placed
+suchthateither Q1
+iisnestedinside Q1
+i−1(inward)orviceversa(outward).Withoutloss
+ofgenerality,we assumethelatter(see Fig.21(c)).
+Consider now a defect in a channel c, with1< c < k, at a certain repetition Qh
+i.
+Since the path is moving outward, the connection between cha nnelsc−1andc+ 1
+blocks visibility for the following repetitionsto the part of the channel segment where
+verticesofchannel cwereplacedtill thatrepetition(seeFig.22(a)foranexamp lewith
+c= 3).
+A possible placementforthe verticesof cin the followingrepetitionsthat doesnot
+increasethedepthofthenestingcouldbeinthesamepartoft hechannelsegmentwhere
+vertices of a channel c′, withc′/\e}atio\slash=c, were placed till that repetition. We call shiftsuch
+a move. However, in order to place vertices of cand ofc′in the same zone, all the
+vertices of cbelonging to the current cell have to be placed there (see das hed lines in
+Fig. 22(b), where c′=c+ 1), which implies that a further defect in channel cat one
+of the following repetitions encloses all the vertices of ea ch of the previously drawn
+cells, hence separating them with a straight line from the fo llowing cells. Hence, also
+theverticesof c′havetoperformashifttoachannel c′′,withc/\e}atio\slash=c′′/\e}atio\slash=c′.Again,ifthe
+vertices of c′and ofc′′lie in the same zone, we have two cells that are separated by a
+straightlineandhencealsotheverticesof c′′havetoperformashift.Byrepeatingsuch
+an argument we conclude that the only possibility for not hav ing vertices of different
+channels lying in the same zone is to shift all the channels c,...,xand to go back
+to channel 1for starting the following repetition in a completely diffe rent region (see
+Fig. 23, where the following repetition is performed comple tely below the previous
+one). However, this implies that there exist two repetition s in one configuration that
+have to be separated by a straight line and therefore are inde pendent, in contradiction
+to ourassumption.Therefore,we canassume that, after 3·x+1repetitions,we arrive
+atanestingofdepth1.Byrepeatingthisargumentwearrivea fter3·x·6repetitionsat
+thenestingofdepth 6claimedinthelemma. /square
+Lemma 9 .If an extended formation lies in a part of the channel that con tains only
+1−side connections, then TandPdo not admit any geometric simultaneous embed-
+ding.
+Proof:First observe that, by Lemma 8, there exists a k-nesting with k≥6in any
+extendedformation EF.
+Consider two nested formations F,F′∈EFbelonging to the k-nesting. Such
+formations, by definition, belong to the same channel. Consi der now the formation
+F′′∈EFbelonging to a different channel such that Fis nested in F′′andF′′is
+nested in F′. Since each pair of channel segments have a 1−side connection, we have
+thatF′′blocks visibility for F′on the channel segment used by Ffor the nesting (see
+Fig.25).Hence, F′hastouseadifferentchannelsegmenttoperformitsnesting ,which
+increases by one the numberof used channel segmentsfor each level of nesting.Since
+thetreesupportsatmost 4channelsegments,thestatementfollows. /square
+26FF’F’’
+Fig.25.Illustrationforthe caseinwhichonly 1-sideconnectionsarepossible.
+Lemma 10 .Consider two channels chp,chqwith the same intersections. Then, none
+of channels chi, wherep < i < q , have an intersection that is disjoint with the inter-
+sectionsof chpandofchq.
+Proof:The statement follows from the fact that the channel borders ofchpand
+chqdelimit the channel for all joints between pandq. So, if any channel chi, with
+p < i < q , had an intersection different from the one of chpandchq, it would either
+intersectwithoneofthechannelbordersof chporchqoritwouldhavetobendaround
+oneofthechannelborders,hencecrossinga straightline tw ice. /square
+Lemma 11 .Consider an x-nesting of a sequence of extended formations on an inter-
+sectionI(a,b), witha≤2. Then, there exists a triangle tin the nesting that separates
+someofthetrianglesnestingwith tfromthebendingarea b(a,a+1)(orb(a−1,a)).
+Proof:Consider three extended formations EF1(H1),EF2(H1),EF3(H1)lying
+in a channel ch1and two extended formations EF1(H2),EF2(H2)lying in a chan-
+nelch2such that all the channels of the sequence of extended format ions are be-
+tweench1andch2and there is no formation F/\e}atio\slash∈EF(H1),EF(H2)nesting be-
+tweenEF1(H1),EF2(H1),EF3(H1)andEF1(H2),EF2(H2).Suppose,withoutloss
+ofgenerality,thatthe bendingpointof ch1isenclosedintothebendingpointof ch2.
+Consideraformation F1∈EF1(H1)nestingwithaformation F′
+1∈EF1(H2).We
+havethat the connectionsfrom F′
+1to channelsegment aandback hasto goaroundthe
+vertex placed by F1on channel segment a. Therefore, at least one of the connections
+ofF′
+1cuts all the channels between ch1andch2, that is, all the channels where the
+sequenceof extendedformationsis placed.Sucha connectio nseparatesthe verticesof
+F1from the vertices of a formation F2∈EF2(H1)on channel segment a. Therefore,
+at least one of the connections of F2to channel segment acuts either all the channels
+in channel segment aor all the channels in channel segment a+ 1(ora−1), hence
+becoming a blocking cut for such channels. It follows that al l the formations nesting
+insideF2on such channels can not place vertices in the bending area b(a,a+ 1)(or
+b(a−1,a))outsideF2. /square
+27Lemma12 .InasituationasdescribedinProposition2,notalltheexte ndedformations
+in a sequence of extended formations can place turning verti ces in the same channel
+segment.
+Proof:Assume, for a contradiction, that all the turning vertices a re in the same
+channel segment. Consider a sequence of extended formation sSEFand the extended
+formationsin SEFusingoneofthesetsofchannels {H1,...,H 4}.
+We first show that in SEFthere exist some extended formations using connec-
+tionsinαconfigurationandsomeusingconnectionsin βconfigurationonthechannels
+{H1,...,H 4}.Considerthecontinuoussubsequenceofextendedformatio nsEF(H1),
+...,EF(H3)inSEF. Assume that all the turning vertices of these extended form a-
+tions are in αconfiguration. Consider a further subsequence of SEFon the same set
+ofchannelswitha defectat H2. Then,the connectionbetween H1andH3crossesH2,
+therebyblockingany further EF(H2)from beingin αconfiguration.Therefore,when
+considering another subsequence of SEFon the same set of channelswhich does not
+contain defects at H1,...,H 3, either the extended formation EF(H2)is inβconfig-
+uration or it uses another channel segment to place the turni ng vertex, as stated in the
+lemma.
+So, consider two channels H1,H2such that there exists an extended formation
+EF(H1)inαconfiguration and an extended formation EF(H2)inβconfiguration.
+Sincealltheextendedformationscontainatriangleopenon onesidethatisnestedwith
+trianglet, we consider five of such triangles, one for each set of channe lsH2,H3,H4
+andtwoforset H1,suchthatfouroftheconsideredextendedformations EF(H1),...,
+EF(H4)are continuousin SEFand the other one EF′(H1)is the first extendedfor-
+mationontheset ofchannels H1following EF(H4)inSEF.
+(a) (b)
+Fig.26.(a)Twotrianglesfromthesamechannelhavetousedifferent channelsegments
+if a triangle of another channel is between them. Turning ver tices are represented by
+blackcircles.(b)Whenadefectat H2inencountered,theconnectionbetween EF(H1)
+andEF(H3)does not permit the following EF(H2)to respect the ordering of trian-
+gles.
+28Noticethat,ifatriangleofanextendedformation EF(Hk)isnestedinatriangleof
+an extendedformation EF(Hs)andthe triangleof EF(Hs)is nestedinside a triangle
+of an extended formation EF′(Hk), withk < s, thenEF(Hk)has to use a different
+channel segment to place its turning vertex (see Fig. 26(a)) . Hence, the triangles have
+to be ordered accordingto the order of the used channels. Als o, if the continuouspath
+connecting two triangles t1= (u,v,w),t2= (u′,v′,w′)of consecutive extended for-
+mationsEF(Hs),EF(Hs+1)connectsvertex uto vertex w′(oru′tow) via the outer
+area,thenatriangleof EF(H1)thatoccurspriorto EF(Hs)andatriangleof EF′(H1)
+that occursafter EF(Hs+1)are nested with the triangle given by the connection of t1
+andt2inanorderingthatis differentfromtheorderofthe channel s.
+Consider now the followingsubsequenceof SEFhaving a defect at H2. The con-
+nection of EF(H1)toEF(H3)in this subsequence blocks access for the following
+EF(H2)totheareawhereit wouldhavetoplaceverticesinordertore specttheorder-
+ing of triangles (see Fig. 26(b)). Therefore, after 3 full re petitions of the sequence in
+SEF, at least one extended formation has to use a different chann el segment to place
+itsturningvertex. /square
+Lemma 13 .In a situation as described in Proposition 2, TandPdo not admit any
+geometricsimultaneousembedding.
+Proof:Consider two extended formations EF(Hx),EF(H1)that are consecutive
+inSEF. First note that the connection between EF(Hx)andEF(H1)cuts all chan-
+nels2,...,x−1in either channel segment cs1orcs2. Since both of these extended
+formations are also connected to the bending area between ch annel segments cs3and
+cs4, it is not possible for an extended formation EF(s), withs∈ {2,...,x−1}, to
+connectfromverticesabovetheconnectionbetween EF(Hx)andEF(H1)tovertices
+below it by following a path to the bending area. Note, furthe r, that if all the extended
+formations EF(s), withs∈ {2,...,x−1}, are in the channel segment that is not
+cut by the connection between EF(1)andEF(x), then a connection is needed from
+cs1tocs2inchannel x.However,byLemma12,afterthreedefectsinthesubsequenc e
+of{2,...,x−1}it is no longer possible for some extended formation EF(s), with
+s∈ {2,...,x−1},toplaceitsturningvertexinthesamechannelsegment.The refore,
+different channel segments have to be used by the extended fo rmationEF(s), with
+s∈ {2,...,x−1}. However, since the path is continuous and since the connect ion
+betweenEF(Hx)andEF(H1)isrepeatedafteracertainnumberofsteps,we canfol-
+low that the path creates a spiral. Also, we note that, in orde r to respect the order of
+the sequence, it will be impossible for the path to reverse th e direction of the spiral.
+Hence,once a directionof the spiral hasbeenchosen,either inwardor outward,all the
+connectionsintheremainingpartofthesequencehavetofol lowthesame.Thisimplies
+that,if a connectionbetween EF(s)andEF(s+1)changeschannelsegment,that is,
+it is performed in a different channel segment than the one be tweenEF(s−1)and
+EF(s),thenalltheconnectionsofthistypehavetochange.Howeve r,whenadefectat
+channels+1isencountered,alsotheconnectionbetween EF(s)andEF(s+2)hasto
+changechannelsegment,therebymakingimpossibleforanyf utureconnectionbetween
+EF(s)toEF(s+1)tochangechannelsegment.Therefore,afterawholerepetit ionof
+the sequence of SEFcontaining defects at each channel, all the extended format ions
+havetoplacetheirturningverticesin thesamechannelsegm ent,whichisnotpossible,
+29by Lemma 12. This concludes the proof that no valid drawing ca n be achieved in this
+configuration. /square
+Lemma 14 .If a shape contains an intersection I(1,3)and does not contain any other
+intersectionthatisdisjointwith I(1,3),thenTandPdonotadmitanygeometricsimul-
+taneousembedding.
+Proof:Firstobservethatonlytheintersections I(2,4)andI(1,4)arenotdisjointwith
+I(1,3)and could occur at the same time as I(1,3). By Lemma 8, there exists at least
+a nesting greater than, or equal to, 6. Each of such nestings h as to take place either
+at intersections I(1,3),I(2,4)or atI(1,4). Remind that, by Property 4, 1-vertices can
+only be placed in cs1orcs2. Also, the sorting of head vertices to avoid a region-level
+nonplanartrees can only be done by placing vertices into cs3orcs4. This implies that
+the stabilizers have to be placed in cs1orcs2. Note that the stabilizers also work as
+1-vertices in the tails of other cells. This means that if ther e exist seven sets of tails
+that can be separated by straight lines, then there exist a re gion-level nonplanar tree,
+by Lemma 6. Observe that, by nesting them according to the seq uence, the previous
+conditionwouldbefulfilled.Thismeansthatwehaveeithera sortingorothernestings.
+We first show that there exist at most two x-nestings with x≥6. Everyx-nesting has
+to take place at either I(1,3),I(2,4)orI(1,4). We assume, w.l.o.g., to have to deal with
+thegreatestpossiblenumberofintersections.
+Consider the case Ih
+(2,4)(see Fig 27(a)). Observe that intersections I(1,4)andI(1,3)
+are either both high or both low and use channel segment cs1. Also, every connection
+fromcs1tocs4cuts either cs2orcs3and, if one of these connections cuts cs2, then
+everynestingcutting cs1closertob(1,2)hastocut cs2.Hence,wecanconsiderallthe
+connections to cs4as connections to cs2orcs3. Also, since any connection cutting a
+channelsegmentismorerestrictivethanaconnectioninsid ethesamechannelsegment,
+such two nestings can be considered as one. Finally, since su ch a nesting connects to
+b(2,3),it isnotpossibleto haveat the sametimea nestingtakingpl aceatIh
+(2,4).Hence,
+we concludethatonlyonenestingispossibleinthiscase.
+1441
+(a) (b)
+Fig.27.(a)CaseI(1,3)Ih
+(2,4). (b)Case I(1,3)Il
+(2,4).
+30Consider the case Il
+(2,4)(see Fig 27(b)). Observe that 1-vertices can be placed at
+most incs2and2-vertices can be placed at most in cs3. This means that the extended
+formationsin everynestinghavetovisit thesevertices.Th erefore,if thereexistsbotha
+nestingat I(1,3)andatI(1,4),thentheconnectionstothe1-and2-verticesinthebending
+areasb(2,3)andb(3,4)are such that every EF nesting at I(1,4)makes a nesting with
+the extended formations nesting at I(1,3). Hence, also in this case only one nesting is
+possible.
+So we consider the unique nesting of depth x≤6and we show that any way of
+sortingthenestingformationsinthechannelswillcausese paratedcells,henceproving
+the existence of a nonplanarregion-leveltree. Consider fo ur consecutiverepetitionsof
+the sequence of formations. It is clear that these formation s are visiting areas of cs1
+and are separated by previously placed formations from othe r formations on the same
+channels.Thiswillresultinsomecellstobecomeseparated incs1.Since,byProperty2,
+the numberof monotonicallyseparated cells in cs1cannotbe largerthan 3, for any set
+of four such separated formations there exists a pair of form ationsF1,F2that change
+their order in cs1. These connections have to be made on either side of the nesti ng. If
+between this pair of formations there is a formation of a diff erent channel, then this
+formation has to choose the other side to reorder with a forma tion outside F1,F2. We
+furthernote that, if there are two such connections F1,F4andF2,F3on the same side
+that are connecting formations of one channel, nested in the orderF1,F2,F3,F4, and
+another connection on the same side between F′
+1,F′
+2such that F′
+1is nested between
+F1,F2andF′
+2betweenF3,F4, then this creates a 1-nesting.In the followingwe show
+thata nestingofdepthatleast 6 isreached.
+Assume the repetitionsof formationsin the extendedformat ionto be placed in the
+ordera,b,c,d,e .Ifthisorderisnotcoherentwiththeorderinwhichthechan nelsappear
+in the sequence of formations inside the EF, then we have already some connections
+that are closing either side of the nesting for some formatio ns. So we assume them to
+be in the order given by the sequence. Then, consider a repeti tion of formations with
+a defect at some channel Ci. We have that there exists a connection closing off at one
+side all the previously placed formations of Ci. However, there are sequences with
+defects also at channels Ci+1andCi−1, which can not be realized on the same side
+as the defects at Ci. We generalize this to the fact that all the defects at odd cha nnels
+are to one side, while the defects at even channels are to the o ther side. Since the path
+is continuous and has to reach from the last formation in a seq uence again to the first
+one, the continuation of the path can only use either the odd o r the even defects. This
+implies that, when considering three further repetitions o f formations, the first and the
+thirdhavingadefectatachannel Ciandthesecondhavingnodefectat Ci,therewillbe
+anestingofdepthonebetweenthesethreeformations.Since ,byLemma9,therecannot
+be a nesting of depth greater than 5 at this place, we conclude that after 6 repetitions
+of such a triple of formations there will be at least two forma tions that are separated
+fromeachother.Byrepeatingthisargumentwearriveafter 7·6·2repetitionsateither
+the existence of 7 formations that are separated on cs1andcs2or at the existence of
+a nesting of depth 6, both of which will not be drawable withou t the aid of another
+intersectionthatis abletosupportthesecondnestingofde pthgreaterthan5. /square
+31Lemma15 .Ifthereexistsasequenceofextendedformationinanyshape containingan
+intersection I(3,1),thenTandPdonotadmitanygeometricsimultaneousembedding.
+Proof:Consider a sequence of extended formation in a shape contain ing an inter-
+sectionI(3,1).Weshowthat TandPdonotadmitanygeometricsimultaneousembed-
+ding.Observethatthereexistseveralpossibilitiesforch annelsegment cs4tobeplaced.
+Eitherthereexistsnointersectionofanelongationofonec hannelsegmentwithanother
+channel segment or there exists at least one of the intersect ionsI(1,4),I(4,2),I(4,1)or
+I(2,4).Iftherearemorethanoneofsuchintersections,thenitisp ossibletohaveseveral
+nestingsofdepth x. We notethat,ifthereexiststheintersection I(3,1),thenat leastone
+ofcs1,cs2,andcs4arepartoftheconvexhull(see Fig.28).
+1331
+(a) (b)
+Fig.28.If channel segment four is not part of the convexhull then eit hercs1orcs2is
+partoftheconvexhull.(a)Case Il
+(1,3).(b)Case Ih
+(1,3).
+First,we showthatthereexistsa nestingat I(3,1).
+Consider case Ih
+(3,1). We have that cs2is on the convex hull restricted to the first
+three channel segments and cs4can force at most one of cs2orcs1out of the convex
+hull.Hence,oneofthemispartoftheconvexhull.We disting uishthetwocases.
+Supposecs2to be part of the convex hull. Assume there exists a nesting at I(1,4).
+Fromcs4the only possible connection without a 1-side connection is the one to cs2,
+which, however, is on the convex hull. Hence, an argument ana logous to the one used
+inLemma14provesthatthenestingat I(2,4)hassizesmallerthan 7∗12,whichimplies
+thatthe restofthenestinghasto takeplaceat I(3,1).
+Supposecs1to be part of the convex hull. Assume that there exists a nesti ng at
+I(2,4). Everyconnectionfrom cs4hastobeeitherto cs1ortocs2,byProperty4.Since
+cs2is already part of the nesting, we have connections to cs1. However, cs1is on the
+convexhull,henceallowingonly 1-sideconnections.Therefore,anargumentanalogous
+totheoneusedinLemma14provesthatthenestingat I(2,4)hassizesmallerthan 7∗12,
+whichimpliestherest ofthenestinghastotake placeat I(3,1).
+Consider case Il
+(3,1). Sincecs2is not part of the convexhull, either cs1orcs4are.
+Ifcs1is on the convexhull, then the same argumentas beforeholds, while ifcs4is on
+theconvexhull,thennoreorderingispossible.
+Clearly,ifthereisnointersectionotherthan I(3,1),anestingintheintersection I(3,1)
+hasto beperformed.
+32Hence,weconcludethatanestingwithadepthof 7∗12ineveryextendedformation
+hasto takeplaceat I(3,1)(oratI(4,1),whichcanbeconsideredasthesame case).
+ByLemma11,thenestinginthebendingareaislimited.Every extendedformation
+EFwhich has at least one vertex either in cs3or incs4has a vertex in the bending
+area. Consider a sequence of extended formations SEFwhich uses only channels in
+this particular shape. It’s obvious that all of these EFinSEFhave to do a nesting
+atI(3,4,1). Observethat thereexist two consecutiveedgeswhich are fo rminga triangle
+withcs1,cs2, andcs3by simply placing vertices inside the channel segments. Sin ce
+everyEF createssuch triangles, there exists a triangle whi ch is not in the bendingarea
+and such that there exists no other triangle between the bend ing area and this triangle.
+This triangle is separating the nesting area from the bendin g area in all but sextended
+formations.However,sinceeveryEFhastousebothofsuchar eas,theinnerareaof cs3
+(orcs4) has to connect to the outer area of cs3(orcs4). Ifcs1is on the convex hull,
+then there exist only 1-sided connections, which implies the statement, by Lemma 1 3.
+On the other hand, if cs1is not on the convexhull, then there exists I(1,4)andcs4can
+be also used to perform connections from the inner to the oute r area. However, since
+cs4is on the convex hull, such connectionsare only 1-side. Hence, by Lemma 13, the
+statementfollows. /square
+Lemma 16 .Letcsiandcsi+1be two consecutivechannelsegments. If there exists an
+orderedset S:= (1,2,...,5)3of extremal doublecutscutting csiandcsi+1such that
+the order of the intersections of the double cuts with csi(withcsi+1) is coherent with
+theorderof S,thenTandPdonotadmitanygeometricsimultaneousembedding.
+Proof:Suppose, for a contradiction, that such a set Sexists. Assume first that csi
+andcsi+1are such that the bendpoint of channel 5encloses the bendpoint of all the
+other channels. Hence, any edge creating a double cut at a cha nnelchas to cut all the
+channelsc′withc′> c,eitherin csiorincsi+1. RefertoFig.29.
+Consider the first repetition (1,2,...,5). Lete1be an edge creating a double cut
+at channel 1. Assume, without loss of generality, that e1cuts channel segment csi.
+Observethat,forchannel 1,thevisibilityconstraintsdeterminedinchannels 2,...,5in
+csiandincsi+1by the doublecut created by e1do not dependon whetherit is simple
+ornon-simple.Indeed,byProperty6,edge e1blocksvisibilityto b(i,i+1)forthepart
+ofcsiwhere edges creating doublecuts at channels 2,...,5following e1inShave to
+placetheirend-vertices.
+Then, consider an edge e3creating a double cut at channel 3in the first repetition
+of(1,2,...,5).
+Ife3cutscsi(seeFig.29(a)),thenithastocreateeitheranon-simpledo ublecutor
+a simple one.However,in the latter case, an edge e′
+3betweencsiandcsi+1in channel
+3,whichcreatesablockingcutinchannel 2,isneeded.Hence,inbothcases,channel 2
+iscutbothin csiandincsi+1,eitherby e3orbye′
+3.Itfollowsthatanedge e2creatinga
+doublecut at channel 2in the secondrepetitionof (1,2,...,5)hasto cut csi+1, hence
+blockingvisibility to b(i,i+1)for the part of csi+1where edgescreating doublecuts
+at channels 3,...,5following it in Shave to place their end-vertices, by Property 6.
+Further,consideran edge e5creatinga doublecut at channel 5in the secondrepetition
+of(1,2,...,5). Since visibility to b(i,i+ 1)is blocked by e1ande3incsiand bye2
+incsi+1,e2hastocreateanon-simpledoublecut(orasimpleoneplusabl ockingcut),
+335e
+3e’2e
+3e
+i+1csics1e
+13553
+15e3e
+i+1csics1e
+31553
+1
+(a) (b)
+Fig.29.ProofofLemma16. (a) e3cutscsi. (b)e3cutscsi+1.
+hencecuttingchannel 4bothincsiandincsi+1.Itfollowsthat,byProperty5,anedge
+e4creatingadoublecutatchannel 4inthethirdrepetitionof (1,2,...,5)canplaceits
+end-vertexneitherin csinorincsi+1.
+Ife3cutscsi+1(see Fig. 29(b)), then it has to create a simple double cut. Ag ain,
+by Property6, edge e3blocksvisibility to b(i,i+1)for the part of csi+1where edges
+creatingdoublecutsfollowing e3inShaveto placetheir end-vertices.Hence,an edge
+e5creatingadoublecutatchannel 5inthefirstrepetitionof (1,2,...,5)cannotcreate
+a simple double cut, since its visibility to b(i,i+1)is blocked by e1incsiand bye3
+incsi+1. This implies that e5creates a non-simple double cut (or a simple one plus
+a blocking cut) at channel 5, cutting either csiorcsi+1, hence cutting channel 4both
+incsiand incsi+1. It follows that, by Property 5, an edge e4creating a double cut at
+channel4inthesecondrepetitionof (1,2,...,5)canplaceitsend-vertexneitherin csi
+norincsi+1.
+Thecaseinwhich csiandcsi+1aresuchthatthebendpointof 1enclosesthebend-
+point of all the other channelscan be provedanalogously.Na mely,the same argumen-
+tation holds with channel 5playing the role of channel 1, channel 1playing the role
+of channel 5, channel 3having the same role as before, channel 4playing the role of
+channel2, and channel 2playing the role of channel 4. Observe that, in order to ob-
+taintheneededorderinginthissetting, 3repetitionsof (1,2,...,5)areneeded.Infact,
+we consider channel 5in the first repetition, channels 3and4in the second one, and
+channels1and2in thethirdone. /square
+Lemma 17 .Each extended formation in shape Ih
+(1,3)Ih
+(4,2)creates double cuts in at
+least onebendingarea.
+Proof:Refer to Fig. 30(a). Assume, without loss of generality, tha t the first bend-
+point of channel c1encloses the first bendpoint of all the other channels. This i mplies
+that the second and the third bendpointsof channel c1are enclosed by the second and
+thethirdbendpointsofall theotherchannels,respectivel y.
+Suppose,foracontradiction,thatthereexistsnodoublecu tinb(2,3)andinb(3,4).
+Hence, any edge econnecting to b(2,3)(tob(3,4)) is such that eand its elongation
+cut each channel once. Consider an edge connecting to b(2,3)in a channel ci. Such
+341
+432
+1c rr
+43
+12
+(a) (b)
+Fig.30.(a)Shape Ih
+(1,3)Ih
+(4,{1,2})hastoconnectat leastonebendwithdoublecuts.(b)
+ShapeIh
+(1,3)Il
+(4,2)hastoconnectbend b(2,3)withdoublecuts.
+an edge creates a triangle together with channel segments 3and4of channel ciwhich
+enclosesthe bendingareas b(3,4)ofall the thechannels chwithh < ibycuttingsuch
+channels twice. Hence, a connection to such a bending area in one of these channels
+hastobeperformedfromoutsidethetriangle.However,sinc einshape Ih
+(1,3)Ih
+(4,2)both
+thebendingareas b(2,3)andb(3,4)areontheconvexhull,thisisonlypossiblewitha
+doublecut,whichcontradictsthe hypothesis. /square
+Lemma 18 .Every sequenceof extendingformationsin shape Ih
+(1,3)Ih,l
+(4,2)containsan
+ordered set (1,2,...,5)3of extremal double cuts with respect to bending area either
+b(2,3)orb(3,4).
+Proof:ShapeIh
+(1,3)Ih
+(4,2)issimilartoshape Ih
+(1,3)Ih
+(4,1),depictedinFig.30(a),with
+theonlydifferenceontheslopeofchannelsegment 4,whichissuchthatitselongation
+crosseschannelsegment 2andnotchannelsegment 1.ShapeIh
+(1,3)Il
+(4,2)isdepictedin
+Fig.30(b).
+Assume,withoutlossofgenerality,thatthefirstbendpoint ofchannel c1isenclosed
+by the first bendpointof all the other channels. This implies that the second bendpoint
+ofchannel c1enclosesthesecondbendpointofall theotherchannels.
+First observe that bending area b(2,3)is on the convex hull, both in shape Ih
+(1,3)
+Ih
+(4,2)andinshape Ih
+(1,3)Il
+(4,2).
+Also, observe that all the extendedformationshave some ver tices inb(2,3)and in
+b(3,4), and hence all the extended formations have to reach such ver tices with path-
+edges.
+In shapeIh
+(1,3)Ih
+(4,2), by Lemma 17, there exist double cuts either in b(2,3)or in
+b(3,4), while in shape Ih
+(1,3)Il
+(4,2)there exist double cuts in b(2,3), since the only
+possible connections to b(2,3)are from channel segments 1and4, which are both
+creating double cuts (see Fig. 30(b)). Hence, we consider th e extremal double cuts of
+eachextendedformationwith respecttooneof b(2,3)orb(3,4),sayb(2,3).
+35Considertwosetsofextendedformationscreatingdoublecu tsinb(2,3)atchannels
+1,...,5, respectively.Observe that the extended formationsin the se two sets could be
+placedinsuchawaythattheorderingoftheirextremaldoubl ecutsis(1,1,2,2,...,5,5).
+The same holds for the following occurrences of extended for mations creating dou-
+ble cuts in b(2,3)at channels 1,...,5, respectively. Clearly, in this way an ordering
+(1n,2n,...,5n)could be achieved and hence an ordered set (1,2,...,5)3of double
+cutswouldbeneverobtained(seeFig.31(a)).
+However, every repetition of extended formations inside a s equence of extended
+formationscontainsa doubledefect at some channel.We show ,with an argumentsim-
+ilar to the one used in Lemma 8, that the presence of such doubl e defects determines
+an ordering (1,2,...,5)3of extremal doublecuts after a certain numberof repetition s
+of extended formations inside a sequence of extended format ions. Namely, consider
+a double defect at channel iin a certain repetition. The connection between channels
+i−1andi+2cannotbeperformedinthesameareaastheconnectionbetwee nchannels
+i−1andiandbetweenchannels iandi+1wasperformedinthepreviousrepetition.
+Hence,suchaconnectionhastobeperformedeitherinthesam eareaastheconnection
+betweenchannels i+1andi+2wasperformed(seeFig.31(b)),orinchannelsegment
+4(this is only possible in shape Ih
+(1,3)Il
+(4,2), see Fig. 31(c)). Observe that, going to
+channelsegment 4tomaketheconnection,thentochannelsegment 1,andfinallyback
+tob(2,3),hencecreatingaspiral,impliesthattheconsidereddoubl ecutisnotextremal
+(see Fig. 31(d)).Therefore,the only possibility to consid er when channelsegment 4is
+usedistomaketheconnectionbetweenchannels i−1andi+2thereandthentocome
+backtob(2,3)with a doublecut.Hence, independentlyon whetherchannels egment4
+isusedornot,theconnectionbetweenchannels i−1andi+2blocksvisibilityforthe
+following repetitions to the areas where the connections be tween some channels were
+performedin thepreviousrepetition.Thisimpliesthat the ordering(1n,2n,...,5n)of
+extremal double cuts cannot be respected in the following re petitions. In fact, a partial
+order(i,i+1,i+2)2is obtained in a repetition of formationscreating extremal dou-
+blecutsatchannels 1,...,5.Also,whentwodifferentdoubledefectshavingachannel
+in commonare considered,the effect of such defects is combi ned.Namely, consider a
+double defect at channel 3in a certain repetition. The connection between channels 2
+and5blocksvisibility to the areas wherethe connectionbetween 2and3andbetween
+3and4were performed at the previous repetitions (see Fig. 32(a)) . Then, consider a
+double defect at channel 1in a following repetition. We have that the connection be-
+tween channels 0and3can not be performed where the connection between 2and3
+was performed in the previous repetitions, since such an are a is blocked by the pres-
+enceofthe connectionbetweenchannels 2and5.Hence,a doublecutat channel 3has
+to be placed after the double cut at channel 5created in the previous repetition (see
+Fig. 32(b)). Consider now a further repetition with a defect not involvingany of chan-
+nels1,...,5. We have that the area where the connection from 1to2was performed
+in the previous repetitions is blocked by the connection bet ween0and3and hence a
+doublecutat channel 1hastobe placedafterthedoublecut atchannel 3createdinthe
+previousrepetition,which,initsturn,wascreatedaftert hedoublecutatchannel 5(see
+Fig.32(c)).
+36135
+2
+31135
+2
+31
+(a) (b)
+31
+4135
+2
+31
+4135
+2
+(c) (d)
+Fig.31.(a)The orderingof theextremaldoublecutsis (1,1,2,2,...,5,5). (b)and(c)
+Whenadoubledefectisencountered,theconnectionbetween channelsi−1andi+2
+cannot be performed in the same area as the connection betwee n channels i−1and
+iand between channels iandi+ 1was performed in the previous repetition: (b) The
+connection is performed in the same area as the connection be tween channels i+ 1
+andi+2was performed.(c) Theconnectionis performedinchannelse gment4. (d)If
+channelsegmentfourisusedto spiral,theconsidereddoubl ecutwasnotextremal.
+371
+2
+315
+3
+1
+2
+315
+3
+(a) (b)
+1
+2
+315
+3
+1
+2
+315
+3
+2345
+112345
+(c) (d)
+Fig.32.(a)Arepetitionwithadoubledefectinchannel 2isconsidered.(b)Arepetition
+with a double defect in channel 0is considered. (c) A repetition without any double
+defectinchannels 1,...,5is considered.(d)An orderedset (1,...,5)isobtained.
+38Also, all the double cuts at channels 2,...,5have to be placed after the double
+cut at1, and hence a shift of the whole sequence 1,...,5after the double cut at 5is
+performed and an ordered set (1,2,...,5)2is obtained (see Fig. 32(d)). Observe that
+at most two sets of repetitions of extended formation inside a sequence of extended
+formations such that each set contains a double defect at eac h channel are needed to
+obtain such a shift. By repeating such an argument we obtain a nother shifting of the
+whole sequence (1,...,5), which results in the desired ordered set (1,2,...,5)3. We
+have that a set of repetitions of extended formation contain ing a double defect at each
+channel is needed to obtain the first sequence (1,2,...,5)2, then two of such sets are
+neededto getto (1,2,...,5)2, andtwomoreareneededtoget to (1,2,...,5)3, which
+provesthestatement.
+Observethat,ifitwerepossibletopartitionthedefectsin totwosetssuchthatthere
+existsnopairofdefectsinvolvingacommonchannelinsidet hesameset,thensuchsets
+could be independently drawn inside two different areas and the effects of the defects
+could not be combined to obtain (1,2,...,5)3. However, since each double defect in-
+volvestwoconsecutivechannels,atleastthreesetsarenee dedtoobtainapartitionwith
+such a property. In that case, however, an ordered set (1,2,...,5)3could be obtained
+bysimplyconsideringarepetitionof (1,2,...,5)ineachofthesets. /square
+Lemma 19 .If channel segment cs2is part of the convex hull, then TandPdo not
+admitanygeometricsimultaneousembedding.
+Proof:Firstobservethat,withanargumentanalogoustotheoneuse dinLemma14,
+itispossibletoshowthatthereexistsanestingatintersec tionI(4,1,2).Then,byProp-
+erty 4, every vertex that is placed in cs4is connected to two vertices that are placed
+either in cs1or incs2. Hence, the continuous path connecting to a vertex placed in
+cs4creates a triangle, having one corner in cs4and two corners either in cs1or in its
+elongation,whichcuts cs4intotwoparts,theinnerandtheouterarea.
+By Lemma 11, not all of these triangles can be placed in the ben ding area b(3,4).
+Hence, every extended formation, starting from the second o f the sequence, have to
+place their vertices in both the inner and the outer area of th e triangle created by the
+first one.
+Observethat,inordertoconnecttheinnertotheouterarea, theextendedformations
+canonlyuse 1-sideconnections.Namely, cs1createsa1-sideconnection.Channelseg-
+mentcs2is on the convexhull. Since, by Property4, every vertexthat is placed in cs3
+is connected to two vertices that are placed either in cs1or incs2, alsocs3creates a
+1-sideconnection.
+Fromthis we concludethat in this configurationthe precondi tionsof Proposition2
+aresatisfied, andhencethestatementfollows. /square
+6 An Algorithm for the Geometric Simultaneous Embedding of a
+Tree ofDepth 2and aPath
+Inthissectionwedescribeanalgorithmforconstructingag eometricsimultaneousem-
+beddingofanytree Tofdepth2andanypath P. RefertoFig. 33.
+39Startby drawingthe root rofTonthe originin a coordinatesystem. Choosea ray
+R1emanating from the origin and entering the first quadrant, an d a rayR2emanating
+from the origin and entering the fourth quadrant. Consider t he wedge Wdelimited by
+R1andR2and containing the positive x-axis. Split WintotwedgesW1,...,W t, in
+this clockwise order around the origin, where tis the number of vertices adjacent to r
+inT,byemanating t−2equispacedraysfromtheorigin.
+Then,considerthetwosubpaths P1andP2ofPstartingat r.Assignanorientation
+toP1andP2suchthatthetwoedges (r,u)∈ P1and(r,v)∈ P2incidentto rinPare
+exitingr.
+Finally,considerthe tsubtreesT1,...,TtofTrootedat a nodeadjacentto r, such
+thatu∈ T1andv∈ Tt.
+3uy
+rx
+v
+RWR1
+2
+21
+W
+W
+Fig.33.A tree with depth two and a path always admit a geometric simul taneous em-
+bedding.
+Theverticesofeachsubtree Tiaredrawninsidewedge Wi,in suchawaythat:
+1. vertex uisthe vertexwith thelowest x-coordinateinthedrawing,exceptfor r;
+2. vertices belonging to P1are placed in increasing order of x-coordinate according
+to theorientationof p1;
+3. vertex visthevertexwiththe highest x-coordinatein thedrawing;
+4. verticesbelongingto P2\rareplacedindecreasingorderof x-coordinateaccording
+to the orientation of p2, in such a way that the leftmost vertex of P2\ris to the
+rightoftherightmostvertexof P1;and
+5. novertexisplacedbelowsegment rv.
+40SinceThas depth 2, each subtree Ti, withi= 1...,t, is a star. Hence, it can be
+drawn inside its own wedge Wiwithout creating any intersection among tree-edges.
+Observe that the same holds even for subtree Tt, where the wedge to consider is the
+partofWtabovesegment rv.
+SinceP1andP2\{r}aredrawninmonotonicorderof x-coordinateandaresepa-
+rated from each other, and edge (r,v)connectingsuch two pathsis on the convexhull
+ofthepoint-set,nointersectionamongpath-edgesiscreat ed.
+Fromthediscussionabove,we havethefollowingtheorem.
+Theorem2. A tree of depth 2and a path always admit a geometric simultaneousem-
+bedding.
+7 Conclusions
+Inthispaperwehaveshownthatthereexistatree Tandapath Ponthesamesetofver-
+ticesthatdonotadmitanygeometricsimultaneousembeddin g,whichmeansthatthere
+exists no set of points in the plane allowing a planar embeddi ngof both TandP. We
+obtained this result by extending the concept of level nonpl anar trees [8] to the one of
+region-levelnonplanartrees.Namely,weshowedthatthere existtreesthatdonotadmit
+anyplanarembeddingiftheverticesareforcedtolieinside particularlydefinedregions.
+Then, we constructed TandPso that the path creates these particular regions and at
+leastoneofthemanyregion-levelnonplanartreescomposin gThasitsverticesforced
+to lie inside them in the desired order. Observe that our resu lt also implies that there
+exist twoedge-disjointtreesthatdonotadmitanygeometri csimultaneousembedding,
+whichanswersanopenquestionposedin[12],wherethecaseo ftwonon-edge-disjoint
+treeswassolved.
+It is important to note that, even if our counterexample cons ists of a huge number
+of vertices, it can also be considered as “simple”, in the sen se that the depth of the
+tree is just 4. In this direction, we proved that, if the tree has depth 2, then it admits
+a geometric simultaneous embedding with any path. This give s raise to an intriguing
+open question about whether a tree of depth 3and a path always admit a geometric
+simultaneousembeddingornot.
+References
+1. U. Brandes, C. Erten, J. Fowler, F. Frati, M. Geyer, C. Gutw enger, S.-H. Hong, M. Kauf-
+mann,S.Kobourov, G.Liotta,P.Mutzel,andA.Symvonis. Col oredsimultaneousgeometric
+embeddings. In G. Lin, editor, International Computing and Combinatorics Conference
+(COCOON) ,volume 4598 of LNCS,pages 254–263, 2007.
+2. P.Brass,E.Cenek,C.Duncan,A.Efrat,C.Erten,D.Ismail escu,S.Kobourov,A.Lubiw,and
+J.Mitchell. Onsimultaneousplanargraphembeddings. ComputationlGeometry ,36(2):117–
+130, 2007.
+3. E. Di Giacomo, W. Didimo, M. van Kreveld, G. Liotta, and B. S peckmann. Matched draw-
+ings of planar graphs. In S.-H. Hong, T. Nishizeki, and W. Qua n, editors, International
+Symposium onGraph Drawing(GD) ,volume 4875 of LNCS,pages 183–194, 2007.
+414. C. Erten and S. Kobourov. Simultaneous embedding of plana r graphs with few bends. In
+J. Pach, editor, International Symposium on Graph Drawing (GD) , volume 3383 of LNCS,
+pages 195–205, 2004.
+5. A. Estrella-Balderrama, J. Fowler, and S. Kobourov. Char acterization of unlabeled level
+planar trees. In M. Kaufmann and D. Wagner, editors, International Symposium on Graph
+Drawing(GD) ,volume 4372 of LNCS,pages 367–379, 2006.
+6. J. Fowler, M. Jünger, S. G. Kobourov, and M. Schulz. Charac terizations of restricted pairs
+of planar graphs allowingsimultaneous embedding withfixed edges. InH.Broersma, T.Er-
+lebach,T.Friedetzky,andD.Paulusma,editors, InternationalWorkshoponGraphTheoretic
+Concepts inComputer Science (WG) ,volume 5344 of LNCS,pages 146–158, 2008.
+7. J.FowlerandS.Kobourov. Characterizationofunlabeled levelplanargraphs. InS.-H.Hong,
+T. Nishizeki, andW. Quan, editors, International Symposium onGraph Drawing (GD) ,vol-
+ume 4875 of LNCS,pages 37–49, 2007.
+8. J. Fowler and S. Kobourov. Minimum level nonplanar patter ns for trees. In S.-H. Hong,
+T. Nishizeki, andW. Quan, editors, International Symposium onGraph Drawing (GD) ,vol-
+ume 4875 of LNCS,pages 69–75, 2007.
+9. F.Frati. Embedding graphs simultaneously withfixed edge s. InM. Kaufmann andD. Wag-
+ner, editors, International Symposium on Graph Drawing (GD) , volume 4372 of LNCS,
+pages 108–113, 2006.
+10. F. Frati, M. Kaufmann, and S. Kobourov. Constrained simu ltaneous and near simultaneous
+embeddings. In S.-H. Hong, T. Nishizeki, and W. Quan, editor s,International Symposium
+onGraph Drawing(GD) ,volume 4875 of LNCS,pages 268–279, 2007.
+11. E.Gassner, M.Jünger,M. Percan,M.Schaefer,andM. Schu lz. Simultaneous graphembed-
+dings with fixed edges. In F. V. Fomin, editor, International Workshop on Graph Theoretic
+Concepts inComputer Science (WG) ,volume 4271 of LNCS,pages 325–335, 2006.
+12. M. Geyer, M. Kaufmann, and I. Vrt’o. Two trees which are se lf-intersecting when drawn
+simultaneously. Discrete Mathematics , 309(7):1909 –1916, 2009.
+13. R. L. Graham, B. L. Rothschild, and J. H. Spencer. Ramsey Theory . John Wiley & Sons,
+1990.
+14. J. Pach and R. Wenger. Embedding planar graphs at fixed ver tex locations. Graphs and
+Combinatorics , 17(4):717–728, 2001.
+15. C.Thomassen. Embeddings of graphs. Discrete Mathematics , 124(1-3):217–228, 1994.
+16. W. T. Tutte. How to draw a graph. In London Mathematical Society , volume 13, pages
+743–768, 1962.
+42
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+arXiv:1001.0556v1  [math.NA]  4 Jan 2010THE DISCRETE ANALOGUE OF THE OPERATORd2m
+dx2mAND
+ITS PROPERTIES
+Kh.M.Shadimetov
+Abstract
+In this paper the discrete analogue Dm[β] of the differential operator d2m/dx2mis constructed
+and its some new properties are proved.
+Key words and phrases: Discrete function, discrete analogue of the differential op erator, Euler
+polynomial.
+1 Main results.
+First S.L.Sobolev [1] studied construction and investigat ed properties of the operator D(m)
+hH[β], which
+is inverse of the convolution operator with function G(m)
+hH[β] =hnGm(hHβ). The function D(m)
+hH[β] of
+discrete variable, satisfying the equality
+hnD(m)
+hH[β]∗G(m)
+hH[β] =δ[β]
+is called by the discrete analogue of the polyharmonic operator ∆m. S.L.Sobolev suggested an algo-
+rithm for finding function D(m)
+hH[β] and proved several properties of this function. In one dime nsional
+case, i.e. the discrete analogue of the operatord2m
+dx2mwas constructed by Z.Zh.Zhamalov [2, 3]. But
+there the form of this function was written with m+ 1 unknown coefficients. In works [4, 5] these
+coefficients were found, hereunder the discrete analogue of t he operatord2m
+dx2mwas constructed com-
+pletely.
+In this paper we give the results of works [4-7], concerning t o construction of the discrete analogue
+Dm[β] of the operatord2m
+dx2m, and discovery of its properties, which early were not known .
+Following statements are valid.
+Theorem 1. The discrete analogue of the differential operatord2m
+dx2mhave following form
+Dm[β] =(2m−1)!
+h2m
+
+m−1/summationdisplay
+k=1(1−λk)2m+1λ|β|
+k
+λkE2m−1(λk)for|β| ≥2,
+1+m−1/summationdisplay
+k=1(1−λk)2m+1
+E2m−1(λk)for|β|= 1,
+−22m−1+m−1/summationdisplay
+k=1(1−λk)2m+1
+λkE2m−1(λk)forβ= 0,(1)
+whereEα(λ)is the Euler polynomial of degree α,λkare the roots of the Euler polynomial E2m−2(λ),
+in module less than unity, i.e. |λk|<1,his the step of the lattice.
+1Property 1. The discrete analogue Dm[β]of the differential operator of order 2mhave represen-
+tation
+Dm[β] =(2m−1)!
+h2m∆[m]
+2[β]∗m−1/summationdisplay
+k=1λ|β|+m−2
+k
+E′
+2m−2(λk),
+where∆[m]
+2[β] =m/summationdisplay
+k=−m(−1)k+m/parenleftbigg2m
+m+k/parenrightbigg
+δ[β−k]is symmetric difference of order 2m.
+Property 2. The operator Dm[β]and monomials [β]k= (hβ)kare connected as
+/summationdisplay
+βDm[β][β]k=/braceleftbigg0 for 0 ≤k≤2m−1,
+(2m)! fork= 2m,(2)
+/summationdisplay
+βDm[β][β]k=
+
+0 for 2 m+1≤k≤4m−1,
+h2m(4m)!B2m
+(2m)!fork= 4m.(3)
+Property 3. The operator Dm[β]and the function exp(2πihpβ)connected as
+/summationdisplay
+βDm[β]exp(2πihpβ) =
+(−1)m22m(2m−1)!h−2msin2m(πhp)
+2m−2/summationtext
+k=0a(2m−2)
+kcos2πhp(m−1−k)+a(2m−2)
+m−1,
+where
+a(2m−2)
+k=k/summationdisplay
+j=0(−1)j/parenleftbigg2m
+j/parenrightbigg
+(k+1−j)2m−1
+are the coefficients of the Euler polynomial E2m−2(λ).
+2 Lemmas.
+As known, Euler polynomials Ek(λ) have following form
+λEk(λ) = (1−λ)k+2Dkλ
+(1−λ)2, (4)
+where
+D=λd
+dλ, Dk=λd
+dλDk−1.
+In [8] was shown, that all roots λ(k)
+jof the Euler polynomial Ek(λ) are real, negative and different:
+λ(k)
+1<λ(k)
+2<...<λ(k)
+k<0. (5)
+2Furthermore, the roots, equal standing from the ends of the c hain (5) mutually inverse:
+λ(k)
+j·λ(k)
+k+1−j= 1. (6)
+If we denote Ek(λ) =k/summationtext
+s=0a(k)
+sλs, then the coefficients a(k)
+sof Euler polynomials, as this was shown by
+Euler himself, are expressed by formula
+a(k)
+s=s/summationdisplay
+j=0(−1)j/parenleftbiggk+2
+j/parenrightbigg
+(s+1−j)k+1.
+From the definition Ek(λ) follow following statement.
+Lemma 1. For polynomial Ek(λ)following recurrence relation is valid
+Ek(λ) = (kλ+1)Ek−1(λ)+λ(1−λ)E′
+k−1(λ), (7)
+whereE0(λ) = 1,k= 1,2,....
+Lemma 2. The polynomial Ek(λ)satisfies the identity
+Ek(λ) =λkEk/parenleftbigg1
+λ/parenrightbigg
+, (8)
+or otherwise a(k)
+s=a(k)
+k−s, s= 0,1,2,...,k.
+Proof of lemma 1. From (4) we can see, that
+Ek−1(λ) =λ−1(1−λ)k+1Dk−1λ
+(1−λ)2. (9)
+Differentiating by λthe polynomial Ek−1(λ), we get
+E′
+k−1(λ) =−(1−λ)kλ−2(kλ+1)Dk−1λ
+(1−λ)2+Ek(λ)
+λ(1−λ).
+Hence and from (9) we obtain, that
+(kλ+1)Ek−1(λ)+λ(1−λ)E′
+k−1(λ) = (kλ+1)λ−1(1−λ)k+1Dk−1λ
+(1−λ)2−
+−(1−λ)k+1λ−1(kλ+1)Dk−1λ
+(1−λ)2+Ek(λ) =Ek(λ).
+So, lemma 1 is proved.
+Proof of lemma 2. The lemma we will prove by induction method. When k= 1 from (4) we
+find
+E1(λ) =λ+1.
+We suppose, that when k≥1 the equality a(k−1)
+n=a(k−1)
+k−1−n,n= 0,1,...,k−1 is fulfilled. We assume,
+thata(k−1)
+n= 0 forn<0 andn>k−1.
+3From (7) we have
+a(k)
+s= (s+1)a(k−1)
+s+(k−s+1)a(k−1)
+s−1,
+then, using assumption of induction, we get
+a(k)
+k−s= (k−s+1)a(k−1)
+k−s+(s+1)a(k−1)
+k−s−1= (k−s+1)a(k−1)
+s−1+(s+1)a(k−1)
+s=a(k)
+s,
+and lemma 2 is proved.
+3 Proof of theorem 1
+For this we will use function
+Gm(x) =x2m−1signx
+2·(2m−1)!.
+To this function we correspond following function of discre te argument:
+Gm[β] =(hβ)2m−1sign(hβ)
+2·(2m−1)!.
+Here we must find such function Dm[β], which satisfies the equality
+hDm[β]∗Gm[β] =δ[β]. (10)
+According to the theory of periodic generalized functions a nd Fourier transformation in them
+instead of function Dm[β] it is convenient to search harrow shaped function [1]
+↽ ⇁
+Dm(x) =/summationdisplay
+βDm[β]δ(x−hβ).
+The equality (10) in the class of harrow shaped functions goe s to equation
+h↽ ⇁
+Dm(x)∗↽ ⇁
+Gm(x) =δ(x), (11)
+where↽ ⇁
+Gm(x) =/summationdisplay
+βGm[β]δ(x−hβ).
+It is known [1], that the class of harrow shaped functions and the class of functions of discrete vari-
+ables are isomorphic. So instead of function of discrete arg umentDm[β] it is sufficiently to investigate
+the function↽ ⇁
+Dm(x), defining from equation (11).
+Later on we need following well known formulas of Fourier tra nsformation:
+F[f(p)] =/integraldisplay
+f(x)exp(2πipx)dx,
+F−1[f(p)] =/integraldisplay
+f(x)exp(−2πipx)dx,
+4F[f(x)∗ϕ(x)] =F[f(x)]·F[ϕ(x)],
+F[δ(x)] = 1.
+Applying to both parts of (11) Fourier transformation, we ge t
+F[↽ ⇁
+Dm(x)]·F[h↽ ⇁
+Gm(x)] = 1. (12)
+Fourier transform of h↽ ⇁
+Gm(p) is well known periodic function, given in Rwith period h−1
+F[h↽ ⇁
+Gm(x)] =(−1)m
+(2π)2m/summationdisplay
+β1
+|p−h−1β|2m, p/negationslash=h−1β. (13)
+This formula is obtained from the equalities
+F[Gm(p)] =(−1)m
+(2π)2m1
+|p|2m( [1, p. 729])
+and↽ ⇁
+Gm(x) =Gm(x)/summationdisplay
+βδ(x−hβ).
+Hence, taking into account (12), we get
+F[↽ ⇁
+Dm(p)] =
+(−1)m
+(2π)2m/summationdisplay
+β1
+|p−h−1β|2m
+−1
+. (14)
+Themain propertiesof this function in multidimensional ca se, appearinginconstruction of discrete
+analogue of the polyharmonic operator, were investigated i n [1].
+We give some of them, which we will use later on.
+1.Zeros of the function F[↽ ⇁
+Dm(p)]are the points p=h−1β.
+2.The function F[↽ ⇁
+Dm(p)]is periodic with period h−1, real and analytic for all real p.
+The function F[↽ ⇁
+Dm(p)] can be represented in the form of Fourier series
+F[↽ ⇁
+Dm(p)] =/summationdisplay
+βˆDm[β]exp(2πihβp), (15)
+where
+ˆDm[β] =h−1/integraldisplay
+0F[↽ ⇁
+Dm(p)]exp(−2πihβp)dp. (16)
+Applying inverse Fourier transformation to the equality (1 5), we get harrow shaped function
+↽ ⇁
+Dm(x) =/summationdisplay
+βˆDm[β]δ(x−hβ). (17)
+5Thus,ˆDm[β] is searching function Dm[β] of discrete argument or discrete analogue of the operator
+d2m
+dx2m. For finding the function ˆDm[β] calculation of the integral (16) inadvisable. We will find i t by
+following way.
+By virtue of known formula
+/summationdisplay
+β1
+(p−β)2=π2
+sin2πp
+and from the formula (13) we get
+F[h↽ ⇁
+G1(p)] =−1
+(2π)2/summationdisplay
+β1
+(p−h−1β)2=−h2
+4sin2πph.
+Hence by differentiating we have
+d
+dpF[h↽ ⇁
+G1(p)] =2
+(2π)2/summationdisplay
+β1
+(p−h−1β)3.
+Thus continuing further, we obtain
+d2m−2
+dp2m−2F[h↽ ⇁
+G1(p)] =−(2m−1)!
+(2π)2m/summationdisplay
+β1
+(p−h−1β)2m=
+= (−1)m−1(2m−1)!(2π)2m−2F[h↽ ⇁
+Gm(p)].
+So,
+F[h↽ ⇁
+Gm(p)] =(−1)mh2
+22mπ2m−2(2m−1)!d2m−2
+dp2m−2/parenleftbigg1
+sin2πhp/parenrightbigg
+.
+Consider, the expression
+d2m−2
+dp2m−2/parenleftbigg1
+sin2πhp/parenrightbigg
+.
+Using
+sinπhp=exp(πihp)−exp(−πihp)
+2i,
+we have
+d2m−2
+dp2m−2/parenleftbigg−4
+(exp(πihp)−exp(−πihp))2/parenrightbigg
+=−4d2m−2
+dp2m−2/parenleftbiggexp(2πihp)
+(exp(2πihp)−1)2/parenrightbigg
+.
+We will do change of variables λ= exp(2πihp), then in view of that
+d
+dp=dλ
+dpd
+dλandd
+dp= 2πihλd
+dλ,
+we get
+d2m−2
+dp2m−2= (2πih)2m−2D2m−2,
+6where
+D=λd
+dλ, D2m−2=λd
+dλD2m−3.
+Thus,
+F[h↽ ⇁
+Gm(p)] =h2m
+(2m−1)!D2m−2λ
+(1−λ)2.
+Hence in virtue of (4) we have
+F[h↽ ⇁
+Gm(p)] =h2m
+(2m−1)!λE2m−2(λ)
+(1−λ)2m. (18)
+From (18), according to (12), we obtain
+F[↽ ⇁
+Dm(p)] =(2m−1)!
+h2m(1−λ)2m
+λE2m−2(λ). (19)
+Now in order to obtain Fourier-series expansion, we will do f ollowing.
+We divide the polynomial (1 −λ)2mto the polynomial λE2m−2(λ):
+(1−λ)2m
+λ2m−2/summationtext
+s=0a(2m−2)
+sλs=λ−2m−a(2m−2)
+2m−3+P2m−2(λ)
+λE2m−2(λ), (20)
+whereP2m−2(λ) is a polynomial of degree 2 m−2. It is not difficult to see, that the rational fraction
+P2m−2(λ)
+λE2m−2(λ)is proper fraction, i.e. degree of the polynomial P2m−2(λ) is less than degree of the
+polynomial λE2m−2(λ). Since the roots of the polynomial E2m−2(λ) are real and different, then the
+rational fractionP2m−2(λ)
+λE2m−2(λ)is expanded to the sum of elementary fractions. Searching ex pansion has
+following form
+P2m−2(λ)
+λE2m−2(λ)=A0
+λ+m−1/summationdisplay
+k=1A1,k
+λ−λ1,k+m−1/summationdisplay
+k=1A2,k
+λ−λ2,k, (21)
+whereA0, A1,k, A2,kare unknown coefficients, λ1,kare the roots of the polynomial E2m−2(λ), in
+modulus less than unity, and λ2,kare the roots of the polynomial E2m−2(λ), in modulus greater than
+unity. By (21) the equality (20) takes the form
+(1−λ)2m
+λ2m−2/summationtext
+s=0a(2m−2)
+sλs=λ−2m−a(2m−2)
+2m−3+A0
+λ+
++m−1/summationdisplay
+k=1A1,k
+λ−λ1,k+m−1/summationdisplay
+k=1A2,k
+λ−λ2,k. (22)
+Reducing to the common denominator and omitting it, we get
+(1−λ)2m=λ2E2m−2(λ)−λ(2m+a(2m−2)
+2m−3)E2m−2(λ)+
+7+A0E2m−2(λ)+m−1/summationdisplay
+k=1A1,kλE2m−2(λ)
+λ−λ1,k+m−1/summationdisplay
+k=1A2,kλE2m−2(λ)
+λ−λ2,k. (23)
+Assuming in the equality (23) consequently λ= 0,λ=λ1,kandλ=λ2,k, we find
+1 =E2m−2(0)A0; (1−λ1,k)2m=λ1,kE′
+2m−2(λ1,k)A1,k;
+(1−λ2,k)2m=λ2,kE′
+2m−2(λ2,k)A2,k.
+Hence
+A0= 1;A1,k=(1−λ1,k)2m
+λ1,kE′
+2m−2(λ1,k);A2,k=(1−λ2,k)2m
+λ2,kE′
+2m−2(λ2,k).
+Using (6), we have
+A1,k=(1−1
+λ2,k)2m
+λ−1
+2,kE′
+2m−2(1
+λ2,k)=(λ2,k−1)2m
+λ2m−1
+2,kE′
+2m−2(1
+λ2,k).
+In virtue of (7) we obtain
+1
+λ2,k/parenleftbigg
+1−1
+λ2,k/parenrightbigg
+E′
+2m−2(1
+λ2,k) =E2m−1(1
+λ2,k),
+λ2,k(1−λ2,k)E′
+2m−2(λ2,k) =E2m−1(λ2,k),
+hence
+E′
+2m−2(1
+λ2,k) =E2m−1(1
+λ2,k)λ2
+2,k
+λ2,k−1,
+E′
+2m−2(λ2,k) =E2m−1(λ2,k)
+λ2,k(1−λ2,k).
+From here application of the lemma 2 gives
+A1,k=−A2,k
+λ2
+2,k, A1,k=(1−λ1,k)2m+1
+E2m−1(λ1,k). (24)
+Since|λ1,k|<1 and|λ2,k|>1, then
+m−1/summationdisplay
+k=1A1,k
+λ−λ1,kandm−1/summationdisplay
+k=1A2,k
+λ−λ2,k
+can be represented as Laurent series on the circle |λ2|= 1:
+m−1/summationdisplay
+k=1A1,k
+λ−λ1,k=1
+λm−1/summationdisplay
+k=1A1,k
+1−λ1,k
+λ=1
+λm−1/summationdisplay
+k=1A1,k∞/summationdisplay
+β=0/parenleftbiggλ1,k
+λ/parenrightbiggβ
+, (25)
+m−1/summationdisplay
+k=1A2,k
+λ−λ2,k=−m−1/summationdisplay
+k=1A2,k
+λ2,k(1−λ
+λ2,k)=−m−1/summationdisplay
+k=1A2,k
+λ2,k∞/summationdisplay
+β=0/parenleftbiggλ
+λ2,k/parenrightbiggβ
+. (26)
+8Putting (25), (26) to (22) and taking into account λ= exp(2πihp) from (19), (20), we obtain
+F[↽ ⇁
+Dm(p)] =(2m−1)!
+h2m/bracketleftBigg
+exp(2πihp)−2m−a(2m−2)
+2m−3+
++exp(−2πihp)+m−1/summationdisplay
+k=1/parenleftBigg
+A1,k∞/summationdisplay
+β=0λβ
+1,kexp(−2πihp(β+1))−
+−A2,kλ−1
+2,k∞/summationdisplay
+β=0/parenleftbigg1
+λ2,k/parenrightbiggβ
+exp(2πihpβ)/parenrightBigg/bracketrightBigg
+.
+Thus, searching Fourier series for F[↽ ⇁
+Dm(p)] have following form
+F[↽ ⇁
+Dm(p)] =/summationdisplay
+βDm[β]exp(2πihβp),
+where
+Dm[β] =(2m−1)!
+h2m
+
+m−1/summationtext
+k=1A1,kλ−β−1
+1,kforβ≤ −2,
+1+m−1/summationtext
+k=1A1,k forβ=−1,
+−22m−1−m−1/summationtext
+k=1A2,kλ−1
+2,kforβ= 0,
+1−m−1/summationtext
+k=1A2,kλ−2
+2,kforβ= 1,
+−m−1/summationtext
+k=1A2,kλ−β−1
+2,kforβ≥2.
+With the help (24) the function Dm[β] we rewrite in the form
+Dm[β] =(2m−1)!
+h2m
+
+m−1/summationdisplay
+k=1(1−λ1,k)2m+1λ|β|
+1,k
+λkE2m−1(λ1,k)for|β| ≥2,
+1+m−1/summationdisplay
+k=1(1−λ1,k)2m+1
+E2m−1(λ1,k)for|β|= 1,
+−22m−1+m−1/summationdisplay
+k=1(1−λ1,k)2m+1
+λ1,kE2m−1(λ1,k)forβ= 0.
+We note, that
+Dm[β] =Dm[−β].
+Theorem 1 is proved completely.
+94 Proofs of properties.
+Proof of property 1.
+Following is takes placed
+F[↽ ⇁
+∆[1]
+2(p)] =−4sin2πph.
+Indeed,
+↽ ⇁
+∆[1]
+2(x) =↽ ⇁
+δ(x+1)−2↽ ⇁
+δ(x)+2↽ ⇁
+δ(x−1) =δ(x+h)−2δ(x)+δ(x−h).
+By definition of Fourier transformation we have
+F[↽ ⇁
+∆[1]
+2(p)] =/integraldisplay
+exp(2πipx)↽ ⇁
+∆[1]
+2(x)dx=−4sin2πph.
+Hence consequently we obtain
+F[↽ ⇁
+∆[m]
+2(p)] =F[mtimes/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright
+↽ ⇁
+∆[1]
+2(p)∗↽ ⇁
+∆[1]
+2(p)∗...∗↽ ⇁
+∆[1]
+2(p)] = (−4)msin2mπph. (27)
+Immediately we have/bracketleftbigg(1−λ)2
+−4λ/bracketrightbiggm
+= sin2mπhp. (28)
+By virtue of (27) and (28) the formula (19) takes form
+F[↽ ⇁
+Dm(p)] =(2m−1)!
+h2mλm−1
+E2m−2(λ)F[↽ ⇁
+∆[m]
+2(p)]. (29)
+Now expanding rational fractionλm−1
+E2m−2(λ)to the sum of elementary fractions, we have
+λm−1
+E2m−2(λ)=m−1/summationdisplay
+k=1/bracketleftbiggB1,k
+λ−λ1,k+B2,k
+λ−λ2,k/bracketrightbigg
+, (30)
+where
+B1,k=λm−1
+1,k
+E′
+2m−2(λ1,k), B2,k=λm−1
+2,k
+E′
+2m−2(λ2,k).
+Since|λ1,k|<1 and|λ2,k|>1, then expandingB1,k
+λ−λ1,kandB2,k
+λ−λ2,kto the Laurent series on the circle
+|λ|= 1, we find
+B1,k
+λ−λ1,k=1
+λ·B1,k
+1−λ1,k
+λ=B1,k
+λ∞/summationdisplay
+β=0/parenleftbiggλ1,k
+λ/parenrightbiggβ
+, (31)
+B2,k
+λ−λ2,k=−B2,k
+λ2,k(1−λ
+λ2,k)=−B2,k
+λ2,k∞/summationdisplay
+β=0/parenleftbiggλ
+λ2,k/parenrightbiggβ
+. (32)
+10On the strength of (6), (7), (8), (31), (32) the equality (30) takes the form
+λm−1
+E2m−2(λ)=m−1/summationdisplay
+k=1λm−2
+k
+E′
+2m−2(λk)/summationdisplay
+βλ|β|
+kλβ, λk=λ1,k.
+So, from (29) we get
+F[↽ ⇁
+Dm(p)] =(2m−1)!
+h2mm−1/summationdisplay
+k=1λm−2
+k
+E′
+2m−2(λk)/summationdisplay
+βλ|β|
+kλβF[↽ ⇁
+∆[m]
+2(p)]. (33)
+Applying to (33) inverse Fourier transformation, after som e simplifications we obtain
+↽ ⇁
+Dm(x) =(2m−1)!
+h2m/summationdisplay
+β∆[m]
+2[β]∗m−1/summationdisplay
+k=1λ|β|+m−2
+k
+E′
+2m−2(λk)δ(x−hβ) =
+=/summationdisplay
+βDm[β]δ(x−hβ). (34)
+According to definition of harrow shaped functions from (34) we get the statement of property 1.
+Proof of property 2.
+The equality (2) proved in [1].
+Here we will prove (3).
+From (15), (17) we obtain
+F[↽ ⇁
+Dm(p)] =/summationdisplay
+βDm[β]exp(2πihβp). (35)
+Using expansion of exp(2 πihpβ) and by formula (2) in cases 0 ≤k≤2m−1 andk= 2m, we will have
+F[↽ ⇁
+Dm(p)] =/summationdisplay
+βDm[β]∞/summationdisplay
+k=0(2πihβp)k
+k!=
+=/summationdisplay
+βDm[β]∞/summationdisplay
+k=2m+1(2πihpβ)k
+k!+(2πip)2m. (36)
+On the other hand, from (14) we get
+F[↽ ⇁
+Dm(p)] = (2πi)2m
+/summationdisplay
+β1
+(p−h−1β)2m
+−1
+. (37)
+Thus, on the strength of (36) and (37) we have
+/summationdisplay
+βDm[β]∞/summationdisplay
+k=2m+1(2πihpβ)k
+k!+(2πip)2m= (2πi)2m
+/summationdisplay
+β1
+(p−h−1β)2m
+−1
+,
+11or
+/summationdisplay
+βDm[β]∞/summationdisplay
+k=2m+1(2πihpβ)k
+k!=−(2πip)2m
+1−/bracketleftBigg/summationdisplay
+γ(ph)2m
+(ph−γ)2m/bracketrightBigg−1
+. (38)
+Consider
+ψ(h,p,m) = 1−/bracketleftBigg/summationdisplay
+γ(ph)2m
+(ph−γ)2m/bracketrightBigg−1
+=
+= 1−
+1+(ph)2m∞/summationdisplay
+γ=1/bracketleftbigg1
+(ph−γ)2m+1
+(ph+γ)2m/bracketrightbigg
+−1
+=
+= 1−
+1+(ph)2m∞/summationdisplay
+γ=1γ−2m/bracketleftbigg
+(1−ph
+γ)−2m+(1+ph
+γ)−2m/bracketrightbigg
+−1
+.
+Hence choosing psuch, that
+Q= (ph)2m∞/summationdisplay
+γ=1γ−2m/bracketleftbigg
+(1−ph
+γ)−2m+(1+ph
+γ)−2m/bracketrightbigg
+<1,
+expanding the fraction1
+1+Qto the series of geometric progression, we have
+ψ(h,p,m) = (ph)2m∞/summationdisplay
+γ=1γ−2m/bracketleftbigg
+(1−ph
+γ)−2m+(1+ph
+γ)−2m/bracketrightbigg
++
++O((hp)4m) = 2(ph)2m∞/summationdisplay
+γ=1γ−2m+O(h2m+1). (39)
+The left part of (38) we rewrite in the form
+/summationdisplay
+βh2mDm[β]∞/summationdisplay
+k=2m+1hk−2m(2πipβ)k
+k!=∞/summationdisplay
+n=1hnµ(p,n,m), (40)
+whereµ(p,n,m) =/summationdisplay
+βh2mDm[β](2πipβ)2m+n
+(2m+n)!doesnot dependon h, sinceh2mDm[β] does not depend
+h, that clear from (1).
+Comparing right hand sides of (39) and (40), from (38) we have µ(p,n,m) = 0 forn= 1,2,...,2m−
+1, i.e. fork= 2m+1,...,4m−1
+µ(p,2m,m) =/summationdisplay
+βh2mDm[β](2πipβ)4m
+(4m)!= (−1)m+12(2π)2mp4m∞/summationdisplay
+γ=1γ−2m.
+12Hence after some calculations and by virtue of the equality
+∞/summationdisplay
+γ=1γ−2m=(−1)m−1(2π)2mB2m
+2·(2m)!
+we get
+/summationdisplay
+βDm[β][β]4m=h2m(4m)!B2m
+(2m)!,
+which proves the property 2.
+Proof of property 3. From (19) and (35) we have
+/summationdisplay
+βDm[β]exp(2πihpβ) =(2m−1)!(1−λ)2m
+h2mλE2m−2(λ).
+Using (8), (40) and λ= exp(2πihpβ), after some simplifications we get property 3.
+REFERENCES
+1. Sobolev S.L. Introduction to the Theory of Cubature Formu las. -Moskow.: Nauka, 1974. - 808
+p.
+2. Zhamalov Z.Zh. About one problem of Wiener-Hopf appearin g in optimization of quadrature
+formulas. - In the Book: Boundary problems for differential eq uations. - Tashkent: Fan, 1975.
+- pp. 129-150.
+3. Zhamalov Z.Zh. About one difference analogue of operatord2m
+dx2mand its construction. - In the
+Book: Direct and inverse problem for differential equations w ith partial derivatives and their
+properties. -Tashkent: Fan, 1978. - pp. 97-108.
+4. Shadimetov Kh.M. Optimal quadrature formulas in the L(m)
+2(Ω) andL(m)
+2(R1) // Dokl. Akad.
+Nauk. Uzbekistan. - 1983. - No.3. - pp. 5-8.
+5. Shadimetov Kh.M. Discrete analogue of thed2m
+dx2mand its construction // Questions of Compu-
+tational and Applied Mathematics. - Tashkent, 1985. - pp. 22 -35.
+6. Shadimetov Kh.M. About one explicit representation of di screte analogue of the differential
+operator of order 2 m// Dokl. Akad. Nauk. Uzbekistan. - 1996. - No.9. - pp. 5-7.
+7. Shadimetov Kh.M. About one method of solution of difference equation with convolution for
+computation of optimal coefficients of Sobolev’s quadrature formulas // Uzbek Mathematical
+Journal. - 1998. - No.4. - pp. 68-76.
+8. Frobenius. Uber Bernoullische Zahlen und Eulersche Poly nomen Sitzungsberichte der Preus-
+sischen Akademic de Wissenschaften. 1910.
+Shadimetov Kholmat Mahkambaevich
+Head of the Department of Computational Methods
+Institute of Mathematics and Information Technologies
+Tashkent, Uzbekistan
+13
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+arXiv:1001.0557v2  [math.CT]  5 Jan 2010EXTENDING BINARY OPERATIONS TO FUNTOR-SPACES
+TARAS BANAKH AND VOLODYMYR GAVRYLKIV
+Abstract. Given a continuous monadic functor T:Comp→Compin the category of compacta and a
+discrete topological semigroup Xwe extend the semigroup operation ϕ:X×X→Xto a right-topological
+semigroup operation Φ : TβX×TβX→TβXwhose topological center Λ Φcontains the dense subsemigroup
+TfXconsisting of elements a∈TβXthat have finite support in X.
+One of powerful tools in the modern Combinatorics of Numbers is the method of ultrafilters based on the
+fact that each binary operation ϕ:X×X→Xdefined on a discrete topological space Xcan be extended
+to a right-topological operation Φ : βX×βX→βXon the Stone- ˇCech compactification βXofX, see [13],
+[16]. The extension of ϕis constructed in two step. First, for every x∈Xextend the left shift ϕx:X→X,
+ϕx:y/mapsto→ϕ(x,y), to a continuous map ϕx:βX→βX. Next, for every b∈βX, extend the right shift
+¯ϕb:X→βX, ¯ϕb:x/mapsto→¯ϕx(b), to a continuous map Φb:βX→βXand put Φ( a,b) = Φb(a) for every
+a∈βX. The Stone- ˇCech extension βXis the space of ultrafilters on X. In [11] it was observed that the
+binary operation ϕextends not only to βXbut also to the superextension λXofXand to the space GX
+of all inclusion hyperspaces on X. IfXis a semigroup, then GXis a compact Hausdorff right-topological
+semigroup containing λXandβXas closed subsemigroups.
+In this note we show that an (associative) binary operation ϕ:X×X→Xon a discrete topological
+spaceXcan be extended to an (associative) right-topological oper ation Φ :TβX×TβX→TβXfor any
+monadic functor Tin the category Compof compact Hausdorff spaces. So, for the functors β,λorG, we
+get the extensions of the operation ϕdiscussed above.
+1.Monadic functors and their algebras
+Let us recall [14, VI], [17, §1.2] that a functor T:C → Cin a category Cis called monadic if there are
+natural transformations η: Id→Tandµ:T2→Tmaking the following diagrams commutative:
+TηT/d47/d47
+Tη
+/d15/d151T
+/d32/d32/d66/d66/d66/d66/d66/d66/d66/d66T2
+µ
+/d15/d15T3µT/d47/d47
+Tµ
+/d15/d15T2
+µ
+/d15/d15
+T2µ/d47/d47TT2µ/d47/d47T
+In this case the triple T= (T,η,µ) is called a monad, the natural transformations η: Id→Tandµ:T2→T
+are called the unitandmultiplication of the monad T, and the functor Tis thefunctorial part of the monad
+T.
+A pair (X,ξ) consisting of an object Xand a morphism ξ:TX→Xof the category Cis called a
+T-algebraifξ◦ηX= idXand the square
+T2X
+µ
+/d15/d15Tξ/d47/d47TX
+ξ
+/d15/d15
+TXξ/d47/d47X
+is commutative. For every object Xof thecategory Cthe pair(TX,µ) is aT-algebra called the freeT-algebra
+overX.
+1991Mathematics Subject Classification. 18B30; 18B40; 20N02; 20M50; 22A22; 54B30; 54H10.
+Key words and phrases. Functor, monad, algebra, binary operation, semigroup, rig ht-topological semigroup, topological
+center.
+12 TARAS BANAKH AND VOLODYMYR GAVRYLKIV
+For twoT-algebras (X,ξX) and (Y,ξY) a morphism h:X→Yis called a morphism of T-algebras if the
+following diagram is commutative:
+TX
+ξX
+/d15/d15Th/d47/d47TY
+ξY
+/d15/d15
+Xh/d47/d47Y
+The naturality of the multiplication µ:T2→Tof the monad Timplies that for any morphism f:X→Y
+inCthe morphism Tf:TX→TYis a morphism of free T-algebras.
+Each morphism h:TX→Yfrom the free T-algebra into a T-algebra (Y,ξ) is uniquely determined by
+the composition h◦η.
+Lemma 1.1. Ifh:TX→Yis a morphism of a free T-algebraTXinto aT-algebra (Y,ξ), thenh=
+µ◦T(h◦η) =µ◦Th◦Tη.
+Proof.Consider the commutative diagram
+X
+η
+/d15/d15η/d47/d47TX
+ηT
+/d15/d15h/d47/d47Y
+TX
+T(h◦η)/d60/d60Tη/d47/d47T2Xµ/d111/d111µ/d79/d79
+Th/d47/d47TYξ/d79/d79
+and observe that
+h=h◦µ◦ηT=ξ◦Th◦ηT=ξ◦Th◦Tη◦µ◦ηT=ξ◦T(h◦η).
+/square
+By atopological category we shall understand a subcategory of the category Topof topological spaces
+and their continuous maps such that
+•for any objects X,Yof the category Ceach constant map f:X→Yis a morphism of C;
+•for any objects X,Yof the category Cthe product X×Yis an object of Cand for any object Zof
+Cand morphisms fX:Z→XandfY:Z→Ythe map (fX,fY) :Z→X×Yis a morphism of
+the category C.
+A discrete topological space Xis called discrete in CifXis an object of Cand each function f:X→Y
+into an object Yof the category Cis a morphism of C. It is clear that any bijection f:X→Ybetween
+discrete objects of the category Cis an isomorphism in C.
+From now on we shall assume ( T,η,µ) is a monad in a topological category Csuch that for any discrete
+objectsX,YinCthe product X×Yis discrete in C.
+2.Binary operations and their T-extensions
+By abinary operation in the category Cwe understand any function ϕ:X×Y→ZwhereX,Y,Zare
+objects of the category C. For anya∈Xandb∈Ythe functions
+ϕa:Y→Z, ϕa:y/mapsto→ϕ(a,y)
+and
+ϕb:X→Z, ϕb:x/mapsto→ϕ(x,b),
+are called the leftandright shifts , respectively.
+A binary operation ϕ:X×Y→Zis calledright-topological if for every y∈Ythe right shift ϕy:X→Z,
+ϕy:x/mapsto→ϕ(x,y), is continuous. The topological center of a right-topological binary operation ϕ:X×Y→Z
+is the set Λ ϕof all elements x∈Xsuch that the left shift ϕx:Y→Zis continuous.
+Definition 2.1. Letϕ:X×Y→Zbe a binary operation in the category C. A binary operation
+Φ :TX×TY→TZis defined to be a T-extension ofϕifEXTENDING BINARY OPERATIONS TO FUNTOR-SPACES 3
+(1) Φ(ηX(x),ηY(y)) =ηZ(ϕ(x,y)) for anyx∈Xandy∈Y;
+(2) for every b∈TYthe right shift Φb:TX→TZ, Φb:x/mapsto→Φ(x,b), is a morphism of the free
+T-algebrasTY,TZ;
+(3) for every x∈Xthe left shift Φ η(x):TY→TZ, Φη(x):y/mapsto→Φ(η(x),y), is a morphism of the free
+T-algebrasTX,TZ.
+This definition implies that for any binary operation ϕ:X×Y→ZitsT-extension Φ : TX×TY→TZ
+is a right-topological binary operation whose topological center Λ Φcontains the set η(X)⊂TX.
+Theorem 2.2. Letϕ:X×Y→Zbe a binary operation in the category C.
+(1)The binary operation ϕhas at most one T-extension Φ :TX×TY→TZ.
+(2)IfX,Yare discrete in C, thenϕhas a unique T-extension Φ :TX×TY→TZ.
+Proof.1. Let Φ,Ψ :TX×TY→TZbe twoT-extensions of the operation ϕ. By the condition (3) of
+Definition 2.1, for every x∈Xanda=ηX(x)∈TX, the left shifts Φ a,Ψa:TY→TZare morphisms of
+the freeT-algebras.
+By the condition (1) of Definition 2.1,
+Φa◦ηY=ηZ◦ϕx= Ψa◦ηY.
+Then Lemma 1.1 implies that
+Φa=µ◦T(Φa◦ηX) =µ◦T(ηZ◦ϕx) =µ◦T(Ψa◦ηX) = Ψa.
+The equality Φ = Ψ will follow as soon as we check that Φb= Ψbfor everyb∈TY. Since Φb,Ψb:TX→
+TZare morphisms of the free T-algebrasTXandTZ, the equality Φb= Ψbfollows from the equality
+Φb◦η(x) = Φη(x)(b) = Ψη(x)(b) = Ψb◦η(x), x∈X
+according to Lemma 1.1.
+2. Now assumingthat the spaces X,Yare discrete in C, we show that the binary operation ϕ:X×Y→Z
+has aT-extension. For every x∈Xconsider the left shift ϕx:Y→Z. SinceYis discrete in C, the
+functionϕxis a morphism of the category C. Applying the functor Tto this morphism, we get a morphism
+Tϕx:TY→TZ. Now for every b∈TYconsider the function ϕb:X→TZ,ϕb:x/mapsto→Tϕx(b). Since
+the object Xis discrete, the function ϕbis a morphism of the category C. Applying to this morphism
+the functor T, we get a morphism Tϕb:TX→T2Z. Composing this morphism with the multiplication
+µ:T2Z→TZof the monad T, we get the function Φb=µ◦Tϕb:TZ→TZ. Define a binary operation
+Φ :TX×TY→TZletting Φ(a,b) = Φb(a) fora∈TX.
+Claim 2.3. Φ(η(x),b) =Tϕx(b)for everyx∈Xandb∈TY.
+Proof.The commutativity of the diagram
+X
+η
+/d15/d15ϕb/d47/d47TZ
+η
+/d15/d15
+TX
+Tϕb/d47/d47Φb/d59/d59/d120/d120/d120/d120/d120/d120/d120/d120/d120
+T2Zµ/d92/d92
+implies the desired equality
+Φ(η(x),b) =µ◦Tϕb(η(x)) =ϕb(x) =Tϕx(b).
+/square
+Now we shall prove that Φ is a T-extension of ϕ.
+i) For every x∈Xandy∈Ywe need to prove the equality
+Φ(ηX(x),ηY(y)) =ηZ◦ϕ(x,y).
+By Claim 2.3,
+Φ(ηX(x),ηY(y)) =Tϕx◦ηY(y) =ηZ◦ϕx(y) =ηZ◦ϕ(x,y).4 TARAS BANAKH AND VOLODYMYR GAVRYLKIV
+The latter equality follows from the naturality of the trans formationη: Id→T.
+ii) The definition of Φ implies that for every b∈TYthe right shift Φb=µZ◦Tϕbis a morphism of
+freeT-algebras, being the compositions of two morphisms Tϕb:TX→T2ZandµZ:T2Z→TZof free
+T-algebras.
+iii) Claim 2.3 guarantees that for every x∈Xthe left shift Φ η(x)=Tϕx:TY→TZis a morphism of
+the freeT-algebras. /square
+Proposition 2.4. Letϕ:X×Y→Z,ψ:X′×Y′→Z′be two binary operations in C,Φ :TX×TY→TZ,
+Ψ :TX′×TY′→TZ′be theirT-extensions, and hX:X→X′,hY:Y→Y′,hZ:Z→Z′be morphisms
+inC. Ifψ(hX×hY) =hZ◦φ, thenTΨ(ThX×ThY) =ThZ◦Φ.
+Proof.Observe that for any x∈Xandx′=hX(x), the commutativity of the diagrams
+Yϕx/d47/d47
+hY
+/d15/d15Z
+hZ
+/d15/d15
+Y′
+ψx′/d47/d47Z′TYTϕx/d47/d47
+ThY
+/d15/d15TZ
+ThZ
+/d15/d15
+TY′
+Tψx′/d47/d47TZ′
+imply that ThZ◦Tϕx(b) =Tψx′(b′) for every b∈TYandb′=ThY(b)∈TY′.
+It follows from Lemma 2.3 that Φ η(x)=Tϕx:TY→TZand Ψη(x′)=Tψx′:TY′→TZ′. Consequently,
+ThZ◦Φb(η(x)) =ThZ◦Φη(x)(b) =ThZ◦Tϕx(b) =Tψx′(b′) = Ψη(x′)(b′) = Ψb′(η(x′))
+and hence
+ThZ◦Φb◦η= Ψb′◦η◦hX.
+Applying the functor Tto this equality, we get
+T2hZ◦T(Φb◦η) =T(Ψb′◦η)◦ThX.
+Since Φb:TX→TZand Ψb′:TX′→TZ′are homomorphisms of the free T-algebras, we can apply
+Lemma 1.1 and conclude that Φb=µ◦T(Φb◦η) and hence
+ThZ◦Φb=ThZ◦µZ◦T(Φb◦η) =µZ′◦T2hZ◦T(Φb◦η) =µZ′◦T(Ψb′◦η)◦ThX= Ψb′◦ThX.
+Then for every a∈TXwe get
+ThZ◦Φ(a,b) =ThZ◦Φb(a) = Ψb′◦ThX(a) = Ψ(ThX(a),ThY(b)).
+/square
+3.Binary operations and tensor products
+In this section we shall discuss the relation of T-extensions to tensor products. The tensor product is a
+function ⊗:TX×TY→T(X×Y) defined for any objects X,Y∈ Csuch thatXis discrete in C.
+For everyx∈Xconsider the embedding ix:Y→X×Y,ix:y/mapsto→(x,y). The embedding ixis a
+morphism of the category Cbecause the constant map cx:Y→ {x} ⊂Xand the identity map id : Y→Y
+are morphisms of the category and Ccontains products of its objects. Applying the functor Tto the
+morphismix, we get a morphism Tix:TY→T(X×Y) of the category C. Next, for every b∈TYconsider
+the function Tib:X→T(X×Y),Tib:x/mapsto→Tix(b). SinceXis discrete in C, the function Tibis a morphism
+of the category C. Applying the functor Tto this morphism, we get a morphism TTib:TX→T2(X×Y).
+Composing this morphism with the multiplication µ:T2(X×Y)→T(X×Y) of the monad T, we get the
+morphism ⊗b=µ◦TTib:TX→T(X×Y). Finally define the tensor product ⊗:TX×TY→T(X×Y)
+lettinga⊗b=⊗b(a) fora∈TX.
+The following proposition describes some basic properties of the tensor product. For monadic functors in
+the category Compof compact Hausdorff spaces those properties were establish ed in [17, 3.4.2].EXTENDING BINARY OPERATIONS TO FUNTOR-SPACES 5
+Proposition 3.1. (1)The diagram X×Yη
+/d43/d43
+η×η/d47/d47TX×TY⊗/d47/d47T(X×Y)is commutative for any dis-
+crete object Xand any object YofC;
+(2)the tensor product is natural in the sense that for any morphi smshX:X→X′,hY:Y→Y′ofC
+with discrete X,Y, the following diagram is commutative:
+TX×TY
+ThX×ThY
+/d15/d15⊗/d47/d47T(X×Y)
+T(hX×hY)
+/d15/d15
+TX′×TY′⊗/d47/d47T(X′×Y′)
+(3)the tensor product is associative in the sense that for any di screte objects X,Y,ZofCthe diagram
+TX×TY×TZ
+id×⊗
+/d15/d15⊗×id/d47/d47T(X×Y)×TZ
+⊗
+/d15/d15
+TX×T(Y×Z)⊗/d47/d47T(X×Y×Z)
+is commutative, which means that (a⊗b)⊗c=a⊗(b⊗c)for anya∈TX,b∈TY,c∈TZ.
+Proof.1. Fix any y∈Yand consider the element b=ηY(y)∈TY. The definition of the right shift ⊗b
+implies that the following diagram is commutative:
+X
+η
+/d15/d15Tib/d47/d47T(X×Y)
+TX
+TTib/d47/d47⊗b/d57/d57/d115/d115/d115/d115/d115/d115/d115/d115/d115/d115
+T2(X×Y)µ/d79/d79
+Consequently, for every x∈Xwe get
+η(x)⊗η(y) =⊗b◦η(x) =Tib◦η(x) =Tix(η(y)) =η(ix(y)) =η(x,y).
+The latter equality follows from the diagram
+Y
+η
+/d15/d15ix/d47/d47X×Y
+η
+/d15/d15
+TYTix/d47/d47T(X×Y)
+whose commutativity follows from the naturality of the tran sformation η: Id→T.
+2. LethX:X→X′andhY:Y→Y′be any functions between discrete objects of the category C. Let
+Z=X×Y,Z′=X′×Y′andhZ=hX×hY:Z→Z′. Given any point b∈TY, consider the element
+b′=ThY(b)∈TY′. The statement (2) will follow as soon as we check that ThZ◦ ⊗b=⊗b′◦ThX. By
+Lemma 1.1, this equality will follow as soon as we check that ThZ◦⊗b◦ηX=⊗b′◦ThX◦ηX=⊗b′◦ηX′◦hX.
+The last equality follows from the naturality of the transfo rmationη: Id→T. As we know from the proof
+of the preceding item, ⊗b′◦ηX′(x′) =Tix′(b′) for anyx′∈X′. For every x∈Xandx′=hX(x) we can
+apply the functor Tto the commutative diagram
+Y
+hY
+/d15/d15ix/d47/d47Z
+hZ
+/d15/d15
+Y′
+ix′/d47/d47Z′6 TARAS BANAKH AND VOLODYMYR GAVRYLKIV
+and obtain the equality ThZ◦Tix=Tix′◦ThYwhich implies the desired equality:
+⊗b′◦ηX′◦hX(x) =⊗b′◦ηX′(x′) =Tix′(b′) =ThZ◦Tix(b) =ThZ◦⊗b◦η(x).
+3. The proof of the associativity of the tensor product can be obtained by literal rewriting the proof of
+Proposition 3.4.2(4) of [17]. /square
+Theorem 3.2. Letϕ:X×Y→Zbe a binary operation in the category CandΦ :TX×TY→TZbe its
+T-extension. If Xis a discrete object in C, thenΦ(a,b) =Tϕ(a⊗b)for any elements a∈TXandb∈TY.
+Proof.Our assumptions on the category Cguarantee that the product X×Yis a discrete object of C
+and hence ϕ:X×Y→Zis a morphism of the category C. So, it is legal to consider the morphism
+Tϕ:T(X×Y)→TZ. We claim that the binary operation
+Ψ :TX×TY→TZ,Ψ : (a,b) =Tϕ(a⊗b),
+is aT-extension of ϕ.
+1. The first item of Definition 2.1 follows Proposition 3.1(1) and the naturality of the transformation
+η: Id→T:
+Ψ(ηX(x),ηY(y)) =Tϕ(ηX(x)⊗ηY(y)) =Tϕ◦ηX×Y(x,y) =ηZ◦ϕ(x,y).
+2. For every b∈TYthe morphism
+Ψb=Tϕ◦⊗b=Tϕ◦µ◦TTib
+is a morphism of the free T-algebrasTXandTZ.
+3. For every x∈Xwe see that
+Ψη(x)(b) =Tϕ(⊗b(η(x))) =Tϕ◦µ◦TTib◦η(x) =Tϕ◦µ◦η◦Tib(x) =Tϕ◦Tib(x)
+is a morphism of the free T-algebrasTYandTZ.
+Thus Ψ is a T-extension of the binary operation ϕ. By the Uniqueness Theorem 2.2(1), Ψ coincides with
+Φ and hence Φ( a,b) = Ψ(a,b) =Tϕ(a⊗b). /square
+4.The topological center of T-extended operation
+Definition 2.1 guarantees that for a binary operation ϕ:X×Y→ZinCanyT-extension Φ : TX×TY→
+TZofϕis a right-topological operation whose topological center Λϕcontains the subset ηX(X). In this
+section we shall find conditions on the functor Tand the space Xguaranteeing that the topological center
+ΛΦis dense in TX.
+We shall say that the functor Tiscontinuous if for each compact Hausdorff space Kthat belongs to the
+category Cand any object ZofCthe mapT: Mor(K,Z)→Mor(TK,TZ),T:f/mapsto→Tf, is continuous with
+respect to the compact-open topology on the spaces of morphi sms (which are continuous maps).
+Theorem 4.1. Letϕ:X×Y→Zbe a binary operation in CandΦ :X×Y→Zbe itsT-extension. If the
+objectXis finite and discrete in C,TXis locally compact and Hausdorff, and the functor Tis continuous,
+then the operation Φis continuous.
+Proof.Since the space Xis discrete, the condition (2) of Definition 2.1 implies that the map Φ η:X×TY→
+TZ, Φη: (x,b)/mapsto→Φ(η(x),b) is continuous. Since Xis finite, the induced map
+Φ(·)
+η:TY→Mor(X,TZ),Φ(·)
+η:b/mapsto→Φb
+ηwhere Φb
+η:x/mapsto→Φ(η(x),b),
+is continuous. By the continuity of the functor T, the mapT: Mor(X,TZ)→Mor(TX,T2Z),T:f/mapsto→Tf,
+is continuous and so is the composition T◦Φ(·)
+η:TY→Mor(TX,T2Z). SinceTXis locally compact and
+Hausdorff, we can apply [9, 3.4.8] and conclude that the map
+TΦ(·)
+η:TX×TY→T2Z, TΦ(·)
+η: (a,b)/mapsto→TΦb
+η(a)
+is continuous and so is the composition Ψ = µ◦TΦ(·)
+η:TX×TY→TZ. Using the Uniqueness Theo-
+rem 2.2(1), we can prove that Ψ = Φ and hence the binary operati on Φ is continuous. /squareEXTENDING BINARY OPERATIONS TO FUNTOR-SPACES 7
+LetXbean object of the category C. We say that an element a∈FXhasdiscrete (finite) support if there
+is a morphism f:D→Xfrom a discrete (and finite) object Dof the category Csuch thata∈Ff(FD).
+ByTdX(resp.TfX) we denote the set of all elements a∈TXthat have discrete (finite) support. It is clear
+thatTfX⊂TdX⊂TX.
+Theorem 4.2. Letϕ:X×Y→Zbe a binary operation and Φ :TX×TY→TZbe aT-extension of ϕ.
+If the functor Tis continuous, and for every finite discrete object DofCthe spaceTDis locally compact
+and Hausdorff, then the topological center ΛΦof the binary operation Φcontains the subspace TfXofTX.
+IfTfXis dense in TX, then the topological center ΛΦofΦis dense in TX.
+Proof.We need to prove that forevery a∈TfXtheleft shiftΦ a:TY→TZ, Φa:b/mapsto→Φ(a,b), is continuous.
+Sincea∈TfX, there is a finite discrete object Dof the category Cand a morphism f:D→Xsuch that
+a∈Ff(FD). Fix an element d∈FDsuch thata=Ff(d).
+Consider the binary operations
+ψ:D×Y→Z, ψ: (x,y)/mapsto→φ(f(x),y),
+and
+Ψ :TD×TY→TZ,Ψ : (a,b)/mapsto→Φ(Ff(a),b).
+It can be shown that Ψ is a T-extension of ψ.
+By Theorem 4.1, the binary operation Ψ is continuous. Conseq uently, the left shift Ψ d:TY→TZ,
+Ψd:b/mapsto→Ψ(d,b), is continuous. Since Ψ d= Φa, the left shift Φ ais continuous too and hence a∈ΛΦ./square
+5.The associativity of T-extensions
+In this section we investigate the associativity of the T-extensions. We recall that a binary operation
+ϕ:X×X→Xisassociative ifϕ(ϕ(x,y),z) =ϕ(x,ϕ(y,z)) for anyx,y,z∈X. In this case we say that X
+is asemigroup .
+A subsetAof a setXendowed with a binary operation ϕ:X×X→Xis called a subsemigroup ofXif
+ϕ(A×A)⊂Aandϕ(ϕ(x,y),z) =ϕ(x,ϕ(y,z)) for allx,y,z∈A.
+Lemma 5.1. Letϕ:X×X→Xbe an associative operation in CandΦ :TX×TX→TXbe its
+T-extension.
+(1)for any morphisms fA:A→X,fB:B→Xfrom discrete objects A,BinC, the mapϕAB=
+ϕ(fA×fB) :A×B→Xis a morphism of Csuch that Φ(TfA(a),TfB(b)) =TϕAB(a⊗b)for all
+a∈TAandb∈TB;
+(2) Φ(TdX×TdX)⊂TdXandΦ(TfX×TfX)⊂TfX;
+(3) Φ((a,b),c) = Φ(a,Φ(b,c))for anya,b,c∈TdX.
+Proof.1. LetfA:A→X,fB:B→Xbe morphisms from discrete objects A,BofCandϕAB=
+ϕ(fA×fB) :A×B→X. By our assumption on the category C, the product A×Bis a discrete object
+inCand henceφABis a morphism in C. Consider the binary operation Φ AB:TA×TB→TXdefined by
+ΦAB(a,b) = Φ(TfA(a),TfB(b)). The following diagram implies that Φ ABis aT-extension of ϕAB:
+TX×TXΦ/d47/d47TX
+X×Xη×η/d102/d102/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78ϕ/d47/d47Xη/d61/d61/d123/d123/d123/d123/d123/d123/d123/d123
+A×B
+η×η/d120/d120/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112fA×fB/d79/d79
+ϕAB/d47/d47X/d15/d15id/d79/d79
+η/d33/d33/d67/d67/d67/d67/d67/d67/d67/d67
+TA×TBTfA×TfB/d79/d79
+ΦAB/d47/d47TX/d15/d15id/d79/d79
+By Theorem 3.2,
+Φ(TfA(a),TfB(b)) = ΦAB(a,b) =TϕAB(a⊗b)8 TARAS BANAKH AND VOLODYMYR GAVRYLKIV
+for alla∈TAandb∈TB.
+2. Given elements a,b∈TdX, we need to show that the element Φ( a,b)∈TXhas discrete support.
+Find discrete objects A,BinCand morphisms fA:A→X,fB:B→Xsuch thata∈FfA(FA) and
+b∈fB(FB). Fix elements ˜ a∈FA,˜b∈FBsuch thata=FfA(˜a) andb=FfB(˜b). Our assumption on the
+category Cguarantees that A×Bis a discrete object in C.
+Consider the binary operations ψ:A×B→Xand Ψ :FA×FB→FZdefined by the formulae
+ψ=φ◦(fA×fB) and Ψ = Φ ◦(TfA×TfB). Let ˜c= ˜a⊗˜b∈T(A×B). By the first statement,
+Φ(a,b) =Tψ(˜a⊗˜b) =Tψ(˜c)∈Tψ(A×B) witnessing that the element Φ( a,b) has discrete support and
+hence belongs to TdX.
+By analogy, we can prove that Φ( TfX×TfX)⊂TfX.
+3. Given any points a,b,c∈TdX, we need to check the equality
+Φ(Φ(a,b),c) = Φ(a,Φ(b,c)).
+Find discrete objects A,B,CinCand morphisms fA:A→X,fB:B→X,fC:C→Xsuch that
+a∈TfA(TA),b∈TfB(TB), andc∈TfC(TC). Fix elements ˜ a∈TA,˜b∈TB, and ˜c∈TCsuch that
+a=TfA(˜a),b=TfB(˜b), andc=TfC(˜c).
+Consider the morphisms ϕAB=ϕ(fA×fB) :A×B→X,ϕBC=ϕ(fB×fC) :B×C→Xand
+ϕABC=ϕ(ϕAB×fC) =ϕ(fA×ϕBC) :A×B×C→X. Consider the following diagram:
+TX×TX×TX
+id×Φ
+/d15/d15Φ×id/d47/d47TX×TX
+Φ
+/d15/d15TA×TB×TCTfA×TfB×TfC/d105/d105/d83/d83/d83/d83/d83/d83/d83/d83/d83/d83/d83/d83/d83/d83/d83
+id×⊗
+/d15/d15⊗×id/d47/d47T(A×B)×TC
+⊗
+/d15/d15TϕAB×TfC/d54/d54/d109/d109/d109/d109/d109/d109/d109/d109/d109/d109/d109/d109/d109
+TA×T(B×C)
+TfA×TϕBC /d117/d117/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107⊗/d47/d47T(A×B×C)
+TϕABC/d40/d40/d81/d81/d81/d81/d81/d81/d81/d81/d81/d81/d81/d81/d81/d81
+TX×TXΦ/d47/d47TX
+In this diagram the central square is commutative because of the associativity of the tensor product ⊗. By
+the item (1) all four margin squares also are commutative. No w we see that
+Φ(Φ(a,b),c)) = Φ(Φ(TfA(˜a),TfB(˜b)),TfC(˜c)) =
+Φ(TϕAB(˜a⊗˜b),TfC(˜c)) =TϕABC((˜a⊗˜b)⊗˜c) =TϕABC(˜a⊗(˜b⊗˜c)) =
+Φ(TfA(˜a),TϕBC(˜a⊗˜b)) = Φ(TfA(˜a),Φ(TfB(˜b),TfC(˜c))) = Φ(a,Φ(b,c)).
+/square
+Combining Lemma 5.1 with Theorem 4.2, we get the main result o f this paper:
+Theorem 5.2. Assume that the monadic functor Tis continuous and for each finite discrete space FinC
+the spaceTFis Hausdorff and locally compact. Let ϕ:X×X→Xbe an associative binary operation in C
+andΦ :X×X→Xbe itsT-extension. If the set TfXof elements with finite support is dense in TX, then
+the operation Φis associative.
+Proof.By Theorem 4.2, the set TfXlies in the topological center Λ Φof the operation Φ and by Lemma 5.1,
+TfXis a subsemigroup of ( TX,Φ). Now the associativity of Φ follows from the following gen eral fact. /square
+Proposition 5.3. A right topological operation ·:X×X→Xon a Hausdorff space Xis associative if its
+topological center contains a dense subsemigroup SofX.
+Proof.Assume conversely that ( xy)z/ne}ationslash=x(yz) for some points x,y,z∈X. SinceXis Hausdorff, the
+points (xy)zandx(yz) have disjoint open neighborhoods O((xy)z) andO(x(yz)) inX. Since the right
+shifts inXare continuous, there are open neighborhoods O(xy) andO(x) of the points xyandxsuch thatEXTENDING BINARY OPERATIONS TO FUNTOR-SPACES 9
+O(xy)·z⊂O((xy)z) andO(x)·(yz)⊂O(x(yz)). We can assume that O(x) is so small that O(x)·y⊂O(xy).
+Take any point a∈O(x)∩S. It follows that a(yz)∈O(x(yz)) anday∈O(xy). Since the left shift
+la:βS→βS,la:y/mapsto→ay, is continuous, the points yzandyhave open neighborhoods O(yz) andO(y) such
+thata·O(yz)⊂O(x(yz)) anda·O(y)⊂O(xy). We can assume that the neighborhood O(y) is so small that
+O(y)·z⊂O(yz). Choose a point b∈O(y)∩Sand observe that bz∈O(y)·z⊂O(yz),ab∈a·O(y)⊂O(xy),
+and thus (ab)z∈O(xy)·z⊂O((xy)z). The continuity of the left shifts lbandlaballows us to find an open
+neighborhood O(z)⊂βSofzsuch thatb·O(z)⊂O(yz) andab·O(z)⊂O((xy)z). Finally take any point
+c∈S∩O(z). Then (ab)c∈ab·O(z)⊂O((xy)z) anda(bc)⊂a·O(yz)⊂O(x(yz)) belong to disjoint sets,
+which is not possible as ( ab)c=a(bc). /square
+6.T-extension for some concrete monadic functors
+In this section we consider some examples of monadic functor s in topological categories. Let Tych
+denote the category of Tychonov spaces and their continuous maps and Compbe the full subcategory of
+the category Tych, consisting of compact Hausdorff spaces.
+Discrete objects in thecategory Tychare discrete topological spaces whilediscrete objects in t he category
+Compare finite discrete spaces.
+Considerthefunctor β:Tych→CompassigningtoeachTychonov space XitsStone- ˇCechcompactifica-
+tion and to a continuous map f:X→Ybetween Tychonov spaces its continuous extension βf:βX→βY.
+The functor βcan be completed to a monad Tβ= (β,η,µ) whereη:X→βXis the canonical embedding
+andµ:β(βX)→βXis the identity map. A pair ( X,ξ) is aTβ-algebra if and only if Xis a compact space
+andξ:βX→Xis the identity map.
+Combining Theorems 2.2, 5.2 we get the following well-known
+Corollary 6.1. Each binary right-topological operation ϕ:X×Y→ZinTychwith discrete Xcan be
+extended to a right-topological operation Φ :βX×βY→βZcontaining Xin its topological center ΛΦ. If
+X=Y=Zand the operation ϕis associative, then so is the operation Φ.
+Now let T= (T,η,µ) be a monad in the category Comp. Taking the composition of the functors
+β:Tych→CompandT:Comp→Comp, we obtain a monadic functor Tβ:Tych→Comp.
+Theorem 6.2. Each binary right-topological operation ϕ:X×Y→Zin the category Tychwith discrete X
+can be extended to a right-topological operation Φ :TβX×TβY→TβZthat contain the set η(X)⊂TβXin
+its topological center ΛΦ. If the functor Tis continuous, then the set TfXof elements a∈TβXwith finite
+support is dense in TβXand lies in the topological center ΛΦof the operation Φ. Moreover, if X=Y=Z
+and the operation ϕis associative, the so is the operation Φ.
+Proof.By Theorem 2.2, the binary operation ϕhas a unique T-extension Φ : TX×TY→TZ. By
+Definition 2.1, the set η(X)⊂TβXlies in the topological center Λ ϕofϕ.
+Now assume that the functor Tis continuous. First we show that the set TfXis dense inTβX. Fix any
+pointa∈FβXandan openneighborhood U⊂TβXofa. Then[a,U] ={f∈Mor(FβX,FβX ) :f(a)∈U}
+is an open neighborhood of the identity map id : FβX→FβXin the function space Mor( FβX,FβX )
+endowed with the compact-open topology. The continuity of t he functorTyields a neighborhood U(idβX)
+of the identity map id βX∈Mor(βX,βX) such that Tf∈[a,U] for anyf∈ U(idβX). It follows from the
+definition of the compact-open topology, that there is an ope n coverUofβXsuch that a map f:βX→βX
+belongs to U(idβX) iffisU-near to id βXin the sense that for every x∈βXthere is a set U∈ Uwith
+{x,f(x)} ⊂U. SinceβXis compact, we can assume that the cover Uis finite. Since Xis discrete, the space
+βXhas covering dimension zero [9, 7.1.17]. So, we can assume th at the finite cover Uis disjoint. For every
+U∈ Uchoose an element xU∈U∩X. Those elements compose a finite discrete subspace A={xU:U∈ U}
+ofX. leti:A→Xbe the identity embedding and f:X→Abe the map defined by f−1(xU) =Ufor
+U∈ U. It follows that i◦f∈ U(idβX) and thus T(i◦f)∈[a,U] andTi◦Tf(a)∈U. Now we see that
+b=Tf(a)∈TAandc=Ti(b)∈TfX∩U, soTfXis dense in βX.
+By Theorem 4.2, the set TfXlies in the topological center Λ Φof Φ.
+Now assume that the operation ϕis associative. By Lemma 5.1, TfXis a subsemigroup of ( X,Φ). Since
+TfXis denseand lies in the topological center Λ Φ, we may derive the associativity of Φ from Proposition 5.3.
+/square10 TARAS BANAKH AND VOLODYMYR GAVRYLKIV
+Problem 6.3. Given a discrete semigroup Xinvestigate the algebraic and topological properties of th e
+compact right-topological semigroup TβXfor some concrete continuous monadic functors T:Comp→
+Comp.
+This problem was addressed in [10], [11] for the monadic func torGof inclusion hyperspaces, in [2]–[5] for
+the functor of superextension λ, in [1], [12], [15] for the functor Pof probability measures and in [6], [7], [8],
+[18] for the hyperspace functor exp.
+In [19] it was shown that for each continuous monadic functor T:Comp→Company continuous
+(associative) operation ϕ:X×Y→ZinCompextends to a continuous (associative) operation Φ :
+TX×TY→TZ.
+Problem 6.4. For which monads T= (T,η,µ) in the category Compeach right-topological (associative)
+binary operation ϕ:X×Y→ZinCompextends to a right-topological (associative) binary opera tion
+Φ :TX×TY→TZ? Are all such monads power monads?
+References
+[1] T.Banakh, M.Cencelj, O.Hryniv, D.Repovs, Characterizing compact Clifford semigroups that embed into c onvolution and
+functor-semigroups // preprint (http://arxiv.org/abs/0811.1026).
+[2] T. Banakh, V. Gavrylkiv, O. Nykyforchyn, Algebra in superextensions of groups, I: zeros and commutat ivity// Algebra
+Discrete Math. no.3 (2008), 1-29.
+[3] T. Banakh, V. Gavrylkiv, Algebra in superextension of groups, II: cancelativity and centers// Algebra Discrete Math. no.4
+(2008) 1-14.
+[4] T. Banakh, V. Gavrylkiv, Algebra in the superextensions of groups, III: minimal left ideals// Mat. Stud. 31:2 (2009) 142-148.
+[5] T. Banakh, V. Gavrylkiv, Algebra in the superextensions of groups, IV: representati on theory // preprint
+(http://arxiv.org/abs/0811.0796).
+[6] T.Banakh, O.Hryniv, Embedding topological semigroups into the hyperspaces ove r topological groups //ActaUniv.Carolinae,
+Math. et Phys. 48:2 (2007), 3–18.
+[7] S.G. Bershadskii, Imbeddability of semigroups in a global supersemigroup ove r a group // in: Semigroup varieties and
+semigroups of endomorphisms, Leningrad. Gos. Ped. Inst., L eningrad, 1979, 47–49.
+[8] R.G. Bilyeu, A. Lau, Representations into the hyperspace of a compact group // Semigroup Forum 13(1977), 267–270.
+[9] R.Engelking, General Topology , PWN, Warsaw, 1977.
+[10] V.Gavrylkiv, The spaces of inclusion hyperspaces over noncompact spaces // Mat. Stud. 28:1(2007), 92–110.
+[11] V. Gavrylkiv, Right-topological semigroup operations on inclusion hype rspaces// Mat. Stud. 29:1 (2008), 18–34.
+[12] H. Heyer, Probability Measures on Locally Compact Groups , Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 94 .
+Springer-Verlag, Berlin-New York, 1977.
+[13] N.Hindman, D.Strauss, Algebra in the Stone- ˇCech compactification, de Gruyter, Berlin, New York, 1998.
+[14] S. MacLane, Categories for working mathematicican, Sp ringer, 1971.
+[15] K.R. Parthasarathy, Probability Measures on Metric Spaces , AMS Bookstore, 2005.
+[16] I. Protasov, Combinatorics of Numbers, VNTL, Lviv, 199 7.
+[17] A. Teleiko, M. Zarichnyi, Categorical Topology of Comp act Hausdofff Spaces, VNTL, Lviv, 1999.
+[18] V. Trnkova, On a representation of commutative semigroups // Semigroup Forum 10:3(1975), 203–214.
+[19] M. Zarichnyi, A. Teleiko, Semigroups and monads // in: Algebra and Topology, Lviv Univ. Press, 1996, P.84–93 (in
+Ukrainian).
+Ivan Franko National University of Lviv, Ukraine
+Vasyl Stefanyk Precarpathian National University, Ivano- Frankivsk, Ukraine
+E-mail address :t.o.banakh@gmail.com
+E-mail address :vgavrylkiv@yahoo.com
\ No newline at end of file
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+arXiv:1001.0558v2  [gr-qc]  5 Jan 2010The Effect of the Tides on the LIGO Interferometers
+A. C. Melissinos, for the LIGO Scientific Collaboration
+Department of Physics and Astronomy, University of Rochest er,
+Rochester NY 14627, USA
+To appear in the Proceedings of the 12th Marcel Grossman Meet ing, Paris, July 2009
+LIGO Document P0900257-v3
+The LIGO interferometers [1] can record the dark port signal at a sampling rate as high as
+262.144 kHz. Of interest is the region around the 1st free spe ctral range (fsr) of the 4 km inter-
+ferometers, ffsr=c/2L= 37.52 kHz. The dominant optical field in the interferometer, E0, is at
+the carrier frequency f0and has narrow width ∆ f0∼1−2 Hz. In addition in the interferometer
+circulate sideband fields E±(f) centered at f±=f0±ffsr. The sideband fields have width typical
+of the arm cavity resonance, ∆ f±∼200 Hz, and peak amplitude density E±(f)∼10−7E0/√
+Hz.
+When the interferometer is locked, the field E0is on a dark fringe, and at the antisymmetric port
+(AS), the phaseshift at the carrier frequency, ∆ φ0= 0. However this does not hold for the sideband
+fieldsE±(f) because the lengths LxandLyof the two arms are not equal. Instead the phase shift
+at frequency f±=f0±ffsris finite and given by ∆ φ±=±2π(Lx−Ly)/L, whereLx−Ly∼2
+cm. The fields E±(f) are mixed with the radio frequency sidebands, and the photo current is de-
+modulated in the usual way. This gives rise to a signal center ed atffsrand of amplitude density
+h(f) proportional to the phase shift ∆ φ±and the field density E±(f),h(f) =C|∆φ±||E±(f)|. The
+uncalibrated frequency spectrum in counts/√
+Hz , in the region of the fsr frequency is shown in
+Fig.1. The enhancement follows the spectral shape of E±(f).
+The data are available as a time series sampled at 1024 Hz and r estricted to the frequency range
+37,504±1024 Hz. The data are from the S5 science run and were written i n 64 s long frames. For
+each frame we obtain the frequency spectrum at a resolution B W = 0.125 Hz and integrate the
+power spectral density (PSD) in the region ffsr±200 Hz. A filter proportional to the interferometer
+transfer function at the fsr is applied. This gives a time ser ies of the power at the fsr, sampled
+every 64 s. The amplitude of this signal is proportional to th e sum of the “biasing” phase shift
+∆φ±given above, and of any time-dependent phase shift ∆ φtat the sideband frequencies f±. Thus
+the observed (integrated) power is
+P=/integraldisplay
+(PSD)df=/braceleftbig
+(∆φ±)2+2∆φ±∆φt+(∆φt)2/bracerightbig
+|C|2/integraldisplay
+|E±(f)|2df (1)
+Since ∆φtis less than few percent of ∆ φ±, the last term is negligible and the time-dependent
+modulation of the power is
+∆P/P= 2|∆φt|/|∆φ±| (2)
+Measurement of ∆ Pcombined with a knowledge of ∆ φ±, (i.e. of ∆ L), suffices to obtain ∆ φt.
+A plot of the power for a 16 month period (April 2006 to August 2 007) for the Hanford H1
+interferometer is shown in Fig.2. The modulation at the dail y and twice daily frequencies is clearly
+1seen in the inset. The 16-month time series was spectrally an alyzed and contains the tidal frequen-
+cies as shown in Figs 3(a,b). To within the measurement resol ution of ∆ fres= 6×10−9Hz there
+is exact agreement between the measured and expected values . The modulation of the power is
+shown as a function of time for a five day period in December 200 6 in Fig.4a, and for April 2007 in
+Fig.4b. The blue curves give the relative amplitude of the ho rizontal component of the tidal forces
+at the Hanford site for the same time period.
+References
+[1] B.Abbott et al. Nucl. Instrum. Methods, A 517154 (2004).
+3.663.683.73.723.743.763.783.83.823.84
+x 10410−1100101102
+Frequency (Hz)h(f), 1/sqrt(Hz)
+Figure 1: Uncalibrated amplitude spectral density for the H 1 interferometer in the region of its
+free spectral range.
+2Figure 2: Integrated power in the free spectral range (fsr) r egion as a function of time, April 2006
+to July 2007. The data are for the H1 interferometer and are sa mpled every 64 s. Note the daily
+and twice-daily modulation that can be seen in the inset.
+31.05 1.1 1.15 1.2 1.25 1.3
+x 10−50500100015002000250030003500
+Frequency (Hz)PowerX: 1.157e−005
+Y: 3268
+X: 1.076e−005
+Y: 1067
+Figure 3: Frequency spectrum of the integrated fsr power in t he diurnal region. Note the fine
+structure.
+2.1 2.15 2.2 2.25 2.3 2.35 2.4
+x 10−505001000150020002500
+X: 2.315e−005
+Y: 990.2
+Frequency (Hz)PowerX: 2.236e−005
+Y: 2322
+X: 2.192e−005
+Y: 263.4
+Figure 4: Frequency spectrum of the integrated fsr power in t he twice daily region. Note the fine
+structure.
+48.4918.4915 8.4928.4925 8.4938.4935 8.4948.4945
+x 108−0.1−0.0500.050.1Relative Amplitude
+  
+GPS Time
+Figure 5: Modulation of the integrated fsr power for five days in December 2006. The blue curves
+give the relative amplitude of the horizontal component of t he tidal forces at the Hanford site for
+the same time period.
+8.5685 8.5698.5695 8.578.5705 8.5718.5715 8.5728.5725
+x 108−0.1−0.0500.050.1Relative Amplitude
+  
+GPS Time
+Figure 6: Modulation of the integrated fsr power for five days in March 2007. The blue curves give
+the relative amplitude of the horizontal component of the ti dal forces at the Hanford site for the
+same time period.
+5
\ No newline at end of file
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+arXiv:1001.0559v1  [math.CV]  4 Jan 2010JULIUS AND JULIA:
+MASTERING THE ART OF THE SCHWARZ LEMMA
+HAROLD P. BOAS
+Abstract. Thisarticlediscussesclassical versionsoftheSchwarzle mma
+at the boundary of the unit disk in the complex plane. The expo sition
+includes commentary on the history, the mathematics, and th e applica-
+tions.
+1.Introduction.
+Despiteteachingcomplex analysisforaquartercentury, Is tilldidn’tknow
+Jack about the Schwarz lemma. That’s what I found out after a g raduate
+student buttonholed me at the door of the printer room.
+“Oh, professor, I noticed while studying for the qualifying exam that the
+bookof Bak andNewmanattributes the‘anti-calculus propos ition’ toErd˝ os,
+but your father’s book cites Loewner. How come?”
+Since I was on the verge of sending to the production staff the s ource
+files of the new edition of my father’s Invitation to Complex Analysis [8],
+the question demanded my instant attention. I soon located t he following
+statement in Complex Analysis [6] by the late Donald J. Newman (1930–
+2007) and his student Joseph Bak.
+Anti-calculus Proposition ([6, p. 78], attributed to Erd˝ os) .Supposefis
+analytic throughout a closed disc and assumes its maximum mo dulus at the
+boundary point α. Thenf′(α)/ne}ationslash= 0unlessfis constant.
+Here the word “analytic” (I shall use the classical synonym “ holomor-
+phic”) means that the function fis represented locally by its Taylor series
+about each point. To say that fis holomorphic at a boundary point of the
+domain means that fextends to be holomorphic in a neighborhood of that
+point.
+Bak and Newman’s title for the proposition refers to the fami liar principle
+fromrealcalculusthatthederivativemustvanishataninte riorpointwherea
+differentiable function attains a maximum. But the name seems inapposite
+to me, for the derivative need not vanish at a boundary extremum. The
+attribution too is suspect; perhaps Newman learned the resu lt from Paul
+Erd˝ os (1913–1996), but the statement seems to have been kno wn several
+years before Erd˝ os was born.
+Date: 23 December 2009.
+2010Mathematics Subject Classification. Primary: 30-03; Secondary: 01A60.
+12 HAROLD P. BOAS
+I donot mean to criticize the book of Bak and Newman. Indeed, t he book
+is an excellent one according to the following criteria: the glowing review in
+thisMonthly [50], theexistenceofaGreek translation, andtheobservat ion
+that the copy in my university’s library is falling apart fro m overuse.
+In any case, the proposition is an arresting statement—thou gh far from
+optimal—and I learned a lot trying to discover who first prove d it, and
+why. My aim here is to clarify both the history and the mathema tics of
+the proposition and its thematic offshoots by taking a tour thr ough some
+variationsoftheSchwarzlemma. Bytheendofthetour, youwi llunderstand
+both my title1and the pun in my opening sentence.
+Butthestoryisnotentirely lighthearted. Wemathematicia ns areengaged
+in a social enterprise, and world events can shake the founda tions of our
+ivory tower. I include some biographical remarks about the p articipants
+in my tale, for they lived in “interesting times,” and the non mathematical
+aspects of their lives should be remembered along with their theorems.
+2.How the lemma got its name.
+Some preliminary normalizations will expose the key issue i n the proposi-
+tion from the introduction. Replacing f(z) byf(z+c), wherecis the center
+of the initial disk, one might as well assume from the start th at the disk is
+centered at the origin. Having made that simplification, con sider the func-
+tionFthat sendsapoint zinthestandardclosed unitdisk {z∈C:|z| ≤1}
+to thevalue f(αz)/f(α). This function Fmaps the unit disk into itself; fixes
+the boundary point 1, where the maximum modulus of Fis attained; and
+has the property that
+(1) F′(1) =αf′(α)/f(α).
+Thusthepropositionreducestoapropertyofthederivative ofaholomorphic
+self-mapping of the unit disk at a boundary fixed point.
+Every beginning course in complex analysis contains a versi on of the fol-
+lowing statement about the derivative of a holomorphic self -mapping of the
+unit disk at an interior fixed point.
+Schwarz lemma. Supposefis a holomorphic function in the open unit
+disk,f(0) = 0, and the modulus of fis bounded by 1. Then either |f′(0)|<
+1and|f(z)|<|z|when0<|z|<1, or elsefis a rotation (in which case
+|f′(0)|= 1and|f(z)|=|z|for allz).
+Thisunassumingstatementhasturnedouttobeextremelyfru itful,spawn-
+ing numerous generalizations, at least two books [5, 21], an d research that
+continues to this day. Influential extensions by Ahlfors [2] and by Yau [49] to
+complex manifolds satisfying suitable curvature hypothes es are widely cited
+in the current literature. MathSciNet and Zentralblatt MAT H list twenty
+1Note to readers from future generations: When I wrote this ar ticle, movie theaters
+were showing the motion picture Julie and Julia , an exaltation of American cultural icon
+Julia Child, author of Mastering the Art of French Cooking .JULIUS AND JULIA: MASTERING THE ART OF THE SCHWARZ LEMMA 3
+articles in the recent five-year period 2001–2005 with “Schw arz lemma” in
+the title.
+The lemma is named for the German mathematician Hermann Aman -
+dus Schwarz (1843–1921), who is the eponym of various other a nalytical
+objects, including the Cauchy–Schwarz inequality, the Sch warz–Christoffel
+formula in conformal mapping, the Schwarzian derivative, a nd the Schwarz
+reflection principle. Every calculus student ought to know a bout Schwarz’s
+example [41, pp. 309–311] showing that the surface area of a c ylinder is not
+equal to the supremum of the areas of inscribed polygonal sur faces.
+TheGreek–Germanmathematician ConstantinCarath´ eodory (1873–1950)
+named and codified the Schwarz lemma in [14, p. 110] (having pr eviously
+usedthelemmain[12] and[13]withoutgivingit atitle). The autobiographi-
+cal notes in Carath´ eodory’s collected papers include the f ollowing account of
+his rediscovery of the lemma [17, p. 407]; the date of the indi cated events is
+1905, the year after Carath´ eodory defended his dissertati on on the calculus
+of variations.
+Shortly after the end of the semester, P. Boutroux came to
+G¨ ottingen for a few days, and during a walk he told me
+about his attempt to simplify Borel’s proof of Picard’s the-
+orem, which was all the rage then. Boutroux had noticed
+that this proof works only because of a remarkable rigid-
+ity in conformal mapping, but he was unable to quantify
+the effect. I could not get Boutroux’s observation out of
+my thoughts, and six weeks later, I was able to prove Lan-
+dau’s sharpened version of Picard’s theorem in a few lines by
+means of the propositionthat nowadays is called the Schwarz
+Lemma. I derived this lemma by using the Poisson integral;
+then I learned from Erhard Schmidt, to whom I communi-
+cated my discovery, not only that the statement is already
+in Schwarz but that it can be proved in a very elementary
+way. Indeed, the proof that Schmidt told me can hardly be
+improved. This is how I arrived at another line of research
+besides the calculus of variations.2
+Just reading the names in this paragraph sends shivers down m y spine:
+truly, giants walked the earth in those days. Carath´ eodory himself subse-
+quently became a leading mathematician in Germany, just beh ind David
+Hilbert (1862–1943) in stature. Indeed, at the time of the bl ow-up in 1928
+between Hilbert and L. E. J. Brouwer (1881–1966) that forced the reorgani-
+zation of the editorial board of Mathematische Annalen [45], Carath´ eodory
+was one of the chief editors of the journal, the others being H ilbert himself,
+Albert Einstein (1879–1955), and the managing editor, Otto Blumenthal
+(1876–1944).
+2The free translation from German into English is mine.4 HAROLD P. BOAS
+Thus Carath´ eodory had no need to drop names, but what a list o f names
+is contained in his little story! ´Emile Borel (1871–1956) has a lunar crater
+namedafterhimandismemorialized bysuchtermsasBorelset , Heine–Borel
+theorem, and Borel–Cantelli lemma; ´Emile Picard (1856–1941) has over six
+thousand mathematical descendants listed in the Mathemati cs Genealogy
+Project and is remembered for his penetrating applications of Schwarz’s
+methodof successiveapproximations, forPicard’stheorem s incomplexanal-
+ysis, for the Picard group, and so on; EdmundLandau (1877–19 38) has been
+an idol to generations of analysts andnumbertheorists for h iselegant books;
+Erhard Schmidt (1876–1959), a great friend of Carath´ eodor y from their stu-
+dent days together in Berlin (where they attended lectures b y Schwarz), has
+both Hilbert–Schmidt operators and the Gram–Schmidt ortho normalization
+process named after him.
+And what of the French scholar Pierre Boutroux? His consider able scien-
+tific contributions are largely overshadowed by the work of t wo prominent
+relatives: his father was the distinguished philosopher an d historian of sci-
+ence´Emile Boutroux (1845–1921), while his uncle was the incompa rable
+mathematician Henri Poincar´ e (1854–1912). Nonetheless, Pierre Boutroux
+(1880–1922) was himself a sufficiently prominent mathematic ian that Henry
+Burchard Fine (1858–1928) hired him at Princeton Universit y in 1913, al-
+though Boutroux soon left to join the French army. He survive d World
+War I but died young, outliving his father by only a few months .
+Now the special case of the lemma that Schwarz handled appear s in the
+notes [41, pp. 109–111] from his lectures during 1869–70, bu t as far as I
+know, these notes did not see print until the publication of S chwarz’s col-
+lected works in 1890. In 1884, Poincar´ e too proved a special case (using an
+argument similar to the modern one), which he singled out as a fundamental
+lemma in a paper on fuchsian groups [38, p. 231]. Schmidt’s el egant proof
+of the Schwarz lemma, now canonical, is to apply the maximum p rinciple to
+the function f(z)/zafter filling in the removable singularity at the origin.
+A standard illustration [18, §69] of the power of the Schwarz lemma is the
+following proof of Liouville’s theorem stating that the only bounded entire
+functions (functions holomorphic in the entire complex plane) are constants .
+After making affine linear transformations of the variables i n both the do-
+main and the range,3one can assume that the entire function ffixes the
+point 0 and has modulus bounded by 1; the goal now is to show tha t such
+a function is identically equal to 0. For an arbitrary positi ve numberR, the
+functionthat takes ztof(Rz)satisfies thehypothesisoftheSchwarz lemma,
+so|f(Rz)| ≤ |z|when|z|<1. Replace Rzbywto see that |f(w)| ≤ |w|/R
+when|w|< R. Keeping wfixed, letRtend to infinity to deduce that
+f(w) = 0 for every point win the complex plane.
+3The corresponding argument in [35] elides the translation i n the range.JULIUS AND JULIA: MASTERING THE ART OF THE SCHWARZ LEMMA 5
+3.Change of base point.
+The original version of the Schwarz lemma applies to a holomo rphic func-
+tion from the unitdisk into itself that fixes theorigin. If in stead the function
+fixes some other interior point ζ, can something still be said?
+The answer is simple to find if one knows that the disk is “symme tric” in
+the sense that there is a self-inverse holomorphic mapping o f the disk that
+interchanges two specified points inside the disk. For prese nt purposes, it
+suffices to have an explicit formula when one of the two points i s the origin.
+Let thesecond point insidethediskbe ζ, anddefineamapping ϕζas follows:
+ϕζ(z) =ζ−z
+1−ζz.
+Evidentlyϕζinterchanges 0with ζ, andϕζmapstheunitdiskintoitself(and
+the boundary of the disk to the boundary) because, as a routin e calculation
+shows,
+(2)/vextendsingle/vextendsingle/vextendsingle/vextendsingleζ−z
+1−ζz/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+= 1−(1−|ζ|2)(1−|z|2)/vextendsingle/vextendsingle1−ζz/vextendsingle/vextendsingle2.
+The composite function ϕζ◦ϕζmaps the unit disk into itself, fixing both 0
+andζ, so the case of equality in the Schwarz lemma implies that ϕζ◦ϕζis
+the identity function; that is, the function ϕζis its own inverse.
+The function ϕζis the required symmetry of the disk that swaps the
+points 0 and ζ. To interchange two general points ηandζof the unit disk,
+one can use the composite function ϕζ◦ϕϕζ(η)◦ϕζ.
+Now iffmaps the unit disk into itself, fixing ζ, thenϕζ◦f◦ϕζmaps the
+unit disk into itself, fixing 0. The Schwarz lemma implies tha t the derivative
+(ϕζ◦f◦ϕζ)′(0) has modulus no greater than 1. By the chain rule, this
+derivative equals ϕ′
+ζ(ζ)f′(ζ)ϕ′
+ζ(0). Butϕ′
+ζ(ζ)ϕ′
+ζ(0) = 1, since ϕζ◦ϕζis
+the identity function. The conclusion is that |f′(ζ)| ≤1, regardless of the
+location of the interior fixed point ζ.
+More generally, if the point ζis arbitrary (not necessarily fixed by f),
+then the Schwarz lemma applies to the composite function ϕf(ζ)◦f◦ϕζ, so
+|ϕ′
+f(ζ)(f(ζ))f′(ζ)ϕ′
+ζ(0)| ≤1,
+and computing the derivatives shows that
+(3) |f′(ζ)| ≤1−|f(ζ)|2
+1−|ζ|2.
+The other conclusion of the Schwarz lemma is that
+|ϕf(ζ)◦f◦ϕζ(z)| ≤ |z|when|z|<1,
+or, sinceϕζis self-inverse,
+(4) |ϕf(ζ)◦f(z)| ≤ |ϕζ(z)|when|z|<1.6 HAROLD P. BOAS
+Written out explicitly, this inequality says the following :
+(5)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(ζ)−f(z)
+1−f(ζ)f(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleζ−z
+1−ζz/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
+Carath´ eodoryalreadymentionsthisideaofcomposingwith aholomorphic
+automorphism of the disk in the paper where he names the Schwa rz lemma
+[14,§6], but the first source I know where inequality (5) is written out is [29,
+p. 71] by the French mathematician Gaston Julia (1893–1978) . In Julia’s
+long essay (about which more later), the inequality is part o f a preliminary
+section in which he collects various auxiliary results.
+Inequality (5) is commonly called the “Schwarz–Pick lemma” after the
+Austrian mathematician Georg Pick (1859–1942), who wrote a n influential
+paper [37] on the subject two years before Julia’s work. Pick ’s innovation is
+not so much the inequality (which he does not write explicitl y) as the point
+of view, decisive for subsequent developments in the field: h is interpreta-
+tion is that holomorphic mappings decrease the hyperbolic, non-Euclidean
+distance in the unit disk (the so-called Poincar´ e metric). A monotonically
+increasing function of the quantity on the right-hand side o f inequality (5),
+the hyperbolic distance is invariant under rotations and un der all the map-
+pingsϕζ.
+Pick is remembered today also for Nevanlinna–Pick interpol ation; for
+Pick’s theorem about the area of lattice polygons; and for hi s numerous
+mathematical descendants, the majority of them through his mathematical
+grandson Lipman Bers (1914–1993), who finished his disserta tion in Prague
+in1938justintimetofleethecountrybeforetheGermanoccup ationof1939.
+At that time Pick had long since retired, and he was trapped in Prague; he
+died soon after being interned in the Theresienstadt concen tration camp.
+Whenζis a fixed point of f, Pick’s observation provides a simple pictorial
+representation of inequality (5). Setting |ϕζ(z)|equal to some constant less
+than 1 restricts the point zto lie on a certain Euclidean circle. This circle
+is also a non-Euclidean circle, and its non-Euclidean cente r is the point ζ.
+Whenf(ζ) =ζ, inequality (5) says that fmaps points inside this circle to
+other points inside the circle: that is, fcontracts the disk bounded by the
+circle. Figure 1 illustrates several such circles that fcontracts.
+4.Straightening out the problem.
+There is nothing essential about working on the standard uni t disk; one
+can easily transform the preceding results to apply to a disk with an ar-
+bitrary center and an arbitrary radius. It may be less obviou s that one
+can equivalently work on the upper half-plane (thought of as a disk with a
+boundary point at infinity); it will be useful to know that one can switch
+back and forth between the two settings.JULIUS AND JULIA: MASTERING THE ART OF THE SCHWARZ LEMMA 7
+01ζ
+Figure 1. Some circles with non-Euclidean center ζ.
+One learns in a first course in complex analysis that if ζis a point in the
+upper half-plane (that is, Im ζ >0), and
+ψζ(z) =z−ζ
+z−ζ,
+thenψζis an invertible holomorphic function of zmapping the upper half-
+plane onto the unit disk, taking ζto 0. In fact,
+(6)/vextendsingle/vextendsingle/vextendsingle/vextendsinglez−ζ
+z−ζ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+= 1−2(Imz)(Imζ)/vextendsingle/vextendsinglez−ζ/vextendsingle/vextendsingle2,
+soψζdoes map the upper half-plane into the unit disk; one can see t hat the
+map is bijective by exhibiting the inverse:
+ψ−1
+ζ(w) =ζ−ζw
+1−w,and Imζ−ζw
+1−w=(Imζ)(1−|w|2)
+|1−w|2.
+Now iffis a holomorphic function that maps the upper half-plane int o
+itself, andζis an arbitrary point in the upperhalf-plane, then the compo site
+functionψf(ζ)◦f◦ψ−1
+ζmaps the unit disk to itself, fixing the point 0.
+The derivative ( ψf(ζ)◦f◦ψ−1
+ζ)′(0) equalsψ′
+f(ζ)(f(ζ))f′(ζ)/ψ′
+ζ(ζ), and one
+computes that ψ′
+ζ(ζ) = (2iImζ)−1, so the Schwarz lemma implies that
+(7) |f′(ζ)| ≤Imf(ζ)
+Imζwhen Imζ >0.
+Absolute-value signs are not needed on the right-hand side, for both the
+numerator and the denominator are positive real numbers. If ζhappens to
+be a fixed point of f, then|f′(ζ)| ≤1, just as in the case of a disk of finite
+radius.
+The bound (7) on |f′(ζ)|looks a bit simpler than the corresponding
+bound (3) in the case of the unit disk; an analogous simplifica tion occurs for8 HAROLD P. BOAS
+the inequality involving the function fitself. In the setting of the preceding
+paragraph, the Schwarz lemma implies additionally that
+(8)|ψf(ζ)◦f(z)| ≤ |ψζ(z)|,or/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(z)−f(ζ)
+f(z)−f(ζ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsinglez−ζ
+z−ζ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+whenζandzlie in the upper half-plane. As far as I know, Julia was the
+first to write this half-plane version of the Schwarz–Pick le mma explicitly
+[29, p. 71]; an equivalent form, in view of equation (6), is th at
+(9)(Imf(z))(Imf(ζ))/vextendsingle/vextendsingle/vextendsinglef(z)−f(ζ)/vextendsingle/vextendsingle/vextendsingle2≥(Imz)(Imζ)/vextendsingle/vextendsinglez−ζ/vextendsingle/vextendsingle2.
+Raised in Algeria, Julia went to Paris on a scholarship, but W orld War I
+interrupted his education. Serving on the front lines in 191 5, he was griev-
+ously wounded, shot in the middle of his face. Julia finished h is dissertation
+during the long recuperation from his injury and the subsequ ent unsuccess-
+ful operations to repair the damage (for the rest of his life, he wore a leather
+strapacross his face to hidethemissingnose[34]). Thenhes tudied iteration
+of rational functions in the plane and wrote a monograph on th e subject for
+a competition announced by the Acad´ emie des Sciences; this work won the
+1918 Grand Prix. Julia’s prize-winning essay subsequently fell into relative
+obscurity until it was revived in the 1970s by Benoit Mandelb rot in his cel-
+ebrated work on fractals. Nowadays computer-generated ima ges of “Julia
+sets” are ubiquitous.
+Of primary interest here is the section in Julia’s article de voted to the
+Schwarz lemma [29, pp. 67–78], which is only a small part of th e nearly
+200-page work. Julia perhaps recognized the independent si gnificance of
+this part of his treatise, for he later published a version of this section as a
+stand-alone note [30].
+5.Fixed point on the boundary.
+The discussion so far is, for Julia, child’s play. But now he i s ready to
+presenthisextension oftheSchwarzlemma. Hesupposesthat anonconstant
+holomorphic function fmaps a disk into itself and has a fixed point on
+the boundary . For this situation to make sense, the function fought to
+be defined on the boundary, or at least at the fixed point. Julia assumes
+initially that fis holomorphic on the closed disk, and he finds it convenient
+to work on the half-plane model of a disk, as in the preceding s ection. After
+making a translation, one might as well take the boundary fixe d point to be
+the origin.
+Nowf(z) =f′(0)z+O(z2). Evidently f′(0) cannot be equal to 0, for if
+f(z) were to behave locally near the origin like a square (or a hig her power),
+thenfwould multiply angles by a factor of 2 (or more), contradicti ng the
+hypothesisthat fmapstheupperhalf-planeintoitself. Multiplication byth e
+nonzerocomplex number f′(0) correspondsto a dilation by |f′(0)|composedJULIUS AND JULIA: MASTERING THE ART OF THE SCHWARZ LEMMA 9
+with a rotation by arg f′(0). Sincefmaps the upper half-plane into itself,
+the rotation must be the identity. Thus f′(0) is actually a positive real
+number. By the method of the preceding section, this conclus ion carries
+over to the setting of the standard unit disk.
+The preceding observations prove the proposition from the i ntroduction,
+andmore. Inviewofthechain-rulecomputation(1), notonly doesanoncon-
+stant holomorphic function fon a disk centered at the origin have nonzero
+derivative at a boundary point αwhere the modulus |f|takes a maximum,
+but in addition the expression αf′(α)/f(α) is a positive real number.
+Thequantity αf′(α)/f(α) hasacompellinggeometric interpretation. The
+functionfmaps the boundary circle onto a curve that lies inside a cir-
+cle centered at the origin with radius |f(α)|and touches this circle at the
+pointf(α). The complex number iαf′(α) represents a vector tangent to the
+curve atf(α), and an outward normal vector is −itimes this quantity, or
+αf′(α). But the curve is tangent to the circle at f(α), so the normal vector
+to the curve points in the same direction as the radius vector of the circle,
+which isf(α). Thus the positivity of the ratio αf′(α)/f(α) expresses the
+tangency of the image curve with the enclosing circle.
+Although Julia carries out most of his analysis in the upper h alf-plane,
+he states explicitly that the positivity of the derivative a t a boundary fixed
+point holds for an arbitrary disk. Actually, the whole devel opment so far is
+as easy as pie for Julia, but now comes the pi` ece de r´ esistan ce.
+TheSchwarz–Pick lemmafortheupperhalf-planeyieldsthei nequality (9)
+for values of the function f; what more can be said if the boundary point 0
+is a fixed point? Simply taking the limit in (9) as ζapproaches 0 yields a
+trivial inequality, for both sides reduce to 0. But useful in formation can be
+extracted from(9) by consideringthe rateat whichthetwo sidesapproach 0.
+Keepingzfixed, replace f(ζ) byf′(0)ζ+O(ζ2), divide both sides of (9)
+by Imζ, and letζapproach 0 along the imaginary axis. What results is the
+following fundamental inequality of Julia:
+(10) f′(0)Imf(z)
+|f(z)|2≥Imz
+|z|2when Imz >0.
+Julia actually writes the equivalent statement that
+(11) Im1
+f(z)≤Im1
+f′(0)z.
+Since Im(1/z) =−Im(z)/|z|2, with a minus sign, inequality (11) has the
+direction reversed relative to inequality (10). Julia’s fo rm of the inequality
+reveals that if equality holds at even one point in the upper h alf-plane, then
+the harmonic function
+Im/parenleftbigg1
+f(z)−1
+f′(0)z/parenrightbigg
+attains a maximum and so is constant; therefore fis a M¨ obius transforma-
+tion (a rational function of degree 1).10 HAROLD P. BOAS
+ζf(ζ)
+rR
+pImζImf(ζ)
+Figure 2. Two congruent non-Euclidean disks.
+Notice that the preceding derivation of Julia’s inequality does not really
+require the function fto be defined on the boundary. As long as there
+is a sequence of points ζnin the upper half-plane tending to 0 such that
+f(ζn)→0 and the fraction Im( f(ζn))/Im(ζn) stays bounded above, the
+same argument implies that inequality (10) holds with f′(0) replaced by
+liminf n→∞{Im(f(ζn))/Im(ζn)}.
+The proof that I have given is not Julia’s. His illuminating g eometric
+argument is the following. Consider in the upper half-plane two disks with
+equal non-Euclidean radius, one centered at a point ζand the other centered
+atf(ζ); see Figure 2. The Schwarz–Pick lemma implies that the imag e of
+the firstdisk under the holomorphic function flies inside the second disk. A
+suitable dilation with center at the indicated point pon the real axis takes
+the pointζto the point f(ζ). Being an invertible holomorphic mapping,
+this dilation preserves the non-Euclidean distance and so m aps the first disk
+precisely onto the second disk. Accordingly, the ratio R/rof the Euclidean
+radii equals the dilation factor (Im f(ζ))/(Imζ). Now suppose ζlies on
+the imaginary axis, and let ζmove vertically down to the origin while the
+Euclidean radius rstays fixed. The variable quantity R/rthen tends to
+a limit that equals the limit of (Im f(ζ))/(Imζ), orf′(0). See Figure 3,
+where the point ζhas disappeared in the limit, and I have introduced a new,
+arbitrary point z. After mapping each point to the negative of its reciprocal,
+one sees in the right-hand part of the figure that dilating the point−1/f(z)
+by thefactor R/rgives a point that lies above the horizontal linewith height
+equal to Im( −1/z). ButR/r=f′(0), sof′(0)Im(−1/f(z))≥Im(−1/z),
+which is Julia’s inequality again.
+Julia has no need to write out the equivalent inequality for a holomor-
+phic function fmapping the unit disk into itself, fixing the point 1, but aJULIUS AND JULIA: MASTERING THE ART OF THE SCHWARZ LEMMA 11
+rR
+zf(z)
+0via inversion−−−−−−−→
+w/mapsto→−1/w1
+2r
+1
+2R
+0−1/z
+−1/f(z)
+Figure 3. Inverting the limiting configuration.
+routine computation (conjugating with the mapping −ψiin the notation of
+Section 4) leads to the following result:
+(12) f′(1)≥1−|z|2
+|1−z|2/slashbigg1−|f(z)|2
+|1−f(z)|2when|z|<1.
+Carath´ eodory makes this inequality explicit in [15].
+For more on Carath´ eodory’s life, consult the meticulously researched
+biography [23], but watch out for the author’s occasional la pses in Eng-
+lish. For instance, the author says that Carath´ eodory marr ied “his aunt”
+[23, p. 51], when actually the relationship was that of secon d cousins once
+removed. (Carath´ eodory’s great-grandfather Antonios wa s the brother of
+Carath´ eodory’s wife’s grandfather Stephanos.)
+6.Multiple fixed points.
+The case of equality in the Schwarz lemma implies that the onl y holo-
+morphic self-mapping of the unit disk with two interior fixed points is the
+identity function. On the other hand, there are nontrivial h olomorphic self-
+mappingsofdisksthatfixaninteriorpointtogether withone ormorebound-
+ary points. For example, the mapping that sends ztoz3takes the unit disk
+into itself and fixes both the interior point 0 and the boundar y points 1
+and−1. Transferring such a situation to the upper half-plane and apply-
+ing Julia’s fundamental inequality (10) with zequal to the interior fixed
+point shows immediately that the derivative at each boundar y fixed point
+is greater than or equal to 1; in fact, the derivative is stric tly greater than 1
+unless the function is the identity.
+In the case of a disk centered at the origin, the rescaling tri ck of Section 2
+leaves the origin fixed, so the chain-rule calculation (1) im plies the following
+statement.
+Theorem (afterJulia) .If a holomorphic function f(not identically 0) maps
+a closed disk centered at 0into itself, fixing 0, and if the modulus |f|attains12 HAROLD P. BOAS
+a maximum on the closed disk at the boundary point α, thenαf′(α)/f(α)is
+a real number greater than or equal to 1.
+That this theorem follows immediately from Julia’s inequal ity is not a
+new observation [40, p. 524], but I have not determined to my s atisfaction
+where the theorem was first written down in this form.
+The unscaled case when α= 1 =f(α) appears as problem 291 in the
+famous problem book of P´ olya and Szeg˝ o [39], the first editi on of which was
+published in German in 1925. In the solution, the authors ind icate that
+the problem was contributed4by Charles Loewner.5It is no surprise that
+Loewner was thinking along these lines, for he was a student o f Pick, and
+Loewneraddressesrelated questionsinthearticle [32] inw hichheintroduces
+his famous differential equation for studying schlicht funct ions.
+W. K. Hayman, writingafter World War II,makes therelated ob servation
+that “the points where f(z) attains its maximum modulus lie among the
+points where z0f′(z0)/f(z0) is real” [25, p. 137], and he attributes the idea
+(thoughnottheformula) toBlumenthal’s remarkable1907 pa per[7], astudy
+of the differentiability properties of the maximum-modulus f unction. I can
+extractatleastthepropositionintheintroductionfromth atpaper[7,p.223,
+formula (7)].
+I learned from thenotes in R. B. Burckel’s encyclopedic trea tise [9, p. 216]
+that the scaled formulation of the theorem, essentially as I have given it
+above, appears in a 1971 paper by I. S. Jack [27, Lemma 1]; the p aper
+is the unique item listed in MathSciNet for this author. The s ubsequent
+literature on univalent functions commonly refers to the th eorem as “Jack’s
+lemma.”6Onthebasisof thiscontribution, I. S.Jack hasanimpressiv eratio
+of citations to published articles (more than 250 hits in Goo gle Scholar).
+The theorem can be sharpened by invoking Schmidt’s trick. If fmaps
+the unit disk into itself, fixing both 0 and 1, then introduce t he function g
+such thatg(z) =f(z)/zwhenz/ne}ationslash= 0 andg(0) =f′(0). For a suitable choice
+of angleθ, the function eiθϕg(0)◦gagain maps the unit disk into itself, fixing
+both 0 and 1, so the derivative at 1 is no smaller than 1. Transl ating this
+conclusion to a statement about fby a routine calculation shows that
+f′(1)≥2
+1+|f′(0)|.
+4Intheauthorindex,theentrythatpointstothesolutionhas thepagenumberin italics,
+and a note at the beginning of the index indicates that this ty pographical convention
+indicates “original contributions.”
+5They actually say “K. L¨ owner,” his name at the time. Loewner was caught in Prague
+when the Germans occupied Czechoslovakia in 1939, but nonet heless he succeeded in
+emigrating. His student Lipman Bers explains [33, p. vii] th at Loewner’s “decision to
+Americanize a name already well known to mathematicians the world over was a sign of
+the 46-year-old scholar’s determination to begin a new life in the United States.”
+6Some authors use the name “Clunie–Jack lemma” on the strengt h of Jack’s comment,
+“we use a method due to Professor Clunie.” It seems that Cluni e himself accepted this
+designation while also acknowledging antecedents [19, p. 1 39].JULIUS AND JULIA: MASTERING THE ART OF THE SCHWARZ LEMMA 13
+Since|f′(0)| ≤1, this inequality refines the assertion that f′(1)≥1. This
+improvement was obtained in [44] by Helmut Unkelbach (1910– 1968); it was
+rediscovered more than half a century later in [36].
+7.Julius has a point.
+A pleasing counterpart to the preceding section occurs when all the fixed
+points lie on the boundary. In this case there is a distinguis hed fixed point
+at which the derivative is a positive number less than or equa l to 1; at all
+the other fixed points, the derivative is larger than 1 and mor eover at least
+as large as the reciprocal of the derivative at the special fix ed point.
+This proposition is a corollary of results of the Dutch mathe matician7
+Julius Wolff (1882–1945). Here is a concrete example:
+f(z) =/parenleftBigg
+z−2
+3
+1−2
+3z/parenrightBigg3
+.
+This function fmaps the unit disk into itself, fixing the four boundary
+points−1, +1, and (9 ±5i√
+7)/16; computing shows that
+f′(−1) = 3/5, f′(1) = 15,andf′((9±5i√
+7)/16) = 12/5.
+The easy part of the proposition is the statement that the pro duct of
+derivatives at two distinct boundary fixed points must be at l east 1. After
+transferring the problem to the upper half-plane, one can as sume that the
+two fixed points are 0 and 1. Now invoke Julia’s inequality (10 ), multiplying
+both sides by the positive quantity |f(z)|2/Imzand taking the limit as
+zapproaches 1 along a vertical line. Since f(1) (which equals 1) and f′(1)
+are real numbers,
+Imf(z)
+Imz=Im[f(z)−f(1)]
+Im(z−1)=f′(1)Im(z−1)+O((z−1)2)
+Im(z−1),
+so it follows that f′(0)f′(1)≥1. I will show at the end of the proof that the
+derivatives f′(0) andf′(1) cannot both be equal to 1.
+Oneof the objectives of Wolff’s theory is to eliminate the ass umption that
+the function is defined on the boundary. To accommodate this g eneralized
+setting, one can declare a boundary point αto be a fixed point of fwhen
+αis equal to the limit of f(z) aszapproaches αalong the direction normal
+to the boundary.
+Suppose, then, that fis a holomorphic function mapping the open upper
+half-plane into itself. Wolff observes [47] that the quotien t (Imf(z))/(Imz)
+increaseswhen zmoves downavertical line(whereRe zisconstant). Indeed,
+letzbe the point in Figure 2 at the top of the circle with Euclidean radiusr.
+As observed in Section 5, the image point f(z) lies inside or on the indicated
+7This Julius Wolff should not be confused with the German ortho pedic surgeon of the
+same name who lived 1836–1902, nor with the German novelist a nd lyricist who lived
+1834–1910.14 HAROLD P. BOAS
+circle with Euclidean radius R. Hence the ratio (Im f(z))/(Imz) is less than
+or equal to the dilation ratio (Im f(ζ))/(Imζ). Alternatively, one can verify
+Wolff’s observation by a calculus argument. If ydenotes the imaginary part
+ofz, then
+∂
+∂y/parenleftbiggImf(z)
+Imz/parenrightbigg
+=1
+Imz/parenleftbigg
+Ref′(z)−Imf(z)
+Imz/parenrightbigg
+.
+But Ref′(z)≤ |f′(z)|, so inequality (7) implies that the derivative on the
+left-hand side is negative or zero. Consequently, when Im zdecreases to 0
+with Rezfixed, the ratio (Im f(z))/(Imz) increases either to a positive limit
+or to +∞.
+Next suppose that there are no interior fixed points. For this part of the
+argument, it is convenient to work on the unit disk. Wolff [46] adapts an
+elegant idea that Arnaud Denjoy (1884–1974) used [20] to sim plify the proof
+of an earlier result of Wolff. When fis a holomorphic function mapping the
+open unit disk into itself, and nis a positive integer, consider the function fn
+defined byfn(z) = (1−1
+n)f(z). There are various ways to see that fnhas a
+fixed point αninside the unit disk: Brouwer’s fixed-point theorem (apply i t
+to a closed disk of radius close to 1), the contraction-mappi ng theorem (the
+non-Euclidean disk is a complete metric space), or Rouch´ e’ s theorem (the
+functionfn(z)−zhas the same number of zeroes inside the disk as zdoes:
+namely, one). Since αn=fn(αn), no subsequence of the αncan converge
+to an interior point αof the disk, else αwould equal f(α), contrary to the
+hypothesis that fhas no interior fixed point. Thus |αn| →1 asn→ ∞.
+Version (4) of the Schwarz–Pick lemma says that
+|ϕαn(fn(z))| ≤ |ϕαn(z)|when|z|<1.
+AsobservedattheendofSection3, whatthisinequalitymean sgeometrically
+isthat foreach z, thepointfn(z) lies insideoron thecircle Cn(z) that passes
+throughzand has non-Euclidean center αn. Some subsequence of the αn
+converges to a boundary point α(after the fact, one can deduce that the
+whole sequence converges), and passing to the limit leads to the conclusion
+that the point f(z) lies inside or on the circle C(z) passing through zand
+tangent to the unit circle at α. In other words, the function fcontracts
+circles that are tangent to the unit circle at α. Figure 4 illustrates some of
+these circles, called horocycles. By identity (2), a point wlies on the circle
+Cn(z) if and only if
+1−|w|2
+|1−αnw|2=1−|z|2
+|1−αnz|2,
+sowlies on the horocycle C(z) if and only if
+1−|w|2
+|1−αw|2=1−|z|2
+|1−αz|2.
+The special boundary point αis known as the Denjoy–Wolff point. The
+linking of the names seems appropriate: Wolff had two notes in theComptes
+Renduson January 4 and January 18 of 1926, Denjoy’s follow-up was onJULIUS AND JULIA: MASTERING THE ART OF THE SCHWARZ LEMMA 15
+01α
+Figure 4. Some horocycles.
+January 25, and Wolff revisited the subject on April 12 and Sep tember 13.
+Although French, Denjoy held a professorship for five years a t Utrecht Uni-
+versity, and Wolff succeeded him in that position.
+Now it is convenient to switch back to the equivalent setting of the upper
+half-plane, placing the Denjoy–Wolff point at 0. Since fcontracts horocy-
+cles tangent to the real axis at 0, the limit of f(z) aszmoves down the
+imaginary axis to 0 equals 0; that is, the origin is a boundary fixed point in
+the generalized sense. Moreover, Im f(z)≤Imzwhenzis on the imaginary
+axis. Hence the limit of the ratio (Im f(z))/(Imz) at 0 along the imaginary
+axis (shown to exist at the beginning of the discussion) is a p ositive num-
+berβless than or equal to 1. If fextends to be continuously differentiable
+at 0, then β=f′(0). In any case, the remark preceding inequality (12)
+shows that Julia’s inequality (10) holds with βin the role of f′(0).
+One technical point needs attention to complete the proof of the proposi-
+tion that I stated at the beginning of this section. Could the re be two fixed
+points on the real axis, say 0 and 1, such that both of the deriv ativesf′(0)
+andf′(1) (more generally, the corresponding limiting ratios β) are equal
+to 1? The answer is no, because Julia’s argument from Section 5 implies
+that theholomorphicfunction fwould contract horocycles basedat 1as well
+as horocycles based at 0. Two such horocycles having Euclide an radius 1/2
+touch at one point, and this point would be an interior fixed po int off,
+contradicting the assumption that all the fixed points lie on the boundary.
+Wolff actually shows considerably more than I have stated. In particular,
+heproves in [47] that the difference quotient ( f(z)−f(α))/(z−α) has a limit
+whenzapproaches the distinguished boundary point αwithin an arbitrary
+triangle contained in the disk and with a vertex at α. This property was
+soon rediscovered by Landau and Georges Valiron (1884–1955 ) in [31] and
+by Carath´ eodory, who systematized the theory in [15] and in troduced the16 HAROLD P. BOAS
+terminology “angular derivative” (Winkelderivierte) for arbitrary boundary
+fixed points (not just the Denjoy–Wolff point).
+In summary, a holomorphic self-mapping of a disk (not equal t o the iden-
+tity function) has a distinguished fixed point—either in the interior or on
+the boundary—at which the derivative has modulus less than o r equal to 1.
+The (angular) derivative exceeds 1 at all the other fixed poin ts.
+Wolff returned to the topic in 1940, proving in [48] a refined st atement
+about theproduct of angular derivatives at two boundaryfixe d points. That
+year the German army invaded the Netherlands, and Wolff was fo rced out
+of the university six months later. He continued publishing nonetheless and
+collaborated with Blumenthal, who had taken refuge in the Ne therlands in
+1939 after six years of persecution in Germany. (All four of B lumenthal’s
+grandparentswereJewish.) ButinAprilof1943, theorderca metodeportto
+theconcentration camps all remainingpersonsin Utrecht of Jewish heritage.
+Blumenthal perished in November 1944 in Theresienstadt [22 ], and Wolff
+three months later in Bergen-Belsen [28].
+8.Boundary uniqueness.
+The case of equality in the Schwarz lemma can be viewed as sayi ng that if
+aholomorphic mappingfroma disk into itself looks like the i dentity function
+to first order at an interior point, then the mapping isthe identity function.
+Theanalogous statement ataboundarypointfails: thefunct ionthatsends z
+toz−1
+4(z−1)3maps the unit disk into itself (as a straightforward calculu s
+calculation shows), looks like the identity at the boundary point 1 not only
+to first order but even to second order, but is not the identity .
+Apparently nobody thought of looking for a general positive result about
+uniqueness at the boundary during the first half of the twenti eth century. It
+turns out that third-order tangency to the identity suffices. The following
+statementcomesfromapapermainlydevotedtomultidimensi onalboundary
+uniqueness problems.
+Theorem (Burns and Krantz [11, Theorem 2.1, p. 662]) .Letφbe a holo-
+morphic function mapping the unit disk into itself such that
+φ(z) =z+O((z−1)4)asz→1.
+Thenφis the identity function.
+Theproofin[11]isshortbutsophisticated; theauthorsinv okebothHopf’s
+lemma and the Herglotz representation of a positive harmoni c function as
+the Poisson integral of a positive measure. Incidentally, t he proposition
+in the introduction is a very special case of Hopf’s lemma, wh ich itself is
+a small part of an influential paper [26] by the Austrian mathe matician
+Eberhard Hopf (1902–1983) on the maximum principle for subs olutions of
+elliptic partial differential equations.
+Remarkably, Julia’s fundamental inequality yields a short and elementary
+proofof theuniquenesstheorem. Theargumentis most conven iently writtenJULIUS AND JULIA: MASTERING THE ART OF THE SCHWARZ LEMMA 17
+in the equivalent setting of a holomorphic function fmapping the upper
+half-plane into itself such that f(z) =z+O(z4) asz→0. Letg(z) denote
+z−1−f(z)−1. Theng(z) =O(z2) asz→0, and Julia’s inequality (11)
+implies that Im g(z)≥0 for allzin the upper half-plane. It is easy to see
+that these two conditions force gto be the 0 function.
+Ifghappens to extend holomorphically to a neighborhood of the p oint 0,
+then the argument is the same as in Section 5: if gis not identically 0, then
+gis multiplying angles at the origin by a factor of 2 (or more), sogmaps
+some points of the upper half-plane into the lower half-plan e, contradicting
+that Img(z)≥0.
+Whenglives only in the open upper half-plane (the general case), t he
+conclusion follows directly from Harnack’s inequality, a s tandard tool in
+potential theory, which says in particular that a positiveharmonic function
+inadiskcannottendto0at theboundaryalongaradiusatafas terratethan
+the distance to the boundary. Since Im g(z) tends to 0 at least quadratically
+aszgoes down the imaginary axis toward the origin, Harnack’s in equality
+implies that the harmonic function Im g, although nonnegative, cannot be
+strictly positive. Thus Im gis a harmonic function that assumes its extreme
+value 0, so the maximum principle (applied to the function −Img) shows
+that Img(z) is identically equal to 0. Hence g(z) too is identically equal
+to 0, sof(z) is identically equal to z.
+Harnack’s inequality appears in a book [24, §19, p. 62] by the Baltic
+German mathematician Axel Harnack (1851–1888). Harnack’s statement
+is that ifuis a harmonic function of fixed sign in a disk of radius r, and
+u0is the value of uat the center point, then the value of uat a point at
+distanceρfrom the center is between u0(r+ρ)/(r−ρ) andu0(r−ρ)/(r+ρ).
+This inequality follows easily from the Poisson integral re presentation for
+harmonic functions.
+9.Conclusion.
+In this article I have discussed the classical Schwarz lemma , the bound-
+ary versions discovered by Gaston Julia and Julius Wolff, and s ome appli-
+cations. This circle of ideas is known as Julia–Wolff theory o r Julia–Wolff–
+Carath´ eodory theory. Readers who wish to go further with th e mathematics
+might start with the expository papers [10] and [35], Carath ´ eodory’s book
+for a treatment of angular derivatives [16, §§298–304], the book [1] concern-
+ing implications for iteration theory both in one dimension and in higher
+dimensions, and the book [21] for applications in infinite-d imensional analy-
+sis. Some entry points to the historical literature are the b ooks [42] and [43]
+for the lives of mathematicians in a desperate time; the book [4] for an in-
+depth treatment of the 1918 Grand Prix, its antecedents, and its aftermath;
+and the book [3] for the early development of iteration theor y.
+Remember that graduate student who asked me the question? He passed
+the qualifying exam with flying colors.18 HAROLD P. BOAS
+References
+[1] M. Abate, Iteration Theory of Holomorphic Maps on Taut Manifolds , Mediterranean
+Press, Rende, Italy, 1989.
+[2] L. V. Ahlfors, An extension of Schwarz’s lemma, Trans. Amer. Math. Soc. 43(1938)
+359–364.
+[3] D. S. Alexander, A History of Complex Dynamics: From Schr¨ oder to Fatou and Ju lia,
+Vieweg, Braunschweig, Germany, 1994.
+[4] M. Audin, Fatou, Julia, Montel, le Grand Prix des Sciences Math´ emati ques de 1918,
+et apr` es ... , Springer, Berlin, 2009.
+[5] F. G. Avkhadiev and K.-J. Wirths, Schwarz–Pick Type Inequalities , Birkh¨ auser,
+Basel, 2009.
+[6] J. Bak and D. J. Newman, Complex Analysis , 2nd ed., Springer, New York, 1997.
+[7] O. Blumenthal, Sur le mode de croissance des fonctions en ti` eres,Bull. Soc. Math.
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+[8] R. P. Boas, Invitation to Complex Analysis (rev. H. P. Boas), 2nd ed., Mathematical
+Association of America, Washington, DC, 2010.
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+New York, 1979.
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+407.
+[11] D. M. Burns and S. G. Krantz, Rigidity of holomorphic map pings and a new Schwarz
+lemma at the boundary, J. Amer. Math. Soc. 7(1994) 661–676.
+[12] C. Carath´ eodory, Sur quelques g´ en´ eralisations dut h´ eor` eme de M. Picard, C. R. Acad.
+Sci. Paris 141(1905) 1213–1215.
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+Paris144(1907) 1203–1206.
+[14] , Untersuchungen ¨ uber die konformen Abbildungen von feste n und
+ver¨ anderlichen Gebieten, Math. Ann. 72(1912) 107–144.
+[15] ,¨Uber die Winkelderivierten von beschr¨ ankten analytische n Funktionen,
+Sitzungsberichte der Preussischen Akademie der Wissensch aften. Physikalisch-
+Mathematische Klasse (1929) 39–54.
+[16] ,Theory of Functions of a Complex Variable (trans. F. Steinhardt), vol. 2,
+Chelsea, New York, 1954.
+[17] ,Gesammelte Mathematische Schriften , vol. 5, C. H. Beck, Munich, 1957.
+[18] ,Conformal Representation , 2nd ed., Dover, Mineola, NY, 1998.
+[19] J. G. Clunie, Some remarks on extreme points in function theory, in Aspects of Con-
+temporary Complex Analysis , Proc. NATO Adv. Study Inst., University of Durham,
+Durham, UK, 1979, Academic Press, London, 1980, 137–146.
+[20] A.Denjoy, Surl’it´ eration desfonctions analytiques ,C. R. Acad. Sci. Paris 182(1926)
+255–257.
+[21] S. Dineen, The Schwarz Lemma , Oxford University Press, New York, 1989.
+[22] V. Felsch, Der Aachener Mathematikprofessor Otto Blum en-
+thal, revised transcript of a lecture (2003–2008), availab le at
+http://www.math.rwth-aachen.de/ ~Blumenthal/Vortrag/ .
+[23] M. Georgiadou, Constantin Carath´ eodory: Mathematics and Politics in Tur bulent
+Times, Springer, Berlin, 2004.
+[24] A. Harnack, Die Grundlagen der Theorie des Logarithmischen Potentiale s und
+der Eindeutigen Potentialfunktion in der Ebene , Teubner, 1887; also avail-
+able from the Cornell University Library Historical Math Mo nographs at
+http://digital.library.cornell.edu/m/math/ .
+[25] W. K. Hayman, A characterization of the maximum modulus of functions regular at
+the origin, J. Anal. Math. 1(1951) 135–154.JULIUS AND JULIA: MASTERING THE ART OF THE SCHWARZ LEMMA 19
+[26] E. Hopf, Elementare Bemerkungen ¨ uber die L¨ osungen pa rtieller Differentialgleichun-
+gen zweiter Ordnung vom elliptischen Typus, Sitzungsberichte der Preussischen
+Akademie der Wissenschaften. Physikalisch-Mathematisch e Klasse (1927) 147–152.
+[27] I. S. Jack, Functions starlike and convex of order α,J. London Math. Soc. (2) 3
+(1971) 469–474.
+[28] Jewish Historical Museum, Digital Monument to the Jewi sh Commu-
+nity in the Netherlands, biographical note on Julius Wolff, a vailable at
+http://www.joodsmonument.nl/person/446320/en .
+[29] G. Julia, M´ emoire sur l’it´ eration des fonctions rati onnelles, J. Math. Pures Appl. (8)
+1(1918) 47–245.
+[30] , Extension nouvelle d’un lemme de Schwarz, Acta Math. 42(1920) 349–355.
+[31] E. Landau and G. Valiron, A deduction from Schwarz’s lem ma,J. London Math. Soc.
+4(1929) 162–163.
+[32] K. L¨ owner (C. Loewner), Untersuchungen ¨ uber schlich te konforme Abbildungen des
+Einheitskreises. I, Math. Ann. 89(1923) 103–121.
+[33] C. Loewner, Collected Papers , Birkh¨ auser, Boston, 1988.
+[34] M. Mend` es France, Nevertheless, Math. Intelligencer 10, no. 4 (1988) 35.
+[35] R. Osserman, From Schwarz to Pick to Ahlfors and beyond, Notices Amer. Math.
+Soc.46(1999) 868–873.
+[36] , A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128
+(2000) 3513–3517.
+[37] G. Pick, ¨Uber eine Eigenschaft der konformen Abbildung kreisf¨ ormi ger Bereiche,
+Math. Ann. 77(1916) 1–6.
+[38] H. Poincar´ e, Sur les groupes des ´ equations lin´ eaire s,Acta Math. 4(1884) 201–312.
+[39] G. P´ olya and G. Szeg˝ o, Problems and Theorems in Analysis. I (trans. D. Aeppli),
+Springer, Berlin, 1998; reprint of the 1978 English transla tion.
+[40] S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81
+(1981) 521–527.
+[41] H. A. Schwarz, Gesammelte Mathematische Abhandlungen , vol. 2, Springer, Berlin,
+1890.
+[42] S. L. Segal, Mathematicians under the Nazis , Princeton University Press, Princeton,
+NJ, 2003.
+[43] R. Siegmund-Schultze, Mathematicians Fleeing from Nazi Germany , Princeton Uni-
+versity Press, Princeton, NJ, 2009.
+[44] H.Unkelbach, ¨UberdieRandverzerrungbeikonformerAbbildung, Math. Z. 43(1938)
+739–742.
+[45] D. van Dalen, The war of the frogs and the mice, or the cris is of the Mathematische
+Annalen, Math. Intelligencer 12, no. 4 (1990) 17–31.
+[46] J. Wolff, Sur une g´ en´ eralisation d’un th´ eor` eme de Sc hwarz,C. R. Acad. Sci. Paris
+182(1926) 918–920.
+[47] , Sur une g´ en´ eralisation d’un th´ eor` eme de Schwartz (Sch warz),C. R. Acad.
+Sci. Paris 183(1926) 500–502.
+[48] , Sur les fonctions holomorphes dont l’ensemble des valeurs est soumis ` a cer-
+taines restrictions, Proc. Nederl. Akad. Wetensch. 43(1940) 1018–1022.
+[49] S. T. Yau, A general Schwarzlemma for K¨ ahler manifolds ,Amer. J. Math. 100(1978)
+197–203.
+[50] L. Zalcman, review of Complex Analysis by J. Bak and D. J. Newman and Invitation
+to Complex Analysis by R. P. Boas, Amer. Math. Monthly 97(1990) 262–266.
+Department of Mathematics, Texas A&M University, College S tation, TX
+77843-3368
+E-mail address :boas@math.tamu.edu
+URL:http://www.math.tamu.edu/~boas/
\ No newline at end of file
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+arXiv:1001.0560v1  [math.FA]  4 Jan 2010Convolution operators defined by singular
+measures on the motion group
+Luca Brandolini Giacomo Gigante
+Sundaram Thangavelu Giancarlo Travaglini
+Abstract
+This paper contains an Lpimproving result for convolution oper-
+ators defined by singular measures associated to hypersurfa ces on the
+motion group. This needs only mild geometric properties of t he sur-
+faces, and it extends earlier results on Radon type transfor ms onRn.
+The proof relies on the harmonic analysis on the motion group .
+1 Introduction
+The classical Radon transforms satisfy Lpimproving properties (see [7]) and
+theyarecloselyrelatedtocertainconvolutionoperatorsassociat edtosingular
+measures (seee.g. [13]). Theaboveresults havebeenextendedin manyways,
+not necessarily related to convolution structures, see e.g. [5], [12 ], [15] and
+the references therein.
+Our starting point is the following result, proved in [10].
+Theorem 1 LetΓbe a convex compact curve in the plane and let γbe the
+arc-length measure of Γ. We identify θ∈[0,2π]witheiθ∈S1(the unit cir-
+cle). Letγθbe the rotated measure, i.e./integraltext
+R2f(x)dγθ(x) =/integraltext
+R2f/parenleftbig
+eiθx/parenrightbig
+dγ(x).
+Consider the operator Tdefined by
+Tf(x,θ) = (f∗R2γθ)(x),
+wherex∈R2and∗R2denotes the convolution in R2. Then
+/ba∇dblTf/ba∇dblL3(R2×S1)≤c/ba∇dblf/ba∇dblL3/2(R2).
+The proof of this theorem relies on an estimate for the average dec ay
+of the Fourier transform /hatwideγproved by A.N. Podkorytov in [8] (see also [3]),
+which has been extended to several variables in [2]. The following sta tement
+is different from the one in [2], but it can be proved by a mild variation of
+the original argument.
+1Theorem 2 LetΓbe a compact convex submanifold of codimension 1inRn
+(i.e.Γcan be seen as the graph of a convex function defined in a convex
+domain in Rn−1). Letγ=χσwhereσis the surface measure on Γandχis
+a smooth cutoff supported in the interior of Γ. Then
+/integraldisplay
+Sn−1|/hatwideγ(Rω)|2dω≤cR−(n−1),
+wheredωis the normalized measure on the unit sphere Sn−1. Moreover the
+constantcdepends only on χand the diameter of Γ.
+The above theorem easily implies the following extension of Theorem 1
+(see [1]). For k∈SO(n) andγa measure on Rn, letγkbe defined by/integraltext
+Rnfdγk=/integraltext
+Rnf(ky)dγ(y), so that /hatwideγk(ξ) =/hatwideγ(k−1ξ).
+Theorem 3 LetΓbe a compact convex submanifold of codimension 1inRn
+and letγ=χσwhereσis the surface measure on Γandχis a smooth cutoff
+function supported in the interior of Γ. Consider the operator Tdefined by
+Tf(x,k) = (f∗Rnγk)(x),
+wherex∈Rn,k∈SO(n)and∗Rndenotes the convolution in Rn. Then
+/ba∇dblTf/ba∇dblLn+1(Rn×SO(n))≤c/ba∇dblf/ba∇dblL(n+1)/n(Rn).
+Theoperator TinTheorem3canbeseenasaconvolutionoperatoronthe
+motion group Mn, which is Rn×SO(n) equipped with the group product
+(x,k)(y,h) = (x+ky,kh) and unit (0 ,e). Indeed the convolution of two
+functionsFandGonMnis defined by
+(F∗MnG)(x,k) =/integraldisplay
+MnF/parenleftbig
+x−kh−1y,kh−1/parenrightbig
+G(y,h)dydh,
+wheredhis the Haar measure on SO(n).
+Note that if F(x,k) =f(x) andµdenotes the measure on Mndefined
+by /integraldisplay
+MnG(x,k)dµ(x,k) =/integraldisplay
+MnG(x,k)dγk(x)dk
+2we have
+F∗Mnµ(x,k) =/integraldisplay
+MnF/parenleftbig
+x−kh−1y,kh−1/parenrightbig
+dµ(y,h) (1)
+=/integraldisplay
+Mnf/parenleftbig
+x−kh−1y/parenrightbig
+dγh(y)dh
+=/integraldisplay
+Mnf(x−ky)dγ(y)dh
+=/integraldisplay
+Rnf(x−ky)dγ(y) = (f∗Rnγk)(x).
+The above family {γk}of hypersurfaces in Rnturns out to be a manifold
+inRn×SO(n). Indeed for any k0∈SO(n) the coset {(x,k0) :x∈Rn}
+contains the n−1 dimensional manifold Γ k0, i.e. the manifold Γ rotated by
+k0. The union of the manifolds Γ k0is a hypersurface XinRn×SO(n).
+Whenn= 2, the Γ’s are convex curves and their union can be seen as
+a 2-dimensional surface in R2×S1; the picture shows this surface in the
+particular case Γ( t) = (t,t2+1), together with the plane θ=π:
+In this paper we want to replace the above manifold Xwith a more
+general manifold YinRn×SO(n), so that the action of Γ as a convolution
+operator on Rnis averaged not only on rotations, but on a wider family of
+transformations. In order to deal with this more general setting it is natural
+to work in the Euclidean motion group Mnrather than in Rn×SO(n) and
+take advantage of the representation theory of Mn.
+32 Main result
+The following is our main result. By (1) it is an extension of Theorem 3.
+Theorem 4 Letn≥2and letYbe aC1submanifold of codimension 1
+inMn. Assume that Ycan be locally represented as F(x,k) = 0with
+∇xF(x,k)/ne}ationslash= 0. Assume furthermore that for every k0∈SO(n)the in-
+tersectionY∩ {(x,k0) :x∈Rn}is a convex hypersurface1inRn. Choose
+χ∈ C1
+c(Mn)and letµbe the measure on Mngiven by/integraltext
+Mnfdµ=/integraltext
+Yfχdσ,
+whereσis the surface measure on Y. Then, ifTf(x,k) = (f∗Mndµ)(x,k),
+we have
+/ba∇dblTf/ba∇dblLn+1(Mn)≤cn/ba∇dblf/ba∇dblL(n+1)/n(Mn) (2)
+Proof.Without loss of generality, we may assume that Yis the graph of
+the function
+x1= Φ(x′,k),
+where we use the notation x′= (x2,...,x n). Thus
+/integraldisplay
+Mnfdµ=/integraldisplay
+Rn−1/integraldisplay
+SO(n)f(Φ(x′,k),x′,k)ν(x′,k)dkdx′,
+whereνis the product of χby a Jacobian term. For every z∈C, letizbe
+the distribution on Rdefined by
+/an}b∇acketle{tiz,η/an}b∇acket∇i}ht=1
+Γ(z)/integraldisplay+∞
+0η(t)tz−1dt.
+We define the family of distributions µzby
+µz=µ∗MnIz,
+whereIzis the distribution defined by
+Iz(x,k) =iz(x1)⊗δ0(x2)⊗···⊗δ0(xn)⊗δe(k).
+For anyk∈SO(n) define the measure µkonRnby
+/integraldisplay
+Rngdµk=/integraldisplay
+Rn−1g(Φ(x′,k),x′)ν(x′,k)dx′.
+Then define the distribution EzonRnby
+Ez(x) =iz(x1)⊗δ0(x2)⊗···⊗δ0(xn)
+1This means that the above intersection is the graph of a convex fun ction on a convex
+set (after choosing suitable coordinates in Rn−1).
+4and letµz
+k=µk∗RnEz. Then it can be easily shown that
+/integraldisplay
+Mnf(x,k)dµ(x,k) =/integraldisplay
+SO(n)/integraldisplay
+Rnf(x,k)dµk(x)dk
+/an}b∇acketle{tµz,f/an}b∇acket∇i}htMn=/integraldisplay
+SO(n)/an}b∇acketle{tµz
+k,f(·,k)/an}b∇acket∇i}htRndk.
+We introduce the analytic family of operators
+Tzf=f∗µz.
+Then the proof follows from Stein’s complex interpolation theorem an d the
+following result.
+Lemma 5 For every real swe have
+T1+is:L1(Mn)−→L∞(Mn), (3)
+T−(n−1)/2+is:L2(Mn)−→L2(Mn). (4)
+Proof of the Lemma. Let us prove (3) first. Indeed for g∈L1(Mn)
+/angbracketleftbig
+µ1+is,g/angbracketrightbig
+Mn=/an}b∇acketle{tµ∗MnI1+is,g/an}b∇acket∇i}htMn=/an}b∇acketle{tI1+is,g∗Mn/tildewideµ/an}b∇acket∇i}htMn
+=1
+Γ(1+is)/integraldisplay+∞
+0(g∗Mn/tildewideµ)(x1,0,...,0,e)xis
+1dx1
+=1
+Γ(1+is)/integraldisplay+∞
+0xis
+1/integraldisplay
+Mng/parenleftbig
+(x1,0,...,0,e)(y1,...,y n,k)−1/parenrightbig
+d/tildewideµ(y1,...,y n,k)dx1
+=1
+Γ(1+is)/integraldisplay+∞
+0xis
+1/integraldisplay
+Mng((x1,0,...,0,e)(y1,...,y n,k))
+dµ(y1,...,y n,k)dx1
+=1
+Γ(1+is)/integraldisplay+∞
+0xis
+1/integraldisplay
+Mng(x1+y1,y2,...,y n,k)dµ(y1,...,y n,k)dx1
+=1
+Γ(1+is)/integraldisplay+∞
+0/integraldisplay
+Rn−1/integraldisplay
+SO(n)g(x1+Φ(y′,k),y′,k)xis
+1ν(y′,k)dkdy′dx1
+(where/tildewideµis defined by/integraltext
+Mnf(y,k)d/tildewideµ(y,k) =/integraltext
+Mnf/parenleftbig
+(y,k)−1/parenrightbig
+dµ(y,k) ).
+The substitution y1=x1+ Φ(y′,k), along with the boundedness of ν,
+immediately gives /vextendsingle/vextendsingle/vextendsingle/angbracketleftbig
+µ1+is,g/angbracketrightbig
+Mn/vextendsingle/vextendsingle/vextendsingle≤c/ba∇dblg/ba∇dblL1(Mn),
+5so thatµ1+is∈L∞(Mn). This proves (3).
+Now we turn to the proof of (4). We need first to recall a few facts from
+the representation theory of Mn.
+The unitary dual /hatwiderMn(n≥2) can be described in the following way (here
+[11] is a reference for the representation theory of Mn, see also [14]). Let L=
+SO(n−1), considered as a subgroup of SO(n). For each σ∈/hatwideLrealised on a
+Hilbert space Vσof dimension dσconsider the space L2(SO(n),σ) consisting
+of functions ϕonSO(n)taking values in Cdσ×dσ, thespace of dσ×dσcomplex
+matrices, satisfying the condition
+ϕ(ℓk) =σ(ℓ)ϕ(k), ℓ ∈L, k∈SO(n)
+which are also square integrable on SO(n):
+/integraldisplay
+SO(n)/ba∇dblϕ/ba∇dbl2dk=/integraldisplay
+SO(n)tr(ϕ(k)∗ϕ(k))dk<∞.
+Note thatL2(SO(n),σ) is a Hilbert space under the inner product
+(ϕ,ψ) =/integraldisplay
+SO(n)tr(ϕ(k)∗ψ(k))dk.
+For eachλ >0 andσ∈/hatwideLwe define a representation πλ,σofMnon
+L2(SO(n),σ) as follows. For ϕ∈L2(SO(n),σ) and (x,k)∈Mnlet
+πλ,σ(x,k)ϕ(ℓ) = exp/parenleftbig
+2πiλℓ−1e1·x/parenrightbig
+ϕ(ℓk),
+wheree1= (1,0,...,0) andℓ∈SO(n). Ifϕj(k) are the column vectors
+ofϕ∈L2(SO(n),σ) thenϕj(ℓk) =σ(ℓ)ϕj(k) for allℓ∈L. Therefore
+L2(SO(n),σ) can be written as a direct sum of dσcopies ofH(SO(n),σ)
+which is defined to be the space of square integrable ϕ:SO(n)→Cdσ
+satisfying
+ϕ(ℓk) =σ(ℓ)ϕ(k), ℓ ∈L.
+It can be shown that πλ,σrestricted to H(SO(n),σ) is an irreducible repre-
+sentation of Mn. Moreover, any infinite dimensional irreducible unitary rep-
+resentation of Mnis unitarily equivalent to one and only one πλ,σ. Finite di-
+mensional irreducible unitary representations of SO(n) also yield irreducible
+unitary representations of Mn. As they do not appear in the Plancherel
+formula we neglect them. We remark that when n= 2 the unitary dual /hatwideL
+contains only the trivial representation.
+Givenf∈L1(Mn)∩L2(Mn) we define the group Fourier transform of f
+by
+πλ,σ(f) =/integraldisplay
+Mnf(x,k)πλ,σ/parenleftbig
+(x,k)−1/parenrightbig
+dxdk.
+6It can be shown that πλ,σ(f) is a Hilbert-Schmidt operator on H(SO(n),σ)
+and we have the Plancherel formula
+/summationdisplay
+σ∈/hatwideL/integraldisplay+∞
+0/ba∇dblπλ,σ(f)/ba∇dbl2
+HSλn−1dλ=ωn/integraldisplay
+Mn|f(x,k)|2dxdk,
+where/ba∇dbl·/ba∇dblHSdenotes the Hilbert-Schmidt norm.
+Applying Plancherel formula to T−(n−1)/2+isfwe get
+/vextenddouble/vextenddoubleT−(n−1)/2+isf/vextenddouble/vextenddouble2
+L2(Mn)
+=/vextenddouble/vextenddoublef∗Mnµ−(n−1)/2+is/vextenddouble/vextenddouble2
+L2(Mn)
+=ωn/summationdisplay
+σ∈/hatwideL/integraldisplay+∞
+0/vextenddouble/vextenddoubleπλ,σ/parenleftbig
+µ−(n−1)/2+is/parenrightbig
+πλ,σ(f)/vextenddouble/vextenddouble2
+HSλn−1dλ
+≤ωn/summationdisplay
+σ∈/hatwideL/integraldisplay+∞
+0/vextenddouble/vextenddoubleπλ,σ/parenleftbig
+µ−(n−1)/2+is/parenrightbig/vextenddouble/vextenddouble2
+OP/ba∇dblπλ,σ(f)/ba∇dbl2
+HSλn−1dλ,
+where/ba∇dbl·/ba∇dblOPis the operator norm on H(SO(n),σ). We shall show below
+that/vextenddouble/vextenddoubleπλ,σ/parenleftbig
+µ−(n−1)/2+is/parenrightbig/vextenddouble/vextenddouble
+OP≤cn (5)
+uniformly in λandσ, so that
+/vextenddouble/vextenddoubleT−(n−1)/2+isf/vextenddouble/vextenddouble2
+L2(Mn)≤cnωn/summationdisplay
+σ∈/hatwideL/integraldisplay+∞
+0/ba∇dblπλ,σ(f)/ba∇dbl2
+HSλn−1dλ=cn/ba∇dblf/ba∇dbl2
+L2(Mn).
+We now prove (5). For ϕ,ψ∈H(SO(n),σ) we have
+πλ,σ/parenleftbig
+(x,k)−1/parenrightbig
+ϕ(u) = exp/parenleftbig
+−2πiλu−1e1·k−1x/parenrightbig
+ϕ/parenleftbig
+uk−1/parenrightbig
+.
+Assume for a moment Re z >0, thenµzis a measure and
+πλ,σ(µz)ϕ(u)
+=/integraldisplay
+Mnπλ,σ/parenleftbig
+(x,k)−1/parenrightbig
+ϕ(u)dµz(x,k)
+=/integraldisplay
+Mnexp/parenleftbig
+−2πiλu−1e1·k−1x/parenrightbig
+ϕ/parenleftbig
+uk−1/parenrightbig
+dµz(x,k).
+7Therefore
+/an}b∇acketle{tπλ,σ(µz)ϕ,ψ/an}b∇acket∇i}htH(SO(n),σ)
+=/integraldisplay
+SO(n)/an}b∇acketle{tπλ,σ(µz)ϕ(u),ψ(u)/an}b∇acket∇i}htCdσdu
+=/integraldisplay
+SO(n)/integraldisplay
+Mnexp/parenleftbig
+−2πiλu−1e1·k−1x/parenrightbig/angbracketleftbig
+ϕ/parenleftbig
+uk−1/parenrightbig
+,ψ(u)/angbracketrightbig
+Cdσdµz(x,k)du
+=/integraldisplay
+SO(n)/integraldisplay
+SO(n)/integraldisplay
+Rnexp/parenleftbig
+−2πiλku−1e1·x/parenrightbig/angbracketleftbig
+ϕ/parenleftbig
+uk−1/parenrightbig
+,ψ(u)/angbracketrightbig
+Cdσ
+×dµz
+k(x)dudk
+=/integraldisplay
+SO(n)/integraldisplay
+SO(n)/hatwiderµz
+k/parenleftbig
+λku−1e1/parenrightbig/angbracketleftbig
+ϕ/parenleftbig
+uk−1/parenrightbig
+,ψ(u)/angbracketrightbig
+Cdσdudk
+=/integraldisplay
+SO(n)/integraldisplay
+SO(n)/hatwiderµk/parenleftbig
+λku−1e1/parenrightbig/hatwiderEz/parenleftbig
+λku−1e1/parenrightbig/angbracketleftbig
+ϕ/parenleftbig
+uk−1/parenrightbig
+,ψ(u)/angbracketrightbig
+Cdσdudk.
+By analytic continuation, the equality
+/an}b∇acketle{tπλ,σ(µz)ϕ,ψ/an}b∇acket∇i}htH(SO(n),σ)
+=/integraldisplay
+SO(n)/integraldisplay
+SO(n)/hatwiderµk/parenleftbig
+λku−1e1/parenrightbig/hatwiderEz/parenleftbig
+λku−1e1/parenrightbig/angbracketleftbig
+ϕ/parenleftbig
+uk−1/parenrightbig
+,ψ(u)/angbracketrightbig
+Cdσdudk
+holds also for z=−(n−1)/2+is. By Cauchy-Schwarz inequality
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+SO(n)/hatwiderµk/parenleftbig
+λku−1e1/parenrightbig/hatwideE−(n−1)/2+is/parenleftbig
+λku−1e1/parenrightbig/angbracketleftbig
+ϕ/parenleftbig
+uk−1/parenrightbig
+,ψ(u)/angbracketrightbig
+Cdσdu/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤/vextenddouble/vextenddouble/angbracketleftbig
+ϕ/parenleftbig
+uk−1/parenrightbig
+,ψ(u)/angbracketrightbig
+Cdσ/vextenddouble/vextenddouble
+L2(SO(n),du)
+×/vextenddouble/vextenddouble/vextenddouble/hatwiderµk/parenleftbig
+λku−1e1/parenrightbig/hatwideE−(n−1)/2+is/parenleftbig
+λku−1e1/parenrightbig/vextenddouble/vextenddouble/vextenddouble
+L2(SO(n),du)
+By [6, Ch. 2] we know that
+/vextendsingle/vextendsingle/vextendsingle/hatwideE−(n−1)/2+is/parenleftbig
+λku−1e1/parenrightbig/vextendsingle/vextendsingle/vextendsingle≤Cλ(n−1)/2.
+8Now
+/hatwiderµk(ξ) =/integraldisplay
+Rnexp(−2πiξ·x)dµk(x)
+=/integraldisplay
+Rn−1exp(−2πiξ·(Φ(x′,k),x′))ν(x′,k)dx′
+=/integraldisplay
+Rn−1exp(−2πiξ·(Φ(x′,k),x′))ν(x′,k)/radicalBig
+1+|∇x′Φ(x′,k)|2
+×/radicalBig
+1+|∇x′Φ(x′,k)|2dx′
+=/integraldisplay
+Rnexp(−2πiξ·x)ν(x′,k)/radicalBig
+1+|∇x′Φ(x′,k)|2dζk(x),
+wheredζkis the surface measure of the convex hypersurface in Rngiven by
+the intersection Y∩{(x,k) :x∈Rn}. By Theorem 2 we get
+/vextenddouble/vextenddouble/vextenddouble/hatwiderµk/parenleftbig
+λku−1e1/parenrightbig/hatwideE−(n−1)/2+is/parenleftbig
+λku−1e1/parenrightbig/vextenddouble/vextenddouble/vextenddouble2
+L2(SO(n),du)
+≤cλn−1/integraldisplay
+SO(n)/vextendsingle/vextendsingle/hatwiderµk/parenleftbig
+λku−1e1/parenrightbig/vextendsingle/vextendsingle2du≤c.
+To end the proof we observe that
+/integraldisplay
+SO(n)/ba∇dbl/an}b∇acketle{tϕ(uk),ψ(u)/an}b∇acket∇i}htCdσ/ba∇dblL2(SO(n),du)dk
+=/integraldisplay
+SO(n)/braceleftbigg/integraldisplay
+SO(n)|/an}b∇acketle{tϕ(uk),ψ(u)/an}b∇acket∇i}htCdσ|2du/bracerightbigg1/2
+dk
+≤/braceleftbigg/integraldisplay
+SO(n)/integraldisplay
+SO(n)|/an}b∇acketle{tϕ(uk),ψ(u)/an}b∇acket∇i}htCdσ|2dudk/bracerightbigg1/2
+≤/braceleftbigg/integraldisplay
+SO(n)/integraldisplay
+SO(n)|ϕ(uk)|2|ψ(u)|2dudk/bracerightbigg1/2
+.
+By Fubini’s theorem and the invariance of the Haar measure on SO(n) we
+get
+/integraldisplay
+SO(n)/ba∇dbl/an}b∇acketle{tϕ(uk),ψ(u)/an}b∇acket∇i}htCdσ/ba∇dblL2(SO(n),du)dk≤ /ba∇dblϕ/ba∇dblH(SO(n),σ)/ba∇dblψ/ba∇dblH(SO(n),σ).
+This ends the proof of the Lemma. Hence Theorem 4 is proved.
+9Remark 6 For functions on Mnwhich are independent of the rotational
+variable, i.e. for functions Fsuch thatF(x,k) =f(x), Theorem 4 can be
+obtained from Theorem 3. Indeed
+/braceleftbigg/integraldisplay
+SO(n)/integraldisplay
+Rn|F∗Mnµ(x,k)|n+1dxdk/bracerightbigg1/(n+1)
+=/braceleftBigg/integraldisplay
+SO(n)/integraldisplay
+Rn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+SO(n)/integraldisplay
+Rnf/parenleftbig
+x−kτ−1y/parenrightbig
+dµτ(y)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsinglen+1
+dxdk/bracerightBigg1/(n+1)
+≤/integraldisplay
+SO(n)/braceleftBigg/integraldisplay
+SO(n)/integraldisplay
+Rn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Rnf/parenleftbig
+x−kτ−1y/parenrightbig
+dµτ(y)/vextendsingle/vextendsingle/vextendsingle/vextendsinglen+1
+dxdk/bracerightBigg1/(n+1)
+dτ
+=/integraldisplay
+SO(n)/braceleftBigg/integraldisplay
+SO(n)/integraldisplay
+Rn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Rnf(x−σy)dµτ(y)/vextendsingle/vextendsingle/vextendsingle/vextendsinglen+1
+dxdσ/bracerightBigg1/(n+1)
+dτ
+=/integraldisplay
+SO(n)/braceleftbigg/integraldisplay
+SO(n)/integraldisplay
+Rn|f∗Rnµτ,σ(x)|n+1dxdσ/bracerightbigg1/(n+1)
+dτ
+≤/integraldisplay
+SO(n)c/ba∇dblf/ba∇dblLn+1
+n(Rn)dτ=c/ba∇dblF/ba∇dblLn+1
+n(Mn),
+whereµτ,σdenotes the measure µτrotated byσ. This yields the following
+weakerversionof Theorem4. Fora general Flet/tildewideF(x,k) = supτ∈SO(n)|F(x,kτ)|
+then
+/ba∇dblF∗µ/ba∇dblLn+1(Mn)≤/vextenddouble/vextenddouble/vextenddouble/tildewideF∗µ/vextenddouble/vextenddouble/vextenddouble
+Ln+1(Mn)≤c/vextenddouble/vextenddouble/vextenddouble/tildewideF/vextenddouble/vextenddouble/vextenddouble
+Ln+1
+n(Mn)=c/ba∇dblF/ba∇dbl
+Ln+1
+nx(L∞τ(Mn)).
+The above seems to be the best we can get by using earlier resul ts such as the
+ones in [1].
+Remark 7 A familiar example (the characteristic function of a small b all)
+and the previous remark can be used to show that the indices in (2) cannot
+be improved.
+Remark 8 It is interesting to compare Theorem 4 with Theorem 1.1 in [9]
+where it is shown that the Lpimproving property of a measure is related to
+the fact that the supporting manifold generates the full gro up.
+Remark 9 The techniques in our paper are L2in nature and they seem
+to provide only Lp−Lp′results. We do not know how to get mixed norm
+estimates similar to the ones which have been proved in [10] t hrough certain
+Lrestimates for the average decay of Fourier transforms (note that in general
+theseLrestimates cannot be obtained by interpolation between L2andL∞,
+see e.g. [4]).
+10The authors wish to thank the referee for her/his suggestions an d for
+pointing out some references.
+References
+[1] L. Brandolini, A. Greenleaf, G. Travaglini, Lp−Lp′estimates for overde-
+termined Radon transforms , Trans. Amer. Math. Soc. 359(2007), 2559–
+2575.
+[2] L. Brandolini, S. Hofmann and A. Iosevich, Sharp rate of average decay
+of the Fourier transform of a bounded set, Geom.Funct.Anal. 13(2003),
+671–680.
+[3] L. Brandolini, A. Iosevich, G. Travaglini, Spherical means and the re-
+striction phenomenon, J. Fourier Anal. Appl. 7(2001), 359–372.
+[4] L. Brandolini, M. Rigoli, G. Travaglini, Average decay of Fourier trans-
+forms and geometry of convex sets , Rev. Mat. Iberoamer. 14(1998),
+519-560.
+[5] M. Christ, A. Nagel, E. Stein, S. Wainger, Singular and maximal Radon
+transforms: analysis and geometry , Ann. of Math. 150(1999), 489–577.
+[6] I.M. Gel’fand and G.E. Shilov, Generalized functions, Vol. 1, Academ ic
+Press (1964).
+[7] D.M. Oberlin, E.M. Stein, Mapping properties of the Radon transform ,
+Indiana Univ. Math. J. 31(1982), 641–650.
+[8] A.N. Podkorytov, The asymptotic of a Fourier transform of a convex
+curve, Vest. Leningr. Univ. Mat. 24 (1991), 57–65.
+[9] F. Ricci, E. M. Stein, Harmonic analysis on nilpotent groups and singu-
+lar integrals. III. Fractional integration along manifold s, J. Funct. Anal.
+86(1989), 360–389.
+[10] F. Ricci and G. Travaglini, Convex curves, Radon transforms and con-
+volution operators defined by singular measures, Proc. Amer. Math. Soc.
+129(2001), 1739–1744.
+[11] R. P. Sarkar and S. Thangavelu, On theorems of Beurling and Hardy for
+the Euclidean motion group, Tohoku Math, J. 57(2005), 335–351.
+11[12] A. Seeger, Radon transforms and finite type conditions , J. Amer. Math.
+Soc.11(1998), 869–897.
+[13] R.S. Strichartz, Convolutions with kernels having singularities on a
+sphere,Trans. Amer. Math. Soc. 148(1970), 461–471
+[14] M.Sugiura, UnitaryrepresentationsandHarmonicAnalysis, Jo hnWiley
+& Sons (1975).
+[15] T. Tao, J. Wright, Lpimproving bounds for averages along curves , Jour.
+Amer. Math. Soc. 16(2003), 605–638.
+Dipartimento di Ingegneria dell’Informazione e Metodi Mat e-
+matici
+Universit `a di Bergamo, Viale Marconi 5
+24044 Dalmine, Bergamo, Italy
+luca.brandolini@unibg.it
+Dipartimento di Ingegneria dell’Informazione e Metodi Mat e-
+matici
+Universit `a di Bergamo, Viale Marconi 5
+24044 Dalmine, Bergamo, Italy
+giacomo.gigante@unibg.it
+Department of Mathematics
+Indian Institute of Science
+Bangalore 560012, India
+veluma@math.iisc.ernet.in
+Dipartimento di Statistica, Edificio U7
+Universit `a di Milano-Bicocca
+Via Bicocca degli Arcimboldi 8
+20126 Milano, Italy
+giancarlo.travaglini@unimib.it
+12
\ No newline at end of file
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+arXiv:1001.0561v1  [astro-ph.SR]  4 Jan 2010Preprint typeset using L ATEX style emulateapj v. 11/10/09
+FABRY-PEROT VERSUS SLIT SPECTROPOLARIMETRY OF PORES AND AC TIVE NETWORK. ANALYSIS
+OF IBIS AND HINODE DATA
+Philip G. Judge
+High Altitude Observatory, National Center for Atmospheri c Research, P.O. Box 3000, Boulder CO 80307-3000, USA; judge @ucar.edu
+Alexandra Tritschler, Han Uitenbroek
+National Solar Observatory/Sacramento Peak, P.O. Box 62, S unspot, NM-88349, U.S.A.; ali@nso.edu, huitenbroek@nso. edu
+Kevin Reardon, Gianna Cauzzi
+INAF – Ossevatorio Astrofisico di Arcetri, I-50125 Firenze, Italy; kreardon@arcetri.astro.it, gcauzzi@arcetri.ast ro.it
+and
+Alfred de Wijn
+High Altitude Observatory, National Center for Atmospheri c Research, P.O. Box 3000, Boulder CO 80307-3000, USA; dwijn @ucar.edu
+ABSTRACT
+We discuss spectropolarimetric measurements of photospheric (F eI630.25nm) and chromospheric
+(CaII854.21nm) spectral lines in and around small magnetic flux concentr ations, including a pore.
+Our long-term goal is to diagnose properties of the magnetic field ne ar the base of the corona. We
+compare ground-based two-dimensional spectropolarimetric mea surements with (almost) simultane-
+ous space-based slit spectropolarimetry. We address the questio n of noise and cross talk in the
+measurements and attempt to determine the suitability of Ca IImeasurements with imaging spec-
+tropolarimeters for the determination of chromospheric magnetic fields. The ground-based observa-
+tions were obtained May 20, 2008, with the Interferometric Bidimen sional Spectrometer (IBIS) in
+spectropolarimetric mode operated at the Dunn Solar Telescope (D ST) at Sunspot, New Mexico. The
+space observations were obtained with the Spectro-Polarimeter o f the Solar Optical Telescope (SOT)
+aboard the Japanese Hinodesatellite. The agreement between the near-simultaneous co-spat ial IBIS
+andHinodeStokes-Vprofilesat 630.25nm is excellent, with V/Iamplitudes compatible with to within
+1%. The IBIS QUmeasurements are affected by residual crosstalk from V, arising from calibration
+inaccuracies, not from any inherent limitation of imaging spectrosco py. We use a PCA analysis to
+quantify the detected cross talk. QUprofiles with Vcrosstalk subtracted are in good agreement
+with the Hinodemeasurements, but are noisier owing to fewer collected photons. C hromospheric
+magnetic fields are notoriously difficult to constrain by polarization of CaIIlines alone. However, we
+demonstrate that high cadence, high angular resolution monochro matic images of fibrils in Ca IIand
+Hα, seen clearly in IBIS observations, can be used to improve the magn etic field constraints, under
+conditions of high electrical conductivity. Such work is possible only w ith time series datasets from
+two-dimensional spectroscopic instruments such as IBIS, under conditions of good seeing.
+Subject headings: Sun: photosphere, chromosphere, magnetic fields, spectropola rimetry
+1.INTRODUCTION
+This is first of several planned papers attempting to
+derive chromospheric magnetic structure using imaging
+spectropolarimetry, a new tool with considerable poten-
+tial. Our motivation is to constrain the magnetic free
+energy in the solar atmosphere, the cause of many of the
+Sun’s most interesting observable phenomena, yet mea-
+surements of this free energy are notoriously difficult to
+obtain. Line-of-sight (LOS) components of photospheric
+magnetic fields have been measured using circularly po-
+larized light routinely for over 50 years. Such measure-
+ments set no constraints on the free magnetic energy. It
+was not until credible measurements of the full polariza-
+tion vector became available (Baur et al. 1980) that the
+freecomponent of the magnetic energy became some-
+thing amenable to observation.
+But several difficulties arise. The magnetic virial the-
+orem relates the total magnetic free energy in a vol-
+ume overlying a surface to the vector field measuredat that surface, provided the surface is in a force-free
+state (e.g. Low & Lou 1990). Most vector field mea-
+surements are made in photospheric lines where, out-
+side of sunspot umbrae, the fields are far from force-free.
+Thus one must either extrapolate the field into the over-
+lying atmosphere, or make measurements higher in the
+atmosphere where the field is close to force-free (the ra-
+tio of gas to magnetic pressure, plasma β,≪1). The
+first option has been pursued by many researchers with
+mixed success (Schrijver et al. 2008; Wiegelmann et al.
+2008; De Rosa et al. 2009), in essence because the prob-
+lem is non-linear (Sakurai 1979), is usually cast into
+force-free form incompatible with photospheric condi-
+tions, and is also ill-posed (sensitive to boundary con-
+ditions, Low & Lou 1990).
+In this paper we begin pursuit of the second option
+using an imaging spectropolarimeter to observe chro-
+mospheric lines. Previous polarimetric studies of chro-
+mospheric lines have been successful in recovering mag-2 Judge et al.
+netic fields (e.g. Metcalf et al. 1995; Solanki et al. 2003;
+Socas-Navarro 2005; Harvey 2006; Centeno et al. 2006;
+Pietarila et al. 2007), but are based on slit spectroscopy.
+Aswewillshow, imagingspectropolarimetryisespecially
+suited to studying the chromosphere because it has the
+spatial coverage and high temporal cadence needed to
+follow dynamic fibril motions which mostly dominate the
+upperchromosphere(e.g.Judge2006). Suchworkisdiffi-
+cultwithconventionalslitspectrographs. Outsideofvery
+quiet regions of the Sun, the plasma βis certainly <1
+in the upper chromosphere, as can be seen by inspection
+of Hαspectroheliograms (e.g. Kiepenheuer 1953; Athay
+1976) dominated by fibril structures resulting from the
+entrainment of plasma by strong magnetic fields. By us-
+ing measurements of the chromospheric vector field
+•the magnetic free energy in the overlying corona
+follows directly from the virial theorem;
+•extrapolations into the overlying atmosphere can
+be made using a boundary condition which is itself
+force-free, unlike the photospheric case;
+•the change in regime from a forced (photo-
+spheric) to force-free state (upper chromosphere
+and corona) can be probed.
+Thethirdpointisofmorethanacademicinterest. Recent
+work by Leake & Arber (2006); Arber et al. (2007) has
+shown that ion-neutral collisions damp components of
+the current density j= curlBwhich are perpendicular
+to the magnetic field B. A significant component of the
+free energy is thus dissipated in the chromosphere before
+it reaches the corona.
+Below, we present vector polarimetry of the
+photosphere (Fe I630.25nm) and chromosphere
+(CaII854.21nm) using the imaging spectropolarimeter
+IBIS (Cavallini 2006; Reardon & Cavallini 2008). How-
+ever, while provenfor earlierinstruments (e.g., the Imag-
+ing Vector Magnetograph IVM, Mickey et al. 1996), the
+technique needs to be proven for IBIS, and so we com-
+bine the IBIS data with (nearly) simultaneous measure-
+ments of the same Fe Iline from the slit spectropolarime-
+ter aboard the Hinodesatellite, to assess the spectropo-
+larimetric capabilities of IBIS. Later work will present a
+physical analysis of these data together with H α, EUV,
+X-ray and G band data obtained simultaneously.
+2.OBSERVATIONS
+Joint observations with the Dunn Solar Telescope
+(DST), Hinode, and TRACE were obtained from 13 to
+21 May 2008. Only data from 20 May 2008 were of suf-
+ficient quality to merit further study and are reported
+here. We targeted the strongest magnetic features on
+the solar disk, but only small pores were observed. Two
+regions were observed: the first (NOAA 10996), centered
+near solar latitude and longitude N8.7, E10.2, was also
+observed by Hinode. The second (NOAA 10994, S12.4,
+W46.1) was observed only at the DST.
+Table1summarizestheobservations. Theheliographic
+pointings listed between instruments were determined
+from a SOLIS full disk magnetogram. The SOLIS scan
+was obtained between 15:11:27 and 15:23:23UT, the re-
+gion shown in Figure 1 being observed by SOLIS near
+15:16:15UT over a period of 1 minute. The figureshows the context of the NOAA 10996 IBIS and Hinode
+spectropolarimetric measurements, in a SOLIS 630.2nm
+magnetogram and a TRACE 19.5nm image. TRACE
+160nm data obtained at 15:11UT were used to deter-
+mine the co-alignment with the SOLIS magnetogram,
+by eye. The SOLIS- TRACE alignment is accurate to
+±1′′. The 19.5nm data shown have been corrected for
+the 160-19.5nm offset using standard SolarSoft TRACE
+software,andarefrom15:12UT. Notethatrelativeinter-
+instrument co-registrationcan be made at sub-arcsecond
+scales, smaller than the uncertainties in absolute helio-
+graphic coordinates.
+It can be seen that only the upper half ( ≈40′′×40′′) of
+the IBIS field of view (FOV) overlapped the HinodeSP
+FOV. It is also clearthat IBIS observeda regionpredom-
+inantly of negative polarity following some 50′′behind
+a positive polarity region. These small concentrations
+of magnetic flux are connected by relatively short coro-
+nal loops seen in the 19.5nm TRACE data. There is
+some 19.5nm “moss” emission associated with the pho-
+tospheric field concentrations. Moss is the bright foot-
+point emission at the base of an overlying hotter corona
+(Fletcher & de Pontieu 1999; Berger et al. 1999).
+2.1.Spectropolarimetric Observations from IBIS
+The IBIS observations reported here were obtained be-
+tween 14:29 and 15:19UT. IBIS is a Fabry-P´ erot based
+filtergraph in a classical mount with a circular FOV of
+diameter 80′′, rapidly tunable in wavelength (Cavallini
+2006; Reardon & Cavallini 2008). IBIS was operated
+in dual-beam spectropolarimetric mode (Tritschler et al.
+2009) in which two liquid crystal variable retarders
+(LCVRs) are placed in a collimated beam in front of
+theinstrument, and apolarizingbeam-splitterisinserted
+in front of the detector. The FOV is then reduced to
+≈40′′×80′′, because the beam splitter produces a pair
+ofsimultaneous I+SiandI−Siimageswith Si= Stokes
+Q,UorV1. At each wavelength six modulation states
+were acquired sequentially, with the first beam acquired
+in the following order: I+Q,I−V,I−Q,I+V,
+I−U, andI+U. This sampling order was chosen to
+maximize the efficiency of the LCVR modulation. The
+FeI, CaIIand Hαlines were sampled with 14, 21 and 22
+non-equidistant wavelength points, respectively. In H α
+only unpolarized measurements were acquired. Thus, a
+full sequence consists of (14 + 21) ×6 + 22 = 232 ex-
+posures, requiring 70s in total. For the Fe I630.2nm
+and Ca II854.2nm lines the scanning was performed by
+jumping sequentially between the blue and red sides of
+the line. To avoid detector saturation the exposure time
+for each individual image was set to 50ms, which lim-
+its the polarimetric sensitivity obtainable. The instru-
+ment performed electronic 2 ×2 binning prior to writ-
+ing to disk which resulted in a detector image scale of
+0.16arcsecpixel−1.
+The narrowband observations are supplemented with
+simultaneous wide band data (WB, 721nm) covering the
+sameFOVwith adetectorimagescaleof ∼0.08′′pixel−1.
+The WB data were used for alignment purposes and the
+correction of the effects of anisoplanatism (to first or-
+1We will interchangeably use the notations S, (IQUV) and
+(S0,S1,S2,S3) as convenient henceforth, to describe the Stokes
+vector and its components.3
+der) during scanning via a de-stretch algorithm using
+speckle reconstructed WB images as reference images.
+The speckle reconstructions were calculated from bursts
+of 77 images each (burst durations of 23s) using the im-
+plementation by W¨ oger et al. (2008). G-band data cov-
+ering a FOV of twice the width but the same height as
+the WB data were also obtained at a cadence of 0.2s on
+a 1024×1024 detector with ≈0.09′′pixel−1. These data
+are not discussed here. All observations were performed
+in conjunction with the high-order adaptive optics (AO)
+system(Rimmele2004). Seeingconditionsweregoodbut
+variable. Full calibration data sets including flats, darks,
+resolution targets and polarization calibration measure-
+ments were obtained before and after science observa-
+tions.
+The IBIS I±Sidata frames were reduced following
+procedures described by Cauzzi et al. (2008), including
+darksubtraction, flatfielding, co-alignmentwithWBim-
+ages, a blueshift correction needed because of the classi-
+cal etalon mountings, de-stretching, and co-registration
+ofall imagesin each sequence. The reduced I±Siframes
+were then combined and corrected for instrumental po-
+larization to determine the solar Stokes vector S. The
+telescope calibrationdata used werefrom February2007.
+A more recent calibration dataset was acquired in Octo-
+ber 2008, but the results were not ready at the time of
+writing. The measured Sis in a frame of reference de-
+fined by the elevation mirror of the telescope (see e.g.
+Skumanich et al. 1997; Beck et al. 2005). In the solar
+reference frame ( Qpositive in the E-W direction on the
+Sun) a final rotation in the Q−Uplane is required.
+However, prior to this rotation some care is needed be-
+cause residual crosstalk from VtoQandUis evident in
+the data. Thus an empirical correction to QUusing the
+Vdata was applied, as described below, to try to derive
+theactual QUenteringthetelescope, beforeapplyingthe
+rotation. The appendix describes the different origins of
+crosstalk in IBIS and the HinodeSP.
+Forthe firsttarget, 30IBISscansof70seachwerecom-
+pleted, for the second, just 5 scans were obtained before
+clouds intervened. Figure 2 shows typical IBIS filter-
+grams of NOAA 10996 taken in the blue wing of Fe I,
+CaIIand Hα, at the nominal line core position of Ca II
+and Hα, and a magnetogram constructed by subtracting
+V(+56m˚A) fromV(-74m˚A) forthe Fe Iline. The seeing
+was examined using a 10′′square region centered near (-
+150′′,120′′), free of measurable magnetic influences. Fig-
+ure 3 shows the rms intensity contrast measured with
+IBIS and the HinodeSP in the continuum near 630 nm,
+as a fraction of the mean intensity. Also shown is the
+630 nm continuum rms contrast corresponding to 1′′res-
+olution derived by Lites (2002). The AO-corrected rms
+contrastwasvariable, and so wasnot dominated by stray
+light. The seeing was clearly worse during the first and
+16th-24th IBIS scans. The seeing-limited resolution of
+these IBIS observations, at best, corresponds to some-
+where between 1′′and the resolution of HinodeSP. (Note
+that “resolution” of Hinodehere refers to the character-
+istic width of the point spread function, PSF, modified
+by the pixel size. In terms of granularcontrast statistics,
+the smaller HinodeSP PSF corresponds to an effective
+resolution close to 0 .′′315.)2.2.Spectropolarimetric Observations from Hinode
+TheSpectro-Polarimeter(SP, Lites et al.2001) aboard
+Hinode(Kosugi et al. 2007) obtained “fast maps” of the
+630nm region. In typical operation mode, 112 wave-
+length samples of the SP CCD detector are read, each
+pixel having a spectral width of 0.00215 nm and a width
+along the slit of 0 .16′′(Centeno et al. 2009). The 12-
+micron wide SP slit presents an effective width of 0 .16′′
+to the solar image. The fast map mode reduces teleme-
+try volume via an effective 2x2 binning of the image pro-
+duced by the SP mapping process: the image pixel wells
+are binned in the direction along the slit during each
+read of the spectral/spatial CCD image, then successive
+integrations of one full rotation (1.6 sec) of the retarder
+polarization modulator are summed onboard. Successive
+integrations correspond to one step of the image perpen-
+dicular to the CCD slit: an average step size of 0 .149′′.
+To further reduce the overall SP data rate for these co-
+ordinated observations as a consequence of the failure in
+late 2007 of the Hinode onboard X-band telemetry sys-
+tem (Shimizu 2009), one of the two CCD polarimetry
+images was not downlinked, and for the remaining data,
+we retained only the central half of the full 164′′length
+of the SP CCD along the slit. The resulting SP fast
+maps for these observations were then 335 ×256 pixels,
+or approximately 99 .6′′×81.9′′. These maps required
+≈21 minutes to execute, and have an effective (square)
+pixel aperture of about 0 .3′′. With the Hinode rotating
+retarder modulator, demodulation was accomplished via
+onboard summing of images corresponding to 4 phases
+over a half-rotation of the modulator. The Stokes vector
+was derived using the level 1 data product from the Hin-
+odeproject which includes the 630.15nm and 630.25nm
+lines of Fe I.
+While the SP observations are seeing-free, they do ex-
+perience spacecraft jitter. The amplitude of the jitter
+is remarkably small (1 σ <0.01′′, Shimizu et al. 2008),
+whichhastwoimplications. Firsttheimagesarefarmore
+stable than can be obtained from the ground. Second,
+the influence of jitter on the spectropolarimetry is small,
+whenweunderstandjittertohavethesameeffectas“see-
+ing” in the sense modeled by Lites (1987); Judge et al.
+(2004). This issue is reviewed in the Appendix.
+3.ANALYSIS
+IBIS obtains narrow-band 2D images but must scan
+through multiple wavelengths to build up the spectra;
+the SP obtains spectra along the slit but must scan spa-
+tially perpendicular to the slit, to build up maps of the
+solar surface. The instruments also differ in the way the
+polarimetry is performed. IBIS isa “Stokesdefinition po-
+larimeter”, i.e. during the IBIS integrations only I+Si
+forSi=QUorVis acquired. Other than inevitable
+I→Sicrosstalk, which is mostly removed by the dual
+beam, there is no other “seeing induced crosstalk”. The
+combined modulation/demodulation matrices ( ˜H′
+ri(ν) of
+Lites 1987) are diagonal. Yet such a polarimeter is still
+susceptible to cross-talk because the telescope mixes the
+four polarization states prior to entering the polarimeter
+in a fashion which may be imperfectly calibrated, and
+the modulation/demodulation is not perfect. Crosstalk
+induced by calibration errors varies far more slowly than
+seeing, and could be “calibrated out” if accurate calibra-4 Judge et al.
+tion data were available. Difficulties arise because the
+calibration matrices are imprecisely known. Below, we
+will apply a simple correction for the V→QUcrosstalk
+by simply requiring the average QUprofiles to be sym-
+metric around line center. The SP data show this to be
+a good assumption.
+Every SP exposure is a linear combination of Iand
+at least two other Stokes parameters. Such measure-
+ments have significant off-diagonal ˜H′
+ri(ν) terms. The
+SP is therefore subject to the systematic errors induced
+by “seeing-induced crosstalk” (again, for the SP, “image
+motion” means jitter). But the residual image motion
+is very small: the factor βiin Lites’ (1987) formula (15)
+is far less than it would have been if observing with the
+SP through the atmosphere. It is in this sense that we
+can use the SP data as fiducial values against which we
+compare the data from IBIS.
+3.1.Polarization of Fe I630.2 nm
+In Figure 4, direct comparisons of Stokes parameters
+between IBIS and HinodeSP are shown. These data are
+typical of the entire dataset. The intensity images and
+magnetogramsshowthecontextoftheStokesparameters
+shown in the rightmost panels, which for the IBIS data
+shown were were extracted from the near-vertical line
+representingthepositionoftheSPslit. Firstconsiderthe
+StokesIVprofiles. The spatial variation of Stokes I,V
+profiles along the SP slit position is remarkably similar
+in the two instruments. The higher angular and spectral
+resolution of the Hinodeobservations is evident. Closer
+scrutiny reveals that Stokes Vto continuum intensity
+ratios,V/Ic, measured by Hinodeare larger than those
+from IBIS, by a factor of typically 2.4. These differences
+can ariseprimarily because the magnetic field is spatially
+intermittent and not fully resolved. Consider V/Icwhich
+measures the net LOS flux per unit area in each resolu-
+tion element, in the weak field limit of the Zeeman effect
+(e.g. Lites 2000). This approximation not wildly incor-
+rect for these data. Consider just one magnetic element
+ofareaaand LOSfield strength Blos. The magnetic field
+generates an intrinsic Vactual. But when measured by an
+instrument iintegrating over area Ai≥a, the Stokes V
+is diluted by the (instrument-dependent) “filling factor”
+fi=a/Ai≤1. Then for instrument i,V/Icis
+Vi/Ic∝fiBlos, (1)
+Thus the lowerthe resolution (larger Ai), the smaller the
+V/Icsignal, when the magnetic field is not resolved. The
+presence of unpolarized stray light can also reduce V/Ic,
+but this effect is clearly smaller judging by the measured
+granular contrast variations. Similar arguments apply to
+QUdata. The data then suggest that the IBIS Vprofiles
+sample areas∼<2.4 times those sampled by the Hinode
+SP instrument, thus the effective resolution of these IBIS
+observations is∼<1.5×worse than that of Hinode. Tak-
+ing, as noted above, the latter to be 0 .′′3 (and not twice
+this which is the Nyquist limit), we get 0 .′′49 for IBIS.
+This number is compatible with the seeing-limited res-
+olution derived independently from Figure 3. Further-
+more, the Stokes V/Icintegrated over unipolar areas of
+several Mm2observed by IBIS and HinodeSP give the
+same LOS magnetic flux(not flux density) to within 1%,
+validatingthe aboveanalysis, no unresolvedflux ofoppo-site sign being significant. Below, we will show that the
+IBISand HinodeStokesVdata, asreducedintoprincipal
+components,haveverysimilarleadingordereigenvectors.
+IBISQUprofiles are shown twice in Figure 4. Those
+marked “Orig.” are profiles obtained in the telescope
+reference frame. These data are clearly not symmet-
+ric about line center, and appear V-like throughout.
+Rotation of these profiles alone to the solar reference
+frame only involves linear combinations of the QUdata
+which will yield similar profiles with large asymmetric,
+V-like components. The entire dataset is similarly con-
+taminated by V-like profiles. Some extra retardance
+has not been accounted for in the telescope calibration
+whichconvertsincoming VtoQUbeforeenteringthepo-
+larimeter. Requiring that QUbe symmetric about the
+(Doppler-shifted)line centers,wefound thatthesignand
+amount of V→QUcrosstalk is not a strong function of
+time, position on the detector etc.. Thus this crosstalk
+can be accounted for by a fixed Mueller matrix, of which
+the important components lead to the corrections:
+Q=Q(Orig)−(0.085±0.004)V,and
+U=U(Orig)−(0.051±0.003)V,
+using 1σstatistical uncertainties. Figure 4 shows these
+corrected QUprofiles also rotated to the solar refer-
+ence frame. The agreement with HinodeSP data for
+QUis now remarkable. These corrections are just
+the largest terms arising from extra retardance 6◦ori-
+ented at about 30◦to the IBIS reference direction, in
+whichQUVbecome mixed via the Mueller algebra (e.g.
+Seagraves & Elmore 1994). This retardance appears to
+arise from a thin film of oil noted on the exit port of
+the telescope. The corrections above were applied to the
+lines of both Fe and Ca.
+TheHinodeSP data for QUin the magnetic net-
+work and pore are significantly above the designed
+sensitivity limit of the instrument. They have es-
+sentially the symmetric profiles expected from the
+Zeeman effect. The heritage of Hinode’s SP is in
+the Advanced Stokes Polarimeter (Elmore et al. 1992;
+Skumanich et al. 1994) and Diffraction Limited Po-
+larimeter (Sankarasubramanian et al. 2006). Such pro-
+files are clearly of solar origin (e.g. Lites 2000) and free
+oflargesystematicerrorsintroduced, forexample, by the
+seeing-induced V→QU(Lites 1987; Judge et al. 2004).
+3.2.Polarization of Ca II854.2 nm
+Figure 5 shows a comparison of Fe Iand Ca IIprofiles
+for NOAA 10996, plotted as in figure 4, and from the
+same pixels. At the peak of the Vprofile near y= 179′′
+the signal-to noise ratio of Vis∼7 for the Ca IIdata
+shown. No significant QUsignal was detected in the
+CaIIdata. These data differ qualitatively from those
+of Fe I, as found from the analysis of Fe I 849.7 and
+853.8 nm lines observed with SPINOR by Pietarila et al.
+(2007). Beyond 0.04 nm from line center, i.e. at wave-
+lengths where at least the intensity is formed in the up-
+perphotosphere(Cauzzi et al.2008), the Ca IIVprofiles
+are similar to those of Fe Iin their signs and spatial dis-
+tribution along the slit. Thus the nominal calibration of
+IBIS leads to credible Stokes Vsignals in the parts of the
+CaIIline profiles whoseintensities originatein the upper
+photosphere. Hence, the (stronger) Stokes Vsignals in5
+the cores are also credible signals, formed in the chro-
+mosphere. Within 0.04 nm of the line core, the profiles
+are spatially much more diffuse, reflecting an expansion
+of magnetic flux with height, but the core profiles them-
+selves are more complex than their photospheric coun-
+terparts.
+The “magnetograms” in the second column are con-
+structed by subtracting the blue lobes of each line’s V
+profile from the red lobes, taking no account of Doppler
+shifts, at ±0.0064and ±0.017nmfromlinecenterforFe I
+and Ca IIrespectively. Vprofiles show that Ca II“mag-
+netograms” are a mix of Doppler, thermal and magnetic
+signals which cannot be interpreted straightforwardly in
+termsofLOScomponentsofthe chromosphericmagnetic
+field. Nevertheless chromospheric magnetic fields are the
+origin of Stokes Vin CaIIin this region, but there are
+also large influences from NLTE radiative transfer and
+chromospheric dynamics (Pietarila et al. 2007). In par-
+ticular, the Ca IIlineintensity forms over many scale
+heights (e.g., see figure 5 of Cauzzi et al. 2008). Hence
+the inner wings form in the upper photosphere where the
+magnetic field is relatively strong (compare the Fe Iline
+core data with the Ca IIline wing data in Figure 2). But
+the core (within roughly ±0.02 nm of line center) forms
+some 1Mm ( 6-7 pressure scale heights) higher, in the
+middle to upper chromosphere. The core intensity data
+(see also figure 4 of Cauzzi et al. 2008) are dominated by
+magnetically-dominatedfibril structures, overconcentra-
+tions of photospheric magnetic flux. The question of the
+formation heights of Stokes Vfor the Ca II854.2 nm
+line in such regions is complex, and will be addressed
+in later work. For now we simply note that contribu-
+tions toVcan arise from the bulk of the chromosphere,
+spanning several pressure scale heights and including the
+plasmaβ= 1 surface. Also the Vprofiles below y= 178
+have two peaks and are correlated with strong line core
+emission, but that from y= 179 to 184 the Vprofile is
+simpler and the cores are dark. Evidently the relation-
+ship between bright Ca IIemission and magnetic field is
+not linear in ourdata. Similar resultshavebeen reported
+elsewhere (e.g. Socas-Navarro et al. 2006).
+Calibrated data for NOAA 10994 are shown in Fig-
+ure 6, including profilesextracted from two columns (“a”
+and “b”). The local vertical of this regionis at 47◦to the
+LOS, thus strong vertical fields are seen partially as QU
+signals from the transverse Zeeman effect, as well as V.
+Credible QUprofiles are observed in the Ca II854.2 nm
+line aswellas the Fe I630.2nm line. These showperhaps
+some residual VtoQUcrosstalk. This is not surprising
+given the difference in wavelength between the Fe I and
+Ca II lines and the unknown optical properties of the oil
+film.
+3.3.Principal Components
+Principal Component Analysis (PCA), a pat-
+tern/shape recognition method, is useful in application
+to solar Stokes profiles because principal components
+(PCs) are often directly related to underlying atmo-
+spheric parameters (Skumanich & L´ opez Ariste 2002),
+for example to magnetic field strength, direction, line of
+sight motions, and thermal properties. Here we use it
+also to quantify crosstalk. PCs are the eigenvectors ofthe covariance matrix with components defined as
+Cij=∝an}b∇acketle{tSiSj∝an}b∇acket∇i}htx,t, (2)
+whereSiis the Stokes vector component (one of IQUV)
+at wavelength i, and the angle brackets imply an average
+over all spatial pixels xand/or time t. Denoting eias
+theitheigenvalue of Candvi,jas thejthcomponent of
+the eigenvector belonging to ei, any data point dk,j,k=
+spatial and/or temporal index, is represented by
+dk,j=nλ/summationdisplay
+i=1ck,ivi,j (3)
+wherenλis the number of wavelengths, and
+ck,i=nλ/summationdisplay
+j=1dk,jvi,j. (4)
+Assume that the eigenvalues/ vectors are ordered in
+decreasing magnitude. When the eigenvalues eidrop
+steeply with increasing i, then most properties of the
+data are described by the first few eigenvectors in the
+sum. Thedatapointsacrossthelineprofilesarenotcom-
+pletely independent of one another, and so the data can
+be represented by truncating the sum to values smaller
+thannλ. If however, the spectrum is shallow, there is
+much independence between data points and a larger
+number of vectors are needed. In our IBIS data, Stokes
+IVhave steep spectra, but QandUare shallower. This
+is because QUare noisy. Noise introduces linear inde-
+pendence between the different wavelengths across the
+lines, thereby flattening the eigenvalue spectrum.
+Figure 7 shows the first three eigenvectors derived for
+SP and the IBIS dataset obtained near 14:43UT, cov-
+ering the FOV shown in 4. The IBIS QUdata shown
+are calibrated but uncorrected for cross talk because our
+aim is to quantify the cross talk here using PCA. Con-
+sider first the QUVdata from Hinodein the figure. The
+PCs for HinodeQandUare almost symmetric around
+the line center and resemble the second derivative of the
+intensity profile, the principal Vcomponent being anti-
+symmetric. This is as expected because the HinodeV
+andQUprofiles arise from the first and second order
+Zeeman effect, respectively. While the IBIS principal
+Vcomponent is asymmetric, so too are the PCs of QU.
+PCAhas thus revealedthe previouslyidentified crosstalk
+inQUas the principal signal, but it is to be noted that
+the more symmetric second and third eigenvectors have
+eigenvalues just 0.6 and 0.2 dex below the PC’s eigen-
+value. There is therefore significant signal of the appro-
+priate symmetry in these IBIS data: the second PCs of
+IBISQUresemble those seen with SP.
+Using PCA, V→QUcross-talk can be quantified by
+projection of profiles onto the leading order PCs. Fig-
+ure 8 shows scatter plots of leading order coefficients ck1
+forQandUwith those for V, for typical observations
+kfrom both instruments. There is no significant corre-
+lation for the HinodeSP data, but the IBIS coefficients
+are correlated, the figure lists Spearman rank correlation
+coefficients. The dashed lines show c1(Q) = 0.085c1(V)
+andc1(U) = 0.051c1(V), the numerical coefficients being
+those of section 3.1. If the QUdata were pure cross-talk
+from an antisymmetric V, then all points would be dis-6 Judge et al.
+tributed along the plotted lines. The actual distributions
+show significant scatter, data for Uhaving a broader,
+skewed distribution. This behavior, arising from an un-
+known source of systematic error, highlights the limita-
+tion of the simple post-facto corrections of section 3.1.
+The corrected Uprofiles are probably reliable to at best
+±50%, given the plotted scatter. Nevertheless, the PCA
+analysis lends support for our empirical corrections.
+PCA should also help disentangle the complex Ca II
+QUVprofiles, in terms of identifying the physical pa-
+rameters most directly related to the Stokes profiles
+(Skumanich & L´ opez Ariste 2002). Figure 9 shows the
+eigenvalue spectra and first few eigenvectors of the PCA
+expansion, for a 10′′×10′′area centered on NOAA 10994
+(see Figure 6), a region chosen because it has measur-
+able linear polarization in the Ca II854.2 nm line. The
+QUprofiles are mostly dominated by noise (as shown
+by the shallow gradient in the eigenvalue spectrum, and
+noisy first eigenvector). But the other eigenvectors show
+that genuine QUsignals are present for this small pore.
+StokesVcontainsrealsignalbecausethefirsteigenvector
+is predominantly of asymmetric form, and the eigenvalue
+spectrum is steeper than QandU. ThisVcomponent
+has no obvious net amplitude or area asymmetry, in con-
+trast with Pietarila et al. (2007), who found a net red-
+asymmetry in the 854.2nm line for some active network.
+3.4.Fibrils and their motions constrain the magnetic
+field
+Figure 5 also shows core and wing intensities of the
+Hαline from IBIS. The wing and core behavior is sim-
+ilar to the 854.2 nm line of Ca II, reflecting conditions
+at deeper layers of the photosphere and top of the chro-
+mosphere respectively, consistent with the picture of line
+formation of Athay (1976). The fibril structures in the
+core intensity images are similar to, not identical with,
+those for the 854.2 nm Ca IIline. Hαimages at a given
+wavelength setting of IBIS are remarkably structured.
+The motions of “blobs” of emission or absorption seem
+to trace the fibril- and hence magnetic field line- orienta-
+tions. Curiously, on the red side of H α, blobs appear to
+converge on the underlying photospheric flux concentra-
+tions. On the blue side, they diverge from them. To try
+to determine if these “blobs” correspond to real material
+motion, we show in figure 10 H αdata from three snap-
+shots in the time series. The leftmost panel is the line
+core intensity image of the middle snapshot. The three
+other panels show data using PCA. The particular eigen-
+vector which corresponds to line of sight Doppler shift,
+i.e. having a profile corresponding to the first derivative
+of the intensity profile with respect to wavelength, was
+identified. This component was then projected onto the
+data themselves at each point and time and images dis-
+played,yieldingDopplermaps. (TheseDopplervelocities
+aresimilarto thosederivedfromthe first wavelengthmo-
+mentofthe profiles). Darker(lighter)profilescorrespond
+to blue (red) shifted components. Examining the region
+between the two plotted circles, severaldarkblobs ofma-
+terial move away from the circle centers with time. The
+Doppler shifts of these blobs are towards the observer
+(vlos<0). (While less clear, movies of the red-shifted
+materialshowcomponentsconvergingontothecirclecen-
+ters). This behavior is compatible with simple flows of
+absorbing material which diverge from or converge toa magnetic structure originating from the photospheric
+flux concentrationand expandingoutwards, perhapsinto
+the corona. Hence, using proper motions (determined by
+eye in this case) and Doppler shifts, one can trace out
+the 3D vector velocity field of the entrained fibril mate-
+rial. In the particular blobs shown, the vector velocity
+has components ( vx,vy,vz)≈(−7,−7,−3) km s−1, i.e.
+the velocity is inclined at 17◦to the plane of the sky,
+only 16◦from the local solar horizontal plane.
+These measurements are more than a curiosity. Given
+the strong collisional coupling between the neutral and
+ionized components, and the high electrical conductivity,
+the 3D velocityfield tracesout the direction ofthe vector
+magnetic field, which is therefore also only ≈16◦to the
+horizontal plane. Such observations offer observers the
+important opportunity to augment spectropolarimetric
+measurements of chromospheric vector magnetic fields,
+which will always suffer from weak Zeeman-induced QU
+signals. Measurementof the LOS component ofthe mag-
+netic field from Stokes V, coupled with kinematic fluid
+velocities, determines the vector magnetic field. Chro-
+mospheric fibril observations in Ca II or H αare thus far
+richer than suggested by inspection of Figure 5. They
+will be discussed in the context of constraining the chro-
+mospheric magnetic fields elsewhere.
+4.DISCUSSION
+We have demonstrated the fidelity of a two-
+dimensional filtergraph instrument, IBIS for accurate
+StokesVmeasurements at high angular resolution, by
+verification through almost simultaneous measurements
+from the Spectro-Polarimeter on board the Hinodesatel-
+lite. IBIS measurements of Stokes QUprofiles of the Fe I
+630.2 nm line are subject, as expected, to systematic er-
+rors (cross talk), which we have quantified both empir-
+ically and using Principal Component Analysis. IBIS
+is a Stokes Definition Polarimeter so that just one po-
+larization state is measured during each camera integra-
+tion. Therefore, the cross talk among QUandVis of a
+slowly varying character, originating from errors in the
+telescopecalibrationdata. Anempiricalcorrectionyields
+QUprofiles in remarkable agreement with the SP data,
+yielding credible QUprofiles both in the Fe I630.2nm
+line, and, for a pore far from disk center at a viewing
+angle of 47◦, credible observations of Stokes QUin the
+chromospheric Ca II854.2 nm line.
+We believe this is the first time that a direct compar-
+ison of spectropolarimetry has been made between slit
+and two-dimensional filtergraph instruments using data
+of the same region obtained within a few seconds of each
+other. (The Advanced Stokes Polarimeter had this capa-
+bility by use of a slit-jaw but the results were not very
+good- Lites 2009, privatecommunication). Giventhe in-
+trinsic difficulty of making spectropolarimetric measure-
+mentsfromthe ground,the excellentagreementofStokes
+profiles found with near-simultaneous observations from
+IBIS and the HinodeSP provides strong support for the
+credibility of measurements made with an IBIS-like in-
+strument. We note that the magnetic features observed
+hereareratherweakcomparedwith strongsunspots, and
+the IBIS camera is to be updated to permit a larger duty
+cycle and increased signal to noise.
+Our goal is to diagnose properties of the magnetic field
+near the base of the coronausing, in part, the Ca IIchro-7
+mospheric lines. At face value, the Ca IIdata appear
+to be of limited use in that under many conditions the
+noise is dominant in the QUdata and corrections must
+be applied for inaccuracies in the telescope polarization
+calibration data. Other complications include difficulties
+concerning non-LTE transfer of in the Ca IIlines. Yet
+IBIS has some significant advantages over slit spectropo-
+larimetric measurements. IBIS images can be corrected
+usingimagereconstructiontechniques. Thisisimportant
+because an angular resolution of 1′′or better, covering
+fields of tens of seconds of arc with a spectral resolution
+of some tens of m ˚A is required to see clearly the fib-
+ril structure of the upper chromosphere (see Figure 2).
+A cadence of 70 seconds is barely enough to track kine-
+matic motion along the fibrils,∼<20 seconds being highly
+desirable. Such criteria may not be achieved using slit
+instruments. The observation of fibril kinematics, even
+in unpolarized light, appears important for diagnosing
+chromospheric magnetic fields because fibrils not only
+traceoutmagneticlinesofforce, but thepartiallyionized
+plasmaisrequiredtoflowalongtheselines becauseofthe
+large electrical conductivity. From our IBIS data for H α,
+it appears possible to measure the velocity vector of fib-
+ril “blobs” in the plane of the sky, and along the LOS,
+yielding the direction of the magnetic field (to within
+an ambiguity of 180◦). Coupled with the Stokes Vsig-
+nals which may in principle yield the LOS field strength,
+there is thus hope that the vector magnetic field might
+be constrained from such observations.
+In conclusion, this dataset provides, for the first time,
+credible imaging spectropolarimetry from photosphere
+through the chromosphere together with a useful time
+series of resolved fibril motions in the upper chromo-
+sphere. These data can be used to place observational
+constraints on the vector magnetic field throughout the
+chromosphere. In the case of NOAA 10996, we have
+found that the magnetic field is highly inclined to the
+LOS, as in a traditional magnetic “canopy” (Giovanelli
+1982), and that the chromospheric heating rates are not
+simply related to the Stokes Vprofiles of Ca II. We will
+explore the possibilities further using the data described
+here in later publications. More generally, IBIS holds
+promise as a tool for chromospheric vector polarimetry
+in the Ca II854.2 nm line, following the work on a larger
+sunspot by Socas-Navarro (2005).
+PGJ gratefully acknowledges the help of F. Woeger
+in speckle reconstruction, and discussions with B. W.
+Liteswhosecommentshelpedtoimprovethemanuscript.
+Hinode is a Japanese mission developed and launched
+by ISAS/JAXA, collaborating with NAOJ as a domes-
+tic partner, NASA and STFC (UK) as international
+partners. Scientific operation of the Hinode mission
+is conducted by the Hinode science team organized at
+ISAS/JAXA. This team mainly consists of scientists
+from institutes in the partner countries. Support for the
+post-launch operation is provided by JAXA and NAOJ
+(Japan), STFC (U.K.), NASA (U.S.A.), ESA, and NSC
+(Norway). This research has also made use of NASA’s
+AstrophysicsDataSystem (ADS). An anonymousreferee
+helped us improve the presentation.
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+APPENDIX
+ORIGINS OF CROSSTALK
+Considerforsimplicity thecaseofaperfect polarimeter. Let Mbe the Muellermatrixdescribingtheknowntelescope
+calibration parameters. The Stokes vector Sentering the polarimeter is R=MSwhereSis the (desired) solar Stokes
+vector.Ris inferred from the modulation and analysis of instrument i. Then the estimate of the solar Stokes vector
+from this instrument is
+Si=M−1R
+With (fictional) perfect knowledge of the Mueller matrix, the solar St okes vector is
+S=M−1
+0R
+We can write M=M0+Ewhere the (unknown) elements of the error matrix Eare assumed small compared with
+M0- the telescope calibration matrix is close to the actual matrix. Keep ing first order terms only,
+S−Si=EM−1R≡ESi,
+which is the error in the inferred solar Stokes vector resulting from an inaccurate telescope calibration. Off-diagonal
+(unknown) terms in Econvert the inferred Sivalues into other Stokes components. For a stable instrument, Ecan be
+considered constant or very slowly varying with time. This kind of err or is present in all practical polarimeters.
+In the presence of atmospheric seeing, the Stokes vector enter ing the telescope is a rapidly varying distortion of Sin
+which neighboring points on the solar surface are spatially smeared. LetTα(x,y;t) represent the αcomponent of the
+Stokes vector entering the telescope from apparent position ( x,y) on the sky at time t. Then, including just the lowest
+order (tip/tilt) distortions at any time t,Tarises from a different place on the sky ( x′,y′) = (x+δx(t),y+δy(t)).
+ExpandTα(x,y;t) as a Taylor series in the solar Stokes component Sαon the sky:
+Tα(x,y;t) =Sα(x+δx,y+δy;t)≈Sα(x,y;t)+∇Sα·s(t)
+to first order (Judge et al. 2004). Here the ∇operator and vector s(t) are vectors in the ( x,y) plane, s(t) =
+(δx(t),δy(t))Tis the tip-tilt component of the seeing. Let Pνbe the power spectrum of the seeing. All measure-
+ments require a finite integration time τ, during which the seeing s(t) varies. At all frequencies νwherePντ≥1,
+seeing detrimentally affects the measurements. When integrated o ver time τ, detectors record energy
+/integraldisplayτ
+03/summationdisplay
+α=0aαTα(x,y;t)dt=/integraldisplayτ
+03/summationdisplay
+α=0aα{Sα(x,y;t)+∇Sα·s(t)}dt, (A1)
+wherea0= 1 (S0=I) andaα/negationslash=0depends on the polarimeter’s particular modulation and integration s cheme. With
+many realizations of the seeing, the equation becomes
+3/summationdisplay
+α=0aα∝an}b∇acketle{tTα∝an}b∇acket∇i}ht=3/summationdisplay
+α=0aα∝an}b∇acketle{tSα∝an}b∇acket∇i}ht, (A2)
+because∝an}b∇acketle{t∇Sα·s(t)∝an}b∇acket∇i}htaveragestozerowhen s(t), arisingfromatmosphericturbulence, isstatisticallyspatiallysymm etric.
+The∝an}b∇acketle{tSα∝an}b∇acket∇i}htcomponents have variances arising from the seeing:
+σ2
+α=∝an}b∇acketle{t|∇Sα·s(t)|2∝an}b∇acket∇i}ht (A3)
+which Lites (1987) has shown can be evaluated from the seeing powe r spectrum using the assumption that the seeing
+is a random phenomenon. Equation (A2) is a linear system for the des ired solar components ∝an}b∇acketle{tSα∝an}b∇acket∇i}htin terms of the
+measurements ∝an}b∇acketle{tTα∝an}b∇acket∇i}ht, subject to statistical variations described by σα.
+For a Stokes definition polarimeter like IBIS, aα=1,2,3is non-zero only for one Stokes parameter, βsay. Then the
+above equations relate ∝an}b∇acketle{tS0∝an}b∇acket∇i}ht+aβ∝an}b∇acketle{tSβ∝an}b∇acket∇i}htonly to∝an}b∇acketle{tT0∝an}b∇acket∇i}htandaβ∝an}b∇acketle{tTβ∝an}b∇acket∇i}ht. The second beam yields simultaneous measurements
+of equation ∝an}b∇acketle{tT0∝an}b∇acket∇i}ht−aβ∝an}b∇acketle{tTβ∝an}b∇acket∇i}ht, and this two equation system is solved for all components Sβin terms of Tαas usual, with
+uncertainties propagated via σ0andσβ.
+But for other polarimeters, like the SP on Hinode,aαis non-zero for at least two of QUVduring each measurement.
+The above equation becomes a system of more than two equations. After eliminating the intensity from the equation9
+Table 1
+Log of observations for 20 May 2008
+Observatory Instrument Observable Center FOV Time UT
+DST IBIS Fe I 630.2, Ca II 854.2, H α (-161,171) 40 ×80 14:29-15:05
+WL white light 40×80 ditto
+G band G band 90×90 ditto
+Hinode SP Fe I 630.2 fast map (-144,210) 107 ×82 14:30-16:39
+FG Na I D, Ca II H, G band (-137,204) ) 82 ×82 14:30-16:41
+EIS 9 lines (-126,189) 520 ×512 14:30-16:37
+XRT 3 bands (-139,169) 130 ×169 14:00:16:36
+TRACE WL, 160nm, 19.5nm (-166,199) 512 ×512 14:01-16:42
+DST IBIS Fe I 630.2, Ca II 854.2, H α (666,-193) 40 ×80 15:12-15:19
+WL white light ditto
+G band G band ditto
+Note. — Solar coordinates are relative to the SOLIS magnetogram o btained near 15:16 UT and are reliable to about ±1′′. The center
+of the SP and FG images reported in the headers before 15:05 UT were quite different from those in the table, being (-151,183 ) and
+(-145,176) respectively. The TRACE image center is for the 19.5 nm band. The IBIS commanded point ing was (-165,158). The pointing
+is approximate for the EIS instrument.
+using the second beam we see that Stokes parameter Sβis also influenced though the terms σα/negationslash=β. If the variances
+σα(e.g. Stokes V) are larger than the desired terms Sβ(e.g. Stokes Q), then the linear system yields estimates for
+Sβwhich are dominated in any particular realization by terms of order σα. In words, crosstalk arises because, during
+the exposure for modulation state S0+Sβ, fluctuations in time occur in Sβ/negationslash=αresulting from the motion of the solar
+image with contains structure. The great advantage of the SP is its remarkably small rms image motion, so that
+image-motion induced cross talk is practically negligible.10 Judge et al.
+Figure 1. A context image from SOLIS (red) and TRACE (cyan) showing the areas observed around NOAA 10996 with the Hinode
+(spectropolarimeter- “SP”, filtergram- “FG”) and with IBIS , before 15:06 UT on 20 May 2008. The SOLIS scan began at 15:11 U T, the
+TRACE image shown was obtained at 15:12 UT. The image coordinates r efer to the SOLIS coordinate system.11
+Figure 2. Typical IBIS data of NOAA 10996 observed on 20 May 2008 from sc an number 11. Almost the entire IBIS FOV is shown. The
+“magnetogram” is simply V(-74 m˚A) minus V(+56 m˚A) for the Fe Iline.12 Judge et al.
+Figure 3. RMS granular contrasts are shown for the quietest regions of the IBIS and HinodeSP FOV, as measured in the continuum
+near 630.2 nm. Also shown is the contrast corresponding to an effective 1′′resolution derived by Lites (2002).13
+Figure 4. Typical IBIS and HinodeSP polarimetric measurements of the 630.2 nm line. The upper panels show IBIS data, the lower
+showHinodedata. The leftmost column shows intensity images, the next c olumn magnetograms, from IBIS (( Vλ0+6pm−Vλ0−6pm)/IC,
+λ0= rest wavelength.) and the FG instrument on Hinode(in Mx cm−2). The near-vertical line shows the position of the Hinodeslit at
+14:36:52 UT, from which the six rightmost images of Stokes pr ofiles as a function of wavelength and position along the SP sl it are taken.
+QUdata marked “Orig.” are before subtraction of crosstalk and a final rotation in the polarization plane, the unflagged QUdata include
+these corrections. IBIS profiles are extracted from the data closest in time and space to the SP measurements. Color scale s are shown at
+the top of each image, where x.y(z)≡x.y×10z.14 Judge et al.
+Figure 5. A figure similar to figure 4, except that the lower panels are IB IS data for the 854.2 nm line of Ca II and H α, theV/ICframe
+for Ca II shows ( Vλ0+17pm−Vλ0−17pm)/IC.15
+Figure 6. A figure similar to figure 4, showing the pore region associate d with NOAA 10994 at S12.4, W46.1.16 Judge et al.
+Figure 7. The first three eigenvectors in the PCA expansion for each of t he four Stokes parameters of the Fe I 630.2 nm line are shown,
+for the IBIS data from 14:43 UT on 20 May 2008, and the correspo ndingHinodeSP data. The first row is the eigenvector with the largest
+eigenvalue, the second and third show the eigenvectors for t he second and third largest eigenvalues. The second and thir d rows lists log 10
+of the modulus of the eigenvalue (the largest is set to 1). The columns are labelled with the instrument and Stokes paramet er. The abscissa
+is wavelength index.17
+Figure 8. Scatter plot of the leading coefficients in the principal comp onent expansion for Stokes QUand forV. The dashed lines show
+the relations c1(Q) = 0.085c1(V) andc1(U) = 0.051c1(V) derived empirically using the symmetry properties discus sed in the text.18 Judge et al.
+Figure 9. Eigenvalues and the first three eigenvectors from a PCA analy sis of the 854.2nm line of Ca II. The data are constructed from
+the central 10′′×10′′area of the pore, NOAA 10994, where significant linear polari zation was measured.19
+Figure 10. Hαdata shown for NOAA 10996. The leftmost image is a simple inte nsity image, the others show the time sequence of
+three snapshots taken roughly 70s apart, of Doppler shifts d erived using the PCA technique. Bright features are redshif ted, dark features
+blue shifted. The dark blobs near 8 o’clock (see the arrow) pr opagate roughly outwards from the centers of the circles mar king a bright
+concentration of magnetic flux. These data are qualitativel y compatible with magnetic field-directed flows along the mag netic fields
+originating from the flux concentration and expanding into t he overlying corona. The velocity vectors can in principle y ield the direction
+of the magnetic field.
\ No newline at end of file
diff --git a/1001.0562.txt b/1001.0562.txt
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@@ -0,0 +1,2177 @@
+arXiv:1001.0562v2  [math.AP]  23 Feb 2010A new dynamical approach of Emden-Fowler equations and syst ems
+Marie Fran¸ coise BIDAUT-VERON∗Hector GIACOMINI†
+.
+Abstract
+We give a new approach on general systems of the form
+(G)/braceleftbigg−∆pu=−div(|∇u|p−2∇u) =ε1|x|ausvδ,
+−∆qv=−div(|∇v|q−2∇u) =ε2|x|buµvm,
+whereQ,p,q,δ,µ,s,m, a,b are real parameters, Q,p,q/\e}atio\slash= 1,andε1=±1, ε2=±1.In the
+radial case we reduce the problem to a quadratic system of four co upled first order autonomous
+equations,ofKolmogorovtype. Itallowstoobtainnewlocalandglob alexistenceornonexistence
+results. We consider in particular the case ε1=ε2= 1.We describe the behaviour of the ground
+states in two cases where the system is variational. We give a result o f existence of ground states
+for a nonvariational system with p=q= 2 ands=m >0,that improves the former ones.
+It is obtained by introducing a new type of energy function. In the n onradial case we solve
+a conjecture of nonexistence of ground states for the system w ithp=q= 2,δ=m+ 1 and
+µ=s+1.
+Keywords Ellipticquasilinearsystems. Variationalornonvariationalproblems. Autonomous
+and quadratic systems. Stable manifolds. Heteroclinic orbits.
+A.M.S. Subject Classification 34B15, 34C20, 34C37; 35J20, 35J55, 35J65, 35J70; 37J45.
+.
+∗Laboratoire de Math´ ematiques et Physique Th´ eorique, CNR S UMR 6083, Facult´ e des Sciences, 37200 Tours
+France. E-mail address:veronmf@univ-tours.fr
+†Laboratoire de Math´ ematiques et Physique Th´ eorique, CNR S UMR 6083, Facult´ e des Sciences, 37200 Tours
+France. E-mail address:Hector.Giacomini@lmpt.univ-tou rs.fr
+11 Introduction
+In this paper we consider the nonnegative solutions of Emden -Fowler equations or systems in
+RN(N≧1),
+−∆pu=−div(|∇u|p−2∇u) =ε1|x|auQ, (1.1)
+(G)/braceleftbigg−∆pu=−div(|∇u|p−2∇u) =ε1|x|ausvδ,
+−∆qv=−div(|∇v|q−2∇u) =ε2|x|buµvm,(1.2)
+whereQ,p,q,δ,µ,s,m,a,b are real parameters, Q,p,q/\e}atio\slash= 1,andε1=±1,ε2=±1.These problems
+are the subject of a very rich litterature, either in the case of source terms ( ε1=ε2= 1) or
+absorption terms ( ε1=ε2= 1) or mixed terms ( ε1=−ε2).In the sequel we are concerned by the
+radial solutions, except at Section 9 where the solutions ma y be nonradial.
+In this article we we give a new way of studying the radial solutions . In Section 2 we reduce
+system (G) to a quadratic autonomous system:
+(M)
+
+Xt=X/bracketleftBig
+X−N−p
+p−1+Z
+p−1/bracketrightBig
+,
+Yt=Y/bracketleftBig
+Y−N−q
+q−1+W
+q−1/bracketrightBig
+,
+Zt=Z[N+a−sX−δY−Z],
+Wt=W[N+b−µX−mY−W],
+wheret= lnr,and
+X(t) =−ru′
+u, Y(t) =−rv′
+v, Z(t) =−ε1r1+ausvδu′
+|u′|p, W(t) =−ε2r1+buµvmv′
+|v′|q.(1.3)
+This system is of Kolmogorov type. The reduction is valid for equations and systems with source
+terms , absorption terms , or mixed terms .It is remarkable that in the new system, pandqappear
+only assimple coefficients , which allows to treat any value of the parameters, even porq<1,and
+s,m,δorµ<0.
+InSection 3werevisitthewell-known scalar case(1.1), whe re(G) becomestwo-dimensional. We
+show that the phase plane of the system gives at the same time the behaviour of the two equations
+−∆pu=|x|auQand−∆pu=−|x|auQ,
+which is a kind of unification of the two problems , with source terms or absorption terms. For the
+case of source term ( ε1= 1),we find again the results of [2], [19], showing that the new dyn amical
+approach is simple and does not need regularity results or en ergy functions. Moreover it gives a
+model for the study of system ( G). Indeed if p=q,a=bandδ+s=µ+m,system (G) admits
+solutions of the form ( u,u),whereuis a solution of (1.1) with Q=δ+s.
+In the sequel of the article we study the case of source terms, i.e. (G) = (S),where
+(S)/braceleftbigg−∆pu=|x|ausvδ,
+−∆qv=|x|buµvm.(1.4)
+This system has been studied by many authors, in particular t he Hamiltonian problem s=m= 0,
+in the linear case p=q= 2,see for example [20], [31], [29], [9], [33], [14], and the pot ential system
+2whereδ=m+1,µ=s+1 anda=b,see [7], [34], [35]; the problem with general powers has been
+studied in [3], [39], [40], [41] in the linear case and [6], [1 2], [42] in the quasilinear case, see also [1],
+[10], [13].
+Here we suppose that δ,µ>0, so that the system is always coupled, s,m≧0,and we assume
+for simplicity
+1<p,q<N, min(p+a,q+b)>0, D=δµ−(p−1−s)(q−1−m)>0.(1.5)
+Wesaythatapositivesolution( u,v) in(0,R) isregularat 0ifu,v∈C2(0,R)∩C([0,R)). Condition
+min(p+a,q+b)>0guaranties theexistence oflocal regularsolutions. Then u,v∈C1([0,R)). when
+a,b >−1,andu′(0) =v′(0) = 0. The assumption D >0 is a classical condition of superlinearity
+for the system.
+We are interessed in the existence or nonexistence of ground states , called G.S ., that means
+global positive ( u,v) in (0,∞) and regular at 0 .We exclude the case of ”trivial” solutions, ( u,v) =
+(0,C) or (C,0),whereCis a constant, which can exist when s>0 orm>0.
+In Section 4 we give a series of local existence or nonexistence results concerning system ( S),
+which complete the nonexistence results found in the litter ature. They are not based on the fixed
+point method, quite hard in general, see for example [19], [2 7]. We make a dynamical analysis of
+the linearization of system ( M) near each fixed point, which appears to be performant, even f or
+the regular solutions. For a better exposition, the proofs a re given at Section 10.
+In Section 5 we study the global existence of G.S. This problem has been often compared with
+thenonexistence of positive solutions of theDirichlet pro blem in aball, see [29], [30], [12], [13]. Here
+we use a shooting method adapted to system ( M), which allows to avoid questions of regularity of
+system (S).We give a new way of comparison, and improve the former result s:
+Theorem 1.1 (i) Assume s<N(p−1)+p+pa
+N−pandm<N(q−1)+q+qb
+N−q.If system (S)has no G.S., then
+(i) there exist regular radial solutions such that X(T) =N−p
+p−1andY(T) =N−q
+q−1for someT >0,
+with0<X <N−p
+p−1and0<Y <N−q
+q−1on(−∞,T).
+(ii) there exists a positive radial solution (u,v)of the Dirichlet problem in a ball B(0,R).
+This result is a key tool in the next Sections for proving the e xistence of a G.S. It gives also
+new existence results for the Dirichlet problem, see Coroll ary 5.3. We also give a complementary
+result:
+Proposition 1.2 Assumes≧N(p−1)+p+pa
+N−pandm≧N(q−1)+q+qb
+N−q.Then all the regular radial
+solutions are G.S.
+In Section 6 we study the radialsolutions of the well known Hamiltonian system
+(SH)/braceleftbigg−∆u=|x|avδ,
+−∆v=|x|buµ,
+corresponding to p=q= 2<N,s=m= 0, a>−2,which is variational. In the case a=b= 0,a
+main conjecture was made in [32]:
+3Conjecture 1.3 System(SH)witha=b= 0admits no (radial or nonradial) G.S. if and only if
+(δ,µ)is under the hyperbola of equation
+N
+δ+1+N
+µ+1=N−2.
+The question is still open; it was solved in the radial case in [26], [29], then partially in [31], [9],
+and up to the dimension N= 4 in [33], see references therein. Here we find again and exte nd to
+the casea,b/\e}atio\slash= 0 some results of [20] relative to the G.S., with a shorter pr oof. We also give an
+existence result for the Dirichlet problem improving a resu lt of [14].
+Theorem 1.4 LetH0be the critical hyperbola in the plane (δ,µ)defined by
+N+a
+δ+1+N+b
+µ+1=N−2. (1.6)
+Then
+(i) System (SH)admits a (unique) radial G.S. if and only if ( δ,µ)is above H0or onH0.
+(ii) The radial Dirichlet problem in a ball has a solution if an d only if (δ,µ)is under H0.
+(iii) On H0the G.S. has the following behaviour at ∞:assuming for example δ >N+a
+N−2,then
+limr→∞rN−2u(r) =α>0,and
+lim
+r→∞r(N−2)µ−(2+b)v=β >0ifµ<N+b
+N−2,
+lim
+r→∞rN−2v=β >0ifµ>N+b
+N−2,
+lim
+r→∞rN−2|lnr|−1v=β >0ifµ=N+b
+N−2.
+Our proofs use a Pohozaev type function; in terms of the new va riablesX,Y,Z,W , it contains
+a quadratic factor
+EH(r) =rN/bracketleftBigg
+u′v′+rb|u|µ+1
+µ+1+ra|v|δ+1
+δ+1+N+a
+δ+1vu′
+r+N+b
+µ+1uv′
+r/bracketrightBigg
+=rN−2uv/bracketleftbigg
+XY−Y(N+b−W)
+µ+1−(N+a−Z)X
+δ+1/bracketrightbigg
+. (1.7)
+As observed in ([20]) the G.S. can present a non-symmetric be haviour. This non-symmetry phe-
+nomena has to be taken in account for solving conjecture (1.3 ).
+In Section 7 we consider the radial solutions of a nonvariational system :
+(SN)/braceleftbigg−∆u=|x|ausvδ,
+−∆v=|x|auµvs,
+wherep=q= 2< N, a=b >−2 andm=s >0.For smallsit appears as a perturbation of
+system (SH).In the litterature very few results are known for such nonvar iational systems. Our
+main result in this Section is a new result of existence of G.S . valid for any s:
+4Theorem 1.5 Consider the system (SN),withN >2, a>−2.We define a curve Csin the plane
+(δ,µ)by
+N+a
+µ+1+N+a
+δ+1=N−2+(N−2)s
+2min(1
+µ+1,1
+δ+1), (1.8)
+located under the hyperbola defined by (1.6) .If(δ,µ)is above Cs,system(SN)admits a G.S.
+This result is obtained by constructing a new type of energy f unction which contains two terms in
+X2,Y2:
+Φ(r) =rN/bracketleftbigg
+u′v′+rbuµ+1vs
+µ+1+rausvδ+1
+δ+1+N+a
+δ+1vu′
+r+N+b
+µ+1uv′
+r+s
+2(δ+1)vu′2
+u+s
+2(µ+1)uv′2
+v/bracketrightbigg
+=rN−2uv/bracketleftbigg
+XY−Y(N+b−W)
+µ+1−(N+a−Z)X
+δ+1+s
+2(δ+1)X2+s
+2(µ+1)Y2/bracketrightbigg
+.(1.9)
+In Section 8 we consider the radialsolutions of the potential system
+(SP)/braceleftbigg−∆pu=|x|ausvm+1,
+−∆qv=|x|aus+1vm,
+whereδ=m+ 1,µ=s+ 1 anda=b,which is variational, see [34], [35]. Using system ( M) we
+deduce new results of existence:
+Theorem 1.6 LetDbe the critical line in the plane (m,s)defined by
+N+a= (m+1)N−q
+q+(s+1)N−p
+p.
+Then
+(i) System (SP)admits a radial G.S. if and only if ( m,s)is above or on D.
+(ii) OnDthe G.S. has the following behaviour: suppose for example q≦p.Letλ∗=N+a−
+(s+1)N−p
+p−1−mN−q
+q−1.Thenlimr→∞rN−p
+p−1u(r) =α>0,and
+lim
+r→∞rN−q
+q−1v(r) =β >0ifλ∗<0, (1.10)
+lim
+r→∞rN−p
+p−1µ−(q+b)
+q−1−mv(r) =β >0ifλ∗>0, (1.11)
+lim
+r→∞rN−q
+q−1|lnr|−1
+q−1−mv(r) =β >0ifλ∗= 0. (1.12)
+In particular (1.10) holds if p=q,orq≦m+1.
+(iii) The radial Dirichlet problem in a ball has a solution if a nd only if ( m,s)is under D.
+5In that case we use the following energy function, which dese rves to be compared with the one
+of Section 6 , since it has also a quadratic factor:
+EP(r) =rN/bracketleftBigg
+(s+1)(|u′|p
+p′+N−p
+pu|u′|p−2u′
+r)+(m+1)(|v′|q
+q′+N−q
+qv|v′|q−2v′
+r)+raus+1vm+1/bracketrightBigg
+=rN−2−a|u′|p−1|v′|q−1
+usvm/bracketleftbigg
+ZW−(s+1)W(N−p−(p−1)X)
+p−(m+1)Z(N−q−(q−1)Y)
+q/bracketrightbigg
+.
+(1.13)
+Finally in Section 9 we deduce a nonradial result for the potential system in the case of two
+Laplacians:
+(SL)/braceleftbigg−∆u=|x|ausvm+1,
+−∆v=|x|aus+1vm.
+Our result proves a conjecture proposed in [7], showing that in the subcritical case there exists no
+G.S.:
+Theorem 1.7 Assumea>−2ands,m≧0.If
+s+m+1<min(N+2
+N−2,N+2+2a
+N−2), (1.14)
+then system (SL)admits no (radial or nonradial) G.S.
+Our proof uses the estimates of [7], which up to now are the onl y extensions of the results of
+[18] to systems. It is based on the construction of a nonradia l Pohozaev function extending the
+radial one given at (1.13) for p=q= 2, different from the energy function used in [7].
+The case of the system ( G) with absorption terms ( ε1=ε2=−1) or mixed terms ( ε1=−ε2=
+1), studied in [4], [5], will be the subject of a second articl e. Our approach also extends to a system
+with gradient terms,/braceleftBigg
+−∆pu=ε1|x|ausvδ|∇u|η|∇v|ℓ,
+−∆qv=ε2|x|buµvm|∇u|ν|∇v|κ,(1.15)
+which will be studied in another work.
+Acknoledgment The authors are grateful to Raul Manasevich whose stimulati ng discussions
+encouraged us to study system ( G).
+2 Reduction to a quadratic system
+2.1 The change of unknowns
+Here we consider the radial positive solutions r/mapsto→(u(r),v(r)) of system ( G) on any interval
+(R1,R2), that means
+
+
+/parenleftBig
+|u′|p−2u′/parenrightBig′
++N−1
+r|u′|p−2u′=r1−N/parenleftBig
+rN−1|u′|p−2u′/parenrightBig′
+=−ε1rausvδ,
+/parenleftBig
+|v′|q−2v′/parenrightBig′
++N−1
+r|v′|q−2v′=r1−N/parenleftBig
+rN−1|v′|p−2v′/parenrightBig′
+=−ε2rbuµvm.
+6Near any point rwhereu(r)/\e}atio\slash= 0,u′(r)/\e}atio\slash= 0 andv(r)/\e}atio\slash= 0, v′(r)/\e}atio\slash= 0 we define
+X(t) =−ru′
+u, Y(t) =−rv′
+v, Z(t) =−ε1r1+ausvδ/vextendsingle/vextendsingleu′/vextendsingle/vextendsingle−pu′, W(t) =−ε2r1+buµvm/vextendsingle/vextendsinglev′/vextendsingle/vextendsingle−qv′,
+(2.1)
+wheret= lnr.Then we find the system
+(M)
+
+Xt=X/bracketleftBig
+X−N−p
+p−1+Z
+p−1/bracketrightBig
+,
+Yt=Y/bracketleftBig
+Y−N−q
+q−1+W
+q−1/bracketrightBig
+,
+Zt=Z[N+a−sX−δY−Z],
+Wt=W[N+b−µX−mY−W].
+This sytem is quadratic , and moreover a very simple one, of Kolmogorov type : it admits four
+invariant hyperplanes: X= 0,Y= 0,Z= 0,W= 0. As a first consequence all the fixed points
+of the system are explicite. The trajectories located on the se hyperplanes do not correspond to a
+solution of system ( G); they will be called nonadmissible .
+We suppose that the discriminant of the system
+D=δµ−(p−1−s)(q−1−m)/\e}atio\slash= 0. (2.2)
+Then one can express u,vin terms of the new variables:
+u=r−γ(|X|p−1|Z|)(q−1−m)/D(|Y|q−1|W|)δ/D, v=r−ξ(|X|p−1|Z|)µ/D(|Y|q−1|W|)(p−1−s)/D,
+(2.3)
+whereγandξare defined by
+γ=(p+a)(q−1−m)+(q+b)δ
+D, ξ=(q+b)(p−1−s)+(p+a)µ
+D,(2.4)
+or equivalently by
+(p−1−s)γ+p+a=δξ,(q−1−m)ξ+q+b=µγ. (2.5)
+Since system ( M) is autonomous, each admissible trajectory Tin the phase space corresponds to
+a solution ( u,v) of system ( G) unique up to a scaling: if ( u,v) is a solution, then for any θ >0,
+r/mapsto→(θγu(θr),θξv(θr)) is also a solution.
+2.2 Fixed points of system (M)
+System (M) has at most 16 fixed points. The main fixed point is
+M0= (X0,Y0,Z0,W0) = (γ,ξ,N−p−(p−1)γ,N−q−(q−1)ξ), (2.6)
+corresponding to the particular solutions
+u0(r) =Ar−γ,v0(r) =Br−ξ, A,B > 0, (2.7)
+7when they exist, depending on ε1,ε2.The values of AandBare given by
+AD=/parenleftbig
+ε1γp−1(N−p−γ(p−1))/parenrightbigq−1−m/parenleftbig
+ε2ξq−1(N−q−(q−1)ξ)/parenrightbigδ,
+BD=/parenleftbig
+ε2ξq−1(N−q−(q−1)ξ)/parenrightbigp−1−s/parenleftbig
+ε1γp−1(N−p−(p−1)γ)/parenrightbigµ.
+The other fixed points are
+0 = (0,0,0,0), N0= (0,0,N+a,N+b), A0= (N−p
+p−1,N−q
+q−1,0,0),
+I0= (N−p
+p−1,0,0,0), J0= (0,N−q
+q−1,0,0), K0= (0,0,N+a,0), L0= (0,0,0,N+b),
+G0= (N−p
+p−1,0,0,N+b−N−p
+p−1µ), H0= (0,N−q
+q−1,N+a−N−q
+q−1δ,0),
+and ifm/\e}atio\slash=q−1,
+P0= (N−p
+p−1,N−p
+p−1µ−(q+b)
+q−1−m,0,(q−1)(N+b−N−p
+p−1µ)−m(N−q)
+q−1−m),
+C0=/parenleftbigg
+0,−q+b
+q−1−m,0,(N+b)(q−1)−m(N−q)
+q−1−m/parenrightbigg
+,
+R0=/parenleftbigg
+0,−q+b
+q−1−m,N+a+δb+q
+q−1−m,(N+b)(q−1)−m(N−q)
+q−1−m/parenrightbigg
+,
+and by symmetry, if s/\e}atio\slash=p−1,
+Q0= (N−q
+q−1δ−(p+a)
+p−1−s,N−q
+q−1,(p−1)(N+a−N−q
+q−1δ)−s(N−p)
+p−1−s,0),
+D0=/parenleftbigg
+−p+a
+p−1−s,0,(N+a)(p−1)−s(N−p)
+p−1−s,0/parenrightbigg
+,
+S0=/parenleftbigg
+−p+a
+p−1−s,0,(N+a)(p−1)−s(N−p)
+p−1−s,N+b+µa+p
+p−1−s/parenrightbigg
+.
+2.3 First comments
+Remark 2.1 This formulation allows to treat more general systems with si gned solutions by re-
+ducing the study on intervals where uandvare nonzero. Consider for example the problem
+−∆pu=ε1|x|a|u|s|v|δ−1v,−∆qv=ε2|x|b|v|m|u|µ−1u.
+0n any interval where uv >0,the couple (|u|,|v|)is a solution of (G).On any interval where
+u>0>v,the couple (u,|v|)satisfies (G)with(ε1,ε2)replaced by (−ε1,−ε2).
+Remark 2.2 There is another way for reducing the system to an autonomous f orm: setting
+U(t) =rγu,V(t) =rξv,H(t) =−r(γ+1)(p−1)/vextendsingle/vextendsingleu′/vextendsingle/vextendsinglep−2u′,K(t) =−r(ξ+1)(q−1)/vextendsingle/vextendsinglev′/vextendsingle/vextendsingleq−2v′,
+8witht= lnr,we find
+/braceleftbigg
+Ut=γU−|H|(2−p)/(p−1)H,Vt=ζU−|K|(2−q)/(q−1)K,
+Ht= (γ(p−1)+p−N)H+ε1UsVδ,Kt= (ζ(q−1)+q−N)K+ε2UµVm.(2.8)
+It extends the well-known transformation of Emden-Fowler in the scalar case when p= 2,used also
+in [2] for general p,see Section 3. When p=q= 2we obtain
+/braceleftbiggUtt+(N−2−2γ)Ut−γ(N−2−γ)U+ε1UsVδ= 0,
+Vtt+(N−2−2ξ)Vt−ξ(N−2−ξ)V+ε2UµVm= 0,(2.9)
+which was extended to the nonradial case and used for Hamilto nian systems ( s=m= 0),with
+source terms in [9] ( ε1=ε2= 1)and absorption terms in [4] ( ε1=ε2=−1). Our system is more
+adequated for finding the possible behaviours: unlike syste m (2.8)it has no singularity, since it is
+polynomial, also its fixed points at ∞are not concerned when we deal with solutions u,v>0.
+Remark 2.3 In the specific case p=q= 2,setting
+z=XZ=ε1r2+a|u|s−2u|v|δ−1v, w =YW=ε2r2+b|u|µ−1u|v|m−2v,
+we get the following system
+/braceleftbiggXt=X2−(N−2)X+z, Y t=Y2−(N−2)Y+w,
+zt=z[2+a+(1−s)X−δY], w t=w[2+b−µX+(1−m)Y].
+It has been used in [20] for studying the Hamiltonian system (SH). Even in that case we will show
+at Section 6 that system (M)is more performant, because it is of Kolmogorov type.
+Remark 2.4 Assumep=qanda=b.Settingt=kˆtand/parenleftBig
+ˆX,ˆY,ˆZ,ˆW/parenrightBig
+=k(X,Y,Z,W ), we
+obtain a system of the same type with N,areplaced by ˆN,ˆa,with
+ˆN−p
+N−p=k=ˆN+ˆa
+N+a.
+It corresponds to the change of unknowns
+r= ˆrk,ˆu(ˆr) =C1u(r),ˆv(ˆr) =C2u(r), C 1=kp(p−1−m+δ)/D, C2=k(p(p−1−s+µ))/D.
+From (2.3) and (2.4), we get ˆγ/γ=ˆξ/ξ=k=p+ˆa
+p+a.There is one free parameter. In particular
+1) we get a system without power ( ˆa= 0),by taking
+ˆN=p(N+a)
+p+a, k=p
+p+a;
+2) we get a system in dimension ˆN= 1,by taking
+k=−p−1
+N−p<0,,ˆa=p+a−(N+a)p
+N−p.
+93 The scalar case
+We first study the signed solutions of two scalar equations wi th source or absorption:
+−∆pu=−r1−N/parenleftBig
+rN−1/vextendsingle/vextendsingleu′/vextendsingle/vextendsinglep−2u′/parenrightBig′
+=ε|x|a|u|Q−1u, (3.1)
+withε=±1,1<p<N, Q /\e}atio\slash=p−1 andp+a>0.
+We cannot quote all thehuge litterature concerning its solu tions, supersolutionsor subsolutions,
+from the first studies of Emden and Fowler for p= 2,recalled in [16]; see for example [2] and [37],
+for anyp>1,and references therein. We set
+Q1=(N+a)(p−1)
+N−p, Q 2=N(p−1)+p+pa
+N−p, γ=p+a
+Q+1−p.
+From Remark 2.4 we could reduce the system to the case a= 0, in dimension ˆN=p(N+a)/(p+a).
+However we do not make the reduction, because we are motivate d by the study of system ( G), and
+also by the nonradial case.
+3.1 A common phase plane for the two equations
+Near any point rwhereu(r)/\e}atio\slash= 0 (positive or negative), and u′(r)/\e}atio\slash= 0 setting
+X(t) =−ru′
+u, Z(t) =−εr1+a|u|Q−1u/vextendsingle/vextendsingleu′/vextendsingle/vextendsingle−pu′, (3.2)
+witht= lnr,we get a 2-dimensional system
+(Mscal)/braceleftBigg
+Xt=X/bracketleftBig
+X−N−p
+p−1+Z
+p−1/bracketrightBig
+,
+Zt=Z[N+a−QX−Z].
+and then |u|=r−γ(|Z||X|p−1)1/(Q+1−p).This change of unknown was mentioned in [11] in the case
+p= 2,ε= 1 andN= 3.It is remarkable that system ( Mscal) is the same for the two casesε=±1,
+the only difference is that X(t)Z(t) has the sign of ε:
+The equation with source ( ε= 1) is associated to the 1stand 3rdquadrant. It is well known that
+any local solution has a unique extension on (0 ,∞).The 1stquadrant corresponds to the intervals
+where|u|is decreasing, which can be of the following types (0 ,∞),(0,R2),(R1,∞),(R1,R2), 0<
+R1<R2<∞.The 3rdquadrant corresponds to the intervals ( R1,R2) where|u|is increasing.
+The equation with absorption ( ε=−1) is associated to the 2ndand 4thquadrant. It is
+known that the solutions have at most one zero, and their maxi mal interval of existence can be
+(0,R2),(R1,∞),(R1,R2) or (0,∞).The 2ndquadrant corresponds to the intervals ( R1,R2) where
+|u|is increasing. The 4thquadrant corresponds to the intervals (0 ,R2) or (R1,∞) where |u|is
+decreasing.
+The fixed points of ( Mscal) are
+M0= (X0,,Z0) = (γ,N−p−(p−1)γ),(0,0), N0= (0,N+a), A0= (N−p
+p−1,0).
+10In particular M0is in the 1stquadrant whenever γ <N−p
+p−1,equivalently Q > Q 1,and in the 4th
+quadrant whenever Q<Q 1. It corresponds to the solution
+u(r) =Ar−γ,forε= 1,Q>Q 1,orε=−1,Q<Q 1,
+whereA=/parenleftbig
+εγp−1(N−p−γ(p−1))/parenrightbig1/(Q−p+1).
+3.2 Local study
+We examine the fixed points, where for simplicity we suppose Q/\e}atio\slash=Q1,and we deduce local results
+for the two equations:
+•Point (0,0) : it is a saddle point, and the only trajectories that conve rge to (0,0) are the
+separatrix, contained in the lines X= 0,Y= 0,they are not admissible.
+•PointN0: it is a saddle point: the eigenvalues of the linearized syst em arep
+p−1and−N. the
+trajectories ending at N0at∞are located on the set Z= 0,then there exists a unique trajectory
+starting from −∞atN0; it corresponds to the local existence and uniqueness of reg ular solutions,
+which we obtain easily.
+•PointA0: the eigenvalues of the linearized system areN−p
+p−1andN−p
+p−1(Q1−Q). IfQ < Q 1,
+A0is an unstable node. There is an infinity of trajectories star ting fromA0at−∞; thenX(t)
+converges exponentially toN−p
+p−1,thus lim r→0rN−p
+p−1u=α>0.The corresponding solutions usatisfy
+the equation with a Dirac mass at 0 .There exists no solution converging to A0at∞.IfQ>Q 1,A0
+is a saddle point; the trajectories starting from A0at−∞are not admissible; there is a trajectory
+converging at ∞,and then lim r→∞rN−p
+p−1u=α>0.
+•PointM0: the eigenvalues λ1,λ2of the linearized system are the roots of equation
+λ2+(Z0−X0)λ+Q−p+1
+p−1X0Z0= 0.
+Forε= 1,M0is defined for Q > Q 1; the eigenvalues are imaginary when X0=Z0,equivalently
+γ= (N−p)/p, Q=Q2. WhenQ < Q 2, M0is a source, there exists an infinity of trajectories
+such that lim r→0rγu=A. WhenQ>Q 2, M0is a sink, and there exists an infinity of trajectories
+such that lim r→∞rγu=A. WhenQ=Q2, M0is a center, from [2] For ε=−1, M0is defined
+forQ < Q 1,it is a saddle-point. There exist two trajectories T1,T′
+1converging at ∞,such that
+limr→∞rγu=Aand two trajectories T2,T′
+2,converging at 0 ,such that lim r→0rγu=A.
+3.3 Global study
+Remark 3.1 System(Mscal)has no limit cycle for Q/\e}atio\slash=Q2. It is evident when ε=−1.When
+ε= 1,as noticed in [19], it comes from the Dulac’s theorem: settin gXt=f(X,Z), Zt=g(X,Z),
+and
+B(X,Z) =XpQ/(Q+1−p)−2Z(p/(Q+1−p)−1, M=BXXt+BZZt+B(fX+gZ),
+thenM=KBwithK= (Q2−Q)γ(N−p)/p,thusMhas no zero for Q/\e}atio\slash=Q2.
+11Then from the Poincar´ e-Bendixson theorem, any trajectory bounded near ±∞converges to one
+of the fixed points. Thus we find again global results:
+•Equation with source ( ε= 1). IfQ < Q 1,there is no G.S.: the regular trajectory Tissued
+fromN0cannot converge to a fixed point. Then Xtends to ∞and the regular solutions uare
+changing sign, there is no G.S..
+IfQ1< Q < Q 2,the regular trajectory Tcannot converge to M0; if it converges to A0,it is
+the unique trajectory converging to A0; the set delimitated by TandX= 0,Z= 0 is invariant,
+thus it contains M0; and the trajectories issued from M0cannot converge to a fixed point, which
+is contradictory. then again Xtends to ∞onTand the regular solutions uare changing sign..
+The trajectory ending at A0converges to M0at−∞; then there exist solutions u >0 such that
+limr→0rγu=Aand lim r→0rN−p
+p−1u=α>0.
+IfQ > Q 2,the only singular solution at 0 is u0,and the regular solutions are G.S., with
+limr→∞rγu=A.IndeedM0is a sink; the trajectory ending at A0cannot converge to N0at−∞,
+thusXconverges to 0 ,andZconverges to ∞,thenucannot be positive on (0 ,∞).The trajectory
+issued from N0converges to M0.
+•Equation with absorption ( ε=−1). IfQ>Q 1,all the solutions udefined near 0 are regular;
+indeed the trajectories cannot converge to a fixed point.
+IfQ<Q 1,we find again easily a well known result: there exists a positi ve solution u1,unique
+up to a scaling, such that lim r→0rN−p
+p−1u1=α >0,and lim r→∞rγu1=A.Indeed the eigenvalues
+atM0satisfyλ1<0<λ2. There are two trajectories T1,T′
+1associated to λ1,and the eigenvector
+(X0+|λ1|,−X0
+p−1).The trajectory T1satisfiesXt>0> Ztnear∞,andX >N−p
+p−1,sinceZ0<0,
+andXcannot take the valueN−p
+p−1because at such a point Xt<0; thenN−p
+p−1< X < X 0and
+Xt>0 as long as it is defined; similarly Z0<Z <0 andZt<0; thenT1converge to a fixed point,
+necessarily A0,showing the existence of u1.The trajectory T′
+1corresponds to solutions usuch that
+limr→∞rγu=Aand lim r→Ru=∞for someR>0.There are two trajectories T2,T′
+2,associated
+toλ2,defining solutions usuch that lim r→0rγu=Aand changing sign, or with a minimum point
+and lim r→Ru=∞for someR>0.The regular trajectory starts from N0in the 2ndquadrant, it
+cannot converge to a fixed point, then lim r→Ru=∞for someR>0.
+•Critical case Q=Q2: it is remarkable that system ( Mscal) admits another invariant line ,
+namelyA0N0,given by
+X
+p′+Z
+Q2+1−N−p
+p= 0. (3.3)
+It precisely corresponds to well-known solutions of the two equations
+u=c(K2+r(p+a)/(p−1))(p−N)/(p+a),forε= 1;u=c/vextendsingle/vextendsingle/vextendsingleK2−r(p+a)/(p−1)/vextendsingle/vextendsingle/vextendsingle(p−N)/(p+a)
+,forε=−1,
+whereK2=cQ−p+1(N+a)−1((N−p)/(p−1))1−p.
+Remark 3.2 The global results have been obtained without using energy fu nctions.The study of
+[2] was based on a reduction of type of Remark 2.2, using an ene rgy function linked to the new
+unknown. Other energy functions are well-known, of Pohozae v type:
+Fσ(r) =rN/bracketleftBigg
+|u′|p
+p′+εra|u|Q+1
+Q+1+σu|u′|p−2u′
+r/bracketrightBigg
+=rN−p|u|p|X|p−2X/bracketleftbiggX
+p′+Z
+Q+1−σ/bracketrightbigg
+,
+12withσ=N−p
+p,satisfying F′
+σ(r) =rN−1+a/parenleftBig
+N+a
+Q+1−N−p
+p/parenrightBig
+|u|Q+1,or withσ=N+a
+Q+1, leading to
+F′
+σ(r) =rN−1/parenleftBig
+N+a
+Q+1−N−p
+p/parenrightBig
+|u′|p.In the critical case Q=Q2, all these functions coincide and
+they are constant, in other words system (Mscal)has a first integral. We find again the line (3.3):
+the G.S. are the functions of energy 0.
+4 Local study of system (S)
+In all the sequel we study the system with source terms: ( G) = (S). Assumption (1.5) is the most
+interesting case for studying the existence of the G.S.
+We first study the local behaviour of nonnegative solutions ( u,v) defined near 0 or near ∞.It
+is well known that any solution ( u,v) positive on some interval (0 ,R) satisfiesu′,v′<0 on (0,R).
+Any solution ( u,v) positive on ( R,∞),satisfiesu′,v′<0 near∞.We are reduced to study the
+system in the region RwhereX,Y,Z,W > 0,and consider the fixed points in ¯R.Then
+X(t) =−ru′
+u, Y(t) =−rv′
+v, Z(t) =r1+ausvδ
+|u′|p−1, W(t) =r1+bvmuµ
+|v′|q−1; (4.1)
+and (X,Y,Z,W ) is a solution of system ( M) inRif and only if ( u,v) defined by
+u=r−γ(ZXp−1)(q−1−m)/D(WYq−1)δ/D, v=r−ξ(WYq−1)(p−1−s)/D(ZXp−1)µ/D(4.2)
+is a positive solution with u′,v′<0.Among the fixed points, the point M0defined at (2.6) lies in
+Rif and only if
+0<γ <N−p
+p−1and 0<ξ <N−q
+q−1. (4.3)
+The local study of the system near M0appears to be tricky, see Remark 4.2. A main difference
+with the scalar case is that there always exist a trajectory c onverging to M0at±∞:
+Proposition 4.1 (PointM0)Assume that (4.3) holds. Then there exist trajectories conve rging to
+M0asr→ ∞,and then solutions (u,v)being defined near ∞,such that
+lim
+r→∞rγu=α>0,lim
+r→∞rξv=β >0. (4.4)
+There exist trajectories converging to M0asr→0, and thus solutions (u,v)being defined near 0
+such that
+lim
+r→0rγu=α>0,lim
+r→0rξv=β >0. (4.5)
+Proof. HereM0∈ R; settingX=X0+˜X,Y=Y0+˜Y,Z=Z0+˜Z,W=W0+˜W,the
+linearized system is 
+
+˜Xt=X0(˜X+1
+p−1˜Z),
+˜Yt=Y0(˜Y+1
+q−1˜W),
+˜Zt=Z0(−s˜X−δ˜Y−˜Z),
+˜Wt=W0(−µ˜X−m˜Y−˜W).
+13The eigenvalues are the roots λ1,λ2,λ3,λ4,of equation
+/bracketleftbigg
+(λ−X0)(λ+Z0)+s
+p−1X0Z0/bracketrightbigg/bracketleftbigg
+(λ−Y0)(λ+W0)+m
+q−1Y0W0/bracketrightbigg
+−δµ
+(p−1)(q−1)X0Y0Z0W0= 0.
+(4.6)
+This equation is of the form
+f(λ) =λ4+Eλ3+Fλ2+Gλ−H= 0,
+with 
+
+E=Z0−X0+W0−Y0,
+F= (Z0−X0)(W0−Y0)−s+p−1
+p−1X0Z0−m+q−1
+q−1Y0W0,
+G=−q−1−m
+q−1Y0W0(Z0−X0)−p−1−s
+p−1X0Z0(W0−Y0),
+H=D
+(p−1)(q−1)X0Y0Z0W0.
+From (1.5) we have H >0,thenλ1λ2λ3λ4<0.There exist two real roots λ3<0< λ4,and
+two rootsλ1,λ2, real with λ1λ2>0, or complex .Therefore there exists at least one trajectory
+converging to M0at∞and another one at −∞.Then (4.4) and (4.5) follow from (4.2). Moreover
+the convergence is monotone for X,Y,Z,W.
+Remark 4.2 There exist imaginary roots, namely Reλ1= Reλ2= 0,if and only if there exists
+c>0such thatf(ci) = 0,that means Ec2−G= 0,andc4−Fc2−H= 0,equivalently
+E=G= 0,orEG>0andG2−EFG−E2H= 0.
+Condition E=G= 0means that
+(i) eitherZ0=X0andW0=Y0,i.e.
+(γ,ξ) = (N−p
+p,N−q
+q), (4.7)
+in other words (δ,µ) = (q(N(p−1−s)+p(1−s+a))
+p(N−q),p(N(q−1−m)+q(1−m+b))
+q(N−p)).
+(ii) or(p−1−s)(q−1−m)>0and(γ,ξ)satisfies
+/braceleftbigg2N−p−q=pγ+qξ,
+(1−m
+q−1)ξ(N−q−(q−1)ξ) = (1−s
+p−1)γ(N−p−(p−1)γ).(4.8)
+This gives in general 0,1 or 2 values of (γ,ξ). For example, in the casem
+q−1=s
+p−1/\e}atio\slash= 1,and
+(p−2)(q−2)>0andN >pq−p−q
+p+q−2we find another value, different from the one of (4.7) for p/\e}atio\slash=q:
+(γ,ξ) = (Nq−2
+pq−p−q−1,Np−2
+pq−p−q−1). (4.9)
+Moreover the computation shows that it can exist imaginary r oots withE,G/\e}atio\slash= 0.
+In the case p=q= 2 ands=mthe situation is interesting:
+14Proposition 4.3 Assumep=q= 2ands=m<N
+N−2,withδ+1−s>0,µ+1−s>0.In the
+plane(δ,µ),letHsbe the hyperbola of equation
+1
+δ+1−s+1
+µ+1−s=N−2
+N−(N−2)s, (4.10)
+equivalently γ+ξ=N−2.ThenHsis contained in the set of points (δ,µ)for which the linearized
+system atM0has imaginary roots, and equal when s≦1.
+Proof.The assumption D >0 implyδ+1−s >0 andµ+1−s >0; condition E=G= 0
+impliess<N/(N−2) and reduces to condition (4.10). Moreover if s≦1,all the cases are covered.
+Indeed 2G= (s−1)E[Y0Z0+X0W0],henceGE≦0.
+Next we give a summary of the local existence results obtaine d by linearization around the
+other fixed points of system ( M) proved in Section 10. Recall that t→ −∞asr→0 andt→ ∞
+asr→ ∞.
+Proposition 4.4 (PointN0)A solution (u,v)is regular if and only if the corresponding trajectory
+converges to N0whenr→0. For anyu0,v0>0,there exists a unique local regular solution (u,v)
+with initial data (u0,v0).
+Proposition 4.5 (PointA0)IfsN−p
+p−1+δN−q
+q−1> N+aandµN−p
+p−1+mN−q
+q−1> N+b,there exist
+(admissible) trajectories converging to A0whenr→ ∞. IfsN−p
+p−1+δN−q
+q−1< N+aandµN−p
+p−1+
+mN−q
+q−1<N+b,the same happens when r→0. In any case
+limrN−p
+p−1u=α>0,limrN−q
+q−1v=β >0. (4.11)
+IfsN−p
+p−1+δN−q
+q−1< N+aorµN−p
+p−1+mN−q
+q−1< N+b,there exists no trajectory converging when
+r→ ∞;ifsN−p
+p−1+δN−q
+q−1>N+aorµN−p
+p−1+mN−q
+q−1>N+b,there exists no trajectory converging
+whenr→0.
+Proposition 4.6 (PointP0)1) Assume that q > m+1andq+b <N−p
+p−1µ < N+b−mN−q
+q−1.If
+γ <N−p
+p−1there exist trajectories converging to P0whenr→ ∞(and not when r→0).Ifγ >N−p
+p−1
+the same happens when r→0(and not when r→ ∞).
+2) Assume that q < m+ 1andq+b >N−p
+p−1µ > N+b−mN−q
+q−1andqN−p
+p−1µ+m(N−q)/\e}atio\slash=
+N(q−1)+(b+1)q.Ifγ <N−p
+p−1there exist trajectories converging to P0whenr→0(and not when
+r→ ∞).Ifγ >N−p
+p−1there exist trajectories converging when r→ ∞(and not when when r→0).
+In any case, setting κ=1
+q−1−m(N−p
+p−1µ−(q+b)),there holds
+limrN−p
+p−1u=α>0,limrκv=β >0. (4.12)
+Remark 4.7 This result improves the results of existence obtained by the fixed point theorem in
+[27] in the case of system (RP)withp=q= 2,a= 0,N= 3,2s+m/\e}atio\slash= 3.The proof is quite
+simpler..
+15Proposition 4.8 (PointI0)IfN−p
+p−1s>N+aandN−q
+q−1µ>N+b,there exist trajectories converging
+toI0whenr→ ∞,and then
+lim
+r→∞rN−p
+p−1u=β,lim
+r→∞v=α>0. (4.13)
+For anys,m≧0,there is no trajectory converging when r→0.
+Proposition 4.9 (PointG0)SupposeN−p
+p−1µ<N+b.Ifq+b<N−p
+p−1µandN+a<N−p
+p−1s,there
+exist trajectories converging to G0whenr→ ∞.IfN−p
+p−1µ < q+bandN−p
+p−1s < N+a,the same
+happens when r→0. In any case
+limrN−p
+p−1u=β,limv=α>0. (4.14)
+Proposition 4.10 (PointC0)SupposeN+b<N−q
+q−1m(henceq<m+1)withm/\e}atio\slash=N(q−1)+(b+1)q
+N−q,
+andδ>(N+a)(m+1−q)
+q+b. Then there exist trajectories converging to C0whenr→ ∞(and not when
+r→0),and then
+limu=α>0,limrkv=β, (4.15)
+wherek=q+b
+m+1−q.
+Proposition 4.11 (PointR0)Assume that N+b <N−q
+q−1m(henceq < m + 1)withm/\e}atio\slash=
+N(q−1)+b+bq
+N−q,andδ <(N+a)(m+1−q)
+q+b.If(p+a)(m+1−q)
+q+b< δ,there exist trajectories converging to
+R0whenr→ ∞(and not when r→0). Ifδ <(p+a)(m+1−q)
+q+b,there exist trajectories converging
+whenr→0(and not when r→ ∞), and then (4.15) holds again.
+We obtain similar results of convergence to the points Q0,J0,H0,D0,S0by exchanging p,δ,s,a
+andq,µ,m,b. There is no admissible trajectory converginf to 0 ,K0,L0,see Remark 10.1.
+5 Global results for system (S)
+We are concerned by the existence of global positive solutio ns. First we find again easily some
+known results by using our dynamical approach.
+Proposition 5.1 Assume that system (S)admits a positive solution (u,v)in(0,∞).Then the
+corresponding solution (X,Y,Z,W )of system (M)stays in the box
+A=/parenleftbigg
+0,N−p
+p−1/parenrightbigg
+×/parenleftbigg
+0,N−q
+q−1/parenrightbigg
+×(0,N+a)×(0,N+b), (5.1)
+in other words
+ru′+N−p
+p−1u>0, rv′+N−q
+q−1v>0, r1+ausvδ<(N+a)/vextendsingle/vextendsingleu′/vextendsingle/vextendsinglep−1, r1+buµvm<(N+b)/vextendsingle/vextendsinglev′/vextendsingle/vextendsingleq−1.
+(5.2)
+16and then
+us−p+1vδ≦C1r−(p+a), uµvm−q+1≦C2r−(q+b),in(0,∞), (5.3)
+whereC1= (N+a)(N−p
+p−1)p−1,C2= (N+b)(N−q
+q−1)q−1,and
+lim
+r→0rN−p
+p−1u=c1≧0,lim
+r→0rN−q
+q−1v(r) =c2≧0,lim inf
+r→∞rN−p
+p−1u>0,lim inf
+r→∞rN−q
+q−1v>0.(5.4)
+As a consequence if s≦p−1orm≦q−1,we have
+u≦K1r−γ, v≦K2r−ξ,in(0,∞), (5.5)
+withK1=C(q−1−m)/D
+1Cδ/D
+2,K2=Cµ/D
+1C(p−1−s)/D
+2.
+Proof.The solution of system ( M) inRdefined on R.On the hyperplane X=N−p
+p−1we have
+Xt>0,the field is going out. If at some time t0, X(t0) =N−p
+p−1,thenX(t)>N−p
+p−1fort > t0,
+in turnXt≧X/bracketleftBig
+X−N−p
+p−1/bracketrightBig
+>0,since since Z >0,thusX(t)>2N−p
+p−1fort > t1> t0,then
+Xt≧X2/2,which implies that Xblows up in finite time; thus X(t)<N−p
+p−1onR; in the same way
+Y(t)<N−q
+q−1.On the hyperplane Z=N+awe haveZt<0,the field is entering. If at some time
+t0, Z(t0) =N+athenZ(t)>N+afort<t0,thenZt≦Z(N+a−Z),sincesX+δY >0,and
+Zblows up in finite time as above; thus Z(t)<N+aonR,in the same way W(t)<N+b.Then
+(5.2),(5.3) and (5.5) follows. By integration it implies th atr(N−p)/(p−1)u(r) is nondecreasing near
+0 or∞, hence (5.4) holds.
+Next we prove Theorem 1.1.
+Proof of Theorem 1.1. (i) The trajectories of the regular solutions start from N0= (0,0,N+
+a,N+b), from Proposition 4.4, and the unstable variety Vuhas dimension 2, from (10.1), (10.2).
+It is given locally by Z=ϕ(X,Y),W=ψ(X,Y) for (X,Y)∈B(0,ρ)\{0} ⊂R2.
+To any (x,y)∈B(0,ρ)\{0}we associate the unique trajectory Tx,yinVugoing through this
+point. IfT∗is the maximal interval of existence of a solution on Tx,y, then lim t→T∗(X(t)+Y(t)) =
+∞.IndeedZ,andWsatisfy 0< Z < N +a,0< W < N +bas long as the solution exists,
+because at a time TwhereZ(T) =N+a,we haveZt<0.If there exists a first time Tsuch that
+X(T) =N−p
+p−1orY(T) =N−q
+q−1,thenT <T∗.We consider the open rectangle Nof submits
+(0,0), ̟1=/parenleftbiggN−p
+p−1,0/parenrightbigg
+, ̟2=/parenleftbigg
+0,N−q
+q−1/parenrightbigg
+, ̟=/parenleftbiggN−p
+p−1,N−q
+q−1/parenrightbigg
+.
+LetU={(x,y)∈B(0,ρ) :x,y>0}; thenU=S1∪S2∪S3∪S,where
+/braceleftbiggSi={(x,y)∈ U:Tx,yleavesNon (̟i,̟)}, i= 1,2,
+S3={(x,y)∈ U:Tx,yleavesNat̟},S={(x,y)∈ U:Tx,ystays inN}.
+Any element of Sdefines a G.S. Assume s <N(p−1)+p+pa
+N−p.Let us show that S1is nonempty .
+Consider the trajectory T¯x,0onVuassociated to (¯ x,0),with ¯x∈(0,ρ),going through ¯M= (¯x,0,
+17ϕ(¯x,0),ψ(¯x,0)); it isnot admissible for our problem, since it is in the hyperplane Y= 0: it satisfies
+the system
+
+Xt=X/bracketleftBig
+X−N−p
+p−1+Z
+p−1/bracketrightBig
+,
+Zt=Z[N+a−sX−Z],
+Wt=W[N+b−µX−W],
+which is not completely coupled . The two equations in X,Zcorresponds to the equation
+−∆pU=raUs. (5.6)
+The regular solutions of (5.6) are changing sign, since sis subcritical, see Section 3. Consider the
+solution ( ¯X,¯Y,¯Z,¯W) of system ( M),of trajectory T¯x,0, going through ¯Mat time 0; it satisfies
+¯Y= 0,and¯X(t)>0,¯Z(t)>0 tend to ∞in finite time T∗, then for any given C≧N−p
+p−1,there
+exist a first time T <T∗such that ¯X(T) =C, and¯Y(T) = 0. We have lim t→−∞¯W=N+b,and
+necessarily 0 <¯W <N+b,in particular 0 <¯W(T)<N+b; and¯Wtis bounded on ( −∞,T∗),then
+¯Whas a finite limit at T∗.The field at time Tis transverse to the hyperplane X=N−p
+p−1: we have
+¯Xt≧CZ(T)
+p−1>0,since¯Z(T)>0. From the continuous dependance of the initial data at time 0,
+for anyε>0,there exists η>0 such that for any ( x,y)∈B((¯x,0),η) and for any ( X,Y,Z,W ) on
+Tx,y,there exists a first time Tεsuch thatX(Tε) =C,and|Y(t)|≦εfor anyt≦Tε, in particular
+for any (x,y)∈B((¯x,0),η) withy>0,and then 0 <Y(t)≦εfor anyt≦Tε.Let us take C=N−p
+p−1.
+Then (x,y)∈ S1. The same arguments imply that S1is open. Similarly assuming m<N(q−1)+q+qb
+N−q
+implies that S2is nonempty and open. By connexity Sis empty if and only if S3is nonempty.
+(ii) Here the difficulty is due to the fact that the zeros of u,vcorrespond to infinite limits
+forX,Y,and then the argument of continuous dependance is no more ava ilable. We can write
+U=M1∪M2∪M3∪S,where
+
+
+M1={(x,y)∈ UandTx,yhas an infinite branch in XwithYbounded },
+M2={(x,y)∈ U:Tx,yhas an infinite branch in YwithXbounded },
+M3={(x,y)∈ U:Tx,yhas an infinite branch in ( X,Y)}.
+In other words, M1is the set of ( x,y)∈ Usuch that for any ( X,Y,Z,W ) onTx,y,there exists
+aT∗such that lim t→T∗X(t)) =∞,andY(t) stays bounded on ( −∞,T∗),that means the set of
+(x,y)∈ Usuch that for any solution ( u,v) corresponding to Tx,y,uvanishes before v; similarly for
+M2.Otherwise M3is the set of ( x,y)∈ Usuch that there exists a T∗such that lim t→T∗X(t) =
+limt→T∗Y(t) =∞,that means ( u,v) vanish at the same R∗=eT∗. In that case, from the H¨ opf
+Lemma, lim r→Ru′
+(r−R)u= 1,then lim t→T∗X
+Y= 1.
+We are lead to show that M1is nonempty and open for s <N(p−1)+p+pa
+N−p. We consider again
+the trajectory ¯Tand takeClarge enough: C= 2(N−p
+p−1+N+|b|
+q−1).Letε∈/parenleftbig
+0,C
+2/parenrightbig
+.For any (x,y)∈
+B((¯x,0),η) withy>0,and any (X,Y,Z,W ) onTx,y,there is a first time Tεsuch thatX(Tε) =C,
+and 0< Y(t)≦εfor anyt≦Tε.AndXis increasing and Xt≧X(X−C),thus there exists T∗
+such that lim t→T∗X(t) =∞.Settingϕ=X/Y,we find
+ϕt
+ϕ=X−Y+Z
+p−1−W
+q−1+N−q
+q−1−N−p
+p−1≧X−Y−C
+2
+18thenϕt(Tε)>0. Letθ= sup{t>Tε:ϕt>0}; suppose that θis finite; then ϕ(θ)> ϕ(Tε) =
+C/ε >2 andX(θ)≦Y(θ)+C < X(θ)/2 +C,which is contradictory. Then ϕis increasing up
+toT∗; if lim t→T∗Y(t) =∞,then lim t→T∗ϕ= 1,which is impossible. Then ( x,y)∈ M1,thusM1
+is nonempty. In the same way M1is open. Indeed for any (¯ x,¯y)∈ M1there exists M >0 such
+that 0<¯Y(t)≦M/2 onT¯x,¯y. To conclude we argue as above, with (¯ x,0) replaced by (¯ x,¯y),and
+Creplaced by C+M.
+Proof of Proposition 1.2. Assumes≧N(p−1)+p+pa
+N−p.Consider the Pohozaev type function
+F(r) =rN/bracketleftBigg
+|u′|p
+p′+raus+1
+s+1vδ+N−p
+pu|u′|p−2u′
+r/bracketrightBigg
+=rN−pup/bracketleftbiggX
+p′+1
+s+1Z−N−p
+p/bracketrightbigg
+.(5.7)
+We findF(0) = 0 and
+F′(r) =rN−1+a/bracketleftbigg/parenleftbiggN+a
+s+1−N−p
+p/parenrightbigg
+vδus+1+δ
+s+1rus+1vδ−1v′/bracketrightbigg
+=rN−1+avδus+1/bracketleftbiggN+a
+s+1−N−p
+p−δY
+s+1/bracketrightbigg
+(5.8)
+From our assumption, Fis decreasing, and Z >0,thusX <N−p
+p−1.ThenS1,S3are empty. If
+moreoverm≧N(q−1)+q+qb
+N−qthenS2is empty, therefore S=U.
+Remark 5.2 Let us only assume that s≧N(p−1)+p+pa
+N−p.If one function has a first zero, it is
+v. Indeed if there exists a first value Rwhereu(R) = 0,andv(r)>0on[0,R),thenF(R) =
+RN
+p′|u′(R)|p>0.
+As a first consequence we obtain existence results for the Dir ichlet problem. It solves an open
+problem in the case s>p−1 orm>q−1,and extends some former results of [12] and [42]. Our
+proof, based on the shooting method differs from the proof of [1 2], based on degree theory and
+blow-up technique. Our results extend the ones of [3, Theore m 2.2] relative to the case p=q= 2,
+obtained by studying the equation satisfied by a suitable fun ction ofu,v.
+Corollary 5.3 system(S)admits no G.S. and then there is a radial solution of the Diric hlet
+problem in a ball in any of the following cases:
+(i)p<s+1,q<m+1,andmin(sN−p
+p−1+N−q
+q−1δ−(N+a),N−p
+p−1µ+mN−q
+q−1−(N+b))≦0;
+(ii)p<s+1, q>m+1andsN−p
+p−1+N−q
+q−1δ−(N+a)≦0orγ−N−p
+p−1>0;
+(iii)p>s+1,q>m+1andmax(γ−N−p
+p−1,ξ−N−q
+q−1)≧0;
+(iv)p≧s+1,q≧m+1andmax(γ−N−p
+p−1,ξ−N−q
+q−1)>0.
+Proof.From Theorem 1.1, we are reduced to prove the nonexistence of G.S.
+(i) Assume p<s+1,andsN−p
+p−1+N−q
+q−1δ−(N+a)<0.We have −∆pu≧Cra−N−q
+q−1δusfor larger.
+From [6, Theorem 3.1], we find u=O(r−(p+a−N−q
+q−1δ)/(s+1−p)), and then sN−p
+p−1+N−q
+q−1δ−(N+a)≧0,
+19from (5.4), which contradicts our assumption. In case of equ ality, we find −∆pu≧Cr−Nfor large
+r,which is impossible. Then there exists no G.S. This improves ythe result of [12] where the
+minimum is replaced by a maximum.
+(ii) Assume p<s+1, q >m+1 andγ−N−p
+p−1>0; thenu=O(r−γ),which contradicts (5.4). If
+γ−N−p
+p−1= 0,then limrN−p
+p−1u=α>0,andξ>N−q
+q−1.Hence−∆qv≧Crb−N−p
+p−1µvmfor larger,then
+v≧Cr(q+b−N−p
+p−1δ)/(q−1−m)=Cr−ξ.There exists C1>0 such that C1≦rξv≦2C1for larger,from
+[6, Theorem 3.1] and (5.5), then −∆pu≧Cr−Nfor someC >0,which is again contradictory.
+(iii) (iv) The nonexistence of G.S is obtained by extension o f the proof of [12] to the case a,b/\e}atio\slash= 0.
+Moreover (iii) implies the nonexistence of positive soluti on (u,v), radial or not, in any exterior
+domain (R,∞)×(R,∞),R>0 from [6].
+Corollary 5.4 Assume (4.3) with p=q= 2.Ifδ+s≧N+2+2a
+N−2andµ+m≧N+2+2b
+N−2,then system
+(S)admits a G.S.
+Proof.It was shown in [28], [41] by the moving spheres method that th e Dirichlet problem has
+no radial or nonradial solution. Then Theorem 1.1 applies ag ain.
+We aso extend and improve a result of nonexistence of [10] for the casep=q= 2,a= 0,s>1:
+Proposition 5.5 Assumes+1>porγ >N−p
+p,and
+s+p(N−q)
+(q−1)(N−p)δ<N(p−1)+pa+p
+N−p(5.9)
+Then system (S)admits noG.S.and then there is a solution of the Dirichlet problem. The same
+happens by exchanging p,s,δ,a,γ withq,m,µ,b,ξ.
+Proof.Consider the function Fdefined at (5.7). Suppose that there exists a G.S. Then from
+(5.1) and (5.9) we find
+N+a
+s+1−N−p
+p−δY
+s+1>N+a
+s+1−N−p
+p−δ
+s+1N−q
+q−1≧0.
+From (5.8), we deduce that Fis nondecreasing. First suppose s+1> p.From (5.3) and (5.4),it
+follows that u=O(r−k)at∞,withk= (p+a−δN−q
+q−1)/(s−p+1).InturnrN−pup=O(r(N−p)−kp) =
+o(1) from (5.9), then F(r) =o(1) near ∞.Next assume s+1≦pandγ >N−p
+p. ThenrN−pup=
+O(rN−p−γp),henceF(r) =o(1) near ∞. In any case we get a contradiction.
+6 The Hamiltonian system
+Hereweconsiderthenonnegative solutionsof thevariation al Hamiltonian problem( SH) inΩ⊂RN
+(SH)/braceleftbigg−∆u=|x|avδ,
+−∆v=|x|buµ,
+20wherep=q= 2<N, s=m= 0, a>b> −2,andD=δµ−1>0.For this case we find
+γ=(2+a)+(2+b)δ
+D, ξ=2+b+(2+a)µ
+D, γ+2+a=δξ, ξ+2+b=µγ.
+The particular solution ( u0(r),v0(r)) = (Ar−γ,Br−ξ) exists for 0 < γ < N −2,0< ξ < N −2.
+HereX,Y,Z,W are defined by
+X(t) =r|u′|
+u, Y(t) =r|v′|
+v, Z(t) =r1+avδ
+|u′|, W(t) =r1+buµ
+|v′|,
+witht= lnr,and system ( M) becomes
+(MH)
+
+Xt=X[X−(N−2)+Z],
+Yt=Y[Y−(N−2)+W],
+Zt=Z[N+a−δY−Z],
+Wt=W[N+b−µX−W]
+This system has a Pohozaev type function, well known at least in the case a=b= 0, given at (1.7):
+EH(r) =rN/bracketleftBigg
+u′v′+rb|u|µ+1
+µ+1+ra|v|δ+1
+δ+1+N+a
+δ+1vu′
+r+N+b
+µ+1uv′
+r/bracketrightBigg
+=rN−2uv/bracketleftbigg
+XY−Y(N+b−W)
+µ+1−(N+a−Z)X
+δ+1/bracketrightbigg
+=rN−2−γ−ξ(ZX)(µ+1)/D(WY)(δ+1)/D/bracketleftbigg
+XY−Y(N+b−W)
+µ+1−(N+a−Z)X
+δ+1/bracketrightbigg
+.
+It can also be found by a direct computation, and EHsatisfies
+E′
+H(r) =rN−1u′v′/parenleftbiggN+a
+δ+1+N+b
+µ+1−(N−2)/parenrightbigg
+.
+We define the critical case as the case where ( δ,µ) lie on the hyperbola H0given by
+N+a
+δ+1+N+b
+µ+1=N−2,equivalently γ+ξ=N−2. (6.1)
+In this case γ=N+b
+µ+1,ξ=N+a
+δ+1,andE′
+H(r)≡0. It corresponds to the existence of a first integral of
+system (M),which can also be expressed in the variables U= rγu,V=rξvof Remark 2.2:
+EH(r) = UtVt−γξUV+Uµ+1
+µ+1+Vδ+1
+δ+1=C.
+The supercritical case is defined as the case where ( δ,µ) is above H,equivalently γ+ξ < N−2
+and the subcritical case corresponds to ( δ,µ) underH.
+Remark 6.1 The energy EH,0of the particular solution associated to M0is always negative, given
+byEH,0=−D
+(µ+1)(δ+1)rN−2−γ−ξX0Y0(Z0X0)(µ+1)/D(W0Y0)(δ+1)/D.
+21Remark 6.2 In the case a=b= 0,it is known that there exists a solution of the Dirichlet prob lem
+in any bounded regular domain ΩofRN,see for example [15], [20]; for general a,b,some restrictions
+on the coefficients appear, see [23] and [14].
+Next consider the critical and supercritical cases. When a=b= 0,there exists no solution if
+Ω is starshaped, see [36]. Here we show the existence of G.S. f or generala,b. The existence in the
+critical case with a=b= 0 was first obtained in [22], then in the supercritical case i n [29], and
+uniqueness was proved in [20], [29]. The proofs of [29] are qu ite long due to regularity problems,
+whenδorµ<1,which play no role in our quadratic system.
+Remark 6.3 The particular case δ=µanda=bis easy to treat. Indeed in that case u=v
+is a solution of the scalar equation ∆u+|x|a|u|δ−1u= 0,for which the critical case is given by
+δ= (N+2+2a)/(N−2).Moreover if system (SH)admits a G.S., or a solution of the Dirichlet
+problem in a ball, it satisfies u=v,from [3]. Then we are completely reduced to the scalar case. In
+particular, in the critical case, the G.S. are given explici tely by:u=v=c(K+r(2+a))(2−N)/(2+a),
+whereK=cδ−1/(N+a)(N−2);in other words they satisfy (3.3) with X=YandZ=W,i.e.
+X(t)
+N−2+Z(t)
+N+a−1 = 0.
+Near∞,the G.S. is (obviously) symmetrical: it joins the points N0andA0.
+Remark 6.4 Consider the case δ= 1, a=b= 0,which is the case of the biharmonic equation
+∆2u=uµ.
+Recall that it is the only case where the conjecture (1.3) was completely proved by Lin in [21]. In
+the critical case µ= (N+4)/(N−4),the G.S. are also given explicitely, see [20]:
+u(r) =c(K+r2)(4−N)/2, K=cµ−1/(N−4)(N−2)N(N+2).
+They satisfy the relation XY=N−Z
+2X+N−4
+2N(N−W)Y,and moreover we find that they are on an
+hyperplane , of equation
+(N−2)X(t)
+N(N−4)+Z(t)
+N−1 = 0.
+Observe also that the G.S. is not symmetrical near∞:ubehaves like r4−Nandvbehaves like
+r2−N.The trajectory in the phase space joins the points N0andQ0= (N−4,N−2,2,0).
+Proof of Theorem 1.4. 1)Existence or nonexistence results:
+•In the supercritical or critical case we apply any of the two c onditions of Theorem 1.1: Here
+EH(0) = 0,andEHis nonincreasing; there does not exist solutions of ( M) such that at some time
+T, X(T) =Y(T) =N−2, because at the time T,
+XY−Y(N+b−W)
+µ+1−(N+a−Z)X
+δ+1= (N−2)/bracketleftbigg
+N−2−N+a
+δ+1−N+b
+µ+1+W
+µ+1+Z
+δ+1/bracketrightbigg
+>0
+22sinceW >0,Z >0,thusEH(eT)>0,which is impossible. Otherwise there exists no solution of t he
+Dirichlet problem in a ball B(0,R),becauseEH(R) =RNu′(R)v′(R)>0 from the H¨ opf Lemma.
+Then there exists a G.S. The uniqueness is proved in [20].
+•In the subcritical case there is no radial G.S.: it would sati sfyEH(0) = 0,andEHis non-
+decreasing, EH(r)≦CrN−2−γ−ξfrom (5.1), and γ+ξ >(N−2),then lim r→∞EH(r) = 0. From
+Theorem 1.1, there exists a solution of the Dirichlet proble m.
+2)Behaviour of the G.S.in the critical case.
+It is easy to see that the condition (1.6) implies µ >2+b
+N−2andδ >2+a
+N−2,and thatδ≦N+a
+N−2
+andµ≦N+b
+N−2cannot hold simultaneously. One can suppose that δ >N+a
+N−2.LetTbe the unique
+trajectory of the G.S.. Then EH(0) = 0,thusTlies on the variety Vof energy 0, defined by
+X(N+a−Z)
+δ+1+Y(N+b−W)
+µ+1=XY. (6.2)
+From (5.2) Tstarts from the point N0,and from (5.1) Tstays in
+A=/braceleftbig
+(X,Y,Z,W )∈R4: 0<X <N −2,0<Y <N −2,0<Z <N +a,0<W <N +b/bracerightbig
+.
+(i) Suppose that Tconverges to a fixed point of the system in ¯R. Then the only possible points
+areA0,P0,Q0which are effectively on V. IndeedI0, J0,G0,H0/\e}atio\slash∈ V.ButQ0= ((N−2)δ−(2+
+a),N−2,N+a−(N−2)δ,0)/\e}atio\slash∈¯R, sinceδ>N+a
+N−2. AndP0∈¯Rif and only if µ≦N+b
+N−2.
+Ifµ>N+b
+N−2,thenTconverges to A0. Ifµ<N+b
+N−2,notrajectoryconvergesto A0,fromProposition
+4.5, thus Tconverges to P0. Ifµ/\e}atio\slash=N+b
+N−2the convergence is exponential, thus the behaviour of
+u,vfollows. Ifµ=N+b
+N−2,thenTconverges converges to A0,=P0; the eigenvalues given by (10.3)
+satisfyλ1=λ2=N−2, λ3=N+a−δ(N−2)<0 andλ4= 0; the projection of the trajectory
+on the hyperplane Y=N−2 satisfies the system
+Xt=X[X−(N−2)+Z], Z t=Z[N+a−δ(N−2)−Z]
+which presents a saddle point at ( N−2,0), thus the convergence of XandZis exponential, in
+particular we deduce the behaviour of u.The trajectory enters by the central variety of dimension
+1,and by computation we deduce that Y−(N−2) =−t−1+O(t−2+ε) near∞,and the behaviour
+ofvfollows.
+(ii) Let us show that Tconverges to a fixed point. We eliminate Wfrom (6.2) and we get a still
+quadratic system in ( X,Y,Z) :
+
+
+Xt=X[X−(N−2)+Z],
+Yt=Y[Y+b+2−(µ+1)X]+µ+1
+δ+1X(N+a−Z),
+Zt=Z[N+a−δY−Z].(6.3)
+We haveXt≧0,andYt≧0near−∞.Supposethat Xhas amaximumat t0followed by aminimum
+att1.At these times Xtt=XZt, thus we find Zt(t0)<0<Zt(t1).There exists t2∈(t0,t1) such
+thatZt(t2) = 0,andt2is a minimum. At this time Z(t2) =N+a−δY(t2),Ztt(t2) =−δ(ZYt)(t2)
+hence
+Yt(t2) =Y(t2)/bracketleftbigg
+Y(t2)+b+2−µ+1
+δ+1X(t2)/bracketrightbigg
+<0
+23andXt(t2)<0,hence (X+Z)(t2)<N−2,and
+N−2−X(t2)>Z(t2)>N+a−δ(µ+1
+δ+1X(t2)−b−2)
+(a+2)+δ(b+2)<(δµ+1
+δ+1−1)X(t2) =δ(2+b)+(2+a)
+(N−2)δ−(2+a)X(t2)
+butX(t2)<X(t0)<δ(N−2)−(2+a),which is contradictory. Then Xhas at most one extremum,
+which is a maximum, and then it has a limit in (0 ,N−2] at∞.In the same way, by symmetry, Y
+has at most one extremum, which is a maximum, and has a limit in (0,N−2] at∞.ThenZhas
+at most one extremum, which is a minimum. Indeed at the points whereZt= 0,−Ztthas the sign
+ofYt. ThusZhas a limit in [0 ,N+a), similarly Whas a limit in [0 ,N+b).
+Open problems: 1) For the case δ=µ,in the critical case it is well known that there exist
+solutions (u,v) of system ( SH) of the form ( u,u),such thatrγuis periodic in t= lnr.They
+correspond to a periodic trajectory for the scalar system ( Mscal) withp= 2,and it admits an
+infinity of such trajectories. If δ/\e}atio\slash=µ,does there exist solutions ( u,v) such that ( rγu,rξv) is
+periodic in t,in other words a periodic trajectory for system ( MH)?
+2) In the supercritical case, we cannot prove that the regula r trajectory Tconverges to M0,
+that means lim r→∞rγu=A,limr→∞rξv=B.HereEH(0) = 0,EHis nonincreasing, then EHis
+negative. The only fixed points of negative energy are M0, G0,H0,but a G.S. satisfies (5.5), then
+it tends to (0 ,0) at∞,henceTcannot converge to G0orH0from Proposition 4.9; but we cannot
+prove that Tconverges to some fixed point.
+7 A nonvariational system
+Here we consider system ( S) withp=q= 2,a=bands=m/\e}atio\slash= 0.
+(SN)/braceleftbigg−∆u=|x|ausvδ,
+−∆v=|x|auµvs,
+whereD=δµ−(1−s)2>0.In order to prove Theorem we can reduce the system to the case
+a= 0,by changing NintoˆN=2(N+a)
+2+a,from Remark 2.4; thus we assume a= 0 in this Section.
+Here
+X=−ru′
+uY=−rv′
+v, Z(t) =−rusvδ
+u′, W(t) =−ruµvs
+v′,
+and system ( M) becomes
+(MN)
+
+Xt=X[X−(N−2)+Z],
+Yt=Y[Y−(N−2)+W],
+Zt=Z[N−sX−δY−Z],
+Wt=W[N−µX−sY−W].
+We have chosen this system because it is not variational, and different hyperbolas in the plane
+(δ,µ):
+24•the hyperbola Hsfor which the linearized system at M0has two imaginary roots, given by
+(Hs)1
+δ+1−s+1
+µ+1−s=N−2
+N−(N−2)s
+whenevers<N
+N−2,andδ+1−s>0, µ+1−s>0,from Proposition 4.3;
+•the hyperbola H0defined by
+(H0)1
+δ+1+1
+µ+1=N−2
+N; (7.1)
+it was shown in [26] that above H0there exists no solution of the Dirichlet problem;
+•an hyperbola Zsintroduced in [38] in case s<N
+N−2,and min(δ,µ)>|s−1|:
+(Zs)1
+δ+1+1
+µ+1=N−2
+N−(N−2)s, (7.2)
+•we introduce the new curve Csdefinedfor anys>0by
+(Cs)N
+µ+1+N
+δ+1=N−2+(N−2)s
+2min(1
+µ+1,1
+δ+1),
+We first extend and complete the results of [38] and [26]:
+Proposition 7.1 (i) Assume s<N
+N−2,andδ+1−s>0, µ+1−s>0.Under the hyperbola Zs,
+system(SN)admits no G.S., and then there is a solution of the Dirichlet p roblem in a ball.
+(ii) Above H0there exists no solution of the Dirichlet problem. Thus there exists a G.S.
+Proof.(i) We consider an energy function with parameters α,β,σ,θ :
+EN(r) =rN/bracketleftbigg
+u′v′+αuµ+1vs+βvδ+1us+σ
+rvu′+θ
+ruv′/bracketrightbigg
+(7.3)
+=rN−2uvΨ0=rN−2−γ−ξ(ZX)ξ/2(WZ)γ/2Ψ0, (7.4)
+from (4.2), where
+Ψ0(X,Y,Z,W ) =XY+αWY+βZX−σX−θY. (7.5)
+We get
+r1−N(uv)−1E′
+N(r) = (σ+θ−(N−2))XY+(Nα−θ)YW+(Nβ−σ)XZ
+−(α(µ+1)−1)XYW−(β(δ+1)−1)XYZ−αsY2W−βsX2Z.
+Takingα=1
+µ+1,β=1
+δ+1,we find
+r3−N(uv)−1E′
+N(r) = (σ+θ−(N−2))XY+(Nα−θ−αsY)YW+(Nβ−σ−βsX)XZ.(7.6)
+If there exists a G.S., from (5.1) it satisfies X,Y <N −2,hence
+r3−N(uv)−1E′
+N(r)>(σ+θ−(N−2))XY+((N−(N−2)s)α−θ)YW+((N−(N−2)s)β−σ)XZ.(7.7)
+25Takingθ=N−(N−2)s
+µ+1,σ=N−(N−2)s
+δ+1,we deduce that E′
+N>0 underZs. Moreover Zsis under Hs,
+thusγ+ξ>N−2.ThenEN(r) =O(rN−2−γ−ξ) tends to 0 at ∞,which is contradictory.
+(ii) Taking α=1
+µ+1=θ
+N,β=1
+δ+1=σ
+N,it comes from (7.6)
+r3−N(uv)−1E′
+N(r) = (N
+δ+1+N
+µ+1−(N−2))XY−αsY2W−βsX2Z
+henceE′
+N<0 when (7.1) holds. At the value Rwhereu(R) =v(R) = 0,we findEN(R) =
+RNu′(R)v′(R)>0,which is a contradiction.
+Remark 7.2 (i)When the four curves are simultaneously defined, they are in the following order,
+from below to above: Zs,Hs,Cs,H0.They intersect the diagonal δ=µrepectively for
+δ=N+2
+N−2−2s, δ=N+2
+N−2−s, δ=N+2
+N−2−s
+2, δ=N+2
+N−2.
+(ii) Forδ=µ,system(SN)has a G.S. for δ≧N+2
+N−2−s.Indeed it admits solutions of the form
+(U,U), whereUis a solution of equation −∆U=Us+δ.Suppose moreover s≦δ.If1−s < δ <
+N+2
+N−2−s,then there exists no G.S; indeed all such solutions satisfy u=v,from [3, Remark 3.3].
+Then the point Ps=/parenleftBig
+N+2
+N−2−s,N+2
+N−2−s/parenrightBig
+appears to be the separation point on the diagonal; notice
+thatPs∈ Hs.
+Next we prove our main existence result of existence of a G.S. valid without restrictions on s.
+The main idea is to introduce a new energy function Φby adding two terms in X2andY2to the
+energyENdefined at (7.3). It is constructed in order that Φ′does not contain YandZ.Then
+we consider the set of couples ( X,Y) such that Φ′has a sign, which is bounded by a cubic curve.
+When (δ,µ) is above Cs, the cubic curve is exterior to the square
+K= [0,N−2]×[0,N−2], (7.8)
+and then we can apply Theorem 1.1.
+Proof of Theorem 1.5. From Theorem 1.1, if s≧N+2
+N−2,all the regular solutions are G.S..
+Thus we can assume s<N+2
+N−2.Letj,k∈Rbe parameters, and
+Φ(r) =EN(r)+rN/bracketleftbigg
+ks
+2vu′2
+u+js
+2uv′2
+v/bracketrightbigg
+=rN/bracketleftbigg
+u′v′+αuµ+1vs+βvδ+1us+σ
+rvu′+θ
+ruv′+ks
+2vu′2
+u+js
+2uv′2
+v/bracketrightbigg
+=rN−2uvΨ =rN−2−γ−ξ(ZX)ξ/2(WY)γ/2Ψ,
+where
+Ψ(X,Y,Z,W ) =XY+αWY+βZX−σX−θY+ks
+2X2+js
+2Y2.
+26Then
+r3−N(uv)−1Φ′(r) = (σ+θ−(N−2))XY+(Nα−θ)YW+(Nβ−σ)XZ
+−(α(µ+1)−1)XYW−(β(δ+1)−1)XYZ+(j−α)sY2W+(k−β)sX2Z
++ksX2[X−(N−2)]+jsY2[Y−(N−2)]+(N−2−X−Y)(ks
+2X2+js
+2Y2).
+We eliminate the terms in Z,Wby takingj=α=1
+µ+1, k=β=1
+δ+1, θ=Nα, σ=Nβ.Then we
+get the function Φ defined at (1.9). Computing its derivative , we obtain after reduction
+B(X,Y) :=−2
+sr3−N(uv)−1Φ′(r)
+=βX2(N−2−X)+αY2(N−2−Y)+XY/bracketleftbigg
+βX+αY+2
+s(N−2−Nα−Nβ)/bracketrightbigg
+.
+From Proposition 7.1 we can assume that N(α+β)−(N−2)>0. We determine the sign of
+Bon the boundary ∂Kof the square Kdefined at (7.8). We have B(0,Y) =αY2(N−2−Y)≧0
+andB(X,0) =βX2(N−2−X)≧0. In particular B(0,0) = 0.Otherwise B(N−2,Y) =YΘ(Y)
+with
+Θ(Y) =αY[2(N−2)−Y]+(N−2)((N−2)β+2
+s(N−2−Nα−Nβ).
+On the interval [0 ,N−2],there holds Θ( Y)>Θ(0). By hypothesis, ( δ,µ) is above Cs,or equiva-
+lently
+(α+β)N
+N−2−1≦s
+2min(α,β); (7.9)
+consequently B(N−2,Y)≧0 and similarly B(X,N−2)≧0.ThenBis nonnegative on ∂Kand is
+zero at (0,0),(0,N−2),(N−2,0). The curve B(X,Y) = 0 is a cubic with a double point at (0 ,0),
+which is isolated under the condition (7.9): B(X,Y)>0 near (0,0),except at this point. Then
+B(X,Y)>0 on the interior of K.
+Suppose that there exists a regular solution such that X(T) =Y(T) =N−2 at the same time
+T.Indeed up to this time ( X,Y) stays inK, thus the function Φ is decreasing. We have Φ(0) = 0 ,
+and at the value R=eT,we find
+Φ(R) =RN−2−γ−ξ(N−2)ξ+γ+2/bracketleftbiggαW+βZ
+N−2+1−(β+α)(N
+N−2−s
+2)/bracketrightbigg
+then Φ(R)>0, since min( α,β)<α+β. Therefore from Theorem 1.1, there exists a G.S.
+Remark 7.3 We wonder if the limit curve for existence of G.S. would be Hs, or another curve Ls
+defined by
+(Ls)1
+δ+1+1
+µ+1=N−2
+N−(N−2)s
+2,
+which ensures that Φ(R)>0,and also B(N−2,N−2)>0.This curve cuts the diagonal at the
+same point Ps=/parenleftBig
+N+2
+N−2−s,N+2
+N−2−s/parenrightBig
+asHs.Notice that Lsis under Hs.
+278 The radial potential system
+Here we study the nonnegative radial solutions of system ( SP) :
+(SP)/braceleftbigg−∆pu=|x|ausvm+1,
+−∆qv=|x|aus+1vm,
+witha=b,δ=m+1,µ=s+1,and we assume (1.5). System ( M) becomes
+(MP)
+
+Xt=X/bracketleftBig
+X−N−p
+p−1+Z
+p−1/bracketrightBig
+,
+Yt=Y/bracketleftBig
+Y−N−q
+q−1+W
+q−1/bracketrightBig
+,
+Zt=Z[N+a−sX−(m+1)Y−Z],
+Wt=W[N+b−(s+1)X−mY−W].
+For this system D, γandξare defined by
+D=p(1+m)+q(1+s)−pq,(p−1−s)γ+p+a= (m+1)ξ,(q−1−m)ξ+q+b= (s+1)γ,
+thusγandξare linked independtly of s,mby the relation
+p(γ+1) =q(ξ+1) =pq(m+s+2+a)
+D. (8.1)
+The system is variational . It admits an energy function, given at (1.13), which can als o can be
+obtained by a direct computation in terms of X,Y,Z,W :
+EP(r) =ψ/bracketleftbigg
+ZW−s+1
+pW((N−p)−(p−1)X)−m+1
+qZ((N−q)−(q−1)Y)/bracketrightbigg
+,(8.2)
+where
+ψ=rN−2−a|u′|p−1|v′|q−1
+usvm=rN−(γ+1)p/bracketleftBig
+Xq(s+1)(p−1)Yp(m+1)(q−1)Zp(q−m−1)Wq(p−s−1)/bracketrightBig1/D
+.
+Then we find
+E′
+P(r) = (N+a−(s+1)N−p
+p−(m+1)N−q
+q)rN−1+aus+1vm+1.
+Thus we define a critical line Das the set of ( δ,µ) = (m+1,s+1) such that
+N+a= (m+1)N−q
+q+(s+1)N−p
+p, (8.3)
+equivalent to pq(m+s+2+a) =ND,orN+a= (m+1)ξ+(s+1)γ,or
+(γ,ξ) = (N−p
+p,N−q
+q)
+The subcritical case is given by the set of points under D, equivalently γ >N−p
+p,ξ >N−q
+qor
+(s+1)γ+(m+1)ξ >N+a.The supercritical case is the set of points above D.
+28Remark 8.1 The energy (EP)0of the particular solution associated to M0is still negative : (EP)0=
+−D
+pqrN+a−(γ+1)p/bracketleftBig
+Xq(p−1)
+0Yp(q−1)
+0Zq(s+1)
+0Wp(m+1)
+0/bracketrightBig1/D
+.
+Remark 8.2 Whenp=q= 2, another energy function can be associated to the transform ation
+given at Remark 2.2: the system (2.9) relative to u(r) =r−γU(t), v(r) =r−ξV(t)is
+/braceleftbiggUtt+(N−2−2γ)Ut−γ(N−2−γ)U+UsVm+1= 0
+Vtt+(N−2−2γ)Vt−γ(N−2−γ)V+Us+1Vm= 0(8.4)
+and the function
+EP(t) =s+1
+2(U2
+t−γ(N−2−γ)U2)+m+1
+2(V2
+t−γ(N−2−γ)V2+Us+1Vm+1(8.5)
+satisfies
+(EP)t=−(N−2−2γ)/bracketleftbig
+(s+1)U2
+t+(m+1)V2
+t/bracketrightbig
+It differs from EP,even in the critical case. This point is crucial for Section 9.
+It has been proved in [34], [35], that in the subcritical case witha= 0, there exists a solution of
+the Dirichlet problem in any bounded regular domain Ω of RN; and in the supercritical case there
+exists no solution if Ω is starshaped. Here we prove two resul ts of existence or nonexistence of G.S.
+which seem to be new:
+Proof of Theorem 1.6. 1)Existence or nonexistence results.
+•In the supercritical or critical case there exists a G.S. Fro m Theorem 1.1, if it were not true,
+then therewould exist regular positive solutions of ( MP) such that X(T) =N−p
+p−1andY(T) =N−q
+q−1.
+It would satify EP≦0.Then at time T,we findEP(R)>0,from (8.2), since W >0,Z >0,which
+is impossible.
+•In the subcritical case, there exists no G.S. Suppose that th ere exists one. Now EPis nonde-
+creasing, hence EP≧0.Its trajectory stays in the box Adefined by (5.1), thus it is bounded. If
+q≧m+1 andp≧s+1,we deduce that , EP(r) =O(rN−(γ+1)p) from (8.2), then EPtends to 0 at
+∞,which is contradictory. Next consider the general case. We h ave
+EP(r)≦rN−(γ+1)p/bracketleftBig
+Xq(p−1)Yp(q−1)Zp(q−m−1)Wq(p−s−1)/bracketrightBig1/D
+ZW
+=rN−(γ+1)p/bracketleftBig
+Xq(p−1)Yp(q−1)Zq(1+s)Wp(1+m)/bracketrightBig1/D
+,
+then the same result holds. Consequently, from Theorem 1.1, there exists a solution of the Dirichlet
+problem
+2)Behaviour of the G.S.in the critical case.
+LetTbe the trajectory of a G.S.; then EP(0) = 0,thusTlies on the variety Vof energy 0, also
+defined by
+qW[(s+1)((p−1)X−(N−p))+pZ] =p(m+1)Z[(N−q)−(q−1)Y)] (8.6)
+29andY <N−q
+q−1,hence (s+ 1)((p−1)X−(N−p)) +pZ >0. From (5.2), Tstarts from N0=
+(0,0,N+a,N+b) and stays in A. Eliminating Win system ( M),we find a system of three
+equations
+
+Xt=X/bracketleftBig
+X−N−p
+p−1+Z
+p−1/bracketrightBig
+,
+Yt=YF,
+Zt=Z[N+a−sX−(m+1)Y−Z],
+where
+F(X,Y,Z) =1
+q/bracketleftbiggN−q
+q−1−Y/bracketrightbiggp(m+1−q)Z+q(s+1)((N−p)−(p−1)X)
+(s+1)((p−1)X−(N−p))+pZ.
+(i) IfTconverges to a fixed point of the system in ¯R, the possible points on VareA0,I0,J0, P0,
+Q0,G0,H0, R0,S0.The eigenvalues of the linearized problem at A0,given by (10.3) satisfy
+λ1,λ2>0,λ3=N+a−sN−p
+p−1−(m+1)N−q
+q−1≦λ4=λ∗=N+a−(s+1)N−p
+p−1−mN−q
+q−1,
+sinceq≦p,andλ3<λ∗forq/\e}atio\slash=p,andλ3=λ∗<0 forq=p,from (8.3). Then A0can be attained
+only whenλ∗≦0,from Proposition 4.5. And P0can be attained only if
+q>m+1, λ∗≧0 andq+a<(s+1)N−p
+p−1, (8.7)
+from Proposition 4.6, because γ=N−p
+p<N−p
+p−1.We observe that the condition λ∗≧0 joint to (8.3)
+impliesm+1<q<pand is equivalent to (8.7). Indeed it implies
+N−p
+p−1(s+1)≦N+a−mN−q
+q−1=N+a−q
+q−1(N+a−N−q
+q−(s+1)N−p
+p);
+then
+(s+1)N−p
+p−1q−p
+p≦−(a+q),
+thusq<p.From (8.3) we obtain
+(N−q)(m+1
+q−1) =q+a−(s+1)N−p
+p≦(s+1)N−p
+p(p−q
+p−1−1)<0,
+hencem+1<qand (8.7) follows. By symmetry, Q0cannot be attained since q≦p.ThenA0and
+P0are incompatible, unless A0=P0, andP0is not attained when p=q.
+(ii) Next we show that Tconverges to A0or toP0. Iftis an extremum value of Y, then
+(m+1
+q−1)Z(t)+s+1
+p((N−p)−(p−1)X(t)) = 0. (8.8)
+This relation implies q>m+1 and
+Xt(t) =X(t)Z(t)
+p−1/bracketleftbigg
+1+p(m+1−q)
+q(s+1)/bracketrightbigg
+=DX(t)Z(t)
+(p−1)q(s+1)>0.
+30In the same way, if tis an extremum value of X,thenp > s+ 1 andYt(t)>0.Near−∞,there
+holdsXt,Yt≧0,andZt,Wt≦0,from the linearization near N0.Suppose that Xhas a maximum
+att0followed by a minimum at t1.Thenp>s+1,andYis increasing on [ t0,t1]. At timet0we
+have (p−1)X(t0) +Z(t0) =N−pandXtt(t0)≦0,thusZt(t0)≦0; eliminating Zwe deduce
+p+a+(p−1−s)X(t0)≦(m+1)Y(t0) and similarly ( m+1)Y(t1)≦p+a+(p−1−s)X(t1);
+henceY(t1)<Y(t0), which is a contradiction. Thus XandYcan have at most one maximum, and
+in turn they have no maximum point. Therefore XandYare increasing, and they are bounded,
+henceXhas a limit in/parenleftBig
+0,N−p
+p−1/bracketrightBig
+andYhas a limit in/parenleftBig
+0,N−q
+q−1/bracketrightBig
+. ThenZ,Ware decreasing; indeed
+at each time where Zt= 0,we haveZtt=Z(−sXt−(m+1)Yt)<0,thus it is a maximum, which
+is impossible.
+ThenTconverges to a fixed point of the system. Moreover, since XandYare increasing, it
+cannot be one of the points I0,J0,G0,H0,R0,S0.It is necessarily A0orP0.We distinguish two
+cases:
+•Caseq≦m+1. Then Tconverges to A0,andλ3,λ∗<0,then (1.10) follows.
+•Caseq > m+ 1.ThenTconverges to A0(resp.P0) whenλ∗≦0 (resp.λ∗≧0). If the
+inequalities are strict, we deduce the convergence of uandvfrom Propositions 4.5 and 4.6, and
+(1.11) follows. If λ∗= 0,thenP0=A0,andλ3=(N−1)(q−p)
+(p−1)(q−1)<0.The projection of the trajectory
+TinR3on the plane Y=N−q
+q−1satisfies the system
+Xt=X/bracketleftbigg
+X−N−p
+p−1+Z
+p−1/bracketrightbigg
+, Z t=Z/bracketleftbigg
+N+a−sX−(m+1)N−q
+q−1−Z/bracketrightbigg
+which presents a saddle point at (N−p
+p−1,0), thus the convergence of XandZis exponential, in
+particular we deduce the behaviour of u.The trajectory enters by the central variety of dimension
+1,and by computation we deduce that Y=N−q
+q−1−1
+q−1−mt−1+O(t−2+ε),then (1.12) follows.
+9 The nonradial potential system of Laplacians
+Here we study the possibly nonradial solutions of the system of the preceeding Section when
+p=q= 2 :
+(SL)/braceleftbigg−∆u=|x|ausvm+1,
+−∆v=|x|aus+1vm,
+withD=s+m.We solve an open problem of [7]: the nonexistence of (radial o r nonradial) G.S.
+under condition (1.14).
+It was shown in [7] in the case N+a≧4.The problem was open when N+a <4,and
+m+s+1>(N+a)/(N−2),which implies N <6.Indeed in the case m+s+1≦(N+a)/(N−2),
+there are no solutions of the exterior problem, see [6, Theor em 5.3]. Recall that the main result of
+[7] is the obtention of apriori estimates near 0 or ∞,by using the Bernstein technique introduced
+in [18] and improved in [8]. Then the behaviour of the solutio ns is obtained by using the change of
+unknown
+u(r,θ) =r−γU(t,θ), v(r,θ) =r−γV(t,θ), t= lnr,
+31extending the transformation of Remark 8.2 to the nonradial case (in fact here tis−tin [7]); it
+leads to the system
+Utt+(N−2−2γ)Ut+∆SU−γ(N−2−γ)U+UsVm+1= 0,
+Vtt+(N−2−2γ)Vt+∆SV−γ(N−2−γ)V+Us+1Vm= 0,
+where ∆ Sis the Laplace-Beltrami operator on SN−1.A corresponding energy is introduced in [7]:
+EL(t) =s+1
+2/integraldisplay
+SN−1(U2
+t−|∇SU|2−γ(N−2−γ)U2)dθ
++m+1
+2/integraldisplay
+SN−1(V2
+t−|∇SV|2−ξ(N−2−ξ)V2)dθ+/integraldisplay
+SN−1Us+1Vm+1dθ,
+extending (8.5) to the nonradial case; it satisfies
+(EL)t=−(N−2−2γ)/integraldisplay
+SN−1/bracketleftbig
+(s+1)U2
+t+(m+1)V2
+t/bracketrightbig
+dθ
+Here we construct another energy function , extending the Pohozaev function defined at (1.13) to
+the nonradial case.
+Lemma 9.1 Consider the function EL(r)defined by
+r−NEL(r) =s+1
+2/integraldisplay
+SN−1/bracketleftBig
+u2
+r−r−2|∇Su|2+(N−2)uur
+r/bracketrightBig
+dθ
++m+1
+2/integraldisplay
+SN−1/bracketleftbigg
+(∂v
+∂ν)2−r−2|∇Sv|2+(N−2)vvr
+r/bracketrightbigg
+dθ+ra/integraldisplay
+SN−1us+1vm+1dθ.
+Then the following relation holds:
+r1−NE′
+L(r) = (N+a−(s+1)N−2
+2−(m+1)N−2
+2)ra/integraldisplay
+SN−1us+1vm+1dθ.
+Proof.In terms of t,we find
+EL(t) =EL,1(t)+EL,2(t)+EL,3(t),with
+EL,1(t) =s+1
+2e(N−2)t/integraldisplay
+SN−1/bracketleftBig
+u2
+t−|∇Su|2+(N−2)uut/bracketrightBig
+dθ,
+EL,2(t) =m+1
+2e(N−2)t/integraldisplay
+SN−1/bracketleftBig
+v2
+t−|∇Sv|2+(N−2)vvt/bracketrightBig
+dθ,EL,3(t) =e(N+a)t/integraldisplay
+SN−1us+1vm+1dθ,
+andusatisfies the equations
+utt+(N−2)ut+∆Su+e(2+a)tusvm+1= 0, (9.1)
+(e(N−2)tut)t+e(N−2)t∆Su+e(N+a)tusvm+1= 0, (9.2)
+32andvsatisfies symmetrical equations. Multiplying (9.2) by uand (9.1) by ( s+ 1)e(N−2)tut,we
+obtain
+0 =/integraldisplay
+SN−1u(e(N−2)tut)t+e(N−2)t/integraldisplay
+SN−1u∆Su+e(N+a)t/integraldisplay
+SN−1us+1vm+1
+=d
+dt/integraldisplay
+SN−1ue(N−2)tut−e(N−2)t/integraldisplay
+SN−1(u2
+t+|∇Su|2)+e(N+a)t/integraldisplay
+SN−1us+1vm+1
+d
+dt/integraldisplay
+SN−1s+1
+2(N−2)ue(N−2)tut−s+1
+2(N−2)e(N−2)t/integraldisplay
+SN−1(u2
+t+|∇Su|2)
+=−s+1
+2(N−2)e(N+a)t/integraldisplay
+SN−1us+1vm+1,
+and symmetrically for v,and adding the equalities we deduce
+0 = (s+1)(e(N−2)td
+dt/integraldisplay
+SN−1(u2
+t−|∇Su|2
+2+(N−2)e(N−2)t/integraldisplay
+SN−1u2
+t
++(m+1)(e(N−2)td
+dt/integraldisplay
+SN−1v2
+t−|∇Sv|2
+2+(N−2)e(N−2)t/integraldisplay
+SN−1v2
+t
++d
+dt(e(N+a)t/integraldisplay
+SN−1us+1vm+1)−(N+a)e(N+a)t/integraldisplay
+SN−1us+1vm+1
+d
+dt
+e(N−2)t
+2/integraldisplay
+SN−1((s+1)(u2
+t−|∇Su|2)+(m+1)(v2
+t−|∇Sv|2))+e(N+a)t/integraldisplay
+SN−1us+1vm+1
+
++N−2
+2e(N−2)t/integraldisplay
+SN−1((s+1)(u2
+t+|∇Su|2)+(m+1)(v2
+t+|∇Sv|2))
+= (N+a)e(N+a)t/integraldisplay
+SN−1us+1vm+1,
+hence
+(EL)t(t) = (N+a−(s+1)N−2
+2−(m+1)N−2
+2)e(N+a)t/integraldisplay
+SN−1us+1vm+1dθ.
+Proof of Theorem 1.7. Supposethat thereexists aG.S. Since s+m+1<(N+2+2a)/(N−2)
+we deduce that ELandELare increasing and start from 0, then they stay positive. Fro m [7,
+Corollary 6.4], since s+m+1<(N+2)/(N−2),three eventualities can hold. The first one is that
+(u,v) behaves like the particular solution ( u0,v0); it cannot hold because ELhas a negative limit,
+33see [7, Remark 6.3]. The second one is that ( u,v) is regular at ∞,that means lim |x|→∞|x|N−2u=
+α >0,lim|x|→∞|x|N−2v=β >0; it cannot hold because lim t→∞EL(t) = 0.It remains a third
+eventuality: when for example m>(N+a)/(N−2), and (u,v) has the following behaviour at ∞:
+lim
+r→∞u=α>0,and lim
+|x|→∞|x|kv=β >0 or 0,withk= (2+a)/(m−1).(9.3)
+Thecondition on mimplies that N <4−afrom assumption(1.14) .Inthat case lim t→∞EL(t) =∞,
+which gives no contradiction. Here we show that a contradict ion holds by using the new energy
+function EL.
+First recall the proof of (9.3). Making the substitution
+u(r,θ) =u(t,θ), v(r,θ) =r−kV(t,θ), t= lnr,θ∈SN−1,
+we get/braceleftbiggutt+(N−2)ut+∆Su+e−2ktusVm+1= 0,
+Vtt+(N−2−2k)Vt+∆SV−k(N−2−k)V+us+1Vm= 0.(9.4)
+Thenu,Vare bounded near ∞,and from [7, Proposition 4.1] uconverges exponentially to the
+constantα,more precisely
+/ba∇dbl|u−α|+|ut|+|∇Su|/ba∇dblC0(SN−1)=O(e−(N−2)t), (9.5)
+becausek/\e}atio\slash= (N−2)/2 and all the derivatives of Vup to the order 2 are bounded. The equation
+inVtakes the form
+Vtt+(N−2−2k)Vt+∆SV−k(N−2−k)V+αs+1Vm+ϕ= 0
+whereϕand its derivatives up to the order 2 are O(e−(N−2)t).From [7, Theorem 4.1], the function
+Vconverges to βor to 0 inC2(SN−1).
+Next we define
+f(t) =e(N−2)t/integraldisplay
+SN−1utdθ=rN−1/integraldisplay
+SN−1urdθ.
+Then
+EL,1(t) = (N−2)s+1
+2αf(t)+O((e−(N−2)t))
+from (9.5). Moreover from (9.4),
+ft(t) =−e(N−2−2k)t/integraldisplay
+SN−1usVm+1dθ<0.
+Sinceuis regular at 0, f(t) = 0(e(N−1)t) at−∞,in particular lim t→−∞f(t) = 0.Andft(t) =
+O(e(N−2−2k)t) =O(e−t) at∞,thenf(t) has a finite negative limit −ℓ2; and
+lim
+t→∞EL,1(t) =−(N−2)s+1
+2αℓ2.
+Moreoverv=e−ktV,andVand its derivatives up to the order 2 are bounded, thus
+EL,2(t) =O(e(N−2−2k)t) =O(e−t)
+Finally
+EL,3(t) =O(e(N+a−k(m+1))t)
+andN+a−k(m+1)<2−N
+m−1<0.ThenELhas a finite limit θ<0 at∞,which is contradictory.
+3410 Analysis of the fixed points
+Here we make the local analysis around the fixed points.
+Proof of Proposition 4.4. (i) Consider a regular solution ( u,v) with initial data ( u0,v0).
+When when r→0,we have
+(−rN−1/vextendsingle/vextendsingleu′/vextendsingle/vextendsinglep−2u′)′=rN−1+aus
+0vδ
+0(1+o(1)),−/vextendsingle/vextendsingleu′/vextendsingle/vextendsinglep−2u′=1
+N+ar1+aus
+0vδ
+0(1+o(1)),
+thus from (2.1), when t→ −∞
+X(t) = (1
+N+aus+1−p
+0vδ
+0)1/(p−1)e(p+a)t/(p−1)(1+o(1)),
+Y(t) = (1
+N+buµ
+0vm+1−q
+0)1/(q−1)e(q+b)t/(q−1)(1+o(1)),
+and lim t→−∞Z=N+a,limt→−∞W= (N+b).In particular the trajectory tends to N0=
+(0,0,N+a,N+b).
+(ii) Reciprocally, consider a trajectory converging to N0.SettingZ=N+a+˜Z,W=N+b+˜W,
+the linearized system is
+Xt=p+a
+p−1X, Y t=q+b
+q−1Y,˜Zt= (N+a)/bracketleftBig
+−sX−δY−˜Z/bracketrightBig
+,˜Wt= (N+b)/bracketleftBig
+−µX−mY−˜W/bracketrightBig
+.
+(10.1)
+The eigenvalues are
+λ1=p+a
+p−1>0, λ2=q+b
+q−1>0, λ3=−(N+a)<0, λ4=−(N+b)<0.(10.2)
+The unstable variety Vuand the stable variety Vshave dimension 2 .Notice that Vsis contained
+in the setX=Y= 0,thus no admissible trajectory converges to N0whenr→ ∞, and there
+exists an infinity of admissible trajectories in R,converging to N0whenr→0. Moreover we get
+limt→−∞e−(p+a)/(p−1)tX(t) =κ >0 and lim t→−∞e−(q+b)/(q−1)tY(t) =ℓ >0. Thus (u,v) have a
+positive limit ( u0,v0) = ((N+a)κp−1)(q−1−m)/D((N+b)ℓq−1)δ/Dfrom (4.2), (4.1), hence ( u,v) is
+a regular solution.
+Next we show that for anyκ >0,ℓ >0 there exists a unique local solution such that
+limt→−∞e−(p+a)t/(p−1)X(t) =κand lim t→−∞e−(q+b)/(q−1)tY=ℓ.OnVu, we get a system of
+two equations of the form
+Xt=X(λ1+F(X,Y)), Yt=Y(λ2+G(X,Y)),
+whereF=AX+BY+f(X,Y),wherefis a smooth function with fX(0,0) =fY(0,0) = 0,
+similarly for G.SettingX=eλ1t(κ+x), Y=eλ2t(ℓ+y),and assuming λ2≧λ1and setting
+ρ=eλ1twe obtain
+xρ=1
+ρ(κ+x)F(ρ(κ+x),ρλ2/λ1(ℓ+y)), y ρ= (ℓ+y)G(ρ(κ+x),ρλ2/λ1(ℓ+y)),
+35withx(0) =y(0) = 0.Then we get local existence and uniqueness. Hence for any u0,v0>0 there
+exists a regular solution ( u,v) with initial data ( u0,v0).Moreoveru,v∈C1([0,R)) whena,b>−1.
+Proof of Proposition 4.5. The linearization at A0=/parenleftBig
+N−p
+p−1,N−q
+q−1,0,0/parenrightBig
+gives, with X=
+N−p
+p−1+˜X,Y=N−q
+q−1+˜Y,
+˜Xt=N−p
+p−1/bracketleftbigg
+˜X+Z
+p−1/bracketrightbigg
+,˜Yt=N−q
+q−1/bracketleftbigg
+˜Y+W
+q−1/bracketrightbigg
+, Zt=λ3Z, W t=λ4W.
+The eigenvalues are
+λ1=N−p
+p−1>0, λ2=N−q
+q−1>0, λ3=N+a−sN−p
+p−1−δN−q
+q−1, λ4=N+b−µN−p
+p−1−mN−q
+q−1.
+(10.3)
+•Convergencewhen r→∞:Ifλ3>0,orλ4>0,thenthestablevariety Vshasat mostdimension1 ,
+it satisfiesW= 0orZ= 0,hence thereis noadmissibletrajectory converging to A0at∞.Ifλ3<0,
+andλ4<0,thenVshas dimension 2 .Moreover Vs∩{Z= 0}has dimension 1 : the corresponding
+system inX,Y,W has the eigenvalues λ1,λ2,λ4; similarly Vs∩ {W= 0}has dimension 1. Then
+there exist trajectories in Vssuch thatZ >0 andW >0,included in Rand thus admissible. They
+satisfy lime−λ3tZ=C3>0,lime−λ4tW=C4>0,then (4.11) follows from (4.2).
+•Convergence when r→0 : Ifλ3<0,orλ4<0,the unstable variety Vuhas at most dimension
+3, and it satisfies W= 0 orZ= 0.Therefore there is no admissible trajectory converging at −∞.
+Ifλ3,λ4>0,thenVuhas dimension 4; in that case there exist admissible traject ories, and (4.11)
+follows as above.
+Proof of Proposition 4.6. We setP0=/parenleftBig
+N−p
+p−1,Y∗,0,W∗/parenrightBig
+,with
+Y∗=N−p
+p−1µ−(q+b)
+q−1−m, W ∗=(q−1)(N+b−N−p
+p−1µ)−m(N−q)
+q−1−m,
+form+1/\e}atio\slash=q.The linearization at P0gives, with X=N−p
+p−1+˜X,Y=Y∗+˜Y, W=W∗+˜W,
+˜Xt=N−p
+p−1/bracketleftbigg
+˜X+Z
+p−1/bracketrightbigg
+,˜Yt=Y∗/bracketleftBigg
+˜Y+˜W
+q−1/bracketrightBigg
+, Zt=λ3Z,˜Wt=W∗/bracketleftBig
+−µ˜X−m˜Y−˜W/bracketrightBig
+The eigenvalues are
+λ1=N−p
+p−1>0, λ3=N+a−sN−p
+p−1−δY∗=D
+q−1−m(γ−N−p
+p−1),
+and the roots λ2,λ4of equation
+λ2−(Y∗−W∗)λ+m+1−q
+q−1Y∗W∗= 0
+Then ifλ3<0 (resp.λ3>0) there is no admissible trajectory converging when r→0 (resp.
+r→ ∞).IndeedVu=Vu∩{Z= 0}(resp.Vs=Vs∩{Z= 0}).
+361) Suppose that q > m+ 1.Sinceq+b <N−p
+p−1µ < N+b−mN−q
+q−1,we haveP0∈ R,and
+λ2λ4<0. First assume λ3<0,that means γ <N−p
+p−1. ThenVshas dimension 2, and Vs∩{Z= 0}
+has dimension 1, there exists trajectories with Z >0,which are admissible, converging when
+r→ ∞.Next assume λ3>0. Then Vuhas dimension 3, and Vu∩{Z= 0}has dimension 2. Thus
+there exist admissible trajectories converging when t→ −∞.
+2) Supposethat q<m+1. Sinceq+b>N−p
+p−1µ>N+b−mN−q
+q−1,we haveP0∈ R,andλ2λ4>0.
+We assume qN−p
+p−1µ+m(N−q)/\e}atio\slash=N(q−1)+(b+1)q,that means Y∗/\e}atio\slash=W∗.First suppose λ3>0,
+that means γ <N−p
+p−1.If Reλ2>0,thenVuhas dimension 4, or Re λ2<0 thenVuhas dimension
+2 andVu∩ {Z= 0}has dimension 1 .In any case, there exist admissible trajectories convergin g
+whenr→0. Next assume λ3<0. If Reλ2>0,thenVshas dimension 1, and Vs∩{Z= 0}=∅.
+If Reλ2<0,thenVshas dimension 3. In any case Vscontains trajectories with Z >0,which are
+admissible, converging when r→ ∞.
+Those trajectories satisfy lim e−λ3tZ=C3>0,limX=N−p
+p−1,limY=Y∗and limW=W∗,
+thus (4.12) follows from (4.2) and (2.5).
+Proof of Proposition 4.8. The linearization at I0= (N−p
+p−1,0,0,0) gives, with X=N−p
+p−1+˜X,
+˜Xt=N−p
+p−1(˜X+Z
+p−1), Yt=−N−q
+q−1Y, Z t= (N+a−sN−p
+p−1)Z, W t= (N+b−µN−p
+p−1)W.
+The eigenvalues are
+λ1=N−p
+p−1>0, λ2=−N−q
+q−1<0, λ3=N+a−sN−p
+p−1, λ4=N+b−µN−p
+p−1.
+•Convergence when r→ ∞: Ifλ3>0 orλ4>0,thenVs=Vs∩{Z= 0}orVs=Vs∩{W= 0}.
+There is no admissible trajectory converging at ∞. Next suppose that λ3,λ4<0.ThenVshas
+dimension 3; it contains trajectories with Y,Z,W > 0,which are admissible. They satisfy lim X=
+N−p
+p−1,lime−λ2tY=C2>0,lime−λ3tZ=C3>0,lime−λ4tW=C4>0,then (4.13) follows from
+(4.2) and (2.4).
+•Convergence when r→0 : Sinceλ2<0 we have Vu=Vu∩{Y= 0},hence there is no admissible
+trajectory converging when r→0.
+Proof of Proposition 4.9. ThepointG0= (N−p
+p−1,0,0,N+b−N−p
+p−1µ)∈ RsinceN−p
+p−1µ<N+b.
+The linearization at G0gives, with X=N−p
+p−1+˜X,W=N+b−N−p
+p−1µ+˜W,
+˜Xt=N−p
+p−1/bracketleftbigg
+˜X+Z
+p−1/bracketrightbigg
+, Yt=Y
+q−1(q+b−N−p
+p−1µ),
+Zt= (N+a−sN−p
+p−1)Z, W t= (N+b−N−p
+p−1µ)/bracketleftBig
+−µ˜X−mY−˜W/bracketrightBig
+The eigenvalues are
+λ1=N−p
+p−1>0, λ2=1
+q−1(q+b−N−p
+p−1µ), λ3=N+a−sN−p
+p−1, λ4=N−p
+p−1µ−N−b<0.
+•Convergence when r→ ∞: Ifλ2>0,orλ3>0,thenVs=Vs∩{Y= 0}orVs=Vs∩{Z= 0},
+there is no admissible trajectory converging at ∞. Assumeλ2,λ3<0,thenVshas dimension 3, it
+contains trajectories with Y,Z >0,which are admissible.
+37•Convergence when r→0 : Ifλ3<0,orλ2<0 there is no admissible trajectory. If λ2,λ3>0
+thenVshas dimension 3 ,it contains admissible trajectories.
+In any case lim X=N−p
+p−1,lime−λ2tY=C2>0,lime−λ3tZ=C3>0,limW=N+b−N−p
+p−1µ,
+hence (4.13) still follows from (4.2) and (2.4).
+Proof of Proposition 4.10. We setC0=/parenleftbig
+0,¯Y,0,¯W/parenrightbig
+,with
+¯Y=q+b
+m+1−q,¯W=m(N−q)−(N+b)(q−1)
+m+1−q. (10.4)
+ThenC0∈ RifN−q
+q−1m>N+b,implyingq<m+1.The linearization at C0gives, with Y=¯Y+˜Y
+andW=¯W+˜W
+Xt=−N−p
+p−1X,˜Yt=¯Y/bracketleftBigg
+˜Y+˜W
+q−1/bracketrightBigg
+, Zt=λ3Z, W t=¯W/bracketleftBig
+−µX−m˜Y−˜W/bracketrightBig
+.
+The eigenvalues are
+λ1=−N−p
+p−1, λ3=N+a−δ¯Y,
+and the roots λ2,λ4of equation
+λ2−(¯Y−¯W)λ+m+1−q
+q−1¯Y¯W= 0 (10.5)
+thenλ2λ4>0.We assume m/\e}atio\slash=N(q−1)+(b+1)q
+N−q,that means ¯Y/\e}atio\slash=¯W.
+•Convergence when r→ ∞: ifλ3>0 we have Vs=Vs∩{Z= 0},hence there is no admissible
+trajectory.Next assume that λ3<0, that means δ >(N+a)m+1−q
+q+b.If Reλ2<0 (resp.>0) then
+Vshas dimension 4 (resp. 2) and Vs∩{X= 0}andVs∩{Z= 0}have dimension 3 (resp. 1) then
+there exist trajectories with X,Z >0,which are admissible.
+In any case lim e−λ1tX=C1>0,limY=¯Y,lime−λ3tZ=C3>0,limW=¯W,then (4.15)
+follows.
+•Convergence when r→0 : Sinceλ1<0 we have Vu=Vu∩{X= 0},hence there is no admissible
+trajectory.
+Proof of Proposition 4.11. We setR0=/parenleftbig
+0,¯Y,¯Z,¯W/parenrightbig
+,where¯Y,¯Ware defined at (10.4), and
+¯Z=N+a−δb+q
+m+1−q.Underourassumptionsitliesin R. SettingY=¯Y+˜Y,Z=¯Z+˜Z,W=¯W+˜W,
+the linearization at R0gives
+Xt=λ1X,˜Yt=¯Y/bracketleftBigg
+˜Y+˜W
+q−1/bracketrightBigg
+, Zt=¯Z/bracketleftBig
+−sX−δ˜Y−˜Z/bracketrightBig
+, Wt=¯W/bracketleftBig
+−µX−m˜Y−˜W/bracketrightBig
+;
+the eigenvalues are
+λ1=1
+p−1(p+a−δb+q
+m+1−q), λ3=−¯Z <0;
+and the roots λ2,λ4of equation of equation (10.5).
+38•Convergence when r→ ∞: Ifλ1>0, that means ( p+a)m+1−q
+q+b<δ,thenVs=Vs∩{X= 0},
+hence there is no admissible trajectory. Next assume λ1<0; if Reλ2<0 (resp.>0) thenVs
+has dimension 4(resp. 2) and Vs∩{X= 0}has dimension 3 (resp. 1) then there exist admissible
+trajectories.
+•Convergence when r→0 : Ifλ1<0, thenVu=Vu∩ {X= 0},hence there is no admissible
+trajectory. Next assume λ1>0.If Reλ2= Reλ4<0 (resp.>0) thenVshas dimension 4 (resp.
+2) andVs∩{X= 0}has dimension 3 (resp. 1) then there exist admissible trajec tories.
+In any case lim e−λ1tX=C1>0,limY=¯Y,limZ=¯Z,limW=¯W,then (4.15) holds again.
+Remark 10.1 Finally there is no admissible trajectory converging to 0 = (0,0,0,0),orK0=
+(0,0,N+a,0),orL0= (0,0,0,N+b). Indeed the linearization at 0gives
+Xt=−N−p
+p−1X, Y t=−N−q
+q−1Y, Z t= (N+a)Z, W t= (N+b)W
+ThenVsandVuhave dimension 2,henceVsis contained in {Z=W= 0},andVuin{X=Y= 0}.
+The linearization at K0gives, with Z=N+a+˜Z,
+Xt=p+a
+p−1X, Y t=−N−q
+q−1Y, Z t= (N+a)/bracketleftBig
+−sX−δY−˜Z/bracketrightBig
+, Wt= (N+b)W.
+The eigenvalues arep+a
+p−1,−N−q
+q−1,−(N+a), N+b.ThenVsandVuhave dimension 2,henceVsis
+contained in {Z=W= 0}, andVuin{Y= 0}.The case of L0follows by symmetry.
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+42
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+arXiv:1001.0563v2  [hep-th]  4 Jan 2010Conformal blocks as Dotsenko-Fateev Integral
+Discriminants
+A.Mironov∗,A.Morozov†andSh.Shakirov‡FIAN/TD-01/10
+ITEP/TH-01/10
+Abstract
+As anticipated in [1], elaborated in [2, 3, 4], and explicitl y formulated in [5], the Dotsenko-Fateev integral
+discriminant coincides with conformal blocks, thus provid ing an elegant approach to the AGT conjecture,
+without any reference to an auxiliary subject of Nekrasov fu nctions. Internal dimensions of conformal
+blocks in this identification are associated with the choice of contours: parameters of the DV phase of the
+corresponding matrix models. In this paper we provide furth er evidence in support of this identity for the 6-
+parametric family of the 4-point spherical conformal block s, up to level 3 and for arbitrary values of external
+dimensions and central charges. We also extend this result t o multi-point spherical functions and comment
+on a similar description of the 1-point function on a torus.
+Contents
+1 Introduction 1
+2 Conformal block as Dotsenko-Fateev integral [5] 2
+3 Evidence in support of (8) 6
+3.1 The case of β= 1 and all αivanishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
+3.2 The case of β= 1 and non-vanishing α1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
+3.3 The case of β= 1 and non-vanishing α2orα3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
+3.4 The case of β= 1 and all αinon-vanishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
+3.5 The case of arbitrary βand allαinon-vanishing . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
+3.6 Determination of the central charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
+3.7 The lesson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
+3.8 A problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
+4 Multipoint functions on a sphere 12
+5 Dotsenko-Fateev integral on a torus 16
+6 Conclusion 18
+1 Introduction
+Non-Gaussian integrals are slowly attracting a growing attention, a nd start to play a role in modern theoretical
+studies. For quantum field theory they have an especially important property: non-Gaussian integrals are not
+fully defined by the action, they also depend on the discrete choice o f integrationcontour, and this freedom itself
+depends on the shape (degree) of the action. Accordingly, the Wa rd identities, reflecting the freedom to change
+integration variables and describing the coupling constant (RG) dep endence of the integral [6], in non-Gaussian
+case have several solutions, in one-to-one correspondence with the possible choice of integration contours. The
+closest subjects in pure mathematics are Picard-Fucks equations and emerging theory of motivic integration
+[7], and they are actively used, say, in Seiberg-Witten theory [8]-[11] and closely related string models. Initial
+∗Lebedev Physics Institute andITEP, Moscow, Russia ; mironov@itep.ru; mironov@lpi.ru
+†ITEP, Moscow, Russia ; morozov@itep.ru
+‡ITEP, Moscow, Russia andMIPT, Dolgoprudny, Russia ; shakirov@itep.ru
+1steps in a more direct approach to the study of non-Gaussian integ rals were recently described in [12], where
+emphasize is put on their similarity to the ordinary resultant theory. Following those papers we use the term
+integral discriminants for non-Gaussian integrals considered from this perspective. The re is of course a parallel
+development in the field of matrix models, where above-mentioned am biguity (contour dependence) could not
+remain unnoticed. There is a long chain of matrix model papers, devo ted to this issue, which finally culminated
+in the theory of Dijkgraaf-Vafa phases [13] and check-operator s [14].
+A new boost to a variety of prominent research directions was rece ntly given by the AGT conjecture [15]-
+[47],[1]-[5], which unified them all in a single entity and allowed to study the p roblems of one direction by means
+of the others. The present paper is devoted to a particular study of this type: representation of conformal
+blocks in terms of the matrix model integral discriminants with the Do tsenko-Fateev action (also known as
+conformal matrix models [48, 6] and β-ensembles [49]), which was suggested in [15, 27, 1], further investig ated
+in [2, 3, 4] and finally put into a clear and explicit form in [5]. As emphasized in [27, 5] this also resolves
+the old puzzle in conformal field theory (CFT) [51]: how arbitrary con formal blocks are described in terms
+of the Dotsenko-Fateev free field correlators [52, 53]. The proble m was that the number of free parameters
+in conformal block (dimensions of internal and external lines) exce eds the number of parameters in the free
+field conformal theory (there is no obvious room for internal-line dim ensions and also the choice of external
+dimensions is constrained by the peculiar free field conservation law/summationtext/vector αi=/vectorQ). As we now know from [1] and
+[5], the lacking parameters are exactly those of the DV phases, whic h parameterize the choice of contour of the
+eigenvalue integration in matrix integral, i.e. the choice of contours in the Dotsenko-Fateev screening operators,
+with conservation condition omitted. This was of course always expe cted in CFT, the change is that now one
+has a clear and unambiguous description of the phenomenon. In this paper we provide further evidence and
+further details about this description, by checking this new version of the AGT relation, between conformal
+blocks and Dotsenko-Fateev integral discriminants, at more levels and for more dimensions than in [5]. A proof
+of this relation, which contains no reference to the still mysterious Nekrasov functions (though some clarity
+seems to emerge here as well, see [34, 37, 5]), and should be, theref ore, self-contained and straightforward,
+remains beyond the scope of the present paper.
+2 Conformal block as Dotsenko-Fateev integral [5]
+In conformalfield theory, the simplest correlatorwith non-trivial coordinatedependence is the 4-point correlator
+/angbracketleftBig
+V∆1(z1,¯z1)V∆2(z2,¯z2)V∆3(z3,¯z3)V∆4(z4,¯z4)/angbracketrightBig
+where ∆ iare the dimensions of 4 primary fields on a sphere. With SL(2) transformations z/ma√sto→az+b
+cz+dwith
+ad−bc= 1, one can always put the four coordinates to 0 ,q,1 and∞. Hence the correlator depends (up to a
+simple factor) on a single variable q=(z1−z2)(z3−z4)
+(z1−z3)(z2−z4). By usual CFT arguments, this 4-point correlator can be
+written as a sum over a single intermediate dimension ∆ of a product of the holomorphic and anti-holomorphic
+conformal blocks:
+/angbracketleftBig
+V∆1(0,0)V∆2(q,¯q)V∆3(1,1)V∆4(∞,∞)/angbracketrightBig
+=/summationdisplay
+∆C/parenleftbig
+∆1,∆2,∆/parenrightbig
+C/parenleftbig
+∆,∆3,∆4/parenrightbig
+× F/parenleftbig
+∆1,∆2,∆3,∆4,∆,c/vextendsingle/vextendsingleq/parenrightbig
+(1)
+with a shorthand notation
+F/parenleftbig
+∆1,∆2,∆3,∆4,∆,c/vextendsingle/vextendsingleq/parenrightbig
+=q∆−∆1−∆2q¯∆−¯∆1−¯∆2B/parenleftbig
+∆1,∆2,∆3,∆4,∆,c/vextendsingle/vextendsingleq/parenrightbig
+B/parenleftbig¯∆1,¯∆2,¯∆3,¯∆4,¯∆,c/vextendsingle/vextendsingle¯q/parenrightbig
+(2)
+In the r.h.s., C’s are the 3-point functions (they do not depend on qand play the role of normalization constants
+needed to make B(0) = 1), cis the central charge and B(q) is the 4-point conformal block. Note that the bar
+here means not the complex conjugation, but the parameters of t he anti-holomorphic conformal block, see [51]
+for more details.
+The function B(q) is now widely recognized as the simplest representative of a family of important special
+functions of string theory, which appear not only in 2 dconformal field theory, but also in 4 dsupersymmetric
+field theory, according to the AGT conjecture [15]. These special f unctions generalize in a clever way the
+hypergeometric functions [27], and should of course possess vario us complementary representations, the most
+important one being seriesandintegralrepresentations. The latter one constitutes the subject of our paper.
+Historically, a series representation for B(q) was found the first. It is obtained by the decomposition of
+correlatorsviaiteratingtheoperatorproductexpansions. Thisp rocedureisextensivelyreviewedintheliterature
+2V∆1(0)V∆2(q) V∆3(1)
+V∆4(∞)∆
+Figure 1: Feynman-like diagram for a 4-point conformal block.
+[18, 27, 44], and, in the 4-point case shown in Fig.1, it gives the following series expansion in powers of q
+B/parenleftBig
+∆1,∆2,∆3,∆4,∆,c/vextendsingle/vextendsingle/vextendsingleq/parenrightBig
+=/summationdisplay
+|Y|=|Y′|q|Y|γ∆1∆2∆(Y)Q−1
+∆(Y,Y′)γ∆∆3∆4(Y′) (3)
+where the sum goes over the Young diagrams Y= (k1≥k2≥...) andY′= (k′
+1≥k′
+2≥...) of equal size
+|Y|=k1+k2+...=k′
+1+k′
+2+...=|Y′|, parameterizing the Virasoro descendants in the intermediate cha nnel.
+The relevant values of the Virasoro triple vertices (the structure constants of operator product expansion) are
+given by
+γ∆1∆2∆3(Y) =/productdisplay
+i/parenleftbig
+k1∆1+∆3−∆2+k1+...+ki−1/parenrightbig
+(4)
+and
+Q∆(Y,Y′) =<∆|LYL−Y′|∆>= (5)
+Y/Y′∅[1] [2] [11] [3] [21] [111] ...
+∅0
+[1] 2∆
+[2]1
+2(8∆+c)6∆
+[11] 6∆ 4∆(1+2∆)
+[3] 6∆+2c 2(8∆+c) 24∆
+[21] 2(8∆+c)8∆2+(34+c)∆+2c36∆(∆+1)
+[111] 24∆ 36∆(∆+1) 24∆(∆+1)(2∆+1)
+...
+is the Shapovalov matrix of the Virasoro algebra. Both γandQ−1are straightforwardlycalculable, what makes
+(3) an explicit and useful series expansion.
+Less investigated is the second chapter of the reference book of special functions, that is, integralrepresen-
+tation of the conformal blocks. Such a representation was actua lly proposed by V.Dotsenko and V.Fateev [52]
+(see also [53]) in terms of the free field correlators
+/angbracketleftBig/angbracketleftBig
+:e˜α1φ(z1):...:e˜αmφ(zm):/angbracketrightBig/angbracketrightBig
+=/productdisplay
+1≤i<j≤m(zj−zi)2˜αi˜αj(6)
+integrated over a part of the variables ziwithsomechoice of contours Ci. Integrated can be only operators of
+unit dimensions, i.e. the corresponding ˜ αare fixed to particular values, called bor−1/b. Integrated operators
+are often called ”screening charges” or simply ”screenings”. Gene rically, the screenings of only one type (say, b)
+are involved [48], though in rationalconformal models the both do es sentially contribute [52]. The precise choice
+ofintegrationcontoursremained a mystery forquite a long time, un til the recentbreakthrough[1]-[5], motivated
+by the AGT conjecture [15]. Following [5], we make a very simple choice of contours for the Dotsenko-Fateev
+partition function
+ZDF=/angbracketleftBigg/angbracketleftBigg
+:e˜α1φ(0): :e˜α2φ(q): :e˜α3φ(1): :e˜α4φ(∞):/parenleftbigg/integraldisplayq
+0:ebφ(z):dz/parenrightbiggN1/parenleftbigg/integraldisplay1
+0:ebφ(z):dz/parenrightbiggN2/angbracketrightBigg/angbracketrightBigg
+=
+=qα1α2
+2β(1−q)α2α3
+2βN1/productdisplay
+i=1q/integraldisplay
+0dziN1+N2/productdisplay
+i=N1+11/integraldisplay
+0dzi/productdisplay
+i<j(zj−zi)2β/productdisplay
+izα1
+i(zi−q)α2(zi−1)α3(7)
+3❥
+N1❥
+N2V∆1(0)V∆2(q) V∆3(1)
+V∆4(∞) ∆
+˜α1˜α2 ˜α3
+˜a= ˜α1+ ˜α2+N1b ˜α4∼=˜a+ ˜α3+N2b
+Figure 2: Feynman-like diagram for a 4-point conformal block. Here ∆ = ˜a/parenleftbig
+˜a+1
+b−b/parenrightbig
+and ∆ j= ˜αj/parenleftbig
+˜αj+1
+b−b/parenrightbig
+are the single
+internal and four external dimensions, respectively. The r ole of the screenings is to modify the free field selection rul e at the vertices:
+instead of ˜ a= ˜α1+ ˜α2and ˜α4∼=˜a+α3one has ˜ a= ˜α1+ ˜α2+bN1and ˜α4=b−1/b−˜α1−˜α2−˜α3−b(N1+N2)∼=˜a+ ˜α3+bN2.
+whereβ=b2andαi= 2b˜αi. Our main statement, which we check in detail, is that the Dotsenko-Fateev
+integral with this simple choice of integration contours pr ecisely reproduces the 4-point conformal
+block:
+ZDF/parenleftBig
+α1,α2,α3,N1,N2,β/vextendsingle/vextendsingle/vextendsingleq/parenrightBig
+=CDF·q∆−∆1−∆2·B/parenleftBig
+∆1,∆2,∆3,∆4,∆,c/vextendsingle/vextendsingle/vextendsingleq/parenrightBig
+(8)
+whereCDFis the Dotsenko-Fateev normalization constant, which does not de pend on q(but depends on all the
+other parameters).
+Theaimofpresentpaperistostudyrelation(8)morethoroughlyan dfindanexplicitcorrespondencebetween
+the six parameters of the conformal block ∆ 1,∆2,∆3,∆4,∆,c, and the six parameters of the Dotsenko-Fateev
+integralα1,α2,α3,N1,N2,β. Our result in the 4-point sector is
+
+
+∆1=α1(α1+2−2β)
+4β,∆2=α2(α2+2−2β)
+4β,∆3=α3(α3+2−2β)
+4β
+∆4=(2β(N1+N2)+α1+α2+α3)(2β(N1+N2)+α1+α2+α3+2−2β)
+4β
+∆ =(2βN1+α1+α2)(2βN1+α1+α2+2−2β)
+4β
+c= 1−6/parenleftbigg√β−1√β/parenrightbigg2(9)
+or in terms of the free field variables
+
+
+∆1= ˜α1/parenleftbigg
+˜α1+1
+b−b/parenrightbigg
+,∆2= ˜α2/parenleftbigg
+˜α2+1
+b−b/parenrightbigg
+,∆3= ˜α3/parenleftbigg
+˜α3+1
+b−b/parenrightbigg
+∆4=/parenleftBig
+b(N1+N2)+ ˜α1+ ˜α2+ ˜α3/parenrightBig/parenleftbigg
+b(N1+N2)+ ˜α1+ ˜α2+ ˜α3+1
+b−b/parenrightbigg
+∆ =/parenleftBig
+bN1+ ˜α1+ ˜α2/parenrightBig/parenleftbigg
+bN1+ ˜α1+ ˜α2+1
+b−b/parenrightbigg
+c= 1−6/parenleftbigg
+b−1
+b/parenrightbigg2(10)
+Clearly the rules are simple, see Fig.2:
+1) Insertion of N1screening integrals is needed to satisfy the free field conse rvation law for
+the first (left) vertex, ˜a= ˜α1+ ˜α2+bN1so that the internal dimension ∆ = ˜a/parenleftbig
+˜a+1
+b−b/parenrightbig
+becomes
+unrelated to the free field value ∆free= (˜α1+ ˜α2)((˜α1+ ˜α2+1
+b−b). Integrals in these N1screenings
+are around positions of V∆1(0)andV∆2(q).
+2) Insertion of N2screening integrals is needed to satisfy the free field conse rvation law for
+the second (right) vertex, ˜α4∼=˜a+ ˜α3+bN2.
+43) Alternatively, these additional N2screening integrals are needed to satisfy the free field
+conservation law
+4/summationdisplay
+i=1˜αi+b(N1+N2) = (g−1)/parenleftbigg1
+b−b/parenrightbigg
+(11)
+(gis the genus, g= 0for the sphere), by putting ˜α4=−˜α1−˜α2−˜α3−b(N1+N2)−1
+b+b.
+Notethat, in the free field formalism, therearetwodifferent ˜ α-parametersassociatedwith a givendimension:
+∆[˜α] = ∆[b−1
+b−˜α]. We denote this reflection in the ˜ α-space by ∼=: ˜α∼=b−1
+b−˜α.
+We derive relations (10) by expanding the both sides of the main relat ion (8) into power series
+ZDF/parenleftBig
+α1,α2,α3,N1,N2,β/vextendsingle/vextendsingle/vextendsingleq/parenrightBig
+=CDF·qdegDF·/bracketleftBigg
+1+∞/summationdisplay
+k=1qkJk/parenleftBig
+α1,α2,α3,N1,N2,β/parenrightBig/bracketrightBigg
+(12)
+B/parenleftBig
+∆1,∆2,∆3,∆4,∆,c/vextendsingle/vextendsingle/vextendsingleq/parenrightBig
+= 1+∞/summationdisplay
+k=1qkBk/parenleftBig
+∆1,∆2,∆3,∆4,∆,c/parenrightBig
+(13)
+and comparing the newly derived coefficients Jkwith the known coefficients Bk(see, for example, [18]):
+B1/parenleftBig
+∆1,∆2,∆3,∆4,∆,c/parenrightBig
+=(∆+∆ 2−∆1)(∆+∆ 3−∆4)
+2∆
+B2/parenleftBig
+∆1,∆2,∆3,∆4,∆,c/parenrightBig
+=(∆+∆ 2−∆1)(∆+∆ 2−∆1+1)(∆+∆ 3−∆4)(∆+∆ 3−∆4+1)
+4∆(2∆+1)+
++/bracketleftbig
+(∆1+∆2)(2∆+1)+∆(∆ −1)−3(∆1−∆2)2/bracketrightbig/bracketleftbig
+(∆3+∆4)(2∆+1)+∆(∆ −1)−3(∆3−∆4)2/bracketrightbig
+2(2∆+1)/parenleftBig
+2∆(8∆−5)+(2∆+1) c/parenrightBig =
+=1
+2∆/parenleftBig
+16∆2+2(c−5)∆+c/parenrightBig/bracketleftBig
+4∆(∆+2∆ 2−∆1)(∆+2∆ 3−∆4)(2∆+1)+
++(∆+∆ 2−∆1)(∆+∆ 2−∆1+1)(∆+∆ 3−∆4)(∆+∆ 3−∆4+1)(4∆+ c/2)−
+−6∆(∆+2∆ 2−∆1)(∆+∆ 3−∆4)(∆+∆ 3−∆4+1)
+−6∆(∆+∆ 2−∆1)(∆+∆ 2−∆1+1)(∆+2∆ 3−∆4)/bracketrightBig
+B3/parenleftBig
+∆1,∆2,∆3,∆4,∆,c/parenrightBig
+=1
+2∆(3∆2+c∆−7∆+2+ c)/bracketleftbigg
+(∆+3∆ 2−∆1)(∆2+3∆+2)(∆+3∆ 3−∆4)−
+−2(∆+3∆ 2−∆1)(∆+1)(∆+2∆ 3−∆4)(∆+∆ 3−∆4+2)+
++(∆+3∆ 2−∆1)(∆+∆ 3−∆4)(∆+∆ 3−∆4+1)(∆+∆ 3−∆4+2)−
+−2(∆+2∆ 2−∆1)(∆+∆ 2−∆1+2)(∆+1)(∆+3∆ 3−∆4)+
++(∆+∆ 2−∆1)(∆+∆ 2−∆1+1)(∆+∆ 2−∆1+2)(∆+3∆ 3−∆4)+
++2(∆+2∆ 2−∆1)(∆+∆ 2−∆1+2)(6∆3+9∆2−9∆+2c∆2+3c∆+c)(∆+2∆ 3−∆4)(∆+∆ 3−∆4+2)
+16∆2+2(c−5)∆+c
+−(∆+2∆ 2−∆1)(∆+∆ 2−∆1+2)(9∆2−7∆+3c∆+c)(∆+∆ 3−∆4)(∆+∆ 3−∆4+1)(∆+∆ 3−∆4+2)
+16∆2+2(c−5)∆+c
+−(∆+∆ 2−∆1)(∆+∆ 2−∆1+1)(∆+∆ 2−∆1+2)(9∆2−7∆+3c∆+c)(∆+2∆ 3−∆4)(∆+∆ 3−∆4+2)
+16∆2+2(c−5)∆+c
+5+(∆+∆ 2−∆1)(∆+∆ 2−∆1+1)(∆+∆ 2−∆1+2)(∆+∆ 3−∆4)(∆+∆ 3−∆4+1)(∆+∆ 3−∆4+2)×
+×(24∆2−26∆+11 c∆+8c+c2)
+12/parenleftBig
+16∆2+2(c−5)∆+c/parenrightBig
+
+The next section is devoted to detailed checks of the equality of coe fficients JkandBk. Here we briefly
+comment about the equality of normalization constants CDFandC/parenleftbig
+∆1,∆2,∆/parenrightbig
+C/parenleftbig
+∆,∆3,∆4/parenrightbig
+. Let us make a
+change (rescaling) of the integration variables: zi=quifori≤N1andzi=vifori > N1. After this rescaling,
+the Dotsenko-Fateev integral takes the form
+ZDF=qα1α2
+2β(1−q)α2α3
+2β1/integraldisplay
+0du1...1/integraldisplay
+0duN11/integraldisplay
+0dv1...1/integraldisplay
+0dvN2/productdisplay
+i<j(quj−qui)2β/productdisplay
+i<j(vj−vi)2β×
+N1/productdisplay
+i=1N2/productdisplay
+j=1(vj−qui)2βN1/productdisplay
+i=1(qui)α1(qui−q)α2(qui−1)α3N2/productdisplay
+j=1vα1
+i(vi−q)α2(vi−1)α3(14)
+Since the integration limits no longer depend on q, it is now easy to find the overall q-degree of the integral
+degDF=βN1(N1−1)+N1+α1N1+α2N1+α1α2
+2β(9)= ∆−∆1−∆2 (15)
+in accordance with (8).
+It is also easy to find the normalization constant:
+CDF=CN1(α1,α2)CN2(a,α3) (16)
+wherea=α1+α2+2βN1and
+CN(x,y) =N/productdisplay
+i=11/integraldisplay
+0dui/productdisplay
+i<j(uj−ui)2βN/productdisplay
+i=1ux
+i(ui−1)y=
+=N/productdisplay
+k=1Γ(x+1+β(k−1))Γ(y+1+β(k−1))Γ(1+ βk)
+Γ(x+y+2+(N+k−2)β)Γ(β+1)(17)
+is the so-called Selberg integral [1, 50, 4]. As one can see, CDFis a product of two factors, which are associated
+with the twoverticesofthe diagramanddepend onrespectivescre eningmultiplicities ( N1andN2) andincoming
+dimensions ( α1,α2andα,α3) of these vertices. As demonstrated in s.4.1 of ref.[4], CN(x,y) coincides with the
+perturbative Nekrasov function and is similar (though not literally eq ual to, see s.3.8 below) to the three-point
+function on a sphere C(∆1,∆2,∆3) also known as DOZZ three-point function [54]. Therefore, we conc lude
+that the Dotsenko-Fateev integral reproduces the conformal block modulo a constant factor. Hence we do not
+consider this factor in what follows, and concentrate on the coeffic ients of the series expansions (12) and (13).
+3 Evidence in support of (8)
+3.1 The case of β= 1and allαivanishing
+This is the simplest possible situation, where the equivalence between the Dotsenko-Fateev partition function
+and the 4-point conformal block can be seen. For the sake of brev ity, we denote X=N2
+2+ 2N1N2. Direct
+calculation gives:
+J1(0,0,0,N1,N2,β= 1) =−X
+2(18)
+J2(0,0,0,N1,N2,β= 1) =−1−7N1+12N2
+1
+4(4N2
+1−1)2X+1−3N2
+1+8N4
+1
+4(4N2
+1−1)2X2(19)
+6J3(0,0,0,N1,N2,β= 1) =−4−26N1+40N2
+1
+24(4N2
+1−1)2X+6−15N1+36N2
+1
+24(4N2
+1−1)2X2−2−N1+8N2
+1
+24(4N2
+1−1)2X3(20)
+The integral at level 4 is also easily calculated (the corresponding co nformal block is directly obtained too):
+J4(0,0,0,N1,N2,β= 1) =−3888−27540N2
+1+56292N4
+1−35040N6
+1+6720N8
+1
+384(4N2
+1−1)2(4N2
+1−9)2X+
++7101−22305N2
+1+50180N4
+1−33904N6
+1+6848N8
+1
+384(4N2
+1−1)2(4N2
+1−9)2X2−
+−3834−4794N2
+1+13632N4
+1−10656N6
+1+2304N8
+1
+384(4N2
+1−1)2(4N2
+1−9)2X3+
++621−201N2
+1+1132N4
+1−1088N6
+1+256N8
+1
+384(4N2
+1−1)2(4N2
+1−9)2X4(21)
+Comparing these expressions with the conformal block with vanishin g ∆1,∆2and ∆ 3, one finds
+Jk(0,0,0,N1,N2,β= 1) =Bk/parenleftBig
+0,0,0,(N1+N2)2,N2
+1,c= 1/parenrightBig
+, (22)
+at least for k= 1,2,3. Thus, in this particular case the Dotsenko-Fateev partition fun ction correctly reproduces
+the conformal block up to level 3, at least. This is the first evidence in support of (8). This particular relation
+(22) has been already verified up to level 2 in [5].
+3.2 The case of β= 1and non-vanishing α1
+Asαiare switched on, the computation of coefficients Jkbecomes harder. Currently, the two levels are calcu-
+lated:
+J1(α1,0,0,N1,N2,β= 1) =−2N1N2(N1+α1)(2N1+N2+α1)
+(2N1+α1)2(23)
+J2(α1,0,0,N1,N2,β= 1) =N1N2(N1+α1)(2N1+N2+α1)
+(α1+1+2N1)2(2N1+α1)2(α1−1+2N1)2×/parenleftBig
+−1+2α2
+1+N2α1+N2
+2+
++7N1α1+2N1N2+7N2
+1−α4
+1−N2α3
+1−N2
+2α2
+1−7N1α3
+1−5N1N2α2
+1−
+−3N1N2
+2α1−19N2
+1α2
+1−9N2
+1N2α1−3N2
+1N2
+2−24N3
+1α1−6N3
+1N2−
+−12N4
+1+2N1N2α4
+1+2N1N2
+2α3
+1+14N2
+1N2α3
+1+10N2
+1N2
+2α2
+1+
++36N3
+1N2α2
+1+16N3
+1N2
+2α1+40N4
+1N2α1+8N4
+1N2
+2+16N5
+1N2/parenrightBig
+(24)
+Comparing these expressions with the conformal block with vanishin g ∆2,∆3(and ∆ 1non-vanishing) one finds
+7Jk(α1,0,0,N1,N2,β= 1) =Bk/parenleftbiggα2
+1
+4,0,0,/parenleftBig
+N1+N2+α1
+2/parenrightBig2
+,/parenleftBig
+N1+α1
+2/parenrightBig2
+,c= 1/parenrightbigg
+, k= 1,2 (25)
+One can see that, in this particular case, the Dotsenko-Fateev pa rtition function reproduces correctly the
+conformal block up to level 2. This is yet another evidence in suppor t of (8).
+3.3 The case of β= 1and non-vanishing α2orα3
+Similarly, for non-vanishing α2one finds
+J1(0,α2,0,N1,N2,β= 1) =−N2(2N2
+1+2N1α2+α2
+2)(2N1+N2+α2)
+(2N1+α2)2(26)
+J2(0,α2,0,N1,N2,β= 1) =N2(2N1+N2+α2)
+2(1+α2+2N1)2(2N1+α2)2(−1+α2+2N1)2×/parenleftBig
+−α2
+2−2N1α2−2N2
+1+
++2α4
+2+N2α3
+2+N2
+2α2
+2+12N1α3
+2+4N1N2α2
+2+2N1N2
+2α2+26N2
+1α2
+2+6N2
+1N2α2+2N2
+1N2
+2+28N3
+1α2+
++4N3
+1N2+14N4
+1−α6
+2−2N2α5
+2−2N2
+2α4
+2−10N1α5
+2−14N1N2α4
+2−10N1N2
+2α3
+2−40N2
+1α4
+2−36N2
+1N2α3
+2−
+−16N2
+1N2
+2α2
+2−84N3
+1α3
+2−44N3
+1N2α2
+2−12N3
+1N2
+2α2−102N4
+1α2
+2−30N4
+1N2α2−6N4
+1N2
+2−72N5
+1α2−12N5
+1N2
+−24N6
+1+N2α7
+2+N2
+2α6
+2+10N1N2α6
+2+8N1N2
+2α5
+2+44N2
+1N2α5
+2+28N2
+1N2
+2α4
+2+112N3
+1N2α4
+2+56N3
+1N2
+2α3
+2+
++180N4
+1N2α3
+2+68N4
+1N2
+2α2
+2+184N5
+1N2α2
+2+48N5
+1N2
+2α2+112N6
+1N2α2+16N6
+1N2
+2+32N7
+1N2/parenrightBig
+(27)
+Comparing these expressions with the conformal block with vanishin g ∆1,∆3(and ∆ 2non-vanishing) one finds
+Jk(0,α2,0,N1,N2,β= 1) =Bk/parenleftbigg
+0,α2
+2
+4,0,/parenleftBig
+N1+N2+α2
+2/parenrightBig2
+,/parenleftBig
+N1+α2
+2/parenrightBig2
+,c= 1/parenrightbigg
+, k= 1,2 (28)
+For non-vanishing α3one finds
+J1(0,0,α3,N1,N2,β= 1) =−1
+2N2α3−1
+2N2
+2−1
+2N1α3−N1N2 (29)
+J2(0,0,α3,N1,N2,β= 1) =1
+4(1+2N1)2(1−2N1)2/parenleftBig
+−N2α3−N2
+2−N1α3−2N1N2+N2
+2α2
+3+2N3
+2α3+
++N4
+2+2N1N2α2
+3+6N1N2
+2α3+4N1N3
+2+11N2
+1N2α3+11N2
+1N2
+2+7N3
+1α3+14N3
+1N2−3N2
+1N2
+2α2
+3−
+−6N2
+1N3
+2α3−3N2
+1N4
+2−6N3
+1N2α2
+3−18N3
+1N2
+2α3−12N3
+1N3
+2−2N4
+1α2
+3−24N4
+1N2α3−24N4
+1N2
+2−
+−12N5
+1α3−24N5
+1N2+8N4
+1N2
+2α2
+3+16N4
+1N3
+2α3+8N4
+1N4
+2+16N5
+1N2α2
+3+48N5
+1N2
+2α3+
++32N5
+1N3
+2+8N6
+1α2
+3+32N6
+1N2α3+32N6
+1N2
+2/parenrightBig
+8Comparing these expressions with the conformal block with vanishin g ∆1,∆2(and ∆ 3non-vanishing) one finds
+Jk(0,0,α3,N1,N2,β= 1) =Bk/parenleftbigg
+0,0,α2
+3
+4,/parenleftBig
+N1+N2+α3
+2/parenrightBig2
+,N2
+1,c= 1/parenrightbigg
+, k= 1,2 (30)
+Relations (28) and (30) provide yet another evidence in support of (8).
+3.4 The case of β= 1and allαinon-vanishing
+For non-vanishing α1,α2andα3one finds
+J1(α1,α2,α3,N1,N2,β= 1) = −(2N2
+1+2N1α1+2N1α2+α2α1+α2
+2)
+2(2N1+α1+α2)2×
+×/parenleftBig
+4N1N2+2N1α3+2N2
+2+2N2α3+α3α1+2N2α1+α3α2+2N2α2/parenrightBig
+(31)
+The formula for J2(α1,α2,α3,N1,N2,β= 1) is a little lengthy, even for β= 1. For the sake of completeness it
+is presented in the separate Appendix at the end of this paper. Com paring these expressions with the conformal
+block, one finds
+Jk/parenleftBig
+α1,α2,α3,N1,N2,β= 1/parenrightBig
+=Bk/parenleftBigg
+α2
+1
+4,α2
+2
+4,α2
+3
+4,/parenleftbigg
+N1+N2+α1+α2+α3
+2/parenrightbigg2
+,/parenleftbigg
+N1+α1+α2
+2/parenrightbigg2
+, c= 1/parenrightBigg
+(32)
+at least, for k= 1,2. This relation is quite important: the equivalence between the conf ormal block and the
+Dotsenko-Fateev integral continues to hold, when the external dimensions are non-vanishing and arbitrary.
+Given this relation, one obtains the following correspondence betwe en the parameters:
+
+
+∆1=α2
+1
+4,∆2=α2
+2
+4,∆3=α2
+3
+4
+∆4=/parenleftbigg
+N1+N2+α1+α2+α3
+2/parenrightbigg2
+∆ =/parenleftbigg
+N2+α1+α2
+2/parenrightbigg2
+(β= 1)↔(c= 1)(33)
+Of course, in this particular form it is valid only for β= 1; we now proceed to the generalization to β/negationslash= 1.
+3.5 The case of arbitrary βand allαinon-vanishing
+To generalize the above formulas to an arbitrary β, it is instructive first to look at the level 1 coefficient J1(β)
+with all αinon-vanishing. The corresponding expression is still not very comp licated and, most importantly,
+factorized:
+J1(α1,α2,α3,N1,N2,β) =−2βN1−2β2N1+2N2
+1β2+2N1βα1+2α2−2βα2+2N1βα2+α2α1+α2
+2
+2β(2βN1+α1+α2)(2βN1+α1+α2−2β+2)×
+/parenleftBig
+4N1N2β2+2N1βα3−2N2β2+2β2N2
+2+2βN2α1+2N2βα3+2βN2+2N2βα2+α3α1+α3α2/parenrightBig
+(34)
+It is easy to recognize here the level 1 conformal block:
+J1(α1,α2,α3,N1,N2,β) =B1(∆1,∆2,∆3,∆4,∆,c) (35)
+9with the following relation between parameters:
+
+
+∆1=α1(α1+2−2β)
+4β
+∆2=α2(α2+2−2β)
+4β
+∆3=α3(α3+2−2β)
+4β
+∆4=(2β(N1+N2)+α1+α2+α3)(2β(N1+N2)+α1+α2+α3+2−2β)
+4β
+∆ =(2βN2+α1+α2)(2βN2+α1+α2+2−2β)
+4β(36)
+As one can see, up to level 1 the Dotsenko-Fateev partition funct ion coincides with the conformal block even
+forβ/negationslash= 1 (or, what is the same, for c/negationslash= 1). However, the precise correspondence between βandcis not seen
+from the above relations. We establish this correspondence below.
+3.6 Determination of the central charge
+The central charge cis not constrained by any of the relations above: it can not be found from level 1 consid-
+erations, since B1does not depend on c. To find c, one needs to look at
+J2(0,0,0,N1,N2,β) =N2(1+βN2+2βN1−β)
+4(3+2βN1−2β)(2N1+1)(2+2 βN1−3β)(−1+2βN1)×
+×/parenleftBig
+6−12N4
+1β3+19β2N2
+1+11β2N1−6N1β3−7β3N2
+2N2
+1−21N2
+1N2β3+16N2N5
+1β4+8N2
+2N4
+1β4+6β3N2−
+4N1N2β2+9N1N2β3+9N1N2β−46β3N2N3
+1+6N1−6N2+15N1N2
+2β2−15N1N2
+2β3+21N2
+1N2β2+
+2β4N2
+2N2
+1−13β+6β2−11βN1+40N2N4
+1β3+20β4N2N3
+1−6N1N2β4+10N2
+1N2β4+6N1N2
+2β4−6N1N2−
+6N2
+1β3+24β3N3
+1−6N2
+1β−24N3
+1β2−6N2
+2β3−19N2β2−10N2
+1N2β+20N3
+1N2β2−40N4
+1N2β4−6N2
+2β−
+6N1N2
+2β+2N2
+1N2
+2β2−16N3
+1N2
+2β4+16N3
+1N2
+2β3+19βN2+13β2N2
+2/parenrightBig
+This has to be compared with
+B2(0,0,0,∆4,∆,c) =(∆−∆4)(8∆3+∆2c+8∆2−8∆2∆4+2c∆−∆4∆c+4∆4∆−8∆+c−∆4c)
+4(16∆2−10∆+2c∆+c)(37)
+where we put
+∆4=β(N1+N2)/parenleftbigg
+N1+N2+β+1
+β/parenrightbigg
+,∆ =βN2/parenleftbigg
+N2+β+1
+β/parenrightbigg
+(38)
+After that it is easy to see that the difference J1−B1is divisible by 6 β2+βc−13β+6. Therefore,
+c(β) = 13−6β−6
+β= 1−6/parenleftbigg/radicalbig
+β−1√β/parenrightbigg2
+(39)
+This completes our check of the relation (8). Of course, much more evidence can be gathered: say, one can
+calculate the whole J2(α1,α2,α3,N1,N2,β) and compare it with B2(∆1,∆2,∆3,∆4,∆,c), and then continue to
+level3andhigher. However,the summaryofevidence aboveavailab leatthe momentisalreadyquite convincing.
+Instead of doing further more involved computer calculations, it is d esirable to find a clever theoretical proof of
+(8).
+3.7 The lesson
+The main lesson so far is that, at least, the 4-point spherical confo rmal block for arbitrary values of the five
+conformal dimensions is given by the free field correlator (7), with two new parameters N1andN2, which are
+10Figure 3:
+Figure 4:
+used to lift the two restrictions of the naive free field: on the interm ediateα-parameter and on the sum of the
+four external α-parameters. Introduction of N1andN2extends the 3-dimensional space of the naive free field
+conformal 4-point blocks and converts it into the 5-dimensional sp ace of arbitrary 4-point conformal blocks.
+The central charge is of course arbitrary. In order to describe t he entire moduli space of conformal blocks, the
+parameters N1andN2should be arbitrary, not obligatory positive integers, thus, the pr ocedure includes an
+analytical continuation in N1andN2.
+As we shall see in s.4, this result can be straightforwardly extended to spherical conformal blocks with
+arbitrary number of legs. Extension to higher genera requires mor e work, only a preliminary result for 1-point
+function on a torus will be presented in s.5 below. Extensions in two ot her directions: from the Virasoro to W
+chiral algebras (from U(2) quivers to U(N) quivers) and from generic to degenerate Verma modules should be
+straightforward, but are not considered in the present paper.
+3.8 A problem
+Despite undisputable success, there remains a problem, which still n eeds to be resolved: it concerns the issue of
+integration contours. The thing is that in our approachwe integrat epolynomials and then perform an analytical
+continuation to arbitrary values of parameters ˜ α,b,N. A polynomial can be integrated only along segments ,
+Fig.3: while integrals along the closed contours vanish because of the factors like 1 −e2πiαin
+/integraldisplay
+around a cut
+between 0 and 1[z(1−z)]α=/parenleftbig
+1−e2πiα/parenrightbig/integraldisplay1
+0[z(1−z)]α(40)
+Of course, after the analytical continuation the answer should be represented by a closed-contour integral, but
+our technique can not distinguish, for example, between the twoop tions in Fig.4. The choice in favorof the right
+picture is obvious in the case of the 4-point function, but the (corr ect) choice of Fig.5 for analytical continuation
+of the right picture in Fig.3 is somewhat less motivated. Even harder p roblems arise in the case of a torus, Fig.6,
+because there we need to integrate a holomorphic Green function, which is not periodic along the B-cycle.
+11Figure 5:
+Figure 6:
+This lack of understanding does not allow us to provide a complete des cription of the structure constants C
+at the end of s.2 (note that in [4] also only a segment integral is consid ered, and this contributes to a difference
+between CN(α1,α2) and the DOZZ function).
+Even more important, this obscures generalization to higher gener a: only the case of N2= 0 for a torus will
+be briefly considered in s.5. An additional problem for genus g >1 is a mismatch between the number 3 g−3 of
+internal dimensions in the vacuum diagram and the number 2 gof non-homological cycles. It can happen that
+the Dotsenko-Fateev integrals are taken along all the 3 g−3 non-homotopic cycles, see Fig.7, but again the issue
+of analytical continuation in the language of contours should be bet ter understood to justify this hypothesis.
+An additional difficulty here is the lack of knowledge about conformal blocks beyond genus one.
+These problems are essential and their resolution is one of the prima ry tasks in further development along
+the lines of the present paper.
+4 Multipoint functions on a sphere
+A natural generalization of relation (8) is to multipoint correlators
+/angbracketleftBig
+V∆1(z1)... V∆m(zm)/angbracketrightBig
+where ∆ i’s are the dimensions of mprimary fields on a sphere. With the help of SL(2) transformations, one
+can always put the coordinates z1,...,z mto positions 0 ,x1,x2,...,x m−3,1,∞. It is often convenient (for the
+reasons clarified below) to choose another parametrization of coo rdinates:
+x1=q1q2...qm−3, x 2=q2...qm−3, x i=m−3/productdisplay
+j=iqj, ..., x m−3=qm−3 (41)
+Accordingly, in this parametrization the m-point correlator on a sphere depends on m−3 variables qiand can
+be written as a sum over m−3 intermediate dimensions δi:
+/angbracketleftBigm−3/productdisplay
+i=1V∆i(xi)·V∆m−2(0)V∆m−1(1)V∆m(∞)/angbracketrightBig
+=/summationdisplay
+δ1...δm−3C/parenleftbig∆1,∆2,δ1/parenrightbigm−2/productdisplay
+j=1C/parenleftbigδj,∆j+2,δj+1/parenrightbigC/parenleftbigδm−3,∆m−1,∆m/parenrightbig×F(42)
+12Figure 7: An example of a genus three Riemann surface with 3 g−3 = 6 non- homotopic non-contractable contours and the
+associated Feynman-like vacuum diagram for the conformal b lock with 3 g−3 = 6 internal dimensions and no external legs.
+... V∆1(0)V∆2(x1)V∆3(x2) V∆m−2(xm−3)V∆m−1(1)
+V∆m(∞)δ1 δ2 δm−4 δm−3
+❥
+N1❥
+N2❥
+Nm−3❥
+Nm−2
+˜a1= ˜α1+ ˜α2+bN1˜a2= ˜a1+ ˜α3+bN2=
+= ˜α1+ ˜α2+ ˜α3+b(N1+N2)✂✂ ✍ ❇❇ ▼
+Figure 8: Feynman-like diagram for a comb-like conformal block. Here δj= ˜aj/parenleftbig
+˜aj+1
+b−b/parenrightbig
+and ∆ j= ˜αj/parenleftbig
+˜αj+1
+b−b/parenrightbig
+are the
+m−3 internal and mexternal dimensions, respectively, and xi=/producttextm−3
+j=iqj. The role of the screenings is to modify the free field
+selection rule at the vertices: instead of ˜ aj= ˜aj−1+ ˜αj+1one has ˜ aj= ˜aj−1+ ˜αj+1+bNj.
+with a shorthand notation
+F=m−3/productdisplay
+i=1qδi−∆1−...−∆i+1
+i B/parenleftbig
+∆1,...,∆m,δ1,...,δ m−3, c/vextendsingle/vextendsingleq1,...,q m−3/parenrightbig
+×
+×m−3/productdisplay
+i=1¯q¯δi−¯∆1−...−¯∆i+1
+i B/parenleftbig¯∆1,...,¯∆m,¯δ1,...,¯δm−3, c/vextendsingle/vextendsingle¯q1,...,¯qm−3/parenrightbig
+(43)
+As before, C’s are the 3-point functions (which do not depend on qand play the role of normalization constants
+forB(0,...,0) = 1),cis the central charge and B(q) is the m-point conformal block on a sphere. The order of
+contractions of the 3-point functions in eq. (42) is most convenien tly represented by a comb-like diagram Fig.8.
+In analogy with the 4-point conformal block, the functions B/parenleftbig
+∆1,...,∆m,δ1,...,δ m−3, c/vextendsingle/vextendsingleq1,...,q m−3/parenrightbig
+possess explicit series representations in positivepowers of q1,...,q m−3, which are described in [44]. This is
+actually the reason to use the parametrization (41): in terms of th e variables x1,...,x m−3, the series would
+contain negative powers of variables, which is less convenient. In th is paper we are interested more in integral
+representations for these conformal blocks, provided by the mu lti-point Dotsenko-Fateev partition functions
+ZDF=/angbracketleftBiggm−2/productdisplay
+a=0:e˜αa+1φ(xa):m−2/productdisplay
+a=1/parenleftbigg/integraldisplayxa
+0:ebφ(z):dz/parenrightbiggNa/angbracketrightBigg
+=
+=/productdisplay
+a<b(xb−xa)αaαb
+2βm−2/productdisplay
+a=1Na/productdisplay
+i=1xa/integraldisplay
+0dzN1+...+Na−1+i/productdisplay
+i<j(zj−zi)β/productdisplay
+im−2/productdisplay
+a=0(zi−xa)αa+1(44)
+13whereβ=b2,αa= 2b˜αaandq0=x0= 0,qm−2=xm−2= 1. It is natural to propose the following integral
+representation of spherical conformal blocks, a multipoint count erpart of the relation (8):
+ZDF/parenleftBig
+α1,...,α m−1,N1,...,N m−2,β/vextendsingle/vextendsingle/vextendsingleq1,...,q m−3/parenrightBig
+=CDF·m−3/productdisplay
+i=1qdegi
+i·B/parenleftbig
+∆1,...,∆m,δ1,...,δ m−3, c/vextendsingle/vextendsingleq1,...,q m−3/parenrightbig
+(45)
+where degi=δi−∆1−...−∆i+1. Again, CDFis the Dotsenko-Fateev normalization constant, which does
+not depend on q(but depends on all the other parameters) and is a product of m−2 factors
+CDF=CN1(α1,α2)CN2(δ1,α3)...CNm−2(δm−3,αm−1) (46)
+which are associated with the m−2 vertices of the diagram and depend on respective screening multip licities
+(N1,...,N m−2) and incoming dimensions of these vertices. The relation between th e 2m−2 Dotsenko-Fateev
+parameters α1,...,α m−1,N1,...,N m−2,βand the 2 m−2 parameters of the conformal block ∆ 1,...,∆m,
+δ1,...,δ m−3,cis
+
+
+∆i=αi(αi+2−2β)
+4β, i= 1,...,m−1
+∆m=/parenleftbig
+α1+...+αm−1+2β(N1+...+Nm−2)/parenrightbig/parenleftbig
+α1+...+αm−1+2β(N1+...+Nm−2)+2−2β/parenrightbig
+4β
+δi=/parenleftbig
+α1+...+αi+1+2β(N1+...+Ni)/parenrightbig/parenleftbig
+α1+...+αi+1+2β(N1+...+Ni)+2−2β/parenrightbig
+4β
+c= 1−6/parenleftbigg√β−1√β/parenrightbigg2
+(47)
+or, in terms of free field variables,
+
+
+∆i= ˜αi/parenleftbigg
+˜αi+1
+b−b/parenrightbigg
+, i= 1,...,m−1
+∆m=/parenleftBig
+˜α1+...+ ˜αm−1+b(N1+...+Nm−2)/parenrightBig/parenleftbigg
+˜α1+...+ ˜αm−1+b(N1+...+Nm−2)+b−1
+b/parenrightbigg
+δi=/parenleftBig
+˜α1+...+ ˜αi+1+b(N1+...+Ni)/parenrightBig/parenleftbigg
+˜α1+...+ ˜αi+1+b(N1+...+Ni)+b−1
+b/parenrightbigg
+c= 1−6/parenleftbigg
+b−1
+b/parenrightbigg2(48)
+The rules, which clearly stand behind these relations, are just the s ame as in the 4-point case, see Fig.8:
+1) Insertion of the first m−3screening integrals with multiplicities N1,...,N m−3is needed
+to satisfy the free field conservation law for the vertices, ˜ai= ˜a1+...+ ˜ai+1+bN1+...+bNi
+so that the internal dimensions δi= ˜γi/parenleftbig
+˜γi+1
+b−b/parenrightbig
+becomes unrelated to the free field values
+∆free= (˜α1+...+ ˜αi+1)/parenleftbig
+˜α1+...+ ˜αi+1+1
+b−b/parenrightbig
+. TheNiintegrals in these screenings are around
+positions of V∆1(0)andV∆i+1(xi).
+2) Insertion of the last Nm−2screening integrals is needed to satisfy the free field conse rvation
+law for the most right vertex ˜αm∼=˜am−3+ ˜αm−1+bNm−2.
+3) Alternatively, these additional Nm−2screening integrals are needed to satisfy the free field
+conservation law
+m/summationdisplay
+i=1˜αi+bm−2/summationdisplay
+i=1Ni= (g−1)/parenleftbigg1
+b−b/parenrightbigg
+(49)
+(gis the genus, g= 0for the sphere), by putting ˜αm=−˜α1−...−˜αm−1−b(N1+...+Nm−3)−1
+b+b.
+14These rules naturally follow from the general logic of conformal field theory, supplied by convincing evidence
+from direct calculations in the 4-point sector. Of course, for m >4 they still need to be carefully checked and/or
+rigorously proved. Here we consider just the simplest check of (45 ) for the 5-point Dotsenko-Fateev integral:
+ZDF=qα1(α2+α3)
+2β
+1 qα1α2
+2β
+2(1−q1)α3α4
+2β(1−q1q2)α2α4
+2β(1−q2)α2α3
+2β×
+×N1/productdisplay
+i=1q1q2/integraldisplay
+0dziN2/productdisplay
+i=1q2/integraldisplay
+0dzN1+iN3/productdisplay
+i=11/integraldisplay
+0dzN1+N2+i/productdisplay
+i<j(zj−zi)2β/productdisplay
+izα1
+i(zi−q1q2)α2(zi−q1)α3(zi−1)α4(50)
+The check of relation (45) for arbitrary values of α1,...,α 4is straightforward, but tedious. To provide simple
+and clear evidence, we perform such a check in the particular case o f vanishing αiand forβ= 1. In this case,
+the power series expansion of the Dotsenko-Fateev partition fun ction has the form
+ZDF=q(N1+N2)2
+1 qN2
+2
+2/bracketleftBigg
+1+/summationdisplay
+k1+k2>0Jk1,k2/parenleftBig
+N1,N2,N3/parenrightBig
+qk1
+1qk2
+2/bracketrightBigg
+(51)
+We find at level 1
+J1,0/parenleftBig
+N1,N2,N3/parenrightBig
+=−N1N3(N1+2N2)(2N1+2N2+N3)
+2(N1+N2)2, J0,1/parenleftBig
+N1,N2,N3/parenrightBig
+=−N1(N1+2N2)
+2(52)
+and at level 2
+J2,0/parenleftBig
+N1,N2,N3/parenrightBig
+=N1N3(2N1+2N2+N3)(N1+2N2)
+(2N1+2N2−1)2(2N1+2N2)2(2N1+2N2+1)2×
+×(−1+16N3N5
+1+2N3N1−3N2
+3N2
+1−6N3N3
+1+8N2
+3N4
+1−8N3
+2N3−4N2
+2N2
+3+2N2N3−56N1N3
+2−48N3
+1N2−
+76N2
+1N2
+2+14N1N2+32N4
+2N3N1+16N3
+2N2
+3N1+112N3
+2N3N2
+1+40N2
+2N2
+3N2
+1+144N2
+2N3N3
+1−20N2
+2N3N1−
+18N2N3N2
+1−6N2N2
+3N1+32N2N2
+3N3
+1+80N2N3N4
+1+8N2
+2+7N2
+1+N2
+3−12N4
+1−16N4
+2)
+J1,1/parenleftBig
+N1,N2,N3/parenrightBig
+=N3(2N1+2N2+N3)(−N2
+1−2N2
+2−2N1N2+N4
+1+4N3
+1N2+4N2
+1N2
+2)
+(2N1+2N2)2(53)
+J0,2/parenleftBig
+N1,N2,N3/parenrightBig
+=N1(N1+2N2)(N2
+1−12N4
+2+7N2
+2−1+8N2
+1N4
+2+16N1N5
+2−3N2
+1N2
+2−6N1N3
+2+2N1N2)
+4(2N2−1)2(2N2+1)2(54)
+These expressions need to be compared with the 5-point conforma l block
+B/parenleftBig
+∆1,∆2,∆3,∆4,∆5,δ1,δ2,c|q1,q2/parenrightBig
+= 1+/summationdisplay
+k1+k2>0Bk1,k2/parenleftBig
+∆1,∆2,∆3,∆4,∆5,δ1,δ2,c/parenrightBig
+qk1
+1qk2
+2(55)
+which was calculated explicitly in [44] at level 1:
+B10/parenleftBig
+∆1,∆2,∆3,∆4,∆5,δ1,δ2,c/parenrightBig
+=(δ1+∆5−δ2)(δ1+∆4−∆1)
+2δ1(56)
+B01/parenleftBig
+∆1,∆2,∆3,∆4,∆5,δ1,δ2,c/parenrightBig
+=(δ2+∆5−δ1)(δ2+∆2−∆3)
+2δ2(57)
+and at level 2:
+B20/parenleftBig
+∆1,∆2,∆3,∆4,∆5,δ1,δ2,c/parenrightBig
+=2(δ1+2∆5−δ2)(δ1+2∆4−∆1)(2δ1+1)
+16δ2
+1−10δ1+2cδ1+c−
+−3(δ1+2∆5−δ2)(δ1+∆4−∆1)(δ1+∆4−∆1+1)
+16δ2
+1−10δ1+2cδ1+c−
+−3(δ2−∆5−δ1)(δ2−∆5−δ1−1)(δ1+2∆4−∆1)
+16δ2
+1−10δ1+2cδ1+c+
++(δ2−∆5−δ1)(δ2−∆5−δ1−1)(δ1+∆4−∆1)(δ1+∆4−∆1+1)(8δ1+c)
+4δ1(16δ2
+1−10δ1+2cδ1+c)(58)
+15B11/parenleftBig
+∆1,∆2,∆3,∆4,∆5,δ1,δ2,c/parenrightBig
+=((δ1+∆5−δ2)(δ2+∆5−δ1−1)+2δ1)(δ1+∆4−∆1)(δ2+∆2−∆3)
+4δ1δ2(59)
+B02/parenleftBig
+∆1,∆2,∆3,∆4,∆5,δ1,δ2,c/parenrightBig
+=2(δ2+2∆5−δ1)(2δ2+1)(δ2+2∆2−∆3)
+16δ2
+2−10δ2+2cδ2+c−
+−3(δ2+2∆5−δ1)(δ2+∆2−∆3)(δ2+∆2−∆3+1)
+16δ2
+2−10δ2+2cδ2+c−
+−3(δ2+∆5−δ1)(δ2+∆5−δ1+1)(δ2+2∆2−∆3)
+16δ2
+2−10δ2+2cδ2+c+
++(δ2+∆5−δ1)(δ2+∆5−δ1+1)(8δ2+c)(δ2+∆2−∆3)(δ2+∆2−∆3+1)
+4δ2(16δ2
+2−10δ2+2cδ2+c)(60)
+Comparing these expressions, one finds
+Jk1,k2/parenleftBig
+N1,N2,N3/parenrightBig
+=Bk1,k2/parenleftBig
+0,0,0,0,(N1+N2+N3)2,(N1+N2)2,(N2)2,c= 1/parenrightBig
+(61)
+at least, for ( k1,k2) = (1,0),(0,1),(2,0),(1,1),(0,2). This relation can be considered as an evidence that
+Dotsenko-Fateev integrals continue to provide a relevant descrip tion of spherical (genus zero) conformal blocks
+beyond four primaries.
+5 Dotsenko-Fateev integral on a torus
+Since the Dotsenko-Fateev integrals (7) and (44) are made from t he correlators of free fields, one can easily
+consider their generalizations on higher genus Riemann surfaces. T here is no an evident way to associate these
+integrals with the corresponding conformal blocks, and we shall no w see that, indeed, the most naive attempt
+leadstoanexpressionwhichissurprisinglysimilar, but stilldifferent. T husgeneralizationoftheAGTconjecture
+in this direction still remains to be found. This short subsection only d escribes the setting.
+If a Riemann sphere is substituted by a torus, then at the r.h.s. of ( 7) and (44) the role of the holomorphic
+Green functions is played by the odd theta-functions
+θ∗(x)∼sinx
+2−qsin3x
+2+q3sin5x
+2−q6sin7x
+2+...=∞/summationdisplay
+n=0(−1)nqn(n+1)/2sin(2n+1)x
+2(62)
+and the typical Dotsenko-Fateev integral looks like
+ZDF(α,β,N,q ) =2π/integraldisplay
+0dz1...2π/integraldisplay
+0dzN/productdisplay
+i<jθ∗(zi−zj)2β/productdisplay
+iθ∗(zi)α= const·(1+J1q+J2q2+...) (63)
+There are many questions to ask about the r.h.s. of this formula, to mention just a few: the integrand is not
+double periodic; the conservation law ˜ α+Nb= 0 is not satisfied (and there is no infinitely remote point to hide
+the compensating insertion at); the integral is taken only along the A-cycle; it is unclear if the argument of [48],
+allowing to neglect the second set of screening charges, is applicable and so on. Not to repeat that there are
+no basic reasons to identify this integral or any of its modifications w ith the toric conformal block, except for a
+certain similarity to the AGT conjecture in the form of (8) and (45).
+For good or for bad, the coefficients at the r.h.s. of (63) can be str aightforwardly calculated. To do this,
+notice that the Green function can be represented as a product
+θ∗(x)∼sinx
+2/braceleftBig
+1+/parenleftBig
+1−4cos2x
+2/parenrightBig
+q+/parenleftBig
+1−12cos2x
+2+16cos4x
+2/parenrightBig
+q2+.../bracerightBig
+(64)
+16Using this formula, it is easy to find
+J1=αN/summationdisplay
+i=1/angbracketleftBig
+1−4cos2zi
+2/angbracketrightBig
++2β/summationdisplay
+i<j/angbracketleftBig
+1−4cos2zi−zj
+2/angbracketrightBig
+(65)
+where the Dotsenko-Fateev correlators are defined as
+/angbracketleftBig
+f(z1,...,z N)/angbracketrightBig
+=2π/integraltext
+0dz1...2π/integraltext
+0dzNf(z1,...,z N)/producttext
+i<j/parenleftbigg
+sinzi−zj
+2/parenrightbigg2β/producttext
+i/parenleftBig
+sinzi
+2/parenrightBigα
+2π/integraltext
+0dz1...2π/integraltext
+0dzN/producttext
+i<j/parenleftbigg
+sinzi−zj
+2/parenrightbigg2β/producttext
+i/parenleftBig
+sinzi
+2/parenrightBigα(66)
+Direct calculations give
+/angbracketleftBig
+cos2zi
+2/angbracketrightBig
+=1+(N−1)β
+2((N−1)β+α/2+1)(67)
+and
+/angbracketleftBig
+cos2zi−zj
+2/angbracketrightBig
+=(N−1)(N−2)β2+β/parenleftBig
+(N−2)α+(2N−3)/parenrightBig
++(1+α+α2/2)
+2/parenleftBig
+(N−1)β+α/2+1/parenrightBig2(68)
+Note that the r.h.s. does not depend on i,jbecause of the permutation symmetry. Consequently,
+J1=αN/angbracketleftBig
+1−4cos(z1)2/angbracketrightBig
++βN(N−1)/angbracketleftBig
+1−4cos(z1−z2)2/angbracketrightBig
+= (69)
+=−Nβ(N−1)(βN−β+1)(βN−3β+1)+N(3βN−3β+2β2N2+1+4β2−6β2N)α+3
+4Nβ(N−1)α2−1
+4Nα3
+/parenleftbig
+1−β+α
+2+βN/parenrightbig2
+The integral ZDFneeds to be compared with the toric 1-point function
+B= 1+B1q+B2q2+... (70)
+where (see, e.g., [35, 38]),
+B1=∆2
+ext−∆ext+2∆
+∆(71)
+Since ∆ and ∆ extare given by formulas like (10)
+∆ =a(a+2−2β)
+4β,∆ext=α(α+2−2β)
+4β(72)
+which are not full squares, it is clear already from a look at the denom inators that (69) does not match (70).
+At the same time the difference is not quite as drastic as it could be. It looks like the most severe discrepancy
+between ZDFand the toric conformal block is just in the number of free paramet ers. In the Dotsenko-Fateev
+integral, αis actually fixed by the free field conservation law
+α+2βN= 0 (73)
+The most straightforward way to relax this restriction is to introdu ceN2additional integrals over the B-cycle.
+In terms of variables zi, this would result in the following modification of the Dotsenko-Fatee v integral:
+ZDF(α,β,N 1,N2,q) =2π/integraldisplay
+0dz1...2π/integraldisplay
+0dzN1/integraldisplay
+BdzN1+1.../integraldisplay
+BdzN1+N2/productdisplay
+i<jθ∗(zi−zj)2β/productdisplay
+iθ∗(zi)α(74)
+and the new conservation law would take form
+α+2β(N1+N2) = 0 (75)
+17There is also a selection rule which states
+a+α+2βN1= 2β−2−a (76)
+in accordance with
+∆[a] =a(a+2−2β)
+4β= ∆[2β−2−α] (77)
+Thus
+a= (N2+1)β−1 (78)
+The exact formula (69) for J1is sufficient to describe only the case when N1=NandN2= 0. In this
+situation, one gets from (78) and (75)
+a=β−1, α=−2βN (79)
+so that ∆ and ∆ extare equal to
+∆ =−(β−1)2
+4β,∆ext=N(βN+β−1) (80)
+Substituting these expressions into B1andJ1, one obtains
+B1=2β3N4−4β2N3+4β3N3−6β2N2+2β3N2−2β2N+2βN2+2βN−2β+β2+1
+(β−1)2(81)
+and
+J1=−Nβ(N+1)(2β2N2−4βN+2β2N−1+4β−3β2)
+(β−1)2=B1+3∆ext−1+3N (82)
+As one can see, the discrepancy between these two quantities is qu ite moderate. In principle, the extra terms
+3∆ext−1+3Ncan be easily absorbed into U(1) factors, but it does not seem natural to introduce U(1) factors
+which depend explicitly not only on dimensions, but also on N. A straightforward solution may be to divide
+each screening integral by (1 −q)3or byq−1/8η(q)3, where
+η(q) =q1/24∞/productdisplay
+n=1(1−qn) (83)
+is the Dedekind eta-function. However, it remains to be checked if s uch a prescription can correctly reproduce
+the toric conformal block at level 2 and at higher levels. The problem clearly deserves further investigation. As
+already explained in s.3.8, an even bigger problem is to adequately switc h onN2/negationslash= 0.
+6 Conclusion
+In this paper, we provide some further evidence in support of the m odified AGT conjecture [1, 5], identifying
+conformal blocks with matrix integrals in the DV phase, and making no any direct reference to the Nekrasov
+functions. This evidence, together with the earlier results in [2, 3, 4 , 45], seems to be sufficient to establish the
+relation at the spherical level. As a byproduct, we found explicit for mulas for the analytical continuation of
+the Dotsenko-Fateev integrals, which appear parallel to the form ulas [13] for the CIV superpotential. However,
+nothing like a straightforward proof of the modified AGT conjectur e is yet available, and it can not yet be used
+to prove neither the original AGT hypothesis [15] along the lines of [1, 5], nor its BS/SW version [9, 34, 37].
+Moreover, generalization to higher genera also runs into certain pr oblems, briefly described in the.s.3.8 above.
+The future proof should clarify these subtle points and establish a d irect relation between the decomposition
+formulas [55] for matrix model partition functions and the conform al block expansions [51, 15, 18] into the triple
+vertices and the inverse Shapovalov matrices.
+18Appendix. Explicit expression for J2(α1,α2,α3,N1,N2,β= 1)
+J2(α1,α2,α3,N1,N2,β= 1) =1
+(α1+α2+2N1+1)2(2N1+α1+α2)2(α1+α2+2N1−1)2×
+/parenleftBig
+−2α3
+2α3−4α1α2
+2α3−2α2
+1α2α3−4N2α2
+2α3−4N2α3
+2−4N2α1α2α3−8N2α1α2
+2−4N2α2
+1α2−4N2
+2α2
+2−4N2
+2α1α2−
+8N1α2
+2α3−12N1α1α2α3−4N1α2
+1α3−8N1N2α2α3−16N1N2α2
+2−8N1N2α1α3−24N1N2α1α2−8N1N2α2
+1−
+8N1N2
+2α2−8N1N2
+2α1−12N2
+1α2α3−12N2
+1α1α3−8N2
+1N2α3−24N2
+1N2α2−24N2
+1N2α1−8N2
+1N2
+2−8N3
+1α3−
+16N3
+1N2+α4
+2α2
+3+4α5
+2α3+2α1α3
+2α2
+3+16α1α4
+2α3+α2
+1α2
+2α2
+3+24α2
+1α3
+2α3+16α3
+1α2
+2α3+4α4
+1α2α3+4N2α3
+2α2
+3+
+12N2α4
+2α3+8N2α5
+2+8N2α1α2
+2α2
+3+36N2α1α3
+2α3+32N2α1α4
+2+4N2α2
+1α2α2
+3+36N2α2
+1α2
+2α3+48N2α2
+1α3
+2+
+12N2α3
+1α2α3+32N2α3
+1α2
+2+8N2α4
+1α2+4N2
+2α2
+2α2
+3+12N2
+2α3
+2α3+12N2
+2α4
+2+4N2
+2α1α2α2
+3+24N2
+2α1α2
+2α3+
+36N2
+2α1α3
+2+12N2
+2α2
+1α2α3+36N2
+2α2
+1α2
+2+12N2
+2α3
+1α2+8N3
+2α2
+2α3+8N3
+2α3
+2+8N3
+2α1α2α3+16N3
+2α1α2
+2+8N3
+2α2
+1α2+
+4N4
+2α2
+2+4N4
+2α1α2+4N1α3
+2α2
+3+32N1α4
+2α3+4N1α1α2
+2α2
+3+100N1α1α3
+2α3+112N1α2
+1α2
+2α3+52N1α3
+1α2α3+
+8N1α4
+1α3+16N1N2α2
+2α2
+3+72N1N2α3
+2α3+64N1N2α4
+2+24N1N2α1α2α2
+3+160N1N2α1α2
+2α3+200N1N2α1α3
+2+
+8N1N2α2
+1α2
+3+112N1N2α2
+1α2α3+224N1N2α2
+1α2
+2+24N1N2α3
+1α3+104N1N2α3
+1α2+16N1N2α4
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+1024N7
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+1N2
+2/parenrightBig
+Acknowledgements
+A.Morozov is indebted for the hospitality and support to the Univers ity of Tours, where part of this work was
+done.
+Our work is partly supported by Russian Federal Nuclear Energy Ag ency, Russian Federal Agency for
+Science and Innovations under contract 02.740.11.5029,by RFBR g rants 07-02-00878(A.Mir.), and 07-02-00645
+(A.Mor. & Sh.Sh.), by joint grants 09-02-90493-Ukr, 09-02-9310 5-CNRSL, 09-01-92440-CE, 09-02-91005-ANF
+andbyRussianPresidentsGrantofSupportfortheScientificScho olsNSh-3035.2008.2. TheworkofSh.Shakirov
+is also supported in part by Moebius Contest Foundation for Young S cientists.
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+arXiv:1001.0564v1  [astro-ph.IM]  4 Jan 2010Title : will be set by the publisher
+Editors : will be set by the publisher
+EAS Publications Series, Vol. ?, 2018
+A DIFFERENT GLANCE TO THE SITE TESTING ABOVE
+DOME C
+Elena Masciadri1, Franck Lascaux1, Jeffrey Stoesz1, Susanna Hagelin1
+and Kerstin Geissler2
+Abstract. Due to the recent interest shown by astronomers towards the
+Antarctic Plateau as a potential site for large astronomica l facilities,
+we assisted in the last years to a strengthening of site testi ng activities
+in this region, particularly at Dome C. Most of the results co llected
+so far concern meteorologic parameters and optical turbule nce mea-
+surements based on different principles using different inst ruments. At
+present we have several elements indicating that, above the first 20-30
+meters, the quality of the optical turbulence above Dome C is better
+than above whatever other site in the world. The challenging question,
+crucial to know which kind of facilities to build on, is to est ablish how
+much better the Dome C is than a mid-latitude site. In this con tri-
+bution we will provide some complementary elements and stra tegies of
+analysis aiming to answer to this question. We will try to con centrate
+the attention on critical points, i.e. open questions that s till require
+explanation/attention.
+1 Introduction
+Most of the site testing done so far above Dome C has been carried o ut with mea-
+surements taken with different instruments and based on different techniques. Our
+approach to the site testing in this context is different. We intend to estimate the
+optical turbulence ( C2
+N) with the employment of atmospherical models at meso-
+scale, more precisely we plan to use the Meso-Nh model (Masciadri & Jabouille
+2001, Masciadri, Avila & Sanchez 2004 , Masciadri & Egner 2006). Th is approach
+provides the possibility to reconstruct 3D C2
+Nmaps (not only vertical profiles)
+and to forecast the optical turbulence. The flexible scheduling and optimization
+of scientific observational programs for new class of ground-bas ed telescopes pass
+1INAF-Osservatorio Astrofisico di Arcetri, Firenze, Italy; e-mail: masciadri@arcetri.astro.it
+2European Southern Observatory, Alonso de Cordova 3107, San tiago, Chile
+c/circlecopyrtEDP Sciences 2018
+DOI: (will be inserted later)2 Title : will be set by the publisher
+necessarily by the forecast of the optical turbulence; this topic is therefore strictly
+correlated to the success and the survival of the ground-based astronomy of next
+decades. We recently set-up a research group at the Osservato rio Astrofisico di
+Arcetri (Florence)1whose aim is to carry out a project (FOROT) centered on the
+characterization and forecast of the Optical Turbulence. The co re project includes
+the study of two sites: the Mt. Graham, site of the Large Binocular Telescope
+(LBT) and the Internal Antarctic Plateau (Dome C, South Pole and Dome A).
+The final goal of our work is to employ this technique to other sites in the world
+mainly for application to the ELTs. The orographicmaps ofthe Inter nal Antarctic
+Plateau, as reconstructed by the Meso-Nh model, is shown in Fig.1. Simulations
+will be carried out with different horizontal resolution depending on s ite and sci-
+entific goals that we intend to reach.
+Why Antarctica ?
+•Because the whole International Community showed great interes t with re-
+spect to this site.
+•Because several site testing campaigns are on-going and this mean s that a
+large number of measurements will be soon available. This means a pos si-
+bility to constrain models outputs.
+•Because we will have the possibility to answer to critical scientific que stions
+among those: (1) ability of meso-scale models in discriminating betwee n
+sites located on the same plateau but characterized by different tu rbulence
+features, (2) characterizationof an uncontaminated site such a s Dome A, (3)
+ability of meso-scale models in forecasting the optical turbulence.
+2 Challenges at Dome C
+We can summarize the promising potentialities of Dome C for future As tronomy
+in this Antarctic Continent with the following statement: at Dome C, above
+20-30 m the quality of the atmosphere is better than above wha tever
+other site at mid-latitude. What does it mean ? Above a good mid-latitude
+site, the free atmosphere seeing εFA(by definition for h≥1 km) is typically
+larger than 0.4 arcsec. Above Dome C, the free atmosphere seeing is≤0.4 arcsec
+and the thickness of the boundary layer is as thin as a few tens of me ters (∼30
+m). An exact estimate of such a thickness is obviously important but we would
+like to highlight that it has been shown (Ragazzoni et al. 2004) that, using 2 DMs
+opportunely conjugated in the low part of the atmosphere, it is pos sible to correct
+the first 200 m reaching 15 arcmin FOV with reasonable good Strehl R atio. This
+already does of Dome C an extremely interesting site. The residual t urbulence in
+the free atmosphere, that is not corrected with the system desc ribed by Ragazzoni
+1http://forot.arcetri.astro.itGive a shorter title using \runningtitle 3
+Fig. 1.Orographic map centred above South Pole and extended on 3000 km×3000 km.
+et al. (2004), is critical to assure the predicted performances. F or this reason we
+think that the main priority in a list of studies to be carried out, is to co nfirm the
+upper limit for εFAmeasured so far. As we will explain in the next section, these
+results need to be confirmed.
+3 The different “glance”
+Fundamental requirement for our study is to collect as many as pos sible elements
+(included measurements) to constrain simulations. Here we list a few critical
+points telling us that we still need further studies in the fields of site t esting and
+a different strategic approaches in the organization of site testing campaigns.
+(1)The vertical profilers: MASS, SSS and Balloons never ran contempo ra-
+neously. We can not, at present, know the dispersion between inst ruments. (2)
+Some instruments, such as the SSS, are prototypes and need car eful validation.
+A comparison made on statistical base between measurements fro m SSS and an-
+other profiler taken as a reference need to be done. Results publis hed so far only
+concern the comparison between one C2
+Nprofile from SSS and balloon. (3)Mea-
+surements show discrepancies sometimes. In Agabi et al. (2006), for example, the
+median isoplanatic angle measured by the DIMM and the Balloons is resp ectively
+2.7 and 4.7 arcsec. This corresponds to a discrepancy of ∼50%. What we suspect
+is that most of the balloons measurements reach the height of 12 km and above
+this height the turbulence is not measured due to balloons explosion. As a conse-4 Title : will be set by the publisher
+quence, the isoplanatic angle measured by balloons is larger than the isoplanatic
+angle measured by a DIMM.
+To supply further elements that might comply with our needs we decid ed to
+followadifferentapproach: tostudytheanalysesprovidedbytheE uropeanCenter
+for Medium Weather Forecasts (ECMWF) General Circulation Models . These are
+vertical profiles of absolute and potential temperature, wind spe ed and direction
+extracted in the grid point (75◦S, 123◦E), horizontal resolution 0.5◦, extended
+up to 0.1 hPa, 60 vertical levels with resolution decreasing with height . Data
+have been treated statistically on the 2003-2004 period at heights h≥30 m. The
+scientific goals of this study are: (1) to provide a yearly meteorolog ic parameters
+analysis, (2) to calculate the probability to trigger optical turbulen ce in different
+slabs of atmosphere and in different periods of the year and (3) to e stimate the
+quality of ECMWF analyses in perspective to use these data to initialise Meso-Nh.
+Analyses reliability has been proved in Geissler & Masciadri (2006). In that
+paper the main results of this study are extensively described. Figu re 2 shows
+the median wind speed in summer and winter time above 2 sites: Dome C a nd
+San Pedro M´ artir. The latter has been selected as representativ e of a mid-latitude
+site. It is possible to observe that above Dome C the wind speed is par ticularly
+weak in summer time on the whole troposphere; a peak appears at ar ound 5 km
+above the ground (tropopause). In winter time the wind speed incr eases mono-
+tonically above 10 km above the ground reaching 30 m/sec above 20 k m. Such
+a strong wind speed should be considered with attention for at least 2 reasons.
+Firstly it might trigger dynamic instabilities in the high free atmosphere , secondly
+it might affect the wavefront coherence time τ0. This parameter indeed scales as
+C2
+N(h)·|V(h)|5/3, i.e. is particularly sensitive to large wind speed values.
+Fig.3 shows the median wind speed vertical profile in winter time (June a nd July)
+above South Pole and the median wind speed vertical profile in winter t ime above
+Dome C. Both results are retrieved by statistical estimates of ECM WF analyses.
+It is interesting to note that the wind speed at high altitudes is much w eaker at
+South Pole than at Dome C.
+How we can connect meteorological and astroclimatic parameters ? The study of
+the Richardson number Ri=Ri(∂θ/∂h,∂V/∂h), parameter indicating the proba-
+bility to triggerturbulence, shows(Geissler& Masciadri2006,Fig.14 )amonotonic
+increasingof the turbulence activity above 15km from summer towa rdsand across
+the winter time. One would expect therefore a more intense turbule nce activity at
+high altitudes during the winter than during the summer time. The tre nd iden-
+tified by the Richardson maps is confirmed by measurements of θ0obtained with
+the same instrument in summer and winter time. θ0is much larger in summer
+time (6.8 arcsec - Aristidi et al. (2005)) than in winter time (2.7 arcse c - Agabi
+et al. (2006)). This means that turbulence at high altitudes is weake r in summer
+than in winter time. We should therefore increase the statistic of me asurements
+in August, September and October to better characterize this ph enomenon.
+A further result related to the seeing in the free atmosphere ( εFA) requiringGive a shorter title using \runningtitle 5
+Fig. 2.Comparison between the median wind speed profile estimated d uring the winter
+time (full bold line) and summer time (dashed line line) (200 3) above Dome C and the
+median wind speed profile estimated above the San Pedro M´ art ir Observatory in summer
+(dotted line) and winter (full thin line) time. San Pedro M´ a rtir is taken as representative
+of a mid-latitude site.
+Fig. 3.ECMWF analyeses estimates. Left: median wind speed vertica l profile in winter
+time (June and July) above South Pole (dotted line). Right: m edian wind speed vertical
+profile in winter time above Dome C (dotted lines are first and t hird quartiles). See
+Geissler & Masciadri (2006) for further details.
+attention and careful verification is the following. We can retrieve f rom the Fig.5
+(Agabi et al. 2006) that seeing measured by a DIMM placed at 20 m ab ove the
+ground grows in a monotonic way reaching values greater than 1 arc sec in August.
+Where this turbulence come from if the seeing above 30 m has been es timated not
+greater than 0.4 arcsec during the winter time ? One could think that this is the
+contribution associated to the thin layer between 20 and 30 m. If we look at the
+typicalC2
+Nvalues at 20-30 meters (10−14≤C2
+N≤5×10−14) (Fig.3, Agabi et
+al. 2006), we calculate the correspondent seeing in the 20-30 m slab and we finally6 Title : will be set by the publisher
+subtract2the result from the seeing reported in Fig.5, we find a seeing in the ran ge
+0.45 - 0.92 arcsec (at h ≥30 m). We therefore obtain a seeing above 30 m that
+is greater than the 0.4 arcsec indicated by preliminary results. This e xhorts us in
+pushing on further studies of seeing in this region of the atmospher e.
+4 Conclusion
+We summarize the main points discussed in this contribution: (1)We indicated
+an alternative method (ECMWF analyses) to check measurements, to constrain
+simulations and characterize climatology above Dome C. (2)Further optical tur-
+bulence measurements are necessary to carry out all scientific pr ograms related
+to site testing. However it would be suitable that future site testing campaigns
+are planned conjointly with all teams working on site testing taking int o account
+requirements of all scientific programs. This is the only way to maximiz e the
+scientific outputs of activities supported by an international logist ic (IPEV and
+PNRA) and funded by national and international structures such as the EC. (3)
+We recommend to run the SSS in contemporary way to the DIMM (N◦1) adapted
+to isoplanatic angle measurements placed at the same height than SS S (8 m) and
+to the DIMM (N◦2) placed at 20 m. It would be fundamental to collect a large
+statistic employing this instruments configuration. In this way we will have3 inde-
+pendent ways to measure the free atmosphere and it will be able to c ross-correlate
+results in this particular critical region of the atmosphere. This app roach is par-
+ticularly important for the FOROT activities and also, as explained in Se ction 2,
+to design the future of Astronomy at Dome C.
+5 Acknowledgments
+This work has been funded by the Marie Curie Excellence Grant (FORO T) -
+MEXT-CT-2005-023878
+References
+Aristidi, E., Agabi, K., Fossat, E., Azouit, M., Martin, F., Sadibekova, T., Travouillon,
+T., Vernin, J., Ziad, A. 2005c, A&A, 444, 2, 651
+Agabi, K., Aristidi, E., Azouit, M., Fossat, E., Martin, F., Sadibekova, T., Vernin, J.,
+Ziad, A. 2006, PASP, 118, 344
+Geissler, K. & Masciadri, E. 2006, PASP, 118, 845, 1048
+Masciadri & Jabouille, 2001, A&A, 2001, 376, 727
+Masciadri, E., Avila, R., Sanchez, L., 2004, RMxAA, 40, 3
+Masciadri, E. & Egner, S., 2006, PASP, 118, 848, 1604
+Ragazzoni et al. 2004, 16 April 2004 - Workshop at LUAN, Nice, France
+2The seeing subtraction is done following the/parenleftbig/summationtext
+(εi)5/3/parenrightbig3/5rule.
\ No newline at end of file
diff --git a/1001.0565.txt b/1001.0565.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2f4e54b38f688250c43afee7ab703578f46219e2
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+++ b/1001.0565.txt
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+arXiv:1001.0565v1  [hep-th]  4 Jan 2010New probe ofmodified gravity
+A. Boyarsky1,2and O. Ruchayskiy1
+1Ecole Polytechnique F´ ed´ erale de Lausanne, FSB/ITP/LPPC , BSP 720, CH-1015, Lausanne, Switzerland
+2Bogolyubov Institute of Theoretical Physics, Kyiv, Ukrain e
+(Dated: January 4, 2010)
+We suggest a new efficient way to constrain a certain class of l arge scale modifications of gravity. We show
+that the scale-free relation between density and size of Dar k Matter halos, predicted within the ΛCDM model
+withNewtonian gravity, gets modifiedina wide classof theor ies of modifiedgravity.
+Models with the large scale modificationof gravity are ac-
+tively discussed in the recent years in connection with the
+observedaccelerated expansionsof the Universeand becaus e
+theycanberelatedtotheexistenceofextradimensions[1,2 ].
+However, the general principles of gauge invariance and uni -
+tarity strongly constrain possible theories of gravity, mo dify-
+ing the Newton’s law at large distances (see analysis of [3]) .
+Thus, in addition to its phenomenological applications, th is
+problem is related to the fundamental questions of particle
+physics, field theory and gravity. It is therefore important to
+searchforlargescalemodificationsofgravityexperimenta lly.
+A possible set of consistent (as a spin-2 field theory) large
+scale modifications of gravity is described by two parame-
+ters – scale rcand a number 0≤α <1[3–5].rcmarks
+thedistancesatwhichat the linearizedlevel gravitationallaw
+changes from 1/r2to some other power 1/rn, and the pa-
+rameterαdetermines the value of n. Phenomenologically,
+deviationsfromNewton’slawwearelookingformayberep-
+resentedin thisparameterspace. Asignificantfractionoft his
+spaceisexcludedbyprecisionmeasurementsoftheMoonor-
+bit[6]. Othernaturalprobesofsuchmodificationsarecosmo -
+logicalobservables(see e.g.[7–12] andrefs. therein).
+In this work we identify a new observable sensitive to the
+largescalemodificationsofgravity. We demonstratethatun i-
+versal properties of individual dark matter halos are also a f-
+fectedby the modificationsof gravity,and providenovelway
+toprobethem. Namely,we showthatthescale-invariantrela -
+tion between density and size of dark matter halos, predicte d
+byNewtoniangravitywithinthe ΛCDMmodel[13]andfound
+toholdtoagoodprecisioninobserveddarkmatterhalos[14] ,
+may receive non-universal (size-dependent) corrections f or a
+widerangeofparameters rcandα.
+Formation of structures in the Universe is an interplay
+between gravitational (Jeans) instability and overall Fre ed-
+man expansion. The gravitational collapse does not start un -
+til the potential energy Uof a gravitating dark matter sys-
+tem overpowers the kinetic energy of the Hubble expansion
+K ∼1
+2H2R2. Once the gravitational collapse has began, at
+any moment of time ta dark halo is confined within a sphere
+of zero velocity or a turn-around sphere . As Hubble expan-
+sion rate H(t)decreases with time, the turn-around radius
+Rta(t)grows. In the Newtonian cosmology with potential
+φN(r) =−GM/rthe turn-aroundradius Rtais
+Rta∝/parenleftbiggGM
+H2/parenrightbigg1/3
+(1)
+(today for masses ∼1012M⊙the turn-around radius is ∼1Mpc). Noticethatatanymomentoftimetheaveragedensity
+within a turn-around radius (1) is proportional to the cosmo -
+logicaldensityandisthesameforhalosofall masses:
+ρta∝H2
+G∝¯ρtot(t) (2)
+It was shown in [13] that the property (2) leads to a uni-
+versal relation between characteristic scales and densiti es of
+dark matter halos. This relation holds in wide class of dark-
+matterdominatedobjects(fromdwarfgalaxiestogalaxyclu s-
+ters) [14] (see also [15–17]). The relation is in a very good
+agreement with pure dark matter simulations [18, 19], sug-
+gesting that baryonic feedback can be neglected in this case .
+Therefore, this relation can serve as a new tool of probing
+propertiesofdarkmatterandgravityatlargescales.
+The relation (2) continues to hold in the Universe where
+gravityismodifiedbythecosmologicalconstant Λ. Thegrav-
+itational energy of a body of mass Mat distance rbecomes
+UΛ=−GM
+r−Λr2
+6. Comparing it with the kinetic energy of
+theHubbleflow Konearrivesonceagainto therelation(2)
+ρta(t)∝Λ
+G(3)
+(c.f.[13]). Therelation(1)stillholdsandisagainindepe ndent
+onthemassofthe halo.
+Whatisthemostgeneralformofgravitationalpotential,fo r
+which the property (2) remains true? Clearly, it will hold fo r
+all thegravitationalpotentialsoftheform
+φ(r) =−GM
+rF/parenleftbiggρ(r)
+ρ⋆/parenrightbigg
+(4)
+whereρ⋆is some constant with dimensionof density. In par-
+ticular,Λ-term obeysthis property(with ρ⋆∝Λ/G). All the
+theories of the form (4) obey the property that relative cor-
+rection to the Newtonian potential φNdependsonlyon the
+densityρ(r)withinaradius r(andnotonthemassorthesize
+ofobjects).
+Next, we consider the modifications of gravity [3, 4]. The
+gravitational potential of a spherically symmetric system of
+massMtherehastheform
+φα(r) =−GM
+rπ(r
+rV) (5)
+where0≤α <1. Here the characteristic ( Vainstein) radius
+rVisdefinedas[3, 4, 20]
+rV=/parenleftBig
+2GMrβ
+c/parenrightBig1
+1+βwhereβ= 4(1−α)(6)2
+If the scale rcis of the order of ∼H−1
+0, such modifications
+ofgravitycanprovideanexplanationforthelate-timecosm o-
+logical expansion of the Universe [1, 2]. The corrections to
+Newton’slawbecomenegligibleas r→0(π(0) = 1)andthe
+radius (6) characterizes the scale where the deviations fro m
+Newton’spotentialbecomeoforderunity. Usingtherelatio n
+r
+rV=/parenleftbiggr1+β
+2GMrβ
+c/parenrightbigg1
+1+β
+∝Mβ−2
+3(1+β)1
+ρ1/3/parenleftbig
+2Grβ
+c/parenrightbig1
+1+β(7)
+(whereρ=M/r3) we find that among the theories of mod-
+ified gravity (5) only β= 2(α=1
+2, the DGP model [1])
+possesstheproperty(4) andconsequently(2).
+The property (2) can be probed experimentally. Exten-
+sive catalog of DM-dominatedobjects of all scales, collect ed
+in [14], exhibits a simple scaling relation of the propertie s of
+the DM halos. Dark matter distributionin the majorityof ob-
+served objects can be described by one of the universal DM
+profiles (such as e.g. NFW [21] or Burkert [22]). Such pro-
+files may be parametrized by two numbers, directly related
+to observations – a characteristic radius rC(off-center dis-
+tance where the rotation curve becomes approximately flat,
+equal e.g. to rsfor NFW) and a DM central mass density
+¯ρC, averaged inside a ball with the size rC. It was shown
+that DM column density S ∝¯ρCrC, (see [13, 14] for a de-
+tailed definition) changes with the mass as S ∝Mκ, where
+κ≈0.22−0.33. It was demonstrated in [13] that in the
+simplest self-similar model (i.e. assuming that rC/Rtais the
+same for the DM halos of all masses) property (1–2) implies
+a scaling S ∝M1/3,compatiblewith observations. Observa-
+tions(see e.g.[23–27])demonstratethat theratioof rCtothe
+virialradiusdependsweakly(as M≈−0.1)onthemassofDM
+halos. The Fig. 1 shows ratio of the virial radius of a halo to
+itsrCforDMdensityprofilesfromthecatalogof[14]. These
+results are in perfect agreement with the ΛCDM numerical
+simulations (see e.g. [18, 28]). Due to this slight deviatio n
+from the self-similarity, the best fit value of the scaling pa -
+rameterκ= 0.23(see[13]fordiscussion). Forthequalitative
+discussion of this work, it is important that S(M)is a fea-
+turelesspower-lawdependence,whoseslopedoesnotdepend
+on mass and that the deviation from the slope1
+3is small (as
+followsfromobservations).
+We can conclude that in the theories, satisfying the con-
+dition (5) (e.g. in DGP model) the properties of DM halos
+theory will follow the same scaling relation S ∝M1/3and
+thedifferencewiththe ΛCDMcase willonlybeina different
+normalization of this scaling relation. For other theories , de-
+scribedin[3,4],with α∝negationslash=1
+2,wecanseefromtheEq.(7))that
+the potential φα(r)is not of the form (4) and we can expect
+deviationfromtheuniversalscalinglaw.
+I.S −MRELATIONFORGENERAL α
+Let us work out the S −Mfor a general αin details. A
+generalexpression,relatingtheturn-aroundtime,turn-a round
+radius and mass within this radius follows from energy con- 0.1 1 10 100
+ 1e+09  1e+11  1e+13  1e+15cvir
+Mvir, MSun
+FIG. 1: Comparison of cvir=Rvir/rCas a function of DM halo
+mass for observed DMdensity profilesfrom the catalog of [14] .
+servationandis givenby
+t0=1√
+2/integraldisplayRta
+0dr/radicalbig
+φα(r)−φα(Rta)(8)
+Using the general form (5) we can rewrite the expression (8)
+inthefollowingform:
+t0=/parenleftbiggπ2R3
+ta
+8GM/parenrightbigg1/2
+I(xta) (9)
+where dimensionless ratio xta≡Rta/rVand the function
+I(xta)isgivenby
+I(xta) =2
+π/integraldisplay1
+0dx
+/parenleftBig
+π(xxta)
+x−π(xta)/parenrightBig1/2(10)
+The solution of this equation gives us the “density” ρta≡
+M/R3
+taas a function of M. Whenrc→ ∞the turn-around
+densityρtabecomes
+ρta=M
+R3
+ta=π2
+8Gt2
+0≡ρ0 (11)
+Here theconstantρ0is a function of lifetime of the Universe
+onlyanddoesnotdependonparametersofadarkmatterhalo.
+This gives a desired relation between a turn-around density
+and the life-time of the Universe in the pure Newtonian cos-
+mology (without cosmological constant). The function I(x)
+isdefinedinsuchaway thatintheNewtonianlimit π(x) = 1
+onegetsI(x) = 1.
+The derivation of Eq. (9) demonstrates that for all theories
+of gravity of the form (4) (including Λ-term and the DGP
+model)ρtais a function of ρ0onlyand does not depend on
+the mass/size of a particular halo. As a result S ∝M1/3
+(see[13]fordetails).
+FurtheranalysisofEq.(9)dependsontheformofthefunc-
+tionπ(x). Its exact form is not known (apart from the DGP
+case). Letusstartwith analyzingseverallimitingcases.3
+0.2 0.4 0.6 0.8 1.0Α
+/Minus0.50.51.0I1/LParen1Α/RParen1
+FIG.2: Function I1(α)definedinEq.(13)
+If the Vainstein radius rV≪Rtafor halos of all masses
+that are experimentally observed (roughly from ∼108M⊙
+to∼1016M⊙), then for distances r≫rVthe corrections
+to the Newtonian potential reduce either to the order one
+renormalization of the gravitational constant (on the “nor mal
+branch”)orbecomeindistinguishablefromthe Λ-term(“self-
+accelerated branch”). In both cases S(M)∝M1/3with the
+normalization,differentfromthe pureNewtoniancase.
+In the opposite case rV≫Rta, one can utilize the per-
+turbative expansion of the function π(x). The gravitational
+potentialwell insidetheVainsteinradiusisgivenby[3,4]
+π(x≪1)≈1+c1xa;a=β+1
+2=5−4α
+2(12)
+wherec1∼ O(1)andispositiveforthe “normalbranch”and
+negativefor the “self-acceleratingbranch”[5] in full ana logy
+with the DGP model. Notice that a >1forα <3
+4. Using
+expansion(12)onearrivesto
+I(xta≪1)≈1+c1
+2xa
+ta
+2
+π/integraldisplay1
+0dx1−xa−1
+/parenleftBig
+1
+x−1/parenrightBig3/2
+
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+≡I1(α)(13)
+wherethefunction I1(α)isshownontheFig. 2.
+Substitutingthe expression(13) back into equation(9) and
+using(7),we obtain
+/parenleftbiggρ0
+ρta/parenrightbigg1/2/parenleftBig
+1+c1
+2xa
+taI1(α)/parenrightBig
+= 1 (14)
+Asxta≪1anda >1,onefindsthat
+ρta≃ρ0/parenleftBig
+1+c1I1(α)xa
+ta(ρ0)/parenrightBig
+(15)
+where to compute xtawe use Eq. (7) with ρ0instead of ρta.
+From Eqs. (7) and (15) we see once again that for all α∝negationslash=1
+2
+(i.e.β∝negationslash= 2) the turn-around density ρtaloses its univer-
+sality and becomes the function of the halo mass M. The
+turn-around radius Rta(M)is related to ρtaviaRta(M) =
+(M/ρta)1/3. 0 1 2 3 4 5 6 7 8
+1071081091010101110121013101410151016DM column density, log10[S/Msun pc-2]
+DM halo mass [Msun]Clusters of galaxies
+Groups of galaxies
+Spiral galaxies
+Elliptical galaxies
+dSphs
+Norm. branch, α = 0   ; rc = 150 Mpc
+Self-acc. branch, α = 0   ; rc = 150 Mpc
+Norm. branch, α = 1/4 ; rc = 300 Mpc
+Self-acc. branch, α = 1/4 ; rc = 300 Mpc
+Best-fit model S ∝ M0.23
+FIG. 3: Examples of S −Mrelation in theories with α <1/2
+together withthe data from [14].
+Under the assumption of exact self-similarity, discussed
+above (i.e. rC/Rta= const) one arrives to the following
+expressionfor S(recallthat β= 4(1−α)):
+S(M) =ρCrC∝M1/3ρ2/3
+ta (16)
+∝M1/3ρ2/3
+0/parenleftBigg
+1+2
+3c1I1(α)/parenleftbiggM
+Mlim/parenrightbigg1−2α
+3/parenrightBigg
+(17)
+where
+Mlim≡1
+G/bracketleftBigg/parenleftBigrc
+2/parenrightBig3β/parenleftbiggπ
+t0/parenrightbigg2(1+β)/bracketrightBigg1
+β−2
+(18)
+Anexampleoftherelation(17)forseveral α’sandrcisshown
+inFig.3.
+Clearly, the most interesting case is when rV≈Rtafor
+some range of observed halo masses. In this regime the de-
+viations from Newtonian gravity become the strongest. The
+range of values rcfor which this happens is shown in Fig. 4
+(the value of t0is chosen to be the lifetime of the Universe
+t0≃1.3×1010years). We expect that for rcin the region
+Fig.4 the slope of the relation S ∝Mκwill change. Anal-
+ysis of this case requires however an exact solution of the
+non-linear analog of the Poisson equation in theories with α
+(see e.g. [4]), i.e. the knowledge of properties of the func-
+tionπ(r/rV)in the range of radii, where the perturbative
+expansion (12) breaks down. Notice, that the region where
+rV≈Rtashrinks toward the value rc=2
+πt0asα→1
+2. For
+thisvalueof rctheVainsteinradiusintheDGPmodelisequal
+to the turn-aroundradius for all halo masses .1Knowledgeof
+thepotential π(x)wouldalsoallowtoprobethemodifications
+ofgravitydirectlyintheLocalGroupandothernearbygalax -
+ies, by studying the infall trajectory around the turn-arou nd
+radius(see e.g.[29]).
+1In general for rc∼H−1
+0the turn-around radius is smaller than rVfor
+α <1
+2and bigger than rVforα >1
+2.4
+0.1 0.2 0.3 0.4 0.5Α5010020050010002000rc/LBracket1Mpc/RBracket1
+FIG. 4: Range of rcfor which rV≈Rtafor some halo masses in
+the range 1010−1015M⊙.
+Anotherwaytoprobethe S−Mrelationforgeneral rcand
+αis to do numerical simulations in the theories of modified
+gravity (see examples in [11, 12]) and compare directly with
+observationsboththescalingofthecentralvaluesof SandM
+andthescatteraroundit.
+Conclusion. The main purpose of this work was to iden-
+tify a new observable that can be used to constrain the large
+scale modifications of gravity. We see that the scaling prop-
+ertiesofdarkmatterhalosaresensitivetosuchmodificatio ns.We demonstratedthat the models with α∝negationslash=1
+2predict the de-
+viation from a simple power-law scaling in the S(M)rela-
+tion. Comparisonofpredictionsofsuchmodelswiththedata ,
+collected in [14] potentially allows to restrict the values of
+rcfrom below for a given α. The improved data-processing
+andnew observationaldata onDM distributionswill allow to
+strengthenthese boundsandmakethemquantitative.
+Inthisworkwehaveanalyzedonlythecasewhentheturn-
+aroundsphere is well inside the Vainstein radius, rV≫Rta.
+To analyze a general case, a better theoretical understandi ng
+of the function π(r)(definedvia Eq. (5)) is needed. Together
+with betterqualityofdata thiswill allow toextendouranal y-
+sisto awiderrangeofparameters.
+Inthe case α=1
+2(theDGP model)the S(M)dependence
+remains featureless. In this case one has rV≈Rtafor all
+masses (for rc∼H−1
+0) and the deviations from the Newto-
+nian gravity at turn-around radius will be strong. Therefor e
+thismodel (in general,all modelsthat have rV∼Rtaforha-
+los ofM∼1012M⊙) can be probed by studying the infall
+trajectoriesaroundthe turn-aroundradius in the Local Gro up
+andnearbygalaxies[29] usingthe availabledataandthe dat a
+fromforthcomingsurveysoftheMilkyWay aswell asGAIA
+mission.
+Acknowledgments. We would like to thank G. Dvali for
+manyuseful andilluminatingdiscussionsand carefulreadi ng
+of the manuscript. This work was supported in part by the
+SwissNationalScienceFoundation.
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+arXiv:1001.0566v1  [astro-ph.CO]  4 Jan 2010Preprint typesetusingL ATEX styleemulateapjv. 05/04/06
+ANUPPER LIMITTOTHEDRY MERGERRATE AT < Z >∼0.55
+ROBERTODEPROPRIS1, SIMONP. DRIVER2, MATTHEW COLLESS3, MICHAELJ. DRINKWATER4,
+JONLOVEDAY5, NICHOLAS P. ROSS6,7, JOSSBLAND-HAWTHORN3,8, DONALDG. YORK9,
+KEVINPIMBBLET4
+ABSTRACT
+We measure the fraction of Luminous Red Galaxies (LRGs) in dy namically close pairs (with projected
+separation less than 20 h−1kpc and velocity difference less than 500 km s−1) to estimate the dry merger
+rate for galaxieswith −23< M(r)k+e,z=0.2+5logh <−21.5and0.45< z <0.65in the 2dF-SDSS LRG
+and QSO (2SLAQ) redshift survey. For galaxieswith a luminos ity ratio of 1 : 4or greater we determine a 5σ
+upper limit to the merger fraction of 1.0% and a merger rate of <0.8×10−5Mpc−3Gyr−1(assuming that
+all pairs mergeon the shortest possible timescale set by dyn amicalfriction). This is significantly smaller than
+predicted by theoretical models and suggests that major dry mergersdo not contribute to the formation of the
+redsequenceat z <0.7.
+Subjectheadings: galaxies: interactions— galaxies: formationandevolutio n
+1.INTRODUCTION
+In the standard ΛCold Dark Matter ( ΛCDM) picture,
+galaxies are assembled via the progressive (hierarchical)
+merger of increasingly more massive subunits (see, e.g.,
+Coleet al. 2008; Neistein &Dekel 2008 for recent reviews).
+About1/2ofthestellarmassinpresent-daymassive( L > L∗)
+galaxiesisexpectedtobeaccretedviamajormergersat z <1
+(e.g.,De Luciaetal. 2006; De Lucia& Blaizot 2007). Merg-
+ers should therefore be a common occurrence in the life
+of galaxies and have a profound influence on their proper-
+ties (such as star formation, morphologyand nuclear activi ty
+amongothers).
+Observationaland theoretical evidence suggests that an in -
+creasing fraction of mergers at lower redshifts should take
+place between spheroidal or gas-poor galaxies (‘dry’ merg-
+ers). In their study of the morphology of merging pairs in a
+CDM universe, Khochfar&Burkert (2003, 2005) found that
+elliptical galaxies brighter than L∗were mainly formed via
+major dry mergers and that gas-rich mergers were only im-
+portant at lower luminosities. Above a ‘threshold’ mass of
+∼6.3×1010M⊙, galaxies are not expected to grow their
+stellar mass by (induced) star formation, but their primary
+mode of mass accretion at z <1is via gas-poor (stellar-
+dominated) mergers (DeLuciaet al. 2006; Khochfar&Silk
+2008). Unlikegas-richmergers,drymergersarenotexpecte d
+to disturb the observed tight scaling relations for early-t ype
+galaxies (Boylan-Kolchinet al. 2005, 2006) and to better re -
+produce the internal structure of local ellipticals (Naabe tal.
+1Cerro Tololo Inter-American Observatory, Casilla 603, LaS erena, Chile
+2School of Physics and Astronomy, University of St. Andrews, North
+Haugh, StAndrews, Fife KY16 9SS,UK
+3Anglo-Australian Observatory, PO Box 296, Epping, NSW 1710 , Aus-
+tralia
+4DepartmentofPhysics,University ofQueensland, Brisbane , Queensland
+4072, Australia
+5Astronomy Centre, University of Sussex, Falmer, Brighton, BN1 9QJ,
+UK
+6Physics Department, Durham University, South Road, Durham , DH1
+3LE,UK
+7Department of Astronomy and Astrophysics, The Pennsylvani a State
+University, 525 Davey Laboratory, University Park, PA1680 2, USA
+8Institute of Astronomy, School of Physics, University of Sy dney, NSW
+2006, Australia
+9Department of Astronomy and Astrophysics, University of Ch icago,
+5640 South Ellis Avenue, Chicago, IL60637, USA2006,2007).
+The first examples of dry mergers, in significant numbers,
+were observed as close pairs of galaxies on the red sequence
+in deep images of high redshift clusters (vanDokkumet al.
+1999, 2001), where approximately50% of galaxies were ex-
+pected to have undergonea dry merger to the present epoch.
+In the local universe, vanDokkum (2005) estimated that ∼
+30%ofz <0.1bright ellipticals in the MUSYC and ND-
+FWSsurveysshowedresidualstructuralfeaturesindicativ eof
+adrymergerinthe ‘recent’past.
+Bell etal.(2006a)usedclosepairsintheCombo-17survey
+havingsimilarphotometricredshifts andshowingevidenceof
+interactionstoinferanintegratedmergerrateof ∼80%since
+z <0.8for red galaxies with MB<−20.5+5log h, while
+Linetal. (2008) used dynamically close pairs in the DEEP2
+survey to derive an integrated dry merger rate of 24%atz <
+1.2forgalaxieswith −21< MB−5logh <−19. However,
+Hsiehet al.(2008)identifyclose pairsinthe RCS surveyand
+derive an integrated merger rate of only 6%per Gyr since
+z= 0.8for galaxies with −25< Mr−5logh <−20, and
+Wenet al.(2008)usethesameapproachasBell etal.(2006a)
+on LRGs in the SDSS survey to determine a merger rate of
+0.8%per Gyr at z <0.12for galaxies with Mr<−21.2+
+5logh. Bundyet al.(2009)findevidenceoffewtozeropairs
+ofredgalaxiesat z <1.2in theGOODSdata.
+The small scale correlation function of LRGs in the SDSS
+atz <0.36analyzed by Masjediet al. (2006, 2008) is con-
+sistent with an upper limit of <1.7%per Gyr to the dry
+merger rate for galaxies with Mi<−22.75 + 5log hat
+0.16< z <0.30. Bell et al. (2006b) used this method to de-
+rive a merger rate of 4%per Gyr for MB<−20.5+5log h
+galaxies at 0.4< z <0.8. White et al. (2007) obtain an in-
+tegrated merger rate of ∼30%for LRGs at 0.5< z <0.9
+in the NDFWS survey. Finally, Wake et al. (2008) apply the
+correlationfunctionmethodto galaxiesthe 2SLAQ surveyto
+derivea mergerrate of 2.4%perGyrat 0.19< z <0.55.
+The galaxy merger rate has been usually measured via the
+fraction of galaxies in close pairs, often with the added re-
+quirement of closeness in velocity space to cull interlop-
+ers (e.g., Pattonetal. 2000, 2002 – hereafter, P00,P02), an d
+the fraction of galaxies showing significant asymmetries in
+their light distribution (e.g., Conselice 2003; Conselice et al.
+2003). The rationale behind the former approach is that if a2 De Propriset al.
+merger is to occur, a companion must be present, and there-
+fore the close pair fractionis relatedto the futuremergerrate
+for the galaxies being considered. The latter method relies
+on the observation that if a galaxy has undergone a recent
+merger,itislikelytobemorphologicallydisturbed,andth ere-
+foreanasymmetriclightdistributionwouldbeasignpostof a
+recent merger. A critical discussion of these approachesma y
+befoundin DePropriset al.(2007)andGeneletal. (2008a).
+In this paper we derive the dynamically close pair frac-
+tion for galaxies in the 2SLAQ survey and determine an up-
+per limit to their merger rate. The benefit of using close
+pairs is that we are able to select major merger candidates
+between galaxies in a specified mass range (by appropriately
+choosingtheluminosityofandmagnitudedifferencebetwee n
+participating galaxies), derive a merger rate within a spec i-
+fied timescale (dependenton the chosen projected and veloc-
+ity separations for the pair members and theoretical simula -
+tions) andidentifyongoingmergercandidatesfor later stu dy.
+Galaxy asymmetries tend to be more difficult to interpret in
+this fashion and usually require better quality imaging tha n
+we have available in our survey, especially for dry mergers
+(Bell etal. 2006a;Wen et al.2008),to whichthe2SLAQ sur-
+vey is most sensitive. Unlike asymmetries, galaxy pairs are
+sensitive to the mergerfraction of ‘progenitor’halos and t his
+quantitymaybemoredirectlycomparedtotheoreticalmodel s
+(Genelet al. 2008a).
+The structure of this paper is as follows: in the next sec-
+tionwepresentthedata,describethemethodologyandderiv e
+the pair fraction and an upper limit to the dry merger rate at
+the intermediateredshiftssampledbythe 2SLAQ survey. We
+then discuss and compareour resultsin the contextof galaxy
+formationmodelsandrecentworkonthedrymergerrate. We
+adopt the latest cosmological parameters with ΩM= 0.27,
+ΩΛ= 0.73andH0= 100km/s/Mpc. Unless otherwise
+stated,allabsolutemagnitudesquotedinthefollowingare in-
+tended as including a term of +5 loghand all distance mea-
+suresneedtobereferredto h(totheappropriatepower).
+2.THE2SLAQ SURVEYDATA
+The data used in the 2SLAQ survey consist of LRGs with
+i <19.8mag. from the original sample of Eisensteinetal.
+(2001), selected by their g−randr−icolors to lie at
+0.45< z <0.65(see Fukugitaet al. 1996 for a description
+of the SDSS filter system). Photometry and astrometry for
+the target LRGs are derived from the SDSS Data Release 1
+(Yorketal. 2000; Abazajianet al. 2003) with improvements
+from the latest release available at the time of the spectro-
+scopic observations(DR4 – Adelman-McCarthyet al. 2006).
+The objects lie in two long strips on the celestial equator, d i-
+vided into several disconnectedpatches, each of which is be -
+tween 10 and 30 deg2in area, for a total survey coverage of
+182deg2.
+Spectroscopy for candidate LRGs was carried out at
+the Anglo-Australian telescope using the 2dF facility
+(Lewiset al. 2002). For objects in the main sample (Sam-
+ple 8), it was found that most galaxies are within the speci-
+fiedredshiftlimitsandwereobservedwithhighspectroscop ic
+completeness(typically 87%). A completedescription of the
+data can be found in the general survey paper by Cannon et
+al. (2006; hereafter C06). Because we have highly complete
+spectroscopy, we can confirm that at least 90% of our galax-
+ies have K-type or LRG spectra, with no sign of star forma-
+tion, while less than 1% of the sample shows emission lines
+(Roseboomet al. 2006). We can therefore use this sample tomeasurethedrymergerrate.
+Following Wake et al. (2006), we computed the absolute
+magnitudefor objectswith reliable redshifts, k+ecorrected
+to the SDSS rband atz= 0.2. This facilitates comparison
+with previous work on galaxy evolution and the merger rate
+fromtheSDSS(e.g.,Masjediet al.2006,2008). Notethatno
+correction for internal extinction was applied to these gal ax-
+ies.
+3.METHODOLOGY
+We calculate close pair statistics following the formalism
+developed by Patton et al. (2000, 2002; hereafter P00, P02)
+for the SSRS2 and CNOC2 surveys. Here, we give an ‘algo-
+rithmic’descriptionoftheproceduresused,andshowhowwe
+applyweightstocorrectforsourcesofincompletenessandt he
+fluxlimited natureof the2SLAQ survey. A fullerdescription
+ofthe methodcanbefoundinP00,P02.
+3.1.Luminosityof thepairsample
+Let us consider a sample of N1primary galaxies brighter
+than a limiting absolute magnitude M1in some volume of
+space, and, in the same volume, a sample of N2secondary
+galaxiesbrighterthanalimitingabsolutemagnitude M2. The
+two samples may coincide (i.e., M1=M2), as is often the
+case for redshift surveys: we then study ‘major’ mergers be-
+tween galaxies of approximately similar luminosity. We are
+interestedinknowingthefractionofgalaxiesinthesecond ary
+sample that are dynamically close to galaxies in the primary
+sample. We define two galaxies to be dynamically close if
+they have a projected separation rp<20h−1kpc and a ve-
+locity difference of <500km s−1, as used by P00, P02 and
+insubsequentwork.
+Of course, pair statistics are usually computed from red-
+shift surveys, which are flux-limited rather than volume lim -
+ited. We thereforeneedto accountforthe dependenceofpair
+counts on the clustering properties and the mean density of
+galaxies in the sample, and to correct for sources of spatial
+and spectroscopic incompleteness. Because clustering is l u-
+minositydependent(e.g.,Norbergetal. 2002)wefollowP00
+andrestrict the analysisto a fixedrangein luminosity,with in
+which clustering properties are not expected to vary signif -
+icantly, by imposing additional bright ( Mbright) and faint
+(Mfaint) limits on the sample. This means that we derive a
+pair fraction (and mergerrate) for galaxies within a specifi ed
+rangeandratioin luminosity.
+In our case, we select galaxies with 0.45< z < 0.65
+(wherethesurveyismostcomplete)andwith Mbright=−23
+andMfaint=−21.5mag. Becausetherangeofluminosities
+we survey is small, we let Mfaintcoincide with M2. We
+also let the primary and secondary samples coincide, to se-
+lect major mergers between galaxies of approximately equal
+luminosities(andmakefull use of the spectroscopicsample ).
+Figure 1 shows the distribution of 2SLAQ galaxies in abso-
+lute magnitude vs. redshift, together with selection lines in
+magnitudeandredshift(seebelowfordetails).
+The faint limit allows us to include most 2SLAQ galaxies
+atz >0.45andreachesslightlybelowthe luminosityofnor-
+malL∗galaxiesatthemeansurveyredshift(or,similarly,be-
+low the turnover in the 2SLAQ LRG luminosity function of
+Wakeet al. 2006). We choose the bright limit of Mr=−23
+to avoid the rarer and most luminous galaxies in 2SLAQ,
+which are likely to be more biased, and to mitigate the ef-
+fectsofluminosity-dependentclustering(asdescribedab ove).
+With these choices, we study majormergers,with luminosityDrymergerrateat < z >∼0.55 3
+0.2 0.3 0.4 0.5 0.6 0.7 0.8
+Redshift-25
+-24
+-23
+-22
+-21
+-20M(r)k+e,z=0.2
+FIG. 1.— The distribution of absolute magnitudes for 2SLAQ gala xies
+vs. redshift. The thick solid line is the redshift-dependen t absolute magni-
+tude limit for selection Mlim(z)(see equation (1) and related explanation
+in text). The thick dashed lines mark the ideal volume limite d box to which
+the observed sample is corrected using the weights describe d in equations
+(2)-(9).
+ratio≥1 : 4, where the secondary is at least 4 times less
+luminousthanthe primarygalaxy10, forMr<−21.5.
+We can translate our luminosity range into a stellar mass
+range using the conversion between rband absolute magni-
+tudeandstellarmassbyBaldryetal.(2006),fortypicalLRG
+colors (u−r >3) and with assumptions as to stellar pop-
+ulations as in Baldryet al. (2006). With these choices our
+−23< Mr<−21.5luminosity range correspondsto a stel-
+larmassrangeof 16–5×1010M⊙.
+3.2.Densityweighting S(z)
+The pair fraction that we actually measure needs to be
+corrected to the value that would be observed in an ideal
+volume-limited survey. In a flux-limited sample, primary
+galaxies at lower redshifts will have a greater likelihood o f
+having a secondary companionwithin the survey than galax-
+ies at higher redshift. We correct for this bias by assigning
+a greater weight to the rarer companions found at the high
+redshift end of the survey. This is carried out by comput-
+ing for each galaxy a weight that renormalizes the sample to
+the density correspondingto a volume limited sample within
+Mbright< M < M 2. Theweightiscalculatedbyintegrating
+the luminosity function over the appropriate ranges in abso -
+lutemagnitude.
+Ateachredshift,wethensearchforpairsofgalaxieswithin
+Mbrightand a redshift-dependent absolute magnitude limit
+Mlim(z), whichisdefinedas:
+Mlim(z) = max[ Mfaint,19.8−5logdL(z)−25−k(z)−e(z)]
+(1)
+wherei= 19.8mag. istheapparentmagnitudelimitofthe
+survey,dL(z)istheluminositydistanceand k(z)ande(z)are
+thekandevolutionarycorrections. Recallthathere Mfaintis
+10Note that of course no sample can be made complete for a given
+luminosity ratio without discarding a large fraction of the data (e.g.,
+Patton &Atfield 2008), so theoretical comparisons need to ta ke into account
+the observational limitations of this technique.set to coincide with M2. Thek+ecorrections are taken to
+be the maximal corrections for a galaxy formed at high red-
+shift and undergoingpure passive evolution, as other choic es
+would allow galaxies to fall in and out of the sample accord-
+ingtotheirstarformationhistories(P00,P02). Notethath ere
+maxmeans ‘the brightest of’ rather than the (arithmetically)
+largerquantity. Fig. 1 showsthis selection limit as applie dto
+2SLAQdata.
+Eachsecondarygalaxyisweightedbytheinverseofaselec-
+tionfunction S(z)definedastheratioofdensitiesinvolume-
+limitedvs. flux-limitedsamples:
+SN(z) =/integraltextMlim(z)
+MbrightΦ(M)dM
+/integraltextM2
+MbrightΦ(M)dM(2)
+SL(z) =/integraltextMlim(z)
+MbrightΦ(M)L(M)dM
+/integraltextM2
+MbrightΦ(M)L(M)dM(3)
+whereL(M) = 100.4(M−M⊙)L⊙andΦ(M)is the LRG
+luminosity function from Wake etal. (2006). The integrals
+runfrom Mbrightto either the faint absolute limit M2(in the
+denominator)ortotheredshiftdependentabsolutemagnitu de
+limitMlim(z)(in the numerator) set by the apparent magni-
+tudelimit ofthesurvey.
+Galaxies in the primary sample at low redshift will also
+have the largest number of observed companions, while pri-
+maries at higher redshifts will have fewer observed compan-
+ions. Thiseffectiscorrectedinasimilarfashionasforgal ax-
+ies in the secondary sample, by applying the inverse of the
+secondaries’weighttoprimaries,i.e SN(z)andSL(z).
+3.3.Spatialincompleteness wb,wv
+We need to account for pairs missed because one of the
+galaxiesfallsoutsideofthesurveyfootprint. Apotential com-
+panionmaylie beyondthesurveylimitsonthe skyorbehid-
+deninthe ‘shadow’ofa brightstar,wheregalaxiescannotbe
+detected or the spectra are contaminated11. For each primary
+galaxywecomputethefraction 1−fboftheπr2
+pareathatmay
+lie outside of the effective survey area. The weight to be ap-
+plied to the secondary galaxies is then wb2= 1/fb. Primary
+galaxies may similarly be lost in the survey boundaries and
+the appropriateweight to be applied to the primarysample is
+wb1=fb=w−1
+b2.
+For objects with small separations, the SDSS photometric
+pipeline tends to merge pairs into a single galaxy. For the
+LRGsamplestudiedbyMasjediet al.(2006)the pipelinebe-
+comesunreliable for rp<3′′. The20h−1kpc search radius
+we use corresponds to angular separations of 4.91′′to4.07′′
+at0.45< z <0.65. We inspectedallour7889imagestover-
+ify whether the SDSS pipeline correctly identifies photomet -
+ric companions,usingthe SDSS ’ImageList/Navigate’ tools .
+Theresultsarethatthepipelinecertainlyfindsalltargets with
+projected separations between 3′′and5′′. For smaller sepa-
+rationsthe pipeline is more erratic, especially for fainte r sys-
+tems. We therefore exclude an area of 3′′around each target
+from our search for companions and correct for the missed
+region via the wbweight, where we assume that the distribu-
+tion of pairs is uniform with projected separation over thes e
+11This is calculated using a formula developed by I. Strateva ( unpub-
+lished) for the SDSSsurvey4 De Propriset al.
+small scales. The size of the added factor lies between 37%
+and54%dependingonredshift.
+The SDSS pipeline also tends to ‘double count’ light for
+close pairs, making these galaxies brighter than they would
+otherwise be. Because fainter galaxies have more merg-
+ers and minor mergers are more common (Patton&Atfield
+2008), this has the effect of artificially raising the pair fr ac-
+tion. Masjediet al. (2006) estimate that this should decrea se
+the actual merger rate for galaxies with rp<5′′by a factor
+of about 5. Since this is an uncertain correction and we are
+interestedinanupperlimittothemergerrate,wedonotcon-
+sider this effect here (the effect cannot be well modelled fo r
+oursample,astheSDSS pipelinecannotberunlocally).
+Wealsocorrectforpossiblecompanionsmissedbythered-
+shift ‘cuts’ we apply to the survey. If the primary lies withi n
+500kms−1oftheredshiftboundaries,weignoreallcompan-
+ionsbetween the primaryand the bordersand applya weight
+wv2= 2to all other companions in the opposite ‘direction’.
+We also need to apply a similar weight to account for poten-
+tial primaries lost in the redshift boundary. This weight is
+the reciprocal of the weight applied to the secondaries, i.e .,
+wv1= 1/2
+3.4.Spectroscopicincompleteness wθ
+As all redshift surveys, 2SLAQ is not complete to its flux
+limit. Our success rate is 87%for all galaxieswithin Sample
+8ofC06butthespectroscopicincompletenessmaybedepen-
+dent on the separation between galaxies. Fibers in the 2dF
+positioner cannot be placed closer than about 25′′from each
+other in every single configuration. However, this effect is
+compensated by the overlap between individual 2SLAQ tiles
+and by the fact that fiber configurations were generally re-
+touchedhalfwaythrougheach(typically4hours)exposuret o
+place fibers assigned to galaxies for which a reliable redshi ft
+had already been obtained on to a nearby target (see C06 for
+adescriptionoftheobservations).
+In order to correct for this source of bias we need to es-
+timate the relative incompleteness for close pairs over the
+rangeofseparationsofinterest(correspondingto20 h−1kpc)
+and compare it with the incompleteness at large separations ,
+wherefiberinteractionsarenotimportant. WefollowP02and
+estimatethisweightbycomputingtheratiobetweenthenum-
+berofpairsbetweengalaxieswithredshiftinformation( Nzz)
+within the redshift range of interest and the number of pairs
+inthe inputphotometricsample( Npp), whichisbydefinition
+complete,asafunctionofangularseparation θTheweightto
+be applied is the ratio between the spectroscopic and photo-
+metricpaircompletenessateachseparationnormalizedtot he
+valueat largeseparations.
+In 2SLAQ we have higher overall completeness than
+CNOC2(byaboutafactorof2)butsampleasmallerrangeof
+projected separations (because of our higher mean redshift ).
+We plotNzz(θ)/Npp(θ)vs.θin Fig. 2 for Sample 8 targets;
+the error bars are assumed to be Poissonian. Although the
+dataarenoisy(becausetherearerelativelyfewpotentialp airs
+to start with), the value of Nzz/Nppatθ <6′′is consistent
+with the value of this ratio at θ >100′′, arguing that we are
+not systematically more incomplete at small separationsth an
+at largeoneswherewe arenotaffectedbyfibercollisions.
+P02modeltheincompletenessintheCNOC2surveybyfit-
+tingapolynomialtotheratio Nzz(θ)/Npp(θ)asafunctionof
+projectedseparation θ. Giventhe small numberstatistics and
+noisiernatureof ourdata, it isnot fullyjustified to modelt he
+completenessas a functionof θwith a polynomialasdonein0 50 100 150 200 250 300
+Separation (arcsec)00.20.40.60.811.21.4Nzz(θ)/Npp(θ)
+FIG. 2.—Theratio ofgalaxies in pairs between galaxies with red shifts and
+galaxies in pairs in the photometric sample, as a function of pair separation
+θ.
+P02. We therefore adopt an uniform weight of 1 for 2SLAQ
+galaxies over the separation of interest, as we do not appear
+to be systematically more incomplete than at larger separa-
+tions. Although pair completeness drops at 30′′–60′′separa-
+tions, because of fiber collisions, these large separations are
+not relevant to our study (as pairs with rp>50h−1kpc or
+∆v <1000kms−1arelargelyspurious– P00,De Propriset
+al. 2007).
+Given the small number statistics, and the fact we are ulti-
+mately derivingan upper limit, we eventually decided to fol -
+low the approach by De Proprisetal. (2005) to estimate the
+contribution from close pairs missed because of fiber colli-
+sions. We search around each of our main targets (Sample 8
+galaxieswith −23.0< M(r)<−21.5and0.45< z <0.65)
+for a companion (lying within rp) in the sample of galaxies
+(from the input photometric sample) for which we did not
+obtain a valid redshift. If such a companion exists, we as-
+sign to it the same redshift as the primary galaxy and require
+that the companion lies within the selection lines in Fig. 1.
+This identifies all pairs (a total of 3) which are potentially
+missed because of redshift incompleteness. Pairs where bot h
+membersaremissedbythespectroscopicsurvey,willsharei n
+thegeneral2SLAQincompleteness,withoutabiasforincom-
+pleteness at small angular separations. We can assume that
+all these 3 ‘extra’ pairs are real and treat them as a source of
+systematicerroronourdeterminationofthe pairfractiona nd
+the mergerrate. Figure 3 shows postage stamp images of the
+twodynamicalpairswefindandofthethreepossiblepairs.
+P02alsouseaweight wswhichaccountsforthelocalmag-
+nitudeincompletenessaroundeachgalaxy,ageometriceffe ct
+duetoslitplacementandlimitingfiltersintheCNOC2survey ,
+a color term and a term that depends on the evolution of the
+galaxy luminosity function over the redshifts covered by th e
+survey. This weight is specific to the methods employed by
+the CNOC2 survey (Yeeet al. 1996). In our case, we have a
+veryhomogeneoussample,thereisnogeometriceffect(othe r
+thantheonecorrectedby wθ),allgalaxieshavesimilarcolors
+andthecolorselectionisveryefficientinselectinggalaxi esof
+the appropriate magnitude and redshift. Even in the CNOC2
+survey most of these weights are equal to 1 (Yeeetal. 1996)
+for most galaxies. For these reasons, we do not adopt this
+weightin ourstudy.
+3.5.Calculationofpairfraction
+FollowingP00 thenumberof close companionspergalaxy
+ina flux-limitedsamplecanbeexpressedas:Drymergerrateat < z >∼0.55 5
+FIG. 3.— Images (from the SDSS) of pairs found in our survey. The t wo
+left-hand ones are the real ones, while the three images on th e right hand
+show the ‘possible’ pairs
+Nc=/summationtextN1
+iwi
+N1Nci/summationtextN1
+iwi
+N1(4)
+andthe totalcompanionluminosity:
+Lc=/summationtextL1
+iwi
+L1Lci/summationtextL1
+iwi
+L1L⊙ (5)
+The sums run overthe i= 1,...,N1primarygalaxies: Nci
+andLciare the number and summed luminosity of galaxies
+fromthesecondarysample that aredynamicallyclose (asde-
+finedabove)tothe ithprimarygalaxyandareexpressedas:
+Nci=/summationdisplay
+jwj
+N2=/summationtext
+jwj
+b2wj
+v2
+SN(zj)(6)
+Lci=/summationdisplay
+jwj
+N2Lj=/summationdisplay
+jwj
+b2wj
+v2
+SL(zj)Lj (7)
+wherethesumsrunoverthose j= 1,...,N2galaxiesinthe
+secondarysamplethatfulfillthecriteriaforbeingdynamic ally
+closeto the ithgalaxyintheprimarysample.
+The weights to be applied to the secondary sample correct
+for spatial incompleteness ( wj
+b,wj
+v) and the flux-limit of the
+survey (S(zj)) for each jthcompanion. We do not apply the
+spectroscopic incompleteness weight wθbecause we use the
+alternatemethoddescribedabovetocalculatethecontribu tion
+from missed close pairs. The weights are defined in the pre-
+vioussubsections. Similar weights also need to be appliedt o
+the primarysample, to correctfor spatial incompletenessa nd
+themeandensity:
+wi
+N1=wi
+b1wi
+v1SN(zi) (8)wi
+L1=wi
+b1wi
+v1SL(zi) (9)
+wherewi
+b1=fi
+bandwi
+v1= 1/2(i.e., thereciprocalsofthe
+weights applied to the secondary sample). No spectroscopic
+completenessweightis neededfortheprimarysample.
+3.6.Mergertimescales
+In order to convert a pair fraction into a merger rate the
+timescale forthe mergereventneedsto be estimated. Merger
+timescalesaredifficulttodetermineanddependonthepoorl y
+known details of the merger process and how the poten-
+tial of the individual galaxies reacts to the merger episode .
+The more commonly used timescales in previous work and
+semi-analytic models are based on dynamical friction argu-
+ments. The dynamicalfrictiontimescale can be calculated a s
+(Pattonet al.2000):
+Tfric=2.64×105r2vc
+MlnΛ(10)
+where r is initial physical pair separation in kpc, vcis the
+circular velocity in km s−1,Mis the mass and Λis the
+Coulomb logarithm. Assuming r= 20h−1kpc,vc= 260
+kms−1andlnΛ = 2forequalmassmergers,wegetatypical
+dynamical friction timescale between 0.1 and 0.3 Gyrs over
+therangeofmasseswesample.
+The calibration by Kitzbichler&White (2009) for the
+merger timescale of pairs in N-body simulations with M >
+5×109M⊙andavelocityseparationof300kms−1yields0.9
+Gyr, while the estimated merger timescale from the N-body
+simulations of Boylan-Kolchinetal. (2008) is ∼0.8Gyr for
+the mass ratios we consider. A comparison between merger
+timescalebydynamicalfrictionandinN-bodysimulationsb y
+Boylan-Kolchinetal. (2008) shows that the dynamical fric-
+tion formula underestimatesthe mergertimescale by a facto r
+between 1.7 and 3.3 for mergers of mass ratio 1:3 to 1:10
+respectively,and leadsto a 40% overestimateof the massac-
+cretionrate(mainlyviaminormergers).
+4.DISCUSSION
+4.1.Results
+Wecannowapplytheaboveproceduretothe2SLAQsam-
+ple. Out of 7889 galaxies we find a total of one dynamically
+close pair (we find a second pair, but it lies within the 3′′ex-
+clusion circle). This yields a pair fraction Nc= 0.047%and
+a luminosity accretionrate Lc= 2.5×107L⊙. For galaxies
+withinz <0.55this isNc= 0.041%. Normally, errors on
+these quantities can be estimated by bootstrap resampling o r
+jack-knifing,butthisisnotfeasiblewithasampleofonlytw o
+objects. Wethenproceedtoestimateanupperlimittothepai r
+fractionandthemergerrate.
+Inordertoderiveanupperlimit tothemergerrate,wetake
+theobserveddetectionsandusethebinomialdistribution. For
+large samples ( N >100) the error distribution is given by
+(Burgasseret al.2003):
+(ǫU
+b−ǫb)/ǫb= (ǫb−ǫL
+b)/ǫb=/radicalbig
+1/n+1/N(11)
+whereǫbis the pair fraction, nis the number of pairs, N
+is the number of objects in the sample, ǫU
+bis the upper 1σ
+probability limit to the pair fraction and ǫL
+bis the lower 1σ
+probability limit to the pair fraction. The region between ǫU
+b
+andǫL
+bcorrespondstothe68%confidenceintervalfor ǫbfora6 De Propriset al.
+Gaussian distribution. In order to obtain aconservativeup per
+limit we decide to use both pairs that we actually find (even
+though one of these is not within the valid range of separa-
+tions).
+Forn= 2andN= 7889thisyields Nc= 0.094±0.066%.
+The systematic error due to the three possible extra pairs is
+0.14±0.08%. Wecanthereforequotea5 σupperlimittothe
+pair fractionof 0.44%(random)plus 0.54%(systematic),for
+atotal of 1.0%.
+As a check, we estimate the pair fraction in each of the
+separate 2SLAQ survey patches: while most regionshave no
+pairs, in one area we find a pair fraction of 0.30%, which
+is consistent with the upper limit we derived above. Fi-
+nally, we can also consider pairs with wider separation in
+both projected distance and velocity: rp<50h−1kpc and
+∆v <1000km s−1. This considerably increases the num-
+ber of contaminants(unphysicalpairs), as shown by P00 and
+DePropriset al.(2007). However,thispairfractionmaypro -
+vide a further interesting constraint. For the 2SLAQ sample
+wefindNc= 0.13%.
+Our upper limit to the pair fraction can be translated into a
+5σupperlimit tothe drymergerrate usingtheexpression:
+Rmg=Ncn(z)0.5pmergT−1
+mg (12)
+wheren(z)is the space density of galaxies in our sample,
+0.5isafactorintroducedtoavoiddoublecountinggalaxies (1
+pair contains 2 galaxies), pmergis the probability that pairs
+will merge (assumed to be 1 here) and Tmgis the merger
+timescale, for which we take the shortest possible timescal e
+set by dynamical friction. We find that a robust 5σupper
+limit to the mergerrate is: <0.8×10−5Mpc−3h−3Gyr−1
+(including the systematic contribution due to the three ext ra
+photometric pairs) for galaxies with −23< Mr<−21.5
+at0.45< z <0.65(a period of 1.4 Gyrs in the history of
+the Universe). A more realistic merger timescale and merger
+fractionmaydecreasethisestimatebymorethanoneorderof
+magnitude.
+4.2.Comparisonwith PreviousMeasurements
+Thelowdrymergerratewe measurehereisin goodagree-
+mentwith anumberofotherestimates: locally,Masjedi etal .
+(2006, 2008) obtain an upper limit of <1.7%per Gyr for
+the dry merger rate of SDSS LRGs with Mi<−22.75and
+z <0.36, a value which is consistent with ours, albeit for
+more luminous(massive) galaxies. This is also similar to th e
+dry merger rate of 0.8%measured by Wen et al. (2008) for
+SDSS LRGs with Mr<−21.5andz <0.12. The inte-
+grated dry merger rate for galaxies in the Red Sequence Sur-
+vey is∼6%sincez= 0.8over−25< Mr<−20, while
+Bundyet al. (2009) find very low to zero likely dry mergers
+inGOODS dataat z <1.2.
+Neverthelessthere are some discrepant estimates in the lit -
+erature. Apparently, the most worrying is the value of 2.4%
+per Gyr for 0.19< z <0.55LRGs from 2SLAQ derived
+by Wake et al. (2008) using the small scale correlation func-
+tion. However, Wake etal. (2008) use LRGs over the en-
+tire range of absolute luminosities in the 2SLAQ survey, and
+their sample therefore includes both minor mergers (lumi-
+nosity ratio greater than 1:4) and less luminous objects. As
+shown by Patton& Atfield (2008) and deRavel et al. (2008),
+the mergerrate increases significantly for minor mergersan d
+less luminous galaxies. Therefore the larger value derivedby Wake etal. (2008) is not necessarily in disagreementwith
+ours.
+Linetal.(2008)applythesamemethodaswedo(dynami-
+cally close pairs) to galaxiesin the DEEP2 surveyand derive
+an integrated merger rate of 24%sincez <1.2for galaxies
+with−21< MB<−19. Assuming a flat evolution of the
+merger rate (Linetal. 2008), this is equivalent to ∼6%per
+Gyr. Here,thedifferentluminosityrangessampled,theban d-
+passdifference,andthefactthatdrymergerswereselected by
+morphology,ratherthanbycolorsandspectralfeaturesasw e
+do,mayplayaroleinexplainingthe difference.
+Bell etal. (2006a) obtain an integrated dry merger rate of
+∼80%sincez= 0.8for galaxies in the Combo-17 survey
+withMB<−20.5, while applying the small scale corre-
+lation function to the same data, Bell etal. (2006b) derive a
+merger rate of 4%per Gyr at 0.4< z <0.8. As noted by
+Wakeet al. (2008), the space density of objects in these sur-
+veys is 20 times greater than ours, and therefore Bell et al.
+(2006a,b) are sampling considerably less luminous objects
+thanwedo. Giventhedependenceofthemergerrateonlumi-
+nosity(Patton&Atfield 2008; de Ravelet al. 2008) this does
+not mean that our results are in disagreement. In addition,
+Khochfar&Silk (2008) show that the abovesample includes
+both wet and dry mergers, unlike ours where we confirm by
+spectroscopy that the vast majority of the sample has LRG-
+typespectra(Roseboometal. 2006).
+Finally, Whiteet al. (2007) measure an integrated dry
+mergerrate of30%forLRGsinthe NDFWS surveybetween
+0.5< z <0.9. Thisisequivalenttoamergerrateof3.4%per
+Gyr,butneedstobecorrectedforthehighermeanredshiftan d
+different space density than the 2SLAQ sample. Wakeet al.
+(2008) estimate that this downward revision is about a facto r
+of 10, which places the result by White etal. (2007) in good
+agreementwithours.
+In addition, there are several estimates of the merger rate
+forall galaxiesusing asymmetriesforthe DEEP2 (Lotzet al.
+2008)andCOSMOSsurveys(Kampczyket al.2007),aswell
+asgalaxiesintheGOODSfields(Lopez-Sanjuanet al.2009),
+and pairs in the COSMOS survey (Kartaltepeet al. 2007).
+The measured merger rate is approximately 2–4%per Gyr
+for the entire population of L > L∗galaxies, of which dry
+mergers are only a subset, which is in agreement with our
+measurement.
+Our result is also consistent with the more indirect merger
+fraction derived from the evolution of the red galaxy lumi-
+nosity function. Note that, in this case, ‘growth’ may take
+place via the transformation of blue galaxies into quiescen t
+objects,movingontotheredsequence,butwithout(necessa r-
+ily) any major merging. Wake et al. (2006) used the 2SLAQ
+data to show that the LRG luminosity function evolves pas-
+sively at z <0.6, a result confirmed by several other stud-
+ies(Bundyet al.2006;Caputi etal.2006;Cimatti et al.2006 ;
+Scarlataetal. 2007). For ∼4L∗LRGs Brownetal. (2007,
+2008) and Cooletal. (2008) show that these galaxies evolve
+essentiallypassivelyat z <1.
+Some massive red mergers are observed in distant clus-
+ters and groups (e.g., vanDokkumet al. 1999, 2001). Us-
+ingthe pairfraction,vanDokkumet al.(1999)estimatedtha t
+∼50%of massive red galaxies in the cluster MS1054-0321
+atz= 0.82have undergone a merger to the present epoch.
+McIntoshetal. (2008) calculate that, for groups and clus-
+ters in the SDSS at z <0.12, the merger rate for mas-
+sive red galaxies is 2–9%per Gyr. These are much higher
+merger rates than we measure, albeit in a much denser en-Drymergerrateat < z >∼0.55 7
+vironment. Assuming that the merger rate scales with den-
+sity, and since clusters are about 100 times denser than the
+field, the values derived by vanDokkumetal. (1999, 2001)
+andMcIntoshet al.(2008)areprobablynotfullyinconsiste nt
+withour∼0.3%upperlimit forfield LRGs.
+4.3.ImplicationsforGalaxyFormation
+At face value, our results are in considerable ten-
+sion with models of galaxy formation in ΛCDM cos-
+mologies where most massive red galaxies grow via dry
+mergers (Khochfar&Burkert 2003; De Luciaetal. 2006;
+Khochfar& Silk 2008). Khochfar&Silk (2008) use a semi-
+analyticmodeltomodelthequenchingofstarformationanda
+standardmergertree,topredictthedrymergerrate forgala x-
+iesaboveacharacteristicmass(themassforwhichstarform a-
+tion is shut off) at z <1. For galaxieswith M >6.3×1010
+M⊙Khochfar&Silk (2008) predict a dry merger rate of
+6×10−5Mpc−3h−3Gyr−1, and a typical merger fraction
+ofabout10–20%perGyr,almostindependentofredshift.
+Our luminosity range of −23< Mr<−21.5translates
+into a mass range of 16 to 5 ×1010M⊙. The upper limit to
+themergerratewe derive( <0.8×10−5Mpc−3h−3Gyr−1,
+including the ‘worst case scenario’ for systematic errors a nd
+assumuingall pairsmergeon the shortest possible timescal e)
+is at least a factor of 5 below these predictions. It is there-
+foreunlikelythat drymergersat z <0.7are importantin the
+buildup of the red sequence (Bundyet al. 2007; Geneletal.
+2008b; Bundyet al. 2009)
+On the other hand, the mass fraction in ≤L∗galaxies
+on the red sequence appears to grow by about 50%since
+z= 1(Bellet al. 2004; Borchetal. 2006; Faberet al. 2007;
+Coolet al. 2008). Scarlataet al. (2007) also note a deficit of
+fainterearly-typegalaxiesin their COSMOS data at z= 0.7.
+Themainmodeofgrowthoftheredsequenceat z <1maybe
+viacessationofstarformationandmorphologicalevolutio nin
+lowermassgalaxies(Bell etal. 2004; Faberet al.2007).
+We therefore favor a scenario where most massive galax-
+iesareformedquasi-monolithicallyathighredshiftandma jor
+mergersarerelativelyunimportantat z <1(e.g.,Boweretal.
+2006; Naabetal. 2007 – cf., Conselice 2006 for an observa-
+tionalperspective). ThisisconsistentwithrecentCDMsim u-
+lations,thatdownplaytheroleofmajormergersingalaxyfo r-
+mation overthe past half of the Hubble time and favor minor
+mergersand internal processes as drivers of galaxy evoluti on
+in the last ∼8Gyr (e.g., Cattaneoet al. 2008; Guo&White
+2008; Parryet al. 2008). However, even in these models themore massive LRGs should grow primarily by major dry
+mergers, a result that appears to be in some conflict with the
+slow growth observed for these galaxies (Brownet al. 2007,
+2008;Cool etal. 2008).
+It is important to distinguish between the merger rate of
+dark matter halos and that of their galaxy tracers. Strictly
+speaking, theory can only predict the former, while the lat-
+ter depends, at least in part, on complex details of baryonic
+physics and dynamical friction (cf. Berrieretal. 2006). Al -
+though our results appear to favor a particular realization of
+ΛCDM models, interpretation must await more detailed and
+realisticsimulationsofthestellarcomponentsofgalaxie sand
+theirfateduringmergersandinteractions.
+WewouldliketothankRobertC.Nicholforsomevaluable
+commentsonthispaper.
+We warmly thank all the present and former staff of the
+Anglo-Australian Observatory for their work in building an d
+operatingthe2dFfacility. The2SLAQSurveyisbasedonthe
+observations made with the Anglo-Australian Telescope and
+for the SDSS. Funding for the SDSS and SDSS-II has been
+providedbythe Alfred P. SloanFoundation,the Participati ng
+Institutions, the National Science Foundation, the U.S. De -
+partmentof Energy,the National Aeronauticsand Space Ad-
+ministration,the JapaneseMonbukagakusho,theMaxPlanck
+Society, and the Higher Education Funding Council for Eng-
+land. TheSDSS Web Site ishttp://www.sdss.org/.
+TheSDSS ismanagedbytheAstrophysicalResearchCon-
+sortium for the Participating Institutions. The Participa ting
+InstitutionsaretheAmericanMuseumofNaturalHistory,As -
+trophysical Institute Potsdam, University of Basel, Unive r-
+sity of Cambridge, Case Western Reserve University, Uni-
+versity of Chicago, Drexel University, Fermilab, the Insti tute
+for Advanced Study, the Japan Participation Group, Johns
+Hopkins University, the Joint Institute for Nuclear Astro-
+physics,theKavliInstituteforParticleAstrophysicsand Cos-
+mology, the Korean Scientist Group, the Chinese Academy
+of Sciences (LAMOST), Los Alamos National Laboratory,
+the Max-Planck-Institute for Astronomy (MPIA), the Max-
+Planck-Institute for Astrophysics (MPA), New Mexico State
+University, Ohio State University, University of Pittsbur gh,
+University of Portsmouth, Princeton University, the Unite d
+StatesNavalObservatory,andtheUniversityofWashington .
+Facilities: AAT (2dF),SDSS.
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+1 An Intermediate-Mass Black Hole of ≥ 500 Solar Masses in the Galaxy ESO 243-49  Sean A. Farrell1,2†, Natalie A. Webb1,2, Didier Barret1,2, Olivier Godet3 & Joana M. Rodrigues1,2  Ultra-luminous X-ray sources are extragalactic objects located outside the nucleus of the host galaxy with bolometric luminosities1 >1039 erg s-1. These extreme luminosities – if the emission is isotropic and below the theoretical (i.e. Eddington) limit, where the radiation pressure is balanced by the gravitational pressure – imply the presence of an accreting black hole with a mass of ~102–105 times that of the Sun. The existence of such intermediate mass black holes is in dispute, and though many candidates have been proposed, none are widely accepted as definitive. Here we report the detection of a variable X-ray source with a maximum 0.2–10 keV luminosity of up to 1.2 x 1042 erg s-1 in the edge-on spiral galaxy ESO 243-49, with an implied conservative lower limit of the mass of the black hole of ~500 Msun. This finding presents the strongest observational evidence to date for the existence of intermediate mass black holes, providing the long sought after missing link between the stellar mass and super-massive black hole populations. Apart from the super-Eddington luminosities of ultra-luminous X-ray (ULX) sources, other circumstantial evidence exists for intermediate mass black holes. Low-temperature disc blackbody spectral components and quasi-periodic oscillations (QPOs) believed to originate at the inner regions of the accretion disc are seen in some ULXs, implying black hole masses above the                                                 1Université de Toulouse, UPS, CESR, 9 Avenue du Colonel Roche, F-31028 Toulouse Cedex 9, France. 2CNRS, UMR5187, F-31028 Toulouse, France. 3Department of Physics and Astronomy, University of Leicester, University Road, Leicester, LE1 7RH, UK. †Present address: Department of Physics and Astronomy, University of Leicester, University Road, Leicester, LE1 7RH, UK.  stellar range2,3. However, neither can be conclusively linked to the innermost disc radius, precluding a definitive mass estimate. In the globular cluster ω Cen the presence of an intermediate mass black hole was derived from the rise in the velocity dispersion profile towards the cluster centre, although a concentration of stellar remnants or a radially biased orbital structure are also valid explanations4. Modelling showed that the cluster dynamics and the current mass are consistent with an accreted stripped dwarf galaxy nucleus5 with a mass of ~107 Msun.  The blackbody X-ray spectrum and bolometric luminosity of ~1039 erg s-1 of the ultra-luminous super-soft-source in M81 may imply the presence of an intermediate mass black hole6, but the source properties are also consistent with a massive white dwarf accreting at a high rate. The luminosity can alternatively be explained through mild super-Eddington accretion and/or beaming7. In addition, the black hole mass was derived by estimating the shape of the X-ray spectrum from widely spaced optical observations. This assumes a smooth continuation between the optical and X-ray spectra, and that the spectra were stable over the ~6 year time-span of the observations. The latter assumption is inconsistent with observational data, as black holes of all sizes demonstrate significant spectral variability over a wide range of timescales8. Whilst investigating the 2XMM serendipitous source catalogue9, we identified 2XMM J011028.1-460421 (HLX-1 for simplicity) at right ascension 01h 10min 28.2s, declination –46° 04’ 22.2” (J2000) with a 3σ positional uncertainty of 0.83’’, observed on 2004 November 23 as part of Obs-Id 0204540201. The position of HLX-1 is coincident with the absorption line galaxy ESO 243-49, at a redshift10 of z = 0.0224 ± 0.0001, 8” from the nucleus yet within the confines of the galaxy (Figure 1). The position and error in each observation were derived in 2 accordance with the methods used for the creation of the 2XMM catalogue9, with the final position and error determined by taking the weighted mean of the two positions. No indication for disruption of the galactic disc, as expected following a galaxy merger, is visible. Contrary to ω Cen, HLX-1 is unlikely to represent the stripped nucleus of an accreted dwarf galaxy. The XMM-Newton European Photon Imaging Camera spectra (Figure 2) of HLX-1 are best fitted by an absorbed power-law model, with parameters consistent with other ULXs1 (Table 1). The derived luminosity (assuming the ESO 243-49 distance) is ~400 times larger than the Eddington value for a 20 Msun black hole, and almost an order of magnitude greater than that of the previously reported most luminous ULX11 (0.2–10 keV unabsorbed Lx < 2 x 1041 erg s-1). A deeper follow-up observation performed with XMM-Newton on 2008 November 28 (Obs-Id: 0560180901) showed a change in the luminosity and the spectral shape (Figure 2). Fourier power spectra were computed using the pn light curves for this observation (6 ms resolution) to test for high-frequency QPOs. No significant modulation was found at frequencies <83 Hz. Fitting of the spectrum required the addition of a soft disc blackbody component (Δχ2 = 155 for 2 degrees of freedom less, Table 1), consistent with other ULXs. Adding this component to the fit of observation 1 did not improve the χ2 significantly (Δχ2 = 2.1 for 2 degrees of freedom less).  Using the posterior predictive p-values method12, we simulated 5,000 data sets with exposure times, background, and energy response identical to the first observation using the best fit spectral parameters of the second observation, and fitted them with an absorbed power law. In less than 1% of the case the power law fit is acceptable and the fitted parameters are consistent with the values reported in Table 1 for the first observation (within the 90% confidence errors). Further, using the same method and an additional 5,000 simulations, we evaluated the significance of the disc blackbody component in the first observation to be ~70%, insufficient to claim it is real. These simulations demonstrate unambiguously that HLX-1 has varied between the two observations. The case for the hyper-luminous nature of HLX-1 depends on its association with ESO 243-49. The probability that any of the real 2XMM sources surrounding ESO 243-49 could be randomly aligned with any of the background galaxies in the field, calculated by 1,000,000 Monte Carlo simulations, is ~9%. However, by taking into account the X-ray spectral shape we can confidently rule out a foreground source. The X-ray spectrum lets us exclude coronal emission from a star, which is typically fitted by multi-temperature thermal models13. Non-thermal emission has been detected from supernovae and from shock-boundaries in shell-type supernova remnants14 with X-ray luminosities between 1037–1041 erg s-1. However, the non-thermal flux decays rapidly after the supernova explosion14, with thermal emission dominating after ~100 days. A Galactic white dwarf accreting from a low mass companion could be visible in X-rays but not show up in our optical image. The spectra of these objects are consistent with emission from a thermal plasma with temperatures >2 keV and non-zero elemental abundances15,16. The spectra from both observations are inconsistent with this model. A quiescent neutron star low mass X-ray binary is also inconsistent with our data, as the spectrum should be thermal (representing emission from the neutron star atmosphere) possibly with the addition of a flat power law component17. The steep power-law spectrum of HLX-1 during observation 1 is not consistent with a neutron star in either the high or low states, and instead indicates the presence of an accreting black hole in the high/soft state18. The derived luminosity for HLX-1 at 10 kpc is only ~1034 erg s-1, too low for a Galactic black hole binary undergoing near-Eddington accretion. Therefore HLX-1 cannot be a foreground Galactic object. Some blazars (a sub-class of radio-loud active galactic nuclei with the jet pointing close to our line of sight) have steep power-law spectra19. However, even the faintest blazars have flux densities >2 mJy at 1.4 GHz (ref.20). Previous radio observations10 detected a source consistent with the core of ESO 243-49 with a 1.4 GHz flux density of 0.16 ± 0.03 mJy, inconsistent with the 3 position of HLX-1 (Figure 1). The relatively low intrinsic absorption also rules-out observing a blazar through the disc of ESO 243-49, yet is consistent with a number of other ULXs21. The association of HLX-1 with ESO 243-49 is almost certain, and thus our derived luminosity valid. The observed variability between observations indicates that HLX-1 cannot be a superposition of multiple lower-luminosity sources.  The maximum luminosity derived from the first observation therefore implies a mass lower limit of ~5,400 Msun, assuming Eddington accretion and isotropic emission. Relativistic jets have been observed in a number of Galactic black hole X-ray binaries during the low/hard state8, where the accretion rate and luminosity decrease and the spectrum flattens. If the source is viewed at angles close to the jet axis, relativistic boosting could amplify the flux so that the luminosity appears to exceed the Eddington value. Modelling of stellar mass black hole binaries has shown that relativistic boosting could produce apparent luminosities five times the Eddington limit22, implying a black hole mass of >1,000 Msun for HLX-1. However, relativistic boosting is unlikely as a flat spectrum – as opposed to the steep spectrum we observe in the first observation – is predicted22, and still requires an intermediate mass black hole for luminosities >1041 erg s-1. Geometric beaming whereby a thick accretion disc collimates the X-ray emission coupled with super-Eddington accretion can explain apparent luminosities up to ~1040 erg s-1, with higher luminosities for hydrogen-poor accretion7. A 20 Msun black hole with an apparent luminosity of 1042 erg s-1 implies a low beaming factor of b ≈ 0.01, with a mass accretion rate of ~2.6 times the Eddington value for hydrogen-poor accretion7. However, the derived luminosity is high enough above the Eddington limit that matter would be easily accelerated to Lorentz factors of ~5–10, leading to relativistic jets and flat spectra23. The combination of the derived luminosity and steep spectrum of HLX-1 in the first observation thus rules out relativistic and geometric beaming. Accreting stellar mass black holes may exceed the Eddington luminosity by up to a factor of ten24, implying an Eddington luminosity of ~1041 ergs s-1 (0.2–10 keV), and a lower mass limit of ~500 Msun for HLX-1. This limit is derived assuming a bolometric luminosity, while our derived luminosity is in the 0.2–10 keV band. The bolometric luminosity is likely to exceed our value, so our mass lower limit is a conservative estimate. HLX-1 therefore presents the strongest case to date for the existence of intermediate mass black holes.  Received 25 August 2008: accepted 20 April 2009.   1. Roberts, T. P. X-ray observations of ultraluminous X-ray sources. Astrophys. Space Sci. 311, 203–212 (2007). 2. Miller, J. M., Fabbiano, G., Miller, M. C. & Fabian, A. C. X-ray spectroscopic evidence for intermediate-mass black holes: cool accretion disks in two ultraluminous X-ray sources. Astrophys. J. 585, L37–L40 (2003). 3. Casella, P., Ponti, G., Patruno, A., Belloni, T., Miniuitti, G. & Zampieri, L. Weighing the black holes in ultraluminous X-ray sources through timing. Mon. Not. R. Astron. Soc. 387, 1707–1711 (2008). 4. Noyola, E., Gebhardt, K. & Bergmann, M. Gemini and Hubble space telescope evidence for an intermediate-mass black hole in ω Centauri. Astrophys. J. 676, 1008–1015 (2008). 5. Bekki, K. & Freeman, K. C. Formation of ω Centauri from an ancient nucleated dwarf galaxy in the young Galactic disc. Mon. Not. R. Astron. Soc. 346, L11–L15 (2003). 6. Liu, J. & Di Stefano, R. An ultraluminous supersoft X-ray source in M81: an intermediate-mass black hole? Astrophys. J. 674, L73–L76 (2008). 7. King, A. R. Accretion rates and beaming in ultraluminous X-ray sources. Mon. Not. R. Astro. Soc. 385, L113–L115 (2008). 8. Fender, R. Jets from X-ray binaries, in Compact stellar X-ray sources (eds. Lewin, W. & van der Klis, M.) 381–419 (Cambridge University Press, Cambridge, 2006). 9. Watson, M. G. et al. The XMM-Newton serendipitous survey. VI. The second XMM-Newton serendipitous source catalogue. Astron. Astrophys. 493, 339–373 (2009).  10. Afonso, J. et al. The phoenix deep survey: spectroscopic catalog. Astrophys. J. 624, 135–154 (2005). 11. Miniutti, G. et al. Have we detected the most luminous ULX so far? Mon. Not. R. Astron. Soc. 373, L1–L5 (2006). 12. Hurkett, C. P. et al. Line searches in Swift X-ray spectra. Astrophys. J. 679, 587–606 (2008). 13. Güdel, M. X-ray astronomy of stellar coronae. Astron. Astrophys. Rev. 12, 71–237 (2004). 14. Immler, S. & Lewin, W. H. G. X-ray supernova, in Supernovae and Gamma-Ray Bursters (ed. Weiller, K.) 91–111 (Springer-Verlag, Berlin, 2003). 4 15. Baskill, D. S., Wheatley, P. J. & Osborne, J. P. The complete set of ASCA X-ray observations of non-magnetic cataclysmic variables. Mon. Not. R. Astron. Soc. 357, 626–644 (2005).  16. Hilton, E. J. et al. XMM-Newton observations of the cataclysmic variable GW Librae. Astron. J. 134, 1503–1507 (2007). 17. Jonker, P. G. Constraining the neutron star equation of state using quiescent low-mass X-ray binaries. AIP Conf. Proc. 983, 519–526 (2008). 18. Done, C., Gierliński, M. & Kubota, A. Modelling the behaviour of accretion flows in X-ray binaries. Everything you always wanted to know about accretion but were afraid to ask. Astron. Astrophys. Rev. 15, 1–66 (2007). 19. Giommi, P. et al. A catalog of 157 X-ray spectra and 84 spectral energy distributions of blazars observed with BeppoSAX, in Blazar Astrophysics with BeppoSAX and Other Observatories (eds Giommi, P., Massaro, E. & Palumbo, G.) 63–99 (ESA-ESRIN, Frascati, 2002). 20. Turriziani, S., Cavazzuti, E. & Giommi, P. ROXA: a new multi-frequency large sample of blazars selected with SDSS and 2dF optical spectroscopy. Astron. Astrophys. 472, 669–704 (2007). 21. Berghea, C. T., Weaver, K. A., Colbert, E. J. M. & Roberts, T. P. Testing the paradigm that ultra-luminous X-ray sources as a class represent accreting intermediate-mass black holes. Astrophys. J. 687, 471–487 (2008). 22. Freeland, M., Kuncic, Z., Soria, R. & Bicknell, G. V. Radio and X-ray properties of relativistic beaming models for ultraluminous X-ray sources. Mon. Not. R. Astro. Soc. 372, 630–638 (2006). 23. Abramowicz, M. A., Ellis, G. F. R & Lanza, A. Relativistic effects in superluminal jets and neutron star winds. Astrophys. J. 361, 470–492 (1990). 24. Begelman, M. C. Super-Eddington fluxes from thin accretion disks? Astrophys. J. 568, L97–L100 (2002). 25. Kalberla, P. M. W. et al. The Leiden/Argentine/Bonn (LAB) survey of Galactic HI. Final data release of the combined LDS and IAR surveys with improved stray-radiation corrections. Astron. Astrophys. 440, 775–782 (2005).  Acknowledgements We thank N. Schartel for granting an observation under the XMM-Newton project scientist discretionary time program. We thank R. Belmont, A. King, J-P. Lasota, K. Mukai, T. Roberts, S. Rosen, S. Sembay and M. Watson for useful discussions. S.A.F acknowledges funding from the CNES. S.A.F. and O.G. acknowledge STFC funding. This work made use of the 2XMM Serendipitous Source Catalogue constructed by the XMM-Newton Survey Science Centre on behalf of ESA. We thank the Swift team for performing a TOO observation which provided justification for an additional observation with XMM-Newton. This work was based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.   Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to S.A.F. (saf28@star.le.ac.uk) or N.A.W (natalie.webb@cesr.fr).  
+  Figure 1 | R-band optical image of ESO 243-49 obtained with the Very Large Telescope. The centre of the red circle (X) indicates the position of HLX-1, with the radius representing a 3σ positional uncertainty of 0.83”. The centre of the white ellipse (R) indicates the position of the radio detection from the Phoenix Deep Survey10, with the radii representing the 3σ uncertainty. The blue ellipse indicates the elliptical confines of the galaxy.   
+  Figure 2 | European Photon Imaging Camera X-ray spectra of HLX-1. Top panel: the unfolded pn (green), MOS1 (red) and MOS2 (blue) spectra of the first observation fitted with the best-fit absorbed power-law model (solid black line). Bottom panel: the unfolded spectra of the second observation, fitted with the best-fit absorbed power law (black dot-dashed line) plus disc blackbody (dashed line) model. The error bars indicate the 90% confidence limits. 
+5  Table 1 | X-ray spectral parameters of HLX-1 Parameter Observation 1 Observation 2 Unit + 0.03 + 0.01 nHtotal 0.08 - 0.03 0.04 - 0.01 1022 atoms cm-2 + 0.03 + 0.01 nHintrinsic 0.06 - 0.03 0.02 - 0.01 1022 atoms cm-2  + 0.01 Tin …  0.18 - 0.01 keV  + 6 NormDBB …  29 - 5  + 0.3 + 0.4 Γ 3.4 - 0.3 2.2 - 0.3  + 1 + 0.8 NormPL 9 - 1 2.2 - 0.6 10-5 photons cm-2 s-1 keV-1 χ2/dof 113.4/108 333.9/329  + 0.1 + 0.6 Lx  11 - 4.0 6.4 - 0.6 1041 erg s-1 nHtotal = total neutral hydrogen column density; nHintrinsic = neutral hydrogen column density intrinsic to the source, after taking into account the Galactic absorption25 of 0.0179 x 1022 atoms cm-2; Tin = temperature at inner disc radius; NormDBB = normalisation of disc blackbody component; Γ = power-law photon index; NormPL = normalisation of power law component; χ2/dof = χ2 statistic and associated degrees of freedom (dof) for the spectral fitting; Lx = unabsorbed X-ray luminosity in the 0.2–10 keV band.  
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+arXiv:1001.0568v1  [astro-ph.HE]  4 Jan 2010Astronomy & Astrophysics manuscript no. aa13404 c/circlecopyrtESO 2018
+October 30, 2018
+The peculiar high-mass X–ray binary 1ES 1210 −646⋆
+N. Masetti1, R. Landi1, V. Sguera1, F. Capitanio2,
+L. Bassani1, A. Bazzano2, A.J. Bird3, A. Malizia1and E. Palazzi1
+1INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica di Bo logna, Via Gobetti 101, I-40129 Bologna, Italy
+2INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica di Ro ma, Via Fosso del Cavaliere 100, I-00133 Rome, Italy
+3School of Physics & Astronomy, University of Southampton, S outhampton, Hampshire, SO17 1BJ, United Kingdom
+Received 5 October 2009; accepted 7 December 2009
+ABSTRACT
+Using data collected with the BeppoSAX ,INTEGRAL andSwiftsatellites, we report and discuss the results of a study
+on the X–ray emission properties of the X–ray source 1ES 1210 −646, recently classified as a high-mass X–ray binary
+through optical spectroscopy. This is the first in-depth ana lysis of the X–ray spectral characteristics of this source. We
+found that the flux of 1ES 1210 −646 varies by a factor of ∼3 on a timescale of hundreds of seconds and by a factor of
+at least 10 among observations acquired over a time span of se veral months. The X–ray spectrum of 1ES 1210 −646 is
+described using a simple powerlaw shape or, in the case of INTEGRAL data, with a blackbody plus powerlaw model.
+Spectral variability is found in connection with different fl ux levels of the source. A strong and transient iron emission
+line with an energy of ∼6.7 keV and an equivalent width of ∼1.6 keV is detected when the source is found at an
+intermediate flux level. The line strength seems to be tied to the orbital motion of the accreting object, as this feature
+is only apparent at the periastron. Although the X–ray spect ral description we find for the 1ES 1210 −646 emission
+is quite atypical for a high-mass X–ray binary, the multiwav elegth information available for this object leads us to
+confirm this classification. The results presented here allo w us instead to definitely rule out the possibility that 1ES
+1210−646 is a (magnetic) cataclysmic variable as proposed previo usly and, in a broader sense, a white dwarf nature
+for the accretor is disfavoured. X–ray spectroscopic data a ctually suggest a neutron star with a low magnetic field as
+the accreting object in this system.
+Key words. Stars: binaries: general — X–rays: binaries — Stars: neutro n — Techniques: spectroscopic — X–rays:
+individuals: 1ES 1210 −646
+1. Introduction
+X–ray binaries are interacting systems in which a compact
+object, neutron star (NS) or black hole (BH) is accreting
+matter from a companion star. Depending on the mass of
+the latter, these systems arebroadlydivided into high-mass
+X-raybinaries(HMXBs; ifthe companionisamassiveearly
+type star) and low mass X-ray binaries (LMXBs; in the
+cases in which the secondary star is of late spectral type,
+in general with mass <∼1 M⊙).
+Several X–ray binaries relatively bright in X–rays could
+not be properly identified and classified until recently due
+to the lack of a precise (arcsecond-sized) X–ray position:
+indeed, the X–ray characteristics may not provide enough
+elements for the correct classification of an X–ray binary
+as a HMXB or as a LMXB. This can therefore only be
+achieved through a multiwavelength approach, mostly with
+the synergic use of information acquired in X–ray and opti-
+calbands(althoughX–raytimingproperties,suchasbursts
+or pulsations, can give precise indications on the nature of
+Send offprint requests to : N. Masetti
+(masetti@iasfbo.inaf.it)
+⋆Partly based on X-ray observations with INTEGRAL, an
+ESA project with instruments and science data centre funded
+by ESA member states (especially the PI countries: Denmark,
+France, Germany, Italy, Switzerland, Spain), Czech Republ ic
+and Poland, and with the participation of Russia and the USA.the accretor). The subject of this paper, 1ES 1210 −646, is
+indeed an object which fits the above description.
+The X-ray source 1ES 1210 −646 was first detected by
+theUhurusatellite and reported (as 4U 1210 −64) in the
+4thUhuruCatalogue (Forman et al. 1978) as a relatively
+bright and variablesourcewith a 2–6keVflux of8.9 ×10−11
+erg cm−2s−1, assuming a Crab-like spectrum (below we
+will always assume this spectral shape for our X–ray flux
+estimates unless explicitly stated otherwise). It was also
+detected in the slew surveys performed with the Einstein
+(Elvis et al. 1992) and EXOSAT (Reynolds et al. 1999)
+satellites, at fluxes 1.7 ×10−11erg cm−2s−1(0.16–3.5 keV)
+and∼7×10−10erg cm−2s−1(1–8 keV), respectively. More
+recently, this source was detected by the wide-field cameras
+(WFCs; Jager et al. 1997) onboard BeppoSAX (Boella et
+al. 1997) at an average flux of 2.57 ×10−10erg cm−2s−1in
+the 2–10 keV (Verrecchia et al. 2007), and at hard X–rays
+above 20 keV in the surveys (Bird et al. 2007; Krivonos et
+al. 2007) obtained with the IBIS instrument (Ubertini et
+al. 2003) onboard the INTEGRAL satellite (Winkler et al.
+2003), at a flux of 1.1 ×10−11erg cm−2s−1in the 20–100
+keVband(Birdetal.2007).High-energyemissionfrom1ES
+1210−646 was also detected by Swift/BAT in the 14–195
+keV band with a flux ∼2×10−11erg cm−2s−1(Tueller et
+al. 2009; Cusumano et al. 2009).
+Unfortunately, the above X–ray detections had posi-
+tional errors of the order of several arcminutes or worse,
+which severely hampered the search for longer-wavelength2 N. Masetti et al.: 1ES 1210 −646, a peculiar high-mass X–ray binary
+counterparts and in turn the determination of the nature
+of this source.
+Things changed thanks to a pointed observation per-
+formed with the X–ray telescope (XRT, Burrows et al.
+2005) onboard the Swiftsatellite (Gehrels et al. 2004): ac-
+cording to Revnivtsev et al. (2007), these data revealed a
+clear counterpart of 1ES 1210 −646 at coordinates (J2000)
+RA = 12h13m14.s702, Dec = −64◦52′30.′′89, with a posi-
+tionuncertaintyof ∼4′′. Revnivtsevetal.(2007)alsostated
+that this source may be an intermediate polar (i.e. mag-
+netic) cataclysmic variable (CV) on the basis of its X–ray
+spectral appearance, in particular because of the presence
+of a strong iron emission line at ∼6.7 keV.
+To firmly identify the nature of this source, Masetti et
+al. (2009) included the single optical object consistent with
+the XRT position of 1ES 1210 −646 in their spectroscopic
+follow-up of INTEGRAL sources: from its spectrum they
+concluded that it is more likely a HMXB rather than a CV.
+This was corroborated by the discovery of a periodicity of
+6.7 d (Corbet & Mukai 2008) from the analysis of the long-
+term X-ray light curve of the source as seen by the All-Sky
+Monitor (ASM) onboard the RXTEsatellite: if this modu-
+lation is interpreted as the orbital period of 1ES 1210 −646,
+as Corbet & Mukai (2008) did, it would strengthen the
+HMXB nature for this source. These authors also reported
+that the 2.5–20 keV X–ray spectrum of the source acquired
+close to the orbital maximum is typical of a HMXB; they
+also confirmed the presence of the iron emission line.
+Despite the number of multiwavelength information
+gathered in the past few years on 1ES 1210 −646, a deeper
+study of this source in X–rays is still lacking. In order
+to fill this gap and better constrain and define the ob-
+servational X–ray properties of 1ES 1210 −646, we col-
+lected and analysed the three available Swift/XRT observa-
+tions of this source, along with the INTEGRAL /IBIS data
+from the 3rdIBIS survey (Bird et al. 2007) and the pub-
+licly available data collected with the two-unit coded-mask
+X-ray monitor JEM-X (Lund et al. 2003), also onboard
+INTEGRAL ; we also performed a deeper analysis of the
+archivalBeppoSAX /WFC data.
+The paperisstructuredasfollows.Section 2willpresent
+the X–ray observations of 1ES 1210 −646 considered in this
+work; in Sect. 3 the results of this multi-observatory X-
+ray campaign will be given, and in Sect. 4 a discussion on
+the source will be presented. Finally, in Sect. 5 we will out-
+line the conclusions.Throughout the paper, if not indicated
+otherwise, uncertainties and limits are given at a 90% con-
+fidence level.
+2. Observations and data analysis
+2.1.Swift: XRT data
+The field of 1ES 1210 −646 was observed three times with
+XRT. All pointings were performed in photon counting
+mode (see Burrows et al. 2005 for details on this observing
+mode). The log of these observations is reported in Table 1.
+The data reduction was performed using the XRTDAS
+v2.0.1 standard data pipeline package ( xrtpipeline
+v0.10.6) to produce the final cleaned event files. As in ob-
+servations #1 and #3 the XRT count rate of the source
+was high enough to produce data pile-up, we extracted the
+events in an annulus centred on the source and with an
+external radius of 57′′and 68′′, respectively. The size ofTable 1. Log of the Swift/XRT observations used in this
+paper.
+Obs. Start day Start time On-source Avg. 0.3–10 keV
+number (UT) time (ks) countrate (cts s−1)
+#1 10 Dec 2006 17:48:48 9.9 0.939 ±0.014
+#2 07 Feb 2008 15:05:24 2.9 0.369 ±0.011
+#3 24 Apr 2008 07:49:26 6.9 1.755 ±0.031
+the inner circle was determined following the procedure de-
+scribed in Romano et al. (2006) and was 9′′for observation
+1 and 21′′for observation 3. In observation #2 the pile-
+up was not an issue, so the data were extracted from a
+circle of a radius of 47′′. The source background was mea-
+sured within a circular region with a radius of 95′′located
+far from the source. The ancillary response file was gen-
+erated with the task xrtmkarf (v0.5.2) within FTOOLS1
+(Blackburn 1995), and accounts for both extraction region
+and PSF pile-up correction. We used the latest spectral re-
+distribution matrices in the calibration database2(CALDB
+2.3) maintained by HEASARC.
+2.2.INTEGRAL : IBIS and JEM-X data
+We extracted the spectral and time-series data of this
+source collected with ISGRI (Lebrun et al. 2003), which
+is the low-energy coded-mask detector of the IBIS instru-
+ment onboard INTEGRAL . The ISGRI data set considered
+in this analysis consists of events in the 17–300 keV band
+coming from both fully-coded and partially-coded observa-
+tions of the field of view of 1ES 1210 −646. The time res-
+olution for these data was that typical of the IBIS science
+windows (ScWs; ∼2 ks). Details on the whole procedure
+can be found in Bird et al. (2007). Hard X–ray long-term
+light curves and a time-averaged spectrum were then ob-
+tained fromthe availabledataand bythe method described
+in Bird et al. (2006, 2007), for a total of 1168 ks on-source
+collected in the time interval between October 2002 and
+April 2006.
+Publicly-available 3–35 keV band JEM-X data of 1ES
+1210−646were also collected. They were reduced and anal-
+ysed as well with the OSA v7.0 software (for details about
+the JEM-X data analysis see Westergaard et al. 2003). We
+searched the entire JEM-X public data archive (from rev-
+olution 46 to 574) for pointings where 1ES 1210 −646 was
+within the JEM-X fully coded field of view ( ∼2.◦4 radius).
+As a result, a total of 70 ScWs were selected, spanning the
+period from 28 May 2003 (Rev. 76) to 26 Jun 2007 (Rev.
+574).
+It was found specifically that 1ES 1210 −646 entered
+the fully-coded field of view of Unit 2 of JEM-X for 18 ks
+between 28 May 2003 and 3 Jun 2003, and for 116 ks in
+the one of Unit 1 between 20 Dec 2004 and 26 Jun 2007.
+In the first case, the source was not detected, whereas in
+the second instance it was in the 3–10 keV range; a more
+in-depth inspection of the latter JEM-X data set revealed
+that the source displayed two different states: a high flux
+1available at:
+http://heasarc.gsfc.nasa.gov/ftools/
+2availableat: http://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/cal dbintro.htmlN. Masetti et al.: 1ES 1210 −646, a peculiar high-mass X–ray binary 3
+Fig.1.Background-subtracted 2–10 keV X–ray light curves of 1ES 1210 −646as observed with XRT during observations
+#1 (upper panel), #2 (central panel) and #3 (lower panel). All curv es are rebinned at 300 s. Times are expressed in
+seconds from the start time of each observation (see Table 1 for d etails). Marked variability on timescales of the order of
+hundreds of seconds can be seen, especially during observation #1 .
+state from Rev. 322 (2 Jun 2005) to Rev. 324 (8 Jun 2005)
+for a total exposure of ∼22 ks (26- σdetection) and a low
+flux state, fainter by afactor of ∼4,in the remainingpartof
+the data set, for a total exposure of ∼94 ks (8-σdetection).
+The statistics were sufficient to extract a meaningful JEM-
+Xspectruminthe3–10keVbandonlyduringthehighstate
+of the source. Information on the source fluxes as detected
+with JEM-X are reported in Sect. 3.2.2.
+2.3.BeppoSAX : WFCs data
+The two WFCs on board BeppoSAX were sensitive in the
+2–28 keV range and were mounted 180◦apart and per-
+pendicular to the pointing direction of the satellite, thus
+looking at two different sky zones during any pointing. In
+this way the WFCs secondary mode observations covered
+almost all of the sky with at least one of their serendipity
+pointings (typically with 100 ks duration) during the six
+years of the BeppoSAX operational life.We searchedfor 1ES 1210 −64detections in the archive3
+of all available BeppoSAX WFC pointings. The source has
+been observed for a total of 1.5 Ms between April 1996
+and April 2002, although it was detected in single pointing
+observations only between July 1996 and December 1997,
+in agreement with the highest source flux detections of the
+RXTE/ASM monitor4(see also Corbet & Mukai 2008).
+Unfortunately, the WFCs were not sensitive enough in
+terms of spectral capability, and 1ES 1210 −646 was too
+faint in each single pointing and also in the sum of the
+observations to permit the extraction of meaningful spec-
+tra. Thus, only information on the source flux could be
+extracted from the WFC data (see Sect. 3.2.3 for the re-
+sults).
+3The WFCs data archive collected by the INAF-IASF of
+Rome (Italy) has been used for this analysis
+4http://xte.mit.edu/ASM lc.html4 N. Masetti et al.: 1ES 1210 −646, a peculiar high-mass X–ray binary
+3. Results
+3.1. Light curves
+Figure1 reportsthe 2–10keVlight curvesof1ES1210 −646
+acquired during the three XRT pointings on this source.
+The data are background-subtracted and rebinned at 300
+s. Variability (up to a factor of ∼3 in flux) of the order of
+hundreds of seconds can be noticed. Apart form this, one
+can see that the average 2–10 keV source countrate varies
+notably among the observations, going from ∼0.15 cts s−1
+of XRT pointing #2 to ∼1 cts s−1in XRT pointing #3.
+In order to see if these erratic behaviours implied spec-
+tral changes as a function of the source intensity within
+each XRT observation, we constructed an X–ray ‘colour-
+intensity diagram’ (see Fig. 2) using the total 0.3–10 keV
+intensity and the hardness ratio between the 2–10 keV and
+the 0.3–2 keV count rates. In the figure, points from differ-
+ent observations are indicated with different symbols. From
+the diagram, it appears that the source gets slightly harder
+asits intensityrises,althoughit seems that within the same
+XRT observation 1ES 1210 −646 shows a fairly constant
+hardness ratio: ∼1 for XRT pointing #1 and #2, and ∼1.6
+for XRT pointing #3.
+Fig.2.X–ray colour-intensity diagram for 1ES 1210 −646
+constructedusingthethreeXRT observationsofthe source.
+PointsareobtainedfromthesourceX–raylightcurves,sub-
+tracted of the background and rebinned at 300 s. Triangles,
+squares and circles refer to XRT pointings #1, #2 and #3,
+respectively. Error bars show 1- σconfidence level uncer-
+tainties for each data point.
+Unfortunately, the wide gaps in the temporal coverage
+do not allow us to perform a meaningful timing analysis on
+any of the XRT pointings. For the same reasonand because
+of sensitivity limits, no long-term periodicity analysis on
+the WFC, IBIS and JEM-X light curves of 1ES 1210 −646
+could reasonably be performed to confirm the modulationfound byCorbet&Mukai(2008).We notehoweverthat the
+source variability appears to be stronger at lower energies
+(<20 keV), given that in the IBIS data it is continuously
+detected as a weak persistent object.
+3.2. Spectra
+In order to perform the spectral analysis on the X–ray data
+collected for 1ES 1210 −646, the spectra from the detectors
+of all spacecraft but IBIS were rebinned to oversample by
+a factor of 3 the full width at half maximum of the en-
+ergy resolution and to have a minimum of 20 counts per
+bin to reliably use the χ2statistics; for IBIS, the rebinning
+was chosen to maximise the signal-to-noise ratio (S/N) for
+each bin. Data were then selected for all detectors when
+a sufficient number of counts were obtained in the energy
+rangeswhere the instrument responsesarewell determined.
+For theSwift/XRT observations, we considered the spectra
+averagedovereach pointinggiven that no substantialvaria-
+tions in the spectral shape during each pointed observation
+were suggested by the inspection of the colour-intensity di-
+agram of the source (See Sect. 3.1). For INTEGRAL the
+data were accumulated over the periods in which each in-
+strument could secure spectra with acceptable S/N, that
+is, between 2 and 8 Jun 2005 for JEM-X and the whole
+interval of the third survey for IBIS. Due to the tradeoff
+between spectral coverage and S/N, the spectral analysis
+was performed over the 0.3–9 keV, 3–10 keV and 20–100
+bands for the XRT, JEM-X and IBIS data, respectively.
+To analyse the X–ray spectra we used the package
+XSPEC (Dorman & Arnaud 2001) v12.4.0ad. Several
+simple models like blackbody (BB), powerlaw, thermal
+bremsstrahlung, and hot diffuse gas emission ( mekalin
+XSPEC; Mewe et al. 1985), were tested for the description
+of the spectral data. Spectral fitting which used more com-
+plex models like a powerlaw modified with a cutoff did not
+improve the results over those obtained with the aforemen-
+tioned simpler desciptionsbecause ofthe relativelylowS/N
+of the X–ray spectra presented here.
+To all models tested in this paper (the best-fit mod-
+els for each satellite pointing are reported in Table 2)
+we applied a photoelectric absorption column, modelled
+with the cross sections of Morrison & McCammon (1983;
+wabsin XSPEC notation) and with solar abundances as
+given by Anders & Grevesse (1989), to describe the line-
+of-sight Galactic neutral hydrogen absorption towards 1ES
+1210−646. For clarity, the luminosities listed in Table 2 re-
+fer to only one ofthe reportedmodels (see Note in Table 2);
+it can however be said that they are basically independent
+of the considered best-fit model. Below, the acronym “dof”
+means “degrees of freedom”.
+3.2.1. XRT data
+The spectral analysis of the three XRT observations (see
+Table 2) indicates that the models which best describe
+the data are those with the powerlaw and the thermal
+bremsstrahlung, absorbed by a neutral hydrogen column
+with density NH∼4-5×1021atoms cm−2. This value is
+comparable to the optical reddening along the line of sight
+as derived by Schlegel et al. (1998), once the empirical for-
+mula of Predehl & Schmitt (1995) is applied; this indicates
+that the measured absorption is most likely of interstellarN. Masetti et al.: 1ES 1210 −646, a peculiar high-mass X–ray binary 5
+origin and none is locally present in 1ES 1210 −646. The
+same conclusion was reached by Masetti et al. (2009) on
+the basis of the properties of the optical counterpart of this
+source.
+Looking at Table 2 one also notes that with the power-
+law description the spectrum is relatively hard at the low-
+est X–ray fluxes (XRT pointing #2), gets softer at an in-
+termediate flux level (XRT pointing #1) and eventually
+turns harder when the source is brightest (XRT pointing
+#3); similarly the temperature steadily rises in the case
+of bremsstrahlung fits from ∼7.3 to∼22 keV from XRT
+pointing #1 to XRT pointing #3.
+The most fascinating spectral feature is the presence
+of a strong and variable iron emission line around 6.7 keV
+which is evidently present only on XRT observation #1 (as
+already noted by Revnivtsev et al. 2007) at an intermedi-
+ate source flux and which is not detected at low and high
+flux levels (see Fig. 3). Assuming a Gaussian shape for the
+iron emission, the probability of a chance improvement of
+the fit over a simple powerlaw description in the spectrum
+of XRT pointing #1 is 5.9 ×10−10according to the F-test
+distribution (Press et al. 1992). Although we are aware of
+the caveats and limitations of the F-test application in as-
+trophysical context (e.g., Protassov et al. 2002), the above
+value for the chance improvement probability leaves little
+doubt that the detection of this emission line is quite ro-
+bust. However, no emission is present in the X–ray spectra
+extracted from XRT observations #2 and #3 (see central
+and lower panels of Fig. 3).
+Table 2 also presents the best-fit parameters describing
+this emission line. As one can see, they are largely indepen-
+dent of the model assumed for the description of the X–ray
+spectral continuum. Table 3 reports values or upper limits
+for the equivalent width (EW) of any iron emission line at
+∼6.7 keV in the spectrum of each pointed observation. As
+is evident, the line EW has a variability of at least a factor
+of 8.
+For the sake of completeness, we report that the BB
+emission does not provide a satisfactory description of the
+spectral data; the mekalmodel is not a viable option ei-
+ther, as it implies a variation of the plasma metal abun-
+dances of a factor of 100 among the three XRT observa-
+tions due to the presence of the highly variable iron line
+described above. Likewise, any combination of two simple
+models (e.g., BB plus powerlaw) does not produce signifi-
+cant improvementsofthe spectral fitting in anyofthe three
+XRT pointings.
+3.2.2. JEM-X and IBIS data
+Next, we consider the combined JEM-X and IBIS spectra
+between 3 and 100 keV (Fig. 4). We introduced a constant
+(Ccalib) to allow for intercalibration differences between the
+two instruments; this constant was left free to vary in the
+fits. We are aware that since the IBIS spectrum is accu-
+mulated over a much longer time span with respect to the
+JEM-X one (see Sect. 2.2), the flux variations seen from
+1ES 1210 −646in particular below 10 keV may produce dif-
+ferent normalisations for the spectra acquired by the two
+instruments.
+Given that ourpast experience tells us that the intercal-
+ibration constant between these two instruments is ∼1, we
+suggest that different values for this constant are likely to
+be due to the flux variability mentioned above. In any case,0.010.1Counts s−1 keV−1
+1 0.5 2 5−4−202Residuals ( σ)
+Energy (keV)
+0.010.1Counts s−1 keV−1
+1 0.5 2 5−202Residuals ( σ)
+Energy (keV)
+0.010.1Counts s−1 keV−1
+1 0.5 2 5−202Residuals ( σ)
+Energy (keV)
+Fig.3.X–ray spectra of 1ES 1210 −646 acquired during
+XRTobservations#1(upperpanel),#2(centralpanel)and
+#3 (lower panel), together with one of the best-fit models
+(an absorbed powerlaw in all cases; see Table 2 for details)
+and the corresponding fit residuals.
+given that 1ES 1210 −646 is more likely to be detected by
+IBIS during its high state, we consider that the JEM-X and
+IBIS spectra are representative of the same (high-intensity)
+state for the source and that it is therefore meaningful to
+combine them.
+In the fits we fixed the neutral hydrogen column den-
+sity to 5×1021atoms cm−2, given that the lack of spectral
+coverage below 3 keV (that is, the spectral range which is6 N. Masetti et al.: 1ES 1210 −646, a peculiar high-mass X–ray binary
+Table 2. Best-fit parameters for the X–ray spectra of 1ES 1210 −646 from the observations described in this paper.
+Model XRT #1 XRT #2 XRT #3 JEM-X+IBIS
+parameter (0.3–9 keV) (0.3–9 keV) (0.3–9 keV) (3–100 keV)
+powerlaw (+Fe line) :
+χ2/dof 207.3/182 69.8/51 188.6/147 —
+NH(×1021cm−2) 5.2±0.4 5.0+0.8
+−0.7 5.1±0.5 —
+Γ 1.80±0.07 1.63+0.15
+−0.141.45±0.08 —
+Normalisation∗11.5+0.9
+−1.00.47+0.09
+−0.07 33±3 —
+E(keV) 6.71+0.05
+−0.03 — — —
+σ(keV) 0.16+0.05
+−0.04 — — —
+I(×10−4ph cm−2s−1) 5.8+1.3
+−1.2 — —
+bremsstrahlung (+Fe line) :
+χ2/dof 221.7/182 72.3/51 186.3/147 —
+NH(×1021cm−2) 4.3±0.3 4.3+0.7
+−0.6 4.7±0.4 —
+kTbr(keV) 7.3+1.4
+−1.1 12+9
+−4 22+11
+−6 —
+Normalisation∗12.1+0.6
+−0.50.56±0.04 48±2 —
+E(keV) 6.72±0.04 — — —
+σ(keV) 0.17+0.06
+−0.04 — — —
+I(×10−4ph cm−2s−1) 6.2+1.2
+−1.3 — —
+BB+powerlaw :
+χ2/dof — — — 99.7/86
+NH(×1021cm−2) — — — [5.0]
+kTBB(keV) — — — 1.54+0.06
+−0.08
+R†
+BB(km) — — — 5.9+1.5
+−1.9
+Γ — — — 2.1±0.7
+N∗
+pow — — — 2.9+46.7
+−2.8
+Ccalib — — — 2.0+3.5
+−1.7
+bremsstrahlung +powerlaw :
+χ2/dof — — — 120.4/86
+NH(×1021cm−2) — — — [5.0]
+kTbr(keV) — — — 8.7+2.7
+−1.7
+N∗
+br — — — 72+10
+−25
+Γ — — — 1.5+0.7
+−4.3
+N∗
+pow — — — 1.7+32.1
+−1.7
+Ccalib — — — 0.13+0.10
+−0.06
+X–ray luminosity ‡:
+0.5–2 keV 0.2 0.1 0.7 —
+2–10 keV 0.4 0.2 1.9 2.0
+20–100 keV — — — 0.1
+Note: the above luminosities, corrected for interstellar Galac tic absorption, are computed
+assuming a distance d= 2.8 kpc (Masetti et al. 2009) and are expressed in units of
+1035erg s−1. In the cases in which the X–ray data could not completely cov er
+the X–ray interval of interest for the luminosity determina tion, an extrapolation of
+the best-fit model is applied.
+∗: in units of 10−3
+†: computed assuming a distance d= 2.8 kpc to the source
+‡: computed using the powerlaw model for the XRT spectra
+and the BB+powerlaw model for the JEM-X+IBIS spectrum
+most sensitive to the N Habsorption effects) does not allow
+us to determine this parameter through the spectral fits.
+The first remarkable issue of the JEM-X+IBIS spec-
+tral analysis is that no simple spectral model is able to
+fit the whole spectrum simultaneously, as none of them
+produces fits with reduced χ2<2. We thus tried combi-
+nations of two models; the best descriptions are obtained
+with BB+powerlaw and bremsstrahlung+powerlaw models
+(see Table 2).Moreover, no iron emission line is apparent in the com-
+bined spectrum, with an upper limit on the EWasreported
+in Table 3. This is actually not surprising, given that the
+source emission level during which the considered JEM-X
+spectrum was accumulated was similar to the one of XRT
+observation#3, in which no iron emission line was detected
+either.
+Concerning instead the JEM-X observations for which
+no spectral information could be obtained, we found thatN. Masetti et al.: 1ES 1210 −646, a peculiar high-mass X–ray binary 7
+10−30.010.1Counts s−1 keV−1
+10 100 5 20 50−202Residuals ( σ)
+Energy (keV)
+Fig.4.Combined averaged JEM-X+IBIS X–ray spectrum
+of 1ES 1210 −646, plotted together with one of the best-fit
+models(absorbedBBpluspowerlaw;seeTable2fordetails)
+and the corresponding fit residuals.
+Table 3. Values and upper limits to the EW of the iron
+emission line in 1ES 1210 −646.
+Observation EW (keV)
+XRT #1 1.6 ±0.3
+XRT #2 <0.9
+XRT #3 <0.2
+JEM-X <0.4
+Note:A line width ( σ) of 0.16 keV
+(see Table 2) was assumed to
+evaluate the above upper limits.
+Unit 1 of JEM-X detected the source at a flux of 4.7 ×10−11
+ergcm−2s−1in the 3–10keVband, while Unit 2 could only
+provide a loose upper limit of 5.6 ×10−11erg cm−2s−1to
+the source flux, again in the 3–10 keV band.
+3.2.3. WFC data
+As reported Sect. 2.3, no spectrum of 1ES 1210 −646 could
+be extracted from the WFC data. The source was clearly
+detected however, as Fig. 5 shows, in the mosaic maps of
+the summed WFCs data (for details on the mosaic produc-
+tion see Capitanio et al. 2008). The source is present in
+both the 3–17 keV and the 17–28 keV mosaics maps at 50
+and 20σconfidence levels, respectively. 1ES 1210 −646 is
+detected by the WFC at a flux of ∼2 mCrab in the 3–28
+keVband, correspondingto ∼5.5×10−11ergcm−2s−1; this
+value is compatible with the low-stateJEM-Xand the XRT
+pointing #1 detections. Outside the period in which it was
+detected, following Verrecchia et al. (2007) we can assume
+a 3–28 keV detection limit of <3×10−11erg cm−2s−1for
+1ES 1210 −646.
+4. Discussion
+We here performed and presented, to the best ofour knowl-
+edge, the first detailed spectral analysis of the X–ray emis-
+sion from the X–ray binary 1ES 1210 −646. We found that
+0 200 400 600 800 1000 1200 1400
+11:50:00.012:00:00.010:00.020:00.030:00.040:00.0-66:00:00.0-65:00:00.0-64:00:00.0-63:00:00.0-62:00:00.0
+1H 1249-637
+1ES 1210-64GX 301-2
+Fig.5.WFCs mosaic image in the 3–17 keV band of the
+1ES 1210 −646 sky region.
+the source shows marked variability (of a factor of at least
+10) on several timescales, from hundreds of seconds to
+months. Moreover, a strong transient iron emission line at
+6.7 keV is detected at intermediate X–ray fluxes only, and
+disappears when 1ES 1210 −646 is at low and high inten-
+sity levels. Below we will discuss the characteristics of this
+X–ray system, the nature of its accretor, and the origin of
+its large and variable iron emission line in the light of the
+results shown in this paper.
+4.1. 1ES 1210 −646: a peculiar HMXB
+Our results indicate that the 0.3–9 keV X–ray spectrum
+of 1ES 1210 −646 is described with a simple powerlaw, or
+a bremsstrahlung, while the INTEGRAL broadband data
+(3–100 keV) are best fit with a combination of BB plus
+powerlaw or bremsstrahlung plus powerlaw (see Table 2).
+Because of the relatively low S/N of the data we pre-
+fer the powerlaw (and the BB+powerlaw) description be-
+cause i) the latter generally shows lower reduced χ2values
+and ii) the JEM-X/IBIS intercalibration constant for the
+bremsstrahlung+powerlaw model is quite small ( ∼0.1).
+All the considered spectral models appear to be ab-
+sorbedbythe neutralinterstellarhydrogenalongthe source
+line of sight only, indicating that the accreting material is
+likely not dense and/or neutral enough to contribute to the
+measured absorption acting on the X–ray emitted by the
+source.
+As recalled in Sect. 1, Revnivtsev et al. (2007) put for-
+ward the identification of this X–ray source as a CV due
+to the presence of a large iron emission line at 6.7 keV
+in the X–ray spectrum. However, as already remarked in
+Masetti et al. (2009), a number of HMXBs also show iron8 N. Masetti et al.: 1ES 1210 −646, a peculiar high-mass X–ray binary
+emission lines at this energy. Besides, the very large EW of
+the line ( ∼1.6 keV, possibly due to the blend of iron lines,
+combined with the relatively low XRT spectral resolution)
+and its strong variability disfavour the CV interpretation
+for 1ES 1210 −646 (for a comparison, see Landi et al. 2009
+for a study of a sample of hard X–ray emitting CVs us-
+ing XRT and IBIS). Also the high-state X–ray luminosity
+of 1ES 1210 −646 is∼1-2 orders of magnitude larger than
+those typical of X–ray emitting CVs (see e.g. Barlow et al.
+2006; Revnivtsev et al. 2008; Brunschweiger et al. 2009).
+Therefore we confidently rule out a CV interpretation for
+this system.
+The possibility (see Masetti et al. 2009) that 1ES
+1210−646 might be similar to the peculiar transient X–
+ray binary CI Cam (=XTE J0421+560), thought to host a
+white dwarf (e.g., Orlandini et al. 2000) can be ruled out
+as well: the X–ray variability of the latter source and its
+optical spectrum (see Orlandini et al. 2000 and references
+therein) are indeed markedlydifferent with respect to those
+of 1ES 1210 −646. Moreover, the presence of a BB emission
+with a temperature of ∼1.5 keV in the X–ray spectrum
+of 1ES 1210 −646 when the source shows high flux levels
+suggests that the accretor is more compact than a white
+dwarf.
+We are also inclined to rule out a LMXB nature be-
+cause these systems have quite different optical (and, to a
+lesser extent, X–ray) spectra with respect to that of 1ES
+1210−646. The huge variability of the iron emission line
+found in 1ES 1210 −646 is never seen, as far as we know, in
+persistent LMXBs.
+Besides, the length of the orbital period also disfavours
+the LMXB interpretation. While searching in the LMXB
+catalogues of Liu et al. (2007) and Ritter & Kolb (2003),
+we found that very few systems have orbital periods of the
+order of days; and those that do actually show very dif-
+ferent emission properties with respect to 1ES 1210 −646.
+For instance, V395 Car (=2S 0921 −630) with a period of
+9.0 d (Branduardi-Raymont et al. 1983) has optical (e.g.,
+Shahbaz et al. 1999) and X–ray (e.g., Kallman et al. 2003)
+spectra quite at variance with those of 1ES 1210 −646. The
+same holds for Cyg X-2 (see Elebert et al. 2009, Di Salvo
+et al. 2002 and references therein), which has a period of
+9.8 d (Casares et al. 1998). Likewise, LMXBs V404 Cyg
+(=GS 2023+338), with an orbital period of 6.4 d (Casares
+et al. 1992), is completely different from 1ES 1210 −646 in
+the sense that the latter is a persistent although variable
+source, while the former is one of the best-known X–ray
+transient LMXBs hosting a dynamically-confirmed BH (cf.
+Kitamoto et al. 1989; Casares & Charles 1994).
+The binary 1ES 1210 −646might actually be more simi-
+lartothe fast HMXB transientSAX J1819.3 −2525,at least
+in terms of optical spectroscopic characteristics (Orosz et
+al. 2001; Maitra & Bailyn 2006); however this system also
+displays a transient X–ray behaviour which is not seen in
+1ES 1210 −646 and moreover it too hosts a dinamically-
+confirmed BH (Orosz et al. 2001). Possibly 1ES 1210 −646
+will eventually evolve into a LMXB as Cyg X-2, as sug-
+gested for the HMXB Cir X-1 by Tauris & Savonije (1999)
+and by Podsiadlowski et al. (2002).
+All things considered,the HMXB interpretationfor1ES
+1210−646 is the one best suited to explain the multiwave-
+length properties of this source. Nevertheless, the optical
+and X–ray spectra of 1ES 1210 −646 are also anomalous
+for a persistent (albeit variable) HMXB. Indeed, the X–rayspectrum is atypical for an HMXB in the sense that we
+do not detect any break in its powerlaw shape below 10
+keV and below 100 keV in the XRT and IBIS spectra, re-
+spectively, whereas HMXBs generally show spectral breaks
+in the 5–20 keV range (e.g., White et al. 1983). Actually,
+Corbet & Mukai (2008) found a break around 6 keV in the
+source X–ray spectrum: if our different description can be
+explained for XRT data as due to their narrower spectral
+coverage with respect to that of Corbet & Mukai (2008),
+theINTEGRAL spectrum is possibly representative of a
+spectral state which is different from that reported by the
+above authors.
+4.2. The nature of the accretor
+The analysis presented here also indicates that a BB emis-
+sion appears in the X–ray spectrum of 1ES 1210 −646when
+the sourceisatits highstate.This latterfactishowevernot
+surprisingbecause accordingto Hickoxet al. (2004)soft ex-
+cessesarequite ubiquitous in HMXB spectra. The BB com-
+ponent detected in the X–ray spectrum of 1ES 1210 −646
+is consistent with an emission from an accreting NS sur-
+face, both in terms of the emitting area and temperature.
+This agrees with the conclusions of Hickox et al. (2004),
+according to which a soft excess in HMXBs with X–ray
+luminosities of less than 1036erg s−1can be produced by
+thermal emission from the surface of the accreting NS (as
+in the case of the low-luminosity HMXB X Per; Coburn
+et al. 2001). That we see this BB component only at the
+highest emission levels for this source is likely a selection
+effect in the sense that it is more apparent when the NS
+accretion rate is highest, whereas during lower flux levels
+the non-thermal (i.e., powerlaw) component dominates.
+Moreover, as we detect a BB component with an emit-
+ting area comparable to the surface of a NS (or at least a
+large fraction of it; see Table 2) in 1ES 1210 −646, the ac-
+creting object is probably an NS with a relatively low mag-
+netic field; in this case, no accretion column would form,
+and the detection of pulsed X–rayemission would be rather
+difficult. Consequentlythe accretorin 1ES1210 −646would
+be ananalogueofthe one in the HMXB CirX-1 (see Jonker
+et al. 2007 and references therein), which is believed to be
+a low magnetic field NS.
+It is also worth pointing out that this source is interest-
+ing from a further point of view, as it seems to be one of
+the very rareBe/X-raybinaries in the Galaxy with spectral
+type beyond B2 (Negueruela 1998). Actually, according to
+the Corbet diagram (Corbet 1986), a 6.7 day orbital pe-
+riod is quite short for a Be/X-ray binary and would be
+better suited to a supergiant system (which is ruled out by
+Masetti et al. 2009), unless the hosted NS is rapidly spin-
+ning (Pspin<∼10−2s). Otherwise, the NS could be rotating
+substantially slower, with Pspin∼102s, and accretes from
+the stellar wind of the companion star.
+4.3. Origin of the iron emission line
+The possible origin of the 6.7 keV emission line generally
+observedintheX–raybinariesisthe Kαtransitionofhighly
+ionised He-like iron. This can be produced either in a thin
+hot plasma region with an electron temperature of sev-
+eral keV (as usually observed in disc-accreting X–ray bina-
+ries; cf. White et al. 1995), or by radiative recombinationN. Masetti et al.: 1ES 1210 −646, a peculiar high-mass X–ray binary 9
+followed by electron cascade transition in a photoionised
+plasma with a relatively low temperature (see Liedahl &
+Paerels 1996 and references therein).
+The latter mechanism is at work in HMXBs like SMC
+X-1 (Vrtilek et al. 2001), Vela X-1 (Goldstein et al. 2004)
+and 4U 1700 −37 (van der Meer et al. 2005): it makes the
+emission lines more easily detectable during eclipses, when
+the X–ray beam irradiating the stellar wind is not directly
+visible but the reprocessed emission from the photoionised
+stellar wind is. However, this effect does not explain the
+iron line variability that we detect in 1ES 1210 −646, as in
+thiscasethelineisapparentfarfromthepossibleeclipsesof
+the system (see below), assuming the ephemeris of Corbet
+& Mukai (2008).
+The X–ray binary Cyg X-3 also shows an iron emission
+lineat6.7keV,likelyemitted fromthe accretedstellarwind
+surrounding the accretor and strongly photoionised by its
+high-energyradiation(see e.g.Szostek et al. 2008and refer-
+encestherein);butinthisobjecttheintensityofthe6.7keV
+iron line as a function of the source flux (Hjalmarsdotter
+et al. 2009) reaches its lowest levels when Cyg X-3 is at in-
+termediate flux levels, thus at variance with respect to the
+behaviour shown by 1ES 1210 −646.
+This accordingly suggests that the iron emission de-
+tected in this latter source is produced in a highly ionised,
+hot and thin accretion stream rather than by a relatively
+cold photoionised plasma.
+In addition, the line variability behaviour of 1ES
+1210−646 is reminiscent of that displayed by GX 301 −2,
+an HMXB showing an iron emission at 6.4 keV with EW
+varying by a factor of ∼6 depending on the orbital phase
+of the NS (Nagase 1989; Endo et al. 2002). In particular, in
+this source the iron emission line is stronger when the ac-
+cretingNS is close to the periastron(Endo et al. 2002).Can
+this effect be present in 1ES 1210 −646 as well? Indeed, ac-
+cording to the orbital ephemeris of Corbet & Mukai (2008),
+we find that XRT observation #1 falls at orbital phase φ∼
+0 (when the orbital modulation of the source flux reaches
+its maximum, that is, at or near the periastron), while the
+other XRT pointings and our JEM-X spectrum were ac-
+quired at orbital phases of ∼0.3,∼0.8 and ∼0.3, respec-
+tively. Likewise, Corbet & Mukai (2008) also find an iron
+emission in an X–ray spectrum acquired near an orbital
+phase of 0.
+We notehoweverthat the line energyin GX301 −2indi-
+cates that the emission in this source is due to fluorescence
+of neutral iron irradiated by the accreting NS, whereas (as
+already remarked) in 1ES 1210 −646 the line at 6.7 keV
+comes from a highly ionised medium. Nevertheless, the line
+intensity behaviour in these two sources as a function of
+the orbital phase is apparently very similar.
+Therefore, the presence of a variable emission line in
+1ES 1210 −646 as a function of its orbital period with a
+detectable intensity as the accretor approaches periastron
+suggests that this feature is tied to the increase of the ac-
+cretion rate from the secondary star (likely from its wind)
+onto the NS due to its orbital motion. The line thus possi-
+bly forms in a small, transient and highly ionised accretion
+figure around the NS itself.
+5. Conclusions
+From our X–ray data analysis carried out with several in-
+strumentsonboardofdifferentspacecraft,weconcludethatthe best interpretation for the nature of the X–ray source
+1ES1210 −646isthatitisanHMXBhostingalowmagnetic
+field NS, although it is a system with several peculiarities
+like its optical and X–ray spectra and the large variability
+of the iron emission line at ∼6.7 keV. This variable line
+is apparently produced in a highly ionised and transient
+accretion figure around the accreting NS during the pe-
+riastron passage. In order to further explore the peculiar
+nature of 1ES 1210 −646 and its behaviour as a function
+of the orbital phase, higher S/N optical and X–ray spectra
+of the source along with multiwavelength monitoring and a
+deepertiminganalysistosearchforshort-termmodulations
+(from milliseconds to hours), are definitely needed.
+Acknowledgements. We thank Domitilla de Martino and Margaretha
+Pretorius for preliminar comments and discussions on the na ture of
+this source, Mauro Orlandini for discussions on HMXB system s and
+X–ray data analysis, and Vanessa McBride for comments and su g-
+gestions. We also thank the anonymous referee for useful rem arks
+which helped us to improve the paper. This research has made u se
+of the ASI Science Data Center Multimission Archive, of the W FC
+archive at INAF/IASF di Roma, of the SIMBAD database operate d
+at CDS, Strasbourg, France, and of the NASA Astrophysics Dat a
+System Abstract Service. Some of the authors acknowledge th e ASI
+and INAF financial support via grant No. I/008/07.
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+arXiv:1001.0569v2  [hep-ph]  8 Jul 2010Collider signals of a composite Higgs in the Standard Model w ith four
+generations
+Shaouly Bar-Shaloma∗, Gad Eilama†, Amarjit Sonib‡
+aPhysics Department, Technion-Institute of Technology, Ha ifa 32000, Israel
+bTheory Group, Brookhaven National Laboratory, Upton, NY 11 973, USA
+Abstract
+Recentfitsofelectroweak precisiondatatotheStandardMod el(SM)witha4thsequential
+family (SM4) point to a possible “three-prong composite sol ution”: (1) the Higgs mass is
+at the TeV-scale, (2) the masses of the 4th family quarks t′,b′are ofO(500) GeV and (3)
+the mixing angle between the 4th and 3rd generation quarks is of the order of the Cabibbo
+angle,θ34∼O(0.1). Such a manifestation of the SM4 is of particular interest as it may
+suggest that the Higgs is a composite state, predominantly o f the 4th generation heavy
+quarks. Motivated by the above, we show that the three-prong composite solution to the
+SM4 can have interesting new implications for Higgs phenome nology. For example, the Higgs
+can decay to a single heavy 4th generation quark via the 3-bod y decays (through an off-shell
+t′orb′)H→¯t′t′⋆→¯t′bW+andH→¯b′b′⋆→¯b′tW−. These flavor diagonal decays can
+be dramatically enhanced at the LHC (by several orders of mag nitudes) due to the large
+width effects of the resonating heavy Higgs in the processes gg→H→¯t′t′⋆→¯t′bW+and
+gg→H→¯b′b′⋆→¯b′tW−, thus yielding a viable signal above the corresponding cont inuum
+QCD production rates. In addition, the Higgs can decay to a si nglet′andb′in the loop-
+generated flavor changing (FC) channels H→b′¯b,t′¯t. These FC decays are essentially
+“GIM-free” and can, therefore, have branching ratios as lar ge as 10−4−10−3.
+∗Electronic address: shaouly@physics.technion.ac.il
+†Electronic address: eilam@physics.technion.ac.il
+‡Electronic address: soni@bnl.gov
+11 Introduction
+The SM4 is one of the simplest new physics scenarios, in which the SM is e nlarged by a complete
+sequential 4th family of chiral matter: a new ( t′,b′) and (ν′,ℓ′) heavy doublets in the quark and
+leptonsectors, respectively (forreviews see [1, 2,3]). Formany y ears, since theLEPI measurement
+of the Z-width and up to the early 2000’s, the SM4 was thought to be ruled out by experimental
+data [4]; however, this interpretation no longer appears to be corr ect. Indeed recent fits of the
+SM4 to precision EW observables [5, 6, 7] and to flavor data [8, 9, 10, 11] have clarified that the
+SM4 is not in disagreement with the present data. According to thes e recent studies, the masses
+of the 4th generation quark doublet may lie in the range 300 −700 GeV and should be slightly
+spilt, i.e., mt′−mb′∼45−75 GeV (consistent also with the current CDF limits on the 4th
+generation quark masses: mt′>∼310 GeV [12] and mb′>∼340 GeV [13]). In turn, in order to be
+consistent with electroweak (EW) precision data, this favors a hea vier Higgs, thereby, completely
+removing the tension between the LEPII bound mH>∼115 GeV and the central (best fitted) Higgs
+mass value mH∼85 GeV obtained from a fit of the SM to the data. This effect is mainly du e
+to an interesting interplay between the contributions of the 4th fa mily fermions to the oblique
+parameters S and T, as was noted in [5, 7].
+Indeed, anextremely interesting outcomeofthese newstudies ist hatthebest fittedHiggsmass
+in the SM4 can be driven up to its unitarity bound, mH<∼1 TeV, if the mixing angle between
+the 4th and 3rd generation quarks, θ34, is of the size of the Cabibbo angle (i.e., θ34∼O(0.1))
+and ifmt′, mb′∼O(500) GeV [6]. The combination of a heavy O(500) GeV 4th generation
+doublet with a TeV-scale Higgs might be a hint that the Higgs is not a fun damental particle but
+is, rather, a composite state, primarily, of the heavy 4th generat ion quarks [1, 14, 15, 16, 17]. In
+particular, within the SM4, such a heavy Higgs and heavy 4th genera tion doublet can drive the
+Landau pole down to the TeV-scale, in which case, the simplest compo siteness condition (used in
+top-condensate type models) which relates the masses of the hea vy fermions to that of the Higgs,
+suggests that mH∼√
+2mQ, whereQis the heavy fermion that forms the composite state [1, 15].
+Another interesting possibility which was recently raised in [16] is that if the masses of t′,b′are
+sufficiently heavy, then the Higgs quartic and 4th generation Yukaw a couplings can have a fixed
+point at a scale of several TeV (rather than a Landau pole), aroun d which the 4th generation
+heavy quarks can form the composite state.
+These recent theoretical developments and experimental indicat ions in favor of the SM4 with
+a heavy Higgs and a heavy 4th generation doublet, as well as the pos sibility of an alternative
+non-SUSY solution to the hierarchy problem (see [16]), lead us to stu dy what we henceforward
+name the “three-prong composite solution” of the SM4:
+mH<∼1 TeV
+ր ց
+θ34∼O(0.1)←− mt′,b′∼O(500) GeV
+2which addresses the composite Higgs scenario.
+We will show that this three-prong composite solution of the SM4 can have interesting new
+implications on Higgs phenomenology at the LHC. In particular, we will f ocus on the case of
+600 GeV<∼mH<∼1000 GeV and 400 GeV<∼mt′,mb′<∼600 GeV so that mt′, mb′< mH<
+2mt′,2mb′.1This range of Higgs and 4th generation quark masses opens up the in teresting
+possibilities of a heavy (composite) Higgs decaying to a single 4th gene ration heavy quark via an
+off shellt′orb′in the flavor diagonal channels:
+H→¯t′t′⋆→¯t′bW++h.c.&H→¯b′b′⋆→¯b′tW−+h.c. . (1)
+or via the 1-loop FC channels:
+H→t′¯t+h.c.&H→b′¯b+h.c. , (2)
+As we will show below, the rates for the 3-body decays in (1), which, within the three-prong
+composite solution to the SM4, are the only probe of the Higgs Yukaw a couplings to the 4thfamily
+quarks, can be dramatically enhanced due to the large width effects in the heavy Higgs resonance
+production processes
+gg→H→¯t′t′⋆→¯t′bW++h.c.&gg→H→¯b′b′⋆→¯b′tW−+h.c. , (3)
+and can reach a detectable level above the QCD production rate.
+In addition, the FC decays in (2) can have branching ratios (BR) as la rge as 10−4−10−3,
+owing to the rather large θ34∼O(0.1) and to the large t′andb′masses which render these decays
+essentially GIM-free.
+Throughout our analysis to follow we set the masses of the 4th gene ration leptons (which
+enter in the calculation of the total Higgs width) to mℓ′= 250 GeV and mν′= 200 GeV. We
+furthermore assume that Vtb=Vt′b′= 1 and that the mixing of the 4th generation quarks with
+the first two generations ones is much smaller than θ34, in particular, that θ34≡Vtb′=Vt′b>>
+Vt′d, Vt′s, Vub′, Vcb′, consistent with current experimental constraints [5, 8, 10].
+2 Heavy Higgs production at the LHC
+The Higgsproduction rateat theLHC isknown to increase by almost a norder ofmagnitude in the
+SM4, depending on the Higgs and 4th family quark masses [5, 18]. This is manifest in the gluon-
+gluon fusion hard process gg→Hand is caused by the enhancement of the ggH 1-loop coupling
+due to the extra heavy 4th family quarks that run in the loop. This en hancement comes in handy
+in particular for the production of a TeV-scale Higgs, where the SM r ate might be marginal. In
+Fig. 1 we plot the gluon-gluon fusion cross-section convoluted with t he PDF for pp→H+Xin
+1Our study complements the case mH<∼500 GeV with mt′, mb′<∼400 GeV which was discussed in Ref.[5].
+3the SM and in the SM4 for three values of the t′mass,mt′= 400,500 and 600 GeV. Here and
+henceforward we always set mb′=mt′−70 GeV, which is the value of mb′appropriate for Higgs
+massesmH>∼600 GeV.2That is, in order for the SM4 to be consistent with EW precision data
+the mass splitting in the 4th family quarks is constrained by [5, 19]:
+mt′−mb′≈/parenleftBigg
+1+lnmH
+115 GeV
+5/parenrightBigg
+×50 GeV. (4)
+As can be seen in Fig. 1, for mH∼800 GeV, the gluon-fusion cross-section increases from
+O(100) fb in the SM to O(1000) fb in the SM4, thus expecting the LHC to produce about 105
+heavy Higgs with a luminosity of O(100) inverse fb.
+Note that, when mH<2mt′,2mb′, the decay pattern of the SM4 heavy Higgs is similar to that
+of the SM Higgs, up to its possible decays to the 4th family leptons. In particular, the leading
+decay channels of the heavy SM4 Higgs remain H→ZZ,WW,t ¯t. Thus, as was recently noted
+in [5], the expected enhancement in the gluon-fusion production cha nnel implies that the “golden
+mode”H→ZZ→4µholds up as a useful Higgs discovery channel throughout the Higgs mass
+range.
+100 200 300 400 500 600 700 800 900 1000
+mH [GeV]101102103104105106σ(pp−>H+X) [fb]LHC with Ecm=10 TeV
+σ(SM)
+σ(SM4; mt’=400 GeV)
+σ(SM4; mt’=500 GeV)
+σ(SM4; mt’=600 GeV)mb’=mt’−70 GeV
+Figure 1: σ(pp→H+X)from the gg→Hhard process at the LHC with a c.m. energy of 10
+TeV, in the SM (solid curve) and in the SM4 for several values o f(mt′,mb′), as a function of
+the Higgs mass. We take K= 1.5for the K-factor to account for the QCD corrections to the
+gluon-gluon fusion cross-section [20].
+2Note that according to Eq. 4 one has 65 GeV<∼mt′−mb′<∼72 GeV for 500 GeV<∼mH<∼1000 GeV, which is
+the range of Higgs masses of interest in this work.
+43 Flavor conserving heavy Higgs decays to a single 4th
+generation quark
+Within the three-prong composite solution of the SM4, where mt′< mH<2mt′andmb′< mH<
+2mb′, the 2-body Higgs decays H→t′¯t′, b′¯b′are kinematically forbidden. In this case, the only
+probe of the Higgs Yukawa couplings to the 4th family quarks are the Higgs 3-body decays in (1)
+which proceed through an off-shell heavy 4th family quark.
+Here we are interested in the case of vanishing mixings between the 4 th generation quarks
+with the 1st two generation ones, in which case BR(t′→bW), BR(b′→tW)∼O(1), so that the
+flavor diagonal 3-body Higgs decays in (1) will lead to:
+BR(H→t′¯t′⋆)∼BR/parenleftBig
+H→t′¯bW−→(bW+)t′¯bW−/parenrightBig
+,
+BR(H→b′¯b′⋆)∼BR/parenleftBig
+H→b′¯tW+→(tW−)b′¯tW+/parenrightBig
+. (5)
+and the charged conjugate channels.
+Noteinparticularthesignatureofthe b′Yukawacouplingin(5): t¯tW−W+→b¯bW+W+W−W−,
+which, with the appropriate kinematical cuts, is expected to be rat her distinct, e.g., leading
+to a signal of same-sign leptons + missing energy accompanied by 2 b- jets and 4 light-jets:
+H→b′¯b′⋆→ℓ+ℓ+bbjjjj+/negationslashET(and the charged conjugate signal).
+Naively one may expect these 3-body decays to have very small rat es, with BR’s at the level
+of 10−6−10−5whenmH<2mt′,2mb′. However, as we will show below, these decays can be
+dramatically enhanced due to finite width effects of the heavy Higgs. Indeed, the non-zero width
+of heavy particles is known to cause substantial enhancement whe n these particles appear in the
+final state of a decay which occurs just around its kinematical thr eshold [21]. Alternatively, when
+the decaying heavy particle emerges as an intermediate state the e ffects of its width close to
+threshold is usually handled with the Breit-Wigner prescription. In th is case, special attention
+is required for the computation of the overall resonant productio n and subsequent decay of the
+heavy particle.
+Here we consider the leading resonance production of the heavy Hig gs via the gluon-fusion
+process(which isfurther enhancedintheSM4, seeprevious sectio n), followed byitsdecays H→F
+as in (5). Using the relativistic Breit-Wigner resonance formula, we e stimate the corresponding
+hard cross-sections by:
+ˆσH(ˆs) = ˆσgg(ˆs)·ˆs·ˆBW(ˆs)·BR(H→F)(ˆs), (6)
+with
+ˆσgg(ˆs) =π2
+8m3
+HΓ(H→gg)(ˆs) (7)
+andˆBWis the normalized Breit-Wigner function:
+ˆBW(ˆs) =mHΓH/π
+[(ˆs−m2
+H)2+m2
+HΓ2
+H], (8)
+5where, in order to incorporate the appropriate kinematic depende ncies away from the resonance,
+we replace mHΓH→√
+ˆsΓH(ˆs) and set Γ H(ˆs) =√
+ˆsΓH/mH(ΓHbeing the total Higgs width) [4].
+We thus get (for the hard process gg→H→F):
+ˆσH(ˆs) =π(ˆs/m2
+H)Γ(H→gg)(ˆs)Γ(H→F)(ˆs)
+8[(ˆs−m2
+H)2+(ˆsΓH/mH)2], (9)
+which, after being convoluted with the gluons distribution functions (PDF’s):
+dL
+dτ(ˆs=τs)≡/integraldisplay1
+τdx
+xg(x,ˆs)g(τ
+x,ˆs), (10)
+gives:
+σH=/integraldisplayτmax
+τmindL
+dτˆσHdτ , (11)
+In the Narrow Width Approximation (NWA), where ˆBW(ˆs)≈δ(ˆs−m2
+H), we obtain:
+σH(NWA) = ˆσggτHdL
+dτBR(H→F), (12)
+whereτH=m2
+H/sand all the other terms in (12) are evaluated at ˆ s=m2
+H.
+Using the Breit-Wigner formula in (9), we have estimated the width eff ects of the heavy Higgs
+on the cross-sections for producing a single 4th family quark throu gh the 3-body Higgs decays:
+σH(t′bW)≡σ(pp→H+X→¯t′t′⋆+h.c.+X→¯t′bW++h.c.+X),
+σH(b′tW)≡σ(pp→H+X→¯b′b′⋆+h.c.+X→¯b′tW−+h.c.+X), (13)
+by integrating the corresponding hard cross-sections over some finite range around the resonance,
+depending on the Higgs mass and width.
+In Fig. 2 we give a sample of our results which show the dramatic enhan cement near the
+threshold for the 3-body Higgs decays H→¯t′bW+,¯b′tW−, due to the large width of the decaying
+Higgs when it is produced at resonance at the LHC. Notice that, due to the large Higgs width,
+the NWA is not adequate for estimating these cross-sections (as e xpected). This can be seen in
+the figure for mHmasses sufficiently away from the threshold of the decay to on-she ll pair of
+heavy 4th generation quarks H→t′¯t′,b′¯b′. For example, for mH= 1000 TeV and mt′= 400 GeV,
+we findσH(BW)/σH(NWA)∼3, where σH(BW) is the Breit-Wigner estimate of (9) for the
+corresponding cross-sections. This difference between the NWA a nd the BW approaches indicates
+the extent to which the NWA is valid in this case.
+We see that around the threshold the enhancement due to the finit e width effect of the res-
+onating Higgs can reach several orders of magnitudes, elevating t hese cross-sections to O(1000)
+[fb], which should be within the detectable level at the LHC. This enhan cement is practically
+independent of the mixing angle θ34, since as long as the mixings of the 4th generation with the
+1st and the 2nd generation quarks are much smaller than θ34the decays t′→bWandb′→tW
+occur with a BR of order one.
+6500 600 700 800 900
+mH [GeV]10−410−310−210−1100101102103104σH(t’bW) [fb]
+σ(NWA)
+σ(mmin<ECM<mH+ΓH/2)
+σ(mmin<ECM<mH+ΓH)
+σ(mmin<ECM<mH+3ΓH/2)
+600 700 800 900 1000
+mH [GeV]σH(b’tW) [fb]
+mt’=400 GeV
+V34=0.15
+Figure 2: σH(t′bW)andσH(b′tW)(see Eq. 13), and the corresponding cross-sections in the NW A
+(see Eq. 12), as a function of mH, formt′= 400GeV (mb′=mt′−70GeV) and θ34= 0.15.
+The hard c.m. energy,√
+ˆs, is integrated from mmintomH+δm, whereδm= ΓH/2(dashed line),
+δm= ΓH(dotted line) and δm= 3ΓH/2(dotted-dashed line), and mminis the largest of either the
+kinematic threshold or the value of mH−δm.
+In Table 1 we evaluate these cross-sections for mH= 800 GeV, mt′= 500 GeV and we list
+the potential (leading) background to these single 4th generation quark signals from the QCD
+continuum production.3We see that with a luminosity of the O(100) inverse fb, one expects
+104−105such single t′andb′events with a (naive) signal to background ratio of S/B∼1/10−1/3
+and a corresponding statistical significance of S/√
+B∼50−200, depending on the range of
+integrationofthehardcross-sections. Nonetheless, thedetec tionoftheseHiggsdecaysignalsatthe
+LHCisratherchallenging. Inparticular, afterthe t′andb′decay(promptly)via t′→bW, b′→tW
+these signals will give two distinct final state topologies:
+pp→H→¯t′t′⋆→(¯bW−)t′bW+[+h.c.+X],
+pp→H→¯b′b′⋆→(¯tW+)b′tW−→(¯bW+W−)b′bW+W−[+h.c.+X],(14)
+which lead tolarge multiplicity events with leptons, jets and missing tra nsverse energy (the t′¯t′one
+resembling thecase of t¯t+jets events intheSM). The reconstruction ofsuch events will, th erefore,
+require a detailed and rather demanding study of both the signal an d background samples.
+For example, it is possible to use subsets of the above signatures wh ich can be distinguishable
+even without a full reconstruction of the event. In fact, such an alysis for b′andt′-pair production
+3σQCD(t′bW) andσQCD(b′tW) were calculated by means of CalcHep [22].
+7pp→t′bW+h.c.+Xpp→b′tW+h.c.+X
+Integrated range σH(t′bW)σQCD(t′bW)σH(b′tW)σQCD(b′tW)
+[fb] [fb] [fb] [fb]
+700 GeV≤√
+ˆs≤mH+1
+2ΓH0.11 1.3 96 634
+700 GeV≤√
+ˆs≤mH+ΓH58 280 504 1716
+700 GeV≤√
+ˆs≤mH+3
+2ΓH274 708 872 2358
+750 GeV≤√
+ˆs≤1050 GeV 14 116 354 1406
+σH(NWA) 4.8·10−35.9·10−3
+Table 1: Cross-sections for the flavor-diagonal single t′andb′production in the Higgs resonance
+channels: σH(t′bW) andσH(b′tW) (see Eq. 13) and the corresponding cross-sections from the
+QCD continuum: σQCD(t′bW) andσQCD(b′tW). The cross-sections are for the LHC with a c.m.
+energy of 10 TeV, for mH= 800 GeV (for which we have Γ H∼300 GeV), mt′= 500 GeV,
+mb′= 430 GeV and θ34= 0.15, and are integrated over several mass ranges of the c.m. ener gy
+of the hard processes as indicated. The corresponding cross-se ctionsσH(NWA) in the NWA are
+also given.
+topologies similar to (14) was already performed by CDF in [12, 13], and may be used as a
+starting point for the search of our heavy Higgs resonance signals H→¯t′t′⋆,¯b′b′⋆at the LHC.
+In particular, for the detection of b′viab′¯b′→t¯tW+W−→b¯bW+W+W−W−, [13] required
+the signature ℓ±ℓ±bj/negationslashET: two same-charge reconstructed leptons, at least two jets one of them
+b-tagged and missing transverse energy (with cuts on the transv erse energy and rapidity of the
+jets/leptons, see [13]). This same-sign leptons signature was foun d to be particularly sensitive
+to the new physics signal b′¯b′→t¯tW+W−→b¯bW+W+W−W−(the main background coming
+fromt¯tand W+jets production) and was used to obtain a bound on the corr esponding b′mass
+mb′>∼340 GeV [13]. Such an analysis can be similarly applied to our signal (at th e LHC)¯b′b′⋆→
+(¯tW+)b′tW−→(¯bW+W−)b′bW+W−, in particular since it does not rely on the full reconstruction
+of theb′¯b′system.
+In thet′¯t′→b¯bW+W−case CDF required a signature of one lepton, four or more jets and
+missing transverse energy ℓ+4j+/negationslashET[12], which allowed a useful discrimination of the t′¯t′signal
+from the background (here also the SM background mostly consist s oft¯tand W+jets events as
+well as multi hadronic jet events from QCD). An analysis in this spirit m ight also be useful for
+searching for our H→¯t′t′⋆→(¯bW−)t′bW+signal at the LHC.
+However, we should stress that the event topologies and the back ground problems are expected
+to be more serious at the LHC and are likely to represent a serious ch allenge. Nonetheless, we are
+cautiously optimistic that it can be handled.
+84 Flavor changing Higgs decays to a single 4th generation
+quark
+In the SM the FC decay H→t¯cis severely suppressed by the GIM-mechanism, leading to an
+un-observably small BRSM(H→t¯c)∼10−13[23]. On the other hand, in the SM4, we expect this
+decay mode and also the FC decays involving the 3rd and 4th generat ion quarks, H→t′¯tand
+H→b′¯b, to be significantly enhanced due to the extra heavy 4th generatio n quarks in the loops
+which essentially removes the GIM suppression. This can be seen by e stimating the width for the
+1-loop FC decay H→U¯uin the limit that only the heaviest down-type quark ( D) runs in the
+loop and its contribution is multiplied by the appropriate GIM-suppres sion factor (see e.g., [24]):
+Γ(H→U¯u)∼/parenleftBigg|V⋆
+UDVuD|
+16π2/parenrightBigg2/parenleftBiggg2
+4π/parenrightBigg3/parenleftbiggmD
+mW/parenrightbigg4
+mH. (15)
+For example, we can use (15) to estimate the ratio between Γ SM(H→t¯c) (where D=b) and
+ΓSM4(H→t′¯t) (where D=b′):
+ΓSM4(H→t′¯t)
+ΓSM(H→t¯c)∼/parenleftBiggm2
+b′|V⋆
+t′b′Vtb′|
+m2
+b|V⋆
+tbVcb|/parenrightBigg2
+. (16)
+Thus, taking mb′= 500 GeV, Vt′b′=Vtb= 1 and Vtb′≡θ34∼3Vcb∼0.15, and assuming that
+the total width of the heavy Higgs in the SM and in the SM4 is the same ( this roughly holds
+formH<2mb′,2mt′), we expect BRSM4(H→t′¯t)∼109·BRSM(H→t¯c)∼10−4, which, as we
+will show below, is indeed the case. On the other hand, for the case o fH→t¯cin the SM4, we
+expect (using the above values for the CKM elements involved and fo rmb′)BRSM4(H→t¯c)∼
+V2
+cb′·109·BRSM(H→t¯c), which at best gives BRSM4(H→t¯c)∼106·BRSM(H→t¯c)∼10−7
+onceVcb′is set to its largest allowed value Vcb′∼0.03−0.04, see e.g. [10].
+This expected enhancement in FC transitions involving the 4th family h eavy quarks of the
+SM4, has ignited a lot of activity in the past two decades. However, p revious studies of FC
+effects in the SM4 have assumed that mt′,mb′> mHand, therefore, focused on the FC decays
+b′→bH,bV[9, 10, 25, 26] and t′→tH,tV[9, 10], where V=g,γ,Z. The BR’s for these t′andb′
+FC decays were found to be typically within the range of 10−4−10−2. The authors of [9, 10] have
+also studied the (SM-like) FC decays t→cH,cVwithin the SM4 and found that, in spite of the
+many orders of magnitudes enhancement, these FC top decays re main below the LHC sensitivity,
+i.e., having a BR typically smaller than 10−7.
+Here, motivated by the estimate in (16), we wish to study the FC hea vy Higgs decays in (2):
+H→t′¯tandH→b′¯b. Recall that our working assumption is that Vt′b∼Vtb′>> Vt′d,Vt′s,Vub′,Vcb′
+andmb′=mt′−70 GeV>∼350 GeV, in which case the 4th generation quarks are expected to
+decay mainly via t′→bWandb′→tW. Thus, both FC Higgs decays will lead to the final state
+b¯bW+W−, but with different kinematics:
+H→t′¯t+h.c.→(bW+)t′(¯bW−)t+h.c. ,
+H→b′¯b+h.c.→(tW−)b′¯b+h.c.→(bW+W−)b′¯b+h.c. . (17)
+9We base our results below on the explicit analytical expressions that were given for the decay
+H→b′¯bin [25]. In Fig. 3 we plot the BR(H→t′¯t+h.c.) andBR(H→b′¯b+h.c.), as a function
+ofmt′, forθ34= 0.15 and for Higgs masses between 600 to 900 GeV. We see, for examp le, that for
+mt′∼500 GeV these BR’s are at the level of few ×10−4ifmH∼800 GeV and θ34∼0.15, which is
+the value of θ34required in order for the SM4 to fit EW precision measurements when mH∼800
+GeV [6]. Note that in Fig. 3 we did not consider the finite width effects of the heavy Higgs. We
+thus expect the BR’s in Fig. 3 to be enhanced by an additional factor of a few, i.e., reaching 10−3,
+when these FC signals are integrated over some finite range around mHin the Higgs production
+process, e.g., integrated over mH−ΓH<√
+ˆs < m H+ ΓHingg→H→t′¯t,b′¯b(see discussion
+in the previous section). Given that the LHC with a luminosity of O(100) fb−1will be able to
+produce about 105TeV-scale Higgs particles (see Fig. 1), one naively expects tens of s uch events
+a year, if indeed θ34∼0.15.
+300 400 500 600 700
+mt’ [GeV]  0  1  2  3  4  5  6  7  8BR÷104H−>tt’
+mH=600 GeV
+mH=700 GeV
+mH=800 GeV
+mH=900 GeV
+400 500 600 700 800
+mt’ [GeV]H−>bb’
+mb’=mt’−70 GeVθ34=0.15
+Figure 3: BR(H→t′¯t+h.c.)(left) and BR(H→b′¯b+h.c.)(right), as a function of mt′for
+θ34= 0.15,mb′=mt′−70GeV and for mH= 600,700,800and900GeV.
+105 Summary and discussion
+We have explored some of the phenomenological implications for heav y Higgs physics in the
+SM4 within what we named the “three-prong composite solution” to E W precision data, which
+accommodates the possibility of a composite Higgs: (1) mH∼O(1) TeV, (2) mt′,b′∼O(500) GeV
+and (3)Vt′b,Vtb′of the order of the Cabibbo angle, i.e., Vt′b∼Vtb′∼θ34∼O(0.1), and we have
+assumed that mH<2mb′,2mt′in accordance with compositeness.
+We focused on the flavor diagonal H→¯t′t′⋆→¯t′bW+andH→¯b′b′⋆→¯b′tW−3-body decay
+channels and the flavor changing (one-loop) H→t′¯t,b′¯bdecays of the composite Higgs to a single
+heavy 4th family quark. We are cautiously optimistic that these signa ls can be observed at the
+LHC, in spite of the decrease in the production cross-section for s uch a heavy Higgs. In particular,
+the one-loop FC decay channels are essentially “GIM-free” and can , therefore, reach a branching
+ratio as large as BR∼10−4−10−3ifθ34∼O(0.1), while the rate of the flavor diagonal channels
+is dramatically enhanced due to large width effects of the decaying Hig gs - potentially yielding
+104−105such events with a luminosity of O(100) inverse fb, which is comparable to the number
+of events expected from the QCD continuum production of ¯t′t′⋆and¯b′b′⋆.
+These mechanisms for producing single heavy 4th family fermions are particularly interesting
+for the LHC when applied to the leptonic sector, due to the absence of the QCD background. As
+noted in [5] (see also [27]), the Higgs decay to a pair of on-shell ν′followed by ν′→ℓW, can yield a
+rateto a four lepton final state (via H→ν′¯ν′→4ℓ+/negationslashET) which is comparable to the rateexpected
+from the “golden mode” H→ZZ→4ℓ, if the mixing between the 4th generation leptons with
+the lighter leptons (via the charged current Ui4ℓ±
+iν′W∓,i= 1,2 or 3) is not exceedingly small.
+However, if mH<2mν′,2mℓ′, then the decays H→ν′¯ν′,ℓ′¯ℓ′are kinematically not open and, as
+in the quark sector, the Higgs Yukawa couplings to the 4th generat ion leptons can be probed only
+through its decays to a single ν′andℓ′. For instance, applying our results above to the leptonic
+channels, e.g., taking mH∼800 GeV and mν′∼500 GeV we expect the BR to the flavor diagonal
+3-bodydecay H→ν′¯ν′⋆, followed by ν′→ℓW, tobedramatically enhanced duetothelargewidth
+effects around the heavy Higgs resonance in gg→H→ν′ℓW(independent of the mixing angle
+Ui4, see section 3), thus yielding about 104H→ν′ℓW→(ℓW)ν′ℓWevents at the LHC with an
+integrated luminosity of O(100) inverse fb. When combined with the BR of the W’s into leptons,
+this will again yield a 4 ℓ+/negationslashETsignal comparable to that of the golden mode H→ZZ→4ℓ.
+The FC decays H→ν′¯νiandH→ℓ′¯ℓiare expected to have a BR of O(10−4) ifUi4∼O(0.1),
+i.e., similar to the corresponding FC decays in the quark sector. Howe ver, the leptonic mixing
+angles between the 4th generation leptons and the SM light leptons a re unfortunately expected
+to be at best Ui4<∼O(0.01) [1, 4].
+Finally, letuscomment onthepossibilityofsearching fornewCP-violat ioneffectsviaourHiggs
+decays to single 4th generation fermions. As is well known, the exte nsion of a 4th generation of
+fermions to the SM adds two new phases to the (now 4x4) CKM, which provide new sources of
+CP-violation [28]. Indeed, as was recently shown in [31], the SM4 may be a natural framework
+11for accommodating the observed flavor and CP structure in natur e, and for addressing the CP-
+properties associated with the 4th generation quarks. Understa nding the new CP-violating sector
+of the SM4 may also shed light on CP-anomalies in K and b-quark system s [9, 11, 29] and on
+baryogenesis [30]. Effects of the new SM4 CP-violating phases can be searched for directly in t′
+andb′systems as was recently noted in [10, 32]. In particular, studying CP -violation in these
+heavy 4th generation quark systems may help to pin down the single d ominating CP-violating
+quantity associated with the 4th generation quarks at high-energ ies [33]. In this respect our single
+t′andb′production channels via a heavy Higgs resonance, gg→H→t′bW, b′tW, may be rich in
+exhibiting various types of CP-asymmetries in analogy to CP-violation in single-top production
+[34].
+Acknowledgments: The research of AS is supported in part by the US DOE contract #
+DE-AC0298CH10886 (BNL). SBS and GE acknowledge research sup port from the Technion.
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+14
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+arXiv:1001.0570v1  [hep-ph]  4 Jan 2010Background magnetic field stabilizes QCD string against bre aking
+M. N. Chernodub∗
+Laboratoire de Math´ ematiques et Physique Th´ eorique, Uni versit´ e Fran¸ cois-Rabelais Tours,
+F´ ed´ eration Denis Poisson - CNRS, Parc de Grandmont, 37200 Tours, France
+DMPA, University of Gent, Krijgslaan 281, S9, B-9000 Gent, B elgium
+(Dated: January 3, 2010)
+The confinement of quarks in hadrons occurs due to formation o f QCD string. At large separation
+between the quarks the QCD string breaks into pieces due to li ght quark-antiquark pair creation.
+We argue that there exist a critical background magnetic fiel deB≃16m2
+π, above which the string
+breaking is impossible in the transverse directions with re spect to the axis of the magnetic field.
+Thus, at strong enough magnetic field a new, asymmetrically c onfining phase may form. The effect
+– which can potentially be tested at LHC/ALICE experiment – l eads to abundance of u-quark rich
+hadrons and to excess of radially excited mesons in the nonce ntral heavy-ion collisions compared to
+the central ones.
+PACS numbers: 25.75.-q,12.38.Aw
+Motivation. Noncentral heavy-ion collisions create
+intense magnetic fields with the magnitude of the order
+of the QCD scale. According to [1] the strength of the
+emerging magnetic field Bmay reach
+eBmax
+RHIC∼m2
+π, eBmax
+LHC∼15m2
+π (1)
+at the Relativistic Heavy Ion Collider (RHIC) and at
+the Large Hadron Collider (LHC). Here e=|e|is the
+absolute value of the electron charge.
+There are various potentially observable QCD effects
+associated with the presence of the strong magnetic field
+background. One can mention a CP-odd generation of
+an electric current of quarks along the axis of the mag-
+netic field (“the chiral magnetic effect”) [2] and enhance-
+ment of the chiral symmetry breaking (“the magnetic
+catalysis”) [3]. The latter is related to the fact that
+the background magnetic field makes the chiral conden-
+sate larger [4]. Acting through the chiral condensate,
+the magnetic field also shifts and strengthens the chiral
+transition [5].
+Recently, observation of certain signatures of the chi-
+ral magnetic effect was reported by the STAR collabora-
+tion at the RHIC experimental facility [6]. Some effects
+were also found in numerical simulations of lattice QCD.
+There exists numerical evidence in favor of existence of
+both the chiral magnetic effect [7] and the enhancement
+of the chiral symmetry breaking [8]. In addition, a chi-
+ral magnetization of the QCD vacuum – discussed first
+in Ref. [9] – was calculated numerically [10]. The lattice
+simulations have also revealed that due to CP-odd struc-
+ture of the QCD vacuum the quark’s magnetic dipole
+moment in a strong magnetic field gets a large CP-odd
+piece, the electric dipole moment [11].
+In addition to the chiral properties, the background
+magnetic field should also be important for confining
+features of QCD despite the photons are not interact-
+ing with the gluons directly. A strong enough magnetic
+field affects the dynamics of the gluons through the in-fluence on the quarks, because the quarks are coupled
+to the both gauge fields [12]. And, indeed, thermody-
+namicargumentsofRef.[13]suggestthatthebackground
+magnetic field should shift and weaken the confinement-
+deconfinementphasetransitionintheQCDvacuum. The
+confininganddeconfiningregionsatzerotemperatureare
+separated by a smooth crossover which is located at [13]
+eBcross[T= 0]∼(700MeV)2∼25m2
+π.(2)
+Due to the crossover nature of the transition, the differ-
+ence between confining and deconfining regions is some-
+what obscure. Therefore the confining interaction be-
+tween the quarks may also be visible at the magnetic
+fields that are stronger than the crossover scale (2).
+Although in our paper we discuss QCD vacuum effects
+ofthestrongmagneticfieldsbelowthecrossoverscale(2),
+it is interesting to mention that at stronger magnetic
+fields,eB/greaterorsimilar150m2
+πan exotic condensation of the u¯u
+pairs should occur [14]. Effects induced by magnetic field
+at finite quark density are also dramatic. For example,
+ateB/greaterorsimilar3m2
+πa stack of parallel π0domain walls should
+become more favorable (from an energy-related point of
+view) than nuclear matter at the same density [15].
+Confinement and string breaking at B =0.The
+quark confinement exists because the chromoelectric flux
+(in, say, a test quark-antiquark pair) is squeezed into a
+stringlike structure, the QCD string, due to nonpertur-
+bative vacuum effects. The QCD string has a nonzero
+tensionσ, and therefore the energy of the long enough
+string is proportional to its length R,
+Vstr(R) =σR, (3)
+In a quarkless QCD (i.e., in Yang-Mills theory) the en-
+ergy of the QCD string (3) rises infinitely with the sepa-
+ration between test quark and antiquark. Inevitably, the
+unboundedness of the potential leads to the quark con-
+finement. However, in the real QCD the string breaks2
+into pieces above certain critical separation Rbrbecause
+of the light quark-antiquark pair creation. The pair cre-
+ation leads to the collapse of the string, and, as a con-
+sequence, to formation of mesonic heavy-light bounds
+states,
+Q¯Q→Q¯Q+q¯q→Q¯q+q¯Q. (4)
+Indeed, at large separations it is more favorable to create
+the light pair because of the energy-related considera-
+tions: the energy of unbroken state [given by the total
+mass 2mQof the two heavy (or, test) quark sources Q
+and¯Qand the energy of the string (3)] is larger com-
+pared to the energy of the broken string state (given by
+the mass 2 mQ¯qof the created light-heavy mesons Q¯qand
+¯Qq). The critical string-breakingdistance Rbris thus de-
+termined by the energy balance:
+2mQ+σRbr= 2mQ¯q (5)
+(here we neglect a weak Coulomb interaction between
+the quarks in the unbroken state and we also disregard
+exponentially suppressed van der Waals interaction be-
+tween the heavy-light mesons in the broken state). If
+R > R br, then the string breaks and light-heavy mesons
+areformed, Fig.1(weconsiderhereasimplestcaseignor-
+ing a multiple pair production via string fragmentation).
+FIG. 1. The conventional string breaking: the QCD string
+spanned between the static quark Qand the static antiquark
+¯Qbreaks due to light q¯qpair creation.
+The stringbreakingwasobservedin lattice simulations
+of QCD with two flavors of equal-mass quarks [16]. An
+extrapolation of the lattice results to the real QCD gives
+the following string breaking distance [16]:
+Rbr≈1.13fm, (6)
+where statistical and systematical errors are of the order
+of 0.1fm each.
+QCD string breaking at nonzero magnetic field.
+A sufficiently strong background magnetic field shouldmodify the dynamics ofthe quarks, affecting not only the
+chiral features, but also influencing the confining proper-
+ties of the system. An anisotropic effect of the magnetic
+field on the confining scales of QCD was first pointed out
+by Miransky and Shovkovy in Ref. [12].
+The energy spectrum of a freerelativistic quark in a
+uniform magnetic field Bfollows a typical Landau pat-
+tern [17]:
+ωn,s/bardbl(p/bardbl) =±/radicalBig
+p2
+/bardbl+m2q+(2n+1−2sz)|eq|B,(7)
+wheremqandeqare, respectively, the mass and the elec-
+tric charge of the fermion, p/bardbl≡pzis the momentum
+of the fermion along the direction of the magnetic field,
+ands/bardbl=±1/2 is the projection of the fermion’s spin
+onto the axis of the magnetic field. The integer num-
+bern= 0,1,2,...labels the Landau levels. The signs
+“±” in front of the square root in Eq. (7) refer to, re-
+spectively, particle and antiparticle branches of the en-
+ergy spectrum. The electric charges of the light uandd
+quarks are, respectively:
+qu= +2e
+3, q d=−e
+3. (8)
+In a strong magnetic field the lowest Landau level
+(LLL) with n= 0 and sz= 1/2 plays a dominant role
+in the quark’s motion because the excited states are too
+heavy. Indeed, forthe soft (low-momentum)fermionsthe
+spin flips, sz→ −szand jumps to the higher states with
+n/greaterorequalslant1 are energetically suppressed by the typical gap
+δEq∼√
+2/lq, (9)
+where
+lq(B) = 1//radicalBig
+|eqB|, q=u,d (10)
+is the magnetic length of the quark q.
+Thus, at the LLL the motion of quarks becomes es-
+sentially (1+1) dimensional. The quark moves along the
+axis of the magnetic field and the longitudinal dynamics
+ofafree quarkis governedbyone-dimensionalrelativistic
+dispersion relation:
+ωLLL(p/bardbl) =±/radicalBig
+p2
+/bardbl+m2q.
+The transverse dynamics (i.e., the motion of the quark in
+the plane orthogonal to the magnetic field) is restricted
+to a region of a typical size of the order of the magnetic
+length (10),
+|δr|q/lessorsimilarlq(B). (11)
+In QCD the influence of the magnetic field should be-
+come significant when the magnetic length (10) becomes
+comparable with a typical QCD length scale Λ QCD∼
+1fm−1∼200MeV,
+lq(B)≃Λ−1
+QCD≃Rbr. (12)3
+Here we quoted the string breaking distance Rbrwhich
+was determined by the lattice simulations [16], Eq. (6).
+The condition (12) is satisfied at
+eBu≃6m2
+π, eB d= 12m2
+π, (13)
+foruanddquarks, respectively.
+Notice that the magnitude of the magnetic field ex-
+pected to be created at noncentral heavy-ion collisions at
+ALICE/LHC(1) is evengreaterthan the thresholds(13),
+so that effects of the transversequark localization may in
+principle be accessible in the LHC experiment. The lo-
+calization effects should be much less visible at the RHIC
+experiment due to the much weaker magnetic fields (1).
+What is the effect of the strong magnetic field
+on the process of the QCD string breaking? The
+breaking occurs due to the light q¯qpair creation. In the
+absence of the magnetic field the created light quark (an-
+tiquark) gets attracted to the heavy antiquark (quark)
+and the long string disappears completely, Fig. 1. How-
+ever, in a sufficiently strong magnetic field Bthe dis-
+tance between the light quark and antiquark in the B-
+transverse plane is limited by the B–dependent magnetic
+lengthlq, Eq. (11). Therefore if the string is located in
+theB-transverse plane, then the created q¯qpair cannot
+destroy the whole string, because the light quark and an-
+tiquark are bounded together by the magnetic field at
+the mutual separation lq, Fig. 2.
+FIG. 2. The (partial) breaking of the QCD string in the back-
+ground magnetic field. The distance between created quark
+qand antiquark ¯ qis restricted by the magnetic length (11).
+The axis of the magnetic field is perpendicular to the plane.
+In principle, the distance between the light quark and
+the light antiquark can be increased in the transverse di-
+rections by jumping to higher Landau levels. This is,
+however, an energetically costly process because of the
+presence of the magnetic gap (9). Therefore the creation
+of theq¯qpair from the vacuum may only remove a piece
+of a sufficiently long QCD string, Fig. (2). Moreover,
+due to the (1 + 1) dimensional character of the motionof the light quarks in the external magnetic field, the q¯q
+pair creation leads to appearance of two nonlocal (elon-
+gated) heavy-light mesons, Fig. 2. On the contrary, in
+the absence of the background magnetic field the string
+breaking leads to formation of the tightly-bound heavy-
+light mesons, Fig. 1.
+An increase in the magnetic field causes a decrease
+in the length (11) of the string piece that can be “re-
+moved” by the pair creation. Thus, as the magnetic field
+increases the system gets less gain in energy due to the
+string breaking. Intuitively it is clear that at certain
+strength of the magnetic field the pair creation become
+energetically unfavorable.
+Let us estimate the critical magnetic field which makes
+the string breaking impossible. Consider a static dis-
+tantly separated quark-antiquark pair Q¯Qin theB-
+transverse plane (i.e., with R≡R⊥andR/bardbl= 0). The
+energy of the unbroken state is
+Estring= 2mQ+σ(B)R, (14)
+where the string tension is, generally, a function of the
+strength of the magnetic field B.
+The emerging q¯qpair removes a segment of the string
+of the length lq(B), and the energy of the broken state is
+Ebroken= 2mQ+2mq(B)+σ(B)[R−lq(B)],(15)
+wheremqis the mass of the light quark.
+The condition for the string breaking not to occur is
+given by the inequality Ebroken/greaterorequalslantEstring. Using Eqs.
+(14) and (15), one gets
+2mq(B)/greaterorequalslantσ(B)lq(B) [no string breaking] .(16)
+The equality in (16) defines the lowest possible magnetic
+field (“critical no-breaking field”) for which the string
+breaking does not occur.
+Let us consider the condition (16) at zero temperature.
+The dynamical masses of q=u,dquarks in the QCD
+vacuum without a background magnetic field are
+mdyn
+q≃300MeV . (17)
+The external magnetic field of the practical scale (1)
+makes negligible contribution to the quarks’ masses (a
+significant contribution to the masses of the quarks is
+expected at the strength eB/greaterorsimilar(10TeV)2, Ref. [12]).
+Assuming that the critical no-breaking magnetic field is
+lower than the crossover strength (2), and taking into
+account the fact that at the crossover all observables
+are smooth, we ignore an essentially nonperturbative B-
+dependence of the string tension. We set the string ten-
+sion to its phenomenological value at B= 0,
+σ(B)≈σ(B= 0) = (440MeV)2.(18)
+Using Eqs. (16), (17) and (18) one finds that the QCD
+string fails to break via creation of u¯uandd¯dpairs if the
+magnetic fields reach the following values, respectively
+eB(u)
+cr≃8m2
+π, eB(d)
+cr= 16m2
+π.(19)4
+The critical fields (19) belong to the confinement region
+below the T= 0 crossover scale, Eq. (2), as anticipated.
+At the background magnetic field 0 < B < B(u)
+crthe
+string breaking in the B-transverse plane may proceed
+via the creation of both uanddquarks. In the field
+windowB(u)
+cr< B < B(d)
+crthe creation of the u¯upairs be-
+comes energetically unfavorable, while the string break-
+ing via the d¯dpair creation can still occur. At B > B(d)
+cr
+the QCD string cannot be broken neither by u¯u- or by
+d¯d-pair creation.
+The difference in the critical values (19) can be un-
+derstood as follows. According to Eqs. (8) and (10), the
+magnetic length ldof thed-quarkis√
+2 times longer then
+the magnetic length luof theu-quark. Consequently, the
+gain in the energy due to the d¯d-pair creation is larger
+compared to the energy gain achieved due to emergence
+of theu-quark pair. Thus, one needs a stronger magnetic
+field to suppress the d¯d-pair creation compared to a field
+neededtosuppressthecreationofa u¯u-pair,B(u)
+cr< B(d)
+cr.
+Proposal for numerical simulations. The effect of
+“freezing” of the QCD string breaking can be tested in
+the lattice simulations of QCD with dynamical fermions.
+The magnetic fields of the order of (19) and higher were
+already achieved in the (quenched) lattice simulations [7,
+8, 10, 11]. There are no principal obstacles to extend
+these simulations to QCD with dynamical fermions and
+check the prediction (19).
+Possible experimental consequences. The freez-
+ing of the QCD string breaking in the magnetic field may
+also have experimental consequences. Firstly, according
+to the estimation (1) of Ref. [1], the fields of the order of
+the critical value (19) may emerge in noncentral heavy-
+ion collisions at the LHC. However, our considerations
+do not apply to the first moments of the created quark-
+gluon fireball because the system stays in the deconfine-
+ment phase due to hot environment of the collision. As
+the plasma expands and cools down, it hadronizes in the
+presence of the decaying magnetic fields. The hadroniza-
+tion process in the magnetic field background involves
+string breaking events which are more favorable for the
+uquarks compared to the dquarks. Thus, the flavor-
+dependent freezing of the string breaking in the external
+magnetic field leads to the u-quark-rich content of the
+hadrons created in noncentral heavy-ion collisions com-
+pared to the central ones.
+Secondly, the background magnetic field stabilizes
+highly excited mesons by (i) freezing the quarks at the
+end-points of QCD string and (ii) suppressing the pro-
+cess of the light pair creation in the B-transverse plane.
+Since in the noncentral collisions the magnetic field axis
+is perpendicular to the reaction plane, we expect that
+the magnetic field reveals itself via abundance of radially
+excited mesons in the noncentral collisions compared to
+the central ones. The excited mesons should dominantly
+be polarized in the reaction plane, as the QCD stringin such mesons should tend to be perpendicular to the
+magnetic field axis. In each noncentral collision the ori-
+entation of the reaction plane can easily be determined
+from the elliptic flow of the emitted particles.
+Thus, we are coming to a qualitative prediction that
+can in principle be checked experimentally in the heavy-
+ion collision experiments. The long QCD strings should
+break into pieces by multiple q¯qpair creation leading to
+formation of generally unstable hadrons. These primary
+hadronslaterdecayintomorestablehadrons,leptonsand
+photons, eventually forming jets [18]. Since the particles
+in the jets come out essentially aligned along the original
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+reaction plane of the noncentral heavy-ion collisions.
+∗On leave of absence from ITEP, Moscow, Russia.
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\ No newline at end of file
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+arXiv:1001.0571v2  [hep-th]  25 Mar 2010JHEP03(2010)120 arXiv:1001.0571
+Quantum field theory, gravity and cosmology
+in a fractal universe
+Gianluca Calcagni
+Max Planck Institute for Gravitational Physics (Albert Eins tein Institute)
+Am M¨ uhlenberg 1, D-14476 Golm, Germany
+E-mail:calcagni@aei.mpg.de
+Abstract: We propose a model for a power-counting renormalizable field theor y
+living in a fractal spacetime. The action is Lorentz covariant and equ ipped with a
+Stieltjes measure. The system flows, even in a classical sense, fro m an ultraviolet
+regime where spacetime has Hausdorff dimension 2 to an infrared limit c oinciding
+withastandard D-dimensional fieldtheory. Wediscuss thepropertiesofascalarfield
+model at classical and quantum level. Classically, the field lives on a fra ctal which
+exchanges energy-momentum with the bulk of integer topological d imensionD. Al-
+thoughanobserver experiences dissipation, thetotalenergy-m omentum isconserved.
+The field spectrum is a continuum of massive modes. The gravitationa l sector and
+Einstein equations arediscussed indetail, also oncosmological backg rounds. We find
+ultraviolet cosmological solutions and comment on their implications fo r the early
+universe.
+Keywords: Models of Quantum Gravity, Cosmology of Theories beyond the SM.Contents
+1. Introduction 1
+2. Fractal universe 3
+2.1 Renormalization 6
+3. Scalar field theory 8
+3.1 Equation of motion and Hamiltonian 8
+3.2 Propagator: configuration space 14
+3.3 Propagator: momentum space 19
+4. Gravity 23
+4.1 Einstein equations 25
+4.2 Cosmology 26
+5. Conclusions and future developments 34
+1. Introduction
+The search for a consistent theory of quantum gravity is one of th e main issues in the
+present agenda of theoretical physics. Beside major efforts suc h as string theory and
+loopquantumgravity, otherindependent linesofinvestigationhave recentlyattracted
+some attention. Among these, Hoˇ rava–Lifshitz (HL) gravity [1, 2 ] is a proposal
+for a power-counting renormalizable model [3, 4] which is not Lorent z invariant.
+Coordinates scale anisotropically, i.e., [ t] =−zand [xi] =−1 in momentum units,
+wherez≥3 is a critical exponent typically fixed at z= 3. Because of this, the total
+actioncanbeengineeredsothattheeffectiveNewtonconstantbe comesdimensionless
+in the ultraviolet (UV) and higher-order spatial derivatives improve the short-scale
+behaviour of particle propagators. Due to the presence of releva nt operators, the
+system is conjectured to flow from the UV fixed point to an infrared (IR) fixed point
+where, effectively, Lorentzanddiffeomorphisminvarianceisrestor edatclassical level.
+Another property of HL gravity stemming from the running of the c ouplings
+effective dimension is that the spectral dimension dS[5, 6, 7, 8] at short scales is
+dS∼2 [2]. This is in intriguing accordance with other proposals for quantum gravity
+such as causal dynamical triangulations [9], asymptotically safegra vity [10] and spin-
+foammodels [11] (see also [12]). Systems whose effective dimensionalit y changes with
+– 1 –the scale can show fractal behaviour, even if they are defined on a smooth manifold.1
+All the above examples incarnate the popular notion that “the Unive rse is fractal”
+at quantum scales.
+Despite the beautiful physics emerging from the HL picture, inspire d by critical
+andcondensed-matter systems, itpotentiallysuffersfromatleas tonemajorproblem.
+Lorentz invariance, one of the best constrained symmetries of Na ture, is surrendered
+at fundamental level. As argued on general grounds [15, 16], even if deviations
+from Lorentz invariance are classically negligible, loop corrections to the propagator
+of fields lead to violations several orders of magnitude larger than t he tree-level
+estimate, unless the bare parameters of the model are fine tuned . This expectation
+[14] is indeed fulfilled for Lifshitz-type scalar models [17]. Although sup ersymmetry
+might relax the fine tuning [18], the present version of HL gravity is cle arly under
+strong pressure, also for other independent reasons.
+Motivated by the virtues and problems of HL gravity, it is the purpos e of this
+paper to formulate an effective quantum field theory with two key fe atures. The
+first is that power-counting renormalizability is obtained when the fr actal behaviour
+is realized at structural level, i.e., when it is implemented in the very defi nition of
+the action rather than as an effective property. In other words, we will require not
+only the spectral dimension of spacetime, but also its UV Hausdorff d imension [19]
+(which will coincide with dSin our case) to be dH∼2. Secondly, we wish to maintain
+Lorentz invariance.
+Therefore, this proposal is (a) defined on a fractal (in a sense ma de precise
+below), (b) Lorentz invariant, (c) power-counting renormalizable , (d) UV finite with
+no ghost or other obvious instabilities, and (e) causal. A condensed overview of the
+model was given in [20].
+Some of the ingredients we shall use are similar to those found in othe r recipes
+(e.g., scalar-tensor theories or models with fractional operators ). Their present mix-
+ing, however, will hopefully give fresh insight into some aspects of qu antum gravity.
+For example, a running cosmological constant naturally emerges fr om geometry as a
+consequence of a deformation of the Poincar´ e algebra.
+The plan of the paper is the following. The main idea is introduced in sect ion
+2. With particular reference to a scalar field theory, a dimensional a nalysis of the
+coupling constants is given in section 2.1. Section 3 is devoted to a sca lar field on a
+Minkowski fractal: itsclassical equationofmotionanddynamics are presented insec-
+tion 3.1, where the Hamiltonian formalism is shown to admit both a dissipa tive and
+conservative interpretation. The causal propagator of the fre e field in configuration
+space is calculated in section 3.2, while its Fourier–Stieltjes transfor m in momentum
+space is discussed in section 3.3. We outline the gravitational sector in section 4.
+1HL gravity with detailed balance possesses a natural fractal stru cture also because of the ap-
+pearance of fractional pseudo-differential operators [13, 14]. T his version of the theory, however,
+seems to be unviable.
+– 2 –Einstein and cosmological equations are derived in sections 4.1 and 4.2 , respectively,
+where cosmological solutions are found and analyzed. Section 5 con tains concluding
+remarks and a discussion on open issues and future developments.
+2. Fractal universe
+In HL gravity, one requires that time and space coordinates scale a nisotropically. On
+one hand, this leads to a running scaling dimension of the couplings and an effective
+two-dimensional phase in the UV. On the other hand, anisotropic sc aling gives rise
+to higher-order spatial operators and a non-Lorentz-invariant action. It turns out
+that we can achieve the first result (and avoid the second) by maint aining isotropic
+scaling,
+[xµ] =−1, µ= 0,1,...,D−1, (2.1)
+while replacing the standard measure with a nontrivial Stieltjes meas ure,
+dDx→d̺(x),[̺] =−Dα∝ne}ationslash=−D. (2.2)
+HereDis the topological (positive integer) dimension of embedding (abstra ct) space-
+time andα >0 is a parameter.2What kind of measure can we choose? A two-
+dimensional small-scale structure is a desirable feature of renorma lizable spacetime
+models of quantum gravity, and the most na¨ ıve way to obtain it is to le t the effective
+dimensionality of the universe to change at different scales. A simple r ealization of
+this feature is via fractional calculus and the definition of a fraction al action.3
+To begin with, we quote the following results in classical mechanics. In [26],
+empirical evidence was given that the Hausdorff dimension of a rando m process
+(Brownian motion) described by a fractional differintegral is propo rtional to the
+orderαof the differintegral; the same relation holds for deterministic fract als, and in
+general the fractional differintegration of a curve changes its Ha usdorff dimension as
+dH→dH+α(see also [27]). Moreover, integrals on net fractals can be approxim ated
+by the left-sided Riemann–Liouville fractional integral of a function L(t) [28, 29, 30,
+2The Hausdorff dimension of a set is greater than or equal to its topo logical dimension but
+the situation one has in mind here is a physical spacetime (the fracta l) embedded in an ambient
+D-dimensional manifold M. All physics takes place in the fractal and there are no observers in the
+“bulk”M. Given this picture, one can interpret the present model as “diffus ion of spacetime” in
+an embedding manifold.
+3Another route, which we shall not follow here, is to define particle ph ysics directly on a fractal
+set with general Borel probability measure ̺. This was done in [21] (and [22, 23] on Sierpinski
+carpets) for a quantum field theory on sets with Hausdorff dimensio n 4−ǫvery close to 4. The
+model in [21] has many aspects of dimensional regularization [24, 25], one difference being that the
+parameter ǫis taken to be physical.
+– 3 –31, 32],
+/integraldisplay¯t
+0d̺(t)L(t)∝0Iα
+¯tL(t) (2.3)
+≡1
+Γ(α)/integraldisplay¯t
+0dt(¯t−t)α−1L(t), (2.4)
+̺(t) =¯tα−(¯t−t)α
+Γ(α+1), (2.5)
+where¯tis fixed and the order αis (related to) the Hausdorff dimension of the set
+[28, 33]. The approximation in eq. (2.3) is valid for large Laplace moment a and can
+be refined to better describe the full structure of the Borel mea sure̺characterizing
+the fractal set. In the latter case, integration on the set is appr oximated by a sum
+of fractional integrals [30].
+Different values of 0 <α≤1 mediate between full-memory ( α= 1) and Markov
+processes (α= 0), and in fact αroughly corresponds to the fraction of states pre-
+served at a given time ¯tduring the evolution of the system [28, 32, 33]. Applications
+of fractional integrals range from statistics, diffusing or dissipativ e processes with
+residual memory [32], such as weather and stochastic financial mod els [34], to sys-
+tem modeling and control in engineering [35].
+Noticing that a change of variables t→¯t−ttransforms eq. (2.4) into the form
+1
+Γ(α)/integraldisplay¯t
+0dttα−1L(¯t−t), (2.6)
+the Riemann–Liouville integral can be mapped onto a Weyl integral [36 ] in the limit
+¯t→+∞. The limit is formal if the Lagrangian Lin eq. (2.6) is not autonomous.
+We assume otherwise, so that lim ¯t→∞L(¯t−t)≡L[q(t),˙q(t)].
+This form will be the most convenient for defining a Stieltjes field theo ry action.
+When¯t→+∞, eq. (2.6) is proportional to the usual formula in αdimensions em-
+ployed in dimensional regularization. After constructing a “fractio nal phase space”
+[37, 38, 39, 40], this analogy confirms the interpretation of the ord er of the fractional
+integral as the Hausdorff dimension of the underlying fractal [38].
+Alltheaboveresultsinonedimensioncanbeeasilygeneralizedtoa D-dimensional
+Euclidean space (e.g., [41, 42]), thus opening a possibility of application s in space-
+time. We entertain the possibility of formulating a scalar field theory w ith Stieltjes
+action, for the purpose of controlling its properties in the ultraviole t.4The scalar
+field model is interesting in its own right but also as a simple example wher eon to
+4Introductions on the Lebesgue–Stieltjes integralcan be found in [43, 44, 45]. A neat geometrical
+interpretation of the Riemann–Stieltjes one-dimensional integral as the projected “shadow of a
+fence” is given in [46, 47]. Projection is a tool sometimes employed to d etermine the Hausdorff
+dimension of a fractal [19].
+– 4 –work out the physics. After that, we shall explore the gravitation al sector. In D
+dimensions, we consider the action
+S=/integraldisplay
+Md̺(x)L(φ,∂µφ), (2.7)
+whereLis the Lagrangian density of the scalar field φ(x) and
+d̺(x) =D−1/productdisplay
+µ=0f(µ)(x)dxµ(2.8)
+is some multi-dimensional Lebesgue–Stieltjes measure (actually a Le besgue measure,
+if̺is absolutely continuous, which we assume to be the case) generalizin g the triv-
+ialD-dimensional measure dDx. We denote with ( M,̺) the metric spacetime M
+equipped with measure ̺. We shall consider the situation where Mis a manifold
+but this may not be the case in general.
+Equation (2.7) resembles a field theory with a dilaton or conformal re scaling
+v=/producttext
+µf(µ)of the Minkowski determinant. As in these other models, one will
+obtain an extra friction term in the equation of motion, although the physics will
+be radically different both at microscopic and macroscopic level. This is because
+the measure weight must scale in a certain way, while dilaton solutions in effective
+actions of string theory typically enjoy much more freedom. We sho uld stress at least
+two more reasons why thepresent model is not just anexotic refo rmulationof dilaton
+scenarios.5First, the dilaton of string theory couples differently in different sec tors,
+thus leading to a violation of the strong equivalence principle; in our ca se, the scalar
+fieldvis still of geometric origin but appears as a global rescaling. Second, a change
+in the measure is accompanied by a new definition of functional variat ions and Dirac
+distributions, in turn leading to an unfamiliar propagator and the def ormation of the
+Poincar´ e algebra.
+If̺isnot invariant under theLorentzgroup SO(D−1,1), theequipped manifold
+is not isotropic even if Mis the Minkowski flat manifold. If only global Poincar´ e
+invariance is broken, as it typically happens in fractals, ( M,̺) is not homogeneous.
+Since we wish the Lorentz group SO(D−1,1) to be part of the symmetry group of
+the action, the Lagrangian density Land theDweightsf(µ)must be Lorentz scalars
+separately. The former can be taken to be the usual scalar field La grangian,
+L=−1
+2∂µφ∂µφ−V(φ), (2.9)
+whereVis a potential and contraction of Lorentz indices is done via the Minko wski
+metricηµν= (−+···+)µν. As for the Stieltjes measure, we make the spacetime
+isotropic choice
+f(µ)=f, µ = 0,1,...,D−1. (2.10)
+5These reasons do not forbid the fractal model to admit alsoa “dilaton-like” reformulation (see
+below).
+– 5 –This should eventually correspond to a fractal in time and space. Th ere are many
+otherAns¨ atze, for instance an isotropic nontrivial measure
+f(0)= 1, f (i)=f, i = 1,...,D−1, (2.11)
+or an anisotropic measure of the form
+f(µ)=/braceleftbigg1, µ= 0,...,i−1
+f, µ=i,...,D−1. (2.12)
+These measures will correspond to different dynamics but, by cons truction, to the
+same UV Hausdorff dimension dH∼2. We take eq. (2.10) with
+v≡fD,[v] =D(1−α), (2.13)
+which generalizes eq. (2.6).6The scalar field action reads
+S=−/integraldisplay
+dDxv/bracketleftbigg1
+2∂µφ∂µφ+V(φ)/bracketrightbigg
+. (2.14)
+We now pause and discuss the interpretation of the measure. Class ically, one can
+boost solutions of the equation of motion to a Lorentz frame where v=v(x) (space-
+like fractal) or v=v(t) (timelike fractal). These two cases will lead to different
+classical physics but at quantum level all configurations should be t aken into ac-
+count, so there is no quantum analogue of space- or timelike fracta ls.7In any case,
+we shall see that the theory on theDα-dimensional fractal is expected to be dissipa-
+tive, i.e., nonunitary. This conclusion is in line with the known results of frac tional
+mechanical systems. Fortunately this will not be a problem because , from the point
+of view of the manifold MwithDtopological dimensions, energy and momentum
+are indeed conserved.
+2.1 Renormalization
+The scaling dimension of φis
+[φ] =Dα−2
+2, (2.15)
+6v=v(x) is a coordinate-dependent Lorentz scalar. An alternative gener alization of non-
+relativistic fractals might have been to choose vto be also metric dependent, e.g., v=
+|gµνχµχν|D(α−1)
+2for some vector χµ. In this case, however, one does not obtain a consistent set of
+equations of motion.
+7We have seen that, approximately, fractional integrals can repre sent systems living on a certain
+class of fractal sets. Since we will assume the existence of a nontr ivial renormalization group flow
+entailing integrals of different orders α1,α2,..., the complete all-scale picture beyond classical level
+will not be a fractal with scale-independent Hausdorff dimension but amultifractal . For this and
+the reason stated in the text, the fractal interpretation of fra ctional integrals is more involved in
+the quantum theory. To our purposes it is not necessary to stick w ith it, although we shall do so
+with a slight abuse of terminology. We shall see that the spectral dim ension of the universe changes
+in a precise, α-dependent way, thus justifying the term “fractal” a posteriori .
+– 6 –which is zero if, and only if,
+α=2
+D. (2.16)
+Thenα= 1/2 in four dimensions. This value can change for the other measures
+defined in eqs. (2.10) and (2.12). In the example (2.12) with [ f] = 1−α, one has
+α= 2/(D−i), and in order for the integral to be properly fractional (rather than
+multiple) it must be i≤D−3. In four dimensions, there can be at most one ordinary
+direction. If i= 1, thenα= 2/3.i= 0 corresponds to eq. (2.13), which we shall
+adopt from now on.
+Let the scalar field potential be polynomial,
+V=N/summationdisplay
+n=0σnφn, (2.17)
+and letNbe the highest (positive) power. The coupling σNhas engineering dimen-
+sion
+[σN] =Dα−N(Dα−2)
+2. (2.18)
+For the theory to be power-counting renormalizable [ σN]≥0, implying
+N≤2Dα
+Dα−2ifα>2
+D, (2.19)
+N≤+∞ ifα≤2
+D. (2.20)
+Whenα= 1, one gets the standard results [ φ] = (D−2)/2,N≤2D/(D−2); in four
+dimensions, the φ4theory is renormalizable. In two dimensions, Nis unconstrained.
+Theseconsiderationsleadustotrytohavetheparametersrunfr omanultraviolet
+nontrivial fixed point where α= 2/Dto an infrared fixed point where, effectively ,
+α=αIR. Thedimension ofspacetime iswell constrained tobe4 fromparticle p hysics
+tocosmologicalscales andstartingatleast fromthelast scatterin gera[48,49,50,51].
+Therefore,αIR= 1 ifD= 4. To actually realize this particular flow, one should add
+relevant operators to the action corresponding to terms with triv ial measure weight.
+The total scalar action then is
+S=/integraldisplay
+dDx/bracketleftBig
+vL+MD(1−α)˜L/bracketrightBig
+, (2.21)
+whereMis a constant mass term ([ M] = 1) and ˜LisL, eq. (2.9), with all different
+bare couplings ( σn→˜σn). We symbolically represent this modification of the action
+as
+v(x)→v(x)+MD(1−α). (2.22)
+The constant term is anyway expected in the most general Lorent z-invariant defini-
+tion of the measure weight.
+– 7 –A fractal structure must shortly evolve to a smooth configuratio n. Oscillations
+of neutralBmesons can constrain the typical UV mass scale to be larger than ab out
+[52]
+M >300÷400 GeV. (2.23)
+Here we do not attempt to place constraints on this scale with other high-energy
+observations.
+As one will see in section 3.3, convergence of the Feynman diagrams is better
+than in four dimensions, as one can check by looking at the superficia l degree of
+divergence, which is the same as for a Dα-dimensional theory [21]. In the case of
+gravity, in fact, the usual configuration-space results in 2+ ǫdimensions should apply
+near the UV fixed point [53, 54, 55, 56, 57, 58, 59].
+Needless to say, the above construction and remarks fall short o f demonstrating
+the existence and effectiveness of such a flow, which should be verifi ed by explicit
+calculations. Our attitude will be to introduce the model and first se e its character-
+istic features and possible advantages, leaving the issue of actual renormalizability
+for the future.
+At any rate, classically the system willflow from a lower-dimensional fractal
+configuration to a smooth D-dimensional one. This is clear from the definition (2.22)
+of the measure weight and its scaling properties when α <1. At small space-time
+scales, the weight v∼ |x|D(α−1)dominates over the constant term, while at large
+scales it is negligible. This is true simply by construction, and independe ntly from
+renormalization issues.
+Therefore, the phenomenological valence of the model is guarant eed, at least.
+In our framework both the Newton’s coupling and the cosmological c onstant will
+vary with time already at classical level. On one hand, in minisuperspac e models
+motivated by other approaches to quantum gravity, the running o f the couplings
+can be implemented at the level of the equations of motion, thus obt aining a high-
+energy “improved” dynamics. This strategy is adopted, for instan ce, in the Planck-
+ian cosmology of asymptotically safe gravity thanks to its renormaliz ation properties
+[60, 61, 62]. On the other hand, a phenomenological time-varying dim ension can
+be considered for constraining the transition scale from fractal t o four-dimensional
+physics [52]. The couplings running is then also obtained in fractal-rela ted cosmo-
+logical toy models with variable dimension [63].8
+3. Scalar field theory
+3.1 Equation of motion and Hamiltonian
+The Euler–Lagrange and Hamilton equations of classical mechanical systems with
+8All these scenarios differ in philosophy with respect to [64], where the spacetime dimension is
+promoted to a dynamical field.
+– 8 –(absolutely continuous) Stieltjes measure have been discussed in [6 5, 66, 67] in the
+one-dimensional case and [68, 69, 70] in many dimensions. The Euler– Lagrange
+equation of scalar field theory can be found in [71].
+We can easily adapt the same procedure in our case. From now on we c onsider
+only the UV part of the action, setting M= 0. Any result in the infrared can then
+be obtained by going to the effective limit α→1.
+The metric space is equipped with a nontrivial measure and caution sh ould be
+exercised when performing functional variations. For instance, t he correct Dirac
+distribution is
+1 =/integraldisplay
+d̺(x)δ(D)
+v(x), (3.1)
+as was also noticed in [21]. Invariance of the action under the infinites imal shift
+φ→φ+δφ (3.2)
+yields the equation of motion (for a generic weight v)
+0 =∂L
+∂φ−/parenleftbigg∂µv
+v+d
+dxµ/parenrightbigg∂L
+∂(∂µφ). (3.3)
+From eq. (2.14) we get
+/squareφ+∂µv
+v∂µφ−V′= 0, (3.4)
+where/square=∂µ∂µand a prime denotes differentiation with respect to φ.
+The above friction term is characteristic of dissipative systems and one would
+expect energy not to be conserved. In fact, the Hamiltonian is no lo nger an integral
+of motion. Let us define the momentum
+πφ≡δS
+δ˙φ=˙φ, (3.5)
+where dots indicate (total) derivatives with respect to time and we h ave taken
+eq. (3.1) into account.
+Defining the Lagrangian
+L≡/integraldisplay
+dxvL, (3.6)
+the Hamiltonian is
+H≡/integraldisplay
+dxv(πφ˙φ)−L (3.7)
+=/integraldisplay
+dxv/parenleftbig1
+2π2
+φ+1
+2∂iφ∂iφ+V/parenrightbig
+≡/integraldisplay
+dxvH, (3.8)
+– 9 –wheredx=dx1...dxD−1. The definition of the equal-time Poisson brackets (time
+dependence implicit)
+{A(x),B(x′)}v≡/integraldisplay
+d̺(y)/bracketleftbiggδA(x)
+δφ(y)δB(x′)
+δπφ(y)−δA(x)
+δπφ(y)δB(x′)
+δφ(y)/bracketrightbigg
+(3.9)
+yields the Hamilton equations
+˙φ={φ,H}v, (3.10)
+˙πφ={πφ,H}v−˙v
+vπφ, (3.11)
+equivalent to eqs. (3.5) and (3.4), respectively. Therefore, time e volution of an ob-
+servableO(φ,πφ,x) is
+˙O=∂tO+{O,H}v−˙v
+vπφ∂O
+∂πφ. (3.12)
+Equations (3.11) and (3.12) signal dissipation. Nonconservation of the Hamiltonian
+is expected from the definition of the Lagrangian (3.6). The measur e factorvis both
+time and space dependent, so it was not possible to factorize it in ord er to write the
+action as the correct Stieltjes time integral of L. However, one can exploit Lorentz
+invariance and pick a frame where v=v(|x|). In this frame Hwould be restored as
+the generator of time translations; for a timelike fractal it is not po ssible to pick this
+frame and there is always energy dissipation. Conversely, physical momentum will
+be dissipated in a spacelike fractal.
+Is there a problem with that? Whatever the choice of classical frac tal model,
+one would have to face the issue of unitarity at quantum level. Moreo ver, we need
+a physical interpretation of dissipation. It turns out that the latt er helps to address
+the above concern.
+Consider a ( D−1)-dimensional box of size land spatial volume lD−1. At the
+scalel, particles live effectively in Dαspacetime dimensions. If α= 1, they occupy
+the whole phase space in the box. Otherwise, they must dissipate en ergy, since the
+energy of the configuration filling the entire topological volume is diffe rent from that
+of a configuration limited to the effective Dα-dimensional world. The total energy of
+the system EinDtopological dimensions is conserved, but the energy Hmeasured
+by aDα-dimensional observer is not (eq. (3.12) with O=Hvand integrated in
+space):
+˙H=∂tH−/integraldisplay
+dx˙vπ2
+φ
+=−/integraldisplay
+dx˙vL. (3.13)
+– 10 –In fact, a conserved quantity is
+E(t) =H(t)+Λ(t), (3.14)
+Λ(t) =/integraldisplayt
+dt/integraldisplay
+dx˙vL, (3.15)
+which we are going to obtain also from Noether’s theorem (see [67, 72 ] for a com-
+putation in classical mechanics). It should be easy to see that Λ can be regarded as
+thecomplementary function of initialized fractional calculus [35, 73, 74, 75]. Phys-
+ically, it is a running cosmological constant of purely geometric origin. For this
+reason, dissipation might eventually prove to be an asset rather th an a liability of
+the theory.
+Unlike standard scalar field theory, the Noether current associat ed with the
+usual Lagrangian continuous symmetries is not covariantly conser ved. On the other
+hand, one can easily find generalized conserved currents. Take a g eneric infinitesimal
+transformation of the field, eq. (3.2), and coordinates, xµ→xµ+δxµ. We consider
+symmetries of the autonomous Lagrangian density Land define “quasi invariance”
+of the action as done in [67, 71, 72]. Then δLis a total divergence,
+L → L+dJµ
+dxµ. (3.16)
+Combining this equation with
+δL=∂L
+∂φδφ+∂L
+∂(∂µφ)δ(∂µφ)
+=∂L
+∂φδφ+∂L
+∂(∂µφ)∂µ(δφ)
+=/parenleftbigg∂L
+∂φ−d
+dxµ∂L
+∂∂µφ/parenrightbigg
+δφ+d
+dxµ/parenleftbigg∂L
+∂∂µφδφ/parenrightbigg
+,
+(3.17)
+one obtains on shell
+−d
+dxµ/bracketleftbigg
+v/parenleftbigg
+Jµ−∂L
+∂∂µφδφ/parenrightbigg/bracketrightbigg
++Jµ∂µv= 0. (3.18)
+Choose a coordinate translation
+xµ′=xµ+δxµ, δxν=−aν, (3.19)
+so thatδφ=aν∂νφandδL=aνdL/dxν. Then
+−d
+dxµ(vTµ
+ν)+L∂νv= 0, (3.20)
+– 11 –where
+Tµ
+ν≡δµ
+νL+∂µφ∂νφ (3.21)
+is the usual energy-momentum tensor. Integrating eq. (3.20) in s pace, one gets
+˙Pν+/integraldisplay
+dx∂νvL= 0, (3.22)
+where
+Pν≡ −/integraldisplay
+dxvT0
+ν. (3.23)
+Theν= 0componentyields P0=Handeq.(3.13)followssuit. The ν=icomponent
+gives the physical momentum
+Pi=/integraldisplay
+dxvπφ∂iφ, (3.24)
+and its conservation law
+˙Pi+˙Λi≡˙Pi+/integraldisplay
+dx∂ivL= 0. (3.25)
+Pigenerates spatial translations in the field but not in its conjugate m omentum,
+since the covariant counterparts of eqs. (3.10) and (3.11) are
+∂µφ={φ,Pµ}v, (3.26)
+∂µπφ={πφ,Pµ}v−∂µv
+vπφ. (3.27)
+The covariant version of eq. (3.12) is
+dO
+dxµ=∂µO+{O,Pµ}v−∂µv
+vπφ∂O
+∂πφ, (3.28)
+yielding
+dPν
+dxµ={Pν,Pµ}v−δ0
+ν/integraldisplay
+dx∂µvL. (3.29)
+The(µ,ν)componentsofthisequationofferconsistencychecksandnewco mmutation
+relations. Component (0 ,0) corresponds to eq. (3.13), while ( i,j) gives
+{Pi,Pj}v= 0. (3.30)
+Components (0 ,i) and (i,0) and eq. (3.25) are consistent among each other if, and
+only if,
+{H,Pi}v=/integraldisplay
+dx∂ivL, (3.31)
+finally showing that the Poincar´ e algebra is now noncommutative unle ssvis only
+time dependent (timelike fractal). One can check that also the Lore ntz algebra is
+– 12 –deformed. Consider the Noether current associated with boost/ rotation transforma-
+tions
+δφ=aνσ(xν∂σ−xσ∂ν)φ, (3.32)
+δL=aνσ(xν∂σ−xσ∂ν)L=aνσ∂µ(gµσxνL−gµνxσL) =∂µJµ.(3.33)
+Substituting in eq. (3.18), we get
+−d
+dxµ(vMµνσ)+L(xν∂σv−xσ∂νv) = 0, (3.34)
+where
+Mµνσ≡xνTµσ−xσTµν. (3.35)
+The algebra of the Lorentz generators Jνσ≡/integraltext
+dxvM0νσ, which includes the Lorentz
+boostsKi≡J0iand the angular momenta Li≡1
+2ǫi
+jkJjk(ǫi
+jkis the Levi-Civita
+symbol), is deformed because of the nontrivial weight v.
+We have ended up with a classical field theory living ona Dα-dimensional fractal
+breaking Poincar´ e invariance. Depending whether the fractal is “ timelike” or “space-
+like,” there will be a momentum or energy transfer between the world -fractal and
+theD-dimensional embedding.
+TheD-dimensional side of the picture can be actually made more precise. S o
+far we have interpreted the function vin the action (2.14) as (the derivative of) a
+Stieltjes measure defined on a fractal of Hausdorff dimension Dα. Of course one
+can regard it as a “dilaton” field coupled with the Lagrangian density Lliving on a
+D-dimensional manifold. Then it is natural to consider the usual δof Dirac,
+1 =/integraldisplay
+dDxδ(D)(x). (3.36)
+Consequently, the momentum conjugate to the field φis
+πv=vπφ. (3.37)
+It is easy to convince oneself that the Poisson brackets are the us ual
+{·,·}1, (3.38)
+and that all vdependence disappears in the D-momentum
+Pµ=Pµ(φ,πv). (3.39)
+Poincar´ e (and Lorentz) invariance is preserved. This completes t he proof that, at
+least at classical level, dissipation occurs relatively between parts o f aconservative
+system. Quantization would follow through, although an UV observe r would experi-
+ence an effective probability flow from or into his world-fractal (see also [76]).
+– 13 –To summarize, because the universe is associated with a “fractal” s tructure one
+typically expect to have breaking of Lorentz invariance at small sca les. The geom-
+etry of the problem is not standard and this modifies the usual defin ition of Dirac
+distribution and Poisson brackets. As a result, from the point of vie w of an observer
+living in the fractal and measuring geometry with weight v, Lorentz and translation
+invariance are broken inasmuch as the action itself is Lorentz and Po incar´ e invariant,
+but the algebra of the Poincar´ e group is deformed. In other word s, when talking
+about fractals embedded in Minkowski spacetime we mean the geometries defined
+by the deformed Poincar´ e group . However, from the point of view of the ambient
+D-dimensional manifold Poisson brackets and functional variations n o longer feature
+the nontrivial measure weight, which is now regarded as an independ ent matter field
+rescaling. In that case, the full Poincar´ e group is preserved. On the other hand, the
+properties of nonrelativistic fractals are more intuitive: they break translation in-
+variance because the measure weight introduces explicit coordinat e dependence and
+the system is not autonomous.
+3.2 Propagator: configuration space
+The theory is Lorentz invariant, ghost free and causal at all scale s. We can check
+this explicitly by computing the causal propagator G(x), which is (proportional to)
+the propagator in two and Ddimensions at the UV and IR fixed points, respectively.
+As usual, we define the vacuum-to-vacuum amplitude (partition fun ction)
+Z[J] =/integraldisplay
+[dφ]ei/integraltextd̺(L+φJ), (3.40)
+whereJis a source and we have already integrated out momenta. Integrat ion by
+parts in the exponent allows us to write the Lagrangian density for a free field as
+L=1
+2φ/parenleftbigg
+/square+∂µv
+v∂µ−m2/parenrightbigg
+φ≡1
+2φCφ. (3.41)
+The propagator is the Green function solving
+CG(x) =δ(D)
+v(x). (3.42)
+By virtue of Lorentz covariance, the Green function Gmust depend only on the
+Lorentz interval
+s2=xµxµ=xixi−t2, (3.43)
+wheret=x0andi= 1,...,D−1. In particular, v=v(s) with the correct scaling
+property is v(s)∼ |s|D(α−1). This definition guarantees reality of the measure and
+avoids problems with unitarity (in particular, the action is real).
+One might be worried that the measure blows up on the light cone, but this is
+an integrable singularity: the check that/integraltext
+AdDxv <∞is done on a compact set A
+– 14 –(for instance, a D-ball of radius R) and in Euclidean signature. Below we will also
+see that there is nothing pathological in the propagator on the light -cone, even if v
+is singular in s= 0. Measures describing fractals may be very irregular and it would
+be worth investigating the physical interpretation of their singular ities, especially in
+Lorentzian signature. For the time being, we notice that one can als o define the
+measure weight to be
+v(s) =/braceleftbigg|s|D(α−1), s∝ne}ationslash= 0
+1, s = 0. (3.44)
+This definition may be in contrast with the original assumption that th e measure be
+absolutely continuous. This would mean that the simplified model with a n overall
+measure weight does not come from a most general fractal model with Lebesgue–
+Stieltjes measure ̺: in other words, d̺(x)∝ne}ationslash=v(x)dDx. This is not a problem for two
+reasons. Ononehand, thesimple model with measure weight visstill abletocapture
+much of the physics of the general Stieltjes model, by virtue of the scaling argument;
+as a matter of fact, one could even take measures which are Lebes gue–Stieltjes only
+asymptotically, and yet obtain a modelization of a fractal quantum fi eld theory in
+certain regimes. On the other hand, one can devise other measure profiles with the
+same scaling properties and regular behaviour. As an example, inste ad of eq. (3.44)
+one could take
+v(s) =1
+2ℓD(1−α)+1
+|s|D(1−α)+2ℓD(1−α), (3.45)
+whereℓ= 1/M. At small s(near the light cone) or large space/time scales, v→
+const; at intermediate scales, vhas the power-law behaviour which we will assume
+from now on.
+Without risk of confusion, we use the symbol ∂to denote total derivatives.
+Noting that
+∂µ=xµ
+s∂s,/square=∂2
+s+D−1
+s∂s, (3.46)
+the inhomogeneous equation (3.42) reads
+/parenleftbigg
+∂2
+s+Dα−1
+s∂s−m2/parenrightbigg
+G(s) =δ(D)
+v(x), (3.47)
+where we added a mass term, [ m] = 1,m2>0. We first consider the Euclidean
+propagator and denote with r=√xixi+t2the Wick-rotated Lorentz invariant.
+In the massless case, the solution of the homogeneous equation is
+G=C(r2)1−Dα
+2, (3.48)
+– 15 –whereCis a normalization constant. The right-hand side of eq. (3.47) is not t heδ
+defined in radial coordinates. To find the latter, one notices that
+1 =/integraldisplay
+dDxvδ(D)
+v(x)
+= ΩD/integraldisplay
+drvrD−1δ(D)
+v(x)
+=/integraldisplay
+drδ(r),
+where Ω D= 2πD/2/Γ(D/2) is the volume of the unit D-ball. Therefore,
+/integraldisplay
+d̺(x)δ(D)
+v(x) =/integraldisplay
+d̺(x)/bracketleftbiggr1−D
+ΩDv(r)δ(r)/bracketrightbigg
+. (3.49)
+Hence, to find the propagator also for r= 0 one can take some test function ϕand
+compute
+ΩD∝an}b∇acketle{tKG,ϕ∝an}b∇acket∇i}ht= lim
+ǫ→0ΩD/integraldisplay+∞
+ǫdrKG(r)ϕ(r), (3.50)
+where
+K=v(r)rD−1C
+=∂r/parenleftbig
+rDα−1∂r/parenrightbig
+−rDα−1m2. (3.51)
+Therefore,
+ΩD∝an}b∇acketle{tKG,ϕ∝an}b∇acket∇i}ht= ΩD∝an}b∇acketle{tG,Kϕ∝an}b∇acket∇i}ht
+= lim
+ǫ→0ΩD/integraldisplay+∞
+ǫdrG(r)∂r/parenleftbig
+rDα−1∂rϕ/parenrightbig
+=CΩD(2−Dα)ϕ(0),
+where we have used eq. (3.48) and integrated by parts once (boun dary terms vanish).
+The last line must be equal to ∝an}b∇acketle{tδ,ϕ∝an}b∇acket∇i}ht, thus fixing C. Then, the Green function for
+m= 0 reads
+G(r) =1
+ΩD(2−Dα)(r2)1−Dα
+2. (3.52)
+This result enjoys several consistency checks. When α= 1, it is the usual Green
+functionGDfor the Laplacian in Ddimensions with standard Lebesgue measure:
+lim
+α→1G(r) =GD(r) =−Γ/parenleftbigD
+2−1/parenrightbig
+4πD/2(r2)1−D
+2.
+In the UV limit α→2/D, one can expand eq. (3.52) as rǫ/ǫ= 1/ǫ+ lnr+O(ǫ).
+Up to a divergent constant, this is the logarithmic propagator in two dimensions,
+rescaled with a volume ratio:
+G∗(r)≡lim
+α→2/DG(r) =Ω2
+ΩDG2(r) =1
+ΩDlnr. (3.53)
+– 16 –As a side remark, notice that Gcan be written as
+G(r)∝d−β
+dr−βGD(r), (3.54)
+where
+β≡D(1−α), (3.55)
+and the fractional derivative can be defined as the Liouville derivativ e:
+d−β
+dr−βrγ=Γ(γ+1)
+Γ(γ+β+1)rγ+β. (3.56)
+The order of the derivative is negative, so that it is actually a fractio nal integral with
+clear meaning: Starting from the problem /squareGD=δ(D)and inserting (heuristically)
+the identity ∂β∂−β, it replaces the second-order operator on GDwith a fractional
+differentiation of order 2+ βonG. We can conclude that
+G(r)∝G(1+β/2)(r), (3.57)
+i.e., the (massless) propagator is proportional to the Green funct ion solving the
+pseudodifferential equation
+/square1+β/2G(1+β/2)=δ(D), (3.58)
+which was calculated and discussed in [77, 78, 79, 80, 81, 82] and rea ds
+G(1+β/2)∝(s2+iε)1+β−D
+2. (3.59)
+Indeed, after Wick rotation eqs. (3.48) and (3.59) agree up to the normalization.
+In this sense, in configuration space our field theory on a fractal is equivalent to a
+certain class of nonlocal models represented by eq. (3.58) [81, 82, 83].
+We now consider the massive case (Helmholtz equation). The solution of the
+homogeneous equation CG= 0 is
+G(r) =/parenleftBigm
+r/parenrightBigDα
+2−1/bracketleftBig
+C1KDα
+2−1(mr)+C2IDα
+2−1(mr)/bracketrightBig
+, (3.60)
+whereC1,2are constants and KandIare the modified Bessel functions. Since for
+smallmthe solution must agree with the massless case (3.48), we can set C2= 0.
+In fact, this is true only for α∝ne}ationslash= 2/D, as one can see from the asymptotic formulæ
+Kν(z)≈
+
+−ln/parenleftbigz
+2/parenrightbig
+−γifν= 0
+Γ(ν)
+2/parenleftbig2
+z/parenrightbigνifν >0, (3.61)
+Iν(z)≈1
+Γ(ν+1)/parenleftBigz
+2/parenrightBigν
+, (3.62)
+– 17 –whereγis the Euler–Mascheroni constant. We shall discuss the case α= 2/D
+separately.9
+To find the solution of the inhomogeneous equation, one exploits the fact that
+the mass term does not contribute near the origin. Expanding eq. ( 3.60) atmr∼0
+whenDα>2 (C2= 0),
+G(r)∼C12Dα
+2−2Γ/parenleftbigDα
+2−1/parenrightbig
+(r2)1−Dα
+2,
+which must coincide with eq. (3.52). This fixes the coefficient C1and the propagator
+reads
+G(r) =−1
+2πD
+2Γ/parenleftbigD
+2/parenrightbig
+Γ/parenleftbigDα
+2/parenrightbig/parenleftBigm
+2r/parenrightBigDα
+2−1
+KDα
+2−1(mr), (3.63)
+in agreement with the Helmholtz propagator ( α= 1).
+Here an important remark is mandatory. In usual quantum field the ory, the
+propagator G(r) is defined up to an immaterial constant C. By “immaterial” we
+obviously mean that /square[G(r) +C] =/squareG(r). This is true in any dimension Dand
+regardless the functional form of G(r), being it just a property of ordinary differen-
+tiation,/squareC= 0. Our model is still characterized by ordinary differential operat ors,
+so the set/braceleftbig
+G(r)+C/vextendsingle/vextendsingleC= const/bracerightbig
+(3.64)
+is an equivalence class defining the propagator. However, the nont rivial measure ̺
+will associate elements of this equivalence class with different Fourier transforms in
+momentum space. In the critical case α= 2/D, the difference will be in unusual
+terms
+B(C1,C2,C)1
+k2
+which resemble massless poles. If the theory is well defined, these t erms will have
+to correspond to generalized Dirac distributions, and notto particle modes with
+arbitrarily chosen residue B(C1,C2,C). Keeping this in mind will prevent us to fall
+into a false paradox when calculating the propagator in momentum sp ace.
+Taking this issue on board, the case α= 2/D,m∝ne}ationslash= 0 is straightforward: it is
+safe to just set α= 2/Din the noncritical propagator (3.63),
+G∗(r) =−1
+ΩDK0(mr). (3.65)
+9The comparison with field theory with fractional powers of the d’Alem bertian holds also when
+m∝ne}ationslash= 0. Recalling that
+/parenleftbigg1
+zd
+dz/parenrightbiggn
+[z−νKν(z)] = (−1)nz−ν−nKν+n(z), n∈N,
+one can continue this formula to any nand write the propagator as
+G(r)∝/parenleftbigg1
+rd
+dr/parenrightbigg−β
+2
+GD(r).
+– 18 –Now to the Lorentzian theory. The fractal field models of [21] and [2 2] both
+meet the Osterwalder–Schrader conditions. This encourages the expectation that a
+generic field theory on a Euclidean fractal, if well defined, should adm it an analytic
+continuation to a theory in Lorentz spacetime. In fact, the Euclide an partition
+functioneq.(3.40)isaSchwingerfunctionendowedwithalltheprop ertiesrequiredby
+the Osterwalder–Schrader theorem: it is analytic, symmetric unde r the permutation
+of arguments, Euclidean covariant, and satisfies cluster decompo sition and reflection
+positivity. Consequently, we can analytically continue the Helmholtz p ropagator
+(3.63) to the Klein–Gordon propagator according to the prescript ions: (i) multiply
+Gtimes the imaginary unit i, due to Wick rotation of the time direction; (ii) replace
+r2withs2+iε, where the positive sign of the extra infinitesimal term correspond s
+to thecausalFeynman propagator. Summarizing,
+G(s) =−i
+2Dα
+2πD
+2Γ/parenleftbigD
+2/parenrightbig
+Γ/parenleftbigDα
+2/parenrightbig/parenleftbiggm2
+s2+iε/parenrightbiggDα
+4−1
+2
+KDα
+2−1/parenleftBig
+m√
+s2+iε/parenrightBig
+, s> 0,(3.66)
+while the massless propagator is
+G(s) =i
+ΩD(2−Dα)(s2+iε)1−Dα
+2, (3.67)
+in accordance with eq. (3.59). Equations (3.53) and (3.65) are cont inued similarly.
+The propagator for timelike intervals is just the analytic continuatio n of the
+former. In the massive case, it is proportional to the Hankel func tion of the first
+kindH(1)
+Dα/2−1. One last thing to check is what happens on the light cone. The
+propagator for Dα= 2, eq. (3.53), is (proportional to) the usual propagator in two
+dimensions, so nothing special occurs. Taking instead the definition (3.44), setting
+α= 1 in eq. (3.67) (contribution of mnegligible), for even D= 4,6,...one obtains
+([84], p. 94)
+G(s)∝1
+(s2+iε)D
+2−1
+=(−∂s2)D
+2−2
+(D/2−2)!/bracketleftbigg
+PV/parenleftbigg1
+s2/parenrightbigg
+−iπδ(s2)/bracketrightbigg
+,
+(3.68)
+where PV denotes the principal value. In D= 4 this reduces to the Plemelji–
+Sokhotski formula. Translated into momentum space, the δstates, as usual, that
+masslessparticlespropagateatthespeedoflightonthelightcone( Huygens’principle
+[78, 80]).
+3.3 Propagator: momentum space
+SinceGhas been argued to be proportional, in configuration space, to the Green
+function of another well-known problem (the functional inverse of a fractional power
+– 19 –of the d’Alembertian), we can already guess its pole structure in mom entum space:
+in general, it will exhibit a branch cut with branch point at k2=−m2.
+Tocalculatethepropagatorinmomentumspace, wecanstartfrom theEuclidean
+one and then analytically continue the result as usual. In the Lorent zian propagators
+the substitution k2→ |k|2−(k0)2−iεis understood.
+In the presence of a Lebesgue–Stieltjes measure, the Fourier tr ansform must be
+modified so that it is consistent with the definition of the Dirac distribu tion eq. (3.1).
+The Fourier–Stieltjes transform Fvof a function G(x) and its inverse are defined as
+[21]
+˜G(k) =/integraldisplay
+d̺(x)G(x)e−ik·x≡Fv[G(x)], (3.69)
+G(x) =1
+(2π)D/integraldisplay
+d̺(k)˜G(k)eik·x. (3.70)
+The measure in eq. (3.70) is such that momentum and configuration s pace have the
+same dimensionality. The Fourier–Stieltjes transform of a function Gis the Fourier
+transform of vG:
+Fv[G] =F[vG] =F[rD(α−1)G(r)]. (3.71)
+In particular, the Fourier–Stieltjes transform of δ(D)
+vis 1. When α∝ne}ationslash= 1 one has
+Fv[1] =F[rD(α−1)]
+= 2DαπD/2Γ/parenleftbigDα
+2/parenrightbig
+Γ/bracketleftBig
+D(1−α)
+2/bracketrightBig1
+kDα
+= (2π)Dδ(D)
+v(k), (3.72)
+soδ(D)
+v, the source in eq. (3.42), is a power-law distribution.
+Equation (3.71) tells us the form of the massless propagator when α∝ne}ationslash= 2/D
+(transform of eq. (3.52)) [84]:
+˜G(k) =1
+ΩD(2−Dα)F[rD(α−1)r2−Dα]
+=−D−2
+Dα−21
+k2. (3.73)
+Notice that:
+•The(Lorentzian)propagatorhasa k2= 0poleinthe(Re k0,Imk0)plane. From
+the world-fractal point of view (that is to say, looking at the pole st ructure of
+˜G(k) rather than of v(k)˜G(k)), the spectrum has the usual support at k2= 0.
+•Whenα= 1, one obtains ˜G(k) =−1/k2and the free wave solution.
+– 20 –•For general α>2/D, the sign of the residue is always negative, which ensures
+the absence of ghosts. However, its value is not 1 but given by a geo metric
+factor. Thisisexpectedastheeffectivetheoryintheworld-fract alisnotunitary
+and some probability is exchanged with the D-dimensional topological bulk.
+In the critical case α= 2/Deq. (3.73) is ill-defined; this is not a problem, since one
+should start from eq. (3.53):
+˜G∗(k) =1
+ΩDF[r2−Dlnr]
+?=/parenleftbiggD
+2−1/parenrightbiggB−lnk2
+k2, (3.74)
+whereB=ψ(D/2−1)−γ+ln4 andψis the digamma function. Some remarks:
+•ForD∝ne}ationslash= 2, the Lorentzian propagator has a branch point at k2= 0.
+•In the limit D→2, the propagator is G∼ −1/k2.
+•The termB/k2does not represent a particle mode. In fact, it is nothing but
+the fractal Dirac distribution eq. (3.72) when α= 2/D. In other words, the
+Fourier–Stieltjes transform of the equivalence class (3.64) is uniqu e up to a
+δ(D)
+v(k) term. The spectrum consists in a quasiparticle continuum of modes
+with momentum k2≤0.
+In eq. (3.64) there exists a particular element such that B= 0 identically, which we
+typically call thepropagator. Equation (3.74) is then replaced by the unambiguous
+expression
+˜G∗(k) =−/parenleftbiggD
+2−1/parenrightbigglnk2
+k2. (3.75)
+The massive case is slightly more complicated and as an exercise we will c alculate it
+explicitly. The transform of the propagator in radial coordinates is
+˜G(k) =/integraldisplay
+dΩDdrrD−1v(r)G(r)e−ik·x.
+The integrand is not radial but one can choose a frame where kµxµ=−krcosθ,
+k≡ |kµ|, and the angular integral reads
+/integraldisplay
+dΩDe−ik·x= ΩD−1/integraldisplayπ
+0dθ(sinθ)D−2eikrcosθ
+= ΩD−1√πΓ/parenleftbiggD−1
+2/parenrightbigg/parenleftbigg2
+kr/parenrightbiggD
+2−1
+JD
+2−1(kr).
+In the last line we used formula 3.915.5 of [85]. Then,
+˜G(k) = Γ/parenleftbiggD
+2/parenrightbigg/parenleftbigg2
+k/parenrightbiggD
+2−1/integraldisplay+∞
+0drrDα−D
+2[ΩDG(r)]JD
+2−1(kr).(3.76)
+– 21 –Now we take the massive propagator (3.63) ( α∝ne}ationslash= 2/D):
+˜G(k) =−Γ/parenleftbigD
+2/parenrightbig
+Γ/parenleftbigDα
+2/parenrightbig/parenleftbigg2
+k/parenrightbiggD
+2−1/parenleftBigm
+2/parenrightBigDα
+2−1
+×/integraldisplay+∞
+0drrD(α−1)
+2+1KDα
+2−1(mr)JD
+2−1(kr)
+=−F/parenleftBig
+Dα
+2,1;D
+2;−k2
+m2/parenrightBig
+m2, (3.77)
+where we used formula 6.576.3 of [85] and Fis the hypergeometric function
+F(a,b;c;z) =∞/summationdisplay
+n=0Γ(a+n)Γ(b+n)
+Γ(a)Γ(b)Γ(c)
+Γ(c+n)zn
+n!. (3.78)
+Whenα= 1, one obtains the usual massive propagator in Ddimensions (formula
+9.121.1 of [85]):
+˜G(k) =−1
+k2+m2. (3.79)
+Inthelimit m→0(whichdoescommutewiththeanalyticcontinuationofeq.(3.77)),
+one recovers eq. (3.73) up to a term O(mDα−2), which is negligible by virtue of the
+no-ghost condition α>2/D.
+Whenα= 2/D, the transform of the critical propagator eq. (3.65) is
+˜G∗(k) =−F/parenleftBig
+1,1;D
+2;−k2
+m2/parenrightBig
+m2(3.80)
+=−/parenleftbiggD
+2−1/parenrightbigglnk2
+k2+O(m2lnm2), (3.81)
+where in the second line we have dropped O(k−2) terms and considered the massless
+limit for comparison with eq. (3.75). When D= 4, by virtue of formula 9.121.6 of
+[85], eq. (3.80) gives exactly (i.e., no δvterms)
+˜G∗(k) =−1
+k2ln/parenleftbigg
+1+k2
+m2/parenrightbigg
+, (3.82)
+which is precisely the generalization of eq. (3.75). G∗(k) has a branch cut for Re k0<
+−/radicalbig
+|k|2+m2. Therefore, the spectrum of the theory has a continuum of mode s with
+rest mass ≥m.
+Interestingly, the convergence domain of eq. (3.77) gives a bound on the topo-
+logical dimension D. The propagator is a hypergeometric series with convergence in
+the unit circle |z|<1, wherez=−k2/m2. Let
+p≡Dα
+2−D
+2+1. (3.83)
+On the unit circle |z|= 1, there are three cases:
+– 22 –•The series diverges if p≥1. As a constraint on αit yieldsα≥1. This case is
+excluded by construction, although the limit case α= 1 is well-defined.
+•The series converges absolutely if p<0, corresponding to α<1−2/D.
+•The series converges except at z= 1 if 0≤p<1, giving 1 −2/D≤α<1.
+•Whenα= 2/D, theseries diverges if D≤2, converges absolutely if D>4, and
+converges except at z= 1 if 2<D≤4. These bounds are basically unchanged
+if one considers the analytic continuation of Fto the massless case.
+The standard physical setting (convergence of the propagator in the unit disk and
+on its boundary except at the singularity on the real k0axis) is naturally recovered
+only for
+2≤D≤4, (3.84)
+where we included both extrema of the interval by analytic continua tion.
+Before concluding the section, we reconsider the issue of the supe rficial degree of
+divergence of Feynman graphs in the UV (see, e.g., [3, 4] for an intro duction to the
+subject andreferences). Consideraone-particle-irreduciblesu bdiagramwith Lloops,
+I≥Linternal propagators and Vvertices. The superficial degree of divergence δis
+the canonical dimension of all these contributions.
+Each loop integral gives [ d̺(k)] =Dα, while the propagator, in any dimension
+and for any value of α, has [˜G] =−2. For the scalar field theory, interaction vertices
+do not carry dimensionality. Overall,
+δ=L(Dα−2)−2(I−L)≤L(Dα−2). (3.85)
+Whenα= 1, one gets the standard result in Ddimensions. In the critical case
+α= 2/D,δ≤0 and one has at most logarithmic divergences. When α <2/Dthe
+theory is superrenormalizable. In the case of gravity also vertices contribute [3, 4],
+each with a factor of 2 (number of derivatives). Then, δis bounded by the dimension
+of operators which already appear in the bare action.
+4. Gravity
+Having studied the properties of a scalar field on an effective fracta l spacetime, we
+turn to gravity. Our conventions for the Levi-Civita connection, R iemann and Ricci
+tensors, and Ricci scalar are
+Γα
+µν≡1
+2gαβ[∂µgνβ+∂νgµβ−∂βgµν], (4.1)
+Rα
+µβν≡∂βΓα
+µν−∂νΓα
+µβ+Γσ
+µνΓα
+βσ−Γσ
+µβΓα
+νσ, (4.2)
+Rµν≡Rα
+µαν, R≡Rµνgµν. (4.3)
+– 23 –TheAnsatzfor the gravitational action is
+Sg=1
+2κ2/integraldisplay
+d̺(x)√−g(R−2λ−ω∂µv∂µv), (4.4)
+wheregis the determinant of the dimensionless metric gµν,κ2= 8πGis Newton’s
+constant,λis a bare cosmological constant, and the term proportional to ωhas been
+added because v, like the other geometric field gµν, is now dynamical. The couplings
+have dimension
+[κ2] = 2−Dα, [λ] = 2,[ω] = 2D(α−1)+4. (4.5)
+In spacetime with D= 2 topological dimensions and trivial measure weight v= 1,
+the Einstein–Hilbert action is a topological invariant and there are no dynamical
+degrees of freedom. This is not the case of eq. (4.4).
+To describe the flow from the UV to the IR fixed point, we should add r elevant
+operators also into the gravitational action. (The relevant opera tors in the matter
+sector are minimally coupled with gravity and they would not be enough .) This is
+done in the same way as for the matter sector, eqs. (2.21) and (2.2 2), with possibly
+differentMg∝ne}ationslash=Mφ. The effective Newton constant then runs from a UV bare
+dimensionless constant to an IR value
+κ2
+IR∼κ2
+UVM2−D
+g. (4.6)
+Note thatκ2
+IRis not necessarily the observed Newton constant κ2
+obs, in which case
+Mg∼mPlis the Planck mass. As one can see from the equations of motion, κ2
+obswill
+depend on the background as well as on the scale of the problem.
+Those in eq. (2.21) arenot all possible relevant operators. Higher- order Riemann
+invariants Riemn([Riemn] = 2n,n >1) are irrelevant, but one could introduce
+lowest-order terms of the form (1 + Riem)nwithn <1. These terms might be
+expected in the context of fractal models, where fractional der ivatives can (but not
+necessarily) find their natural setting.10Considering all these operators, one ends up
+with a term /integraldisplay1
+0dnc(n)/parenleftbigg
+1+Riem
+mass2/parenrightbiggn
+, (4.7)
+wherec(n) are arbitrary dimensionless coefficients. In the unrealistic case wh ere
+they are all equal to 1 and Riem = R, the integral can be summed explicitly to
+a nonpolynomial functional which admits Minkowski as a vacuum and y ields the
+Einstein–Hilbert Lagrangian with cosmological constant at small R(in mass units):
+R
+ln(1+R)= 1+R
+2−R2
+12+O(R3).
+10If they were regarded as necessary, all ∂βterms should make their appearance in any sector.
+This is a different definition of the theory which we will not consider.
+– 24 –This particular f(R) model might be of some cosmological interest. However, it is a
+toy model and we shall not continue its discussion. In fact, we are in terested in the
+equations of motion near the UV fixed point, so we will ignore relevant operators
+from now on ( Mg=Mφ= 0).
+4.1 Einstein equations
+Assuming that matter is minimally coupled with gravity, the total actio n is
+S=Sg+Sm, (4.8)
+whereSgis eq. (4.4) and Sm=/integraltext
+d̺√−gLmis the matter action. The derivation of
+the Einstein equations is almost as in scalar-tensor models. We shall r epeat it here
+to make the presentation self-contained. To find the equations of motion we need
+the variations
+δ√−g=−1
+2gµν√−gδgµν, (4.9)
+δR= (Rµν+gµν/square−∇µ∇ν)δgµν, (4.10)
+where∇νVµ≡∂νVµ−Γσ
+µνVσis the covariant derivative of a vector Vµand the curved
+d’Alembertian on a scalar φis
+/squareφ=1√−g∂µ(√−g∂µφ). (4.11)
+The Einstein equations δS/δgµν= 0 read
+Σµν=κ2Tµν, (4.12)
+Σµν≡Rµν−1
+2gµν(R−2λ)+gµν/squarev
+v−∇µ∇νv
+v
++ω/parenleftbigg1
+2gµν∂σv∂σv−∂µv∂νv/parenrightbigg
+, (4.13)
+Tµν≡ −2√−gδSm
+δgµν=−2∂Lm
+∂gµν+gµνLm. (4.14)
+Taking the trace of eq. (4.12) gives
+−/parenleftbiggD
+2−1/parenrightbigg
+R+Dλ+(D−1)/squarev
+v+/parenleftbiggD
+2−1/parenrightbigg
+ω∂µv∂µv=κ2Tµ
+µ.(4.15)
+When taking into account the variation of the total action with resp ect to the scalar
+v,
+R−2λ=−2κ2Lm−ω(2v/squarev+∂µv∂µv), (4.16)
+eq. (4.15) becomes
+R+(D−1)/squarev
+v+ω[Dv/squarev+(D−1)∂µv∂µv] =κ2/parenleftbig
+Tµ
+µ−DLm/parenrightbig
+=−2κ2Tr∂Lm
+∂gµν.(4.17)
+– 25 –Tµνis the stress-energy tensor of matter. Its definition determines the continuity
+equation [86]. In fact, let
+δSm=1
+2/integraldisplay
+dDxv√−gTµσδgµσ+/integraldisplay
+dDx√−gLmδv (4.18)
+be the infinitesimal variation of the matter action with respect to th e external fields
+δgµσandδv. For the infinitesimal coordinate transformation (3.19), one has
+δgµσ=gνσ∂µaν+gµν∂σaν+aν∂νgµσ, (4.19)
+δv=aν∂νv, (4.20)
+where we used the definition of the Lie derivative for rank-2 and ran k-0 tensors.
+Plugging eqs. (4.19) and (4.20) into (4.18) and integrating by parts, we get
+δSm=−/integraldisplay
+dDxaν/bracketleftbigg
+∂µ(v√−gTµ
+ν)−1
+2v√−gTµσ∂νgµσ−∂νv√−gLm/bracketrightbigg
+.(4.21)
+δSmmust vanish on shell (i.e., when the dynamical equations are satisfied ). Using
+the properties of the Levi-Civita connection (4.1) and the definition of the covariant
+derivative of a rank-2 tensor,
+∇µTµ
+ν=∂µTµ
+ν+Γµ
+µσTσ
+ν−Γσ
+µνTµ
+σ
+=1√−g∂µ/parenleftbig√−gTµ
+ν/parenrightbig
+−1
+2(∂νgµσ)Tµσ,
+one finally obtains the continuity equation
+∇µ(vTµ
+ν)−∂νvLm= 0, (4.22)
+which generalizes eq. (3.20).
+If matter is a scalar field, it is straightforward to see that its equat ion of motion
+δSm/δφ= 0 is eq. (3.4) with /squaregiven by eq. (4.11), in agreement with eq. (4.22).
+The continuity and Einstein equations are not independent because of the con-
+tracted Bianchi identities 2 ∇µRµν=∇νRandeq. (4.16). The divergence of( vtimes)
+eq. (4.12) correctly reproduces eq. (4.22). The check takes into account that in the
+absence of torsion the covariant derivative commutes on a scalar, [∇µ,∇ν]v= 0,
+while on a vector [ ∇µ,∇ν]Vσ=Rτ
+µνσVτ.
+4.2 Cosmology
+With the notable difference that matter is nonminimally coupled with the scalarv,
+the equations of motion are similar to those of Brans–Dicke theory [8 7], which is
+well constrained by large-scale observations [88]. This fact and the foreign physical
+setting lead to an altogether different dynamics.
+– 26 –In this section we specialize to a Friedmann–Robertson–Walker (FRW ) line ele-
+ment
+ds2=gµνdxµdxν=−dt2+a(t)2˜gijdxidxj, (4.23)
+wheretis synchronous time, a(t) is the scale factor and
+˜gijdxidxj=dr2
+1−kr2+r2dΩ2
+D−2 (4.24)
+is the line element of the maximally symmetric ( D−1)-dimensional space ˜Σ of
+constant sectional curvature k(equal to −1 for an open universe, 0 for a flat universe
+and+1foracloseduniversewithradius a). Infourdimensions, dΩ2
+2=dθ2+sin2θdϕ2.
+Quantities built up with the spatial metric ˜ gijwill be decorated with a tilde. On this
+background, the only nonvanishing Levi-Civita and Ricci component s are
+Γ0
+ij=Hgij,Γj
+i0=Hδj
+i,Γk
+ij=˜Γk
+ij, (4.25)
+and
+R00=−(D−1)(H2+˙H), (4.26)
+Rij=˜Rij+[(D−1)H2+˙H]gij,˜Rij=2k
+a2gij, (4.27)
+R= (D−1)/parenleftbigg2k
+a2+DH2+2˙H/parenrightbigg
+, (4.28)
+where
+H≡˙a
+a(4.29)
+is the Hubble parameter (not to be confused with the Hamiltonian Hof section 3.1)
+and we have exploited the symmetries of ˜Σ [86].
+For simplicity we consider a perfect fluid (zero heat flow and anisotro pic stress)
+as the only content of the universe:
+Tµν= (ρ+p)uµuν+pgµν, (4.30)
+whereρ=T00andp=Ti
+i/(D−1) are the energy density and pressure of the fluid
+anduµ= (1,0,...,0)µis the unit timelike vector ( uµuµ=−1) tangent to a fluid
+element’s worldline. We also take a timelike fractal v=v(t).
+The 00 component of the Einstein equations (4.12) is
+/parenleftbiggD
+2−1/parenrightbigg
+H2+H˙v
+v−1
+2ω
+D−1˙v2=κ2
+D−1ρ+λ
+D−1−k
+a2,(4.31)
+while combining that with the trace equation (4.15) one obtains
+/squarev
+v−(D−2)/parenleftbigg
+H2+˙H−H˙v
+v+ω
+D−1˙v2/parenrightbigg
++2λ
+D−1=κ2
+D−1[(D−3)ρ+(D−1)p].
+(4.32)
+– 27 –Other useful expressions can be found by suitable combinations of the dynamical
+equations. From eq. (4.17),
+R+(D−1)/squarev
+v+ω/bracketleftbig
+Dv/squarev−(D−1)˙v2/bracketrightbig
+=−κ2(ρ+p),(4.33)
+and eq. (4.16) one gets
+2λ+(D−1)/squarev
+v+(D−2)ω(v/squarev−˙v2) =−κ2(ρ−p), (4.34)
+while from the 00 component of (4.12) and eq. (4.16),
+H2+˙H−H˙v
+v+ω
+D−1v/squarev=−κ2
+D−1(ρ+p). (4.35)
+Ifρ+p∝ne}ationslash= 0, one can combine eqs. (4.33) and (4.35) to get a purely gravitatio nal
+constraint:
+˙H+(D−1)H2+2k
+a2+/squarev
+v+H˙v
+v+ω(v/squarev−˙v2) = 0. (4.36)
+The continuity equation (4.22) contracted with −uνis
+˙ρ+/bracketleftbigg
+(D−1)H+˙v
+v/bracketrightbigg
+(ρ+p) = 0, (4.37)
+where we used the definition of proper-time derivative, uµ∇µ= ˙, and the Hubble
+expansion Θ ≡ ∇µuµ= (D−1)Hin covariant formalism [89, 90, 91]. Note that this
+is not the volume expansion as in standard general relativity, as the latter is actually
+˜Θ =dln(aD−1v)
+dt, (4.38)
+whichisthesquarebracket ineq.(4.37). Forabarotropicfluid p=wρ, thecontinuity
+equation is solved by ρ∼(aD−1v)−(1+w), up to some dimensionful prefactor. In
+generalwis not a constant and we define the effective barotropic index
+w(t)≡p
+ρ. (4.39)
+The scalar field is a particular case of perfect fluid, with p=Lφ=˙φ2/2−V,
+ρ=˙φ2/2+V, anduµ=−∂µφ/˙φ[92]. Equation (4.37) becomes eq. (3.4),
+¨φ+/bracketleftbigg
+(D−1)H+˙v
+v/bracketrightbigg
+˙φ+V′= 0. (4.40)
+Whenv= 1 andD= 4, we recover the standard Friedmann equations in four
+dimensions, eqs. (4.31) and (4.32) (no gravitational constraint):
+H2=κ2
+3ρ+λ
+3−k
+a2, (4.41)
+H2+˙H=−κ2
+6(3p+ρ)+λ
+3. (4.42)
+– 28 –On the other hand, for the measure weight
+v=t−β, (4.43)
+whereβis given by eq. (3.55), the gravitational constraint is switched on. T hen, the
+above equations should be taken cum grano salis . The UV regime, in fact, describes
+short scales at which inhomogeneities should play some role. If these are small, the
+modified Friedmann equations define a background for perturbatio ns rather than a
+self-consistent dynamics.
+Modulo this caveat, we can look at flat ( k= 0) background solutions in the deep
+UV regime with no cosmological constant. The gravitational constr aint (4.36) is a
+Riccati equation in Hwhich, together with the useful formulæ
+H˙v
+v=−Hβ
+t,/squarev
+v=β
+t/bracketleftbigg
+(D−1)H−1+β
+t/bracketrightbigg
+, (4.44)
+fixes almost completely the background expansion, regardless the matter content. A
+direct consequence of this overdetermination of the dynamics is th at there are no
+vacuum solutions ( ρ= 0 =p). This might not happen for other fractal profiles than
+eq. (4.43), but at early times the measure must scale as eq. (4.43). This feature,
+therefore, is robust.
+Let us consider the cases ω= 0 andω∝ne}ationslash= 0 separately. When β=D−2 = 2
+(UV regime), the ω= 0 solution is
+a(t) =(t9+c)1/3
+t2, (4.45a)
+H(t) =t9−2c
+t(t9+c), (4.45b)
+ǫ≡ −˙H
+H2=[t9−(14+3√
+22)c][t9−(14−3√
+22)c]
+(t9−2c)2,(4.45c)
+wherecis an integration constant. The energy density and pressure which solve all
+the equations simultaneously are
+ρ=−3
+κ2(t9+4c)(t9−2c)
+t2(t9+c)2, (4.45d)
+p=−3
+κ2[t9+(√
+10−3)√
+10c][t9+(√
+10+3)√
+10c]
+t2(t9+c)2. (4.45e)
+These expressions are sufficient to characterize three cases:
+•c >0: The scale factor decreases ( H <0) fromt= 0 untilt=t∗≡(2c)1/9,
+where the universe bounces ( H∗= 0,a∗= 31/32−2/9c1/9). Fromt=t∗to
+t=t1≡[(14+3√
+22)c]1/9, the universe expands in superacceleration ( ǫ<0),
+while fort > t1the expansion is only accelerated. The energy density ρis
+negative for t>t∗, while the pressure pis always negative.
+– 29 –•c= 0: Linear (decelerating) expansion, a=t, whileρ=p<0 always.
+•c <0: The universe expands in deceleration from a big bang event at t=
+t0≡ |c|1/9. The energy density and pressure are negative for t >|4c|1/9and
+t>|(√
+10+3)√
+10c|1/9, respectively.
+All these scenarios need a matter component with non-positive defi nite energy den-
+sity, so they are excluded if only ordinary matter is allowed. We envisa ge four simple
+modifications of this result. One is to change the flat prescription k= 0. Another is
+to consider the above formulæ only asymptotically, since the simple me asure profile
+eq. (4.43) is certainly valid only at early times. The only case where the universe
+expands at small twithρ>0 isc<0, for which one hassuperstiff matter ( w(t)>1).
+A third option is to allow for a nonzero geometric contribution U(v), a potential for
+v.
+A fourth possibility is that ω∝ne}ationslash= 0, which does lead to interesting cosmology.
+There is only one real solution to the gravitational constraint, nam ely,
+a(t) =1
+t2Φ/parenleftbigg11
+4;13
+4;3ω
+2t4/parenrightbigg1/3
+, (4.46)
+H(t) =−2
+t−22ω
+13t5Φ/parenleftbig15
+4;17
+4;3ω
+2t4/parenrightbig
+Φ/parenleftbig11
+4;13
+4;3ω
+2t4/parenrightbig, (4.47)
+where Φ (also denoted as 1F1orM) is Kummer’s confluent hypergeometric function
+of the first kind:
+Φ(a;b;z)≡Γ(b)
+Γ(a)+∞/summationdisplay
+n=0Γ(a+n)
+Γ(b+n)zn
+n!. (4.48)
+The expressions for ρandpare
+ρ=2(2ω+3t4)(3ω+4t4)
+t10+48ω2
+132t10Φ/parenleftbig11
+4;17
+4;3ω
+2t4/parenrightbig2
+Φ/parenleftbig11
+4;13
+4;3ω
+2t4/parenrightbig2
+−24ω(2ω+3t4)
+13t10Φ/parenleftbig11
+4;17
+4;3ω
+2t4/parenrightbig
+Φ/parenleftbig11
+4;13
+4;3ω
+2t4/parenrightbig, (4.49)
+p=−2(ω+3t4)(6ω+5t4)
+t10+48ω2
+132t10Φ/parenleftbig11
+4;17
+4;3ω
+2t4/parenrightbig2
+Φ/parenleftbig11
+4;13
+4;3ω
+2t4/parenrightbig2. (4.50)
+We can use the asymptotic forms of Φ to have some semi-analytic insig ht of the
+system. When z→ −∞,
+Φ(a;b;z)z→−∞∼Γ(b)
+Γ(b−a)(−z)−a+(−z)−aN/summationdisplay
+n=1Γ(a+n)
+Γ(a)Γ(b)
+Γ(b−a−n)z−n
+n!,(4.51)
+– 30 –whereNis some finite order. On the other hand,
+Φ(a;b;z)z→+∞∼Γ(b)
+Γ(a)ezza−b+ezza−bN/summationdisplay
+n=1Γ(b−a+n)
+Γ(b−a)Γ(b)
+Γ(a−n)(−z)−n
+n!.(4.52)
+At late times the scale factor decreases and the fluid behaves effec tively as phantom
+matter (w<−1):
+at→+∞∼1
+t2, H∼ −2
+t, (4.53a)
+ǫ∼ −1
+2, (4.53b)
+ρ∼24
+t2, p∼ −30
+t2, (4.53c)
+w∼ −5
+4. (4.53d)
+At early times we must distinguish between positive and negative ω. Forω>0, the
+universe is contracting and the fluid behaves like an effective cosmolo gical constant:
+at→0∼const×eω
+2t4
+t4/3, H∼ −2ω
+t5, (4.54a)
+ǫ∼ −5t4
+2ω, (4.54b)
+ρ∼12ω2
+t10, p∼ −12ω2
+t10, (4.54c)
+w∼ −1. (4.54d)
+These quantities are plotted in figures 1 and 2 for ω= +1.
+Forω<0, atearlytimestheuniverse expands andaccelerates, even ifthe perfect
+fluid is stiff:
+at→0∼const×t5/3, H∼5
+3t, (4.55a)
+ǫ∼3
+5, (4.55b)
+ρ∼2|ω|
+t6, p∼2|ω|
+t6, (4.55c)
+w∼1. (4.55d)
+These are plotted in figures 3 and 4 for ω=−1.
+The null energy condition is violated, so none of these scenarios can be realized
+by an ordinary scalar field, for which ρ+p=˙φ2≥0. The system is invariant under
+time reversal, so the model with ω >0, run backwards in time, also describes an
+expanding superaccelerating ( ǫ<0) universe filled with phantom matter. Contrary
+to standard general relativity, the energy density increases . This can be explained
+– 31 –0 1 2 3 402468
+ta
+0 1 2 3 4/Minus35/Minus30/Minus25/Minus20/Minus15/Minus10/Minus50
+tH
+0 1 2 3 4/Minus0.8/Minus0.6/Minus0.4/Minus0.20.0
+tΕ
+0 1 2 3 4/Minus80/Minus60/Minus40/Minus200204060
+t
+Figure 1: The scale factor a, Hubble parameter H, slow-roll parameter ǫ, energy density
+ρ(thick line) and pressure p(dashed line) for ω= +1.
+0 1 2 3 4/Minus1.4/Minus1.3/Minus1.2/Minus1.1/Minus1.0
+tw
+Figure 2: The equation of state w=p/ρforω= +1.
+by recalling that the fractal geometry causes dissipation. The obs erver, in this case,
+experiences an incoming flux of energy from the four-dimensional b ulk.
+The model with ω <0 is a universe which expands in acceleration. At some
+point the expansion is quasi de Sitter (notice the small plateau of Hin figure 3),
+and after a brief period of superacceleration the cosmic expansion decelerates, stops,
+and reverts. Again, the fluid behaves unusually: during the acceler ated expansion
+– 32 –0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.10.20.30.40.50.6
+ta
+0.0 0.5 1.0 1.5 2.0 2.5 3.00246810
+tH
+0.0 0.5 1.0 1.5 2.0 2.5 3.0/Minus0.50.00.51.01.52.02.53.0
+tΕ
+0.5 1.0 1.5 2.0 2.5 3.0/Minus10010203040
+t
+Figure 3: The scale factor a, Hubble parameter H, slow-roll parameter ǫ, energy density
+ρ(solid line) and pressure p(dashed line) for ω=−1.
+0.5 1.0 1.5 2.0 2.5 3.0/Minus1.0/Minus0.50.00.51.0
+tw
+Figure 4: The equation of state w=p/ρforω=−1. The maximum, which lies beyond
+the frame, is w≈17.383 att≈0.851.
+the equation of state is between dust and stiff matter (0 < w(t)<1), while the
+energy density decreases. Then, after a non-monotonic transit ory period around
+the inversion point, the equation of state violates first the strong and then the null
+energycondition. Eventuallytheenergydensitystilldecreasesev enduringthecosmic
+contraction, because the world-fractal dissipates into the bulk.
+– 33 –The overall cosmological picture may be regarded as problematic fo r any value
+ofω. On one hand, there are no vacuum solutions. On the other hand, s olutions
+with matter require fluids violating most or all energy conditions, thu s signalling an
+unstable or unrealistic field (however, see below). The ω<0 case does so explicitly,
+sincevis a ghost.
+This might be cured by allowing a nonzero intrinsic curvature, more co mplicated
+matterprofiles, oranontrivialpotentialfor v.11However, themostnaturalpossibility
+isthataclassical FRWbackground, eitherexactorlinearlyperturb ed, isnotrealistic.
+Then, one would have to treat the UV limit as highly inhomogeneous. Th is is not
+at all unexpected, as we are dealing with quantum scales where the m inisuperspace
+equations (maximal symmetry) are likely to fail.
+5. Conclusions and future developments
+There are several avenues of investigation left to explore. Here w e mention just three.
+•Quantum field theory. Many aspects of the field theory have yet to be fully
+understood: among the most important are renormalization, the h ierarchy
+problem and the physical significance of the UV propagator and the “natural”
+bound eq. (3.84). Even in the Minkowski embedding, the fractal st ructure may
+have interesting properties we have not discussed here. For insta nce, because
+of violation of translation invariance the Fermi frame does not exist and, as in
+general relativity, parity is strictly a local symmetry.
+Other formulations of the theory including fractional derivatives w ould modify
+the propagator and dissipation properties and, pending a suitable d efinition
+of the kinetic terms, they could lead to a more transparent physics . Also,
+for simplicity we have defined an action on an embedding spacetime with out
+boundaries, but a very interesting alternative is to consider scena rios with
+boundaries; for example, a field theory defined on RD
++would be closer in spirit
+to the unilateral fractional and Weyl integrals of fractal classica l mechanics,
+eq. (2.6) with ¯t→+∞. The propagator and several other features should
+change accordingly but, mutatis mutandis , the main idea of a fractal universe
+would still be valid and maintain the same motivations.
+•Cosmology. If the system quickly flows to the IR fixed point, dissipative effects
+might be negligible on cosmological spacetime scales, with a notable exc eption.
+At late times, an imprint of the nontrivial short-scale geometry migh t survive
+as a cosmological constant of purely gravitational origin (pressur eless matter
+11Different geometric profiles would not work. For ω >0, the null energy condition is violated
+at early times ( |t|>1 in the backwards model) where eq. (4.43) holds. For ω <0,vis a ghost by
+definition.
+– 34 –does not contribute to it). A detailed study may reveal whether th e behaviour
+of the effective cosmological constant (3.15) is compatible with obse rvations.
+On the other hand, the UV regime may be relevant in the early univers e, espe-
+cially during inflation. We have seen that the flat background dynamic s cannot
+be realized by a scalar field. It would be interesting to study the cons equences
+of a nonvanishing intrinsic curvature or potential U(v). If, even in that case,
+ordinary matter (in particular, a scalar field) were not allowed, then one would
+have to abandon maximal symmetry and standard perturbative te chniques of
+inflationary cosmology.12However, an appealing alternative is to assume that
+matter is actually a condensate field stemming from a fermionic secto r. It is
+known that a condensate violates the null energy condition, its mas s-gap ef-
+fective energy density being negative in certain regimes [93, 94]. The physics
+of condensation is far from being exotic and is under good control. I t would
+be interesting to see whether a Dirac sector with four-fermion inte raction is
+renormalizable on a fractal and undergoes a condensation phase.
+•Extra dimensions. Inapplicationsofthemodelweassumedthatthetopological
+dimension of embedding spacetime is D= 4. An interesting alternative is a
+universe with extra topological dimensions, D >4. In this scenario the value
+ofαwould change and the IR limit should be realized in combination with a
+suitable compactification mechanism.
+Acknowledgments
+The author thanks R. Gurau for discussions and is grateful to G. N ardelli for invalu-
+able comments and discussions.
+Open Access. This article is distributed under the terms of the Creative Commons
+Attribution Noncommercial License which permits any noncommercia l use, distribu-
+tion, and reproduction in any medium, provided the original author( s) and source
+are credited.
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\ No newline at end of file
diff --git a/1001.0572.txt b/1001.0572.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f6efd1c59844217ed0891a78e9bac0427a47a412
--- /dev/null
+++ b/1001.0572.txt
@@ -0,0 +1,1169 @@
+arXiv:1001.0572v1  [astro-ph.EP]  5 Jan 2010Frequency of Solar-Like Systems and of Ice and Gas Giants
+Beyond the Snow Line from High-Magnification Microlensing
+Events in 2005-2008
+A. Gould1,2, Subo Dong1,3,4, B.S. Gaudi1,2, A. Udalski5,6, I.A. Bond7,8, J. Greenhill9,10,
+R.A. Street11,12,13, M. Dominik9,11,14,15,16, T. Sumi7,17, M.K. Szyma´ nski5,6, C. Han1,18,
+and
+W. Allen19, G. Bolt20, M. Bos21, G.W. Christie22, D.L. DePoy23, J. Drummond24,
+J.D. Eastman2, A. Gal-Yam25, D. Higgins26, J. Janczak2, S. Kaspi27,28, S. Koz/suppress lowski2,
+C.-U. Lee29, F. Mallia30, A. Maury30, D. Maoz27, J. McCormick31, L.A.G. Monard32,
+D. Moorhouse33, N. Morgan2, T. Natusch34, E.O. Ofek35,36, B.-G. Park29, R.W. Pogge2,
+D. Polishook27, R. Santallo37, A. Shporer27, O. Spector27, G. Thornley33, J.C. Yee2
+(TheµFUN Collaboration),
+and
+M. Kubiak6, G. Pietrzy´ nski6,38, I. Soszy´ nski6, O. Szewczyk38, /suppress L. Wyrzykowski39, K.
+Ulaczyk6, R. Poleski6
+(The OGLE Collaboration),
+and
+F. Abe17, D.P. Bennett40,9, C.S. Botzler41, D. Douchin41, M. Freeman41, A. Fukui17,
+K. Furusawa17, J.B. Hearnshaw42, S. Hosaka17, Y. Itow17, K. Kamiya17, P.M. Kilmartin43,
+A. Korpela44, W. Lin8, C.H. Ling8, S. Makita17, K. Masuda17, Y. Matsubara17,
+N. Miyake17, Y. Muraki45, M. Nagaya17, K. Nishimoto17K. Ohnishi46, T. Okumura17,
+Y.C. Perrott41, L. Philpott41, N. Rattenbury41,11, To. Saito47, T. Sako17, D.J. Sullivan44,
+W.L. Sweatman8, P.J. Tristram43, E. von Seggern41, P.C.M. Yock41
+(The MOA Collaboration),
+and
+M. Albrow42, V. Batista48, J.P. Beaulieu48, S. Brillant49, J. Caldwell50, J.J. Calitz51,
+A. Cassan48, A. Cole10, K. Cook52, C. Coutures53, S. Dieters48, D. Dominis Prester54,
+J. Donatowicz55, P. Fouqu´ e56, K. Hill10, M. Hoffman51, F. Jablonski57, S.R. Kane58,
+N. Kains15,11, D. Kubas49, J.-B. Marquette48, R. Martin59, E. Martioli57, P. Meintjes51,
+J. Menzies60, E. Pedretti15, K. Pollard42, K.C. Sahu61, C. Vinter62, J. Wambsganss63,14,
+R. Watson10, A. Williams59, M. Zub63,64
+(The PLANET Collaboration),
+and
+A. Allan65, M.F. Bode66, D.M. Bramich67,9, M.J. Burgdorf68,69,14, N. Clay66, S. Fraser66,
+E. Hawkins12, K. Horne15,9, E. Kerins70, T.A. Lister12, C. Mottram66, E.S. Saunders12,65,
+C. Snodgrass49,71,14, I.A. Steele66, Y. Tsapras12,72,9
+(The RoboNet Collaboration),– 2 –
+and
+U. G. Jørgensen62,73,9, T. Anguita63,74, V. Bozza75,76, S. Calchi Novati75,76, K. Harpsøe62, T.
+C. Hinse62,77, M. Hundertmark78, P. Kjærgaard62, C. Liebig15,63, L. Mancini75,76, G. Masi79,
+M. Mathiasen62, S. Rahvar80, D. Ricci81, G. Scarpetta75,76, J. Southworth82, J. Surdej81, C.
+C. Th¨ one62,83
+(The MiNDSTEp Consortium)– 3 –
+1Microlensing Follow Up Network ( µFUN)
+2Department of Astronomy, Ohio State University, 140 W. 18th Ave ., Columbus, OH 43210, USA;
+gould,gaudi,jdeast,jyee,pogge,simkoz@astronomy.ohio-state.edu, nick.morgan@alum.mit.edu
+3Institute for Advanced Study, Einstein Drive, Princeton, NJ 0854 0, USA; dong@ias.edu
+4Sagan Fellow
+5Optical Gravitational Lens Experiment (OGLE)
+6Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warsz awa, Poland; e-mail: udalski, msz,
+mk, pietrzyn, soszynsk, kulaczyk, rpoleski@astrouw.edu.pl
+7Microlensing Observations in Astrophysics (MOA)
+8Institute of Information and Mathematical Sciences, Massey Univ ersity, Private Bag 102-904, North
+Shore Mail Centre, Auckland, New Zealand; i.a.bond,l.skuljan,w.lin,c.h.ling,w .sweatman@massey.ac.nz
+9Probing Lensing Anomalies NETwork (PLANET)
+10University of Tasmania, School of Mathematics and Physics, Privat e Bag 37, GPO Hobart, Tas 7001,
+Australia; John.Greenhill,Andrew.Cole@utas.edu.au
+11RoboNet
+12Las Cumbres Observatory Global Telescope network, 6740 Corto na Drive, suite 102, Goleta, CA 93117,
+USA
+13Dept. of Physics, Broida Hall, University of California, Santa Barbar a CA 93106-9530, USA
+14Microlensing Network for the Detection of Small Terrestrial Exopla nets (MiNDSTEp)
+15SUPA School of Physics and Astronomy, Univ. of St Andrews, Scot land KY16 9SS, United Kingdom;
+md35,nk87,ep41,kdh1@st-andrews.ac.uk
+16Royal Society University Research Fellow
+17Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya, 464-8601, Japan;
+sumi,abe,afukui,furusawa,itow,kkamiya,kmasuda,ymatsu,nmiyake,mnag aya,okumurat,sako@stelab.nagoya-
+u.ac.jp
+18Department of Physics, Institute for Basic Science Research, Ch ungbuk National University, Chongju
+361-763, Korea; cheongho@astroph.chungbuk.ac.kr
+19Vintage Lane Observatory, Blenheim, New Zealand; whallen@xtra.co .nz
+20Perth, Australia; gbolt@iinet.net.au
+21Molehill Astronomical Observatory, Auckland, New Zealand; molehill@ ihug.co.nz
+22Auckland Observatory, Auckland, New Zealand; gwchristie@christ ie.org.nz
+23Department of Physics and Astronomy, Texas A&M University, Colle ge Station, TX, USA; de-
+poy@physics.tamu.edu– 4 –
+24Possum Observatory, Patutahi, New Zealand; john drummond@xtra.co.nz
+25Department of Particle Physics and Astrophysics, Weizmann Instit ute of Science, 76100 Rehovot, Israel;
+avishay.gal-yam@weizmann.ac.il
+26Hunters Hill Observatory, Canberra, Australia; dhi67540@bigpon d.net.au
+27School of Physics and Astronomy and Wise Observatory, Tel-Aviv U niversity, Tel-Aviv 69978, Israel;
+shai,dani,david,shporer,odedspec@wise.tau.ac.il
+28Department of Physics, Technion, Haifa 32000, Israel
+29Korea Astronomy and Space Science Institute, Daejon 305-348, Korea; leecu,bgpark@kasi.re.kr
+30Campo Catino Austral Observatory, San Pedro de Atacama, Chile; francoma-
+llia@campocatinobservatory.org,alain@spaceobs.com
+31Farm Cove Observatory, Centre for Backyard Astrophysics, Pa kuranga, Auckland, New Zealand; farm-
+coveobs@xtra.co.nz
+32Bronberg Observatory, Centre for Backyard Astrophysics, Pr etoria, South Africa; lagmonar@nmisa.org
+33Kumeu Observatory, Kumeu, New Zealand; acrux@orcon.net.nz,gu y.thornley@gmail.com
+34AUT University, Auckland, New Zealand; tim.natusch@aut.ac.nz
+35Palomar Observatory, California, USA; eran@astro.caltech.edu
+36Einstein Fellow
+37Southern Stars Observatory, Faaa, Tahiti, French Polynesia; ob s930@southernstars-observatory.org
+38Universidad de Concepci´ on, Departamento de Fisica, Casilla 160–C, Concepci´ on, Chile; e-mail:
+szewczyk@astro-udec.cl
+39Institute of Astronomy, University of Cambridge, Madingley Road, CB3 0HA Cambridge, United King-
+dom; e-mail: wyrzykow@ast.cam.ac.uk
+40University of Notre Dame, Department of Physics, 225 Nieuwland Sc ience Hall, Notre Dame, IN 46556-
+5670 USA; bennett@nd.edu
+41Department of Physics, University of Auckland, Private Bag 92019 , Auckland, New Zealand;
+c.botzler,p.yock@auckland.ac.nz,yper006@aucklanduni.ac.nz
+42University of Canterbury, Department of Physics and Astronomy , Private Bag 4800, Christchurch 8020,
+New Zealand
+43Mt. John Observatory, P.O. Box 56, Lake Tekapo 8780, New Zealan d
+44School of Chemical and Physical Sciences, Victoria University, Wellin gton, New Zealand;
+a.korpela@niwa.co.nz,denis.sullivan@vuw.ac.nz
+45Department of Physics, Konan University, Nishiokamoto 8-9-1, Ko be 658-8501, Japan
+46Nagano National College of Technology, Nagano 381-8550, Japan– 5 –
+47Tokyo Metropolitan College of Industrial Technology, Tokyo 116-8 523, Japan
+48Institut d’Astrophysique de Paris, Universit´ e Pierre et Marie Curie , CNRS UMR7095, 98bis Boulevard
+Arago, 75014 Paris, France; beaulieu,marquett@iap.fr
+49European Southern Observatory, Casilla 19001, Santiago 19, Chile ; sbrillan,dkubas@eso.org
+50McDonald Observatory, 16120 St Hwy Spur 78 #2, Fort Davis, TX 79 734 USA; cald-
+well@astro.as.utexas.edu
+51University of the Free State, Faculty of Natural and Agricultural Sciences, Department of Physics, PO
+Box 339, Bloemfontein 9300, South Africa; HoffmaMJ.SCI@mail.uovs.a c.za
+52Lawrence Livermore National Laboratory, Institute of Geophys ics and Planetary Physics, P.O. Box 808,
+Livermore, CA 94551-0808 USA; kcook@llnl.gov
+53CEA/Saclay, 91191 Gif-sur-Yvette cedex, France; coutures@ia p.fr
+54Department of Physics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia
+55TechnischeUniversitaetWien, WiederHauptst. 8-10,A-1040Wienn a, Austria; donatowicz@tuwien.ac.at
+56LATT, Universit´ e de Toulouse, CNRS, France; pfouque@ast.obs- mip.fr
+57Instituto Nacional de Pesquisas Espaciais, Sao Jose dos Campos, S P, Brazil
+58NASA Exoplanet Science Institute, Caltech, MS 100-22,770 south Wilson Avenue, Pasadena, CA 91125,
+USA; skane@ipac.caltech.edu
+59Perth Observatory, Walnut Road, Bickley, Perth 6076, WA, Austr alia;
+rmartin,andrew@physics.uwa.edu.au
+60South African Astronomical Observatory, PO box 9, Observator y 7935, South Africa
+61Space Telescope Science Institute, 3700 San Martin Drive, Baltimor e, MD 21218, USA
+62Niels Bohr Institutet, Københavns Universitet, Juliane Maries Vej 3 0, 2100 København Ø, Denmark
+63Astronomisches Rechen-Institut, Zentrum f¨ ur Astronomie der Universit¨ at Heidelberg (ZAH),
+M¨ onchhofstr. 12-14, 69120 Heidelberg, Germany
+64Institute of Astronomy, University of Zielona G´ ora, Lubuska st. 2, 66-265 Zielona G´ ora, Poland
+65School of Physics, University of Exeter, Stocker Road, Exeter E X4 4QL, United Kingdom
+66Astrophysics Research Institute, Liverpool John Moores Univer sity, Liverpool CH41 1LD, United King-
+dom
+67European Southern Observatory, Karl-Schwarzschild-Straße 2 , 85748 Garching bei M¨ unchen, Germany
+68Deutsches SOFIA Institut, Universit¨ at Stuttgart, Pfaffenwald ring 31, 70569 Stuttgart, Germany
+69SOFIA Science Center, NASA Ames Research Center, Mail Stop N21 1-3, Moffett Field CA 94035, USA
+70Jodrell Bank Centre for Astrophysics, The University of Manches ter, Oxford Road, Manchester M13– 6 –
+ABSTRACT
+We present the first measurement of the planet frequency beyon d the “snow
+line”, for the planet-to-star mass-ratio interval −4.5<logq <−2, corresponding
+to the range of ice giants to gas giants. We find
+d2Npl
+dlogqdlogs= (0.36±0.15) dex−2
+at mean mass ratio q= 5×10−4with no discernible deviation from a flat ( ¨Opik’s
+Law) distribution in log projected separation s. The determination is based on
+a sample of 6 planets detected from intensive follow-up observation s of high-
+magnification ( A > 200) microlensing events during 2005-2008. The sampled
+host stars have a typical mass Mhost∼0.5M⊙, and detection is sensitive to
+planets over a range of planet-star projected separations ( s−1
+maxRE,smaxRE), where
+RE∼3.5 AU (Mhost/M⊙)1/2is the Einstein radius and smax∼(q/10−4.3)2/3.
+This corresponds to deprojected separations roughly 3 times the “snow line”.
+9PL, United Kingdom; Eamonn.Kerins@manchester.ac.uk
+71MaxPlanckInstituteforSolarSystemResearch,Max-Planck-Str . 2,37191Katlenburg-Lindau,Germany
+72Astronomy Unit, School of Mathematical Sciences, Queen Mary, U niversity of London, Mile End Road,
+London, E1 4NS, United Kingdom
+73Centre for Star and Planet Formation, Københavns Universitet, Ø ster Voldgade 5-7, 1350 København
+Ø, Denmark
+74Departamento de Astronom´ ıa y Astrof´ ısica, Pontificia Universida d Cat´ olica de Chile, Santiago, Chile
+75Universit` adegli Studi di Salerno, Dipartimentodi Fisica“E.R.Caian iello”, ViaPonteDon Melillo, 84085
+Fisciano (SA), Italy
+76INFN, Gruppo Collegato di Salerno, Sezione di Napoli, Italy
+77Armagh Observatory, College Hill, Armagh, BT61 9DG, United Kingdom
+78Institut f¨ ur Astrophysik, Georg-August-Universit¨ at, Friedr ich-Hund-Platz 1, 37077 G¨ ottingen, Germany
+79Bellatrix Astronomical Observatory, Via Madonna de Loco 47, 0302 3 Ceccano (FR), Italy
+80Department of Physics, Sharif University of Technology, P. O. Box 11155–9161, Tehran, Iran
+81Institut d’Astrophysique et de G´ eophysique, All´ ee du 6 Aoˆ ut 17 , Sart Tilman, Bˆ at. B5c, 4000 Li` ege,
+Belgium
+82Astrophysics Group, Keele University, Staffordshire, ST5 5BG, Un ited Kingdom
+83INAF, Osservatorio Astronomico di Brera, 23806 Merate (LC), I taly– 7 –
+Despite the frenetic nature of these observations, we show that they have the
+properties of a “controlled experiment”, which is what permits meas urement of
+absolute planet frequency. High-magnification events are rare, b ut the survey-
+plus-followup high-magnification channel is very efficient: half of all h igh-mag
+events were successfully monitored and half of these yielded planet detections.
+The planet frequency derived from microlensing is a factor 7 larger t han the
+one derived from Doppler studies at factor ∼25 smaller star-planet separations
+[i.e., periods 2 – 2000 days]. However, this difference is basically consist ent with
+the gradient derived from Doppler studies (when extrapolated well beyond the
+separations from which it is measured). This suggests a universal s eparation
+distribution across 2 dex in planet-star separation, 2 dex in mass ra tio, and
+0.3 dex in host mass. Finally, if all planetary systems were “analogs” o f the
+Solar System, our sample would have yielded 18.2 planets (11.4 “Jupite rs”, 6.4
+“Saturns”, 0.3 “Uranuses”, 0.2 “Neptunes”) including 6.1 systems with 2 or more
+planet detections. This compares to 6 planets including one two-plan et system in
+the actual sample, implying a first estimate of 1 /6 for the frequency of solar-like
+systems.
+Subject headings: gravitational lensing – planetary systems
+1. Introduction
+To date, 10 microlensing-planet discoveries have been published, wh ich permits, at least
+in principle, a measurement of planet parameter distribution functio ns. Of course, the size
+of this sample is small, both absolutely and relative to the dozens of pla nets that have been
+discovered from transit surveys and the hundreds from Doppler ( radial velocity, hereafter
+RV) surveys. However, microlensing probes a substantially differen t part of parameter space
+from these other methods. The majority of RV planets, and the ov erwhelming majority of
+transiting planets are believed to have reached their present locat ions, generally well within
+the “snow line”, by migrating by a factor /greaterorsimilar10 inward from their birthplace. By contrast,
+microlensing planets are generally found beyond the snow line, where gas giants (analogs of
+Jupiter and Saturn) and ice giants (analogs of Uranus and Neptune ) are thought to form.
+Thus it would be of substantial interest to compare the properties of microlensing planets,
+which have not suffered major migrations, to other planets that ha ve.
+In the present incarnations of microlensing experiments, planet de tection occurs through
+several different routes, with various degrees of “human interve ntion”. At one extreme, plan-
+ets can be detected simply on the basis of survey data, without eve n the knowledge that a– 8 –
+planet was present until after the event is over. This is the plan for so-called second gener-
+ation microlensing planet surveys, which will permit rigorous, particle -physics-like objective
+analysis of a “controlled experiment”. In fact, Tsapras et al. (200 3) and Snodgrass et al.
+(2004) have already carried out such analyses of first-generatio n survey-only data. How-
+ever, to date only one secure microlensing planet has been discover ed from survey-only data,
+MOA-2007-BLG-192Lb (Bennett et al. 2008).
+Rather, most microlensing planet detections have taken place thro ugh a complex inter-
+play of survey and followup observations. Only a year after the firs t microlensing events were
+discovered, the MACHO collaboration issued an IAU circular urging fo llow-up observations
+of a microlensing event, and the OGLE collaboration initiated its Early W arning System
+(EWS) (Udalski et al. 1994; Udalski 2003), which regularly issued “a lerts” of ongoing mi-
+crolensing events, usually well before peak, as soon as they were r eliably detected. EWS,
+together with a similar program soon initiated by the MACHO collaborat ion (Alcock et al.
+1996) with their much larger camera, enabled the formation in 1995 o f the first microlensing
+follow-up teams, PLANET (Albrow et al. 1998) and MACHO/GMAN (Alco ck et al. 1997),
+and soon thereafter, MPS (Rhie et al. 1999). The EROS team issued only a few alerts, but
+one of these led to the first mass measurement of an unseen objec t (An et al. 2002), and the
+first spatially resolved high-resolution spectrum of another star ( Castro et al. 2001). OGLE-
+II inaugurated wide-field observations as well in 1998. In order to c over large areas of the sky
+(even with wide-field cameras), the survey teams would generally ob tain only ∼1 point per
+night per field. Since typical planetary deviations from “normal” (po int lens) microlensing
+light curves last only of order a day or less, this was not generally ade quate to detect a
+planet. Hence, Gould & Loeb (1992) already advocated the format ion of follow-up teams
+that would choose several favorable events to monitor more freq uently, using telescopes on
+several continents to permit 24-hour coverage. In 2000, MACHO ceased operations, but a
+new survey group, MOA, had already begun survey observations.
+Once this synergy between survey and follow-up teams was establis hed, it evolved
+quickly on both sides. Both types of teams developed the capacity f or “internal alerts”,
+whereby real-time photometry was quickly analyzed to find hints of a n anomaly. If these
+hints were regarded as sufficiently interesting, they would trigger a dditional observations by
+the team. The first such alert was by MACHO/GMAN in 1996 and sever al followed the next
+year from PLANET. In 2003, OGLE developed the Early EWS, which au tomatically alerted
+the observer to possible anomalies, who then would make additional o bservations and, if
+these were confirming, publicize them to the community. OGLE also pio neered making their
+data publicly available, which greatly facilitated follow-up work. These developments gener-
+ally evolved into a system of mutual alerts, open transfer of data b etween teams, and active
+ongoing email discussion of developing events.– 9 –
+The first secure microlensing planet, OGLE-2003-BLG-235Lb (Bon d et al. 2004) was
+discovered by means of such an internal alert. The MOA team notice d deviations in their
+data, and initiated intensive follow-up of this event. In retrospect , one can see that even
+without this internal alert, the combined OGLE and MOA data would ha ve been sufficient
+to show that the star had a companion. However, the normal surv ey data would not have
+permitted one to tell whether this companion was a planet or a low-ma ss star (or possibly a
+brown dwarf).
+Detections of this type are at the opposite extreme from the pure -survey detection,
+which can be modeled as a controlled experiment. Without understan ding the efficiency at
+which the observers issue their real time alerts, one cannot measu re the absolute rate of
+planets from observations that are triggered by the presence of the planet itself. Another
+planet, OGLE-2007-BLG-368Lb (Sumi et al. 2010) falls partly into t his category. In this
+case, the alert was triggered by ARTEMIS (Dominik et al. 2008), who se ultimate goal is
+to communicate such alerts directly to the followup telescopes witho ut human interference
+(and so bring such alerts into the fold of “rigorous controlled exper iments”), although at this
+stage ARTEMIS alerts were still vetted by humans. A further anom aly was soon detected
+by the MOA observer. But even without followup observations trigg ered by these alerts,
+this companion probably could have been constrained to be planetar y, although with larger
+errors, so this detection could still be integrated into the “contro lled experiment” framework
+even without trying to model the alert process.
+However, if follow-up observations are carried out without being sig nificantly influenced
+by the possible presence of a planet, then these also can be treate d as a controlled experi-
+ment, and absolute rates can be derived using either the method of Gaudi & Sackett (2000)
+or of Rhie et al. (2000). Such an analysis was carried out for the firs t 43 events monitored by
+the PLANET Collaboration (Albrow et al. 2001; Gaudi et al. 2002). In particular, since no
+planets were detected (and so only upper limits obtained), there wa s no possibility of recog-
+nition of a possible planet playing any role in the observation strategy . But, as emphasized
+above, once planets started to be discovered, the situation beca me more nuanced.
+One path toward obtaining followup observations that are triggere d on potentially plan-
+etary anomalies, without compromising the “controlled experiment” ideal, is to establish
+robotic triggering programs that communicate directly to robotic t elescopes. This has been
+the goal of the RoboNet Collaboration since its inception in 2004 (Tsa pras et al. 2009). Al-
+gorithms for detecting anomalies in robotically acquired real-time dat a (Dominik et al. 2007,
+2008) and directing robotic telescopes (Horne et al. 2009) have be en devised and are being
+further developed, and will expand their scope as the Las Cumbres Observatory Global
+Telescope Network (LCOGT) itself expands. But to date, the most effective RoboNET– 10 –
+observations have been “hand triggered” (e.g., for OGLE-2007-B LG-349Lb).
+In spite of this nuanced situation, Sumi et al. (2010) were neverth eless able to extract
+relative frequencies of planets from the ensemble of all 10 microlens ing planets. They argued
+that each of these planet detections was characterized by a sens itivityg(q)∝qm, where
+qis the planet/star mass ratio. While mvaries from event to event, they argued that
+m= 0.6±0.1 was an appropriate average value. They then fit the ensemble of d etections to
+a power-law mass distribution f(q)dlogq∝qndlogqand found n=−0.68±0.20. But they
+did not attempt to extract absolute frequencies from their sample .
+Here, we analyze an important subclass of microlensing planet searc hes: high-magnification
+events that are intensively monitored over the peak. The observa tions of these events are
+always frenetic, sometimes even comical, so it may seem surprising th at they nevertheless
+constitute a “controlled experiment”, or very nearly so, and henc e are subject to rigorous
+analysis.
+More than a decade ago Griest & Safizadeh (1998) pointed out that high-mag events are
+much more sensitive to planets than their far more common cousins, the low-magnification
+events. The reason is fairly simple. For point-lens/point-source mic rolensing, the magnifi-
+cation is very nearly A→u−1foru/lessorsimilar1/3, where uis the source-lens separation normalized
+to the Einstein radius. “High-magnification” ( Amax≫1) therefore implies impact param-
+eteru0=A−1
+max≪1, i.e., that the source is projected very close to the lensing star. I f the
+lensing star has a planet, the planet’s gravitational field will perturb the otherwise circularly
+symmetric gravitational field of the star, and so induce a tiny “cent ral caustic” (contour of
+infinite magnification) around it. If the source passes over or near this caustic, the lightcurve
+is perturbed, revealing the planet’s presence. The key point is that the planet can be at a
+broad range of separations and at virtually any angle relative to the source trajectory and
+still create a caustic that will perturb the light curve.
+However, following this seminal paper, microlensing follow-up monitor ing groups con-
+tinued to focus primarily on garden variety microlensing events, whic h are only sensitive to
+so-called “planetary caustics”. These are formed by the action of the primary-star grav-
+itational field on the planet gravitational field, and as such are bigge r and so have larger
+cross sections than central caustics. At first sight, this makes t hem more favorable targets,
+but this conclusion only holds if one has unlimited telescope time for mon itoring. Then
+one would monitor all available events (and hence primarily low-mag eve nts) and would find
+most planets in these events (just because the caustics have a lar ger cross section). But
+if observing resources are limited, then one should focus these on h igh-mag events because
+these can be predicted (at least in principle) from the pre-peak par t of the light curve and
+have individually higher sensitivity to planets. By contrast, there is n o way to predict that– 11 –
+a source is approaching a planetary caustic.
+Nevertheless, within the context of continued focus on planetary caustics in normal
+events, there were significant efforts to take advantage of high- mag events as well. In 1998
+MPS issued the first high-mag alert for MACHO-98-BLG-35, which re ceived an enthusiastic
+response from the MOA group, leading to the possible detection of a planet (Rhie et al. 2000;
+Bond et al. 2002b), but with too-low significance to be confident. Fr om the inception of its
+survey, MOA made real-time alerts of high magnification events a prio rity and attempted
+to organize follow-up from other continents, with 10 such alerts th e first year (Bond et al.
+2002a). The most spectacular success of this program was MOA-2 003-BLG-32/OGLE-2003-
+BLG-219, which was densely sampled over its Amax= 520 peak by the Wise observatory in
+Israel after such an alert, and which yielded the best upper limits on planetary companions to
+a lens to that date (Abe et al. 2004). Theoretical work was also don e to optimize observations
+of high-mag events for planet sensitivity (Rattenbury et al. 2002) .
+When the Microlensing Follow Up Network ( µFUN) began operations in 2001, it followed
+the already-established model of follow-up observations, which did include high-mag events
+(Yoo et al. 2003), but did not emphasize them. However, three thin gs happened to change its
+orientation toward concentrating on high-mag events. First, the OGLE-III survey came on
+line in 2002 with a discovery rate of 350 events per year, moving up to 600 events per year in
+2004. This compared with 40–80 events per year discovered by OGL E-II in 1998–2000. The
+number of events alerted per year has a direct impact on whether o r not one is in the regime
+of “limited” or “unlimited” resources. Before 2002, if one restricte d oneself to high-mag
+events, one would mostly be sitting on one’s hands. For example, whe n Albrow et al. (2001)
+and Gaudi et al. (2002) analyzed 5 years of PLANET collaboration da ta, they reported only
+two events with Amax>100. OGLE-III dramatically changed that situation. More recently ,
+the MOA collaboration inaugurated MOA-II (2007 first full season) , which has had the net
+effect of increasing the total rate of reported events by about 5 0%.
+Second,µFUN began attracting the intrinsically “limited” observing resources of ama-
+teur astronomers. These contrast with the larger dedicated pro fessional observatories in two
+key ways. First, the observers generally cannot observe all night , every night, or they will
+be unable to keep their day jobs. Second, the smaller apertures of their telescopes restrict
+them to relatively brighter targets. Both “limitations” naturally driv e amateurs to high-mag
+events, which have a bigger chance of science payoff and are bright er (because highly mag-
+nified). Moreover, there is one crucial dimension in which amateurs a re not limited: they
+have completely free and almost instantaneous access to their tele scopes at any time. Thus,
+while dedicated follow-up telescopes are typically operated only in the 3-4 month core of the
+season (when microlensing targets are observable for at least half the night), amateurs can– 12 –
+react to alerts deep into the wings of the season, close to doubling t he number of high-mag
+events that can be monitored. In 2004, one amateur began obser vingµFUN targets on her
+own initiative. She requested regular alerts and began organizing ot her amateurs to join in,
+who in turn self-organized a network. About half of the µFUN authors of this paper are
+amateurs.
+Third,µFUN had to become aware of this changed situation, i.e., that it had mo ved from
+the domain of unlimited to limited resources. This transition was partly aided by the fact that
+µFUN access to two of its professional telescopes (Wise and SMARTS CTIO) was limited
+by their being shared resources. But throughout 2003-2004, µFUN “straddled two horses”,
+focusing on high-mag events when available, but trying to keep to th e old planetary-caustic
+strategy most of the time. Preparations for the 2005 season wer e significantly influenced by
+preliminary work (ultimately, Dong et al. 2006) showing that unless hig h-mag events were
+intensively monitored over their peak, much of their sensitivity to pla nets is compromised.
+Then in April 2005, µFUN intensively followed the (by those days’ standards) high-
+mag event OGLE-2005-BLG-071, which resulted in the detection of the second microlensing
+planet (Udalski et al. 2005). This detection led to µFUN consciously changing its orientation,
+procedures, recruitment, etc., with the aim of focusing primarily on high-mag events.
+From 2005 onwards, considerable effort has gone into identifying po tential high-mag
+events, and in some cases obtaining additional data to improve the p rediction of Amax. If an
+event is deemed a plausible high-mag candidate, then observers are notified by email, without
+necessarily being urged to observe, just to put them on alert. Onc e high magnification
+seems probable, observations are requested with various degree s of urgency. The urgency is
+conditioned by the fact that peak sensitivity is usually less than a day (the normal human
+cycle time) but more than a few hours (so requiring observations fr om multiple continents).
+Many factors enter into the quality of the final light curve, including weather conditions on 6
+continents plus Pacific Islands, observer availability, communication problems, etc. Indeed,
+it is difficult to convey the level of chaos during one of these events.
+Despite (and also because of) this chaos, the resulting data strea m generally retains the
+character of a “controlled experiment”. In the “ideal incarnation ” of this search mode, the
+event is recognized (with greater or lesser certainty) to be appro aching a high-magnification
+peak, and an alert is issued to interested observers urging them to observe it intensively over
+the predicted peak. The observations take place regardless of wh ether the planet is present or
+not. The very chaos, remoteness of observing locations and comm unication problems make
+it difficult to gain knowledge of planetary perturbations until after t he key observations are
+over. In Section 2, we discuss how well the real events conform to this ideal, but assuming
+for the moment that they do, the ensemble of high-magnification ev ents with and without– 13 –
+detections can be analyzed in exactly the same way as can be done fo r the “pure survey”
+mode. In Section 3, we derive an analytic expression for the sensitiv ity of an ensemble of
+high magnification events monitored densely over their peak, and in S ection 4, we calculate
+numerical sensitivities and use these to measure the frequency of ice giants and gas giants
+beyond the snow line. Finally, in Section 5, we discuss some implications o f our results.
+2. Selection Criteria and Data
+We begin by designing criteria for selecting events and planets to be in cluded in the
+analysis that enable the detections and non-detections to be analy zed on the same footing.
+This is essential to the goal of defining a data sample that can be tre ated as a “controlled
+experiment”.
+2.1. Selection Criteria for Events
+E1) High-cadence ( <10 min)µFUN data over some portion of the peak
+E2) High-cadence data covering at least one wing of the peak, |t−t0|< teff, with no major
+gaps
+E3) High signal-to-noise ratio (S/N) data (/summationtext
+iσ−2
+i>50000) covering the other wing, where
+theσiare the data-point errors in magnitudes
+E4) Light-curve is not dominated by binary (non-planetary) featu res
+We define teff≡u0tEfor point-lens/point-source events, where tEis the Einstein crossing
+time. For point-lens events that suffer finite-source effects, we g eneralize this definition to
+the time interval during which the magnification is within√
+2 of the peak. And for planetary
+events, we further generalize it to the time interval that the magn ification would have been
+within√
+2 of the peak if the lensing star had lacked planets.
+We now justify these criteria.
+In addition to reflecting the fact that we are summarizing µFUN work, criterion (E1)
+ensures that we can rigorously review the available data on ∼3000 events discovered during
+2005-2008, and reduce them to a manageable subset of “only” 315 that can be investigated
+usingµFUN files. These 315 were then quickly pared down to a few dozen eve nts that are
+consistent with high-magnification and actually meet criterion (E1).
+Criteria (E2) and (E3) should be considered together. Our underly ing requirement
+is to have enough coverage of the event so that planetary deviatio ns that give rise to a
+∆χ2= 500 deviation (for the rereduced data set), have a high probabilit y of yielding a– 14 –
+unique scientific interpretation. See criterion (P2), below. These t wo criteria are basically
+derived from the experience analyzing the event OGLE-2005-BLG- 169 (Gould et al. 2006),
+which does somewhat better than barely satisfy both. The analysis of this event was already
+fairly difficult because of the multiple χ2minima and would have been quite degenerate if,
+for example, it lacked high S/N data on the rising side. Hence it would no t have led to a
+publishable planet with a reasonably well-defined mass ratio.
+Criterion (E4) is adopted for two reasons. First, in contrast to po int-lens events, most
+binary events are not modeled with sufficient precision to measure th eu0parameter well
+enough to construct a well-defined sample of events with maximum ma gnification greater
+than some threshold Amax. Second, the problem of detecting planets in the presence of light-
+curve features dominated by a binary is not well-understood. Henc e, the results we derive
+here really apply to stars not giving rise to strong binary features, which excludes roughly
+3–6% of all stars (Alcock et al. 2000; Jaroszy´ nski 2006).
+Table 1 lists all events from 2005-2008 that satisfy these four crit eria and that had
+peak magnifications Amax>100. Column 1 is the event name, column 2 is the maximum
+magnification, column 3 is the time of closest approach between the s ource and lens t0,
+column 4 is the Einstein timescale tE, column 5 gives the mass of the lens star for cases that
+it is known, and column 6 gives the method by which it is derived. These le ns masses and
+methods are discussed in Section 4.2. The events are listed in inverse order ofAmax.
+Figure 1 shows the cumulative distribution A−1
+maxfor the 15 events in Table 1. It displays
+a clear break at Amax= 200. Below this value, the distribution is uniform in A−1
+max, which
+is the expected behavior for a complete sample (ignoring finite sourc e effects), i.e., uniform
+inu0, which is the impact parameter in units of the Einstein radius. The das hed line, with
+slope ofdNev/dA−1
+max= 2600 is a good match to the Amax>200 data. Cohen et al. (2010)
+found such uniformity for the underlying sample of OGLE events in 20 08, with a slope of
+dNev/du0= 1080 for u0<0.05. From this comparison, we learn two things. First, µFUN was
+aggressive enough to achieve a uniform subsample only for events w ithAmax>200. Second
+µFUN was able to intensively monitor half of all events in the Amax>200 subsample.
+That is, assuming that OGLE found similar numbers of high-mag event s in 2005-2007, and
+accounting for the fact that MOA found 50% more events (not fou nd by OGLE) in 2007-
+2008, the full sample of high-mag events was about dNev/du0∼5×1080 = 5400, of which
+µFUN effectively monitored about 48%.
+Although it may not be immediately obvious, Figure 1 implies that we must impose a
+fifth criterion.– 15 –
+E5)Amax>200
+Figure 1 demonstrates that µFUN was substantially less enthusiastic about events
+Amax<200 than Amax>200, whether because it simply did not act on events known
+in advance to be in the former category or just became less enthus iastic about observations
+once these events were recognized near peak not to be extremely magnified. This bias is
+a natural consequence of µFUN’s limited observing resources (as discussed in Section 1):
+there are 4 times as many events with Amax>50 asAmax>200 and their duration of peak,
+2teff∼2tEA−1
+maxlasts 4 times as long, so 16 times more observing resources would be r equired
+to follow them all. Hence, if an event proved midway to be one that the objective evidence
+demonstrates µFUN cared less about, there would be a tendency on the part of obs ervers
+to slacken efforts (whether or not the internal alert was officially ca lled off). Then the event
+would have less chance of meeting the selection criteria. But if a plane t were detected during
+the peak observations (and there is a greater chance of recogniz ing a planet in real time for
+lowerAmaxbecause the peak lasts longer) then observations would not slacke n, but rather
+intensify. Since this bias cannot be rigorously quantified, planets an d non-detections from
+Amax<200 events must both be excluded from the sample.
+2.2. Selection Criteria for Planets
+P1) Planet must be discovered in an event that satisfies (E1)-(E5)
+P2) Planetary fit yields improvement ∆ χ2>500
+P3) Planet-star mass ratio qmust lie in the range
+q−< q < q +, q −= 10−4.5, q −= 10−2. (1)
+Criterion (P1) is self-evident but is stated explicitly for completenes s and emphasis.
+Criterion (P2) may appear at first sight somewhat draconian, but it is realistic. To explain
+this, we first note that among the six planets listed in in Table 2, the “w eakest” detection
+is ∆χ2= 880, which is for MOA-2008-BLG-310Lb (Dong et al. 2009b). Now, it is certainly
+possible to recognize systematic residuals from a point-lens fits “by eye” at a much lower
+level, even ∆ χ2= 100. Indeed, Batista et al. (2009) argued that no systematic re siduals
+were present in the fit to OGLE=2007-BLG-050 at a much lower level, ∆χ2= 60. But
+if such deviations had been observed in an event, this would not have necessarily enabled
+discovery of a planet, where “discovery” here means “publication” . First, ∆ χ2= 100 in the
+final, rereduced and carefully cleaned data implies something like ∆ χ2= 50 in the standard
+pipeline data, and systematic deviations due to a planet at this level w ould probably not be
+recognized as significant, i.e., clearly distinguishable from systematic s that appear in dozens
+of other events, and which just reflect instrumental, weather, o r data-reduction problems.– 16 –
+But more to the point, even if the unrereduced-data were ∆ χ2= 100, triggering strenuous
+efforts to clean and rereduce the dataset, resulting in, say, a ∆ χ2= 200 improvement, it
+is far from clear that this deviation (even if strongly believed to be re al) would lead to a
+publishable planet detection. This is because, in addition to obtaining a n acceptable fit to
+the data, such a paper would have to demonstrate that there cou ld be no acceptable fits
+to the data to non-planetary solutions. We have already designed c riteria (E2) and (E3)
+to eliminate those events for which very high ∆ χ2is possible without leading to a unique
+interpretation (due to incomplete coverage of the deviation). But we still must set the
+threshold high enough so that if an anomalous event survives criter ia (E2), (E3), and (P2),
+it has a small chance of being ambiguous in its interpretation. Our bes t estimate of this,
+from experience fitting events, is ∆ χ2= 500. However, we regard other values in the range
+350-700 as also being plausible candidates for this threshold. We will s how in Section 4 that
+our basic conclusions are robust to changes within this range.
+The upper boundary in Equation (1), Criterion (P3), is necessary b ecause at high mass
+ratiosq, one cannot be confident that the event will not be rejected (con sciously or un-
+consciously) as a “brown dwarf” or “low-mass-star”, and theref ore not be monitored as
+intensively as it might be [and so not pass criteria (E2) and (E3).] To illus trate this, we
+review how OGLE-2008-BLG-513Lb (Yee et al., in preparation) was “ almost rejected” as a
+binary, even though it is probably a planet. This event has a large, st rong, resonant caustic,
+that was initially mistaken for a binary. During the long intra-caustic p eriod, it was realized
+that the companion might be a planet, and that intensive observatio ns of the caustic exit
+would be necessary to resolve this question. Such observations we re obtained, and from these
+we know that the impact parameter was u0∼0.027, soAmax∼37, which means that the
+event fails criterion (E5) and so is excluded from our sample. But if th ese data had not been
+obtained, then u0for this event would not be known, and it would not be known whether
+the event was in the sample or not, and if it were, whether the compa nion was a planet or
+not.
+Why does this example then not just prove that the whole concept o f “controlled exper-
+iment” is unviable? The answer is given by Figures 2 and 3, below. One se es from these that
+at high mass ratios, q∼10−2, there is an extremely wide range of sfor which the planet is
+“detectable”. Only a small fraction of these “detectable” events haves∼1, which produce
+large resonant caustics that might be mistaken for binaries. Hence , while in principle some
+of these “detections” might be lost to this confusion, the great ma jority would not cause
+any confusion. The reason that planets like OB08513Lb make their w ay into the detections
+at all, despite their relative rarity, is that the caustics are so large t hat they are detectable
+over a wide range of u0, only a small fraction of which would pass the “high-mag” criteria
+of Section 2.1. Nevertheless, as qgrows, this potential problem grows with it. We adopt– 17 –
+logq+=−2, but recognize that values ranging from −2.3 to−1.8 might also have been
+plausible choices.
+The lower boundary q−= 10−4.5is established because of concerns of the real “de-
+tectability” of low mass planets in the presence of higher mass planet s in the same system.
+In the method of Rhie et al. (2000), which we employ in Section 4.1, plan et sensitivity is de-
+termined by fitting simulated star-planet light curves that are cons tructed to have the same
+error properties as the actual data, to point-lens models. If the best such model increases χ2
+by more than a given threshold (say ∆ χ2>500), then the planet is said to be detectable.
+Since this method directly mimics the process of planet detection for single-planet systems,
+it is a good way to characterize the detectability of such systems. B ut high-magnification
+events are particularly sensitive to multiple planets (Gaudi et al. 1998) and why should this
+approach tell us anything about the detectability of a second plane t in a system already
+containing one planet? As first shown by Bozza et al. (1999), the ne t perturbations of such
+two-planet high-magnification lightcurves usually “factor” into the sum of perturbations in-
+duced by each planet separately. For example, the only published tw o-planet system has this
+property (Gaudi et al. 2008; Bennett et al. 2010). Of course, th e factoring is not perfect, but
+in this real case (and in many simulated cases), once the dominant-p lanet perturbation is
+removed, the secondary perturbation is easily recognized, leading to an excellent starting
+point for a combined fit to both planets simultaneously. For reasona bly comparable planet
+mass ratios, the only exception to this is if the planet-star axes are closely (within /lessorsimilar20◦)
+aligned (Rattenbury et al. 2002; Han 2005). In this case, the single -planet fit still fails, but
+the residuals to this fit are not easily recognizable. While such difficultie s might impede
+recognition of the second planet, the required alignment is so close, that such cases would
+be a small minority of two planet systems.
+However, this factoring has only been studied in detail for planets w ith relatively com-
+parable masses (Han 2005). The situation may not be as simple when t he mass ratio of the
+two planets is extreme. Based on analysis of the events listed in Table 2, below, we cannot
+be confident of excluding all “second planets” with q <10−5. To be “conservative”, we have
+moved the boundary to q−= 10−4.5.
+Because both q−andq+have some uncertainty, we must ask how robust our conclusions
+are to changes in these parameters within a reasonable range. We s how in Section 4 that
+our basic conclusions about planet frequency are not seriously affe cted by uncertainty in q−
+andq+. However, these uncertainties will prevent us from deriving a slope of the mass-ratio
+function.
+Six planets satisfy criteria (P1)-(P3). Table 2 displays their charac teristics. Columns
+2–5 are the parameters ( u0,ρ,q,|logs|) measured from the event, where ρis the source– 18 –
+radius in units of the angular Einstein radius θE. In several cases, there is an unresolved
+s↔s−1degeneracy, which is irrelevant to the current study, so we just d isplay the absolute
+value of the log. The final column is the maximum detectable value of |logs|according to
+calculations reported in Section 4.1.
+2.3. Were Discovery Observation Cadences Really Independe nt of the Planet?
+Were the observations that led to the discovery of the six planets in fact carried out
+independent of the presence of the planet? For three of these, O GLE-2005-BLG-169Lb
+(Gould et al. 2006), MOA-2007-BLG-400Lb (Dong et al. 2009b), an d MOA-2008-BLG-310Lb
+(Janczak et al. 2010), the planet was not recognized until after t he event had returned to
+baseline. OGLE-2007-BLG-349 (Dong et al., in preparation) was rec ognized to have a sig-
+nificant deviation possibly due to a planet based on observations in Ch ile, 36 hours after the
+call for intensive observations based on its high-mag trajectory, and roughly 7 hours after
+observations had begun in South Africa. While it is true that reports of this potential planet
+heightened excitement, and may perhaps have increased the comm itment of observers to get
+observations, the density of observations (from 4 continents plu s Oceania) did not qualita-
+tively change from before till after the potentially-planetary anom aly was recognized. This is
+the only one of the six planets in our sample that is not yet published. T he reason is that the
+system contains a third body, which has proven difficult to fully chara cterize. However, the
+characteristics of OGLE-2007-BLG-349Lb are very well establish ed, and the third body is
+certainly not in the mass range being probed in the current analysis. Hence we feel confident
+including this planet in the sample.
+The two planets OGLE-2006-BLG-109Lb,c (Gaudi et al. 2008; Benn ett et al. 2010) re-
+quire closer examination. OGLE-2006-BLG-109 was recognized to b e an interesting event
+almost 10 days prior to peak due to detection of what turned out to be the resonant caustic
+of OGLE-2006-BLG-109Lc, the Saturn mass-ratio planet in this sy stem. This anomaly did
+indeed trigger some additional observations, which did help charact erize this planet. But
+a review of email communications that initiated follow-up observation s during the event
+shows that far more intensive observations were triggered sever al days later, after the event
+had appeared to return to “normal” (point-lens-like) microlensing, exactly by its high-mag
+trajectory. Although these emails remark on the possible presenc e of a planet, they place
+primary emphasis on this being an otherwise “normal” microlensing eve nt that was reaching
+extreme magnification. (Note that the appearance of extreme ma gnification was not itself
+an artifact of the presence of planets(s), but was simply due to low source-lens impact pa-
+rameter.) It was the intensive observations from New Zealand, trig gered by these emails,
+that captured the “central structure” of the caustic due to OG LE-2006-BLG-109c. These
+would have enabled basic characterization of this planet even withou t the flurry of followup– 19 –
+observations 10 days earlier. Moreover, it was the same email that triggered intensive ob-
+servations from two widely separated locations (Israel and Chile), that enabled detection of
+the “central caustic” due to OGLE-2006-BLG-109Lb, the Jupite r mass-ratio planet. Thus,
+all detections were in reasonable accord with the “controlled exper iment” ideal.
+3. Analytic Treatment
+3.1. Triangle Diagrams
+Batista et al. (2009) and Yee et al. (2009) recently analyzed the se nsitivity to planets
+of two events in our sample from Table 1, OGLE-2007-BLG-050 and O GLE-2008-BLG-279
+respectively. Figures 2 and 3 are versions of their results, but with the detection thresh-
+old used in this paper, ∆ χ2= 500. In contrast to the wide range of detection-sensitivity
+morphologies shown in Figure 8 Gaudi et al. (2002), both of these dia grams have a simple
+triangular appearance, which is basically described by a two-parame ter equation,
+|logs|max(q) =ηlogq
+qmin, (2)
+whereqis the planet/star mass ratio, sis the planet-star projected separation in units of the
+Einstein radius, ηis the slope of the triangle, and qmindefines the “bottom” of the triangle.
+Planets lying inside the triangle are detected with 100% efficiency and t hose lying outside
+are undetectable. The boundary region is quite narrow. In principle , planet detection is
+a function not only of ( s,q), but also α, the angle of the source-lens trajectory relative
+to the planet-star axis. The narrowness of the boundary reflect s that detection is almost
+independent of α. See Batista et al. (2009) and Yee et al. (2009) for concrete illustr ations.
+It is also striking that the slopes η∼0.32 andη∼0.35 are nearly the same for the
+two diagrams, leading to the conjecture that ηis very nearly constant for well-monitored
+high-magnification events. Indeed, of the 43 events analyzed by A lbrow et al. (2001) and
+Gaudi et al. (2002), two are relatively high-mag ( Amax>100) and both have the same tri-
+angular appearance and very similar slope η, as does the extreme Amax= 3000 event OGLE-
+BLG-2004-343 with simulated coverage analyzed by Dong et al. (200 6). If truly generic,
+this would mean that high-mag event sensitivities have an extremely s imple triangular form
+characterized by a single parameter, qmin.
+Moreover, while qmin, i.e. the depth of the triangles in Figures 2 and 3, obviously depends
+on the intensity, quality, and uniformity of coverage, one expects the fundamental scaling to
+be
+qmin=ξA−1
+max, (3)
+whereξis a parameter that depends on the data quality, etc. The reason f or this expected
+scaling is that the size of the central caustic is proportional to q, andA−1
+maxmeasures how– 20 –
+closely the source probes the center, which is roughly the maximum o f the impact parameter
+u0and the source size ρ(Han & Kim 2009; Batista et al. 2009)
+Hence, armed with an empirical estimate of ξ, one can quickly gage the sensitivity of one
+event or an ensemble of events to planets, which is quite useful bot h to guide observations and
+as a check on “black-box” simulations of event sensitivity. Indeed a s we will show below, one
+can approximately “read off” the frequency of planets by just cou nting the number detected
+and the number of high-mag events surveyed.
+Based on this handful of published analyses, we estimated ξ∼1/70 for a ∆ χ2=
+500 threshold. Since these analyses (naturally) focused on event s with better-than-average
+coverage, we estimate that ξ∼1/50 is more appropriate for a sample such as ours.
+3.2. Analytic Estimate From Triangles
+We now assume that planets are distributed uniformly in log s[¨Opik’s law (Opik 1924)]
+in the neighborhood of the Einstein ring1. We will show further below that this assumption
+is consistent with current microlensing data. Then, assuming η= 0.32 to be universal, we
+have
+Pi(q)dlogq= 2ηfi(q) logq
+qmin,iΘ(q−qmin,i)dlogq (4)
+wheref(q) is the number of planets per dex of projected separation per dex of mass ratio,
+and Θ is a step function. The expected number of planets detected in high-mag events is
+then just the sum of Equation (4) over all high-mag events with goo d coverage
+dNpl
+dlogq=Nev/summationdisplay
+i=1Pi(q). (5)
+Figure 1 demonstrates that the µFUN sample is uniform for 0 < A−1
+max< ǫ, whereǫ= 0.005.
+Hence, we can turn this sum into an integral,
+dNpl
+dlogq=Nev/summationdisplay
+i=1Pi(q)→Nev
+ǫ/integraldisplayǫ
+0PAmax(q)dA−1
+max= 2ηf(q)Nev
+ǫ/integraldisplayǫ
+0log[q/(ξ/Amax)]Θ(q−ξ/Amax)])dA−1
+max
+=2η
+ξf(q)Nev
+ǫ/integraldisplayξǫ
+0log(q/qmin)Θ(q−qmin)dqmin=2η
+ξln 10f(q)Nev
+ǫ/integraldisplaymin(q,ξǫ)
+0ln(q/qmin)dqmin,
+(6)
+1If in fact planets are distributed dN/dlogs∝sp, then Eq. (4) is in error by sinh x/xwithx=
+ηplnq/qmin. Forη= 0.32,p= 0.4 andq/qmin= 100, this is still only a factor 1.06.– 21 –
+which may be evaluated,
+dNpl
+dlogq=2ηNev
+ln 10g(q)f(q) (7)
+g(q) =q
+qthr(q < ξǫ );g(q) = 1 + lnq
+qthr(q > ξǫ ) (8)
+whereqthr≡ξǫ. Below this threshold, detection efficiency falls linearly with q. Above
+the threshold, it rises logarithmically with q. Note that the appearance of “ln 10” in these
+formulae is an artifact of our having chosen to express the density of planets in units of dex
+of separation, rather than the “natural unit” of an e-folding.
+Both the break at qthrin the normalized survey sensitivity g(q) and the functional forms
+ofg(q) on either side of this break are easily understood from the triangu lar form of the
+individual-event sensitivity diagrams. For q > qthr, allNevevents contribute sensitivity. If
+we compare two mass ratios, log qand logq+dlogq, the latter is sensitive to a log-separation
+interval on the triangle that is larger by exactly 2 ηdlogqforeach individual event , so the
+sensitivity of the ensemble of events is simply 2 ηlogq+const. On the other hand, for q < qthr,
+only a fraction q/qthrof the events contribute, which breaks the logarithmic form of g(q).
+Note that our estimate ξ= 1/50 implies that our Amax< ǫ−1= 200 survey has
+qthr=ǫξ= 10−4, (9)
+i.e., twice the mass ratio of Neptune. We discuss the implications of this threshold in
+Section 5.
+4. Numerical Evaluation
+4.1. Sensitivities for the Full Sample
+To more accurately determine the sensitivity of our survey and to in fer the frequency
+of planets, we carry out a detailed sensitivity analysis of all 13 of the Amax>200 events in
+Table 1 (except the two that were already done). We use the metho d of Rhie et al. (2000)
+(outlined in Section 2.2) except that we take full account of finite so urce effects (Dong et al.
+2006), which are much more important for the present sample of ev ents because of their
+higher magnification. For the last three events above the cut in Tab le 1,θE(and hence
+ρ≡θ∗/θE) is not well measured. For these, we follow the procedure of Gaudi et al. (2002)
+and adopt ρ=θ∗/(µtyptE), where θ∗is determined from the instrumental color-magnitude
+diagram in the standard way (Yoo et al. 2004), tEis the measured Einstein timescale, and
+µtyp= 4 mas yr−1is the typical source-lens proper motion toward the Galactic bulge. Figure– 22 –
+4 is a “portrait album” of the resulting triangle diagrams (only one side shown to conserve
+space) and Figure 5 shows the integrated sensitivity of each event as a function of mass ratio
+q. That is, it is the integral of the sensitivity (in Figs. 2, 3, and 4) over horizontal slices.
+Hence, if the sensitivity were truly a triangle, the curves in Figure 5 w ould be perfectly
+straight lines, with slope 2 η(illustrated by the bold black line segment) and x-intercepts at
+qmin. Most of the curves do have this behavior over the range −4/lessorsimilarlogq/lessorsimilar−2, which is the
+main range of sensitivity of this technique and also where the planets in Table 2 are located.
+Moreover, the inferred intercepts of the straight-line portion of these curves do generally
+reach to lower mass ratio qminfor higher Amaxevents, although with considerable scatter.
+However, while some events (like OGLE-2008-BLG-279) are almost p erfectly straight down
+to zero, others (like OGLE-2005-BLG-169 and OGLE-2006-BLG-1 09) show a sharp flattening
+toward lower mass ratios. There are two reasons for this. Events like OGLE-2005-BLG-169
+have non-uniform coverage over peak, which makes detectability a strong function of angle.
+The contours in the triangle diagram separate, so that while the 50% -sensitivity contour is
+fairly straight, there is still substantial sensitivity below qmin, which is defined by where the
+two 50% contours meet, creating a long tail of sensitivity below this t hreshold. Events like
+OGLE-2006-BLG-109 have very small source size relative to impact parameter, ρ/u0≪1,
+which enables detection of small mass-ratio planets that are very c lose to Einstein ring
+because the relatively large, but very weak, caustics of these plan ets are then not “washed
+out” as they would be for larger ρ(Bennett & Rhie 1996). Hence the entire “triangle” has
+a curved appearance, although the contours are tightly packed t ogether.
+The black bold dashed curve in Figure 5 is the combined sensitivity, i.e., t he sum of the
+sensitivities for all 13 events (divided by 10, so it fits on the same plot ), which we call G(q).
+The curves in Figure 5 allow us to compare the observed log projecte d separation s,
+with the maximum detectable separation. See Table 2. Because some planets suffer from the
+s↔s−1degeneracy, we only show the absolute value of log s. The cumulative distribution
+of the ratios of these quantities is shown in Figure 6. The separation s are consistent with
+being uniform in log s, with Kolmogorov-Smirnov probability of 20%. (Moreover, in genera l,
+the high-magnification events with the greatest values of |logs|– and so the greatest po-
+tential leverage for probing the distribution as a function of s– are also the most severely
+affected by the s↔s−1degeneracy. This applies, for example, to MOA-2007-BLG-400Lb.
+Thus the dependence on swill be much better explored using “planetary caustics” in low-
+magnification events, for which the s↔s−1degeneracy is easily resolved. Gould & Loeb
+1992; Gaudi & Gould 1997)
+4.2. Masses of Host Stars
+Now,f(q) may in principle be a function of the host mass M(and perhaps other– 23 –
+variables as well). With only six detections, we are obviously in no positio n to subdivide
+our sample. Nevertheless, it is important to assess what host mass range we are actually
+probing. There do exist mass estimates or limits for all 5 hosts of the planets that have
+been detected, and there are also mass estimates for 5 of the 8 len sing stars in the sample
+for which no planet was detected, which are given in Table 1 together with the method
+of estimation. For 5 of the events, both the angular Einstein radius θEand the “microlens
+parallax” πEare measured, which together permits a mass measurement M=θE/κπE, where
+κ≡4G/(c2AU)∼8.1 masM−1
+⊙(Gould 2000b). The measurement for OGLE-2007-BLG-349
+is preliminary (Dong et al., in preparation) but the others are secure . For four of the events,
+there is a measurement of θE=√κMπrel, whereπrelis the source-lens relative parallax,
+but notπE. This measurement of the product of Mandπrelcombined with the lens-source
+relative proper motion µ=θE/tEand a Galactic model (GM) permit a Bayesian estimate
+ofM. Finally there are two events for which adaptive optics (AO) observ ations provide
+information on the host. For OGLE-2006-BLG-109, AO resolution o f the host confirms the
+microlensing mass determination from θEandπE. For MOA-2008-BLG-310, AO observations
+detect excess light (not due to the source) but it is not known whet her this excess is due to
+the lens or another star. So only an upper limit on the mass is obtained .
+Seven of these 10 measurements are in the range of middle M to middle K stars, while
+one is a brown dwarf and another is likely to be a late M dwarf. They cov er a fairly broad
+range approximately centered on 0 .5M⊙, with a tail toward lower mass. Of course, there are
+also 3 lenses in the Amax>200 sample for which there is no mass measurement or estimate.
+These have timescales tEof 10, 26, and 59 days, which is much more typical of microlensing
+events than the sample having mass measurements (with 3 events h avingtE>100 days). If
+these are otherwise typical events, then the lenses probably lie mo stly in the Galactic bulge
+(Kiraga & Paczy´ nski 1994), in which case their mean mass is roughly 0.4M⊙(Gould 2000a).
+Given that this is a minority of the sample, that the information about this minority is far
+less secure, and that the difference from the sample with harder inf ormation is not very large,
+we adopt
+M∼0.5M⊙ (10)
+for the typical mass of the sample. However, we note that the implic ations discussed in
+Section 5 would not be greatly affected if we had adopted M∼0.4M⊙.
+4.3. Likelihood Analysis
+To evaluate the mass-ratio distribution function f(q) =Aqn, we maximize the likelihood:
+L=−Nexp+Nobs/summationdisplay
+i=1lnG(qi)f(qi);Nexp=/integraldisplayq+
+q−dqG(q)f(q) (11)– 24 –
+and find
+f(q) =dNev
+dlogqdlogs= (0.36±0.15)×/parenleftBigq
+5×10−4/parenrightBig−0.60±0.20
+dex−2. (12)
+However, as we now argue, while the normalization of Equation (12) is robust, the slope is
+not.
+In Section 2.2, we summarized why there must be some boundaries q±beyond which
+the experiment is seriously degraded, but argued that there is no “ impartial algorithm” for
+deciding exactly where those boundaries should be. We find that if we varyq+between
+−2.3 and−2., and we vary q−between −5 and−4.5, that the normalization in Equation
+(12) varies by only ±10%, which is much smaller than the statistical errors. However, th e
+power-law index varies between −0.6 and−0.2. If we had a much larger sample of planets,
+then we could set the boundaries at various places within the range o f our detections, thereby
+simultaneously reducing both the number of detections and Nexpin Equation (11). For an
+infinite sample, such a procedure should lead to no variation in either s lope or normalization.
+For a finite sample, the variation would provide an estimate of the err or in these quantities
+due to the uncertainty in knowledge of these boundaries. However , when we apply this
+procedure to our small sample, we find quite wild variations, implying th at we cannot derive
+a reliable slope from these data.
+In Section 2.2 we mentioned that the threshold value ∆ χ2>500 also had some intrinsic
+uncertainty. However, in this case the effect is very small. For exam ple, if we decrease the
+threshold to ∆ χ2>350, then the normalization in Equation (12) decreases by only 7%,
+much less than the Poisson error. Given that the normalization in Equ ation (12) is robust
+but the slope is not, we give our final result as:
+dNev
+dlogqdlogs= (0.36±0.15) dex−2atq∼5×10−4(13)
+Figure 7 summarizes the principal inputs to the modeling. The top pan el shows the
+cumulative distribution of the detections. The bottom panel shows the sensitivity of the
+survey as function of planet mass, both the analytic approximation derived in Section 3.2
+and the numerical determination derived in Section 4.1. These hardly differ.
+The density given in Equation (13), can be obtained by a very simple ar gument. The
+“triangle” for each event has sensitivity to (1/2 base ×height) = η[log(0.01Amax/ξ)]2∼
+1.7[1+0.42 log(Amax/400)]2dex2of planet parameter space. We observed N= 13 events and– 25 –
+found six planets, so 6 /(13×1.7)∼0.27 dex−2, i.e. correct within the statistical error.
+5. Discussion
+We have presented the first measurement of the absolute freque ncy of planets beyond
+the snow line over the mass-ratio range −4.5/lessorsimilarlogq/lessorsimilar−2. The resulting planet frequency,
+Equation (13), can be understood directly from the data and the “ triangle” sensitivity dia-
+grams. The result is applicable to a range of host masses centered n earM∼0.5M⊙. The
+distribution is consistent with being flat in log projected separation s.
+5.1. Comparison with Previous Microlensing Results
+Gould et al. (2006) had earlier concluded that “cool Neptunes are c ommon” based on
+one of the planets analyzed here (OGLE-2005-BLG-169Lb) and an other planet with similar
+mass ratio, OGLE-2005-BLG-390Lb (Beaulieu et al. 2006), which ha d been detected through
+another channel: followup observations of low magnification events . OGLE-2005-BLG-390Lb
+was actually recognized as a possible planetary event during the plan etary deviation, but
+detailed review of these communications and their impact on the obse rving schedule shows
+that this “feedback” was not critical to robust detection. Moreo ver, at that time there
+had been no other detections through this channel, so the Beaulieu et al. (2006) detection
+could also be treated as a “controlled experiment”. These could the n be combined to obtain
+an absolute rate for “cool Neptunes”, albeit with large errors. Ho wever, with only two
+detections, Gould et al. (2006) were not able to specify the mass ra nge of “cool Neptunes”
+and hence were not able to express their results in units of dex−2as we have done here. If
+we nevertheless, somewhat arbitrarily, say that the Beaulieu et al. (2006) and Gould et al.
+(2006) result applies to 1 dex in log q, centered at the q= 8×10−5of their two detections,
+and adopt their 0.4 dex interval in log s, their estimated density of cool Neptunes can be
+translated to 0 .95+0.77
+−0.55dex−2(1σ). To make a fair comparison with Equation (13), it is
+necessary to adopt some slope for the mass function in order to ac count for the factor ∼6
+difference in mass. If we adopt the Sumi et al. (2010) slope of n=−0.68 from microlensing,
+then our prediction for this mass range would be 1 .25 dex−2. If we adopt the Cumming et al.
+(2008) slope of n=−0.31 from RV, it would be 0 .64 dex−2. Either way, these are consistent.
+Sumi et al. (2010) analyzed all 10 published microlensing planets, inclu ding the 5 that
+we analyze here. They approximated the sensitivity functions of th is heterogeneous sample
+by a single power law ( ∝qm,m= 0.6±0.1) and derived a power-law mass-ratio distribution
+dNpl/dlogq∼qn,n=−0.68±0.2. Since we are unable to derive a slope from our analysis,– 26 –
+and they do not derive a normalization, there can be no direct compa rison of results.
+5.2. Comparison with Radial Velocity Results
+Based on an analysis of RV planets, Cumming et al. (2008) derive a nor malization of
+0.035 dex−2, a factor 10 ±4 smaller than the one found here. A factor (5 ×10−4/1.66×
+10−3)−0.31= 1.5 of this difference is due to the fact that they normalize at higher ma ss
+ratio. The remaining factor 7 ±3 difference is most likely due to the different star-planet
+separations probed by current microlensing and RV experiments. T he Cumming et al. (2008)
+study targets stars with periods of 2–2000 days, corresponding to a mean semi-major-axis of
+a= 0.31 AU. Microlensing probes a factor ∼3 beyond the snow line (Fig. 8 from Sumi et al.
+2010)2. To make contact between microlensing observations of primarily low er mass stars
+with RV observations of typically solar type stars, we should conside r planets in similar
+physical conditions, which we choose to normalize by the snow line. Th at is, we should
+compare to G-star planets at 3 “snow-line radii”, i.e., a∼8 AU. Hence, the inferred slope
+between the RV and microlensing measurements is dlnN/dlna= log(7±3)/log(8/0.31) =
+0.56±0.16, which is consistent with the slope of dlnN/dlna= 0.39±0.15 derived by
+Cumming et al. (2008) for RV stars within their period range. Thus, simple extrapolation
+of the RV density profile derived from planets thought to have migra ted large distances,
+adequately predicts the microlensing results based on planets beyo nd the snow line that are
+believed to have migrated much less. See Figure 8. Figure 9 compares microlensing and RV
+detections as a function of mass ratio q.
+5.3. Prospects for Sensitivity to Very Low Mass Planets
+Equation (8) and Figure 5 show a break in sensitivity at qthr≃10−4. For a power-law
+mass-ratio distribution, the ratio of planets expected above and b elow this threshold is
+Npl(q > qthr)
+Npl(q < qthr)=n+ 1
+n/bracketleftBig
+znln(z) +/parenleftBign−1
+n/parenrightBig
+(zn−1)/bracketrightBig
+, (14)
+wherez≡0.01/qthr. This ratio is an extremely strong function of the adopted slope of
+the mass function, n. Forn=−0.68 (Sumi et al. 2010), it is 1.0, whereas for n=−0.31
+2Just as RV measurements respond to a projected stellar velocities , and so measure msiniof the planet
+which is always less than or equal to the planet mass m, so microlensing observations measure the projected
+separation s, which for circular orbits is related to the semi-major axis by REs=asinγwhereγis the angle
+between the star-planet axis and the line of sight. The statistical d istribution sin γis exactly the same as for
+siniin RV. Hence, except for rare cases when the orbit is constrained b y higher-order effects (Dong et al.
+2009a; Bennett et al. 2010), amust be statistically estimated from s(andRE), which is what is done in Fig.
+8 of Sumi et al. (2010).– 27 –
+(Cumming et al. 2008), it is 4.7. The fraction of planets within the lower domain that lies
+belowqis simply qn+1. Hence, while several authors have shown that individual planets
+at or near Earth mass ratio are detectable in high-magnification eve nts (Abe et al. 2004;
+Dong et al. 2006; Yee et al. 2009; Batista et al. 2009), the actual r ate of detection will be
+strongly influenced by the actual value of n. As discussed in Section 2.2, probing to lower
+masses will require technical advances to robustly identify low mass planets in the presence
+of higher mass planets. But it will also require increasing the number o f events that are
+monitored.
+There is some potential to do this. First, as shown in Section 4, only a bout half of
+events that are announced by search teams are intensively monito red. Hence, there is room
+to double the rate by more aggressive monitoring. This will be aided by inauguration of
+OGLE-IV, which will have much higher time sampling and so will permit mor e accurate
+prediction of high-mag events. Second, it is possible that the more in tensive OGLE-IV
+survey will increase the underlying sample of high-mag events. Finally , systematic analysis
+of high-mag events could bring down the effective ∆ χ2threshold from 500 to, say, 200,
+which would decrease ξ(and soqthr) by a factor (500 /200)2/3∼1.8. This would bring only a
+modest (logarithmic) increase in the sensitivity in the range q > qthrbut would aid linearly
+forq < qthr.
+5.4. Constraints on Migration Scenarios
+We showed in Section 5.2 that the planet density derived here, dNpl/dlogqdlogs=
+0.36 dex−2is consistent with the density derived from RV studies, if the latter a re extrap-
+olated to ∼25 times the semi-major axis where the measurement is made. Regar dless of
+the details of this comparison, the fact that the density of planets beyond the snow line is
+7 times higher than that at 0.3 AU, indicates that most giant planets do not migrate very
+far. Moreover, the fact that the slope found in RV studies at small sadequately predicts the
+density at large s, would seem to imply that whatever is governing the amount of migrat ion
+is acontinuous parameter. That is, it is not the case that there are two classes of planetary
+systems: those with migration and those without. Rather, all syst ems have migration, but
+by a continuously varying amount. This picture would be in accord with the evolving view
+of the Solar System, that even though the giant planets are in the g eneral area of their birth
+“beyond the snow line”, they have migrated to a modest degree.
+5.5. Comparison to Solar System
+Another interesting point of comparison is to the planet density in th e Solar System,
+where there are 4 planets in the mass-separation regime that micro lensing currently ex-
+plores, Jupiter, Saturn, Uranus, and Neptune. How common are “ Solar-System analogs”,– 28 –
+i.e., systems with several giant planets out beyond the snow line?
+To address this question, we ask what the result of our study would have been if every
+microlensed star possessed a “scaled version” of our own Solar Sys tem in the following sense:
+4 planets with the same planet-to-star mass ratios and same ratios of semi-major axes as the
+outer Solar-System planets, but with the overall scale determined by the “snow line”. While
+there is observational evidence from the asteroid belt that the So lar-System snow line is
+nearRsnow,⊙= 2.7 AU (Morbidelli et al. 2000), there is considerable uncertainty on ho w this
+scales with stellar mass (Sasselov & Lecar 2000; Kennedy & Kenyon 2 008). We therefore
+parametrize this relation by
+Rsnow(M) =Rsnow,⊙/parenleftBigM
+M⊙/parenrightBigν
+, (15)
+and consider a range 0 .5< ν < 2, adopting ν= 1 for our fiducial value. We consider that
+the typical Einstein radius is RE= 3.5 AU (M/M⊙)1/2and the typical lensed star is M=
+0.5M⊙. Then scaling down the semi-major axis of a Jupiter analog so aJup−analog/Rsnow=
+aJup/Rsnow,⊙implies
+aJup−analog
+RE=aJup
+3.5 AU/parenleftBigM
+M⊙/parenrightBigν−0.5
+, (16)
+and similarly for the other 3 planets. We then imagine that these syst ems are viewed at
+random orientations, with the individual planets in random phases. F rom a Monte Carlo
+simulation, we find that we would then have expected (for our fiducia lν= 1) to have detected
+18.2 planets (11.4 “Jupiters”, 6.4 “Saturns”, 0.3 “Uranuses”, 0.2 “ Neptunes”) including 6.1
+systems with 2 or more planet detections. See Figure 10. For 1 < ν < 2, the planet totals and
+multi-planet detections barely change. However they fall somewha t for smaller ν, reaching
+15.3 planets and 3.8 two-planet systems at ν= 0.5.
+These results compare to 6 planets including 1 two-planet system in t he actual sample.
+Hence, our Solar System appears to be 3 times richer in planets than other stars along
+the line of sight toward the Galactic bulge. The single detection of a mu lti-planet sys-
+tem (Gaudi et al. 2008) allows the first estimate of the frequency o f stars with “solar-like
+systems”, defined as having multiple giants in the snow zone: 1/6, alb eit with large errors.
+Work by AG was supported in part by NSF grant AST 0757888 and in pa rt by NASA
+grant 1277721 issued by JPL/Caltech. Work by SD was performed u nder contract with the
+California Institute of Technology (Caltech) funded by NASA throu gh the Sagan Fellowship
+Program. This work was supported in part by an allocation of comput ing time from the– 29 –
+Ohio Supercomputer Center. The OGLE project is partially support ed by the Polish MNiSW
+grant N20303032/4275 to AU.The MOA collaboration was supported by the Marsden Fund
+of New Zealand. The MOA collaboration and a part of authors are sup ported by the Grant-
+in-Aid for Scientific Research, JSPS Research fellowships and the Glo bal COE Program
+”Quest for Fundamental Principles in the Universe” from JSPS and M EXT of Japan. TS
+acknowledges support from grants JSPS18749004, JSPS207401 04, and MEXT19015005. The
+RoboNet project acknowledges support from PPARC (Particle Phy sics and Astronomy Re-
+search Council) and STFC (Science and Technology Facilities Council). CH was supported
+by Creative Research Initiative Program (2009-0081561) of Natio nal Research Foundation
+of Korea. AGY’s activity is supported by a Marie Curie IRG grant from the EU, and by
+the Minerva Foundation, Benoziyo Center for Astrophysics, a res earch grant from Peter and
+Patricia Gruber Awards, and the William Z. and Eda Bess Novick New Scie ntists Fund at
+the Weizmann Institute. DPB was supported by grants AST-07088 90 from the NSF and
+NNX07AL71G from NASA. Work by SK was supported at the Technion by the Kitzman
+Fellowship and by a grant from the Israel-Niedersachsen collaborat ion program. Mt Canopus
+observatory is financially supported by Dr David Warren.
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+Table 1: Monitored Events with Magnification A > 100
+Name Amaxt0(HJD) tEM/M⊙ Method
+OGLE-2007-BLG-224 2424 4233.7 7 0 .056±0.004M=θE/κπE
+OGLE-2008-BLG-279 1600 4617.3 101 0 .64±0.10M=θE/κπE
+OGLE-2005-BLG-169 800 3491.9 43 0 .49+0.23
+−0.29 GM⊕θE⊕tE
+MOA-2007-BLG-400 628 4354.6 14 0 .30+0.19
+−0.12 GM⊕θE⊕tE
+OGLE-2007-BLG-349 525 4348.6 121 ∼0.4 M=θE/κπE
+OGLE-2007-BLG-050 432 4222.0 68 0 .50±0.14M=θE/κπE
+MOA-2008-BLG-310 400 4656.4 11 ≤0.67±0.14 AO
+OGLE-2006-BLG-109 289 3831.0 127 0 .51+0.05
+−0.04 M=θE/κπE,AO
+OGLE-2005-BLG-188 283 3500.5 14 0 .16+0.21
+−0.08 GM⊕θE⊕tE
+MOA-2008-BLG-311 279 4655.4 18 0 .20+0.26
+−0.09 GM⊕θE⊕tE
+MOA-2008-BLG-105 267 4565.8 10
+OGLE-2006-BLG-245 217 3885.1 59
+OGLE-2006-BLG-265 211 3893.2 26
+OGLE-2007-BLG-423 157 4320.3 29
+OGLE-2005-BLG-417 108 3568.1 23
+Table 2: Planets in Densely Monitored High-mag Events
+Name log u0logρlogq|logs| |logsmax|
+OGLE-2005-BLG-169Lb -2.9 -3.4 -4.1 0.009 0.19
+OGLE-2006-BLG-109Lb -2.5 -3.5 -2.9 0.20 0.39
+OGLE-2006-BLG-109Lc -2.5 -3.5 -3.3 0.017 0.27
+OGLE-2007-BLG-349Lb -2.7 -3.3 -3.5 0.099 0.48
+MOA-2007-BLG-400Lb -3.6 -2.5 -2.6 0.47 0.55
+MOA-2008-BLG-310Lb -2.5 -2.3 -3.5 0.035 0.14– 34 –
+1/Amaxcumulative number
+0 .002 .004 .006 .008 .01051015
+Fig. 1.— Cumulative distribution of inverse maximum magnification A−1
+maxfor high-mag
+events observed by µFUN during the years 2005-2008. The distribution is uniform in A−1
+max
+for high-mag events Amax>200, with a slope of dNev/dA−1
+max= 2600 ( dashed line ). A
+“controlled experiment” therefore requires a selection criterion Amax>200. See Section 2.1.– 35 –
+Fig. 2.— Triangle diagram for OGLE-2007-BLG-050 (adapted from Fig . 9 of Batista et al.
+2009) showing sensitivity to planets of planet/star mass ratio qand planet-star projected
+separation d(aliassin the current paper). Planets within the contours would be detect able
+at ∆χ2>500 for 25%, 50%, 75% and 90% of source-lens trajectories. Since these trajectories
+are random, the contours reflect the probability of detecting the planet at a given sandq.
+The slope of the contours is η= 0.32.– 36 –
+Fig. 3.— Triangle diagram of planet sensitivities for OGLE-2008-BLG-2 79 (adapted from
+Fig. 7 of Yee et al. 2009). Similar to Fig. 3 except with contours of ∆ χ2>500 detectability
+with 10%, 25%, 50%, 75% 90% and 99% probability. Contour slope is η= 0.35.– 37 –
+0 1 0 1 0 1 0 1
+Fig. 4.— New “triangle diagram” sensitivities for all events in our sample except the two
+shown in Figures 2 and 3. Sensitivity is the fraction of all trajectory anglesαthat a planet
+would have been detected at ∆ χ2>500, if a planet of mass ratio qhad been present at
+projected separation s(in units of the Einstein radius). Only half of the diagram (which is
+almost perfectly symmetric) is shown here to conserve space. Pos itions (log q,|logs|) of all
+detected planets from Table 2 are shown as gold stars.– 38 –
+log qsensitivity
+−6 −5 −4 −3 −20.511.52
+OB07224 2424
+OB08279 1600
+OB05169  800
+MB07400  628
+OB07349  525
+OB07050  432
+MB08310  400
+OB06109  289
+MB05188  283MB08311  279
+MB08105  267
+OB06245  217
+OB06265  211
+Total/10η=0.32
+Fig. 5.— Calculated sensitives for each of the 13 microlensing events. These are integrals
+over horizontal cuts (at fixed q) in Figures 2, 3, and 4. Events are shown in “rainbow order”
+according to magnification, with the first 7 in solid lines and the last 6 in d ashed lines. The
+bold dashed black curve represents the total sensitivity of the sa mple (divided by 10).– 39 –
+|log s|/|log s|maxcumulative number
+0 .2 .4 .6 .8 10246
+Fig. 6.— Cumulative distribution of the (absolute value of the log of the ) measured projected
+separation, |logs|, in units of the maximum value of this parameter, derived in Section 4.1
+The observed distribution is consistent with planets being distribute d uniformly in log s,
+with Kolmogorov-Smirnov probability of 20%.– 40 –Cumulative
+−4.5 −4 −3.5 −3 −2.5 −20246
+log qG(q)
+−4.5 −4 −3.5 −3 −2.5 −205101520
+NumericalAnalytic
+Fig. 7.— Principal inputs to the modeling. Top panel: cumulative distribu tion of planet
+detections in 4 years of intensive monitoring of high-magnification ev ents. Bottom panel:
+Sensitivity of the survey as a function of planet/star mass ratio q, both the analytic approx-
+imation derived in Section 3.2 and the numerical determination derived in Section 4.1 and
+shown in Figure 5. These hardly differ.– 41 –
+Fig. 8.— Planet frequencies determined from microlensing (this paper ) and RV
+(Cumming et al. 2008) at different semi-major axes. The RV result is s caled to mass ra-
+tioq= 5×10−4, using the RV-derived slope, n=−0.31. In order to take account of the
+different host star masses ( M∼1M⊙for RV,M∼0.5M⊙for microlensing) we have placed
+the microlensing point at 8 AU, i.e., 3 times the solar-system “snow line” distance. This
+is because microlensing planets are typically detected at 3 times the d istance of their own
+systems’ snow line (which is of course much closer than 8 AU).– 42 –
+Fig. 9.— Planet frequencies as a function of mass ratio for microlensin g (this paper) and RV
+(Cumming et al. 2008; Mayor et al. 2009; Johnson et al. 2010) detec tions. As illustrated in
+Fig. 8, the Cumming et al. (2008) RV sample and the microlensing sample are consistent with
+each other (despite different frequencies in this figure) because t hey are at different distances.
+The Cumming et al. (2008) and Mayor et al. (2009) RV samples are dire ctly comparable on
+this diagram because both are G stars, and they are consistent. H owever, there is some ten-
+sion between the Johnson et al. (2010) RV measurement and the mic rolensing measurement,
+since both are similar type stars. This is because Cumming et al. (2008 ) and Johnson et al.
+(2010) are “inconsistent” with each other, if one assumes that th e frequency of planets as
+a function of mass ratio is independent of host mass, as we have don e in arguing for the
+consistency of microlensing with Cumming et al. (2008). Also shown is t he (unnormalized)
+slope derived from microlensing observations by (Sumi et al. 2010).– 43 –
+Fig. 10.— Results of Monte Carlo simulation of the anticipated results o f our microlensing
+survey assuming that every star toward the Galactic bulge had a “s caled solar system”, as
+specified by Eqs. (15) and (16). A total of 18 planets would have be en detected, including
+6 two-planet systems, compared to the actual detections of 6 pla nets including 1 two-planet
+system. This seems to indicate that the solar system is overdense in planets, especially
+multiple planets. Also shown are the frequency of various specific co mbinations.
\ No newline at end of file
diff --git a/1001.0573.txt b/1001.0573.txt
new file mode 100644
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@@ -0,0 +1,461 @@
+arXiv:1001.0573v1  [cond-mat.supr-con]  4 Jan 2010Theory of time-resolved spectral function in high-tempera ture superconductors with
+bosonic modes
+Jianmin Tao and Jian-Xin Zhu
+Theoretical Division & CNLS, Los Alamos National Laborator y, Los Alamos, New Mexico 87545
+(Dated: November 21, 2018)
+We develop a three-temperature model to simulate the time de pendence of electron and phonon
+temperatures in high-temperature superconductors displa ying strong anistropic electron-phonon
+coupling. This model not only takes the tight-binding band s tructure into account, but also is
+valid in superconducting state. Based on this model, we calc ulate the time-resolved spectral func-
+tion via the double-time Green’s functions. We find that the d ip-hump structure evolves with the
+time delay. More interestingly, new phononic structures ar e obtained when the phonons are excited
+by a laser field. This signature may serve as a direct evidence for electron-vibration mode coupling.
+PACS numbers: 74.25.Jb, 74.72.-h, 71.38.-k, 79.60.-i
+The discovery of high-temperature superconductors
+(HTSC) has raised an important issue on the mechanism
+leading to the formation of Cooper pairs, which is still
+under debate. To address this issue, a number of spec-
+troscopy techniques [1–3] have been used to study the
+role and nature of bosonic modes, to which electrons are
+strongly coupled. In particular, due to technological ad-
+vances and the improved sample quality, angle-resolved
+photoemission spectroscopy (ARPES) [1] has been used
+to probe details of the energy and momentum struc-
+ture of single-particle excitations via the measurement
+of photoemission intensity. However, different interpre-
+tations of the same data may lead to completely different
+mechanisms [4, 5]. For example, the dip-hump structure
+observed in ARPES [6–13] could be interpreted [14] as
+naturnallyoccurringintheinteractingsystembuthaving
+little effect on the superconducting pairing mechanism.
+Itcouldalsobeinterpreted[15]intermsofphononmodes
+that could drive d-wave pairing. Recent theoretical anal-
+ysis [16] on the spectral function of thermally excited
+electrons in the cuprates has shown that the out-of-plane
+and out-of-phase buckling mode strongly couples to the
+electronicstatesnearthe anti-nodal Mpoints inthe Bril-
+louin zone, while the in-plane breathing mode couples
+strongly to the electronic states near the d-wave nodal
+points. The signature of the anistropic electron-phonon
+(el-ph) coupling seems to get enhanced in the supercon-
+ducting state, suggesting the significant role of the el-ph
+interaction in the superconducting mechanism.
+The information extracted from the conventional
+ARPES is limited. Time-resolved ARPES offers the
+capability to simultaneously capture the single-particle
+(frequency domain) and collective (time domain) infor-
+mation, thus making it possible to directly probe the link
+between the collective modes and single-particle states.
+In this setting, either electrons or lattice vibrational
+modes can be selectively excited with an ultrafast laser
+pulse. Recent applications of this technique include the
+studies of transient electronic structure in Mott insula-
+tors [17, 18] and high- Tccuprates [19] with optical pump.Furthermore, direct pumping of vibrational mode has
+also been realized in manganites [20], though not yet in
+the cuprates. Motivated by the thrust of this experi-
+mental technique, in this Letter, we aim to provide a
+theoretical underpinning of transient electronic structure
+for HTSC, in a hope to better understand the nature of
+bosonic modes in these systems. As such, the time evolu-
+tion of the el-ph coupling is investigated for both normal
+and superconducting states with the time-resolved spec-
+tral function. Our theory consists of two parts. First, we
+develop a three-temperature model to simulate the time
+dependence of the electron and phonon temperatures.
+Then, based on the three-temperature model, we calcu-
+latethetime-resolvedspectralfunction. Ourresultsshow
+a kink-structure in the time dependence of the electronic
+temperature at the superconducting transition temper-
+ature. Accordingly, we find from the spectral density
+that the energy position of the phonon mode is offset by
+a time-dependent gap function. More interestingly, new
+signatures of the el-ph coupling, which are absent when
+electrons are excited, can be observed in the case of se-
+lective excitation of phonons. This signature may serve
+as a direct evidence for the electron-vibration mode cou-
+pling as opposed to the coupling between electrons and
+spin fluctuations.
+Three-temperature model : Consider a two-dimensional
+superconductor exposed to a laser field. The model
+Hamiltonian for a vibrational mode νcan be written as
+H=/summationdisplay
+kσξkc†
+kσckσ+/summationdisplay
+k(∆kc†
+k↑c†
+−k↓+h.c.)+/summationdisplay
+q/planckover2pi1Ωνq
+×/parenleftbigg
+b†
+νqbνq+1
+2/parenrightbigg
++1√NL/summationdisplay
+kqσgν(k,q)c†
+k+q,σckσAνq
++Hfield(τ), (1)
+wherec†
+kσ(b†
+νq) andckσ(bνq) are the creation and anni-
+hilation operators for an electron with momentum kand
+spinσ(phonon with momentum qand vibrational mode
+ν),Aνq=b†
+−νq+bνq, the quantity ξkis the normal-state
+energy dispersion, µthe chemical potential, ∆ kthe gap2
+function, and gνthe coupling matrix.
+By performing the Bogoliubov-de Gennes transforma-
+tion [21], ck↑=ukαk−vkβ†
+kandc−k↓=ukβk+vkα†
+k,
+we obtain Ee=/summationtext
+kEk(/an}bracketle{tα†
+kαk/an}bracketri}ht − /an}bracketle{tβkβ†
+k/an}bracketri}ht), where Ek=/radicalbig
+ξ2
+k+∆2
+k. The time evolution of /an}bracketle{tα†
+kαk/an}bracketri}htand/an}bracketle{tβkβ†
+k/an}bracketri}htis
+calculated using the Heisenberg equations-of-motion ap-
+proach. They are found to be
+∂/an}bracketle{tα†
+kαk/an}bracketri}ht
+∂τ=2π
+NL/summationdisplay
+qg2
+ν(ukuk−q−vkvk−q)2(δ2eβeΩ0−δ1)
+×/bracketleftbigg
+e(βph−βe)Ω0−1/bracketrightbigg
+(1−fk)fk−qNΩ0,(2)
+and∂/an}bracketle{tβkβ†
+k/an}bracketri}ht/∂τ=−∂/an}bracketle{tα†
+kαk/an}bracketri}ht/∂τ, wherefk=f(Ek) =
+1/(eβeEk+ 1),NΩ0=N(Ω0) = 1/(eβphΩ0−1),δ1=
+δ(Ek−q−Ek+Ω0), andδ2=δ(Ek−q−Ek−Ω0). Here
+we have specifically set Ω ν= Ω0. Differentiation of both
+sides of the expression for Eeand substitution of Eq. (2)
+leads to the rate of energy exchange
+∂Ee
+∂τ=4π
+NL/summationdisplay
+qg2
+ν(ukuk−q−vkvk−q)2δ(Ek−q−Ek−Ω0)
+×Ω0/bracketleftbigg
+e(βph−βe)Ω0−1/bracketrightbigg
+fk(1−fk−q)NΩ0. (3)
+Recent experiment [19] shows that there exists cold lat-
+tice, which is negligibly coupled to the electrons but dis-
+sipates the energy of hot phonons via anharmonic cool-
+ing. Considering this observation, we write a set of rate
+equations for the electron, hot phonon, and lattice tem-
+peratures as
+∂Te
+∂τ=1
+Ce∂Ee
+∂τ+Pe
+Ce, (4)
+∂Tph
+∂τ=−1
+Cph∂Ee
+∂τ+Pph
+Cph−Tph−Tl
+τβ,(5)
+∂Tl
+∂τ=/parenleftbiggCph
+Cl/parenrightbiggTph−Tl
+τβ, (6)
+wherePeis the power for pumping electrons and Pphthe
+power for pumping hot phonons. The specific heat of
+electrons can be calculated from the Boltzmann entropy
+Se=−2kB/summationtext
+k{[1−f(Ek)]ln[1−f(Ek)]+f(Ek)lnf(Ek)}
+byCe=Te∂Se/∂Te, while the specific heat of hot
+phonons for one-vibrationalmode can be calculated from
+the simple relationship Cph=/planckover2pi1Ω∂N(Ω)/∂Tph|Ω=Ω0.
+The results are given by
+Ce=βkB/summationdisplay
+k/parenleftbigg
+−∂f(Ek)
+∂Ek/parenrightbigg/parenleftbigg
+2E2
+k+β∆k∂∆k
+∂β/parenrightbigg
+,(7)
+Cph=kB
+4/parenleftbigg/planckover2pi1Ω0
+kBTph/parenrightbigg2/bracketleftbigg
+coth2/parenleftbigg/planckover2pi1Ω0
+2kBTph/parenrightbigg
+−1/bracketrightbigg
+,(8)
+respectively, with coth x≡(ex+e−x)/(ex−e−x). Equa-
+tions (3)-(8) constitute our three-temperature model. 0 0.2 0.4 0.6 0.8 1 1.2
+-100 0 100 200 300 400 500 600Temperature (T)
+τTe
+Tph
+Tl(a)
+ 0 0.05 0.1 0.15 0.2
+-4.5-3.5-2.5-1.5
+τTeTph
+TlTc=0.06
+ 0 0.2 0.4 0.6 0.8
+-600-200 200 600 1000Temperature (T)
+τTe
+TphTl
+Tc = 0.06(b)
+FIG. 1: (Color) Time evolution of electron ( Te), phonon
+(Tph), and lattice ( Tl) temperatures for selectively exciting
+electrons (a) and phonons (b) with a laser pulse.
+The original version of this model was phenomeno-
+logically proposed [19] as an extension of the two-
+temperaturemodel[22] forthe normalstate. Thepresent
+model has the following advantages: (i) it incorporates
+the detailed band structure, (ii) it is valid for bothnor-
+mal and superconducting states, and (iii) it includes the
+anistropic effect on the el-ph coupling.
+Now we apply our three-temperature model to sim-
+ulate the time evolution of the electron, hot phonon,
+and lattice temperatures for a d-wave superconductor.
+We use a five-parameter tight-binding model [23] to
+describe the energy dispersion: ξk=−2t1(coskx+
+cosky)−4t2coskxcosky−2t3(cos2kx+ cos2ky)−
+4t4(cos2kxcosky+coskxcos2ky)−4t5cos2kxcos2ky−µ,
+wheret1= 1,t2=−0.2749,t3= 0.0872,t4= 0.0938,
+t5=−0.0857, and µ=−0.8772. Unless specified explic-
+itly, the energy is measured in units of t1and the time
+is measured in units of /planckover2pi1/t1hereafter. (For t1= 150
+meV, it corresponds to 1740 Kelvin in temperature and
+/planckover2pi1/t1to 4.4 femtosecond in time.) The d-wave gap func-
+tion has the form ∆ k= ∆0(Te)(coskx−cosky)/2. The
+temperature-dependent part is given by [24] ∆ 0(Te) =
+∆00tanh{(π/z)/radicalbig
+ar(Tc/Te−1)}, wherez= ∆00/(kBTc)
+and tanh x= 1/cothx. In our calculations, we
+set ∆ 00= 0.2, the critical temperature Tc= 0.06,
+the specific heat jump at Tcisr= ∆Ce/Ce∼
+1.43, and a= 2/3. In this work, we focus on
+the buckling phonon mode, for which [16, 25, 26]
+gB1g=g0{[cos2(qx/2)+cos2(qy/2)]/2}−1/2{φx(k)φx(k+
+q)cos(qy/2)−φy(k)φy(k+q)cos(qx/2)}withφx=
+(i/Nk)[ξktx,k−txy,kty,k],φy= (i/Nk)[ξkty,k−txy,ktx,k],
+Nk= [(ξ2
+k−t2
+xy,k)2+ (ξktx,k−txy,kty,k)2+ (ξkty,k−
+txy,ktx,k)2]1/2,tα,k=−2t1sin(kα/2), and txy,k=
+−4t2sin(kx/2)sin(ky/2). To be consistent with experi-
+ment [16, 19], we set g0= 0.4 and the mode frequency
+Ω0= 0.3. The relaxation time τβ= 200. The pump
+poweris a Gaussianpulse of P=P0e−τ2/(2σ2), forwhich,
+the corresponding FWHM (full width at half maximum)
+is 2.35σ. To pump electrons, we set P0= 0.15 and
+σ= 1. Considering that the energy scale of phonons is
+much smaller than that of electrons, we set P0= 0.006
+andσ= 100 for pumping phonons. The ratio Cph/Clin
+Eq. (6) is set to be 0.2.3
+Figure 1 displays the temporal evolution of three re-
+spective temperatures for pumping electrons (a) and hot
+phonons (b). Before pumping, all three types of de-
+grees of freedom (DoF) are in equilibrium, which is set at
+Te=Tph=Tl= 0.01. As shown in Fig. 1(a), when the
+pump pulse is absorbed by the electrons, the electronic
+temperature rises steeply around τ= 0, and reaches the
+maximum aftera smalltime delay. It then begins to drop
+atatimescaledeterminedbytheel-phcouplingstrength,
+followed by a slower relaxation. Interestngly, we also ob-
+serve a small kink in TeatTe=Tc, which entirely arises
+from the breakup of the Cooper pairs, resulting in the
+dramatic change in the rise rate of Te. This kink was not
+captured in the earlymodel [19]. Simultaneously, the hot
+phononic temperature Tphrises smoothly via the energy
+exchange with electrons and then drops by exchanging
+energy with the cold lattice, causing the slight increase
+ofTl. When directly pumping phonons, the time depen-
+denceofthetemperaturesissimilarlyobserved,including
+the kink in Te, as shown in Fig. 1(b). Due to the large
+pumping width, the kink is more easily visible in this
+case.
+Time-resolved spectral function : The time-resolved
+spectralfunctionisdefinedas A(k,ω)≡ −2
+πImG11(k,ω).
+HereG11is the one-one component of the retarded
+Green’s function ˆG(k,ω), which is related to the self-
+energy by ˆG−1(k,ω) =ˆG−1
+0(k,ω)−ˆΣ(k,ω), with
+ˆG−1
+0(k,ω) =ωˆσ0−∆kˆσ1−ξkˆσ3. ˆσ0,1,2,3are the unit
+and Pauli matrices. As is known [16], in the equilibrium
+state with Te=Tph, the self-energy can be evaluated
+more conveniently within the imaginary-time Green’s
+function approach, in which a key step is to convert
+the Bose-Einstein distribution to the Fermi distribution,
+nB(iωn±Ek−q) =−nF(±Ek−q) (withωn= (2n+1)πT,
+T=Te=Tph). For the current situation, the temper-
+atures of electrons and hot phonons are no longer tied
+to each other and the above conversions are not valid
+any more. To avoid this restriction, here we apply the
+double-time Green’s function approach [27] to calculate
+ˆΣ(k,ω). The result is
+ˆΣ(k,ω) =1
+NL/summationdisplay
+q|gν(k,−q)|2
+{[(ω−Ω0)Φ1−(ω+Ω0)Φ2+Ω0(Φ3+Φ4)]ˆσ0
++[2Ek−q(Φ1−Φ3)+2Ω 0(Φ3−Φ4)]ˆσ1
++[ξk(Φ1−Φ2)+(Ω 0ξk/Ek)(Φ3−Φ4)]ˆσ3},(9)
+where Φ 1=N(Ω0)/[(ω−Ω0+Ek−q)(ω−Ω0−Ek−q)],
+Φ2=N(−Ω0)/[(ω+Ω0+Ek−q)(ω+Ω0−Ek−q)], Φ3=
+f(−Ek−q)/[(ω+Ω0−Ek−q)(ω−Ω0−Ek−q)], and Φ 4=
+f(Ek−q)/[(ω+Ω0+Ek−q)(ω−Ω0+Ek−q)].
+For simplicity, let us first look at the density of states
+(DOS), which is defined by ρ(ω) =1
+NL/summationtext
+kA(k,ω). For
+directpumpingofelectrons,wecalculatetheDOSattime
+sequences τ=−100,−2.7, 2.8, and 600, respectively.
+The results are plotted in Fig. 2 (a). At the initial time 0 0.4 0.8 1.2 1.6 2
+-1-0.5  0 0.5  1ρ(ω)
+ωτ = - 100
+-2.7
+2.8600
+dip 2dip 3 dip 1(a)
+ 0 0.4 0.8 1.2 1.6 2
+-1-0.5  0 0.5  1ρ(ω)
+ωτ = - 1000
+-158
+110800
+dip 2dip 3 dip 1(b)
+ 0.2 0.4 0.6 0.8
+-0.1 0 0.1
+ωτ = - 158
+FIG. 2: (Color) Time evolution of the density of states for se -
+lectively exciting electrons (a) and phonons (b), respecti vely.
+τ=−50 where Te= 0.01 and the system is at supercon-
+ducting state [∆ 0(Te) = 0.19], we observe two signatures
+ofthe el-phcouplinglocatedat ±(Ω0+∆0(Te)), asshown
+in Fig. 2 (dip 1 and dip 2). They are symmetric with re-
+spect to ω= 0, but dip 2 is much stronger. It is related
+to the fact that the normal-state van Hove singularity is
+located below the Fermi energy and as such the coher-
+ent peak at −∆0(Te) has stronger intensity than that at
+∆0(Te). In addition, dip 3 occurs near the characteris-
+tic energy −(EM+ Ω0), where EMis the quasiparticle
+energy at k= (π,0) or equivalent wavevector points in
+Brillouin zone. It arises from the van Hove singularity at
+k= (π,0). The signature at the energy EM+Ω0is much
+weaker also because of the van Hove singularity location
+in the normal state. At τ=−2.7 whereTe= 0.041 and
+∆0(Te)≈0.1, similar behaviors are observed. However,
+atτ= 2.8,Te= 1.05 and thus only the signature of the
+normal-state el-ph coupling (dip 3) can be observed. At
+τ= 600, a similar structure of the normal-state el-ph
+coupling is observed.
+For direct pumping of phonons, we choose different
+time sequences τ=−1000,−158, 110, and 800 to eval-
+uate the DOS. The results are displayed in Fig. 2(b).
+From Fig. 2(b), we observe that the dips suggesting
+the el-ph couplig in both normal and supercondcting
+states becomes stronger than those for selectively pump-
+ing electrons, as expected. In particular, we observe two
+new small dips at ω=±0.05 forτ=−158 (at which
+Te= 0.041 and Tph= 0.163), both of which arise from
+the poles of Σ k(ω) atω=±(Ω0−∆0(Te)), in addition to
+those observed in the case of electrons being excited di-
+rectly. This is because the hot phononic temperature be-
+comes very high while the electronic temperature is still
+cold and the contributions from the terms as weighted by
+N(±Ω0) are significantly enhanced. Figure. 2(b) clearly
+shows that, as time elapses from τ=−1000,−158, to
+110, the peaks move toward the zero energy , while from
+τ= 110 to 800, the locations of these two peaks remain
+unchanged. This can be understood by considering that,
+after pumping, Terises with the time delay, while ∆ 0(Te)
+decreases and rapidly vanishes when TereachesTc.
+Finallywenumericallyevaluatethetime-resolvedspec-
+tral function for the two excitations discussed above at
+those time sequences chosen for calculating the DOS. In4
+FIG. 3: (Color) Time-resolved spectral functionfor select ively
+pumping electrons (a 1-a4forτ=−100,−2.7,2.8,600) and
+phonons (b 1-b4forτ=−1000,−158,110,800), respectively.
+The vertical axis is kx/πat a fixed ky= 0.75πand the hori-
+zontal axis is energy. The black, red, and magenta lines rep-
+resent the energy location, −(EM+Ω0),−(∆0(Te)+Ω0), and
+−(Ω0−∆0(Te)), corresponding to the dip or hump location
+discussed in Fig. 2.
+our calculations, we consider cuts along kx-axis in the
+Brillouin zone at a kychosen near the zone boundary.
+Fig. 3 shows the snapshots of the image of the spec-
+tral function A(k,ω) as a function of kxandω. From
+Fig. 3(a 1,a2), we observe kinks at ω=−(∆0(Te) + Ω0)
+(red lines) and −(EM+ Ω0) (black lines) at τ=−100
+and−2.7. Theredlineisshifted inenergyastimeelapses
+fromτ=−100 toτ=−2.7 because the BCS gap ∆ 0(Te)
+drops significantly from 0.19 to 0.1. These kinks cor-
+respond to the dip structures discussed in Fig. 2. As
+τ >−2.3, the superconducting gap vanishes, so do the
+kinksmarkedwithredlines(seeFig.3(a 3-a4)). Thekinks
+marked with black lines persist through the whole time
+sequence, which can be ascribed to the el-ph coupling
+signature in the normal state. Therefore we have ev-
+ery reason to regard these kinks marked with red lines
+as the signature of the el-ph coupling in the supercon-ducting state. For hot phonon pumping (Fig. 3(b 1-b4)),
+in addition to those signatures of the el-ph coupling
+observed for electron pumping, new kink structure at
+−(Ω0−∆0(Te))(marked with magenta line in Fig. 3(b 2)
+is also observed. These observations are consistent with
+our finding in the DOS. In the viewpoint that the elec-
+tronic spin fluctuations arise from the strong correlation
+within the electronic DoF itself, and if one can assume
+that the spin fluctuations will ride on the electrons and
+thusitseffectivetemperaturewillbetiedtotheelectronic
+temperature, direct pumping of hot phonons will provide
+a unique way to differentiating the bosonic modes be-
+ing of electronic or phononic origin, to which electronic
+quasiparticles are strongly coupled in HTSC.
+In conclusion, we have developed a three-temperature
+model to simulate the time evolution of temperature for
+electrons, hot phonons, and the lattice. Our model takes
+both quasiparticle excitation and relaxation into account
+and thus is valid in both normal and superconducting
+states, because at the early stage of excitation, the sys-
+tem can be still in the supercondcting state. Based on
+this model, we derive a formula for the time-resolved
+spectralfunction, allowingus toinvestigatethe dynamics
+of the el-ph coupling for HTSC materials.
+Acknowledgments. We thank A. V. Balatsky, El-
+bert E. M. Chia, Hari Dahal, J. K. Freericks, M. Graf,
+A. Piryatinski, A. J. Taylor, S. A. Trugman, and D.
+Yarotski for valuable discussions. This work was sup-
+ported by the National Nuclear Security Administration
+of the U.S. DOE at LANL under Contract No. DE-
+AC52-06NA25396, the U.S. DOE Office of Science, and
+the LDRD Program at LANL.
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\ No newline at end of file
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+arXiv:1001.0574v2  [hep-ph]  8 Feb 2010A Two-Higgs Doublet Model With Remarkable CP Properties
+P. M. Ferreiraa,band Jo˜ ao P. Silvaa,c
+aInstituto Superior de Engenharia de Lisboa,
+Rua Conselheiro Em´ ıdio Navarro, 1900 Lisboa, Portugal
+bCentro de F´ ısica Te´ orica e Computacional,
+Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, P ortugal
+cCentro de F´ ısica Te´ orica de Part´ ıculas,
+Instituto Superior T´ ecnico, P-1049-001 Lisboa, Portugal
+October 30, 2018
+Abstract. We analyze generalized CP symmetries of two-Higgs doublet m odels, extending
+them from the scalar to the fermion sector of the theory. We sh ow that, with a single exception,
+those symmetries imply massless fermions. The single model which accommodates a fermionic
+mass spectrum compatible with experimental data possesses a remarkable feature. It displays a
+new type of spontaneous CP violation, which occurs not in the scalar sector responsible for the
+symmetry breaking mechanism but, rather, in the fermion sec tor.
+1 Introduction
+Although the Standard Model (SM) of electroweak interactio ns has had extraordinary exper-
+imental confirmation, the Higgs mechanism of the symmetry br eaking responsible for giving
+mass to the particles remains largely untested. In particul ar, there is no fundamental reason
+why the SM should have only one Higgs. In fact, one of the simpl est extensions of the SM is the
+Two-Higgs Doublet Model (THDM) proposed by Lee [1], where th ere are five scalar particles
+(three neutral and one charged). One of the main features of t his model, and the main reason
+Lee proposed it, is that it allows for the possibility of the v acuum of the theory spontaneously
+breaking the CP symmetry .
+The most general scalar potential in the THDM contains 14 rea l parameters, as opposed to
+the2intheSMpotential. Thereisthusagreatinterestinred ucingthenumberoffreeparameters
+in the THDM, through the imposition of some symmetry in the th eory. One such class of
+symmetriesaretheso-called generalizedCP(GCP)transfor mations-fieldtransformationswhich
+combine CP with unitary transformations. As far as we know, G CP were first discussed in [2].
+Their explicit use for quarks appeared in [3] and GCP in the sc alar sector was initially developed
+by the Viena group in [4, 5, 6]. GCP symmetries are quite inter esting in that they reduce
+immensely to number of free parameters in the potential. Rec ently [7, 8] it was shown that,
+from the point of view of the scalar potential, these GCP symm etries fall unto three categories,
+designated in [8] by CP1, CP2 and CP3. The CP1 class correspon ds to the standard CP
+transformation. The CP3 models, unlike CP1 and CP2, corresp ond to a continuous symmetry.
+In this paper we will show that CP2 cannot be extended from the scalar sector to the
+fermionic one without forcing at least one fermion to have ze ro mass. Likewise, it will be
+shown that, out of the infinite number of possible CP3 models, only one can be extended to the
+1fermionic sector without inducing zero fermion masses. Thu s we are able to drastically curtail
+the number of models which includes an acceptable fermion se ctor.
+Having found only one acceptable GCP THDM, we will then proce ed to analyze its CP
+properties. We conclude that the model has a new type of spont aneous CP breaking. In this
+model, CP violation does not emerge from explicit CP breakin g in the Yukawa sector, as it
+does in the SM; nor does it emerge from spontaneous CP violati on in the scalar sector, induced
+by the vacuum of the model, as in the Lee model. Rather, in the m odel herein presented, the
+lagrangian is explicitly CP conserving and the scalar secto r preserves CP, even after spontaneous
+symmetry breaking. Nonetheless, the vacuum expectation va lues generated by the scalar sector
+induce a CP violating phase in the quark sector. As far as we kn ow, this is the first illustration
+in the literature of this peculiar type of spontaneous CP vio lation.
+This paper is structured as follows: in section 2 we present t he scalar potential of the theory
+and review recent results on its possible symmetries, as wel l as the impact that GCP has on its
+parameters. In section 3 we show how GCP can be extended to the fermion sector, and we prove
+that, with a single exception, GCP always implies massless f ermions; we also perform a fast fit
+to experimental constraints. In section 4 we discuss the CP p roperties of this model, and show
+that it has a new type of spontaneous CP breaking. We present o ur conclusions in section 5.
+2 The scalar potential
+Let us first review the scalar sector of the THDM, and what is kn own of its symmetries. In
+the THDM one introduces two hypercharge Y= 1/2 Higgs scalar doublets. The most general
+THDM scalar potential which is renormalizable and compatib le with the gauge symmetries of
+the SM is given by
+VH=m2
+11Φ†
+1Φ1+m2
+22Φ†
+2Φ2−/bracketleftBig
+m2
+12Φ†
+1Φ2+H.c./bracketrightBig
++1
+2λ1(Φ†
+1Φ1)2+1
+2λ2(Φ†
+2Φ2)2
++λ3(Φ†
+1Φ1)(Φ†
+2Φ2)+λ4(Φ†
+1Φ2)(Φ†
+2Φ1)+/bracketleftBig
+1
+2λ5(Φ†
+1Φ2)2+λ6(Φ†
+1Φ1)(Φ†
+1Φ2)
++λ7(Φ†
+2Φ2)(Φ†
+1Φ2)+H.c./bracketrightBig
+, (1)
+and it involves 14 parameters. Here m2
+11,m2
+22, andλ1,···,λ4are real parameters, while, in
+general,m2
+12,λ5,λ6andλ7are complex. “H.c.” stands for Hermitian conjugation.
+This large number of parameters may be reduced by requiring t hat the model be invariant
+for some type of symmetry. Namely, we may require invariance for field transformations of the
+form
+Φa→SabΦb, (2)
+for anSmatrix belonging to U(2), which are known as Higgs family symmetries. Another pos -
+sibility1consists on the so-called generalized CP symmetries [2, 3, 4 , 5, 6], requiring invariance
+under the following field transformations:
+Φa→XabΦ∗
+b, (3)
+withX∈U(2) as well. Each of these symmetries corresponds to a differen t model, with different
+phenomenology. One may wonder how many different choices of scalar potentials can one make
+in the THDM. Recently it has been shown that applying the symm etries with any possible
+1Both field transformations discussed here obey a basic requi rement, namely leaving the kinetic terms of the
+lagrangian unaffected.
+2choices for SandXleads only to six classes of scalar potentials [7, 8]. The imp act of the six
+classes on the parameters are given in Table 1, for specific ba sis choices. This result is already
+Table 1: Impact of the symmetries on the coefficients of the Hig gs potential in a specific basis.
+See Ref. [8] for more details.
+symmetry m2
+11m2
+22m2
+12λ1λ2λ3λ4 λ5 λ6λ7
+Z2 0 0 0
+U(1) 0 0 0 0
+U(2) m2
+110 λ1λ1−λ3 0 0 0
+CP1 real real real real
+CP2 m2
+110 λ1 −λ6
+CP3 m2
+110 λ1 λ1−λ3−λ4(real) 0 0
+extremely important: there are only six possible different ty pes of physical models in the scalar
+sector. The first three may be obtained from family symmetrie s alone, and are not the subject
+of this work. The next three may be obtained from GCP symmetri es alone, and we will study
+them in detail. Let us begin by explaining what the difference i s between them. Since any basis
+of fields {Φ1,Φ2}has to give the same physical results, we will use that libert y of basis choice
+to simplify the GCP field transformations of Eq. (3). In fact i t has been shown that there is
+always a basis of scalar fields, for which the most general GCP transformation matrix Xmay
+be brought to the form
+Xθ=/bracketleftbiggcθsθ
+−sθcθ/bracketrightbigg
+, (4)
+where 0≤θ≤π/2 [9]. Henceforth, c= cos,s= sin and the subindices indicate the angle. The
+classes CP1, CP2, and CP3, arise respectively from X0,Xπ/2, andXθfor any one (or several)
+θexcluding 0 and π/2. CP1 is the usual CP symmetry. CP2, like CP1, is a discrete sy mmetry,
+although it eliminates far more parameters in the potential than CP1 does. CP3 corresponds to
+an infinite number of possibilities, since the angle θvaries in a continuous interval. We will now
+analyze the extension of these symmetries to the fermion sec tor. Although we will only look at
+the quark sector, the same conclusions can be trivially exte nded to leptons.
+3 Yukawa interactions
+In the previous section we have seen that there are only three classes of models with GCP
+symmetries. However, that study considered only the scalar sector, not the whole lagrangian.
+We will now show how GCP can be extended to the fermion sector, and what consequences arise
+thereof. The scalar-quark Yukawa interactions of the THDM m ay be written as
+−LY= ¯qL(Γ1Φ1+Γ2Φ2)nR+ ¯qL(∆1˜Φ1+∆2˜Φ2)pR+H.c., (5)
+whereqL= (pL,nL)⊤(nRandpR) is a vector in the 3-dimensional generation space of left-
+handed doublets (right-handed charge −1/3 and +2 /3) quarks. Γ 1, Γ2, ∆1, and ∆ 2are 3×3
+matrices.
+We now wish to impose GCP on scalars and also on fermions. For t he quark fields, the GCP
+transformations take the form
+qL→Xαγ0Cq∗
+L,
+nR→Xβγ0Cn∗
+R,
+pR→Xγγ0Cp∗
+R, (6)
+3whereγ0(C) is the Dirac (charge-conjugation) matrix. As was the case w ith the scalar GCP
+transformations, we can also simplify the above with an appr opriate basis choice for the fermion
+fields. In fact, we may take Xto the simple form [9]
+Xα=
+cαsα0
+−sαcα0
+0 0 1
+, (7)
+where 0≤α≤π/2, and similarly for XβandXγ. GCP is a good symmetry of LYif and only
+if Γ∗
+b= (Xθ)abX†
+αΓaXβ, which we may write as
+XαΓ∗
+1−(cθΓ1−sθΓ2)Xβ= 0,
+XαΓ∗
+2−(sθΓ1+cθΓ2)Xβ= 0. (8)
+Since they are complex, this gives us 36 equations in the 36 un known real and imaginary parts
+of the various entries of the Γ matrices. Because of Eq. (7), t hey break into four blocks, which
+we designate by mn,m3, 3n, and 33, where mandncan take the values 1 or 2. In each block we
+have a system of homogeneous linear equations; the paramete rs are zero unless the determinant
+of the system vanishes. For instance, the real (Re) and imagi nary (Im) parts of (Γ 1)33and (Γ 2)33
+in Eq. (8) give the following set of linear equations:
+/bracketleftbigg1−cθsθ
+−sθ1−cθ/bracketrightbigg /bracketleftbiggRe(Γ1)33
+Re(Γ2)33/bracketrightbigg
+= 0,
+/bracketleftbigg1+cθ−sθ
+sθ1+cθ/bracketrightbigg /bracketleftbiggIm(Γ1)33
+Im(Γ2)33/bracketrightbigg
+= 0. (9)
+The determinant of the first of these matrices gives 2(1 −cθ), whereas the second one is equal to
+2(1+cθ). Because of the limited range of variation of θ, the second determinant is never zero -
+which means that the Γ 33coefficients will always be real. As for the first of these deter minants,
+it is only zero if θ= 0 (the standard CP definition). For any other case, the 33 ent ries of the Γ
+matrices will be zero.
+A similar reasoning applies to the m3 entries. Eq. (8) gives us a set of 4 ×4 equations on the
+real and imaginary parts of the Yukawa couplings, with corre sponding determinants given by
+4(cα−cθ)2and 4(cα+cθ)2, respectively. Since αhas the same range of variation than θ, these
+determinants only vanish in very specific situations; they v anish when α=θorα=θ=π/2,
+respectively. In the same manner, the 3 nequations give rise to determinants equal to 4( cβ−cθ)2
+and 4(cβ+cθ)2for the real and imaginary parts, respectively, with analog ous conditions for
+vanishing determinants.
+Finally, the mnblock from Eq. (8) gives us a set of 8 ×8 equations for the real and imaginary
+parts of the corresponding Yukawas. The corresponding dete rminants are
+16(cθ±cα+β)2(cθ±cα−β)2, (10)
+with the “+” signs corresponding to the equations for the ima ginary parts. We performed a
+thorough analysis of the conditions for vanishing determin ants, and the results are summarized
+in Table 2. Similar results hold for the charged +2 /3 quark matrices ∆ 1and ∆ 2, withβ→γ.
+After spontaneous symmetry breaking, the fields acquire the vacuum expectation values
+(vevs)v1/√
+2 andv2eiδ/√
+2; where v1andv2are real, without loss of generality. It is convenient
+to rotate into the so-called Higgs basis {H1,H2}through Φ a=UabHb, where
+U†=1
+v/bracketleftbiggv1v2e−iδ
+v2−v1e−iδ/bracketrightbigg
+, (11)
+4Table 2: Impact of the GCP symmetry on Γ 1and Γ2. We separate real (Re) and imaginary
+(Im) components.
+Γamatrix component condition for
+element vanishing determinant
+33 Im impossible
+Re θ= 0
+13, 23 Im α=θ=π/2
+Re α=θ
+31, 32 Im β=θ=π/2
+Re β=θ
+11, 12, 21, 22 Im θ=π−α−β
+Re θ=α+βorθ=α−β
+orθ=β−α
+is unitary, and v=/radicalbig
+v2
+1+v2
+2= (√
+2GF)−1/2. This rotates the vev into H1allowing us to
+parametrize
+H1=/bracketleftbiggG+
+(v+H0+iG0)/√
+2/bracketrightbigg
+, H2=/bracketleftbiggH+
+(R+iI)/√
+2/bracketrightbigg
+, (12)
+whereG+andG0aretheGoldstonebosons,which, intheunitarygauge, becom ethelongitudinal
+components of the W+and of the Z0, andH0,RandIare real neutral fields. In the new scalar
+basis, the Yukawa coupling matrices become ΓH
+a= ΓbUba, ∆H
+a= ∆bU∗
+ba, and the scalar couplings
+are also rotated. Finally, the fermion mass basis is obtaine d through transformations Uαwith
+α=dL,dR,uL,uRthat diagonalize the quark couplings to H1,
+(v/√
+2)U†
+dLΓH
+1UdR=Dd= diag(md,ms,mb),
+(v/√
+2)U†
+uL∆H
+1UuR=Du= diag(mu,mc,mt), (13)
+while the couplings to H2become (we follow the notation of [10, 11])
+(v/√
+2)U†
+dLΓH
+2UdR=Nd,
+(v/√
+2)U†
+uL∆H
+2UuR=Nu. (14)
+The Yukawa lagrangian may then be written as
+−v√
+2LY= (¯uLV,¯dL)(DdH1+NdH2)dR
++ (¯uL,¯dLV†)(Du˜H1+Nu˜H2)uR+H.c. ,
+(15)
+whereV=U†
+uLUdLis the CKM matrix. Thus NdandNuare responsible for flavor changing
+neutral currents (FCNC) involving the scalars RandI.
+Let us now look back at Table 2. We see that θ= 0, corresponding to the usual definition of
+CP, forces all Yukawa couplings to be real. This is well known , but it requires that CP violation
+arise spontaneously. This theory leads to FCNC, which are co nstrained by experiment. This
+could becuredwith furtherdiscretesymmetries. Itisknown that imposingtheabsenceof FCNC
+is inconsistent with GCP for three quark families and any num ber of Higgs fields [6]. Here we
+5will take the view that the scalar masses may be large enough t o suppress FCNC, and consider
+the most general couplings.
+The case of θ=π/2 corresponds to CP2. As we have seen, this forces (Γ 1)33= (Γ2)33= 0.
+Ifα=β=π/2 then the mnblock vanishes. Thus the determinant of Ddvanishes, forcing one
+quark mass to zero. This is excluded by experiment. If α/ne}ationslash=π/2 and/or β/ne}ationslash=π/2, then the last
+column and/or the last row of Ddvanishes, which leads again to a zero mass quark. As a result,
+it is impossible to extend CP2 to the quark sector in a way consistent with experiment. This
+zero mass problem had already been faced in [12, 13], for a par ticular extension of CP2. We
+note that CP2 could be consistently extended to the fermion s ector provided there were four
+fermion families.
+Now we come to the most interesting case: consider 0 < θ < π/ 2. Here (Γ 1)33= (Γ2)33= 0
+and, in order for the m3 and 3nentries to differ from zero (otherwise there would be at least a
+zero mass quark), we need to have α=β=θ. In this case, the determinants for the mnblock
+in Eq. (10) simplify to
+Imaginary: 256( cθ/2)8(1−2cθ)2(16)
+Real: 256( sθ/2)8(1+2cθ)2. (17)
+The second determinant never vanishes, which implies that t hemnentries will only have imag-
+inary parts. And the first determinant, Eq. (16), only vanish es ifθ=π/3. For all other cases,
+the quark mass matrices will have zero eigenvalues. Notice t hat the fact that the angles α,β
+andθare constrained to be in the interval [0 ,π/2] was fundamental in this demonstration.
+Therefore, out of the infinite number of GCP symmetries, only one survives: the case where
+one hasθ=α=β=π/3. For this model the Yukawa matrices have the form:
+Γ1=
+ia11ia12a13
+ia12−ia11a23
+a31a320
+,
+Γ2=
+ia12−ia11−a23
+−ia11−ia12a13
+−a32a310
+. (18)
+Notice that Γ 1has 2 (4) independent imaginary (real) entries and that Γ 2does not involve
+any new parameter. Similar parametrizations hold for ∆ 1and ∆ 2, involving 6 new parameters
+(a→b). We stress that this is the only possible extension of GCP in to the fermion sector
+consistent with the fact that the quarks have non-vanishing masses, and it leads to a very
+tightly constrained “minimal” model.
+Our theory has only 12 Yukawa parameters and two independent vevs. Indeed, v1may be
+made real without loss of generality; given v2, this fixes v2, and we only need the phase δ. In
+order to perform a numerical fit, we define the matrices
+Hd=v2
+2ΓH
+1(ΓH
+1)†=UdLD2
+dU†
+dL,
+Hu=v2
+2∆H
+1(∆H
+1)†=UuLD2
+uU†
+uL. (19)
+The eigenvalues of these matrices give the square of the quar k masses, providing six constraints.
+There are four other experimental constraints arising from the four independent parameters
+parametrizing the CKM matrix V. We can take these as the eigenvalues of HuHd, and also the
+CP-violating quantity [3]
+J= Tr[Hu,Hd]3= 6i(m2
+t−m2
+c)(m2
+t−m2
+u)(m2
+c−m2
+u)
+×(m2
+b−m2
+s)(m2
+b−m2
+d)(m2
+s−m2
+d)Im(VusVcbV∗
+ubV∗
+cs). (20)
+6Using Eq. (18) in Eq. (20), we have shownthat δ= 0implies J= 0. Said otherwise, spontaneous
+CP violation is required in order to have CP violation in the C KM matrix, and this only occurs
+if we add a soft symmetry breaking term to the scalar potentia l. However, the CP properties of
+this model are far from obvious, and will be discussed in the f ollowing section.
+Using a fast numerical fit, we have found that setting a11= 4.6927×10−6,a12=−5.9799×
+10−4,a13=−2.32×10−2,a23=−6.6×10−3,a31=−8.815×10−5,a32= 5.1193×10−6
+in the down-quark sector and b11= 7.3×10−3,b12= 7.6445×10−5,b13= 9.578×10−1,
+b23= 2.325×10−1,b31= 1.3446×10−4,b32= 5.9491×10−4, in the up-quark sector, and
+v1= 173.944,v2cosδ=−0.8467,v2sinδ=−0.9565, we obtain md= 0.00298,ms= 0.10511,
+mb= 4.19701,mu= 0.00200,mc= 1.27439,mt= 171.451, for the masses, and the magnitudes
+of the CKM matrix elements are
+|V|=
+0.97430 0 .22521 0 .00339
+0.22516 0 .97348 0 .04039
+0.00579 0 .04011 0 .99918
+. (21)
+All masses and vevs are in GeV. The values of v2=v2
+1+v2
+2, the masses, and the magnitudes
+of the CKM matrix elements are in very good agreement with the experimental data [14]. One
+may wonder whether the vevs we used above can be generated by t he scalar potential. As we
+mentioned earlier, we need to introduce soft breaking terms in the scalar potential to generate
+δ, namely by setting m2
+11/ne}ationslash=m2
+22and Re(m2
+12)/ne}ationslash= 0. It is easy to find values for the potential’s
+parameters which generate the desired vevs: they can be repr oduced by m2
+11=−28791,m2
+22=
+5679.5, Re(m2
+12) =−167.8012, Im( m2
+12) = 0,λ1=λ2= 0.9516,λ3= 0.0176, and λ4= 0.3643
+(the quadratic parameters are expressed in units of GeV2).
+The numerical fit is quite successful, but the exception is |Vtd|which, due to the hierarchical
+nature of the CKM matrix elements, becomes much more difficult to fit2. A related problem
+occurs with the CP-violating quantity Jckm= Im(VusVcbV∗
+ubV∗
+cs). For our choice of parameters,
+oneobtains Jckm=−5.9×10−8, whiletheexperimental dataleadsto Jckm= (3.05±0.20)×10−5.
+The reason has to do with the sensitivity of our fit procedure t o the input parameters. We have
+fitZus,Zub,Zcs, andZcb, whereZij=|Vij|2. These four elements can be used to parametrize
+completely the CKM matrix [15, 16, 17]. In particular [18],
+4J2
+ckm= 4ZusZubZcsZcb−(1−Zus−Zub−Zcs−Zcb+ZusZcb+ZubZcs)2.(22)
+Using the experimental values for the CKM matrix elements [1 4], one can show that to first
+order
+∆Jckm
+Jckm∼540.95∆Zus
+Zus+0.7∆Zub
+Zub
++ 10065 .9∆Zcs
+Zcs+18.0∆Zcb
+Zcb. (23)
+This shows how extremely sensitive Jckmis to the exact value of Vcs, explaining why it is easy
+to perform a fast fit to the latter, but thus inducing a large er ror on the former.
+4 A New Type of Spontaneous CP Violation
+We now come to the most remarkablefeature of this model, name ly its spontaneous CP violation
+properties. Wehave alreadymentioned that, toobtainanon- zerovaluefortheJarlskoginvariant
+2Notice that the direct measurement of |Vtd|comes from the mixing in the Bsystem, which occurs in the SM
+through a box diagram. This mixing can receive contribution s involving the second Higgs, thus altering the value
+of|Vtd|.
+7J, Eq. (20), we need the vevs of the scalar fields to have a relati ve phase. Usually, this fact
+by itself is taken to be a sign of spontaneous CP violation in t he scalar sector. The model
+we are now discussing behaves, however, in a very different fas hion. Regardless of its possible
+future experimental relevance as a viable description of na ture, this model serves as a theoretical
+example of a new kind of spontaneous CP violation.
+First, let us demonstrate that the lagrangian of this model i s explicitly CP-conserving, i.e.
+that it is possible to write it in a field basis where all parame ters are real. We see in the CP3
+entry in Table 1 that the scalar potential only has real param eters. As was mentioned above,
+the soft breaking terms we introduced are also real, so there are no complex phases in the scalar
+sector, prior to SESB.
+Now for the Yukawa sector. It might seem, looking at the form o f the Yukawa matrices in
+Eq. (18), that one cannot avoid the presence of complex numbe rs in the Lagrangian. However,
+we are free to choose a different quark basis, by performing tra nsformations of the type
+qL→ULqL, nR→URnR, (24)
+withU(3) matrices ULandUR. This changes the Yukawa matrices according to
+Γk→U†
+LΓkUR. (25)
+By choosing U†
+L=UR= diag(e−iπ/4,e−iπ/4,eiπ/4), we remove all factors of ifrom the Γ matrices
+and are left with
+Γ1→
+a11a12a13
+a12−a11a23
+a31a320
+,
+Γ2→
+a12−a11−a23
+−a11−a12a13
+−a32a310
+, (26)
+where all parameters are real. We stress that the basis choic e that leads to this result involves
+only the fermion fields, and as such does not introduce any pha ses in the scalar potential. An
+identical fermion basis choice may be used to render real the Yukawa matrices in the up-quark
+andleptonicsectors. Hence, wehaveprovedthatitispossib letofindabasiswhereallparameters
+in the lagrangian are real - the model is explicitly CP conser ving.
+After SESB, with the soft breaking we discussed, we introduc e a complex phase δin the
+theory, from the vevs v1/√
+2 andv2eiδ/√
+2. One might wonder whether δ/ne}ationslash= 0 implies the
+presence of spontaneous CP violation in the scalar sector. T his is best investigated with the
+basis-invariant quantities developed by Lavoura and Silva [19] and by Botella and Silva [11].
+Their explicit calculation shows that they all vanish - thus , with the proposed soft breaking of
+the CP3 symmetry there is no CP violation in the scalar sector , whether explicit or spontaneous.
+A simple way to confirm this is by changing to the Higgs basis, a s given by the transformation
+in Eq. (11). In that basis, the potential, with the soft break ing terms already included, becomes
+VH= ¯m2
+11H†
+1H1+ ¯m2
+22H†
+2H2−/bracketleftBig
+¯m2
+12H†
+1H2+H.c./bracketrightBig
++1
+2¯λ1(H†
+1H1)2+1
+2¯λ2(H†
+2H2)2
++¯λ3(H†
+1H1)(H†
+2H2)+¯λ4(H†
+1H2)(H†
+2H1)+/bracketleftBig
+1
+2¯λ5(H†
+1H2)2+¯λ6(H†
+1H1)(H†
+1H2)
++¯λ7(H†
+2H2)(H†
+1H2)+H.c./bracketrightBig
+. (27)
+8All coefficients are real, except ¯ m2
+12,¯λ5,¯λ6, and¯λ7. The stationarity conditions for the vacuum
+impose 2¯ m2
+12=¯λ6v2. Thus, to study CP we need only the phases of
+−¯λ7=¯λ6=2
+v4v1v2(λ1−λ3−λ4)sinδ η,
+¯λ5=−1
+v4(λ1−λ3−λ4)η2, (28)
+where
+η=−i/parenleftBig
+v2
+1eiδ+v2
+2e−iδ/parenrightBig
+. (29)
+The only phases invariant under a trivial rephasing of H2and, thus, possibly signaling CP
+violation are Im( ¯λ6¯λ∗
+7), Im(¯λ∗
+5¯λ2
+6), and Im( ¯λ∗
+5¯λ2
+7). These are precisely the three quantities
+introduced by Lavoura and Silva [19] to probe all possible so urces of CP violation in the scalar
+sector in a basis-invariant way. Since these quantities are all zero, we conclude that there is no
+CP violation in the scalar sector of the theory, even after SE SB. This means, in particular, that
+one can find a field basis where there is no mixing between the CP -even and CP-odd neutral
+scalar particles.
+One may then ask whether the phase δis at all relevant in the model. The answer is yes
+because, due to the interplay between the scalar and Yukawa s ectors of the theory, there is no
+choice of field basis through which one can absorb δ. In fact, the CP-violating quantity Jof
+Eq. (20) is directly proportional to sin δ. Notice, too, that as long as δ/ne}ationslash= 0 the FCNC involving
+NdandNu, in Eq. (14), will also, in general, involve complex phases a nd as such may well serve
+as further sources of CP violation.3But all possible sources of CP violation vanish if δ= 0,
+even though no CP breaking occurs in the scalar sector.
+This, then, is a new type of CP violation:
+•It is not like that of the SM, since there CP is explicitly brok en at the lagrangian level;
+•It is not like the CP violation that occurs in Lee-type mechan isms, since there CP is
+spontaneously broken in the scalar sector.
+In the model herein presented, it is the fermion sector which exhibits the CP violation that
+arises spontaneously. And the CKM matrix is generated throu gh a spontaneous breaking of CP,
+not an explicit one as is the case of the SM. Of course, these re sults were obtained at tree level.
+Still, it is interesting that the scalar sector does the deed (spontaneously break the symmetry)
+but it is the fermion sector which pays the consequence (prov iding CP violation). To the best
+of our knowledge, this type of CP violation is unheard of in th e literature.
+5 Conclusions
+Wehaveshownthat, forthemostpart, thegeneralized CPsymm etriesofthescalarsector cannot
+be extended to the quark sector with three generations, whil e keeping all six quarks massive.
+This reduces dramatically the types of available models. We have found the only exception: the
+only THDM with GCP which leads to an acceptable fermion mass s pectrum. We have shown
+that the model has very few parameters, both in the scalar and Yukawa sectors, and that it
+can give a good fit to the known quark masses and mixings. Due to the hierarchical nature
+3If the masses of the new scalars are large enough, then these s ources of CP violation will have a small impact
+on current experiments, including those currently used to c onstrain V. Our aim in this section is to highlight a
+new scenario for spontaneous CP violation and not a particul ar implementation thereof.
+9of the masses and CKM matrix elements, very precise numerica l fits are needed. This model
+has FCNC which might have considerable impact on its experim ental predictions for hadron
+phenomenology. A detailed study of its impact on such observ ables is therefore needed, and will
+be presented elsewhere [20].
+We have also shown that this model has the peculiar feature th at the scalar sector by itself
+is both explicitly and spontaneously CP-conserving, but th at it generates a symmetry breaking
+which induces CP violation in the quark sector. This constit utes a new type of spontaneous
+CP violation, different from the SM or the usual findings of the T HDM. As such this model,
+regardless of its experimental relevance, serves as the firs t example of a new way of generating
+CPviolation. Itis not uniqueinthat respect: in fact, it isa trivial exercise to buildathree-Higgs
+doublet model which displays exactly the same CP properties .
+There is a further motivation for studying in detail models o f spontaneous CP violation.
+The excellent agreement between the SM and all CP violating e xperiments has been taken as
+a definitive confirmation of the SM’s CP violation mechanism, i.e.the existence of explicitly
+CP violating Yukawa couplings. Models of CP violation, like the one presented here, where J
+could be the dominant source of CP violation raise the tantal izing possibility that, contrary
+to widespread belief, the current experiments on CP violati on confirm the Cabibbo-Kobayashi-
+Maskawa source of CP violation but not its origin in the expli cit CP breaking of the Yukawa
+interactions.
+Acknowledgments
+We are grateful to L. Lavoura for reading and commenting on th e manuscript. The work of
+P.M.F. is supported in part by the Portuguese Funda¸ c˜ ao para a Ciˆ encia e a Tecnologia (FCT)
+under contract PTDC/FIS/70156/2006. The work of J.P.S. is s upported in part by FCT under
+contract CFTP-Plurianual (U777).
+References
+[1] T. D. Lee, “A Theory of Spontaneous T Violation,” Phys. Re v. D8, 1226 (1973), .
+[2] T.D.LeeandG.C.Wick, “SpaceInversion, TimeReversal, AndOtherDiscreteSymmetries
+In Local Field Theories,” Phys. Rev. 148, 1385 (1966).
+[3] J. Bernab´ eu, G. C. Branco, and M. Gronau, “Cp Restrictio ns On Quark Mass Matrices,”
+Phys. Lett. 169B, 243 (1986).
+[4] G. Ecker, W. Grimus, and W. Konetschny, “Quark Mass Matri ces In Left-Right Symmetric
+Gauge Theories,” Nucl. Phys. B191, 465 (1981).
+[5] G. Ecker, W. Grimus and H. Neufeld, “Spontaneous CP Viola tion In Left-Right Symmetric
+Gauge Theories,” Nucl. Phys. B 247, 70 (1984).
+[6] H. Neufeld, W. Grimus and G. Ecker, “Generalized CP invar iance, neutral flavor conserva-
+tion and the structure of the mixing matrix”, Int. J. Mod. Phy s. A3, 603 (1988).
+[7] I. P. Ivanov, “Minkowski space structure of the Higgs pot ential in 2HDM: II. Minima,
+symmetries, and topology,” Phys. Rev. D 77, 015017 (2008) [arXiv:0710.3490 [hep-ph]].
+[8] P. M. Ferreira, H. E. Haber and J. P. Silva, “Generalized C P symmetries and special
+regions of parameter space in the two-Higgs-doublet model, ” Phys. Rev. D 79, 116004
+(2009) [arXiv:0902.1537 [hep-ph]].
+10[9] G. Ecker, W. Grimus and H. Neufeld, “A Standard Form For Ge neralized CP Transforma-
+tions,” J. Phys. A 20, L807 (1987).
+[10] L. Lavoura, “Models of CP violation exclusively via neu tral scalar exchange,” Int. J. Mod.
+Phys. A 9, 1873 (1994).
+[11] F. J. Botella and J. P. Silva, “Jarlskog - like invariant s for theories with scalars and
+fermions,” Phys. Rev. D 51, 3870 (1995) [arXiv:hep-ph/9411288].
+[12] M. Maniatis, A. von Manteuffel and O. Nachtmann, “A new typ e of CP symmetry, family
+replication and fermion mass hierarchies,” Eur. Phys. J. C 57, 739 (2008) [arXiv:0711.3760
+[hep-ph]].
+[13] M. Maniatis and O. Nachtmann, “On the phenomenology of a two-Higgs-doublet model
+with maximal CP symmetry at the LHC,” JHEP 0905, 028 (2009) [arXiv:0901.4341 [hep-
+ph]].
+[14] C. Amsler et al.[Particle Data Group], “Review of particle physics,” Phys. Lett. B667, 1
+(2008).
+[15] G. C. Branco and L. Lavoura, “Rephasing Invariant Param etrization Of The Quark Mixing
+Matrix,” Phys. Lett. B 208, 123 (1988).
+[16] L. Lavoura, “Parametrization of the four generation qu ark mixing matrix by the moduli of
+its matrix elements,” Phys. Rev. D 40, 2440 (1989).
+[17] L. Lavoura, “On The Reconstruction Of The Four Generati on Ckm Matrix From The
+Moduli Of Its Matrix Elements: Existence Of Discrete Ambigu ities,” Phys. Lett. B 223,
+97 (1989).
+[18] G. C. Branco, L. Lavoura, and J. P. Silva, CP Violation (Oxford University Press, Oxford,
+1999).
+[19] L. Lavoura and J. P. Silva, “Fundamental CP violating qu antities in a SU(2) x U(1) model
+with many Higgs doublets”, Phys. Rev. D 50, 4619 (1994) [arXiv:9404276 [hep-ph]].
+[20] P. M. Ferreira and J. P. Silva, in preparation.
+11
\ No newline at end of file
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+arXiv:1001.0575v2  [gr-qc]  24 May 2010Evolutions of Magnetized and Rotating Neutron Stars
+Steven L. Liebling,1Luis Lehner,2,3,4David Neilsen,5and Carlos Palenzuela6,7
+1Department of Physics, Long Island University – C.W. Post Ca mpus, Brookville, NY 11548
+2Perimeter Institute for Theoretical Physics, Waterloo, On tario, Canada
+3Department of Physics, University of Guelph, Guelph, Ontar io, Canada
+4Canadian Institute for Advanced Research, Cosmology & Grav ity Program
+5Department of Physics and Astronomy, Brigham Young Univers ity, Provo, UT 84602
+6Canadian Institute for Theoretical Astrophysics (CITA), T oronto, Ontario, Canada
+7Max-Planck-Institut f¨ ur Gravitationsphysik, Albert-Ei nstein-Institut, Golm, Germany
+(Dated: August 7, 2018)
+We study the evolution of magnetized and rigidly rotating ne utron stars within a fully general
+relativistic implementation of ideal magnetohydrodynami cs with no assumed symmetries in three
+spatial dimensions. The stars are modeled as rotating, magn etized polytropic stars and we examine
+diverse scenarios to study their dynamics and stability pro perties. In particular we concentrate
+on the stability of the stars and possible critical behavior . In addition to their intrinsic physical
+significance, we use these evolutions as further tests of our implementation which incorporates new
+developments to handle magnetized systems.
+I. INTRODUCTION
+Neutron stars play a key role in some of the most in-
+teresting astrophysical events observed, from supernova
+remnantsand pulsarsto aless certainrolein longgamma
+ray bursts (GRBs). As such they have attracted signif-
+icant research into their dynamics (for a recent review
+see [1]). In this paper, we revisit the dynamics of rotat-
+ing, and possibly magnetized, neutron stars modeled as
+polytropicstarswithinafullynonlinear,generalrelativis-
+tic model of ideal magnetohydrodynamics (GR+MHD).
+We study the stability properties of these stars and high-
+light possible critical behavior exhibited by the system.
+Furthermore, our studies serve to demonstrate the effec-
+tiveness of our code and certain new developments dis-
+cussed here. This code has been applied to other astro-
+physical problems [2–7].
+In recentyears, anumber offully relativisticevolutions
+(asopposedtothoseusingNewtoniangravityorotherap-
+proximations to general relativity) of rotating polytropes
+have appeared studying gravitational wave production
+and black hole formation in astrophysically relevant sys-
+tems. To this end, studies beyond spherical symmetry
+are required which are computationally more demand-
+ing. For instance, rotating stars require moving beyond
+spherical symmetry and many interesting axisymmetric
+scenarios can be addressed using 2D implementations al-
+lowing for excellent resolution without requiring major
+resources (e.g. [8–10]). In contrast, studies of the most
+general flows and instabilities require 3D simulations and
+are the most expensive realistically accessible scenarios
+being currently considered. Future efforts, including ra-
+diation transport mechanisms, will move beyond 3D sce-
+narios and require efficient use of petaflop (and beyond)
+resources (e.g. [11]).
+Another important distinction in these studies is
+whether the modeled stars are rigidly rotating or allow
+for differential rotation. Rigidly rotating stars support
+more mass than non-rotatingones (so-called TOV stars),but generally do not support the largest masses achieved
+with differential rotation nor do they demonstrate some
+of the more interesting instabilities such as the bar mode
+instability [12]. Instead, uniformly rotating stars tend to
+demonstrate one of two behaviors; stability or instability
+to collapse to a black hole. Previous studies suggest that
+significant disks do not form from such collapse [13–15].
+Recall that stars rotating differentially are expected to
+settle into rigidly rotating configurations on short time
+scales, and hence a normal neutron star observed today
+is generally expected to be rigidly rotating.
+Achieving ever more realism, models of neutron stars
+have also begun to consider stars with magnetic field [9,
+16–18]. Magnetic fields provide, in particular, an effec-
+tive pressure which generally supports greater mass [19]
+as well as an efficient way to transport angular momen-
+tum. In differentially rotating stars, even small mag-
+netic fields can be amplified by the magnetorotational
+instability. Generally these studies begin with a non-
+magnetized neutron star to which a small, seed magnetic
+field is added. However, in [20], fully consistent, magne-
+tized stars are evolved, although only nonrotating results
+are presented. Another approach is to study the modes
+througha perturbation approach[21, 22] and analyzethe
+growth of these modes with respect to a given stationary
+solution. The dynamical behavior of magnetized stars is
+important also for their role in explaining strong electro-
+magnetic emissions. Indeed, isolated stars with strong
+magnetic fields, so called magnetars , are suspected to be
+the engines powering anomalous X-ray pulsars (AXPs)
+and soft gamma ray repeaters (SGRs) [23]. At least 10%
+of all neutron stars [24] are born as magnetars. In the
+context of single stars, it is thus interesting to exam-
+ine if strong magnetic fields may deform stars away from
+axisymmetry making them strong producers of gravita-
+tional waves [25].
+Here we present results with uniformly rotating neu-
+tron stars which possess a fully consistent magnetic
+dipole moment. That is to say, the initial data used here2
+represents a stationary state of the full GR+MHD equa-
+tions. These evolutions are computed with a general rel-
+ativity code employing the generalized harmonic scheme
+(allowing for black hole excision). Further, no symme-
+try assumptions are made. High resolutions are achieved
+using a distributed adaptive mesh refinement infrastruc-
+ture. These evolutions demonstrate that our code can
+evolve a stable rotating star for many periods accurately.
+Similarly, unstable stars evolve to black holes with no
+evidence of any significant disk forming. Finally, we give
+evidence that unstable, rotating, magnetized stars rep-
+resent minimally unstable solutions which could serve as
+Type I critical solutions.
+In Section II we provide details about the formulation
+of the equations. In Section III we discuss aspects of our
+numerical implementation, and describe the diagnostic
+quantities evaluated in Section IV. In Section V we dis-
+cuss the initial data we use. We present our results in
+Section VI, and conclude in Section VII.
+II. FORMULATION AND EQUATIONS OF
+MOTION
+Neutron stars can be modeled by relativistic fluids
+(possibly with the inclusion of magnetic fields) under the
+action of strong gravitational fields [26]. These systems
+are governed both by the Einstein equations for the ge-
+ometry and by the relativistic equations of magnetohy-
+drodynamics for the matter. We write both systems as
+first order hyperbolic equations. This form of the equa-
+tions is convenient in order to take advantage of several
+rigorous numerical techniques devised for such systems
+to ensure, at the linear level, stability of the implementa-
+tion. More information regarding the motivation for this
+approach can be found in [2, 27, 28]. By way of notation,
+we use letters from the beginning of the alphabet ( a,b,
+c) for spacetime indices, while letters from the middle
+of the alphabet ( i,j,k) range over spatial components.
+We adopt geometric units where c=G= 1. However,
+as discussed in Section III, when appropriate, we rescale
+the value of Gto achieve improved accuracy in the con-
+servative to primitive variable conversion stage.
+A. The Einstein equations
+The Einstein equations can be written as a system of
+ten nonlinear partial differential equations for the space-
+time metric gab. The harmonic formulation of the Ein-
+stein equations exploits the fact that the coordinates xa
+can be chosen satisfying the generalized harmonic condi-
+tion [29, 30]
+∇c∇cxa=−Γa=Ha(t,xi), (1)
+where Γa≡gbcΓabcare the contracted Christoffel sym-
+bols. The arbitrary source functions Ha(t,xi) deter-
+mine the coordinate freedom of Einstein equations. Theoriginal harmonic coordinates correspond to the case
+Ha(t,xi) = 0, which is the choice here. The Einstein
+equations can be expressed in their generalized harmonic
+form [29], in particular
+gcd∂cdgab+∂aHb+∂bHa=−16π/parenleftbigg
+Tab−T
+2gab/parenrightbigg
++2 ΓcabHc+2gcdgef/parenleftbigg
+∂egac∂fgbd−ΓaceΓbdf/parenrightbigg
+.(2)
+The matter is coupled to the geometry by means of the
+stress energy tensor Taband its trace T≡gabTab, which
+will be dictated by the particular model of magnetized
+fluidunderconsideration,detailedin thenext subsection.
+The spacetime can be foliated into hypersurfaces of
+constant coordinate time x0≡t= const. On these
+spacelike hypersurfaces, one defines a spatial 3-metric
+hij=gij. A vector normal to the hypersurfaces is given
+byna≡−∇ at/||∇at||, and coordinates defined on neigh-
+boring hypersurfaces can be related through the lapse, α,
+and shift vector, βi. With these definitions, the space-
+time differential element can then be written as
+ds2=gabdxadxb
+=−α2dt2+hij/parenleftbig
+dxi+βidt/parenrightbig/parenleftbig
+dxj+βjdt/parenrightbig
+.(3)
+Indices on spacetime quantities are raised and lowered
+with the 4-metric, gab, and its inverse, while the 3-metric
+hijand its inverse are used to raise and lower indices on
+spatial quantities.
+We adopt a first order reduction of the second order
+differential equations represented in Eqs. (2). This re-
+duction can be achieved by introducing new independent
+variables related to the time and space derivatives of the
+fields
+Qab≡−nc∂cgab, Diab≡∂igab.(4)
+Within these definitions we can write our evolution equa-
+tions in our GH formalism in the following way [31]
+∂tgab=βkDkab−α Qab, (5)
+∂tQab=βk∂kQab−αhij∂iDjab
+−α ∂aHb−α ∂bHa+2αΓcabHc
++ 2αgcd(hijDicaDjdb−QcaQdb−gefΓaceΓbdf)
+−α
+2ncndQcdQab−α hijDiabQjcnc
+−8πα(2Tab−gabT)
+−2σ0α[naZb+nbZa−gabncZc]
++σ1βi(Diab−∂igab), (6)
+∂tDiab=βk∂kDiab−α ∂iQab
++α
+2ncndDicdQab+α hjkncDijcDkab
+−σ1α(Diab−∂igab). (7)
+This GH formulation includes a number of constraints
+that must be satisfied for consistency. On one hand,3
+therearetwosetsoffirstorderconstraints,obtainedfrom
+Eqs. (4) and defined as
+Ciab≡∂igab−Diab= 0,
+Cijab≡∂iDjab−∂jDiab= 0, (8)
+which were introduced when performing the reduction
+to first order [2, 31]. On the other hand, there are
+the Hamiltonian and momentum constraints, that in the
+Generalized Harmonic formulation show up in terms of a
+four-vector Za, which is defined as
+2Za≡−Γa−Ha(t,xi). (9)
+It can be shown that the Hamiltonian and momentum
+constraints are satisfied if Za=∂tZa= 0 [32]. In order
+to dynamically control the violation of the constraints,
+wehaveincluded certaintermsproportionaltothese con-
+straints (8)–(9). These additional terms depend on free
+parameters σ0andσ1, allowing one to dynamically damp
+constraint—including the Hamiltonian, momentum, and
+first-order constraints (8)—violating modes on a time
+scale proportional to −σi([31, 33]).
+We evolve the gravitational field equations shown in
+Eqs. (5-7). These equations rely on the computation of
+the4-dimensionalChristoffelsymbolsfromthemetric gab
+Γabc=1
+2(Dbca+Dcba−Dabc). (10)
+While we evolve the Diabfunctions, the set D0abare not
+evolved, but are calculated from evolved quantities as
+D0ab=−αQab+βkDkab. (11)
+This description suffices to explain the gravitational
+evolution, and the following section describes the evolu-
+tion of the matter. However, we note here that the MHD
+equations are written in the standard 3+1 decomposi-
+tion of spacetime and thus require the spatial metric hij,
+the lapse α, shiftβi, and ADM extrinsic curvature, Kij.
+These quantities can be written in terms of our evolved
+fields using
+hij=gij, α=/radicalbig
+−1/g00, βi=γijg0j,
+Kij=1
+2Qij+1
+α(D(ij)0−βkD(ij)k). (12)
+Conversely, the Hamiltonian and momentum constraints
+are usually written in terms of spatial derivatives of the
+metricDkijand the extrinsic curvature Kij. In fact, we
+usethese3+1quantities(andsimilarexpressionsfortheir
+derivatives) to calculate the residuals of the Hamiltonian
+and momentum constraints expressed in their standard
+form.
+B. MHD equations
+We now briefly introduce the perfect fluid equations.
+Additional information can be found in our previous
+work [27, 28] as well as in topical review articles [34, 35].The stress-energy tensor for the perfect fluid in the
+presence of a Maxwell field is given by
+Tab= [ρo(1+ǫ)+P]uaub+Pgab
++FacFbc−1
+4gabFcdFcd. (13)
+Thefluid isdescribed byrestmassdensity ρo, the specific
+internal energy density ǫ, the isotropic pressure Pand
+the four velocity of the fluid ua. With these quantities
+we can construct the enthalpy
+he=ρo+ρoǫ+P, (14)
+and construct the standard spatial coordinate velocity of
+the fluid vias
+W≡−naua, vi≡1
+Whijuj, (15)
+whereWis the Lorentz factor between the fluid frame
+and the fiducial ADM observers.
+The Maxwell tensor Fabcan be written as
+Fab=naEb−nbEa+ǫabcdBcnd,(16)
+whereEaandBaare the electric and magnetic fields
+measuredbya“normal”observer na. Consequently,both
+fields are purely spatial, i.e., Eana=Bana= 0.
+The evolution of the magnetized fluid is described by
+different sets of conservation laws. The magnetic field, in
+the ideal MHD limit, follows the Maxwell equation
+∇µ(∗Fµν+gµνΨ) =κnνΨ, (17)
+where∗Fab≡ǫabcdFcd/2 is the dual of the Maxwell ten-
+sorand we haveintroduced arealscalarfield Ψ to control
+the divergence constraint. This technique is known as di-
+vergence cleaning [36] and allows for a convenient way to
+control the constraint violation by inducing a damped
+wave equation for the scalar field Ψ. The other Maxwell
+equation,intheidealMHDlimit, onlygivesthedefinition
+for the current density, since the electric field is given in
+terms of the velocity of the fluid and the magnetic field,
+that is,
+Ei=−ǫijkvkBk. (18)
+Conservation of the stress energy tensor in Eq. (13),
+∇aTab= 0, (19)
+provides 4 evolution equations for the fluid variables,
+namely the velocity and the internal energy. Conserva-
+tion of the baryon number
+∇a(ρoua) = 0 (20)
+leads to the evolution equation of the rest mass density
+ρo. Closure of the equations is achieved by introducing
+an equation of state (EOS) relating the pressure with4
+the other thermodynamical quantities of the fluid, P=
+P(ρo,ǫ).
+High resolution shock capturing schemes (HRSC) are
+robust numerical methods for compressible fluid dy-
+namics. These methods, based on Godunov’s seminal
+work [37], are fundamentally based on expressing the
+fluid equationsasintegralconservationlaws. Tothis end,
+we introduce conservative variables q= (D,Si,τ,Bi)T,
+where
+D=Wρo, (21)
+Si= (heW2+B2)vi−(Bjvj)Bi,(22)
+τ=heW2+B2−P
+−1
+2/bracketleftbigg
+(Bivi)2+B2
+W2/bracketrightbigg
+−D. (23)
+andBiis both a primitive and conservative variable.
+In an asymptotically flat spacetime these quantities are
+conserved, and are related to the total energy, momen-
+tum, and, in the non-relativistic limit, the kinetic en-
+ergy, respectively. The quantities w= (ρo,vi,P,Bi)T
+are called the primitive variables in contrast to the con-
+servative variables. The fluid state can be specified using
+either set of variables, and both sets are required to write
+the MHD evolution equations. Anticipating the form of
+these equations, we also introduce the densitized con-
+served variables
+˜D=√
+hD,˜Si=√
+hSi,˜τ=√
+hτ,˜Bi=√
+hBi,
+(24)
+whereh= det(hij). The fluid equations can now be
+written in balance law form
+∂t˜q+∂kfk(˜q) =s(˜q), (25)
+wherefkare flux functions, and sare source terms. The
+fluid equations in this form are
+∂t˜D+∂i/bracketleftbigg
+α˜D/parenleftbigg
+vi−βi
+α/parenrightbigg/bracketrightbigg
+= 0, (26)
+∂t˜Sj+∂i/bracketleftbigg
+α/parenleftbigg
+˜Sj/parenleftbigg
+vi−βi
+α/parenrightbigg
++√
+hP hi
+j/parenrightbigg/bracketrightbigg
+=α3Γi
+jk/parenleftBig
+˜Sivk+√
+hPhik/parenrightBig
++˜Sa∂jβa
+−∂jα(˜τ+˜D)
+−ζα(˜BiW−2+vivj˜Bj)∂k˜Bk,(27)
+∂t˜τ+∂i/bracketleftbigg
+α/parenleftbigg
+˜Si−βi
+α˜τ−vi˜D/parenrightbigg/bracketrightbigg
+=α/bracketleftbigg
+Kij˜Sivj+√
+hKP−1
+α˜Sa∂aα/bracketrightbigg
+,
+−ζαvj˜Bj∂k˜Bk(28)
+∂t˜Bb+∂i/bracketleftbigg
+˜Bb/parenleftbigg
+vi−βi
+α/parenrightbigg
+−˜Bi/parenleftbigg
+vb−βb
+α/parenrightbigg/bracketrightbigg
+=−α√
+hhbi∂iΨ−ζαvi∂j˜Bj(29)
+∂tΨ =−crαΨ−chα√
+h∂i˜Bi+(βi−αvi)∂iΨ.(30)wherecr=κand we have allowed for different speeds
+than light by introducing the parameter ch. Except in
+the tests, we will use the original prescription (17) with
+ch= 1. Here3Γi
+jkis the Christoffel symbol associated
+with the 3-metric hij, andKis the trace of the extrin-
+sic curvature, K=Kii. Notice that the aforementioned
+system is an extended version of the one employed in our
+earlier works [5, 27, 28] Here we have added additional
+terms toggled by the parameter ζwhich allow for consid-
+eringanextensionofthe“eight-wave”formulationwhich,
+in the absence of the cleaning field Ψ, ensures the strong
+hyperbolicity of the system [36] –at the cost of introduc-
+ing derivative terms in the sources–. Furthermore, by
+settingζ= 1, the propagation speeds of two constraint
+violating modes become non-vanishing and hence viola-
+tions are dragged along by the fluid’s velocity. This is
+numerically convenient as possible violations will propa-
+gate off the grid.
+Finally, we close the system of fluid equations with an
+equation of state (EOS). We choose the ideal fluid EOS
+P= (Γ−1)ρoǫ, (31)
+whereΓistheconstantadiabaticexponent. Nuclearmat-
+ter in neutron stars is relatively stiff, and we set Γ = 2
+in this work. When the fluid flow is adiabatic, this EOS
+reduces to the well known polytropic EOS
+P=κρoΓ, (32)
+whereκis a dimensional constant. We use the polytropic
+EOS only for setting initial data.
+The set of fluid equations, Eqs. (26-29), are used to
+evolve the conservative variables. However, these equa-
+tions also contain the primitive variables which neces-
+sitates a step in the evolution scheme which solves for
+the primitive variables in terms of the conservative ones.
+Given the primitive variables, the conservative variables
+are easily calculated from the algebraic expressions in
+Eqs. (24). Calculating the primitive variables from the
+conservative variables, however, is more delicate, as it re-
+quires the solution of a transcendental equation. To this
+endwefirstdefinethequantity x≡heW2. Wethenwrite
+S2=SiSiin terms of xand solve for W2, obtaining
+W2=x2(x+B2)2
+x2(x+B2)2−x2S2−(2x+B2)(SiBi)2.(33)
+Using the definition of τ, we define a function that is
+identically zero
+f(x) =x−P−1
+2/bracketleftbigg(SiBi)2
+x2+B2
+W2/bracketrightbigg
++B2−τ−D= 0.(34)
+If the enthalpy can be expressed as a simple function of
+the pressure, as can be done for the ideal fluid and a
+generalized EOS with a cold nuclear component, then we
+can express the pressure as a function x. For the ideal
+fluid EOS used here, the enthalpy equation
+he=x
+W2=ρ0+ρ0ǫ+P (35)5
+FIG. 1: Demonstration of the effectiveness of divergence
+cleaning at handling deviations from a divergenceless mag-
+netic field. A calculation of the divergence of the magnetic
+field,x2/bracketleftBig
+/vector∇·/vectorB/bracketrightBig
+, along the x-axis for three times is shown for
+each of three cases: (top) No divergence cleaning; (middle)
+Cleaning with ch= 0.1 andcr= 0.01; and (bottom) Cleaning
+withch= 1.0 andcr= 0.1. These runs were otherwise iden-
+tical for a magnetized, rotating star of coordinate equator ial
+radius of 10 with spin along the z-axis with three levels of re-
+finement (at ±50,±25, and±12.5) and a perturbation to the
+magnetic field to introduce an explicit deviation from diver -
+genceless. Concentrating on the MHD equations, the metric
+terms were frozen at their initial values, e.g. the Cowling a p-
+proximation. The bottom row shows clear wavelike behavior
+as it “cleans” the divergence.
+can be solved for Pby substituting in ρ0=D/Wand
+the EOS to obtain
+P=Γ−1
+Γ/parenleftbiggx
+W2−D
+W/parenrightbigg
+. (36)
+Combining these equations, fis a function of a single
+unknown x. This equation can be solved numerically
+using the Newton-Raphson method. It is useful to note
+thatxhas a minimum physical value, which is found by
+requiring in Eq. (33) that W2≥1.
+III. IMPLEMENTATION DETAILS
+The code is constructed within the hadcomputa-
+tional infrastructure which provides distributed adaptive
+mesh refinement (AMR). The AMR follows in the styleof Berger & Oliger [38], but uses the tapering condi-
+tion for AMR boundaries instead of temporal interpo-
+lation [39]. The combined set of geometric equations and
+fluid equations, Eqs. (5-7) and Eqs. (26-30) respectively,
+is discretized using the method of lines. The geometric
+equations are discretized using operators that satisfy a
+summation by parts property [40, 41]. The fluid equa-
+tions are discretized using the HLLE method [42]. The
+semi-discrete equations are solved using a third order ac-
+curate, total variation diminishing (TVD) Runge-Kutta
+solver [43].
+The fluid equations diverge as the density goes to zero,
+and, asis standardpractice, wedisallowanytruevacuum
+by setting such regions to a floor or atmosphere value.
+The floor is applied after each fluid update as
+D←min(D,δ), τ←min(τ,δ),(37)
+whereδischosento be manyordersofmagnitudesmaller
+than the maximum densities and pressures in the initial
+data. The comparison of otherwise identical runs but
+with different floor values suggest that the use of an at-
+mosphere generally does not affect accuracy.
+We have, however, found certain issues with precision
+occurring within the primitive solver. Typical maximum
+values of the density are about 10−2in geometric units,
+with a floor value of δ= 10−8. We have found it use-
+ful therefore to scale Newton’s constant Gsuch that
+the fluid densities and pressures are close to order unity.
+Thus, rather than using the typical choice of G= 1, we
+might use G= 1/1000. As Gaffects only the coupling
+of the fluid to the geometry, the evolution of the geomet-
+ric equations is not affected by this scaling. Empirically,
+we find that scaling G allows the primitive variables to
+be more easily recovered in low density regions. This im-
+provementappears to be related to finite precision effects
+in the primitive solver. The scaling decreases the effec-
+tive floor value while avoiding the problems associated
+with having a true vacuum.
+The Maxwell equations require that the magnetic field
+be divergenceless. This is the so-called “no monopole”
+constraint. A variety of schemes exist with the goal
+of controlling the growth of the divergence. We choose
+a strategy that ensures flexibility and robustness when
+dealing with multiple grid structure (as in AMR) and al-
+lows, in principle, for a clean boundary treatment [44].
+To this end we have implemented hyperbolic divergence
+cleaning as described in [36] (also see [45]). We thus
+introduce a scalar Ψ( x,y,z,t) which is sourced by the
+negative of the divergence of the magnetic field as shown
+in Eq. (30). As described in [36], this scheme implies
+that the divergence obeys a damped wave equation so
+that constraintviolationspropagateoffthe gridandtheir
+value is reduced.
+For initial data generated with Magstar, the diver-
+gence is around machine precision, and so to test the
+implementation of divergence cleaning, we introduce a
+perturbation to the magnetic field in order to produce a
+significant amount of divergenceto the magnetic field. In6
+particular, this perturbation takes the form of a spher-
+ical, Gaussian shell of radius ro, width δ, and ampli-
+tudeAadded to each component of /vectorB. We expect the
+divergence cleaning to propagate this perturbation as a
+damped wave, and we therefore plot the scaled quantity
+x2/bracketleftBig
+/vector∇·/vectorB/bracketrightBig
+in Fig. 1. As can be seen by comparing the
+cases of no cleaning ( top) and with cleaning ( bottom), the
+divergence propagates with damping through the refine-
+ment boundaries across the grid.
+As typical with codes dealing with linearly degenerate
+hyperbolic systems, like those those employed in numeri-
+calrelativity, adissipationoperatoris appliedtothemet-
+ric variables. This operation uses a high-order derivative
+to serve as a low-pass filter and does not affect the accu-
+racy of the simulation. We have found it useful for keep-
+ingthingssmooth. Althoughthisoperationisnotapplied
+to the fluid variables, we have found it quite important
+in keeping the magnetic field components smooth. The
+magnetic field evolution is coupled tightly with that of
+the velocity, and any nonsmoothness which appears in
+the velocity can easily affect the smoothness of the mag-
+netic field. The addition of dissipation to the magnetic
+field and divergence cleaning field helps control the be-
+havior of the magnetic field.
+At the boundaries of the domain, simple outflow
+boundary conditions are applied to the fluid variables.
+This is accomplished by copying the values of the conser-
+vative variables near the boundary outward. Most of the
+gravitational variables are treated using Sommerfeld-like
+boundary conditions of the form [46]
+/parenleftbigg
+∂t+∂r+1
+r/parenrightbigg
+(gab−ηab) = 0,(38)
+whereηabis just the Minkowski metric. The rest, which
+are not so crucial, are set either by maximally dissipa-
+tive[2] orconstraintpreservingboundaryconditions[31].
+A relevant issue when considering the collapse of stars
+is the formation of a black hole. To deal with such situa-
+tions, we adopt black hole excision where we dynamically
+monitor for the appearanceof trapped surfaces (which lie
+inside an event horizon if cosmic censorship holds) and
+excise cubical region(s) from the computational domain.
+As discussed in [47, 48], this excision introduces inner
+boundaries which are of “outflow”type and so no bound-
+ary condition is required there. However, for a more ro-
+busthandlingofthefluid, wealsoallowforamodification
+of the fluid equations inside the trapped surface [7]. The
+MHD equations are written in balance law form
+˙U+F(U)′=S, (39)
+which we modify to include a damping term near the
+black hole
+˙U+F(U)′=S−f(r)(∆x)p(U−U0).(40)
+Here the function f(r) decreases smoothly with r, from a
+givenvalueattheexcisionregiontozeroattheoutermosttrapped surface (OTS) found, and is zero for r≥rOTS,
+so that the exterior of the BH is causally disconnected
+from the effect of this extra term. U0is set to zero or
+to the value of the atmosphere if the corresponding field
+has one. The coefficient (∆ x)pensures that the damping
+term converges to zero as the grid spacing ∆ xis reduced.
+Aslongasonechooses pgreaterthanorequaltotheorder
+of convergence of the code, this term will not modify the
+convergence rate. We typically adopt a value of p= 4.
+Finally, gravitational radiation is calculated via the
+evaluation of the Newman-Penrose scalar Ψ 4which is
+computed by contracting the Weyl tensor, Cabcd, with a
+suitably defined null tetrad {ℓ,n,m,¯m}
+Ψ4=Cabcdna¯mbnc¯md(41)
+extracted at spherical surfaces Σ ilocated in the wave-
+zone, far from the sources. We also consider possible
+corrections required to deal with gauge ambiguities, as
+discussed in [49]. We refer the reader to that paper for
+details on the adopted tetrad and required corrections.
+IV. DIAGNOSTIC QUANTITIES
+The initial configuration for the stars is axisymmetric,
+and we therefore want to be able to measure any change
+to this structure. For this purpose, we monitor certain
+distortion parameters [50] defined as
+η+=Ixx−Iyy
+Ixx+Iyy(42)
+η×=2Ixy
+Ixx+Iyy(43)
+η=/radicalBig
+η2
+++η2
+× (44)
+in terms of the moment of inertia tensor
+Ijk≡/integraldisplay
+Dxjxkd3x. (45)
+These parameters are computed with respect to the con-
+servative variable D.
+It is also standard practice to display the maximum
+of the (primitive) rest mass density, and we compute the
+fractional change in time as
+∆ρ≡max|ρ0(t)|−max|ρ0(0)|
+max|ρ0(0)|. (46)
+Generally, for stable stars one sees this quantity oscillate
+from inherent numerical perturbations about a stable so-
+lution. For unstable solutions, one expects this quantity
+to change in a significant way. Similarly, we compute the
+relative change in baryon mass as a function of time as
+∆Mbaryon≡Mbaryon(t)−Mbaryon(0)
+Mbaryon(0)(47)7
+where
+Mbaryon≡/integraldisplay
+D dV. (48)
+This mass is related to the expected baryon number and
+should be strictly conserved as long as mass is not leav-
+ing the computational domain. Many HRSC schemes
+explicitly conserve this quantity, but here it is not a pri-
+oriconserved. Our use of an atmosphere entails adding
+a small amount of mass in regions that would otherwise
+become evacuated. Another reason is that we are using
+a finite difference based AMR which does not accom-
+modate as readily a strictly conservative treatment as a
+finite volume method would [51]. Finally, the presence
+of source terms in the evolution equations for the other
+conservative variables also breaks perfect conservation of
+these other quantities.
+We also compute the angular momentum of the fluid.
+Since our stars rotate about the z-axis, we need only
+compute a single such quantity
+Jz=/integraldisplay/parenleftbig
+xnjTjy−ynjTjx/parenrightbig√
+h d3x.(49)
+The use of Cartesian coordinates is typically avoided
+when one expects angular momentum to be conserved
+since cylindrical coordinates are better adapted to the
+respective Killing vector. However, because this project
+is part of a more general effort to model a variety of
+systems with different natural coordinates and no sym-
+metries, we simply monitor the extent to which this is
+conserved by computing a fractional change as
+∆Jz≡Jz(t)−Jz(0)
+Jz(0). (50)
+Finally, we also monitor the extent to which our nu-
+merical solution satisfies the Einstein equations as ex-
+pressed in Eq. (1). That is, we compute the so-called
+residual of these equations, expressed as the four-vector
+Zaas in Eq. (9). In particular, we monitor the norm of
+this vector
+|/vectorZ|≡||Za||2. (51)
+Analytical solutions to Einstein equations must have a
+vanishing residual and thus we monitor this residual for
+solutionsobtained at variousresolutionstocheck forcon-
+vergence to a consistent solution.
+V. INITIAL DATA
+We use initial data generated with the program
+Magstar, part of the Lorene software package. These
+solutions are described in [19] and are rigidly rotating,
+magnetized neutron stars. They are generated as fully
+consistent solutions of Einstein’s equations as opposed
+to taking nonmagnetized stars and adding a seed mag-
+netic field without re-solving the constraints. The initialFIG. 2: Diagram of solution space of stars generated with
+Magstar (using its units). The total gravitational mass is
+plotted versus the central enthalpy. The upper curve (pur-
+ple, open triangles) represents stars with no magnetic field
+at the mass shredding limit. Very close to this are shown
+(green, opencircles) stars rotatingat thesame frequencie s but
+with a radial magnetic field at the pole of 1000 GT= 1016G.
+The masses of these stars are barely changed with respect
+to their nonmagnetic counterparts. The lower curve (blue,
+open squares) represents the static limit with no rotation. A
+sequence of constant baryon mass stars ( MB= 3.6Msol) is
+shown (red stars). This sequence is also shown in the inset
+along with their angular momentum (yellow open circles) and
+frequency of rotation (green stars). The location of the ini tial
+data used in the initial data convergence test of Fig. 3 and
+the long term, stability test of Fig. 4 is also shown (green,
+solid square) although that solution strictly does not belo ng
+here because it has non-vanishing magnetic field.
+magnetic field is dipolar and aligned with the rotation
+axis, produced by a current function f(Aφ) = constant
+whereAφis the toroidal component of the electromag-
+netic potential vector.
+To get a handle on the solutions generated, we first
+turn off the magnetic field and compute the “usual” two-
+parameter solution space as described in [52]. Using the
+terminology of [19], we compute solutions based on the
+two parameters of central enthalpy and frequency. Ex-
+amination of Eq. (13) of [19] shows that the log-central
+enthalpy Hcis related to the entropy used here ( heas
+defined in Eq. (14)) by
+Hc= ln(he)+C (52)
+whereCis constructed by physical constants and heis
+evaluated at the center of the star.8
+FIG. 3: The residuals of two constraint equations, namely
+the Hamiltonian constraint and the y-component of the mo-
+mentum constraint. A slice along the x-axis of the com-
+puted residuals for various unigrid resolutions. Top:That
+the Hamiltonian constraint residual converges to zero with in-
+creasing resolution is taken as evidence that the initial da ta is
+properly constructed and read into the code. Bottom: That
+the momentum constraint residual improves with the first in-
+crease in resolution is similar evidence. However, the next
+increase in resolution fails to bring down this residual. We
+suspect that this remaining error is associated not with tru n-
+cation error but instead with inherent errors in the numeri-
+cal transformation and interpolation from Lorene’s spherical
+basis to our Cartesian one. Note the spikes that arise at the
+stellar boundaries due to discontinuities in the fluid varia bles,
+these are not expected to converge to zero.
+These nonmagnetized solutions are diagrammed in a
+plot of total gravitational mass versus central enthalpy
+in Fig. 2. The lower curve shows the static limit for stars
+which are not rotating. The upper curve is the mass
+shedding limit, represented by the largest frequency for
+whichMagstar returned a solution. These curves serve
+as the upper and lower bounds on the solution space of
+stationary, unmagnetized stars. It should be noted that
+Magstar generates only rigidly rotating stars, and there-
+fore the evolutions are far from the fast rotating regime
+expected to excite large and growing nonaxisymmetric
+modes.
+We also compute and show a sequence of stars at con-
+stant baryon mass. Such sequences are important be-
+cause real stars are expected to conserve baryon mass as
+they evolve and thus the sequences are expected to ap-
+proximate their evolution. Furthermore, along such se-quences it has been shownthat there is a stability change
+at the minimum in the angular momentum [52]. Looking
+at the inset of Fig. 2, one therefore expects the solutions
+on the right to be unstable while those on the left should
+remain stable.
+Note that the addition of a non-vanishing magnetic
+field adds another dimension to this diagram although
+the effect of the magnetic field on the initial data is not
+so dramatic. Certainly there are many ways in which to
+“add”a magnetic field to a stellarsolution. In [19] a non-
+vanishing current is assumed and a solution is obtained
+with the same baryon mass but now with nonvanishing
+magnetic field. In particular, solutions at the mass shed-
+ding limit with no magnetic field are shown in Fig. 2 by
+open triangles. Almost indistinguishable from these solu-
+tions are the those magnetized such that the radial mag-
+netic field at the pole of the star is 1000 gigatesla ( GT)
+or 1016G. shown with open circles.
+We consider different perturbations. To perturb the
+pressure, we decrease it according to real parameters Ap
+andmintermsoftheunperturbed pressure p0depending
+on the azimuthal angle ϕ
+p=p0/bracketleftbig
+1−Apsin2(mϕ)/bracketrightbig
+. (53)
+We also consider perturbations to the rest mass density
+decreasing it everywhere by a fraction Aρ
+ρ=ρ0[1−Aρ]. (54)
+We verify the initial data in our evolution code in a
+numberofways. Weexamineconvergenceofthedatatoa
+unique solution which solvesthe constraints. In Fig. 3 we
+showthe residualofthe Hamiltonian and y-componentof
+the momentum constraint for three different resolutions.
+The Hamiltonian constraint is a good measure of the fi-
+delity of the solution to the Einstein equations, and, as is
+clear from the figure, its residual decreases rapidly with
+resolution. Furthermore, we confirm that the divergence
+of the magnetic field is around machine precision.
+VI. RESULTS
+Stable, Rotating Star Test:
+Before addressing the effects of the magnetic field, we
+verify that the code reproduces the expected behavior as
+described, for example, in [53]. First, we consider evo-
+lutions of stable, rotating stars and find that the code
+evolves such a star as long as desired while maintain-
+ing a stationary solution. This a demanding test as it
+depends on the balance between gravitational and hy-
+drostatic forces in a rotating configuration with both a
+non-conforming grid and variables not adapted to the
+symmetries of the problem.
+One example of the the behavior of the numerical so-
+lution is shown in Fig. 4. As apparent in the top frame of
+the figure, the fractional change in the maximum of the
+density oscillates with a slow overall increase. This oscil-
+lation is characteristic of quasinormal ringing of the star9
+FIG. 4: Convergence results for the evolution of a stable,
+nonmagnetized, rotating star (the particular initial solu tion
+is shown in Fig. 2 as a green solid square). Three FMR
+evolutions are shown, each with a multiple of the coarsest
+resolution resolving the equator of the star with: (magenta ,
+dot-dashed) 30 points/star, (blue, dotted) 60 points/star , and
+(black, solid) 120 points/star. Also shown is a run with the
+same stellar resolution of (red, long-dashed) 60 points/st ar
+but with a coarse grid that extends twice as far in all direc-
+tions. The two highest resolution runs were terminated earl y
+because of the computational cost, not because of any robust -
+ness problems. The insets show the same data but in finer
+detail for the first rotational period. The topframe shows
+the fractional change in maximum density versus rotational
+period. With increasing resolution the solution better ap-
+proximates a stationary solution. The upper middle frame
+shows the fractional change in the baryon mass, Although our
+scheme, using vertex centered AMR, an atmosphere, and full
+general relativity withsources, is notstrictly conservat ive, the
+plot shows that deviations from strict conservation are sma ll
+and decrease with more resolution. The lower middle frame
+shows the fractional change in integrated angular momentum .
+Again, the code demonstrates convergence to conservation.
+Thebottom frame shows the maximum of the norm of the
+Zaconstraint residuals as a function of time which also con-
+verge.
+excited by inherent numerical error. However, the figure
+shows data for a number of resolutions, and the trend as
+resolution improves is toward a flatter curve. This trend
+is particularly apparent in the inset which shows just the
+first rotational period. This behavior suggests that the
+code is converging to the continuum solution.
+In addition to the runs with varying resolution, Fig. 4
+shows another evolution with a coarse level extendingFIG. 5: Collapse to black hole of an unstable, unperturbed,
+magnetized star for multiple of a base resolution. The ini-
+tial data is an unperturbed Magstar solution with central en-
+thalpyHc= 0.8 rotating at a frequency f= 835Hzand with
+polar magnetic field of 1000 GT= 1016G. Thetopframe
+shows the maximum density which increases with time as the
+star collapses. The upper middle frame shows a norm of
+the divergence of the magnetic field. The lowerframes show
+the distortion parameters of the Dfield as functions of time.
+twice as far but with fine levels identical to the medium
+resolution run just discussed. Generally, the results are
+thesameasforthenon-extendeddomain,indicatinglittle
+effect from the boundary for the the first half-period.
+Although the fluid scheme usedhere isnot strictly con-
+servative because of (i) AMR boundaries for our vertex
+centered scheme, (ii) our outer boundary treatment, and
+(iii) our use of a fluid floor, Fig. 4 shows that the frac-
+tional change in the volume-integrated baryon mass re-
+mainsconstanttoahighdegree. Similarly,theintegrated
+angular momentum of the fluid converges to conserva-
+tion.
+Finally, the bottom frame of Fig. 4 shows the maxi-
+mum value of the constraint violation. These values in-
+crease with time as the numerical error accumulates, but
+higher resolution runs demonstrate less violation though
+it saturates due to the intrinsic error of the initial and
+boundary data.
+Unstable, Rotating, Magnetized Star Test:
+Similarly, we evolve a rotating, magnetized star located
+on the unstable side. These stars, perturbed by inherent
+numerical error, collapse to black holes. We see no evi-
+dence of significant disk formation (as in e.g. [15, 16, 54–10
+56]). Notice that even though we do not impose any
+type of symmetry, no asymmetric, unstable modes are
+observed. This behavior is consistent with previous work
+which studied the possible onset of axisymmetric insta-
+bilities [57]. These previous studies found that such in-
+stabilities require high rotation rates characterized by
+T/W >0.25 in contrast to those studied here for which
+T/W≤0.1. Consequently, we see no evidence for defor-
+mations of the star as would be apparent by monitoring
+the distortion parameters moving away from zero.
+In Fig. 5, we show an example of such an evolution at
+three successively finer resolutions. Because the star is
+collapsing, the maximum density increases dramatically.
+The magnetic field remains essentially poloidal through-
+out the collapse and its maximum magnitude increases
+due to the resulting compression of the field lines. De-
+spite this increase, the magnetic field plays no important
+role in the collapse since its associated pressure is still
+several orders of magnitude smaller than the fluid pres-
+sure. While we do observe an increase in the norm of the
+divergence, the growth is not particularly fast and the
+divergence remains small in absolute terms and relative
+to the magnetic field.
+Fig. 6 displays two snapshots of the density and mag-
+netic field strength of the collapsing star tested in Fig. 5.
+The first one illustrates a stageduring the collapse before
+an apparent horizon forms. The second one shows the
+behavior after an apparent horizon is found and its inte-
+rior excised. The apparent horizon appears at t≃0.6P,
+when the maximum of the density is ρmax= 0.271 and
+the minimum of the lapse is αmin= 0.2.
+Perturbations of Unstable Stars :
+Previous work presented in [53] argues that, in general,
+unstablestarsshouldeither collapsetoablackholeorex-
+pandandoscillateaboutastablestellarsolution. Seeking
+to duplicate this behavior, we perturb an unstable star.
+Indeed, as shown in Fig. 7, we find precisely the behav-
+ior described. If our perturbation increases the pressure
+sufficiently, we find a star which expands and oscillates
+about some other, presumably stable solution. However,
+if we choose a perturbation which barely increases the
+pressure, the star collapses to a black hole.
+However, given the interest in black hole critical phe-
+nomena (see [58, 59]) over the past couple of decades,
+we study in detail the separation between these two be-
+haviors. That is, we continue to adjust Apin Eq. (53)
+searching for a value A∗
+pabove which one finds black hole
+formation, and below which one finds an expanding so-
+lution (the way we have parameterized the pressure per-
+turbation, A∗
+p<0).
+As apparent from Fig. 7, the more one continues this
+tuning, thelongertheunstablestarsurvives. Thistypeof
+tuning is reminiscent of a similar analysis of an unstable,
+irregular static solution [60]. What these results suggest
+is that the unstable solution (i) sits at the threshold of
+black hole formation with (ii) a single unstable mode.
+That small perturbations about the solution send it ei-
+FIG. 6: Density ρ0and magnetic field strength on the y= 0
+plane for the same star as shown in Fig. 5. The density is
+shown with respect to the colormap while the contours denote
+themagnetic field, whichisequallyspacedfrom 0to5 ×1015G.
+The top left corresponds to t= 0.5Pwhile the thetop right to
+t= 0.9P(the circle shown represents the apparent horizon.)
+The bottom plot illustrates the normalized mass as a functio n
+of time, along with its rate of change.
+FIG. 7: The maximum density ρmaxas a function of time for
+an unstable, perturbed ( m= 3 in Eq. (53)), non-magnetic,
+rotating star. The (unperturbed) initial star has central e n-
+thalpy 0.8 and rotates at the mass shedding limit. Solutions
+resulting from tuning the amplitude of the perturbation Apto
+roughly one part in 106show the two disparate outcomes (as
+discussed in [53]): collapse to black hole or violent oscill ations
+about a stable stellar solution with equivalent mass. With
+more tuning, the star resembles the initial, unstable solut ion
+for a longer time.11
+ther to collapse or expansion suggest that it sits at the
+threshold. Furthermore that a single parameter is suffi-
+cient to stabilize the solution suggests that there is a sin-
+gle unstable mode. In contrast, sometimes one can tune
+multiple parameters to find a threshold solution [61].
+If these suggestions hold up, these would suggest that
+at least some of these unstable solutions might serve
+as Type I critical solutions. Indeed, previous work [62]
+perturbed stable TOV stars in spherical symmetry
+and found unstable TOV stars at criticality in Type
+I collapse. In that work, they were able to achieve
+phenomenal resolution and tuning. In contrast, while
+they perturbed a self-consistent stablesolution and saw
+the tuned evolution driven to the unstable branch of
+solutions, here we begin with the unstable solution and
+perturb around it. Here, we have only been able to tune
+to about one part in a million, being prevented from
+tuning further because successive evolutions stop the
+trend towards longer lived solutions. That such searches
+are prevented from continuing might indicate some
+new phenomena, or, more likely, that boundary effects
+andnumericalerrorbegintospoilthethresholdbehavior.
+Wehavelookedforascalinglawinthesurvivaltimesof
+these tuned unstable stars. To the extent that this rough
+tuning is representative of the overall behavior, we find
+that the different solutions appear to scale as expected.
+However, because our searches have terminated so far
+from criticality, we cannot have much confidence in a
+precise scaling relationship. There has been recent work
+in the axisymmetric collision of neutron stars [63, 64]
+which appears to demonstrate the same type of critical
+behavior as observed in [62].
+We find the same type of behavior about a magne-
+tized star as shown in Fig. 8. Here we have carried out
+three searches by varying a different parameter all per-
+turbingthesamesolution. Thefiguremakesapparentthe
+same ringing for all three families, although the ampli-
+tude varies across the different tuning families. It seems
+reasonable to take the results of these tunings as further
+evidence that only a single mode is unstable since if there
+were more unstable modes, these different tunings would
+produce solutions more varied from each other. We have
+begun to look at the geometry of the purported unstable
+mode by looking at the difference ∆ = ρ0(t)−ρ0(0) at
+late times for near-critical evolutions. These calculations
+indicate the the mode is likely axisymmetric with differ-
+ences between ∆ on the x= 0 plane and on the y= 0
+plane being at about the 10% level.
+VII. CONCLUSIONS
+We study the evolution of rotating stellar configura-
+tions and examine different phenomenology related to
+stability and magnetic field influence in their dynamical
+behavior. The stars are constructed assuming a poly-
+tropic equation of state using the code Magstar whichFIG. 8: Results of three critical searches for a magnetized, ro-
+tating star with central enthalpy of 0 .8 and a radial magnetic
+field at the pole of 1 ,000GT. One search varied the ampli-
+tude of the pressure perturbation Apwithm= 7 (black and
+cyan), another with m= 3 (blue and magenta), and another
+variedthedensityperturbation Aρ. Insolid line areshownthe
+super-critical evolutions which collapse to black holes, w hile
+sub-critical solutions for the two searches are shown with d ot-
+ted lines. The search over pressure with m= 7 achieved tun-
+ing to about one part in 103, that with m= 3, one part in
+104, and that over density achieved about one part in 105.
+is part of the publicly available Lorene package. We
+present several studies which indicate our code reliably
+and robustly evolves astrophysically relevant scenarios
+including magnetic field effects. We study the dynamical
+effect of perturbations of stars on the unstable branch of
+solutions. We find evidence that these unstable solutions
+may play a similar role as the unstable TOVstars play in
+spherically symmetric evolutions as studied in [62]. This
+is significant because it suggests that the addition of an-
+gular momentum, magnetic field, and three dimensions
+do not allow for a multitude of unstable modes. Needless
+to say, the phenomena associated with the threshold of
+gravitational collapse merit further study.
+Beyond the studies considered in this work, further in-
+teresting phenomenology to consider include the impact
+of magnetic fields in the stability of the star, a thor-
+oughanalysisofthe possiblecriticalphenomenaobserved
+and differences due to more realistic equations of state.
+The investigation ofsuch scenarioswill be presented else-
+where.
+Acknowledgments
+We would like to thank J. Novak for his gracious assis-
+tance with Magstar andLorene. SLL thanks Matthew
+Choptuik and Scott Noble for valuable discussions con-
+cerningtheapparentcriticalbehavior. Wealsowouldlike
+to thank Matthew Anderson, Eric Hirschmann, Miguel
+Megevand, Patrick Motl, Joel Tohline and Travis Gar-12
+ret for comments and suggestions. SLL, DN and CP
+thank the Perimeter Institute for Theoretical Physics
+for hospitality where parts of this work were com-
+pleted. This work was supported by the National Sci-
+ence Foundation under grants PHY-0803629 and PHY-
+0653369toLouisianaState University, CCF-0832966and
+PHY-0803615 to Brigham Young University, and CCF-
+0833090, PHY-0803624 to Long Island University. and
+NSERC through a DiscoveryGrant. Researchat Perime-ter Institute is supported through Industry Canada and
+by the Province of Ontario through the Ministry of Re-
+search& Innovation. This researchwas alsosupported in
+part by the National Science Foundation through Tera-
+Grid resourcesprovidedby SDSC under allocationaward
+PHY-040027. In addition to TeraGrid resources, we
+have employed clusters belonging to the Louisiana Opti-
+cal Network Initiative (LONI), and clusters at LSU and
+BYU.
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\ No newline at end of file
diff --git a/1001.0576.txt b/1001.0576.txt
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+++ b/1001.0576.txt
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+arXiv:1001.0576v1  [astro-ph.GA]  4 Jan 2010The Orbit of the Orphan Stream
+Heidi Jo Newberg1, Benjamin A. Willett1, Brian Yanny2, & Yan Xu3
+ABSTRACT
+We use recent SEGUE spectroscopy and SDSS and SEGUE imaging dat a
+to measure the sky position, distance, and radial velocities of star s in the tidal
+debris stream that is commonly referred to as the “Orphan Stream .” We fit
+orbital parameters to the data, and find a prograde orbit with an a pogalacticon,
+perigalacticon, and eccentricity of 90 kpc, 16.4 kpc and e= 0.7, respectively.
+Neither the dwarf galaxy UMa II nor the Complex A gas cloud have velo cities
+consistent with a kinematic association with the Orphan Stream. It is possible
+that Segue-1 is associated with the Orphan Stream, but no other k nown Galactic
+clusters or dwarf galaxies in the Milky Way lie along its orbit. The detect ed
+portion of the stream ranges from 19 to 47 kpc from the Sun and is a n indicator
+ofthemassinteriortothesedistances. Thereisamarkedincrease inthedensityof
+Orphan Stream stars near ( l,b) = (253◦,49◦), which could indicate the presence
+of the progenitor at the edge of the SDSS data. If this is the proge nitor, then
+the detected portion of the Orphan Stream is a leading tidal tail. We fi nd blue
+horizontal branch (BHB) stars and F turnoff stars associated wit h the Orphan
+Stream. Theturnoffcoloris( g−r)0= 0.22. TheBHBstarshavealowmetallicity
+of [Fe/H] WBG=−2.1. The orbit is best fit to a halo potential with a halo
+plus disk mass of about 2 .6×1011M⊙, integrated to 60 kpc from the Galactic
+center. Our fits are done to orbits rather than full N-body simulations; we show
+that ifN-body simulations are used, the inferred mass of the galaxy would be
+slightly smaller. Our best fit is found with a logarithmic halo speed of vhalo=
+73±24 km s−1, a disk+bulge mass of M(R <60 kpc) = 1 .3×1011M⊙, and
+a halo mass of M(R <60 kpc) = 1 .4×1011M⊙. However, we can find similar
+fits to the data that use an NFW halo profile, or that have smaller disk masses
+and correspondingly larger halo masses. Distinguishing between diffe rent classes
+of models requires data over a larger range of distances. The Orph an Stream
+1Dept. of Physics, Applied Physics and Astronomy, Rensselaer Polyt echnic Institute Troy, NY 12180,
+newbeh@rpi.edu
+2Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, I L 60510
+3National Astronomical Observatories of China, 20 Datun Road, Be ijing, China– 2 –
+is projected to extend to 90 kpc from the Galactic center, and mea surements of
+these distant parts of the stream would be a powerful probe of th e mass of the
+Milky Way.
+Subject headings: Galaxy: structure — Galaxy: halo — Stars: kinematics
+1. Introduction
+The discovery that the Milky Way’s stellar halo is lumpy (Newberg et al. 2 002) has
+sparkedaflurryofactivityinidentifyingtheMilkyWay’sstellarsubstr ucture. Dwarfgalaxies
+andglobularclustersundergotidaldisruption, creatingstreamso fstarsinthehalo. New, low
+surface brightness dwarf galaxies and globular clusters have been identified (Willman et al.
+2005; Koposov et al. 2007; Belokurov et al. 2007a), and several la rge structures have been
+discovered whose nature is unknown or controversial (Yanny et a l. 2003; Majewski et al.
+2004; Belokurov et al. 2007c; Juri´ c et al. 2008). Tidal debris in th e Milky Way’s stellar
+halo teaches us about the accretion history of the Milky Way, and it h as been suggested
+that tidal debris may be the key to measuring the distribution of dar k matter in the Galaxy
+(Combes et al.1999;Murali & Dubinski1999;Helmi2004;Johnston , Law & Majewski 2005;
+Eyre & Binney 2009; Majewski et al. 2009; Odenkirchen et al. 2009) .
+One of the newly discovered substructures is the “Orphan Stream ,” which was indepen-
+dently discovered by Grillmair (2006) and Belokurov et al. (2006). Gr illmair (2006) was the
+first to publish a full discovery paper, showing that the stream was at least 60◦long and 2◦
+wide, and likely the remains of a small dwarf galaxy that has been comp letely disrupted. In
+Grillmair’s paper, the stream was characterized as about 21 kpc dist ant from the Sun along
+the whole length of the stream. Belokurov et al. (2007b) published a n independent discov-
+ery paper naming the Orphan Stream; they found a similar length, wid th, and likely origin.
+However, Belokurov et al. found a strong distance gradient, from 20 kpc at one end of the
+stream to 32 kpc from the Sun at the other end. They also published “suggestive” radial
+velocities from sparse samples of Sloan Digital Sky Survey (SDSS) da ta in two fields, ranging
+fromVR= -40 km s−1at the close end of the stream to ∼100 km s−1at the distant end
+of the stream. They noted that the dwarf galaxy Ursa Major II, t he HI clouds of Complex
+A, and a number of anomalous globular clusters (including Ruprect 10 6 and Palomar I) lie
+near the same great circle as the Orphan Stream stars.
+Followingthediscoverypapers, threefurtherstudies, Fellhauer e t al.(2007),Jin & Lynden-Bell
+(2007), and Sales et al. (2008), attempted to fit an orbit the the O rphan Stream and to de-
+termine whether it was connected to the assortment of objects o n the same great circle.– 3 –
+Fellhauer et al. (2007) fit two tidal disruption models to the available O rphan Stream data,
+with the assumption that the UMa II dwarf galaxy is the progenitor o f the stream. They
+successfully fit the position and distance data, and the velocity for the more distant part of
+the stream. They are only able to fit the near velocity, at VR=−40 km s−1, by assigning
+part of the Orphan Stream to the leading tail and part of the Orpha n Stream to the trailing
+tail. They conclude that UMa II is a likely progenitor of the Orphan Str eam, that some of
+the young halo globular clusters may be associated with the Orphan S tream, that Complex
+A could be matched only if it was a whole orbit ahead of the Orphan Stre am, and that the
+picture made the most sense if all of these objects came from an or iginal object that was
+a larger dwarf galaxy. Shortly after the Fellhauer et al. paper, Jin & Lynden-Bell (2007)
+explored the possibility that Complex A and the Orphan Stream were r elated to each other.
+They used the sky positions of the Orphan Stream and Complex A and the radial velocities
+of Complex A, assumed the two objects were along the same orbit, a nd fit the values for the
+distances to the two objects and the radial velocities along the Orp han Stream. In order to
+make the orbit fit, the distance to Complex A is 4 kpc, the distance to the Orphan Stream
+is assumed to be 9 kpc, and the heliocentric radial velocity of the Orp han Stream ranges
+from -155 km s−1to -30 km s−1. These values are in conflict with the measurements of
+previous authors and this paper, so it seems unlikely that the Orpha n Stream and Com-
+plex A are on the same orbit, and in the same revolution of orbital pha se. More recently,
+Sales et al. (2008) successfully fit an orbit to the distances and velo cities measured in the
+Belokurov et al. (2007b) paper. They concluded that the orbit of t he Orphan Stream does
+not intersect UMa II or Complex A, and that the data is consistent w ith “a single wrap of
+the trailing arm of a fully disrupted satellite.”
+The fits to the Orphan Stream orbit depend on having reliable measur ements of the star
+positions, distances, and (especially) velocities along the Orphan St ream. In this paper, we
+extract the best distances and velocities of Orphan Stream stars in the SDSS Data Release
+7 (DR7; Abazajian et al. 2009), making extensive use of the newly re leased Sloan Exten-
+sion for Galactic Understanding and Exploration (SEGUE) subsample of stellar velocities
+(Yanny et al. 2009a). We show that the suggestive Belokurov et al. (2007b) velocities, which
+were used in the orbit calculations of later authors, were not repre sentative of true Orphan
+Stream members.
+We fit a new orbit to the Orphan Stream data that is consistent with o ur new measure-
+ments, and find a best fit Milky Way mass of M(R<60 kpc) = 2 .65×1011M⊙.If we extrap-
+olate the integrated mass out to a radius 240 kpc (approximately th e virial radius), assuming
+a log potential normalized to vhalo= 73 km s−1, we findM(R <240 kpc) ∼6.9×1011M⊙.
+A recent history of mass estimates for the Milky Way include Wilkinson & Evans (1999),
+who use the orbits of satellite galaxies and globular clusters and find M(R <50 kpc) =– 4 –
+5.4+0.2
+−3.6×1011M⊙andMvirial= 1.9+3.6
+−1.7×1012M⊙. Klypin et al. (2002), on theoretical forma-
+tion grounds predict a virial mass for the Milky Way Mvirial∼1×1012M⊙, almost twice as
+high as our measurement. Other estimates are from Battaglia et al. (2005) with Mvirial=
+0.8+1.2
+−0.2×1012M⊙from halo tracers and Li & White (2008) with Mvirial= 2.43×1012M⊙
+from theory and relative motions of the Milky Way and Andromeda. Ab adi et al. (2006)
+used a careful combination of predicted velocity dispersion theory and observations from
+Battaglia et al. (2005) to arrive at Mvirial∼6.4×1011M⊙, consistent with the value we
+propose here. Furthermore, Sales et al. (2007) propose a relatio n, based on analysis of large
+N-body simulations of satellites and their hosts, between the disper sions of the satellite ve-
+locities and the virial velocity of each host. Applied to the Milky Way, th ey derive a virial
+mass of 7 ×1011M⊙. Our mass estimate is on the low end, but within the range of recent
+measurements of the Milky Way’s mass.
+2. Angular position of the Orphan Stream
+We begin with ananalysis of the positions of stream stars in the sky, a s thesky positions
+of the Orphan Stream are the least controversial measurements . We select stars from SDSS
+and SEGUE photometric data.
+The SDSS Legacy survey imaged one quarter of the sky in ugrizfilters uniformly (pho-
+tometric color errors of <2% tor∼21) to faint depth ( g∼23). Additional imaging was
+carried out as part of the SEGUE survey. Simple color cuts allow one t o isolate large pop-
+ulations and construct density maps of certain selected spectral types. Magnitude-position
+space maps of blue horizontal branch (BHB), F turnoff and red gian t branch (RGB) stars,
+being more or less accurate standard candles, are a tool for isolat ing substructure in stars in
+our Milky Way’s halo (Newberg et al. 2002). We deredden all halo star c andidate photome-
+try (indicated with a ‘0’ subscript on magnitudes and colors), which a regenerally beyond the
+foreground dust screen, using the standard maps of Schlegel, Fin kbeiner, & Davis (1998).
+We selected from the DR7 database of STARs, all objects with 100◦< α < 175◦
+and−25◦< δ <70◦. Later in the paper, we will show that the stars of the Orphan
+Stream are at distances between 15 and 50 kpc from the Sun, and h ave a turnoff color near
+(g−r)0= 0.22. Therefore, to select from this dataset all of the stars that c ould be Orphan
+Stream turnoff stars, we selected only those stars with 0 .12<(g−r)0<0.26, (u−g)0>0.4,
+and 20.0<g0<22.5. Redder stars were not selected because there is a larger backg round of
+field disk and halo stars on the red side of the turnoff than on the blue side. A greyscale plot
+of the sky densities of these stars, presented in Galactic coordina tes, is shown in Figure 1.
+The most prominent excess of F turnoff stars is from the leading tida l tail of the Sagittarius– 5 –
+dwarf spheroidal galaxy, and a fainter stream, almost parallel to t he first that may be a
+bifurcation of the Sagittarius dwarf tidal stream (Belokurov et al. 2006). The narrower,
+more horizontal, tidal debris stream is the Orphan Stream. Note th at the surface density
+of stars in the Orphan Stream increases just before the left edge of the SDSS data. Since
+this portion of the stream is closer to the Galactic center (see below ), we do not expect the
+density to increase as it does at apogalacticon where the velocities a re lower. The increase
+in density may indicate the progenitor is at the edge, near l= 253◦, or just off the edge of
+the data on the left side, towards higher Galactic longitudes.
+We trace the position of the Orphan Stream across the sky, and ta bulate the Galactic
+latitude every ten degrees in Galactic longitude along the stream in Ta ble 1. The center
+of the stream was determined to the nearest half degree pixel, so t he error in latitude is
+estimated at about 0.7 degrees. We then calculated the Euler angles required to transform
+from Galactic coordinates to a Sun-centered coordinate system w here the portion of the
+Orphan Stream that we detected is on the equator. Using a notatio n similar to that used
+by Majewski et al. (2003) to describe the Sagittarius dwarf orbita l plane, we define BOrphan
+and Λ Orphanby the rotation:
+
+cosBOrphancosΛOrphan
+cosBOrphansinΛOrphan
+sinBOrphan
+=M
+cosbcosl
+cosbsinl
+sinb
+,
+where
+M=
+cosψcosφ−cosθsinφsinψcosψsinφ+cosθcosφsinψsinψsinθ
+−sinψcosφ−cosθsinφcosψ−sinψsinφ+cosθcosφcosψcosψsinθ
+sinθsinφ −sinθcosφ cosθ
+.
+The values of ( φ,θ,ψ) = (128.79◦,54.39◦,90.70◦) were determined by using gradient descent
+to find the fit that minimized the sum of the squares of the angular dis tances between the
+equator at BOrphan= 0◦and the first nine stream positions in Table 1 (the last point was
+added after further study discussed in §4). The zeropoint of Λ Orphanwas set at l= 220◦,
+near where the leading tidal tail of the Sagittarius dwarf crosses t he Orphan Stream. In the
+lower panel of Figure 1, we show that the line BOrphan= 0◦passes through the center of the
+Orphan Stream in the region where we detect it. The (Λ Orphan,BOrphan) values for points
+along the Orphan Stream are also tabulated in Table 1. Note that at lo w Galactic longitude,
+the stream departs slightly from BOrphan= 0◦. It is not possible to fit an orbit over the whole
+sky with a simple rotation of coordinates, even if the Galactic potent ial is spherical, since
+we view the sky from a Sun-centered, rather than Galaxy-center ed, viewpoint. In general,
+halo orbits do not follow great circles over their entire length.– 6 –
+3. Distance to the Orphan Stream
+Although the large number of F turnoff stars make them the best tr acers of the position
+of the stream in the sky, their large range of absolute magnitudes a t a given distance make
+them less useful as distance indicators. Instead, we start by pho tometrically selecting blue
+horizontal branch (BHB) stars along the stream. We color-select A stars from all of the
+STARsinSDSSDR7that arewithinonedegreeofthecenter ofthest ream(−1◦<BOrphan<
+1◦), using the color range −0.4<(g−r)0<0.0, 0.8<(u−g)0<1.6, and the magnitude
+range 15<g0<21.5. In Yanny et al. (2000) we showed that low and high surface gravit y A
+stars could be roughly separated from each other using their ugrcolors, as shown in Figure
+10 of that paper; low surface gravity BHB stars are redder in ( u−g)0at a given ( g−r)0.
+We perform this separation on the color-selected A stars.
+In Figure 2 we show g0vs. ΛOrphanfor the 593 stars with colors of BHB stars and within
+one degree of the path of the Orphan Stream in the Λ Orphanrange showed. We argue below
+that an excess of BHBs exists along the line shown, and have added s pectrosopic Orphan
+Stream candidates to the figure here as open circles. Above the loc us of Orphan Stream
+BHBs stretching from g0= 17 at Λ Orphan= 20◦tog0= 19 at Λ Orphan=−25◦, there is a
+set of what appears to be blue straggler stars that, due to imperf ectugrcolor separation,
+leaked into our selection box two magnitudes fainter (Yanny et al. 20 00). Our distances to
+positions along the Orphan Stream agree well with those of Belokuro v et al. (2007b) where
+the data directly overlap.
+We now refine the distance to the stream using SDSS/SEGUE spectr oscopy of BHB
+stars. Because we can rely on the spectroscopy to separate low f rom high surface gravity
+(measured by the tabulated DR7 parameter logg) A stars, we can d etect horizontal branch
+starsthatareredder than( g−r)0= 0, where thephotometricseparationbecomesineffective.
+We selected all spectra from SDSS DR7 that were within 1◦of the Orphan Stream ( −1◦<
+BOrphan<1◦), in the range −30◦<ΛOrphan<30◦where the the stream is visible in F
+turnoff stars, in the color range −0.4<(g−r)0<0.3 and 0.8<(u−g)0<1.6, and
+with 0.1<logg<3.5. We further required that the sppParams table (SSPP, see Sect ion
+6) parameter “elodierverr” be greater than zero, indicating that the pipeline successfully
+fit a stellar template to the spectrum. This removes QSOs and galaxie s from the sample.
+Starting with this sample, we made a series of plots of vgsrvs. ΛOrphanandg0vs. ΛOrphan,
+and removed stars that were not clustered in both distance and ve locity. In other words,
+if stars in an apparent velocity clump were not also clustered in appar ent magnitude, then
+that group was removed from the sample. We also discovered that b y selecting stars with
+low metallicity, [Fe/H] WBG<−1.6 (Wilhelm, Beers, & Gray 1999), we lost fewer stream
+stars than background. The BHB stars that were left after this it erative selection process– 7 –
+are plotted in Figure 2 as open circles.
+We measure the apparent magnitude of BHB stream stars in five plac es where there
+is enough data in Figure 2 to be fairly confident of the stream location ; these magnitudes
+are tabulated in Table 1, along with estimates of the error. These er ror estimates are
+rough estimates based on how much we believed we could vary the mag nitude and still stay
+consistent with the datapoints on the stream. We then estimate th e functiong0(ΛOrphan) for
+Orphan Stream BHB stars by fitting a parabola to the four datapoin ts with lower errors.
+The best fit is
+g0= 0.00022Λ2
+Orphan−0.034ΛOrphan+17.75.
+4. Color-Magnitude Diagram of the Orphan Stream
+We can now use the relationship between the g0magnitude of the horizontal branch
+and Λ Orphanto adjust the apparent magnitudes of stars along the Orphan Str eam so that
+the horizontal branch is at a constant magnitude across its entire extent. We do this by
+calculating gcorr=g0−0.00022Λ2
+Orphan+0.034ΛOrphan. Notethatthepositionofthehorizontal
+branch at Λ Orphan= 0◦remains unchanged. We select all of the STARs from SDSS DR7 that
+are within 2◦of the Orphan Stream ( −2◦<BOrphan<2◦) and in the portion of the stream
+where it is visible in Figure 1 and away from the Sagittarius stream ( −26◦<ΛOrphan<−7◦
+or 10◦<ΛOrphan<40◦). Background stars were selected from the same regions of Λ Orphan,
+but with 2◦<|BOrphan|<4◦.
+In Figure 3, we show a background subtracted Hess diagram of sta rs along the Orphan
+Stream. In order to better see the horizontal branch stars, we include an insert of the BHB
+section of the ‘on stream’ ( |Bcorr,Orphan |<1) color magnitude diagram (without background
+subtraction) in the regions of the diagram in which the color sub-sele ction is valid. Here,
+the Orphan BHBs are clearly seen at gcorr= 17.75 as an overdensity. There is also a large
+overdensity of F turnoff stars at ( g0,(g−r)0) = (21.2,0.22).
+We are now in a position to ‘correct’ the g0magnitudes of the stars in Figure 1 so that
+turnoff stars along the entire length of the Orphan Stream will be at the same magnitude.
+This allows us to make a narrower magnitude cut and see the stream o ver a larger range
+of distances. In Figure 4 we show the ( l,b) distribution of the F turnoff stars with 0 .12<
+(g−r)0<0.26 and 20.7< gcorr<21.7. This allows us to see the Orphan Stream all the
+way tol= 170◦. The position of the stream at l= 171◦was added to Table 1 from analysis
+of Figure 4. The upper panel of Figure 4 shows that the Orphan Str eam is several degrees
+aboveBOrphan= 0◦at low Galactic longitude.– 8 –
+This diagram suggests that for l<200◦, we should search for Orphan Stream stars over
+a higher range of BOrphan. In order to follow the Orphan Stream at lower Galactic latitudes,
+it is thus necessary to correct BOrphan(ΛOrphan), since the stream is not at BOrphan= 0◦in this
+region. We fit a parabola to the to the three points at l= 175◦,185◦,and 195◦in Table 1, to
+findBOrphan=−0.00628Λ2
+Orphan−0.42ΛOrphan−5.00 for−35◦<ΛOrphan<−15◦. We define
+a new variable, Bcorr, for which the Orphan Stream is at Bcorr= 0◦along our entire data set.
+Bcorr=BOrphanfor ΛOrphan≥ −15◦, andBcorr=BOrphan+0.00628Λ2
+Orphan+0.42ΛOrphan+5.00
+for ΛOrphan<−15◦.
+In order to search for the Orphan progenitor in the gap between 2 55◦< l <268◦
+where there is no SDSS data available, we searched the online SuperC OSMOS data set
+(Hambly et al.2001)forfaintturnoffstarsinthegap. Weselecteds tarswith0.5<BJ−R2<
+1.0,20<BJ<21 and added them to Figure 4 with approximately the same scaling and the
+same orientation as the SDSS/SEGUE imaging. A faint,narrow trail, like ly the extension of
+theOrphanstreamtidal tail, isvisible at255◦<l<268,38◦<b<48◦. TheSuperCOSMOS
+data suffers from poor photometric calibration across and betwee n measured photographic
+plates, which makes it impossible to trace the Orphan Stream in the re gion in which we
+detect it in the SDSS data. This makes it difficult for us to calibrate the density of stars in
+the SuperCOSMOS detection with the density observed in the SDSS d ata. We are unable
+to either confirm or rule out the overdensity at ( l,b) = (253◦,49◦).
+5. Density of F turnoff stars along the Orphan Stream
+In Figure 5, we show the number of F star counts above backgroun d, as a function
+of position along the Orphan Stream. The stream stars were select ed to be within −2◦<
+Bcorr<2◦. The background stars were selected from the region of the sky t hat is between
+2◦and 4◦in angular distance from the stream (including stars on both sides of the stream).
+The binning of Figure 5 was chosen so that ‘edge-of-bin’ effects do n ot dominate the relative
+number counts in the ‘outrigger’ SEGUE stripe at l= 270◦, (ΛOrphan= +35◦).
+We note from this density plot that the stellar density is much higher n ear ΛOrphan= 22◦
+thanitisalongtherest oftheobserved tidaltail. Thisisvery interes ting because thisportion
+of the stream is also the closest to the Sun and the Galactic center. It is not expected that
+the stellar density will rise sharply along the tidal stream, especially c loser to the Galactic
+center, except near the progenitor. Stream stars generally pile u p at apogalacticon, where
+they are moving more slowly.
+Therefore, we interpret this as evidence that the progenitor dwa rf galaxy is at or near– 9 –
+the edge of the SDSS footprint, at ( l,b) = (255◦,48◦). If this is correct, and the Orphan
+progenitor is located in theregion 255◦<l<268◦, then the Orphan Stream stars intersected
+on “outrigger” SEGUE stripe 1540 (Yanny et al. 2009a), near ( l,b) = (270◦,39◦) are part of
+the trailing (rather than leading) tail (see below for tail orientation ).
+The density falls sharply for Λ Orphan<−25◦. This could be due to a combination of
+three factors: (1) the stream density may actually drop, (2) the distance to the stream could
+be extrapolated incorrectly in this angular range, causing us to miss stream stars in the
+data outside of our 20 .7< gcorr<21.7 cut, or (3) the falling completeness of the F turnoff
+stars in the SDSS DR7 database at g0>22.5 reduces the contrast of the stream against
+the background. Therefore, we can say little with certainty about the stream density for
+ΛOrphan<−25◦.
+6. Velocities of Orphan Stream stars
+Now that we have a measure of the spatial position of the Orphan St ream, we can im-
+prove our selection of spectroscopically observed stars in the str eam. While initial detection
+of halo substructure is usually best done with deep, wide-field, filled- sky multi-color imaging,
+determination of the orbit of a given structure depends on kinemat ics, either proper motions
+or radial velocities (or both), for identified structure members. S ufficiently accurate proper
+motions are currently difficult to obtain for halo objects more distan t than about 15 kpc
+from the Sun (unless hundreds or thousands of individual proper m otions can be averaged),
+and thus for outer halo structures, we rely primarily on radial veloc ities from spectroscopy
+to constrain structure orbits.
+The SEGUE survey contributed velocities for 240,000 stars, includin g several thousand
+targeted spectra of halo stream candidates with velocity accurac ies of 5-10 km s−1, suffi-
+cient to isolate stream members from the broad background halo an d constrain structure
+velocity dispersions. Spectra are also highly useful in determining st ructure membership for
+individual objects by providing estimates of stellar metallicity and lumin osity class (surface
+gravity). Diffuse halo stream tidal tails are often overwhelmed in num ber by unrelated halo
+stars, but byusing theSEGUESSPPpipeline (Lee et al.2008a,b;Allend e Prieto et al.2008)
+estimates of stellar atmospheric parameters, individual stream ca ndidates may be separated
+from the background.
+We select A stars from all stars with spectra in SDSS DR7 using the co lor selection
+−0.4<(g−r)0<0.3 and 0.8<(u−g)0<1.6. Throughout this paper, all stellar spectra
+are selected from the database with elodierverr >0 to remove galaxies and QSOs from the– 10 –
+sample. We limit the spectra to BHB stars by selecting those with aver age surface gravity
+measurement from the SSPP in the range 0 .1<logg<3.5. We then select spectra within
+twodegreesoftheOrphanstream. ForΛ Orphan<−15◦,weselectstarswith −2◦<Bcorr<2◦.
+And finally, we select BHB stars that have apparent magnitudes con sistent with the distance
+to the Orphan Stream (17 .4<g0−0.00022Λ2
+Orphan+0.034ΛOrphan<18.0).
+The filled circles in Figure 6 show the line-of-sight, Galactic standard o f rest velocities
+(vgsr= RV+10.1cos(b)cos(l)+224cos(b)sin(l)+6 .7sin(b) km s−1) for the spectroscopically
+observed BHB stars with sky positions and distances consistent wit h membership in the
+Orphan Stream. Stars with [Fe/H] WBG<−1.6 are circled. The Orphan Stream stars are
+clearly separated from the background, at velocities near vgsr∼+110 km s−1. Almost all
+of the BHB stars that have Orphan Stream velocities also have [Fe/H ]WBG<−1.6. We
+have highlighted in green the low metallicity stars with −0.0445Λ2
+Orphan−0.935ΛOrphan+95
+km s−1< vgsr<−0.0445Λ2
+Orphan−0.935ΛOrphan+165 km s−1. These are the stars that we
+used to determine the velocity of the stream as a function of Λ Orphan. For comparison, the
+preliminary velocities from Belokurov et al. (2007b) are marked with m agenta squares.
+The stars in Figure 6 are binned by Λ Orphanin 10◦bins starting at Λ Orphan=−35◦.
+Seven bins contain Orphan Stream stars, as shown in Figure 6. In ea ch bin we measure the
+mean Λ Orphan, the mean g0, error in the mean ( δg0), the mean vgsr, error in the mean ( δvgsr),
+and standard deviation of the velocity distribution ( σv). These values, and the number of
+stars in each bin ( N), are tabulated in Table 2.
+Because we now have better information on the path of the stream , it is worth revisiting
+the magnitude measurements that we originally determined from the data in Figure 2. In
+Figure 7 we show g0vs. ΛOrphanfor the photometrically selected BHB stars (selected the
+same way as those in Figure 2) that are now within 2◦of the Orphan Stream. We also show
+the Orphan Stream BHBs with spectra, as selected from Figure 6. T he Orphan Stream can
+be traced about ten degrees further across the sky using the ne w data selection than it could
+be using Figure 2. For each value of Λ Orphanin Table 2, we use Figure 7 to determine a value
+ofg0, and estimate the error in this determination.
+For each value of Λ Orphan, we determine the Galactic longitude for Bcorr= 0◦. Then
+we use the data in Figure 4 to determine the Galactic latitude at that lo ngitude. These
+numbers are also tabulated in Table 2. Note that although the basic d ata is similar, the
+determinations of position and distance were made independently, a nd may not be identical
+to those of Table 1, even for similar values of Galactic longitude.
+Table 2 shows that the typical velocity dispersion in the stream is 10 k m s−1. The
+velocity dispersion of the last point, with only 3 stars, at l= 271◦, is unphysically low,– 11 –
+since it is smaller than the velocity errors, but it is the number that wa s measured using the
+technique described. The position of the stream here was quite diffic ult to pick out from the
+photometry, due to the gap in the data. The unexpected velocity d ispersion, for stars that
+may not be at the same location as the photometric overdensity, lea ds to a large uncertainty
+in the validity of the detection here. Because the detection is uncer tain, and it is not known
+if the stars at this position are in the same tail as the rest of the dat a, this data point was
+excluded from orbit fits for the Orphan Stream (see §10).
+7. Orphan Stream spectra
+Now that we know the velocity of the stream as a function of sky pos ition, we can
+look for stars other than BHB stars that are part of the Orphan S tream. We start by
+selecting all stellar spectra that are now within 2◦of the Orphan Stream, following the
+method outlined in the first paragraph of §6, and are within the portion of the stream that
+we have observed, −35◦<ΛOrphan<40◦. We eliminate sky fibers by choosing only spectra
+withg0<22. We choose a somewhat narrower range of velocities than in our p revious
+velocity cut, to reduce the number of background stars in the sam ple. We select stars with
+−0.0445Λ2
+Orphan−0.935ΛOrphan+112.5<vgsr<−0.0445Λ2
+Orphan−0.935ΛOrphan+147.5.
+Brighter stars that are consistent with Orphan Stream membersh ip are expected to
+be giant stars with low surface gravity, while fainter stars should be higher surface gravity
+main sequence stars. To determine which stars are stream star ca ndidates, we calculated
+gcorr. This moves all of the stream stars to the distance modulus of the O rphan Stream at
+ΛOrphan= 0◦. Then, all of the stars with gcorr<19 + 4(g−r)0should be giants, so stars
+outside of the range 0 .1<logg<3.5 were discarded. We restricted the surface gravities of
+the fainter stars ( gcorr>19 + 4(g−r)0)) to high surface gravities (logg >3.0). This was
+necessary to reduce the number of stars that in the blue straggle r region of the diagram, but
+unfortunately also removed all of the candidate turnoff stars, wh ich were faint enough that
+they did not have measured surface gravities.
+Our final data selection was for low metallicity stars. Since these sta rs are not all BHB
+stars, we used the adopted metallicity measurement (feha) in the S DSS database, rather
+than the [Fe/H] WBGmeasurement that we used in our previous selection. We limited our
+sample to stars with [Fe/H] adopted<−1.6. The final dataset includes 87 stars, which are
+shown as magenta crosses in Figure 8. Figure 8 is a reproduction of F igure 3 with overlays
+described below.
+In order to understand the background spectral distribution, w e also select a set of stars– 12 –
+with spectra that were near, but not on, the Orphan Stream. The on-stream stars were
+selected to be within two degrees of the sky position of the stream, with−36◦<ΛOrphan<
+40◦. Theoff-streamstarswere selected witha distance oftwo to five d egrees fromthestream.
+The sky area of the off-stream stars is 50% larger, but because th e SEGUE spectral coverage
+is not uniform on the sky (there are several SEGUE pointings direct ly on the stream), it
+contains a similar number of SDSS/SEGUE spectra. Using these sky a reas, we selected
+8385 stars in the on-stream sample, and 8726 stars in the off-stre am sample. The off-stream
+sample was then culled using identical velocity, surface gravity, and metallicity criteria. The
+resulting “background” spectra are overlaid in Figure 8 as black circ les.
+Some of the spectroscopically selected candidate Orphan Stream m embers show a rea-
+sonable correspondence to the positions we would expect from CMD s of globular clusters.
+We see the BHB stars at gcorr∼17.75. There are candidate blue straggler stars extending
+from the turnoff at gcorr∼21 towards brighter magnitudes and bluer colors. There is a
+nearly vertical red giant branch at ( g−r)0= 0.5. There are many fewer comparison back-
+ground spectra left after selecting for Orphan Stream candidate parameters; these show no
+horizontal branch, many fewer blue straggler candidates, and th e stars near the giant branch
+do not show the expected trend of redder stars at brighter magn itudes.
+There are two selection effects in this diagram that should be clearly u nderstood. First,
+the surface gravity selection is different for stars below a line that r uns from [g0,(g−r)0] =
+(17,−0.5) to (g0,(g−r)0) = (21,0.5). This selection could visually enhance the appearance
+of the blue straggler separation. The second selection effect is tha t neither SDSS nor SEGUE
+spectroscopic selection evenly samples stars in color or magnitude. In particular, a very large
+number of SEGUE spectra were obtained for G stars with 0 .48<(g−r)0<0.55. This could
+give the appearance of a giant branch at ( g−r)0∼0.5, even if there is not one there.
+However, we think most of the “on-stream” stars on the giant bra nch in Figure 8 are stream
+stars, because they shift slightly with gcorr, as is expected from a real giant branch, and
+they have a very different distribution from the stars that were se lected with all of the same
+criteria except the removal of the velocity cut. Note that there a re many more on-stream
+spectra (87 stars) than off-stream spectra (28 stars) in the fin al samples, even though the
+originalstar samples were thesamesize. If weexclude BHB starsfr omtheon-streamsample,
+then therearestill 51non-BHB stars intheon-stream sample comp aredto 24 non-BHBstars
+in the off-stream sample. This suggests that about half of the spec tra of giant stars, blue
+stragglers, and asymptotic giant branch stars are bona fide stre am members.
+InordertoestimatethemetallicityofOrphanStreamstars, weove rlayfiducialsequences
+of globular clusters M92 ([Fe/H] = -2.3), M3 ([Fe/H] = -1.57), and M13 ([Fe/H] = -1.54)
+on the color-magnitude diagram in Figure 8. These three clusters ha ve blue horizontal– 13 –
+branch stars at g0= 15.59,15.10, and 14.80, respectively (Clem, Vanden Berg, & Stetson
+2008; An et al. 2008). Using the distance modulus from Harris (1996 ), the inferred absolute
+magnitude of the BHB stars is Mg= 0.50 for M3, Mg= 0.52 for M92, and Mg= 0.38 for
+M13. These three clusters have been shifted in apparent magnitud e so that their horizontal
+branches align with gcorr= 17.75. The relatively metal rich cluster M71 ([Fe/H] = -0.73)
+has also been superimposed. Since it does not have a BHB, it has been shifted so that
+if it did have a BHB at the average absolute magnitude of Mg0= 0.45, at its cataloged
+(m−M)0= 13.02 (Harris 1996), then that horizontal branch would be at gcorr= 17.75.
+Since the turnoff color and giant branch stars of our Orphan strea m candidates are
+equally consistent with the M3, M92 and M13 fiducials at our level of ac curacy, we have de-
+cided to use anintermediate horizontal branch absolute magnitude ofMg0= 0.45to compute
+distances in this paper. This is the same as the value used in Newberg, Yanny & Willett
+(2009) to measure the distance to the Cetus Polar Stream in the Ga lactic halo. Since the
+horizontalbranchoftheOrphanStreamisat g0= 17.75atΛ Orphan= 0◦, theimplied distance
+modulus there is µ= 17.30; this is the distance modulus to which all stream stars in the
+CMDs are shifted. All of the distances in Table 2 have been computed assuming the BHBs
+haveMg0= 0.45. The systematic error in our absolute magnitude could be easily as large as
+0.3 magnitudes, which is 15% in distance. A systematic error in the abs olute magnitudes of
+Orphan Stream BHB stars would shift all of the distances closer or f arther away. However,
+since the random errors are the important factor for fitting the G alactic potential, and not
+the systematic errors, no systematic errors have been included in the distance errors in Table
+2.
+The turnoff, giant, and BHB stars in our spectroscopic sample could reasonably fit an
+isochrone between those of M92 and M3 ( −2.3<[Fe/H]<−1.6), which is consistent with
+the measured metallicities of the BHB stars. However, since we do no t spectroscopically
+sample very many stars that are redder than ( g−r)0= 0.55, in either the SDSS or in
+SEGUE-I, we are not sensitive to the giant branches of higher meta llicity populations, even
+if we include stars with higher measured metallicities in the selection. Sin ce we expect that
+the progenitor of the Orphan Stream was a dwarf galaxy, it is reaso nable that there may
+be a range of metallicities in the turnoff and giant branch even if there is not a range of
+metallicities in the BHB stars (Majewski et al. 2003; Chou et al. 2007) .
+As a final test of our position, velocity, and distance selection, we s how in Figure
+9 the Galactic coordinates of spectroscopically selected BHBs. We s elect BHBs from all
+SDSS/SEGUE stellar spectra with the criteria −0.4<(g−r)0<0.3, 0.8<(u−g)0<1.6,
+and 0.1<logg<3.5. We then selected all of the BHBs with apparent magnitudes
+consistent with BHBs at the distance of the stream, as a function o f ΛOrphan(17.4<– 14 –
+g0−0.00022Λ2
+Orphan+ 0.034ΛOrphan<18.0). This selection really only applies within a
+few degrees of the position of the Orphan Stream, but using this se lection criterion over our
+whole dataset allows us to visualize the density of spheroid BHBs and d etermine whether
+there is an overdensity of BHBs with the properties of the Orphan S tream at the expected
+locations in the sky.
+The last two criteria for selecting BHBs in the Orphan Stream are velo city and metal-
+licity, which are not possible to use as photometric target selection c riteria. The BHBs with
+the correct velocity to be in the Orphan Stream ( −0.0445Λ2
+Orphan−0.935ΛOrphan+ 95<
+vgsr<−0.0445Λ2
+Orphan−0.935ΛOrphan+165) and with low metallicity ([Fe/H] WBG<−1.6)
+are shown as filled circles in Figure 9. The larger triangles show the pos itions of the Orphan
+Stream, as determined from photometry of F turnoff stars in Figur e 4 and presented in Table
+2.
+Our spectroscopic sample of BHBs is not uniformly sampled in magnitud e, color, or
+position on the sky, so there could be many selection effects in those variables. The open
+circles in Figure 9 give us an impression of the density of BHB stars sam pled in our color
+and magnitude range across the sky. The BHBs with the correct ve locities and metallicities
+to be stream stars are more abundant along the path of the Orpha n Stream, which supports
+strongly our selection criteria, and our identification of stream mem bers.
+Note that in Figure 9 we have possible detections of the Orphan Stre am at Galactic
+longitudes of l= 160◦or less, suggesting that the stream does extend further than we
+have been able to trace it. Also, there is some confusion about the p osition of the stream
+atl= 270◦. The selection of that position from photometry of F turnoff stars was very
+ambiguous, and the selected point is more than two degrees from th e nominalBOrphan= 0◦
+expected position. The highest density of BHB spectra at the expe cted magnitude, velocity,
+and metallicity for stream membership is much closer to BOrphan= 0◦, but there are no
+spectraatlowerGalacticlatitudessothereisanambiguitybetweent hespectroscopicposition
+at higher Galactic latitude and a broader stream that is centered at lower Galactic latitude.
+We have argued that the Orphan Stream progenitor, or its remnan t, is likely to be near
+l= 253◦. That means there is a reasonable chance that the stream detect ion atl= 270◦is
+part of the trailing tidal tail, while all of the other detections are par t of the leading tidal
+tail. If this is the case, then it would be reasonable that the orbit fit w ould not pass exactly
+through the detections.– 15 –
+8. Proper motions of Orphan Stream stars
+We use DR7 proper motions, which are derived from an astrometric m atch between
+SDSS and UCAC catalogs (Munn et al. 2004), as an additional check o n consistency of the
+proposed fit to the stream. DR7 proper motions have typical erro rs of 3 mas yr−1per
+coordinate, restricting the usefulness to those pieces of the str eam withd <25 kpc, in the
+area of Figure 1 with 230◦<l<275◦. Figure 10 shows the SDSS-UCAC proper motions of
+two sets of stream candidates: 1) spectroscopic stream candida tes, with metallicity, velocity,
+gravity, matching the stream (all the magenta crosses from Figur e 8); and 2) spectroscopic
+BHB stream candidates (all the black + signs from Figure 8), our mos t secure stream
+members. Later, we will fit an orbit to the Orphan Stream, and will sh ow that the orbit
+plausibly fits this proper motion data.
+9. Metallicity of the Orphan Stream
+Throughout this paper we have been identifying Orphan Stream sta rs by selecting low
+metallicity ([Fe/H] <−1.6) stars. In this section we present the statistics of stream star s
+selected using all of the position, velocity, surface gravity, magnit ude, and color information
+available to us, but not using metallicity selection.
+There are 36 spectroscopically selected Orphan Stream BHB stars shown in green in
+Figure 6 and as ‘+’ asterisks in Figure 8. If we use exactly the same se lection criteria
+except omitting the low metallicity cut, we find a sample of 39 stars. Th e metallicities
+of the 37 of these stars which have measured metallicities are histog ramed in Figure 11,
+showing a very low metallicity peak at about [Fe/H] WBG=−2.1. We compared these 37
+stars with the 384 halo BHB stars in Figure 17 of Yanny et al. (2009b) , which have a mean
+of [Fe/H] WBG=−1.6, using a Mann-Whitney test. The sum of the ranks of the 37 Orpha n
+stream BHB stars is 4094, with a z-score of 5.25, and a p-value less t han 0.0001. The
+metallicities of the Orphan stream stars do not match those of the s pheroid at an extremely
+significant level.
+In Figure 11, we also show the metallicity distribution of all of the on-s tream spectro-
+scopic stars in Figure 8, excluding BHB stars and bright stars with gcorr<16, and all of the
+off-stream spectroscopic stars in Figure 8, also excluding BHB star s and bright stars. The
+on-stream stars are primarily giant branch stars, but include a few candidate blue straggler
+and asymptotic giant branch stars. There are 87 on-stream star s with spectra in Figure 8. If
+we remove thelow metallicity cut, there are136stars, ofwhich 39ar eBHBs and3arebright,
+leaving 96 stars whose metallicity (using the average SDSS feha meta llicity measurement)– 16 –
+is histogramed in the lower panel of Figure 11. In comparison, there are only 28 off-stream
+stars in Figure 8, and if the metallicity cut is removed the number of off -stream stars rises
+to 73. After removing bright stars with gcorr<16, and stars that have magnitudes, surface
+gravities, and colors of BHB stars, there are 59 stars left whose m etallicity is histogramed
+in Figure 11. Recall that the size of the off-stream region was adjus ted so that there were a
+similar number of SDSS/SEGUE spectra in the on-stream region and t he off-stream region.
+Because coverage and target selection criteria change as a funct ion of sky position, it is pos-
+sible that the normalization of the on-stream and offstream datase ts may not be identical or
+that there are slightly different selection functions for each.
+There was no significant difference between the metallicity distributio n of the on-stream
+and off-stream stars in the lower panel of Figure 11, using either a M ann-Whitney test or a
+KS test. Both samples have a mean metallicity of [Fe/H] ∼ −1.6. However, it appears to
+the eye that there are a few more very low metallicity stars in the off- stream sample and a
+few extra stars at about [Fe/H]= −1.9 in the on-stream sample. The difficulty here is that
+we are using very small samples, and 50% of the on-stream stars ar e likely contamination
+from background stars, so better and cleaner samples are requir ed to study the metallicity
+distribution of the population as a whole. It is not unexpected for BH B stars, which only
+form from low metallicity populations, to be more metal-poor than gian t branch stars.
+10. Orbit Fitting
+10.1. Methodology and Model Choices
+We now use the combined positions, distance estimates, and line-of- sight velocities to
+fit an orbit to the Orphan Stream. We use the same procedure desc ribed by Willett et al.
+(2009) with two modifications. First, instead of fitting the distance to the stream, we fit the
+g0magnitude. We do this because of the symmetry of the errors in g0. Specifically we adopt
+a reduced chi-squared of
+χ2
+b=/summationdisplay
+i/parenleftbiggbmodel,i−bdata,i
+σb/parenrightbigg2
+, (1)
+χ2
+rv=/summationdisplay
+i/parenleftbiggrvmodel,i−rvdata,i
+σrv,i/parenrightbigg2
+, (2)– 17 –
+χ2
+g0=/summationdisplay
+i/parenleftbiggg0model,i−g0data,i
+σg0,i/parenrightbigg2
+,and (3)
+χ2=1
+η/parenleftbig
+χ2
+b+χ2
+v+χ2
+g0/parenrightbig
+, (4)
+whereη=N−n−1,Nis the number of data points, and nis the number of parameters.
+Distances for the fiducial stream points are calculated from the g0magnitudes for BHBs in
+Figure 7 under the assumption that the absolute magnitude for a st ream BHB is Mg0= 0.45
+at about (g−r)0=−0.15, similar to the globular cluster fiducials overlaid in Figure 8. The
+distance parameter is fitted internally in the gradient descent sear ch, but because the error
+bar as represented in distance space is not symmetric above and be low the fitted value, the
+chi-squared computation is done in magnitude space.
+Second, in addition to the logarithmic halo potential model, we also con sider an NFW
+(Navarro, Frenk, & White 1996) model. Although our data does not span a large enough
+range of Galactocentric distances to determine which model is pref erred, it is useful to
+optimize both models for easy comparison with the literature.
+Likewise, we will use two published disk potentials: a lower mass expone ntial disk from
+Xue et al.(2008)andahighermassMiyamoto-Nagaidiskwithparame tersfromLaw, Johnston, & Majewski
+(2005). While we do not have sufficient data to constrain the disk mod el, it is instructive to
+see the effect of using different, commonly used disk models on the inf erred halo mass.
+To obtain our initial guess for the kinematic parameters, we draw fr om Tables 1 and 2
+a test particle at Point 1: ( l,b,R) = (218◦,53.5◦,30 kpc) and construct a vector to Point 2:
+(l,b,R) = (215◦,54.0◦,31 kpc). By matching the radial velocity at Point 1, we obtain an ini-
+tial parameter starting point of ( R,vx,vy,vz) = (30 kpc ,−125 km s−1,75 km s−1,95 km s−1).
+With this initial kinematic guess we will consider seven specific cases, w hich include
+three potential models published by previous authors, and four mo dels in which the halo
+mass is varied to best match our data (logarithmic and NFW halos are c ompared, as well as
+a low mass exponential disk vs. a high mass M-N disk):
+1. We fit the exact model of (Xue et al. 2008), using an exponential disk and NFW halo.
+The parameters for bulge, disk, and halo were taken from their pap er. the mass of the
+bulge, diskandhalo, integratedoutto60kpc, aretabulatedinTable 3. Themassofthe
+Milky Way using this model is M(R<60 kpc) = 4 .0×1011M⊙, of which 3 .3×1011M⊙
+is in the halo. The value for the scale radius, rs= 22.25 kpc, used for all the NFW
+potentials in this work comes from the top panel of Figure 16 of Xue e t al. (2008). For– 18 –
+a Milky Way virial radius of rvir= 240 kpc (Sales et al. 2007) this corresponds to a
+concentration index of c∼11.
+2. We fit the same model as the previous case, but allow the vc,maxnormalization of the
+NFW model (the halo mass) to vary in amplitude.
+3. We fit the same bulge and exponential disk model as the previous t wo cases, but use
+a logarithmic halo model. We allow the normalization (mass) of the halo to vary as a
+free parameter to be fit.
+4. We fit stream kinematics only, assuming a spherical logarithmic halo as given by
+Law, Johnston, & Majewski (2005). Disk and bulge parameters fo r this case are iden-
+tical to the Law, Johnston, & Majewski (2005) paper. Their mode l, with a disk model
+from Miyamoto & Nagai (1975), yields M(R<60 kpc) = 4 .7×1011M⊙.
+5. We fit the same model as case 4, but allow the halo speed above to v ary as a free
+parameter within the spherical logarithmic halo potential model, while keepingd=
+12 kpc, where dis the halo core softening radius.
+6. Wefitthekinematics withinthespherical NFWhalogiven inNavarro, Frenk, & White
+(1996) and further described by Klypin et al. (1999), normalized to M(R<60 kpc) =
+3.3×1011M⊙. We fit the vc,maxof the NFW model to give a similar total potential
+to that for the logarithmic halo with vhalo= 114 km s−1above. These give rise to
+vc,max= 155 km s−1andrs= 22.25 kpc.
+7. We fit the same model as case 6, but allow the NFW maximum circular s peedvc,max
+to vary while keeping rs= 22.25 kpc.
+We summarize these potential models in Table 3.
+The models were fit to all of the available data in Tables 1 & 2, with the ex ception of
+the last point in Table 2. Attempting to fit the data set with the l= 271◦point included
+resulted in substantially worse χ2values were obtained ( χ2∼4). Additionally, fitting this
+point introduced deviations of the orbit from the observed stream distances, with the high l
+distancesbeingunderestimated, andthelow ldistancesbeingoverestimated. Thissystematic
+differences between the model and data led us to perform all of our best fit models without
+thel= 271◦point, for all six of the orbit fits listed above.
+The datapoint at l= 271◦is less certain because the measurements were problematic,
+and because the continuity of the Orphan orbit is broken by the miss ing SDSS data at
+255◦<l<268◦, and the location of a possible Orphan progenitor dwarf in this region , while– 19 –
+likely, is uncertain. If the progenitor is in this region, then the points withl <255◦are
+in a tidal tail with the opposite sense (i.e. leading) to those with l >268◦(i.e. trailing).
+If this is the case, than a simple orbit fit, (rather than a full N-body simulation), will not
+accurately account for the deviations in distance and velocity that occur on opposite sites
+of a progenitor when that progenitor has an internal velocity dispe rsion that is a significant
+fraction (more than a few percent) of the total potential in ques tion.
+In addition to these cases, we also attempted several more which d id not produce very
+interesting results, butwhichwewillsummarizehere. Weransevera l orbitsfitsinanattempt
+to measure the halo flattening parameter, q. The fits were very insensitive to q; fitting this
+parameter in the logarithmic halo yielded results with a very large erro r inq. Therefore, we
+fixed the flattening parameter at q= 1.
+Wealsoattemptedtosimultaneously fit vhaloandd, andvc,maxandrs, forthelogarithmic
+andNFWhalos, respectively. Ingeneral, fittingthescalelengthsdid notsubstantiallychange
+the halo speed results, and the scale lengths were fit only with very la rge errors, on the order
+of tens of kiloparsecs. Therefore, we did not attempt to fit the sc ale lengths in either model.
+10.2. Results
+For each halo model, we ran five random starts of the fitting algorith m in the same
+manner described by Willett et al. (2009). Once the best fit orbit was found for each model,
+wecomputedtheHessianerrorsbyvaryingtheparameterstepsiz esuntiltheerrorsconverged
+for different step size choices. Using a starting point of ( l,b) = (218◦,53.5◦), the best fit
+parameters and their errors are enumerated in Table 3.
+Figure 12 shows the three orbit fits for the models with the exponen tial disk. The
+position and velocities of the datapoints are well fit by all three expo nential disk models.
+The lower panel of Figure 12 shows that for this stream, the distan ce fit is most sensitive
+to the potential. The weaker potential of the lower mass halo model gives the best fit to
+observed stream distances from Tables 1 and 2 (black points with er ror bars), as it allows the
+stream to escape to larger Galactocentric radii without significant ly changing the observed
+radial velocity gradient. There is little difference between the best fi t NFW halo and the
+best fit logarithmic halo.
+Figure 13 shows the circular velocity curve for models N=1-3 from Ta ble 3, which use
+the same exponential disk parameters as (Xue et al. 2008) in all cas es. The figure shows
+that the orbits that are well fit to the Orphan Stream don’t fit the ( Xue et al. 2008) nor
+the (Koposov, Rix, & Hogg 2009) result; the rotation speed of the model is too low at all– 20 –
+Galactic radii.
+WethereforetryusingaM-Ndiskwithparametersasadoptedby(L aw, Johnston, & Majewski
+2005). This disk is about twice massive than the exponential disk mod el. We note here that
+we have not fully explored all possible disk shapes and masses, and a m ore massive expo-
+nential disk may give similar results to the more massive M-N disk consid ered here. Figure
+14 shows the four fits for the models with the M-N disk. As before, lo wer halo speeds are
+a better fit to the Orphan Stream than those used by previous aut hors, and there is little
+difference between the fits for the best fit NFW and logarithmic halos . Evidently, the Or-
+phan Stream is telling us that the mass of the Milky Way is smaller than pr eviously thought,
+so that distant portions of the stream, which are therefore expe riencing lower gravitational
+attraction to the Galaxy, are pushed further away from the Galax y center at a given energy.
+Figure15 shows thesame circular velocity asFigure13 for models 4-7 withan M-Ndisk.
+This time we find that the low mass halo orbit fits area good fit to the Xue et al. (2008) and
+(Koposov, Rix, & Hogg 2009) result. They actually fit the Xue et al. ( 2008) model better
+than the model fit in that paper. Using the higher velocity leads to a s ystematic deviation in
+the distances, so that the model is further away than all of the da tapoints at high Galactic
+latitude, and closer for low Galactic latitudes. Also, as we will see later , adding velocity
+segregation from an N-body simulation only makes the systematic er rors in the distance
+larger; for example stars in the leading tidal tail have lower total en ergy than the progenitor
+and have smaller Galactocentric distances. The energy difference in creases as a function of
+distance from the progenitor, along the stream. However, we not e that our formal error bar
+onvhalo= 73±24 km s−1is quite large and marginally consistent (within 2 sigma) with the
+highervhalo= 114 km s−1value of Law, Johnston, & Majewski (2005) and others. Although
+the formal error bars are large, the orbit fits suggest that a lowe r halo speed is preferred.
+Of the models that we tried, the three low-mass exponential disk mo dels (1-3) fit either
+the Orphan Stream data or the Xue et al. (2008) rotation curve, b ut not both. The two low
+halo mass models (models 5,7) with a higher mass M-N disk fit the best, w ith a logarithmic
+halo fitting about the same as an NFW profile. The two higher halo mass models (models
+4,6) are poorer fits to both the Orphan Stream and the Xue et al. (2 008) rotation curves.
+Although we cannot formally rule out the halo models fit by Xue et al. (2 008) and
+Law, Johnston, & Majewski (2005), in order to simultaneously fit t he Xue et al. (2008) cir-
+cular velocity data, we prefer a lower total mass of the Milky Way tha n measured in either
+of those two papers. Our total Milky Way mass is 60% of that found b y Xue et al. (2008)
+and Law, Johnston, & Majewski (2005). As mentioned in the introd uction, our masses are
+at the low end, but not out of range of recently published masses. T o estimate the virial
+mass of the Milky Way given our best fit model 5, we multiply the halo mas s within 60– 21 –
+kpc by 4 (for a virial radius of 240 kpc), and add the disk plus bulge ma ss. The result
+isM(R <240 kpc) = 6 .9×1011M⊙. Of interest is the recent result of Odenkirchen et al.
+(2009), who suggested that a lower halo mass might make it easier to fit the kinematics of
+the Pal 5 globular cluster stream. The findings on Milky Way circular ve locity with radius
+to distance of R∼60 kpc by Xue et al. (2008) also are consistent with our low value for
+the halo speed. When the M-N disk model is used, they are also in roug h agreement with
+Koposov, Rix, & Hogg (2009) who measure vc(R= 8kpc) = 224 ±13 km s−1.
+We note that we didn’t explore scaling the mass of the disk (or bulge) b eyond their
+default values of Mdisk= 1011M⊙for the M-N disk or Mdisk= 5×1010for the exponential
+disk. Thus, while we find that the heavier M-N disk (combined with a lowe r amplitude
+halo) fits the joint data sets of the Orphan Stream, and the Xue et al. (2008) BHBs circular
+velocity points, better than models with the lighter exponential disk , it is likely that similar
+fits could be obtained with a heavier exponential disk (though still wit h a lighter halo). If
+more data were available, the disk mass could be varied to find the bes t combination of disk
+and halo mass.
+It isimportanttonotefromFigures12and14thatif wecouldfollowth eOrphanStream
+just a little farther out into the halo, we would have a much better po wer to determine the
+halo mass, since the distances to the stream for each case diverge .
+Figure 16 shows the best fit vhalo= 73 km s−1logarithmic halo model in (X,Y,Z)
+Galactic coordinates with the direction of motion indicated by the arr ows. From the apo-
+and peri-galactic distances of ∼90 kpc and ∼16.4 kpc, respectively, we calculate an orbital
+eccentricity of e= 0.7. The inclination of the Orphan Stream is i∼34◦with respect to
+the Galactic plane as seen from the Galactic center. It will be importa nt to obtain data for
+the Orphan stream at apogalacticon in order to distinguish between different models for the
+Galactic potential.
+The proper motions along the model orbits of Figure 14 are calculate d for the three
+potentials. The implied proper motions are essentially identical in the r egion of the stream
+where it is observed 170◦<l<270◦, and thus only the best fit model (model 5) is overlaid
+as a solid curve on Figure 10. The fact that the stream is moving in pro grade orbit (see
+below) makes the proper motions significantly smaller in magnitude tha n if the stream were
+retrograde (Willett et al. 2009), and thus, while the errors on the p roper motions and stream
+membership uncertainties make it difficult to constrain the Galactic po tential with proper
+motions of this accuracy, it is reassuring that the data and model a re broadly consistent.– 22 –
+10.3. The Orphan Stream is moving in a prograde direction
+The velocities of the fit orbit definitively show that the stream stars are moving in a
+direction fromhigher lto lowerl. As a simple check, if we reverse the sign of all the velocities
+(vx,vy,vz) in the model orbit, the path on the sky that the stream traces ou t is the same, but
+the radial velocities do not match the observations by a wide amount (|δ(v)|>100 km s−1).
+Thus the Orphan Stream is on a prograde orbit around the Milky Way. One may now
+ask if the visible piece of the Orphan tidal stream is a leading or trailing t idal tail. If we
+assume that the ‘progenitor’ of the Orphan Stream lies in the range 248◦<l<268◦as the
+density plot of Figure 5 and the visual impression of Figure 4 suggest s, (with the density
+enhancement visible at ( l,b) = (253◦,49◦)), then the portion of the tidal stream stretching
+froml= 250◦tol= 170◦, combined with the space velocity of stars moving in that same
+direction, implies that we are seeing a leading tidal arm, rather than a trailing tidal arm.
+The piece of stream intercepted at l= 270◦is then a trailing tidal arm.
+10.4. N-Body Realization
+We now comment on the fact that our ‘simple orbit integration and fitt ing’ technique is
+not an N-body simulation, and therefore effects of stars leaving th e progenitor and drifting
+ahead or behind to greater and lower energies, which changes their distance from the central
+body, are not captured as they would be in a full N-body.
+To conduct the N-body simulation, we first integrate the log halo (be st fit model 5)
+orbit back 4 Gyr and place a 10,000 particle Plummer sphere at the pre dicted location. We
+use a Plummer sphere scale radius of rs= 0.2 kpc and total mass of Mtotal∼2.5×106M⊙.
+Using the orbit kinematics from 4 Gyr in the past, we evolve it forward for 3.945 Gyr so
+that the dwarf progenitor ends up at l∼250◦. We conducted the N-body simulation using
+the gyrfalcON tool of the NEMO Stellar Dynamics Toolbox (Teuben 19 95).
+The result of this simulation is given in Figure 17. The top three panels s how the loga-
+rithmic halo orbit with vhalo= 73 km s−1and N-body results for ( l,b),(l,vgsr),and (l,d⊙).
+We see that the N-body stream is a good fit to the sky locations and v elocities of the
+simple orbit (black curve inFigure17). Inthedistance comparison(t hirdpanel), theN-body
+is significantly below the orbit at low l. This is in agreement with our intuition for a stream
+that is moving from high lto lowerl, since energy differentials due to the internal dispersion
+of stars in the progenitor draw stars with lower energy (and lower G alactocentric distance)
+further ahead closer to the Galactic center than those with higher total energy. Note that
+this is independent of the question of whether the tail is leading or tr ailing. We know the– 23 –
+direction that the stream is going from the observed radial velocitie s and distances. We
+expect that we are seeing a leading tidal tail because we think the dw arf remnant is located
+at the high Galactic longitude end of the observed portion of the str eam. But even if we are
+incorrect about the position of the dwarf, the observed slopes in t he lower panels of figures
+12 and 14 indicate a lower mass Milky Way, and a full N-body simulation will only make
+the slope for higher mass estimates fit worse.
+The lowest panel inFigure17shows thedensity ofN-bodysimulated p articlesremaining
+at the current epoch along the stream vs. l. This density distribution may be compared with
+that of the data in Figure 5. The N-body simulation has a lower density of particles near
+l= 240◦,(ΛOrphan= 15◦). These two effects are apparent in the Orphan imaging data in
+Figures 5 and 4, respectively. This suggests that our method of fit ting a simple orbit to the
+tidal debris is a reasonable first approximation, and fitting the dens ity is a useful diagnostic
+tool.
+11. Connection of the Orphan Stream to other Galactic halo st ructures
+Given the Orphan Stream orbit of §10, we can revist the question the Orphan Stream’s
+connection to other halo structures, such as Segue-1, Complex A and Ursa Major II. We add
+position, velocity, and distance using the data from the literature f or these three objects as
+points onto Figures 12 and 14 and 16 for reference.
+11.1. Segue-1
+The distance to Segue-1 is d= 23 kpc from the Sun (Belokurov et al. 2007a), consistent
+with both Sagittarius and Orphan Streams in this part of the sky. A c ase has recently
+been made that Segue-1 is associated with the Sagittarius stream ( Niederste-Ostholt et al.
+2009). It is, however, in the same general area of the sky as the O rphan Stream, and it has
+a positive velocity, vgsr= +114 km s−1, as measured by Geha et al. (2009), very close to
+that of Orphan Stream stars in Table 2 here, and in contrast to the Sgr stars in this area,
+which have vgsr∼ −120 km s−1, as shown in Yanny et al. (2009b). Thus, on the basis of
+kinematics, we believe that it is more likely that Segue-1 is a star cluste r associated with the
+Orphan Stream than that it is associated with Sgr. More extensive p roper motions of both
+objects will help resolve this issue definitively (Geha et al. 2009).– 24 –
+11.2. Ursa Major II
+In Figure 1, Ursa Major II (Zucker et al. 2006) is visible at the right h and side of
+the diagram near ( l,b) = (152◦,37◦), not far from the extension of the projected Sgr orbit.
+Kinematics of UMa II(Martin et al. 2007), show itto have a vradial=−115km s−1(following
+the body and figures of that paper rather than the abstract) wh ich corresponds to vgsr∼
+−35 km s−1. This is in strong disagreement with the Orphan model predictions of vgsr∼+60
+km s−1at the approximate location of UMa II. The distance d⊙to the object is also in
+disagreement with the models in Figures 12 and 13. Additionally, the de nsity of Orphan
+Stream stars, were UMa II the Orphan progenitor, should increas e as one gets closer to UMa
+II, not decrease. For these reasons, given the improved SEGUE d ata points and implied
+Orphan Stream kinematics, the current data do not support the h ypothesis that UMa II is
+the progenitor of the Orphan Stream.
+11.3. Complex A
+As discussed above, and in Sales et al. (2008), Complex A is much too c lose to be
+associated with the Orphan Stream without extensive gas dissipatio n. It would take a
+strongly dissipative interaction and a significant amount of time for t he HI gas to become
+stripped from the Orphan Stream and migrate to lower Galactocent ric distances.
+11.4. Other possible associations
+We check our best fit orbits against known globular clusters and dwa rf galaxies to see
+if there are any possible associations. Of the globular clusters given in Harris (1996), we
+find six that lie within 5 degrees of the orbits: NGC 6752, NGC 6362, Ru p 106, Pal 1, NGC
+7078, and NGC 7089. Of these, only three have radial velocities con sistent with the orbit:
+NGC 6362, NGC 7078, and NGC 7089. However, these are approxima tely 10 kiloparsecs
+from the Sun, while the orbit is consistently above 20 kiloparsecs. Th erefore we rule out
+these as being associated with this stream.
+In addition to Ursa Major II, we find three additional dwarf galaxies that lie within 5
+degrees of the projected orbits: Leo II, Leo A, and Ursa Major Y . Of these, only Leo A has
+a radial velocity that could associate it with the orbit, but its distanc e of 800 kpc from the
+Sun precludes this possibility.– 25 –
+12. Conclusions
+We present substantially better spatial and radial velocity measur ements of Orphan
+Stream stars than previous papers, and are therefore able to fit a better orbit solution,
+which is substantially different from previous papers.
+WhiletheOrphanStreamwassonamedbecauseoftheabsenceofap rogenitor, thereisa
+suggestion from the data in this paper that the progenitor is locate d near (l,b) = (253◦,49◦),
+right at the edge of the SDSS survey footprint. Like previous pape rs, our data suggest a
+small dwarf galaxy to be the likely progenitor, since the velocity dispe rsion of the stream
+stars is∼10 km s−1.
+We summarize our findings as follows:
+1) The Orphan Stream is visible on the sky in SDSS imaging and traces an arc from
+(l,b) = (271◦,38◦) to (l,b) = (171◦,46◦) with avgsr∼+110±20 km s−1for most of this
+length at distances increasing from d⊙= 18 kpc to over d⊙= 46 kpc, where it is no longer
+detectable in our data. The stream is on a prograde orbit with periga lacticon, apogalacticon,
+eccentricity and inclination of 16 .4 kpc,∼90 kpc,0.7,and 34◦, respectively.
+2) We identify F turnoff stars and BHBs in the Orphan Stream, along w ith candidate
+blue stragglers. All of these populations except F turnoff stars als o have representatives with
+SEGUE spectra; there may also be F turnoff stars with spectra, bu t they are so faint that
+surface gravities could not be measured. The turnoff color is ( g−r)0= 0.22.
+3) The metallicity of the Orphan Stream BHBs is very low, [Fe/H] = −2.1±0.1.
+4) The progenitor for the Orphan Stream may be located between ( l,b) = (250◦,50◦)
+and (270◦,40◦). This assumption, combined with the observation that the Orphan Stream
+is on a prograde orbit around the Galaxy implies that the tidal tail at l<250◦is a leading
+tidal tail.
+5) The Orphan Stream data is best fit to a Milky Way potential with a ha lo plus disk
+plus bulge mass of about 2 .6×1011M⊙, integrated to 60 kpc from the Galactic center. Our
+best fit is found with a logarithmic halo speed of vhalo= 73±24 km s−1, a disk+bulge mass
+ofM(R <60 kpc) = 1 .3×1011M⊙, and a halo mass of M(R <60 kpc) = 1 .4×1011M⊙.
+However, we can find similar fits to the data that use an NFW halo profi le. Our halo
+speed ofvhalo= 73±24 km s−1is smaller than that of previous literature. Although our
+fits with smaller disk masses and correspondingly larger halo masses a re not good fits the
+the Xue et al. (2008) rotation curves, we have not tried enough mo dels to rule out this
+possibility. Distinguishing between different classes of models require s data over a larger
+range of distances.– 26 –
+6)The Orphan Stream is projected to extend to 90 kpc from the Ga lactic center, and
+measurements of these distant parts of the stream would be a pow erful probe of the mass of
+the Milky Way.
+7) An N-bodysimulation yields an excellent fit to the observed data, in cluding matching
+the approximate shape of the stellar stream star density along the stream. The N-body mass
+used was about Mtotal,Orphan= 106M⊙, about 10−3the total mass of the Sgr stream and dSph
+system. The total Orphan system mass is not highly constrained.
+8) The list of possible halo objects associated with the Orphan Strea m is reduced to
+one: The ‘dissolved star cluster’ Segue-1. Other possible associat ed objects, namely UMa
+II, the Complex A HI cloud, and the halo’s globular clusters are not clo se to the Orphan
+Stream orbit presented here in distance or velocity (or both).
+Because the orbit fit is not able to significantly constrain the flatten ingqof the halo
+potential, we assume a q= 1.0 throughout. Nevertheless, the analysis of the Orphan Stream
+shows us that we are able to constrain another important halo pote ntial parameter, namely
+the amplitude of the halo potential. A spherical logarithmic potential with 60% of the mass
+of the fit of Xue et al. (2008) provides a reasonable fit to the Orpha n Stream data.
+We acknowledge useful discussions with Steve Kent. We acknowledg e helpful comments
+from the referee which significantly improved the presentation of t he results of this paper.
+This work was supported by the National Science Foundation, gran t AST 06-06618. Funding
+for the SDSS and SDSS-II has been provided by the Alfred P. Sloan F oundation, the Partic-
+ipating Institutions, the National Science Foundation, the U.S. Dep artment of Energy, the
+National Aeronautics and Space Administration, the Japanese Mon bukagakusho, the Max
+Planck Society, and the Higher Education Funding Council for Englan d. The SDSS Web
+Site is http://www.sdss.org/.
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+This preprint was prepared with the AAS L ATEX macros v5.2.– 30 –
+Table 1. Orphan Stream Detections
+l b δb ΛOrphan BOrphan BHBg0δg0
+◦ ◦ ◦ ◦ ◦mag mag
+255 48.5 0.7 22.34 0.08 17.1 0.1
+245 52.0 0.7 15.08 0.56
+235 53.5 0.7 8.86 0.21
+225 54.0 0.7 2.95 -0.23 17.6 0.2
+215 54.0 0.7 -2.93 -0.33 17.9 0.1
+205 53.5 0.7 -8.85 -0.09 18.0 0.1
+195 52.0 0.7 -15.08 0.05
+185 50.5 0.7 -21.42 1.12 18.6 0.1
+175 47.5 0.7 -28.59 1.88
+171 45.8 1.0 -31.81 2.10
+Table 2. Orphan Stream Detections with Velocities
+ΛOrphan l b δb BHBg0δg0vgsr δvgsr σvNd δd
+◦ ◦ ◦ ◦mag mag km s−1km s−1km s−1kpc kpc
+-30 173 46.5 0.7 18.8 0.2 115.5 6.7 11.5 4 46.8 4.5
+-20 187 50.0 1.0 18.5 0.1 119.7 6.9 11.9 4 40.7 1.9
+-9 205 52.5 0.7 18.0 0.1 139.8 4.6 12.9 9 32.4 1.5
+-1 218 53.5 1.0 17.8 0.1 131.5 3.1 8.2 8 29.5 1.4
+8 234 53.5 0.7 17.4 0.1 111.3 11.1 11.1 2 24.5 1.2
+18.4 249.5 50.0 0.7 17.1 0.1 101.4 2.9 9.8 12 21.4 1.0
+36 271 38.0 3.5 16.8 0.1 38.4 1.7 2.5 3 18.6 0.9
+Table 3. Orphan Stream Models
+N M Bulge Disk Mdisk Halo Mhalo d/rsRfit vx vy vz vhalo χ2Ma
+60
+1010M⊙type 1010M⊙type 1010M⊙kpc kpc km s−1km s−1km s−1km s−11010M⊙
+1 1.5 Exp 5 NFW 33 22.25 28 .8−170±11 94 ±2 108 ±10 155 1.55 40
+2 1.5 Exp 5 NFW 20 22.25 28 .5−157±10 78 ±12 107 ±9 120 ±7 1.37 24
+3 1.5 Exp 5 Log 17.6 12 28 .4−152±12 72 ±12 106 ±9 81 ±12 1.35 26.5
+4 3.4 M-N 10 Log 35 12 28 .9−179±11 106 ±2 109 ±10 114 1.98 47
+5 3.4 M-N 10 Log 14 12 28 .6−156±10 79 ±1 107 ±9 73 ±24 1.70 26.4
+6 3.4 M-N 10 NFW 33 22.25 28 .9−178±5 106 ±3 108 ±10 155 1.96 43.5
+7 3.4 M-N 10 NFW 16 22.25 28 .7−161±11 85 ±1 107 ±9 109 ±31 1.73 28.4
+aMass due to sum of bulge, disk, and halo components out to R = 60 kpc– 1 –
+-15-551525354555657585
+150180210240270b
+-15-551525354555657585
+150180210240270Orphan Stream
+SextansSegue I
+UMa IIb
+l
+Fig. 1.— We show the sky position of the Orphan Stream as traced by F turno ff stars with 0 .12<
+(g−r)0<0.26,(u−g)0>0.4, and 20 .0< g0<22.5. Darker areas of the greyscale plot have higher stellar
+density. The most prominent overdensity, going from ( l,b) = (240◦,65◦) to (l,b) = (200◦,15◦), is from the
+Sagittarius dwarf tidal stream. The density falls off at higher latitud es because its stars are too faint for
+our selection. Twenty degrees to the right, and running parallel to Sagittarius, is another stream that is
+generally referred to as a “bifurcation” of the Sagittarius dwarf t idal stream. The Orphan Stream is a nearly
+horizontal arc in this projection, at b∼50◦. There are very many stream stars at l= 255◦. The positions of
+the Sextans and Ursa Major II dwarf galaxies are labeled in the lower panel. The lower panel shows where
+BOrphan= 0◦, which is our initial fit to the sky position of the Orphan Stream.– 2 –
+Fig. 2.— We determine the distance to the Orphan Stream from a plot of Λ Orphanvs.g0for photo-
+metrically selected BHB stars that are within one degree of BOrphan= 0◦. We also show the positions
+of spectroscopically selected BHB stars that are within one degree ofBOrphan= 0◦, have low metallicity
+([Fe/H] WBG<−1.6), and are clustered in distance and velocity consistent with membe rship in a tidal debris
+stream. The black line shows our fit to the distance of the stream as a function of Λ Orphan.– 3 –
+-1.0-0.5-0.00.51.01.52.014
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24Orphan BHBs
+gcorr
+(g-r)o
+Fig. 3.— We present a color-magnitude Hess diagram of the stars within two d egrees of the Orphan
+Stream ( −2◦< BOrphan<2◦), and not near the Sagittarius dwarf tidal stream ( −26◦<ΛOrphan<−7◦or
+10◦<ΛOrphan<40◦). Abackgroundpopulation with the sameΛ Orphanrange,but with 2◦<|BOrphan|<4◦,
+has been subtracted. The apparent magnitudes of the stars hav e been shifted, based on their position
+along the stream, so that BHBs at the distance of the Orphan Stre am will have an apparent magnitude of
+gcorr= 17.75. This is the apparent magnitude of BHB stars at Λ Orphan= 0◦. We identify the turnoff of the
+Orphan Stream in this diagram at ( g0,(g−r)0) = (21.2,0.22), at its bluest point. This figure includes an
+inset in which BHB stars have been color-selected to exclude QSOs an d blue stragglers. Note the excess of
+BHB stars at ( g−r)0=−0.15 andgcorr= 17.75. Since there are very few color-selected BHB stars in the
+background region, background is not subtracted in the inset.– 4 –
+-15-551525354555657585
+150180210240270b
+-15-551525354555657585
+150180210240270Orphan Stream
+SextansSegue I
+b
+l
+Fig. 4.— Similar toFigure1, except that the g0ofstarshavebeen shifted as afunction ofΛ Orphanbasedon
+the fit in Figure 2. Stars with 20 .7< gcorr<21.7, the magnitude centered on the Orphan Stream’s turnoff,
+are plotted in a Hess diagram over the SDSS footprint. The stream c an be seen extending to nearly l= 170◦.
+Additionally, BJ,R2data from SuperCOSMOS is extracted and added to the figure to co ver the region of
+the Orphan Stream where no SDSS data exists. The stars here hav e 20< BJ<21,0.5< BJ−R2<1.0
+(with the default calibration and reddening corrections adopted th roughout). A narrow trail is visible in
+the SuperCOSMOS extension, running from ( l,b) = (268◦,38◦) to (255◦,48◦). The lower panel adds the
+Bcorr= 0 trace of the stream, which differs slightly from BOrphan= 0◦forl <200◦. The location of the halo
+object Segue-1 is indicated. The inset shows a blowup of the region w ith 230◦< l <280◦,30◦< b <70◦.– 5 –
+Fig. 5.— NumbercountsofFturnoffstarswithin ±2◦ofthestreamareplottedasanopenblackhistogram.
+The number of background turnoff stars off-the-stream on eithe r side (2◦<|Bcorr|<4◦) are plotted in red.
+The difference is plotted as a hashed histogram. Note the significant excess of turnoff stars over background
+near Λ Orphan= +23◦, corresponding to ( l,b) = (255◦,49◦). It is possible that the stream progenitor lies in
+this region of sky. The bins at Λ Orphan= 21◦,33◦,and 36◦have been corrected for incompleteness in both
+the data and background counts.– 6 –
+Fig. 6.— Velocities ofBHB starswithin 2◦ofthe OrphanStream. The solid dots indicate allstarswith the
+correct surface gravity, ugrcolors and g0magnitudes (based on Figure 2) to be BHBs and stream members.
+The red circles indicate stars with [Fe/H] WBGthat is lower than -1.6. Points with green centers are our
+Orphan Stream candidate BHBs with spectra which meet our final ve locity cut as well as the metallicity cut.
+The concentration of points at Λ Orphan= 0◦,vgsr∼ −100 km s−1are likely associated with the Sagittarius
+leading tidal stream (Yanny et al. 2009b). The 10◦marks indicate the bins used in calculating the Orphan
+Stream properties at positions given by the lower black points with er ror bars. The magenta boxes indicate
+the earlier, preliminary velocity possibilities for the Orphan Stream fr om Belokurov et al. (2007b), converted
+from radial velocity to vgsr.– 7 –
+Fig. 7.— Similar to Figure 2 except: 1) stars are selected to be within 2◦of the stream instead of just 1◦
+and 2) stars are chosen using the corrected Bcorrfor ΛOrphan<−15◦. Orphan Stream BHBs with spectra
+(green filled circles from Figure 6) are overlaid as open circles. We also overlay the adopted fit to the g0
+magnitude of the BHB stars as a function of Λ Orphan.– 8 –
+-1.0-0.5-0.00.51.01.52.014
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24Orphan BHBs
+gcorr
+(g-r)o
+Fig. 8.— Same as Figure 3, with overlay of globular cluster fiducial sequences , normalized to place
+their HBs at gcorr= 17.75. The clusters are: M92 [Fe /H] =−2.3 (blue), M3 [Fe /H] =−1.57 (red), M13
+[Fe/H] =−1.54(green) and M71[Fe /H] =−0.73(magenta). OrphanStream BHB spectral candidates(filled
+green points from Figure 6) are marked with small black + signs. Othe r spectra within 2◦of the stream,
+and having velocity, metallicity and surface gravity consistent with t he stream are indicated with magenta
+crosses×. Control field spectra, meeting the same velocity, gravity and met allicity conditions but which lie
+more than 2◦and less then 5◦from the stream are indicated with black circles.– 9 –
+Fig. 9.— We show the sky positions for all Orphan Stream spectral BHB cand idates which meet the
+luminosity, velocity, color and magnitude selection conditions of Figur es 6 and 8 as open circles. Filled
+magenta dots show those BHBs with exceptionally low metallicity, [Fe /H]<−1.6. The blue triangles
+indicate the set of chosen fiducial positions along the Orphan Strea m with spectral velocity measurements.
+There is good consistency between the low metallicity stars and the k inematically selected sample, except at
+l= 270◦. The SDSS data in this stripe are ambiguous, but we note that our ex trapolated stream position
+is a reasonable fit to the SuperCOSMOS data.– 10 –
+Fig. 10.— Though proper motions of stream candidate stars are not used in fi tting the Orphan Stream
+orbit in this paper, they can provide a consistency check. We plot he reµraandµdecvslfor two subsets
+of candidate Orphan Stream stars: 1) Spectroscopic stream can didates (magenta crosses from Figure 8),
+are shown as crosses, and 2) Spectroscopically more secure BHB s tream candidates (green filled circles from
+Figure 6, black + signs from Figure 8), are shown as filled black circles s uperimposed on a cross. Errors on
+individual points are typically 3 mas yr−1in bothαandδ. Zero proper motion is indicated with a dashed
+line. A representative Orphan Stream orbit model (model 5) is over laid as a heavy black line.– 11 –
+Fig. 11.— In the upper panel, we show the metallicity distribution of Orphan Str eam BHB candidates,
+without any prior metallicity cut, as measured by the Wilhelm, Beers, & Gray (1999) method of estimating
+[Fe/H] based on Ca II K line strength and ( g−r)0color. The Orphan Stream BHBs have a very low
+metallicity of [Fe /H] =−2.1±0.1. The lower panel shows the ‘adopted’ SSPP [Fe/H], for the stars w hich
+meet the magnitude, color, surface gravity and velocity cuts for s tream membership (BHBs excluded). The
+distribution for stars with |Bcorr|<2◦are shown in light outline, while that for stars with 2◦<|Bcorr|<4◦
+is shown in heavy outline.– 12 –
+Fig. 12.— We present orbit fits ( b,vgsr,anddvs. Galactic longitude) to the Orphan Stream data for three
+different halo potentials (models 1-3), with a fixed bulge and exponen tial disk model with parameters copied
+from Xue et al. (2008). In red (dot dash, model 1) we show the bes t orbit using the NFW halo parameters
+from Xue et al. (2008). In orange (dotted, model 2) we show the b est fit if we allow the vcparameter to
+vary in the NFW profile. In black (solid, model 3) we show the best fit if we instead use a logarithmic
+potential. Note that the best fit NFW and logarithmic potentials give s imilar fits, but both have a lower
+velocity (and therefore halo mass) than found by Xue et al. (2008) . To fit the Orphan Stream distance data,
+it is necessary to reduce the amplitude of the halo potential by abou t 40%. The distance-velocity space
+locations of Segue-1, Ursa Major II, Complex A are indicated as labe led in the top panel. Note that only
+Segue-1 is possibly associated with this tidal stream.– 13 –
+Fig. 13.— We now compare the best fit models from Figure 12 with the Milky Way ro tation curve data
+from Xue et al. (2008) and Koposov, Rix, & Hogg (2009). The red cu rve (dot dash, model 1) is the best fit
+found by Xue et al. (2008), from Figure 16a of that paper, so it is a g ood fit to the data points. The orange
+curve (dotted, model 2) and black curve (solid, model 3), for whic h we allowed the halo mass to vary, are
+significantly below all of the rotation curve data points. The rotatio n curves from individual components of
+the potential (exponential disk, bulge, and NFW halo) used in Xue et al. (2008) are also shown. We also
+show a Miyamoto-Nagai disk, scaled to a total mass similar to that of the exponential disk, to show that the
+rotation curve in the region we probe is not dramatically different for different disk profiles, and the halo
+rotation curves for the lower mass halos.– 14 –
+Fig. 14.— We present orbit fits ( b,vgsr,anddvs. Galactic longitude) to the Orphan Stream data for
+potential models 4-7, with a fixed bulge and Miyamoto & Nagai (1975) disk with parameters copied from
+Law, Johnston, & Majewski(2005). In green (dash, model 4) we showthe fit best orbit using the logarithmic
+halo parameters from Law, Johnston, & Majewski (2005). In red (dot dash, model 6) we show the best fit
+with NFW halo parameters fixed from Xue et al. (2008). Neither of th ese fixed halo potentials from previous
+papers gives a good fit to the relative distances along the Orphan St ream; the slope of the dvs.lplot is too
+shallow for these models. The black (solid, model 5) and orange curv es (dotted, model 7) show the best fit
+orbits if the disk is fixed and the halo masses are allowed to vary for th e logarithmic and NFW profile halos,
+respectively. As in Figure 12, we see very little difference between th e best logarithmic and NFW profile fits,
+and the preferred values of the halo mass is lower than that of prev ious authors. Again, it is necessary to
+reduce the amplitude of the halo potential by about 40% to fit the Or phan Stream distances. The locations
+of Segue-1, Ursa Major II, Complex A are as labeled as in Figure 12.– 15 –
+Fig. 15.— We now compare the best fit models from Figure 14 with the Milky Way ro tation curve data
+from Xue et al. (2008) and Koposov, Rix, & Hogg (2009). All of the m odels use a fixed M-N disk and bulge,
+with a mass about twice as large as the exponential disk in the models in Figures 12 and 13. The green
+(dash, model 4) and red (dot dash, model 6) curves use fixed par ameters from the logarithmic halo potential
+of Law, Johnston, & Majewski (2005) and NFW halo potential of Xu e et al. (2008), respectively. Neither of
+these fit the Orphan Stream distances, and they are not especially good fits to the rotation curve, predicting
+rotation velocities above most of the data points. The black (solid, m odel 5) and orange curves (dotted,
+model 7) are reasonably good fits to the Xue et al. (2008) data, ev en though they were not fit to this data.
+In these models, the halo mass was allowed to vary. Our best fit mode l is model 5 (black, solid curve),
+though model 7 is nearly as good. The rotation curves from individua l components of the potential (M-N
+disk, bulge, and NFW or logarithmic halo) are also shown.– 16 –
+Fig. 16.— The black line shows our preferred fit orbit in the logarithmic halo with vhalo= 73 km s−1
+in right-handed Galactic rectangular coordinates ( X,Y,Z). The Sun is at ( −8,0,0) kpc and the Galactic
+center at the origin. The ( l,b) coordinates and BHB magnitudes are converted to ( X,Y,Z) assuming a
+BHB absolute magnitude of Mg= 0.45. The arrows indicate the forward direction of the orbit and the
+Sun’s motion. Points from Table 2 of observations of the Orphan Str eam data are shown as asterisks. The
+positions of Segue-1, UMa II and Complex A are shown with the same s ymbols as in Figure 14.– 17 –
+Fig. 17.— Theb,vgsranddSunvs.l(top 3 panels) orbit of the preferred halo (logarithmic potential
+withvhalo= 73 km s−1, M-N disk and heavy bulge), simply integrated, is shown as a heavy bla ck curve.
+A 10,000 point N-body simulation of a Plummer sphere dwarf with Mtotal∼106M⊙is integrated in this
+potential from a time 4 Gyr ago forward for 3.945Gyr placing the pro posed progenitor at ( l,b)∼(250◦,50◦).
+The lower panel shows the density distribution of the final (curren t) epoch positions of the N-body points
+as a histogram in l. Note the similarity of the the simulated distribution, including the dip in density at
+l= 240◦(ΛOrphan= +15◦) seen in Figure 5, and the spread (first panel) and density falloff at l <180◦seen
+in Figure 4.
\ No newline at end of file
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new file mode 100644
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--- /dev/null
+++ b/1001.0577.txt
@@ -0,0 +1,1468 @@
+arXiv:1001.0577v2  [hep-th]  14 Jul 2010Particle Physics Implications of F-theory 1
+Particle Physics Implications of F-theory
+Jonathan J. Heckman
+School of Natural Sciences, Institute for Advanced Study, P rinceton, NJ 08540
+Key Words F-theory, Grand Unified Theories, String Phenomenology
+Abstract We review recent progress in realizing Grand Unified Theorie s (GUTs) in a strongly
+coupled formulation of type IIB string theory known as F-the ory. Our main emphasis is on the
+expected low-energy phenomenology of a minimal class of F-t heory GUTs. We introduce the
+primary ingredients in such constructions, and then presen t qualitative features of GUT models
+in this framework such as GUT breaking, doublet-triplet spl itting, and proton decay. Next, we
+review proposals for realizing flavor hierarchies in the qua rk and lepton sectors. We discuss
+possible supersymmetry breaking scenarios, and their cons equences for experiment, as well as
+geometrically minimal realizations of F-theory GUTs which incorporate most of these features.
+CONTENTS
+Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
+GUT-Like Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
+Local Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
+Organization of this Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
+Building Blocks of F-theory GUTs . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
+Brief Review of F-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
+Geometry and Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
+The Four-Dimensional Effective Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 10
+Seven-Brane Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
+GUT Breaking and its Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . 15
+Local Models and GUT Breaking Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . 15
+Proton Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
+Gauge Coupling Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
+Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
+Moduli Dominated Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Annu. Rev. Nuc. Part. Sci. 2010 60
+Gauge Mediation Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
+Flavor in F-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
+Geometry of Flavor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
+Quark Yukawas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
+Neutrino Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
+MinimalE8Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
+Experimental Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
+Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
+1 Introduction
+Of the many vacua of string theory, presumably some are compa tible with ex-
+periment. Even so, not a single string based model has yet bee n found which
+satisfies all known constraints. Extracting predictions fr om this vast list of pos-
+sibilities requires narrowing the search to phenomenologi cally promising vacua.
+Once suitable assumptions are imposed, we can then ask wheth er the remaining
+vacua are compatible with observation, and moreover, what n ew phenomena to
+expect. Correlations between seemingly unrelated physica l features then trans-
+late into predictions of the model.
+For the purposes of particle physics, it is expected that gra vity only plays a
+significant role at energies near the Planck scale Mpl∼1019GeV. As a first step
+in identifying promising vacua, it is thus natural to focus o n a limit where the
+effects of gravity decouple from the rest of particle physics. More formally, this
+can bephrasedas the existence of a local model wheretheparticle physics degrees
+of freedom localize on a spacetime filling brane which can be s eparated from the
+rest of the string compactification. This by itself is a rathe r mild condition
+which can be met in many string constructions [1], again prov iding a landscape
+of potentially interesting vacua.
+As an additional criterion for selecting promising vacua, w e shall also assume
+thatlowenergysupersymmetrywillsoonbefound,andthatmo reover, thisshould
+be viewed as evidence for Grand Unified Theories (GUTs). Thou gh circum-
+stantial, the evidence for GUT-like structures based on SU(4)×SO(4),SU(5),
+orSO(10) gauge groups [2,3,4] includes the unification of the gau ge coupling
+constants in the Minimal Supersymmetric Standard Model (MS SM), and the in-
+triguing fact that the chiral matter unifies into complete GU T multiplets charged
+underSU(5). See [5] for a recent review of some aspects of GUTs.
+2Particle Physics Implications of F-theory 3
+Though the requirements of a local model and GUT-like struct ures can indi-
+vidually be met, combining them turns out to significantly na rrow the available
+options in string based models. A strongly coupled formulat ion of IIB string
+theory known as F-theory [6,7,8] provides a potentially pro mising starting point
+for realizing GUTs in local models. For recent work on F-theo ry GUTs, see for
+example [9] – [59]. Here we investigate what types of F-theor y GUT geome-
+tries realize elements of the Standard Model, and conversel y, how the geometry
+constrains the available model building options.
+1.1 GUT-Like Structures
+Wenowbrieflyreview theaspectsofGUTsweshallaim torealiz e instringtheory.
+Throughout, we work in conventions compatible with N= 1 supersymmetry, so
+that all fermions are taken to be left-handed Weyl spinors.
+In the MSSM, there are three generations of chiral superfield s given by Stan-
+dard Model analogues of the quark doublets Q, anti-upsUand anti-downs D
+transforming in the respective SU(3)C×SU(2)L×U(1)Yrepresentations
+Q: (3,2)1/6, U: (3,1)−2/3, D: (3,1)1/3 (1)
+and three generations of lepton doublets L, and anti-leptons Ein the represen-
+tations
+L: (1,2)−1/2, E: (1,1)1. (2)
+Compared with the Standard Model, the Higgs sector is extend ed to include
+Higgs up and Higgs down chiral superfields in the representat ions
+Hu: (1,2)1/2, Hd: (1,2)−1/2. (3)
+The Higgs fields couple to the chiral matter through the super potential terms
+WMSSM⊃λu
+ijHuQiUj+λd
+ijHdQiDj+λl
+ijHdLiEj(4)
+whichrespectively inducemassesfortheup-type, down-typ e, andcharged leptons
+of the Standard Model.
+A remarkable feature of the MSSM is that when extrapolated up from the
+weak scale, to one loop order the gauge coupling constants un ify at a common
+value ofαGUT=g2/4π∼1/24 at an energy scale of 2 ×1016GeV. This suggests
+unifying all interactions into a gauge group such as SU(5). Indeed, the chiral
+matter of the MSSM also unifies into the SU(5)⊃SU(3)C×SU(2)L×U(1)Y
+representations
+10M→Q⊕U⊕E,5M→D⊕L. (5)4 Heckman
+The Higgs up and Higgs down fields of the MSSM fit in the 5 Hand5H, and the
+interaction terms for the up and down type quarks are respect ively controlled by
+the interaction terms
+5H×10M×10Mand5H×5M×10M. (6)
+This is only thefirststep in realizing aGUT model. For exampl e, a realistic GUT
+must also contain mechanisms for breaking the GUT group; rem oving the Higgs
+tripletsfromthe5 Hand5H; generatingmorerealisticflavortextures; maintaining
+a sufficiently stable proton; and coupling to a supersymmetry breaking sector.
+1.2 Local Models
+Our main focus in this article will be on a class of string theo ry based con-
+structions known as local models [1]. Since gravity is quite weak even at the
+GUT scale, the aim of a local model is to first reproduce detail s of the Standard
+Model, and worry about gravity later. Geometrically, a loca l model corresponds
+to a system where gauge theory localizes on a (4 + k)-dimensional subspace of
+the ten-dimensional string theory. Starting from the ten-d imensional Einstein-
+Hilbert action, and the (4+ k)-dimensional Yang-Mills action, the resulting gauge
+coupling constant and four-dimensional Newton’s constant respectively depend
+on the internal volumes as
+(g4d
+YM)2∝Vol(Mk)−1, G4d
+N∝Vol(M6)−1. (7)
+Decoupling gravity corresponds to a limit where the ratio of the characteristic
+radii (Vol( Mk))1/k/(Vol(M6))1/6becomes parametrically small.
+In perturbative type II superstring theory, this suggests a ssociating the closed
+string sector with gravity, and the open string sector with g auge theory. In this
+case, the Standard Model degrees of freedom localize on Diri chlet- or D-branes.
+As a point of terminology, we shall refer to a Dp-brane as one w hich fills the
+temporal direction and p spatial directions. In F-theory we shall encounter non-
+perturbative generalizations of D-branes which allow us ex tra model building
+options. For reviews of model building efforts with open strin gs, see [60].
+Perturbative openstringdegrees of freedomprovidenearly all of thequalitative
+ingredients necessary to realize the Standard Model. The tr ansfer matrix of open
+strings between such D-branes defines the color space degree s of freedom of the
+gauge theory. For example, nspacetime filling D3-branes at the same point
+inR6realizes four-dimensional N= 4U(n) gauge theory. It is also possibleParticle Physics Implications of F-theory 5
+to engineer SOandUSpgauge theories using unoriented open strings. Open
+strings stretched between different stacks of D-branes corre spond to matter fields
+charged under the fundamental of one gauge group, and the ant i-fundamental of
+the other. With unoriented strings it is also possible to gen erate matter charged
+in the two index symmetric and anti-symmetric representati ons of a gauge group.
+Note that all of the matter content of the Standard Model tran sforms in such
+representations. In string perturbation theory, the gauge invariant interaction
+terms always contract fundamental indices with anti-funda mental indices.
+Thereare, however, drawbackstolocal modelsbasedonpurel yperturbativede-
+greesoffreedom. Muchofthistensionstemsfromtheabsence ofcertainGUT-like
+structures. While it is possible to engineer SU(5), or more precisely U(5) gauge
+theories with the matter content of a GUT, terms such as the 5 H×10M×10M
+interaction cannot be generated in string perturbation the ory because such in-
+teraction terms do not match up an equal number of fundamenta l and anti-
+fundamental indices. Other GUT structures based on SO(10) or exceptional
+groups fare even less well. The spinor 16 of SO(10) cannot be realized in open
+string perturbation theory, and E-type gauge group structu res are also unavail-
+able. All of this points to the need for extra non-perturbati ve ingredients.
+1.3 Organization of this Review
+The rest of this review is organized as follows. In section 2 w e introduce the
+primary building blocks of F-theory GUTs. Next, in section 3 we review how
+this class of models addresses some of the qualitative issue s present in four-
+dimensional GUT models. In section 4 we discuss supersymmet ry breaking sce-
+narios. Section 5 discusses more detailed aspects of flavor p hysics for quarks and
+leptons, and in section 6 we review some of the characteristi cs of minimal models
+based on a single point of E8unification. Section 7 presents our conclusions and
+possible directions for future investigation.
+The intent of this article is to provide a point of entry to the literature. For
+earlier reviews with slightly different emphasis, see [20,21 ,37,55]. In addition,
+some important topics will only be treated briefly, if not at a ll. Most noticeably
+absent is a discussion of recent efforts to include the effects of gravity [24,29,32,
+41,36,43,46,56,59].6 Heckman
+2 Building Blocks of F-theory GUTs
+2.1 Brief Review of F-theory
+A remarkable feature of type IIB string theory is that it is in variant under the
+duality group SL(2,Z), corresponding to all 2 ×2 matrices with integer entries
+with unit determinant. Under the action of the SL(2,Z) duality group, the
+Neveu-Schwarz (NS) and Ramond Ramond (RR) two-forms transf orm as a two-
+component doublet. Moreover, complexifyingthestringcou plinggsby combining
+it with the RR zero-form C0, the axio-dilaton
+τ=C0+i
+gs(8)
+transforms as τ→(aτ+b)/(cτ+d) under integers a,b,c,dsuch thatad−bc= 1.
+Under the weak/strong duality map τ→ −1/τgiven bya=d= 0 and
+b=−c=−1, the fundamental string and D1-brane interchange roles. F or
+more general duality group transformations the fundamenta l string maps to a
+bound state of pfundamental strings and qD1-branes or a ( p,q) string. Since
+fundamental strings end on D-branes, acting by the duality g roup provides us
+with new strongly coupled ( p,q) analogues of the more familiar D-branes on
+which strings of the same ( p,q) type can end. Branes of different ( p,q) type can
+also form non-perturbative bound states.
+The profile of the axio-dilaton is closely connected with the presence of seven-
+branes in the compactification. As an illustrative example, consider a D7-brane
+fillingR7,1and sitting at apoint of the remaining two spatial direction s. Theone-
+form fluxF1sourced by the D7-brane is locally defined by the relation F1=dC0.
+Globally, we cannot write F1in this form. Indeed, passing around a circle sur-
+rounding the D7-brane, C0shifts asC0→C0+1. Similarly, nD7-branes source
+nunits ofF1flux, inducing C0→C0+n. This corresponds to an SL(2,Z) trans-
+formation with a=d= 1,b=nandc= 0. Acting by SL(2,Z) transformations,
+we see that different seven-branes will affect the axio-dilaton in different ways.
+F-theory [6] provides a geometric formulation of strongly c oupled type IIB
+vacua which automatically keeps track of τin the presence of seven-branes. The
+main idea is to specify the profile of τby including two additional geometric
+directions in addition to the usual ten spacetime dimension s of string theory. Let
+us stress that these two extra dimensions are on a somewhat di fferent footing
+from the other ten. In this twelve-dimensional geometry, th e parameter τis to
+be viewed as the shape of a two-torus, or elliptic curve [6]. T his two-torus canParticle Physics Implications of F-theory 7
+be visualized as a parallelogram in the complex plane C, with vertices at 0, 1, τ
+andτ+1, and parallel sides identified. The shape of this torus is i nvariant under
+SL(2,Z) transformations τ→(aτ+b)/(cτ+d), which makes manifest the action
+of theSL(2,Z) duality group of IIB string theory. The presence of seven-b ranes
+will affect the value of τ, and is indicated in the geometry by allowing τto attain
+different values at different points in the spacetime.
+For the purposes of model building, we will be particularly i nterested in com-
+pactifications of F-theory which preserve four real superch arges in four dimen-
+sions. Retaining this amount of supersymmetry imposes the g eometric condition
+that these supercharges be covariantly constant in the four complex-dimensional
+internal space. Much as in other string compactifications, t his condition is met
+for F-theory compactified on a four complex-dimensional Cal abi-Yau space. Un-
+like other string compactifications, the six physical inter nal directions define a
+complex threefold B3which need not be Calabi-Yau.
+The geometry of the Calabi-Yau fourfold automatically incl udes the effects of
+seven-branes. As we have seen, encircling a seven-branelea ds to aSL(2,Z) trans-
+formationof τ. Whilethisisanacceptabledescriptionaway fromtheseven -brane,
+this means that on top of the seven-brane, the shape of the two -torus becomes
+singular, and effectively pinches off. As a hypersurface in C2with coordinates x
+andy, the elliptic curve can be modeled in Weierstrass form as
+y2=x3+fx+g (9)
+wherefandgwill in general have non-trivial position dependence on B3. The
+location of the seven-branes are then given by roots of the di scriminant of the
+cubic inx
+∆ = 4f3+27g2. (10)
+The equation ∆ = 0 corresponds to a one complex dimensional eq uation inB3,
+and so sweeps out a two complex-dimensional subspace. As a po lynomial in the
+coordinates of thethreefold base B3, ∆ may factorize into irreduciblepolynomials
+so that ∆ = ∆ 1···∆n. Each factor ∆ i= 0 then defines the location of seven-
+branes inB3.
+2.2 Geometry and Gauge Theory
+The locations of seven-branes are dictated by where the elli ptic curve of an F-
+theory compactification pinches off. The precise ways that thi s pinching off can
+occur admits a classification in terms of the ADE Dynkin diagr ams familiar from8 Heckman
+gauge theories, which is known as the Kodaira classification of singular fibers.
+Locally, we can describe the eight real-dimensional space o f the Calabi-Yau four-
+fold in terms of two complex directions spanned by the seven- brane, and two
+additional complex directions given by an ADE type singular ity (see table 1).
+As their name suggests, these singularities correspond to g eometries in which a
+numberofS2’s intersecting according to theADE Dynkin diagram have col lapsed
+to zero size.
+Table 1: Dictionary for ADE singularities and seven-brane gauge
+groups.
+ADE Type Equation in C3F-theory Gauge Group
+Any2=x2+zn+1SU(n+1)
+Dny2=x2z+zn−1SO(2n)
+E6y2=x3+z4E6
+E7y2=x3+xz3E7
+E8y2=x3+z5E8
+In M-theory and type IIA string theory, it is known that compa ctifying on an
+ADE singularity realizes the corresponding ADE gauge group SU(n),SO(2n)
+orE6,7,8in the uncompactified directions, and the situation in F-the ory is no
+different. For example, in M-theory, integrating the three-f orm potential Cabc
+over eachS2of the Dynkin diagram yields a U(1) gauge boson. M2-branes
+wrapped over the remaining S2’s are charged under these U(1)’s, and constitute
+the off-diagonal gauge bosons of the ADE gauge group. The duali ty between
+circle compactifications of IIB and IIA string theory lifts t o F-theory and M-
+theory, allowingustotranslatethisderivationovertothe F-theoryside. Although
+apparently unsuitable for GUT phenomenology, it is also pos sible to engineer the
+non-simply laced groups USp(2n),SO(2n+1),F4, andG2.
+Associated with a singular ADE fiber of an F-theory compactifi cation is a cor-
+responding seven-brane with that ADE gauge group. Just as in perturbative
+type IIB string theory, spacetime filling seven-branes whic h occupy the same
+subspace in the internal directions will allow us to enginee r higher-dimensional
+gauge theories. Though our ultimate interest will be in seve n-branes which wrap
+four compact internal directions, let us first consider the m aximally supersym-
+metric gauge theory of seven-branes on R7,1with gauge group GSengineered by
+an ADE singularity of type (by abuse of notation) GS. To arrive at the eight-
+dimensional seven-brane theory in flat space, let us first con sider ten-dimensionalParticle Physics Implications of F-theory 9
+supersymmetric Yang-Mills theory on R9,1with sixteen real supercharges and
+gauge group GS. The field content of this ten-dimensional theory consists o f
+an adjoint-valued gauge boson AI, and an adjoint-valued gaugino, which trans-
+forms as a sixteen component Majorana-Weyl spinor. Dimensi onally reducing
+this ten-dimensional theory to eight dimensions yields the maximally supersym-
+metric eight-dimensional gauge theory. Focussing on the bo sonic field content,
+reduction to eight dimensions yields an eight-dimensional gauge boson transform-
+ing as a vector under SO(7,1) given by those components of AIwith directions
+along the eight spacetime directions. The two additional ga uge bosons of AIthen
+combine to form a complex scalar Φ.
+The eigenvalues of the complexified scalar Φ determine the po sitions of the
+seven-branes in the threefold base B3. Since the order of the singularity de-
+termines the gauge group on the seven-brane, these vevs must also deform the
+geometry of the ADE type singularity. Consider for example a stack ofnseven-
+branes located at z= 0 with gauge group SU(n). This corresponds to an An−1
+singularity
+y2=x2+zn. (11)
+Wecanbreakthegaugegroupdownto U(1)n−1throughthevevΦ = diag( t1,...,tn)
+wheret1+...+tn= 0. In the geometry this translates to the deformation
+y2=x2+n/productdisplay
+i=1(z−ti). (12)
+Note that for generic values of z, each factor in this product behaves roughly as a
+constant. However, near the locus z=ti, we encounter the location of one of the
+seven-branes. This can be visualized as keeping the center o f mass of the brane
+system fixed and separating the nseven-branes.
+More general Higgsing patterns of gauge groups also transla te to complex de-
+formations, or the unfolding of a singularity. See [61] for furtherdiscussion on the
+deformation theory of the ADE singularities. Such deformat ions ofGSare pa-
+rameterized in the gauge theory by the Casimirs of Φ which in t he above example
+with Φ = diag( t1,...,tn) are given by Tr(Φn), which can in turn be expressed in
+terms of the elementary symmetric polynomials in the ti. Thus, the vevs of the
+Casimirs specify a breaking pattern which in turn dictates h ow the seven-branes
+are distributed in the geometry.
+We can also consider more general vevs for Φ with non-trivial position depen-
+dence on the worldvolume spanned by the original stack of sev en-branes. This
+corresponds to rotating some of the seven-branes of the orig inal stack so that10 Heckman
+they now occupy a different set of four internal directions of t he geometry. Such
+seven-branes will then intersect the original stack along t wo real dimensions. To
+illustrate this, consider a U(2) gauge theory described by two coincident seven-
+branes. WhenΦ = diag( φ,−φ), thisbreaks U(2) downto U(1)2. Lettingsdenote
+a local holomorphic coordinate of S, whenφ∼s, we see that for non-zero s, the
+gauge group is broken, but at s= 0, theU(2) symmetry is restored, correspond-
+ing to the meeting of the two seven-branes. Modes Ψ charged un der both stacks
+of branes obey a wave equation obtained from the dimensional reduction of the
+covariant derivative of ten-dimensional Yang-Mills theor y to eight dimensions.
+Though the full system of equations is somewhat more involve d, at a schematic
+level, the mode Ψ obeys
+∂Ψ
+∂s−φ·Ψ = 0, (13)
+so that Ψ ∼exp(−|s|2) exhibits Gaussian falloff away from the locus of symme-
+try enhancement at s= 0. In other words, matter becomes trapped along the
+intersection of the seven-branes. For further discussion o n matter localization in
+F-theory, see [62,10].
+Shifting back and forth between the gauge theory descriptio n and geometric
+dataallowsustotreatsomeofthemoresubtleaspectsofsymm etryenhancements
+in an F-theory compactification in terms of eight-dimension al gauge theory. In-
+deed, though at generic points the singularity type over a co mplex surface will be
+given byGS, over complex one-dimensional curves, this enhancement ca n jump
+to a higher rank GΣ⊃GS, and over points can jump again to Gp⊃GΣ⊃GS.
+As in our discussion near equation (13), we can treat the syst em as aGpgauge
+theory which has been Higgsed down to the lower singularity t ypes. The position
+dependence of Φ then dictates how these matter curves meet in the geometry.
+2.3 The Four-Dimensional Effective Theory
+In eight-dimensional flat space R7,1=R3,1×C2, the components of the eight-
+dimensional gauge boson split up into two four-component ve ctors, each of which
+transforms as a vector under one factor of spacetime, and as a scalar of the
+other factor. The remaining bosonic degrees of freedom corr espond to the scalar
+Φ which controls the position of the seven-brane. The theory of a spacetime
+filling seven-brane which wraps four small directions is qui te similar because
+in a sufficiently small four-dimensional patch of the interna l geometry, we can
+again treat the theory as an eight-dimensional gauge theory onR3,1×C2. The
+main subtlety is how to take this local description and patch it together moreParticle Physics Implications of F-theory 11
+globally inaway consistent withsupersymmetryin R3,1. Thoughafulldiscussion
+would take us too far afield, the main point is that on compacti fications which
+preserve four real supercharges, we can organize all of the fi eld content into
+four-dimensional N= 1 superfields which transform as differential forms with a
+given number of holomorphic and anti-holomorphic indices o n the two complex-
+dimensional surface Swrapped by the seven-brane. In terms of four-dimensional
+N= 1 supermultiplets labelled by points of the internal geome try, the fields of
+the seven-brane theory organize into an adjoint-valued vec tor multiplet W(0,0),
+a chiral multiplet A(0,1)with one anti-holomorphic index, and a chiral multiplet
+Φ(2,0)with two holomorphic indices [9,10].
+Dimensionally reducing the eight-dimensional gauge theor y to four dimensions,
+thevolumeofthetwo complex-dimensional surfacedetermin esthegaugecoupling
+constant of the four-dimensional gauge theory through the r elation
+α−1
+GUT=4π
+g2
+YM∼M4
+∗Vol(S) (14)
+whereM∗is amass scale close to the stringscale M∗∼1017GeV. Roughly speak-
+ing, the radius of the volume factor Vol( S)−1/4∼MGUTsets the characteristic
+scale of unification.
+Our discussion so far has focussed on the theory of an isolate d seven-brane.
+Achieving realistic phenomenology requires intersecting the GUT seven-brane
+with other seven-branes of the compactification. In perturb ative type IIB setups,
+the pairwise and triple intersections of stacks of D7-brane s respectively yield bi-
+fundamental matter and Yukawa couplings in the four-dimens ional theory. In F-
+theory, theanalogousphenomenonisduetootherseven-bran eswrappingcomplex
+surfacesS′with gauge groups GS′which intersect the GUT stack of seven-branes.
+In this context, the gauge theory localizes on complex two-d imensional surfaces
+Swrapped by seven-branes, chiral matter localizes on comple x one-dimensional
+curves defined by the intersection of such surfaces Σ = S∩S′, and cubic inter-
+action terms such as Yukawa couplings localize at points defi ned by the triple
+intersection of matter curves, p= Σ1∩Σ2∩Σ3(see table 2).
+The intersection of seven-branes along a complex one-dimen sional curve Σ cor-
+responds to a further enhancement in the ADE type of the singu larity, which we
+denote byGΣ⊃GS×GS′. As mentioned in subsection 2.2, we can locally model
+this situation in terms of an eight-dimensional gauge theor y with gauge group
+GΣwhich is Higgsed down to GS×GS′[62,63,9,10]. Some of the off-diagonal
+components of the fields A(0,1)and Φ(2,0)then become trapped along the locus12 Heckman
+Table 2: Geometric Ingredients of F-theory GUTs
+Dimension Ingredient
+10 Gravity
+8 Gauge Theory
+6 Chiral Matter (+ Flux)
+4 Cubic Interaction Terms
+wherethegauge groupis restoredto all of GΣ. Theresultingtheory thencontains
+a six-dimensional hypermultiplet charged under both GSandGS′. The adjoint
+representation of GΣdecomposes into irreducible representations of GS×GS′,
+some of which are charged under both seven-branes
+ad(GΣ) = (R,R′)+(R,R′)+··· (15)
+withRandR′both non-trivial. In four-dimensional N= 1 language, this six-
+dimensional field corresponds to a collection of vector-lik e pairs in the ( R,R′)⊕
+(R,R′) trapped along the curve.
+This recovers the usual result from perturbative type IIB st ring theory. For
+example, theintersection ofD7-braneswithgaugegroups GS=SU(n)andGS′=
+SU(m) contains six-dimensional matter in the ( n,m)⊕(n,m) ofSU(n)×SU(m),
+corresponding to open strings stretched between the two sta cks of D7-branes.
+In anSU(5) GUT, matter in the 10 and 5 ofSU(5) are respectively given by
+enhancements in the singularity type to SO(10) andSU(6). F-theory vacua also
+extendtheavailable matter content beyondperturbatively realized states. For ex-
+ample, though the spinor 16 of SO(10) cannot be generated by open string states
+with two ends, it will be present in geometries where GS=SO(10) enhances to
+E6. This is because the decomposition of the adjoint of E6⊃SO(10)×U(1)
+contains the 16
+78→450⊕10⊕16−3⊕16+3. (16)
+The kinetic terms of the matter field wave functions are also d etermined by
+the geometry. Since these fields localize on matter curves, i n a holomorphic basis
+of chiral superfields, the corresponding kinetic terms are o f the form
+M2
+∗Vol(Σ)·/integraldisplay
+d4θΨ†Ψ. (17)
+Inparticular, matter localized on distinctmatter curves w ill have diagonal kinetic
+terms. A canonical normalization of matter fields then corre sponds to performing
+the rescaling Ψ →(M2
+∗Vol(Σ))−1/2Ψ.Particle Physics Implications of F-theory 13
+Further enhancements in the singularity type at points of th e geometry induce
+interaction terms between matter localized on curves. To il lustrate this, consider
+a local point enhancement to E6which is Higgsed down to SO(10) andSU(6)
+along complex curves and SU(5) in the bulk of the seven-brane. The adjoint
+representation of E6decomposesinto irreduciblerepresentations of SU(5)×U(1)2
+as:
+78→240,0⊕10,+2⊕10,0⊕5+6,0⊕10−3,+1⊕10−3,−1⊕c.c.. (18)
+Returningtoour discussionnear equation (13), fields withd ifferentU(1)2charges
+localize on different curves. An interesting feature is that i n the interaction term
+5×10×10, each field has a distinct U(1) charge so that these three matter fields
+automatically localize on three distinct matter curves. In other words, once two
+of the matter curves meet, a third comes along to form the 5 ×10×10 interaction.
+See figure 1 for a depiction of this geometry. Though basicall y correct, we will
+refine this statement later in section 5 when we discuss seven -brane monodromy.
+More abstractly, in a neighborhood pof the enhancement to a singularity of
+typeGp, we can model the configuration of seven-branes in terms of an eight-
+dimensional theory with gauge group GpHiggsed down to GΣialong various
+matter curves, and GSelsewhere on the seven-brane. In the eight-dimensional
+theory, the internal gauge fields and Φ (2,0)interact through the superpotential
+[9,10]
+Wparent=/integraldisplay
+Tr(∂A(0,1)+A(0,1)∧A(0,1))∧Φ(2,0). (19)
+Expanding A(0,1)=∝angbracketleftA∝angbracketright+δAand Φ(2,0)=∝angbracketleftΦ∝angbracketright+δΦ yields cubic superpotential
+terms coupling the δA’s andδΦ’s of the form
+Wp=λ123Ψ1Ψ2Ψ3 (20)
+for chiral superfields Ψ ilocalized on curves of the internal geometry. Here, the
+Yukawa coupling λ123is given by the overlap of the Ψ iwave functions in the
+internal geometry
+λ123=/integraldisplay
+Sψ1ψ2ψ3. (21)
+In order for the coupling of equation (20) to be generated, th e product of the Ψ’s
+must form a gauge invariant operator under the gauge groups o f the seven-branes
+participating in the Yukawa interaction. On the other hand, it may happen that
+the D-term/integraldisplay
+d4θΨ†
+1Ψ2Ψ3
+ΛUV(22)14 Heckman
+is instead gauge invariant. Integrating out Kaluza-Klein m odes of mass Λ UV
+localized on the Ψ 1curve generate this higher dimension operator [19]. Note th at
+here the suppression scale Λ UVis simply the Kaluza-Klein mass scale associated
+with harmonics on the matter curve, which is in turn set by the characteristic
+radius of the seven-brane theory to be near the GUT scale. Thi s means that if
+present, such terms will dominate over similar Mplsuppressed contributions.
+As an aside let us note that although these ingredients appea r to provide the
+most promising route for model building, it is also possible to consider models
+where some of the matter content propagates in the worldvolu me of the seven-
+brane. In this more general context, there are additional in teractions such as
+those between three bulk matter fields, and interactions bet ween one bulk mode
+and two matter fields on the same curve.
+2.4 Seven-Brane Flux
+The Standard Model has three chiral generations of matter. I n an F-theory com-
+pactification, thisdata is encodedin how gauge fieldfluxinth einternal directions
+of the seven-branes couples to the higher-dimensional field s of the seven-brane
+theory.
+A background flux on a seven-brane takes values in a subgroup Γ ⊂GS. Much
+as in compactifications of the heterotic string, this breaks the gauge group in four
+dimensions down to the commutator subgroup of Γ in GS. Note that a flux in
+an abelian subgroup such as U(1)Yhypercharge can then break SU(5) down to
+SU(3)C×SU(2)L×U(1)Y.
+Matter fields localized at the pairwise intersection of seve n-branes with gauge
+groupsGSandGS′will couple to both sets of background fluxes. These fields
+obey the six-dimensional Dirac equation
+/D6Ψ = (/D4+/DΣ)Ψ = 0 (23)
+where/D4and/DΣrespectively denote the Dirac operator in R3,1and the matter
+curve Σ. Since the coupling Ψ/DΣΨ of the higher-dimensional theory descends to
+a mass term of the four-dimensional theory, it follows that t he massless modes
+correspond to zero modes of the Dirac operator along the matt er curve.
+Index theory now determines the number of zero modes. For abe lian field
+strengthsFSandFS′, the net number of chiral generations is
+# of chiral modes = qS/integraldisplay
+ΣFS
+2π+qS′/integraldisplay
+ΣFS′
+2π(24)Particle Physics Implications of F-theory 15
+where theq’s denote the charges of the fields under the respective gauge groups.
+By tuning the discrete parameters of a compactification, it i s in principle pos-
+sible to realize the spectrum of the Standard Model [14]. Eve n so, as of this
+writing, a fully consistent compactification realizing pre cisely the matter curves
+and interaction terms of the Standard Model has yet to be cons tructed.
+3 GUT Breaking and its Consequences
+Having introduced the primary building blocks of an F-theor y GUT, we now
+study some of their consequences for low-energy physics. To frame the discussion
+to follow, we shall often focus on the case of a minimal SU(5) F-theory GUT
+with a seven-brane wrapping a complex surface S. Matter in the 10 Mand5M
+respectively localizes on enhancements to SO(10) andSU(6), with Higgs fields
+localized on additional SU(6) enhancements. To reach the Standard Model,
+SU(5) must be broken to SU(3)C×SU(2)L×U(1)Y. This has consequences for
+the massless spectrum, proton decay, and gauge coupling uni fication.
+3.1 Local Models and GUT Breaking Fluxes
+Breaking the GUT group down to the Standard Model gauge group turns out
+to be surprisingly restrictive in F-theory. In four-dimens ional GUT models and
+higher-dimensional stringy models, the primary means to br eak the GUT group
+are:
+•Vevs for adjoint-valued chiral superfields
+•Discrete Wilson lines.
+Thelocal model condition obstructs both of these options in an F-theory GUT [9,
+10,14,17]. In the seven-brane theory, the vevs of adjoint-v alued chiral superfields
+determine the location of the two complex-dimensional surf ace wrapped by the
+seven-brane Sinside the threefold base B3. Since gravity is decoupled in a local
+model, this seven-brane cannot explore B3, so there are no such zero modes.
+As a brief aside, let us note that in some GUT models such as flip pedSU(5)
+GUTs, GUT breakingcan berealized through the vevs of fields i n representations
+different from the adjoint. We will briefly touch on this point l ater when we
+discuss F-theory GUT implementations of flipped SU(5) GUTs.
+Wilson line breaking is also problematic because it require s a non-trivial fun-
+damental group π1(S)∝negationslash= 0. Assuming there exists a limit where B3remain of16 Heckman
+finite size while Scontracts to a point, this requires Sto be adel Pezzo surface,
+namely a two complex-dimensional surface of positive curva ture. For further dis-
+cussion on del Pezzo surfaces geared towards physicists, se e for example [64]. All
+such surfaces are simply connected, obstructing Wilson lin e breaking. Additional
+discussion on restrictions expected from the local model co ndition can be found
+in [29,32,52,59]. In principle, this condition can be weake ned to allow a milder
+version of decoupling [43].
+GUT breaking by gauge field fluxes provides an alternative to t hese options.
+Recall that non-trivial flux in a subgroup Γ ⊂GSwill break the gauge group
+down to the commutator subgroup. Indeed, setting Γ = U(1)Y⊂SU(5) yields
+the commutator subgroup SU(3)C×SU(2)L×U(1)Y.
+It may appear surprising that this mechanism has not been exp loited in earlier
+work on compactifications of superstringtheory. As shown in [65] this mechanism
+does not work in compactifications of the perturbative heter otic string due to a
+generalized Chern-Simons coupling to the Neveu-Schwarz tw o-formBin the ten-
+dimensional Lagrangian density
+LHet⊃(dB+A∧∝angbracketleftF∝angbracketright)2(25)
+where∝angbracketleftF∝angbracketrightis the field strength in the internal directions of the compac tifica-
+tion. Dualizing Bto an axion, this generates a string scale mass for Avia the
+St¨ uckelberg mechanism. Re-mixing by other gauge bosons is possible, though
+this distorts gauge coupling unification [65] (see [66,67] f or recent discussion).
+The analogous coupling in the seven-brane theory is between the Ramond-
+Ramond four-form potential C4and the gauge field strengths in R3,1and the
+internal directions /integraldisplay
+R3,1×SC4∧Tr(FR3,1∧FS). (26)
+Assuming two of the legs of C4wedge with FS, this leaves a two-form in R3,1,
+which is the analogue of Bin equation (25).
+The global topology of the complex threefold base B3can prevent the gauge
+field from developing a mass. The essential point is that the c oupling of equation
+(26) involves two-forms such as FSdefined purely on S, and a two-form given
+by two legs of C4defined on the entire threefold base B3. Depending on how
+Sembeds in B3, the integral of the wedge product of C4andFScan vanish.
+When this happens, there is no coupling of the gauge field to an axion, and the
+gauge field remains massless [14,17]. This defines a mathemat ical condition on
+the relative cohomology of SandB3, as well as a choice of flux FSonS, andParticle Physics Implications of F-theory 17
+it is known that it can be met in explicit examples. In practic al terms, what
+is required is that the two-cycle Poincar´ e dual to FSinSmust lift to a trivial
+two-cycle in B3. This is quite similar to the topological condition found in [68]
+which prevents the U(1)Yhypercharge gauge boson from developing a mass in
+models based on D3-brane probes of singularities. An intere sting feature of this
+result is that although string dualities connect certain he terotic and F-theoretic
+vacua, when this topological criterion is satisfied, no hete rotic dual exists [14,17].
+3.1.1 SpectrumConstraints Thepresenceofabackgroundinternalgauge
+fieldfluxinthehyperchargedirectionor hyperflux alsoaffects thezeromodespec-
+trum of the theory. Consider again a minimal SU(5) GUT model. The internal
+gauge bosons of SU(5) in the (3 ,2)−5/6⊕(3,2)5/6ofSU(3)C×SU(2)L×U(1)Y
+are in exotic representations of the Standard Model gauge gr oup, and can only
+be avoided for an essentially unique choice of cohomology cl ass for the hyper-
+flux [14]. An important subtlety for this type of GUT breaking flux is the overall
+factor of 5 in the hypercharge of these off-diagonal elements. To actually avoid
+introducing exotics, it is necessary to allow seemingly fra ctional hyperfluxes [14].
+In order for all matter fields to couple to an integral number o f net flux quanta,
+this also requires fluxes to be activated on the other seven-b ranes of the com-
+pactification [14,17].
+Increasing the rank of the GUT group only adds to the number of constraints
+which a background flux must satisfy. In [14] it was found that breakingSO(10)
+toSU(3)C×SU(2)L×U(1)Y×U(1)byfluxesalwaysgeneratesexotics propogating
+in thebulkof theseven-brane. Theredo, however, exist fluxe swhichbreak SU(6)
+down to the Standard Model without introducing such bulk exo tics [54].
+Itisalsopossibletoconstructhybridswhichincorporatefl uxbreakingtoafour-
+dimensional model, and then a further breaking of the GUT gro up down to the
+StandardModel. For example, inaflipped SU(5)×U(1) GUT, avevformatter in
+the10−1⊕10+1leads totheStandardModel gaugegroup. Partial fluxbreakin gof
+SO(10) down to SU(5)×U(1) has been used as a starting point for constructing
+flippedSU(5) F-theory GUTs [14,39,40]. Note that this provides a natu ral
+embedding of a flipped GUT in a higher unified structure. It is a lso possible to
+consider flux breaking of SO(10) down to SU(3)×U(1)×SU(2)L×SU(2)Ras
+in [22]. Suitable vevs for additional fields descending from vector-like pairs in the
+16⊕16 ofSO(10) can then break the GUT group down further.
+Fluxes also affect the spectrum of zero modes localized on matt er curves. For
+example, in an SU(5) GUT with matter in the 5, the decomposition of the 518 Heckman
+reveals that the triplets and doublets couple differently to t he internal hyperflux
+5→(3,1)+1/3⊕(1,2)−1/2 (27)
+In other words, when the net hyperflux through a matter curve i s non-zero, we
+do not retain full GUT multiplets in the zero mode spectrum. A n economical
+way to evade this problem is to demand that flux through chiral matter curves is
+zero [14]. In less minimal SO(10) models, flux from U(1)YandU(1)B−Lcan also
+be chosen so that different components of a GUT multiplet local ize on distinct
+curves [22].
+Doublet-triplet splitting in the Higgs sector by fluxes occu rs when the Higgs
+field localizes on a curve with non-zero hyperflux [14]. Since HuandHdhave
+oppositeU(1)Yquantum numbers, this also means that in order to get a Huand
+Hdzero mode, these fields should localize on different matter cur ves [14].
+3.2 Proton Decay
+Proton decay is a signature of GUT models (see [69] for a revie w) and has been
+studied in F-theory GUT models in [70,14,17,48]. Dangerous dimension four
+and five operators which can induce rapid proton decay must be sufficiently sup-
+pressed in order to evade current experimental bounds. On th e other hand, the
+coefficients of dimension six operators provide a potentiall y exciting window into
+GUT physics.
+The most phenomenologically problematic couplings are cub ic F-terms such
+as5M×5M×10M. Such couplings are absent in the MSSM by assuming the
+existence of a Zmatter
+2matter parity under which chiral matter is odd and the
+Higgs fields are even. Geometrically, we can exclude such con tributions by de-
+manding the absence of such Yukawa enhancement points. The a bsence of such
+interactions is also compatible with symmetry considerati ons, either through em-
+bedding Zmatter
+2⊂U(1)PQ⊂E8as in [19], or by viewing it as a geometric action
+of the compactification [14,38]. E7singularity structures in F-theory GUTs also
+effectively forbid dimension four proton decay operators [70 ].
+Dimension five operators such as
+η5
+MGUT·/integraldisplay
+d2θQQQL (28)
+withη5a numerical coefficient, induce rapid proton decay primarily through the
+channelp→K+ν. Current bounds on the lifetime of the proton require η5<
+10−10. Inafour-dimensionalGUT,doublet-tripletsplittingcan alsoinadvertentlyParticle Physics Implications of F-theory 19
+generate this operator. For example, when the Higgs up and Hi ggs down triplets
+couple to the MSSM matter fields through interactions such as
+WGUT⊃TuQQ+TdQL+MGUTTuTd, (29)
+heavy triplet exchange will generate the operator QQQL/M GUT. This problem
+is potentially worse in higher-dimensional models since th ere is now a whole
+Kaluza-Klein tower of excitations to worry about.
+Doublet-triplet splitting by hyperflux automatically addr esses this issue by
+geometrically sequestering the Higgs up and down on distinc t matter curves.
+This means that the Higgs triplets pair up with other Kaluza- Klein modes, and
+the offending dimension five operator is not generated [14]. In an intersecting
+seven-brane configuration, this can also be traced to the pre sence of additional
+anomalous symmetries under which the MSSM fields are charged , such as aU(1)
+Peccei-Quinn symmetry. As advocated in [19,33,42], this U(1) naturally fits
+inside the unfolding of an E8singularity.
+Dimension six operators such as
+η6
+MGUT·/integraldisplay
+d4θU†E†QQ (30)
+withη6anumericalcoefficient, induceprotondecayprimarilythrou ghthechannel
+p→e+π0, but at a level which impinges on the regions currently being probed
+by experiment. In a four-dimensional GUT, the coefficient η6∼g2
+GUT, and
+is generated by the exchange of off-diagonal gauge bosons of th e broken GUT
+group. A similar effect is present in F-theory GUTs from all of t he off-diagonal
+elements of the broken gauge group. Moreover, because these modes actually fit
+inside an entire Kaluza-Klein tower of heavy states, the ful l contribution from
+such effects is summed up by a two-point function integrated ov er the internal
+directions wrapped by the seven-brane. Due to the presence o f internal volume
+factors for the heavy modes participating in the correspond ing Green’s function,
+there is a relative parametric enhancement in some decay cha nnels, by a factor
+of logα−1
+GUT[21,17]. Unfortunately, the overall ambiguity in the norma lization of
+theinternalGreen’s functionmeansthatextracting apreci secoefficient isdifficult
+with present techniques [17].
+3.3 Gauge Coupling Unification
+Starting from the weak scale, and evolving to higher energie s, we define the GUT
+scale as the energy at which the SU(2)LandU(1)Ygauge couplings (normalized20 Heckman
+to embed in SU(5)) are equal. Unification occurs provided the SU(3)Cgauge
+coupling also meets at this same scale. At one loop order, the MSSM couplings
+unify, and at two loop order, the value of α3is roughly 4% lower than αGUT,
+though the precise value depends on TeV and GUT scale thresho ld corrections.
+Indeed, particularpatternsofsoftbreakingmassescansom etimeshelpwithgauge
+coupling unification [71].
+As in many GUT models, the GUT breakingsector itself can shif t the tree level
+values of the gauge coupling constants. This shift is compar able in magnitude to
+the effects from subleading two loop corrections to the evolut ion of the MSSM
+couplings. These contributions come with various signs, an d rather importantly,
+there exist regimes of parameter space in F-theory GUTs wher e gauge coupling
+unification can be retained [17,25]. Effects from the closed st ring sector con-
+nected with hyperflux breaking also affect the running of coupl ings [44], though
+a complete discussion of such effects is beyond the scope of the present article.
+First consider the contribution from GUT breaking fluxes. In the seven-brane
+theory, the gauge field strength couples to itself through ki netic terms, as well as
+terms quartic in the field strength. The net contribution is g iven by the kinetic
+termforgaugefieldsoftheeight-dimensional gaugetheory, andtheChern-Simons
+like coupling of the seven-brane to the background RR zero-f orm potential C0
+M4
+∗/integraldisplay
+R3,1×STr(F∧∗F)+/integraldisplay
+R3,1×SC0∧Tr(F4) (31)
+where in the above we have suppressedthe order one coefficient s multiplying each
+term. Prior to GUT breaking, the GUT coupling is controlled b y the volume
+wrapped by the seven-brane so that α−1
+GUT=M4
+∗Vol(S). Activating an internal
+field gauge field strength breaks the GUT group and shifts the r elative values of
+the gauge coupling constants by
+αi(MGUT)−1→αi(MGUT)−1+ki (32)
+wherekiis an order one number which depends on the details of the inst anton
+number for the flux, as well as the breaking pattern. The perce ntage change in
+thevalue of thecouplings is thenon theorder of αGUT∼5%. In [17], these effects
+were studied for U(1)Y⊂SU(5) GUT breaking fluxes, and in [25], a different
+embedding of abelian fluxes in U(5)GUTwas studied in related IIB compactifica-
+tions. In both cases it was found that the sign of the correcti on tends to increase
+the mismatch in unification.
+Incomplete GUT multiplets of heavy Kaluza-Klein modes will also induce
+threshold corrections to unification [17,25]. In [25], it wa s found that addingParticle Physics Implications of F-theory 21
+a single vector-like pair of Higgs triplets 1015−1016GeV allows unification to
+be retained. Summing over the entire tower of Kaluza-Klein s tates and including
+the contribution from the TrF4terms, the specific linear combination of one loop
+determinants for the Laplacians of k−forms which enters the threshold correction
+can be recast as a quasi-topological invariant [17,72] know n as the holomorphic
+Ray-Singer torsion [73]
+T≡1
+2/summationdisplay
+k(−1)k+1logdet′∆k (33)
+where ∆ kdenotes the Laplacian of bundle-valued k−forms, the one-loop deter-
+minants are evaluated in zeta-function regularization, an d det′denotes the de-
+terminant with all zero modes omitted. These threshold effect s can in principle
+have either sign, thus preserving precision unification in a region of parameter
+space [17].
+4 Supersymmetry Breaking
+So far, our discussion has focussed on the dynamics of the mod el near the GUT
+scale. Making contact with observation requires a discussi on of supersymmetry
+breaking, and in particular how all of the superpartners dev elop masses com-
+patible current experimental bounds. In keeping with the ph ilosophy of local
+models, in this section we focus on supersymmetry breaking s cenarios which rely
+on dynamics in the vicinity of the seven-brane controlled by the vev of a GUT
+singlet
+∝angbracketleftX∝angbracketright=x+θ2FX. (34)
+Our aim will be to realize scenarios where the parameters FXandxgenerate
+viable soft supersymmetry breaking masses for the MSSM gaug inos and scalars.
+Besides the soft masses, it is also necessary to generate wea k scale coefficients
+for theBµandµterms
+Leff⊃Bµhuhd+/integraldisplay
+d2θµHuHd+h.c. (35)
+wherehdenotes the bosonic component of the superfield H. The values of these
+parameters, in addition to the soft supersymmetry breaking terms in the Higgs
+sector dictate electro-weak symmetry breaking, and conseq uently the value of the
+ratio of the Higgs up and Higgs down vevs, tan β=vu/vd.
+As a brief aside, in many supersymmetricmodels, especially those derived from
+string theory, there are typically many additional nearly fl at scalar directions22 Heckman
+which must also be stabilized. Here we assume that these modu li have been
+stabilized through high scale dynamics which will not inter fere with our present
+discussion. Seeforexample [74] forrecent discussionon co mbiningcertain moduli
+stabilization scenarios with supersymmetry breaking in F- theory GUT models.
+4.1 Moduli Dominated Scenarios
+Though it is natural to decouple the dynamics of many moduli i n a local model,
+the overall volume modulus of the complex surface wrapped by the GUT brane
+remains dynamical, even after decoupling gravity. This sug gests using this mode
+as a source for supersymmetry breaking effects [75]. Since thi s modulus couples
+to all of the MSSM superpartners with particular scaling dim ensions, a vev for
+it will generate a soft supersymmetry breaking pattern of ma sses for the MSSM.
+In [12] the phenomenology of this scenario was studied in gre ater detail, where
+it was found that achieving appropriate electroweak symmet ry breaking requires
+particular scaling dimensions for the fields of the MSSM. Thi s type of scaling is
+most easily achieved in configurations where all of the matte r localizes on matter
+curves. In particular, in such scenarios the Yukawas origin ate from the triple
+intersection of seven-branes, which fits with the main empha sis of models we
+have presented here.
+In this class of models, a bino-like lightest neutralino con stitutes the LSP,
+providing a natural dark matter candidate for this scenario . Generating an ap-
+propriate relic abundance also requires a stau NLSP, with ma ss close to that of
+the bino, and a large value of tan β∼40.
+4.2 Gauge Mediation Scenarios
+In a limit where gravity is decoupled from gauge theory, and t he volume modulus
+is stabilized dueto high scale dynamics, gauge mediated sup ersymmetrybreaking
+scenarios are quite natural (see [76] for a review of gauge me diation). Such sce-
+narios are quite attractive, because transmission of super symmetry breaking by
+gaugefieldsdoesnotintroducenewsourcesofflavorviolatio n, whichispotentially
+problematic in other approaches. Gauge mediated supersymm etry scenarios in
+F-theory GUTs have been studied in [16,19]. See for example [ 77,78] for other
+work on realizing gauge mediation in string based models.
+In minimal gauge mediation, the SUSY breaking GUT singlet Xcouples toParticle Physics Implications of F-theory 23
+vector-like pairs of messenger fields Y⊕Y′through the superpotential term
+W⊃XYY′. (36)
+Messenger andgaugeloopsthengenerate softsupersymmetry breakingmassesfor
+thegauginos andscalars oftheMSSM.Theprecisemassspectr umdependsonthe
+scale of supersymmetry breaking√FX, the messenger scale x, the representation
+content and number of messengers, and the messenger scale va lues for the µ
+andBµterms. As a rule of thumb, generating TeV scale colored super partners
+requiresFX/x∼105GeV.
+This type of scenario has a clean geometric realization in F- theory GUTs [14].
+WhenYandY′localize on distinct matter curves, they will meet a third cu rve,
+normal to the GUT seven-brane, on which Xlocalizes. In principle, YandY′
+can correspond to matter in the 5 ⊕5 or the 10 ⊕10.
+Gauge mediated interactions generate soft masses, but do no t address how the
+µterm of the MSSM is generated. The first issue is to avoid a GUT s cale value
+for theµparameter in the MSSM superpotential
+WMSSM⊃µHuHd. (37)
+This condition is met in models where the Higgs up and down are sequestered on
+distinct matter curves, because the requisite term will not be invariant under the
+gauge symmetries of the other seven-branes intersecting th e GUT stack [14,19].
+In the low-energy effective theory, this corresponds to a near ly exact anomalous
+U(1)Peccei-Quinnsymmetry. SuchaPQsymmetrydescendsfrom anotherseven-
+brane of the compactification which intersects the GUT stack so that all matter
+fields are charged under U(1)PQ. As a point of terminology, we shall refer to this
+as thePQ seven-brane of the compactification.
+In minimal setups achieving a weak scale µ-term determines the required scale
+of supersymmetry breaking. The higher dimension operator
+/integraldisplay
+d4θX†HuHd
+ΛUV(38)
+inducesµ∼FX/ΛUV. Inthegeometry, this operator isgenerated at points where
+theXcurve and Higgs curves meet [19]. The scale Λ UV∼1015GeV is close to
+the GUT scale [14,19], which requires FX∼1017GeV2to generate a weak scale
+µ∼100 GeV. Combined with the requirement FX/x∼105GeV, this implies
+x∼1012GeV [19].
+The analogous operator compatible with U(1)PQwhich generates the Bµterm24 Heckman
+is /integraldisplay
+d4θX†XX†HuHd
+Λ3
+UV(39)
+producing a value for Bµ∼(x/ΛUV)·µ2, which is far smaller than µ2. This is
+the primary contribution to the Bµterm because of localization of the matter
+fields on Riemann surfaces. This yields the messenger scale b oundary condition
+Bµ∼0. In addition, the messenger scale A-terms are also zero due to brane
+localization, so that additional potentially problematic CP violating phases are
+not generated. Finally, at low energies, this scenario lead s to large values of
+tanβ∼20−35.
+Additional contributions to the soft masses will arise from anomalous U(1)
+gauge bosons which couple the supersymmetry breaking secto r to the visible sec-
+tor [79]. In [19], the effects of integrating out an anomalous U(1)PQgauge boson
+were also studied, where it was shown that there are addition al contributions to
+the soft breaking parameters from higher dimension operato rs of the form
+Leff⊃ −g2
+PQeXeΨ/integraldisplay
+d4θX†XΨ†Ψ
+M2
+U(1)PQ(40)
+whereeXandeΨdenote the PQ charges of Xand Ψ,gPQdenotes the gauge
+coupling of the U(1)PQgauge theory, and MU(1)PQis the mass of the U(1)PQ
+gauge boson. In principle MU(1)PQcan range from masses somewhat below the
+GUT scale to much higher values.
+Combining the effects of minimal gauge mediation with the effect s of heavy PQ
+gauge boson exchange induces a PQ deformation of the soft masses away from
+the usual minimal gauge mediation spectrum
+m2
+soft=m2
+mGMSB−q∆2
+PQ (41)
+where ∆ PQ∼g2
+PQF/MU(1)PQ, andq∼ −eXeΨis determined by the product of
+theXand Ψ PQ charge of the field Ψ in question. The PQ charges of the M SSM
+chiral matter is opposite in sign to that of X. This has the effect of lowering the
+massofthescalars, suchasthestau, butdoesnotshiftthema ssesofthegauginos.
+The actual value qdepends on the neutrino sector of the model. This is because
+compatibility with the other interaction terms of the MSSM u niquely determines
+a choice of PQ charge assignments compatible with the neutri no scenarios of [33].
+See figure 3 for a plot of the characteristic mass spectrum of a Majorana neutrino
+scenario at minimal and maximal PQ deformation.
+Generating a hierarchically suppressed supersymmetry bre aking scale is moreParticle Physics Implications of F-theory 25
+challenging, but can be arranged through a Polonyi term
+Weff⊃exp(−Vol(SPQ))X (42)
+induced by Euclidean D3-instantons wrapped on the PQ seven- brane [15,18,16,
+19]. Here, realizing the precise scales x∼1012GeV,FX∼1017GeV2requires
+a certain amount of tuning in the K¨ ahler potentials of Xand other axion-like
+fields which contribute to the QCD axion [19]. In principle, i t is also possible to
+use the supergravity effective potential to generate an appro priate potential for
+X[80,16], though this requires additional assumptions abou t the cosmological
+constant.
+The supersymmetry breaking sector also feeds into the physi cs of axions. The
+phase ofXcorresponds to a Goldstone mode of the nearly exact global U(1)PQ
+symmetry, and constitutes along with other axion-like field s the QCD axion [19].
+Interestingly, the numerology demanded by gauge mediation thatx∼1012GeV
+is consistent with the available axion window. The axion sup ermultiplet also
+influences the cosmology of F-theory GUTs [26]. The bosonic p artner of the ax-
+ion – the saxion – develops a weak scale mass due to supersymme try breaking.
+The fermionic component corresponds to the axino, which is a lso the spin 1/2
+component of the gravitino. As the Goldstino of local supers ymmetry breaking,
+the gravitino has a mass of FX/Mpl∼10−100 MeV set by the supersymmetry
+breaking scale. The saxion field oscillates in the early Univ erse, eventually de-
+caying prior to the start of big bang nucleosynthesis. When i t decays, it releases
+a significant amount of entropy. The diluted relic abundance of gravitinos makes
+it a viable dark matter candidate [26]. In principle, axioni c dark matter can also
+comprise a component of dark matter. Note that saxion decay a lso dilutes the
+relic abundance of other possible dark matter candidates [4 2].
+5 Flavor in F-theory
+Proceeding down in energy scales, we have studied the physic s at the GUT scale,
+andsupersymmetrybreaking. We nowturntorealizations offl avor forthequarks
+and leptons of the Standard Model, and its extension to massi ve neutrinos.
+The masses and mixing angles of the quark sector exhibit stri king and rather
+mysterious flavor hierarchies. For example, the mass of the u,c,tquarks are
+respectively 0 .002,1,170 GeV, and the magnitudes of the CKM quark mixing26 Heckman
+matrix elements are [81]
+|VCKM| ∼
+|Vud| |Vus| |Vub|
+|Vcd| |Vcs| |Vcb|
+|Vtd| |Vts| |Vtb|
+∼
+1 0.2 0.004
+0.2 1 0 .04
+0.008 0.04 1
+.(43)
+By contrast, neutrino oscillation experiments indicate a f ar lower characteristic
+mass scale of order 0 .05 eV, and compared to the CKM matrix, the leptonic
+mixing matrix VPMNSis less hierarchical [81]. Moreover, the magnitude of the
+(1,3) element of the PMNS matrix |V1,3
+PMNS|= sinθ13is still consistent with a
+value of zero, and is bounded above by 0 .2.
+GUT structures naturally explain some aspects of flavor. For example, the
+mass of the bquark and τlepton unify at the GUT scale [83], which fits with
+embedding their Yukawas in the interaction term 5H×5M×10M. Geometrically,
+order one coefficients for both the 5H×5M×10Mand 5H×10M×10Mseem the
+easiest to arrange, and this is thecase wefocus on here. Sinc ethe top and bottom
+quark have different masses, this is most compatible with larg e tanβscenarios.
+The limited representation content for zero modes in an F-th eory GUT elimi-
+nates some common mechanisms in four-dimensional models [8 2] for correlating
+GUT breaking with flavor structure. On the other hand, these s ame geometric
+ingredients suggest new possibilities, both for quarks and leptons. To frame our
+discussion, we again focus on minimal SU(5) F-theory GUTs. We show that min-
+imal geometries required to realize the MSSM superpotentia l also produce rank
+one Yukawas. Subleading corrections to these Yukawas respo nsible for flavor hi-
+erarchies are then induced through violations of local symm etries, either from
+background ambient three-form fluxes in the case of quarks, o r from non-chiral
+massive modes of the compactification, in the case of neutrin os.
+5.1 Geometry of Flavor
+The overlap of matter field wave functions in the internal dir ections of the seven-
+brane determine the up-type Yukawa matrix
+λu
+ij=/integraldisplay
+SψHuψQ
+iψU
+j (44)
+with similar expressions for the down-type and charged lept on Yukawas. The
+maximal overlap between matter localized on different curves occurs where these
+curves geometrically intersect. To leading order, the up-t ype Yukawa matrix isParticle Physics Implications of F-theory 27
+given by a sum over all such intersection points
+λu
+ij=/summationdisplay
+pψHu(p)ψQ
+i(p)ψU
+j(p). (45)
+Notethateachsummandisanouterproductoftwoflavorvector s, andsodefinesa
+rank one 3 ×3matrix. In other words, a single interaction point already generates
+the expected structure of one generation of weak scale mass, and two generations
+which to first approximation are massless [14,23]. In actual compactifications
+there will be additional enhancement points of the geometry [51,52]. Realizing
+a single heavy generation then suggests that either these po ints are tuned to
+be close together [14,51], or that the MSSM fields only partic ipate in one such
+point [14,23].
+The number of heavy generations depends on how many curves pa rticipate
+in a Yukawa interaction. To see this, let us consider a model w here the three
+generations of matter aredistributedonuptothreematter c urves. The5 ×10×10
+interaction requires a point of enhancement from SU(5) to at least E6. Returning
+to our discussion near equation (18), this interaction term couples two 10’s which
+would seem to localize on different matter curves since they ha ve distinct charges
+under the abelian factor of SU(5)×U(1)2⊂E6. This is phenomenologically
+problematic because the Yukawa matrix will then have zeroes along the diagonal.
+Assuming roughly similar normalizations for the kinetic te rms, this corresponds
+to a model with only one hierarchically light generation [14 ].
+Inactualcompactifications ofF-theorythecurvesonwhicht hetwo10’slocalize
+are often just different branches of a single complex one-dime nsional curve [28].
+This is an example of a more general phenomenon known as seven-brane mon-
+odromy, and constitutes an important element in achieving realist ic flavor struc-
+ture in an F-theory GUT.
+5.1.1 Seven-Brane Monodromy Seven-brane monodromy is best moti-
+vated by example. To this end consider the breaking pattern d efinedby unfolding
+anAn−1singularity y2=x2+znas
+y2=x2+zn−2(z2+2βz+γ) =x2+zn−2(z−t+)(z−t−),(46)
+wheret±=−β±/radicalbig
+β2−γ. This corresponds to an SU(n−2) stack of seven-
+branes atz= 0, and another seven-brane at z2+2βz+γ= 0. The branch cuts
+int±identifies the two seemingly different seven-branes at z=t+andz=t−.
+Returningtoourdiscussionnearequation(18), intermsoft heeight-dimensional
+SU(n) gauge theory locally defined at z=t±= 0, the vevs of the Casimirs of28 Heckman
+Φ can be packaged in terms of the coefficients in the unfolding o f the singular-
+ity [10]. When Φ is not diagonalizable, the corresponding ma tter curves exhibit
+branch cuts, reflected in the t±.
+We now define seven-brane monodromy in more general terms. Co nsider again
+a configuration where the singularity type is GSat generic points of the complex
+surface wrapped by the seven-brane, which enhances to Gpat a point of the
+geometry. The unfolding of this singularity is specified by c oordinates tiof the
+Cartan subalgebra of Gp, which are to be thought of as the eigenvalues of Φ. The
+monodromy group of the seven-brane configuration Gmonopermutes the ti’s. In
+a breaking pattern of the form Gp⊃GS×Γ,Gmonois a subgroup of the Weyl
+group of Γ.
+Returning to the example of the 5 ×10×10 interaction, note that under the
+breaking pattern E6⊃SU(6)×SU(2)⊃SU(5)×U(1)×SU(2), the adjoint of
+E6decomposes as
+78→(24,1)⊕(1,3)⊕(5−6,1)⊕(5+6,1)⊕(10−3,2)⊕(10−3,2) (47)
+sothatthe10’stransformasadoubletof SU(2). Indeed, theWeyl groupof SU(2)
+isZ2, and monodromy acts by interchanging the components of the d oublet. The
+geometry then consists of a single E6enhancement point where one 10 and one 5
+curve meet to form the 5 ×10×10 interaction. Thus, monodromy in the seven-
+brane configuration allows us to achieve two hierarchically light generations [28].
+5.2 Quark Yukawas
+We now turn to the Yukawas of the lighter quark generations. O ne common way
+to generate flavor hierarchies is through the Froggatt-Niel sen mechanism [84]. In
+this scenario, the matter fields of the Standard Model are cha rged under an ad-
+ditional horizontal symmetry, where different generations h ave distinct Froggatt-
+Nielsen charges. Spontaneous breaking of this horizontal s ymmetry generates
+additional corrections λij∼εai+bj, producing hierarchical masses and mixing
+angles. This begs the question, however, as to how the parame tersaiandbjare
+to be chosen, as well as how the parameter εis fixed.
+It is possible to engineer F-theory GUTs utilizing this four -dimensional mecha-
+nism[57]. Infact, even withoutintroducingexplicit GUTsi nglets, theingredients
+of an F-theory compactification provide similar, though mor e stringy variants on
+this theme. Matter localized on complex one-dimensional cu rves have similar
+charges induced by the action of the internal Lorentz group [ 23]. More precisely,Particle Physics Implications of F-theory 29
+the matter wave functions satisfy ∂Aψ= 0, and so are locally holomorphic. Or-
+ganizing the wave function solutions according to their ord er of vanishing near
+a Yukawa point, we have ψi∼z3−i+O(z4−i) fori= 1,2,3,in a three genera-
+tion model. The action of the internal Lorentz group corresp onds to a rephasing
+symmetry of z, providing a horizontal symmetry of the chiral generations . Back-
+ground gauge field and three-form fluxes of an F-theory compac tification distort
+the profile of matter field wave functions [23]
+ψ→exp(Mijzizj)ψ (48)
+where the zidenote local holomorphic coordinates in a patch of the Yukaw a
+point. This distortion of wave functions then suggests a pos sible mechanism for
+generating corrections to Yukawas. Note that by dimensiona l analysis, Mscales
+asM2
+GUT.
+Even so, the topological structure of a single non-zero entr y turns out to be
+quite robust against physical perturbations. Gauge field flu xes, for example,
+distort the profile of the physical matter field wave function s, but fail to distort
+the Yukawa structure [47,49]. Yukawa distorting fluxes corr espond to three-
+form fluxes such as H=HR+τHNS, which induce higher dimension operator
+deformations of the seven-brane gauge theory. The effects of t hree-formH-flux
+are locally captured in a patch of the Yukawa enhancement poi nt in terms of
+a non-commutative deformation of the seven-brane theory [4 7]. In principle,
+instanton effects can also generate similar non-commutative deformations [53],
+the effects of which can also be captured in terms of an effective c ontribution
+fromH-fluxes [85].
+Having different hypercharges, the different components of a GU T multiplet
+will couple to the background fluxes differently, leading to di fferent wave function
+and Yukawa distortions for the various generations. This is quite important,
+because while the bottom quark and τlepton mass unify near the GUT scale,
+the masses of other particles mc∝negationslash=mµandmd∝negationslash=medo not. In other words,
+the geometry of the compactification preserves a leading ord er notion of matter
+unification, which is violated by flux distortion of the wave f unctions [14,23,47].
+Expandinginsuccessivepowersof zinM, eitherbyexpandinginhigherpowers
+ofMin the exponential of equation (48), or by including higher o rder derivative30 Heckman
+corrections yields the structures [23]
+λDER∼
+ε5ε4ε3
+ε4ε3ε2
+ε3ε21
+,λFLX∼
+ε8ε6ε4
+ε6ε4ε2
+ε4ε21
+(49)
+where each entry is multiplied by an order one complex coeffici ent which we
+suppress. As the name suggests, the λDERexpansion is given by expanding to
+first order in the flux and then expanding in successive gradie nts or derivatives
+of the flux. The λFLXexpansion is given by expanding to leading order in the
+gradientsoftheflux,andthenexpandinginsuccessivepower softhefirstgradient.
+Because the matter field wave functions couple differently to t he background
+fluxes, theprecisevalueof εandthedominanceoftheFLXorDER expansioncan
+depend on the matter field in question [23]. Identifying up-t ype quark Yukawas
+withλFLXand down-type quarks with λDER, this yields hierarchical up-type
+quark masses
+mu:mc:mt∼ε8
+u:ε4
+u: 1 (50)
+and down-type quark masses
+md:ms:mb∼ε5
+d:ε3
+d: 1. (51)
+Though this is somewhat heuristic, it reproduces the observ ed masses and mixing
+angles surprisingly well [23]. Numerically, achieving a ma tch to the observed
+quark masses works best when εu∼0.26 andεd∼0.27. This is quite close
+to the value expected based on the relative scaling of Mto the string scale
+ε∼M2
+GUT/M2
+∗∼√αGUT∼0.2 [23,47].
+Achieving a hierarchical CKM matrix is more subtle. The CKM m atrix mea-
+sures the mismatch between the gauge and mass quark eigensta tes. Since the
+profile of matter field wave functions varies over the geometr y of the seven-brane,
+achieving a roughly diagonal CKM matrix requires near align ment of the up
+and down type Yukawa points. Assuming that this further cond ition for point
+unification has been met, the resulting CKM matrix is then [23 ,47]
+VF−th
+CKM∼
+1ε ε3
+ε1ε2
+ε3ε21
+∼
+1 0.2 0.008
+0.2 1 0 .04
+0.008 0.04 1
+(52)Particle Physics Implications of F-theory 31
+at the GUT scale. In principle, this should be evolved to lowe r energy scales, but
+thisturnsouttomultiplyvariousentriesbyorderonecoeffic ients, whichisalready
+beyond the approximation scheme adopted here. Returning to equation (43), it
+is surprising how well these crude numerical estimates matc h with observation.
+From the perspective of Yukawa enhancements, putting E6andSO(12) to-
+gether can be accomplished through a higher unification to ei therE7orE8.
+Including neutrinos pushes this further to E8[42]. We will return to this theme
+later when we discuss minimal unification at a point of E8in section 6.
+5.3 Neutrino Models
+Thenumerology oftheseesaw mechanism mν∼M2
+weak/ΛUV, forΛUVclosetothe
+GUT scale strongly suggests a connection between GUTs and ne utrinos. Itis well
+known that Majorana and Dirac neutrino masses can respectiv ely be generated
+by the higher dimension operators
+/integraldisplay
+d2θ(HuL)2
+ΛUV,/integraldisplay
+d4θH†
+dLNR
+ΛUV. (53)
+The Higgs vevs Hu∼MweakandHd∼θ2FHd∼θ2M2
+weakinduce the mass terms
+/integraldisplay
+d2θM2
+weak
+ΛUVNLNL,/integraldisplay
+d2θM2
+weak
+ΛUVNLNR. (54)
+Right-handed singlets which interact with the MSSM fields ca n originate from
+matter localized on curves normal to the GUT stack [14,34,33 ], but can also in
+principle originate from moduli fields [38]. Right-handed n eutrinos in the spinor
+16 of anSO(10)⊃flippedSU(5) F-theory GUT are also possible [14,39,40].
+The flavor structure of SU(5) models with neutrinos localized on singlet curves
+has been developed more, and so we focus on this case. In such s cenarios, the
+interaction with the MSSM fields localizes at a point of enhan cement from SU(5)
+to at leastSU(7). Decomposing the adjoint of SU(7) toSU(5)×U(1)2yields
+48→240,0⊕10,0⊕1−2,0⊕12,0⊕5+1,−7⊕5−1,−7⊕5−1,+7⊕5+1,+7(55)
+sothat theMSSMfieldscorrespondto apairingof the5and 5withaGUT singlet
+1±2,0, correspondingto the right-handed neutrino. Dependingon theU(1) charge
+assignments of the matter fieldsand theneutrinoscenario in question, integrating
+outmassivemodesofthecompactification willthengenerate thehigherdimension
+operators of equation (53). For Majorana scenarios, seven- brane monodromy is
+especially important because it breaks U(1) symmetries which would otherwise
+forbid Majorana mass terms.32 Heckman
+The participation of massive non-chiral modes dilutes the fl avor hierarchies in
+comparison to what is present in the quark and charged lepton Yukawas yielding
+neutrino masses
+mν1:mν2:mν3∼ε2:ε: 1, (56)
+which is known as a normal mass hierarchy scenario. Up to orde r one factors,
+the predicted ratio of atmospheric and solar neutrino mass s plittings is then [31]
+∆m2
+sol
+∆m2
+atm∼ε2∼αGUT∼0.04 (57)
+which is numerically quite close to the experimental value o f 0.03 [81]. Again,
+let us stress that this numerology works surprisingly well c onsidering all of the
+crude approximations performed.
+Large neutrino mixing angles will generically be present in models where the
+neutrino interaction point is geometrically separated fro m the lepton interaction
+point. This is in accord with the observed large mixing angle sθatmandθsol, but
+leads to a certain amount of tension with the current upper bo und on the mixing
+angleθ13<0.2.
+As inthequarksector, realizing ahierarchical mixingmatr ix naturally suggests
+unifying the charged lepton and neutrino interaction point s. Even when the
+neutrino and charged lepton interaction points are close to gether, the Kaluza-
+Klein dilution of flavor hierarchies still generates large m ixing angles with PMNS
+mixing matrix [33]
+VPMNS∼
+Ve1ε1/2ε
+ε1/2Vµ2ε1/2
+ε ε1/2Vτ3
+(58)
+where theV’s are constrained by unitarity of VPMNS. Using the same parametric
+scalingε∼√αGUTas in the quark sector yields ε1/2∼0.45, leading to order one
+solar and atmospheric neutrino mixing. In addition, the mod el also predicts
+θ13∼α1/2
+GUT∼0.2. (59)
+In other words, the expectation is that θ13is close to the current experimen-
+tal bound. This appears to also be a common feature of neutrin o models with
+additional interaction points [34].
+Majorana and Dirac neutrino scenarios can in principle be di stinguished by
+neutrinoless double beta decay experiments. The correspon ding decay amplitudeParticle Physics Implications of F-theory 33
+is only generated in Majorana scenarios, and is proportiona l to a mass term
+mββ, which in F-theory GUTs is on the order of 6 meV [33,34]. This v alue is
+quite small, and may only be within reach of experiments on th e horizon of a
+decade [86].
+More options are available in less minimal scenarios. Anarc hic neutrino mixing
+and mass matrices can be realized either by separating the ch arged lepton and
+neutrino interaction points [33], or by including multiple neutrino enhancement
+points [34]. Tuning the locations of the neutrino interacti on points, neutrino
+models with an inverted mass hierarchy can also be realized [ 34].
+6 Minimal E8Unification
+In this article, we have demanded that the geometry of an F-th eory GUT satisfy
+a long list of phenomenological constraints. These include a list of matter curves
+where GUT multiplets localize, interaction terms for the MS SM superpotential,
+neutrino sector, and supersymmetry breaking sector (for ga uge mediation mod-
+els). From this perspective, each requirement would seem to add an additional
+degree of arbitrariness to the models we have considered. In addition, it is not
+altogether clear whether all of these ingredients can in fac t consistently combine
+in a single, unified framework.
+The requirements of flavor physics that the quark and leptons exhibit hier-
+archies in the mixing matrices suggests unifying the E6,SO(12) andSU(7)
+interaction points into a single point of E8enhancement [33,42]. This E8en-
+hancement breaks down to SU(6) andSO(10) along the GUT matter curves,
+and is also compatible with the supersymmetry breaking sect or introduced in
+section 4. The matter curves and interaction terms are then d etermined by the
+breaking pattern for the adjoint of E8⊃SU(5)GUT×SU(5)⊥
+248→(24,1)⊕(1,24)⊕(5,10)⊕(5,10)⊕(5,10)⊕(5,10).(60)
+Given the large set of independent directions in the Cartan o fSU(5)⊥, such an
+unfolding might appear to generate a large number of extrane ous matter curves.
+Seven-brane monodromy, which already figures prominently i n the physics of
+flavor propagates to other sectors of the model, constrainin g the ways that addi-
+tional matter can be added. In fact, it is possible to classif y the possible breaking
+patterns of E8with seven-brane monodromy, subject to the physical condit ions
+•Hierarchical CKM and PMNS matrix34 Heckman
+•Kaluza-Klein seesaw for neutrinos
+•U(1)PQto forbid a bare µterm
+•µterm from either/integraltext
+d4θX†HuHd
+ΛUVor/integraltext
+d2θSHuHd
+whereXandSare to be thought of as GUT singlets which develop suitable
+vevs to induce a µterm. In addition to identifying many matter curves, the
+monodromy group also identifies the U(1)’s inSU(5)⊥, so that only U(1)PQ
+survives for Majorana neutrino scenarios, and in Dirac neut rino scenarios, only
+U(1)PQandU(1)B−Lsurvive. See figure3 for a depiction of a minimal E8model.
+There are typically just enough matter curves to accommodat e the Standard
+Model, and a very constrained messenger sector. In all Major ana neutrino sce-
+narios, and all but one Dirac neutrino scenario, the messeng ers transform in
+vector-like pairs in the 10 ⊕10. Moreover, the matter curve for the 10 Malso
+supports the messenger [42]. In loop exchange diagrams invo lving such messen-
+ger fields, increasing the number of messengers or equivalen tly the dimension of
+the representation tends to lower the masses of the scalar pa rtners relative to the
+gauginos. In particular, once the effects of the PQ deformatio n are included, the
+mass of the lightest stau tends to be lowered relative to the m ass of the bino [42].
+6.1 Experimental Signatures
+The qualitative experimental signatures of this minimal ga uge mediation model
+are dictated by the scale of supersymmetry breaking, and the identity of the
+NLSP. The NLSP decays to its Standard Model counterpart and t he gravitino
+with decay rate F2
+X/m5
+NLSP, leading to a lifetime on the order of one second
+to an hour. This means that on timescales probed by colliders , the NLSP is
+quasi-stable, which is somewhat different from other gauge me diation scenarios.
+In bino NLSP scenarios, the characteristic missing energy p lus two prompt
+photon signature of low scale gauge mediation models is not r eproduced here.
+Rather, the bino will leave the detector as missing energy, m uch as in gravity
+mediated supersymmetrybreaking. Nevertheless, the differe nt particle spectrum,
+especially for colored states allows such scenarios to be di stinguished from F-
+theory GUTs [31].
+A characteristic feature of quasi-stable stau NLSP scenari os is that once pro-
+duced in a collider, it will appear as a heavy charged particl e which registers in
+a detector as a “fake muon”. Compared to the signatures of oth er supersymmet-
+ric models with a bino LSP or quasi-stable NLSP, this is a clea n signature withParticle Physics Implications of F-theory 35
+low Standard Model background. The expected collider pheno menology of such
+F-theory GUT scenarios at the CERN Large Hadron Collider has been studied
+in [87].
+7 Conclusions
+F-theory GUTs providea geometric framework for connecting stringscale physics
+to phenomena of the Standard Model. In this article we have in troduced the
+primary ingredients which enter into such constructions, e mphasizing the tight
+interplay between these ingredients in phenomenological m odels. While flexible
+enough to accommodate many aspects of the Standard Model, th e geometry is
+rigid enough to favor particular phenomenological scenari os.
+Minimal scenarios which make use of only ingredients found w ithin a single
+E8point seem particularly rigid and predictive. What is parti cularly interesting
+is that within this specific class of models, there are variou s avenues by which
+these models can be falsified and thus indirectly tested by th e LHC, and other
+upcoming experiments for neutrinos, dark matter and possib ly proton decay.
+The construction of explicit geometries realizing all of th ese ingredients re-
+mains an active area of investigation, and such considerati ons will likely provide
+an important guide for future model building efforts. Consist ently coupling such
+models to gravity remains an important avenue of investigat ion, potentially pro-
+viding new constraints which cannot be seen in purely local m odels. Moreover,
+extracting the collider and flavor physics signatures of bro ader classes of such
+vacua may help to illuminate whether F-theory GUTs are reali zed in Nature.
+While any one aspect of a model could be viewed as suggestive, the fact that
+constraints from the geometry propagate to many aspects of a n F-theory GUT
+makes the prospects of non-trivial correlations and the pro spect of contact with
+distinct experiments especially exciting.
+Acknowledgements.
+We thank C. Vafa for a very stimulating collaboration which l ed to much of the
+work reviewed here. In addition, we also thank our collabora tors on related F-
+theory projects. We also also thank J. Marsano, S. Raby, C. Va fa, B. Wecht and
+E. Witten for helpful comments on the draft. The research of J JH is supported
+by NSF grant PHY-0503584.36 Heckman
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+E6
+10M10M
+5HSO(10)SO(10)SU(5)
+SU(6)
+Figure 1: Depiction of an E6singularity enhancement. The ellipsoidal shape
+denotestheinternaldirectionsoftheseven-brane, withga ugegroupSU(5). Along
+complex one dimensional curves, the singularity enhances t oSO(10) andSU(6),
+where matter in the 10 and 5 of SU(5) respectively localize. There is a further
+enhancement to E6at a point of the geometry, where the Yukawa interaction
+5H×10M×10Mlocalizes.Particle Physics Implications of F-theory 41
+W/PlusMiΝush0A0H0,H/PlusMiΝusu/OverTilde
+L,c/OverTilde
+L,d/OverTilde
+L,s/OverTilde
+L
+b/OverTilde
+1
+t/OverTilde
+1
+e/OverTilde
+L,Μ/OverTilde
+L
+Ν/OverTilde
+e,Ν/OverTilde
+Μ
+Τ/OverTilde
+1g/OverTilde
+Ν/OverTilde
+Τ
+Χ/OverTilde0
+1Χ/OverTilde0
+2Χ/OverTilde/PlusMiΝus
+1Χ/OverTilde0
+3Χ/OverTilde0
+4Χ/OverTilde/PlusMiΝus
+2u/OverTilde
+R,c/OverTilde
+R,d/OverTilde
+R,s/OverTilde
+Rb/OverTilde
+2,t/OverTilde
+2
+e/OverTilde
+R,Μ/OverTilde
+RΤ/OverTilde
+2
+020040060080010001200GeV
+Figure 2: Plot of the sparticle mass spectrum (indicated by /tildewide’s) in an F-theory
+GUT with one vector-like pair of messengers in the 10 ⊕10 andFX/x= 5×104
+GeV for minimal (left, red columns) and maximal (right, blue columns) PQ
+deformation. At ∆min
+PQ= 0, a bino-like lightest neutralino /tildewideχ0
+1is the NLSP. For
+moderate values of ∆ PQ∼100 GeV, the NLSP transitions to the lightest stau
+/tildewideτ1, which persists until a tachyon develops near ∆max
+PQ∼200 GeV.42 Heckman
+SU(5) GUT
+E8NR
+10M
+5H5H5MY10
+10Y'XD
+Figure3: Depictionofaminimalmodelwith E8pointunification. Intheproposed
+model, all oftheinteraction termsdescendfromasingle E8pointofenhancement.
+Here, matter curves inside the seven-brane are depicted by s traight lines and
+curves normal to the seven-brane are depicted by cigar-shap ed tubes. In these
+minimal models, there is typically just enough room to accom modate the MSSM
+spectrum, and a minimal messenger sector ( Y’s) in the 10 ⊕10.
\ No newline at end of file
diff --git a/1001.0578.txt b/1001.0578.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6b9e474c5d820cb9d44fe1e05d0484c4f32d1e76
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@@ -0,0 +1,939 @@
+arXiv:1001.0578v2  [cond-mat.mes-hall]  10 Aug 2011Influence of a Random Telegraph Process on the Transport thro ugh a Point Contact
+Fabian Hassler,1Gordey B. Lesovik,2and Gianni Blatter1
+1Theoretische Physik, ETH Zurich, CH-8093 Zurich, Switzerlan d
+2L.D. Landau Institute for Theoretical Physics RAS, 117940 M oscow, Russia
+(Dated: November 18, 2018)
+We describe the transport properties of a point contact unde r the influence of a classical two-level
+fluctuator. We employ a transfer matrix formalism allowing u s to calculate arbitrary correlation
+functions of the stochastic process by mapping them on matri x products. The result is used to
+obtain the generating function of the full counting statist ics of a classical point contact subject to
+a classical fluctuator, including extensions to a pair of two -level fluctuators as well as to a quantum
+point contact. We show that the noise in the quantum point con tact is a sum of the (quantum)
+partitioning noise and the (classical) noise due to the two- level fluctuator. As a side result, we
+obtain the full counting statistics of a quantum point conta ct with time-dependent transmission
+probabilities.
+PACS numbers: 73.23.-b, 73.63.Nm, 73.50.Bk, 05.60.Gg
+I. INTRODUCTION
+The impact of mobile impurities on transport through
+a quantum conductor attracted a lot of attention soon
+after it was realized that the conductance of a dirty co-
+herent sample is sensitive to the position of a single im-
+purity; this discovery formed the basis for the explana-
+tion of flicker noise as it appears due to the presence
+of bistable mobile impurities.1,2A detailed characteriza-
+tion of charge transport through a quantum conductor
+is provided by the full counting statistics (FCS).3Dur-
+ing the past decade, this description has been applied to
+numerous systems4and a first attempt to describe the
+influence of a two-level fluctuator on the FCS of quan-
+tum transport has been given in Ref. 5. In the present
+paper, we calculate the full counting statistics of charge
+transport through a (classical or quantum) point contact
+coupled to a classical two-level fluctuator; we go beyond
+previous studies by considering the combined effects of
+the presence of one or many mobile impurities, as well
+as quantum-partitioning and the Fermi statistics on the
+FCS. Also, we reconsider carefully the case when parti-
+tioning is neglected and correct previous findings which
+are flawed when calculating the fourth or higher cumu-
+lants.
+The influence of a fluctuating (uncontrollable) envi-
+ronment on a (controllable) device is a generic problem6
+and our work is related to other studies, e.g., the trans-
+port statistics through a quantum dot in the Coulomb-
+blockade regime7,8or the effect of a bistability on the
+transport through a quantum point contact9,10. Another
+example is the dephasing of qubits due to a classical11or
+quantum12two-level system, or the studies on 1 /f-noise
+originating from telegraph noise due to classical13,14or
+quantum11,15–17two-levelfluctuators. Thetwoproblems,
+full counting statistics of charge transport and dephasing
+of a quantum system (qubits) are related through the
+equivalence of fidelity18and full counting statistics3, as
+has been pointed out recently.19
+In our analysis below, we describe the time evolutionof the two-level fluctuator by rate equations which can
+be solved explicitly. We then study the full counting
+statistics of a wire with a conductance depending on the
+state of the fluctuator. The fluctuator induces noise in
+the transport currentthroughthe wire which we evaluate
+using a mapping of correlation functions on matrix prod-
+ucts. Usingthismapping, weareabletocalculatethe full
+counting statistics of a classical wire coupled to a two-
+level fluctuator. Furthermore, we discuss the situation of
+a second (independent) fluctuator and show that a non-
+linear interaction with the wire can lead to correlations
+in the noise even though the fluctuators evolve indepen-
+dent of each other. Finally, we apply our method to the
+case of a quantum wire which exhibits intrinsic partition-
+ing (shot) noise.20,21We derive a formula which incorpo-
+rates both classical- (due to the two-level fluctuator) and
+quantum- (due to the point contact) noise. Thereby, we
+give an explicit expression for the full counting statistics
+of a quantum point contact whose transmission probabil-
+ities change with time. As two-level fluctuators seem to
+be a major obstacle for achieving solid state implemen-
+tations of qubits with long coherence times, being able
+to characterize the influence of a two-level fluctuator on
+transport through a point contact offers the possibility
+to learn about the fluctuating environment by measuring
+the full counting statistics through a nearby quantum
+point contact. The (partial) overlap of our results with
+previous work7,9,11will be discussed below.
+II. SINGLE TWO-LEVEL FLUCTUATOR
+Consideraclassicalparticlewhichcanbetrappedinan
+external potential at two positions denoted by x1andx2,
+cf. Fig. 1. The potential is characterized by the energies
+E1,2associated with the two valleys, which are shifted
+by the amount ∆ = E2−E1with respect to each other,
+and the height of the barrier U. We assume the dynam-
+ics to be given by thermally activated hopping over the
+barrier.22The particle performs a random (Brownian)2
+∆Uγ12
+x1 x2 x
+IGx V
+FIG. 1: Sketch of the setup: A two-level system fluctuating
+incoherently between states x1,2with rates γ12andγ21is
+coupled to a wire with a constriction. The conductance G
+of the wire changes according to the state of the fluctuator,
+inducing noise in the current flowing through the device.
+motion where the probabilities P1,2to be in either valley
+obey the rate equations
+˙P1(t) =−γ21P1(t)+γ12P2(t),
+˙P2(t) =γ21P1(t)−γ12P2(t), (1)
+where the rate to escape from x1tox2is given by γ21=
+γ1→2∝e−U/ϑand the reverse process from x2tox1is
+governed by the rate γ12=γ2→1∝e(∆−U)/ϑ, withϑ
+the temperature. In equilibrium, the probabilities Peq
+1,2
+are such as to obey the balance equation dN21=dN12
+which equates the number of particles dN21=γ21Peq
+1dt
+going from x1tox2during the time dt, with the number
+of particles dN12=γ12Peq
+2dtgoing the opposite way.
+The equilibrium probabilities therefore satisfy the Gibbs
+weight
+Peq
+2/Peq
+1=e−∆/ϑ(2)
+and together with the probability conservation Peq
+1+
+Peq
+2= 1, we obtain
+Peq
+1=γ12/Γ, Peq
+2=γ21/Γ,(3)
+where we have introduced the total rate Γ = γ12+γ21.
+Introducing the vector P(t) with components P1,2(t),
+the rate equation (1) can be written as ˙P(t) =−hP(t),
+with the Fokker-Planck Hamiltonian23given by
+h=/parenleftbigg
+γ21−γ12
+−γ21γ12/parenrightbigg
+. (4)
+Note that the Hamiltonian his not Hermitian (left and
+right eigenvalues are not simply adjoint to each other).
+Nevertheless, its eigenvalues are real (and even positive).
+To make this point clear, we write the Hamiltonian hin
+a new basis h′=s−1hsusing the transformation matrix
+s= diag(/radicalbig
+Peq
+1,/radicalbig
+Peq
+2), such that
+h′=/parenleftbigg
+γ21−Γ/radicalbig
+Peq
+1Peq
+2
+−Γ/radicalbig
+Peq
+1Peq
+2γ12/parenrightbigg
+.(5)It is now visible that the matrix h′(and thereforealsothe
+matrixh) has real (the matrix is symmetric) and positive
+(thedeterminantandthetraceofthematrixarepositive)
+eigenvalues. The evolution conserves the probability as
+∂t[P1(t) +P2(t)] = 0, as is evident from Eq. (1). This
+relation implies that (1 ,1)·˙P(t) = 0 and it follows that
+∝an}bracketle{t0|= (1,1) is a left eigenvector of the Hamiltonian hto
+the eigenvalue 0. To every left eigenvector there exists a
+correspondingright eigenvectorwith the same eigenvalue
+which we will denote by |0∝an}bracketri}ht. The right eigenvector to
+the eigenvalue 0 is given by the equilibrium distribution
+|0∝an}bracketri}ht=Peq,h|0∝an}bracketri}ht= 0. The second eigenvalue is given by Γ
+with the corresponding right [left] eigenvectors assuming
+the form |Γ∝an}bracketri}ht= (1,−1)T[∝an}bracketle{tΓ|= (Peq
+2,−Peq
+1)]; note that
+the eigenvectorsare normalized such that ∝an}bracketle{ta|b∝an}bracketri}ht=δaband/summationtext
+a|a∝an}bracketri}ht∝an}bracketle{ta|=112witha,b∈ {0,Γ}. Using the eigenbasis
+ofh, it is possible to compute the evolution operator
+P(t >0) = exp( −ht) =/summationtext
+ae−at|a∝an}bracketri}ht∝an}bracketle{ta|. The matrix ele-
+mentPmn(t) denotes the conditional probability for the
+particle to be transferred from state ntomin the time
+t. The evolution only depends on the time difference as
+the Hamiltonian his time-independent. An explicit cal-
+culation yields the expression
+P(t) =/parenleftbigg
+Peq
+1+Peq
+2e−ΓtPeq
+1(1−e−Γt)
+Peq
+2(1−e−Γt)Peq
+2+Peq
+1e−Γt/parenrightbigg
+(6)
+for the propagator during the time t. As the stochastic
+process is Markovian, the propagator (6) incorporates
+all the information needed in order to calculate general
+correlation functions of the stochastic process,24see also
+below.
+III. CLASSICAL WIRE
+We consider a classical wire coupled to the two-level
+fluctuator. We assume the two-level system to be a
+charge impurity which interacts with the wire, e.g., via
+Coulomb forces. The net effect of the charge impurity
+is to change the conductance of the wire G1,2depending
+on the position x(t) =x1,2of the two-level system. The
+wire is biased by a constant voltage Vsuch that the cur-
+rent is determined by I1,2=VG1,2. The current in the
+wire jumps between I1andI2in a random way given by
+the dynamics ofthe two-levelfluctuatorwhich we assume
+to be in thermal equilibrium, i.e., in the state |0∝an}bracketri}ht. The
+fluctuations of the two-level system induce current noise.
+In our discussion, any kind of back-action of the wire on
+the two-level fluctuator is neglected.
+A. Correlation Functions – Mapping on Matrices
+With our focus on the full counting statistics, we
+are interested in obtaining the moments of the charge
+Q=/integraltextt
+0dt1I(t1) transmitted through the point contact
+during the time t; here,I(t1) denotes I1,2depending on3
+the state x(t1) =x1,2of the two-level fluctuator at time
+t1. AsI(t1) =VG(t1), we first concentrate on correla-
+tion functions of G(t), where statistical averagesover the
+stochastic process (1) will be denoted by ∝an}bracketle{t·∝an}bracketri}ht. In quan-
+tum mechanics, it is well-known that correlation func-
+tions can be evaluated either in the operator or in the
+path-integral formalism.25Likewise, we have the choice
+to apply a stochastic path-integral approach7,26or to use
+the operator formalism. Here, we stay with the opera-
+tor formalism introduced in the previous section. Note
+that the propagator Pmn(t2−t1), cf. Eq. (6), denotes the
+conditional probability (the transfer matrix) to find the
+system in state xmat timet2, given that it resided in
+xnat timet1,Pmn(t2−t1) =∝an}bracketle{tx(t2)=xm|x(t1)=xn∝an}bracketri}ht; i.e.,
+within a path-integral formulation, P(t) involves already
+an integration over all possible paths between t1andt2.
+The average conductance ∝an}bracketle{tG(t1)∝an}bracketri}htis readily calculated.
+For a system residing in a stationary state given by
+∝an}bracketle{tx(t1)=xn∝an}bracketri}ht=Peq
+n, we obtain
+G(t1) =/summationdisplay
+n=1,2Gn∝an}bracketle{tx(t1)=xn∝an}bracketri}ht=G1Peq
+1+G2Peq
+2.(7)
+The calculation of the conductance correlator
+∝an}bracketle{tG(t2)G(t1)∝an}bracketri}htis more involved. We proceed slowly
+in order to motivate our general mapping between the
+calculation of correlation functions and the evaluation
+of matrix products of specific matrices. In order to
+calculate ∝an}bracketle{tG(t2)G(t1)∝an}bracketri}ht, we assume first that t2> t1;
+classical correlators are symmetric so that the opposite
+ordering of times reduces to the same quantity. Using
+the fact that the stochastic process is Markovian,
+we expand the correlation function ∝an}bracketle{tG(t2)G(t1)∝an}bracketri}ht=/summationtext
+m,nGm∝an}bracketle{tx(t2)=xm|x(t1)=xn∝an}bracketri}htGn∝an}bracketle{tx(t1)=xn∝an}bracketri}ht;24this
+expansion can be seen as a transfer-matrix expansion of
+the correlation function. We obtain the mapping for the
+correlator ( t2> t1)
+∝an}bracketle{tG(t2)G(t1)∝an}bracketri}ht=/summationdisplay
+mnGmPmn(t2−t1)GnPeq
+n
+=/summationdisplay
+klmnGkδkmPml(t2−t1)GlδlnPeq
+n
+=∝an}bracketle{t0|Ge−h(t2−t1)G|0∝an}bracketri}ht (8)
+with the diagonal matrix Gmn=Gnδmn. Introducing
+the “interaction representation”
+GI(t) =ehtGe−ht(9)
+of the matrix G, the correlation function Eq. (8) can be
+further simplified to
+∝an}bracketle{tG(t2)G(t1)∝an}bracketri}ht=T ∝an}bracketle{t0|GI(t2)GI(t1)|0∝an}bracketri}ht,(10)
+where the time-ordering operator Thas been included in
+order to relieve the restriction t2> t1. It is easy to see
+that the above derivation is not restricted to the second
+ordercorrelationfunction, butcanbeappliedin thesameway to higher order correlationfunctions. We thus arrive
+at the mapping
+∝an}bracketle{tG(tN)···G(t1)∝an}bracketri}ht=T ∝an}bracketle{t0|GI(tN)···GI(t1)|0∝an}bracketri}ht,(11)
+where the left hand side is a correlation function for the
+classical stochastic process involving the two-level fluctu-
+ator and the right hand side is a matrix element involv-
+ing the matrices GI(tn) and the vectors |0∝an}bracketri}ht=Peqand
+∝an}bracketle{t0|= (1,1).
+B. Full Counting Statistics
+We are now in the position to calculate the generating
+function
+χ(λ) =∝an}bracketle{teiλ/integraltextt
+0dt′I(t′)∝an}bracketri}ht (12)
+for the zero-frequency current-correlation functions (mo-
+ments) of a classical point contact coupled to a two-
+level fluctuator. The moments are obtained as the Tay-
+lor coefficients ∝an}bracketle{tQn∝an}bracketri}ht= (−i∂λ)nχ|λ=0. Alternatively, the
+stochastic process can be characterizedby irreducible cu-
+mulants which are given by the expansion coefficient of
+the logarithm of the characteristic function
+∝an}bracketle{t∝an}bracketle{tQn∝an}bracketri}ht∝an}bracketri}ht=/parenleftigd
+idλ/parenrightign
+logχ(λ)/vextendsingle/vextendsingle/vextendsingle
+λ=0. (13)
+Thecharacteristicfunction χ(λ)canberecastinthe form
+χ(λ) =T ∝an}bracketle{t0|eiλV/integraltextt
+0dt′GI(t′)|0∝an}bracketri}ht (14)
+using the mapping Eq. (11). This equation can be
+rewritten using the well-known mapping between the
+“Schr¨ odinger” and the “interaction” representation,23
+e−(h+v)t=e−htTexp[−/integraltextt
+0dt′vI(t′)] with vI(t) =
+ehtve−ht. Here, we apply the relation in the opposite
+direction to arrive at an expression without the awkward
+time-ordering,
+χ(λ) =∝an}bracketle{t0|e(−h+iλVG)t|0∝an}bracketri}ht. (15)
+This formula was derived before by Bagrets and Nazarov
+using a stochastic path integral formulation of the prob-
+lem (the Fokker-Planck Hamiltonian hand the counting
+fieldλare denoted by ˆLandλin their paper7). We
+believe that the present approach using transfer matri-
+ces and the mapping of the interaction picture onto the
+Schr¨ odinger picture is more transparent. Note though,
+that the counting field λenters differently in their work
+comparedtoours. Here, λcouplestotheclassicalcurrent
+Iwhereas Bagrets and Nazarov discuss the transport of
+individual (quantum) particles such that λmay only en-
+ter in the combination exp( iλ) due to the quantization
+of charge.
+For further convenience, we subtract the average
+charge∝an}bracketle{tQ∝an}bracketri}ht=V∝an}bracketle{tG∝an}bracketri}httin order to obtain the reduced full
+counting statistics
+ˆχ(λ) =χ(λ)e−iλ/angbracketleftQ/angbracketright=∝an}bracketle{t0|e−ˆht|0∝an}bracketri}ht,(16)4
+with the matrix ˆh=h−iλV(G−∝an}bracketle{tG∝an}bracketri}ht). Apart from the
+average charge (which is zero for ˆ χ), both log χand log ˆχ
+generate the same cumulants. The explicit calculation of
+the characteristic function of the full counting statistics
+ˆχinvolves the eigenvalues
+ˆh±=Γ
+2/bracketleftig
+1+iλ∆g∆Peq±/radicalbig
+1+2iλ∆g∆Peq−λ2(∆g)2/bracketrightig
+of the matrix ˆh, where we have introduced the difference
+in the equilibrium population ∆ Peq=Peq
+2−Peq
+1and
+the difference in the (dimensionless) conductance ∆ g=
+V(G2−G1)/Γ.
+C. Asymptotic long-time limit
+For long times Γ t≫1, the matrix exponential (16) is
+dominated by the eigenvalue ˆh−ofˆhwith the smallest
+real part.7,27,28We obtain an explicit expression for the
+generating function
+log ˆχ≫(λ) =−ˆh−t. (17)
+All cumulants become linear in t, due to the fact that
+the autocorrelation time in the system is given by Γ−1
+and every state decays to the equilibrium state after this
+time. The fluctuations forΓ t≫1 can be seen asasum of
+independent stochastic processes and the cumulant gen-
+erating function log ˆ χbecomes extensive in t. Interest-
+ingly, it is possible to obtain an explicit relation for the
+cumulants ( n≥2)
+∝an}bracketle{t∝an}bracketle{tQn∝an}bracketri}ht∝an}bracketri}ht≫=n!Vn(G1−G2)nΓ1−nt (18)
+×n−1/summationdisplay
+k=1Nn−1,k(−1)k+1(Peq
+1)k(Peq
+2)n−k,
+with the Narayana numbers Nn,m=/parenleftbign
+m/parenrightbig/parenleftbign
+m−1/parenrightbig
+/n.29The
+cumulants in (18) grow factorially in magnitude with n
+and oscillate as a function of ∆ Peqwhich are generic
+features of high-order cumulants.30The first couple of
+cumulants assume the form
+∝an}bracketle{t∝an}bracketle{tQ2∝an}bracketri}ht∝an}bracketri}ht≫= 2Peq
+1Peq
+2V2(G1−G2)2t
+Γ, (19)
+∝an}bracketle{t∝an}bracketle{tQ3∝an}bracketri}ht∝an}bracketri}ht≫= 6Peq
+1Peq
+2(Peq
+2−Peq
+1)V3(G1−G2)3t
+Γ2.
+Note that all the odd cumulants vanish if the process
+is symmetric γ12=γ21, cf. Eq. (3). The cumulants in
+(18) and (19) agree with the result in Ref. 5. However,
+startingwith the 4-thcumulantadiscrepancyarises; e.g.,
+for the 4-th cumulant we obtain
+∝an}bracketle{t∝an}bracketle{tQ4∝an}bracketri}ht∝an}bracketri}ht≫=
+24Peq
+1Peq
+2[(Peq
+2−Peq
+1)2−Peq
+1Peq
+2]V4(G1−G2)4t
+Γ3,which is different from the result5
+∝an}bracketle{t∝an}bracketle{tQ4∝an}bracketri}ht∝an}bracketri}ht≫= 24Peq
+1Peq
+2(Peq
+2−Peq
+1)2V4(G1−G2)4t
+Γ3.
+The latter result is incorrect as it misses terms due to
+the implicit assumption in Ref. 5 that the reduced con-
+ductance correlators ∝an}bracketle{t∝an}bracketle{tG(tn)···G(t1)∝an}bracketri}ht∝an}bracketri}htmay only depend
+on the largest time difference tn−t1. Even though this
+hypothesis is correct for correlators up to n= 3, it fails
+for higher-order correlators.
+D. Short times
+For short times, t≪Γ−1there is no evolution of the
+two-levelsystemand wecan set ht= 0. The full counting
+statistics reads
+ˆχ≪(λ) =∝an}bracketle{t0|eiλV(G−/angbracketleftG/angbracketright)t|0∝an}bracketri}ht (20)
+=Peq
+2eiλV(G2−G1)Peq
+1t+Peq
+1eiλV(G1−G2)Peq
+2t,
+with the cumulants ( n≥2) given by
+∝an}bracketle{t∝an}bracketle{tQn∝an}bracketri}ht∝an}bracketri}ht≪=Vn(G1−G2)ntn(21)
+×n−1/summationdisplay
+k=1En−1,k−1(−1)k+1(Peq
+1)k(Peq
+2)n−k,
+where the Eulerian numbers En,mare defined through
+En,m=/summationtextm
+k=0(−1)k/parenleftbign+1
+k/parenrightbig
+(m+1−k)n.29Inthe shorttime
+limit, the cumulants grow like ∝an}bracketle{t∝an}bracketle{tQn∝an}bracketri}ht∝an}bracketri}ht ∝tn, i.e., higher
+order cumulants are suppressed at short times. The first
+couple of cumulants are explicitly given by
+∝an}bracketle{t∝an}bracketle{tQ2∝an}bracketri}ht∝an}bracketri}ht≫=Peq
+1Peq
+2V2(G1−G2)2t2, (22)
+∝an}bracketle{t∝an}bracketle{tQ3∝an}bracketri}ht∝an}bracketri}ht≫=Peq
+1Peq
+2(Peq
+2−Peq
+1)V3(G1−G2)3t3.
+E. Arbitrary times
+Expanding the matrix exponential in Eq. (16) in its
+eigenbasis, the generator for the full counting statistics
+reads
+ˆχ=ˆh+e−ˆh−t−ˆh−e−ˆh+t
+ˆh+−ˆh−(23)
+for arbitrary times. This result has been first derived in
+Ref. 11 in the context of dephasing of a qubit due to the
+interaction with a classical two-level fluctuator, where λ
+denotes the interaction of the qubit with the fluctuator.
+Here, we are interested in the transport properties of a
+pointcontactcharacterizedbycumulantswhich aregiven
+by the Taylor expansion of log χaroundλ= 0; the rela-
+tion between these two problems is a consequence of the
+generic equivalence between full counting statistics and5
+fidelity, see Ref. 19. The first couple of cumulants are
+given by
+∝an}bracketle{t∝an}bracketle{tQ2∝an}bracketri}ht∝an}bracketri}ht= 2Peq
+1Peq
+2V2(G1−G2)2[(Γt−1)+e−Γt]
+Γ2,
+∝an}bracketle{t∝an}bracketle{tQ3∝an}bracketri}ht∝an}bracketri}ht= 6Peq
+1Peq
+2(Peq
+2−Peq
+1) (24)
+×V3(G1−G2)3[(Γt−2)+(Γt+2)e−Γt]
+Γ3.
+F. Symmetric fluctuator
+A special situation is given when the two-level fluctu-
+ator is symmetric, ∆ = 0, i.e., Peq
+1=Peq
+2= 1/2. Then
+the characteristic function assumes the simple form
+logχ≫(λ) =Γt
+2/bracketleftig/radicalbig
+1−λ2V2(G1−G2)2/Γ2−1/bracketrightig
+(25)
+for long times. In the short time limit, the generating
+function
+χ≪(λ) = cos[λV(G1−G2)t/2] (26)
+becomes periodic. In both cases, due to the symmetry
+of the states x1andx2, only the even cumulants are
+nonvanishing.
+IV. A PAIR OF TWO-LEVEL FLUCTUATORS
+Needless to say, the mapping of Sec. IIIA is not re-
+stricted to a single two-level fluctuator. It can be gen-
+eralized to an arbitrary number of states whose dynam-
+ics is governed by classical rate equations described by
+a Fokker-Planck Hamiltonian h. To illustrate this con-
+cept, the example of two classical, uncorrelated two-level
+fluctuators coupled to a wire is discussed in the follow-
+ing. We restrict ourselves to the case where the dy-
+namics of the two-level systems is completely indepen-
+dent of each other such that h=hα+hβ; here and
+in the following, we denote quantities involving only
+the first (second) fluctuator with a superscript α(β),
+e.g.,hα=hα⊗11β. Written explicitly in the basis
+{|1∝an}bracketri}htα⊗ |1∝an}bracketri}htβ,|2∝an}bracketri}htα⊗ |1∝an}bracketri}htβ,|1∝an}bracketri}htα⊗ |2∝an}bracketri}htβ,|2∝an}bracketri}htα⊗ |2∝an}bracketri}htβ}, the
+Fokker-Planck Hamiltonian reads
+h=
+γα
+21+γβ
+21−γα
+12−γβ
+12 0
+−γα
+21γα
+12+γβ
+210−γβ
+12
+−γβ
+21 0γα
+21+γβ
+12−γα
+12
+0−γβ
+21−γα
+21γα
+12+γβ
+12
+.(27)
+One is tempted to think that the independent dynam-
+ics of the two subsystems would lead to independent
+statistics, such that the characteristic function of the
+full counting statistics is given by the product of the
+individual characteristic functions. Indeed, this is the
+generic case for two-level fluctuators coupling to a qubit,
+where the combined effect leads to 1 /fnoise.11,13,15–17However, here, this argument is only valid when the in-
+teraction with the wire is “linear” such that the effects
+of the individual subsystems simply add up, in formula
+G=Gα+Gβ. Having the model of the Fig. 1 in mind,
+this assumptionisincorrectasa(quantum) point contact
+does not react linearly on changes in the gate potential
+and, therefore, the noise of individual fluctuators does
+not simply add up. In the following, we first treat the
+(simple) case of linear interaction and then comment on
+the correlation which arises in the general case by apply-
+ing perturbation theory in the nonlinearity.
+Introducing the reference conductance G0=Gα
+0+Gβ
+0
+as well as the induced changes ∆ Gx=Gx
+1−Gx
+0due to
+thefluctuator x=α,β, theconductancematrixforlinear
+interaction is given by
+G=G0114+diag(0,∆Gα,∆Gβ,∆Gα+∆Gβ); (28)
+the increase of conductance in the case when both fluctu-
+atorsare in state x2is the sum of the respective increases
+when only one of the fluctuators is in state x2, that is to
+say, the effects of the two fluctuators simply add up. In
+this case, the characteristic function assumes the form
+χ(λ) =χα(λ)χβ(λ) and the cumulants ∝an}bracketle{t∝an}bracketle{tQn∝an}bracketri}ht∝an}bracketri}htbecome a
+sum of cumulants generated by the two individual sub-
+systems.
+In the case of a general (diagonal) matrix G, the so-
+lution of the problem involves the determination of the
+roots of a polynomial of fourth degree and the charac-
+teristic function does not separate, even though the time
+evolution of the two fluctuators is completely indepen-
+dent of each other. To be more explicit, we want to show
+how a small perturbation ∆ G≪∆Gα+ ∆GβinG44
+destroying the linearity (additivity) leads to correlations
+which can be arbitrary large for long times. We define
+χcorr(λ) =χ(λ)/χα(λ)χβ(λ) as the part of the charac-
+teristic function which describes the correlation between
+the action of the individual subsystems. In the long-
+time limit, to first order in ∆ G, we can apply standard
+perturbation theory to find the correction to the lowest
+eigenvalueofEq.(27). The cumulantgeneratingfunction
+for the correlation is given by
+logχcorr
+≫(λ) =
+iλV∆Gt/productdisplay
+x=α,β(Γx+iλV∆Gx)Peq,x
+2−ˆhx
+−
+ˆhx
++−ˆhx
+−.(29)
+The average transmitted charge changes according to
+∆∝an}bracketle{tQ∝an}bracketri}ht=Peq,α
+2Peq,β
+2V∆Gt, (30)
+withPeq,α
+2Peq,β
+2the probability to be in the state |2∝an}bracketri}htα⊗
+|2∝an}bracketri}htβandV∆Gthe change in the current. Less trivial,
+the correlation contribution to the noise
+∆∝an}bracketle{t∝an}bracketle{tQ2∝an}bracketri}ht∝an}bracketri}ht= 4V∆∝an}bracketle{tQ∝an}bracketri}ht/bracketleftigg
+Peq,α
+1
+Γα∆Gα+Peq,β
+1
+Γβ∆Gβ/bracketrightigg
+(31)
+depends both on ∆ Gαand ∆Gβ.6
+V. QUANTUM WIRE
+Considering a quantum rather than a classical wire,
+additional noise appears due to the probabilistic nature
+of the charge transport (due to partitioning) even in the
+absence of a fluctuating environment. Given a quantum
+wire with Nchannels characterizedby their transmission
+eigenvalues Tγ,γ= 1,...,N, andbiasedbyavoltagepo-
+tentialV, the characteristic function of the full counting
+statistics is given by3
+logχq(λ) =qVt
+2π/planckover2pi1/summationdisplay
+γlog[1+(eiqλ−1)Tγ],(32)
+withqthe chargeofthe electron; this resultis validin the
+asymptotic limit qVt//planckover2pi1≫1, for low-temperatures ϑ≪
+qV, and with the proviso that the energy-dependence of
+the transmission eigenvalues is negligible in the energy
+interval set by the voltage. The quantum nature of the
+fermions leads to the noise
+∝an}bracketle{t∝an}bracketle{tQ2∝an}bracketri}ht∝an}bracketri}htq=q∝an}bracketle{tQ∝an}bracketri}htq/summationtext
+γTγ(1−Tγ)/summationtext
+γTγ(33)
+whichdisappearsprovidedthatallthechannelsareeither
+closedTγ= 0 or completely open Tγ= 1. Note that the
+noise in Eq. (33) is sub-Poissonian, i.e., the Fano factor
+F=∝an}bracketle{t∝an}bracketle{tQ2∝an}bracketri}ht∝an}bracketri}htq/q∝an}bracketle{tQ∝an}bracketri}htqis less than 1.
+Here, we are interested in the case where the quan-
+tum wire is capacitively coupled to a two-level fluctuator
+such that the transmission eigenvalues Tγchange over
+time; note that we neglect a possible energy dependence
+of the transmission eigenvalues, which correspondsto the
+fact that we assume that the scattering center does not
+produce any time delay due to the scattering event. The
+characteristic function log χq(λ) = detQis given by the
+determinant of the matrix31
+Qkγ,k′γ′=∝an}bracketle{tφkγ(t)|eiλqQt|φk′γ′(t)∝an}bracketri}ht(34)
+with the counting operator Qt=/integraltext
+Idx|x∝an}bracketri}ht∝an}bracketle{tx|integrated
+over the interval I= [0,vFt] and
+φkγ(x,t) =/braceleftigg
+eik(x−vFt)+rγ(t+x/vF)e−ik(x+vFt)x <0
+τγ(t−x/vF)eik(x−vFt)x >0
+(35)
+the single-particle solution of the time-dependent
+Schr¨ odinger equation involving a scattering center at
+x= 0 with time-dependent transmission [reflection] am-
+plitudeτγ(t) [rγ(t)]; note that we have suppressed the
+transverse part of the wave function belonging to the
+channel index γ. Equation (35) is valid in the linear-
+spectrum approximation, where vFis the Fermi velocity
+and 0≤k≤kF.32As the matrix Qis block-diagonal in γ
+and constitutes a Toeplitz matrix with respect to the in-
+dexk, its determinant can be shown (using the technique
+discussed in Ref. 31) to have the form
+logχq(λ) =qV
+2π/planckover2pi1/integraldisplayt
+0dt′/summationdisplay
+γlog[1+(eiqλ−1)Tγ(t′)] (36)forlongtimes qVt//planckover2pi1≫1; theexpressionEq.(36)reduces
+to Eq. (32) when the transmission probability does not
+change in time. We now add the classical two-level fluc-
+tuator to the system: Depending on the state x=x1,2of
+the nearby two-level fluctuator, the quantum wire is de-
+scribed by one of the two sets of transmission eigenvalues
+Tγ
+1,2. The total generating function χqtl(λ) is an average
+over the individual contributions
+χqtl(λ) =∝an}bracketle{tχq(λ)∝an}bracketri}ht
+=/angbracketleftbig
+e(qV/2π/planckover2pi1)/integraltextt
+0dt′/summationtext
+γlog[1+(eiqλ−1)Tγ(t′)]/angbracketrightbig
+,(37)
+where the average ∝an}bracketle{t·∝an}bracketri}htis over the stochastic process of
+the two-level fluctuator. Equation (37) can be calculated
+explicitly with the method outlined in Sec. IIIA. Indeed,
+Eq. (12) goes over to Eq. (37) via replacing iλI(t) with
+(qV/2π/planckover2pi1)/summationtext
+γlog[1+(eiqλ−1)Tγ(t)]. Inthismapping,the
+conductances Gnare changed to ( q/2πi/planckover2pi1λ)/summationtext
+γlog[1 +
+(eiqλ−1)Tγ
+n] withTγ
+nthe transparency of the channel γ
+when the two-level fluctuator is in the state n. Inserting
+iλ∆g→µ=qV
+2π/planckover2pi1Γ/summationdisplay
+γlog/bracketleftigg
+1+(eiqλ−1)Tγ
+2
+1+(eiqλ−1)Tγ
+1/bracketrightigg
+(38)
+into Eq. (17) and using Eq. (16), we obtain
+logχqtl
+≫(λ) =/summationdisplay
+n,γPeq
+nlogχq
+n(λ) (39)
+−Γt
+2/bracketleftbigg
+1+µ∆Peq−/radicalbig
+1+2µ∆Peq+µ2/bracketrightbigg
+;
+here, log χq
+n(λ) = (qVt/2π/planckover2pi1)/summationtext
+γlog[1 +(eiqλ−1)Tγ
+n] is
+the characteristic function for the FCS, Eq. (32), depen-
+dent on the system’s state ( n= 1,2) andγruns over the
+channel index. The expression (39) coincides with the re-
+sult obtained by Jordan and Sukhorukov who considered
+the case of rare transitions ( /planckover2pi1Γ≪qV), see Ref. 9. In
+their work, Jordan and Sukhorukov describe the trans-
+port of a conserved charge in a classical bistable system
+where large charge fluctuations drive transitions between
+stablestateswith different transportcharacteristics. The
+generic fluctuations present in the system’s stable states
+then combine with the fluctuations of the bistable charge
+to generate the transport statistics of the bistable sys-
+tem. While describing a very general situation, the spe-
+cific analysis in Ref. 9 is limited to those cases where the
+charge-switching events are rare on the time scale of the
+underlying fluctuations in the stable states. The transla-
+tion to our model may not be obvious from the start and
+is done by choosing the process of charge partitioning
+for generating the underlying fluctuations (with different
+transmission rates Tγ
+n,n= 1,2 for the two stable states),
+i.e., the generators log χq
+n(λ) correspond to the long time
+generators Hntin Ref. 9. The non-linearity leading to
+switching between stable states generates the transition
+probabilitiesΓ 1,2; the latter aredetermined by an instan-
+ton trajectory and have to be small in the case of Ref. 9.7
+F
+T2= 0.1
+T2= 0.25
+T2= 0.5
+qV//planckover2pi1ΓT2= 0
+0.1
+10 0.1 110
+1
+FIG. 2: Plot of the Fano factor Fas a function of the bias
+voltageVfor a single mode wire with Peq
+1=Peq
+2= 1/2,
+T1= 1, and T2= 0,0.1,0.3, and 0.5 . The Fano factor starts
+of at a value T2(1−T2)/(1+T2)<1 for small voltages. In the
+opposite regime, it is given by [(1 −T2)2/(1 +T2)]qV/2π/planckover2pi1Γ
+which becomes superpoissonian for Vlarge enough.
+In our model these rates are given as the basic input pa-
+rameters γijdefining our two-level fluctuator. The con-
+dition of rare charge switching events corresponds to the
+requirementthat /planckover2pi1Γ≪qV. In this limit Eq.(39) reduces
+to the result of the classical point contact, Eq. (17), with
+the conductances given by
+G1,2=q2
+2π/planckover2pi1/summationdisplay
+γTγ
+1,2. (40)
+We observe that the statistics is dominated by the fluc-
+tuations of the impurity and the results derived in the
+previous sections remain valid in the case of a quantum
+wire when the classical conductance is replaced by the
+Landauer formula (40).
+However, our result Eq. (39) is also valid in the op-
+posite regime /planckover2pi1Γ≫qV, i.e., when the two-level fluctu-
+ator noise acts on a timescale which is fast compared to
+the partitioning noise. Then only the first term in (39)
+contributes and the cumulant generating function is the
+average of the expressions (32) for the quantum point
+contact, to be taken over the positions x1,2with weights
+given by the probabilities Peq
+1,2. Having access to both
+regimes, it is possible to study the crossover from clas-
+sical noise (due to the two-level fluctuator) to quantum-
+partitioning noise (due to the quantum point contact).
+To this end, we calculate the long-time asymptotics of
+the first two moments of (37),
+∝an}bracketle{tQ∝an}bracketri}htqtl
+≫=V(Peq
+1G1+Peq
+2G2)t (41)
+for the average charge and
+∝an}bracketle{t∝an}bracketle{tQ2∝an}bracketri}ht∝an}bracketri}htqtl
+≫=qVtq2
+2π/planckover2pi1/summationdisplay
+n,γPeq
+nTγ
+n(1−Tγ
+n)
++2Peq
+1Peq
+2V2(G1−G2)2t
+Γ(42)for the noise. The noise is simply given by the sum of
+the quantum partitioning noise (first term) and the noise
+due to the dynamics of the impurity (second term). Note
+the crossoverofthe noise(42) from sub-Poissonian F≤1
+for a fast fluctuator with /planckover2pi1Γ≫qV[the first term in (42)
+dominates] to super-Poissonian F≥1 when the fluctua-
+tor is slow /planckover2pi1Γ≪qV[the second term in (42) dominates],
+provided that G1∝ne}ationslash=G2, see Fig. 2.
+VI. EXPERIMENTAL TEST
+It is difficult to experimentally confirm the crossover
+from sub- to super-Poissonian noise as described in the
+previoussectionasthequantum-partitioningnoiseistyp-
+ically small and thus the noise of the classical two-level
+fluctuator will dominate. As a promising setup, we envi-
+sion coupling a quantum point contact in GaAs/AlGaAs
+toadoubledot, e.g., inanInAsnanowire,whichservesas
+tunable two-level fluctuator. Such a system was studied
+recently, see Ref. 33.
+To observe the crossover, both the noise Stl(0) from
+the classical two-level fluctuator and the quantum-
+partitioning noise Sqp(0) (at zero frequency) have to
+dominate over the thermal Nyquist-Johnson noise which
+is given by
+SNJ(0)≈2q2kBT
+π/planckover2pi1≃10−28A2s (43)
+at 50mK. The quantum-partitioning noise [first term in
+Eq. (42)] can be estimated as
+Sqp(0)≈q3V
+8π/planckover2pi1≃V[mV]10−27A2s (44)
+when the system is tuned in the middle of a conductance
+step with T≈1/2. Note that for bias voltages V≥
+0.1mV the quantum-partitioning noise is larger than the
+thermal noise floor.
+Tunneling of an electron between the two quantum
+dots with a rate Γ leads to a change in the conduc-
+tance. InRef. 33, thischangewasoftheorderof0 .1q2//planckover2pi1.
+However, in order to being able to observe the crossover
+the capacitive coupling of the double dot to the quan-
+tum point contact should be reduced to a level such that
+G1−G2≈0.001q2//planckover2pi1. This providesus with the estimate
+(usingPeq
+1=Peq
+2= 1/2)
+Stl(0) =V2(G1−G2)2
+Γ≃V2[mV2]
+Γ[MHz]10−26A2s (45)
+for the zero-frequency noise due to the two-level fluc-
+tuator [second term in Eq. (42)]. At a bias voltage
+V≃1mV with a rate Γ ≃100MHz, the noise due to the
+fluctuator is given by Stl(0)≃10−28A2s dominated by
+the quantum-partitioning noise with Stl(0)≃10−27A2s.
+Notethatinexperimentsratesoftheorderof10 −100kHz
+have been observed.33,34We expect that rates in the8
+100MHz regime to be realistic due to the exponential
+dependence of the tunneling rate on the potential bar-
+rier. Increasing the bias voltage Vto values of a few mV
+the noise due to the classical two-level fluctuator starts
+to dominate and the crossover as depicted in Fig. 2 can
+be observed.
+VII. CONCLUSION
+Wehavedeterminedtheinfluenceofathermallydriven
+two-level fluctuator on a point contact through the cal-
+culation of the full counting statistics of transported
+charge. Both, classical and quantum point contacts have
+been considered and extensions to multiple fluctuators
+have been discussed. In our analysis, we have made
+use of a mapping between correlation functions of classi-
+cal stochastic processes and simple time-ordered matrixproducts. For the case of a quantum point contact, we
+have shown that the partitioning noise and the noise due
+the two-levelfluctuator add up and the noise crossesover
+from sub- to super-Poissonian depending on the applied
+voltagebias. To extend the present formalismto the case
+ofcurrentcorrelatorsatfinite frequencies, whichprovides
+additionalinsightsintothedynamicsofthetwo-levelfluc-
+tuator, is an interesting problem for future studies.
+Acknowledgments
+We acknowledgefruitful discussions with Misha Suslov
+and financial support from the Pauli Center at ETH
+Zurich, the Swiss National Foundation, the Russian
+Foundation for Basic Research (11-02-00744-a), the pro-
+gram‘Quantumphysicsofcondensedmatter’oftheRAS,
+and the Computer Company NIX (F1025/10-04).
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\ No newline at end of file
diff --git a/1001.0579.txt b/1001.0579.txt
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@@ -0,0 +1,612 @@
+arXiv:1001.0579v3  [hep-ph]  26 Aug 2010MADPH-10-1554
+Constraining WIMP magnetic moment from CDMS II experiment
+Won Sang Cho(a), Ji-Haeng Huh(a), Ian-Woo Kim(b), Jihn E. Kim(a), and Bumseok Kyae(a)
+(a)Department of Physics and Astronomy and Center for Theoreti cal Physics,
+Seoul National University, Seoul 151-747, Korea and
+(b)Department of Physics, University of Wisconsin, Madison, W I 53706, USA
+We consider a degenerate or a nearly degenerate dark matter s ector where a sizable magnetic
+moment of an almost Dirac type neutral dark matter candidate Nis anticipated. Then, due to soft
+photon exchange, the cross-section in direct detection of Ncan be enhanced at low Q2region. We
+discuss the implication of this type of models in view of the r ecent CDMS II report.
+PACS numbers: 13.40.Ks, 95.35.+d, 14.80.Nb
+Keywords: Dark matter, Magnetic moment, WIMP, CDMS II exper iment
+I. INTRODUCTION
+Recent cosmological and astrophysical observations
+strongly suggests that a significant portion of energy
+density of the universe exists in a form of dark matter
+(DM). DM has been speculated after the observation of
+the galaxy matter velocity distribution. The angular fre-
+quency distribution of the cosmic microwave background
+radiation, recent surveys of the expansion rate of remote
+galaxies and the simulation of the structure formation
+support around 25% of energy density in a form of non-
+relativistic matter while only one sixths of such matter
+has been identified as visible one.
+Among many plausible theories of DM [1] or equiva-
+lent modification of gravity [2], weakly interacting mas-
+sive particles (WIMP) are of particular interest [3]. The
+scenario with WIMP assumes that DM particles were
+produced in the early Universe by thermal processes and
+the DM relic density has been frozen out by decoupling
+since that epoch. The DM energy density from the cur-
+rentobservationis wellmatched with such athermal pro-
+duction scenario with an electroweak scale DM mass and
+appropriate electroweak scale interactions. It is also very
+interesting if DM has some interesting connection with
+the origin of the electroweak symmetry breaking as in
+the minimal supersymmetric standard model (MSSM)
+[4]. Above all, this WIMP scenario motivates searches
+for DM particles by detecting and/or producing them
+directly in the laboratory.
+Direct detection ofDM has been carriedout for several
+decades and the experimental sensitivity has been drasti-
+cally improved in recent years [5, 6]. The abundant DM
+particles that flows through Earth may sporadically col-
+lide with ordinary matter nuclei, resulting in the recoil
+of a nucleus. The current WIMP search through the en-
+ergy deposit in the cryogenic detector device has reached
+the DM cross-section with ordinary matter nuclei at the
+order of ∼10−44cm2, and the upgraded CDMS II ex-
+periment [6] has reached to this level.
+In most DM models, fermionic DM is a Majorana par-
+ticle, such as neutralino of the MSSM [7]. However, wecannot rule out a Dirac fermionic DM Nat present, and
+hence it is very important to consider physical effects
+of such Dirac fermionic nature of DM. In this regard,
+we consider an appreciable magnetic moments of DM.
+In fact, the magnetic moments of neutral fermions were
+considered for a long time since the time of weak neutral
+currents [8] up to the present age of DM [9].
+One notable feature in the scattering through mag-
+netic moment of neutral fermion is that it has a larger
+cross section for a lower momentum photon exchange.
+The trend observed by two CDMS II candidates [6] in-
+deed show this behavior: the energy deposit at 12.3 keV
+and 15.5 keV (just above the threshold of 10 keV) while
+much largerenergydeposit is allowedin that experiment.
+At this time with two possible low Q2candidates of the
+CDMS II experiment [6], therefore, it is appropriate to
+scrutinize the Dirac DM aspect more closely. In partic-
+ular, we will pay attention to the magnetic moment f
+ofN.1In principle, this study includes the effects of
+the electric dipole moment also, but we will not specify
+them explicitly which would have needed an additional
+assumption about CP violation.
+There have been several works on the DM dipole mo-
+ments [9, 10]. In these works, various observational con-
+straints ( e.g.the cosmic microwave background radi-
+ation, the cosmic γ–ray detection,the DM relic abun-
+dance) have been considered. Although the authors of
+Ref. [9, 10] commented other possibilities, they mainly
+considered DM models in which the thermal DM relic
+density is determined only by annihilation due to mag-
+netic dipole interactions. However, as pointed out by
+[9, 10], models that give rise to the DM magnetic mo-
+ment interaction with the photon exchange usually have
+other annihilation channels. Here, we consider the cases
+inwhichtherelicdensitydoesnotconstraintheDMmag-
+netic dipole moment. Nonetheless, DM direct detection
+can constrain the DM magnetic dipole moment.
+1fis the magnetic moment of Nin units of the NBohr magneton,
+e/2mN.2
+As a prototype, we consider supersymmetic (SUSY)
+models with extra charged singlets while direct interac-
+tion between DM and nuclei can be forbidden in the
+leading order. The magnetic dipole moment is generi-
+cally induced by one-loop diagram. Such models have
+been discussed in a certain class of leptophilic scenarios
+for explaining recent cosmic ray anomalies, where DM
+particle has suppressed coupling with colored particles.
+We also emphasize that our analysis fully includes nu-
+clear anomalous magnetic moment interactions. With-
+out the photon exchange as in the neutralino case, the
+F2form factor effects of such nucleus are usually neg-
+ligible since the range of the DM–nucleus interaction is
+much smaller than the size of nucleus. However, the in-
+teraction through photon exchange can make the contri-
+bution comparable. The nucleus magnetic moment effect
+has not been considered properly in the previous works.
+Here, we present the DM detection rates including such
+effects and compare them with the recent CDMS II data.
+This paper is organized as follows. In Sec. II, we cal-
+culate the effective magnetic moment operator from a
+simple class of supersymmetric models. In generic mod-
+els beyond the SM, one-loop induced magnetic moment
+should usually have a similar structure calculated in this
+section. In Sec. III, we calculate the decoupling tem-
+perature of Nby solving the Boltzmann equation. We
+also use the ( g−2)ebound to constrain the hypothetical
+Yukawa coupling λ. In Sec. IV, we calculate the direct
+detection cross-section and show the Q2-dependence due
+to the magnetic moment, which can be observed by the
+recoil energy distribution. The behavior due to the mag-
+netic moment is distinguishable from the predictions of
+other DM models. Next, the CDMS II experimental re-
+portisdiscussedinthecontextofDMmodelswithalarge
+magnetic moment. In this section, we also comment on
+the collider phenomenology of this class of models. Sec.
+V is a brief conclusion, summarizing the allowed mag-
+netic moment fof Dirac or almost-Dirac DM models.
+II. MAGNETIC MOMENT OF N
+A neutral Dirac fermion Ncan acquire a magnetic mo-
+ment as shown in Fig. 1 if it couples to charged par-
+ticles, fermion ψand boson φ, by a Yukawa coupling,
+λ¯ψRNLφ+h.c. In the MSSM, the lightest supersymmet-
+ric particle (LSP) is not a Dirac fermion and thus the
+LSP DM cannot acquire a magnetic moment. However,
+in the extensions of MSSM with extra singlet chiral su-
+perfield, which have been proposed for many reasons :
+to addressµ-problem or to raise the mass of the lightest
+Higgs boson, the LSP χof the MSSM sector and the ad-
+ditional singlet Nmay form a single Dirac fermion state
+[11, 12].
+For a specific calculation, we consider the following
+superpotential which has been discussed in [11],
+W=λNecE+mNN¯N+mEEEc+ρN3.(1)In this model, there is no tree level interaction between
+Nand nucleus because it couples only to leptons and its
+scalar partners. For the magnetic moment, the existence
+of the coupling of the type λNecEis the essential one,
+which can arise in many other models extending the SM.
+For DM magnetic moment, it is required that Nmust be
+a DM component, presumably by an exact Z2symme-
+try. Actually Eq.(1) is the simplest model in which the
+magnetic moment can play an essential role in the DM
+detection. More complicated cases can be obtained by
+its proper extensions.
+γ
+N Nλ λ∗ φ φ
+ψ
+(a)
+γ
+N Nλ λ∗ ψ ψ
+φ
+(b)
+FIG. 1: One-loop diagrams for the magnetic moment of N.
+However, in the extension of MSSM, the interaction
+typeλ¯ψRNLφ+h.c. is present through NHuHdwhich
+however has to be avoided by the following reasons.
+Firstly, after the Higgs fields HuandHddevelop VEVs,
+they give a VEV to N, which is harmful because of the
+mass mixing with the electron e. Second, it splits masses
+ofNandN, and the anomalous magnetic moment inter-
+action is a transition type between two mass eigenstates,
+and hence the direct detection rate as a function of re-
+coil energy must be considered more carefully if the mass
+difference is of order 1–10 keV.
+For a Dirac-type neutral fermion, its electromagnetic
+interaction is dominated by the magnetic moment, as
+pointed out for the case of neutrinos [8]. Let us define
+the magnetic moment of Nas
+ef
+2mNNiσµνNFµν. (2)
+For Eq. (1), fis estimated from Fig. 1 [13],
+f=|λ|2m2
+N
+16π2/integraldisplay1
+0dx/braceleftBigqφ(x2−x3)
+m2
+Nx2+(m2
+ψ−m2
+N)x+m2
+φ(1−x)
+−qψ(x2−x3)
+m2
+Nx2+(m2
+φ−m2
+N)x+m2
+ψ(1−x)/bracerightBig
+(3)3
+wheremψis the mass of the fermion ( ecorE) andmφis
+the mass of the boson (˜ ecor˜E) in the one-loop diagram.
+This calculation is an illustration for a sizable magnetic
+momentofahypotheticalDMparticle N. Asshownhere,
+a large magnetic moment is not unreasonable since Nis
+considered to be heavy.
+If mass eigenstates splits the mass by a tiny amount,
+then the magnetic moment is of transition type with the
+initialNand the final Nof Eq. (2) considered different.
+Then, the lifetime of the heavier component is
+1
+Γ= 3.6×10−5/parenleftbigg10−6
+f/parenrightbigg2/parenleftBigmN
+100 GeV/parenrightBig2/parenleftbigg10 keV
+∆mN/parenrightbigg3
+s
+(4)
+where ∆mNis the mass difference between two mass
+eigenstates of this almost-Dirac fermion. If the lifetime
+falls in the 10−10s region with parameters chosen appro-
+priately, then the decay products of Ndeposit energy in
+the cryogenic detector. This happens for f≤O(10−4) in
+which case the real signal from the cryogenic data must
+be revamped.
+III. RELIC DENSITY
+Let us now proceed to discuss the effects of a large
+magneticmoment ofan extrasingletDM Nin cosmology
+and in particle phenomenology. The DM relic density is
+given in terms of the velocity averaged annihilation cross
+section (σannv). The dominant annihilation channel of
+the Dirac DM Nallowed by Eq. (1) is N+N→e−+e+,
+which is mediated by exchange of the scalar component
+ofE. It is straightforward to evaluate σannv, which is
+approximately given by
+σannv=a+bv2+O(v4). (5)
+aandbare
+a=3|λ|4
+8πm2
+N(1+B)2
+b=|λ|4
+48πm2
+N(5B2−16B−7)
+(B+1)4,(6)
+whereB=m2
+˜E/m2
+N. Although there is also the annihi-
+lation by the magnetic dipole moment, Eq. (2), we will
+neglect it because its order of magnitude is estimated as
+σdipole
+annv∼1
+4π/parenleftbiggef
+mN/parenrightbigg2
+∼|λ|4
+4πm2
+N/parenleftBigg
+1
+16π2m2
+N
+Max(m2
+ψ,m2
+φ)/parenrightBigg2
+<10−4σannv.whereσannis theN¯Nannihilation cross section.
+M˜E[GeV]λ
+0 600 100 200 300 400 5000.20.40.60.81.0
+(g−2)µ
+Over ClosuremN>mE
+mN>mE50 GeV
+150 GeV
+FIG. 2: The allowed region in the λvs.M˜Eplane from the
+relic density and ( g−2)µbounds for mN= 100 GeV. The
+boundaries for mN= 50 GeV and 150 GeV are shown as the
+lavender line and the green dash line, respectively.
+In Fig. 2 for mN= 100 GeV, we show the allowed
+regionfromthe constraintonthe cosmicrelicdensityand
+also from the ( g−2)µbound. The excluded regions are
+the lower right wedge and the upper left wedge. Toward
+theNDM scenario, we also excluded the kinematically
+forbidden region mN> mE. For the muon magnetic
+moment, we use the formula given in Eq. (1) with µcNE
+and Eq. (3). The electron magnetic moment bound is
+buried in the muon magnetic moment bound. For the
+cases ofmN= 50 GeV and 150 GeV, the boundaries
+are shown as the lavender line and the green dash line,
+respectively.
+Sincefis of order |λ|2/16π2∼0.6×10−2|λ|2, the
+condition |f|<10−4, as commented below Eq. (4), is
+satisfied only in a small skyblue dashed-arc region in the
+lower-left allowed region of Fig. 2. Except in this small
+area, we need not consider the possibility of a heavier
+component decay in the cryogenic detector.
+IV. DIRECT DETECTION RATE
+At lowQ2, the photon wavelength exceeds the nuclear
+size and hence the whole nucleus, with charge and mag-
+netic moment, acts as a target. Thus, the elastic scatter-
+ing cross section for the case of a spin1
+2nucleus target
+becomes, viz. Fig. 3,4
+γN
+Nf (Ze, µ anom) •
+FIG. 3: The elastic scattering of Nwith a nucleus through
+the magnetic moment of N.
+dσ
+dErec=2πα2
+emf2
+Mm2
+N|/vector p|2/bracketleftbigg
+Z2/braceleftbiggΛ−(s,m2
+N,M2)
+2MErec+(2m2
+N+M2−s)/bracerightbigg
++ 2ZF2(4m2
+N−MErec)+F2
+2/braceleftbiggΛ+(s,m2
+N,M2)
+M2−2sErec
+M+E2
+rec/bracerightbigg/bracketrightbigg
+, (7)
+whereMis the mass of the nucleus with atomic weight
+A,Erecis the nuclear recoil energy, Qem=Zeis the
+nuclear charge,
+Λ∓(s,m2
+N,M2) = (m2
+N+M2−s)2∓4m2
+NM2,(8)
+andF2=1
+2Fst
+2=1
+2/parenleftBig
+µ
+µNm(Z,A)
+mp−Z/parenrightBig
+for spin1
+2case,
+whereFst
+2is the conventional notation. Our F2is a fac-
+tor1
+2of the conventional definition, Fst
+2. The first two
+terms (‘Zterms’) in the right hand side of Eq. (7) result
+from the electromagnetic interaction between the tensor
+operatorof N(NσµνN) in Eq. (2) and the vectorcurrent
+of a spin1
+2target nucleus ( ψγµψ). On the other hand,
+the last three terms (‘ F2terms’) come from the interac-
+tion between the tensor operators of Nand the nucleus.
+As seen in Eq. (2), the tensor operator of a given nucleus
+defines the spin of the nucleus ( ψσµνψ).
+For the case of a higher spin target nucleus, ‘ γµ’ and
+‘σµν’ in the vector and tensor operators of the nucleus
+should be replaced by larger dimensional representations
+of the Dirac gamma matrix and the Lorentz generator,
+respectively (see e.g. Ref. [14]). Using Appendix, we
+can deduce that F2=1
+2Fst
+2=1
+2/parenleftBig
+µ
+µNm(Z,A)
+mp/radicalBig
+sN+1
+3sN−Z/parenrightBig
+from Eq. (7) with a higher spin generalization. In the
+non-relativistic limit ( Erec≪ {M, mN}andv≪1), the
+differential cross section is given by
+dσ
+dErec=4πα2
+emf2
+m2
+NErec/bracketleftbigg
+Z2(1−Erec
+2Mv2−Erec
+mNv2)
++/parenleftbiggµ
+µN/parenrightbigg2sN+1
+3sNMErec
+m2pv2/bracketrightBigg
+.(9)The direct-detection rate (per unit detector mass) in a
+detector with nucleus is given by
+dR
+dErec=ρN
+mNM/integraldisplay
+|/vector v|>vmind3/vector vf(/vector v)dσ
+dErec.(10)
+Here, we assume that the WIMP of mass mNaccounts
+for the local DM density ρNand have a local velocity
+distribution f(/vector v) with the normalization/integraltext
+d3/vector vf(/vector v) = 1.
+Using a simple Maxwell-Boltzmann velocity distribution
+andρN≃0.3GeV/cm3, in Fig. 4 we plot the expected
+direct-detection rates of an almost-Dirac DM N(solid
+line) andthe LSP χ(dashline) in the MSSM forthespin-
+independent(SI) interactions which always dominates for
+nuclei with A≥30 in surveys of the SUSY parameter
+spaces [15]. The SI cross section is given by
+dσ
+dErec=2M
+πv2(Zfp+(A−Z)fn)2F2(Erec),(11)
+wherefp≃fn≃10−8GeV−2are the SI cou-
+plings of WIMPs to protons and neutrons, respec-
+tively, the SI form factor F(Erec) = 3e−κ2s2/2(sin(κr)−
+κrcos(κr))/(κr)3, withs= 1 fm,r=√
+R2−5s2,R=
+1.2A1/3fm, andκ=√2mnErec. In Fig. 4, we present
+the differential event detection rate for mN= 70 GeV,
+Eq. (10), as a solid line above the CDMS II threshold
+of about 10 keV. The neutralino DM case with mχ= 70
+GeV is shown as a dashed line. To show the recoil energy
+dependence, we choose an arbitrary normalization such
+that both of these lines match at 10 keV. The target is
+Ge withZ= 32 andA= 73. Its anomalous magnetic
+moment isF2=−36.32 as shown in Table I. The 1 /Erec
+dependence in the first term of Eq. (7) generically leads
+thedRMDM/dErecto divergein low Ereclimit. However,5
+O(8,16)3Na(11.23)3
+2Si(14,28)2Ge(32,73)9
+2I(53,127)5
+2Xe(54,131)3
+2Cs(55,133)7
+2W(74,183)1
+2
+µ/µN1.66812 2.21752 1.1218 −0.879467 2.81327 0.692 2.58 0.117785
+F24.83 13.36 4.02 −36.3294.56 6.52 83.93 −26.30
+TABLE I: Magnetic moments µof several target nuclei used in cryogenic detectors. Here, µNis the proton Bohr magneton
+µN=e/2mp. The nuclear spin is denoted as superscripts. F2is the half of the conventional definition, F2=1
+2Fst
+2, which is
+related to the effective anomalous magnetic moment correspo nding to the equivalent particle that has the same mass, char ge
+and magnetic dipole moment as the target nucleus.
+tmN= 70 GeV 
+t32 Ge 73 for CDMS II
+tf= 10−5, F 2=−36.32 
+tDash :σSI, Solid :σMDM
+Erec[keV]dR/dE rec(Arbitrary Unit)
+10 20 30 40 50 60 70 80 90 100 12345
+FIG. 4: WIMP( N)–nucleus detection rate as a function of
+the recoil energy, normalized at Erec= 10 keV. The solid
+line corresponds to the magnetic moment and the dashed line
+corresponds to the SI interaction. The MDM graph and the
+SI graph meet at two points due to our normalization to their
+equality at 10 keV. So, at very low Q2, the MDM is bigger
+than SI as expected.
+such a divergent effect is diminished by small velocity of
+WIMP,v2∼10−6which appears in the coefficient of the
+1/Erec-term as follows:
+Λ−
+2MErec=2M|/vector p|2
+Erec≃2M3v2
+Erec
+≃2M310−6
+10(−5∼−4)(GeV).(12)
+Due to the suppression of the IR divergent feature
+in non-relativistic scattering, the dRMDM/dErecdiver-
+gently larger than dRSI/dEreconly in the region below
+Erec∼10keV, while the slope of the dRMDM/dErec
+aboveErec∼10keV becomes eventually more flatter
+than the SI. This is why the dRMDM/dErecdistribu-
+tion looks larger than dRSI/dErecaboveErec10keV in
+Fig. (4). This non-relativistic v2- suppression in the
+1/Erec-term might result in further interesting possibil-
+ity. Due to the suppression, ‘ Zterms’ can be comparable
+to each other, even with ‘ F2terms’. In this regard, we
+obtain the upper-bound on the DM magnetic moment
+from the recent CDMS II data for both of the F2= 0
+andF2∝ne}ationslash= 0 cases. The “maximal gap method”[16] isFIG. 5: The allowed region of the DM magnetic moment fvs.
+the DM mass mN. The upper colored regions are excluded
+forF2= 0 and F2/negationslash= 0, respectively, with the 90 % confidence
+level for the CDMS II data [6].
+used to estimate the proper allowed region with a 90%
+confidence level. The most stringent bound appears as
+f/lessorsimilar2.88×10−3formN≃21 GeV and F2∝ne}ationslash= 0. When
+we ignore the contributions of nucleus anomalous mag-
+netic moment, F2= 0, then the upper-bound of fis
+1.16×10−2formN∼100GeV. Taking into account the
+non-zeroF2, the upper-bound goes down to 1 .04×10−2
+formN∼100GeV. The event rate becomes larger
+with non-zero F2so that the allowed region is more con-
+strained, producing 1 −10% of difference in f. However,
+we can easily expect that depending on the materials in
+direct detection experiments, the difference can be sig-
+nificantly amplified due to the enhanced magnetic dipole
+moment effect. Thus, it is worthwhile to study the DM
+multi-pole interactions with nuclei more carefully.
+V. CONCLUSION
+WeconsideredaDiracoranalmost-DiracDM Nwhich
+mayacquirealargemagneticmoment. Usingthepossible
+dipole interactions, we estimated the signal event rate
+which is expected in the CDMS II experiment. Using
+the recent report of the CDMS II experiment, we present
+the upper-bound of the magnetic dipole moment. The
+most stringent bound appears for mN≃21 GeV and
+F2∝ne}ationslash= 0:f/lessorsimilar1.4×10−3with a sizable nucleus anomalous6
+dipole moment F2contribution. We point out that this
+sizableF2contribution is possible in the non-relativistic
+scattering of WIMP and nucleus, leading to a several
+factor improvement in the exclusion plot. In the future
+refined direct DM search experiments, the possibility of
+DM magnetic moment of Ncan be probed with another
+independent information on its mass.
+Acknowledgments
+We thank J. H. Yoo for useful discussions. WSC,
+JHH, JEK and BK are supported in part by the Ko-
+rea Research Foundation, Grant No. KRF-2005-084-
+C00001. B.K. is also supported in addition by the
+FPRD of the BK21 program and the KICOS Grant No.
+K20732000011-07A0700-01110 of Ministry of Education
+and Science. IWK is supported by the U.S. Department
+of Energy under grant No. DE-FG02-95ER40896.
+Appendix
+In the calculation of scattering cross-section of DM
+with target nuclei, we treated the nuclei as a spin1
+2particle. However, many nuclei used in the direct de-
+tection of DM have spin different from1
+2. Moreover,
+our parametrization of the magnetic moment using F2,
+in which spin1
+2for nuclei has been assumed, should be
+matched to produce a realistic value. For that, we con-
+sider two body wave function, in non-relativistic quan-
+tum mechanics with non-local interaction, which is ob-
+tained after integrating out the photon field,
+Ψαi(/vector x,/vector y) =Nα(/vector x)⊗χi(/vector y) (13)
+whereNα(/vector x) is the nuclear wave function with S(N)
+z=α
+andχi(/vector y) is the DM wave function with S(χ)
+z=i. The
+Scr¨ odinger equation for this two body system reads
+HΨ =/parenleftBigg
+−∇2
+x
+2M−∇2
+y
+2mN/parenrightBigg
+Ψ+gDMZe2
+2mN|/vector x−/vector y|∇x·/vectorS(N)Ψ
++gDMgnucleie2
+2mN2M|/vector x−/vector y|3/vectorS(N)·/vectorS(χ)Ψ
+= (H0+Hmo-di+Hdi-di)Ψ
+whereHmo-diandHdi-diare the monopole–dipole and
+dipole–dipole interactions, respectively. Here, we ne-
+glected unimportant normalization for each terms be-
+cause we only need to match the dependency on the nu-
+cleon spin.
+Since the initial state of the problem is unpolar-
+ized, its density matrix is proportional to the iden-
+tity in (2sN+ 1)(2sχ+ 1)-dimensional space, that is,
+ρ=1/(2sN+1)(2sχ+1), where sNandsχare the spin
+of nuclei and DM, respectively. Therefore, after averag-
+ing the polarizations, the cross-section is proportional to
+tr[ρM†M], whereMis
+M=Hint+Hint1
+E−H+iǫHint.(14)For the dipole-dipole interaction contribution, the cross-
+section in the leading order becomes
+σ∝tr[ρM†M]
+∝|gDMgnuclei|2
+(2sN+1)(2sχ+1)tr/bracketleftBig
+/vectorS(N)·/vectorS(χ)/vectorS(N)·/vectorS(χ)/bracketrightBig
+=|gDMgnuclei|2
+9sN(sN+1)sχ(sχ+1)(15)
+where the dot product is over the SO(3) space and Tr
+is over the spin multiplicities. Here, if we use gnuclei=
+2Mµ/e/radicalbig
+sN(sN+1), we can see that the term propor-
+tional toµ2is independent of the nucleus spin. Here, if
+we usegnuclei= 2Mµ/esNwe see that the term involving
+µ2is proportional to ( sN+1)/3sN. It confirms that our
+parametrization of F2in the text is appropriate.7
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+[16] J. Kopp, T. Schwetz and J. Zupan, arXiv:0912.4264
+[hep-ph]. S. Yellin, Phys. Rev. D 66(2002) 032005
+[arXiv:physics/0203002].
\ No newline at end of file
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+arXiv:1001.0580v2  [physics.optics]  18 Mar 2010PhotonicCrystalSpectrometer
+NadiaK.Pervez,1,∗WarrenCheng,1ZhangJia,1Marshall P.Cox,1
+HassanM. Edrees,1,2andIoannisKymissis1
+1Columbia Laboratory forUnconventional Electronics, Depa rtment ofElectrical Engineering,
+Columbia University, New York, 10027, USA
+2Department of Electrical and Computer Engineering,
+TheCooper Union for the Advancement ofScience and Art,New Y ork, 10003, USA
+∗nadia.pervez@gmail.com
+Abstract: We demonstratea new kindof optical spectrometeremploying
+photonic crystal patterns to outcouple waveguided light fr om a transparent
+substrate. This spectrometer consists of an array of photon ic crystal pat-
+terns, nanofabricated in a polymer on a glass substrate, com bined with a
+camera.Thecameracapturesanimageofthelightoutcoupled fromthepat-
+terned substrate; the array of patterns produces a spatiall y resolved map of
+intensitiesfordifferentwavelengthbands.Theintensity mapoftheimageis
+convertedinto a spectrum using the photonic crystal patter n response func-
+tions.Wepresentaproofofconceptbycharacterizingawhit eLEDwithour
+photoniccrystal spectrometer.
+Referencesandlinks
+1. N.Savage, “Spectrometers,” Nature Photon. 3,601–602 (2009).
+2. X-Rite, Incorporated. “MatchRight iVue Spectrophotome ter Sell Sheet, ”
+http://www.xrite.com/documents/literature/en/L3-186 _iVueSellSheet.pdf .
+3. Y. Maruyama, K. Sawada, H. Takao, and M. Ishida, “A novel fil terless fluorescence detection sensor for DNA
+analysis,” IEEETrans.Electron Devices 53,553–558 (2006).
+4. D. S. Goldman, P. L. White, and N. C. Anheier, “Miniaturize d spectrometer employing planar waveguides and
+grating couplers for chemical analysis,” Appl. Opt. 29,4583–4589 (1990).
+5. D.Sander, M.-O.Duecker, O.Blume, and J. Mueller, “Optic al microspectrometer in SiON slab waveguides,” in
+Integrated Optics and Microstructures III ,M. Tabib-Azar, ed., Proc.SPIE 2686,100–107 (1996).
+6. B. Momeni, E. S. Hosseini, and A. Adibi, “Planar photonic c rystal microspectrome-
+ters in silicon-nitride for the visible range,” Opt. Expres s17,17060–17069 (2009),
+http://www.opticsinfobase.org/oe/abstract.cfm?URI=o e-17-19-17060 .
+7. G. Schweiger, R. Nett, and T. Weigel, “Microresonator arr ay for high-resolution spectroscopy,” Opt. Lett. 32
+2644–2646 (2007).
+8. E.Yablonovitch, “Inhibited Spontaneous Emission in Sol id-State Physics and Electronics,” Phys.Rev. Lett. 58,
+2059–2062 (1987).
+9. J.D.Joannopoulos, S.G.Johnson,J.N.Winn,and R.D.Mead e,.Photonic Crystals: Molding theFlowofLight,
+Second Edition (Princeton Univ. Press,Princeton, 2008).
+10. M. Rattier, H. Benisty, E. Schwoob, C. Weisbuch, T. F. Kra uss, C. J. M. Smith, R. Houdre and U. Oesterle,
+“Omnidirectional and compact guided light extraction from Archimedean photonic lattices,” Appl. Phys. Lett.
+83,1283–1285 (2003).
+11. M.Boroditsky,T.F.Krauss,R.Coccioli,R.Vrijen,R.Bh at,andE.Yablonovitch, “Lightextractionfromoptically
+pumped light-emitting diode by thin-slab photonic crystal s,” Appl. Phys.Lett. 75,1036–1038 (1999).
+12. J.J.Wierer,A.David,andM.M.Megens,“III-nitride pho tonic-crystal light-emitting diodeswithhighextraction
+efficiency,” Nature Photon. 3,163–169 (2009).
+13. C.Wiesmann,K.Bergenek, N.Linder, and U.T.Schwarz, “P hotonic crystal LEDs–designing lightextraction,”
+Laser &Photon. Rev. 3,262–286 (2009).
+14. D. Zwillinger D. CRC Standard Mathematical Tables and Formulae, 31st editio n(Chapman & Hall/CRC, New
+York,2003).
+15. seeforexampleSparkfunElectronics-Light/Imaging http://www.sparkfun.com/commerce/categories.php?c=1 02.16. K. Ishihara, M. Fujita, I. Matsubara, T. Asana, and S. Nod a, “Direct fabrication of photonic crystal on glass
+substrate by nanoimprint lithography,” Jpn.J.Appl. Phys. 45,L210–L212 (2006).
+1. Introduction
+The invention of compact, low cost spectrometer modules has broadened the traditional spec-
+trometryapplicationbasewellbeyondbasicscientificmeas urementtoincludeconsumerappli-
+cations such as monitoring and feedback in color printing an d color paint matching [1, 2]. In
+aconventionalspectrometertherelationshipbetweendiff ractionangleandwavelengthcouples
+module size and resolution. Here, we present a new kind of spe ctrometer which breaks that
+paradigm. The photonic crystal spectrometer measures a spe ctrum through the projection on
+to a basis formed by photonic crystal pattern response funct ions rather than binning spatially
+resolveddiffractedlight.Wepresentaproofofconceptoft hisphotoniccrystalspectrometerby
+characterizinga whiteLEDusinganarrayofninephotoniccr ystalpatterns.
+A number of methods for making compact spectrometers have be en previously proposed,
+includingusingthechangeinabsorptionofamaterialmediu masafunctionofwavelength[3],
+andmonolithicintegrationofgratingsontopof[4]oratthe endof[5]thinplanarwaveguides.
+Another kind of spectrometer which also takes advantage of p hotonic crystals has also been
+reported [6]; that device employs the enhanced dispersion o f the superprism effect to create
+a spatially resolved pattern that can be read by a single dete ctor. The superprism effect-based
+spectrometer is well suited to narrow bandwidth, high resol ution applications. A two dimen-
+sional array based spectroscopy scheme using dielectric mi crosphere resonators has also been
+reported[7]; that array hasa similar geometricscalingas t he device presentedhere.Threeim-
+portantdifferencesbetweenthisworkand[7]aredetermini sticcontroloverthearrayresponse,
+size – the entire array presented here is smaller than a singl e microresonator array element –
+and the calculationof a broadbandspectrumbased rathertha n use of the arrayto matchsingle
+wavelengthfingerprints.The work presentedhere demonstra tesan extremelysimple, inexpen-
+sive, and reproducible broadband spectrometer based upon t he ability of photonic crystals to
+wavelength-selectivelyoutcouplewaveguidedlightfroma substrate.
+Fig. 1. Illustration of spatially resolved, wavelength-se lective outcoupling of waveguided
+light using and array of photonic crystals. Light waveguide d into the edge of the substrate
+isoutcoupledbyphotoniccrystalpatternscorrespondingt odifferentbandsofwavelengths.
+Figure 1 illustrates the conceptof spatially resolved,wav elength-selectiveoutcouplingused
+in the spectrometer. In the illustration, light is introduc edinto the edge of the transparent sub-
+strate. Photonic crystals with four different lattice cons tants are patterned in the layer above
+the substrate. Each of these patternsoutcouplesa differen tband of wavelengths, illustrated bythe arrowswith differentcolors.A camera takes a picture of the substrate, which is used as an
+intensitymaptoquantitativelyassess theresponseofthed ifferentphotoniccrystalpatterns.
+2. Photoniccrystalsforoutcouplingofwaveguidedlight
+In a photonic crystal, a periodic potential formed by spatia l variation in the relative permit-
+tivityεrof a medium interacts with electromagnetic radiation resul ting in partial or complete
+bandgaps [8, 9]. The band structure is determined by the choi ce of lattice, the basis formed
+by the shape and size of the holes, the thickness of the patter ned layer, and the contrast in the
+spatial variationin εr.Theenergyscale forthebandstructureisset bythelattice constanta.
+Similar two dimensional photonic crystal structures have b een extensively investigated as
+a way to improve light extraction efficiency in light emittin g diodes. Schemes involving the
+outcoupling of waveguided light using weak photonic crysta ls as well as the disruption of
+waveguided modes in the emissive layer using strong photoni c crystals have been reported
+in the literature [10, 11, 12, 13]. The extraction scheme emp loyed here is an extension of the
+waveguidedlightoutcouplingschemeusedforLEDenhanceme nt.
+When light of wavelength λis introduced into a dielectric slab, the light has |kkk|=nk0
+wherek0=2π
+λandnis the slab’s index of refraction. In the slab, the magnitude of the trans-
+verse component of kkkis|kkk/bardblm|=/radicalBig
+(nk0)2−k2
+⊥mwherek⊥m≈mπ
+d,dis the slab thickness and
+m=1,2,3.... In a photoniccrystalpatternedregionwith lattice consta ntathe periodicpoten-
+tial opens up gaps, allowing the transverse component of the wavevector to be scattered by a
+reciprocallattice vector GGG. Figure2illustratesthe effectofthis scattering,whicha llowsprevi-
+ouslyguidedmodestobeextractedafterascatteringevent. Themodesscatteredto |kkk/bardblm|>nk0
+areevanescentinboththeslabandair.Themodesscatteredt ok0<|kkk/bardblm|<nk0remainguided
+in the slab, and those scattered to |kkk/bardblm|<k0can propagatein the air. An additional constraint
+inoursystemistheabilityofthecameratocapturethelight ,limitedto |kkk/bardblm|<NAk0whereNA
+isthe numericalapertureofthecameraobjective.Onlyligh tthat isimagedbythe cameracon-
+tributes to the measured photonic crystal response functio n. Thus the NAaffects the response
+function profile, with larger NAobjectives yielding wider response functions. The most effi -
+cient extractionoccurswhen |kkk/bardbl|≈|GGG|. Inourdevicean0.96mm thickglasscoverslipisused
+as a substrate; for all but very high modes in our system nk0≫mπ
+d. Therefore |kkk/bardbl|≈nk0, and
+optimumextractionoccursfor nk0≈|GGG|orλ
+n≈a.
+Inthedevicepresentedhere,thephotoniccrystalsarefabr icatedinpolymethylmethacrylate
+(PMMA)ontopoftheglasssubstrate.Theindexofrefraction nofPMMAoverthewavelengths
+of interest (450 – 700nm) is 1.499 – 1.512 which is slightly lo wer than the glass ( n≈1.52);
+the contrasting medium is air. The photonic crystal pattern s employed are all square lattices,
+with fill factors ( r/a, whereris the hole radius) of approximately 0.34. With these design
+parameters, the periodic potential should not be strong eno ugh to open a complete bandgap
+[9].Thewavelengthselectivityoftheoutcouplingisrobus ttodisorderintheperiodicpotential
+since the interaction is distributed over several periods. SEM images of patterns similar to the
+smalleronesusedinthiswork(datanotshown)shownon-unif ormholesizeaswellasmissing
+holesduetodosefluctuationsin theelectronbeamlithograp hytool.
+3. Experimental
+Arrays of photonic crystal patterns were fabricated in PMMA (MicroChem 495PMMA
+A series) on glass via electron beam lithography (EBL). The g lass substrates were
+22mm×22mm×0.96mmcoverslips.Inthisworksquarelatticesphotoniccr ystalpatternswere
+chosen over hexagonallattices for ease of fabrication. All of the lattice points written as large
+dots during EBL. An 8–10nm layer of gold was used to prevent ch arging during the EBLFig. 2. Scattering of three guided modes with |kkk/bardbl|=/radicalBig
+k2x+k2y=k1(solid blue), k2(solid
+green), and k3(solid red) by a reciprocal lattice vector GGGinto|kkk1+GGG|(dashed blue),
+|kkk2+GGG|(dashed green), and |kkk3+GGG|(dashed red). Modes scattered (i) outside of the
+dashed gray circle ( |kkk/bardbl|>nk0) are evanescent in the slab, (ii) into the annulus bounded
+by the dashed gray and solid gray circles ( nk0>|kkk/bardbl|>k0) are guided in the slab and
+evanescent in air, (iii) into the annulus bounded by the soli d gray and black circles ( k0>
+|kkk/bardbl|>NAk0)propagate inairbutcannot beimagedbythecamera,and(iv) insidethesolid
+black circle ( |kkk/bardbl|<NAk0) propagate inthe airand canbe imaged bythe camera.
+process.Atypicalpatternexposuredosewas320 µC/cm2.
+To determinetheoptimalparametersforthe photoniccrysta larrayelementsdifferentlattice
+constants, fill factors,and PMMA thicknesseswere evaluate din the contextof light extraction
+measurements. Lattice constants from 250 to 650nm were inve stigated to confirm response
+function peaks at a≈λ
+n. Fill factors ranging from 0.25 to 0.5 were assessed in the co ntext of
+response function peak intensity and uniformityover the pa tterned regions (particularlyin the
+contextofproximityeffectsinEBLdosearrays).PMMAthick nessesfrom233to509nmwere
+also evaluated for response function peak intensity but thi s process did not identify a single
+optimum value for all lattice constants. Based on these expe riments, an array of nine square
+lattices with lattice constants from 300 to 420nm and approx imate fill factors of 0.34 were
+fabricatedin396nmPMMA.ThePMMAthicknesswasmeasuredvi aspectroscopicellipsom-
+etryandfillfactorswereestimatedfromSEMimages.Eachspe ctrometerpatternconsistedofa
+3×3arrayof 30 µm×30µmphotoniccrystal patterns,with a total array size of appro ximately
+100µm×100µm.
+Photonic crystal array measurements were performed using w aveguided light and a USB
+camera.Lightwaswaveguidedintoallfouredgesofasampleu singacustomfiberopticlineof
+lightassembly(LumitexInc.OptiLine).Thelineoflightas semblyconsistedofabundleof310
+0.010” plastic fibers affixed into four 0.80” linear array seg ments. Each linear array segment
+was butt-coupled to an edge of the glass substrate. Mineral o il was used to improve coupling
+between the line of light assemblyand the glass. Althoughth e line of light assemblywas usedfor all of the array measurements in this work, we have couple d to similar structures using a
+single 1mm plastic fiber. The line of light was required for th e measurements presented here
+because the sample contained an EBL dose array of spectromet er patterns; the spectrometer
+patternused in this workwas thereforeone of manyfabricate don the same substrate.The line
+oflightassemblyallowedforsimultaneouscouplingintoal l patternsonthe substrate.
+RGB imagesofthe arrayweretakenusinga USBcamera(MatrixV isionmvBlueFox-120c)
+using a 4×objective with a numerical aperture of 0.096 (Infinity Photo -Optical Infinistix). As
+discussedintheprevioussection,thenumericalaperturep laysanimportantroleindetermining
+the response function widths. The pattern spectral respons es were characterized using a pro-
+grammablelightsource(NewportApexXearclampandmonochr omator).Theprogrammable
+light source (PLS) output was measured using a calibrated ph otodetector (Newport UV-818).
+TheUSB camerawascharacterizedbytakingimagesofthePLS o utputreflectedfromateflon
+block.Theteflonblockreflectivitywascharacterizedusing thePLSandthecalibratedphotode-
+tector.
+The performance of the photonic crystal spectrometer was ch aracterized by replacing the
+PLS input with a 5mm white LED driven at 5mA. The same white LED at the same drive
+current was used for comparison measurement with a commerci al spectrometer (Ocean Op-
+tics USB4000). The tip of the LED package was placed against S MA input for the compari-
+son measurement. The response of the commercial spectromet er was characterized using the
+PLS. Allmeasurementswereperformedinthedark.Imageanal ysiswasperformedusingLab-
+view Vision (National Instruments). Nine 8 ×8 pixel regions of interest (ROIs) were defined
+for the 3×3 spectrometerarray.The neutralweight grayscaleluminan ce,rather than the green
+weighted NTSC grayscale luminance, is calculated for each R OI by averaging the mean red,
+green,andbluevaluesin eachregion.
+4. ResultsandDiscussion
+Figure 3 shows the response of the array of nine photonic crys tal patterns to four different
+light inputs. Figure 3(a)–(c) are narrow-band inputs; the m ultiple illuminated regions in each
+of these pictures illustrate the overlap between the respon se functions of the photonic crystal
+patterns. The full set of images is used to calculate the neut ral weight grayscale luminance
+for each pattern. In this manner the response of each pattern to each measured wavelength is
+obtained,as shownin Figure 4. While the implementationpre sentedhere uses colorimagesto
+obtain pattern intensity information,the same informatio ncan be readily obtained throughthe
+analysisofgrayscaleimages.
+The photonic crystal spectrometer works by mapping intensi ties in pattern space to wave-
+lengthspaceviatheresponsefunctionsforeachpattern.Us ingamatrixformalism,theresponse
+ofthesystem canbedescribedas
+AAAxxx=bbb (1)
+whereAAAis anm×nmatrixconsisting of the intensities of mpatternsat nwavelengths, xxxis the
+wavelength space representation of the input, and bbbis the pattern space representation of the
+input. Since the mpatterns peak at different wavelengths, AAAis full rank therefore its Moore-
+Penrosepseudoinverseisa validrightinverse[14] whichca nbe usedto solveEquation(1)for
+the spectral response of the input. The projection operator PPP, on to the subset of wavelength
+spacerealizablegiven AAA, is
+PPP=AAA−1
+rightAAA. (2)
+Therecoveredspectrum ˜xxx=PPPxxxistherefore
+˜xxx=AAA−1
+rightbbb. (3)Fig. 3. Response of a 3 ×3 array of photonic crystal patterns to different inputs. (a ) 470nm
+waveguided light is primarily outcoupled by two array eleme nts in the bottom row, (b)
+530nm waveguided light is primarily outcoupled by two array elements in the middle row
+and one in the bottom row, (c) 630nm waveguided light is prima rily outcoupled by two
+array elements in the top row, and (d) a white light emitting d iode (LED) spectrum is
+outcoupled by array elements in all rows. Each photonic crys tal array element measures
+approximately 30 µm×30µm ;the total arraysize is approximately 100 µm×100µm .
+Fig.4.Spectralresponseofphotoniccrystalpatterns.Spe ctralresponseofthe3 ×3arrayof
+photonic crystal patterns pictured in Figure 3 measured as m ean neutral-weight grayscale
+intensity vs. wavelength. Peak wavelengths are determined by the photonic crystal lattice
+constants.Usingthisformalism,asinglephotographcontainingmulti plephotoniccrystalpatterns,such
+asFigure3(d),isusedtomeasurethespectralcontentofali ghtsource.Therecoveredspectrum
+in Equation (3) is a linear combination of pattern response f unctions. The performance of the
+spectrometeris determinedby PPPwhichisa functionofboththenumberof patternelementsin
+thisbasissetandthenumberofwavelengthsusedtocharacte rizethepatterns.Theresolutionis
+constrainedbythenumberofwavelengthsusedtocharacteri zethepatterns,whiletheaccuracy
+isconstrainedbytheabilityofthebasistorepresenttheme asuredspectrum.
+Figure5comparestheperformanceofaninepatternphotonic crystalspectrometertoacom-
+mercialspectrometer.Themajorfeaturesinthespectrumha vebeenreproducedusingasimple
+and inexpensivesystem: nine patterns in PMMA on a glass subs trate and a camera. The spec-
+trum using the photonic crystal spectrometer is obtained th rough the analysis of grayscale in-
+tensitiesofthepatternsinFigure3(d),usingEquation(3) .Theprojectedspectrum,alsoshown
+in Figure 5, represents the performance limit for the photon ic crystal spectrometer given the
+pattern response functions shown in Figure 4; it is obtained by projecting the LED spectrum
+measuredwith the commercialspectrometeron to the basis fu nctionsshown in Figure 4 using
+Equation(2).
+Fig. 5. Comparison of photonic crystal spectrometer and com mercial spectrometer. Com-
+parison of white LED spectrum measurement using commercial ly available spectrometer,
+calculated spectrum projected on to the pattern spectral re sponse function basis shown in
+Figure 4, and measurement usingthe photonic crystal spectr ometer.
+Theaccuracyoftherecoveredspectrumcanbeimprovedwitht heuseofmorephotoniccrys-
+tal patterns. Figure 6 shows the effect of an increased densi ty of response functions (similarly
+shaped to those in Figure 4) on a white LED spectrum projectio n. The 11 element response
+functionbasisinFigure6(b)issimilartothatoftheexperi mentalsystem(Figure4).Increasing
+the numberofresponsefunctionsto 30,asin Figure6(d),gre atlyimprovestheaccuracyofthe
+projected spectrum. The project spectrum in Figure 6(e) is a faithful reproductionof the unal-
+tered spectrum in Figure 6(a). The experimental realizatio n presented here is an array of nine
+patterns, but the same measurement setup can handle more tha n 600 similarly sized patterns
+without any modification. Thus a higher performance photoni c crystal spectrometer is readily
+attainable.
+5. Conclusion
+Wehavedemonstratedasimple,lowcostspectrometerbasedo ntheabilityofphotoniccrystals
+to wavelength selectively outcouple waveguided light from a glass slide. The response of anFig. 6. Effect of number of basis elements on photonic crysta l spectrometer response. Cal-
+culated projections of ideal white LED spectrum on to differ ent bases, with 2nm wave-
+length step size. (a) ideal white LED spectrum (no projectio n), (b) 11 response function
+basis similar to pattern responses in Figure 4, (c) projecti on of white LED spectrum on
+to 11 function basis in (b), (d) 30 response function basis, ( e) projection of white LED
+spectrum onto30 functionbasis in(d).array photonic crystal patterns on the glass slide to a light source is captured using a camera.
+The resulting image is used to calculate the spectrum of the l ight source, via the response
+functionsofthephotoniccrystalpatterns.
+Themostexpensivecomponentofthephotoniccrystalspectr ometeristhecamera.Compact
+cameras,ubiquitousincellphonesandlaptops,areextreme lyinexpensive;CMOScameramod-
+ules retail for less than $20 [15]. The photonic crystal patt erns presented here were fabricated
+usingelectronbeamlithography,howevernanoimprintlith ographyofpatternswithsimilarfea-
+tures to the highest aspect ratio patterns presented here ha s previously been demonstrated in
+the literature with excellent uniformity over 3mm ×3mm areas [16]. Thus the photonic crys-
+tal arrays can be manufactured using an extremely low cost an d high throughput technique.
+With a high quality imprint master, the only limitation on th e number of patterns that can be
+incorporatedintothespectrometeristhe cameraresolutio n.
+Oneadvantageofthephotoniccrystalspectrometeristhati tcanbeeasilytailoredtospecific
+applicationsthroughtheselectionoftheincorporatedpho toniccrystalpatterns.Thisallowsfor
+the use of lower resolution cameras in applications where br oadband high resolution perfor-
+manceisnotrequired.Forexample,inalowcostspectrophot ometerdesignedforthedetection
+ofaspecificcompound,thephotoniccrystalpatternscanbel imitedtothoserequiredfordetec-
+tionofwavelengthsofinterestandrejectionoffalse posit ivecompounds.
+6. Acknowledgments
+TheauthorswishtoacknowledgesupportfromeMaginundersu bcontractfromtheU.S.Army
+Night Vision & Electronic Sensors Directorate. This work ha s used the shared experimental
+facilities that are supported primarily by the MRSEC Progra m of the National Science Foun-
+dation under Award Number DMR-0213574 and by the New York Sta te Office of Science,
+TechnologyandAcademicResearch(NYSTAR).Theauthorswou ldliketothankM.S.Foster
+andL.A. Fetterforhelpfuldiscussions.
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+arXiv:1001.0581v2  [astro-ph.EP]  27 Jan 2010Draft version October 30, 2018
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+SECOND GENERATION PLANETS
+Hagai B. Perets
+Harvard-Smithsonian Center for Astrophysics, 60 Garden St .; Cambridge, MA, USA 02138
+Draft version October 30, 2018
+Abstract
+Planets are typically thought to form in protoplanetary dis ks left over from protostellar disk of
+their newly formed host star. However, an additional planet ary formation route may exist in old
+evolved binary systems. In such systems stellar evolution c ould lead to the formation of symbiotic
+stars, where mass transferred from the expanding evolved st ar to its binary companion could form
+an accretion disk around it. Such a disk could provide the nec essary environment for the formation
+of a second generation of planets in both circumstellar or ci rcumbinary configurations. Pre-existing
+first generation planets surviving the post-MS evolution of such systems may serve as seeds for,
+and/or interact with, the second generation planets, possi bly forming atypical planetary systems.
+Second generation planetary systems should be typically fo und in white dwarf binary systems, and
+may show various observational signatures. Most notably, s econd generation planets could form in
+environment which are inaccessible, or less favorable, for first generation planets. The orbital phase
+space available for the second generation planets could be f orbidden (in terms of the system stability) to
+first generation planets in the pre-evolved progenitor bina ries. In addition planets could form in metal
+poor environments such as globular clusters and/or in doubl e compact object binaries. Observations of
+planets in such forbidden or unfavorable regions may serve t o uniquely identify their second generation
+character. Finally, we point out a few observed candidate se cond generation planetary systems,
+including PSR B1620-26 (in a globular cluster), Gl 86, HD 274 42 and all of the currently observed
+circumbinary planet candidates. A second generation origi n for these systems could naturally explain
+their unique configurations.
+1.INTRODUCTION
+Currently, over 400 extra-Solar planetary systems have
+been found; most of hem around main sequence stars,
+with a few tens found in wide binary systems. Typi-
+cally, such planets are thought to form in a protoplan-
+etary disk left over following the central star formation
+in a protostellar disk (e.g. Armitage 2007, for a re-
+cent review). Several studies explored the later effects of
+stellar evolution on the survival and dynamics of plan-
+ets around an evolving star (Debes & Sigurdsson 2002;
+Villaver & Livio 2007) or the possible formation of plan-
+ets around neutron stars (NSs; see Phillips & Thorsett
+(1994); Podsiadlowski (1995) for reviews). Others stud-
+ied the formation and stability of planets in binary sys-
+tems (see Haghighipour 2009, for a review). In this study
+we focus on the implications of stellar evolution in bina-
+rieson the formation and growth of planets.
+One of the most likely outcomes of stellar evolution
+in binaries is mass transfer from an evolved donor star
+[which later becomes a compact object; a white dwarf
+(WD), NS or a black hole (BH); here we mostly focus
+on low mass stars which evolve to become WDs] to its
+binary companion. If the binary separation is not too
+large, this process could result in the formation of an ac-
+cretion disk containing a non-negligible amount of mass
+around the companion. Here we suggest that such a disk
+could resemble in many ways a protoplanetary disk, and
+could therefore produce a second generation of planets
+and debris disks around old stars. In addition, the re-
+newed supply of material to a pre-existing (’first gener-
+ation’) planetary system (if such exists after surviving
+hperets@cfa.harvard.eduthe post-MS evolution of the host star), is likely to have
+major effects on this system, possibly leading to the re-
+growth/rejuvenation of the planets and planetesimals in
+the system as well as possibly introducing a second epoch
+of planetary migration.
+The later evolution of evolved binaries could, in some
+cases, even lead to a third generation of planet formation
+in the system. When the previously accreting star (the
+lower mass star in the pre-evolved system) goes through
+its stellar evolution phase, it too can expand to become a
+mass donor to its now compact object companion. A new
+disk of material then forms around the compact object
+and planet formation may occur again, this time around
+the compact object (see also Beer et al. 2004 which tries
+to explain the formation the pulsar planet system PSR
+B1620-26). One could even suggest further next genera-
+tions of planets in multiple systems (triples, quadruples
+etc.), in which the post-MS evolution in each of the stel-
+lar components could provide new material for a another
+generation of planet formation. Such further generation
+of planets, however, is less likely, requiring much more
+fine tuned conditions than those existing more robustly
+in binary systems.
+In the following we discuss the role of binary stellar
+evolution in the formation of a second and third gener-
+ation of planets in evolving binary systems (a schematic
+overview of this scenario is given in Fig. 1). We begin by
+discussing the conditions for the formation of a second
+generation protoplanetary disk and its properties (§2).
+We then explore the role of such disks in the formation
+of new planets (§3), their effects on pre-exiting planetary
+systems (§4), and on the formation of planets around
+compact objects (§5) . We then review the observational2 Perets, H. B.
+

+
+I
+II
+III
+	
+	
+ 
+
+
+
+
+
+
+

+

+


+




+I I II III
+

+

+Fig. 1.— Second (and third) generation planet formation. The
+various stages of second generation planet formation are sh own
+schematically. (a) The initial configuration: a binary MS sy stem,
+possibly having a first generation (I) circumstellar planet around
+the lower mass on an allowed (stable) orbit. (b) The higher ma ss
+stars evolves to the AGB phase, and sheds material which is ac -
+creted on the secondary, and forms a protoplanetary disk. Th e bi-
+nary orbit expands, and the allowed stability region expand s with
+it. The existing first generation planet may or may not surviv e
+this stage (see text). (c) Second generation debris disk and planets
+are formed, in regions previously forbidden for planet form ation (in
+the pre-evolved system; panel a). (d) The secondary evolves off the
+MS, and sheds material to its now WD companion. A protoplane-
+tary disk forms. The binary orbit and the planetary allowed r egion
+(around the WD) further expand. Second generation planets m ay
+or may not survive this stage (see text). (e) Third generatio n debris
+disk and planets are formed around the WD, in regions previou sly
+forbidden for planet formation (in the pre-evolved system; panel
+a).
+expectations for such second generation planetary sys-
+tems as suggested from our discussion (§6). Finally we
+suggest several planetary systems as being candidate sec-
+ond generation planetary systems (§7) and then conclude
+(§8).
+2.SECOND GENERATION PROTOPLANETARY DISKS
+The first stage in planet formation would be setting the
+initial conditions of the protoplanetary disk in which the
+planets could form. We therefore first try to understand
+whether appropriate disks could be produced following
+a mass transfer epoch during the stellar evolution of a
+binary. Studies of planet formation in binary stars sug-
+gest that the binary separation should be large, in order
+for planets to be formed and produce a planetary sys-
+tem around one of the binary stellar components (see
+Haghighipour 2009, for a review). This is also consistent
+with the observational picture in which the smallest sep-arations for a planets hosting binaries are of the order of
+~20 AU (Eggenberger et al. 2004). We therefore mostly
+focus on relatively wide wind accreting binaries in §2.1.
+Nevertheless, mass transfer in close binaries could pro-
+duce circumbinary disks of material, possibly serving as
+an environment for formation of circumbinary planets,
+we briefly discuss this latter more complicated possibil-
+ity in §2.2. We then shortly also discuss the composition
+of second generation disks.
+2.1.Circumstellar disks in wide binaries
+In order to understand the conditions for the formation
+of circumstellar disks from wind accreted material, we
+follow Soker & Rappaport (2000). For a disk to form
+around a star (or compact object), one requires that that
+ja> j2, wherejais the specific angular momentum of the
+accreted material, and j2= (GM2R2)1/2is the specific
+angular momentum of a particle in a Keplerian orbit at
+the equator of the accreting star of radius R2. For typical
+values for a MS accretor and a mass-losing terminal AGB
+star they find the following condition for the formation
+of a disk
+1<ja
+j2≃7.2/parenleftBigη
+0.2/parenrightBig/parenleftbiggM1+M2
+2.5M⊙/parenrightbigg/parenleftbiggM2
+M⊙/parenrightbigg3/2
+×/parenleftbiggR2
+R⊙/parenrightbigg−1/2/parenleftBiga
+10AU/parenrightBig−3/2/parenleftBigvr
+15kms−1/parenrightBig−4
+(1)
+whereM1andM2are the masses of the donor and accret-
+ing stars, respectively. R2is the radius of the accreting
+star, and ais the semi-major axis of the binary. vris the
+relative velocity of the wind and the accretor, ηis the pa-
+rameter indicating the reduction in the specific angular
+momentum of the accreted gas caused by the increase in
+the cross-section for accretion from the low-density side
+(whereη∼0.1andη∼0.3for isothermal and adiabatic
+flows, respectively; Livio et al. 1986), and where they ap-
+proximate vrto be a constant equal to 15km s−1. From
+this condition, it turns out that a disk around a main
+sequence star with R2∼R⊙, could be formed up to or-
+bital separations of a∼37AU; for a WD accretor an
+orbital separation of even a∼60AU is still possible.
+Note, however, the strong dependence on the wind ve-
+locity, which could produce a few times wider or shorter
+accreting systems for the range of 5−20km s−1pos-
+sible in these winds (e.g. Kenyon 1986; Habing 1996).
+Since the peak of the typical binary separation distri-
+bution is at this separation range (Duquennoy & Mayor
+1991), we conclude that a non negligible fraction of all bi-
+naries should evolve to form accretion disks during their
+evolution.
+AGB stars can lose a large fraction of their mass during
+to their companion. Given a simple estimate, a fraction
+of(Ra/2a)2of the mass, where Rais the radius of the
+Bondi-Hoyle accretion cylinder (i.e. gas having impact
+parameter b < R a= 2GM2/v2
+r), is transferred to com-
+panion accretion disk. The total mass transferred from
+the AGB could therefore range between ∼0.1−20per-
+cents of the total mass lost from the AGB star (for wind
+velocities of 5−15kms−1and separation between 10−40
+AU; with the smallest separation and largest mass corre-Second generation planets 3
+sponding to the largest fraction and vice verse). The to-
+tal mass going through the accretion disk could therefore
+range between ∼10−3−1M⊙(for the range of param-
+eters and donor stars masses of 1−7M⊙). Accordingly
+the accretion rate on to the star could have a wide range,
+with rates as high as ∼10−4M⊙yr−1and as low as 10−7
+yr−1(See also Winters et al. 2000, for detailed discussion
+of mass loss rates). Red giants before the AGB stage
+lose mass at slower rates, with ˙M∼10−10−10−8M⊙
+yr−1(Kenyon 1986). Such a range of accretion rates at
+the different stellar evolutionary stages is comparable to
+the range expected and observed in regular (’first gen-
+eration’) protoplanetary disks at different stages of their
+evolution (e.g. Alexander 2008, for a review).
+2.2.Circumbinary disks around close binaries
+Formation of circumbinary disks in wide binaries is
+of less interest for the the formation of second genera-
+tion planets. Dynamically stable configuration of disks
+and/or planets would require them to have very wide
+separations (a few times the binary orbits; Haghighipour
+2009). Disks around wide binaries would have even wider
+configurations, at which regions planet formation is less
+likely to occur (Dodson-Robinson et al. 2009).
+The evolution of close binaries could be quite compli-
+cated. Close binaries could evolve through a common en-
+velope stage, which is currently poorly understood. Such
+binaries, and even wider binaries which do not go through
+a common envelope stage, but tidally interact, are likely
+to inspiral and form shorter period binaries during the
+binary post-MS evolution (see 3.1). Formation of cir-
+cumbinary disks from the material lost from the evolving
+component of the binary, have hardly been discussed in
+the literature, and suggested models are highly uncertain
+(Akashi & Soker 2008, ; also Soker, private communica-
+tion, 2010)). Nevertheless, the higher mass of the bi-
+nary relative to the evolving AGB star, always provides
+a deeper gravitational potential than that of the AGB
+star. In such a configuration material from the slow AGB
+wind which escapes the AGB star potential would still
+be bound to the binary, and therefore fall back and ac-
+crete onto the binary. Indeed, many circumbinary disks
+are observed around evolved binary systems (see section
+7).
+Another possibility exists in which a third companion
+in a hierarchical triple (with masses M1=m1+m2,for
+the close binary and M2for the distant third companion,
+and semi-major axis a1anda2for the inner and outer
+system, respectively) evolves and shed its material on
+the binary. This possibility is more similar to the case
+of the formation of a circumstellar disk in a binary, in
+which case Eq. 1 above could be applied. In this case
+the radius of the accreting star is replaced by the binary
+separation of the close binary, i.e. R2→a1and the
+mass of the accretor is taken to be to be that of the close
+binary system etc.
+2.3.Disks composition
+AGB stars are thought to have a major contri-
+bution to the chemical enrichment of the galaxy
+(van den Hoek & Groenewegen 1997; Tosi 2007). Such
+stars could chemically pollute their surrounding and cre-
+ate an environment with higher metallicity in which latergenerations of stars form as higher metallicity stars (see
+e.g. Tosi 2007, for a recent review). In addition, large
+amounts of dust are formed and later on ejected from the
+atmospheres of evolved stars and their winds (Gail et al.
+2009; Gail 2009, and references therein). Given the high
+metallicity and dust abundances expected from, and ob-
+served in the ejecta of evolved stars, the accretion disk
+formed from such material is expected to be metal and
+dust rich. The composition of such disks may therefore
+serve as ideal environment for planet formation, as re-
+flected in the correlation between the metallicity of stars
+and the frequency of exo-planetary systems around them
+(e.g. Fischer & Valenti 2005).
+3.SECOND GENERATION PLANETS
+Studies of first generation planet formation suggest
+that the accretion rates in regular protoplanetary disks,
+even in close binaries (Jang-Condell et al. 2008), could be
+very similar to those expected in accretion disks formed
+following mass transfer. The composition of such disks
+may be even more favorable for planet formation due to
+their increased metallicity. Observationally, first and se c-
+ond generation disks (formed from the protostellar disks
+of newly formed stars, and from mass transfer in bina-
+ries, respectively) seem to be very similar in appearance,
+raising the possibility of planet formation in these disks,
+as already mentioned in several studies (Jura & Turner
+1998; Zuckerman et al. 2008; Melis et al. 2009). The life-
+time of AGB stars is of the order of up to a few Myrs (e.g.
+Marigo & Girardi 2007), comparable to that of regular,
+first generation, protoplanetary disks. In view of the dis-
+cussion above, it is quite plausible that new, second gen-
+eration planets, could form in the appropriate timescales
+in second generation disks around relatively old stars, in
+much the same way as planets form in the protoplan-
+etary disk of young stars. In the following we discuss
+the implications and evolution of such second generation
+planetary systems.
+Although very similar, the environments in which sec-
+ond generation planets form has several differences from
+the protoplanetary environment of first generation plan-
+ets. As mentioned above the composition of the disk
+material is likely to be more metal rich than typical pro-
+toplanetary disk, and these planets should all form and
+be observed in WD binaries (or other evolved binaries or
+compact objects such as NSs or BHs ), with the youngest
+possibly already existing in post-AGB binaries. Both the
+binarity and disk composition properties of second gener-
+ation planets may not be unique, and could be similar to
+those in which first generation planets form and evolve.
+Indeed many planets are found in binary systems or have
+metal rich host stars. Nevertheless, second generation
+planets form in much older systems than first genera-
+tion ones and have a different source of material. Such
+differences could have important effects on the formation
+and evolution of second generation planets which are also
+likely to be reflected in their observational signatures, as
+we now discuss.
+Second generation planets should be exclusively found
+in binaries with compact objects (most likely WDs), in-
+cluding binaries in which both components are compact.
+A basic expectation would be that the fraction of planet
+hosting WD binary systems should differ from that in MS
+binary systems with similar dynamical properties. More-4 Perets, H. B.
+over, double compact object binaries (WD-WD, WD-NS
+etc.) may show a larger frequency of planets than single
+compact objects1.
+Correlation between planetary companions and their
+host star metallicity could show a difference between
+WD binary systems and MS binary systems. This may
+be a weak signature, given that the host star itself
+may accrete metal rich material from the companion.
+Nevertheless, the latter case could be an advantage for
+targeting second planet searches, through looking for
+them around chemically peculiar stars in WD systems
+(e.g. Jeffries & Smalley 1996; Jorissen 1999; Bond et al.
+2003). The high metallicity of second generation disks
+originate from the evolved star that produced them and
+need not be related to the metallicity of their larger scale
+environments. For this reason, second generation planets
+can form even in metal poor environments such as glob-
+ular clusters, or more generally around metal poor stars.
+This possibility could be tested by searching for WD
+companions to planet hosting, but metal poor, stars (e.g.
+Santos et al. 2007) . Reversing the argument, one can di-
+rect planet searches around metal poor stars (which typ-
+ically do not find planets; Sozzetti et al. 2009) to look for
+them in binary systems composed of a metal poor star
+and a WD. Similarly, planet hosting stars in the metal
+poor environments of globular clusters are likely to be
+members of binary WD systems. Note, however, that
+given their formation in binaries, second generation plan-
+ets are not expected to exist in the globular cluster cores,
+where dynamical interactions may destroy even relatively
+close binaries. Such second generation planetary systems
+should still be able to exist at the outskirts of globular
+clusters. Interestingly, the only planet found in a globu-
+lar cluster (PSR B1620−26; Backer et al. 1993) is a cir-
+cumbinary planet around a WD-NS binary, found in the
+outskirts of a globular cluster, as might be expected from
+our discussion (see section 7 for further discussion of this
+system).
+The age of second generation planets, if could be mea-
+sured, should be inconsistent with and much younger
+than their stellar host age (such implied inconsistencies
+could be revealed in some cases, e.g. WASP-18 plane-
+tary system; Hellier et al. 2009). Their typical composi-
+tion is likely to show irregularities and be more peculiar
+and metal rich, relative to that of typical first generation
+planets.
+The second generation protoplanetary disk in which
+second generation planets form does not have to be
+aligned with the original protostellar disk of the star and
+its rotation direction, but is more likely to be aligned
+with the binary orbit (but not necessarily; the accretion
+disk from the wind of a wide binary may form in a more
+arbitrary inclination).
+Studies on the formation and stability of planets in
+circumbinary orbits (see Haghighipour 2009 for a re-
+1Planets around an evolving donor star could be engulfed by
+the star and be destroyed. We are not interested in comparing the
+planet frequency around these donor stars, but rather the pl anet
+frequency around the accretor. Specifically, the frequency of plan-
+ets around the MS star in MS-WD systems should be compared
+with the planet frequency around either MS stars in MS-MS sys -
+tems; similarly, the planet frequency around the WDs in WD-W D
+systems should be compared with planet frequency around the WD
+in WD-MS systems.view) suggest that they can form and survive in such sys-
+tems. Recently, several circumbinary planets candidates
+have been found (Qian et al. 2009, 2010a,b; Lee et al.
+2009), possibly confirming such theoretical expectations.
+Circumbinary second generation disks such as discussed
+above could therefore serve, in principle, as nurseries for
+the formation and evolution of second generation planets.
+The evolution of second generation planets in circum-
+stellar and circumbinary disk could be very different. In
+the following we highlight some important differences be-
+tween the types of orbits possible for second generation
+planets, and the way in which these could serve as a
+smoking gun signature for the effects and evolution of
+second generation planets and/or disks.
+3.1.Orbital phase space of second generation planets
+3.1.1. Circumstellar planets
+Stable circumstellar planetary systems in binaries
+could be limited to a close separation to their host star,
+since planets may be prohibited from forming at wider
+orbits or become unstable at such orbits which are more
+susceptible to the perturbations from the stellar binary
+companion (Haghighipour 2009, and references therein).
+During the post-MS evolution of a wide binary, its orbit
+typically widens (due to mass loss), therefore allowing for
+planets to form and survive at wider circumstellar orbits.
+This larger orbital phase space, however, is open only to
+second generation planets formed after the post-MS evo-
+lution of the binary (see also Thébault et al. 2009, for
+a somewhat related discussion on a dynamical evolution
+of such forbidden zone due to perturbations in a stellar
+cluster).
+Let us illustrate this by a realistic example. Con-
+sider a MS binary with stellar components of 1.6 and
+0.8M⊙and a separation of ab= 12 AU on a an or-
+bit of 0.3 eccentricity. The secondary protoplanetary
+disk in this binary would be truncated at about 2-
+2.5 AU in such a system (Artymowicz & Lubow 1994),
+and a planetary orbit would become unstable at simi-
+lar separations (Holman & Wiegert 1999). Specifically,
+Holman & Wiegert find
+ac/ab= (0.464±0.006)+(−0.38±0.01)µ
++(−0.631±0.034)eb+(0.586±0.061)µeb
++(0.15±0.041)e2
+b+(−0.198±0.047)µe2
+b,(2)
+whereacis critical semi major axis at which the orbit is
+still stable, m=M2/(M1+M2),abandebare the semi
+major axis and eccentricity of the binary, and M1 and
+M2 are the masses of the primary and secondary stars,
+respectively. Giant planets are not likely to form in such
+a system, since such planets are thought to form far from
+their host star (although they may migrate later on to
+much smaller separations, e.g. forming hot Jupiters),
+where icy material is available for the initial growth of
+their planetary embryos (Pollack et al. 1996). In fact, it
+is not clear if any type of planet could form under such
+hostile conditions in which strong disk heating is induced
+by perturbations from the stellar companion. The small-
+est binary separations in which planets could form are
+thought to be ~20 AU for giant planets (Haghighipour
+2009, and references therein), although some simulationsSecond generation planets 5
+suggest that terrestrial planets may form even in closer
+systems, near the host star (up to ~0.2 qb, whereqbis the
+stellar binary pericenter distance; Quintana et al. 2007).
+Indeed the smallest separation observed for planet host-
+ing MS binaries is of ~20 AU. We can conclude that
+first generation circumstellar giant planets are not likely
+to form in the binary system considered here. In our
+example the pericenter distance of the initial system is
+q=ab(1−e) = 8.4AU, i.e. planets, and especially gas
+giants, are not likely to form in such system.
+Back to our example, at a later epoch, the more mas-
+sive stellar component in this system evolves off the main
+sequence to end its life as a WD of ~0.65 M⊙(see e.g.
+Lagrange et al. 2006). Due to the adiabatic mass loss
+from the system it may evolve to a larger final separa-
+tion, given by (e.g. Lagrange et al. 2006, and references
+therein)
+af=mi
+mfai, (3)
+wheremf(= 0.8+0.65 = 1.45M⊙)is the final mass of
+the system after its evolution, and mi(= 1.6 + 0.8 =
+2.4M⊙)andai(=ab= 12 AU ) are the initial mass
+and separation of the system, respectively (where the ec-
+centricity does not change in this case). We now find
+af= 2.4/1.45ai= 19.8AU [more detailed calculations,
+using the binary evolution code by Hurley et al. (2002)
+give a similar scenario]. At such a separation even cir-
+cumstellar giant planets could now form in the system
+(Kley & Nelson 2008), i.e. second generation planets
+could form in in this binary either at a few AU around
+the star, or closer if they migrated after their forma-
+tion). Therefore, any circumstellar giant planet observed
+around the MS star in this system would imply that such
+a planet must be a second generation planet, since it
+could not have formed as a first generation planet in
+the pre-evolved system. Thus, such cases could serve
+as a smoking gun signature and a unique tracer for the
+existence and identification of second generation plan-
+ets. In fact, the example chose here is not arbitrary.
+The final configuration of the system in this example is
+very similar to that of the planetary system observed
+in the WD binary system Gl 86. A giant planet of
+4MJuphave been found at ∼0.1AU from its MS host
+star, which has a WD companion, at a separation of
+∼18AU (Mugrauer & Neuhäuser 2005; Lagrange et al.
+2006). The configuration of this system possibly indi-
+cates that this system is indeed a Bone fide second gen-
+eration planetary system (see also section 7) .
+In Fig. 2, we use more detailed stellar evolution cal-
+culations (using the binary stellar evolution code BSE;
+Hurley et al. 2002) to show a range of final vs. ini-
+tial semi-major axis of circular binary systems with ini-
+tial masses M1= 0.8M⊙andM2= 1.6M⊙in which
+the higher mass star evolves to become a WD (of mass
+∼0.6M⊙). Also shown in this figure is the critical semi-
+major axis acin both the initial (pre-evolved MS-MS
+binary) and the final (evolved; MS-WD binary) configu-
+ration. As can be seen, the pre and post evolution critical
+semi-major axis differ significantly, presenting a region of
+orbital phase space open only to second generation plan-
+ets but forbidden for first generation ones.
+3.1.2. Circumbinary planetsInitial Semi−major Axis (AU)Final Semi−major Axis (AU)
+123456789101112131415161718190246810121416182022242628303234
+First Generation
+Forbidden Zone
+Fig. 2.— The configuration of evolved binary systems vs.
+their pre-evolved configuration. The pre-evolved binary is a MS-
+MS binary on a circular orbit with component masses M1=
+0.8M⊙andM2= 1.6M⊙, an the evolved binary is a MS-
+WD binary on a circular orbit with component masses M1=
+0.8M⊙andM2= 0.6M⊙. Solid thick line shows the final semi-
+major axis of binary systems vs. their initial, pre-evolved , semi-
+major axis. Solid thin line shows the critical semi-major ax isac
+below which circumstellar planetary orbits around M2are stable
+in the pre-evolved MS-MS system. Dashed line shows the criti -
+cal semi-major axis acbelow which circumstellar planetary orbits
+aroundM2are stable in the evolved MS-WD system. The region
+between these lines is forbidden for first generation planet ary or-
+bits in the pre-evolved system, but allowed for second gener ation
+orbits in the evolved system.
+In a sense, the requirements from a stable circumbi-
+nary planetary system present a mirror image of those
+needed for a circumstellar system in a binary. The
+orbital separation of circumbinary planets should be
+typically a few (>2-4) times the binary separation
+(Holman & Wiegert 1999; Moriwaki & Nakagawa 2004;
+Pierens & Nelson 2007, 2008; Haghighipour 2009), in or-
+der not to be perturbed by the binary orbit. Specifically,
+Holman & Wiegert find
+ac/ab= (1.6±0.04)+(5.1±0.05)eb+(4.12±0.09)µ
++(−2.22±0.11)eb+(−4.27±0.17)ebµ
++(−5.09±0.11)µ2+(4.61±0.36)e2
+bµ2,+(−4.27±0.17)ebµ
++(−5.09±0.11)µ2+(4.61±0.36)e2
+bµ2(4)
+whereacis critical semi major axis at which the orbit
+is still stable, m=M2/(M1+M2),abandebare the
+semi major axis and eccentricity of the binary, and M1
+and M2 are the masses of the primary and secondary
+stars, respectively. Circumbinary planets are therefore
+not expected to be observed very close to their host stars.
+As with circumstellar second generation planets, the or-
+bital phase space available for second generation plan-
+ets differs from that of pre-existing first generation stars.
+The post-MS orbit of a relatively pre-evolved close binary
+(e.g. separation of 1-2 AU) could be shrunk through it’s
+evolution due to angular momentum loss in a common
+envelope or circumbinary disk phase [e.g. Ritter (2008)
+in the context of forming cataclysmic variables]. Second
+generation planets could therefore form much closer to
+such evolved binaries than any pre-existing first genera-
+tion planets. Therefore, similar to the circumstellar case6 Perets, H. B.
+Initial Semi−major Axis (AU)Final Semi−major Axis (AU)
+00.5 11.5 22.5 33.5012345678
+First Generation
+Forbidden Zone
+Fig. 3.— The configuration of evolved binary systems vs.
+their pre-evolved configuration. The pre-evolved binary is a MS-
+MS binary on a circular orbit with component masses M1=
+0.8M⊙andM2= 1.6M⊙, an the evolved binary is a MS-
+WD binary on a circular orbit with component masses M1=
+0.8M⊙andM2= 0.6M⊙. Solid thick line shows the final semi-
+major axis of binary systems vs. their initial, pre-evolved , semi-
+major axis. Solid thin line shows the critical semi-major ax isac
+above which circumbinary planetary orbits are stable in the pre-
+evolved MS-MS system. Dashed line shows the critical semi-m ajor
+axisacabove which circumstellar planetary orbits are stable in th e
+evolved MS-WD system. The region between these lines is forb id-
+den for first generation planetary orbits in the pre-evolved system,
+but allowed for second generation orbits in the evolved syst em.
+(but now looking on inward migration of the binary),
+circumbinary planetary systems observed to have rela-
+tively close orbits around their evolved host binary would
+point to their second generation origin. As described be-
+low (section 7), such candidate planetary systems have
+been recently observed. We note, however, that a caveat
+for such an argument is that pre-exiting first generation
+planets, if they survive the post-MS evolution, could mi-
+grate inward (together with the binary, or independently)
+in the second generation disk. Nevertheless, even the lat-
+ter case would reflect the existence and importance of a
+second generation protoplanetary disk and its interaction
+with first generation planets, as briefly discussed in the
+next section. Whether such inward migration is a likely
+consequence presents an interesting question for further
+studies.
+Fig. 3, is similar to Fig. 2, but now showing the
+critical semi-major axis acfor a circumbinary orbit in
+both the initial (pre-evolved MS-MS binary) and the fi-
+nal (evolved; MS-WD binary) configuration. Again, the
+pre and post evolution critical semi-major axis differ sig-
+nificantly, presenting a region of orbital phase space open
+only to second generation planets but forbidden for first
+generation ones.
+We note in passing, that even non-evolved (low
+mass) MS close binaries with orbits of a few days
+were most likely evolved from wider binaries in triple
+systems, that shrunk to their current orbit due
+to the processes of Kozai cycles and tidal friction
+(Mazeh & Shaham 1979; Eggleton & Kisseleva-Eggleton
+2006; Tokovinin et al. 2006; Fabrycky & Tremaine 2007).
+Formation of circumbinary close planetary systems even
+around short period MS binaries is therefore likely tobe dynamically excluded, or would require a compli-
+cated and fine tuned dynamical history. Similarly, ex-
+istence of close planetary systems around blue straggler
+stars, which were likely formed through similar processes
+(Perets & Fabrycky 2009) is also unlikely.
+4.FIRST GENERATION PLANETS IN SECOND
+GENERATION DISKS
+At the formation epoch of the second generation disk,
+the accreting star may already host a planetary system.
+The first generation planetesimals and/or planets could
+therefore serve as “seeds” for a much more rapid growth
+of second generation planets. In this case second gen-
+eration planet formation might behave quite differently
+than regular planet formation, suggesting a stage where
+large planetesimals and planets co-exist with and are
+embedded in a large amount of gaseous material. The
+first generation planets and planetesimals could now go
+through an epoch of regrowth (rejuvenation) through ac-
+cretion of the replenished material. The first generation
+planets could now grow to become much more massive
+than typical planets. Moreover, such planetary seeds
+could induce a more efficient planet formation and pro-
+duce more massive planets on higher eccentricity orbits
+(Armitage & Hansen 1999). A possible observational sig-
+nature of these planets could therefore be their relative
+larger masses, and possibly higher eccentricity. Whether
+such regrowth could even lead to a core accretion forma-
+tion of exceptionally massive planets, effectively becom-
+ing brown dwarfs (in this case such brown dwarfs could
+now form much more often in the so called brown dwarf
+desert regime), is an interesting possibility. Note that in
+newly formed young binaries late infall of material from
+the protostellar disk of one star to its companion (the
+protoplanetary disk of one star may form earlier than
+that of its companion), may drive similar processes. Such
+a possibility could explain the existence of more massive
+close in planets in planet hosting binary systems. If pro-
+toplanetary disks preferentially form earlier around the
+more (less) massive component of the binary, more mas-
+sive close in planets should preferentially form around
+these components, providing an observational signature
+for this process. In such regrowth scenarios, more mas-
+sive planets are likely to form in binaries with closer sep-
+aration, which could typically form more massive disks.
+The replenished gas may also drive a new epoch of
+planetary migration. Together with the change in plane-
+tary masses and/or the formation of additional new plan-
+ets in the system, the previously steady configuration of
+the planetary systems may now dynamically evolve into
+a new configuration. Such late dynamical reconfigura-
+tion could lead to ejection of planets and planetesimals
+from the system as well as to inward migration and/or
+possibly infall into their host stars. Ejected planets coul d
+become free floating planets. Alternatively these could
+still be bound to the binary system, in which case fur-
+ther interactions with either the original host star or the
+companion would produce a more complicated dynamical
+history. Several possibilities are opened in this case; the
+dynamical evolution could lead to a later ejection from
+the system, infall into one of the stars or a recapture into
+a more stable circumstellar or circumbinary orbit. All
+of these possibilities, including the possibility of plane t
+exchange between the binary stars, are not restricted,Second generation planets 7
+however, to second generation exo-planets, and could be
+possible in first generation planetary systems hosted by
+binaries (e.g. Marzari et al. 2005). Infall of planets into
+their host stars may spin them up; stars in WD binary
+systems may therefore show higher rotational velocities.
+This possible signature, however, could also be induced
+by the material accreted from the second generation disk,
+rather than a planet infall.
+Given the possible misalignment between the second
+generation disk and the pre-existing planetary system,
+unique disk-planets configurations could be produced.
+Planets misaligned with the gaseous disk may both af-
+fect the disk through warping, and could be affected by
+it (Marzari & Nelson 2009). Small relative inclinations
+are likely to be damped through the planets interactions
+with the disk, re-aligning the planets (Cresswell et al.
+2007). Observations of two co-existing misaligned plan-
+etary systems or even counter rotating planets in the
+same system could, in principle, serve as a spectacular
+example of second generation planetary formation with
+pre-existing first generation planets. However, dynami-
+cal interactions in multiplanetary systems could possibly
+produce somewhat similar effects, although likely not the
+counter rotating configurations (Chatterjee et al. 2008;
+Jurić & Tremaine 2008).
+Mass loss from the donor star in the binary can result
+in the dynamical evolution of relatively wide orbits into
+even wider orbits, due to the reduced gravitational poten-
+tial of the system (see section 3.1). Alternatively, close
+binaries could become even closer due to gas drag and
+common envelope evolution. The change in the dynami-
+cal configuration of the binary could therefore have major
+impact on the evolution of pre-existing planets, not dis-
+cussed above (where it was assumed that the binary sys-
+tem have already evolved to a stable configuration, and
+only planetary formation and evolution processes take
+place). Pre-existing stable planetary systems could be
+destabilized at this stage due to the migration of the host-
+ing stars, involving direct scattering by the stars and/or
+going through resonant interactions, similar, but with a
+greater amplitude, to asteroids and Kuiper belt objects
+in the Solar system, thought to scatter resonantly due to
+planet migration. These complex issues are beyond the
+scope of this paper, and will be explored elsewhere.
+Finally, we note in passing that in binaries of small
+separations (a few AU) even very low mass companions
+such as brown dwarfs or even massive planets (rather
+than a stellar binary companion) could develop accre-
+tion disks, under favorable conditions, such as low wind
+velocities. Such disks would have very low masses; nev-
+ertheless, they might suffice for the formation of a new
+generation of moons or rings around these planets.
+5.SECOND/THIRD GENERATION PLANETS FORMATION
+AROUND COMPACT OBJECTS AND EVOLVED STARS
+Few planetary systems and debris disks have been
+found to exist around compact objects, such as the pulsar
+planets(Wolszczan & Frail 1992), debris disks around the
+neutron star (Wang et al. 2006), planets around evolved
+stars (Silvotti et al. 2007; Lee et al. 2009) and the pos-
+sible planets observed around WDs (Qian et al. 2009,
+2010a,b). These systems are generally thought to ei-
+ther form in a fall back disk of material following the
+post-MS evolution of the progenitor star (e.g. fall backmaterial from a supernova; Lin et al. 1991, bus see also
+Livio et al. 1992); or form around the MS star and sur-
+vive the post-MS evolution stage of the star.
+Mass transfer in binaries poses an alternative and ro-
+bust way of providing disk material to compact objects.
+Second (or possibly third) generation planets could there-
+fore naturally form around many compact objects, and
+should typically exist around compact objects in doubly
+compact binary systems. Such systems may show differ-
+ences relative to planets formed around MS stars due to
+the quite different radiation from the hosting (compact)
+star, as well as the difference of their magnetic fields.
+Some studies explored planet formation around neutron
+stars (see Sigurdsson et al. 2008, for a recent review)
+and WDs (Livio et al. 1992), but these issues and the is-
+sue of binarity in these systems are yet to be studied in
+more depth. In this context Tavani & Brookshaw (1992);
+Banit et al. (1993) made some pioneering efforts relating
+to planet formation in NS binaries, although focusing on
+pulsar planets and the evaporation of stellar companion
+to the NS as a source for protoplanetary disk material.
+Much more directly related are the studies done later
+by Livio et al. (1992) and Beer et al. (2004) in order to
+explain the pulsar planet PSR B1620−26 (see section 7).
+This alternative scenario for the formation of plane-
+tary systems around compact objects suggests a differ-
+ent interpretation for planets in compact object system;
+observations of such planetary systems might not reflect
+the survival of planets through post-MS evolution of their
+host star, but rather their formation at even a later stage
+from the ashes of a companion star. Such systems (e.g.
+Schmidt et al. 2005; Maxted et al. 2006; Silvotti et al.
+2007; Geier et al. 2009) should therefore be targets for
+searches of compact object companions. Note, however,
+that formation of sdB stars may require the existence
+of a close companion (see Heber 2009, and references
+therein), such as these observed planets. This, in turn,
+would suggest that a planet have already been in place
+prior to the sdB star formation in these systems.
+6.OBSERVATIONAL EXPECTATIONS
+From the above discussion we can try to summarize
+and formulate basic expectations regarding second gen-
+erations planetary systems and their host stars. These
+could serve both to identify the second generation origin
+of observed planetary systems and to provide guidelines
+and directions for targeted searches for second generation
+planets.
+1. Second generation planets should exist in evolved
+binary systems, most frequently in WD-MS or WD-
+WD binaries. Such systems should therefore be the
+prime targets for second generation planetary sys-
+tems searches. Since WD binaries are relatively
+frequent(Holberg 2009), second generation planets
+could be a frequent phenomena (with the caution
+that our current understanding of planet forma-
+tion, especially in such systems, is still very lim-
+ited). Evolved binary systems may show a differ-
+ent frequency of hosting planetary systems (around
+the MS star in a binary MS-WD systems) than
+MS binaries with similar orbital properties. Planet
+hosting compact objects are more likely to be part
+of double compact object binaries rather than be8 Perets, H. B.
+singles.
+2. The host binaries separation are likely to be be-
+tween 20 AU to 200 AU for binaries hosting cir-
+cumstellar planets, and up to a few AU for binaries
+hosting circumbinary planets.
+3. Planets in post-MS binaries could reside in orbital
+phase space regions inaccessible to pre-existing first
+generation planets in such systems (section 3.1).
+Observations of planets in such orbits could serve as
+a smoking-gun signature of their second generation
+origin.
+4. Second generation planets could form even around
+metal poor stars and/or in metal poor environ-
+ments such as globular clusters; planetary systems
+in such environment are likely to be part of evolved
+binary systems.
+5. Planets showing age inconsistency with (i.e. being
+younger than) their parent host star could be possi-
+ble second generation planets candidates. In such
+systems one may therefore search for a compact
+object companion to the host star.
+6. Stars in WD binaries showing evidence for mass
+accretion from a companion star (e.g. chemically
+peculiar stars such as Barium and CH stars) could
+be more prone to have second generation planetary
+companions (for adequate binary separations)2
+7. Second generation planets could be more massive
+than regular exo-planets. This may be suggestive
+that old planet hosting stars with extremely mas-
+sive planets (or brown dwarf companions found in
+the brown dwarf dessert) are more likely to be sec-
+ond generation planetary systems. Again this could
+point to the existence of a compact object (most
+likely WD) companion at the appropriate separa-
+tions.
+8. Planets around compact objects could be second
+generation planets from mass transfer accretion.
+These systems are therefore more likely to have a
+compact object companion. Also, polluted WDs,
+thought to reflect accretion of asteroids, might be
+more likely to have WD companions.
+9. The spin-orbit alignment between second genera-
+tion planets and their host stars is likely to differ
+from that of first generation planets, and second
+generation planets might be more likely to show
+high relative inclinations between different planets
+in multiple planetary system.
+10. Statistical correlation between planetary compan-
+ions and their host star metallicity could show a
+difference between WD binary systems and MS bi-
+nary systems (but this may be a weak signature,
+2Note that material from second generation disks and/or plan -
+ets which was processed in the AGB star would not pollute the
+star with6Li as would first generation planetary material [e.g.
+see Gonzalez (2006) for a review]. It is not clear to us, howev er,
+whether such differences could be detected, and what are the r e-
+quired statistics.given that the host star itself accretes metal rich
+material). Chemically peculiar stars in WD bina-
+ries may serve as good second generation planetary
+hosts candidates.
+11. The MS stars in WD-MS binaries may also show
+higher rotational velocities, due to planetary infall,
+although such phenomena could also be induced by
+the re-accretion of matter from the second genera-
+tion disk. In either case, stars with high rotational
+velocities in WD binaries could be better candidate
+hosts of second generation planets.
+7.CANDIDATE SECOND GENERATION PLANETARY
+SYSTEMS
+Observationally, many post-AGB binary stars show
+evidence for having disks surrounding them either in
+circumstellar or circumbinary disks (Van Winckel
+2004; de Ruyter et al. 2006; Hinkle et al. 2007;
+van Winckel et al. 2009). Such disks have many
+similarities with protoplanetary disks as observed in
+T-Tauri stars, suggesting that new generations of planets
+could form there(Zuckerman et al. 2008; Melis et al.
+2009).
+As discussed above, there are several observational
+expectations for second generation planetary systems.
+Currently, several observed planetary systems and plan-
+etary candidates may be consistent with such expec-
+tations, and may therefore be considered as candidate
+second generation planets. These systems are found
+in evolved binary systems, and include both circum-
+stellar planets such as in GL 86 (Queloz et al. 2000;
+Lagrange et al. 2006; Mugrauer & Neuhäuser 2005) and
+HD 27442 (Mugrauer et al. 2007) as well as several
+circumbinary planets and candidates: PSR B1620−26
+(Backer et al. 1993), HW Vir (Lee et al. 2009), NN
+Ser (Qian et al. 2009) and DP Leo (Qian et al. 2010b).
+These systems are consistent with being second gener-
+ation planetary systems, as their host binary separa-
+tion is the region where second generation disks could
+be formed.
+Interestingly, according to Lagrange et al. (2006) the
+planetary orbit of GL 86 could have been in the forbid-
+den region for first generation planets (i.e. where they
+could have not formed and survived) for the most plau-
+sible parameters for the WD progenitor in this system,
+which are consistent with its cooling age (Lagrange et al.
+2006; Desidera & Barbieri 2007). Although this discrep-
+ancy could possibly be circumvented by taking different
+age and mass estimates for the progenitor, under some
+assumptions (Desidera & Barbieri 2007), a second gener-
+ation origin for this planet presents an alternative simple
+and natural explanation. Besides its consistency with
+being a second generation planet, Gl 86 may therefore
+show the smoking gun signature of a second generation
+planetary system.
+Similarly the circumbinary planets candidates in HW
+Vir, NN Ser and DP Leo are found to be in regions
+likely to be inaccessible or less favorable for first gen-
+eration planets in the progenitor pre-evolved binary sys-
+tems. These planets orbit their evolved binary host at
+separations of only a few AU, where the pre-stellar evolu-
+tion orbits of these binaries might have as wide as a few
+AU (e.g. Ritter 2008), i.e. these planets could not haveSecond generation planets 9
+formed in-situ as first generation planets in their current
+position. However, these orbits are accessible for in-situ
+formation of second generation planets are the already
+evolved and much shorter period binary observed today.
+The globular cluster pulsar planet PSR B1620−26
+(Backer et al. 1993) is another highly interesting second
+generation candidate. It is the only planet found in a
+globular cluster (metal poor environment) and it is a
+circumbinary planet around a WD-NS binary, found in
+the cluster outskirts of a globular cluster, as might be
+expected from our discussion above of second genera-
+tion planets in globular clusters (see section 3). Sev-
+eral studies suggested scenarios for the formation of
+this system and its unique and puzzling configuration
+(Sigurdsson & Thorsett 2005; Sigurdsson et al. 2008).
+These usually require highly fine tuned and complex dy-
+namical and evolutionary history. A second generation
+origin could give an alternative and more robust expla-
+nation [as also suggested by Livio & Pringle (2003) and
+studied by Beer et al. (2004)]. The high relative inclina-
+tion observed for this planets suggested that even in the
+second generation scenario some an encounter with an-
+other star is required to explain this system (Beer et al.
+2004). However, recent studies show that high in-
+clinations could also occur from planet-planet scatter-
+ing in a multiple planet system (Chatterjee et al. 2008;
+Jurić & Tremaine 2008). Note that the existence of an-
+other planet in this system is highly unlikely for the favor-
+able formation scenarios discussed by Sigurdsson et al.
+(2008), but could be a natural consequence for the sec-
+ond generation scenario. This could motivate further
+observational study of this system. Similarly, findings
+of additional globular cluster planets preferably in WD
+binary systems would strongly support the second gener-
+ation scenario origin of globular cluster planets (see also
+Sigurdsson & Thorsett 2005).
+Other possibly weaker candidate second generation
+planets would be those planets found around metal poor
+stars (e.g. Santos et al. 2007). Finding a WD compan-
+ions for such stars, however, would give a strong support
+for the second generation scenario and the identification
+of these systems as second generation planets.
+We conclude that several second generation candidate
+planets have already been observed, and show proper-
+ties consistent with the scenario discussed in this pa-
+per. Moreover, the current properties of these systems
+may pose problems for our current (poor) understand-
+ing of first generation planets formation and evolution in
+evolved binary systems, which could be naturally solved
+in the context of second generation planets. Observed
+WD-MS binary systems (e.g. Rebassa-Mansergas et al.
+2009) should therefore serve as potentially promising tar-
+gets for exo-planets searches.
+8.CONCLUSIONS
+In this paper we suggested and discussed the formation
+of second generation planetary systems in mass transfer-
+ring binaries. We found that this is a possible route for
+planet formation in old evolved systems and around com-
+pact objects in double compact object binaries. We pre-
+sented possible implications for this process and the plan-
+etary systems it could produce, and detailed the possible
+observational signatures of second generation planetary
+systems. We also pointed out a few currently observedplanetary systems with properties suggestive of a second
+generation origin.
+The possibility of second generation planets may open
+new horizons and suggest new approaches and targets for
+planetary searches and research. It suggests that stel-
+lar evolution processes and stellar deaths may serve as
+the cradle for the birth and/or rejuvenation of a new
+generation of planets, rather than just being the death
+throes or hostile hosts for pre-existing planets. In par-
+ticular such processes could provide new routes for the
+formation of habitable planets, opening the possibilities
+for their existence and discovery even in (the previously
+thought) less likely places to find them. The environ-
+ments of old stars and more so of compact objects could
+be very different from that of young stars. Such dif-
+ferent environments can strongly affect the formation of
+second generation planets and possibly introduce unique
+processes involved in their formation and evolution. The
+discovery and study of second generation planets could
+therefore shed new light on our understanding of both
+planet formation and binary evolution, and drive further
+research on the wide range of novel processes opened up
+by this possibility.
+I am most greatful to Scott Kenyon for many iluminat-
+ing discussions. I would also like to thank Scott Kenyon
+and Noam Soker for helpful comments on an earlier ver-
+sion of this manuscript.10 Perets, H. B.
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+arXiv:1001.0582v1  [astro-ph.CO]  4 Jan 2010ASTE Simultaneous HCN(4–3) and HCO+(4–3)
+Observations of the Two Luminous Infrared Galaxies
+NGC 4418 and Arp 220
+Masatoshi Imanishi∗
+National Astronomical Observatory, 2-21-1, Osawa, Mitaka , Tokyo 181-8588, Japan
+masa.imanishi@nao.ac.jp
+Kouichiro Nakanishi†
+National Astronomical Observatory, 2-21-1, Osawa, Mitaka , Tokyo 181-8588, Japan
+MasakoYamada
+Institute of Astronomy and Astrophysics Academia Sinica, P. O. Box 23-141, Taipei 10617, Taiwan,
+R.O.C.
+YoichiTamura
+Nobeyama Radio Observatory, Minamimaki, Minamisaku, Naga no, 384-1305, Japan
+and
+KotaroKohno
+Institute of Astronomy, University of Tokyo, 2-21-1, Osawa, Mitaka, Tokyo, 181-0015, Japan
+(Received ; accepted )
+Abstract
+We report the results of HCN( J=4–3) and HCO+(J=4–3) observations of two lu-
+minous infrared galaxies (LIRGs), NGC 4418 and Arp 220, made using the Atacama
+Submillimeter Telescope Experiment (ASTE). The ASTE wide-band cor relator pro-
+vided simultaneous observations of HCN(4–3) and HCO+(4–3) lines, and a precise
+determination of their flux ratios. Both galaxies showed high HCN(4– 3) to HCO+(4–
+3) flux ratios of >2, possibly due to AGN-related phenomena. The J= 4–3 to J=
+1–0transitionfluxratiosforHCN(HCO+)aresimilar tothoseexpected forfullyther-
+malized (sub-thermally excited) gas in both sources, in spite of HCN’s higher critical
+density. If we assume collisional excitation and neglect an infrared r adiative pumping
+process, our non-LTE analysis suggests that HCN traces gas with significantly higher
+density than HCO+. In Arp 220, we separated the double-peaked HCN(4–3) emis-
+sion into the eastern and western nuclei, based on velocity informat ion. We confirmed
+that the eastern nucleus showed a higher HCN(4–3) to HCN(1–0) fl ux ratio, and thus
+contained a larger amount of highly excited molecular gas than the we stern nucleus.
+Key words: galaxies: active — galaxies: nuclei — galaxies: ISM — radio lines:
+1galaxies — galaxies: individual (NGC 4418 and Arp 220)
+1. Introduction
+Luminous infrared galaxies (LIRGs) radiate the bulk of their large lum inosities ( L>∼
+1011L⊙) as infrared dust emission (Sanders & Mirabel 1996). Their large inf rared luminosities
+indicatethat(1)powerfulenergysourcesarepresent, hiddenb ehinddust; (2)energeticradiation
+from the hidden energy sources is absorbed by the surrounding du st; and (3) the heated dust
+grains re-emit this energy as infrared thermal radiation. To under stand this LIRG population,
+it is essential to clarify the hidden energy sources, whether starb ursts (i.e., nuclear fusion inside
+stars) are dominant, or active galactic nuclei (AGNs; active mass a ccretion onto a central
+compact supermassive black hole [SMBH] with >106M⊙) are also energetically important.
+Unlike AGNs surrounded by a torus-shaped (toroidally-shaped) du sty medium, which
+are classified optically as Seyferts (Veilleux & Osterbrock 1987), LI RGs have a large amount of
+concentrated molecular gas and dust in their nuclei (Sanders & Mira bel 1996). For this reason,
+the putative compact AGNs can easily be buried(i.e., obscured in virtually all directions),
+making the optical detection of AGN signatures very difficult. Howev er, investigating the
+energetic importance of such optically elusive buriedAGNs is crucial to understand the true
+nature of the LIRG population.
+A starburst (nuclear fusion) and a buried AGN (mass accretion ont o a SMBH) have very
+different energy-generation mechanisms. Specifically, while UV is the predominant energetic
+radiation in a starburst, an AGN emits strong X-ray emission in additio n to UV. The radiative
+energy generation efficiency of a nuclear fusion reaction in a normal starburst is only ∼0.5% of
+Mc2(M is the mass of material used in the nuclear fusion reaction). Thus , the emission surface
+brightness of a starburst region is modest and has both observat ional (Werner et al. 1976; Soifer
+et al. 2000) and theoretical (Thompson et al. 2005) upper limits ( ∼1013L⊙kpc−2). An AGN,
+however, achieves high radiative energy generation efficiency (6–4 2% of Mc2, where M is the
+mass of accreting material; Bardeen 1970; Thorne 1974), produc ing avery highemission surface
+brightness with >1013L⊙kpc−2(Soifer et al. 2000). These differences between a starburst and
+an AGN could create differences in the properties of the molecular ga s and dust around the
+energy sources, allowing us to distinguish the hidden energy source s if, for example, different
+emission-line flux ratios are observed from the surrounding materia l.
+Molecular gas emission lines in the sub-millimeter and millimeter wavelength ranges are
+∗Department of Astronomy, School of Science, Graduate Univers ity for Advanced Studies (SOKENDAI),
+Mitaka, Tokyo 181-8588
+†Department of Astronomy, School of Science, Graduate Univers ity for Advanced Studies (SOKENDAI),
+Mitaka, Tokyo 181-8588
+2a potentially effective tool for probing this distinction, because the effects of dust extinction
+are so small in these wavelength ranges that signatures of AGNs de eply buried in gas and
+dust are detectable. It was found observationally that HCN( J=1–0) emission (rest frequency
+νrest= 88.632 GHz or rest wavelength λrest= 3.385 mm) is systematically stronger, relative
+to HCO+(J=1–0) (νrest= 89.189 GHz or λrest= 3.364 mm), in AGN-dominated galaxy nuclei
+than in starburst galaxies (Kohno 2005; Krips et al. 2008). The tre nd of a strong HCN(1–0) to
+HCO+(1–0)emission ratio was further confirmed inluminous buried AGN can didates (Imanishi
+et al. 2004; Imanishi et al. 2006b; Imanishi & Nakanishi 2006; Iman ishi et al. 2007b; Imanishi
+et al. 2009). These results demonstrated that this HCN(1–0)/HC O+(1–0) method functions as
+a diagnostic tool to distinguish the hidden energy sources of dusty galaxies. Extension to more
+distant, faint LIRGs is expected to play an important role for the co mprehensive understanding
+of the LIRG population in the forthcoming Atacama Large Millimeter/s ubmillimeter Array
+(ALMA) era. However, during the first operation phase of ALMA, b and 3 (84–116 GHz or
+2.59–3.57mm) is the longest available wavelength coverage, so that t his HCN(1–0)/HCO+(1–0)
+method canbeapplied onlytoLIRGsat z≤0.05, hampering further systematic investigation of
+LIRGs. Energy diagnostic methods using other transition lines and/ or other molecular species
+must be exploited.
+HCN(J=4–3) (νrest= 354.505 GHz) and HCO+(4–3) (νrest= 356.734 GHz) lines exist in
+Earth’s 350 GHz atmospheric window, and can be one of the main freq uency ranges exploited
+with ALMA. Because the frequency coverage of ALMA band 7 is 275– 373 GHz, HCN(4–3) and
+HCO+(4–3) lines can be studied in LIRGs at z >0.05. Thus it is important to investigate
+whether AGNs show high HCN(4–3) to HCO+(4–3) flux ratios, similar to the high HCN(1–0)
+to HCO+(1–0) flux ratios.
+In this paper, we present the results of simultaneous HCN(4–3) an d HCO+(4–3) obser-
+vations of two nearby, well-studied LIRGs, NGC 4418 and Arp 220, b oth of which show high
+HCN(1–0) to HCO+(1–0) flux ratios (Imanishi et al. 2004; Imanishi et al. 2007b). Thr oughout
+this paper, we adopt H 0= 75 km s−1Mpc−1, ΩM= 0.3, and Ω Λ= 0.7, to be consistent with
+our previously published papers.
+2. Targets
+NGC 4418, at z= 0.007, is a nearby LIRG with an infrared luminosity of L IR∼9
+×1010L⊙(Table 1). Although it shows no clear Seyfert signature in the optica l spectrum
+(Armus, Heckman, & Miley 1989; Lehnert & Heckman 1995), it is cons idered one of the closest
+and strongest buried AGN candidates, based on the AGN-like infrar ed spectral shape (Roche
+et al. 1991; Dudley & Wynn-Williams 1997; Spoon et al. 2001) and the hig h HCN(1–0) to
+HCO+(1–0) flux ratio in the millimeter wavelength range (Imanishi et al. 200 4).
+Arp 220 (z = 0.018) is one of the best-studied nearby ultraluminous in frared galaxies
+(LIR>1012L⊙; Sanders et al. 1988). It has two nuclei, the eastern (Arp 220 E) a nd western
+3(Arp 220 W) nuclei, separated by ∼1′′(Scoville et al. 2000; Soifer et al. 2000). The combined
+optical spectrum of both nuclei is classified as a LINER (i.e., non-Sey fert) (Veilleux et al. 1999).
+Starburst activity hasbeendetected, but wasenergetically insuffi cient toaccount quantitatively
+for the observed infrared luminosity (Imanishi et al. 2006a; Armus et al. 2007; Imanishi et al.
+2007a), indicating an energy source deeply buried in Arp 220’s nuclea r core (Spoon et al.
+2004; Gonzalez-Alfonso et al. 2004). The energy source in Arp 220 W has a very high emission
+surface brightness, for which a buried AGN is a plausible (if not definit e) explanation (Downes
+& Eckart 2007; Sakamoto et al. 2008).
+3. Observations and Data Reduction
+The HCN(4–3) andHCO+(4–3)observations of NGC 4418and Arp220 were madeusing
+the Wideband and High dispersion Spectrometer system with FFX cor relator (WHSF) (Okuda
+& Iguchi 2008; Iguchi & Okuda 2008) on the Atacama Submillimeter T elescope Experiment
+(ASTE), a 10-m antenna located at Pampa La Bola in the Atacama Des ert of Chile, at an
+altitude of 4800 m (Ezawa et al. 2004; Ezawa et al. 2008). Because t he WHSF has 4-GHz
+frequency coverage in each side band, both the HCN(4–3) ( νrest= 354.505 GHz) and HCO+(4–
+3) (νrest= 356.734 GHz) lines were simultaneously observable. Figure 1 shows t he raw ASTE
+WHSF spectra of NGC 4418 and Arp 220.
+The observations were made in 2008 May, remotely from an ASTE ope ration room
+in the Mitaka campus of the National Astronomical Observatory of Japan, using the network
+observation system N-COSMOS3 developed by Kamazaki et al. (200 5). Table 2 summarizes the
+detailed observation log. The weather conditions were excellent. Ty pical system temperatures
+were 180–300 K in the single side band. We employed the position-switc hing mode, so that the
+target position and off-sky position, 4 arcmin away from the target , were switched after each
+10 s exposure. Telescope pointing was checked every 2 h. The tota l net on-source integration
+times were 200 min and 155 min for NGC 4418 and Arp 220, respectively . Flux calibration
+was made every observation day. By changing the frequency sett ing of the WHSF, we observed
+a strong emission line in W28 (IRC10216) at the frequency of the red -shifted HCN(4–3) and
+HCO+(4–3) lines of NGC 4418 (Arp 220), to estimate the possible sensitivit y variation along
+the WHSF bandpass. This variation was found to be <20%. The half-power beam width of
+the ASTE 10-m dish was 22′′atν∼345 GHz. At this frequency, the main beam efficiency is
+0.6, and 1 K in antenna temperature corresponds to 78.5 Jy.
+Data reduction was carried out with NEWSTAR, a package developed at Nobeyama
+Radio Observatory. A fraction of poor-quality data was flagged. T o estimate the strength of
+HCN(4–3) and HCO+(4–3) emission, we subtracted baselines from the obtained raw spe ctra.
+We first divided the spectra into two parts at the center, the lower -frequency HCN(4–3) part
+and higher-frequency HCO+(4–3) part. The baseline was estimated independently in each part,
+based on the data points free from emission lines.
+4For NGC 4418, ten data sets with 20 min net on-source exposure tim e for each were
+obtained. Thebehaviorofthebaselinewasextremelygood,andthe HCN(4–3)emissionlinewas
+easily recognizable in individual data sets. First-order linear baseline s were adopted both for
+the HCN(4–3) and HCO+(4–3) emission lines, and the baseline-subtracted individual spectr a
+were summed to obtain final spectra.
+For Arp 220, final data consist of seven data sets with 20 min net on -source exposure
+and one data set with 15 min net on-source exposure. For the HCN( 4–3) line, first-order linear
+baselines were adopted in individual data sets, and the baseline-sub tracted individual spectra
+were summed. The HCN(4–3) emission line was detected in the final sp ectrum (see Appendix
+for more detail). Although we tried several baseline fits, the HCN(4 –3) emission line was
+always visible, and its profile was virtually unchanged in the final spect rum, under reasonable
+baseline choices. For theHCO+(4–3)line, when we applied a first-order linear baseline, winding
+patterns remained in the final spectrum, and HCO+(4–3) line was not clearly detected. We
+thus attempted a second-order baseline, but the HCO+(4–3) emission line remained undetected
+in the final spectrum.
+4. Results
+Figure 2 presents the final spectra around the HCN(4–3) and HCO+(4–3) emission lines,
+after baseline subtraction. In both NGC 4418 and Arp 220, HCN(4– 3) emission was clearly
+detected, with peak antenna temperatures of 7–15 mK, but HCO+(4–3) emission was not. For
+NGC 4418, HCN(4–3) and HCO+(4–3) emission line spectra are published here for the first
+time.
+ForNGC4418, wefitthedetectedHCN(4–3)emissionwithasingleGau ssianprofile. For
+Arp220HCN(4–3)emission, adouble-peaked profilewasclearly seen . Thus wefit it withatwo-
+component Gaussian profile. Table 3 summarizes the fitting results. The estimated HCN(4–3)
+fluxes were 220 Jy km s−1and 410 Jy km s−1for NGC 4418 and Arp 220, respectively, with
+the uncertainties of ∼20%. For the undetected HCO+(4–3) emission, we adopted the lowest
+plausible continuum level in the spectra, and estimated conservativ e upper limits, by counting
+the possible flux excess above the adopted continuum levels. Table 3 summarizes these upper
+limits for the HCO+(4–3) emission.
+For Arp 220, HCN(4–3) measurements have been made previously b y two groups
+(Wiedner et al. 2002; Greve et al. 2009). Our ASTE measurement of the HCN(4–3) flux
+(410±82 Jy km s−1) fell between the previously reported values of 260 Jy km s−1(Wiedner
+et al. 2002) and 587 ±118 Jy km s−1(Greve et al. 2009). The velocity difference in the two
+components in the double-peaked HCN(4–3) profile was ∼300 km s−1(Table 3), similar to
+that of HCN(1–0) emission (Imanishi et al. 2007b), further stren gthening the detection of the
+HCN(4–3) emission line in Arp 220. Based on the argument of Imanishi et al. (2007b), we
+ascribe the lower-velocity (blue-shifted) and higher-velocity (red -shifted) components to the
+5Arp 220 W and E nuclei, respectively.
+The HCN(4–3) to HCO+(4–3) flux ratios1were>2.7 and>3.0 for NGC 4418 and Arp
+220, respectively, where the ratio for Arp 220 used the combined e mission from both nuclei.
+The non-detection of HCO+(4–3) emission lines hampered the spectral separation into Arp 220
+E andW andrendered independent estimates ofthe line flux ratio for these nuclear components
+difficult. HCN(4–3)/HCO+(4–3) flux ratios significantly higher than unity were confirmed in
+both galaxies.
+We emphasize that our simultaneous HCN(4–3) and HCO+(4–3) observations us-
+ing ASTE WHSF are crucial for obtaining reliable HCN(4–3) to HCO+(4–3) flux ratios.
+Submillimeter and millimeter observations still present non-negligible un certainties in abso-
+lute flux calibrations. In fact, even for measurements obtained in t he last several years, the
+HCN(4–3) and HCN(1–0) fluxes of Arp 220 differed by a factor of 2 a t maximum as observed
+by different groups with different telescopes under different weath er conditions and using dif-
+ferent calibration methods (Wiedner et al. 2002; Imanishi et al. 200 7b; Gracia-Carpio et al.
+2008; Greve et al. 2009). This level of absolute flux calibration unce rtainty can be fatal to
+an investigation of the HCN(4–3) to HCO+(4–3) flux ratios because we are discussing the
+difference of the ratios by about a factor of 2. By simultaneously ob serving HCN(4–3) and
+HCO+(4–3) lines, the possible absolute flux calibration uncertainty does n ot propagate to the
+HCN(4–3)/HCO+(4–3) flux ratios, which are then dominated only by statistical noise s and
+fitting errors.
+5. Discussion
+5.1. HCN(4–3) to HCO+(4–3) flux ratios
+In this subsection, we explore HCN(4–3) and HCO+(4–3) emission from NGC 4418 and
+Arp 220, in relation to HCN(1–0) and HCO+(1–0). For this purpose, we used our ASTE
+WHSF and Nobeyama Millimeter Array (NMA) data (Imanishi et al. 2004 ; Imanishi et al.
+2007b), because (1) simultaneous HCN and HCO+observations by both telescopes make the
+comparison of HCN to HCO+flux ratios reliable, and (2) HCN(1–0) and HCO+(1–0) absolute
+fluxes in the NMA data have been confirmed to agree with measureme nts from other single-dish
+telescopes (Imanishi et al. 2004; Imanishi et al. 2007b). The NMA m easurements of HCN(1–0)
+and HCO+(1–0) fluxes are summarized in Table 4.
+For NGC 4418, we obtained an HCN(4–3) to HCO+(4–3) flux ratio >2.7, confirming
+the high HCN to HCO+flux ratio previously found for the J= 1–0 transition line ( ∼1.8;
+Imanishi et al. 2004). The HCN(4–3) to HCN(1–0) flux ratio was est imated to be 20. This is
+1By definition, flux is proportional to brightness-temperature ×ν2, whereνis frequency. Because HCN(4–3)
+(νrest= 354.505 GHz) and HCO+(4–3) (νrest= 356.734 GHz) have very similar frequencies, their flux ratios
+are virtually identical to their brightness-temperature ratios. In this paper, we use a flux ratio.
+6comparable to the ν2scaling (∼16) expected for fully thermalized optically thick gas (Riechers
+et al. 2006). The HCO+(4–3) to HCO+(1–0) flux ratio was <15.
+For Arp 220, the HCN(4–3) and HCO+(4–3) flux ratio was estimated to be >3.0. The
+J= 4–3 to J= 1–0 flux ratio for Arp 220 total emission was 12 and <8 for HCN and HCO+,
+respectively. When we separated the emission from individual nuclei, the HCN(4–3) to HCN(1–
+0) flux ratios were 21 and 9 for Arp 220 E and W nuclei, respectively. T he ratio is higher in
+Arp 220 E than Arp 220 W, further supporting the previous argume nt that a larger fraction of
+highly excited molecular gas is present in Arp 220 E (Greve et al. 2009) .
+The critical density of HCN is 2.3 ×105cm−3forJ= 1–0 and 8.5 ×106cm−3forJ=
+4–3. That of HCO+is 3.4×104cm−3forJ= 1–0, and 1.8 ×106cm−3forJ= 4–3 (Table 3
+of Greve et al. 2009). Because the critical density of HCO+is a factor of ∼5–7 smaller than
+HCN for both the J= 1–0 and J= 4–3 transitions, HCO+can be collisionally excited more
+easily than HCN in single-phase molecular gas. Thus, the stronger HC N emission in NGC 4418
+and Arp 220 requires (1) an HCN abundance that is significantly highe r than HCO+, and/or
+(2) the presence of some physical mechanism that enhances HCN e mission.
+Regarding the first HCN abundance enhancement scenario, Yamad a et al. (2007) con-
+cluded, based on three-dimensional line transfer simulations, that HCN abundance must be an
+order of magnitude higher than HCO+to account for the observed high HCN to HCO+flux
+ratio (∼2). While the typical HCN to HCO+abundance ratio in our Galactic molecular cloud
+is∼1 (Blake et al. 1987; Pratap et al. 1997; Dickens et al. 2000), HCN ov erabundance, relative
+to HCO+, by an order of magnitude, is predicted in some parameter range fo r dense molecular
+gas illuminated by an X-ray emitting energy source (Meijerink & Spaan s 2005). Since an AGN
+is a much stronger X-ray emitter than starburt activity, this HCN o verabundance scenario
+is a possibility for the observed high HCN(4–3)/HCO+(4–3) flux ratios in the buried AGN
+candidates, NGC 4418 and Arp 220.
+Strong HCN emission in luminous buried AGN candidates may be explained by the
+infrared radiative pumping scenario (Aalto et al. 1995; Garcia-Burillo et al. 2006; Guelin et
+al. 2007; Weiss et al. 2007; Aalto et al. 2007a). In an AGN, the emissio n surface brightness is
+much higher than in a starburst, so that the surrounding dust is he ated to a high temperature
+(several hundredK)andproducesstrongmid-infrared10–20 µmcontinuumemission. TheHCN
+molecule has a line at infrared 14.0 µm. Molecular gas around an AGN can be vibrationally
+excited by absorbing these infrared 14.0 µm photons, and through the subsequent cascade
+process, can enhance HCN rotational lines in the sub-millimeter and m illimeter range.
+Guelin et al. (2007) suggested that this infrared pumping scenario m ay work also for
+HCO+and HNC, because they have lines at 12.1 µm (HCO+) and 21.7 µm (HNC). However,
+for this infrared pumping scenario to affect particular molecules, inf rared photons at the wave-
+lengths of these molecular lines must be absorbed. Namely, absorpt ion features must be de-
+tected in infrared spectra at 14.0 µm (HCN), 12.1 µm (HCO+), and 21.7 µm (HNC). While
+7the infrared HCN 14.0 µm absorption feature is clearly detected in highly obscured ultralumi-
+nous infrared galaxies, including NGC 4418 and Arp 220 (Lahuis et al. 2 007; Veilleux et al.
+2009), absorption features at HCO+12.1µm and HNC 21.7 µm are not (Veilleux et al. 2009).
+Therefore, we see observational signatures that this infrared r adiative pumping scenario could
+be at work for HCN, but not for HCO+or HNC. This may explain the strong HCN emission
+found in luminous AGN candidates.
+Since boththeHCNoverabundance andinfraredradiativepumping s cenarioscouldwork
+more effectively in an AGN than starburst activity, the enhanced HC N(4–3) emission may
+originate in putative buried AGNs in NGC 4418 and Arp 220. Three-dime nsional modelings
+which realistically incorporate actual physical parameters of molec ular gas around an AGN are
+needed for more detailed quantitative comparison with observation s.
+5.2. Molecular gas properties
+ForNGC4418, theHCN(4–3)toHCN(1–0)fluxratio(20)washigher thantheHCO+(4–
+3) to HCO+(1–0) flux ratio ( <15). For Arp 220 as well, the former ratio (12) was higher than
+the latter ratio ( <8). Thus, our ASTE observations suggest that J= 4–3 transition lines are
+excited more efficiently in HCN than HCO+, in spite of the higher critical density of HCN(4–
+3) compared to HCO+(4–3) (a factor of ∼5 [§5.1]). As mentioned in §5.1, in a single-phase
+molecular gas, molecular species with lower critical density are more e asily collisionally excited.
+Thus, HCO+should be excited more than HCN. Some mechanisms are needed to ex plain the
+strong HCN(4–3) emission. Examples include a multiple phase molecular gas (Greve et al.
+2009), and the infrared radiative pumping scenario. The second sc enario will be investigated
+elsewhere, using a three-dimensional model (Yamada et al. 2009, in prep). Here, we explore
+the first possibility. Namely, we consider only collisional excitation and neglect the infrared
+radiative pumping process.
+We used RADEX (van der Tak et al. 2007) to investigate molecular gas properties that
+could account for our ASTE and NMA results. The RADEX software a dopts the non-LTE
+analysis of molecular line spectra, solving radiative transfer in simple g eometry gas. We varied
+the H2number density and kinetic temperature from 103–108cm−3and 10–200 K, respectively.
+Auniformspherewithalinewidthof200kms−1wasassumed, basedontheHCN(4–3)emission
+measurements (Table 3). X-ray observations suggested that bo th NGC 4418 and Arp 220 have
+highly-obscured, Compton thick (N H>1024cm−2) AGNs (Maiolino et al. 2003; Iwasawa et
+al. 2005). Taking HCN and HCO+abundances relative to H 2of 10−9−10−8(Blake et al.
+1987; Dickens et al. 2000), column densities with 1016cm−2were adopted for both HCN and
+HCO+. The T = 2.73 K background emission was also added for the calculation . Figure 3
+shows the HCN(4–3) to HCN(1–0) and HCO+(4–3) to HCO+(1–0) flux ratios in Jy km s−1,
+as a function of molecular gas number density and kinetic temperatu re. In the 10–200 K
+temperature range, the molecular gas number density was estimat ed to be n H>106cm−3to fit
+8the observed HCN(4–3) to HCN(1–0) flux ratios for NGC 4418 ( ∼20) and Arp 220 ( ∼12).
+Now, the HCO+(4–3) to HCO+(1–0) flux ratios were <15 and<8 for NGC 4418 and
+Arp 220, respectively. For T >25 K (>101.4K), molecular gas with a number density n H
+<106cm−3reproduced the observed HCO+flux ratios. Thus, assuming collisional excitation,
+we confirmed that in both NGC 4418 and Arp 220, HCN emission traces gas with a number
+density significantly higher than that traced by HCO+ 2, as previously argued for Arp 220
+based on the measurements of HCN(4–3), HCO+(4–3), and other transition lines (Greve et al.
+2009). Even given HCN and HCO+column densities of 1015cm−2and a line width of 300 km
+s−1, our main conclusion is unchanged.
+Three-dimensional hydrodynamic simulations of galactic disks have s hown that the frac-
+tion of molecular gas with n H>106cm−3is miniscule (Wada & Norman 2007). Yet, the strong
+HCN(4–3) emission line detected in NGC 4418 and Arp 220 suggests th e presence of a large
+amount ofsuch high-density (n H>106cm−3) molecular gas, if theHCN(4–3) lineiscollisionally
+excited. It is likely that the high nuclear concentration of molecular g as in LIRGs increases the
+fraction of high-density molecular gas.
+6. Summary
+We presented the results of simultaneous HCN(4–3) and HCO+(4–3) observations of two
+luminous infrared galaxies, NGC 4418 and Arp 220, using ASTE WHSF. O ur main conclusions
+are as follows:
+1. Strong HCN(4–3) emission was detected in both NGC 4418 and Arp 220, but HCO+(4–3)
+emission provided only upper limits.
+2. In both sources, we found HCN(4–3) to HCO+(4–3) flux ratios of >2, even higher than
+the HCN(1–0) to HCO+(1–0) emission line flux ratios. The high HCN/HCO+flux ratios
+could be explained by HCN overabundance and/or infrared radiative pumping scenarios,
+both of which may work more effectively in an AGN than starburst act ivity.
+3. HCN(4–3) to HCN(1–0) flux ratioswere higher than HCO+(4–3) to HCO+(1–0) flux ratios
+in both NGC 4418 and Arp 220, suggesting that higher J-transitions were excited more
+efficiently in HCN than HCO+. If we neglect an infrared radiative pumping mechanism,
+and consider only collisional excitation, these results are incompatib le with a one-zone
+molecular gas model, because the critical density of HCN is higher tha n HCO+at these
+transitions. A multi-phase molecular gas in which HCN selectively probe s higher-density
+molecular gas than HCO+is required. We applied RADEX models, and found that in
+both NGC 4418 and Arp 220, gas with a number density n H>106cm−3and nH<106
+2For NGC 4418, the small HCN(3–2) to HCO+(3–2) flux ratio (less than unity) reported by Aalto et al.
+(2007b) is not compatible with our model. The HCN(3–2) flux itself is als o significantly lower than that
+expected for thermalized, optically thick gas. We have no clear inter pretation, as long as only collisional
+excitation is considered.
+9cm−3reproduced the observed HCN and HCO+J= 4–3 to 1–0 flux ratios, respectively,
+in the 25–200 K temperature range.
+4. For Arp 220, HCN(4–3) to HCN(1–0) flux ratio is higher in Arp 220 E than in Arp 220
+W, suggesting that Arp 220 E contains a larger amount of highly excit ed molecular gas.
+We thank T. Okuda for the support of our ASTE WHSF observing run , and the anony-
+mous referee for his/her very useful comments. The ASTE proje ct is driven by Nobeyama
+Radio Observatory (NRO), a branch of National Astronomical Obs ervatory of Japan (NAOJ),
+in collaboration with University of Chile, and Japanese institutes includ ing University of
+Tokyo, Nagoya University, Osaka Prefecture University, Ibarak i University, and Hokkaido
+University. Observations with ASTE were in part carried out remote ly from Japan by us-
+ing NTT’s GEMnet2 and its partner R&E (Research and Education) ne tworks, which are
+based on AccessNova collaboration of University of Chile, NTT Labor atories, and NAOJ.
+M.I. is supported by Grants-in-Aid for Scientific Research (197401 09). This study utilized
+the NASA/IPAC Extragalactic Database (NED) operated by the Je t Propulsion Laboratory,
+California Institute of Technology, under contract with the Nation al Aeronautics and Space
+Administration.
+10Appendix 1. Analysis of the Arp 220 HCN(4–3) emission line
+We explain in detail how the HCN(4–3) emission line of Arp 220 becomes d etectable in
+our analysis. We took eight independent data sets. Single-order line ar baselines were employed
+in individual data sets (Figure 4) and eight baseline-subtracted spe ctra were summed to obtain
+a final HCN(4–3) spectrum. The HCN(4–3) emission was clearly dete cted in the final spectrum
+of Arp 220, as seen in Figure 2.
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+13Table 1. Detailed information on the observed LIRGs
+Object Redshift f12f25f60f100logLIR(logLIR/L⊙)
+(Jy) (Jy) (Jy) (Jy) (ergs s−1)
+(1) (2) (3) (4) (5) (6) (7)
+NGC 4418 0.007a0.9 9.3 40.7 32.8 44.5 (10.9)
+Arp 220 0.018b0.48 7.92 103.33 112.40 45.7 (12.1)
+Notes.
+Col.(1): Object name.
+Col.(2): Redshift.
+Cols.(3)–(6): f 12, f25, f60, and f 100areIRAS FSC fluxes at 12, 25, 60, and 100 µm,
+respectively.
+Col.(7): Decimallogarithmoftheinfrared(8 −1000µm)luminosity inergss−1calculated
+as follows: LIR=2.1×1039×D(Mpc)2×(13.48×f12+ 5.16×f25+ 2.58×f60+f100) ergs
+s−1(Sanders & Mirabel 1996). The values in parentheses are the decim al logarithms of the
+infrared luminosities in units of solar luminosities.
+a: 1 arcsec corresponds to 130 pc.
+b: 1 arcsec corresponds to 340 pc.
+Table 2. Observing log of ASTE observations
+Object Observing Date Central Pointing Flux
+(UT) frequency calibrator calibrator
+(1) (2) (3) (4) (5)
+NGC 4418 2008 May 22, 23, 24, 27 353.15 V Hya, IRC10216 W28
+Arp 220 2008 May 24, 27 349.21 RX Boo IRC10216
+Notes.
+Col.(1): Object name.
+Col.(2): Observation date in UT.
+Col.(3): Central frequency of ASTE WHSF used for the observatio ns.
+Col.(4): Pointing calibrator object name.
+Col.(5): Flux calibrator object name.
+14Table 3. Properties of HCN(4–3) and HCO+(4–3) emission lines
+Object Line LSR velocity Line width Flux Flux Luminosity Rat io
+[km s−1] [km s−1] [K km s−1] [Jy km s−1] 107[K km s−1pc2]
+(1) (2) (3) (4) (5) (6) (7) (8)
+NGC 4418 HCN(4–3) 2060 335 4.7 220 4.7 >2.7
+HCO+(4–3) — — <1.7 <80 <1.7
+Arp 220 HCN(4–3) 5265, 5560 220,200 8.7 410 57 >3.0
+3.4 (W) + 5.3 (E) 160 (W) + 250 (E) 22 (W) + 35 (E)
+HCO+(4–3) — — <2.9 <135 <19
+Notes.
+Col.(1): Object name.
+Col.(2): HCN(4–3) or HCO+(4–3) line.
+Col.(3): LSR velocity {vopt≡(ν0
+ν−1)×c}of the HCN(4–3) and HCO+(4–3) in [km
+s−1].
+Col.(4): Line width in FWHM of the HCN(4–3) and HCO+(4–3) emission lines in [km
+s−1].
+Col.(5): FluxoftheHCN(4–3)andHCO+(4–3)emission linesinmainbeamtemperature
+in [K km s−1]. For ASTE, the main beam efficiency is η∼0.6 atν∼345 GHz. For the HCN(4–
+3) line of Arp 220, individual contributions from Arp 220 E and W are als o shown. The flux
+uncertainty is estimated to be <20%, based on the combination of the empirical one of ASTE
+WHSF (10–15%) and systematic one of flux calibrators ( ∼10%) (Wang et al. 1994).
+Col.(6): Fluxofthe HCN(4–3) andHCO+(4–3)emission lines in[Jykm s−1]. ForASTE,
+1 [K] in main beam temperature corresponds to 47 [Jy] at ν∼345 GHz.
+Col.(7): Luminosity of the HCN(4–3) and HCO+(4–3) emission lines in ×107[K km s−1
+pc2].
+Col.(8): HCN(4–3)/HCO+(4–3) flux ratio (flux ∝ν2×brightness-temperature).
+Because both the HCN(4–3) and HCO+(4–3) data are taken simultaneously, the ratio is not
+affected by possible absolute flux calibration uncertainties in the AST E observations (see §4).
+15Table 4. HCN and HCO+data atJ= 4–3 and J= 1–0 transition lines for NGC 4418 and Arp 220
+Object Line Flux Reference
+(1) (2) (3) (4)
+NGC 4418 HCN(4–3) 220 ±44 This work
+HCO+(4–3) <80 This work
+HCN(1–0) 10.3 Imanishi et al. (2004)
+HCO+(1–0) 5.5 Imanishi et al. (2004)
+Arp 220 HCN(4–3) 410 ±82 This work
+250 (E), 160 (W) This work
+260 Wiedner et al. (2002)
+587±118 Greve et al. (2009)
+HCO+(4–3) <135 This work
+106±23 Greve et al. (2009)
+80 Sakamoto et al. (2009)
+HCN(1–0) 34 Imanishi et al. (2007b)
+12 (E), 18 (W) Imanishi et al. (2007b)
+HCO+(1–0) 18 Imanishi et al. (2007b)
+5.5 (E), 6.5 (W) Imanishi et al. (2007b)
+Notes.
+Col.(1): Object name.
+Col.(2): HCN or HCO+line.
+Col.(3): Flux in [Jy km s−1]. For Arp 220, individual fluxes from Arp 220 E and W are
+also shown. For HCN(1–0) and HCO+(1–0) fluxes from Arp 220 E and W, only the spatially
+unresolved core emission is extracted.
+Col.(4): Reference.
+16Fig. 1. ASTE WHSF wide-band (4 GHz) spectra of NGC 4418 and Arp 220, bef ore baseline subtraction.
+The abscissa is observed frequency in [GHz] and the ordinate is ante nna temperature in [K]. The spectrum
+is smoothed to a velocity resolution of 60 km s−1and 50 km s−1for NGC 4418 and Arp 220, respectively.
+The expected frequency of the HCN(4–3) and HCO+(4–3) lines are indicated as arrows. HCN(4–3) and
+HCO+(4–3) emission lines are simultaneously covered.
+17Fig. 2. HCN(4–3) and HCO+(4–3) spectra of NGC 4418 and Arp 220, after baseline subtractio n.
+First-order linear fits were basically adopted as the baselines, exce pt for the Arp 220 HCO+(4–3) line, for
+which a second-order fit was employed. The abscissa is velocity in [km s−1] and the ordinate is antenna
+temperature in [K]. Spectra of NGC 4418 and Arp 220 are smoothed t o velocity resolutions of 60 km s−1
+and 50 km s−1, respectively. Gaussian fits are over-plotted as dashed lines for t he detected HCN(4–3) lines
+in NGC 4418 and Arp 220. For the Arp 220 HCN(4–3) line, we ascribe th e blue (lower velocity) and red
+(higher velocity) components to Arp 220 W (denoted as “W”) and Ar p 220 E (“E”), respectively, based
+on Nobeyama Millimeter Array interferometric HCN(1–0) and HCO+(1–0) data (Imanishi et al. 2007b)
+(see§4). The horizontal solid line indicates the zero flux level.
+18Fig. 3. HCN(4–3) to HCN(1–0) ( Left) and HCO+(4–3) to HCO+(1–0) (Right) flux ratios in [Jy km s−1]
+as a function of gas number density and kinetic temperature, calcu lated with RADEX. The abscissa and
+ordinate are the decimal logarithms of kinetic temperature in [K] and H2gas number density in [cm−3],
+respectively. The HCN and HCO+column density is assumed to be 1016cm−2(see§5.2). A velocity
+width of 200 km s−1was adopted (see §5.2). Contours are 0.1, 1, 5, 12, 20, 50, and 100 for HCN, and 0.1,
+1, 5, 8, 15, 50, and 100 for HCO+. The measured ratios for NGC 4418 and Arp 220 are indicated as thic k
+lines. In the calculated parameter range, the line optical depth is 0– 100 (i.e., reliable range in RADEX
+calculations).
+1920Fig. 4. Adopted linear baseline fits for individual unbinned HCN(4–3) spectr a of Arp 220 (dashed lines),
+and smoothed spectra with velocity resolutions of 50 km s−1(solid lines). Signatures of the HCN(4–3)
+emission lines are seen in most of the individual spectra.
+21
\ No newline at end of file
diff --git a/1001.0583.txt b/1001.0583.txt
new file mode 100644
index 0000000000000000000000000000000000000000..92f37b023895b7f7de77fcd7515c1eb11258716a
--- /dev/null
+++ b/1001.0583.txt
@@ -0,0 +1,680 @@
+Structural Disjoining Potential for Grain Boundary Premelting
+and Grain Coalescence from Molecular-Dynamics Simulations
+Saryu Fensin,1David Olmsted,2Dorel Buta,1Mark Asta,3, 1Alain Karma,2, 4and J. J. Hoyt5
+1Department of Chemical Engineering and Materials Science,
+University of California, Davis, CA
+2Department of Physics, Northeastern University, Boston, MA
+3Department of Materials Science and Engineering,
+University of California, Berkeley, CA
+4Center for Interdisciplinary Research on Complex Systems,
+Northeastern University, Boston, MA
+5Department of Materials Science and Engineering,
+McMaster University, Hamilton, ON, Canada
+Abstract
+We describe a molecular dynamics framework for the direct calculation of the short-ranged struc-
+tural forces underlying grain-boundary premelting and grain-coalescence in solidication. The
+method is applied in a comparative study of (i) a 9 h115i120otwist and (ii) a 9 h110if411g
+symmetric tilt boundary in a classical embedded-atom model of elemental Ni. Although both
+boundaries feature highly disordered structures near the melting point, the nature of the tempera-
+ture dependence of the width of the disordered regions in these boundaries is qualitatively dierent.
+The former boundary displays behavior consistent with a logarithmically diverging premelted layer
+thickness as the melting temperature is approached from below, while the latter displays behavior
+featuring a nite grain-boundary width at the melting point. It is demonstrated that both types
+of behavior can be quantitatively described within a sharp-interface thermodynamic formalism
+involving a width-dependent interfacial free energy, referred to as the disjoining potential. The
+disjoining potential for boundary (i) is calculated to display a monotonic exponential dependence
+on width, while that of boundary (ii) features a weak attractive minimum. The results of this work
+are discussed in relation to recent simulation and theoretical studies of the thermodynamic forces
+underlying grain-boundary premelting.
+1arXiv:1001.0583v1  [cond-mat.mtrl-sci]  4 Jan 2010INTRODUCTION
+At high homologous temperatures the atomic structure of a grain boundary often dis-
+plays pronounced disorder. In some cases this structural disorder can involve the formation
+of nanometer-scale intergranular lms with liquid-like properties below the bulk melting
+point, a phenomenon commonly referred to as grain-boundary premelting. The interfacial
+thermodynamic driving forces underlying grain-boundary premelting are understood to be
+an important factor in
uencing grain coalescence behavior during the late stages of solidi-
+cation (e.g., [1]). Specically, when premelting is thermodynamically favored there exists an
+associated repulsive \disjoining pressure" which hinders the coalescence of two misoriented
+grains at nanometer-scale distances. In such cases a signicant undercooling may be required
+for dendrite arms to merge to form solid \bridges" that are capable of sustaining thermal-
+contraction stresses without grain sliding or rupture. The quantitative characterization of
+grain-boundary disjoining forces is thus an important issue in the context of modeling the
+formation of solidication defects, known as \hot tears," which occur deep within the mushy
+zone during casting or welding [1{4].
+Despite its practical importance in the context of solidication defects, the magnitude and
+spatial extent of grain-boundary disjoining forces in metals and alloys remain incompletely
+understood. As in the case of surface premelting [5], these forces can in principle be probed
+experimentally through measurements of the extent of equilibrium premelting as a function
+of temperature near the bulk melting point. In comparison to surface premelting, however,
+the challenges inherent in characterizing the structure of \buried" internal interfaces at
+high homologous temperatures have signicantly limited the number of direct experimental
+studies of grain-boundary premelting [6{13]. This situation has provided motivation for
+several recent theoretical studies based on conventional phase-eld [14{17] and phase-eld-
+crystal (PFC) [18, 19] methods, which have led to new insights into the rich variety of
+possible premelting behavior that may be exhibited by grain boundaries as a function of
+their bi-crystallography.
+Due to the diculties inherent in validating theoretical models for grain-boundary pre-
+melting directly from experimental measurements, the authors recently proposed an in-
+dependent methodology for calculating grain-boundary disjoining forces from equilibrium
+molecular-dynamics (MD) simulations [20]. The technique represents an extension of the
+2numerous previous MD studies of grain-boundary premelting performed over the past three
+decades [21{34]. The technique for calculating disjoining potentials by MD is based on an
+analysis of the equilibrium distribution of the widths of premelted intergranular lms, which
+are related to the underlying disjoining forces through the 
uctuation-dissipation theorem.
+The purpose of this paper is to describe the technical details surrounding the implementa-
+tion of this method, and to demonstrate its application in the study of two distinct classes
+of grain boundary premelting behavior. Specically, we consider the premelting behavior of
+a high and intermediate energy boundary in elemental Ni where the widths of the premelted
+layers continuously increase or remain nite, respectively, as the melting point is approached
+from below. It is shown that the former behavior is consistent with an interfacial free energy
+(	) that decreases exponentially with increasing width ( w) of the premelted layer, while the
+latter behavior can be quantitatively modeled with a dependence of 	 on wthat features
+a weak attractive minimum at nanometer-scale widths. This minimum is accompanied by
+a grain boundary structure with alternating solid bridges and disordered regions that bears
+strong similarities to the type of structure found to be associated with such a minimum in
+a recent phase-eld-crystal study [19].
+The remainder of this paper is outlined as follows. In the next section we review a
+continuum, sharp-interface formalism for the thermodynamic properties underlying grain-
+boundary premelting and coalescence, based on the denition of the so-called \disjoining
+potential," the (negative) derivative of which is the disjoining pressure referred to above.
+Section III discusses the technical details underlying the calculation of the disjoining poten-
+tial by MD, as well as the details of the simulations undertaken in the current application
+of the method to the study of high-temperature grain boundaries in fcc Ni. The results are
+presented in section IV, followed by a summary and discussion of the ndings in the context
+of previous theoretical studies in section V.
+DISJOINING POTENTIAL
+The equilibrium width of a premelted grain-boundary is governed by a competition be-
+tween bulk and interfacial thermodynamic factors, which can be represented mathematically
+as follows:
+G(w) = 
Gfw+ 	(w): (1)
+3In Eq. 1,G(w) represents the total excess free energy of a premelted grain boundary of width
+w, which is composed of two contributions: 
 Gfrepresents the bulk free energy dierence
+between liquid and solid phases (positive below the melting point), per unit volume, while
+	(w) represents the width-dependent interfacial free energy, which we refer to as the dis-
+joining potential. The function 	( w) takes the limits of 
GB(the interfacial free energy of a
+hypothetical \dry" grain boundary) and 2 
SL(twice the solid-liquid interfacial free energy)
+for small and innite values of w, respectively. For intermediate values of the width 	( w)
+can display a complex dependence on w.
+The dependence of 	( w) onwis generally governed by distinct short and long-ranged
+contributions. An attractive interaction between solid-liquid interfaces [35, 36] arises due
+to dispersion forces which are dominant at large w, and are predicted to give rise to nite
+interfacial widths at TM[35]. For systems where the wetting condition, 
GB>2
SLholds, a
+repulsive contribution to 	( w) arises from short-ranged structural interactions (	 sr) associ-
+ated with the overlap of the density waves in the diuse regions of the solid-liquid interfaces.
+Mean-eld arguments [37], as well as lattice-gas models (e.g., [28]), yield an exponentially
+decaying form for this short-ranged contribution:
+	sr(w) = 2
SL+ 

exp[w=] (2)
+where 

=
GB2
SLandis an interaction length on the order of the atomic spacing.
+In the absence of long-ranged dispersion forces, insertion of Eq. 2 into Eq. 1 leads to the pre-
+diction of an equilibrium grain boundary width that diverges logarithmically as the melting
+temperature ( TM) is approached from below. In previous work [20] we have argued, based
+on previous estimates of the dispersion forces for surface premelting, that the structural con-
+tributions to the disjoining potential for grain-boundary premelting in metals are expected
+to dominate the long-ranged dispersion contributions for nanometer-scale grain-boundary
+widths.
+Recent theoretical results demonstrate that 	 sr(w) can generally display more complex
+dependencies on wthan suggested by Eq. 2. Diuse-interface phase eld models [14{16],
+which neglect dispersion forces and thus model 	 srdirectly, have shown that depending on
+grain-boundary misorientation and the detailed choice of interfacial thermodynamic param-
+eters, the premelting transition can exhibit either the continuous behavior associated with
+4Eq. 2, a hysteretic rst-order character, or an intermediate behavior where the boundary
+width increases with increasing temperature but remains nite at TM(these three types of
+behavior are referred to as types 2, 1 and 3 in Ref. [34]). The PFC method was recently used
+to study grain boundary premelting for both three-dimensional body-centered-cubic (bcc)
+[18] and two-dimensional hexagonal [19] systems. The latter study involved a systematic
+investigation of symmetric tilt boundaries as a function of misorientation. Well beyond a
+critical misorientation angle (where 
GB>2
SL) 	srwas found to exhibit a purely \repul-
+sive" behavior, monotonically decreasing with increasing w, consistent with Eq. 2. However,
+well below the critical angle, 	 srexhibited an attractive minimum, giving rise to a nite
+area-averaged liquid-layer width at the melting temperature, re
ecting the presence of local-
+ized premelting within the cores of the grain-boundary dislocations in low-angle boundaries.
+The possibility that the short-ranged contributions to the disjoining potential can exhibit
+an attractive minimum was considered in earlier theoretical studies of wetting transitions
+[38]. In these studies a double-exponential ansatz was used to model such behavior. For the
+purposes of the present study this double-exponential form can be written as follows:
+	sr(w) = 2
SL+ [
 1exp (w= 1)
2exp (w= 2)]; (3)
+where 
 1and 
 2, which are both positive quantities, represent the strengths of repulsive
+and attractive exponential contributions with decay lengths 1and2, respectively. Al-
+though Eq. 3 has not been derived directly from a microscopic theory, it provides a useful
+phenomenological form to model the MD data, as shown below. This equation gives rise
+to a predicted value of the quantity 
GB2
SLequal to 
 1
2, where again 
GBis the
+interfacial free energy of a hypothetical dry grain boundary given by the limit of wgoing to
+zero in Eq. 3. Choosing 2> 1, and if 
 1>
2, as in the results given below, this disjoining
+potential is repulsive at short distances, attractive at large w, and features a minimum at a
+widthwmgiven as:
+wm=12
+21ln [
12
+
21]: (4)
+With the form for the disjoining potential given by Eq. 3 the width of the premelted boundary
+is predicted to be nite, with a value wm, at the melting point.
+In the MD results presented below we demonstrate the disordering behavior for two grain
+boundaries with relatively high and intermediate energy which we show can be quantitatively
+5modeled by Eqs. 2 and 3, respectively. To compute quantitative values for the disjoining
+potentials from these simulations, we make use of the following relation between G(w) and
+the equilibrium distribution P(w) of grain-boundary widths sampled by the MD systems at
+a temperature T:P(w)/exp [AG(w)=kBT], whereAis the interfacial area. Calculations
+ofP(w) by MD, combined with an accurate knowledge of the bulk melting properties un-
+derlying 
Gfallows one to extract 	( w), as described below. Since the MD simulations
+are based on a short-ranged classical interatomic-potential model, they do not include the
+long-ranged dispersion-force contributions to 	( w), and thus sample 	 sr(w) directly.
+METHODOLOGY
+In this study the simulations were based on an embedded-atom method (EAM) inter-
+atomic potential model for Ni developed by Foiles, Baskes and Daw (FBD) [39]. This po-
+tential was chosen as we have in previous work characterized both the bulk melting properties
+and solid-liquid interfacial free energies for FBD Ni; both sets of properties are prerequisites
+for a quantitative study of grain-boundary premelting. As reviewed in Ref. [20] the melting
+temperature for the FBD Ni potential has been bracketed to lie in the range T M=1709-1710
+K, and the value of the solid-liquid interfacial free energy has been calculated to be 
SL=285
+mJ/m2[40]. The latent heat for the potential was also calculated to be 0.015 eV=A3.
+In the present study we considered the following three dierent grain boundary struc-
+tures: a 9 symmetric h011if411g38.9otilt boundary (hereafter referred to as the 9 tilt
+boundary), a 11 symmetric h011if311g50.5otilt boundary (hereafter referred to as the 11
+tilt boundary), and a 9 h115i120otwist boundary (hereafter referred to for convenience as
+the 9 twist boundary, even though we recognize that this boundary can also be described
+as a symmetric tilt boundary). These were selected from a large library of grain boundary
+structures that t into moderately-sized periodic simulation cells, developed in the work of
+Olmsted et al.[41]. As shown in Table I, the selected boundaries have zero-temperature en-
+ergies (
0
+GB) that span a range of values relative to 2 
SL, and were thus expected to feature
+a range of premelting behavior.
+6Grain Boundary type Lx(A)Ly(A)Lz(A)
0
+GB(mJ=m2)
0
+GB=2
SL
+9h011if411g38.9osymmetric tilt 237.86 32.36 31.68 909 1.5
+9h115i120otwist 253.23 37.34 38.80 1440 2.5
+11h011if311g50.5osymmetric tilt 246.55 32.36 33.02 450 0.79
+TABLE I: The grain boundary types used in the MD simulations along with the simulation cell
+sizes, reported at zero temperature. For reference the lattice constant of FBD Ni potential at
+zero temperature is 3.52 A. These boundaries span a large range of 
0
+GB=2
SL, where
0
+GBis the
+zero-temperature grain-boundary energy, and 
SLis the solid-liquid interfacial free energy.
+Fixed atomsGBFixed atomsxyz
+FIG. 1: Computational cell used for molecular dynamics simulations at zero temperature. The
+boundary plane is in the y-z directions. The cell is periodic in y and z.
+Energy Minimization
+The optimization of grain-boundary structures at zero temperature made use of a simu-
+lation block with two grains separated by a 
at grain boundary as shown in Fig 1. The cell
+was periodic in the plane of the boundary and non-periodic perpendicular to it. The grains
+were sandwiched between two slabs parallel to the grain-boundary plane. The atoms in each
+of these slabs were xed relative to each other and could only undergo a rigid body motion.
+The slabs could move normal to the boundary, allowing volume expansion to maintain zero
+normal stress, and in the plane of the boundary, avoiding any resistance from the surfaces
+to translational movement of the grains relative to each other. Multiple trial congurations
+were built in order to minimize the boundary energy with respect to relative translations of
+the grains and with respect to the number of atoms in the grain boundary, as described in
+Ref. [41]. The energy of each trial conguration was minimized using the conjugate gradient
+method. After minimization, the energy of the grain boundary was computed as the total
+energy of the unconstrained atoms, less the bulk crystal energy for the same number of
+7atoms, divided by the area of the boundary. The dimensions of the computational cell used
+in these energy minimizations and in the subsequent MD simulations are given in Table I.
+Molecular-Dynamics Simulations
+Finite temperature simulations were performed by MD employing the LAMMPS code
+[42]. All simulations were performed with a constant number of atoms, constant volume
+and temperature (NVT ensemble). The temperature was maintained by using a Nose-
+Hoover thermostat [43] with a thermostat relaxation time of 0.1 ps, employing a time-step
+of 1 fs. The simulations began with the optimized zero-temperature geometry, which was
+equilibrated at dierent temperatures. Prior to each simulation at a given temperature, the
+periodic lengths parallel to the grain boundary plane were expanded according to the nite-
+temperature value of the lattice constant for the bulk crystal (determined separately). These
+periodic lengths were then held constant in all of the NVT MD simulations. The atoms in
+the slabs that were held xed during the energy minimization procedure were made dynamic
+for the nite-temperature simulations, and the boundary conditions were modied such that
+both the grains terminated in free surfaces.
+As discussed in the next sub-section, two types of analysis were conducted on the simu-
+lated systems to study their premelting behavior. The rst was a calculation of the excess
+volume as a function of temperature. In the simulations used for this analysis equilibration
+times were a few ns. The system volume was sampled over a total time of at least 10 ns at
+a frequency of 10 ps.
+The second analysis involved the calculation of equilibrium grain boundary width his-
+tograms. For these analyses, the simulations started at a given temperature with an equili-
+bration lasting 10 ns. Statistics were then obtained for the boundary width histograms from
+a total of 4000 snaphsots, sampled at a frequency of 10 ps. As discussed below, this number
+of snapshots and sampling rate ensured that at least a hundred statistically independent
+samples were obtained at each of the temperatures studied for the histogram analysis.
+8Calculation of Excess Volume and Width Histograms
+Prior to performing detailed analyses of width histograms, we performed an analysis of
+premelting behavior based on the temperature dependence of the excess volume. The excess
+volume was calculated by time averaging over values for single congurations, computed as
+follows. For a single snapshot the excess volume was computed from the distance between
+two lattice planes, one in each grain. These planes were chosen in each grain such that they
+were far from the boundary and the free surfaces. The volume of material that would lie
+between the two planes in a perfect crystal can be computed by counting the total number
+of atoms between the two planes (including half of the atoms in each of the two planes) and
+multiplying that number by the volume per atom of the bulk crystal at the same temperature
+(and zero pressure). The dierence between the actual volume and the expected bulk volume,
+divided by the area of the boundary, is the excess volume. A slight linear dependence of
+this measured excess volume on the distance between the planes was found, presumably
+the result of the numerical error in the lattice constant at high temperature. However, any
+consistent choice was adequate for our purposes here, as the excess volume results were used
+mainly to determine the qualitative nature of the premelting behavior, as discussed below.
+The specic planes chosen for the excess volume were 1/4 and 3/4 of the way through the
+simulation cell.
+In order to compute equilibrium grain-boundary width distributions, P(w;T i), for a given
+interface temperature Ti, we proceeded as follows. For each snapshot the grain-boundary
+widthwwas determined using a scheme developed by Hoyt et al. [40] for the analysis
+of solid-liquid interface capillary 
uctuations. In this approach each atom is assigned a
+structural order parameter, i=1
+12P
+jj~ rij~ rc
+ijj2, whererijare the actual positions of the
+12 nearest neighbors of atom iandrc
+ijare the atom sites for the corresponding neighbors
+in the perfect crystal. The ivalues are then averaged in bins along the direction normal
+to the boundary and the point of in
ection in the average order parameter prole is taken
+as the position of the grain boundary. In the case of grain boundaries two separate proles
+are required. The rst uses rc
+ijfor the crystal orientation of one of the two grains in the
+bicrystal, and for the second rc
+ijis chosen based on the other grain. After these two order
+9parameter proles ( 1and2) are obtained each is t to the following function:
+(x) =a+btanh [(xx0)=d]; (5)
+where a,b,x0anddare tting parameters, and the boundary width is derived from the
+dierence in values of x0determined from the 1and2proles.
+The analysis used to compute grain-boundary widths is illustrated graphically in Fig. 2
+for a snapshot of the 9 twist boundary at an undercooling 2 K below the bulk melting
+temperature. In Fig. 2 the top panel shows the atoms, color coded according to the magni-
+tude of the dierence in the order parameter values, lighter colors (red in the on-line gure)
+denoting atoms in an environment corresponding to either of the two grains, and darker
+colors (blue in the on-line gure) indicating an environment that does not correspond to
+either grain. The corresponding order parameter proles with the ts to Eq. 5 (solid lines)
+are shown in the bottom of Fig. 2. The width is dened as the dierence in the center
+positions of the ts to 1and2. In this example, a width of 13.5 Ais obtained by the
+analysis.
+It should be emphasized that the choice of an order parameter used to compute the
+grain boundary width is not unique (this point was discussed also in a recent related study
+by Williams and Mishin [33]). Other choices for the structural order parameter used to
+dene an interface width would include the so-called centrosymmetry parameter [44], the
+order-parameter used by Morris in studies of solid-liquid interfaces [45], excess mass used
+by Mellenthin[19], a parameter introduced by Williams and Mishin [33] based on the local
+structure factor, and an order parameter introduced by von Alfthan et al. [34] based on
+structural units. Dierent choices for the order parameter are expected to lead to slightly
+dierent values for the interface width, and the quantitative values of the disjoining potential
+derived from them. For example, in the sharp-interface formalism we describe the limit of
+	(w) for smallwas the interfacial free energy of a hypothetical \dry" grain boundary - we
+would expect that this limiting value will depend on the way in which the width is dened,
+which is not unreasonable since with any choice of this denition a real boundary at zero
+temperature will have some small nite width. The ambiguities associated with the dierent
+choices for the denition of width is inherent in the use of a sharp-interface theoretical
+formalism to describe the properties of systems such as these with diuse interfaces. In
+10w FIG. 2: A snapshot of a premelted grain boundary at an undercooling of 2 Kand the corresponding
+average order parameter versus distance across the grain boundary. Atoms in the top panel are
+colored based on the order parameter value with lighter color (red in the on-line gure) representing
+atoms in either of the two grains and darker color (blue in the on-line gure) showing atoms in the
+grain boundary. The construction to determine the grain-boundary width wis illustrated in the
+bottom panel and a width of w13:5Ais obtained.
+applications of such sharp-interface models, however, the ambiguity presents no problem in
+practice provided that each of the bulk and interfacial thermodyamic quantities are dened
+consistently.
+In order to determine whether the simulation times employed in this study were sucient
+to give adequate statistics for the width histograms, we estimated the correlation time for
+width 
uctuations based on an analysis of the decay of a width autocorrelation function
+C(
t) dened as follows:
+C(
t) =hw(t)w(t+ 
t)ihwi2; (6)
+In Eq. 6,w(t) denotes the instantaneous width of the grain boundary at a time t, andh:::i
+denote ensemble (time) averages. The time required for the autocorrelation function to
+decay by a fraction of 1 =ewas taken as a measure of the correlation time.
+The correlation times were found to vary signicantly with boundary type and temper-
+ature. The maximum value of the correlation time was obtained as 100 ps for the 9 tilt
+11boundary at a temperature 2 K below TM. Given that 4000 snapshots were sampled in
+the MD simulations, at a frequency of 10 ps, at least a hundred independent samples were
+obtained for each of the histograms presented below.
+Calculation of Disjoining Potentials from Width Histograms
+As discussed in the previous section, the probability of observing a given boundary width
+at an interface temperature T iis related to the total free energy of the system as follows:
+P(w;T i) =Ciexp[AG(w;T i)=kBTi]; (7)
+whereG(w;T i) is given by Eq. 1. In the above expression the subscript idenotes one of the
+seven dierent temperatures for which the width histograms have been calculated by MD:
+T1= 1710,T2= 1708,T3= 1705,T4= 1700,T5= 1690,T6= 1680,T7= 1650K, which
+represent undercoolings relative to the bulk melting temperature, TM=1710 K, ranging
+between zero and 60 K. Equation 7 emphasizes the fact that each undercooling yields a
+histogram that spans a dierent range of widths, and the unknown normalization constants
+at each temperature are denoted as Ci.
+As discussed in the next section, analyses of the calculated width histograms were un-
+dertaken to compute disjoining potentials for two dierent boundaries, one of which (the
+9 twist) featured a diverging interface width as TMis approach from below, and another
+(the 9 tilt) whose width remains nite at TM. In the former case the disjoining potential
+was modeled using the form given by Eq. 2, while for the latter use was made of Eq. 3.
+The unknown parameters in these expressions can be obtained in two ways based on the
+MD-calculated width histograms using Eq. 7. The rst is to t the histogram data at each
+temperature independently. This gives a range of values for the disjoining potential param-
+eters, which are expected to be more accurate at the higher temperatures where one has
+access to the largest range of width values. A rened estimate of the potential parameters
+can be obtained from a histogram analysis using all of the data at once, as follows.
+From Eq. 1 the disjoining potential can be written as
+	(w) =G(w;T i)w
Gf(Ti) (8)
+12where 
Gfis the bulk free energy dierence between liquid and solid per unit volume. For
+the EAM Ni system studied here the temperature dependence of 
 Gfis known accurately
+from previous studies. For the small range of temperatures considered in the MD simulations
+of width histograms, which are close to the bulk melting temperature, 
 Gfcan be accurately
+computed from the following expression:
+
Gf=L(
Ti=TM) (9)
+whereLis the latent heat per unit volume, and 
 Ti=TMTiis the interface undercool-
+ing. From Eq. 7, the disjoining potential can be written in terms of the equilibrium width
+distribution as:
+	(w) =(kBTi)lnP(w;T i)=Ai
Gfw+ai (10)
+where theaiare unknown constants related to Ciin Eq. 7. These parameters lead to constant
+osets when the quantity - kBTilnP(w;T i)=Ai
Gfwis plotted versus wfor each interface
+temperature. To construct the disjoining potential a least squares t is used to rene the
+values of the shift parameters ( ai) along with the potential parameters (
 
andin Eq. 2
+or 
 1,1, 
 2, and2in Eq. 3). In this procedure the initial values for the shift parameters
+were estimated \by eye" to give maximal overlap between the data, and the initial values
+for the potential parameters were obtained from an average of the independent ts to the
+data at separate temperatures (described above). The values of the parameters were then
+iteratively rened to minimize the square of the dierences between the MD data and the
+analytical expressions for 	( w).
+Eects of Thermostat, System Size and Boundary Conditions
+In the course of this work additional studies were also conducted to investigate the eects
+of the details of the simulation methodology on the resulting disjoining potentials. These
+additional analyses were performed for the 9 twist boundary in Ni as well as a series of
+h100itilt boundaries in Ni and bcc Fe.
+The rst issue addressed in these additional simulations was the eect of thermostat.
+Equation 7 assumes that the measured grain-boundary widths sample a canonical ensem-
+ble, and in order to achieve this by MD simulations we employed a standard Nose-Hoover
+13thermostat [43] in the present work. An alternative choice would be to use a Langevin
+thermostat [46], which would relax temperature 
uctuations faster locally and produce dif-
+ferent dynamics in the system, but would be intended to sample the same ensemble. To
+test for unexpected eects of dynamics on the equilibrium width histograms and disjoining
+potential, two boundaries showing continuous premelting behavior (i.e., diverging widths as
+the melting temperature is approached from below), one in bcc Fe and one in fcc Ni, were
+simulated using both a Nose-Hoover thermostat and a Langevin thermostat with a time
+constant of 0.25 ps. No statistically signicant dierences in the disjoining potential were
+found with the two thermostats. Further, we tested a range of reasonable time constants
+for both thermostats, nding that statistically signicant eects on the calculated disjoining
+potentials were again absent unless the relaxation time for the Langevin thermostat was
+reduced to extremely short time scales, on the order of the inverse of the vibrational fre-
+quencies in the system. Interestingly, in simulations which used the same time constant of
+0.25 ps, the correlation time for width 
uctuations was found to be substantially reduced in
+the simulations with the Langevin relative to those with the Nose-Hoover thermostat. The
+Langevin thermostat thus oers the potential advantage of providing improved statistics
+related to width histograms for a given simulation time.
+The second issue investigated was the eect of system size on the calculated disjoining
+potential. For atomically rough grain boundaries, such as those investigated in this work,
+previous theoretical studies have demonstrated that capillary 
uctuations should give rise
+to a weak dependence of the disjoining potential on cross-sectional area [14, 28]. To check
+for any unexpected large system-size eects, simulations were performed for the 9 twist
+boundary using both the cross-sectional area given in Table I, as well as a value that was
+four times larger (i.e, with periodic lengths that were doubled in each of directions parallel
+to the boundary). Although the width histograms obtained for the larger system were much
+narrower, as expected from Eq. 7, it was still possible to obtain data over a suciently large
+range of widths to extract a disjoining potential. The results for the disjoining potential were
+found to be consistent with those obtained from the smaller system provided that a small
+eect of system size on the bulk melting temperature (amounting to a roughly one-degree
+lowering of TMfor the larger system, which is within the accuracy of the known value of
+the bulk melting point) was accounted for in the analysis. Similar results were obtained in
+analyses of system size eects on calculated disjoining potential for a high-energy h100itilt
+14boundary in Ni.
+A nal issue that was investigated in these studies concerns the eect of boundary condi-
+tions on the calculated width distributions. For a high-angle h100isymmetric tilt boundary
+in Fe this was investigated by computing width distributions at 10 K undercooling using
+both the free-surface boundary conditions employed in the current study, as well as a full pe-
+riodic boundary condition in a simulation cell containing two grain boundaries and a periodic
+length normal to the boundaries that was allowed to be dynamic to maintain zero normal
+stress. No statistically signicant dierences were found between the width distributions
+derived with the two dierent boundary conditions.
+RESULTS
+The eect of temperature on the structure of the three boundaries considered in this
+work is illustrated in Figs. 3 and 4. Figure 3 plots the calculated excess volume versus the
+logarithm of the interface undercooling, and displays three qualitatively dierent types of
+behavior. The 9 twist boundary has an excess volume (represented by the red circles)
+/s49/s46/s52
+/s49/s46/s50
+/s49/s46/s48
+/s48/s46/s56
+/s48/s46/s54
+/s48/s46/s52
+/s48/s46/s50/s69/s120/s99/s101/s115/s115/s32/s86/s111/s108/s117/s109/s101/s32/s40/s129/s41
+/s49/s48 /s49/s48/s48 /s49/s48/s48/s48
+/s85/s110/s100/s101/s114/s99/s111/s111/s108/s105/s110/s103/s32/s40/s75/s41/s32/s183/s57/s32/s60/s49/s49/s53/s62/s32/s49/s50/s48/s111
+/s32/s116/s119/s105/s115/s116
+/s183/s57/s32/s115/s121/s109/s109/s101/s116/s114/s105/s99/s32/s60/s48/s49/s49/s62/s32/s123/s52/s49/s49/s125/s32/s51/s56/s46/s57/s111/s116/s105/s108/s116
+/s32/s183/s49/s49/s32/s115/s121/s109/s109/s101/s116/s114/s105/s99/s32/s60/s48/s49/s49/s62/s32/s123/s51/s49/s49/s125/s32/s53/s48/s46/s53/s111/s32/s116/s105/s108/s116
+FIG. 3: Excess volume versus temperature plot for three dierent grain boundaries in Nickel. The
+red circles correspond to a 9 h115itwist grain boundary, the green diamonds correspond to a 9
+symmetrich011if411gtilt grain boundary and the black triangles correspond to a 11 symmetric
+h011if311gtilt grain boundary. The three boundaries show very dierent behavior.
+that continuously increases as the melting temperature is approached from below. The
+behavior displayed by this boundary in Fig. 3 is consistent with a logarithmic divergence of
+the interface width, as would be expected if the disjoining potential is of the form given by
+15Eq. 2. At the opposite extreme, the 11 tilt boundary (represented by the black triangles)
+has an excess volume that is relatively small and depends only weakly on temperature. The
+9 tilt boundary (represented by the green diamonds) displays an intermediate behavior.
+The excess volume rises with increasing temperature at a rate that initially tracks that of
+the continously premelting 9 twist boundary. At high temperature, the rate of increase of
+the excess volume decreases and the boundary maintains a nite excess volume at TM.
+Representative snapshots and calculated width histograms are shown in Fig. 4 for each of
+the three boundaries at 1708 K (two degrees below TM). The width histogram corresponding
+to the 9 twist boundary is much broader than those for the other two boundaries. This
+boundary samples relatively large widths, and P(w) shows pronounced asymmetry with
+an extended tail to large wvalues. The 11 tilt boundary, by contrast, features a width
+histogram that is very narrow and is centered on relatively small values of w. The width
+distribution for this boundary is roughly Gaussian in shape, with no detectable asymmetry
+towards large values of w. The width distribution for the 9 tilt boundary shows features
+intermediate between the other two. The average width is larger than that of the 11
+boundary, and the width distribution is considerably broader. Compared to the 9 twist
+boundary, however, the asymmetry and the tail extending to larger widths is not nearly as
+pronounced in the width distribution for the 9 tilt boundary.
+The MD snapshots in Fig. 4 next to each histogram provide additional insights into the
+nature of the high-temperature structural disorder present in each of the three boundaries.
+As in Fig. 2, the atoms in these snapshots have been colored based on the dierence between
+values of the structural order parameters ( 1and2) dened above. Lighter colors (red in the
+on-line gure) indicate atoms with an environment corresponding to one of the two grains,
+while darker color (blue in the on-line gure) denotes a disordered environment distinct
+from those in either of the grains. The snapshots and width histograms show that the 9
+twist boundary at 1708 K displays thick premelted layers, consistent with the continuous
+premelting behavior suggested by the excess volume. The 11 tilt boundary is seen to
+display appreciable disorder only over a width that is on the scale of one atomic plane;
+this boundary is observed in the MD simulations to remain highly ordered up to TM. The
+intermediate behavior of the 9 tilt boundary is characterized by disorder that extends over
+a distance of several atomic planes. An important feature of the disorder observed in this
+boundary is the nonuniform character of the widths along the area of the boundary. As
+16FIG. 4: The distribution function P(w) vswfrom the MD simulations for the three dierent
+boundaries at 1708 K. The green diamonds and the red circles correspond to the 9 tilt and twist
+grain boundaries respectively. The11 tilt boundary is represented by the black triangles. The
+lines represent a least square t of the premelting models of Eqs. 1 and 2 for the 9 twist grain
+boundary, Eqs. 1 and 3 for the 9 tilt boundary and a gaussian t for the 11 tilt boundary.
+The snapshots next to each gure correspond to the widest grain boundary for each boundary and
+are color coded according to the ivalues. Lighter color (red in the on-line gure) indicates an
+atom with an environment corresponding to one of the two grains, darker color (blue in the on-line
+gure) represents atoms in some other environment.
+illustrated in the snapshot of this boundary in Fig. 4, thick and extended regions of disorder
+up to a nanometer or more in thickness are often observed in part of the boundary, with
+much narrower, more ordered regions in between. For the remainder of this section we will
+focus on the behavior of the two 9 boundaries, showing that the temperature dependence
+of the width histograms can be accurately described by the disjoining-potential formalism
+described above.
+FIG. 5: Snapshots from a MD simulation at an undercooling 2 K illustrating the dynamic nature of
+the grain-boundary width. The snapshots corresponding to the 9 tilt grain boundary are on the
+right of the panel. The snapshots to the left are from the 9 twist boundary discussed elsewhere
+[20]. The atoms are colored by the dierence between the two order parameters as described above.
+17FIG. 6: The distribution function P(w) vswfrom the MD simulations. The function corresponding
+to the 9 tilt grain boundary is in the right of the panel. The function to the left is from the 9
+twist boundary discussed elsewhere [20]. The lines represent least square ts of the premelting
+model of Eqs. 1 and 2 for the 9 twist grain boundary and Eqs. 1 and 3 for the 9 tilt boundary.
+Figure 5 further illustrates the nature of the disorder present in the 9 twist and tilt
+boundaries. The snapshots were obtained from a 40 ns MD simulation at 1708 K, and
+represent the largest, smallest and an average value for the interface widths. Results for the
+9 twist boundary are shown on the left and those for the 9 tilt boundary are on the right
+of Fig. 5. The snapshots illustrate the highly dynamic nature of the boundary structures at
+1708 K. The large 
uctuations in interface width displayed by these boundaries forms the
+basis for the histogram analysis underlying calculations of the disjoining potential.
+The calculated width histograms for both 9 boundaries are displayed over a range of
+interface undercoolings in Fig. 6. The histograms on the left and right in Fig. 6 correspond
+to the 9 twist and tilt boundaries, respectively. For both of the boundaries the histograms
+become broader as the bulk melting temperature is approached (i.e., with decreasing un-
+dercooling). For the 9 twist boundary the system is observed to melt completely over the
+course of the 20 ns runs at the bulk melting temperature, and histograms can be obtained
+for this boundary only for nite values of the undercooling. By contrast, the width of the
+9 tilt boundary remains nite at TMand the width histogram can be calculated at zero
+undercooling for this boundary, as shown in Fig. 6. The inset in each of the panels shows
+the quantitykBTilog [P(w;T i)]=A
Gfwversuswfor all of the interface temperatures.
+The data for the 9 tilt boundary shows clear evidence of an attractive minimum in the
+disjoining potential, as will be discussed further below.
+18The solid lines in Fig. 6 represent the least square ts of the MD data for P(w) to the
+disjoining-potential formulas given by Eqs. 7, 1 and 2 for the 9 twist boundary, and Eqs. 7,
+1 and 3 for the 9 tilt boundary. For the 9 twist boundary the potential parameters
+obtained from the separate ts to the data for the individual temperatures span the range
+=0.25 to 0.29 nm, and 
 
=101 to 150 mJ/m2. For the 9 tilt boundary the tted values
+for1and2ranged between 0.142 to 0.144 and 0.143 to 0.144 nm, respectively, 
 1
2
+spanned the values 103 to 163 mJ/m2, and 
 2=
1took values between 1.003 and 1.006.
+With these tted potential parameters, the disjoining potential for the 9 tilt boundary
+exhibits a weak minimum at a width wmwith values ranging between 0.32 and 0.36 nm,
+and a depth relative to 2 
SLvarying between -7 and -11 mJ/m2. The fact that the MD
+calculated width histograms can be accurately described by disjoining potentials with a
+relatively narrow range of tted potential parameter values indicates that the formalism
+described in the previous section represents a valid model for describing the temperature
+dependence of the structural disorder observed in these boundaries.
+In order to rene the calculation of the disjoining potential for the 9 twist and tilt
+boundaries, we employ the histogram analysis described in the previous section, involving
+a renement of both the shift parameters aiand the potential parameters in Eqs. 2 and 3
+for the twist and tilt boundaries, respectively. The resulting ts are shown in Fig. 7 and
+correspond to the following values for the potential parameters: = 0:25 nm and 
 
= 156
+mJ/m2for the 9 twist boundary, and 
 1
2= 103 mJ/m2, 
 2=
1= 1:003,1= 0:1471
+nm,2= 0:1474 nm for the 9 tilt boundary. These parameter values are consistent with
+the values given above from the independent ts. The excellent agreement of the ts with
+the MD data in Fig. 7 again indicates that the disjoining-potential formalism represents
+an accurate framework for modeling the premelting behavior of these 9 boundaries. The
+analysis used to obtain the results in ts in Fig. 6 assumed a melting point of TM= 1710
+K. If the melting temperature is changed even by one degree,i.e., TM= 1709Kthen poor
+ts toP(w) are obtained for the data at the lowest undercoolings.
+The calculated disjoining potentials in Fig. 7 are characterized by the following features.
+For the 9 tilt boundary the disjoining potential has a minimum at a nite width, wm,which
+corresponds to the average equilibrium grain boundary width at the melting temperature.
+The potential is repulsive for w<w mand attractive for w>w m. In contrast, the 9 twist
+boundary features a purely repulsive disjoining potential that is well modeled by exponential
+19/s49/s53
+/s49/s48
+/s53
+/s48
+/s45/s53/s89/s40/s119/s41/s32/s45/s32/s50 /s103/s115/s108/s32/s32/s40/s109/s74/s47/s109/s50/s41
+/s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53
+/s71/s114/s97/s105/s110/s32/s66/s111/s117/s110/s100/s97/s114/s121/s32/s87/s105/s100/s116/s104/s44/s32/s119/s32/s40/s110/s109/s41/s183/s57/s32/s220/s49/s49/s53/s221/s32/s116/s119/s105/s115/s116/s32/s98/s111/s117/s110/s100/s97/s114/s121
+/s183/s57/s32/s220/s49/s49/s48/s221/s32/s123/s52/s49/s49/s125/s32
+/s115/s121/s109/s109/s101/s116/s114/s105/s99/s32/s116/s105/s108/s116/s32/s98/s111/s117/s110/s100/s97/s114/s121FIG. 7: An illustration of the histogram method used to extract the disjoining potential. It shows
+the merged data from individual histogram data used to reproduce the complete disjoining potential
+	(w) for the 9 twist and the 9 tilt boundaries. The solid line is the best t to the exponential
+decay given in Eq. 2 for the 9 twist boundary, and a double exponential function in Eq. 3 for the
+9 tilt grain boundary.
+form given in Eq. 2, as expected based on the logarithmic divergence of the excess volume
+calculated for this boundary (see Fig. 3).
+SUMMARY AND DISCUSSION
+In this paper we have presented a detailed description of a method for computing the
+disjoining potential for grain-boundary premelting and grain coalescence from an analysis of
+width 
uctuations measured in equilibrium MD simulations. The approach has been applied
+to two grain boundaries in an EAM model of elemental Ni. For the 9 twist boundary,
+which features an excess volume that increases logarithmically as TMis approached from
+below, the measured width histograms are consistent with a disjoining potential that decays
+exponentially, with a decay length of approximately 0 :25 nm and a maximum value at zero
+width of approximately 160 mJ/m2relative to 2 
SL. For the 9 tilt boundary, which features
+an excess volume that remains nite at TM, the measured width histograms are shown to be
+consistent with a disjoining potential that features a weak attractive minimum at wm0:34
+nm, with a depth relative to 2 
SLof approximately -8 mJ/m2.
+It is interesting to compare the results of the present work with previous studies of grain-
+20boundary premelting based on MD simulations, phase-eld theory and PFC calculations.
+The exponential form of the disjoining potential for the 9 twist boundary presented here
+and in Ref. [20] is qualitatively consistent with the results of previous investigations where
+a diverging grain-boundary width has been observed for high-energy boundaries in a variety
+of dierent systems (e.g., [25, 26, 28, 34]). The disjoining potential calculated here for the
+9 tilt boundary features a weak attractive minimum at wm, and is replusive and attractive
+for smaller and larger values of w, respectively. This form for the disjoining potential
+corresponds to a grain-boundary whose width increases with Tat low temperature, but
+remains nite at and above TM(up to some maximum superheating). This type of behavior
+for the temperature dependence of the grain boundary width is qualitatively similar to that
+found in recent MD studies for one of three twist boundaries in Si [34] and a symmetric tilt
+boundary in Cu [33].
+As discussed in the introduction section disjoining potentials with attractive minima,
+qualitatively similar to that calculated here for the 9 tilt boundary, were obtained in PFC
+calculations by Mellenthin et al.[19] for low-angle tilt boundaries in a model two-dimensional
+hexagonal system. The boundaries which displayed this type of disjoining potential had
+highly non-uniform interface structures, containing premelted regions localized on grain-
+boundary dislocation cores and separated by well ordered regions which we will refer to
+as solid \bridges". For these structures, the width corresponding to the minimum in the
+disjoining potential, which represents an average over the area of the boundary, corresponds
+to the thickness of an equivalent uniform layer with the same amount of \liquid" as contained
+in the premelted cores. This area-averaged width can be quite small (e.g., on the order of
+the atomic spacing) even though the radius of the premelted cores is much larger.
+It is interesting to compare the structures observed in the PFC calculations of Mellenthin
+et al. [19] with those for the 9 tilt boundary considered in the present study. Visual
+inspection of the numerous snapshots showed that the disorder in this boundary is indeed
+often highly non-uniform, as illustrated in Fig. 8.
+This gure shows a representative snapshot from a simulation at T= 1708 K, viewed
+down the tilt axis. In Fig. 8, the structural disorder in the boundary is clearly nonuniform
+- the region of high structural disorder is highlighted by the grey ellipse, while the red lines
+connected across the boundary plane highlight the ordered solid bridges. Thus, even though
+the 9 boundary studied in the present work has a misorientation angle (38.9 degrees)
+21FIG. 8: Snapshot of the 9 tilt boundary at 1708 showing the non-uniform structural disorder
+in the grain boundary. The grey ellipse shows the regions with high disorder while the red lines
+highlight the ordered bridges.
+which is far too large for its structure to be described in terms of separated grain-boundary
+dislocation cores, the non-uniform nature of the structural disorder and the presence of solid
+\bridges" is qualitatively very similar to the types of structures observed in Ref. [19].
+It should be emphasized, however, that the structure of the 9 boundary observed in
+the MD simulations is highly dynamic (c.f., Fig. 5) such that the solid bridges form and
+disappear rapidly on the time scale of the simulations. This behavior could have interesting
+consequences for the shear response of such boundaries, as the solid bridges are expected to
+oer enhanced resistance to shear which would otherwise be expected to be very limited for
+a premelted grain boundary (e.g., [26]). The shear response of such boundaries would thus
+be an interesting topic for future MD studies.
+ACKNOWLEDGMENTS
+Work at Northeastern, UC Davis and Berkeley was supported by the Director, Oce of
+Science, Oce of Basic Energy Sciences, Materials Sciences and Engineering Division, of
+the U. S. Department of Energy (DOE), under contracts DE-FG02-07ER46400, DE-FG02-
+06ER46282, and DE-AC02-05CH11231. JJH acknowledges nancial support from a Natural
+Sciences and Engineering Research Council (NSERC) of Canada Discovery grant. All the
+22authors acknowledge support from the DOE Computational Materials Science Network pro-
+gram. This research used resources of the National Energy Research Scientic Computing
+Center, which is supported by the Oce of Science of the U. S. Department of Energy under
+Contract No. DE-AC02-05CH11231. We also acknowledge helpful discussions with R. G.
+Hoagland.
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+arXiv:1001.0584v3  [hep-ph]  18 May 2010Preprint typeset in JHEP style - HYPER VERSION MSUHEP–100104
+Radiative Electroweak Symmetry Breaking
+in a Little Higgs Model
+Roshan Foadi, James T. Laverty, Carl R. Schmidt and Jiang-Hao Yu
+Department of Physics and Astronomy, Michigan State Univer sity
+East Lansing, MI 48824, USA
+E-mail:foadiros@msu.edu ,laverty1@msu.edu ,schmidt@pa.msu.edu ,
+yujiangh@msu.edu
+Abstract: WepresentanewLittleHiggsmodel, motivated bythedeconst ructionofafive-
+dimensional gauge-Higgs model. The approximate global sym metry isSO(5)0×SO(5)1,
+breaking to SO(5), with a gauged subgroup of [ SU(2)0L×U(1)0R]×O(4)1, breaking to
+SU(2)L×U(1)Y. Radiative corrections produce an additional small vacuum misalignment,
+breaking the electroweak symmetry down to U(1)EM. Novel features of this model are: the
+only un-eaten pseudo-Goldstone boson in the effective theory is the Higgs boson; the model
+contains a custodial symmetry, which ensures that ˆT= 0 at tree-level; and the potential
+for the Higgs boson is generated entirely through one-loop r adiative corrections. A small
+negative mass-squared in the Higgs potential is obtained by a cancellation between the
+contribution of two heavy partners of the top quark, which is readily achieved over much
+of the parameter space. We can then obtain both a vacuum expec tation value of v= 246
+GeV and a light Higgs boson mass, which is strongly correlate d with the masses of the
+two heavy top quark partners. For a scale of the global symmet ry breaking of f= 1 TeV
+and using a single cutoff for the fermion loops, the Higgs boso n mass satisfies 120 GeV
+/lessorsimilarMH/lessorsimilar150 GeV over much of the range of parameter space. For fraised to 10 TeV,
+these values increase by about 40 GeV. Effects at the ultraviol et cutoff scale may also raise
+the predicted values of the Higgs boson mass, but the model st ill favorsMH/lessorsimilar200 GeV.
+Keywords: Beyond Standard Model , Spontaneous Symmetry Breaking.Contents
+1. Introduction 1
+2. Gauge Sector 3
+3. Fermion Sector 8
+4. Effective Potential 12
+5. Electroweak Constraints 19
+6. Conclusions 21
+A.SO(5)Generator Matrices 22
+B. Gauge Boson Masses and Mixing 23
+B.1 The Charged Sector 23
+B.2 The Neutral Sector 24
+C. Fermion Masses and Mixing in the Top Quark Sector 26
+D. Higgs Potential for small |H|/f 28
+E. Fermion Sector with Complete SO(5)Multiplets and Decoupled SM
+Partners 29
+1. Introduction
+The mechanism of electroweak symmetry breaking and the stab ilization of the weak scale
+are two of the most important unresolved questions in partic le physics. The Standard
+Model (SM) Higgs boson offers the simplest answer to the first qu estion, but it leaves
+the second question unresolved. In fact, the SM Higgs boson i s unstable under quantum
+corrections, as its mass is naturally driven to the ultravio let cutoff scale. Over the past
+decade a class of theories known as Little Higgs (LH) models h as been proposed as a
+way to extend and stabilize the SM [1]–[23]. In LH models the H iggs boson is a pseudo-
+Goldstone boson of an approximate and spontaneously broken global symmetry. The latter
+is explicitly and collectively broken by extended gauge and Yukawa sectors, in such a
+way that the Higgs acquires a potential only if two or more couplings in the gauge or
+Yukawa sector are simultaneously switched on. Since quadra tically divergent one-loop
+contributions to the Higgs mass can only arise from diagrams involving onecoupling, it
+– 1 –follows that these have to cancel. This is very similar to the supersymmetric scenario, in
+which the superpartners cancel the SM quadratic divergence s. However in LH models the
+cancellation occurs between particles with the samespin, with interesting and extensively-
+studied collider signatures [24]–[27].
+Clearly, for a LH model to be realistic the generated Higgs po tential must have a
+nonzerovacuumexpectation value(vev). Furthermore, thee lectroweak vev vmustbemuch
+smaller than the vev fassociated with the spontaneous breaking of the larger symm etry
+group, since the main goal of any LH model is to naturally gene rate a hierarchy of scales
+betweenvand the new-physics scale f. This implies that the ratio of the negative mass-
+squared,m2, to the quartic coupling, λ, in the Higgs potential must be small in magnitude
+compared to f2. Typically in LH models, m2receives its dominant contribution from
+loops with the heavy partner of the top quark (which is requir ed in the theory to cancel
+the quadratic divergence from the top-quark loop). However , the dominant contribution
+toλis also typically generated by loops of the same heavy top qua rk partner, so that a
+sufficiently large λis not generated radiatively. For this reason, other effectiv e operators
+are introduced into the theory, whose coefficients depend on t he details of the ultraviolet
+completion, but whose size can be estimated by naive dimensi onal analysis. For instance,
+in the Moose-type models, such as the Minimal Moose [4], the q uartic coupling arises from
+plaquette operators; in the Littlest Higgs [5] the quartic c oupling arises from a hard mass-
+squared for the additional scalars in the theory, which are t hen integrated out by equations
+of motion; and in the Simplest Little Higgs [19] model it aris es from a small mass term
+for the scalars. One disadvantage of this approach is that th e unspecified coefficient of the
+new operator introduces an additional degree of unpredicta bility in the effective theory.
+Furthermore, even with the new contribution to λ, there must still be some amount of
+cancellation of the contribution to m2of the heavy top quark partner if one is to obtain a
+reasonably light Higgs boson [28].
+A second requirement fortheHiggs sector is theabsenceof la rge isospin violation. This
+is usually achieved by enlarging the overall global symmetr y group to include SU(2)L×
+SU(2)R, which in aLH model can bedone minimally by imposingan SO(5) symmetry [10].
+This can create some problems in models with two Higgs double ts, with a potential which
+requires their vev’s to be misaligned. This misalignment is a source of custodial isospin
+violation, which shows up in the form of dimension-six opera tors when the heavy states are
+integrated out. In Ref. [12] this problem is avoided by const ructing a model with a single
+Higgs doublet and an approximate custodial SU(2)C, an extension of the Littlest Higgs
+with a coset SO(9)/SO(5)×SO(4). The electroweak constraints can also be weakened
+by introducing “T-parity”, a new discrete symmetry under wh ich the heavy fields are odd
+and the SM fields are even [14, 17, 20]. Then no effective operato rs are generated from
+tree-level exchanges of a single heavy field, since a vertex m ust contain an even number of
+these.
+In this paper we introduce a LH model in which the only un-eate n scalar field is
+the Higgs boson, electroweak symmetry breaking is fully rad iative, and an approximate
+custodial symmetry suppresses the sources of nonstandard i sospin violation. The model is
+based on an SO(5)0×SO(5)1global symmetry, of which the [ SU(2)0L×U(1)0R]×O(4)1
+– 2 –subgroup is gauged. The global and gauged symmetry structur e is similar to that of
+the Custodial Minimal Moose model [10]; however, in our mode l there is only one non-
+linear sigma field, with the result that the Higgs boson is the only spin-zero field in the
+theory and there are no plaquette operators. The gauge secto r of this model has also
+been considered in Ref. [29]. Our model is inspired from the d econstruction of an SO(5)×
+U(1)Xgauge-Higgs model [30], which uses the fact that the SO(5) structure is the minimal
+way to accommodate a gauge-Higgs and custodial symmetry. In addition, it suggests the
+inclusion of fermions in terms of SO(5) multiplets, with a simple implementation of the LH
+mechanism in the Yukawa sector. The novel feature of this fer mion sector is that a second
+heavy top quark partner produces canceling contributions t o them2term in the Coleman-
+Weinberg potential, so that it can easily be made small and ne gative. As a consequence,
+the radiative Higgs quartic coupling, although small, is la rge enough to trigger spontaneous
+symmetry breaking with v≪f, and the effective theory is more predictive than in LH
+models in which the quartic coupling arises from additional operators. In particular, the
+Higgs boson is naturally light in this model, with a mass that depends predominantly on
+a single mixing angle, sin2θt, in the top quark sector. For f= 1 TeV and 10 TeV, we find
+MH/lessorsimilar150 GeV and MH/lessorsimilar190 GeV, respectively, over most of the range of sin2θt. Even
+after including effects of unknown fermion dynamics at the cut off scale, the assumption
+that the Higgs potential is dominated by calculable contrib utions at one loop leads to a
+light Higgs boson over much of the parameter space.
+The remainder of this paper is organized as follows. The gaug e and fermion sectors
+of the theory are introduced in Sec. 2 and 3, respectively. In Sec. 4 we compute the
+Coleman-Weinberg potential and analyze the parameter spac e in which we can obtain
+bothv= 246 GeV and a light Higgs boson mass. In Sec. 5 we compute the t ree-level
+electroweak parameters, and derive the experimental bound s on theSO(4)1coupling (g1)
+andf. Finally in Sec. 6 we offer our conclusions. Detailed calculat ions for the mass
+matrices and the Higgs potential can be found in the appendic es.
+2. Gauge Sector
+The gauge symmetry of our model is SU(2)3×U(1), which is embedded in an approximate
+SO(5)×SO(5) global symmetry. The global symmetry is then broken spon taneously to
+the diagonal SO(5) by a non-linear sigma field. This symmetry structure is re presented
+in Fig. 1 by a moose diagram consisting of two sites, 0 and 1, wh ere the outer circles
+are the global SO(5)’s and the inner ellipses are the gauged subgroups. In ter ms of the
+moose site indices, the global symmetry can be written SO(5)0×SO(5)1, while the gauged
+subgroup is [ SU(2)0L×U(1)0R]×[SU(2)1L×SU(2)1R]. In this description the LandR
+subscripts indicate the two commuting SU(2) subgroups of SO(5), whileU(1)0Ris aU(1)
+subgroup of SU(2)0R. Note that the model can be considered a severe deconstructi on of
+the 5-dimensional SO(5)×U(1)XGauge-Higgs model of Ref. [30], where the extra U(1)X
+symmetry has been removed. In terms of this deconstruction, the sites 0 and 1 are the two
+end-branes of the 5-dimensional interval, while the non-li near sigma field plays the role of
+the fifth component of the gauge fields in the bulk.
+– 3 –SU(2)0LSU(2)1L
+SU(2)1R U(1)0R
+SO(5)0SO(5)1Σ
+Figure 1: Moose diagram for the model. The approximate global symmetry is SO(5)0×SO(5)1,
+with an embedded gauge symmetry of [ SU(2)0L×U(1)0R]×O(4)1∼=[SU(2)0L×U(1)0R]×
+[SU(2)1L×SU(2)1R×P1LR].
+The non-linear sigma field is parametrized by
+Σ =e√
+2iπATA/f, (2.1)
+where we have chosen the normalization, tr/parenleftbig
+TATB/parenrightbig
+=δAB, so that the gauged SU(2)
+sub-matrices have the conventional normalization. A conve nient basis for the ten SO(5)
+generator matrices is {Ta
+L,Ta
+R,T1,T2,T3,T4}, given in Appendix A in Eq. (A.3). Under
+anSO(5)0×SO(5)1transformation, the sigma field transforms as Σ →U0ΣU†
+1, whereU0,1
+areSO(5) matrices in the fundamental representation. Gauging th e [SU(2)0L×U(1)0R]×
+[SU(2)1L×SU(2)1R] subgroup leads to the following covariant derivative
+DµΣ =∂µΣ−ig0LWaµ
+0LTa
+LΣ−ig0RBµ
+0RT3
+RΣ+ig1LWaµ
+1LΣTa
+L+ig1RWaµ
+1RΣTa
+R.(2.2)
+With this we can write the Lagrangian density for the gauge an d sigma fields as
+Lgauge=−1
+4Waµν
+0LWa
+0Lµν−1
+4Bµν
+0RB0Rµν−1
+4Waµν
+1LWa
+1Lµν−1
+4Waµν
+1RWa
+1Rµν
++f2
+4tr/bracketleftig
+(DµΣ)(DµΣ)†/bracketrightig
+. (2.3)
+In this paper we shall write g1Landg1Ras if distinct. However, in models similar to ours
+it has been found that promoting an SU(2)L×SU(2)Rgauge symmetry to O(4) turns out
+to protect the tightly constrained ZbL¯bLcoupling from large loop corrections [31, 32, 35].
+For this reason, and for simplicity, we will choose g1L=g1R≡g1for any computations,
+imposing the L-Rexchange symmetry P1LRnecessary for the full O(4)1∼SU(2)1L×
+SU(2)1R×P1LR. However, we will not compute the ZbL¯bLcoupling, as well as other
+electroweak observables at one loop, leaving this for futur e work [36].
+With the gauged subgroups embedded in the global SO(5)0×SO(5)1as given by
+Eq. (2.2), a vacuum alignment of /an}bracketle{tΣ/an}bracketri}ht= 1 spontaneously breaks the gauge symmetry
+– 4 –[SU(2)0L×SU(2)1L]×[U(1)0R×SU(2)1R] down to the SM electroweak group SU(2)L×
+U(1)R=Y. Thereare6exact Goldstonebosons, whichwill beeaten by6l inear combinations
+of the gauge fields, giving them masses of order the symmetry b reaking scale f. The
+remaining 4 dynamical fields contained in Σ have exactly the r ight quantum numbers to
+play theroleofthestandardmodelHiggs doublet H. Although HisnotanexactGoldstone
+boson, we note that the gauge sector of the model has the colle ctive symmetry breaking
+necessary to forbid any quadratic divergences to the Higgs e ffective potential at one loop.
+If we set the couplings to zero at either site 0 or at site 1, the globalSO(5) symmetry at
+that site becomes exact, and all 10 pion fields, including the Higgs doublet, become exact
+Goldstone bosons. Thus, any field-dependent term in the Higg s effective potential must
+have contributions collectively from both the couplings at site 0 and at site 1, which can
+only contain quadratic divergences at two loops or higher.
+Working in unitary gauge, where we set the eaten Goldstone bo son fields to zero, we
+can identify Hin Σ by letting
+Π≡√
+2πATA=
+04×4/parenleftigg
+H
+˜H/parenrightigg
+/parenleftig
+H†˜H†/parenrightig
+0
+, (2.4)
+where
+H=/parenleftigg
+h1
+h2/parenrightigg
+and˜H=−iσ2H∗=/parenleftigg
+−h∗
+2
+h∗
+1/parenrightigg
+, (2.5)
+with
+h1=1√
+2(π1+iπ2), (2.6)
+h2=1√
+2(π3+iπ4).
+Expanding and re-organizing the Σ field, we obtain
+Σ =eiΠ/f= 1+iΠ√
+2|H|s−Π2
+2|H|2(1−c), (2.7)
+where
+s= sin/parenleftigg√
+2|H|
+f/parenrightigg
+andc= cos/parenleftigg√
+2|H|
+f/parenrightigg
+, (2.8)
+and|H|= (h2
+1+h2
+2)1/2.
+Any further misalignment of the vacuum will result in a vacuu m expectation value for
+the Higgs doublet,
+/an}bracketle{tH/an}bracketri}ht=1√
+2/parenleftigg
+v
+0/parenrightigg
+, (2.9)
+breaking the gauge symmetry completely down to U(1)EM. Determining the value of v
+requires an analysis of the effective potential, which we do at one loop in this paper. For
+– 5 –this we need the mass terms for the gauge bosons, as a function of the Higgs field, which
+we can take to be along the direction of its vacuum expectatio n value, without loss of
+generality. Using the expression Eq. (2.7) for Σ in the gauge Lagrangian, Eq. (2.3), we
+obtain
+Lmass=f2
+4/braceleftigg
+g2
+0LWaµ
+0LWa
+0Lµ+g2
+0RBµ
+0RB0Rµ+g2
+1LWaµ
+1LWa
+1Lµ+g2
+1RWaµ
+1RWa
+1Rµ
+−2(1−a)g0Lg1LWaµ
+0LWa
+1Lµ−2ag0Lg1RWaµ
+0LWa
+1Rµ
+−2ag0Rg1LBµ
+0RW3
+1Lµ−2(1−a)g0Rg1RBµ
+0RW3
+1Rµ/bracerightigg
+, (2.10)
+where
+a=1
+2(1−c) = sin2/parenleftbigg|H|√
+2f/parenrightbigg
+. (2.11)
+Fora= 0 the mass matrices can be easily diagonalized. The charged gauge boson
+masses are
+M2
+W±= 0
+M2
+W±
+L=1
+2/parenleftbig
+g2
+0L+g2
+1L/parenrightbig
+f2(2.12)
+M2
+W±
+R=1
+2g2
+1Rf2,
+and the neutral gauge boson masses are
+M2
+W3= 0
+M2
+B= 0
+M2
+ZL=1
+2/parenleftbig
+g2
+0L+g2
+1L/parenrightbig
+f2(2.13)
+M2
+ZR=1
+2/parenleftbig
+g2
+0R+g2
+1R/parenrightbig
+f2.
+The massless states, WaandB, correspond to the unbroken SU(2)L×U(1)Ygauge sym-
+metry.
+For a nonzero vacuum expectation value, /an}bracketle{t|H|/an}bracketri}ht=v/√
+2, it is also straightforward to
+solve for the mass eigenvalues exactly. There is one massles s neutral boson, corresponding
+to the photon, and the remaining neutral and charged gauge bo son masses can be obtained
+as the solutions to two cubic characteristic equations. In F ig. 2 we plot the light Wand
+Zboson masses and in Fig. 3 we plot the heavy gauge boson masses as a function of v/f
+for representative choices of the parameters: g2
+1= 6 andf= 1 TeV. Clearly, for f= 1
+TeV the only allowed value of v/fis∼0.246, but it is nonetheless interesting to note the
+symmetry of the solutions under the exchange of ( v/f)↔(π−v/f). This is a result of
+the parity symmetry, P1LR, which holds when g1L=g1R. Under this symmetry:
+Waµ
+1L↔Waµ
+1R
+Σ→Σ′= ΣP,
+– 6 –0.00.51.01.52.02.53.00.00.10.20.30.4
+v/Slash1fM/LParen1TeV/RParen1
+Figure 2: Light gauge boson masses ( WandZ) as a function of v/f, forg2
+1= 6 andf= 1 TeV.
+The upper curve is MZand the lower curve is MW.
+0.00.51.01.52.02.53.01.721.741.761.781.80
+v/Slash1fM/LParen1TeV/RParen1
+Figure 3: Heavy gauge boson masses as a function of v/f, forg2
+1= 6 andf= 1 TeV. The curves
+from top to bottom are MZL,MWL,MZR, andMWR.
+with
+P=
+0 0 0−1 0
+0 1 0 0 0
+0 0 1 0 0
+−1 0 0 0 0
+0 0 0 0 −1
+. (2.14)
+The matrix PsatisfiesPTa
+L,RP=Ta
+R,L. It can be shown that the transformed field Σ′
+is related to the original field Σ by a shift of v/f→v/f+π, up to an overall O(4)1
+transformation. This, coupled with the discrete H↔ −Hsymmetry of the model, results
+in the symmetry of the mass solutions.
+As required by a little Higgs model, we will want v/fto be small. Thus, it is useful
+to solve for the masses and mixings perturbatively in a≈[v/(2f)]2. At leading nonzero
+– 7 –order inv/f, the massless charged gauge bosons, W±, gain a mass
+M2
+W±≈1
+4g2
+Lv2, (2.15)
+while the massless neutral gauge bosons, W3andB, mix exactly as in the standard model
+to give the photon Aand theZboson with masses
+M2
+A= 0
+M2
+Z≈1
+4/parenleftbig
+g2
+L+g2
+R/parenrightbig
+v2, (2.16)
+where we have defined the couplings gLandgRby
+1
+g2
+L=1
+g2
+0L+1
+g2
+1L
+1
+g2
+R=1
+g2
+0R+1
+g2
+1R. (2.17)
+Note thatgLandgRplay the roles of the standard model SU(2)LandU(1)Ygauge cou-
+plings, respectively. Of course, the photon is exactly mass less, being associated with the
+unbrokenU(1)EM, with coupling constant egiven by
+1
+e2=1
+g2
+L+1
+g2
+R=1
+g2
+0L+1
+g2
+1L+1
+g2
+0R+1
+g2
+1R. (2.18)
+More details of the gauge boson masses and mixings are given i n Appendix B.
+3. Fermion Sector
+Inthissection, wewillconsideronlyonegenerationofquar ks,althoughmultiplegenerations
+of quarks and leptons can be incorporated as well. We are moti vated by the deconstruction
+ofthe5-dimensional SO(5)×U(1)XGauge-Higgs modelofRef.[30], buttheimplementation
+of fermions in ourmodel benefits fromthe additional flexibil ity afforded by thegeneral non-
+linear sigma model method. In particular, we shall let all of the fermion fields transform as
+non-trivial representations of the global SO(5)0symmetry at site 0 only, and as non-trivial
+representations of the corresponding gauge symmetries, SU(2)0L×U(1)0R.
+For each generation of quarks in the standard model, we will h ave three multiplets of
+SO(5)0, (ψA,ψB,ψC), one each for the left-handed quark doublet QL, the right-handed
+up quarkuR, and the right-handed down quark dR, respectively.∗The multiplets are Dirac
+multiplets, in that each comes in a right-handed and left-ha nded pair,
+ψ≡/parenleftigg
+ψL
+ψR/parenrightigg
+, (3.1)
+exceptthat the standard model fields within the multiplet are missi ng their Dirac partners.
+For example, the QLfield resides in the multiplet ψA
+L, which transforms as the fundamental
+∗Due to our unfortunate choice of notation, we will be using th e subscripts LandRto label the chirality
+of the fermion fields, as well as the two gauged subgroups of SO(5). When applied to a fermion field, the
+subscripts always denote the chirality. Everywhere else th ey label the subgroup of SO(5).
+– 8 –5 ofSO(5), while the corresponding ψA
+Rmultiplet is missing the QRfield. In terms of
+component fields we have
+ψA
+L=
+Q
+χ
+u
+A
+L, ψA
+R=
+0
+χ
+u
+A
+R, (3.2)
+where
+Q=/parenleftigg
+Qu
+Qd/parenrightigg
+andχ=/parenleftigg
+χy
+χu/parenrightigg
+(3.3)
+transform as doublets under SU(2)0Landutransforms as a singlet. Under U(1)0Rthe
+fields transform with a charge given by Y=T3
+R+qX, whereqX= +2/3 for quarks and
+qX= 0for leptons†. Inthisway, we findthat theelectromagnetic charge of each c omponent
+field is given by
+qEM=T3
+L+T3
+R+qX=T3
+L+Y , (3.4)
+a result which holds for the component fields in each SO(5) multiplet. Throughout this
+paper, we will use the symbols y,u, anddto indicate the electromagnetic charges of the
+fields byqEM(y) = +5/3,qEM(u) = +2/3, andqEM(d) =−1/3.
+The right-handed up quark field uRresides in the multiplet ψB
+R, which also transforms
+as the fundamental 5 of SO(5), and has a corresponding Dirac partner multiplet ψB
+L, which
+is missing the uLfield. In terms of component fields we have
+ψB
+L=
+Q
+χ
+0
+B
+L, ψB
+R=
+Q
+χ
+u
+B
+R. (3.5)
+As with the previous multiplets, the Qandχcomponents transform as doublets under
+SU(2)0L, theucomponent transforms as a singlet, and all component fields t ransform
+with charge Y=T3
+R+qXunderU(1)0R.
+Finally, the right-handed down quark field dRresides in the multiplet ψC
+R, which trans-
+forms as the adjoint 10 of SO(5), and has a corresponding Dirac partner multiplet ψC
+L,
+which is missing the dLfield. In terms of component fields we have
+ψC
+L=1√
+2
+−u−φy0 0Qu
+φd−u+0 0Qd
+−y0u+φyχy
+0−y φdu−χu
+χu−χy−QdQu0
+C
+L,
+†In the extra-dimensional gauge-Higgs model the charge qXarises from the extra U(1)Xbulk gauge
+symmetry. In our model, we are free to give the fermion fields a ny charge Yunder the U(1)0R, and so qX
+corresponds to the difference between Yand the canonical charge T3
+R.
+– 9 –ψC
+R=1√
+2
+−u−φy−d0Qu
+φd−u+0−d Qd
+−y0u+φyχy
+0−y φdu−χu
+χu−χy−QdQu0
+C
+R, (3.6)
+where
+u±=1√
+2(u±φu). (3.7)
+UnderSU(2)0L, the fieldsφtransform as triplets, the fields Qandχtransform as doublets,
+and the fields y,u, anddtransform as singlets. Under U(1)0Rthe fields transform with
+a charge given by Y=T3
+R+qX(withT3
+Rin the adjoint representation for ψC), so that
+Eq. (3.4) holds for all fields.
+The Lagrangian density for the fermion fields with Dirac mass es can be written
+LDirac=i¯ψAD /ψA−λAf¯ψAψA+i¯ψBD /ψB−λBf¯ψBψB
++itr/parenleftbig¯ψCD /ψC/parenrightbig
+−λCftr/parenleftbig¯ψCψC/parenrightbig
+, (3.8)
+where the covariant derivatives are
+Dµψ(A,B)=/bracketleftbig
+∂µ−ig0LWaµ
+0LTa
+L−ig0RBµ
+0R/parenleftbig
+T3
+R+qX/parenrightbig/bracketrightbig
+ψ(A,B)
+DµψC=∂µψC−ig0LWaµ
+0L/bracketleftbig
+Ta
+L,ψC/bracketrightbig
+−ig0RBµ
+0R/parenleftbig/bracketleftbig
+T3
+R,ψC/bracketrightbig
++qXψC/parenrightbig
+.(3.9)
+With this Lagrangian all ψAfields have a Dirac mass MA=λAf, allψBfields have a
+Dirac mass MB=λBf, and allψCfields have a Dirac mass MC=λCf,exceptfor the
+fields with missing partners, which are massless. For each ge neration of quarks there will
+be five heavy charge +5/3 fermions: one with mass MA, one with mass MBand three with
+massMC. There will be three heavy charge -1/3 fermions: one with mas sMBand two
+with massMC. There will be eight heavy charge +2/3 fermions: two with mas sMA, two
+with massMBand four with mass MC. The fields QA
+L,uB
+R, anddC
+Rremain massless at this
+point.
+Let us consider how to give the light fermions a mass, by notin g the symmetries of
+the Dirac mass terms in Eq. (3.8). They are written to appear s ymmetric under the
+SO(5)0transformation ψ(A,B)→U0ψ(A,B)andψC→U0ψCU†
+0; however, this symmetry is
+explicitly broken by the missing partners in the SO(5) multiplets. On the other hand, the
+SO(5)1symmetry is preserved by default. In addition, there is a glo balU(1) symmetry
+associated with each of the ψA,ψB, andψCfields, which must be broken to give the light
+fermions a mass.
+We can create objects that transform under the SO(5)1symmetry, by multiplying the
+complete fermion multiplets by the Σ field: ψA′
+L= Σ†ψA
+L,ψB′
+R= Σ†ψB
+R, andψC′
+R= Σ†ψC
+RΣ.
+Since theSO(5)1symmetry is broken explicitly by the gauge interactions to O(4)1, we can
+include this breaking by projecting onto O(4) invariant subspaces, using the O(4)-invariant
+– 10 –vector,
+E=
+0
+0
+0
+0
+1
+(3.10)
+It is useful to think of this vector as a spurion field which tra nsforms as E→U1Eunder
+theSO(5)1transformation. In this way, we can write three Yukawa terms for the fermions
+that have the SO(5)1symmetry broken purely by the vector E. They are
+LYukawa=−/bracketleftig
+λ1f/parenleftbig¯ψA
+LΣ/parenrightbig
+EE†/parenleftig
+Σ†ψB
+R/parenrightig
++√
+2λ2f/parenleftbig¯ψA
+LΣ/parenrightbig/parenleftig
+1−EE†/parenrightig/parenleftig
+Σ†ψC
+RΣ/parenrightig
+E
++λ3f/parenleftbig¯ψA
+LΣ/parenrightbig/parenleftig
+1−EE†/parenrightig/parenleftig
+Σ†ψB
+R/parenrightig
++h.c./bracketrightig
+=−/bracketleftig
+λ1f/parenleftbig¯ψA
+LΣ/parenrightbig
+EE†/parenleftig
+Σ†ψB
+R/parenrightig
++√
+2λ2f/parenleftbig¯ψA
+LψC
+RΣ/parenrightbig
+E (3.11)
++λ3f/parenleftbig¯ψA
+LΣ/parenrightbig/parenleftig
+1−EE†/parenrightig/parenleftig
+Σ†ψB
+R/parenrightig
++h.c./bracketrightig
+,
+where we have used the SO(5) transformation properties of the adjoint representati on to
+simplify the second term. Note that these three terms corres pond directly to the three
+“brane” mass terms in the 5-dimensional SO(5)×U(1)XGauge-Higgs model of Ref. [30].
+In addition we note that the Yukawa terms of Eq. (3.12) preser ve theSO(5)0symmetry,
+while the Dirac mass terms of Eq. (3.8) preserve the SO(5)1symmetry, so that the fermion
+interactions also exhibit the collective symmetry breakin g that is necessary to cancel the
+one-loop quadratic divergences to the Higgs potential.
+According to Refs. [32, 37], the term with λ3results in a large negative correction to
+theTparameter in extra-dimensional models. Furthermore, we ca n forbid this term if we
+assume that the terms that simultaneously break the SO(5)1and the global U(1)’s in the
+fermion sector must be proportional to E. Thus, we will follow the lead of Ref. [30] and
+setλ3= 0. Expanding in terms of component fields, we obtain
+LYukawa=−/bracketleftbiggiscλ1f√
+2|H|/parenleftig
+¯QA
+LH/parenrightig
+uB
+R−isλ2f√
+2|H|/parenleftig
+¯QA
+L˜H/parenrightig
+dC
+R+...+h.c./bracketrightbigg
+,(3.12)
+which contains the same Yukawa terms for the light fermions a s in the standard model. If
+we assume that λ(1,2)≪λ(A,B,C), then this results in masses for the up and down quarks
+of
+Mu≈λ1v/√
+2
+Md≈λ2v/√
+2, (3.13)
+while the heavy fermions get only small shifts from their mas ses ofMA,MB,MC. In
+general,λ1andλ2will be matrices in generation space, leading to weak mixing and the
+CKM matrix.
+– 11 –0.00.51.01.52.02.53.00.00.51.01.52.0
+v/Slash1fMT/LParen1TeV/RParen1
+Figure 4: Charged +2/3 fermion masses, in the top quark sector, as a funct ion ofv/f, forf= 1
+TeV,λA=λ1=√
+2λtandλB= 0.981λt. The curves from top to bottom are MTA,MTB, andMt.
+The only quark for which the approximation λ1≪λ(A,B,C)may not hold is the top
+quark. If we take λ1for the top quark sector of the same order as λ(A,B,C)we find that the
+charge +2/3 fermionsof ψCand onelinear combination of each of thecharge +2/3 fermion s
+ofψAandψBhave mass eigenvalues unaffected by the Yukawa term. The remai ning three
+linear combinations mix dueto the Yukawa term and have masse s, to leading nonzero order
+inv/f, of
+Mt≈λtv/√
+2
+MTA≈/radicalig
+λ2
+A+λ2
+1f (3.14)
+MTB≈λBf,
+where we have defined1
+λ2
+t=1
+λ2
+1+1
+λ2
+A. (3.15)
+We see that even for λ1not small, the top quark mass is down by a factor of v/fcompared
+to the heavy quarks. It is possible to obtain these three mass eigenvalues exactly as the
+solution of a cubic characteristic equation. The three mass es are plotted as a function of
+v/fin Fig. 4. More details of the fermion masses and mixings in th e top quark sector are
+given in Appendix C.
+4. Effective Potential
+In our model, the vacuum expectation value of the Higgs doubl et is driven entirely by
+the radiatively-produced effective potential. The potentia l depends on 7 independent pa-
+rameters: {f,g1,g0L,g0R,λA,λB,λ1}. Here, we have chosen to equate the gauge couplings
+at site 1:g1=g1L=g1R. The fermion parameters λA,λB, andλ1are those for the
+third-generation quark sector. We note that the additional fermion parameters λ2andλC
+can be neglected in the limit of zero bottom quark mass; λ2is directly proportional to the
+– 12 –bottom quark mass, while the heavy states in the ψCmultiplet do not mix in this limit.
+Finally, we must include a cutoff Λ for our theory. Using naive dimensional analysis, we
+choose this to be proportional to the symmetry-breaking sca lefby Λ = 4πf.
+Thesevenparameterslistedabovearenotentirelyunconstr ained,sincewemustrecover
+the standard model at low energies. In particular we must rec over the electroweak gauge
+couplingsg≡gLandg′≡gR, the top Yukawa coupling λt≡√
+2Mt/v, and the Higgs
+vacuum expectation value v. This results in four constraints on the above parameters.
+Three of these relations have been given previously in Eqs. ( 2.17) and Eq. (3.15). Using
+Eqs. (2.17), it is possible to treat g1as independent, while fixing g0Landg0Rby the
+relations
+1
+g2
+0L=1
+g2
+L−1
+g2
+1
+1
+g2
+0R=1
+g2
+R−1
+g2
+1. (4.1)
+Note that these equations imply that g1>gL,R. We impose Eq. (3.15) by defininga mixing
+angle in the top sector,
+sinθt=λ1/radicalig
+λ2
+1+λ2
+A, (4.2)
+so that the top mass parameters are given in terms of θtbyλA=λt/sinθtandλ1=
+λt/cosθt. The fourth constraint is that the minimum of the effective pot ential for the
+Higgs doublet is at /an}bracketle{t|H|/an}bracketri}ht=v/√
+2. In the following, we find it convenient to choose the set
+{f,g1,sinθt}as our free parameters, while varying λBto minimize the effective potential
+at the correct value of v.
+The gauge and fermion contributions to the Higgs potential a re generated at the one-
+loop level and can be expressed by the formulae of Coleman and Weinberg [38]. Because
+of the collective symmetry breaking, there are no quadratic divergences at this order;
+however, there are logarithmic divergences, which must be c utoff at the scale Λ = 4 πf.
+The Coleman-Weinberg potential for our model can be written
+V=Vgauge+Vfermion, (4.3)
+where
+Vgauge=3
+64π2/braceleftbigg
+2 Tr/bracketleftbigg
+M4
+CC(Σ)ln/parenleftbiggM2
+CC(Σ)
+Λ2/parenrightbigg/bracketrightbigg
++Tr/bracketleftbigg
+M4
+NC(Σ)ln/parenleftbiggM2
+NC(Σ)
+Λ2/parenrightbigg/bracketrightbigg/bracerightbigg
+,
+Vfermion=−3
+16π2Tr/bracketleftbigg/parenleftig
+M†Mtop(Σ)/parenrightig2
+ln/parenleftbiggM†Mtop(Σ)
+Λ2/parenrightbigg/bracketrightbigg
+, (4.4)
+whereM2
+CC,M2
+NC, andMtopare given in the appendices in Eq. (B.3), Eq. (B.9), and
+Eq. (C.5), respectively. In general, the logarithm of the cu toff, lnΛ2, may be accompanied
+by a scheme-dependent additive constant, which can only be d etermined within the high-
+energy completed theory. In this paper, we will set these to z ero.
+We are now ready to explore the parameter space of the Coleman -Weinberg potential.
+Using the masses MW,MZ,Mtand the Fermi constant GFas inputs, we impose the
+– 13 –0.00.51.01.52.02.53.00.0950.1000.1050.1100.1150.120
+v/Slash1fV/LParen1TeV4/RParen1
+Figure 5: Coleman-Weinberg Potential as a function of v/f, forg2
+1= 6,f= 1 TeV,λA=λ1=√
+2λtandλB= 0.981λt. This choice of parameters gives v= 246 GeV and MH= 130 GeV.
+0.0 0.1 0.2 0.3 0.40.093650.093700.093750.093800.093850.09390
+v/Slash1fV/LParen1TeV4/RParen1
+Figure 6: Same as Figure 5, but plotted with v/franging from 0 to .4 to show the minimum in
+detail.
+constraints with g2
+L=.426,g2
+R=.122,λ2
+t=.990, and require a minimum of the potential
+atv= 246 GeV. We consider the following range of parameters:
+.5≤g2
+1≤4π
+.1≤sin2θt≤.9 (4.5)
+1 TeV≤f≤10 TeV,
+which assumes that none of the dimensionless parameters in t he set{g1,g0L,g0R,λA,λ1}
+are too large. Within this range of parameters, we find that it is always possible to obtain
+two values of λBfor each choice of {f,g1,sinθt}that give the correct vev. In Figs. 5 and 6
+we plot the potential for a typical set of parameters {f= 1 TeV,g2
+1= 6, sin2θt= 1/2}
+withλB= 0.981λt, that gives v= 246 GeV and MH= 130 GeV.
+Before discussing the two different branches of solutions for λBfurther, it is useful to
+consider the Coleman-Weinberg potential, expanded for sma ll values of the Higgs field H.
+– 14 –0.0 0.2 0.4 0.6 0.8 1.00.80.91.01.11.2
+sin2ΘtΛB/Slash1/LParen1ΛA/Slash121/Slash12/RParen1
+Figure 7: The “small- MH” branch of solutions for λB/(λA/√
+2) as a function of sin2θtforf= 1
+TeVandforthreedifferentvaluesof g1. Fromtoptobottomthethreecurvescorrespondto g2
+1= 0.5,
+g2
+1= 2π, andg2
+1= 4π, respectively.
+We have
+V=m2H†H+λ(H†H)2+···. (4.6)
+The full expressions for m2andλare given in Appendix D; however, we find that the
+qualitative features of the two solutions can be understood from the dominant fermion-
+loop contributions to m2=m2
+gauge+m2
+fermion. We obtain
+m2
+fermion=3
+8π2/braceleftigg
+/parenleftbig
+2M2
+TBλ2
+1−M2
+TAλ2
+t/parenrightbig/parenleftigg
+lnΛ2
+M2
+TA−1
+2/parenrightigg
++2M4
+TBλ2
+1
+M2
+TA−M2
+TBlnM2
+TB
+M2
+TA/bracerightigg
+,(4.7)
+withM2
+TA= (λ2
+A+λ2
+1)f2andM2
+TB=λ2
+Bf2.
+Notethatm2
+fermioncanbeeitherpositiveornegative, duetothecollaboration ofthetwo
+heavy fermions. In fact, in order to find a Higgs vacuum expect ation value with v≪f, it is
+necessary that thecontributionsto m2
+fermioncancel tosomedegree. Assuggested above, this
+can happen in two different ways. Firstly, one could cancel the coefficient of the divergent
+logarithm lnΛ2, which is proportional to (2 M2
+TBλ2
+1−M2
+TAλ2
+t) =λ2
+1f2(2λ2
+B−λ2
+A). This
+cancels exactly for λB=λA/√
+2, giving a completely finite fermion contribution to the ful l
+Coleman-Weinberg potential at one loop. The choice λB≈λA/√
+2 also gives a reasonable
+approximation to the first (“small- MH”) branch of solutions for λB. This can be seen in
+Fig. 7, where we plot λB/(λA/√
+2) for this branch as a function of sin2θtforf= 1 TeV
+and for three different values of g2
+1. Over most of the range of sin2θt, we findλB≈λA/√
+2
+within 10%. As we shall see later in this section, the simple r elation between λAandλB
+is in general modified by ultraviolet effects, but it is still po ssible to find a choice of λB
+that givesv= 246 GeV and a light Higgs boson for most of the parameter spac e. The
+predictions for the Higgs boson mass that correspond to the s olutions given here are shown
+in Fig. 8 for f= 1 TeV and f= 10 TeV for the same three values of g2
+1. For the range of
+parameters given in Eq. (4.5) we find 120 GeV /lessorsimilarMH/lessorsimilar320 GeV, with the lighter values of
+– 15 –0.0 0.2 0.4 0.6 0.8 1.0100150200250300
+sin2ΘtMH/LParen1GeV/RParen1
+Figure 8: The “small- MH” branch predictions for the Higgs boson mass as a function of sin2θt.
+The upper three curves are for f= 10 TeV, while the lower three curves are for f= 1 TeV. Within
+each set of three, the curves correspond from top to bottom to g2
+1= 0.5,g2
+1= 2π, andg2
+1= 4π,
+respectively.
+MHcorresponding to smaller values of λAand larger values of λ1. In particular, for f= 1
+TeV, we obtain MH/lessorsimilar150 GeV over a large range of sin2θt. Interestingly, the predictions
+forMHshow very little dependence on the gauge coupling g1, withMHvarying by only a
+few GeV for 0 .5≤g2
+1≤4π. Furthermore, the predictions show only modest dependence
+onf, withMHincreasing by about 40 GeV as fis increased from 1 TeV to 10 TeV.
+The second (“large- MH”) branch of solutions for λBcan also be identified with a
+cancellation in m2
+fermion. In this case the cancellation occurs for large MTB, with the result
+M2
+TB≈Λ2e−1/2. The exact solutions have 7 /lessorsimilarλB/λt/lessorsimilar9, with corresponding values of
+the Higgs boson mass of 380 GeV /lessorsimilarMH/lessorsimilar910 GeV. As with the other branch of solutions,
+we find that the values of λBandMHdepend mostly on sin2θt, with little dependence on
+g1andf. On the other hand, this branch of solutions is probably not s atisfactory, since it
+requires the mass MTBof one of the heavy fermions to be of the same size as the cutoff Λ .
+In addition, this solution will be strongly affected by the inc lusion of a scheme-dependent
+constant, lnΛ2→lnΛ2+δF, which again shows that the theory with this choice of λB
+will be strongly influenced by unknown dynamics at the cutoff. F inally, the larger values
+ofMHobtained for this branch of solutions also makes it less viab le phenomenologically,
+as we will see in the next section. For these reasons, we focus on the “small- MH” branch
+of solutions in the remainder of this paper.
+One may wonder whether the “small- MH” branch of solutions is also strongly affected
+byultraviolet physicsatthecutoffscale. Forinstance, ift hereisadifferentcutoff associated
+with theψAfermions and the ψBfermions, one might expect that the factor
+/parenleftbig
+2M2
+TBλ2
+1−M2
+TAλ2
+t/parenrightbig
+lnΛ2
+M2
+TA=λ2
+1f2(2λ2
+B−λ2
+A)lnΛ2
+M2
+TA,
+– 16 –/LParen12, 1/Slash12/RParen1/LParen11/Slash12,1/Slash12/RParen1/LParen11,1/RParen1
+/LParen12,2/RParen1
+/LParen11/Slash12,2/RParen1
+0.0 0.2 0.4 0.6 0.8 1.0100150200250300
+sin2ΘtMH/LParen1GeV/RParen1
+Figure 9: Sensitivity of the “small- MH” branch predictions for the Higgs boson mass to non-
+identical fermion cutoffs. All four curves are for f= 1 TeV,g2
+1= 2π. The curves are labeled
+by (ΛA/Λ,ΛB/Λ), where Λ = 4 πf. The dashed curves are the corresponding“large- MH” branch
+predictions for the Higgs boson mass, which lie in the mass-range of t his plot for Λ B/Λ = 1/2.
+which is strongly canceled in this branch of solutions, woul d be replaced by
+λ2
+1f2/parenleftigg
+2λ2
+BlnΛ2
+B
+M2
+TA−λ2
+AlnΛ2
+A
+M2
+TA/parenrightigg
+.
+In Appendix E we present a modification of the fermion sector t hat leaves the fermion
+contribution to the one-loop Coleman-Weinberg potential f or the Higgs boson finite, and
+has exactly the effect just described above. In this case there is an additional term in the
+potential,
+∆Vfermion=−3
+16π2f4λ2
+1s2/braceleftbigg
+2λ2
+BlnΛ2
+Λ2
+B−λ2
+AlnΛ2
+Λ2
+A/bracerightbigg
+, (4.8)
+whichexactly cancels thedependenceontheUVcutoffΛin VfermionofEq.(4.4), exchanging
+it for the dependence on the two large mass parameters, Λ Aand ΛB.
+For ΛA/ne}ationslash= ΛB, the “small- MH” solutions now occur for
+λ2
+B≈λ2
+A
+2/parenleftigg
+ln(Λ2
+A/M2
+TA)
+ln(Λ2
+B/M2
+TA)/parenrightigg
+. (4.9)
+This implies that MTB=λBfis no longer completely determined by MTA(or equivalently,
+byλAor sinθt), since the relationship is modified by the ratio of logarith ms of the unknown
+cutoffs, Λ Aand ΛB. However, the Higgs boson mass is still strongly correlated with the two
+heavy fermion masses MTAandMTB. In Fig. 9 we investigate the sensitivity of the Higgs
+boson mass to UV effects by plotting MHas a function of sin2θt, while varying Λ Aand ΛB
+together and independently between Λ /2 and 2Λ, where Λ = 4 πf. We usef= 1 TeV and
+g2= 2πas representative values in this plot. As expected, and in co ntrast to the “large-
+MH”branchofsolutions, thepredictionfor theHiggs massis ve ry insensitivetovaryingthe
+– 17 –scales together from (Λ A/Λ,ΛB/Λ) = (1/2,1/2) to (2,2), at least for 0 .3/lessorsimilarsin2θt/lessorsimilar0.9.
+On the other hand, for (Λ A/Λ,ΛB/Λ) = (1/2,2) the predictions for MHdecrease by about
+25-40 GeV, while for (Λ A/Λ,ΛB/Λ) = (2,1/2) the predictions for MHincrease by about
+80 GeV. For this latter choice of cutoffs, it can be seen from the figure that a solution for
+v= 246 GeV is only obtained for 0 .6/lessorsimilarsin2θt/lessorsimilar0.8.This is related to the fact that the
+“large-MH” solutions decrease in energy for smaller Λ B, as displayed by the dashed curves
+in Fig. 9. The sensitivity of the Higgs boson mass to non-iden tical fermion cutoffs can be
+understood largely in terms of the residual dependence of th e Higgs quartic coupling λon
+the heavy fermion mass ratio MTB/MTA(see Eq. (D.5) in Appendix D), which in turn is
+affected by Eq. (4.9). Thus, fixing the two heavy fermion masses largely determines the
+Higgs boson mass, with larger values of MHcorrelated with larger values of MTB/MTAfor
+a given sin2θt. In addition, we note that over much of the parameter space th e predicted
+Higgs boson mass is still below 200 GeV for a significant porti on of the range of sin2θt.
+To conclude this section, we comment on the size of the fine-tu ning‡that is needed
+in this model to obtain a Higgs vacuum expectation value with v2≪f2. We have in-
+vestigated this issue by analyzing the fine-tuning of v2with respect to the parameters
+pi∈ {g1L,g1R,g0L,g0R,λA,λB,λ1,ΛA,ΛB}, where the fine-tuning with respect to piis de-
+fined by ∆ pi= (pi/v2)(∂v2/∂pi), following Barbieri and Giudice [34]. We then let the
+total fine-tuning be the combination of each of the separate fi ne-tunings in quadrature,
+∆ = (/summationtext
+i∆2
+pi)1/2, subject to the constraints, (3.15) and (4.1). Details of th e formalism
+that we have followed can be found in Ref. [28]. For f= 1 TeV,g2
+1L=g2
+1R= 2π, and
+ΛA= ΛB= Λ = 4πf, we find values of ∆ of ∼100−140 for Higgs masses between 120
+and 160 GeV, with the dominant contributions coming from ∆ λBand ∆ λA(including the
+associated constraint). These values are comparable to the minimium values obtained for
+the Simplest [19] and Littlest [5] Little Higgs models, whic h are displayed in Fig. 13 of
+Ref. [28]. The fact that the fine-tuning is of similar size in o ur model is not surprising,
+since all of the Little Higgs models considered in Ref. [28], as well as our model, contain
+the exact same large negative contribution to m2from a heavy partner of the top quark:
+δm2=−3λ2
+t
+8π2M2
+TlnΛ2
+M2
+T. (4.10)
+The different models have different mechanisms for (partially) canceling this term to obtain
+a light Higgs boson, but since the size of this term is compara ble in all of the Little Higgs
+models considered, one would expect the amount of fine-tunin g to also be comparable. We
+donote, however, thattheamountoffine-tuningcanbereduce dinourmodelifweallow Λ A
+and ΛBto become as low as Λ /3, which reduces the logarithmic enhancement of the above
+term. In this case we can obtain values of ∆ of ∼40−50 for Higgs masses between 120 and
+160 GeV, withthedominantcontributions nowcomingfrom∆ ΛAand∆ΛB. Theseamounts
+of fine-tuning are typically below the values for the Minimal Supersymmetric Standard
+Model in the same range of Higgs masses as shown in Fig. 13 of Re f. [28]. Given the
+‡We have not considered here the “hidden” fine-tuning necessa ry to maintain the global symmetry of
+the fermion couplings against non-symmetric running, as di scussed in Ref. [33]. Our model, like other Little
+Higgs models, is not obviously immune to this effect.
+– 18 –ambiguities in precisely quantifying the amount of fine-tun ing, we prefer to be conservative
+in our conclusions from this investigation, taking away fro m it simply that the amount
+of fine-tuning in our model is comparable and typically no wor se than other Little Higgs
+models.
+5. Electroweak Constraints
+Thefirstplacetoconsiderfortestingtheexperimentalviab ilityofanybeyond-the-standard-
+model theory is in constraints from electroweak precision m easurements. In our model, the
+electroweak observables receive tree-level corrections f rom the new gauge fields. In fact,
+although the standard model light fermions couple to all of t he massive gauge fields, which
+are mixtures of the gauge fields at site 0 and site 1, they are on ly charged under the
+SU(2)0L×U(1)0Rgauge symmetry. As a result, the corrections to low-energy o bservables
+occur only through electroweak gauge current correlators, and are thus “universal” in the
+sense of Barbieri et al.[39]. The correlators can be easily computed from the quadra tic
+Lagrangian by inverting the subsetof the propagator matrix involving the site-0 fields only.
+This leads to the following expressions for the electroweak parameters [39], to leading order
+inv2/f2:
+ˆS=v2
+4f2/parenleftbig
+sin2φL+ cot2θsin2φR/parenrightbig
+(5.1)
+ˆT= 0 (5.2)
+Y=v2
+2f2cot2θsin4φR (5.3)
+W=v2
+2f2sin4φL. (5.4)
+Here sinφL=gL/g1Land sinφR=gR/g1Rare defined in Eq. (B.4) and Eq. (B.11),
+respectively, and θis the weak mixing angle defined in Eq. (B.13). We can express t he
+couplingsgL≡gandgR≡g′in terms of α(M2
+Z),MZ, andGF, and in addition we have
+v2= 1/(√
+2GF) and
+sin2θ=/bracketleftigg
+4πα(M2
+Z)√
+2GFM2
+Z/bracketrightigg1/2
+. (5.5)
+Notice that the corrections to the electroweak observables are not oblique, since nonzero
+values forYandWsignal the presence of direct corrections, corresponding t o four-fermion
+operator exchanges at zero momentum [39, 40]. Notice also th at the custodial symmetry
+of the model ensures that ˆT= 0 at tree-level.
+The observables of Eqs.(5.1)-(5.4) depend on three unknown parameters: f,g1Land
+g1R. In anO(4)1theory the two couplings are identical, g1L=g1R≡g1, and thus
+we can nicely constrain the model in a two-parameter space ( f,g1). The global fit in
+Ref. [39] to the experimental data implies that a heavy Higgs boson is only compatible with
+positive ˆT; therefore, we only consider the “small- MH” branch of solutions. The combined
+– 19 –0 1 2 3 4 501234
+f/LParen1TeV/RParen1g1
+Figure 10: Bounds on g1andffrom combined experimental constraints on ˆS,Y, andW, at the
+95% confidence level.
+experimental constraints on ˆS,Y, andW, taken from Ref. [39] with the light Higgs fit, give
+the bounds of Fig. 10, where the colored area is excluded at th e 95% confidence level. The
+representative values used in the plots in the previous sect ions,f= 1 TeV and g2
+1= 6, are
+within the allowed region. The bounds in Fig. 10 are not expec ted to be strongly affected
+by loop corrections; however, there may be constraints on th e heavy top quark sector
+coming from one loop contributions to the ˆTparameter. An analysis of these contributions
+is currently underway [36].
+Finally, wemustcommentonthefactthatthecouplingsofthe standardmodelfermions
+to the gauge boson eigenstates, given in Eqs. (3.8) and (3.9) , are not unique, in the sense
+that one can always add operators that correspond to renorma lizations of the broken cur-
+rents:
+∆LDirac=iκA¯ψA
+L/parenleftig
+ΣD /Σ†/parenrightig
+ψA
+L+iκB¯ψB
+R/parenleftig
+ΣD /Σ†/parenrightig
+ψB
+R
++iκC1tr/bracketleftig
+¯ψC
+R/parenleftig
+ΣD /Σ†/parenrightig
+ψC
+R/bracketrightig
++iκC2tr/bracketleftig
+¯ψC
+RγµψC
+R(DµΣ)Σ†/bracketrightig
+.(5.6)
+In the main discussion we have assumed that all of the fermion s act as fundamental point
+particles, charged only under the SU(2)0L×U(1)0Rgauge symmetry. In that case, the
+κicoefficients would arise only perturbatively through loop di agrams, and we can assume
+them to be small. On the other hand, it is possible to imagine a more general scenario
+where these coefficients are of order one. In fact, in the decon struction of the gauge-Higgs
+model of Ref. [30] the fundamental fields that naturally appe ar are actually ψA′
+L= Σ†ψA
+L,
+ψB′
+R= Σ†ψB
+R, andψC′
+R= Σ†ψC
+RΣ, which are charged under the SU(2)1L×SU(1)1Rgauge
+– 20 –symmetry. This corresponds to the case where κA=κB=κC1=κC2= 1. In this case
+the electroweak corrections are not “universal”, and in add ition, there will be a nonzero
+contribution to ˆT. For these reasons, we have chosen the simpler fermion imple mentation
+of Sec. 3, and we assume that the κiare negligible.
+6. Conclusions
+Inthisarticle, wehavepresentedanewLittleHiggs model, m otivated bythedeconstruction
+of a five-dimensional gauge-Higgs model [30]. It is based on t he approximate global sym-
+metry breaking pattern SO(5)0×SO(5)1f→SO(5), with gauged subgroups spontaneously
+breaking under the pattern [ SU(2)0L×U(1)0R]×O(4)1f→SU(2)L×U(1)Yv→U(1)EM,
+where we have made the simplifying assumption of g1L=g1R. The novel features of this
+model are these: the only physical scalar in the effective theo ry is the Higgs boson; the
+model contains a custodial symmetry, which ensures that ˆT= 0 at tree-level; and the
+potential for the Higgs boson is generated entirely through one-loop radiative corrections.
+Due to the collective symmetry breaking in the model these co rrections have no quadratic
+divergences, depending only logarithmically on the cutoff o f the effective theory.
+The fact that the electroweak symmetry breaking is fully rad iatively-generated is a
+unique and intriguing feature of this model. In particular, it implies that the model is more
+constrained, and arguably more predictive, than other Litt le Higgs models. For instance,
+if we use a single cutoff Λ for the fermion logarithmic diverge nces, then once the scale fis
+chosen and the correct value of the Higgs boson vev, v, is imposed, we find that the Higgs
+boson mass, as well as the masses of the heavy partners of the t op quark, depend almost
+exclusively on a single fermion mixing parameter, sin2θt. For the “small- MH” branch, we
+find forf= 1 TeV that the Higgs boson mass satisfies 120 GeV /lessorsimilarMH/lessorsimilar150 GeV over
+most of the range of sin2θt. Forfraised to 10 TeV, these values increase by about 40
+GeV. If we take into account possible UV effects in the fermion s ector by introducing two
+distinct fermion cutoffs Λ Aand ΛB, we still find that the Higgs boson mass is correlated
+with the masses of the heavy top quark partners, and it lies be low 200 GeV for much of
+the parameter space.
+The radiative symmetry breaking is achieved in this model wi th an amount of fine-
+tuning that is of similar size as in other Little Higgs models . Therelation v≪fis obtained
+by a cancellation between the contributions of two different h eavy top quark partners to
+the Higgs boson mass-squared. Once this cancellation is ach ieved, the Higgs boson is
+automatically light in the “small- MH” branch of solutions, with the phenomenologically-
+viable range of masses given above. This contrasts with othe r little Higgs models, where
+an additional operator is included to give a large Higgs quar tic coupling and v≪f, but
+a similar cancellation of contributions to m2is still necessary to keep the Higgs boson
+light [28].
+Wehaveanalyzed thetree-level constraints onthemodelfro melectroweak precisionex-
+perimentsandfoundthatthemodelisviableforareasonably largeandphenomenologically-
+interesting range of fandg1≡g1L=g1R. The model introduces a number of new states,
+which may be probed at the LHC. In addition to the Higgs boson, there are two heavy
+– 21 –neutral vector bosons and two heavy charged vector bosons, w hose masses and couplings
+depend directly on fandg1. In the third-generation fermion sector, there are eight ne w
+heavy up-like quarks, three new heavy down-like quarks, and five new heavy charge 5/3
+quarks. The masses and mixings of some portion of these heavy top quarks will satisfy
+relations required by the radiative symmetry breaking and w hich depend on the Higgs
+boson mass. If the other generations of quarks follow the sam e multiplet structure, which
+is probably necessary to avoid flavor-changing neutral curr ents, this heavy fermion zoo will
+be multiplied by the number of generations. In addition, sim ilar heavy partners for the
+leptons should exist. Since the decay rates of these heavy fe rmions to the SM fermions are
+proportional to mixing angles, which in turn are proportion al to the light fermion masses,
+it is possible that some of these heavy particles may have lon g lifetimes, with interesting
+decay signatures. We expect there to bea rich phenomenology at the LHC, which demands
+a more detailed study [36].
+Acknowledgments
+This work was supported by the US National Science Foundatio n under grant PHY-
+0555544. J.H.Y. would also like to acknowledge the support o f the U.S. National Science
+Foundation under grant PHY-0555545 and PHY-0855561.
+A.SO(5)Generator Matrices
+Here we give a basis for the ten SO(5) generator matrices that is particularly useful for
+our purposes. The 5 ×5 generator matrices in the standard basis are
+/parenleftig
+Tab/parenrightig
+ij=−i√
+2(δa
+iδb
+j−δa
+jδb
+i), (A.1)
+wherea,b= 1...5 (witha < b) are the generator labels, i,j= 1...5 are the row and
+column indices, and we have chosen the normalization, tr/parenleftbig
+TabTcd/parenrightbig
+=δacδbd, so that the
+gaugedSU(2) sub-matrices have the conventional normalization.
+Itispossibletoperformasimilarity transformationonthe sematrices, T′=S†TS, such
+that two of them are simultaneously diagonal. For example, i t is possible to diagonalize
+T′12andT′34by the matrix
+S=1√
+2
+1 0 0−1 0
+i0 0i0
+0 1 1 0 0
+0i−i0 0
+0 0 0 0√
+2
+. (A.2)
+Applyingthissimilarity transformationtoall ofthematri ces andchoosingconvenient linear
+combinationsofthem, weobtainthefollowingsetofbasisma trices:{Ta
+L,Ta
+R,T1,T2,T3,T4},
+– 22 –where
+Ta
+L=
+/parenleftigg
+I⊗/parenleftbig1
+2σa/parenrightbig/parenrightigg0
+0
+0
+0
+0 0 0 0 0
+, Ta
+R=
+/parenleftigg
+−/parenleftbig1
+2σa/parenrightbigT⊗I/parenrightigg0
+0
+0
+0
+0 0 0 0 0
+,
+T1=1
+2
+0 0 0 0 1
+0 0 0 0 0
+0 0 0 0 0
+0 0 0 0 1
+1 0 0 1 0
+, T2=1
+2
+0 0 0 0i
+0 0 0 0 0
+0 0 0 0 0
+0 0 0 0 −i
+−i0 0i0
+, (A.3)
+T3=1
+2
+0 0 0 0 0
+0 0 0 0 1
+0 0 0 0 −1
+0 0 0 0 0
+0 1−1 0 0
+, T4=1
+2
+0 0 0 0 0
+0 0 0 0i
+0 0 0 0i
+0 0 0 0 0
+0−i−i0 0
+.
+In the above expressions, Iis the 2×2 identity matrix and σaare the 2 ×2 Pauli matrices
+fora= 1,2,3.
+B. Gauge Boson Masses and Mixing
+From Eq. (2.10), we can obtain the mass terms for the neutral a nd charged gauge bosons
+of the following form:
+Lmass=Wµ†M2
+CCWµ+1
+2Zµ†M2
+NCZµ, (B.1)
+where the vectors WµandZµare given by:
+Wµ=
+W+µ
+0L
+W+µ
+1L
+W+µ
+1R
+,Zµ=
+W3µ
+0L
+W3µ
+1L
+W3µ
+1R
+Bµ
+0R
+, (B.2)
+withW±µ= (W1µ∓iW2µ)/√
+2 for each of the SU(2) groups.
+B.1 The Charged Sector
+We first consider the charged gauge boson sector. The mass mat rix in this sector takes the
+form:
+M2
+CC=f2
+2
+g2
+0L −(1−a)g0Lg1L−ag0Lg1R
+−(1−a)g0Lg1Lg2
+1L 0
+−ag0Lg1R 0 g2
+1R
+. (B.3)
+– 23 –Fora= 0 this mass matrix can be diagonalized in terms of the mixing angleφL, given by
+sinφL=g0L/radicalig
+g2
+0L+g2
+1L,
+cosφL=g1L/radicalig
+g2
+0L+g2
+1L. (B.4)
+Recalling the coupling gL, defined in Eq. (2.17), this implies
+gL=g0LcosφL=g1LsinφL. (B.5)
+For nonzero vacuum expectation value we can solve perturbat ively in the small pa-
+rameter,
+a= sin2/parenleftbigg|H|√
+2f/parenrightbigg
+=|H|2
+2f2−|H|4
+12f4+···, (B.6)
+There will be one light eigenstate, W±µ, which we will identify as the standard model W±,
+and two heavy eigenstates, W±µ
+LandW±µ
+R. ToO(a2), the masses are
+M2
+W≈f2
+2/bracketleftbig
+2ag2
+L−a2g2
+L/parenleftbig
+cos22φL+1/parenrightbig/bracketrightbig
+M2
+WL≈f2
+2/bracketleftbigg
+(g2
+0L+g2
+1L)−2ag2
+L+a2/parenleftbigg
+g2
+Lcos22φL+g2
+0Lg2
+1Rsin2φL
+g2
+0L+g2
+1L−g2
+1R/parenrightbigg/bracketrightbigg
+M2
+WR≈f2
+2/bracketleftbigg
+g2
+1R+a2/parenleftbigg
+g2
+L−g2
+0Lg2
+1Rsin2φL
+g2
+0L+g2
+1L−g2
+1R/parenrightbigg/bracketrightbigg
+. (B.7)
+Expanding the gauge eigenstates in terms of the mass eigenst ates, toO(a), we obtain
+W±µ
+0L≈W±µ/parenleftig
+cosφL+a
+4sin4φLsinφL/parenrightig
++W±µ
+L/parenleftig
+−sinφL+a
+4sin4φLcosφL/parenrightig
++W±µ
+R/parenleftbigg
+−agL
+g1RcosφL+ag0Lg1Rsin2φL
+g2
+0L+g2
+1L−g2
+1R/parenrightbigg
+W±µ
+1L≈W±µ/parenleftig
+sinφL−a
+4sin4φLcosφL/parenrightig
++W±µ
+L/parenleftig
+cosφL+a
+4sin4φLsinφL/parenrightig
++W±µ
+R/parenleftbigg
+−agL
+g1RsinφL−ag0Lg1RsinφLcosφL
+g2
+0L+g2
+1L−g2
+1R/parenrightbigg
+(B.8)
+W±µ
+1R≈W±µ/parenleftbigg
+agL
+g1R/parenrightbigg
++W±µ
+L/parenleftbigg
+ag0Lg1RsinφL
+g2
+0L+g2
+1L−g2
+1R/parenrightbigg
++W±µ
+R.
+B.2 The Neutral Sector
+The mass matrix for the neutral gauge fields takes the form:
+M2
+NC=f2
+2
+g2
+0L−(1−a)g0Lg1L−ag0Lg1R 0
+−(1−a)g0Lg1Lg2
+1L 0 −ag1Lg0R
+−ag0Lg1R 0 g2
+1R−(1−a)g1Rg0R
+0 −ag1Lg0R−(1−a)g1Rg0Rg2
+0R
+.
+(B.9)
+– 24 –Fora= 0 the mass matrix is block diagonal, so that the SU(2)0L×SU(2)1Land the
+SU(2)0R×SU(2)1Rsub-matrices can be diagonalized separately in terms of the anglesφL,
+defined in Eq. (B.4), and φR, defined similarly by
+sinφR=g0R/radicalig
+g2
+0R+g2
+1R, (B.10)
+cosφR=g1R/radicalig
+g2
+0R+g2
+1R. (B.11)
+The angleφRis related to the coupling gR, from Eq. (2.17), by
+gR=g0RcosφR=g1RsinφR. (B.12)
+After diagonalizing the two sub-matrices, there are two mas sless neutral states, which
+correspond to the standard model W3µandBµ. One linear combination of these is the
+photon, which is massless for arbitrary values of the parame tera. It can be separated out
+in terms of a third angle θ(essentially the weak mixing angle), which is defined by
+sinθ=gR/radicalig
+g2
+L+g2
+R,
+cosθ=gL/radicalig
+g2
+L+g2
+R. (B.13)
+The coupling to the photon is
+1
+e2=1
+g2
+L+1
+g2
+R=1
+g2
+0L+1
+g2
+1L+1
+g2
+0R+1
+g2
+1R, (B.14)
+so thate=gLsinθ=gRcosθ.
+For nonzero vacuum expectation value, there will be four neu tral states: the photon
+Aµ, which is exactly massless, the light eigenstate Zµ, and two heavy eigenstates, ZLand
+ZR. We can solve perturbatively in the parameter afor the masses and mixings of these
+states. To O(a2), the masses are
+M2
+A= 0 (exact)
+M2
+Z≈f2
+2/bracketleftbig
+2a(g2
+L+g2
+R)−a2(g2
+L+g2
+R)/parenleftbig
+cos22φL+cos22φR/parenrightbig/bracketrightbig
+M2
+ZL≈f2
+2/bracketleftbigg
+(g2
+0L+g2
+1L)−2ag2
+L+a2/parenleftbigg
+(g2
+L+g2
+R)cos22φL+G2
+LR
+∆g2/parenrightbigg/bracketrightbigg
+M2
+ZR≈f2
+2/bracketleftbigg
+(g2
+0R+g2
+1R)−2ag2
+R+a2/parenleftbigg
+(g2
+L+g2
+R)cos22φR−G2
+LR
+∆g2/parenrightbigg/bracketrightbigg
+,(B.15)
+where we have defined for compactness:
+GLR=g0Lg1RsinφLcosφR+g1Lg0RcosφLsinφR
+∆g2=g2
+0L+g2
+1L−g2
+0R−g2
+1R. (B.16)
+– 25 –Expanding the gauge eigenstates in terms of the mass eigenst ates, we obtain
+W3µ
+0L≈Aµ(sinθcosφL)+Zµ/parenleftbigg
+cosθcosφL+asin4φLsinφL
+4cosθ/parenrightbigg
++Zµ
+L/parenleftig
+−sinφL+a
+4sin4φLcosφL/parenrightig
++Zµ
+R/parenleftbigg
+−asin4φRcosθcosφL
+4sinθ+aGLRsinφL
+∆g2/parenrightbigg
+W3µ
+1L≈Aµ(sinθsinφL)+Zµ/parenleftbigg
+cosθsinφL−asin4φLcosφL
+4cosθ/parenrightbigg
++Zµ
+L/parenleftig
+cosφL+a
+4sin4φLsinφL/parenrightig
++Zµ
+R/parenleftbigg
+−asin4φRcosθsinφL
+4sinθ−aGLRcosφL
+∆g2/parenrightbigg
+W3µ
+1R≈Aµ(cosθsinφR)+Zµ/parenleftbigg
+−sinθsinφR+asin4φRcosφR
+4sinθ/parenrightbigg
+(B.17)
++Zµ
+L/parenleftbigg
+−asin4φLsinθsinφR
+4cosθ+aGLRcosφR
+∆g2/parenrightbigg
++Zµ
+R/parenleftig
+cosφR+a
+4sin4φRsinφR/parenrightig
+Bµ
+0R≈Aµ(cosθcosφR)+Zµ/parenleftbigg
+−sinθcosφR−asin4φRsinφR
+4sinθ/parenrightbigg
++Zµ
+L/parenleftbigg
+−asin4φLsinθcosφR
+4cosθ−aGLRsinφR
+∆g2/parenrightbigg
++Zµ
+R/parenleftig
+−sinφR+a
+4sin4φRcosφR/parenrightig
+,
+where the coefficients of Aµare exact, while the other coefficients are correct to O(a).
+C. Fermion Masses and Mixing in the Top Quark Sector
+The mass terms for the fermions can be obtained from Eqs. (3.8 ) and (3.12). We are
+assuming that λ3= 0, and that λ1andλ2are small for all fermions, except for the top
+quark. Thus, the only Yukawa coupling that is non-negligibl e isλ1for the top quark sector,
+and the only fermions for which therewill besubstantial mix ing are in thetop quark sector.
+In addition, this Yukawa term only mixes charge +2 /3 quarks, so that we need only be
+concerned with them.
+There are nine charge +2 /3 quarks of each chirality in the top quark sector. Their
+mass terms in the Lagrangian are
+Ltop sector =−λAf/parenleftbig
+¯χtA
+LχtA
+R+¯tA
+LtA
+R/parenrightbig
+−λBf/parenleftbig¯QtB
+LQtB
+R+ ¯χtB
+LχtB
+R/parenrightbig
+−λCf/parenleftbig¯QtC
+LQtC
+R+ ¯χtC
+LχtC
+R+¯φtC
+LφtC
+R+¯tC
+LtC
+R/parenrightbig
+(C.1)
+−λ1f/parenleftbigg
+¯tA
+Lc+is√
+2/parenleftbig¯QtA
+L+χtA
+L/parenrightbig/parenrightbigg/parenleftbigg
+¯tB
+Rc−is√
+2/parenleftbig¯QtB
+R+χtB
+R/parenrightbig/parenrightbigg
++h.c. ,
+wheres= sin(√
+2|H|/f) andc= cos(√
+2|H|/f). The fields that come from the ψCmulti-
+plets are not mixed by the λ1Yukawa-term. They combine to form four Dirac states with
+massesMC=λCf. In addition, we can diagonalize one linear combination of e ach of the
+ψAandψBfields that do not appear in the λ1Yukawa-term. Introducing the new linear
+combinations,
+QtB=1√
+2/parenleftbig
+TB+KtB/parenrightbig
+– 26 –χtB=1√
+2/parenleftbig
+TB−KtB/parenrightbig
+tA=1/radicalbig
+1−s2/2/parenleftbigg
+cTA+is√
+2KtA/parenrightbigg
+(C.2)
+χtA=1/radicalbig
+1−s2/2/parenleftbigg
+cKtA+is√
+2TA/parenrightbigg
+,
+we find that the Dirac field KtA= (KtA
+L,KtA
+R) decouples with mass MA=λAf, and the
+Dirac field KtB= (KtB
+L,KtB
+R) decouples with mass MB=λBf.
+The remaining set of three left-handed and right-handed fer mions mix with a mass
+lagrangian given by
+Ltop mass=−¯TLMtopTR+ h.c., (C.3)
+where
+TL=
+TA
+L
+TB
+L
+QtA
+L
+,TR=
+TA
+R
+TB
+R
+tB
+R
+, (C.4)
+and
+Mtop=f
+λA−iλ1s/radicalig
+1−s2
+2λ1c/radicalig
+1−s2
+2
+0λB 0
+0λ1s2√
+2iλ1sc√
+2
+. (C.5)
+Thisfermionmassmatrixcanbediagonalized withabiunitar ytransformation, VMU†.
+To simplify the following expressions, we recall the definit ion for the top Yukawa coupling,
+Eq. (3.15),
+λ2
+t=λ2
+Aλ2
+1
+λ2
+A+λ2
+1. (C.6)
+We also define
+∆λ2=λ2
+A+λ2
+1−λ2
+B. (C.7)
+Then, to O(s2), we obtain the mass of the light eigenstate (the top quark):
+m2
+t=λ2
+tf2
+2s2+/bracketleftbiggλ6
+tf2
+4λ2
+Aλ2
+1−λ4
+tf2
+2λ2
+1/bracketrightbigg
+s4, (C.8)
+and the masses of the heavy eigenstates:
+mTA′= (λ2
+A+λ2
+1)f2+/bracketleftbigg
+−λ2
+tf2
+2+λ2
+Bλ2
+1f2
+∆λ2/bracketrightbigg
+s2
++/bracketleftbigg
+−λ6
+tf2
+4λ2
+Aλ2
+1+λ4
+tf2
+2λ2
+1−λ4
+Bλ4
+1f2
+(∆λ2)3−λ2
+Aλ2
+1(λ2
+A−λ2
+B)f2
+2(∆λ2)2/bracketrightbigg
+s4(C.9)
+and
+mTB′=λ2
+Bf2+/bracketleftbigg
+−λ2
+Bλ2
+1f2
+∆λ2/bracketrightbigg
+s2+/bracketleftbiggλ4
+Bλ4
+1f2
+(∆λ2)3+λ2
+Aλ2
+1(λ2
+A−λ2
+B)f2
+2(∆λ2)2/bracketrightbigg
+s4.(C.10)
+– 27 –ToO(s2), the left-handed gauge eigenstates in terms of mass eigens tates are
+QtA
+L=/parenleftbigg
+1−s2
+4λ4
+t
+λ4
+A/parenrightbigg
+tL+is√
+2λ2
+t
+λ2
+ATA′
+L+s2
+√
+2λ1
+λBλ2
+A−λ2
+B
+∆λ2TB′
+L, (C.11)
+TA
+L=/parenleftbigg
+1−s2
+4λ4
+t
+λ4
+A−s2
+2λ2
+1λ2
+B
+(∆λ2)2/parenrightbigg
+TA′
+L+is√
+2λ2
+t
+λ2
+AtL+isλ1λB
+∆λ2TB′
+L,(C.12)
+TB
+L=/parenleftbigg
+1−s2
+2λ2
+1λ2
+B
+(∆λ2)2/parenrightbigg
+TB′
+L−s2
+√
+2λ2
+t
+λBλ1tL+isλ1λB
+∆λ2TA′
+L, (C.13)
+while the right-handed gauge eigenstates in terms of mass ei genstates are
+tB
+R=−iλt
+λ1/parenleftbigg
+1+s2
+4λ2
+1(λ2
+1+3λ2
+A)
+(λ2
+1+λ2
+A)2/parenrightbigg
+tR+isλ2
+1
+∆λ2TB′
+R
++λt
+λA/parenleftbigg
+1−s2
+2λ2
+1(λ2
+1+λ2
+A)
+(∆λ2)2−s2
+4λ2
+A(λ2
+1+3λ2
+A)
+(λ2
+1+λ2
+A)2/parenrightbigg
+TA′
+R, (C.14)
+TA
+R=λt
+λ1/parenleftbigg
+1−s2
+2λ2
+1(λ2
+1+λ2
+A)
+(∆λ2)2+s2
+4λ2
+1(λ2
+1+3λ2
+A)
+(λ2
+1+λ2
+A)2/parenrightbigg
+TA′
+R
++iλt
+λA/parenleftbigg
+1−s2
+4λ2
+A(λ2
+1+3λ2
+A)
+(λ2
+1+λ2
+A)2/parenrightbigg
+tR+isλ1λA
+∆λ2TB′
+R, (C.15)
+TB
+R=/parenleftbigg
+1−s2
+2λ2
+1(λ2
+1+λ2
+A)
+(∆λ2)2/parenrightbigg
+TB′
+R+isλt(λ2
+1+λ2
+A)
+λA(∆λ2)TA′
+R. (C.16)
+D. Higgs Potential for small |H|/f
+At small values of the Higgs field H, the one-loop Coleman-Weinberg potential can be
+expanded as
+V=m2H†H+λ(H†H)2+···, (D.1)
+wherethecoupling λwillalsohavelogarithmicdependenceon H†H. Lettingm2=m2
+gauge+
+m2
+fermion, we have
+m2
+gauge=3
+64π2/braceleftigg
+3M2
+WLg2
+L/parenleftigg
+lnΛ2
+M2
+WL−1
+2/parenrightigg
++M2
+ZRg2
+R/parenleftigg
+lnΛ2
+M2
+ZR−1
+2/parenrightigg/bracerightigg
+,(D.2)
+withM2
+WL=M2
+ZL= (g2
+0L+g2
+1L)f2/2,M2
+WR=g2
+1Rf2/2 andM2
+ZR= (g2
+0R+g2
+1R)f2/2, and
+m2
+fermion=3
+8π2/braceleftigg
+/parenleftbig
+2M2
+TBλ2
+1−M2
+TAλ2
+t/parenrightbig/parenleftigg
+lnΛ2
+M2
+TA−1
+2/parenrightigg
++2M4
+TBλ2
+1
+M2
+TA−M2
+TBlnM2
+TB
+M2
+TA/bracerightigg
+,(D.3)
+withM2
+TA= (λ2
+A+λ2
+1)f2andM2
+TB=λ2
+Bf2.
+Expressing the ( H†H)2coupling as λ=λgauge+λfermion, we have
+λgauge=−3
+256π2/braceleftigg
+g2
+0L/parenleftbig
+g2
+1L+g2
+1R/parenrightbig/parenleftigg
+lnΛ2
+M2
+WL+M2
+WR
+M2
+WR−M2
+WLlnM2
+WL
+M2
+WR−1
+2/parenrightigg
++2g4
+L/parenleftigg
+lnM2
+WL
+M2
+W(H)−1
+2/parenrightigg
++/bracketleftigg
+4g2
+LM2
+WLM2
+WR/f2
+M2
+WL−M2
+WR/bracketrightigg
+lnM2
+WL
+M2
+WR
+– 28 –+1
+2(g2
+0L+g2
+0R)(g2
+1L+g2
+1R)/parenleftigg
+lnΛ2
+M2
+ZL+M2
+ZR
+M2
+ZR−M2
+ZLlnM2
+ZL
+M2
+ZR−1
+2/parenrightigg
++g4
+L/parenleftigg
+lnM2
+ZL
+M2
+Z(H)−1
+2/parenrightigg
++g4
+R/parenleftigg
+lnM2
+ZR
+M2
+Z(H)−1
+2/parenrightigg
+(D.4)
++2g2
+Lg2
+R/parenleftigg
+lnM2
+ZL
+M2
+Z(H)+M2
+ZL
+M2
+ZL−M2
+ZRlnM2
+ZR
+M2
+ZL+1
+2/parenrightigg
++/bracketleftigg
+2(g2
+L+g2
+R)M2
+ZLM2
+ZR/f2
+M2
+ZL−M2
+ZR/bracketrightigg
+lnM2
+ZL
+M2
+ZR/bracerightigg
+−m2
+gauge
+6f2.
+and
+λfermion=3
+4π2/braceleftigg
+λ4
+t
+4/parenleftigg
+lnM2
+TA
+M2
+t(H)−1
+2/parenrightigg
+−ln(1−x)/bracketleftbiggλ4
+1(2−x)
+x3+λ2
+1λ2
+t(1−x)
+x2+λ2
+1λ2
+A
+x/bracketrightbigg
+−/bracketleftbigg2λ4
+1
+x2+λ2
+1λ2
+t
+x/bracketrightbigg/bracerightbigg
+−2m2
+fermion
+3f2, (D.5)
+wherex= 1−M2
+TA/M2
+TB. In addition, in the above formulae, we use the field-depende nt
+masses for the light fields: M2
+W(H) =g2
+L(H†H)/2,M2
+Z(H) = (g2
+L+g2
+R)(H†H)/2, and
+M2
+t(H) =λ2
+t(H†H).
+E. Fermion Sector with Complete SO(5)Multiplets and Decoupled SM
+Partners
+In order to probe the sensitivity of the model to UV completio n of the fermion sector,
+we consider a modification that leaves the fermion contribut ion to the effective potential
+completely finite at one loop§. First, we make the fields ψA
+RandψB
+Linto complete SO(5)
+multiplets by reinstating the missing SM partners, QA
+RanduB
+L, in Eqs. (3.2) and (3.5).
+Then we decouple them by adding two new fermions, Q′A
+Landu′B
+R, which mix via large
+mass terms,
+∆Lmass=−Λ′
+A¯Q′A
+LQA
+R−Λ′
+B¯uB
+Lu′B
+R+h.c. . (E.1)
+With this modification, the Dirac mass terms proportional to λAandλBof Eq. (3.8) now
+preserve both the SO(5)0andSO(5)1symmetries, since the Dirac fields ψAandψBare in
+completeSO(5) multiplets. Instead, the collective symmetry breaking occurs through the
+Yukawa terms of Eq. (3.12), which break the SO(5)1symmetry, and the decoupling mass
+terms of Eq. (E.1), which break the SO(5)0symmetry. However, these two symmetry-
+breaking terms contain no fermion fields in common; therefor e, any one-loop diagram
+that contributes to the Higgs potential and breaks both SO(5) symmetries must contain
+Dirac mass insertions to mix the fermion fields (in addition t o the two symmetry-breaking
+insertions). The requirement of the three separate contrib utions to the one-loop diagrams
+renders them completely finite.
+§We are grateful to an anonymous referee for suggesting this m odification of the fermion sector.
+– 29 –With the modified fermion sector, the masses of all of the orig inal eigenstates are un-
+changed, uptocorrections of O(f2/Λ′2
+A,B). Inaddition, therearetwo newheavy eigenstates
+with Higgs-field-dependent masses given by
+M2
+ΛA= Λ′2
+A+λ2
+Af2+λ2
+1λ2
+Af4
+Λ′2
+As2
+2+···
+M2
+ΛB= Λ′2
+B+λ2
+Bf2+λ2
+1λ2
+Bf4
+Λ′2
+Bc2+···. (E.2)
+wheres= sin(√
+2|H|/f)andc= cos(√
+2|H|/f), andwehaveneglectedtermsof O(f6/Λ′4
+A,B).
+Including the effects of the heavy mass eigenstates in the Cole man-Weinberg effective po-
+tential gives a new contribution of
+∆Vfermion=−3
+16π2f4λ2
+1s2/braceleftbigg
+2λ2
+B/parenleftbigg
+lnΛ2
+Λ′2
+B−1
+2/parenrightbigg
+−λ2
+A/parenleftbigg
+lnΛ2
+Λ′2
+A−1
+2/parenrightbigg/bracerightbigg
+.(E.3)
+Redefining Λ′
+A,B=e−1/4ΛA,B, we obtain
+∆Vfermion=−3
+16π2f4λ2
+1s2/braceleftbigg
+2λ2
+BlnΛ2
+Λ2
+B−λ2
+AlnΛ2
+Λ2
+A/bracerightbigg
+, (E.4)
+which is exactly the modified potential studied in section 4. As expected, the dependence
+on the UV cutoff Λ in Eq. (E.4) exactly cancels with that from th e other fermion fields,
+exchanging it for a dependence on the scales Λ Aand ΛB.
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+– 32 –
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+arXiv:1001.0585v3  [math.AC]  5 Mar 2014FILTERING FREE RESOLUTIONS
+DAVID EISENBUD, DANIEL ERMAN, AND FRANK-OLAF SCHREYER
+Abstract. A recent result of Eisenbud-Schreyer and Boij-S¨ oderberg prov es that the Betti
+diagram of any graded module decomposes as a positive rational linea r combination of pure
+diagrams. When does this numerical decomposition correspond to a n actual filtration of
+the minimal free resolution? Our main result gives a sufficient condition for this to happen.
+We apply it to show the non-existence of free resolutions with some p lausible-looking Betti
+diagramsand to study the semigroupof quiver representationsof the simplest “wild” quiver.
+1.Introduction
+Letkbe a field, let S:=k[x1,...,x n] be the polynomial ring, and let Mbe a finitely
+generated graded S-module. We write
+FM: 0/d47/d47FM
+pφp/d47/d47...φ2/d47/d47FM
+1φ1/d47/d47FM
+0
+for the graded minimal free resolution of M. We define βi,j(FM) =βi,j(M) by the formula
+FM
+i=/circleplusdisplay
+j∈ZS(−j)βi,j(M).
+The underlying question of this paper is as follows.
+Question 1.1. When does a knowledge of the numbers βi,jimply that the module Mdecom-
+poses as a direct sum? More generally, when can we deduce from the Betti numbers that the
+Mhas a submodule M′whose free resolution FM′is a summand, term by term, of FM?
+We will say that a submodule M′⊂Miscleanly embedded if it satisfies the condition in
+the last sentence of the question—that is, if the natural map
+TorS
+i(M′,k)→TorS
+i(M,k)
+is a monomorphism for every i. Of course any summand is cleanly embedded.
+Here is a well-known example where knowledge of the βi,jallows us to predict a summand.
+Suppose that Mis zero in negative degrees, that is, β0,j(M) = 0 for j <0. Ifβn,n(M) =b
+thenMcontains/parenleftbig
+S/(x1,...,x n)/parenrightbigbas a direct summand. (Reason: βn,n(M) is, by local
+duality, equal to the component of the socle of Min degree 0.)
+Question 1.1 has a special interest in light of Boij-S¨ oderberg Theor y: The conjecture
+of Boij and S¨ oderberg, proven by Eisenbud and Schreyer in [ES09 ] and then extended in
+2010Mathematics Subject Classification. 13D02, 13C05, 13C14.
+The first author was partially supported by an NSF grant, and the s econd author was partially supported
+by an NDSEG fellowship and an NSF postdoctoral fellowship.
+1[BS12], says that the Betti diagram of Mcan be written uniquely as a positive rational
+linear combination
+β(M) =s/summationdisplay
+t=0ctπdt
+ofpureBetti diagrams πdtwhere the degree sequences dtsatisfyd0< d1< ... < ds. Here a
+degree sequence is an element
+d= (d0,...,d n)∈(Z∪{∞})n+1withdi+1≤di+1for alli,
+and the (rational) Betti diagram πdis given by
+(1) βi,j(πd) =/braceleftigg
+0 j/n⌉}ationslash=di/producttext
+k/negationslash=i,dk<∞1
+|di−dk|j=di,
+anddt≤dt+1means that dt
+i≤dt+1
+ifor every i. (see§2 for the definition of a pure diagram
+and a summary of the necessary part of Boij-S¨ oderberg theory ).
+With this result in mind it is natural to refine Question 1.1 and ask:
+Question 1.2. When does the decomposition of the Betti diagram of a graded m oduleM
+into pure diagrams arise from some filtration of Mby cleanly embedded submodules?
+In particular, when does the Betti diagram c0πd0correspond to the resolution of a cleanly
+embedded submodule M′⊂M?
+Certainly such a submodule M′does not always exist: often the numbers βi,j(c0πd0) are
+not even integers, and there are subtler reasons as well (see Exa mple 1.7 and§6). However,
+our main result says such a module M′doesexist when d0is “sufficiently separate” from the
+rest of the dt. To make this precise, we write
+(*) d0≪d1ifd0< d1,d0
+2≤d1
+1and ifd0
+i≤d1
+2+i−1 fori >2.
+Theorem 1.3 ((Existence of a cleanly embedded pure submodule)) .Letdim(S)≥2and
+letMbe a finite length graded S-module with Boij-S¨ oderberg decomposition
+β(M) =s/summationdisplay
+i=0ciπdi.
+(1) Ifd0≪d1, then there is a cleanly embedded submodule M′⊂Mwithβ(M′) =c0πd0.
+In particular, the diagram c0πd0has integer entries.
+(2) Ifd0≪d1andd0
+n−n < d1
+1, thenM′is a direct summand of M.
+Withcorrespondinghypothesesonall di,weobtainafullcleanfiltration(asinDefinition2.4).
+Corollary 1.4. If, with hypotheses as in Theorem 1.3, d0≪d1≪···≪ ds, thenMadmits
+a filtration 0 =M0⊂···⊂ Ms⊂Ms+1by cleanly embedded submodules Misuch that
+β(Mi+1/Mi) =ciπi
+d.
+In the following, and in the rest of the paper, we write the Betti diagram ofM,β(M),
+as a matrix whose entry in column iand row i+jisβi,j(M). In examples, we follow the
+convention that the upper left entry of β(M) corresponds to β0,0(M).
+2Example 1.5.LetS=k[x,y,z]. IfMis any module with
+β(M) =/parenleftbigg
+4 8 6−
+−6 8 4/parenrightbigg
+=/parenleftbigg
+3 8 6−
+− − − 1/parenrightbigg
++/parenleftbigg
+1− − −
+−6 8 3/parenrightbigg
+,
+then, since the corresponding degree sequences are d0= (0,1,2,4) andd1= (0,2,3,4),
+Theorem 1.3(2) implies that Msplits as M=M1⊕M2with
+β(M1) =/parenleftbigg
+3 8 6−
+− − − 1/parenrightbigg
+andβ(M2) =/parenleftbigg
+1− − −
+−6 8 3/parenrightbigg
+.
+The technique we develop to prove Theorem 1.3 actually yields the res ult in more general
+(but harder to formulate) circumstances; see §6.
+Application: The Insufficiency of Integrality. One application of Theorem 1.3 is to
+prove the non-existence of resolutions having otherwise plausible- looking Betti diagrams:
+Proposition 1.6. Letp∈Zbe any prime. Then there exists a diagram Dwith integral
+entries, such that cDis the Betti diagram of a module if and only if cis divisible by p.
+This result simultaneously strengthens parts (2) ,(3) and (4) of [Erm09, Thm. 1.6]. Its proof
+is given in§7. The following question, posed in [EFW11, Conjecture 6.1], remains o pen:
+do all but finitely many integral points on a ray of purediagrams correspond to the Betti
+diagram of a module?
+Example 1.7.There is no graded module Mof finite length with Betti diagram
+D:=
+2 3 2−
+−3 3−
+−2 3 2
+.
+Reason: The Boij-S¨ oderberg decomposition of Dis
+D=1
+5
+6 15 10−
+− − − −
+− − − 1
++3
+5
+1− − −
+−5 5−
+− − − 1
++1
+5
+1− − −
+− − − −
+−10 15 6
+
+The corresponding degree sequences are d0= (0,1,2,5),d1= (0,2,3,5) andd2= (0,3,4,5),
+soTheorem 1.3impliesthatamodulewithBettidiagram Dwouldadmit acleanly embedded
+submodule M′with betti diagram
+β(M′) =1
+5
+6 15 10−
+− − − −
+− − − 1
+=
+6
+53 2−
+− − − −
+− − −1
+5
+.
+This is absurd, since the entries of the diagram are not integers.
+Now consider the diagrams cD, wherecis a rational number. The same argument implies
+that these are not Betti diagrams of modules of finite length unless cis an integral multiple
+of 5. On the other hand, if R:=k[x,y,z]/(x,y,z)3,ωR(3) is the twisted dual of R, and
+R′:=k[x,y,z]/(x2,y2,z2−xy,xz,yz ), then
+(2) β(R⊕ωR(3)⊕R′⊕3) =
+10 15 10−
+−15 15−
+−10 15 10
+= 5D.
+3We conclude that cDis the Betti diagram of a module of finite length if and only if cis an
+integral multiple of 5. /square
+Application: invariants of the representations of •/d47/d47/d47/d47/d47/d47•It was proven in [Erm09,
+Thm 1.3] that the semigroup of all Betti diagrams of modules with bou nded regularity
+and generator degrees is finitely generated, and the generators were worked out in some
+small examples. In those cases the semigroup coincides with the set of integral points in
+the positive rational cone generated by the Betti diagrams of mod ules. With the added
+power of Theorem 1.3 we can determine the generators in the first c ase where this does not
+happen: the case of modules over k[x,y,z] having only two nonzero graded components,
+M=M0⊕M1.
+This case has an interpretation in the representation theory of qu ivers. Consider repre-
+sentations over kof the quiver with three arrows:
+Q:•/d47/d47/d47/d47/d47/d47•
+The problem of classifying representations of Qup to isomorphism (or, equivalently, clas-
+sifying triples of matrices up to simultaneous equivalence) is famously of “wild type”; the
+variety of classes of representations with a given dimension vector D:= (dimM0,dimM1)
+has dimension that grows with D, and many components.
+The Betti diagram of Mprovides a discrete invariant of such a representation. The
+(Castelnuovo-Mumford) regularity of Mis 1, so the Betti diagram has the form
+β(M) =/parenleftbigg
+β0,0β1,1β2,2β3,3
+β0,1β1,2β2,3β3,4/parenrightbigg
+.
+Some of the numbers in this diagram are easy to understand: for ex ample,β3,3is the dimen-
+sion of the common kernel of the three matrices, and β0,1is the dimension of M1modulo
+the sum of the images of the matrices. Passing to an obvious subquo tient, therefore, we
+may assume that β3,3=β0,1= 0. In this case β0,0= dimM0andβ1,1= dimM1−3β0,0are
+determined by the dimension vector D, as areβ3,4andβ2,3and the difference β1,2−β2,2.
+However, the value of β2,2is a more subtle invariant, semicontinuous on the family of
+equivalence classes of representations. In §8 we determine the semigroup of Betti diagrams
+β(M) that come from representations of Q.
+A monotonicity principle and the proof of Theorem 1.3. In order to prove Theo-
+rem 1.3, we must construct an appropriate submodule of Mbased only on the information
+contained in the Betti diagram of M. Our construction is based on the notion of a numerical
+subcomplex .
+Definition 1.8. Anumerical subcomplex of a minimal free resolution FMis a subcomplex
+G“whose existence is evident from the Betti diagram β(FM)” in the sense that there is a
+sequence of integers αisuch that each Giconsists of all the summands of FM
+igenerated in
+degrees< αi, and each FM
+i/Giis generated in degrees > αi+1.
+For instance, in the example in (2), the linear strand
+S10←S15(−1)←S10(−2)←0.
+ofFMis a numerical subcomplex of FMdetermined by α= (1,2,3,4).
+For the proof of Theorem 1.3, we use a numerical subcomplex FMto construct a submod-
+uleM′⊆M, whereβ(M′) =c0πd0. Defining the appropriate numerical subcomplex and
+4the submodule M′will be relatively straightforward. However, since numerical comple xes
+generally fail to be exact, it is not a priori clear that we should be able to determine the Betti
+diagram of the submodule M′. This computation relies on a monotonicity principle about the
+Betti numbers of pure diagrams.
+Theorem 1.9 (Monotonicity principle) .Suppose that d,eare degree sequences with di=ei
+anddi+1=ei+1. Ifd < ethen
+βi,di(πd)
+βi+1,di+1(πd)<βi,ei(πe)
+βi+1,ei+1(πe)
+This theorem turns out to be surprisingly powerful, and we apply it to compute the Betti
+diagram of our submodule M′⊆M. This Monotonicity Principle is related to some of the
+numerical inequalities for pure diagrams from [McC11, Lemma 4.1] and [Erm10,§3].
+This paper is organized as follows. In the next section we provide the necessary back-
+ground on Boij-S¨ oderberg theory. In §3–§5, we develop our technique for producing cleanly
+embedded submodules. We then discuss some limitations and extensio ns of our main result
+in§6. The last two sections are devoted to the applications described a bove.
+2.Notation and Background on Boij-S ¨oderberg Theory
+Throughout, all modules are assumed to be finitely generated, gra dedS-modules. We use
+(FM,φM) to refer to the minimal resolution of a module M, though we may omit the upper
+indexMin cases where confusion is unlikely.
+Definition 2.1. Fix a module M, a minimal free resolution (FM,φM)ofM, and a sequence
+of integers f= (f0,...,f n)∈Zn+1. We define (F(f)M,φ(f)M)to be a sequence of free
+modules and maps
+···/d47/d47F(f)M
+iφ(f)M
+/d47/d47F(f)M
+i−1/d47/d47···
+as follows. Let ιi:F(f)M
+i→FM
+ibe the inclusion of the graded free submodule consisting of
+all free summands of FM
+igenerated in degrees < fi, and let πi:FM
+i→F(f)M
+ibe a splitting
+ofιiwhose kernel consists of free summands generated in degrees ≥fi. Finally, set
+φ(f)M
+i=πi−1◦φM
+i◦ιi:F(f)M
+i→F(f)M
+i−1.
+Note that F(f)Mis not necessarily a complex (see Example 2.3).
+Example 2.2.Let
+β(FM) =
+12 26 16−
+− − − 1
+−5−1
+− −12 17
+.
+ThenF((1,3,5,6))Mis a numerical subcomplex with Betti diagram
+β(F(1,3,5,6)M) =
+12 26 16−
+− − − 1
+− − − 1
+− − − −
+.
+This is the largest numerical subcomplex containing only the linear firs t syzygies.
+5Example 2.3.ForS=k[x,y,z], letM=S/(x,y,z2). Then
+β(M) =/parenleftbigg
+1 2 1−
+−1 2 1/parenrightbigg
+.
+We have
+β(F(1,2,4,5)M) =/parenleftbigg
+1 2 1−
+− −2 1/parenrightbigg
+,
+butF(1,2,4,5)Mis not a complex since
+φ(1,2,4,5)M
+1=/parenleftbigx y/parenrightbig
+andφ(1,2,4,5)M
+2=/parenleftbigg
+0z2−y
+−z20x/parenrightbigg
+do not compose to 0.
+Wethink of a Betti diagram β(M) as anelement ofthe infinite dimensional Q-vector space
+V:=⊕n
+i=0⊕j∈ZQ. Thesemigroup of Betti diagrams Bmodis the subsemigroup of Vgenerated
+byβ(M) for all modules M. We define the cone of Betti diagrams BQas the positive cone
+spanned by BmodinV, and we define Bintas the semigroup of lattice points in BQ. See
+[Erm09] for comparisons between BintandBmod.
+Boij-S¨ oderberg theory describes the cone BQ.1As conjectured in [BS08] and proven
+in [ES09,BS12], the extremal rays of BQare spanned by pure diagrams πd(as defined above
+in (1)) where d= (d0,...,d n)∈(Z∪{+∞})n+1is a degree sequence, i.e. di+1≤di+1. We
+will also use the notation /tildewideπdfor the smallest integral point on the ray spanned by πd. So
+/tildewideπd=mπdwithm=lcm(/producttext
+k/negationslash=i,k≤c|di−dk|,i= 0,...,t) wheret= max{i|di<∞}is the
+length of the degree sequence.
+The cone BQhas the structure of a simplicial fan: if we partially order the sequen cesd
+termwise, then there is a unique decomposition of any β(M)∈BQas
+(3) β(M) =s/summationdisplay
+i=0ciπdi
+withci∈Q≥0andd0<···< ds. We refer to this as the Boij-S¨ oderberg decomposition of
+β(M). For an expository account of Boij-S¨ oderberg theory, see on e of [SE10,Flø12].
+If ∆ = ( d0,...,ds) is a chain of degree sequences d0< d1<···< ds, then we use
+the notation BQ(∆),Bint(∆) and Bmod(∆) for the restrictions of BQ,Bint,andBmodto the
+simplicial cone generated by the pure diagrams whose degree seque nces lie in ∆. When
+D∈BQ(∆) with ∆ = ( d0,...,ds), thetop strand ofDconsists of the entries parametrized
+byd0, namely/parenleftig
+β0,d0(D),β1,d0
+1(D),...,β n,d0n(D)/parenrightig
+. We refer to c0πd0as the first step of the
+Boij-S¨ oderberg decomposition, and so on. We will repeatedly use t he fact that the algorithm
+for decomposing any such Dproceeds as a greedy algorithm on the top strand of D∈BQ.
+See [ES09,§1] for details. ‘
+Definition 2.4. Afull clean filtration of a finitely generated graded S-moduleMis a sequence
+of cleanly embedded submodules
+M=M0/supersetnoteqlM1/supersetnoteqlM2/supersetnoteql···/supersetnoteqlMt= 0
+1There is also a“dual” side of the theory that describes the cone of c ohomologydiagrams of vector bundles
+and coherent sheaves on Pn; see [ES09,ES10].
+6such that each Mi/Mi+1has a pure resolution.
+It is immediate that we can put together full clean filtrations in exten sions:
+Lemma 2.5. LetM′⊆Mbe a cleanly embedded submodule, and let M′′=M/M′. IfM′
+andM′′admit full clean filtrations, then so does M. /square
+Many numerical invariants of Mmay be computed in terms of the Betti diagram of M,
+including the projective dimension of M, the depth of M, the Hilbert polynomial of M, and
+more. We extend all such numerical notions to arbitrary diagrams D∈V. For instance, we
+say that the diagram
+D=/parenleftbigg
+18
+32−
+− − −1
+3/parenrightbigg
+has projective dimension 3.
+WhenMhasfinitelength, weusethenotation M∨forthegradeddualmoduleHom( M,k).
+3.The North fork of FM
+We begin the construction of cleanly embedded submodules by study ing the maximal
+numerical subcomplex of FMthat contains only the first syzygies of minimal degree. For
+instance, let Mbe any module such that
+(4) β(M) =
+10 15 10−
+−15 15−
+−10 15 10
+.
+Mis generated entirely in degree 0, and Mhas some linear first syzygies. In this case,
+the maximal numerical subcomplex of FMcontaining these linear first syzygies is the linear
+strand of FM, which corresponds to F(f)Mwheref= (1,2,3,5):
+F(f)M:S10←S(−1)15←S(−2)10←0.
+This type of numerical subcomplex plays an important role for us, an d we refer to it as
+theNorth fork of FM. This name is meant to suggest that F(f)Mconsists of the part of
+the complex that “flows through” the minimal degree first syzygies . The following definition
+states this more formally.
+Definition 3.1. TheNorth fork of FMis(F(f)M,φ(f)M), wherefis defined as follows: Let
+f0be one more than the maximal degree of a generator of Mand letf1be one more than
+the minimal degree of a first syzygy of M. Fori >1, set
+(5) fi:= min{j|j > fi−1,andβi,j(M)/n⌉}ationslash= 0}.
+Note that fi> fi−1with the possible exception that f1might be smaller than or equal to
+f0. Allowing f1≤f0slightly streamlines our argument in the case of a module generated
+in multiple degrees. Namely, since all generators of F0have degree < f0, it follows that φM
+1
+has the block form φM
+1=/parenleftbigφ(f)M
+0bM
+1/parenrightbig
+.
+Lemma 3.2. The North fork of FMis a complex.
+7Proof.By splitting the inclusions F(f)M
+i→FM
+i, we may decompose each φM
+iasφM
+i=/parenleftbigaM
+ibM
+i/parenrightbig
+, where the source aM
+iisF(f)M
+i. SinceF(f)M
+iconsists of all summands generated
+in degree < fi, the image of aM
+idoes not depend on the choice of basis for Fi.
+From the definition of the fiit follows that aM
+ifactors through the inclusion F(f)M
+i−1→
+FM
+i−1. As in Definition 1.8, we use φ(f)M
+ito denote the induced map φ(f)M
+i:F(f)M
+i→
+F(f)M
+i−1. We may thus rewrite φM
+ias a block upper triangular matrix:
+(6) φM
+i=/parenleftbiggdeg< fideg≥fi
+deg< fi−1φ(f)M
+i∗
+deg≥fi−10∗/parenrightbigg
+for alli >1. It follows immediately that ( F(f)M,φ(f)M) is actually a complex. /square
+Example 3.3.LetMbe as in (4). The Betti diagram of N:= coker( φ(f)M
+1) has the form
+β(N) =
+10 15 10−
+− − ∗ −
+− − ∗ ∗
+− −......
+,
+where∗indicates and unknown entry. The β3,3andβ3,4entries of β(N) are zero because any
+low-degree third syzygy of Nwould lift to a third syzygy of M(see Lemma A.2 for more
+details).
+In Section 5 we shall show that, under the hypotheses of Theorem 1.3, the cleanly em-
+bedded submodule of Mwhose existence is asserted by the theorem is the module H0
+m(N),
+whereN:= coker( φ(f)M
+1). The following lemma relates the resolution of a module defined
+via the North fork or a similar construction, like the module coker( φ(f)M
+1), to the resolution
+of the original module M. (The unusual numbering of the following lemma matches the
+published version of this paper, where this lemma appeared in a separ ate corrigendum.)
+Lemma A.2. LetMbe a module and let f= (f0,...,f n)∈Zn+1be any sequence such that
+f0is greater than the maximal degree of a generator of M, and such that
+f2:= min{j|j > f1andβ2,j(M)/n⌉}ationslash= 0}.
+LetNbe the cokernel of φ(f)M, as defined in Definition 2.1, and let Kbe the kernel of the
+natural surjection N→M.
+(1)Tori(K,k)is generated in degree ≥f2+i−1.
+(2) Ife≤f2+i−2then we get an injection: Tori(N,k)e→Tori(M,k)e
+(3) Ife < f2+i−2then we also get a surjection: Tori(N,k)e→Tori(M,k)e
+Proof.By definition of N, we have that Tor 1(N,k)⊆Tor1(M,k), so the long exact sequence
+in Tor induces:
+···→Tor2(M,k)→Tor1(K,k)0→Tor1(N,k)→...
+Letmbe the minimal degree of a generator of Tor 1(K,k). Since Kis generated in degree
+≥f1(see Definition 2.1), it follows that m > f 1; the surjectivity of Tor 2(M,k)→Tor1(K,k)
+then implies that β2,m(M)/n⌉}ationslash= 0. Hence m≥f2and (1) follows immediately.
+For (2) and (3), we consider:
+···→Tori(K,k)e→Tori(N,k)e→Tori(M,k)e→Tori−1(K,k)e→···
+8Ife≤f2+i−2, then (1) implies that Tor i(K,k)e= 0, which proves (2). If e < f2+i−2,
+then (1) implies Tor i−1(K,k)e= 0, which yields (3). /square
+4.The Monotonicity Principle and its application
+Proof of Theorem 1.9. Ifdandehavethesamelengthasdegreesequences, then, byinserting
+a maximal chain of degree sequences between dande, we see that it is enough to treat the
+case where dk=ekfor all but one value of k, which cannot be equal to ior toi+ 1, and
+whereek=dk+1. In view of the Herzog-K¨ uhl equations (1), the desired inequa lity is
+|dk−di+1|
+|dk−di|<|1+dk−di+1|
+|1+dk−di|.
+Ifk > i+1 then 0 < dk−di+1< dk−di, so the result has the form
+a
+b<a+1
+b+1
+where 0 < a < b, and this is immediate. In the case k < i, on the other hand, we have
+di+1−dk> di−dk> di−dk−1>0, so the result has the form
+a
+b<a−1
+b−1
+witha > b >1, and again this is immediate.
+Ifdandehave different lengths as degree sequences, then we can immediate ly reduce to
+the case d= (d0,...,d t)∈Zt+1ande= (d0,...,d t−1,∞)∈(Z∪{∞})t+1. In this case, we
+setdℓ:= (d0,d1,...,d t−1,dt+ℓ) for allℓ∈N. A direct computation via (1) yields:
+πe= lim
+ℓ→∞ℓ·πdℓ.
+Since all of the degree sequences dℓhave length t, we conclude that
+βi,di(πd)
+βi,di+1(πd)<βi,di(πd1)
+βi,di+1(πd1)<···<βi,di(πdℓ)
+βi,di+1(πdℓ)<βi,di(πdℓ+1)
+βi,di+1(πdℓ+1)<···<βi,ei(πe)
+βi,ei+1(πe).
+/square
+The next example shows how the Monotonicity Principle can be used to determine Betti
+diagrams.
+Example 4.1.Consider MandNas in Example 3.3. Recall that the Betti diagram of Nhas
+the form
+(7) β(N) =
+10 15 10−
+− − ∗ −
+− − ∗ ∗
+− −......
+.
+Can one determine the remaining entries of the above Betti diagram from the given infor-
+mation?
+Since we know that F(f)Mis a numerical subcomplex of the minimal free resolution of
+N, we at least know something about the top strand of β(N). One can thus attempt to
+compute the first Boij-S¨ oderberg summand of β(N). With the Monotonicity Principle this
+approach leads to a complete determination of β(N) as follows.
+9Ifπdis a diagram that could appear in the Boij-S¨ oderberg decomposition ofβ(N) and
+which contributes to either the β1,1orβ2,2entry, then dmust have the form (0 ,1,d2,d3) with
+2≤d2and 5≤d3. The minimal such disd= (0,1,2,5), and by applying the formula (1),
+we see
+β1,1(π(0,1,2,5))
+β2,2(π(0,1,2,5))=15
+10.
+Note that this equals the ratioβ1,1(N)
+β2,2(N).
+Now, the Monotonicity Principle implies that if e= (0,1,2,d3) withd3>5 then
+β1,1(πe)/β2,2(πe)>15/10.If we allow eto have the form e= (0,1,d2,d3) withd2>2,
+thenπedoes not have any β2,2entry, and so the ratio would be ∞. We conclude that every
+pure diagram πdwhich could conceivably contribute to β1,1(N) satisfiesβ1,1(πd)
+β2,2(πd)≥15
+10,with
+equality if and only if d= (0,1,2,5).
+Since the decomposition algorithm implies that we cannot eliminate β1,1before we elimi-
+nateβ2,2, it follows that we must eliminate both entries simultaneously. Thus, t he first step
+of the Boij-S¨ oderberg decomposition of β(N) is given by 1·/tildewideπd0= 1·/tildewideπ(0,1,2,5).
+Continuing to apply the decomposition algorithm, we next consider th e diagram β(N)−
+1·/tildewideπd0, which has the form
+β(N)−1·/tildewideπd0=
+4− − −
+− − ∗ −
+− − ∗ ∗
+− −......
+.
+Since the second column consists of all zeroes, this diagram must be 4π(0). Hence,
+β(N) =/tildewideπ(0,1,2,5)+4/tildewideπ(0)=
+10 15 10−
+− − − −
+− − − 1
+.
+/square
+We will generally apply the monotonicity principle via the following corollar y. However,
+as illustrated by Example 6.2 and by the computations in §8, the principle can be useful in
+more general situations.
+Corollary 4.2. LetMbe a module satisfying the hypotheses of Theorem 1.3 (1), and let
+F(f)Mbe the North fork of FM. SetN:= coker( φ(f)M
+1). We may write
+β(N) =c0πd0+Dfree
+whereDfreeis the Betti diagram of a free module.
+Proof of Corollary 4.2. We combine definition 3.1 and the fact that d0
+2≤d1
+1(which is part
+of the definition d0≪d1) to conclude that f2=d1
+2. Lemma A.2 then implies that
+βi,d0
+i(M)=βi,d0(N)fori= 1,2.
+Next, again since d0≪d1, we have d0
+1< d0
+2≤d1
+1andd0
+2≤d1
+1< d1
+2. It follows that πd0is
+the only pure diagram from the Boij-S¨ oderberg decomposition of β(M) that contributes to
+10BQβ(N)•
+P
+Figure 1. This figure is a sketch of the situation of Remark 4.3. Our partial
+information about β(N) corresponds to a polyhedron P. SinceP∩BQconsists
+of a single point, this actually determines all of β(N).
+the Betti numbers β1,d0
+1(M) andβ2,d0
+2(M). This implies the second equality of
+(8)β1,d0
+1(N)
+β2,d0
+2(N)=β1,d0
+1(M)
+β2,d0
+2(M)=β1,d0
+1(πd0)
+β2,d0
+2(πd0),
+and the first equality follows from the first paragraph of this proof .
+Now, let eibe the minimal degree of a generator of the ith syzygy module of N, or∞if
+this syzygy module is 0. We claim that the degree sequence e= (e0,...,e n) is at least d0.
+Since all first syzygies of Nlie in degree d0
+1, this would imply that e1=d0
+1ore1=∞. For
+contradiction, assume that ei< d0
+ifor some i≥2. The definition of d0≪d1in (*) implies
+that
+ei< d0
+i≤d1
+2+i−1 =f2+i−1.
+Sinceei≤f2+i−2, Lemma A.2(2) implies that Tor i(M,k)eiis nonzero. This yields a
+contradiction, since Tor i(M,k)eiis nonzero by definition of ei, and hence eicannot be small
+thand0
+i, yielding the claim.
+Now, ife2=d0
+2bute/n⌉}ationslash=d0, then by Theorem 1.9 combined with (8), we have that
+β1,d0
+1(N)
+β2,d0
+2(N)<β1,d0
+1(πe)
+β2,d0
+2(πe).
+Ife2/n⌉}ationslash=d0
+2, then the denominator on the right would be 0.
+By convexity, the only sums/summationtext
+eaeπewhich satisfy (8) are rational linear combinations
+ofπd0and of projective dimension 0 pure diagrams π(e0,∞,...,∞). Finally, since β1,d0
+1(N) =
+β1,d0
+1(M), we conclude that the coefficient of πd0in the Boij-S¨ oderberg decomposition of
+β(N) equalsc0. This completes the proof. /square
+Remark 4.3.The idea behind Example 4.1 and Corollary 4.2 may be illustrated by conve x
+geometry. Our goal is to understand where in the cone BQthe diagram β(N) lies. As
+illustrated in (7), we only have partial knowledge about β(N). We can think of this partial
+information as cutting out a polyhedron Pin the vector space V, and the diagram β(N)
+must lie in the intersection of PandBQ. The computation in Example 4.1 then shows that
+P∩BQconsists of a single point (see Figure 1), which is how we determine the remaining
+entries of β(N).
+115.Proof of Theorem 1.3 and Corollary 1.4
+Webeginbyshowing thatundersuitablehypotheses theconclusiono fCorollary4.2implies
+an actual splitting of N:
+Lemma 5.1. IfNis a module such that
+β(N) =D≥2+Dfree
+whereD≥2is a diagram of codimension ≥2andDfreeis a diagram of projective dimension
+0, and such that
+min{j|β0,j(Dfree)/n⌉}ationslash= 0}≥max{j|β0,j(D≥2)/n⌉}ationslash= 0},
+thenNsplits as a direct sum N∼=N≥2⊕Nfreewithβ(N≥2) =D≥2andβ(Nfree) =Dfree.
+Informally, the displayed inequality above says that the minimum degr ee of a “generator”
+ofDfreeis at least as large as the maximum degree of a “generator” of D≥2.
+Proof.Leta:= max{j|β0,j(N)/n⌉}ationslash= 0}, the maximal degree of a minimal generator of N. Let
+Kbe the quotient field of S. By considering the Hilbert polynomial of N, we see that
+N⊗SKhas rank≥1, and thus some minimal generator of degree ainNgenerates a free
+submodule. This gives us an exact sequence
+0→S(−a)→N→Q→0.
+The map S(−a)→Nlifts to a map S(−a)→FN
+0whose image is a free summand, so
+β(Q) satisfies the same hypothesis as β(N). By induction on the number of generators, we
+see that Qis a direct sum of a free module Gand a module Hof codimension≥2. Since
+Ext1(G,S) = Ext1(H,S) = 0, the sequence splits. /square
+Example 5.2.The inequality appearing in Lemma 5.1 is necessary. For instance, let S=
+k[x,y] and let N:=S(−1)⊕S/(x2,xy). Then
+β(N) =/parenleftbigg
+1− −
+1 2 1/parenrightbigg
+=/parenleftbigg
+− − −
+1 2 1/parenrightbigg
++/parenleftbigg
+1− −
+− − −/parenrightbigg
+.
+Thusβ(N) has the form D≥2+Dfree. ButN≇S⊕S(−1)/(x,y).
+Example 5.3.The conclusion of Lemma 5.1 may fail without the hypothesis “codimen sion
+≥2”. For instance, if S=k[x,y] andm= (x,y), then
+β(m) =/parenleftbig2 1/parenrightbig
+=/parenleftbig1 1/parenrightbig
++/parenleftbig1−/parenrightbig
+,
+butmdoes not split.
+We are now ready to complete the proof of our main result:
+Proof of Theorem 1.3. We first prove part (1). We let F(f)Mbe the North fork of FM, and
+we define N:= coker( φ(f)M
+1). Since Msatisfies the hypotheses of Theorem 1.3, we may
+apply Corollary 4.2 and Lemma 5.1, and conclude that N=M′⊕Gwhereβ(M′) =c0πd0
+andGis free.
+12We may then rewrite φ(f)M
+1as a block matrix φ(f)M
+1=/parenleftbigg
+/tildewidea1
+0/parenrightbigg
+, where/tildewidea1is a minimal
+presentation matrix of M′. This enables us to rewrite φ1in upper triangular form:
+φ1=/parenleftbigg
+/tildewidea1/tildewideb1
+0/tildewidec1/parenrightbigg
+for some matrices /tildewideb1and/tildewidec1. SinceMis presented by a block triangular matrix, we obtain
+a right exact sequence:
+M′→M→coker(/tildewidec1)→0.
+To finish the proof, we will apply Lemma A.2 to show that this sequence is exact on the
+left and that M′is a cleanly embedded submodule. Note that, by the definition of Nwe
+havef2=d1
+2, which satisfies the conditions of Lemma A.2 by definition of d0≪d1(see (*)
+on page 2). In fact, the definition of d0≪d1further implies that d0
+i≤d1
+2+i−2 for all
+i≥1. Fori >2 this is part of the definition, and the fact that d0< d1andd0
+2≤d1
+1imply
+thatd0
+i≤d1
+2+i−2 fori≤2 as well. Thus, for any i≥1, Lemma A.2(2) implies that
+Tori(N,k)d0
+i(which equals Tor i(M′,k)d0
+i) injects into Tor i(M,k)d0
+i. SinceM′andMareboth
+finite length and since M′has a pure resolution, the inclusion in the case i=nimplies that
+M′→Mis injective as claimed; the other inclusions on Tor groups imply that M′⊆Mis
+cleanly embedded, completing the proof of (1).
+For (2), since d0≪d1we obtain a cleanly embedded submodule M′⊆Mwithβ(M′) =
+c0πd0. SetM′′:=M/M′. The sequence
+0→M′→M→M′′→0
+corresponds to an element α∈Ext1(M′′,M′), which then corresponds to a cocycle α0∈
+Hom(FM′′
+1,M′). Since FM′′
+1is generated in degree at least d1
+1, it follows that the image of
+the map α0is generated in degree at least d1
+1. However, since β(M′) =c0πd0, we see that
+M′has regularity d0
+n−n, and thus is zero in degrees > d0
+n−n. By our assumption
+d1
+1> d0
+n−n,
+so the imageof α0is0. Weconclude that αcorresponds to thezero element of Ext1(M′′,M′),
+and thus that M∼=M′⊕M′′, as desired. /square
+Proof of Corollary 1.4. With notation as in Theorem 1.3, we choose M1=M′. The proof of
+Theorem 1.3 shows that, for degree reasons, the induced map FM1
+j→FM
+jis a split injection
+for allj. It follows that
+β(M/M1) =s/summationdisplay
+i=1ciπdi,
+so we may iterate the construction. /square
+Example 5.4.There exist cases covered by Corollary 1.4 where a full clean filtratio n exists,
+but where that filtration is not a splitting: Let S=k[x,y,z] and let Φ be a generic 9 ×9
+skew-symmetric matrix of linear forms. Let I⊆Sbe the ideal generated by the 8 ×8
+principal Pfaffians of Φ, and let R=S/I. ThenRhas a pure resolution of type (0 ,4,5,9).
+We claim that if Mis a generic extension
+0→R→M→R(−2)→0,
+13thenMadmits a full clean filtration which is not a splitting.
+Note first that, for any such extension, R→Mis cleanly embedded for degree reasons.
+Namely, if we construct a resolution of Mby combining theresolutions of RandR(−2), then
+there is no possibility of cancellation. It thus suffices to show that Ex t1(R,R)2/n⌉}ationslash= 0. Such
+an extension corresponds to a nonzero map α0:FR(−2)
+1=S(−6)9→Rsuch that α0◦Φ = 0.
+SinceRhas regularity 6 and im( α0◦Φ)⊆R7= 0, we see that α0◦Φ is automatically 0.
+One may easily check that there exists such an α0that is not a coboundary.
+Example 5.5.Forn >2, fix any e≥2, and let Mbe any module such that β(M) decomposes
+as a sum of the pure diagrams π(0,e,e+1,e+2,...,e+n−2,e+n−1)andπ(0,1,2,...,n−1,e+n−1).ThenMhas
+a Betti diagram of the form:
+β(M) =
+∗ ∗ ∗ ...∗ −
+− − − ...− −
+............
+− − − ...− −
+− ∗ ∗ ...∗ ∗
+.
+Theorem 1.3(2) implies that Msplits as M=M′⊕M′′whereM′has a pure resolution
+of type (0 ,e,e+ 1,e+ 2,...,e+n−2,e+n−1) andM′′has a pure resolution of type
+(0,1,2,...,n−1,e+n−1). Note that every S-module with a pure resolution of type
+(0,e,e+1,e+2,...,e+n−2,e+n−1) is a direct sum of copies of R:=S/me. It follows
+thatM′isisomorphictoadirectsumofcopiesof R. Byasimilarargument, M′′isisomorphic
+to a number of copies of ωR(n). Hence, any such Mdecomposes as M=Ra⊕ωR(n)bfor
+somea,b.
+6.Beyond Theorem 1.3
+Since the Boij-S¨ oderberg decomposition of a module may involve pur e diagrams with non-
+integral entries, it is clear that there exist many graded modules wh ich do not admit full
+clean filtrations.
+Example 6.1.Letn= 2,R=k[x,y]/(x,y)2,andM=k[x,y]/(x,y2). Then:
+β(M) =/parenleftbigg
+1 1−
+−1 1/parenrightbigg
+=1
+3β(R)+1
+3β(ωR(4)).
+ClearlyMcannot admit a full clean filtration. Though we might hope that M⊕3admits
+such a filtration, this is not the case either [SW11, Ex. 4.5].
+However, there does exist a flat deformation M′ofM⊕3such that M′admits a full clean
+filtration:
+0→R→M′→ωR(4)→0.
+Namely, wemayset M′= (S/(x,y2))⊕(S/(x2,y))⊕(S/(x+y,(x2−2y+y2)). Thissuggests
+a more subtle possible affirmative answer to our Question 1.2. /square
+Each result of§3–5 can be extended to situations that are not covered by the hyp otheses
+of Theorem 1.3.
+14Example 6.2.LetE:=/tildewideπ(0,2,3,4,5,8)+ 2/tildewideπ(0,2,3,5,6,8)+/tildewideπ(0,3,4,5,6,8)+/tildewideπ(0,3,4,6,7,8), and let Mbe a
+module such that β(M) =E. We have
+E=
+11− − − − −
+−60 128 90 32 −
+−144 300 128 60 −
+− − − 280 240 69
+.
+Note that the degree sequences do not satisfy the conditions of C orollary 1.4. Nevertheless,
+we will see that that Madmits a full clean filtration.
+We first construct a cleanly embedded (but not pure) submodule of M. We let F(f)M
+be the North fork of FMand we let N:= coker( φ(f)M
+1). The proof of Corollary 4.2 applies
+nearly verbatim to yield
+β(N) =/tildewideπ(0,2,3,4,5,8)+2/tildewideπ(0,2,3,5,6,8)+6/tildewideπ(0).
+By Lemma 5.1, we obtain a splitting N=M′⊕GwhereGis a free module. The arguments
+used in the proof of Theorem 1.3 then imply that M′is a cleanly embedded submodule of
+M. We thus have a short exact sequence
+0→M′→M→M′′→0
+whereβ(M′) =/tildewideπ(0,2,3,4,5,8)+2/tildewideπ(0,2,3,5,6,8)andβ(M′′) =/tildewideπ(0,3,4,5,6,8)+/tildewideπ(0,3,4,6,7,8).
+Repeating the same argument for ( M′)∨and for ( M′′)∨, and then applying Lemma 2.5,
+we conclude that Madmits a full clean filtration.
+One of the key features of our proof of Theorem 1.3 is that the diag ramsd0,...,dsthat
+arise in the Boij-S¨ oderberg decomposition are separated from ea ch other in the poset of
+degree sequences. In particular, d0andd1always differ in at least two consecutive positions.
+This is essential to our proof of Corollary 4.2, and it suggests some in teresting examples to
+explore.
+Consider, for example, the diagrams D=/tildewideπ(0,1,3,5)+/tildewideπ(0,2,4,5)andD′=/tildewideπ(0,1,2,3,5,6)+
+/tildewideπ(0,1,3,4,5,6), so that
+D=
+11 15− −
+−10 10−
+− −15 11
+and D′=/parenleftbigg
+3 12 15 10− −
+− −10 15 12 3/parenrightbigg
+.
+Question 6.3. Letβ(M)be a scalar multiple of either DorD′. DoesMadmit a cleanly
+embedded submodule with a pure resolution?
+Remark6.4.Although many aspects of our technique apply to modules of dimensio n greater
+than 0, there is one obstacle to extending our results to such modu les. LetMbe a module of
+nonzeroKrulldimensionthatotherwisesatisfiesthehypotheses o fTheorem1.3, anddefine N
+via theNorth forkasin the proofof Theorem 1.3. It is possible that t he projective dimension
+ofNcould be larger than the projective dimension of M, and this possibility undermines our
+application of the monotonicity principle. It would thus be interesting to produce a positive
+answer to Question 1.2 for some case where the dimension of Mis nonzero.
+157.Application: Pathologies of Bmod
+Example 1.7 illustrates the existence of a ray of BQwhere only1
+5of the lattice points
+correspond to Betti diagrams of modules. We now prove Propositio n 1.6, which implies that
+there are rays where the true Betti diagrams are arbitrarily spar se among the lattice points.
+The proof will show that such pathologies already arise in codimension 3.
+Proof of Proposition 1.6. LetS=k[x1,x2,x3] and let p≥5 prime. Set d0= (0,1,2,p),d1=
+(0,⌊p/2⌋,⌈p/2⌉,p) andd2= (0,p−2,p−1,p). Consider the diagram
+D=1
+p/tildewideπd0+α
+p/tildewideπd1+1
+p/tildewideπd2
+whereαis any positive integer such α+1+/parenleftbigp−1
+2/parenrightbig
+≡0 modp. We claim that Dhas integral
+entries but that cD∈Bmodif and only if cis divisible by p.
+We first check the integrality of D. Observe that each Betti number of /tildewideπd0is divisible by
+pexcept for the 0th Betti number; each Betti number of /tildewideπd2is divisible by pexcept for the
+3rd Betti number; and the Betti numbers of /tildewideπd1are (1,p,p,1). Hence, we only need to check
+thatβ0,0(D) andβ3,p(D) are integral. We compute
+β0,0(D) =1
+pβ0,0(/tildewideπd0)+α
+pβ0,0(/tildewideπd1)+1
+pβ0,0(/tildewideπd2) =1
+p+α
+p+/parenleftbigp−1
+2/parenrightbig
+p.
+Our assumption on αthen implies that β0,0(D) is integral. A symmetric computation works
+forβ3,p(D).
+LetcD∈Bmod, and we will show that cis divisible by p. LetMbe any module such
+thatβ(M) =cD. LetNbe the cokernel of the submatrix of the presentation matrix of
+Mcontaining all of the degree 1 and degree ⌊p/2⌋columns (so we throw away the degree
+p−2 columns). Lemma A.2(2) and (3) then imply that βi,j(N) =βi,j(M) fori= 2,3 and
+j < p+i−3.
+In particular, the top strand of β(N) is at least (0 ,1,2,p), and so we may use the mono-
+tonicity principle to conclude that the first step of the Boij–S¨ oder berg decomposition of
+β(N) is given byc
+p/tildewideπd0. The top strand of the resulting diagram β(N)−c
+p/tildewideπd0is then at least
+(0,⌊p/2⌋,⌈p/2⌉,p), and an additional application of the monotonicity principle yields the
+full Boij–S¨ oderberg decomposition of β(N) to be:
+β(N) =c
+p/tildewideπd0+cα
+p/tildewideπd1+c
+pπ(0,∞,...,∞).
+By Lemma 5.1, Nsplits off a free summand Sc
+p, and hence pdividesc.
+It now suffices to show that pD∈Bmod. This follows from the fact that /tildewideπdi∈Bmodfor
+i= 0,1 or 2. In particular, /tildewideπd2=β(R) whereR:=S/(x1,x2,x3)p−2, and/tildewideπd0=β(R∨(p−3)).
+To see that /tildewideπd1∈Bmod, letAbe ap×pskew-symmetric matrix of generic linear forms.
+By [BE77], the principal Pfaffians of Adefine an ideal I⊆Ssuch that β(S/I) =/tildewideπd1.
+This completes the proof when p≥5. For the cases p= 2 (respectively 3), we may choose
+the diagram D=1
+2/tildewideπ(0,1,2,4)+1
+2/tildewideπ(0,2,3,4)(respectively D=1
+3/tildewideπ(0,1,2,5)+2
+3/tildewideπ(0,3,4,5)) and apply
+similar arguments as above. /square
+168.Application: Quiver representations
+In this section, we determine all Betti diagrams corresponding to q uiver representations
+of the form•/d47/d47/d47/d47/d47/d47•. As discussed in the introduction, this is equivalent to computing the
+possible Betti diagrams of finite length modules of the form:
+β(M) =/parenleftbigg
+β0,0β1,1β2,2β3,3
+β0,1β1,2β2,3β3,4/parenrightbigg
+.
+Throughout this section, we thus set d0= (0,1,2,3),d1= (0,1,2,4),d2= (0,1,3,4),d3=
+(0,2,3,4) andd4= (1,2,3,4) and we let /tildewide∆ = (d0,d1,d2,d3,d4).
+Our goal is to compute the minimal generators of Bmod(/tildewide∆). In addition to the connec-
+tion with quiver representations, this computation provides the fir st detailed and nontrivial
+example of the generators of Bmod(/tildewide∆). Further, this computation illustrates that the Mono-
+tonicity Principle and some of the other techniques introduced in §3-5 can be extended to
+more situations, but at the cost of wrestling with integrality conditio ns and precise numerics.
+As noted in the introduction, if β3,3(M) (orβ0,1(M)) is nonzero, then a copy of the
+residue field k(ork(−1)) splits from M. It is therefore equivalent to restrict to the case
+whereβ3,3=β0,1= 0 and to compute the generators for Bmod(∆) where ∆ = ( d1,d2,d3).
+The result of this computation is summarized in the following propositio n.
+Proposition 8.1. The semigroup Bmod(∆)has ten minimal generators. These consist of
+the following ten Betti diagrams:
+/parenleftbigg
+3 8 6−
+− − − 1/parenrightbigg
+,/parenleftbigg
+1− − −
+−6 8 3/parenrightbigg
+,/parenleftbigg
+1 2 1−
+−1 2 1/parenrightbigg
+,/parenleftbigg
+1 1− −
+−3 5 2/parenrightbigg
+,/parenleftbigg
+2 5 3−
+− −1 1/parenrightbigg
+,
+/parenleftbigg
+2 4 1−
+−1 4 2/parenrightbigg
+,/parenleftbigg
+3 7 3−
+− −3 2/parenrightbigg
+,/parenleftbigg
+2 3− −
+−3 7 3/parenrightbigg
+,/parenleftbigg
+2 4− −
+− −4 2/parenrightbigg/parenleftbigg
+3 6− −
+− −6 3/parenrightbigg
+.
+See Figure 2. Before proving this proposition, we introduce some sim plifying notation.
+Every element of Bint(∆) can be represented as:
+D= 4rπ(0,1,2,4)+2sπ(0,1,3,4)+4tπ(0,2,3,4)
+with (r,s,t)∈Z3
+≥0(c.f. [Erm09, pp. 347–9].) The necessary and sufficient conditions fo r a
+triplet (r,s,t)∈Z3
+≥0to yield an integral point are:
+•r+s≡0 mod 3
+•s+t≡0 mod 3
+•r+s+t≡0 mod 2.
+For the rest of this section, we use triplets ( r,s,t) to refer to diagrams in Bint(∆), and we
+only consider triplets ( r,s,t) that satisfy the above congruency conditions. In this notation,
+Proposition8.1 amounts tothe claimthat thefollowing ten ( r,s,t)triplets arethegenerators
+ofBmod:
+(6,0,0),(0,0,6),(1,2,1),(3,3,0),(0,3,3),
+(1,8,1),(3,9,0),(0,9,3),(0,12,0),(0,18,0).
+174cπ(0,1,2,4)4cπ(0,2,3,4)
+2cπ(0,1,3,4)The region s <5 The region s≥5
+Figure 2. Proposition 8.1 can be illustrated by considering a slice of the cone
+BQ(∆) where r+s+t=cfor some c≫0. In the region where s <5, roughly
+half of the lattice points in the cone belong to Bmod. In the region where s≥5,
+every lattice point in the cone belongs to Bmod.
+Proof of Proposition 8.1. WefirstnotethateachofthetendiagramslistedinProposition8.1
+is the Betti diagram of an actual module. When β0,0= 1 orβ3,4= 1, such examples are
+straightforward to construct. Next, we have
+β/parenleftbigg
+coker/parenleftbigg
+x y0 0z2
+0x y z x2/parenrightbigg/parenrightbigg
+=/parenleftbigg
+2 4 1−
+−1 4 2/parenrightbigg
+.
+LetLbe any 2×3 matrix of linear forms whose columns satisfy no linear syzygies, and let
+N:= coker( L). Then
+β(N/m2N) =/parenleftbigg
+2 3− −
+−3 7 3/parenrightbigg
+The Betti diagram of ( N/m2N)∨then yields the dual diagram. Finally, examples corre-
+sponding to (0 ,12,0) and (0 ,18,0) are given in [Erm09, Proof of Thm. 1.6(1)].
+We must now show that every diagram in Bmod(∆) may be written as a sum of our ten
+generators. We proceed by analyzing cases based on the different possible values of sin our
+(r,s,t) representation of diagrams.
+The case s= 0.Based on Example 5.5 in the case n= 3 and e= 2, we conclude that
+(r,0,t) corresponds to an element of Bmod(∆) if and only if both randtare divisible by 6.
+The case s= 1.There are two families of triplets ( r,1,t) satisfying the congruency condi-
+tions. The first family is parametrized by (2+ 6 γ,1,5+6α) for some γ,α∈Z≥0, and the
+second family is parametrized by (5+6 γ,1,2+6α). To prove that none of these diagrams
+belongs to Bmod(∆), it suffices (by symmetry under M/mapsto→M∨) to rule out the first family.
+We thus assume, for contradiction, that there exists Msuch that β(M) corresponds to
+the triplet (2+6 γ,1,5+6α) for some α,γ∈Z≥0. We let F(f)Mbe the North fork of FM.
+We then set N:= coker( φ(f)M
+1), and we have
+β(N) =
+2+α+3γ3+8γ2+6γ−
+− − β2,3(N)β3,4(N)
+− − β2,4(N)β3,5(N)
+− −......
+.
+18ToproducetheBoij-S¨ oderbergdecomposition, webeginbysubtr actingc1πd1forsome c1≥0.
+Note that
+c1πd1=c1/parenleftbigg1
+81
+31
+4−
+− − −1
+24/parenrightbigg
+.
+Assume first that c1<24γ, so thatc1
+4<6γ. In this case, the greedy decomposition
+algorithm must eliminate the β3,4first (orc1= 0 andβ3,4= 0 to begin with). The resulting
+diagram β(N)−c1πd1hasβ3,4= 0 and the ratio β1,1/β2,2is strictly larger than 3 /2 =
+β1,1(π(0,1,2,5))/β1,1(π(0,1,2,5)). The monotonicity implies that this is impossible. Hence, we
+must have c1≥24γ.
+If nowc1= 24γ, then we may again apply the monotonicity principle to β(N)−24γπd1to
+conclude that the next step of the Boij-S¨ oderberg decompositio n must be precisely1
+5π(0,1,2,5).
+This would leave nothing left in column 1, and thus β(N)−γπd1−1
+5π(0,1,2,5)would be a
+diagram of projective dimension 0. But this would contradict the inte grality of β(N), since
+it would imply that β3,5(N) =1
+5.
+The final possibility is that c1>24γ. Sincec1<48γ, this implies that we must eliminate
+theβ2,2entry first and hence that c1must equal 8+24 γ. After subtracting (8+24 γ)πd1, we
+are left with:
+β(N)−(8+24γ)π(d1)=
+1+α1
+3− −
+− − β2,3(N)β3,4(N)−(1
+3+γ)
+− −......
+.
+Sinceβ3,4(N)−(1
+3+γ) is nonzero (it is not an integer), the next step of the Boij-S¨ oder berg
+decomposition must eliminatethis entry. Thismeans thatthenext st ep ofthedecomposition
+must be1
+6πd2. However, this would leave a 0 in column 1 and a nonzero entry in column 2,
+which is impossible.
+The case s= 2.There are two families of triplets ( r,2,t) satisfying the congruency con-
+ditions. The first family has the form (1 + 6 γ,2,1 + 6α) and the second family has the
+form (4 + 6 γ,2,4 + 6α), where γ,α∈Z≥0. Every element of the first family is a sum of
+our proposed generators, so we must show that no element of the second family belongs to
+Bmod(∆). We obtain a contradiction by essentially the same analysis as in th e cases= 1.
+The case s= 4.There are two families of triplets ( r,4,t) satisfying the congruency con-
+ditions, namely (2+ 6 γ,4,2+6α) and (5+ 6 γ,4,5+ 6α). Since every element of the first
+family is a sum of our proposed generators, we must show that no ele ment of the second
+family belongs to Bmod(∆). A similar, though more involved, analysis as in the case s= 1
+then illustrates that there are no such diagrams.
+The cases s= 3,5,6.We claim that if D∈Bint(∆) corresponds to an ( r,s,t)-triplet where
+s= 3,5,or 6, then D∈Bmod(∆), with the exception of (0 ,6,0). There are six families to
+consider in total: (3+ 6 γ,3,6α),(6γ,3,3+ 6α),(4+6γ,5,1+6α),(1+6γ,5,4+ 6α),(3+
+6γ,6,3+6α),and (6γ,6,6α). Any element from any of these families may be written as a
+sum of our proposed generators, except for (0 ,6,0). The diagram corresponding to (0 ,6,0)
+does not belong to Bmodby [Erm09, Proof of Thm. 1.6(1)].
+The cases s >6.One may directly check that all elements of Bint(∆) with s >6 can be
+written as an integral sum of the proposed generators. /square
+19Acknowledgements
+We thank Christine Berkesch, Jesse Burke, Courtney Gibbons, St even Sam and Jerzy
+Weyman for useful conversations. We thank the anonymous refe ree for a careful reading
+that improved the paper. We also thank Amin Nematbakhsh and to Gu nnar Fløystad for
+bringing to our attention a mistake in the proof of Corollary 4.2 in a pre vious version. The
+computer algebra system Macaulay2 [GS] provided valuableassistance instudying examples.
+References
+[BE77] David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and
+some structure theorems for ideals of codimension 3, Amer. J. Math. 99(1977), no. 3, 447–485.
+MR0453723 (56 #11983)
+[BS08] Mats Boij and Jonas S¨ oderberg, Graded Betti numbers of Cohen-Macaulay modules and the mul-
+tiplicity conjecture , J. Lond. Math. Soc. (2) 78(2008), no. 1, 85–106. MR2427053 (2009g:13018)
+[BS12] ,Betti numbers of graded modules and the multiplicity conjec ture in the non-cohen-macaulay
+case, Algebra Number Theory 6(2012), no. 3, 437–454.
+[EFW11] David Eisenbud, Gunnar Fløystad, and Jerzy Weyman, The existence of equivariant pure free
+resolutions , Ann. Inst. Fourier (Grenoble) 61(2011), no. 3, 905–926. MR2918721
+[Erm09] Daniel Erman, The semigroup of Betti diagrams , AlgebraNumber Theory 3(2009),no. 3,341–365.
+MR2525554 (2010k:13022)
+[Erm10] ,A special case of the Buchsbaum-Eisenbud-Horrocks rank con jecture, Math. Res. Lett. 17
+(2010), no. 6, 1079–1089. MR2729632 (2012a:13023)
+[ES09] David Eisenbud and Frank-Olaf Schreyer, Betti numbers of graded modules and cohomology of
+vector bundles , J. Amer. Math. Soc. 22(2009), no. 3, 859–888. MR2505303 (2011a:13024)
+[ES10] ,Cohomology of coherent sheaves and series of supernatural b undles, J. Eur. Math. Soc.
+(JEMS) 12(2010), no. 3, 703–722. MR2639316 (2011e:14036)
+[Flø12] GunnarFløystad, Boij-s¨ oderberg theory: Introduction and survey , Progressin commutativealgebra
+1: Combinatorics and homology, 2012, pp. 1–54.
+[GS] Daniel R. Graysonand Michael E. Stillman, Macaulay2, a software system for research in algebraic
+geometry .
+[McC11] JasonMcCullough, A polynomial bound on the regularity of an ideal in terms of ha lf of the syzygies ,
+2011. arXiv:1112.0058.
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+ent sheaves , Proceedings of the International Congress of Mathematicians. Volume II, 2010,
+pp. 586–602. MR2827810
+[SW11] Steven V. Sam and Jerzy Weyman, Pieri resolutions for classical groups , J. Algebra 329(2011),
+222–259. MR2769324 (2012e:20102)
+Department of Mathematics, University of California, Berk eley, CA 94720, USA
+E-mail address :eisenbud@math.berkeley.edu
+Department of Mathematics, University of Wisconsin-Madis on, Madison, WI 53706, USA
+E-mail address :derman@math.wisc.edu
+URL:http://www.math.wisc.edu/~derman/
+Mathematik und Informatik, Universit ¨at des Saarlandes, Campus E2 4, D-66123 Saarbr ¨ucken,
+Germany
+E-mail address :schreyer@math.uni-sb.de
+20
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+arXiv:1001.0587v1  [physics.atom-ph]  4 Jan 2010Parity violation in atomic ytterbium: experimental sensit ivity and systematics
+K. Tsigutkin,1,∗D. Dounas-Frazer,1A. Family,1J. E. Stalnaker,1,†V. V. Yashchuk,2and D. Budker1,3
+1Department of Physics, University of California at Berkele y,
+Berkeley, CA 94720-7300
+2Advanced Light Source Division, Lawrence Berkeley Nationa l Laboratory, Berkeley CA 94720
+3Nuclear Science Division, Lawrence Berkeley National Labo ratory, Berkeley, California 94720
+(Dated: October 30, 2018)
+We present a detailed description of the observation of pari ty violation in the 6s2 1S0→5d6s3D1
+408-nmforbiddentransitionofytterbium, abriefreportof whichappearedearlier. Linearlypolarized
+408-nm light interacts with Yb atoms in crossed E- andB-fields. The probability of the 408-
+nm transition contains a parity violating term, proportion al to (E·B)[(E×E)·B], arising from
+interference between the parity violating amplitude and th e Stark amplitude due to the E-field ( E
+is the electric field of the light). The transition probabili ty is detected by measuring the population
+of the3P0state, to which 65% of the atoms excited to the3D1state spontaneously decay. The
+population of the3P0state is determined by resonantly exciting the atoms with 64 9-nm light to
+the 6s7s3S1state and collecting the fluorescence resulting from its dec ay. Systematic corrections
+due to E-field and B-field imperfections are determined in aux iliary experiments. The statistical
+uncertainty is dominated by parasitic frequency excursion s of the 408-nm excitation light due to
+imperfect stabilization of the optical reference with resp ect to the atomic resonance. The present
+uncertainties are 9% statistical and 8% systematic. Method s of improving the accuracy for the
+future experiments are discussed.
+PACS numbers: 11.30.Er, 32.90.+a
+I. INTRODUCTION
+In an earlier paper [1], we reported on observa-
+tion of the atomic parity violation (PV) effect in the
+6s2 1S0→5d6s3D1408-nm forbidden transition of
+174Yb. We measured the PV induced transition matrix
+element to be 8 .7±1.4×10−10ea0, which confirms the
+theoretically anticipated PV enhancement in Yb [2] and
+constitutes the largest atomic parity violation effect ob-
+served so far. However, the measurement accuracy is
+not yet sufficient for the observation of the isotopic and
+hyperfine differences in the PV amplitude, the study of
+which is the main goal of the present experiments. Here
+we describe the impact of the apparatus imperfections
+and systematic effects on the accuracy of the measure-
+ments and discuss ways of improving it.
+During the initial stage of the experiment, an effort
+was invested into measuring various spectroscopic prop-
+erties of the Yb system of direct relevance to the PV
+measurement, including determination of radiative life-
+times, measurementofthe Stark-inducedamplitudes, hy-
+perfine structure, isotope shifts, and dc-Stark shifts of
+the 6s2 1S0→5d6s3D1transition [3]. In addition, the
+6s2 1S0→3D2transition at 404 nm has been observed,
+andtheelectricquadrupoletransitionamplitudeandten-
+sor transition polarizability have been measured [4]. The
+forbidden magnetic-dipole (M1) amplitude of the 408 nm
+transition was measured to be 1 .33×10−4µBusing the
+∗Electronic address: tsigutkin@berkeley.edu
+†Present address: Department of Physics and Astronomy, Ober lin
+College, Oberlin, OH 44074M1-(Stark-induced)E1 interference technique [5]. The
+ytterbium atomic system, where transition amplitudes
+and interferences are well understood, has proven use-
+ful for gaining insight into the Jones-dichroism effects
+that had been studied in condensed-matter systems at
+extreme conditions and whose origin had been a matter
+of debate (see Ref. [6] and references therein).
+An experimental and theoretical study of the dynamic
+(ac) Stark effect on the 6s2 1S0→5d6s3D1forbid-
+den transition was also undertaken [7]. A model was de-
+veloped to calculate spectral line shapes resulting from
+resonant excitation of atoms in an intense standing light
+wavein the presence ofoff-resonantac-Starkshifts. A bi-
+product of this work was an independent determination
+(from the saturation behavior of the 408-nm transition)
+of the Stark transition polarizability, which was found to
+be in agreement with the earlier measurement [4].
+The present Yb APV experiment involves a measure-
+ment using an atomic beam. An alternative approach
+would involve working with a heat-pipe-like vapor cell.
+Various aspects of such an experiment were investigated,
+including measurements of collisional perturbations of
+relevant Yb states [8], nonlinear optical processes in a
+dense Yb vapor with pulsed UV-laser excitation [9], and
+an altogether different scheme for measuring APV via
+optical rotation on a transition between excited states
+[10].
+The present paper addresses the issues of sensitivity
+and systematics in the Yb APV experiment. In Sec-
+tions II and III the experimental technique and its ap-
+plication in the present experiment are discussed. In
+Section IV a method of analyzing the impact of various
+apparatus imperfections is described based on theoret-2
+ical modeling of signals recorded by the detection sys-
+tem in the presence of imperfections. In Section V a de-
+tailed description of the experimental apparatus is given,
+alongwithadiscussionoftheoriginsoftheimperfections,
+which is followed by an account of the measurements of
+the imperfections in Section VI. In Sections VII and VIII
+we discuss measurements and analysis of the PV ampli-
+tude and systematic effects, and ways of improving the
+accuracy of the PV measurements to better than 1% in
+order to measure the difference in the APV effects be-
+tween different isotopes and hyperfine components.
+II. EXPERIMENTAL TECHNIQUE FOR THE
+APV MEASUREMENT
+As discussed in Ref. [1], the idea of the experiment is
+to excite the forbidden 408-nm transition (Fig. 1) with
+resonant laser light in the presence of a quasi-static elec-
+tric field. The PV amplitude of this transition arises
+due to PV mixing of the 5d6s3D1and 6s6p1P1states.
+The purpose of the electric field is to provide a refer-
+ence transition amplitude, which is due to Stark mixing
+of the same states interfering with the PV amplitude.
+In such an interference method [11, 12], one is measur-
+ing the part of the transition probability that is linear in
+both the reference Stark-induced amplitude and the PV
+amplitude. In addition to enhancing the PV dependent
+signal, the Stark-PV interference technique provides for
+all-important reversals that separate the PV effects from
+the systematics.
+FIG. 1: (color online) Low-lying energy eigenstates of Yb an d
+transitions relevant to the APV experiment.
+Even though the APV effect in Yb is relatively large,
+and the M1 transition is strongly suppressed, the M1
+transition amplitude is still three orders of magnitude
+larger than the PV amplitude. As a result, the geometry
+of the experiment was designed to suppress spurious M1-Stark interference. In addition, this effect is minimized
+bythe useofa power-build-upcavitytogenerateastand-
+inglightwave. Sinceastandingwavehasnonetdirection
+of propagation any transition rate which is linear in the
+M1 amplitude, will cancel out (see below).
+The advantages of the present experimental configura-
+tion can be demonstrated by consideringYb atomsin the
+presence of static electric, E, and magnetic, B, fields in-
+teracting with a standing monochromatic wave produced
+by two counter-propagating coherent waves in an optical
+cavity. Theelectricfieldinthestandingwave, E, isasum
+of the fields of the two waves. For resonant atoms, the
+transition rate from the ground state1S0to the excited
+state3D1is (see e.g. [13], Eq. (3.127))
+RM=4
+/planckover2pi12Γ|AM|2, (1)
+where Γ is the natural linewidth of the transition, AM
+is the transition amplitude, and M= 0,±1 is the mag-
+netic quantum number of the excited state. Here and
+in the rest of this section it is assumed that the individ-
+ual magnetic sublevels of the3D1state are resolved. For
+convenience, we set /planckover2pi1= 1 henceforth and measure the
+transition rate in units of Γ.
+Thetransitionamplitude AMisthesumoftheelectric-
+(E1) and magnetic- (M1) dipole transition amplitudes:
+AM=A(E1)
+M+A(M1)
+M. (2)
+The E1 amplitude has two contributions corresponding
+to the Stark- and PV- mixing of the3D1and1P1states.
+That is,
+A(E1)
+M=A(Stark)
+M+A(APV)
+M
+=iβ(−1)M(E×E)−M+iζ(−1)ME−M,(3)
+whereβis the vector transition polarizability, ζis re-
+lated to the reduced matrix element of the Hamilto-
+nian describing the weak interaction, and E0,±1are the
+spherical components of the vector E. Although Stark-
+induced transitions are generally characterized by scalar,
+vector, and tensor polarizabilities [4, 11], for the case of
+aJ= 0→1 transition, only the vector polarizability
+contributes. Equation (3) is derived in Appendix A.
+Similarly, the M1 transition amplitude has two com-
+ponents: one for each of the two counter-propagating
+laser beams. Let E+=E+ˆEandE−=E−ˆEdenote the
+electric fields of the beams traveling in the kand−kdi-
+rections, respectively. Then E=E++E−and the M1
+amplitude is given by
+A(M1)
+M=M(−1)M(k×E+)−M+M(−1)M(−k×E−)−M
+=M(−1)M(δk×E)−M, (4)
+whereMis the reduced matrix element of the M1 tran-
+sition and kis a unit vector in the direction of the
+wavevector. Here we have introduced δk=δkkwith
+δk= (E+−E−)/E. For a perfect standing wave, E+=E−3
+and hence δk= 0 and the M1 transition is completely
+suppressed. In practice, E−=E+−δEdue to the small
+but nonzerotransmissionofthe backmirrorinthe cavity.
+Since|δE| ≪ E,|δk| ≈ |δE/E| ≪1. Thus, although the
+M1 transition amplitude is not strictly zero, it is highly
+suppressed.
+Without loss of generality, the quantities β,ζ, andM
+are assumed to be real. In general, the rate RMgiven
+by Eq. (1) includes terms proportional to βM(Stark-M1
+interference) and βζ(Stark-PV interference).
+A careful choice of field geometry allows for additional
+suppression of undesirable Stark-M1 interference. From
+Eq. (3), it is evident that the Stark-PV interference is
+proportional to the rotational invariant
+(E/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+PV·B)[(E×E)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+Stark·B], (5)
+which is P-odd and T-even. In the present experimental
+apparatus the electric field, E, is applied orthogonally to
+the magnetic field, B, and collinearly with the axis of the
+linearly-polarizedstandinglight wave,as shownin Fig. 2.
+FIG. 2: (color online) Orientation of fields for PV-Stark in-
+terference experiment and schematic of the present APV ap-
+paratus. Not shown is the vacuum chamber containing all the
+depicted elements, except the photomultiplier (PMT) and th e
+photodiode (PD). PBC–power buildup cavity. Light is ap-
+plied collinearly with x.
+This geometry is such that the M1 and Stark-induced
+amplitudes are out of phase. Thus, they do not interfere
+and therefore do not produce spurious PV-mimicking ef-
+fects (see Section IV).
+III. PV SIGNATURE: IDEAL CASE
+In the ideal case where we neglect the apparatus im-
+perfections, the static magnetic and electric fields are
+B=BˆzandE=Eˆx, respectively, and the light stand-
+ing wave has an electric field
+E=E(sinθˆy+cosθˆz). (6)With this field orientation (see Fig. 2), Eqs. (1) through
+(4) yield
+R0= 4E2/parenleftbig
+β2E2sin2θ+2ζβEsinθcosθ/parenrightbig
+,(7)
+R±1= 2E2/parenleftbig
+β2E2cos2θ−2ζβEsinθcosθ/parenrightbig
+,(8)
+where terms of order ζ2and higher are neglected, and
+δk= 0 is assumed.
+In orderto isolate the Stark-PVinterferenceterm from
+the dominant Stark-inducedtransitionrate, wemodulate
+the electric field: E=Edc+˜E0cos(ωt), where ˜E0is the
+modulation amplitude, ωis the modulation frequency,
+andEdcprovides a DC bias. Then Eqs. (7) and (8)
+become
+RM=R[0]
+M+R[1]
+Mcos(ωt)+R[2]
+Mcos(2ωt),(9)
+whereR[n]
+Mis the amplitude of the nth harmonic of the
+transition rate RM. The dominant Stark-induced con-
+tribution, which oscillates at twice the modulation fre-
+quency, is
+R[2]
+0= 2β2˜E2
+0E2sin2θ, (10)
+R[2]
+±1=β2˜E2
+0E2cos2θ. (11)
+On the other hand, the amplitude R[1]
+Mcontains the
+Stark-PV interference term:
+R[1]
+0= 8E2/parenleftBig
+β2˜E0Edcsin2θ+ζβ˜E0sinθcosθ/parenrightBig
+,(12)
+R[1]
+±1= 4E2/parenleftBig
+β2˜E0Edccos2θ−ζβ˜E0sinθcosθ/parenrightBig
+.(13)
+The term R[0]
+Mis a constant “background”:
+R[0]
+0= 4E2/parenleftBig
+β2(˜E2
+0+E2
+dc)sin2θ+4ζβEdcsinθcosθ/parenrightBig
+,
+R[0]
+±1= 2E2/parenleftBig
+β2(˜E2
+0+E2
+dc)cos2θ−4ζβEdcsinθcosθ/parenrightBig
+.
+For an arbitrary polarization angle θ, all three Zee-
+man components of the transition, as shown in Fig. 3a,
+are present while scanning over the spectral profile of the
+transition. The first-harmonicsignal due to Stark-PVin-
+terference has a characteristic signature: the sign of the
+oscillating terms for the two extreme components of the
+transition is opposite to that of the central component.
+The second-harmonic signal provides a reference for the
+lineshape since it is free from interference effects linear
+inE(Fig. 3b). With a non-zero DC component present
+in the applied electric field, a signature identical to that
+in the second harmonic will also appear in the first har-
+monic, Fig. 3c. The latter can be used to increase the
+first-harmonic signal above the noise, which makes the
+profile analysis more reliable.
+To obtain the PV term from the measured first-
+and second-harmonic transition rates, we first normalize
+the first-harmonic signals R[1]
+Mby their second-harmonic4
+FIG. 3: Discrimination of the PV effect by E-field modulation
+under static magnetic field. The Zeeman pattern is shown for
+the polarization angle θ=π/4.
+counterparts R[2]
+Mand combine the results in the follow-
+ing way:
+K=R[1]
+−1
+R[2]
+−1+R[1]
++1
+R[2]
++1−2R[1]
+0
+R[2]
+0=∓16ζ
+β˜E0.(14)
+Here we take θ=±π/4, which are optimal polarization
+angles for the PV measurements (see next section). This
+method has the advantage that Kis independent of Edc,
+so that the E-field bias may be chosen based on technical
+requirements of the experimental apparatus.
+IV. PV SIGNATURE: IMPACT OF
+APPARATUS IMPERFECTIONS
+While the current Yb-APV apparatus has been de-
+signed to minimize systematic effects, the PV mimicking
+systematics may be a result of a combination of multiple
+apparatus imperfections. In order to understand the im-
+portance of these effects, the electric and magnetic field
+misalignmentsandstrayfields wereincluded inatheoret-
+ical model of the transition rates as well as the excitation
+light’s deviations from linear polarization. In addition,
+we relax the assumption that δk= 0 and include the
+effects of the residual M1 transition.
+The quantization axis is defined along ˆz, and following
+the ideal case model, the axis of the standing light wave
+is collinear with ˆx. We added a small ellipticity to the
+light polarization by taking
+E=E(ˆysinθ+ˆzeiφcosθ), (15)
+whereφis a small phase. For |φ| ≪1, the ellipticity
+of the light is 2 φsin(2θ). The electric field imperfections
+are included by taking
+E=˜E+E′,
+where
+˜E= (˜E0ˆx+ ˜eyˆy+ ˜ezˆz)cos(ωt)
+E′=Edcˆx+eyˆy+ezˆz,
+are the AC and DC components of the electric field. It is
+assumedthat they-and z-componentsofthe AC fieldare
+in phase with the leading oscillating E-field. The impactof the out-of-phase AC components was analyzed within
+a complete model of the systematics and found to be
+negligible. TheACcomponentsaredue tomisalignments
+of the applied E-field with respect to the light wave axis
+aswellastothequantizationaxis ˆz. TheDCcomponents
+arise due to a misalignment of the DC-bias field and also
+due to stray electric fields in the interaction region.
+The magnetic-field imperfections are defined within
+the same frame of reference by taking analogously
+B=˜B+B′,
+where
+˜B=˜bxˆx+˜byˆy+Bˆz
+B′=b′
+xˆx+b′
+yˆy+b′
+zˆz,
+where˜BandB′are reversing and stray non-reversing
+magnetic fields, respectively.
+Equations (1) through (4) apply when the quantiza-
+tion axis is along the magnetic field, thus a rotation Dis
+applied to each of the vectors E,B,E, andksuch that
+DB∝ˆz. That is, we take
+B→DB,E→DE,E→DE,andk→Dk,(16)
+where
+D=D(−αy,ˆy)D(αx,ˆx). (17)
+Here the matrix D(α,ˆn) represents a rotation about an
+axisˆnthrough angle α. The angles αxandαyare given
+by
+αx,y= (B−b′
+z)(b′
+y,x+˜by,x)/B2. (18)
+Thus, the electric field Eand the polarization vector
+Eacquire additional components after the rotation (be-
+sides, for example, eyand ˜ey).
+Due to the imperfections, the normalized-rate modu-
+lation amplitudes now include additional terms besides
+the Stark- and the PV effects:
+R[1]
+M
+R[2]
+M≡rM=r(Stark)
+M+r(APV)
+M+r(M1)
+M+r(φ)
+M,(19)
+wherer(Stark)
+Mis the Stark contribution due to the DC-
+bias and the field imperfections, r(APV)
+Mis the PV-Stark-
+interferenceterm, r(M1)
+Misthe M1-Stark-interferencecon-
+tribution, and r(φ)
+Mis a contribution due to the distorted
+linear polarization of the light (which is a Stark contri-
+bution, but we explicitly separate the contribution linear
+inφ). Expressions for the lowest-order terms are sum-
+marized in the Table I.
+The normalized-amplitude combination (14) has been
+chosen to determine the PV asymmetry. Since the M1
+and ellipticity terms have opposite signs for M=±1,
+their contributions to Kcancel, while the contributions
+fromr(APV)
+Madd.5
+TABLE I: Lowest-order terms contributing to the normalized
+transition-rate modulation amplitudes rM.
+r(APV)
+M r(M1)
+M r(φ)
+M
+M= 0+4ζcotθ
+β˜E00 0
+M=−1−4ζtanθ
+β˜E0+4δkM(˜ey−˜eztanθ)
+β˜E2
+0+4ezφtanθ
+˜E0
+M= +1−4ζtanθ
+β˜E0−4δkM(˜ey−˜eztanθ)
+β˜E2
+0−4ezφtanθ
+˜E0
+The Stark-contribution, r(Stark)
+M, hasseveraltermsthat
+are produced due to different imperfections and impacts
+all three Zeeman components, M= 0,±1. In order to
+determine which terms could potentially mimic the PV
+asymmetry in K, we discriminate the PV contribution
+toKwith respect to the B-field reversal and flip of the
+polarization angle, θ. Switching to a different Zeeman
+component ofthe transition is alsoareversal, which isin-
+corporatedintheexpressionfortheasymmetry, K. Anal-
+ysis of the noise affecting the accuracy of PV-asymmetry
+measurements demonstrate that the highest signal-to-
+noise ratio is achieved when θ=±π/4, and therefore,
+the polarization flip is a change of the polarization an-
+gle byπ/2. Thus, the normalized-amplitude combina-
+tion (14) must be determined for four different combina-
+tions of the B-field directions and light-polarization an-
+gles:K(+B,+π/4),K(−B,+π/4),K(+B,−π/4), and
+K(−B,−π/4), so that terms having different symmetries
+with respect to the reversals can be isolated:
+
+K1
+K2
+K3
+K4
+=1
+4
+−1−1 +1 +1
+−1 +1 +1 −1
++1−1 +1−1
++1 +1 +1 +1
+·
+K(+B,+θ)
+K(−B,+θ)
+K(+B,−θ)
+K(−B,−θ)
+.
+(20)
+The result of this procedure is summarized in Table II.
+TABLE II: List of the lowest-order terms contributing to the
+asymmetry Kfor|θ|=π/4 sorted with respect to their re-
+sponse to the reversals. K4corresponding to a rather long list
+of terms that are invariant with respect to all reversals, is not
+shown in the table.
+K1 K2 K3
+8(˜eyez+ ˜ezey)
+˜E2
+0+16˜bxey
+B˜E0+16ζ
+β˜E016b′
+xey
+B˜E016b′
+xez
+B˜E0
+The PV asymmetry contributing to K1is B-field even,
+θ-flip odd. It competes with the second-order terms that
+areacombinationoftheE-fieldandB-fieldalignmentim-
+perfections and stray fields. Using the theoretical value
+ofζ≃10−9ea0[14, 15] combined with the measured
+|β|= 2.24+0.07
+−0.12×10−8ea0/(V/cm) [4, 7], the expectedPV asymmetry, 16 ζ/β˜E, is∼4·10−4, forθ=π/4 and
+˜E0= 2 kV/cm. For a typical value of misalignments
+and “parasitic” fields, ey,z/˜E0and˜bx/B(on the order of
+10−3in the present apparatus), the contribution of the
+“parasitic” terms may be up to a few percent of the total
+value ofK1. Ways of measuring the contribution of these
+imperfections are discussed in the following sections.
+V. EXPERIMENTAL APPARATUS
+The forbidden 408-nmtransitionis excited by resonant
+laser light coupled into the power-buildup cavity in the
+presence of the magnetic and electric fields. The transi-
+tion rates are detected by measuring the population of
+the3P0state, where 65% of the atoms excited to the
+3D1state decay spontaneously (Fig. 1). This is done
+by resonantly exciting the atoms with 649-nm light to
+the 6s7s3S1state downstream from the main interac-
+tion region, and by collecting the fluorescence resulting
+from the decay of this state to the3P1and3P2states
+and subsequently, from the decay of the3P1state to the
+groundstate1S0(556nm transition). As longas the 408-
+nm transition is not saturated, the fluorescence intensity
+measured in the probe region is proportional to the rate
+of that transition.
+A schematicof the Yb-APV apparatusis shown in Fig.
+2. A beam of Yb atoms is produced (inside of a vacuum
+chamberwitharesidualpressureof ≈3×10−6Torr) with
+an effusive source: a stainless-steel oven loaded with Yb
+metal, operating at 500 −600◦C. The oven is outfitted
+with a multi-slit nozzle, and there is an external vane
+collimator reducing the spread of the atomic beam in the
+horizontal direction. The resulting Doppler width of the
+408-nm transition is ≈12MHz [7].
+FIG. 4: (color online) Schematic of the power buildup cavity .
+Downstream from the collimator, the atoms enter the
+maininteractionregionwheretheStark-andPV-induced
+transitions take place. Up to 80 mW of light at the tran-
+sition wavelength of 408 .345nm in vacuum is produced
+by frequency doubling the output of a Ti:Sapphire laser
+(Coherent 899) using the Wavetraincwring-resonator
+doubler. After shaping and linearly polarizing the laser
+beam,≈10mWofthe 408-nmradiationiscoupledinto a
+power buildup cavity (PBC) inside the vacuum chamber.6
+FIG. 5: (color online) Schematic of the optical setup. Light at 408-nm is produced by frequency doubling the output of a
+Ti:Sapphire laser (Coherent 899) using the Wavetraincwring resonator doubler. The laser is locked to the PBC using t he
+FM-sideband technique. The PBC is locked to a confocal Fabry -P´ erot ´ etalon. This scannable ´ etalon provides the maste r
+frequency. The 649-nm excitation light is derived from a sin gle-frequency diode laser (New Focus Vortex). The diode las er is
+locked to a frequency-stabilized He-Ne laser using another scanning Fabry-P´ erot ´ etalon.
+The cavity was designed to operate as an asymmet-
+ric cavity with flat input mirror and curved back mirror
+with a 25-cm radius of curvature and 22-cm separation
+between the mirrors. The atomic beam intersects the
+cavity mode in the middle of the cavity, where the 1 /e2
+radius of the mode in intensity is 172 µm. The asymmet-
+ric configuration has the advantage of larger mode radius
+at the interaction position compared to a symmetric cav-
+ity. Alargermodeallowsus toreduce the ac-Starkshifts,
+consequentlyreducingthewidthofthe408nmtransition.
+Alternatively,thecavitycanbemodifiedtooperateinthe
+symmetric confocal condition where multiple transverse
+modes can be excited, thereby increasing the effective
+“mode” size. However, we were unable to obtain high
+power and stable lock for the confocal configuration.
+ThecavitymirrorswerepurchasedfromResearchElec-
+troOptics,Inc. Fortheflatinputmirrorthetransmission
+is 350 ppm with the absorption and scattering losses of
+150 ppm total at 408 nm. The curved back mirror is de-
+signed to have a lower transmission of 50 ppm in order
+to additionally suppress the net light wave vector and,
+therefore, the M1 transition amplitude. The absorption
+and scattering losses in the curved mirror are 120 ppm.
+The finesse and the circulating power of the PBC are
+up toF= 9000 and P= 8 W. These parameters wereroutinely monitored during the PV measurements. De-
+tails of the characterization of the PBC are addressed in
+Appendix B.
+We found that the use of the 408-nm-PBC in vacuum
+isaccompaniedbysubstantialdegradationofthemirrors.
+Typically after 6 hours of operation, the finesse drops by
+a factor of two. This is a limiting factor for the duration
+of the measurement run. The degradation of the finesse
+is due to the increased absorption and scattering losses.
+This effect is reversible: the mirror parameters can be
+restored by operating the PBC for several minutes in
+air, which makes performing a number of runs possible
+without replacing the mirrors. However, it takes several
+hours with the present apparatus to reach the desired
+vacuum after exposing the PBC to air. Presently, this
+effect is under investigation aiming for longer-duration
+experiments and shorter breaks in between.
+A schematic of the PBC setup is presented in Fig. 4.
+The mirrors are mounted on precision optical mounts
+(Lees mounts) with micrometer adjustments for the hor-
+izontal and vertical angles and the pivot point of the
+mirror face. The mirror mounts are attached to an Invar
+rod supported by adjustable table resting on lead blocks.
+The input mirror is mounted on a piezo-ceramic trans-
+ducer allowing cavity scanning.7
+FIG. 6: (color online) Schematic of the E-field electrodes as -
+sembly, and a result of the E-field modeling showing an X-Z
+slice of the amplitude of the E-field z-component, ez, in a
+midplane (Y=0) of the assembly normalized by the total E-
+field amplitude, E. The voltage is applied to electrode 1, and
+electrode 2 and the correction electrodes are grounded.
+The laser is locked to the PBC using the FM-sideband
+technique [16]. In order to remove frequency excursions
+of the PBC in the acoustic frequency range, the cavity
+is locked to a more stable confocal Fabry-P´ erot ´ etalon,
+once againusingthe FM-sideband technique. This stable
+scannable cavity provides the master frequency, with the
+power-build-upcavityservingasthesecondarymasterfor
+the laser. A schematic of the optical system is presented
+in Fig. 5.
+The magnetic field is generated by a pair of rectangu-
+lar coils designed to produce a magnetic field up to 100G
+with a 1% non-uniformity over the volume with the di-
+mensions of 1 ×1×1 cm3in the interaction region. Addi-
+tional coils placed outside of the vacuum chamber com-
+pensate for the external magnetic fields down to 10 mG
+at the interaction region. The residual B-field of this
+magnitude does not have an impact on the PV measure-
+ments since its contribution is discriminated using the
+field reversals.
+Theelectricfieldisgeneratedwith twowire-frameelec-
+trodes separated by 2.1 cm (see Fig. 6). The copper elec-
+trode frames support arrays of 0.2-mm diameter gold-plated wires. This design allows us to reduce the stray
+chargesaccumulated on the electrodes by minimizing the
+surface area facing the atomic beam, thereby minimizing
+stray electric fields. AC voltage of up to 10 kV at a
+frequency of 76.2 Hz is supplied to the electrodes via a
+high-voltage amplifier. An additional DC bias voltage of
+up to 100 V can be added.
+The result of the electric field non-uniformity calcula-
+tions is shown in Fig. 6. These calculations demonstrate
+that errorsin the centeringofthe light beamwith respect
+to the E-field plates may induce substantial parasitic
+components as large as, for example, |ez| ∼5×10−3˜E0,
+producing parasitic effects comparable to the PV asym-
+metry. In order to measure and/or compensate the
+impact of the parasitic fields, additional electrodes de-
+signed to simulate stray E-field components were added
+to the interaction region. By applying high-voltage to
+these electrodes (“correction electrodes” in Fig. 6), the
+parasitic-field components may be exaggerated and ac-
+curately measured as described in the following sections.
+Light at 556 nm emitted from the interaction region
+is collected with a light guide and detected with a pho-
+tomultiplier tube. This signal is used for initial selec-
+tion of the atomic resonance and for monitoring pur-
+poses. Fluorescent light from the probe region is col-
+lected onto a light guide using two optically polished
+curved aluminum reflectors and registered with a cooled
+photodetector (PD). The PD consists of a large-area
+(1×1 cm2) Hamamatsu photodiode connected to a 1-GΩ
+transimpedance pre-amplifier, both contained in a cooled
+housing (temperatures down to −15◦C). The pre-amp’s
+bandwidth is 1 kHz and the output noise is ∼1 mV
+(rms). The 649-nm excitation light is derived from a
+single-frequency diode laser (New Focus Vortex) produc-
+ing≈1.2 mW of cw output, high enough to saturate
+the 6s6p3P0→6s7s3S1transition. Due to the sat-
+uration of this transition, ∼3 fluorescence photons per
+atom exited to the3P0state are emitted at the probe
+region. The natural width of the 649-nm transition is
+70 MHz, thus, its profile covers all transverse velocity
+groups(vx) in the atomic beam ( ≈8 MHz Doppler width
+at 649 nm). A drift of the laser frequency ( ∼100 MHz
+per minute) is eliminated by locking the diode laser to a
+frequency-stabilized He-Ne laser using a scanning Fabry-
+P´ erot ´ etalon with the scanning rate of 25 Hz. The spec-
+tral distance between the ´ etalon transmission peaks from
+the two lasers is measured during each scan and main-
+tained constant within an accuracy of ±3 MHz, good
+enough to eliminate any degradation of the probe-region
+signal.
+The signals from the PMT and PD are fed into lock-
+in amplifiers for frequency discrimination and averaging.
+A typical time of a single spectral-profile acquisition is
+20 s. The signals at the first and the second harmonic of
+the electric-field modulation frequency are registered si-
+multaneously, which reduces the influence of slow drifts,
+such as instabilities of the atomic-beam flux. The mod-
+ulation frequency is limited by several factors. Thermal8
+distribution of atomic velocities in the beam causes a
+spread (of ≈2 ms) in the time of flight between the in-
+teraction region and the probe region. This, along with
+the finite bandwidth of the PD, leads to a reduction of
+the signal-modulation contrast (see below). The choice
+of the modulation frequency of 76.2 Hz is a tradeoff be-
+tween this contrast degradation and the frequent E-field
+reversal.
+VI. RESULTS AND ANALYSIS
+In Fig. 7 a profile of the B-field-split 408-nm spectral
+line ofthe174Yb is shown. The 649-nm-light-inducedflu-
+orescence was recorded during a single profile scan. Sta-
+tistical error bars determined directly from the spread of
+data are smaller than the points in the figure. The pecu-
+liar asymmetric line shape of the Zeeman components is
+a result of the dynamic Stark effect [7]. During a typical
+FIG. 7: (color online) A profile of the B-field-split 408-nm
+spectral line of174Yb recorded at 1st- and 2nd-harmonic of
+themodulation. Alsoshown is asimulated PVcontributionin
+the first-harmonic signal. ˜E=5 kV/cm; DC offset=40 V/cm;
+θ=π/4; the effective integration time is 200 ms per point.
+experimental run 100 profiles are recorded for each com-
+bination of the magnetic field and the polarization angle
+(400 profile scans in total). In order to compute the nor-
+malized amplitude, rq, of a selected Zeeman component,
+the actual first-harmonic signal near the Zeeman peak
+is divided by the respective second-harmonic signal and
+then averaged over a number of the data points1. Then,
+the combination Kof Eq. (14) is computed for each pro-
+filescanfollowedbyaveragingtheresultoverallthescans
+at a given B- θconfiguration. This procedure is repeated
+for all four reversals, and all B- θsymmetrical contribu-
+tions,K1−4, are determined. In the present experiment,
+the values of K2,3,4-terms are found to be consistent with
+1In the normalized rate calculations only data points having in-
+tensity higher that 1/3 of the respective Zeeman peak are use d
+to avoid excessive noise from spectral regions with low sign al
+intensity.zero within the statistical uncertainty, which is the same
+as that of the PV-asymmetry (see below).
+As can be seen from Table II, terms in K1associated
+with the fields imperfection are of crucial importance:
+16
+˜E0/bracketleftBigg
+ey/parenleftBigg
+˜ez
+2˜E0+˜bx
+B/parenrightBigg
++ez˜ey
+2˜E0/bracketrightBigg
+.
+In order to measure the contribution of these terms, arti-
+ficially exaggerated E-field imperfections both static and
+oscillating, eex
+z,eex
+y, ˜eex
+yand ˜eex
+z, are imposed by use of
+the “correction electrodes” (see Fig. 6), and two sets of
+the experiments were performed. In the first one, a DC-
+voltage was applied to the correction electrodes, and the
+measurements were done reversing eex
+yandeex
+z. These
+experiments yield values of ˜ eyand ˜ez+2˜E0˜bx/B. In the
+secondset, anAC-voltagemodulated synchronouslywith
+the main E-field is applied to the correction electrodes.
+In order to reverse the sign of the parasitic terms a π-
+phase-shift of ˜ eex
+yand ˜eex
+zwith respect to the modulation
+signal is employed by switching the wiring of the correc-
+tion electrodes. Thus, values of the DC-imperfections, ey
+andez, are determined. The magnitudes of the applied
+electric fields and their distributions are calculated using
+a 3D-numerical-model of the interaction region. The re-
+sults of the experiments are presented in Table III. The
+TABLE III: Results of measurements of the electric field
+imperfections using artificially exaggerated AC- and DC-
+components, ˜ eex
+y,zandeex
+y,z. These fields were generated by
+use of the correction electrodes, Fig. 6. ˜E0= 2000(2) V/cm.
+DC-Set AC-Set
+Exaggerated imperfections (V/cm)
+eex
+y=−140(2) ˜ eex
+y=−120(2)
+eex
+z= 20(2) ˜ eex
+z= 30(2)
+Measurements (mV/cm)
+˜eyeex
+z
+2˜E0= 16(10) ey˜eex
+z
+2˜E0= 4(5)
+(2˜E0˜bx
+B+ ˜ez)eex
+y
+2˜E0= 442(10) ez˜eex
+y
+2˜E0= 40(5)
+Parasitic fields (V/cm)
+˜ey= 3.2(2) ey= 0.5(0.6)
+(2˜E0˜bx
+B+ ˜ez) =−12.6(0.3) ez=−1.3(0.2)
+net contribution of these imperfections to K1in the ab-
+sence of the exaggerated fields is found to be2:
+ey/parenleftBigg
+˜ez
+2˜E0+˜bx
+B/parenrightBigg
++ez˜ey
+2˜E0=
+=−2.6(1.6)stat.(1.5)syst.mV/cm. (21)
+2Compare with the PV asymmetry parameter ζ/β≈40 mV/cm.9
+The systematic uncertainty comes from a sensitivity of
+the numerical model of the E-field, which is used for cal-
+culating the exaggerated fields in the interaction region,
+to an imperfect approximation of the electrode-system
+geometry. These experiments suggest that this field’s
+imperfection cannot mimic the PV-effect entirely, never-
+theless, it appears to be a major source of systematic un-
+certainty impacting the accuracy of the PV-asymmetry
+measurements. Themostprominentcontributionisgiven
+by a combined effect of the parasitic components of the
+electric field and the non-zero projection of the lead-
+ing magnetic field on the direction of the electric field:
+ey(˜ez/2˜E0+˜bx/B). The PV-asymmetry parameter, ζ/β
+is obtained from the measuredvalue of K1by compensat-
+ing for the influence of these magnetic- and electric-field
+imperfections, Eq. (21).
+There is another effect that cannot, by itself, mimic
+the PV-asymmetry, but needs to be taken into account
+for proper calibration. This effect is related to the E-field
+modulation implemented in the present experiment. The
+atoms are excited to the metastable state, 6s6p3P0, by
+the light beam in the interaction region and then travel
+∼20 cm until they are detected downstream in the probe
+region. Due to the spread in the time-of-flight between
+the interaction and probe regions, the phase mixing leads
+to a reduction of the signal modulation contrast at the
+probe region, and it depends on the signal-modulation
+frequency. Since the signal comprises two time-scales of
+interest, first- and second-harmonic of the E-field mod-
+ulation, the contrast reduction is different for the two.
+Therefore, the ratio of the signal modulation amplitudes,
+rM, on which we base the PV-asymmetry observation,
+appear altered in the probe region compared to what it
+would be at the interaction region. The amplitude com-
+bination, K, and, therefore, the PV-parameter, ζ/β, are
+similarly affected. In our data analysis, a correction co-
+efficient, C0, is introduced, which has been calculated
+theoretically:
+/bracketleftbiggζ
+β/bracketrightbigg
+probe reg.=C0/bracketleftbiggζ
+β/bracketrightbigg
+real.
+Under present experimental conditions, this coefficient,
+C0, is found to be 1.028(3), and the measured PVparam-
+eter is corrected accordingly. Principles of its derivation
+are given in Appendix C.
+In Fig. 8, the PV interference parameter ζ/βis shown
+as determined in 19 separate runs ( ∼60 hours of integra-
+tion in total). Its mean value is 39(4) stat.(3)syst.mV/cm,
+which is in agreement with the theoretical predictions
+[14, 15]. The value of the PV parameter was extracted
+using the expression given in the first column of Ta-
+ble II, taking into account the calibration correction
+C0. Thus, |ζ|= 8.7±1.4×10−10ea0, which is the
+largest APV amplitude observed so far (here we used
+|β|= 2.24+0.07
+−0.12×10−8ea0/(V/cm) [4, 7]).
+The signofthe PVinterferenceparameter ζ/βisfound
+by comparing the measurements with the theoreticalFIG. 8: (color online) The PV interference parameter ζ/β.
+Mean value: 39(4) stat.(3)syst.mV/cm, |ζ|= 8.7±1.4×
+10−10ea0.
+model of the transition rates employing the field geome-
+try shown in Fig. 2. The direction and, thus, the signs of
+the electricandmagnetic fieldsaswell asthe polarization
+angleθwere calibrated prior to the PV measurements.
+Special care was taken of detecting parasitic phase shifts
+in the lock-in amplifier. A signal from an arbitrary func-
+tion direct digital synthesis (DDS) generator simulating
+the output of the probe region photodetector was fed
+into the amplifier. The signal is comprised of a sum of
+twosinusoidalwaveforms,onefrequencydoubled, attenu-
+ated, andphaseshifted with respecttotheother. Results
+of the signal parameters measurement from the lock-in,
+such as the first-to-second harmonic amplitude ratio, rel-
+ative phase shift and its sign, are compared to those used
+in the DDS generator to simulate the signal. The differ-
+ence in the measured and generated amplitude ratio is
+found to be below 0.01%, and the relative phase shift is
+detected within ±1.5◦. No relative sign flips between the
+first- and second-harmonic amplitudes were detected.
+VII. ERROR BUDGET
+The present measurement accuracyis not yet sufficient
+to observe the isotopic and hyperfine differences in the
+PV amplitude, which requires an accuracy better than
+≈1% for PV amplitude in a single transition [17–19]. In
+the present apparatus the signal levels achieved values
+high enough to reach the signal-to-noise ratio (SNR) of
+2//radicalbig
+τ(s) for the PV asymmetry if the noise were domi-
+nated by the fluorescence-photon shot-noise ( τis the in-
+tegration time). This is good enough to reach the sub-
+percent accuracy in a few hours of integration. However,
+a number of additional factors limit the accuracy.
+One of the most important noise sourcesis the fluctua-
+tions of the modulating- and DC-field parameters during
+the experiment. The first- and the second-harmonic sig-
+nals depend differently on the modulating electric field
+amplitude, ˜E0, and the DC-bias, thus, a noise in the
+electronics controlling the fields contaminates the first-10
+to-second harmonic ratio directly. A substantial effort
+was made to cope with this problem: a home-built high-
+voltage amplifier used in the first 13 runs was replaced
+by a commercial Trek 609B amplifier and a circuit con-
+trolling the DC-bias was upgraded. This allowed us to
+control the DC-bias and ˜E0with mV-scale accuracy that
+would make the SNR to approach the shot-noise limit if
+this were the only source of the noise. As seen in Fig. 8,
+the last six measurements exhibit higher accuracy than
+the rest. These are the runs after the HV-system up-
+grade. However, the present SNR of ≈0.03//radicalbig
+τ(s) is
+worse by almost two orders of magnitude than the shot-
+noise limit.
+There are other fluctuations in the system parameters,
+such as light intensity fluctuations in the PBC, fluctua-
+tions of the spectral position of the PBC resonance with
+respect to the frequency reference, and noise in the de-
+tection system. All of them contribute to the noise in
+the first- and the second- harmonic signals but we found
+that such noise largely canceled in the ratio rM.
+However, there is a noise source, which is not canceled
+in the ratio. The following experiments demonstrated
+that this noise source is related to frequency excursions
+of the Fabry-P´ erot ´ etalon serving as the frequency ref-
+erence for the optical system. In these experiments the
+excitation light was frequency tuned to a wing of the
+atomicresonance,andthefirst-andsecond-harmonicsig-
+nals were recorded without scanning over the resonance.
+Then, the same was done when the spectral position was
+set atthe peak ofthe resonance, anda changeofthe SNR
+for the harmonics ratio was determined. These experi-
+ments were performed using the upgraded HV-system.
+Results of the measurements are presented in Fig. 9. For
+FIG. 9: (color online) Impact of the frequency excursions of
+the Fabry-P´ erot ´ etalon on the noise level in the harmonics
+ratio. A change in the noise level when the optical system
+was tuned from the wing of the atomic resonance to its peak
+is shown. In the inserts above the excitation light spectral
+position is shown schematically with respect to the atomic
+resonance. Arrows denote the fluctuations.
+a shot-noise-limied signal, the SNR at the peak of theresonance is expected to be a factor of about√
+2 higher
+than at the wing due to larger signal. It was found,
+however, that the SNR went up by a factor of 4 by tun-
+ing from the wing to the peak of the resonance. This
+demonstrates that the main source of noise is not photon
+statistics but fluctuations in the spectral reference. In-
+deed, the frequencyexcursionsatthe wingofthe spectral
+line produces substantially more intensity noise due to a
+steeper spectral slope than that at the peak, where the
+slope is nominally zero. It must be emphasized, that in
+the case of slow frequency excursions (compared to the
+E-field modulation period), the noise in the first- and the
+second-harmonicchannelswould be canceled in the ratio.
+However, fast excursions can generate noise in the signal
+ratio.
+The factors affecting the measurement accuracy men-
+tioned above have an impact on the statistical error of
+the result. The present systematic errors(summarized in
+Table IV) has nearly the same significance as the statis-
+tical one and also comprises a number of factors.
+One of the most significant factors is the uncertainty
+in the field-imperfection contributions, Eq. (21). This
+uncertainty is mostly due to statistical factors such as
+laser drifts, nevertheless, it provides an offset to the PV-
+parameter. Since the measurement of this contribution
+is actually the same measurement as the PV-effect, any
+improvements of the stability reduces the overall system-
+atic uncertainty. We would like to emphasize also that
+Eq. (21) represents a mean value of the imperfection con-
+tribution over numerous experiments averagingover pos-
+sible fluctuations of the field-imperfection contribution.
+These fluctuations may be partially responsible for the
+variance in the PV-parameter, and, thus, the statisti-
+cal uncertainty of its value. This fact demonstrates that
+the elimination of the field-imperfections is an essential
+requirement for improving the overall accuracy of the ex-
+periments.
+Another significant source of the systematic uncer-
+tainty is the uncertainty in the value of the electric field
+in the interaction region. While the voltage applied to
+the E-field plates and the correction electrodes is con-
+trolled precisely, the actual E-field value used in the PV
+parameter determination depends on the accuracy of the
+numericalmodelingoftheelectric-fielddistributioninthe
+particular geometry. There are two factors in the model
+contributing to the uncertainty: finite accuracy of mea-
+surements of the interaction region geometrical parame-
+ters, and the imperfect approximation of the geometry in
+the numerical simulation.
+However, while this systematic uncertainty plays a sig-
+nificant role for measurements of the PV parameter of a
+single isotope, for the isotope ratios this uncertainty will
+cancel (or will be substantially reduced), if the measure-
+ments observingdifferent isotopes are performed without
+changing the E-field geometry. The same is true for the
+calibration parameter, C0, which also cancels in the iso-
+tope ratios.
+There are other, rather minor, factors contributing to11
+the systematic uncertainty, for example, a finite accuracy
+of the polarization angle flip, errors in the lock-in ampli-
+fiers, a finite dynamic range of the lock-ins etc. The net
+contribution of these factors is found to be /lessorsimilar1%.
+A summary ofthe systematic errorbudget is presented
+in Table IV.
+TABLE IV: List of factors contributing to the systematic un-
+certainty of the PV parameter, ζ/β.
+Factor Uncertainty (%)
+˜Evalue:
+geometry 5
+numerical modeling 3
+E-field imperfections 5
+Phase mixing 0.5
+Other 1
+Total(in quadrature) 8
+VIII. TOWARDS MEASUREMENT OF THE
+ISOTOPE RATIOS
+As pointed out above, a goal of the future measure-
+ments of the parity-violation effects in ytterbium is ob-
+serving a difference in the PV effect between different
+isotopes. The net uncertainty of the PV parameter of a
+single isotope must be better than 1% based on the the-
+oretical predictions. To this end, a program of the appa-
+ratus upgrades and improvements is developed. Besides
+general improvements of the stability of the system pa-
+rameters, increase of the signal levels, suppression of the
+electronics noise etc., the main focus is on elimination
+of the frequency excursions of the frequency reference,
+whichisamajorsourceofthestatisticalnoise. Improving
+the statistical uncertainty will contribute to more precise
+measurementand controlofthe E-field-imperfection con-
+tribution to the systematic part of the uncertainty. The
+latter is another high-priority improvement essential for
+reaching the goal.
+In the future apparatus, the referencing of the opti-
+cal system to the Fabry-P´ erot ´ etalon will be replaced
+by locking the system to a femtosecond frequency comb
+that will be available shortly. The impact of the E-field
+imperfection is planned to be substantially suppressed
+by redesigning of the interaction region to provide more
+uniform and controlled electric field distribution. Until
+now, no scientific or technical obstacles were discovered
+preventing us to improve the apparatus to the desired
+level of sensitivity.
+IX. ACKNOWLEDGEMENTS
+The authors acknowledge helpful discussions with and
+important contributionsofM. A. Bouchiat, C. J. Bowers,E. D. Commins, B. P. Das, D. DeMille, A. Dilip, S. J.
+Freedman, J. S. Guzman, G. Gwinner, M. G. Kozlov, S.
+M. Rochester, and M. Zolotorev. This work has been
+supported by NSF.
+Appendix A: Derivation of transition amplitudes
+The total Hamiltonian (before including light-atom in-
+teractions and assuming Bis alongˆz) can be written as
+H=HAtomic+HZeeman+HStark+HAPV,(A1)
+whereHAtomicis the atomic Hamiltonian and HZeeman,
+HStark, andHAPVrepresent the contributions from the
+static magnetic field B, the static electric field E, and
+the parity non-conserving weak interaction, respectively.
+Here
+HZeeman=−µ·B=gµBJ·B=gµBJzB,(A2)
+whereµ=−gµBJis the magnetic dipole moment of the
+atom,gis the Land´ e factor, µBis the Bohr magneton,
+andJis the angular-momentum operator. Similarly,
+HStark=−d·E=−diEi, (A3)
+wheredis the atomic electric-dipole operator. Finally,
+HAPV=iH0
+0, (A4)
+whereH0
+0is a scalar operator. Summation over repeated
+indices is assumed.
+In the presence of a strong magnetic field, that is,
+when Zeeman splitting dominates Stark-shifts, it is use-
+ful to think of H1≡HStark+HAPVas a perturbation to
+H0≡HAtomic+HZeeman. In this case, the LS-coupled
+states|2S+1LJ;M∝angbracketright, such as |3D1;M∝angbracketrightand|1S0;0∝angbracketright, are
+eigenstates of the unperturbed Hamiltonian H0. Then
+the first-order perturbation theory can be used to deter-
+mine the eigenstates of the total Hamiltonian:
+|a∝angbracketright=|a∝angbracketright+/summationdisplay
+a′|a′∝angbracketright∝angbracketlefta′|H1|a∝angbracketright
+ω(a)−ω(a′), (A5)
+whereω(a) is the energy of state |a∝angbracketright. (Perturbed eigen-
+states are denoted using an overbar.)
+The electric-dipole amplitude for the optical transition
+of interest is
+A(E1)
+M=∝angbracketleft3D1;M|(−d·E)|1S0∝angbracketright ≡A(Stark)
+M+A(APV)
+M,
+(A6)
+where
+A(Stark)
+M=/summationdisplay
+a′∝angbracketleft3D1;M|d·E|a′∝angbracketright∝angbracketlefta′|d·E|1S0∝angbracketright
+ω(3D1)−ω(a′)
++/summationdisplay
+a′∝angbracketleft3D1;M|d·E|a′∝angbracketright∝angbracketlefta′|d·E|1S0∝angbracketright
+ω(1S0)−ω(a′),(A7)12
+and
+A(APV)
+M=/summationdisplay
+a′∝angbracketleft3D1;M|iH0
+0|a′∝angbracketright∝angbracketlefta′|d·E|1S0∝angbracketright
+ω(3D1)−ω(a′)
+−/summationdisplay
+a′∝angbracketleft3D1;M|d·E|a′∝angbracketright∝angbracketlefta′|iH0
+0|1S0∝angbracketright
+ω(1S0)−ω(a′).(A8)
+The Stark amplitude can be written as
+A(Stark)
+M=Tij∝angbracketleft3D1;M|Uij|1S0∝angbracketright, (A9)
+whereTij=EiEjand
+Uij=/summationdisplay
+a′di|a′∝angbracketright∝angbracketlefta′|dj
+ω(3D1)−ω(a′)+dj|a′∝angbracketright∝angbracketlefta′|di
+ω(1S0)−ω(a′).(A10)
+LetTk
+qandUk
+qrepresent the irreducible spherical com-
+ponents of the tensors TijandUij. Then TijUij=
+(−1)qTk
+−qUk
+qand Eq. (A7) becomes
+A(Stark)
+M= (−1)qTk
+−q∝angbracketleft3D1;M|Uk
+q|1S0∝angbracketright
+= (−1)qTk
+−q(3D1||Uk||1S0)√
+3∝angbracketleft0,0;k,q|1,M∝angbracketright
+=iβ(−1)q(E×E)−q∝angbracketleft0,0;1,q|1,M∝angbracketright.(A11)
+Hereβis the vector Stark transition polarizability and
+defined by
+β≡1√
+6(1S0||U1||3D1). (A12)
+To derive Eq. (A11), we used ∝angbracketleft0,0;k,q|1,M∝angbracketright=δk1δqM
+andT1
+−q=/summationtext
+q1,q2∝angbracketleft1,q1;1,q2|1,−q∝angbracketrightEq1Eq2= (i/√
+2)(E×
+E)−q.
+For the parity-violating contribution to the E1 transi-
+tion amplitude we can likwise write
+A(APV)
+M=iEi∝angbracketleft3D1;M|Wi|1S0∝angbracketright,(A13)
+where
+Wi=/summationdisplay
+a′H0
+0|a′∝angbracketright∝angbracketlefta′|di
+ω(3D1)−ω(a′)−/summationdisplay
+a′di|a′∝angbracketright∝angbracketlefta′|H0
+0
+ω(1S0)−ω(a′).(A14)
+LetE1
+qandW1
+qrepresent the spherical components of
+the vectors EiandWi, respectively. Then EiWi=
+(−1)qE1
+−qW1
+qand we have
+A(APV)
+M=i(−1)qE1
+−q∝angbracketleft3D1;M|W1
+q|1S0∝angbracketright
+=i(−1)qE1
+−q(3D1||W1||1S0)√
+3∝angbracketleft0,0;1,q|1,M∝angbracketright
+=iζ(−1)qE1
+−q∝angbracketleft0,0;1,q|1,M∝angbracketright. (A15)
+Hereζis given by
+ζ≡1√
+3(3D1||W1||1S0). (A16)Appendix B: Characterization of the PBC mirrors
+The finesse of the cavity is measured using the cavity-
+ring-down method [20]. The laser beam is sent through a
+Pockels cell (Cleveland Crystals Inc. QX 1020 Q-Switch)
+and a polarizer before entering the cavity. The polarizer
+is aligned with the laser polarization so that the light is
+transmittedwhen there isnovoltageapplied tothe Pock-
+els cell. A high-voltage pulse generator is used to send
+a fast step signal (30-ns wavefront) to the Pockels cell
+which rotatesthe polarizationofthe light sothat it is not
+transmitted through the polarizer. The laserfrequencyis
+locked to the resonance frequency of the cavity, and then
+the Pockels cell is switched into the non-transmitting
+state, causing a fast interruption of the laser power. The
+subsequent decay of the light inside the cavity is moni-
+tored with a fast photodiode (50-MHz bandwidth) mea-
+suring the power transmitted through the back mirror of
+the cavity. The signal is fit to an exponential decay. The
+decay time is related to the finesse of the cavity ( F) by
+F=πc
+Lτ,
+wherecis the speed of light, Lis the cavity length, and
+τis the intensity decay time. An example of the PBC
+transmission signal and its fit are shown in Fig. 10.
+FIG. 10: (color online) Application of the cavity-ring-dow n
+method for the determination of the finesse of PBC.
+Following the analysis discussed in [21], if we denote
+the transmission of mirrors 1 and 2 by T1andT2, re-
+spectively, and the absorption+scatter loss per mirror
+asl1,2= (A+S)1,2, then the total cavity losses L=
+T1+T2+l1+l2determine the finesse F:
+F=2π
+T1+T2+l1+l2. (B1)
+Information on the transmission of the mirrors discrim-
+inated from the A+S losses can be obtained using the13
+measured value of the finesse and the power transmitted
+trough PBC, Ptr:
+Ptr
+ǫPin= 4T1T2/parenleftbiggF
+2π/parenrightbigg2
+, (B2)
+wherePinis the input power, and ǫis a mode-matching
+factor. For two arbitrary mirrors, for which neither T1,2
+norl1,2are known the Eq. (B1,B2) do not provide a
+solution, since a number of variables exceeds the number
+ofequations. Nevertheless,fortwomirrorsfromthe same
+coating run when one can assume that T1=T2=Tand
+l1=l2=l, the equations (B1,B2) become
+F=π
+T+l.
+Ptr
+ǫPin= 4T2/parenleftbiggF
+2π/parenrightbigg2
+,
+and for known mode-matching factor ǫthe parameters of
+the mirrors ( Tandl) can be determined. The factor ǫ
+depends on the geometry of the cavity, and is assumed
+to stay constant upon replacing of the mirrors, if the ge-
+ometry of the input laser beam and the configuration of
+the PBC are unchanged. This gives the possibility to
+calibrate this factor by using a mirror set for which the
+transmission is known. We used for this purpose the mir-
+ror set purchased from Advanced Thin Films, Inc., for
+which reliable data on the transmission of the mirrors
+is provided by the supplier. By measuring the finesse
+of the PBC comprised of these mirrors and the ratio of
+the transmitted-to-input power, the mode-matching fac-
+tor and the A+S mirror losses lare found. This set is
+not an actual mirror set that was used in the PV experi-
+ment, nevertheless, the parameters of other mirrors were
+determined by replacing one mirror in the “reference”
+set by the “test” mirror, parameters of which are sought.
+The geometry of the cavity was unchanged during the
+replacement. This tactic allows for the measurement of
+parameters of any arbitrary mirror.
+Appendix C: Impact of the phase mixing effect on
+the harmonics ratio
+Atoms undergo the 6s2 1S0→5d6s3D1transition
+in the interaction region where they are illuminated by
+408-nm light and are exposed to the static magnetic field
+and the oscillating electric field E(t). Excited atoms
+then spontaneously decay from the 5d6s3D1state to the
+metastable 6s6p3P0state. The population of 6s6p3P0
+is proportional to the transition rate RMforM= 0,±1.
+Without loss of generality, we assume that the constant
+of proportionality is equal to one.
+The rate RMis measured in the probe region. The
+probe region is located a distance d≈20 cm away from
+the interaction region. Therefore, an atom that arrives
+at the detection region at time texperienced an electric
+fieldwithmagnitude E(t−d/vz)intheinteractionregion,wherevzis the atom’s speed and d/vzis the amount of
+time required for the atom to travel a distance d.
+Because some atoms travel faster or slower than oth-
+ers, the detection region is full of atoms that have each
+experienced a different electric field while in the inter-
+action region. Each atom contributes to the total rate
+and hence the observed rate RMis the thermal average
+of every contribution:
+RM(t;ω,d,v0) =/integraldisplay∞
+0RM(t−d/vz)f(vz;v0)dvz,(C1)
+where
+f(vz;v0)dvz= 2(vz/v0)3e−(vz/v0)2dvz/v0,(C2)
+is the probability for an atom to have speed between vz
+andvz+dvz. Herev0=/radicalbig
+2kBT/m= 2.9×104cm/s is
+the thermal speed, T≈873 K is the oven temperature,
+andm= 161 GeV /c2is the atomic mass of Yb.
+It is convenient to introduce the dimensionless vari-
+ablesx=vz/v0andτ=ωt, and the dimension-
+less parameter α=ωd/v0. Then the average rate
+RM(t;ω,d,v0)→RM(τ;α) depends only on the dimen-
+sionless quantities αandτ, and Eq. (C1) becomes
+RM(τ;α) =R[0]
+M+R[1]
+M|I(α)|cos(τ+Arg[I(α)])
++R[2]
+M|I(2α)|cos(2τ+Arg[I(2α)]),(C3)
+with
+I(α)≡/integraldisplay∞
+0e−iα/xf(x;1)dx. (C4)
+Note that |I(α)| →0 asα→ ∞whereas|I(α)| ≈1 when
+α <1. This places a limit on the modulation frequency:
+We require that ω < v 0/d= 2π×230 Hz in order to
+avoid a significant decrease in signal.
+The lock-in amplifier receives an input signal propor-
+tional to RMand returns two output signals S[1]
+MandS[2]
+M
+corresponding to the first and second harmonic compo-
+nents, respectively. This process can be modeled as
+S[n]
+M(φn;α) =1
+π/integraldisplay2π
+0RM(τ;α)cos(nτ+φn)dτ
+=R[n]
+M|I(nα)|cos(Arg[I(nα)]+φn),(C5)
+where the phases φ1,2of the lock-in amplifier are chosen
+to maximize the signals S[1,2]
+M. That is,
+φn=φn(α)≡ −Arg[I(nα)]. (C6)
+Ourmeasurement sMis the ratioofthe first-and second-
+harmonic signals:
+sM=S[1]
+M(φ1;α)
+S[2]
+M(φ2;α)=R[1]
+M|I(α)|
+R[2]
+M|I(2α)|=rM×C(α),(C7)14
+whereC(α)≡ |I(α)|/|I(2α)|is thecorrection factor .
+Therefore, we must further divide the ratio sMof ob-
+served output signals by C(α) to measure the ratio rM.
+The correction factor C(α) and the optimal lock-in
+phasesφ1,2(α) inherit dependence on the modulation fre-
+quency ( ω= 2π×76.2 Hz), the distance between inter-
+action and detection regions ( d≈20 cm), and the oven
+temperature ( T≈873 K) through the parameter α:
+α=ωd/radicalbig
+2kBT/m= 0.33(2), (C8)
+where the uncertainty in αis given by
+δα=α/radicalbig
+(δT/2T)2+(δd/d)2, (C9)
+forδT≈50 K and δd≈1 cm. The correction factor can
+be computed numerically and has a value
+C0=C(α) = 1.028(3), (C10)
+with uncertainty given by δC0=|C′(α)|δα. Likewise,
+the lock-in phases have the following values
+φ10=φ1(α) = 16(1)◦, φ20=φ2(α) = 33(2)◦,(C11)
+whereδφn0=|φ′
+n(α)|δα.
+In order to understand the impact of imperfect phase
+selections, we include the effects of slight deviations from
+the optimal phase φn(α) by taking
+φn→φn(α)+ϕn, (C12)whereϕn≈0 represents a small deviation. Then the
+correction factor becomes
+C(α)→˜C(α,ϕ1,ϕ2) =C(α)×cos(ϕ1)
+cos(ϕ2),(C13)
+and hence ˜C0=˜C(α,0,0) =C(α) =C0. The uncer-
+tainty in the correction factor becomes
+δC0→δ˜C0=/radicalBig
+δC2
+0+δϕ4
+1+δϕ4
+2,(C14)
+whereδϕnis the uncertainty in the deviation ϕn. To de-
+rive this expression, we estimated the partial uncertainty
+in˜C0due toϕnby∂2
+ϕn˜C(α,ϕ1,ϕ2)δϕ2
+n.
+To estimate the uncertainty δϕn, we assume that we
+are within about 1◦of the optimal phase. This choice
+is consistent with the magnitude of the uncertainty in
+the optimal phases φ10andφ20. Therefore, we will take
+δϕn=δφn0to be the accuracy with which we can select
+the lock-in phases. Then δϕ1= 0.02,δϕ2= 0.03, and
+δ˜C0= 0.0031≈0.0029 =δC0. (C15)
+Hence small deviations (on the order of 1◦) have a negli-
+gible effect on the uncertainty in the correction factor.
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+nal of the Optical Society of America B-Optical Physics
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+36, 493 (1975).
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+L. Hunter, Physical Review Letters 42, 343 (1979).
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+An exploration through problems and solutions. Second
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\ No newline at end of file
diff --git a/1001.0588.txt b/1001.0588.txt
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@@ -0,0 +1,654 @@
+arXiv:1001.0588v1  [astro-ph.CO]  4 Jan 2010
+THEASTROPHYSICAL JOURNAL. ACCEPTED 2010 JANUARY 04
+Preprint typesetusingL ATEX styleemulateapjv. 11/10/09
+THEHARD X-RAY VIEWOF REFLECTION,ABSORPTION, ANDTHE DISK- JETCONNECTION INTHE
+RADIO-LOUDAGN3C33
+D. A. E VANS1,2, J. N. R EEVES3, M. J. H ARDCASTLE4, R. P. K RAFT2, J. C. L EE2, S. N. V IRANI5
+TheAstrophysical Journal. Accepted 2010 January 04
+ABSTRACT
+We present results from SuzakuandSwiftobservations of the nearby radio galaxy 3C33, and investiga te
+the nature of absorption, reflection, and jet production in t his source. We model the 0.5–100 keV nuclear
+continuumwith a power law that is transmitted either throug hone or morelayersof pc-scaleneutral material,
+orthroughamodestlyionizedpc-scaleobscurer. Thestanda rdsignaturesofreflectionfroma neutralaccretion
+diskareabsentin3C33: thereisnoevidenceofarelativisti callyblurredFeK αemissionline,andnoCompton
+reflection hump above 10 keV. We find the upper limit to the neut ral reflection fraction is R <0.41for an
+e-folding energy of 1 GeV. We observe a narrow, neutral Fe K αline, which is likely to originate at least
+2,000 R sfrom the black hole. We show that the weakness of reflection fe atures in 3C33 is consistent with
+two interpretations: either the inner accretion flow is high ly ionized, or the black-hole spin configuration is
+retrogradewithrespecttotheaccretingmaterial.Subjectheadings: galaxies: active– galaxies: jets–galaxies: individual(3 C33)–X-rays: galaxies
+1.INTRODUCTION
+The origin of jets in active galactic nuclei (AGN) is one
+of the most important unsolved problems in extragalactic as -
+trophysics. While 90% of all AGN (Seyfert galaxies andradio-quiet quasars) show little or no jet emission, the re-maining 10% (the radio-loud AGN and radio-loud quasars)launch powerful, relativistic twin jets of particles from t heir
+cores. Since jets transport a significant fraction of the mas s-
+energy liberated during the accretion process, sometimes o ut
+to∼Mpc distances, understanding how they are produced is
+keyto acompletepictureofaccretionandfeedbackinAGN.
+X-ray observations of the nuclei of radio-loud and radio-
+quiet AGN are essential for establishing the connection be-tweentheaccretionflow,blackhole,andjet. Instandardmod -
+els of X-ray emission in AGN, quasi-isotropic power-law X-ray emission is generated in a corona that lies above a high
+˙Maccretion disk (Krolik& Kallman 1987; Ghisellini etal.
+1994). A fraction of this hard X-ray emission is reflected bycold gas in the accretion disk, producing an observed spec-trum imprinted with the features of photoelectric absorpti on,
+fluorescent 6.4 keV Fe K αline emission, and a Compton
+backscattered reflection continuum that produces a bump inthe spectrum above 10 keV (George&Fabian 1991). TheFe Kαline may be relativistically broadened, which is the
+signature of disk reflection near the innermost stable circu lar
+orbit(ISCO)oftheblackhole(e.g.,Fabianet al.1989). Inr a-
+diogalaxies,thereexistsan additional power-lawcomponent
+of X-ray emission that is associated with the parsec-scale j et
+(e.g.,Evanset al. 2006;Balmaverdeet al. 2006)
+X-ray observations reveal systematic differences between
+1Massachusetts Institute of Technology, Kavli Institute fo r Astro-
+physics and Space Research, 77 Massachusetts Avenue, Cambr idge, MA
+02139
+2Harvard-Smithsonian Center for Astrophysics, 60 Garden St reet,
+Cambridge, MA02138
+3Astrophysics Group, School of Physical and Geographical Sc iences,
+Keele University, Keele, ST55BG, UK
+4School of Physics, Astronomy & Mathematics, University of H ert-
+fordshire, College Lane, Hatfield AL10 9AB,UK
+5Department of Astronomy, Yale University, P.O. Box 208101, New
+Haven, CT 06520the spectra of radio-loud and radio-quiet AGN. Diskreflection is commonly detected in radio-quiet galaxies(Reeves& Turner2000;Reeveset al.2006): Comptonreflec-tion bumps are prominent, and over 50% of local Seyfertgalaxies show broadened Fe K αemission from within 25
+Schwarzschild radii. This result may hold even when theeffects of complex X-ray absorption are taken into ac-count (Nandraetal. 2007). On the other hand, radio-loud AGN tend to show weak, or absent neutral Comp-ton reflectionhumps(Grandiet al. 2006; Larssonet al. 2008;Sambrunaet al.2009;Kataokaet al.2007),andinmostcases,unresolvedneutralFeK αlines(Balmaverdeetal. 2006).
+Continuum X-ray observations of radio-loud AGN have
+mostly been restricted to bright broad-line radio galaxies(BLRGs) and quasars, which are oriented relatively closeto the line of sight with respect to the observer. In thesesources,unabsorbednon-thermalemission fromthe jet coul d
+potentially contaminate the unabsorbed accretion-relate d X-
+ray spectrum and thus dilute the apparent strength of theCompton reflection component. Narrow-line radio galaxies
+(NLRGs), on the other hand, which are oriented at low tointermediate angles, have the distinct advantage that (una b-
+sorbed) jet-related X-ray emission can be readily spectral ly
+separatedfrom(heavilyabsorbed)accretion-relatedemis sion,
+allowingadirectmeasurementofthestrengthofComptonre-flection. However, narrow-line radio galaxies tend to be rel -
+atively faint X-ray sources compared to broad-line objects ,
+meaning that Suzakuobservations of these sources are par-
+ticularly useful, owing to the high effective area of the X-r ay
+ImagingSpectrometer(XIS)andHardX-rayDetector(HXD)instruments.
+Here, we present the results from SuzakuandSwiftobser-
+vations of one of the brightest NLRGs, 3C33,. It is the onlyNLRG to show potential evidence for Compton reflection inits 2–10 keV X-ray spectrum (Kraftet al. 2007). The sourcemay also show evidence of photoionized emission lines initsX-rayspectrum(Torresiet al. 2009),whichwouldmakeitoneofthefewradio-loudAGNtodoso. Inthispaper,wean-alyze the SuzakuandSwiftspectra (§3), and show that reflec-
+tionisnotrequiredin the sourceonce the >10keV spectrum2 EVANSETAL.is taken into account ( §4). We perform detailed diagnostics
+of the Fe K αbandpass( §5), and demonstrate that there is no
+evidenceforanappreciablyvelocitybroadenedcomponentt o
+theline. We combinedatafromnon-contemporaneousobser-vations with Suzaku,Swift,Chandra, andXMM-Newton , in
+order to search for variability in the source ( §6). We also use
+these data to search foremission lines in the soft X-ray spec -
+trum (§7). We end with a detailed interpretation of the prop-
+erties of absorption, reflection, and the disk-jet connecti onin
+3C33(§8).
+2.OBSERVATIONSAND DATAREDUCTION
+2.1.Suzaku
+We observed 3C33 with Suzakuon 2007 December 26
+(OBSID 702059010)fora nominalexposureof 100 ks. BoththeX-rayImagingSpectrometer(XIS)andHardX-rayDetec-tor (HXD) were operated in their normal modes. The sourcewas positioned at the nominal aimpoint of the HXD instru-ment. The data were processed using v. 2.1.16of the Suzaku
+processing pipeline, which includes the latest Charge Tran s-
+ferInefficiency(CTI)correctionappliedfortheXIS.Weuse d
+the standard cleaned events files, which are screened to re-move periods during which the satellite passed through theSouthAtlanticAnomaly(SAA),hadapointingdirection <5◦
+abovetheEarth,orhadEarthday-timeelevationangles <20◦.
+We describe in detail our analysis of the XIS and HXD databelow.
+2.1.1.XIS
+We used data from the two operational front-illuminated
+(FI) CCDs (XIS0 and XIS3), together with the back-illuminated XIS1 detector. The three XIS CCDs were op-erated with a frame time of 8 s. For our analysis, we useddata taken in the 3 ×3 and 5×5 edit modes. We selected only
+eventscorrespondingtogrades0, 2,3,4, and6,andremovedhotandflickeringpixelswith the CLEANSIS tool.
+Weextractedthespectrumof3C33fromtheXISCCDsus-
+ing a source-centered circle of radius 2.5′, with background
+sampled from an adjacent region free from any unrelatedsources, as well as the55Fe calibrationsourcesat the corners
+of each detector. We generatedresponse matrix files (RMFs)foreachdetectorusingv. 2007-05-14ofthe XISRMFGEN soft-
+ware,andancillaryresponsefiles(ARFs)usingv. 2008-04-0 5
+oftheXISSIMARFGEN software.
+The net exposuretimes and countrates for the three CCDs
+are shown in Table 1. We co-added the two FI spectra us-ing the ADDASCASPEC program, and grouped the resulting
+spectrum and that of the XIS1 detector to a minimum of 200counts per bin in order to use χ2statistics. We restricted the
+energyrangeforourspectralfittingto 0.5–10keV.
+2.1.2.HXD
+We extracted the source spectrum from the HXD/PIN de-
+tector, using the cleaned PIN events files described above.The source was not detected with the GSO instrument. ThePIN non X-ray background (NXB) spectrum was generatedfrom the latest ‘tuned’ time-dependent instrumental back-groundevent file providedby the SuzakuGuest ObserverFa-
+cility. We extracted the source and backgroundspectra usin g
+a common Good Time Interval (GTI) criterion. The sourcespectrumwasalso dead-timecorrectedusingthe HXDDTCOR
+tool.10 20 500 0.02 0.04 0.06normalized counts/sec/keV
+channel energy (keV)
+FIG. 1.—SuzakuHXD/PINspectrumof3C33 (black)andtotalCXB+NXB
+background spectrum ( red). The source spectrum has been binned to 3 σ
+above the background level.
+10 20 501 1.1 1.2 1.3
+Energy (keV)Source Spectrum / CXB+NXB spectrum
+FIG. 2.— Ratio of SuzakuHXD/PIN spectrum of 3C33 to the total
+CXB+NXB background spectrum. Both spectra have been binned using the
+3σcriterion described in the text.
+We estimated the contribution from the Cosmic X-ray
+Background (CXB) using the epoch-dependent PIN re-sponse for a flat emission distribution. We used a typi-cal CXB spectrum found using HEAO-1 observations (Boldt1987) of the form 9.412×10−3×(E/1 keV)−1.29×
+exp(−E/40keV)photonscm−2s−1FOV−1keV−1, where
+Eis the photon energy. We added the NXB and CXB
+spectra together using the MATHPHA tool. Using the ap-
+propriate response file for epoch 4 of Suzakuobservations,
+ae
+hxdpinhxnome4 20080129.rsp, we binned the source
+spectrum to 3 σabove the total (NXB + CXB) background
+level. Thebinnedsourcespectrumandtotalbackgroundspec -
+trum are shown in Figure 1. In Figure 2, we plot the ratio ofthe source spectrum to the NXB + CXB spectrum, using our3σbinning criterion. It can be seen that the last bin is only
+marginally significant. We therefore restrict our subseque nt
+spectralanalysisto15–30keV.
+2.2.Swift BAT
+3C33isdetectedinthe SwiftBAT 22-monthall-skysurvey
+(Tuelleret al. 2009). We used the preprocessed,background -
+subtractedBATspectrumandstandarddiagonalresponsema-TheHardX-RayView of3C33 3trix made available to the community6. The uncertaintiesare
+calculated from the RMS noise, and include an additionalmultiplicative factor to account for systematic noise, as d e-
+scribed by Tuelleret al. (2009). Only the first 6 of the 8 totalchannelsare abovethe background,so we discardedthe finaltwo bins, leaving an energy range for our spectral fitting of14–100keV.
+2.3.Chandra
+3C33 was observed for 20 ks (PI S. S. Murray) with the
+Chandra ACIS CCD camera on 2005 November 08 (OBSID
+6910), and for another 20 ks on 2005 November 12 (OBSID7200). Both observations were performed in FAINT mode,with the source placed near the standard aimpoint of the S3chip. We reprocessed the data using CIAO v4.1.2 with theCALDB v4.1.4 to create a new level-2 events file filtered forthegrades0,2,3,4,and6andwiththe0.5′′pixelrandomiza-
+tion removed. To check for periods of high background, weextracted lightcurves for the entire S3 chip, excluding poi nt
+sources. No periodsof highbackgroundwere foundin eitherdataset. TheobservationlogisshowninTable1.
+We extracted nuclear spectra from both Chandra data sets
+using the CIAO routine PSEXTRACT . We chose a source-
+centeredregionofradius 5′′,andsampledbackgroundfroma
+surroundingannulus of inner radius 5′′and outer radius 10′′.
+Finally,we groupedthespectrato25countsperbin.
+2.4.XMM-Newton
+XMM-Newton observed3C33on2004January21withthe
+EPICCCDsfor9ks(PIM.J.Hardcastle). WereprocessedtheXMM-Newton data using SAS version 5.4.1, and generated
+calibrated event files using the EMCHAIN and EPCHAINscripts. We imposed the additional filtering criteria of se-lecting events with only the PATTERN ≤4 and FLAG=0 at-
+tributes. We searchedforperiodsofhighparticlebackgrou nd
+by extracting light curves from the whole field of view, ex-cluding a circle centered on the source, and selected onlyeventswithPATTERN=0andFLAG=0attributesandoveranenergy range of 10–12 keV (MOS) cameras and 12–14 keV(pn). Nointervalsofhighbackgroundwerefound.
+For our spectral analysis we used data from the EPIC pn
+camera only, owing to its greater effective area with respec t
+to the MOS cameras. We extractedthe source spectrumfromcircle of radius 35′′and extracted the background spectrum
+froma large off-sourceregionon the same chip. We groupedthe resulting spectrum to 30 counts per bin. Finally, we gen-erated correspondingRMF and ARF files using the RMFGEN
+andARFGENtools.
+3.SPECTRALANALYSIS
+3.1.OverviewandMotivationofModels
+Inthissection,wedescribeourspectralfittingtothebroad -
+band X-ray spectrum of 3C33. Since the main goal of thispaper is to constrain the properties of Compton reflection in3C33, we only consider the SuzakuandSwiftdata sets here.
+We introduce the Chandra andXMM-Newton spectra in §6,
+when we perform time-variability analysis of the continuumandFe Kαline.
+We briefly highlight our spectral fitting methods here. In
+Section 3.2, we show that models with single zones of neu-tral absorption ( Models I and II ) fail to provide a good fit
+6http://swift.gsfc.nasa.gov/docs/swift/results/bs22m on/
+10−710−610−510−410−30.01Counts s−1 keV−1
+1 10 100−2 0 2χ
+ Energy (keV)
+FIG. 3.—SuzakuXIS FI(black), XIS BI (red), HXD PIN (green), and
+SwiftBAT(blue)spectra and residuals of 3C33. The model fit is the sum
+of a heavily absorbed power law, a soft, unabsorbed power law (to take into
+account thesoftexcess),and aGaussian FeK αemission line (ModelI) .The
+noticeable residuals at ∼3 keV motivates us to consider more complicated
+spectral models.to the spectrum of 3C33. This motivates us to introducemorecomplexspectralmodels,suchasmultiplezonesofneu-tral absorption ( Models III and IV , described in §3.3) and a
+partially ionized absorbing medium ( Models V and VI , de-
+scribedin §3.4). Oncea baselinemodelhasbeenestablished,
+we attempt to constrain the properties of Compton reflection(Models VII and VIII ,§4). The casual reader may simply
+wishtoconsultTable3forasummaryofthebest-fittingmod-els.
+In all spectral fits, we linked the parameters of the model
+components across the SuzakuXIS FI, XIS BI, HXD PIN,
+andSwiftBAT spectra. We tied the normalizations of the
+FI and BI XIS spectra together, but applied a constant cross-normalization factor of 1.09 for the PIN spectrum relative t o
+the XIS as described in the SuzakuData Reduction Guide7.
+We also included a Gaussian of initial centroid 6.4 keV torepresent the Fe K αemission, and allowed its energy, line
+widthandnormalizationtovaryfreely. Wereportindetailt he
+propertiesoftheFe K αline in§5.
+We performed our spectral fitting using v.1.5.0-13 of the
+ISIS spectral fitting package8, which has been modified to
+run on a highly parallelized compute platform. All resultspresented here use a cosmology in which Ωm,0= 0.3,ΩΛ,0
+= 0.7, and H 0= 70 km s−1Mpc−1. At the redshift of
+3C33,z=0.0597, the luminosity distance is 267.3 Mpc. Er-
+rors quoted are 90 per cent confidence for one parameter ofinterest (i.e., χ2min+ 2.7), unless otherwise stated. All spec-
+tral fitsincludethe Galactic absorptionto 3C33 of NH,Gal=
+3.03×1020cm−2(Kalberlaet al.2005).
+3.2.SingleZonesofNeutralAbsorption: ModelsIandII
+We first fitted to 3C33 the canonical model for the X-ray
+spectrum of NLRGs: the combination of a heavily absorbedpower law and a soft, unabsorbed power law (Evanset al.(2006);Balmaverdeetal.(2006)) (ModelI) .Wetiedtogether
+the photon indices of both power laws. The model resultedin a poor fit to the spectrum ( χ2= 242for 152 dof), with
+noticeable residuals at energies ∼3 keV (see Fig. 3), and an
+atypicallylowphotonindex( Γ=1.39±0.05). Anidenticalfit
+7http://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/a bc/
+8http://space.mit.edu/cxc/isis/4 EVANSETAL.10−710−610−510−410−30.01Counts s−1 keV−1
+1 10 100−2 0 2χ
+ Energy (keV)
+FIG. 4.— As Fig. 3, but fitted to a double partially covered primar y law
+(Model III) . This model provides a good fit to the spectrum ( χ2= 176for
+150 dof).is, of course, achieved if we allow the power law to be par-tiallycoveredbyneutralmaterial (ModelII) ;inthiscase,the
+coveringfractionis 96±1%.
+3.3.MultipleZonesofNeutralAbsorption: ModelsIIIandIV
+We next investigated the possibility that an additional
+layer of cold absorption is required to fit the broadbandspectrum of 3C33 adequately. We adopted a doublepartial-covering model, implemented in XSPEC/ISIS asphabs×zpcfabs(1) ×zpcfabs(2) ×(powerlaw+zgauss), where
+ZPCFABS represents the partial coverer ( Model III ). The in-
+dividual column densities NH,1andNH,2, respectively have
+coveringfractions f1,f2.
+Table 3 shows the best-fitting parameters for this model,
+and Figure 4 shows the counts spectrum, model, and residu-als. In short, the model provides a good fit to the spectrum(χ2=176 for 150 dof), and represents a substantial improve-
+ment over Models I and II. Further, the photon index of thepowerlaw ( Γ1= 1.71±0.09)is significantlysteeperthanin
+thepreviousmodel,andis nowconsistentwith typicalvalue s
+found in narrow-line radio galaxies (e.g., Balmaverdeetal .
+2006). Again,an identicalfit is foundif we choosenotto ex-plicitly associate the heavily absorbed and unabsorbed spe c-
+tralcomponents (Model IV) .
+3.4.Warm Absorption: ModelsVandVI
+An alternative method of modeling the observed spectral
+complexity in 3C33 is to allow the absorber to be partiallyionized: such an absorber transmits more of the low-energycontinuumemission than does a neutral obscurerof identica l
+columndensity. We modeledthe Suzakuspectrumasapower
+law, partially covered by a single zone of partially photoio n-
+izedabsorption( Model V). The ionizedabsorbedis modeled
+bythe Z XIPCFmodel.
+This model provideda good fit to the spectrum ( χ2= 184
+for 151 dof), with a column density NH= (5.2+0.3
+−0.2)×
+1023cm−2of modestly ionized gas [log (ξ)=1.51+0.17
+−0.20
+ergs cm s−1] as its best-fitting absorption parameters. The
+spectrum and model are shown in Figure 5. Again, an iden-tical fit is found if we choose not to explicitly associatethe heavily absorbed and unabsorbed spectral components(ModelVI) .
+3.5.Definitionof baselinemodel
+10−710−610−510−410−30.01Counts s−1 keV−1
+1 10 100−2 0 2χ
+ Energy (keV)
+FIG. 5.— As Fig. 3, but fitted to a power law partially covered by a s ingle
+zone of warm absorption (Model V) . The model provides a good fit to the
+spectrum ( χ2= 184for 151 dof).
+We have demonstrated that the broadband X-ray spectrum
+of 3C33 can be described with a power law that is partiallycoveredby multiple zones of neutral absorption (Model III) ,
+or by a power law that is modified by a partially-coveringwarm absorber (Model V) . Both fits include neutral Fe K α
+emission. We have also shown that mathematically identical
+fitsare achieved if we do not associate the hard and soft X-
+ray components with the same physical process (Model IV
+andModel VI) .At thisstage,we defineasourbaselinebest-
+fitting spectral models a Model IV, since it is the most statis -
+tically acceptable. We emphasize that this does not changethe results of our spectral fitting in subsequent sections. W e
+providea detaileddiscussionofthe meritsofboththeneutr al
+andionizedabsorptionmodelsin §8.
+4.CONSTRAINTSON REFLECTION
+We determined the strength of any Compton reflection in
+the spectrum of 3C33 by replacingthe primarypowerlaw inour baseline model with a PEXRAVcomponent( Model VII ).
+This model calculates the sum of the direct power-law emis-sion plus the fraction, R, reflected from an infinite slab of
+neutral material. We initially froze the e-folding energy of
+thePEXRAVcomponent at 1000 keV (i.e., we chose there to
+be no spectral cutoff in our Suzakuspectrum), adoptedan in-
+clinationangle of 60◦, and fixed the elemental abundancesat
+theirsolar values.
+This neutral reflection model provided no significant im-
+provement in the fit ( χ2=175 for 149 dof). The best-fitting
+value of the reflection fraction is R= 0, with a 90% confi-
+denceupperlimit of R <0.41. The columndensities, cover-
+ingfractions,andpower-lawphotonindexareallconsisten tat
+the 1σlevel with the valuesin Model IV. The best-fitting pa-
+rametersare givenin Table 3. Since the strengthof reflectio n
+correlateswiththee-foldingenergy,werepeatedourfitswi th
+different values of Ecut, ranging from 100 to 1000 keV. We
+show the upper limits to Ras a function of e-folding energy
+inFigure6.
+We also attempted to fit the spectrum with an ionized
+Compton reflection model, using the the R EFLIONX table
+model (Ross&Fabian 2005) ( Model VIII ). This model re-
+sulted in a similar fit to the spectrum ( χ2=174 for 148 dof),
+andtheionizationparameterpeggedatitshardlimitof ξ=104
+ergscms−1.
+5.FLUORESCENTLINEEMISSIONTheHardX-RayView of3C33 50 200 400 600 800 10000.4 0.45 0.5 0.55
+e−folding energy (keV)R
+FIG. 6.—90%confidenceupperlimittotheneflectionfractionasa function
+of e-folding energy.
+4 4.5 5 5.5 6 6.5 7 7.5 80.5 1 1.5 2ratio
+channel energy (keV)
+FIG. 7.— Data/model residuals for SuzakuXISFI(black)andXISBI (red)
+spectra in the energy range 4–8 keV, after fitting Model IV to t he FI, BI,
+and PIN spectra to the broadband spectrum, but excluding the energy range
+6.2–6.5 keV (rest frame). Prominent residuals at ∼6.4 keV are observed,
+indicating the need for a narrow Fe K αline. No red wing to the Fe K α
+emission isobserved.
+0.05 0.1Sigma   keV
+FIG. 8.— Energy and line width confidence contours for the Fe K αline
+measured from the SuzakuXIS FI and BI spectra. Shown are the 68%, 90%,
+and 99% contours.
+We next measured the properties of the fluorescent Fe K α
+emission. Togivea qualitativesense ofthelineprofile,wefit-
+ted the continuum-only components of Model IV to the XISFI, XIS BI, and HXD PIN spectra in the energy range 2–70keV. We excluded the rest frame Fe K αbandpass of 6.2–6.5
+keV.Thecontinuumparametersofthisfitareconsistentatth e
+1σlevel to those found in Model IV. We then extended the
+energyrangeto includethe Fe K αbandpass. The data/model
+residuals between 4 and 8 keV are shown in Figure 7. Thereare clear residuals at a rest-frame energy of 6.4 keV, corre-spondingtoneutralFe K αemission.
+Our best-fitting model (Model IV), which we discussed in
+§3.3 includes Fe K αemission with the following properties:
+its centroidis 6.385+0.021
+−0.023keV and the upperlimit to the line
+width is<65 eV (both values are quoted at 90% confidence
+fortwointerestingparameters). Thisislessthanthe ∼100eV
+FWHMresolutionoftheXISat ∼6keV.Confidencecontours
+for the energy and line width are shown in Figure 8. Theequivalent width of the Fe K αline is129±43eV. The pa-
+rametersoftheFeK αlinearesummarizedinTable2,andare
+consistent with those foundwith Chandra andXMM-Newton
+observations(Kraftetal. 2007; Evansetal. 2006). The addi -
+tionofa Fe K βlineisstatistically insignificant.
+There is no indication of a velocity broadened component
+ofthe Fe K αemission, either in the formof a broadunderly-
+ingGaussianora relativisticallyblurred‘diskline’. How ever,
+giventhatsomeradiogalaxiesmayhavemodestlybroadened,ionizedFe Kαlines(e.g.,Kataokaet al. 2007),we calculated
+theupperlimittotheequivalentwidthofaGaussianoffroze n
+linewidth σ=0.5 keVandenergy6.8keV.We foundthe 90%
+confidenceupperlimitto be <65eV.
+6.SEARCHING FOR VARIABILITY:JOINTFITTINGTOSUZAKU,
+SWIFT,CHANDRA,AND XMM-NEWTONSPECTRA
+Wesearchedforinter-epochspectralvariabilityof3C33by
+simultaneously fitting the SuzakuXIS FI, PI, and PIN, Swift
+BAT,Chandra ACIS (two observations), and XMM-Newton
+EPIC/pn spectra. We adoptedour baseline fit (Model IV) forour spectral fits. We used this model instead Model III be-cause the normalizations of the soft power laws are likelyto differ considerably. This is due to the different extrac-tion regions used for our SuzakuXIS (2.5′),Chandra (5′′),
+andXMM-Newton (35′′) analyses, which sample dissimilar
+amounts of extended, soft X-ray emission. We restricted theenergyrangeofthefitsto0.5–10keV( SuzakuXISFIandBI),10−4 10−3 0.01 0.1normalized counts s−1 keV−1
+FIG. 9.— Counts spectra, best fitting model, and residuals for al l data.
+Shown are the SuzakuXIS FI(black), XIS BI(red),HXD PIN (green),Swift
+BAT(blue),Chandra OBSID 6910 (light blue) ,Chandra OBSID 7200 (ma-
+genta), andXMM-Newton pn(orange). The model fit shown is the sum of a
+partially covered power law, a soft, unabsorbed power law, a nd a Gaussian
+FeKαline (Model IV).6 EVANSETAL.0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.55×10 −3 0.01 0.015 
+ Energy (keV)Counts s−1  keV −1 
+N VI RRC 
+O VII RRC 
+Ne IX RRC 
+Na X RRC N VII RRC 
+O VIII RRC 
+Ne X RRC Fe XXO VIII Ly α1
+Ne X Ly α1
+Na XI Ly α1
+Mg XII Ly α1O VII w 
+Ne IX w 
+Na X w 
+Mg XI w O VII x+y 
+Ne IX x+y 
+Na X x+y 
+Mg XI x+y O VII z 
+Ne IX z 
+Na X z 
+Mg XI z 
+Al XII z 
+FIG. 10.— Combined Chandra,XMM-Newton , andSuzakuCCD spectrum of 3C33 in the energy range 0.5–1.5 keV. Also plo tted are the spectral transitions
+identified in thephotoionized plasma ofthe canonical Seyfe rt 2 galaxy, NGC 1068.
+15–30keV( SuzakuPIN),14-100keV( SwiftBAT),0.5–7keV
+(Chandra),and0.5–8keV(XMM-Newton).
+Forourspectralfitting,welinkedthecolumndensities,cov -
+ering fraction and photon index associated with the primarypower law across all five data sets, and tied the photon indexof the unabsorbed power law to that of the absorbed com-ponent. We allowed the normalizations of the Swift,Chan-
+dra,XMM-Newton , and (combined) Suzakupower laws to
+vary independently, and applied the SuzakuPIN/XIS cross-
+normalizationfactorof1.09. Finally,weaddedanunresolv ed
+GaussianFe K αlinetoourmodel.
+The model provided a good description of all the spectra
+(χ2= 292for 261 dof). All the parametersare shown in Ta-
+ble 4. In summary,the best-fittingparametersof the partial ly
+coveredpowerlaware NH=(4.9+0.6
+−0.5)×1023cm−2withacov-
+ering fraction of 92+3−4%, and photon index Γ=1.67±0.10.
+The unabsorbed2-100keV (extrapolated)luminositiesof th e
+primary power law are consistent with no inter-epoch vari-ability. In Figure 9 we show the counts spectra and modelresidualsforthesevendatasets.
+7.SOFT-BANDX-RAY EMISSION
+In order to search for emission lines that might indi-
+cate emission from a thermal plasma, as first reported byTorresietal. (2009), we combined the Chandra,XMM-
+Newton, andSuzakuspectra together using the ISIS spectral
+fitting package9. We rebinned each data set to a common
+grid, namely the HWHM of the SuzakuXIS CCDs. In Fig-
+ure10,weshowthefluxedspectrumbetween0.5and1.5keV,and plot the key photoionization transitions identified in t he
+Chandra HETGS spectrum of the canonical Seyfert 2 galax-
+iesNGC1068(Ogleet al. 2003)andNGC2110(Evansetal.2006). We interpretthese resultsin §8.3.
+8.INTERPRETATION
+We have shown that the nuclear X-ray spectrum of 3C33
+is well described by a power law that is transmitted eitherthrough one or more layers of neutral material, or through apartially ionized obscurer. There is no evidence for a Comp-tonized reflection continuum above ∼10 keV, which means
+wehavenowresolvedthequestionofreflectionraisedbypre-viousChandra observations (Kraftet al. 2007). In this sec-
+tion, we use the Fe K αline to diagnose the physical location
+andstateofthefluorescentemission( §8.1),discusstheorigin
+of the intrinsic absorption in 3C33 ( §8.2), consider various
+9http://space.mit.edu/cxc/isis/originsforthesoftX-rayemission( §8.3),andfinallyinterpret
+theaccretion,reflectionandpossiblejet-productionprop erties
+of 3C33 in the context of other radio-loud and radio-quietAGN(§8.4).
+8.1.FeKαDiagnostics
+The strong Fe K αline detected with Suzakuallows us
+to place constraints on the location and physical state ofthe fluorescing material. The energy of the line core,6.385+0.021
+−0.023keV,isconsistentwithFefluorescencefromneu-
+tral or near-neutral (up to ∼Fe XVIII) species. The unre-
+solvedFeK αlinewidthandtheestimatedblackholemassof
+∼4×108M⊙(Smithet al.1990;Bettoniet al.2003)provide
+a lower limit to the emission radius of >∼0.1pc (∼2000 R s),
+using Keplerian arguments. Finally, the equivalent width o f
+the Fe Kαline is consistent with transmission throughan ab-
+sorbing column of relative iron abundance AFe=1.0 and col-
+umndensitysimilartothatmeasuredfromourspectralfittin g
+(Ghiselliniet al. 1994;Miyazakiet al.1996).
+There is no evidence for an additional velocity broadened
+component of the Fe K αline, which rules out the presence
+ofrelativisticallyblurredfluorescentemissionfromthei nner-
+most portions of the accretion flow. This may imply that astandard, cold Shakura-Sunyaev accretion disk is truncate d
+at a large distance from the central supermassive black hole .
+We cannot rule out the presence of a modestly broadened(R∼100Rs), ionized Fe K αline in 3C33 but nonetheless
+it seems clearthat fluorescentemission fromtheinner diski s
+absent. We willreturnto thispointin §8.4.
+8.2.Originof Absorption
+We considered in detail several absorption models that
+can account for the nuclear spectrum. In one (Model III),two zones of neutral material (a partial coverer of columnNH,1∼6×1023cm−2withacoveringfractionof ∼80%,plus
+a second column of NH,2∼2×1023cm−2) obscure the pri-
+maryemission. Thesecondcolumntakestheformofapatchyabsorber if we explicitly choose to associate the soft, unab -
+sorbed, X-ray emission with the primary power law (ModelIV). Alternatively, we can model the absorber as a singlezoneofmodestlyionizedmaterial(ModelV),whichtransmit s
+moreofthelow-energycontinuumemissionthananeutralob-scurerandthusreproducesthespectralcomplexityinthe2– 10
+keVspectrum. Therelativelymildionizationofthewarmab-sorberisalsoconsistentwiththeneutralornear-neutralF eKα
+emissionobserved. Again,the warmabsorberis patchyif weTheHardX-RayView of3C33 7choose to associate the soft X-ray continuum with the samephysicalprocessasthe primarypowerlaw.
+Both models have distinct advantages over simple param-
+eterizations of the nuclear continuum: they adequately ex-plainthelow-energyspectrumandreconcilethephotoninde x
+of the primary power law with canonical values found in ra-dio galaxies. However, we are unable to distinguish betweenthese models due to the lack of photon statistics at lower en-ergies, where discrete ionized absorption features would b e
+observedinthe caseofthewarmabsorber.
+Ifwe chooseto associate theunresolvedfluorescentFe K α
+emission with the circumnuclearabsorber then, as calculat ed
+above,thelowerlimittotheradiusoftheobscureris ∼0.1pc.
+We can also calculate the upper limit to the radius, r, in the
+case of the warm absorber model, since the ionizing lumi-nosity,L=ξnr2, whereξis the ionization parameter and
+nis the electron number density. Assuming that the thick-
+ness of the absorber is less than the distance to the absorberfrom the source of ionization, i.e., ∆r/r <1, we can estab-
+lish the inequality r < L/N Hξ. We find R<∼20pc, which
+is consistent with the parsec-scale absorber invoked in AGNunificationmodels.
+8.3.Origin ofSoftX-rayEmission
+Thereisclearevidenceofa‘softexcess’intheX-rayspec-
+trum of 3C33. We consider several possibilities for its ori-gin here. First, the emission could be the result of patchycircumnuclearabsorption(eitherneutralorpartiallyion ized),
+which allows a small fraction of continuum emission to betransmittedfreefromobscuration. Suchamodelisconsiste nt
+with observations of numerous AGN, including NGC2110(Evanset al. 2007). The lack of variability of the source ontimescales of years ( §6) suggests that such a patchy absorber
+would subtend a large solid angle at the source of the X-rayemission.
+Second, the soft emission could represent the scattering of
+the primary (heavily obscured) power-law emission in to theline of sight, as is predicted by unified models of AGN (e.g.,Urry& Padovani 1995). Here, the emission is scattered byfreeelectronsinthevicinityofthebroad-lineregionandm ay
+beassociated with the(possiblyphotoionized)circumnucl ear
+regions of the AGN. The ‘efficiency’ of scattering is simplygiven by the ratio of the normalizations of the primary andsoft power laws — for 3C33, we find the efficiency to be∼2%. Thisnumberis consistentwith the typical rangefound
+in Seyfert galaxies of 1–10% (Turneretal. 1997). Furthersupportforthismodelcomesfromthespatiallyextendedsof t
+X-rayand[O III]onscalesof ∼5kpcemissionseenin Chan-
+draandHSTobservations of the source (Kraftet al. 2007).
+The emission is elongated along the axis of the large-scaleradio emission in the source, and therefore likely along theaxis of the circumnuclear obscurer described by unificationmodels. In this case, we might expect a contribution fromphotoionizedemission lines in the soft X-rayspectrumin th e
+case of 3C33, as argued by Torresiet al. (2009). Our com-bined soft X-ray spectrum (Fig. 10) shows possible hints ofHe-likespeciesofOVIIandMgXI,butthereareinsufficientphoton statistics to determine if we are witnessing photoio n-
+izedgas.
+A third and final possibility is that the soft X-ray emis-
+sion isnotassociated with the primary X-ray emission re-
+lated to the accretion flow, and instead originates in an unre -
+solved parsec-scale jet. Given that 3C33 is a strongly radio -
+loud AGN, a contribution to the X-ray emission from theFIG. 11.— 1-keV X-ray luminosity of the soft X-ray emission comp onent
+asafunction of5-GHzradiocoreluminosity inallobserved3 CRRsourcesat
+z <0.5(Evans et al. 2006; Hardcastle etal. 2006). Open circles are LERG,
+filled circles NLRG, open stars BLRG, and filled stars quasars . Large sur-
+rounding circles indicate that a source is an FRI. 3C33 is mar ked by a box.
+Dotted lines show the regression line to all data and its 1 σconfidence range.
+Thesimilarity of 3C33 to the other sources suggests that its softX-ray com-
+ponent originates in an unresolved jet, although we cannot r ule out other
+interpretations.jet is not unexpected, and has been argued for extensivelyin other radio galaxies by, e.g., Hardcastle& Worrall (1999 )
+and Balmaverdeet al. (2006). The key to these authors’ in-terpretations is the strong observed correlation between t he
+X-ray and radio core luminosities in large samples of radio-loudAGN,whichtheyarguesupportsacommonoriginofthetwocomponentsatthebaseofajet. InFigure11,weshowthe1-keVand 5-GHz luminosity densitiesof all observed z<0.5
+radio galaxies (Hardcastleetal. 2006), together with 3C33 .
+This figure shows that 3C33 has a similar measured X-rayluminosity to that predicted from the Hardcastleet al. (200 6)
+relation.
+In summary, we cannot distinguish between these three
+physical scenarios. Future observations with the gratings in-
+struments on board Chandra andXMM-Newton would allow
+ustodeterminewhichiscorrect.
+8.4.AccretionandReflectionin3C33
+Our results demonstrate that 3C33 shows no signs of
+Compton reflection from neutral material in the inner re-gions of an accretion disk: there is no reflection hump at en-ergies>10 keV, and no evidence for relativistically broad-
+ened Fe K αemission (Model VII). Indeed, the narrow Fe
+Kαline is likely to originate in a much farther region, at
+least 2,000 R sfrom the black hole. Further, as 3C33 is a
+narrow-line radio galaxy, meaning we can spectrally sepa-
+rate the X-ray emission associated with the accretion flowand jet, we rule out the idea that X-ray emission from thejet dilutes the reflection spectrum of the source at high ener -
+gies. ThelackofaComptonhumpin3C33isconsistentwith8 EVANSETAL.growingnumbersofradio-loudAGNobservedwith Suzakuto
+show systematically weaker reflection than their radio-qui et
+counterparts(e.g., Larssonetal. 2008; Sambrunaetal. 200 9;
+Kataokaet al.2007). Therearetwo principalmodelsthatcanexplainthisdichotomy,whichbasicallystemfromdifferen ces
+in either the accretion flow geometry, or the black-hole spinconfiguration. We examine both in the context of 3C33 be-low.
+8.4.1.Ionized AccretionModels
+Ballantyneet al. (2002) interpret the weak reflection fea-
+tures observed in radio-loud AGN such as 3C33 to be theresult of an ionized inner accretion flow. Since the reflectedspectrumofanincidentpowerlawontohighlyionizedmate-rial is itself a power law, they demonstrate that the reflecti on
+fraction,R, can remain arbitrarilyhigh. Our attempt to fit an
+ionized reflection model (Ross&Fabian 2005) to 3C33 ledto the ionizationparameterpeggingat its hard limit of ξ=104
+ergs cm s−1(Model VIII). For 3C33, therefore, we cannot
+distinguishbetweena veryhighlyionizedreflector,orsimp ly
+an absent inner disk. We further note that the lack of ionizedFe Kαemissionisconsistentwiththishypothesis.
+8.4.2.Retrograde Black-Hole Spin
+TheX-rayspectrumof3C33isalsoconsistentwiththepic-
+ture of Garofalo,Evans,&Sambruna(2009). Theyargue thejet power of high-excitation radio galaxies such as 3C33 isthe result of retrograde black-hole spin with respect to theaccreting material. This configuration results in a larger i n-
+nermost circular stable orbit than for a prograde black hole .
+The weak or absent signatures of reflection in 3C33, then,are simply a consequenceof the large inner disk radius. Fur-thermore,Garofalo,Evans,&Sambruna(2009)showthattheretrogradespinconfigurationproducesbothstrongBlandfo rd-
+Payne and Blandford-Znajek jets (Blandford&Payne 1982and Blandford&Znajek 1977, respectively). They point outthat this highly relativistic motion away from the inner ac-cretion flow can easily wash out reflection features, as firstdemonstratedbyReynolds&Fabian(1997)andBeloborodov(1999). Our X-ray observations of 3C33 support this sce-nario.
+In summary, we have presented two, equally plausible,
+models that can explain the paucity of a Compton reflectionhumpin3C33. Future,high-sensitivityobservationsof3C3 3
+withIXOwill beneededto distinguishbetweenthem.
+9.CONCLUSIONS
+We have presented results from a new 100-ks Suzakuob-
+servation of the radio galaxy 3C33. The Suzakudata have
+allowed us to break degeneracies in the reflection and ab-sorptionpropertiesof3C33thatwerepresentinourpreviou s
+Chandra andXMM-Newton observations. Wehaveshownthe
+following:
+1. The nuclearX-ray spectrum of 3C33 is well described
+by a power law of photon index ∼1.7 that is absorbed
+either through one or more layers of pc-scale neutralmaterial, or through a partially ionized pc-scale ob-scurer. Bothmodelshavedistinctadvantagesoversim-ple parameterizations of the nuclear continuum: theyadequatelyexplainthelow-energyspectrumandrecon-cile the photon index of the primary power law withcanonicalvaluesfoundin radiogalaxies.2. There is evidencefor a ‘soft excess’ in the X-ray spec-
+trumof 3C33. Itsoriginis consistentwith severalpos-sibilities, including patchy circumnuclear absorption,photoionizedgas,andanunresolvedparsec-scalejet.
+3. 3C33 showsno signs of Comptonreflectionfromneu-
+tral material in the inner regions of an accretion disk:theupperlimit tothereflectionfractionforneutralma-terialisR <0.41forane-foldingenergyof1GeV.
+4. The observed Fe K αline originates in neutral or near-
+neutral material at least 2,000 R sfrom the black hole.
+We cannot rule out the presence of a modestly broad-ened (R∼100Rs), ionized Fe K αline in 3C33 but
+nonetheless it seems clear that fluorescent emissionfromtheinnerdiskisabsent.
+5. The absence of a Compton reflection hump in 3C33
+is consistent with either an ionizedaccretion flow
+(Ballantyneet al. 2002), or with retrograde black-holespin (Garofalo,Evans,& Sambruna 2009). Future ob-servations with IXOwill allow us to distinguish be-
+tweenthesetwo models.
+We wish to thank the anonymous referee for comments
+which greatly improvedthis paper. DAE gratefully acknowl-edgesfinancialsupportforthisworkfromNASAundergrantnumber NNX08AI59G. MJH thanks the Royal Society fora Research Fellowship. This research has made use of dataobtained from the Suzaku satellite, a collaborative missio n
+between the space agencies of Japan (JAXA) and the USA(NASA). This work has also made use of the Chandra X-
+rayObservatory,whichisoperatedbytheSmithsonianAstro -
+physical Observatory for and on behalf of NASA under con-tract NAS8-03060. Thiswork is additionallybased on obser-vationsobtainedwith XMM-Newton ,anESAsciencemission
+with instruments and contributions directly funded by ESAMember States and NASA. Finally, this research has madeuse of the NASA/IPAC ExtragalacticDatabase (NED) whichisoperatedbytheJet PropulsionLaboratory,CaliforniaIn sti-
+tuteofTechnology,undercontractwiththeNASA.TheHardX-RayView of3C33 9REFERENCES
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+C.R. 2007, ApJ,669, 83010 EVANSETAL.TABLE1
+OBSERVATION LOG
+Observatory Dateof observation Obs. ID Instrument Screene dexposure time Net nuclear count rate (s−1)
+Suzaku 2007 Dec 26 702059010 XIS0 127 ks (5.47±0.14)×10−2
+2007 Dec 26 XIS1 127 ks (5.07±0.14)×10−2
+2007 Dec 26 XIS3 127 ks (5.17±0.08)×10−2
+2007 Dec 26 HXD/PIN 99ks (5.22±0.26)×10−2
+Swift 2004 Dec 15- 2006 Nov01 SWIFTJ0109.0+1320 BAT 2.8Ms (effect ive) (6.31±0.89)×10−4
+Chandra 2005 Nov08 6910 ACIS-S 20ks (6.38±0.16)×10−2
+Chandra 2005 Nov12 7200 ACIS-S 20ks (6.55±0.18)×10−2
+XMM-Newton 2004 Jan21 0203280301 EPIC/pn 6.3ks (1.37±0.05)×10−1
+TABLE2
+FEKαPROPERTIES FROM MODELIV
+Parameter Value
+Rest-frame energy 6.385+0.021
+−0.023keV
+Line width <65eV [XISFWHM= 100eV]
+Intensity (2.40±0.04)×10−5photons cm−2s−1
+Equivalent width 129 ±43 eVTheHardX-RayView of3C33 11TABLE3
+CONTINUUM SPECTRAL PARAMETERS FOR JOINTMODELFITS TOSUZAKU AND SWIFTSPECTRUM . LUMINOSITIES QUOTED ARE UNABSORBED AND EXTRAPOLATED TO 2-100KEV.
+L(2−100keV)
+Absorber Covering Reflector (Power Law)
+Overview Model Description of spectrum NH(cm−2) log( ξ) fraction Γ R log(ξ) (ergs s−1) χ2/dof
+(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
+Single I Abs(PL(1)+Gauss)+PL(2) (3.7±0.20)×1023··· 100% Γ1=1.39±0.05··· ··· (3.9+0.5
+−0.6)×1044242/152
+absorption ··· ··· ··· Γ2=Γ1 ··· ··· (1.7±0.1)×1043
+Multiple II PCAbs(PL+Gauss) (3.7±0.20)×1023··· 96±1%1.39±0.05 ··· ··· (4.1+0.5
+−0.6)×1044242/152
+absorption III PCAbs(1) ×PCAbs(2) ×(PL+Gauss) NH,1=(6.2+0.9
+−1.0)×1023··· 84+6−10%1.71±0.09 ··· ··· (4.5+1.1
+−1.6)×1044176/150
+NH,2=(1.6±0.4)×1023··· 91+5−3%··· ··· ··· ···
+IV PCAbs(1) ×Abs(2)×(PL(1)+Gauss)+PL(2) NH,1=(6.0+0.9
+−1.0)×1023··· 85+6−4%Γ1=1.71±0.09··· ··· (4.4+1.1
+−1.6)×1044176/150
+NH,2=(1.7±0.4)×1023··· 100% ··· ··· ··· ···
+··· ··· ··· Γ2=Γ1 ··· ··· (6.5±0.4)×1042
+Warm V PCWarmAbs(PL(1)+Gauss) (5.2+0.3
+−0.2)×10231.51+0.17
+−0.2098±1%Γ1=1.67±0.07 ··· ··· (4.2+0.7
+−0.1)×1044184/151
+absorption ··· ··· ··· ··· ··· ··· ···
+VI WarmAbs(PL(1)+Gauss)+PL(2) (5.2+0.3
+−0.2)×10231.49+0.17
+−0.20100% Γ1=1.69±0.07 ··· ··· (4.1+0.7
+−0.8)×1044184/151
+··· ··· ··· Γ2=Γ1 ··· ··· (6.9±0.4)×1042
+Neutral VII PCAbs(1) ×Abs(2)×(RefPL(1)+Gauss)+PL(2) NH,1=(5.7±1.0)×1023··· 86±6%Γ1=1.75±0.08<0.41··· (4.0+1.4
+−1.0)×1044175/149
+reflection NH,2=(1.6+0.4
+−0.6)×1023··· 100% ··· ··· ··· ···
+··· ··· ··· Γ2=Γ1 ··· ··· (5.8±0.4)×1042
+Ionized VIII PCAbs(1) ×Abs(2)×(PL(1)+Reflion(2)+Gauss)+PL(3) NH,1=(6.6±0.2)×1023··· 83±1%Γ1=1.58+0.03
+−0.02··· ··· (1.9±0.2)×1044174/148
+reflection NH,2=(2.0±0.1)×1023··· 100% Γ2=Γ1 ···4 (pegged) (2.5±0.1)×1044
+··· ··· ··· Γ3=Γ1 ··· ··· (9.6±0.5)×1042
+Col. (1): Model number. Col. (2): Description of spectrum (A bs=Neutral absorption, PCAbs=Partially covering neutral absorption, WarmAbs=WarmAbsorption ( ξis the ionization parameter and is quoted in units
+ofergscms−1),PCWarmAbs=Partially covering warmabsorption, PL=Powe rLaw,RefPL=Sumofdirect andreflected power-law continuum fromneutral material, Reflion=Ross &Fabian (2005)ionized reflection
+continuum, Gauss=Redshifted Gaussian line. Col. (3): Intr insic neutral hydrogen column density. Galactic absorptio n has also been applied. Col. (4): Ionization parameter of ab sorber. Col. (5): Covering fraction.
+Col. (6): Power-law photon index. Col. (7): Reflection fract ion. Col. (8): Ionization parameter for Ross &Fabian (2005) ionized reflection model. Col. (9): 2–100 keV (extrapolated ) unabsorbed luminosity of
+primary power law. Col. (10): Value of χ2and degrees of freedom.12 EVANSETAL.TABLE4
+BEST-FITTING PARAMETERS FOR JOINT FIT TO SUZAKU, SWIFT,CHANDRA,ANDXMM-N EWTONSPECTRA. LUMINOSITIES QUOTED ARE
+UNABSORBED AND EXTRAPOLATED TO 2-100KEV.
+L(2−100keV)
+Covering (PowerLaw)
+Component NH(cm−2) fraction Γ (ergss−1)
+Partiallycovering (9.8+3.4
+−2.3)×1022100% 1.67±0.10LChandra ,6910=(3.7+1.2
+−0.8)×1044
+primarypower law (4.9+0.6
+−0.5)×102392+3−4%··· LChandra ,7200=(3.5+1.0
+−0.7)×1044
+LXMM=(3.3+1.0
+−0.9)×1044
+LSuzaku=(3.8+1.2
+−0.8)×1044
+LSwift=(4.1+1.9
+−1.2)×1044
+Softpower law ··· 100% = Γ1 LChandra ,6910=(2.1±0.3)×1042
+LChandra ,7200=(2.5±0.4)×1044
+LXMM=(3.5±0.5)×1044
+LSuzaku=(7.0±0.5)×1044
\ No newline at end of file
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diff --git a/1001.0590.txt b/1001.0590.txt
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+++ b/1001.0590.txt
@@ -0,0 +1,649 @@
+arXiv:1001.0590v1  [cond-mat.str-el]  4 Jan 2010Influence of Thermal Fluctuations of Spin Density Wave Order Parameter on the
+Quasiparticle Spectral Function.
+M. Khodas and A.M. Tsvelik
+Department of Condensed Matter Physics and Materials Scien ce,
+Brookhaven National Laboratory, Upton, NY 11973-5000, USA
+The two-dimensional model of itinerant electrons coupled t o an anti-ferromagnetic order parame-
+ter is considered. In the mean field solution the Fermi surfac e undergoes reconstruction, and breaks
+into disconnected “pockets”. We have studied the effect of th e thermal fluctuations of the order
+parameter on the spectral density in such system. These fluct uations lead to a finite width of the
+spectral line scaling linearly with temperature. Due to the thermal fluctuations the quasi-particle
+spectral weight is transfered into a magnetic Brillouin zon e. This can be interpreted as restoration
+of “arcs” of the non-interacting Fermi surface.
+PACS numbers: 71.27.+a, 75.30.Fv, 71.10.-w
+I. INTRODUCTION
+Fluctuations play a prominent role in systems of re-
+duced dimensionalities leading to a complete or partial
+suppression of long range order and rendering mean field
+approximation inapplicable. Calculation of correlation
+functions then becomes an arduous task. The question
+is whether in the absence of true long range order these
+functions display features of the ordered state and if yes,
+to what extent. The correlation function we are con-
+cerned with in this paper is the single electron spec-
+tral function. Measurements of this function constitute
+the most powerful experimental probes in the physics of
+strongly correlated systems. We discuss the situation
+when the system is close to being antiferromagnetically
+ordered and study the effect of thermal order parameter
+fluctuations.
+There is a considerable literature addressing the influ-
+ence of quantum fluctuations (we refer the reader to [1]
+and [2] which also provide references to the related pa-
+pers). However, at finite temperatures one has to take
+into account thermal fluctuations which brings specific
+problems. In our previous publication [3] we consid-
+ered the influence of thermal fluctuations on the spectral
+function of a two dimensional (2D) superconductor. 2D
+superconductors have quasi long range order such that
+phase fluctuations are critical in the entire temperature
+region below Tc. We have found that at least as far as
+the thermal fluctuations were concerned, the frequently
+used approach based on the conversion of this problem
+into a gauge theory turned out to be inadequate. The
+latter approach includes a gauge transformation of the
+fermion fields with a subsequent attempts to treat the
+problem as a gauge field theory one (as, for instance,
+in [2]). The difficulty comes from the fact that the re-
+sulting calculational scheme contains strong ultraviolet
+divergencies and hence requires knowledge about states
+located far from the Fermi surface. As an alternative we
+have suggested the direct perturbation expansion in the
+order parameter. This procedure contains only infrared
+divergencies and is therefore universal.Inthepresentpaperweapplytheapproachof[3]tocal-
+culate the spectral function in the presence of a fluctuat-
+ingcommensurateSpin DensityWave(SDW). Thisprob-
+lem has a potential relevance to the problem of cuprates.
+There is a significant experimental evidence in favor of
+Fermi surface reconstruction taking place in the under-
+doped phase of the copper oxide superconductors (see,
+for example, [4],[5]). On the other hand, it is still un-
+clearwhetheroneneedsareallongrangeordertoobserve
+such reconstruction or a short range one will suffice. In
+the present paper we will address this question in the
+context of the spectral function. We consider only clas-
+sical (thermal) fluctuations of the order parameter. This
+is more than an academic excercise since thermally fluc-
+tuating SDW phase has been suggested to occupy a part
+of the cuprate phase diagram in the strongly underdoped
+regime [6].
+II. DESCRIPTION OF THE MODEL AND THE
+RESULTS.
+We consider a popular spin-fermion model in two-
+dimensions where electrons interact with a commensu-
+rate SDW [1]. The antiferromagnetic ordering open gaps
+at the points of the Fermi surface (FS) connected by the
+vector of anti-ferromagnetic fluctuation Q= (π,π), see
+Fig. 1. As a result the FS undergoes reconstruction into
+disconnected pockets. The Hamiltonian for quasiparti-
+cles located near two FS points connected by Qis
+H=/summationdisplay
+kαξ(k)ψ†
+kαψkα+J/summationdisplay
+kSψ†
+k+Qασαβψkβ,(1)
+wherekis a momentum vector in the Brillouin zone, and
+αis the spin index. The kinetic energy close to the ”hot
+spots” can be approximated as
+/summationdisplay
+kαξ(k)ψ†
+kαψkα≈/summationdisplay
+i=1,2/summationdisplay
+kαvikψ†
+ikαψikα(2)
+with the index ienumerating two subbands related by
+the vector Q. The sum in Eq. (2) is over small mo-
+menta and two subbands can be combined together (see2
+inset in Fig. 1). We assume that there is no nesting; to
+simplify the calculations we consider the case when the
+corresponding Fermi velocities are perpendicular to each
+other:v1=vˆx, andv2=vˆy. At the mean field level the
+spectrum is determined by the equation
+ω2−v(kx+ky)ω+v2kxky−J2= 0 (3)
+with the solutions
+ω1,2=v(kx+ky)/2±/radicalBig
+[v(kx−ky)/2]2+J2(4)
+signifying tips of electron- and hole-like Fermi pockets.
+v1v2
+i=2
+i=1
+FIG. 1: Formation of a Fermi pockets (thick solid line) in
+the mean field Fermi surface reconstraction caused by the
+SDW ordering. The bare Fermi surface (thin solid line) is
+not nested. The dashed line is the magnetic Brillouin zone
+boundary. Two subbands with i= 1,2 are connected by the
+anti-ferromagnetic wave vector Q= (π,π). Fermi velocities
+v1andv2are assumed to be perpendicular. Inset shows two
+subbands brought together.
+Fluctuations of the SDW order parameter transform
+rigid energy gaps into pseudogaps and smear the sharp
+peaks in the spectral function. Below we will study this
+process in detail.
+It has been demonstrated in [1] that the feedback from
+the quasiparticles onto the spin Hamiltonian makes sig-
+nificant changesin the spin dynamics, but does not affect
+zero frequency modes. Since we consider only classical
+(thermal) fluctuations, such feedback will be neglected.
+For the sake of simplicity we also assume that the vec-
+tor of spin polarization lies either (i) in the XY-plane or
+(ii) directed alongthe zaxis. In both cases the transition
+occurs at finite temperature. In the first case the order
+parameterhasU(1) symmetryandthe transitionis ofthe
+Kosterlitz-Thouless type. The spin fluctuations below Tc
+are critical. This power law behavior will also hold above
+Tc, though only at distances smaller than the correlation
+lengthξ. However,sincethelatterlengthisexponentially
+large in (T−Tc), there is a range of temperatures T >T c
+and energies where the obtained expressionsfor the spec-
+tral function remain valid. In all this region where the
+magnetic correlation length is either infinite ( T < T c)or exponentially large, the order parameter fluctuations
+can be considered as classical. This is essential for our
+approach. In the second case the order parameter has
+Z2symmetry and below Tcit acquires finite expectation
+value. Hence T <T cregion is trivial and we will be con-
+cerned only with T≥Tcregion. The correlation length
+in this region is ξ∼(T−Tc)−1; to neglect quantum fluc-
+tuations we need it to be much larger than T−1meaning
+that we need to stay close to Tc.
+We will start with the easy plane anisotropy case; the
+easy axis case can be obtained as a simple generalization.
+The fluctuating order parameter is staggered magnetiza-
+tionS, it lies in a plane and forms an angle φwith the
+fixed direction in the plane. Under the assumptions de-
+scribed above the quasiparticle Lagrangian is simplified:
+L=/summationdisplay
+i,α¯ψi,α[∂τ+ξi(−i∇)]ψi,α
++J/summationdisplay
+αβeiφ¯ψ1,ασ−
+αβψ2,β+c.c., (5)
+whereξi(k) =vik. We assume that the free energy for
+the classical phase field φis Gaussian:
+F
+T=ρs
+2T/integraldisplay
+dxdy/bracketleftbig
+(∂xφ)2+(∂yφ)2/bracketrightbig
+.(6)
+Now the problem looks similar to the one of the thermal
+fluctuations in superconductors considered in our previ-
+ous paper [3].
+To be definite we consider the propagator of the spin-
+up particles. The spectral weight reaches its maximal
+value in the vicinity of the bare mass shell, ω∼kx
+and also close to the mass shell of the spin-down particle
+ω∼ky(the shadow mass shell). These two regions form
+two complementary parts of the Fermi pocket. As the
+spectral weight is small at the magnetic Brillouin zone
+boundary,kx∼kywe have studied the Green function
+separately at ω∼kxandω∼ky.
+The summary of our results is as follows. At the mass
+shell we got the following expression for the Green func-
+tion,
+G−1=G−1
+mf+2da4dΓ2(2−2d)J4
+(−i(ω−ky))4−4dG−1
+mfln/parenleftbiggG−1
+mf
+ω−ky/parenrightbigg
+2dia6dΓ2(2−2d)Γ(1−2d)J6
+(−i(ω−ky))5−6d, (7)
+whered=T/4πρsis the scaling dimension of the order
+parameter, ais the lattice constant, and the mean field
+Green function is
+G−1
+mf=ω−kx+iJ2a2dΓ(1−2d)
+(−i(ω−ky))1−2d.(8)
+We would like to emphasise that the expression (7) does
+not rely on the smallness of the parameter d<1/2.3
+At the shadow mass shell, ω∼kywe get
+G=J2eiπdΓ(1−2d)
+(ω−kx)2
+×/bracketleftbigg
+ω−ky+iJ2a2dΓ(1−2d)
+(−i(ω−kx))1−2d/bracketrightbigg−1+2d
+.(9)
+Our analytical results, are presented graphycally in
+Fig. 2. The area of validity of these expressions is con-
+trolled by the energy scale
+TK=J(Ja)d/(1−d). (10)
+Equation (7) is valid for |ky|> TK/v. Equation (9) is
+valid for |kx|>TK/v.
+The remaining part of the paper contains a deriva-
+tion of the results, Eqs. (7), (9). Following the approach
+of [3] we develop a perturbation theory in the coupling
+constantJ. Though this perturbation theory is free of
+ultra-violetsingularitiesit containsinfra-redsingularities
+atω∼kx(y)which we sum up.
+III. BEHAVIOR AT THE MASS SHELL: ω∼kx
+In this section we derive the result (7). It is useful to
+consider the self-energy Σ ω(k) defined in the standard
+way by the Dyson equation
+Gω(k) = [ω−kx−Σω(k)]−1. (11)
+As is shown below, the self-energy is a regular function
+of frequency at the mass shell, and only weakly logarith-
+mically non-analityc in the coupling constant.
+TheJ2contribution to the self-energy is
+Σ(2)
+ω(k) =−i(Jad)2Γ(1−2d)
+(−i(ω−ky))1−2d. (12)
+We notice that in the limit of infinite phase stiffness, d=
+0, Eq. (12) reproduces the mean field spectrum, Eq. (4),
+as expected. Note that the self energy Eq. (12) is regular
+atthemassshell. This,however,isnotthecaseforhigher
+order contribution. In fact, the fourth order contribution
+has a weak logarithmic singularity (see App. A1),
+Σ(4)= 2d−i(Jad)4Γ2(2−2d)
+(−i(ω−ky))3−4dαlogα,(13)
+where
+α=ω+i0−kx
+ω+i0−ky. (14)
+The aforementioned analyticity of the self-energy at the
+mass shell is restored once the leading on-shell singulari-
+ties in all orders in J2are summed up. The reminder of
+the present section is devoted to this task.0.5
+1
+1.5
+20.511.52
+0246
+kyAω=0(k)
+kx(a)
+0.5
+1
+1.5
+20.511.52
+0246
+kyAω=0(k)
+kx(b)
+FIG. 2: (color online) The spectral density at the Fermi sur-
+face,Aω=0(k), as given by Eqs. (7) and (9) at ky> kxand
+kx> kyrespectively. These two graphs are separated by the
+area where the presented derivation ceases to be valid. The
+parameter dis (a)d= 0.08, and (b) d= 0.2.
+Using the expressions for the bare (retarded) Green
+functions
+iG(0)
+1(2)(ω,r) =θ(rx(y))δ(ry(x))eiωrx(y).(15)
+we write the self-energy at the order J2nwithn≥3 in
+the form (see Fig.3)
+Σ(2n)
+ω(k) =i(−iJ)2n/integraldisplay∞
+0dxnei(ω−kx)xn/integraldisplay∞
+0dynei(ω−ky)yn
+×n−1/productdisplay
+i=2/integraldisplayxi+1
+0dxin−1/productdisplay
+i=1/integraldisplayyi+1
+0dyi
+×C(2n)(r1,...,rn;p1,...,pn), (16)4
+whereri= (xi,yi−1),pi= (xi,yi),x1=y0= 0 (see
+Fig. 3) and the cumulant in the last line
+C(2n)(r1,...,rn;p1,...,pn) =δ1,n
+−n−1/summationdisplay
+i=1δ1,iδi+1,n+...+(−1)nδ1,1···δn,n(17)
+is expressed in terms of averages of the exponents of the
+phase fields:
+δi,i+l=/angbracketleftBig
+eiφ(ri)+...+iφ(ri+l)e−iφ(pi)−...−iφ(pi+l)/angbracketrightBig
+.(18)
+Y
+nx2p
+X...
+1p
+1r2rnp
+nyY
+X...
+1pnp
+ny (a) (b)
+nx1r2rnr
+FIG. 3: (color online) Graphical representation of the self -
+energy correction of the order 2 nin coupling constant,
+Eq. (16). Solid vertical and horizontal lines represent seg -
+ments of a real space electron trajectory for (a) particle cl ose
+to the mass shell, ω≈kx, (b) particle close to the shadow
+mass shell, ω≈ky. Incoming (blue) and outgoing (red) ar-
+rowed skew dashed lines represent exponential factors eiφ(ri)
+and e−iφ(pj)respectively.
+The latter average with free energy, Eq. (6) is well
+known,
+δ1,n=a2dn(n−2)/braceleftBigg/tildewide/producttextn
+i,j=1|ri−rj|/tildewide/producttextn
+i,j=1|pi−pj|/producttextn
+i,j=1|ri−pj|/bracerightBigg2d
+,
+(19)
+where tilde over the product sign excludes i=j.
+In what follows we sum up the most singular terms
+in expansion (16). Our solution is based on the physical
+idea that at the mass shell the horizontal segments in the
+staircase diagram in Fig. 3(a) are parametrically longer
+then the vertical ones. In other words, the mass shell
+singularity comes from the integration region yi≪xj.
+Accordingly, we introduce new variables ξi,
+yn−yn−1=ξnxn,...,y 1−y0=ξ1xn(20)
+and argue that the important domain of integration is
+ξi≪1. We expand the cumulant (17) in powers of ξis
+and retainthe lowestpowerterm to getthe most singular
+contribution. It can be shown by inspection that thisexpansion gives
+C(2n)≈a2dnn/productdisplay
+i=1(yi−yi−1)−2d
+×/bracketleftbigg[x2
+n+(yn−1−y0)2]d[x2
+n+(yn−y1)2]d
+[x2n+(yn−y0)2]d[x2n+(yn−1−y1)2]d−1/bracketrightbigg
+≈ −2da2dn(xn)nξ1ξnn/productdisplay
+i=1ξ−2d
+i. (21)
+We substitute Eqs. (20) and (21) in Eq. (16), and per-
+formintegrationsover xn. Wenowanalyzetheremaining
+integrations over ξis,
+Σ(2n)=2d(−i)2n+1(Jad)2nΓ[n(2−2d)−1]
+(n−2)!(−i(ω−ky))n(2−2d)−1In(α),(22)
+where
+In(α)=/integraldisplay∞
+0n/productdisplay
+i=1dξiξ1−2d
+1ξ−2d
+2···ξ−2d
+n−1ξ1−2d
+n
+(α+ξ1+ξ2+...+ξn)n(2−2d)−1.(23)
+We notice that for n= 2,3 the integral in Eq. (23) di-
+verges at the upper limit. In this case the expansion in
+Eq. (21) is not justified. These values of nhave to be
+treated separately, (see App. A for details). For n= 2
+the most singular part is given in Eq. (13) and for n= 3
+we obtain, (see App. A2).
+Σ(6)= 2d−i(Jad)6Γ(1−2d)Γ2(2−2d)
+(−i(ω−ky))5−6dlogα.(24)
+Forn≥3 the integral in Eq. (23) is
+In(α) = (1−2d)2α3−nΓn(1−2d)Γ(n−3)
+Γ[n(2−2d)−1].(25)
+The contributions of the orders n≥3 give
+/summationdisplay
+n≥3Σ(2n)= 2dα−i(Jad)4Γ2(2−2d)
+(−i(ω−ky))3−4dlog(1+x)
+2d−i(Jad)6Γ(1−2d)Γ2(2−2d)
+(−i(ω−ky))5−6d[1−log(1+x)],(26)
+where
+x=α−1(Jad)2Γ(1−2d)
+(−i(ω−ky))(2−2d). (27)
+The sum of contributions (12), (13), (24) and (26) yields
+the final result Eq. (7).
+IV. GREEN FUNCTION AT THE SHADOW
+MASS SHELL, ω∼ky.
+In this section we turn to the analysis of the behavior
+of the Green function at the “shadowside” of the pocket,5
+ω∼ky. As this region is separated from the mass shell,
+instead of the self-energy it is more convenient to study
+the Green function itself. It is also convenient to discuss
+the amputated propagator, ¯Σω(k) =Gω(k)/bracketleftBig
+G(0)
+ω(k)/bracketrightBig−2
+.
+To the second order in J,¯Σω(k) is given by Eq. (12),
+and is singular at ω=ky. At the next, fourth order, the
+correction (see App. B) has stronger singularity
+¯Σ(4)≈i(Jad)4Γ(1−2d)Γ(2−2d)
+(−i(ω−kx))3−4dα2−2d.(28)
+In what follows we resum the most singular terms in
+the expansion of ¯Σ to all orders in J. Forn≥2 we
+introduce new variables,
+Yξ1=x1−0,Yξ2=x2−x1,...
+... ,Yξ n−1=X−xn−2, (29)
+and integrating over yis variables we write the correction
+of order (2n) to the amputated Green function as
+¯Σ(2n)=(−i)2n−1(Jad)2n
+(−i(ω−kx))2n(1−d)−1
+×Γ[2n(1−d)−1]
+(n−1)!¯In(α),(30)
+where the remaining integrals
+¯In(α)=n−1/productdisplay
+i=1/integraldisplay∞
+0dξiξ−2d
+1···ξ−2d
+n−1
+(α−1+ξ1+...+ξn−1)2n(1−d)−1(31)
+are convergent at the upper limit, and can be evaluated
+as
+¯In(α) =αn−2dΓn−1(1−2d)Γ(n−2d)
+Γ(2n(1−d)−1).(32)
+As a result we obtain for the singular part
+¯Σ(2n)=i(−iJad)2nαn−2dΓn−1(1−2d)Γ(n−2d)
+(n−1)!(−i(ω−kx))2n(1−d)−1.(33)
+Forn= 1 the last expression reduces to the second order
+corrections, Eq. (12). The sum of singular contributions
+Eq. (33) yields the result of Eq. (9).
+V. CONCLUSIONS
+First we would like to make a remark about the easy
+axis anisotropy regime where the phase transition is in
+the Ising model universality class. From our calculations
+it is easy to see that as far as the singular terms are
+concerned, the results remain unchanged provided one
+considers only one particular value for the scaling dimen-
+sion:d= 1/8. This is despite the fact that multipoint
+correlation functions of the Ising model order parame-
+ter fields are more complicated than (19). However, thesingularities are determined by more simple correlators,
+namely by the diagrams where pairs of the operators are
+very close to each other (see Fig. 3) resulting in a fusion
+of two order parameter fields. In the Ising model such
+fusion generates the energy density operator and in the
+XY-model it generates the gradient of φfield. Both op-
+erators have the same multi-point correlation functions.
+Now we can discuss the results. They are well illus-
+trated by Figs. 1,4. The region of applicability of our
+calculationsinvolvestheenergyscale TK=J(Ja)d/(d−1).
+The result at the mass shell is valid for |ky|>TKand the
+result at the shadow mass shell holds at |kx|>TK. With
+increasing temperature the spectral weight is transferred
+towardsthe bare Fermi surface and the shadow band fea-
+ture quickly fades away as is clearly seen on figure (4)
+where the darker areas correspond to large values of the
+spectral density.
+Although our model does not include all the features
+ascribed to the cuprates, the results obtained may serve
+as a good qualitative guide to the problem. For instance,
+we see that critical thermal fluctuations give rise to the
+characteristic linear temperature dependence ∼Tof the
+spectral peak width. This is an indication that such fluc-
+tuations are responsible of this feature in the cuprates.
+Ourresultsdemonstratethatwiththeriseoftemperature
+the renormalized mass shell identified as the maximum
+intensity line in Fig. 4 approachesthe bare Fermi surface,
+while the peak becomes rather incoherent. The back-
+side of the Fermi pockets fades away so that the pockets
+now look like arcs. These effects are qualitatively simi-
+lar to that of the quantum fluctuations studied in Ref. [6]
+though the intensity of quantum fluctuations is regulated
+not by temperature, but by the interactions.
+In summary, we have presented systematic study of
+the thermal fluctuations effects in two-dimensional sys-
+tem of electrons in interaction with SDW order param-
+eter. In particular the spectral density has been found
+to be strongly sensitive to these fluctuations. Fluctua-
+tions tend to restore the non-interacting Fermi surface
+topology thus overriding the effects of the SDW order.
+Acknowledgments
+We are grateful to A. Chubukov for encouraging dis-
+cussions and interest to the work. We acknowledge sup-
+port by the US DOE under contract number DE-AC02-
+98 CH 10886. This research was also supported as part
+of the Center for Emerging Superconductivity funded by
+the U.S. Department of Energy, Office of Science. M.
+Khodas acknowledges support from BNL LDRD grant
+08-002.
+Appendix A: Perturbation theory at the mass shell.
+In the present appendix we evaluate the most singular
+corrections to the self-energy at the mass shell, ω∼kx.6
+kxky(a)
+00.511.5200.511.52
+kxky(b)
+00.511.5200.511.52
+FIG. 4: False color plot representing the spectral density a t
+the Fermi surface, Aω=0(k), as given by Eqs. (7) and (9) at
+ky> kxandkx> kyrespectively. The region kx,ky< TK
+where the results are inapplicable is not shown. The param-
+eterdis (a)d= 0.04, and (b) d= 0.13. Dashed line shows
+the mean field Fermi surface as given by Eq. (4).
+1. Fourth order contributions
+In the fourth order the general expression (16) takes
+the form
+Σ(4)
+ω(k) =iJ4/integraldisplay∞
+0dx2ei(ω−kx)x2/integraldisplay∞
+0dy2ei(ω−ky)y2
+/integraldisplayy2
+0dy1C(4)(r1,r2;p1,p2), (A1)wherer1= (0,0),r2= (x2,y1),p1= (0,y1), andp2=
+(x2,y2) (see Fig. 3). Equation (21) gives
+C(4)=a4d
+y2d
+1(y2−y1)2d
+×/bracketleftBigg/parenleftbig
+(y2−y1)2+x2
+2/parenrightbigd/parenleftbig
+y2
+1+x2
+2/parenrightbigd
+(x2
+2+y2
+2)dx2d
+2−1/bracketrightBigg
+.(A2)
+We expect the main contribution to come from the re-
+giony1,y2−y1≪x2it is convenient to introduce new
+variables,y2=ξx2,y1=ηξx2. Toisolatetheleadingsin-
+gularity in Eq. (A1) we expand the cumulant in Eq. (A2)
+forξ/lessorsimilar1
+C(4)≈ −2da4dξ2−4dη1−2d(1−η)1−2d.(A3)
+This expansion holds for any d<1/2.
+Σ(4)=−2di(Jad)4Γ(3−4d)
+×/integraldisplay∞
+0dξ/integraldisplay1
+0dηξ3−4dη1−2d(1−η)1−2d
+[−i((ω−ky)ξ+(ω−kx))]3−4d.(A4)
+The integral in Eq. (A4) is not convergent at the upper
+limit. Thisisanartifactoftheapproximation(A3)which
+is not justified for ξ/greaterorsimilar1. To overcome this we differenti-
+ate equation (A1) twice with respect to the parameter α
+defined by Eq. (14),
+∂2Σ(4)
+∂α2=−2diΓ(5−4d)(Jad)4(−i(ω−ky))4d−3
+×/integraldisplay∞
+0dξ/integraldisplay1
+0dηξ3−4dη1−2d(1−η)1−2d
+(ξ+α)5−4d.(A5)
+The integrations in Eq. (A5) are easily done with the
+result
+∂2Σ(4)
+∂α2=−2diα−1Γ2(2−2d)(Jad)4
+(−i(ω−ky))3−4d.(A6)
+Integrating Eq. (A6) back we finally get
+Σ(4)=−2diΓ2(2−2d)(Jad)4(−i(ω−ky))4d−3
+×(αlogα+Aα+B) (A7)
+withA,Bconstants. In the last equation the most sin-
+gular term is presented in Eq. (13).
+2. Order J6
+In this case the general expression (16) with n= 3
+reduces to
+Σ(6)
+ω(k) =i(−iJad)6/integraldisplay∞
+0dx3ei(ω−kx)x3/integraldisplay∞
+0dy3ei(ω−ky)y3
+×/integraldisplayx3
+0dx2/integraldisplayy3
+0dy2/integraldisplayy2
+0dy1
+×ABC−B−C+1
+y2d
+1(y2−y1)2d(y3−y2)2d, (A8)7
+where
+A=(x2
+3+y2
+2)d|x2
+3+(y3−y1)2|d
+(x2
+3+(y2−y1)2)d(x2
+3+y2
+3)d,(A9)
+B=[(x3−x1)2+(y2−y1)2]d
+(x3−x1)2d
+×[(x3−x1)2+(y3−y2)2]d
+[(y3−y1)2+(x3−x1)2]d, (A10)
+and
+C=|x2
+1+y2
+1|d|x2
+1+(y2−y1)2|d
+|x1|2d|y2
+2+x2
+1|d. (A11)
+We write
+ABC−B−C+1= (A−1)BC+(B−1)(C−1).(A12)
+It is apparentform Eq. (A12) that the leading singularity
+originates from the first term, ( A−1)BC≈A−1≈
+−2d(y1−0)(y3−y2). Introducing new variables as in the
+Sec. III and integrating over xis we obtain
+Σ(6)= 2di(Jad)6Γ(5−6d)
+(−i(ω−ky))5−6d
+×/integraldisplay∞
+03/productdisplay
+i=1dξiξ1−2d
+1ξ−2d
+2ξ1−2d
+3
+(α+ξ1+ξ2+ξ3)5−6d.(A13)
+Hereagain,theintegralsaredivergentontheupper limit.
+Similarly to the previous section we differentiate it once
+with respect to the variable αintroduced in Eq. (14) in
+order to isolate the leading logarithmic singularity,
+∂Σ(6)
+∂α=−2diα−1(Jad)6Γ(6−6d)
+(−i(ω−ky))5−6d
+×/integraldisplay∞
+03/productdisplay
+i=1dξiξ1−2d
+1ξ−2d
+2ξ1−2d
+3
+(1+ξ1+ξ2+ξ3)6−6d.(A14)
+The remaining integrals are easily evaluated. The sub-
+sequent integration over αrestores the singularity in theself energy correction,
+Σ(6)=2d−i(Jad)6Γ(1−2d)Γ2(2−2d)
+(−i(ω−ky))5−6d
+×(logα+C), (A15)
+whereCis an integration constant. The singular part of
+Eq. (A15) is given by Eq. (24).
+Appendix B: Leading singularities at the shadow
+mass shell, ω∼kyto the fourth order.
+In this appendixwe evaluatethe singularcontributions
+to the Green function at the shadow mass shell, ω∼ky
+to fourth order in the coupling constant. We start with
+the expression (A1) introduced in App. A1. In contrast
+to the discussion in App. A1 we anticipate the singular-
+ity atω=kyto come from the region y2≫x2, and
+introduce new variables accordingly, x2=ξy2,y1=ηy2.
+Performing integration over y2we obtain
+Σ(4)=i(Jad)4/integraldisplay∞
+0dξ/integraldisplay1
+0dηΓ(3−4d)η−2d(1−η)−2d
+[(−i((ω−ky)+(ω−kx)ξ)]3−4d
+×/bracketleftbigg|(1−η)2+ξ2|d|η2+ξ2|d
+|1+ξ2|d|ξ|2d−1/bracketrightbigg
+. (B1)
+We notice that the singularity at ω∼kycomes from the
+region of small ξ. Therefore we keep only the first term
+in the square brackets in Eq. (B1). After performing
+remaining integrations over ξis we obtain
+Σ(4)=i(Jad)4Γ(1−2d)Γ(2−2d)α2−2d
+(−i(ω−kx))3−4d.(B2)
+We stress that contrary to the mass shell singularities
+discussed in App. A, where it was important to compute
+the self-energy, at the shadow mass shell it is enough to
+consider the Green function itself.
+[1] Ar. Abanov, A. V. Chubukov and J. Schmalian, Adv.
+Phys.52, 119 (2003) and references therein.
+[2] S. Sachdev, M. A. Metlitski, Y. Qi and C. Xu, arXiv:
+0907.3732 and references therein.
+[3] M. Khodas and A. M. Tsvelik, arXiv:0910.3967.[4] N. Doiron-Leyraud et.al., Nature (London) 447, 565
+(2007).
+[5] E. A. Yelland et.al., Phys. Rev. Lett. 100, 047003 (2008).
+[6] S. Sachdev, arXiv:0907.0008.
\ No newline at end of file
diff --git a/1001.0591.txt b/1001.0591.txt
new file mode 100644
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+++ b/1001.0591.txt
@@ -0,0 +1,1032 @@
+arXiv:1001.0591v2  [cs.CG]  13 Mar 2011Comparing Distributions and Shapes using the Kernel Distanc e
+Sarang Joshi
+sjoshi@sci.utah.eduRaj Varma Kommaraju
+rajvarma@cs.utah.eduJeff M. Phillips
+jeffp@cs.utah.eduSuresh Venkatasubramanian
+suresh@cs.utah.edu
+Abstract
+Starting with a similarity function between objects, it is p ossible to define a distance metric on pairs of
+objects, and more generally on probability distributions o ver them. These distance metrics have a deep basis
+in functional analysis, measure theory and geometric measu re theory , and have a rich structure that includes
+an isometric embedding into a (possibly infinite dimensiona l) Hilbert space. They have recently been applied
+to numerous problems in machine learning and shape analysis .
+In this paper, we provide the first algorithmic analysis of th ese distance metrics. Our main contributions
+are as follows: (i) We present fast approximation algorithm s for computing the kernel distance between two
+point sets PandQthat runs in near-linear time in the size of P∪Q(note that an explicit calculation would
+take quadratic time). (ii) We present polynomial-time algo rithms for approximately minimizing the kernel
+distance under rigid transformation; they run in time O(n+poly(1/ǫ,logn)). (iii) We provide several general
+techniques for reducing complex objects to convenient spar se representations (specifically to point sets or
+sets of points sets) which approximately preserve the kerne l distance. In particular, this allows us to reduce
+problems of computing the kernel distance between various t ypes of objects such as curves, surfaces, and
+distributions to computing the kernel distance between poi nt sets. These take advantage of the reproducing
+kernel Hilbert space and a new relation linking binary range spaces to continuous range spaces with bounded
+fat-shattering dimension.
+11 Introduction
+LetK:/Rbbbd×/Rbbbd→Rbe a kernel function, such as a Gaussian kernel; K(p,q)describes how similar two points
+p,q∈/Rbbbdare. For point sets P,Qwe can define a similarity function κ(P,Q)=/summationtext
+p∈P/summationtext
+q∈QK(p,q). Then the
+kernel distance is defined as
+DK(P,Q)=/radicalbig1
+κ(P,P)+κ(Q,Q)−2κ(P,Q). (1.1)
+By altering the kernel K, and the weighting of elements in κ, the kernel distance can capture distance between
+distributions, curves, surfaces, and even more general obj ects.
+Motivation. The earthmover distance (EMD) takes a metric space and two pr obability distributions over the
+space, and computes the amount of work needed to "transport" mass from one distribution to another. It has
+become a metric of choice in computer vision, where images ar e represented as intensity distributions over a
+Euclidean grid of pixels. It has also been applied to shape co mparison[8], where shapes are represented by
+point clouds (discrete distributions) in space.
+While the EMD is a popular way of comparing distributions ove r a metric space, it is also an expensive one.
+Computing the EMD requires solving an optimal transportati on problem via the Hungarian algorithm, and
+while approximations exist for restricted cases like the Eu clidean plane, they are still expensive, requiring either
+quadratic time in the size of the input or achieving at best a c onstant factor approximation when taking linear
+time[38, 2]. Further, it is hard to index structures using the EMD for per forming near-neighbor, clustering
+and other data analysis operations. Indeed, there are lower bounds on our ability to embed the EMD into
+well-behaved normed spaces [4].
+The kernel distance has thus become an effective alternativ e to comparing distributions on a metric space.
+In machine learning, the kernel distance has been used to bui ld metrics on distributions [42, 23, 41, 39, 32 ]
+and to learn hidden Markov models [40]. In the realm of shape analysis [45, 18, 19, 17], the kernel distance
+(referred to there as the current distance ) has been used to compare shapes, whether they be point sets, curves,
+or surfaces.
+All of these methods utilize key structural features of the k ernel distance. When constructed using a positive
+definite1similarity function K, the kernel distance can be interpreted through a lifting ma pφ:/Rbbbd→Hto a
+reproducing kernel Hilbert space (RKHS), H. This lifting map φis isometric; the kernel distance is precisely
+the distance induced by the Hilbert space ( DK({p},{q})=/bardblφ(p)−φ(p)/bardblH). Furthermore, a point set Phas
+an isometric lifted representation Φ(P)=/summationtext
+p∈Pφ(p)as a single vector in HsoDK(P,Q)=/bardblΦ(P)−Φ(Q)/bardblH.
+Moreover, by choosing an appropriately scaled basis, this b ecomes a simple ℓ2distance, so all algorithmic tools
+developed for comparing points and point sets under ℓ2can now be applied to distributions and shapes.
+Dealing with uncertain data provides another reason to stud y the kernel distance. Rather than thinking
+ofK(·,·)as a similarity function, we can think of it as a way of capturi ng spatial uncertainty; K(p,q)is the
+likelihood that the object claimed to be at pis actually at q. For example, setting K(p,q)= exp(−/bardblp−
+q/bardbl2/σ)/(/rad〉callow
+2πσ)gives us a Gaussian blur function. In such settings, the kern el distance D2
+K(P,Q)computes the
+symmetric difference |P△Q|between shapes with uncertainty described by K.
+Our work. We present the first algorithmic analysis of the kernel dista nce. Our main contributions are as
+follows:
+(i) We present fast approximation algorithms for computing the kernel distance between two point sets P
+andQthat runs in near-linear time in the size of P∪Q(note that an explicit calculation would take quadratic
+time). (ii) We present polynomial-time algorithms for appr oximately minimizing the kernel distance under
+rigid transformation; they run in time O(n+poly(1/ǫ,logn)). (iii) We provide several general techniques for
+reducing complex objects to convenient sparse representat ions (specifically to point sets or sets of points sets)
+which approximately preserve the kernel distance. In parti cular, this allows us to reduce problems of computing
+1A positive definite function generalizes the idea of a positi ve definite matrix; see Section 2.
+1the kernel distance between various types of objects such as curves, surfaces, and distributions to computing
+the kernel distance between point sets.
+We build these results from two core technical tools. The firs t is a lifting map that maps objects into a finite-
+dimensional Euclidean space while approximately preservi ng the kernel distance. We believe that the analysis
+of lifting maps is of independent interest; indeed, these me thods are popular in machine learning [41, 39, 40]
+but (in the realm of kernels) have received less attention in algorithms. Our second technical tool is an theorem
+relatingǫ-samples of range spaces defined with kernels to standard ǫ-samples of range spaces on {0,1}-valued
+functions. This gives a simpler algorithm than prior method s in learning theory that make use of the γ-fat
+shattering dimension, and yields smaller ǫ-samples.
+2 Preliminaries
+Definitions. For the most general case of the kernel distance (that we will consider in this paper) we associate
+a unit vector U(p)and a weighting µ(p)with every p∈P. Similarly we associate a unit vector V(p)and
+weightingν(q)with every q∈Q. Then we write
+κ(P,Q)=/integraldisplay
+p∈P/integraldisplay
+q∈QK(p,q)/angleleftbig1U(p),V(q)/anglerightbig1dµ(p)dν(q), (2.1)
+where〈·,·〉is the Euclidean inner product. This becomes a distance DK, defined through (1.1).
+WhenPis a curve in/Rbbbdwe let U(p)be the tangent vector at pandµ(p)=1. When Pis a surface in/Rbbb3we
+letU(p)be the normal vector at pandµ(p)=1. This can be generalized to higher order surfaces through t he
+machinery of k-forms and k-vectors[45, 35].
+WhenPis an arbitrary probability measure2in/Rbbbd, then all U(p)are identical unit vectors and µ(p)is the
+probability of p. For discrete probability measures, described by a point se t, we replace the integral with a sum
+andµ(p)can be used as a weight κ(P,Q)=/summationtext
+p∈P/summationtext
+q∈QK(p,q)µ(p)ν(q).
+From distances to metrics. When Kis a symmetric similarity function (i.e. K(p,p)=maxq∈/RbbbdK(p,q),
+K(p,q)=K(q,p), and K(p,q)decreases as pandqbecome “less similar”) then DK(defined through (2.1)
+and (1.1)) is a distance function, but may not be a metric. How ever, when Kis positive definite, then this is
+sufficient for DKto not only be a metric3, but also for D2
+Kto be of negative type [16].
+We say that a symmetric function K:/Rbbbd×/Rbbbd→/Rbbbis asymmetric positive definite kernel if for any nonzero
+L2function fit satisfies/integraltext
+p∈P/integraltext
+q∈Qf(q)K(p,q)f(p)dpdq>0. The proof of DKbeing a metric follows by con-
+sidering the reproducing kernel Hilbert space Hassociated with such a K[5]. Moreover, DKcan be expressed
+very compactly in this space. The lifting map φ:/Rbbbd→Hassociated with Khas the “reproducing property”
+K(p,q)=〈φ(p),φ(q)〉H. So by linearity of the inner product, Φ(P)=/integraltext
+p∈Pφ(p)dµ(p)can be used to re-
+trieve DK(P,Q)=/bardblΦ(P)−Φ(Q)/bardblHusing the induced norm /bardbl·/bardblHofH. Observe that this defines a norm
+/bardblΦ(P)/bardblH=/radicalbig1
+κ(P,P)for a shape.
+Examples. IfKis the “trivial” kernel, where K(p,p)=1 and K(p,q)=0 for p/\e}at〉o\slas〈=q, then the distance between
+any two sets (without multiplicity) P,QisD2
+K(P,Q)=|P∆Q|, whereP∆Q=P∪Q\(P∩Q)is the symmetric
+difference. In general for arbitrary probability measures ,DK(P,Q)=/bardblµ−ν/bardbl2. IfK(p,q)=〈p,q〉, the Euclidean
+dot product, then the resulting lifting map φis the identity function, and the distance between two measu res
+is the Euclidean distance between their means, which is a pse udometric but not a metric.
+2We avoid the use of the term ’probability distribution’ as th is conflicts with the notion of a (Schwarz) distribution that itself plays
+an important role in the underlying theory.
+3Technically this is not completely correct; there are a few s pecial cases, as we will see, where it is a pseudometric [41].
+2Gaussian properties. To simplify much of the presentation of this paper we focus on the case where the
+kernel Kis the Gaussian kernel; that is K(p,q)= e−||p−q||2/h. Our techniques carry over to more general
+kernels, although the specific bounds will depend on the kern el being used. We now encapsulate some useful
+properties of Gaussian kernels in the following lemmata. Wh en approximating K(p,q), the first allows us to
+ignore pairs of points further that/radicalbig1
+hln(1/γ)apart, the second allows us to approximate the kernel on a gri d.
+Lemma 2.1 (Bounded Tails) .If||p−q||>/radicalbig1
+hln(1/γ)then K(p,q)<γ.
+Lemma 2.2 (Lipschitz) .Forδ∈/Rbbbdwhere/bardblδ/bardbl<ǫ, for points p ,q∈/Rbbbdwe have|K(p,q)−K(p,q+δ)|≤ǫ//rad〉callow
+h.
+Proof. The slope for ψ(x)=e−x2/his the function ψ′(x)=−(2/h)xe−x2/h.ψ′(x)is maximized when x=/radicalbig1
+h/2, which yields ψ′(/radicalbig1
+h/2)=−/radicalbig1
+2/he<1//rad〉callow
+h. Thus|ψ(x)−ψ(x+ǫ)|<ǫ//rad〉callow
+h. And since translating by
+δchanges/bardblp−q/bardblby at mostǫ, the lemma holds.
+2.1 Problem Transformations
+In prior research employing the kernel distance, ad hoc discretizations are used to convert the input objects
+(whether they be distributions, point clouds, curves or sur faces) to weighted discrete point sets. This process
+introduces error in the distance computations that usually go unaccounted for. In this subsection, we provide
+algorithms and analysis for rigorously discretizing input objects with guarantees on the resultin g error. These
+algorithms, as a side benefit, provide a formal error-preser ving reduction from kernel distance computation on
+curves and surfaces to the corresponding computations on di screte point sets.
+After this section, we will assume that all data sets conside redP,Qare discrete point sets of size nwith
+weight functions µ:P→/Rbbbandν:Q→/Rbbb. The weights need not sum to 1 (i.e. need not be probability
+measures), nor be the same for PandQ; we will set W=max(/summationtext
+p∈Pµ(p),/summationtext
+q∈Qν(q))to denote the total
+measure. This implies (since K(p,p)=1) thatκ(P,P)≤W2. All our algorithms will provide approximations
+ofκ(P,Q)within additive error ǫW2. Since without loss of generality we can always normalize so thatW=1,
+our algorithms all provide an additive error of ǫ. We also set∆=( 1/h)maxu,v∈P∪Q/bardblu−v/bardblto capture the
+normalized diameter of the data.
+Reducing orientation to weights. The kernel distance between two oriented curves or surfaces can be
+reduced to a set of distance computations on appropriately w eighted point sets. We illustrate this in the case of
+surfaces in/Rbbb3. The same construction will also work for curves in /Rbbbd.
+For each point p∈Pwe can decompose U(p)/defines(U1(p),U2(p),U3(p))into three fixed orthogonal components
+such as the coordinate axes {e1,e2,e3}. Now
+κ(P,Q)=/integraldisplay
+p∈P/integraldisplay
+q∈QK(p,q)/angleleftbig1U(p),V(q)/anglerightbig1dµ(p)dν(q) =/integraldisplay
+p∈P/integraldisplay
+q∈QK(p,q)3/summationdisplay
+i=1(Ui(p)Vi(q))dµ(p)dν(q)
+=3/summationdisplay
+i=1/integraldisplay
+p∈P/integraldisplay
+q∈QK(p,q)(Ui(p)Vi(q))dµ(p)dν(q) =3/summationdisplay
+i=1κ(Pi,Qi),
+where each p∈Pihas measureµi(p)=µ(p)/bardblUi(p)/bardbl. When the problem specifies Uas a unit vector in/Rbbbd, this
+approach reduces to dindependent problems without unit vectors.
+Reducing continuous to discrete. We now present two simple techniques (gridding and sampling ) to
+reduce a continuous Pto a discrete point set, incurring at most ǫW2error.
+3We construct a grid Gǫ(of size O((∆/ǫ)d)) on a smooth shape P, so no point4p∈Pis further than ǫ/rad〉callow
+h
+from a point g∈Gǫ. Let Pgbe all points in Pcloser to gthan any other point in Gǫ. Each point gis assigned a
+weightµ(g)=/integraltext
+p∈Pg1dµ(p). The correctness of this technique follows by Lemma 2.2.
+Alternatively , we can sample n=O((1/ǫ2)(d+log(1/δ))points at random from P. If we have not yet
+reduced the orientation information to weights, we can gene ratedpoints each with weight Ui(p). This works
+with probability at least 1 −δby invoking a coreset technique summarized in Theorem 5.2.
+For the remainder of the paper, we assume our input dataset Pis a weighted point set in /Rbbbdof size n.
+3 Computing the Kernel Distance I: WSPDs
+The well-separated pair decomposition (WSPD) [9, 21]is a standard data structure to approximately compute
+pairwise sums of distances in near-linear time. A consequen ce of Lemma 2.2 is that we can upper bound the
+error of estimating K(p,q)by a nearby pair K(˜p, ˜q). Putting these observations together yields (with some
+work) an approximation for the kernel distance. Since D2
+K(P,Q)=κ(P,P)+κ(Q,Q)−2κ(P,Q), the problem
+reduces to computing κ(P,Q)efficiently and with an error of at most (ǫ/4)W2.
+Two sets Aand Bare said to be α-separated[9]if max{diam( A),diam( B)}≤αmina∈A,b∈B||a−b||. Let
+A⊗B={{x,y}|x∈A,y∈B}denote the set of all unordered pairs of elements formed by AandB. Anα-WSPD
+of a point set Pis a set of pairs W=/braceleftbig1{A1,B1},... ,{As,Bs}/bracerightbig1such that
+(i)Ai,Bi⊂Pfor all i,
+(ii)Ai∩Bi=/emptysetstressfor all i,
+(iii) disjointly/uniontexts
+i=1Ai⊗Bi=P⊗P, and
+(iv) AiandBiareα-separated for all i.
+For a point set P⊂/Rbbbdof size n, we can construct an α-WSPD of size O(n/αd)in time O(nlogn+n/αd)[21, 14].
+We can use the WSPD construction to compute D2
+K(P,Q)as follows. We first create an α-WSPD of P∪Q
+inO(nlogn+n/αd)time. Then for each pair {Ai,Bi}we also store four sets Ai,P=P∩Ai,Ai,Q=Q∩Ai,
+Bi,P=P∩Bi, and Bi,Q=Q∩Bi. Let ai∈Aiand bi∈Bibe arbitrary elements, and let Di=/bardblai−bi/bardbl. By
+construction, Diapproximates the distance between any pair of elements in Ai×Biwith error at most 2 αDi.
+In each pair{Ai,Bi}, we can compute the weight of the edges from PtoQ:
+Wi=/parenleftbig/summationdisplay
+p∈Ai,Pµ(p)/parenrightbig/parenleftbig/summationdisplay
+q∈Bi,Qν(q)/parenrightbig
++/parenleftbig/summationdisplay
+q∈Ai,Qν(q)/parenrightbig/parenleftbig/summationdisplay
+p∈Bi,Pµ(p)/parenrightbig
+.
+We estimate the contribution of the edges in pair (Ai,Bi)toκ(P,Q)as
+/summationdisplay
+(a,b)∈Ai,P×Bi,Qµ(a)ν(b)e−D2
+i/h+/summationdisplay
+(a,b)∈Ai,Q×Bi,Pµ(b)ν(a)e−D2
+i/h=Wie−D2
+i/h.
+Since Dihas error at most 2 αDifor each pair of points, Lemma 2.2 bounds the error as at most Wi(2αDi//rad〉callow
+h).
+In order to bound the total error to (ǫ/4)W2, we bound the error for each pair by (ǫ/4)Wisince/summationtext
+iWi=/summationtext
+p∈P/summationtext
+q∈Qµ(p)ν(q)= W2. By Lemma 2.1, if Di>/radicalbig1
+hln(1/γ), then e−D2
+i/h< γ. So for any pair with
+Di>2/radicalbig1
+hln(4/ǫ), (forα<1/2) we can ignore, because they cannot have an effect on κ(P,Q)of more than
+(1/4)ǫWi, and thus cannot have error more than (1/4)ǫWi.
+Since we can ignore pairs with Di>2/radicalbig1
+hln(4/ǫ), each pair will have error at most Wi(2α(2/radicalbig1
+hln(4/ǫ)//rad〉callow
+h)
+=Wi(4α/radicalbig1
+ln(4/ǫ)). We can set this equal to (ǫ/4)Wiand solve forα<(1/4)ǫ//radicalbig1
+ln(4/ǫ). This ensures that
+each pair with Di≤2/radicalbig1
+hln(4/ǫ)will have error less than (ǫ/4)Wi, as desired.
+4For distributions with decaying but infinite tails, we can tr uncate to ignore tails such that the integral of the ignored a rea is at most
+(1−ǫ/2)W2and proceed with this approach using ǫ/2 instead ofǫ.
+4Theorem 3.1. By building and using an ((ǫ/4)//radicalbig1
+ln(4/ǫ))-WSPD, we can compute a value U in time O (nlogn+
+(n/ǫd)logd/2(1/ǫ)), such that/barex/barexU−D2
+K(P,Q)/barex/barex≤ǫW2.
+4 Computing the Kernel Distance II: Approximate Feature Map s
+In this section, we describe (approximate) feature represe ntationsΦ(P)=/summationtext
+p∈Pφ(p)µ(p)for shapes and
+distributions that reduce the kernel distance computation to anℓ2distance calculation/bardblΦ(P)−Φ(Q)/bardblHin
+an RKHS, H. This mapping immediately yields algorithms for a host of an alysis problems on shapes and
+distributions, by simply applying Euclidean space algorit hms to the resulting feature vectors.
+The feature map φallows us to translate the kernel distance (and norm) comput ations into operations in a
+RKHS that take time O(nρ)ifHhas dimension ρ, rather than the brute force time O(n2). Unfortunately , His in
+general infinite dimensional, including the case of the Gaus sian kernel. Thus, we use dimensionality reduction
+to find an approximate mapping ˜φ:/Rbbbd→/Rbbbρ(where ˜Φ(P)=/summationtext
+p∈P˜φ(p)) that approximates κ(P,Q):
+/barex/barex/barex/barex/summationdisplay
+p∈P/summationdisplay
+q∈QK(p,q)µ(p)ν(q)−/summationdisplay
+p∈P/summationdisplay
+q∈Q/angleleftbig˜φ(p),˜φ(q)/anglerightbig/barex/barex/barex/barex≤ǫW2.
+The analysis in the existing literature on approximating fe ature space does not directly bound the dimension
+ρrequired for a specific error bound5. We derive bounds from two known techniques: random project ions[36]
+(for shift-invariant kernels, includes Gaussians) and the Fast Gauss Transform [48, 20](for Gaussian kernel).
+We produce three different features maps, with different bo unds on the number of dimensions ρdepending on
+logn(nis the number of points), ǫ(the error),δ(the probability of failure), ∆(the normalized diameter of
+the points), and/ord(the ambient dimension of the data before the map).
+4.1 Random Projections Feature Space
+Rahimi and Recht [36]proposed a feature mapping that essentially applies an impl icit Johnson-Lindenstrauss
+projection from H→/Rbbbρ. The approach works for any shift invariant kernel (i.e one t hat can be written as
+K(p,q)=k(p−q)). For the Gaussian kernel, k(z)=e−/bardblz/bardbl2/2, where z∈/Rbbbd. Let the Fourier transform of
+k:/Rbbbd→/Rbbb+isg(ω)=( 2π)−d/2e−/bardblω/bardbl2/2. A basic result in harmonic analysis [37]is that kis a kernel if and
+only if gis a measure (and after scaling, is a probability distributi on). Letωbe drawn randomly from the
+distribution defined by g:
+k(x−y)=/integraldisplay
+ω∈/Rbbbdg(ω)eι〈ω,x−y〉dω=Eω[/angleleftbig1ψω(x),ψω(y)/anglerightbig1],
+whereψω(z)=( cos(〈ω,z〉),sin(〈ω,z〉))are the real and imaginary components of eι〈ω,z〉. This implies that/angleleftbig1ψω(x),ψω(y)/anglerightbig1is an unbiased estimator of k(x−y).
+We now consider a ρ-dimensional feature vector φΥ:P→/Rbbbρwhere the(2i−1)th and(2i)th coordi-
+nates are described by µ(p)ψωi(p)/(ρ/2)=( 2µ(p)cos(/angleleftbig1ωi,z/anglerightbig1)/ρ,2µ(p)sin(/angleleftbig1ωi,z/anglerightbig1)/ρ)for someωi∈Υ=
+{ω1,... ,ωρ/2}drawn randomly from g. Next we prove a bound on ρusing this construction.
+Lemma 4.1. ForφΥ:P∪Q→/Rbbbρwithρ=O((1/ǫ2)log(n/δ)), with probability≥1−δ
+/barex/barex/barex/barex/summationdisplay
+p∈P/summationdisplay
+q∈QK(p,q)µ(p)ν(q)−/summationdisplay
+p∈P/summationdisplay
+q∈Q/angleleftbig1φΥ(p),φΥ(q)/anglerightbig1/barex/barex/barex/barex≤ǫW2.
+5Explicit matrix versions of the Johnson-Lindenstraus lemm a[24]cannot be directly applied because the source space is itsel f
+infinite dimensional, rather than /Rbbbd.
+5Proof. We make use of the following Chernoff-Hoeffding bound. Give n a set{X1,... , Xn}of independent ran-
+dom variables, such that |Xi−E[Xi]|≤Λ, then for M=/summationtextn
+i=1Xiwe can bound Pr [|M−E[M]|≥α]≤
+2e−2α2/(nΛ2). We can now bound the error of using φΥfor any pair(p,q)∈P×Qas follows:
+Pr/bracketleftbig/barex/barex/angleleftbig1φΥ(p),φΥ(q)/anglerightbig1−µ(p)ν(q)k(p−q)/barex/barex≥ǫµ(p)ν(q)/bracketrightbig
+=Pr/bracketleftbig/barex/barex/angleleftbig1φΥ(p),φΥ(q)/anglerightbig1−EΥ/bracketleftbig1/angleleftbig1φΥ(p),φΥ(q)/anglerightbig1/bracketrightbig1/barex/barex≥ǫµ(p)ν(q)/bracketrightbig
+≤Pr
+/barex/barex/barex/barex/barex/summationdisplay
+i2
+ρµ(p)ν(q)/angleleftbig
+ψωi(p),ψωi(q)/anglerightbig
+−EΥ
+/summationdisplay
+i2
+ρµ(p)ν(q)/angleleftbig
+ψωi(p),ψωi(q)/anglerightbig
+/barex/barex/barex/barex/barex≥ǫµ(p)ν(q)
+
+≤2e−2(ǫµ(p)ν(q))2/(ρΛ2/2)≤2e−ρǫ2/64,
+where the last inequality follows by Λ≤2 maxp,q(2/ρ)µ(p)ν(q)/angleleftbig1ψω(p),ψω(q)/anglerightbig1≤8(2/ρ)µ(p)ν(q)since for
+each pair of coordinates /bardblψω(p)/bardbl≤2 for all p∈P(orq∈Q). By the union bound, the probability that this
+holds for all pairs of points (p,q)∈P×Qis given by
+Pr/bracketleftbig
+∀(p,q)∈P×Q/barex/barex/angleleftbig1φΥ(p),φΥ(q)/anglerightbig1−µ(p)ν(q)k(p−q)/barex/barex≥ǫµ(p)ν(q)/bracketrightbig
+≤(n2)2e−ρǫ2/64.
+Settingδ≥n22e−ρǫ2/64and solving for ρyields that for ρ=O((1/ǫ2)log(n/δ)), with probability at least
+1−δ, for all(p,q)∈P×Qwe have|µ(p)ν(q)k(p−q)−/angleleftbig1φΥ(x),φΥ(y)/anglerightbig1|≤ǫµ(p)ν(q). It follows that with
+probability at least 1−δ
+/barex/barex/barex/barex/summationdisplay
+p∈P/summationdisplay
+q∈Qµ(p)ν(q)K(p,q)−/summationdisplay
+p∈P/summationdisplay
+q∈Q/angleleftbig1φΥ(p),φΥ(q)/anglerightbig1/barex/barex/barex/barex≤ǫ/summationdisplay
+p∈P/summationdisplay
+q∈Qµ(p)ν(q)≤ǫW2.
+Note that the analysis of Rahimi and Recht [36]is done for unweighted point sets (i.e. µ(p)=1) and actually
+goes further, in that it yields a guarantee for any pair of poi nts taken from a manifold Mhaving diameter ∆.
+They do this by building an ǫ-net over the domain and applying the above tail bounds to the ǫ-net. We can
+adapt this trick to replace the (logn)term inρby a(dlog(∆/ǫ))term, recalling∆=( 1/h)max p,p′∈P∪Q/bardblp−p′/bardbl.
+This leads to the same guarantees as above with a dimension of ρ=O((d/ǫ2)log(∆/ǫδ)).
+4.2 Fast Gauss Transform Feature Space
+The above approach works by constructing features in the fre quency domain. In what follows, we present an
+alternative approach that operates in the spatial domain di rectly . We base our analysis on the Improved Fast
+Gauss Transform (IFGT) [48], an improvement on the Fast Gauss Transform. We start with a b rief review of
+the IFGT (see the original work [48]for full details).
+IFGT feature space construction. The goal of the IFGT is to approximate κ(P,Q). First we rewrite κ(P,Q)
+as the summation/summationtext
+q∈QG(q)where G(q)=ν(q)/summationtext
+p∈Pe−/bardblp−q/bardbl2/h2µ(p). Next, we approximate G(q)in two
+steps. First we rewrite
+G(q)=ν(q)/summationdisplay
+p∈Pµ(p)e−/bardblq−x∗/bardbl2
+h2e−/bardblp−x∗/bardbl2
+h2e2/bardblq−x∗/bardbl·/bardblp−x∗/bardbl
+h2,
+where the quantity x∗is a fixed vector that is usually the centroid of P. The first two exponential terms can
+be computed for each pandqonce. Second, we approximate the remaining exponential ter m by its Taylor
+expansion ev=/summationtext
+i≥0vi
+i!. After a series of algebraic manipulations, the following e xpression emerges:
+G(q)=ν(q)e−/bardblq−x∗/bardbl2
+h2/summationdisplay
+α≥0Cα/parenleftbigq−x∗
+h/parenrightbigα
+6where Cαis given by
+Cα=2|α|
+α!/summationdisplay
+p∈Pµ(p)e−/bardblp−x∗/bardbl2
+h2/parenleftbigp−x∗
+h/parenrightbigα
+.
+The parameter αis amultiindex , and is actually a vector α=(α1,α2,... ,αd)of dimension d. The expression
+zα, for z∈/Rbbbd, denotes the monomial zα1
+1zα2
+2...zαd
+d, the quantity|α|is the total degree/summationtext
+αi, and the quantity
+α!=Π i(αi!). The multiindices are sorted in graded lexicographic order , which means that αcomes beforeα′if
+|α|<|α′|, and two multiindices of the same degree are ordered lexicog raphically .
+The above expression for G(q)is an exact infinite sum, and is approximated by truncating th e summation at
+multiindices of total degree τ−1. Note that there are at most ρ=/parenleftbig1τ+d−1
+d/parenrightbig1=O(τd)such multiindices. We
+now construct a mapping ˜φ:/Rbbbd→/Rbbbρ. Let ˜φ(p)α=/radicalbig
+2|α|
+α!µ(p)e−/bardblp−x∗/bardbl2
+h2/parenleftbig
+p−x∗
+h/parenrightbigα
+. Then
+G(q)=/summationdisplay
+α˜φ(q)α/summationdisplay
+p∈P˜φ(p)α
+andS=/summationtext
+q∈QG(q)is then given by
+S=/summationdisplay
+p∈P/summationdisplay
+q∈Q/summationdisplay
+α˜φ(q)α˜φ(p)α=/summationdisplay
+p∈P/summationdisplay
+q∈Q〈˜φ(q),˜φ(p)〉.
+IFGT error analysis. The error incurred by truncating the sum at degree τ−1 is given by
+Err(τ)=/barex/barex/summationdisplay
+p∈P/summationdisplay
+q∈QK(p,q)µ(p)ν(q)−/summationdisplay
+p∈P/summationdisplay
+q∈Q/angleleftbig˜φ(p),˜φ(q)/anglerightbig/barex/barex≤/summationdisplay
+p∈P/summationdisplay
+q∈Qµ(p)ν(q)2τ
+τ!∆2τ=W22τ
+τ!∆2τ.
+SetǫW2=Err(τ). Applying Stirling’s approximation, we solve for τin log(1/ǫ)≥τlog(τ/4∆2). This yields
+the boundsτ=O(∆2)andτ=O(log(1/ǫ)). Thus our error bound holds for τ=O(∆2+log(1/ǫ)). Using
+ρ=O(τd), we obtain the following result.
+Lemma 4.2. There exists a mapping ˜φ:P∪Q→/Rbbbρwithρ=O(∆2d+logd(1/ǫ))so
+/barex/barex/barex/barex/summationdisplay
+p∈P/summationdisplay
+q∈QK(p,q)µ(p)ν(q)−/summationdisplay
+p∈P/summationdisplay
+q∈Q/angleleftbig˜φ(p),˜φ(q)/anglerightbig/barex/barex/barex/barex≤ǫW2.
+4.3 Summary of Feature Maps
+We have developed three different bounds on the dimension re quired for feature maps that approximate κ(P,Q)
+to withinǫW2.
+IFGT :ρ=O(∆2d+logd(1/ǫ)). Lemma 4.2. Advantages: deterministic, independent of n, logarithmic depen-
+dence on 1/ǫ. Disadvantages: polynomial dependence on ∆, exponential dependence on d.
+Random-points :ρ=O((1/ǫ2)log(n/δ)). Lemma 4.1. Advantages: independent of ∆andd. Disadvantages:
+randomized, dependent on n, polynomial dependence on 1 /ǫ.
+Random-domain :ρ=O((d/ǫ2)log(∆/ǫδ)). (above) Advantages: independent of n, logarithmic dependence
+on∆, polynomial dependence on d. Disadvantages: randomized, dependence on ∆andd, polynomial
+dependence on 1 /ǫ.
+For simplicity , we (mainly) use the Random-points based result from Lemma 4.1 in what follows. If appro-
+priate in a particular application, the other bounds may be e mployed.
+7Feature-based computation of DK.As before, we can decompose D2
+K(P,Q)=κ(P,P)+κ(Q,Q)−2κ(P,Q)
+and use Lemma 4.1 to approximate each of κ(P,P),κ(Q,Q), andκ(P,Q)with errorǫW2/4.
+Theorem 4.1. We can compute a value U in time O ((n/ǫ2)log(n/δ))such that|U−D2
+K(P,Q)|≤ǫW2, with
+probability at least 1−δ.
+A nearest-neighbor algorithm. The feature map does more than yield efficient algorithms for the kernel
+distance. As a representation for shapes and distributions , it allows us to solve other data analysis problems on
+shape spaces using off-the-shelf methods that apply to poin ts in Euclidean space. As a simple example of this,
+we can combine the Random-points feature map with known results on approximate nearest-neig hbor search
+in Euclidean space [3]to obtain the following result.
+Lemma 4.3. Given a collection of m point sets C={P1,P2,... ,Pm}, and a query surface Q, we can compute
+the c-approximate nearest neighbor to QinCunder the kernel distance in time O (ρm1/c2+o(1))query time using
+O(ρm1+1/c2+o(1))space and preprocessing.
+5 Coresets for the Kernel Distance
+The kernel norm (and distance) can be approximated in near-l inear time; however, this may be excessive for
+large data sets. Rather, we extract a small subset (a coreset )Sfrom the input Psuch that the kernel distance
+between SandPis small. By triangle inequality , Scan be used as a proxy for P. Specifically , we extend the
+notion ofǫ-samples for range spaces to handle non-binary range spaces defined by kernels.
+Background on range spaces. Letξ(P)denote the total weight of a set of points P, or cardinality if no
+weights are assigned. Let P⊂/Rbbbdbe a set of points and let Abe a family of subsets of P. For examples of
+A, letBdenote the set of all subsets defined by containment in a ball a nd letEdenote the set of all subsets
+defined by containment in ellipses. We say (P,A)is arange space . Let ¯ξP(A)=ξ(A)/ξ(P). Anǫ-sample (or
+ǫ-approximation ) of(P,A)is a subset Q⊂Psuch that
+max
+A∈A/barex/barex¯ξQ(Q∩A)−¯ξP(P∩A)/barex/barex≤ǫ.
+To create a coreset for the kernel norm, we want to generalize these notions of ǫ-samples to non-binary
+((0,1)-valued instead of{0,1}-valued) functions, specifically to kernels. For two point s etsP,Q, define ¯κ(P,Q)=
+(1/ξ(P))(1/ξ(Q))/summationtext
+p∈P/summationtext
+q∈QK(p,q), and when we have a singleton set Q={q}and a subset P′⊆Pthen we
+write ¯κP(P′,q)=( 1/ξ(P))/summationtext
+p∈P′K(p,q). Let K+=maxp,q∈PK(p,q)be the maximum value a kernel can take
+on a dataset P, which can be normalized to K+=1. We say a subset of S⊂Pis anǫ-sample of(P,K)if
+max
+q/barex/barex¯κP(P,q)−¯κS(S,q)/barex/barex≤ǫK+.
+The standard notion of VC-dimenion [47](and related notion of shattering dimension) is fundamenta lly
+tied to the binary ({0,1}-valued) nature of ranges, and as such, it does not directly a pply toǫ-samples of
+(P,K). Other researchers have defined different combinatorial di mensions that can be applied to kernels [15,
+25, 1, 46]. The best result is based on γ-fat shattering dimension Fγ[25], defined for a family of (0,1)-valued
+functions Fand a ground set P. A set Y⊂Pisγ-fat shattered byFif there exists a function α:Y→[0,1]
+such that for all subsets Z⊆Ythere exists some FZ∈Fsuch that for every x∈Z F Z(x)≥α(x)+γand
+for every x∈Y\Z F Z(x)≤α(x)−γ. Then Fγ=ξ(Y)for the largest cardinality set Y⊂Pthat can be
+γ-fat shattered. Bartlett et al.[7]show that a random sample of O((1/ǫ2)(Fγlog2(Fγ/ǫ)+log(1/δ))elements
+creates anǫ-sample (with probability at least 1 −δ) with respect to (P,F)forγ=Ω(ǫ). Note that the γ-
+fat shattering dimension of Gaussian and other symmetric ke rnels in/Rbbbdisd+1 (by settingα(x)=.5 for
+8allx), the same as balls Bin/Rbbbd, so this gives a random-sample construction for ǫ-samples of(P,K)of size
+O((d/ǫ2)(log2(1/ǫ)+log(1/δ)).
+In this paper, we improve this result in two ways by directly r elating a kernel range space (P,K)to a similar
+(binary) range space (P,A). First, this improves the random-sample bound because it us es sample-complexity
+results for binary range spaces that have been heavily optim ized. Second, this allows for all deterministic
+ǫ-sample constructions (which have no probability of failur e) and can have much smaller size.
+Constructions for ǫ-samples. Vapnik and Chervonenkis [47]showed that the complexity of ǫ-samples is
+tied to the VC-dimension of the range space. That is, given a r ange space(X,A)a subset Y⊂Xis said to
+beshattered byAif all subsets of Z⊂Ycan be realized as Z=Y∩RforR∈A. Then the VC-dimension of
+a range space(X,A)is the cardinality of the largest subset Y⊂Xthat can be shattered by A. Vapnik and
+Chervonenkis[47]showed that if the VC-dimension of a range space (X,A)isν, then a random sample Yof
+O((1/ǫ2)(νlog(1/ǫ)+log(1/δ))points from Xis anǫ-sample with probability at least 1 −δ. This bound was
+improved to O((1/ǫ2)(ν+log 1/δ))by Talagrand[44, 26].
+Alternatively , Matousek [28]showed thatǫ-samples of size O((ν/ǫ2)log(ν/ǫ))could be constructed de-
+terministically , that is there is no probability of failure . A simpler algorithm with more thorough runtime
+analysis is presented in Chazelle and Matousek [12], which runs in O(d)3d|X|(1/ǫ)2νlogν(1/ǫ)time. Smaller
+ǫ-samples exist; in particular Matousek, Welzl, and Wernisc h[31]and improved by Matousek [29]show
+thatǫ-samples exist of size O((1/ǫ)2−2/(ν+1)), based on a discrepancy result that says there exists a label ing
+χ:X→{− 1,+1}such that maxR∈A/summationtext
+x∈R∩Xχ(x)≤O(|X|1/2−1/2νlog1/2|X|). It is alluded by Chazelle [11]
+that if an efficient construction for such a labeling existed , then an algorithm for creating ǫ-samples of size
+O((1/ǫ)2−2/(ν+1)log(ν/ǫ)2−1/d+1)would follow, see also Phillips [34]. Recently , Bansal [6]provided a ran-
+domized polynomial algorithm for the entropy method, which is central in proving these existential coloring
+bounds. This leads to an algorithm that runs in time O(|X|·poly(1/ǫ)), as claimed by Charikar et al.[10, 33].
+An alternate approach is through the VC-dimension of the dual range space (A,¯A), of (primal) range space
+(X,A), where ¯Ais the set of subsets of ranges Adefined by containing the same element of X. In our context,
+for range spaces defined by balls (/Rbbbd,B)and ellipses of fixed orientation (/Rbbbd,E)their dual range spaces have
+VC-dimension ¯ ν=d. Matousek[30]shows that a technique of matching with low-crossing number [13]along
+with Haussler’s packing lemma [22]can be used to construct a low discrepancy coloring for range spaces where
+¯ν, the VC-dimension of the dual range space is bounded. This te chnique can be made deterministic and runs
+inpoly(|X|)time. Invoking this technique in the Chazelle and Matousek [12]framework yields an ǫ-sample of
+sizeO((1/ǫ)2−2/(¯ν+1)(log(1/ǫ))2−1/(¯ν+1))inO(|X|·poly(1/ǫ))time. Specifically , we attain the following result:
+Lemma 5.1. For discrete ranges spaces (X,B)and(X,E)for X∈/Rbbbdof size n, we can construct an ǫ-sample of
+size O((1/ǫ)2−2/(d+1)(log(1/ǫ))2−1/d+1)in O(n·poly(1/ǫ))time.
+For specific range spaces, the size of ǫ-samples can be improved beyond the VC-dimension bound. Phi llips[34]
+showed for ranges Rdconsisting of axis-aligned boxes in /Rbbbd, thatǫ-samples can be created of size O((1/ǫ)·
+log2d(1/ǫ)). This can be generalized to ranges defined by kpredefined normal directions of size O((1/ǫ)·
+log2k(1/ǫ)). These algorithms run in time O(|X|(1/ǫ3)polylog(1/ǫ)). And for intervals Iover/Rbbb,ǫ-samples of
+(X,I)can be created of size O(1/ǫ)by sorting points and retaining every ǫ|X|th point in the sorted order [27].
+ǫ-Samples for kernels. Thesuper-level set of a kernel given one input q∈/Rbbbdand a value v∈/Rbbb+, is the set
+of all points p∈/Rbbbdsuch that K(p,q)≥v. We say that a kernel is linked to a range space (/Rbbbd,A)if for every
+possible input point q∈/Rbbbdand any value v∈/Rbbb+that the super-level set of K(·,q)defined by vis equal to
+some H∈A. For instance multi-variate Gaussian kernels with no skew a re linked to(/Rbbbd,B)since all super-
+level sets are balls, and multi-variate Gaussian kernels wi th non-trivial covariance are linked to (/Rbbbd,E)since
+all super-level sets are ellipses.
+Theorem 5.1. For any kernel K :M×M→/Rbbb+linked to a range space (M,A), anǫ-sample S of(P,A)for S⊆M
+is aǫ-sample of(P,K).
+9A (flawed) attempt at a proof may proceed by considering a series of app roximate level-sets, within which each
+point has about the same function value. Since S is an ǫ-sample of(P,A), we can guarantee the density of S
+and P in each level set is off by at most 2ǫ. However, the sum of absolute error over all approximate level-sets
+is approximately ǫK+times the number of approximate level sets. This analysis fails becaus e it allows error to
+accumulate; however, a more careful application of the ǫ-sample property shows it cannot. A correct analysis
+follows using a charging scheme which prevents the error from accum ulating.
+Proof. We can sort all pi∈Pin similarity to qso that pi<pj(and by notation i<j) ifK(pi,q)>K(pj,q).
+Thus any super-level set containing pjalso contains pifori<j. We can now consider the one-dimensional
+problem on this sorted order from q.
+We now count the deviation E(P,S,q)=¯κP(P,q)−¯κS(S,q)from p1topnusing a charging scheme. That is
+each element sj∈Sis charged toξ(P)/ξ(S)points in P. For simplicity we will assume that k=ξ(P)/ξ(S)is
+an integer, otherwise we can allow fractional charges. We no w construct a partition of Pslightly differently , for
+positive and negative E(P,S,q)values, corresponding to undercounts and overcounts, resp ectively .
+Undercount of ¯κS(S,q).For undercounts, we partition Pinto 2ξ(S)(possibly empty) sets {P′
+1,P1,P′
+2,
+P2,... , P′
+ξ(S),Pξ(S)}of consecutive points by the sorted order from q. Starting with p1(the closest point to
+q) we place points in sets P′
+jorPjfollowing their sorted order. Recursively on jandi, starting at j=1 and
+i=1, we place each piinP′
+jas long as K(pi,q)>K(sj,q)(this may be empty). Then we place the next k
+points piintoPj. After kpoints are placed in Pj, we begin with P′
+j+1, until all of Phas been placed in some set.
+Lett≤ξ(S)be the index of the last set Pjsuch thatξ(Pj)=k. Note that for all pi∈Pj(for j≤t) we have
+K(sj,q)≥K(pi,q), thus ¯κS({sj},q)≥¯κP(Pj,q).
+We can now bound the undercount as
+E(P,S,q)=ξ(S)/summationdisplay
+j=1/parenleftbig
+¯κP(Pj,q)−¯κS({sj},q)/parenrightbig
++ξ(S)/summationdisplay
+j=1¯κP(P′
+j,q)≤t+1/summationdisplay
+j=1¯κP(P′
+j,q)
+since the first term is at most 0 and since ξ(P′
+j)=0 for j>t+1. Now consider a super-level set H∈A
+containing all points before st+1;His the smallest range that contains every non-empty P′
+j. Because (for j≤t)
+each set Pjcan be charged to sj, then/summationtextt
+j=1ξ(Pj∩H)=k·ξ(S∩H). And because Sis anǫ-sample of(P,A),
+then/summationtextt+1
+j=1ξ(P′
+j)=/parenleftbig/summationtextt+1
+j=1ξ(P′
+j)+/summationtextt
+j=1ξ(Pj∩H)/parenrightbig
+−k·ξ(S∩H)≤ǫξ(P). We can now bound
+E(P,S,q)≤t+1/summationdisplay
+j=1¯κP(P′
+j,q)=t+1/summationdisplay
+j=1/summationdisplay
+p∈P′
+jK(p,q)
+ξ(P)≤1
+ξ(P)t+1/summationdisplay
+j=1ξ(P′
+j)K+≤1
+ξ(P)(ǫξ(P))K+=ǫK+.
+Overcount of ¯κS(S,q):The analysis for overcounts is similar to undercounts, but w e construct the partition
+in reverse and the leftover after the charging is not quite as clean to analyze. For overcounts, we partition P
+into 2ξ(S)(possibly empty) sets {P1,P′
+1,P2,P′
+2,... , Pξ(S),P′
+ξ(S)}of consecutive points by the sorted order from q.
+Starting with pn(the furthest point from q) we place points in sets P′
+jorPjfollowing their reverse-sorted order.
+Recursively on jandi, starting at j=ξ(S)andi=n, we place each piinP′
+jas long as K(pi,q)<K(sj,q)(this
+may be empty). Then we place the next kpoints piintoPj. After kpoints are placed in Pj, we begin with P′
+j−1,
+until all of Phas been placed in some set. Let t≤ξ(S)be the index of the last set Pjsuch thatξ(Pj)=k(the
+smallest such j). Note that for all pi∈Pj(forj≥t) we have K(sj,q)≤K(pi,q), thus ¯κS({sj},q)≤¯κP(Pj,q).
+We can now bound the (negative) undercount as
+E(P,S,q)=t/summationdisplay
+j=ξ(S)/parenleftbig
+¯κP(Pj,q)−¯κS({sj},q)/parenrightbig
++1/summationdisplay
+j=t−1/parenleftbig
+¯κP(Pj,q)−¯κS({sj},q)/parenrightbig
++ξ(S)/summationdisplay
+j=1¯κP(P′
+j,q)
+≥/parenleftbig1¯κP(Pt−1,q)−¯κS({st−1},q)/parenrightbig1−1/summationdisplay
+j=t−2¯κS({sj},q),
+10since the first full term is at least 0, as is each ¯ κP(Pj,q)and ¯κP(P′
+j,q)term in the second and third terms. We
+will need the one term ¯ κP(Pt−1,q)related to Pin the case when 1≤ξ(Pt−1)<k.
+Now, using that Sis anǫ-sample of(P,A), we will derive a bound on t, and more importantly (t−2). We
+consider the maximal super-level set H∈Asuch that no points H∩Pare in P′
+jfor any j. This is the largest
+set where each point p∈Pcan be charged to a point s∈Ssuch that K(p,q)>K(s,q), and thus presents the
+smallest (negative) undercount. In this case, H∩P=∪s
+j=1Pjfor some sandH∩S=∪s
+j=1{sj}. Since t≤s, then
+ξ(H∩P)=(s−t+1)k+ξ(Pt−1)=(s−t+1)ξ(P)/ξ(S)+ξ(Pt−1)andξ(H∩S)=s. Thus
+ǫ≥¯ξS(H∩S)−¯ξP(H∩P)=s
+ξ(S)−(s−t+1)ξ(P)/ξ(S)
+ξ(P)−ξ(Pt−1)
+ξ(P)≥t−1
+ξ(S)−ξ(Pt−1)
+ξ(P).
+Thus(t−2)≤ǫξ(S)+ξ(Pt−1)(ξ(S)/ξ(P))−1. Letting pi=mini′∈Pt−1K(pi′,q)(note K(pi,q)≥K(st−1,q))
+E(P,S,q)≥κ(Pt−1,q)
+ξ(P)−K(st−1,q)
+ξ(S)−/parenleftbig
+ǫξ(S)+ξ(Pt−1)ξ(S)
+ξ(P)−1/parenrightbigK+
+ξ(S)
+=−ǫK++K+/parenleftbigk−ξ(Pt−1)
+ξ(P)/parenrightbig
+−k·K(st−1,q)−κ(Pt−1,q)
+ξ(P)
+≥−ǫK++K+/parenleftbigk−ξ(Pt−1)
+ξ(P)/parenrightbig
+−K(pi,q)/parenleftbigk−ξ(Pt−1)
+ξ(P)/parenrightbig
+≥−ǫK+.
+Corollary 5.1. For a Gaussian kernel, any ǫ-sample S for(P,B)(or for(P,E)if we consider covariance) guarantees
+that for any query q∈/Rbbbdthat/barex/barex¯κP(P,q)−¯κS(S,q)/barex/barex≤ǫK+.
+Coreset-based computation of kernel distance. For convenience here we assume that our kernel has
+been normalized so K+=1. Let Pbe anǫ-sample of(P,K), and all points p∈Phave uniform weights so
+ξ(P)=ξ(P)=W. Then for any q∈/Rbbbd
+ǫ≥|¯κP(P,q)−¯κP(P,q)|=/barex/barex/barex/barexκ(P,q)
+ξ(P)−κ(P,q)
+ξ(P)/barex/barex/barex/barex.
+and hence/barex/barexκ(P,q)−κ(P,q)/barex/barex≤ǫξ(P)=ǫW.
+It follows that if Qis also anǫ-sample for(Q,K), then/bardblκ(P,Q)−κ(P,Q)/bardbl≤2ǫW2. Hence, via known con-
+structions ofǫ-samples randomized [47, 44]or deterministic[28, 12, 31, 29, 43, 34 ](which can be applied to
+weighted point sets to create unweighted ones [28]) and Theorem 5.1 we have the following theorems. The
+first shows how to construct a coreset with respect to DK.
+Theorem 5.2. Consider any kernel K linked with (/Rbbbd,B)and objects P,Q⊂M⊂/Rbbbd, each with total weight W,
+and for constant d. We can construct sets P ⊂Pand Q⊂Qsuch that|DK(P,Q)−DK(P,Q)|≤ǫW2of size:
+•O((1/ǫ2−1/(d+1))log2−1/d+1(1/ǫ)), via Lemma 5.1; or
+•O(1/ǫ2)(d+log(1/δ)))via random sampling (correct with probability at least (1−δ)).
+We present an alternative sampling condition to Theorem 5.2 in Appendix A. It has larger dependence on ǫ,
+and also has either dependence on ∆or on log n(and is independent of K+). Also in Appendix A we show that
+it is NP-hard to optimize ǫwith a fixed subset size k.
+The associated runtimes are captured in the following algor ithmic theorem.
+Theorem 5.3. When K is linked to (/Rbbbd,B), we can compute a number ˜D such that|DK(P,Q)−˜D|≤ǫin time:
+•O(n·(1/ǫ2d+2)logd+1(1/ǫ)); or
+•O(n+(logn)·((1/ǫ2)log(1/δ))+( 1/ǫ4)log2(1/δ))that is correct with probability at least 1−δ.
+Notice that these results automatically work for any kernel linked with(/Rbbbd,B)(or more generally with
+(/Rbbbd,E)) with no extra work; this includes not only Gaussians (with n on-trivial covariance), but any other
+standard kernel such as triangle, ball or Epanechnikov.
+116 Minimizing the Kernel Distance under Translation and Rota tion
+We attack the problem of minimizing the kernel distance betw eenPandQunder a set of transformations:
+translations or translations and rotations. A translation T∈/Rbbbdis a vector so Q⊕T={q+T|q∈Q}. The
+translation T∗=arg minT∈/RbbbdDK(P,Q⊕T), applied to Qminimizes the kernel norm. A rotation R∈SO(d)can
+be represented as a special orthogonal matrix. We can write R◦Q={R(q)|q∈Q}, where R(q)rotates qabout
+the origin, preserving its norm. The set of a translation and rotation(T⋆,R⋆)=arg min(T,R)∈/Rbbbd×SO(d)DK(P,R◦
+(Q⊕T))applied to Qminimizes the kernel norm.
+Decomposition. In minimizing DK(P,R◦(Q⊕T))under all translations and rotations, we can reduce this
+to a simpler problem. The first term κ(P,P)=/summationtext
+p1∈P/summationtext
+p2∈Pµ(p1)µ(p2)K(p1,p2)has no dependence on Tor
+R, so it can be ignored. And the second term κ(Q,Q)=/summationtext
+q1∈Q/summationtext
+q2∈Qν(q1)ν(q2)K(R(q1+T),R(q2+T))can
+also be ignored because it is invariant under the choice of TandR. Each subterm K(R(q1+T),R(q2+T))only
+depends on||R(q1+T)−R(q2+T)||=||q1−q2||, which is also independent of TandR. Thus we can rephrase
+the objective as finding
+(T⋆,R⋆)=arg max
+(T,R)∈/Rbbbd×SO(d)/summationdisplay
+p∈P/summationdisplay
+q∈Qµ(p)ν(q)K(p,R(q+T))=arg max
+(T,R)∈/Rbbbd×SO(d)κ(P,R◦(Q⊕T)).
+We start by providing an approximately optimal translation . Then we adapt this algorithm to handle both
+translations and rotations.
+6.1 Approximately Optimal Translations
+We describe, for any parameter ǫ>0, an algorithm for a translation ˆTsuch that D2
+K(P,Q⊕ˆT)−D2
+K(P,Q⊕T∗)≤
+ǫW2. We begin with a key lemma providing analytic structure to ou r problem.
+Lemma 6.1.κ(P,Q⊕T∗)≥W2/n2.
+Proof. When T∈/Rbbbdaligns q∈Qsoq+T=pforp∈Pit ensures that K(p,q)= 1. We can choose
+the points pandqsuch thatµ(p)andν(q)are as large as possible. They must each be at least W/n, so
+K(p,q)µ(p)ν(q)≥W2/n2. All other subterms in κ(P,Q⊕T)are at least 0. Thus κ(P,Q⊕T)≥W2/n2.
+Thus, ifκ(P,Q⊕T∗)≥W2/n2, then for some pair of points p∈Pandq∈Qwe must haveµ(p)ν(q)K(p,q+
+T∗)≥µ(p)ν(q)/n2, i.e. K(p,q+T∗)≥1/n2. Otherwise, if all n2pairs(p,q)satisfyµ(p)ν(q)K(p,q+T∗)<
+µ(p)ν(q)/n2, then
+κ(P,Q⊕T∗)=/summationdisplay
+p∈P/summationdisplay
+q∈Qµ(p)ν(q)K(p,q+T∗)</summationdisplay
+p∈P/summationdisplay
+q∈Qµ(p)ν(q)/n2=W2/n2.
+Thus some pair p∈Pandq∈Qmust satisfy||p−(q+T∗)||≤/radicalbig1
+ln(n2), via Lemma 2.1 with γ=1/(n2).
+LetGǫbe a grid on/Rbbbdso that when any point p∈/Rbbbdis snapped to the nearest grid point g∈Gǫ, we
+guarantee that||g−p||≤ǫ. We can define an orthogonal grid Gǫ={(ǫ//rad〉callow
+d)z|z∈/Zbbbd}, where/Zbbbdis the
+d-dimensional lattice of integers. Let G[ǫ,p,Λ]represent the subset of the grid Gǫthat is within a distance Λ
+of the point p. In other words, G[ǫ,p,Λ]={g∈Gǫ|||g−p||≤Λ}.
+Algorithm. These results imply the following algorithm. For each point p∈P, for each q∈Q, and for each
+g∈G[ǫ/2,p,/radicalbig1
+ln(n2)]we consider the translation Tp,q,gsuch that q+Tp,q,g=g. We return the translation
+Tp,q,gwhich maximizes κ(P,Q⊕Tp,q,g), by evaluating κat each such translation of Q.
+Theorem 6.1. The above algorithm runs in time O ((1/ǫ)dn4logd/2n), for fixed d, and is guaranteed to find a
+translationˆT such that D2
+K(P,Q⊕ˆT)−D2
+K(P,Q⊕T∗)≤ǫW2.
+12Proof. We know that the optimal translation T∗must result in some pair of points p∈Pandq∈Qsuch
+that||p−(q+T∗)||≤/radicalbig1
+ln(n2)by Lemma 6.1. So checking all pairs p∈Pand q∈Q, one must have
+||p−q||≤/radicalbig1
+ln(n2). Assuming we have found this closest pair, p∈Pandq∈Q, we only need to search in the
+neighborhood of translations T=p−q.
+Furthermore, for some translation Tp,q,g=g−qwe can claim that κ(P,Q⊕T∗)−κ(P,Q⊕Tp,q,g)≤ǫ. Since
+||T∗−Tp,q,g||≤ǫ/2, we have the bound on subterm |K(p,q+T∗)−K(p,q+Tp,q,g)|≤ǫ/2, by Lemma 2.2. In
+fact, for every other pair p′∈Pandq′∈Q, we also know|K(p′,q′+T∗)−K(p′,q′+Tp,q,g)|≤ǫ/2. Thus the
+sum of these subterms has error at most (ǫ/2)/summationtext
+p∈P/summationtext
+q∈Qµ(p)ν(q)=(ǫ/2)W2.
+Since, the first two terms of D2
+K(P,Q⊕T)are unaffected by the choice of T, this provides an ǫ-approximation
+forD2
+K(P,Q⊕T)because all error is in the (−2)κ(P,Q⊕T)term.
+For the runtime we need to bound the number of pairs from PandQ(i.e. O(n2)), the time to calculate
+κ(P,Q⊕T)(i.e. O(n2)), and finally the number of grid points in G[ǫ/2,p,/radicalbig1
+ln(n2)]. The last term re-
+quires O((1/ǫ)d)points per unit volume, and a ball of radius/radicalbig1
+ln(n2)has volume O(logd/2n), resulting in
+O((1/ǫ)dlogd/2n)points. This product produces a total runtime of O((1/ǫ)dn4logd/2n).
+For a constant dimension d, using Theorem 5.2 to construct a coreset, we can first set n=O((1/ǫ2)log(1/δ))
+and now the time to calculate κ(P,Q⊕T)isO((1/ǫ4)log2(1/δ))after spending O(n+(1/ǫ2)log(1/δ)logn)
+time to construct the coresets. Hence the total runtime is
+O(n+logn(1/ǫ2)(log(1/δ))+( 1/ǫd+8)·logd/2((1/ǫ)log(1/δ))log4(1/δ)),
+and is correct with probability at least 1 −δ.
+Theorem 6.2. For fixed d, in
+O(n+logn(1/ǫ2)(log(1/δ))+( 1/ǫd+8)·logd/2((1/ǫ)log(1/δ))log4(1/δ))
+time we can find a translation ˆT such that D2
+K(P,Q⊕ˆT)−D2
+K(P,Q⊕T∗)≤ǫW2, with probability at least 1−δ.
+6.2 Approximately Optimal Translations and Rotations
+For any parameter ǫ>0, we describe an algorithm to find a translation ˆTand a rotationˆRsuch that
+D2
+K(P,ˆR◦(Q⊕ˆT))−D2
+K(P,R⋆◦(Q⊕T⋆))≤ǫW2.
+We first find a translation to align a pair of points p∈Pandq∈Qwithin some tolerance (using a method
+similar to above, and using Lemma 2.1 to ignore far-away pair s) and then rotate Qaround q. This deviates
+from our restriction above that ˆR∈SO(d)rotates about the origin, but can be easily overcome by perfo rming
+the same rotation about the origin, and then translating Qagain so qis at the desired location. Thus, after
+choosing a q∈Q(we will in turn choose each q′∈Q) we let all rotations be about qand ignore the extra
+modifications needed to ˆTandˆRto ensureˆRis about the origin.
+Given a subset S⊂Qof fewer than dpoints and a pair (p,q)∈P×Qwhere q/∈S, we can define a rotational
+grid around p, with respect to q, so that Sis fixed. Let Rd,Sbe the subset of rotations in d-space under which the
+setSis invariant. That is for any R∈Rd,Sand any s∈Swe have R(s)=s. Letτ=d−|S|. Then (topologically)
+Rd,S=SO(τ). Let RS,p,q=minR∈Rd,S||R(q)−p||and letˆq=RS,p,q(q). LetH[p,q,S,ǫ,Λ]⊂Rd,Sbe a set
+of rotations under which Sis invariant with the following property: for any point q′such that there exists a
+rotation R′∈Rd,Swhere R′(q)=q′and where||q′−ˆq||≤Λ, then there exists a rotation R∈H[p,q,S,ǫ,Λ]
+such that||R(q)−q′||≤ǫ. For the sanity of the reader, we will not give a technical con struction, but just note
+that it is possible to construct H[p,q,S,ǫ,Λ]of size O((Λ/ǫ)τ).
+13Algorithm. For each pair of ordered sets of dpoints(p1,p2,... , pd)⊂Pand(q1,q2,... , qd)⊂Qconsider the
+following set of translations and rotations. Points in Pmay be repeated. For each g∈G[ǫ/d,p1,/radicalbig1
+ln(max{1/ǫ,n2})]
+consider translation Tp1,q1,gsuch that q1+Tp1,q1,g=g. We now consider rotations of the set (Q⊕Tp1,q1,g).
+LetS={q1}and consider the rotational grid H[p2,q2+Tp1,q1,g,S,ǫ/d,/radicalbig1
+ln(1/ǫ)]. For each rotation R2∈
+H[p2,q2+Tp1,q1,g,S,ǫ/d,/radicalbig1
+ln(1/ǫ)]we recurse as follows. Apply R2(Q⊕Tp1,q1,g)and place R2(q2+Tp1,q1,g)
+inS. Then in the ith stage consider the rotational grid H[pi,Ri−1(...R2(q2+Tp1,q1,g)...),S,ǫ/d,/radicalbig1
+ln(1/ǫ)].
+Where Riis some rotation we consider from the ith level rotational grid, let ¯R=Rd◦Rd−1◦...◦R2. Let(ˆT,ˆR)
+be the pair(Tp,q,g,¯R)that maximize κ(P,¯R◦(Q⊕Tp,q,g)).
+Theorem 6.3. The above algorithm runs in time
+O(n2d+2(1/ǫ)(d2−d+2)/2log(d2−3d+2)/4(1/ǫ)logd/2(max{n2,1/ǫ})),
+for a fixed d, and is guaranteed to find a translation and rotation pa ir(ˆT,ˆR), such that
+D2
+K(P,ˆR◦(Q⊕ˆT))−D2
+K(P,R⋆◦Q⊕T⋆)≤ǫW2.
+Proof. We compare our solution (ˆT,ˆR)to the optimal solution (T⋆,R⋆). Note that only pairs of points (p,q)∈
+P×Qsuch that||p−R⋆(q+T⋆)||</radicalbig1
+ln(1/ǫ)need to be considered.
+We first assume that for the ordered sets of dpoints we consider (p1,p2,... , pd)⊂Pand(q1,q2,... ,qd)⊂Q
+we have (A1)||pi−R⋆(qi+T⋆)||≤/radicalbig1
+ln(1/ǫ), and (A2) for S={q1,... , qi−1}, letqi∈Qbe the furthest point
+from Ssuch that||pi−(qi+T⋆)||≤/radicalbig1
+ln(1/ǫ). Note that (A2) implies that for any rotation R∈Rd,Sthat
+||qi−R(qi)||>||q′−R(q′)||for all q′∈Qthat can be within the distance threshold under (T⋆,R⋆). In the case
+that fewer than dpairs of points are within our threshold distance, then as lo ng as these are the first pairs
+in the ordered sequence, the algorithm works the same up to th at level of recursion, and the rest does not
+matter. Finally , by Lemma 6.1 we can argue that at least one pa ir must be within the distance threshold for our
+transition grid.
+For each point q∈Qwe can show there exists some pair (T,R)considered by the algorithm such that
+||R⋆(q+T⋆)−R(q+T)||≤ǫ. First, there must be some translation T=Tp1,q1,gin our grid that is within a
+distance ofǫ/dofT⋆. This follows from Lemma 2.2 and similar arguments to the pro of for translations.
+For each qiwe can now show that for some Ri∈H(the rotational grid) we have ||Ri(Ri−1(...R2(qi+
+Tp1,q1,g)...))−R⋆(qi+T⋆)||≤ǫ. By our assumptions, the transformed qimust lie within the extents of H.
+Furthermore, there is a rotation R′
+jthat can replace each Rjforj∈[2,i]that moves qiby at mostǫ/dsuch
+thatR′
+i(R′
+i−1(...R′
+2(qi)...))= R⋆(qi). Hence, the composition of these rotations affects qiby at mostǫ/(i−1),
+and the sum effect of rotation and translation errors is at mo stǫ.
+Since each qiis invariant to each subsequent rotation in the recursion, w e have shown that there is a pair
+(T,R)considered so||R(qi+T)−R⋆(qi+T⋆)||≤ǫforqiin the ordered set (q1,q2,... ,qd). We can now use
+our second assumption (A2) that shows that at each stage of th e recursion qiis the point affected most by the
+rotation. This indicates that we can use the above bound for a ll points q∈Q, not just those in our ordered set.
+Finally , we can use Lemma 2.2 to complete the proof of correct ness. Since if each K(p,q)has error at most
+ǫ, then
+/barex/barex/barex/barex/barex/barex/summationdisplay
+p∈P/summationdisplay
+q∈Qµ(p)ν(q)K(p,ˆR(q+ˆT))−/summationdisplay
+p∈P/summationdisplay
+q∈Qµ(p)ν(q)K(p,R∗(q+T∗))/barex/barex/barex/barex/barex/barex≤/summationdisplay
+p∈P/summationdisplay
+q∈Qµ(p)ν(q)ǫ=ǫW2.
+We can bound the runtime as follows. We consider all d!/parenleftbig1n
+d/parenrightbig1=O(nd)ordered sets of points in Qand all nd
+ordered sets of points from P. This gives the leading O(n2d)term. We then investigate all combinations of grid
+14points from each grid in the recursion. The translation grid has size O((∆/ǫ)d)=O((1/ǫ)dlogd/2(max{1/ǫ,n2})).
+The size of the ith rotational grid is O((/radicalbig1
+log(1/ǫ)/ǫ)d−i, starting at i=2. The product of all the rota-
+tional grids is the base to the sum of their powers/summationtextd−1
+i=1(d−i)=/summationtextd−1
+i=1i=(d−1)(d−2)/2=(d2−3d+
+2)/2, that is O((1/ǫ)(d2−3d+2)/2log(d2−3d+2)/4(1/ǫ)). Multiplying by the size of the translational grid we get
+O((1/ǫ)(d2−d+2)/2log(d2−3d+2)/4(1/ǫ)logd/2(max{n2,1/ǫ})). Then for each rotation and translation we must
+evaluateκ(P,R◦(Q⊕T))inO(n2)time. Multiplying these three components gives the final bou nd of
+O(n2d+2(1/ǫ)(d2−d+2)/2log(d2−3d+2)/4(1/ǫ)logd/2(max{n2,1/ǫ})).
+The runtime can again be reduced by first computing a coreset o f size O((1/ǫ2)log(1/δ))and using this
+value as n. After simplifying some logarithmic terms we reach the foll owing result.
+Theorem 6.4. For fixed d, in
+O(n+logn(1/ǫ2)(log(1/δ))+( 1/ǫ)(d2+7d+6)/2(log(1/ǫδ))(d2+7d+10)/4),
+time we can find a translation and rotation pair (ˆT,ˆR), such that
+D2
+K(P,ˆR◦(Q⊕ˆT))−D2
+K(P,R⋆◦Q⊕T⋆)≤ǫW2,
+with probability at least 1−δ.
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+A Coresets for the Kernel Distance
+A.1 Alternative Coreset Proof
+In Section 5 we presented a construction for a coreset for the kernel distance, that depended only on 1 /ǫfor a
+fixed kernel. This assumed that K+=maxp,qK(p,q)was bounded and constant. Here we present an alternative
+proof (also by random sampling) which can be made independen t ofK+. This proof is also interesting because
+it relies onρthe dimension of the feature map.
+Again, we sample kpoints uniformly from Pand reweightη(p)=W/k.
+Lemma A.1. By constructing S with size k =O((1/ǫ3)log(n/δ)log((1/ǫδ)logn))we guarantee D2
+K(P,S)≤ǫW2,
+with probability at least 1−δ.
+Proof. The error in our approximation will come from two places: (1) the size of the sample, and (2) the
+dimensionρof the feature space we perform the analysis in.
+Letφ:P×/Rbbb+→Hdescribe the true feature map from a point p∈P, with weight µ(p), to an infinite
+dimensional feature space. As before, set Φ(P)=/summationtext
+p∈Pµ(p)φ(p), and recall that DK(P,Q)=/bardblΦ(P)−Φ(Q)/bardblH,
+for any pair of shapes PandQ.
+By the results in the previous section, we can construct ˜φ:P×/Rbbb+→/Rbbbρ(such asφΥdefined for Lemma 4.1)
+such that ˜Φ(P)=/summationtext
+p∈P˜φ(p)and for any pair of shapes PandSwith weights W=/summationtext
+p∈Pµ(p)=/summationtext
+p∈Sη(p),
+we have/barex/barex/bardblΦ(P)−Φ(S)/bardbl2
+H−/bardbl˜Φ(P)−˜Φ(S)/bardbl2/barex/barex≤(ǫ/2)W2, with probability at least 1 −δ/2. This bounds the
+error in the approximation of the feature space.
+We now use the low dimension ρof this approximate feature space to bound the sampling erro r. Specifically ,
+we just need to bound the probability that /bardbl˜Φ(P)−˜Φ(S)/bardbl2=/bardblE[˜Φ(S)]−˜Φ(S)/bardbl2≥(ǫ/2)W2, since E[˜Φ(S)]=
+˜Φ(P). This is always true if for each dimension (or pair of dimensi ons if we alter bounds by a factor 2)
+m∈[1,ρ]we have/barex/barex˜Φ(S)m−E[˜Φ(S)m]/barex/barex≤/radicalbig1
+ǫW2/(2ρ), so we can reduce to a 1-dimensional problem.
+We can now invoke the following Chernoff-Hoeffding bound. G iven a set{X1,... , Xr}of independent random
+variables, such that/barex/barexXi−E[Xi]/barex/barex≤Λ, then for M=/summationtextr
+i=1Xiwe can bound Pr [|M−rE[Xi]|≥α]≤2e−2α2/(rΛ2).
+18By lettingα=/radicalbig1
+W2ǫ/(2ρ)andXi=˜φ(pi)m, the mth coordinate of ˜φ(pi)forpi∈S,
+Pr/bracketleftbig/bardblex/bardblex˜Φ(S)−˜Φ(P)/bardblex/bardblex2≥(ǫ/2)W2/bracketrightbig
+=Pr/bracketleftbig/bardblex/bardblex/bardblex˜Φ(S)−E/bracketleftbig˜Φ(S)/bracketrightbig/bardblex/bardblex/bardblex2
+≥(ǫ/2)W2/bracketrightbig
+≤ρPr/bracketleftbig/barex/barex/barexΦ(S)m−kE/bracketleftbig˜φ(pi)m/bracketrightbig/barex/barex/barex≥/radicalbig1
+ǫW2/(2ρ)/bracketrightbig
+≤ρ2e−2ǫW2
+2ρ/(kΛ2)≤ρ2e−ǫW2
+ρkk2
+4W2=ρ2e−kǫ/(4ρ),
+where the last inequality follows because Λ= max p,q∈S||˜φ(p)−˜φ(q)||≤2W/ksince for any p∈Swe have
+||˜φ(p)||=W/k. By settingδ/2≥ρ2e−kǫ/(4ρ), we can solve for k=O((ρ/ǫ)log(ρ/δ)). The final bound
+follows usingρ=O((1/ǫ2)log(n/δ))in Lemma 4.1.
+Again using the feature map summarized by Lemma 4.1 we can com pute the norm in feature space in O(ρk)
+time, after sampling k=O((1/ǫ3)log(n/δ)log((1/ǫδ)logn))points from Pand withρ=O((1/ǫ2)log(n/δ)).
+Theorem A.1. We can compute a vector ˆS=˜Φ(S)∈/Rbbbρin time O(n+(1/ǫ5)log2(n/δ)log2((1/ǫδ)logn)))such
+that/bardblex/bardblex˜Φ(P)−ˆS/bardblex/bardblex2≤ǫW2with probability at least 1−δ.
+A.2 NP-hardness of Optimal Coreset Construction
+In general, the problem of finding a fixed-size subset that clo sely approximates the kernel norm is NP-hard.
+Definition A.1 (KERNEL NORM).Given a set of points P={Xi}n
+i=1, a kernel function K, parameter k and a
+threshold value t, determine if there is a subset of points S⊂Psuch that|S|=k and DK(S,P)≤t.
+Theorem A.2. KERNEL NORMis NP-hard, even in the case where k =n/2and t=0.
+Proof. To prove this, we apply a reduction from P ARTITION : given a set Q={xi}n
+i=1of integers with sum to/summationtextn
+i=1=2m, determine if there is a subset adding to exactly m. Our reduction transforms Qinto a set of points
+P={x′
+i}n
+i=1which has subset Sof size k=n/2 such that||S−P||≤tif and only if Qhas a partition of two
+subsets Q1andQ2of size n/2 such that the sum of integers in each is m.
+Letc=1
+n/summationtext
+xi=2m/nandx′
+i=xi−cand let t=0. Let the kernel function Kbe an identity kernel defined
+K(a,b)=〈a,b〉, where the feature map is defined φ(a)=a. This defines the reduction.
+Lets=(n/k)/summationtext
+x′
+i∈Sx′
+iandp=/summationtext
+x′
+i∈Px′
+i. Note that p=0 by definition. Since we have an identity kernel so
+φ(a)=a,DK(S,P)=/bardbls−p/bardbl. Thus there exists an Sthat satisfies DK(S,P)≤0 if and only if s=0.
+We now need to show that scan equal 0 if and only if there exists a subset Q1⊂Qof size n/2 such that its
+sum is m. We can write
+s=n
+k/summationdisplay
+x′
+i∈Sx′
+i=2/summationdisplay
+x′
+i∈S/parenleftbig
+xi−2m
+n/parenrightbig
+=−2m+2/summationdisplay
+x′
+i∈Sxi.
+Thus s=0 if and only if/summationtext
+x′
+i∈Sxi=m. SinceSmust map to a subset Q1⊂Q, where x′
+i∈Simplies xi∈Q1,
+then s=0 holds if and only if there is a subset Q1⊂Qsuch that/summationtext
+xi∈Q1=m. This would define a valid
+partition, and it completes the proof.
+19
\ No newline at end of file
diff --git a/1001.0592.txt b/1001.0592.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7af203c7c7118185683c67756c3f59a04067a13c
--- /dev/null
+++ b/1001.0592.txt
@@ -0,0 +1,1853 @@
+Information Asymmetries in Pay-Per-Bid Auctions
+How Swoopo Makes Bank
+John W. Byers
+Boston University
+byers@cs.bu.eduMichael Mitzenmachery
+Harvard University
+michaelm@eecs.harvard.eduGeorgios Zervasz
+Boston University
+zg@bu.edu
+November 5, 2018
+Abstract
+Innovative auction methods can be exploited to increase prots, with Shubik's fa-
+mous \dollar auction" [12] perhaps being the most widely known example. Recently,
+some mainstream e-commerce web sites have apparently achieved the same end on a
+much broader scale, by using \pay-per-bid" auctions to sell items, from video games
+to bars of gold. In these auctions, bidders incur a cost for placing each bid in addition
+to (or sometimes in lieu of) the winner's nal purchase cost. Thus even when a win-
+ner's purchase cost is a small fraction of the item's intrinsic value, the auctioneer can
+still prot handsomely from the bid fees. Our work provides novel analyses for these
+auctions, based on both modeling and datasets derived from auctions at Swoopo.com,
+the leading pay-per-bid auction site. While previous modeling work predicts prot-free
+equilibria, we analyze the impact of information asymmetry broadly, as well as Swoopo
+features such as bidpacks and the Swoop It Now option specically, to quantify the
+eects of imperfect information in these auctions. We nd that even small asymmetries
+across players (cheaper bids, better estimates of other players' intent, dierent valua-
+tions of items, committed players willing to play \chicken") can increase the auction
+duration well beyond that predicted by previous work and thus skew the auctioneer's
+prot disproportionately. Finally, we discuss our ndings in the context of a dataset of
+thousands of live auctions we observed on Swoopo, which enables us also to examine
+behavioral factors, such as the power of aggressive bidding. Ultimately, our ndings
+show that even with fully rational players, if players overlook or are unaware any of
+these factors, the result is outsized prots for pay-per-bid auctioneers.
+John W. Byers is supported in part by NSF grant CNS-0520166. Computer Science Dept., Boston
+University, Boston, MA 02215 & Adverplex Inc., Cambridge, MA 02139.
+ySupported in part by NSF grants CCF-0915922, CNS-0721491, and CCF-0634923, and in part by
+research grants from Yahoo!, Google, and Cisco Systems. School of Engineering and Applied Sciences,
+Harvard University, Cambridge, MA 02138.
+zComputer Science Dept., Boston University, Boston, MA 02215 & Adverplex Inc., Cambridge, MA
+02139.
+1arXiv:1001.0592v3  [cs.GT]  30 Mar 2010Contents
+1 Introduction 4
+1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
+1.2 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
+2 A Symmetric Pay-Per-Bid Model 10
+3 Asymmetries in the Perceived Number of Bidders 12
+3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
+3.2 Analysis for Fixed-Price Auctions . . . . . . . . . . . . . . . . . . . . . . . . 13
+3.3 Analysis for Ascending-Price Auctions . . . . . . . . . . . . . . . . . . . . . 15
+3.4 Incorporating Uncertainty Into Population Estimates . . . . . . . . . . . . . 16
+4 Extended Models for Asymmetries 18
+4.1 Modeling Information Asymmetries . . . . . . . . . . . . . . . . . . . . . . . 18
+4.2 A Markov Chain Approach for Analyzing Asymmetries . . . . . . . . . . . 19
+5 Asymmetries in Bid Fees 21
+5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
+5.2 Fixed-Price Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
+5.3 A Single Player with Access to Cheap Bids ( k= 1) . . . . . . . . . . . . . . 26
+5.4 Ascending-Price Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
+6 Varying Object Valuations 28
+6.1 Fixed-Price Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
+7 Hidden Information: Collusion and Shill Bidders 30
+7.1 Collusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
+7.1.1 Fixed-Price Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
+7.1.2 Ascending-Price Auctions . . . . . . . . . . . . . . . . . . . . . . . . 33
+7.2 Shill Bidding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
+8 Playing Chicken and the Impact of Aggression 36
+8.1 Swoop It Now and Chicken . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
+8.2 Aggression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
+9 Conclusions and Future Work 41
+10 Acknowledgments 43
+A Computing the Steady State of a Markov Chain 44
+2B Dataset Description 46
+B.1 Outcomes Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
+B.2 Trace Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
+31 Introduction
+One of the more interesting commercial web sites to appear recently from the standpoint
+of computational economics is Swoopo. Swoopo runs an auction website, using a nontradi-
+tional \pay-per-bid" auction format. Although we provide a more formal description later,
+the basic framework is easy to describe. As with standard eBay auctions, pay-per-bid
+auctions for items begin at a reserve price (generally 0), and have an associated countdown
+clock. When a player places a bid, the current auction price is incremented by a xed
+amount, and some additional time is added to the clock. When the clock expires, the last
+bidder must purchase the item at the nal auction price. The pay-per-bid enhancement is
+that each time a player increments the price and becomes the current leader of the auc-
+tion, they must pay a bid fee. At Swoopo.com, the price increment typically ranges from
+1 cent to 24 cents, and placing a bid typically costs 60 cents. An important variation is
+a xed-price auction, where the winner obtains the right to buy the item at a xed price
+p. Whenp= 0, such an auction is referred to as a 100% o auction; in this case Swoopo
+derives all of its revenue from the bids.
+While there are other web sites using similar auctions, Swoopo has become the leader
+in this area, and recently has inspired multiple papers that attempt to analyze the char-
+acteristics of the Swoopo auction [2, 4, 9]. These models share the same basic framework,
+based on assuming players decide whether or not to bid in a risk-neutral fashion, which we
+explain in detail in Section 2. Some of these papers then go further, and attempt to justify
+their model by analyzing data from monitoring Swoopo auctions.
+One of the most interesting things about the nearly identical analyses undertaken thus
+far is that the simple versions of the model predict negligible prots for Swoopo, in that the
+expected revenue matches the value of the item sold. This fails to match the results from
+datasets studied in these papers, other anecdotal evidence [14,15], as well as hard evidence
+we compiled from a dataset comprising over one hundred thousand auction outcomes that
+we collected, which show Swoopo making dramatic prots (see Figure 1).1Some sugges-
+tions in previous work have been made to account for this, including the relaxation of the
+assumption that players are risk-neutral [9], or the addition of a regret cost to model the
+impact of sunk costs [2].
+In this paper, we take the previous analysis as a starting point, but we focus on whether
+there are intrinsic aspects of the pay-per-bid auction framework that can derive prot from
+even rational, risk-neutral players who correctly model sunk costs. Specically, previous
+work has modeled the game as inherently symmetric, with all players adopting identi-
+1We estimate Swoopo's net prots for an auction by summing up estimated bid fees plus the nal
+purchase price and subtracting the stated retail value for the item. While we know the nal purchase price
+exactly, we can only estimate bid fees, as some bidders have access to discounted bids, for reasons we discuss
+in detail in Section 5. We therefore overestimate bid fees by assuming all bidders pay the standard bid fee.
+On the other hand, Swoopo's stated retail value for the item tends to be above market rate, so by using the
+stated retail value for our calculation we underestimate Swoopo's prot. We do not suggest these eects
+simply cancel each other out, but we believe our estimate provides a suitable ballpark gure.
+40 200 400 600 800 10000.00.20.40.60.81.0
+Profit margin (%)Auctions (%)Overall profit margin: 85.97%Figure 1: Empirical estimate of prot margins for 114,628 Swoopo ascending-price auc-
+tions. The overall prot margin is computed by summing the prot across all auctions and
+dividing it by the total cost.
+cal randomized strategies. However, there are natural asymmetries that can arise in the
+Swoopo auction, particularly asymmetries in information . A rational player's strategy re-
+volves around his assessment of the probability of winning the auction outright by bidding,
+based on the current bid, the number of bidders, the bid fee, and the value of the item. Let
+us focus on one of these parameters, the number of players n. Although previous models
+assume that nis known to all players in advance, in practice, there is no way to know
+exactly how many players are actively participating or monitoring the auction at any time.
+In Section 3, we show that even small asymmetries in beliefs about the number of active
+players can lead to dramatic changes in overall auction revenue, and these changes can
+grow sharply as the estimates vary from the true number of players. We also quantify a
+similar eect due to uncertain beliefs, as opposed to asymmetry across beliefs.
+As a related example, previous analyses assume that all players both pay the same fee
+to place a bid in an auction and ascribe an identical value to an item. The latter is clearly
+not the case, and we discuss the implications in Section 6. Less obviously, not all bidders on
+Swoopo are paying the same price per bid, since one item available for auction at Swoopo is
+a bidpack, which is eectively an option to make a xed number of bids in future auctions
+for free (\freebids"). Players that win bidpacks at a discount on face value therefore have
+the power to make bids at a cheaper price than other players. As many players may not
+factor in this eect (and are unlikely in any case to be able to accurately estimate either
+5the number of players using freebids, or the nominal bid fee those players pay), this again
+creates an information gap that tends to lead to increased prots for Swoopo. In this
+case, players using less expensive bids have a decided advantage, as we show in Section 5.
+Pushing this to the extreme, we have the case of shill bidders, who bid on behalf of the
+auctioneer, and can be modeled as bidders who incur no cost to bid (but also never claim
+an item). While we do not suggest shill bidders are present in online pay-per-bid auctions,
+our analysis in Section 7.2 nevertheless shows that they would have a striking impact on
+protability.
+Previous analyses also assume that no players have any available side information that
+they can exploit. However, two ways in which this side information may be present are
+when coalitions of bidders form, a case we address in Section 7.1, and the setting in which
+a Swoopo bidder is determined to buy the item, which we analyze in Section 8. This
+interesting second case is facilitated by Swoopo's Swoop It Now feature, introduced in the
+US around July 2009, which enables any bidder to use all of their bid fees in an auction as
+a down payment for the item from Swoopo at full retail value, up until one hour after the
+auction concludes. Auctions containing one or more committed players who will use the
+Swoop It Now feature if needed ultimately reduce to games of chicken [10], an exceptionally
+protable outcome for Swoopo that closely resembles auctions involving shill bidders.
+Finally, our framework allows us to examine other interesting aspects of these types of
+auctions that are dicult to model analytically, but which can be studied via empirical
+observations. One question that we are particularly interested in is whether certain bidder
+behavior, such as aggressive bidding and bullying, is eective, as earlier work speculates [2].
+In Section 8.2, we formulate a new denition of bidder aggression, and demonstrate that
+bidders range widely across the aggression spectrum. While aggressive bidders win more
+often, analysis of our dataset shows that the most aggressive bidders contribute the lion's
+share of prots to Swoopo, and successful strategies are most frequently associated with
+below-average aggression.
+We believe that modeling and analyzing these information asymmetries are interesting
+in their own right, although we also argue that they provide a more realistic framework
+and possible explanation for Swoopo prots than previous work. Indeed, our work reveals
+the previously hidden complexity of this auction process in the real-world setting.
+We emphasize that while we provide data in an attempt to justify these additions to
+the model, in contrast to previous work, we eschew eorts to t existing data to our model
+to parameterize and validate it. At a high level, we feel that at this stage validation
+attempts based on data tting are unwarranted. Indeed, the attempt is reminiscent of
+similar attempts in the area of power laws in computer networks, where after many initial
+works attempted to justify their model of power law growth by showing it t the data, it
+has been widely argued that tting data is an improper approach for validation, as many
+models with very dierent characteristics lead to power law behaviors [6,7,16].
+61.1 Related Work
+Several recent working papers have studied pay-per-bid auctions [2, 4, 9]. While there are
+some dierences among the papers, they all utilize the same basic framework, which is
+based on nding an equilibrium behavior for the players of the auction. We describe this
+framework in Section 2, and use it as a starting point. The key feature of this framework
+from our standpoint is that it treats the players as behaving symmetrically, with full infor-
+mation. Unsurprisingly, in such a setting the expected prot for Swoopo is theoretically
+zero.
+Our key deviation from past work is to consider asymmetries inherent in such auctions,
+with a particular focus on information asymmetry. Information asymmetry broadly refers
+to situations where one party has better information than the others, and has become a
+key concept in economics, with thousands of papers on the topic. The pioneering work
+of Akerlof [1], Spence [13], and Stiglitz [11], for which the authors received a Nobel Prize
+in 2001, established the area. Typical examples of information asymmetry include insider
+trading, used-car sales, and insurance. Interestingly, the study of information asymmetry in
+auctions appears signicantly less studied. We believe that our analysis of Swoopo auctions
+provides a simple, natural example of the potential eects of information asymmetry (as
+well as other asymmetries) in an auction setting, and as such may be valuable beyond
+the analysis itself. Indeed, our rst example shows how information asymmetry about a
+basic parameter of an auction { the number of participants { can signicantly aect its
+protability.
+1.2 Datasets
+Where appropriate, we motivate our work or provide evidence for our results via data from
+Swoopo auctions. We have collected two datasets. One dataset is based on information
+published directly by Swoopo, which contains limited information about an auction. Infor-
+mation provided includes basic features such as the product description, the retail price,
+the nal auction price, the bid fee, the price increment, and so on. This dataset covers
+over 121,419 auctions. We refer to this as the Outcomes dataset.
+Our second dataset is based on traces of live auctions that we have ourselves recorded
+using our own recording infrastructure. Our traces include the same information from
+the Swoopo auctions as well as detailed bidding information for each auction, specically
+the time and the player associated with each bid. This dataset spans 7,353 auctions and
+2,541,332 bids. We refer to this as the Trace dataset. Our methodology to collect bidding
+information entailed continuous monitoring of Swoopo auctions. We probe Swoopo accord-
+ing to a varying probing interval that is described in detail in Appendix B; in particular,
+when the auction clock is at less than 2 minutes, we probe at least once a second. Swoopo
+responds with a list of up to ten tuples of the form ( username;bidnumber ) indicating the
+players that placed a bid since our previous probe and the order in which they did so. Often
+7this list would include more than one tuple. In these cases we ascribe the same timestamp
+to all of these bids; given the high probing frequency, this is a reasonable approximation.
+A more signicant limitation from our methodology arises when more then ten players bid
+between successive probes. In these cases, Swoopo responded with just the ten latest bids.
+In particular this happened when more than ten players were using BidButlers, automatic
+bidding agents provided by the Swoopo interface, to bid at a given level. Overall, we cap-
+tured every bid from 4,328 auctions. The remaining 3,025 auctions had a total of 491,360
+missing bids; we did not consider these in our study.
+As two examples of the kind of information that can be derived from our dataset
+we present Figures 2 and 3, derived from the Outcomes dataset. The former presents
+Swoopo's monthly prot margin for a set of what we refer to as regular auctions, grouped
+by price increment. Regular auctions exclude \NailBiter" auctions which do not allow the
+use of BidButlers, beginner auctions which are for players who have not won an auction
+previously, and xed-price auctions. The latter gure displays Swoopo's prot margin by
+item, for items that have been auctioned o at least 200 times in regular auctions.
+MonthProfit margin
+2008−08 2008−11 2009−02 2009−05 2009−08 2009−11−50%0%50%100% 200%
+●●●
+●●
+●●●●
+●●●●●
+11
+1
+11
+11
+11
+1
+111
+1●●●
+●●222
+22●5●●●●●●
+666
+666
+●●●●●●111
+111 222
+222●●
+●●●●●●●
+●●● 11
+1111111
+11155
+5555555
+555
+●●
+●
+●●22
+2
+2244
+4
+44
+Figure 2: Swoopo monthly prot margins by price increment for regular auctions from
+theOutcomes dataset. The dotted line represents the prot margin across all price
+increments.
+More details regarding the datasets can be found in Appendix B. These datasets are
+publicly available.2
+2Available at: http://cs-people.bu.edu/zg/swoopo.html . Please contact zg@bu.edu for questions
+regarding this dataset.
+8−100% 0% 100% 200% 300% 400%Kingston Micro SD Reader & 2GB SD CardSanDisk Cruzer Micro 16 GB USB 2.0 Flash DriveKaspersky Internet Security 2009Corsair Voyager Mini 4 GB USB FlashTranscend TS8GJFV10 USB 2.0 8GB Flash DriveTranscend JetFlash V10 USB 2.0 16GB Flash DriveMicrosoft Windows 7 Home PremiumLogitech G7 Laser Cordless MouseLogitech Cordless Wave Keyboard and MouseGarmin nüvi 770 GPS NavigatorKingston Flash 16GB DataTraveler DT100 (Black)Kingston DataTraveler 32 GB USB 2.0 Flash DriveTranscend 16 GB SDHC Class 6 Flash Memory CardPanasonic KX−TG8232B Cordless Phone SystemApple MacBook Pro MB991LL/A 13.3−Inch LaptopPanasonic KX−TG9332T Cordless Phone SystemWii Play (includes Wii Remote Control)Acer AS8730−6951 18.4−Inch LaptopMicrosoft Xbox 360 console 60 GB HDMIWii Sports Resort with Wii MotionPlusCanon Digital Rebel XSi + 18−55mm Lens BlackWii Music (Wii)Samsung NC10 10.2" Mini−NotebookDualShock 3 PS3 controller (PS3)TomTom Go 930TApple iPod touch 32 GB (new generation)Sony PSP 3000Sony PS3 160 GB + Uncharted: Drakes FortuneSony PlayStation 3 Slim Console (120GB)Mario Kart with Wheel (Nintendo Wii)Nintendo DSi Console (Blue)Canon EOS Rebel T1i 15MP DSLR + 18−55mm lensesNintendo DSi Console (Black)Nikon D90 12.3 MP DSLR Camera with 18−105mm KitNokia N97 Unlocked Phone1 Ounce Gold Bar (31.10g)Apple iPod touch 8 GB (new generation)Sony PlayStation 3 80 GBSony VAIO VGN−AW110J/H 18.4" Laptop (Vista)Apple MacBook MB466LL/A 13.3−Inch LaptopWii | Nintendo Console + Wii SportsDS | Nintendo DS Lite (silver)Acer AS8730−6918 18.4−Inch LaptopGeorge Foreman Deep FryerDS | Nintendo DS Lite BlackWii Fit | Nintendo WiiFit + Balance BoardDS | Nintendo DS Lite WhiteDS | Nintendo DS Lite (Crimson Red and Black)DS | Nintendo DS Lite (Cobalt Blue and Black)300 Bids VoucherWii Play with Wii Remote50 Bids Voucher
+−61.6%−47.7%−35.8%−34.2% −0.5%  4.8%  8.2% 11.1% 16.5% 31.1% 41.4% 44.6% 52.3% 53.4% 59.5% 61.5% 66.1% 67.7% 70.8% 74.4% 81.6% 83.4% 89.8% 90.6% 94.6% 96.7% 99.5%105.3%111.4%118.4%118.5%120.7%124.4%126.8%129.0%134.3%138.7%143.9%144.5%149.4%155.7%158.8%166.1%167.2%180.6%186.1%193.3%193.8%211.4%294.8%364.8%373.8%Figure 3: Prot margin for items that have been auctioned o at least 200 times in regular
+auctions from the Outcomes dataset.
+92 A Symmetric Pay-Per-Bid Model
+We start with a basic model and analysis of Swoopo auctions from previous work, following
+the notation and framework of [9], although we note that essentially equivalent analyses
+have also appeared in other work [2,4]. This serves to provide background and context for
+our work.
+We consider an auction for an item with an objective value of vto all players. There are
+nplayers throughout the auction. The initial price of the item is 0. In the ascending-price
+version of the auction, when a player places a bid, he pays an up-front cost of bdollars, and
+the price is incremented by sdollars.3The auction has an associated countdown clock;
+time is added to the clock when a player bids to allow other players the opportunity to bid
+again. When an auction terminates, the last bidder pays the current price of the item and
+receives the item. In a variant called a xed-price auction, the winner buys the item for
+a xed price p; bids still cost bdollars but there is no price increment. When p= 0, this
+is called a 100%-o auction. In our analysis, we simplify players' strategies by removing
+the impact of timing.4Instead of bidding at a given time, players choose to bid based on
+the current price; if multiple players choose to bid based on the current price, we generally
+break ties by assuming a random bidder bids rst. A player that chooses not to bid at
+some price may bid later at a higher price.
+The basic formulation for analyzing this game is that a player who makes the qth bid
+is bettingbthan no future player will bid. Let jbe the probability that somebody makes
+thejth bid (given that j1 previous bids have been made). Then the expected payo
+for the player that makes the qth bid is (vsq)(1q+1); a player will only bid if this
+payo is non-negative. Note that when q > Qbvb
+scit is clear that no rational player
+will bid, as the item price plus bid fee exceeds the value. For convenience in the analysis
+we will assume thatvb
+sis an integer, to avoid technical issues when this does not hold
+(see [2] for a discussion); this assumption ensures that a player that makes the Qth bid is
+indierent to the outcome (the expected payo is 0). In the xed-price variant, the payo
+is (vp)(1q+1); as long as v>p , bidding may occur.
+The equilibrium behavior is found by determining the probability that a player should
+bid so that the expected payo is zero whenever qQ, leaving the players indierent as
+to the choice of whether to bid or not to bid. (Alternative equilibria that are not germane
+to our analyses are discussed in [2].) Hence the indierence condition is given by
+b= (vsq)(1q+1);
+or
+q+1= 1b
+vsq
+3The case of descending-price models could equally be modeled and studied using our techniques.
+4We study aspects of timing empirically in Section 8.2, where we consider the repercussions of aggressive
+bidding in live auctions.
+10in the ascending-price auction, and
+q= 1b
+vp
+at all steps in the xed-price auction.
+In what follows it is helpful to let qbe the probability that each player chooses to
+make theqth bid given that the ( q1)st bid has been made and that the player is not the
+current leader. Note that by symmetry each player bids with the same probability. Hence,
+forq>1, for ascending-price auctions we must have
+1q= (1q)n1
+q= 1(1q)1=(n1)
+q= 1b
+vs(q1)1=(n1)
+:
+Similarly, we have
+q= 1b
+vp1=(n1)
+for the xed-price auction.
+We point out that the rst bid is a special case, since at that point there is no leader.
+To maintain consistency, we want the indierence condition to hold for the rst bid; that
+is, players still bid such that their expected prot is zero. This requires a simple change,
+since at the rst bid there are nplayers who might bid instead of n1, giving for the
+ascending-price auction
+q=8
+<
+:1b
+v
1
+nforq= 1,
+1
+b
+vs(q1)1
+n1forq>1.(1)
+Similar equations hold for the xed-price variant:
+q=8
+><
+>:1
+b
+vp1
+nforq= 1,
+1
+b
+vp1
+n1forq>1.(2)
+The expected revenue for the auction can easily be calculated directly using the above
+quantities. However, we suggest a simple argument (that can be formalized in various
+ways, such as by dening an appropriate martingale) that demonstrates that Swoopo's
+expected revenue is vif there is at least one bid, and zero if no player bids. (A similar
+argument appears in [2].) First note that in auctions where there is at least one bid,
+an item of value vis transferred to some player at the end of the auction. Also, by the
+11indierence condition, the expected gain to the player that places any bid is zero. (Think
+of a bidbas counterbalanced by the auctioneer putting an expected value bat risk.)
+Therefore, by linearity of expectations, the auctioneer recoups a sum of payments equal to
+vin expectation over the course of the auction, conditioned on there being at least one bid.
+The probability that no player bids is (1 1)nby denition of 1, and thus the expected
+revenue isv(1(11)n) =vb.
+To be clear, in what follows, we will always consider revenue conditioned on the auction
+having had at least one bid, since otherwise, the auction is essentially a non-operation for
+the auctioneer. We call such auctions successful .
+As we noted earlier, in practice the revenue Swoopo earns per auction is much larger
+thanvon average. To help explain this, we consider the assumptions behind this model.
+This model assumes that the key parameters n,v, andbare the same for each player and
+known to all. These assumptions seem hard to justify in practice: players may misestimate
+the total population, some may have access to cheaper bids, or they may each value the
+auctioned object dierently. We provide some general extensions to the model for these
+types of variations in Section 4, but before that, we motivate our general considerations by
+considering the specic example where the number of players nis not known.
+3 Asymmetries in the Perceived Number of Bidders
+3.1 Motivation
+The analysis of Section 2 makes the assumption that the number of players is xed and
+known throughout. This assumption has been questioned in previous work; for example,
+in [9], they propose a simple alternative to the standard model where participants enter
+and leave the auction over time. However, even in this variation, the expected number
+of players at each time step is known and the distribution is assumed to be Poisson, so
+that the end result is a small variation on the previous analysis. Here we take a dierent
+approach and consider the original model but without the assumption that every player
+has the same estimate of n, the number of players in the game.
+Before diving into the analysis, we provide some motivating data from our datasets.
+During an auction, Swoopo provides some limited information regarding the number of
+players participating in the auction. Specically, it provides a list of the bidders that have
+been active over the last 15 minute period. Analysis of our Trace dataset suggests this is
+insucient information for determining the number of players in the auction, and in fact
+we suspect it can lead players to signicantly underestimate the number of other players
+in the auction. This will prove to have dramatic impact on the analysis, and in particular
+on the expected revenue to Swoopo.
+Following Swoopo, we dene an active bidder as someone who has bid in the last fteen
+minutes. Using our Trace dataset, we observed each auction at one minute intervals, and
+at each time instant we computed the number of active bidders as a percentage of the total
+1212 51020 50100200
+Minutes before end of auctionActive bidders
+0% 20% 40% 60%Figure 4: Percentage of active bidders using a 15-minute sliding window. The x-axis is in
+log scale.
+number of players who ultimately participated in the auction. The points on the curve in
+our plot of these values in Figure 4 can be interpreted as the percentage of all participants
+accounted for in Swoopo's active bidder list as a function of time. Our plot shows that
+auction participation builds to a crescendo at the end of the auction, so a typical report
+ten minutes from the end of the auction only reports 20% of all bidders, and when that
+number doubles in a ve minute span, it still re
ects only 40% of the population. Also,
+note that due to the nature of the auction, there is no xed time at which the auction ends,
+so even bidders making predictions based on past observations are using a certain degree of
+guesswork. We therefore suggest that players relying on active bidder information may well
+be misguided about the size of the playing eld. In the following sections we analytically
+quantify the eect such a misestimation has on Swoopo's expected revenues.
+3.2 Analysis for Fixed-Price Auctions
+For simplicity we begin with the case of xed-price auctions. To initially frame the analysis,
+we further assume that the true number of players is n, but allplayers perceive the number
+of players as nkfor somekin the range [1 ;n2]. In this case, there is still symmetry
+among the players, but they choose to bid based on incorrect information. Following the
+previous analysis, to maintain the indierence condition that the expected revenue for a
+player that bids at each point should be equal to their bid fee, we have ( vp)(1q) =b,
+where now qis the perceived probability that someone else will place the qth bid. As
+13before,
+q= 1b
+vp: (3)
+Again we let qbe the probability that a player chooses to make the qth bid. For q >1
+we have (1q) = (1q)nk1, orq= 1(1q)1
+nk1:
+Crucially,qis not equal to q, the true probability that will make the qth bid. Since
+(1q) equals the probability that nobody makes the qth bid, we have
+1q= (1q)n1
+q= 1
+(1q)1
+nk1n1
+q= 1b
+vpn1
+nk1
+: (4)
+Remember that the above holds for q > 1, as for the rst bid the bidding probabilities
+are slightly dierent, as explained in Section 2. To simplify the math, as an unsuccessful
+auction with zero bids is uninteresting, we assume the rst bid has been placed. Then q
+is the same at all points, so we simply call the value . The probability that the auction
+lasts another rbids, after the rst, is given by r(1). If we let Rbe the revenue from
+a successful auction, we calculate Swoopo's expected revenue as:
+E[R] =b+p+b1X
+r=0rr(1): (5)
+In the simple case where p= 0 the expected revenue is:
+E[R] =bv
+bn1
+nk1: (6)
+Notice that when k= 0, the expected revenue from a successful auction is indeed vas
+had been demonstrated in previous works. Also, as kappears in the exponent of the v=b
+term, even small values of kcan have a signicant eect on the revenue. This dramatic
+impact on revenue as kvaries is depicted for a representative auction for $100 in cash with
+a bid fee of $1 and 50 players in Figure 5(a). These will be our default parameters for
+xed-price auctions throughout this work.
+Conversely, one could consider what happens when players overestimate the population,
+that is to say k<0. As expected, the revenue for Swoopo then shrinks, incurring an overall
+loss as demonstrated in the left half of Figure 5(a). From our standpoint, the key feature of
+this graph is the asymmetry: underestimates of the player population size have signicantly
+larger revenue eects than overestimates. If players tend to underestimate the number of
+players, as our empirical evidence suggests is likely, auction revenues increase dramatically.
+Indeed, even if the average estimated number of players is correct, when there is variation
+14−15 −10 −5 0 5 10 15
+kRevenue ($)
+100 300 500 700
+●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●(a) All players underestimate the population by k.
+Negative values of kstand for overestimates.
+0 5 10 15
+kRevenue ($)
+100200300400500600700
+●●●●●●●●●●●●●●●●
+●Underestimation by k
+Over− and underestimation by k(b) Half the players underestimate the population
+bykand half overestimate it by an equal amount.
+Figure 5: Expected revenue for Swoopo in a successful 100% o auction for two dierent
+population misestimation settings; n= 50,v= 100 andb= 1.
+across estimates, Swoopo gains. For example, we can consider a simple case where half the
+players underestimate the population by kand half overestimate it by the same amount.
+Denote by q;kthe probability with which the former bid and similarly by q;+kthe
+probability with which the latter do. Then we have:
+q;k= 1(1q)1
+nk1; (7)
+q;+k= 1(1q)1
+n+k1: (8)
+Computing the revenue for this mixed case involves more complicated machinery which
+we describe in detail in Section 4. For now, we observe that even though Swoopo has far
+more to gain by pure underestimation of the player population, a mix of overestimation and
+underestimation in equal measures still yields markedly increased revenues, as depicted in
+Figure 5(b). (This can also be seen as a consequence of convexity of the revenue curve as
+the estimate of the number of players varies.)
+Similar analyses can be made for dierent settings.
+3.3 Analysis for Ascending-Price Auctions
+Recall that in an ascending-price auction an auction lasts at most Q=bvb
+scbids sub-
+sequent to the rst. Following reasoning similar to that for xed-price auctions, we can
+express the expected revenue of a successful ascending-price auction where the population
+is underestimated by kby
+E[R] = (b+s) +QX
+q=1(b+s)q(1q+1)qY
+j=1j (9)
+15−40 −20 0 20 40
+kRevenue ($)
+100200300400500
+●●●●●●●●●●●●●●●●●●●●●●●●●Figure 6: Expected revenue as the underestimation error kvaries in an ascending price
+auction;n= 50,v= 100,b= 1, ands= 0:25.
+wherej= 1
+b
+vs(j1)n1
+nk1. Following the same steps as [9], Equation 9 can be
+simplied to yield:
+E[R] = (b+s) +Q1X
+q=00
+@(b+s)q+1Y
+j=1j1
+A: (10)
+In an ascending-price auction, the revenue is trivially upper bounded by ( Q+ 1)(b+s),
+regardless of k. Figure 6 displays the expected revenue for Swoopo in successful auctions as
+the population estimation error kvaries. Here our auction is for $100 in cash with a bid fee
+of $1, 50 players, and a price increment of $0.25. Again, these will be our default parameters
+for ascending-price auctions throughout this work. The two dashed lines correspond to the
+true value of the auctioned item and the trivial upper bound on revenue of ( Q+ 1)(b+s).
+The plot conrms that for large enough nandk, whenq1 throughout most of the
+auction, this upper bound is nearly tight. Therefore, despite the asymptotic leveling o of
+revenues, Swoopo can still capitalize signicantly from misinformed players.
+3.4 Incorporating Uncertainty Into Population Estimates
+Throughout the paper, we focus on the case where players have xed beliefs about relevant
+quantities, such as the number of players participating in the game. However, it is also
+straightforward to extend our techniques to settings where there is underlying uncertainty
+within beliefs, either instead of or in addition to asymmetry across beliefs. As a concrete
+example, we consider the case where players are symmetric and all perceive the population
+size as being drawn from a distribution such that there are iplayers with probability pi.
+We do not focus on beliefs governed by distributions in the rest of the paper, because our
+main thematic points can be made using simpler point beliefs. Moreover, beliefs governed
+by distributions also raise challenging questions, such as how players determine an initial
+16belief distribution and whether they can update their beliefs as the auction proceeds, that
+we do not consider in this work. However, it should intutitively be clear that uncertainty,
+as well as asymmmetry, can lead to situations that benet the auctioneer. We formalize
+one such situation now.
+Consider a xed-price auction auction with nplayers, where the players are symmetric
+and believe that the auction population size is governed by a distribution where the number
+of players is iwith probability zisuch thatP
+iizi=n. In other words, the expectation
+of players' estimates is correct, but they do not know the exact number of players. We
+demonstrate that players bid more frequently because of this uncertainty, leading to extra
+revenue for the auctioneer.
+If all players think that there are nplayers, we have the indierence condition
+b= (vp)(1q);
+where here qis the the perceived probability that any other player bids. Let 1be the
+probability that a specic player who is not the leader bids. As 1 q= (11)n1, we
+have1is the solution to
+(11)n1=b
+vp:
+In the setting where players believe the number of players is governed by a distribution,
+we have the same indierence condition. However, because of the uncertainty, if we let 2
+be the probability that a specic player who is not the leader bids, we nd
+1q=X
+izi(12)i1;
+since the probability that no other player bids is now given by a mixture based on the
+probability distribution. Hence in this setting
+X
+izi(12)i1=b
+vp:
+We now give a convexity argument to show that 21; that is, there is more bidding
+with uncertainty. Consider
+f() =X
+izi(1)i1:
+and
+g() = (1)n1:
+Note that both fandgare decreasing in . Furthermore, (1 )xis a convex function in
+xfor2[0;1]. Hence
+f() =X
+izi(1)i1(1)P
+izi(i1)= (1)n1=g():
+170 5 10 15
+kRevenue ($)
+100200300400500600700
+●●●●●●●●●●●●●●●●
+●Pure underestimation by k
+Over− and underestimation by k with probability 1/2
+Half the players overestimate by k, half underestimateFigure 7: Expected revenue for Swoopo when players are uncertain about the population.
+Now by denition of 2and1,f(2) =b=(vp) =g(1)f(1). Asfis decreasing in
+, fromf(2)f(1), we have21, as desired.
+Figure 7 provides a comparison of three settings, where in all cases the number of players
+isn= 50: 1) all players underestimate the number of players by k; 2) with probability 1 =2,
+all players believe that there are nkplayers, otherwise all believe that there are n+k
+players; and 3) half the players believe there are nkplayers and half believe there are
+n+kplayers. We see that for large misestimates in this setting, uncertainty provides more
+benet for the auctioneer than asymmetry.
+4 Extended Models for Asymmetries
+We now consider a general framework in which full information is not available to all
+players and there are asymmetries. We describe the underlying model in Section 4.1, and
+in Section 4.2, we describe the analysis approach based on Markov chains that we utilize
+throughout the paper. We brie
y discuss a full information model as an issue for future
+work in the conclusion.
+4.1 Modeling Information Asymmetries
+We now consider variations of the auction where there are asymmetries in information. For
+this we need to extend the symmetric model and make a crucial distinction between the
+true values of the game's parameters { v,bandn{ and the way players perceive them.
+(We motivate the misperception of each these parameters in the appropriate sections.) For
+simplicity, we will assume henceforth that there are two groups of players: A, of sizek,
+andB, of sizenk. We can extend our approach to a larger number of groups naturally,
+but with increased complexity.
+Players in group Aperceive the value of the item as vA, the bid fee as bA, and the
+number of participants in the game as nA. DenevB,bBandnBsimilarly. Initially,
+we assume that each player is asymmetry-unaware, i.e. each player assumes all players
+18have identical parameters and thus the groups are not aware of each other. We will be also
+interested in cases where one group is aware of the split and therefore has an advantage over
+the other group. That setting will utilize the same basic structure; we develop it in later
+sections. In both settings, except in the special case of collusion (studied in Section 7.1),
+members of the groups are not aware of the identities of individuals in either group.
+The parameters determine both the perceived and the true probability of the qth bid
+being placed. So, for group A, letA
+qbe the perceived probability that anyone { in either
+group { will place the qthbid. In other words, A
+qis an estimate of qfrom the perspective
+of players in group A. Also, dene A
+qas the true probability that one or more players in
+groupAplaces theqth bid and similarly dene B
+qandB
+qfor groupB. Ifqis the true
+probability of the qth bid being placed then we have 1 q= (1A
+q)(1B
+q).
+Players in group Awill bid according to their perceived indierence condition, which
+for ascending-price auctions is now ( vAs(q1))(1A
+q) =bA, and similarly for group B.
+(Similar derivations hold for xed-price auctions.) Using the fact that 1 A
+q= (1A
+q)n1
+we can easily derive the individual bidding probability for group Aplayers:
+A
+q= 1bA
+vAs(q1) 1
+nA1
+: (11)
+The derivation for group Bplayers is identical. Using the individual bidding probabilities
+we can compute the probability of a bid being placed by anyone in group Aas
+1A
+q= (1A
+q)k(12)
+A
+q= 1bA
+vAs(q1)k
+nA1
+: (13)
+Note that generally q6=A
+q6=B
+q.
+4.2 A Markov Chain Approach for Analyzing Asymmetries
+To compute various quantities of interest when we have asymmetric behaviors requires a
+bit of work, primarily because the probability of a bid at any given time depends in part
+on what group the current auction leader belongs to. In the models we have described,
+however, the auction itself is memoryless, in that, given the leader and the current number
+of bids, the history to reach the current state is unimportant to the future of the auc-
+tion. Essentially all of our models have this form. Hence, we can place these auctions in
+the setting of Markov chains in order to eciently calculate the distribution of auction
+outcomes.
+Specically, the general case for two groups of players can be captured by an absorb-
+ing, time-inhomogeneous Markov chain as shown in Figure 8. (Recall that in a time-
+inhomogeneous Markov chain, the transition probabilities can depend on the current time
+19A B
+WA WBpAB(q)
+pBA(q)pAA(q) pBB(q)
+1pAA(q)pAB(q) 1pBA(q)pBB(q)
+Figure 8: A state-machine for an asymmetric game with two groups of players.
+as well as the state, but not on the history of the chain.) The chain contains four states:
+in stateAa member of the rst group is leading the auction, while in state WAthe auc-
+tion has been won by a member of the rst group. We dene states BandWBsimilarly.
+Observe that WAandWBare absorbing states. Also observe we overload the notation A
+andBto refer both to the sets of players and a state of the Markov chain; this should not
+cause confusion as the meaning will be clear by context. Finally, note that there is no state
+corresponding to the initial setting prior to the rst bid. While we could have a special
+initial state, we instead choose the starting state probabilistically from AorBaccording
+to the appropriate probabilities for the rst bid, recalling that we assume our auction is
+successful.
+We usepAB(q) to denote the transition probability of going from state Ato stateB
+after theqth bid, and similarly we can dene pBA(q),pAA(q), and so on. For example,
+pAB(1) is the transition probability from state Ato stateBwhen one bid has already been
+placed. When considering xed-price auctions, the bidding probabilities, and hence the
+state transition probabilities, are invariant from bid to bid. In this special case the Markov
+chain becomes time-homogeneous, and given the distribution on the initial state we can
+derive analytical expressions for the probability of terminating in state AorB. A good
+description of this approach can be found in many standard texts; we provide a summary
+based on [3] in Appendix A.
+For ascending-price auctions, which are time-inhomogeneous, we resort to numerical
+methods employing simple recurrence relations. This can also be useful to obtain more
+specic information in the case of xed-price auctions (or as an alternative approach for
+calculating various quantities). For example, let PA(q) be the probability of being in state
+Aafterqbids; herePWA(q) represents the probability that a player from Ahas won the
+20auction at some point up to bid q, so thatPWA(q) +PWB(q) becomes 1 for an ascending-
+price auction when qis suciently large and converges to 1 for a xed-price auction in the
+limit asqgoes to innity. Then we have
+PA(q+ 1) =PA(q)pAA(q) +PB(q)pBA(q); (14)
+and other similar recurrences, including
+PWA(q+ 1) =PA(q)pAWA(q) +PWA(q): (15)
+Given these various equations, it is easy to compute quantities such as the expected
+revenue. For example, in an ascending-price auction, assuming all players have a bid fee
+ofb, every time Ais in the lead, he has paid a bid of bfor this, and the price has gone up
+bys. LettingRbe the revenue, we easily nd
+E[R] = (b+s) 
+1 +QX
+i=1PA(i) +QX
+i=1PB(i)!
+: (16)
+Notice that the simple nature of the Markov chain framework allows us to derive all
+the important quantities, such as the expected revenue for Swoopo, directly from the
+appropriate transition probabilities. Hence, in the rest of the paper, we focus on nding
+these probabilities. Where suitable, we explicitly derive corresponding quantities, but in
+other cases we implicitly use this Markov chain characterization and leave the details to
+the reader.
+5 Asymmetries in Bid Fees
+5.1 Motivation
+We now consider asymmetries that arise when players have dierent bid fees. While it seems
+clear that in pay-per-bid auctions players may fail to properly estimate the population of
+bidders, it is less clear why the cost per bid may vary among players, so we now motivate
+this direction.
+Among other items oered on auction at Swoopo are bidpacks . A bidpack, as its name
+suggests, is a set of prepaid bids. As of this writing they come in ve sizes: 40, 75, 150,
+400, and 1000 bids. Their corresponding retail values are $24, $45, $90, $240 and $600, but
+if won in an auction they can potentially be had for a substantial discount. Players who
+win bidpack auctions and participate in later auctions can eectively enjoy lower bidding
+fees compared to other participants, generally without the other participants' knowledge.
+To provide evidence that bidpacks can lead to varying bid fees, we look to our data.
+Based on our Outcomes dataset, we derive the average cost of a bidpack as a percentage
+of the nominal retail cost for winners of bidpack auctions in Figure 9(a); this includes
+21Percentage of retail priceFrequency
+05001000150020002500
+0% 50% 100% 150% 200% 250% 300%Mean: 34.72%(a) Winners' total cost of bidpacks as a percentage
+of retail price ( Outcomes dataset).
+Percentage of retail priceFrequency
+0 50 100 150
+0% 50% 100% 150% 200% 250% 300%Mean: 45.09%(b) Winners' total cost of bidpacks as a percentage
+of retail price when accounting also for lost auctions
+(Trace dataset).
+Figure 9: Measuring the acquisition cost of bidpacks
+the winners' bid costs and the price they paid. As can be seen, bids can be had at a
+substantial discount, with the winner's total cost at just over 1 =3 of the retail cost on
+average. With the Outcomes dataset, however, we can only determine the cost of bids
+made by the auction winners; this clearly underestimates the costs to players who may
+also lose auctions for bidpacks, raising their average cost per bid. To attempt to account
+for this, we similarly examine our Trace dataset, and compute the average cost for any
+winner of a bidpack auction, including the cost for bidpack auctions where the player has
+lost. These results appear in Figure 9(b). Naturally, this leads to a smaller estimated
+average discount, although the discount is still over 1 =2 of the retail cost. While this may
+still be an underestimate of bidpack costs (as we cannot take into account auctions we have
+not captured, and our results are biased towards winners) it strongly suggests that winners
+of bidpack auctions enjoy a substantial discount in bid fees when applying those bids to
+other auctions.
+Cheaper bids are also available through seasonal promotions that Swoopo conducts. For
+example, Swoopo has had promotions oering more bids for the same price. A screenshot
+showing such a promotion is given in Figure 10. A further cause for variation in bid fees
+relates to the remarkable fact that Swoopo auctions take place with bidders bidding in
+dierent currencies. An example of this is shown in Section 6.
+Between bidpack auctions, seasonal promotions, and currency dierences, the possibil-
+ity that players are paying varying bid fees becomes reality. In the following we analyze
+and quantify the impact this has on Swoopo auctions.
+5.2 Fixed-Price Auctions
+We rst consider the simpler case of xed-price auctions with price p. We assume that the
+nbidders are divided in two groups AandB, of sizekandnkrespectively. We will
+22Figure 10: A recent seasonal bidpack promotion.
+assume that k2; the case where k= 1 can be handled similarly but the case structure of
+the analysis is slightly dierent. Group Aincurs a bid fee of bAwhile group Bincurs a bid
+fee ofbBwithbA<bB. In context, we may presume that group Ais the set of bidders who
+are bidding at a discount whereas group Bis the set of players who are charged regular
+bid fees. In what follows we also assume Aplayers are aware of the two groups while B
+players perceive everyone as belonging to the same group as themselves. This creates an
+information asymmetry. We believe this choice of model is natural; we suspect many (less
+sophisticated) players may not recognize that others are obtaining cheaper bids. It also
+provides an example of how our Markov chain approach of Section 4.2 applies to such a
+setting.
+Intuitively, players in A, having access to lowered bid fees, would outlast players in
+groupBin expectation. Much less obviously, it also turns out that the expected length
+of the auction has nodependence on the bid fee of the Bplayers { the auction length is
+solely determined by the bid fees incurred by the Aplayers.
+LetA
+qbe the collective probability that some player in group Amakes theqth bid,
+and similarly dene B
+q. Then the probability that no player makes the qth bid is:
+(1q) = (1A
+q)(1B
+q) (17)
+whereqis dened to be the true collective probability that anyone, in either group, bids.
+Next, consider the game from the point of view of Bplayers. Remember that, according
+to them, everyone belongs to a single group incurring the same bid fee. Dene B
+qas the
+perceived probability that anyone, in either group, makes the qth bid according to the
+23information available to Bplayers. From the indierence condition for Bplayers we have:
+(vp)(1B
+q) =bB
+B
+q= 1bB
+vp: (18)
+We derive the true probability B
+qthat aBplayer bids as:
+(1B
+q) = (1B
+q)n1
+B
+q= 1(1B
+q)1
+n1
+B
+q= 1bB
+vp 1
+n1
+: (19)
+We can then write the probability of group Bbidding as:
+1B
+q=(
+(1B
+q)nkif groupAis leading,
+(1B
+q)nk1if groupBis leading;
+which after manipulation becomes:
+B
+q=8
+><
+>:1
+bB
+vpnk
+n1if groupAis leading,
+1
+bB
+vpnk1
+n1if groupBis leading.(20)
+Remember that B
+qis the true collective probability with which group Bplayers bid.
+Furthermore, notice that players in group Aare aware of this probability.
+Assuming the leader before the qth bid was from group Band using the indierence
+condition for group Awe can derive an expression for A
+q:
+(vp)(1A
+q)(1B
+q) =bA(21)
+A
+q= 1bA
+(vp)(1Bq)(22)
+A
+q= 1bA
+vpvp
+bBnk1
+n1
+(23)
+A
+q= 1bA
+bBbB
+vpk
+n1
+: (24)
+The derivation for group Aleading is similar, leading to:
+A
+q=8
+><
+>:1bA
+bB
+bB
+vpk1
+n1if groupAis leading,
+1bA
+bB
+bB
+vpk
+n1if groupBis leading.(25)
+24bARevenue ($)
+0.05 0.20 0.40 0.60 0.80 1.001003005007009001100●
+●
+●
+●
+●
+●●●●●●●●●●●●●●●●k=5
+k=25
+k=45(a) Revenue as the price of cheap bids varies.
+bAGroup A advantage
+0.05 0.20 0.40 0.60 0.80 1.000x1x2x3x4x5x6x●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●●●●●●●●●●●k=5
+k=25
+k=45 (b) Relative likelihood of a specic player in group
+Awinning the auction when compared with a spe-
+cic player in group B.
+Figure 11: A xed-priced auction with kplayers provisioned with cheap bids; n= 50,
+v= 100 andbB= 1.
+Next, using Equation 17 we can derive an expression for q, the true probability that a
+qth bid is placed:
+q= 1bA
+vp; (26)
+which holds irrespectively of who is the current leader. It seems counterintuitive that nei-
+ther the number of Bplayers nor their bid fee play any role in determining the probability
+q. However, this is similar to the original setting where all players pay the same bid fee,
+andqwas independent of n.
+We can also write an expression for A
+q:
+A
+q=8
+><
+>:1
+bA
+bB1
+k1
+bB
+vp1
+n1if groupAis leading,
+1
+bA
+bB1
+k
+bB
+vp1
+n1if groupBis leading.(27)
+Having computed the individual bid probabilities for each player we can compute
+Swoopo's expected revenue in successful auctions using the framework we developed in
+Section 4.2. Consider our usual xed-price auction with n= 50,b= 1 andp= 0. Some
+bidders have access to a discounted bid fee bA, while the rest pay the regular rate of $1
+per bid. Figure 11(a) displays Swoopo's excepted revenue as the fee bAcharged to group
+Abidders for bidding varies.
+The expected revenue per successful Swoopo auction actually increases, superlinearly,
+in the gap between bid fees. This is somewhat surprising, given that the amount of revenue
+from each bid from group Aisdecreasing . We provide a high-level explanation of what
+appears to be happening. Group Bbidders not only pay full price for their bids, but are
+also participating in an auction that tends to last substantially longer than they expect.
+25Furthermore, they bid with higher probability than they would if they had complete in-
+formation. Consequently Swoopo's revenues explode. Of course our analysis hinges on the
+assumption that group Bbidders never realize that they have been dealt a losing hand; re-
+call for xed-price auctions the underlying bidding behavior is memoryless. But, during an
+actual auction, Swoopo does not reveal whether a player makes a discounted bid, making
+it hard for players to assess how level the playing eld is, and thus also making our model
+plausible. (We leave extensions to a setting where players change their beliefs about other
+auction players as the auction proceeds, and change their strategy accordingly, as future
+work.)
+Also of interest is the advantage gained by a specic player having access to cheap bids.
+Using the same example as above, Figure 11(b) displays the relative likelihood of a specic
+Aplayer winning the auction compared to a specic Bplayer winning the auction as a
+function of the discounted bid fee. Observe that there is a clear synergy here: provisioning
+of cheaper bids helps the players who receive them and the auctioneer alike.
+5.3 A Single Player with Access to Cheap Bids (k= 1)
+The analysis above depends on the size of the Agroup of bidders being larger than one. If
+k= 1, then Equations 20 and 25 do not hold any more. For completeness we provide the
+details for this case. (The uninterested reader may skip ahead.) In this case we have:
+B
+q=8
+<
+:1
+bB
+vp
+if groupAis leading,
+1
+bB
+vpn2
+n1if groupBis leading,(28)
+and
+A
+q=8
+<
+:0 if group Ais leading,
+1bA
+bB
+bB
+vp1
+n1if groupBis leading.(29)
+Using equation 17 we can calculate the probability that a bid is placed as:
+q=(
+1bB
+vpif groupAis leading,
+1bA
+vpif groupBis leading.(30)
+Notice that, unlike the case where k > 1, the probability of the auction ending depends
+on which group is currently leading the auction. This aects the duration of the auction
+and consequently Swoopo's revenues. Auctions are substantially shorter when there is
+only one player with access to cheap bids, and similarly yield much less revenue. Figure
+12(a) displays Swoopo's revenue when k= 1, all other parameters being unchanged. The
+relative advantage of a sole player with cheap bids increases dramatically, however, an
+eect demonstrated in Figure 12(b).
+26bARevenue ($)
+0.05 0.20 0.40 0.60 0.80 1.00100 200 300●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●●●●●●●●(a) Swoopo's revenue as the price of cheap bids
+varies.
+bAGroup A advantage
+0.05 0.20 0.40 0.60 0.80 1.000x50x100x150x200x●
+●
+●
+●
+●
+●●●●●●●●●●●●●●●(b) Relative likelihood of the single player in group
+Awinning the auction when compared with a spe-
+cic player in group B.
+Figure 12: A xed-priced auction with a single player provisioned with cheap bids; n= 50,
+v= 100 andbB= 1.
+5.4 Ascending-Price Auctions
+Using the methods of Section 4.1, we can also compute the expected revenue of an ascending-
+price auction with varying bid fees. Sample results using a price increment of s= 0:25
+are shown in Figure 13(a). Figure 13(b) displays the relative likelihood of a specic player
+in groupAwinning the auction compared to a specic player in group Bas the price of
+the cheaper bid varies. The results are similar, although as one might expect, the gains to
+Swoopo are smaller in this setting.
+bARevenue ($)
+0.05 0.20 0.40 0.60 0.80 1.00050100150200250300
+●
+●
+●
+●
+●
+●
+●
+●
+●●●●●●●●●●●●●k=5
+k=25
+k=45
+(a) Swoopo's revenue as the price of cheap bids
+varies.
+bAGroup A advantage
+0.05 0.20 0.40 0.60 0.80 1.000x2x4x6x8x10x12x●
+●
+●
+●
+●
+●
+●
+●●●●●●●●●●●●●●k=5
+k=25
+k=45(b) Relative likelihood of a specic player in group
+Awinning the auction when compared with a spe-
+cic player in group B.
+Figure 13: Ascending-price auctions with varying bid fees; n= 50,v= 100,bB= 1 and
+s= 0:25.
+276 Varying Object Valuations
+We now consider the impact of varying the perceived value vof the auctioned item among
+the players. There are multiple motivations for this consideration. A rst motivation is that
+people may simply value the item dierently. In particular, Swoopo provides a nominal
+retail value for the auction item, which is generally signicantly higher than the purchase
+price one could easily obtain elsewhere (such as on Amazon). There may therefore be na ve
+players who base their valuation on the nominal retail price and more sophisticated players
+who know the actual retail price of an item. (This motivation is touched on brie
y in [2],
+although our analysis is markedly dierent.)
+Another more surprising motivation is that Swoopo runs its auctions simultaneously
+in multiple countries. That is, the participants in an auction often correspond to players
+in dierent countries, bidding on the local version of the Swoopo site. This necessarily
+introduces small inconsistencies due to the use of dierent currencies, so bids as well as
+valuations are likely to dier in some small degree. (One dollar is not a xed whole
+number of euros.) But larger variations often occur because the auction corresponds to
+dierent items in dierent countries. This is not without justication { a certain type
+of TV, or monitor, or other electronic device generally would only work in its country of
+origin, so close substitutes have to be found for dierent countries. Figure 14 shows an
+actual example, with screenshots of an auction with the same auction ID and the same
+bidders at the US and German Swoopo sites, with dierent currencies and dierent items
+with dierent valuations up for auction. This strongly suggests that the case of dierent
+valuations is present in real auctions.
+6.1 Fixed-Price Auctions
+Motivated by the example of the same auction in dierent countries, we consider the
+following model. There are two groups of bidders. Group Aof sizekvalues the auctioned
+object atvwhere2(0;1). GroupBof sizenkvalues the same object at v.
+Furthermore, the members of each group perceive everyone as having the same valuation
+as themselves. The number of players, n, is globally known.
+The indierence condition for group Ais (1A
+q)(vp) =bwhereA
+qis again the
+perceived probability that anyone makes the qth bid according to players in group A. This
+implies that
+A
+q= 1b
+vp: (31)
+Similarly, for group Bwe have
+B
+q= 1b
+vp: (32)
+We know that (1 A
+q) = (1A
+q)n1, and similarly for group B. This implies that
+28(a) US auction.
+(b) German auction.
+Figure 14: The same auction as seen from the US and German versions of the Swoopo
+website.
+29ααRevenue ($)
+0.05 0.50 1.00 1.50 2.00050100 150 200
+●●●●●●●●●●●●●●●●●●●●
+●k=5
+k=25
+k=45Figure 15: Revenue for Swoopo as the valuation parameter varies. Shown for three
+dierent values of kandn= 50,v= 100,b= 1 in a xed-price auction.
+thetrue probabilities with which individual players in the two groups bid are
+A
+q= 1b
+vp 1
+n1
+; (33)
+B
+q= 1b
+vp 1
+n1
+: (34)
+Furthermore, the true collective probabilities of a bid being placed by either group are
+A
+q=(
+1(b
+vp)k
+n1 if the current leader is in group A,
+1(b
+vp)k1
+n1 if the current leader is in group B,(35)
+B
+q=(
+1(b
+vp)nk1
+n1 if the current leader is in group A,
+1(b
+vp)nk
+n1 if the current leader is in group B,(36)
+Then the auction termination probability is (1 q) = (1A
+q)(1B
+q).
+Figure 15 displays Swoopo's revenue as the valuation parameter varies from 5% to
+200% for three dierent values of kwithn= 50 andb= 1. The results are natural;
+the more players that overvalue an item, the better for Swoopo. Unlike some of the
+other variations we have studied, the eects on the revenue are naturally bounded: if the
+maximum valuation of an item among all players is 2 v, the expected revenue in this model
+will not exceed 2 v. Similar results hold for ascending-price auctions.
+7 Hidden Information: Collusion and Shill Bidders
+The analysis for varying the number of players gives us a simple framework for analyzing
+two other standard situations with information asymmetry, namely where certain players
+30have hidden information. In the rst setting, we consider the case where a subset of players
+collude to form a bidding coalition. In the second, we consider shill bidders , or bidders in
+the employ of the auctioneer.
+7.1 Collusion
+In the setting of collusion, the natural approach is for the members of the coalition to agree
+to not bid against each other, so that if one of them is currently leading the auction the
+others bid with zero probability. The agents can either behave independently, or we can
+think of the coalition as a single player or agent acting on behalf of the coalition members.
+A benet of colluding here is that the coalition can potentially intimidate other players
+from bidding, by making it appear that there are more players in the auction than there
+actually are, thereby reducing the probability that other players bid. We wish to quantify
+the advantage gained by this form of collusion in terms of the size of the coalition.
+7.1.1 Fixed-Price Auctions
+For our analysis, we assume that there is a group Aofkplayers in a coalition, and a group
+Bofnkother players not in the coalition. To these nkplayers, there appear to
+benidentical players in the auction. Again, the coalition players bid as usual, provided
+a coalition member is not the leader. Proceeds and expenses are shared equally between
+coalition members.
+Non-coalition members bid according to their perceived indierence condition:
+B
+q= 1b
+vp: (37)
+This yields
+1B
+q= (1B
+q)n1(38)
+B
+q= 1b
+vp 1
+n1
+: (39)
+From this we can derive the true probability of a bid by group B:
+B
+q=8
+><
+>:1
+b
+vpnk
+n1if groupAis leading,
+1
+b
+vpnk1
+n1if groupBis leading.(40)
+We observe that players in group B, just as a consequence of overestimating the total
+population, bid less frequently than they should. This fact alone is enough for the coalition
+of players in group Ato gain an edge in winning the auction.
+31Next, we look at the indierence condition for a player in group Awhen someone from
+groupBis leading the auction, as otherwise group Aplayers do not bid. Remember that
+both costs and proceeds are shared and that players in Ahave full information:
+1
+kb=1
+k(vp)(1q) (41)
+b= (vp)(1q) (42)
+b= (vp)(1A
+q)(1B
+q) (43)
+A
+q= 1b
+(vp)(1Bq)(44)
+A
+q= 1b
+vpk
+n1
+: (45)
+We can derive the individual bidding probabilities for coalition members at equilibrium.
+Recall as we stated earlier when group Bis leading the auction group Aplayers act inde-
+pendently. Hence
+A
+q=8
+<
+:0 if group Ais leading,
+1
+b
+vp1
+n1if groupBis leading.(46)
+Finally, using the fact that 1 q= (1A
+q)(1B
+q), we can derive the following
+expression for the probability of a bid being placed by either group:
+q=8
+<
+:1
+b
+vpnk
+n1if groupAis leading,
+1b
+vpif groupBis leading.(47)
+The increased chances of group Awinning the auction are apparent, as the auction is more
+likely to end when Aleads.
+Equations 39 and 46 are nearly sucient to determine the probabilities for our Markov
+chain analysis. The only remaining issue regards our choice of tie-breaking rule. In our
+description thus far, members of the coalition behave independently when bidding, and
+hence if several members of the coalition bid, we would expect each to have a chance
+to become the leader. We refer to this as the coalition having many bidders . We could
+instead imagine the coalition acting essentially as a single player controlling many identities
+and only selecting a single one to use at each opportunity to bid (albeit with an upwards
+adjusted probability of bidding, i.e., A
+qinstead ofA
+q). In this case, the coalition would
+be less likely to win in case of ties.
+One would expect two consequences of collusion. First, a coalition of kbidders should
+have more than ktimes the probability of a non-colluding bidder to win. Second, the
+320 10 20 30 40 50
+Coalition size, kRevenue ($)
+20406080100●●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●●●●●●●●●●Many bidders
+Single bidder(a) Revenue for Swoopo when collusion occurs.
+0 10 20 30 40 50
+Coalition size, kCoalition advantage
+1x200x400x600x800x
+●●●●●●●●●●●●●●●●●●●●●●●●●Many bidders
+Single bidder (b) Relative likelihood of the colluding group win-
+ning the auction vs. any specic outsider.
+Figure 16: Fixed-price auctions with a coalition of size k;n= 50,v= 100 andb= 1.
+overestimation of the actual player population should negatively impact Swoopo's revenues.
+We conrm both of these consequences empirically.
+Figure 16(a) displays the revenue Swoopo can expect in the presence of a coalition
+of sizekfor both tie-breaking rules. As can be seen, revenue declines signicantly when
+large coalitions are present. Figure 16(b) displays the relative likelihood of the coalition
+winning the auction compared to any particular outsider. Even small groups of colluding
+players can gain a very large advantage in winning the auction, superlinear in the size of
+the coalition, oering a signicant incentive to collude.
+7.1.2 Ascending-Price Auctions
+When considering ascending-price auctions we are faced with a time-inhomogeneous Markov
+chain as the bid probabilities vary in the price of the item. A closed-form solution is not
+as easily attainable as for time-homogeneous, xed-price auctions. Instead we resort to
+the numerical evaluation methods we have described in Section 4.1. We consider our usual
+auction for $100 in cash with a bid fee of $1 and a price increment of 25 cents. Figures 17(a)
+and 17(b) respectively display the the revenue for Swoopo and the relative likelihood of a
+colluding bidder winning as the size of the coalition grows. The results are quite similar
+in this setting to the xed-price case; a coalition can dramatically lower prots, and have
+a competitive advantage that grows superlinearly in their size.
+7.2 Shill Bidding
+A further consideration is the eect of shill bidders, or bidders under the employ of the
+auction site who attempt to drive up revenue by bidding in order to prevent auctions from
+terminating early. This is not a theoretical problem; pay-per-bid auction sites other than
+Swoopo have been accused of using shill bidding [8]. In the working paper [9], shill bidders
+330 10 20 30 40 50
+Coalition size, kRevenue ($)
+020406080100●
+●
+●
+●
+●
+●
+●
+●
+●
+●●(a) Revenue for Swoopo as the size kof the coalition
+grows in an ascending price auction.
+0 10 20 30 40 50
+Coalition size, kCoalition advantage
+1x200x400x600x800x
+● ●●●●●●●●●●(b) Relative likelihood of the colluding group win-
+ning the auction vs. any specic outsider.
+Figure 17: Ascending-price auctions with a coalition of size k;n= 50,v= 100,b= 1 and
+s= 0:25.
+were considered, but it was assumed that they would behave equivalently to other players
+in the auction. This assumption was necessary to maintain the symmetry of the analysis,
+and was justied by the argument that if shill bidders behave exactly as other players,
+they would be more dicult to detect. We argue that sites employing shill bidding may
+be willing to shoulder the increased detection risk associated with increased shill bidding
+as long as it is accompanied by increased prot.
+There are several possible ways of introducing shill bidders. Here we focus on the
+following natural one: we dene a ( ;L)-shill as one that enters the auction with probability
+and bids with probability one at each opportunity when they are not the leader until L
+bids have been made (in total, by all players), at which stage he drops out of the auction.
+Such an approach provides useful tradeos; increasing orLincreases the probability of
+detection, but oers the potential for increased prot.
+To analyze shill bidding we employ our usual framework. When there is no shill (with
+probability 1), we assume we have a standard auction with nplayers; with probability
+the shill enters and the auction has n+ 1 players. To analyze auctions with a shill,
+we separate the bidders into two groups: group Aconsists of the lone shill, and group B
+consists of the nlegitimate players who are not informed of the shill's presence.
+This information is sucient to determine the transition probabilities in order to use
+our Markov chain analysis. Recall that shill bidders produce no revenue for the auctioneer,
+so the expected revenue is determined by the expected number of times a legitimate player
+is the leader. For convenience we adopt our usual tie-breaking rule, so the leader is picked
+uniformly at random from the players, including the shill, who decide to make a bid at
+each step. (One could imagine other rules { such as the shill is never picked in case of a
+tie, to maximize bid fees obtained.)
+Before presenting some example data, we provide some intuition. We rst consider a
+34020406080100
+LProfit ($)
+4080120160200240280320360●●●●●● ● ● ●●ρρ=5%
+ρρ=15%
+ρρ=25%(a) Fixed-price auction, p= 0.
+020406080100
+LProfit ($)
+4080120160200240280320360●●●● ● ● ● ● ●●ρρ=5%
+ρρ=15%
+ρρ=25% (b) Ascending-price auction.
+Figure 18: Expected prot for Swoopo in the presence of a single-identity ( ;L)-shill;
+n= 50,v= 100,b= 1 ands= 0:25.
+xed-price auction with a shill bidder. The auction can be thought of as proceeding in two
+stages: in the rst stage that lasts for up to Lbids the shill participates, and in the second
+stage the shill abstains. It should be clear that if the auction reaches the second stage
+the expected revenue is nearly v; the only issue is that there are only nlegitimate players,
+but they each behave as though there are n+ 1 players. For large enough nthe revenue
+remains close to vin the second stage, and the auctioneer's marginal prot from using a
+shill is therefore essentially equal to the bid fees from other players in the rst stage of the
+auction. The case of ascending-price auctions is slightly more interesting due to the fact
+that legitimate players bid with decreasing probability as the price of the item goes up. As
+a consequence, it is more likely for the shill to win the item as Lincreases, in which case
+the auctioneer only earns revenue from the bid fees.
+In order to see the eect of shills more clearly, rather than plot the per-auction revenue
+with shill bidders, we instead plot the per-auction prot . We do this for two reasons. First,
+since a symmetric, full-information auction results in zero expected prot for the auctioneer
+in our model, all prot in our plots can be attributed to the presence of the shill. Second,
+in this setting, there is some chance that the shill will win the auction, in which case the
+auctioneer's revenue is all prot, a fact not well captured by a revenue plot. Figures 18(a)
+and 18(b) display the expected prot for Swoopo in the presence of a ( ;L)-shill for xed
+and ascending-price auctions respectively. Shill participation in ascending-price auctions
+has diminishing returns with L, which is to be expected; even though the shill is forcibly
+extending the expected length of the auction, as the price of the item goes up, legitimate
+players become less willing to participate.
+As an example of another possible shill model, we have also considered a variation
+where the shill player uses two distinct identities, so that he can bid with a second identity
+when his rst identity is the leader, and vice versa. In this way, the shill can guarantee
+thatLbids will occur before dropping out of the auction, at the expense of introducing
+35020406080100
+LProfit ($)
+4080120160200240280320360●●●●●●●●●●ρρ=5%
+ρρ=15%
+ρρ=25%(a) Fixed-price auction, p= 0.
+020406080100
+LProfit ($)
+4080120160200240280320360●●●●●●●●●●ρρ=5%
+ρρ=15%
+ρρ=25% (b) Ascending-price auction.
+Figure 19: Expected prot for Swoopo in the presence of a double-identity ( ;L)-shill;
+n= 50,v= 100,b= 1 ands= 0:25.
+another perceived player. This approach seems riskier in terms of likelihood of detection,
+but leads to additional revenue, since it prevents the auction from ending early with the
+shill winning and thereby eectively provides more opportunities for other players to bid.
+While the results for the double-identity shill plotted in Figure 19 are similar to those
+depicted for the single-identity shill, the prot gains due to extra bids are more substantial
+and the plots exhibit only minimal diminishing returns.
+8 Playing Chicken and the Impact of Aggression
+In this section we address a recently added feature to Swoopo's interface, Swoop It Now,
+that appears to have not been analyzed previously. This feature has the potential to sig-
+nicantly change the dynamics of Swoopo auctions. Our suggestion is that this feature
+may lead to a subclass of players whose strategy makes some Swoopo auctions resemble
+the game of chicken [10], in contrast to the Markovian games we have modeled in previous
+sections.5In games of chicken, it is generally understood that it can be useful for players
+to signal their intentions, explicitly or implicitly, to other players, in order to cause them
+to give up and allow the signaling player to win. A natural signaling approach in the
+timed auction context is to bid both frequently and quickly after another player bids. This
+bidding strategy has been noted previously, in the work of [2], where the author dubs this
+\bidding aggressively" and nds that aggressive bids have higher expected prot. Here
+we undertake an independent study, making several new contributions. Besides presenting
+how this behavior can be viewed as a signaling mechanism for a game of chicken embed-
+ded in Swoopo, we provide a novel and natural denition of aggression for pay-per-bid
+5One might argue that the resemblance is more to a war of attrition auction [5] than to chicken; we nd
+the chicken nomenclature easier to use, but the underlying idea is the same.
+36auctions. Then, using our trace data, we analyze auctions for signs of aggressiveness, and
+estimate how aggressiveness correlates with winning auctions and protability for players.
+A surprising nding is that both too little and too much aggression appear to be losing
+strategies. We emphasize that here our analysis is based less on formal models of player
+behavior than in previous sections; our analysis is therefore necessarily more speculative
+and worthy of future study.
+8.1 Swoop It Now and Chicken
+Swoopo recently added the Swoop It Now option to auctions on its site, which gives each
+player the ability to purchase the item at a given price even if one loses the auction.6That
+is, in many auctions, Swoopo provides a nominal retail value for the auction item, call it
+r. At the end of the auction, a player who has spent a total of on bids during the course
+of the auction can buy the item at a price of r; that is, the bids are transformed from
+otherwise unrecoverable sunk costs to a partial payment for the auction item. As argued in
+previous work [2,9], the nominal retail price provided by Swoopo is generally signicantly
+higher than the price for which one could buy the item online, and is surely well above
+Swoopo's price for most items.
+Unfortunately we do not currently know how often Swoop It Now is used; to our knowl-
+edge such information is neither given by Swoopo, nor derivable from any data Swoopo
+makes available. We suspect the feature is often overlooked, or that players are not in-
+terested, given the high nominal retail value. On the other hand, after a large losing
+investment in an auction, this option may become attractive to certain players.
+Let us consider the behavior this additional feature introduces and its consequences in
+two settings: the case where only one committed player is willing to exercise this option,
+and the case where multiple players potentially are. Our assumption here is that r=v,
+wherevis a common value of the item for all players and >1. Our key nding is that
+when multiple players consider taking advantage of this option as a backstop, the game
+becomes a variant of chicken.
+Let us rst suppose that a single player has the opportunity to buy at the price r,
+including the amount they spend on bids. This player may believe the odds of winning
+the auction early are sucient to keep bidding at every possible step, nding that the
+expected gain from winning early dominates the maximum possible loss of ( 1)v. This
+player will therefore continue to bid until either winning the auction or spending enough
+so that it is cheaper to buy the item using Swoop It Now than to win it at the current
+auction price; the other n1 players will play as usual. In eect, in this situation, the
+player is essentially equivalent to a shill bidder, except here the player keeps bidding until
+spending a certain amount, rather than until a certain number of bids have been made.
+The approach of Section 7.2 can be applied with minor variations. (In such settings, the
+6While we do not know the exact date of its introduction, based on various Web postings, deployment
+on the US site appears to have occurred around July 2009, before we began taking traces of auctions.
+371.1 1.2 1.3 1.4 1.50100200300400
+ααProfit ($)
+● ● ● ● ● ● ● ● ● ●●Committed player
+Swoopo(a) Fixed-price auction.
+1.1 1.2 1.3 1.4 1.50100200300400
+ααProfit ($)
+● ● ● ● ● ● ● ● ● ●●Committed player
+Swoopo (b) Ascending-price auction.
+Figure 20: Prot for Swoopo and a committed player as varies;n= 50,v= 100,b= 1
+ands= 0:25.
+Markov chain state space must be expanded, so this player can keep the amount spent thus
+far as part of the state; this is easy to accomplish.) Also, note in this case Swoopo always
+sells at least one copy of the item, as opposed to the setting with a shill bidder. The main
+outcome, naturally, is that having a player who intends to use the Swoop It Now feature
+increases Swoopo's prot by prolonging the auction, assuming the presence of this player
+does not change other players' strategies. Figure 20 displays the prot earned by the single
+committed participant as well as Swoopo. Here, as when studying shill bidders, it makes
+more sense to consider the prot for Swoopo. In the case where the player uses the Swoop
+It Now feature, if they have bid so far, they will have to pay an additional side payment
+ofrto complete the purchase. Also, we decrement Swoopo's prot by an additional v
+to account for the transfer of a second item to auction winner. The higher is the longer
+the committed player stays in the game. As a result, prot for the committed player is
+decreasing in while the reverse holds for Swoopo. As usual, for ascending-price auctions
+Swoopo's prot is bounded - no one will bid after Q+ 1 rounds. On the other hand,
+if the player who is committed to the Swoop It Now purchase can signal their intention
+through aggressive bidding so that other players drop out of the auction, the end result is
+signicantly less prot for Swoopo, as the auction will end early.
+In the case of two (or more) players who are interested in using the Swoop It Now
+feature, the resulting game instead resembles the game-theoretic version of chicken. For
+convenience we consider a xed-price auction with a price of 0. Suppose that two players
+plan to continue to bid until either obtaining the item or spending rin bids and then using
+the Swoop It Now feature. If both exhaust their bids, they will both lose ( 1)vin value.
+But if, instead, one of them backs o, allowing the other player to win, that player will
+lose only what they have bid so far { call this { and the other player will purchase the
+item at a discount { call their gain 
v, on average. Table 8.1 displays the chicken game in
+the standard payo matrix notation.
+38Quit Play Till End
+Quit;;
v
+Play Till End 
v;v;v
+Table 1: Payouts for chicken strategies
+Obviously, we have simplied things considerably in this description; there may be
+more than two players willing to use the Swoop It Now feature, there may be other players
+involved, and it may be unclear which players intend to treat the auction as a game of
+chicken. This is clearly a subject in need of further study. However, we do show that the
+Swoop It Now feature, by keeping individual losses bounded, does embed the potential for
+games of chicken to erupt within Swoopo auctions. As aggressive bidding is a natural way
+to signal intent in this setting, (and may be a sound tactic in its own right), we turn to a
+study of aggression, making use of our Trace dataset.
+8.2 Aggression
+In earlier work [2], Augenblick has suggested that aggressive bidding, including bidding
+immediately after another player has bid and bidding frequently in the same auction, leads
+to higher expected value for a player. His analysis is based on individual bids rather
+than bidders; that is, he considers for each bid how the time since the previous bid and the
+number of bids by the bidder for that item correlate to the expected prot, using regression
+techniques.
+We adopt a dierent approach, by looking instead at how aggressive bidding aects the
+protability of a player within an auction, and by providing a single aggression metric to
+measure the aggressiveness of a strategy. As a bidder may vary his strategy across auctions,
+we dene aggressiveness in the context of a given auction. We believe that considering the
+eects of aggressiveness at the level of player protability oers important insights as it
+views the merits of an aggressive strategy holistically. (Also, it is clear that having bid
+many times previously will aect the expected prot of a single bid, simply because having
+bid many times means the auction has gone on longer, which aects the current probability
+the auction terminates.)
+We dene an aggressive strategy as one which consists of placing many bids in rapid
+succession to preceding bids. Specically, let the response time for a bid be the number of
+seconds since the preceding bid. Aggression should be inversely proportional to response
+time and proportional to the number of bids a bidder places within an auction. Hence we
+dene the aggression of a bidder in a given auction as:
+Aggression =Number of bids
+Average response time (seconds/bid): (48)
+To investigate whether aggressive bidding is a successful strategy we look at the traces
+390 2 4 6 8 100.00.20.40.60.81.0
+xPr((Aggression>>x))All
+In the black
+In the red
+Won auction(a) Empirical CCDF of aggression.
+0 10000 20000 30000 40000
+RankCumulative player profit ($)
+−40000 −20000 010000
+010203040
+Aggression ((bids2sec))Cumulative profit
+Aggression (b) Cumulative prot vs. aggression rank.
+Figure 21: Characterizing he protability of aggressive strategies.
+of 3,026 complete (no missing bids) \NailBiter" auctions (Swoopo auctions which do not
+permit the use of automated bids by a \BidButler") in our Trace dataset. Figure 21(a)
+displays the empirical CCDF of aggression across all bidders and for three classes of strate-
+gies:
+\Won auction": strategies that resulted in winning the auction.
+\In the black": strategies which resulted in winning the auction protably.
+\In the red": strategies which resulted in losing money irrespective of whether the
+auction was won or lost.
+(For Figure 21(a), when calculating the prot of a bidder, we assume a xed bid cost of
+60 cents, as we cannot determine the true cost. This could aect our interpretation of the
+results.)
+Our rst observation is that aggression follows a highly skewed distribution: the major-
+ity of players display little aggression, while a small number of players are highly aggressive.
+Also, not surprisingly, those players winning the auction were bidding much more aggres-
+sively than others. More interestingly, we see that successful players, i.e., those who not
+only won the auction, but did so protably, are more aggressive than average, but less
+aggressive than those who win auctions. Arguably, aggression is successful in moderation.
+Figure 21(b) provides more insight into this latter point. For all bidder-auction pairs
+in our dataset, we compute the aggression and protability of each outcome, and rank
+these outcomes by aggression (least aggressive rst). We then plot the cumulative prot
+for all outcomes through a given rank with dark shading. For reference, we also plot the
+aggression of bidders at a given rank using light shading and the scale depicted on the
+right-hand side of the plot. We see that successful strategies are mostly concentrated at
+aggression ranks lower than average. More interestingly, a fact not evident in Figure 21(a),
+the highly aggressive players are responsible for most of Swoopo's prots.
+40Aggressive Number of Auction revenue Mean winner
+bidders auctions (as % of retail price) prot margin
+0 1,699 62% 77%
+1 493 135% 51%
+2 834 246% 26%
+Table 2: Evidence of chicken
+In Table 8.2, we give empirical evidence that Swoopo prots are strongly and posi-
+tively impacted by the presence of multiple aggressive players, dened as players with an
+aggression level of at least 3 bids2/sec. Strikingly, when two or more aggressive players are
+present, Swoopo's per-auction revenues are in excess of 2.4 times the stated retail price of
+the good. At the same time, and as expected, as the number of aggressive players in an
+auction increases the prot margin of the auction winner decreases. The fourth column of
+Table 8.2 displays the mean of winner prot margins.
+Finally, having provided empirical evidence regarding aggressive behavior, we now re-
+visit the question of whether games of chicken are also taking place within Swoopo. To do
+that we turn to the Trace dataset and look at the 3,026 \NailBiter" auctions for evidence
+ofduels : auctions culminating in long sequences of back-and-forth bidding between two
+opponents. We nd that 9% of all auctions culminated in a duel lasting at least 10 bids, 5%
+lasted at least 20 bids, and 1% lasted at least 50 bids. The longest duel we observed was
+201 bids long and somewhat humorously took place between users Cikcik andThedduell .
+We believe this provides further evidence that at least some auctions are becoming essen-
+tially games of chicken, and reiterate our supposition that aggressive bidding is used as a
+signaling method in such settings.
+9 Conclusions and Future Work
+Swoopo provides a fascinating case study in how new, non-trivial auction mechanisms
+perform in real-world situations, and it also provides a focal point for developing the
+theory of auction mechanisms in the context of human behaviors. Here we have focused on
+the key issue of asymmetry, and in particular, how various manifestations of information
+asymmetry may be responsible in large part for the signicant prots Swoopo appears to
+enjoy today. At the same time, we have also shown that the protability of these auctions is
+potentially fragile, especially in cases where signaling by committed players willing to play
+a game of chicken or collusion between players can end the auction early. In these settings,
+information asymmetry can potentially reduce the prots enjoyed by the auctioneer.
+There are clearly many interesting directions to follow from here. One area we have
+started to examine is asymmetric models of pay-per-bid auctions with full information.
+For example, players could have diering bid fees or valuations of the item, but with these
+41fees and valuations known in advance to all players. Although a full information setting
+may not be as immediately practical as that considered in our work, asymmetries expose
+a richer set of issues than the symmetric, full information case, and have not yet been
+considered in any depth in previous work.
+Let us consider what such a model might look like. For convenience, let us consider a
+xed-price auction. Suppose that there are players A1;A2;:::;An, with player Aihaving
+bid feebAiand valuevAi. LetAibe the probability that Aishould bid when Aiis not in
+the lead in equilibrium. The indierence condition for player Aiis now given simply by:
+bAi= (vAip)0
+@1Y
+j6=iAj1
+A;
+orY
+j6=iAj= 1bAi
+(vAip):
+It is helpful to substitute i= lnAiandi= ln
+1bAi
+(vAip)
+. Then we can readily solve
+the family of simple linear equations:
+X
+j6=iAj=i:
+This assumes an equilibrium based on the indierence condition (and, for example, that
+Ai6= 0). It would be interesting to consider under what conditions such equilibria exist,
+as well as if they are the only equilibria. In some cases, the auction might for example
+reduce to a game of chicken, in which case there may be multiple equilibria. For the case
+of ascending-price auctions, the calculations are more challenging: one can use backward
+induction, nding the probabilities for the Qth bid and then calculating downward.
+It is worth noting that with varying bids, in a full information auction where all players
+ascribe the same value vto an item and use a strategy based on the indierence condition,
+Swoopo's expected revenue for successful auctions remains vfollowing the same argument
+as given in Section 2. This highlights the key role of information asymmetry in our result
+showing that varying bids can lead to large prots for Swoopo. Further comparisons with
+a full information model should be similarly enlightening.
+Another broad topic for future work regards more extensive study of user behavior on
+Swoopo. While our study considers the impact of one natural tactic, aggression, in Swoopo
+auctions, there are many others that could also be considered. The impact of timing has
+generally been abstracted away, but it is probably important to user behavior, as our study
+of aggression suggests. More understanding of the impact of timing considerations appears
+important for further study. In particular, perhaps there are timing-based or other active
+signaling mechanisms that a strategic player could leverage to maximize expected return
+42in these auctions. A possible direction related to signaling is the issue of learning. Can
+players dynamically change their beliefs about underlying auction parameters, such as the
+bid fee other players are charged or the intentions of other players, based on how the
+auction proceeds in order to improve their performance? As another example, one feature
+that we have not yet studied is the impact of bidders who use automatic bidding agents,
+such as BidButlers: how do they in
uence the auction, and can such bidders be detected?
+And nally, there remains the thorny problem of attempting to quantify the impact of how
+specic auction characteristics we have considered { misestimates of the number of players,
+varying bid fees, varying valuations of items, the ability for players to use the BidButler
+and the Swoop It Now features { on real-world protability. Such an eort would likely
+require more detailed information regarding bids currently available only to Swoopo, and
+better models of user behavior.
+10 Acknowledgments
+Michael Mitzenmacher would like to thank Maher Said for several helpful discussions re-
+garding Swoopo. Georgios Zervas would like to thank Azer Bestavros for helpful discussions
+regarding collusion.
+References
+[1] G. A. Akerlof. The market for \lemons": Quality uncertainty and the market mecha-
+nism. The Quarterly Journal of Economics , 84(3):488{500, 1970.
+[2] N. Augenblick. Consumer and Producer Behavior in the Market for Penny Auctions:
+A Theoretical and Empirical Analysis. Unpublished manuscript. Available at www.
+stanford.edu/ ~ned789 , 2009.
+[3] C. M. Grinstead and J. L. Snell. Introduction to Probability . American Mathematical
+Society, 1997.
+[4] T. Hinnosaar. Penny Auctions. Unpublished manuscript at http://toomas.
+hinnosaar.net/ , 2009.
+[5] V. Krishna and J. Morgan. An analysis of the war of attrition and the all-pay auction.
+Journal of Economic Theory , 72(2):343{362, 1997.
+[6] M. Mitzenmacher. A brief history of generative models for power law and lognormal
+distributions. Internet mathematics , 1(2):226{251, 2004.
+[7] M. Mitzenmacher. Editorial: The future of power law research. Internet Mathematics ,
+2(4):525{534, 2005.
+43[8]http://www.pennyauctionwatch.com/tag/shills/ .
+[9] B. C. Platt, J. Price, and H. Tappen. Pay-to-Bid Auctions. Unpublished manuscript.
+Available at http://econ.byu.edu/Faculty/Platt , 2009.
+[10] A. Rapoport and A. Chammah. The game of chicken. American Behavioral Scientist ,
+10, 1966.
+[11] M. Rothschild and J. Stiglitz. Equilibrium in competitive insurance markets: An essay
+on the economics of imperfect information. The Quarterly Journal of Economics ,
+90(4):629{649, 1976.
+[12] M. Shubik. The dollar auction game: a paradox in noncooperative behavior and
+escalation. Journal of Con
ict Resolution , 15(1):109{111, March 1971.
+[13] M. Spence. Job market signaling. The Quarterly Journal of Economics , 87(3):355{374,
+1973.
+[14] B. Stone. Sites ask users to spend to save. New York Times , August 17, 2009.
+[15] R. H. Thaler. Paying a price for the thrill of the hunt. New York Times , November
+15, 2009.
+[16] W. Willinger, D. Alderson, and J. Doyle. Mathematics and the internet: A source
+of enormous confusion and great potential. Notices of the American Mathematical
+Society , 56(5):586{599, 2009.
+A Computing the Steady State of a Markov Chain
+Letf(k;n;p) be the probability mass function of the binomial distribution where nis the
+number of trials, kis the number of successes and pit the probability of success. The
+transition probabilities in Figure 8, presented in Section 4.2, are given by:
+pAA=k1X
+i=1nkX
+j=0f(i;k1;A
+q)f(j;nk;B
+q)i
+i+j(49)
+pAB=k1X
+i=0nkX
+j=1f(i;k1;A
+q)f(j;k1;B
+q)j
+i+j(50)
+pBB=kX
+i=0nk1X
+j=1f(i;k;A
+q)f(j;nk1;B
+q)j
+i+j(51)
+pBA=kX
+i=1nk1X
+j=0f(i;k;A
+q)f(j;nk1;B
+q)i
+i+j(52)
+44whereA
+qandB
+qare the individual bidding probabilities of players in group Aand groupB
+respectively. Their exact values will depend on the specic asymmetry we are considering.
+We will be interested in computing the steady state probability distribution of the
+Markov chain. The goal of nding closed-form solutions for time-inhomogeneous Markov
+chains for ascending price auctions is beyond the scope of our work; hence for these chains
+we resort to numerical methods, simply calculating the probability of being in each state
+in each stage, as described in Section 4.2. But for xed-price auctions the analysis is far
+more straightforward and we present it here following the framework of [3]. First dene
+the labeled transition matrix Pas:
+P=0
+BBBB@A B W A WB
+ApAApAB 1pAApAB 0
+BpBApBB 0 1pBBpBA
+WA 0 0 1 0
+WB 0 0 0 11
+CCCCA(53)
+Pis in canonical form with the transient states coming before the absorbing ones. Let Q
+be the submatrix of Pcontaining solely the transient states AandB. For an absorbing
+Markov chain Pthe matrix N= (IQ)1is called its fundamental matrix. The entry
+nijofNgives the expected number of times the process is in state jgiven that it started
+in statei. Notice that Nexists and is equal to I+Q2+Q3+


. We can express Nin
+terms of the transition probabilities:
+N= 1pBB
+pBBpAApAApBBpABpBA+1pAB
+pBBpAApAApBBpABpBA+1
+pBA
+pBBpAApAApBBpABpBA+11pAA
+pBBpAApAApBBpABpBA+1!
+: (54)
+For any given starting state we can compute the time to absorption by computing the
+vectort=Nc, wherecis a column vector whose entries are 1. Eectively, by adding
+together the elements of Nrow-wise we are summing up the time spent in each of the
+transient states. In particular we have
+t= pAB+pBB1
+pAA(pBB)+pAA+pABpBA+pBB1
+pAA+pBA+1
+(pAA1)(pBB1)pABpBA!
+: (55)
+LetpAandpBbe the probabilities of the game starting in states AandBrespectively
+and letp0be a vector containing them. It is worth highlighting here that pA+pB<1 as
+there is always the probability that nobody will place the rst bid. We can compute the
+expected number of bids the game will last by calculating tp0. Alternatively, as we often
+do, if we wish to assume that rst bid always occurs p0has to be scaled by 1 =(pA+pB) so
+thatpA+pB= 1.
+We will also be interested in computing the revenue of the auction. When players are
+charged a single bid fee the revenue can be easily obtained from expected number of bids.
+45If the two groups of players are charged dierently the revenue of is given by p0Nbwhere
+bis a vector of bid fees.
+Next, dene Ras the upper-right, two-by-two submatrix of P. LetSbe a matrix whose
+(i;j)thelement is the probability that the chain will be absorbed in state jgiven that it
+started in state i. Then
+S=NR= (pAA+pAB1)(pBB1)
+(pAA1)(pBB1)pABpBApAB(pBB+pBA1)
+pABpBA(pAA1)(pBB1)
+(pAA+pAB1)pBA
+(pAA1)(pBB1)pABpBA(pAA1)(pBB+pBA1)
+(pAA1)(pBB1)pABpBA!
+: (56)
+We can also compute the unconditional probabilities wof the chain being absorbed in each
+of the sink states assuming the rst bid is placed (using the scaled version of p0):
+w=p0S= (pAA+pAB1)(pBB1)pApAB(pBB+pBA1)pB
+(pAA1)(pBB1)pABpBA
+(pAA1)(pBB+pBA1)pB(pAA+pAB1)pBApA
+(pAA1)(pBB1)pABpBA!0
+: (57)
+This framework will allow us to characterize a series of asymmetries by computing the
+vectorstandw, analytically where possible and computationally otherwise.
+B Dataset Description
+In this section we provide more information about the two datasets we collected and used
+throughout this paper. The dataset is publicly available at http://cs-people.bu.edu/zg/swoopo-dataset.tar.gz .
+B.1 Outcomes Dataset
+OurOutcomes dataset comprises 121,419 auctions conducted between September 8, 2008
+and December 12, 2009. For each auction we collected the following information:
+46Field name Description Example value
+auction id A unique numerical id for the auction. 259070
+product id A unique product id. 10011706
+item A text string describing the product. 300-bids-voucher
+desc More information about the product. 300 Bids Voucher
+retail The stated retail value of the item, in dol-
+lars.180
+price The price the auction reached, in dollars. 31.26
+finalprice The price charged to the winner in dol-
+lars.731.26
+bidincrement The price increment of a bid, in cents. 6
+bidfee The cost incurred to make a bid, in cents. 60
+winner The winner's username. Schonmir1500
+placedbids The number of paid bids placed by the
+winner.106
+freebids The number of free bids place by the win-
+ner.0
+endtime str The auction's end time. 13:29 PDT 12-12-2009
+flgclick only A binary 
ag indicating a \NailBiter" auc-
+tion.1
+flgbeginnerauction A binary 
ag indicating a beginner auc-
+tion.0
+flgfixedprice A binary 
ag indicating a xed-price auc-
+tion.0
+flgendprice A binary 
ag indicating a 100%-o auc-
+tion.0
+B.2 Trace Dataset
+OurTrace dataset comprises 7,352 auctions conducted between October 1, 2009 and
+December 12, 2009. In addition to the information contained in the Outcomes dataset,
+ourTrace dataset also contains information about the actual bids that were placed during
+the auction. We probed Swoopo at semi-regular time-intervals - more frequently near
+the end of the auction when bids are placed in rapid succession, less frequently near the
+beginning of the auction when bidding is sparse. Specically, to avoid inundating Swoopo
+with useless requests we implemented the following back-o procedure. Our initial probing
+interval is set to one second. When the countdown clock had more than 10 minutes left
+on it, and there were no bids reported in our last request, we would increase our probing
+interval by half a second, up to a maximum of a minute. A new request including at
+7This can dier from the price eld in the case of xed-price auctions.
+47least one bid would reset the probing interval to one second. When the countdown clock
+had at least 2 and less than 10 minutes left on it, our maximum probing interval was 10
+seconds. Finally, when the countdown clock had fewer than 2 minutes left on it we probed
+at the rate Swoopo suggested but at least once every second (every probe response was
+associated with a Swoopo dened update-interval that Swoopo uses to instruct the user's
+browser when to next update). Swoopo would respond with a line of the following format:
+ct=15|cs=1|ra=0|cw=Schonmir1500|cp=3126|bh=521:Schonmir1500:1:3126:0:#|lui=4#1#0#0
+Each line was timestamped on our end to provide the actual time of the bid(s). In
+plain English the above line tells us that the 521st bid of the auction was placed by
+user Schonmir1500 . It raised the price of the item to $31.26 and it was not placed by a
+BidButler. Finally, when the bid was placed another 4 seconds were added to countdown
+clock to reset it to its current value of 15. Specically, the elds, which are separated using
+vertical bars, carry the following information:
+Field name Description
+ct Current time on the countdown clock.
+cs Current state, 1 means the auction is still active, 20 that is has ended.
+Swoopo denes more status codes which we did not encounter.
+ra Technical use. When set to 1 forces the browser to reload the page,
+irrelevant to the auction.
+cw Username of current winner.
+cp Current price in cents.
+bh Bid history, #-separated tuples of the form:
+bidnumber:username:bidtype: price:yourbid . A bidtype of
+1 indicates a player bid whereas a bidtype of 2 indicates a BidButler
+bid. The eld yourbid is set to 1 when the observer has placed the
+corresponding bid.
+lui Last update index, four #-separated numbers: seconds added to clock
+from player bids, number of player bids, seconds added to clock by Bid-
+Butler bids, number of BidButler bids.
+48
\ No newline at end of file
diff --git a/1001.0593.txt b/1001.0593.txt
new file mode 100644
index 0000000000000000000000000000000000000000..318a95c5d7fd84732787c39973e55cbfb58e16ee
--- /dev/null
+++ b/1001.0593.txt
@@ -0,0 +1,459 @@
+arXiv:1001.0593v1  [astro-ph.IM]  4 Jan 2010Recent GRBs observed with the 1.23m CAHA
+telescope and the status of its upgrade
+Javier Gorosabel1, Petr Kub´ anek1,2, Martin Jel´ ınek1, Alberto J.
+Castro-Tirado1, Antonio de Ugarte Postigo3, Sebasti´ an Castillo Carri´ on4,
+Sergey Guziy1, Ronan Cunniffe1, Matilde Fern´ andez1, Nuria Hu´ elamo5,
+V´ ıctor Terr´ on1, Nicol´ as Morales1, Jos´ e Luis Ortiz1, Stefano Mottola6, and
+Uri Carsenty6
+1Instituto de Astrof´ ısica de Andaluc´ ıa (IAA-CSIC), 18008 Granada, Spain.
+2Imaging Processing Laboratory (IPL), Universidad de Valen cia, Valencia, Spain.
+3INAF/Brera Astronomical Observatory, Via Bianchi 46, 2380 7, Merate(LC),
+Italy.
+4Universidad de M´ alaga, M´ alaga, Spain.
+5LAEX-CAB (INTA-CSIC); LAEFF, PO Box 78, 28691 Villanueva de la Ca˜ nada,
+Madrid, Spain.
+6Institute of Planetary Research, DLR, Berlin, Germany.
+We report on optical observations of Gamma-Ray Bursts (GRBs) f ollowed
+up by our collaboration with the 1.23m telescope located at the Calar A lto
+observatory. The 1.23m telescope is an old facility, currently under going up-
+grades to enable fully autonomous response to GRB alerts. We discu ss the
+current status of the control system upgrade of the 1.23m teles cope. The up-
+grade is being done by our group7based on the Remote Telescope System,
+2nd Version (RTS2), which controls the available instruments and int eracts
+with the EPICS database of Calar Alto. Currently the telescope can run fully
+autonomouslyor under observersupervision using RTS2. The fast reactionre-
+sponse mode for GRB reaction (typically with response times below 3 m inutes
+from the GRB onset) still needs some development and testing. The telescope
+is usually operated in legacy interactive mode, with periods of superv ised au-
+tonomous runs under RTS2. We show the preliminary results of seve ral GRBs
+followed up with observer intervention during the testing phase of t he 1.23m
+control software upgrade.
+7Ourgroupis called ARAE(RoboticAstronomy&High-EnergyAs trophysics)and
+is based onmembers ofIAA(InstitutodeAstrof´ ısica deAnda luc´ ıa). Currentlythe
+ARAEgroup is responsible to develop the BOOTES network of ro botic telescopes
+[14]. See http://www.iaa.es/arae/ andhttp://www.iaa.es/bootes/index.php
+for more details.2 J. Gorosabel, P. Kub´ anek, M. Jel´ ınek, A. J. Castro-Tirad o et. al.
+Fig. 1. View of the Calar Alto observatory. The picture shows the five telescopes
+of Calar Alto and the staff building (lower-right corner). Th e arrow shows the local-
+ization of the 1.23m telescope.
+1 Introduction
+The 1.23m telescope is at the German-Spanish observatory of Calar Alto
+(CAHA) in the province of Almer´ ıa, South-East of Spain. The obser vatory’s
+altitude (2168 m), mean seeing (0.9”; see [23]) and a large fraction (6 5%)
+of clear nights, make Calar Alto one of the most competitive observa tories
+in Europe in the optical and near-infrared bands. The observator y harbours
+five telescopes, but only three are currently operative: the 1.23m , 2.2m and
+3.5m telescopes. Fig. 1 shows a view of the Calar Alto observatory. T he arrow
+indicates the position of the 1.23m telescope.
+The 1.23m telescope is a Ritchey-Cr´ etien telescope built in 1975 by Ca rl
+Zeiss. The focal ratio of the telescope is f/8 with a total field of view of 90’ and
+a focal plane scale of ∼20.9”/mm. The aberration free field of view is limited
+to∼15′. The German mount of the telescope is driven by a mechanical and
+hydraulic system renovated in 2008 by the observatory staff. Fig. 2 shows a
+drawing and a picture of the of 1.23m CAHA telescope.
+Two instruments are available for the 1.23m, the MAGIC near-infrar ed
+camera and the 2kx2k SITE#2b optical CCD. Currently, most of th e time
+(∼95%) only the optical CCD is used, as the MAGIC camera, one of theGRBs observed with the CAHA 1.23m and upgrade of its control s ystem 3
+Fig. 2. The 1.23m telescope of Calar Alto. Left panel: the drawing shows the me-
+chanics and the hydraulic system of the 1.23m telescope of Ca lar Alto. One of the
+mechanical peculiarities of the telescope is the presence o f a steel sphere (indicated
+by a red arrow) which, by means of a high-pressure hydraulic s ystem, supports all
+the weight of the telescope. Right panel: Picture of the 1.23m inside the dome. As
+seen the instrumentation is always located in the Cassegrai n focus. Note the German
+mount of the telescope.
+first near-infrared cameras ever built, was superseded by moder n instruments
+on larger telescopes. In addition to the offering of the MAGIC near- infrared
+camera, CAHA is considering the possibility of installing visitor instrume nts
+on the 1.23m telescope.
+The field of view of the optical CCD camera is 17’ ×17’ with a pixel size
+of 24µm. That translates to a pixel scale of 0.5” per pixel. The read-out tim e
+of the whole chip in 1x1 binning mode is very long, close to 4 minutes. For
+this reason rapid follow-up observations of transients like Gamma-R ay Bursts
+(GRBs) areusuallycarriedouttrimmingthe windowand/orbinning the CCD
+by 2x2 pixels. Using this technique, the readout time is usually kept be low
+one minute.
+The CCD chip is refrigerated using liquid nitrogen, which makes dark
+current negligible, fewer than 2 electrons per hour. The read-out noise is also
+low, about 7 electrons. However, the chip has several bad columns which
+RTS2 (see next Section and [16]) must avoid at the time of a GRB alert ( see
+the vertical lines on the image of GRB 090628, Fig. 7). The optical ca mera is
+equipped witha BVRIfilterwheel.AWollastonprismcanbealsomountedin
+the filterwheel, which makesthe 1.23mtelescope suitablealsoforpola rimetric
+studies.4 J. Gorosabel, P. Kub´ anek, M. Jel´ ınek, A. J. Castro-Tirad o et. al.
+Fig. 3.The MAGIC near-infrared and optical CCD cameras of the 1.23m telescope,
+both in the Cassegrain focus of the telescope. Left panel: MAGIC near-infrared
+camera. The MAGIC detector is a 256x256 pixel HgCdTe array, p roviding a field
+of view of 4’x4’. The filter wheel of MAGIC allows JHKbroad-band imaging. The
+instrument is only mounted by special request, so is not usua lly available. Right
+panel:Optical CCD camera running on the 1.23m telescope of Calar Al to. The
+optical CCD camera is based on a 2kx2k SITE#2b chip. The green tube is the
+dewar and the white box is the electronic system associated t o the camera. The
+filter wheel is located in the black flange just on the detector dewar.
+2 Upgrading the 1.23m control software with RTS2
+A call for proposals was issued by CAHA in 2008 for the use of the 1.23 m
+telescopeover4years.Sixteamsobtainedobservingtime,havingt hefollowing
+scientific drivers: Solar System bodies, Binary stars, Transits of e xoplanets,
+T Tauri stars, and GRBs. The 1.23m is also scheduled for public outre ach
+purposes, usually associated to high-schools.
+The ARAE (Robotic Astronomy & High-Energy Astrophysics) group of
+IAA8agreed to contribute to CAHA by providing the control system of t he
+1.23mtelescope.The ARAE groupwasgranted26GRB triggersper y earwith
+an averagedurationof0.5nights per trigger.The upgradeofthe t elescope and
+its instrument control system is currently being carried out based on RTS2
+(see [16] and referencestherein). One of the key requirementso fthe upgradeis
+to keep the existing telescope software and hardware untouched . The system
+must remain operational every night throughout the upgrade.
+As none of the telescope instruments, nor their control electron ics and
+software was developed by our group, we are kept away from the c omplex
+details of their construction and internal operation. As described later, we
+exclusively communicate with the control interfaces of the existing software.
+The existing control system is an adapted version of that current ly used
+by the 3.5m Calar Alto telescope. It allows the observer to control t he in-
+struments through graphical user interface (GUI) programs ru nning on two
+8Instituto de Astrof´ ısica de Andaluc´ ıa, partner in the CAH A operations.GRBs observed with the CAHA 1.23m and upgrade of its control s ystem 5
+main observatory computers - one for the telescope control, the other for the
+camera. The observer is responsible for preparing the observing p lan, opening
+and closing of the dome, taking care of executing exposures, inspe cting im-
+ages for good pointing and their quality, and synchronizing the teles cope and
+the filter wheel movements with the CCD exposures. The major dra wbacks of
+this approachare obvious:the observerspends most ofthe night workinghard
+to get the data and keep the system running, the system is prone t o human
+errors when the observer is tired. Moreover, the observing logs a re either hard
+to extract from the technical log or are not created by the syste m at all, and
+the interruption of the observation to react on a quickly evolving ta rget of
+opportunity requires observer presence and attention.
+In contrast, RTS2 was designed to create an autonomous observ atory en-
+vironment. The observer is allowed to interact with the system, and at worst
+case to take full manual control. The system is able to guide the obs ervatory
+through the night, taking care of closing and opening the dome, acq uiring
+sky flats, darks, and last, but not least, keeping detailed logs of th e images
+acquired, judging pointing accuracy and producing preliminary resu lts.
+RTS2 device drivers are responsible for handling any errors that oc cur
+during their operation. If possible, the device is reset, and anothe r attempt to
+get it operating is made. The RTS2 rts2-xmlrpcd component is responsible for
+communicatingthe errorsto the usersand customscripts,which a llowsforthe
+execution of more complicated scenarios. One of the core principles is to avoid
+restarting drivers that have failed. If the driver produces a core dump, that is
+safely removed from the system and it is up to the observer to rest art it. This
+feature also allows RTS2 to be quite flexible – for example a configurat ion
+without any telescope, just with a camera and other instrumentat ion, can be
+made without changing a single line of code. More information on handlin g
+the errors and other related issues is provided in [18].
+The autonomouscapabilities of the RTS2 system resides in a generic la yer,
+with underlyinghardware-specificdriversandcommunicationviathe TCP/IP
+network stack. So in an ideal world, once the provided skeleton driv ers are
+used to write low level RTS2 drivers for the hardware, every telesc ope can
+be made fully autonomous. To our knowledge, this is a big step forwar d from
+the traditional, incremental way of how observatory control sof tware has been
+developed. Instead of being written primarily as a set of controls fo r the hard-
+ware, with some subsequent autopilot features, RTS2 was designe d from the
+start to provide autonomous capabilities. Secondarily RTS2 also pro vides a
+way for the observer to interact directly with the hardware.
+2.1 Current state of the upgrade
+Development of the first version of the RTS2 drivers was a question of a few
+days (and nights), since the RTS2 drivers were being written by the main
+author of RTS2 (with the assistance of the CAHA staff). The major obstacle
+which we had to face was running RTS2 on an old Solaris operating syst em,6 J. Gorosabel, P. Kub´ anek, M. Jel´ ınek, A. J. Castro-Tirad o et. al.
+which is used to run the 1.23m control computers. As RTS2 was writt en in
+quite portable C++ on Linux, and using GNU Autotools9for build control,
+the porting process involved changing a few unavailable functions10. The sys-
+tem is now able to operate the same as any other observatory using RTS2.
+The remaining obstacle is the lack of a natural incorporation of the a uto-
+guider in the RTS2 environment. This fact prevents us from taking im ages
+with long exposure times. Tests performed with the 1.23m showed th at ex-
+posures longer than 300s produce elongated Point-Spread-Func tions (PSFs),
+especially under sub-arc-second seeing conditions. The guider is an old in-
+strument, with a quite complicated interface, without any autonom ous capa-
+bilities. Thus, some time will be needed before we will be able to perform
+observations with the guider smoothly integrated in RTS2.
+Currently the 1.23m is able to respond to GRB alerts generated by th e
+GRB Coordinates Network (GCN). The maximum slew time of the 1.23m
+telescope is 4 minutes for the most unfavourable move. Usually the r esponse
+time is below 3 minutes. The response time is limited by the speed of the
+current engines/mechanics moving the dome and the telescope. Giv en that
+the existing telescope/dome mechanical parts are strong and relia ble, CAHA
+does not plan to renew them, so we do expect to overcome the slew- time
+limitation in the near future. The RTS2 Gamma-Ray Burst Daemon ( rts2-
+grbd) of the 1.23m is continuously linked via TCP/IP network socket to the
+GCN server at Goddard Space Flight Center (GSFC). In order to re act to
+GCN alerts, it was necessary to accommodate the GCN connection in the
+CAHA firewall. Fig. 4 shows a working scheme of the 1.23m response mo de
+to GCN alerts under RTS2.
+When ahigh-energysatellite(usually the Swiftmission with its BAT[1] de-
+tector)localizesaGRB,thepositionisdumpedinafewsecondsfromt heGRB
+occurrenceto ground-trackingstations and distributed by the G CNto its sub-
+scribers.In ourcasetheGCNpacketreachestheCAHA rts2-grbd server.Then
+the ongoingobservationsare interrupted and the 1.23mis pointed t owardsthe
+GRB position in order to start a series of short exposures (usually w ith the
+CCD windowed and binned in order to save read-out time). Interrup ting an
+observing run is a standard mechanism in RTS2 (see details in [18]). The in-
+terruption can either be hard, which interrupts the current expo sure, or soft,
+which waits the current exposure to finish. The rules governing this choice
+will be decided by the observers once the system is fully operational.
+TheRTS2controlsystemstoresmostofthedataina Postgresql database11.
+Itincludestoolsforretrievingandmanipulatinginformationfromthe database.
+New targets can be entered through command line tools, autonomo usly from
+various triggering systems, or from the RTS2 Web interface.
+9GNU Autotools web site, http://www.gnu.org/software/autoconf
+10and renaming the variables called ”sun” (”sun” is a defined sy mbol on Solaris
+systems).
+11Postgresql web site, http://www.postgresql.orgGRBs observed with the CAHA 1.23m and upgrade of its control s ystem 7
+Fig. 4. Flow diagram of the GCN network and the response mode of the 1. 23m
+telescope. The whole process can take up to 4 minutes, depend ing on the position of
+the GRB on sky. The time needed to receive the coordinates fro m the high-energy
+satellite in the RTS2 server takes just a few seconds (usuall y less than 15 seconds).
+Currently most of the delay is due to the slow pointing of the 1 .23m, which could
+take up to 4 minutes in the worst case.
+The system has a powerful internal scripting engine, which enables users
+to define the observing strategy. By tracking devices states, th e system han-
+dles synchronization among instruments, so the user does not hav e to know
+the details of the execution of the underlying observation. Scriptin g enables
+the user to specify all image parameters, dithering strategy, sele cted filters,
+and much more. For example the following script runs observations in I-band,
+based on a series of 9 dithered images with a exposure time (per fram e) of 240
+seconds, with a 1x1 CCD binning and trimming the CCD to a sub-window o f
+500x400 pixels [x1=600, x2=1100, y1=300, y2=700]:
+FW0.filter=I READT=(600,500,300,400) binning=1 E 240
+T0.WOFFS=(-72s,+36s)
+for 3{for 3{E 240 T0.woffs+=(16s,-54s) }T0.woffs+=(0,+3m) }
+Note that the syntax for the sub-window in the above example is (x1 ,
+x2-x1, y1, y2-y1). We have incorporated in RTS2 a call to the auto matic8 J. Gorosabel, P. Kub´ anek, M. Jel´ ınek, A. J. Castro-Tirad o et. al.
+astrometric calibration package described in [7]. This ”on-the-fly” a strometric
+calibration corrects the telescope pointing, so the object can be p laced on the
+desired (x,y) coordinates of the detector. The optical distortion on the CCD
+is negligible (less than one tenth of a pixel), so we decided to make a simp le
+(and hence fast) fit considering a rotation term and a center field s hift. The
+astrometric calibration is done on every frame. Polynomial fit is not u sed.
+The typical astrometric error (using ∼20 stars) is below 0 .5′′, good enough
+to correct the 1.23m pointing. The typical standard deviation in the inferred
+rotation angle is ∼0.1 degrees when ∼20 field stars are used in the fit.
+The success rate of the “on-the-fly” calibration is above 95%. The astrometric
+calibration of the remaining images is done manually afterwards.
+In the near future we plan to provide also a rough photometric calibr a-
+tion based on the USNO-B catalogue [22]. Simultaneously to the telesc ope
+triggering an e-mail is sent to the 1.23m users to warn them on the oc cur-
+rence of the GRB. Additionally the GCN alert triggers the creation of an
+e-mail and a SMS for the ARAE group with relevant information of the GRB
+(coordinates, Galactic reddening, finding chart, elevation curves , and other
+information). Please see [4] for more details about this system.
+Further improvements on this status are likely doable, so we feel co nfident
+that, under the current limitations (for instance the telescope sle w speed),
+we could reduce the GRB response times. This might allow us to detect the
+prompt optical emission associated to the gamma-ray event.
+Another potential upgrade of the 1.23m telescope could be the inco rpo-
+ration of MAGIC in RTS2. Rapid response observations with MAGIC wo uld
+make the 1.23m telescope very competitive in the GRB field. Currently this
+is beyond our scopes (and also beyond the CAHA man-power mainten ance
+capabilities), but we do not discard it since the flexibility of RTS2 to acc om-
+modate new devices would allow us to integrate MAGIC (and new possib le
+visitor instruments) quickly.
+2.2 Integration of the legacy interfaces to RTS2
+The following subsections provide descriptions of the legacy interfa ces and
+their interaction with RTS2.
+Interaction of RTS2 with the Calar Alto EPICS
+The individual operations of the CAHA telescopes are coordinated, and con-
+trolled,bytheExperimentalPhysicsandIndustrialControlSyst em(EPICS12).
+ThetaskofEPICSistoprovideaccessandcontrolofalltheCAHAt elescopes,
+aswellasdatafromthecentralweatherstationandfromthesee ingandextinc-
+tion monitors.The seeingand extinction monitorsarebasedon small aperture
+telescopes located in CAHA whose values are available through EPICS .
+12EPICS web site, http://www.aps.anl.gov/epics/index.phpGRBs observed with the CAHA 1.23m and upgrade of its control s ystem 9
+The 1.23m telescope mount, its focuser and the camera filter wheel are
+all fully controllable through EPICS. The legacy telescope interface accepts
+objects coordinates in the J2000 coordinate system, and handles all the re-
+quired calculations internally, including precession, aberration, refl ection and
+telescope pointing model offsets. Both the filter wheel and the foc user can
+be controlled through their own EPICS channels. Other devices in th e 1.23m
+telescope system do not exist in the EPICS universe: the CCD detec tor, the
+auto-guider camera, the dome itself, and various auxiliary switches (for ex-
+ample the dome lights).
+Weusethe informationprovidedthroughEPICStoautomatesever altasks
+of the 1.23m. For instance, the meteorological information available from the
+EPICS is used to trigger the bad weather state, and hence to close the dome13
+and stop observations. Bad weather is also triggered when both ot her tele-
+scopes are closed - this is the rule imposed on the 1.23moperations by CAHA.
+For discussion on how the weather state voting is integrated into RT S2, in-
+cluding its fail-safe capabilities14, please see [18].
+Yet another possible application lies in coordinating observations with the
+other Calar Alto telescopes. From the EPICS system, RTS2 can lear n what
+the targetsofthe othertelescopesare,whether they arein the RTS2 database,
+and depending on the target information can either start their mon itoring or
+remove them from the list of targets which should be observed.
+CCD integration within RTS2
+The optical CCD detector is connected to its own control compute r. The con-
+trol computer communicates with the control software, running on a master
+workstation, over the network. From the available C source code, we created
+a RTS2 device driver. All the major settings are supported, includin g binning
+and partialchip readout.The camerabehavesasjust anotherRT S2 supported
+CCD camera, visible in monitoring software and available for scripting.
+Guider camera and its integration in RTS2
+The guider is built from a video camera with an image intensifier, fed fro m
+a pick-off mirror on a two-axis stage. This allows to place the guider ima ge
+anywhere in the telescope field of view (FoV), thus eliminating the sign ificant
+disadvantages of autoguiding-by-astrometry using the main came ra: limited
+detectorFoV, lowgainandarequirementforshortexposuretimes . Theguider
+camera has its own control computer, which assumes the observe r is sitting in
+13Based on calculated Sun position through libnova,http://libnova.sf.net
+14Network crashes and other failures (including the EPICS one s) are properly han-
+dled inside RTS2. A detailed description on how the weather v oting system works
+is beyond the scope of this paper. We refer the reader to the ex tended discussion
+given in [18].10 J. Gorosabel, P. Kub´ anek, M. Jel´ ınek, A. J. Castro-Tira do et. al.
+front of it (e.g. the video output is sent directly to the screen). Pa rtial remote
+control has since been provided, in part by placing a web camera in fr ont of
+the guiding screen.
+We would like to improve this setup with a fully integrated RTS2 device,
+which would transparently provide automatic search for bright sta rs in the
+FoV, and guiding capabilities. In principle, RTS2 is able to do this, and so me
+promising tests were already carried out on the other RTS2 contro lled tele-
+scopes. Currently the biggest problem is to figure out how to commu nicate
+with the guider, and to implement a full auto-guiding loop.
+Dome control and other switches
+As the dome is a critical component, its control is separate from th e EPICS
+system (although dome status is reported to EPICS). In order to change the
+dome state, special commands must be run on the dome control co mputer.
+Similar commands are available to turn off and on dome lights, to contro l
+the telescope drives, the hydraulics, the tracking and the mirror c over. Those
+commands are fully interfaced in RTS2, so RTS2 is able to control all t hose
+switches.
+3 Preliminary results
+Although not fully autonomous, the 1.23m has already performed fo llow up
+optical observations of GRBs. None of the results below itemized we re ac-
+quired by the automatic response mode of the RTS2 package, but s ome data
+were manually acquired by using RTS2 as the observingtool. Most of t he data
+were taken by in situobserves, using the currently available GUI.
+AllthebelowlistedGRBsshowedX-rayafterglowswhichwerelocalized by
+theXRTX-raytelescopeonboard Swift(see[2]fordetailedinformationonthe
+XRT instrument). So their X-ray afterglows were localized with unce rtainties
+of only a few arc-seconds, making the corresponding optical stud ies much
+more efficient.
+GRB 090313:
+The optical afterglow of this GRB [5] was observed with the 1.23m in th e
+R-band during two consecutive nights one week after the gamma-ra y event.
+The observations were accompanied with K-band observations carried out
+with the 3.5m telescope of Calar Alto equipped with Omega 2000. The data of
+the afterglow are currently being analyzed and are part of an inter national
+monitoring which will be published in [21].GRBs observed with the CAHA 1.23m and upgrade of its control s ystem 11
+Fig. 5. The figure shows the optical afterglow of GRB 090424 as detect ed with
+the 1.23m CAHA telescope. The image has been created by combi ningBVRI-band
+images taken with the 2kx2k CCD camera currently in use. The u pper right box
+shows a magnified picture of the afterglow region. The magnit ude of the afterglow
+wasR= 19.3. The mean observing epoch of the image is April 24.87 UT. Nor th is
+up and East right. The field of view of the images is 8’x8’.
+GRB 090424:
+We detected the afterglow of GRB 090424 [3] in BVRIon April 24.87 UT
+with a magnitude of R= 19.3±0.1. The preliminary results were reported in
+[9]. In the days following GRB 090424, a long term monitoring was perfo rmed
+inRandIbands in order to search for the underlying supernova. Fig. 5
+shows a colored image constructed with the BVRI1.23m images taken on
+April 24.87 UT. The final results have been included in [15].
+GRB 090621B:
+We detected the two afterglow candidates reported for this GRB [6 ] by Levan
+et al. and Galeev et al. [19, 8]. The observations were done in the I-band, 4.14
+- 4.75 hours after the GRB. A deep second epoch observation is pen ding in
+order to search for photometric variability of the candidates. The preliminary
+results were reported in [10]. Fig. 6 shows both candidates.12 J. Gorosabel, P. Kub´ anek, M. Jel´ ınek, A. J. Castro-Tira do et. al.
+Fig. 6.The picture shows the co-added I-band image taken for GRB 090621B with
+the 1.23m telescope of Calar Alto. The faintest of the two obj ects (the most upper
+one indicated with tick-marks) represents the candidate re ported in [19]. Approx-
+imately 10′′towards the South-West it is marked the candidate reported b y [8],
+brighter than the one of [19]. The cyan circles shows the USNO -A2 catalogue stars
+used for the astrometric and photometric calibration. The fi eld of view of the image
+is 70′′×50′′, with North up and East left.
+GRB 090628:
+R-bandobservationsofthe XRT position[24] werecarriedout1.43–3 .20hours
+after the gamma-ray event. No counterpart was found down to R= 22 in the
+XRT errorbox.Simultaneousnear-infraredobservationswouldha vebeen very
+helpful in order to discriminate the possible high-redshift nature of this GRB
+but unfortunately they were not possible. The results of the obse rvations were
+reported in [17].
+GRB 090727:
+This GRB was observed on July 27.9646-28.0063UT in the I-band, 0.45-1.45
+hours after the gamma-rayevent. The afterglow reported by [25 ] was detected
+with a magnitude of I∼19.4, using as calibrator the USNO B1.0 star with
+I= 18.72andcoordinatesRA J2000=21:03:49.099,DEC J2000=+64:55:58.23.
+The results can be found in [11].GRBs observed with the CAHA 1.23m and upgrade of its control s ystem 13
+Fig. 7. Co-added R-band image of GRB 090628 taken with the 1.23m telescope.
+The large circle represents the preliminary XRT error circl e [24], whereas the small
+one shows the refined one [20] reported for GRB 090628. The fiel d of view of the
+image is 125′′×100′′. The total exposure time is 3120 s. The individual images
+were taken on June 28.9486 - 29.0225 UT (1.43 - 3.20 hours afte r the gamma-ray
+event). No objects were found in both error circles down to R= 22. More detailed
+information can be found in [13].
+The discovery of the optical afterglow of GRB 090813:
+GRB 090813 represents the first optical afterglow discovered wit h the 1.23m
+CAHA telescope. I-band observations of the XRT position were initiated 437s
+after the GRB trigger, revealing an object with a rough magnitude o fI∼17
+coincident with the XRT position. The lack of the object on the DSS st rongly
+suggested its association with the optical afterglow of GRB 090813 . The pre-
+liminary results were reported in [12].
+GRB 090814B:
+This is the only GRB detected by the INTEGRAL satellite to date that w e
+have followed up with the 1.23m telescope. We carried out a series of I-band
+observations with different exposure times ranging from 180s to 70 0s. The
+total exposure time invested for this GRB was 9520s, with a mean ob serving14 J. Gorosabel, P. Kub´ anek, M. Jel´ ınek, A. J. Castro-Tira do et. al.
+epoch of Aug 14.1259 UT. No optical object was found within the refi ned
+X-ray error circle provided hours later by XRT. The 3 σlimiting magnitude
+of the co-added image is I= 20.6. A more extended description can be found
+in [13].
+4 Conclusions
+The 1.23m is a wide-purpose telescope which is currently used by six int erna-
+tional teams to perform long-term projects with a duration of fou r years. The
+ARAE group of IAA is responsible for automating the telescope oper ations so
+that suchteamscanperformtheir (usuallylong)observingcampaig nswithout
+errors, yet enabling quick override observations of GRBs.
+The GRB results obtained to date have been mostly taken by night ob -
+servers. Use of the fully autonomous mode, provided by RTS2, is pe nding
+non-trivial integration of the guider. After this is done, we have re asonable
+hopes to believe that telescope will be able to react to GRB alerts in a f ew
+minutes. This could allow us to detect the optical emission at the first stages
+of the explosion, making the associated GRB science much more attr active
+for the GRB community.
+We acquired images for seven GRBs, detecting the optical afterglo ws of
+four of them. The typical reaction time of these observations ran ged from ∼8
+minutes up to a few hours. When fully implemented, the autonomous s ystem
+should be able to react to triggers within 4 minutes. We have reasona ble
+hopes to obtain much more interesting and world–competitive result s in the
+near future.
+Acknowledgments
+The research of JG, AJCT, RC and MJ is supported by the Spanish pr o-
+grammes AYA2008-03467/ESP,AYA2009-14000-C03-01and AYA 2007-06377.
+We are very grateful to all the CAHA staff and in particular to Ulli Thie le for
+his excellent support with the 1.23m telescope. PK would like to acknow ledge
+generous financial support provided by Spanish Programa de Ayudas FPI del
+Ministerio de Ciencia e Innovaci´ on (Subprograma FPI-MICI NN)and Euro-
+peanFondo Social Europeo . We also would like to thank the two anonymous
+referees for their helpful comments.GRBs observed with the CAHA 1.23m and upgrade of its control s ystem 15
+References
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+Telescope (BAT) on the SWIFT Midex Mission , Space Science Reviews, Vol. 120,
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+14. M. Jel´ ınek, A.J. Castro-Tirado, A. de Ugarte Postigo, e t al. 2009, Realtime GRB
+followup by BOOTES-1B , this volume.
+15. D.A. Kann, S. Klose, B. Zhang, et al. 2007, ApJ in press, arXiv:0712.2186v2 .
+16. P. Kub´ anek, et al. 2006, RTS2: a powerful robotic observatory manager , Society
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+18. P. Kub´ anek, 2009, RTS2 – the Remote Telescope System , this volume.
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+20. V. Mangano, B. Sbarufatti, et al. 2009, GCN Circ. 9599.
+21. A. Melandri, et al. A&A in preparation.
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\ No newline at end of file
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+Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 27 October 2018 (MN L ATEX style file v2.2)
+Analytic and numerical realisations of a disk galaxy
+M. J. Stringer1;3, A .M. Brooks2;3, A. J. Benson2;3, F. Governato4
+1. Department of Physics, South Road, Durham, DH1 3LE
+2. Theoretical Astrophysics, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, U.S.A.
+3. Kavli Institute of Theoretical Physics, Santa Barbara, CA 93106, U.S.A.
+4. Department ofAstronomy, University of Washington, Box 351580, Seattle, WA98195, U.S.A.
+27 October 2018
+ABSTRACT
+Recent focus on the importance of cold, unshocked gas accretion in galaxy formation –
+not explicitly included in semi-analytic studies – motivates the following detailed com-
+parison between two inherently different modelling techniques: direct hydrodynamical
+simulation and semi-analytic modelling. By analysing the physical assumptions built
+into the gasoline simulation, formulae for the emergent behaviour are derived which
+allow immediate and accurate translation of these assumptions to the galform semi-
+analytic model. The simulated halo merger history is then extracted and evolved using
+these equivalent equations, predicting a strikingly similar galactic system. This exer-
+cise demonstrates that it is the initial conditions and physical assumptions which are
+responsible for the predicted evolution, not the choice of modelling technique. On this
+level playing field, a previously published galform model is applied (including addi-
+tional physics such as chemical enrichment and feedback from active galactic nuclei)
+which leads to starkly different predictions.
+1 INTRODUCTION
+Two different approaches are widely used to test theories
+of galaxy formation. Both make use of developing compu-
+tational resources to integrate a set of coupled differential
+equationsforwardintime,whereeachequationappliesphys-
+ical constraints, or empirical laws, to some of the system’s
+properties.
+The semi-analytic method (White & Frenk 1991) can
+be characterised by the intent to encapsulate as much of the
+physical behaviour as possible in the equations themselves,
+chosen to describe systematic properties (virialisation, con-
+servation of angular momentum, etc.) and thus leaving the
+minimum to numerical calculation.
+Inherently different to this is the approach taken by
+simulations, where the ethos is to apply equations which are
+known to govern the particular properties (e.g. the gravi-
+tational force between two particles) and demonstrate, by
+skilfully programmed numerical calculation, that these re-
+produce macroscopic properties.
+In practice, adjustment will always need to be made to
+account for the fact that each simulated particle represents
+many, many real particles. In traditional simulations which
+consideredonlygravity,thisadjustmentispurelymathemat-
+ical. When gravity is the dominant interaction, results are
+therefore very robust and simulations are mostly accepted
+as being a faithful realisation of the Universe at large scales,
+where this is the case. A notable measure of this acceptance
+is that semi-analytic models have taken to using samples
+of halos taken directly from simulations (Kauffmann et al.
+1999; Helly et al. 2003; Hatton et al. 2003), rather than sam-
+ples created statistically (Bower et al. 1993; Lacey & Cole1993) using methods based on the analytic arguments of
+Press & Schechter (1974).
+Ingeneral,though,theremustcomeascalebelowwhich
+the governing equations are no longer describing particle in-
+teractions but emergent phenomena. At these smaller scales,
+the two largely contrasting approaches begin to look more
+similar. As an immediate example, simulations which en-
+compass an entire galaxy have not attempted to resolve in-
+dividual stars and must therefore apply an equation which
+approximates the conversion rate of large bodies of gas into
+stars, based on locally averaged density. An equivalent and
+sometimes identical equation appears in semi-analytic mod-
+els (§3.7).
+Both approaches have had some success in accounting
+forthe observed properties of galaxies. Predictive power, on
+the other hand, relies on surety in the underlying equations,
+oraclearunderstandingofhowdifferingunderlyingassump-
+tions map on to final observable quantities.
+To bolster this crucial understanding, it is extremely
+helpfultostudy,indetail,thepredictedevolutionofasingle,
+largegalaxyanditssatellites,asfollowedbythesetwodiffer-
+ent techniques. If the analytic model begins from the precise
+halo merger history found in the simulation, and applies the
+same assumptions about the physics at smaller scales, will
+it predict the same observable properties and, if not, where
+and why do the two calculations diverge?
+1.1 Previous research
+Existing quantitative comparisons between simulations and
+semi-analytic models have focussed on samplesof galaxies.
+c
0000 RASarXiv:1001.0594v1  [astro-ph.GA]  4 Jan 20102M. J. Stringer, A. M. Brooks, A. J. Benson, F. Governato
+The approach which is adopted here – adapting the semi
+analytic model to emulate the assumptions made by the
+simulation—was also used by Helly et al. (2003) to demon-
+strate that the cooled mass predicted by the two approaches
+can be consistent. In particular, consistency was demon-
+strated over two orders of magnitude in halo mass. Earlier
+works by Benson et al. (2001) and Yoshida et al. (2002), us-
+ing similar techniques, reached broadly similar conclusions.
+A broader comparison was made by Cattaneo et al.
+(2007) of two predicted galaxy populations: one by an SPH
+model and the other by version of the GalICS semi-analytic
+model, modified to emulate the SPH method. Again, consis-
+tency is demonstrated between the two techniques once the
+semi-analytic model is appropriately “stripped down”, but
+this consistency comes at the expense of agreement with
+observational constraints, which has been met by the full
+version of GalICS.
+1.2 Outline
+To extend this previous research, in this present work we
+will compare in detail the formation of a single galaxy in
+agasoline numerical simulation and in the galform
+semi-analytic model. The structure of this paper is as
+follows:
+§2:A review of the broader aspects of the gasoline code
+and of this particular simulation.
+§3:Detailed analysis of the way that gasoline models each
+of the following key physical processes, together with an ex-
+plaination of how the particular approach in question can
+be emulated in the galform model.
+§3.1: Merging
+§3.2: Accretion
+§3.3: UV heating
+§3.4: Gas distribution
+§3.5: Cooling
+§3.6: Disk formation
+§3.7: Star formation
+§3.8: Feedback
+§3.9: Disk stability
+Particularly relevant aspects of the galform model are
+reviewed (or referenced) when appropriate. General aspects
+of the model can be found in Cole et al. (2000) and Bower
+et al. (2006).
+§4.1:By applying these same underlying physical assump-
+tions to galform , the two modelling techniques are shown
+to make broadly consistent predictions.
+§4.2:Having demonstrated this consistency, galform is
+then used to find how this halo would evolve under the
+physical assumptions made in previous studies (which have
+been shown to successfully match the collective properties
+of galaxies).
+§5:Summary.Table 1. Resolution details for the gasoline simulation
+Dark Matter particles: 7:6105M
+Gas particles: 1:3105M
+Star particles: 3:8104M
+Force resolution: 0.3 kpc
+2 GASOLINE
+The “MW1” simulation analyzed in this paper was gener-
+ated using Gasoline , an N–Body+Smoothed Particle Hy-
+drodynamics (SPH) Parallel Treecode (Wadsley, Stadel &
+Quinn2004),withinaflat, -dominatedWMAP1cosmology
+(
+0= 0:3,=0.7,h= 0:7,8= 0:9,
+b= 0:039). A mod-
+erate resolution version of this run was originally discussed
+in detail in Governato et al. (2007), and the higher resolu-
+tion version analised here has subsequently been used for a
+number of studies (Zolotov et al. 2009; Read et al. 2009).
+High mass and force resolution (see Table 1) was achieved
+using the volume renormalization technique (Katz & White
+1993).
+This simulated galaxy matches the observed mass–
+metallicity relationship at varying redshifts (Brooks et al.
+2007; Maiolino et al. 2008). This success demonstrates that
+these simulations overcome the historic “overcooling prob-
+lem” (Mayer, Governato & Kaufmann 2008), in which gas
+in simulations cools rapidly, forming stars too quickly and
+early, and thus producing a mass–metallicity relation that
+is too enriched at a given stellar mass. Brooks et al. (2007)
+showed that reproducing the mass–metallicity relationship
+was a result of the adopted feedback prescription. This feed-
+back mechanism, combined with high numerical resolution1,
+avoids spurious loss of angular momentum. As a conse-
+quence, simulated galaxies also match the observed Tully-
+Fisher relationship (Mayer, Governato & Kaufmann 2008;
+Governato, Mayer & Brook 2008; Governato et al. 2009).
+Gasoline ’s star formation and supernovae feedback
+scheme is described in detail in Stinson et al. (2006), and
+parameters from Governato et al. (2007) are adopted for
+MW1. The star formation recipe reproduces the Kennicutt-
+Schmidt law, and each star particle is born with a Kroupa
+initial mass function (Kroupa 1993). Combined with a pre-
+scription for a cosmic UV background (Haardt & Madau
+1996) that mimics reionization and unbinds non-collapsed
+baryons from simulated halos with total masses below about
+109M,thisfeedbackschemedrasticallyreducesthenumber
+of luminous satellites containing a significant mass of stars,
+avoiding the well known “substructure problem” (White &
+Frenk 1991; Kauffmann, White & Guiderdoni 1993; Quinn,
+Katz & Efstathiou 1996; Moore et al. 1999).
+As described in Stinson et al. (2006), supernova energy
+is deposited into the ISM mimicking the blast-wave phase of
+a supernova (following the Sedov-Taylor solution). Cooling
+is turned off for gas particles within the supernova blastwave
+radius for the duration that the remnant is expected to adi-
+abatically expand (§3.8). Supernova feedback regulates star
+formation efficiency as a function of halo mass, resulting
+in the mass–metallicity relationship described above. This
+regulation of star formation also leads to realistic trends in
+1With4:8106total particles within the virial radius at z=0.
+c
0000 RAS, MNRAS 000, 000–000Analytic and numerical realisations of a disk galaxy 3
+gas fractions, with the lowest mass galaxies being the most
+gas rich (Brooks et al. 2007), and reproducing the observed
+incidence rate and column densities of Dampled Lyman 
+systems at z=3 (Pontzen et al. 2008).
+MW1 and its most massive progenitor halo were identi-
+fied at each simulation output step using AHF2(Gill, Knebe
+&Gibson2004;Knollmann&Knebe2009).AHFdetermines
+the virial radius, rv, for each halo at each output time step
+using the overdensity criterion for a flat universe following
+Gross (1997). The full gas accretion history of MW1 is de-
+scribed in detail in Brooks et al. (2009), and used in this
+study for comparison to Galform .
+3 TRANSLATION FROM SIMULATION TO
+SEMI-ANALYTICS
+3.1 The merger tree
+The common point between the two different models of
+galaxy formation is the halo merger tree. This is gener-
+ated by finding bound structures in the “MW1” simulation
+of Brooks et al. (2009) using AHF. A virial mass is assigned
+to each structure and its fate at each timestep is categorised,
+being either:
+1. Isolated;
+2. Inside the virial radius of a larger system; or,
+3. Merged.
+Two criteria are used to classify a galaxy as “merged”:
+(a) If the DM mass decreases by more than 50% in a single
+timestep; or,
+(b) If the number of DM particles falls below 64.
+These are both significantly different from the definition
+adopted by galform (see Cole et al. 2000), so a detailed
+comparison of merger times, along the lines of (Jiang et al.
+2008), is deferred for another study.
+When a structure changes from category 1 to category
+2, the galform code updates its status to a “satellite” and
+predicts its subsequent evolution based on its approach ve-
+locity, vand position rat this juncture. These two proper-
+ties are combined into the dimensionless parameter (§A1),
+which characterises the orbit. The distribution of values for
+in this simulation was found to be consistent with the
+most recent assumption applied in galform (Fig. 1).
+This orbital information is then used to estimate a
+merger time based on the standard Chandrasekhar formula
+(A6) as defined by Lacey & Cole (1993). Details of this part
+of the model can be found in Appendix A and in Cole et al. (
+2000; §4.3).
+The resulting dark and baryonic components are shown
+in Fig. 2. Being generated purely from the derived merger
+tree, the two realisations might be expected to be identical.
+One reason this is not quite the case is because the total
+virial mass, Mv, is measured directly, but the relative com-
+position of baryons and dark matter may vary a little from
+halo to halo, and from one realisation to the other.
+2AMIGA’s Halo Finder, available for download at
+http://popia.ft.uam.es/AMIGA
+0 1 2 3 400.40.81.2
+00.40.81.2
+0 0 1 2 30246Figure 1. Orbital parameters randomly selected from the distri-
+bution (A5) of Cole et al. (2000) are shown in the dashed line,
+while the dotted line shows the distribution from Benson (2005).
+The solid line shows the distribution found in the simulation of
+Brooks et al. (2009) (which results from applying equation (A4)
+to the positions and velocities of the satellites).
+Previously, galform required that the halo mass was
+monotonically increasing as a function of time along each
+branch of the merger tree (i.e. each halo should be more
+massive than the sum of the masses of its progenitor halos).
+While this condition is always fulfilled for merger trees gen-
+erated using the extended Press-Schechter theory, it is not
+necessarily fulfilled for merger trees extracted from N-body
+simulations.
+For this reason, previous use of N-body merger trees in
+galform has required some adjustment of halo masses3to
+enforce this monotonic behaviour (Helly et al. 2003). Un-
+fortunately, this adjustment is somewhat arbitrary and can
+lead to significant changes in halo mass when adjustments
+are propagated through the merger tree.
+Monotonicity in halo masses along branches of the
+merger tree is not enforced in this work. Instead, they are
+allowed to follow whatever evolution the N-body simula-
+tion provides. This requires some modification of the way
+in which galform assigns baryonic masses to halos. Previ-
+ously, each halo was assigned an initial mass of hot gas equal
+to:
+Mhot=
+b
+
+M"
+Mv(new)X
+progenitorsMv(progenitor)#
+;(1)
+whereMvwas the total mass of the halo and “progenitors”
+refers to halos that exist on a previous timestep but have,
+at some point, merged into the current “parent” (or its most
+massive progenitor). If halo masses do not monotonically
+increase, this could lead to negative gas masses. To ensure
+against this, the initial mass of gas is assigned to be the
+greater of (1) and zero.
+3The standard approach which was taken begins at the earliest
+times in the tree, checks whether each halo is more massive than
+the sum of the masses of its progenitor halos and, if it is not, sets
+its mass to be very slightly larger than that sum. This process
+was repeated for all halos in the tree.
+c
0000 RAS, MNRAS 000, 000–0004M. J. Stringer, A. M. Brooks, A. J. Benson, F. Governato
+0 5 10020406080100
+0 5 100510
+Figure 2. Confirmation that the mass evolution of the “MW1”
+system which emerged in the gasoline simulation of Brooks et
+al. (2009) (solid lines) matches that contained in the extracted
+merger history used by galform (dashed lines).
+The results are identical to the original implementation
+in cases where the halo masses are monotonically increasing,
+but this can result in the total baryonic mass not being pre-
+cisely equal to the fraction (
+b=
+M)of the total mass of the
+final halo. Fortunately, this difference is found to be small
+and, in any case, the simulated halos do not have precisely
+this ratio either.
+3.2 Accretion
+The temperaturehistory ofaccreted gas was investegated by
+Kereš et al. (2005) in a simulation containing 1120 galaxies.
+The distribution of gas particles in terms of their maximum
+past temperature is bimodal, suggesting that classification
+into hot and cold modes of accretion is appropriate. The
+contribution from each of these modes is then found to de-
+pend strongly on halo mass and on environment. Notably,
+the halo mass ( Mv31011M) which separates low-
+mass galaxies dominated by cold accretion and higher mass
+galaxies dominated by the hot mode, is close to that found
+by analytic arguments (Birnboim & Dekel 2003).
+The decomposition of the accreting gas into different
+modes is a principle feature of the analysis by Brooks et al.
+(2009). The three modes of accretion are defined as follows
+(along with their analogues in the semi-analytic realisation):
+(a)Clumpy acccretion is defined to include “any parti-
+cles that have, at any previous output step, belonged to a
+galaxy halo other than the main halo we are considering.”
+(Brooks et al. 2009)
+The information needed to evaulate this component in
+the semi-analytic model is readily available from the merger
+tree, being the virial masses, Mv, of the existing, progenitor
+halos:
+Mclumpy (new) =
+b
+
+MX
+progenitorsMv(progenitor) : (2)
+The subsequent fate of any baryons within these substruc-
+tures is decided by their inbound trajectory and consider-
+ations of dynamical friction, as described in §3.1 and Ap-
+pendix A.
+(b)Smooth accretion: Shocked component
+012345
+0 5 100510Figure 3. AnanalysisofaccretionontotheMW1galaxyin gaso-
+line, in the light of assumptions associated with galform . The
+top panel shows the cooling time for a gas particle at the virial ra-
+dius (5) at two relevant temperatures, together with the timescale
+for freefall under gravity (6). In the lower panel, the solid lines
+show how accretion is divided amongst these three modes in the
+simulation. The shaded areas show how the equivalent compo-
+nents would be estimated in galform . As a simple attempt to
+emulate (3a), shocked and unshocked are distinguished by com-
+paring the cooling time at T=9=16Tvto the the freefall time
+(as explained in §3.2b).
+The following criteria are required for a gas particle in the
+simulation to be categorised as shocked:
+(a)T3
+8Tv and (b) 
>43Tv
+3T3
+21;(3)
+whereTvis the virial temperature defined in gasoline
+kBTvGM vmH
+3rv: (4)
+and
is the fractional increase in density since the pre-
+vious timestep. The latter is an entropy criterion and the
+factor of 4 is motivated by the Rankine-Huginot shock jump
+conditions for a singular isothermal sphere. Though low res-
+olution can artificially broaden shocks, Brooks et al. (2009)
+verifythatthisrunisofsufficientlyhighresolutiontoresolve
+shock discontinuities.
+The virial temperature in galform is defined differently,
+being a factor of 3=2higher,Tv3
+2Tv, so (3a) corresponds
+toT9
+16Tv.
+c
0000 RAS, MNRAS 000, 000–000Analytic and numerical realisations of a disk galaxy 5
+Finding an appropriate analogue to this condition within
+theframeworkof galform isdifficult.However,onerelevant
+calulation that ismade compares the cooling timescale for
+gas at temperature T:
+tcool(r) =3mHkBT
+2gas(r)(T;Z); (5)
+and the free fall time from a radius, r, enclosing mean den-
+sity:
+t(r) =r
+3
+8G(r)=rv
+2r
+mH
+kBTv(6)
+The cooling function is found for given temperature T
+and metallicity, Z, by reference to the published tables of
+Sutherland & Dopita (1993). gasis the gas density, (r)
+the mean density of all the mass enclosed by radius r, and
+mHis the mean particle mass.
+So, in the context of galform accreting gas is designated
+as shocked if:
+tcool(Tv;rv)>t(Tv;rv); (7)
+this being a reasonable, simple equivalent to (3a).
+(c)Smooth accretion: Unshocked component
+Gas in the simulation which fails to meet either of
+the criteria in (3) is categorised as unshocked. Similarly,
+accreting gas in galform is designated as unshocked if
+tcool<t.
+The relative contributions of these three components
+are shown in Fig. 3 for both realisations. Being determined
+directly from the merger tree, it is unsurprising that the
+clumpy components are quite consistent.
+That the shocked component is overestimated by the
+semi-analytic model is entirely as anticipated by Brooks et
+al. (2009), though the eventual impact this has on the pre-
+dicted properties of the galaxy may not be quite as severe,
+as will be seen in the forthcoming sections.
+3.2.1 Alternative criterion
+Birnboim & Dekel (2003) investigate the conditions for the
+existence of virial shocks in galactic halos, culminating in
+the following expression for the existence of a shock:
+gas(rv)rv(Tv)
+4 (mH)2ju0j3<0:0126:
+u2
+0=16kBTv
+3mH
+(8)
+Referring to equations (4), (5) and (6) above, this condition
+can be re-written:
+t<0:2tcool: (9)
+As can be seen from Fig.3, the cooling time does not exceed
+the free-fall time by a factor of 5 until about 8 Gyr. The cri-
+terion (9) would therefore imply that all smoothly accreted
+gas will be unshocked before that point, corresponding to
+an unshocked:shocked ratio of 2 or 3, as opposed to 1=2in
+gasoline or 1=6according to (7).
+So,forthisparticularmassaggregationhistory,onesim-
+pleestimate(7),overestimatestheshockedcomponentinthe
+simulation and another, more thorough estimate (8), gives
+an underestimate. This suggests that the error in assumingspherical symmetry may unfortunately exceed any accuracy
+that can be gained by analysis of shock conditions. Calibrat-
+ing some of these analytic models to simulations may help
+quantitatively account for the complex geometry.
+3.3 UV heating
+A UV background is incorporated by Brooks et al. (2009)
+which is based on an updated model by Haardt & Madau
+(1996). This background “turns on at z= 6” and heats the
+gas in halos so that only those with “ 100 or more dark
+matter particles are of sufficient mass to retain bound gas
+particles”. For the resolution in this case, 100 DM particles
+corresponds to Mcut= 7:6107M.
+Thegalform model is build with two corresponding
+parameters: A redshift, zcut, below which the UV back-
+ground heating is effective (set to zcut= 6for consistency
+with the above) and a halo velocity, vcutbelow which all
+cooling is suppressed. For a virial overdensity, 
v= 300,
+the mass cut-off, Mcut, mentioned above translates to a cut-
+off halo velocity of:
+vcut= (
 v=2)1
+6(1 +z)1
+2(
+MGMcutH0)1
+330 km s1(10)
+These values are duly applied, but one key difference be-
+tween the modelling of this process remains. The UV back-
+ground in the simulation can slowcooling in larger halos
+without necessarily completely suppressing it, while in gal-
+formsuppression is all or nothing (below and above vcut
+respectively).
+3.4 Hot gas distribution
+The distribution of hot gas that is assumed in the galform
+model was recently updated, in the light of work by Sharma
+& Steinmetz (2005), to the following density profile for hot
+gas of mass MH:
+(r) =(0)
+(1 +r=rH)3:"
+(0)MH=4r3
+HRrv=rH
+0x2(1 +x)3dx#
+(11)
+This form was assumed in recent work on disk formation
+and structure (Stringer & Benson 2007), and will also be
+used here.
+The reason for this choice is clear from Fig. 4, which
+demonstrates that it is a satisfactory description of the dis-
+tribution of diffuse gas in the outer parts of this simulated
+system. In this Figure, the analytic form for the hot halo gas
+(11)andcolddiskgas(21)havebeenfittedtothesimulation
+results, yielding the respective scalelengths:
+Halo core radius, rH.
+Disk scalelength, rD.
+(with the initial constraint on each line that it integrates
+to the correct total mass for that component.) These scale-
+lengths, found from fitting the simulation results, can be
+scrutinised against those deduced from the usual analytic
+assumptions.
+Firstly, the halo core radius is typically assumed to be
+a constant fraction of the virial radius, which is not quite
+the case in this particular simulation (Fig. 4). However, in
+the absence of any obvious pattern in the evolution of rH,
+the usual simple scaling is applied, with the choice:
+c
0000 RAS, MNRAS 000, 000–0006M. J. Stringer, A. M. Brooks, A. J. Benson, F. Governato
+0 0.4 0.800.20.4
+0 0.4 0.800.20.400.20.4
+00.20.400.20.4
+00.20.4
+Figure 4. The total gas density profile from the simulation of Brooks et al. (2009) at six different redshifts, shown as solid lines. The
+dotted lines show the analytic forms which are assumed by the galform model for the disk gas (eqn. 21, blue line) and diffuse halo gas
+(eq. 11, red line). The sum total of these two components is also shown (magenta line). Each of these depend on a single parameter,
+fitted respectively by the disk scalelength, rDand the core radius of the hot gas profile, rH. (The integral under each analytic component
+is already fixed to match the total hot and cold gas mass within the virial radius.) Because there are so many more points to fit the hot
+component, these are much better matches than the cold gas. As it is the hot scalelength ( notthe cold gas scalelength) which is actually
+an input to galform , this convenient bias has not been corrected.
+rH=rv=3; (12)
+being the best approximation. Fig. 5 compares this to val-
+ues found from fitting to the true gas distributions in Fig.
+4, the result looking favourable. This straightforward scal-
+ing with the virial radius (12) will therefore be used in the
+matched galform realisation. The cold gas scalelengths,also shown in Fig.5, are not inputs to galform , but are
+calculated within the model as described in §3.6.
+To understand the relevance of the hot gas distribution
+on the system’s evolution as a whole, it is worth reviewing
+the precise treatment of gas cooling in galform .
+c
0000 RAS, MNRAS 000, 000–000Analytic and numerical realisations of a disk galaxy 7
+0 5 100123
+0 5 10110100
+Figure 5. The evolution of three key scalelenghts of the sys-
+tem: Virial radius, rv, halo gas scalelength, rH(11), and disk
+scalelength, rD(21). Dashed lines show the values that will be
+assumed by galform . Points show the values which best fit the
+gas profile of the simulation (Fig. 4).
+3.5 Cooling
+ThecoolingmodeldescribedbyColeetal.(2000)determines
+the mass of gas able to cool in any timestep by following
+the propagation of the cooling radius in a notional hot gas
+density profile4. This profile is fixed when a halo is flagged
+as “forming” and is only updated when the halo undergoes
+another formation event. The mass of gas able to cool in any
+given timestep is equal to the mass of gas in this notional
+profile between the cooling radius at the present step and
+that at the previous step. The cooling time is assumed to
+be the time since the formation event of the halo. Any gas
+which is reheated into or accreted by the halo is ignored
+until the next formation event, at which point it is added to
+the hot gas profile of the newly formed halo. The notional
+profile is constructed using the properties (e.g. scale radius,
+virial temperature etc.) of the halo at the formation event
+and retains a fixed metallicity throughout, corresponding to
+the metallicity of the hot gas in the halo at the formation
+event.
+This work makes use of a new cooling model, which
+will be described in full detail in Benson et al. (in prep.).
+Rather than arbitrary “formation” events, this model uses
+a continuously updating estimate of cooling time and halo
+properties.Thepropertiesdescribedin§3.4(densitynormal-
+ization, core radius) are reset at each timestep. The previous
+infall radius5(i.e. the radius within which gas was allowed
+to infall and accrete onto the galaxy) is computed by finding
+the radius which encloses the mass previously removed from
+the hot component in the current notional profile.
+The new model must supply an alternative estimate
+4We refer to this as a “notional” profile since it is taken to repre-
+sent the profile before any cooling can occur. Once some cooling
+occurs presumably the actual profile adjusts in some way to re-
+spond to this and so will no longer look like the notional profile,
+even outside of the cooling radius.
+5defined as the lesser of the cooling and freefall radii.of the time available for cooling in the halo, tavail, from
+which the cooling radius can be computed in the usual way
+(i.e. by finding the radius in the notional profile at which
+tcool=tavail). This is done by considering the energy radia-
+tion rate per particle ,_rand the thermal energy per particle ,
+th, which are estimated by making standard assumptions:
+th=3
+2kBTv and _r= (Tv;Z)n : (13)
+In terms of these quantities, the cooling time (5) is simply:
+tcool(r;t) =th(t)
+_r(rcool;t): (14)
+At any time, the model needs to identify some radius in the
+hot halo,rcool, where the gas has had just enough time to
+radiate all its thermal energy. This as yet undetermined ra-
+dius is defined to satisfy the condition that the cooling time
+for gas particles at this point is equal to the time available
+for them to cool, which is estimated in terms of the mean
+energy radiation rate per particle6,_r:
+tavail(t) =Rt
+0_r(ti)dti
+_r(t): (16)
+Equating (14) and (16) then gives:
+tcool =tavail
+3
+2kBTv
+(t)n(rcool;t)=Rt
+0(ti)n(ti)dti
+(t)n(t); (17)
+where nis the mean number density inside the virial radius.
+This approach must account for the hierarchical assem-
+bly of a halo, so the denominator in the right hand side
+strictly includes a sum over all progenitors, i, of this system
+which exist at time ti. Incorporating this sum into (17), and
+rearranging, then yields a final expression to be solved for
+the cut-off density, cool(rcool):
+cool(t)
+(t)=3
+2kBTvRt
+0P
+ii(ti)ni(ti)
+dti: (18)
+This equation is notably independent of halo structure,
+a factor which only comes in when the limiting density is
+used to compute the total cooled mass. The scalelength, rH,
+appears in the normalisation of the hot gas profile (11), and
+hence effects the mass enclosed by the cut-off density, cool.
+This can be appreciated graphically. Solutions to (18)
+are illustrated in Fig. 6 for three choices of rHand three
+examples of cut-off density. For the higher cut-off density,
+a smallerrH(more centrally concentrated) will lead to a
+greater cooled mass. For the case of a low cut-off density, a
+centrally concentrated profile will actually lead to a lesser
+cooled mass, contrary, perhaps, t
+6This is proportional to the mean number density n:
+_rRN
+0n(N0)dN0
+N=(rH)n (15)
+The factordoes depend on halo structure (i.e. on rH) but, if the
+distribution of hot gas relative to the virial radius is the same for
+each halo, as implied in this case by (12), then this factor cancels
+out in (17).
+c
0000 RAS, MNRAS 000, 000–0008M. J. Stringer, A. M. Brooks, A. J. Benson, F. Governato
+0 0.2 0.4 0.6 0.8 10120.1110
+0 0.2 0.4 0.6 0.8 10.1110
+Figure 6. Three normalised hot gas distributions (solid lines),
+corresponding to three different values of the scalelength rHap-
+plied to the profile (11). Intersections between a solid line and
+one of the dotted lines correspond to solutions to equation (18)
+for a particular value of rHand a particular cut-off density. The
+cooled fraction, Mcool=Mhot, is given by the integral under the
+solid curve up to the intersection with the relevant cut-off density.
+3.6 Disk formation
+Ingalform , galactic disks are assumed to form from the
+cooled gas, conserving the net angular momentum that it
+possessed when it was part of the halo. The assumed dis-
+tribution for the specific angular momentum, j, of the hot
+halo gas is taken from Sharma & Steinmetz (2005):
+dM(j)
+dj/j1exp
+j
+hji
+: (19)
+The value of was found to lie between 0.5 and 1.5, but the
+mean value = 0:89is adopted here. This is normalised to
+enclose the correct total mass and the mean specific angular
+momentum of the hot gas, hji, which is found directly from
+the spin parameter,
+hjijEj1=2
+GM3=2
+v:
+hji=1
+MvZMv
+0jdM(j)
+(20)
+Details of the calculation of the halo energy, E, can be found
+in Cole et al. ( 2000; §3.2.1). The disk scalelength, rD, is re-
+calculated after each timestep such that angular momentum
+is conserved in the system. The calculation is based on the
+following surface density distribution (applied to each disk
+component, i):
+i(R) = i(0)er
+rD;
+soMi= 2i(0)r2
+D
+(21)
+and a circular speed, vc, which is constant with radius. The
+subsequent evolution of the disk is then followed by apply-
+ing physical assumptions which are as close as possible to
+thosemadeinthesimulation,asisexplainedinthefollowing
+sections.
+As with the sub-halo trajectories, the parameter can
+ordinarily be assigned at random, from a specified distri-
+bution. In this case, its value is measured for each of the
+simulated halos and passed on to galform , along with the
+rest of the merger tree information (§3.1).
+Thegoalhereisnottoprovethatparameterslike ,and
+must necesarily be set by simulations in order to validate
+the results of any semi-analytic model. Rather, it is an ex-
+ampleofthecorrespondencebetweentheseinitialconditionsand the properties of the galaxy which they produce. This
+should reassure theorists that, when such models are applied
+to large samples of galaxies, their individual properties will
+be representative as long as these governing, random, but
+physicalparameters are drawn from the correct natural dis-
+tributions.
+3.7 Star formation
+At face value, gasoline andgalform would appear to be
+applying the same assumptions regarding the conversion of
+cold gas to stars. These respective assumptions can be found
+fromequation(4)ofStinsonetal.(2006)andequations(4.4)
+and (4.14) of Cole et al. (2000)7:
+(a)d?
+dt=c?cold
+tdynand ( b) =?Mcold
+disk:(22)
+where is the total instantanoues star formation rate in the
+whole disk, and ?is the local value per unit volume. The
+timescales which appear in these expressions have similar
+physical motivation, but are not identical. One, tdyn, is the
+time for gravitational collapse of a spherically symmetric
+region of local density cold. The other, disk, is the angular
+period of the whole system at the half-mass radius, r1
+2:
+(a)tdyn1pGcoldand ( b)diskr1=2
+vc:(23)
+So, despite the apparent similarity of the two expressions in
+(22), they can lead to quite different instantaneous totalstar
+formation rates,  . To illustrate this, one can consider the
+surface density distribution (21) which galform assumes,
+together with some decrease in density away from the plane
+of the disk:
+i(R;z) =i(R;0)ejzj
+h: [so i(R) = 2i(R;0)h](24)
+Applying the local star formation assumption of (22a) to
+this distribution gives:
+ a= 2p
+Gcold(0;0)3
+2Z1
+1e3jzj
+2hdzZ1
+0e3r
+2rDRdR
+=c?2
+33
+M3=2
+coldr
+G
+4hr2
+D: (25)
+Comparison between this and the usual prescription in gal-
+formexposes the big physical difference between the two
+star formation models; (22a) considers the gravitational col-
+lapse of the gas due to its own gravity while (22b) includes
+the gravity of all components8, including the stars and dark
+matter (Mtotrather than just Mcold).
+The derived analytic equivalent (25) is shown in Fig.7
+(using the fitted values of rDfrom §3.4 and the illustrative
+approximation9of a constant scaleheight, h= 200pc) along-
+side the full simulation results. Discrepancies are larger at
+7An additional factor involving the circular speed appears in
+(4.11) of Cole et al. (2000). This is omitted here (equivalent to
+setting?= 0).
+8The dark matter does influence the creation of stars due to its
+effect on the overall stability, but this is a separate part of the
+analytic treatment (§3.9).
+9This choice is not too critical since happears only to the one
+half power in eqn. 25).
+c
0000 RAS, MNRAS 000, 000–000Analytic and numerical realisations of a disk galaxy 9
+0 5 100123
+0 5 10110100
+Figure 7. The star formation efficiency from the simulation is
+shown as a solid line. The points show the result of applying the
+corrected star formation rate (25) to the disk scalelengths, rD
+which best fit the simulation (Fig.4). The illustrative approxi-
+mation of a constant scaleheight, h= 200pc, is used. The feint
+dotted line shows the efficiency found in empirical studies, which
+is given by eqn. (22b) with ?= 0:017(Kennicutt 1998).
+early times when the assumed cold gas gas density distribu-
+tion is tenuous, but the match improves as the disk settles
+and is excellent at late times, showing that (25) is as consis-
+tent with the assumptions in Gasoline as is possible within
+this particular analytic framework.
+3.7.1 Burst star formation
+During merger events, and if the disk is deemed to be un-
+stable (§3.9), cold disk gas in the galform model is as-
+sumed to funnel to the centre of the new system and be
+converted to stars within a timestep, which in this case is
+about 0.2Gyr (Cole et al.2000; §4.3). The incorporation of
+this non-equilibrium effect should broadly correct the total
+star formation rate on occasions when the assumed den-
+sity profile (21) is not such a good description of the true
+gas distribution. This is relevant here in the early stages of
+the simulation (as apparent when comparing the points and
+solid line in Fig. 7).
+3.7.2 Schmidt-Kennicutt Law
+Equation (22b) can be recognised as a version of the estab-
+lishedempiricalrelationbetweenstarformationrateandgas
+supply (Kennicutt 1998), for which the constant of propor-
+tionality has the value ?0:017. The corresponding star
+formation efficiency for this simulated system can be cal-
+culated from its approximate half-mass radius and circular
+velocity (Fig. 10) and this is shown in Fig. 7 as a guide.
+3.8 Feedback
+3.8.1 Heating of disk gas
+The MW1 simulation treats each type II supernova event as
+a blast wave which expands to a maximum radius, rmax, and
+the enclosed volume is assumed to be unable to cool for as
+long as the shell can be expected to survive, tmax. If the star
+formation rate varies little over this timescale, and the shell
+subsequently cools rapidly, this will correspond to a heated
+mass,
+0 5 1000.050.1
+0 5 1000.20.40.60.81Figure 8. A simple illustration of the correlation between star
+formationrateandthemassofheatedgaspresentinthedisk.The
+effective timescale from equation (28), theat= 7106years. The
+inset panel gives an indication of the almost negligible fraction of
+the total disk gas which this represents.
+Mheat(t)_M?tmax
+mSN
+4coldr3
+max
+3
+_M?(t)theat;(26)
+wheremSNis the average mass of stars formed per super-
+nova. Now, (26) conceals the fact that both rmaxandtmax
+(and hence theat) are, rightly, functions of the local gas den-
+sity,and pressure, P. The precise dependence assumed is
+found in equations 9 and 11 of Stinson et al. (2006). Com-
+bining this into (26) gives:
+theat(0:16P0:2)30:34P0:70:86P1:3:(27)
+Referring to the phase diagram in figure 4 of Stinson
+et al. (2006), gas in star forming regions appears to be ap-
+proximately isothermal at T104K (which is expected due
+to the steep gradient in the cooling curve at this temper-
+ature). This leads to the suggestion that the environmen-
+tal dependence in (27) will be minimal, the pressure and
+density terms almost cancelling for constant temperature.
+Though tenuous, this provides at least a rough explaina-
+tion for the extremely simple dependence which does indeed
+emerge from the full calculation, as shown in Fig. 8:
+Mheat(t)_M?(t)theat:
+theat7106yrs
+(28)
+The precise definition of Mheatused in the figure is the mass
+ofdiskgaswhichfailstosatisfythetemperaturecriterionfor
+star formation ( T < 30;000K). Thus, the integrated effect
+of feedback will be to slightly reduce the fraction of the total
+disk gas which is available for star formation10.
+_M?=McoldMheat
+?=Mcold
+?+theat(29)
+where?is the particular timescale derived from equation
+(25). This revision is duly applied in galform , being a fair
+reflection of the physical assumptions in the simulation.
+10For a massive galaxy such as this, the star-forming timescale is
+typically?>108years, so the correction to allow for re-heating
+actually makes very little difference. Quick examination of the
+inset to Fig. 8 confirms this; the fraction of the total gas which
+gets heated above the star-forming limit is tiny.
+c
0000 RAS, MNRAS 000, 000–00010 M. J. Stringer, A. M. Brooks, A. J. Benson, F. Governato
+0 5 10012
+0 5 100246810
+Figure 9. The dimesionless ratio which is used as a stability
+critereon in the galform model, shown for this particular re-
+alisation (solid line) and for a relisation where the collapse of
+“unstable” disks has not been enforced (dotted line).
+3.8.2 Outflow
+To discover the extent of outflow from the galaxy in the
+simulation, all particles were identified which have moved
+from the inside to the outside of a radius of 30 comoving
+kpc. The total mass of such material turned out to be even
+less than the heated disk mass (see Fig. 8) and can safely be
+ignored for the purposes of this study. The outflow of gas as
+a result of supernovae in the central galaxy is therefor set
+to zero in this version of galform , for consistency with the
+Gasoline simulation.
+It is important to note that this feedback model is more
+effective at lower halo masses, due to shallower potential
+wells(Brooksetal.2007).Thisdependenceisalsoaccounted
+for in the usual galform feedback model, where the rate
+of mass outflow is assumed to be at least equal to the star
+formation rate, and thousands of times higher for smaller
+systems (Bower et al. 2006). The consequence of this differ-
+ence is investigated in §4.3.
+3.9 Disk stability
+The analytic condition which is used to assess the stability
+of disks in the galform model follows Efstathiou, Lake &
+Negroponte (1982). It deems disks to be unstable to bar
+formation if:
+GM disk=rD
+v2c>1:56: (30)
+When this inequality is met, all disk material is transferred
+to a central bulge.
+In the usual galform model, some set fraction of disk
+mass is also transferred to a central black hole (Bower et al.
+2006). This was used to arrive at the value in (30) which
+was found to best match the Magorrian relation between
+bulge mass and black hole mass, MBHM1:12
+bulge, observed
+by Häring & Rix (2004). In this study, the black hole ac-
+cretion fraction is set to zero for the purposes of consitency
+with the simulation, where this aspect of evolution is not
+considered. Being physically well motivated, the condition
+(30) has been applied throughout this study. Conveniently,
+the precise formalism used need not cause too much concern
+here; the change that its application produces in the compo-
+nent masses (in Fig. 11, for example) is barely enough to be
+noticed. This can be understood from Fig. 9, which shows
+0100200300
+0 5 10024Figure 10. The evolution of the disk rotation velocity, vc, (top
+panel) and the disk scalelength, rD, (lower panel) for both realisa-
+tions. The circular velocity of the simulated disk (black squares)
+is taken at 5 comoving kpc from the centre. The approximate gas
+scalelengths (blue squares) are found by fitting (21) to the simu-
+lated gas distributions in Fig. 4. The stellar scalelengths (red tri-
+angles) are taken from fitting (21) the the stellar profiles, shown
+later in Fig. B1. Dashed lines are the values calculated in gal-
+formfrom conservation of angular momentum (§3.6 - gas and
+stars are assumed to have the same scalelength).
+that only a very limited adjustment to the disk mass is re-
+quired to prevent the limit in (30) from being reached. The
+mass aggregation of this particular system, and the fact that
+thespecificangularmomentumofthegasisrelativelyhigh11
+( > 0:07for the latter half of the halo’s history) leads to
+the stabilisation of the system, even if the redistribution of
+mass is not enforced.
+4 RESULTS
+4.1 Matched Model
+4.1.1 Structure
+The size and rotation speed are shown for both realisations
+in Fig. 10. The predicted galform value is based on con-
+serving the initial angular momentum (19) of in-falling gas;
+an assumption which is particularly effective in this case
+where the distributions for the initial and final mass were
+appropriate (Fig.4). This correspondence is encouraging for
+the approach of calculating disk sizes from the spin param-
+eter,, of the halo from which the gas cooled.
+11The criterion (30) can also be written:
+Mdisk>1:2[MDM(r1=2) +Msph(r1=2)]; (31)
+whereMDMandMsphare the masses of the dark matter and
+spheroid components. The disk’s eventual half mass radius will
+enclose more dark matter if the original spin parameter is higher.
+c
0000 RAS, MNRAS 000, 000–000Analytic and numerical realisations of a disk galaxy 11
+0 5 10020406080100
+0 5 100510
+Figure 11. The evolution of hot halo gas, cold disk gas and stars
+according to both modeling approaches. The star formation rate
+ingalform is calculated using assumptions identical to those
+in the simulation (22a), but taking into account their integrated
+effect over the whole disk profile (21).
+4.1.2 Cooled Mass
+The total cooled mass (stars and cold gas), calculated using
+the methods of §3.2, can be seen from Fig. 11. Agreement
+withthesimulationisexcellent,giventheinherentdifference
+in the methods of calculation, and bolsters confidence in the
+rather intricate calculation of the energy radiated by the
+system throughout its complex merger history (13-18).
+The division of the cooled mass into stars and gas are
+also shown in Fig. 11. The evident agreement can be under-
+stood by reference to Fig’s 4, 7 and 8, which demonstrate
+that the considerable complexities of the simulation reduce
+to relatively simple relationships when integrated over the
+entire disk.
+4.1.3 Discrepancies
+There is a minor discrepancy at early times ( t < 8Gyr)
+in the cooled mass generated by the two modelling tech-
+niques. This is unlikely to be due to differences in cooling
+from the hot halo: given the similar density distributions
+(Fig. 4) and cooling function, the two models should yield
+similar amounts of cooled gas, all else being equal.
+The discrepancy is also not due to a difference in the
+time of development of a hot gas halo. The top panel of Fig.
+4 shows that accreted gas in the galform model transitions
+from unshocked, cold gas to shocked, hot gas at an age of
+about 2 Gyr, while the bottom panel shows that this time is
+also when the gasoline model begins to develop a hot gas
+halo.
+This demonstrates that the adopted analytic separation
+of unshocked and shocked gas (free fall limited regime versus
+cooling limited regime) is an excellent analog to the transi-
+tion between cold gas accretion and shocked gas accretion
+seeninsimulationsasagalaxycrossesthethresholdmassca-
+pable of developing a hot halo. It is important to note that
+0246
+0 5 100510Figure 12. Having focussed on modifications to galform , this
+figure shows the effects of modifying the SPH code gasoline to
+emulate the semi-analytic technique. The solid black line is the
+star formation rate (lower panel) and its integral (upper panel)
+whichoccursinthestandard gasoline simulationwhichhasbeen
+under discussion. The dotted red lines show the consequence of
+delaying star formation from cold accreted gas is delayed by a
+cooling time.
+this transition has always been included in analytic mod-
+els of galaxy formation, and thus cold gas accretion at low
+galaxy mass does not alter the standard picture of galaxy
+formation.
+The discrepancy in cooled mass arises after the devel-
+opment of this hot halo. The simulated galaxy continues to
+accrete cold gas via filaments that penetrate within the hot
+gas halo (see Fig. 3 here, figure 5 of Brooks et al. (2009)
+and also Ocvirk et al. (2008), Agertz et al. (2009) and Dekel
+et al. (2009)), while cold gas accretion ends in the analytic
+model.
+4.2 Adapting gasoline to emulate semi-analytics
+Brooks et al. (2009) demonstrated that the inclusion of cold
+gas accretion along filaments leads to an earlier phase of star
+formationthanpredictedifallgaswasinitiallyshockheated,
+due to the shorter cooling times onto the central galaxy of
+the cold gas. Fig. 12 shows that the simulated galaxy does
+indeed form stars earlier than the analytic model, though
+the overall discrepancy is never more than a factor of two.
+In an attempt to adapt the simulations to include simi-
+lar physics as the analyticmodel, Fig. 12 shows the effect on
+the stellar growth in the simulations if all of the cold gas had
+instead been shocked. The hot gas has cooling times several
+Gyr longer than the cold gas. This is quantified, and a delay
+has been added to the formation time of the stars spawned
+from the cold accreted gas (Brooks et al. 2009). As seen in
+Fig. 12, this delay leads to a slower build of up stellar mass
+in the simulations. Comparison with Fig.11 shows this to
+be in even better agreement with the stellar growth of the
+c
0000 RAS, MNRAS 000, 000–00012 M. J. Stringer, A. M. Brooks, A. J. Benson, F. Governato
+012
+0 5 10051015
+Figure 13. The results of applying the unadjusted version of
+galform (Bower et al. 2006) to the same merger tree as used
+for Fig. 11 and indeed the whole paper. To bring out the contrast
+betweenthetwomodels,theshadingnowrepresentsthe galform
+predictions and the lines show the simulation results. The upper
+panel is the equivalent to Fig. 9, showing the stability of the disk
+in this version for this model.
+analytic model, which neglects this cold gas accretion along
+filaments.
+4.3 Comparison with standard galform
+assumptions
+In order to represent the same physical assumptions as this
+particular simulation, the semi-analytic code had to be sig-
+nificantly modified from the version which has been used
+most recently to study the collective properties of galaxy
+samples (Bower et al. 2006; Stringer et al. 2009). There
+are three very significant differences between this version of
+galform and the simulation studied here, all of which are
+manifest in the comparison of the history of the simulation
+and the Bower et al. (2006) model (Fig. 13).
+4.3.1 Inclusion of AGN feedback
+When there are instabilities (§3.9) or merger events in this
+model which trigger disk collapse, 5% of the available disk
+gas is assumed to accrete onto a central black hole. If thefollowing criteria are satisfied, it is assumed that no hot halo
+gas will be able to cool onto the disk.
+(1) The resulting Eddington luminosity of the black hole is
+at least 4% of the cooling luminosity of the halo.
+(2) The cooling time (5) exceeds the freefall time (6) by
+some chosen factor (in this case, tcool>0:58t).
+This effect is visible in Fig. 13 as a period of rapidly decreas-
+ing cold gas mass; during such phases it is no longer being
+replenished by gas cooling in from the halo.
+4.3.2 Stronger supernova feedback
+This is assumed to be extremely strong in the Bower et al.
+(2006) model, the justification being that a more modest
+conversion of supernova energy to gas outflow would allow
+the formation of too many low-mass galaxies. (Indeed, it
+appears that the predicted number may still be too high
+(Stringer et al. 2009) even with this generous allocation of
+feedback energy.)
+In conjunction with the inclusion of AGN heating, this
+assumption leads to a system with a much greater hot gas
+component: As soon as the supply of gas to the disk can no
+longer be replenished by cooling, all the gas in the system
+quickly ends up in the hot component. This effect is ad-
+ditionally enforced by the low star formation efficiency (see
+below), which prevents disk mass being locked up into stars.
+4.3.3 Inefficient star formation
+The star formation assumptions in Gasoline and the Cole
+et al. (2000) model have already been contrasted in §3.7.
+The assumed efficiency in the Bower et al. (2006) model is
+to be so low ( ?= 0:0029) that, even after allowing for the
+structure and gas fraction, the final conversion rate is about
+a factor of 10 lowerthan in the simulation. ( _M?=Mcold0
+to 0.2/Gyr as opposed to the rate of 0.5 to 3/Gyr seen in
+Fig. 7).
+This stands out in Fig 13. The large mass of cold gas
+which is supported by the disk in the semi-analytic realisa-
+tion (except when cooling is shut off by AGN) cannot be
+sustained in the simulation due to much more effective con-
+version to stars.
+5 SUMMARY
+The merger history of a simulated, Milky Way-like galaxy
+has been used to recreate the same system using the gal-
+formsemi-analytic model.
+The incoming trajectories of satellite halos were inves-
+tigated and found to be consistent with the usual, assumed
+distributions. Information on the different modes of accre-
+tion onto the main halo is shown to be inherently contained
+in the merger tree (and would be similarly contained in a
+statistically generated merger tree which may be used in the
+study of larger volumes).
+The subsequent fate of accreted gas is not modelled
+equivalentlybythetwomethods,buttheanalyticframework
+doesaccount for both shocked and unshocked gas. Further-
+more, the estimated formation time of the hot halo is in
+excellent agreement with the simulation.
+c
0000 RAS, MNRAS 000, 000–000Analytic and numerical realisations of a disk galaxy 13
+The single-parameter analytic distributions which are
+assumed in galform are found to be a satisfactory descrip-
+tion of the radial distribution of gas and stars inside the
+simulated halo. It was also shown that the choice of hot
+halo scalelength is of minor importance in the calculation of
+cooled gas mass.
+However, the enforcement of spherical symmetry in
+these analytic distributions means that any subsequent cold
+accretion along filaments (persisting in spite of shocks else-
+where) is neglected. Thus, the totalquantity of unshocked
+gas onto the central galaxy is underestimated, as expected
+by Brooks et al. (2009). Some gas is consequently delayed in
+itsarrivalatthecentralgalaxy,thoughitisfoundtoproceed
+at later times.
+The disk structure is predicted analytically by conserv-
+ing the angular momentum of the cooled gas. Resulting
+scalelengths are within a factor of two of the simulation re-
+sults and the circular velocities even closer. This is quite
+satisfactory for the case of a single halo, but more work is
+needed to establish whether the two techniques will con-
+sistently agree to this level for an arbitrary set of initial
+conditions (and whether any disagreement is biased).
+With regard to the subsequent evolution of the disk,
+thegalform code was altered to apply the same funda-
+mental assumptions as the simulation by considering their
+integrated effect over the assumed disk structure . Subsequent
+agreement was excellent given this immense reduction in
+complexity from many thousands of SPH particles to a few
+one-dimensional equations.
+So, by assuming the same physics and using the same
+initial conditions, the two techniques predict a final system
+which is recognisably the same, despite fundamental differ-
+ences in the way that these assumptions are applied and the
+evolution followed. This suggests that equations which at-
+tempt to characterise the emergent behaviour of the system
+may, with sufficient understanding, become as reliable as
+those which emulate the properties of particles themselves.
+With such potential consistency thus established, the
+same system was then evolved under physical assumptions
+which have been more frequently adopted within galform
+(Bower et al. 2006). The resulting system (at any time) is
+notrecognisably the same as that predicted by the simula-
+tion: having less than a quarter of the stellar mass, barely
+half the rotational velocity and with negligible star forma-
+tion at z=0. Though not all aspects of a system need to be
+correctly represented to learn from the modelling exercise,
+this divergence from the same initial conditions proves that
+at least one of these published models poorly represents the
+true nature of galaxy formation.
+It is hoped that this detailed comparison between two
+verydifferenttheoreticalapproacheswillpromotetheaware-
+ness of their respective limitations and help lead them both
+towards a truer description of real galaxies.
+ACKNOWLEDGMENTS
+The authors would primarily like to thank the KITP, Santa
+Barbara for their hospitality, supported in part by the Na-
+tional Science Foundation under Grant No. PHY05-51164,
+without which this interdisciplinary study would not have
+taken place.AMBacknowledgessupportfromtheShermanFairchild
+Foundation. AJB acknowledges support from the Gordon &
+Betty Moore Foundation. FG acknowledges support from
+the HST GO-1125, NSF AST-0607819 and NASA ATP
+NNX08AG84G grants.
+Thanks go to Richard Bower, Shaun Cole and Car-
+los Frenk for their very helpful comments. Also, they and
+their collegues Carlton Baugh, John Helly, Cedric Lacey
+and Rowena Malbon allowed us to use the galform semi-
+analytic model of galaxy formation ( www.galform.org ) in
+this work, for which we are extremely grateful.
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+arXiv:0904.3333
+APPENDIX A: SATELLITES
+The orbits of sub halos around their host can be charac-
+terised (Lacey & Cole 1993) by the dimensionless parameter
+ =
+j
+jc(E)0:78
+rc(E)
+rv2
+; (A1)
+whererc(E)andjc(E)are the radius and specific angular
+momentum of a circular orbit with energy E. For a potential
+with a flat rotation curve; where the circular velocity at all
+radii isvc,
+(r) = (rv) +v2
+clnr
+rv
+: (A2)
+Hence,forasubhaloatposition, rwithvelocity v,theradius
+of a circular orbit with the same energy is:
+rc(v;r) =jrjexp
+v2
+2v2c1
+2
+: (A3)
+The orbital parameters are found by simply applying equa-
+tions (A1) to (A3) to the position and velocity vectors of
+each satellite halo (after moving to a frame where these vec-
+tors are both 0for the host halo):
+ =j
+vcrv0:78
+rc(E)
+rv1:22
+0 1 2 3 400.40.81.2
+00.40.81.2
+0 0 1 2 30246Figure A1. Left panel: The velocities and radii of the satellite
+halos in the simulation of Brooks et al. (2009), at the timestep
+immediately before they are deemed to have merged with their
+host. The dots are those which appear in the distribution of 
+in Fig. 1, crosses are those that have >4:5and hence do not
+appear. The line shows the constant value of corresponding
+to that cut-off, for the given approach angle. Right panel: The
+distribution of approach angles of these satellites alongside the
+distribution found from the study by Benson (2005).
+=
+jvrj
+vcrv0:78
+jrj
+rv1:22
+exp
+0:61
+v2
+v2c1
+:(A4)
+Thegalform model assigns each satellite a value for this
+parameter, , drawn at random from the log-normal distri-
+bution that it was found to follow in simulations by Tormen
+(1997):
+d lnn
+d ln =1
+p
+2exp
+(ln  + 0:14)2
+22
+; (A5)
+with= 0:26. The sample of orbit parameters produced by
+this random assignment is shown in Fig. A1, together with
+the actual values from the Brooks et al. (2009) simulation.
+The fraction of the halo free-fall time that it takes for
+the satellite to merge through dynamical friction is assumed
+to be proportional to :
+mrg
+[Gv(z)]1
+2= 0:3722
+ln (Mhost=Msat)Mhost
+Msat: (A6)
+ThisisbasedonthestandardChandrasekharformulaforthe
+dynamical friction and appears originally in Lacey & Cole
+(1993).vis the mean density at the virial radius (deter-
+mined by the cosmology, not the specific halo’s properties).
+The distribution of angles at which sub structures en-
+ter their host halo has been studied by Benson (2005) for
+the VLS and VIRGO simulations and and found to have
+a repeatable distribution. This distribution can be applied
+in the galform model to generate an alternative set of or-
+bital parameters, , which are shown in Fig. A1 alongside
+the standard assumption of the log-normal distribution, and
+the results from gasoline .
+Merger times predicted by (A6) have been compared by
+Jiangetal.(2008)withtheresultsofGADGET2simulations
+(Springel 2005). The agreement found was of the order of a
+factor of two, which is deemed acceptable for the continued
+use of this formula in galform . However, since (A6) was
+found to consistently underestimate the simulated merger
+time, the improved fitting formula proposed byJiang et al.
+(2008) may well be adopted in future.
+c
0000 RAS, MNRAS 000, 000–000Analytic and numerical realisations of a disk galaxy 15
+Unfortunately, due to the very different definition of
+mergingadoptedbythehalofinderusedhere(§3.1),amean-
+ingful comparison of the respective times in the two realisa-
+tion has not been possible.
+APPENDIX B: DISK PROFILES
+The stellar disk radii that appear in Fig. 5 are generated
+from analysis of the distribution of stars in the simulation.
+Fig. B1 compares the distribution of stellar mass from the
+simulation with the analytic forms assumed by galform ,
+both for disk stars (21) and for the mass of stars in the
+bulge, assumed to be distributed such that the projected
+surface density profile is given by:
+bulge/exp
+r
+rb1
+4
+: (B1)
+Though the agreement is not comprehensive, the ana-
+lytic forms are an adequate description of the global distri-
+butionofsimulatedstarsforthemajorityofthesystemsevo-
+lution. Unsurprisingly, the simple profiles of (21) and (B1)
+fail to describe the simulated system at early times, before
+an ordered galactic system has formed.
+The fact that galform adheres, at alltimes, to the fit-
+ting forms that are really only appropriate to recent times
+(z <2) is clearly a valid criticism of this technique. How-
+ever, as long as such forms are indeed a good description
+of low-redshift systems, then the only way their earlier mis-
+judgements can significantly mislead the predictions of the
+model isintheearlytimestar formation rates(Fig.7),which
+are based on this assumed structure.
+0 5 0 500.20.400.20.400.20.4Figure B1. The stellar mass profile from the simulation of
+Brooks et al. (2009) at six different redshifts (solid black lines).
+The dotted blue lines show the analytic forms which are assumed
+by the galform model for the disk (21). Dotted red lines show
+the bulge components (B1). The total stellar contribution, shown
+by dotted magenta lines, is constrained to integrate to the same
+mass as the soild lines. The two lines are therefore fitted using
+three free parameters: The relative fraction of mass in the two
+components, the disk scalelength, rDand the scalelength of the
+bulge profile, rb.
+c
0000 RAS, MNRAS 000, 000–000
\ No newline at end of file
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+arXiv:1001.0595v4  [q-bio.PE]  27 Sep 2011FINDING THE PROBABILITY OF INFECTION IN AN SIR
+NETWORK IS NP-HARD
+MICHAEL SHAPIRO AND EDGAR DELGADO-ECKERT
+Abstract. A common approach in epidemiology is to study the transmissi on
+of a disease in a population where each individual is initial ly susceptible (S),
+may become infective (I) and then removed or recovered (R) an d plays no
+further epidemiological role. Much of the recent work gives explicit consid-
+eration to the network of social interactions or disease-tr ansmitting contacts
+and attendant probability of transmission for each interac ting pair. The state
+of such a network is an assignment of the values {S,I,R}to its members.
+Given such a network, an initial state and a particular susce ptible individual,
+we would like to compute their probability of becoming infec ted in the course
+of an epidemic. It turns out that this and related problems ar eNP-hard. In
+particular, it belongs in a class of problems for which no effic ient algorithms
+for their solution are known. Moreover, finding an efficient al gorithm for the
+solution of any problem in this class would entail a major bre akthrough in
+theoretical computer science.
+1.Introduction
+Mathematical modelling of epidemics is often traced to the celebrate d SIR model
+of Kermack and McKendrick [1]. This model posits a population of cons tant size
+whose members fall into one of three classes: susceptible (S), infe ctive (I) and
+removed (R). Approximating these as continuous and assuming well- mixing, i.e.,
+each individual is in equal contact with and equally likely to infect each o ther
+individual, allows for an approximate description of the infection dyna mics using
+ordinary differential equations (ODE).
+Clearly, as it has been argued by many in theoretical [2, 3, 4] as well as exper-
+imental studies [5], the well-mixing assumption is not an accurate repr esentation
+of real contact patterns. Thus, much recent work has focused on the role of the
+network of disease-transmitting contacts. (Reviewed in [6]. See a lso, [7, 8]. For a
+comparison of well-mixed and network-based models, see [9].) Indee d, Kermack’s
+and McKendrick’s ODE model arises as the limiting case of a simplistic net work
+model in which each individual has an equal chance of infecting every other. How-
+ever, real-worldsocial contact networksexhibit complex pattern s ofinterconnection
+between individuals. Further, the probability of transmitting diseas e from one indi-
+vidual to another depends on the nature, frequency and duratio n of the contact as
+well as the immune competence of the target individual. This leads to a modelling
+formalism of social networks as a probabilistic graph G= (G,Pr). Here Gis the
+graphG= (V,E), each vertex u∈Vis an individual, each edge e= (u,v)∈E
+records the fact that umight infect vand Pr : E→[0,1] gives the probability that
+Key words and phrases. epidemics, SIR networks, contact network, network reliabi lity, NP-
+hard.
+The first author wishes to acknowledge support from NIH Grant K25 AI079404-02.
+12 MICHAEL SHAPIRO AND EDGAR DELGADO-ECKERT
+uinfectsvifubecomes infective while vis susceptible. In this formalism, Gis a
+fixed graph Gwith labelling Pr.
+This relativelynew modelling paradigm has triggeredan enormous amou nt ofre-
+search in theoretical epidemiology. The field has greatly benefited f rom approaches
+that range from applications of bond percolation theory and other techniques from
+statistical physics [10, 11, 3, 12, 13, 14, 15, 16, 17, 18] to large s cale simulation en-
+deavours[4, 19, 20, 21, 22]. Given that this mathematical formalism seems accurate
+andpowerfultodescribethespreadofinfectiousdiseases,then aturalquestionarises
+as to whether calculations performed within this formalism can be use d in practical
+situations to make useful predictions. Such calculations are based on potentially
+measurable parameters such as network topology and transmissio n probabilities
+[23]. For instance, one could attempt to calculate the probability tha t, given a
+social contact network G, an epidemic starting with a set Pof infectives results in
+the infection of an initially susceptible individual u. Are there any computational
+limitationswhentryingtocalculatesuchmagnitudes? Ifyes, howlimitin g arethey?
+Fortunately, to address the computational issues associated wit h this and similar
+calculations, we don’t need to start from scratch, given that netw ork engineers have
+already studied since the 1970s problems that are essentially the sa me.
+Intheeraofelectronicallydigitalizedinformationanddigitalcompute rs,commu-
+nications networks have become the biggest and count among the m ost important
+networks. The size of these networks is exponentially increasing. F or instance, the
+size of the Internet shows exponential growth since its creation in the early nineties
+(http://www.isc.org/). As the components of such networks are s ubject to failure,
+engineers face the problem of designing, constructing and operat ing networks that
+meet the required standards of reliability. Of particular interest is t he estimation of
+how reliable a given networkis in performing its function, provided som e knowledge
+about the reliability of its components is available. In many cases, the function-
+ality of the network can be expressed as the ability of its topology to support the
+network’s operation. In other words, the network is functional if and only if certain
+connectivity properties are fulfilled. Consider a network of comput ers which use
+this network to transmit messages. Let us suppose that each of t hese computers is
+reliable, but that each communication link has some chance of failure w hen called
+upon to transmit a message. We then encounter the same formalism explained
+above for social networks. A communications network is given by G= (G,Pr)
+where each vertex u∈Vis a computer, each edge e= (u,v)∈Eis a communica-
+tion link and Pr : E→[0,1] is the reliability of the communication link from uto
+v. One might ask, given a communications network G, a set of computers Pand
+a computer u /∈P, if the computers in Pall send a message, what is the chance
+it will reach u? We will see that this is the same problem we stated above in the
+context of epidemics on social contact networks.
+It has long been known in the communications network literature tha t this prob-
+lem is computationally intractable. A standard benchmark of comput ational com-
+plexity is the class of NP-complete problems. This class has the following proper-
+ties:
+•At present, no algorithm for an NP-complete problem is known to have a
+running time which is bounded by a polynomial. Indeed, many algorithms
+forNP-complete problems have exponential running time. It is unknownPROBABILITY OF INFECTION IN AN SIR NETWORK IS NP-HARD 3
+whether any NP-complete problem can be solved by an algorithm with
+polynomial running time.
+•If any problem in this class can be solved by an algorithm whose running
+time is bounded by a polynomial, then every problem in this class can be
+solved by an algorithm whose running time is bounded by a polynomial.
+In view of the second, it is considered unlikely that any NP-complete problem
+has a polynomial time solution. The communication among computers p roblem
+(and hence the epidemiology problem) listed above is known to be as ha rd as any
+NP-complete problem. Such problems are termed NP-hard. This is not the first
+problem in network epidemiology known to be NP-hard. Previously known exam-
+ples include the following: Given a social contact network and limited re sources
+•What is the optimal strategy for vaccinating a limited number of individ -
+uals?
+•What is the optimal strategy for quarantining a limited number of indiv id-
+uals?
+•What is the optimal strategy for placement of a limited number of sen sors
+for monitoring the course of an epidemic?
+(See [24, 25, 26, 27].) These problems involve the search for an optim um among
+subsets of the vertices or edges of the given social contact netw ork. It might be
+hoped that finding the probability of infection of a single individual wou ld be com-
+putationally less demanding. As the engineers have taught us, this is not so. While
+this result has been recently reported in the physics and operation s research com-
+munity [28], it seems almost unknown among epidemiologists.
+This article is organized as follows: In Section 2 we give a very brief ove rview
+of the relevant concepts and methods in computational complexity . This provides
+the unacquainted reader with the basic tools for understanding th e main message
+of this paper. Section 3 provides the elementary formal mathemat ical framework
+for studying SIR epidemics on networks, including the connection wit h percolation
+theory. In Section 4 we present a series of problems that have bee n studied in
+network engineering and demonstrate their structural isomorph ism with certain
+problems concerning SIR epidemics on networks. Section 5 is devote d to studying
+the computational complexity of extended/generalized epidemiolog ical problems.
+We finish in Section 6 with some concluding remarks.
+2.Computational complexity
+In this section we give a brief account of the class NP-complete. This class is a
+common benchmark for describing problems which are algorithmically s oluble but
+computationally intractable. For those wishing a fuller account we re commend [29].
+In describing the class NP-complete, it is useful to describe the class P, and
+necessary to describe the class NP. These classes of problems are defined in terms
+of computational complexity.
+The computational complexity of a problem Π is measured in terms of t he run-
+ning time necessary for an algorithm which solves Π. Defining these te rms requires
+somepreliminaries. First, notethata problemΠconsistsofacollectionof instances ,
+DΠ. Thus, “Determinewhether18iscomposite”isaninstanceofthe pr oblem, “For
+any integer n, determine whether nis composite.” This is an example of a decision
+problem, that is, for each instance, the answer is either “yes” or “no”. A d ecision
+problem Π can be formalized as the pair ( DΠ,YΠ), where YΠ⊂DΠconsists of4 MICHAEL SHAPIRO AND EDGAR DELGADO-ECKERT
+theyes instances . In this example, DΠis the set of integers and YΠis the set of
+composite integers. We will refer to this problem as Π composite.
+Notice that each instance π∈Π has asize,ℓ(π) and that the computational cost
+of solving the problem grows with the size of the problem. In this exam ple, the size
+ℓ(n) of the instance nis the number of digits in n. If we then have an algorithm
+Mwhich solves Π, we can consider the running time rM(π) required by Mwhen
+applied to the instance π. This could be measured in elapsed time or in terms of
+the number of steps carried out by Min this computation. We can then define the
+running time ofMto be
+rM(n) =
+
+0
+if{π|ℓ(π) =n}=∅
+max{rM(π)|ℓ(π) =n}
+otherwise
+The class Pconsists of those decision problems which can be solved with a poly-
+nomial running time. Stated formally, a decision problem Π belongs to t he class
+Pif there is an algorithm Mwhich solves Π and a polynomial p(n) such that
+rM(n)≤p(n). An example of a problem in the class Pis Πmult. An instance of
+Πmultis three integers, a,bandc. The size of an instance is the total number of
+digits in a,bandc. These constitute a yes instance if a×b=c.
+The class NPconsists of non-deterministic polynomial time problems. That is,
+a decision problem is NPif a machine which is allowed to guess can verify a yes
+instance in polynomial time. Π composite provides and example of a problem which
+isNP. Given an instance of Π composite, i.e., an integer c, ifcis, in fact, composite, a
+correct guess as to its factors aandb, can be verified in polynomial time by calling
+Πmult. One can define this class in terms of the operation of non-determin istic
+Turing machines. See, for example, [30]. Clearly P⊆NP. In view of the perceived
+complexity of many problems in NP, it is generally believed that P/\e}atio\slash=NP.
+The class NP-complete consists of the hardest problems in NP. The problems
+inNP-complete have the following property: Suppose that Π 1isNP-complete.
+SupposethatΠ 2isNP.Thenthereisanalgorithm Mwhichtranslatesanyinstance
+π2of Π2into an instance π1of Π1such that π1is a yes instance of Π 1if and only
+ifπ2is a yes instance of Π 2. Further, both the computational cost of translating
+π2intoπ1and the size ℓ(π1) are bounded by a polynomial in ℓ(π2). It follows
+that if any NP-complete problem can be solved (deterministically) in polynomial
+time, then every NPproblem can be solved in polynomial time. Put another way,
+if anyNP-complete problem can be solved in polynomial time, we will then have
+P=NP.
+Hundredsofproblemsareknowntobe NP-complete[29]. Thesecomefromfields
+such as graph theory, number theory, scheduling, code optimizat ion and many oth-
+ers. They are widely believed to be intrinsically intractable, but this re mains an
+open question. Other problems which are not necessarily NP-complete (e.g., be-
+cause they are not decision problems) are known to be at least as ha rd. This is
+because for such a problem, say Γ ,there is an NP-complete problem Π that can be
+reducedto Γ,where the computational cost of this reduction is bounded by a poly -
+nomial in the length of the instance problem considered. Thus, Γ can be used to
+solve Π.These problems are called NP-hard. Since NP-complete problems trans-
+form to each other, all NP-complete problems can be solved by a reduction to anPROBABILITY OF INFECTION IN AN SIR NETWORK IS NP-HARD 5
+NP-hardproblem. NP-hardproblemsarefoundinfieldsasdiverseasepidemiology
+and origami [31].
+3.SIR epidemics on networks
+We start by describing a network SIR model in which both the populat ion and
+the individual transmission probabilities are constant with respect t o time.
+A state of this system is the assignment of each individual to one of t he classes
+S, I or R. The transmission probabilities determine who can infect who m and con-
+sequently which states can follow a given state. Indeed, they also d etermine the
+probability that any one of these states follows the given state. An epidemic is a
+sequence of states each of which is a possible successor of the pre vious state. Con-
+sequently, given an initial state, we can speak of the probability tha t an epidemic
+evolves through a given sequence of states and the probability tha t it arrives at a
+particular state. Let us formalize this.
+As above, a social contact network is a pair G= (G,Pr) where Gis the graph
+with vertex set Vand edge set E. Each edge has the form ( u,v) withu,v∈Vand
+u/\e}atio\slash=v. The function Pr assigns a probability to each edge, that is Pr : E→[0,1].
+ThestatesofGare given by1
+St(G) ={ϕ|ϕ:V→ {S,I,R}}.
+Given states ϕ1andϕ2, the state ϕ2is apossible successor ofϕ1if it satisfies
+the following conditions:
+(1) Ifϕ1(u) =R, thenϕ2(u) =R. (Recovered individuals stay recovered.)
+(2) Ifϕ1(u) =I, thenϕ2(u) =R. (Infected individuals recover in one step.)
+(3) Ifϕ1(u) =S, thenϕ2(u)∈ {S,I}. (Susceptible individuals either stay
+susceptible or become infected.)
+(4) Ifϕ2(u) =I, thenϕ1(u) =Sand there is a vertex q∈V\{u}and an
+edge (q,u) withϕ1(q) =I. (Infected individuals were susceptible and were
+infected by a neighbour.)
+The requirement that individuals recover in exactly on time-step migh t appear
+to be a drastic oversimplification. However, the formalism is rich enou gh to ac-
+commodate patterns of latency and extended periods of infectivit y. This can be
+done by replacing the individual represented by vertex uby a sequence of vertices
+u1,u2,...representing uon day 1, uon day 2, etc. See, e.g., [32].
+Anepidemic Φ is a sequence of states ϕ1,...,ϕ kwhereϕi+1is a possible suc-
+cessor of ϕifori= 1,...,k−1. The length of this epidemic is ℓ(Φ) =k. Since
+individuals recover after one step, infection must be transmitted o r die out. As a
+consequence, no epidemic can be longerthan the longest self-avoid ingpath in G, for
+otherwise, it must infect some vertex twice. If we assume that eac h edge transmits
+or fails to transmit independently, then it is not hard to compute the probability
+that a susceptible individual is infected by its infected neighbours. T his, in turn,
+allows one to compute the probability that a state ϕ1is followed by a particular
+successor state ϕ2. Let us denote this probability by Pr( ϕ2|ϕ1). This system en-
+joys the Markov property, that is, the probability of a given state depends only on
+1In particular, a state ϕcan be seen as a subset of the Cartesian product V× {S,I,R},and
+therefore, it is meaningful to speak of the probability of a s tate or of a collection of states.6 MICHAEL SHAPIRO AND EDGAR DELGADO-ECKERT
+the previous state. Thus given an initial state ϕ1, the probability of the epidemic
+Φ =ϕ1,...,ϕ n, is
+Pr(Φ|ϕ1) =n/productdisplay
+i=2Pr(ϕi|ϕi−1).
+The probability that ubecomes infected at the nthstep in the courseof an epidemic
+starting with ϕ1is
+Pr(ϕn(u) =I|ϕ1) =/summationdisplay
+{ϕ1,...,ϕn|
+ϕn(u)=I}Pr(ϕ1,...,ϕ n|ϕ1).
+Abusing notation, we denote the probability that ubecomes infected in the course
+of some epidemic starting with ϕ1by
+Pr(u|ϕ1) =n/summationdisplay
+j=1Pr(ϕj(u) =I|ϕ1).
+Note that since an infected individual becomes recovered at the ne xt stage, no
+epidemic appearing in this sum is an initial sub-epidemic of another. Acc ordingly,
+these are disjoint cases.
+We will be interested in initial states ϕ1consisting only of infectives and suscep-
+tibles. In this case, we can identify ϕ1with the set of infectives P=ϕ−1
+1(I). This
+gives the notation Pr( u|P).
+Let us formalize the problem Π epidemicof finding Pr( u|P). An instance πof
+this problem consists of
+•A graph G= (V,E).
+•A labelling2Pr :E→[0,1]∩Q.
+•An initial infective set P⊂V.
+•An individual u∈V\P.
+A solution to πis the value Pr( u|P).
+We take ℓ(Π) =|V|.
+The epidemiological viewpoint we have just described follows the evolu tion of
+probabilities over time. If we ignore the order of events, we come to the simpler
+viewpoint of percolation . Percolation methods have been used in epidemiology.
+(See, for example, [34, 35, 32, 3, 36, 37]. The latter two contain ex tensive refer-
+ences.) Since an individual is only infected for one time step in the cour se of any
+epidemic, an edge can transmit at most once in the course of an epide mic. This al-
+lowsus to consider a random variable that takesas values subgraph sofG. GivenG,
+we takeGto be the random variable which takes values in {G′= (V,E′)|E′⊆E}.
+The probability that Gtakes the value G′is given by
+Pr(G′) =/parenleftBigg/productdisplay
+e∈E′Pr(e)/parenrightBigg/parenleftBigg/productdisplay
+e/∈E′(1−Pr(e))/parenrightBigg
+.
+We may think of E′as determining whether e= (u,v) transmits in the course of
+an epidemic if that epidemic has a state ϕwithϕ(u) =Iandϕ(q) =S. Given a
+2There are technical issues here concerning the values of the se probabilities. To avoid these
+issues they are usually assumed to be rational numbers and bo unds are placed on the sizes of their
+denominators. For details, see [33]. Since Qis dense in R, this is not a limitation on the possible
+probability values relevant in real applications.PROBABILITY OF INFECTION IN AN SIR NETWORK IS NP-HARD 7
+pathτinG, we will abuse notation by writing τ⊂Gande∈τfor the edges of τ.
+Given a path τ, the probability that it appears in G′= (V,E′) is
+(1) Pr( {G′|τ⊂G′}) =/productdisplay
+e∈τPr(e).
+For a proof of the following theorem, see, e.g., [32].
+Theorem 1. Suppose Gis social contact network. Then
+Pr(u|P) =Pr({G′|G′contains a path from Ptou}).
+In particular, Pr (u|P)is a finite sum of terms of the form (1). Accordingly, it is
+a polynomial in the values Pr (e)with integer coefficients and degree |E|.
+This theorem provides the link between epidemiology and communicatio ns net-
+works.
+4.NP-hard problems on communications networks: Consequences f or
+epidemiological calculations
+We assume that a communications network consists of a set of comp uters, each
+of which is reliable and a set of communication links each of which has a kn own
+likelihoodoffailureandthatthecommunicationlinksfunctionorfailind ependently.
+There is no loss of generality in regarding each node as infallible, since a fallible
+computer can be modelled as a pair of nodes with a fallible link connecting its
+input to its output. Once again, we can formalize this as G= (G,Pr), where
+G(V,E) represents installed capacity ( Vbeing the set of computers and Ethe
+set of communication links), Pr : E→[0,1] the reliability of each link and G
+is the random variable assuming values in {G′= (V,E′)|E′⊆E}.EachG′=
+(V,E′) is the subnetwork of functioning links left after the failure of the e dges
+e∈E\E′. Successful transmission of a message on this network depends o n the
+connectivity of the subgraph realized by G. Network engineers focus on several
+kinds of connectivity. We first examine two of the simplest.
+Thetwo-terminal reliability problem is defined as the calculation of the probabil-
+ity that there is at least one correctly functioning path in the netwo rk connecting a
+predefined source node to a predefined target node. An instance πof Πtwo terminal
+consists of the following:
+•A graph G= (V,E).
+•A labelling Pr : E→[0,1]∩Q.
+•A source terminal u⊂V.
+•A target terminal v∈V\{u}.
+A solution to πis the value Pr( v|u).
+By Theorem 1, this value is an integer polynomial in the values Pr( E). Thus, if
+we restrict to the case where Pr( E) takes a single value, this becomes an integer
+polynomial in one variable called the reliability polynomial . Thus a related problem
+is the following:
+An instance πof Πrel polyis
+•A graph G= (V,E).
+•A source terminal u⊂V.
+•A target terminal v∈V\{u}.8 MICHAEL SHAPIRO AND EDGAR DELGADO-ECKERT
+A solution to πis the coefficients of the reliability polynomial. A number of
+additional network reliability problems have been studied (see [33], a n excellent
+introduction to this field). These include
+•kterminal reliability. This requires that kchosen terminals are mutually
+pair wise connected.
+•Broadcasting, also known as all terminal reliability: This requires tha t all
+terminals are pair wise connected.
+Naturally, in addition to the network reliability problems presented ab ove, many
+other reasonable problems can be defined or could arise from pract ical applica-
+tions. Formally, once a model G= (G,Pr) of the network has been chosen, a
+general mechanism to define a reliability problem is the following: A netw ork op-
+eration is specified by defining a set Op(G)⊆ {G′= (V,E′)|E′⊆E}of states
+considered to be functional. The set Op(G) is sometimes called a stochastic binary
+system; the elements of Op(G) are termed pathsets. Specifying the pathsets for G
+determines the whole stochastic binary system, and therefore de fines the network
+operation. The reliability problem consists of finding the probability Pr (Op(G))
+that the probabilistic graph Gassumes values in the set Op(G).
+A first naive algorithm to solve a network reliability problem formulated in this
+general manner is to enumerate all states of G(i.e., the cardinality of the set {G′=
+(V,E′)|E′⊆E}), determine whether a given state is a pathset or not using some
+predesigned recognition procedure3, and sum the occurrence probabilities of each
+pathset. Due to the statistical independence assumed, the prob abilityof occurrence
+of a pathset is simply the product of the operation probabilities of th e edges in
+the pathset and the failure probabilities of the edges not present in the pathset.
+Complete state enumeration requires the generation of all 2|E|states of G,implying
+that the running time of this algorithm would exponentially depend on t he number
+of links in the network.
+A substantialamountofeffort has been put intofinding moreefficien t algorithms
+for exact calculation of network reliability problems (see [33]). Howev er, efficient
+exact solutions seem unlikely:
+Theorem 2. The problems Πtwo terminal andΠrel polyareNP-hard.
+For a proof of this result, see, for instance, Theorem 1 in [38]. The se prob-
+lems belong to the class #P-complete [39, 40, 41, 42, 43, 33]. #Pis the set of
+the counting problems associated with the decision problems in the se tNP. Thus,
+while adecisionproblem mightaskwhether something exists(e.g., an as signmentof
+truth values to a set of variables which satisfies a given formula), th e corresponding
+enumeration problem asks how many of these there are. Solving the enumeration
+problemsolvesthe correspondingdecision problem since knowingwhe ther the num-
+ber ofthese things is positive tells us whether one exists. In particu lar, the counting
+version of any problem is always at least as hard as the correspondin g existence
+problem. In analogy to NP-completeness, a problem is #P-complete if and only if
+it is in#P, and every problem in #Pcan be reduced to it by a polynomial-time
+counting reduction (see [29] for more details).
+Corollary 1. The problem ΠepidemicisNP-hard.
+3Such recognition procedures generally boil down to path-fin ding or spanning tree methods,
+which are efficient (i.e., of polynomial running time) and wel l-know procedures in algorithmic
+graph theory and computer science.PROBABILITY OF INFECTION IN AN SIR NETWORK IS NP-HARD 9
+To see this, notice that every instance of Π two terminal is an instance of Π epidemic,
+namely, an instance in which Pconsists of a single vertex.
+More generally, despite dedicated efforts, no algorithm of polynomia l running
+timehasbeenfoundthatallowsfortheexactcalculationoftheprob abilityPr( Op(G))
+of a given set of pathsets Op(G),unless very specific assumptions are made on the
+topology of the underlying probabilistic network ([33, 44]). We consid er it an open
+question as to which (if any) of these more general networkreliabilit y problems (de-
+fined through the choice of a suitable stochastic binary system Op(G)) correspond
+to epidemiological problems.
+5.NP-hardness of extended problems in epidemiology
+Epidemic on networks with time-varying transmission proba bilities. As
+we have seen in the previous section, the seemingly simple problem of fi nding an
+individual’s chances of infection is NP-hard. This is even so in the case where the
+set of initial infectives is a single individual.
+We can generalize Π epidemicby allowing transmission probabilities to vary over
+time. We have seen that the length of any epidemic is at most the lengt h of the
+longest self-avoiding path in G. Consequently, time-varying transmission probabil-
+ities can be encoded as
+Pr :E×{1,...,|E|} →[0,1].
+In this case, percolation methods no longer apply. However, every instance of
+Πepidemiccan be mapped into an instance of this extended problem. Thus, the
+time-varying version of this problem is NP-hard.
+Epidemicon networks with diseaselatency. OnemightalsogeneralizeΠ epidemic
+to allow patterns of latency and extended periods of infectivity4. We will take I
+to be a sequence of distinct states, {I1,I2,...,I N}. We assume that for each stage
+Iithere is an infectivity µiand a probability of recovery ρi. We take ρN= 1. We
+now consider a social contact network Gand infectivity pattern I. We refer to this
+as an{S,I,R}network. The states of this network are
+{ϕ|ϕ:V→ {S}∪I ∪{ R}}.
+We modify the definition of possible successor states so that the allo wable transi-
+tions are from StoI1, fromIitoIi+1fori= 1,...,N−1 and from IitoRfor
+i= 1,...,N. Ifϕ(u) =Ii,utransitions to state Rwith probability ρiand to state
+Ii+1with probability 1 −ρi. Ife= (u,v)∈Eandϕ(u) =Ii, andϕ(v) =S, thenu
+infectsvwith probability Pr I(e,i) =µiPr(e). We assume that Iis non-trivial in
+the sense that there is iwithµi/\e}atio\slash= 0 and ρj/\e}atio\slash= 1. This ensures that an infected in-
+dividual has a positive probability of reaching an infective state. As b efore, under
+the assumption that transmissions and recoveries happen indepen dently, we can
+develop an expression for Pr I(u|P).
+FixI. An instance of Π Iis an instance of Π epidemic.
+A solution to Π Iis the value Pr I(u|P)
+Theorem 3. Given an non-trivial infectivity pattern I,ΠIisNP-hard.
+Lemma 1. GivenG= (G,Pr)andI, there is G′= (G,Pr′)so that for each P⊂V
+andu /∈P, PrI(u|P) =Pr′(u|P).
+4For a more general version of this see [32].10 MICHAEL SHAPIRO AND EDGAR DELGADO-ECKERT
+Proof.Consider an edge e= (u,v). Suppose that ϕ1(u) =Iiandϕ1(v) =S.
+What are the chances that vremains uninfected by u? (We assume for the moment
+thatvis not infected by some other neighbour during the next Nsteps.) We take
+µ= Pr(e). Let us denote by νithe probability that uremains infected for isteps,
+but noti+1 steps. We then have
+νi=ρii/productdisplay
+j=1(1−ρj).
+The probability that vremains uninfected by uis
+τI(µ) =N/summationdisplay
+i=1νji/productdisplay
+j=1(1−µjµ)
+We now define G′= (G,Pr′) by taking
+Pr′(e) = 1−τI(Pr(e)).
+This does what is required. /square
+Proof of Theorem 3. We will show that Π rel polyis polynomially reducible to Π I.
+FixIto be a non-trivial pattern of infectivity. Suppose we are given an in stance
+πof Πrel poly. This consists of a graph Gand source and target vertices uandv.
+Suppose also that we have a polynomial time algorithm for solving Π I. We choose
+N+ 1 arbitrary probabilities p0,...,p N+1. These give us N+ 1 instances of Π I
+by taking Gi= (G,Pri), where Pr itakes the constant value pi. By the previous
+lemma, solving these N+ 1 instances of Π IsolvesN+ 1 distinct instances of
+Πepidemicwhich consist of the graph Gand differing constant functions Pr′
+i. These
+N+1 values give us N+1 independent linear equations whose unknowns are the
+coefficients of the reliability polynomial. Solving for these is a polynomial time
+problem. /square
+Expected number of total infections. One might hope that while computing
+anindividual’sprobabilityofinfectionis NP-hard,theremightbeawaytocompute
+the expected number of infections. This, too, is NP-hard. Let us formalize this.
+An instance πof Πexpectedis
+•A graph G= (V,E).
+•A labelling Pr : E→[0,1]∩Q.
+•An initial infective set P⊂V.
+A solution to πis the expected number of infections,
+/summationdisplay
+u∈VPr(u|P).
+The following theorem was proved in [28]. For the sake of completenes s, we provide
+a proof here.
+Theorem 4. ΠexpectedisNP-hard.
+Proof.We will show that Π epidemiccan be polynomially reduced to Π expected. Sup-
+pose we are given an instance πof Πepidemic. Let/tildewideπbe the instance of Π epidemic
+which is formed from πby appending a single edge from utov /∈Vand assigning
+Pr(u,v) = 1. It is clear that the expected number of infections in /tildewideπdiffers from
+the number of expected infections in πby exactly Pr( u|P). Thus, if we had aPROBABILITY OF INFECTION IN AN SIR NETWORK IS NP-HARD 11
+polynomial time algorithm for finding the expected number of infectio ns, we could
+find the probability of any individual becoming infected. /square
+The fact that Π rel polyisNP-hard suggests that the difficulty lies not in the
+probabilitiesPrbutinthetopologyof G. Oneproblemwhichwehavenotaddressed
+hereisthequestionofcalculatingtheprobabilityofinfectioninan {S,I,R}network
+whereG=Gtchangesovertime due to stochastic births and deaths. It seems lik ely
+that this will also provide a source of NP-hard problems. However, this requires
+a reformulation of the underlying problem.
+6.Discussion and conclusions
+It has been the purpose of this paper to draw the attention ofnet work epidemiol-
+ogiststo resultsin communicationsnetworkreliabilitywhichshed light o nquestions
+regarding the computational aspects of epidemiology of {S,I,R}networks.
+Theorem 1 and Theorem 4 tell us that generally, in the absence of a m ajor
+break-through in computer science we cannot expect to be able to compute exact
+probabilities of infection or expected number of infection in large soc ial contact
+networks. As [29] points out, problems do not go away simply becaus e we have
+deemed them NP-hard.
+Since the network engineers have been here before us, it is temptin g to ask
+whether their solutions will work for epidemiologists. While we consider the case
+open, the prospects seem mixed. Network engineers are often in t he position of
+being able to choosethe classof networksunder consideration. As opposed to scale-
+free [8, 45] and small-world network structures [46, 47, 8], which fr equently arise
+from a self-organization process during the spontaneous growth of a network, engi-
+neered or purposefully designed networks show rather different s tructures. Some of
+the classesthat allow efficient calculations(exact or approximate)in clude trees, full
+graphs, series-parallel graphs [33], and channel graphs [44]. Unfo rtunately, these
+classes of networks seem unrealistic as models of social contact ne tworks.
+Network engineers have turned to Monte Carlo simulation for the ca lculation of
+estimates of network reliability. We would like to give pointers into their literature
+[48, 49, 50, 51, 52, 53, 54, 28]. This approachhas received increas ed attention in the
+last decade due to the power of modern computers and computing c lusters. While
+Monte Carlo simulation only calculates an unbiased point estimator for reliability
+probabilities, increasing the number of simulated samples causes the se estimates to
+converge to the actual value.
+The fact that efficient and precise algorithms for computing infectio n proba-
+bilities are out of reach (see Theorems 1, 3 and 4) has real-world con sequences.
+Designing a response to an emerging epidemic can depend on determin ing the kind
+of epidemiological probabilities we have been discussing [55]. The effect iveness of
+interventions during an emerging epidemic often crucially depends on timely im-
+plementation. Our results and those of [24] and [25] place an empha sis on the
+search for efficient and quick methods that give good approximation s when applied
+to real-world social networks.
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+arXiv:1001.0596v1  [math.GN]  4 Jan 2010GO-SPACES AND NOETHERIAN SPECTRA
+DAVID MILOVICH
+Abstract. The Noetherian type of a space is the least κfor which the space
+has aκop-like base, i.e., a base in which no element has κ-many supersets.
+We prove some results about Noetherian types of (generalize d) ordered spaces
+and products thereof. For example: the density of a product o f not-too-many
+compact linear orders never exceeds its Noetherian type, wi th equality possible
+only for singular Noetherian types; we prove a similar resul t for products of
+Lindel¨ of GO-spaces. A countable product of compact linear orders has an
+ωop
+1-like base if and only if it is metrizable, and every metrizab le space has
+anωop-like base. An infinite cardinal κis the Noetherian type of a compact
+LOTS if and only if κ/\egatio\slash=ω1andκis not weakly inaccessible. There is a
+Lindel¨ of LOTS with Noetherian type ω1and there consistently is a Lindel¨ of
+LOTS with weakly inaccessible Noetherian type.
+1.Introduction
+The Noetherian type of a topological space is an order-theoretic a nalog of its
+weight.
+Definition 1.1. Given a cardinal κ, define a poset to be κop-likeif no element is
+belowκ-many elements.
+In the context of families of subsets of a topological space, we will a lways im-
+plicitly order by inclusion. For example, a descending chain of open set s of type ω
+isωop-like; an ascending chain of open sets of type ωisωop
+1-like but not ωop-like.
+Definition 1.2. Given a space X,
+•theweightofX, orw(X), is the least κ≥ωsuch that Xhas a base of size
+at mostκ;
+•theNoetherian type ofX, orNt(X), is the least κ≥ωsuch that Xhas a
+base that is κop-like.
+Equivalently, Nt(X) is the least κ≥ωsuch that Xhas a base Bsuch that/intersectiontextA
+has empty interior for all A ∈[B]κ.
+Noetherian type was introduced by Peregudov [10]. Preceding this in troduction
+are several papers by Peregudov, ˇSapirovski˘ ı and Malykhin [5, 8, 9, 11] about the
+topological properties Nt(·) =ωandNt(·)≤ω1. More recently, the author has
+extensively investigated the Noetherian type of βN\N[7] and the Noetherian types
+of homogeneous compacta and dyadic compacta [6]. (See Engelking [1 ], Juh´ asz [3],
+and Kunen [4] for all undefined terms.)
+Asurprisingresultfrom[6] isthat nodyadiccompactumhasNoether iantypeω1.
+In other words, given an ωop
+1-like base of a dyadic compactum X, one can construct
+2000Mathematics Subject Classification. Primary 54F05; Secondary 54A25, 03E04.
+Some of these results are from the author’s thesis, which was partially supported by an NSF
+graduate fellowship.
+12 DAVID MILOVICH
+anωop-like base of X. This result does not generalize to all compacta. In that same
+paper, it wasshown how to construct a compactum with Noetherian typeκ, for any
+infinite cardinal κ. It is still an open problem whether any infinite cardinals other
+thanω1are excluded from the spectrum of Noetherian types of dyadic com pacta,
+althoughitwasshownthattherearedyadiccompactawithNoether ianωanddyadic
+compacta with Noetherian type κ+, for every infinite cardinal κwith uncountable
+cofinality.
+Question 1.3.Ifκis a singular cardinal with cofinality ω, then is there a dyadic
+compactum with Noetherian type κ+? Is there a dyadic compactum with weakly
+inaccessible Noetherian type?
+The above two questions are typical of the “sup=max” problems of set-theoretic
+topology. See Juh´ asz [3] for a systematic study of these problem s.
+Though the above two questions remain open problems, we can now a nswer
+the corresponding questions for compact linear orders. The spec trum of Noether-
+ian types of linearly ordered compacta includes ω, excludes ω1, includes all singular
+cardinals, includes κ+for all uncountable cardinals κ, and excludes all weak inaces-
+sibles. In the process of proving this claim, we will prove a general te chnical lemma
+which says roughly that if Xis a product of not-too-many µ-compact GO-spaces
+for some fixed cardinal µ, thend(X)≤Nt(X) and in most cases d(X)< Nt(X).
+Definition 1.4.
+•A spaceXisκ-compact if κis a cardinal and every open cover of Xhas a
+subcover of size less than κ.
+•AGO-space , or generalized ordered space, is a subspace of a linearly or-
+dered topological space. Equivalently, a GO-space is a linear order w ith a
+topology that has a base consisting only of convex sets.
+•Thedensityd(X) of a space Xis the least infinite cardinal κsuch that X
+has a dense subset of size at most κ.
+It isnaturaltoaskwhathappenstothe spectrumofNoetheriant ypesofcompact
+linear orders if we gently relax the assumption of compactness. It t urns out that
+there are Lindel¨ of linear orders with Noetherian type ω1, and, less expectedly, that
+it is consistent (relative to existence of an inaccessible cardinal) tha t there is a
+Lindel¨ of linear order with weakly inaccessible Noetherian type. Howe ver, it is not
+consistent for a Lindel¨ of GO-space to have strongly inacessible No etherian type.
+We also consider the relationship between metrizability and Noetheria n type,
+focusing on GO-spaces. Every metric space has Noetherian type ω. For a Lindel¨ of
+GO-space X,Xis metrizable if and only if Nt(X) =ωif and only if Nt(X) =ω1
+andXis separable. For a countable product Xof compact linear orders, Xis
+metrizable if and only if Nt(X) =ωif and only if Nt(X) =ω1. (Note that every
+Lindel¨ of metric space is separable, and every compact GO-space is a compact linear
+order.)
+2.Small densities and large Noetherian types
+Definition 2.1. Theπ-weightπ(X) of a space Xis the least infinite cardinal κ
+such that a space has π-base of size at most κ;
+Proposition2.2. [10]IfXis a space and π(X)<cfκ≤κ≤w(X), thenNt(X)>
+κ.GO-SPACES AND NOETHERIAN SPECTRA 3
+Proof.Suppose Ais a base of XandBisπ-base of Xof size at most π(X). We
+then have |A| ≥κ; hence, there exist U ∈[A]κandV∈ Bsuch that V⊆/intersectiontextU.
+Hence, there exists W∈ Asuch that W⊆V⊆/intersectiontextU; hence,Ais notκop-like./square
+Note that if Xis a product ofat most d(X)-manyGO-spaces, then π(X) =d(X)
+is witnessed by the following construction. For any D(topologically) dense in X
+and of minimal size, collect all the finitely supported products of top ologically open
+intervals with endpoints from the union of {±∞}and the set of all coordinates of
+points from D.
+Trivially, Nt(X)≤w(X)+for all spaces X. The next example shows that this
+upper bound is attained.
+Example 2.3. [6] The double-arrow space, defined as ((0 ,1]×{0})∪([0,1)×{1})
+ordered lexicographically, has π-weightωand weight 2ℵ0. By Proposition 2.2, it
+has Noetherian type/parenleftbig
+2ℵ0/parenrightbig+.
+3.Lindel¨of GO-spaces
+Theorem 3.1. Every metric space has an ωop-like base.
+Proof.LetXbe a metric space. For each n < ω, letAnbe a locally finite open
+refinement of the (open) balls of radius 2−ninX. SetA=/uniontext
+n<ωAn\{∅}. The
+setAis a base of Xbecause if p∈Xandn < ω, then there exists U∈ An+1such
+thatp∈UandUis contained in the ball of radius 2−nwith center p. Let us show
+thatAisωop-like. Suppose that m < ω,U∈ A,V∈ Am, andU/subsetnoteqlV. There
+then exist p∈Uandǫ0> ǫ1>0 such that the ǫ0-ball with center pis contained
+inUand theǫ1-ball with center pintersects only finitely many elements of Anfor
+alln < ωsatisfying 2−n> ǫ0/2. If 2−m≤ǫ0/2, thenVis contained in the ǫ0-ball
+with center p, in contradiction with U/subsetnoteqlV. Hence, 2−m> ǫ0/2; hence, there are
+only finitely many possibilities for mandVgivenU, forVintersects the ǫ1-ball
+with center p. /square
+Lemma 3.2. LetXbe a Lindel¨ of GO-space with open cover A. The cover Ahas
+a countable, locally finite refinement consisting only of cou ntable unions of open
+convex sets.
+Proof.Let{An:n < ω}be a countable refinement of Aconsisting only of open
+convex sets. For each n < ω, setBn=An\/uniontext
+m<nAm; setB={Bn:n < ω}. The
+setBis a locally finite refinement of A. LetCbe the set of open convex subsets
+ofXwhich intersect only finitely many elements of B. LetDbe the set of U∈ C
+satisfying U⊆Vfor some V∈ C. Let{Dn:n < ω}be a countable subcover of D.
+For each n < ω, setEn=Dn\/uniontext
+m<nDm; setE={En:n < ω}. The set Eis a
+locallyfinite refinementof C. Foreach n < ω, setFn=An\/uniontext{E∈ E:Bn∩E=∅},
+which is a countable union of convex sets; set F={Fn:n < ω}. SinceEis locally
+finite, each Fnis open. Hence, each Fnis a countable union of open convex sets.
+Moreover, Bn⊆Fn⊆Anfor alln < ω; hence,Fis a refinement of A.
+Thus, it suffices to show that Fis locally finite. Since Eis a locally finite cover of
+X, it suffices to show that each element of Eonly intersects finitely many elements
+ofF. Leti < ωand choose V∈ Csuch that Ei⊆V. Suppose j < ωand
+Ei∩Fj/\e}atio\slash=∅. We then have Ei∩Bj/\e}atio\slash=∅by definition of Fj. Hence, V∩Bj/\e}atio\slash=∅;
+hence, there are only finitely many possibilities for Bj; hence, there are only finitely
+many possibilities for Fj. /square4 DAVID MILOVICH
+Definition 3.3. LetHθdenote the set of all sets that are hereditarily of size less
+θ, whereθis a regular cardinal sufficiently large for the argument at hand. The
+relationM≺Hθmeans that /a\}bracketle{tM,∈/a\}bracketri}htis an elementary substructure of /a\}bracketle{tHθ,∈/a\}bracketri}ht.
+Lemma 3.4. LetXbe a nonseparable, Lindel¨ of GO-space. The space Xdoes not
+have anωop-like base.
+Proof.LetAbe a base of X. Let us show that Ais notωop-like. First, let us
+construct sequences of open sets /a\}bracketle{tAn,k/a\}bracketri}htn,k<ωand/a\}bracketle{tBn,k/a\}bracketri}htn,k<ω. Our requirements
+are that Bn,i⊆An,i∈ A, thatBn,iis a countable union of open convex sets, that
+{Bn,k:k < ω}is a locally finite cover of Xand pairwise ⊆-incomparable, and that
+{Ai,k:k < ω}∩{Aj,k:k < ω} ⊆[X]1for alli < j < ω andn < ω.
+Suppose n < ωand we are given /a\}bracketle{tAm,k/a\}bracketri}htk<ωand/a\}bracketle{tBm,k/a\}bracketri}htk<ωfor allm < nand
+they meet our requirements. Let p∈X. SetVp=/intersectiontext{Bm,k:m < nandk <
+ωandp∈Bm,k}. The set Vpis open. If |Vp|= 1, then set Up=Vp. If|Vp|>1,
+then choose Up∈ Asuch that p∈Up/subsetnoteqlVp. SetU={Up:p∈X}. By
+Lemma 3.2, there exists a countable, locally finite refinement BnofUconsisting
+only of countable unions of open convex sets. Since Bnis locally finite, it has no
+infinite ascending chains; hence, we may assume Bnis pairwise ⊆-incomparable
+because we may shrink Bnto its maximal elements. Let {Bn,k:k < ω}=Bn.
+For each k < ω, setAn,k=Upfor some p∈Xsatisfying Bn,k⊆Up. Suppose
+m < nandi,j < ωandAm,i=An,j/\e}atio\slash∈[X]1. Choose p∈Xsuch that An,j=Up;
+choosek < ωsuch that p∈Bm,k. We then have Bm,i⊆Am,i=Up/subsetnoteqlVp⊆Bm,k,
+in contradiction with the pairwise ⊆-incomparability of {Bm,l:l < ω}. Thus,
+{Am,l:l < ω}∩ {An,l:l < ω} ⊆[X]1for allm < n. By induction, /a\}bracketle{tAn,k/a\}bracketri}htn,k<ω
+and/a\}bracketle{tBn,k/a\}bracketri}htn,k<ωmeet our requirements.
+Let{X,≤,A} ⊆M≺Hθand|M|=ω. SinceXis nonseparable, there must
+be a nonempty open convex set Wdisjoint from M. SinceXis Lindel¨ of, every
+isolated point of Xis inM. Hence, Wmust be infinite. Choose {a < c < e } ⊆
+W. Choose U∈ Asuch that U⊆(a,e). By elementarity, we may assume that
+/a\}bracketle{t/a\}bracketle{tAn,k,Bn,k/a\}bracketri}ht/a\}bracketri}htn,k<ω∈Mfor alln,k < ω . For each n < ω, choose in< ωsuch
+thatc∈Bn,in. To complete the proof, it suffices to show that Uhas infinitely
+many supersets in A. Fixn < ω. Sincec/\e}atio\slash∈M, we cannot have An,in={c}; hence,
+An,in/\e}atio\slash=Am,imforallm < n. Hence, itsufficestoshowthat U⊆An,in. Thereexists
+/a\}bracketle{tIj/a\}bracketri}htj<ω∈Msuch that Bn,in=/uniontext
+j<ωIjand each Ijis convex and open. Hence,
+there exists j < ωsuch that c∈Ij. SinceU⊆(a,e) andIj⊆Bn,in⊆An,in,
+it suffices to show that ( a,e)⊆Ij. Seeking a contradiction, suppose ( a,e)/\e}atio\slash⊆Ij.
+Sincec∈Ij, we must have a < b < I jfor some b∈XorIj< d < e for some
+d∈X. By symmetry, we may assume we have the former. Since Xis Lindel¨ of,
+{p∈X:p < Ij}has a countable cofinal subset Y. SinceIj∈M, we may assume
+Y∈M. Hence, Y⊆M; hence, there exists y∈Msuch that b≤y < Ij. Hence,
+Mintersects ( a,e); hence, Mintersects W, which is absurd. /square
+Theorem 3.5. LetXbe a Lindel¨ of GO-space. The following are equivalent.
+(1)Xis metrizable.
+(2)Xhas anωop-like base.
+(3)Xis separable and has an ωop
+1-like base.
+Proof.By Theorem 3.1, (1) implies (2). By Lemma 3.4, (2) implies (3). Hence, it
+sufficesto showthat (3) implies (1). Suppose Xhasacountabledense subset DandGO-SPACES AND NOETHERIAN SPECTRA 5
+anωop
+1-like base. We then have π(X) =ω; hence, by Proposition 2.2, w(X) =ω;
+hence,Xis metrizable. /square
+See Example 6.1 for a nonseparable Lindel¨ of linear order that has No etherian
+typeω1.
+4.Small Noetherian types and smaller densities
+For compact linearly ordered topological spaces, the theorem at t he end of this
+section strengthens Theorem 3.5. To prepare for this theorem, w e first prove our
+main technical lemma, which we state in very general terms.
+Lemma 4.1. Suppose κis a regular uncountable cardinal, µis an infinite cardinal,
+|λ<µ|< κfor allλ < κ,Xis a product of fewer than κ-manyµ-compact GO-spaces,
+andNt(X)≤κ. We then have d(X)< κ.
+Proof.LetX=/producttext
+i<νXiwhereν < κand each Xiis aµ-compact subspace of a
+linearly ordered topological space Yi. Seeking a contradiction, suppose d(X)≥κ.
+LetUbe aκop-like base of X,{/a\}bracketle{tYi,≤Yi,Xi:i < ν/a\}bracketri}ht,U} ∈M≺Hθ,|M|< κ,
+M∩κ∈κ, andM<µ⊆M. (We can construct Mas the union of an appropriate
+elementary chain of length ρ, whereρis the least regular cardinal ≥µ. Such an M
+is not too large because ρ < κ, a fact that follows from µ≤ |2<µ|< κand cf(µ)<
+µ⇒µ+≤ |µcf(µ)|< κ). Since d(X)>|M|, there is a finite subproduct/producttext
+i∈σXi
+ofXthat has a nonempty open subset disjoint from M. We may choose this open
+subset to be the interior of a set of the form B=/producttext
+i∈σBiwhere each Biis maximal
+among the convex subsets of Xidisjoint from M. SetAi={p∈Xi:p < B i}and
+Ci={p∈Xi:p > B i}. Since{p:Ai< p < C i}=Bi, which is nonempty but
+disjoint from M, we have {Ai,Ci} /\e}atio\slash⊆Mby elementarity.
+Claim. max{cf(Ai),ci(Ci)} ≥µfor alli∈σ.
+Proof.Seeking a contradiction, suppose cf( Ai)< µand ci(Ci)< µ. We then have
+Ai∩M,Ci∩M∈M. SinceAi∩Mis cofinal in Ai,Ai={p:∃q∈Ai∩M p≤
+q} ∈M. Likewise, Ci∈M, in contradiction with the fact that {Ai,Ci} /\e}atio\slash⊆M./square
+Therefore, we may assume that cf( Ai)≥µfor alli∈σ(by symmetry). Since
+Xisµ-compact, there exists xi= supYi(Ai) = min( Bi)∈Xifor alli∈σ.
+Claim. There exists yi= supYi(Bi)∈(xi,∞)for alli∈σ, with the understanding
+that in this proof all intervals are intervals of Xi(soyi∈Xi).
+Proof.If ci(Ci)≥µ, then, by µ-compactness, there exists yi= infYi(Ci)∈Xi. In
+this case, yiis alsomax( Bi) because yi/\e}atio\slash∈Ci. Moreover, yi= max(Bi)>min(Bi) =
+xibecause otherwise the interior of Bwould be empty, for xi= supYi(Ai), which
+is not an isolated point in Xi. If ci(Ci)< µ, thenCi∈M, just as in the previous
+claim’s proof, so there exists Di∈Msuch that Diis a cofinal subset of {p∈
+Xi:p < C i}of minimal size. In this case, Diincludes a cofinal subset of Bi, so
+Di/\e}atio\slash⊆M, so|Di| ≥κ, soµ < κ≤ |Di|= cf(Bi), so there exists yi= supYi(Bi)∈Xi
+byµ-compactness. Also, cf( Bi)≥κimplies supYi(Bi)>min(Bi) =xi. Thus, in
+any case there exists yi= sup(Bi)∈(xi,∞) for alli∈σ. /square
+LetU∈ Usatisfyxi∈πi[U]⊆(−∞,yi) for alli∈σ. Since cf( Ai)≥µ >1 for
+alli∈σ, we then have /a\}bracketle{txi:i∈σ/a\}bracketri}ht ∈/producttext
+i∈σWi⊆πσ[U] where each Wiis of the form
+(ui,vi) or (ui,vi] for some ui< xiandvi≤yi; we may assume ui∈M. Moreover,6 DAVID MILOVICH
+there exist pi,qi∈Xi∩Msuch that ui< pi< qi< xi. SinceUis aκop-like base, it
+includes fewer than κ-many supersets of/intersectiontext
+i∈σπ−1
+i[(ui,qi)] as members. Since the
+set of supsersets of/intersectiontext
+i∈σπ−1
+i[(ui,qi)] inUis a set in Mand a set of size less than
+κ, it is also a subset of M. In particular, U∈M.
+Fix an arbitrary i∈σ. If cf(πi[U])< µ, thenMwould include a cofinal subset of
+πi[U], in contradiction with BimissingM. Therefore, cf( πi[U])≥µ. Hence, there
+existsz= supYi(πi[U]). By elementarity, z∈M, soBi< z, soz= min(Ci) =
+supYi(Bi) =y. Because of the freedom in how we chose U, it follows that every
+neighborhood of xiincludes a neighborhood that, like πi[U], has supremum yi
+(inYi) and has cofinality at least µ. Therefore, there is an infinite increasing
+sequence of points between xiandyithat are contained in every neighborhood
+ofxi, in contradiction with Xibeing a subspace of the ordered space Yi. Thus,
+d(X)< κ. /square
+Corollary 4.2. Suppose that Xis a product of at most 2ℵ0-many Lindel¨ of GO-
+spaces such that Nt(X)≤/parenleftbig
+2ℵ0/parenrightbig+. We then have d(X)≤2ℵ0.
+Corollary 4.3. Suppose that κis a regular uncountable cardinal, Xis a product
+of less than κ-many linearly ordered compacta, and Nt(X)≤κ. We then have
+d(X)< κ.
+The last corollary fails for singular κ. As we shall see in Theorem 5.1, if λis an
+uncountablesingularcardinal, then Nt(λ+1) =λ, despite the factthat d(κ+1) =κ
+for all infinite cardinals κ. Moreover, the space ( κ+1)κhas Noetherian type ωand
+densityκfor all infinite cardinals κ, so we cannot weaken the above hypothesis that
+Xhas less than κ-many factors. The equation Nt((κ+ 1)κ) =ωfollows from a
+general theorem of Malykhin.
+Theorem 4.4. [5]LetX=/producttext
+i∈IXiwhere each Xihas a minimal open cover of
+size two (e.g., XiisT1). Ifsupi∈Iw(Xi)≤ |I|, thenNt(X) =ω.
+Proof.For each i∈I, let{Ui,0,Ui,1}be a minimal open cover of Xi. Sincew(X) =
+supi∈Iw(Xi), we may choose Ato be a base of Xof size at most |I|and choose
+an injection f:A →I. LetBdenote the set of all nonempty sets of the form
+V∩π−1
+f(V)/bracketleftbig
+Uf(V),j/bracketrightbig
+whereV∈ Aandj <2. Since fis injective, every infinite
+subset of Bhas empty interior. Hence, Bis anωop-like base of X. /square
+Theorem 4.5. LetXbe a product of countably many linearly ordered compacta.
+The following are equivalent.
+(1)Xis metrizable.
+(2)Xhas anωop-like base.
+(3)Xhas anωop
+1-like base.
+(4)Xis separable and has an ωop
+1-like base.
+Proof.By Theorem 3.1, (1) implies (2), which trivially implies (3). By Corol-
+lary 4.3, (3) implies (4). Finally, (4) implies (1) because if Xis separable, then
+π(X) =ω, sow(X) =ωby Proposition 2.2. /square
+5.The Noetherian spectrum of the compact orders
+Theorem4.5impliesthatnolinearlyorderedcompactumhasNoetheria ntypeω1.
+What is the class of Noetherian types of linearly ordered compacta? We shall proveGO-SPACES AND NOETHERIAN SPECTRA 7
+that an infinite cardinal κis the Noetherian type of a linearly ordered compactum
+if and only if κ/\e}atio\slash=ω1andκis not weakly inaccessible.
+Theorem 5.1. Letκbe an uncountable cardinal and give κ+1the order topology.
+Ifκis regular, then Nt(κ+1) =κ+; otherwise, Nt(κ+1) =κ.
+Proof.Using Corollary 4.3, the lower bounds on Nt(κ+ 1) are easy. We have
+d(κ+ 1)≥λfor all regular λ≤κ, soNt(κ+ 1)> λfor all regular λ≤κ. It
+follows that Nt(κ+ 1)≥κandNt(κ+ 1)>cfκ. We can also prove these lower
+bounds directly using the Pressing Down Lemma. Let Abe a base of κ+1 and let
+λbe a regular cardinal ≤κ. Let us show that Ais notλop-like. For every limit
+ordinalα < λ, choose Uα∈ Asuch that α= maxUα; choose η(α)< αsuch that
+[η(α),α]⊆Uα. By the Pressing Down Lemma, ηis constant on a stationary subset
+Sofλ. Hence, A ∋ {η(minS) + 1} ⊆Uαfor allα∈S; hence, Ais notλop-like.
+Once again, it follows that Nt(κ+1)≥κandNt(κ+1)>cfκ.
+Trivially, Nt(κ+1)≤w(κ+1)+=κ+. Hence, it suffices to show that κ+1 has
+aκop-like base if κis singular. Suppose E∈[κ]<κis unbounded in κ. LetFbe
+the set of limit points of Einκ+1. Define Bby
+B={(β,α] :E∋β < α∈For sup(E∩α)≤β < α∈κ\F}.
+The setBis aκop-like base of κ+1. /square
+Definition 5.2. Given a poset Pwith ordering ≤, letPopdenote the set Pwith
+ordering ≥.
+Theorem 5.3. Suppose κis a singular cardinal. There then is a linearly ordered
+compactum with Noetherian type κ+.
+Proof.Setλ= cfκandX=λ++1. Partition the set of limit ordinals in λ+into
+λ-many stationary sets /a\}bracketle{tSα/a\}bracketri}htα<λ. Let/a\}bracketle{tκα/a\}bracketri}htα<λbe an increasing sequence of regular
+cardinals with supremum κ. For each α < λandβ∈Sα, setYβ= (κα+1)op. For
+eachα∈X\/uniontext
+β<λSβ, setYα= 1. Set Y=/uniontext
+α∈X{α}×Yαordered lexicographi-
+cally. We then have Nt(Y)≤w(Y)+≤ |Y|+=κ+. Hence, it suffices to show that
+Yhas noκop-like base.
+Seeking a contradiction, suppose Ais aκop-like base of Y. For each α < λ, let
+Uαbe the set of all U∈ Athat have at least κα-many supersets in A. For all
+isolated points pofY, there exists α < λsuch that {p} /\e}atio\slash∈ Uα; whence, p/\e}atio\slash∈/uniontextUα.
+Since/a\}bracketle{tα+1,0/a\}bracketri}htis isolated for all α < λ+, there exist β < λand a set Eof successor
+ordinals in λ+such that |E|=λ+and (E×1)∩/uniontextUβ=∅. LetCbe the closure
+ofEinλ+. The set Cis closed unbounded; hence, there exists γ∈C∩Sβ+1. Set
+q=/a\}bracketle{tγ,κβ+1/a\}bracketri}ht. We then have q∈E×1; hence, q/\e}atio\slash∈/uniontextUβ. Sinceqhas coinitiality
+κβ+1, any local base Batqwill contain an element Usuch that Uhasκβ-many
+supersets in B. Hence, there exists U∈ Uβsuch that q∈U; hence,q∈/uniontextUβ, which
+yields our desired contradiction. /square
+Theorem 5.4. No linearly ordered compactum has weakly inaccessible Noet herian
+type. More generally, for every weakly inaccessible κ, products of fewer than κ-many
+linearly ordered compacta do not have Noetherian type κ.
+Proof.Suppose κis weakly inaccessible, Xis a product of fewer than κ-many
+linearly ordered compacta, and Nt(X)≤κ. It suffices to prove Nt(X)< κ.
+By Corollary 4.3, we have d(X)< κ; hence, each factor of Xhasπ-weight less8 DAVID MILOVICH
+thanκ; hence, π(X)< κ. Ifw(X)≥κ, thenNt(X)> κby Proposition 2.2, in
+contradiction with our assumptions about X. Hence, w(X)< κ; hence,Nt(X)≤
+w(X)+< κ. /square
+6.The Lindel ¨of spectrum
+The spectrum of Noetherian types of Lindel¨ of linearly ordered top ologicalspaces
+trivially includes the spectrum of Noetherian types of compact linear ly ordered
+topological spaces. More interestingly, the inclusion is strict, as th e next example
+shows.
+Example 6.1. Theorem 4.5 fails for Lindel¨ of linearly ordered topological spaces.
+LetXbe (ω1×Z)∪({ω1}×{0}) ordered lexicographically. The space Xis Lindel¨ of
+and nonseparable and {{/a\}bracketle{tα,n/a\}bracketri}ht}:α < ω1andn∈Z}∪{X\(α×Z) :α < ω1}is an
+ωop
+1-like base of X. Moreover, Xhas noωop-like base because every local base at
+/a\}bracketle{tω1,0/a\}bracketri}htincludesadescending ω1-chainofneighborhoods. Thus, Nt(X) =ω1. Easily
+generalizing this example, if κis a regular cardinal and Xis (κ×Z)∪({κ}×{0})
+ordered lexicographically, then Xisκ-compact and Nt(X) =κ.
+A consequence of Lemma 4.1 is that Lindel¨ of linearly ordered topolog ical spaces
+cannot have strongly inaccessible Noetherian type, just as in the c ompact case.
+More generally, we have the following theorem, which is proved just a s Theorem 5.4
+was proved.
+Theorem 6.2. Supposeκis a weakly inaccessible cardinal, |λ<µ|< κfor allλ < κ,
+andXis aXis a product of fewer than κ-manyµ-compact GO-spaces. We then
+haveNt(X)/\e}atio\slash=κ.
+Proof.Suppose that Nt(X)≤κ. Let us show that Nt(X)< κ. By Lemma 4.1, we
+haved(X)< κ; hence, each factor of Xhasπ-weight less than κ; hence, π(X)<
+κ. Ifw(X)≥κ, thenNt(X)> κby Proposition 2.2, in contradiction with our
+assumptions about X. Hence, w(X)< κ; hence,Nt(X)≤w(X)+< κ. /square
+Corollary 6.3. Ifκis strongly inaccessible, then the class of Noetherian type s of
+µ-compact GO-spaces excludes κif and only if µ < κ.
+Proof.“If”: Theorem 6.2. “Only if”: Example 6.1. /square
+On the other hand, it is consistent (relative to the consistency of a n inaccessi-
+ble), that some Lindel¨ of linearly ordered topological space has wea kly inaccessible
+Noetherian type. To show this, we first force 2ℵ0≥κwhereκis weakly inaccessible
+(say, by adding κ-many Cohen reals). Next, we construct the desired linear order
+in this forcing extension using the following theorem.
+Theorem 6.4. Ifκis a weak inaccessible and 2ℵ0≥κ, then there is a Lindel¨ of
+linear order Zsuch that Nt(Z) =κ.
+Proof.LetBbe a Bernstein subset of X= [0,1],i.e.,Bincludes some point in P
+and misses some point in P, for all perfect P⊆X. Letf:B→κbe surjective.
+For each x∈B, setYx=ωop+ωf(x)+ω, which is Lindel¨ of. For each x∈X\B, set
+Yx={0}. SetZ=/uniontext
+x∈X({x} ×Yx) ordered lexicographically. First, let us show
+thatZis Lindel¨ of. Let Ube an open cover of Z. For every x∈X\B,/a\}bracketle{tx,0/a\}bracketri}hthas
+neighborhoods OxandUxsuch that U ∋Ux⊇Ox=/uniontext
+a<b<c({b}×Yb) wherea,c∈
+(X∩Q)∪{±∞}. Therefore, there is a countable D⊆Xsuch that {Ox:x∈D}GO-SPACES AND NOETHERIAN SPECTRA 9
+covers (X\B)×{0}. SetV={Ux:x∈D}andC={x∈X:Yx/\e}atio\slash⊆/uniontext
+x∈DOx}.
+The setCis closed in Xand a subset of the Bernstein set B, soCis countable.
+Therefore,/uniontext
+x∈C({x}×Yx) is Lindel¨ of; hence, it is covered by a countable W ⊆ U,
+makingV ∪Wa countable subcover of U.
+Finally, let us show that Nt(Z) =κ. For every α < κandx∈f−1[{α+ 1}],
+Yxhas a point with cofinality ωα+1, soNt(Z)≥ωα+1. Therefore, it suffices to
+construct a κop-like base of Z. LetAdenote the countable set of all sets of the
+form/uniontext
+a<b<cYbwherea,c∈(X∩Q)∪{±∞}, which includes a local base at /a\}bracketle{tx,0/a\}bracketri}ht
+for every x∈X\B. Since each Yxforx∈Bhas no maximum or minimum, we can
+combine Awith a copy of a base BxofYxfor eachx∈Bin order to produce a base
+BofZ. We may choose each Bxto have size less than κ, soBmust be κop-like./square
+References
+[1] R. Engelking, General Topology , Heldermann-Verlag, Berlin, 2nd ed., 1989.
+[2] S. Jackson and R. D. Mauldin, On a lattice problem of H. Steinhaus , J. Amer. Math. Soc.
+15(2002), no. 4, 817–856.
+[3] I. Juh´ asz, Cardinal functions in topology—ten years later , Mathematical Centre Tracts 123,
+Mathematisch Centrum, Amsterdam, 1980.
+[4] K. Kunen, Set theory. An introduction to independence proofs , Studies in Logic and the Foun-
+dations of Mathematics 102. North-Holland Publishing Co., Amsterdam-New York, 1980.
+[5] V. I. Malykhin, On Noetherian Spaces , Amer. Math. Soc. Transl. 134(1987), no. 2, 83–91.
+[6] D. Milovich, Noetherian types of homogeneous compacta and dyadic compac ta, Topology
+Appl.156(2008), 443–464.
+[7] D. Milovich, Splitting families and the Noetherian type of βω\ω, J. Symbolic Logic 73(2008),
+no. 4, 1289–1306.
+[8] S. A. Peregudov, Certain properties of families of open sets and coverings , Moscow Univ.
+Math. Bull. 31(1976), no. 3–4, 19–25.
+[9] S. A. Peregudov, P-uniform bases and π-bases, Soviet Math. Dokl. 17(1976), no. 4, 1055–
+1059.
+[10] S. A. Peregudov, On the Noetherian type of topological spaces , Comment. Math. Univ. Car-
+olin.38(1997), no. 3, 581–586.
+[11] S. A. Peregudov and B. ´E.ˇSapirovski˘ ı, A class of compact spaces , Soviet Math. Dokl. 17
+(1976), no. 5, 1296–1300.
+Texas A&M InternationalUniversity, Dept. of Engineering, Ma thematics, andPhysics,
+5201 University Blvd., Laredo, TX 78041
+E-mail address :david.milovich@tamiu.edu
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+arXiv:1001.0597v2  [stat.ME]  21 Jan 2011Inference ofglobal clusters from locallydistributeddata
+XuanLongNguyen
+DepartmentofStatistics
+UniversityofMichigan
+Abstract
+Weconsidertheproblemofanalyzingtheheterogeneity ofcl usteringdistributions formul-
+tiple groups of observed data, each of which is indexed by a co variate value, and inferring
+global clusters arising from observations aggregated over the covariate domain. We propose
+a novel Bayesian nonparametric method reposing on the forma lism of spatial modeling and
+a nested hierarchy of Dirichlet processes. We provide an ana lysis of the model properties,
+relating and contrasting the notions of local and global clu sters. We also provide an efficient
+inference algorithm, and demonstrate the utility of our met hod in several data examples, in-
+cludingtheproblemofobjecttrackingandaglobalclusteri nganalysisoffunctionaldatawhere
+the functional identity information isnot available.1
+Keywords: global clustering, local clustering, nonparametric Bayes , hierarchical Dirichlet process, Gaus-
+sian process, graphical model, spatial dependence, Markov chain Monte Carlo, model identifiability
+1 Introduction
+In many applications it is common to separate observed data i nto groups (populations) indexed
+by some covariate u. A particularly fruitful characterization of grouped data is the use of mixture
+distributions to describe the populations in terms of clust ers of similar behaviors. Viewing ob-
+servations associated with a group as local data, and the clu sters associated with a group as local
+clusters, it is often of interest to assess how the local hete rogeneity is described by the changing
+values of covariate u. Moreover, in some applications the primary interest is to e xtract some sort
+ofglobalclusteringpatternsthatariseout oftheaggregat edobservations.
+Consider, for instance, a problem of tracking multiple obje cts moving in a geographical area.
+Using covariate uto indexthe timepoint,at a giventimepoint uwe are provided witha snapshot
+ofthelocationsoftheobjects,whichtendto begroupedinto local clusters. Overtime,theobjects
+may switch their local clusters. We are not really intereste d in the movement of each individual
+object. Itisthepathsoverwhichthelocalclustersevolvet hatareourprimaryinterest. Such paths
+are the global clusters. Note that the number of global and lo cal clusters are unknown, and are to
+beinferred directlyfrom thelocallyobservedgroupsofdat a.
+The problem of estimating global clustering patterns out of locally observed groups of data
+also arises in the context of functional data analysis where the functional identity information
+is not available. By the absence of functional identity info rmation, we mean the data are not
+actually given as a collection of sampled functional curves (even if such functional curves exist
+in reality or conceptually), due to confidentiality constra ints or the impracticality of matching the
+1This work was partially supportedby NSF-CDI grantNo. 09406 71. The author wishes to thank the refereesand
+theAssociate Editorforvaluablecommentsthathelpimprov ethepresentationofthiswork.
+1identityofindividualfunctionalcurves. Asanotherexamp le,theprogesteronehormonebehaviors
+recorded by a number of women on a given day in their monthly me nstrual cycle is associated
+with a local group, which are clustered into typical behavio rs. Such local clusters and the number
+of clusters may evolve throughout the monthly cycle. Moreov er, aggregating the data over days
+in the cycle, there might exist one or more typical monthly (“ global” trend) hormone behaviors
+duetocontraception ormedicaltreatments. Theseare thegl obal clusters. Duetoprivacyconcern,
+the subject identity of the hormone levels are neither known nor matched across the time points
+u. In other words, the data are given not as a collection of horm one curves, but as a collection of
+hormonelevelsobservedovertime.
+Intheforegoingexamples,thecovariate uindexesthetime. Inotherapplications,thecovariate
+might index geographical locations where the observations are collected. More generally, obser-
+vationsassociatedwithdifferentgroupsmayalsobeofdiff erentdatatypes. Forinstance,consider
+the assets of a number of individuals (or countries), where t he observed data can be subdivided
+intoholdingsaccording to differentcurrency types (e.g., USD, gold,bonds). Here, each uisasso-
+ciated with a currency type, and a global cluster may be taken to represent a typical portforlio of
+currency holdings by a given individual. In view of a substan tial existing body of work drawing
+fromthespatialstatisticsliteraturethatweshalldescri beinthesequel,throughoutthispaperaco-
+variatevalue uissometimesreferredtoasaspatiallocationunlessspecifi edotherwise. Therefore,
+the dependence on varying covariatevalues uof the local heterogeneity of data is also sometimes
+referred to asthespatialdependenceamonggroupsofdataco llected at varyinglocal sites.
+We propose in this paper a model-based approach to learning g lobal clusters from locally dis-
+tributeddata. Becausethenumberofbothglobalandlocalcl ustersareassumedtobeunknown,and
+becausethelocalclustersmayvarywiththecovariate u,anaturalapproachtohandlingthisuncer-
+taintyisbasedonDirichletprocessmixturesandtheirvari ants. ADirichletprocessDP (α0,G0)de-
+finesadistributionon(random)probabilitymeasures,wher eα0iscalledtheconcentrationparam-
+eter, and parameter G0denotes the base probability measure or centering distribu tion (Ferguson,
+1973). A random draw Gfrom the Dirichlet process (DP) is a discrete measure (with p robability
+1), whichadmitsthewell-known“stick-breaking”represen tation(Sethuraman, 1994):
+G=∞/summationdisplay
+k=1πkδφk, (1)
+where theφk’s are independent random variables distributed according toG0,δφkdenotes an
+atomic distribution concentrated at φk, and the stick breaking weights πkare random and depend
+only on parameter α0. Due to the discrete nature of the DP realizations, Dirichle t processes and
+their variants have become an effective tool in mixture mode ling and learning of clustered data.
+The basic idea is to use the DP as a prior on the mixture compone nts in a mixture model, where
+each mixture component is associated with an atom in G. The posterior distribution of the atoms
+provides the probability distribution on mixture componen ts, and also yields a probability distri-
+bution of partitions of the data. The resultant mixture mode l, generally known as the Dirichlet
+processmixture,waspioneeredbytheworkofAntoniak(1974 )andsubsequentiallydevelopedby
+manyothers(e.g., (Lo, 1984; Escobarand West, 1995; MacEac hern and Mueller, 1998)).
+A Dirichlet process (DP) mixture can be utilized to model eac h group of observations, so
+a key issue is how to model and assess the local heterogeneity among a collection of DP mix-
+tures. In fact, there is an extensiveliterature in Bayesian nonparametrics that focuses on coupling
+2multiple Dirichlet process mixture distributions (e.g., M acEachern (1999); Muelleretal. (2004);
+DeIorio et al. (2004); Ishwaran and James (2001); Teh etal. ( 2006)). A common theme has been
+to utilize the Bayesian hierarchical modeling framework, w here the parameters are conditionally
+independentdraws froma probabilitydistribution. In part icular, supposethatthe u-indexedgroup
+is modeled using a mixingdistribution Gu. We highlightthe hierarchical Dirichlet process (HDP)
+introducedbyTeh et al.(2006),aframeworkthatweshallsub sequentiallygeneralize,whichposits
+thatGu|α0,G0∼DP(α0,G0)for some base measure G0and concentration parameter α0. More-
+over,G0is also random, and is distributed according to another DP: G0|γ,H∼DP(γ,H). The
+HDP model and other aforementioned work are inadequate for o ur problem, because we are in-
+terested in modeling the linkage among the groups notthrough the exchangeability assumption
+amongthegroups,butthroughthemoreexplicitdependence o nchangingvaluesofacovariate u.
+CouplingmultipleDP-distributedmixturedistributionsc anbedescribedunderageneralframe-
+work outlined by MacEachern (1999). In this framework, a DP- distributed random measure can
+be represented by the random “stick” and “atom” random varia bles (see Eq. (1)), which are gen-
+eral stochastic processes indexed by u∈V. Starting from this representation, there are a num-
+ber of proposals for co-varying infinite mixture models (Dua n et al., 2007; Petroneet al., 2009;
+Rodriguezet al., 2010; Dunson, 2008; Nguyenand Gelfand, 20 10). These proposals were de-
+signed for functional data only, i.e., where the data are giv en as a collection of sampled functions
+ofu,andthusnotsuitableforourproblem,becausefunctionali dentityinformationisassumedun-
+knowninoursetting. Inthisregard,theworkofGriffin and St eel(2006);DunsonandPark(2008);
+Rodriguezand Dunson (2009) are somewhat closer to our setti ng. These authors introduced spa-
+tial dependency of the local DP mixtures through the stick va riables in a number of interesting
+ways, while Rodriguezand Dunson (2009) additionally consi dered spatially varying atom vari-
+ables, resulting in a flexible model. These work focused most ly on the problem of interpolation
+and prediction, not clustering. In particular, they did not consider the problem of inferring global
+clustersfrom locallyobserveddatagroups,whichis ourpri mary goal.
+Todrawinferencesaboutglobalclusteringpatternsfromlo callygroupeddata,inthispaperwe
+will introduce an explicit notion of and model for global clu sters, through which the dependence
+amonglocallydistributedgroupsofdatacanbedescribed. T hisallowsustonotonlyassessthede-
+pendenceoflocalclustersassociatedwithmultiplegroups ofdataindexedby u,butalsotoextract
+the global clusters that arise from the aggregated observat ions. From the outset, we use a spatial
+stochastic process, and more generally a graphical model Hindexed over u∈Vto characterize
+the centering distribution of global clusters. Spatial sto chastic process and graphical models are
+versatileand customary choice for modeling of multivariat edata (Cressie, 1993; Lauritzen, 1996;
+Jordan, 2004). To “link” global clusters to local clusters, we appeal to a hierarchical and non-
+parametric Bayesian formalism: The distribution Qof global clusters is random and distributed
+according to a DP: Q|H∼DP(γ,H). For each u, the distribution Guof local clusters is as-
+sumed random, and is distributed according to a DP: Gu|Qindep∼DP(αu,Qu), whereQudenotes
+themarginaldistributionat uinducedbythestochasticprocess Q. Inotherwords,inthefirststage,
+the Dirichlet process Qprovides support for global atoms , which in turn provide support for the
+local atoms of lower dimensions for multiple groups in the second stage. Due to the use of hier-
+archy and the discreteness property of the DP realizations, there is sharing of global atoms across
+thegroups. Because differentgroups may shareonly disjointcomponentsoftheglobalatoms,the
+3spatialdependencyamongthegroupsisinducedbythespatia ldistributionoftheglobalatoms. We
+shall refer to the described hierarchical specification as t he nested Hierarchical Dirichlet process
+(nHDP)model.
+The idea of incorporating spatial dependence in the base mea sure of Dirichlet processes goes
+backtoCifarelli and Regazzini(1978);Muliereand Petrone (1993);Gelfand et al.(2005),although
+not in a fully nonparametric hierarchical framework as is co nsidered here. The proposed nHDP
+is an instantiation of the nonparametric and hierarchical m odeling philosophy eloquently advo-
+cated in Teh andJordan (2010), but there is a crucial distinc tion: Whereas Teh and Jordan gener-
+ally advocated for a recursive construction of Bayesian hierarchy, as exemplified by the po pular
+HDP (Teh et al., 2006), the nHDP features a richer nestedhierarchy: instead of taking a joint dis-
+tribution, one can take marginal distributions of a random d istribution to be the base measure to
+a DP in the next stage of the hierarchy. This feature is essent ial to bring about the relationship
+between global clusters and local clusters in our model. In f act, the nHDP generalizes the HDP
+model in the following sense: If Hplaces a prior with probability one on constant functions (i .e.,
+ifφ= (φu)u∈V∼Hthenφu=φv∀u,v∈V)then thenHDPisreduced to theHDP.
+MostcloselyrelatedtoourworkisthehybridDPofPetroneet al.(2009),whichalsoconsiders
+global and local clustering, and which in fact serves as an in spiration for this work. Because
+the hybrid DP is designed for functional data, it cannot be ap plied to situations where functional
+(curve) identity information is not available, i.e., when t he data are not given as a collection of
+curves. When such functional id information is indeed avail able, it makes sense to model the
+behavior of individual curves directly, and this ability ma y provide an advantage over the nHDP.
+Ontheotherhand,thehybridDPisarathercomplexmodel,and inourexperiment(seeSection5),
+ittendstooverfitthedataduetothemodelcomplexity. Infac t, weshowthatthenHDPprovidesa
+moresatisfactoryclusteringperformancefortheglobalcl ustersdespitenotusinganyfunctionalid
+information,whilethehybridDPrequiresnotonlysuchinfo rmation,italsorequiresthenumberof
+global clusters (“pure species”) to be pre-specified. It is w orth notingthat in theproposed nHDP,
+by not directly modeling the local cluster switching behavi or, our model is significantly simpler
+from bothviewpointsofmodelcomplexityand computational efficiencyofstatisticalinference.
+The paper outline is as follows. Section 2 provides a brief ba ckground of Dirichlet processes,
+the HDP, and we then proceed to define the nHDP mixture model. S ection 3 explores the model
+properties,includingastick-breakingcharacterization ,ananalysisoftheunderlyinggraphicaland
+spatial dependency, a P´ olya-urn sampling characterizati on. We also offer a discussion of a rather
+interesting issue intrinsic to our problem and the solution , namely, the conditions under which
+global clusters can be identified based on only locally group ed data. As with most nonparametric
+Bayesianmethods,inferenceisanimportantissue. Wedemon strateinSection4thattheconfluence
+of graphical/spatial with hierarchical modeling allows fo r efficient computations of the relevant
+posterior distributions. Section 5 presents several exper imental results, including a comparison to
+arecent approach in theliterature. Section 6concludes the paper.
+42 Model formalization
+2.1 Background
+We start with a brief background on Dirichlet processes (Fer guson, 1973), and then proceed to
+hierarchicalDirichletprocesses(Teh et al.,2006). Let (Θ0,B,G0)beaprobabilityspace,and α0>
+0. ADirichletprocessDP (α0,G0)isdefinedtobethedistributionofarandomprobabilitymeas ure
+Gover(Θ0,B)suchthat,foranyfinitemeasurablepartition (A1,...,Ar)ofΘ0,therandomvector
+(G(A1),...,G(Ar))is distributed as a finite dimensional Dirichlet distributi on with parameters
+(α0G0(A1),...,α 0G0(Ar)).α0is referred to as the concentration parameter, which govern s the
+amount of variability of Garound the centering distribution G0. A DP-distributed probability
+measureGis discrete with probability one. Moreover, it has a constru ctive representation due
+toSethuraman(1994): G=/summationtext∞
+k=1πkδφk,where(φk)∞
+k=1areiiddrawsfrom G0,andδφkdenotesan
+atomic probability measure concentrated at atom φk. The elements of the sequence π= (πk)∞
+k=1
+are referred to as “stick-breaking” weights, and can be expr essed in terms of independent beta
+variables:πk=π′
+k/producttextk−1
+l=1(1−π′
+l), where(π′
+l)∞
+l=1are iid draws from Beta (1,α0). Note thatπ
+satisfies/summationtext∞
+k=1πk= 1with probability one, and can be viewed as a random probabity measure
+on the positive integers. For notational convenience, we wr iteπ∼GEM(α0), following Pittman
+(2002).
+A useful viewpoint for the Dirichlet process is given by the P ´ olya urn scheme, which shows
+that draws from the Dirichlet process are both discrete and e xhibit a clustering property. From
+a computational perspective, the P´ olya urn scheme provide s a method for sampling from the
+random distribution G, by integrating out G. More concretely, let atoms θ1,θ2,...are iid ran-
+dom variables distributed according to G. BecauseGis random, θ1,θ2,...are exchangeable.
+Blackwell and MacQueen (1973) showed that the conditional d istribution of θigivenθ1,...,θi−1
+has thefollowingform:
+[θi|θ1,...,θi−1,α0,G0]∼i−1/summationdisplay
+l=11
+i−1+α0δθl+α0
+i−1+α0G0.
+This expression shows that θihas a positive probability of being equal to one of the previo us
+drawsθ1,...,θi−1. Moreover, the more often an atom is drawn, the more likely it is to be drawn
+in the future, suggesting a clustering property induced by t he random measure G. The induced
+distributionoverrandompartitionsof {θi}isalsoknownastheChineserestaurantprocess(Aldous,
+1985).
+A Dirichlet process mixture model utilizes Gas the prior on the mixture component θ. Com-
+bining with a likelihoodfunction P(y|θ) =F(y|θ),the DP mixturemodel is given as: θi|G∼G;
+yi|θiind∼F(·|θi). Such mixture models have been studied in the pioneering wor k of Antoniak
+(1974)andsubsequentiallybyanumberofauthors(Lo,1984; EscobarandWest,1995;MacEachern and Mueller,
+1998), For more recent and elegant accounts on the theories a nd wide-ranging applications of DP
+mixturemodeling,seeHjortet al. (2010).
+Hierarchical Dirichlet Processes. Next, we proceed giving a brief background on the HDP
+formalism of Teh et al. (2006), which is typically motivated from the setting of grouped data.
+5Underthissetting,theobservationsareorganizedintogro upsindexedbyacovariate u∈V,where
+Vis the index set. Let yu1,yu2,...,yunube the observations associated with group u. For each
+u, the{yui}iare assumed to be exchangeable. This suggests the use of mixt ure modeling: The
+yuiare assumed identically and independently drawn from a mixt ure distribution. Specifically,
+letθui∈Θudenote the parameter specifying the mixture component asso ciated withyui. Under
+the HDP formalism, Θuis the same space for all u∈V, i.e.,Θu≡Θ0for allu, andΘ0is
+endowed with the Borel σ-algebra of subsets of Θ0.θuiis referred to as local factors indexed by
+covariateu. LetF(·|θui)denote the distribution of observation yuigiven the local factor θui. Let
+Gudenote a prior distribution for the local factors (θui)nu
+i=1. We assume that the local factors θui’s
+are conditionallyindependentgiven Gu. As aresultwehavethefollowingspecification:
+θui|Guiid∼Gu;yui|θuiiid∼F(·|θui),foranyu∈V;i= 1,...,nu. (2)
+UndertheHDPformalism,tostatisticallycouplethecollec tionofmixingdistributions Gu,we
+positthatrandomprobabilitymeasures Guareconditionallyindependent,withdistributionsgiven
+by aDirichletprocesswithbaseprobabilitymeasure G0:
+Gu|α0,G0iid∼DP(α0,G0).
+Moreover,theHDPframeworktakesafullynonparametricand hierarchicalspecification,byposit-
+ingthatG0isalsoarandomprobabilitymeasure,whichisdistributeda ccordingtoanotherDirich-
+let processwithconcentrationparameter γandbaseprobabilitymeasure H:
+G0|γ,H∼DP(γ,H).
+An interestingpropertyof theHDPis thatbecause Gu’sare discreterandom probabilitymeasures
+(with probability one) whose support are given by the suppor t ofG0. Moreover, G0is also a
+discrete measure, thus the collection of Guare random discrete measures sharing the samecount-
+able support. In addition, because the random partitions in duced by the collection of θuiwithin
+each group uare distributed according to a Chinese restaurant process, the collection of these
+Chinese restaurant processes are statistically coupled. I n fact, they are exchangeable, and the
+distribution for the collection of such stoschastic proces ses is known as the Chinese restaurant
+franchise(Teh et al., 2006).
+2.2 Nestedhierarchy of DPsfor globalclustering analysis
+Settingandnotations. Inthispaperweareinterestedinthesamesettingofgrouped dataasthatof
+the HDP that is described by Eq. (2). Specifically, the observ ationsyu1,yu2,...,yunuwithin each
+groupuareiiddrawsfromamixturedistribution. Thelocalfactor θui∈Θudenotestheparameter
+specifyingthemixturecomponentassociatedwith yui. The(θui)nu
+i=1are iid drawsfrom themixing
+distribution Gu.
+ImplicitintheHDPmodelistheassumptionsthat thespaces Θuallcoincide, andthat random
+distributions Guare exchangeble. Both assumptions will be relaxed. Moreove r, our goal here is
+the inference of global clusters, which are associated with global factors that lie in the product
+spaceΘ :=/producttext
+u∈VΘu. To this end, Θis endowed with a σ-algebraBto yield a measurable space
+6(Θ,B). Within this paper and in the data illustrations, Θ =RV, andBcorresponds to the Borel
+σ-algebra of subsets of RV, Formally,a global factor , which are denoted by ψorφin the sequel,
+isahighdimensionalvector(orfunction)in Θwhosecomponentsareindexedbycovariate u. That
+is,ψ= (ψu)u∈V∈Θ, andφ= (φu)u∈V∈Θ. As a matterof notations,wealways use itodenote
+the numbering index for θu(so we have θui). We always use tandkto denote the number index
+for instances of ψ’s andφ’s, respectively (e.g., ψtandφk). The components of a vector ψt(φk)
+are denotedby ψut(φuk). Wemayalso useletters vandwbesideuto denotethegroupindices.
+Modeldescription. Ourmodelinggoalistospecifyadistribution Qontheglobalfactors ψ,and
+to relateQto the collection of mixing distributions Guassociated with the groups of data. Such
+resultant model shall enable us to infer about the globalclusters associated with a global factor ψ
+on the basis of data collected locallyby the collection of groups indexed by u. At a high level,
+the random probability measures Qand theGu’s are “glued” together under the nonparametric
+and hierarchical framework, while the probabilistic linka ge among the groups are governed by a
+stochastic process φ= (φu)u∈Vindexed byu∈Vand distributed according to H. Customary
+choices ofsuchstochasticprocesses includeeitheraspati alprocess,oragraphical model H.
+Specifically, let Qudenote the induced marginal distribution of ψu. Our model posits that for
+eachu∈V,GuisarandommeasuredistributedasaDPwithconcentrationpa rameterαu,andbase
+probability measure Qu:Gu|αu,Q∼DP(αu,Qu). Conditioning on Q, the distributions Guare
+independent, and Guvaries around the centering distribution Qu, with the amount of variability
+given byαu. The probability measure Qis random, and distributed as a DP with concentration
+parameterγand base probability measure H:Q|γ,H∼DP(γ,H), whereHis taken to be a
+spatial process indexed by u∈V, or more generally a graphical model defined on the collectio n
+of variables indexed by V. In summary, collecting the described specifications gives thenested
+HierarchicalDirichletprocess (nHDP)mixturemodel:
+Q|γ,H∼DP(γ,H),
+Gu|αu,Qindep∼DP(αu,Qu),forallu∈V
+θui|Guiid∼Gu, yui|θuiiid∼F(·|θui)forallu,i,
+As weshall see in thenext section,the φk’s, which are drawsfrom H, providethesupport for
+globalfactorsψt∼Q,whichinturnprovidethesupportforthelocalfactors θui∼Gu. Theglobal
+andlocalfactorsprovidedistinctrepresentationsforbot hglobalclustersandlocalclustersthatwe
+envision being present in data. Local factors θui’s provide the support for local cluster centers at
+eachu. Theglobalfactors ψinturnprovidethesupportforthelocalclusters,buttheya lsoprovide
+thesupportforglobalclustercentersinthedata,whenobse rvationsareaggregatedacrossdifferent
+groups.
+Relations to the HDP. Both the HDP and nHDP are instances of the nonparametric and h ierar-
+chical modelingframework involvinghierarchy ofDirichle t processes (Teh andJordan, 2010). At
+a high-level, the distinction here is that while the HDP is a r ecursive hierarchy of random prob-
+ability measures generally operating on the same probabili ty space, the nHDP features a nested
+hierarchy, in which the probabilityspaces associated with different levelsin the hierarchy are dis-
+tinctbutrelatedinthefollowingway: theprobabilitydist ributionassociatedwithaparticularlevel,
+7sayGu, has support in the support of the marginal distribution of a probability distribution (i.e.,
+Q) in the upperlevel in thehierarchy. Accordingly,for u∝ne}ationslash=v,GuandGvhavesupportin distinct
+components of vectors ψ. For a more explicit comparison, it is simple to see that if Hplaces
+distribution for constantglobal factors φwith probability one (e.g., for any φ∼Hthere holds
+φu=φv∀u,v∈V), then weobtaintheHDP ofTeh et al. (2006).
+3 Model properties
+3.1 Stick-breaking representationand graphical orspatia ldependency
+Giventhatthemultivariatebasemeasure QisdistributedasaDirichletprocess,itcanbeexpressed
+usingSethuraman’sstick-breakingrepresentation: Q=/summationtext∞
+k=1βkδφk. Eachatomφkismultivariate
+anddenotedby φk= (φuk:u∈V). Theφk’sareindependentdrawsfrom H,andβ= (βk)∞
+k=1∼
+GEM(γ). Theφk’s andβare mutuallyindependent. The marginal induced by Qat each location
+u∈Vis:Qu=/summationtext∞
+k=1βkδφuk. Since each Quhas support at the points (φuk)∞
+k=1, eachGu
+necessarily hassupportat thesepointsas well,and can bewr ittenas:
+Gu=∞/summationdisplay
+k=1πukδφuk;Qu=∞/summationdisplay
+k=1βkδφuk. (3)
+Letπu= (πuk)∞
+k=1. SinceGu’s are independent given Q, the weightsπu’s are independent
+givenβ. Moreover, because Gu|αu,Q∼DP(αu,Qu)it is possible to derive the relationship
+between weights πu’s andβ. Following Teh etal. (2006), if His non-atomic, it is necessary
+and sufficient for Gudefined by Eq. (3) to satisfy Gu∼DP(αuQu)that the following holds:
+πu∼DP(αu,β), whereπuandβare interpreted as probability measures on the set of positi ve
+integers.
+The connection between the nHDP and the HDP of Teh et al. (2006 ) can be observed clearly
+here: Thestick-breakingweightsof thenHDP-distributed Guhavethesamedistributionsas those
+of the HDP, whilethe atoms φukare linked by a graphical model distribution,or more genera lly a
+stochasticprocess indexedby u.
+The spatial/graphical dependency given by base measure Hinduces the dependency between
+theDP-distributed Gu’s. We shallexplorethisindetailsby consideringspecifice xamplesofH.
+Example 1 (Graphical model H).For concreteness, we consider a graphical model Hof three
+variablesφu,φv,φwwhich are associated with three locations u,v,w∈V. Moreover, assume the
+conditionalindependencerelation: φu⊥φw|φv. Letψ= (ψu,ψv,ψw)be arandom draw from Q.
+BecauseQ∼DP(γ,H),ψalso hasdistribution HonceQis integratedout. Thus, ψu⊥ψw|ψv.
+At each location u∈V, the marginal distribution Quof variableψuis random and Qu|γ,H
+∼DP(γ,Hu). Moreover, in general the Qu’s are mutually dependent regardless of any (condi-
+tional)independencerelationsthat Hmightconfer. ThisfactcanbeeasilyseenfromEq.(3). With
+probability 1, all Qu’s share the same β. It follows that Qu⊥Qw|Qv,β. Becauseβis random,
+theconditionalindependencerelationnolongerholdsamon gQu,Qw,Qvingeneral. Fromamod-
+eling standpoint, the dependency among the Qu’s is natural for our purpose, as Qprovides the
+distribution for the global factors associated with the glo bal clusters that we are also interested in
+inferring.
+8PSfrag replacements
+HHuHvHw
+QQuQvQw
+GuGvGw
+θuiθviθwiγ
+β
+φukφvkφwkπuπvπw∞{φk}
+{ψt}
+ψutψvt{θui}
+{θvi}PSfrag replacements
+HHuHvHwQ
+QuQvQwGuGvGw
+θuiθviθwiγ
+βφukφvkφwk
+πuπvπw∞
+{φk}
+{ψt}
+ψutψvt{θui}
+{θvi}
+Figure1. Left: ThenHDPisdepictedasagraphical model,whereeachun shaded noderepresents a
+random distribution. Right: Agraphical model representat ion ofthenHDPusingthestick-breaking
+parameterisation.
+Turningnowto distributions Guforlocal factors θui, wenotethat Gu,Gv,Gwareindependent
+givenQ. Moreover, for each u∈V, the support of Guis the same as that of Qu(i.e.,θuifor
+i= 1,2,...take value among (ψut)∞
+t=1). Integrating over the random Q, for any measurable
+partitionA⊂Θu, there holds: E[Gu(A)|H] =E[E[Gu(A)|Q]|H] =E[Qu(A)|H] =Hu(A). In
+sum, the global factors ψ’s take values in theset of (φk)∞
+k=1∼H, and providethe support set for
+the local factors θui’s at eachu∈V. The prior means of the local factors θui’s are also derived
+from thepriormean oftheglobalfactors.
+Example 2 (Spatial model H).To quantify more detailed dependency among DP-distributed
+Gu’s,letVbeafinitesubset of RrandHbea second-orderstochasticprocess indexedby v∈V.
+A customary choice for His a Gaussian process. In effect, φ= (φu:u∈V)∼N(µ,Σ), where
+thecovariance Σhas entriesoftheexponentialform: ρ(u,v) =σ2exp−{ω∝bardblu−v∝bardbl}.
+For any measurable partitions A⊂Θu, andB⊂Θv, we are interested in expressions for
+variation and correlation measures under QandGu’s. LetHuv(A,B) :=p(φu∈A,φv∈B|H).
+Defineg(γ) = 1/(γ+ 1). Applying stick-breaking representation for Qu, it is simple to derive
+that:
+Proposition1. Foranypairofdistinctlocations u,v),thereholds:
+Cov(Qu(A),Qv(B)|H) =g(γ)(Huv(A,B)−Hu(A)Hv(B)), (4)
+Var(Qu(A)|H) =g(γ)(Hu(A)−Hu(A)2), (5)
+Corr(Qu(A),Qv(B)) :=Cov(Qu(A),Qv(B)|H)
+Var(Qu(A)|H)1/2Var(Qv(B)|H)1/2
+=(Huv(A,B)−Hu(A)Hv(B))
+(Hu(A)−Hu(A)2)1/2(Hv(B)−Hv(B)2)1/2.(6)
+Foranypairoflocations u,v∈V, if∝bardblu−v∝bardbl → ∞, itfollowsthat ρ(u,v) = Cov(φu,φv|H)
+→0. Due to standard properties of Gaussian variables, φuandφvbecome less dependent of each
+other,andHuv(A,B)−Hu(A)Hv(B)→0,sothatCorr(Qu(A),Qv(B))→0. Ontheotherhand,
+ifu−v→0,weobtainthat Corr(Qu(A),Qv(A))→1,as desired.
+Turningtodistributions Gu’s forthelocal factors, thefollowingresult can beshown:
+9Proposition2. Foranypairof u,v∈V, thereholds:
+Var(Gu(A)|H) =E[Var(Gu(A)|Q)|H]+Var(E[Gu(A)|Q]|H)
+= (g(γ)+g(αu)−g(γ)g(αu))(Hu(A)−Hu(A)2), (7)
+Corr(Gu(A),Gv(B)) =g(γ)Corr(Qu(A),Qv(B)|H)
+(g(γ)+g(αu)−g(γ)g(αu))1/2(g(γ)+g(αv)−g(γ)g(αv))1/2.
+whereg(αu) = 1/(αu+1).
+Eq.(8)exhibitsaninterestingdecompositionofvariance. Notethat Var(Gu(A)|H)≥Var(Qu(A)|H).
+That is,thevariationofalocal factoris greaterthan thato ftheglobalfactor evaluatedat thesame
+location, where theextravariationis governedby concentr ationparameter αu. Ifαu→ ∞so that
+g(αu)→0, the local variation at udisappears, with the remaining variation contributed by th e
+globalfactorsonly. If αu→0sothatg(αu)→1,thelocalvariationcontributedby Gucompletely
+dominatestheglobalvariationcontributedby Qu.
+Finally,turningtocorrelationmeasuresinthetwostagesi nourhierachicalmodel,wenotethat
+Corr(Gu(A),Gv(B)|H)≤Corr(Qu(A),Qv(B)|H). That is, the correlation across the locations
+inVamong the distributions Gu’s of the local factors is bounded from above by the correlati on
+among the distribution Qu’s for the global factors. Note that Corr(Gu(A),Gv(B))vanishes as
+∝bardblu−v∝bardbl → ∞. Thecorrelationmeasureincreasesaseither αuorαvincreases. Thedependenceon
+γis quite interesting. As γranges from 0to∞so thatg(γ)decreases from 1to0, and as a result
+thecorrelationmeasureratio Corr(Gu(A),Gv(B))/Corr(Qu(A),Qv(B))decreases from 1to0.
+3.2 P´olya-urn characterization
+TheP´ olya-urn characterization of thecanonical Dirichle t process is fullyretained by thenHDP. It
+is also useful in highlightingboth local clustering and glo bal clustering aspects that are described
+by thenHDPmixture. In thesequel,theP´ olya-urn character ization is givenas a samplingscheme
+for both the global and local factors. Recall that the global factorsφ1,φ2,...are i.i.d. random
+variablesdistributedaccordingto H. Wealsointroducedrandomvectors ψtwhicharei.i.d. draws
+fromQ. Bothφkandψtaremultivariate,denotedby φk= (φuk)u∈Vandψt= (ψut)u∈V. Finally,
+foreach location u∈V, thelocalfactorvariables θuiare distributedaccordingto Gu.
+Notethateach ψtisassociatedwithone φk,andeachθuiisassociatedwithone ψut. Lettuibe
+the index of the ψutassociated with the local factor θui, andktbe the index of the φkassociated
+withtheglobalfactor ψt. LetKbethepresentnumberofdistinctglobalfactors φk. Thesampling
+process starts with K= 0and increases Kas needed. We also need notations for counts. We
+use notation nutto denotethepresent numberof local factors θultaking value ψut.nudenotes the
+number of local factors at group u(which is also the number of observations at group u).nu·kis
+the number of local factors at utaking value φuk. Letmudenote the number of factors ψtthat
+provide supports for group u. The notation qkdenotes the number of global factors ψt’s taking
+valueφk, whileq·denotesthetotalnumberofglobalfactors ψt’s. To beprecise:
+nut=/summationdisplay
+iI(tui=t);nu·k=/summationdisplay
+tnutI(kt=k);nu=/summationdisplay
+tnut;
+mu=/summationdisplay
+tI(nut>0);qk=/summationdisplay
+tI(kt=k);q·=/summationdisplay
+kqk.
+10First, consider the conditional distribution for θuigivenθu1,θu2,...,θu,i−1, andQ, where the
+Guisintegratedout:
+θui|θu1,...,θu,i−1,αu,Q∼mu/summationdisplay
+t=1nut
+i−1+αuδψut+αu
+i−1+αuQu. (8)
+Thisisamixture,andarealizationfromthismixturecanbeo btainedbydrawingfromthetermson
+the right-hand side with probabilitiesgiven by the corresp onding mixing proportions. If a term in
+thefirstsummationischosen,thenweset θui=ψutforthechosen t,andlettui=t,andincrement
+nut. If the second term is chosen, then we increment muby one, drawψumu∼Qu. In addition,
+wesetθui=ψumu, andtui=mu.
+Now we proceed to integrate out Q. SinceQappears only in its role as the distribution of the
+variableψt, we only need to draw sample ψtfromQ. The samples from Qcan be obtained via
+theconditionaldistributionof ψtas follows:
+ψt|{ψl}l/negationslash=t,γ,H∼K/summationdisplay
+k=1qk
+q·+γδφk+γ
+q·+γH. (9)
+Ifwedrawψtviachoosingaterminthesummationontheright-handsideof thisequation,weset
+ψt=φk, and letkt=kfor the chosen k, and increment qk. If the second term is chosen then we
+incrementKby one, drawφK∼Hand setψt=φK,kjt=K, andqK= 1.
+TheP´ olya-urncharacterizationofthenHDPcanbeillustra tedbythefollowingculinarymetaphor.
+Supposethatthere arethree groupsofdishes(e.g., appetiz er, main courseand dessert)indexedby
+u,vandw. View a global factor φk’s as a typical meal box where each φuk,φvkandφwkis asso-
+ciated with a dish group. In an electic eatery, the dishes are sold in meal boxes, while customers
+comein, buy dishes and share among oneanotheraccording to t hefollowingprocess. A new cus-
+tomer can join either one of the three groups of dishes. Upon j oining the group, she orders a dish
+to contributetothegroup,i.e., alocal factor θui, based onits popularitywithinthegroup. She can
+also choose to order a new dish, but to do so, she needs to order the entire meal box, i.e. a global
+factorψt. A mealbox ischosen basedon itspopularityas awhole, acros s alleating groups.
+The “sharing” of global factors (meal box) across indices ucan be seen by noting that the
+“pool”ofpresentglobalfactors {ψl}hassupportinthediscretesetofglobalfactorvalues φ1,φ2,....
+Moreover, the spatial (graphical) distribution of the glob al factors induces the spatial dependence
+amonglocal factors associatedwitheach group indexedby u. See Fig.2 foran illustration.
+3.3 Modelidentifiability and complexity
+This section investigates the nHDP mixture’s inferential b ehavior, including issues related to the
+model identifiability. It is useful to recall that a DP mixtur e model can be viewed as the infinite
+limitoffinitemixturemodels(Neal,1992;Ishwaran and Zare pour,2002b). ThenHDPcanalsobe
+viewed as the limit of a finite mixture counterpart. Indeed, c onsider the following finite mixture
+11PSfrag replacements{φk}{ψt}
+φukφvkψutψvt{θui} {θvi}
+Figure 2. Illustration of the assignments of mixture component membe rship via global and local
+factor variables for twogroups indexed by uandv.
+model:
+β|γ∼Dir(γ/L,...γ/L )πu|αu,β∼Dir(αuβ)φk∼H
+QL=L/summationdisplay
+k=1βkδφkGL
+u=L/summationdisplay
+k=1πukδφuk. (10)
+It is a known fact that as L→0,QL⇒Qweakly, in the sense that for any real-valued bounded
+and continuous function g, there holds/integraltext
+g dQL→/integraltext
+g dQin distribution (MuliereandSecchi,
+1995).2Because for each u∈V, there holds GL
+u∼DP(αuQL), it also follows that GL
+u⇒Gu
+weakly. Theabovecharacterizationprovidesaconvenientm eansofunderstandingthebehaviorof
+thenHDPmixturebystudyingthebehaviorofitsfinitemixtur ecounterpartwith Lglobalmixture
+components,as L→ ∞.
+Information denseness of nHDP prior. For concreteness in this section we shall assume that
+for anyu∈Vthe likelihood F(yu|φu)is specified by the normal distribution whose parameters
+such as mean and variance are represented by φu. Writeφu= (µu,σ2
+u)∈(R×R+). Recall that
+conditionally on Q,Gu’s are independent across u∈V. GivenGu, the marginal distribution on
+observation yuhasthefollowingdensity:
+fu(yu|Gu) =/integraldisplay
+F(yu|φu)dGu(φu). (11)
+Thus,eachfuisthedensityofalocation-scalemixtureofnormaldistrib ution. Thefu’sarerandom
+due to the randomness of Gu’s. In other words, the nHDP places a prior distribution, whi ch we
+denote by Π, over the collection of random measures (Gu)u∈V. This in turn induces a prior over
+the joint density of y:= (yu)u∈V, which we call Πas well. Replacing the mixing distributions Q
+andGuby thefinite mixture QLandGL
+u’s (as specified by Eq. (10)), we obtain the corresponding
+2A stronger result was obtained by IshwaranandZarepour (200 2b), Theorem 2, in which convergence holds for
+anyintegrablefunction gwithrespectto H.
+12marginaldensity:
+fL
+u(yu|Gu) =/integraldisplay
+F(yu|φu)dGL
+u(φu). (12)
+LetΠLtodenotetheinduced priordistributionfor {fL
+u}u∈V. From theabove, ΠL⇒Πweakly.
+We shall show that for each u∈Vthe prior ΠLis information dense in the space of finite
+mixtures as L→ ∞. Indeed, for any group index u, consider any finite mixture of normals fu,0
+associatedwithmixingdistributions Q0andGu,0oftheform:
+Q0=d/summationdisplay
+k=1βk,0δφk,0, Gu,0=d/summationdisplay
+k=1πuk,0δφuk,0, (13)
+Proposition 3. Suppose that the base measure Hplaces positive probability on a rectangle con-
+taining the support of Q0, then the prior ΠLplaces a positive probability in arbitrarily small
+Kullback-Leibler neighborhoodof fu,0forLsufficientlylarge. That is, for any ǫ>0, there holds:
+ΠL(fu:D(fu,0||fu)<ǫ)>0foranysufficientlylarge L.
+At a high level, this result implies that the nHDP provides a p rior over the space of mixture
+distributions that is “well spread” in the Kullback-Leible r topology. A proof of this result can be
+obtained using the same proof techniques of Ishwaran and Zar epour (2002a) for a similar result
+applied to (non-hierarchical) finite-dimensional Dirichl et distributions, and is therefore omitted.
+An immediate consequence of the information denseness prop erty is the weak consistency of the
+posteriordistributionof yuforanyu∈V, thankstotheasymptotictheory ofSchwartz (1965).
+Identifiability of factors φ.The above results are relevant from the viewpointof density estima-
+tion (for the joint vector y). From a clustering viewpoint, we are also interested in the ability of
+the nHDP prior in recovering the underlying local factors φuk’s, as well as the global factors φk’s
+for the global clusters. This is done by studyingthe identifi abilityof the finite mixtures that lie in
+the union of the support of ΠLfor allL <∞. This is the set of all densities (fL
+u)u∈V;L<∞whose
+correspondingmixingdistributionsare givenby Eq.(10).
+Recall that each marginal fL
+uis a normal mixture, and the Lmixture components are param-
+eterised by φuk= (µuk,σ2
+uk)fork= 1,...,L. Again, let fu,0be the “true” marginal density
+of a mixture distribution for group uthat hasdmixture components, and the associated mixing
+distributions Q0andGu,0are given by Eq. (13). The parameter for the k-th component for each
+k= 1,...,dis denoted by φuk,0= (µuk,0,σ2
+uk,0). The following is a direct consequence of Theo-
+rem 2 ofIshwaran and Zarepour(2002a):
+Proposition 4. Supposethat for any u∈V,fu(yu) =fu,0(yu)foralmostall yu. In addition,the
+mixingdistributions GL
+usatisfythefollowingcondition:
+/integraldisplay
+R×R+exp/parenleftbiggµ2
+u
+2(σ∗u−σu)/parenrightbigg
+GL
+u(dφu)<∞,
+foranyu∈V, whereσ∗
+u= min{σu1,0,...,σuk,0}. Then,Gu=Gu,0forallu∈V.
+In other words, this result claims that it is possible to iden tify all local clusters specified by
+φukandπukfork= 1,...,d, up to the ordering of the mixture component index k. A more
+13substantialissueistheidentifiabilityofglobalfactors. Underadditionalconditionsof“true”global
+factorsφk,0’s,andthedistributionofglobalfactors QL,theidentificationofglobalfactors φk,0’sis
+possible. Viewing a global factor φk= (φuk)u∈V(likewise,φk,0) as a function of u∈v, a trivial
+example is that when φk,0are constant functions, and that base measure H(and consequentially
+QL) places probability 1 on such set of functions, then the iden tifiability of local factors implies
+the identifiability of global factors. A nontrivial conditi on is that the “true” global factors φk,0as
+a function of ucan be parameterised by a small number of parameters (e.g. a l inear function, or
+an appropriately defined smooth function in u∈V). Then, it is possible that the identifiability of
+local factors also implies the identifiability of global fac tors. An in-depth theoretical treatment of
+thisimportantissueis beyondthescopeofthepresent paper .
+Theaboveobservationssuggestseveralprudentguidelines forpriorspecifications(viathebase
+measureH). To ensure good inferential behavior for the local factors φu’s, it is essential that the
+base measure Huplaces sufficiently small tail probabilities on both µuandσu. In addition, if it is
+believed the underlying global factors are smooth function in the domain V, placing a very vague
+priorHovertheglobalfactors(such asafactorialdistribution H=/producttext
+u∈VHubyassumingthe φu
+are independent across u∈V) may not do the job. Instead, an appropriate base measure Hthat
+putsmostofitsmassonsmoothfunctionsisneeded. Indeed,t heseobservationsarealsoconfirmed
+by ourempiricalexperimentsinSection 5.
+4 Inference
+In this section we shall describe posterior inference metho ds for the nested Hierarchical Dirichlet
+process mixture. We describe two different sampling approa ches: The “marginal approach” pro-
+ceeds by integrating out the DP-distributed random measure s, while the “conditional approach”
+exploitsthestick-breakingrepresentation. Theformerap proacharisesdirectlyfromtheP´ olya-urn
+characterization of the nHDP. However its implementation i s more involved due to book-keeping
+of the indices. Within this section we shall describe the con ditional approach, leaving the details
+ofthemarginal approach to thesupplementalmaterial. Both samplingmethodsdraw from theba-
+sicfeatures ofthesamplingmethodsdevelopedfortheHiera rchical DirichletProcess ofTeh et al.
+(2006),inadditiontothecomputationalissuesthatarisew henhigh-dimensionalglobalfactorsare
+sampled.
+Forthereader’sconvenience,werecallkeynotationsandin troduceafewmoreforthesampling
+algorithms. tuiis the index of the ψutassociated with the local factor θui, i.e.,θui=ψutui; andkt
+is the index of the φkassociated with the global factor ψt, i.e.,ψt=φkt. The local and global
+atomsarerelatedby θui=ψutui=φuktui. Letzui=ktuidenotethemixturecomponentassociated
+with observation yui. Turning to count variables, n−ui
+utdenotes the number of local atoms θul’s
+that are associated with ψt, excluding atom θui.n−ui
+u·kdenotes the number of local atoms θulthat
+such thatzul=k, leaving out θui.t−uidenotes the vector of all tul’s leaving out element tui.
+Likewise,k−tdenotesthevectorofall kr’sleavingoutelement kt. Inthesequel,theconcentration
+parametersγ,αu,andparametersfor Hareassumedfixed. Inpractice,wealsoplacestandardprior
+distributionsontheseparameters,followingtheapproach esofEscobarand West(1995);Teh et al.
+(2006) forγ,αu, and, e.g., Gelfandet al. (2005)for H’s.
+The main idea of the conditional sampling approach is to expl oit the stick-breaking represen-
+14tation ofDP-distributed Qinstead of integratingit out. Likewise, we also considerno t integrating
+over the base measure H. Recall that a priori Q∼DP(γ,H). Due to a standard property of the
+posteriorof a Dirichlet process, conditioningon the globa l factorsφk’s and the index vector k,Q
+is distributed as DP (γ+q·,γH+/summationtextK
+k=1qkδφk
+γ+q·). Note that vector qcan be computed directly from k.
+Thus, an explicit representation of QisQ=/summationtextK
+k=1βkδφk+βnewQnew, whereQnew∼DP(γ,H),
+and
+β= (β1,...,βK,βnew)∼Dir(q1,...,qk,γ).
+Conditioning on Q, or equivalently conditioning on β,φk’s in the stick breaking representa-
+tion, the distributions Gu’s associated with different locations u∈Vare decoupled (indepen-
+dent). In particular, the posterior of GugivenQandk,tand theφk’s is distributed as DP (αu+
+nu,αuQu+/summationtextK
+k=1nu·kδφuk
+αu+nu). Thus, an explicit representation of the conditional distr ibution ofGuis
+givenasGu=/summationtextK
+k=1πukδφuk+πunewGnew
+u,whereGnew
+u∼DP(αuβnew,Qnew
+u)and
+πu= (πu1,...,πuK,πunew)∼Dir(αuβ1+nu·1,...,αuβk+nu·K,αuβnew).
+In contrast to the marginal approach, we consider sampling d irectly in the mixture component
+variablezui=ktui, and in doing so we bypass the sampling steps involving kandt. Note that
+the likelihood of the data involves only the zuivariables and the global atoms φk’s. The mixture
+proportionvector βinvolvesonlycountvectors q= (q1,...,qK). ItsufficestoconstructaMarkov
+chain onthespaceof (z,q,β,φ).
+Samplingβ.As mentionedabove, β|q∼Dir(q1,...,qK,γ).
+Samplingz.Recall that a priori zui|πu,β∼πuwhereπu|β,αu∼DP(αu,β). Letn−ui
+u·kdenote
+thenumberofdataitemsinthegroup u,exceptyui,associatedwiththemixturecomponent k. This
+can bereadily computedfromthevector z.
+p(zui=k|z−ui,q,β,φk,Data) =/braceleftBigg
+(n−ui
+u·k+αuβk)F(yui|φuk)ifkpreviouslyused
+αuβnewf−yui
+uknew(yui) ifk=knew.(14)
+where
+f−yui
+uk(yui) =/integraltext
+F(yui|φuk)/producttext
+u′i′/negationslash=ui;zu′i′=kF(yu′i′|φu′k)H(φk)dφk/integraltext/producttext
+u′i′/negationslash=ui;zu′i′=kF(yu′i′|φu′k)H(φk)dφk. (15)
+Notethatifzuiistakentobe knew,thenweupdate K=K+1. (Obviously, knewtakesthevalueof
+theupdated K).
+Samplingq.To clarify the distribution for vector q, we recall an observation at the end of Sec-
+tion3.2thatthesetofglobalfactors ψt’scanbeorganizedintodisjointsubsets Ψu, each ofwhich
+isassociatedwithalocation u. Moreprecisely, ψt∈Ψuifandonlyif nut>0. Withineach group
+u,letmukdenotethenumberof ψt’stakingvalue φk. Then,qk=/summationtext
+u∈Vmuk.
+Conditioning on zwe can collect all data items in group uthat are associated with mixture
+componentφk, i.e., item indices uisuch thatzui=k. There are nu·ksuch items, which are
+distributedaccordingtoaDirichletprocesswithconcentr ationparameter αuβk. Thecountvariable
+15mukcorrespondstothenumberofmixturecomponentsformedbyth enu·kitems. Itwasshownby
+Antoniak(1974)that thedistributionof mukhas theform:
+p(muk=m|z,m−uk,β) =Γ(αuβk)
+Γ(αuβk+nu·k)s(nu·k,m)(αuβk)m,
+wheres(n,m)are unsigned Stirling number of the first kind. By definition, s(0,0) =s(1,1) =
+1,s(n,0) = 0forn>0,ands(n,m) = 0form>n. Forotherentries,thereholds s(n+1,m) =
+s(n,m−1)+ns(n,m).
+Samplingφ.Thesamplingof φ1,...,φkfollowsfromthefollowingconditionalprobabilities:
+p(φk|z,Data)∝H(φk)/productdisplay
+ui:zui=kF(yui|φuk)foreachk= 1,...,K.
+Let usindextheset Vby1,2,...,M, where|V|=M. Wereturn toourtwoexamples.
+As the first example, supposethat φkis normally distributed,i.e., under H,φk∼N(µk,Σk),
+and that the likelihood F(yui|θui)is given as well by N(θui,σ2
+ǫ), then the posterior distributionof
+φkisalsoGaussianwithmean ˜µkand variance ˜Σk, where:
+˜Σ−1
+k=Σ−1
+k+1
+σ2
+ǫdiag(n1·k,...,nM·k),
+˜µk=˜Σk/parenleftbigg
+Σ−1
+kµk+1
+σ2ǫ/bracketleftbigg/summationdisplay
+iy1iI(z1i=k).../summationdisplay
+iyMiI(zMi=k)/bracketrightbiggT/parenrightbigg
+. (16)
+Forthesecondexample,weassumethat φkisvery highdimensional,and thepriordistribution H
+is not tractable (e.g., a Markov random field). Direct comput ationis no longerpossible. A simple
+solution is to Gibbs sample each component of vector φk. Suppose that under a Markov random
+fieldmodelH,theconditionalprobability H(φuk|φ−u
+k)issimpletocompute. Then,forany u∈V,
+p(φuk|φ−u
+k,z,Data)∝H(φuk|φ−u
+k)/productdisplay
+i:zui=kF(yui|φuk).
+Computation of conditional density of data A major computational bottleneck in sampling
+methods for the nHDP is the computation of conditional densi ties given by Eq. (15) and (18). In
+general,φisvery highdimensional,andintegratingover φ∼Hisintractable. Howeveritispos-
+sibletoexploitthestructureof Htoalleviatethissituation. Asanexample,if Hisconjugateto F,
+thecomputationoftheseconditionalscan beachievedinclo sedform. Alternatively,if Hisspeci-
+fied as a graphical model where conditional independence ass umptionscan be exploited, efficient
+inference methodsingraphical modelscan bebroughttobear onourcomputationalproblem.
+Example 1. Suppose that the likelihood function Fis given by a Gaussian distribution, i.e.,
+yui|θui∼N(θui,σ2
+ǫ)forallu,i, and thattheprior Hisconjugate,i.e., His alsoa Gaussiandistri-
+bution:φk∼N(µk,Σk). Due to conjugacy,the computations in Eq. (18) are readily a vailablein
+closedforms. Specifically,thedensityinEq. (18)takesthe followingexpression:
+f−yui
+uk(yui) =1
+(2π)1/2σǫ|Ck+|
+|Ck|exp/parenleftbigg
+−1
+2σ2ǫy2
+ui+1
+2µ−ui
+k+TC−1
+k+µ−ui
+k+−1
+2µ−ui
+kTC−1
+kµ−ui
+k/parenrightbigg
+,
+160 5 10 15−2−10123Spatially varying mixture distributions
+Locations uY
+0 5 10 15−2−10123Spatially varying mixture distributions
+Locations uY
+Figure 3. Left: Data set A illustrates a simulated problem of tracking particles organized into
+clusters, which move in smooth paths. Right: Data set B illus trates bifurcating trajectories. In both
+cases, data are given not astrajectories, but only asindivi dual points denoted by circles at each u.
+where
+C−1
+k+=Σ−1
+k+1
+σ2ǫdiag(n−ui
+1·k,...,1+n−ui
+u·k,...,n−ui
+M·k),
+µ−ui
+k+=Ck+/parenleftbigg
+Σ−1
+kµk+1
+σ2
+ǫ/bracketleftbigg
+···/summationdisplay
+i′:zu′i′=kyu′i′+yuiI(ui=u′i′)···/bracketrightbiggT/parenrightbigg
+,
+C−1
+k=Σ−1
+k+1
+σ2
+ǫdiag(n−ui
+1·k,...,n−ui
+u·k,...,n−ui
+M·k),
+µ−ui
+k=Ck/parenleftbigg
+Σ−1
+kµk+1
+σ2ǫ/bracketleftbigg
+···/summationdisplay
+i′:zu′i′=k;u′i′/negationslash=uiyu′i′···/bracketrightbiggT/parenrightbigg
+. (17)
+It is straightforward to obtain required expressions for f−yt
+k(yt),f−yui
+uknew(yui), andf−yt
+knew(yt)– the
+lattertwoquantitiesare givenin theAppendix.
+Example 2. IfHis a chain-structured model, the conditional densities defi ned by Eq. (18) are
+not available in closed forms, but we can still obtain exact c omputation using an algorithm that is
+akin to the well-known alpha-beta algorithm in the Hidden Ma rkov model (Rabiner, 1989). The
+running time of such algorithm is proportional to the size of the graph (i.e., |V|). For general
+graphical models, one can apply a sum-product algorithm or a pproximate variational inference
+methods(WainwrightandJordan, 2008).
+5 Illustrations
+Simulation studies. We generate two data sets of spatially varying clustered pop ulations (see
+Fig. 3 for illustrations). In both data sets, we set V={1,...,15}. For data set A, K= 5
+1734567020406080100120140160Num of global clusters
+0 5 10 15−2−10123Posterior distributions of global cluster centers
+Locations uY
+Figure 4. Data set A. Left: Posterior distribution of the number of glo bal clusters. Right: Poste-
+rior distributions of the global atoms. Dashed lines denote the mean curve and (.05,.95) credible
+intervals.
+12345050100150Num of global clusters
+0 5 10 15−2−10123Posterior distributions of global cluster centers
+Locations uY
+Figure 5. Data set B. Left: Posterior distribution of the number of glo bal clusters (atoms). Right:
+Posterior distributions of the global atoms. Dashed lines d enote the mean curve and the (.05,.95)
+credible intervals.
+global factors φ1,...,φ5are generated from a Gaussian process (GP). These global fac tors pro-
+videsupportfor15spatiallyvaryingmixturesofnormaldis tributions,eachofwhichhas5mixture
+components. The likelihood F(θui)is given by N(θui,σ2
+ǫ),σǫ= 0.1. For eachuwe generated
+independently 100 samples from the corresponding mixture ( 20 samples from each mixture com-
+ponents). Note that each circle in the figures denote a data sa mple. This kind of data can be
+encountered in tracking problems, where the samples associ ating with each covariate ucan be
+viewed as a snapshot of the locations of moving particles at t ime pointu. The particles move in
+clusters. Theymay switch clusters at any time,but theident ificationof each particleis notknown
+as they movefrom one timestep to the next. The clusters thems elvesmovein relativelysmoother
+paths. Moreover, the number of clusters is not known. It is of interest to estimate the cluster
+centers, as well as their moving paths.3For data set B, to illustrate the variation in the number
+of local clusters at different locations, we generate a numb er of global factors that simulate the
+3Particle-specifictrackingispossibleiftheidentityofth especificparticleismaintainedacrosssnapshots.
+18345670102030405060Num of global clusters
+0 5 10 15−2−10123Posterior distributions of global cluster centers
+Locations uY
+Figure 6. Effects of vague prior for Hresults in weak identifiability of global clusters, even as t he
+local clusters are identified reasonably well.
+12345020406080100Num of local clusters at loc=1
+12345020406080100Num of local clusters at loc=3
+12345020406080100120Num of local clusters at loc=5
+12345020406080100120Num of local clusters at loc=8
+12345020406080100120Num of local clusters at loc=10
+12345020406080100120140Num of local clusters at loc=11
+12345020406080100120140160Num of local clusters at loc=13
+12345020406080100120140160Num of local clusters at loc=15
+Figure7. DatasetB:Posteriordistribution ofthenumberoflocalclu stersassociating withdifferent
+group index (location) u.
+bifurcation behavior in a collection of longitudinaltraje ctories. Here a trajectory corresponds to a
+global factor. Specifically, we set V={1,...,15}. Starting at u= 1there is one global factor,
+which is a random draw from a relatively smooth GP with mean fu nctionµ(u) =βµu, where
+βµ∼Unif(−0.2,0.2)and theexponentialcovariancefunctionparameterised by σ= 1,ω= 0.05.
+Atu= 5,theglobalfactorsplitsintotwo,withthesecond onealso a n independentdraw fromthe
+same GP, which is re-centered so that its value at u= 4is the same as the value of the previous
+global factor at u= 4. Atu= 10, the second global factor splits once more in the same manner .
+These three global factors provide support for the local clu sters at each u∈V. The likelihood
+F(·|θui)is given by a normal distribution with σǫ= 0.2. At eachuwe generated 30 independent
+observations.
+Althoughit is possibleto perform clustering analysisfor d ata at each location u, it is not clear
+howtolinktheseclustersacrossthelocations,especially giventhatthenumberofclustersmightbe
+differentfordifferent u’s. ThenHDPmixturemodelprovidesanaturalsolutiontothi sproblem. It
+190 5 10 15 20 25−3−2−101234Progresterone hormone curves
+Locations uY
+Figure 8: Progeresterone hormone curves.
+0 5 10 15 20 25−2−10123Posterior distributions of global cluster centers
+Locations uY
+0 5 10 15 20 25−1.5−1−0.500.511.522.5Posterior distributions of global cluster centers
+Locations uY
+Figure 9. Clustering results using the nHDP mixture model (Left), and the hybrid-DP
+of Petrone et al. (2009) (Right). Mean and credible interval s of global clusters (in dashed lines)
+are compared to sample mean curves of the contraceptive grou p and no contraceptive group in
+black solid withsquare markers.
+isfitforbothdatasetsusingessentiallythesamepriorspec ifications. Theconcentrationparameters
+are given by γ∼Gamma(5,.1)andα∼Gamma(20,20).His taken to be a mean-0 GP using
+(σ,ω) = (1,0.01)for data set A, and (1,0.05)for data set B. The variance σ2
+ǫis endowed with
+priorInvGamma (5,1). Theresultsofposteriorinference(viaMCMCsampling)for bothdatasets
+are illustrated by Fig. 4 and Fig. 5. With both data sets, the n umber global clusters are estimated
+almost exactly (5 and 3, respectively, with probability >90%). The evolution of the posterior
+distributions on the number of local clusters for data set B i s given in Fig. 7. In both data sets,
+the local factors are accurately estimated (see Figs. 4 and 5 ). For data set B, due to the varying
+number of local clusters, there are regions for u, specifically the interval [5,10]where multiple
+globalfactors alternatetheroleofsupportinglocal clust ers,resultinginwidercrediblebands.
+In Section 3 we discussed the implications of prior specifica tions of the base measure Hfor
+the identifiability of global factors. We have performed a se nsitivity analysis for data set A, and
+20  
+10 20 30 40 50 6010
+20
+30
+40
+50
+60 0.750.80.850.90.951
+  
+10 20 30 40 50 6010
+20
+30
+40
+50
+60
+0.30.40.50.60.70.80.91
+Figure 10. Pairwise comparison of individual hormone curves. Each ent ry in the heatmap depicts
+the posterior probability that the two curves share the same localclusters, averaged over a fixed
+interval ([1,20] in the left, and [21,24] inthe right figure) inthe menstrual cycle.
+1 2 301020304050607080Num of global clusters
+1 2 3010203040506070Num of local clusters at loc=5
+1 2 301020304050607080Num of local clusters at loc=18
+1 2 3020406080100Num of local clusters at loc=22
+1 2 3020406080100Num of local clusters at loc=24
+Figure 11. The leftmost panel shows the posterior distribution of the n umber of global clusters,
+while remaining panels show the the number of local clusters associating withgroup index u.
+found that the inference for global factors is robust when ωis set to be in [.01,.1]. Forω= 0.5,
+forinstance,whichimpliesthat φuareweaklydependentacross u’s,wearenotabletoidentifythe
+desired global factors (see Fig. 6), despite the fact that lo cal factors are still estimated reasonably
+well.
+Theeffectsofpriorspecificationfor σǫontheinferenceofglobalfactorsaresomewhatsimilar
+tothehybridDPmodel: asmaller σǫencourageshighernumbersofandlesssmoothglobalcurves
+toexpandthecoverageofthefunctionspace(seeSec. 7.3ofN guyenand Gelfand(2010)). Within
+our context, the prior for σǫis relatively more robust than that of ωas discussed above. The prior
+forconcentration parameter γis extremelyrobustwhilethepriorsfor αu’sare somewhatless. We
+believe the reason for this robustness is due to the modeling of the global factors in the second
+stageofthenestedhierarchyofDPs,andtheinferenceabout thesefactorshastheeffectofpooling
+data from across the groups in the first stage. In practice, we take allαu’s to be equal to increase
+therobustnessoftheassociatedprior.
+Progesterone hormone clustering. We turn to a clustering analysis of Progesterone hormone
+data. This data set records the natural logarithm of the prog esterone metabolite, measured by
+urinary hormoneassay, during a monthlycycle for 51 female s ubjects. Each cycle ranges from -8
+21  
+10 20 30 40 50 6010
+20
+30
+40
+50
+60
+0.40.50.60.70.80.91
+  
+10 20 30 40 50 6010
+20
+30
+40
+50
+60
+0.20.30.40.50.60.70.80.91
+Figure 12. Pairwise comparison of individual hormone curves using the hybrid-DP (Petrone et al.,
+2009). Each entry in the heatmap depicts the posterior proba bility that the two curves share the
+samelocalclusters, averaged over afixedinterval ([1,20] inthe left, and [21,24] inthe right figure)
+in the menstrual cycle.
+to15(8dayspre-ovulationto15dayspost-ovulation). Wear einterestedinclusteringthehormone
+levels per day, and assessing the evolution over time. We are also interested in global clusters,
+i.e., identifying global hormone pattern for the entire mon thly cycle and analyzing the effects on
+contraception on the clustering patterns. See Fig. 8 for the illustration and Brumback andRice
+(1998) formoredetailson thedataset.
+Forpriorspecifications,weset γ∼Gamma(5,0.1),andαu= 1forallu. Letσǫ∼InvGamma (2,1).
+ForH, we setµ= 0,σ= 1andω= 0.05. It is found that the there are 2 global clusters with
+probability close to 1. In addition, the mean estimate of glo bal clusters match very well with the
+sample means from the two groups of women, a group of those usi ng contraceptives and a group
+thatdonot(seeFig.9). Examiningthevariationsoflocalcl usters,thereisasignificantprobability
+of having only one local cluster during the first 20 days. Betw een day 21 and 24 the number of
+local clustersis2 withprobabilitycloseto1.
+To elaborate the effects of contraception on the hormone beh avior (the last 17 female subjects
+are known to use contraception), a pairwise comparison anal ysis is performed. For every two
+hormone curves, we estimate the posterior probability that they share the same local cluster on a
+givenday,whichisthenaveragedoverdaysinagiveninterva l. Itisfoundthatthehormonelevels
+among these women are almost indistinguishable in the first 2 0 days (with the clustering-sharing
+probabilitiesintherangeof 75%),butinthelast4days,theyaresharplyseparatedintotwod istinct
+regimes(withtheclustering-sharingprobabilitybetween thetwogroupsare droppedto 30%).
+WecompareourapproachtothehybridDirichletprocess(hyb rid-DP)approach (Petroneet al.,
+2009; Nguyen and Gelfand, 2010), perhaps the only existing a pproach in the literature for joint
+modeling of global and local clusters. The data are given to t he hybrid-DP as the replicates of
+a random functional curve, whereas in our approach, such fun ctional identity information is not
+used. In other words, for us only a collection of hormone leve ls across different time points are
+given (i.e., the subject ID of hormone levels are neither rev ealed nor matched with one another
+acrosstimepoints). Forasensiblecomparison,thesamepri orspecificationforbasemeasure Hof
+22the global clusters were used for both approaches. The infer ence results are illustrated in Fig. 9.
+A close look reveals that the global clusters obtained by the hybrid-DP approach is less faithful
+to the contraceptive/no contraceptive grouping than ours. This can be explained by the fact that
+hybrid-DPisamorecomplexmodelthatdirectly specifiesthe localclusterswitchingbehaviorfor
+functional curves. It is observed in this example that an ind ividual hormone curve tends to over-
+switch the local cluster assignments for u≥20, resulting in significantly less contrasts between
+the two group of women (see Fig. 10 and 12). This is probably du e the complexity of the hybrid-
+DP,whichcanonlybeovercomewithmoredata(seePropositio ns7and8ofNguyenand Gelfand
+(2010) for a theoretical analysis of this model’s complexit y and posterior consistency). Finally, it
+is also worth noting that the hybrid-DP approach practicall y requires the number of clusters to be
+specifiedapriori(asintheso-called k-hybrid-DPinPetroneet al.(2009)),whilesuchinformatio n
+isdirectly infered fromdatausingthenHDPmixture.
+6 Discussions
+We have described a nonparametric approach to the inference of global clusters from locally dis-
+tributed data. We proposed a nonparametricBayesian soluti onto thisproblem, by introducingthe
+nested Hierarchical Dirichlet process mixture model. This model has the virtue of simultaneous
+modeling of both local clusters and global clusters present in the data. The global clusters are
+supported by a Dirichlet process, using a stochastic proces s as its base measure (centering distri-
+bution). The local clusters are supported by the global clus ters. Moreover, the local clusters are
+randomly selected using another hierarchy of Dirichlet pro cesses. As a result, we obtain a col-
+lection of local clusters which are spatially varying, whos e spatial dependency is regulated by an
+underlying spatial or a graphical model. The canonical aspe cts of the nHDP (because of its use
+of the Dirichlet processes) suggest straightforward exten sions to accomodate richer behaviors us-
+ing Poisson-Dirichlet processes (also known as the Pittman -Yorprocesses), where they have been
+found to be particularly suitable for certain applications , and where our analysis and inference
+methods can be easily adapted. It would also be interesting t o consider a multivariate version of
+thenHDPmodel. Finally,themannerinwhichglobalandlocal clustersarecombinedinthenHDP
+mixture model is suggestive of ways of direct and simultaneo us global and local clustering for
+variousstructureddatatypes.
+7 Appendix
+7.1 Marginalapproach tosampling
+The P´ olya-urn characterization suggests a Gibbs sampling algorithm to obtain posterior distribu-
+tions of the local factors θui’s and the global factors ψt’s, by integrating out random measures Q
+andGu’s. Rather than dealing with the θui’s andψtdirectly, we shall sample index variables tui
+andktinstead, because θui’s andψt’s can be reconstructed from the index variables and the φk’s.
+This representation is generally thought to make the MCMC sa mpling more efficient. Thus, we
+construct a Markov chain on thespace of {t,k}. Although thenumber of variables is in principle
+unbounded,onlyfinitelymanyareactuallyassociated todat aand represented explicitly.
+23A quantity that plays an important role in the computation of conditional probabilities in this
+approachistheconditionaldensityofaselectedcollectio nofdataitems,giventheremainingdata.
+Forasingleobservation i-thatlocation u,definetheconditionalprobabilityof yuiunderamixture
+componentφuk, givent,kand alldataitemsexcept yui:
+f−yui
+uk(yui) =/integraltext
+F(yui|φuk)/producttext
+u′i′/negationslash=ui;zu′i′=kF(yu′i′|φu′k)H(φk)dφk/integraltext/producttext
+u′i′/negationslash=ui;zu′i′=kF(yu′i′|φu′k)H(φk)dφk. (18)
+Similary, for a collection of observationsof all data yuisuch thattui=tfor a chosen t, which we
+denotebyvector yt,letf−yt
+k(yt)betheconditionalprobabilityof ytunderthemixturecomponent
+φk,givent,kandall dataitemsexcept yt.
+Samplingt.Exploitingtheexchangeabilityofthe tui’swithinthegroupofobservationsindexedby
+u, we treattuias the last variable being sampled in the group. To obtain the conditional posterior
+fortui, we combine the conditional prior distribution for tuiwith the likelihood of generating
+datayui. Specifically, the prior probability that tuitakes on a particular previously used value tis
+proportionalto n−ui
+ut,whiletheprobabilitythatittakesonanewvalue tnew=mu+1isproportional
+toαu. The likelihood due to yuigiventui=tfor some previously used tisf−yui
+uk(yui). Here,
+k=kt. Thelikelihoodfor tui=tnewiscalculated by integratingoutthepossiblevaluesof ktnew:
+p(yui|t−ui,tui=tnew,k,Data) =K/summationdisplay
+k=1qk
+q·+γf−yui
+uk(yui)+γ
+q·+γf−yui
+uknew(yui),(19)
+wheref−yui
+uknew(yui) =/integraltext
+F(yui|φu)Hu(φu)dφuis thepriordensityof yui. As aresult, theconditional
+distributionof tuitakes theform
+p(tui=t|t−ui,k,Data)∝/braceleftBigg
+n−ui
+utf−yui
+ukt(yui) iftpreviouslyused
+αup(yui|t−ui,tui=tnew,k)ift=tnew.(20)
+Ifthesampledvalueof tuiistnew, weneed toobtainasampleof ktnewby samplingfrom Eq.(19):
+p(ktnew=k|t,k−tnew,Data)∝/braceleftBigg
+qkf−yui
+uk(yui)ifkpreviouslyused ,
+γf−yui
+uknew(yui)ifk=knew.(21)
+Samplingk.Aswiththelocalfactorswithineachgroup,theglobalfacto rsψt’sarealsoexchange-
+able. Thus we can treat ψtfor a chosen tas the last variable sampled in the collection of global
+factors. Notethatchangingindexvariable ktactuallychangesthemixturecomponentmembership
+forrelevantdataitems(acrossallgroups u)thatareassociatedwith ψt,thelikelihoodobtainedby
+settingkt=kis given byf−yt
+k(yt), whereytdenotes the vector of all data yuisuch thattui=t.
+So, theconditionalprobabilityfor ktis:
+p(kt=k|t,k−t,Data)∝/braceleftBigg
+qkf−yt
+k(yt)ifkpreviouslyused ,
+γf−yt
+knew(yt)ifk=knew,(22)
+wheref−yt
+knew(yt) =/integraltext/producttext
+ui:tui=tF(yui|φu)H(φ)dφ.
+24Samplingof γandα.WefollowthemethodofauxiliaryvariablesdevelopedbyEsc obarand West
+(1995) and Teh et al. (2006). Endow γwith a Gamma (aγ,bγ)prior. At each sampling step, we
+drawη∼Beta(γ+1,q·). Then the posterior of γis can be obtained as a gamma mixture, which
+canbeexpressedas πγGamma(aγ+K,bγ−log(η))+(1−πγ)Gamma(aγ+K−1,bγ−log(η)),
+whereπγ= (aγ+K−1)/(aγ+K−1 +q·(bγ−log(η))). The procedure is the same for each
+αu, withnuandmuplaying the role of q·andK, respectively. Alternatively, one can force all αu
+tobeequal and endowitwithagammaprior, asin Teh et al.(200 6).
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+27
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+arXiv:1001.0598v2  [astro-ph.HE]  24 Oct 2010Preprint typeset using L ATEX style emulateapj v. 11/10/09
+A NEW CLASSIFICATION METHOD FOR GAMMA-RAY BURSTS
+Hou-Jun L ¨u1, En-Wei Liang1,2, Bin-Bin Zhang2, and Bing Zhang2
+ABSTRACT
+RecentSwiftobservations suggest that the traditional long vs. short GRB clas sification scheme does
+not always associate GRBs to the two physically motivated model typ es, i.e. Type II (massive star
+origin) vs. Type I (compact star origin). We propose a new phenome nological classification method
+of GRBs by introducing a new parameter ε=Eγ,iso,52/E5/3
+p,z,2, whereEγ,isois the isotropic gamma-ray
+energy (in units of 1052erg), and Ep,zis the cosmic rest frame spectral peak energy (in units of 100
+keV). For those short GRBs with “extended emission”, both quant ities are defined for the short/hard
+spike only. With the current complete sample of GRBs with redshift an dEpmeasurements, the
+εparameter shows a clear bimodal distribution with a separation at ε∼0.03. The high- εregion
+encloses the typical long GRBs with high-luminosity, some high- z“rest-frame-short” GRBs (such as
+GRB 090423 and GRB 080913), as well as some high- zshort GRBs (such as GRB 090426). All
+these GRBs have been claimed to be of the Type II origin based on oth er observational properties
+in the literature. All the GRBs that are argued to be of the Type I or igin are found to be clustered
+in the low- εregion. They can be separated from some nearby low-luminosity long GRBs (in 3 σ)
+by an additional T90criterion, i.e. T90,z/lessorsimilar5 s in the Swift/BAT band. We suggest that this new
+classification scheme can better match the physically-motivated Ty pe II/I classification scheme.
+Subject headings: gamma-ray bursts: general — methods: statistical
+1.INTRODUCTION
+Phenomenologically, gamma-ray bursts (GRBs) are
+classified as long vs.short with a division line at the
+observed duration T90∼2 s (Kouveliotou et al. 1993).
+Robust associations of the underlying supernovae (SNe)
+with some long GRBs (Galama et al. 1998; Stanek et al.
+2003; Hjorth et al. 2003; Malesani et al. 2004; Modjaz
+et al. 2006; Pian et al. 2006) and the fact that long
+GRB host galaxies are typically irregular galaxies with
+intense starformation(Fruchter et al. 2006)suggestthat
+they are likely related to the deaths of massive stars,
+and the “collapsar” model has been widely recognized
+as the standard scenario for long GRBs (Woosley 1993;
+Paczy´ nski 1998; Woosley & Bloom 2006). Observational
+breakthroughs led by the Swiftmission (Gehrels et al.
+2004) suggest that at least some short GRBs are associ-
+ated with nearby host galaxies with little star formation
+(Tanvir et al. 2005), and that they are not associated
+with an underlying SN (Gehrels et al. 2005; Villasenor
+et al. 2005; Fox et al. 2005; Hjorth et al. 2005; Berger
+et al. 2005), favoring the idea that they are produced by
+mergers of two compact stellar objects, such as NS-NS
+and NS-BH mergers (Eichler et al. 1989; Narayan et al.
+1992; Nakar 2007).
+However, several lines of observational evidence in the
+Swiftera suggest that duration is not necessarily a re-
+liable indicator of the physical nature of a GRB. (1)
+The non-detection of a SN signature associated with the
+nearby long GRBs 060614 and 060505 (Gehrels et al.
+2006; Gal-Yam et al. 2006; Fynbo et al. 2006; Della
+Valle et al. 2006) disfavors the conventional collapsar
+scenario of long GRBs (Woosley & Bloom 2006). Some
+1Department of Physics, Guangxi University, Nanning
+530004, China; lew@gxu.edu.cn
+2Department of Physics and Astronomy, University of
+Nevada, Las Vegas, NV 89154. zhang@physics.unlv.eduobservationalproperties of the ∼100 s long GRB 060614
+are very similar to those of some short GRBs (Gehrels et
+al. 2006; Zhang et al. 2007), making it likely associated
+with the physical category that most short GRBs belong
+to. (2) Significant soft “extended” gamma-ray emission
+and late X-ray flares are observed in a handful of “short”
+GRBs (Barthelmy et al. 2005; Norris & Bonnell 2006;
+Perley et al. 2009), suggesting that they are not neces-
+sarily short. A new physically motivated classification
+scheme, i.e., Type II (massive star origin) vs. Type I
+(compact star origin) was proposed (Zhang 2006; Zhang
+etal. 2007). Kannetal(2010,2008)systematicallystud-
+ied the optical afterglow emission properties of Type II
+and Type I GRBs, and suggested that these properties
+carry important information about the nature of GRB
+progenitors. A full definition of Type I/II GRBs as well
+as the multiple observational criteria are presented in
+Zhang et al. (2009). (3) Two high-redshift gamma-ray
+bursts, GRB 080913 at z= 6.7 (Greiner et al. 2009;
+Perez-Ramirez et al. 2010) and GRB 090423 at z= 8.2
+(Tanvir et al. 2009; Salvaterra et al. 2009), appear as
+intrinsically short GRBs. Their observed durations are
+(8±1)s and∼10.3sin the Swift/BAT band, respectively.
+Correctedredshift, the durationsof twohigh- zGRBs are
+shorter than 2 seconds in the rest frame. The physical
+origin of these high- zGRBs have been subject to debate
+(Greiner et al. 2009; Perez-Ramirez er al. 2010; Tanvir
+et al. 2009; Salvaterra et al.2009). Although compact
+star mergers may occur at such a high redshift (Belczyn-
+ski et al. 2010), various observational properties of these
+two GRBs point towards the massive star origin (Zhang
+et al. 2009; Lin et al. 2009; Belczynki et al. 2010).
+(4) A more striking case is GRB 090426, whose observed
+BAT band T90is only 1.2±0.3s, and the rest frame du-
+ration is only ∼0.33 s at z= 2.609 (Levesque et al.
+2009). It greatly exceeds the previous short GRB red-
+shift record, i.e. z= 0.923for GRB 070714B(Graham et2 L¨ u et al.
+al. 2009). Phenomenologically, this is an unambiguous
+short-durationGRB,buttheavailableafterglowandhost
+galaxypropertiespointtowardsadifferentpicture: it has
+a blue, very luminous, star-forming putative host galaxy
+with a small angular offset of the afterglow location from
+the center, and a similar medium density as typical Type
+II GRBs. All these suggest that the burst is more closely
+related to the core collapse of a massive star (Levesque
+et al. 2009; Antonelli et al. 2009; Xin et al. 2010). (5)
+Monte Carlo simulations suggest that the compact star
+merger model cannot interpret both the Swiftz-known
+short GRB sample and the BATSE short GRB sample.
+Instead one may need a mix of GRBs from compact star
+mergers and from massive star core collapses to account
+for the observed short GRB population (Virgili et al.
+2009; Cui et al. 2010).
+In summary, the traditional long vs.short GRB
+scheme no longer always correspondingly match the two
+distinct physical origins, i.e., collapse of massive stars
+(Type II GRBs) vs.mergers of compact stars (Type I
+GRBs) (Zhang 2006; Zhang et al. 2007, 2009; Bloom et
+al. 2008). Zhang et al (2009) proposed a procedure (a
+flow chart, see their Fig.8) invoking a full set of observa-
+tional characteristics, including supernova(SN)-GRB as-
+sociation, specific star-forming rate (SFR) of the host
+galaxy, burst location offset, burst duration, hardness,
+spectral lag, statistical correlations, energetics, and af-
+terglow properties, to judge the physical category of a
+GRB. On the other hand, for most GRBs one may not
+immediately retrieve all the information needed for such
+a classification. It would be interesting to search for ad-
+ditional phenomenological classification schemes to see
+whether there exist other quantities, especially those in-
+voking prompt GRB emission properties only, that can
+give a good indication of the physical nature of a GRB.
+This motivates us to explore a new phenomenological
+classification method. Besides the burst duration, we
+think both the burst energy and the spectral properties
+in the burst rest frame would carry important informa-
+tion about the nature of the burst. In this paper, we
+propose a new discriminator based on the isotropic burst
+energy and rest frame peak energy ( Ep) of theνfνspec-
+trum of the prompt gamma-rays. We will show that this
+classification method is more closely connected to the
+Type II/I physical classification scheme.
+2.SAMPLE
+In order to develop our classification method, we use a
+sample of all GRBs with both redshift and spectral pa-
+rameters known up to Mar, 2010. Our sample includes
+137 GRBs. They are detected by BeppoSAX ,HETE-
+2,Swift,Suzaku, andFermi. Most of them are taken
+from Amati et al. (2008, 2009), Kann et al. (2010),
+Krimm et al. (2009), Zhang et al. (2009) and references
+therein. Their isotropic gamma-ray energy ( Eγ,iso) is
+calculated in the GRB rest-frame 1 −104kev band with
+the spectral parameters in order to avoid instrument se-
+lection effect. The soft XRF 080109 (Soderberg et al.
+2008), whose emission is in the XRT band (0.3-10kev)
+instead of BAT band, is also included in our sample.
+The XRT and UVOT observations place a limit of its
+Epin the range 0 .037 kev < Epeak<0.3 kev. We
+adoptEpeak≈0.12+0.23
+−0.089kev (Li 2008). Eight burstsin our sample (GRBs 050709, 050724, 051210, 060614,
+061006, 061210, 070714B, and 071227) are the so-called
+short GRBs with “extended emission” discussed in lit-
+erature. For these GRBs, both parameters ( Eγ,isoand
+Ep,z) arederivedfor the initial hardspike only(extended
+emission excluded)3. We tabulate only those GRBs with
+T90,z=T90/(1 +z)<2 s (Twenty-nine GRBs) in our
+sample in Table 1.
+3.A NEW GRB CLASSIFICATION PARAMETER
+Firstly, we show the distribution of T90,zin Fig. 1(a).
+Similar to that shown in Lin et al. (2009), T90,zfor the
+current GRB sample with redshift measurements has a
+tentative bimodal distribution, with peaks at log T90,z=
+−0.5 and 1.25, respectively. Due to the overlap between
+the two classes (long vs. short), those GRBs having
+T90,z= 1∼5 seconds cannot be unambiguously cate-
+gorized into either group4. We test the bimodality with
+the KMM algorithm presented by Ashman et al. (1994),
+and obtain a chance probability p∼10−2. The correct
+allocation rate for short GRBs estimated with the KMM
+algorithm is 73 .3%.
+We develop our classification method by defining a pa-
+rameter,
+ε(κ)≡Eγ,iso,52/Eκ
+p,z,2, (1)
+whereEγ,iso,52=Eγ,iso/1052ergs and Ep,z=Ep(1 +
+z)/102keV. Note that εdoes not depend on any spec-
+trum - energy correlation or theoretical assumption. We
+explore various κvalues, and found that given a same
+observed GRB (known fluence and Ep) the implicit z-
+dependence of ε(κ)∝D2
+L(z)/(1+z)1+κessentially van-
+ishes atz >2, ifκ= 5/3 (see Fig. 2 for a comparison
+of thez-dependence of εfor different κ). We therefore
+define
+ε≡Eγ,iso,52/E5/3
+p,z,2, (2)
+which removes the redshift dependence for high- zGRBs.
+This parameter may be more suitable for the classifica-
+tion of high- zGRBs, such as GRBs 090423 and 080913.
+We calculate εfor the bursts in our sample and show
+the logεdistribution in Fig.1 (b). A clear bimodal fea-
+ture is found, with a division line at ∼0.03 (high- εvs.
+low-ε). The high- εportion follows a log normal distri-
+bution with ε= 0.80±0.11 (1σ). For the low- εit is
+0.003±0.002. We test the bimodality of the εdistri-
+bution with the KMM algorithm and obtain a chance
+3We note that T90depends on the detector’s sensitivity and
+energy band. In our analysis, we define T90using the BAT energy
+band and sensitivity. In the Swiftera, it is found that some “short”
+GRBs (as observed in the BATSE band) have extended emission.
+It is the convention of the Swiftteam and the community to define
+a “short” GRB based on the duration of the short spike only, wi th
+the extended emission excluded. Observationally, the spec trum of
+the short spike is much harder than that of the extended emiss ion,
+the latter shares many properties with X-ray flares. For a sam e
+burst, the “extended emission” depends on redshift, since i t may
+be buried within the background if the redshift is high enoug h.
+We therefore only use the information of the short/hard spik e to
+perform our classification.
+4It was proposed that those GRBs having T90,z= 1∼5 sec-
+onds may be an intermediate population based on the excess of
+the GRBs over the bimodal distribution of T90,z. Although these
+GRBs are less energetic and have dimmer afterglows than typi cal
+long GRBs, they share many similar properties of typical GRB s,
+such as spectral lag and spectrum-energy correlations. The y may
+be a sub-group of the long GRBs (de Ugarte Postigo 2010).A New Classification Method For GRBs 3
+probability p <10−4. The overall correct allocation rate
+of GRBs in the two categories (high- εvs. low-ε) is as
+high as 99.8%, indicating that εis a new parameter for
+GRB classification.
+ItisgenerallybelievedthatGRBsoriginateeitherfrom
+core collapses of massive stars (Type II) or mergers of
+compact stars (Type I). One may ask whether this phe-
+nomenal classification matches the physical scheme of
+the Type I and Type II. The origins of most GRBs in
+our sample have been extensively discussed in literature
+(Zhang et al. 2009; Kann et al. 2010, 2008). We there-
+fore examine how the members in our two new categories
+(high-εvs. low- ε) are associated with the two physi-
+cal model types. To define the physical category of the
+GRBs, we apply the flow chart of Zhang et al. (2009,
+their Fig.8). We denote all “Type II” or “Type II can-
+didates” as “Type II”, and “Type I” or “Type I can-
+didates” as “Type I” (which include the Type I Gold
+sample and most other short/hard GRBs in Zhang et al.
+2009). We note that these definitions are fully consistent
+with those adopted in Kann et al. (2010, 2008). Fig. 1
+displaysallthe GRBsin oursamplein the ε−T90,zplane.
+Different symbols are used to mark different groups. For
+example, triangles denote the GRBs that are believed to
+be of the Type II origin, among which the black triangles
+denote the traditional long GRBs, and the blue triangles
+denote the GRBs thosewith T90,z’s shorterthan 2 s. The
+red circles denote those GRBs that are believed to have
+the Type I origin. Six green stars denote nearby low
+luminosity GRBs with supernova associations. It is re-
+markable to see that the εclassification clearly separates
+Type II GRBs from Type I GRBs.
+In order to quantitatively assess the clustering fea-
+ture among different types of GRBs, we derive the GRB
+distribution probability p(logε,logT90,z) from the ra-
+tio between the number of GRBs in the grid log ε+
+dlogε,logT90,z+dlogT90,zand the total number of
+GRBs in our sample. Since the distribution of εis con-
+cerned in our analysis, we take the error of log εinto
+account in our calculation of p. For a given GRB with
+logε±δlogε, we assume that the probability distribu-
+tion of this GRB is a normalized log-normal distribution
+centering at log εwith a width of δlogε. The proba-
+bility (pi) of the GRB in a given rage [log ε−dlogε/2,
+logε+dlogε/2] is calculated by integrating the proba-
+bility distribution function in the range. The probability
+p(logε,logT90,z) is then obtained by summing up piover
+the GRBs in our sample. We take dlogT90,z= 0.35
+anddlogε= 0.60 in our calculation. The contours
+of the probability distribution are shown in Fig. 1(c).
+One can clearly identify two clustering regions [with
+p(logε,logT90,z)>0.1]centeredinthehigh- εregimeand
+the low-ε, low-T90,zregime. At the 3 σlevel, the high- ε
+group includes typical long GRBs with high-luminosity,
+someintrinsicshortdurationGRBs (including thehigh- z
+GRB 090423 and GRB 080913), as well as some high-
+zshort GRBs (e.g. GRB 090426). In contrast, the
+low-εgroup contains the extensively discussed Type I
+GRBs, including those with and without extended emis-
+sion. Based on the probability contours, a low- εGRB
+withT90,z/lessorsimilar5 s in the Swift/BAT band would be
+identified as a Type I GRB at the 3 σconfidence level.
+GRBs 060614 and 060505 are marginally included in the
+p(logε,logT90,z)>0.003 region (the light grey region inFig. 1c)oftheTypeIGRBs. Nearbylowluminositylong
+GRBs (LL-GRBs), e.g. GRBs 980425, 031203, 050826,
+and 060218 (green stars in Fig. 1c) are out of the 3 σ
+contoursofthe Type I and Type II GRBs. This may hint
+a distinct GRB population as suggested by Liang et al.
+(2007; see also Soderberg et al. 2004; Cobb et al. 2006).
+4.CONCLUSIONS AND DISCUSSION
+By introducing a new parameter ε, we have proposed
+a new phenomenological classification method for GRBs.
+We demonstrated that GRBs with known redshifts are
+cozily classified into two classes with a separation ε=
+0.03. Due to the clear bimodal distribution of εwith lit-
+tle overlap, the overall correct allocation rate of a GRB
+to a particular category (high- εvs. low- ε) is as high
+as>99.8%, which is much better than the traditional
+duration classification method. More importantly, this
+classification scheme is more closely related to the two
+physically motivated model classes - Type II vs. I. In
+particular, the high- εcategory is a good representation
+of the Type II GRBs, while the low- εcategory, with an
+additional duration criterion T90,z/lessorsimilar5 s, is a good rep-
+resentation of the Type I GRBs. We suggest that the
+εparameter can be evaluated for future GRBs with z
+measurements,andthe ε-basedclassificationmaybeper-
+formed regularly to infer the physical origin of a GRB.
+The well-separated bimodal distribution of log εde-
+pends on our selection of κin Eq. (1). As shown in Fig.
+2,εis insensitive to zatz >2 for a given burst by taking
+κ= 5/3, but it is not at low redshift. One may suspect
+that if the bimodal distribution is due to the redshift ef-
+fect. To examine this issue, we re-plot the Figure 1 by
+separating GRBs into the z >2 and the z <2 samples
+(Figure 3, left and middle panels, respectively). We also
+show the GRB distributions in the log(1 + z)−log(ε)
+plane for all the bursts in our sample in Figure 3. It
+is found that the the clustering feature of log εis not
+caused by the redshift selection effect. Therefore, εis a
+GRB discriminator for both GRBs at z <2 andz >2.
+Another concern about our classification method may
+be sample uniformity. The GRBs in our sample were
+detected with different instruments with different instru-
+mental threshold. However, our classification method
+is based on the global energy and spectral information.
+Different from the burst duration, a well-measured Ep
+essentially does not depend on the detector. So is Eγ,iso.
+TheEpvalues of the GRBs in our sample, which expand
+almost four orders of magnitude, are well within the en-
+ergyband ofprevious and current GRB missions, so they
+are well measured. The Eisovalues expand almost eight
+orders of magnitude. Therefore, our sample could be re-
+garded as a complete sample with known EpandEisofor
+current missions.
+We should note that our classification scheme is par-
+ticularly helpful to diagnose the physical origin of GRBs
+with a rest-frame short duration (e.g. T90,z<2 s, Table
+1). For example, it convincingly classifies GRBs 090423,
+080913 and 090426 into the high- ε, and therefore, the
+TypeIIcategory,whichisotherwisedifficult withthe du-
+rationcriterion5. The distinction ofthe T90,z<2sbursts
+5As shown in Figure 2, a GRB with the same observed fluence
+andEpat higher redshift may have a higher ε. However, the
+bimodal distribution may not be due to the redshift effect (se e4 L¨ u et al.
+with different εvalues is also evident from their X-ray
+afterglow properties. Kann et al. (2008) compared the
+optical afterglows Type I GRBs with Type II GRBs, and
+found that those of Type I GRBs have a lower average
+luminosityand showalargerintrinsic spreadofluminosi-
+ties. In Figure 4, we compare the XRT lightcurves and
+the 12-hour X-ray luminosities between the two types of
+T90,z<5ss GRBs. The low- ε T90,z<2ss GRBs (Type
+I, red) are systematically fainter than the high- εones
+(Type II, blue)6, whose properties are rather similar to
+other high- εlong GRBs (black).
+Physically it is not obvious why the parameter ε(cor-
+respondingto κ= 5/3)givesacozyclassificationscheme.
+We have tried other κvalues in Eq.(1), but the classifi-
+cations defined by other ε(κ) are not as clean as the one
+defined by Eq.(2). As shown in Figure 2, an apparent
+advantage of κ= 5/3 is to diminish the z-dependence at
+high-zfor a given observed GRB.
+One can also discuss the connection between the log-
+normal distribution of εin the high- εregime and some
+empirical relations of Type II GRBs. With the Amati-
+relationEp,z∝E0.54
+γ,iso(Amati et al. 2009), we find that
+ε∝E0.185
+p,z, insensitive to Ep,z. Liang & Zhang (2005)
+discovered an empirical relation among Eγ,iso,Ep,z, and
+tb,z, namely, Eγ,iso∝E1.94±0.17
+p,zt−1.24±0.23
+b,z, where tb,z
+is the rest-frame break time of the optical afterglow
+lightcurve. This gives ε∝E0.27±0.17
+p,zt−1.24±0.23
+b,z, which
+is sensitive to tb,z. A clustering in εtherefore corre-
+sponds to a clustering in tb,z. Interpreting the tb,zas the
+jet break time, the Liang-Zhang relation may be trans-lated into the Ghirlanda-relation, Eγ,iso,52(1−cosθj)≃
+AE∼1.7
+p,z,2, whereA∼0.01−0.03 (Ghirlanda et al. 2004;
+Dai et al. 2004). Given ε≡Eγ,iso,52/E1.7
+p,z,2, we can get
+ε≈A(1−cosθj)−1, whereθjis the jet opening angle.
+The lognormal εdistribution ( ε= 0.80±0.11, at 1σ)
+therefore corresponds to a lognormal jet opening angle
+distribution for Type II GRBs, i.e., θj∼0.16±0.03 rad
+(assuming A∼0.01). We note that the jet opening an-
+gle derived by Ghirlanda et al. (2004) indeed clusters in
+a small range. Since the Ghirlanda-relation is very dif-
+ficult to understand physically, we suspect that it may
+be related the fact that εis quasi-universal for Type II
+GRBs. Future data with very early or very late opti-
+cal break times may test whether the Ghirlanda/Liang-
+Zhangrelationsorthefactofaquasi-universal εaremore
+fundamental.
+We acknowledge the use of the public data from the
+SwiftandFermidata and GCN circulars archive. We
+thank helpful comments from the referee. This work is
+supported by the National Natural Science Foundation
+of China (Grants 11025313, 10873002, 11078008), the
+National Basic Research Program (“973” Program) of
+China (Grant 2009CB824800), Guangxi SHI-BAI-QIAN
+project (Grant 2007201), Guangxi Science Foundation
+(2010GXNSFC013011), the program for 100 Young and
+Middle-aged Disciplinary Leaders in Guangxi Higher
+Education Institutions, and the research foundation of
+Guangxi University. It is also partially supported by
+NASA NNX09AT66G, NNX10AD48G, and NSF AST-
+0908362.
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+TABLE 1
+The sample of 29 GRBs for T90,z<2 s in the rest frame.
+GRB z T 90,zEE Ep Eγ,iso logε Type Afterglow Refs.
+name redshift (s) (s) (KeV) (1052erg) X/O(IR)
+050406 2.44 1.57 n/a 25+25
+−190.14+0.13
+−0.03−0.74+0.73
+−0.55high-ε(II) Y / Y (1)
+050416A 0.6535 1.51 n/a 15+2.5
+−2.50.1+0.01
+−0.010.00+0.12
+−0.12high-ε(II) Y / Y (2)
+050509B 0.2248 0.06 n/a 82+611
+−802.4+4.4
+−4×10−4−3.62+1.21
+−0.51low-ε(I) N / N (1,3)
+050709 0.1606 0.09 130 ±7 83+18
+−122.7+1.1
+−1.1×10−3−2.54+0.16
+−0.11low-ε(I) N / N (1,3)
+050724*0.2576 2.39 154 ±1 110+400
+−459+19
+−2×10−3−2.28+0.74
+−0.30low-ε(I) Y / N (1,3)
+050813 ∼0.72 ∼0.26 n/a 210+710
+−1301.5+2.5
+−0.8×10−2−2.75+1.12
+−0.45low-ε(I) N / N (1,3)
+050922C 2.198 1.41 n/a 130+35
+−355.3+1.7
+−1.7−0.31+0.19
+−0.19high-ε(II) Y / Y (2)
+051221A 0.5464 0.91 n/a 402+72
+−930.28+0.21
+−0.1−1.88+0.13
+−0.17low-ε(I) Y / N (3)
+060206 4.048 1.51 n/a 78+9
+−94.3+0.9
+−0.9−0.36+0.10
+−0.10high-ε(II) Y / Y (2)
+060502B 0.287 0.10 n/a 340+720
+−1903+5
+−2×10−3−3.59+1.14
+−0.41low-ε(I) N / N (1,3)
+060614*0.1254 ∼4.44 106 ±3 302+214
+−850.24+0.04
+−0.04−1.68+0.51
+−0.21low-ε(I) Y / Y (2,3)
+060801 1.131 0.23 n/a 620+1070
+−3400.17+0.021
+−0.021−2.64+1.01
+−0.40low-ε(I) Y / N (1,3)
+060926 3.208 1.90 n/a 19+8
+−180.42+0.59
+−0.040.16+0.31
+−0.69high-ε(II) Y / N (1,4)
+061006 0.4377 ∼0.35 120 640+144
+−2270.22+0.12
+−0.12−2.26+0.16
+−0.26low-ε(I) Y / N (1,3)
+061201 0.111 0.54 n/a 873+458
+−2840.018+0.002
+−0.015−3.39+0.38
+−0.24low-ε(I) Y / N (1,3)
+061210 0.4095 0.57 85 ±5 540+760
+−3100.09+0.16
+−0.05−2.51+0.82
+−0.42low-ε(I) N / N (1,3)
+061217 0.827 0.22 n/a 400+810
+−2100.03+0.04
+−0.02−2.96+0.47
+−0.38low-ε(I) N / N (1,3)
+070429B 0.9023 0.25 n/a 120+746
+−660.03+0.01
+−0.01−2.13+0.75
+−0.40low-ε(I) N / N (1,3)
+070506 2.31 1.30 n/a 71+95
+−250.26+0.17
+−0.05−1.20+0.67
+−0.26high-ε(II) Y / N (1,5)
+070714B 0.9225 ∼1.56 ∼100 1120+780
+−3801.16+0.41
+−0.22−2.16+0.51
+−0.25low-ε(I) Y / Y (1,3)
+070724A 0.457 0.27 n/a ∼68 0 .003+0.001
+−0.001−2.51+0.24
+−0.24low-ε(I) Y / N (3)
+071020 2.142 1.34 n/a 323+51
+−519.5+4.3
+−4.3−0.70+0.11
+−0.11high-ε(II) Y / N (2)
+071227 0.394 1.30 ∼100 1000+200
+−2000.22+0.08
+−0.08−2.56+0.14
+−0.14low-ε(I) Y / N (3)
+080520 1.545 1.10 n/a ∼30 0 .07+0.02
+−0.02−0.94+0.16
+−0.16high-ε(II) Y / N (6)
+080913 6.7 1.04 n/a 121+232
+−397+1.81
+−1.81−0.77+1.39
+−0.23high-ε(II) Y / Y (4)
+090423 8.1 1.13 n/a 54+22
+−2210+3
+−3−0.11+0.30
+−0.30high-ε(II) Y / Y (7,8)
+090426 2.609 0.33 n/a 45+57
+−430.42+0.59
+−0.04−0.72+0.92
+−0.69high-ε(II) Y / Y (9)
+090510 0.903 0.26 n/a 4414+420
+−4203.8+2.5
+−2.5−2.63+0.07
+−0.07low-ε(I) Y / N (10)
+100206 0.41 0.12 n/a 439+73
+−606.15+0.39
+0.39×10−2−2.61+0.07
+−0.07low-ε(I) Y / N (11)
+References . — (1) Butler et al. (2007); (2) Amati et al. (2008); (3) Zhang et al. (2009); (4) Piranomonte et al. (2006); (5) Thoene
+et al. (2007); (6) Sakamoto et al. (2008); (7) Tanvir et al. (2 009); (8) Salvaterra et al. (2009); (9) Levesque et al. (2009 ); (10) Abdo et
+al. (2009); (11)Cenko et al. (2010)
+*GRB 050724 and GRB 060614 have T90,zslightly longer than 2 s. They are included due to the close an alogy with other GRBs in
+the sample.
+Soderberg, A. M., et al. 2004, Nature, 430, 648
+Soderberg, A. M. et al. 2008, Nature, 453, 469
+Stanek, K.Z., Matheson, T., Garnavich, P.M., et al. 2003, Ap J,
+591, L17
+Tanvir, N. R., Chapman, R., Levan, A. J., & Priddey, R. S. 2005 ,
+Nature, 438, 991
+Tanvir, N. R., et al. 2009, Nature, 461, 1254
+Thoene, C. C., et al. 2007, GCN, 6379, 1Villasenor, J.S., Lamb, D.Q., Ricker, G.R., et al. 2005, Nat ure,
+437, 855
+Virgili, F. J., Zhang, B., O’Brien, P., & Troja, E. 2009,
+arXiv:0909.1850
+Woosley, S. E. 1993, ApJ, 405, 273
+Woosley, S. E., & Bloom, J. S. 2006, ARA&A, 44, 507
+Xin, L., et al. 2010, MNRAS,tmp. 1404 (arXiv:1002.0889)
+Zhang, B. 2006, Nature, 444, 1010
+Zhang, B., Zhang, B.-B., Liang, E.-W., et al. 2007, ApJ, 655, L25
+Zhang, B., Zhang, B.-B., Virgili, F.J., et al. 2009, ApJ, 703 , 16966 L¨ u et al.
+/s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s54/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49/s48/s49/s48/s49/s48/s50/s48/s51/s48
+/s49/s48/s48/s51/s49/s54/s68
+/s48/s56/s48/s49/s48/s57
+/s40/s99/s41/s32/s84/s121/s112/s105/s99/s97/s108/s32/s84/s121/s112/s101/s32/s73/s73
+/s32/s83/s104/s111/s114/s116/s32/s84
+/s57/s48/s44/s122 /s44/s32/s72/s105/s103/s104/s45 /s32/s84/s121/s112/s101/s32/s73/s73
+/s32/s76/s111/s119/s45 /s32/s84/s121/s112/s101/s32/s73
+/s32/s76/s111/s119/s45 /s32/s84/s121/s112/s101/s32/s73/s32/s119/s105/s116/s104/s32/s69/s69
+/s32/s76/s76/s45/s71/s82/s66/s115/s48/s51/s49/s50/s48/s51
+/s48/s54/s48/s53/s48/s53/s48/s54/s48/s54/s49/s52/s48/s56/s48/s57/s49/s51/s48/s57/s48/s52/s50/s51
+/s48/s57/s48/s52/s50/s54/s48/s54/s48/s50/s49/s56
+/s32/s32/s108/s111/s103/s32
+/s108/s111/s103/s32 /s84
+/s57/s48/s44/s32/s122/s32/s32/s40/s115/s41/s57/s56/s48/s52/s50/s53/s48/s53/s48/s56/s50/s54/s40/s97/s41
+/s78/s78
+/s32/s32
+/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s40/s98/s41
+/s32/s32
+Fig. 1.— The 1-D and 2-D distributions for the bursts in our sample in t he logT90,zvslogεplane along with the bimodal fits ( solid
+lines, in panel a, b ) as well as the 2-D distribution for the GRBs in our sample in t he logε−logT90,zand logε−log(1+z) plane ( panel
+c). Thetriangles denote the Type II GRB candidates as discussed in literature . Among them the traditional long GRBs are denoted as
+blackand the GRBs with T90,z<2 s are denoted as blue. Thered solid (open) circles represent the Type I GRBs with (without) extended
+emission as discussed in the literature. The diamond denotes the special burst GRB 060505. Nearby LL-GRBs are den oted as green stars.
+The possibility contours for GRB clustering are marked, wit hp(logε,logT90,z)>0.1 (dark grey ) andp(logε,logT90,z)>0.003 (light
+grey), respectively. The dashed vertical and horizontal lines a re the divisions of ε= 0.03 andT90,z= 5 seconds, respectively.
+/s49 /s49/s48/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48
+/s32/s107 /s61/s32/s53/s47/s51
+/s32/s107 /s61/s32/s52/s47/s51
+/s32/s107 /s61/s32/s50
+/s32/s32/s40 /s41
+/s49/s43 /s122 
+Fig. 2.— Dependence of ε(κ) on (1 + z) forκ= 4/3,5/3,2.0 for a given burst with the same measured fluence and Ep.A New Classification Method For GRBs 7
+/s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s54/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49/s48/s49
+/s40/s99/s41/s32/s84/s121/s112/s105/s99/s97/s108/s32/s84/s121/s112/s101/s32/s73/s73
+/s32/s72/s105/s103/s104/s45 /s32/s84/s121/s112/s101/s32/s73/s73
+/s32/s32/s108/s111/s103/s32
+/s108/s111/s103/s32 /s84
+/s57/s48/s44/s32/s122/s32/s32/s40/s115/s41/s48/s52/s56/s49/s50
+/s40/s97/s41
+/s32/s32/s78
+/s48 /s53 /s49/s48 /s49/s53/s40/s98/s41
+/s32/s32
+/s78/s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s54/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49/s48/s49/s48/s53/s49/s48/s49/s53/s50/s48
+/s49/s48/s48/s51/s49/s54/s68
+/s48/s56/s48/s49/s48/s57
+/s40/s99/s41/s32/s84/s121/s112/s105/s99/s97/s108/s32/s84/s121/s112/s101/s32/s73/s73
+/s32/s72/s105/s103/s104/s45 /s32/s84/s121/s112/s101/s32/s73/s73
+/s32/s76/s111/s119/s45 /s32/s84/s121/s112/s101/s32/s73
+/s32/s76/s111/s119/s45 /s32/s84/s121/s112/s101/s32/s73/s32/s119/s105/s116/s104/s32/s69/s69
+/s32/s76/s76/s45/s71/s82/s66/s115/s48/s51/s49/s50/s48/s51
+/s48/s54/s48/s53/s48/s53/s48/s54/s48/s54/s49/s52/s48/s54/s48/s50/s49/s56
+/s32/s32/s108/s111/s103/s32
+/s108/s111/s103/s32 /s84
+/s57/s48/s44/s32/s122/s32/s32/s40/s115/s41/s57/s56/s48/s52/s50/s53/s48/s53/s48/s56/s50/s54
+/s48 /s49/s48 /s50/s48 /s51/s48/s40/s98/s41
+/s32/s32
+/s78/s78/s40/s97/s41
+/s32/s32
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s54/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49/s48/s49/s48/s49/s48/s50/s48/s51/s48
+/s32/s84/s121 /s112/s105/s99/s97/s108/s32/s84/s121 /s112/s101/s32/s73/s73
+/s32/s83/s104/s111/s114/s116/s32 /s84
+/s57/s48/s44/s122/s44/s32/s72/s105/s103/s104/s45 /s32/s84/s121 /s112/s101/s32/s73/s73
+/s32/s76/s111/s119/s45 /s32/s84/s121 /s112/s101/s32/s73/s32/s119/s105/s116/s104/s32/s69/s69
+/s32/s76/s111/s119/s45 /s32/s84/s121 /s112/s101/s32/s73
+/s32/s76/s76/s45/s71/s82/s66/s115/s48/s56/s48/s57/s49/s51
+/s40/s99/s41
+/s32/s32/s108/s111/s103/s32
+/s108/s111/s103/s32/s40 /s49/s43/s122 /s41 /s32/s48/s57/s48/s52/s50/s51
+/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48
+/s32/s78/s40/s98/s41
+/s32/s32/s32/s78
+/s32/s32
+/s40/s97/s41
+Fig. 3.— The 1-D ( panels a and b ) and 2-D ( panel c) distributions for the bursts in our sample in the log T90,z−logεplane for GRBs
+forz >2 (left group of panels ) andz <2 (middle group of panels ). The GRB distributions in the log(1 + z)−logεplane are also shown
+in theright group of panels . The symbol style is the same as Figure 1. The contours are the same as that in Figure 1 for all GRBs in our
+sample.
+/s49 /s50 /s51 /s52 /s53 /s54 /s55/s52/s49/s52/s51/s52/s53/s52/s55/s52/s57/s53/s49
+/s32/s32/s108/s111/s103/s32 /s76
+/s120/s32/s40/s101/s114/s103/s32/s47/s32/s115/s41
+/s108/s111/s103/s32 /s116/s32 /s47/s40/s49/s43 /s122 /s41/s32/s40/s115/s41/s116/s39/s61/s49/s50/s32/s104/s111/s117/s114/s115/s40/s97/s41
+/s52/s56 /s52/s57 /s53/s48 /s53/s49 /s53/s50 /s53/s51/s52/s50/s52/s51/s52/s52/s52/s53/s52/s54/s52/s55
+/s32/s32/s108/s111/s103/s32 /s76
+/s120/s44/s32/s49/s50/s104/s32/s32/s47/s32/s40/s101/s114/s103/s32/s115/s45/s49
+/s41
+/s108/s111/s103/s32 /s69
+/s105/s115/s111/s44/s32/s47/s32/s101/s114/s103/s40/s98/s41
+Fig. 4.— Comparisons of (a) the XRT lightcurves and (b) the LX,12h−Eisodistribution among the low- εGRBs with T90,z<5 s (red),
+high-εGRBs with T90,z<2 s (blue), and 44 long-duration high- εGRBs from the Liang et al. (2009) sample ( black). Here LX,12his the
+X-ray luminosity in the XRT band at the rest-frame 12 hours po st the GRB trigger.
\ No newline at end of file
diff --git a/1001.0599.txt b/1001.0599.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5aa53416ff817ae8e17ab6a7b92af25f3d932a9b
--- /dev/null
+++ b/1001.0599.txt
@@ -0,0 +1,707 @@
+1 
+ GPU-based Fast Low-dose Cone Beam CT 
+Reconstruction via Total Variation 
+Xun Jia 
+Department of Radiation Oncology, University of California San Diego, La 
+Jolla, CA 92037-0843  5 
+Yifei Lou 
+Department of Mathematics, University of California Los Angeles, Los Angeles, 
+CA 90095  
+John Lewis, Ruijiang Li, Xuejun Gu, Chunhua Men, William Y. Song, and 
+Steve B. Jianga) 10 
+Department of Radiation Oncology, University of California San Diego, La Jolla, CA 92037-0843  
+Purpose : Cone-beam CT (CBCT) has been widely used in image guided 
+radiation therapy (IGRT) to acquire updated volumetric anatomical information 
+15 
+before treatment fractions for accurate patient alignment purpose. However, the 
+excessive x-ray imaging dose from serial  CBCT scans raises a clinical concern 
+in most IGRT procedures. The excessi ve imaging dose can be effectively 
+reduced by reducing the number of x-ray projections and/or lowering mAs levels in a CBCT scan. It is the goal of this work to develop a fast GPU-based 
+20 
+algorithm to reconstruct high quality  CBCT images from undersampled and 
+noisy projection data so as to lower the imaging dose.  
+Methods:  The CBCT is reconstructed by minimizing an energy functional 
+consisting of a data fidelity term and a total variation regularization term. We developed a GPU-friendly version of the forward-backward splitting algorithm 
+25 
+to solve this model. A multi-grid technique is also employed. We test our CBCT 
+reconstruction algorithm on a digital NCAT phantom and a head-and-neck 
+patient case. The performance under low mAs is also validated using a physical Catphan phantom and a head-and-neck Rando phantom. Results:  It is found that 40 x-ray projections are sufficient to reconstruct CBCT 
+30 
+images with satisfactory quality for IG RT patient alignment purpose. Phantom 
+experiments indicated that CBCT images can be successfully reconstructed with 
+our algorithm under as low as 0.1 mAs/projection level. Comparing with 
+currently widely used full-fan head-and-neck scanning protocol of about 360 projections with 0.4 mAs/projection, it is estimated that an overall 36 times dose 
+35 
+reduction has been achieved with our algorithm. Moreover, the reconstruction time is about 130 sec on an NVIDIA Te sla C1060 GPU card, which is estimated 
+~100 times faster than similar iterative reconstruction approaches.  
+                                                      
+a) Electronic mail: sbjiang@ucsd.edu  2             X. Jia et al.  
+ Conclusions: The developed GPU-based CBCT reconstruction algorithm is 
+capable of lowering imaging dose considerably. The high computational 40 
+efficiency makes this iterative CBCT reco nstruction approach feasible in real 
+clinical environments.  
+Key words: Cone beam CT, reconstruction, total variation, GPU 
+ 3             X. Jia et al.  
+ I. INTRODUCTION 45 
+ 
+Cone Beam Computed Tomography (CBCT) plays an important role in cancer 
+radiotherapy. In CBCT, the patient’s volumetr ic information is retrieved from its x-ray 
+projections in cone beam geometry along a number of directions. In the past, the CBCT reconstruction algorithms ha ve undergone significant advances. Successful algorithms 50 
+include the well known FDK algorithm
+1 and a large number of iterative reconstruction 
+algorithms, such as EM2 and ART/SART3-4, to name a few.  
+Among its various applications, CBCT is par ticularly convenient for accurate patient 
+setup in image guided radiation therapy (IGRT). Yet, the high imaging dose to healthy organs (a few cGy per scan)
+5-7 in CBCT scans is a clini cal concern, especially when 55 
+CBCT is performed on a daily basis before each treatment fraction. The imaging dose in 
+CBCT can be reduced by reducing the number of x-ray projections and/or lowering mAs 
+levels (tube current and pul se duration). In this approach , however, the consequent CBCT 
+images reconstructed with conventional F DK algorithms are highly degraded due to 
+insufficient and noisy projections. It is therefore desirable to develop new techniques to 60 
+perform CBCT reconstructions with a greatly  reduced number of noisy projections, while 
+maintaining high image quality.  
+Reconstructing CBCT images from only few num ber of x-ray projections for IGRT 
+is feasible, though quite difficult. First, for the purpose of patient setup before a radiotherapy treatment, it is adequate to onl y retrieve major anatomic features from a 65 
+CBCT image, such as tumor location and shap e. This fact considerably reduces the 
+complexity of the reconstruction problem. Second, recent development in mathematics 
+provides us new perspectives of solving prob lems of this kind. Recently, a burst of 
+research in compressed sensing
+8-13 have demonstrated the feasibility of recovering signals 
+from incomplete measurements through optimization methods in many different 70 
+mathematical situations. In particular, To tal Variation (TV) method has presented its 
+tremendous power in CT reconstruction problems, where the CT images can be successfully reconstructed by minimizing the TV semi-norm under the constraints posed by the x-ray projections along a few directions
+14-15. This approach has also been 
+extensively applied into many other imaging modalities16-19 and its efficacy has been 75 
+improved by combining with other techniques, such as incorporating prior information20-
+21. 
+Despite the great power of TV method and its  promising applications in the problem 
+of CBCT reconstruction, the computation is very time-consuming due to the high 
+complexity of currently available algorithms a nd the extremely large data size involved in 80 
+real clinical problems. It usually takes sev eral hours or even longer for current TV-based 
+reconstruction approaches to produce a decen t CBCT image. This fact prevents them 
+from practical applications in real clinical environments. Generally speaking, the 
+reconstruction process can be sped up by u tilizing a better optimiza tion algorithm and a 
+more powerful computational hardware. Acco mpanied with the recent development of 85 
+compressed sensing, a number of efficient minimization algorithms targeting at problems of this type have been invented such as graph cut
+22-23, TV-operator splitting24-25, and 4             X. Jia et al.  
+ Bregman iterations26-27. Meanwhile, general purpose gr aphic processing units (GPUs) 
+have offered us a promising prospect of in creasing efficiencies of heavy duty tasks in 
+radiotherapy, such as CBCT FDK reconstruction28-32, deformable image registration30, 33-90 
+34, dose calculation35-38, and treatment plan optimization39. Taking advantages of these 
+developments, the computation efficiency of TV-based CBCT reconstruction is expected to be enhanced considerably.  
+We have recently reported some preliminary results on the development of a GPU-
+based CBCT reconstruction algorithm in a short letter
+40, where we have reconstructed 95 
+CBCT images by minimizing an energy functi onal consisting of a TV  regularization term 
+and a data fidelity term posed by the x-ray projections. By developing new minimization algorithms with mathematical structures suita ble for GPU parallelization, we can take 
+advantage of the massive computing power of GPU to dramatically improve the efficiency of the TV-based CBCT reconstr uction. In this paper, we will present a 100 
+comprehensive description and validation of our algorithm.   
+II. METHODS 
+ 
+A. Model 105 
+ 
+Let us consider a patient volumetri c image represented by a function ݂ሺݖ,ݕ,ݔሻ , where 
+ሺݖ,ݕ,ݔሻࡾאଷ is a vector in 3-dimensional Euclidean space. A projection operator ܲఏ 
+maps ݂ሺݖ,ݕ,ݔሻ  into another function on an x-ray imager plane along a projection angle 
+ߠ : 110 
+ܲఏሾ݂ሺݖ,ݕ,ݔሻሿሺݒ,ݑሻൌ׬d݈݂ሺݔௌ൅݊ଵݕ,݈ௌ൅݊ଶݖ,݈ௌ൅݊ଷ݈ሻ௅ሺ௨,௩ሻ
+଴ , (1) 
+where ሺݔௌ,ݕௌ,ݖௌሻ is the coordinate of the x-ray source S and ሺݒ,ݑሻ  is the coordinate for 
+point P on the imager,  ࢔ൌሺ ݊ ଵ,݊ଶ,݊ଷሻ being a unit vector along the direction SP. Fig. 1 
+illustrates the geometry. The upper integration limit ܮሺݒ,ݑሻ is the distance from source S 
+to point P. Denote the observed projection image at the angle ߠ by ܻఏሺݒ,ݑሻ . A CBCT 
+reconstruction problem is formulated as to  retrieve the volumetric image function 115 
+݂ሺݖ,ݕ,ݔሻ based on the observation of ܻఏሺݒ,ݑሻ  at various angles given the projection 
+mapping in Eq. (1). 
+In this paper, we attempt to reconstruc t the CBCT image by minimizing an energy 
+functional consisting of a data fide lity term and a regularization term: 
+݂ൌa r g m i n   ܧ ଵሾ݂ሿ൅ܧߤଶሾ݂ሿ,  s.t.  ݂ሺݖ,ݕ,ݔሻ൒0  for ׊ሺݖ,ݕ,ݔሻࡾאଷ, (2) 
+where the energy functionals are ܧଵሾ݂ሿൌଵ
+௏ԡ݂׏ԡଵ and ܧଶሾ݂ሿൌଵ
+ேഇ஺∑ฮܲఏሾ݂ሿെܻఏฮଶଶ
+ఏ . 120 
+Here ܸ is the volume in which the CBCT image is reconstructed. ܰఏ is the number of 
+projection used and ܣ is the projection area on each x-ray imager. ԡ…ԡ௣ denotes ݈௣-norm 
+of functions. In Eq. (2), the data fidelity term ܧଶሾ݂ሿ ensures the consistency between the 
+reconstructed volumetric image ݂ሺݖ,ݕ,ݔሻ  and the observation ܻఏሺݒ,ݑሻ . The first term, 
+ܧଵሾ݂ሿ, known as TV semi-norm, has been shown to be extremely powerful41 to remove 125 5             X. Jia et al.  
+ artifacts and noise from ݂ while preserving its sharp edg es to a certain extent. The CBCT 
+image reconstruction from few projections is a well known underdetermined problem in 
+that there are infinitely many functions ݂ such that ܲఏሾ݂ሺݖ,ݕ,ݔሻሿሺݒ,ݑሻൌܻఏሺݒ,ݑሻ . By 
+performing the minimization as in Eq. (2) the projection condition will be satisfied to a good approximation. The TV regularization term serves the purpose of picking out the 130 
+one with desired image properties, namely smooth while with sharp edges, among those 
+infinitely many candidate solutions. The dimensionless constant ߤ൐0  controls the 
+smoothness of the reconstructed images by  adjusting the relative weight between ܧ
+ଵሾ݂ሿ 
+and ܧଶሾ݂ሿ. In the reconstruction results shown in  this paper, we choose the value of ߤ 
+manually for best image quality. 135 
+ 
+B. Minimization Algorithm 
+  
+The main obstacle encountered while solv ing Eq. (2) is due to the projection 
+operator ܲఏ. In the matrix representation, this  operator becomes a sparse matrix. 140 
+However, it contains approximately 4ൈ10ଽ non-zero elements for a typical clinical case 
+studied in this paper, occupying about 16 GB  memory space. This matrix is usually too 
+large to be stored in typical computer me mory and therefore it has to be computed 
+repeatedly whenever necessary. This rep eated work consumes a large amount of the 
+computational time. For example, if we were  to solve Eq. (2) with a simple gradient 145 
+descent method, ܲఏ would have to be evaluated repeatedly while computing the search 
+direction,  i.e. negative gradient, and while calculating the functional value for step size 
+search as well. This would significantly limit the computation efficiency.  
+In order to overcome this difficulty, we utilize the forward-backward splitting 
+algorithm42-44. This algorithm splits the minimization of the TV term and the data fidelity 150 
+term into two alternating steps, while the computation of x-ray projection ܲఏ is only in 
+one of them. The computation efficiency can  be improved owing to this splitting. As  
+FIG. 1 . The geometry of x-ray projection. The operator ܲఏ maps ݂ሺݖ,ݕ,ݔሻ   in ࡾଷ onto 
+another function ܲఏሾ݂ሺݖ,ݕ,ݔሻሿሺݒ,ݑሻinࡾଶ, the x-ray imager plane, along a projection angle 
+ߠ .ܮሺݒ,ݑሻ  is the length of SP and ݈ሺݖ,ݕ,ݔሻ  is that of SV. The source to imager distance is ܮ଴. 
+ x yz
+uv
+0P
+SY(u,v)
+f(x,y,z)
+n0V6             X. Jia et al.  
+ such, let us consider the optimality condition of Eq. (2) by setting the functional variation 
+with respect to function ݂ሺݖ,ݕ,ݔሻ to zero: 
+ఋ
+ఋ௙ሺ௫,௬,௭ሻܧଵሾ݂ሿ൅ߤఋ
+ఋ௙ሺ௫,௬,௭ሻܧଶሾ݂ሿൌ0 . (3) 
+If we split the above equation into the following form  155 
+ߤఋ
+ఋ௙ሺ௫,௬,௭ሻܧଶሾ݂ሿൌఉ
+௏ሺ݂െ݃ሻ , 
+ఋ
+ఋ௙ሺ௫,௬,௭ሻܧଵሾ݂ሿൌെఉ
+௏ሺ݂െ݃ሻ, (4) 
+where ߚ൐0  is a parameter and ݃ሺݖ,ݕ,ݔሻ  is an auxiliary function used in this splitting 
+algorithm, the minimization problem can be acco rdingly split, leading to the following 
+iterative algorithm to solve Eq. (2):  
+Algorithm  A1: 
+ Do the following steps until convergence 
+1. Update: ݃ൌ݂െఓ௏
+ఉఋ
+ఋ௙ሺ௫,௬,௭ሻܧଶሾ݂ሿ; 
+2. Minimize: ݂ൌa r g m i nܧ ଵሾ݂ሿ൅ఉ
+ଶܧଷሾ݂ሿ; 
+3. Correct: ݂ሺݖ,ݕ,ݔሻൌ0, i f݂ሺݖ,ݕ,ݔሻ ൏ 0 . 
+ 160 
+Here a new energy functional is defined as ܧଷሾ݂ሿൌଵ
+௏ԡ݂െ݃ԡଶଶ. The Step 1 in Algorithm 
+A1 is a gradient descent update with respect  to the minimization of energy functional 
+ܧଶሾ݂ሿ. It is also interesting to observe that th e Step 2 here is just an Rudin-Osher-Fatemi 
+(ROF) model, which has been shown to be ex tremely efficient and capable of removing 
+noise and artifacts from a degraded image ݃ሺݖ,ݕ,ݔሻ  while still preserving the sharp edges 165 
+and main features41. In addition, since ݂ represents the x-ray attenuation coefficients, it is 
+necessary to ensure its non-negativeness by a si mple truncation as in Step 3. These three 
+steps are iterated until a solution to the problem in Eq. (2) is achieved. 
+The choice of ߚ is crucial in this algorithm. On one hand, a small value of ߚ will lead 
+to a large step size of the gradient descent update in Step 1, causing instability of the 170 
+forward-backward splitting algorithm. On the other, a large ߚ will tend to make the TV 
+semi-norm ܧଵሾ݂ሿ unimportant in Step 2, reducing the its efficacy in removing artifacts. In 
+practice, an empirical value ߚ~10ߤ  is found to be appropriate. 
+For the sub-problem in Step 2, optimizing an energy functional  ܧோைிሾ݂ሿ ൌ ܧ ଵሾ݂ሿ൅
+ఉ
+ଶܧଷሾ݂ሿ, we solve it with a simple gradient descen t method. At each iteration step of the 175 
+gradient descent method, the functional value  ܧ଴ൌܧோைிሾ݂ሿ is first evaluated, as well as 
+the functional gradient ݀ሺݖ,ݕ,ݔሻ ൌఋ
+ఋ௙ሺ௫,௬,௭ሻܧோைிሾ݂ሿ. An inexact line search is then 
+performed along the negative gradient direction with an initial step size ߣൌߣ଴. The trial 
+functional value ܧ௡௘௪ൌܧோைிሾ݂െ݀ߣሿ  is evaluated. If Amijo’s rule is satisfied45,  
+namely 180 7             X. Jia et al.  
+ ܧሾ݂െ݀ߣሿ൑ܧ଴െߣܿଵ
+௏׬dݔdݕd݀ݖሺݖ,ݕ,ݔሻଶ, (5) 
+where ܿ is a constant, the step size ߣ is accepted and an update of the image ݂՚݂െ݀ߣ  
+is applied; otherwise, the search step size is reduced by a factor ߙ with אߙሺ 0 , 1 ሻ . This 
+iteration is repeated until the rela tive energy decrease in a step |ܧ௡௘௪െܧ଴|/ܧ଴  is less 
+than a given threshold  ߝ .The parameters relevant in th is sub-problem are chosen as 
+empirical values of  ܿ ൌ 0.01 , ߙൌ0 . 6  and ߝ ൌ 0.1% . The computation results are found 185 
+to be insensitive to these choices. 
+Boundary condition is another important issue during the implementation. For 
+simplicity, zero boundary condition is imposed in our computation along the anterior-
+posterior direction and the lateral direction,  while reflective boundary condition is used 
+along the superior-inferior direction. 190 
+ 
+C. Further improvements 
+ 
+1. GPU implementation 
+ 195 
+Computer graphic cards, such as the NVIDI A GeForce series and the GTX series, are 
+conventionally used for display purpose on desktop computers. Recently, NVIDIA 
+introduced special GPUs dedicated for scien tific computing, for example the Tesla C1060 
+card that is used in this paper. Such a GPU card has a total number of 240 processor cores 
+(grouped into 30 multiprocessors with 8 cores each), each with a clock speed of 1.3 GHz. 200 
+The card is also equipped with 4 GB DDR3 memory, shared by all processor cores. 
+Utilizing such a GPU card with tremendous pa rallel computing ability can considerably 
+elevate the computation efficiency of our algorithm. In fact, a number of components can be easily accomplished in a parallel fashion. For instance, the evaluating of functional 
+value ܧ
+ோைிሾ݂ሿ  in Step 2 of Algorithm A1 can be performed by first evaluating its value 205 
+at each ሺݖ,ݕ,ݔሻ  coordinate and then sum over all c oordinates. The functional gradient 
+݀ሺݖ,ݕ,ݔሻ  can be also computed with each GPU thread responsible for one coordinate.  
+ 
+2. Closed-form gradient 
+ 210 
+A straightforward way of implementing Algorithm A1 is to interpret ܲఏሾ݂ሿ as a matrix 
+multiplication and then ܧଶሾ݂ሿ as a vector norm ∑ฮܲఏ݂െܻఏฮଶଶ
+ఏ . This leads to a simple 
+form ∑ܲఏ்ሺܲఏ݂െܻఏሻ ఏ  for the functional variation ܧߜଶሾ݂ሿ ݂ߜሺݖ,ݕ,ݔሻ⁄  in Step 1, apart 
+from some constants, where ·் denotes matrix transpose. In practice ܲఏ  is too large to be 
+stored in computer memory. Note that ܲఏ݂ is simply a forward projection calculation, it 215 
+can be easily computed by a ray-tracing algorithm, such as Siddon’s algorithm46-48. Due 
+to the massive parallelization ability of GPU, multiple threads can compute projections of 
+a large number rays simultaneously and high e fficiency can be achieved. On the other 
+hand, we lack an efficient algorithm to perform the operation of ܲఏ். In fact, ܲఏ் is a 
+backward operation in that we tend to update  voxel values along a ray line. If we were to 220 8             X. Jia et al.  
+ perform this backward operation by Siddon’s algorithm in the GPU implementation with 
+each thread responsible for updating voxels along a ray line, a memory conflict problem would take place due to the possibility of updating a same voxel value by different GPU threads. As a consequence, one thread will have to wait until another thread finishes 
+updating. It is this fact that severely limits the exploitation of GPU's massive parallel 225 
+computing power.  
+To overcome this difficulty, we analytically compute the functional variation 
+ఋ
+ఋ௙ሺ௫,௬,௭ሻܧଶሾ݂ሿ in Step 1 of Algorithm A1 and a closed-form equation is obtained 
+ఋ
+ఋ௙ሺ௫,௬,௭ሻܧଶሾ݂ሿൌଵ
+ேഇ஺∑ଶ௅యሺ௨,כ௩כሻ
+௅బ௟మሺ௫,௬,௭ሻఏ ሾܲఏሾ݂ሺݖ,ݕ,ݔሻሿሺכݒ,כݑሻെܻఏሺכݒ,כݑሻሿ . (6) 
+Here ሺכݒ,כݑሻ is the coordinate for a point P on imager, where a ray line connecting the 
+source Sൌሺ ݔ ௦,ݕ௦,ݖ௦ሻ and the point Vൌሺݖ,ݕ,ݔሻ  intersects with the imager. ܮ଴ is the 230 
+distance from the x-ray source S to the imager. ݈ሺݖ,ݕ,ݔሻ  and ܮሺכݒ,כݑሻ are the distance 
+from S to the point V and from S to the point P on imager, respectively. See Fig. 1 for the 
+geometry. The derivation of Eq. (6) is briefly shown in the Appendix. 
+The form of Eq. (6) suggests a very effi cient way of evaluating this functional 
+variation and hence accomplishing Step 1 in Al gorithm A1 in a parallel fashion. For this 235 
+purpose, we can first perform the forward pr ojection operation and compute an auxiliary 
+function defined as ܶఏሺݒ,ݑሻ ؠ ሾܲఏሾ݂ሺݖ,ݕ,ݔሻሿሺݒ,ݑሻെܻఏሺݒ,ݑሻሿ  for all ሺݒ,ݑሻ  and ߠ .For 
+a point ሺݖ,ݕ,ݔሻ where we try to evaluate the functional variation, it suffices to take the 
+function values of ܶఏሺכݒ,כݑሻ at a ሺכݒ,כݑሻ coordinate corresponding to the ሺݖ,ݕ,ݔሻ, 
+multiply by proper prefactors, and finally sum over ߠ .In numerical computation, since 240 
+we can only evaluate ܶఏሺݒ,ݑሻ  at a set of discrete ሺݒ,ݑሻ coordinates and ሺכݒ,כݑሻ does not 
+necessarily coincide with these discrete coordina tes, a bilinear interpolation is performed 
+to get ܶఏሺכݒ,כݑሻ. Because the computation of ܶఏሺݒ,ݑሻ can be achieved in a parallel 
+manner with each GPU thread responsible for a ray line and the evaluation of 
+ఋ
+ఋ௙ሺ௫,௬,௭ሻܧଶሾ݂ሿ can be performed with each thread for a voxel ሺݖ,ݕ,ݔሻ, extremely high 245 
+efficiency in Step 1 is expected given the vast parallelization ability of GPU. 
+ 3. Multi-grid method 
+ 
+Another technique we employed to increase computation efficiency is multi-grid 250 
+method
+49. Because of the convexity of the energy functional in Eq. (2), the minimization 
+problem is equivalent to solving a set of nonlinear differential equations posed by the 
+optimality condition as in Eq. (3). It h as long been known that, when solving a 
+differential equation with a certain kind of finite difference scheme, the convergence rate 
+of an iterative approach largely depends on the mesh grid size. In particular, the 255 
+convergence rate is usually wor sened when a very fine grid size is used. Moreover, finer 
+grid also implies more number of unknown variables, significantly increasing the size of the computation task.  
+A well known multi-grid approach can be utilized to resolve these problems. Suppose 
+it is our ultimate purpose to reconstruct a volumetric CBCT image ݂ሺݖ,ݕ,ݔሻ on a fine 260 9             X. Jia et al.  
+ grid ߗ௛ of size ݄ ,we could start with solving the problem on a coarser grid  ߗଶ௛ of size 
+2݄ with the same iterative approach as in Algorithm A1. Upon convergence, we could 
+smoothly extend the solution ݂ଶ௛ on ߗଶ௛ to the fine grid ߗ௛ and use it as the initial guess 
+of the solution. Because of the decent quality of this initial guess, only few iteration steps 
+of Algorithm A1 are required to achieve the final solution on ߗ௛. Moreover, this two- 265 
+level idea can be used recursively. In practi ce, we performed a 4-level multi-grid scheme, 
+i.e. the reconstruction is sequentially achieved on grids ߗ଼௛՜ߗସ௛՜ߗଶ௛՜ߗ௛. A 
+considerably efficiency gain is observed in our implementation.  
+Finally, we summarize our algorithm of CBCT reconstruction, including all 
+techniques mentioned previously, in Fig. 2. 270 
+ 
+III. Results 
+ 
+A. Digital phantom 
+ 275 
+We test our reconstruction algorithm with a digital NURBS-based cardiac-torso (NCAT) 
+phantom50, which maintains a high level of anatomical realism ( e.g., detailed bronchial 
+trees).  The phantom is generated at thorax region with a size of 512ൈ512ൈ70  voxels FIG. 2 . The flow chart of our GPU-based CBCT reconstr uction. Step 6 in left panel is enlarged 
+in detail in right panel.  No
+YesNo
+No
+YesYes
+No
+Yes6b: Compute  gradient d =▽EROF[f] 
+6c: Set search step length λ =λ0
+6d: Evaluate  Enew= EROF[f‐λd]6a: Evaluate  E0= EROF[f]
+6f: Update image 
+f = f‐λd6e: Satisfy Armijo rule ?6g: λ =αλ1: Start
+2: Load data and transfer to GPU
+8: Enough iterations  ?
+12: Transfer data to CPU and output
+13: End3: Set grid to coarsest scale h
+5: Step 1 in Algorithm  A1 4: initialize fhwith FDK algorithm
+6h: |Enew‐E0|/E 0< ε?9: Reach finest scale?10: Set grid scale 
+h=h/211: Upsample
+fh→fh/26: Step 2 in Algorithm  A1 
+7: Step 3 in Algorithm  A1 10             X. Jia et al.  
+ and the x-ray imager is mo deled to be an array of 512ൈ384 . The source to axes distance 
+is 100 cm and the source to detector distance is 150 cm. X-ray projections are computed 280 
+along various directions with Siddon's ray tracing algorithm46-48. Three slice cuts of the 
+phantom are displayed in Fig. 3, as well as a typical x-ray projection along the AP direction. 
+We first reconstruct CBCT images based on different number of x-ray projections 
+ܰ
+ఏ. In all cases, the projections are taken al ong equally spaced angles covering an entire 285 
+360 degree rotation. The reconstruction results under ܰఏൌ 5, 10, 20, and 40 projections 
+are listed in Fig. 4. For the purpose of co mparison, we also show here the images 
+reconstructed from conventional FDK algorithm1 with same experimental setting. Clearly, 
+the reconstructed CBCT image from our method based on 40 projections is almost visually indistinguishable from the ground-truth. On the other hand, the one produced by 290 
+the conventional FDK algorithm is full of str eak artifacts due to the insufficient number 
+of projections. Moreover, the required number of projections can be further lowered for 
+some clinical applications. For example, 20 projections may suffice for patient setup 
+Our 
+algorithm  
+ 
+FDK 
+  
+# of 
+projections  ܰఏൌ5 ܰఏൌ1 0  ܰఏൌ2 0  ܰఏൌ4 0  
+FIG. 4.  Axial view of the reconstructed CBCT image under ܰఏൌ 5, 10, 20, and 40 x-ray 
+projections are shown. The top row is obtained from our new algorithm, while the bottom one 
+is from the FDK algorithm.  
+(a) (b) 
+(c) 
+(d) 
+FIG. 3 . The NCAT phantom used in our CBCT recons truction is shown in (a) axial view, (b) 
+coronal view, and (c) sagittal view. The x-ray pr ojection along the AP direction is also shown 
+in (d).  
+11             X. Jia et al.  
+ purposes in radiotherapy, where major anatomi cal features have already been retrieved. 
+As far as radiation dose is concerned, the r esults shown in Fig. 4 implies a 9 times or 295 
+more dose reduction compared with the currently widely used FDK algorithm, where 
+about 360 projections are usually taken in a full-fan head-and-neck scanning protocol.  
+ܰఏ Error 
+݁ )%( Correlation 
+c Computation 
+Time ݐ( sec) 
+5 28.63 0.9386 38.7 
+10 15.96 0.9813 51.2 
+20 11.38 0.9906 77.8 
+40 7.10 0.9963 130.3 
+ 
+TABLE. 1:  The relative error  ݁ ,the correlation coefficient ܿ ,and the computation time  ݐ as 
+functions of the number of x-ray projections ܰఏ used in our CBCT reconstruction. 
+In order to quantify our reconstruction accu racy, we use the correlation coefficient 
+ؠܿ corrሺ݂,݂ ଴ሻ and the relative error ؠ݁ԡ݂െ݂଴ԡଶ/ԡ݂଴ԡଶ as metrics to measure the 
+closeness between the ground truth image ݂଴ሺݖ,ݕ,ݔሻ  and the reconstruction results 300 
+݂ሺݖ,ݕ,ݔሻ . The relative error ݁ and the correlation coefficient ܿ corresponding to the 
+results shown in Fig. 4 are summarized in Table 1. As expected, the more projections 
+used, the better reconstruction quality will be obtained, leading to smaller relative error ݁ 
+and higher correlation coefficient ܿ .More importantly, we emphasize here that the total 
+reconstruction time is short enough for real clinical applications. As we can see from 305 
+Table 1, the reconstructions can be accom plished within 77~130 seconds on a NVIDIA 
+Tesla C1060 GPU card, when 20~40 projec tions are used. Comparing with the 
+computational time of several hours in currently similar reconstruction approaches, our algorithm has achieved a tremendous effici ency enhancement (~100 times speed up). 
+ 
+FIG. 5. The reconstructed images from 40 projections  are shown in left column. From top to 
+bottom are axial, coronal, and sagittal view . In the right column, the image intensity profiles 
+through the centers of the im ages in three directions ar e shown in red circles. The 
+corresponding ground-truth profiles are indicated by solid lines.
+Finally, we provide a complete visualiz ation of the reconstructed CBCT image in 310 
+Fig. 5, where three orthogonal planes of the final reconstructed image with ܰఏൌ4 0  
+projections are drawn. To have a clear vi sual comparison between the reconstructed 
+images and the ground truth, we also plot the image profiles along the central lines in x, y, 
+12             X. Jia et al.  
+ and z directions. The corresponding profiles in the ground-truth image are indicated by 
+solid lines. These plots clearly show that the reconstructed image, though containing 315 
+small fluctuations, is close to the ground -truth data to a satisfactory extent.  
+ 
+B. Catphan phantom  
+To demonstrate our algorithm's performance in a real physical phantom, we performed 320 
+CBCT reconstruction on a CatPhan 600 phantom (The Phantom Laboratory, Inc., Salem, 
+NY). 379 projections within 200 degrees are ac quired using a Varian On-Board Imager 
+system (Varian Medical Systems, Palo Alto, CA) at 2.56 mAs/projection under a full-fan mode. A subset of equally spaced 40 projections is used to perform the reconstruction. In 
+Fig. 6, we show the reconstruction results of  the CatPhan phantom at four axial slices 325 
+obtained from our algorithm and from the conventional FDK algorithm. Clearly, the 
+images obtained from our method are smooth a nd major features of the phantom are 
+captured. On the other hand, the FDK algorith m leads to images contaminated by serious 
+streaking artifacts.  
+To further test the capability of handling noi sy data, we have also performed CBCT 330 
+scans under different mAs levels and reconstr uctions are then conducted using our TV-
+Our algorithm 
+FDK 
+FIG. 6. The axial views of the reconstructe d images of a CatPhan phantom from ܰఏൌ4 0  
+projections based on our method (top row) and the FDK method (bottom row) at 2.56 
+mAs/projection. Different columns correspo nd to different layers at the phantom.  
+Our algorithm  
+FDK 
+mAs/projection 0.10 0.30 1.00 2.56 
+FIG. 7. Axial view of the reconstructed CBCT imag es of a CatPhan phantom at various mAs 
+levels for ܰఏൌ4 0  projections. 
+13             X. Jia et al.  
+ based algorithm and the FDK algorithm, as in Fig. 7. Again, the images produced by our 
+method are smooth and free of streaking artifacts, undoubtedly outperforming those from the FDK algorithm. In particular, under an extremely low mAs level of 0.1 mAs/projection, our method is still able to  capture major features of the phantom. 335 
+Comparing with the currently widely used full-fan head-and-neck scan protocol of 0.4 
+mAs/projection, this performance implies a dose reduction by a factor of ~4 due to lowering the mAs level. Taking into account  the dose reduction by reducing the number 
+of x-ray projections, an overall 36 times dose reduction has been achieved.  340 
+C. Head-and-neck Case 
+ 
+We have also validated our CBCT reconstr uction algorithm on realistic head-and-neck 
+anatomical geometry. We take a patient head-and-neck CBCT scan using a Varian On-
+Board Imager system with 363 projections  in 200 degrees and 0.4 mAs/projection. A 345 
+subset of only 40 projections is selected for the reconstruction. Fig. 8 shows the 
+reconstruction results in this case from our  algorithm as well as from the convential FDK 
+algorithm. Due to the complicated geometry and high contrast between bony structures 
+and soft tissues in this head-and-neck region,  streak artifacts are extremely severe in the 
+images obtained from FDK algorithm under the undersampling case. In contrast, our 350 
+algorithm successfully leads to decent CBCT im age quality, where artifacts are hardly 
+observed and high image contrast is maintained . In particular, when a metal dental filling 
+exists in the patient, our algorithm can still capture it with high contrast, whereas FDK algorithm produces a number of streaks in the CBCT image.   
+We have also performed CBCT scans on an  anthropomorphic skeleton Rando head 355 
+phantom (The Phantom Laboratory, Inc., Salem, NY) to validate our algorithm under low mAs levels in such a complicated anatomy. 363 projections within 200 degrees are acquired using a Varian On-Board Imager sy stem at various mAs/projection levels. The 
+CBCT reconstruction uses only a subset of equa lly spaced 40 projections. In Fig. 9, we 
+show the reconstruction results of this pha ntom under 0.4 mAs/projection, the current 360 
+standard scanning protocol for head-and-neck patient. These results are presented in axial 
+view at three different slices as well as in segittal view. In addition, we present the CBCT 
+Our algorithm  
+FDK 
+FIG. 8. The reconstructed CBCT images of a patient from ܰఏൌ4 0  x-ray projections based 
+on our algorithm (top row) and the FDK algorithm (bottom row). The first three columns 
+correspond to axial views at different laye rs and the last column is sagittal view. 
+14             X. Jia et al.  
+ reconstruction results under different mAs le vels ranging from 0.1 mAs/projection to 2.0 
+mAs/projection in Fig. 10.  In all of these testing cases, our method can reconstruct high quality CBCT images, even under low mAs levels at low number of projections. This 365 
+demonstrates the advantages of our algorithm over the conventional FDK algorithm. As far as the dose reduction, a factor of 36 has been achieved in this head-and-neck case. 
+Our algorithm  
+ 
+FDK 
+ 
+FIG. 9. The reconstructed images of a Rando head phantom from ܰఏൌ4 0  x-ray projections 
+based on our method (top row) and the FDK method (bottom row). The first three columns 
+correspond to axial views at different layers and the last column is sagittal view. The horizontal 
+dark lines in the segittal view are separations betw een neighboring slice sections of the phantom.  
+ 
+Our algorithm  
+FDK 
+mAs/projection 0.10 0.20 0.40 2.00 
+FIG. 10. Axial view of the reconstructed CBCT images  of a head phantom at various mAs levels 
+for ܰఏൌ4 0  projections. 
+ 
+IV. Conclusion 370 
+ 
+In this paper, we have developed a fast iterative algorithm for CBCT reconstruction. We 
+consider an energy functional consisting of a data fidelity term and a regularization term 
+of TV semi-norm. The minimization problem is solved with a GPU-friendly forward-backward splitting method together with a multi-grid approach on a GPU platform, 375 
+leading to both satisfactory accuracy and efficiency.  
+Reconstructions performed on a digital NC AT phantom and a real patient at the 
+head-and-neck region indicate that images with  decent quality can be reconstructed from 
+40 x-ray projections in about 130 seconds. We  have also tested our algorithm on a 
+CatPhan 600 phantom and Rando head phantom  under different mAs levels and found 380 
+that CBCT images can be successfully rec onstructed from scans with as low as 0.1 
+15             X. Jia et al.  
+ mAs/projection . All of these results indicate that  our new algorithm has improved the 
+efficiency by a factor of 100 over existing similar iterative algorithms and reduced imaging dose by a factor of at least 36 compar ed to the current clinical standard full-fan 
+head and neck scanning protocol. The high computation efficiency achieved in our 385 
+algorithm makes the iterative CBCT reconstruction approach applicable in real clinical 
+environments.
+ 
+ 
+Acknowledgement  
+ 390 
+This work is supported in part by the University of California Lab Fees Research 
+Program. We would like to thank NVIDIA for providing GPU cards for this project.   
+Appendix 
+Derivation of Eq. (6) 395 
+ 
+Without losing of generality, assume the x-ray projection is taken along positive x-
+direction. With the help of delta f unctions, we could rewrite Eq. (1) as 
+ܲఏሾ݂ሺݖ,ݕ,ݔሻሿሺݒ,ݑሻൌ׬d݈dݔdݕdݖ݂ ሺݖ,ݕ,ݔሻ  
+    ·ߜሺݔെݔௌെ݊ଵ݈ሻߜሺݕെݕ ௌെ݊ଶ݈ሻߜሺݖെݖ ௌെ݊ଷ݈ሻ. (A1)
+From Fig. 1, the unit vector ࢔ൌሺ ݊ ଵ,݊ଶ,݊ଷሻ can be expressed: 
+     ݊ଵൌ௅బ
+௅ሺ௨,௩ሻ,     ݊ଶൌെ௨
+௅ሺ௨,௩ሻ,݊ଷൌ௩
+௅ሺ௨,௩ሻ， (A2)
+where  ܮሺݒ,ݑሻൌሾ ܮ଴ଶ൅ݑଶ൅ݒଶሿଵ/ଶ. Now for the functional ܧଶሾ݂ሿ in Eq. (2), we are 400 
+ready to take variation with respect to  ݂ሺݖ,ݕ,ݔሻ , yielding   
+ఋ
+ఋ௙ሺ௫,௬,௭ሻܧଶሾ݂ሿൌଵ
+ேഇ஺∑2׬dݑdݒd݈ఏ ൣܲఏሾ݂ሺݖ,ݕ,ݔሻሿሺݒ,ݑሻെܻఏሺݒ,ݑሻ൧  
+      ·ߜሺݔെݔௌെ݊ଵ݈ሻߜሺݕെݕ ௌെ݊ଶ݈ሻߜሺݖെݖ ௌെ݊ଷ݈ሻ. (A3)
+Let us define functions  
+݄ଵሺ݈,ݒ,ݑሻൌݔെݔ ௌെ݊ଵ݈ൌݔെݔ ௌെ௅బ
+௅ሺ௨,௩ሻ݈ ,
+݄ଶሺ݈,ݒ,ݑሻൌݕെݕ ௌെ݊ଶ݈ൌݕെݕ ௌ൅௨
+௅ሺ௨,௩ሻ݈ ,
+݄ଷሺ݈,ݒ,ݑሻൌݖെݖ ௌെ݊ଷ݈ൌݖെݖ ௌെ௩
+௅ሺ௨,௩ሻ݈( .A4)
+From the properties of delta function, it follows that 
+ఋ
+ఋ௙ሺ௫,௬,௭ሻܧଶሾ݂ሿൌ  
+      ଵ
+ேഇ஺∑2ఏൣܲఏሾ݂ሺݖ,ݕ,ݔሻሿሺכݒ,כݑሻെܻఏሺכݒ,כݑሻ൧ଵ
+ቚങሺ೓భ,೓మ,೓యሻ
+ങሺೠ,ೡ,೗ሻቚ
+ሺೠ,כೡ,כ೗כሻ , (A5)
+where ሺכ݈,כݒ,כݑሻ is the solution to Eq. (A4). Explicitly, we have 
+כݑൌെሺݕെݕ௦ሻܮ଴/ሺݔ െݔ ௦ሻ, 
+כݒൌሺݖെݖ௦ሻܮ଴/ሺݔ െݔ ௦ሻ, 
+ כ݈ൌܮሺכݒ,כݑሻሺݔ െݔ௦ሻ/ܮ଴. (A6)
+The geometric meaning of this solution is clear. כ݈ is the distance from x-ray source to the 405 
+point Vൌሺ ݖ,ݕ,ݔ ሻ . ሺכݒ,כݑሻ is the coordinate for a point P on imager, where a ray line 
+connecting the source Sൌሺ ݔ ௦,ݕ௦,ݖ௦ሻ and the point Vൌሺݖ,ݕ,ݔሻ  intersects with the 16             X. Jia et al.  
+ imager. Finally, we need evaluate the Jacobian ቚడሺ௛భ,௛మ,௛యሻ
+డሺ௨,௩,௟ሻቚ in Eq. (A5). This somewhat 
+tedious work yields a simple results: 
+ቚడሺ௛భ,௛మ,௛యሻ
+డሺ௨,௩,௟ሻቚ
+ሺ௨,כ௩,כ௟כሻൌ௅బ௟మሺ௫,௬,௭ሻ
+௅యሺ௨,כ௩כሻ. (A7)
+Substituting (A7) into (A5) leads to Eq. (6).   410 17             X. Jia et al.  
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\ No newline at end of file
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+arXiv:1001.0600v1  [math.CO]  4 Jan 2010Finite irreflexive homomorphism-homogeneous
+binary relational systems✩
+Dragan Maˇ sulovi´ c, Rajko Nenadov, Nemanja ˇSkori´ c
+Department of Mathematics and Informatics, University of N ovi Sad
+Trg Dositeja Obradovi´ ca 4, 21000 Novi Sad, Serbia
+Abstract
+A structure is called homogeneous if every isomorphism between finit e sub-
+structures of the structure extends to an automorphism of the structure.
+Recently, P. J. Cameron and J. Neˇ setˇ ril introduced a relaxed ve rsion of ho-
+mogeneity: we say that a structure is homomorphism-homogeneou s if ev-
+ery homomorphism between finite substructures of the structur e extends to
+an endomorphism of the structure. In this paper we characterize all finite
+homomorphism-homogeneous relational systems with one irreflexiv e binary
+relation.
+Keywords: finite digraphs, homomorphism-homogeneous structures
+2000 MSC: 05C20
+1. Introduction
+A structure is homogeneous if every isomorphism between finite substruc-
+tures of the structure extends to an automorphism of the struc ture. For
+example, finite and countably infinite homogeneous directed graphs were de-
+scribed in [2]. In their recent paper [1] the authors discuss a genera lization
+of homogeneity to various types of morphisms between structure s, and in
+particular introduce the notion of homomorphism-homogeneous st ructures:
+✩Supported by the Grant No 144017 of the Ministry of Science of the republic of Serbia
+Email address: –dragan.masulovic, rajko.nenadov,
+nemanja.skoric ˝@dmi.uns.ac.rs (Dragan Maˇ sulovi´ c, Rajko Nenadov, Nemanja ˇSkori´ c)
+Preprint submitted to Novi Sad Journal of Mathematics Novem ber 16, 2018Definition 1.1 (Cameron, Neˇ setˇ ril [1]) A structure is called homomor-
+phism-homogeneous if every homomorphism between finite substructures of
+the structure extends to an endomorphism of the structure.
+In this short note we characterize all finite homomorphism-homoge neous
+relational systems with one irreflexive binary relation.
+2. Preliminaries
+A binary relational system is an ordered pair ( V,E) whereE⊆V2is
+a binary relation on V. A binary relational system ( V,E) isreflexive if
+(x,x)∈Efor allx∈V,irreflexive if (x,x)/∈Efor allx∈V,symmetric if
+(x,y)∈Eimplies (y,x)∈Efor allx,y∈Vandantisymmetric if (x,y)∈E
+implies (y,x)/∈Efor all distinct x,y∈V.
+Binary relational systems can be thought of in terms of digraphs (h ence
+the notation ( V,E)). Then Vis the set of verticesandEis the set of edges
+of the binary relational system/digraph ( V,E). Edges of the form ( x,x) are
+calledloops. If (x,x)∈Ewe also say that xhas a loop . Instead of ( x,y)∈E
+we often write x→yand say that xdominates y, or that yis dominated
+byx. Byx∼ywe denote that x→yory→x, whilex⇄ydenotes that
+x→yandy→x. Ifx⇄y, we say that xandyform adouble edge . We
+shall also say that a vertex xisincident with a double edge if there is a vertex
+y/ne}ationslash=xsuch that x⇄y.
+Digraphs ( V,E) whereEis a symmetric binary relation on Vare usually
+referred to as graphs.Proper digraphs are digraphs ( V,E) whereEis an
+antisymmetric binary relation. In this paper, digraphs ( V,E) where Eis
+neither antisymmetric norsymmetric will bereferredtoas improper digraphs .
+In an improper digraph there exists a pair of distinct vertices xandysuch
+thatx⇄yand another pair of distinct vertices uandvsuch that u→v
+andv/ne}ationslash→u.
+A digraph D′= (V′,E′) is asubdigraph of a digraph D= (V,E) if
+V′⊆VandE′⊆E. We write D′/lessorequalslantDto denote that D′is isomorphic
+to a subdigraph of D. For∅ /ne}ationslash=W⊆VbyD[W] we denote the digraph
+(W,E∩W2) which we refer to as the subdigraph of Dinduced by W.
+Verticesxandyareconnected in Dif there exists a sequence of vertices
+z1,...,z k∈Vsuch that x=z1∼...∼zk=y. A digraph Disweakly
+connected if each pair of distinct vertices of Dis connected in D. A digraph
+Disdisconnected if it is not weakly connected. A connected component of
+2Dis a maximal set S⊆Vsuch that D[S] is weakly connected. The number
+of connected components of Dwill be denoted by ω(D).
+Verticesxandyaredoubly connected in Dif there exists a sequence of
+verticesz1,...,z k∈Vsuch that x=z1⇄...⇄zk=y. Define a binary
+relationθ(D) onV(D) as follows: ( x,y)∈θ(D) if and only if x=yor
+xandyare doubly connected. Clearly, θ(D) is an equivalence relation on
+V(D) andω(D)/lessorequalslant|V(D)/θ(D)|. We say that a digraph Disθ-connected if
+ω(D) =|V(D)/θ(D)|, and that it is θ-disconnected ifω(D)<|V(D)/θ(D)|.
+Note that a θ-connected digraph need not be connected, and that a θ-
+disconnected digraph need not be disconnected; a digraph Disθ-connected
+if every connected component of Dcontains precisely one θ(D)-class, while it
+isθ-disconnected if there exists a connected component of Dwhich consists
+of at least two θ(D)-classes. In particular, every proper digraph with at least
+two vertices is θ-disconnected, and every graph is θ-connected.
+LetKndenote the complete irreflexive graph on nvertices. Let 1denote
+the trivial digraph with only one vertex and no edges, and let 1◦denote the
+digraph with only one vertex with a loop. An oriented cycle with nvertices
+is a digraph Cnwhose vertices are 1, 2, ..., n,n/greaterorequalslant3, and whose edges are
+1→2→...→n→1.
+For digraphs D1= (V1,E1) andD2= (V2,E2), byD1+D2we denote
+thedisjoint union ofD1andD2. We assume that D+O=O+D=D,
+whereO= (∅,∅) denotes the empty digraph . The disjoint union D+...+D/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+k
+consisting of k/greaterorequalslant1 copies of Dwill be abbreviated to k·D. Moreover, we
+let 0·D=O.
+LetD1= (V1,E1) andD2= (V2,E2) be digraphs. We say that f:V1→
+V2is ahomomorphism between D1andD2and write f:D1→D2if
+x→yimpliesf(x)→f(y),for allx,y∈V1.
+Anendomorphsim is a homomorphism from Dinto itself. A mapping f:
+V1→V2is anisomorphism between D1andD2iffis bijective and
+x→yif and only if f(x)→f(y),for allx,y∈V1.
+Digraphs D1andD2areisomorphic ifthereisanisomorphism between them.
+We write D1∼=D2. Anautomorphsim is an isomorphism from Donto itself.
+A digraph Dishomomorphism-homogeneous if every homomorphism
+f:W1→W2between finitely induced subdigraphs of Dextends to an
+endomorphism of D(see Definition 1.1).
+33. Finite irreflexive binary relational systems
+Cameron and Neˇ setˇ ril have shown in [1] that a finite irreflexive gra ph is
+homomorphism-homogeneous if and only if it is isomorphic to k·Knfor some
+k,n/greaterorequalslant1. It was shown in [3, Theorem 3.10] that a finite irreflexive proper
+digraph is homomorphism-homogeneous if and only if it is isomorphic to
+k·1for some k/greaterorequalslant1 ork·C3for some k/greaterorequalslant1. In this section we show
+that these are the only finite homomorphism-homogeneous irreflex ive binary
+relational systems by showing that no finite irreflexive improper digr aph is
+homomorphism-homogeneous.
+Lemma 3.1 LetDbe a finite homomorphism-homogeneous irreflexive im-
+proper digraph. Then every vertex of Dis incident with a double edge.
+Proof.Letx⇄ybe a double edge in Dand letvbe an arbitrary vertex
+ofD. The mapping
+f:/parenleftbiggx
+v/parenrightbigg
+is a homomorphism between finitely induced subdigraphs of D, so it extends
+to an endomorphism f∗ofDby the homogeneity requirement. Then x⇄y
+impliesv=f∗(x)⇄f∗(y). /square
+Lemma 3.2 LetDbe a finite homomorphism-homogeneous irreflexive im-
+proper digraph and let S∈V(D)/θ(D)be an arbitrary equivalence class of
+θ(D). ThenD[S]∼=Knfor some n/greaterorequalslant2.
+Proof.Lemma 3.1 implies that |S|/greaterorequalslant2 for every S∈V(D)/θ(D).
+Suppose that there is an S∈V(D)/θ(D) such that D[S] is not a com-
+plete graph. Then there exist u,v∈Ssuch that u/ne}ationslash→vorv/ne}ationslash→u. Let
+z1,z2,...,z k∈V(D) be the shortest sequence of vertices of Dsuch that
+u=z1⇄z2⇄...⇄zk=v.
+Thenk/greaterorequalslant3 sinceu/ne}ationslash⇄v, and the fact that z1,z2,...,z kis the shortest such
+sequence implies that z1/ne}ationslash⇄z3. The mapping
+f1:/parenleftbiggz1z3
+z2z3/parenrightbigg
+4is a homomorphism between finitely induced subdigraphs of D, so it extends
+to an endomorphism f∗
+1ofDby the homogeneity requirement. Let x1=
+f∗
+1(z2). It is easy to see that x1/∈ {z1,z2,z3}andx1⇄yfor ally∈ {z2,z3}.
+Consider now the mapping
+f2:/parenleftbiggz1z3x1
+z2z3x1/parenrightbigg
+.
+which is clearly a homomorphism between finitely induced subdigraphs o f
+D. It extends to an endomorphism f∗
+2ofD. Letx2=f∗
+2(z2). Again, it is
+easy to see that x2/∈ {z1,z2,z3,x1}and that x2⇄yfor ally∈ {z2,z3,x1}.
+Analogously, the mapping
+f3:/parenleftbiggz1z3x1x2
+z2z3x1x2/parenrightbigg
+is a homomorphism between finitely induced subdigraphs of D, so it extends
+to an endomorphism f∗
+3ofD. Letx3=f∗
+3(z2). Again, x3/∈ {z1,z2,z3,x1,x2}
+andx2⇄yfor ally∈ {z2,z3,x1,x2}. And so on. We can continue with this
+procedure as many times as we like, which contradicts the fact that Dis a
+finite digraph. /square
+Proposition 3.3 There doesnotexist afinitehomomorphism-homogeneous
+irreflexive improper digraph.
+Proof.Suppose that Dis a finite homomorphism-homogeneous irreflexive
+improper digraph. Then there exist vertices x,y∈V(D) such that x→y
+andy/ne}ationslash→x. LetS=x/θ(D) andT=y/θ(D). Clearly, S∩T=∅.
+LetT={y,t1,...,t k}. SinceD[T] is a complete graph (Lemma 3.2), the
+mapping
+f:/parenleftbiggx t1... tk
+y t1... tk/parenrightbigg
+is a homomorphism between finitely induced subdigraphs of D, so it extends
+to an endomorphism f∗ofDby the homogeneity requirement. Let us com-
+putef∗(y). From f∗(t1)∈Tit follows that f∗(T)⊆T. Moreover, f∗|Tis
+injective since there are no loops in D. Therefore, f∗|T:T→Tis a bijec-
+tion. But f∗(ti) =tifor alli∈ {1,...,k}, so it follows that f∗(y) =y. Now,
+x→yimpliesf∗(x)→f∗(y), that is, y→y, which is impossible since there
+are no loops in D. /square
+5Corollary 3.4 LetDbe a finite irreflexive binary relational system. Then
+Dis homomorphism-homogeneous if and only if it is isomorphic to one of
+the following:
+(1)k·Knfor some k,n/greaterorequalslant1;
+(2)k·C3for some k/greaterorequalslant1.
+References
+[1] Cameron, P. J., Neˇ setˇ ril, J.: Homomorphism-homogeneous rela tional
+structures. Combinatorics, Probability and Computing Vol. 15, 200 6,
+91–103
+[2] Cherlin, G. L.: The Classification of Countable Homogeneous Direct ed
+Graphs and Countable Homogeneous n-tournaments. Memoirs of the
+American Mathematical Society, Vol. 131, No. 621, 1998
+[3] Maˇ sulovi´ c, D.: Towards the characterization of finite homomo rphism-
+homogeneous digraphs. (submitted)
+6
\ No newline at end of file
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+arXiv:1001.0601v1  [math.GR]  4 Jan 2010ZARISKI TOPOLOGIES ON GROUPS
+TARAS BANAKH, IGOR PROTASOV
+Abstract. Then-th Zariski topology ZGn[x]on a group Gis generated by the sub-base con-
+siting of the sets {x∈G:a0xε1a1xε2···xεnan/\e}atio\slash= 1}wherea0,...,a n∈Gandε1,...,ε n∈
+{−1,1}. We prove that for each group Gthe 2-nd Zariski topology ZG2[x]is not discrete and
+present an example of a group Gof cardinality continuum whose 2-nd Zariski topology has
+countable pseudocharacter. On the other hand, the non-topolo gizable group Gconstructed by
+Ol’shanskii has discrete 665-th Zariski topology ZG665[x].
+In this paper we study topological properties of groups endowed w ith the Zariski topologies
+ZGn[x]. These topologies are defined on each group Gas follows. Let G[x] =G∗/a\}bracketle{tx/a\}bracketri}htbe the free
+product of the group Gand the cyclic group /a\}bracketle{tx/a\}bracketri}htgenerated by an element x /∈G. The group
+G[x] can be written as the countable union
+G[x] =/uniondisplay
+n∈ωGn[x].
+Here for a subset A⊂Gwe put
+A0[x] =AandAn+1[x] =An[x]∪{wxa,wx−1a:w∈An[x], a∈A}forn∈ω.
+The elements of the subset An[x] are called monomials of degree ≤nwith coefficients in the
+setA.
+Each monomial w∈G[x] can be thought as a map w(·) :G→G, w:g/mapsto→w(g), wherew(g)
+is the image of wunder the group homomorphism G[x]→Gthat is identical on Gand maps
+the element x∈G[x] ontog∈G. The subset
+Zw={x∈G:w(x)/\e}atio\slash= 1}
+is called the co-zeroset of the monomial w. Here 1 is the neutral element of G.
+Any family of monomials W⊂G[x] induces a topology ZWonG, generated by the sub-base/braceleftbig
+Zw:w∈W/bracerightbig
+consisting of co-zero sets of the monomials from W. The topology ZG[x]is
+called the Zariski topology onGby analogy with the Zariski topology well-known in algebraic
+geometry. This topology was studied in [6], [2]. For n∈ωthe topology ZGn[x]will be referred
+to as then-th Zariski topology onG. It is clear that the Zariski topologies form a chain
+ZG0[x]⊂ZG1[x]⊂ZG2[x]⊂ ··· ⊂ZG[x].
+The 0-th Zariski topology ZG0[x]is antidiscrete while the 1-st Zariski topology ZG1[x]coincides
+with the cofinite topology on Gand hence satisfies the separation axiom T1. The same is true
+for Zariski topologies ZGn[x]withn≥1.
+It is easy to see that for every n∈ωthe groupGendowed with the nth Zariski topology
+ZGn[x]is a quasi-topological group. The latter means that the group oper ation is separately
+continuous and the inversion is continuous with respect to the topo logyZGn[x]onG.
+1991Mathematics Subject Classification. 20F70, 22A30.
+Key words and phrases. Zariski topology, topologizable group.
+12 TARAS BANAKH, IGOR PROTASOV
+As we already know, for an infinite group G, the topology ZG1[x], being cofinite, is non-
+discrete. The same is true for the topology ZG2[x].
+Theorem 1. For every infinite group Gthe 2-nd Zariski topology ZG2[x]is not discrete.
+Proof.Assuming the converse, we would find a finite subset W⊂G2[x] such that {1}=/intersectiontext
+w∈WZw. Find a countable subgroup H⊂Gsuch thatW⊂H∗ /a\}bracketle{tx/a\}bracketri}ht=H[x]. By [17], the
+groupHis a quasitopological group with respect to some non-discrete Haus dorff (even regular)
+topologyτ.
+We claim that for each monomial w=a0xε1a1xε2a2∈W⊂H2[x] withε1,ε2∈ {−1,1}the
+cozero set Zwcontains a neighborhood Uw∈τof 1. Since 1 ∈Zw, we geta01ε1a11ε2a2/\e}atio\slash= 1 and
+hencea01ε1a1/\e}atio\slash=a−1
+21−ε2. Sincethetopology τonHisHausdorff, thepoints a01ε1a1anda−1
+21−ε2
+have disjoint neighborhoods O(a01ε1a1),O(a−1
+21−ε2)∈τ. Since (H,τ) is a quasitopological
+group, there is a neighborhood Uw∈τof 1 such that a0Uε1wa1⊂O(a01ε1a1) anda−1
+2U−ε2w⊂
+O(a−1
+21−ε2). For this neighborhood Uwwe get 1/∈a0Uε1wa1Uε2wa2andUw⊂Zw. Now we see
+from/intersectiontext
+w∈WUw⊂/intersectiontext
+w∈WZw={1}that 1 is an isolated point of Hin the topology τ, which is
+a contradiction. /square
+Theorem 1 implies that for each infinite group Gthe quasitopological group ( G,ZG2[x]) has
+infinite pseudocharacter. By the pseudocharacter ψx(X) of a topological T1-space at a point
+x∈Xwe understand the smallest cardinality |Ux|of a family Uxof neighborhoods of the point
+xsuch that ∩Ux={x}. The cardinal ψ(X) = supx∈Xψx(X) is called the pseudocharacter of
+the topological space X.
+Writing Theorem 1 in terms of the pseudocharacter, we get
+Corollary 1. For any infinite group Gwe get
+ψ(G,ZG1[x]) =|G|andψ(G,ZG2[x])≥ ℵ0.
+It is interesting to observe that the lower bound ψ(G,ZG2[x])≥ ℵ0in this corollary cannot
+be improved to ψ(G,ZG2[x]) =|G|.
+Theorem 2. There is a group Gof cardinality continuum such that ψ(G,ZG2[x]) =ℵ0. In fact,
+the groupGcontains two disjoint countable subsets A,B⊂Gsuch that {1}=/intersectiontext
+w∈WZwwhere
+W={xax−1b−1:a∈A, b∈B}.
+Proof.LetT=/uniontext
+n∈ω{0,1}nbe the binary tree (consisting of finite binary sequences) and
+Aut(T) be its automorphism group(which has cardinality of continuum). Th e action of Aut( T)
+onTinducesanactionofAut( T)onthefreegroup F(T)overT. Inthefreegroup F(T)consider
+the setLof words of the form alwherea∈Tandl=aˆ0 is the “left” successor of ain the
+binarytree T. Bysymmetry, let R⊂F(T)bethesetofwords arwherea∈Tandr=aˆ1isthe
+“right”successor of ainT. Itiseasytocheckthatthesets LF(T)={wxw−1:x∈L, w∈F(T)}
+andRF(T)={wxw−1:x∈R, w∈F(T)}are disjoint.
+LetF(T)\{1}={wn:n∈ω}be an enumeration of the set of non-unit elements of F(T).
+By induction for every n∈ωwe can find points an,bn∈F(T) such that
+•bn=wnanw−1
+n;
+•an/∈ {bi:i≤n};
+•bn/∈ {ai:i≤n};
+•an/∈RF(T);
+•bn/∈LF(T).ZARISKI TOPOLOGIES ON GROUPS 3
+Consider the countable disjoint subsets
+A=LF(T)∪{an:n∈ω}andB=RF(T)∪{bn:n∈ω}
+of the free group F(T).
+LetGbe the semidirect product of the groups F(T) and Aut(T). The elements of the group
+Gare pairs (w,f), wherew∈F(T) andf∈Aut(T). The group operation on Gis given by
+the formula:
+(w,f)·(u,g) = (w·f(u),fg).
+The groups F(T) and Aut( T) are identified with the subgroups {(w,1) :w∈F(T)}and
+{(1,f) :f∈Aut(T)}ofG.
+We claim that for every non-unit element x∈G\{1}we getxAx−1∩B/\e}atio\slash=∅. Ifx∈F(T),
+thenx=wnforsomen∈ωandthenxAx−1∩B∋bn=wnanw−1
+n. Ifx /∈F(T), thenx= (w,f)
+for some non-identity automorphism f∈Aut(T). It follows that f(t)/\e}atio\slash=tfor some node t∈T.
+We can assume that thas the smallest possible height. The node tis not the root of the tree T
+because the automorphism fofTdoes not move the root. So, thas an immediate predecessor
+ain the tree T. The minimality of tguarantees that f(a) =aand hence {t,f(t)}={l,r}
+wherel=aˆ0 andr=aˆ1 are the “left” and “right” successors of ain the binary tree T. It
+follows that al∈L⊂Aandf(al) =f(a)f(l) =ar∈R. Consequently,
+x·al·x−1= (w,f)·(al,1)·(f−1(w−1),f−1) =
+= (w·f(al),f·1)·(f−1(w−1),f−1) = (w·ar·w−1,1)∈RF(T)⊂B,
+witnessing that xAx−1∩B/\e}atio\slash=∅and hence 1 ∈xAx−1B−1. /square
+Theorem 3. IfGis an infinite abelian group or a free group, then ψ(G,ZG[x]) =|G|. Moreover,
+for any family U ⊂ZG[x]of neighborhoods of 1 with |U|<|G|the intersection ∩Ucontains a
+subgroupH⊂Gof cardinality |H|=|G|.
+Proof.Letκ=|G|. In order to prove that ψ(G,ZG[x]) =|G|, take any family of monomials
+W⊂G[x] of size|W|< κwith 1∈/intersectiontext
+w∈WZw. It follows that w(1)/\e}atio\slash= 1 for every w∈Wand
+hence the set W(1) ={w(1) :w∈W}is a subset of G\ {1}and has cardinality |W(1)| ≤
+|W|<κ.
+If the group Gis abelian, then Gcontains a subgroup Sof cardinality |S|=κthat is
+isomorphic to the direct sum ⊕α∈κSαof cyclic groups, see [3, §16]. Since the set S∩W(1)
+has cardinality < κ, there is a subset A⊂κof cardinality |A|< κsuch thatS∩W(1)⊂
+⊕α∈ASα. Then the subgroup H=⊕α∈κ\ASαhas cardinality |H|=κand is disjoint with W(1).
+Consequently, H·W(1)⊂G\ {1}. We claim that H⊂/intersectiontext
+w∈WZw. Indeed, for every w∈W
+andx∈H, by the commutativity of Gwe getw(x) =w(1)xnfor somen∈Zand hence
+w(x)∈w(1)·H⊂W(1)·H⊂G\{1}witnessing that H⊂Zw.
+Next, assume that G=F(A) is a free group over an infinite alphabet A⊂G. It follows that
+the setW(1) of cardinality <κlies in the subgroup F(B) for some subset B⊂Aof cardinality
+<κ. Consider the subgroup H=F(A\B) and observe that w(x)/\e}atio\slash= 1 for every x∈H. This
+means that H⊂/intersectiontext
+w∈WZw.
+IfG=F(A) is a non-commutative free group over a finite alphabet A, thenGcontains
+a free subgroup F⊂Gwith infinitely many generators, see [7, II.1.2]. By the preceding
+case the free group Fcontains a subgroup H⊂Fof cardinality |H|=ℵ0=|G|such that
+H⊂F∩/intersectiontext
+w∈WZω. /square
+It is interesting to compare Corollary 1 and Theorem 2 with4 TARAS BANAKH, IGOR PROTASOV
+Theorem 4. For a finite subset Aof an infinite group Gand the family
+W=G1[x]∪{bxax−1c,bx−1axc:a∈A, b,c∈G},
+the groupGendowed with the topology ZWis a quasitopological group with pseudocharacter
+ψ(G,ZW) =|G|.
+Proof.It follows from the definition of the family Wthat the inversion ( ·)−1:G→Gand
+left shiftslb:G→G,b∈G, are continuous with respect to the topology τW. So, (G,τW)
+is a left-topological group with continuous inversion. The continuity of right shifts follows
+from the continuity of the left shifts and the continuity of the inver sion. Thus ( G,τW) is a
+quasitopological group.
+Inordertoshowthat ψ(G,ZW) =|G|, fixafamily V⊂Wofsize|V|<|G|with1∈/intersectiontext
+v∈VZv.
+Each monomial v∈Vis of the form bvxavx−1cvorbvx−1avxcvfor someav∈Aandbv,cv∈G.
+Letdv=b−1
+vc−1
+vand observe that bvxavx−1cv/\e}atio\slash= 1 if and only if xavx−1/\e}atio\slash=dv. Since 1 ∈Zv, we
+getav/\e}atio\slash=dvfor everyv∈V.
+LetA={a1,...,a n}be an enumeration of the finite set A. For every k≤nletDk={dv:
+v∈V, av=ak}. It follows that ak/∈Dkand|Dk| ≤ |V|<|G|.
+LetG0=Gand by induction for every k<ndefine a subgroup Gk+1ofGlettingGk+1=Gk
+if the centralizer
+ZGk(ak+1) ={x∈Gk:xak+1=ak+1x}
+has size<|G|andGk+1=ZGk(ak+1) otherwise. So, ( Gk)k≤nis a decreasing sequence of
+subgroups of Ghaving size |G|.
+LetKbe the set of positive numbers k≤nsuch that |ZGk−1(ak)|<|G|. For every k∈K
+andd∈Dkconsider the set Xk,d={x∈Gn:xakx−1=d}. We claim that |Xk,d|<
+|ZGn(ak)| ≤ |ZGk−1(ak)|<|G|. Indeed, for any x,y∈Xk,dwe getxakx−1=d=yaky−1and
+thusy−1xak=aky−1x, which implies y−1x∈ZGn(ak) and thusXk,d∈yZGn(ak). Then the set
+Xk=/uniondisplay
+d∈DkXk,d∪X−1
+k,d
+has cardinality |Xk| ≤2·|Dk|·|ZGn(ak)|<|G|. Finally, consider the set Y=Gn\/uniontext
+k∈KXk
+and observe that Y=Y−1and|Y|=|Gn|=|G|.
+Weclaimthat Y⊂Zvforeveryv∈V. Intheoppositecase v(y) = 1forsome v∈Vandsome
+y∈Y. It follows that bvyavy−1cv= 1 orbvy−1avycv= 1. SinceY=Y−1, we loose no generality
+assuming that bvyavy−1cv= 1 and hence yavy−1=dv. Findk≤nsuch thatav=kand then
+dv∈Dk. Ifk /∈K, theny∈Gn⊂Gk=ZGk−1(ak) and then Dk∋dv=yavy−1=yaky−1=ak,
+which contradicts ak/∈Dk. Ifk /∈K, thenyaky−1=dv∈Dkimplies that y∈Xkwhich
+contradicts the choice of y∈Y=Gn\/uniontext
+k∈KXk. /square
+Proposition 4 implies the following fact, proved in [1] by the technique of ultrafilters:
+Corollary 2. For any disjoint finite subsets A,Bof an infinite group Gthe set{x∈G:
+xAx−1∩B=∅}is infinite.
+Remark 1. According to [11] or [13] each group Gwithψ(G,ZG[x]) =|G|is topologizable,
+that is, admits a non-discrete Hausdorff group topology.
+Remark 2. In [4] groups Gwithψ(G,ZG[x])≥κare calledκ-ungebunden .
+GroupsGwith non-discete n-th Zariski topology ZGn[x]have the following property:ZARISKI TOPOLOGIES ON GROUPS 5
+Theorem 5. If then-th Zariski topology ZGn[x]on a group Gis non-discrete, then any finite
+subsetsA0,A1,...,A n⊂Gwith1/∈A0A1···Anthere is an infinite symmetric subset X=
+X−1∋1such that 1/∈A0XA1X···XAn.
+Proof.By induction on k∈ωwe can construct a sequence ( xk)k∈ωof points of the group G, a
+sequence (Ck)k∈ωof finite subsets of G, and a sequence ( Wk)k∈ωof finite subsets of Gn[x] such
+that for every k∈Nthe following conditions are satisfied:
+(1)Ck=C−1
+k= (Ck−1∪{1,xk−1,x−1
+k−1})n⊂G;
+(2)Wk={w∈Cn
+k[x] :w(1)/\e}atio\slash= 1};
+(3)w(xk)/\e}atio\slash= 1 for any w∈Wk;
+(4)xk/∈ {xi:i<k}.
+We start the inductive construction letting x0= 1 andC0=C−1
+0be any finite set containing
+A0∪···∪An. For eachkthe choice of the point xkis possible since the nth Zariski topology
+ZGn[x]is not discrete.
+The conditions (1)–(3) of the inductive construction guarantee t hat the set X={xk,x−1
+k:
+k∈ω}has the desired property: 1 /∈A0XA1X···XAn. /square
+LookingatTheorem 1 onemayaskwhat happens withtheZariski top ologiesZGn[x]forhigher
+n. For sufficiently large nthe topology ZGn[x]can be discrete.
+The following criterion is due to A.Markov [6]:
+Theorem 6 (Markov) .For a countable group Gthe following conditions are equivalent:
+(1)the topology ZG[x]is discrete;
+(2)the topology ZGn[x]is discrete for some n∈ω;
+(3)the groupGis non-topologizable.
+We recall that a group Gisnon-topologizable ifGadmits no non-discrete Hausdorff group
+topology. For examples of non-topologizable groups, see [14], [9], [1 0], [8], [15], [16], [5]. In
+particular, the non-topologizable group from [9] yields the following e xample mentioned in [5]:
+Example 1. There is a countable group Gwhose 665-th Zariski topology ZG665[x]is discrete.
+What happens for n<665, in particular, for n= 3?
+Problem 1. Is the 3-d Zariski topology ZG3[x]non-discrete on each infinite group G?
+Also Corollary 1 and Example 2 suggest:
+Problem 2. Isψ(G,ZG2[x])≥log|G|for every infinite group G?
+Remark 3. By [4] for every infinite cardinal κthere is a topologizable group Gwith|G|=κ
+andψ(G,ZG[x]) =ℵ0.6 TARAS BANAKH, IGOR PROTASOV
+References
+[1] T.Banakh, N.Lyaskovska, Weakly P-small not P-small subsets in groups , Internat. J. Algebra Comput. 18:1
+(2008), 1-6.
+[2] D. Dikranjan, D. Shakhmatov, Reflection principle characterizing groups in which uncond itionally closed
+sets are algebraic , J. Group Theory 11:3 (2008), 421–442.
+[3] L. Fuchs, Infinite abelian groups. Vol. I. Pure and Applied Mathem atics, Vol. 36 Academic Press, New
+York-London, 1970.
+[4] G. Hesse, Zur Topologisierbarkeit von Gruppen , Ph.D. Dissertation, Univ. Hannover, Hannover, 1979.
+[5] A.A. Klyachko, A.V. Trofimov, The number of non-solutions of an equation in a group , J. Group Theory
+8:6 (2005), 747–754.
+[6] A.A. Markov, On unconditionally closed sets , Mat. Sb. 18:1 (1946), 3–28.
+[7] O.V. Mel’nikov, V.N. Remeslennikov, V.A. Roman’kov, L.A. Skornyako v, I.P. Shestakov, General Algebra.
+Vol.1, “Nauka”, Moscow, 1990 (in Russian).
+[8] S. Morris, V. Obraztsov, Nondiscrete topological groups with many discrete subgrou ps, TopologyAppl. 84:1-3
+(1998) 105–120.
+[9] A.Yu. Ol’shanskii, A remark on a countable nontopologized group , Vestnik Moscow Univ. Ser. I. Mat. Mekh,
+no.3 (1980), p.103.
+[10] A.Yu. Ol’shanskii, The geometry of defining relations in groups, Na uka, Moscow, 1989.
+[11] K.-P. Podewski, Topologisierung algebraischer Strukturen , Rev. Roumaine Math. Pures Appl. 22 (1977),
+no. 9, 1283–1290.
+[12] O.V.Sipacheva, A class of groups in which all unconditionally closed sets ar e algebraic , Fundam. Prikl.
+Mat. 12 (2006), no. 8, 217–222; translation in: J. Math. Sci. (N. Y .) 152 (2008), no. 2, 288–291
+[13] O.V. Sipacheva, Unconditionally τ-closed and τ-algebraic sets in groups , Topology Appl. 155:4 (2008),
+335–341.
+[14] S. Shelah, On a problem of Kurosh, Jnsson groups, and applications , in: Word problems, II (Conf. on
+Decision Problems in Algebra, Oxford, 1976), pp. 373–394,Stud. L ogic Foundations Math., 95, North-Holland,
+Amsterdam-New York, 1980.
+[15] A.V. Trofimov, A theorem on embedding into a nontopologizable group , Vestnik Moskov. Univ. Ser. I Mat.
+Mekh. (2005), no. 3, 60–62.
+[16] A.V. Trofimov, A perfect nontopologizable group , Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2007), no.1,
+7–13; transl. in Moscow Univ. Math. Bull. 62:1 (2007) 5–11.
+[17] E. Zelenyuk, On topologies on groups with continuous shifts and inversio n, Visn. Kyiv. Univ. Ser. Fiz.-Mat.
+Nauki. 2000, no. 2, 252–256 (in Ukrainian).
+Ivan Franko National University of Lviv, Ukraine and
+Uniwersytet Humanistyczno-Pryrodniczy Jana Kochanowski ego w Kielcach, Poland
+Taras Shevchenko National University of Kyiv, Ukraine
+E-mail address :T.O.Banakh@gmail.com
+E-mail address :I.V.Protasov@gmail.com
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+arXiv:1001.0602v2  [physics.class-ph]  20 Mar 2010On the locus formed by the maximum heights of projectile moti on with air resistance
+H. Hern´ andez-Salda˜ na∗
+Departamento de Ciencias B´ asicas,
+Universidad Aut´ onoma Metropolitana-Azcapotzalco,
+Av. San Pablo 180, M´ exico 02200 D.F., Mexico.
+(Dated: May 5, 2018)
+We present an analysis on the geometrical place formed by the set of maxima of the trajectories
+of a projectile launched in a media with linear drag. Such a pl ace, the locus of apexes, is written
+in term of the Lambert Wfunction in polar coordinates, confirming the special role p layed by this
+function in the problem. In order to characterize the locus, a study of its curvature is presented in
+two parameterizations, in terms of the launch angle and in th e polar one. The angles of maximum
+curvature are compared with other important angles in the pr ojectile problem. As an addendum,
+we find that the synchronous curve in this problem is a circle a s in the drag-free case.
+I. INTRODUCTION
+An amazing characteristic of some old fashion prob-
+lems is their endurement. The projectile motion is one
+of them. Being one of the main problems used to teach
+elementary physics, variations and not well known facts
+about it appear in the physics literature of the XXI cen-
+tury. A search in the web [1] or in the Science Citation
+Indexgivesanideaofthisfact. Someoftherecentstudies
+deal with the problem of air resistance in the projectile
+motion and its pedagogical character made of it an ex-
+cellent example to introduce the Lambert Wfunction, a
+special function.[2] The Lambert Wfunction is involved
+in many problems of interest for physicist and engineers,
+from the solution of the jet fuel problem to epidemics[2]
+or, even, Helium atom eigenfunctions.[3] One of those
+problems is the solution for the range Rin the case that
+the air resistance has the from /vectorf=−mb/vector v.[4, 5]
+In this paper we analyze the not well known fact of the
+geometricalplaceformedbythemaximaofalltheprojec-
+tile trajectories at launch angle αand in the presence of
+a drag force proportional to the velocity, we shall denote
+this locus as Cm(ε). The resulting locus becomes a Lam-
+bertWfunction of the polar coordinate r(θ) departing
+from the origin. This problem raises as a natural contin-
+uation from the nice fact that in the drag-free case such
+a locus is an ellipse[6–8] with an universal eccentricity
+e=√
+3/2.[6]
+The paper is organized as follows. In section II, the
+set of maxima for projectile trajectories moving under in
+the presence of air resistance is presented. In section III
+we find a closed form, in polar coordinates, to express
+such a geometrical place, Cm. In section IV we present
+a numerical calculation of the curvature of Cmusing the
+polar angle and the launch angle as parameterizations.
+Additionally, we demonstratethat the synchronouscurve
+is a circle as in the drag-free case in section V. In section
+VI we conclude.
+∗Electronic address: hhs@correo.azc.uam.mxII. THE PROJECTILE PROBLEM WITH AIR
+RESISTANCE
+Several approximations in order to consider the air re-
+sistance exist in the literature, the simplest is the linear
+case. In such a case the force is given by
+/vectorF=−mb/vector v−mgˆ , (1)
+wheremis the mass of the projectile and bis the drag
+coefficient. The units of bares−1. The velocity compo-
+nents are labeled as /vector v=uˆ ı +wˆ , withu=dx/dtand
+w=dy/dt. The solutions for the position and velocity
+are obtained trough direct integration of Eq.(1) yielding
+x(t) =u0
+b[1−exp(−bt)], (2)
+y(t) =w0+g/b
+b[1−exp(−bt)]−gt/b, (3)
+for the coordinates, and
+u(t) =u0exp(−bt), (4)
+w(t) = (w0+g/b)exp(−bt)−g/b, (5)
+for the speeds. We used the initial conditions x(0) =
+y(0) = 0 and u0=V0cosαandw0=V0sinα. Noticing
+that the terminal speed is g/bin theyaxis.
+For the same initial speed V0these solutions are func-
+tion of the launch angle αand the locus formed by the
+apexes is obtained if time is eliminated between the so-
+lutions in time in Eqs. (2) and (3), giving the equation
+y(x) =w0+g/b
+u0x−g
+b2ln/parenleftbigg
+1−bx
+u0/parenrightbigg
+,(6)
+and considering the value at the maximum, via dy/dx=
+0. The corresponding solution is
+xm
+ρ= cosαsinα1
+1+εsinα, (7)
+ε2ym
+ρ=εsinα−ln(1+εsinα), (8)2
+-10 -5 0 5 10x012345y
+FIG. 1: (color online) Locus Cm(ε) formed by the apexes of
+all the projectile trajectories (continuous line in blue) g iven
+by Eqs. (7) and (8) in rectangular coordinates or by Eq. (13),
+the last one express Cmin polar coordinates and in term of
+the Lambert Wfunction. The dashed red line is the ellipse
+of eccentricity e=√
+3/2 which represents the drag-free case,
+i.e.Cm(0). The parameters are V0= 10 and ε= 0.1.
+where we introduce the dimensionless perturbative pa-
+rameter ε≡bV0/g, the dimensionless length ρ=V2
+0/g,
+and noticing thatb2
+gcan be expressed asε2
+ρ. An alter-
+native procedure consists in set the derivative dy/dtto
+zero to obtain the time of flight to the apex of the tra-
+jectory and, evaluate the coordinates at that time. The
+points (xm,ym) conform the locus of apexes Cm(ε) for
+all parabolic trajectories as a function of the launch an-
+gleα. In Fig. 1 we plot Cm(ε) described by Eqs. (7)
+and (8), for the drag-free case (in dashed red line) and
+forε= 0.1 in continuous blue line. Several projectile
+trajectories are plotted in thin black lines. The locus of
+apexesCm(ε) defined by ( xm,ym) of Eqs. (7) and (8)
+is described parametrically by the launch angle αand it
+changes for different values of ε≡bV0/g. In the next
+section we shall find a description of Cm(ε) in terms of
+polar coordinatesand in a closed form using the Lambert
+Wfunction.
+III. THE LOCUS CmAS A LAMBERT W
+FUNCTION
+In order to obtain an analytical closed form of the lo-
+cus we change the variables to polar ones, i.e., xm=
+rmcosθmandym=rmsinθm. The selection of a de-
+scription departing from that origin instead of the center
+orthe focus ofthe ellipse is because the resultinggeomet-
+rical place is no longer symmetric and the only invariant
+point is just the launching origin. We substitute the po-
+lar forms of xmandyminto equations (7) and (8) andrearranging terms it must be expressed as
+rm(θm)
+ρcosθmexp/parenleftBig
+−ε2sinθmrm(θm)
+ρ/parenrightBig
+= cosαsinαexp(−εsinα).(9)
+The lhs depends on rmandθmmeanwhile the rhs de-
+pends on α, however the last angle is a function of θm
+and reads as
+tanθm=1
+ε2(εsinα−ln(1+εsinα))
+cosαsinα1
+1+εsinα,(10)
+by making tan θm=ym/xmfrom Eqs. (7) and (8).
+In order to obtain ˜ r(θm)≡rm(θm)/ρwe set
+f(α(θm))≡cosαsinαexp(−εsinα),(11)
+since Eq. (10) allows us to have, implicitly, α(θm). We
+shall return to this point later. Hence, we can write Eq.
+(9) as
+−ε2sinθm˜r(θm)exp/parenleftbig
+−ε2sinθm˜r(θm)/parenrightbig
+=−ε2tanθmf(α).
+(12)
+Where we multiplied both sides of Eq. (9) by
+−ε2sinθm. Setting z=−ε2tanθmf(α) andW(z) =
+−ε2sinθm˜r(θm) in Eq. (12), it shall have the familiar
+Lambert Wfunction form, z=W(z)exp(W(z)), from
+which we can obtain ˜ ras
+˜r(θm) =−1
+ε2sinθmW(−ε2tanθmf(α)).(13)
+It is important to note that the argument of the Lam-
+bertfunction inthis equationisnegativeforallthe values
+ε >0.W(x) remains real in the range x∈[−1/e,0) and
+have the branches denoted by 0 and −1.[2] We select the
+principal branch, 0, since it is the bounded one, however,
+for valuesof ε >1.1 there is a precisionproblem since the
+required argument values are near to −1/e≡ −exp(−1).
+It is important to stress that in Eq. (13) the independent
+variable is the angle θand, it constitutes the parameter-
+ization of the curve Cm.
+Thepolarexpressionof Cmcanalsobewritteninterms
+of the tree function T(z) =−W(−z), giving
+˜r(θm) =1
+ε2sinθmT(ε2tanθmf(α(θm))).(14)
+We recover the drag-free result
+˜r= 2sinθm
+1+3sin2θm(15)
+whenε→0. An explanation of this unfamiliar form of
+an ellipse is given in appendix A followed by a discussion
+about the ε→0 limit of expression (13) in appendix B.
+Formula (13) exhibits the deep relationship between
+the Lambert Wfunction and the linear drag force pro-
+jectile problem, since not only the range is given as this
+function [4, 5]. The problem open the opportunity to
+study the W function in polar coordinates, that, almost3
+0 0.5 1 1.500.511.5
+θ(α), ε = 0 
+α(θ), ε = 0
+θ(α), ε = 1
+α(θ), ε = 1
+FIG. 2: (color online) The angle θas a function of the launch
+angleαfor two different values of εand their inverses.
+in the review of referencei [2], is absent. Even when it is
+possible to write the locus in terms of ym(xm) this form
+does not shows the formal elegance of relation (13).
+Now we return to equation (10) since we need to solve
+explicitly it in order to have the function α(θm). This
+task is not trivial since even when we approximate the
+rhs in expression (10) up to first order in ε,
+tanθm=tanα
+2(1+ε
+3sinα), (16)
+the inversion is not easy. A way to do the inversion is
+to expand in a Taylor series the rhs and then invert the
+series term by term.[9] Using Mathematica to perform
+this procedure up to O(18), we obtain as a result
+α(θ)≈arctan(2tan θ)−1
+3ε(2tanθ)2+2
+9ε2(2tanθ)3
++1
+54(27ε−10ε3)(2tanθ))4+···(17)
+Theε-independent terms had been resumated to yield
+arctan(2tan θ). However, the series does not converge
+for values in the argument larger than 1. The reason is
+the small convergence ratio for the Taylor expansion of
+arcsin(·).
+An easier way to perform the inversion is to evaluate
+θm(α) using Eq. (10) and plot the points ( θm(α),α), the
+result is shown in figure 2. The result is in agreement
+with the plot of Eq. (17) up to its convergence ratio
+and it is not shown. Notice that this method is exact in
+the sense that we can obtain as many pair of numbers
+as we need, a function is, finally, a relation one to one
+between two sets of real numbers. Another result is to
+obtain the derivative dα/dθ, since it shall be needed in
+the following sections. To this end, we note that both
+functionsincreasemonotonicallyandtheirderivativesare
+not zero, except at the interval end. Hence, we can use
+the inverse function theorem in order to obtain
+dα
+dθ=1
+dθ
+dα. (18)0 0.25 0.5 0.75 1 1.25 1.5
+θ0.60.811.21.41.61.82d/dθ α(θ)ε =0.1
+ε = 1.0
+ε = 10.0
+ε = 0.0
+0 0.25 0.5 0.75 1 1.25 1.5
+θ-5-4-3-2-10d2 /dθ2 α(θ)(a)
+(b)
+FIG. 3: (color online) (a) First and (b) second derivatives o f
+αas function of θfor various values of parameter ε. Note that
+major changes occur for θ < π/4.
+The result is shown in Fig. 3(a) as well as the second
+derivative in Fig. 3(b). The second derivative is calcu-
+lated using an approximation to the slope to the func-
+tion previously calculated and using 10000 points in the
+interval [0 ,π/2]. A smaller number of points could be
+considered.
+IV. THE CURVATURE OF Cm.
+A. Polar angle parameterization.
+In the drag-free situation, Cmis an ellipse and its de-
+scription is well know, however, in the presence of linear
+drag this is not the case. We do not expect that the lo-
+cus could be a conic section and henceforth we need to
+characterize it. It is usual to consider curvature, radius
+of curvature or the length of arc in order to characterize
+a locus. In the present case we consider the curvature
+ofCmin both parameterizations, first with the polar an-
+gleθmand secondly with the launch angle α. We left the
+calculus of the length ofarc to a posteriorwork, since the
+calculations became increasingly complex and the goal of
+the present section is to start the understanding of Cm
+and to illustrate the way it can be done using the Lam-
+bert W function. Here and in the rest of the section we
+drop, for clearness, the subindex min ˜rmandθm.
+The corresponding formula for the curvature Kfor po-
+lar coordinates is [10]
+K=1
+ρ˜r2+2˜r2
+θ−˜r˜rθθ
+(˜r2+ ˜r2
+θ)3/2, (19)
+inordertouseEq. (13). Herethesubindex θcorresponds
+a derivative respect to that variable.
+A direct calculation on the drag-free ˜ r(θ) of Eq.(15)4
+yields to
+K0=1
+2√
+2ρ(5−3cos2θ)6
+(1+3sin2θ)3(47−60cos2θ+21cos4 θ)3/2,
+(20)
+which have a maximum at θ= 1/2arctan(4 /3)≈0.464.
+A graph of this results appears in figure 4(red line). Note
+that this value is different from that we obtain if we eval-
+uateθ(α=π/4) = 1/2, the launch angle of maximum
+range. The maximum curvature happens at a smaller
+angle than the angle of maximun range. It is interesting
+to note that the angle 2 θ= arctan(4 /3) corresponds to
+a triangle which sides fulfill the relation 32+42= 52, a
+Pythagoras’ triple.
+Using the numerical results for α(θ) from the previous
+section, it is possible to carry on the calculation of K
+(see figure 4) performing the derivatives of ˜ rfrom Eq.
+(13) in a direct form and evaluating numerically the re-
+quired values of αand its derivatives. For the required
+derivatives of W(z) we used the expressions [2]
+d
+dxW(x) =W(x)
+x(1+W(x)), (21)
+and
+d2W(x)
+dx2=−exp(−2W(x))(W(x)+2)
+(1+W(x))3.(22)
+Using this method, we obtained good results for values of
+εup to≈1 but we require to calculate arguments of the
+Lambert Wfunction near the limit z=−1/efor larger
+values of ε. The reliability of our numerical result was
+done comparing the first and second derivatives of r(θ)
+with those corresponding to the ellipse.
+As can be seen in figure 4, Kpresent a maximum in
+all the cases which can be calculated as well. We left to
+an ulterior work the analysis of the maxima distribution
+as a function of the pertubative parameter, that is not
+the case for the curvature with αparameterization as we
+shall see in the next section.
+B. Launch angle parameterization.
+For the launch angle parameterization of Cmwe shall
+use expression [10]
+κ=x′y′′−y′x′′
+(x′2+y′2)3/2, (23)
+for calculate the curvature from the rectangular form of
+equations(7) and(8), wherethe primes denotederivative
+respect the parameterization variable, αin this case.
+A direct calculation yields
+κ=√
+2
+ρP1(α)
+(√
+P2(α))3×(1+εsinα)2.(24)0 0.5 1 1.5
+ θ020406080 ρ Κ(θ) ε = 0.0
+ ε = 0.5
+ ε = 1.0
+ ε = 1.5
+ ε = 2.0
+FIG. 4: (color online) Curvature of Cm(ε) using the polar
+angle as the parameter from Eq. (19). In red line appears the
+corresponding result for the ellipse, Eq. (20). The maximum
+happens at θ= 1/2arctan(4 /3), different from the value of
+maximum range α=π/4.
+With
+P1(α) = 16+6 ε2−8ε2cos2α+2ε2cos4α
++30εsinα−εsin3α+εsin5α,(25)
+and
+P2(α) = 5+3cos4 α+3ε2−4ε2cos2α+
+ε2cos4α+10εsinα−5εsin3α+
+εsin5α.(26)
+In the limit ε→0, we recover the drag-free curvature
+κ0=16√
+2
+ρ1
+(5+3cos4 α)3/2, (27)
+which have a maximum at α=π/4 in the interval α∈
+[0,π/2], as expected. A plot of ρκ(α) for several values
+ofεbeginning at zero and ending at ε= 10 appears in
+Figure 5(a). In red appears the drag-free case. Note that
+both extremal values increase for increasing εvalue as
+ρκ(0)≈(16+6ε−6ε2) andρκ(π/2)≈ε. Notice that for
+smallε, theκcrosses the drag-free curvature κ0.
+The angles α∗at which κattain their maxima are ob-
+tained in the usual way and requires to solve, numerical
+or graphically, the equation
+3(1+εsinα∗)sin2α∗×Q1(α∗)Q2(α∗)+
+εcosα∗Q3(α∗)Q4(α∗) = 0,(28)
+with
+Q1= 4 (3+ ε2)cos2α∗+ε(−4ε−15sinα∗+
+5sin3α∗);(29)
+Q2= 16+6 ε2−8ε2cos2α∗+2ε2cos4α∗+
+30εsinα∗−εsin3α∗+εsin5α∗;(30)5
+0 0.5 1 1.5α0204060 ρ κ(α) ε= 0
+ ε = 0.1
+ ε = 1.0
+ ε = 10.0
+0 50 100 150 200 250 300 ε00.10.20.30.40.5  α(a)
+(b)
+FIG. 5: (color online) (a) Curvature as a function of the
+launch angle αfor increasing values of parameter εaccording
+to equation (24). The values of εare indicated in the inset.
+(b) Several important angles as a function of dimensionless
+parameter εare plotted. In lines and diamonds appears the
+angle,α∗, at which the curvature is maximum. In red circles
+the angle at which the range is maximum according to exact
+solution, Eq. (34), and in blue crosses the same angle accord -
+ing to Eq. (35) (see text for discussion). In a dashed blue lin e
+the angle at which skewness is maximum is plotted.
+Q3= 5+3cos4 α∗+3ε2−4ε2cos2α∗+
+ε2cos4α∗+10εsinα∗−
+5εsin3α∗+εsin5α∗;(31)
+and
+Q4= 70+36 ε2−16(1+3ε2)cos2α∗+
+2(5+6ε2)cos4α∗+154εsinα∗−31εsin3α∗
++7εsin5α∗.(32)
+In Figure 5(b) the calculated values of α∗as a function
+ofεappear. This angle is between the optimal angle for
+maximum range (red circles and blue crosses) and the
+angle for the greatest forward skew (dashed line).[12]
+αskew= arcsin/bracketleftbigg1
+3ε/parenleftBig
+(D+/2)1/3+(D−/2)1/3−2/parenrightBig/bracketrightbigg
+,
+(33)
+whereD±=±3ε/radicalbig
+3(27ε2−32) + 27 ε2−16, valid for
+ε >/radicalbig
+32/27.[13] In Figure 5(b) the optimal angles are
+drawn, in red circles the exact result in terms of Lambert
+Wfunction[5, 11]
+αmax,S= arcsin/bracketleftBigg
+ε
+exp/parenleftbig
+W/parenleftbigε2−1
+e/parenrightbig
++1/parenrightbig
+−1/bracketrightBigg
+,(34)
+and the approximated result[4]
+αmax,W=W(ε2/e)
+ε. (35)
+Both expressions are equivalent for large εbut differ at
+smallε, as expected.0 0.05 0.1x00.050.1y
+FIG. 6: (color online) Orbits launched at different angles an d
+the corresponding geometrical place Cm(ε) forε= 80. See
+text for explanation.
+Meanwhile the difference between these anglesat small
+pertubative parameter is unimportant, at large εthe be-
+havior of the corresponding trajectories is different. One
+reason is the large asymmetry in the locus formed by the
+set of apexes. In Fig. 6(a) we plotted Cm(ε) and the
+corresponding trajectories for the different launch angles
+forε= 80. The blue line corresponds to Cm(ε), note that
+the maximum height is y∼0.12 in contrast to y∼5.0
+for the drag-free case, however, this can be the case of
+small friction parameter band large initial velocity V0
+giving a large εvalue. In a black line appears the orbit
+launched at α∗, in red line the corresponding to attain
+the maximum range and in blue dashed line the orbit
+with maximum skewness.
+V. THE SYNCHRONOUS CURVE
+In MacMillan’s book[7] the calculation of the syn-
+chronous curve was done for the drag-free case. This
+curve is formed if many projectiles were fired simulta-
+neously from the same point, each one with at different
+launch angle and same initial speed V0. The locus will be
+a circle of radius V0tand center in the point (0 ,−1
+2gt2),
+i.e.
+x2+(y+1
+2gt2)2=V2
+0t2. (36)
+Here, we shall demonstrate that a circle is the syn-
+chronous curve in the linear drag case as well. Following
+reference [7], we eliminate the launch angle αfrom the
+position solutions, in the present case they are equations
+(2) and (3). We write down cos αand sinαand rearrange6
+the terms to give,
+cosα=b
+V0x
+1−exp(−b), (37)
+sinα=b
+V0y−g
+b2(1−bt−exp(−bt))
+1−exp(−bt).(38)
+Substituting these expressions in the identity cos2α+
+sin2α= 1 we obtain
+x2+(y−yc(t))2=R2(t). (39)
+With
+yc(t) =g
+b2(1−bt−exp(−bt)), (40)
+the center and
+R(t) =V0
+b(1−exp(−bt)), (41)
+the radius. In order to recover the case where b→0,
+we consider a Taylor expansion for the exponential up to
+second order in the exponential in Eq. (40) and up to
+first order in Eq. (41). The fact that this circle exists in
+the presence of a drag force is remarkable.
+VI. CONCLUSIONS
+We obtained an explicit form for the locus Cmcom-
+posed by the set of maxima of all the trajectories of a
+projectile launched at an initial velocity /vectorV0, and in the
+presence of a linear drag force, −mb/vector v, i.e.Cmis the lo-
+cus of the apexes. In polar coordinates, Cmis written in
+terms of the principal branch of the Lambert Wfunction
+for negative values. This represents the parameterization
+of the curve by the polar angle θmonly and gives Cmin
+a closed form and exhibits the deep relationship between
+the Lambert Wfunction and the linear drag problem.
+The curvature of Cmwas calculated for different values
+of the dimensionless parameter ε≡bV0/gin two param-
+eterizations. The first one, the polar parameterization,
+shows a maximum that slightly departs from the drag-
+free case in θ= 1/2arctan(4 /3). A wider exploration of
+the functional dependence respect to εis pending due to
+numerical accuracy in the calculation of the Lambert W
+function near the limit at x=−1/e. In the case of a pa-
+rameterization using the launch angle αthere is not such
+a restriction. In this case, the curvature was calculated
+for a wide range of the parameter εyielding maximum
+at angle values larger than those corresponding to max-
+imum range. Comparison with the maximum skewness
+angle [12] was also done and the difference is larger than
+the previous one. As an addendum, we demostrate that
+the synchronous curve, in this case, is a circle as in the
+drag-free case.Acknowledgments
+This work was supported by PROMEP 2115/35621.
+HHS thanks to M. Olivares-Becerrilfor useful discussions
+and encouragement.
+Appendix A: Polar form of an Ellipse with origin at
+bottom.
+Ellipsecanonicalformorthe polarformwith theorigin
+considered in one of the focus are standard knowledge.
+In the present case, however, we require to consider the
+origin of coordinates located in the bottomof the ellipse,
+since, in the presenceofa dragforce, the launching origin
+is the only invariant point when we change the drag force
+value. To obtain the ellipse form, we depart from the
+drag-free solutions at the locus of the apexes,
+xm=ρsinαcosα, (A1)
+and
+ym=ρ
+2sin2α, (A2)
+beingρ≡V2
+0
+g. With the help of the trigonometric re-
+lations sin2 α= 2sinαcosαand 2sin2= 1−cos2αwe
+transform the upper equations into
+sin2α=2xm
+ρ, (A3)
+cos2α= 1−4ym
+ρ. (A4)
+Taking the squares in both expressions, summing them
+and arranging terms, we arrive to
+4rm
+ρ/parenleftbiggrm
+ρ(1+3sin2θm)−2sinθm/parenrightbigg
+= 0.(A5)
+Where we used the polar coordinates xm=rmcosθm
+andym=rmsinθm. The solutions are rm= 0 and
+rm(θm) =2ρsinθm
+1+3sin2θm. (A6)
+The second one is the required form for the ellipse.
+Appendix B: Drag-free limit for ˜r
+In order to obtain the drag-free limit for the locus Cm
+given in Equation (13), we note that
+tanθ= (1/2)tanα (B1)
+and that f(α) = sinα(θm)cosα(θm). The first expres-
+sion is obtainable from the drag-free solutions Eqs. (A1)
+and (A2), and the second is obtained by setting b→0 in
+Eq. (11).7
+The expansion in a power series of the Lambert W
+functionuptofirstorderisjusttheidentity[2]andhence,
+˜r(θm) =−1
+ε2sinθmW(−ε2tanθmf(α))
+≈ −1
+ε2sinθm(−ε2tanθmf(α))
+.=sinαcosα
+cosθm.
+= 2sinθcos2α
+cos2θm.(B2)
+Where we used relation (B1) in order to obtain the lastline. Using trigonometricidentity sec2α−tan2α= 1and
+Eq. (B1) we obtain that
+cos2θm
+cos2α= 1+3sin2θm. (B3)
+Using this result in the expression of ˜ rwe obtain the
+desired result, Eq. (15).
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+culus. 11th Ed. (Addisson-Wesley, 2004).
+[9] G.B. Arfken. Mathematical Methods for physicist . 5thEd.(Academic Press, 2000).
+[10] E. Kreyzig. “Principal, Normal, Osculating Cir-
+cle” in Diffential Geometry . (Dover,N.Y., 1991).
+E.W. Wiesstein “Curvature” From MathWorld.
+http://mathworld.wolfram.com/Curvature.html .
+[11] S.M. Steward, “A little introductory and intermediate
+physics with the Lambert function”, Proc. of the 16th
+Biennial Congress of the Australian Institute of Physics.
+M. Colla ed. Vol 2. pp 194-197. Australian Institute of
+Physics. Parville, VIC (2005).
+[12] S.M. Steward, “Characteristics of the trajectory of a p ro-
+jectile in a linear resisting medium and the Lambert
+Wfunction”, Australian Inst. of Physics. 17th. National
+Congress 2006. Paper No. WC0035.(2006).
+[13] In Ref. 12, the author comment that for ε= 1 we
+obtain the special value αskew= sin−1(1/φ) withφ=
+(1+√
+5)/2, the golden ratio. However, the solution that
+appears in the article is not longer valid for ε= 1. If the
+solution corresponds to another real root of the equation
+this special value corresponds to the case when the initial
+speed is equal to the limit speed b/g. This makes much
+more intriguing this fact.
\ No newline at end of file
diff --git a/1001.0603.txt b/1001.0603.txt
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index 0000000000000000000000000000000000000000..841b253276c6b03d667455cd2e4f0f40f252cfca
--- /dev/null
+++ b/1001.0603.txt
@@ -0,0 +1,296 @@
+arXiv:1001.0603v1  [cond-mat.str-el]  4 Jan 2010Neutron spectra of Herbertsmithite Materials: Observatio n of a Valence Bond Liquid
+phase?
+R. R. P. Singh
+University of California Davis, CA 95616, USA
+(Dated: November 19, 2018)
+We argue that neutron spectra in a short-range ordered Valen ce Bond state is dominated by two-
+spinon states localized in small spatial regions such as a pi nwheel. These excitations lead to angle
+averaged dynamic structure factor that is spread over a wide frequency range up to about 2 .5J,
+whereas its wavevector dependence at all frequencies remai ns very close to that of isolated dimers.
+These results are in excellent agreement with recent Neutro n scattering data in the Herbertsmithite
+materials ZnCu 3(OH)6Cl2.
+PACS numbers:
+Recent experimental studies[1–6] of the Herbert-
+smithite material ZnCu 3(OH)6Cl2with structurally per-
+fect Kagome planes has brought renewed interest in the
+study of Quantum Spin-Liquid phases in the Kagome
+Lattice Heisenberg Model.[7–12] The issues of Quantum
+Spin-Liquid versus Valence Bond Crystal Order,[13] of
+a possible gap in the spin excitation spectra and of de-
+confinement of fractional spin excitations continue to be
+subjects of intense theoretical debate.[14–22] The exper-
+imental studies provide a complimentary perspective to
+this long standing problem.
+In a recent letter de Vries et al presented neutron scat-
+tering data on these materials overa rangeof momentum
+and frequency transfers.[23] They discuss their data pri-
+marily in the context of Algebraic Spin-Liquid and other
+theories.[11, 12] Here, we would like to argue that the
+data is much better understood in terms of a Valence
+Bond phase[13, 16] with well developed short-ranged Va-
+lence Bond order but no long-range Valence Bond Crys-
+tal order. Such a finite temperature phase, lacking long-
+range quantum coherence, may appropriately be called a
+classical Valence Bond Liquid.
+The powder-diffraction neutron data of de Vries et al
+covers a temperature range from 2K to 120 K, an energy
+transfer of up to 30 meV and the full range of momen-
+tum transfer values. Their key findings can be summa-
+rized as follows: After allowance is made for phonons as
+well as for some impurity spins, the magnetic behavior
+intrinsic to the system, is rather well described by an
+angle averaged q-dependent form-factor which is essen-
+tially identical to that of a single spin-dimer. However,
+unlike a single dimer, where such a spectrum would be
+strongly peaked at the singlet-triplet energy gap ( Jfor
+an isolated dimer), the spectral weight is spread nearly
+uniformly overawide rangeofenergiesextending at least
+down to 4meV and up beyond 30meV. Furthermore, this
+behavior is evident at 2K but persists also at 120 K. The
+frequency dependence makes the behavior clearly incon-
+sistent with that of isolated dimers and the temperature
+dependence makes it inconsistent with a Valence Bond
+Crystal order (and any other quantum ground state withFIG. 1: Proposed Valence Bond Phase of the Kagome Lattice
+HeisenbergModel consists of aHoneycomb Lattice of resonat -
+ing hexagons (H), where six hexagons surround a pinwheel
+(P). The pinwheels are isolated from empty triangles leadin g
+to substantially reduced quantum fluctuations on their dime r
+bonds. Triplets on dimers represented by black thick lines
+are heavy nearly immobile particles, whereas triplets on gr ey
+thick lines represent particles mobile throughout the latt ice.
+an ordering scale much smaller than 100K as expected in
+the model).
+We first note that the exchange constant for the ma-
+terial has been estimated[2, 24, 25] to be in the range
+170K-190K, which translates to about 15meV. There is
+increasing theoretical support for a Valence Bond Crys-
+tal (VBC) ground state of the Kagome-Lattice Heisen-
+berg Model with a 36-site unit cell.[16, 20] One of the
+distinctive features of the VBC state is a pin-wheel (See
+Fig.1)ineachunit cell, adefectfreestructureofValence-
+Bond containing triangles, where the bonds remain al-
+most completely dimerized. An isolated pinwheel is an
+example of a Delta chain,[21, 26, 27] a system of corner
+sharingtriangles, where the Hamiltonian is minimized by2
+FIG. 2: Kink-antikink or 2-spinon states for the pinwheel. A
+triplet excitation is created by breaking a singlet bond in t he
+ground state. The Kink spinon lies on the inner hexagon,
+while the more mobile antikink spinon can move on the out-
+side vertices of the pinwheel through states (a) through (f)
+shown in the figure. When the two spinons are together as in
+Fig. (a), the kink spin can also move by going through higher
+energy intermediate states.
+dimerization.[28] In the VBC state, their is significant
+quantum fluctuations. In particular, the dimerization in-
+side the resonating hexagons is strongly reduced. But,
+the pinwheel region is geometrically protected against
+quantum fluctuations and hence remains essentially fully
+dimerized.[29]
+Different Valence Bond phases have energy difference
+of only 0 .001 J per site.[16] Thus one expects any phase
+transition to such a state to occur at a very low tem-
+perature. However, short-range Valence Bond order is
+set byJand can develop at significantly higher tem-
+peratures. Once this short-range order is a few lattice
+constants, pin-wheel like structures should begin to form
+locally. Because they have extremely low local energies,
+they should be stable up to higher temperatures.
+The triplet excitations of the pinwheel are kink-
+antikink pairs.[21, 26, 27] In an infinite Delta chain, the
+kinks are immobile, where as the antikink can hop from
+one triangle to another. The kink has zero excitation
+energy, whereas the energy of the antikink can be ap-
+proximated by
+ǫ(k) = 5/4−cosk. (1)
+Note that this means that excitations extend over the
+energy range J/4< ǫ <9J/4. For the infinite sys-
+tem more detailed analytical and numerical calculations
+show[26, 27] that the lowest energy antikink is roughly at
+0.219J, whereas the upper energy may extend up to as
+much as 3 J. For the finite system, spin excitation from
+the ground state creates a pair of parallel spins on one of
+the singlet bonds of the ground state. While the kink an-
+tikink description is roughly valid, because both the kink
+and antikink remain in each other’s vicinity, both free0 1 2 3 4
+Omega00.511.522.53Structure FactorPinwheel Spectra 
+Different qs, Broadening=0.01
+FIG. 3: Angle averaged dynamic structure factor versus fre-
+quency of the Pinwheel state at T= 0 with small broadening.
+0 1 2 3 4
+omega00.10.20.30.40.5Structure FactorPinwheel Spectra
+Different qs, Broadening=0.1
+FIG. 4: Angle averaged dynamic structure factor versus fre-
+quencyof the Pinwheel state at T= 0 with larger broadening.
+spins or spinons become mobile. Fig. 2, shows the con-
+figurations that correspond to the mobile antikink. On
+the other hand the kink can also move by going through
+higher energy intermediate configuration, when the two
+spinsarenext toeachother. In the Sz= 1sector, assum-
+ingtwofreespinsandtherestofthesystemintheground
+stateleads to66states, ofwhich 36states areofthe kink-
+antikink type, where one spin is in the inner hexagon,
+whereas the other spin is on the outside vertices of the
+pinwheel. One expects the spectral weight to be primar-
+ily spread over these states, giving rise to spectral weight
+spread roughly over the energy range J/4< ǫ <9J/4.3
+0 5 10
+Wavenumber00.51Structure Factor
+Single-Dimer Structure FactorQ-dependent structure factor for all frequencies
+Compared with single-dimer
+FIG. 5: Angle averaged dynamic structure factor at differ-
+ent frequencies, scaled to have a maximum of unity, versus
+wavenumber compared with results of a single dimer.
+This can be easily confirmed by exact diagonalization
+of the 12-site Heisenberg model on a pinwheel. One finds
+that the lowest triplet state has an excitation energy of
+0.260J. The spectral weight is spread over a large num-
+ber of states. The highest spectral weight of any one
+single state is only about 5 percent. The states with the
+highest 36 spectral weights are spread over the energy
+range 0.260J < ǫ <2.039Jand they contribute above 80
+percent to the spectral weight. If we look at total spec-
+tralweightup tosomeenergy, roughly95 .5percentofthe
+weight extends up to an energy of 2 .25 J, 97.7 percent of
+the weight extends up to an energy of 2 .5 J and roughly
+99.7 percent of the weight extends up to an energy of
+3 J. This strongly confirms that the dominant spectral
+contributions come from the kink-antikink states.
+In the Valence Bond Crystal state, these triplets can
+ultimately decay to still lower lying light triplets,[29]
+though those decay times are likely to be very long, be-
+cause the pinwheels are surrounded by dimerized trian-
+glesandhavenoemptytrianglesintheirimmediatevicin-
+ity, which strongly reduces quantum fluctuations. In the
+liquid phase, these excitations should have a shorter fi-
+nite lifetime, which one could represent by a Lorentzian
+broadening. The frequency dependence of the angle av-
+eraged dynamic structure factor, for several q-values, at
+a small Lorentzian broadening (0.01 J) is shown in Fig
+3, where as at a larger broadening (0.10 J) it is shown
+in Fig. 4. The latter may be more representative of the
+liquid state. One finds that at higher broadening one has
+a spectral weight that is spread roughly continuously up
+to an energy of about 3 J.
+Foranypairofspinsatadistance r, the angleaveraged
+value of exp( i/vector q·/vector r) is given by sin( qr)/qr. This can beused to calculate the angle averaged dynamic structure
+factor for any given q. For all frequencies, the struc-
+ture factor as a function of wavenumber shows depen-
+dence, which is very close to that of an isolated dimer
+(See Fig. 5). In an isolated pinwheel, the equal-time
+correlation function is strictly that of a dimer, but ex-
+citations are extended over the full pin-wheel. Thus the
+energy integrated structure factor is strictly that of an
+isolated dimer, but not the spectral weight at a given en-
+ergy. However, what the calculations show is that while
+the spectral weight is spread out over a wide range of
+frequencies, the q-dependence is always near that of an
+isolated dimer. Since delta chains are likely to be ubiq-
+uitous in the short-range Valence Bond ordered phase of
+the Kagome Lattice Heisenberg model,[21] this spectral
+feature may persist up to energy comparable to J, that
+is, as long as the system has short range Valence Bond
+order.
+This picture implies that the equal-time spin-spin cor-
+relations in the system are essentially only nearest neigh-
+bor. However, the arrangement of dimerized triangles
+makes any triplet excitation break into a kink-antikink
+pairs. These lead to spectral weight spread over a wide
+frequency range.
+In fact, the spectra obtained by exact-diagonalization
+of24, 30and36-siteclustersbyLaeuchliandLhuillier[19]
+show very similar frequency dependence, where much of
+the spectral weight is nearly uniformly spread between
+J/4 to about 2 .5J. The finite size system has only short-
+range Valence bond order. This is further evidence that
+almost all states with short range Valence Bond order
+have these features.
+One also knows that the Herbertsmithite materi-
+als have a small Dzyaloshinski-Moria anisotropy.[30–32]
+These anisotropies would strongly influence the low en-
+ergy spectra and the nature of long-range order without
+significantly altering the high energy spectral properties.
+In the experiments, the reduction in spectral weight at
+low temperatures below an energy of 4 meV may well be
+related to the DM anisotropy which would cause the low
+energy spectral weight to move to even lower energies.
+Furthermore, the true signature of long-range VBC
+order, would be the observation of low energy light
+triplets, which live on the perfect hexagon and bridging
+dimers.[17, 29] In the VBC phase these provide an ex-
+tended network for the triplets to move around, whereas
+the pinwheels form isolated pristine regions which are
+nearly protected from quantum fluctuations, and have
+only localized triplets. The honeycomb VBC state has
+50 percent of the spectral weight in the non-fluctuating
+dimer-like heavy excitations and the rest of the 50 per-
+cent of the spectral weight in the light mobile triplets,
+which with long-range VBC order have an energy gap of
+orderJ/20.[14, 15]
+In conclusion, we have argued that the observation of
+dimer-likeq-dependence combined with aspectral weight4
+spread over a wide frequency range and remaining nearly
+temperature independent over a wide range of temper-
+ature is strongly suggestive of a Valence Bond Liquid
+phase, with short-range Valence Bond Order exceeding a
+couple of lattice constants. These excitations can be re-
+garded as a kink-antikink pair or two spinons, which are
+confined in a very small spatial region. They extend up
+to fairly high temperatures of order J, where such excita-
+tions are necessarily broadened by the underlying classi-
+cal liquid environment. Whether at low enough tempera-
+tures, the development of long-range quantum coherence
+leads to truly delocalized spinons as in algebraic quan-
+tum spin-liquid phases or to the development of much
+sharper localized and extended triplet excitations as in a
+Valence Bond Crystal remains to be seen.
+We would like to thank Mark de Vries for valuable
+discussions.
+[1] M. P. Shores et al J. Am. Chem. Soc. 127, 13462 (2005).
+[2] J. S. Helton et al, Phys. Rev. Lett. 98, 107204 (2007).
+[3] S. H. Lee et al, Nat. Mat. 6, 853 (2007).
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+[5] T. Imai et al, Phys. Rev. Lett. 100, 077203 (2008).
+[6] M. A. De Vries et al, Phys. Rev.Lett. 100, 157205 (2008).
+[7] P. W. Anderson, Mater. Res. Bull. 8, 153 (1973).
+[8] C. Zeng and V. Elser, Phys. Rev. B 51, 8318 (1995).
+[9] F. Mila, Phys. Rev. Lett. 81, 2356 (1998).
+[10] G.Misguich, D.SerbanandV.Pasquier, Phys.Rev.Lett.
+89, 137202 (2002).
+[11] Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen, Phys.
+Rev. Lett. 98, 117205 (2007).
+[12] M. Hermele, T. Senthil, and M. P. A. Fisher, Phys. Rev.B 72, 104404 (2005).
+[13] J. B. Marston and C. Zeng, J. Appl. Phys. 69, 5962
+(1991); A. V. Syromyatnikov and S. V. Maleyev, Phys.
+Rev.B66, 132408 (2002); P.Nikolic andT. Senthil, Phys.
+Rev. B68, 214415 (2003); R. Budnik and A. Auerbach,
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+Sindzingre and C. Lhuillier, EPL 88, 27009 (2009).
+[15] H. C. Jiang et al, Phys. Rev. Lett. 101, 117203 (2008).
+[16] R. R. P. Singh and D. A. Huse, Phys. Rev. B 76, 180407
+(2007).
+[17] B.-J. Yang, Y. B. Kim, J. Yu and K. Park, Phys. Rev. B
+77, 224424 (2008).
+[18] G. Misguich and P. Sindzingre, J. Phys.-Cond. Matt. 19,
+145202 (2007).
+[19] A. Laeuchli, C. Lhuillier, arXiv:0901.1065.
+[20] G. Evenbly, G. Vidal, arXiv:0904.3383.
+[21] Z. Zhao and O. Tchernyshyov, Phys. Rev. Lett. 103,
+187203 (2009).
+[22] D. Poilblanc, M. Mambrini, D. Schwandt,
+arXiv:0912.0724.
+[23] M. A. de Vries et al Phys. Rev. Lett. 103, 237201 (2009).
+[24] M. Rigol and R. R. P. Singh, Phys. Rev. Lett. 98, 207204
+(2007).
+[25] G. Misguich and P. Sindzingre, Eur. Phys. J. B 59, 305
+(2007).
+[26] T. NakamuraandK.Kubo, Phys.Rev.B53, 6393(1996).
+[27] D. Sen et al., Phys. Rev. B 53, 6401 (1996).
+[28] C. K. Majumdar, J. Phys. C 3, 911 (1970).
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+(2007).
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\ No newline at end of file
diff --git a/1001.0604.txt b/1001.0604.txt
new file mode 100644
index 0000000000000000000000000000000000000000..20d8db29475dac8ea672a21398c962b4fdeb5816
--- /dev/null
+++ b/1001.0604.txt
@@ -0,0 +1,641 @@
+arXiv:1001.0604v1  [astro-ph.EP]  4 Jan 2010Kepler-4b: Hot Neptune-Like Planet of a G0 Star Near
+Main-Sequence Turnoff†
+William J. Borucki,1David G. Koch,1Timothy M. Brown,2Gibor Basri,3Natalie M.
+Batalha,4Douglas A. Caldwell,1William D. Cochran,5Edward W. Dunham,7Thomas N.
+Gautier III,9John C. Geary,8Ronald L. Gilliland,12Steve B. Howell,11Jon M. Jenkins,6,1
+David W. Latham,8Jack J. Lissauer,1Geoffrey W. Marcy,3David Monet,10Jason F. Rowe,
+1and Dimitar Sasselov8
+ABSTRACT
+Early time-series photometry from NASA’s Keplerspacecraft has revealed a planet transiting
+the star we term Kepler-4, at RA= 19h02m27s.68,δ= +50◦08′08′′.7. The planet has an orbital
+period of 3 .213 days and shows transits with a relative depth of 0 .87×10−3and a duration of
+about3.95hours. Radialvelocitymeasurementsfromthe KeckHIRESspec trographshowareflex
+Doppler signal of 9 .3+1.1
+−1.9ms−1, consistent with a low-eccentricity orbit with the phase expected
+from the transits. Various tests show no evidence for any compan ion star near enough to affect
+the light curve or the radial velocities for this system. From a trans it-based estimate of the host
+star’s mean density, combined with analysis of high-resolution spect ra, we infer that the host star
+is near turnoff from the main sequence, with estimated mass and rad ius of 1.223+0.053
+−0.091M⊙and
+1.487+0.071
+−0.084R⊙. We estimate the planet mass and radius to be {MP,RP}={24.5±3.8M⊕,
+3.99±0.21R⊕}. The planet’s density is near 1 .9 gcm−3; it is thus slightly denser and more
+massive than Neptune, but about the same size.
+Subject headings: planetary systems — stars: fundamental parameters — stars: individual (Kepler-4,
+KIC 11853905, 2MASS 19022767+5008087)
+†Some of the data presented herein were obtained at the
+W.M. Keck Observatory, which is operated as a scientific
+partnership among the California Institute of Technology,
+the University of California and the National Aeronautics
+and Space Administration. The Observatory was made
+possible by the generous financial support of the W.M.
+Keck Foundation.
+1NASA/Ames Research Center, Moffett Field, CA
+94035
+2Las Cumbres Observatory Global Telescope, Goleta,
+CA, 93117
+3University of California-Berkeley, Berkeley, CA 94720
+4San Jose State University, San Jose, CA 95192
+5University of Texas at Austin, Austin, TX 78712
+6SETI Institute, Mountain View, CA 94043
+7Lowell Observatory, Flagstaff, AZ 86001
+8Harvard-Smithsonian Center for Astrophysics, Cam-
+bridge, MA 02138
+9Jet Propulsion Laboratory/California Institute of
+Technology, Pasadena, CA 91109
+10United States Naval Observatory, Flagstaff, AZ 860021. Introduction
+Transiting extrasolar planets provide unpar-
+alleled opportunities for detailed study of the
+physical characteristics of distant solar systems.
+Since the first transiting planet detection a decade
+ago, ground-based surveys such as TrES, HAT,
+andSuper-WASP(Alonso et al.2004;Bakos et al.
+2004; Pollacco et al. 2006) and the spaceborne
+telescope CoRoT (Baglin et al. 2007) have lo-
+cated more than 50 transiting planets, spanning a
+large range in size and mass. With the advent of
+NASA’s Kepler Mission , we have a new and ex-
+traordinarily sensitive tool for studying transiting
+11National Optical Astronomy Observatories, Tucson,
+AZ 85726
+12Space Telescope Science Institute, Baltimore, MD
+21218
+1planets. Keplerhas enough sensitivity to detect
+Earth-size planets orbiting in the habitable zones
+of Sun-like stars during its planned 3.5-year mis-
+sion (Koch et al. 2010; Borucki et al. 2010). The
+first science data to return after Kepler’s launch
+in Mar 2009 were time series from a 9.7-day com-
+missioning run, followed after a 1.6-day gap by a
+33.5-day science run. Here we describe the tran-
+siting planet Kepler-4b, one of several transiting
+planets discoveredduring these first two observing
+intervals.
+2. Observations, Analysis, Tests for False
+Positives
+Observations of the Keplertarget field com-
+menced 1 May 2009; the data that we describe
+here are long cadence (LC) photometry, which
+correspond to integration times of 29.426 minutes.
+For a full description of the Keplerfield of view,
+observing modes, and data processing pipeline,
+see Jenkins et al. (2010); Caldwell et al. (2010).
+ForKeplertarget stars brighter than r= 13, the
+RMS photometric precision attained for relative
+flux time series is typically better than 2 ×10−4
+per 29-minute integration (Gilliland et al. 2010).
+We detrended photometry from the mission data
+reduction pipeline and searched it for significant
+transit-like events using the procedures described
+by Jenkins et al. (2010) and by Batalha et al.
+(2010).
+One of the transiting planet candidates iden-
+tified by the process just described was the star
+we now term Kepler-4. This star is uncommonly
+bright by Keplerstandards. Its Kepler magnitude
+(AB magnitude averaged over the Keplerband-
+pass) is Kepmag = 12 .211. Characteristics of this
+star are given in Table 1; briefly, it is a somewhat
+metal-rich ([ Fe/H] = +0.17±0.06) star of nearly
+solar temperature (5857 ±120 K), evidently seen
+near the end of its main-sequence lifetime, as ex-
+plained below.
+Figure 1 shows the light curve for Kepler-4,
+folded with a period of 3 .21346 d.1Since the
+transit signal was clear, we proceeded with follow-
+up observations as described in Gautier et al.
+1Time series of the photometry and of ra-
+dial velocity data presented here may be re-
+trieved from the MAST/HLSP data archive at
+http://archive.stsci.edu/prepds/kepler hlsp.Fig. 1.— The phased light curve of Kepler-4b
+containing13 transits observedby the Keplerpho-
+tometer between 1 May 2009 and 15 Jun 2009.
+The upper panel shows the full 44-day time se-
+ries after detrending. The bottom panel shows the
+light curve folded with the orbital period; differ-
+ent plot symbols denote odd- and even-numbered
+transits. The lower curve shows transit data over-
+plotted on the fitted transit model. The upper
+curvecoverstheexpectedtimeofoccultation, with
+the fitted (essentially constant) model overplot-
+ted. This model assumes a circular orbit. The full
+folded light curve (not shown here) gives no evi-
+dence for an occultation at any phase. For transit
+and orbital parameters, see Table 1.
+2(2010); Batalha et al. (2010). Experience with
+both ground-andspace-basedtransit observations
+shows that a fairly large fraction of events that re-
+semble planetary transits are actually caused by
+eclipses involvingonly stars, or eclipses with prop-
+erties that are significantly confused by diluting
+light from one or more stellar companions, either
+physicalorprojected. Forthisreason, wedesigned
+our follow-up observations to determine whether
+the light curve dips can be ascribed to a transiting
+planet; if not, what eclipsing star or other process
+might be responsible for them; and if so, what the
+properties of the host star and its planet might
+be.
+Ground-based visible-light speckle imaging
+from the WIYN Telescope, NIR adaptive-optics
+imaging from the Mt. Palomar 5m telescope, and
+also with the NIRC2 camera on the Keck tele-
+scope all show Kepler-4 as an isolated star, apart
+from a neighborthat is 3.5magnitudes fainter at a
+distance of 11.9′′. We estimate that this star con-
+tributes2 ±2%tothefluxwemeasureforKepler-4,
+which dilutes the transit by a factor of 1.02 ±0.02.
+We take this dilution and its uncertainty into ac-
+count in computing the the planetary radius RP
+and its probable errors. Limits on nearby com-
+panions are described by Batalha et al. (2010);
+the imagery rules out background eclipsing bi-
+naries that might simulate the observed transits,
+up to a magnitude difference of 9.8 in H band,
+for companions between 0.12 and 3.0 arcsec from
+Kepler-4. We have also searched for motion of the
+image centroid that is correlatedwith the transits,
+finding no such motion with a limit of 4 ×10−4
+arcsec.
+Two reconnaissance spectra obtained with the
+TRES spectrograph on the 1.5-m Tillinghast Re-
+flector at the Whipple Observatory showed a ve-
+locity variationof lessthan 150m/s overfive days.
+Accordingly we obtained radial velocity measure-
+ments with the HIRES spectrograph on the Keck
+I telescope (Vogt et al. 1994). Figure 2 shows the
+radial velocity data for Kepler-4, folded with the
+transit period and shifted so that transit center
+occurs at zero phase. Assuming zero eccentricity,
+we obtained a radial velocity variation with phase
+that is consistent with the observed light curve,
+with areflexvelocityamplitude Kof9.3+1.1
+−1.9ms−1
+and velocity residuals of only 3.6 ms−1. We also
+performed a fit in which we allowed the eccentric-ityeto float. This gave e= 0.22±0.08 andK=
+10.0 ms−1; the reduced χ2improved only slightly.
+Finally, a Monte Carlo bootstrap (described be-
+low) that simultaneously fits all photometry and
+RV values, and that is consistent with stellar evo-
+lution models, gives esinω=−0.003±0.058 and
+ecosω=−0.005±0.057. There is thus only
+marginal evidence for a noncircular orbit. In the
+analysisbelow, we dohoweveraccountthroughout
+forthepossiblerangeof eandωinourestimatesof
+stellar and planetary parameters and their uncer-
+tainties. Given the integration times of the spec-
+tra and Kepler-4’s brightness, Teff, and slow ro-
+tation (vsini= 2.2±1.0 kms−1), the residuals
+are consistent with expectations. We also used
+the HIRES spectra to search for variations in the
+shapes of line bisectors; the component of the bi-
+sectorspan that is in phase with the orbitalperiod
+has amplitude −1.2±4.2 ms−1(with uncertainty
+estimated from the bisector scatter), offering no
+support for the presence of a blended eclipsing bi-
+nary star.
+We can reject many photometric blend scenar-
+ios, but not all. Kepler-4b’s long transit duration
+rules out hierarchical triple systems in which the
+primary of the eclipsing pair contributes less than
+about 25% of the system luminosity. Near-twin
+star systems permitted by this constraint would
+lead to errors in the planet radius and mass by
+factors of a few – these would be serious, but not
+large enough that transits by stars could be con-
+fused with those by planets. Moreover, virtually
+all transiting planet parameter estimates are vul-
+nerable to confusion of this sort. An unresolved
+background eclipsing binary star might accurately
+simulate the transit light curve, but such a sys-
+tem would not likely produce the very small ob-
+served RV variation, nor the small line bisector
+variation. The most plausible remaining blend
+scenario involves an unresolved background tran-
+siting Jupiter-size planet. Such a blend could be
+consistent with all current observations, though
+the required coincidence in apparent position is
+very unlikely a priori.
+Thus, although some kinds of confusing pho-
+tometric blends are possible, there is no evidence
+for them. In what follows, we assume that the ob-
+served light curve results from transits by a rather
+smallextrasolarplanetacrossthe face ofanormal,
+single, Sun-like star.
+3      -1001020Radial velocity (m s-1)
+       -25025 Residual (m s-1)
+0.0 0.2 0.4 0.6 0.8 1.0
+Phase -25025 Bisector
+var. (m s-1)
+Fig. 2.— Top panel: The phased radial velocity
+curveofKepler-4,consistingof19epochsobserved
+using the Keck/HIRES spectrometer, spanning 69
+days. The dashed overplotted curve is a fit as-
+suming a circular orbit, phased to match the tran-
+sit photometry. The solid curve shows the best-
+fit eccentric orbit, with e= 0.22. Middle panel:
+O-C residuals of velocities relative to the circular
+orbit fit. Bottom panel: Bisector span for each
+epoch, measured between the 40% and 90% depth
+points in the cross-correlation against a represen-
+tative spectrum.3. Properties of the Planet and Host Star
+3.1. Method for Estimating Host Star
+Masses and Radii
+TheKeplerphotometer produces a light curve
+that implies the planetary radius in units of the
+host star radius. Likewise, groundbased followup
+Dopplerspectroscopyyields the massofthe planet
+in terms of the host star’s mass. Thus, having
+accurate values for the host star’s fundamental
+parameters – mass, radius, effective temperature,
+and composition – will be essential to carryingout
+Kepler’s mission to characterize planets circling
+distant stars.
+But it is also possible to reverse this logical
+flow, and use transiting planets as probes of their
+host stars. Seager & Mall´ en-Ornelas (2003) de-
+rived expressions relating the light curve to the
+mean density ρ∗of a transiting planet’s host star.
+They showed that ρ∗can be expressed in terms
+of the planet’s orbital period, the fractional flux
+obstructed by the planet, and two different mea-
+sures of the transit duration. All of these quanti-
+ties are directly measurable from the light curve,
+and their interpretation depends only on Newto-
+nian mechanics and geometry.
+Themethodweuse(henceforth,the ρ∗method)
+for estimating stellar radii and other parame-
+ters is an implementation of that described by
+Sozzetti et al. (2007) and subsequently used by,
+e.g., Bakos et al. (2007); Winn et al. (2007) and
+Charbonneau et al. (2007). This technique has
+become the most trusted way of obtaining infor-
+mation about transiting planets’ host stars, as il-
+lustrated by Torres et al. (2008).
+The basis for the ρ∗method is that, although
+stellar models are ordinarily taken to depend
+on 5 parameters {mass, age, metallicity, ini-
+tial helium abundance, and mixing length }=
+{M∗,A∗,[Z],Y0,α}, the last two of these do not
+vary much among stars, and for practical pur-
+poses are often taken as known. In particular, the
+much-used Yonsei-Yale (henceforth YY) model
+grid (Yi et al. 2001) uses specified values for the
+mixing length and for the initial helium abun-
+dance. In this approximation, stars are described
+by a 3-dimensional grid of models, parameterized
+by mass, age, and metallicity. Every transiting
+planet discovered by Keplerand followed up with
+4reconnaisance spectroscopy has available a transit
+light curve that constrains ρ∗, and also spectro-
+graphic estimates of Teffand [Z]. Moreover, for
+main-sequence stars, the problem of inferring stel-
+lar structure parameters from the 3 observables
+{ρ∗, Teff,[Z]}is usually well-posed, allowing pre-
+cise conclusions without important degeneracies.
+Thus, the quantities that we may readily observe
+usually suffice to isolate a single set of model pa-
+rameters {mass, age, metallicity }; from these, we
+may compute any other global property (e.g. ra-
+dius, luminosity, or surface gravity).
+We implement the ρ∗method by searching a
+precomputedgridofmodels(interpolatedfromthe
+YY models) to find the one that best matches the
+observationsina χ2sense. Theoptimizationprob-
+lem involves interpolating within the given model
+gridand performing aconventional(eg “amoeba”)
+hill-climbing optimization search. Estimating er-
+rorsand verifyingthat the searchhasconvergedto
+the global optimum is done using a Markov Chain
+Monte Carlo (MCMC) process. To allow for pos-
+sible systematic errors in the spectrographic anal-
+ysis, we took uncertainties in [ Z] to be the larger
+of 0.06 dex and the quoted value, and twice the
+quoted value for Teff. Forρ∗, we used a param-
+eterization of the (often highly asymmetric) er-
+ror distribution estimated from a jack-knife anal-
+ysis of the photometry. Not all parameter choices
+{M⋆,A∗,[Z]}permitted by a given ρ∗are consis-
+tent with the photometry, and vice versa. More-
+over, in the light curve analysis, the estimated
+ρ∗depends somewhat on the initial guess for M⋆
+(Koch et al. 2010) and on the range of orbital ec-
+centricity allowed by the observations. To account
+for the different information supplied by these two
+kinds of model fitting, we iterated the solution by
+putting the MCMC probability distributions back
+into the light curve analysis. The most likely pa-
+rameter values and uncertainty distributions we
+report below are the result of this once-iterated
+fit.
+Brown (2009) has tested the ρ∗method against
+a sample of169stars(mostly members of eclipsing
+binariestakenfromthecompilationbyTorres et al.
+(2009)) that have accurately known masses and
+radii. This comparison shows that the method’s
+systematic errors do not exceed 2-3% in radius,
+and about 6-9% in mass. Random errors ap-
+pear to arise about equally from uncertaintiesin the estimated stellar Teffand metallicity, and
+from uncertainties in the masses and radii in-
+ferred for the eclipsing binary components using
+traditional (but largely model-independent) tech-
+niques. Systematic errors arise mostly because
+rapidly-rotating and hence magnetically active
+stars are seen to have larger radii than slowly-
+rotating stars with otherwise similar properties
+(eg, Torres et al. (2006); L´ opez-Morales (2007);
+Morales et al. (2008)). The sample of eclipsing
+binary stars consists almost entirely of such fast
+rotators; this causesan excessof up to severalper-
+cent in the actual radii of stars of sub-solar mass,
+relative to the YY models. On the other hand, the
+host starsof confirmed transitingplanets found by
+Keplerare mostly slow rotators (if only because it
+is difficult to measure precise radial velocities of
+rapidly-rotating stars). Errors in these estimates
+should therefore be dominated by measurement
+errors, especially in [ Z] and inρ∗.
+3.2. The Star Kepler-4
+Simultaneous fits to the transits of Kepler-4b
+andto the RVdata(allowingnonzeroeccentricity)
+yieldtheratiooforbitalsemimajoraxis atostellar
+radiusR∗, namely a/R⋆= 6.47+0.26
+−0.28, which with
+the orbital period gives ρ∗= 0.50±0.16 gcm−3.
+This is a low value, roughly 1/3 that of the Sun,
+suggesting that Kepler-4 must have evolved con-
+siderably away from the zero-age main sequence.
+Applying the ρ∗method confirms this conclusion,
+and in fact finds a range of model parameters, all
+of which fit the observations almost equally well,
+but which imply distinct evolutionary states for
+the star. Figure 3 shows evolution tracks for three
+models having masses between 1.13 M⊙and 1.28
+M⊙. In this mass range, old main-sequence stars
+have small convective cores. Because of efficient
+mixing, these cores suffer hydrogen exhaustion all
+at once, following which the entire star must com-
+press and heat until the central temperature rises
+enough to start shell hydrogen burning. The blue-
+going hook at main-sequence turnoff is a result of
+this compression. In the case of Kepler-4, the ob-
+servedρ∗andTeff(indicated by the error box in
+Figure3) are closelymatched byeach ofthe evolu-
+tion tracks, albeit at different ages and with mod-
+els occupying different evolutionary states. Thus,
+for the lowest-mass star to fit the observations, it
+must have just finished its core-contraction phase,
+5and be on its way to becoming a shell-burning
+subgiant. The highest-mass star to fit the same
+observations must still be on the main sequence
+but nearing hydrogen exhaustion in its core, just
+poised to begin core contraction.
+It is not possible to distinguish among these
+possibilities using existing observations. Although
+the masses of acceptable models differ by as much
+as 13%, the radii differ by only 1/3 as much (in
+order to give the same observed mean density).
+This radius difference would cause a difference in
+log(g) that is too small to measure with current
+methods. Likewise, the luminosity difference of
+about 8% implies a parallax difference of only 4%,
+also too small to discern at this star’s likely dis-
+tance of about 550 ±80pc. Asteroseismology may
+however offer a way out of this quandry. Because
+of the star’s relatively large luminosity and ap-
+parent brightness, it may be possible to measure
+the frequencies of its pulsation modes, and hence
+distinguish among the feasible evolution scenarios.
+Kepler-4isnowbeingobservedwith Kepler’s short
+(60 s) cadence; the mission will report pulsation
+properties estimated from these data if and when
+the pulsation signals rise above the noise.
+We list the inferred properties of Kepler-4 in
+Table 1. For the quantities M⋆andR⋆we give un-
+certainties that include the effects of observational
+uncertainties,ofuncertaintyinorbitaleccentricity,
+and of uncertainty in the host star’s evolutionary
+state.
+3.3. The Planet Kepler-4b
+Given the properties of its host star, the depth
+of transits due to Kepler-4b implies a planetary
+radius of 0 .357±0.019RJ= 3.99±0.21R⊕, and
+a mass of 0 .077±0.012MJ= 24.5±3.8M⊕. As
+withM⋆andR⋆, these values and uncertainties
+shouldbeinterpretedasdescribingthecentersand
+68%-probabilitypointsofthe marginalprobability
+distributions, accounting for all of the uncertain-
+ties. Kepler-4b is therefore slightly more massive
+than Neptune, and about the same size. Its mean
+density is about 1 .9 gcm−3, greater than Jupiter’s
+and considerably larger than Saturn’s. Assuming
+a 10% Bond albedo and efficient heat redistribu-
+tion to the planet’s night side, its equilibrium tem-
+peratureis1650 ±200K(Koch et al. 2010). Radial
+velocity data provide no evidence for other mas-
+sive planets in small orbits. Because of the limitedtime span and precision of the extant RV obser-
+vations, it is however impossible to rule out such
+planetary companions.
+4. Discussion
+Figure 4 shows Kepler-4 in a mass-radius di-
+agram that spans the range so far occupied by
+small transiting exoplanets. Kepler-4b is the
+third known transiting Neptune-like planet, to-
+gether with GJ436b and HAT-P-11b; all three
+have masses and radii equal or larger than Nep-
+tune and Uranus. The most important differences
+between the 3 exoplanets are their host stars and
+the incident flux the planets receive (GJ436b:
+M dwarf, Teq= 650 K; HAT-P-11b: K dwarf,
+880 K; Kepler-4b: G subgiant, 1650 K). On the
+mass-radius diagram Kepler-4b and GJ436b both
+lie below the Z= 0.9 Baraffe et al.(2008) non-
+irradiated model. In these models, Z= 0.9 de-
+notes that the planet has a 10% by mass envelope
+of H and He, and ’non-irradiated’ corresponds
+roughly the case of GJ436b and cooler. The fact
+that Kepler-4b and GJ436b have essentially iden-
+tical radii(3.88 ±0.15R⊕for GJ436bfromNASA
+EPOXI (Ballard et al. 2009) as compared to 3.99
+±0.21R⊕for Kepler-4) at the same mass, de-
+spite Kepler-4b’s high Teq, implies a difference in
+bulk composition. We suggest that Kepler-4b has
+a H/He envelope of about 4-6% by mass and a
+correspondingly higher water and rock fraction.
+However, the inherent degeneracy in this part of
+the mass-radius diagram means that the H/He
+envelope might be slightly more massive if the
+rock-to-water ratio in the interior of Kepler 4b is
+unusually high (i.e., significantly higher than in
+Neptune or Uranus). Nevertheless, we can state
+withameasureofconfidencethattherearenopos-
+sible interior models for Kepler-4b with no H/He
+envelope and neither it nor GJ436b is compact
+enough to be a water-rich super-Earth.
+The slightly eccentric orbits of the two previ-
+ously known transiting Neptune-like planets, in
+similar potentials and at similar ages, suggest that
+their interiors are somewhat less dissipative to
+stellar tides than the giants in our Solar System.
+Therefore, it would be very interesting to resolve
+whether the orbit of Kepler-4b has comparable ec-
+centricity or is circular; the current observations
+are still inconclusive.
+6Funding for this Discovery mission is provided
+by NASA’s Science Mission Directorate. We are
+grateful first to the entire Keplerteam, past and
+present. Their tireless efforts were all essential to
+the success of the mission. For special advice and
+assistance,wethankLarsBuchhave,DavidCiardi,
+MeganCrane, Willie Torres,Mike Haas, and Riley
+Duran.Facilities: The Kepler Mission.
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+This 2-column preprint was prepared with the AAS L ATEX
+macros v5.2.
+7Table 1
+System Parameters for Kepler-4
+Parameter Value Notes
+Transit and orbital parameters
+Orbital period P(d) 3 .21346±0.00022 A
+Midtransit time E(HJD) 2454956 .6127±0.0015 A
+Scaled semimajor axis a/R⋆ 6.47+0.26
+−0.28A
+Scaled planet radius RP/R⋆ 0.02470+0.00031
+−0.00030A
+Impact parameter b≡acosi/R⋆ 0.022+0.234
+−0.022A
+Orbital inclination i(deg) 89 .76+0.24
+−2.05A
+Orbital semi-amplitude K(ms−1) 9 .3+1.1
+−1.9A,B
+Orbital eccentricity e 0 (adopted) A,B
+Center-of-mass velocity γ(ms−1) −1.27±1.1 A,B
+Observed stellar parameters
+Effective temperature Teff(K) 5857 ±120 C
+Spectroscopic gravity log g(cgs) 4 .25±0.10 C
+Metallicity [Fe/H] +0 .17±0.06 C
+Projected rotation vsini(kms−1) 2 .2±1.0 C
+Mean radial velocity (kms−1) −61.0±0.10 B
+Derived stellar parameters
+MassM⋆(M⊙) 1 .223+0.053
+−0.091C,D
+Radius R⋆(R⊙) 1 .487+0.071
+−0.084C,D
+Surface gravity log g⋆(cgs) 4 .17±0.04 C,D
+Luminosity L⋆(L⊙) 2 .26+0.66
+−0.48C,D
+Absolute V magnitude MV(mag) 4 .00±0.28 D
+Age (Gyr) 4.5±1.5 C,D
+Distance (pc) 550 ±80 D
+Planetary parameters
+MassMP(MJ) 0 .077±0.012 A.B,C,D
+Radius RP(RJ, equatorial) 0 .357±0.019 A.B,C,D
+Density ρP(gcm−3) 1 .91+0.36
+−0.47A,B,C,D
+Surface gravity log gP(cgs) 3 .16+0.06
+−0.10A,B,C,D
+Orbital semimajor axis a(AU) 0 .0456±0.0009 E
+Equilibrium temperature Teq(K) 1650 ±200 F
+Note.—
+A: Based primarily on the photometry.
+B: Based on the radial velocities.
+C: Based on spectrum analysis (FIES/MOOG or HIRES/SME).
+D: Based on the Yale-Yonsei evolution tracks.
+E: Based on Newton’s version of Kepler’s Third Law.
+F: Assumes Bond albedo = 0.1 and complete redistribution.
+86200 6100 6000 5900 5800 5700
+Teff (K)0.0-0.1-0.2-0.3-0.4-0.5log(Mean Density)
+Mass=1.22Mass=1.28
+Mass=1.13
+Fig. 3.— Yonsei-Yaleevolutionarytracksforstars
+with [Z] = 0.17 and masses between 1.13 and 1.28
+M⊙, plotted in the ρ∗−Teffplane. The box in-
+dicates our adopted observational constraints on
+ρ∗andTeff. For this range of masses, all tracks
+lie partly inside the error box, showing that mul-
+tiple solutions are possible that satisfy these con-
+straints.5 10 20 30405060708090.  100 
+Planetary Mass (Earth Masses)1234567Radius (Earth Radii)Kepler-4b
+Uranus
+Neptune
+GJ 436bBaraffe Z=0.5, NI, 7Gyr
+HAT-P-11bHD 149026
+Baraffe Z=0.9, I, 3,5,7 Gyr
+Baraffe Z=0.9, NI, 3,5,7 Gyr
+Valencia Z=1, 50% water
+Valencia Z=1, rock
+CoRoT-7bGJ1214b
+Fig. 4.— Kepler-4b is shown in a mass-
+radius plot, along with other hot Neptunes,
+the hot super-Earths CoRoT-7b (L´ eger et al.
+2009; Queloz et al. 2009) and GJ1214b
+(Charbonneau et al. 2009), and model curves
+by Baraffe, Charbrier, & Barman (2008) and by
+Valencia, Sasselov, & O’Connell (2007), showing
+increasing heavy-element fractions toward the
+bottom of the Figure. Multiple dashed and solid
+curves show model results for differing ages;
+“I” indicates irradiated and “NI” non-irradiated
+models. Kepler-4b appears to be denser than
+HAT-P-11b but similar to GJ 436b, denser
+than the non-irradiated Z= 0.9 model by
+Baraffe, Charbrier, & Barman (2008), but much
+less dense than is expected for a water or rocky
+planet.
+9
\ No newline at end of file
diff --git a/1001.0605.txt b/1001.0605.txt
new file mode 100644
index 0000000000000000000000000000000000000000..de92838a539d8927cf2f7801d3a576db046e3fcd
--- /dev/null
+++ b/1001.0605.txt
@@ -0,0 +1,1428 @@
+Historical Costs of Coal-Fired Electricity and Implications for the Future
+James McNerneya,b, J. Doyne Farmerb,c, Jessika E Trancik,d,b
+aDepartment of Physics, Boston University, Boston, MA 02215, USA
+bSanta Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
+cLUISS Guido Carli, Viale Pola 12, 00198, Roma, Italy
+dEngineering Systems Division, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
+Abstract
+We study the costs of coal-red electricity in the United States between 1882 and 2006 by decomposing it in terms of
+the price of coal, transportation costs, energy density, thermal eciency, plant construction cost, interest rate, capacity
+factor, and operations and maintenance cost. The dominant determinants of costs have been the price of coal and plant
+construction cost. The price of coal appears to 
uctuate more or less randomly while the construction cost follows
+long-term trends, decreasing from 1902 - 1970, increasing from 1970 - 1990, and leveling o since then. Our analysis
+emphasizes the importance of using long time series and comparing electricity generation technologies using decomposed
+total costs, rather than costs of single components like capital. By taking this approach we nd that the history of
+coal-red electricity costs suggests there is a 
uctuating 
oor to its future costs, which is determined by coal prices.
+Even if construction costs resumed a decreasing trend, the cost of coal-based electricity would drop for a while but
+eventually be determined by the price of coal, which 
uctuates while showing no long-term trend.
+Key words: coal, electricity, historical cost
+Contents
+1 Introduction 1
+2 Historical data 2
+2.1 Data sources . . . . . . . . . . . . . . . . . 2
+2.2 Decomposition formula . . . . . . . . . . . . 2
+2.3 Fuel cost . . . . . . . . . . . . . . . . . . . 3
+2.3.1 Coal price . . . . . . . . . . . . . . . 3
+2.3.2 Transportation costs . . . . . . . . . 3
+2.3.3 Coal energy density . . . . . . . . . 4
+2.3.4 Thermal eciency . . . . . . . . . . 4
+2.3.5 Fuel cost . . . . . . . . . . . . . . . 5
+2.4 Operation and maintenance cost . . . . . . 5
+2.5 Capital cost . . . . . . . . . . . . . . . . . 6
+2.5.1 Specic construction cost . . . . . . 6
+2.5.2 Capacity factor . . . . . . . . . . . . 8
+2.5.3 Interest rate . . . . . . . . . . . . . . 8
+2.5.4 Capital cost . . . . . . . . . . . . . . 9
+2.6 Total cost . . . . . . . . . . . . . . . . . . . 9
+3 Analysis of cost trends 9
+4 Analysis of cost variation 11
+5 Future Implications 12
+5.1 Commodities vs. pure technologies . . . . . 12
+5.2 Modeling a commodity . . . . . . . . . . . . 13
+Corresponding author. E-mail address: trancik@mit.edu5.3 Modeling a technology . . . . . . . . . . . . 13
+5.4 Application to coal-red electricity . . . . . 14
+6 Conclusions 14
+7 Acknowledgments 15
+A Change decomposition 15
+1. Introduction
+Coal generates two-fths of the world's electricity [54]
+and almost a quarter of its carbon dioxide emissions [2].
+The impact of any market-based eort to reduce carbon
+emissions will be highly sensitive to future costs of coal-
+red electricity in comparison to other energy technolo-
+gies. The relative cost of technologies will, for example,
+determine the carbon reductions resulting from a partic-
+ular carbon tax or the cost of a given cap on emissions.
+However, no study exists that examines total generation
+costs and component contributions using data over a time
+span comparable to that of the forecasts needed. This
+study attempts to fulll this need. We also aim to make
+methodological advances in the analysis of historical en-
+ergy costs and implications for future costs.
+To do this, we build a physically-accurate model of
+the total generation cost in terms of the price of coal, coal
+transportation cost, coal energy density, thermal eciency,
+plant construction cost, interest rate, capacity factor, and
+operations and maintenance cost. This contrasts with the
+Preprint submitted to Energy Policy October 24, 2012arXiv:1001.0605v4  [physics.soc-ph]  23 Oct 2012approach taken in econometrics where generation costs,
+or more typically plant costs, are broken down using a
+regression model [30, 42, 7, 4, 26, 49, 40, 23, 27, 35, 25].
+We also focus on the long term evolution of quanti-
+ties averaged across plants in the United States, going
+back to the earliest coal-red power plant in 1882 through
+2006, rather than cross-sections or panels of plants. Thus,
+our data set sacrices cross-sectional richness to examine
+a longer time span (over a century). This is important for
+characterizing the factors driving the long term evolution
+of costs.
+The data suggest a qualitative dierence between the
+behavior of fuel and capital costs, the two most signicant
+contributors to total cost. Coal prices have 
uctuated and
+shown no overall trend up or down; they became the most
+important determinant of fuel costs when average ther-
+mal eciencies ceased improving in the U.S. during the
+1960s. This 
uctuation and lack of trend are consistent
+with the fact that coal is a traded commodity and there-
+fore it should not be possible to make easy arbitrage prots
+by trading it. According to standard results in the theory
+of nance, this implies that it should follow a random walk.
+In contrast, plant construction costs, the most important
+determinant of capital costs, followed a decreasing trajec-
+tory until 1970, consistent with what one expects from a
+technology. After 1970, construction costs dramatically
+reversed direction, at least in part due to pollution con-
+trols.
+Analysis of historical trends suggests a 
uctuating 
oor
+on the total costs of coal-red electricity which is deter-
+mined by coal prices. Under a scenario in which plant
+costs return to their pre-1970 rate of decrease, fuel costs
+would rise in their relative contribution to the total. This
+would lead to higher uncertainty in total generation costs,
+and create a 
oor below which the generation cost would
+be unlikely to drop. Even under a scenario in which car-
+bon capture and storage (CCS) capabilities are added to
+plants, the same qualitative behavior would be expected.
+Our analysis makes several methodological advances.
+It calculates total costs of generation, rather than the
+costs of individual components like capital, and over a
+long (100 year) time span. We build on an approach
+developed for photovoltaics and nuclear ssion [41, 29] to
+decompose changes in cost. The renement eliminates ar-
+ticial residuals arising from the cross-eects of variables
+in
uencing the cost. Physically accurate models like the
+one presented here may sometimes allow (data permitting)
+for more reliable decompositions of cost than regression
+models.
+The paper is organized as follows: in Section 2, we
+present the historical record of costs for coal-red electric-
+ity, as represented by time series for a number of important
+variables in
uencing generation cost. The generation cost
+consists of three main components | fuel, capital, and op-
+eration and maintenance. The rst two are further decom-
+posed; the fuel component into the coal price, transporta-
+tion cost, coal energy density, and thermal eciency; thecapital component into plant construction costs, capacity
+factor, and interest rate. In Section 3, we determine the
+variables contributing most to the historical changes in
+generation costs, focusing on their long term trends rather
+than their short term variation. In Section 4, we examine
+the eect their short term variation has on total genera-
+tion costs. In Section 5, we use the experience gained from
+analyzing the data to examine future implications.
+2. Historical data
+2.1. Data sources
+Sources for data are given in the captions of the rele-
+vant gures. The data comes from the U.S. Energy Infor-
+mation Administration, Census Bureau, Bureau of Mines,
+Federal Energy Regulatory Commission (formerly the Fed-
+eral Power Commission), Federal Reserve, the Edison Elec-
+tric Institute, and the Platts UDI world electric power
+plants database, with some minor contributions from a
+few other databases and technical reports. Assumptions
+made to ll in for missing data are noted in the relevant
+gures and text.
+All data presented here are for the United States be-
+tween 1882 (the approximate beginning of the electricity
+utility industry in the U.S.) and 2006. All data are esti-
+mates for averages over U.S. coal utility plants. All prices
+and costs are presented in real 2006 currency, de
ated us-
+ing the GDP de
ator.1The data is made available at
+www.santafe.edu/les/coal electricity data.
+2.2. Decomposition formula
+We seek to build up the total generation cost (TC)
+from the following variables,2for which data is available:
+OM = total operation and maintenance cost( ¢/kWh)
+FUEL = total fuel cost ( ¢/kWh)
+CAP = total capital cost ( ¢/kWh):
+1Given the existence of other industry-specic de
ators (e.g.
+PPIs, the Handy-Whitman indices), it is worth explaining why we
+de
ate all prices by the GDP de
ator. To study how the total cost
+of electricity is aected by the direct inputs to electricity production,
+later given in equation (1) | O&M, the coal price, transportation
+price, and the construction price | we de
ate prices in a way that
+preserves their ratios, while removing the eect of changes in the
+overall price level of the economy. Such ratios represent a meaning-
+ful quantity | the relative economic scarcity of two goods | and
+the real prices should preserve them. Using a single de
ator for all
+these inputs guarantees that their price ratios are unchanged. To
+remove changes in the overall price level of an economy, the GDP
+de
ator is appropriate.
+2By total generation cost, we mean all production costs of elec-
+tricity up to the busbar, the point at which electricity leaves the
+plant and enters the grid.
+2These three major cost components are in turn decom-
+posed further into
+COAL = price of coal ( $/ton)
+TRANS = price of transporting coal to plant ( $/ton)
+= energy density of coal (Btu/lb)
+= plant eciency
+SC = specic construction cost ( $/kW)
+r= nominal interest rate
+CF = capacity factor :
+Specic construction cost (SC) here means the construc-
+tion cost of a plant per kilowatt of capacity. Each variable
+is given a subscript tto denote its value in year t. The
+total generation cost (TC) in year tis
+TCt= OMt+ FUEL t + CAP t (1)
+= OMt+COALt+ TRANS t
+tt+SCtCRF(rt;n)
+CFt8760 hrs:
+The three main terms are the three major cost components
+{ operation and maintenance, fuel, and capital. The fuel
+component accounts for the cost of coal and its delivery
+to the plant, the amount of energy contained in coal, and
+the eciency of the plant in converting stored chemical
+energy to electricity. The capital component levelizes the
+construction cost of the plant using the capital recovery
+factor, CRF( rt;n), dened as
+CRF(rt;n) =rt(1 +rt)n
+(1 +rt)n1; (2)
+wherenis the plant lifetime in years. The capital recov-
+ery factor is the fraction of a loan that must be payed
+back annually, assuming a stream of equal payments over
+nyears and an annual interest rate rt. For a plant of ca-
+pacityK, the capital component is the annuity payment
+on money borrowed for construction, KSCtCRF(rt;n),
+divided by the yearly electricity production of the plant,
+KCFt8760 hrs. Note that Kcancels out. Thus, the
+capital component is the annuity payment on borrowed
+plant construction funds perkilowatt-hour of annual elec-
+tricity production by the plant. We follow convention and
+levelize over an assumed plant lifetime of n= 30 years.
+Note that the total cost for year tuses the capital cost
+of plants built that year, while the actual 
eet of plants
+existing in year talso includes plants built in previous
+years. Since we do not have data on the retirement dates
+of plants, we use the cost of the plants built in year tto
+estimate the cost of electricity generated in that year. Av-
+erage capital costs of the whole 
eet existing in a particular
+year have the same basic evolution as that of new plants
+alone, but tend to lag the latter, since they include capital
+costs from previous years.
+In Sections 2.3 { 2.5, we present and discuss the time-
+series data for each of these variables separately. In Section
+2.6 we combine the variables to compute the total cost,
+Figure 1: The price of coal at the mine, the price of transporting it
+to the plant, and their sum. The price of coal has 
uctuated with
+no clear trend up or down. Transportation contributed a signicant
+while decreasing expense to fuel costs. Source: [31, 19, 46, 52, 9]
+and examine the eect that individual variables had on
+it. The ensuing sections are an extensive discussion of the
+data; readers more interested in nal analyses may wish
+to skip to Section 3.
+2.3. Fuel cost
+2.3.1. Coal price
+Real coal prices (COAL) have varied over the past 130
+years (Fig. 1.) The largest change occurred between 1973
+and 1974. A government study [6] found this was due to
+the OPEC oil embargo started in December 1973, which
+raised prices for substitutes coal and natural gas, and an-
+ticipation of a strike by the United Mine Workers union,
+which later occurred at the end of 1974. Wage increases
+starting in 1970 also played a somewhat less important
+role.
+Coal prices at the mine were derived from price time-
+series for individual varieties of coal. For the period 1882-
+1956, the coal price is a production-weighted average over
+anthracite and bituminous varieties, and for 1957-2006
+over anthracite, bituminous, subbituminous, and lignite.
+No data could be found to weight prices by the quanti-
+ties in which they were consumed by plants, and therefore
+we relied on the quantities in which they were produced.
+The resulting average price series closely resembles that of
+bituminous coal by itself.
+2.3.2. Transportation costs
+Transportation costs (TRANS) were derived mainly
+from the cost of coal delivered to power plants and the
+price of coal at the mine, though some direct data also ex-
+ists. Transportation costs have added a signicant though
+decreasing amount to the cost of coal delivered to power
+plants (Fig. 1). Transportation costs before 1940 were on
+par with the price of coal at the mine; since 1940, they
+have dropped to 20 { 40% of the delivered cost on aver-
+age, though there is considerable variation from plant to
+plant. The introduction of unit trains in the 1950s may
+3Figure 2: The cost of coal-stored energy, (COAL t+ TRANS t)=t.
+The left axes show cost in dollars per million BTU, the right axes in
+cents per kWh. The inset shows the energy density of coal in Btu
+per pound. The dashed line indicates an assumed value of 12,000
+Btu per pound. Coal energy densities have come down about 16%
+in the last 40 years. Source: [11, 19]
+account for some of the decrease in transportation costs.
+A unit train carries a single commodity from one origin to
+one destination, shortening travel times and eliminating
+the confusion of separating cars headed for dierent desti-
+nations. About 50% of coal shipped in the United States
+(90% of which is used by utilities) is carried by unit trains
+[10]. Other means of transporting coal are barges, collier
+ships, trucks, and conveyor belts in cases where plants are
+built next to mines. Although we focus on average trans-
+portation costs, we note that the local cost of coal varies
+considerably between regions.
+Several acts during the 1970s partially deregulated the
+U.S. rail industry, culminating in the 1980 Staggers Rail
+Act which completed deregulation [32]. Rail rates before
+1980 were determined by the Interstate Commerce Com-
+mission; after the Staggers Act, railroads were allowed to
+determine their own rates. The deregulation is believed to
+have contributed to lower rates in the following years [32].
+Many utilities responded to environmental regulations
+by switching to higher priced, low-sulfur coals [23]. Switch-
+ing to low sulfur coals also extended transportation dis-
+tances, which may explain the increase in transportation
+costs seen around 1970. The 1990 Clean Air Act Amend-
+ments tightened regulations again, extending transporta-
+tion distances [18]. However, a coincident decrease in rail
+rates per ton-mile caused rates per ton to continue de-
+creasing.
+2.3.3. Coal energy density
+Since 1960, the average energy density of coal ( ) has
+dropped steadily (Fig. 2 inset.) The lower the energy
+density, the more coal needed by plants to consume equal
+amounts of primary energy. Thus, the eect of lower en-
+ergy densities has been to increase fuel costs through in-
+creased purchase and transportation costs.
+Changes in the energy density re
ect changes in the
+overall mixture of coal species used by the industry, as well
+Figure 3: Average eciency of coal plants and cost of the fuel com-
+ponent, (COAL t+ TRANS t)=(tt). Eciency increases drove de-
+creases in fuel costs until 1960. Since then eciency has remained
+stable around 32 { 34%. Source: [39, 19]
+as variation of energy density within species. One cause of
+the decrease in energy density may be the increased use of
+subbituminous coal, which burns more cleanly than previ-
+ously used bituminous due to a lower sulfur content, but
+also contains less energy per pound. No data for energy
+density could be found before 1938, and a value of 12,000
+Btu per pound was assumed (the 1938 value).
+2.3.4. Thermal eciency
+The thermal eciency of a fossil-steam plant ( ) is the
+fraction of stored chemical energy in fuel that is converted
+into electrical energy (Fig. 3). It accounts for every eect
+which causes energy losses between the input of fuel and
+the bus-bar of the plant, the point where electricity enters
+the electric grid. These eects include incomplete burn-
+ing of fuel, radiation and conduction losses, stack losses,
+excess entropy produced in the turbine, friction and wind
+resistance in the turbine, electrical resistance, power to
+run the boiler feed pump and other plant equipment, and
+the thermodynamic limits of the heat cycle used.
+The earliest coal plants obtained eciencies below 3%.
+Over the rst 80 years, average eciency grew by more
+than a factor of 10. Improving eciencies meant that less
+coal was required to produce equal amounts of electricity,
+lowering fuel costs. For the last 50 years, the average e-
+ciency of coal-red plants has stayed approximately level
+at 32{34%, although individual plants obtained eciencies
+as high as 40% during the 1960s [24].
+The factors in
uencing eciency of plants fall into two
+major categories: thermodynamic factors, which pertain
+to the specic heat cycle employed, and operational fac-
+tors, which re
ect the mechanical and electrical eciencies
+of individual stages and components. The thermodynamic
+limit eciency is determined by the details of the steam
+cycle used { the maximum and minimum temperatures,
+the maximum and minimum pressures, and the exact type
+of cycle followed (e.g. the number of reheat stages used.)
+Current plant designs typically have thermodynamic limit
+4eciencies around 46%.3The operational eciencies of
+the boiler, turbine, generator, and other components fur-
+ther reduce the eciency achievable in practice (Table 1.)
+The net eect of the operational eciencies is to reduce
+total eciency by a factor of about 0.7.
+The capacity factor aects the operating parameters
+of the plant (e.g. pressures, throttles) and therefore in
u-
+ences plant eciency. Typically, a lower capacity factors
+will mean higher variability and lower eciencies. The
+presence of pollution controls can also lower eciency, be-
+cause they increase auxiliary power requirements.
+Historical increases in eciency came from improve-
+ments in both the thermodynamic and operational cat-
+egories. Higher steam temperatures, higher steam pres-
+sures, and changes to the heat cycle relaxed the thermo-
+dynamic constraint, while better designed parts reduced
+operational losses. However, operational factors are now
+working at high eciencies (Table 1), while changing the
+parameters of the heat cycle faces highly diminishing re-
+turns for the thermodynamic limit eciency [24]. This
+latter problem reduces incentives to develop economical
+materials that could withstand higher pressures and tem-
+peratures. The overall result has been a standstill in av-
+erage eciency for the last 50 years in the U.S. (Although
+the average eciency of U.S. plants has been static in the
+last 50 years, that of European and Japanese plants has
+continued to grow in the same period.) Given the present
+technological options, the current average eciency pre-
+sumably represents an optimum after balancing the higher
+fuel cost of a less ecient plant against the higher con-
+struction costs of a more ecient plant.
+2.3.5. Fuel cost
+Over the entire time period (1882-2006) the factors
+most responsible for changes in the fuel cost are the price
+of coal and the eciency. Changes in energy density and
+transportation costs had relatively minor eects. As can
+be seen in Fig. 3, the net eect of varying coal price, en-
+ergy density, transportation cost, and eciency on the fuel
+component of electricity costs was a long term decrease in
+the cost of the fuel component until about 1970. The de-
+crease was mainly due to improving eciency. After 1970,
+3We assume a Rankine cycle with superheating and one reheat
+cycle operating at 1000F, 2400 psi.
+Table 1: Operational sources of energy loss. Source: [24]
+Source of energy loss Eciency
+Stack losses, radiation
+& conduction from boiler 87%
+Excess entropy produced in turbine 92%
+Windage, friction, & elec. resistance 95%
+Boiler feed pump power requirement 95%
+Auxiliary power requirements 97%
+Total (product) 70%
+Figure 4: The cost of operation and maintenance of coal plants. A
+value equal to 21% of the overall fuel cost for a given year was used
+to ll in missing data. Source: [11, 12, 14, 15, 3, 43]
+coal prices increased dramatically during a time when ef-
+ciency was 
at, increasing overall fuel costs 70% between
+1970 and 1974.
+2.4. Operation and maintenance cost
+The O&M cost (OM) is the least signicant of the three
+cost components (fuel, O&M, capital), representing about
+5-15% of total generation costs during the last century
+(Fig. 11). Although it is less signicant than fuel or cap-
+ital, we attempted to reconstruct a time-series for O&M.
+Reliable historical data for O&M costs was dicult to
+acquire. Sources were frequently in con
ict with each other
+due to dierences in denition of O&M costs. We use data
+from the Energy Information Administration (EIA) be-
+cause it uses a consistent denition over the longest time
+period. To ll in missing years, we took O&M costs to
+be 21% of the value of the overall fuel cost, based on the
+empirical observation that O&M costs were consistently
+about 21% of overall fuel costs between 1938 and 1985 in
+all years for which we had data.4Later O&M costs break
+from this pattern. This assumption seemed the best choice
+to avoid introducing discontinuities or other articial be-
+haviors in the total costs. We do not have a theoretical
+explanation for why the "21% rule" replicates the O&M
+costs so well for the years where data is available. However
+we note that based on empirical analysis, the O&M costs
+are more strongly correlated with the fuel costs than with
+capital or construction costs.5
+Based on our reconstruction, O&M costs decreased un-
+til 1970 (Fig. 4). For pulverized coal plants, historical de-
+clines \are attributed mainly to the introduction of single-
+4OM's ratio to FUEL was always between 19 and 24%, with an
+average of 21%, during three periods for which data was available:
+1938-1947, 1956-1963, and 1978-1985.
+5The correlation coecient between OM and FUEL is 0.87, while
+that between OM and CAP is 0.69 and that between OM and SC is
+0.33.
+5Figure 5: Specic capital cost of coal plants. Plant costs generally de-
+creased until 1970, when pollution regulation, increased construction
+times, and possibly other factors caused a rise. Source: [16, 17, 47]
+boiler designs, automatic controls, and improved instru-
+mentation" [56, 48]. From the data available, we infer that
+an increase occurred between 1960 and 1980. This is sup-
+ported by observations that pollution controls introduced
+to plants during this time raised O&M costs [28, 27].
+O&M costs are also in
uenced by the capacity factor.
+Higher production rates lower O&M costs by amortizing
+cost over greater electricity production [28].
+2.5. Capital cost
+2.5.1. Specic construction cost
+Construction cost data were obtained from Historical
+Plant Cost and Annual Production Expenses 1982 [16],
+published by the Energy Information Administration. A
+potential source of bias with this data is that plant costs
+were given as accounts which accumulate a utility's nomi-
+nal expenditure on a given plant. These accounts therefore
+include costs of additional units since its construction, and
+sum together nominal expenditures from dierent years.
+The specic construction cost (SC) is the cost of build-
+ing a plant per kW of capacity installed. Specic con-
+struction costs decreased from the beginning of the indus-
+try until about 1970, then doubled between 1970 and 1987
+(Fig. 5.) They appear to have been roughly 
at since
+then, though we lack data for the period 1988-1999.
+The specic construction cost is a leaf on our cost tree
+(a very important leaf), but could easily be the subject
+of a separate decomposition study unto itself, as Joskow
+& Rose [27] have done. Such a study would have a dif-
+ferent scope from the present study, and involve collection
+of dierent data. Nevertheless, we can discuss the deter-
+minants of specic construction cost and in some cases
+provide numerical estimates of their impact. Four fac-
+tors are frequently mentioned as important: economies of
+scale, add-on environmental controls, thermal eciency,
+and construction inputs (both prices and quantities.) Note
+that
+Economies of scale appear to have lowered construc-tion costs as unit capacity grew (Fig. 13).6Joskow &
+Rose study the eects of size on construction cost, and we
+combine their results with our size data to attempt a rough
+estimate of the cost reduction coming from unit capacity
+increases. They regress the real specic construction cost
+of U.S. coal units built between 1960 and 1980 onto a log
+linear function of unit capacity and other variables (such as
+the regional labor cost, the presence of pollution controls,
+and year dummies). In their model, SC depends on unit
+capacityk(t) as SC/k(t)a. Under the simplest specica-
+tion of their model, they nd a=:183. To estimate the
+size eect, we calculate the ratio of the size factor between
+two years, t1andt2:k(t2)a=k(t1)a. This is the factor
+by which specic construction costs would have changed
+due to capacity changes, all else being equal. Results are
+shown for 3 periods in Table 2.
+Period k(t2)a=k(t1)aSC(t2)=SC(t1)
+1908 { 1970 0.585 0.205
+1970 { 1989 1.410 2.058
+1989 { 2006 0.613 1.453
+Table 2: 2nd column: Factor by which specic construction costs
+would have changed due to capacity changes, ceteris paribus, using
+Joskow & Rose scaling exponent a=0:183. 3rd column: Factor
+by which specic construction costs actually changed.
+There are at least three limitations to calculating the
+size eect this way. One is that the estimate of ais based
+upon units built between 1960 and 1980, and amay be
+dierent at other times. Another is that measurements
+of scaling exponents depend sensitively on how samples of
+coal units are grouped; e.g. should one measure a single
+scaling coecient for all coal units, or group them by pres-
+sure class, or by vintage, or by some other characteristic?7
+The third limitation is more fundamental: it is not clear
+that specic construction cost actually has a capacity de-
+pendence of the form SC/k(t)a. Log-linear forms are
+common in the literature because they allow linear regres-
+sion methods to be applied and because scaling phenom-
+ena often follow power laws. Nevertheless, no theory yet
+supports any particular functional form.
+Increases in thermal eciency can raise or lower the
+construction costs of plants [33], by an amount that de-
+pends on the trade-o between higher materials and build-
+ing costs but greater power. As engineering knowledge
+of a technology increases, it may be possible to obtain a
+higher thermal eciency for the same materials and build-
+ing costs. This results in lower construction costs (per unit
+power). However achieving a large jump in eciency over a
+short period of time may lead to higher construction costs.
+6A unit is a boiler plus turbogenerator. A plant may have one or
+more units.
+7Joskow and Rose demonstrate this sensitivity using dierent
+multiple regression models, in which units were either all pooled
+together or grouped into 4 pressure classes. Among the various pres-
+sure classes and regression models, scaling coecients varied from
+0:454 to +0:199.
+6Still, such a jump may still be attractive to investors if the
+plant is expected to have a high capacity factor and/or
+coal prices are high. The choice of thermal eciency at
+a single point in time typically carries with it a tradeo
+between higher eciency/higher capital costs (where cap-
+ital costs are in
uenced by the capacity factor) and lower
+eciency/higher fuel costs.
+At any given time in history, a range of eciencies were
+accessible to the designers of plants and the utilities pur-
+chasing them. Utilities attempt to optimize the eciency
+of the plants they purchase with respect to total costs,
+given their expectations about future fuel prices and the
+future load prole. The eciency time series of Fig. 3
+already accounts for this tradeo, and depicts a change in
+the average believed-optimal eciency over time.
+Add-on environmental controls also raise plant costs,
+by an amount that may decrease over time with increased
+engineering knowledge. Joskow & Rose estimate that sul-
+fur scrubbers and cooling towers add about 15% and 6%
+respectively to the cost of plants. Besides raising plant
+costs directly by requiring new equipment, there is evi-
+dence that pollution controls also raised costs indirectly by
+increasing the complexity of the plant, which now required
+greater planning and longer construction times [27, 5].
+Finally, changes in the price or quantity of construc-
+tion inputs will aect plant costs. Required quantities of
+materials and labor may decrease over time with increas-
+ing engineering knowledge, while changes in the rest of the
+economy may alter prices.
+A long-standing trend of decreasing specic construc-
+tion costs reversed direction around 1970 when costs be-
+gan increasing, and the cause of this uptick has been a
+focus of interest. Several contributing factors have been
+identied. One is the introduction of pollution controls.
+Starting with the Clean Air Act in 1970, several laws were
+passed in the US requiring plants to add pollution controls
+to reduce the level of NO 2, SOx, and particulates from
+
ue gas emissions [27, 23]. The Clean Air Act of 1970
+was followed by a number of similar acts between 1970
+and 2003. While some utilities initially responded to pol-
+lution controls by switching to low-sulfur coals, others re-
+sponded by installing de-sulfurization equipment known as
+sulfur scrubbers. Eventually all new plants were required
+to install scrubbers [23]. This new equipment raised O&M
+costs and decreased eciency slightly, but mainly raised
+construction costs.
+Other causes cited for the increase in costs are in
ation,
+interest rates, decreasing construction productivity and in-
+creasing construction times [27, 5], reversed economies of
+scale (plants got smaller after 1970), and diminished op-
+portunities and incentives to improve plant construction
+due to design variation and principal-agent problems [35].
+However, the relative contribution of each factor is unclear,
+and the cause of the uptick is only partially understood.
+[27, 35, 34]. Joskow & Rose in particular nd a large resid-
+ual of unaccounted-for changes in cost after controlling for
+unit size, regional wages, utility experience, industry expe-
+Figure 6: Experience curves for plant construction cost, using
+two possible experience measures. Experience curves are a popu-
+lar framework for predicting future costs of technologies. Source:
+[16, 17, 47, 44, 51, 19]
+rience, pollution controls, indoor vs. outdoor construction,
+and \rst-unit eects."8We refer the reader to several
+studies that examine more thoroughly the contribution of
+various factors [28, 23, 27, 24, 5].
+A common representation of technological improvement
+is the experience curve or performance curve, a plot of the
+cost versus cumulative production of some item. There is a
+large literature debating the value of experience curves as
+a model for technological improvement, which we review
+in Section 5.1. In Fig. 6 we show the experience curve for
+plant construction costs. We use two dierent measures of
+experience: the number of coal units built, and the total
+capacity installed. Note that the cluster of points on the
+right side of Fig. 5 is separated from the last point in 1987
+by a gap due to missing data. However, in both plots of
+Fig. 6, this cluster of points joins up with those from the
+1980s. This is consistent with experience curve hypothesis
+that costs are more correlated with changes in experience
+(as represented by cumulative production) than with time.
+8The rst unit eect on a plant site is often more expensive than
+subsequent units because of one-time plant site costs.
+7Figure 7: Capacity factor of coal plants. Dashed lines are linear
+interpolations. Coal plants are typically base load plants, serving
+the persistent part of electricity demand at high capacity factor.
+The growth of the capacity factor, indicating greater utilization of the
+plant's capital, was a substantial factor reducing coal-red generation
+costs. Source: [51, 9, 28, 44, 19]
+2.5.2. Capacity factor
+The capacity factor (CF) is the amount of electricity
+produced divided by total potential production:
+CFt=kWh production in year t
+kW capacity8760 hrs/year:
+The capacity factor measures the utilization of a plant's
+capacity, and is bounded between 0 and 1.
+Electricity generation incurs large xed costs from plant
+construction, making high capacity factor { high utiliza-
+tion of the plant's capital { desirable to spread costs over
+the greatest possible production. The capacity factor is
+largely determined in advance, by the choice to build a
+base load plant or peaking plant. Demand for electricity
+varies hourly and seasonally. Building a coal plant with
+a high enough capacity to meet peak demand would leave
+the plant underused during periods of low demand, eec-
+tively raising capital costs. The cheapest way to meet
+varying electricity demand is with a combination of base
+load plants that have low operating cost (O&M plus fuel)
+and run at high output rate to cover the persistent portion
+of electricity demand, and peaking plants that are rela-
+tively inexpensive to build and cover the excess portion of
+demand unmet by the base load plant.
+Modern coal plants usually serve as baseload plants
+and therefore tend to have high capacity factors, around
+.7 { .8 (Fig. 7), though over the last 50 years the capacity
+factor has varied between .50 and .82. Capacity factors
+before 1940 were much lower. This may be because early
+plants required frequent maintenance and few devices ex-
+isted that required electricity, resulting in less consistent
+demand for electricity throughout the day. The rst ma-
+jor use of electricity was for lighting, particularly street
+lighting, which was only necessary a few hours each day.
+Figure 8: The capital cost of coal plants. The capital cost accounts
+for the specic construction cost (Fig. 5), the capacity factor (Fig.
+7), and the interest rate (inset.) This series resembles that of the
+specic construction costs but is steeper due to increases in capac-
+ity factor. The inset shows the historical return-on-investment of
+electric utilities, used as the eective interest rate for amortizing
+the construction cost. The Aaa corporate bond rate is shown for
+comparison. Source: [21, 51, 13, 9]
+2.5.3. Interest rate
+As mentioned earlier, we amortize the specic con-
+struction cost of plants (SC) to obtain the capital cost
+(CAP) using
+CAPt=SCtCRF(rt;n)
+CFt8760 hours: (3)
+CRF is the capital recovery factor dened in Eq. (2).
+We take the plant lifetime nto be 30 years for amorti-
+zation purposes. This is both conventional and matches
+the longest bond maturities, the period over which capital
+payments would be made.9The resulting CRF for a given
+year was 1-2% higher than the annual interest rate for that
+year.
+The eective interest rate rused to calculate the cap-
+ital recovery factor was the average return-on-investment
+(ROI) of the electric utility industry in each year (Fig. 8
+inset.) The ROI is dened as the sum of annual inter-
+est and dividend payments made to investors divided by
+the industry's gross plant value. (Note that ris a nomi-
+nal interest rate.) This data was gathered from combined
+income statements and balance sheets for the electric util-
+ity industry as a whole [51, 13, 9]. The data is therefore
+not coal-specic, but we do not expect that interest rates
+charged to coal utilities would dier signicantly from the
+average rate charged to utilities as a whole. The ROI var-
+ied between 4 and 8%.
+Interest rates had an important but transient eect;
+they contributed to the rise in costs seen during the 1970s
+and 1980s.
+9However, as a point of interest, it is quite possible that plant
+lifetimes are not 30 years, and moreover have changed with time.
+Longer plant lifetimes would eectively reduce electricity costs by
+spreading capital costs out over a longer period. One study found
+that most coal plants currently in use have lifetimes between 31 and
+60 years, with an average of 49 [55].
+8Figure 9: The total cost of coal-red electricity. The top curve is the
+sum of the fuel, capital, and O&M curves below it.
+2.5.4. Capital cost
+The capital cost (CAP) given by Eq. 3 combines con-
+struction costs, plant usage, and interest rates to obtain
+the contribution of capital to total generation costs (Fig.
+8 main axes.) Its shape resembles that of the specic con-
+struction cost of Fig. 5. The most important factors re-
+ducing capital costs were decreasing construction costs and
+increasing plant usage.
+2.6. Total cost
+Figure 9 shows the total generation cost of coal-red
+electricity (TC), along with the three major cost compo-
+nents. To check that the cost history constructed was ap-
+proximately correct, we sought independent data to vali-
+date this series. To our knowledge, neither cost nor price
+data for coal-red electricity exists, so we use the average
+price of electricity from all types of generation. Coal pro-
+vided about half of all annual electricity production for the
+US throughout its history, so we expect the average price
+to be heavily in
uenced by the price of coal-red electric-
+ity. In addition, we use historical data for transmission
+and distribution losses, taxes, and retained earnings (i.e.
+\post-cost" adjustments) to estimate the average cost of
+electricity to obtain a theoretically closer series for com-
+parison. The all-sources cost and price series described
+above, along with the reconstructed coal-red cost, are
+shown in Fig. 10. These series compare favorably and
+validate the decomposition.
+Between 1970 and 1985, costs increased due to a num-
+ber of unrelated developments that happened nearly si-
+multaneously. Coal prices increased from high oil prices,
+anticipation of strikes, and increasing wages; O&M costs
+increased because of added pollution controls; and plant
+construction costs increased from added pollution controls,
+diminishing productivity in the construction sector, and
+high in
ation-driven interest rates.
+In the next three sections we analyze the data pre-
+sented above. In Section 3 we look at the long term cost
+trends of each variable. In Section 4 we look at the short
+term variation caused by each variable.
+Figure 10: The total cost of coal-red electricity shown in Fig. 9,
+along with the average price and cost of electricity from all generating
+sources. The cost from all sources, which largely consists of coal-red
+generation, was used as a point of comparison to validate our built-up
+cost for coal-red electricity. The former is expected to be somewhat
+higher because it includes electricity from more expensive sources.
+3. Analysis of cost trends
+We analyze the trends in total generation costs in two
+ways. First, we show the contribution of each major cost
+component to total generation costs (TC). Second, we
+show the contribution of each variable to changes in the
+total generation cost.
+The major contributors to total generation cost are the
+fuel and capital components. The contribution from fuel
+has usually been 50-65%, and that from capital 25-40%.
+However, during two periods this ordering failed { from
+about 1925 to 1939, when the roles of fuel and capital
+swapped places, and from 1984 to the present, where they
+contribute about evenly. The contribution of O&M costs
+appears to have been relatively steady around 5-15% dur-
+ing the whole history (Fig. 11), though as noted earlier
+O&M costs before 1938 are not based on data but inferred
+from fuel costs (Fig. 4). This estimated breakdown is
+similar to ones given in other sources [28, 5].
+Figure 11: The share of total generation costs contributed by each
+component. Fuel and capital have traded o in dominance of total
+generation costs over time, and are currently close in size.
+9Table 3: Decomposition of the change in cost of coal-red electricity. The rst two columns indicate the dollar amount of cost changes
+contributed by the individual variables of equation (1). The last two columns indicate what percentage of the change each variable is
+responsible for. Negative percent contributions indicate that a variable opposed the change that was actually realized.
+Variablei Eect on generation cost, 
TCi % of 
TC caused by i
+1902 { 1970 1970 { 2006 1902 { 1970 1970 { 2006
+TC27:24a¢/kWh 1 :764b¢/kWh 100.0 % 100.0 %
+OM 2:94 0 :171 10.8 % 9.7 %
+FUEL14:08 0:104 51.7 % {5.9 %
+CAP10:22 1 :697 37.5 % 96.2 %
+OM 2:94 0 :171 10.8 % 9.7 %
+COAL 0 :65 0:088 {2.4 % {5.0 %
+TRANS 0:71 0:286 2.6 % {16.2 %
+15:04 0 :021 55.2 % 1.2 %
+ 1:01 0 :249 {3.7 % 14.1 %
+SC 3:38 1 :544 12.4 % 87.5 %
+r 0:20 0 :235 {0.8 % 13.3 %
+CF 7:03 0:081 25.8 % {4.6 %
+aThis is a 90% drop from the 1902 generation cost.
+bThis is a 57% rise from the 1970 generation cost.
+The second way we break down total costs is to decom-
+pose the change in total cost contributed by each variable.
+That is, we calculate the change 
TC iin the total gener-
+ation cost caused by each variable i. These contributions
+from individual variables sum up to the total change 
TC:
+
TC = 
TC OM+ 
TC FUEL + 
TC CAP
+= 
TC OM+ 
TC COAL + 
TC TRANS + 
TC
++ 
TC+ 
TC SC+ 
TCr+ 
TC CF
+=X
+i
TCi
+The method for calculating each 
TC iis a generalization
+of Nemet's method [41]. Our generalization is based on
+partial derivatives and is described in Appendix A.
+Table 3 displays the results of this second decomposi-
+tion during two periods, 1902{1970 and 1970{2006. The
+change in the generation cost 
TC is given in the rst row
+of the table; e.g. the generation cost dropped 27 ¢/kWh
+between 1902 and 1970 and increased 1.8 ¢/kWh between
+1970 and 2006. The next three rows indicate the contri-
+butions to this change coming from the three major cost
+components: 
TC OM, 
TC FUEL , and 
TC CAP. These
+contributions sum up to 
TC. The next eight rows de-
+compose these contributions still further. Together these
+eight contributions also sum up to the total change in gen-
+eration cost. Appropriate combinations of them will also
+sum to equal the contributions from OM, FUEL, and CAP.
+The last two columns of the table give the results in
+percentage terms. The percent change in generation cost
+eected by variable ibetween years t1andt2is
+% of change caused by i= 100
TCi(t1;t2)
+
TC(t1;t2)(4)where 
TC i(t1;t2) denotes the change caused by variable
+ibetween years t1andt2. By construction all percent con-
+tributions sum to 100%. An implication of Eq. (4) is that
+variables which oppose a given change in generation cost
+appropriately have negative percent contributions. For
+example, while in the rst period total generation costs
+dropped about 27 ¢/kWh (representing 100% of the total
+change), coal prices nevertheless increased slightly, and by
+themselves would have increased generation costs by .65
+¢/kWh (thus representing {2.4 % of the total change.)
+During the rst period, two factors stand out most:
+increasing capacity factor, responsible for 26% of the de-
+crease in costs, and improving eciency, responsible for
+55%. Following distantly are the specic capital cost (12%)
+and O&M cost (11%). The fuel component as a whole was
+responsible for 52% of the decrease in cost, and the capital
+component for 38%.
+Surprisingly, the construction cost was only responsi-
+ble for 12% of the change, despite decreasing by a fac-
+tor of 5.4 during this time. However, it is important to
+realize that the magnitude of individual changes cannot
+be considered independently of other changes. If other
+cost-decreasing changes had not occurred simultaneously,
+the contribution of specic capital cost would have been
+much greater. Equally, if construction costs had not come
+down, it would have held up progress of the technology and
+changes in other factors would have been less important.
+Note that this is not a bug of the decomposition method
+used, but a fact occurring for any decomposition because
+changes in variables cannot be considered independently
+of other variables.10
+During the second period, increases in plant construc-
+tion costs contributed dramatically to the increase in gen-
+10Consider for example, some quantity ywhich depends on three
+10eration cost. The capital component as a whole rose suf-
+ciently by itself to double total costs. The O&M cost
+made a much smaller contribution to the increase, while
+a small change in fuel costs mitigated the total change in
+generation costs somewhat.
+So far, we have studied the long term evolution of each
+variable and its eect on the generation cost. In addition
+to long term trends, though, some variables show signi-
+cant short term variation. In the next section, we examine
+the in
uence of this variation on the generation cost.
+4. Analysis of cost variation
+We now look at the in
uence of short time (10 year)
+variations on the generation cost. We can do this using
+the same decomposition technique used above. First, we
+calculate 
TC ifor each variable ibetween two given years
+t1andt2, as before. Then, instead of comparing 
TC ito
+
TC, we compare it to the value of TC in year t1:
+% variation caused by i= 100
TCi(t1;t2)
+TC(t1):
+This measures how much the change in iactually raised
+the generation cost between years t1andt2. Note that
+the previous section asked \how much of the generation
+cost change 
TC does 
TC irepresent?," regardless of
+the actual size of 
TC. Now we are asking whether 
TC i
+actually changed the cost signicantly.
+Rather than showing the variation in generation cost
+(TC) contributed by all the variables presented in the pre-
+vious section, we focus on the three largest contributors of
+variation: the price of coal (COAL), the specic construc-
+tion cost of plants (SC), and the capacity factor (CF). Fig.
+12 shows the in
uence these variables had on the gener-
+ation cost. The height of the bar at each year tshows
+the value of 
TC i(t10;t)=TC(t10); i.e. the size of
+the price change caused by variable iover the previous 10
+years. Thus, Fig. 12 uses a sliding window which considers
+changes over every possible 10-year period.
+The largest increase over any 10-year span came from
+coal price changes between 1968 and 1978, when coal prices
+alone caused generation costs to increase by 64.3%. The
+total increase in generation cost during this same period
+was 131.1%. The remainder of the increase was driven
+mostly by increasing O&M and capital costs.
+Nevertheless, the construction cost and the capacity
+factor also contributed signicant variation. Whereas coal
+variablesx1,x2, andx3:
+y(x1;x2;x3) =x1x2+x3
+Suppose that x2changes by 100%, and x3changes by 5%. Does this
+mean thatx2's contribution to the change is greater? Not necessar-
+ily; ifx1is suciently small, then the change in the productx1x2
+may be tiny compared with the 5% change in x3. Thus, there are
+important \cross eects" of variables, which cannot be avoided and
+are a general fact of decomposing changes.
+Figure 12: The variation in total generation cost, TC, caused by
+changes in the coal price, COAL (top), specic construction cost,
+SC (middle), and capacity factor, CF (bottom). Heights of vertical
+lines estimate how large a change in the total generation cost was
+caused by the change in the coal price over the preceding 10 years.
+Time intervals with historically large changes are highlighted in red.
+These variables caused the largest cost changes out of those given in
+equation (1).
+price variation has tended to be smooth, with changes
+from consecutive years reinforcing each other, variation
+from construction costs is more noisy from year to year
+(although still following a long term trend.) Unlike coal,
+which is more or less the same product every year, each
+year's batch of new coal plants has unique characteristics
+that may make it more or less expensive than those of
+neighboring years. Thus, some of the spikes seen in the
+second plot of Fig. 12 are due to two batches of plants
+10 years apart that happen to have signicantly dierent
+average costs.
+11Figure 13: Number of coal units produced (top), and geometric av-
+erage capacity of new units (bottom). Source: [44]
+Finally, although the capacity factor also contributed
+large variation, it is mostly an exogenous factor. It de-
+pends on industry-wide trends, rather than on any factors
+unique to coal-red electricity, as mentioned before.
+5. Future Implications
+The history of coal-red electricity suggests there is a
+
uctuating 
oor to its future costs, which is determined
+by coal prices. In the following sections, we elaborate on
+this point.
+To motivate our discussion, we note that the driving
+variable behind the capital cost has been the specic con-
+struction cost, while the driving variable behind the fuel
+cost since eciencies stopped improving has been the price
+of coal. The specic construction cost has tended to fol-
+low steady long term trends. In contrast, the price of coal
+seems to vary randomly, with no clear trends. As discussed
+in the next section, we hypothesize that this dierence in
+price behavior is due to a fundamental distinction between
+acommodity and a technology . First, we suggest a frame-
+work for discussing these qualitative dierences in price
+evolution, and then we apply the framework to the present
+example of coal-red electricity.
+5.1. Commodities vs. pure technologies
+In this section we present the conjecture that the prices
+of commodities and technologies evolve in fundamentallydierent ways. The meaning of these terms will be clari-
+ed momentarily, but the relevance to coal-generated elec-
+tricity is that coal is essentially a commodity, whereas the
+construction cost of a plant is closer to (though not purely)
+a technology.
+A commodity is a raw material or primary agricultural
+product that can be bought and sold. A standard assump-
+tion in the theory of nance is that markets are ecient;
+roughly speaking, this means it should not be possible to
+make consistent prots by arbitrage of commodities using
+simple strategies. If the price of coal were too predictable
+using simple methods, such methods would become com-
+mon knowledge, and the buying and selling activity of
+speculators would aect prices in a way that would de-
+stroy their predictability. More specically, one expects
+prices to follow a random walk to some approximation.
+Furthermore, the activity of speculators should bound the
+long-term rate of growth or decline of the price. Otherwise
+it would be possible to make unreasonable prots { more
+reliably than could be made in alternative investments {
+by either buying coal and hoarding it or by short selling
+coal.11
+A pure technology is a body of knowledge, such as
+knowledge of a manufacturing process. We will say a
+product behaves like a pure technology when accumulated
+knowledge, re
ected in changes to the technology's design,
+is a greater determinant of its cost evolution than spec-
+ulation. For example, the fuel component of coal-red
+plants was initially more pure technology-like. Changes
+to boiler and plant design [53] resulted in growth of ther-
+mal eciency and pure technology-like cost evolution in
+the fuel component (which depends on the thermal e-
+ciency.) Although the input cost of coal could and did
+change, its changes had less impact on the fuel component
+cost than these pure design changes. In the present day,
+with many of the eciency improving design changes al-
+ready exploited, changes to input costs determine changes
+in the fuel component to a greater extent, making the fuel
+component more commodity-like.12
+Commodities and pure technologies are ends of a spec-
+trum, and probably no real products are completely one
+or the other. The specic construction cost, for example,
+depends on construction technologies { that knowledge of
+11Other ways to speculate on the price of coal are to buy or sell
+coal companies and to buy or sell land containing coal deposits.
+12It's also reasonable to wonder the extent to which coal itself
+is a pure technology versus a commodity. This question is further
+complicated by the fact that coal is in
uenced by scarcity factors
+as well as technological factors. Anecdotal evidence makes it clear
+that both of these are in
uential. Both of these factors can, and
+probably did, cause changes in the cost/price of coal. (Although
+it is a bit interesting that they should stay so balanced with each
+other over a 130 year period. We say this noting that the dramatic
+rise in the 70s was largely due to exceptional circumstances having
+little to do with scarcity or technology factors.) Nevertheless, the
+theory of nance bounds the anticipated long-term changes that can
+occur. This allows for unanticipated changes that may have any size.
+It also allows for anticipated long term changes that are suciently
+slow that they are not worth taking advantage of.
+12construction process and design which separates ecient
+uses of materials and labor input from inecient uses.
+However, specic construction cost also depends on mate-
+rial commodities, such as steel. Thus specic construction
+cost is partly a technology and partly a commodity. Like-
+wise coal is not completely a commodity, since knowledge
+of coal mining methods also aect its price, in addition to
+the speculation mechanisms mentioned above. A third ex-
+ample of this mixed composition is a photovoltaic system.
+The price of a photovoltaic (PV) system depends in part
+on commodities, such as metals and silicon. Increases in
+the price of silicon, for example, caused increases in the
+cost of PV cells in 2005-2006. Nonetheless, silicon is far
+from a raw material { processing dominates the cost of
+mono and polycrystalline silicon, rather than commodity
+input costs. Thus, while the price of PV cells is partially
+commodity-driven, it may behave more like a pure tech-
+nology than a commodity. Its historical behavior suggests
+this is true (see the silicon price time series in Nemet [41]),
+though this possibility deserves further investigation.
+5.2. Modeling a commodity
+The price of a commodity is usually modeled using time
+series methods. Time series models form a class of nested
+models that can be arbitrarily complex, but the simplest
+model and conventional starting point is an autoregressive
+model of order 1 (AR(1).)
+The AR(1) model is dened by the equation
+pt=
pt1++"t:
+Hereptis the logarithm of the price in year t,"tis a
+random variable, and 
andare parameters. The noise
+term"tis assumed to be uncorrelated in time and normally
+distributed with variance 2
+". The three parameters 
,,
+and", combined with an initial condition p(0), determine
+a given AR(1) process.13
+The parameter 
determines the long-term behavior of
+the process. When 
= 1, the process is a random walk
+with drift, and when  > 1 the mean of the distribution
+of future prices grows without bound. When 
 < 1, the
+process is stationary and the distribution of future prices
+asymptotically has a xed variance. The closer 
is to
+1, the larger the variance of the limiting distribution be-
+comes. For 
= 1, the parameter generates drift in the
+random walk, but for 
 <1 it sets a non-zero mean value
+around which the price moves.
+To see how closely the coal price resembles a random
+walk, we t the historical coal price to the AR(1) model
+using least squares regression. The parameter values ob-
+tained are
:956:053,:144:174, and":080.
+Note that
has an error interval that includes 
= 1. Since
+
= 1 has qualitatively dierent behavior than 
 <1, it is
+13\Order one" refers to the number of earlier prices the model
+regresses the current price onto. An AR(3) model, for example,
+would be specied as pt=
1pt1+
2pt2+
3pt3++"t.important to know the likelihood that the underlying pro-
+cess actually has 
= 1 and is therefore a random walk.
+We apply various Dickey-Fuller tests of the null-hypothesis
+that the underlying process is a random walk, and nd
+that in no test can this possibility be rejected. Thus, our
+hypothesis is validated in two dierent ways: First, to
+within statistical error 
1, suggesting a random walk,
+and second, to within statistical error the drift rate 0,
+suggesting no long term trend in coal prices.
+5.3. Modeling a technology
+To complete our discussion contrasting commodities
+and technologies, we describe the two most common mod-
+els, experience curves (which are typically assumed to be
+power law) and extrapolation of time trends (which are
+typically assumed to be exponential).
+The cost of a technology is often modeled in terms of
+an experience curve, a plot of its cost versus its cumula-
+tive production. The motivation for plotting cost against
+cumulative production often uses the following chain of
+reasoning: cumulative production measures \experience"
+with a technology; as a rm or industry gains experience it
+makes improvements to the technology, which cause cost
+reductions; therefore there may be a simple, predictable
+relationship between cumulative production and cost. Al-
+ternatively, one can argue that cumulative production is
+an indicator of prot-making potential, which drives the
+level of eort directed at improving a given technology [22].
+In fact, a simple relationship is frequently observed:
+empirical experience curves often appear to obey power
+laws. There is a large literature attesting to this regu-
+larity [8, 50]. Partly for this reason, the experience curve,
+combined with the assumption of a power law form, has be-
+come the prevailing method for extrapolating future costs
+[20]. The power law functional form can also be derived
+from theoretical models [37, 36].
+We nd it useful to distinguish the data from any par-
+ticular hypothesis about its shape. We therefore use the
+term experience curve to mean the plot of cost versus cu-
+mulative production, whatever its shape. We reserve the
+term Wright's law for the hypothesis that an experience
+curve should follow a power law. While we acknowledge
+that many researchers implicitly include the power law
+functional form with the denition of an experience curve,
+we prefer to refer to the data in a theory-neutral way.
+Alternatively, the cost of technologies has been mod-
+eled as decreasing exponentially with time; this evolu-
+tion can be viewed as a generalization of Moore's law.14
+Moore's law (an exponential decrease in cost with time)
+14See Nagy et al. [38]. Moore's law originally states that the
+number of transistors per integrated circuit doubles about every two
+years on average. This regularity can be restated in terms of the cost
+per transistor, which also decreases exponentially over time. The
+new statement of Moore's law has the advantage of being expressed
+in general terms comparable across technologies (i.e. cost versus
+time).
+13and Wright's law (a power law decrease in cost with cumu-
+lative production ) are not necessarily incompatible. They
+could simultaneously hold when cumulative production grows
+exponentially with time.15
+Two kinds of criticisms have been brought against the
+program of plotting cost data and tting it to curves.16
+One is that it buries too much detail about the processes
+driving cost reductions [22]. Another criticism questions
+the predictive power of dierent functional forms, since
+they have not been rigorously compared against each other.
+Further progress in this area requires both careful com-
+parison of the predictive ability of dierent eective mod-
+els and study of the causes underlying observed relation-
+ships. Nagy et al. have recently compared the historical
+prediction accuracy of various functional forms proposed
+by researchers to describe cost evolution [38], including
+Wright's law and Moore's law.
+We note that any mathematical description of a tech-
+nology's cost evolution is likely to fail when the technol-
+ogy itself substantially changes. A famous example is the
+Model T Ford, which dropped smoothly in cost from its in-
+troduction in 1909, when Henry Ford announced he would
+make a car that the common could aord, to 1929, when he
+ceased production. During this period the cost of Model
+T's follow Wright's law quite well [1]. After that Ford
+produced other models with better performance and, not
+surprisingly, higher costs. This is an important point; in
+general an extrapolation may only hold when the prod-
+uct faces xed performance criteria. When a product is
+called upon to meet criteria it did not previously meet,
+such as lower pollution or higher safety, its cost evolution
+may show discontinuous behavior.
+The two models presented are the most common mod-
+els for what we call technology-like cost evolution. The
+important property they share is that both involve pre-
+dictable, long-term, decreasing costs. With this in mind,
+we now return to the specic case of coal-red electricity.
+5.4. Application to coal-red electricity
+Plant construction costs have not dropped signicantly
+in 40 years. Pollution controls have redened coal plants
+to such a degree that they are not really the same product
+as that produced in the rst 80 years. But let us imagine
+what would happen if coal plant construction costs were
+to revert to the same technology-like cost evolution they
+followed during the rst 80 years. Plant costs would drop
+again, and dropping plant costs would cause fuel { under-
+going commodity-like evolution { to become the dominant
+cost. The dominance of fuel costs would in turn lead to
+greater sensitivity of generation costs to 
uctuations in the
+price of coal. Thus, the price of coal would determine a
+
uctuating 
oor on coal-red generation costs.
+15See, for example, Sahal [45].
+16These criticisms have often been directed against experience
+curves in particular, but could apply to any low-dimensional model
+of cost evolution.This scenario assumes no dramatic change in thermal
+eciency. As mentioned in Section 2.3.4, eciency in the
+U.S. has remained 
at the last 50 years for several rea-
+sons, while it has increased elsewhere. However, it would
+take a substantial increase in eciency to yield a substan-
+tial reduction in fuel costs | of a magnitude comparable
+to historical reductions. For example, if the average e-
+ciency increased from 33% to 43%, the fuel cost would be
+decreased by a factor of (1 =2)=(1=1) = (1=:43)=(1=:33) =
+0:77, a far cry from the (physically unrepeatable) factor-
+of-10 improvement from the start o the industry.
+For any plant design, once eciency improvements have
+been exhausted, the price of coal will set a 
oor on total
+costs. If we consider the possibility of carbon capture and
+storage| which is likely to increase capital costs initially|
+the potential reduction in total costs over time would simi-
+larly be limited by the 
uctuating 
oor dened by the coal
+price.
+6. Conclusions
+We make several methodological advancements in this
+paper. We consider total generation costs (decomposed
+into components) rather than costs of single components
+alone, use data over a relatively long time span (over a
+century and covering the lifetime of the industry), and use
+a physically accurate bottom-up model of costs. As part
+of this decomposition, we model coal prices as a random
+walk while modeling construction costs and O&M costs as
+an improving technology.
+We nd evidence that the fuel and capital costs of coal-
+red electricity evolve dierently due to dierent behaviors
+in coal prices and plant construction costs. Coal prices
+have 
uctuated with no trend up or down, while plant
+construction costs have followed long-term trends. The
+behavior of coal prices is consistent with the facts that
+coal is a freely traded commodity. Plant construction cost
+may follow a pattern of long term reduction because it is
+only weakly in
uenced by these eects and is able to realize
+improvements typically seen in technologies over time.
+Such a dierence in the behavior of coal prices and
+plant construction costs would yield dierent behavior for
+the fuel and capital cost components. Although histori-
+cally both the fuel and capital components improved at
+similar rates, the main driver of improvement in fuel costs
+{ thermal eciency { has been unchanged since the early
+1960s. Without a substantial improvement in the average
+thermal eciency, the main driver of change in fuel costs
+would be the coal price. Coal prices, in contrast, are sta-
+tistically neither decreasing nor increasing, and so provide
+a statistically 
uctuating 
oor on the overall generation
+cost, with no clear long-term trend.
+Although all the analysis in this paper is specic to
+coal, we hypothesize that a similar analysis might apply to
+other fossil fuel-based sources of electricity, such as natural
+gas. In fact, because natural gas and oil are both traded on
+exchanges, with standardized futures contracts, it is easier
+14to speculate in them than it is in coal. Thus we would
+expect that the commodity part of our model will apply
+to them even more strongly. This would suggest that the
+cost of the major fossil fuel-based sources of electricity are
+all constrained by a noisy 
oor determined by fuel prices.
+7. Acknowledgments
+We gratefully acknowledge nancial support from NSF
+Grant SBE0738187. All conclusions presented here are
+those of the authors and do not necessarily re
ect those
+of the NSF. We are grateful to Margaret Alexander and
+Tim Taylor for substantial help obtaining source materials.
+We thank Arnulf Gr ubler for helpful conversations and
+suggestions. We thank Charlie Wilson for several valuable
+conversations and extensive feedback on an early draft,
+and Dan Schrag for a challenge that led to the idea for
+this paper. We also thank anonymous reviewers for their
+insightful comments.
+A. Change decomposition
+Consider the following problem. We have a function
+f=f(x;y), and during some period of time fchanges
+as a result of simultaneous changes to xandy. We want
+to know how much of the change to feach variable is
+\responsible for."
+To be more precise, let 
 fbe the change in f. We
+would like to decompose 
 finto 2 terms, corresponding
+to the change contributed by each variable:
+
f= 
fx+ 
fy;
+where 
fidenotes the change in fresulting from the
+change ini. A way to do this decomposition is suggested
+by the calculus identity
+df=@f
+@xdx+@f
+@ydy:
+The trick is to generalize this expression to nite rather
+than innitesimal changes. More to the point, we need
+to be able to take common combinations of variables {
+e.g. products, quotients { and express changes to these
+combinations in ways appropriate for nite dierences.
+For example, consider a product of two variables, f(x;y) =
+xy. The dierential fis
+df=xdy +ydx
+suggesting the decomposition for nite dierences is 
 f=
+x
y+y
x. However, the correct expression is
+
f=x
y+y
x+ 
x
y:
+In addition to the two expected terms, a third cross term
+containing both dierences appears. As the dierences
+become small, the cross term will vanish more quicklythan the other terms, recovering the calculus limit df=
+xdy+ydx. However, for nite dierences, the cross term
+potentially introduces a large residual if ignored.
+In order to decompose the change in finto just 2
+pieces, we evenly split the cross term into the other terms:
+
f=x
y+1
+2
x
y+y
x+1
+2
x
y
+= (x+1
+2
x)
y+ (y+1
+2
y)
x
+The rst term may be interpreted as the change in fdue
+to change in y; the second as the change in fdue to change
+inx. We therefore have our desired decomposition for the
+case of products of 2 variables:
+
fx(y+1
+2
y)
x
+
fy(x+1
+2
x)
y
+which by construction has the desired property 
 f=
+
fx+ 
fy. Similar rules can be derived for products of 3
+or more variables, for quotients, and other expressions.
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\ No newline at end of file
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+arXiv:1001.0606v3  [cond-mat.mtrl-sci]  22 Mar 2010Spin dynamicsin semiconductors
+M.W. Wu∗, J. H.Jiang,M.Q. Weng
+Hefei National Laboratory for Physical Sciences atMicrosc ale, University ofScience and Technology ofChina, Hefei, A nhui, 230026, China
+Department ofPhysics, University ofScience and Technolog y of China, Hefei, Anhui, 230026, China
+Abstract
+This article reviews the current status of spin dynamics in s emiconductors which has achieved a lot of progress
+in the past years due to the fast growingfield of semiconducto rspintronics. The primary focus is the theoretical and
+experimentaldevelopmentsofspinrelaxationanddephasin ginbothspinprecessionintimedomainandspindi ffusion
+andtransportinspacial domain. A fullymicroscopicmany-b odyinvestigationon spindynamicsbasedon thekinetic
+spinBlochequationapproachisreviewedcomprehensively.
+Keywords: spindynamics;spin-orbitcoupling;spinrelaxation;spin dephasing;spindiffusion;spintransport;
+electron-electronCoulombscattering;magneticsemicond uctors;magnetizationdynamics;ultrafastphenomena;
+quantumwell; quantumwire;quantumdot;many-bodye ffect; highfieldeffect.
+1. Introduction
+Since the pioneering works by Lampel [1] and Parsons [2] and t he following extensive experimental and theo-
+retical works at Ioffe Institute in St. Petersburg and Ecole Polytechnique in Par is in 1970s and early 1980s, a lot of
+understanding on spin dynamics in semiconductors has been a chieved. Some basic spin relaxation and dephasing
+mechanismshavealso beenproposedatthat time. Anicerevie wonthesefindingscanbefoundinthebook “Optical
+Orientation” [3].
+Starting from the late 1990s, there is a big revival of resear ch interest in the spin dynamics of semiconductors,
+jump-started by some nice experimental works by Awschalom a nd co-workers [4, 5]. An extensive number of ex-
+perimental and theoretical investigations have been carri ed out on all aspects of spin properties. Properties of spin
+relaxation and dephasing in both the time domain (in the spac ial uniform system) and the spacial domain (in spin
+diffusion and transport) have been fully explored in di fferent materials at various conditions (such as temperature ,
+externalfield,dopingdensity /materialandstrain)anddimensions(includingbulkmateri als,quantumwells,quantum
+wires and quantum dots). These materials include III-V and I I-VI zinc-blende and wurtzite semiconductorsand sil-
+icon/germanium, as well as diluted magnetic semiconductors. The spin relaxation and dephasing mechanisms have
+been rechecked and reinvestigated at di fferent conditions. The spin-related material properties ha ve also achieved
+muchprogressalongwiththeseinvestigationswhichfacili tatethebetterunderstandingoftheobservedphenomenaas
+wellasthemanipulationofthespincoherence. Manynovelsp in-relatedproperties,suchasthespinHalle ffect[6–12],
+spin Coulomb drag e ffect [13–15], spin photogalvanic e ffect [16, 17] and persistent spin helix e ffect [18–20], have
+been discovered. For the sake of realizing the spintronic de vices such as spin transistors, there have been extensive
+investigationsonspininjectionanddetection. Muchprogr esshasbeenachievedonthespininjectionfromferromag-
+netic materials into semiconductors. Nevertheless, a sati sfactory realization of spin transistor which is crucial to the
+applicationofsemiconductorspintronicsisyetto come.
+Despitedecadesofstudies,theoreticalunderstandingoft hespinrelaxationanddephasingwhichisoneofthepre-
+requisitesoftherealizationofspintronics,inbothspaci aluniformandnonuniformsystemsisstill notfullyachieve d.
+∗Corresponding author.
+Emailaddress: mwwu@ustc.edu.cn (M. W.Wu)
+Preprintsubmitted to Physics Reports March 24,2010Manynewexperimentalfindingsgofarbeyondthepreviousthe oreticalunderstandings,thankstothefastdevelopment
+of experimental techniques including the sample preparati ons and ultrafast optical techniques. Some of the physics
+requires a many-body and /or non-equilibrium (even far away from the equilibrium) the ory. However, the previous
+theories [3] are all based on the single-particle approach a nd for systems near the equilibrium which thus have their
+strict limits in applying to the nowadays experiments. Some even give totally incorrect qualitative predictions. Wu
+et al. developeda fully microscopic many-bodykinetic spin Bloch equation approach to study the spin dynamicsin
+semiconductorsandtheirnanostructures. Unlikeotherapp roacheswidelyused inthe literaturewhichtreatscatterin g
+using the relaxation time approximation,the kinetic spin B loch equation approach treats all scattering explicitly an d
+self-consistently. Especially, the carrier-carrierCoul omb scattering is explicitly includedin the theory. This al lowed
+them to study spin dynamics not only near but also far away fro m the equilibrium, for example, the spin dynamics
+in the presence of high electric field (hot-electronconditi on)and/orwith large initial spin polarization. They applied
+this approachto study spin relaxation /dephasingand spin di ffusion/transportin variouskinds of semiconductorsand
+their nanostructures under diversified conditions and have offered a complete and systematic understanding of spin
+dynamicsinsemiconductors. Manynovele ffectswerepredictedandsomeofthemhavebeenverifiedexperi mentally.
+In this review, we try to provide a full coverage of the latest developmentsof spin dynamics in semiconductors.
+Both experimental and theoretical developmentsare summar ized. In theory, we first review the studies based on the
+single-particleapproach. Thenweprovideacomprehensive reviewoftheresultsbasedonthekineticspinBlochequa-
+tionapproach. Weorganizethepaperasfollows: Wefirstrevi ewthespininteractionsinsemiconductorsinSection2.
+Then we briefly discuss the physics of spin dynamics in semico nductors in Section 3. Here we review the previous
+understandingonspinrelaxationanddephasingmechanisms . Differingfromtheexistingnicereviewsonthesetopics
+[21–23], we try to providea full and up-to-datepicture of sp in dynamicsof itinerant carriers in semiconductors. We
+thenreviewthelatestexperimentaldevelopmentsonspinre laxationanddephasingintimedomainaswellasthelatest
+theoreticalinvestigationsbasedonthesingle-particlea pproachinSection4. Thetheoreticalinvestigationsofthe spin
+relaxation and dephasing in the spacial uniform systems bas ed on the kinetic spin Bloch equation approach together
+withtheexperimentalverificationsarereviewedinSection 5. InSection6we reviewthelatest experimentalandthe-
+oretical(single-particletheory)results on spin di ffusionand transport. Thenin Section 7 we review the investig ation
+on spin diffusion and transport using the kinetic spin Bloch equation ap proach and the related experimental results.
+We summarizeinSection8.
+2. Spin interactionsin semiconductors
+2.1. Ashort introductionofspin interactions
+Before introducing the spin interactions, it should be ment ioned that the “spins” in semiconductors are not pure
+spins, or in other words,they are angularmomentumswhich co nsist of bothspin and orbitalangularmomentums. A
+directconsequenceisthat the g-factorisnolonger g0=2.0023ofpureelectronspins.
+There are various spin interactions in semiconductors. Bes ides the Zeeman interaction, there are spin-orbit cou-
+plingduetothespaceinversionasymmetryorstrain, s(p)-dexchangeinteractionwithmagneticimpurities,hyperfine
+interaction with nuclear spins, spin-phonon interaction, electron-hole and electron-electron exchange interactio ns.
+Among these interactions, spin-orbit couplingusually pla ys the most important role in spin dynamics. For electrons
+in conduction band, spin-orbit coupling consists of the Dre sselhaus, Rashba and strain-induced terms. For valence
+band electrons, besides these terms, there is an important s pin-orbit coupling from the spin-dependent terms in the
+Luttinger Hamiltonian in cubic semiconductors. These spin -orbit couplings make the motion of carrier spin couple
+with the orbital motion and give rise to many novel e ffects in coherent (ballistic) and dissipative (di ffusive) electron
+spin/chargedynamics. A direct consequenceis that the spin polar izationof a wave packet oscillates during its prop-
+agation [24–31]. Recent studies indicate that as the preces sion during the electron propagation correlates the spin
+polarization with real-space trajectory, the spin lifetim e is enhanced at certain spacial inhomogeneousspin polariz a-
+tion states, such as, some spin grating states [18, 20, 32–37 ]. Another important spin interaction is the exchange
+interaction between carriers due to the permutative antisy mmetry of the many-body fermion wavefunctions. It has
+beendemonstratedtobeimportantforspindynamicsofelect ronsinquantumdots[38]andlocalizedelectronsbound
+to impurities [39, 40]. The e ffect of the exchange interactions between freeelectrons on spin dynamics was only
+observed in experiments very recently and has attracted int erest [41–43], though it has been predicted much earlier
+[44].
+2Spin interactions together with the induced spin structure s are the physical foundations of the vital topic of co-
+herent spin manipulation which is crucial for spin-based qu antum information processing [45–52] and spintronic
+deviceoperation[22,53–56]. Forexample,spin-orbitcoup lingsenablecoherentmanipulationofspinbyelectricfield
+[53, 57–76]. As electric field is more easily accessible and c ontrollable in genuine electronic devices at nanoscale,
+theelectricalmanipulationofspinsisimportantforsemic onductorspintronics. Theideahasbeenextendedtoa large
+varietyofschemesof electricalor opticalmanipulationof spincoherence: coherentcontrolvia electrical modulatio n
+ofg-tensorinnanostructureswithspacial-dependent g-tensor[77–81];andsimilarschemebasedonspatiallyvary ing
+magnetic field [82, 83], or hyperfine field [70, 84, 85]; cohere nt manipulation based on tuning the exchange inter-
+action between electrons in a double quantum dot by gate-vol tage [38, 86]; coherent spin manipulation via detuned
+opticalpulseswhichactase ffectivemagneticpulsesastheStarkshiftisspin-dependent duetospin-dependentvirtual
+transition of the circularly polarized light [87–95]. The p-dors-dexchange interactionsand the induced spin struc-
+tures further enable the above schemes to control Mn ion spin s in paramagnetic phase [96–98] or magnetization in
+ferromagneticphase[99, 100]inmagneticsemiconductorsv iamanipulatingcarrierspins.
+Oneofthekeyobstaclesforspintronicdeviceoperationand spin-basedquantuminformationprocessingisthespin
+relaxationandspin dephasing(i.e., thedecayof the longit udinalandtransversespincomponents,respectively). Spi n
+relaxationandspindephasing,inprinciple,alsooriginat efromthespininteractions: thefluctuationorinhomogenei ty
+inspin interactionsleadsto spinrelaxationandspindepha sing.1Forexample,the spin-orbitcouplingis theoriginof
+oneof the most efficient spin relaxationmechanisms–theD’yakonov-Perel’me chanism[101, 102]. Spin mixingdue
+to conduction-valenceband mixing together with momentum s cattering give rise to the Elliott-Yafet spin relaxation
+mechanisms[103,104]. Otherspinrelaxationmechanismsar ethes(p)-dexchangemechanism,theBir-Aronov-Pikus
+mechanism [105, 106] due to the electron-holeexchangeinte raction [107], the hyperfineinteraction mechanism and
+the anisotropicexchangeinteractionsbetweenlocalizede lectrons[39, 40]. Spin relaxationand dephasingin genuine
+semiconductorstructuresaretheconsequenceofthecoacti onofthesemechanisms. Henceitisimportanttodetermine
+thedominantspinrelaxationmechanismundervariouscondi tions[3,23, 39,108–112].
+Finally, as the focus of this review is spin dynamicsin semic onductors,we only provide an overviewfor various
+spin interactions and explain the related physics using int uitive and qualitative pictures. For details, we suggest th e
+salient worksin the existing literature: for spin-orbitco uplingin semiconductors,the readerscould referto the boo k
+byWinkler[113];forknowledgeofthe s(p)-dexchangeinteractions,wesuggestthereviewarticlebyJun gwirthetal.
+[114]andthatbyFurdyna[115]andreferencestherein;theh yperfineinteractioninsemiconductorshasbeenreviewed
+recently by Fischer et al. [116]; details of the spin-phonon interaction can be found in Ref. [117]; the electron-hole
+exchangeinteractionhasbeendiscussedin detailinRefs. [ 107, 118, 119].
+2.2. Spin-orbitinteractionsin semiconductorsandtheirn anostructures
+A consequenceof the relativistic e ffect in atomic and condensed matter physics is that an electro n movingin the
+atomicpotential V(r)feelsaneffectivemagneticfieldactingonitsspin
+HSO=1
+4m2
+0c2p·[σ×(∇V(r))], (1)
+wherem0andcrepresent the free electron mass and the velocity of light in vacuum, respectively, σis the Pauli
+matrices, and pstands for the canonical momentum. V(r) is the total potential generated by other charges. In semi-
+conductors, V(r)includestheperiodicpotentialsgeneratedbytheion-cor e,thedeviationoftheperiodicpotentialdue
+todefectsandphonons,andtheexternalpotential. Thisist heoriginofvariousspin-orbitinteractionsinsemiconduc -
+tors.
+2.2.1. Electronspin-orbitcouplingterm in semiconductor sdueto spaceinversionasymmetry
+Insemiconductorswithspaceinversionsymmetry,theelect ronspectrumsatisfiesthefollowingrelation, εk↑=εk↓.
+This is the consequence of the coaction of the time-reversal symmetry and the space-inversion symmetry. In the
+absence of space inversion symmetry, εk↑/nequalεk↓. In this case, there should be some term which breaks the spin
+1A practical operating scheme should both control the spin in teractions and avoid their fluctuation and inhomogeneity ca refully. Optimization
+of theability of coherent control and spin lifetime is often needed.
+3degeneracyof thesame kstates. Thetermshouldbe anoddfunctionofboth kandσasit breaksthe space inversion
+symmetry whereas keeps the time reversal symmetry. This ter m is the so-called the spin-orbit coupling term. For
+electronwith spin S=1/2,asσiσj=δi,j+iǫijkσk(i,j,k=x,y,zandǫijkis Levi-Civitatensor),onlythe linear form
+ofσicanappearinthe spin-orbitcouplingterm. Hence,theonlyp ossibleformofthespin-orbitcouplingtermis
+HSO=1
+2Ω(k)·σ, (2)
+whereΩ(k) is an odd function of k, which acts as an e ffective magnetic field. In a reversedlogic, any electron spin -
+orbitcouplingwhichcanbewrittenintheform HSO=/summationtext
+n,m,i,jCij
+nmσn
+ikm
+j,canbereducedtotheformofEq.(2). Hence
+thebreakingofthespaceinversionsymmetryisaprerequisi te2fortheappearanceofelectronspin-orbitcouplingterm
+insemiconductors.3
+Insemiconductors,thereare severalkindsofspaceinversi onasymmetry: (i)thebulkinversionasymmetrydueto
+structureslackinginversioncenter[123],suchaszinc-bl endeIII-VorII-VIsemiconductors;(ii)thestructureinve rsion
+asymmetryresultingfromtheinversionasymmetryofthepot entialsinnanostructuresincludingexternalgate-voltag e
+and/orbuilt-inelectric field [124, 125]; and(iii) interfacein versionasymmetryassociated with the chemicalbonding
+within interfaces [126, 127]. Besides, strain can also indu ce inversion asymmetry and leads to spin-orbit coupling.
+The explicit form of the spin-orbit couplings due to these sp ace inversion asymmetries can be obtained, e.g., by the
+k·ptheory[113].
+2.3. Explicitformof thespin-orbitcouplingterms insemic onductors
+The explicitformof thespin-orbitcouplingin semiconduct orscanbe derivedfromthe k·ptheory. Forcommon
+III-V and II-VI semiconductors, such as GaAs, InAs, CdTe and ZnSe, the minimum k·ptheory to describe carrier
+andspindynamicsistheeight-bandKanemodel[113,128]. In thefollowing,weintroducethespin-orbitcouplingin
+theframeworkofsuchmodel.
+2.3.1. KaneHamiltonianandblock-diagonalization
+Inzinc-blendestructures,the Kanemodelisofthefollowin gform[117]
+HKane=HcHc,vHc,sv
+H†
+c,vHvHv,sv
+H†
+c,svH†
+v,svHsv. (3)
+The subscript c,v,svdenote the conduction, valence and the split-o ffvalence bands respectively. As Hc,v,Hv,sv
+andHc,svare nonzero, the electron motions in these bands are coupled together, which are obviously too complex.
+Fortunately, the couplings between these bands are much sma ller than their separations. By a perturbative block-
+diagonalization,one can decouple the motions in these band s. The block-diagonalizationis achieved by the L¨ owdin
+partitionmethod[129]whichisactuallyaunitarytransfor mation. Forexample,theeigen-energy En(k)andeigen-state
+Ψn(k)oftheoriginalKaneHamiltoniansatisfy thefollowingeig en-equation
+HKaneΨn(k)=En(k)Ψn(k). (4)
+Afteraunitarytransformation,
+˜HKane˜Ψn(k)=En(k)˜Ψn(k), (5)
+where˜HKane=UHKaneU†is block-diagonal. As the couplings between bands Hc,v,Hv,svandHc,svare small near
+the center of the concernedvalley (For most of the III-V and I I-VI semiconductors,the concernedvalley is Γvalley,
+2Note that, it was found that there is intersubband spin-orbi t coupling even in symmetric quantum wells [120, 121], which seems to contradict
+the above conclusion. However, even in symmetric quantum we lls, the space inversion symmetry is brokenat the boundary of the quantum well
+[120, 121].
+3Differently, hole has spin J=3/2, where J2
+i/nequal1. Hence, terms which are quadratic in Jcan appear in the spin-orbit coupling. These terms
+shouldbeeven in kinordertokeepthetime-reversal symmetry. Hence,holecan havespin-orbitcoupling termswhichdonotbreakspaceinve rsion
+symmetry. Actually, the Luttinger Hamiltonian [122] only c ontains spin-dependent terms quadratic in k.
+4wherek=0 is the valley center.), the results can be given in a perturb ative expansion series. Below, we give the
+resultswithonlythelowestordernontrivialtermskept,wh ichareenoughforthediscussionofspindynamicsinmost
+cases. Besides the transformationof the Hamiltonian,the w avefunctionalso changesslightly, ˜Ψn(k)=UΨn(k). This
+modificationof the wavefunctionalways leads to the mixing o f different spin states. In the presence of spin-mixing,
+anymomentum scattering can lead to spin flip and spin relaxation , which gives the Elliott-Yafet spin relaxation
+mechanism[103,104].
+Inthefollowing,wepresenttheconduction(electron)andv alence(hole)bandpartsoftheblock-diagonalHamil-
+tonian˜HKane.
+2.3.2. ElectronHamiltonianandspin-orbitcouplingin bul ksystem
+By choosing x,y,zaxes along the crystal axes of [100], [010] and [001] respect ively, the electron Hamiltonian
+canbewrittenas
+He=k2
+2me+1
+2ΩD·σ+1
+2ΩS·σ, (6)
+wheremeis the electron (conduction band) e ffective mass. The second and third terms in the right hand side of
+the above equation describe the Dresselhaus spin-orbit cou pling [123] and strain-induced spin-orbit coupling [117]
+respectively.
+ΩDx=2γDkx(k2
+y−k2
+z),ΩSx=2C3(ǫxyky−ǫxzkz)+2Dkx(ǫyy−ǫzz), (7)
+withothercomponentsobtainedbycyclicpermutationofind ices. HereγDistheDresselhauscoe fficient,whichcanbe
+expressedformallyas γD=2η/[3mcv/radicalbig2meEg(1−η/3)][117],where η=∆SO/(Eg+∆SO)withEgand∆SObeingthe
+bandgapand the spin-orbitsplitting of the valence band res pectively. mcvis a parameter with mass dimension and is
+relatedtotheinteractionbetweentheconductionandvalen cebands[117]. Thecoe fficientsγD,C3andDarecrucialfor
+spin dynamics. There have been a lot of studies on these quant ities, especially γD, in commonIII-V semiconductors
+(such as GaAs, InAs, GaSb and InSb) both theoretically and ex perimentally [117, 130–141]. Interestingly, these
+studies give quite di fferent values of γDin GaAs from 7.6 to 36 eV ·Å3. Detailed lists of γDin GaAs in the literature
+are presented in Refs. [139–141]. Latest theoretical advan cementsinclude the first principle calculations [139, 141]
+and full-Brillion zone investigations [141, 142]. Recentl y, from fitting the magnetotransport properties in chaotic
+GaAs quantum dots [140] and from fitting the spin relaxation i n bulk GaAs via the fully microscopic kinetic spin
+Bloch equation approach [110], γDin GaAs was found to be 9 and 8 .2 eV·Å3respectively, which are close to the
+valueγD=8.5 eV·Å3fromab initiocalculations with GW approximation[139].4The coefficient for strain-induced
+spin-orbit coupling is C3=2c2η/[3/radicalbig2meEg(1−η/3)], where c2is the interband deformation-potential constant.
+Dcomes from higher order corrections which is usually smalle r thanC3. The strain-induced spin-orbit coupling
+was studied experimentally in Refs. [77, 144], indicating t hat the strain induced spin-orbit coupling can be more
+importantthan the Dresselhausspin-orbit couplingand tha t both the C3andDterms can be dominant. Recently, the
+coefficientsC3andDfromab initiocalculationsshowed goodagreementwith the experimentalr esults [145], where
+C3=6.8eV·ÅandD=2.1eV·Å.
+2.3.3. Electronspin-orbitcouplingin nanostructures
+Electric field can also break the space inversion symmetry an d lead to additional spin-orbit coupling [124, 125].
+In III-V and II-VI heterostructures the induced spin-orbit coupling, named the Rashba spin-orbit coupling, can be
+as important as (or even more important than) the Dresselhau s spin-orbit coupling due to bulk inversion asymmetry
+[146, 147]. The effect is causedbythe total electric field, includingthe exter nalelectric field dueto gate-voltage,the
+electric field due to built-in electrostatic potential and t he electric field due to interfaces [113]. The leading e ffect of
+interfaces is to produce a spacial variation of band edge, wh ich thus leads to an e ffective electric field. Via L¨ owdin
+partitionmethod,oneobtains
+HR=α0σ·(k×Ev(r)). (8)
+Hereα0=eη(2−η)P2/(3m2
+0E2
+g) withPbeingthe interbandmomentummatrix[113], Ev(r)=∇Vv(r)/|e|whereVv(r)
+isthepotentialfelt byvalenceelectron. Averaging Ev(r)over,e.g.,the subbandwavefunctionofaquantumwell,one
+4There areother recent papers [141,143] with γDclose to 8.5 eV·Å3.
+5has
+HR=αRσ·(k׈n), (9)
+whereαR=α0∝an}b∇acketle{tEv(r)∝an}b∇acket∇i}htisthe Rashbaparameterand ˆnisa unitvectoralongthegrowthdirectionofthequantumwel l.
+Historically, there is a paradox about the Rashba parameter . According to the Ehrenfest Theorem, the average
+of the force acting on a bound state is zero. Therefore, the Ra shba parameter should be very small. This paradox is
+resolvedby Lassnig [148], who pointed out that for the Rashb a spin-orbitcouplingin conductionband, αRis related
+with the average of the electric field in valence band over the conduction band (subband) wavefunction. As pointed
+outabove,theelectricpotentialonthevalenceorconducti onbandcomprisesthreeparts,
+Vv(r)=Vext(r)+Vbuilt(r)+Ev(r),Vc(r)=Vext(r)+Vbuilt(r)+Ec(r), (10)
+whereEc(r) andEv(r) are the positiondependentconductionand valenceband edg es.Vbuilt(r) is the built-inelectro-
+static potential and Vext(r) is the external electrostatic potential. Let VCoul(r)=Vext(r)+Vbuilt(r) [total electrostatic
+(Coulomb)potential]. Accordingto the Ehrenfesttheorem, onehas∝an}b∇acketle{t∇VCoul(r)∝an}b∇acket∇i}ht=−∝an}b∇acketle{t∇Ec(r)∝an}b∇acket∇i}ht,as theaverageoverthe
+(conduction)electronwavefunction(denotedas ∝an}b∇acketle{t...∝an}b∇acket∇i}ht)ofthenetforcefeltbyelectroniszero. Iftheratioofthev alence
+bandoffsettothe conductionbandoneis rvc, then
+∝an}b∇acketle{t∇Vv(r)∝an}b∇acket∇i}ht=∝an}b∇acketle{t∇(VCoul(r)+rvcEc(r))∝an}b∇acket∇i}ht=(1−rvc)∝an}b∇acketle{t∇VCoul(r)∝an}b∇acket∇i}ht. (11)
+For example, in GaAs /AlxGa1−xAs quantum wells rvc≃−0.5, hence 1−rvc≃1.5. Therefore,the Rashba spin-orbit
+coupling is proportionalto the average of the sum of the exte rnal and built-in electric field. The above discussion is
+just a simple illustration on the existence of the Rashba spi n-orbit coupling, where a few factors have been ignored.
+Nevertheless, the indication that the Rashba spin-orbit co upling can be tuned electrically inspired the community.
+Manyproposalsof spintronicdevicesbased on electrical ma nipulationof spin-orbitcoupling,such as, the spin field-
+effecttransistors[54],wereproposed.5Inthisbackground,theRashbaspin-orbitcouplingsinvari ousheterostructures
+have been studied extensively. Theoretical investigation s on the Rashba spin-orbit coupling in quantum wells were
+performedin Refs. [151–157]. Specifically, it was found tha t even in symmetric quantum wells, the built-in electric
+field contributes an intersubband Rashba spin-orbit coupli ng, though it does not contribute to the intrasubbandspin-
+orbitcoupling[120,121].
+Experimental methods to determine the spin-orbit coupling coefficients consist of magnetotransport (including
+boththeShubnikov-deHaasoscillation[158–168]andweak( anti-)localization[137,169–176]),opticallyprobedspi n
+dynamics (spin relaxation [177–179] and spin precession [6 0]), electron spin resonance [180] and spin-flip Raman
+scattering [133, 181], etc. Recently, it was proposed that t he radiation-inducedoscillatory magnetoresistance can b e
+used as a sensitive probe of the zero-field spin-splitting [1 82]. The experimental results indicated that the largest
+Rashba parameter αRcan be 30×10−12eV·m [161, 163] or even 40 ×10−12eV·m [165] for InAs and 14 ×10−12eV·m
+forGaSb[180]quantumwells, whilethe smallestRashba para metercanbenegligible[178, 183].6
+In asymmetric quantum wells, both the Rashba and Dresselhau s spin-orbit couplingsexist. After averaging over
+thelowestsubbandwavefunctionin(001)quantumwell,the D resselhausspin-orbitcouplingbecomes,
+HD=βD(−kxσx+kyσy)+γD(kxk2
+yσx−kyk2
+xσy), (12)
+whereβD=γD∝an}b∇acketle{tˆk2
+z∝an}b∇acket∇i}ht=/integraltext
+dzφ1(z)∗(−∂2
+z)φ1(z) withφ1(z) being the lowest subband wavefunction. In narrow quantum
+wells,thelinear- ktermdominates. TheratioofthestrengthoftheDresselhaus spin-orbitcouplingtotheRashbaone
+βD/αRcan be tuned by the well width or the electric field along the gr owth direction [37, 113]. The electric field
+dependenceof the Dresselhaus and Rashba spin-orbit coupli ngs in GaAs/AlxGa1−xAs quantum wells for varies well
+widthswasstudiedintheworkbyLauandFlatt´ e[146],indic atingthatunderhighbiastheRashbaspin-orbitcoupling
+can exceedthe Dresselhaus one.7The interferenceof the two spin-orbit couplingsleadsto an isotropicspin splitting,
+5For recent advancement in experiment on the spin field-e ffect transistors, see, e.g.,Refs. [55,149, 150].
+6The channel width dependence [167], density dependence [15 9–162, 165, 166, 169–173], gate-voltage dependence [158, 1 59, 163, 164, 173,
+178], temperature dependence [177] and interface e ffect [162, 165, 172, 174, 184] were investigated in various ma terials from InAs, InGaAs and
+GaAsto GaSb. Rashba spin-orbit coupling in quantum wire was studied in Refs. [185–191].
+7Theeffects ofstructure inversion asymmetry,heterointerface an d gate-voltage onthe Dresselhaus and Rashba spin-orbit cou plings in quantum
+wells were studied experimentally in Refs. [37, 55,137,162 ,175, 178, 179,183,192,193].
+6whichhence resultsin anisotropicspin precession[64–67] , spin relaxation[194–199], spin di ffusion[29, 32–34, 37]
+andspinphotocurrent[200–202]. Reversely,theratioof βD/αRcanbeinferredfromtheanisotropyofspinprecession
+[64, 203],spin relaxation[198,199],spindi ffusion[29,33] andspinphotocurrent[200,201].
+2.3.4. HoleHamiltonianandholespin-orbitcouplingin bul ksystem
+TheholeHamiltoniancanbewrittenas[113]
+Hh=HL+Hh
+SO+Hhsp, (13)
+whereHListheLuttingerHamiltonian, Hh
+SOisthespin-orbitcouplingduetothespaceinversionasymme tryandHhsp
+describesthe hole-strainand hole-phononinteractionswh ich can be foundin Ref. [117]. The LuttingerHamiltonian
+HLisgivenby[122](inthe orderofholespinstate Jz=−3
+2,−1
+2,1
+2,3
+2withzaxisalong[001]direction)
+HL=P+Q L M 0
+L∗P−Q0M
+M∗0P−Q−L
+0M∗−L∗P+Q, (14)
+where
+P=γ1
+2m0k2,Q=γ2
+2m0(k2
+x+k2
+y−2k2
+z),L=−2√
+3γ3
+2m0(kx−iky)kz,M=−√
+3
+2m0[γ2(k2
+x−k2
+y)−2iγ3kxky],(15)
+withγ1,γ2andγ3beingtheLuttingerparameters.
+Hh
+SO=1
+η(ΩD+ΩS)·J (16)
+which is similar to the spin-orbit coupling in the conductio n band except differed by a factor of 1 /η. HereJ=
+(Jx,Jy,Jz) istheholespinoperatorwith J=3/2.
+Undersphericalapproximation,theLuttingerHamiltonian canbe writteninacompactform,
+Hsp
+L=γ1
+2m0k2+¯γ2
+2m0[5
+2k2−2(k·J)2], (17)
+where ¯γ2=(2γ2+3γ3)/5. Underthisapproximation,thespectrumofholebecomespa rabolic,
+Eλ1/λ2(k)=k2
+2m0(γ1±2¯γ2). (18)
+Theindices λ1=±1
+2,λ2=±3
+2denotetheholespin. Thefourbrancheswithdi fferenteffectivemass m0/(γ1±2¯γ2)are
+thelightholeandheavyholebranchesrespectively. Thetwo folddegeneracyofthe spectrumisaconsequenceofthe
+coexistenceofthe time-reversalsymmetryandthe spaceinv ersionsymmetryoftheLuttingerHamiltonian.
+Inzinc-blendeIII-VorII-VIsemiconductors, γ1and ¯γ2areusuallyofthesameorderofmagnitude(e.g.,inGaAs,
+γ1=6.85 and ¯γ2=2.5 [204]). Thereforethe spin-dependentterm in Eq. (17) (the spin-orbit couplingterm) is com-
+parablewiththespin-independentones. Thisindicatestha tthespin-orbitcouplingisverystronginbulkholesystem,
+whichisoneofthemaindi fferencesbetweenbulkelectronandholesystems. Themaincon sequencesarethat: change
+in orbit motion(states) can influencestronglyon spin dynam ics[23, 113, 205, 206]; the spin relaxation /dephasingis
+usuallymuchfaster inholesystem( ∼100fs)thaninelectronsystem[207–210].
+2.3.5. Holespin-orbitcouplinginnanostructures
+As mentioned above the hole spin-orbit coupling is importan t in determining hole spectrum and its spin state.
+In nanostructures, it is always necessary to diagonalize th e hole Hamiltonian Eq. (13) including the confining po-
+tential [113, 143]. The dispersion and spin states can vary m arkedly with the confinement geometry [113]. It was
+7demonstratedthatin p-GaAsquantumwellsthein-planeandout-of-planee ffectivemassesand g-factorsofthelowest
+heavy-holesubbandcanbequitedi fferentfordifferentgrowthdirections[113].
+Similartotheelectroncase,theelectricfieldcanalsoindu cetheRashbaspin-orbitcouplinginholesystem,which
+iswrittenas[113]
+Hh
+R=αh
+0J·(k×Ec). (19)
+In[001]grownquantumwells, it isreducedto
+Hh
+R=αh
+0∝an}b∇acketle{tEc∝an}b∇acket∇i}ht(kyJx−kxJy), (20)
+whereαh
+0is a material dependent parameter [113] and ∝an}b∇acketle{tEc∝an}b∇acket∇i}htis the average of the electric field (along the growth
+direction)actingonconductionbandovertheholesubbande nvelopefunction. Inpractice,toobtaintheRashbaspin-
+orbit coupling in, e.g., the lowest heavy-hole subband, one must further block-diagonalizethe subband Hamiltonian
+via the L¨ owdin partition method [113]. Quite di fferent from the electron case, the subband block-diagonaliz ation
+gives another Rashba spin-orbit coupling which originates from the electric field on other hole subband rather than
+the conduction band. Practical calculation indicated that such kind of Rashba spin-orbit coupling is more important
+than that due to the valence-conduction band coupling [Eq. ( 20)] [113]. The expression of such kind of Rashba
+spin-orbitcouplinginthelowest heavy-holesubbandisgiv enby[113],
+HHH
+R=αhh∝an}b∇acketle{tEz∝an}b∇acket∇i}hti(k3
++σ−−k3
+−σ+), (21)
+wherek±=kx±iky,σ±=σx±iσywithσbeingthe Paulispinmatricesinthe manifoldofholespin Jz=−3/2,3/2.
+∝an}b∇acketle{tEz∝an}b∇acket∇i}htis the electric field acting on the valence band averagedover the lowest hole subbandenvelopefunction. For an
+infinitedepthrectangularquantumwell [113],
+αhh=e
+m2
+064
+9π2γ3(γ2+γ3)1
+∆hl
+11(1
+∆hl
+12−1
+∆hh
+12)+1
+∆hl
+12∆hh
+12, (22)
+where∆hl
+ij(i,j=1,2...)is the splitting between the i-th heavy-holesubbandand the j-th light-holesubband. Similar
+to the case in electron system, the hole Rashba spin-orbit co upling can also be tuned by gate-voltage and structure
+asymmetrybut now αhhis alsoelectricfielddependent[113,211–214].
+2.4. Spin-orbitcouplingin nanostructuresdueto interfac einversionasymmetry
+Theinterfaceinversionasymmetryis akindofinversionasy mmetrycausedbytheinversionasymmetricbonding
+ofatomsattheinterfacesofnanostructures.8ItwasfirstpointedoutbyAle ˘inerandIvchenko[215]thatthesymmetry
+at the interface may be di fferent from the symmetry away from the interface, and hence ma y introduce a new spin-
+orbitcoupling[127, 216, 217]. The interface-inducedspin -orbitcouplingwas usuallystudied in quantumwells. The
+underlyingphysicsis similarforothernanostructures.
+Let us start by examiningthe case of GaAs /AlAs quantumwells. Without the interface e ffect, the quantumwells
+possess a D2dsymmetry. It is easy to find that due to the di fference of Al and Ga atoms, the symmetry at the
+interface is reduced to C2v[113]. For quantum wells which possess a mirror symmetry, th e asymmetry induced by
+thetwointerfacescancelsout[113]. However,inthesituat ionthatthematerialsinthewell andbarrierdonotsharea
+commonatom, the interface e ffect at the two interfacescan not be canceledout as theyare di fferentinterfaces[126].
+For example, in InAs /GaSb quantum wells, the two interfaces are Sb-In-As and In-A s-Ga. The symmetry of the
+quantumwell isthenreducedto C2vandresultsinnewspin-orbitcouplingtermsforholesandel ectrons. It inducesa
+term∼(JxJy+JyJx) andindependentof kin the hole Hamiltonian[113], which mixesthe heavy-holean dlight-hole
+spinseffectively in narrowquantumwells [216, 217] and largely supp ressesthe hole spin lifetime in these structures
+[216]. Theinducedelectronspin-orbitcouplingisoftheli near-Dresselhaus-type ∼(kxσx−kyσy)[218].
+The experimental test of the importance of the interface-in duced electron spin-orbit coupling was performed by
+comparingthespindynamicsinInGaAs-AlInAsquantumwells (whichshareonecommonatom)withthatinInGaAs-
+InP quantum wells (which have no common atom) with similar pa rameters and overall quality [219]. The observed
+8Theinterface inversion asymmetry induced spin-orbit coup ling is also reviewed in Ref. [113].
+8huge difference in spin relaxation times between these two samples in dicates that the interface inversion asymmetry
+inducedspin-orbitcouplingisimportantinnarrowquantum wellswithoutcommonatominwell andbarrier.
+It should be pointed out that in [110] grown quantum wells, ev en in the case without common atom in well and
+barrier,thetwo interfacescanbesymmetric.9Hencethe interfaceinversionasymmetrydoesnotcontribut e. Bycom-
+paring the spin relaxation in InAs /GaSb quantum wells with di fferent growth directions, with the help of theoretical
+calculation,itwasfoundthattheinterfaceinversionasym metryplaysanimportantroleinnarrowInAs /GaSbquantum
+wells[221,222].
+In genuine nanostructures, as interfaces can not be perfect , the interface inversion asymmetry exists in most
+samples. A quantitative analysis of such e ffect requires the knowledge of specific interfaces, which is u sually very
+difficulttoacquire. Ingeneral,sinceitisaninterfacee ffect,theinterfaceinversionasymmetryshouldbeimportant to
+spin-orbit coupling in narrow quantum wells with no common a tom in the barrier and well, whereas unimportantin
+othercases.
+2.5. Zeemaninteractionwith externalmagneticfield
+TheZeemaninteractionforelectroniswrittenas
+HZ=µBS·ˆg·B, (23)
+where ˆgistheg-tensorand µBistheBohrmagneton. The g-tensordeterminesthespinprecession(boththefrequency
+and the direction) in the presence of an external magnetic fie ld. It is hence an important quantity for spin dynamics.
+For typical III-V semiconductors, such as GaAs, the conduct ion band g-tensor is isotropic ˆ ge=geˆ1. In GaAs, at
+conduction band edge, ge=−0.44, whereas for higher energy gfactor increases [223–225]. An empirical relation
+for the energy εdependenceof the gfactor in GaAs was found from experimentsas ge(ε)=−0.44+6.3ε(εin unit
+of eV) [225–227]. Experiments indicated that the gfactor varies from −0.45 to−0.3 in the temperature range of
+4∼300K [228–232]. Recent calculations[232] indicatedthat t he temperaturedependenceof g-factoris mainlydue
+tothecombinedeffectsof(i)theenergy-dependent gfactorand(ii)thethermaldilatationoflattice.
+In quantum wells the gfactors of the barrier material and well material are usuall y different or even of opposite
+signs. The gfactor can then be tunedby well width. For example,in GaAs /AlAs quantumwells: at large well width
+the electronic property is GaAs-like and ge<0; at small well width the subband wavefunction is largely pe netrated
+into the AlAs barrier and the electronic property is AlAs-li ke, hence ge>0. This phenomenon has been observed
+in experiment[233]. Moreover,due to the confinementin quan tumwell, which reducesthe symmetryof the system,
+the electron g-factor usually becomes anisotropic: the in-plane g-factorg∝ba∇dblis different from the out-of-plane one g⊥
+[113,234–236].
+In parabolic GaAs /AlGaAs quantum wells, as the Al concentration is tuned almos t continuously with position
+in the growth direction, the g-tensor becomes position dependent ˆ ge(z). The effectiveg-tensor is an average over
+position ˆg∗=/integraltext
+dzˆge(z)|ψe(z)|2whereψe(z) is the electron subband wavefunction. This fact enables e fficient tuning
+ofelectron g-tensorby modulatingthe subbandwavefunctionvia, e.g., g ate-voltage[67, 78, 237]. A time-dependent
+gate-voltagecaninduceamodificationtotheZeemaninterac tionδHZ=µBS·δˆg·B(δˆgisthechangeof g-tensordue
+to gate-voltage). In propergeometry,the time-dependentg ate-voltageinducesa spin precession perpendicularto the
+unperturbed spin precession. In such a way, electron spin re sonance is achieved with time-dependent gate voltage.
+Such a spin resonance,which is called g-tensormodulationresonance,was first proposedand realiz ed by Kato et al.
+[77].
+The hole Zeeman interaction is complicated. In bulk system t here are two terms which contribute to the hole
+Zeemaninteraction,
+Hh
+Z=2κµBB·J+2qµBB·T. (24)
+Hereκis the isotropic valence band gfactor, whereas qis the anisotropic one. T=(J3
+x,J3
+y,J3
+z). In GaAs κ=1.2,
+whereasq=0.01 is much smaller and can always be neglected. It is noted tha t there is no direct coupling between
+spin-up and -down heavy holes ( Jz=3
+2,−3
+2) in the Zeeman interaction unless in the much smaller anisot ropic term
+[thesecondterminEq.(24)].
+9Thestudy on thequantum well growth direction dependence of the interface induced spin-orbit coupling is presented in R ef. [220].
+9In (001) quantum wells, the lowest subband is the heavy-hole subband if no strain is applied. According to the
+above analysis, for an in-plane magnetic field, the coupling of spin-up and -down states is rather small. Within the
+subband quantization, L¨ owdin partition method gives the d ominant terms of the Zeeman interaction in the lowest
+heavy-holesubbandas
+HHH
+Z=3κµB
+m0∆hl
+11/braceleftig
+σx/bracketleftig
+Bxγ2/parenleftig
+k2
+x−k2
+y/parenrightig
+−2Byγ3kxky/bracketrightig
++σy/bracketleftig
+2Bxγ3kxky+Byγ2/parenleftig
+k2
+x−k2
+y/parenrightig/bracketrightig/bracerightig
+, (25)
+where∆hl
+11denotesthe splitting betweenthe lowest heavy-holesubban dand the lowest light-holesubband. Note that
+the in-plane gfactor of the lowest heavy-hole subband is k-dependent and the average of the gfactor is zero. The
+in-planeg-factorofthe lowestheavy-holesubbandvariessignificant lywith thegrowthdirectionofthe quantumwell
+aspointedoutbyWinkleret al. [113,238].
+Zeeman interaction of holes in quantum wires has been invest igated only recently. Experimentally, Danneau et
+al. foundthat the g-factorofholesinquantumwiresalong[ ¯233]directionis largelyanisotropic[239]. Theoreticall y,
+Csontos and Z¨ ulicke reported that the hole g-factorin quantumwire can vary largely with subband[240] a nd can be
+tunedeffectivelyby confinement[241]. These predictionswere partl yconfirmedrecentlyin the experimentby Chen
+et al. [242]. Similar results were found in hole quantum dots [243, 244]. The large anisotropy and variation of the
+holeg-factorin these quantumconfinedsystemsare notsurprising ,as(i) the symmetryof thesestructuresisreduced
+and(ii)the holespin-orbitcouplingisverystrong.
+2.6. Spin-orbitcouplingin Wurtzite semiconductorsandot hermaterials
+In previous discussions, the spin-orbit coupling and g-factor in bulk and nanostructures of the zinc-blende III-V
+and II-VI compoundsare reviewed. Besides these two importa nt kinds of materials, there are several other kinds of
+materialswith different structureswhich have also attracted muchattention o f the semiconductorspintroniccommu-
+nity.
+One of these materials is the wurtzite structure semiconduc tors, such as GaN, AlN and ZnO.10In contrast to the
+zinc-blende semiconductorssuch as GaAs, the existence of h exagonal c-axis in wurtzite semiconductorsleads to an
+intrinsic wurtzite structure inversion asymmetry in addit ion to the bulk inversion asymmetry [246, 247]. Therefore,
+the electron spin splittings include both the Dresselhaus e ffect [245, 248] (cubic in k) and Rashba effect (linear in
+k) [124, 125, 249–254]. In a recent work by Fu and Wu [245], a Kan e-type Hamiltonian was constructed and the
+spin-orbitcouplingforelectronandholebandswereinvest igatedin thefullBrillionzonein bulkZnOandGaN.
+Forthestatesnearthe Γpoint(k=0),bychoosingthe zaxisalongthe caxisofthecrystal,theelectronspin-orbit
+couplingintheconductionbandreads[245]
+He
+SO=[ΩR
+e(k)+ΩD
+e(k)]·σ/slashbig2 (26)
+with
+ΩR
+e(k)=2αe(ky,−kx,0),ΩD
+e(k)=2γe(bk2
+z−k2
+∝ba∇dbl)(ky,−kx,0). (27)
+HereΩR
+e(k) andΩD
+e(k) are the Rashba-type and Dresselhaus terms, respectively. αeandγeare the corresponding
+spin-orbit coupling coe fficients.bis a coefficient originating from the anisotropy induced by the lattic e structure. It
+wasfoundthat in ZnOandGaN, b=3.855and3.959respectively[245]. Thevalueof αeandγeare listed in Table 1
+[245].
+In wurtzite semiconductors,the valence bands consist of th ree separated bands: Γ9v(heavy-hole,spin±3/2),Γ7v
+(lighthole,spin±1/2)andΓ7′v(split-offhole,spin±1/2). Thespin-orbitcouplingsinthesebandsatsmall kcanalso
+bewritteninthegeneralformof1
+2[ΩR
+i(k)+ΩD
+i(k)]·σ,whichincludesboththelinearRashba-typeterm1
+2ΩR
+i(k),and
+the cubic Dresselhausterm1
+2ΩD
+i(k), withi=e,9,7,7′being the bandindex. For heavy-holeband Γ9v, the spin-orbit
+couplingfieldsarewrittenas[245]
+ΩR
+9(k)=0,ΩD
+9(k)=2γ9(ky(k2
+y−3k2
+x),kx(k2
+x−3k2
+y),0). (28)
+10GaN also has zinc-blende structure phase. Thespin-orbit co upling in zinc-blende GaN was studied in Ref. [245]
+10Forvalencebands Γ7v(light-hole)andΓ7′v(split-offhole),the spin-orbitcouplingfields are the same as that giv enin
+Eq. (27) with only the coe fficientsαeandγebeing replaced by αiandγi(i=7,7′) (the coefficients forΓ7vandΓ7′v
+bands)respectively[245]. Thecoe fficientsαiandγi(i=e,9,7,7′)arelisted inTable 1[245].11
+Table1: Rashba-type( αi)(inmeV·Å)andDresselhaus( γi)(ineV·Å3)spin-orbitcouplingcoe fficientsinwurtziteZnOandGaNatsmallmomentum
+withi=e,9,7,7′. FromFu and Wu [245]. (afrom Ref. [249];bfrom Ref. [250])
+αeα9α7 α7′γeγ9γ7γ7′
+ZnO: 1.1a0 35a(21b) 51a0.33 0.09 6.3 6.1
+GaN: 9.0b0 45b32b0.32 0.07 15.3 15.0
+Interestingly,in two-dimensionalelectron system (quant umwells or heterojunctions)where the growth direction
+is along the c-axis, the linear- kspin-orbit coupling is of the same form for both the Dresselh aus and Rashba-type
+terms due to the crystal field. Additional contribution to th e spin-orbit coupling comes from the electric field across
+the two-dimensional electron system, which is just the Rash ba spin-orbit coupling [124, 125]. Therefore, all the
+linear inkterms are of the Rashba-type,12which makes them indistinguishablein experiments. Indeed ,experiments
+revealedthatthespin-orbitcouplingisdominatedbytheRa shba-typespin-orbitcoupling,wherenoe ffectthatsignals
+spin-orbitcouplingof the formof ( kxσx−kyσy) was observed[247, 256–260]. The measuredRashba paramete rlies
+intherangeof0.6to8 ×10−12eV·mforvariousconditions[251–254,261–268].
+Another kind of material of special interest is the so-calle d gaplesssemiconductor,such as HgTe, partly because
+thatinHgTe/CdTequantumwellsthequantumspinHalle ffectwasobserved[269]. HgTe,whichisalsoazinc-blende
+II-VI semiconductor, has the same symmetry as CdTe and GaAs. However, theΓ8band, which consists of heavy
+and light holes in GaAs, is higher than the conduction band Γ6. Moreover, the heavy-hole band is inverted, i.e., the
+effective mass of electron in the heavy-hole band is positive. O n the other hand, the conduction band Γ6, becomes
+hole-like. In HgTe /CdTe quantum wells, the relative position of the Γ6andΓ8bands can be tuned by well width.
+At the crossing band point a massless Dirac spectrum is reali zed [270]. The spin-orbit coupling in these bands was
+investigated both theoretically [271] and experimentally [147, 272–278], where the Rashba spin-orbit coupling was
+foundtobe verylarge( ≃4×10−11eV·m)intwo-dimensionalelectronsystem.
+Spin dynamics in silicon has attracted renewed interest rec ently due to the experimental advancement in silicon
+spintronic devices [149, 150, 279–282] and also due to recen t development in spin qubits based on silicon quantum
+dots[283–285]ordonorboundelectrons[286–289]. Silicon hasadiamondstructurewithspaceinversionsymmetry,
+hencebulksilicondoesnothavetheDresselhausspin-orbit coupling. Intwo-dimensionalorothernanostructures,the
+Rashba spin-orbit coupling due to structure inversion asym metry emerges [124, 125, 290]. Moreover, the interface
+inversion asymmetry also contributes to new spin-orbit cou pling, which can be either the Rashba-type or the linear-
+Dresselhaus-type or their combinations depending on the sy mmetry of the interface [291, 292]. The e ffect of the
+electric field on electron spin-orbit coupling in Si /SiGe quantum well was discussed in Ref. [293]. In practice, t he
+doping inhomogeneity can induce a random (in-plane positio n dependent) electric field along the growth direction
+even in nominally symmetrically doped quantum wells. This r andom electric field can induce a random Rashba
+spin-orbit coupling [294]. Similarly, the surface roughne ss can also induces a random spin-orbit coupling [291].
+Experimentally, the electron spin-orbit coupling in silic on quantum wells was determined via the anisotropy of the
+electron spin resonance frequencyand line-width [295–297 ]. Hole spin-orbit coupling in silicon quantumwells was
+studiedin Refs.[298, 299].
+11Using these coefficients, Buß et al. found reasonable agreement between calcu lated spin relaxation times and the experimentally measure d
+ones recently [255].
+12Thesymmetry of the cubic term is also the sameas the linear on e.
+112.7. Hyperfineinteraction
+Theoriginofthehyperfineinteractionisthatthenuclearsp insfeelthemagneticfieldgeneratedbycarriersdueto
+theirspinsandorbitalangularmomentums. Forconductionb andelectron,whichhas s-wavesymmetry,themagnetic
+fieldgeneratedbyelectronorbitalmotionisnegligible. Th eelectronhyperfineinteractionisoftheform[3,300]
+Hhf=/summationdisplay
+i,ν2µ0
+3βνγeγνδ(r−Ri,ν)S·Ii,ν. (29)
+Hereµ0is vacuumpermeability, γe=2µBandiisthe indexofthe Wigner-Seitzcell. ForIII-VandII-VIcom pounds
+(such as GaAs), there are two atoms (such as Ga and As) in each W igner-Seitz cell. And the nuclei of certain atoms
+have their isotopes (such as69Ga and71Ga). The index νlabels both the atomic site in the Wigner-Seitz cell and
+the isotopes. βνis the abundance and/summationtext
+νβν=NatmwithNatmbeing the number of atoms within a Wigner-Seitz
+cell.Natm=2 for zinc-blende structures. γν=gνµNwithgνandµNrepresenting the gfactor of the νnucleus and
+the nuclear magneton respectively. Ri,νandIi,νrepresent the position and spin of the νnucleus in i-th Wigner-Seitz
+cell. In practical calculation it is assumed that the isotop es are distributed uniformly. Within the envelope function
+approximation,the electronhyperfineinteractioniswritt enas
+Hhf=/summationdisplay
+i,νAνv0|ψ(Ri)|2S·Ii,ν=h·S, (30)
+whereAν=2µ0
+3γeβνγν|uc(Rν)|2N0andh=/summationtext
+i,νAνv0|ψ(Ri)|2Ii,ν. Herev0is the volume of the Wigner-Seitz cell and
+N0=1/v0,ucis Bloch wave amplitude. A=/summationtext
+νAνmeasures the strength of the hyperfine interaction. In GaAs,
+possible isotopes are69Ga,71Ga and75As, which all have spin I=3/2. Their natural abundances are β69Ga=0.6,
+β71Ga=0.4 andβ71As=1. The average strength of the hyperfine interaction is A≃90µeV. In silicon, the possible
+isotopes are28Si and29Si with natural abundances β28Si=0.9533 andβ29Si=0.0467. In silicon, the main isotope
+28Si has nonuclearspinand onlythe minorisotope29Si has spin I=1/2. Hencethe averagehyperfineinteractionin
+siliconA∼0.2µeV ismuchsmaller thanthat inGaAs.
+For holes, as the valence band Bloch wavefunction is very sma ll at the nuclei position (which is a property of
+p-wavesymmetry),the contacthyperfineinteractionis negli gible. However,the long rangedipole-dipoleinteraction
+and the interaction between electron orbital angular momen tum and nuclear spin do not vanish for holes.13Within
+theenvelopefunctionapproximation,Fischer et al. showed thatthe hyperfineinteractionforheavy-holesinquantum
+wellisofthe Isingform[301]
+Hh
+hf=/summationdisplay
+i,νAh
+νv0|ψ(Ri)|2SzIz
+i,ν, (31)
+whereSzis thez-component of the heavy-hole spin and Ah
+νis the hole hyperfine coupling constant. Fischer et al.
+estimatedthattheholehyperfinecouplingconstant Ah
+νisabout−10µeVinGaAs[301],aboutoneorderofmagnitude
+smallerthanthe electronone.
+Besides inducing spin relaxation, the hyperfine interactio n can also transfer the nonequilibrium carrier spin po-
+larization to nuclear spin system, which is called the dynam ic nuclear polarization [3, 302–310]. Via the hyperfine
+interaction, periodic optical pumping of electron spin pol arization acts as a periodic magnetic field on nuclear spin
+andinducesanuclearspinresonance[311]. Asnuclearspinr elaxesmuchslowerthanelectronspin,nuclearspinwas
+proposed as long-lived “quantum memory” for spin qubits [31 2, 313]. Such proposal was realized in experiment in
+31P donorsinisotopicallypure28Si crystal[314].
+2.8. Exchangeinteractionwithmagneticimpurities
+The exchangeinteractionbetweencarrierand magneticimpu rities,such asMn, is veryimportantindilutedmag-
+neticsemiconductors[315],asitisthephysicaloriginofv ariousmagneticordersinthesematerials. Forexample,the
+dorbitalelectronsofMnimpuritiesmixwith s(conductionband)or p(valenceband)electronsinthesemiconductor
+13In contrast, these interactions are zero for (conduction ba nd) electrons dueto s-wave symmetry.
+12matrix, which leads to the exchange interactions between ca rriers and Mn impurities [315, 316]. These interactions,
+knownasthe s-dandp-dexchangeinteractions,canbe writtenas
+Hsd=−/summationdisplay
+iJsdS·S(i)
+dδ(r−Ri),Hpd=−/summationdisplay
+iJpdJ·S(i)
+dδ(r−Ri). (32)
+HereS(i)
+dis theith Mn spin at Ri.SandJrepresent the electron and hole spins respectively. JsdandJpdstand for
+thes-dandp-dexchange constants respectively. The exchange constants, which are basic parameters of magnetic
+semiconductors, have been extensively studied both theore tically and experimentally [for review in (III,Mn)V, see
+Ref. [114] and references therein; for (II,Mn)VI, see Ref. [ 115]]. To illustrate the strength of the s(p)-dexchange
+interaction, we give two examples: in GaMnAs, N0Jsd≈−0.1 eV [233, 317], N0Jpd≈−1 eV [114]; in CdMnTe,
+N0Jsd≈0.22eV,N0Jpd≈−0.88eV [115] where N0=1/v0withv0beingthevolumeofthe unitcell.
+2.9. Exchangeinteractionbetweencarriers
+Theexchangeinteractionbetweencarriersoriginatesfrom thecombinationofthecarrier-carrierCoulombinterac-
+tionandthepermutativeantisymmetryofthewavefunctiono fthefermioniccarriersystem. Theexchangeinteraction
+usually increases with the overlap between the wavefunctio n of the carriers. Therefore, it is usually strong in local-
+ized carrier system, such as carriers in quantum dots or thos e bound to ionized impurities. Experimentshave shown
+that the exchange interaction between electrons in quantum dots can be tuned via the magnetic field14[319, 320]
+or by the gate voltage [38, 86, 319], varying from several meV to very small. The exchange interaction between
+electronsbound to adjacent donorsis believed to be respons ible for the spin relaxation of the donor boundelectrons
+[39, 40, 321–324, 324, 325]. The underlying physics is that d ue to the spin-orbit coupling, the exchange interaction
+becomesanisotropic which does not conserve the total spin o f the electron system any more and hence leads to spin
+relaxation.
+Besides, the exchange interaction between extended carrie rs also shows up in spin dynamics. For example, the
+exchangeinteractionbetweenelectronsandholes15leadsto electronspin relaxation,whichiscalled the Bir-A ronov-
+Pikusmechanism. Specifically,thisexchangeinteractionc onsistsoftwoparts: theshort-rangepartandthelong-rang e
+part. Theshort-rangepartiswrittenas[118]
+HSR=−1
+V1
+2∆ESR
+|φ3D(0)|2ˆJ·ˆSδK,K′, (33)
+where∆ESRis the exchange splitting of the exciton ground state and |φ3D(0)|2=1/(πa3
+0) witha0being the exciton
+Bohrradius. K=ke+khis the sumofthe electron keandholekhwave-vectorsof theinteractingelectron-holepair.
+Vis thevolumeofthesample. Thelong-rangepartreads[118]
+HLR=1
+V3
+8δK,K′∆ELT
+|φ3D(0)|2(ˆMzˆSz+1
+2ˆM−ˆS++1
+2ˆM+ˆS−), (34)
+where∆ELTis the longitudinal-transversesplitting; ˆMz,ˆM−andˆM+(=ˆM†
+−) are operators in hole spin space. ˆS±=
+ˆSx±iˆSyare the electronspin ladderoperators. Theexpressionsfor ˆMzandˆM−are givenbelow (in the orderof hole
+spinJz=3
+2,1
+2,−1
+2,−3
+2),
+ˆMz=1
+K2−K2
+∝ba∇dbl2√
+3KzK−1√
+3K2
+− 0
+2√
+3KzK+(1
+3K2
+∝ba∇dbl−4
+3K2
+z)−4
+3KzK−−1√
+3K2
+−
+1√
+3K2
++−4
+3KzK+(4
+3K2
+z−1
+3K2
+∝ba∇dbl)2√
+3KzK−
+0−1√
+3K2
++2√
+3KzK+K2
+∝ba∇dbl, (35)
+14This was firstpredicted by Burkard et al. [318].
+15Theelectron-hole exchange interaction is in fact from thec onduction-valence band mixing [107].
+13ˆM−=1
+K20 0 0 0
+−2√
+3K2
+∝ba∇dbl4
+3KzK−2
+3K2
+−0
+4√
+3KzK+−8
+3K2
+z−4
+3KzK−0
+2K2
++−4√
+3KzK+−2√
+3K2
+∝ba∇dbl0. (36)
+HereK±=Kx±iKyandK2
+∝ba∇dbl=K2
+x+K2
+y. Theshort-rangeandlong-rangepartsoftheelectron-hole exchangeinteraction
+canbewrittenina compactform( i=z,+,−),
+Hex=1
+VδK,K′(ˆJz(K)ˆSz+1
+2ˆJ−(K)ˆS++1
+2ˆJ+(K)ˆS−) withJi(K)=1
+|φ3D(0)|2/bracketleftigg3
+8∆ELTˆMi(K)−1
+2∆ESRˆJi/bracketrightigg
+.(37)
+In semiconductor nanostructures, e.g., in quantum wells, t he above electron-hole exchange interaction should be
+writteninthesubbandbases. Suchexpressionshavebeender ivedinRef.[118]. Besides,itisshownthatthequantum
+confinementcanenhancetheelectron-holeexchangeinterac tionin quantumwells[119].
+Unlike the electron-hole exchange interaction, the exchan ge interaction between free electrons has almost been
+ignored by the community until several years ago. It was first pointed out by Weng and Wu that the exchange
+interaction between free electrons(the Coulomb Hartree-F ockterm), which acts as an e ffective magnetic field along
+the spin polarization direction, plays important role in sp in dynamics [44]. For example, if the spin polarization is
+alongthe zaxis,the effectivemagneticfieldfromtheHartree-Focktermreads[44, 1 10],
+BHF(k)=/summationdisplay
+qVq/parenleftig
+fk−q↑−fk−q↓/parenrightig/slashig
+geµB, (38)
+wherefk−q↑andfk−q↓are the distributions on the spin-up and spin-down bands res pectively. geis electron g-factor
+andVq=e2/[ǫ0κ0q2ǫ(q)] in bulk with ǫ(q) being the dielectric function. At large spin polarization ,BHFcan be as
+largeasafewtensofTeslaforanelectrondensityof4 ×1011cm−2at120KinGaAsquantumwell[44]. Recently,the
+effect of the exchange interaction between electrons was obser ved in experiments in high mobility two-dimensional
+electronsystem[41,42, 326]. TheHartree-Focke ffectivemagneticfield hasalsobeenobservedinotherexperim ents
+recently[327]. Detailedreviewonthe Hartree-Focke ffectivemagneticfield onspindynamicsis giveninSec.5.4.3.
+3. Spin relaxationandspin dephasingin semiconductors
+After briefly introducingvarious spin interactions, we now move to one of the central issues in this review: spin
+relaxation and dephasing, i.e., the dissipative part of the spin dynamics. As stated before, anyfluctuation or inho-
+mogeneity in spin interactions can lead to spin relaxation a nd spin dephasing. From the spin interactions, one can
+identify the possible spin relaxation /dephasingmechanisms. In what follows, we first introduceth e conceptsof spin
+relaxationanddephasingin time domainas well asin spacial domain. We thenintroducethe relevantspin relaxation
+mechanismsinsemiconductorsandtheirnanostructures.
+3.1. Spinrelaxationandspindephasing
+Simply speaking, spin relaxation is related to the nonequil ibrium population decay of the spin-resolved energy
+eigenstates, whereas spin dephasing is related to the destr uction of phase coherence of these eigenstates. For single
+spin system with isotropic g-factor,spin relaxationis related to the decay of spin pola rizationparallel to the external
+magnetic field, whereas spin dephasing is related to the deca y of spin polarization transverse to the magnetic field.
+Spinrelaxationtime(denotedas T1)andspindephasingtime ( T2)arequantitiestocharacterizethetimescale ofspin
+relaxation and spin dephasing.16Microscopically, spin relaxation and spin dephasing are us ually related to different
+processes. Althoughspinrelaxationalsoinevitablyleads tospindephasing,thereareprocesseswhichonlycontribut e
+to spin dephasing, called pure dephasing processes [328]. S pin relaxation and spin dephasing in single spin system
+are caused by the unavoidable coupling with the fluctuating e nvironment, such as the phonon system, nuclear spin
+systemandotherelectronsnearby. Theseprocessesarealwa ysirreversible.
+16Spin decay does not have to beexponential. However, in mostc ases, acharacteristic decay timecan beidentified.
+14Oftentherearemanyelectronspinsformingaspinensemble. Foranensembleofspins,spinpolarizationcandecay
+without coupling to any environment: when spin precession f requencies or directions vary from part to part within
+the ensemble, the total spin polarizationgets a free-induc tion-decaydue to destructiveinterference. This variatio n is
+called the inhomogeneous broadening [329, 330]. For a finite ensemble with independent spins, the decay of total
+spin polarization can be reversible: after some time, spins in all parts of the ensemble rotate to the same direction.17
+However,quiteoften,theensembleisverylarge,e.g.,ther eare∼1016cm−3electronsin n-dopedGaAs. Inthisregime,
+the spin decay due to inhomogeneous broadening can not be rev ersed automatically. However, the inhomogeneous
+broadening induced spin decay can still be removed by the tec hnique of spin echo. Standard spin echo is achieved
+by aπ-pulse magnetic field which is perpendicular to the inhomoge neous broadened (e ffective) magnetic field, to
+invert the sign of the phase gained through spin precession s o that the gained phase can be gradually canceled out
+in the subsequent evolution. The ensemble spin relaxation a nd spin dephasing times including the inhomogeneous
+broadeningare usually denoted as T∗
+1andT∗
+2respectively,whereas the irreversible(echoed)spin rela xation and spin
+dephasingtimesaredenotedas T1andT2respectively. Inthefollowing,whenweneednottodiscrimi natethetwo,we
+simply denote the spin relaxationtime and dephasingtime as τsand called it spin lifetime. There are a lot of factors
+which contribute to inhomogeneousbroadening,such as, the k-dependentspin-orbit coupling[332] or the energy(or
+k)-dependent g-factor[333,334]. Inquantumdots,duetothespin-orbitco upling,electronorhole g-factorissizeand
+geometrydependent[335]. The variationsof the size and geo metryof quantumdotsin a quantumdot ensemblealso
+lead to the inhomogeneousbroadening[331]. Finally, the ti me and/or space variations of the nuclear hyperfinefield
+alsoleadto theinhomogeneousbroadening.
+Spin diffusion and transport also su ffers spin relaxation. In the absence of the inhomogeneousbro adening, spin
+diffusionlength Lsis related to spin lifetime τsasLs=√Dsτs, whereDsis the spin diffusion constant. However,in
+semiconductorsithasbeenshownthattheinhomogeneousbro adeningusuallydominatesspindi ffusion[25,28,336],
+which breaks down the above relation [32, 35]. The inhomogen eous broadening in spin di ffusion and transport is
+different from that in spin relaxation in time domain: electron a t differentkstate has different velocity and/or spin
+precession frequency Ω(k) (e.g., due to the spin-orbit coupling), making the spin pro pagation frequency along the
+spin diffusion direction ( ˆn)k-dependent∼Ω(k)/(k·ˆn) [25, 28, 29, 336, 337]. This inhomogeneousbroadeningcan
+not be removed by traditional spin-echo. However, it can be t uned via controlling spin-orbit coupling [18, 29, 32,
+34, 337, 338]. For example, when both the Rashba and Dresselh aus spin-orbit couplings exist, by tuning the two to
+be comparable, spin di ffusion length can be largely increased [18, 29, 32, 34, 337–34 0]. Such enhancement of spin
+diffusion length has been achieved in a recent experiment, where the control over both the doping asymmetry and
+well width was shown to be e ffective to manipulate the relative strength of the Rashba and Dresselhaus spin-orbit
+couplingsandhencethespindi ffusionlength[37]. Studiesonspindi ffusionintheliteraturearereviewedinSection6
+and 7. Below we concentrateon spin relaxation and spin depha singin time domain,where the basic physicsand the
+mechanismsresponsibleforspinrelaxationandspindephas ingare reviewed.
+3.1.1. Singlespinrelaxationandspindephasingduetofluct uatingmagneticfield
+As stated above, fluctuations in spin interaction due to the c oupling with the environment are the origin of spin
+relaxation and spin dephasing in single spin system. In this subsection, we discuss generally the spin relaxation and
+spindephasinginsinglespinsystem,wherethefluctuations inspininteractionaresimplycharacterizedasfluctuating
+magnetic fields. These fluctuating magnetic fields are assume d to have a correlation time τcwhich is much shorter
+than the spin relaxation /dephasingtime, so that the Markovianapproximationis just ified [341]. We also assume that
+thefluctuationsareweaksothattheycanbetreatedwithinth eBornapproximation. Thespindynamicsisthensolved
+bythe standardBorn-Markovmethod[341].18
+TheHamiltonianissimplygivenby
+H=1
+2[ω0ˆnz+ω(t)]·σ, (39)
+whereˆnzisaunitvectoralong zdirection.ω0andω(t)arespinprecessionfrequenciesduetothestaticandfluctu ating
+magnetic fields respectively. The correlation of the fluctua ting magnetic fields is assumed to be (for i,j=x,y,z)
+17This phenomenon was recently observed in experiments in spi n dephasing in an ensemble of self-assemble quantum dots [33 1].
+18Itis noted that discussions in this subsection aresimilar t o those in Ref. [22]in Sec. IVB2.
+15ωi(t)ωj(t′)=δi,jω2
+ie−|t−t′|/τc, with the overline denoting the ensemble average. The requi rement of the shortness of
+the correlation time τcand the weakness of the fluctuating magnetic fields is equival ent to/radicalig/summationtext
+iω2
+iτc≪1. Spin
+relaxationtimeandspindephasingtimeareobtainedbysolv ingtheBorn-Markovequationofmotion. Afterstandard
+derivationoneobtainsthat[22],
+1
+T1=(ω2x+ω2y)τc
+1+ω2
+0τ2c,1
+T2=ω2zτc+(ω2x+ω2y)τc
+2(1+ω2
+0τ2c). (40)
+From these results, one may find several features of spin rela xation/dephasing. First, a trivial result, only the noise
+magneticfields perpendicularto the spin polarizationlead to spin decay as only these fields can rotate spin. Second,
+in the absence of static magnetic field, spin relaxation /dephasing rate is proportional to the noise correlation tim e
+τc. That is to say that the more fluctuating the noise (the shorte r correlation time) is, the more ine ffective it is.
+This counter-intuitive result is known as motional narrowi ng [329]. Third, a static magnetic field suppresses spin
+relaxation/dephasing caused by the fluctuating magnetic field perpendic ular to it. The underlying physics is better
+understood in the fashion of what was presented in the review article by Fabian et al. [22]: In the presence of
+magneticfield, the spin-flip scatteringtermsacquirea dyna micphase. However,the scatteringtermswhichconserve
+thespinalongthemagneticfielddonotacquiresuchdynamicp hase. Thespin-fliptermisthenproportionalto
+/integraldisplayt≫τc
+0dt′ωi(0)ωi(t′)exp(−iω0t′)+H.c.∝ω2
+iτc
+ω2
+0τ2c+1. (41)
+Thereforetheeffectoffluctuatingmagneticfieldperpendiculartothestatic magneticfield issuppressed.
+Inthefollowing,weillustrateseverallimitswhicharefre quentlyencounteredinspinrelaxationinsemiconductors.
+Eq.(40)canbe writtenas
+1/T2=1/T′
+2+1/(2T1), (42)
+where 1/T′
+2=ω2zτc. Therefore, in general case, T2≤2T1. In the isotropic noise case, ω2x=ω2y=ω2z, at very low
+static magnetic field ω0τc≪1, spin relaxation time is equal to spin dephasing time, T1=T2[342]. At very large
+static magneticfield, spinrelaxationvanishes1 /T1→0,whereasthe spindephasingrate isfinite, 1 /T2=1/T′
+2. This
+is a pure dephasing case. The noise can be anisotropic. For ex ample, when there are only transverse noises, ωx/nequal0,
+ωy/nequal0andωz=0,T2=2T1.19Anotherexample,thereareonlylongitudinalnoises, ωx=ωy=0andωz/nequal0. Inthis
+case,thereis nospinrelaxation1 /T1=0 butspin dephasingrateis still finite, i.e.,1 /T2=ω2zτc. Thisisanotherpure
+dephasingcase. If ω2x+ω2y≪ω2z, thenT2≪T1.
+3.1.2. Ensemblespin relaxationandspindephasing: asimpl e model
+To illustrate the role of the inhomogeneousbroadeningin th e ensemble spin relaxation and spin dephasing, here
+weintroduceasimplemodelwhichactuallyhasbeenwellstud iedinthecontextofsemiconductoroptics. Denotethe
+conduction band as “spin up”, the valence band as “spin down” and their distributions as f↑
+kandf↓
+k. The interband
+coherence, like spin coherence, is a complex variable ρk=ρcv(k)e−iωt(ωis the laser light frequency). Hence the
+model system is equivalent to a spin ensemble. The optical re laxation and dephasing is described by the Bloch
+equation[345, 346]
+∂tSk=ωk×Sk−1
+T20 0
+01
+T20
+0 01
+T1(Sk−S0
+k). (43)
+Insemiconductoroptics, ωk=(−dcvE,0,δk) wheredcvis the interbandopticaldipole, Eis the laser electric field and
+δk=εc,k−εv,k−ω=k2/2mR+Eg−ωis the detuning with the frequencyof light. For simplicity, takeω=Egand
+19An example isthe spin relaxation in GaAs quantum dots due tot he spin-orbit coupling and phonon scattering as demonstrat ed byGolovach et
+al. [343]. This conclusion was later generalized to spin rel axation due to arbitrary phonon scattering at low temperatu re when only the lowest two
+Zeeman levels are relevant by Jiang etal. [344].
+16henceδk=k2/(2mR).mRis the reduced effective mass of electron-hole pair. Sk=(Reρk,−Imρk,[f↑
+k−f↓
+k]/2) and
+theequilibriumpolarizationis S0
+k=(0,0,−1/2).T1andT2representtheirreversibledecayduetofluctuations. Inthe
+absenceoflaser field E=0,the solutionoftheequationgivesthetransversepolariz ationas
+/bracketleftiggSx
+k(t)
+Sy
+k(t)/bracketrightigg
+=/bracketleftiggcos(δkt)−sin(δkt)
+sin(δkt) cos(δkt)/bracketrightigg/bracketleftiggSx
+k(0)
+Sy
+k(0)/bracketrightigg
+e−t/T2. (44)
+Astheoscillationfrequency δkvarieswith k,the totaltransversepolarizations Sx(t)=/summationtext
+kSx
+k(t) andSy(t)=/summationtext
+kSy
+k(t)
+get an additional decay due to inhomogeneous broadening bes idese−t/T2. In two-dimensional system, assuming a
+simplifiedinitialcondition: Sx
+k(0)=0,Sy
+k(0)=−1fork≤kmandSy
+k(0)=0fork>km, oneobtains
+Sx(t)=e−t/T2/summationdisplay
+ksin(δkt)=e−t/T2k2
+m
+4π1−cos(δkmt)
+t/T′′
+2. (45)
+Theinhomogeneousbroadeninginducesapowerlawdecaywith acharacteristictime T′′
+2=2mR/k2
+m. Nowtheoptical
+dephasingrateis
+1/T∗
+2=1/T2+1/T′′
+2. (46)
+The inhomogeneousbroadeninginduced dephasing1 /T′′
+2can be removedby photon echo. Exertinga π-pulse along
+they-axisat time T,the transversepolarizationafterthepulseis
+/bracketleftiggSx
+k(T)
+Sy
+k(T)/bracketrightigg
+=/bracketleftiggcos(δkT) sin(δkT)
+−sin(δkT) cos(δkT)/bracketrightigg/bracketleftiggSx
+k(0)
+Sy
+k(0)/bracketrightigg
+e−T/T2. (47)
+Thetransversepolarizationat time t=2Tisgivenby
+/bracketleftiggSx
+k(2T)
+Sy
+k(2T)/bracketrightigg
+=/bracketleftiggcos(δkT)−sin(δkT)
+sin(δkT) cos(δkT)/bracketrightigg/bracketleftiggSx
+k(T)
+Sy
+k(T)/bracketrightigg
+e−T/T2=/bracketleftiggSx
+k(0)
+Sy
+k(0)/bracketrightigg
+e−2T/T2. (48)
+Nowtheinhomogeneousbroadeninginduceddephasingisremo vedandonlytheirreversibledephasing1 /T2remains.
+Theoretically,thedecayrateoftheincoherentlysummedop ticalcoherence P(t)=/summationtext
+k|ρk(t)|=/summationtext
+k|Sx
+k(t)+iSy
+k(t)|,can
+beusedtoobtainthe irreversibleopticaldephasingrate1 /T2[347],asthe phaseof ρk(t)is removed.
+Similar to the case of optical dephasing, the inhomogeneous broadening of the spin precession induces spin de-
+phasinginspinensemble[334]. Thisinhomogeneousbroaden ingcanberemovedbyspinecho[348]. Aspointedout
+by Wu and co-workers, the irreversible spin dephasing can be obtained from the decay of the incoherently summed
+spincoherence[334, 336, 349, 350],similar tothatin optic aldephasing.
+In system with isotropic g-tensor, the spin polarization parallel to the magnetic fiel d does not precess. However,
+ingeneral,the g-tensorcanbeanisotropicandspinpolarizationparallelt othemagneticfieldcanalso precess. Inthis
+case, spin polarization along the magnetic field may also su ffer the inhomogeneous broadening induced decay. We
+then have both T∗
+1andT∗
+2in such situation.20An example of such ensemble is the spins in self-assembled qu antum
+dots[352–354].
+3.2. Spinrelaxationmechanisms
+In this subsection, we introduce spin relaxation mechanism s in semiconductors. Generally speaking, any fluc-
+tuation or inhomogeneity of spin interaction can induce spi n relaxation and dephasing. However, there are several
+mechanisms which are more e fficient than others. For the materials widely used in spintron ics, such as III-V and
+II-VI semiconductors, there are only a few relevant spin rel axation/dephasingmechanisms, such as the Elliott-Yafet
+mechanism [103, 104], the D’yakonov-Perel’ mechanism [101 , 102], the Bir-Aronov-Pikus mechanism [105, 106]
+and theg-tensor inhomogeneity mechanism [333, 334]. These mechani sms dominate spin relaxation and dephasing
+inmetallicregime. Intheinsulatingregime,othermechani smssuchastheanisotropicexchangeinteraction[40,322]
+and the hyperfineinteraction [3] prevail. In semiconductor quantumdots, spin relaxation /dephasingis dominatedby
+the electron-phonon scattering and the hyperfine interacti on as well as their combinations. In magnetically doped
+semiconductors,the exchangeinteractionwith magneticim puritiescan also play an importantrole in spin relaxation
+anddephasing. Below wegiveabriefintroductiontotheseme chanisms.
+20A simpleexample demonstrating thesame physics is presente d in Ref. [351].
+173.2.1. Elliott-Yafetmechanism
+The Elliott-Yafet mechanism was first proposed by Elliott [1 04] and Yafet [103] during their study on spin re-
+laxation in silicon and alkali metals. It was pointed out by E lliott that due to the spin-orbit interaction the electroni c
+eigenstates(Blochstates) mixspin-upandspin-downstate s. Forexample[21, 104],
+Ψkn↑(r)=[akn(r)|↑∝an}b∇acket∇i}ht+bkn(r)|↓∝an}b∇acket∇i}ht]eik·r, (49)
+Ψkn↓(r)=[a∗
+−kn(r)|↓∝an}b∇acket∇i}ht−b∗
+−kn(r)|↑∝an}b∇acket∇i}ht]eik·r, (50)
+whereakn(r) andbkn(r) possess the lattice periodicity. These two Bloch states ar e connected by time reversal and
+space inversion operators [21, 104]. Usually the spin mixin g is very small, i.e., |b|≪1 and|a|≈1. When the spin-
+orbitinteractionismuchsmaller than the bandsplitting,a n estimation of|b|fromperturbationtheorygives[21, 355]
+|b|∼max{LSO/∆E}, whereLSOis the spin-orbit interactions with other bands and ∆Eis the band distances. In the
+presenceofsuchspinmixing,anyspin-independentscatter ingcancausespinflipandhencespinrelaxation. Itshould
+be emphasized that without scattering the spin-mixing alon e can not lead to any spin relaxation. Besides, there is
+another process leading to spin relaxation: the phonon modu lation of the spin-orbit interaction. As the spin-orbit
+interaction is induced by the periodic lattice ions, the lat tice vibration can then directly couple to spin and lead to
+spin-flip. Such kindof processwas first consideredby Overha userwithin jullium model for metal [356] and then by
+Yafetwithspecificbandstructure[103]. Thisprocessiscal ledtheYafetprocess,whereasthepreviousonedueto the
+spinmixingiscalledtheElliott process.
+Elliottobservedarelation,the“Elliottrelation”,betwe enspinlifetime τsandthedeviationoftheelectron g-factor
+fromthatofthefreeelectron g0=2.0023[104]: 1 /τs≈(∆g)2/τpwhere∆g=g−g0andτpistheaveragemomentum
+scattering time. The relation is based on the observation th at∆g∼|b|by perturbation theory. The Elliott relation
+was verified experimentallyby Monodand Beuneu, who observe dan empirical factor of 10, i.e., 1 /τs≃10(∆g)2/τp
+[357]. Ofcoursetheserelationsarerough,thespecificrati oofτs/τpdependsonspecificscatteringandbandstructure.
+Systematic theoretical study was given by Yafet [103], wher e the microscopic band structure and electron-phonon
+scattering were considered to obtain the spin lifetime. Yaf et gave a relation between the spin lifetime τsand the
+resistivityρ: 1/τs∼∝an}b∇acketle{tb2∝an}b∇acket∇i}htρ[103]. The Yafet relation was tested experimentally by Mono d and Beuneu [358]. It was
+foundthat in many materials the Yafet relation agrees well w ith experiments[21, 358]. However,it is not consistent
+withtheexperimentalresultsinMgB 2wherethespinrelaxationrateisnotproportionaltotheres istivityabove150K.
+The problemwas solved by Simon et al. [355] recently via gene ralizingthe Elliott-Yafet theoryto the regime where
+thescattering-inducedspectralbroadeningofthequasi-p article1/τpiscomparablewiththebandgap. Thiscondition
+issatisfiedinMgB 2becauseoneofthebandacrosstheFermisurfaceisveryclose tothenearestband( ≃0.2eV)and
+theelectron-phononinteractionisstrong. Inthisregime, thespin lifetimeisgivenby,
+τ−1
+s=L2
+effτp
+1+∆ω2
+effτ2p, (51)
+where∆ωeffis the average band gap and Leffis the interband spin-orbit interaction. In the weak scatte ring or large
+band-gap regime, ∆ωeffτp≫1, the above equation returns to the Yafet relation, 1 /τs=(Leff/∆ωeff)2/τp. Spin
+relaxation due to the Elliott-Yafet mechanism in polyvalen t metals (such as aluminum) was studied by Fabian and
+Das Sarma [359, 360]. Through realistic calculation, they f ound that the electron spin relaxation is significantly
+enhancedattheBrillouinzoneboundaries,specialsymmetr ypointsandlinesofaccidentaldegeneracy. Thetotalspin
+relaxationrate isthendeterminedbythe spinrelaxationra tesat these“spin-hot-spots”[359, 360].
+In III-V or II-VI semiconductors zinc-blende structure, fo r realistic calculation of the Elliott-Yafet spin relax-
+ation, one should start from the Kane Hamiltonian. By L¨ owdi n partitioning (block-diagonalization),the conduction
+band wavefunction is transformed to ˜Ψc(k)=UΨn(k) whereUis the unitary matrix for L¨ owdin partitioning. This
+transformationmixesspin-upand-downstates alittle, whi chenablesspinflip byspin-independentfluctuation. After
+the transformation U, the conduction band matrix element of a spin-independent fl uctuation of the lattice potential
+(including the electron-electron interaction) Vck,ck′=∝an}b∇acketle{tck|V|ck′∝an}b∇acket∇i}ht, changes into ˜Vck,ck′=∝an}b∇acketle{tck|U†VU|ck′∝an}b∇acket∇i}ht, which may
+contain spin-flip part. The spin-flip interaction consists o f long-range and short-range parts. The long-range part
+comes from the first order perturbation of the wavefunction ( both inUandU†) and the intraband spin-independent
+fluctuation. Theshort-rangepartcomesfromthecombinatio nofU(orU†)andtheinterbandelectron-phononinterac-
+tions. Forexample,for U=eSk,thelowestorderlong-rangepartisgivenby ∝an}b∇acketle{tck|1
+2(S†
+kS†
+kV+2S†
+kVSk′+VSk′Sk′)|ck′∝an}b∇acket∇i}ht,
+18whereas the lowest order short-range part is given by ∝an}b∇acketle{tck|S†
+kV+VSk′|ck′∝an}b∇acket∇i}ht. The long-range interaction is the Elliott
+process,whereastheshort-rangeinteractionistheYafetp rocess.
+Afterthetransformation,theleadingtermofthelong-rang epartisgivenby[117]
+˜Vck,ck′=Vck,ck′[1−iλc(k×k′)·σ], (52)
+whereλc=η(1−η/2)/[3meEg(1−η/3)] withη=∆SO/(∆SO+Eg). The spin lifetime due to long-range part of the
+Elliott-Yafetmechanismis
+τ−1
+s=A∝an}b∇acketle{tεk∝an}b∇acket∇i}ht2
+E2gη2/parenleftigg1−η/2
+1−η/3/parenrightigg2
+τ−1
+p, (53)
+wherethenumericalfactor A∼1dependingonthespecificscattering. Theshort-rangeinte ractionduetotheinterband
+electron–optical-phononinteractionisgivenby
+˜VOP
+k,k′=−1
+3ηd2[U×(k+k′)]·σ
+2meEg(1−η/3)]1/2. (54)
+HereUistherelativedisplacementofthetwoatomsinaunitcellwh ichcanbeexpressedascombinationsofphonon
+creation and annihilation operators [117]. Assuming that t he longitudinal and transversal optical phonons have the
+samefrequency ωLO=ωTO, thespinrelaxationduetotheshort-rangeinteractionrea ds[117]
+τ−1
+s=Asr∝an}b∇acketle{tεk∝an}b∇acket∇i}ht2
+EgE0η2
+1−η/3τ−1
+p. (55)
+HereE0=C2/(4d2me) withC=eωLO/radicalig
+4πD(κ−1∞−κ−1
+0) (Dis the volume density, κ0andκ∞are the static and high
+frequencydielectricconstants)and Asr∼1. Theshort-rangeinteractionduetotheinterbandelectro n–acoustic-phonon
+interaction can be derived similarly. The explicit form is g iven in Ref. [117]. Usually the long-range interaction is
+moreimportantthanthe short-rangeoneinIII-Vsemiconduc tors[21, 117].
+The Elliott-Yafet spin relaxation in in bulk InSb at low temp erature was first studied by Chazalviel [361], where
+only the electron-impurity scattering is considered. Very recently, Jiang and Wu studied the Elliott-Yafet spin re-
+laxationin bulk III-Vsemiconductorswith all relevantsca tterings(i.e., electron-impurity,electron-phonon,ele ctron-
+electronCoulombandelectron-holeCoulombscatterings)i ncludedin a fullymicroscopicfashion[110]. TheElliott-
+Yafetspin relaxationin bulk silicon wasstudied systemati callybyCheng et al. [362], wherethe Elliott andthe Yafet
+processesinterferedestructivelyandthe spinrelaxation islargelysuppressed[362].
+Similarly,boththeadmixtureofdi fferentspinstatesduetotheL¨ owdinpartitioningandthepho nonmodulationof
+spin-orbitinteractionleadtothe Elliott-Yafetspinrela xationforholesandsplit-o ffholes.
+3.2.2. D’yakonov-Perel’mechanism
+In III-V and II-VI semiconductors, due to the bulk inversion asymmetry in lattice structure, spin-orbit coupling
+emergesinconductionband. Insemiconductornanostructur es,thestructureandinterfaceinversionasymmetryfurthe r
+contributesadditionalspin-orbitcoupling. Spin-orbitc ouplingis equivalentto a k-dependenteffectivemagneticfield
+HSO=1
+2Ω(k)·σ, whereΩ(k) is the spin precession frequency. In the presence of moment um scattering, electron
+changesitsmomentum krandomly,hencespinprecessesrandomlybetweenadjacents catteringevents. Thisrandom-
+walk-like evolution of spin phase leads to spin relaxation. This spin relaxation mechanism is called the D’yakonov-
+Perel’ mechanism [101, 102]. There are two regimes for the D’ yakonov-Perel’spin relaxation: (i) strong scattering
+regime where∝an}b∇acketle{tΩ∝an}b∇acket∇i}htτp≪1 (∝an}b∇acketle{t...∝an}b∇acket∇i}htstands for the average over the electron ensemble), and (ii) weak scattering regime
+where∝an}b∇acketle{tΩ∝an}b∇acket∇i}htτp/greaterorsimilar1. Hereτpis the momentum scattering time. The spin lifetime in regime (i) can be estimated
+as 1/τs=∝an}b∇acketle{tΩ2∝an}b∇acket∇i}htτpaccording to the random walk theory. This spin relaxation ha s the salient feature of motional
+narrowing, i.e., stronger momentum scattering leads to lon ger spin lifetime. In regime (ii) the momentumscattering
+no longer impedes spin relaxation. In contrast, via the spin -orbit coupling, momentum scattering provides a spin
+relaxation channel [334, 363]. In this regime, stronger mom entum scattering leads to shorter spin lifetime [363].
+Analytic results with only the electron-impurity scatteri ng give that the irreversible spin lifetime is τs=2τp[363–
+365]. As∝an}b∇acketle{tΩ∝an}b∇acket∇i}htτp/greaterorsimilar1, the spin precession due to the spin-orbit coupling is not i nhibited by the scattering. As a
+19consequence, the ensemble spin polarization oscillates at zero magnetic field [363, 364]. Besides, such prominent
+spin precession leads to the free induction decay. The ensem ble spin lifetime is then limited by both the irreversible
+decay and the free induction decay τ−1
+s≃/radicalbig
+∝an}b∇acketle{tΩ2∝an}b∇acket∇i}ht[21, 363, 365]. Both regime (i) and regime (ii) have been real ized
+experimentallyin GaAs quantum wells [366, 367]. Especiall y, the crossover from regime (ii) to regime (i) has been
+observedbyBrandet al. [368].
+The characteristic of the D’yakonov-Perel’ spin relaxatio n is that during adjacent momentum scatterings spins
+precesscoherently. Henceaspin-echoatatimescalecompar abletoorsmallerthanthemomentumscatteringtime τp
+canefficientlysuppresstheD’yakonov-Perel’spinrelaxation. Su chaproposalwasgivenina recentworkbyPershin
+[348].
+In most cases the system is in the strong scattering regime. I n this regime, some analytical results about spin
+lifetime can be obtained for isotropic band, if one assumes t hat (1) the carrier-carrier scattering can be neglected,21
+(2)the system is near equilibrium22and (3) the electron-phononscattering can be treated in the elastic scattering ap-
+proximation. Within these assumptionsand approximations ,for three-dimensionalcase, followingPikus and Titkov,
+theevolutionofspin polarizationalong zdirectionis[117]
+∂tSz=−˜τl[Sz∝an}b∇acketle{tΩl
+xΩl
+x+Ωl
+yΩl
+y∝an}b∇acket∇i}ht−Sx∝an}b∇acketle{tΩl
+xΩl
+z∝an}b∇acket∇i}ht−Sy∝an}b∇acketle{tΩl
+yΩl
+z∝an}b∇acket∇i}ht], (56)
+where∝an}b∇acketle{t...∝an}b∇acket∇i}htrepresentstheaverageoverthedirectionof k. Theequationsfortheevolutionof SxandSycanbeobtained
+byindexpermutation. Themomentumscatteringtime ˜ τlisgivenby
+˜τ−1
+l=/integraldisplay1
+−1W(θ)[1−Pl(cosθ)]dcosθ. (57)
+Theinteger lis determinedby theangulardependenceofthe spin-orbitfie ld. The aboveresultsassume thatthe spin-
+orbitcouplingHamiltoniansatisfies ˆHSO=1
+2Ωl·σ=/summationtext
+mˆClmYl
+m(θk,φk),whereθkandφkare theangularcoordinates
+ofkinsphericalcoordinates, Yl
+mstandsforthesphericalhamonicsandthecoe fficientsˆClmare2×2matrices. Herethe
+summationistakenover m,whereas lisfixed. Forexample, l=1forlinearspin-orbitcouplingduetostrainand l=3
+forDresselhauscubicspin-orbitcoupling. Iftherearebot hlinearandcubicspin-orbitcouplings,thesummationover
+lshouldalsobeaddedintotheaboveequations. AccordingtoE q.(56),withthesummationover l,thespinrelaxation
+tensorisgivenby[3, 21,22]
+(τ−1
+s)ij=/summationdisplay
+l˜τp
+γl[∝an}b∇acketle{t|Ωl|2∝an}b∇acket∇i}htδij−∝an}b∇acketle{tΩl
+iΩl
+j∝an}b∇acket∇i}ht], (58)
+whereγl=˜τp/˜τland ˜τp=˜τ1. Fortwo-dimensionalcase, expandingthe spin-orbitcoupl ingasˆHSO=/summationtext
+l1
+2Ωleilθk·σ,
+oneobtainsasimilar result[369, 370]
+(τ−1
+s)ij=/summationdisplay
+l˜τp
+γl[|Ωl|2δij−Ωl
+iΩ−l
+j], (59)
+with ˜τ−1
+l=/integraltext2π
+0W(θ)(1−coslθ)dθ. Togiveanexample,considerelectronspinrelaxationduet otheD’yakonov-Perel’
+mechanisminbulkIII-Vsemiconductors. Intheabsenceofst rain,theelectronspin-orbitcouplingcomessolelyfrom
+theDresselhausterm,
+Ω(k)=2γD[kx(k2
+y−k2
+z),ky(k2
+z−k2
+x),kz(k2
+x−k2
+y)]. (60)
+Inthiscase spinrelaxationisisotropic
+(τ−1
+s)ij=δij˜τp
+γ38
+105(2γD)2k6. (61)
+Hereγ3dependsonrelevantscattering: forionized-impurityscat teringγ3≃6;foracoustic-phononscattering γ3≃1;
+foroptical-phononscattering γ3≃41/6. Afterthe averageoverelectrondistribution,the spinre laxationrateis given
+by[21],
+τ−1
+s≃Q′τmα2∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}ht/Eg. (62)
+21Itwasfirstpointed outbyWuandNing [334]that although norm al(notumklapp) carrier-carrier scattering does notcontr ibute tothe mobility,
+itcontributes to the D’yakonov-Perel’ spin relaxation as i trandomizes the momentum.
+22Thatis, electron distribution is close to the equilibrium d istribution and the spin polarization is very small.
+20α=2γD/radicalig
+2m3eEgis a dimensionlessparameter. τm=∝an}b∇acketle{t˜τp(εk)εk∝an}b∇acket∇i}ht/∝an}b∇acketle{tεk∝an}b∇acket∇i}ht.23Q′∼0.1 is a numericalconstantdepending
+on the relevant momentum scattering [21] Q′=128
+3675γ−1
+3(ν+7
+2)(ν+5
+2), where the power law ˜ τp∼εν
+kis assumed for
+eachkindofmomentumscattering. Fornondegenerateelectr onsystem,oneobtains( Q=105
+8Q′)
+τ−1
+s≃Qτmα2(kBT)3/Eg, (63)
+whereQ∼1dependingonrelevantscattering: Q≃1.5forionizedimpurityscattering, Q≃3forlongitudinaloptical
+phonon scattering [371], Q≃0.8 for piezoelectric acoustic phonon scattering, and Q≃2.7 for acoustic phonon
+scatteringduetodeformationpotential[3, 21].
+Finally, it should be noted that cautionmust be taken on the assumptions and approximations used in t he above
+results. For example, in intrinsic bulk GaAs at temperature below 100 K, the electron-electron and electron-hole
+scatterings dominate the momentum scattering. The assumpt ion that the carrier-carrier scattering can be neglected
+does not hold. Therefore, the above results fail [110]. It ha s been shown that the electron-electron scattering is
+importantforspinrelaxationinhighmobilitytwo-dimensi onalsystemwheretheaboveresultsalsofail [366, 372].
+The influence of the magnetic field on the D’yakonov-Perel’ sp in relaxation comes from two factors: the Zee-
+man interaction and the orbital e ffect.24The Zeeman interaction leads to the Larmor spin precession a nd induces a
+slowdownofthe relaxationforspincomponentparalleltoth eLarmorspinprecessiondirection[117],
+τs(B)≃τs(0)[1+(ωLτc)2], (64)
+whereωLis theLarmorfrequency. τc=γ−1
+l˜τpis thecorrelationtime ofthe randomspin precessiondueto s pin-orbit
+coupling. This result is similar to Eq. (40). The Larmor spin precession also mixes the relaxation rates of the spin
+componentsperpendicular to the Larmor spin precession dir ection. This may lead to considerable e ffect on the spin
+relaxationinsemiconductornanostructureswherethespin relaxationtensorisusuallyanisotropic[199,377,388]. I n
+non-quantizingmagnetic field, the orbital e ffect induces the cyclotron motion where the electron velocit y (hence its
+k) rotates under the Lorentz force. As the spin-orbit e ffective magnetic field Ω(k) iskdependent, the rotation in k
+leadsto the rotationin the spin-orbitfield. Thisleadsto th e reductionof the D’yakonov-Perel’spin relaxationwhich
+isrelatedto therotatingcomponentof Ω(k)[117, 376]
+τs(B)≃τs(0)[1+(ωcτc)2], (65)
+whereωc=eB/meisthecyclotronfrequency. NotethattheratiooftheLarmor frequencyandthecyclotronfrequency
+isωL/ωc=|ge|me/(2m0). In semiconductors, as meis usually smaller than m0and often|ge|<2, the ratio is always
+muchsmallerthan1(inGaAs,itis ≃0.015). Hencethecyclotrone ffectisstrongerthantheZeemaninteraction[117].
+The reductionof spin relaxation rate is saturated at high ma gnetic field, where Landaulevel quantizationdominates.
+Atsuchhighmagneticfield,electronspinrelaxessimilarto thatinlocalizedstatesorthatinquantumdot[382–387].
+The hole D’yakonov-Perel’spin relaxation is di fferent from the electron one mainly in two aspects: (1) the hol e
+spin-orbitcouplingisusuallymuchstronger,and(2)holeh asspinJ=3/2. Duetothestrongspin-orbitcoupling,the
+D’yakonov-Perel’mechanismisverye fficientinholesystemandthespinlifetimeisusuallyverysho rt. Forexample,
+hole spin lifetime in intrinsic bulk GaAs is observed to be ≃0.1 ps [207]. As the spin-orbit coupling is strong, the
+holesystemisusuallyintheweakscatteringregimeoftheD’ yakonov-Perel’spinrelaxation[206,208,363]. L¨ uetal.
+found that in two-dimensional hole system the spin polariza tion oscillates without an exponentialdecay in the weak
+scatteringregime[363]. In thiscase,the spin lifetimeisn ot easilycharacterized. However,the incoherentlysummed
+spin coherence P(t)=/summationtext
+k|ρk(t)|(ρkdenotes the spin coherence) shows good exponential decay an d from its decay
+ratethe irreversiblespin dephasingrate is obtained[363] . Anothersalient featureofholespin system isthat holehas
+spinJ=3/2. As a consequence the spin density matrix of hole system is 4 ×4 which contains spin coherence that
+doesnot correspondto anyspinpolarization. Also, the hole spin-orbitcouplingmay notbe equivalentto ane ffective
+magnetic field as it can be high powers of the hole spin operato rJ(such as ( k·J)2in the Luttinger Hamiltonian
+23In the remaining part of thepaper, wedenote all these quanti ties (˜τl, ˜τpandτm) asτpin qualitative discussions, if wecan.
+24Effectof magnetic field on theD’yakonov-Perel’ spin relaxatio n was studied in the semiclassical limit in Refs. [117,333,3 73–381]as well as
+in the quantum limit in Refs. [382–387].
+21[Eq. (17)]). In the work of Winkler [389], the hole spin densi ty matrix is decomposed by the spin multipole series
+to provide a more systematic understanding on the spin-depe ndent phenomena in hole system. Winkler found that
+the hole spin-orbit coupling can induce transfer between di fferent spin multipoles, while the magnetic field can only
+induce spin precession of the spin dipole (i.e., the spin pol arization) in bulk hole system [389]. The transfer of the
+hole spin multipoles under the hole spin-orbit coupling was found to influence the hole spin polarization relaxation
+dueto theD’yakonov-Perel’mechanism[208].
+3.2.3. Bir-Aronov-Pikusmechanism
+It was proposedby Bir, AronovandPikusthatthe electron-ho leexchangescatteringcan leadto e fficient electron
+spin relaxation in p-type semiconductors [105, 106]. According to the electron -hole exchange interaction given in
+Eq. (37), within elastic scattering approximation, spin li fetime limited by the Bir-Aronov-Pikusmechanism is given
+bythe FermiGoldenrule,
+1
+τs(k)=4π/summationdisplay
+q,k′
+m,m′δ(εk+εh
+k′m′−εk−q−εh
+k′+qm)|J(+)k′m′
+k′+qm|2fh
+k′m′(1−fh
+k′+qm). (66)
+Hereεh
+kmis hole energywith spin index m,Jis the electron-holeexchangeinteraction matrix element [ see Eq. (37)
+in Sec. 2.9] and fhis hole distribution function. As hole spin and momentum rel ax very fast, fhis taken as hole
+equilibriumdistribution. Inbulksemiconductors,therea reseveralfactswhichcouldhelptoreducetheabovecompli-
+cated equation: (i) the heavy-hole density of states is much larger than the light-hole one (thanks to the much larger
+heavy-holeeffective mass), which makes the contributionto the spin relax ationrate mainly from the heavy hole; (ii)
+theheavy-holeeffectivemassismuchlargerthantheelectronone,whichenabl estheelasticscatteringapproximation.
+Using these approximationsand including only the short-ra ngeexchange interaction, one obtains a simple result for
+nondegenerateholesystem[117]
+τ−1
+s=2
+τ0nha3
+B∝an}b∇acketle{tvk∝an}b∇acket∇i}ht
+vB, (67)
+whereaBistheexcitonBohrradius,1 /τ0=(3π/64)∆E2
+SR/(EB)withEBbeingtheexcitonBohrenergy, nhisthehole
+density,∝an}b∇acketle{tvk∝an}b∇acket∇i}ht=∝an}b∇acketle{tk/me∝an}b∇acket∇i}htis the average electron velocity, and vB=1/(mRaB) withmR≈mebeing the reduced mass of
+theinteractingelectron-holepair. Inthe presenceofloca lizedholes,theequationisimprovedto be[117]
+τ−1
+s=2
+τ0NAa3
+B∝an}b∇acketle{tvk∝an}b∇acket∇i}ht
+vB/parenleftiggnh
+NA+5
+3NA−nh
+NA/parenrightigg
+, (68)
+whereNAis theacceptordensity. Fordegenerateholesystem [117],
+τ−1
+s≃3
+τ0nha3
+B∝an}b∇acketle{tvk∝an}b∇acket∇i}ht
+vBkBT
+Eh
+F, (69)
+withEh
+Fdenoting the hole Fermi energy. In an interacting electron- hole plasma, electron and hole attract each other
+and there is an enhancement of the electron-hole exchange in teraction due to this attraction. This enhancement is
+describedbytheSommerfeldfactor. ForunscreenedCoulomb potential,theSommerfeldfactoris |ψ(0)|2=2π
+εk/EB/slashbig[1−
+exp(−2π
+εk/EB)]. With this factor, spin relaxation is enhanced, 1 /τ′
+s=|ψ(0)|4/τs. However, for a completely screened
+Coulomb potential, there is no enhancement, |ψ(0)|2=1. The effect of the Sommerfeld factor was discussed in
+Refs. [117, 390]. It should be noted that the long-range elec tron-hole exchange interaction can not be neglected,
+often (such as in GaAs) it is more important than the short-ra nge one [110]. Hence the above analytical formulae
+is quitelimited, unlessESRis substituted by some proper average of the whole (both the s hort-range and the long-
+range)electron-holeexchangeinteraction. Theanalytica lexpressionsfortheBir-Aronov-Pikusspinrelaxationinb ulk
+materialsandin quantumwellswithbothshort-rangeandlon g-rangeinteractionsaregiveninRef. [390].
+Spin relaxation rate due to the Bir-Aronov-Pikus mechanism is usually calculated via Eq. (66), which actually
+implies the elastic scattering approximation [118, 390, 39 0, 391]. Recently Zhou and Wu [109] reinvestigated the
+Bir-Aronov-Pikusspinrelaxationwithoutsuchapproximat ionfromthefullymicroscopickineticspinBlochequation
+approach [44, 334, 350]. They found that spin relaxation rat e was largely overestimated at low temperature in the
+22previous theories. The underlying physics is that at low tem perature the Pauli blocking impedes spin-flip in fully
+occupiedstates[109]. Thespin-flipisonlyallowedaroundt he chemicalpotential,whichsuppressestheBir-Aronov-
+Pikusspinrelaxationat lowtemperature. Amoetal. foundsi milar argumentsandresults[392].
+Finally, holes also su ffer the Bir-Aronov-Pikus spin relaxation. However, as hole s pin-orbit coupling is strong,
+the D’yakonov-Perel’ mechanism is very e fficient and the Bir-Aronov-Pikus mechanism rarely shows up. T he Bir-
+Aronov-Pikusmechanismmaybecomeimportantforholesinhe avilyn-dopedquantumwells,wheretheD’yakonov-
+Perel’mechanismissuppressedbythehole-electronandhol e-impurityscatteringsandthespinlifetimecanberather
+long(/greaterorsimilar500ps)[393–395].
+3.2.4. g-tensorinhomogeneity
+Under a given magneticfield, spin precession(both directio nand frequency)is determinedby the g-tensor. Mar-
+gulisandMargulisfirstproposedaspinrelaxationmechanis minthepresenceofmagneticfieldduetothemomentum-
+dependent g-factor[333]. LaterWuandNingalsopointedoutthatenergy -dependent g-factorgivesrisetoaninhomo-
+geneousbroadening [334] in spin precession and hence resul ts in spin relaxation. Without scattering the inhomoge-
+neousbroadeningonlyleadstoareversiblespinrelaxation . Anyscattering,includingtheelectron-electronscatter ing,
+which randomizes spin precession, results in irreversible spin relaxation [334]. In the irreversible regime, under a
+magneticfield Balongjdirection,therelaxationrateofthespincomponentalongt heidirection[ i,j=x,y,z]canbe
+obtainedin analogywith Eq.(40),
+1
+τs,ii=(µBB)2/summationdisplay
+l/nequali(g2
+lj−glj2)τc
+1+(µBB)2/summationtext
+n/nequallgnj2τ2c. (70)
+Herel,n=x,y,z,gijistheg-tensorand τcisthecorrelationtime ofspinprecessionlimitedbyscatte ring.gljdenotes
+theensembleaverageof glj. Theaboveequationholdsonlyin themotionalnarrowingreg ime,i.e.,
+(µBB)/radicalig/summationtext
+l/nequali(g2
+lj−glj2)τc≪1 for any i. In III-V or II-VI semiconductors and their nanostructures , the electron
+g-tensor is k-dependent [223, 224], hence τcis limited by momentum scattering ( τc≃τp) [333, 374].25In the case
+of isotropic g-tensor, the g-tensor inhomogeneity only leads to spin relaxation transv erse to the magnetic field. For
+example,inIII-VorII-VIbulksemiconductors,the k-dependent g-factorlimitedelectronspindephasingtimeis
+T−1
+2≃(µBB)2(g2−g2)τp. (71)
+Itis notedthat theinducedspindephasingrateincreaseswi thincreasingmagneticfield. The g-tensorinhomogeneity
+mechanismusuallydominatesat highmagneticfield [374].
+Another limit is that there is no scattering or the scatterin g is very weak. In this case, the spin lifetime is limited
+by free induction decay.26In this regime, if the inhomogeneity of the g-tensor can be characterized by a Gaussian
+distribution,the coherentspinprecessionleadstoa Gauss iandecay∼exp(−t2/τ2
+s) withthespinlifetimebeing
+1
+τs,ii≃µBB√
+2/radicaligg/summationdisplay
+l/nequali(g2
+lj−glj2). (72)
+In self-assembled quantum dot ensemble, the g-tensor inhomogeneity mechanism leads to e fficient spin relaxation,
+masking the intrinsic irreversible one [331, 396, 397]. How ever, such spin relaxation can be removed by spin echo
+or mode-lockingtechniques in experiments [331]. The g-tensor inhomogeneitymechanism is also important for the
+spin relaxation of localized electrons in n-type quantum wells [398] and localized holes in p-type quantum wells
+[399,400].27
+25If theg-tensor is only energy dependent, then τcis only limited by inelastic scattering.
+26Out of the strong scattering and no scattering limits, the sp in lifetime is limited by both the free induction decay time a nd the irreversible spin
+relaxation time.
+27Thek-dependent hole g-factor in [001] quantum wells is given by Eq.(25).
+233.2.5. Hyperfineinteraction
+Recallingthatthe hyperfineinteractionbetweenelectrons pinandnuclearspinsis
+Hhf=/summationdisplay
+i,νAνv0|ψ(Ri)|2S·Ii,ν=h·S. (73)
+Hereνlabels both the atomic site in the Wigner-Seitz cell and the i sotopes, while iis the index of the cell. Aνis the
+coupling constant of the hyperfine interaction. v0is the volume of the Wigner-Seitz cell and N0=1/v0.ψ(r) is the
+envelopefunctionoftheelectronwavefunction. histheeffectivehyperfinefieldactingonelectronspin,whichisalso
+called the Overhauser field. The hyperfine interaction excha nges electron spin with nuclear spin and hence leads to
+electronspinrelaxation.
+Therearedifferentscenariosofthehyperfineinteractioninducedelectr onspinrelaxationactinginseveralregimes.
+There are four regimes: (i) the non-interacting spin ensemb le without spin echo [38]; (ii) the non-interacting spin
+ensemble with spin echo [38]; (iii) the hoppingregime where electronsat different local sites are weakly connected;
+and(iv)themetallicregimewheremostelectronsareinexte ndedstates.
+In regime (i), spins are separated in space or time and do not i nteract with each other, i.e., they evolve indepen-
+dently. Examples are spins in a number of singly charged self -assembled quantum dots [401] and single spin in a
+quantum dot (or two spins in a double quantum dot) measured at different times [38, 402]. In this regime, the spin
+relaxationis determinedby the free inductiondecay due to t he randomlocal Overhauserfield. For the quantumdots
+ensemble, each electron spin in a quantum dot interacts with about 103∼106nuclear spins. According to central
+limittheorem,thedistributionoftheOverhauserfield isGa ussian
+P(h)=1
+(2πσ2
+h)3/2exp(−h2/2σ2
+h). (74)
+Thevariance σhisgivenby
+σh=/radicaligg
+1
+3/summationdisplay
+νA2νIν(Iν+1)v0/integraldisplay
+d3r|ψ(r)|4=h1//radicalbig
+NL, (75)
+whereh1=/radicalig
+2
+3/summationtext
+νA2νIν(Iν+1) andNL=2/[v0/integraltext
+dr|ψ(r)|4] is the effective number of the nuclei. Assume that the
+Overhauser field is quasi-static, i.e., the Overhauser field fluctuates at the time scale much longer than the electron
+spin lifetime [403]. Under such an approximation, the elect ron spin polarization decays as exp( −t2/τ2
+s), where the
+spinlifetimeis
+τ−1
+s=√
+2σh. (76)
+ForNL∼105, calculation gives a short spin lifetime on the time scale of 10 ns (1µs) in GaAs (silicon) quantum
+dots.28Electron spin relaxation under external (static or time-de pendent) magnetic field can be di fferent from the
+zero field case, but is still well described by averaging over spin precession under the coaction of the external and
+Overhauser field with the distribution P(h) [301, 403, 405].29It was proposed that high nuclear spin polarization
+[406]orstate-narrowingofthenucleardistribution[407, 408]canmarkedlynarrowthedistributionoftheOverhauser
+field and suppress the spin relaxation. Recent experiment in double quantum dot demonstrated the enhancement of
+theelectronspinlifetimebya factorof ∼70throughpolarizingthenuclearspins[409].
+In regime (ii), the inhomogeneousbroadening of the Overhau ser field is removed by spin-echo. The spin relax-
+ationisthencausedby thetemporalfluctuationofthe Overha userfield. The fluctuationiscausedby thenuclearspin
+dynamics due to the hyperfine interaction and nuclear spin di pole-dipole interaction. As nuclear spin relaxes much
+slower than electron spin, the electron spin dynamics due to the hyperfine interaction is non-Markovian. Further-
+more, the fluctuation in nuclear spin system is a many-bodypr oblem, where the collective excitation is not obvious.
+28This timescale is much shorter than both the time scale of the nuclear spin precession under the hyperfine field generated b y electron∼1µs
+(∼100µs) and nuclear spin-flip time due to the nuclear spin dipole-d ipole interaction∼100µs (∼6000µs) in GaAs (silicon) [303, 404], which
+justifies the quasi-static treatment of the Overhauser field [403].
+29Themagnetic field may also change the electronic wavefuncti on and hence the spin relaxation as NLdepends on the wavefunction [301].
+24These factorsmake it di fficult to calculate the spin lifetime limited by the hyperfinei nteraction. Nevertheless, recent
+developments attacked the problem via the equation of motio n approach [344, 410–413], Green function approach
+[410, 414, 415] and the quantum cluster expansion approach [ 289, 416–422]. It was predicted theoretically that via
+designed magnetic pulses, the electron spin coherence lost through the hyperfine interaction can be restored [423–
+431]. Recent experimentconfirmedthe theoretical predicti on[432]. Other experimentalinvestigationsin regime (ii)
+involve: the study of the decay of spin echo of a single spin in a quantum dot [402], two spins in a double quantum
+dot[38,433]andspinsofdonorboundelectronsinsilicon[2 88];thedevelopmentofanopticalmode-lockingmethod
+tostudytheirreversibleelectronspindephasinginself-a ssembledquantumdots[331, 434].
+Regime (iii) is the hopping regime where electrons at di fferent local sites are weakly connected. The hopping
+between the local sites induces the exchange interaction be tween electrons. The exchange interaction interrupts the
+spin precession inducedby the random local Overhauser field . If the exchange interaction is weak, the electron spin
+ensemblesuffersboththefreeinductiondecayandtheirreversiblespind ecayduetotherandomizationcausedbythe
+exchange interaction. If the exchange interaction is stron g, the spin precession around the random local Overhauser
+field is motional narrowed [39, 40, 323, 435]. An example of re gime (iii) is the localized electron spin ensemble in
+theinsulatingphaseofa n-dopedsemiconductor.
+Theexchangeinteractionbetweenelectronslimitsthecorr elationtime τcofthespinprecessionduetotherandom
+Overhauserfield. Inthefollowing,wefocusonthemotionaln arrowinglimit,whichisalsothefocusintheliterature.
+Inthislimit,∝an}b∇acketle{tωhfτc∝an}b∇acket∇i}ht≪1 (∝an}b∇acketle{t...∝an}b∇acket∇i}htdenotingthe electronensembleaverage)with ωhfbeingthespin precessionfrequency
+dueto thehyperfineinteraction. Inthisregime,thespinlif etimeisgivenby[3, 435]
+τ−1
+s=2
+3∝an}b∇acketle{tω2
+hf∝an}b∇acket∇i}htτc. (77)
+Heretheprefactor2
+3isduetothefactsthatonlytransversefluctuationcanleadt ospinrelaxationandthattheangular
+distribution of the fluctuation field is uniform. Roughly τc≃1/∝an}b∇acketle{tJij∝an}b∇acket∇i}ht, where∝an}b∇acketle{tJij∝an}b∇acket∇i}htis the averaged exchange coupling
+strengthwith iandjbeingthesite indices[39, 40].
+In regime (iv), the metallic regime, most of the electrons ar e extended and kis a good quantum number. In this
+regime,thespin relaxationduetothe hyperfineinteraction isdescribedasa spin-flipscattering. For example,in bulk
+semiconductors,thespin lifetimedueto thehyperfineinter actionisgivenby[436],
+τ−1
+s=∝an}b∇acketle{t1
+τs(εk)∝an}b∇acket∇i}ht=me∝an}b∇acketle{tk∝an}b∇acket∇i}ht
+3πν0/summationdisplay
+νβνA2
+νIν(Iν+1). (78)
+Here∝an}b∇acketle{tk∝an}b∇acket∇i}ht=/summationtext
+kkfk(1−fk)//summationtext
+kfk(1−fk)withfktheFermidistribution. CalculationbyFishmanandLampel[ 436]in
+bulk GaAs shows that the spin lifetime is on the order of 103∼104ns in the metallic regime, whereas the measured
+spin lifetime is less than 300 ns [21, 39]. Therefore the hype rfine interaction mechanism is irrelevant to the spin
+relaxationin themetallicregimein bulkGaAs.
+Hole spin relaxation due to the hyperfineinteraction attrac ted a lot of interest recently [116, 301, 437–440]. For
+a long time, it was believed that holes interact weakly with n uclear spin and the hyperfine interaction is negligible
+to hole spin relaxation [3]. However,recent calculationsi ndicated that the hole-nucleushyperfineinteractionis onl y
+one order of magnitude smaller than the electron one [301], w hich still provides considerable e ffects on hole spin
+relaxation [301]. In principle, the above discussions on el ectron spin relaxation due to the hyperfine interaction are
+alsoapplicabletoholesystem. Di fferently,theheavy-holehyperfineinteractioninquantumwe llsisoftheIsingform
+Hh
+hf=/summationtext
+i,νAh
+νv0|ψ(Ri)|2SzIz
+i,ν(zaxis is along the growth direction). Consequently, the hole spin relaxation due to
+the hyperfine interaction is di fferent from the electron one: it is anisotropic [301]. Furthe r studies on the hole spin
+relaxation due to the hyperfine interaction, especially in q uantum dots where hole spin is a promising candidate for
+qubits[441,442],arerequired.
+3.2.6. Anisotropicexchangeinteraction
+The anisotropic exchange interaction is an e fficient spin relaxation mechanism in the insulating phase of d oped
+semiconductors [39, 40, 322–324, 324, 325, 443, 444]. The in teraction comes from a correction to the Heisenberg
+exchange interaction between carriers bound to adjacent do pants due to the spin-orbit coupling. It can be written as
+25[40, 322],
+Hanex≈−Jij/bracketleftig
+sin(γij)/vector γij
+γij·(Si×Sj)+(1−cosγij)/parenleftbig/vector γij
+γij·Si/parenrightbig/parenleftbig/vector γij
+γij·Sj/parenrightbig/bracketrightig
+, (79)
+in which Jijis the isotropic exchange constant. /vector γijis a vector due to spin-orbit coupling. It is estimated as /vector γij≈
+meΩSO(√2meEBrij
+rij)rij/√2meEB, whereΩSO(k) is the spin precession frequency, EB>0 is the electron binding
+energyand rijis the vector connectingthe positions of site iandj. The first term is the Dzhyaloshinskii-Moriyain-
+teractionandthesecondoneisthescalarexchangeinteract ion. Thescalarexchangeinteractiondoesnotconservethe
+spinperpendicularto /vector γij,whereastheDzhyaloshinskii-Moriyainteractiondoesnot conservespinalonganydirection.
+Consequently the anisotropic exchange interaction leads t o decay of total carrier spin polarization [322]. As γijis
+usuallyvery small in insulatingphase, sin( γij)≈γijand 1−cos(γij)≈1
+2γ2
+ij. One shouldnote that γijis proportional
+tothestrengthofthe spin-orbitcoupling.
+Ininsulatingphase,electronspinprecessesaroundtheani sotropicexchangefieldproducedbyarandomlyoriented
+adjacent electronspin, which results in a random-walk-lik espin precession. The correlationtime of spin precession,
+is limited by hopping time as well as the isotropic exchange i nteraction with adjacent electrons. The latter perturbs
+the spin precession of individual electron but conserves th e total spin polarization. Studies indicate that the latter is
+muchmoreefficientthantheformerinGaAs[40]. Henceanestimationofthe correlationtimeis τc≈1/∝an}b∇acketle{tJij∝an}b∇acket∇i}ht,andthe
+spinprecessionfrequencyis Ω≈∝an}b∇acketle{tJijγij∝an}b∇acket∇i}ht. Thereforespinrelaxationtime reads[40]
+τ−1
+s≈2
+3∝an}b∇acketle{tJij∝an}b∇acket∇i}ht∝an}b∇acketle{tγij∝an}b∇acket∇i}ht2. (80)
+Heretheprefactor2
+3isduetothefactsthatonlytransversefluctuationcanleadt ospinrelaxationandthattheangular
+distribution of the random exchange field is uniform (as the a djacent electron spins orient randomly). It should be
+noted that the spin relaxation rate, which ∝ ∝an}b∇acketle{tγij∝an}b∇acket∇i}ht2, is proportional to the square of the strength of the spin-or bit
+coupling.
+The anisotropic exchange interaction mechanism has also be en reviewed by Kavokin in Ref. [40], where many
+aspectsofspinrelaxationoflocalizedelectronsweredisc ussed. However,wenotethattherewasnotheoreticalstudy
+ontheanisotropicexchangeinteractionforholesininsula tingphaseuptillnow. Asthespin-orbitcouplingisstronge r,
+one would expectthat the anisotropicexchangeinteraction is more efficient in hole system. Recently there are a few
+experimentalstudies [400, 445, 446] on hole spin relaxatio nin the insulating phase in two-dimensionalhole system.
+Theoreticalinvestigationontheroleoftheanisotropicex changeinteractioninholespinrelaxationunderexperimen tal
+conditionsisdesired.
+3.2.7. Exchangeinteractionwith magneticimpurities
+In magnetic semiconductors, the exchange interaction with magnetic impurities can be an important source of
+the spin relaxation. In (II,Mn)VI magnetic semiconductors , thes-dexchange interaction has been recognized to be
+responsibleforelectronspinrelaxation[447–450]. Recen tly,itwasfoundthatin p-typeparamagneticGaMnAsquan-
+tum wells, the s-dexchange interaction dominateselectron spin relaxation a t low temperature [111]. In general, the
+spin relaxationdueto the s-d(p-d) exchangeinteractionis similar to that dueto the hyperfine interaction. There can
+be several regimes, where di fferent scenarios take place. However, there are only two regi mes which are commonly
+encountered: the insulatingregimewheremost carriersare localizedand the metallic regimewheremost carriersare
+extended. Intheinsulatingregime,the exchangeinteracti onwithrandomlyorientedmagneticimpurityspinsinduces
+randomspinprecessions. Thereareseveralprocessesinter ruptsuchrandomspinprecessions: (i)exchangeinteractio n
+between carriers; (ii) carrier hopping; (iii) fluctuation o r diffusion of the magnetic impurity spins. These processes
+limit the correlationtime τcof the random spin precession. If τcis small, so that∝an}b∇acketle{tΩs(p)-d∝an}b∇acket∇i}htτc≪1 (Ωs(p)-dis the spin
+precessionfrequencyduetothe s(p)-dexchangeinteractionand ∝an}b∇acketle{t...∝an}b∇acket∇i}htrepresentstheensembleaverage). Inthisregime,
+thespinrelaxationratereads
+τ−1
+s=2
+3∝an}b∇acketle{tΩ2
+s(p)-d∝an}b∇acket∇i}htτc. (81)
+In metallic regime, spin relaxationrate is given by the Ferm i Golden rule. For example, in bulk system, the electron
+spinrelaxationrateis[451, 452]
+τ−1
+s=me∝an}b∇acketle{tk∝an}b∇acket∇i}ht
+3πJ2
+sdNMSM(SM+1), (82)
+26whereJsdisthes-dexchangeconstant, NMisthedensityofmagneticionswithspin SM,∝an}b∇acketle{tk∝an}b∇acket∇i}ht=/summationtext
+kkfk(1−fk)//summationtext
+kfk(1−
+fk) withfkbeing the Fermi distribution. Hole spin relaxation rate due to thep-dexchange interaction can also be
+calculatedviatheFermiGoldenrule,onlythatthespectrum andtheeigen-spinoraremorecomplicated. Inprinciple,
+thesameschemeisapplicabletoelectrons(holes)innanost ructures[448].
+3.2.8. Otherspin relaxationmechanisms
+The main remaining spin relaxation mechanisms are various s pin-phonon interactions (except those which have
+beengroupedintotheElliott-Yafetmechanism). Generally theoriginofthespin-phononinteractionsisthatwhenspin
+interactionsdependonthepositionsofatomicions(bothho standdopant),latticevibrationcouplesdirectlywithspi n.
+There are several such kinds of spin interactions, such as th e strain-induced spin-orbit coupling and the hyperfine
+interaction. The strain also modifies g-tensor [453]. All such lattice dependent spin interaction s can be the origin
+of the spin-phonon interactions [453–458]. In nanostructu re, the gate-voltage and structure induced modification of
+spin-orbit coupling [113] and g-tensor [81, 113] further induce more spin-phonon interact ions [344, 457]. Besides,
+spin mixing associated with spin-independent phonon scatt ering also leads to the spin-flip phonon scattering. For
+example, for localized electrons which are boundto impurit ies or confined in quantumdots, spin-orbit coupling and
+hyperfineinteractioncaninducespin mixingandenablesspi n-flipelectron-phononscattering[344, 459].
+In existing literature, there are a lot of studies on the spin -phonon interaction induced spin relaxation. Most of
+themfocusonspinrelaxationofdonor-boundelectrons[324 ,453,460–463]andelectrons /holesinquantumdots[343,
+344, 454–460, 464–492]. A comprehensive study of electron s pin relaxation/dephasingdue to various spin-phonon
+interactionsin GaAs quantumdotsis presentedin Ref. [344] . The spin-phononscatteringsare usually limited by the
+electron-phononscattering rate and the e ffect of the spin-orbit interaction. It was believed that they are unimportant
+inmetallicregime. Therelativeimportanceofdi fferentspin-phononinteractionsarecomparedforvariousco nditions
+in GaAs quantum dots [344] and in Si /Ge quantum dots [486]. Among those spin-phonon interaction s, the phonon
+inducedg-tensorfluctuationwasfoundtobe irrelevant inGaAsquantumdots[344],whereasitis important inSi/Ge
+quantumdots [486]. Finally, the gate noise can also lead to s pin relaxation in gated quantumdots in the presence of
+thehyperfineinteraction,spin-orbitinteractionorelect ron-electronexchangeinteraction[493,494].
+3.2.9. Relativeefficiencyof spinrelaxationmechanismsin bulksemiconductor s
+In this subsection, we briefly discuss and review the relativ e efficiency of various spin relaxation mechanisms.
+We give some rough discussions based on the analytical formu lae which are based on some approximations.30For
+simplicity, we restrict the discussion here on the bulk III- V semiconductors. The situation in III-V semiconductor
+nanostructuresor II-VIsemiconductorsand their nanostru cturesmay be different. Nevertheless, some insight can be
+gainedfromthesediscussions. Wealsocommentonholespinr elaxationandcarrierspinrelaxationincentrosymmet-
+ricsemiconductorssuchassiliconandgermanium.
+For spin relaxation in bulk III-V semiconductors in the insu lating regime, as stated before, the anisotropic ex-
+change interaction dominates at high doping density, where as the hyperfine interaction dominates at low doping
+density. In metallic regime, the spin relaxation mechanism s are the D’yakonov-Perel’,the Elliott-Yafet and the Bir-
+Aronov-Pikusmechanisms. In n-dopedsemiconductorstheBir-Aronov-Pikusmechanismisi neffectiveduetolackof
+holes. Hence the relevant mechanisms are the Elliott-Yafet and the D’yakonov-Perel’mechanisms. From Eqs. (53)
+and (62), the ratio of the spin lifetimes due to the Elliott-Y afet and the D’yakonov-Perel’mechanisms in bulk III-V
+semiconductorsis givenby[110, 117],
+τEY
+τDP≈A1(me
+mcv)2∝an}b∇acketle{tεk∝an}b∇acket∇i}htτ2
+pEg(1−η/3)
+(1−η/2)2, (83)
+where the prefactor A1∼1. It is inferred from the above expression that the Elliott- Yafet spin relaxation is more
+importantinsemiconductorswithsmaller Eg,suchasInAsandInSb. Thefactor ∝an}b∇acketle{tεk∝an}b∇acket∇i}htτ2
+pindicatesthattheElliott-Yafet
+mechanismismoreimportantatlowertemperaturewith highe rimpuritydensity.
+30Besides competition, there are also some cooperation betwe en different spin relaxation mechanisms. For example, spin-flip mo mentum
+scatterings due to the Elliott-Yafet, the Bir-Aronov-Piku s, thes(p)-dexchange interaction and other mechanisms also lead to the r andomization of
+momentum, they thus contribute to the D’yakonov-Perel’ spi n relaxation as well. However, as these spin-flip momentum sc atterings are usually
+much weaker than the spin-conserving ones, they only lead to asmall modification of the D’yakonov-Perel’ spin relaxatio n [109,111].
+27Inp-typeandintrinsicsemiconductorstherelativee fficienciesoftheBir-Aronov-PikusandtheD’yakonov-Perel’
+mechanismsarealwayscompared. TheBir-Aronov-Pikusmech anismisbelievedtobeimportantforhighholedensity
+at low temperature [21, 117]. In such case the hole system is d egenerate, thus it is meaningful to compare the
+D’yakonov-Perel’andthe Bir-Aronov-Pikusmechanismsfor degenerateholes. FromEqs.(62)and(69)onefinds
+τBAP
+τDP≈A2τpτ0∝an}b∇acketle{tεk∝an}b∇acket∇i}ht5/2E1/2
+B
+Eg(me
+mcv)2η2
+1−η/31
+nha3
+BEh
+F
+kBT, (84)
+where the prefactor A2∼1 andEh
+Fis hole Fermi energy. It is seen from the above formula that th e Bir-Aronov-
+Pikus mechanism is more important in semiconductors with la rger band gap Eg, smaller spin-orbit interaction (i.e.,
+smallerη) and strongerelectron-holeexchangeinteraction(i.e., s mallerτ0). Fromthe aboveequation,onefinds four
+controllingfactors: τp,∝an}b∇acketle{tεk∝an}b∇acket∇i}ht,nhandT(Note that Eh
+F∼n2/3
+h). The Bir-Aronov-Pikusmechanismis more importantat
+higherhole/impuritydensityand lowerelectron density. For nondegene rateelectron, as∝an}b∇acketle{tεk∝an}b∇acket∇i}ht∼kBT, the Bir-Aronov-
+Pikus mechanism is more important at low temperature. Howev er, for degenerate electron, the Bir-Aronov-Pikus
+mechanismismoreimportantathightemperature,as ∝an}b∇acketle{tεk∝an}b∇acket∇i}ht∼EF. Suchqualitativedi fferentbehaviorfornondegenerate
+anddegenerateelectronwas illustratedin arecentstudy[1 10](see alsoSec.5.6).
+The Elliott-Yafet mechanism can be comparable to the Bir-Ar onov-Pikusmechanism in p-type semiconductors.
+Theratioofthe twospinlifetimesfordegenerateholeis
+τEY
+τBAP≈A3τp
+τ0nha3
+BE2
+g
+E1/2
+B∝an}b∇acketle{tεk∝an}b∇acket∇i}ht3/21
+η2(1−η/3
+1−η/2)2kBT
+Eh
+F, (85)
+whereA3∼1. According to the above equation, the Elliott-Yafet mecha nism is more important in semiconduc-
+tors with larger τ0(i.e., weaker electron-hole exchange interaction), small er band-gap Egand largerη(i.e., larger
+spin-orbit interaction). Hence the Elliott-Yafet mechani sm may exceed the Bir-Aronov-Pikus mechanism in semi-
+conductors with small band-gap and large spin-orbit intera ction, such as InSb and InAs. For a given material, the
+Elliott-Yafetmechanismis moreimportantat strongermome ntumscattering(i.e.,smaller τp)andlowerholedensity.
+For degenerateelectrons, the Elliott-Yafetmechanism is m ore importantat largerelectron density and lowertemper-
+ature, as∝an}b∇acketle{tεk∝an}b∇acket∇i}ht∼EF. However, for nondegenerateelectrons, the Elliott-Yafet mechanism is more important at higher
+temperature,as∝an}b∇acketle{tεk∝an}b∇acket∇i}ht∼kBT.
+Inbrief,someunderstandingofthetopiccanbeobtainedfro mtheabovediscussions. Nevertheless,astherearea
+lotofapproximationsintheseformulae,theycan notgiveapictureofwholeparameterrange. Insomecasethemany -
+body effects which are absent in the above analysis plays an importan t role. Moreover, the relative e fficiency relies
+stronglyon the genuinematerial parameters. Only a realist ic calculationcan fullyresolvethe problem. A systematic
+calculationfromthefullymicroscopickineticspinBloche quationapproachbyJiangandWu[110]indicatedthatthe
+Elliott-Yafet mechanism is lessimportant than the D’yakonov-Perel’ mechanism in allparameter regime in n-type
+bulk III-V semiconductorsin metallic regime. In the same wo rk, the Bir-Aronov-Pikusmechanism was found to be
+unimportantinintrinsicsamples,whereasit dominatesspi nrelaxationin p-typesamplesatlowtemperatureandhigh
+hole density when the electron density is low. However, the B ir-Aronov-Pikus mechanism is ine ffective when the
+electrondensityishigh[110]. Itwasalsofoundviathesame approachthatinintrinsicor p-typeGaAsquantumwells
+when electron density is comparablewith hole density, the B ir-Aronov-Pikusmechanism is unimportant[109]. In a
+recentwork[112],therelativeimportanceoftheBir-Arono v-PikusmechanismandtheD’yakonov-Perel’mechanism
+inp-type GaAs quantum wells was compared comprehensively. Sys tematic comparisons of various spin relaxation
+mechanisms in paramagnetic GaMnAs quantum wells was perfor med in Ref. [111] also via the kinetic spin Bloch
+equationapproach.
+Finally, in centrosymmetric semiconductors, such as silic on and germanium, the situation is totally di fferent. In
+bulksystem,asthespin-orbitcouplingterminconductionb andvanishes,theD’yakonov-Perel’mechanismisabsent
+in electron spin relaxation. Hence in n-type bulk silicon and germanium, the Elliott-Yafet mechan ism dominates
+electron spin relaxation in metallic regime [21, 22, 362]. I n insulating regime, electron spin relaxation is dominated
+by other mechanisms such as the hyperfine interaction and the spin-phonon interaction [289, 453, 461, 463]. In
+nanostructures, which breaks the centro-inversion symmet ry, spin-orbit coupling term shows up [291, 294, 295].
+Experiment indicated that in silicon quantum wells the D’ya konov-Perel’ mechanism is dominant in high mobility
+28regime, whereas the Elliott-Yafet mechanism is more import ant in low mobility regime [297]. Due to the nature of
+indirect band, optical spin orientation is not easily acces ible. Hence the electron spin dynamics was studied only
+inn-type system, whereas the hole spin dynamics in p-type system. In p-type system, like the situation in III-V
+semiconductors,hole spin-orbitcouplingin the Luttinger Hamiltonianis largeandthe D’yakonov-Perel’mechanism
+is believed to be dominant. The D’yakonov-Perel’mechanism also dominates hole spin relaxation in nanostructures
+ofsiliconandgermanium[299].
+4. Spin relaxation: experimentalstudiesand single-parti cletheories
+The study of spin lifetime and spin di ffusion length is one of the central focuses in spintronics. A l arge quantity
+of paperscan be found in the past decades. Recent developmen tof spin grating measurement givesnew insight into
+spin diffusion [13, 33, 495], while spin noise spectroscopyenables m easurement of the intrinsic carrier spin lifetime
+with negligible disturbance on the carrier system [496–499 ]. Also the latest technique of tomographic Kerr rotation
+achievesspinstatetomography[500]. Weexpectthatthesen ewexperimentaltechniqueswillinspireandenablemore
+importantfindings.
+Inthissectionwereviewexperimentalstudiesandsingle-p articletheoriesonspinrelaxationanddephasingtimes.
+Here, the term “single-particle theory” refers to the theor y where the carrier-carrier scattering is not considered. I t
+will be shown in next section that the carrier-carrier scatt ering plays an important role in spin relaxation. We focus
+onspinrelaxationinIII-VandII-VIsemiconductorswheret hebulkinversionasymmetryinducedspin-orbitcoupling
+plays an important role. Spin relaxation in centrosymmetri c semiconductors,such as silicon and germanium, is also
+reviewed. Thediscussionsinthe previoussectionwill beus edfrequently.
+4.1. Carrierspinrelaxationat lowtemperatureininsulati ngregime
+Carriersin semiconductorsare mostlyionizedfromdopants . At low temperature,dopantscantrap thosecarriers.
+Whendopingdensityislow,carriersareboundtoisolateddo pantsandthesystembehavesasaninsulator. Atelevated
+doping density, carrier system becomes metallic after the m etal-insulator transition at some critical density nc. In
+metallic regime carrier transport is band-like, whereas in insulating regime it is dominated by carrier hopping. It is
+then not difficult to understandthat the relevantcarrier spin relaxatio n mechanismsin insulating regime are di fferent
+from those in the metallic regime. In this subsection, we rev iew spin relaxation at low temperature in insulating
+regime. Remarkably,thelongestspinlifetimesinGaAswasr eportedasτs≃300ns[501,502]atzeromagneticfield.
+At high magnetic field in lightly dopedGaAs, the spin lifetim e can reach τs≃19µs [503]. In high purity GaAs, the
+spinrelaxationtimeofdonor-boundelectronscanbeaslong asseveralmilliseconds[504].31
+It has been identified that spin relaxation in insulating reg ime is dominated by anisotropic exchange interaction
+mechanism at high doping density whereas by the hyperfine int eraction mechanism at low doping density, at zero
+magneticfield[39,40, 323]. If/summationtext
+f/radicalig
+∝an}b∇acketle{tω2
+f∝an}b∇acket∇i}htτc≪1,spinrelaxationrateisgivenby[3, 435]
+τ−1
+s≃/summationdisplay
+f2
+3∝an}b∇acketle{tω2
+f∝an}b∇acket∇i}htτc, (86)
+whereωfis the electron spin precession frequencydue to the random e ffective magnetic field. The index fdenotes
+the mechanism leading to the random fields: it can be the hyper fine interaction with the nuclei and the anisotropic
+exchange interaction with adjacent electrons. ∝an}b∇acketle{t...∝an}b∇acket∇i}htrepresents ensemble average. τcis the correlation time of the
+random spin precessions. τccan be limited by disturbance of electron spin state or the ra ndom field ωf. Realistic
+calculation indicated that in GaAs τcis mainly limited by the former due to the isotropic exchange interaction [39,
+40].32Thereforeτc≃1/∝an}b∇acketle{tJij∝an}b∇acket∇i}htwhereJijis the isotropic exchange coupling constant between iandjdonor-bound
+31The ensemble spin dephasing time ( T∗
+2≃5 ns [505]) and irreversible spin dephasing time ( T2=7µs [506]) are much shorter under the same
+condition.
+32In fact such disturbance of electron spin state can be viewed as spin diffusion in anetwork coupled by the isotropic exchange interac tion [40].
+Fromthe random-walk theory, τcis related to the spin di ffusion constant Dsasτc≃(3n2/3
+DDs)−1[40].
+29electrons. The average of Jijis taken over the donor configuration. Spin relaxation rate d ue to the anisotropic
+exchangeinteractionis thengivenby[seealso Eq.(80)]
+τ−1
+s≈2
+3∝an}b∇acketle{tJij∝an}b∇acket∇i}ht∝an}b∇acketle{tγij∝an}b∇acket∇i}ht2, (87)
+whereγijis the spin precession angle duringthe tunnel hopingfrom itojdonor site (see Sec. 3.2.6). The hyperfine
+interaction-limitedspin lifetimeis
+τ−1
+s≃2
+3σ2
+h1
+∝an}b∇acketle{tJij∝an}b∇acket∇i}ht, (88)
+whereσh[givenbyEq.(75)]isthevarianceoftheOverhauserfield. It isnotedthatthespinrelaxationratesduetothe
+twomechanismshavedi fferentdependenceson ∝an}b∇acketle{tJij∝an}b∇acket∇i}ht. Naively,itisexpectedthat ∝an}b∇acketle{tJij∝an}b∇acket∇i}htincreaseswithdecreasinginter-
+donordistance(i.e.,increasingdopingdensity). Itisthe neasytounderstandthattheanisotropicexchangeinteract ion
+dominatesspin relaxation at high doping density, whereas t he hyperfineinteraction dominatesat low dopingdensity
+[39, 40,323]. Forhydrogen-likecenters,theexchangecoup lingconstantisgivenby[325]
+Jij≃0.82e2
+4πǫ0κ0a(rij
+a)5/2exp(−2rij
+a), (89)
+whereaistheBohrradiusofthehydrogen-likewavefunction. Itisn otedthatthedependenceoninter-donordistance
+rijisexponential.33
+Usually both localized and free (extended) electrons coexi st and they are also coupled by exchange interactions.
+It was found that the exchange coupling between the localize d and free electrons is so e fficient that the two share a
+commonspinlifetime[40,324,508]. Thespinrelaxationrat eofthewholeelectronsystemisthengivenby[324],
+τ−1
+s=τ−1
+lsnl/(nl+nf)+τ−1
+fsnf/(nl+nf), (90)
+wherenlandnfare the densities of localized and free electrons, respecti vely.τlsandτfsare the spin lifetimes
+of localized and free electrons, separately. In many cases s pin relaxation is dominatedby localized electronsas they
+havemuchfasterspinrelaxationrate[40,324]. Besides,th eexchangeinteractionbetweenfreeandlocalizedelectron s
+can also limit τcand elongate the spin lifetime [501, 502]. The e ffect of electron localization on spin relaxation in
+n-doped quantum wells, especially in symmetric (110) quantu m wells, was discussed in a recent theoretical work
+[509].
+In the presence of magnetic field, the situation becomes much more complicated [40, 510]. First, magnetic field
+reduces the isotropic exchange coupling (as the wavefuncti on becomes more localized) and hence increases τc[40].
+The anisotropic exchange interaction is also reduced. Seco nd, spin relaxation due to the random spin precession is
+slowed down by the magnetic field parallel to spin polarizati on [see Eq. (40)]. Third, spin relaxation due to g-tensor
+inhomogeneityincreaseswith the magneticfield. Fourth, at high magneticfield, spin-flip electron-phononscattering
+may come into play [383]. All these factors complicate the ma gnetic field dependence of the spin relaxation. Up
+till now, there is little discussion on the problem [40, 510] . Experimental results on magnetic field dependence of
+spin lifetime are available in Ref. [511]. Yet the observed m agnetic field dependence is to be explained by theory
+[511]. Before turning to experimental studies, it should be mentioned that until now there are only a few theoretical
+workson spin relaxationin insulatingregime[39, 40, 322–3 24, 324, 325, 443, 507, 510, 512, 513], morestudies are
+needed. Especiallyholespinrelaxationininsulatingregi mewasdiscussedonlyin Ref.[399]whereonlythe g-factor
+fluctuationmechanismwasstudied.
+We nowreviewthe experimentalstudiesoncarrierspin relax ationat lowtemperatureininsulatingregime. Many
+efforts have been devoted to the spin relaxation in n-GaAs at low temperature [4, 39, 144, 321, 497, 501–503, 511,
+33As stated in the previous footnote, τcis actually limited by spin di ffusion due to exchange coupling in the network of randomly dis tributed
+donors. In someclusters where the inter-donor distance is s horter than the average one, spin di ffusion becomes much faster due to the exponential
+dependence. Therefore this kind of cluster serves as easy pa ssages of spin diffusion. At low doping density spin di ffusion is dominated by such
+kind of small cluster. In this case the spin di ffusion should be treated with percolation theory [40]. Such k ind of studies were performed in
+Refs. [40,507].
+30514–521]. InoneoftheseminalworkbyKikkawaandAwschalom [4],asurprisinglylongspinlifetime(130ns)was
+observed in n-GaAs, which sheds light on the possibility of the semicondu ctor spintronics. This, together with the
+observation of a very long spin di ffusion length (100 µm) in GaAs [5] as well as the progress on spin injection into
+semiconductors [522, 523], excited the society and led to th e rapid rising of the field of semiconductor spintronics.
+Interestingly,most oftheseworksweredoneinbulk n-GaAswitha dopingdensityof1016cm−3.
+Dzhioevetal. [39]measured,inawiderange(1014∼1019cm−3),thedopingdensitydependenceofspinlifetime
+at low temperature T≤5 K.34The results are presented in Fig. 1. Spin lifetime first incre ases before reaching its
+first maximum at doping density nD≃3×1015cm−3; it then decreases with doping density; around the metal-
+insulatortransitionpointatdopingdensity nD=2×1016cm−3,spin lifetimeincreasesrapidlyandreachesitssecond
+maximum;afterthat(i.e.,inmetallicregime)spinlifetim edecreasesmonotonicallywithdopingdensity. Thisintric ate
+behaviorisduetothedi fferentspinrelaxationmechanismsinmetallicandinsulatin gregimes. Inmetallicregime,spin
+relaxationis dominatedby the D’yakonov-Perel’mechanism whereτ−1
+s∼∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}htτp. At such low temperature,electron
+system is degenerate. Hence τp∼nD/k3
+F∼n0
+D(according to Brooks-Herringformula) and ∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}ht∼n2
+D. Therefore τs
+decreases with doping density nDasτs∼n−2
+D. In insulating regime, the D’yakonov-Perel’mechanism is i rrelevant.
+The relevant spin relaxation mechanisms are the anisotropi c exchange interaction (at high doping density) and the
+hyperfine interaction (at low doping density). As indicated by Eqs. (87) and (88), spin relaxation due to these two
+mechanismshavedi fferentdopingdensitydependences(as ∝an}b∇acketle{tJij∝an}b∇acket∇i}htincreaseswithincreasingdopingdensity). Therefore,
+their competition leads to the nonmonotonicdoping density dependence: spin lifetime first increases then decreases
+withincreasingdopingdensity. Itisnotedinthefigurethat themeasured τcdecreaseswithincreasingdopingdensity,
+whichagreeswiththe theoryas τc≃1/∝an}b∇acketle{tJij∝an}b∇acket∇i}ht.
+Figure 1: Spin lifetime τs(/square,∝ci∇cleco√†∇t,♦and/medbullet) and spin correlation time τc(△) as function of donor concentration nDinn-GaAs. Solid curves:
+theoretical calculation of τs. Dotted curve: theoretical calculation of τcdue to exchange interaction between adjacent electrons. Th e vertical
+dashed lines indicate the positions of the peaks of the spin l ifetime. Especially, the dashed line at nD=2×1016cm−3also denotes the metal-
+insulator transition. The hyperfine interaction, the aniso tropic exchange interaction and the D’yakonov-Perel’ mech anisms dominate the spin
+relaxation in the low, medium, high doping density regimes r espectively, as indicated in the figure. From Dzhioev etal. [ 39].
+It was demonstratedby Dzhioevet al. that di fferentspin relaxation scenarioscan be invokedby optical pu mping
+[501, 502]. At low concentration spin relaxation is limited by the hyperfine interaction mechanism, where the spin
+lifetime isτs=5 ns. The authors showed that by injecting additional electr ons, the exchange interaction between
+thoseelectronsandthedonor-boundelectronsmotionallyn arrowstherandomspinprecessioncausedbythehyperfine
+interaction. Thespinlifetimeisthenelongatedto τs=300ns.35
+34Similar investigation in CdTe was performed in arecent work [524].
+35Asthephoto-excited electrons maya ffectthespinrelaxation significantly atlowtemperature ini nsulating regime[40,501,502],itisimportant
+31Recently, Schreiber et al. studied energy resolved electro n spin relaxation in n-GaAs near the metal-insulator
+transition [519]. They found that there are three component sin the time-resolved Kerr rotation under magnetic field
+parallel to the surface by examining the spin precession fre quency as well as spin lifetime. The di fferent precession
+frequenciesindicatethat thosecomponentsshouldbe attri butedto electronsat di fferentenergystates. Thesequential
+emergenceofthethreecomponentsconfirmssuchhypothesis. Theauthorsattributedthesethreestatesas: delocalized
+states in donor band, localized states in the tail of donor ba nd and free electron states in the conduction band. The
+authorsalsofoundthatabsolutevalueofthe g-factorinlocalizedstatesinthedonorbandtailsissmalle rthanthosein
+theothertwo states.
+The magnetic field and temperaturedependencesof spin relax ationtime in lightlydoped n-GaAs were measured
+systematically byColton et al. in Refs. [503, 511]. The main results are shown in Fig. 2. Surprisinglya complicated
+nonmonotonic magnetic field dependence was observed: spin l ifetime first increases ( B<2-2.5 T) then decreases
+(2-2.5<B<3-4 T) and further increases ( B>3-4 T) with increasing magnetic field [511]. Moreover,spin l ifetime
+decreases rapidly with increasing temperature. For 1 <T<20 K and 0 .5≤B≤2.5 T, it exhibits a power law
+τs∼T−1.57[511]. The underlying physics is still unclear [40, 511]. Re cently, the temperature dependence of spin
+lifetime in quantum wells was studied carefully in Refs. [44 5, 525, 526]. In these works, differently, it was found
+that the temperature dependence of electron (hole) spin lif etime is quite weak for T<5 K (T<2 K). At higher
+temperature, the spin lifetime decreases dramatically wit h increasing temperature. It was proposed that the excited
+localizedstatesfromthermalactivation,whichhavemuchs horterspinlifetimesthanthegroundstates,isresponsibl e
+fortheobservedphenomena[526]. Thetemperaturedependen ceofspinlifetimeininsulatingregimewasalsostudied
+theoreticallyin Refs.[40, 507].
+Figure 2: Left: Spin lifetime T1as function of magnetic field at various temperatures (a) T=1.5 and 5 K; (b) T=1.5, 5, 6.5, 8 and 12 K. Right:
+Spin lifetime T1as function of temperature for various magnetic field. All te mperature dependences can be fitted to T1∼T−1.57, as indicated by
+thethin solid lines. Doping density is nD=1015cm−3. FromColton etal. [511].
+The electric field dependence of the spin lifetime in insulat ing regime also reflects characteristics of electron
+localization. Furis et al. measured such dependence[527]. Their results showed that the spin lifetime varies slowly
+with electric field in the linear transportregime. However, after the thresholdfield ofimpact ionization,spin lifetim e
+drops rapidly with electric field. At such threshold, the fre e electrons get enough energy from the electric field to
+to reexamine the spin relaxation times obtained by Hanle, or Faraday/Kerr measurements via the spin noise spectroscopy method. S uch study has
+been done by R¨ omer et al. in Ref. [520], where the spin relaxa tion times measured by di fferent methods are compared in very dilute doped
+(nD=1014cm−3), low doped ( nD=2.7×1015cm−3),and doping close to metal-insulator transition ( nD=1.8×1016cm−3)samples.
+32ionizethelocalizedelectrons[527]. Afterthelocalizede lectronsbecomefreeandtheelectronsystemacquiresenoug h
+energytobecomea hotelectronsystem,the D’yakonov-Perel ’spinrelaxationisaccelerated. Thespinlifetimehence
+dropsrapidly.
+Density dependenceof spin lifetime in two-dimensional ele ctron system was studied by Sandhu et al. [528] and
+Chen et al. [398] (see Fig. 3), where metal-insulator transi tion is involved. In the work of Sandhu et al., metal-
+insulatortransition was demarcatedby conductivity[528] . Differently Chen et al. revealedmetal-insulatortransition
+by determining the density of states at band edge via g-factor measurement [398]. In GaAs g-factor has an energy
+dependence g(ε)=gc+βε. Thereforeatzero(low)temperaturethedensityofstates D(ε)iswrittenas
+D(ε)=β(dg∗
+dn)−1, (91)
+whereg∗isthemeasured(ensembleaveraged) g-factorand niselectrondensity. Bymeasuringthedensitydependence
+ofg∗, Chen et al. obtained the density of states near band edge (se e Fig. 3) where the tail reveals the existence of
+localized states. Electron localization also manifests it self in the magnetic field dependence of spin lifetime: spin
+dephasingratein insulatingregimewasfoundto follow
+1
+T∗
+2(B)=1
+T∗
+2(0)+∆gµBB√
+2, (92)
+due to the g-factor inhomogeneity and localization.36Similar dependence was also found in other two-dimensional
+structures at low temperature [529–531], two-dimensional hole system [400, 531] and in ZnSe epilayers close to
+GaAs/ZnSe interface [532]. In localized regime, the electron (ho le) spin lifetime in two-dimensional system is very
+long. Usually electron spin lifetime is of the order of nanos econds to microseconds [445, 531, 533, 534], whereas
+hole spin lifetime can reach nanoseconds [445, 446]. As temp erature increases, spin lifetime decreases due to de-
+localization and enhancement of the D’yakonov-Perel’ spin relaxation which increases with electron kinetic energy
+[400, 445, 525, 535, 536]. Pump density dependence of spin li fetime was studied in Refs. [534, 535] where spin
+lifetime was found to decrease with pump density at T<2 K, similar to that in bulk materials in insulating regime
+[4, 39, 518]. The underlyingphysicsis similar to the temper ature dependence: the increasing pump density leads to
+theincreaseofelectrondensityaswell asthethermalizati onoftheelectronsystem,andenhancesspin relaxation.
+Spin relaxationin type-II quantumwells was studied in both GaAs and ZnSe quantumwells at very low temper-
+ature [537, 538]. As electrons and holes are spatially separ ated, the electron-hole exchange interaction is weakened
+andspinrelaxationtime iselongated. Especially,inGaAst ype-IIquantumwell, electronspinlifetimecanbeaslong
+as7µsat 1.7K [537].
+Recently, Korn et al. observed extremely long hole spin life time,≃70 ns, in two-dimensional hole system in
+(001)quantum wells at 0 .4 K [446]. They also observedthe decrease of hole spin lifeti me with increasing magnetic
+field roughlyfollowing Eq. (92). The low-field ( ∼0.2 T) spin lifetime was observed to decrease by about one order
+ofmagnitudewhentemperatureincreasesfrom0.4K to4.5K.I nterestingly,spin relaxationoflocalizedholesdueto
+the hyperfineinteraction can be inhibited by a verysmall in- plane magneticfield ( ∼10 mT) due to the Ising form of
+the hole hyperfineinteraction [301]. Also, the g-tensor inhomogeneitymechanism is not dominantat small (b ut still
+much larger than 10 mT) magnetic field [446]. Hence the most li kely spin relaxation mechanism at such magnetic
+fieldistheanisotropicexchangeinteractionmechanism.
+Finally, it should be mentioned that dynamic nuclear polari zation is more efficient in insulating regime than in
+metallicregimebecausethehyperfineinteractionplaysamo reimportantroleinelectronspinrelaxationin insulating
+regime. The dynamic nuclear spin polarization, in turn, a ffects the electron spin dynamics. Experimental studies
+involvecontinuous[3, 302] and time-resolved[311, 539–54 1] measurementsof electron spin dynamicsto reveal the
+nuclearspindynamics.
+4.2. Carrierspinrelaxationin metallicregime
+At high temperature and /or high density, carrier system is in metallic regime where m ost of the carriers are in
+extended states in the conduction /valence band with kbeing a good quantum number. The relevant spin relaxation
+36Inmetallic regimespinrelaxation rateeither decreases wi th increasing magnetic fieldorvaries as[1 /T∗
+2(B)−1/T∗
+2(0)]∼B2duetothe g-factor
+inhomogeneity.
+33Figure3: Up: Spinrelaxation timeasfunctionofHeNelasere xcitation intensity(HeNeintensity). Insetindicates mag neticfielddependenceofspin
+relaxation ratefordi fferentHeNelaser intensities [solid linesdenotethefitting saccording toEq.(92)]. Down: The(a)measured and(b)calcu lated
+g-factor as function of HeNe laser intensity (electron densi ty). Inset of (b) shows the calculated density of states (DOS ) of the two-dimensional
+electron gas with alocalization tail (solid curve) and that of theunperturbed two-dimensional electron gas (dashed cu rve). Thetemperature is 4K.
+FromChen et al. [398].
+mechanismsinthemetallicregimearetheElliott-Yafet,th eD’yakonov-Perel’andtheBir-Aronov-Pikusmechanisms.
+In the presenceof magneticfield, the g-tensorinhomogeneitymechanismcan also be important[374 ]. Theirrelative
+efficiencies differ in various materials and structures. The di fferent temperature, mobility and carrier density depen-
+dences of these mechanisms together with their competition result in various behaviors observed in experiments. In
+thissubsection,wereviewelectron /holespinrelaxationinmetallicregimeinbothbulksemicon ductorsandsemicon-
+ductor nanostructures.37We focus on the experimentsand single-particle theories, w hereas the many-bodytheory is
+also mentioned partly when it is necessary to understand the experimental results. Review of the many-bodytheory
+anditssalient predictionsandresultsarepresentedinthe nextsection.
+37The experiments and the corresponding theory of electron sp in relaxation in p-type bulk semiconductors are summarized in the seminal boo k
+“Optical Orientation” [3]. Wewill notreview this topic here.
+344.2.1. Electronspinrelaxationinn-typebulkIII-VandII- VIsemiconductors
+As there are few holes, the Bir-Aronov-Pikusmechanism is ir relevant in n-type III-V and II-VI semiconductors.
+Relevant mechanisms are the D’yakonov-Perel’ and Elliott- Yafet mechanisms. As has been shown in recent work
+[110], the Elliott-Yafet mechanism is less important than t he D’yakonov-Perel’mechanism in n-type III-V semicon-
+ductors,eveninthenarrowband-gapsemiconductorssuchas InAsandInSb. WeneglecttheElliott-Yafetmechanism
+in the subsequent discussions. A simple theory of the D’yako nov-Perel’ spin relaxation in bulk system has been
+presentedin Sec.3.2.2[21,117]. Themainresultis
+τ−1
+DP≃Qτpα2∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}ht/Eg. (93)
+Q∼1. Inthepresenceofmagneticfield,the g-tensorinhomogeneitymechanismalsoworks[374],
+τ−1
+s≃(µBB)2(g2−g2)τp. (94)
+Theg-tensor is assumed to be isotropic. The above equations shou ld be taken only qualitatively, as they are based
+on a series of approximations and assumptions which are to be justified. From these equations the qualitative be-
+havior of spin relaxation can be understood. To achieve such understanding, the knowledge of momentum scat-
+tering time τpis needed. In n-type III-V and II-VI semiconductors, momentum scattering mainly consists of the
+impurity, phonon, electron-electron scatterings. In earl ier theories based on single particle approach, the electro n-
+electronscatteringisconsideredirrelevantto spinrelax ation.38Moreover,theacousticphononandtransverse-optical
+phonon scatterings are negligible. Therefore, the importa nt scattering mechanisms are the ionized-impurity scatter -
+ing and the longitudinal-optical-phonon scattering. Acco rding to the Brooks-Herring formula, the temperature and
+density dependence of ionized-impurity scattering time is [21]τei
+p∼T3/2n−1
+D(nDis doping density) for T≫TF
+(nondegenerate regime) and τei
+p∼T0n0
+DforT≪TF(degenerate regime). Based on these dependences, for the
+D’yakonov-Perel’spin relaxation due to the ionized-impur ityscattering, τs∼T−9/2nDin nondegenerateregime and
+τs∼T0n−2
+Din degenerateregime. For spin relaxation due to the g-tensorinhomogeneitymechanism associated with
+the ionized-impurity scattering, roughly, τs∼B−2T−7/2nDin nondegenerate regime and τs∼B−2T0n−4/3
+Din degen-
+erate regime. The longitudinal-optical-phononscatterin g is only important at high temperature kBT/greaterorsimilarωLO/4 (ωLO
+is longitudinal-optical-phonon frequency). For example, in GaAs the longitudinal-optical-phonon scattering is im-
+portant only at T/greaterorsimilar100 K. The temperature and density dependences of longitudi nal-optical-phononscattering are
+complex. Roughly,the longitudinal-optical-phononscatt eringrate increaseswith temperaturebut variesweaklywit h
+density.39
+We first review experimental works. Let us start with tempera ture dependence of spin lifetime. Spin lifetime
+was measured as functionof temperaturein GaAs [4, 371, 520, 543], GaSb [544], GaN [545–548], InAs[549–552],
+InSb [551, 553], InP [554], ZnSe [555], ZnO [556], CdTe [524] and HgCdTe [557, 558]. It was found to decrease
+with increasing temperature at high temperature (nondegen erateregime) in all these works [e.g., see Fig. 4].40This
+is consistent with the discussion in above paragraph: τs∼T−9/2for ionized-impurity scattering. For longitudinal-
+optical-phononscattering,as τ−1
+pvariesslowerthan1 /∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}ht∼T−3,τsstilldecreaseswithtemperature. Itwasobserved
+that at high density in degenerate regime the spin lifetime v aries slowly with temperature [544, 555, 560]. This is
+understood as that in degenerate regime the ionized-impuri ty scattering and longitudinal-optical-phonon scatterin g
+timesaswell as∝an}b∇acketle{tεk∝an}b∇acket∇i}htvaryslowly withtemperature[see Eq.(93)].
+The magnetic field dependence of spin lifetime was studied in GaAs [4, 517, 561] and GaN [255, 545]. As
+indicated in Sec. 3.2.2, spin lifetime due to the D’yakonov- Perel’ mechanism increases with increasing magnetic
+field. On the other hand, spin lifetime due to the g-tensor inhomogeneity mechanism decreases with increasin g
+magnetic field. These two mechanisms compete with each other , hence spin lifetime first increases then decreases
+with increasingmagneticfield [374]. Thepeak magneticfield Bproughlysatisfies α2∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}ht/Eg≃(µBBp)2(g2−g2). In
+38In fact, electron-electron scattering plays important rol e in the D’yakonov-Perel’ spin relaxation in n-type bulk III-V semiconductors, as was
+shown by Jiang and Wuin Ref. [110].
+39DysonandRidley calculated thelongitudinal-optical-pho non scattering timefornondegenerate electron system. The yfoundthatinnondegen-
+erate regime the longitudinal-optical-phonon scattering rate increases with electron energy and temperature [542].
+40Atlow temperature thesystem may be insulating where spin re laxation is governed by localized electrons [324,559].
+35Figure 4: Temperature dependence of spin lifetime for bulk n-GaAs at
+B=0 andB=4 T with doping density nD=1016cm−3. The dotted and
+dashedcurvesrepresentthecalculated D’yakonov-Perel’ ( DP)andElliott-
+Yafet (EY) spin lifetimes respectively. From Kikkawa and Aw schalom
+[4].
+0 1 2 3 4 5 6
+B (tesla)101102103104T*2  (ps)undoped
+Nd=1×1016cm-3
+Nd=1×1018 cm-3
+Nd=5×1018cm-3
+Figure 5: Magnetic field Bdependence of ensemble spin dephasing time
+T∗
+2for bulk GaAs at various doping density at temperature T=5 K.
+Experimental data from Kikkawa and Awschalom [4](re-plott ed byˇZuti´ c
+etal. [21]).
+Figure 6: Mobility dependence of spin relaxation rate in
+bulkn-GaAs at temperature T=77 K. Experimental re-
+sults are represented by dots. Solid lines: calculation fro m
+1/τs=Q1τpα2(kBT)3/EgwithQ1=3 (scattering by
+phonons, upper straight line) and Q1=1.5 (scattering by
+ionized impurities, lower straight line). Triangles: calc u-
+lation with 1 /τs=Q1τpα2(kBT)3/Egusingτpmeasured
+from spin-relaxation suppression in Faraday geometry, as-
+suming scattering by phonons ( ▽), and ionized impurities
+(△). Inset: Density dependence of the spin relaxation rate
+(s. r. rate). NdandNadenote the donor and acceptor den-
+sities respectively. Data point ( ◦) is taken from Ref. [4]
+atT=100 K. Solid curve is a guide to the eyes. From
+Dzhioev etal. [371].
+III-V and II-VI bulk semiconductorsthe electron g-factor is energy-dependent g≃gc+βεk, hence (g2−g2)∼∝an}b∇acketle{tε2
+k∝an}b∇acket∇i}ht.
+Therefore, the peak magnetic field Bp∼∝an}b∇acketle{tεk∝an}b∇acket∇i}htwhich increases with the average energy. Typical experimen tal results
+are shown in Fig. 5. It is seen that for the undoped sample the s pin lifetime first increases and then decreases with
+increasingmagneticfield. For B>1 T,at lowdensitythespinlifetime decreaseswithmagnetic field whereasathigh
+density it increases with magnetic field. This is easily unde rstood since∝an}b∇acketle{tεk∝an}b∇acket∇i}htis larger at high density, the peak Bp
+movestohighermagneticfield[374].
+Thevariationof spinlifetimewith mobilityhasbeeninvest igatedin GaAs[371] andInAs[550] at zeromagnetic
+field.41InGaAs,thespinrelaxationratewasfoundtoincreasewithm obility,whichisconsistentwiththeD’yakonov-
+Perel’ mechanism. However,the simple formula,e.g., Eq. (6 3) in Sec. 3.2.2,can not explain the results qualitatively
+[371] (see Fig. 6). Momentumscatteringthat doesnot contri buteto mobility was suggestedto be responsiblefor the
+smallerspinrelaxationrateat highmobilityregime[371].
+The photo-excitation density dependence of spin lifetime w as investigated in n-type GaAs [4, 562, 563], GaSb
+[544]andInAs[552]. Inallthesemeasurements,thespinlif etimewasfoundtodecreasewithexcitationdensityin n-
+dopedsamples. Thedecreasewasunexplainedinsomeworks[4 ,562]orwasassignedtotheElliott-Yafetmechanism
+41Themagnetic field is always zero in this subsection unless ot herwise specified.
+36[552]. Other works explained the decrease as a result of ther malization of the electron system via photo-excitation
+[544]. Thedecreaseofspinlifetimemaybeduetotheincreas eof∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}htastheelectrondensityincreases. However,the
+thermalizationeffectmayalsoplayanimportantrole.42
+ThedopingdensitydependencewasstudiedinGaAs[4,39,371 ]andInAs[549]. Inalltheseworksspinrelaxation
+timewas foundtodecreasewith dopingdensityin degenerate regime. Thereisfew datain the nondegenerateregime
+up till now [4, 549]. The decrease of spin lifetime in degener ate regime is understood as following: in degenerate
+regimetheionized-impurityscatteringtimeandlongitudi nal-optical-phononscatteringtimevaryslowlywithdensi ty,
+whereas∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}ht∼n2
+Dincreaseswithdensity. Thereforethespinlifetime τs∼1/(∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}htτp)decreaseswithdensity.
+Electric field dependence was studied in GaAs [527, 567], ZnO [568] and GaN [548]. Usually, spin lifetime
+decreases with increasing electric field in metallic regime in GaAs [527, 567]. This is because the electric field
+induces electron drift and the hot-electron e ffect which increase the average electron energy and then enha nce the
+D’yakonov-Perel’spin relaxation [110]. However, the hot- electron effect also gives rise to the enhancement of mo-
+mentum scattering [569] which leads to the suppression of th e D’yakonov-Perel’spin relaxation. These two e ffects
+compete with each other [569]. It was found that in the case of cubic-kspin-orbit coupling, the former e ffect domi-
+nates because the spin lifetime varies rapidly with the elec tron energy[110, 570]. The spin lifetime hence decreases
+with increasing electric field [110, 570]. This is the case fo r bulk GaAs. The situation is more complicated in the
+case of linear- kspin-orbitcoupling: the spin lifetimefirst increasesduet o the enhancementofmomentumscattering
+then decreases due to the increases of electron energy [569] . This is the case for bulk wurzite ZnO where Ghosh
+et al. observed that the spin lifetime first increases then de creases with increasing electric field [568]. Usually the
+effect of electric field on spin relaxation is more pronouncedat low temperature[110, 372, 527, 548, 568]. The spin
+lifetime can also be reducedby applyingstrain which induce sadditionalspin-orbit coupling[144, 145, 571, 572]. It
+was foundthat 1 /τs∼ǫ2, whereǫis the appliedstress [572] when the strain-inducedspin-or bitcouplingdominates.
+Under strain field the temperature and density dependence of spin relaxation becomes weaker as the strain-induced
+spin-orbitcouplingis linear in k[see Eq.(7)]. It is also foundthat the spin relaxationcan be anisotropicunderstrain
+[572].
+In wurzite GaN and ZnO, the spin-orbitcouplingis di fferent from that in zinc-blendesemiconductors. The spin-
+orbitcouplinginconductionbandreads
+He
+SO=[ΩR
+e(k)+ΩD
+e(k)]·σ/slashbig2 (95)
+with
+ΩR
+e(k)=2αe(ky,−kx,0),ΩD
+e(k)=2γe(bk2
+z−k2
+∝ba∇dbl)(ky,−kx,0). (96)
+It is seen that the symmetry of the spin-orbit coupling is qui te different from that of the Dresselhaus one. Unlike in
+bulk GaAs, electron spin relaxation in bulk GaN is anisotrop ic. Lifetime of spins along [0001] direction is smaller
+thanthosealongotherdirections. Theanisotropyofspinre laxationwasdemonstratedinrecentexperiment[255].
+Spin lifetime τsat room temperature is also of concern due to interest in poss ible device application. It was
+measuredinGaAs[543,573]with τsintherangeof15-110ps,inInAs[552,574,575]with τsintherangeof2-24ps
+andin InSb[551, 553, 558] with τsinthe rangeof2-300psdependingonthedopingdensity.
+Wenowreviewthesingle-particletheoryonelectronspinre laxationin n-typebulkIII-VandII-VIsemiconductors.
+Besides the early works in 1970s and early 1980s, which have b een summerized in the book “Optical Orientation”
+[3],therearealsoalotoftheoreticalstudiesinthepastde cadeaftertherisingofthesemiconductorspintronics[576 ].
+These studies try to go beyond the widely used formulae for th e D’yakonov-Perel’[Eqs. (62) and (63)] and Elliott-
+Yafet [Eq. (53)] spin relaxation [117]. These developments go in the same paradigm: calculate spin-orbit coupling
+andscatteringviamicroscopictheoryandthenaveragethes pinrelaxationrateat di fferentkas
+τ−1
+s=/integraltext
+dk(f↑
+k−f↓
+k)τ−1
+s(k)
+/integraltext
+dk(f↑
+k−f↓
+k). (97)
+42The thermalization e ffect was shown to be crucial in a recent fully microscopic calc ulation [564]. This is also supported by the experimental
+results that the spin lifetime was observed to decrease with photon energy of the pump laser [543,565, 566].
+37Thespin relaxationrate atdi fferentk,τ−1
+s(k),duetotheD’yakonov-Perel’mechanismisgivenin Eq.(58) ,whilethat
+dueto theElliott-Yafetmechanismisgivenby
+τ−1
+s(k)=/summationdisplay
+k′[Γ(k↑,k′↓)+Γ(k↓,k′↑)], (98)
+withΓ(k↑,k′↓) andΓ(k↓,k′↑) denoting the spin-flip scattering rates. Such a paradigm wa s first applied to n-
+type bulk GaAs, InAs and GaSb by Lau et al. where the spin-orbi t coupling was calculated via the fourteen-band
+k·ptheory[577]. The momentumscattering rates due to the elect ron-impurityand electron-phononscattering were
+calculated via standard methods in transport theory. Simil ar schemes were then applied to bulk GaAs and GaN by
+Krishnamurthy et al. [578]. Later the scheme was developed t o the case of arbitrary band structure by Yu et al.
+[209].43The role of scattering by dislocations in both the D’yakonov -Perel’ and Elliott-Yafet spin relaxation was
+studied by Jena within such scheme [579]. A systematic study on spin relaxation in III-V semiconductorsbased on
+such paradigm was given by Song and Kim [108]. However, in the ir work the Boltzmann statistics was assumed
+which makes the discussion at low temperature and /or high density meaningless. The Elliott-Yafet spin relaxa tion
+duetotheelectron-electronscatteringwasstudiedinRefs .[580,581]withinsuchparadigm. Recentlyspinrelaxation
+in ZnO due to the D’yakonov-Perel’ and Elliott-Yafet mechan isms was also studied in such paradigm [559]. An
+improvementoftheparadigmwasgiveninRef.[582]. Inthatw ork,it wasarguedthatthestrongscatteringcondition
+∝an}b∇acketle{tΩ(k)∝an}b∇acket∇i}htτp≪1 may not be satisfied for large kas both∝an}b∇acketle{tΩ(k)∝an}b∇acket∇i}htandτpincrease with k. Hence the spin relaxation rate
+shouldbemodifiedtoincludethe freeinductiondecayas[582 ]
+1
+τ′s(k)=τs(εk)+/radicalbig128Eg
+αε3/2
+k−1
+. (99)
+Spinrelaxationrateoftheelectronensembleisthenobtain edbyaveragingover kviaEq.(97). A closerexamination
+onthe D’yakonov-Perel’spinrelaxationdueto theelectron -longitudinal-optical-phononscatteringfromtheequati on
+of motionwas givenin Ref. [542], whereit was shown that the e lastic scatteringapproximationcan have problemin
+treating the longitudinal-optical-phononscattering to s pin relaxation. The e ffect of the electron-longitudinal-optical-
+phononscatteringontheD’yakonov-Perel’spinrelaxation maynotbecharacterizedbyasinglemomentumscattering
+time used in the paradigm. Spin relaxation due to the D’yakon ov-Perel’mechanism under electric field was studied
+viatheMonteCarlomethodinRef.[583],wherespinlifetime wasfoundtodecreasewiththeelectricfield.
+4.2.2. Electronspinrelaxationinintrinsic bulkIII-Vand II-VIsemiconductors
+Inintrinsicsemiconductorselectronsaregeneratedtoget herwithequallnumberofholesbyphotoexcitation. Due
+to the large number of holes, the Bir-Aronov-Pikusmechanis m is now relevant to the spin relaxation. As have been
+pointed out in Ref. [110] that the Elliott-Yafet mechanism i s unimportant in intrinsic III-V semiconductors. Similar
+conclusion should also hold for II-VI semiconductors. Henc e, the relevant spin relaxation mechanisms are the Bir-
+Aronov-Pikus and the D’yakonov-Perel’ mechanisms. It was d ebated about the relative importance of these two
+mechanismsin intrinsicIII-VandII-VIsemiconductors[21 ,390, 392, 584–586]. Single-particletheorybased onthe
+elasticscatteringapproximationcomestotheconclusiont hatspinrelaxationduetotheBir-Aronov-Pikusmechanism
+can be more important at low temperature and high hole densit y [390]. However, as pointed out by Zhou and Wu
+[109], the Pauli blocking of electrons at low temperature an d/or high hole density largely reduces the Bir-Aronov-
+Pikus spin relaxationfrom a fully microscopickinetic spin Bloch equationapproach[44, 334, 350]. After that Jiang
+and Wu showed that in intrinsic bulk III-V semiconductorsth e Bir-Aronov-Pikusmechanism is unimportant via the
+same approach [110], where similar conclusion should also h old for II-VI semiconductors. Therefore, the relevant
+spin relaxation mechanism is the D’yakonov-Perel’ mechani sm. Spin relaxation in intrinsic bulk III-V and II-VI
+semiconductorscanthenbeunderstoodeasily.
+Experimental studies on spin relaxation in intrinsic semic onductors involve temperature dependence [229, 582,
+587, 588] and density dependence [566, 582, 585]. The temper ature dependence coincides with what predicted by
+43In this work, the approach that the authors used to calculate hole spin relaxation rate seems to have some problem, as they concluded that the
+hole spin relaxation is dominated by the Elliott-Yafet mech anism which isnot true.
+38theory: spin lifetime varies slowly in degenerate regime an d then decreases rapidly in nondegenerate regime with
+increasing temperature. The density dependence, however, indicates an interesting behavior: spin lifetime increase s
+withelectrondensity[582]. Thisistotallydi fferentfromwhatobservedin n-typesemiconductorswherespinlifetime
+decreases with electron density [4, 39, 371]. However, in in trinsic semiconductors, the main scattering mechanisms
+are the carrier-carrierscattering andthe longitudinal-o ptical-phononscattering, whichmakesspin relaxationdi fficult
+to understand via the single-particle theory. From the many -body kinetic spin Bloch equation approach [44, 334,
+350], Jiang and Wu found a nonmonotonic density dependence o f spin lifetime due to the nonmonotonic density
+dependenceof the carrier-carrierscattering rate: in nond egenerateregime the carrier-carrierscattering rate incr eases
+withincreasingdensitywhereasindegenerateregimeitdec reaseswithincreasingdensity[110]. Aftertheirpredicti on
+Teng et al. observed in intrinsic bulk GaAs that at high excit ation density ( ne>1017cm−3) spin lifetime decreases
+with density [585]. However, Teng et al. assigned the decrea se of spin lifetime to the Bir-Aronov-Pikusmechanism
+which is wrong (see comment [589]). Later Ma et al. observed s uch a nonmonotonic density dependence of spin
+lifetimeinbulkCdTe[566].44
+4.2.3. Electronspinrelaxationinn-typeandintrinsicIII -VandII-VIsemiconductortwo-dimensionalstructures
+As was found in n-type and intrinsic bulk III-V and II-VI semiconductors tha t spin relaxation is dominated by
+the D’yakonov-Perel’mechanism, it is reasonableto believ e that the same conclusionholdsfor the two-dimensional
+case. Indeed, calculations showed that the Elliott-Yafet m echanism is unimportant in GaAs and InGaAs quantum
+wells[111, 591, 592].45Fullymicroscopicmany-bodykineticspin Bloch equationca lculationindicatedthatthe Bir-
+Aronov-Pikusmechanismis ine ffectivein intrinsicGaAs quantumwells [109].46These evidencesenableusto focus
+onspin relaxationduetotheD’yakonov-Perel’mechanism.
+Anisotropicspin relaxation
+One ofthesalient featuresof theD’yakonov-Perel’spinrel axationin two-dimensionalstructures(quantumwells
+and heterostructures)is that it varies significantly with t he growth direction and the structure. The former originate s
+from tailoring the symmetry of the Dresselhaus spin-orbit c oupling by the growth direction. The latter partly comes
+from the fact that the strength of the linear Dresselhaus ter m can be tuned by quantum confinement. Moreover, the
+Rashba spin-orbit coupling also appears in asymmetric two- dimensional structures. The joint e ffect of the Dressel-
+haus and the Rashba spin-orbit couplingsleads to anisotropic spin relaxation. The strength of the Rashba spin-orbit
+coupling can be controlled by the gate voltage. The tunabili ty of spin relaxation opens route to a variety of new
+phenomenaandfunctionalities[22, 596].
+Let us start with the spin-orbitcoupling which is crucial to the D’yakonov-Perel’spin relaxation. Assuming that
+theconfinementisstrongenoughsothatonlythelowestsubba ndisinvolved. Considerasimplecase: (001)quantum
+well,wheretheDresselhausspin-orbitcouplingis
+HD=βD(−kxσx+kyσy)+γD(kxk2
+yσx−kyk2
+xσy). (100)
+βD=γD∝an}b∇acketle{tψ1|ˆk2
+z|ψ1∝an}b∇acket∇i}htwhereψ1is the envelope function of the lowest subband and ˆkz=−i∂z.βD∼γD(π/a)2witha
+beingthewell width. TheRashbaspin-orbitcouplingreads
+HR=αR(kyσx−kxσy). (101)
+Fornarrowquantumwellwhen k2≪(π/a)2[k=(kx,ky) isthein-planewavevector],the lineartermdominates
+HSOC=(αRky−βDkx)σx+(−αRkx+βDky)σy. (102)
+44They, however, assigned the decrease of the spin lifetime wi th increasing density to the Elliott-Yafet mechanism, whic h is incorrect (see
+comment[590]).
+45In narrow bandgap semiconductor InSb quantum wells, experi ment gives some evidences that the Elliott-Yafet spin relax ation can be more
+importantthan theD’yakonov-Perel’ oneatlow mobility sam ples asrevealed byLitvinenko etal. inRef.[593]. Neverthe less, thetwo mechanisms
+arestill comparable in their experiment.
+46However, in some wide bandgap semiconductors which have wea k spin-orbit coupling and strong electron-hole exchange in teraction, the
+Bir-Aronov-Pikus mechanism can be important in intrinsic s amples, especially when excitons are formed. Such as the sit uation in ZnSe [594].
+Also in some special cases where the D’yakonov-Perel’ mecha nism is suppressed, the Bir-Aronov-Pikus or the Elliott-Ya fet mechanisms may be
+important, such as in (110) quantum wells [595]. Wewillretu rn to these special cases after the discussion in common case s.
+39It was first pointed out by Averkiev and Golub [194] that the sp in relaxation is anisotropic when the Rashba and
+Dresselhausspin-orbitcouplingscompetewitheachother. Thespin relaxationtensoristhen
+τ−1
+zz=2τ−1
+xx=2τ−1
+yy=8m∝an}b∇acketle{tτpεk∝an}b∇acket∇i}ht(α2
+R+β2
+D), τ−1
+xy=−8m∝an}b∇acketle{tτpεk∝an}b∇acket∇i}htαRβD. (103)
+Diagonalizingthespinrelaxationtensor,oneobtains
+τ−1
+zz=8m∝an}b∇acketle{tτpεk∝an}b∇acket∇i}ht(α2
+R+β2
+D), τ−1
+±±=4m∝an}b∇acketle{tτpεk∝an}b∇acket∇i}ht(αR∓βD)2, (104)
+wherethe principalaxesare n±=1√
+2(1,±1,0)andz-axis. It is meaningfulto rewrite the spin-orbitcouplingi n these
+principalaxis,
+HSOC=(αR−βD)k+σ−−(αR+βD)k−σ+. (105)
+Recall that τ−1
+ij∼[Ω2δij−ΩiΩj]. As averaging over angle k+k−=0 andk2
++=k2
+−, one readily finds that 1 /τ++∼
+(αR−βD)2,1/τ−−∼(αR+βD)2,1/τ+−=0and1/τzz∼2(α2
+R+β2
+D). IfαR=βD(αR=−βD),1/τ++=0(1/τ−−=0),i.e.,
+spinlifetimeisinfinitealong n+(n−). However,takingintoaccountofthecubicDresselhauster m,spinlifetimealong
+thisparticulardirectionis finite but still muchlargertha n that alongotherdirections[597]. Thereforespin relaxat ion
+ishighlyanisotropywhen αR≃±βD[196, 370].
+Experimentally as only the spin polarization along the quan tum well growth direction can be measured,47spin
+relaxation anisotropy is revealed indirectly by comparing spin relaxation rates along zdirection in the two cases
+with magnetic field along [110] and [1 ¯10] directions. When a moderate magnetic field is along [110] direction, spin
+polarizationalongthegrowthdirectionevolves
+Sz(t)=Sz(0)exp[−1
+2(τ−1
+zz+τ−1
+−−)t]cos(ω0t+φ0). (106)
+Theobservedenvelopedecayrateis1
+2(τ−1
+zz+τ−1
+−−). Applyingmagneticfieldalong[1 ¯10]direction,oneobtainsadecay
+rate of1
+2(τ−1
+zz+τ−1
+++). Comparing these two, one can obtain spin lifetimes τ++andτ−−and the ratio|αR/βD|can be
+extracted.48The spin relaxation anisotropy in (001) quantum wells was fir st observed by Averkiev et al. [198] via
+the Hanle measurements, and then by several groups via time- resolved measurements [66, 199, 598, 599]. Among
+these works, the spin relaxation anisotropy was demonstrat ed to be controlled /affected by density [598], magnetic
+field[199],gate-voltage(seeFig.7)[599,600]andtempera ture[66,598]. Thesymmetryofthespin-orbitcouplingin
+other two-dimensionalstructures with di fferent growth directions also has marked e ffect on spin relaxation. In some
+particularcases,suchas(111)and(110)quantumwells,spi nrelaxationcanbelargelysuppressed. Astherearealsoa
+lotofpapersonthosetopics,we will discussthemindetail l aterwhilebelowweonlyfocuson(001)quantumwells.
+Spin relaxationin (001)quantum wells
+Widely concerned topics are the temperature, density, mobi lity, excitation density, magnetic field, gate-voltage
+and quantum subband quantization energy dependencesof spi n relaxation. These dependencesreveal intriguing un-
+derlyingphysicsandprovideimportantknowledgesforspin tronicdevicedesign. Belowwefirst reviewexperimental
+studies and then the single-particle theories. To benefite t he understanding, we first present some simple analytical
+resultswhichassumeonlyelastic scatteringinthestrongs catteringregime[196]
+τ−1
+z=4/angbracketleftig/bracketleftig
+˜τ1(α2
+R+˜β2
+D)k2+˜τ3γ2
+Dk6/16/bracketrightig/angbracketrightig
+, τ−1
+±=2/angbracketleftig/bracketleftig
+˜τ1(αR∓˜βD)2k2+˜τ3γ2
+Dk6/16/bracketrightig/angbracketrightig
+, (107)
+Hereτi(i=z,+,−)denotestherelaxationtimeofthespincomponentalong idirectionand∝an}b∇acketle{t...∝an}b∇acket∇i}htrepresentstheaverage
+definedbyEq.(97). ˜βD=βD−1
+4γDk2. Itisnotedthatwhenthelinearspin-orbitcouplingdomina tes,thespinlifetime
+varies as 1/∝an}b∇acketle{tτpεk∝an}b∇acket∇i}htwhich changes with temperature and density slowly. However , when the cubic term becomes
+important, the spin lifetime can vary as 1 /∝an}b∇acketle{tτpε2
+k∝an}b∇acket∇i}htor 1/∝an}b∇acketle{tτpε3
+k∝an}b∇acket∇i}ht. In the latter case, spin relaxation varies rapidly with
+densityandtemperature.
+•Temperature dependence. In experiments, the temperature dependence was measured in metallic regime in
+Refs. [66, 367, 591, 593, 601–604]. In heavily-doped (low mo bility) quantum wells, spin lifetime decreases with
+47Note that arecently developed technique of tomographic Ker r rotation can measure spin polarization in other direction s [500].
+48Actually, via such method, one can determine the ratio butno t surewhether it is |αR/βD|or|βD/αR|[198].
+40Figure7: Polar plotofthespindephasing timemeasured asaf unction oftheangle(indegree) between themagneticfield an dthe[110]axisatfour
+applied biases Uin coupled double quantum wells. The spin dephasing time at e ach angle is represented by the distance to the center of the p olar
+coordinates. Experimental data are shown by points; theore tical values are presented by solid curves. The ratio of the R ashba spin-orbit coupling
+parameterαto the Dresselhaus one β,α/β, extracted from the anisotropy of spin dephasing time is als o presented in each figure. The values ∆E
+represent theexciton Stark shifts for the given biases U. Temperature is T=2 K.FromLarionov and Golub [599].
+increasing temperature. A good example is the experiment of Ohno et al. [602] [Fig. 8]. In heavily-dopedquantum
+wells, the momentumscattering is dominatedby the electron -impurityscattering except that at high temperature the
+electron-longitudinal-optical-phononscattering may be come more important. When the electron-impurityscatterin g
+dominates,asboth τpand∝an}b∇acketle{tεk∝an}b∇acket∇i}htincreasewithtemperature,spinlifetimedecreaseswithin creasingtemperaturerapidly.
+Athightemperaturetheriseofelectron-longitudinal-opt ical-phononscatteringslowsdownthedecreaseorevenlead s
+toanincreaseofthespinlifetimewithincreasingtemperat ure.
+Inundopedquantumwells, Malinowskiet al. [603] presented a systematic studyonelectronspin relaxation. The
+main results are shown in Fig. 9. It is seen that the temperatu re dependence of spin relaxation rate varies largely
+with quantum well width. At small well width, spin relaxatio n rate changes slowly with temperature, whereas at
+large well width, it increases rapidly with temperature. Th e temperature dependence can be fitted roughly as ∼T0
+for narrow quantum well and ∼T2for wide quantum well. The authors explained that as the long itudinal-optical-
+phonon scattering dominates at high temperature for undope d quantum wells, roughly τp∼T−1[178]. For narrow
+quantum wells, spin relaxation rate τ−1
+s∼τp∝an}b∇acketle{tεk∝an}b∇acket∇i}ht∼T0, whereas for wide quantum wells τ−1
+s∼τp∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}ht∼T2. It is
+noted that in the wide quantum well, the increase of spin rela xation rate is still slower than that in bulk samples at
+lowtemperature,whichindicatesthecrossoveroftheleadi ngspin-orbitcouplingfromlineartermto cubictermwith
+increasingtemperature.
+In high mobility two-dimensional electron system at su fficiently low temperature the momentum scattering can
+be very weak. In this situation, the D’yakonov-Perel’ spin r elaxation is in the weak scattering regime where spin
+polarization shows precessional decay [367, 368]. Brand et al. observed the transition from the precessional decay
+regime to the motional-narrowing regime [368]. As spin life time changes from τs∼τ−1
+pin motional-narrowing
+regime toτs∼τpin precessional decay regime, the temperature dependence o f spin lifetime has a turning point
+around the transition.49This was observed in experiment by Leyland et al. (see Fig. 10 ) [367]. Several works on
+temperaturedependenceofspinrelaxationinhighmobility two-dimensionalelectronsystemrevealedtheimportance
+oftheelectron-electronscatteringtospinrelaxation[36 7,591,604,605]. Thisimportantissuewillbereviewedinth e
+49Such behavior also exists in hole spin relaxation in Si /Gequantum wells as reported by Zhang and Wuin Ref. [299].
+41Figure 8: Temperature Tdependence of electron spin relaxation
+timeτsforn-doped (001) GaAs /AlGaAs quantum wells with well
+width 7.5 nm. Electron density is 4 ×1010cm−2in each quantum
+well. Dotted curve is the calculated result of spin lifetime based
+on theD’yakonov-Perel’ theory. From Ohno et al. [602].
+Figure 9: Temperature dependence of spin relaxation rate fo r
+undoped (001) GaAs /AlGaAs quantum wells with various well
+widths.♦: well width 6 nm; /medbullet: well width 10 nm; /square: well width
+15 nm;/triangledownsld: well width 20 nm. Solid curve is a fit with 1 /τs∼T2
+dependence. Dashed curve represents the data in intrinsic b ulk
+GaAs by Maruschak et al. (as reproduced in Ref. [3]). From Ma-
+linowski et al. [603].
+frameworkofthemany-bodytheoryin Section5.
+•Excitationdensitydependence. Excitationdensitydependenceofspinrelaxationwasinves tigatedinRefs.[66,
+514, 606–612]. Interestingly, it was discovered that the sp in lifetime decreases with excitation density at low tem-
+perature in n-doped quantum wells [66, 608, 610]. At room temperature in u ndoped quantum wells, Aleksiejunas
+et al. found that at low density spin lifetime increases with excitation density whereas at high density it decreases
+(see Fig. 11) [607]. Moreover, Teng et al. observed a peak in d ensity dependence at room temperature in undoped
+quantumwell [609]. As in intrinsicquantumwellsthe domina ntscatteringsare themany-bodyelectron-electronand
+electron-holeCoulombscatterings,theresultscannotbee xplainedintheframeworkofsingle-particleapproach. Nev -
+ertheless,many-bodytheorieshaverevealedtheunderlyin gphysics: theelectron-electronandelectron-holeCoulom b
+scatterings increase with density in nondegenerate (low de nsity) regime, whereas decrease in degenerate (high den-
+sity)regime[110,111,299,613]. Thenonmonotonicdensity dependenceofthesemany-bodycarrier-carrierCoulomb
+scatteringresultsinthenonmonotonicdensitydependence ofspinlifetime[110, 111, 299, 613].
+•Mobility dependence. Spin lifetime as function of mobility µ(∝τp) was studied in Refs. [607, 614] at room
+temperaturewherethequalitativerelation τs∼µ−1was observedwhichsignalsthe D’yakonov-Perel’spinrelax ation
+mechanism. However, Brand et al. found that in high mobility two-dimensional electron system at low temperature
+(T<100 K), spin lifetime deviates from τs∼µ−1largely [368]. Electron-electron scattering is proved to b e the
+key to understand the observed results.50In high mobility two-dimensional electron system at low tem perature,
+the electron-electron scattering is the dominant scatteri ng as other scatterings (the electron-impurity and electro n-
+phonon scatterings) are weak. As it randomizes the momentum , the electron-electron scattering also contributes to
+theD’yakonov-Perel’spinrelaxation. However,sincethee lectron-electronscatteringdoesnotcontributetomobili ty,
+τs∼µ−1doesnot hold anymore. The total momentumscattering time wi th the electron-electronscatteringincluded,
+τ∗
+p, is then much smaller than that deduced from mobility, τp(see Fig. 12). At room temperature, the electron-
+longitudinal-optical-phonon scattering becomes the stro ngest momentum scattering which also limits the mobility,
+hencetherelation τs∼µ−1isrecovered.
+•Quantum well width dependence. The quantum well width dependence of spin relaxation was inv estigated
+in Refs. [592, 602, 603, 614, 616–619]. All these studies wer e performed at room temperature in nondegenerate
+regime. In this regime, by neglecting the cubic Dresselhaus term and the Rashba term and assuming infinite well
+50This was firstpredicted theoretically by Wu and Ning [332, 33 4] and later by Glazov and Ivchenko [615].
+42Figure 10: Temperature dependence of spin lifetime for samp le (a) (well width 20 nm, electron density 1 .8×1011cm−2) and (b) (well width
+10 nm, electron density 3 .1×1011cm−2). The curves are guides for the eyes. Open circles and square s are spin-lifetimes measured in the high
+temperature regimewherespinevolution isexponential. So lid symbolsarevalues of( |Ω(kF)|2τ∗
+p)−1(denoted in thefigureas( ∝an}b∇acketle{tΩ2∝an}b∇acket∇i}htτ∗
+p)−1)(Ω(kF)is
+obtained fromanalysis of the spin evolution in the low-temp erature oscillatory regime. τ∗
+pis themomentum scattering timeincluding the electron-
+electron scattering.). Open triangles are the decay time co nstant of the oscillatory spin evolution, t0, obtained by fitting the experimental data with
+oscillatory exponential decay Sz(t)=S0exp(−t/t0)cos(ωt+φ). Arrows indicate the value of |Ω(kF)|−1for each sample, corresponding to the
+condition|Ω(kF)|τp≃1. From Leyland et al. [367].
+Figure 11: Excitation density dependence of spin lifetime τsin undoped (001) InGaAs multiple quantum wells at room tempe rature. From
+Aleksiejunas et al. [607].
+depth, spin lifetime is given by [369] τs=Eg/(2α2E2
+1ekBTτp) withα=2γD/radicalig
+2m3eEgandE1ebeing the electron
+subband quantization energy. It hence gives τs∼E−2
+1e. Most experimental data roughly agree with the relation. Fo r
+example,Tackeuchi et al. fitted τs∼E−2.2
+1eformultiple GaAs/AlGaAs quantumwell [616]. However,there are some
+situations where results deviate largely from the above rel ation [614]. Britton et al. reported that for small E1e, the
+relation is largely deviated whereas for large E1e, the spin lifetime generally shows the quadratic behavior [ 618]. A
+similar result was obtained by Malinowski et al. [603] which is shown in Fig. 13. It is seen that for medium E1e,
+the relation agrees well with experimental result. At small E1eas the cubic Dresselhaus spin-orbit coupling plays
+important role, the relation deviates from the data. At larg eE1eas the wavefunction penetration is unnegligible and
+theinfinite-depth-wellassumptionnolongerholds,therel ationalso deviatesfromexperimentalresult.
+•Magnetic field dependence. Spin lifetime has also been measured as function of magnetic field in quantum
+wells in metallic regime in Refs. [66, 199, 326, 525, 620, 621 ]. There are two special configurations of magnetic
+field: one that the magnetic field lies in the quantumwell plan e (the Voigt configuration)and the otherthat magnetic
+field is perpendicular to the well plane (the Faraday configur ation). In the Voigt configuration, the orbital e ffect of
+magnetic field is negligible. In this case the magnetic field h as two consequences on spin relaxation: first it mixes
+43Figure 12: Momentum scattering time τ∗
+p(the total momentum
+scattering including the electron-electron scattering) a ndτp(the
+momentum scattering time extracted from mobility) as funct ion
+of temperature. The electron density is 1 .9×1011cm−2. Dotted
+curve is a guide to the eyes for τp. Dashed curve represents an
+empirical fit of electron-electron scattering time as ∼T−1at high
+temperature. Solid curve sums these two contributions (1 /τpand
+the electron-electron scattering rate) as a guide curve to τ∗
+p. The
+data points of τ∗
+pis obtained from Monte Carlo simulation of spin
+evolution. FromBrand et al. [368].
+Figure 13: Spin relaxation rate as function of the subband qu an-
+tization energy E1ein GaAs/AlxGa1−xAs quantum wells at room
+temperature. The solid curve is a fit to the data points of the f orm
+τ−1
+s=a+bE2
+1e. From Malinowski et al. [603].
+thein-planeandout-of-planespinrelaxationsduetotheLa rmorprecession;secondtheLarmorspinprecessionslows
+downspinrelaxationbyafactorof(1 +ω2
+Lτ2
+p). Inusualcondition ωLτp/lessorsimilar0.1(e.g.,B=2Tandτp=1psinGaAsyield
+ωLτp=0.05),thustheseconde ffectisweak. Thefirste ffectisusuallymoreimportantasthein-planeandout-of-pla ne
+spinlifetimesdifferlargely. Theeffectofmixingsaturatessoonatalowmagneticfieldaround0.1 T.Thisisconsistent
+with experiments: the magnetic field dependenceis usually m ore pronouncedat low field [66, 199, 620], whereas at
+high field the dependence becomes weak [525]. In asymmetric q uantum wells the in-plane spin relaxation can be
+quite anisotropic. In this case the e ffect of magnetic field depends largely on its direction [66, 19 8, 199, 598, 599].
+Interestingly,Stich et al. foundthat the magneticfield dep endenceofspin lifetimefor B∝ba∇dbl[110]exhibitsa minimum
+whereasthe magnetic dependencefor B∝ba∇dbl[1¯10] shows a maximumfor an asymmetric (001) quantum well [199 ]. In
+theFaradayconfigurationwithmagneticfieldperpendicular toquantumwellplane,theorbitale ffectinducescyclotron
+motionwhich has an important e ffect on spin relaxation. As the cyclotronfrequencyis much la rgerthan the Larmor
+frequency (in GaAs ωc≃70ωL), the cyclotron motion e ffectively suppresses the spin relaxation. This phenomena
+was observed by Sih et al. in InGaAs /GaAs quantum well (See Fig. 14) [621].51At higher magnetic field and low
+temperature, spin lifetime oscillates with magnetic field, as the Landau level filling a ffects both spin precession and
+momentum scattering. Finally in materials with strong ener gy dependence of the g-factor (e.g., in narrow band-gap
+semiconductors)athighmagneticfield,the g-tensorinhomogeneitymechanismcanbeimportant. InInGaA squantum
+wells, spin relaxation was observed to first increase and the n decrease with increasing magnetic field for an in-plane
+magnetic field [623]. The increase is due to the mixing of out- of-plane and in-plane spin relaxations, whereas the
+decreasecanbeattributedtothe g-factorinhomogeneityspinrelaxationmechanism.
+•Gate-voltage dependence. The spin relaxation can also be tuned by the gate-voltage [52 5, 599]. The gate-
+voltage may have several consequences on spin relaxation. F irst it changes electron density. Second it changes the
+Rashbaspin-orbitcoupling. Themodificationoftheenvelop efunctionalongthegrowthdirectionfurtherchangesthe
+linear Dresselhaus spin-orbit coupling. Third it may chang e the mobility. Hence the underlying mechanism for the
+gate-voltagedependenceofthespin lifetimeiscomplex.
+•Initial spin polarization dependence. Initial spin polarization dependence was studied by Stich e t al. [41,
+42, 326] where spin relaxation rate decreases with initial s pin polarization as predicted by Weng and Wu [44]. The
+underlying physics will be reviewed in the next section on th e kinetic spin Bloch equation approach [44, 334, 350]
+where the Coulomb Hartree-Fock term acts as an e ffective magnetic field in spin precession [44]. This e ffective
+magneticfieldisalwaysalongspinpolarizationdirection. RealisticcalculationsfromthekineticspinBlochequatio ns
+51Also observed in Si /Gequantum wellby Wilamowski and Jantsch [622].
+44Figure 14: Up panel: Spin dephasing time T∗
+2measured (symbols) as a function of magnetic field BatT=2 K, 5 K, and 20 K and calculated
+(curves) from a spin relaxation model at T=2 K, 4 K, and 12 K. Down panel: the longitudinal resistivity Rxxas a function of B at T=2 K, 5
+K, and 20 K. Measurement is performed in single n-modulation doped InGaAs /GaAs quantum well. Also shown are dotted curves indicating t he
+position of Bnfor Landau-level index n=16, 14,12, 10, 8and 6 ( nlabels the filled Landau levels). FromSih et al. [621].
+indicated that the induced magnetic field can be as large as te ns of Tesla [44]. Such large longitudinal e ffective
+magnetic field largely suppresses the spin relaxation. Besi des spin relaxation, experimentalists also found evidence
+of the effective magnetic field in the sign change of the Kerr rotation [ 327]. Recently the Coulomb Hartree-Fock
+effectivemagneticfieldwasagaindiscussedintheissue ofspin accumulation[43].
+•Excitationphotonenergydependenceandothers. Theexcitationphotonenergydependenceofspinrelaxation
+was investigated in Ref. [605] where the energy-dependent m omentum scattering time due to the electron-electron
+scattering was revealed. Spin dynamics in higher subband wa s studied in Refs. [624–626], indicating the important
+roleofinter-subbandmomentumscattering. Spindynamicsu ndermicrowavedrivingfieldwasdiscussedinRef.[627].
+•Interface inversion asymmetry induced spin relaxation. The interface inversion asymmetry induced spin-
+orbit coupling and its consequence to spin relaxation were s tudied in Refs. [219, 221, 222].52As was reviewed
+in Section 2.4, in the case where the quantum well and the barr ier materials share no common atom, the interface
+inversion asymmetry exhibits even in symmetric quantum wel ls. The interface inversion asymmetry induced spin-
+orbit coupling is important in narrow quantum wells. By comp aring with the case with common atom of the same
+well width, it was discovered that spin relaxation in quantu m wells without common atom is always shorter [219,
+221,617]. Thiseffectisespeciallymarkedinnarrowquantumwells. Itcannotb eexplainedbytheD’yakonov-Perel’
+theory taking into account only the Dresselhaus spin-orbit coupling [617]. As in [110]-grown quantum well, the
+two interfaces could be symmetric even in the case without co mmon atom, which gives a special case withoutthe
+interface-inducedspin-orbit coupling [222]. By comparin g spin relaxation in [110]-grownquantum well and [001]-
+grown quantum well with same well width, one can also estimat e the value of interface-inducedspin-orbit coupling
+[222].
+•Spin relaxation in diluted nitride materials. Spin relaxation in diluted nitride materials such as GaAsN a nd
+InGaAsNwasstudied inRefs. [629–631]. Interestingly,the spin lifetime increaseswith increasingtemperatureupto
+roomtemperaturefor T>40K,butdecreaseswithincreasingtemperaturebelow40K.I twasfoundthatlessthan1%
+dopingofnitrideinGaAsandInGaAsmakesthespinlifetimea troomtemperatureincreasebymorethanoneorderof
+magnitudein as-grownsamplesbeforeannealing[629]. Afte r annealingthe spin lifetime is drasticallydroped[629].
+The above unusual behavior indicates that localized electr ons bound to nitride dopants play an important role. Even
+for delocalized electrons, scattering with nitride dopant s greatly reduces τpand hence increases the spin lifetime.
+52Thewellwidthdependence ofspinlifetime innarrowquantum wellswherethewellandbarrier materials donotshareanyco mmonatom(e.g.,
+in InGaAs/InP quantum well) may also contain the information of the int erface inversion asymmetry [617,628].
+45Annealingimprovesthecrystalqualityofdilutednitridem aterialwhichhenceincreasesthemobilityandreducesthe
+spin lifetime. Spin lifetime at room temperature as functio n of subband quantization energy E1egivesτs∼E−1
+1ein
+unannealedInGaAsN /GaAsmultiplequantumwells[631].
+Single-particletheoriesforspin relaxationin (001)quan tum wells
+Wenowturntoreviewthesingle-particletheoryofelectron spinrelaxationin n-typeandintrinsictwo-dimensional
+electron system. As in bulk system, the single-particle the ory assumes that for all kthe strong scattering criteria
+Ωkτp≪1isfulfilled. Itisalsoassumedthatthecarrier-carriersc atteringisirrelevant.53Withintheelasticscattering
+approximation,54spinlifetimeforsystemneartheequilibrium(alsoimplyin gthatthespinpolarizationis verysmall)
+can be calculated via Eq. (107). Such paradigm has been widel y applied to study spin relaxation. Often the spin-
+orbit coupling in the two-dimensional structure is calcula ted via the k·pmethod within the multiband envelope-
+functionapproximationtogetherwith the Schr¨ odinger-Po issonequation of the heterostructure(see, e.g., Ref. [633 ]).
+The momentum scattering times are calculated via the formul ae developed for calculating the mobility due to the
+electron-impurityandelectron-phononscatterings. It sh ouldbe mentionedthat in heterostructuresmanyfactorssuc h
+asstructure,dopingandgate voltagecan a ffect the spin-orbitcoupling. To quantitativelydeterminet he spin lifetime,
+one has to quantitatively determine the spin-orbit couplin g. In theoretical calculation, the spin-orbit coupling is
+determined by the multiband envelope-function calculatio n with realistic structure parameters. In experiments, the
+Rashba and linear Dresselhaus spin-orbit couplings can be d etermined quantitatively via several methods, such as
+electric-field-inducedspin precession[64, 66]andcurren t-inducedmodificationof g-factor[634].
+A good success of such paradigm is that it well reproduces the subband quantization energy E1edependence of
+the electron spin lifetime [577] which can not be explained b y the simple relation of τs∼E−2
+1efrom the D’yakonov-
+Kachorovskiitheory[seeFig.15(b)]. Thisisduetothefact thatthisparadigmtakesfullaccountofspin-orbitcouplin g
+viadiagonalizingthe multibandenvelope-functionequati on. Inthe sameworkLauet al. also demonstratedthat their
+calculation achieved better agreement with experiments in the mobility dependence of the spin lifetime compared
+with the D’yakonov-Kachorovskiitheory(see Fig. 15). They also demonstratedthe roleof the electron-longitudinal-
+optical-phononscattering to spin relaxation as functiono f temperaturecomparedwith that of the impurityscattering
+asdepictedinFig.15. Systematiccalculationviasuchpara digmofvariousdependencesofelectronspinlifetimecan
+befoundin Ref.[633].
+•Temperature dependence. The temperature dependence of the D’yakonov-Perel’ spin re laxation was calcu-
+lated within such paradigm in Refs. [577, 635, 636]. Kainz et al. attempted to perform a microscopic calculations
+of the temperature-dependentspin-relaxation rates with r ealistic system parameters [636]. By using the parameters
+fromexperiments,theycalculatedthetemperaturedepende nceofspinrelaxationtimeandcompareditwiththeexper-
+imentaldatabyOhnoet al. [602](see Fig.16). Intheircalcu lationthemomentumscatteringtimewas notcalculated
+microscopically but inferred from the Hall mobility. They c lassified three types of scatterings and assumed that the
+Hallmobilityissolelylimitedbyeachtypeofscattering. B ydoingsotheyobtainedthreespinlifetimes. Findingthat
+theexperimentalmeasuredspinlifetimefalls intotheregi ondeterminedbythethreecalculatedones,theyconcluded
+thatthecalculationagreeswith theexperiments. By closee xamination,theyfoundthat theelectron-ionized-impurit y
+scattering dominates at low temperature ( T<100 K), whereas the electron-longitudinal-optical-phono n scattering
+dominatesat hightemperature[636]. The temperaturedepen dencewas foundtobe morepronouncedat low electron
+density[635, 636].
+•Electrondensitydependence. Theelectrondensitydependenceofspinrelaxationwascalc ulatedinRefs.[195,
+197, 635–638]. At zero temperature (degenerate regime), Go lub et al. found that spin relaxation rate increases
+monotonicallywithelectrondensity[195]. Interestingly ,Averkievetal. foundthattheratioofspinlifetime τ−/τ+[τ+
+(τ−)denotesspinlifetimeforspinalong[110]([1 ¯10])direction)hasa peakindensitydependence: atlowdens itythe
+spinrelaxationisdominatedby 1 /τ−whereasat highdensity1 /τ+becomesmoreandmoreimportantfor αR,βD>0
+[195, 635]. From Eq. (107), one can understand the above resu lts by noting that ˜βD=βD−1
+4γDk2can become
+53However, it has been found that in high mobility two-dimensi onal system, electron-electron scattering is the dominant momentum scattering
+to the D’yakonov-Perel’ spin relaxation [44,368,372].
+54Although most of the single-particle theory is based on the e lastic scattering approximation, some works go beyond that . Dyson and Ridley
+developed amethod tocalculate themomentum scattering tim eduetoelectron-longitudinal-optical-phonon scatterin g beyond theelastic scattering
+approximation. Inbulksystem,theyshowedthattheelastic scattering approximation mayhaveproblemintreating thel ongitudinal-optical-phonon
+scattering as it is essentially inelastic [542]. They furth er applied their method to study the D’yakonov-Perel’ spin r elaxation associated with the
+electron-longitudinal-optical-phonon scattering in qua ntum wells and wires in Ref. [632].
+461000500 5000
+Mobility (cm2/Vs)10100T1 (ps)100E1 (meV)
+50
+110T1 (ps)
+100
+T (K)200 300100200300400T1 (ps)
+(a)(b)
+(c)
+Figure 15: Electron-spin lifetime T1as a function of (a) mobility, (b) subband quantization ener gyE1, and (c) temperature, for 75-
+Å GaAs/Al0.4Ga0.6As multiple quantum wells at room temperature. Dots represe nt the results of experiment (Ref. [614]). The theoretical r esults
+with electron-optical-phonon scattering (solid curves) a nd electron-neutral-impurity scattering (dashed curves) are shown, as well as the results
+fromthe D’yakonov-Kachorovskii theory (dot-dashed curve s). FromLau et al. [577].
+negativeat large k. Similar results have beenobtainedin temperaturedepende ncein nondegenerateregime [635]. In
+heterojunction, both the Rashba spin-orbit coupling and th e linear Dresselhaus term depend on the electron density
+as the built-in electric field and the wave-function across t he junction vary with electron density. αRandβDin
+heterojunctioncanbeestimatedas αR≃α0nee2/(2κ0ǫ0)andβD≃γD/bracketleftbig16.5πnee2m∗/(8κ0ǫ0)/bracketrightbig2
+3,whereα0representsthe
+Rashba coefficient,κ0is static dielectric constant and ǫ0stands for the vacuumdielectric constant [635]. Averkieve t
+al. found that the spin relaxation rate 1 /τ+has a minimum in the density dependence due to the cancellati on of the
+Rashba and linear Dresselhaus spin-orbit coupling, wherea s 1/τ−and 1/τzincrease with density monotonically(see
+Fig. 17) [635]. At the electron density where 1 /τ+hasa minimum, the spin relaxationis highlyanisotropic. A m ore
+careful consideration of the same problem within the multib and envelope-functionapproach was given by Kainz et
+al. for various well width in Refs. [197, 636] where the spin- orbit coupling was treated more carefully. Via similar
+approach the Rashba and Dresselhaus spin-orbit couplings i nδ-doped InSb/AlxIn1−xSb asymmetric quantum well
+was calculatedin a rangeof carrier densities[637].55Based on these results, the density dependenceof spin lifet ime
+was calculated where a minimum in the density dependence of t he in-plane spin relaxation rate was also observed
+[637]. The spin-orbit coupling and the density dependence o f spin relaxation time in such quantum well were also
+studied by Li et al. [638], where the calculation was compare d with the experimental results in Ref. [593]. The
+authorsalso showed from the eight-band k·pmodel that the spin-orbit coupling deviates strongly from t he linear- k
+Rashba/Dresselhausmodel[638].
+•Gate-voltagedependence. Thegate-voltagedependenceofspin lifetimefora triangul arquantumwell defined
+byastructurewithaninfiniteheightbarrieratleftboundar yandaconstantelectricfieldatrightboundarywasstudied
+by Averkiev et al. [635]. They found that the in-plane spin li fetimeτ+has a maximum when the Rashba and linear
+Dresselhausspin-orbitcouplingscanceleachother. Thega te-voltagedependenceofthespin-orbitcouplingin δ-doped
+InSb/In1−xAlxSbasymmetricquantumwellwascalculatedinRef.[637]. Itw asalsoshowninRef.[640]thatthespin
+lifetimescanbetunede ffectivelyviatheelectricfieldalongthegrowthdirection. S ystematicstudyonthedependence
+of spin lifetime on the electric field across the GaAs /AlGaAs quantum well at room temperature was performed by
+Lau and Flatt´ e [146]. From a 14-bandenvelope-functionapp roachthey calculated the Rashba and Dresselhausspin-
+orbit couplings as function of electric field for various wel l widths (see Fig. 18). They further calculated the spin
+lifetimeasfunctionofelectricfieldandreportedthatthee lectricfieldeffectisimportantatwidequantumwellwhere
+thesubbandwavefunctioniseasily a ffectedbytheelectric field. Againthespinlifetimealongthe [110]directionhas
+amaximumasfunctionofelectricfield. Fora7.5nmquantumwe llthecontributionoftheRashbaspin-orbitcoupling
+inspin relaxationexceedsthatofthe Dresselhausoneat 150 kV/cm. Forsuch narrowquantumwell, the Dresselhaus
+55Density dependence of spin splitting in such structure was m easured and compared with calculation in Ref. [639].
+47Figure 16: (a) Spin-lifetime τzand (b) measured Hall mobility µHallin experiments as a function of temperature T. In (a) three solid curves
+denote calculation with the mobility attributed to the thre e types of scattering (labeled as III, II and I in the figure.) T he first type of scattering
+(I):phonon scatterings dueto the deformation potential, s cattering by screened ionized impurities, neutral impurit ies, alloy and surface roughness;
+The second type of scattering (II): other kinds of phonon sca tterings; The third type of scattering (III): scattering by weakly screened ionized
+impurities. The dotted curves refer to the calculation in th e degenerate limit (zero temperature) whereas the dashed cu rves denote the calculation
+in the nondegenerate limite (the Boltzmann statistics). Fr om Kainz et al. [636].
+spin-orbit coupling varies little with electric field (up to 200 kV/cm), whereas the Rashba spin-orbit coupling varies
+linearlywithelectricfield. Similarresultswereobtained byYangandChang[156].
+•Nonlinear Rashba spin-orbit coupling to spin relaxation. Usually the multi-band envelope-functioncalcu-
+lation includesthe nonparabolice ffect in the spin-orbit coupling, e.g., at high energy the Rash ba spin-orbitcoupling
+deviates from linearity [146, 156]. The nonlinear e ffect in the Rashba spin-orbit coupling was studied comprehen -
+sively by Yang and Chang [156]. They proposed a model to chara cterize such nonlinear e ffect where the Rashba
+parameterissubstitutedby ˜ αR=αR
+1+ζk2,whereζisaparameterdependingonquantumwellstructureandmater ial. The
+modelfitswellwiththecalculation. Thenonlinearityismor epronouncedinnarrowband-gapsemiconductors. Within
+such model the spin relaxation as function of electric field, well width, density and the ratio αR/βDwas investigated
+comprehensively[156].
+•Magnetic field dependence. The magnetic field dependenceof the spin relaxation in two-d imensionalsystem
+with both the Rashba and Dresselhaus spin-orbit couplings w as discussed by Glazov [377].56The main results are
+thatthespinrelaxationtensorgivenbyEq.(107)isextende dtothecasewithmagneticfieldalongarbitrarydirection.
+A special case is that when the magnetic field is along the grow th direction of the two-dimensional structure, where
+Glazovgave
+1
+τz=4˜τ1k2α2
+R
+1+(ωL−ωc)2˜τ2
+1+β2
+D
+1+(ωL+ωc)2˜τ2
+1+˜τ3k6γ2
+D
+41
+1+(ωL−3ωc)2˜τ2
+3. (108)
+Spinlifetimeasfunctionofthedirectionofmagneticfieldw asdiscussedandcomparedwithexperimentaldata[377].
+Magnetic field effect on spin relaxation in the weak scattering regime where th e scattering frequency is comparable
+with the spin precession frequencywas studied by Glazov [38 0]. In such regime, spin polarization shows zero-field
+oscillationsduetothespin-orbitfieldinducedspinpreces sion. Glazovfoundthatthemagnetic-field-inducedcyclotr on
+rotation (which also rotates the k-dependent spin-orbit field) leads to fast oscillations of s pin polarization around a
+non-zero value and a strong suppression of spin relaxation [ 380]. Recent experiment confirmed such prediction in
+(001)quantumwellsandshowedthatthe e ffectsareabsentin (110)quantumwells asthespin-orbitfield isalongthe
+growth direction which does not lead to any spin precession [ 641]. Spin relaxation in the presence of both electric
+and magnetic fields was discussed theoretically by Bleibaum [378, 379].57In the presence of spin-orbit coupling
+56Similar study for spin relaxation in Si /Gequantum well with only the Rashba spin-orbit coupling was given by Tahan etal. [290].
+57This has been reported earlier by Weng et al. [569] from the ki netic spin Bloch equation approach (see next section).
+48Figure 17: Spin-relaxation rates 1 /τ+(solid curves), 1 /τz(dashed curves) and 1 /τ−(dotted curves) as function of electron concentration for
+Boltzmann electron gas in GaAs /AlAs heterostructure at temperature T=(1) 30, (2) 77,(3) 150 and (4) 300 K.From Averkiev et al. [635] .
+the electric field can induce an e ffective magnetic field due to electron drifting [569]. Spin re laxation is modified by
+such effective magnetic field and the real magnetic field due to the ind uced spin precession. The above e ffects are
+in the Markovian limit. Besides, there could emerge non-Mar kovian spin dynamics under magnetic field. Glazov
+and Sherman considered the case that the spin information ca n be stored in the closed orbits of cyclotron motion
+and transferred to the open orbits [381].58As the closed orbit averaged out the spin-orbit coupling, th e electron
+spin lifetime in the closed orbits can be much longer than tha t in the open orbits. In such a system, the fraction of
+spins in the open orbits soon decays, whereas the fraction of spins in the closed orbits decays slowly exhibiting a
+long-livedtail [381].59Thedecayofthe tail isgovernedbythescatteringprocesses whichtransferelectronsbetween
+closed and open orbits [381]. Based on a semiclassical model , the spin dynamics under weak /strong scattering and
+weak/strong magnetic-field was discussed [381]. Particularly, t hey showed that at strong magnetic field Balong the
+growthdirection,thetaildisappearsandthespinlifetime iselongatedas∼B3[381]. Underhighmagneticfieldinthe
+quantum Hall regime, spin relaxation was studied in Refs. [3 82–387].60It was shown that at zero temperature with
+fillingfactor ν=1,thespinrelaxesasymptoticallywithapowerlawratherth anexponential[384]. Thespinrelaxation
+is sensitive to the filling factor and temperature. Experime ntshave shown that via tuning the filling factor the e ffect
+of the hyperfine interaction can be manipulated [643]. As the magnetic field is high, the g-tensor inhomogeneity
+mechanism can be important [387]. In general, spin relaxati on in quantum Hall regime is quite di fferent from that
+in the classical regime. The spin-flip electron-phononscat tering plays important role. Besides, the electron-electr on
+interactionis crucialto thegroundstate aswellasto spind ynamics[384].
+•Weaklocalizatione ffectand others. Asin the D’yakonov-Perel’mechanism τs∼1/τp, theweak localization
+correction to mobility will lead to correction in spin relax ation as well. The weak localization correction to spin
+relaxationwasstudiedin Refs. [644–646]. Theclassical me moryeffect,whichexistswhenthecharacteristicscale of
+the disorder are comparable with the mean free path, leads to a non-Markovianspin dynamics with nonexponential
+tail∼1/t2for quantum wells with equal Dresselhaus and Rashba spin-or bit coupling strengths [647]. Pershin and
+PrivmanproposedthattheD’yakonov-Perel’spinrelaxatio nintwo-dimensionalsystemcanbesuppressedbyalattice
+ofantidots[648].
+•Spin relaxation in rolled-up two-dimensional electron gas .Spin dynamics in rolled-up two-dimensional
+electrongaswasinvestigatedbyTrushinandSchliemann[64 9]. Itwasshownthatthesymmetryofspin-orbitcoupling
+varies with the radius rolled-up structure [649]. At certai n radius, spin precession and relaxation of a special spin
+58Thenon-Markovian spindynamicscanalsoemergeininsulati ngregime,wherethenuclearspinsstoredthehistoricalele ctron spininformation.
+59As spins in the closed orbits oscillate dueto spin-orbit cou pling, the tail also oscillates.
+60Experimental studies can befound in Refs. [642, 643].
+49Figure 18: Left figure: Electric field F(along the growth direction) dependence of electron spin pr ecession vector for a 75 Å and a
+150 Å GaAs/Ga0.6Al0.4As quantum well at 300 K for di fferent energies E=10 meV (red solid curve), 20 meV (green dashed curve), 50 meV
+(blue dot-dashed curve), 80 meV (purple double-dot-dashed curve) and 100 meV (black dotted curve). x-component of the spin precession vector
+induced by the linear Dresselhaus term Ω(D)
+x,1(1,E) for quantum wells with well width (a) Lw=75 Å and (b) Lw=150 Å.x-component of the
+spin precession vector induced by the Rashba term Ω(R)
+x,1(1,E) for (c)Lw=75 Å and (d) Lw=150 Å quantum wells. x-component of the spin
+precession vector induced by the cubic Dresselhaus term Ω(D)
+x,3(1,E) for (e)Lw=75 Å.x-component of the spin precession vector induced by
+higher order (cubic) Rashba term Ω(R)
+x,3(1,E) for (f)Lw=75 Å (Note that the scale is two orders of magnitude smaller). Right figure: Electric
+fieldF(along the growth direction) dependence of electron spin re laxation time T1, dephasing time T2and dephasing rate 1 /T2forLw=50 Å,
+Lw=75 ÅandLw=150 Å GaAs/Ga0.6Al0.4As quantum wells at 300 K with mobility µ=800 cm2/V s. Note that the relative importance of the
+Dresselhaus (BIA)termandRashba(SIA)termtospinrelaxat ion asfunction oftheelectric field alongthegrowth directi on isdepicted in(f). From
+Lauand Flatt´ e [146].
+component is completely quenched, very similar to that in qu antum wells with αR=βD(consider only the linear- k
+spin-orbitcoupling)[649].
+•Crossover from two-dimension to one-dimension. An interesting problem is that how the spin relaxation
+varies with the channel width of the two-dimensional struct ure. This problem was investigated theoretically first
+by Mal’shukov and Chao [650] and later by Kiselev and Kim [651 , 652], showing that spin relaxation for some
+nonuniform distributed spin polarization and the uniform s pin polarization along certain direction (direction of the
+effective magneticof the linear spin-orbitcouplingwith kalong the unconstraineddirection)is suppressedwhen the
+channel width is smaller than the spin precession length [65 0–652]. Originally, these studies considered only the
+Rashba spin-orbit coupling. Recently Kettemann extended t he theory to include the Dresselhaus spin-orbit coupling
+(both linear and cubic terms) [653]. After that experimenta l studies on submicron InGaAs wires observed that spin
+lifetime first increases and then decreases with decreasing channel width (see Fig. 19) [654, 655]. The suppression
+of spin relaxationwas first explainedby the theory of Mal’sh ukovand Chao, which, however,is doubtfulas the spin
+polarizationdistributioncreatedbyopticalexcitationi snotofthe typepointedoutin theirwork. Detailed theoreti cal
+examinationindicateddi fferentexplanations[656].
+•Non-uniform system: Random spin-orbit coupling to spin rel axation. All the above studies are devoted
+to uniform system. For example, the Rashba spin-orbit coupl ing is considered as HR=αR(σxky−σykx) where
+αR=α0Ezis proportional to a uniform electric field along thegrowth d irection. However, in genuine system, such
+electric field can not be uniformdueto imperfection. Such im perfectioncan arise fromthe the fluctuationduringthe
+growth, makingEzposition-dependent: Ez→Ez(r∝ba∇dbl) withr∝ba∇dbl=(x,y). Spin relaxation due to this random Rashba
+spin-orbitcouplingcan be important when other spin-relax ationsources are ine ffective [294, 657]. This was studied
+comprehensively by Sherman and co-workers in Refs. [294, 65 7–659]. A particular case is the symmetric (110)
+quantumwells,wheretheaverageRashbaspin-orbitcouplin giszeroandtheDresselhausspin-orbitcouplingdoesnot
+lead to any spin relaxation due to symmetry. Glazov and Sherm an obtained that under weak and moderate magnetic
+field along the growthdirectionat rc≫ld(rcis the cyclotronradius and ldis characteristic length of randomdopant
+distribution)and αRkτp≪1 [658],
+τ−1
+z=4∝an}b∇acketle{t(δαR)2∝an}b∇acket∇i}htk2τd+4∝an}b∇acketle{tαR∝an}b∇acket∇i}ht2k2τp
+(1+ω2cτ2p), (109)
+50Figure19: (a)Faraday rotation (FR)at5Kforquantumwell(o pensquares)and750nmwirespatterned along[100](opencir cles) and[110](filled
+circles) asfunction ofdelay time ∆t. Black curves areguides totheeyes,andthedataareo ffsetforclarity. (b)Widthdependence wofspinlifetime
+τSPfor quantum wires patterned along [100] (open circles) and [ 110] (filled circles). The dotted line depicts the spin lifet ime of the unpatterned
+quantum well. Measurements were performed at B=0. From Holleitner etal. [654].
+whereδαR=αR−∝an}b∇acketle{tαR∝an}b∇acket∇i}htis the random Rashba coe fficient with∝an}b∇acketle{tαR∝an}b∇acket∇i}htrepresenting the average Rashba coe fficient.
+τd=ld/vk. TheZeemaninteractionis ignoredas ωL≪ωc. Itis seenthatboththerandomandtheaveragespin-orbit
+couplingsleadtospinrelaxationsimilartotheD’yakonov- Perel’mechanism. As ωcτd≪1(asrc≫ld),themagnetic
+fielddoesnotsuppressthespinrelaxationduetotherandomR ashbaspin-orbitcoupling. Asimpleestimationindicates
+thatin asymmetricallydopedquantumwell, the randomspin- orbitcouplingleadsto a spin relaxationrate two orders
+of magnitude smaller than that due to the averaged one, leadi ng to a spin lifetime of several tens of nanoseconds in
+GaAs quantum wells [294]. The spin relaxation due to the rand om Rashba spin-orbit coupling may be an important
+sourcefor(110)quantumwells[660]. At highmagneticfield, rc≃ld, thespinrelaxationexhibitsnonexponentialtail
+dueto thememorye ffectasspin inclosedorbitsdecaysslowly[658].
+•Decay of non-uniformly distributed spin polarization. The decay of a standing wave of spin polarizationin
+a two-dimensional system with only Rashba spin-orbit coupl ing was studied by Pershin [36]. It was found that the
+spin relaxation dependson the period of the standing wave. T he coherent spin precession of electronsmovingin the
+same direction was shown be responsible for such phenomena. In experimentssuch standing wave can be generated
+byspin-gratingtechnique[495]. Ina seriesof theoretical [18, 20,32, 34, 338]andexperimental[33, 37, 661] works,
+spin relaxation in spin-grating system was studied, where t he spin relaxation in such system is essentially related to
+thespindiffusionlimitedbytheD’yakonov-Perel’mechanism[18,32]. T hespinrelaxationinspin-gratingsystemor
+othernonuniformspindistributionswill bereviewedinSec tion6 andSec. 7.
+Spin relaxationin (110)quantum wells: experimentsandthe ories
+We now turn to spin relaxationin two-dimensionalstructure sgrownalong the [110]direction. The bulkDressel-
+hausspin-orbitcouplingbecomes
+HD=γD[(−k2
+x−2k2
+y+k2
+z)kz,4kxkykz,kx(k2
+x−2k2
+y−k2
+z)]·σ/slashbig2 (110)
+where the three axises are ex=1√
+2(1,−1,0),ey=(0,0,−1) andez=1√
+2(1,1,0). When only the lowest subband is
+considered,theeffectivespin-orbitcouplingis
+HSOC=γD[0,0,kx(k2
+x−2k2
+y−∝an}b∇acketle{tˆk2
+z∝an}b∇acket∇i}ht)]·σ/slashbig2+HR (111)
+inwhich∝an}b∇acketle{tˆk2
+z∝an}b∇acket∇i}htdenotestheaverageoverthelowestsubbandand HR=αR(σxky−σykx)istheRashbaspin-orbitcoupling.
+In symmetric two-dimensionalstructures HR=0. It is noted that the e ffective magnetic field is then along the [110]
+directionforall k. TheD’yakonov-Perel’spinrelaxationforspinpolarizati onalongthe[110]directionisthenabsent.
+Thequestionarises that what kindof mechanismis now respon siblefor spin relaxationalong the [110]direction. To
+51exploretheproblem,Ohnoetal. performedasystematicstud yofthedependenceofspinlifetimeonthecharacteristic
+parameters such as the subband quantization energy, electr on mobility and the temperature (see Fig. 20) [595, 662].
+Afteracarefulspeculationontheobserveddependencesofs pinrelaxationtime,theyconcludedthatthespinrelaxatio n
+in undoped(110)quantumwells is dominatedbythe Bir-Arono v-Pikusmechanism[595]. However,the temperature
+dependence of spin lifetime is anomalous and seems contradi ctory to the Bir-Aronov-Pikus mechanism: the spin
+lifetime increases with increasingtemperature[595, 602] . The anomalousincrease of spin lifetime with temperature
+can be understood as following: the dissociation of exciton s increases with temperature rapidly which reduces the
+electron-holeexchangeinteractionmarkedlyandsuppress estheBir-Aronov-Pikusmechanism[595].61Similareffect
+was also foundin ZnSe quantumwells [594]. However,the situ ation inn-dopedquantumwells is relatively obscure
+[595], although the observed mobility dependence suggests that the Elliott-Yafet mechanism may dominate the spin
+relaxationat roomtemperature(see Fig.20)[595].
+Figure 20: Left: Spin lifetime τsas well as carrier recombination time τrinn-doped GaAs (110) and (100) quantum wells as function of elec tron
+mobilityµat room temperature. Right: Temperature Tdependence of spin relaxation time τsin undoped (110) quantum wells. The inset shows
+theexcitation intensity Iexdependence of τsatroom temperature. From Ohno et al. [595].
+We now focus on the electron spin relaxation in n-doped symmetric (110) quantum wells. Let us reexamine the
+three spin relaxation mechanisms in symmetric (110) quantu m wells more carefully. The D’yakonov-Perel’mecha-
+nismleadsto aspinrelaxationtensor,
+τ−1
+z=0, τ−1
+x=τ−1
+y=∝an}b∇acketle{tτpβ2
+Dk2∝an}b∇acket∇i}ht/slashbig2 (112)
+whenonlythelinearspin-orbitcouplingtermisconsidered . FortheElliott-Yafetmechanism,usuallythein-planespi n
+relaxationrateislargerthantheout-of-planeone[111],b utthetwoaregenerallycomparable[111,370]. However,the
+Bir-Aronov-Pikus mechanism is isotropic [111]. Therefore the spin relaxation anisotropy will reflect the relevance
+of the Bir-Aronov-Pikus mechanism. The experimental inves tigation was first performed by D¨ ohrmann et al. by
+measuringspin decay time in the presence of an in-planemagn eticfield [388]. With an in-planemagnetic field, e.g.,
+B∝ba∇dbley, spindynamicsisgovernedbythefollowingequations,
+∂tSx=−Sx/τx+ωLSz, ∂tSz=−Sz/τz−ωLSx. (113)
+Thesolutionis
+Sz(t)=S(0)e−1
+2(τ−1
+x+τ−1
+z)tcos(ωt−φ)/cos(φ) (114)
+for2ωL>|τ−1
+z−τ−1
+x|. Here tanφ=τ−1
+x−τ−1
+z
+2ωandω=/radicalig
+ω2
+L−(τ−1x−τ−1z)2/4. The observedmagneticfield dependence
+is a step-function-like: for a moderate magnetic field B>0.5 T, spin lifetime changes from τzto 2/(τ−1
+x+τ−1
+z). The
+61This is also confirmed by the increase of photo-carrier lifet ime with temperature [663].
+52measured spin lifetimes at B=0 andB=0.6 T in GaAs (110) modulation n-doped quantum wells as function of
+temperatureareshowninFig.21.62Itisseenthatatlowtemperaturetheanisotropyisquitewea kwhichindicatesthat
+the spin relaxation is dominated by the Bir-Aronov-Pikus me chanism. With increasing temperature, spin relaxation
+anisotropyincreasesas the Bir-Aronov-Pikusmechanism be comesweaker and the D’yakonov-Perel’mechanismfor
+in-planespin relaxation growsstronger.63The decrease of τzat high temperature was explainedby the intersubband
+spin relaxationmechanism[388, 667]. Theintersubbandspi n relaxationmechanismcan be understoodas following:
+whenhighersubbandsareinvolved,thespin-orbitcoupling
+HD=γD[(−k2
+x−2k2
+y+k2
+z)kz,4kxkykz,kx(k2
+x−2k2
+y−k2
+z)]·σ/slashbig2 (115)
+canenableintersubbandspin-flipscatteringas thefirst two termscouplestates with di fferentspin andparity. At high
+temperature when the higher subbands are populated, the int ersubband spin relaxation mechanism can be important
+[388]. This mechanism also explains the mobility dependenc e of spin lifetime in Fig. 20. Recently, Zhou and Wu
+proposeda virtualintersubbandspinrelaxation,wherethe highersubbandsneednottobepopulated[668]. However,
+calculation indicated that both the real and virtual inters ubband spin relaxation mechanisms are ine ffective at low
+temperatureformodulation n-dopedquantumwellsasthescatteringissuppressed[388,6 68].
+The photo excitation of holes in the photoluminescence or Fa raday/Kerr rotation measurement makes the Bir-
+Aronov-Pikusmechanismbecomeinvolved. Thismasksthe int rinsic electronspin relaxationin n-doped(110)quan-
+tum wells at low temperature[388]. Recent advancementin sp in noise spectroscopymethod[497, 498, 669] enables
+probingspin relaxationwithoutphoto-carrierexcitation . The Bir-Aronov-Pikusmechanismis then removed,and the
+intrinsicspinlifetimecanbeapproached. Thismethodwasa ppliedtomodulation n-dopedGaAs(110)quantumwells
+by M¨ uller et al., where a much longer spin lifetime of τz=24 ns at low temperature(20 K) was obtained by careful
+analysis of the data [660]. The authors attributed the spin r elaxation at such low temperature to the random Rashba
+spin-orbit coupling mechanism [294, 670]. Later, theoreti cal calculation of the spin relaxation time limited by the
+randomRashba spin-orbitcouplingby Zhou and Wu via the full y microscopic kinetic spin Bloch equation approach
+[670]agreeswell withtheexperimentalresultsbyM¨ ullere t al. [660].
+Figure 21: Temperature dependence of spin lifetime τsforB=0 (filled circles) and B=0.6 T(open circles) in (110) GaAs quantum wells. Inset:
+Corresponding temperature dependence of spin relaxation a nisotropyγx/γz=τz/τx. From D¨ ohrmann et al. [595].
+Other investigations include the achievement of high tempe rature gate control of spin lifetime [193, 671, 672],
+which can be useful in spin switch devices or spin field-e ffect transistors. The typical results of spin lifetime as
+function of the electric field Ealong the growth direction are shown in Fig. 22. The low field d eviation of τz∼
+E2indicates the relevance of spin relaxation mechanism other than the D’yakonov-Perel’ one associated with the
+Rashba spin-orbit coupling. Bel’kov et al. studied the rela tion of symmetry to spin relaxation in (110) quantum
+62Similar temperature dependence of spin lifetime in (110) In GaAs quantum wells was studied in Ref. [664].
+63In InGaAs quantum wells larger spin relaxation anisotropy w as found [665, 666], which is due to the much stronger Dressel haus spin-orbit
+coupling in InGaAs.
+53wells [183, 663]. The symmetry of the quantum well was probed by the magnetic field induced photo-galvanic
+effect. At certain configuration, the photocurrent is proporti onal to the Rashba spin-orbit coupling coe fficient. The
+authorsobserved that photocurrentindeed vanishes for sym metric quantum wells. It was also observedthat the spin
+lifetime is longest in symmetric quantum wells. Therefore t he experiment demonstrated that the structure inversion
+asymmetry can be tuned down to zero. As only the Rashba spin-o rbit coupling contributes to the D’yakonov-Perel’
+spinrelaxation,the(110)quantumwellscanbeusedasagood platformtoinvestigatetheRashbaspin-orbitcoupling.
+Eldridgeetal. presentedanall-opticalmeasurementsofth eRashbaspin-orbitcoupling: bymeasuringthespinlifetim e
+via polarizedpump-probereflection techniqueand measurin gthe diffusion constant from spin-gratingmethod (from
+whichτpis extracted),they extractedthe Rashba coe fficientα0=(2mkBTτsτp)−1/2/(eE) from the D’yakonov-Perel’
+theory[179]. Atlowtemperature,theyfoundgoodquantitat iveagreementwiththe k·pcalculationandanunexpected
+temperaturedependence. Inundoped(110)quantumwells,El dridgeet al. discoveredthatthe asymmetryofthe band
+edge profile does notcontribute to the Rashba spin-orbit coupling when the elect rostatic potential is absent [178].
+These findings confirm the theory of Lassnig [148] (see Sec. 2. 3.3). Spin relaxation in (110) quantum wells under
+surfaceacousticwaveswasstudiedinRefs. [673, 674].
+Figure22: Measuredspinrelaxationrate vs.biasvoltageandcorrespondingelectricfield(opencircles )comparedwithcalculation forasymmetrical
+quantum well (filled circles). T=170 K.FromKarimov et al. [193].
+Theoretical investigation on spin relaxation in (110) quan tum wells is few up to now [640, 668, 670, 675–678].
+Among these works, Wu and Gonokami first proposed that the D’y akonov-Perel’ mechanism can be e ffective for
+relaxation of spin pointing along growth direction when an i n-plane magnetic field is exerted [675]. Later it was
+shown that the in-plane spin relaxation rate can be tuned by s train which can be induced by the mole fraction of
+gallium in an InGaAs /InP quantum well [676]. E ffect of electric field across quantum well on spin relaxation w as
+studiedinRef.[640]. Recently,ZhouandWuproposedavirtu alintrasubbandspin-flipelectron-phononandelectron-
+impurity scattering due to the intersubband spin-orbit cou pling [668]. Tarasenko gave the spin relaxation tensor in
+asymmetric(110)quantumwells with a finite Rashba spin-orb itcoupling[677]. He foundthat in the presenceof the
+Rashba spin-orbit coupling, the decay of electron spin init ially oriented along the growth direction is characterized
+by two spin lifetimes. Glazov et al. proposed a symmetric mul tiple (110) GaAs quantum well structure to suppress
+the spin relaxation due to the random Rashba e ffect [678]. In such structure, the donor Coulomb potentials s een
+by electrons in the central quantum well is largely screened by the electrons in other quantum wells, which hence
+strongly suppresses the random Rashba spin-orbit coupling and spin relaxation [678]. In this structure, spin-flip
+scatteringbetweenelectronsindi fferentquantumwells, however,leadstoanadditionalspinre laxation. Thespin-flip
+inter-well electron-electron scattering is comparable wi th the random Rashba spin-orbit coupling mechanism in the
+non-degenerateregime,butissuppressedinthe degenerate regime[678].
+Spin relaxationin (111)quantum wells
+Besides (110) quantum wells, spin relaxation in (111) quant um wells also attracted much attention due to its
+particular symmetry. In (111) quantum wells the linear Dres selhaus spin-orbit coupling is of the same form of the
+54Rashbaone. Thelinear- kspin-orbitcouplingtermisthen
+HSO,1=αIA(σxky−σykx), (116)
+whereαIA=αR+2γD∝an}b∇acketle{tˆk2
+z∝an}b∇acket∇i}ht/√
+3. Thecubic- ktermreads
+HSO,3=γD
+2√
+3/bracketleftig
+k2(−kyσx+kxσy)+√
+2(3k2
+x−k2
+y)kyσz/bracketrightig
+. (117)
+Thespin lifetimesarethen
+τ−1
+x=τ−1
+y=/angbracketleftig
+k2˜τ1/bracketleftig
+12α2
+IA−4√
+3γDαIAk2+(1+2˜τ3/˜τ1)γ2
+Dk4/bracketrightig/slashig
+6/angbracketrightig
+, (118)
+τ−1
+z=/angbracketleftig
+k2˜τ1(γDk2−2√
+3αIA)2/slashbig3/angbracketrightig
+. (119)
+It was first proposed by Cartoix` a et al. [340, 679] that by tun ing the gate-voltage, the condition γD∝an}b∇acketle{tk2∝an}b∇acket∇i}ht=2√
+3αIA
+can be achieved, where spin relaxation is suppressed for all spin components (see Fig. 23): τz=∞andτx=τy=
+3/(γ2
+D∝an}b∇acketle{tk6∝an}b∇acket∇i}ht˜τ3). Microscopic calculation using eight-band k·pmethod indicated the feasibility of such scheme in an
+InAlAs/InGaAs/InP quantum well [680]. Similar scheme based on tuning spin- orbit coupling via strain was also
+proposed[676].
+Figure 23: Calculated spin lifetime for (111) quantum wells as function of−αSIA/αBIA(αSIA=αR,αBIA=2γD∝an}b∇acketle{tˆk2
+z∝an}b∇acket∇i}ht/√
+3). From Cartoix` a et al.
+[679].
+Spin relaxationin arbitrarilyorientedquantum wells
+Accounting only the linear- kDresselhaus spin-orbit coupling, spin relaxation tensor f or an arbitrarily oriented
+quantumwell reads
+τ−1
+ij=(δijTrˆν−νij)/τ0
+s(εk). (120)
+Hereτ0
+sis the in-planespin lifetime for (001)quantumwell. The ten sor ˆνdependson the orientationof the quantum
+welln=(nx,ny,nz)64as[369]
+νxx=4n2
+x(n2
+y+n2
+z)−(n2
+y−n2
+z)2(9n2
+x−1), ν xy=nxny[9(n2
+x−n2
+z)(n2
+y−n2
+z)−2(1−n2
+z)],(121)
+withothercomponentsobtainedbycyclicpermutationofthe x,yandzindexes.
+Spin relaxationin (sub)-monolayers: experiments
+Spin relaxation in undoped sub-monolayerand monolayer InA s structures grown in GaAs matrix was studied in
+64Thex,yandzaxis are taking along the [100], [010] and [001] directions r espectively.
+55Refs. [681, 682]. The monolayer structure on (001) surface c an be regarded as an ideal two-dimensional system.
+However,the(311)orientedmonolayerformswire-likeordi sk-likemicrostructuresonGaAsstepsandfacets. Insub-
+monolayer InAs structures, such as 1 /3 monolayer and 1 /2 monolayer, InAs was found to be organized as disk-like
+islandswithlateralsizeoftensofnanometers. However,th ecarriersystemisstilloftwo-dimensionalnaturebutwith
+lower density of states and smaller mobility. It was then fou nd that the spin relaxationis suppressed by reducingthe
+layerthickness(seeFig.24)[681]. Theboundariesanddefo rmationpotentialsareenhancedwithdecreasedcoverage,
+whichleadstothedecreaseofmomentumscatteringtimeands uppressesspinrelaxation[681]. Formonolayerstruc-
+tures, the (311) structure has a much longer spin lifetime th an the (001) one as the surface roughnessis much larger
+in the former [681]. The g-factor and spin dephasing time were also measured as functi on of excitation density for
+thesestructures[681]. Thespindephasingtimedecreasesw ith excitationdensityforallthe 1 /3,1/2and1monolayer
+structures[681]. Thedecreaseofspinlifetimemaybedueto theenhancementoftheD’yakonov-Perel’mechanismas
+∝an}b∇acketle{tk6∝an}b∇acket∇i}htincreasesorduetotheenhancementoftheBir-Aronov-Pikus mechanismastheholedensityincreases. However,
+via examination of the temperature dependence of spin lifet ime, Yang et al. [681] found that the Bir-Aronov-Pikus
+mechanismcanonlybeimportantatlowexcitationdensityin 1/3monolayerstructureduetothestrongelectron-hole
+exchange interaction between the spatially confined electr ons and holes [682]. The temperature dependence of spin
+lifetime at higher excitation density exhibits a peak [681, 682], which signals the D’yakonov-Perel’ spin relaxation
+associated with the electron-electron scattering [110, 37 2]. Finally, the density dependence of g-factor was used to
+analysistheelectronicdensityofstatesinthese structur es[681].
+Figure 24: (a) Electron-spin dephasing time T∗
+2and (b) effectiveg-factor|g∗|for 1/3 monolayer (ML), 1 /2 ML and 1 ML InAs on (100) GaAs
+substrates with various carrier densities. Thetemperatur e is77 K. FromYang et al. [681].
+Spin relaxationin parabolicquantum wells: experiments
+A good candidate for spintronic device structure is the para bolic quantum wells, where the g-factor and spin
+lifetime can be tunede fficientlyby electrical means[67, 78, 683]. Recently,the Ras hba andlinear Dresselhausspin-
+orbit couplings in parabolic quantum wells were measured by monitoring the spin precession frequency of drifting
+electrons via time-resolved Kerr rotation [67]. It was foun d that the Rashba spin splitting can be tuned significantly
+by the gate biases, whereas the Dresselhaus spin-orbit coup ling varies only weakly. The spin relaxation was then
+tunedby gate-voltage(see Fig. 25). It was observedthat the anisotropyof spin relaxationvanishes when the Rashba
+spin-orbitcouplingistunedtozero.
+Spin relaxationin II-VI semiconductortwo-dimensionalst ructures: experiments
+SpinrelaxationinII-VIsemiconductortwo-dimensionalst ructureshasalsobeenwidelystudied. Thefirstremark-
+ableadvancementisthat thespin lifetimecan beincreasedb yseveralordersofmagnitudevia n-typedopingin ZnSe
+quantumwells[584]. Itwasobservedthatthespinlifetimei sontheorderofnanosecondsandisonlyweaklytemper-
+aturedependent[584]. The spin lifetime in undopedZnSe qua ntumwells is on the orderof 10ps at low temperature
+56Figure25: (001)parabolicquantumwell. (a)Leftpanel: the Rashbaspin-orbitcoupling coe fficientαasafunction ofthefront-gate: FG(back-gate:
+BG)voltagewithfixedBG(FG)voltage( △:VFG=0.1V;×:VFG=−0.5V;◦:VBG=0.1V;/square:VBG=−0.5V).Rightpanel: αasafunction ofthe
+external electric field Eext. The red line represents a least-squares fit. (b) Out-of-pla ne spin relaxation rate 1 /τz=1/T∗
+2(θ=90)+1/T∗
+2(θ=180).
+(c) In-plane spin-dephasing asymmetry 1 /T∗
+2(θ=90)−1/T∗
+2(θ=180).θ=90 (θ=180) denotes that the external magnetic field is along [110]
+([¯110]) direction. Thedashed line marks thegate voltage wher eα=0. From Studer etal. [67].
+[584,594]. Inundopedquantumwellsthespinlifetimewasob servedtoincreasefromlessthan10psat20Kto500ps
+at 200 K [594]. As the exciton binding energy is large (19 meV) , at low temperature spin dynamics is governed by
+theexcitonbehavior. Experimentalexaminationontherela tionbetweenspinlifetimeandmomentumlifetimereveals
+a motionalnarrowingnature τs∼τ−1
+p, which coincideswith the picture of Maialle et al. [118]: th e electron-holeex-
+changeinteractionservesasane ffectivemagneticfielddependingonexcitonmomentum. Themom entumdependent
+spin precession leads to a spin relaxation similar to the D’y akonov-Perel’one for electron spin. At high temperature
+the spin relaxation is further suppressed with increasing t emperature due to the ionization of excitons which weak-
+ens the electron-hole exchange interaction [594]. The spin relaxations of electron, exciton and trion were studied in
+n-doped single CdTe quantum wells with di fferent doping density [684]. It was found that the exciton spi n lifetime
+(18-36 ps) is much shorter than the electron one ( ∼180 ps) and the trion spin relaxation is governed by the fast h ole
+spin relaxation [684]. Spin relaxation in (110) ZnSe quantu mwells was studied in Ref. [685] where behaviorsquite
+different from that in GaAs (110) quantum wells were found. Spin l ifetime was reported to decrease monotonically
+with increasingtemperaturein CdTe quantumwells [535]. Th eelectrondensity dependenceof spin lifetime in CdTe
+quantumwellswasfoundto benonmotonic: it exhibitsapeaka t 8×1010cm−2forT=5K [686].
+4.2.4. Electronspinrelaxationin p-typeIII-VandII-VIse miconductortwo-dimensionalstructures
+Thetwomainelectronspinrelaxationmechanismsin p-typetwo-dimensionalstructuresaretheD’yakonov-Perel ’
+andtheBir-Aronov-Pikusmechanisms. Itisbelievedthatat lowtemperatureand /orhighholedensitytheBir-Aronov-
+Pikus mechanism dominates. The Elliott-Yafet mechanism ma y also dominate spin relaxation in heavily p-doped
+narrow bandgap semiconductors at very low temperature wher e the Bir-Aronov-Pikus mechanism is suppressed by
+Pauliblocking[117]. TheD’yakonov-Perel’mechanismdete rminesspin relaxationin otherregimes.
+Experimentalinvestigationsonelectronspinrelaxationi np-typequantumwellsarescarce. Ina6nm p-modulation-
+doped GaAs multiple quantum well with hole density nh=4×1011cm−2, Damen et al. measured an electron spin
+lifetime of 150 ps at 10 K [687]. In δ-dopeddoubleheterostructureswith a dopingdensity 8 ×1012cm−2, extremely
+long spin lifetimes up to 20 ns was observed at 6 K and a decreas e of spin lifetime with temperature τs∼T−0.6was
+found for T=6-60 K (see Fig. 26) [688]. The extremely long spin lifetime w as explained as suppression of the
+electron-holeexchangeinteraction by spatially separate d electronsand holes in the δ-dopeddouble heterostructures.
+57However,the D’yakonov-Perel’spin relaxationshouldalso be suppressedto achievesuchlong spin lifetime, i.e., the
+momentum scattering should be strong in the structure.65Later Gotoh et al. measured spin relaxation in a device
+where the spacial separation between electrons and holes ca n be tuned by a gate-voltage at room temperature [689].
+They showed that the spin relaxation is enhanced when the spa cial electron-hole separation is shortened. From the
+observed results, they concluded that electron spin relaxa tion is dominated by the Bir-Aronov-Pikus mechanism in
+their structures. However, the gate-voltage also modifies t he Rashba spin-orbit coupling, theses e ffects should also
+be taken into account for a close examination. The temperatu re dependence of spin relaxation at low temperature
+(T≤60 K) in p-modulation-doped GaAs quantum wells was investigated in R efs. [690–692] where a decrease of
+spin lifetime was found and the spin relaxation varies from 6 00 ps to 40 ps. Energy-resolved spin dynamics at the
+sufaces of p-GaAs was studied by Schneider et al. using time- and spin- re solved two-photon photoemission [693].
+Thespin lifetime was foundto increasewith decreasingelec tronkinetic energyin agreementwith the corresponding
+theory[390]. Theyalso observeda suppressionof spin relax ationin (001)surfaceswhere holedensity is reducedby
+aboutanorderofmagnitudeduetotheband-bendinge ffect.
+Figure 26: Left: Temperature dependence of the electron-sp in lifetimeτspand luminescence recombination time τrec. The solid line indicates a
+T−0.6power law, the dashed line is drawn as a guide to the eyes. Righ t: Self-consistent potential profile of the structure inves tigated. Subband
+energies and probability densities at k∝ba∇dbl=0 are also shown. “hh” and “lh” denote heavy-hole and light-h ole respectively; “e1” denotes the first
+electron subband. The original structure is a 60 nm wide GaAs /Al0.33Ga0.67As quantum well, where it is δ-doped in its center with Be ( p-type).
+FromWagner etal. [688].
+Theoretically, the role of electron and hole spin relaxatio n dynamics in time-resolvedluminescence spectral was
+studiedbyUenoyamaandSham[694,695]. Theultrafastcarri erandspindynamicswastheninvestigatedfromafully
+microscopickineticspinBlochequationapproach[44,334, 350]includingalsothecarrier-carrierCoulombscatterin g
+byWuandMetiu[350]. CalculationoftheBir-Aronov-Pikuss pinrelaxationinexcitonicsystemandlaterinelectron-
+hole plasma was performed by Maialle et al. [118, 390, 391] wh ere the energy-dependentspin relaxation time and
+theeffectsofthevalence-bandspinmixingaswellastheelectricfi eldalongthegrowthdirectionwerestudied. These
+studies found that (i) spin lifetime decreases with electro n kinetic energy (see Fig. 27); (ii) the valence-band spin
+mixingcanleadtoafactoroftwocorrection;and(iii)spinr elaxationduetotheBir-Aronov-Pikusmechanismcanbe
+tuned substantially by gate-voltage. The relative importa nce of the Bir-Aronov-Pikus, Elliott-Yafet and D’yakonov-
+Perel’ mechanisms was also compared in GaAs quantum wells [3 90]. Maialle suggested that the Bir-Aronov-Pikus
+mechanism is more important than the D’yakonov-Perel’ mech anism at medium energy in contrast to that the Bir-
+Aronov-Pikusmechanismismoreimportantat lowenergyinbu lkGaAsfromthesame calculation(seeFig.27).
+65Theelectron-impurity scattering istheonlypossible cand idate tosuppresstheD’yakonov-Perel’ spinrelaxation int heexperimental condition,
+as other scatterings are limited atsuch low temperature and low electron density.
+58Figure 27: Left: Electron spin lifetime τsdue to the Bir-Aronov-Pikus mechanism in bulk GaAs and GaAs q uantum wells as function of the
+electron kinetic energy for di fferent hole densities Nas labeled in the figure. The dotted curves with open triangle s are the same as the solid
+curves butincluding the Sommerfeld factor. Right: Calcula ted electron spin lifetimes τsdueto the D’yakonov-Perel’ (DP), Elliott-Yafet (EY)and
+Bir-Aronov-Pikus (BAP)mechanisms as functions of theelec tron kinetic energy. Thespin lifetimes dueto the D’yakonov -Perel’ and Elliott-Yafet
+processes are taken from Ref. [436], whereas the Bir-Aronov -Pikus (BAP) spin lifetimes are calculated by assuming both nondegenerate holes
+(label in thefigureas“BAP”) and degenerate holes (label in t hefigureas“BAP deg.”) with (a)hole densities N=4×1018cm−3in bulk GaAsand
+(b)N=1012cm−2in GaAsquantum wells with di fferent well widths L. FromMaialle [390].
+4.2.5. ElectronspinrelaxationinIII-VandII-VIsemicond uctorone-dimensionalstructures
+There are much less works on electron spin relaxation in one- dimensionalstructures compared with the electron
+spinrelaxationintwo-dimensionalstructures. Itisbelie vedthattheD’yakonov-Perel’mechanismisthemoste fficient
+one, unless it is suppressed in certain geometry. Due to the a dditional constrain, the spin-orbit couplings in one-
+dimensionalsystemisquitedi fferentfromthoseinthetwo-dimensionalcase. Bydenotingth eunconstraineddirection
+as ˆz,the Rashbaspin-orbitcouplingforthelowest subbandisth en
+HR=α0|e|−1/bracketleftig
+σx∝an}b∇acketle{t∂yV(x,y)∝an}b∇acket∇i}ht−σy∝an}b∇acketle{t∂xV(x,y)∝an}b∇acket∇i}ht/bracketrightig
+kz, (122)
+whereV(x,y)is the electrostaticpotentialand ∝an}b∇acketle{t...∝an}b∇acket∇i}htstandsfor theaverageoverthe lowest subbandwavefunction . The
+Dresselhausspin-orbitcouplingdependsonthegrowthdire ctionofthequantumwire. If ˆ z∝ba∇dbl[001],then
+HD=γD∝an}b∇acketle{t(ˆk2
+x−ˆk2
+y)∝an}b∇acket∇i}htσzkz. (123)
+One notices that spin precession direction is the same for all kz. Such symmetry leads to an infinite spin lifetimes
+forspinpolarizationalongthe spinprecessiondirection. However,spinlifetime in otherdirectionsare still finite. By
+properarrangementofthegrowthdirectionandtheconfineme nt,theRashbaandDresselhausspin-orbitcouplingscan
+be cancelled, where relaxation for all the spin componentsd ue to the D’yakonov-Perel’mechanism can be inhibited
+[696,697].
+Experimentally, spin relaxation in undoped rectangular Ga As/AlAs quantum wires with small ( <20 nm) lateral
+sizeswasstudiedviatime-resolvedphotoluminescence[69 8]. ThemeasuredspinlifetimeisshowninFig.28together
+with the spin lifetime in quantumwells for comparison. One fi nds that spin lifetime in the narrower(12 nm ×12nm)
+quantum wire can be largerthan that in the wider one (19 nm ×13 nm). This is in contrast to the fact that both the
+D’yakonov-Perel’ and Bir-Aronov-Pikus mechanisms increa se with the confinement, as also indicated by the spin
+lifetimeinquantumwellsfordi fferentwell widths. Thetemperaturedependenceof thewidequ antumwireissimilar
+to that in quantum wells, which indicates the multi-subband effect [696]. In narrow quantum wire, the temperature
+dependenceisnonmonotonic,possiblyindicatingthee ffectoftheelectron-holeCoulombscattering[109, 110]. Wir e
+width dependenceof the spin relaxationin wide quantumwire s was studied in Refs. [654, 655], where the crossover
+fromtwodimensiontoonedimensionwasdiscussed. Transpor tstudiesofspinlifetimewas reportedinRef. [699].
+Theoretically, the study of spin relaxation in quantum wire s focused on the D’yakonov-Perel’mechanism. Pra-
+manik et al. studied spin relaxation in a wide quantum wire (4 nm×30 nm) using a semiclassical approach (via
+59Figure 28: Temperature dependence of the spin relaxation ti me in quantum wires and quantum wells with various confinemen ts (as labeled in the
+figure). FromSogawa et al. [698].
+the Monte Carlo simulation) [701]. In their work spin relaxa tion was found to be anisotropic as analyzed above.
+They showed that electric fields which drive electrons to hig hkstates lead to faster spin relaxation [701]. Later
+Dyson and Ridley studied electron spin relaxation in quantu m wires due to the electron-longitudinal-optical-phonon
+scattering [632]. Unlike previous studies concerning the e lectron-longitudinal-optical-phononscattering, no ela stic
+scatteringapproximationwasmade. Theygaveanalyticalex pressionsfortheenergydependentscatteringtimewhere
+the inelastic nature of the collision was fully taken into ac count. They found that the spin relaxation rate increases
+with confinement asymmetry as indicated by Eq. (123) where th e Dresselhaus spin-orbit coupling is proportionalto
+∝an}b∇acketle{t(ˆk2
+x−ˆk2
+y)∝an}b∇acket∇i}ht. Spin lifetime due to the electron-longitudinal-optical- phononscattering was also found to increase with
+temperature [632]. In the presence of both the Rashba and Dre sselhaus spin orbit couplings, the spin relaxation be-
+comesanisotropic. Consideraspecialcase,wherethestren gthoftheRashbaspin-orbitcouplingisequaltothatofthe
+linear Dresselhaus spin-orbit coupling: In two-dimension al case, this leads to a zero spin-splitting for kalong [110]
+direction,whereasamaximumspin-splittingfor kalong[1¯10]direction. Bysupperimposingadditionalconstrainmen t
+toformquantumwireswiththegrowthdirectionalong[110]d irectionandkeepingtheconditionthatthestrengthsof
+the Rashba and linear Dresselhaus spin-orbit couplings are equal, it is easy to understand that the spin-orbit field is
+zeroand spin relaxationis inhibited. Such conditionholds for narrowquantumwires whereonly the lowest subband
+is relevant. Recently, Liu et al. showed that it also holds fo r quasi-one-dimensional quantum wires where the wire
+width is comparable to the spin precession length or mean fre e path (see Fig. 29) [697]. The underlying physics is
+that electrons with a large transverse momentum component t o the wire orientation almost do not contribute to the
+spin-dephasingbecause of motional narrowingas they su ffer strongboundaryscattering. Only electronswith a large
+momentum component parallel to wire orientation contribut e significantly to spin relaxation. Hence spin relaxation
+depends largely on the wire orientation for equal Rashba and Dresselhaus spin-orbit coupling strengths when the
+cubic Dresselhaus term is neglected66(see Fig. 29) [697]. The authors also considered the realist ic conditions of
+imperfectorientationalong[110]direction,thecubicDre sselhaustermandinequalityoftheRashbaandlinearDres-
+selhaus spin-orbit coupling strengths [697]. Recent exper imental results confirmed the above theoretical prediction s
+(see Fig. 30) [700]. In the experiment, the strength of the Ra shba and linear Dresselhaus spin-orbit coupling is not
+equal. From the wire orientation dependence, the authors es timated that the ratio of the strengths of the Rashba and
+linearDresselhausspin-orbitcouplingisabout2.
+4.2.6. Holespinrelaxationin metallicregime
+Becauseofthecomplexityofthevalencebands,holespinrel axationisquitedifferentfromtheelectronspinrelax-
+ation. In bulkIII-VandII-VIsemiconductors,dueto the str ongspin-orbitcouplingand theheavy-light-holemixing,
+66Note that the dependence of the spin relaxation on quantum wi re orientation was also shown by Holleitner et al. [654, 655] where a shorter
+spin lifetime in wires along [110] direction than that in wir es along [100] direction was found. This may be because of an o pposite sign of the
+Rashba spin-orbit coupling in those wires according to the t heory by Liu et al. [697].
+60Figure 29: The spin-dephasing time T∗
+2of a spin ensemble is
+plotted as a function of the wire orientation with respect to the
+[100] lattice direction. The spin polarization is initiall y oriented
+along the [001] direction. T∗
+2is strongly enhanced for a quasi-
+one-dimensional wire oriented in the [110] direction ( black) as
+compared to an ensemble in a two-dimensional system ( gray). In
+the calculation, the Rashba and Dresselhaus spin-orbit cou pling
+strengths are taken to be equal, and the cubic Dresselhaus te rm is
+ignored. Thetwoupperinsetsshowtheresultsnearthetwope aks.
+The lower inset shows the evolution of the spin polarization for
+variouswireorientation (withrespecttothe[100]directi on). From
+Liuet al. [697].
+Figure 30: Spin-dephasing time τ1measured via Kerr rotation
+in quasi-one-dimensional structures with di fferent wire orienta-
+tion together with spin relaxation in two-dimensional elec tron gas
+(2DEG).FromDenega et al. [700].
+the D’yakonov-Perel’ spin relaxation is very e fficient, yielding very short spin lifetime τs∼100 fs [207, 210]. In
+nanostructures,the heavy-light-holedegeneracyat k=0 is lifted and the e ffect of the spin-orbitcouplingis reduced.
+Spin relaxation is hence slowed down. In quantum wells, hole spin relaxation is in the picosecond regime, usu-
+ally longer at low temperature [400, 445, 446]. Hole spin rel axation in nanostructures is more complicated, as hole
+spin-orbit coupling and hole subband structure can vary lar gely with the geometry (e.g., size and growth direction)
+of the nanostructures [113]. Besides the D’yakonov-Perel’ mechanism, other mechanisms may also be important.
+For example, consider a unstrained symmetric p-type quantum well with only the lowest heavy-hole subband b eing
+relevant. The Rashba spin-orbit coupling vanishes as the st ructure is symmetric. The Luttinger Hamiltonian only
+induceshole-spin-mixingbut no zero-field spin splitting i n the lowest heavy-holesubband due to its space-inversion
+symmetry. The zero-field spin splitting can only originate f rom the Dresselhaus spin-orbit coupling, which is much
+weaker than the spin-orbit couplingin bulk system due to the LuttingerHamiltonian. The D’yakonov-Perel’mecha-
+nismisthenlargelysuppressed. Ontheotherhand,thespin- mixinginducedbytheLuttingerHamiltonianleadstoan
+effective“spin”-flip scatteringassociated with anymomentum scattering (the Elliott-Yafet-typemechanism),such as
+hole-phononscattering [694, 695, 702–704]. Such “spin”-fl ip processes can be very e fficient, especially near a sub-
+band anti-crossing point [705]. In general both the D’yakon ov-Perel’ and Elliott-Yafet-type mechanisms should be
+considered. FerreiraandBastardcomparedthetwo spinrela xationmechanismsandfoundthatthe D’yakonov-Perel’
+mechanism is usually more important in asymmetric quantum w ells [706]. Finally, in heavily n-doped samples, the
+electron-holeexchangeinteraction(theBir-Aronov-Piku smechanism)mayalsoberelevant.
+Belowwefirstreviewholespinrelaxationinbulkmaterials. ExperimentsinundopedGaAsrevealedthatholespin
+lifetime is about100 fs at room temperature[207].67Thereticalcalculationof the ultrafast nonequilibriumdy namics
+under optical excitation including the spin-orbit couplin g as well as hole-phononand hole-hole scatterings repeated
+theobservedresults[210]. Nearthe equilibrium,calculat ionreporteda spin lifetimeabout200fsfor heavy-holeand
+less than 100fs for light-holeat roomtemperature[209]. Ho le spin relaxationwas also consideredin the framework
+of non-Markovian stochastic theory [708]. Spin lifetime of theΓ7band hole in wurtzite GaN was measured to be
+120 ps [709], much longer than that in GaAs. This is because un like the zinc-blende structures, valence bands in
+67Hole spin relaxation in split-o ffbands was studied by Kauschke et al. [707].
+61Figure 31: Pump-probe measurements of hole spin dynamics in bulk undoped GaAs at di fferent probe wavelengths 3200 nm (top), 3000 nm
+(middle) and 3800 nm(bottom). T: transmision,∆T: differential transmision. Di fference in differential transmision probed by right ( ∆T+)and left
+(∆T−) circular polarizations ( ∆T+−∆T−)/Tis related to hole spin polarization. From Hilton and Tang[2 07].
+wurtzitesemiconductorsaresplittedintothree separated bands,Γ9,Γ7andΓ7′. Thebandseparationreducesthee ffect
+ofthespin-orbitcouplingandsuppressestheD’yakonov-Pe relspinrelaxation.
+Holespinrelaxationinbulksemiconductorunderstrainwas studiedbyD’yakonovandPerel’[710]. Itwasfound
+thattheholespinalongthestrainaxisrelaxesmuchslowert hantheunstrainedsamples. Thisisbecausestrainliftsthe
+heavy-holeandlight-holedegeneracyat Γpointandlargelyreducesthespin-orbite ffectinbothheavy-andlight-hole
+bands.
+Morestudiesweredevotedto holespinrelaxationintwo-dim ensionalstructures. Experimentally,Ganichevet al.
+measured hole spin relaxation by spin-sensitive bleaching of intersubbandhole spin orientation in a p-type quantum
+wellwithwellwidth Lw=15nm. Atlowtemperature( T<50K)holespinlifetimevariesas ∼T−1/2forholedensity
+=2×1011cm−2[711]. Subsequent studies via the same method extended the d ata toT<140 K and to different
+well widths (see Fig. 33) [712]. Strikingly, spin relaxatio n is more efficient in wider quantum wells, in contrast to
+the inversetendencyin electroncase. Thisisa specific feat ureoftwo-dimensionalhole systemswherethe spin-orbit
+couplingisdeterminedbyheavy-lightholemixing,whichis strongerinwiderquantumwells[seeEq.(22),theRashba
+coefficient is proportional to the inverse of the hole subband spli tting, i.e., the Rashba coe fficient decreases with
+increasingwell width][206]. Minkovet al. foundthatthe ho lespinlifetime (deducedfromthe weakantilocalization
+measurements)decreasesrapidlywith holedensity at veryl ow temperature T=0.44K for3<nh<10×1011cm−2
+butvariesslowlywithtemperaturefor T<5Katnh≃8×1011cm−2inp-InGaAs/GaAsquantumwells[713]. Thisis
+consitent with the D’yakonov-Perel’spin relaxationin deg enerateregime, where temperature(hole density) changes
+both hole distribution and hole-momentumscattering margi nally(markedly). In undoped1.5 monolayerInAs /GaAs
+quantumwellat10K,Lietal. observedlongholespinlifetim e(τh
+s≃200ps)whichcanbecomparabletotheelectron
+spin lifetime τe
+s≃350 ps in the same structure [714]. This may be because that th e hole spin-orbitcoupling is weak
+in such ultrathin layers, as hole spin-orbit coupling decre ases with quantum well width [206]. Interestingly, it was
+found that the hole spin lifetime has a peak in the temperatur e dependence [715], which resembles the temperature
+dependenceofelectronspinlifetimeinsimilarstructure[ 681,682]. ThisisconsistentwiththeD’yakonov-Perel’spi n
+relaxationduetothe hole-holeCoulombscattering[299]. B aylac et al. observedthat theholespin lifetimedecreases
+with photo-excitation energy at very low temperature (1.7 K ) [716]. The result was explained by the fact that hole
+spinrelaxationrateincreaseswithholekineticenergy[71 6]. BoththeD’yakonov-Perel’andElliott-Yafetmechanism s
+givesuch dependence[363, 702, 706]. Ina designeddoublequ antumwell structure,Lu et al. demonstratedthat hole
+spin relaxation in a narrow well with width Lw=4.5 nm at room temperature can be suppressed ( τh
+s≃100 ps) if
+electrons tunnel out to an adjacent quantum well rapidly [61 1]. This gives an evidence that the Bir-Aronov-Pikus
+62mechanism plays important role in hole spin relaxation. Hol e spin relaxation time in n-doped quantum wells was
+first measured by Damen et al., finding that τh
+s≃4 ps at 10 K [687]. Long hole spin lifetime up to nanoseconds
+was observed in later experiments in n-doped quantum wells at low temperature [393, 394, 445, 717, 718] which is
+assigned to localized holes. Hole spin relaxation in n-type quantum wells was reported to decrease with magnetic
+field[717]whichisbecausethatthe g-tensorinhomogeneitymechanismisimportantintwo-dimen sionalholesystem
+[399]. Also hole spin lifetime decreases drastically with i ncreasing temperature in n-type quantum wells [393, 445]
+(see Fig. 32) as spin relaxation rate increases with increas ing hole kinetic energy. Hole spin relaxation in type-II
+quantum wells was studied by Kawazoe et al. [719] where two sp in decay componentswere observed: a faster one
+withτh
+s∼20-100ps and a slower one with τh
+s∼20 ns. The slower decaywas attributedto the localized holes [719].
+In II-VI semiconductor quantum wells, such as CdTe quantum w ells, hole spin lifetime is on the order of tens of ps
+[531,684,720].
+Figure 32: Temperature dependence of hole-spin lifetime
+inn-doped GaAs/AlGaAs quantum wells. Reproduced
+fromBaylac et al. [393].
+Figure 33: Temperature dependence of hole-spin relaxation time inp-
+doped GaAs/AlGaAs quantum wells. The curves are the calculated re-
+sults. Theinset showsthe holespin-orbit coupling constan tβ(in meVÅ)
+as function of quantum well width. FromSchneider et al. [712 ].
+Theoretically, hole spin relaxation and its relation to pho toluminescencespectra were studied by Uenoyama and
+Shaminn-doped,undopedand p-dopedquantumwells [694, 695], where the hole-phononscat tering was examined.
+Specificcalculationsof“spin”-flipscattering,whichlead stotheElliott-Yafet-likespinrelaxation,wereperforme dby
+FerreiraandBastard[702,706]andbyVervoortetal. [216], takingintoaccountoftheinterfaceinversionasymmetry.
+The D’yakonov-Perel’ spin relaxation due to the “spin”-con serving scattering via standard single particle paradigm
+was given by Ferreira and Bastard [706]. The role of the Bir-A ronov-Pikus mechanism in hole spin relaxation in
+quantum wells was discussed by Maialle [395]. It was found th at in narrow quantum wells, the Bir-Aronov-Pikus
+mechanismiscomparablewithothermechanisms.
+Finally, it should be mentioned that up till now the hole spin relaxation in one-dimensional structures was only
+investigated by L¨ u et al. theoretically [705]. They studie d the topic comprehensively via the kinetic spin Bloch
+equationapproach[44,334, 350]. Theirworkwill beintrodu cedin Section5.
+4.3. Carrierspinrelaxationin III-VandII-VIparamagneti cdilutedmagneticsemiconductors
+Semiconductorsdilutedlydopedwithmagneticimpurities, suchasMn,areinterestingmaterialsbecausethemag-
+netism and electrics can be incorporated together. Underst anding carrier spin dynamics in such kind of materials is
+important for the application. Also the spin dynamics of the magnetic impurity or the magnetization dynamics is
+closely related to the carrier spin dynamics. Most of the wor ks in the literature focus on Mn doped III-V and II-VI
+diluted magnetic semiconductors. In this subsection we rev iew carrier spin dynamics in these materials. We restrict
+ourreviewonspindynamicsin theparamagneticphase,where thedescriptionismucheasier.
+Adirectconsequenceofthe s(p)-dexchangeinteractionisthatitcontributestothecarriers pin-flipscatteringand
+hence carrier spin relaxation. The s(p)-dexchange interaction is believed to dominate spin relaxati on in (II,Mn)-VI
+63semiconductors[449]. Besides spin relaxation, the s(p)-dexchangeinteraction also leads to spin precession: carrie r
+spin precesses in the mean field of the s(p)-dexchange interaction. Under external magnetic field, carri er spins
+precessinthecoactionoftheexternalfieldandthemeanfield ofMnspins. Forexample, B∝ba∇dblˆx,electronspinLarmor
+precessionfrequencyisthen
+ωL=Ee
+Z=|geµBB−xN0Jsd∝an}b∇acketle{tSd
+x∝an}b∇acket∇i}ht|. (124)
+∝an}b∇acketle{tSd
+x∝an}b∇acket∇i}ht=−5
+2B5/2[5gMnµBB/(2kBT)] is the average Mn spin polarization at equilibrium where B5/2(x) is the spin-5/2
+Brillouinfunctionand gMnistheMnspin g-factor. WhentheMndensityishigh,theexchangeinteracti oncanbemuch
+larger than the Zeeman interaction geµBB, leading to the giant Zeeman splitting. In ZnSe /MnSe heterostructures, a
+largeg-factor of 400 was observed by Crooker et al. [447]. The contr ibution from the exchange interaction to spin
+precession leads to nonlinear magnetic field dependence of L armor frequency, which can be used to determine the
+strength of the exchange coupling in experiment [233, 317, 7 21, 722]. The exchange mean field is largely reduced
+at elevated temperature. This leads to a drastic decrease in spin precession frequency with increasing temperature
+[233, 317, 450]. The precession of carrier spins and Mn spins can be stronglycoupledwhen their frequenciesmatch
+(see Fig. 34) [721, 723]. Besides, nonequilibriumcarrier s pin polarization can be transferred to the Mn spin system
+viaexchangeinteraction[98,724,725]and viceversa [726–731]. MnspinbeatswereobservedbyCrookeretal. (see
+Fig.35)[724, 725]andtheoreticallystudiedbyLinderandS ham[732].
+Figure34: Ramanshiftforthespin-fliptransitionsforelec tronand
+Mn ion as function of magnetic field. The expected behavior fo r
+noninteracting subsystems calculated is indicated by thes olid and
+dashed curves. ES
+MnandES
+edenote Mn and electron spin splitting
+respectively. The anti-crossing splitting is 2 δ. The temperature is
+T=2K.InsetshowstheRamanspectra(every0.1T)intheregion
+of theanti-crossing. FromTeran et al. [721].
+Figure 35: Induced Faraday rotation ΘF, showing the final oscillations
+of the electron spins superimposed on an induced precession of the Mn
+spins. From Crooker et al. [724].
+Historically, carrier spin dynamics was first studied in (II ,Mn)VI diluted magnetic semiconductors. In these ma-
+terials no charge doping is invoked, as Mn ⇒Mn2++2e−. At high doping density and low temperature the Mn-Mn
+interaction leads to a spin-glass order. However, at high te mperature or low doping density it is paramagnetic. The
+bandgapalso changes with Mn doping. For example in CdMnTe, t he bandgapincreases with Mn doping density. A
+widely studied system is Cd 1−xMnxTe/Cd1−yMnyTe quantumwell, where the layer with higher Mn density serve s as
+barrier. Aparticularcaseis x=0,thes(p)-dexchangeinteractionisthenincorporatedthroughthewave functionpen-
+etratedinto the barrierlayers. The s(p)-dexchangeinteractioncan then be engineeredvia tuningthe s tructure[733].
+Another example of such structure is to insert MnSe layers in to ZnCdSe quantum wells [447]. Crooker et al. found
+that thes(p)-dexchangeinteractionchangessignificantly with the number of MnSe layers [in their experiment,they
+used 1(×3 atomic layer), 3( ×1 atomic layer) and 24 ( ×1/8 atomic layer) layers of MnSe] in ZnCdSe quantum well
+while keeping the overall Mn doping density unchanged[447] . The modification of the s(p)-dexchange interaction
+comes from tuning the overlapbetween the confined electrons /holesand Mn ions by structure engineering. Another
+reason is that the Mn-Mn antiferromagneticcoupling betwee n neighboring Mn spins which “locks” the Mn spins is
+largely suppressed in 1 /8 MnSe sub-monolayers as the average distance between Mn spi ns increases. Interestingly
+64Smyth et al. found that the spin subband of the ZnMnSe /ZnSe superlattice can be significantly modified by external
+magnetic field [734]: Due to the giant Zeeman splitting, the s pin-down subband is lowered and spin-up subband is
+raised. At thethresholdmagneticfield Bc, spin-upsubbandenergyisas highas thebarrierenergy. Abo veBcspin-up
+electrons switch into the ZnSe layers. The spin and electron ic structure is then tuned by the external magnetic field.
+InparabolicZnSe/ZnCdSequantumwell with MnSe monolayersinserted,Myerset al. demonstratedthat the s-dex-
+changeinteractioncan be manipulatedby gate voltage[735] . As thes(p)-dexchangeinteractionis tuned,the carrier
+spindynamicsisalsomanipulated[447,734–736].
+Femtosecond pump-probe measurements were first performed b y Freeman et al. [733], revealing a very short
+spin relaxation time of ∼3 ps in Cd 1−xMnxTe/Cd1−yMnyTe quantumwells [733, 737]. After that, Bastard and Chang
+[448], Bastard and Ferreira[703] presented theoreticalst udies on spin relaxationdue to the s-dexchangeinteraction
+in such kind of quantum wells. They obtained comparable spin lifetime with experiments. They also predicted well
+widthdependenceofspinlifetimeandproposedabiasdouble -quantum-wellstructuretomanipulatethespinlifetime.
+Interestingbehaviorwas observedunder magneticfield. As t he neighborMn ions are coupledantiferromagnetically,
+Mn ions can link together to form clusters which have small sp in momentums and become ine ffective in spin-flip
+scattering. Hence,only isolatedMnionscount. BastardandChangtookane ffectiveMndensityas neff=x(1−x)12N0
+to warrant only Mn ionswithout any nearest neighbors. Akimo to et al. [736] foundthat electron spin relaxation rate
+inCdTe/CdMnTequantumwellsisproportionalto theprobabilityoffi ndingelectroninthebarrierlayerofCdMnTe,
+which is consistent with the theory of Bastard and Chang [448 ]. As stated in previous paragraph, spin-up subband
+is raised higher than the spin-down one. Hence spin-flip scat tering from up-spin to down-spin is more favorable
+than the opposite one at low temperature. Smyth et al. report ed that spin-flip time of spin-up electron decreases
+with increasing magnetic field whereas that of spin-down ele ctron increases: the spin-flip times vary with magnetic
+field as two branches [734]. Similar behavior was observed in other diluted magnetic heterostructures (see inset
+of Fig. 36) [447, 738–742]. Such two-branch behavior is weak ened with increasing excess photo-carrier energy
+or decreasing exchange interaction [738, 741]. Both electr on and hole spin lifetimes were found to be smaller in
+dilutedmagneticheterostructurescomparedtothenonmagn eticones[724],whichindicatestherelevanceofthe s(p)-
+dexchangeinteractiontothe spinrelaxation.
+Figure 36: Calculated electron spin-flip times τtheory
+sf/2 as a function of the magnetic field B. The inclusion of phase-space-filling (PSF) enhances
+the two-branch feature of the τsfvs.Bcurves. Different structures are considered: heterostructures with 18 ×1/6 monolayer (ML), 15 ×1/5
+ML, 12×1/4 ML and diluted magnetic semiconductor (DMS) quantum well. Inset: experimental results by Crooker et al. [447] with var ious
+heterostructures: 24 ×1/8 ML (triangle), 6 ×1/2 ML (square) and 3 ×1 ML (circle). The two branches of spin-flip time correspond t oτ↑→↓and
+τ↓→↑. Thecalculation is notintended to compare quantitatively with the experimental results. From Egues and Wilkins [743] .
+Theoretically,spinrelaxationratecanbecalculatedfrom theFermiGoldenrule[451, 452],
+Γk↑,k′↓=2π|∝an}b∇acketle{tk↑|Hs−d|k′↓∝an}b∇acket∇i}ht|2δ(εk↑−εk′↓). (125)
+Thespin-flipratesare
+τ−1
+↑→↓=∝an}b∇acketle{tΓk↑,k′↓(1−fk′↓)∝an}b∇acket∇i}ht, τ−1
+↓→↑=∝an}b∇acketle{tΓk↓,k′↑(1−fk′↑)∝an}b∇acket∇i}ht. (126)
+65Here↑→↓(↓→↑) denotes the spin-up to spin-down (spin-down to spin-up) tr ansition.∝an}b∇acketle{t...∝an}b∇acket∇i}htmeans average over the
+electronensemble. Thetotalspinrelaxationrateis τ−1
+s=τ−1
+↑→↓+τ−1
+↓→↑. Forexampleinquantumwellsatzeromagnetic
+field[111, 744, 745],
+τ−1
+s=IsxN0J2
+sdme∝an}b∇acketle{t∝an}b∇acketle{tSd
+−Sd
+++Sd
++Sd
+−∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht/(4Lw), (127)
+whereJsdis thes-dexchange constant, xstands for the mole fraction of Mn and N0denotes the density of the unit
+cells.Sd=(Sd
+x,Sd
+y,Sd
+z) is Mn spin operator and Sd
+±=Sd
+x±iSd
+y.∝an}b∇acketle{t∝an}b∇acketle{t...∝an}b∇acket∇i}ht∝an}b∇acket∇i}htstands for average over Mn spin distribution.
+HereIs=Lw/integraltext
+|ψ↑
+e(z)|2|ψ↓
+e(z)|2dzwithψ↑(↓)
+ebeing the spin-resolved electron subband wavefunction and Lwdenoting
+thewellwidth. At equilibriumstate, ∝an}b∇acketle{t∝an}b∇acketle{tSd
+−Sd
+++Sd
++Sd
+−∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht=4
+3Sd(Sd+1). Astothetwospin-fliptransitionrates, τ−1
+↑→↓∝
+∝an}b∇acketle{t∝an}b∇acketle{tSd
+−Sd
++∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht=Sd(Sd+1)−∝an}b∇acketle{t∝an}b∇acketle{t(Sd
+z)2∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht−∝an}b∇acketle{t∝an}b∇acketle{tSd
+z∝an}b∇acket∇i}ht∝an}b∇acket∇i}htwhereasτ−1
+↑→↓∝∝an}b∇acketle{t∝an}b∇acketle{tSd
++Sd
+−∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht=Sd(Sd+1)−∝an}b∇acketle{t∝an}b∇acketle{t(Sd
+z)2∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht+∝an}b∇acketle{t∝an}b∇acketle{tSd
+z∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht. Inthepresence
+of a magnetic field along the zaxis, the Mn spin polarization ∝an}b∇acketle{t∝an}b∇acketle{tSd
+z∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht<0. This enhances the spin-up to spin-down
+transition but suppresses the inverse process. The calcula tion of Egues and Wilkins indicated that the phase-space-
+fillingeffect(Pauliblockingfactor(1 −fk′↑(↓)))furtherenhancesthedi fferencebetweenthetwospin-flipscatteringrates
+(seeFig.36)[743]. Itisnotedthatthetotalspinrelaxatio nrateisproportionalto Sd(Sd+1)−∝an}b∇acketle{t∝an}b∇acketle{t(Sd
+z)2∝an}b∇acket∇i}ht∝an}b∇acket∇i}htwhichdecreases
+withincreasingmagneticfield. Theunderlyingphysicsisth atthemagneticfieldpinstheMnspinandthensuppresses
+thespin-flipprocessesandelectronspinrelaxation[746,7 47]. Magneticfielde ffectonelectron/hole/excitonspin-flip
+scattering rates at low temperature in CdMnTe quantum wells was investigated comprehensivelyby Tsitsishvili and
+Kalt[745]. Inmostcasesthemagneticfielddependenceofspi nrelaxationrateisweak. Unlikelongitudinalmagnetic
+field, a transverse magnetic field always leads to faster spin relaxation (for both electrons and holes) (see Fig. 37)
+[449, 724, 725, 748–750]. This is because that the magnetic fi eld pinning of Mn spins along transverse direction
+increases the factor Sd(Sd+1)−∝an}b∇acketle{t∝an}b∇acketle{t(Sd
+z)2∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht=∝an}b∇acketle{t∝an}b∇acketle{t(Sd
+x)2+(Sd
+y)2∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht[111, 744]. Electron spin relaxation is enhanced
+for exciton bound electrons as the center-of-mass e ffective mass is larger for exciton. This was first predicted by
+Bastard and Ferreira [703] and then confirmed experimentall y by Smits et al. [749]. Temperature dependence of
+electron, hole and exciton spin relaxation has been studied in Refs. [450, 749, 751, 752]. Smits et al. presented a
+systematic experimentalstudy on excitonicenhancementof electron/holespin relaxationin CdMnTe based quantum
+wellstructures[749].
+Figure 37: Electron (left) and hole (right) spin lifetime as function of transverse magnetic field for nonmagnetic (NM) ( Zn0.77Cd0.23Se/ZnSe)
+quantum well as well asthat for quantum wellwith 1 ×3,3×1 and 24×1/8 MnSemonolayers inserted (for the details of the heterostr uctures see
+Ref. [724]). T=4.6K. FromCrooker etal. [724].
+Recently, R¨ onnburg et al. presented a systematic experime ntal investigation on electron spin relaxation in bulk
+CdMnTe. Their major results are shown in Figs. 38 and 39. In th e experiment, the photon energy was chosen to
+be centered on the 1s exciton absorption lines and the photo- excitation intensity was kept low in order to minimize
+sample heating and carrier-density-dependent e ffects. A salient feature is that electron spin lifetime incre ases with
+temperature, which signals a motional-narrowingspin rela xation mechanism. This is quite di fferent from the previ-
+ouslyconsidered s-dexchangescatteringmechanismwhereelectronspinrelaxat ionrateinbulksystemisproportional
+to∝an}b∇acketle{tk∝an}b∇acket∇i}ht, whichincreaseswithtemperature. R¨ onnburget al. thenpr esenteda newtheoryofspinrelaxationbasedonthe
+66thermalandspacialfluctuationofthe s-dmeanfield. Thetransversespinrelaxationrateisgivenby
+1
+T2=γ2τ0∝an}b∇acketle{t(δMz)2∝an}b∇acket∇i}ht+/parenleftig
+∝an}b∇acketle{t(δMx)2∝an}b∇acket∇i}ht+∝an}b∇acketle{t(δMy)2∝an}b∇acket∇i}ht/parenrightig
+/2
+1+(geµBB+γ∝an}b∇acketle{tδMz∝an}b∇acket∇i}ht)2τ2
+0, (128)
+whereγ=Jsd/(gMnµB),M=(Mx,My,Mz)isthe(Mnspin)magnetizationand δMi(i=x,y,z)denotesthefluctuation
+of magnetization felt by optically excited electrons. τ0stands for the correlation time of the fluctuations. geis the
+intrinsicg-factor of the itinerant electrons. By working out the therm al and spacial fluctuations, the spin relaxation
+rateisgivenexplicitlyas[450]
+1
+T2=γ2τ0
+Nexn0xkBTχ(B,T)+∝an}b∇acketle{tMz∝an}b∇acket∇i}ht2+n0xkBT[3χ(0,T)−χ(B,T)]−∝an}b∇acketle{tMz∝an}b∇acket∇i}ht2
+2+2(geµBB+γ∝an}b∇acketle{tδMz∝an}b∇acket∇i}ht)2τ2
+0. (129)
+Hereχ(B,T)=∂∝an}b∇acketle{tMz(B,T)∝an}b∇acket∇i}ht
+∂Bis the susceptibility. ∝an}b∇acketle{t...∝an}b∇acket∇i}htdenotesthe thermalaverageof Mn spins. By assumingthe spac ial
+fluctuationtobeaPoissondistribution,itgivestheamplit udeofthespacialfluctuation ∝an}b∇acketle{t(δMi)2∝an}b∇acket∇i}htsf=∝an}b∇acketle{tMi∝an}b∇acket∇i}ht2(i=x,y,z).
+Accordingtothecentrallimittheorem,thefluctuationisre ducedbythenumberofMnspinsfeltbytheelectronwave-
+packetNex. InintrinsicbulkCdMnTeatlowtemperatureelectronsandh olesareboundtogethertoformexcitons. The
+spacialextensionofexciton-boundelectronwave-packeti scontrolledbytwofactors: theexcitonBohrradiusandthe
+thermal length Lthof the center-of-mass motion. The latter is estimated as the inverse of the thermal fluctuation of
+the center-of-mass wave-vector, Lth=0.37/√mkBTwithm=me+mh. Taking into account these two factors, one
+obtains the volume of the wave-packet Vexand then the number of Mn spins within the volume Nex=xn0Vex. The
+correlation time also consists of two contributions, 1 /τ0=1/τprop+1/τMn
+sf.τMn
+sfis the Mn spin-flip time which is at
+least several hundredsof picoseconds[725] and hence ine ffective.τpropdenotes the excitonpropagationtime, which
+is the time for an exciton to see a new environment of Mn spins. It is approximated as the time for an exciton to
+propagate a distance equal to its spacial extension with a th ermal average velocity. In such a way τ0is determined.
+By taking parameters in the literature, R¨ onnburg et al. cal culated the transverse spin lifetime and found that the
+calculated results agree remarkably well with experimenta l data both qualitatively and quantitatively (see Fig. 39).
+TheMndoping xdependenceof T2atsmallxshows1/xbehavioras Mz,χandNexareproportionalto x. AthighMn
+doping,theantiferromagneticinteractionbetweenneighb oringMnionsshouldbetakenintoaccountwithacorrection
+xeff=x(1−x)12[743]. Thisleadstotheobservedincreaseof T2at largexin Fig.39. Themagneticfielddependence
+also agrees well with the calculation (not shown). The intia l decrease of T2with magnetic field is mainly due to the
+increase of∝an}b∇acketle{tMz∝an}b∇acket∇i}ht2. At high magnetic field, χ(B,T) and∝an}b∇acketle{tMz∝an}b∇acket∇i}htsaturate. The magnetic field dependence then mainly
+comes from the factor 1 /[2+2(geµBB+γ∝an}b∇acketle{tδMz∝an}b∇acket∇i}ht)2τ2
+0] which makes T2increase with B(Similar behavior was also
+observedin the experimentof Cronenbergeret al. [753]). Th e magneticfield dependenceis morepronouncedat low
+temperaturewhere τ0islargerandthe magneticfielddependenceof ∝an}b∇acketle{tMz∝an}b∇acket∇i}htisstronger.
+The Mn density dependence is informative and reveals the und erlying spin relaxation mechanisms. In CdMnTe
+quantum wells, spin relaxation rate was found to be proporti onal toxforx>4×10−3, indicating spin relaxation
+due to the s-dexchangescattering (see Fig. 40) [449]. However,in GaMnAs quantumwells Poggioet al. foundthat
+spin lifetime first increasesand thendecreaseswith increa singMn density. A peak appearsaround x∼10−4for well
+width from 3 to 10 nm (see Fig. 41) [233]. It was speculated tha t the dominant spin relaxation mechanisms for Mn
+densitiesbelowandabovethepeakdensityaredi fferent. However,calculationbasedonrealisticparameters isneeded
+todeterminetherelevanceofvariousmechanisms. Suchcalc ulationhasbeendonebyJianget al. recently[111].
+Relaxation dynamicsof spin-polarizedelectronsexcited a t higher energy subbandwas revealed by probe-energy
+dependenceof the spin lifetime in Ref. [754]. The probe-ene rgydependenceis understoodby the intrabandelectron
+energyrelaxationandsubsequentspinrelaxation. Energy- resolvedspinrelaxationinwurtzitemagneticsemiconduct or
+CdMnSewasstudiedin Ref.[755].
+At low temperature, Mn ion can bind a hole to form a neutral cen ter. Hence optically excited electrons interact
+with the neutralcentersrather thanthe Mn ionsand holessep arately. The groundstate of the neutral center is a two-
+fold degeneratestate with angularmomentum mJ=±1 as thep-dexchangeinteraction is anti-ferromagnetic.68The
+68Such a spin structure can be utilized to establish optical in itialization and readout of Mn spin [98] as circularly polar ized light (with spin
+σJ=±1) selectively excites the mJ=±1 state dueto the optical selection rules.
+67Figure38: Results oftime-resolved Faraday rotation exper iments onCd 1−xMnxTe(x=0.0073) at8Tandforvarious temperatures: (a)Temperature
+dependence of the Faraday-rotation transients; (b) temper ature and magnetic-field dependence of the electron spin pre cession frequency extracted
+fromthe transients; (c) fitted electron spin dephasing time τeas afunction of temperature and magnetic field. FromR¨ onnbu rg et al. [450].
+Figure 39: Left: experiment. Right: theory. Doping density and temperature dependence of electron spin dephasing time . Magnetic field is fixed
+at 5 T. Inset in right figure: the picture of exciton moving thr ough the local moments of magnetic dopants. Inset in the left figure: experimental
+set-up. From R¨ onnburg et al. [450].
+exchangeinteractionbetweenelectronandtheneutralcent erconsistsoftwoorigins,the s-dexchangeinteractionand
+the electron-hole exchange interaction. Interestingly, b oth interactions are ferromagnetic. Due to opposite Mn and
+holespinorientations,thewholeexchangeinteractionisw eakened.69Thecancellationofthetwoexchangeinteraction
+leadstosuppressionofspinrelaxation. AsGaMnAsispartia llycompensatedsemiconductors,i.e.,interstitialMn’sa ct
+as donors whereas substitutional Mn’s act as acceptors, hol e density is smaller than the Mn acceptor density. Hence
+therearestillsomechargedMnacceptors. Recently,Astakh ovetal. foundthattheexchangeinteractioncanbefurther
+reducedbygeneratingmoreholesvia opticalpumping. Theyf oundthatbelowsomethresholdexcitationpower,spin
+relaxationissuppressedbyincreasingtheopticalexcitat ionpower(seeFig.42)[757]. Apictureoflocalizedelectro ns
+was presented in their paper to explain the results at low tem perature. In such picture, the spin relaxation is due to
+randomspin precessionwhen electronspassby the neutralce nters. The spin relaxationrate is 1 /τs=2
+3∝an}b∇acketle{tω2∝an}b∇acket∇i}htτcwhere
+τcisthecorrelationtime. Inthepresenceofalongitudinalma gneticfield, spinrelaxationissuppressedbyafactorof
+1/[1+(ωLτc)2]. By measuring the longitudinalmagnetic field dependence, Akimov et al. found that τcin GaMnAs
+is much longer than that in GaAs:Ge with similar acceptor con centration. They explained that the strong exchange
+couplingbetweenMnacceptorandholeprotectsholespinfro mdissipationandthecorrelationtimeofrandomelectron
+69This leads to a deviation of the s-dexchange constant deduced from the measured electron spin p recession frequency from the genuine s-d
+exchange constant [756].
+68Figure 40: Electron spin relaxation rates vs.Mn effective
+contentxefffor aQWwidth Lw=8nm: experimental data
+(crosses), calculated values (solid line). From Camilleri et
+al. [449].
+Figure 41: Thetransverse electron spin lifetime T∗
+2atT=5 Kversus the
+percentage of Mn xfor four quantum well sample sets of varies widths.
+The plotted values of T∗
+2are the mean values from magnetic field B=0
+to 8 T, and the error bars represent the standard deviations. From Poggio
+et al. [233].
+spinprecessionisonlylimitedbyelectronhopping,leadin gto alarge τc[758].
+Figure 42: (a) Spin relaxation time τsvs.excitation power for GaAs:Mn ( NMn=8×1017cm−3) (dots) and for p-GaAs containing nonmagnetic
+acceptors ( Na=6×1017cm−3)(open squares). (b)Spin lifetimes vs.temperature fordifferent excitation powers in GaAs:Mn. Curves aretoguide
+theeyes. FromAstakhov et al. [757].
+Hole spin relaxationin dilutedmagneticquantumwells andh eterostructureswas studiedin Refs. [449, 724, 725,
+749]. For heavy-holes,the p-dexchangeinteractionis not able to flip holespin unlessfaci litated byheavy-lighthole
+mixing [702]. It was reported that spins of exciton-boundho les relax faster than free holes due to enhanced heavy-
+light hole mixing in excitons[749]. Also the mixing is incre ased by the mean field of the p-dinteraction. Camilleri
+observeda relation for hole spin relaxationrate 1 /τh
+s≃a(xeff∝an}b∇acketle{tM∝an}b∇acket∇i}ht)2+bwherexeffis the effective (isolated) Mn mole
+fraction,∝an}b∇acketle{tM∝an}b∇acket∇i}htdenotes the magnetization and bstands for the heavy-light hole mixing independent of ∝an}b∇acketle{tM∝an}b∇acket∇i}htand other
+spin relaxation sources [449]. Hole spin relaxation always decreases with increasing magnetic field [449, 724, 725,
+749]. Theoreticalinvestigationofholespin relaxationis presentedin Refs.[702,745].
+4.4. SpinrelaxationinSiliconandGermaniumin metallicre gime
+Spin relaxation in silicon and germanium is an old topic. Spi n relaxation of donor bound electrons has been
+studied intensively since the middle of last century [453, 7 59–764] and revisited in recent decades in the context of
+69quantum computation [289, 416, 417, 460, 765, 766]. In this s ubsection, we focus on spin relaxation in silicon and
+germaniuminthemetallicregime.
+Let us first start with bulk silicon.70As silicon possesses space inversionsymmetry,the spin-or bit couplingterm
+of the conduction band is zero. Hence the D’yakonov-Perel’ m echanism is absent. In n-doped silicon, the only
+relevant spin relaxation mechanism is the Elliott-Yafet me chanism. Experimentally, electron spin relaxation in bulk
+silicon was mostly studied via electron spin resonance as th e optical orientation and detection method is forbidden
+due to its indirect band gap nature. Most of the investigatio ns focused on the low temperature regime [767–770],
+whereasonly a few coveredthe high temperature regime[763, 771]. Among these works, a systematic investigation
+was performed by L´ epine [763]. Analysis indicated that the spin relaxation scenario varies with temperature.71At
+high temperature ( T>150 K) most of electrons are in extendedstates and spin relax ation is dominated by the band
+electrons other than donors. Theoretically, conduction ba nd electron spin relaxation in silicon was first studied by
+Elliott [104] and Yafet [103] separately. The Elliott-Yafe t spin relaxation consists of both the Elliott process where
+spin-flip is due to the spin-mixingof conductionband and the Yafet processwhere spin-flip is caused by direct spin-
+phononcoupling. By taking into account of the Yafet process due to intravalley electron-acousticphononscattering,
+Yafet gave a qualitative relation of τs∼T−5
+2[103]. Recently, Cheng et al. found that the Elliott process and the
+Yafetprocessinterferedestructivelyinsilicon. Thetota lspinrelaxationrateismuchslowerthanthatpredictedbyt he
+individualYafetorElliottprocess[362]. Theygavenewqua litativerelationof τs∼T−3,whichagreesquitewellwith
+experiments[362] [seeFig.43].
+Figure 43: Spin relaxation time T1in bulk silicon as a function of temperature T. The solid curve is the calculation (Pseudo), the symbols ar e
+the spin injection (SI Appelbaum) [150] and the electron spi n resonance (ESR Lepine [763] and ESR Lancaster [771]) data ( see Ref. [22]). The
+dashed-dotted curveisthespinlifetimecalculated withon lytheintravalley scattering (Intra). Theinsetshowsthec ontourplotofthespinrelaxation
+rateT−1
+1of hotelectrons, as afunction of the electron energy εand lattice temperature T. From Cheng et al. [362].
+Two-dimensionalstructure can introducestructure and /or interface inversion asymmetrywhich induces the spin-
+orbit coupling in conduction band. Spin-orbit coupling in s ilicon or germanium quantum wells was studied in
+Refs. [290–293, 772]. Correspondingly, the D’yakonov-Per el’ spin relaxation in two-dimensional structure was in-
+vestigatedinRef.[290]. Experimentally,Tyryshkinetal. measuredspinechoesanddeducedaspincoherencetimeup
+to3µs forahighmobilitytwo-dimensionalelectronsystemforme dinSi/SiGequantumwells[773]. Theangularde-
+pendenceof spin relaxationindicatedthat the D’yakonov-P erel’mechanismdue to the Rashba spin-orbitcouplingis
+therelevantoneinhighmobilitysamples[295,296,773]. Li kethatinbulksilicon,spinrelaxationintwo-dimensional
+siliconstructureswasalsostudiedmostlyviaelectronspi nresonance[295–297,622,774–780]. Recently,electrical ly
+detectedspinresonancehasalso beendevelopedandapplied tosilicontwo-dimensionalstructures[287,781, 782].
+Thedependenceofspinlifetimeontheorientationofextern almagneticfieldwasstudiedinRefs.[296,622,776],
+revealing the relevance of the D’yakonov-Perel’ spin relax ation mechanism in high mobility samples. Examination
+70There is few study in bulk germanium in metallic regime.
+71Theanalysis has been summerized in Ref. [21].
+70ofthe relationbetweenthe spinlifetimeandthe momentumsc atteringrateindicatedthat inlowmobilitysamplesthe
+spin relaxation is dominated by the Elliott-Yafet mechanis m [297].72Temperature and density dependences of spin
+relaxation rate were studied in Refs. [295, 622, 774] and [29 5] respectively. Spin relaxation can be suppressed by
+cyclotronmotion as demonstrated by Wilamowski et al. [622] . Spin relaxation in quantum Hall regime was studied
+byMatsunamietal. [782].
+Theoretically,Shermanpointedout that evenin nominallys ymmetricsilicon quantumwells, the dopinginhomo-
+geneity can cause a random spin-orbit coupling [294]. Such e ffect may play an important role in spin relaxation in
+symmetric silicon quantum wells [294, 657]. Similarly, the surface roughness can also induces a random spin-orbit
+coupling due to interface inversion asymmetry [291]. Hole s pin-orbit coupling in silicon quantum wells was dis-
+cussed in Refs. [298, 299]. The e ffect of carrier-carrier scattering on the D’yakonov-Perel’ spin relaxation in n-type
+siliconquantumwells[783]and p-typesilicon(germanium)quantumwells[299]hasalsobeen reportedintherecent
+literature.
+4.5. Magnetizationdynamicsinmagneticsemiconductors
+Magneticsemiconductors,especiallytheMndopedIII-Vmag neticsemiconductors,haveattractedmuchattention
+for decades. The invention of ferromagneticIII-V semicond uctorsopens the perspective of integrating the magnetic
+recording and electronic circuit on one chip and has inspire d a lot of studies [114, 784–787]. As the ferromagnetic
+order in III(Mn)-V magnetic semiconductors originates fro m the carrier mediated Mn spin-spin interactions via the
+strongs(p)-dexchangeinteractions[114],themanipulationofcarriers wouldleadtoefficientcontroloverthemagne-
+tization. Theefficientmanipulationofmagnetizationviaoptical[788–808] andelectrical[809–812]meanshavebeen
+realized.
+In this subsectionwe review recent studieson opticallyind ucedmagnetizationdynamicsin III(Mn)-Vferromag-
+netic semiconductors. Historically, the optically induce d magnetization dynamics was first discovered in ferromag-
+neticmetal(nickel)[813]. Thestudiesinmetalhavesomere lationwiththoseinferromagneticsemiconductors,which
+hence shouldalso be mentioned. After more than a decadestud y, the picture of the underlyingphysicshas been dis-
+coveredandsome usefullmodelshavebeenraised. However,s tudieswhichcanquantitativelyrelate themicroscopic
+carrierdynamicswiththe observedmacroscopicmagnetizat iondynamicsarestill absent.
+We firstreviewopticallyinducedmagnetizationdynamicsin ferromagneticmetals. Letusstartfromtheobserved
+phenomena. In experiments,strong femtosecondlaser pulse s (usually∼1 mJ cm−2) are used to pumpthe ferromag-
+netic materials. The magnetization is then monitored by pro be pulses via magneto-optical Kerr e ffect. The relative
+magnetization M(t)/M(0)isassumedtobeproportionaltotherelativeKerrrotati onθ(t)/θ(0)whereθ(0)[M(0)]isthe
+Kerr rotation (magnetization) before pumping. A typical cu rve is shown in Fig. 44. The magnetization dynamics is
+characterizedby three stages: (i) an ultrafast demagnetiz ationwithin 1 ps; (ii) a recoveryof magnetization via equi-
+libration processes such as electron-phononscattering fr om 1 ps to∼20 ps; (iii) after∼20 ps a damped precession
+of the magnetization due to the e ffective exchange field [814, 815]. The characteristic time sc ale of these stages are
+the ultrafast demagnetization time τM, the energy lifetime τEand the Landau-Lifschitz-Gilbert damping time τLLG.
+In Nickel,τM∼100 fs,τE∼0.5 ps, andτLLG∼700 ps [815]. Many works have been attributed to study the mag -
+netizationdynamicsbyvariousmethodstorevealtheunderl yingmechanismandthedependencesofthedynamicson
+the external conditions [813–820]. It was found that the mag netization in stage (ii) is governed by the hot-electron
+temperature, which agrees well with the static M(T) curve [816]. The damped precession of the magnetization in
+stage(iii)wasfoundtobeduetothe deviationofthe magneti zationdirectionfromtheequilibriumonecausedbythe
+laser pumping [815]. Among these processes, the ultrafast d emagnetization and the magnetization precession have
+attracteda lotofattentionto uncovertheunderlyingmicro scopicmechanisms.
+The ultrafast demagnetization in ferromagnetic metals is t he first story which closely grasped the attention of
+the society. It was first discovered by Beaurepaire et al. in n ickel in 1996 [813]. The magnetization decreases by
+50%within /lessorsimilar100femtoseconds,whichisthefastestmagnetizationdynam icseverfound. Thelarge“instantaneous”
+demagnetizationhas been puzzling researchers, as previou s measurement yielded a spin-lattice lifetime of ∼100 ps
+[821]. Purely electronic mechanisms such as electron-magn onscattering was also suggested to explain the ultrafast
+demagnetization.However,althoughthepureelectronicpr ocessescanflipindividualelectronspinsrapidly[822],th ey
+72Therelation between spin lifetime and momentum scattering timewas also studied in Refs. [295,622].
+71Figure44: Main: experimental development oftheinduced pe rpendicular component of M,∆Mz/Mz,after laser heating anickel thin filmat t=0.
+Insets: precessionofanindividualspinintheexchangefiel d(left)andof Mintheeffectivefield(right). τM,τEandτLLGdenotethedemagnetization
+time, the energy relaxation time and the Landau-Lifschitz- Gilbert damping time, respectively. The three stages of the magnetization dynamics is
+clearly seen. From Koopmans etal. [815].
+conservethe total spinofthe electronsystem andhencecann otleadto demagnetization[814, 817]. Koopmanset al.
+first suggested that the measured optical signal at the first p icosecond does not reflect the magnetization dynamics,
+butonlyreflectstheopticaldynamics[814]. Theyfoundthat therelativechangeintheKerrrotationdoesnotcoincide
+with that in Kerr ellipticity within the first 0.5 ps, which di stincts with the magnetizationinducedKerr e ffect. Hence
+the “instantaneous” decrease of the Kerr rotation or ellipt icity can not be directly related to the demagnetization
+[814]. A close examination indicated that the photo-excite d electrons thermalize also within the first picosecond
+[814]. A moredirectandreliablemeasurementofthespin dyn amicsvia X-raymagneticcirculardichroismobserved
+a demagnetization time of ∼120 fs [818]. In the same experiment, the photo-electron the rmalization time was
+identifiedas∼640fs. Atthisstage,thetimescaleofthemagnetizationdyn amicsisfaithfullycharacterized. However,
+the demagnetizationmechanismis still unclear. Althoughs everal mechanismsand modelshavebeen proposed[813,
+815, 823–825],theyhavenotyet beenconfirmedbyexperiment sunambiguously.
+The focus of this subsection is the photo-inducedmagnetiza tion dynamics in ferromagnetic III(Mn)-V semicon-
+ductors. Oneofthesignificantdi fferenceisthatthemagnetizationisprovidedbytheMnspin,h encethemagnetization
+and carrier degrees of freedom are separated. Furthermore, unlike the complex energy bands in metals, the energy
+bands in ferromagnetic III(Mn)-V semiconductors are well d escribed by the Kane model with the s(p)-dexchange
+interaction [114], which greatly simplifies the theoretica l study. Theoretical studies can then help to identify the
+underlyingmechanismsofthephoto-inducedmagnetization dynamics[99,826–828].
+Experimentally, laser-induced demagnetization in (III,M n)V ferromagnetic semiconductors was first studied by
+Kojima et al. [807] in GaMnAs. The magnetizationdynamics wa s monitoredvia the magneto-opticalKerr rotation.
+They observed a slow demagnetization time of ∼500 ps, which was attributed to the possible thermal isolati on be-
+tween the chargeand spin system in GaMnAs[807]. The sub-pic osecondultrafast demagnetizationin ferromagnetic
+semiconductorswasfirst demonstratedbyWang et al. [788] in InMnAswithmuchhigherintensity /greaterorsimilar1 mJ/cm−2per
+pumppulsethantheoneintheexperimentofKojimaetal. (45 µJ/cm−2). Atypicaltraceofthetime-resolvedKerrro-
+tationisillustratedinFig.45. Thephenomenaaresimilart othatinferromagneticmetals: asub-picosecond( ∼200fs)
+demagnetizationandalongtime recovery. However,itisnot edthat themagnetizationfurtherdecreasesafterthefirst
+1ps,incontrastwiththerapidrecoveryinferromagneticme tals[815]. InInMnAs,thisslowdemagnetizationprocess
+can last for several hundreds of picosecond at low pump fluenc e. With increased pump fluence the duration of the
+slow demagnetizationprocessdecreaseswhereasthe decayr ate increases. Meanwhilethe degreeofdemagnetization
+due to the fast process increases. At high pump fluence /greaterorsimilar10 mJ/cm2, the magnetization is completely quenched
+after the fast processand the slow demagnetizationdiminis hes. The slow demagnetizationprocess is assigned to the
+Mn spin-lattice relaxationin the existing literature [788 ]. The argumentsare as follows: In III(Mn)-Vferromagnetic
+semiconductorsthe large hole spin polarizationblocks the carrier-Mn spin-flip scattering, hence the Mn spin system
+isthermallyisolated[807]. Ontheotherhand,thecarriers ystemisefficientlycoupledtophononbathviathecarrier-
+phononinteractionat a time scale of /lessorsimilar1 ps. The thermalized phononbath then leads to a Mn spin-latt ice relaxation
+72Figure 45: (a) The first 3 ps of demagnetization dynamics in In MnAs. The shadowed region denotes the cross correlation of t he pump and probe
+pulses. (b) Demagnetization dynamics covering the entire t ime range of the experiment. ∆θKis the derived difference Kerr rotation which is
+supposed to beproportional to the induced magnetization ch ange along thegrowth direction. FromWang et al. [788].
+atthe timescale of100ps,whichresultsin theslowdemagnet izationprocess[807].
+Figure 46: Illustration of the microscopic process of ultra fast demagnetization. ×denotes the p-dexchange scattering. Thin arrow denotes the
+spin polarization direction of photo-excited holes. Bold a rrow represents theinitial magnetization direction. Each p-dexchange scattering reduces
+themagnetization and transfers thespin polarization into the hole system. Holes losetheir spin polarization quickly through the D’yakonov-Perel’
+(DP) and Elliott-Yafet (EY) mechanisms. During these proce sses, the efficient hole-longitudinal-optical-phonon scattering [den oted as blue arrow
+(LO) in the figure] relaxes the excess energy of the photo-exc ited holes. After the energy relaxation, hole spin relaxati on slows down (the hole-
+longitudinal-optical-phonon scattering also slows down) and becomes less e fficient than the p-dexchange scattering. Hole spin polarization then
+saturates and the cascade processes terminate.
+We now focus on the fast demagnetization process. The underl ying physics of the ultrafast demagnetization is
+illustrated in Fig. 46. The key process is that via the strong p-dcoupling the Mn spin polarization is e fficiently
+transferred to the hole spin polarization, which is similar to the inverse Overhauser e ffect [826]. However, such
+processesissuppressediftheholespinrelaxationrateiss mallerthantherateatwhichthespinpolarizationisinject ed
+from the Mn spin system via the p-dexchange scattering. E fficient hole spin relaxation is required for a ultrafast
+demagnetization. Itisknownthatholespinrelaxationisve ryfastinbulk p-GaAs,atatimescaleof100fs,duetothe
+strong spin-orbit coupling [207, 210]. However, in III(Mn) -Vferromagneticsemiconductors, the large spin splitting
+duetothemean p-dexchangefieldremovesthe Γpointdegeneracyandsuppressesthee ffectofthespin-orbitcoupling
+at lowkineticenergy[3, 21]. Fortunatelyforphoto-excite dholes,ofwhichthe kineticenergyis high[788],the spin-
+orbitsplitting is largerthan the exchangesplitting. The h olespin relaxationcan againbe verye fficient. The fast spin
+73relaxationofphoto-excitedholesfacilitatesthe“invers eOverhausereffect”andleadstotheultrafastdemagnetization.
+Duetothehole-phononandhole-holescattering,thephoto- excitedholessoonlosetheirenergyandthermalizewithin
+1ps. Aftertheenergyrelaxationprocess,theultrafastdem agnetizationisterminated. Suchscenariohasbeenlaidout
+inrecentstudies[790, 826].73
+In Ref. [826], Cywinski et al. developed a mean field theory to describe the above scenario of ultrafast de-
+magnetization. In their theory, the magnetization was desc ribed by the mean Mn spin polarization which was then
+characterizedbyasingleMnspindensitymatrix. TheMn-Mnc orrelationsbeyondthemeanfieldlevelwerearguedto
+beaveragedoutinsuchastrongcarrierexcitationregime. T heinstantaneousMnandholespinsplittingwasprovided
+by the mean field of the p-dexchange interaction. The hole dynamics was then described by the spin-resolved dis-
+tributionfunction fnkwithndenotingthe (spin)bandindex. Themagnetizationdynamics was obtained,in principle,
+by solving the coupled kinetic equations of holes and Mn spin . The “inverse Overhauser e ffect”, i.e., the transfer of
+spin polarization from Mn spin system to hole spin system due to the nonequilibriumhole distribution triggered by
+the photo-excitation,was thenincludedin the kineticequa tion. Within such a model,the Mn spin-fliprate dueto the
+“inverseOverhausere ffect”(from mstatetom±1statewith−5/2≤m≤5/2beingtheMnspinindex)canbewritten
+as[826]
+Wm,m±1=β2
+42π|∝an}b∇acketle{tm|S∓|m+1∝an}b∇acket∇i}ht|2/summationdisplay
+nn′/integraldisplaydk
+(2π)3dk′
+(2π)3|∝an}b∇acketle{tn′k′|s±|nk∝an}b∇acket∇i}ht|2fnk(1−fn′k′)δ(˜εnk−˜εn′k′±∆M),(130)
+whereβis thep-dexchange constant and S±(s±) are the Mn (hole) spin ladder operators. ∆Mrepresents the Mn
+instantaneous mean field spin splitting produced by the hole s. ˜εnkis the instantaneous band energy with the mean
+field exchange interaction included. By considering the hol e energy- and spin-relaxation in a phenomenological
+way, the magnetization dynamics was solved within a simplifi ed two band hole approximation [826]. The results
+reproducedthe main features of the ultrafast demagnetizat ion: (i) the process is terminated at the time scale of hole
+energyequilibrationtime τE[826]; (ii)the demagnetizationrateandthe degreeofthema gnetizationchangeincrease
+with the pump fluence and hole spin relaxation rate [826]. The obtained demagnetization time is on the order of
+1 ps [826]. The feature of the magnetization dynamics also qu alitatively agrees with the observation in experiments
+[790, 826]. Further calculations based on the realistic ban d structure and microscopic interactions, such as what has
+been done in paramagneticphase in Ref. [111], is needed to el ucidate the underlying physics unambiguouslyand to
+comparewith theexperimentalresults.
+At lowpumpfluence,suchas ∼10µJcm−2(comparedtothefluence /greaterorsimilar1mJcm−2usedindemagnetizationmea-
+surements),richmagnetizationdynamicsistriggeredbyph otoillumination. Dampedprecessionofthemagnetization
+was observed,where the drivingforce was attributedto the p hoto-inducedchangein the magnetic anisotropyenergy
+[751, 797, 800–802, 804, 805, 829, 830]. It was also discover edthat the magnetizationprecessioncan be coherently
+manipulated by photo pulses [798, 799]. Besides, with circu larly polarized light, the magnetization is rotated due
+to thep-dexchange mean field produced by the photo-injected hole spin polarization [751, 797, 803]. The photo-
+inducedchangeinthemagneticanisotropyenergyhasbeenar guedtooriginatefrom: (i)thechangeofthetemperature
+of the hole gas due to hot photo-excited holes [800]; (ii) the change of the hole density [829]; (iii) the nonthermal
+effect [805]. The dependence of the precession frequency (clos ely related to the magnetic anisotropy energy) and
+dampingconstant(theGilbertconstant)werehencestudied forvariousphotointensity,temperatureandholedensity,
+and compared with theoretical results [827, 828, 831]. It sh ould be mentioned that there is other mechanism which
+can also trigger such magnetization precession, such as the photo-induced spin precession due to the interband po-
+larization proposed by Chovan and Perakis [99]. Very recent ly Kapetanakiset al. [832] developeda nonequilibrium
+theorybasedontheoriginalideaofRef.[99]andwellexplai nedtheexperimentalresultsofmagnetizationprecession
+[791]. In fact, within the first picosecond, the hot photo-ex cited holes can also modify the magnetization direction
+besideschangingthemagnetizationamplitude(demagnetiz ation)duetothefactthatinthepresenceofholespin-orbit
+couplingthe total (hole +Mn)spin polarization is not conserved. Once the magnetizat ion direction deviatesfrom the
+equilibrium one, a damped precession is motivated by the e ffective magnetic field due to the external and internal
+(anisotropy)magneticfields[814].
+73Note that asimilar picture of Mn ion spin relaxation in ZnMnS e/ZnBeSe paramagnetic quantum well was also depicted in Ref. [ 730].
+745. Spin relaxationanddephasing based onkinetic spin Bloch equationapproach
+Inthissection,weintroduceafullymicroscopicmany-body approachnamedkineticspinBlochequationapproach
+developed by Wu et al. [44, 332, 349, 350]. Unlike the approac hes reviewed in the previous sections which treat
+scatterings using the relaxation time approximation, the k inetic spin Bloch equation approach treats all scatterings
+explicitlyandself-consistentlytoallorders. Especiall y,thecarrier-carrierCoulombscatteringisexplicitlyin cludedin
+thetheory. Thisallowsthemtostudyspindynamicsnotonlyn earbutalsofarawayfromtheequilibrium,forexample,
+the spin dynamics in the presence of high electric field (hot- electron condition) [569] and with large initial spin
+polarizations[41,42,44]. Thissectiononlyfocusesonthe resultsinspacialuniformsystemswhilethoseinthespacia l
+non-uniform systems are given in Section 7. We organize this section as follows: In Section 5.1 we first introduce
+the kinetic spin Bloch equationapproach. Then we address th e effect of electron-electronCoulomb scatteringon the
+spindynamicsinSection5.2. ThegeneralkineticspinBloch equationsin n-orp-dopedconfinedstructuresaregiven
+in Section 5.3 where we even include the case of spacial gradi ent.74The results of spin relaxation and dephasing in
+quantumwells, quantumwires,andbulksamplesarereviewed inSections5.4,5.5and5.6,respectively.
+5.1. KineticspinBlochequationapproach
+5.1.1. ModelandHamiltonian
+We first use a four-spin-band model to demonstrate the establ ishment of the kinetic spin Bloch equations and
+applytheseequationstoinvestigatethespinprecessionan drelaxation/dephasing. Byconsideringa(001)zincblende
+quantum well with its growth axis in the z-direction and a moderate magnetic field along the x-direction (Voigt
+configuration),wewritethe Hamiltonianofelectronsinthe conductionandvalencebandsas
+H=/summationdisplay
+µkσεµkc†
+µkσcµkσ+/summationdisplay
+µkσσ′gµB[B+Ωµ(k)]·Sµσσ′c†
+µkσcµkσ′+HE+HI, (131)
+withµ=candvstanding for the conduction and valence bands, respectivel y. Here we only include the lowest
+subbands of the conduction and valence bands. This is valid f or quantum wells with su fficiently small well widths.
+εvk=−Eg/2−k2/2mh=−Eg/2−εhkandεck=Eg/2+k2/2me=Eg/2+εekwithmhandmedenotingeffectivemasses
+of the hole and electron, respectively. Egis the band gap. For quantum well without strain, the heavy ho le band is
+abovethe light hole one and hence the valence band to be consi deredis the heavy-holeband with Svσσ′representing
+spin3/2matricesand σ=±3/2.Scσσ′standsforthespin1 /2matricesforconductionelectrons. Ωµ(k)representsthe
+effectivemagneticfieldfromtheD’yakonov-Perel’termofthe µ-band.
+HEin Eq. (131) denotes the dipole coupling with the light field E−σ(t) withσ=±representing the circularly
+polarizedlight. Duetothe selectionrule
+HE=−d/summationdisplay
+k[E−(t)c†
+ck1
+2cvk3
+2+H.c.]−d/summationdisplay
+k[E+(t)c†
+ck−1
+2cvk−3
+2+H.c.], (132)
+in which ddenotes the optical-dipole matrix element and Eσ(t)=E(0)
+σ(t)cosωtwithωbeing the central frequency
+of the light pulse. E(0)
+σ(t) denotes a Gaussian pulse with pulse width δt.HIstands for the interactionHamiltonian. It
+contains both the spin-conserving scattering, such as the e lectron-electron Coulomb, electron-phonon and electron-
+impurityscatterings, andthe spin-flipscattering,such as the scatteringdueto the Bir-Aronov-Pikusand Elliott-Yaf et
+mechanisms.
+It can be seen from HEthat the laser pulse introducesthe optical coherencebetwe en the conductionand valence
+bandsPk1
+23
+2≡eiωt∝an}b∇acketle{tc†
+vk3
+2cck1
+2∝an}b∇acket∇i}htandPk−1
+2−3
+2≡eiωt∝an}b∇acketle{tc†
+vk−3
+2cck−1
+2∝an}b∇acket∇i}ht. Inthemeantime,duetothepresenceofthemagneticfield
+in the Voigt configuration, these optical coherences may fur ther transfer coherence to Pk−1
+23
+2≡eiωt∝an}b∇acketle{tc†
+vk3
+2cck−1
+2∝an}b∇acket∇i}htand
+Pk1
+2−3
+2≡eiωt∝an}b∇acketle{tc†
+vk−3
+2cck1
+2∝an}b∇acket∇i}ht, ofwhichthe directopticaltransitionis forbidden. Dueto the presenceofthe magneticfield
+and/oreffectivemagneticfield,ifthereisanyspinpolarization,i.e .,fµkσ≡∝an}b∇acketle{tc†
+µkσcµkσ∝an}b∇acket∇i}ht/nequalfµk−σ,electronscanflipfrom
+74For the case of spin di ffusion and transport, seeSection 7.
+75σ-band to−σ-band, and hence induce the correlation between two spin ban ds, i.e.,ρµkσ−σ≡∝an}b∇acketle{tc†
+µk−σcµkσ∝an}b∇acket∇i}ht≡ρ∗
+µk−σσ.
+Similartotheopticalcoherencewhichisrepresentedby Pkσσ′,thespincoherencecanbewellrepresentedby ρµkσ−σ.
+Inthefollowingwecallitspincoherence. Thesecoherences aswellaselectron/holedistributionsillustratedinFig.47
+arethequantitiestobedeterminedself-consistentlythro ughthe kineticspinBloch equations.
+1/2 -1/2
+σ -σfe1/2fe-1/2
+fhσfh-σρe1/2,-1/2
+P1/2,σP1/2,-σP-1/2,σ
+P-1/2,-σ
+ρhσ,-σ
+Figure47: Illustration ofthefour-spin-band model: twoco nduction bandswithspin ±1/2andtwovalence bandswithspin ±σ. Thespincoherence
+ρ,optical and forbidden optical coherences Pas well as carrier distributions of the four bands are also il lustrated.
+5.1.2. Kinetic spinBlochequations
+We construct the kinetic spin Bloch equations using the none quilibriumGreen functionmethod [833] as follows
+[350]
+˙ρµν,k,σσ′=˙ρµν,k,σσ′|coh+˙ρµν,k,σσ′|scat. (133)
+Hereρµν,k,σσ′represents the single-particle density matrix with µandν=corv. (µandνcan be subband indices
+and/or valley indices also.) The diagonal elements describe the carrier distribution functions ρµµ,k,σσ=fµkσand
+the off-diagonal elements either represent the optical coherence ρcv,k,σσ′=Pkσσ′e−iωtfor the inter conduction band
+and valence band polarization, or the spin coherence ρµµ,k,σ−σfor the inter spin-band correlation. The coherent
+terms ˙ρµν,k,σσ′|cohare composed of the contributions from the energy spectrum, electric pumping field, magnetic
+and/oreffectivemagneticfieldaswell astheCoulombHartree-Fockter m. Thescatteringterms ˙ ρµν,k,σσ′|scatconsistof
+the spin-conservingscatteringssuch as the electron-elec tronCoulomb, electron-phononand electron–non-magnetic -
+impurityscatteringsand /orthespin-flipscatteringssuchasthescatteringsduetoth eBir-Aronov-Pikusand /orElliott-
+Yafetmechanismsandthehyperfineinteraction. Thedetaile dexpressionsofthesetermswillbegiveninthefollowing
+subsectionsfordifferentspecificproblems.
+By solvingthekineticspinBlochequations(133) withthein itial conditions:
+ρµν,k,σσ′(0)=/braceleftiggfµkσ(0) withµ=νandσ=σ′
+0 with µ/nequalνand/orσ/nequalσ′, (134)
+oneobtainsthetimeevolutionofthedensitymatrix.
+5.1.3. Faradayrotationangle,spindephasingandspin rela xation
+The Faraday rotation angle can be calculated for two degener ate Gaussian pulses with a delay time τ. The first
+pulse (pump)is circularly polarized,e.g., E0
+pump(t)=E0
+±(t), and travelsin the k1-direction. The second pulse (probe)
+is linear polarized and is much weaker than the first one, e.g. ,E0
+probe(t)≡χ[E0
+−(t−τ)+E0
++(t−τ)] withχ≪1. The
+probepulsetravelsinthe k2direction. TheFaradayrotationangleisdefinedas[732,834 ]
+ΘF(τ)=C/summationdisplay
+k/integraldisplay
+Re[¯Pk1
+23
+2(t)E0
+−∗(t−τ)−¯Pk−1
+2−3
+2(t)E0
++∗(t−τ)]dt, (135)
+with¯Pkσσ′standing for the optical transition in the probe direction. Cis a constant. The spin relaxation /dephasing
+can be determined from the slope of the envelope of the Farada y rotation angleΘF(τ). However, calculation of this
+76quantityisquitetime consuming. Inreality,weuse theslop eoftheenvelopeof
+∆Nµ=/summationdisplay
+k(fµk|σ|−fµk−|σ|), ρµ=/summationdisplay
+k/vextendsingle/vextendsingle/vextendsingleρµµ,k,|σ|−|σ|/vextendsingle/vextendsingle/vextendsingle, ρ∗
+µ=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+kρµµ,k,|σ|−|σ|/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (136)
+to substract the spin relaxation time T1, spin dephasing time T2and ensemble spin dephasing time T∗
+2, respectively
+[350,835]. Similarly,theopticaldephasingisdescribedb ythe incoherentlysummedpolarization[346,833],
+Pσσ(t)=/summationdisplay
+k/vextendsingle/vextendsingle/vextendsinglePk,σσ(t)/vextendsingle/vextendsingle/vextendsingle. (137)
+ThelaterwasfirstintroducedbyKuhnandRossi[347]. Itisun derstoodthatbothtruedissipationandtheinterference
+amongthe kstatesmaycontributetothedecay. Theincoherentsummatio nisthereforeusedtoisolatetheirreversible
+decayfromthedecaycausedbyinterference.
+5.2. Electron-electronCoulombscatteringtospindynamic s
+Ithaslongbeenbelievedthatthespin-conservingelectron -electronCoulombscatteringdoesnotcontributetothe
+spinrelaxation/dephasing[836]. ItwasfirstpointedoutbyWuandNing[334]t hatinthepresenceofinhomogeneous
+broadening in spin precession, i.e., the spin precession fr equencies are k-dependent, any scattering, including the
+spin-conserving scattering, can cause irreversible spin d ephasing. This inhomogeneous broadening can come from
+the energy-dependent g-factor [225, 226, 334, 837–840], the D’yakonov-Perel’ ter m [332], the random spin-orbit
+coupling[294], and even the momentumdependenceof the spin diffusionrate alongthe spacial gradient[336]. This
+can be illustrated from the kinetic spin Bloch equationsof a four-spinbandmodelin a quantumwell with the lowest
+valence band being the heavy-hole band and ρcv,k,σσ′=Pkσσ′e−iωt. When the D’yakonov-Perel’term Ω(k)=0, the
+coherentpartsofthekineticspinBlochequationsare given by[334]
+∂fekσ
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoh=2/summationdisplay
+qVqIm/parenleftbig/summationdisplay
+σ′P∗
+k+qσσ′Pkσσ′+ρcc,k+q,−σσρcc,k,σ−σ/parenrightbig−gµBBImρcc,k,σ−σ,(138)
+∂ρcc,k,σ−σ
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoh=i/summationdisplay
+qVq/bracketleftbig(fek+qσ−fek+q−σ)ρcc,k,σ−σ−(fekσ−fek−σ)ρcc,k+q,σ−σ
++Pk+qσσ1P∗
+k−σσ1−P∗
+k+q−σσ1Pkσσ1/bracketrightbig+i
+2gµBB(fekσ−fek−σ), (139)
+∂Pkσσ′
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoh=−iδσσ′(k)Pkσσ′−i
+2gµBBPk−σσ′−i/summationdisplay
+qVq/bracketleftbigPk+qσσ′(1−fhkσ′
+−fekσ)−Pk+q−σσ′ρcc,k,σ−σ+Pk−σσ′ρcc,k+q,σ−σ/bracketrightbig, (140)
+for electron distribution function fekσ≡fckσ, spin coherenceand optical coherence, respectively. fhkσ≡1−fvkσis
+the hole distribution function. The first term on the right ha nd side of Eq. (138) is the Fock term from the Coulomb
+scattering with Vq=2πe2
+qdenotingthe Coulomb matrix element. The first term of Eq. (14 0) gives the free evolution
+ofthepolarizationcomponentswiththe detuning
+δσσ′(k)=εehk−∆0−/summationdisplay
+qVq(fek+qσ+fhk+qσ′), (141)
+inwhichεehk=εek+εhkand∆0=ω−Eg.∆0isthedetuningofthecenterfrequencyofthelightpulseswi threspect
+tothe unrenormalizedbandgap. Thelast terminEq.(140)des cribestheexcitoniccorrelationswhereasthe first term
+inEq.(139)describestheHartree-Fockcontributionstoth espincoherence.
+Forspin-conservingscattering,
+/summationdisplay
+k∂ρµν,k,σσ′
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglescat=0. (142)
+77By performing the summation over kfrom both sides of the kinetic spin Bloch equations and furth er noticing that
+theHartree-FockcontributionsfromtheCoulombinteracti oninthecoherentpartsofthekineticspinBlochequations
+satisfy
+/summationdisplay
+k∂ρµν,k,σσ′
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleHF
+coh=0, (143)
+onehas
+∂2
+∂t2Im/summationdisplay
+kρcc,k,σ−σ=−g2µ2
+BB2Im/summationdisplay
+kρcc,k,σ−σ, (144)
+∂
+∂tRe/summationdisplay
+kρcc,k,σ−σ=0. (145)
+Thisshowsthat there is no spin dephasingin the absence of in homogeneousbroadeningand spin-flip scattering. For
+opticaldephasing, δσσ′(k)inEq.(140)is k-dependent,i.e.,hereexiststheinhomogeneousbroadenin g. Thereforeany
+optical-dipoleconservingscattering can lead to irrevers ible optical dephasing[833]. Similarly, if there is inhomo ge-
+neous broadening in the spin precession, any spin-conservi ng scattering can lead to irreversible spin dephasing. As
+electron-electronCoulombscatteringis a spin-conservin gscattering,of courseit contributesto the spindephasing in
+thepresenceoftheinhomogeneousbroadening[334].
+WuandNingfirstshowedthatwiththeenergy-dependent g-factorasaninhomogeneousbroadening,theCoulomb
+scatteringcanleadtoirreversiblespindephasing[334]. I n[001]-grown n-dopedquantumwells,theimportanceofthe
+Coulomb scattering for spin relaxation /dephasing was proved by Glazov and Ivchenko [615] by using pe rturbation
+theoryandbyWeng andWu[44]fromthefullymicroscopickine ticspinBlochequationapproach. Ina temperature-
+dependentstudyofthespindephasingin[001]-oriented n-dopedquantumwells,Leylandetal. experimentallyverifie d
+the effects of the electron-electronCoulomb scattering [368, 591 ]. Later Zhou et al. even predicted a peak from the
+Coulombscatteringinthetemperaturedependenceofthespi nrelaxationtimeinahigh-mobilitylow-density n-doped
+[001]quantumwell[372]. ThiswaslaterdemonstratedbyRua net al. experimentally[604].
+5.3. KineticspinBlochequationsin n-or p-dopedconfinedst ructures
+In this section we present the spin dynamics in n- orp-type confined semiconductor structures. In this case, we
+onlyneed to considera simplified single-particledensity m atrixρkwhich consistsonly the electron /holedistribution
+functionsandtheinter-spin-bandpolarization(spincohe rence). Intheeffective-massapproximation,theHamiltonian
+ofelectronsinthe confinedsystem reads
+H=/summationdisplay
+i[He(Ri)+eE·ri]+HI, (146)
+He(R)=P2
+2m∗+Hso(P)+V(rc)+1
+2g∗µBB·σ, (147)
+inwhichrcrepresentsthecoordinatealongtheconfinementand R=(r,rc).P=−i/planckover2pi1∇−eA(R)isthemomentumwith
+B=∇×A.HsoinEq.(147)istheHamiltonianofthespin-orbitcouplingwh ichconsiststheDresselhaus /Rashbaterm
+aswell asthe strain-inducedspin-orbitcoupling. V(rc) representstheconfinementpotential. Theenergyspectrum of
+single-particleHamiltonian(147) reads
+He|k,n∝an}b∇acket∇i}ht=εn(k)|k,n∝an}b∇acket∇i}ht (148)
+withndenotingtheindexforbothsubband[subjectedtothe confine mentV(rc)] andspin. Hence |k,n∝an}b∇acket∇i}ht=|k∝an}b∇acket∇i}ht|n∝an}b∇acket∇i}htk.
+Due to the spin-orbit coupling, k∝an}b∇acketle{tn|n′∝an}b∇acket∇i}htk′usually is not diagonal if k′/nequalk. The Hilbert space spanned by |k,n∝an}b∇acket∇i}ht
+is helix space [597, 841]. The kinetic spin Bloch equations i n this space are very complicated. Usually we use the
+wavefunctions|n∝an}b∇acket∇i}htkat a fixed k0as a completebasis, i.e., {|n∝an}b∇acket∇i}ht0}. Therefore,the Hilbertspace becomes {|k∝an}b∇acket∇i}ht|n∝an}b∇acket∇i}ht0}which
+iscompleteandorthogonal. Wecallthisspacecollinearspa ce. Theeigenfunction |k,n∝an}b∇acket∇i}htcanbeexpandedinthisspace
+as
+|k,n∝an}b∇acket∇i}ht=/summationdisplay
+m0∝an}b∇acketle{tm|∝an}b∇acketle{tk|k,n∝an}b∇acket∇i}ht|k∝an}b∇acket∇i}ht|m∝an}b∇acket∇i}ht0. (149)
+78Thematrixelementofthe Hamiltonian Hein thecollinearspacereads
+Enn′
+k=0∝an}b∇acketle{tn|∝an}b∇acketle{tk|He|k∝an}b∇acket∇i}ht|n′∝an}b∇acket∇i}ht0. (150)
+As|k∝an}b∇acket∇i}ht|n∝an}b∇acket∇i}ht0in generalis notthe eigenfunctionof He,Enn′
+kis not diagonal. eE·rin Eq.(146) is the drivingterm of the
+electricfieldalongtheconfinement-freedirections. HIistheinteractionHamiltonian.
+Inthecollinearspace,the densityoperatorcanbewrittena s
+ρ(Q)=/summationdisplay
+k,n1n2In1n2(qc)c†
+k+qn1ckn2, (151)
+inwhichIn1n2(qc)=0∝an}b∇acketle{tn1|eiqc·rc|n2∝an}b∇acket∇i}ht0istheformfactor. TheinteractionHamiltonian HIreads
+Hee=/summationdisplay
+Qkk′,n1n2n3n4V(Q)In1n2(qc)In3n4(qc)c†
+k′+qn3c†
+k−qn1ckn2ck′n4, (152)
+Hep=/summationdisplay
+Qk,n1n2,λMλ(Q)In1n2(qc)c†
+k+qn1ck,n2(aQλ+a†
+−Qλ), (153)
+Hei=/summationdisplay
+Qk,n1n2Vi(Q)In1n2(qc)ρi(Q)c†
+k+qn1ckn2, (154)
+forelectron-electron,electron-phononandelectron-imp urityinteractions,respectively.
+By solving the non-equilibriumGreenfunctionwith general izedKadanoff-BaymAnsatz and the gradientexpan-
+sion[833],thekineticspinBlochequationsread
+∂ρk(r,t)
+∂t=∂
+∂tρk(r,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledr+∂
+∂tρk(r,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledif+∂
+∂tρk(r,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoh+∂
+∂tρk(r,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglescat. (155)
+Thefirst termontherighthandside ofthe aboveequationis
+∂
+∂tρk(r,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledr=1
+2{∇r¯εk(r,t),∇kρk(r,t)} (156)
+with{A,B}=AB+BArepresenting the anti-commutator. (¯ εk(r,t))n1n2=En1n2
+k+eE·rδn1n2+Σn1n2
+HF(k,r,t), in which
+Σn1n2
+HF(k,r,t)=−/summationtext
+QI(qc)ρk−q(r,t)V(Q)I(−qc)is theHartree-Fockterm. Thesecondtermdescribesthe di ffusion
+∂
+∂tρk(r,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledif=−1
+2{∇k¯εk(r,t),∇rρk(r,t)}. (157)
+Thecoherenttermisgivenby
+∂
+∂tρk(r,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoh=−i[¯εk(r,t),ρk(r,t)], (158)
+where[A,B]=AB−BAisthe commutator. ThescatteringterminEq.(155) reads
+∂
+∂tρk(r,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglescat=−{Sk(>,<)−Sk(<,>)+Sk(>,<)†−Sk(<,>)†}, (159)
+with
+Sk(>,<)=/summationdisplay
+QNi/integraldisplayt
+−∞dτVi(Q)I(qc)e−iEk−q(t−τ)ρ>
+k−q(τ)Vi(−Q)I(−qc)ρ<
+k(τ)eiEk(t−τ)
++/summationdisplay
+Q/integraldisplayt
+−∞dτMλ(Q)I(qc)e−iEk−q(t−τ)ρ>
+k−q(τ)Mλ(−Q)I(−qc)ρ<
+k(τ)eiEk(t−τ)[N<(Q)eiωQ(t−τ)+N>(Q)e−iωQ(t−τ)]
++/summationdisplay
+Qk′q′c/integraldisplayt
+−∞dτV(Q)I(qc)e−iEk−q(t−τ)ρ>
+k−q(τ)V(−q′
+c)I(−q′
+c)ρ<
+k(τ)eiEk(t−τ)Tr[I(−qc)e−iEk(t−τ)ρ>
+k′(τ)
+×I(q′
+c)ρ<
+k′−q(τ)e−iEk′−q(t−τ)]. (160)
+79HereN<(Q)=N(Q) is the phonon distribution, N>(Q)=N(Q)+1,ρ>
+k=1−ρk, andρ<
+k=ρk.Niis the impurity
+density. As Ek=/summationtext
+nεn(k)Tk,nwithTk,n=|k,n∝an}b∇acket∇i}ht∝an}b∇acketle{tk,n|,eiEkt=/summationtext
+neiεn(k)tTk,n. By furtherassumingthe spin precession
+period is much longer than the average momentum scattering t ime and applying the Markovian approximation, i.e.,
+replacingρk(τ)in theintegrandofEq.(160)by
+ρk(τ)≈eiEk(t−τ)ρk(t)e−iEk(t−τ), (161)
+thetimeintegralcanbecarriedoutandonearrivesat theene rgyconservation:
+Sk(>,<)=π/summationdisplay
+Q,n1n2NiVi(Q)I(qc)ρ>
+k−q(t)Tk−q,n1Vi(−Q)I(−qc)Tk,n2ρ<
+k(t)δ(εn1(k−q)−εn2(k))
++π/summationdisplay
+Q,n1n2Mλ(Q)I(qc)ρ>
+k−q(t)Tk−q,n1Mλ(−Q)I(−qc)Tk,n2ρ<
+k(t)
+×[N<(Q)δ(εn1(k−q)−εn2(k)−ωQ)+N>(Q)δ(εn1(k−q)−εn2(k)+ωQ)]
++π/summationdisplay
+Qk′q′c,n1n2n3n4V(Q)I(qc)ρ>
+k−q(t)Tk−q,n1V(−Q)I(−q′
+c)Tk,n2ρ<
+k(t)
+×Tr[I(−qc)ρ>
+k′(t)Tk′,n3I(q′
+c)Tk′−q,n4ρ<
+k′−q(t)]δ(εn1(k−q)−εn2(k)+εn3(k′)−εn4(k′−q)).(162)
+Theexpressionof Sk(<,>)canbeobtainedbyinterchanging <and>in Eqs.(161)and(162).
+5.4. Spinrelaxationanddephasingin n-or p-typequantumwe lls
+Inthissection,wereviewthespindynamicsin n-orp-typequantumwellsundervariousconditions. Wefirstwrite
+the kinetic spin Bloch equationsin both the helix and collin ear spin spaces in quantumwells in Sec. 5.4.1. Then we
+reviewthespindynamicsneartheequilibriumandfarawayfr omtheequilibriuminSec.5.4.2and5.4.3respectively.
+We highlight the e ffect of the Coulomb scattering to the spin relaxation and deph asing in Sec. 5.4.4. After that we
+reviewthenon-Markoviane ffectofholespindynamicsinSec.5.4.5. Wereviewtheelectro nspinrelaxationduetothe
+Bir-Aronov-PikusmechanisminSec.5.4.6. Thespindynamic sinthepresenceofastrongTHzlaserfieldisreviewed
+in Sec. 5.4.7. Spin relaxation in paramagnetic GaMnAs quant um wells,75GaAs (110) quantum wells, Si /SiGe and
+Ge/SiGequantumwellsandwurtziteZnO(001)quantumwellsarer eviewedinSecs. 5.4.8-5.4.11,respectively.
+5.4.1. Kinetic spinBlochequationsin (001)quantumwells
+We first considera quantumwell with a small well width sothat onlythelowest subbandis needed. The electron
+HamiltonianofEq.(147) nowreads
+He=k2
+2m∗+h(k)·σ+ε0. (163)
+Hereε0is the energy of the lowest subband. It can be calculated from the Schr¨ odinger equation with confinement
+potential V(rc)=V(z).h(k)=1
+2gµB[B+Ω2D(k)] withΩ2D(k) being the Dresselhaus and /or Rashba terms. The
+eigenenergy and eigenfunction of Eq. (163) are εξ(k)=k2
+2m∗+ξ|h(k)|+ε0and|k,ξ=±∝an}b∇acket∇i}ht=Tk,ξ|↑∝an}b∇acket∇i}ht
+∝an}b∇acketle{t↑|Tk,ξ|↑∝an}b∇acket∇i}ht, respectively.
+The projector operator reads Tk,ξ=1
+2[1+ξh(k)
+|h(k)|·σ]. The space spanned by {|k,ξ∝an}b∇acket∇i}ht}is the helix space. As |k,ξ∝an}b∇acket∇i}htis
+k-dependent,thekineticspinBlochequationsaremorecompl icatedinthisspace. Meanwhile,onemayusethespace
+spannedby{|k∝an}b∇acket∇i}ht|n∝an}b∇acket∇i}ht0}. Here we choose k0=0,i.e., theΓ-pointandthen|n∝an}b∇acket∇i}ht0isthe eigenvectorof σz,|σ∝an}b∇acket∇i}ht=|↑∝an}b∇acket∇i}htor|↓∝an}b∇acket∇i}ht,
+whichisk-independent. Thisspaceisthecollinearspace. Inthe coll inearspinspace,thedensitymatrixis
+ρk=/parenleftiggfk↑ρk↑↓
+ρk↓↑fk↓/parenrightigg
+. (164)
+Whenanelectricfield Eisappliedalongthequantumwell, thekineticspinBlochequ ationsaregivenby[28,569]
+∂ρk(t)
+∂t=eE·∇kρk(t)−i[h(k)·σ+ΣHF(k,t),ρk(t)]+ρk(t)
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglescat. (165)
+75RecentlyShen etal.havefurtherextendedthekineticspinBlochequationtheor ytostudytheGilbertdampinginferromagneticsemiconduct ors
+[842].
+80HereΣHF(k,t)=−/summationtext
+qVqρk−q(t),andSk(>,<)inthescatteringterms[Eq.(162)]reads
+Sk(>,<)=πNi/summationdisplay
+q,η1η2|Uq|2ρ>
+k−q(t)Tk−q,η1Tk,η2ρ<
+k(t)δ(εη1(k−q)−εη2(k))
++π/summationdisplay
+Q,η1η2|gQ,λ|2ρ>
+k−q(t)Tk−q,η1Tk,η2ρ<
+k(t)[N<(Q)δ(εη1(k−q)−εη2(k)−ωQ)
++N>(Q)δ(εη1(k−q)−εη2(k)+ωQ)]
++π/summationdisplay
+qk′,η1η2η3η4V2
+qρ>
+k−q(t)Tk−q,η1Tk,η2ρ<
+k(t)Tr[ρ>
+k′(t)Tk′,η3Tk′−q,η4ρ<
+k′−q(t)]
+×δ(εη1(k−q)−εη2(k)+εη3(k′)−εη4(k′−q)), (166)
+with|Uq|2=/summationtext
+qzVi(Q)Vi(−Q)I(qz)I(−qz) and|gQ,λ|2=Mλ(Q)I(qz)Mλ(−Q)I(−qz) andVq=/summationtext
+qzV(Q)I(qz)I(−qz)
+being the matrix elements with the form factors. It is noted t hatIσ1σ2(qz) is diagonal in the collinear spin space |σ∝an}b∇acket∇i}ht.
+In the calculation, one either uses the static screening or t he screening under the random-phaseapproximation[372]
+forVqandUq[843], according to the di fferent conditions of investigation. In the collinear spin sp ace, the screened
+Coulombpotentialandelectron-impurityinteractionpote ntialin therandom-phaseapproximationread
+Vq=/summationdisplay
+qzvQ|I(iqz)|2/ǫ(q), (167)
+|Uq|2=/summationdisplay
+qzu2
+Q|I(iqz)|2/ǫ(q)2, (168)
+wherevQ=4πe2
+Q2isthe bareCoulombpotential, u2
+Q=Z2
+iv2
+QwithZithechargenumberofimpurity,and
+ǫ(q)=1−/summationdisplay
+qzvQ|I(iqz)|2/summationdisplay
+kσfk+qσ−fkσ
+ǫk+q−ǫk. (169)
+Eq. (165) is valid in both the collinear and helix spin spaces . By performinga unitary transformation ρh
+k=U†
+kρc
+kUk,
+onemaytransferthedensitymatrixfromthecollinearspace ρc
+ktothehelixone ρh
+k, with
+Uk=/parenleftigg∝an}b∇acketle{t↑|k,+∝an}b∇acket∇i}ht ∝an}b∇acketle{t↑|k,−∝an}b∇acket∇i}ht
+∝an}b∇acketle{t↓|k,+∝an}b∇acket∇i}ht ∝an}b∇acketle{t↓|k,−∝an}b∇acket∇i}ht/parenrightigg
+. (170)
+Finally,wepointoutthattheenergyspectrum εη(k)inthescatteringEq.(166)containsthespin-orbitcoupli ng. When
+thecouplingismuchsmallerthanthe Fermienergy,onemayne glectthecouplingandhence
+δ(εη(q)−εη′(k))≈δ(ε(q)−ε(k)). (171)
+Byfurtherutilizingthe relation/summationdisplay
+ηTk,η=1, (172)
+thescatteringbecomes
+Sk(>,<)=πNi/summationdisplay
+q|Uq|2ρ>
+k−q(t)ρ<
+k(t)δ(ε(k−q)−ε(k))
++π/summationdisplay
+Q|gQ,λ|2ρ>
+k−q(t)ρ<
+k(t)[N<(Q)δ(ε(k−q)−ε(k)−ωQ)+N>(Q)δ(ε(k−q)−ε(k)+ωQ)]
++π/summationdisplay
+qk′V2
+qρ>
+k−q(t)ρ<
+k(t)Tr[ρ>
+k′(t)ρ<
+k′−q(t)]δ(ε(k−q)−ε(k)+ε(k′)−ε(k′−q)).
+This is the form used in the electron systems [25, 27–29, 32, 4 1, 42, 44, 69, 111, 199, 336, 372, 569, 570, 609, 835,
+844–846].
+81Before discussingthe results fromthe full kinetic spin Blo ch equations,we first show a simplest case by keeping
+onlythe electron-impurityscatteringin the scatteringte rm,whereonecanobtainananalyticalsolutionin thespacia l
+homogeneous system by further neglecting the Coulomb Hartr ee-Fock contribution [363]. The kinetic spin Bloch
+equationsthenread
+∂
+∂tρk(t)=−i[α(kyσx−kxσy),ρk(t)]−2πNi/summationdisplay
+q|Uq|2δ(εk−q−εk)[ρk(t)−ρk−q(t)]. (173)
+Byexpanding ρkasρk=/summationtext
+lρl(k)eiθkl, onehas
+∂
+∂tρl(k,t)=αk/braceleftbig[S+,ρl+1(k,t)]−[S−,ρl−1(k,t)]/bracerightbig−|Ul(k)|2ρl(k,t), (174)
+inwhich|Ul(k)|2=m∗Ni
+2π/planckover2pi12/integraltext2π
+0dθ|U(/radicalbig
+2k2(1−cosθ))|2(1−coslθ) with|U0(k)|2=0and|U−1(k)|2=|U1(k)|2≡1
+τp(k). By
+multiplying1
+2σandthenperformingtrace frombothsidesofEq.(174),oneco mesto
+∂
+∂tSl(k,t)=αk[F†Sl+1(k,t)−FSl−1(k,t)]−|Ul(k)|2Sl(k,t), (175)
+withSl(k,t)=1
+2Tr[ρl(k,t)σ]andF=0 0 1
+0 0−i
+−1i0. InderivingEq.(175),wehaveusedtherelation1
+2Tr([S+,ρl]σ)=
+F†Sl(k,t) and1
+2Tr([S−,ρl]σ)=FSl(k,t). By keepingonly the lowest threeordersof Slin the strongscattering limit
+(xk≪1),i.e.,l=0,±1,onehas
+∂
+∂t−αk−1
+αkτp(k)−F 0
+F†0−F
+0F†−1
+αkτp(k)S1
+S0
+S−1=0. (176)
+ThesolutionofEq.(176)reads
+S1(k,t)=−xk
+2/radicalbig
+1−x2xe−t
+2τp(k)sinht
+2τp(k)//radicalig
+1−x2
+k1
+i
+0f(εk−µ), (177)
+S0(k,t)=e−t
+2τp(k)sinht
+2τp(k)/√
+1−x2
+k/radicalig
+1−x2
+k+cosht
+2τp(k)//radicalig
+1−x2
+k0
+0
+1f(εk−µ), (178)
+S−1(k,t)=−xk
+2/radicalbig
+1−x2xe−t
+2τp(k)sinht
+2τp(k)//radicalig
+1−x2
+k1
+−i
+0f(εk−µ), (179)
+with the initial conditions being S1(k,0)=S−1(k,0)=0 andS0(k,0)=f(εk−µ)ez, i.e., the initial spin polarization
+being along the z-axis.xkin Eqs. (177-179) is xk=4αkτp(k). From Eqs. (177-179), electron spin at momentum k
+reads
+Sk(t)=S0(k,t)+S1(k,t)eiθk+S−1(k,t)e−iθk, (180)
+andhencethecomponentalongthe z-axisis givenby
+Sz
+k(t)=Sz
+0(k,t)=1
+2f(εk−µ)/bracketleftbigg/parenleftbigg
+1+1/slashbig/radicalig
+1−x2
+k/parenrightbigg
+e−t
+2τp(k)(1−√
+1−x2
+k)+/parenleftbigg
+1−1/slashbig/radicalig
+1−x2
+k/parenrightbigg
+e−t
+2τp(k)(1+√
+1−x2
+k)/bracketrightbigg
+.(181)
+From Eq. (181), one can see the di fferent time evolutions of the spin polarization at di fferent regimes. When xk>
+1, i.e.,αkτp(k)>1/4, the system is in the weak scattering regime and the terms e±t
+2τp(k)√
+1−x2
+kgive just the spin
+oscillations. Hencethespinrelaxationisgivenby
+1/τs(k)=1/[2τp(k)]. (182)
+82However,thespinoscillationfrequencyis ωk=/radicalbig
+(2αk)2−1/[2τp(k)]2. Onecanseefrom ωkthatthescatteringtends
+to suppress the spin oscillation frequency. This indicates the counter-effect of the scattering to the inhomogeneous
+broadening. When xk<1, thespinpolarizationdecaysaccordingto
+1/τs,±(k)=/parenleftbigg
+1±/radicalig
+1−[4αkτp(k)]2/parenrightbigg/slashbig[2τp(k)]. (183)
+Inthestrongscatteringlimit,i.e., xk≪1,
+1/τs,+(k)=1/τp(k), (184)
+1/τs,−(k)=(2αk)2τp(k). (185)
+Asxk≪1,τs,−(k)≫τs,+(k) and the spin relaxation is determined by τs,−(k). It is noted that τs,−(k) is exactly the
+resultintheliterature[3, 101].
+Finally, one can see from Eqs. (183) and (185) that in the weak scattering regime, a stronger scattering leads
+to a faster spin relaxation. Nevertheless, in the strong sca ttering regime, a stronger scattering leads to a weaker spin
+relaxation. Thiscanbeunderstoodasfollowing. Inthepres enceofinhomogeneousbroadening,thescatteringhasdual
+effects to the spin relaxation: (i) It gives a spin relaxation ch annel; (ii) It has a counter-e ffect to the inhomogeneous
+broadening by making the system more homogeneous. In the wea k scattering limit, the counter e ffect is weak and
+hence adding a new scattering always leads to an additional r elaxation channel and hence a fast spin relaxation. In
+thestrongscatteringlimit,thecountere ffectbecomessignificantandhenceaddinganewscatteringalw aysleadstoa
+longerspin relaxationtime.
+5.4.2. Spinrelaxationanddephasinginn-type(001)GaAsqu antumwells neartheequilibrium
+By numericallysolving the kinetic spin Bloch equationswit h all the relevant scatteringsincluded,Weng and Wu
+studiedthespindephasingin n-typeGaAsquantumwellsat hightemperature( ≥120K),wheretheelectron-acoustic
+phononscatteringisunimportant,first forsmall well width [44]with onlythelowest subbandandthenforlargewell
+width [844] with mutisubband e ffect considered. They further investigated the hot-electro n effect in spin dephasing
+byapplyingalargein-planeelectricfield[569],wheretheh ot-electroneffectisinvestigated. Byfurtherincreasingthe
+in-plane electric field, electrons can populate higher subb and and/or higher valleys. These e ffects were investigated
+by Weng and Wu [844] for mutisubbandcase and Zhanget al. [845 ] for multivalleycase. The spin relaxationat low
+temperature was first investigated by Zhou et al. [372] from k inetic spin Bloch equation approach by including the
+electron-acousticphononscattering. JiangandWustudied theeffectofstrainonspinrelaxation[570]. Spinrelaxation
+forsystem with competingDresselhausandRashbatermswasi nvestigatedtheoreticallybyChengandWu[597]and
+bothexperimentallyandtheoreticallybyStichetal. [199] . Spinrelaxationwithlargeinitialspinpolarizationwasfi rst
+studied by Weng and Wu [44] and many predictionswere verified experimentallywith good agreement between the
+experimentaldata and theoreticalcalculationsby Stich et al. [41, 42]. The density dependenceof the spin relaxation
+wasalsoinvestigatedboththeoreticallyandexperimental ly[609]. Inthisandnextsectionswereviewthemainresults
+oftheaboveinvestigations.
+It was revealed that the spin relaxation time based on the D’y akonov-Perel’ mechanism has a rich temperature
+dependencedependingondi fferentimpuritydensities,carrierdensitiesandwell width s.
+Figure48showsthetemperaturedependenceofthe spinrelax ationtimeofa 7.5nmGaAs /Al0.4Ga0.6Asquantum
+well at different electron and impurity densities [372]. For this small well width, only the lowest subband is needed
+inthe calculation. It isshowninthefigurethat whenthe elec tron-impurityscatteringisdominant,the spinrelaxation
+time decreases with increasing temperature monotonically . This is in good agreement with the experimental finding
+asshowninFig.49,wherethedotsareexperimentaldatafrom Ohnoetal. [602],andthetheoreticalcalculationbased
+onthekineticspinBlochequationswellreproducestheexpe rimentalresultsfrom20Kto300K[372]. However,itis
+shownthatforsamplewithhighmobility,i.e.,lowimpurity density,whentheelectrondensityislowenough,thereisa
+peakatlowtemperature. Thispeak,locatedaroundtheFermi temperatureofelectrons Te
+F=EF/kB,isidentifiedtobe
+solelyduetotheCoulombscattering[372,847]. Itdisappea rswhentheCoulombscatteringisswitchedo ff,asshown
+by the dashed curvesin the figure. This peak also disappearsa t high impurity densities. It is also noted in Fig. 48(c)
+thatforelectronsofhighdensitysothat Te
+Fishighenoughandthecontributionfromtheelectron–longi tudinaloptical-
+phononscatteringbecomesmarked,thepeakdisappearseven forsamplewithnoimpurityandthespinrelaxationtime
+83101102
+ 0  50  100  150  200  250  300 
+ τ (ps)
+T (K)n=4 × 1010/cm2(a)
+101102
+ 0  50  100  150  200  250  300 
+ τ (ps)
+T (K)n=1 × 1111/cm2(b)
+101102
+ 0  50  100  150  200  250  300 
+ τ (ps)
+T (K)n=2 × 1011/cm2(c)
+Figure 48: Spin relaxation time τvs.the temperature Tin GaAs (001) quantum well with well width a=7.5 nm and electron density nbeing (a)
+4×1010cm−2,(b)1×1011cm−2,and (c) 2×1011cm−2,respectively. Solid curves with triangles: impurity dens ityni=n;solid curves with dots:
+ni=0.1n;solid curves with circles: ni=0; dashed curves with dots: ni=0.1nand no Coulomb scattering. From Zhou etal. [372].
+101102103
+ 10  100 
+ τ (ps)
+T (K)(a)
+ 
+experiment
+γ=0.9γ0γ=1.0γ0γ=1.1γ0 10
+ 9
+ 8
+ 7
+ 6
+ 5
+ 4
+ 3
+ 10  100 
+ µHall (103 cm2/Vs)
+T (K)(b)
+Figure 49: (a) Spin relaxation time τvs.temperature Tfor GaAs (001) quantum well with well width a=7.5 nm and electron density n=
+4×1010cm−2atthreedifferent spin-splitting parameters. Dots: experimental data ; dot-dashed curve: γ=0.9γ0;solidcurve: γ=γ0;dashedcurve:
+γ=1.1γ0.γ0=0.0114 eV·nm3. (b) Hall mobility µHallvs.temperature T(Ref. [602]), from which the temperature dependence of the i mpurity
+density is determined. From Zhouet al. [372].
+increases with temperature monotonically. The physics lea ding to the peak is due to the crossover of the Coulomb
+scattering from the degenerate to the non-degeneratelimit . AtT<Te
+F, electrons are in the degenerate limit and the
+electron-electronscatteringrate1 /τee∝T2. AtT>Te
+F,1/τee∝T−1[843,848]. Therefore,atlowelectrondensityso
+thatTe
+Fis lowenoughandthe electron-acousticphononscatteringi sveryweak comparingwith the electron-electron
+Coulombscattering, the Coulombscattering is the dominant scatteringfor highmobilitysample. Hence the di fferent
+temperature dependence of the Coulomb scattering leads to t he peak. It is noted that the peak is just a feature of
+the crossoverfrom the degenerateto the non-degeneratelim it. The location of the peak also dependson the strength
+of the inhomogeneousbroadening. When the inhomogeneousbr oadening depends on momentum linearly, the peak
+tendstoappearattheFermitemperature. Asimilarpeakwasp redictedintheelectronspinrelaxationin p-typeGaAs
+quantumwellandtheholespinrelaxationin(001)straineda symmetricSi/SiGequantumwell,wheretheelectronand
+hole spin relaxationtimes both show a peak at the hole Fermi t emperature Th
+F[112, 299]. When the inhomogeneous
+broadening depends on momentum cubically, the peak tends to shift to a lower temperature. It was predicted that a
+peakinthetemperaturedependenceoftheelectronspinrela xationtimeappearsatatemperatureintherangeof( Te
+F/4,
+Te
+F/2) in the intrinsic bulk GaAs [110] and a peak in the temperatu re dependence of the hole spin relaxation time at
+Th
+F/2 inp-type Ge/SiGe quantum well [299]. Ruan et al. demonstrated the peak ex perimentally in a high-mobility
+low-densityGaAs/Al0.35Ga0.65As heterostructure[604] as shownin Fig. 50 where a peakappe arsatTe
+F/2 in the spin
+relaxationtime versustemperaturecurve.
+It is also seen from Fig. 48 that at high temperature ( >100 K), except for the case where the electron-impurity
+84Figure 50: (a) Measured electron g-factor as a function of temperature for two-dimensional el ectron gas (squares) in GaAs /Al0.35Ga0.65As het-
+erostructure and bulk GaAs (circles). (b) Measured electro n spin relaxation/dephasing time as a function of temperature for two-dimensi onal
+electron gas (squares) and bulk GaAs (circles). All of the da ta were taken at B=0.5 T and powers of pump:probe =200:20µW. From Ruan et al.
+[604].
+scattering dominates the scattering, the spin relaxation t ime increases with increasing temperature. Weng and Wu
+also showed this in a 15 nm quantum well with a magnetic field B=4 T in the Voigt configuration [44]. They
+evencomparedthespinrelaxationtimesobtainedfromtheki neticspinBlochequationapproachandfromthesingle-
+particlemodel[196,577]1 /τ=/integraltext∞
+0dEk(fk1
+2−fk−1
+2)Γ(k)//integraltext∞
+0dEk(fk1
+2−fk−1
+2),inwhichΓ(k)isthespinrelaxationrate.
+It isseen fromthe inset ofFig. 51 thatthe spin relaxationti me based onthe previoussingle-particlemodeldecreases
+rather than increases with increasing temperature. As the i mpurity density is set to zero in both computations, the
+maindifferencecomesfromtheCoulombscatteringwhichis missingin thesingleparticlecalculation.
+However,even for sample without any impurity,the temperat uredependenceof the spin relaxation time can also
+decreasewithincreasingtemperatureathightemperatures . ThiswasshownbyWengandWuinGaAs(001)quantum
+wellwith widerwellwidth[844]. Inthecalculation,theyco nsideredtwo wellwidths, i.e.,17.8nmand12.7nm,and
+showed the spin relaxation times as function of temperature with both the lowest subband and multisubband e ffects
+considered. One cansee fromthe dashedcurves(onlythe lowe stsubbandiscalculated)inFig. 52thatforsmall well
+width, the spin relaxation time increases with increasing t emperature but for larger well width, the spin relaxation
+timefirstincreasesandthendecreaseswithtemperature. It isalsonotedthatwhenthemultisubbande ffectisincluded
+(solidcurves),similarresultsarealso obtained.
+The physics leading to aboverich behaviorsoriginates from the competition between the inhomogeneousbroad-
+eningandthe scattering. With theincreaseof temperature, electronsare distributedtohighermomentumstates. This
+leads to a larger inhomogeneous broadening from the D’yakon ov-Perel’ term and hence a shorter spin relaxation
+time. Inthemeantime,highertemperaturealsocausesscatt ering(especiallytheelectron-phononscattering)strong er.
+The scattering tends to make electrons distribute more homo geneouslyand suppresses the inhomogeneousbroaden-
+ing. Therefore the scattering tends to cause a longer spin re laxation time with the increase of temperature. When
+the electron-impurity scattering dominates the whole scat tering, the spin relaxation time decreases with increasing
+temperaturemonotonically,thanksto the increase of the in homogeneousbroadeningtogether with the weak temper-
+ature dependence of the electron-impurityscattering. For samples with high mobility, at low temperature where the
+scattering is determined by the electron-electron Coulomb scattering, a peak appears due to the crossover from the
+degeneratetonon-degeneratelimits. Athightemperaturew hereelectronsareinnon-degeneratelimit,thescattering is
+determinedbytheelectron-electronandelectron-phonons catterings. Thecalculationsshowthat forsmallwell width
+when only the linear Dresselhaus term is dominant, the tempe rature dependenceof the scattering is stronger and the
+spin relaxation time increases with temperature. However, for wide quantum well, at certain temperature the cubic
+Dresselhaus term becomesdominant. The fast increase of the inhomogeneousbroadeningfrom the cubic term over-
+comes the effect from the scattering and the spin relaxation time decreas es with temperature. Jiang and Wu further
+85Figure 51: Spin dephasing time τvs.temperature Twith spin polarization P=2.5 % (a) and P=75 % (b) under two di fferent impurity levels
+in GaAs (001) quantum well. Curves with dots: Ni=0; curves with squares: Ni=0.1Ne. The spin dephasing times predicted by the simplified
+treatment of D’yakonov-Perel’ term (solid curve) and the ki netic spin Bloch equation approach (dots) for Ni=0 are plotted in the inset for
+comparison. FromWeng and Wu[44].
+introduced strain to change the relative importance of the l inear and cubic D’yakonov-Perel’ terms and showed the
+differenttemperaturedependencesofthespinrelaxationtime[ 570]. Thispredictionhasbeenrealizedexperimentally
+by Holleitner et al. in Ref. [655], where they showed that in n-type two-dimensional InGaAs channels, when the
+linear D’yakonov-Perel’ term is suppressed, the spin relax ation time decreases with temperature monotonically, as
+showninFig.53. Itisnotedfortheunstructuredquantumwel linFig.53,wherethelineartermisimportant,thespin
+relaxationtimeincreaseswithtemperature. Similarfindin gswerereportedbyMalinowskietal. [603],whomeasured
+the spin relaxation in intrinsic GaAs /AlxGa1−xAs quantum wells with di fferent well widths, for temperature higher
+than80KasshowninFig.54(a). Oneclearlyseesanincreaseo fthespinrelaxationtimewithincreasingtemperature
+for small well width and a decrease of the spin relaxation tim e for large well width. Figure 54(b) shows the theo-
+retical calculation based on the kinetic spin Bloch equatio ns for three small well widths, where the lowest subband
+approximation is valid for the small well widths (6 and 10 nm) and barely valid for the 15 nm well width. All the
+relevant scatterings, such as the electron-electron, elec tron-hole, electron-phonon and electron-impurity scatte rings
+are included in the computation. The peaksin the cases of 10 a nd 15 nm well widths originate fromthe competition
+ofthelinearandcubicD’yakonov-Perel’termsaddressedab ove.
+Similar situation also happens in the density dependence of the spin relaxation time. A peak in the density
+dependence of the spin relaxation time was predicted theore tically and realized experimentally [609], as shown in
+Fig. 55 where the spin relaxation time is plotted against the photoexcited carrier density in a GaAs quantum well at
+roomtemperature. Thispeakcan beeasily understoodfromth e relationτs∝[∝an}b∇acketle{tΩ2(k)∝an}b∇acket∇i}htτ∗
+p]−1, whereτ∗
+pshouldinclude
+theeffectfromtheCoulombscattering[334,591,615]. Whenthesys temisinthenon-degeneratelimit,theaverageof
+∝an}b∇acketle{tΩ2(k)∝an}b∇acket∇i}htis performedat the Boltzmanndistribution and is therefore independentof the carrier density. Consequently
+τsincreases with increasing carrier density as τ∗
+pdecreases with the density. However, when the carrier densi ty is
+high enoughand the average should be performedusing the Fer mi distribution,∝an}b∇acketle{tΩ2(k)∝an}b∇acket∇i}htbecomes density dependent.
+Especiallyinthestrongdegeneratelimit, ∝an}b∇acketle{tk∝an}b∇acket∇i}ht∼kF∝√
+Nand∝an}b∇acketle{tΩ2(k)∝an}b∇acket∇i}htbecomesstronglydensitydependent. Moreover,
+the carrier-carrier Coulomb scattering decreases with inc reasing density at strong degenerate limit. As a result, the
+spinrelaxationtimedecreaseswithincreasingcarrierden sity. Thepeakpositiondependsonthecompetitionbetween
+the inhomogeneous broadening and the scattering. Also the l inear and cubic D’yakonov-Perel’ terms influence the
+peakposition.
+By includingthe intersubbandscattering, especially the i ntersubbandCoulomb scattering, Weng and Wu studied
+themultisubbande ffectinspinrelaxation[844]. Itwasdiscoveredthatalthoug hthehighersubbandhasamuchlarger
+inhomogeneousbroadening,thespinrelaxationtimesoftwo subbandsareidenticalthankstothestrongintersubband
+Coulomb scattering [844]. This prediction was later verifie d experimentally by Zhang et al. [625], who studied the
+spin dynamics in a single-barrier heterostructure by time- resolved Kerr rotation. By applying a gate voltage, they
+860200400600
+100 150 200 250 300τs(ps)
+T (K)
+Figure 52: Spin dephasing time τsvs. the background temperature Tfor two GaAs (001) quantum wells with width a=17.8 nm (•) and 12.7 nm
+(/diamondsolid). Thesolid curves arethespindephasing timecalculated wi th thelowesttwo subbandsincluded andthedashed curvesare thosecalculated with
+only the lowest subband. From Wengand Wu[844].
+effectivelymanipulatedtheconfinementofthesecondsubbanda ndthemeasuredspinrelaxationtimesofthefirstand
+second subbands are almost identical at large gate voltage a s shown in Fig. 56(c). The big deviations at small gate
+voltages are because that the wavefunctions of the second su bband are extended due to the weak confinement and
+hencetheintersubbandCoulombscatteringbecomesweaker.
+Finally we address spin relaxation in systems with both the D resselhaus and Rashba spin-orbit couplings. It has
+beenfirstpointedoutbyAverkievandGolub[194]thatfor(00 1)quantumwellwithidenticalDresselhausandRashba
+spin-orbitcouplingstrengths,whenthe cubicDresselhaus termis ignored,the e ffectivemagneticfield becomesfixed
+and is aligned along the (110) or ( ¯110) direction depending on the sign of the Rashba field. Ther efore one obtains
+infinite spin lifetime if the spin polarization is parallel t o the effective magnetic field direction. In reality, due to the
+presenceofthecubicDresselhaustermand /orthedifferencebetweentheDresselhausandRashbaspin-orbitcoupl ing
+strengths, the spin life time along the (110)or ( ¯110) directionis still finite. But there is strong anisotrop yof the spin
+lifetime along different spin polarizations. Cheng and Wu studied this anisotr opy under identical linear Dresselhaus
+and Rashba coupling strengths, but with the cubic Dresselha us term included [597]. Stich et al. applied a magnetic
+field parallel to the (110) and ( ¯110) directions and obtained large magnetoanisotropyof el ectron spin dephasing in a
+high mobility (001) GaAs quantumwell [199]. The initial spi n polarization was obtained by optical pumping and is
+thereforealong the growth direction of the quantum well. Th e experimentalsetup is illustrated in Fig. 57(a). Due to
+the mixing of the di fferent anisotropic spin orientationsby the magnetic field, t hey observed different magnetic field
+dependencesof the spin dephasingtime along the (110)and ( ¯110) directions, as shown in Fig. 57(d). The maximum
+and minimum are determined by the relative strengths of the D resselhaus and Rashba terms. It is also seen that
+calculationbasedonthefullymicroscopickineticspinBlo chequationscanwellreproducetheexperimentalfindings.
+5.4.3. Spinrelaxationanddephasinginn-type(001)GaAsqu antumwells farawayfrom theequilibrium
+Due to the full account of the Coulomb scattering, the kineti c spin Bloch equation approach can be applied to
+study spin system far away from the equilibrium. By so called far away from the equilibrium, one refers to the spin
+dynamicswith large spin polarization and /or with large in-plane electric field where the hot-electron effect becomes
+significant.
+Weng and Wu first studied the spin relaxation with large initi al spin polarization [44] and discovered marked
+increaseofthespinrelaxation /dephasingtimewithincreasingspinpolarizationasshowni nFig.58(a). Itisnotedthat
+theplottedspin dephasingtime is theinverseofthe spin dep hasingrate. Thereforeoneexpectsa markedlyincreased
+total spin lifetime. This enhancement is more pronounced in samples with larger mobility, as shown in Fig. 58(b).
+The physics leading to such an increase was identified from th e Coulomb Hartree-Fock term. In the presence of the
+87Figure 53: Measured temperature dependence of the spin rela xation time for two-dimensional n-InAsGa channels with a width of 1.5 µm, where
+thecubicin ktermsoftheDresselhaus termdominatethespinrelaxation. Openandfilledsquaresrepresentdataofchannels along[100 ]and[110],
+while the open circles depict the spin relaxation time of the unstructured quantum well. From Holleitner etal. [655].
+spinpolarization,theCoulombHartree-Focktermprovides an effectivemagneticfieldalongthe z-axis:
+BHF
+z=1
+gµB/summationdisplay
+qVq(fk+q1
+2−fk+q−1
+2). (186)
+This effective magnetic field can be as large as 20 T and e ffectively blocks the spin precession as the energies of
+the spin-up and -down electrons are no longer detuned. This c an be seen clearly from Fig. 59 that by removing the
+longitudinal component of the Hartree-Fock term, the polar ization dependence of the spin relaxation time becomes
+pretty mild. It is noted that unlike a real magnetic field whic h breaks the time-reversal symmetry, the Coulomb
+interaction does not. This can be seen from the fact that the H artree-Fock term cancels each other after performing
+the summation over all the kstates [Eq. (143)]. Therefore the spin relaxation time with a large effective magnetic
+fieldis still finite. As the e ffectivemagneticfield decreaseswith temperature,Wengand W u pointedoutthat the spin
+relaxation time should decrease with increasing temperatu re at large spin polarization, in contrast to the case with
+smallspinpolarization[asshowninFig.51(b)].
+These predictionshavebeen realizedexperimentallyby Sti ch et al. [41, 42] and Zhanget al. [327]. By changing
+theintensityofthecircularlypolarizedlasers,Stichet a l. measuredthespindephasingtimein ahighmobility n-type
+GaAsquantumwellasafunctionofinitialspinpolarization asshowninFig.60(a). Indeedtheyobservedanincrease
+ofthespindephasingtimewiththeincreasedspinpolarizat ion,andthetheoreticalcalculationbasedonthekineticsp in
+Bloch equationsnicely reproducedthe experimentalfinding swhen the Hartree-Fockterm was included[Fig. 60(b)].
+It was also shown that when the Hartree-Fock term is removed, one does not see any increase of the spin dephasing
+time[Fig. 60(c)]. Later,theyfurtherimprovedthe experim entby replacingthecircular-polarizedlaser pumpingwith
+the elliptic polarized laser pumping. By doing so, they were able to vary the spin polarization without changing the
+carrier density. Figure. 61 shows the measured spin dephasi ng times as function of initial spin polarization under
+two fixedpumpingintensities, togetherwith the theoretica lcalculationswith and withoutthe CoulombHartree-Fock
+term. Again the spin dephasing time increases with the initi al spin polarization as predicted and the theoretical
+calculationswiththeHartree-Focktermareingoodagreeme ntwiththeexperimentaldata. Moreover,Stichetal. also
+confirmedthepredictionofthetemperaturedependencesoft hespindephasingtimeatlowandhighspinpolarizations
+[44]. Figure 62(a) shows the measured temperature dependen ces of the spin dephasing time at di fferent initial spin
+polarizations. As predicted, the spin dephasingtime incre ases with increasingtemperatureat small spin polarizatio n
+but decreases at large spin polarization. The theoretical c alculations also nicely reproduced the experimental data.
+Thehot-electrontemperaturesin thecalculationweretake nfromthe experiment[Fig.62(b)]. Thee ffectivemagnetic
+88Figure 54: Electron spin relaxation in intrinsic GaAs /Al0.35Ga0.65As quantum wells of a variety of widths. (a) Experimental res ults from Mali-
+nowski et al. [603]. The solid line represents a quadratic de pendence of relaxation rate on temperature. The dashed curv e is data on electron spin
+relaxation in bulk GaAs from Maruschak et al. (as reproduced by Meier and Zakharchenya [3]). From Malinowski et al. [603] . (b) Theoretical
+calculation via the kinetic spin Bloch equation approach (c urves). The dots are data from (a). The parameters are Ne=Nh=4×1011cm−2,
+Ni=0.1Ne,γD=0.0162 eV·nm3. HereNeandNhare the electron and hole densities respectively, Niis the impurity density, and γDis the
+spin-orbit splitting parameter.
+fieldfromtheHartree-FocktermhasbeenmeasuredbyZhanget al. [327]fromthesignswitchoftheKerrsignaland
+thephasereversalofLarmorprecessionswitha biasvoltage ina GaAsheterostructure.
+Kornetal. [326]alsoestimatedtheaveragee ffectbyapplyinganexternalmagneticfieldintheFaradayconfi gura-
+tion,asshownin Fig.63(a)forthesame samplereportedabov e[41,42]. Theycomparedthespindephasingtimesof
+bothlargeandsmallspinpolarizationsasfunctionofexter nalmagneticfield. Duetothee ffectivemagneticfieldfrom
+the Hartree-Fock term, the spin relaxation times are di fferent under small external magnetic field but become iden-
+tical when the magnetic field becomes large enough. From the m erging point, they estimated the mean value of the
+effective magnetic field is below 0.4 T. They further showed that this effective magnetic field from the Hartree-Fock
+termcannotbecompensatedbytheexternalmagneticfield,be causeitdoesnotbreakthetime-reversalsymmetryand
+isthereforenotagenuinemagneticfield,assaidabove. This canbeseenfromFig.63(b)thatthespinrelaxationtime
+at large spin polarization shows identical external magnet ic field dependences when the magnetic field is parallel or
+antiparalleltothegrowthdirection.
+The spindynamicsin the presenceofa highin-planeelectric field was first studiedbyWeng et al. [569] in GaAs
+quantumwellwithonlythelowestsubbandbysolvingthekine ticspinBlochequations. Toavoidthe“runaway”e ffect
+[849,850],theelectricfieldwascalculatedupto1kV /cm. ThenWengandWufurtherintroducedthesecondsubband
+into the model and the in-plane electric field was increased u pto 3 kV/cm [844]. Zhang et al. included Lvalley and
+theelectricfieldwasfurtherincreasedupto7kV /cm[845].
+The effect of in-plane electric field to the spin relaxation in syste m with strain was investigatedby Jiang and Wu
+[570]. Zhouet al. also investigatedtheelectric-fielde ffectat lowlattice temperatures[372].
+The in-plane electric field leads to two e ffects. (i) It shifts the center-of-massof electrons to kd=m∗vd=m∗µE
+withµrepresenting the mobility, which further induces an e ffective magnetic field via the D’yakonov-Perel’ term
+[569]. Theinducede ffectivemagneticfieldcanbeestimatedby
+Btot=B+B∗=B+1
+gµB/integraldisplay
+dk(fk1
+2−fk−1
+2)Ω(k)/slashig/integraldisplay
+dk(fk1
+2−fk−1
+2). (187)
+(ii) The in-plane electric field also leads to the hot-electr on effect [851]. By taking the electron distribution function
+asthedriftedFermifunction fkσ={exp[((k−m∗vd)2/2m∗−µσ)/kBTe]+1}−1inthecasewithonlythelowestsubband
+included,theeffectivemagneticfieldforsmallspinpolarizationcanberoug hlyestimatedas[569]
+B∗=γDm∗2vd{Ef/[2(1−e−Ef/kBTe)]−Ec}/gµB, (188)
+89 60 80 100 120 140 160
+ 0  0.5  1  1.5  2  2.5  3  3.5  4 0  2  4  6  8  10τ (ps)
+ N (1011 cm-2) N (1017 cm-3)
+Experiment (2D)
+Experiment (3D)
+Calculation (2D)
+Calculation (2D, without HF)
+Calculation (2D, without SFEHS)
+Calculation (2D, without SCEHS)
+Figure 55: Carrier density dependence of electron spin rela xation time τat room temperature. Dots: experimental data in a GaAs (001) quantum
+well (2D) with a width of 10 nm; open squares: experimental da ta in bulk material (3D). Solid curve: full theoretical calc ulation; chain curve:
+theoretical calculation without the spin-flip electron-ho le scattering (SFEHS); dashed curve: theoretical calculat ion without the spin-conserving
+electron-hole scattering (SCEHS);dotted curve: theoreti cal calculation without the Coulomb Hartree-Fock term. Not e the scale of the bulk data is
+on thetop frameof the figure. From Teng et al. [609].
+withEfandEc, the Fermi and confinement energies of the quantum well, resp ectively. Teis the hot-electron tem-
+perature. Figure 64(a) shows the spin precession in the pres ence of a high electric field in a 15 nm quantum well at
+T=120K.Onefindsthatthespinprecessesevenintheabsenceofa nyexternalmagneticfieldandthespinprecession
+frequencychanges with the direction of the electric field in the presence of an external magnetic field. The e ffective
+magnetic field deduced from the spin precessions in Fig. 64(a ) is plotted in Fig. 64(b), which is in good agreement
+withthecorrespondingresultfromEq.(188).
+The spin relaxation time can be e ffectively manipulated by the in-plane electric field. When on ly the lowest
+subband is considered, the electric field influences the spin relaxation via concurrent e ffects of the increase of the
+inhomogeneousbroadeningbydrivingelectronsto highermo mentumstatesandtheincrease /decrease(dependingon
+the type of scattering and also the degenerate /nondegenerate limit) of scattering from the hot-electron t emperature.
+Weng et al. reportedrich electric field dependenceof the spi n relaxationtime undervariousconditions[569]. When
+the electric field is high enough so that higher subbands and /or valleys are involved, due to the di fferent spin-orbit
+coupling strengths and /or effective masses in di fferent subbands and valleys, the spin relaxation time can be m anip-
+ulated more effectively [844, 845]. This can be seen from Fig. 65(a) and (b) w here spin relaxation times are plotted
+against the in-plane electric field in GaAs quantum wells, wi th the second subband and Lvalley being populated at
+high electric field, respectively. The inhomogeneous broad ening comes from the Dresselhaus terms. In Γvalley, it
+reads
+hΓ(kΓ)=γ
+2/parenleftbigkΓx(k2
+Γy−∝an}b∇acketle{tk2
+Γz∝an}b∇acket∇i}ht),kΓy(∝an}b∇acketle{tk2
+Γz∝an}b∇acket∇i}ht−k2
+Γx),∝an}b∇acketle{tkΓz∝an}b∇acket∇i}ht(k2
+Γx−k2
+Γy)/parenrightbig, (189)
+andinL-valley,it reads[106, 142]
+hLi(kLi)=β/parenleftbigkLix,kLiy,∝an}b∇acketle{tkLiz∝an}b∇acket∇i}ht/parenrightbig׈ni. (190)
+Hereˆniis the unit vector along the longitudinal principle axis of Livalley. Note the coordinate system of above
+equationshas beengivenin Ref. [852]. kΓ=kandkLi=k−K0
+LiwithK0
+Li=π
+a0(1,±1,±1).a0is the lattice constant.
+∝an}b∇acketle{tkλz∝an}b∇acket∇i}ht(∝an}b∇acketle{tk2
+λz∝an}b∇acket∇i}ht) representsthe averageofthe operator −i∂/∂z−K0
+λz[(−i∂/∂z−K0
+λz)2] overthe electronstates in λ-valley.
+Under the infinite-depth assumption, ∝an}b∇acketle{tn|k2
+λz|n∝an}b∇acket∇i}ht=n2π2
+a2withn, the subband index and a, the well width. Therefore,
+when the linear- kterms are dominant, the inhomogeneousbroadeningof the sec ond subband in hΓ(kΓ) is four times
+as large as the first subband. Also a tight-bindingcalculati onby Fu et al. [142] indicates β=0.026eV·nm, which is
+much larger than1
+2γD∝an}b∇acketle{tk2
+Γz∝an}b∇acket∇i}ht≈0.001 eV·nm when only the lowest subband of the Γ-valley is considered and the well
+90Figure 56: (a), (b) and (c) Measured spin coherence times for the 1st and 2nd subbands of a single-barrier GaAs /AlAs heterostructure, as the
+wavelength varies for three di fferent conditions:−0.3 V, 0 T;−0.3 V, 2 T and−0.6 V, 2 T. The points are experimental results, and the curves a re
+bestfittings tothepoints. (d)and(e)E ffectiveg∗factors forthe1stand2ndsubbandsplotted asafunction ofw avelength. FromZhangetal. [625].
+widthissettobe7.5nm[845]. Therefore,theinhomogeneous broadeningsofthehighersubbandandvalleyaremuch
+larger. It is therefore easy to understand that the spin rela xation time decreases with increasing electric field when
+moreelectronsare excitedtohighersubbandand /orvalley.
+Bysimplylookingattheinhomogeneousbroadeningsof ΓandLvalleys,onemaynaivelycometotheconclusion
+that the spin relaxation rate of electronsin Lvalley should be much faster than that in Γvalley. Also by noticing the
+g-factorof LvalleygL=1.77[853], whichis also muchlargerthan that of Γvalley,gΓ=−0.44[204], one mayalso
+speculate that the spin precession under a magnetic field in t he Voigt configuration should be much faster in the L
+valleythanin theΓvalley. Zhanget al. showedbothare in fact incorrect[845]. Theycalculatedthe spin precessions
+ofelectronsinΓandLvalleysinGaAsquantumwellwithbothhighin-planeelectri cfieldandamagneticfiledinthe
+Voigtconfiguration,anddiscoveredthatthespinprecessio nfrequenciesaswellasspindephasingtimesofbothvalleys
+are identical to each other, as shown in Fig. 66, and the “e ffective”g-factor of the Lvalley from the spin precession
+is 0.44, which is in the same magnitude as that of the Γvalley. The physics leading to these unexpected results was
+revealeddue to the strong intervalley electron-phononsca ttering. As shown in Fig. 67 where the spin precessions of
+ann-type(001)GaAsquantumwellunderelectricfield E=2kV/cmareplottedwithall thescatteringsincluded(a),
+without the intervalley electron-phonon scattering (b), w ithout the intervalley electron-electron Coulomb scatter ing
+(c),andwithoutintervalleyscattering(d),respectively . It isseenthatwhentheintervalleyelectron-phononscatt ering
+is switched off, the spin polarizationin the Lvalleys decayspretty fast. This is in contrast to the multi- subbandcase
+addressed in the previous subsection where the inter-subba nd Coulomb scattering is the cause of the identical spin
+relaxationtimesofeachsubband.
+5.4.4. Effectof theCoulombscatteringtothespin dephasingandrelax ation
+As addressedin the previoussubsections,the Coulombscatt eringcan contributeto the spin dephasingandrelax-
+ation due to the D’yakonov-Perel’ mechanism. A natural ques tion would be how the Coulomb scattering changes
+the spin dephasing /relaxation time. Glazov and Ivchenko pointed out that the Co ulomb scattering can prolong the
+spin dephasing/relaxationtime limited by the D’yakonov-Perel’mechanism . Weng et al. comparedthe electron spin
+dephasingtimesin (001)GaAsquantumwell withandwithoutt heCoulombscatteringinthepresenceofanin-plane
+electricfield [569]. AsshowninFig.68(a),inwhichthespin dephasingtimeswithandwithouttheCoulombscatter-
+ing are plotted against in-plane electric field. It is seen th at adding Coulomb scattering always leads to a longer spin
+dephasing time both near (at E∼0) and far away from (with hot-electron e ffect) the equilibrium. This looks quite
+contradictory to the previous understanding of optical dep hasing [833] where adding a Coulomb scattering always
+leadsto a shorteropticaldephasingtime. Also asstated in S ec. 5.2that inthe presenceof theinhomogeneousbroad-
+ening,addinga newscatteringleadsto a new dephasingchann el. Howcan theadditional“channel”leadsto a longer
+spin dephasing time? To understand this, L¨ u et al. [363] put a scaling coefficientγin front of the D’yakonov-Perel’
+91Figure 57: (a) Schematic of the time-resolved Faraday rotat ion experiment. In-plane magnetic fields are applied either in the [110] or [1 ¯10]
+directions. (b) Sketch of the precession of optically orien ted spins about a [1 ¯10] in-plane magnetic field. (c) Sketch of the precession of o ptically
+oriented spins about a [110] in-plane magnetic field. (d) Com parison of the experimental (solid symbols) and theoretica lly calculated (open
+symbols)spindephasingtimesfordi fferentin-planemagnetic-field directions inGaAs(001)quan tumwell.αandβaretheRashbaandDresselhaus
+spin-orbit coefficients, respectively. FromStich etal. [199].
+term and compared the spin dephasing times of electrons, hea vy holes and light holes in (001) GaAs quantum well
+with and without the Coulomb scattering as function of γ. As shown in Fig. 68(b) and (c), for larger γ, i.e., in the
+weak scattering limit, adding Coulomb scattering always le ads to a shorter spin dephasing time. However, for small
+γ, i.e., in the strong scattering limit, adding Coulomb scatt ering leads to a longer spin dephasing time. This can be
+understoodas follows: In the weak scattering limit, the cou nter effect of the scattering to the inhomogeneousbroad-
+ening is pretty weak, and the only e ffect of the scattering is an additional spin dephasing channe l. However, in the
+strong scattering limit, the counter e ffect of the scattering is more pronouncedand hence the spin de phasing time is
+increasedbyaddingtheCoulombscattering. Itisnotedthat γ=1correspondstothegenuinecaseandfromthefigure
+one notices that electronshappento be in the strong scatter ign limit. Thereforeaddingthe Coulomb scattering leads
+to a longerspin dephasingtime. However,both heavy and ligh t holesare in the weak scattering limit. Consequently
+addingthe Coulombscatteringmakes the spin dephasingtime shorter. In fact, foroptical dephasingproblem,the in-
+homogeneousbroadeningin Eq. (140) happensto be in the weak scattering limit. Therefore,the Coulombscattering
+alwaysbringsashorteropticaldephasingtime[833].
+The strong/weakscattering limit can be measured by ξ≡|g∗µBΩ(k)|τ∗
+p. Whenξ≫1 (≪1), the system is in the
+weak (strong) scattering limit. Here τ∗
+pis the effective momentum scattering time from not only the carrier-p honon
+and carrier-impurityscatterings, but also from the Coulom b scattering [591, 615]. It is also noted that the regime of
+strong and weak scattering can be changed even for the same sa mple by temperature. It has been shown by Harley
+group[368,591]andalsolaterbyStichetal. [41,199]thata tlowtemperatureelectronsinhighmobility(001)GaAs
+quantumwell can bein the weak scatteringlimit. Also forhol esinp-type(001)GaAsquantumwells, L¨ uet al. have
+shown that the system can be changed from the weak scattering limit to the strong one by impurity density76and
+temperature, etc. [363]. In fact, in the strong (weak) scatt ering limit, besides the Coulomb scattering, adding any
+scattering,cancausea longer(shorter)spindephasing /relaxationtime.
+5.4.5. Non-Markoviane ffect intheweak scatteringlimit
+As stated in Sec. 5.3, the scattering terms of the kinetic spi n Bloch equations are given by Eqs. (159) and (160),
+which are time-integrals. By applying the Markovianapprox imation[Eq. (161)], one can carry out the time-integral
+andthe scatteringtermsaresimplifiedtoEq.(162), whereon ehastheenergyconservation[ δ-functionsin Eq.(161)]
+and the trace of the history disappears. This approximation is valid only in the strong scattering limit where the
+spin precession between two adjacent scatterings is neglig ible. However, in the weak scattering limit, electron spin
+76Itis noted that the impurity density should be1 /4 of the value presented in Ref. [363].
+92Figure 58: (a) Spin dephasing time τvs.the initial spin polarization Pof electrons in GaAs (001) quantum well at di fferent temperatures. The
+impurity density Ni=0. (b) Spin dephasing time τvs.the initial spin polarization PatT=120 K for different impurity levels. Dots ( •):Ni=0;
+Diamonds ( /diamondsolid):Ni=0.01Ne;Squares ( /squaresolid):Ni=0.1Ne. Thecurves areplotted for theaid of eyes. From Weng and Wu[4 4].
+can experience many precessions between two scattering eve nts, and the Markovian approximation is invalid. In
+this circumstance one has to trace back the history by carryi ng out the time integral. Hence the dynamics becomes
+non-Markovian.
+Glazov and Sherman first investigated the electron spin rela xation in GaAs quantum wells under a strong mag-
+netic field by the Monte Carlo simulation [381]. They reporte d a longer non-Markovian spin relaxation time [381].
+Zhang and Wu studied the non-Markovian hole spin dynamics in p-type GaAs quantum wells using kinetic spin
+Bloch equation approach[854]. By performingtime derivati veof the time integral in Eq. (160), they transferred the
+integrodifferentialkinetic spin Bloch equationsin non-Markovianlim it to a larger set of di fferential equations. They
+comparedthe time evolutionsof the incoherentlysummed spi n coherenceof heavy holes in both the Markovianand
+non-Markovianlimitsat di fferentRashba strengths. Fromthedecayoftheincoherentlys ummedspincoherence,one
+mayextract T2.77AsshowninFig.69(d),inthestrongscatteringlimit,botha pproachesyieldthesamespindephasing
+time. Nevertheless,in the weakscatteringlimit, thenon-M arkoviankineticsgivesa longerholespin dephasingtime,
+which is in good agreement with the prediction of Glazov and S herman [381]. The physics behind this phenomena
+is easily understood from the nature of non-Markovianscatt ering, which keeps some memory of the coherence dur-
+ing the scattering. What is most important is that they predi cted quantum spin beats when the spin precession time
+is comparable to the momentum scattering time. The beats are solely from the non-Markovian e ffect (or memory
+effect), with the beating frequency being exactly the longitud inal optical-phononfrequency [see Fig. 69(b) and (c)].
+The quantum spin beats are similar to the longitudinal optic al-phonon quantum beats in optical four wave mixing
+signal[855]. However,a measurementofthese quantumspin b eatscannotbeperformedbythe regularFaraday /Kerr
+rotationmeasurement. A possibleway isthroughthespinech omeasurement.
+5.4.6. ElectronspinrelaxationduetotheBir-Aronov-Piku smechanisminintrinsic and p-typeGaAsquantumwells
+It has long been believed in the literature that the Bir-Aron ov-Pikus mechanism is dominant at low temperature
+inp-type samples and has important contributionto intrinsic s ample with high photo-excitation[574, 687–689, 693,
+856,857]. TheseconclusionsweremadebasedonEq.(66)from theFermiGoldenrule,whichin(001)quantumwell
+reads[109]
+1
+τBAP(k)=4π/summationdisplay
+k′qδ(εe
+k−q−εe
+k+εh
+k′−εh
+k′−q)|M(K−q)|2fh
+k′−q(1−fh
+k) (191)
+77Instead of T∗
+2which is measured from the Faraday /Kerr rotation experiment.
+93Figure 59: Spin dephasing time τvs.the initial spin polarization PforT=120 K and Ni=0 in GaAs (001) quantum well. Dots ( •): With the
+longitudinal component of the Hartree-Fock term included; Diamonds ( /diamondsolid): Without the longitudinal component of the Hartree-Fock t erm. The
+curves areplotted for the aid of eyes. From Weng and Wu[44].
+withK=k+k′and
+|M(k−q)|2=9∆E2
+LT
+16|φ3D(0)|4/summationdisplay
+qzfex(qz)(K−q)2
+q2z+(K−q)22
+. (192)
+Here∆ELTis the longitudinal-transversesplitting in bulk. |φ3D(0)|2=1/(πa3
+0) denotes the bulk exciton state at zero
+relativedistance. fex(qz)istheformfactor[109,118]. Inthemeantime,thespinrela xationtimefromtheD’yakonov-
+Perel’ mechanismused for comparisonwas calculatedwithou tthe Coulombscatteringwhich hasbeen demonstrated
+tobeimportantintheprevioussections.
+Zhou and Wu reexamined the problem using the fully microscop ic kinetic spin Bloch equation approach [109].
+TheyconstructedthekineticspinBlochequationsinintrin sicandp-type(001)GaAsquantumwellswiththeelectron-
+phonon, electron-impurity, electron-electron Coulomb an d electron-heavy hole Coulomb scatterings explicitly in-
+cluded. Theelectron-heavyholeCoulombscatteringinclud esthespin-conservingscatteringandthe spin-flipscatter -
+ing(theBir-Aronov-Pikusterm). Theyalso extendedthescr eeningundertherandomphaseapproximationEq.(169)
+toincludethecontributionfromheavyholes:
+ǫ(q)=1−/summationdisplay
+qzvQfe(qz)/summationdisplay
+k,σfk+q,σ−fk,σ
+εe
+k+q−εe
+k−/summationdisplay
+qzvQfh(qz)/summationdisplay
+k′,σfh
+k′+q,σ−fh
+k′,σ
+εh
+k′+q−εh
+k′, (193)
+wherefe(qz) andfh(qz) are the form factors [109]. The scattering terms from the sp in-flip electron-heavy hole ex-
+changeinteraction(Bir-Aronov-Pikusterm)read
+∂fk,σ
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleBAP=−2π/summationdisplay
+k′,qδ(εe
+k−q−εe
+k+εh
+k′−εh
+k′−q)|M(K−q)|2
+×[(1−fh
+k′,σ)fh
+k′−q,−σfk,σ(1−fk−q,−σ)−fh
+k′,σ(1−fh
+k′−q,−σ)(1−fk,σ)fk−q,−σ],(194)
+∂ρk
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleBAP=−π/summationdisplay
+k′,q,σδ(εe
+k−q−εe
+k+εh
+k′−εh
+k′−q)|M(K−q)|2(195)
+×[(1−fh
+k′,σ)fh
+k′−q,−σ(1−fk−q,−σ)ρk+fh
+k′,σ(1−fh
+k′−q,−σ)fk−q,σρk]. (196)
+94Figure60: (a)Normalized time-resolved Faradayrotation t racesfordifferentdegreesofinitialelectron spinpolarization, PinGaAs(001)quantum
+well. Thetotal densities of electrons, ntot=ne+ntot
+ph, are2.19, 2.66, 3.83,8.39 (in units of 1011cm−2) forP=1.6%,8%, 18%, 30%, respectively.
+(b) Calculated spin decay curves for the experimental param eters, like initial spin polarization, total electron dens ities, electron mobility, and
+temperature T=4.5K(solidlines). Inthecalculation, aphenomenological de cay timeisincorporated asasinglefitting parameter τ(seetext). The
+dashedcurvefor P=30%iscalculated forahotelectron temperature of Te=120K.(c)Sameas(b)butcalculated withouttheHartree-Foc k term.
+In particular for large Pthe decay is much faster than in the experiments [cf. (a)]. Th e inset shows the data, displayed in (b), in a semilogarithmi c
+plot. Atlow Pthezero-field oscillations aresuperimposed. FromStich et al. [41].
+It is noted from Eq. (194) that there are quadratic terms in th e sense of the electron distribution function fkσ. This
+immediatelyimpliesthatonecannotrecoverEq.(191)fromt heFermiGoldenrule. TheonlywaytorecoverEq.(191)
+isundertheelasticscatteringapproximation: εe
+k−q≈εe
+k,andεh
+k′≈εh
+k′−qwherethequadratictermsexactlycanceleach
+other and the spin relaxation time is given by Eq. (191). It is well known that the elastic scattering approximationis
+valid only in the non-degenerateregime at high temperature s. When the temperatureis low enoughso that electrons
+are degenerate,the elastic scattering approximationis no t validany more. Thereforethe quadratictermsbecomeun-
+negligible. Infact,theterms(1 −fk−q,−σ)and(1−fkσ)inEq.(194)comefromthePauliblocking,whichreducesthe
+spin relaxation rate. Therefore the Bir-Aronov-Pikusmech anism becomes less important at low temperatures when
+the Pauli blocking becomes important. This e ffect was missing in the literature [574, 687–689, 693, 856, 85 7]. In
+Fig. 70(a), the spin relaxation times due to the Bir-Aronov- Pikusmechanism in intrinsic GaAs quantum well calcu-
+lated from the full spin-flip electron-holeexchange scatte ring [Eq. (194)] are compared with those without the Pauli
+blockingat different pumping densities. It is seen that the previouscalcul ations based on Eq. (191) always overesti-
+matetheimportanceoftheBir-Aronov-Pikusmechanismatlo wtemperatures. InFig.70(b),thespinrelaxationtimes
+of intrinsic GaAs quantum wells due to the Bir-Aronov-Pikus mechanism and the D’yakonov-Perel’mechanism are
+comparedatdifferentphoto-excitations. Itisseenthatinsteadofbeingim portantatlowtemperature,theBir-Aronov-
+Pikus mechanism is even negligible at low temperature when t he Pauli blocking becomes important. Two errors in
+the previous treatment may cause the incorrect conclusion t hat the Bir-Aronov-Pikusmechanism is dominant at low
+temperature: (i) The previous theory overlooked the Pauli b locking and hence overestimated the importance of the
+Bir-Aronov-Pikusmechanism; (ii) It overlookedthe contri butionof the Coulombscattering to the D’yakonov-Perel’
+mechanismandthusunderestimatedtheimportanceoftheD’y akonov-Perel’mechanism. Itisalsonotedthatthetem-
+perature at which the Bir-Aronov-Pikus may have some contri bution is around the hole Fermi temperature. Similar
+conclusionswerealso obtainedin p-typeGaAsquantumwells[109].
+Later, Zhou et al. performed a thorough investigation of ele ctron spin relaxation in p-type (001) GaAs quantum
+wellsbyvaryingimpurity,holeandphoto-excitedelectron densitiesoverawiderangeofvalues[112],undertheidea
+that very high impurity density and very low photo-excitede lectron density may e ffectively suppressthe importance
+of the D’yakonov-Perel’ mechanism and the Pauli blocking. T hen the relative importance of the Bir-Aronov-Pikus
+andD’yakonov-Perel’mechanismsmaybereversed. Thisinde edhappensasshowninthephase-diagram-likepicture
+95/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s49/s56/s48/s50/s48/s48/s50/s50/s48/s50/s52/s48/s50/s54/s48/s50/s56/s48/s51/s48/s48/s51/s50/s48/s51/s52/s48
+/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116
+/s32/s102/s117/s108/s108/s32/s99/s97/s108/s99/s117/s108/s97/s116/s105/s111/s110
+/s32/s99/s97/s108/s99/s117/s108/s97/s116/s105/s111/s110/s32
+/s32/s32/s32/s32/s32/s32/s32/s32/s32/s119/s105/s116/s104/s111/s117/s116/s32/s72/s70/s32/s116/s101/s114/s109
+/s32/s32/s83/s112/s105/s110/s32/s100/s101/s112/s104/s97/s115/s105/s110/s103/s32/s116/s105/s109/s101/s32/s40/s112/s115/s41
+/s73/s110/s105/s116/s105/s97/s108/s32/s115/s112/s105/s110/s32/s112/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s40/s37/s41/s40/s97/s41
+/s48 /s53 /s49/s48 /s49/s53/s57/s48/s49/s48/s48/s49/s49/s48/s49/s50/s48/s49/s51/s48
+/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116
+/s32/s102/s117/s108/s108/s32/s99/s97/s108/s99/s117/s108/s97/s116/s105/s111/s110
+/s32/s99/s97/s108/s99/s117/s108/s97/s116/s105/s111/s110/s32
+/s32/s32/s32/s32/s32/s32/s32/s32/s32/s119/s105/s116/s104/s111/s117/s116/s32/s72/s70/s32/s116/s101/s114/s109/s40/s98/s41/s32/s32/s83/s112/s105/s110/s32/s100/s101/s112/s104/s97/s115/s105/s110/s103/s32/s116/s105/s109/s101/s32/s40/s112/s115/s41
+/s73/s110/s105/s116/s105/s97/s108/s32/s115/s112/s105/s110/s32/s112/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s40/s37/s41
+Figure61: GaAs(001)quantum well: (a)Thespindephasing ti mes asafunction ofinitial spinpolarization forconstant, lowexcitation density and
+variable polarization degree of the pump beam. The measured spin dephasing times are compared to calculations with and w ithout the Hartree-
+Fock (HF) term, showing its importance. (b) The spin dephasi ng times measured and calculated for constant, highexcitation density and variable
+polarization degree. FromStich et al. [42].
+Figure 62: GaAs (001) quantum well: (a) Spin dephasing time a s a function of sample temperature, for di fferent initial spin polarizations. The
+measureddatapointsarerepresented bysolidpoints,while thecalculated dataarerepresented bylinesofthesamecolo ur. (b)Electron temperature
+determined from intensity-dependent photoilluminance me asurements as a function of the nominal sample temperature, for different pump beam
+fluenceand initial spinpolarization, under experimental c onditions corresponding tothemeasurements shownin (a). T hemeasured data points are
+represented by solid points, while the curves serveas guide to the eye. FromStich etal. [42].
+in Fig. 71 where the relative importance of the Bir-Aronov-P ikus and D’yakonov-Perel’ mechanisms is plotted as
+function of hole density and temperature at low and high impu rity densities and photo-excitation densities. For the
+situation of high hole density they even includedmulti-hol e subbands as well as the light hole band. It is interesting
+to see from the figures that at relativelyhigh photo-excitat ions,the Bir-Aronov-Pikusmechanismbecomesmore im-
+portant than the D’yakonov-Perel’mechanism only at high ho le densities and high temperatures(around hole Fermi
+temperature)whentheimpurityisverylow[zeroinFig.71(a )]. ImpuritiescansuppresstheD’yakonov-Perel’mech-
+anism and hence enhance the relative importance of the Bir-A ronov-Pikusmechanism. As a result, the temperature
+regimeisextended,rangingfromthe holeFermitemperature to theelectronFermitemperatureforhighholedensity.
+When the photo-excitation is weak so that the Pauli blocking is less important, the temperature regime where the
+Bir-Aronov-Pikusmechanism is important becomes wider com pared to the high excitation case. In particular, if the
+impurity density is high enough and the photo-excitation is so low that the electron Fermi temperature is below the
+lowesttemperatureoftheinvestigation,theBir-Aronov-P ikusmechanismcandominatethewholetemperatureregime
+of the investigationat su fficiently highhole density, as shown in Fig. 71(d). The corres pondingspin relaxationtimes
+ofeachmechanismunderhighorlow impurityandphoto-excit ationdensitiesare demonstratedin Fig. 72. Theyalso
+discussedthedensitydependencesofspinrelaxationwiths omeintriguingpropertiesrelatedtothehighholesubbands
+96Figure 63: (a) Spin dephasing times as afunction of an extern al magnetic field perpendicular to the quantum well plane for small and large initial
+spin polarization in GaAs (001) quantum well. (b) Same as (a) for large initial spin polarization and both polarities of t he external magnetic field.
+FromKorn etal. [326].
+Figure 64: (a) Electron densities of spin-up and -down elect rons and the incoherently summed spin coherence ρvs. timetfor GaAs (001)
+quantum well (the well width is 15 nm) with initial spin polar izationP=2.5 % under different electric fields E=0.5 kV/cm (solid curves) and
+E=−0.5 kV/cm(dashed curves). Toppanel: B=0T;bottom panel: B=4T.Notethescale ofthe spincoherence ison therightsideof thefigure
+and the scale of the top panel is di fferent from that of the bottom one. (b) Net e ffective magnetic field B∗from the D’yakonov-Perel’ term vs. the
+drift velocity VdatT=120 K with impurity density Ni=0. Thesolid curve is the corresponding result from Eq.(188) . From Weng etal. [569].
+[112].
+The predicted Pauli-blockinge ffect in the Bir-Aronov-Pikusmechanism has been partially de monstrated experi-
+mentallybyYanget al. [681], as shownin Fig. 73. Theyshowed by increasingthe pumpingdensity,the temperature
+dependenceofthespindephasingtimedeviatesfromtheonef romtheBir-Aronov-Pikusmechanismandthepeaksat
+highexcitationsagreewellwith thosepredictedbyZhouand Wu[109].
+5.4.7. Spindynamicsinthepresenceofa strongTHzlaserfiel din quantumwells withstrong spin-orbitcoupling
+TheinfluenceofastrongTHzfield,whichcane ffectivelychangetheelectronorbitalmomentumsandsignific antly
+modify the electron density of states, has been extensively studied in spin-unrelated problems such as the dynamic
+Franz-Keldysheffectandopticalside-bande ffect[58,59,61,82,841,858]. Sofar,theapplicationofstro ngTHzfield
+to the spin systems is very limited and only in theory. Johnse n is the first one who investigated the optical sideband
+generation in systems with spin-orbit coupling and suggest ed the formation of odd sidebands which are otherwise
+absent without the spin-orbit coupling [859]. Cheng and Wu s howed theoretically that in InAs quantum wells, a
+strong in-plane THz electric field can induce a large spin pol arization oscillating at the same frequency of the THz
+driving field [841]. Later Jiang et al. and Zhou suggested sim ilar effects in single charged InAs quantum dots [860]
+andinp-typeGaAs(001)quantumwells[861]. Thesecalculationsin Refs.[841,859–861]areperformedwithoutany
+dissipation. However, in reality, due to the scattering the re is spin dephasing and relaxation. Whether the large spin
+97Figure 65: GaAs (001) quantum wells: (a) Spin dephasing time vs. the applied electric field Eat different temperatures and well widths: •,
+T=120K;/diamondsolid,T=200K;Solidcurves, a=17.8nm;Dashedcurves, a=12.7nm. Inset: Thecorresponding electron temperature Teasafunction
+of theelectric field. FromWeng and Wu[844]. (b) Spin relaxat ion timeτvs. electric filed Ewhen temperature T=300 (solid curves), 200 (dashed
+curves) and 100 K (chain curves) with B=Ni=0. Inset: fraction ofelectrons in Lvalleys against electric field. From Zhang et al. [845].
+Figure 66: Temporal evolution of spin polarization |Pλ|inλvalley (λ=Γ,L) of GaAs (001) quantum well with electric field E=1 kV/cm and
+magnetic field B=4 T.From Zhang et al. [845].
+polarizationoscillationsarestillkeptinthepresenceof thedissipationisunansweredinRefs.[841,860,861]. Also the
+THz field caneffectivelyaffect the densityof states, andalso cause the hot-electrone ffect, whichin turnshouldhave
+pronouncedeffect on the spin relaxation /dephasing. To answer these questions, Jiang et al. extended the kinetic spin
+Blochequationstostudythespinkineticsinthepresenceof strongTHzlaserfieldsviatheFloquet-Markovapproach,
+first in quantum dots [482] then in quantum wells [69]. Their i nvestigation suggested that the dissipation does not
+block large THz spin polarization oscillations and the spin relaxation and dephasing can be e ffectively manipulated
+bythe THzfield.
+In the Coulomb gauge A(t)=Ecos(Ωt)/Ωand the scalar potential Φ=0, the electron Hamiltonian in a (001)
+quantumwell withsmall wellwidthalongthe z-axisreads
+He=/summationdisplay
+kσσ′Hσσ′
+0(k,t)ˆc†
+kσˆckσ′, (197)
+with
+ˆH0(k,t)=[k+eA(t)]2
+2m∗ˆ1+αR{ˆσxky−ˆσy[kx+eA(t)]}
+={εk+γEkxΩcos(Ωt)+Eem[1+cos(2Ωt)]}ˆ1+αR(ˆσxky−ˆσykx)−αRˆσyeEcos(Ωt)/Ω.(198)
+Hereεk=k2
+2m∗,γE=eE
+m∗Ω2andEem=e2E2
+4m∗Ω2.Ωis the Thz frequencyand αRis the Rashba coefficient. It is notedthat
+98Figure67: Spinpolarization |Pλ|vs.tunderelectric field E=2kV/cminGaAs(001)quantumwell. Solidcurve: Γvalley; dashed curve: Lvalley;
+and chain curve: the total. (a): with both the intervalley el ectron-phonon scattering ( He−p
+Γ−L) and intervalley electron-electron Coulomb scattering
+(He−e
+Γ−L);(b): without He−p
+Γ−L;(c): without He−e
+Γ−L;and (d): without He−e
+Γ−LandHe−p
+Γ−L.T=300 K and Ni=B=0. From Zhang etal. [845].
+thelast termmanifeststhattheTHz electricfieldactsasaTH z magneticfieldalongthe yaxis
+Beff=−2αReEcos(Ωt)/(gµBΩ), (199)
+wheregis the electron g-factor. We will show later that this THz-field-induced e ffective magnetic field has many
+importanteffectsonspinkinetics. Thetermproportionalto EemisresponsibleforthedynamicalFranz-Keldyshe ffect
+[841, 862, 863]. This term does not contain any dynamic varia ble of the electron system and thus has no e ffect on
+the kinetics of the electron system. Usually, the largest ti me-periodic term is the term γEkxΩcos(Ωt), where the
+sidebandeffectmainlycomesfrom. UnderanintenseTHzfield,thistermca nbecomparabletoorlargerthan εk. The
+Hamiltonianofthe scatteringremainsall thesameasbefore ,e.g.,Eqs.(152-154).
+TheSchr¨ odingerequationforelectronwithmomentum kreads
+i∂tΨk(t)=ˆH0(k,t)Ψk(t). (200)
+AccordingtotheFloquettheory[864],thesolutiontotheab oveequationis
+Ψkη(t)=eik·r−iεktφ1(z)ξkη(t)e−i[γEkxsin(Ωt)+Eemt+Eemsin(2Ωt)
+2Ω]≡eik·rΦη(z)e−i[γEkxsin(Ωt)+Eemt+Eemsin(2Ωt)
+2Ω],(201)
+withη=±denotingthespinbranchand φ1(z)beingthewavefunctionofthelowestsubband. ξkη(t)=e−iykηt/summationtext
+nσυkη
+nσeinΩtχσ
+whereykηandυkη
+nσaretheeigen-valuesandeigen-vectorsoftheequation
+(ykη−nΩ)υkη
+nσ=iσ
+2ΩαReE(υkη
+n−1,−σ+υkη
+n+1,−σ)+αR(ky+iσkx)υkη
+n,−σ. (202)
+This equation is equivalent to Eq. (2) in Ref. [841]. For each k, the spinors{|ξkη(t)∝an}b∇acket∇i}ht}at any time tform a complete-
+orthogonalbasisofthespinspace[482, 865,866]. Thetimee volutionoperatorforstate kcanbe writtenas
+ˆUe
+0(k,t,0)=/summationdisplay
+η|ξkη(t)∝an}b∇acket∇i}ht∝an}b∇acketle{tξkη(0)|e−i[εkt+γEkxsin(Ωt)]e−i[Eemt+Eemsin(2Ωt)/(2Ω)]. (203)
+99Figure 68: (a) Spin dephasing time in GaAs (001) quantum well s with (close circles) and without (open circles) the Coulom b scattering. From
+Weng et al. [569]. (b) and (c): Spin dephasing time vs.the scaling coefficient of the D’yakonov-Perel’ term γforT=120 K and 300 K,
+respectively. The solid (dashed) curves are the results wit h (without) the Coulomb scattering. /squaresolid: Electrons; /trianglesolid: light holes;•: heavy holes. The
+impurity density Ni=0and magnetic field B=0. From L¨ uet al. [363].
+At zerotemperatureandin theabsenceofanyscattering,the spectralfunctionis[841]
+A(k;t1,t2)=/summationdisplay
+η=±1Φη(k,t1)Φ†
+η(k,t2), (204)
+fromwhichthe densitymatrixreads
+ρ(t1,t2)=1
+(2π)2/integraldisplay
+dkA(k;t1,t2). (205)
+Noteρ(t1,t2) is a 2×2matrix in the spin space. Inthe collinearspin space, ρ↑↑=ρ↓↓. The averagemagneticmoment
+fromtheTHzfield (alongthe x-axis)andthe Rashbafieldisgivenby
+M(T)=/parenleftigg
+0,−gµB
+n↑+n↓/integraldisplayEf(T)
+−∞dωImρ↑,↓(ω,T),0/parenrightigg
+, (206)
+withEf(T) determined from nσ=1
+2π/integraltextEf(T)
+−∞dωρσ,σ(ω,T). HereT=(t1+t2)/2 andωis Fourier transferred from
+t=t1−t2. Figure 74 shows the average magnetic moment Myversus the time Tat different electric fields and THz
+frequencies. Itisseen thata THzmagneticsignalise ffectivelyinducedbytheTHzelectricone.
+To extend the kinetic spin Bloch equations to the situation w ith an intense THz field, Jiang et al. pointed out
+[69] that it is insufficient to include this field only in the driving term as in the pr evious investigation of the hot-
+electron effect where a static electric field is applied [372, 569]. The co rrect way is to evaluate the collision inte-
+gral with the Floquet wavefunctions, i.e., the solution of t he time-dependent Schr¨ odinger equation Eq. (200) [867].
+Moreover, the Markovian approximation should be made with r espect to the spectrum determined by the Floquet
+wavefunctions [867]. These improvements constitute the Fl oquet-Markov approach [865, 867], which works well
+when the driven system is in dynamically stable regime and th e system-reservoir coupling can be treated pertur-
+batively. With the Floquet-Markov approach and by projecti ng the density matrix in the Floquet picture ˆ ρF
+k(t)=
+ˆUe
+0†(k,t,0)ˆρk(t)ˆUe
+0(k,t,0),the kineticspinBloch equationsread
+∂tˆρF
+k(t)=∂tˆρF
+k(t)/vextendsingle/vextendsingle/vextendsinglecoh+∂tˆρF
+k(t)/vextendsingle/vextendsingle/vextendsinglescat, (207)
+where∂tˆρF
+k(t)|cohand∂tˆρF
+k(t)|scatare the coherent and scattering terms, respectively. The co herent terms, which de-
+scribe the coherent precession determined by the electron H amiltonian Heand the Hartree-Fock contribution of the
+electron-electronCoulombinteraction,canbewrittenas
+∂tˆρF
+k(t)/vextendsingle/vextendsingle/vextendsinglecoh=i/summationdisplay
+k′,qzVk−k′,qz|I(iqz)|2/bracketleftbigˆSk,k′(t,0)ˆρF
+k′(t)ˆSk′,k(t,0),ˆρF
+k(t)/bracketrightbig. (208)
+100 0 0.5 1 1.5 2 2.5
+ 0  0.2  0.4  0.6  0.8  1 ρHH (arb.units)
+t (ps)(a) non-MarkovianMarkovian
+ 0 0.5 1 1.5 2 2.5
+ 0  0.2  0.4  0.6  0.8  1 ρHH (arb.units)
+t (ps)(b) non-MarkovianMarkovian
+ 0 0.5 1 1.5 2
+ 0  0.2  0.4  0.6  0.8  1 ρHH (arb.units)
+t (ps)(c) non-MarkovianMarkovian
+ 0 0.5 1 1.5 2
+ 0  0.2  0.4  0.6  0.8  1 ρHH (arb.units)
+t (ps)(d) non-MarkovianMarkovian
+Figure 69: Time evolutions of the incoherently summed spin c oherenceρHHof heavy holes in GaAs (001) quantum wells with Rashba spin-o rbit
+coupling strength (a) γ7h7h
+54Ezm0//planckover2pi12=6.25nm,(b) 2.5nm,(c) 0.625 nmand (d) 0.25nm,respectively .γ7h7h
+54is theRashba coefficient and Ezis the
+electic field from the gate voltage [363]. Correspondingly, the mean spin precession time Ω−1is (a) 0.046 ps, (b) 0.116 ps, (c) 0.463 ps and (d)
+1.160ps. Solid curve: in non-Markovian limit; Dashed curve : in Markovian limit. Thewell width a=5nmand thetemperature T=300 K.From
+Zhang and Wu [854].
+HereˆΣF
+HF(k,t)=−/summationtext
+k′,qzVk−k′,qz|I(iqz)|2ˆρF
+k′(t) is the Coulomb Hartree-Fock self-energy. The scattering t erms are
+composed of terms due to the electron-impurity ( ∂tρF
+k/vextendsingle/vextendsingle/vextendsingleei), electron-phonon ( ∂tρF
+k/vextendsingle/vextendsingle/vextendsingleep) and electron-electron ( ∂tρF
+k/vextendsingle/vextendsingle/vextendsingleee)
+scatterings,respectively. Underthe generalizedKadano ff-BaymAnsatz[833],these scatteringtermsread[69]
+∂tρF(ηη′)
+k(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingleei=−η1η2η3/summationdisplay
+k′,qz,nπniU2
+k−k′,qz|I(iqz)|2/bracketleftbigg/braceleftig
+S(ηη1)
+k,k′(t,0)S(n)(η2η3)
+k′,kδ(nΩ+¯εk′η2−¯εkη3)
+×/parenleftig
+ρ>F(η1η2)
+k′(t)ρ<F(η3η′)
+k(t)−ρ<F(η1η2)
+k′(t)ρ>F(η3η′)
+k(t)/parenrightig/bracerightig
++/braceleftig
+η↔η′/bracerightig∗/bracketrightbigg
+, (209)
+∂tρF(ηη′)
+k(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingleep=−η1η2η3/summationdisplay
+k′,qz,n,λ,±π|Mλ,k−k′,qz|2|I(iqz)|2/bracketleftbigg/braceleftig
+S(ηη1)
+k,k′(t,0)S(n)(η2η3)
+k′,ke∓itωλ,k−k′,qzδ(±ωλ,k−k′,qz+nΩ+¯εk′η2−¯εkη3)
+×/parenleftig
+N±
+λ,k−k′,qzρ>F(η1η2)
+k′(t)ρ<F(η3η′)
+k(t)−N∓
+λ,k−k′,qzρ<F(η1η2)
+k′(t)ρ>F(η3η′)
+k(t)/parenrightig/bracerightig
++/braceleftig
+η↔η′/bracerightig∗/bracketrightbigg
+,(210)
+∂tρF(ηη′)
+k(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingleee=−η1...η7/summationdisplay
+k′,k′′,n,n′π/bracketleftig/summationdisplay
+qzVk−k′,qz|I(iqz)|2/bracketrightig2/bracketleftbigg/braceleftig
+T(ηη1)
+k,k′(t,0)T(n′)(η2η3)
+k′,kT(n−n′)(η4η5)
+k′′,k′′−k+k′T(η6η7)
+k′′−k+k′,k′′(t,0)
+×δ(nΩ+¯εk′η2−¯εkη3+¯εk′′η4−¯εk′′−k+k′η5)/parenleftig
+ρ>F(η1η2)
+k′(t)ρ<F(η3η′)
+k(t)ρ<F(η5η6)
+k′′−k+k′(t)ρ>F(η7η4)
+k′′(t)
+−ρ<F(η1η2)
+k′(t)ρ>F(η3η′)
+k(t)ρ>F(η5η6)
+k′′−k+k′(t)ρ<F(η7η4)
+k′′(t)/parenrightig/bracerightig
++/braceleftig
+η↔η′/bracerightig∗/bracketrightbigg
+. (211)
+Intheseequations, N±
+λ,k−k′,qz=Nλ,k−k′,qz+1
+2(1±1)standsforthephononnumber, niistheimpuritydensity, ˆ ρ>
+k=ˆ1−ˆρk,
+ˆρ<
+k=ˆρk, and ¯εkη=εk+ykη.
+S(η1η2)
+k′,k(t,0)=∝an}b∇acketle{tξk′η1(t)|ξkη2(t)∝an}b∇acket∇i}htei[(εk′−εk)t+γEsin(Ωt)(k′
+x−kx)]=/summationdisplay
+nS(n)(η1η2)
+k′,keit(nΩ+¯εk′η1−¯εkη2), (212)
+with
+S(n)(η1η2)
+k′,k=/summationdisplay
+mσFk′η1∗
+mσFkη2
+n+mσ. (213)
+101Figure70: GaAs (001) quantum wells: (a) Spin relaxation tim edue to theBir-Aronov-Pikus mechanism with fullspin-flip e lectron-hole exchange
+scattering(solidcurves)andwithonlythelineartermsint hespin-flipelectron-holeexchangescattering(dottedcur ves)atdifferentelectrondensities
+againsttemperature T.n0=1011cm−2. (b)Spin relaxation timeduetotheBir-Aronov-Pikus (soli d curves) andD’yakonov-Perel’ (dashed curves)
+mechanisms and the total spin relaxation time (dash-dotted curves)vs.temperature Tin intrinsic quantum wells at di fferent densities ( n=p=2,
+4,6n0) when well width a=20 nmand impurity density ni=n.n0=1011cm−2. From Zhouand Wu[109].
+Figure 71: Ratio of the spin relaxation time due to the Bir-Ar onov-Pikus mechanism to that due to the D’yakonov-Perel’ me chanism,τBAP/τDP,
+in GaAs (001) quantum wells as function of temperature and ho le density with (a) Ni=0,Nex=1011cm−2; (b)Ni=0,Nex=109cm−2; (c)
+Ni=Nh,Nex=1011cm−2; (d)Ni=Nh,Nex=109cm−2. The black dashed curves indicate the cases satisfying τBAP/τDP=1. Note the
+smaller theratio τBAP/τDPis,themoreimportant theBir-Aronov-Pikus mechanism beco mes. Theyellow solid curves indicate thecases satisfying
+∂µh[NLH(1)+NHH(2)]/∂µhNh=0.1. Intheregimeabovetheyellow curvethemulti-hole-subba nd effectbecomessignificant. FromZhouetal. [112].
+HereFkη
+nσ=/summationtext
+mυkη
+n+mσJm(γEkx) withJm(x) standingforthe m-thorderBessel function.
+T(η1η2)
+k′,k(t,0)=∝an}b∇acketle{tξk′η1(t)|ξkη2(t)∝an}b∇acket∇i}htei(εk′−εk)t=/summationdisplay
+nT(n)(η1η2)
+k′,keit(nΩ+¯εk′η1−¯εkη2), (214)
+with
+T(n)(η1η2)
+k′,k=/summationdisplay
+mσυk′η1∗
+mσυkη2
+n+mσ. (215)
+{η↔η′}stands for the same terms as in the previous {}but with the interchange η↔η′. The term of the electron-
+electron scattering is quite di fferent from those of the electron-impurity and electron-pho non scattering, as the mo-
+mentumconservationeliminatesthe termof eiγEsin(Ωt)kx.
+The above equations clearly show the sideband e ffects, i.e., nΩin theδ-functions. The extra energy, nΩ, is
+providedbytheTHzfieldduringeachscatteringprocess. Thi smakestransitionsfromthelow-energystates(small k)
+tohigh-energyones(large k)becomepossible,eventhroughtheelastic electron-impur ityscattering. Theseprocesses
+arethesideband-modulatedscatteringprocesses.
+ThekineticspinBlochequationsaresolvednumerically. Af terthat,ρF(ηη′)
+k(t)foreach kis obtained. From
+ρF(ηη′)
+k(t)=∝an}b∇acketle{tξkη(0)|ˆUe†
+0(k,t,0)ˆρk(t)ˆUe
+0(k,t,0)|ξkη′(0)∝an}b∇acket∇i}ht=∝an}b∇acketle{tξkη(t)|ˆρk(t)|ξkη′(t)∝an}b∇acket∇i}ht, (216)
+102101102103104105
+100101102103104τ  ( ps )
+τBAP / τDP 
+(a)Ni = 0 τtot
+τDP
+τBAP
+τBAP / τDP
+103104105
+ 10  100100101
+T  ( K )τ  ( ps )
+τBAP / τDP 
+(b)Ni = Nh τtot
+τDP
+τBAP
+τBAP / τDP
+Figure 72: Spin relaxation times of electrons in GaAs (001) q uantum wells due to the D’yakonov-Perel’ and Bir-Aronov-Pi kus mechanisms, the
+total spin relaxation time together with the ratio τBAP/τDPvs. temperature TforNex=109cm−2(curves with•), 3×1010cm−2(curves with /trianglesolid)
+and 1011cm−2(curves with /squaresolid) with hole density Nh=5×1011cm−2and impurity densities (a) Ni=0 and (b) Ni=Nh. The electron Fermi
+temperatures for thoseexcitation densities are Te
+F=0.41,12.4and 41.5K,respectively. Thehole Fermitemperature is Th
+F=124 K.Notethescale
+ofτBAP/τDPis on the right-hand side of the frame. FromZhou et al. [112].
+by performing a unitary transformation, one comes to the sin gle particle density matrix ˆ ρk(t) in the collinear basis
+{|σ∝an}b∇acket∇i}ht}whichiscomposedbythe eigen-statesof ˆ σz. Inthisspin space,thespinpolarizationalonganydirecti oncanbe
+obtainedreadily,e.g., Sz=/summationtext
+k1
+2(ρ↑↑
+k−ρ↓↓
+k),Sx=/summationtext
+kRe{ρ↑↓
+k}andSy=−/summationtext
+kIm{ρ↑↓
+k}. Fromthetemporalevolutionof
+Sz, thespinrelaxationtimeisextracted.
+By numericallysolving the kinetic spin Bloch equationswit h all the scattering included,Jiang et al. showedthat
+with dissipation, the THz field can still pump a large (severa l percent) spin polarization which oscillates at the same
+frequencywith the THz field, as shown in Fig. 75. This feature coincides with what predicted by Cheng and Wu in
+the dissipation-free investigation[841]. What di ffers from the previouscase is shown in Fig. 75 that there is a de lay
+ofSywith respect to the THz-field inducedmagneticfield Beff. Thisdelay is due to the retardedresponseof the spin
+polarizationto the spinpumpingcausedbythe THzfield. Thea mplitudeof thesteady-state spinpolarization S0
+y(the
+peak value of Sy) depends on the THz field strength and the THz frequency. Thes e features have been addressed in
+detailinRef. [69].
+Apart from pumping THz spin polarizations, the THz field can e ffectively manipulate the electron density of
+statesandcausethehot-electrone ffect. Bothinturncanleadtothemanipulationofthespinrela xationanddephasing.
+Figure76showsthedependenceofthespinrelaxationtimeon theTHzfieldstrengthatdi fferentimpuritydensities(a)
+and temperatures (b), together with the correspondinghot- electron temperatures (c). Two consequences of the THz
+field lead to the rich behaviorsin the figure: (i) the total THz field-inducedeffectivemagneticfield B[B=Beff+Bav
+withBav(t)=2αR∝an}b∇acketle{tkx∝an}b∇acket∇i}ht/(|g|µB) fromthe Rashbaspin-orbitcoupling]and(ii) thehot-elec troneffect. Effect(i) cangive
+a magnetic field as large as several tesla [2.6 T per 1 kV /cm THz field with ν≡Ω/(2π)=0.65 THz]. This effective
+magnetic field blocks the inhomogeneous broadening from the Rashba spin-orbit coupling and thus elongates the
+spin relaxation time. E ffect (ii) leads to the enhancement of momentum scattering as w ell as the inhomogeneous
+broadening, while the enhancement of the inhomogeneousbro adening tends to shorten the spin relaxation time, the
+boostof the scattering tendsto increase (ordecrease) the s pin relaxationtime in the strong (or weak)scattering limit
+asdiscussedintheprevioussections. Similarlythespinre laxationtimecanalsobemanipulatedbytheTHzfrequency
+asshowninFig. 77.
+103Figure 73: Kerr rotation of the 1 /3 monolayer InAs sample at di fferent temperatures measured at B=0.82 T. The pumping density is about 1.5
+×1017/cm3. Thetemperature dependence of theelectron spin decoheren ce timeT∗
+2and the effectiveg-factor are shown in (b) and (c),respectively.
+TheT∗
+2of 1/3 monolayer InAs sample with various pumping density at low t emperature as specified is also shown for comparison. From Ya ng et
+al. [681].
+T(T0)E=3 kV/cm(b)Ω=0.4 THz
+1
+2
+4
+10.80.60.40.2
+T(T0)Ω=1 THz(a)
+1074E=1kV/cmMy(gµB)
+10.80.60.40.200.3
+0.2
+0.1
+0
+−0.1
+−0.2
+−0.3
+Figure 74: The average magnetic moment Myvs. the time of InAs (001) quantum well at E=1, 4, 7 and 10 kV/cm for fixedΩ=1 THz (a) at
+Ω=0.4,1, 2 and 4 THzfor fixed E=3kV/cm (b). Theelectron density is N=1011cm−2. From Cheng and Wu[841].
+5.4.8. SpinrelaxationanddephasinginGaAs(110)quantumw ells
+In symmetric GaAs (110) quantum wells, the D’yakonov-Perel ’ term mainly comes from the Dresselhaus term
+whichreads
+gµBΩnn′
+x(k)=γD[−(k2
+x+2k2
+y)∝an}b∇acketle{tn|kz|n′∝an}b∇acket∇i}ht+∝an}b∇acketle{tn|k3
+z|n′∝an}b∇acket∇i}ht], (217)
+gµBΩnn′
+y(k)=4γDkxky∝an}b∇acketle{tn|kz|n′∝an}b∇acket∇i}ht, (218)
+gµBΩnn′
+z(k)=γD(k2
+x−2k2
+y−∝an}b∇acketle{tn|k2
+z|n′∝an}b∇acket∇i}ht)δnn′, (219)
+where∝an}b∇acketle{tn|km
+z|n′∝an}b∇acket∇i}ht=/integraltext
+dzφ∗
+n(z)(−i∂/∂z)mφn′(z) withφn(z) representing the envelope function of the electron in n-th
+subband. As∝an}b∇acketle{tn|kz|n∝an}b∇acket∇i}ht=∝an}b∇acketle{tn|k3
+z|n∝an}b∇acket∇i}ht=0, when only the lowest subband is relevant, the e ffective magnetic field Ω(k)
+is along the z-axis. Therefore, it was proposed both theoretically and ex perimentally [574, 577, 595, 662] that the
+D’yakonov-Perel’mechanismcannotcause anyspinrelaxati onifthe spinpolarizationisalongthe z-axis.
+Wu and Kuwata-Gonokami first pointed out that if a magnetic fie ld in the Voigt configuration is applied in the
+system, thereis spin relaxationanddephasingdue to the D’y akonov-Perel’mechanism[675]. Thiscan be seen from
+104Figure75: Spinpolarization along yaxis,Sy,asfunction oftimewithzero initial spinpolarization for E=0.5kV/cm(solid curve) and1.0kV /cm
+(dotted curve) in InAs (001) quantum well. T=50 K and Ni=0.05Ne. The dashed curve is the THz-field-induced e ffective magnetic field Beff
+withE=1.0kV/cm. Note that the scale of the dashed curve is on the right-han d side of theframe. From Jiang etal. [69].
+Figure 76: InAs (001) quantum wells: (a) Dependence of the sp in relaxation time τon THz field strength for impurity densities: Ni=0 (•);
+Ni=0.02Ne(/square);Ni=0.05Ne(△). Solid curves: from full calculation; Dotted curve: from t he calculation without theTHz-field-induced e ffective
+magnetic field Beff. (b) Dependence of the spin relaxation time τon THz field strength for impurity densities: Ni=0 (•);Ni=0.02Ne(/square); and
+Ni=0.05Ne(△).T=100 K(solid curves) and 50 K(dashed curves). (c) Dependence of thehot-electron temperature Teon THzfield strength for
+different impurity densities: Ni=0 (•);Ni=0.02Ne(/square);Ni=0.05Ne(△).T=100 K (solid curves) and T=50 K (dashed curves). From Jiang
+etal. [69].
+thecoherenttermofthekineticspinBlochequations[675]:
+∂fkσ
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoh=−gµBBImρk,σ−σ+2Im/summationdisplay
+qVqρk,σ−σρk+q,−σσ, (220)
+∂ρk,σ−σ
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoh=i
+2gµBB(fkσ−fk−σ)−iσω(k)ρk,σ−σ+i/summationdisplay
+qVq(fk+qσ−fk+q−σ)ρk,σ−σ−i/summationdisplay
+qVq(fkσ−fk−σ)ρk+q,σ−σ,
+(221)
+whereBis along the x-axis andω(k)=−gµBΩ11
+z(k). It is seen that in the presence of the magnetic field, the spi n
+coherenceρk,σ−σis involved and ω(k) in Eq. (221) providesthe inhomogeneousbroadeningwhich l eads to the spin
+relaxationanddephasing[675]. ExperimentallyD¨ ohrmann etal. [388,667]observeda“turnon”ofthespinrelaxation
+byswitchingonthemagneticfield.
+The spin relaxationin the absenceof the magneticfield in sym metricGaAs (110)quantumwells with small well
+width is an interesting problem and has been studied extensi vely [183, 193, 388, 602, 660, 662, 663, 667]. In most
+oftheseworks,themainreasonlimitingthespinrelaxation timeisattributedtotheBir-Aronov-Pikusmechanism. In
+the spin noise spectroscopy measurements by M¨ uller et al. [ 660], the excitation of the semiconductor is negligible
+and hence the Bir-Aronov-Pikus mechanism is avoided. They r eported spin relaxation about 24 ns and attributed it
+totheD’yakonov-Perel’mechanismdueto therandomRashbafi eldscausedbyfluctuationsinthe donordensityfirst
+proposedbySherman[294,657]. ZhouandWufurtherinvestig atedthiseffectusingthekineticspinBlochequations,
+wherethecontributionoftheCoulombscatteringoriginall ymissedinRefs.[294,657]isincluded[670]. Themethod
+105Figure 77: InAs (001) quantum wells: (a) Dependence of the sp in relaxation time τon THz frequency for impurity densities: Ni=0 (•);
+Ni=0.02Ne(/square); andNi=0.05Ne(△). Solid curves: from full calculation; Dotted curve: from t he calculation without the THz-field-induced
+effective magnetic field Beff.E=1 kV/cm andT=50 K. (b) Dependence of the hot-electron temperature Teon THz frequency for impurity
+densities: Ni=0 (•);Ni=0.02Ne(/square);andNi=0.05Ne(△).E=1 kV/cm andT=50 K.FromJiang et al. [69].
+to incorporatethe e ffect of the Random Rashba spin-orbit couplingin the kinetic s pin Bloch equationapproachis to
+calculate the time evolutions of the kinetic spin Bloch equa tions under sufficient Rashba coefficientsαR, which are
+assumedtosatisfytheGaussiandistribution P(αR)=1√
+2π∆e−α2
+R/(2∆2). Thespinrelaxationtimeisobtainedbytheslope
+ofthecoherentlysummedspinpolarization
+Sz=/integraldisplay
+dαRP(αR)/summationdisplay
+k1
+2(fαR
+k↑−fαR
+k↓). (222)
+A calculation based on the kinetic spin Bloch equationscan n icely reproducethe experimentalfindings by M¨ uller et
+al. [660] in Fig. 78(a). Theyalso showed that at low impurity density, the Coulombscattering has a strong influence
+to the spin relaxation, as shown in Fig. 78(b), in which a peak due to the Coulomb scattering in the temperature
+dependenceofthespinrelaxationtime waspredicted.
+BesidestheRandomRashbaspin-orbitcouplingfield,Zhouan dWuproposedavirtualintersubbandspin-flipspin-
+orbitcouplinginducedspin relaxationin GaAs(110)quantu mwells andshowedthismechanismbecomesimportant
+forsampleswith highimpuritydensity[668].
+Figure 78: GaAs (110) quantum wells: (a) Spin relaxation tim esvs.the in-plane magnetic field strength from the KBSE approach a nd from the
+experimental data in Ref. [660] with temperature T=20 K, well width a=16.8 nm, electron density Ne=1.8×1011cm−2and impurity density
+Ni=0.01Ne. The fitting parameters are Ew=4.2 kV/cm,γD=5.02 eV·Å3. (b) Spin relaxation times due to the random Rashba spin-orb it
+coupling induced D’yakonov-Perel’ mechanism vs.temperature Tfor impurity densities Ni=Neand 0.01NewithB=0,a=16.8 nm and
+Ne=1.8×1011cm−2. Thecorresponding Fermitemperature is 75 K.Thesolid (das hed) curves represent the results calculated with (without ) the
+electron-electron Coulomb scattering. From Zhou and Wu [66 8].
+1065.4.9. SpindynamicsinparamagneticGa(Mn)Asquantumwell s
+Jiangetal. furtherextendedthekineticspinBlochequatio nstostudytheelectronspinrelaxationinparamagnetic
+Ga(Mn)As quantum wells [111]. Besides the D’yakonov-Perel ’[101, 102] and Bir-Aronov-Pikus[105, 106] mech-
+anisms, they further included the spin relaxation mechanis m due to the exchange coupling of the electrons and the
+localized Mn spins (the s-d exchange scattering mechanism) [114] and the Elliott-Yafet mechanism [104, 868] due
+to the presence of the heavily doped Mn. In Ga(Mn)As, the Mn do pants can be either substitutional or interstitial;
+thesubstitutionalMnacceptsoneelectronwhereastheinte rstitialMnreleasestwo. Directdopinginlow-temperature
+molecular-beam epitaxy growth gives rise to more substitut ional Mn ions than interstitial ones, which makes the
+Ga(Mn)As a p-type semiconductor [114, 233, 869, 870]. In GaAs quantum we lls near Ga(Mn)As layer, the Mn
+dopantscandiffuseintotheGaAsquantumwells,wheretheMnionsmainlytake theinterstitialpositions,makingthe
+quantumwell n-type[871–873].
+ThekineticspinBlochequationsarethesameasthoseinGaAs quantumwells, excepttheadditionaltermsin the
+coherentandscatteringtermsassociatedwiththeMnspin SandtheElliott-Yafetmechanism. Theadditionalcoherent
+termreads
+∂tˆρk|coh
+Mn=−i[ˆHsd
+mf,ˆρk], (223)
+withˆHmf
+sd=−NMnα∝an}b∇acketle{tS∝an}b∇acket∇i}ht·σ
+2. Here∝an}b∇acketle{tS∝an}b∇acket∇i}htis the averagespin polarizationof Mn ions and αis thes-dexchangecoupling
+constant. For simplicity, Mn ionsare assumed to be uniforml ydistributedwithin and aroundthe quantumwells with
+a bulk density NMn. The additional scattering terms are those from the s-dexchange scattering and the Elliott-Yafet
+mechanism. The s-dexchangescatteringtermisgivenby
+∂tˆρk|scat
+sd=−πNMnα2Is/summationdisplay
+η1η2k′GMn(−η1−η2)δ(εk−εk′)(ˆsη1ˆρ>
+k′ˆsη2ˆρ<
+k−ˆsη2ˆρ<
+k′ˆsη1ˆρ>
+k+H.c.), (224)
+withGMn(η1η2)=1
+4Tr(ˆSη1ˆSη2ˆρMn).ˆSηand ˆsη(η=0,±1) are the spin ladder operators. The equation of motion
+for Mn spin density matrix consists of three parts ∂tˆρMn=∂tˆρMn|coh+∂tˆρMn|scat+∂tˆρMn|rel. The first part de-
+scribes the coherent precession around the external manget ic field and the s-dexchange mean field, ∂tˆρMn|coh=
+−i/bracketleftig
+gMnµBB·ˆS−α/summationtext
+kTr(ˆσ
+2ˆρk)·ˆS,ˆρMn/bracketrightig
+. The second part represents the s-dexchange scattering with electrons
+∂tˆρMn|scat=−πα2
+4/summationtext
+η1η2kδ(εk−εk′)Tr(ˆs−η2ˆρ<
+k′ˆs−η1ˆρ>
+k)/bracketleftig
+(ˆSη1ˆSη2ˆρMn−ˆSη1ˆρMnˆSη2)+H.c./bracketrightig
+. Thethirdpartcharacterizesthe
+Mn spin relaxation due to other mechanisms, such as the p-dexchangeinteraction with holes or Mn-spin–lattice in-
+teraction,with a relaxationtime approximation, ∂tˆρMn|rel=−/parenleftig
+ˆρMn−ˆρ0
+Mn/parenrightig
+/τMn. Here ˆρ0
+Mnrepresentsthe equilibrium
+Mnspindensitymatrix. τMnistheMnspinrelaxationtime,whichistypically0.1 ∼10ns[98].
+After incorporating the Elliott-Yafet mechanism, besides the ordinary spin-conserving terms, there are spin-flip
+terms. Forelectron-impurityscatteringtheseadditional termsare
+∂tˆρk|EY
+ei=−πni/summationdisplay
+k′δ(εk−εk′)/bracketleftig
+U(1)
+k−k′/parenleftbigˆΛ(1)
+k,k′ˆρ>
+k′ˆΛ(1)
+k′,kˆρ<
+k−ˆΛ(1)
+k,k′ˆρ<
+k′ˆΛ(1)
+k′,kˆρ>
+k/parenrightbig
++U(2)
+k−k′/parenleftbigˆΛ(2)
+k,k′ˆρ>
+k′ˆΛ(2)
+k′,kˆρ<
+k−ˆΛ(2)
+k,k′ˆρ<
+k′ˆΛ(2)
+k′,kˆρ>
+k/parenrightbig+H.c./bracketrightig
+, (225)
+whereni=NS
+Mn+4NI
+Mn+ni0withNS
+Mn,NI
+Mnandni0representingthedensitiesofsubstitutionalMn,intersti tialMnand
+non-magnetic impurities, respectively. U(1)
+k−k′=λ2
+c
+4/summationtext
+qzV2
+k−k′,qz|I(iqz)|2q2
+zandU(2)
+k−k′=−λ2
+c/summationtext
+qzV2
+k−k′,qz|I(iqz)|2. Here
+λc=η(1−η/2)
+3mcEg(1−η/3)withη=∆SO
+∆SO+Eg.Egand∆SOare the band-gap and the spin-orbit splitting of the valence band,
+respectively [3]. The spin-flip matrices are given by ˆΛ(1)
+k′,k=[(k+k′,0)׈σ]zandˆΛ(2)
+k,k′=[(k,0)×(k′,0)]·ˆσ. It is
+notedthat ˆΛ(1)
+k′,kandˆΛ(2)
+k′,kcontributetotheout-of-planeandin-planespinrelaxatio ns,respectively. Theyaregenerally
+different and therefore the spin relaxation due to the Elliott-Y afet mechanism in quantum wells is anisotropic. The
+Elliott-Yafetmechanismcanbeincorporatedintoothersca tteringssimilarly[110].
+By solving the kinetic spin Bloch equations, Jiang et al. [11 1] studied the spin relaxation in both n- andp-type
+Ga(Mn)Asquantumwells. For n-typesample, the total electron density is given by Ne=Ni
+e+NMn
+e+NexwithNMn
+e,
+Ni
+eandNexrepresentingthedensitiesfromMndonors,otherdonorsand photoexcitation,respectively. Thecalculated
+spin relaxation times due to various mechanisms are shown as function of Mn concentration in n-type samples with
+107Ni
+e=0(a)and Ni
+e=1011cm−2(b)respectively(Fig.79). It isinterestingtosee that eve nforthestrong s-dexchange
+couplingtaken in the calculation, the spin relaxation due t o thes-dexchangecouplingis still much weaker than that
+due to the D’yakonov-Perel’mechanism. Also the Bir-Aronov -Pikusand Elliott-Yafet mechanisms are irrelevant to
+the spin relaxation. The spin relaxatin is solely determine d by the D’yakonov-Perel’ mechanism. Moreover, they
+predicted a peak in the Mn concentration dependence of the sp in relaxation time. The physics leading to the peak
+is the same as the carrier-densitydependenceof the spin rel axationtime in n-typeGaAs quantumwells addressedin
+Sec.5.4.2, withthepeakbeingaroundtheelectronFermitem peratureTe
+F.
+101102103104105106107
+10-710-610-510-4101102103 40  4  0.4  0.04
+xNeMn  ( 1011 cm-2 )τ  ( ps )(a)
+TFe  ( K )
+101102103104105106107
+10-710-610-510-4101102103 40  4  0.4  0.04
+xNeMn  ( 1011 cm-2 )τ  ( ps )(b)
+TFe  ( K )
+Figure 79: Spin relaxation time τdue to various mechanisms in n-type Ga(Mn)As quantum wells which are (a) undoped or (b) n-doped before
+Mn-doping as function of Mn concentration xat 30 K (•) and 200 K ( /square). Red solid curves: the spin relaxation time due to the D’yak onov-Perel’
+mechanism τDP; Green dotted curves: the spin relaxation time due to the Ell iott-Yafet mechanism τEY; Brown dashed curves: the spin relaxation
+time due to the Bir-Aronov-Pikus mechanism τBAP; Blue chain curve: the spin relaxation time due to the s-dexchange scattering mechanism τsd.
+TheFermi temperature of electrons Te
+Fis ploted as black curve with △(the scale of Te
+Fis on the right hand side of the frame) and Te
+F=Tfor both
+T=30and 200 Kcases areplotted as black dashed curves. Thescal e of theelectron density fromMn donors NMn
+eis also plotted on the top of the
+frame. FromJiang et al. [111].
+Forp-type Ga(Mn)As quantum wells, both substitutional and inte rstitial Mn ions exist in the system. For sim-
+plicity, all the holes are assumed free. Due to the presence o f interstitial Mn ions, the ratio of the hole density Nhto
+the Mn density N Mnwas obtained by fitting the experimental data in Ref. [233], a s shown in Fig. 80. The electron
+spinrelaxationsduetovariousmechanismswerecalculated asfunctionofMnconcentrationatdi fferenttemperatures
+by Jiang et al. [111], as shown in Fig. 81. Unlike the case of n-type samples, due to the presence of large hole
+density at high Mn concentration, the Bir-Aronov-Pikusand s-dexchange scattering mechanisms can be important.
+At very low temperature[Fig. 81(a)], due to the Pauli blocki ng addressedin Sec. 5.4.6, the Bir-Aronov-Pikusmech-
+anism is negligible. The spin relaxationis determinedby th e D’yakonov-Perel’mechanism at low Mn concentration
+and thes-dexchange scattering mechanism at high Mn concentration. At medium temperature [Fig. 81(b)], both
+the Bir-Aronov-Pikusand s-dexchange scattering mechanisms determine the spin relaxat ion at high Mn concentra-
+tions. At high temperature [Fig. 81(c)], the spin relaxatio n is determined by the Bir-Aronov-Pikus mechanism. As
+the D’yakonov-Perel’mechanism determines the spin relaxa tion at low Mn concentrations, the spin relaxation time
+limitedbyitincreaseswithincreasingMnconcentration(i ncreasingelectron-impurityscattering). Moreover,both the
+Bir-Aronov-Pikusand s-dexchangescatteringmechanismsincrease with increasingM n density. As a result, there is
+apeakintheMndensitydependenceoftheelectronspinrelax ationtimeatanytemperature(exceptfortheextremely
+lowtemperaturewherethelocalizationbecomesimportant) .
+The temperature,photo-excitationdensity andmagneticfie ld dependencesof the spin relaxationin paramagnetic
+Ga(Mn)Asquantumwellshavealsobeeninvestigatedindetai l inRef.[111].
+108 0.15 0.2 0.25 0.3 0.35
+ 0 0.02  0.04  0.06  0.08  0.1  0.12 0 1 2 3 4 5
+x  ( 10-2 )Nh / NMn
+Nh  ( 1012 cm2 )
+Figure 80: Ratio of the hole density to the Mn density Nh/NMnvs.the Mn concentration xinp-type Ga(Mn)As quantum wells. The black dots
+represent the experimental data. The red solid curve is the fi tted one. The hole density Nhis also plotted (the blue dashed curve). Note that the
+scale ofNhison the right hand side of theframe. From Jiang etal. [111].
+102103104105106107108109
+10-710-610-510-410-3 200  20  2  0.2  0.02
+xNMn  ( 1011 cm-2 )
+T = 5 Kτ  ( ps )
+(a)τtotτDPτEYτBAPτsd
+102103104105106107108
+10-710-610-510-410-3 200  20  2  0.2  0.02
+xNMn  ( 1011 cm-2 )
+T = 50 Kτ  ( ps )
+(b)τtotτDPτEYτBAPτsd
+102103104105106107108
+10-710-610-510-410-3 200  20  2  0.2  0.02
+xNMn  ( 1011 cm-2 )
+T = 200 Kτ  ( ps )
+(c)τtotτDPτEYτBAPτsd
+Figure81: Spinrelaxation time τdueto various mechanisms andthetotal spinrelaxation time inp-type Ga(Mn)Asquantum wells against theMn
+concentration xat (a)T=5, (b) 50 and (c) 200 K.Thescale of NMnis also plotted on thetop of the frame. FromJiang et al. [111] .
+5.4.10. Hole spindynamicsin(001)strainedasymmetricSi /SiGeandGe/SiGequantumwells
+Among different kindsof hosts for spintronicsdevices, Silicon appea rsto be a particularly promisingone, partly
+due to the high possibility of eliminating hyperfinecouplin gsby isotropic purificationand well developedmicrofab-
+rication technology [874]. Many investigations have been c arried out to understand the electron spin relaxation in
+bulk Silicon and its nanostructures [150, 290, 295, 763, 771 , 776]. The study of hole spin relaxation in Silicon is
+verylimited. Glavinand Kim presenteda first calculationof the spin relaxationoftwo-dimensionalholesin strained
+asymmetricSi/SiGe (Ge/SiGe)quantumwells [298] bymeansof thesingle-particleap proximationwherethe impor-
+tanteffectoftheCoulombscatteringtothespinrelaxationisabsen t. Moreseriously,aspointedoutbyZhangandWu
+[299],thenondegenerateperturbationmethodwithonlythe lowestunperturbedsubbandofeachholestateconsidered
+in the calculation of the subband energy spectrum and envelo pe functionsby Glavin and Kim [298] is inadequate in
+convergingthe calculation. However, when more unperturbe d subbands are included as basis functions, the nonde-
+generate perturbationmethod even fails. By applying the ex act diagonalizing method to obtain the energy spectrum
+and envelope functions, Zhang and Wu studied the hole spin re laxation in (001) strained asymmetric Si /Si0.7Ge0.3
+(Ge/Si0.3Ge0.7) quantum wells in the situation with only the lowest hole sub band being relevant by means of the
+kineticspinBloch equationapproach[299].
+ThestructuresofSiO 2/Si/Si0.7Ge0.3andSiO 2/Ge/Si0.3Ge0.7(001)quantumwellsareillustratedinFig.82,withthe
+109confiningpotential V(z)approximatedbythetriangularpotentialduetothelargeg atevoltage. Thesubbandenvelope
+functionsareobtainedbysolvingtheeigen-equationofthe 6×6LuttingerHamiltonianincludingtheheavyhole,light
+hole and split-offhole states [122, 246, 875] under the confinement V(z), with sufficient basis functions included
+[299]. Due to the biaxial strain [122, 246, 875, 876], the low est subbandin Si/SiGe quantumwells is light hole-like,
+which is an admixture of the light hole and split-o ffhole states, whereas that in Ge /SiGe quantum wells is a pure
+heavy hole state. By using the L¨ owdin partition method [113 ], the effective Hamiltonian of the lowest hole subband
+inSi/SiGe (Ge/SiGe)quantumwellscanbewrittenas[299]
+H(l,h)
+eff=−k2
+2m(l,h)−1
+2σ·Ω(l,h)(kx,ky), (226)
+wherekis the in-planemomentum, m(l)[m(h)] is the in-planeeffectivemass of the lowest light (heavy)hole subband
+in Si/SiGe (Ge/SiGe) quantum wells, σare the Pauli matrices, and Ω(l)[Ω(h)] is the Rashba term of the lowest light
+hole (heavy hole) subband in Si /SiGe (Ge/SiGe) quantum wells. Ω(l)has both the linear and cubic dependences on
+momentum,whereas Ω(h)hasonlythecubicdependence. Forthelowestlightholesubb andinSi/SiGequantumwells,
+m(l)=m0[A−B(λ(l1l1)
+00/2−√
+2λ(l1l2)
+00)]−1, (227)
+Ω(l)=Ω(l)
+1+Ω(l)
+3, (228)
+Ω(l)
+1x,y= Ξkx,y, (229)
+Ω(l)
+3x= ΠBkx(k2
+x+k2
+y)+Θ[3Bkx(k2
+x−k2
+y)+2/radicalbig
+3(3B2+C2)k2
+ykx], (230)
+Ω(l)
+3y= ΠBky(k2
+x+k2
+y)+Θ[3Bky(k2
+y−k2
+x)+2/radicalbig
+3(3B2+C2)k2
+xky], (231)
+with
+Ξ =/planckover2pi1
+m0/radicalbig
+6(3B2+C2)κ(l1l2)
+00, (232)
+Π =−/planckover2pi13
+2m2
+0/radicalbigg
+3(3B2+C2)
+2/summationdisplay
+α=l,s∞/summationdisplay
+n=0(1−δlαδ0n)κ(l1α2)
+0n−κ(l2α1)
+0n
+E(l)
+0−E(α)
+n[√
+2(λ(l1α2)
+0n+λ(l2α1)
+0n)−λ(l1α1)
+0n],(233)
+Θ =−/planckover2pi13
+2m2
+0/radicalbigg
+3B2+C2
+3∞/summationdisplay
+n=0κ(l1h)
+0n−1√
+2κ(l2h)
+0n
+E(l)
+0−E(h)
+n(√
+2λ(l2h)
+0n+λ(l1h)
+0n). (234)
+Forthe lowestheavyholesubbandinGe /SiGequantumwells,
+m(h)=m0(A+B/2)−1, (235)
+Ω(h)=Ω(h)
+3, (236)
+Ω(h)
+3x= Λ[3Bkx(k2
+x−k2
+y)−2/radicalbig
+3(3B2+C2)k2
+ykx], (237)
+Ω(h)
+3y= Λ[3Bky(k2
+x−k2
+y)+2/radicalbig
+3(3B2+C2)k2
+xky], (238)
+with
+Λ =−/planckover2pi13
+2m2
+0/radicalbigg
+3B2+C2
+3/summationdisplay
+α=l,s∞/summationdisplay
+n=01√
+2κ(hα2)
+0n−κ(hα1)
+0n
+E(h)
+0−E(α)
+n(√
+2λ(hα2)
+0n+λ(hα1)
+0n). (239)
+HereA,BandCare the valence band parameters, which relate to the Lutting er parameters γ1,γ2andγ3through
+A=γ1,B=2γ2and√
+3B2+C2=2√
+3γ3.E(α)
+n(α=h,l,s) are the subbandenergylevels. λ(αβ)
+nn′andκ(αβ)
+nn′are defined
+asλ(αβ)
+nn′=/integraltext+∞
+−∞dzχ(α)
+n(z)χ(β)
+n′(z)andκ(αβ)
+nn′=/integraltext+∞
+−∞dzχ(α)
+n(z)dχ(β)
+n′(z)
+dz, withχ(α)
+ntheenvelopefunctinos[299]. Thecalculated
+spin-orbit coupling coe fficients of the lowest subband of Si /SiGe quantum well ( Ξ,ΠandΘ) and Ge/SiGe quantum
+well(Λ)aregivenin Fig.83.
+110-10-8-6-4-2 0 2
+ 0 0.1 0.2 0.3 0.4
+-∞SiO2 Si Si0.7Ge0.3
+(a)
+-10-8-6-4-2 0
+-10 -5  0  5  10  15  20  25  30  35 0 0.1 0.2 0.3                                         Energy (0.1 eV)
+                                        Hole distritution (arb. units)z (nm)-∞SiO2 Ge Si0.3Ge0.7
+(b)
+Figure 82: Schematics of the SiO 2/Si/Si0.7Ge0.3quantum well structure (a) and SiO 2/Ge/Si0.3Ge0.7quantum well structure (b). Two vertical
+dashed lines in each figure represent the two interfaces. The solid curves represent the confining potential V(z) with electric field E=300 kV/cm.
+Thevalence banddiscontinuities attheSi /Si0.7Ge0.3andGe/Si0.3Ge0.7interfaces areneglected inthetriangular potential appro ximation. Thechain
+curves with their scale on the right hand side of the frame res pectively represent the lowest light hole and heavy hole dis tributions in Si/Si0.7Ge0.3
+and Ge/Si0.3Ge0.7quantum wells along the z-axis. FromZhang and Wu[299].
+Theholespinrelaxationtimeiscalculatedbysolvingtheki neticspinBlochequations,withthehole-deformation
+optical/acousticphonon[877],hole-impurityandhole-holeCoulom bscatteringsexplicitlyincluded[299]. Thetypical
+results are shown in Fig. 84 for Si /SiGe quantum wells, where a peak appears in both the temperat ure dependence
+(located around the hole Fermi temperature) and the density dependence (located around the crossover from the
+degenerate to the nondegenerate regimes) of hole spin relax ation time. It is also shown that the hole-hole Coulomb
+scatteringplaysan essential rolein thespin relaxation. A sshownin Fig. 85, byswitchingo ffthehole-holeCoulomb
+scattering, the spin relaxation times di ffer dramatically. Similar behavior also happens in Ge /SiGe quantum wells,
+exceptthat the peakin the temperaturedependenceof the hol e spin relaxationtime is locatedaroundhalf of the hole
+Fermi temperature. The shift of the peak to the lower tempera ture is suggested to be due to the cubic momentum
+dependence of the Rashba term [Eqs. (236-238)], in contrast to the linear one which is important in the case of
+Si/SiGe quantumwells. Thee ffectsofimpurityandgatevoltageonthe holespin relaxation arediscussed in detailin
+Ref.[299].
+Finally, it is noted that unlike holes in GaAs quantum wells w here the system is in the weak scattering limit
+[363], holes in Si/SiGe quantum wells are generally in the strong scattering li mit thanks to the strong hole-hole
+Coulomb scattering and weak Rashba spin-orbit coupling. Ho wever, holes in Ge /SiGe quantum wells can be in the
+weak scattering limit with high density at low temperature d ue to the larger Rashba term [299]. In any case, adding
+impuritiescanshiftthe systemfromweakscatteringlimitt o thestrongone[299, 363].
+5.4.11. Spinrelaxationanddephasinginn-typewurtzite Zn O(0001)quantumwells
+While there are a lot of studies on the spin dynamics in cubic z inc-blende semiconductors, much attention has
+also been devoted to the spin properties of zinc oxide (ZnO) w ith wurtzite structures [556, 559, 568, 878–882].
+Harmonet al. calculatedthespin relaxationtime inbulk mat erialinthe frameworkof singleparticleapproach[559].
+L¨ u and Cheng [613] applied the kinetic spin Bloch equation a pproach and calculated the spin relaxation in n-type
+ZnO(0001)quantumwellsundervariousconditions,includi ngwellwidth,impuritydensityandexternalelectricfield
+(hot-electroneffect). Inwurtzitestructure,theD’yakonov-Perel’termcom esfromtheRashbatermduetotheintrinsic
+wurtzitestructureinversionasymmetry ΩR(k)andtheDresselhausterm ΩD(k)whichcanbewrittenas[245]:
+gµBΩR(k)=αe(ky,−kx,0), (240)
+gµBΩn
+D(k)=γe(b∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}htn−k2
+||)(ky,−kx,0), (241)
+111Figure 83: Spin-orbit coupling coe fficients/planckover2pi1
+2Ξ,/planckover2pi1
+2Πand/planckover2pi1
+2Θfor the lowest light hole subband in Si /SiGe quantum wells and/planckover2pi1
+2Λfor the lowest
+heavy holesubband in Ge /SiGequantum wells against theelectric field E. Thescaleof/planckover2pi1
+2Ξisontheright hand sideoftheframe. FromZhangand
+Wu[299].
+Figure 84: (a) Spin relaxation time τagainst temperature Twith different hole densities. (b) Spin relaxation time τagainst hole density Nhwith
+different temperatures. For both cases the impurity density Ni=0 and the electric field E=300 kV/cm. From Zhang and Wu [299].
+withαe,γeandbstanding for the spin-orbitcouplingcoe fficients.∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}htn=n2π2/a2is the subbandenergyin the hard
+wall potential approximation. By takingthe lowest two subb andsinto account,L¨ u andCheng solvedthe kinetic spin
+Blochequations. ThetypicalresultsareshowninFig. 86.
+BasicallythepropertiesofthespinrelaxationinZnOquant umwellsareallthesamewiththoseinGaAsones. One
+still observesthe peak in the temperature and density depen dences(the former requires low impurity density) of the
+spinrelaxationtime. Thedi fferenceis that forGaAs quantumwells, the temperaturepeako fthe spin relaxationtime
+canonlybeobservedatlowelectrondensity(i.e.,lowtrans itiontemperature)andlowimpuritydensity[109,372,604] ,
+because the electron-phonon scattering becomes strong eno ugh to destroy the nonmonotonic Tdependence of the
+scatteringtimeinducedbytheelectron-electronscatteri ng. SuchcasecanbeavoidedinZnOquantumwells,inwhich
+the electron-phonon scattering is always pretty weak due to the large optical phonon energies ( ∼800 K). Thus the
+temperaturepeakcanbefoundevenforhighelectrondensity samples.
+5.5. Spinrelaxationinquantumwires
+In recent years, progress in nanofabrication and growth tec hniques has made it possible to produce high-quality
+quantum wires and investigate physics in these nanostructu res [883–890]. The energy spectrum of quantum wire
+systemswith strongspin-orbitcouplinghasbeen studiedbo thexperimentally[239, 890, 891] andtheoretically[191,
+240, 241, 890, 892–896]. Unlike quantum wells discussed in S ec. 5.4, there is additional confinement in quantum
+wires and the remaining free degree of freedom is along the wi re growth direction. Therefore, there are stronger
+112Figure 85: Spin relaxation time τagainst hole density Nhwith different scatterings included in Si /SiGe quantum well. T=300 K and E=
+300 kV/cm. Solid curve: with the hole-hole Coulomb (h-h), the hole- optical phonon (h-op) and the hole-acoustic phonon (h-ac) s catterings.
+Dashedcurve: sameasthesolid curvewiththeadditional hol e-impurity scattering ( Ni=1010cm−2)included; Chain curve: with theh-opand h-ac
+scatterings; Dotted curve: sameasthechain curve with thea dditional hole-impurity scattering ( Ni=5×108cm−2)included. FromZhang andWu
+[299].
+Figure 86: ZnO (0001) quantum wells: (a) Spin relaxation tim eτvs. temperature Tat different impurity densities. The dashed curve is obtained
+from the calculation of excluding the electron-phonon scat tering. (b) and (c): Spin relaxation time vs.the electron density with di fferent impurity
+densities and temperatures (20 K and 300 K respectively). /trianglesolid:Ni/Ne=0;/squaresolid:Ni/Ne=0.1;/diamondsolid:Ni/Ne=1. From L¨ uand Cheng [613].
+subbandeffectandalsomarkedanisotropyalongthegrowthdirectionso fwires. Consequentlythisgivesmorechoices
+forthemanipulationofthespin degreeoffreedom.
+Forquantumwires,thespinrelaxationwascalculatedinthe frameworkofsingleparticleapproach[632,698,701,
+897] andMonteCarlo simulations[651, 697, 701, 898]. Cheng et al. first appliedthe kineticspinBloch equationsto
+studythespindynamicsin(001)orientedInAsquantumwires withonlythelowestsubbandinvolved[899]. L¨ uetal.
+studiedtheinfluenceofhighersubbandsandwireorientatio nsonspinrelaxationin n-typeInAsquantumwires[696].
+They reported that the intersubband Coulomb scattering can make an important contribution to the spin relaxation.
+Also due to intersubbandscattering in connectionwith the s pin-orbitcoupling,spin relaxationin quantum wires can
+showdifferentcharacteristicsfromthoseinbulkandinquantumwell s. Holespinrelaxationin p-typeGaAsquantum
+wireswasalso investigatedbyL¨ uet al. fromthekineticspi nBlochequationapproach[705].
+5.5.1. Electronspinrelaxationinn-typeInAsquantumwire s
+By modeling the InAs quantum wire by a rectangular confinemen t potential of infinite well depth along the x-y
+plane (|x|≤ax,|y|≤ay), one can write the Rashba and Dresselhaus Hamiltonian by re placingkx,k2
+x,ky, andk2
+y
+in the bulk Rashba and Dresselhaus Hamiltonian by ∝an}b∇acketle{tψnx|−i∂/∂x|ψn′x∝an}b∇acket∇i}ht,∝an}b∇acketle{tψnx|(−i∂/∂x)2|ψn′x∝an}b∇acket∇i}ht,∝an}b∇acketle{tψny|−i∂/∂y|ψn′y∝an}b∇acket∇i}htand
+113∝an}b∇acketle{tψny|(−i∂/∂y)2|ψn′y∝an}b∇acket∇i}ht,respectively. Here ψnx=/radicalig
+2
+axsinnxπx
+axandψny=/radicalig
+2
+aysinnyπy
+ay. ThebulkRashbatermreads
+HR(k)=γ6c6c
+41σ·k×E=γ6c6c
+41[σx(kyEz−kzEy)+σy(kzEx−kxEz)+σz(kxEy−kyEx)]. (242)
+Fora(100)InAsquantumwire,the x,yandzaxescorrespondtothe[100],[010]and[001]crystallograp hicdirections,
+respectively,andthebulkDresselhaustermcanbewrittena s[113]:
+H100
+D=b6c6c
+41{σx[kx(k2
+y−k2
+z)]+σy[ky(k2
+z−k2
+x)]+σz[kz(k2
+x−k2
+y)]}. (243)
+Fora(110)quantumwire,the x,yandzdirectionscorrespondtothe[ ¯110],[001]and[110]crystallographicdirections,
+andonehas
+H110
+D=b6c6c
+41{σx[−1
+2kz(k2
+x−k2
+z+2k2
+y)]+2σykxkykz+σz[1
+2kx(k2
+x−k2
+z−2k2
+y)]}. (244)
+For a (111) quantum wire, the x,yandzdirections correspond to the [11 ¯2], [¯110], and [111] crystallographicdirec-
+tions,andonehas
+H111
+D=b6c6c
+41{σx[−√
+2√
+3kxkykz−1
+2√
+3k3
+y−1
+2√
+3kyk2
+x+2√
+3kyk2
+z−√
+2
+3k2
+ykz]
++σy[1
+2√
+3k3
+x+1
+2√
+3kxk2
+y−1√
+6k2
+xkz−1√
+6kz(k2
+x+k2
+y)]+σz[√
+3√
+2k2
+xky−1√
+6k3
+y−2
+3kzk2
+y]}.(245)
+ThekineticspinBlochequationsread ˙ ρk=˙ρk|coh+˙ρk|scat, withthe coherenttermsbeing
+˙ρk/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoh=i/bracketleftig/summationdisplay
+QVQIQρk−qI−Q,ρk/bracketrightig
+−i/bracketleftig
+He(k),ρk/bracketrightig
+. (246)
+HereQ≡(qx,qy,q).IQistheformfactor. In( s,s′)space [s=(nx,ny,σ)],it reads
+IQ,s1,s2=∝an}b∇acketle{ts1|eiQ·r|s2∝an}b∇acket∇i}ht=δσ1,σ2F(m1,m2,qy,ay)F(n1,n2,qx,ax), (247)
+where
+F(m1,m2,q,a)=2iaq[eiaqcosπ(m1−m2)−1]/bracketleftigg1
+π2(m1−m2)2−a2q2−1
+π2(m1+m2)2−a2q2/bracketrightigg
+.(248)
+The first term in Eq. (246) is the Coulomb Hartree-Fock term, a nd the second term comes from the single particle
+Hamiltonian He=P2
+2m∗+HR+HD+Vc(r) in (s,s′) space. For small spin polarization, the contribution from the
+Hartree-Focktermin the coherenttermis negligible[41, 42 , 44] andthecoherentspin dynamicsis essentiallydueto
+thespinprecessionaroundthe e ffectiveinternalfieldsdescribedbyEqs.(242)-(245).
+Thescatteringcontributionsto thedynamicequationofthe spin-densitymatrixincludetheelectron-nonmagnetic
+impurity,electron-phononandelectron-electronscatter ings:
+∂ρk
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsinglescat=∂ρk
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingleim+∂ρk
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingleph+∂ρk
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingleee,
+∂ρk
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingleim=πNis1,s2/summationdisplay
+Q|Ui
+Q|2δ(Es1,k−q−Es2,k)IQ[(1−ρk−q)Ts1I−QTs2ρk−ρk−qTs1I−QTs2(1−ρk)]+H.c.,
+∂ρk
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingleph=πs1,s2/summationdisplay
+Q,λ|MQ,λ|2IQ/braceleftbigδ(Es1,k−q−Es2,k+ωQ,λ)[(NQ,λ+1)(1−ρk−q)Ts1I−QTs2ρk−NQ,λρk−qTs1I−QTs2(1−ρk)]
++δ(Es1,k−q−Es2,k−ωQ,λ)[NQ,λ(1−ρk−q)Ts1I−QTs2ρk−(NQ,λ+1)ρk−qTs1I−QTs2(1−ρk)]/bracerightbig+H.c.,
+∂ρk
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingleee=πs1,s2,s3,s4/summationdisplay
+Q,k′V2
+Qδ(Es1,k−q−Es2,k+Es3,k′−Es4,k′−q)IQ/braceleftbig(1−ρk−q)Ts1I−QTs2ρkTr[(1−ρk′)Ts3IQTs4ρk′−qI−Q]
+−ρk−qTs1I−QTs2(1−ρk)Tr[ρk′Ts3IQTs4(1−ρk′−q)I−Q]/bracerightbig+H.c., (249)
+114in which Ts1,s,s′=δs1,sδs1,s′. The statically screened Coulomb potential in the random-p hase approximation reads
+[833]
+Vq=/summationdisplay
+qx,qyvQ|IQ|2/κ(q), (250)
+withthebareCoulombpotential vQ=4πe2/Q2and
+κ(q)=1−/summationdisplay
+qx,qyvQ/summationdisplay
+ks|IQ,s,s|2fk+q,s−fk,s
+Es,k+q−Es,k. (251)
+In Eq. (249), Niis the density of impurities, and |Ui
+Q|2is the impurity potential. Furthermore, |MQ,λ|2andNQ,λ=
+[exp(ωQ,λ/kBT)−1]−1are the matrixelementsofthe electron-phononinteraction andthe Bose distributionfunction,
+respectively. Thephononenergyspectrumforphononwithmo deλandwavevector Qisdenotedby ωQ,λ.
+By numerically solving the kinetic spin Bloch equations, L¨ u et al. [696] investigated the influence of the wire
+size, orientation, doping density, and temperature on the s pin relaxation time. They also showed that the Coulomb
+scattering makes marked contribution to the spin relaxatio n. Some typical results that the spin relaxation time can
+be effectively manipulated in quantum wires with di fferent orientations are summarized in Fig. 87. For (100) quan -
+tum wires [Fig. 87(a)], due to the competition of the longitu dinal and transverse e ffective magnetic fields from the
+Dresselhaus and Rashba terms addressed above, the spin rela xation time can be e fficiently manipulated by the wire
+size. Specifically, there is a minimum in the spin relaxation time when ax=ay, thanks to the cancellation of the
+longitudinalcomponentfromthe Dresselhausterm( ∝an}b∇acketle{tk2
+x∝an}b∇acket∇i}ht−∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht=0) whenonlythe lowest subbandis populated. For
+(110)quantumwire with ax=ay=30nm,theelectronicpopulationismainlyinthelowestsubb and. Inthe presence
+ofanelectricfieldoftheform( Ex,Ey, 0),the relevantRashbaandDresselhaustermsare
+H110
+R=γ6c6c
+41(−σxEykz+σyExkz), (252)
+H110
+D=−1
+2b6c6c
+41σxkz(∝an}b∇acketle{tk2
+x∝an}b∇acket∇i}ht−k2
+z+2∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht). (253)
+Theeffective magneticfield formedbythe Dresselhausterm is along thex-direction,which correspondsto the [ ¯110]
+crystallographic direction, and the e ffective magnetic field formed by the Rashba term is in the x-yplane. If the
+directionofthe total e ffectivemagnetic field formedbythe spin-orbitcouplingis tu nedto be exactly the directionof
+the initial spin polarization, then one can expect an extrem ely long spin relaxation time as pointedout in Refs. [597,
+697]. This is exactly the case as shown in Fig. 87(b). However ,for wider wire size, the spin relaxation time is much
+smaller due to the contribution of higher subbands. For (111 ) quantum wires, again in the case with only the lowest
+subbandbeingrelevant,theRashbaandDresselhaustermsre ad
+H111
+R=γ6c6c
+41(−σxEykz+σyExkz), (254)
+H111
+D=b6c6c
+41(−√
+2
+3σxkz∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht−1√
+6σykz(∝an}b∇acketle{tk2
+x∝an}b∇acket∇i}ht+∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht)−2
+3σzkz∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht). (255)
+Similar to the case of (110) quantumwires, one expectsvery l ong spin relaxation time if the total e ffective magnetic
+field points into the directions of initial spin polarizatio n. For a numerical example of this e ffect, L¨ u et al. chose Ey
+suchthatγ6c6c
+41Ey+(√
+2/3)b6c6c
+41∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht=0 fora small wire width ax=ay=10nm, so that the xcomponentof thetotal
+effective magnetic field is zero. For an initial spin polarizati onalong the z-direction, which correspondsto the [111]
+crystallographic direction, the spin relaxation time as a f unction of Exis shown in Fig. 87(c). It is seen that when
+ax=ay=10 nm, there is a pronounced maximum of the spin relaxation ti me atEx=70 kV/cm, which fulfills the
+relationγ6c6c
+41Ex+1√
+6b6c6c
+41(∝an}b∇acketle{tk2
+x∝an}b∇acket∇i}ht+∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht)≈0. Consequently, for this field strength, the direction of th e total effective
+magneticfieldisexactlyalongthedirectionoftheinitials pinpolarizationandthisleadstoaverylongspinrelaxatio n
+time.
+5.5.2. Holespinrelaxationin p-type(001)GaAsquantumwir es
+Investigation on hole spin relaxation in quantum wires is ve ry limited [698]. Unlike the hole spins in bulk III-V
+materialswhichrelax veryfast dueto the mixtureof the heav y-holeand light-holestates, in confinedstructuressuch
+115Figure 87: (a) Spin relaxation time τvs. the quantum wire width in ydirection ayfor InAs (100) quantum wires at di fferentax. The electron
+density is N=4×105cm−1andT=100 K. The arrows mark the densities at which the electron pop ulations in the second and higher subbands
+areapproximately 30 %. (b) Spin relaxation time τvs.Exfor (110) quantum wires at T=50 K and Ne=4×105cm−1./trianglesolid:ax=ay=30 nm with
+an initial spin polarization along the x-direction; /squaresolid:ax=ay=30 nm with an initial spin polarization along the y-direction;•:ax=ay=50 nm
+with an initial spin polarization along the x-direction. (c) Spin relaxation time τvs.Exfor different wire sizes ax=ayatT=50 K and
+Ne=4×105cm−1. The growth direction of the quantum wire is along the [111] c rystallographic direction. The solid curves are the result s with
+the initial spin polarization along z-direction and the dashed curve is the result with the initia l spin polarization along x-direction. From L¨ u et al.
+[696].
+as quantum wells [363] and quantum dots [491], the degenerac y of the heavy and light hole bands is lifted and the
+mixtureofthesebandscanbetunedbystrain[491]. Therefor ethespinlifetimeofholescanbemuchlongerthanthat
+inthebulk[363,491]. Inquantumwires,similarsituationa lsohappens. L¨ uetal. performedasystematicinvestigatio n
+on the spin relaxationof p-type GaAs quantumwires by numericallysolvingthe kinetic spin Bloch equations[705].
+They reported that the quantum wire size influences the spin r elaxation time effectively by modulating the energy
+spectrumandthe heavy-hole–light-holemixingofwirestat es.
+By consideringa p-doped(001)GaAs quantumwire with rectangularconfinement and hard wall potential,L¨ u et
+al. first obtained the subband structure by diagonalizing th e hole Hamiltonian including the quantum confinement.
+Here the light-hole admixture is dominant in the lowest spin -split subband, but the heavy-hole admixture becomes
+alsoimportantinhighersubbandsduetotheheavy-hole–lig ht-holemixing. Thentheyinvestigatedthetimeevolution
+of holes by numerically solving the fully microscopic kinet ic spin Bloch equations in the obtained subbands, with
+all the scatterings, particularly the Coulomb scattering, explicitly included. They found that the quantum wire size
+influences the spin relaxation time e ffectively because the spin-orbit coupling and the subband st ructure in quantum
+wires dependstronglyon the confinement. When the quantumwi re size increases, the lowest spin-split subbandand
+the second-lowest spin-split subband can be very close to ea ch other at an anticrossing point. If the anticrossing is
+closeto the Fermisurface,the contributionfromthe spin-fl ipscatteringreachesa maximumandcorrespondinglythe
+spin relaxation time reaches a minimum. Moreover, they show ed that the dependenceof the spin relaxation time on
+confinement size in quantum wires behaves oppositely to the t rend found in quantum wells. It was also found that,
+whenthequantumwiresizeisverysmall,thespinrelaxation timecaneitherincreaseordecreasewithincreasinghole
+density, dependingon the spin mixing of the subbands. Howev er, the behavior of holes in quantum wires where the
+spin relaxationtime increasesor decreaseswith hole densi tyis quite differentfromthe oneof light holesin quantum
+wells with small well width [363]. These featuresoriginate fromthe subbandstructureof the quantumwires and the
+spin mixing which give rise to the spin-flip scattering. The s pin mixing and inter-subband scattering are modulated
+more dramatically in quantum wires by changing the hole dist ribution in different subbands. They also investigated
+the effects of temperature and initial spin polarization, showing that the inter-subband scattering and the Coulomb
+Hartree-Fockcontributioncanmakea markedcontributiont othe spinrelaxation.
+By assuming the growth direction of the quantum wire along th ez-axis ([001] crystallographic direction), the
+Hamiltonian of holes in the basis of spin-3 /2 projection ( Jz) eigenstates with quantum numbers +3
+2,+1
+2,−1
+2and−3
+2
+canbewrittenas[113]
+Hh=HhhS R 0
+S†Hlh0R
+R†0Hlh−S
+0R†−S†Hhh+Hr
+8v8v+Hb
+8v8v+Vc(r), (256)
+116whereVc(r)isthehard-wallconfinementpotentialin xandydirectionsand
+Hhh=1
+2m0[(γ1+γ2)[P2
+x+P2
+y]+(γ1−2γ2)P2
+z, (257)
+Hlh=1
+2m0[(γ1−γ2)[P2
+x+P2
+y]+(γ1+2γ2)P2
+z, (258)
+S=−√
+3γ3
+m0Pz[Px−iPy],R=−√
+3
+2m0{γ2[P2
+x−P2
+y]−2iγ3PxPy}, (259)
+Hr
+8v8v=γ8v8v
+41
+/planckover2pi1[Jx(PyEz−PzEy)+Jy(PzEx−PxEz)+Jz(PxEy−PyEx)], (260)
+Hb
+8v8v=b8v8v
+41
+/planckover2pi13{Jx[Px(P2
+y−P2
+z)]+Jy[Py(P2
+z−P2
+x)]+Jz[Pz(P2
+x−P2
+y)]}. (261)
+In these equations, m0denotes the free electron mass, γ1,γ2andγ3are the Luttinger parameters, Eis the electric
+fieldand Jiarespin-3/2angularmomentummatrices[122]. Hr
+8v8visthespin-orbitcouplingarisingfromthestructure
+inversionasymmetryand Hb
+8v8visthespin-orbitcouplingfromthebulkinversionasymmetr y. AsshowninRef.[705],
+these two termsturn out to be one or two ordersof magnitudesm aller than the intrinsic spin-orbitcouplingfrom the
+four-band Luttinger-KohnHamiltonian [the first term in Eq. (256)]. Moreover, from the first term in Eq. (256), one
+can see that the light hole spin-upstates can be directly mix edwith the heavy hole states by SandR, but the mixing
+betweenlight holespin-upstates andlight holespin-downs tates has to be mediatedby the heavyhole states. All the
+mixing is related to the confinement. When the confinement dec reases, the mixing increases due to the decrease of
+theenergygapbetweenthelightholeandheavyholestates.
+The kinetic spin Bloch equations ˙ ρk+˙ρk|coh+˙ρk|scat=0 can be written in either the collinear spin space which
+is constructed by basis {s}, with{s}obtained from the eigenfunctions of the diagonal part of Hh(k).|s∝an}b∇acket∇i}ht=|m,n∝an}b∇acket∇i}ht|σ∝an}b∇acket∇i}ht
+with∝an}b∇acketle{tr|m,n∝an}b∇acket∇i}ht=2√axaysin(mπy
+ay)sin(nπx
+ax)eikzand|σ∝an}b∇acket∇i}htstanding for the eigenstates of Jz. Then the matrix elements in the
+collinear spin space ρc
+k,s1,s2is written as ρc
+k,s1,s2=∝an}b∇acketle{ts1|ρk|s2∝an}b∇acket∇i}ht. Here the superscript “ c” denotes the quantum number
+distinguishingstatesinthecollinearspinspace. Onecana lsoprojectρkinthe“helix”spinspacewhichisconstructed
+bybasis{η}withηbeingtheeigenfunctionsof Hh(k):
+Hh(k)|η∝an}b∇acket∇i}ht=Eη,k|η∝an}b∇acket∇i}ht. (262)
+Thisbasisfunctionisamixtureoflight-holeandheavy-hol estatesandis kdependent. Thenthematrixelementsinthe
+helixspin space ρh
+k,η,η′canbe writtenas ρh
+k,η,η′=∝an}b∇acketle{tη|ρk|η′∝an}b∇acket∇i}ht, with thesuperscript“ h”denotingthe helixspin space. The
+density matrix in the helix spin space can be transformed fro m that in the collinear one by a unitary transformation:
+ρh
+k=U†
+kρc
+kUk,whereUk(i,α)=ηi
+α(k)withηi
+α(k)beingthe ithelementofthe αtheigenvectorafterthediagonalization
+ofHh(k).
+Inhelixspinspace,thecoherenttermsread
+˙ρh
+k|coh=i/bracketleftig/summationdisplay
+QVQU†
+kIQUk−qρh
+k−qU†
+k−qI−QUk,ρh
+k/bracketrightig
+−i/bracketleftig
+U†
+kHh(k)Uk,ρh
+k/bracketrightig
+, (263)
+whereIQis the form factor in the collinear spin space with wave vecto rQ≡(qx,qy,q). The first term in Eq. (263)
+istheCoulombHartree-Focktermandthesecondtermistheco ntributionfromtheintrinsicspin-orbitcouplingfrom
+the Luttinger-KohnHamiltonian. IQcan be written as IQ,s1,s2=∝an}b∇acketle{ts1|eiQ·r|s2∝an}b∇acket∇i}ht=δσ1,σ2F(m1,m2,qy,ay)F(n1,n2,qx,ax),
+withF(m1,m2,q,a)beingexpressedasEq.(248). Forsmallspinpolarization, thecontributionfromtheHartree-Fock
+term in the coherent term is negligible [41, 42, 44] and the sp in precession is determined by the spin-orbit coupling,
+˙ρh
+k,η,η′|coh=−iρh
+k,η,η′(Eη,k−Eη′,k),whichisproportionaltotheenergygapbetween ηandη′subbands.
+The scatteringtermsincludethe hole-nonmagnetic-impuri ty,hole-phononandhole-holeCoulombscatterings. In
+thehelixspinspace,thescatteringtermsaregivenby
+117˙ρh
+k|scat=πNi/summationdisplay
+Q,η1,η2|Ui
+Q|2δ(Eη1,k−q−Eη2,k)U†
+kIQUk−q[(1−ρh
+k−q)Tk−q,η1U†
+k−qI−QUkTk,η2ρh
+k
+−ρh
+k−qTk−q,η1U†
+k−qI−QUkTk,η2(1−ρh
+k)]
++π/summationdisplay
+Q,η1,η2,λ|MQ,λ|2U†
+kIQUk−q{δ(Eη1,k−q−Eη2,k+ωQ,λ)[(NQ,λ+1)(1−ρh
+k−q)Tk−q,η1U†
+k−qI−QUkTk,η2ρh
+k
+−NQ,λρh
+k−qTk−q,η1U†
+k−qI−QUkTk,η2(1−ρh
+k)]+δ(Eη1,k−q−Eη2,k−ωQ,λ)[NQ,λ(1−ρh
+k−q)Tk−q,η1
+×U†
+k−qI−QUkTk,η2ρh
+k−(NQ,λ+1)ρh
+k−qTk−q,η1U†
+k−qI−QUkTk,η2(1−ρh
+k)]}
++π/summationdisplay
+Q,k′/summationdisplay
+η1,η2,η3,η4V2
+QU†
+kIQUk−qδ(Eη1,k−q−Eη2,k+Eη3,k′−Eη4,k′−q){(1−ρh
+k−q)Tk−q,η1U†
+k−qI−QUkTk,η2ρh
+k
+×Tr[(1−ρh
+k′)Tη3,k′U†
+kIQUk−qTk′−q,η4ρh
+k′−qU†
+k−qI−QUk]−ρh
+k−qTk−q,η1U†
+k−qI−QUkTk,η2(1−ρh
+k)
+×Tr[ρh
+k′Tη3,k′U†
+kIQUk−qTk′−q,η4(1−ρh
+k′−q)U†
+k−qI−QUk]}+H.c., (264)
+inwhichTk,η(i,j)=δηiδηj.VQinEq.(264)reads VQ=4πe2/[κ0(q2+q2
+∝ba∇dbl+κ2)],withκ0representingthestaticdielectric
+constant and κ2=4πe2Nh/(kBTκ0a2) standing for the Debye screening constant. Niin Eq. (264) is the impurity
+density and|Ui
+Q|2={4πZie2/[κ0(q2+q2
+∝ba∇dbl+κ2)]}2is the impurity potential with Zistanding for the charge number of
+the impurity.|MQ,λ|2andNQ,λ=[exp(ωQ,λ/kBT)−1]−1are the matrix element of the hole-phonon interaction and
+theBosedistributionfunctionwithphononenergyspectrum ωQ,λat phononmode λandwavevector Q,respectively.
+Herethehole-phononscatteringincludesthehole–longitu dinaloptical-phononandhole–acoustic-phononscatterin gs
+withthe explicitexpressionsof |MQ,λ|2canbe foundin Refs. [44, 372, 844]. Itis notedthatthe energ yspectrum Eη,k
+inthescatteringtermcontainsthespin-orbitcouplingwhi chcannotbeignoredduetothestrongcoupling. Bysolving
+thekineticspinBlochequations,oneobtainsholespin rela xation.
+ 40 60 80 100 120 140 160 180 200
+ 0  0.02  0.04  0.06  0.08  0.11+ 2–3+4+5–
+kz (1/nm)(a) ax = 6 nmE (meV)
+1–2+3–4–5+
+ 30 40 50 60 70 80 90 100
+ 0  0.02  0.04  0.06  0.081–2–3–4–5–
+kz (1/nm)kz (1/nm)(c) ax = 10 nm
+1+2+3+4+5+
+ 40 60 80 100 120 140
+ 0  0.02  0.04  0.06  0.081+2–3+4+5–
+kz (1/nm)kz (1/nm)(b) ax = 8 nm
+1– 2+3–4–5+
+ 20 30 40 50 60 70 80 90
+ 0  0.02  0.04  0.06  0.081–2+3+4–5–
+kz (1/nm)(d) ax = 12 nmE (meV)
+1+2–3–4+5+
+ 20 30 40 50 60 70
+ 0  0.02  0.04  0.061–2+3+4–5–
+kz (1/nm)kz (1/nm)(e) ax = 15 nm
+1+2–3–4+5+
+ 20 30 40 50 60
+ 0  0.02  0.04  0.061–2+3+4–5–
+kz (1/nm)kz (1/nm)(f) ax = 20 nm
+1+2–3–4+5+
+Figure 88: Typical energy spectra of holes in GaAs (001) quan tum wires for (a) ax=6 nm; (b) ax=8 nm; (c) ax=10 nm; (d) ax=12 nm;
+(e)ax=15 nm; and (f) ax=20 nm.ay=10 nm.∝an}b∇acketle{tE∝an}b∇acket∇i}htatT=20 K is also plotted: solid line for Nh=4×105cm−1and dashed line for
+Nh=2×106cm−1. FromL¨ u etal. [705].
+The typical subbandstructure is shown in Fig. 88 for di fferent confinements. Each subbandis denotedas l+(l−)
+if the dominant spin component is the spin-up (-down) state. One can see from Fig. 88 that 1 +and 1−subbands
+are very close to each other, so are the subbands 2 ±. The spin-splitting between them is mainly caused by the spi n-
+orbit coupling arising from the bulk inversion asymmetry, b ecause that the spin-splitting caused by the spin-orbit
+118couplingarisingfromthe structureinversionasymmetryis threeordersofmagnitudesmaller thanthe diagonalterms
+inEq.(256)andcannotbeseeninFig.88. Thespin-splitting causedbythebulkinversionasymmetryisproportional
+to(P2
+x−P2
+y),which disappearswhenthe confinementin xandydirectionsare symmetrical. Therefore, l±are almost
+degenerate when ax=ay=10 nm. If one excludes the spin-orbit coupling from the bulk i nversion asymmetry
+and structure inversion asymmetry, l±are always degeneratebecause of the Kramers degeneracy. On e also observes
+that when axgets larger, the subbands are closer to each other. Especial ly, in the case of ax=ay=10 nm, there
+are anticrossing points due to the heavy-hole–light-hole m ixing in the Luttinger Hamiltonian. When axkeeps on
+increasing,the anticrossingpointat small kbetweenthe 1±and2±graduallydisappears. However,at large kregion,
+thelowesttwosubbandsbecomeveryclosetoeachother. Thes ewillleadtosignificante ffectonspinrelaxationtime.
+InFig.88,a quantity ∝an}b∇acketle{tE∝an}b∇acket∇i}ht,with
+∝an}b∇acketle{tE∝an}b∇acket∇i}ht=/summationtext
+l/integraltext+∞
+−∞dk(ρh
+k,l+,l+−ρh
+k,l−,l−)(El+,k+El−,k)
+2/summationtext
+l/integraltext+∞
+−∞dk(ρh
+k,l+,l+−ρh
+k,l−,l−), (265)
+is introducedto representthe energyregionwhere spin prec essionand relaxationbetween the +and−bandsmainly
+take place. It is seen from the figure that for Nh=4×105cm−1and 2×106cm−1,∝an}b∇acketle{tE∝an}b∇acket∇i}htonly intersects with 1 ±and
+2±subbands. It is also seen that the dominant spin componentin 1+(1−) state is the spin-up(spin-down)light-hole
+state.
+Therearethreemechanismsleadingtospinrelaxation. Firs t,thespin-flipscattering,whichincludesthescattering
+betweenl+andl−subbands and the scattering between l+andl′−subbands ( l/nequall′), can cause spin relaxation.
+The spin relaxationtime decreaseswith the spin-flip scatte ring,with the scattering strengthbeing proportionalto th e
+spin mixing of the helix subbands. Second, because of the coh erent term ˙ρh
+k|coh, there is a spin precession between
+differentsubbands. Thefrequencyofthisspinprecessiondepen dsonkandthisdependenceservesasinhomogeneous
+broadening. AsshowninRefs.[44,332,334,350,569,844],i nthepresenceoftheinhomogeneousbroadening,even
+the spin-conservingscattering can cause irreversible spi n relaxation. As a result, the spin-conservingscattering, i.e.,
+the scattering between l+andl′+and the scattering between l−andl′−, can cause spin relaxation along with the
+inhomogeneousbroadening. Atlast,thespin-flipscatterin galongwiththeinhomogeneousbroadeningcanalsocause
+anadditionalspinrelaxation.
+It is seen from Fig. 88(a) that when Nh=4×105cm−1andax=6 nm,∝an}b∇acketle{tE∝an}b∇acket∇i}htonlyintersects with the 1 ±subbands
+and is far away from the 2 ±subbands. Therefore, holes populate the 1 ±subbands only. As pointed out before,
+the coherent term ˙ ρh
+k,1+,1−|cohis proportional to ( E1+,k−E1−,k). As holes are only populating states in the small k
+region where the spin splitting between 1 ±is negligible, the spin precession between these two states , and thus the
+inhomogeneousbroadening, is very small. Consequently the main spin-relaxation mechanism is due to the spin-flip
+scattering,i.e.,thescatteringbetween1 ±subbands.
+In the case of larger axandNhas shown in Fig. 88(c)-88(f), where ∝an}b∇acketle{tE∝an}b∇acket∇i}htis close to or intersects with the 2 ±
+subbands,holespopulateboththe 1 ±and2±subbands. The spin-flipscattering here includesthe scatte ringbetween
+1±states, the scattering between 2 ±states and the spin-flip scattering between 1 ±and 2±subbands. This spin-flip
+scattering is still found to be the main spin relaxation mech anism. Besides, di ffering from the case of Fig. 2(a), the
+coherentterm ˙ ρh
+k,1±,2±|cohis proportionalto the energygap between 1 ±and 2±, and it is muchlarger than ˙ ρh
+k,1+,1−|coh.
+As a result, there is a muchstronger spin precession between 1±and 2±subbandswith a frequencydependingon k,
+andthe inhomogeneousbroadeningcaused by this precession along with boththe spin-conservingscattering and the
+spin-flipscatteringcanmakea considerablecontributiont o thespinrelaxation.
+A typical hole spin relaxation time as a function of wire widt h is given in Fig. 89 at di fferent temperatures and
+hole densities. The underlying physics can be well understo od from the corresponding subband mixing in Fig. 88.
+Similarly the temperature, hole density, and spin polariza tion dependencesof spin relaxation were also discussed in
+detailinRef. [705].
+5.6. SpinrelaxationinbulkIII-Vsemiconductors
+The study of spin dynamics in bulk III-V semiconductorshas a long history. The theoretical study of spin relax-
+ationandthesystematicexperimentalinvestigationstart edasearlyas1970s[3]. Theearlystudieshavebeenreviewed
+comprehensivelyin the book “Optical Orientation” [3]. In the past decade, the topic gained renewed interest in the
+119 0 1 2 3 4 5 6 7 8 9
+ 6  8  10  12  14  16  18  20 20 25 30 35 40 45 50 55 60
+ax (nm)(a) Nh = 4 × 105 cm-1
+<E> (meV)τ (ps)     
+T = 20 K
+ 100 K
+ 300 K
+<E> at T = 20 K
+ 0 1 2 3 4 5
+ 6  8  10  12  14  16  18  20 20 30 40 50 60 70
+ax (nm)(b) Nh = 2 × 106 cm-1
+<E> (meV)τ (ps)     
+T = 20 K
+<E> at T = 20 K
+Figure 89: Spin relaxation time τvs. the quantum wire width in xdirection axfor (a)Nh=4×105cm−1at different temperatures and (b)
+Nh=2×106cm−1atT=20 K.ay=10 nm. FromL¨ u etal. [705].
+contextofsemiconductorspintronics[576]. Di fferingfromearlyexperimentalstudieswhichmainlyfocuson electron
+spinrelaxationin p-typebulkIII-Vsemiconductors,experimentalinvestigat ionsinthepastdecademainlyfocuson n-
+typeandintrinsic bulkIII-Vsemiconductors[21]. Theoret ically,SongandKim systematically calculatedthe density
+and temperaturedependencesof electronspin relaxationti me in bulk n-typeand p-type GaAs, InAs, GaSb and InSb
+by includingthe D’yakonov-Perel’,Elliott-Yafet and Bir- Aronov-Pikusmechanisms[108]. However,their approach
+was based on the approximateformulaein the book “Optical Orientation” [3] where the momentumscattering time
+is calculated via the approximateformulaefor mobility [10 8]. A key mistake is that they used the formulae that can
+onlybeusedinthenondegenerateregime. Thismakestheirre sultsinthelow-temperatureand /orhigh-densityregime
+questionable. Moreover, the single-particle approach the y used limits the validity of their results. Other theoretic al
+investigationshavesimilarproblems[209,390,559,577, 5 78, 583]. Thereforeasystematic many-bodyinvestigation
+fromthefullymicroscopickineticspinBlochequationappr oachisneeded.
+In this subsection, we review the comprehensive study on the topic via the fully microscopic kinetic spin Bloch
+equationapproachby Jiang andWu [110] wheremanyimportant predictionsand resultsthat cannot be achievedvia
+thesingle-particleapproachwereobtained.78
+The system considered is the bulk III-V semiconductors, whe re the spin interactions have been introduced in
+Sec.2. Specifically,thespin-orbitcouplingconsistsofth eDresselhausterm,
+Ω(k)=2γD[kx(k2
+y−k2
+z),ky(k2
+z−k2
+x),kz(k2
+x−k2
+y)] (266)
+and the strain-induced term which is linear in k. The electron-hole exchange interaction is given by Eq. (37 ) which
+consists of both the short-range interaction [see Eq. (33)] and the long-range one [see Eq. (34)]. The Elliott-Yafet
+mechanism is included in the electron-impurity, electron- phonon, electron-electron and electron-hole scatterings in
+thekineticspinBlochequations[110].
+5.6.1. Comparisonwith experiments
+Jiang and Wu compared their calculation via the kinetic spin Bloch equation approach [110] with experimental
+results measured in GaAs in Refs. [4, 586, 900]. The results a re presented in Fig. 90(a)-(c) where the calculated
+spin lifetimes are plotted as solid curves and the experimen tal results as red dots. It is seen that the calculation
+agrees quite well with experimental data for both n- andp-type GaAs in a wide temperature and density regimes.
+The deviation in the low-temperature regime ( T<20 K) in Fig. 90(a) is due to the rising of electron localizati on.
+Good quantitative agreement with experimental data for int rinsic GaAs at room temperature was also achieved by
+78Review oftheexperimental studiesandthesingle-particle theories onelectron spinrelaxation inbulk n-typeandintrinsicIII-Vsemiconductors
+in metallic regime is presented in Sec. 4.2.1 and 4.2.2, resp ectively. Studies on electron spin relaxation in bulk p-type III-V semiconductors was
+reviewed in thebook “Optical Orientation” [117].
+120103104105106107
+ 10  100τ  ( ps )
+T  ( K )(a)
+ 0 50 100 150 200 250
+ 0  0.5  1  1.5  2τ  ( ps )
+Nex  ( 1018 cm-3 )(b)
+102103
+ 100  150  200  250  300τ  ( ps )
+T  ( K )(c)
+Figure 90: (a) n-GaAs. Spin relaxation times τfrom the experiment in Ref. [4] ( •) and from the calculation via the kinetic spin Bloch equatio n
+approach with only the D’yakonov-Perel’ mechanism (solid c urve) as well as that with only the Elliott-Yafet mechanism ( dashed curve). ne=
+1016cm−3,ni=neandNex=1014cm−3.γD=8.2eV·Å3. (b)p-GaAs. Spinrelaxation times τfromtheexperiment inRef.[586]( •)andfromthe
+calculation via the kinetic spin Bloch equation approach (s olid curve). nh=6×1016cm−3,ni=nhandT=100 K.γD=8.2 eV·Å3. (c)p-GaAs.
+Spin relaxation times τfrom the experiment in Ref. [900] ( •) and from the calculation via the kinetic spin Bloch equatio n approach (solid curve).
+nh=1.6×1016cm−3,ni=nhandNex=1014cm−3.γD=10 eV·Å3. From Jiang and Wu[110].
+Jiang and Wu in Ref. [589]. In those calculations, allthe material parameters are taken from the standard handboo k
+ofLandolt-B¨ ornstein [204]. The only free parameter is the Dresselhaus spin-orbi t coupling constant γDwhich has
+not been unambiguously determined by experiment or theory. Nevertheless the parameter γDfor GaAs used in the
+calculationisclosetothevaluefromrecent abinitiocalculationwithGWapproximation( γD=8.5eV·Å3)[139]and
+that from the recent fitting of the magnetotransportin chaot ic GaAs quantum dots ( γD=9 eV·Å3) [140]. The good
+agreementwith experimentaldata indicatesthat thecalcul ationhasachieved quantitativeaccuracy .
+5.6.2. Electron-spinrelaxationinn-typebulk III-Vsemic onductors
+100101
+101102τEY / τDP
+T  ( K )(a)
+ 
+ne = 1016 cm-3
+3x1016 cm-3
+1017 cm-3
+3x1017 cm-3
+100101102
+101102τEY / τDP
+T  ( K )(b)
+Figure 91: Ratio of the Elliott-Yafet spin relaxation time τEYto the D’yakonov-Perel’ one τDPfor (a)n-InSb (b) n-InAs and n-GaAs as function
+of temperature for various electron densities. In Fig.(b): ne=1016cm−3(curve with•), 2×1017cm−3(curve with /square), 1018cm−3(curve with△)
+for InAs,and ne=1016cm−3(curve with ▽)for GaAs. FromJiang and Wu[110].
+Comparison of di fferent spin relaxation mechanisms. Inn-type bulk III-V semiconductors at low photo-
+excitation density, the Bir-Aronov-Pikus mechanism is irr elevant as the hole density is low. The mechanisms left
+are then the Elliott-Yafetand D’yakonov-Perel’mechanism s. Previously,it was believedthat the Elliott-Yafet mech-
+anism is important in narrow bandgap semiconductors, such a s InSb and InAs. Jiang and Wu compared the relative
+efficiency of the two mechanisms at various conditions for InSb a nd InAs [110]. The results are shown in Fig. 91.
+In contrast to previous understanding, they found that the E lliott-Yafet mechanism is much lessefficient than the
+121D’yakonov-Perel’ one in both InSb and InAs. Although the low -temperature results for the high density cases are
+absentinFig.91, it wasfoundthatinsuchregimethe ratio τEY/τDPvariesslowlywithtemperature[110]. ForGaAs,
+theElliott-Yafetmechanismisstill less importantasindi catedbyFig.91(b).
+To givea qualitativepictureofthe relativeimportanceoft he Elliott-Yafetmechanismin otherIII-Vsemiconduc-
+tors, Jiang and Wu analyzed the problem by using the approxim ate formulae for the D’yakonov-Perel’ and Elliott-
+Yafetspin relaxationtime [Eqs.(53)and(62)]. Fromthosee quations,
+τEY
+τDP=2Q
+3A∝an}b∇acketle{tεk∝an}b∇acket∇i}htτ2
+pΘ. (267)
+HereΘ=8γ2
+Dm3
+eE2
+g(1−η/3)2/[(1−η/2)2η2].QandAare numerical factors around unity. The factor Θ, which is
+solely determined by the material parameters, is listed in T able 2 for various III-V semiconductors. The spin-orbit
+couplingparameter γDis fitted fromexperiments(exceptthat γD’s forInP andGaSb are fromthe k·pcalculationin
+Ref.[138]),whereasotherparametersarefrom Landolt-B¨ ornstein [204]. Onenoticesfromthetable thatthefactor Θ
+is much smaller for InAs and InSb than other III-V semiconduc tors. According to this, the Elliott-Yafet mechanism
+shouldbemuchlesse fficientthantheD’yakonov-Perel’oneinGaSbandInP.Actuall y,onenoticesthatΘ∝E2
+gwhich
+decreases rapidly with decreasing bandgap. In commonly use d III-V semiconductors, InSb has smallest bandgap.
+However,evenforInSbtheElliott-Yafetmechanismismuchl essimportantthantheD’yakonov-Perel’mechanismin
+themetallicregime. Therefore,inotherIII-Vsemiconduct ors,theElliott-Yafetmechanismisalso unimportant .79
+Table2: TheΘfactor for III-V semiconductors. From Jiang and Wu[110].
+GaAs GaSb InAs InSb InP
+Θ(eV) 2.7×10−20.12 2.0×10−39.2×10−40.27
+Very recently, Litvinenko et al. studied the magnetic field d ependence of spin lifetime in n-type InSb and InAs
+experimentally [901]. They found in n-InSb that the spin lifetime increases significantly with in creasing magnetic
+fieldin the Faradayconfiguration(magneticfieldparallel to spin polarization)[see Fig.92]. Asthe Elliott-Yafetspin
+relaxationhas little magnetic field dependence,this resul t implies that the dominantspin relaxationmechanismin n-
+InSbistheD’yakonov-Perel’mechanismatlowmagneticfield . AthighmagneticfieldtheElliot-Yafetspinrelaxation
+dominatesastheD’yakonov-Perel’mechanismisstronglysu ppressedbythelongitudinalmagneticfield. Theyfound
+similar results in n-InAs. These results confirmed the previous prediction by Ji ang and Wu that the Elliott-Yafet
+mechanismis lessefficientthanthe D’yakonov-Perel’onein n-typeInSbandInAs[110].
+The D’yakonov-Perel’spin relaxation. As both the Bir-Aronov-Pikusand Elliott-Yafet mechanisms are unim-
+portantin bulk n-typeIII-V semiconductorsin metallic regime, the only rel evant one is the D’yakonov-Perel’mech-
+anism. Althoughthe D’yakonov-Perel’mechanismhas been st udied for aboutforty years, the understandingon it in
+bulkIII-Vsemiconductorsisyetadequate. Forexample,int hepreviousliterature,theelectron-electronscattering has
+longbeenbelievedtobeirrelevantinbulkIII-Vsemiconduc tors. JiangandWushowedthattheelectron-electronscat-
+teringisimportantfor spinrelaxationin n-GaAsin the nondegenerate regimeexceptwhenthe electron-longitudinal-
+optical-phonondominates momentum scattering. The same co nclusion should also hold for other bulk n-type III-V
+semiconductors.
+The D’yakonov-Perel’ spin relaxation: density dependence .In Fig. 93(a), spin relaxation time as function of
+electrondensityisplottedfor n-GaAsat40K.Thedensityofimpurityistakenasthesameasth atofelectron, ni=ne.
+Remarkably,onenoticesthatthedensitydependenceis nonmonotonic withapeakaroundne=1016cm−3. Previously,
+thenonmonotonicdensitydependenceofspinlifetimewasob servedinlow-temperature( T/lessorsimilar5K)experiments,where
+the localized electrons play a crucial role and the electron system is in the insulating regime or around the metal-
+insulator transition point. Jiang and Wu found, for the first time, that the spin lifetime in metallicregime is also
+79ThisisalsotrueforintrinsicIII-Vsemiconductors andinm ostcasesfor p-typesemiconductors aswellasforsomeII-VIsemiconducto rs (such
+as CdTe, seeRef. [590]).
+122Figure 92: Spin lifetime in n-InSb as function of magnetic field at 100 K. Themagnetic field is parallel to the spin polarization direction (Faraday
+configuration). Theoretical dependences of the Elliott-Ya fet (dashed curve), D’yakonov-Perel’ and the total (solid c urve) spin lifetimes are also
+shown. Reproduced from Litvinenko et al. [901].
+nonmonotonic . Moreover, they pointed out that it is a universal behavior for allbulk III-V semiconductors at all
+temperature where the peak is located at TF∼TwithTFbeing the electron Fermi temperature. From the piont of
+view of spintronicdevice application,as devicesare more f avorableto operate in the metallic regime,this prediction
+givestheimportantinformationthatthelongestspinlifet imeinmetallicregimeisat TF∼T. Theunderlyingphysics
+forthenonmonotonicdensitydependenceinmetallic regime iselucidatedbelow.
+To understand the D’yakonov-Perel’spin relaxation qualit atively, let us first recall the widely used approximate
+formulae [117], τDP≃1/[∝an}b∇acketle{t|Ω(k)|2−Ω2
+z(k)∝an}b∇acket∇i}htτ∗
+p] where∝an}b∇acketle{t...∝an}b∇acket∇i}htdenotes the ensemble average.80The expression contains
+two key factors of the D’yakonov-Perel’ spin relaxation: (i ) the inhomogeneous broadening from the k-dependent
+transverse spin-orbit field ∼∝an}b∇acketle{t|Ω(k)|2−Ω2
+z(k)∝an}b∇acket∇i}ht; (ii) the momentum scattering time τ∗
+p[including the contributions
+of the electron-impurity, electron-phonon, electron-ele ctron and electron-hole (whenever holes exist) scattering s].
+The D’yakonov-Perel’ spin relaxation time increases with i ncreasing momentum scattering rate, but decreases with
+increasinginhomogeneousbroadening.
+To elucidate the underlying physics, Jiang and Wu plotted th e spin relaxation times calculated with only the
+electron-impurity,electron-electronand electron-phon onscatteringsin Fig. 93(a) respectively. It is seen that th e spin
+relaxation time with only one kind of scattering is smaller t han that with all scatterings, which is a consequence of
+the motionalnarrowingnatureof the D’yakonov-Perel’mech anismτs∝1/τp. One noticesthat the electron-electron
+scattering gives important contribution to spin relaxation in the nondegenerate (low d ensity) regime. Interestingly,
+both the electron-electronand electron-impurityscatter ings lead to nonmonotonicdensity dependenceof spin relax-
+ationtime. Onenoticesthattheelectron-phononscatterin gismuchweaker(thecorrespondingspinrelaxationtimeis
+muchshorteras τs∝1/τp)asthe temperatureislow.
+Letusfirstlookatthedensitydependenceoftheelectron-el ectronscatteringtime. Infact,thedensityandtemper-
+ature dependencesof the electron-electronscattering tim e have been widely investigated in spin-unrelated problems
+(see, e.g., Ref. [848]). From the previous works [843, 848], after some approximation, the asymptotic density and
+temperaturedependencesoftheelectron-electronscatter ingtimeτee
+pinthedegenerateandnondegenerateregimesare
+givenby,
+τee
+p∝n2
+3e/T2forT≪TF, (268)
+τee
+p∝T3
+2/neforT≫TF. (269)
+80∝an}b∇acketle{t...∝an}b∇acket∇i}ht=/integraltext
+dk(f↑
+k−f↓
+k)...
+/integraltext
+dk(f↑
+k−f↓
+k).
+123Fromthe aboveequations,one noticesthat the electron-ele ctronscattering time in the nondegenerateand degenerate
+regimeshasdifferentdensitydependence. Inthenondegenerate(lowdensit y)regime,theelectron-electronscattering
+time decreases with electron density [see Eq. (269)], where the inhomogeneous broadening ∼∝an}b∇acketle{t|Ω(k)|2−Ω2
+z(k)∝an}b∇acket∇i}ht∝
+∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}ht(εkis electron kinetic energy) varies slowly with density as th e electron distribution is close to the Boltzmann
+distribution in nondegenerate regime. The spin relaxation time thus increases with electron density. In degenerate
+(high density) regime, both the electron-electronscatter ing time [see Eq. (268)] and the inhomogeneousbroadening
+increase with electron density. Therefore, the spin relaxa tion timeτs≃1/[∝an}b∇acketle{t|Ω(k)|2−Ω2
+z(k)∝an}b∇acket∇i}htτee
+p] decreases with
+electrondensity.
+For spin relaxation associated with the electron-impurity scattering, the scenario is similar: In nondegenerate
+regime, the decrease of the electron-impurity scattering t ime with electron density (1 /τei
+p∝ni=ne) leads to the
+increase of the spin relaxation time with increasing electr on density. In the degenerate regime, the inhomogeneous
+broadeningincreaseswithincreasingelectrondensity,as ∝an}b∇acketle{t|Ω(k)|2−Ω2
+z(k)∝an}b∇acket∇i}ht∝k6
+F∝n2
+e. Ontheotherhand,theelectron-
+impurityscatteringtime variesslowly with electrondensi ty because 1 /τei
+p∼niV2
+kFkF∼nekF/k4
+F∝n0
+e. Consequently,
+thespinrelaxationtimedecreaseswithincreasingelectro ndensity. Forspinrelaxationrelatedto theelectron-phon on
+scattering, the situation, however, is di fferent: In nondegenrate regime both the inhomogeneousbroad ening and the
+electron-phonon scattering rate vary slowly with electron density as electron distribution is close to the Boltzmann
+distribution. In degenerate regime, the increase of the inh omogeneousbroadening is faster than the variation of the
+electron-phononscattering,whichhenceleadstothedecre aseofspinrelaxationtimewithincreasingelectrondensit y.
+In summary, the electron density depdence of spin relaxatio n time is nonmonotonicas the momentumscattering
+time and the inhomogeneous broadening have di fferentqualitative density dependences in the nondegenrate and
+degenerate regimes. The spin relaxation time increases (de creases) in the nondegenrate (degenerate) regime with
+increasing electron density. A peak hence locates in the cro ssover regime, where TFis around T. Such a scenario
+should hold for allthe III-V semiconductors at alltemperature and the nonmonotonic density dependence is thu s a
+universal behavior.81Furthermore,inFig.93(b),JiangandWushowedthatthenonm onotonicdensitydependenceof
+spinrelaxationtime also existsevenwhenthe spin-orbitco uplingis dominatedbythe linear- kterm.82Subsequently,
+similar behaviorwas foundin two-dimensionalsystem [111, 299, 613], where the underlyingphysicsis similar. The
+predictedpeakwaslaterobservedbyKrausset al. [563](see ,also [564]).
+102103
+101510161017τ  ( ps )
+ne  ( cm-3 )(a)102103
+101510161017τ  ( ps )
+ne  ( cm-3 )(b)
+Figure 93: n-GaAs at T=40 K. (a) spin relaxation time τas function of electron density ne(ni=ne) from full calculation (curve with •), from
+calculation with only the electron-electron scattering (c urve with /square), with only the electron-impurity scattering (curve with △), and with only the
+electron-phonon scattering (curve with ▽); (b) spin relaxation time τas function of electron density ne(ni=ne) for the case with strain-induced
+spin-orbitcoupling(curvewith •: withboththelinear-andthecubic- kspin-orbitcoupling; curvewith △: withonlythelinear- kspin-orbitcoupling)
+and the case without strain (curve with /square). From Jiang and Wu[110].
+81The nonmonotonic density dependence should also exist in bu lk II-VI semiconductors with zinc-blende structures which has similar band
+structure with III-VI semiconductors.
+82Fromthispoint, thenonmonotonic density dependence shoul d alsoexistin bulkwurtzite semiconductors with bulk inver sion asymmetry,such
+as GaN,AlN and ZnO.
+1245.6.3. Electron-spinrelaxationinintrinsic bulkIII-Vse miconductors
+In intrinsic semiconductors, the carriers are generated by photo-excitation where the electron density is equal to
+theholedensity ne=nh=Nex(Nexdenotestheexcitationdensity). Asthe impuritydensityis verylow (onecan take
+ni=0), the carrier-carrierscattering is dominant unless at hi gh temperature where the electron-longitudinal-optical-
+phonon scattering becomes more important. Intrinsic bulk s emiconductors thus o ffer a good platform to study the
+many-bodyeffect toelectronspinrelaxation.
+Comparison of di fferent mechanisms. As both the Bir-Aronov-Pikusand D’yakonov-Perel’mechani sms con-
+tribute to electronspin relaxation,the relativee fficiencyof the Bir-Aronov-PikusandD’yakonov-Perel’mecha nisms
+should be compared. In Fig. 94(a) the Bir-Aronov-Pikus and D ’yakonov-Perel’spin relaxation times as function of
+temperature are plotted. It is noted that the Bir-Aronov-Pi kus spin relaxation time is over one order of magnitude
+larger than the D’yakonov-Perel’ one, which indicates that the Bir-Aronov-Pikus mechanism is much less e fficient
+than the D’yakonov-Perel’ one in intrinsic bulk III-V semic onductors. Systematic calculation for various tempera-
+tures and excitation densities confirms that the Bir-Aronov -Pikus mechanism is irrelevant in intrinsic bulk GaAs in
+the metallic regime. Such conclusion also holds for GaSb. Re cent experiments arrived at the same conclusion for
+intrinsicInSb[553]. Therefore,theBir-Aronov-Pikusmec hanismisunimportantinintrinsicGaAs,GaSb andInSb.
+101102103104
+ 0  50  100  150  200  250  300τ  ( ps )
+T  ( K )(a) 
+BAP
+DP
+104105
+ 0  50  100  150  200  250  300τ  ( ps )
+T  ( K )(b)
+101102103104
+ 0  50  100  150  200  250  300τ  ( ps )
+T  ( K )(c) 
+full
+without HF
+without EH
+Figure 94: Spin relaxation time τfor intrinsic GaAs with excitation density Nex=1017cm−3. The initial spin polarization is P=50%. (a) spin
+relaxation time τdue to the Bir-Aronov-Pikus (BAP) and D’yakonov-Perel’ (DP ) mechanisms as function of temperature. (b) The Bir-Aronov -
+Pikusspinrelaxation timecalculated fromEq.(270)(dotte d curvewith△),fromthekineticspinBlochequation approach withbothlo ng-rangeand
+short-range exchange scatterings (solid curve with •) as well as from the kinetic spin Bloch equation approach wit h only the short-range exchange
+scattering (solid curve with /square). (c) The D’yakonov-Perel’ spin relaxation time from full c alculation (solid curve with •), from the calculation
+without the Coulomb Hartree-Fock term (dashed curve with /square), and from the calculation without the electron-hole Coulo mb scattering (dotted
+curve with△). FromJiang and Wu[110].
+TheBir-Aronov-Pikusspinrelaxation: short-range vs.long-rangeinteractionandthePauliblocking. Asto
+the Bir-Aronov-Pikusspin relaxation, it should be mention ed that the long-range part of the electron-hole exchange
+interaction[seeEq.(34)]hasalwaysbeenignoredinthelit erature[3,21,108,436]. JiangandWuexaminedtherela-
+tivecontributionofthelong-rangeandshort-rangeintera ctionstotheBir-Aronov-Pikusspinrelaxation. InFig.94( b)
+theBir-Aronov-Pikusspinrelaxationtime calculatedwith boththe long-rangeandshort-rangeexchangeinteractions
+aswellasthatwithoutthelong-rangeexchangeinteraction areplotted. Itisseenthatthespinrelaxationtimeincreas es
+by about three times when the long-range exchange interacti on is removed. This indicates that the long-range inter-
+actionismoreimportantthantheshort-rangeoneinGaAsandhencecannotb eneglected. Moreover,intheprevious
+literature,theBir-Aronov-Pikusspin relaxationtime was calculatedviathe FermiGoldenrule[3,390],
+1
+τBAP(k)=4π/summationdisplay
+q,k′
+m,m′δ(εk+εh
+k′m′−εk−q−εh
+k′+qm)|J(+)k′m′
+k′+qm|2fh
+k′m′(1−fh
+k′+qm). (270)
+As pointedout by Zhouand Wu in the two-dimensionalsystem [1 09] (see Sec. 5.4.6),such approach,which ignores
+thePauli-blockinge ffect of electrondistribution,failsat low temperature(ind egenerateregime). Thespin relaxation
+time calculated from Eq. (270) with only the short-range exc hange interaction is plotted in Fig. 94(b). One notices
+125that the results from Eq. (270) indeed deviate from the exact results via the kinetic spin Bloch equation approach at
+lowtemperature.
+The D’yakonov-Perel’ spin relaxation: temperature depend ence and the effects of the Coulomb Hartree-
+Fock term and the electron-hole scattering. In Fig. 94(c) the D’yakonov-Perel’ spin relaxation time as f unction
+of temperature is plotted. The initial spin polarization is P=50%, i.e., ideal circularly polarized light excitation.
+To elucidate the role of the Coulomb Hartree-Fock term, the s pin relaxation time calculated without the Coulomb
+Hartree-Fock term is also plotted. It is seen that the Coulom b Hartree-Fock term largely a ffects the spin relaxation
+at low temperature, whereas at high temperature it is ine ffective. The underlying physics is as follows: The spin
+relaxationtime undertheHartree-Focke ffectivemagneticfield isestimatedas[similartoEq.(64)]
+τs(P)=τs(P=0)[1+(gµBBHFτ∗
+p)2] (271)
+whereBHFis the averaged effective magnetic field and τ∗
+pstands for the momentum scattering including the carrier-
+carrier scattering. At low temperature, the carrier-carri er [see Eq. (268)] and electron-phonon scatterings are sup-
+pressed. Therefore, τ∗
+pislargeandtheHartree-Fockmagneticfieldhasstronge ffectonspinrelaxationtime. However,
+athightemperaturethemomentumscatteringisstrongand τ∗
+pissmall. Consequently,theHartree-Fockmagneticfield
+haslittle effectonspinrelaxation.
+It is noted that without the Coulomb Hartree-Fock term the sp in relaxation time has nonmonotonic temperature
+dependence. Usually, the spin relaxationtime without the C oulombHartree-Fockterm is close to the spin relaxation
+time at small initial spin polarization. To confirm that, Jia ng and Wu plotted the spin relaxation time at the same
+condition but for P=2% in Fig. 95. It is seen that the spin relaxation time is indee dnonmonotonic in temperature
+dependence.83In contrast, the spin relaxation time decreases with increa sing temperature in n-type III-V semicon-
+ductorsastheelectron-impurityscatteringdominatesatl owtemperature. Thenonmonotonictemperaturedependence
+of the spin relaxation time originates from the nonmonotoni c temperature dependence of the electron-electron and
+electron-holescatteringtimesasnotedfromEqs.(268)and (269).84Thepeakisthenlocatedinthecrossoverregime.
+Systematic calculation indicates that the peak temperatur e is around TF/3 and lies in the range of ( TF/4,TF/2) for
+variouscarrierdensityinbothGaAsandInAs[110].
+Thespinrelaxationtimecalculatedwithouttheelectron-h olescatteringisalsoplottedinFig.96(c)toindicatethe
+contribution of the electron-hole scattering. It is seen th at at high temperature ( T>60 K), the spin relaxation time
+becomes smaller without the electron-hole scattering. How ever, at low temperature ( T<60 K), the spin relaxation
+time becomes larger. The decrease of the spin relaxation tim e at high temperature indicates the importance of the
+electron-holescattering according to the motional narrow ingτs∝1/τp. However, at low temperature, the Coulomb
+Hartree-Focktermcomplicatesthebehavior. AccordingtoE q.(271),iftheCoulombHartree-Focktermplayssignifi-
+cantrole[i.e.,( gµBBHFτ∗
+p)2>1],thenτs∼[(|Ω|2−Ω2
+z)τ∗
+p]−1(gµBBHFτ∗
+p)2∼τ∗
+p. Thereforeremovingtheelectron-hole
+scattering leads to longer spin relaxation time. These resu lts demonstrate that the electron-hole scattering plays an
+importantroleinbothlowandhightemperatureregimes.
+The D’yakonov-Perel’ spin relaxation: initial spin polari zation dependence. The effect of the Coulomb
+Hartree-Focktermisalsoreflectedintheinitial polarizat iondependenceofthespinrelaxationtime[41, 42,44,326],
+which is plotted in Fig. 95. It is seen that the spin relaxatio n time increases by about one order of magnitude when
+the initial spin polarization increases from 2% to 50%. With out the Coulomb Hartree-Fock term the increment of
+the spin relaxation time is negligible (not shown). It was al so found that, in contrast to the two-dimensional case
+[41, 42, 44, 326], the initial spin polarization dependence inn-type III-V semiconductors is very weak. This is be-
+cause the momentum scattering in n-type III-V semiconductors is strong even at low temperatur e as the impurity
+densityishigh( ni=ne)[110].
+The D’yakonov-Perel’ spin relaxation: density dependence .In Fig. 96(a), the density dependence of the
+spin relaxation time is plotted. It is seen again that the Bir -Aronov-Pikusmechanism is much less e fficient then the
+D’yakonov-Perel’ mechanism. Remarkably, the density depe ndence is nonmonotonic and a peak exhibits. The un-
+derlying physics is similar with that for the n-type case except that the electron-impurityscattering is substituted by
+83The nonmonotonic temperature dependence of spin relaxatio n time has been observed in bulk intrinsic GaAs recently [902 ], which confirms
+theprediction by Jiang and Wu[110].
+84Theelectron-hole scattering time has similar temperature dependence as that of the electron-electron scattering.
+126 0 50 100 150 200
+ 0  50  100  150  200  250  30060 50 40 30 20 10 0τ  ( ps )
+T  ( K )P  ( % )
+Figure 95: Intrinsic GaAs with Nex=2×1017cm−3. Spin relaxation time τas function of temperature for P=2 % (curve with /square) and the spin
+relaxation timeasfunction ofinitial spinpolarization PforT=20K(curvewith•)(notethatthescaleof Pisonthetopoftheframe). FromJiang
+and Wu[110].
+the electron-holescattering. As the electron-holescatte ringhas similar densitydependencewith that of the electro n-
+electronscattering,thedensitydependenceofthespinrel axationtimeisthennonmonotonic. Thepeakisalsolocated
+in the crossover regime, where TFis around T. Such behavior is also universal forallbulk intrinsic III-V semicon-
+ductorsat alltemperature.85
+To elucidate the role of the electron-hole scattering and th e Coulomb Hartree-Fock term in spin relaxation, the
+spin relaxation times calculated without these terms are pl otted together with the spin relaxation time from the full
+calculation in Fig. 96(b). The importance of the electron-h ole scattering is obvious in a wide density range. The
+CoulombHartree-Focktermis showntobe importantonlyinth ehighdensityregime.86
+102103104105106107108
+1014101510161017τ  ( ps )
+Nex  ( cm-3 )(a)
+T = 40 K
+ 
+BAP
+DP
+102103
+1014101510161017τ  ( ps )
+Nex  ( cm-3 )(b)
+T = 40 K 
+full
+without HF
+without EH
+Figure 96: Intrinsic GaAs with P=50% and T=40 K. (a) The Bir-Aronov-Pikus (BAP) and D’yakonov-Perel’ ( DP) spin relaxation times
+τas function of photo-excitation density Nex. (b) The D’yakonov-Perel’ spin relaxation time from the ful l calculation (curve with •), from the
+calculation without the Coulomb Hartree-Fock term (curve w ith/square), and from the calculation without the electron-hole Coulo mb scattering (curve
+with△). FromJiang and Wu[110].
+5.6.4. Electron-spinrelaxationin p-typebulkIII-Vsemic onductors
+Comparison ofthe D’yakonov-Perel’and Bir-Aronov-Pikusm echanismsin GaAs . The mainsourcesof spin
+relaxation have been recognized as the Bir-Aronov-Pikus an d D’yakonov-Perel’ mechanisms [3].87An important
+85Suchdensity dependence shouldalsoholdforstrainedIII-V semiconductors, wurtzitesemiconductors withbulkinvers ion asymmetryandbulk
+II-VI semiconductors.
+86TheHartree-Fock e ffectivemagneticfieldcanbeestimatedas BHF=˜VqneP/(gµB)where˜VqdescribestheaverageCoulombinteraction. Hence
+theHartree-Fock effective magnetic field is strong at high electron density.
+87TheElliott-Yafet mechanism was checked to beunimportant i n metallic regime for both p-GaAs and p-GaSb by Jiang and Wu[110].
+127issueistherelativee fficiencyofthetwomechanismsundervariousconditions. This wasstudiedcomprehensivelyby
+JiangandWu inRef.[110]. Belowwe reviewtheirresults.
+Low photo-excitationcase. Jiang and Wu first discussed the low photo-excitationcase. I n this case, the electron
+densityislowandtheelectronsystemisnondegenerate. The ratiooftheBir-Aronov-Pikusspinrelaxationtimetothe
+D’yakonov-Perel’one is plotted in Fig. 97(a) for variousho le densitiesas functionof temperature. It is seen that the
+D’yakonov-Perel’mechanism dominates at high temperature , whereas the Bir-Aronov-Pikus mechanism dominates
+atlow temperature. Thisisconsistentwiththecommonbelie fintheliterature[3,21, 22,106,108,436].
+The temperaturedependencesof the Bir-Aronov-Pikusand D’ yakonov-Perel’spinrelaxationtimesare plottedin
+Fig. 97(b) for a typical case with nh=3×1018cm−3. It is seen that both the D’yakonov-Perel’spin relaxation t ime
+and the Bir-Aronov-Pikusone decrease with temperature. As electron system is nondegenerate,the inhomogeneous
+broadeningvaries as ∼∝an}b∇acketle{t|Ω|2−Ω2
+z∝an}b∇acket∇i}ht∝T3, which leads to the rapid decrease of the D’yakonov-Perel’s pin relaxation
+time. The Bir-Aronov-Pikus spin relaxation time decreases with temperature partly because of the Pauli blocking
+of holes. To elucidate this, the Bir-Aronov-Pikusspin rela xation time without the Pauli blocking of holes is plotted
+as dotted curve in Fig. 97(b). The results indicate that the P auli blocking of holes e ffectively suppresses the Bir-
+Aronov-Pikus spin relaxation at low temperature and makes t he temperature dependence of the Bir-Aronov-Pikus
+spinrelaxationtimestronger.
+10-1100101102103104
+100101102τBAP / τDP
+T  ( K )(a)
+102103104105106
+101102100τ  ( ps )
+τBAP / τDP
+T  ( K )(b) 
+BAP
+DP
+total
+Figure 97: p-GaAs. Ratio of the Bir-Aronov-Pikus spin relaxation time t o the D’yakonov-Perel’ one τBAP/τDPas function of temperature for
+various hole densities with Nex=1014cm−3andni=nh. (a):nh=3×1015cm−3(curve with•), 3×1016cm−3(curve with /square), 3×1017cm−3
+(curve with△), and 3×1018cm−3(curve with ▽). (b): The spin relaxation times due to the Bir-Aronov-Piku s (BAP) and D’yakonov-Perel’ (DP)
+mechanisms, the total spin relaxation time and the ratio τBAP/τDP(curve with△)vs.the temperature for nh=3×1018cm−3. The dotted curve
+represents the Bir-Aronov-Pikus spin relaxation time calc ulated without the Pauli blocking of holes. Note the scale of τBAP/τDPis on the right
+hand side of the frame. FromJiang and Wu[110].
+High photo-excitationcase. The high photo-excitationcase was also discussed by Jiang a nd Wu. The excitation
+densitywastakenas Nex=0.1nh. TheratiooftheBir-Aronov-Pikusspinrelaxationtimetot heD’yakonov-Perel’one
+as functionof temperature for varioushole densities is plo tted in Fig. 98(a). Interestingly, the ratio is nonmonotoni c
+and has a minimum roughly around the Fermi temperature of ele ctrons,T∼TF. In contrast to the low photo-
+excitation case, the Bir-Aronov-Pikus mechanism no longer dominates the low temperature regime. To understand
+such behavior, the spin relaxation times due to the two mecha nisms and their ratio are plotted in Fig. 98(b) for a
+typical case nh=3×1018cm−3. Before looking at the figure, one should note that the only di fference for the two
+cases it that the electron density is much larger in the high e xcitation density. In the figure, one notices that, quite
+differently,the D’yakonov-Perel’spin relaxationtime satura tesat low temperature,whichis the reason whythe ratio
+of the two spin relaxation times increases at decreasing tem perature. The underlying physics for the saturation of
+the D’yakonov-Perel’ spin relaxation time at low temperatu re is as follows: As the electron density is high, at low
+temperature the electron system enters into the degenerate regime, where the inhomogeneous broadening and the
+momentumscattering(dominatedbytheelectron-impuritys cattering)variesslowlywithtemperature. Thisthusleads
+tothesaturationoftheD’yakonov-Perel’spinrelaxationt ime at lowtemperature.
+RoleofscreeningonD’yakonov-Perel’spinrelaxation. OnemayfindfromFig.98(b)thattheD’yakonov-Perel’
+spinrelaxationtimehas nonmonotonic temperaturedependence. Unlikethenonmonotonictemperat uredependencein
+intrinsicsemiconductors,herethebehaviorisnotcausedb ythecarrier-carrierscattering. Theunderlyingphysicsi sa
+12810-1100101102103104
+100101102τBAP / τDP
+T  ( K )(a)
+102103
+102 2 4 6 8 10τ  ( ps )
+τBAP / τDP
+T  ( K )(b) 
+BAP
+DP
+total
+Figure 98: p-GaAs. Ratio of the Bir-Aronov-Pikus spin relaxation time t o the D’yakonov-Perel’ one τBAP/τDPas function of temperature for
+various hole densities with Nex=0.1nhandni=nh. (a):nh=3×1015cm−3(curve with•), 3×1016cm−3(curve with /square), 3×1017cm−3(curve
+with△), and 3×1018cm−3(curve with ▽). The hole Fermi temperatures for these densities are Th
+F=1.7, 7.7, 36, and 167 K, respectively. The
+electron Fermi temperatures are TF=2.8, 13, 61, and 283 K, respectively. (b): The spin relaxation t imes due to the Bir-Aronov-Pikus (BAP) and
+D’yakonov-Perel’ (DP)mechanisms,thetotalspinrelaxati on timeandtheratio τBAP/τDP(curvewith△)vs.thetemperaturefor nh=3×1018cm−3.
+The dotted (dashed) curve represents the Bir-Aronov-Pikus spin relaxation time calculated without the Pauli blocking of electrons (holes). Note
+thescale of τBAP/τDPis on theright hand side of theframe. From Jiang and Wu [110].
+little bitmorecomplex: itisrelatedtothetemperaturedep endenceofscreening. To elucidatetheunderlyingphysics,
+Jiang and Wu plotted the D’yakonov-Perel’spin relaxation t imes calculated with the Thomas-Fermiscreening [346]
+[which applies in the degenerate (low temperature) regime] , the Debye-Huckle screening [346] [which applies in
+the nondegenerate(hightemperature)regime]and the scree ning with the random-phaseapproximation[346] (which
+applies in the whole temperature regime) in Fig. 99(a). It is noted that with the Thomas-Fermi screening (which
+is temperature-independent)the peak disappears, whereas with the Debye-Huckle screening the peak remains. This
+indicates that the increase of screening with decreasing te mperature is crucial for the appearance of the peak. The
+scenario is as follows: With decreasingtemperature, the el ectron system graduallyenters into the degenerateregime
+andthetemperaturedependenceoftheinhomogeneousbroade ning∼∝an}b∇acketle{t|Ω|2−Ω2
+z∝an}b∇acket∇i}htbecomesmild. However,asthehole
+Fermienergyissmallerthantheelectronone(duetoitslarg eeffectivemass),thereisatemperatureintervalwherethe
+holesystemisstillinthenondegenerateregime. Thescreen ing,whichmainlycomesfromholes(again,duetoitslarge
+effective mass), still increases with decreasing temperature significantly, κ2∼1/T. Therefore, the electron-impurity
+scattering (the dominant one at such temperatures) time, τei
+p∝1/[∝an}b∇acketle{tV2
+q∝an}b∇acket∇i}ht∝∝an}b∇acketle{t(q2+κ2)−2∝an}b∇acket∇i}ht], increases with decreasing
+temperature. Thus the spin relaxation time, τs∼1/(∝an}b∇acketle{t|Ω|2−Ω2
+z∝an}b∇acket∇i}htτei
+p), decreases with decreasing temperature at such
+temperature interval. When the temperature is further lowe red down, both the screening and the inhomogeneous
+broadening vary little with temperature and the spin relaxa tion time saturates. These behaviors, together with the
+decrease of the spin relaxation time with increasing temper ature at high temperature (mainly due to the increase of
+inhomogeneousbroadeningin nondegenerateregime), lead t o thenonmonotonic temperature dependence. The peak
+isroughlyaroundtheelectronFermi temperature, T∼TF.
+It is noted from Fig. 98(b) that the temperature dependence o f the total spin relaxation time is still monotonic:
+it decreases with increasing temperature. This is due to the contribution of the Bir-Aronov-Pikus mechanism. It is
+expectedthat in other materials where the Bir-Aronov-Piku smechanism is less importantthan that in GaAs, such as
+GaSb,thetemperaturedependenceofthetotalspinrelaxati ontimeis nonmonotonic.
+The question arises that whether there is a nonmonotonic tem perature dependence of the spin relaxation time
+inn-type semiconductors due to screening. A simple estimation may help to illustrate the problem: the electron-
+impurityscattering88rate variesas 1 /τei
+p∝∝an}b∇acketle{tkV2
+q∝an}b∇acket∇i}ht∝∝an}b∇acketle{tk(q2+κ2)−2∝an}b∇acket∇i}ht∼∝an}b∇acketle{tk∝an}b∇acket∇i}ht(κ2)−2∼T2.5,89whereasthe inhomogeneous
+broadeningvariesas ∼∝an}b∇acketle{t|Ω|2−Ω2
+z∝an}b∇acket∇i}ht∼∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}ht∼T3. Therefore,the spin relaxationtime τs∼1/[∝an}b∇acketle{t|Ω|2−Ω2
+z∝an}b∇acket∇i}htτei
+p]∼T−0.5,
+stilldecreaseswithincreasingtemperature. Suchestimat ionappliesfornondegenerateregime. Forotherregimes,th e
+88Theelectron-impurity scattering is the cause of the monoto nic temperature dependence of the spin lifetime in n-type semiconductors.
+89The factor∝an}b∇acketle{tk∝an}b∇acket∇i}htcomes from the density of states. This factor varies little i n thep-type case in the above discussion at the relevant temperatu re,
+as electron system is in the degenerate regime.
+129inhomogeneousbroadening still varies with temperature fa ster than the scattering rate does. Therefore the the spin
+relaxationtime alwaysdecreaseswith increasingtemperat ureinn-typeIII-Vsemiconductorsin metallicregime.
+However, the situation may change when the strain-induced s pin-orbit coupling dominates. In such case, the
+inhomogeneous broadening ∼∝an}b∇acketle{t|Ω|2−Ω2
+z∝an}b∇acket∇i}ht∼∝an}b∇acketle{tεk∝an}b∇acket∇i}ht∼Tvaries with temperature slower than the electron-impurity
+scatteringdoes. One readilyobtainsthat τs∼T1.5in nondegenerateregime. As shownin Fig. 99(b),the tempera ture
+dependence is indeed nonmonotonic in strained n-GaAs with ne=2×1015cm−3. However, the screening e ffect is
+onlyimportantforlowelectrondensity ne/lessorsimilar1016cm−3. At highelectrondensity,thescreeningplaysamarginalro le
+in the electron-impurity scattering even at low temperatur e. This is because 1 /τei
+p∝∝an}b∇acketle{tk(q2+κ2)−2∝an}b∇acket∇i}ht. When electron
+densityishigh,the q2factorbecomeslargerthan κ2onaverage.
+102
+102τDP  ( ps )
+T  ( K )(a)
+ 
+RPA
+TF
+DH 600 800 1000 1200 1400 1600
+ 0 1 2 3 4 5 6 7 8 9 10τDP  ( ps )
+T  ( K )(b)
+ 
+RPA
+TF
+DH
+Figure 99: (a) p-GaAs with hole density nh=3×1018cm−3,ni=nhandNex=0.1nh. Temperature dependence of the D’yakonov-Perel’
+spin relaxation times calculated with the Thomas-Fermi (TF ) (curve with /square) and Debye-Huckle (DH) (curve with △) screenings as well as the
+screening with the random-phase-approximation (RPA) (cur ve with•). (b)n-GaAs with electron density ne=2×1015cm−3,ni=neand
+Nex=4×1013cm−3. Thespin relaxation times calculated with theThomas-Ferm iand Debye-Huckle screenings as well as thescreening with t he
+random-phase-approximation vs.temperature. FromJiang and Wu [110].
+Photo-excitation density dependence. After analyzing the relative e fficiency of the D’yakonov-Perel’and Bir-
+Aronov-Pikusmechanismsin the two limiting cases of low and high photo-excitation,one would be eager to see the
+crossover between the two limits. In Fig. 100 the photo-exci tationdensity dependence of the Bir-Aronov-Pikusand
+D’yakonov-Perel’ spin relaxation times as well as their rat io are plotted for two hole densities at 50 K. In the low
+excitation limit, the D’yakonov-Perel’(Bir-Aronov-Piku s)mechanism is more important for the case in Fig. 100 (a)
+[(b)]. For both cases, the Bir-Aronov-Pikus and D’yakonov- Perel’ spin relaxation times first decrease slowly then
+rapidly with increasing photo-excitation density. Moreov er, the D’yakonov-Perel’ spin relaxation time decreases
+faster than the Bir-Aronov-Pikus one. Hence the importance of the Bir-Aronov-Pikus mechanism decreases with
+photo-excitationdensity.
+To understand such behavior, one notices that only the elect ron density varies significantly with the photo-
+excitation density. The photo-excitation density depende nce of the Bir-Aronov-Pikus spin relaxation time mainly
+comesfromthe fact that 1 /τBAP∝∝an}b∇acketle{tvk∝an}b∇acket∇i}ht∝∝an}b∇acketle{tε1/2
+k∝an}b∇acket∇i}ht[see Eq. (69)], whereasthat of the D’yakonov-Perel’spin re laxation
+timeoriginatesfromtheinhomogeneousbroadening1 /τDP∝∝an}b∇acketle{t|Ω|2−Ω2
+z∝an}b∇acket∇i}ht∝∝an}b∇acketle{tε3
+k∝an}b∇acket∇i}ht. Atlowphoto-excitationdensity,the
+electron system is nondegenerate, thus both the Bir-Aronov -Pikus and D’yakonov-Perel’ spin relaxation times vary
+slowly. At higher photo-excitation density, both the Bir-A ronov-Pikus and D’yakonov-Perel’ spin relaxation times
+decreaserapidlywithphoto-excitationdensitybuttheD’y akonov-Perel’spinrelaxationtime decreasesfaster.
+Holedensitydependence. TheholedensitydependenceofboththeD’yakonov-Perel’an dBir-Aronov-Pikusspin
+relaxationtimestogetherwith their ratioare plottedin Fi g. 101. It is notedthat the Bir-Aronov-Pikusspin relaxatio n
+timedecreasesas1 /nhat low holedensity,whichisconsistent with τBAP∝1/nh[see Eq.(67)]. At highholedensity,
+τBAPdecreasesslowerthan1 /nhduetothePauliblockingofholes. Thedensitydependenceof the D’yakonov-Perel’
+spin relaxation time is not so obvious: the spin relaxation t ime first increases, then decreases and again increases
+with the hole density. As the electron distribution (hence t he inhomogeneousbroadening) does not change with the
+hole density,the variationof the D’yakonov-Perel’spin re laxationtime comessolely fromthe momentumscattering
+(dominatedbytheelectron-impurityscattering). Jiangan dWufoundthatthenonmonotonicholedensitydependence
+130103104105
+1014101510161017100101τ  ( ps )
+τBAP / τDP
+Nex  ( cm-3 )(a)T = 50 K
+ 
+DP
+BAP
+total
+102103104
+101410151016101710-1100101τ  ( ps )
+τBAP / τDP
+Nex  ( cm-3 )(b) T = 50 K
+ 
+DP
+BAP
+total
+Figure 100: p-GaAs. Spin relaxation times τdue to the Bir-Aronov-Pikus (BAP) and D’yakonov-Perel’ (DP ) mechanisms together with the total
+spin relaxation time vs.the photo-excitation density Nex. The ratio of the two is plotted as dashed curve (note that the scale is on the right hand
+sideof the frame). (a): ni=nh=3×1017cm−3. (b):ni=nh=3×1018cm−3.T=50 K.From Jiang and Wu[110].
+oftheD’yakonov-Perel’spinrelaxationtimeisagainrelat edtothe screening.
+103104105106
+101610171018101910-1100101102τ  ( ps )
+τBAP / τDP
+nh  ( cm-3 )(a) 
+BAP
+DP
+total
+103104
+101610171018101910-1100101102103104τ  ( ps )
+κ2 / <q2>
+nh  ( cm-3 )(b) 
+RPA
+DH
+TF
+Figure 101: p-GaAs. (a): spin relaxation times τdue to the Bir-Aronov-Pikus (BAP) and D’yakonov-Perel’ (DP ) mechanisms together with the
+total spin relaxation time against hole density nh.Nex=1014cm−3,ni=nh, andT=60 K. The dotted curve denotes a fitting of the curve with
+•using 1/nhscale. The curve with △denotes the ratio τBAP/τDP(note that the scale is on the right hand side of the frame). (b ): spin relaxation
+times due to the D’yakonov-Perel’ mechanism with the Debye- Huckle (DH) (curve with /square), Thomas-Fermi (TF) (curve with △), and the random-
+phase-approximation (RPA) (curve with •) screenings. The ratio κ2/∝an}b∇acketle{tq2∝an}b∇acket∇i}htis plotted as curve with ▽(note that the scale is on the right hand side of
+theframe). FromJiang and Wu [110].
+Toelucidatetheunderlyingphysics,theD’yakonov-Perel’ spinrelaxationtimecalculatedwiththerandom-phase-
+approximationscreeningtogetherwiththosecalculatedwi ththeThomas-Fermiscreening[346]andtheDebye-Huckle
+screening[346] are plotted in Fig. 101(b). From the figure it is seen that the first increase and the decrease is related
+to the Debye-Huckle screening, whereas the second increase is connected with the Thomas-Fermi screening. The
+underlying physics is understood as follows: In the low hole density regime, the screening (mainly from holes)
+is small and the Coulomb potential, Vq∝1/(κ2+q2), changes slowly with the screening constant κas well as
+hole density. Hence the electron-impurity scattering incr eases with nhas 1/τei
+p∝ni=nh. For higher hole density
+(nh>1017cm−3), the screening constant κbecomes larger than the transfered momentum q. Hence the electron-
+impurityscatteringdecreaseswith nhbecause1/τei
+p∝ni∝an}b∇acketle{t(κ2+q2)−2∝an}b∇acket∇i}ht∼nhκ−4∝n−1
+hasκ2∝nhforthe Debye-Huckle
+screening. As the hole density increases, the hole system gr adually enters into the degenerate regime, where the
+Thomas-Fermiscreeningappliesand κ2∝n1/3
+h. Hence,theelectron-impurityscatteringincreaseswitht heholedensity
+as 1/τei
+p∝nhκ−4∝n1/3
+h. Consequently,the D’yakonov-Perel’spin relaxation time first increases, then decreases and
+againincreaseswithincreasingholedensity.
+1316. Spin diffusion andtransportin semiconductors
+6.1. Introduction
+Figure 102: Schematic of Datta-Das transistor. FromDatta a nd Das [54].
+The implementationof spintronicdevicesalso relies on the understandingof spin resolvedtransport phenomena.
+The first prototype of the spin field-e ffect-transistor, proposed by Datta and Das [54] and named aft er them, utilizes
+the Rashba spin-orbit coupling for operation. As shown in Fi g. 102, the so called Datta-Das transistor consists of a
+three-layer structure with magnetic source and collector c onnected by a semiconductor channel, and a voltage gate.
+The spin polarizedcarriersare injected into the conductin gchannel fromthe source and driventowardsanddetected
+by the collector. The carriers can flow freely through the dra in (“on” state) if their spins are parallel to that of drain
+or are blocked (“off” state) if anti-parallel. Similar to the traditional field- effect-transistor, the on /offstates of spin
+field-effect-transistorareswitchedbythegatevoltagewhichcontr olsthespindirectionofthepassingcarriersvia the
+Rashba spin-orbitcouplingacting as an e ffectivemagnetic field, with its strengthcontrolledby the ga te voltage. It is
+believedthatspinfield-e ffect-transistorhastheadvantagesoflowenergyconsumptio nandfastswitchingspeedsince
+it does not involve creating or eliminating the electrical c onducting channel during the switching like the traditiona l
+fieldeffecttransistor[576, 903].90
+As onecansee fromthe operationof thespin field-e ffect-transistorprototype,the topicsof spintransportinc lude
+thegenerationofnonequilibriumspinpolarizedcarriersi nconductors,thetransferofthesecarriersfromoneplacet o
+another insider the conductor and /or across the interfaces into other conductors, as well as th e detection of the spin
+signalofthesecarriers.
+Generationof the spin polarizationcan be achievedby optic al orientationof electronspin throughtransferof an-
+gularmomentumofthecircularlypolarizedphotonstotheca rriers[3]andelectricinjectionofspinpolarizedcarrier s
+from ferromagneticconductorinto non-magneticconductor [904, 905]. When electronstransfer from spin polarized
+regime to unpolarized one, say from ferromagnetic metal or s emiconductors across the interface into non-magnetic
+conductors,nonequilibriumspinpolarizationaccumulate sneartheinterface[23,906]. Duetospinrelaxation,thesp in
+polarizationofthecarriersinthenonmagneticconductori snotspacialuniformbutdecaysascarriersmoveawayfrom
+the interface, characterized by the spin injection length. The spacial inhomogeneityof the spin polarization also re-
+sultsinthespindiffusion,whoserateisdescribedbythespindi ffusioncoefficient. Inthepresenceoftheelectricfield,
+spinsaredraggedbytheelectricfieldalongwiththecarrier s. Theresponsetotheelectricfieldischaracterizedbythe
+conductivityor mobility. An important task is to reveal the relations among these characteristic parametersand how
+these parameters change under di fferent conditions. Spin detection is to extract the informat ion of spin polarization
+by sensing the change of the magnetic, optical and /or electrical signals due to the nonequilibrium spin polari zation
+in the non-magneticconductor. In experiments,optical met hodsare usually more powerful as one can generatehigh
+spin polarization in semiconductor and detect the spin sign als with spacial resolution optically. However, it is more
+desirabletogenerateanddetectthespinsignalelectrical lyinreal spintronicdevices.
+Eventhoughthe Datta-Dastransistoris conceptuallysimpl e, thereare some crucialobstacleswhichmake it hard
+to be realized: The low spin injection /detectionefficiency between the semiconductor and ferromagnetic source and
+90Although operations of the spin field-e ffect-transistor do not create or eliminate conducting chann el, the electrons have to travel between the
+sourceand drain when thetransistor switches on /offstates. Therefore, theswitching timeofspin field-e ffect-transistor islimited by themobility of
+the channel. The energy dissipation and the switching time o f traditional and spin field-e ffect-transistors were compared theoretically by Hall and
+Flatt´ e [222]. They pointed out that the energy dissipation of a typical spin field-e ffect-transistor can be two orders of magnitude smaller than t hat
+of atypical traditional transistor, but theswitching time is usually longer [222].
+132draindue tothe conductancemismatch[907–910](see Sec. 6. 2.4fordetails) andspin-flipscatteringat the interfaces
+[911–914], spin relaxation in the semiconductor channel du e to the joint effect of the spin-orbit coupling and (spin
+conversing) scattering [101, 102, 915] (see Sec. 5 for more d etails), and the precise control of the gate controlled
+spin-orbitcoupling. Duetothesedi fficultiestheDatta-Dastransistorhasnotbeenrealizedunti lKooetal. claimedso
+recently[55], althoughtherearestill controversiesonth eirclaim[916–918].
+Toovercometheseobstaclesonemustfirstunderstandspintr ansportinsidethesemiconductorchannelandacross
+the interfaces. A lot of understandings on spin transport, t o be reviewed in following sections, have been gained
+in the past decade. Based on these understandings, some vari ants of the Datta-Das transistor have been proposed.
+Schliemann et al. first proposed a nonballistic spin field-e ffect-transistor based on the spin-orbit coupling of both
+the Rashba and the Dresselhaus types [339]. Due to the interp lay of these two types of spin-orbit coupling, the spin
+diffusion length in the semiconductor channel becomes larger or even infinity for spin polarized along some special
+directions [194, 196, 203, 919, 920] or for spin polarized cu rrent flowing along some special directions [18, 20, 29,
+37, 339, 921] (see Sec. 7.5 for details). This type of spin fiel d-effect-transistor is robust against the spin conversing
+scatteringandisa morerealistic alternativeto theDatta- Dasspinfield-effect-transistor.
+6.2. Theoryofspininjectionbasedondrift-di ffusionmodel
+Although spin transport in semiconductor is a relatively ne w problem, study of the spin transport in ferromag-
+netic material has much longer history. It is thus natural to borrow the concepts and methods of spin transport in
+ferromagneticmetal to study the spin transport in semicond uctor. In ferromagneticconductors, the carriers and cur-
+rent are spontaneously spin polarized below the Curie tempe rature. The basic transport properties of ferromagnetic
+metal can be understood by two-current model, where majorit y (say, spin-up) and minority (spin-down) carriers
+contribute unequally and independently to the total conduc tance [922–926]. In two-current model, the spin-up and
+-downelectronstransportindependentlyexceptforweaksp in-flipscatteringthatflipscarriersofonespintotheother
+[907, 908, 925, 927]. The carriers of di fferent spins usually have di fferent diffusion coefficientsD↑/↓and mobilities
+µ↑/↓. The electric currentsof the spin-up and -down electronsar e determinedby the drift caused by the electric field
+anddiffusionduetothespacialinhomogeneityofthecarrierdensit ies. Underlinearapproximationandinthedi ffusive
+limit,thecurrentsare [22, 907, 908, 927]
+jη=−enη∝an}b∇acketle{tvη∝an}b∇acket∇i}ht=eµηnηE+eDη∇nη=σηE+eDη∇nη,(η=↑,↓), (272)
+wherenηandση=eµηnηare the electron density and conductance of spin η, respectively. The electric field Eis
+determinedbythePoissonequation
+∇·E=−∇2φ=e(n↑+n↓−n0)/ε, (273)
+withφ,−e,εandn0beingtheelectricpotential,electroncharge,dielectric constantandthebackgroundpositivecharge
+densityrespectively. Since the totalelectronnumberis co nservative,onecan writedownthe continuityequationsfor
+thesetwokindsofelectrons:
+∂nη
+∂t=1
+e∇·jη−δnη
+τη¯η+δn¯η
+τ¯ηη, (274)
+withδnη=nη−n0
+ηandτη¯ηstanding for deviationof the nonequilibriumchargerdensi tynηfromthe equilibriumone
+n0
+ηandtheaveragecarrierspin-fliptime,respectively.
+Combined with different initial and boundary conditions, the above drift-di ffusion equation or its equivalence is
+widely used in the study of spin-related transport in semico nductors, including in the understanding of the existing
+experimentalspininjection /extractionresultsandin theproposingofnewschemesofspi ntronicdevices. However,it
+shouldbepointedoutthatalthoughtheseequationsgivequa litativelycorrectresultsinmanycases,thevalidityofth e
+drift-diffusionmodelshouldbecarefullyexaminedwhenappliedtospi ntransportinsemiconductors. Thisisbecause
+thespintransportinferromagneticmetalandsemiconducto rcanbequitedifferent: First,theappliedelectricfieldand
+spacialgradientofelectrondensityinmetalareusuallysm allduetolargeconductanceandhighcarrierdensity,hence
+electronsareusuallynotfarawayfromtheequilibrium. The reforeEq.(272)describesthecurrentinmetalsaccurately
+in most cases. In semiconductors, however, both the applied electric field and the gradient of carrier density can be
+very large, and electrons can be easily driven to states far a way from the equilibrium. As a result, Eq. (272) may no
+133longer hold. More importantly, the spin-up and -down branch es in ferromagnetic metal are well separated and the
+coherence between these two branches is very small. While in nonmagnetic semiconductor the spin-up and -down
+electronsare usuallydegenerate,an appliedore ffective(fromthe spin-orbitcoupling)magneticfieldcan cau sespins
+toprecessandresultinlargespin coherence.
+Therefore quantitative calculation based on the drift-di ffusion model is questionable when the spin coherence is
+essential to the spin kinetics or when the electrons involve in the transport are far away from the equilibrium. It has
+beenshownthatinthepresenceofthemagneticfield,eithert heexternalappliedoneortheintrinsice ffectiveonefrom
+thespin-orbitcoupling,the drift-di ffusionmodelisinadequateinaccountingforthe spintranspo rtin semiconductors
+[25, 27, 28, 32, 336]. Nevertheless,the drift-di ffusionmodelis still useful in qualitativestudy of spin tran sportsince
+itssimplicityandflexibilitytoaddnewfactorsofphysics.
+6.2.1. Spintransportinnonmagneticsemiconductorusingd rift-diffusionmodel
+Onecaneasilyobtainsomebasicpropertiesofthespintrans portusingdrift-diffusionmodeltostudythetransport
+property of the magnetic momentum inside nonmagnetic semic onductors in the simplest case. Assuming that the
+chargedensity in nonmagneticsemiconductoris uniformand the electronic transport coe fficientsare not affected by
+the spin polarization, one can then write down the transport equation for the magnetic momentum Sin a uniform
+electricfield Eandmagneticfield B[915],
+∂S
+∂t=D∇2S−eµE·∇S+gµBB×S−S
+τs, (275)
+withµ,D, and 1/τsbeing the charge mobility, di ffusion coefficient and spin relaxation time, respectively. Note that
+the drift-diffusion equation here has been modified to include the Larmor pr ecession of the spin around the applied
+magneticfield B.
+Spin accumulation in the steady state. When a semiconductor is in contact with a spin polarization s ource at
+x=0,B=0 and the electric field is along the x-direction, the drift-di ffusionmodel predictsspin accumulationwith
+exponentialdecayinthesemiconductor S(x)=S0exp[−x/Ls(E)]inwhichtheelectric-field-dependentspininjection
+lengthreads[904, 928, 929]
+L−1
+s(E)=1
+Ls/radicaligg
+1+L2
+d
+4L2s−E
+|E|Ld
+2L2s, (276)
+whereLs=√Dτsis the diffusion length, Ld=|eµE|τs=|vd|τsis the drifting length over spin relaxation time with
+drift velocity vddetermined by the electric field. Without the electric field, the spin injection length is the di ffusion
+lengthLs. The applied electric field can significantly change the inje ction length by dragging or pulling the electron
+[928,929]: Forlargedownstreamelectricfield,theelectro nspininjectionisthedistancethattheelectronsmovewith
+driftvelocitywithinthespin lifetimetime, Ls(E)=Ld; Forlargeupstreamfield, Ls(E)=L2
+s/Ld.
+Evolution of the spin polarized electron package. The shape of a spin polarized carrier packet changes with
+timeduetothedriftanddi ffusion. Thetemporalevolutionofa δspinpolarizedpacketofheight S0atx=0is
+S(x,t)=S0√
+2πDtexp/bracketleftigg
+−t
+τs−(x−vdt)2
+4Dt/bracketrightigg
+, (277)
+whichhastheformofGaussianfunctionwhosecenterisdeter minedbythedrifting, vdt, andwidthdeterminedbythe
+diffusion,√
+Dt.
+6.2.2. Hanleeffectinspin transport
+In the spacial homogeneoussystem with a spin excitation sou rce, such as circularly polarized light [2, 311, 930]
+or spin resonant microwave radiation [931, 932], the total s pin is the accumulation of survival spin from the past.
+Whenaperpendicularmagneticfield ispresented,spinunder goesLarmorprecession. Sincespinsexcitedat di fferent
+timehavedifferentprecessionphasesandtendtocanceleachother,thepe rpendicularmagneticfieldreducesthetotal
+accumulatedspin. ThisisknownastheHanlee ffect[915,933]. Inspintransport,theHanlee ffectalsoappears[150]:
+Spinaccumulationat position xis the sum ofthe electronspinstravel fromdi fferentplaces. Since di fferentelectrons
+have different transit time when they reach x, their phases are di fferent and tend to cancel each other. As shown in
+134Fig. 103, for a constant spin pumping /injection starting from t=0 atx=0, when a constant magnetic field Bis
+appliedalongthe ˆ ydirection,the spinaccumulationat time tandposition xis
+Sz(x,t)=/integraldisplayt
+0S0√
+2πt′e−t′/τs−(x−eµEt′)2/(4Dt′)cosωt′dt′, (278)
+Sx(x,t)=/integraldisplayt
+0S0√
+2πt′e−t′/τs−(x−eµEt′)2/(4Dt′)sinωt′dt′, (279)
+withω=gµBBbeing the Larmor frequency. For the steady state spin inject ion under a magnetic field, the spin
+accumulation can be calculated by letting time in Eqs. (278) and (279) to be infinity or can be solved directly from
+Eq.(275). Theresultsare
+Sz(x)=S0e−x/Ls(E,B)cos[x/L0(E,B)], (280)
+Sx(x)=S0e−x/Ls(E,B)sin[x/L0(E,B)], (281)
+where
+L−1
+s(E,B)=E
+|E|Ld
+2L2s+1
+Ls/radicaltp/radicalbt1+L2
+d
+4L2s2
++(ωτs)2cosθ
+2, (282)
+L−1
+0(E,B)=1
+Ls/radicaltp/radicalbt1+L2
+d
+4L2s2
++(ωτs)2sinθ
+2, (283)
+tanθ=ωτs
+1+L2
+d/(4L2s). (284)
+Onecanseethatthespinpolarizationschangewiththeposit ionasadampedoscillationwiththeelectricandmagnetic
+field dependent injection length Ls(E,B) and the spacial oscillation “period” L0(E,B) defined by Eqs. (282–284).
+Evenwhenthere isno spin relaxationmechanism, i.e., τs=∞, the spin injectionlengthis still finite in the presence
+of a perpendicular magnetic field, L−1
+s(0,B)=L−1
+0(0,B)=√ω/2D.91Without the driving of the electric field, the
+oscillation “period” is larger or close to the injection len gth, which means that the oscillation may not be easy to
+detect. Undera strongdownstreamelectricfield, Ls(E,B)≃Ld, andL0(E,B)≃Ld/ωτs,it isthenpossibletoobserve
+manyoscillationsinspinaccumulation.
+O xz
+B/bardblˆyS/bardblˆz
+E/bardblˆx
+Figure 103: Orientations of spin transport and electric /magnetic fields.
+6.2.3. Spininjectiontheory
+Tostudythespininjectionfromferromagneticelectrodeto semiconductor,itisusuallymoreconvenienttousethe
+drift-diffusionequationintheformofelectrochemicalpotential[22 ,907,908,927]. Forthestateneartheequilibrium,
+δnη(r)=gηδ˜ζη(r),withδ˜ζη(r)=˜ζη(r)−˜ζ0
+η(r)standingforthe deviationinthe chemicalpotential ˜ζη(r)awayfromthe
+equilibriumone ˜ζ0
+η(r)oftheelectronwithspin ηatposition r.gη=(∂nη/∂˜ζη)isthetemperaturedependentdensityof
+states. Thecurrentcanberewrittenas[22, 907, 908, 927]
+jη=−ση∇φ(r)+eDηgη∇δ˜ζ=ση
+e∇ζη(r), (285)
+91This effect was firstpredicted by Weng and Wumicroscopically fromth e kinetic spin Bloch equation approach [336].
+135in which we have used the Einstein relation, ση=e2Dηgη, and introduced the electrochemical potential, ζ(r)=
+˜ζ(r)−eφ(r).
+Introducingthenotations, g=g↑+g↓,gs=g↑−g↓,σ=σ↑+σ↓,σs=σ↑−σ↓,ζ=(ζ↑+ζ↓)/2,ζs=(ζ↑−ζ↓)/2,
+D=(D↑+D↓)/2andDs=(D↑−D↓)/2,andenforcingthe chargeneutrality,i.e. n↑+n↓≡n0
+↑+n0
+↓,oneobtains
+ζ(x)+eφ(x)=−gsζs(x)/g. (286)
+Thenon-equilibriumspinaccumulationisexpressedas
+δS=4g↑g↓
+gζs. (287)
+From Eq. (285) one can further write down the charge and spin c urrents expressed in form of the electrochemical
+potential,
+je=j↑+j↓=σ
+e∇ζ+σs
+e∇ζs, (288)
+js=j↑−j↓=σs
+e∇ζ+σ
+e∇ζs, (289)
+∂ζs
+∂t=D∇2ζs−ζs
+τs, (290)
+withD=g(g↑/D↓+g↓/D↑)−1beingthe effectivespindiffusioncoefficient.
+From these equations, one can obtain the spin transport prop erties in conductors. For nonmagnetic conductor,
+the spin-up and -down bands are degenerate, one recovers the results obtained in Sec. 6.2.1 for the spin transport
+in nonmagnetic conductor under weak electric field. In ferro magnetic conductor, the dynamics of nonequilibrium
+spin accumulation is similar to that in nonmagnetic conduct or, i.e., it exponentially decays with the position, with
+the spin diffusion length being Ls=/radicalig
+Dτs. For a uniform ferromagnetic conductor without nonequilib rium spin
+accumulation,the currentpolarizationisdeterminedbyth edifferenceoftheconductivitiesofthetwospinbranches,
+Pσ=js/je=σs/σ. (291)
+Inthepresenceofthespinaccumulation,thecurrentpolari zationis modifiedas
+Pj=Pσ+Ls∇ζs/(jeR), (292)
+whereR=σLs/(4σ↑σ↓) is the effective resistance of the conductor with unit surface area ov er a distance of the
+diffusionlength.
+6.2.4. Spininjectione fficiencyof ferromagneticconductor /nonmagneticconductorjunction
+To discuss the spin injection from ferromagnetic conductor into nonmagnetic conductor, one needs to study the
+transport through their interface. In drift-di ffusion model, the effect of interface is phenomenally described by spin
+selective interface conductance. Assuming that there is no strong spin-flip scattering at the interface, the current of
+each spin branch is conservative. However, the electrochem ical potential can be discontinuous if the contact is not
+ohmic. Introducingthecontactconductance Σηforspinη,theboundaryconditionattheinterface(x =0)isthenwritten
+as[904,905,907,908,910]
+jη(0)=Ση[ζηF(0)−ζηN(0)]. (293)
+Similarly, one can introduce Σ=Σ↑+Σ↓,Σs=Σ↑−Σ↓and rewrite the boundary conditions for charge current and
+spincurrent.
+Usingtheabovedrift-di ffusionequation,thematchingconditionofinterfaceat x=0,andtheboundaryconditions
+ζsF(−∞)=0 andζsN(∞)=0 at the far left endof ferromagneticelectrodeand the far ri ghtendof nonmagneticelec-
+trode,one isable to write downthe spininjectione fficiencyacrossthe ferromagnet /nonmagneticconductorjunction.
+Thecurrentpolarizationatthe interfaceis[907, 908]
+Pj(0)=js(0)
+je(0)=RFPσF+RcPc
+RF+Rc+RN, (294)
+136whereRF,RNandRc=Σ/(4Σ↑Σ↓), are the effective resistances of ferromagnetic conductor, nonmagnet ic conduc-
+tor and contact, respectively. PσFandPc=(Σ↑−Σ↓)/Σare the conductance polarizations of the ferromagnetic
+conductor and contact respectively. Since the ferromagnet ic conductor usually has much larger conductance and
+shorter spin diffusion length, RF≪RN. Therefore, for transparent contact, the polarization of t he injected current
+Pj(0)=PσFRF/(RF+RN)is muchsmaller than PσF, the currentpolarizationinferromagneticconductor,due to the
+conductancemismatch [907–910]. For large contact resista nce, the currentpolarization is determinedby the contact
+polarization Pj(0)≃Pc. Thereforethe conductionmismatch can be reducedby insert ingan spin selective layerwith
+largeresistance[22,907–910,927,934]. Onecanalsoavoid theconductancemismatchbyusingspinfilter[935–937]
+toreplacethe ferromagneticmetalasspininjector.
+6.2.5. Silsbee-Johnsonspin-chargecoupling
+Inelectricalspininjection,thespinpolarizedcurrentflo wsfromaferromagneticelectrodeintononmagneticcon-
+ductorandproducesanon-equilibriumspinaccumulationne artheinterfaceinthenonmagneticconductor. Theinverse
+oftheaboveeffectalsoholds: Thepresenceofnon-equilibriumspinaccumu lationinnonmagneticconductornearthe
+interface will produce an electromotive force in the circui t which drivesa charge current to flow in a close circuit or
+resultsinavoltagedropinanopencircuit. Thise ffectiscalledSilsbee-Johnsonspinchargecoupling[905,93 8]. This
+spin-charge coupling can a ffect the spin transport properties in return. In the spin inje ction, the electromotive force
+causedbythespinaccumulationimpedesthecharge /spincurrentandreducestheoverallconductivity,whichis called
+thespinbottlenecke ffect[905, 938–941].
+Figure 104: Schema ofSilsbee-Johnson e ffect and all electrical spin detection. FromJohnson [940].
+Physically speaking, this spin-charge coupling can be unde rstood by the simplified model shown in Fig. 104,
+proposedbySilsbee[938]. Inthismodel,itisassumedthatt heferromagneticelectrodeisahalf-metalsothatonlyone
+spin branch(say, spin-up)electronscarry the currentand F ermi levelsin ferromagneticand nonmagneticconductors
+are aligned before they are connected. After they are connec ted, the spin accumulation near the interface raises the
+(electro)chemicalpotential of spin-up electrons in nonma gneticelectrode, and lowers that of spin-down electrons to
+ensure the neutrality of charge, as shown in the left part of F ig. 104. As a result of this electrochemical potential
+shifting in nonmagnetic electrode, an electromotive force must be produced to raise the electrochemical potential
+in ferromagnetic electrode so that it aligns with that of spi n-up electrons in nonmagnetic electrode. Quantitative
+calculationofSilsbee-Johnsoncouplingbasedondrift-di ffusionmodelcanbeobtainedfromEqs.(286–293). Inclose
+circuitthe changeintheresistancecausedbythespinbottl eneckeffectis
+δR=RN(RcP2
+c+RFP2
+F)+RFRc(PF−Pc)2
+RF+Rc+RN. (295)
+Formorecomplicatedstructureslikeferromagnetic /nonmagnetic/ferromagneticconductors,theresistancechangedue
+tothespinaccumulationdependsonthelengthofnonmagneti cconductorandthespinpolarizationsofferromagnetic
+electrodes. When the length of nonmagnetic conductor is muc h larger than the spin di ffusion length, the resistance
+change is simply the sum of that of two ferromagnetic /nonmagnetic contacts. When the length of nonmagnetic
+conductoris smaller than or of the order of spin di ffusion length, the change in resistance dependson the alignm ent
+of the magneticmomentumof the two electrodes,as shown in th e rightpart of Fig. 104. Thisis knownas spin valve
+effect. Itcanbequantitativelycalculatedbymatchingthebou ndaryconditionsonthesetwo contacts[908].
+137Inopencircuit,foraspinaccumulationat x=0,thevoltagedropcausedbytheelectromotiveforceoverad istance
+farlongerthanthespin di ffusionlengthisgivenby
+∆φ=/parenleftbiggs/g−σs/σ/parenrightbigζs(0)/e. (296)
+If a second ferromagnetic electrode is attached to the nonma gnetic electrode, it can be used to detect the spin ac-
+cumulation at the interface of the nonmagnetic electrode an d the second ferromagnetic electrode. If there is spin
+accumulation, the voltage drop between the nonmagnetic and the second ferromagnetic electrodes depends on the
+relative alignments of the spin momentums in these two elect rodes, as shown in the right part of Fig. 104. Silsbee-
+Johnsonspin-orbitalcouplingis particularusefulinthe n on-localspindetection[281, 940–946].
+6.3. SpininjectionthroughSchottkycontacts
+In the drift-diffusion model, the contact of ferromagnetic conductor and sem iconductor is usually described as
+a sheet layer with finite or zero resistance. Studies of the sp in injection through Schottky barrier, which appears
+naturally at the interface of ferromagnetic metal and semic onductor, have also been carried out using drift-di ffusion
+model[947, 948] andMonteCarlosimulation[949–954].
+In the framework of the drift-di ffusion model, the tunneling through the Schottky barrier is s till described by
+(spin selective) interface resistance. However, the condu ctivity inside the semiconductor is no longer a constant,
+since it is proportional to carrier concentration which var ies dramatically with position near the Schottky barrier.
+Taking the position dependence of the conductivity into acc ount, it is shown that the existence of a depletion region
+ofthe Schottkybarriergreatly reducesthe spinaccumulati on. Fora ferromagneticmetal /semiconductorcontactwith
+significantdepletionregion,thespinaccumulationisredu cedalmosttozeroinadistancelessthan100nm[947,948].
+Outside the depletion region,the spin accumulationdecays with the distance at a slower rate of spin injection length
+(at the orderof 1 µm). The existenceof the Schottkybarrieralso greatlyreduc esthe spin polarizationof the injected
+current. Although the spin polarization of injected curren t decays at the rate of spin injection length throughout the
+semiconductorside, it dropsrapidly inside the ferromagne ticconductornear the ferromagneticmetal /semiconductor
+surfacewhenahighbarrierispresented.
+In the Monte Carlo simulations, studies focus on the transpo rt inside the semiconductor. The ferromagnetic
+conductor is treated as source that provides spin polarized carriers and is not a ffected by the semiconductor. The
+currentis injected into the semiconductorthroughthe barr ier byboth thermionicemission and direct tunneling[955,
+956]. In this model, the tunneling probability through the b arrier, proportional to the density of states for spin-up
+and -down carriers in the ferromagnetic conductor, is natur ally spin dependent. Therefore, this model is beyond the
+drift-diffusionmodel. Inthesimulation,thespacialprofileoftheSch ottkybarrierisdeterminedbysolvingthecharge
+motionandthePoissonequationself-consistently[957]. T hetransportinthesemiconductorisbasedonsemiclassical
+approximation that includes “drifting” and “scattering” p rocesses. The electron spin is subjected to the influent of
+the Rashba and/or Dresselhaus spin-obit interaction during the “drifting ” process. The Monte Carlo simulations of
+the spin injection from Fe contact into GaAs quantum well sho wed that, although injected spin polarization is large
+rightnexttotheSchottkybarrier,similartotheresultsof thedrift-diffusionmodel,thetotalspinpolarizationdropsto
+nearly zero in a few tens of nanometers,a distance of the orde r of depletion region,due to large unpolarizedcarriers
+in the semiconductor. Moreover,the average magnetic momen tumof the injectedelectronsalso decaysdramatically
+in the depletion region but much more slowly beyond that. Thi s is because the Rashba and Dresselhaus spin-orbit
+couplings depend on the electron momentum, the larger the ki netic energy, the larger the e ffective magnetic field.
+In the depletion region, electrons have much larger kinetic energy, therefore su ffer much faster decay rate. Outside
+the depletion region, the decay rate is much slower as electr ons lose their kinetic energy to overcome the electrical
+potential barrier as well as due to the inelastic scattering [949–954]. Usually, the Rashba e ffect considered in the
+spinkineticsin quantumwells is determinedby theelectric field perpendicularto the quantumwell plane(dueto the
+built-in electric field or the gate voltage). In the presence of the Schottky barrier, there is a strong in-plane electric
+field in the depletion region induced by the barrier. This bar rier induced electric field also leads to the Rashba e ffect
+andfurtherreducesthemagneticmomentumofthe injectedel ectrons[952].
+As one can see from the above studies that, a high Schottky bar rier with significant depletion region is highly
+undesirable for spin injection. In order to increase the spi n injection efficiency, one needs to modify the Schottky
+barrier. A thin heavily doped n+layer, which sharply reduces the height and thickness of the barrier, was proposed
+138to be inserted between the ferromagneticelectrodeand semi conductorto increase the injection e fficiency[958–965].
+Usingthissetup,highlye fficientelectricalspin injectioncanbeenachieved. It hasbe enreportedthatin a wide range
+of temperature, up to 30% of the spin polarization is injecte d from Fe into n-type AlGaAs and GaAs and survives
+overa distance of 100 nm [958, 959, 964]. Up to 50% spin polari zationhas been reportedto survive overa distance
+of300 nmin the experimentsof spin injectionfromFe into n-typeZnSe whenthe temperatureis aroundor less than
+100K [966].
+Theoreticalstudiesofspininjectionthroughtheinterfac eofferromagneticconductorandsemiconductorwitha δ-
+dopedlayerneartheinterfacewerecarriedoutinRefs.[960 –963]. Inthesestudies,it wasassumedthatthedepletion
+region is only a few nanometers and the tunneling electrons d o not lose their spin polarization over this region. By
+assuming that the depletion region is of triangular shape, a direct tunnel current was calculated and the transport
+outside the depletion region was performed using the drift- diffusion model. Unlike the traditional drift-di ffusion
+model, the spin polarization of the tunneling carriers depe nds on the band structure of ferromagnetic conductor, but
+is independent on the transport properties of the ferromagn etic metal. Hence, there is no conductance mismatch in
+this model, and it is possible to achieve high spin polarizat ion at the interface. Moreover, it was shown that the spin
+injection efficient is not a constant but strongly depends on the injection current, the larger the current amplitude,
+the higher the injection e fficiency. In order to achieve high e fficiency, a high electric field is required and therefore
+injection length also increases with the injection current as predicted by Eq. (276) [960–963]. However, the applied
+electricfieldalsoenhancestheRashbaspin-orbitcoupling andreducestheinjectedspinpolarizationaspointedoutby
+Wang and Wu [952]. Recent experimentby Kum et al. showed that the competingeffects of the electric field cancel
+eachother,leadingto analmostnegligibledecreaseofmagn etoresistancewith biascurrent[967].
+6.4. Spindetectionandexperimentalstudyofspintranspor t
+Experimentalstudy of spin transport also requiresan e ffective methodto extract the spin informationfrom semi-
+conductors. In some senses, spin detectionand extractiona re reverseof the spin generationand injection. Each spin
+generation/injectionmechanismcan also be reverselyused to extractth e spin information. Generallyspeaking,there
+are two categories of spin detection: optical and electrica l spin detections. Optical spin detection, such as the Fara-
+day/Kerrrotation and circular polarizationof electrolumines cence,providesreliable informationof spin polarization
+in semiconductor and has been proven to be a powerful experim ental tool in the study of the spin transport. While
+highefficiencyelectricalspindetectionisessential tothespin tr ansistor.
+6.4.1. Spindetectionusingelectroluminescence
+The electroluminescence is the reverse of the optical orien tation. In the usual electroluminescence experiment
+setup, the electrons (holes) to be detected are driven to a st ructure like a light emitting diode where they recombine
+with unpolarized holes (electrons) and emit photons. Due to the selection rule, the emitted photons are circularly
+polarizedif the carriersto be detected are spin polarized. By measuring the polarization of the electroluminescence,
+oneobtainsthespinpolarization. Thistechniquewasemplo yedinthefirst experimentalobservationofspininjection
+from ferromagnetic nickel tip of a scanning tunneling micro scope into nonmagnetic p-type GaAs, and the injected
+spin polarizationof about 40% at the surface was reported[9 68]. However, the early attempts on high spin injection
+efficiency in the real devices were not so successful due to spin- flip scattering at the interface and the conductance
+mismatch [907, 908]. Spin polarization injected from tradi tional ferromagnetic metals, such as Fe, Ni, Co and their
+alloys,intoGaAsorInAswasusuallyonlyafewpercents[969 –971]. Byvariousmeans,thespininjectione fficiency
+has been improved over the years. Here we list some important results in the following, loosely cataloged by the
+structuresusedinthe experiments.
+•Spin injection from ferromagnetic or diluted magnetic semi conductors into GaAs: Ferromagnetic or diluted
+magnetic and nonmagnetic semiconductors have similar cond uctance, therefore spin injection in these struc-
+tures surfers less conductance mismatch. However, since fe rromagnetic semiconductor has low Curie tem-
+perature, the experiments are usually performed at low temp erature. Spin injection from n-type II-V diluted
+magnetic semiconductor into n-GaAs under applied magnetic field is very e fficient. Injected spin polarization
+from Be xMnyZn1−x−ySe into GaAs achieves90% at low temperature( <5 K), but dropsto 20% when the tem-
+perature rises to 35 K [522, 972]. Spin injection from p-type GaMnAs or MnAs into n-type GaAs is more
+difficult. Theefficiencyof directspininjectionfromGaMnAsor MnAsinto n-typeGaAsusuallyis onlya few
+139percentevenat lowtemperature[973–976]. InjectionfromG aMnAsorMnAsviainterbandtunnelingin Esaki
+diodeis muchmoree fficient [977–984]. 80% injected spin polarizationin Esaki di ode hasbeen realizedwhen
+temperatureis10K [983]. Molenkampgrouphasproposedandd emonstratedto use amagneticdoublebarrier
+resonant tunneling diode as spin injector [985–989]. A spin polarization up to 80% has been injected from
+BeTe/Zn1−xSe/BeTe intoGaAsunder1.6Ktemperatureand7.5Tmagneticfield [985].92
+•Spin injection from traditional ferromagneticmetal throu gha Schottky barrier into GaAs or AlGaAs: By con-
+structinga Schottkybarrierwith narrowdepletionregimea t the interfaceofthe ferromagneticmetaland semi-
+conductor,spininjectione fficiencyisgreatlyimproved[974,990–994]. Injectedspinpo larizationof30%from
+Fe intoGaAsatroomtemperaturewasreported[993].
+•Spin injection from traditional ferromagnetic metal throu gh an insulator layer into GaAs or AlGaAs: By
+inserting an insulator tunneling layer, the injected spin p olarization can be further increased. The tunneling
+layer improvesthe surface conditionby preventingthe magn etic atoms from diffusing into semiconductorand
+thus reduces the spin-flip scattering at the interface. More over,this tunneling layer can be regardedas a spin-
+selective contactwith lowconductanceandhencereducesth e conductancemismatch [22, 907, 908, 927, 934].
+TypicaltunnelinglayersarecomposedofAlO x[995–998]andMgO[999–1007],althoughGaO xhasalsobeen
+used to improvethe chargeinjectione fficiency[1008]. Using AlO xastunnelinglayer[995–998], the injection
+efficiencycanachieve40%whenthe temperatureislowerthan180 K [997,998]. MgOtunnelinglayeris even
+more efficient in improving the spin injection. Due to the band struct ures and symmetry, tunneling from Fe,
+Co and their alloy through MgO barrier is strongly spin selec tive. The conductance of majority spin is orders
+of magnitude higher than that of minority spin [1000, 1009–1 012]. In this case, the injected spin polarization
+is determined by the tunneling spin polarization. Injected spin polarization of 70-80% from CoFe electrode
+throughMgOlayerintoGaAsatroomtemperaturewasreported [999, 1002].
+•Spin injection from Heusler alloys into GaAs or AlGaAs: In ad ditional to the spin injection from traditional
+ferromagneticmetal, there are also experimentson using He usler alloys (some of them have been predictedto
+be half-metals[1013, 1014])asspininjectorin hopingthat bothhighspin andchargeinjectione fficienciescan
+berealized[1015–1019]. Over50%injectione fficiencyfromHeusleralloyCo 2FeSiintoGaAsupto100Khas
+beendemonstrated[1018].
+The electroluminescencetechniquehas also been used to dem onstratethat spin polarizedcarrierscan be injected
+andsustain the polarizationovera microscopicdistance by constructingthe light emittingdiodestructureaway from
+the surface of ferromagnetic and nonmagnetic conductors [5 22, 523, 946, 958, 970, 975, 977, 1020, 1021] or away
+fromopticalinjectionposition[1022]. Consideringtheop ticalabsorptionofthesemiconductor,thephotonemitteda t
+positionawayfromthesurfaceshouldbeweightedasexp[ −α(λ)x]. Byfurthercombinedwiththefactthatabsorption
+coefficient is wave-length dependent, experimental measurement of wave-length dependence of the electrolumines-
+cencepolarizationwasusedtoestimatethe spininjectionl ength[1023].
+6.4.2. Spinimaging
+ThespindetectionusingtheFaraday /KerreffectmeasurestheFaraday /Kerrrotationangleoftransmitted /reflected
+linearly polarized photons. These angles are proportional to the spin component(magnetization)along the direction
+definedbythe propagationofprobelightbeam[748],andthus providedirectinformationofthemagnetizationalong
+thisdirection. Usuallytheprobelightisnearlynormaltot hesamplesurface,thereforeonlythemagnetizationnormal
+to the surface can be measured by this technique directly. Th e information of spin coherence can be obtained by
+applyinganin-planemagneticfieldtorotatethein-planesp incomponenttothenormaldirection. Recentlydeveloped
+tomographic Kerr rotation technique further enables the me asurement of the spin coherence at any direction [500,
+1024].
+92It is noted that the direction of the spin polarization throu gh magnetic double barrier resonant diode can be changed by a n external voltage
+which selects the spin dependent resonant tunneling level [ 985–989]. This is quite di fferent from the spin injection from ferromagnetic material
+where the injected spin polarization is the same as the major ity spin of the injector. To change the spin direction inject ed from the ferromagnetic
+injector, a magnetic field should be applied to change its spi n direction. While using magnetic double barrier resonant d iode, it can be achieved
+electrically.
+140Figure 105: Schematic experiment setup for spin image of ele ctrical spin injection. Left: Surface scanning, an in-plan e magnetic field is applied
+torotate theinjected spinto outofplaneforKerr rotation m easurement. FromCrooker etal. [1025];Right: Side-surfac e scanning. FromKotissek
+etal. [1026].
+By moving the focus of the probe light and performing the meas urement at different time, one obtains the space
+andtemporalresolvedinformationofthespinpolarization ,andthusimagesthetransportofspinpolarization. Typica l
+schemesof experimentsetups and results for spin imaging ar e shown in Fig. 105. In the experiments,spin polarized
+carriersareusuallyinjectedfromferromagneticfilmsinto thesemiconductor. Dependingontheexperimentsetup,the
+probelaser can be focused on the top surface of the semicondu ctor(left side of Fig. 105) [5, 30, 31, 527, 910, 1025,
+1027] or on the side-surface (right side of Fig. 105) [1026]. Since the easy axis of the ferromagnetic film is usually
+parallelto thefilmplane,anin-planemagneticfield isrequi redtorotatetheinjectedspintothenormalofthesurface
+[30, 31, 527, 910, 1025, 1027]. In the side surface scanninge xperiments,the easy axis of the ferromagneticfilm can
+be arrangedto be parallelto the probelaser and no magneticfi eld is required[1026]. The spin polarizedelectronsin
+ferromagneticelectrodeare driventhroughinterfaceinto the semiconductorsat low temperature. The polarizationof
+theinjectedspinisestimatedtobeafewpercents. Spintran sportovermacroscopicdistance( >1µm)hasbeenclearly
+demonstratedby these experiments[30, 31, 527, 910, 1025–1 027]. The spin image techniquehas also been adopted
+tostudythespinaccumulationscausedbytheextrinsicspin Halleffect [7–10, 1028–1036].
+6.4.3. Electricalspindetection
+There are also many works on all electrical spin injection /transport/detection. The simplest electrical spin de-
+tection method is to utilize the magnetoresistivee ffect by driving the spin polarized carriers from one ferromag netic
+electrodeononesideacrosstheinterfacetosemiconductor andextractthespinusinganotherferromagneticelectrode
+ontheothersideastheDatta-Dastransistor. Duetospin-va lveeffect,theresistancesshouldbedi fferentwhenthespin
+momentum of left electrode is parallel and anti-parallel to that of the right electrode if the spin polarization injecte d
+fromtheleftelectrodesurvivesattherightelectrode. Thi seffectisquantitativelycharacterizedbymagnetoresistance ,
+therelativechangeofthe resistancewhenthemagneticmome ntumschangefromparallelto anti-parallelalignments.
+Early works using this simple ferromagnetic /nonmagnetic/ferromagneticsandwich structure indeed showed that the
+resistance does have the hysteresis loop when an magnetic fie ld is applied to change the direction of the magnetic
+momentum on the right electrode [969, 1037, 1038]. Due to con ductance mismatch, magnetoresistance of the fer-
+romagnetic/semiconductor/ferromagnetic structure is not very large, usually less tha n 1% for ohmic ferromagnetic
+metal/semiconductorcontact [969, 1037, 1038]. For the system su ffered less from the conductance mismatch, mag-
+netoresistance of 8 .2% in MnAs/GaAs/Ga:MnAs structure has been archived at low temperature [103 9]. However,
+since magnetoresistance in this structure is usually low an d the spin polarized current flows through ferromagnetic
+141Figure 106: (a) A schematic diagram of the non-local experim ent (not to scale). The five contacts have magnetic easy axes a longy-axis. The
+large arrows indicate the magnetizations of the source and d etector. Electrons are injected along the path shown in red. Theinjected spins (purple)
+diffuse in either direction from contact 3. The non-local voltag e is detected at contact 4. (b) Non-local voltage, V4,5, versus in-plane magnetic
+field,By, (swept in both directions) for sample A at a current I1,3=1.0 mAat T=50 K. Theraw data are shown in the upper panel, the lower panel
+showsthedatawiththis background subtracted. (c)Non-loc al voltage, V4,5,versusperpendicular magneticfield, Bz,forthesamecontacts andbias
+conditions as in (b). Thedata in the lower panel have the back ground subtracted. Thedata shown in black are obtained with the magnetizations of
+contacts 3 and 4 parallel, and the data shown in red are obtain ed in the antiparallel configuration. FromLou et al. [944].
+electrode, the spin accumulation signal can be masked by oth er magnetoresistance e ffects such as the anisotropic
+magnetoresistanceofthemagneticelectrodeandlocalHall effectcausedbythefringemagneticfieldnearthecontact
+[1040–1044]. Thesedirectmeasurementsofspin valvee ffectinthe sandwichstructurearenotdecisive.
+A more sophisticated all electrical spin detection is to use non-local spin-valve e ffect [281, 940–946], first pro-
+posedbyJohnsonbasedontheSilsbee-Johnsonspinchargeco upling[905,938,940,941]. Theschematicofnon-local
+spin-valvesetup used in the experimentby Lou et al. is repre sentedin Fig. 106. In the experimentdi fferent contacts
+are grown on top of a GaAs layer. The spin polarized electrons are driven to form the spin polarized current flow
+betweencontacts1and3,whilenon-localvoltagebetweenco ntacts4and5ismeasured. DuetoSilsbee-Johnsonspin
+chargecoupling,ifthereisaspinaccumulationattheinter faceofGaAslayerandcontact4,achangeinthenon-local
+voltagecan bedetectedwhenthe directionofmagnetization incontact4 changes. Thisnon-localgeometryseparates
+the injection and detection paths and thus avoids or reduces the spurious effects caused by the other magnetoresis-
+tance effects. Using this technique, clearer signals of spin accumul ation were demonstrated through hysteresis loop
+behaviorin the resistance changewith the magnetic field [28 1, 942–946, 1045]. More recently, Koo et al. employed
+this method to demonstrate the oscillatory channel conduct ance due to gate voltage controlled spin precession [55].
+This experiment is of particular interest since it is claime d to be the first experimental realization of electric spin
+injection, spin detection and coherent spin manipulation i n one device, although there are some controversieson the
+claim[916–918].
+6.4.4. Spintransportinsilicon
+Spinrelatedphenomenonsinsiliconhasalsoattractedmuch attentionrecentlybecauseofthelongspinlifetimein
+siliconandthe compatibilitytothe currentindustrialsem iconductortechnologies[149, 150, 279, 280, 282, 290, 291,
+295,776,781,1046–1058]. Amajordi fficultyforstudyingspinrelatedpropertiesinsiliconisthe spingenerationand
+detection. Opticalspingeneration /detectionisnoteffectivesincesiliconisanindirect-bandsemiconductor[21 ,1046].
+Early studies on the Faraday /Kerr effects in silicon were limited in very narrow frequency ranges in microwave or
+infraredregion[1059–1061]. Althoughthe limitation was l ifted by using terahertz time domainspectroscopy [1062,
+1063], which covers a wider frequencyrange, spin detection using the Faraday/Kerr rotation is yet to be conducted.
+There are a few experiments using electroluminescence to ob tain the information of the spin polarization in silicon
+142Figure 107: (a) Schematic device structure of the SiGe spin
+light-emitting diode with the Co /Pt ferromagnetic top electrode
+and (b) associated simplified band diagram under bias in the i n-
+jection regime. FromGrenet et al. [279].
+Figure 108: (a) Electroluminescence spectra recorded at 5 K
+and 77 K for a saturated remanent state of the ferromagnetic
+Co/Pt contact. The applied voltage and current are respectivel y
+around 10 V and 10 mA. The no-phonon (NP) peak is clearly
+circularly polarized indicating ane fficient spininjection intothe
+silicon top layer. The transverse optical (TO) phonon repli ca is
+also slightly circularly polarized. (b) Electroluminesce nce cir-
+cular polarization recorded at the maximum of the non-phono n
+line for different remanent states of the Co /Pt layer (red dots).
+FromGrenet etal. [279].
+[279,282]. Byconstructingsurface-emittinglightemitti ngdiodeswithn-i-psiliconlayers,Jonkeretal. demonstra ted
+the electrical spin injection from Fe throughan Al 2O3layer into n-typesilicon. The injected spin polarizationunder
+3 T magnetic field was estimated to be 30% at 5 K, with significan t spin polarization extending to at least 125 K.
+[282]. Using a fullystrained Si 0.7Ge0.3quantumwell to replacethe intrinsic silicon layer,Grenet et al. demonstrated
+the electrical spin injection from CoFe into n-type silicon at zero magnetic field [279]. Their experiment setup and
+results are shown in Figs. 107 and 108. The advantage of the st rained SiGe quantum well over bulk silicon is the
+appearanceof the direct optical transitionthroughthe so c alled non-phonontransition[1064–1066]. Due to the non-
+phonon transition and its phonon replica [1066], radiative recombination in SiGe quantum well is much faster than
+that in silicon. The radiative relaxation time in SiGe quant um well is about tens of nanoseconds [279] compared to
+hundredsofmicrosecondsinsilicon[282,1067,1068]. More overthepresenceofgermaniumandstressalsoenhances
+the spin-orbit coupling and raises the circular polarizati on of the emitted light from the recombination of the spin
+polarizedelectrons with holes [1069–1071]. These factors improvethe efficiency of the electroluminescence. Using
+thisstructureto detect the electrical spin injectionfrom CoFe into silicon, as shownin Fig. 108, circularpolarizati on
+of3%wasachievedat 5K andremainedalmost constantupto200 K at zeromagneticfield[279].
+Most spin related experiments in silicon use electrical inj ection/detection [149, 150, 280, 1048–1058, 1072].
+Robustelectricspininjectionthroughahot-electronspin valve[149,150,279,280,282,1048–1054]orSchottkytun-
+nelingbarrier contact [281, 1058, 1073, 1074] into silicon has been proposedand realized. The schematic electronic
+band diagram and experimental setup of spin injection and de tection using the hot-electron spin valve are shown in
+Fig. 109: Driven by the voltage VE, unpolarized electrons are injected from nonmagnetic emit ter (Al electrode in
+the original experiment) into ferromagnetic base (FM1) thr ough a tunneling barrier. FM1 base acts as a spin filter
+in which electrons with majority spin pass through the Schot tky barrier into silicon ballistically, but electrons with
+minority spin can not pass as they quickly lose energy in the F M1 base due to the spin selective scattering. Under
+thevoltage Vc1,electronstransportinsidethesiliconandarriveatthese condferromagneticlayer(FM2). FM2serves
+as spin detector and makes the current Ic2depend on the current spin polarization by allowing only ele ctrons with
+majorityspinofFM2passthroughintothesecondcollector. ThealignmentofmagneticmomentumofFM1andFM2
+can be changedby an in-planemagnetic field. The success of sp in injection is demonstratedby spin valve e ffect and
+Hanle effect, shown in Fig. 110. Spin valve e ffect, that is the change in the spin momentum alignments of FM1 and
+FM2 results in the change in Ic2, is quantitatively described by magnetocurrent, ( IP
+c2−IAP
+c2)/IAP
+c2, the relative change
+in the current when the alignment of magnetizations in FM1 an d FM2 changes. 2% magnetocurrent was observed
+in the original experiment [150]. By relocating the ferroma gnetic bases away from silicon interfaces to eliminate
+the“magnetically-dead”silicide layer[1075–1077], them agnitudeof magnetocurrentwassubstantiallyincreasedby
+more than one order of magnitude [280, 1050]. To further confi rm that the change in Ic2is indeed caused by spin
+valve effect instead of other spurious e ffects, non-local spin detection was also used [281]. More con clusive results
+showing the spin precession and spin dephasing were obtaine d: A perpendicular magnetic field, which rotates the
+143injected spin, changes the relative alignment of the inject ed spin and spin in FM2. As a result Ic2oscillates with the
+strengthoftheperpendicularmagneticfield [149,150,280– 282,1048–1054].
+Figure 109: Schematic electronic band diagram (a) and exper -
+imental setup (b) of the hot-electron spin valve. From Appel -
+baum etal. [150].
+Figure 110: (a) In-plane spin-valve e ffect for the silicon spin
+transport device with emitter tunnel junction bias VE=−1.6V
+andVC1=0Vat85K.(b) Spin precession and dephasing (Hanle
+effect), measured by applying a perpendicular magnetic field.
+FromHuang etal. [1050].
+Due to the spin relaxation and the interference of the spin pr ecession around the magnetic field [25, 336, 846,
+1052], the injected spin polarization decays with the dista nce. The spin injection length decreases with the increase
+of the magnetic field, and was shown to become very small when t he magnetic field becomes slightly strong [150].
+SinceIc2is proportionalto the spin polarization at FM2, I c2also decreases with the magnetic field. This decay, first
+predictedbyWengandWu [336]fullymicroscopically,wasex plainedbytheHanle e ffectinRef.[150].
+7. Microscopictheoryofspin transport
+InSection6,thephenomenaltheoryofspintransportbasedo nthedrift-diffusionmodelisreviewed. Inthatmodel,
+all characteristic parameters such as the mobility µ, diffusion coefficientDin spin transport, and spin relaxation
+timeτscan not be determined within the framework of model, but rath er need to be experimentally measured or
+calculated from other theoretical model. It is usually assu med thatµandDare simply the charge mobility and
+diffusion coefficient. However, there is no reason that these assumptions sh ould be universally true aside from the
+na¨ ıve intuition, and the relations between the characteri stic parametersof chargetransport and spin transport rema in
+yet to be revealed. In the presence of the spin-orbit couplin g, the charge transport and spin transport are coupled. A
+fullymicroscopictheoryisessentialtounderstandthisco upledtransport. Inthissection,wewillreviewspintransp ort
+insidethe semiconductorfromthe fullymicroscopicpointo fview. Themainresultsreviewedin thissection arealso
+based on the kinetic spin Bloch equation approach. The equat ions have been laid out in Sec. 5, but in Sec. 5 we
+focusesonreviewingtheresultsofthespacialuniformsyst em. Theresultsreviewedinthissectionarefocusedonthe
+spacial nonuniformsystem. There are several related resea rches based on the linear response theory [35, 1078] and
+kinetictheory93[1079–1086],whoseresultswill also bebrieflyaddressed.
+93Equations of this theory are the sameas thekinetic spin Bloc h equations butwithout electron-electron Coulomb interac tion.
+1447.1. KineticspinBlochequationswith spacialgradient
+The kinetic spin Bloch equations for the spin kinetics in the presence of spacial inhomogeneity are given by
+Eqs. (155-162). The electric field Ein these equations is determined from the Poisson equation [ Eq. (273)]. As
+pointed out in Sec. 5.3, the kinetic spin Bloch equations inc lude all the factors important to the spin dynamics,
+including the drifting driven by an electric field, di ffusion due to the spacial inhomogeneity, spin precession aro und
+thetotalmagneticfieldaswellasalltherelevantscatterin g. Bysolvingtheseequations,allthemeasurablequantitie s,
+such as mobility, charge di ffusion length, spin di ffusion length, can be obtained self-consistently without an y fitting
+parameter.
+More importantly, from this fully microscopic approach, so me important issues overlooked by the phenomeno-
+logical drift-diffusion model are recovered. Weng and Wu [336] performed the fir st fully microscopic investigation
+on spin diffusion and transport by setting up and solving the kinetic spi n Bloch equations. Unlike the previous spin
+transporttheories[907,908,927–929,1087–1089]whereon lythediagonalelementsofthedensitymatrix ρσσ(r,k,t)
+areincludedinthetheory,theyshowedthatitisofcruciali mportancetoincludetheo ff-diagonalterms ρσ−σ(r,k,t)in
+studyingthe spin di ffusionand transport[25, 336]. Theypredictedspin oscillat ionsin GaAs quantumwell along the
+spin diffusion in the absence of any applied magnetic field [24, 25]. Th ese oscillations were later observed in exper-
+iments [31, 567]. The spin oscillationswithout any applied magnetic field are beyondthe two-currentdrift-di ffusion
+model widely used in the spin transport study [22, 907, 908, 9 27–929, 1087–1089]. Moreover, by introducing the
+off-diagonalterms, Weng and Wu showed that there is an addition al inhomogeneousbroadeningassociated with the
+spacial gradient of the spin polarization [25, 28, 336]. As s hown in Eq. (157), the coe fficient of the spacial gradient
+is proportional to ∇k¯εk(r,t), which is momentum dependent. It does not lead to any signifi cant results for charge
+transportasidefromthedi ffusioninspace. However,whenthereisspinprecessionthism omentum-dependentcoe ffi-
+cient introducesan additional inhomogeneousbroadenings ince the spacial spin precession is momentum dependent
+[25,28,336]. Thisinhomogeneousbroadening,combinedwit hanyspin-conservingscattering,leadstoanirreversible
+spinrelaxationanddephasinginthespintransport.
+In the followingsubsections, we will first presentsome simp lified caseswhere analyticalresults canbe obtained.
+Thenwe reviewthenumericalresultsfromthekineticspinBl ochequations.
+7.2. Longitudinalspin decoherenceinspindi ffusion
+The importance of the o ff-diagonalterms of the density matrix in the study of spin tra nsport can be qualitatively
+understoodby studyingthe spin di ffusionin GaAs quantumwell in a muchsimplified case where ther e is no electric
+field,noscatteringandnoHartree-Fockself-energy. Assum ingthatthetransportisalongthe x-direction,inthesteady
+state,thekineticspinBlochequations[Eqs.(155-162)]ca nbesimplifiedas
+kx
+m∗∂ρ(x,k,t)
+∂x+i/bracketleftbigh(k)·σ
+2,ρ(x,k,t)/bracketrightbig=0. (297)
+In order to avoid the complexity of the spin injection, it is a ssumed that the electron at x=0 is polarized along the
+z-axis,andthe boundaryconditionsarewrittenas
+ρ(0,k)=/parenleftiggf0
+↑(k) 0
+0f0
+↓(k)/parenrightigg
+, (298)
+wherefσ(0,k)=f0
+σ(k)={exp[(εk−µσ)/T]+1}−1istheFermidistributionfunctionwithchemicalpotential µσ. The
+solutiontothisequationinthesteadystate reads[28]
+ρ(x,k)=e−i(m∗/2kx)h(k)·σxρ(0,k)ei(m∗/2kx)h(k)·σx. (299)
+Thisequationdescribesthespacialspinprecessionaround thetotalmagneticfield h(k). Forelectronwithmomentum
+k,theprecessionperiodischaracterizedby
+ω(k)=h(k)m∗/kx. (300)
+h(k)=gµBB+Ω(k)withΩ(k)beingtheD’yakonov-Perel’term. Eq.(299)clearlyshowst heeffectoftheinhomoge-
+neousbroadeningin thespacialspinprecessiononthe spint ransport. Foreachelectronmovingalongthe x-direction
+145with fixed velocity vkx=kx/m∗, its spin precesses with fixed spacial period without any dec ay. However, electrons
+withdifferentmomentumshavedi fferentspacialprecessionperiodsandcanhavedi fferentprecessionaxesiftheDres-
+selhausand/orRashba terms are present. As a result, spins of di fferentelectronseventuallybecomeoutof phase and
+cancel each other as they movingalong the transport directi on. Due to this interference amongelectrons of di fferent
+k,thetotalspinmomentumdecaysalongthedi ffusiondistance. Comparingwiththespinkineticsinthetime domain,
+where the inhomogeneous broadening is determined by inhomo geneity in the precession frequencies and direction
+determined by h(k) [44, 332, 334, 363, 372], in spin transport it is determined by the inhomogeneousin the spacial
+oscillation “frequency” and direction given by ω(k) in Eq. (300) [25, 28, 336]. Even for a uniform magnetic field
+in the Voigt configuration, which does not induce any inhomog eneous broadening in the time domain, the spacial
+precession “frequencies” ω(k) are still different for different electrons. Therefore, a spacial uniform magnetic fiel d
+alonecan provide an inhomogeneousbroadening in the spin transpo rt as first pointed out by Weng and Wu back to
+2002[25,336]. Thise ffectisillustratedinFig.111wheretheelectrondensities NσinaGaAsquantumwellsubjectto
+auniformin-planemagneticfieldare plottedasfunctionsof positionx. Inordertodemonstratethe e ffectofthisnew
+inhomogeneousbroadening,the spin-orbitcouplingis assu med to be zero. One can clearly see the decay in spin po-
+larizationdue to the inhomogeneousbroadeningcaused byth e differencein spacial precessionperiods. The stronger
+the magnetic field is, the quickerthe spin polarizationdeca ys[25, 336]. It is notedthat althoughthe inhomogeneous
+broadeningalonecanreducethe totalspin momentum,it isju st aninterferencee ffect. Thereisnoirreversiblelossof
+spin coherence since there is no scattering (hence no true di ssipation) in system. This can be seen from the fact that
+the incoherently summed spin coherence ρ, also plotted in Fig. 111, does not decay with the position. I t should be
+pointedoutthatthisadditionalspindecoherenceisbeyond theHanleeffectsinceforthelatterthespinsignaldoesnot
+decaywhenthereisnoscatteringaccordingtoEq.(282).
+Figure 111: Electron densities of up spin and down spin (soli d curves) and incoherently summed spin coherence ρ(dashed curve) versus the
+diffusion length xinn-type GaAs quantum well without spin-orbit coupling but wit hB=1 T. Note the scale of the spin coherence is on the right
+sideof the figure. From Wengand Wu[336].
+Whenthescatteringisturnedon,itprovidesachannelwhich combinedwiththeinhomogeneousbroadeningleads
+to irreversible spin dephasing [44, 332, 334, 350, 569, 844] . As a result both spin polarization and spin coherence
+decaywiththedistance[24,25,27,28,336,846]. Similarto thespinevolutioninthetimedomain,thescatteringalso
+has counter effect that suppresses the inhomogeneous broadening in ω(k) given by Eq. (300) [225, 226, 334, 837–
+840]. The spin dephasing /relaxationmechanismin spin transporthas been realized ex perimentallyby Appelbaumet
+al. [149, 150, 280, 1048–1053] in bulk silicon, where there i s no D’yakonov-Perel’ spin-orbit coupling due to the
+centerinversionsymmetry. Forspintransportinbulksilic oninthepresenceofamagneticfield,thedecayofthespin
+polarization should be mainly caused by the above mentioned inhomogeneous broadening if other spin-relaxation
+mechanisms are ignored. Moreover, the inhomogeneousbroad ening in this situation is particularly simple since all
+electrons have same precession axis, the only di fference is the magnitude of the spacial precession frequenci esωk
+givenbyEq.(300). Thereforebulksiliconprovidesanideal platformtostudythisspinrelaxation /decoherencedueto
+the additionalinhomogeneousbroadening. As shownin Fig. 1 10, indeedthe spin injection lengthdecreaseswith the
+magneticfield,consistentwiththetheoreticalprediction [25,28, 32,846].
+AlthoughthefullkineticspinBlochequationswithscatter ingarecomplicated,onecanstillobtainsomeanalytical
+146results undersome approximations[25, 28, 32, 846, 915]. As suming that there are no applied electric field, inelastic
+scattering and the Hartree-Fock term, the kinetic spin Bloc h equations for spin transport along the x-axis can be
+writtenas[25, 28,32, 846]
+∂ρl(x,k,t)
+∂t+vk
+2∂
+∂x[ρl+1(x,k,t)+ρl−1
+k(x,k,t)]=−i/summationdisplay
+m[hl−m(k)·σ/2,ρm(x,k,t)]−ρl(x,k,t)/τl
+k,(301)
+wherevk=k/m∗isthevelocityoftheelectronwithmomentum k.ρl(x,k,t)andhl(k)arethel-thorderoftheFourier
+components of ρ(x,k,t) andh(k) with respect to the angle of kand thex-axis, respectively. τl
+kis thel-th order
+momentumrelaxationtime dueto theelectron-impurityscat tering. Notedthat1 /τ0
+k=0andτl
+k=τ−l
+k.
+In the diffusive limit where the momentum relaxation time is small, the higher order components ρl(x,k,t) with
+|l|>1 are small and can be neglected. If there is no spin-orbit cou pling and the applied magnetic field is along the
+y-axis,byneglectinghigherorderFouriercomponentswith |l|>1,inthesteadystatethekineticspinBlochequations
+canbesimplifiedas[846]
+∂2ρ0(x,k)
+∂x2=−2/parenleftiggω
+2vk/parenrightigg2
+[σy,[σy,ρ0(x,k)]]+iω
+v2
+kτ1
+k[σy,ρ0(x,k)], (302)
+whereω=gµBBisspinprecessionfrequencyintimedomainundertheunifor mmagneticfield B. Intermofthespin
+momentum,the solutiontothisequationforthespin injecti onatx=0isa dampedoscillation
+Sz(x,k)=Sz(0,k)cosx
+vkτ1
+k∆e−xω∆/vk(303)
+with
+∆=/radicaligg
+1+1
+(ωτ1
+k)2−1−1/2
+. (304)
+At low temperature, when the injected spin only polarized at the Fermi level, the spin injection length L−1
+d=
+ω∆/vf, withvfstanding for the Fermi velocity. The injection length obtai ned here decreases monotonely with the
+increase of magnetic field and scattering strength. Therefo re, the scattering enhances the spin dephasing /relaxation
+inspin diffusion. Thecountere ffectof theelectron-impurityscatteringonthe inhomogeneo usbroadeninghereis not
+significant[846].
+7.2.1. SpindiffusioninSi/SiGequantumwells
+Inordertogainadeeperinsightintothespinrelaxationalo ngthespindiffusioncausedbytheadditionalinhomo-
+geneous broadening addressed in the previous subsection, Z hang and Wu studied the spin transport in a symmetric
+siliconquantumwellwithevennumberofmonoatomicsilicon layersthroughthekineticspinBlochequationapproach
+[846]. ThereisnoD’yakonov-Perel’spin-orbitalcoupling inthiskindofquantumwell [291]. Sincethe Rashbaterm
+is verysmall in the asymmetricSi /SiGe quantumwell [124, 290, 291, 295, 776, 781], it was argue dthat their results
+alsoholdfortheasymmetricSi /SiGe quantumwellsaslongastheD’yakonov-Perel’termiswe akenough[846].
+ThekineticspinBlochequationsin(001)Si /SiGequantumwellaresimilartothoseinGaAsquantumwell[8 46].
+Intheequationsthedriftingundertheelectricfield,di ffusionduetothespacialinhomogeneityandprecessionaroun d
+themagneticfield (includingthe contributionoftheHartre e-Fockterm),aswell asthe spin-conservingscatteringare
+all addressed. Numerical solutions of full kinetic spin Blo ch equations for spin transport in silicon quantum well at
+T=80K withdifferentscatteringmechanismsandunderdi fferentmagneticfieldsareshowninFigs. 112and113.
+Figure112clearlyshowsthatthespininjectionlengthdecr easeswithincreasingmagneticfield,inaccordancewith
+thepredictionby Weng andWu [25, 336]. Thisresultis easy to understandsinceωk=m∗gµBB/kxis proportionalto
+themagneticfield. Increaseoftheappliedmagneticfieldres ultsintheincreaseofinhomogeneousbroadening.
+Figure113showsthee ffectofscatteringonthespindi ffusion. Itisseenfromthefigurethataddinganynewscatter-
+ingleadstoashorterspindi ffusionlength. Thisisquitedi fferenttotheresultsinthesystemwiththeD’yakonov-Perel’
+spin-orbit coupling. When the D’yakonov-Perel’spin-orbi t coupling is present, the scattering not only opens a spin
+147dephasing/relaxationchannel,butalsohasacountere ffectontheinhomogeneousbroadening[25,28,29,363,1090].
+In the strong scattering limit, adding a new scattering supp resses the inhomogeneous broadening and prolongs the
+spindiffusionlength[25,28]. However,forthesystemwithouttheD’ yakonov-Perel’spin-orbitcoupling,thecounter
+effect of the scattering on the inhomogeneousbroadening is mar ginal. The scattering a ffects the spin diffusion only
+throughthe momentumrelaxationtime and hence only reduces the spin injection length as shown by Eqs. (303) and
+(304)[846].
+The electron density dependence of the spin di ffusion was also investigated at di fferent temperatures. It was
+found that at relatively high temperature or low density, wh en the electrons are nondegenerate, the spin di ffusion
+length and spacial precession period are insensitive to the electron density. At low temperate or high density when
+the electronsare degenerate,the spin di ffusionlength and spacial precessionperiodincrease with in creasing density.
+This is understood from the Fermi wavevector dependence of t he damping rate and precession period as shown in
+Eqs.(303)and(304).
+-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
+ 0  0.5  1  1.5  2Sz (1011 cm-2)
+x (µm)T=80 K, Ni=0.1N0B=2.0 T
+1.0 T
+0.5 T
+Figure 112: Spin polarization Szvs. position xin the steady
+state forintrinsic silicon quantum wellunder di fferent magnetic
+field strengths. Solid curve: B=2 T; Dashed curve: B=1 T;
+Dotted curve: B=0.5 T.T=80 K and Ni=0.1N0. From
+Zhang and Wu [846]. 0 0.04 0.08 0.12 0.16 0.2
+ 0  1  2  3  4  510-710-610-510-410-310-210-1100Sz (1011 cm-2)
+|Sz| (1011 cm-2)
+x (µm)T=80 K, B=1 T EE
+EP
+EI
+EI+EE+EP
+Figure113: Thesteady-state spacial distributions ofspin signal
+Szin silicon quantum well with di fferent scatterings included.
+Thecurves labeled with “EE”,“EP”or“EI”stand forthecalcu -
+lations with the electron-electron, electron-phonon or el ectron-
+impurity scattering, respectively, while the curve labele d with
+“EI+EE+EP”stands for the calculation with all the scatterings.
+In order to get a clear view of the decay and precession of Sz,
+the corresponding absolute value of Szagainstxis also plotted
+on alog-scale (Notethe scale ison the right hand of theframe ).
+The dashed curves correspond to the part with Sz<0. From
+Zhang and Wu[846].
+7.3. Spinoscillationsintheabsenceof magneticfieldalong thediffusion
+When the D’yakonov-Perel’ term is present, the problem beco mes more complicated since in this case both the
+spacial precession frequencies and axis are momentum depen dent. Studies of spin transport in the system with the
+spin-orbitcouplingfrom the kineticspin Bloch equationap proachhave predictedmanynovelresults[25, 27, 28, 32,
+336, 846]. Someofthemhavebeenverifiedexperimentally[31 , 567].
+One interesting result is the spin oscillation in the absenc e of any applied magnetic field along the spin di ffusion
+when the Dresselhaus and /orRashba terms are present. In the spacial homogeneoussyst em, spin oscillation without
+any applied magnetic field can only be observed in time domain at very low temperature ( T<2 K), i.e., in the
+weak scattering limit [368]. In high temperature regime whe re scattering is strong enough, the Dresselhaus and /or
+Rashba terms result in a monotonicallydecay of spin polariz ation versus time in the spacial uniform system. If one
+adopts simple relaxation approximation to describe the e ffect of the spin-orbit coupling, one should not expect any
+spin oscillation in the di ffusive limit where the scattering is strong enough. Indeed, f rom the drift-diffusionequation
+Eq. (275), there is no spin oscillation either in spin inject ion in the steady state or in transient spin transport if the
+appliedmagneticfieldisabsent.
+However according to the kinetic spin Bloch equation approa ch, in the spin transport the spacial spin precession
+is characterizedby ωk[Eq. (300)]. If one only considersthe Dresselhaus e ffective magnetic field, the average of ωk
+is∝an}b∇acketle{tωk∝an}b∇acket∇i}ht=m∗γ(∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht−∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}ht,0,0). For electrons in quantum well, this value is not zero. The refore, the spacial spin
+148Figure114: Theoretical predictions oftheabsolute valueo fthespinimbalance |∆N|vs.theposition xandthetime tforaspinpulsein n-typeGaAs
+(001)quantumwellbasedonthekinetic spinBlochequation a pproach (left)anddrift-di ffusion model(right). Theinitialpulsewidth δx=0.15µm.
+FromWeng and Wu[25].
+oscillation due to the Dresselhaus e ffective magnetic field survives even at high temperature when the scattering is
+strongenough.
+By solving the kinetic spin Bloch equations, the spin oscill ation without any applied magnetic field was first
+predicted in the transient di ffusion of a spin pulse in GaAs quantum wells [24, 25, 27]. A typi cal time evolution of
+spin pulse is shown in Fig. 114 where the absolute value of the spin imbalance|∆Nσ(x,t)|is plotted as a function of
+the position xalong the diffusion direction and time t. The initial state is a spin pulse of Gaussian spacial profile ,
+∆N(x,0)=∆N0exp(−x2/δx2) withδx=0.15µm. For comparison, the result based on the drift-di ffusion model is
+alsoshown. OnecanseethatthekineticspinBlochequationa pproachandthedrift-di ffusionmodelgivequalitatively
+similar results for the evolution at the center of the pulse, where it decays monotonically with time as a result of
+spin diffusion and spin dephasing. However, the results from these tw o theories are quite di fferent for the evolution
+outside the original spin pulse. The result from the drift-d iffusion model shows that spin polarization at large xfirst
+increases with time and then decays to zero. The sign of the sp in polarization does not change throughoutthe space
+and time. On the contrary,the kinetic spin Bloch equation th eorypredicts that spin polarizationoscillates with time.
+The oscillation is the combined result of the spin precessio n caused by the Dresselhaus e ffective magnetic field and
+thespindiffusion.
+Figure 115: The contour plots of spin imbalance ∆Nvs.the position xand the time tfor different pulse widths in n-type GaAs (001) quantum
+well. (a):δx=0.1µm;(b):δx=1.0µm.T=200 K.From Jiang etal. [27].
+The evolutionsof spin pulse with di fferentpulse widths are shown in Fig. 115. Since the oscillati on is a result of
+diffusionandspinprecessionaroundthee ffectivemagneticfield,thesystemwithnarrowerspinpulsesh owsstronger
+spin oscillation in the transient spin transport. For the na rrower pulse, the peak of the reversed spin polarization
+appearsinshortertime. Moreover,moreoscillationcanbeo bservedwith narrowerpulse[27].
+The effects of the scattering and the electric field on the oscillati on without any magneticfield were also studied.
+149It was shown that the oscillation is robustagainst the scatt ering. Addingthe electron-impurityscattering slows down
+the diffusion but does not eliminate the oscillation. In some regime , the electron-impurityscattering even boosts the
+oscillation as it helps to sustain the spin coherence[24, 27 ]. The Coulomb scattering has a similar e ffect on the spin
+diffusion and oscillation [27]. Without any electric field, the s pin signals diffuse symmetrically in the space around
+the original pulse center. When an electric field is applied a long the diffusion direction, the di ffusion is no longer
+symmetrical. Thepulseisdraggedagainsttheelectricfield andstrongeroppositespinpolarizationappearsattheside
+againsttheelectricfield [27].
+Figure116: Experimental resultofspinprecession inbulkG aAswithoutmagnetic field. (a)-(c)images oftwo-dimension al spinflow(E=10V/cm)
+at 4 K, showing induced spin precession with increasing [110 ] uniaxial stress. (d) Kerr rotation vs.probe photon energy for the images, showing
+blueshift of GaAs band edge with stress. (e) Line cuts throug h the images. Inset: Spacial frequency of spin precession vs.band edge shift. From
+Crooker and Smith [31].
+The spin oscillation without any applied magnetic field in th e transient spin transport was later observed exper-
+imentally by Crooker and Smith in strained bulk system [31], which is shown in Fig. 116. Unlike two-dimensional
+case, in bulk the average of ωkfrom the Dresselhaus term is zero, since ∝an}b∇acketle{tωk∝an}b∇acket∇i}ht=m∗γ(∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht−∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}ht,0,0)=0 due to the
+symmetryinthe y-andz-directions. Thisisconsistentwiththeexperimentalresu ltthatthereisnospinoscillationfor
+thesystemwithoutstress. However,whenthestressisappli ed,anadditionalspin-orbitcoupling,namelythecoupling
+of electron spins to the strain tensor, appears [3, 31, 130, 2 46, 570, 1091, 1092]. This additional spin-orbit coupling
+also acts as an effective magneticfield. In the experimentsetup, the stress ap pliedis along the [110]axis, the spacial
+oscillationcausedbythisadditionale ffectivemagneticfieldischaracterizedby
+∝an}b∇acketle{tωk∝an}b∇acket∇i}ht=−c3ǫyx(0,1,0), (305)
+wherec3isaconstantdependsontheinterbanddeformationpotentia landǫyxisin-planeshear[3,31,130,246,1091,
+1092]. Therefore,oncethestressisapplied,onecanobserv espacialspinoscillationasshownin Fig.116,evenwhen
+thereisnoappliedmagneticfield.
+Thespinoscillationwithoutanyappliedmagneticfieldwasl atershowntheoreticallytosurviveeveninthesteady
+state spininjection[28,35, 949–954].
+The oscillation without an applied magnetic field can also be understood from the simplified kinetic spin Bloch
+equationsEq. (301). In the di ffusive limit, by neglectinghigh order of the momentumrelaxa tiontime, Eq. (301) can
+befurthersimplifiedas
+∂S(x,t)
+∂t=D∂2S(x,t)
+∂x2−¯h×∂S(x,t)
+∂x−← →ΓS(x,t) (306)
+in (001) GaAs quantum well where the dominant spin-orbit cou pling is the Dresselhaus term. Here S(x,k,t)=/summationtext
+kTr{σρ0(x,k,t)}isthespinmomentum. D=∝an}b∇acketle{tk2τ1
+k/2m∗2∝an}b∇acket∇i}htis thediffusioncoefficient.
+¯h=∝an}b∇acketle{tk2τ1
+k/m∗∝an}b∇acket∇i}ht(α,0,0), (307)
+150is the net Dresselhaus e ffective magnetic field in the spin di ffusion. The last term in Eq. (306) is the spin relaxation
+causedbytheDresselhaustermtogetherwiththespin conser vingscattering. Therelaxationmatrixreads
+← →Γ =1
+τ∝ba∇dbl0 0
+01
+τ∝ba∇dbl0
+0 01
+τ⊥, (308)
+where1/τ∝ba∇dbl=α2∝an}b∇acketle{tk2τ1
+k∝an}b∇acket∇i}ht/2+∝an}b∇acketle{t(γk3)2τ3
+k∝an}b∇acket∇i}ht/32and 1/τ⊥=2/τ∝ba∇dblare the spin relaxationtimes of in-planeand out-of-plane
+componentsrespectively. Similar equationshavealso been obtainedfromlinearresponsetheory[35, 1078]. Onecan
+clearlyseethespinprecessionaroundthenete ffectivemagneticfield ¯hfromEq.(306). Inthetransientspintransport,
+the precession around the e ffective magnetic field results in spin oscillations in both ti me and space domains even
+when there is no applied magnetic field [25, 27, 28, 32, 336]. M oreover, due to the precession, the steady-state spin
+injectionisnolongerasimple exponentialdecaybuta dampe doscillation[28,32, 35,1078].
+7.4. Steady-statespintransportinGaAsquantumwell
+Due to spin oscillation, in the steady state the polarizatio n of injected spin in GaAs quantum well is a damped
+oscillation∆N(x)∝exp(−x/Ld)cos(x/L0+φ). HereLdandL0standforthediffusionlengthandthespacialoscillation
+period,respectively. Howthesetwoparameterschangewith thescatteringmechanism,temperature,appliedmagnetic
+orelectricfieldareimportanttotherealizationofspintro nicdevices.
+7.4.1. Effectof thescatteringonthe spindi ffusion
+Similar to the spin kinetics in the spacial uniform system, t he scattering also a ffects the spin transport in com-
+plicated ways. On one hand, each scattering mechanism provi des additional channel of spin relaxation /dephasing
+in the presence of inhomogeneous broadening. Adding new sca ttering mechanism tends to increase spin relax-
+ation/dephasing.However,scatteringalsoa ffectsthechargeandspintransportparameterssuchasdi ffusioncoefficient
+and mobility. Electron-impurityand electron-phononscat tering can directly change the electron momentum and re-
+duce the spin and charge di ffusion coefficients and mobilities. For electron-electronCoulomb scat tering, although it
+doesnotdirectlychangethechargedi ffusioncoefficientandmobility,it reducesthespindi ffusioncoefficientthrough
+theinhomogeneousbroadening ωkandtheCoulombdrage ffect[13–15,27,1093–1101]. Allofthesee ffectsresultin
+shorterspininjectionlength Ld. Ontheotherhand,thescatteringtendstosuppresstheinho mogeneousbroadeningin
+timedomainaswellasinspacedomain. Thishelpstoprolongt hediffusionlength Ld. Thesecompetinge ffectsonspin
+diffusionandspintransportwereinvestigatedusingthekineti cspinBlochequationapproach[24,25,27,28,32,336].
+Themainresultsarereviewedinthefollowing.
+1.901.941.982.022.062.10
+ 0  2  4  6  8  1010-810-710-610-510-410-310-210-1Nσ (1011 cm—2)
+ρ (a.u.)
+x (µm)(a)
+T=120 K
+EP: ↑
+↓
+EP+EE: ↑
+↓
+EP+EE+EI: ↑
+↓
+1.901.941.982.022.062.10
+ 0  2  4  6  8  1010-810-710-610-510-410-310-210-1Nσ (1011 cm—2)
+ρ (a.u.)
+x (µm)(b)
+T=300 K
+EP: ↑
+↓
+EP+EE: ↑
+↓
+EP+EE+EI: ↑
+↓
+Figure117: Effectofthescattering onspindi ffusioninthesteadystateat(a) T=120Kand(b) T=300KinGaAsquantumwellwith a=7.5nm.
+Red curves: with only the electron–longitudinal-optical- phonon (EP) scattering; Green curves: with both the electro n-electron (EE)and electron–
+longitudinal-optical-phonon scattering; Blue curves: wi th all the scattering, i.e., the electron-electron, electr on–longitudinal-optical-phonon and
+electron-impurity (EI) scattering. The impurity density Ni=Ne. Note the scale of the incoherently summed spin coherence is on the right hand
+sideof the figure. From Cheng and Wu[28].
+1511.901.941.982.022.062.10
+ 0  2  4  6  8  1010-810-710-610-510-410-310-210-1Nσ (1011 cm—2)
+ρ (a.u.)
+x (µm)a=7.5 nm
+T=120 K: ↑
+↓
+T=200 K: ↑
+↓
+T=300 K: ↑
+↓
+Figure 118: Nσandρvs.the position xatT=120, 200
+and 300 K in n-type GaAs (001) quantum well with width
+a=7.5 nm and L=10µm.Ni=0. Note the scale of the
+incoherently summed spin coherence is on the right hand side
+of thefigure. From Cheng and Wu[28]. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
+-4 -2  0  2  4  6 0.4 0.5 0.6 0.7 0.8 0.9 1L0—1 (µm—1)
+Ld (µm)
+B (T)B//xB//y
+Figure 119: Inverse of the period of the spin oscillation L−1
+0
+(curves with•)and thespin diffusion length Ld(curves with /squaresolid)
+forn-type GaAs (001) quantum well vs.the external magnetic
+fieldBwhich is applied either vertical ( ∝ba∇dbly, dashed curves) or
+parallel (∝ba∇dblx, solid curves) to the di ffusion direction. T=120
+K andNi=0. The dashed curves are for the vertical magnetic
+field and the solid curves arefor theparallel one. Note thesc ale
+of the diffusion length Ldis on the right hand side of the figure.
+FromCheng and Wu[28].
+In Fig. 117, the spin-resolved electron density Nσand the incoherently summed spin coherence ρin the steady
+stateareplottedagainsttheposition xbyfirst includingonlytheelectron–longitudinal-optical -phononscattering(red
+curves),thenaddingtheelectron-electronscattering(gr eencurves)andfinallyaddingtheelectron-impurityscatte ring
+(blue curves) with Ni=NeatT=120 K (a) and 300 K (b). In the calculation there is no applied m agnetic field.
+One can see that the scattering a ffects the transport in a complex way: At T=120 K, the spin injection length Ld
+alwaysdecreaseswhenanewscatteringmechanismisadded: LdissignificantlyreducedwhentheCoulombscattering
+is added; adding electron-impurity scattering further red ucesLd, but only slightly. However, when T=300 K,Ld
+becomes slightly shorter when the Coulomb scattering is firs t added, but becomes slightly longer when electron-
+impurity scattering is further added. These results show th at for GaAs quantum well with width a=7.5 nm, all
+scattering mechanisms are not strong enough at T=120 K and the system falls into weak scattering limit. The
+countereffectofthescatteringto theinhomogeneousbroadeningiswea k. Thereforeaddinga newscatteringreduces
+theinjectionlength. At T=300K,thescatteringbecomesstronger. Asaresultthecompe tingeffectsofthescatterings
+canceleachotherandresultin marginalchangeindi ffusionlengthwhena newscatteringisadded.
+In Fig. 118, the results of steady-state spin injection at di fferent temperatures are shown. It is seen that the spin
+injectionlength Ldslightlydecreasesasthetemperaturerises. Thechangein Ldis milddueto thecancellationofthe
+oppositeeffects in scattering. The spacial period L0on the otherhand systematically increaseswith the tempera ture.
+This is because when there is only the Dresselhaus term, L−1
+0≃|∝an}b∇acketle{tωk∝an}b∇acket∇i}ht|=|m∗γ(∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht−∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}ht)|. In small quantum well,
+∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}htislargerthan∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht. Asthetemperatureincreases, ∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}htbecomeslargerandconsequently L0becomeslarger.
+7.4.2. Spintransportinthepresenceof magneticandelectr ic fields
+The magnetic field e ffect on the steady-state spin di ffusion is shown in Fig. 119 where the spin di ffusion length
+Ldand the spin oscillation period L0as functionsof the applied magnetic field at T=120 K are plotted. The direc-
+tion of the magnetic field is either vertical (along the y-axis) or parallel (along the x-axis) to the diffusion direction.
+As the magnetic field increases the inhomogeneous broadenin g in spin diffusion, it leads to additional spin relax-
+ation/decoherenceandthereforethespindi ffusionlengthdecreaseswiththemagneticfield,regardlesso fthedirection
+of the magnetic field. However, it is interesting to see that t he spin diffusion length has di fferent symmetries when
+the direction of the magnetic field changes: When Bis parallel to the y-axis,Ld(B)=Ld(−B); However, when Bis
+paralleltothe x-axis,Ld(B)/nequalLd(−B). Thiscanbeunderstoodfromthefactthatthedensitymatri cesfromthekinetic
+spin Bloch equations have the symmetry ρkx,ky,kz(B)=ρkx,−ky,kz(−B) whenBis along the y-axis. This symmetry is
+brokenif Bisalongthe x-axis.
+Differing from the magnetic field dependence of the spin di ffusion length, the spin oscillation period L0de-
+creases monotonically with the vertical magnetic field, reg ardless the sign of the field. However L0increases with
+1521.901.952.002.052.10
+ 0  2  4  6  8  10 0 1 2 3 4 5 6-1 -0.5  0  0.5  1Nσ (1011 cm—2)
+Ld (µm)
+x (µm)E (kV/cm)
+E=0.5 kV/cm: ↑
+↓
+0: ↑
+↓
+-0.5: ↑
+↓
+-1: ↑
+↓
+Figure 120: Spin-resolved electron density Nσvs.positionx
+at different electric field E=0.5, 0,−0.5 and−1 kV/cm and
+the spin diffusion length Ldforn-type GaAs (001) quantum
+well against the electric field EatT=120 K.Ni=0. It is
+noted that although xis plotted up to 10 µm,L=25µmwhen
+E=−1.0 and−0.75 kV/cm;20µmwhenE=−0.5and−0.25
+kV/cm;10µmwhenE=0;and 5µmwhenE=0.5,0.75 and
+1.0 kV/cm. Note the scales of the spin di ffusion length are on
+the top and the right hand side of the figure. From Cheng and
+Wu[28].
+Figure 121: Spin transport at F=40 V/cm andND=1.4×
+1016cm−3instrained bulkGaAs. (a)measuredFaradayrotation
+for drift along [1 10] and strain along [110]; (b) calculated spin
+polarization with the parameters of (a) (using C3=5.2 eVÅ);
+(c) Spacial oscillation “frequency” Λ−1
+pversus strain ǫ[110], lin-
+ear fit corresponding to C3=8.1 eVÅ, and expected result for
+C3=5.2 eVÅ (dotted line); (d) measured Faraday rotation for
+drift and strain along [100]. FromBeck et al. [567].
+the parallel magnetic field when the field is along the di ffusion direction and less than 4 T (the spin precession dis-
+appears when the field is larger than 4 T) but L0increases with the magnetic field when it is anti-parallel to the
+diffusion direction. In the presence of the Dresselhaus term and the applied magnetic field B, the spacial period
+is determined by m∗γ(∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht−∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}ht,0,0)+m∗gµBB∝an}b∇acketle{t1/kx∝an}b∇acket∇i}ht.∝an}b∇acketle{t1/kx∝an}b∇acket∇i}htrepresents the average of 1 /kx. Due to the spin
+transport it does not vanish and can be roughly estimated by ∝an}b∇acketle{t1/kx∝an}b∇acket∇i}ht−1≈ ∝an}b∇acketle{tkx∝an}b∇acket∇i}ht=/summationtext
+kkx∆fk//summationtext
+k∆fk, which is a
+positive value for spin transport along x-direction. For the vertical magnetic field, the average of t he Dresselhaus
+effective magnetic field and the applied one are perpendicular t o each other, so the magnitude of the spin oscilla-
+tion period is determined by/bracketleftbig(m∗gµBB)2/∝an}b∇acketle{tkx∝an}b∇acket∇i}ht2+m∗2γ2(∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht−∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}ht)2/bracketrightbig−1/2which always decreases with the magnetic
+field. However, for the parallel field, these two vectors are i n the same direction and the period is determined by
+|m∗γ(∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht−∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}ht)+m∗gµBB/∝an}b∇acketle{tkx∝an}b∇acket∇i}ht|−1. These two vectors can enhance or cancel each other dependin gon their relative
+strengthand sign. Forthe particularcase in Fig. 119, the th e inverseof period L−1
+0decreaseswith the magneticfield,
+untilB∼4T when|γ(∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht−∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}ht)+gµBB/∝an}b∇acketle{tkx∝an}b∇acket∇i}ht|∼0. Furtherincreaseofthemagneticfield doesnotleadto clea rspin
+oscillationsince theinhomogeneousbroadeningcausedbyt hemagneticfield becomestoolarge.
+The effect of electric field on the spin transport is shown in Fig. 120 where the spin densities in the steady state
+are plotted against the position xat different electric fields E=0.5, 0,−0.5 and−1 kV/cm. In the calculations, the
+temperature is 120 K and only the intrinsic scattering mecha nisms, i.e. the electron-phonon and electron-electron
+Coulomb scatterings, are included. It is seen from the figure that the spin oscillation period almost does not change
+when the electric field variesfrom −1 kV/cm to 0.5 kV/cm. The null effect of the electric field on spacial precession
+is quite different from the effect of electric field on the precession in time domain [569, 57 0]. In the spacial uniform
+system,dueto thedriftingoftheelectronsdrivenbytheele ctric field,theDresselhausorRashbatermgivesane ffec-
+tivemagneticfieldproportionaltotheelectricfield. Thise ffectivemagneticfieldinturngivesaprecessionfrequency
+proportionalto theelectric field [569, 570]. However,in th espin transport,theaverageoftheinhomogeneousbroad-
+ening∝an}b∇acketle{tωk∝an}b∇acket∇i}ht=m∗γ(∝an}b∇acketle{tk2
+y∝an}b∇acket∇i}ht−∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}ht,0,0) does not depend on the drifting velocity, therefore is not affected by the electric
+field. Thisisconsistentwiththepreviousexperimentalobs ervationinthestrainedbulksystem[567],whichisshown
+inFig.121. Intheexperiment,theoscillationisinducedby theadditionalspin-orbitcouplingcausedbythestrain. As
+showninEq.(305),thespacialoscillationfrequency[deno tedasΓ−1
+pinFig.121(c)]is C3ǫyx,whichisproportionalto
+thestrainbutdoesnotdependonthe driftingvelocityorele ctricfield.
+It is further noted from the figure that the spin di ffusion length is markedly a ffected by the electric field. To
+reveal this effect, the spin diffusion length is plotted as function of electric field in the sa me figure. One finds when
+the electric field varies from −1 kV/cm to+1 kV/cm, the spin diffusion length decreases monotonically, which is
+153-
+injection direction(100)(010)
+(110)(110)(001)
+θ
+xz
+Figure 122: Schematic of the di fferent directions considered for the spin polarizations ([1 10], [¯110] and [001]-axes and spin di ffusion/injection
+(x-axis). From Cheng etal. [29].
+consistentwiththepredictionofthedrift-di ffusionmodel[928,929]. Frominhomogeneousbroadeningpoin tofview,
+thiscanbeeasilyunderstoodfromthefactthatwhenanelect ricfieldisappliedalong −x-direction(+x-direction),the
+driftvelocitycausedbytheelectricfieldisalongthe x-direction(−x-direction),whichenhances(cancels)thevelocity
+driven by the spin gradient. Therefore, ∝an}b∇acketle{tkx∝an}b∇acket∇i}htis increased (reduced) and the inhomogeneousbroadening is decreased
+(enhanced). A longer(shorter)spindi ffusionlengthisthenobserved[28].
+7.5. Anisotropyofspin transportinthe presenceof competi ngRashbaandDresselhausfields
+When the Dresselhaus and Rashba terms are both important in s emiconductor quantum well, the total e ffective
+magnetic field can be highly anisotropic and spin kinetics is also highly anisotropicin regardsto the direction of the
+spin [194]. For some special polarization direction,spin r elaxation time is extremelylarge [194, 196, 203, 339, 919,
+920]. For example, if the coe fficients of the linear Dresselhaus and Rashba terms are equal t o each other in (001)
+quantum well of small well width and the cubic Dresselhaus te rm is not important, the e ffective magnetic field is
+alongthe[110]directionforall electrons. Forthespincom ponentsperpendiculartothe[110]direction,thise ffective
+magneticfieldflipsthespinandleadstoafinitespindephasin gtime. Forspinalongthe[110]direction,thise ffective
+magneticfield can not flip it. Therefore,when the spin polari zationis alongthe [110]direction, the Dresselhausand
+Rashba terms can not cause any spin dephasing. When the cubic Dresselhaus term is taken into account, the spin
+dephasingtimeforspinpolarizationalongthe[110]direct ionisfinitebutstillmuchlargerthanotherdirections[597 ].
+The anisotropy in the spin direction is also expected in spin transport. When the Dresselhaus and Rashba terms
+arecomparable,thespininjectionlength Ldforthespinpolarizationperpendicularto[110]direction isusuallymuch
+shorter than that for the spin polarization along [110] dire ction. In the ideal case when there are only the linear
+DresselhausandRashbatermswithidenticalstrengths,spi ninjectionlengthforspinpolarizationparalleltothe[11 0]
+direction becomes infinity [196, 339, 920]. This e ffect has promoted Schliemann et al. to propose the nonballist ic
+spin-field-effect transistor [339]. In such a transistor, a gate voltage is used to tune the strength of the Rashba term
+andthereforecontrolthespininjectionlength.
+However, spin transport actually involves both the spin pol arization and spin transport directions. The latter has
+longbeenoverlookedintheliteratureofspintransport. In thekineticspinBlochequationapproach,thisdirectionco r-
+respondstothe spacialgradientinthedi ffusionterm[Eq.(157)]andtheelectricfieldinthedriftingt erm[Eq.(156)].
+Theimportanceofthespin transportdirectionhasnot beenr ealizeduntilChenget al. pointedoutthat thespin trans-
+portishighlyanisotropicnotonlyinthesenseofthespinpo larizationdirectionbutalsointhespintransportdirecti on
+when the Dresselhaus and Rashba e ffective magnetic fields are comparable [29]. They even predic ted that in (001)
+GaAsquantumwellwithidenticallinearDresselhausandRas hbacouplingstrengths,thespininjectionalong[ ¯110]or
+[110]94can be infinite regardlessof the directionof the spin polarization[29]. Inthe followin gwe reviewthe results
+onthe anisotropicspintransportfromthe kineticspin Bloc hequationapproach.
+The schematic of spin transport in (001) GaAs quantum well is shown in Fig. 122. The transport direction is
+chosentobealongthe x-axis,theanglebetween x-axisand[100]crystaldirectionis θ. Inthiscoordinatesystem, the
+94[¯110] or [110] depends on the relative signs of the Dresselhau s and Rashba coupling strengths.
+154Dresselhauseffectivemagneticfieldis
+hD(k)=β/parenleftbig−kxcos2θ+kysin2θ,kxsin2θ+kycos2θ,0/parenrightbig+γ(k2
+x−k2
+y
+2sin2θ+kxkycos2θ)/parenleftbigky,−kx,0/parenrightbig,(309)
+whiletheRashbafield reads
+hR(k)=α/parenleftbigky,−kx,0/parenrightbig, (310)
+withβ=γ∝an}b∇acketle{tk2
+z∝an}b∇acket∇i}ht. Forthespecial casewith α=β, thetotalmagneticfield canbewrittenas
+h(k)=2β/bracketleftbigg
+sin(θ−π
+4)kx+cos(θ−π
+4)ky/bracketrightbigg
+ˆn0+γk2
+x−k2
+y
+2sin2θ+kxkycos2θ/parenleftig
+ky,−kx,0/parenrightig
+,(311)
+with the special direction ˆn0=/parenleftbigcos(π/4−θ),sin(π/4−θ),0/parenrightbigrepresenting the crystal direction [110]. As given by
+Eq. (300), the spin precession around the e ffective field magnetic field is characterized by ωk=m∗h(k)/kxfor spin
+transportalongthe x-axis.
+7.5.1. SpininjectionundertheidenticalDresselhausandR ashbastrengths α=β
+Whenα=β,
+ωk=m∗/braceleftbigg
+2β/parenleftigg
+sin(θ−π
+4)+cos(θ−π
+4)ky
+kx/parenrightigg
+ˆn0+γ(k2
+x−k2
+y
+2sin2θ+kxkycos2θ)/parenleftbigky/kx,−1,0/parenrightbig/bracerightbigg
+.(312)
+It can be splitted into two parts: the zeroth-order term (on k) which is always along the same direction of ˆn0and
+the second-order term which comes from the cubic Dresselhau s term. If the cubic Dresselhaus term is omitted, the
+effective magnetic fields for all kstates align along ˆn0(crystal [110])direction. Therefore,if the spin polariza tion is
+alongˆn0,thereisnospinrelaxationeveninthepresenceofscatteri ngsincethereisnospinprecession. Nevertheless,
+it is interesting to see from Eq. (312) that when θ=3π/4, i.e., the spin transport is along the [ ¯110] direction,ωk=
+2m∗βˆn0is independent on kif the cubic Dresselhaus term is neglected. Therefore, in th is special spin transport
+direction, there is no inhomogeneous broadening in the spin transport for anyspin polarization. The spin injection
+length is therefore infinite regardless of the direction of s pin polarization. This result is highly counterintuitive,
+considering that the spin relaxation times for the spin comp onents perpendicular to the e ffective magnetic field are
+finite in the spacial uniform system. The surprisingly contr adictory results, i.e., the finite spin relaxation /dephasing
+time versus the infinite spin injection length, are due to the difference in the inhomogeneous broadening in spacial
+uniform and non-uniformsystems. In spacial uniform system , the inhomogeneousbroadening is determined by the
+D’yakonov-Perel’term,i.e., h(k)[Eq.(311)],whichismomentumdependent. Howeverinthesp intransportproblem
+the inhomogeneousbroadeningis determined by ωk, which is momentum-independentwhen the transport directi on
+is along [ ¯110] here. This prediction has not yet been realized experim entally. However, very recent experimental
+findingsonspinhelix[18, 20,37, 921] haveprovidestrongev idencetosupportthisprediction.
+7.5.2. Spin-diffusion/injection–directionandspin-polarization–directionde pendenceat α=β
+When the cubic Dresselhaus term is taken into account, ωkbecomes momentum dependent again for all spin
+transport directions. This inhomogeneous broadening resu lts in a finite but still highly anisotropic di ffusion length
+forbothspinpolarizationandspintransportdirections. T heanisotropyisshowninFig.123,wherethespindi ffusion
+lengthLdand the inverseof the spin oscillation period L−1
+0are plotted against the spin di ffusion/injectionangle θfor
+spinpolarizationsalong ˆn0,ˆzandˆn1=ˆz׈n0(crystaldirection[ ¯110]),respectively.
+Itisseenthatthespininjectionlengthbecomesfinitebutin dependentonthedirectionofthespindi ffusion/injection
+if the spin polarization is along [110] ( =ˆn0). Meanwhile, the spin polarization along this direction ha s an oscilla-
+tion period approachinginfinity. This can be understoodbec ause the spin polarizationis in the same direction as the
+effective magnetic field given by the zeroth-order term in ωkand, thus, results in no oscillation. It is noted that the
+effective magnetic field from the second-order term of ωkisk-dependent and the average of the second-order term
+overkstates is nearly zero. Therefore the cubic term does not lead to any oscillation at high temperature due to the
+scattering [363, 368, 597]. However, the inhomogeneous bro adening due to the second-order term leads to a finite
+155spin diffusion length. By rewritingthe second-orderterm of ωkintom∗γ
+2/planckover2pi12k2sin(2θk+2θ)/parenleftbigky/kx,−1,0) withkandθk
+denotingthe magnitude and the direction of the wavevector krespectively, one finds that the magnitude of the inho-
+mogeneousbroadeningdoesnotchangewith thespin di ffusion/injectiondirection θ. Consequentlythespin di ffusion
+lengthdoesnotchangewiththespindi ffusion/injectiondirection.
+ˆn1ˆzˆn0
+θ(π)
+L−1
+0(µm−1)Ld(µm)2.5
+2
+1.5
+1
+0.5
+01 0.80.60.40.202.5
+2.0
+1.5
+1.0
+0.5
+0.0
+Figure123: Spindi ffusionlength Ld(solidcurves)andtheinverseofthespinoscillation perio dL−1
+0(dashedcurves)for n-typeGaAs(001)quantum
+well withα=βas functions of the injection direction for di fferent spin polarization directions ˆn0,ˆzandˆn1atT=200 K. It is noted that the scale
+of thespin oscillation period is on the right hand sideof the frame. From Cheng etal. [29].
+When the injected spin momentum is perpendicular to ˆn0,L0andLdbecome anisotropic in regard to the spin
+transportdirection. Thespininjectionlengthissmall exc eptaroundθ=3π/4.L0andLdare almostidenticalfortwo
+spin polarization directions ˆ zandˆn1. Because in this case, as γ∝an}b∇acketle{tk2∝an}b∇acket∇i}ht≪β, the kinetics of systems are determined by
+thezerothordertermsof ωkwhicharethe same forthesetwo directions. The oscillation periodin the diffusionL0is
+determinedby the k-independentcomponentof ωk[29], i.e., 2 m∗βsin(θ−π/4)//planckover2pi12which is in good agreementwith
+the results shown in Fig. 123. Moreover, when the injection d irection is along [ ¯110] (θ=3π/4), the spin diffusion
+lengths for all of the three spin polarization directions ar e almost identical. The anisotropy in the direction of the
+spinpolarizationdisappears. Thisis becausewhenthespin diffusion/injectiondirectionis θ=3π/4,thek-dependent
+componentin the zerothorderterm of ωkdisappears. Thespin injectionlength is determinedbythe i nhomogeneous
+broadeninginthequadratictermsin ωk,whichareidenticalforanyspinpolarizationdirections.
+L0Ld
+N (1011cm−2)
+L0(µm)Ld(µm)0.374
+0.372
+0.37
+0.368
+8765432102.6
+2.4
+2.2
+2.0
+1.8
+1.6
+Figure 124: Spin di ffusion length Ldand spin oscillation period
+L0atT=200 K for spin transport in n-type GaAs (001) quan-
+tum well along θ=3π/4 direction as functions of the electron
+density. Note the scale of the oscillation period is on the ri ght
+hand side of the frame. FromCheng et al. [29].L0N=4×1010/cm2: LdL0N=4×1011/cm2: Ld
+T (K)
+L0(µm)Ld(µm)0.376
+0.374
+0.372
+0.37
+0.368
+0.3663002702402101801501203.6
+3.2
+2.8
+2.4
+2
+1.6
+1.2
+Figure 125: Spin di ffusive length Ldand spin oscillation period
+L0inn-type GaAs (001) quantum well at two di fferent electron
+densities, 4×1011cm−2and 4×1010cm−2vs.thetemperature.
+The spin diffusion/injection direction θ=3π/4. Note the scale
+of the oscillation period is on the right hand side of the fram e.
+From Cheng et al. [29].
+1567.5.3. Temperatureandelectron-densitydependencewith i njectionalong [¯110]andα=β
+The electron-density and temperature dependences of spin i njection along [ ¯110] in the case α=βare shown in
+Figs.124and125respectively. Itisseenthatthedi ffusionlength Lddecreaseswithelectrondensityandtemperature,
+whereas the oscillation period L0increases with them. However, the change observed in the osc illation period is
+almostnegligible(1%)in contrastto thepronouncedchange sobservedinthe di ffusionlength(morethan40%). This
+isquitedifferentfromthespininjectioninGaAsquantumwellwhenthere isonlytheDresselhausterm,where Ldand
+L0onlychangeslightlywiththe electrondensityandtemperat ure[25,28].
+This is understoodby the di fference in the behavior of the inhomogeneousbroadening ωk. Unlike case of α=0
+where the zeroth order term dominates the inhomogeneous bro adening, when α=βandθ=3π/4, the second-
+ordertermalonedeterminestheinhomogeneousbroadening. Thistermincreasese ffectivelywiththetemperatureand
+electron density due to the increase of the average value of k2. This is the reason for the obtained marked decrease
+of the spin diffusion length. The oscillation period L0is determined by the k-independent zeroth-order term of ωk,
+whichdoesnot changewith the electrondensityand thetempe rature. Thereforeone observesonlya slight changein
+theoscillationperiod,originatingfromthe second-order termandthe scattering.
+7.5.4. Gatevoltagedependencewith injectiondirectional ong[¯110]
+Spin transport along [ ¯110] under different gate voltages, i.e., di fferent Rashba coupling strengths αsince it can
+be controlled by the gate voltage, was also studied [29]. It w as pointed out that when the cubic Dresselhaus term is
+present,the longest spin injectionlength no longertakes p lace atα=β[29].95Thiscan also be understoodfrom the
+inhomogeneousbroadening. Forspintransportalong[ ¯110]withDresselhausandRashbaterms, ωkcanbewrittenas
+ωk=2[(−β+α+γk2
+2)ky
+kx−γkxky]ˆn1−2[β+α−γ
+2(k2
+x−k2
+y)]ˆn0. (313)
+It is seen that the e ffect ofαon the inhomogeneous broadening is determined by the first te rm. In order to get the
+longestspininjectionlength, αshouldbetunedtomakethefirst termbecomeminimal. Duetoth ecubicDresselhaus
+term, the optimized condition is α=ˆβ=β−γ∝an}b∇acketle{tk2∝an}b∇acket∇i}ht/2 instead of α=β[29, 34]. This prediction was later confirmed
+bythe experiment[37].
+7.6. Transientspingrating
+The drift-diffusion model tells that the spin signal exponentially decays with the distance in spin injection and
+the injection length Ldis related with the spin di ffusion coefficientDsand spin relaxation time τsby the formula
+Ld=√Dsτs[Eq. (276)]. However, as pointed out by Weng and Wu [336], the drift-diffusion model oversimplifies
+spin transport in semiconductors by neglecting the o ff-diagonal terms of the density matrix. When these terms are
+included, the spin signal in the spin injection was shown to b e a damped oscillation, characterized by spin injection
+lengthLdandspacialspinoscillationperiod L0,insteadofsimpleexponentialdecay[25, 28,32]. Intheext remecase
+ofspintransportalongthe[ ¯110]-axisin(001)GaAsquantumwell undertheidenticallin earDresselhausandRashba
+spinorbitcouplingstrengths,thespininjectionlengthbe comesinfiniteevenwhenthespindi ffusioncoefficientandthe
+spinrelaxationtime are finite [29]. Thisresult is totallyb eyondthe drift-diffusionmodelasboth Dsandτsare finite.
+Therefore, the relations among the characteristic paramet ers given by the drift-di ffusion model need modification
+[29, 32, 34, 35]. Weng et al. presented the correct relations among the characteristic parameters, which hold even
+underthe extremecaselike thecasein Sec.7.5,bystudyingt hetransientspingrating[32].
+Transient spin grating, whose spin polarization varies per iodically in real space, is excited optically by two non-
+collinear coherent light beams with orthogonal linear pola rization [13, 33, 37, 495, 661]. Transient spin grating
+technique is well suited to study the spin transport since it can directly probe the decay rate of nonuniform spin
+distributions. Spin di ffusion coefficientDscan be obtained from the transient spin grating experiments [13, 33, 495,
+661]. In the literature, the drift-di ffusion model was employed to extract Dsfrom the experimental data. With the
+drift-diffusion model, the transient spin grating was predicted to dec ay exponentially with time with a decay rate of
+Γq=Dsq2+1/τs, whereqis the wavevectorof the spin grating [13, 495]. However,thi s result is not accurate since
+95Similar conclusion has also been given by Stanescu and Galit ski in Ref. [34].
+157itneglectsthespinprecessionwhichplaysanimportantrol einspintransport. Indeed,experimentalresultsshowthat
+the decay of transient spin grating takes a double-exponent ial form instead of single exponential one [13, 33, 37].
+Therefore,it isworthytostudytransientspingratingusin gthekineticspinBlochequationapproach.
+7.6.1. Simplifiedsolutionfortransientspin grating
+The transient spin grating can be qualitatively understood in a simplified case where analytical solution can be
+obtained. By neglectingthe Hartree-Fockterm, the inelast ic scattering such as the electron-phononand the electron-
+electron Coulombscatterings and the couplingto the Poisso n equation, in the di ffusive limit, one is able to write the
+kineticspinBloch equationsasEq.(306), with ¯hand← →Γbeing
+¯h=∝an}b∇acketle{tk2τ1
+k/m∝an}b∇acket∇i}ht(−ˆβcos2θ,ˆβsin2θ−α,0), (314)
+and
+← →Γ =1
+2(α2+ˆβ2−2αˆβsin2θ)∝an}b∇acketle{tk2τ1
+k∝an}b∇acket∇i}ht
++∝an}b∇acketle{t(γk3)2τ3
+k∝an}b∇acket∇i}ht/162αˆβ∝an}b∇acketle{tk2τ1
+k∝an}b∇acket∇i}htcos2θ 0
+2αˆβ∝an}b∇acketle{tk2τ1
+k∝an}b∇acket∇i}htcos2θ(α2+ˆβ2+2αˆβsin2θ)∝an}b∇acketle{tk2τ1
+k∝an}b∇acket∇i}ht
++∝an}b∇acketle{t(γk3)2τ3
+k∝an}b∇acket∇i}ht/160
+0 02(α2+ˆβ2)∝an}b∇acketle{tk2τ1
+k∝an}b∇acket∇i}ht
++∝an}b∇acketle{t(γk3)2τ3
+k∝an}b∇acket∇i}ht/8(315)
+forthesystemwithboththeDresselhausandRashbaterms. On ecanobtaintheanalyticalsolutionoftheabovekinetic
+spinBlochequationsfortransientspingratingasfollowin g[32]
+Sz(q,t)=Sz(q,0)(λ+e−t/τ++λ−e−t/τ−), (316)
+withrelaxationrates
+Γ±=1
+τ±=Dq2+1
+2(1
+τs1+1
+τs)±1
+2τs2/radicaligg
+1+16Dq2τ2
+s2
+τ′
+s1, (317)
+inwhich
+λ±=1
+21±1/radicalig
+1+16Dq2τ2
+s2/τ′
+s1. (318)
+Hereτs1(τs2)isthespinrelaxationtimeofthein-planespinwhichmixes (doesnotmix)withtheout-of-planespindue
+totheneteffectivemagneticfield. Forspininjection /diffusionalongthe[110]axis, τs1=∝an}b∇acketle{t(α−ˆβ)2k2τ1
+k∝an}b∇acket∇i}ht/2+γ2∝an}b∇acketle{tk6τ3
+k∝an}b∇acket∇i}ht/32
+andτ′
+s1=∝an}b∇acketle{t(α−ˆβ)2k2τ1
+k∝an}b∇acket∇i}ht/2. For spin injection /diffusion along the [ ¯110] axis,τs1=∝an}b∇acketle{t(α+ˆβ)2k2τ1
+k∝an}b∇acket∇i}ht/2+γ2∝an}b∇acketle{tk6τ3
+k∝an}b∇acket∇i}ht/32
+andτ′
+s1=∝an}b∇acketle{t(α+ˆβ)2k2τ1
+k∝an}b∇acket∇i}ht/2. In the long wavelength limit ( q≪1),Γ+≃1/τs+(1+4τs2/τ′
+s1)Dq2andΓ−≃
+1/τs1+(1−4τs2/τ′
+s1)Dq2become quadratic functions of q, roughly correspond to the out-of-planeand the in-plane
+relaxation rates respectively. In general cases both of the se two decay rates [Eq. (317)] are not simple quadratic
+functions of q. If one uses the quadratic fitting to yield the spin di ffusion coefficient, one gets a value that is either
+larger (forΓ+) or smaller (forΓ−) than the true one. The accurate way to get the information of spin diffusion
+coefficientshouldbefromtheaverageofthese tworates
+Γ=(Γ++Γ−)/2=Dq2+1/τ′
+s (319)
+with1/τ′
+s=(1/τs+1/τs1)/2,whichdiffersfromthecurrentwidelyusedformulabyreplacingthespi n decayrate by
+theaverageofthe in-andout-of-planerelaxationrates. Th edifferenceofthese two decayratesisa linearfunctionof
+thewavevector qwhenqisrelativelylarge:
+∆Γ=cq+d. (320)
+Forthe simplifiedsolution c=2/radicalbigDs/τ′
+s1anddisa valueclose tozero.
+The steady-state spin injection can be extracted from the tr ansient spin grating signal by integratingthe transient
+spin grating signal Eq. (316) over the time from 0 to ∞and the wave-vector from −∞to∞. From the simplified
+158solution [Eqs. (316), (319) and (320)], the integrated tran sient spin grating reads Sz(x)=Sz(0)e−x/Lscos(x/L0+ψ)
+with
+Ls=2Ds//radicalig
+|c2−4Ds(1/τ′s−d)|, (321)
+L0=2Ds/c. (322)
+It is noted that if one only considers the Rashba term or the li near Dresselhaus term one can recover the result
+Ls=2√Dsτsfrom linear response theory [35]. It is seen that the spin pre cession actually prolongsthe out-of-plane
+spin injection length by mixing the fast decaying out-of-pl ane spin with the slow decaying in-plane spin. Equa-
+tions (321) and (322) give the right spin injection length an d the spin oscillation period in the presence of the spin-
+orbitcoupling. Fromthe experimentpointofview,onecan mo nitorthe time evolutionsoftransientspingratingwith
+differentwavevectors qandobtainthecorrespondingdecayrates Γ±. ByfittingΓ±withEqs.(319)and(320),onecan
+then calculate the spin injection length and spin oscillati on period accurately from Eqs. (321) and (322), instead of
+using the inaccurate formula from the drift-di ffusion model. Weng et al. demonstrated that the relations defi ned by
+Eqs.(321)and(322)are trueevenintheextremecase whenthe drift-diffusionmodeltotallyfails[32].
+7.6.2. Kinetic spinBlochequationsolutionfortransients pingrating
+00.10.20.30.4
+0 1 2 300.0250.050.075Γ (ps−1)
+∆Γ (ps−1)
+q(104cm−1)
+Figure 126:Γ =(Γ++Γ−)/2 and∆Γ =(Γ+−Γ−)/2vs.
+qatT=295 K in n-type GaAs (001) quantum well. Open
+boxes/triangles aretherelaxation rates Γ+/−calculated from the
+full kinetic spin Bloch equations. Filled /open circles represent
+Γand∆Γrespectively. Notedthatthescalefor ∆Γisontheright
+hand side of the frame. The solid curves are the fitting to Γand
+∆Γrespectively. The dashed curves are guide to eyes. From
+Wenget al. [32].101102
+100150200250τ(ps)
+T (K)(a)τ+τ−
+100150200250300102103
+τ(ps)
+T (K)(b)
+τ+τ−
+Figure 127: Spin relaxation times τ±vs.temperature in n-type
+GaAs(001)quantumwellfor(a)high-mobility samplewith q=
+0.58×104cm−1and(b)low-mobilitysamplewith q=0.69×104
+cm−1. The dots are the experiment data from Ref. [33]. From
+Weng et al. [32].
+The spin diffusion coefficient together with the relations Eqs. (321) and (322) obtai ned by the above simplified
+kineticspinBlochequationsarederivedwithonlytheelast icscatteringapproximation. Itdoesnottakeintoaccounto f
+the inelastic electron-phononand electron-electron Coul omb scattering. Therefore the di ffusion coefficient obtained
+from Eq. (319) does not include the contribution of the spin C oulomb drag. Moreover, this approach oversimplifies
+thecomplexeffectofthescatteringonthespintransport. Whetherthesere lationsremainvalidinthegenuinesituation
+remainsyet to be checked. To answer this, Weng et al. numeric allysolvedthe full kinetic spin Bloch equationswith
+all the scattering explicitly included [32]. Their numeric al results showed that all of the important qualitative resu lts
+presented in Sec. 7.6.1 are still valid. Especially, the tem poral evolution of the transient spin grating with all the
+scatteringincludedstill showsadouble-exponentialdeca y,whichagreeswiththeexperimentalresult[13,33,37] but
+iscontrarytotheresultfromthedrift-di ffusionmodel. Moreoverthetwodecayratescanstillbefittedw ithEqs.(319)
+and(320) prettywell [32]. Thefittingis showninFig.126,wh eretherelaxationrates Γ±=1/τ±, theiraverageΓand
+difference∆Γofatypicaltransientspingratingareplottedasfunctions ofq. Thenumericalresultssuggestedthatthe
+twodecayratesΓ±=1/τ±arenotquadraticfunctionsof q,buttheiraveragedecayrate Γis. Usingquadraticfunction
+to fit the decay rates Γ±(q), one gets poor result. In contrast, the average decay rate Γfits pretty well by a quadratic
+functionΓ=Dsq2+1/τ′
+s. The resident error of the quadratic fitting for Γis two orders of magnitude smaller than
+those ofΓ±. Moreover, forΓthe constant term τ′
+sis very close to 4 τs/3, inverse of the average of the in-plane and
+159out-of-plane spin relaxation rates. Inspired by Eq. (319), the coefficient of the quadratic term Dscan be reasonably
+assumedtobethespindi ffusioncoefficient. Inthiswayonecancalculatethespindi ffusioncoefficientwiththeeffect
+ofspin Coulombdragincluded. For ∆Γ, it canbe accuratelyfitted by a linearfunctionof q. It is thereforeconcluded
+thatevenwithallthescatteringsincluded,onecanstillex tractthespindiffusioncoefficientDsandobtaintheinjection
+lengthandoscillationperiodusingEqs.(321)and(322) fro mthetransientspingratingexperiments.
+FromEqs.(319)and(320)onecanseethat τ+decreasesmonotonicallyas qincreaseswhile τ−hasapeakatsome
+wavevector q0. Earlier theoretical works, which only considered linear D resselhaus term, predicted that the ratio of
+these two relaxation times τ−/τ+and the position of the peak q0do not vary with temperature [18, 35]. However,
+Wengetal. showedthatoncethecubicDresselhaustermisinc luded,both τ−/τ+andq0dependontemperature. With
+the cubicDresselhausterm, the peakposition q0movesfrom q0=√
+15m∗β/2 (which is independentof temperature)
+toq0=√
+15m∗ˆβ/2=√
+15m∗(β−γk2/2)/2. Since∝an}b∇acketle{tk2∝an}b∇acket∇i}htincreaseswith increasingtemperature,thus q0decreaseswith
+it. The temperature dependenceof τ+/τ−also originates from the contributionof the cubic Dresselh aus term. These
+theoretical results qualitatively agree with experiment o nes [33]. In additional to the correct qualitative agreemen t
+between the theoretical and experimental results, the quan titative accuracy of results from the kinetic spin Bloch
+equation approach is also good. The results are shown in Fig. 127, in which spin relaxation times are plotted as
+function of temperature together with the experimental dat a from Ref. [33]. In the calculation the finite square well
+assumption [372] was used, and all the parameters were chose n to be the experimental values if available. The
+only adjustable parameters were the spin-orbit coupling co efficientsγandα. In the calculation, γwas chosen to be
+11.4 eVÅ3and 13.8 eVÅ3for the high [Fig. 127(a)] and low mobility [Fig. 127(b)] sam ples respectively and αwas
+set tobe0.3β,closetothechoiceintheexperimentalwork[33].
+7.6.3. SpinCoulombdrag
+Figure 128: A representation of electron-electron Coulomb
+scattering that does not conserve spin-current. Before the col-
+lision the spin-current, Jspin, is positive; after, it is negative.
+The charge current, Jc, does not change. Colors correspond to
+spin states. FromWeber etal. [13].200300400500600700
+100 150 200 250 300D (cm2/s)
+T (K)DsDc
+Figure129: Diffusioncoefficientasafunctionoftemperaturein
+n-type GaAs (001) quantum well. Solid circles: Spin di ffusion
+constantDswithCoulomb drag; Opencircles: Chargedi ffusion
+constantDc. FromWeng et al. [32].
+From the evolution of the transient spin grating one can obse rve the Coulomb drag e ffect on the spin diffusion
+directly. The effect ofelectron-electronCoulombscatteringonthe spin di ffusioncoefficientisillustrated in Fig. 128.
+For spin-unpolarizedchargedi ffusion, the electronsmove along the same direction and the Co ulomb scattering does
+notchangethecenter-of-massmotion,thereforeit doesnot changethechargedi ffusioncoefficientdirectly. However,
+for the spin transport, spin-up and -down electrons move aga inst each other. Therefore the Coulomb scattering be-
+tweenthemslowsdowntherelativemotionofthesetwospinsp eciesandthusreducesspindi ffusioncoefficient. This
+istheso-calledspindragorspinCoulombdrage ffect[13–15,27,1093–1101],whichwasfirstproposedbyD’Ami co
+andVignale[14,15, 1094–1098].
+In Fig. 129 the spin di ffusion coefficient calculated in the above mentioned method as a function of temperature
+waspresented. Forcomparison,thechargedi ffusioncoefficient,whichiscalculatedbysolvingthekineticspinBloch
+160equationswiththeinitialconditionbeingthechargegradi entinsteadofthespingradient,isalsoincluded. Itisclea rly
+seenfromthefigurethat Ds<Dc.
+From the figure one can tell that in the relative high temperat ure regime, as the temperature increases, the spin
+and charge diffusion coefficients and their difference all decrease. Therefore the Coulomb drag is stronger in the
+low temperature regime. However, even at room temperature t he Coulomb drag is still strong enough to reduce the
+diffusioncoefficientby30%. Theseresultsquantitativelyagreewiththose ofRefs.[1094,1095]. Itshouldbepointed
+outthatthereductionofspindi ffusioncoefficientmostlycomesfromtheCoulombdrag. Thespin-orbitcou plingonly
+hassmall effect on the diffusioncoefficientsince the spin-orbitcouplingis verysmall comparedt o the Fermi energy.
+Thenumericalresult showsthat the removalof thespin-orbi tcouplingonlychangesspindi ffusioncoefficientbyone
+tenthpercentforthesystem studied,whichisconsistentwi ththeresultfromthelinearresponsetheory[1099].
+Phenomenologically,thespinCoulombdragmodifiesOhm’sla wasEη=/summationtext
+η′ρηη′jη′,inwhich Eηisthe“electric”
+fieldonelectronswithspin η. Thefieldalsoincludesthecontributionfromthegradiento fthelocalchemicalpotential,
+which can be spin dependent. The o ff-diagonal terms of the resistivity ρ↑↓correspond to the spin drag resistivity,
+definedbyρ↑↓=E↑/j↓whenj↑=0. UsingthegeneralizedEinsteinrelationswiththespinCo ulombdrageffecttaken
+intoaccount,thespin di ffusioncoefficientreads[13–15,1094]
+Ds
+Dc=χ0
+χs1
+1+|ρ↑↓|/ρ, (323)
+whereDcandρare the charge diffusion coefficient and mobility, respectively and χ0/χsis the many-bodyenhance-
+ment of spin susceptibility of the electron gas. In normal el ectron density and temperature, χ0/χsis close to 1
+[13, 848, 1102, 1103]. Spin drag resistivity can be calculat ed from the linear response theory. In two dimension, it
+takestheformof[14, 15,1094, 1101]:
+ρ↑↓=−1
+2e2n↑n↓kBT/integraldisplayd2q
+(2π)2q2/integraldisplay∞
+0dω
+2π|V↑↓(q,ω)|2ImΠ0↑(q,ω)ImΠ0↓(q,ω)
+sinh2(ω/2T), (324)
+in which V↑↓(q,ω) is the dynamicallyscreened e ffective Coulomb interaction between spin ↑and↓electrons.Π0ηis
+the non-interacting spin-resolved polarization function . Spin-orbit coupling is shown to have small enhancement to
+spin drag resistivity [1099]. ρ↑↓increases with temperatureas ( T/TF)2in low temperatureregime ( T≪TF) but de-
+creaseswith temperatureas1 /Tin hightemperatureregime( T≫TF) [1093, 1095]. Thequantitativecomparisonof
+ρ↑↓betweentheoreticalcalculationandexperimentresultsis shownin Fig.130. Withthemodificationoffinitequan-
+tum well width and local field correlations, ρ↑↓calculated from Eq. (324) is in good agreement with the exper iment
+dataextractedfromRef.[13]usingEq.(323)[1101].
+Jiang et al. showed that the spin Coulomb drag also plays an im portant role in the transient spin transport [27].
+In Fig. 131, the spacial profiles of the spin imbalance at di fferent times are plotted for a Gaussian spin pulse with
+different scattering. It is clearly seen that when the Coulomb sc attering is included, both the di ffusion of the spin
+pulse and the decay of the spin polarization at the center of t he pulse become much slower. When there is only
+electron-phononscattering,athirdpeakin ∆Nσappearsafterabout15picoseconddi ffusion. TheCoulombscattering
+delays the appearanceof the third peak to 35 picoseconds. Mo reover,since the Coulomb scattering reducesthe spin
+dephasing, the magnitudes of the spin polarization and the p eaks with the Coulomb scattering are higher than those
+without.
+7.6.4. InfinitespininjectionlengthinSec.7.5revisited f rom thepointof viewof transientspingrating
+For most of the cases, Lsdetermined by Eq. (321) is in the same order of√Dsτs, although the former is usually
+larger. However, there are some special cases, say the denom inator in Eq. (321) approaches zero, where Lscan be
+ordersof magnitude larger than√Dsτs. Specifically, according to Cheng et al. [29], reviewed in Se c. 7.5, when the
+spininjection/transportdirectionis along[ ¯110]in (001)quantumwell, Lsbecomeslargerandlargeras αapproaches
+β,regardlessofthedirectionofspinpolarization. Intheli mitofα=β,thespininjectionlengthtendstoinfinitywhen
+the cubic Dresselhausterm is ignoredand the spin polarizat ionoscillates with a spacial periodof 2 π/(2m∗β) without
+anydecay[29].
+The non-decay spin oscillation model can also be understood from the point of view of transient spin grating.
+FromthesimplifiedsolutioninSec.7.6.1,oneobtainsthatt hefittingparametersfor ∆Γared→0andc→2/radicalbig
+Ds/τ′s
+1610 100 200 3000123 
+n = 4.3 1011 cm-2 0
+ρ
+
 [kOhm]
+Ta = 12 nm
+Figure 130: Spin drag resistivity ρ↑↓as a function of temper-
+ature for in n-type GaAs (001) quantum well with the electron
+densityn=4.3×1011cm−2. The dashed and dotted curves
+correspond to ρ↑↓, calculated within the random phase approxi-
+mation for quantum well width a=0 and 12 nm, respectively.
+The solid curve correspond to ρ↑↓, calculated beyond the ran-
+dom phase approximation for a=12 nm. The symbols repre-
+sentρ↑↓, deduced from the experimental data for a=12 nm of
+Ref. [13]. FromBadalyan etal. [1101].10−2
+0 1 2 3 4 5
+x(µm)|∆Nσ|(1011cm-2)
+60 ps10−235 ps10−215 ps10−2t= 5 pswith LO scattering
+with LO & Coulomb scattering
+with LO, Coulomb & impurity scattering
+Figure 131: The absolute values of the spin imbalance |∆Nσ|
+vs.the position xat certain times tinn-type GaAs (001) quan-
+tum well for the cases with only the electron-phonon scatter ing
+(orange); the electron-phonon and electron-electron scat tering
+(blue); and the electron-phonon, electron-electron and el ectron-
+impurity scattering (green). Thered curveshows theinitia l spin
+pulse.∆Nσ(0)=1011cm−2,δx=0.1µm andNi=0.5Ne.
+Solid curves:∆N>0; Dashed curves: ∆N<0. From Jiang et
+al. [27].
+whenα→βinthesystemwithoutthecubicDresselhausterm. Consequen tlyτ±=(√
+Dq±1/√τs)−2whenα=β. It
+isthenstraightforwardtoseethat τ−becomesinfiniteprovided q=q0=1/√Dτs=2m∗β. Thereforethesteady-state
+spin injection along [ ¯110] axis is dominated by this non-decay transient spin grat ing mode which is responsible for
+theinfinitespininjectionlengthandthespacial oscillati onperiod2π/q0=2π/(2m∗β)[32].
+7.6.5. Persistent spinhelix state
+The spin transportmodel with infinite spin injectionlength in the Sec. 7.5.1[29], or the non-decaytransient spin
+grating with wavevector q0=(q0,0,0) in the previous subsection [32], is related to the persist ent spin helix, which
+was first proposedby Bernevig et al. [18, 20, 37, 921]. Persis tent spin helix has an SU(2) symmetry which is robust
+againstthespin-conservingscattering. TheSU(2)symmetr yofthepersistentspinhelixisgeneratedbythefollowing
+operators[18]:
+S−
+q0=/summationdisplay
+kc†
+k;−ck+q0;+,S+
+q0=/summationdisplay
+kc†
+k+q0;+ck;−,Sz
+0=/summationdisplay
+k(c†
+k;+ck;+−c†
+k;−ck;−). (325)
+Herec†
+k;λ(ck;λ) is the creation (annihilation)operator of electron with m omentum kand spinλ.λ=±are the index
+ofspineigenstateswithspin-orbitcoupling. Itisnotedth atnonzero∝an}b∇acketle{tSz
+0∝an}b∇acket∇i}htcorrespondstospinpolarizationalong[110]
+direction. While nonzero ∝an}b∇acketle{tS±
+q0∝an}b∇acket∇i}htcorrespondto spin gratingwith wave-vector q0. The spin profile of nonzero ∝an}b∇acketle{tS±
+q0∝an}b∇acket∇i}htin
+therealspaceishelicalwave. Theout-of-planemodeofthe s pinhelixisillustratedinFig.132c.
+Thegeneratingoperatorsofpersistentspinhelixobeythec ommunicationrelationsforangularmomentum,
+[Sz
+0,S±
+q0]=±2S±
+q0,[S+
+q0,S−
+q0]=Sz
+0. (326)
+Moreover, since|k;+∝an}b∇acket∇i}htand|k+q0;−∝an}b∇acket∇i}htare degenerate, these operators also commute with the elect ron Hamiltonian.
+Thissymmetryis robust againstspin conservingscattering , such as electron-impurity,electron-phononand electron -
+electronCoulombscatterings, since these three operators also commutewith the finite wave-vectorparticle densities
+ρq=/summationtext
+σc†
+k+qσckσ,whicharethecomponentsthatappearin thescatteringHami ltonian. ThisSU(2)symmetrymeans
+that the life time of expectation values of Sz
+0,S±
+q0are infinite, and hence the transient spin grating with q0does not
+decay with time even when there is spin conserving scatterin g. In spin transport, this corresponds to the infinite
+spin injection length proposed by Cheng et al. [20, 29, 32]. W hen the cubic Dresselhaus term is present, the SU(2)
+162Figure 132: a, Transient spin grating decay curves at variou s
+wavevectors, q, for an asymmetrically doped GaAs quantum
+well with a mixture of Rashba and Dresselhaus spin-orbit cou -
+plings. b,Lifetimesforthespin-orbit-enhanced [ τE,correspond-
+ing toτ−in Eq.(317)] and -reduced [ τR,corresponding to τ+in
+Eq(317)]helix modes extracted fromdouble-exponential fit sto
+the data in a. Thesolid lines are a theoretical fit(see text) u sing
+a single set of spin-orbit parameters for both helix modes. E r-
+ror bars (s.d.) are the size of the data points. c, Illustrati on of
+a helical spin wave, which is one of the normal modes. In this
+picture,zisthegrowthdirection [001],andtheaxes x′andy′re-
+spectively refer to the [110] and [1 ¯10] directions in the plane of
+the quantum well. Thegreen spheres represent electrons who se
+spindirectionsaregivenbythearrows. FromKoraleketal.[ 37].
+Figure 133: a, c, Lifetimes of the enhanced helix mode are
+shown for GaAs quantum wells with varying degrees of doping
+asymmetry (a) and well width (c). The normalized asymmetry
+isthedifferencebetween theconcentrations ofdopantsoneither
+side of the well, divided by the total dopant concentration. b, d,
+Plots summarizing the spin-orbit parameters from these fits in
+a and c. In this figure, the spin-orbit coupling strengths are ex-
+pressed asdimensionless quantities bynormalizing totheF ermi
+velocity.β1andβ3are the coefficients of the linear and cubic
+Dresselhaus terms, corresponding to βandγin the text, respec-
+tively. From Koralek etal. [37].
+symmetryisbroken,thelifetimesofthespinhelixbecomefin ite,whichcorrespondstothefinitespininjectionlength
+inthepresenceofthecubicDresselhausterm[20,29, 32].
+Recently, Koralek et al. have observed the emergence of the p ersistent spin helix experimentally in the asym-
+metrically modulation-dopedGaAs quantum wells, where bot h the Dresselhaus and Rashba spin-orbit couplingsare
+important, by using the transient spin-grating spectrosco py [37]. Their results are shown in Figs. 132 and 133. The
+experiment results clearly show that the transient spin gra ting signal is in a double exponential decay form and can
+be fitted by two lifetimes, τEandτR(corresponding to τ−andτ+in the previous subsections). For some special
+wavevectors, τEis much larger than τRwhen the Dresselhaus and the Rashba terms are comparable. Al l of these
+experimentalresultsconsist with thetheoreticalones[32 , 33]. By constructinga series ofquantumwells with di ffer-
+ent well widths (to tune the linear Dresselhaus spin-orbit c oupling) and doping asymmetries (to change the Rashba
+spin-orbit coupling), Koralek et al. demonstrated how the l ifetimes of spin helix depend on the Dresselhaus and
+Rashbatermsaswell astemperature. Theyfoundthat theopti mizedconditionto observethepeakspinhelixlifetime
+isα=ˆβ=β−γk2/2 instead of α=βdue to the existence of the cubic Dresselhaus term. This findi ng is consistent
+withtheprevioustheoreticalpredictions[29, 32,34].
+8. Summary
+The main concern of this review was to give an overview of the l atest developments of the spin dynamics of
+semiconductors. Wefocusedontwomainissues: (i)spinrela xationanddephasingand(ii)spindi ffusionandtransport.
+We reviewed both experimental and theoretical progresses i n these two directions. In theoretical part, besides the
+results based on the single-particle approach, we reviewed our systematic investigation based on fully microscopic
+many-bodykinetic spin Bloch equation approach. Many novel effects predicted by this approach have been realized
+experimentally. Also manywidely adoptedcommonbeliefs ba sed on the single-particletheory in the literature were
+showntobeincorrect. Weprovidedacomprehensiveundersta ndingofthespindynamicsofsemiconductorsandtheir
+163confinedstructuresfroma fullymicroscopicmany-bodypoin tofview.
+Acknowledgements
+This work was supported by the National Natural Science Foun dation of China under Grant No. 10725417, the
+NationalBasicResearchProgramofChinaunderGrantNo.200 6CB922005andtheKnowledgeInnovationProjectof
+ChineseAcademyofSciences. Oneoftheauthors(MWW)wouldl iketothankL.J.ShamandJ.M.Kikkawawhofirst
+broughtthis excitingfield into his attention back to 1999. H e would also like to thank H. Haug, H. Metiu, J. Fabian,
+C. Sch¨ uller, T. Korn, X. Marie and E. L. Ivchenko for many dis cussions and/or collaborations. Many interesting
+discussionswithH.Z.Zheng,T.S.Lai,Y.JiandX.D.Cuionth eexperimentalfindingsarealsoacknowledged. Last
+butnottheleast,hewouldliketothankhisformerandcurren tstudentsM.Q.Weng,J.L.Cheng,J.H.Jiang,K.Shen,
+P.ZhangandY.ZhoufortheirexcellentcollaborationsandP .Zhangforthetypesettingofpartofthismanuscriptand
+theproofreadingreadingofthewholemanuscript.
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+arXiv:1001.0607v1  [math.AG]  5 Jan 2010Regularity of quotients by an automorphism of
+orderp
+Stefan Wewers
+IAZD, Leibniz-Universit¨ at Hannover
+Abstract
+LetBbe a regular local ring and G⊂Aut(B) a finite group of local
+automorphisms. Assume that Gis cyclic of prime order p, where pis
+equal to the residue characteristic of B. We give conditions under which
+the ring of invariants A=BGis again regular.
+Introduction
+LetXbe a regular integral scheme and G⊂Aut(X) a finite group of auto-
+morphisms of X. The quotient scheme Y:=X/Gmay not be regular; its
+singularities are, by definition, quotient singularities . To study the singularities
+ofY, we may localize and assume that X= SpecBandY= SpecA, whereB
+is a local domain and A=BGis the ring of invariants.
+Quotient singularitieshavebeen intensively studied, in connection wit h reso-
+lution of singularities and as objects in their own right. However, mos t results
+concerntamequotient singularities. In the above situation this means that the
+order of Gis prime to the characteristic of the residue field of B.
+In a recent preprint [8], D. Lorenzini has studied the resolution gra phs of
+wildquotient singularities in dimension 2, exhibiting many interesting featu res
+that do not occur for tame quotient singularities. His results rely he avily on
+a detailed combinatorial study of the possible intersection matrices that can
+occur. The present note aims at complementing the methods used in [8]. The
+basic idea is the following.
+LetY=X/Gbe as above, and let Y′→Ybe a resolution of Y. Then
+the fiber product X′:=X×YY′is aG-equivariant modification of Xwhose
+quotient Y′=X′/Gis regular. Conversely, given a G-equivariant modification
+X′→X, the quotient scheme Y′:=X′/Gis a modification of Y– which may or
+may not be regular. From this point of view it is natural to look for con ditions
+onX′under which Y′is regular. However, it is difficult to find such conditions
+in the literature which apply to the case of wild group actions.
+Our main result (Theorem 1.3) gives a sufficient condition for the regu larity
+of the quotient scheme Y=X/GwhenGis a cyclic group of order p. This
+1is admittedly a very modest contribution to our motivating problem. S till, our
+criterion seems to be new, and we hope that it will be useful in the fut ure.
+The motivation for writing this paper grew out of discussions with F. K ir´ aly,
+who discovered a special case of our main result, see Example 1.10. T he author
+thanks him, W. L¨ utkebohmert and M. Raynaud for helpful discuss ions and
+comments on earlier versions of this paper.
+1 The main result
+1.1Let (B,m,k) be a noetherian regular local domain, and let G⊂Aut(B)
+be a finite group of local automorphisms. We are interested in the fo llowing
+question.
+Problem 1.1 Under which condition is the ring of invariants
+A:=BG={x∈B|σ(x) =xfor allσ∈G}
+again regular?
+For the applications we have in mind, the following additional assumptio ns
+seem reasonable and useful:
+(a) The residue field kis algebraically closed.
+(b) The induced G-action on kis trivial.
+(c)Ais noetherian and Bis a finite A-algebra.
+Assumptions(a)and(b)canalwaysbeachievedbypassingtothest ricthenseliza-
+tion and are therefore harmless. They imply that Ais a local domain with
+residue field k. Condition (c) is satisfied in most situations arising from a geo-
+metric context. Suppose, for instance, that Bis the localization of a finitely
+generated algebra over an excellent domain R, and that the action of GonB
+isR-linear. Then (c) holds. See [2] for a general discussion of Condition (c).
+If the order of the group Gis prime to the characteristic of the residue field
+k, a definitive answer to Problem 1.1 is known.
+Theorem 1.2 Suppose that (a)-(c) holds and that the order of Gis prime to
+the characteristic of the residue field k. Then the ring Ais regular if and only
+if the image of GinGL(m/m2)is generated by pseudo-reflections1.
+Proof:See [12] or [14] ✷
+The main result of the present paper (Theorem 1.3 below) gives a suffi cient
+criterion for Ato be regular in the case where Gis cyclic of prime order p. Here
+pmay be equal to the characteristic of k, and hence our result is not covered
+by Theorem 1.2
+1An element σ∈GL(V) is called a pseudo-reflection if rank(σ−1)≤1
+21.2Let us now assume that Gis cyclic of order p, wherepis prime. We
+choose a generator σ∈Gand consider the ideal
+Iσ:=/a\}bracketle{tσ(x)−x|x∈B/a\}bracketri}htB✁B.
+By definition, Iσis the smallest G-invariant ideal such that Gacts trivially on
+B/Iσ. Inparticular,this showsthat Iσdoesnotdepend onthechosengenerator
+σ. Condition (b) says that Iσis contained in m.
+An element x∈Bis called a regular parameter ifx∈m\m2. SinceBis
+regular, this means that xis part of a regular system of parameters. It follows
+thatB/Bxis a regular local ring and that ( x)✁Bis a prime ideal.
+Here is our main result:
+Theorem 1.3 Suppose that there exists a regular parameter π∈m\m2such
+that
+(i)Iσ= (πδ), withδ∈N,
+(ii)we either have Iσ= (σ(π)−π), or elseσ(π)−π∈(πδ+1).
+ThenA=BGis regular.
+After some preliminary work done in Section 2 and 3, the proof of The orem
+1.3 will be given in Section 4. For the rest of this section, we discuss th e scope
+and the limitations of Theorem 1.3 and some open problems.
+Remark 1.4 Let ¯σ∈GL(m/m2) denote the image of σ. By definition of
+Iσ,m/(Iσ+m2) is the largest quotient of m/m2on which ¯ σacts trivially. If
+the hypothesis of Theorem 1.3 is verified, then it follows that ¯ σis a pseudo-
+reflection. So for char( k)/\e}atio\slash=p, Theorem 1.3 is a direct consequence of Theorem
+1.2. Moreover, in this case the natural homomorphism
+G→GL(m/m2)
+is known to be injective (see e.g. [14]). This implies that we must have δ= 1 in
+Condition (i) of Theorem 1.3 and that Iσ= (σ(π)−π) in Condition (ii).
+On the other hand, if char( k) =p,δmay be strictly larger than 1. In
+particular, the hypothesis of Theorem 1.3 is stronger then the con dition ‘¯σis a
+pseudo-reflection’: it is easy to give examples where ¯ σ= 1 and Ais not regular.
+See e.g. [10].
+Remark 1.5 The authorsuspects that Condition (ii) in Theorem1.3 isimplied
+by Condition (i) and could therefore be omitted. However, he did not succeed
+in proving this.
+Remark 1.6 The sufficient condition given by Theorem 1.3 is not necessary
+forAto be regular. See e.g. Example 1.8 below.
+3The Purity Theorem of Zariski-Nagata gives us a necessary conditio n for
+regularity. Namely, if Ais regular, then all minimal prime ideals of Bcontaining
+Iσhave height one, and are therefore principal. This prompts the follo wing
+question:
+Question 1.7 Is it true that Ais regular if and only if Iσis a principal ideal?
+1.3We shall give three examples that illustrate various points. In all thr ee
+examples, the ring Bis a ring of power series over a complete discrete valuation
+ringR. In particular, Bhas dimension 2. We assume moreover that the action
+of the group Gfixes the subring R⊂B.
+ThefirstexampleshowsthatthehypothesisofTheorem1.3isnotan ecessary
+condition for regularity of the ring A.
+Example 1.8 LetRbe a complete discrete valuation ring of mixed character-
+istic (0,p), containing a pth root of unity ζp. SetB:=R[[x]], and let σ:B∼→B
+be theR-automorphism of order pgiven by
+σ(x) =ζp·x.
+Then
+A:=B/angbracketleftσ/angbracketright=R[[xp]]
+is regular. However, the ideal
+Iσ=/parenleftbig
+(ζp−1)x/parenrightbig
+✁B
+is contained in two distinct principal prime ideals, so Condition (i) of The orem
+1.3 fails.
+Remark 1.9 Example 1.8 is a special case of the following situation. Assume
+that the ring Bhas a regular system of parameters ( π1,...,π d−1,x) such that
+σ(πi) =πifori= 1,...,d−1
+and
+σ(x)≡u·xmod (π1,...,π d−1)
+for a unit u∈B×. Then A=BGis regular, see [5], Proposition 7.5.2. In
+the situation of Theorem 1.3 a system of parameters ( π1,...,π d−1,x) as above
+exists, but this is a consequence of the proof of Theorem 1.3, and is not obvious
+beforehand.
+Thenextexample,whichwasfirststudiedbyF.Kir´ aly[6], describesa special
+situation where the answer to Question 1.7 is affirmative.
+4Example 1.10 LetRandB=R[[x]] be as in Example 1.8. Let σ:B∼→
+Bbe an automorphism of order pwhich induces the identity on the residue
+fieldk. Contrary to Example 1.8, we assume that σrestricts to a nontrivial
+automorphism of the subring R⊂B.
+Letv:R→Z∪{∞}denote the discrete valuation on Rand letπ∈Rbe a
+uniformizer, i.e. v(π) = 1. Our assumptions imply that
+σ(π) =π+uπµ,
+withµ≥2 andu∈R×. Write
+fσ:=σ(x)−x=a0+a1x+a2x2+...,
+withai∈Rand set
+ν:= min{v(ai)|i= 0,1,...}.
+Since (π,x) is a system of parameters for B, we have
+Iσ= (πµ,fσ)✁B.
+It follows that Iσis a principal ideal if and only if one of the following cases
+occurs:
+(I)µ≤ν, or
+(II)v(a0) =ν < µ.
+In Case (I) we have Iσ= (πµ) and in Case (II) we have Iσ= (πν). In both
+cases, the hypothesis of Theorem 1.3 holds, and hence A=BGis regular. In
+this special case, the statement of Theorem 1.3 has been proved e arlier by F.
+Kir´ aly, and his results yield somewhat more. In Case (I), the ring of invariants
+Ais actually a power series ring over RG. Moreover, if neither Case (I) nor
+Case (II) holds, then Iσis not a principal ideal and the ring Ais not regular.
+We refer to [6] for more details.
+The distinction of Case (I) and (II) in Example 1.10 illustrates a dichot omy
+which is crucial for the proof of Theorem 1.3. Let L:= Frac( B) andK:=
+Frac(A) denote the fraction fields. Then L/Kis a Galois extension with Galois
+groupG. Assume that the hypothesis of Theorem 1.3 holds. Let vdenote
+the discrete valuation of Lwhich corresponds to the localization of Bat the
+prime ideal ( π). It follows from Condition (i) that vis ramified in L/K. The
+two cases of Condition (ii) correspond to Case (I) and Case (II) in E xample
+1.10. We shall see in Section 3 that in the first case the extension L/Kis
+totally ramified alongv, whereas it is fiercely ramified in the second case (see
+Definition 3.1 and Proposition 3.2).
+51.4Ifwedroptheassumptionthatthegroup Gactingon Biscyclicoforder
+p, the study of the ramification behavior of the extension L/Kbecomes much
+more complicated. The problem is that it is in general difficult to ‘separa te’
+a wildly ramified extension in a canonical way into subextensions which a re
+either totally or fiercely ramified. This seems to be the main reason wh y the
+statement of Theorem 1.3 does not easily generalize to more genera l groups.
+Our last example illustrates this point.
+Example 1.11 LetRandB=R[[x]] be as in Example 1.8 and 1.10. Let
+G⊂Aut(B) be a finite group of automorphisms which fixes the subring R⊂B
+and acts trivially on the residue field k. We assume that Gis an elementary
+abelian group of order p2, i.e.G∼=Z/p×Z/p, with generators σ1,σ2. We also
+assume that the induced action of GonRis faithful. Let Ai:=B/angbracketleftσi/angbracketright,i= 1,2.
+The restriction of σ1toA2is an automorphism of order psuch that Aσ1
+2=A.
+Likewise, Aσ2
+1=A.
+Suppose that A1andA2are regular. Does it follow that Ais regular?
+The statement of Theorem 1.2 may lead one to believe that the answe r to this
+question is yes. However, this is not the case; in the following we shall give an
+explicit counterexample, where p= 2.
+LetK:=Qnr
+2denote the maximal unramified extension of Q2. LetL/Kbe
+the Galois extension generated by the roots of the polynomial ( x2−2)(x2−3).
+HenceL/Kis generated by elements√
+2,√
+3 which are square roots of 2 and 3,
+respectively. Let G= Gal(L/K) denote the Galois group. Then G∼=Z/2×Z/2
+is generated by the two automorphisms σ1,σ2determined by
+σ1(√
+2) =√
+2, σ1(√
+3) =−√
+3,
+σ2(√
+2) =−√
+2, σ2(√
+3) =√
+3.
+It follows that
+K1:=Q2(√
+2) =Lσ1, K2:=Q2(√
+3) =Lσ2.
+Letvdenote the unique extension to Lof the 2-adic valuation, normalized by
+v(2) = 1. Set
+π1:=√
+2, π2:=√
+3−1, π:=π2
+π1−1.
+One checks that
+v(π1) =v(π2) =1
+2, v(π) =1
+4.
+It follows that
+OK1=Znr
+2[π1],OK2=Znr
+2[π2],OK=Znr
+2[π].
+LetB:=OL[[x]] be the ring of power series in OL. We extend the action of
+GonOLtoBby setting
+σ1(x) :=x, σ 2(x) =−x.
+6This is the example announced above. Indeed, it is easy to see that
+A1=Bσ1=OK1[[x]]
+is a power series ring over OK1and hence regular. Similarly, if we set y:=
+(1+π)x, then we find that
+B=OL[[y]],andσ2(y) =y.
+It follows that
+A2=Bσ2=OK2[[y]]
+is regular, too. However, A=BG=Aσ2
+1is not regular. This can be checked
+using the if-and-only-if criterion from Example 1.10. Note that ( π1,x) is a
+regular sequence of parameters for A1. Now
+σ2(π1)−π1=−2π1, σ2(x)−x=−2x,
+and so
+Iσ2= (−2π1,−2x) = (2)·mA1✁A1
+is not a principal ideal. Using the criterion of [6] mentioned in Example 1.1 0,
+we conclude that Ais not regular. The reader is invited to check this by a direct
+calculation of the dimension of mA/m2
+A.
+2 Derivations and p-cyclic inseparable descent
+In this section we prove an auxiliary result (Corollary 2.2) which is a cru cial
+step in the proof of Theorem 1.3. The setup is similar as in the previous section;
+however, we work in equal characteristic p, and instead of an automorphism of
+orderpwe consider a derivation of the ring B.
+Let (B,m,k) denote a noetherian regular local ring of dimension d≥1.
+We assume moreover that Bis complete and has characteristic p. Then it
+follows from [9], Corollary 2 of Theorem 60, that Bis isomorphic to a formal
+power series ring over k. More precisely, if ( x1,...,x d) is a regular system of
+parameters for B, then we get an isomorphism
+B∼=k[[x1,...,x d]].
+We letLdenote the fraction field of B. We write Der k(B) for the p-Lie-algebra
+ofk-derivations θ:B→B. Note that any such derivation extends uniquely to
+a (continuous) derivation of L.
+Lemma 2.1 Letθ∈Derk(B)be ak-derivation of Bsuch that the following
+holds:
+(i)thek-linear map
+¯θ:m/m2→k
+induced by θis not zero,
+7(ii)we have θp=aθ, for some a∈Frac(B).
+Then there exists a regular system of parameters (x1,...,x d)forBsuch that
+θ(x1)≡1 (mod m)
+and
+θ(x2) =...=θ(xd) = 0.
+Proof:Letθbe as in the statement of the lemma. It follows immediately
+from (i) that there exists a system of parameters x1,...,x dsuch that
+θ(x1)≡1 (mod m),
+θ(xi)≡0 (mod m) fori >1.(1)
+Our strategy is to change the xifori >1 step by step in order to improve the
+last congruence modulo arbitrary powers of m. SinceBis complete, we can take
+the limit and find parameters xisuch that θ(xi) = 0 for i >1.
+Suppose that
+θ(xi)≡0 (mod mn) fori >1, (2)
+for some n≥1. We claim that there exist elements ∆ 2,...,∆d∈mn+1such
+that
+θ(∆i)≡ −θ(xi) (mod mn+1), i= 2,...,d. (3)
+Assuming this claim, we can set
+˜xi:=xi+∆i, i= 2,...,d,
+and obtain
+θ(˜xi)≡0 (mod mn+1), i= 2,...,d.
+The lemma then follows by induction and a limit argument.
+To prove the claim we consider the k-linear map
+¯θn:mn+1/mn+2→mn/mn+1
+induced by θ. We have to show that for i >1 the class of θ(xi) inmn/mn+1
+lies in the image of ¯θn. A short calculation, using (1) and (2), shows that the
+image of ¯θnis spanned by the images of the monomials of degree n
+xl1
+1xl2
+2···xld
+d, l 1+...+ld=n,
+such that
+l1/\e}atio\slash≡ −1 (mod p).
+In particular, for n < p−1 the map ¯θnis surjective, and the claim is true.
+Suppose now that n≥p−1 and fix an index i >1. By (2) we can write
+θ(xi)≡n/summationdisplay
+l=0alxl
+1yl(modmn+1),
+8whereal∈kandylis a homogenous polynomial of degree n−lin the variables
+x2,...,x d. A straightforward calculation, using (1) and (2), shows that
+θp(xi)≡n/summationdisplay
+l=p−1l(l−1)···(l−p+2)alxl−p+1
+1yl(modmn−p+2).(4)
+On the other hand, it follows from (ii) and (2) that
+θp(xi) =aθ(xi)≡0 (mod mn). (5)
+(Here we have used that the element a∈Frac(B) occuring in (ii) is of the form
+a=θ(x1)−1θp(x1) and therefore lies in B.) Combining (4) and (5) we find that
+al= 0 ifl≡ −1 (modp). This shows that the class of θ(xi) inmn/mn+1lies in
+the image of ¯θnand proves the claim. The proof of Lemma 2.1 is now complete.
+✷
+Corollary 2.2 Letθ∈Derk(B)be as in Lemma 2.1. Then:
+(i)The subring A:= Ker(θ)⊂Bis regular, and B/Ais finite and flat of rank
+p.
+(ii)Suppose that
+θ(f)∈B,
+for some element f∈L. Then there exists f0∈Bsuch that
+θ(f0) =θ(f).
+(iii)Suppose that
+θ(f)/f∈B,
+for some element f∈L×. Then there exists u∈B×such that
+θ(u)/u=θ(f)/f.
+Proof:Let (x1,...,x d) be a regular system of parameters as in Lemma 2.1,
+and seth:=θ(x1)∈B×. Thenθ=hθ0, where
+θ0(x1) = 1, θ0(x2) =...=θ0(xd) = 0.
+Clearly, we have A= Ker(θ) = Ker(θ0). It is now easy to see that
+A=k[[xp
+1,x2,...,x d]]
+is regular and that
+B=A⊕A·x1⊕...⊕A·xp−1
+1 (6)
+is finite and flat over Aof rankp. So (i) is proved.
+9For the proof of (ii) we write
+f=a0+a1x1+...+ap−1xp−1
+1,
+withai∈K:= Frac(A) =L∩A. Then the hypothesis θ(f)∈Bimplies
+θ0(f) =a1+2a2x1+...+(p−1)ap−1xp−2
+1∈B.
+Now (6) shows that a1,...,a p−1∈A. Therefore,
+θ(f0) =θ(f),withf0:=a1x1+...+ap−1xp−1
+1∈B.
+This proves (ii).
+Assertion (iii) follows from [11], Theorem 2. For the convenience of th e
+reader, we sketch the argument.
+Letf∈L×be given such that θ(f)/f∈B. We claim that there exists a
+Weil divisor don Spec( A) such that
+B·d= (f).
+(This is obviously a ‘descent argument’ and explains the title of this se ction.)
+Assuming the claim we can prove (ii), as follows. By (i), Ais regular and in
+particular factorial. Therefore, d= (f0), for some f0∈K= Frac(A). So
+f=uf0for a unit u∈B×, and we get θ(f)/f=θ(u)/u, as desired.
+To prove the claim, it suffices to show the following: for every prime ide al
+p✁Bof height one we have
+ep|vp(f).
+Herevpis the normalized valuation on Lassociated to pandepdenotes the
+ramification index of pin the field extension L/K. Sinceep∈ {1,p}, we may
+assume that n:=vp(f) is prime to p, and we have to show that ep= 1.
+Lett∈Bpbe a uniformizer for vp. Then we can write
+f=utn,withu∈B×
+p.
+From
+θ(f)/f=θ(u)/u+n·θ(t)/t∈B
+we conclude θ(t)/t∈Bp. This means that θinduces a derivation ¯θon the
+residue field kp. From assumption (i) of Lemma 2.1 it follows that ¯θ/\e}atio\slash= 0. From
+kp∩A⊂k¯θ=0
+p/subsetnoteqlkp
+we conclude
+fp:= [kp:kp∩A] =p.
+Now the inequality p= [L:K]≥ep·fpimpliesep= 1, and the claim is proved.
+✷
+103 Ramification of p-cyclic Galois extensions
+In the situation of Theorem 1.3, we obtain a cyclic Galois extension L/Kof de-
+greepby taking fraction fields: L:= Frac(B),K:= Frac(A) =LG. The present
+section contains some preliminary investigation of the ramification of this ex-
+tension with respect to the discrete valuation corresponding to th e parameter
+π. The main point here is that for p-cyclic Galois extensions it is possible to
+distinguish two types of wild ramification: total ramification andfierce ramifi-
+cation. Accordingly, the proof of Theorem 1.3 will be divided into these two
+cases.
+3.1LetL/Kbe a cyclic Galois extension of degree p(wherepis, as always,
+a prime number). Let vbe a discrete valuation on Lwhich is fixed by G.
+We choose a generator σofG. The valuation rings of KandLwith respect
+tovare denoted by OKandOL, the residue fields by ¯Kand¯L. The letter π
+will always denote a uniformizer for vonL(i.e.πOL=pLis the maximal ideal
+ofOL). We normalize the valuation vsuch that v(π) = 1.
+We set
+eL/K:= [v(L×) :v(K×)], f L/K:= [¯L:¯K].
+These invariants are related by the fundamental equality
+eL/K·fL/K= [L:K] =p.
+See e.g. [7], Corollary XII.6.3.
+Definition 3.1 We classify the ramification behavior of L/Katvas follows:
+I:IfeL/K=pandfL/K= 1, we call L/Ktotally ramified . There are two
+subcases:
+(a) If char( ¯L)/\e}atio\slash=p,L/Kis called tamely ramified .
+(b) If char( ¯L) =p,L/Kis called totally wildly ramified .
+II:Suppose that eL/K= 1 and fL/K=p. There are again two subcases:
+(a) If¯L/¯Kis Galois, then L/Kis called unramified .
+(b) If¯L/¯Kis inseparable, then L/Kis called fiercely ramified .
+The following invariant is useful to distinguish these cases:
+δ:= min{v(σ(x)−x)|x∈ OL} ≥0. (7)
+Proposition 3.2 (i)L/Kis unramified if and only if δ= 0.
+(ii)IfL/Kis tamely ramified, then δ= 1. (Note: the converse of this impli-
+cation is actually false: L/Kmay be fiercely ramified.)
+11(iii)The extension L/Kis totally ramified if and only if
+δ=v(σ(π)−π), (8)
+for some uniformizer π. In this case (8)holds for every uniformizer π.
+Proof:Assertions (i) and (ii) are classical. See e.g. [13], IV, §1. Assertion
+(iii) is certainly well known as well. We will nevertheless give a proof beca use
+it yields some useful insight.
+We may assume that δ >0. Letπbe an arbitrary uniformizer. By the
+definition of δ, the map
+˜θπ:OL→¯L, x /ma√sto→/parenleftBigσ(x)−x
+πδmodπ/parenrightBig
+,
+is well defined and does not vanish. A short calculation shows that ˜θπis a
+derivation (i.e. ˜θπ(x+y) =˜θπ(x) +˜θπ(y) and˜θπ(xy) = ¯x·˜θπ(y) + ¯y·˜θπ(x)),
+and hence we have
+˜θπ(aπ) = ¯a·˜θπ(π), (9)
+for alla∈ OL.
+Suppose that ˜θπ(π) = 0. This is equivalent to the condition
+δ < v(σ(π)−π). (10)
+Then (9) shows that both the equation ˜θπ(π) = 0 and (10) hold in fact for all
+uniformizers π. Moreover, ˜θπinduces a derivation
+θπ:¯L→¯L,¯x/ma√sto→˜θπ(x) modπ.
+We have θπ/\e}atio\slash= 0 by definition of δ. In particular, there exists a unit x∈ O×
+L
+withδ=v(σ(x)−x). It is also clear that
+¯K⊂Ker(θπ)/subsetnoteql¯L.
+Since [¯L:¯K]≤p, it follows that ¯K= Ker(θπ) and [¯L:¯K] =p. In particular,
+L/Kis fiercely ramified (Case II (b)).
+On the other hand, if L/Kis fiercely ramified, then eL/K= 1, which means
+that there exists a uniformizer πinOK=OG
+L. Then we obviously have ˜θπ(π) =
+0. All claims made in Proposition 3.2 have now been proved. ✷
+3.2Suppose that L/Kis fiercely ramified (Case II (b)). In this case the
+proof of Proposition 3.2 yields the following.
+Corollary 3.3 Suppose that L/Kis fiercely ramified. Then for every uni-
+formizer πthe map
+θπ:¯L→¯L,¯x/ma√sto→/parenleftBigσ(x)−x
+πδmodπ/parenrightBig
+,
+is a derivation with the following properties.
+12(i)θπ/\e}atio\slash= 0,
+(ii) Ker( θπ) =¯K,
+(iii)θp
+π=aθπ, for some a∈¯K.
+Proof:(i) and (ii) have been proved above. Property (iii) is true for all
+derivations of a field ¯Lof characteristic pfor which
+[¯L: Ker(θπ)] =p.
+See e.g. [3], Chapter IV.8, Exercise 3. ✷
+The following lemma will be used in Section 4.5.
+Lemma 3.4 Letx∈ OLbe such that
+v(σ(x)−x)≥δ+n,
+for some n≥1. Then there exists y∈ OKwith
+y≡x(modpn
+L).
+Proof:We proceed by induction on n, starting with n= 1. Let π∈ OKbe
+an invariant uniformizer. The assumption v(σ(x)−x)≥δ+1 means θπ(¯x) = 0.
+It follows from Corollary 3.3 (ii) that ¯ x∈¯K. So we can take for y∈ OKany
+representative of ¯ x.
+We may hence suppose n≥2. By induction, there exists an element z∈ OK
+such that z≡x(modπn−1). Write x=z+aπn−1, witha∈ OL. Then
+θπ(¯a)≡σ(x)−x
+πδ+n−1≡0 (mod π).
+As before, it follows that ¯ a∈¯K. Letb∈ OKbe any lift of ¯ aand sety:=
+z+bπn−1∈ OK. ✷
+3.3It is well known that the trace map Tr L/K:L→Kcontains interesting
+information about the ramification of L/K. We will use this only in the case
+whereL/Kis totally wildly ramified.
+As before, we fix a uniformizer π∈ OL,v(π) = 1. For any element a∈L×
+we set
+[a]π:=a·π−v(a)modπ∈¯L×.
+Remark 3.5 The map L×→¯L×,a/ma√sto→[a]π, obeys the following rules:
+(i) [π]π= 1,
+(ii) [a·b]π= [a]π·[b]π,
+(iii) [a+b]π= [a]π+[b]π, ifv(a) =v(b) and [a]π+[b]π/\e}atio\slash= 0.
+13Lemma 3.6 Suppose that L/Kis totally wildly ramified (Case I (b)). Let
+π∈ OLbe a uniformizer (i.e. v(π) = 1).
+(i)The trace induces an ¯L-linear isomorphism
+Tr :p−(p−1)(δ−1)
+L /p−(p−1)(δ−1)+1
+L∼→ OK/pK∼=¯L.
+(ii)There exists an element s(π)
+L/K∈¯L×, uniquely determined by the condition
+TrL/K(xπ−(p−1)(δ−1))≡s(π)
+L/K·x(modπ),
+for allx∈ OL.
+(iii)Ifπ′=uπis another uniformizer, with u∈ O×
+L, then
+s(π′)
+L/K= ¯u−(p−1)(δ−1)·s(π)
+L/K.
+(iv)We have
+s(π)
+L/K=−[σ(π)−π]p−1
+π.
+Proof:(cf. [4], Prop. 2.4) Set λ:=σ(π)−π. By Proposition 3.2 (ii), (iii)
+and the assumption on L/Kwe have v(λ) =δ≥2 and
+σ(λ)≡λ(modπδ+1).
+Also, for i= 1,...,p−1 we have
+σi(π)−π=λ+σ(λ)+...+σi−1(λ).
+Using Remark 3.5 (iii) and the above, one shows easily that
+v(σi(π)−π) =v(λ) =δ (11)
+and
+[σi(π)−π]π=i·[σ(π)−π]. (12)
+LetP∈ OL[X] denote the minimal polynomial of πoverK. By [13], Lemma
+III.6.2, we have
+TrL/K/parenleftbig
+πiP′(π)−1/parenrightbig
+=/braceleftBigg
+0, i= 0,...,p−2,
+1, i=p−1.(13)
+Since
+P′(π) =p−1/productdisplay
+i=1(π−σi(π)),
+14we getv(P′(π)) = (p−1)δfrom (11). To prove (i), note that an element
+a∈p−(δ−1)(p−1)
+L can be uniquely written in the form
+a=p−1/summationdisplay
+i=0aiπi
+P′(π),
+witha0,...,a p−2∈pKandap−1∈ OK. By (13) we have Tr L/K(a) =ap−1;
+furthermore, ap−1∈pKif and only if a∈p−(δ−1)(p−1)+1
+L. This proves (i). The
+Assertions(ii) and (iii) follow easily. Using (13), (12) and the definition ofs(π)
+L/K,
+we get
+s(π)
+L/K= [P′(π)]π= [p−1/productdisplay
+i=1π−σi(π)]π
+= (p−1)!·[σ(π)−π]p−1
+π=−[σ(π)−π]p−1
+π,
+proving (iv). ✷
+4 The proof of Theorem 1.3
+In this section we prove our main result, Theorem 1.3. The assumptio ns made
+in Section 1.1 and 1.2 and Conditions (i) and (ii) of Theorem 1.3 are in forc e
+throughout.
+4.1The ring of invariants A:=BGis a local, integrally closed domain. By
+Assumption (c) of Section 1.1, Ais noetherian and Bis a finite A-algebra.
+Our first step is to show that in the proof of Theorem 1.3 we may assu me
+thatBis complete. This assumption will be used later in the proof of Lemma
+4.4 and Lemma 4.5.
+LetˆAandˆBdenote the completions of AandB, with respect to their
+maximal ideals. These are complete local rings with residue field k. The action
+ofGonBextends uniquely to ˆB, and we have a canonical map ˆA→ˆBG.
+Lemma 4.1 The above map is an isomorphism, ˆA∼=ˆBG.
+Proof:By definition of Aand Assumption (c) we have an exact sequence
+offiniteA-modules
+0→A→Bσ−1−→B. (14)
+SinceBis noetherian we have rad( mAB) =mB, soˆBis the completion of the
+A-module B. SinceAis noetherian, the exact sequence (14) induces an exact
+sequence
+0→ˆA→ˆBσ−1−→ˆB,
+by [9], Thm. 54. This proves the lemma. ✷
+15By [1], Prop. 11.24, ˆAis regular if and only if Ais regular. So all we have to
+show is that the action of GonˆBsatisfies Hypothesis (i) and (ii) of Theorem
+1.3 (if it does for B).
+Letx1,...,x dbe a regular system of parameters for B; it is also a regular
+system of parameters for ˆB. Moreover, it is easy to check that
+Iσ= (σ(x1)−x1,...,σ(xd)−xd)✁B.
+Therefore,
+ˆIσ:= (σ(x)−x|x∈ˆB) =ˆB·Iσ.
+Our claim follows immediately.
+For the rest of this section we may therefore assume that Bis a complete
+local ring.
+4.2We set¯B:=B/Bπ; this is a complete local ring with residue field k.
+Sinceπis a regular parameter of B,¯Bis regular. It follows from Condition (i)
+of Theorem 1.3 that the map
+˜θπ:B→¯B, x /ma√sto→/parenleftBigσ(x)−x
+πδ(modπ)/parenrightBig
+is a well defined derivation. Moreover, the ideal generated by the im age of˜θπ
+is equal to ¯B.
+LetL:= Frac(B) denote the fraction field of B. Letv:L×→Zdenote the
+discrete valuation corresponding to the prime ideal Bπ✁B, normalized such
+thatv(π) = 1. So the residue field of vis the fraction field of ¯B,¯L= Frac(¯B).
+LetK:=LGbe the fixed field. Then L/Kis a Galois extension with Galois
+groupG=/a\}bracketle{tσ/a\}bracketri}ht∼=Z/pZ, andvis fixed by G. We are therefore in the situation of
+Section 3. Clearly, the number δfrom Theorem 1.3 is the same as the invariant
+defined by (7):
+δ= min{v(σ(x)−x)|x∈ OL}.
+Sinceδ >0, the extension L/Kis ramified, i.e. Case II (a) in Definition 3.1 is
+excluded. There are three cases left.
+If˜θπ(π)/\e}atio\slash= 0, then L/Kis totally ramified by Proposition3.2 (iii). Moreover,
+Condition (ii) of Theorem 1.3 implies that ˜θπ(π)∈¯B×is a unit. If δ= 1 then
+L/Kis tamely ramified (Case I (a)); otherwise, L/Kis totally wildly ramified
+(Case I (b)).
+On the other hand, if ˜θπ(π) = 0, then L/Kis fiercely ramified (Case II (b)).
+In this case the derivation ˜θπinduces a nontrivial derivation θπ:¯B→¯B, and
+the unique extension of θπto¯Lis the derivation from Corollary 3.3.
+After these preliminary remarks, the proof of Theorem 1.3 is done b y a
+case-by-case analysis of the three different types of ramification ofvinL/K.
+164.3 Case I (a): L/Kis tamely ramified Suppose that char( ¯L)/\e}atio\slash=p. By
+Proposition 3.2 (ii) we have δ= 1. Therefore, the morphism
+G→Aut(Bπ/Bπ2)
+is injective. It follows that
+σ(π)≡ζp·π(modπ2),
+for some primitive pth root of unity ζp∈¯B. But the assumption Iσ= (π)
+implies that
+σ(π)−π=uπ
+for a unit u∈B×. From the above we get
+u≡ζp−1 (mod m).
+In particular, ζp−1/\e}atio\slash∈m, which shows that char( k)/\e}atio\slash=p. Now the hypothesis of
+Theorem 1.3 implies that Ais regular, see [12] and Remark 1.4.
+4.4 Case I (b): L/Kis totally wildly ramified In this case Condition
+(ii) of Theorem 1.3 implies
+σ(π)−π=uπδ, (15)
+whereu∈B×is a unit and δ≥2. Letp:=A∩Bπ✁A. This is a prime ideal
+of height one.
+Lemma 4.2 The element
+λ:=NL/K(π) =p−1/productdisplay
+i=0σi(π)∈A
+generates p. In particular, pis a principal ideal.
+Proof:It is obvious that λ∈p. Letα∈pbe an arbitrary element. To
+prove that p= (λ), it suffices to show that λ|αinB.
+Wehave π|αbydefinitionof p. Thismeansthat v(α)>0. Sinceα∈A⊂K
+andeL/K=pby assumption, we actually have v(α)≥p. This implies πp|α.
+By (15) we have
+σ(π) =π(1+uπδ−1)∼π.
+Herea∼bmeans:b=vafor a unit v∈B×. It follows that σi(π)∼πfor alli
+and hence λ∼πp. Therefore, λ|α, as desired, and the lemma is proved. ✷
+Let¯A:=A/p. We consider ¯Aas a subring of ¯Bvia the natural embedding
+¯A ֒→¯B=B/Bπ.
+Lemma 4.3 We actually have ¯A=¯B. In particular, ¯Ais regular.
+17Proof:Clearly, the rings ¯Aand¯Bhave the same fraction field ¯L. Consider
+the¯L-linear map
+Tr :¯L→¯L,¯x/ma√sto→/parenleftbig
+TrL/K(xπ−(p−1)(δ−1)) modπ/parenrightbig
+,
+studied in Section 3.3. We claim that Tr(¯B)⊂¯A. Indeed, if xis an element of
+B, then the proof of Lemma 3.6 shows that
+v(TrL/K(xπ−(p−1)(δ−1)))≥0.
+Therefore, Tr L/K(xπ−(p−1)(δ−1))∈(B[π−1]∩OL)G=BG=Aby factoriality
+ofB, proving the claim.
+It follows from (15) and Lemma 3.6 (ii), (iv) that
+Tr(¯x) =−¯up−1¯x,
+where ¯u∈¯B×is the image of uin¯B. We conclude −¯up−1¯B=¯B⊂¯A. So
+¯A=¯B, and the lemma is proved. ✷
+Using the two lemmas it is now easy to see that Ais regular: if ¯ x1,...,¯xd−1
+is a regular sequence of parameters for ¯Aandxi∈Ais a lift of ¯ xi, then
+(λ,x1,...,x d−1) is a regular system of parameters for A.
+4.5 Case II (b): L/Kis fiercely ramified Letθ=θπ∈Der(¯L) be the
+derivation from Corollary 3.3. It follows from the definition of θthat
+θ(¯B)⊂¯B.
+Also, Condition (i) of Theorem 1.3 implies that there exists an element x∈B
+such that
+σ(x)−x
+πδ∈B×.
+This shows that the k-linear map
+¯θ:m¯B/m2¯B→k.
+inducedby θisnotzero, ¯θ/\e}atio\slash= 0. Since θp=aθforsomeelement a∈¯K(Corollary
+3.3 (iii)), the derivation θsatisfies the hypotheses of Lemma 2.1 (with respect
+to the ring ¯B). Therefore, we may (and will) use Corollary 2.2 in the proof of
+the following two lemmas.
+Lemma 4.4 There exists an element λ∈Aof the form λ=uπ, withu∈B×.
+Proof:Byinduction, wewill constructasequenceofelements λ0,λ1,...∈B
+of the form λn=unπ,un∈B×, such that:
+(a)λn≡λn−1(modπn),
+(b)v(σ(λn)−λn)≥δ+n+1,
+18for alln≥0. The limit λ:= lim nλnis then an element λ∈A=BGof the
+formλ=uπ, withu∈B×.
+We start the induction at n= 0 by setting λ0:=π. Then Condition (a) is
+empty and (b) is true by Proposition 3.2 (iii).
+The case n= 1 is special. The uniformizer λ1we are looking for is of the
+formλ1=u1π, withu1∈B×. Condition (a) is still empty, and Condition (b)
+is equivalent to
+0≡σ(λ1)−λ1
+πδ+1≡u1w+θπ(¯u1) (mod π),
+wherew:= (σ(π)−π)π−δ−1∈B. So we have to find an element ¯ u1∈¯B×with
+θπ(¯u1)
+¯u1=−¯w. (16)
+By Lemma 3.4, there exists an element µ∈ OKwithµ≡π(modp2
+L). If we
+writeµ=vπ, withv∈ O×
+L, then the same calculation that resulted in (16)
+shows that
+θπ(¯v)
+¯v=−¯w∈¯B.
+Now it follows from Corollary 2.2 (iii) that there exists ¯ u1∈¯B×such that (16)
+holds. Now the case n= 1 of the lemma is proved.
+We may hence assume that n≥2 and that λ0,...,λ n−1have already been
+constructed. Since λn−1∼π, we may assume that λn−1=π. In particular,
+v(σ(π)−π)≥δ+n. (17)
+The element λnwe are looking for is of the form
+λn=π(1+aπn),
+for some a∈B. Using (17) we get
+σ(λn)−λn
+πδ+n≡c+θπ(¯a) (mod π),
+wherec:= (σ(π)−π)π−δ−n∈B. This means that we have to find an element
+¯a∈¯Bsuch that
+θπ(¯a) =−¯c. (18)
+By (17) and Lemma 3.4, there exists an element µ∈ OKwithµ≡π(modpn
+L).
+Writeµ=π(1+bπn), withb∈ OL. Then the same calculation from which we
+deduced (18) yields
+θπ(¯b) =−¯c.
+Using Corollary 2.2 (ii) we find an element ¯ a∈¯Bsatisfying (18). Now the proof
+of the lemma is complete. ✷
+19Letλ∈Abe as in the lemma. We get a natural embedding
+¯A:=A/Aλ ֒→¯B=B/Bπ.
+It is clear that ¯Ais a complete local ring with residue field kand quotient field
+¯K.
+Lemma 4.5 We have ¯A=¯B∩¯K.
+Proof:The proof is very similar to the proof of the previous lemma. The
+inclusion ¯A⊂¯B∩¯Kis clear. To provethe converse,suppose we havean element
+x∈Bsuch that ¯ x∈¯K. We will inductively construct a sequence y0,y1,...∈B
+such that
+(a)x≡yn(modπ),
+(b)yn≡yn−1(modπn),
+(c)v(σ(yn)−yn)≥δ+n+1,
+for alln≥0. The limit y:= lim nynis then an element y∈A=BGwithx≡y
+(modπ).
+Forn= 0 we set y0:=x. Then (a) is clear, (b) is empty and (c) follows
+from ¯x∈¯K. So we may assume n≥1. The element ynwe are looking for is of
+the form yn=yn−1+aπn, witha∈B. The crucial Condition (c) is equivalent
+to
+σ(yn)−yn
+πδ+n≡c+θπ(¯a)≡0 (mod π),
+wherec:= (σ(yn−1)−yn−1)π−δ−n∈B. So we have to find ¯ a∈¯Bsuch that
+θπ(¯a) =−¯c. (19)
+By Lemma 3.4, there does exist an element z∈ OLsatisfying the properties
+required for yn. If we write z=yn−1+bπn, withb∈ OL, then we get θπ(¯b) =
+−¯c∈¯B. Now Corollary 2.2 (ii) shows that ¯ a∈¯Bsatisfying (19) exists, and we
+are done. ✷
+The lemma shows that ¯A= Ker(θπ|¯B). By Corollary2.2 (i) it follows that ¯A
+is regular. With the same argument used at the end of the previous s ubsection
+we conclude that Ais regular. Now Theorem 1.3 is proved. ✷
+References
+[1] M.F. Atiyah and I.G. Mcdonald. Introduction to Commutative Algebra .
+Addison-Wesley, 1969.
+[2] J. Fogarty. K¨ ahler differentials and Hilbert’s fourteenth proble m for finite
+groups.Amer. J. Math. , 102(6):1159–1175, 1980.
+20[3] N. Jacobson. Lectures in Abstract Algebra III: Theory of Fields and Galois
+Theory. Springer, 1964.
+[4] K. Kato. Swan conductors with differential values. In Galois Representa-
+tions and Arithmetic Algebraic Geometry , number 12 in Advanced Studies
+in Pure Math., pages 315–342, 1987.
+[5] N.M. Katz and B. Mazur. Arithmetic Moduli of Elliptic Curves . Princeton
+Univ. Press, 1985.
+[6] F. Kir´ aly. Quotient singularities and models of curves . PhD thesis, Uni-
+versit¨ at Ulm, expected 2010.
+[7] S. Lang. Algebra. Addison-Wesley, 1993.
+[8] D. Lorenzini. Wild quotient singularities of surfaces. Preprint, 20 09.
+[9] H. Matsumura. Commutative Algebra . Benjamin, 2nd edition, 1980.
+[10] B.R. Peskin. Quotient-singularities and wild p-cyclic actions. J. Algebra ,
+81:72–99, 1983.
+[11] P. Samuel. Classes de diviseurs et d´ eriv´ ees logarithmiques. Topology ,
+3(1):81–96, 1968.
+[12] J.-P. Serre. Groupes finis d’automorphismes d’anneaux locaux r ´ eguliers. In
+Colloque d’Alg` ebre (Paris, 1967), Exp. 8 . Secr´ etariat math´ ematique, 1967.
+[13] J.-P. Serre. Corps locaux . Hermann, 3. edition, 1968.
+[14] J. Watanabe. Some remarks on Cohen-Macaulay rings with many zero
+divisors and an application. J. Algebra , 39:1–14, 1976.
+21
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+arXiv:1001.0608v1  [quant-ph]  5 Jan 2010AnEfficientQuantumAlgorithmfor someInstances
+oftheGroupIsomorphismProblem
+Franc ¸oisLe Gall
+Department of Computer Science
+Graduate School of Information Science and Technology
+TheUniversity of Tokyo
+email: legall@is.s.u-tokyo.ac.jp
+Abstract. Inthispaperweconsidertheproblemoftestingwhethertwofi nitegroups
+are isomorphic. Whereas the case where both groups are abeli an is well understood
+and can be solved efficiently, very little is known about the c omplexity of isomor-
+phism testing for nonabelian groups. Le Gall has constructe d an efficient classical
+algorithm for aclass ofgroups corresponding tooneof themo stnatural waysof con-
+structing nonabelian groups from abelian groups: the group s that are extensions of
+an abelian group Aby a cyclic group Zmwith the order of Acoprime with m. More
+precisely, the running time of that algorithm is almost line ar in the order of the input
+groups. In this paper we present a quantum algorithm solving the same problem in
+time polynomial in the logarithm of the order of the input groups. This algorithm
+worksintheblack-box setting and isthefirst quantum algori thm solving instances of
+the nonabelian group isomorphism problem exponentially fa ster than the best known
+classical algorithms.
+1 Introduction
+Background Testing group isomorphism (the problem asking to decide, fo r two given finite
+groupsGandH, whether there exists an isomorphism between GandH) is a fundamental prob-
+lem in computational group theory but little is known about i ts complexity. It is known that the
+group isomorphism problem (for groups given bytheir multip lication tables) reduces tothe graph
+isomorphism problem [22], and thus the group isomorphism pr oblem is in the complexity class
+NP∩coAM(since the graph isomorphism problem is in this class [2]). M iller [27] has developed
+a general technique to check group isomorphism in time O(nlogn+O(1)), wherendenotes the size
+of the input groups and Lipton, Snyder and Zalcstein [25] hav e given an algorithm working in
+O(log2n)space. However, no polynomial time algorithm is known for th e general case of this
+problem.
+Another lineofresearchisthedesignofalgorithms solving thegroupisomorphism problem for
+particular classes of groups. For abelian groups polynomia l-time algorithms follow directly from
+efficient algorithms for the computation of the Smith normal form of integer matrices [10, 18].
+More efficient methods have been given by Vikas [33] and Kavit ha [20] for abelian groups given
+by their multiplication tables, and fast parallel algorith ms have been constructed by McKenzie
+and Cook [26] for abelian permutation groups. The current fa stest algorithm solving the abelian
+group isomorphism problem for groups given as black-boxes h as been developed by Buchmann
+and Schmidt [7] and works in time O(n1/2(logn)O(1)). However, as far as nonabelian groups are
+concerned, very little is known. For solvable groups Arvind and Tor´ an [1] have shown that the
+group isomorphism problem is in NP∩coNPunder certain complexity assumptions but, until
+This work was done while the author was a researcher at Kyoto U niversity, affiliated with the ERATO-SORST
+Quantum Computation and InformationProject,Japan Scienc e and Technology Agency.
+1recently, the only polynomial-time algorithms testing iso morphism of nontrivial classes of non-
+abelian groups werearesult byGarzon andZalcstein [15],wh ichholds for averyrestricted class,
+andabody ofworks initiated byCooperman et al.[11] onsimpl egroups, whichwill bediscussed
+later.
+Very recently, Le Gall [23] proposed an efficient classical a lgorithm solving the group iso-
+morphism problem over another class of nonabelian groups. S ince for abelian groups the group
+isomorphism problem can besolved efficiently, that workfoc used onone of the most natural next
+targets: cyclic extensions of abelian groups. Loosely spea king such extensions are constructed
+by taking an abelian group Aand adding one element ythat, in general, does not commute with
+the elements in A. More formally the class of groups considered in [23], denot ed by S, was the
+following.
+Definition 1.1. Let G be a finite group. The group G is said to be in the class Sif there exists
+a normal abelian subgroup A in G and an element y ∈G of order coprime with |A|such that
+G=/a\}bracketle{tA,y/a\}bracketri}ht.
+Intechnicalwords Gisanextensionofanabeliangroup Abyacyclicgroup Zmwithgcd(|A|,m)=
+1. This class of groups includes all the abelian groups and ma ny non-abelian groups too, as
+discussed in details in [23]. For example, for A=Z4
+3andm=4, there are exactly 9 isomorphism
+classes in S(1 class of abelian groups and 8 classes of nonabelian groups ). Moreover, the class
+Sincludesseveralgroupsthathavebeenthetargetofquantum algorithms, asdiscussedlater. The
+mainresult in[23] was thefollowing theorem.
+Theorem 1.1 ([23]).There exists a deterministic algorithm checking whether tw o groups G and
+H intheclass S(givenasblack-box groups) areisomorphicand, ifthisisth ecase, computingan
+isomorphism from GtoH. Itsrunning timehasforupper bound n1+o(1),wheren =min(|G|,|H|).
+Statement of our results In the present paper, we focus on quantum algorithms solving the
+group isomorphism problem in the black-box setting. Cheung and Mosca [9] have shown how to
+compute the decomposition of an abelian group into a direct p roduct of cyclic subgroups in time
+polynomial inthelogarithm ofitsorder onaquantum compute r, andthushowtosolvetheabelian
+groupisomorphism problem intimepolynomial inlog nintheblack-box model. (Noticethattheir
+algorithm is actually a generalization of Shor’s algorithm [31], which can be seen as solving the
+group isomorphism problem over cyclic groups.) This then gi ves an exponential speed-up with
+respect to the best known classical algorithms for the same t ask. One can naturally ask whether
+a similar speed-up can be obtained for classes of nonabelian groups. In this paper, we prove that
+this isthe case. Our main result isthe following theorem.
+Theorem 1.2. There exists a quantum algorithm checking with high probabi lity whether two
+groups G and H in the class Sgiven as black-box groups are isomorphic and, if this is the
+case, computing an isomorphism from G to H. Its running time i s polynomial in logn, where
+n=min(|G|,|H|).
+Toour knowledge, thisisthefirstquantum algorithm solving nonabelian instances ofthegroup
+isomorphism problem exponentially faster than the best kno wn classical algorithms. Our algo-
+rithm relies on several new quantum reductions to instances of the so-called abelian Hidden Sub-
+group Problem, a problem that can be solved efficiently on a qu antum computer. Our result can
+then be seen as an extension of the polynomial time library of computational tasks which can be
+2accomplished using Shor’s factoring and discrete logarith m algorithms [31], and further quantum
+algorithms for abelian groups. Wealsomention that groups i ntheclass Sappear atseveral occa-
+sions in the quantum computation literature, mostly connec ted to the Hidden Subgroup Problem
+oversemidirectproductgroups[6,13,16,28]. Ourtechniqu esmayhaveapplications inthedesign
+of further quantum algorithms for this problem, or for other similar group-theoretic tasks.
+Overview of ouralgorithm Ourquantum algorithm followsthesamelineastheclassical algo-
+rithm in [23], but the twomaintechnical parts areboth signi ficantly improved and modified.
+Since a group Gin the class Smay in general be written as the extension of an abelian group
+A1by a cyclic group Zm1and as the extension of an abelian group A2by a cyclic group Zm2
+withA1/\e}atio\slash∼=A2andm1/\e}atio\slash=m2, we use, as in [23], the concept of a standard decomposition o fG,
+which is an invariant for the groups in the class Sin the sense that two isomorphic groups have
+similar standard decompositions (but the converse is false ). A method for computing efficiently
+standard decompositions intheblack-box model wasoneofth emaincontributions of[23],where
+the time complexity of this step was O(n1+o(1))due to the fact that the procedure proposed had
+to try, in the worst case, for each generator gofG, all the divisors of |g|. Instead, in the present
+work we propose a different procedure for this task (Section 3), which can be implemented in
+time polynomial in log non a quantum computer, based on careful reductions to group- theoretic
+problems for which known efficient quantum algorithms are kn own: order finding, decomposing
+abelian groups and constructive membership inabelian grou ps.
+Knowing standard decompositions of GandHallows us to consider only the case where H
+andGare two extensions of the same abelian group Aby the same cyclic group Zm(Proposition
+6.1). Twomatrices M1andM2inthe group GL(r,F)of invertible matrices of size r×rover some
+well-chosen finite field Fcan then be associated to the action of ZmonAin the groups Gand
+Hrespectively. The second main technical contribution of [2 3] showed that, loosely speaking,
+testingisomorphism of GandHthenreduces(whentheorderof Aiscoprimewith m)tochecking
+whether there exists an integer k∈{1,...,m}such that M1andMk
+2are conjugate in GL(r,F)(a
+precise version of this statement is given in Proposition 6. 2 of the present paper). The strategy
+adopted in [23]to solve this problem had time complexity close to nin the worst case (basically,
+all the integers kin{1,...,m}were checked). In the present paper, we give a poly (logn)time
+quantum algorithm for this problem. More generally, we show in Section 5 that the problem of
+testing, for any two matrices M1andM2inGL(r,F)whereris any positive integer and Fis any
+finite field, whether there exists a positive integer ksuch that M1andMk
+2are conjugate in the
+groupGL(r,F)reduces tosolving aninstance of aproblem wecall S ETDISCRETE LOGARITHM .
+Thisquantum reduction isefficient inthat it canbeimplemen ted intimepolynomial inboth rand
+log|F|,and works by considering fieldextensions of Fand matrix invariants of M1andM2.
+Loosely speaking, the problem S ETDISCRETE LOGARITHM asks, given two sets {x1,...,xv}
+and{y1,...,yv}of elements in F, to compute an integer ksuch that{yk
+1,...,yk
+v}={x1,...,xv},
+if such an integer exists. This computational problem is a ge neralization of the standard discrete
+logarithm problem (which is basically the case v=1) but appears to be much more challenging.1
+The quantum algorithm we propose (in Section 4) works in time polynomial in vand log|F|, and
+1To illustrate this point, let us consider the following simp le strategy: for each j∈{1,...,v}, try to find some k
+suchthat yk
+1=xjusingthequantumalgorithmforthestandarddiscretelogar ithmproblembyShor[31],andthencheck
+whether{yk
+1,...,ykv}={x1,...,xv}. Theproblemhereisthata ksuchthat yk
+1=xjwillbeonlydefinedmodulo |y1|,and
+itmaybethecasethat {yk
+1,...,ykv}/\e}atio\slash={x1,...,xv}but{yk′
+1,...,yk′
+v}={x1,...,xv}forsomek′satisfying k′=kmod|y1|.
+Testingall these k′’s cantake exponential time.
+3reliesonareductiontoseveralinstancesoftheabelianHid denSubgroupProblem. Oursolutionto
+the problem S ETDISCRETE LOGARITHM is then an extension of the computational tasks which
+canbe solved efficiently using knownquantum algorithms for abelian groups.
+Other related works To our knowledge, the only other work on polylogarithmic tim e non-
+abelian group isomorphism testing in the back-box setting i s a body of results, initiated by Coop-
+erman et al. [11], focusing on identifying simple groups. Re member that a simple group is a
+groupthat hasnonontrivial normal subgroup. Acelebrated r esult ingrouptheory classifies allthe
+simple finite groups into 26 sporadic groups and a few numbers of infinite classes in which each
+group has a label of some prescribed form. A natural question that arises is, given a black-box
+group guaranteed to be simple, how to compute this label, i.e ., how to identify this group? It is
+known that, based on the mathematical properties of the simp le groups, it is possible to do this
+(classically) in polylogarithmic time whenever the input i s guaranteed to be a so-called classical
+group over a field of known characteristic. We refer to the boo k by Kantor and Seress [19] and
+references therein for anextensive treatment of this subje ct.
+2 Preliminaries
+2.1 Group theory andstandard decompositions
+We assume that the reader is familiar with the basic notions o f group theory and state without
+proofs definitions and properties of groups wewill use inthi s paper.
+Foranypositiveinteger m,wedenoteby Zmtheadditivecyclicgroupofintegers {0,...,m−1},
+and byZ∗
+mthemultiplicative group of integers in {1,...,m−1}coprime with m.
+LetGbe a finite group. For any subgroup Hand any normal subgroup KofGwe denote by
+HKthe subgroup{hk|h∈H,k∈K}={kh|h∈H,k∈K}. Given a set Sof elements of G, the
+subgroup generated by the elements of Sis written/a\}bracketle{tS/a\}bracketri}ht. We say that two elements g1andg2ofG
+are conjugate in Gif there exists an element y∈Gsuch that g2=yg1y−1. For any two elements
+g,h∈Gwe denote by [g,h]the commutator of gandh, i.e.,[g,h] =ghg−1h−1. More generally,
+given two subsets S1andS2ofG, wedefine [S1,S2]=/a\}bracketle{t[s1,s2]|s1∈S1,s2∈S2/a\}bracketri}ht. Thecommutator
+subgroupof Gisdefinedas G′=[G,G]. Thederivedseriesof Gisdefinedrecursively as G(0)=G
+andG(i+1)= (G(i))′. The group Gis said to be solvable if there exists some integer ksuch that
+G(k)={e}. Given twogroups G1andG2, a mapφ:G1→G2is ahomomorphism from G1toG2
+if,for anytwoelements gandg′inG1,therelation φ(gg′)=φ(g)φ(g′)holds. Wesaythat G1and
+G2areisomorphic ifthereexistsaone-one homomorphism from G1toG2,andwewrite G1∼=G2.
+Given any finite group G, we denote by|G|its order and, given any element ginG, we denote
+by|g|theorder of ginG. Foranyprime p, wesaythat agroup isa p-group ifits order isapower
+ofp. If|G|=pei
+1...perrfor distinct prime numbers pi, then for each i∈{1,...,r}the group G
+has a subgroup of order pei
+i. Such a subgroup is called a Sylow pi-subgroup of G. Moreover, if
+Gis additionally abelian, then each Sylow pi-group is unique and Gis the direct product of its
+Sylow subgroups. Abelian p-groups have remarkably simple structures: any abelian p-group is
+isomorphic to adirect product of cyclic p-groupsZpf1×···×Zpfsfor some positive integer sand
+positive integers f1≤...≤fs, and this decomposition is unique. Wesay that a set {g1,...,gt}of
+telements of anabelian group Gisabasis of GifG=/a\}bracketle{tg1/a\}bracketri}ht×···×/a\}bracketle{t gt/a\}bracketri}htandthe order of each giis
+aprimepower.
+Foragivengroup Gintheclass Singeneral manydifferent decompositions asanextension of
+an abelian group by a cyclic group exist. For example, the abe lian group Z6=/a\}bracketle{tx1,x2|x2
+1=x3
+2=
+4[x1,x2]=e/a\}bracketri}htcan be written as/a\}bracketle{tx1/a\}bracketri}ht×/a\}bracketle{tx2/a\}bracketri}ht,/a\}bracketle{tx2/a\}bracketri}ht×/a\}bracketle{tx1/a\}bracketri}htor/a\}bracketle{tx1,x2/a\}bracketri}ht×{e}. That is why we introduce
+thenotion of astandard decomposition, asit wasdone in [23].
+Definition 2.1. Let G be a finite group in the class S. For any positive integer m denote by Dm
+G
+the set (possibly empty) of pairs (A,B)such that the following three conditions hold: (i) A is a
+normal abelian subgroup of G of order coprime with m; and (ii) B is a cyclic subgroup of G of
+order m; and (iii) G =AB. Letγ(G)be the smallest positive integer such that Dγ(G)
+G/\e}atio\slash=∅. A
+standard decomposition of Gisan element of Dγ(G)
+G.
+2.2 Black-box groups andtheabelian HiddenSubgroupProble m
+In this paper we work in the black-box model, first introduced (in the classical setting) by
+Babai and Szemer´ edi [5]. A black-box group is arepresentat ion of agroup Gwhere elements are
+represented by strings, and an oracle is available to perfor m group operations. To be able to take
+advantage of thepower of quantum computation whendealing w ithblack-box groups, theoracles
+performing groupoperations havetobeabletodeal withquan tum superpositions. Thesequantum
+black-box groups have been first studied by Ivanyos et al. [17 ] and Watrous [34, 35], and have
+become the standard model for studying group-theoretic pro blems in the quantum setting.
+More precisely, a quantum black-box group is a representati on of a group where elements are
+represented by strings (of the same length, supposed to be lo garithmic in the order of the group).
+We assume the usual unique encoding hypothesis, i.e., each e lement of the group is encoded by
+a unique string, which is crucial for technical reasons (wit hout it, most quantum algorithms do
+not work). A quantum oracle VGis available, such that VG(|g/a\}bracketri}ht|h/a\}bracketri}ht) =|g/a\}bracketri}ht|gh/a\}bracketri}htfor anygandhinG
+(using strings torepresent thegroup elements), andbehavi ng inanarbitrary wayonother inputs.2
+Wesaythatagroup Gisinputasablack-boxifasetofstringsrepresenting gener ators{g1,...,gs}
+ofGwiths=O(log|G|)is given as input, and queries to the oracle can be done at cost 1. The
+hypothesis on sisnatural since every group Ghasagenerating set of size O(log|G|),and enables
+usto make the exposition of our results easier. Also notice t hat aset of generators of any size can
+beconverted efficiently into aset of generators of size O(log|G|)if randomization isallowed [3].
+Any efficient quantum black-box algorithm gives rise to an ef ficient concrete quantum algo-
+rithm whenever the oracle operations can be replaced by effic ient procedures. Especially, when
+a mathematical expression of the generators input to the alg orithm is known, performing group
+operations can be done directly on the elements in polynomia l time (in log|G|) for many natu-
+ral groups, including permutation groups and matrix groups . This is why the black-box model
+is one of the most general settings to work with when consider ing group-theoretic problems, and
+especially whendesigning sublinear-time algorithms for s uch problems.
+Quantum algorithms areveryefficient forsolving computati onal problems overabelian groups.
+Inthe following theorem, wedescribe themain results wewil l need in this paper.
+Theorem2.1 ([9,17,31]) .Thereexistsquantumalgorithmssolving,intimepolynomia linlog|G|,
+thefollowing computational tasks with probability at leas t1−1/poly(|G|):
+(i) Given a group G given as a black-box (with unique encoding ) and any element g ∈G,
+compute theorder of gin G.
+2A quantum oracle computing the inverse of elements is not nec essary since the inverse of an element can be
+computed ifone knows itsorder — thislatter taskcanbe done e fficientlyas statedinTheorem 2.1.
+5(ii) Given an abelian group G given as a black-box (with uniqu e encoding), compute a basis
+(g1,...,gs)of G.
+(iii) Given anabelian group Ggiven asablack-box (with uniq ue encoding), abasis (g1,...,gs)
+of G,andany g∈G,compute adecomposition of gover (g1,...,gs), i.e.,integers u 1,...,us
+such that g =gu1
+1···guss.
+More precisely, Task(i) canbe solved using ablack-box vers ion of Shor’s algorithm [31],Task
+(ii) can be solved using Cheung and Mosca’s algorithm [9], an d Task (iii) can be solved using
+the quantum algorithm by Ivanyos et al. [17]. The discrete lo garithm problem is the special case
+of task (iii) above when Gis a cyclic group. Moreover, since factoring an integer redu ces to
+computing the order of elements in a cyclic group, the efficie nt solution to Task (iii) implies an
+efficient solution for the integer factoring problem (we ref er to Shor’s paper [31] for a precise
+description of this reduction).
+Actually, all the tasks in Theorem 2.1 can be seen as black-bo xes versions of instances of the
+so-called Hidden Subgroup Problem (HSP) over abelian group s. We now recall the definition of
+this problem, since we will need it in Section 5. Let Gbe a group, Kbe a subgroup of G, andX
+be a finite set. A function f:G→Xis said to be K-periodic if fis constant on each left coset of
+K,withdistinct valueondistinct cosets. Givenasinputs (i) agroupGgivenasasetofgenerators,
+and(ii) afunction fgivenasanoracle, whichis K-periodic for anunknown subgroup KofG,the
+HiddenSubgroup Problem asks tooutput aset of generators fo rK. Theabelian Hidden Subgroup
+Problem is the special case where the underlying group Gis abelian. It is known that the abelian
+HSPcanbesolvedintimepolynomial inlog |G|[21],evenif Gisgivenasablack-box groupwith
+unique encoding [17,29].
+2.3 Invariant factors and elementary divisors of amatrix
+Inthissubsection wereviewthenotionsofinvariant factor sandelementarydivisorsofamatrix.
+Thesearestandardresults, andwerefertoanytextbookonal gebra(e.g.,[12])forproofsandmore
+details. In this subsection Fdenotes a finite field, and GL(r,F)denotes the group of invertible
+matrices of size r×roverFfor somepositive integer r.
+Leta(x) =xk+bk−1xk−1+...+b1x+b0be any monic polynomial in F[x]. Thecompanion
+matrixofa(x), denoted by Ca(x)is thek×kmatrix with 1’s down the first subdiagonal, −b0,
+−b1,...,−bk−1down the last column and zero elsewhere. For example, the com panion matrix of
+x4+b3x3+b2x2+b1x+b0isthe matrix
+
+0 0 0−b0
+1 0 0−b1
+0 1 0−b2
+0 0 1−b3
+.
+LetMbeamatrixin GL(r,F). Thenitisknownthatthereexistsauniquelist (a1(x),...,as(x))of
+monicpolynomialsin F[x],witheachpolynomial ai(x)dividingai+1(x)foreachi∈{1,...,s−1},
+suchthat Missimilartotheblockdiagonal matrix diag(Ca1(x),...,Cas(x)). Thislistofpolynomials
+is called the invariant factors ofM, and this block diagonal matrix is called the rational normal
+formof the matrix Mand is unique. In particular the polynomial as(x)is the minimal monic
+polynomial of M,i.e.,the(unique) monicpolynomial ofsmallestdegreein F[x]suchthat as(M)=
+0. It is known that matrices are conjugate in GL(r,F)if and only if they have the same invariant
+6factors (or equivalently ifthey havethesamerational norm al form). Moreover, these invariant are
+the same if Mis seen as a matrix over a field extension KofF, i.e., two matrices in GL(r,F)are
+similar in GL(r,F)if and only if they are similar in GL(r,K).
+LetKbe a field extension of Fthat splits the minimal polynomial as(x)ofM, i.e.,as(x) =
+(x−λ1)b1···(x−λt)btwherethe λi’saredistinctelementsof Kandthebi’saretheirmultiplicities.
+Eachinvariant factor ai(x)ofMcanthenbewrittenas ai(x)=(x−λ1)ci1···(x−λt)cit,whereeach
+cijisanonnegative integer in {0,...,bj}. Thenthe set of elementary divisors ofMisthe set with
+possible repetitions
+{(x−λj)cij|i∈{1,...,s},j∈{1,...,t}such that cij/\e}atio\slash=0}.
+The set of elementary divisors associated to Mis unique, and it is known that two matrices are
+similar in GL(r,F)if and only if they have the same set of elementary divisors ov erK, whenKis
+anextension fieldof Fsplitting boththeir minimal polynomials. Forexample, sup pose that r=4,
+s=2,a2(x)=(x−λ1)(x−λ2)2, anda1(x)=(x−λ1)for distinct elements λ1andλ2inK. Then
+theset of elementary divisors is {(x−λ1),(x−λ1),(x−λ2)2}.
+The elementary divisors of Mare closely connected to the so-called Jordan normal form of M.
+Letcbe a nonnegative integer and λbe an element in K. The Jordan matrix of size cassociated
+toλ, denoted by J(λ,c), is thec×cmatrix with λalong the main diagonal and 1 along the first
+superdiagonal. For example:
+J(λ,4)=
+λ1 0 0
+0λ1 0
+0 0λ1
+0 0 0 λ
+.
+Itiseasytocheckthat theminimalpolynomial of J(λ,c)is(x−λ)c. Inparticular, thisshowsthat
+theset of elementary divisors of J(λ,c)is{(x−λ)c}.
+Suppose that the set of elementary divisors of a matrix M(inGL(r,F), but seen as a matrix in
+GL(r,K)whereKsplits its minimal polynomial) is {(x−λk)dk|k∈{1,...,ℓ}},where the λk’s
+maynotbedistinct (andnecessarily r=∑ℓ
+k=1di). Thenitisknownthat Missimilar over GL(r,K)
+tothe block diagonal matrix
+diag(J(λ1,d1),...,J(λℓ,dℓ)).
+Thisblockdiagonal matrixiscalledthe Jordannormalform ofMandisunique uptotheordering
+of theλi’s. For example the Jordan normal form for the example consid ered above with the set of
+elementary divisors {(x−λ1),(x−λ1),(x−λ2)2}is
+diag(J(λ1,1),J(λ1,1),J(λ2,2))=
+λ10 0 0
+0λ10 0
+0 0 λ21
+0 0 0 λ2
+.
+3 Computing a Standard Decomposition
+In this section we present a quantum algorithm computing a st andard decomposition of any
+group in the class Sin timepolynomial inthe logarithm of the order of the group.
+73.1 Description of thealgorithm
+The precise description of the algorithm, which wedenote Pr ocedure D ECOMPOSE , is given in
+metacode inFigure 1. Further descriptions on how each step i simplemented follow.
+Procedure D ECOMPOSE
+INPUT: aset of generators {g1,...,gs}of agroup GinSwiths=O(log|G|).
+OUTPUT: apair(U,v)whereUis asubset of Gandv∈G.
+1 compute aset of generators {g′
+1,...,g′
+t}of the derived subgroup G′witht=O(log|G|);
+2 compute κ=lcm(|g1|,...,|gs|);
+3 factorize κand write κ=pe1
+1···perrwherethe prime numbers piare distinct;
+4U←{g′
+1,...,g′
+t};V←∅;Σ←∅;
+5fori=1tor
+6do
+7 Γi←∅;
+8 forj=1tosdoΓi←Γi∪{gκ/pei
+i
+j};
+9 if[Γi,G′]=eandgcd(pi,|G′|)/\e}atio\slash=1thenU←U∪Γi;
+10 if[Γi,G′]=eandgcd(pi,|G′|)=1
+11 then
+12 search for an element γi∈Γisuch that/a\}bracketle{tΓi/a\}bracketri}htG′=/a\}bracketle{tγi,G′/a\}bracketri}ht;
+13 ifnosuch element exists
+14 thenU←U∪Γi
+15 elseΣ←Σ∪{γi};
+16 endthen
+17 if[Γi,G′]/\e}atio\slash=ethen{take an element γi∈Γisuch that|γi|=maxγ∈Γi|γ|;
+18 V←V∪{γi};}
+19 enddo
+20forallwinΣ
+21do
+22 ifthere exists anelement zinΣsuch that [w,z]/\e}atio\slash=e
+23 then{ifzwz−1∈/a\}bracketle{tw/a\}bracketri}htthenU←U∪{w}elseV←V∪{w};}
+24enddo
+25forallw∈Σ\(U∪V)
+26do
+27 if[w,u]={e}for allu∈UthenU←U∪{w}elseV←V∪{w};
+28enddo
+29b←Πg∈V|g|;z←Πg∈Vg;v←z|z|/b;
+30 output (U,v);
+Figure 1: Procedure D ECOMPOSE .
+•At Step 1 a set of generators {g′
+1,...,g′
+t}of the derived subgroup G′witht=O(log|G|)is
+computed in time polynomial in log |G|with success probability 1 −1/poly(|G|)using the
+classical algorithm byBabai et al. [4].
+•The order of G′at Steps 9 and 10, and the orders of elements at Steps 2, 17 and 2 9 are
+computed using the quantum algorithms for Tasks(i) and (ii) inTheorem 2.1.
+8•The least common multiple at Step 2 is computed using standar d algorithms, and is factor-
+ized at Step 3using Shor’s factoring algorithm [31].
+•AtStep12, noticethat [Γi,G′]=eimplies that/a\}bracketle{tΓi/a\}bracketri}htG′isanabelian group. Foreachelement
+γiinΓi(thereare O((log|G|)2)suchelements), thequantumalgorithmsforTasks(i)and(ii )
+in Theorem 2.1 are used to check whether |/a\}bracketle{tΓi/a\}bracketri}htG′|=|/a\}bracketle{tγi,G′/a\}bracketri}ht|. Since necessarily /a\}bracketle{tγi,G′/a\}bracketri}ht≤
+/a\}bracketle{tΓi/a\}bracketri}htG′,this test is sufficient tocheck whether /a\}bracketle{tΓi/a\}bracketri}htG′=/a\}bracketle{tγi,G′/a\}bracketri}ht.
+•ThetestsatSteps9,10to17aredonebynoticingthat [Γi,G′]={e}ifandonlyif [γ,g′
+j]=e
+for eachγ∈Γiand each j∈{1,...,t}.
+•Testing whether zwz−1is in/a\}bracketle{tw/a\}bracketri}htat Step 23 is done by trying to decompose zwz−1over
+/a\}bracketle{tw/a\}bracketri}htusing the quantum algorithm for Task (iii) in Theorem 2.1, an d then checking if the
+decomposition indeed represents zwz−1(since, apriori, thisalgorithm canhave anarbitrary
+behavior when zwz−1/∈/a\}bracketle{tw/a\}bracketri}ht).
+This description, along with Theorem 2.1 and with the observ ation that the sets U,VandΣ
+have size O((log|G|)2), show that all the steps of Procedure D ECOMPOSE can be implemented
+in time polynomial in log |G|. The following theorem states the time complexity of Proced ure
+DECOMPOSE ,and also its correctness.
+Theorem3.1. LetGbeagroupintheclass S,givenasablack-boxgroup(withuniqueencoding).
+The procedure DECOMPOSE on input G outputs, with high probability, a pair (U,v)such that
+(/a\}bracketle{tU/a\}bracketri}ht,/a\}bracketle{tv/a\}bracketri}ht)is a standard decomposition of G. It can be implemented in tim e polynomial in log|G|
+onaquantum computer.
+Before giving a complete proof of Theorem 3.1 in Subsection 3 .2, we first describe its out-
+line below, which we believe is also instructive in that it de scribes what procedure D ECOMPOSE
+actually does.
+Suppose that (A,/a\}bracketle{ty/a\}bracketri}ht)is a standard decomposition of Gwith|y|=m. This decomposition is
+unknown, and the value of mtoo. Suppose that κ=pe1
+1···perrwhere the pi’s are distinct prime
+numbers. The first thing that is done is to convert the set of ge nerators of Ginto a set Γ=∪r
+i=1Γi
+of generators of prime powers (where each Γiconsists of elements of order pki
+iwith0≤ki≤ei).
+The idea of the procedure is then to construct two sets: a set Uwhich will contain generators
+ofAand a set Vwhich will contain elements of prime power order of the form ayαwitha∈A
+andα/\e}atio\slash≡0 modm. More precisely, most elements of Γcan be assigned to either UorVusing
+simple rules (from the properties of groups in the class S): If the order of an element gofΓis
+not coprime with|G′|, thengshould be put in U(Step 9); If at least two elements of Γare in the
+same subset Γibut do not define a cyclic subgroup (up to elements in the commu tator subgroup),
+then they both should be put in U(Step 14); If an element gofΓdoes not commute with all
+the elements of G′, thengshould be put in V(Step 18; for technical reasons, only one element
+satisfying this condition from each Γiisput inV).
+It remains to deal with the set Σof elements satisfying neither of these three conditions. F or
+elements w∈Σnot commuting withatleast oneelement zinΣ,deciding whether wshould beput
+inUor inVcan bedone bychecking whether zwz−1∈/a\}bracketle{tw/a\}bracketri}htor not (Steps22and 23). Thelast part
+of the procedure (Steps 25 to 28) deals with the elements in Σcommuting with all elements in Σ;
+these elements are put asfar aspossible in Utomake/a\}bracketle{tU/a\}bracketri}htaslarge aspossible.
+Finally, at Step 29, the product of all the elements in Vis raised to some well chosen power in
+order to obtain an element vsuch that/a\}bracketle{tv/a\}bracketri}ht∩/a\}bracketle{tU/a\}bracketri}ht={e}. It can be shown that (/a\}bracketle{tU/a\}bracketri}ht,/a\}bracketle{tv/a\}bracketri}ht)is then a
+standard decomposition of G.
+93.2 Proof of Theorem 3.1
+Westart with twolemmas.
+Lemma3.1. LetGbeagroup inthe class Sand suppose that (A,/a\}bracketle{ty/a\}bracketri}ht)∈Dm
+G. Letw=ayαbean
+element of G witha ∈Aandα/\e}atio\slash≡0 modm. If the order of wis aprime power, then a ∈G′.
+Proof.If the order of wis a prime power, then it is necessarily a prime power prdividingm
+sinceα/\e}atio\slash≡0 modm. Nowe= (ayα)pr=xapryαpr=xaprwherexis some element in G′. Thus
+apr∈G′⊆A. Sincepriscoprime with|A|,weconclude that a∈G′.
+Lemma 3.2. Let G be a group in the class Sand suppose that (A,/a\}bracketle{ty/a\}bracketri}ht)∈Dm
+G. LetΣbe a set of
+elements of G of prime power order such that each element of Σhas order coprime with |G′|and
+commutes withall the elements inG′. Let wand zbe twoelements of Σsuch that [w,z]/\e}atio\slash=e. Then
+(1) if zwz−1∈/a\}bracketle{tw/a\}bracketri}htthen w∈Aand z=ayαwitha∈Aandα/\e}atio\slash≡0 modm.
+(2) if zwz−1/∈/a\}bracketle{tw/a\}bracketri}htthen z∈Aand w=ayαwitha∈Aandα/\e}atio\slash≡0 modm.
+Proof.Since[w,z]/\e}atio\slash=e, at least one of wandzis of the form ayαwitha∈Aandα/\e}atio\slash≡0 modm.
+Lemma3.1showsthatexactlyoneamong wandzisofthisform,whiletheotherisin A(remember
+that theelements wandzcommute withall the elements in G′).
+Let us first prove assertion (1). Suppose that z∈A(and thus necessarily w=ayαwitha∈A
+andα/\e}atio\slash≡0 modm). Ifzwz−1∈/a\}bracketle{tw/a\}bracketri}ht, then[z,w] = (zwz−1)w−1is in/a\}bracketle{tw/a\}bracketri}httoo. Since the order of
+wis necessarily coprime with |G′|(remember that|w|is a prime power and thus divides m), we
+conclude that [z,w]=e. This gives acontradiction. Thus, if zwz−1∈/a\}bracketle{tw/a\}bracketri}ht, thenw∈A.
+We now prove assertion (2). Suppose that w∈A(and thus necessarily z=ayαwitha∈Aand
+α/\e}atio\slash≡0 modm). Thenzwz−1isalsoinA. Moreprecisely, zwz−1=[z,w]w. Fromthetheobservation
+thatzwz−1has thesameorder as wand thefact that gcd(|w|,|G′|)=1,weconclude that [z,w]=e
+and thatzwz−1∈/a\}bracketle{tw/a\}bracketri}ht. Thus, if zwz−1/\e}atio\slash∈/a\}bracketle{tw/a\}bracketri}ht,thenz∈A.
+Wenow proceed withthe proof of Theorem 3.1.
+Proof of Theorem 3.1. ThecomplexityofProcedure D ECOMPOSE followsfromthedescriptionof
+theprocedure given inSubsection 3.1. It remains to prove it scorrectness.
+Let(A,/a\}bracketle{ty/a\}bracketri}ht)beastandarddecomposition of Gwith|y|=m. Noticethateachcalltothequantum
+algorithms solving the tasks mentioned in Theorem 2.1 reali zed in the Procedure D ECOMPOSE
+has success probability at least 1 −1/poly(|G|). Then, with high probability, there is no failure at
+those steps. In the following we suppose that this is the case and show that, then, the procedure
+necessarily outputs astandard decomposition of G.
+First, notice that the sets Γiconstructed inthe loop of Steps 5to19 aresuch that G=/a\}bracketle{t∪r
+i=1Γi/a\}bracketri}ht.
+Moreover, they satisfy the following property: If pidividesm, then/a\}bracketle{tΓi/a\}bracketri}htG′=/a\}bracketle{tym/pei
+i,G′/a\}bracketri}htfrom
+Lemma 3.1; If pidoes not divides m, then/a\}bracketle{tΓi/a\}bracketri}htG′=ApiG′, whereApidenotes the Sylow pi-
+subgroup of A(since, in this case, the |G|/pei
+i-th power of an element ayαofGisxa|G|/pei
+iwhere
+xisan element of G′).
+At the end of the loop of Steps 5 to 19, the set U∪V∪Σis a generating set of G(here the fact
+thatG′⊆/a\}bracketle{tU/a\}bracketri}htis important). More precisely, the set Ucontains only elements of A. The set V
+contains only elements of the form ayαim/pei
+ifor some i∈{1,...,r}such that pidividesm, where
+a∈G′(from Lemma3.1)and αiisaninteger such that gcd(αi,pi)=1. Moreover thereisatmost
+10one element of this form in Vfor eachi∈{1,...,r}such that pidividesm. The set Σis a set of
+elements satisfying the conditions of Lemma3.2.
+In the loop of Steps 20 to 24, all the elements w∈Σsuch that [w,Σ]/\e}atio\slash={e}are put in either U
+orV. FromLemma3.1and Lemma3.2,the elements put in Uareelements of Aandthe elements
+put inVareof the form w=ayαfor some a∈G′and some α/\e}atio\slash≡0 modm. Attheend of theloop,
+theelements of Σ\(U∪V)are commuting withall the elements of Σ.
+Finally, the loop of Steps 25 to 28 ensures that all the elemen ts ofΣ\(U∪V)are put in either
+UorVin the following way. The new elements put in Uare precisely those commuting with the
+original set U(since these new elements also commute together, the final su bgroup/a\}bracketle{tU/a\}bracketri}htwill then
+be abelian). The elements put in Vare such that, at the end of the loop, Vcontains again only
+elements of the form ayαim/pei
+iwitha∈G′andgcd(αi,pi) =1 for some i∈{1,...,r}such that
+pidividesm. Moreover there is at most one element of this form in Vfor each such i(from the
+construction oftheset Σ). Thislatter observation impliesthat theelement zconstructed at Step29
+issuch that/a\}bracketle{tz/a\}bracketri}htG′=/a\}bracketle{tV/a\}bracketri}htG′.
+Thefinalsubgroup /a\}bracketle{tU/a\}bracketri}htisabelianand,since G′⊆/a\}bracketle{tU/a\}bracketri}ht,isnormal in G. Since/a\}bracketle{tz/a\}bracketri}htG′=/a\}bracketle{tV/a\}bracketri}htG′,we
+know that/a\}bracketle{tz,U/a\}bracketri}ht=G(remember that G′⊆/a\}bracketle{tU/a\}bracketri}ht). The element vconstructed at Step 29 is of the
+formayα, witha∈G′andαcoprime with m, and then/a\}bracketle{tv,U/a\}bracketri}ht=G, butvsatisfies the additional
+relationvb=e. Since/a\}bracketle{tU/a\}bracketri}htisabelianandeachelementof Uhasordercoprimewith |v|,weconclude
+thatgcd(|v|,|/a\}bracketle{tU/a\}bracketri}ht|) =1. Thus/a\}bracketle{tv/a\}bracketri}ht∩/a\}bracketle{tU/a\}bracketri}ht={e}.
+This shows that the output (U,v)of Procedure D ECOMPOSE is such that that (/a\}bracketle{tU/a\}bracketri}ht,/a\}bracketle{tv/a\}bracketri}ht)∈Dm′
+G
+wherem′=|v|≤m(moreprecisely,|v|dividesmbyconstruction). Since mistheminimalinteger
+such that Dm
+G/\e}atio\slash=∅(because (A,/a\}bracketle{ty/a\}bracketri}ht)is a standard decomposition of G), we conclude that m=m′
+and that Procedure D ECOMPOSE findsastandard decomposition of thegroup G.
+4 Set DiscreteLogarithm
+4.1 Statement of theproblem
+We first introduce the following useful notation. Let Fbe a finite field, and Σ={x1,...,xt}be
+anysubsetof Fwithpossiblerepetitions, i.e.,allthe xi’sareelementsof F,butmaynotbedistinct.
+Forany integer k,wedenote by Σkthe subset of Fwith possible repetitions {xk
+1,...,xk
+t}.
+In this section we consider the following problem. Here uis a positive integer which is a
+parameter of the problem (taking u≥2 does not make the problem significantly harder, but this
+enables usto give amoreconvenient presentation of our resu lts).
+SETDISCRETE LOGARITHM
+INPUT: two lists (S1,...,Su)and(T1,...,Tu)where, for each integer h∈{1,...,u},ShandTh
+are subsets with possible repetitions of some finitefield Fh.
+OUTPUT: apositive integer ksuch that Tk
+h=Shfor allh∈{1,...,u}, if such an integer exists.
+Notice that the case u=1 with|S1|=|T1|=1 is the usual discrete logarithm problem
+over the multiplicative group of the field F1. Actually, our algorithm solving the problem
+SETDISCRETE LOGARITHM will only need the multiplicative structure of the fields, an d then
+also works if we replace in the definition each field Fhby any multiplicative finite group Gh.
+However, since the main applications of our algorithm deal w ith field structures (as described in
+Section 5and Section 6), wedescribe our results in theprese nt slightly less general form.
+11Given an instance of S ETDISCRETE LOGARITHM , letmSdenote the smallest positive integer
+such that xmS=1for allx∈S1∪···∪Su,and letmTdenote the smallest positive integer such that
+ymT=1for ally∈T1∪···∪Tu. The mainresult of this section isthe following theorem.
+Theorem 4.1. There exists a quantum algorithm that solves with high proba bility the prob-
+lemSETDISCRETE LOGARITHM , and runs in time polynomial in u, log(mS+mT), and
+max1≤h≤u(|Sh|+|Th|+log|Fh|).
+4.2 Proof of Theorem 4.1
+We first describe how to compute intersections of cosets of ab elian groups efficiently using a
+quantum computer.
+Proposition 4.1. LetΓbeanabelian group, givenasablack-box, and Γ1,Γ2betwosubgroups of
+Γgivenbygeneratingsets. Letxandybetwoelementsof Γ. Thereexistsaquantumalgorithmthat
+decideswithhighprobability, intimepolynomial in log|Γ|,whetherx Γ1∩yΓ2isempty. Moreover,
+whenthealgorithmdecidesthatx Γ1∩yΓ2/\e}atio\slash=∅,italsooutputsanelement γ∈Γ,andt=O(log|Γ|)
+elementsγ1,...,γtsuch that x Γ1∩yΓ2=γ/a\}bracketle{tγ1,...,γt/a\}bracketri}htwithhigh probability.
+Proof.A standard result of group theory states that the set xΓ1∩yΓ2is either empty, or is a coset
+of the subgroup Γ1∩Γ2(note that this statement is true even if Γis not abelian). Notice that
+xΓ1∩yΓ2/\e}atio\slash=∅if and only if xy−1∈Γ1Γ2. This can be checked efficiently using the quantum
+algorithm byIvanyosetal.[17]testing membershipinabeli an groups, but moreworkisneededto
+findanexplicit element in xΓ1∩yΓ2.
+Let{α1,...,αs}and{β1,...,βt}be bases of Γ1andΓ2respectively. Define the abelian group
+P1=Z|α1|×···×Z|αs|×Z|β1|×···×Z|βt|×Z|xy−1|anddefinethe map f1fromP1toΓasfollows:
+for any(a1,...,as,b1,...,bt,c)inP1,
+f1(a1,...,as,b1,...,bt,c)=αa1
+1···αassβb1
+1···βbttx−cyc.
+Noticethattheset Q1={(a1,...,as,b1,...,bt,c)∈P1|xcy−c=αa1
+1···αassβb1
+1···βbtt}isasubgroup
+ofP1, and that the function f1is constant on cosets of Q1inP1, with distinct values on distinct
+cosets. This is thus an instance of the abelian HSP, and a set o f generators of Q1can be found in
+timepolynomial inlog |P1|=O(log|Γ|). ThesetxΓ1∩yΓ2isnot emptyifandonly if Q1contains
+some element of the form (a1,...,as,b1,...,bt,1), in which case the element γ=xα−a1
+1···α−ass
+isinxΓ1∩yΓ2.
+We now show how to compute a generating set of the subgroup Γ1∩Γ2. This can be done
+using the quantum algorithm by Friedl et al. [14] computing t he intersection of subgroups in
+“smoothlysolvable”groups,butwepresenthereamuchsimpl erquantumalgorithmfortheabelian
+case, inspired by techniques developed in [26]. Let f2be the map from the abelian group P2=
+Z|α1|×···×Z|αs|×Z|β1|×···×Z|βt|toK1K2defined as follows: for any (a1,...,as,b1,...,bt)in
+P2,
+f2(a1,...,as,b1,...,bt)=αa1
+1···αassβb1
+1···βbtt.
+Notice that the set Q2={(a1,...,as,b1,...,bt)∈P2|αa1
+1···αassβb1
+1···βbtt=1}is a subgroup of
+P2, and that the function f2is constant on cosets of Q2inP2, with distinct values on distinct
+cosets. This is thus an instance of the abelian HSP,and a set o f generators{z1,...,zr}ofQ2with
+r=log|Γ|can befound in time polynomial inlog |P2|=O(log|Γ|)using the algorithm described
+in Subsection 2.2 . For each i∈{1,...,r}let us write zi= (ui1,...,uis,vi1,...,vit)and define
+γi=αui1
+1···αuiss. Then it iseasy tocheck that Γ1∩Γ2=/a\}bracketle{tγ1,...,γr/a\}bracketri}ht.
+12Weare now ready to give our proof of Theorem 4.1.
+Proof of Theorem 4.1. For the sake of brevity, let us denote Σ=S1∪···∪Su∪T1∪···∪Tu. We
+first compute the orders of all the elements in Σusing Shor’s algorithm [31]. The value mSis the
+least common multiple of the orders of all the elements in S1∪···∪Su, and the value mTis the
+least common multiple of the orders of all the elements in T1∪···∪Tu. The values mSandmT
+can then be computed in time polynomial in log (mS+mT),|Σ|, and max 1≤h≤ulog|Fh|. Notice
+that, for any positive integer k, the least common multiple of the orders of all the elements i n
+Tk
+1∪···∪Tk
+uismT/gcd(k,mT). Then, if mSdoes not divide mT, then there is no solution to the
+problem S ETDISCRETE LOGARITHM . IfmSdividesmTbutmS/\e}atio\slash=mT,thenasolution(ifitexists)
+can befound by replacing the list (T1,...,Tu)by the list (TmT/mS
+1,...,TmT/mSu). Thus, without loss
+of generality, we suppose hereafter that mS=mTand denote by mthis value. Then a solution k
+canbe searched for in the set Z∗
+m.
+Let{m1,...,mℓ}=∪z∈Σ{|z|}denote the set of orders of the elements in Σ. For each h∈
+{1,...,u}and each i∈{1,...,ℓ}, wedefine thesubsets
+Sh,i={x∈Sh||x|=mi}andTh,i={y∈Th||y|=mi}.
+Letus also define the sets
+Kh,i={k∈Z∗
+m|Tk
+h,i=Sh,i}andKh,i={k∈Z∗
+m|Tk
+h,i=Th,i}.
+It is straightforward to check that the set Kh,iis a subgroup of Z∗
+m, and that the set Kh,iis either
+empty, or isacoset of Kh,iinZ∗
+m.
+LetK⊆Z∗
+mdenote the set of solutions of the instance of S ETDISCRETE LOGARITHM we are
+considering. Then
+K=/intersectiondisplay
+1≤h≤u/parenleftig/intersectiondisplay
+1≤i≤ℓKh,i/parenrightig
+.
+The setKcan be computed efficiently by applying successively the qua ntum algorithm of Propo-
+sition 4.1 if, for each h∈{1,...,u}and each i∈{1,...,ℓ}, the setKh,iis known (more precisely,
+if agenerating set of Kh,iand an element of Kh,iare known).
+The final part of the proof shows how to compute these sets Kh,i. Let us fix an integer h∈
+{1,...,u}andaninteger i∈{1,...,ℓ}. Wesupposethat Sh,iandTh,ihavethesamesize(otherwise
+Kh,i=∅and thus K=∅). Denote Sh,i={x1,...,xv}andTh,i={y1,...,yv}, wherev=|Sh,i|
+depends on handi. Wepresent aquantum procedure computing asetofgenerator s ofKh,i,andan
+elementkh,iinKh,iwhen this set isnot empty, intimepolynomial in v, logm,and log|Fh|.
+We first show how to compute the subgroup Kh,i. Let≺be an arbitrary strict total ordering of
+the elements of Fh. Without loss of generality we can suppose that x1/precedesequalx2/precedesequal···/precedesequalxv. Letµbe
+thefunction from Z∗
+m×{1,...,v}toFhdefined asfollows: for any k∈Z∗
+mand anyj∈{1,...,v},
+µ(k,j)is thej-th element (with respect to the order ≺) of the set Tk
+h,i. Letfbe the function from
+Z∗
+mto(Fh)vsuch that, for any k∈Z∗
+m:
+f(k)=(µ(k,1)y−1
+1,...,µ(k,v)y−1
+v).
+Notice that the set {k∈Z∗
+m|f(k) = (1,...,1)}is precisely the subgroup Kh,iofZ∗
+m. Moreover,
+the function fis constant on cosets of Kh,iinZ∗
+m, with distinct values on distinct cosets (since
+f(k1)=f(k2)implies that Tk1
+h,i=Tk2
+h,iand thusk1∈k2Kh,i). This is thus an instance of the abelian
+13HSP,and aset of generators of Kh,ican befound intimepolynomial in v, logmand log|Fh|using
+the algorithm described in Subsection 2.2 (notice that the u nderlying group is Z∗
+m, and that the
+value of the function fcan becomputed intime v, logmand log|Fh|).
+Wenow show howto compute anelement kh,iinKh,iif this set is not empty. Wefirst trytofind
+an element α∈Z∗
+misuch that Tα
+h,i=Sh,i. This is done by, for each j∈{1,...,v}, trying to find
+an integer αj∈Z∗
+misuch that xαj
+1=yj, if such an integer exists (notice that, for each j, there is at
+most one element αjinZ∗
+misatisfying this condition, which can be computed in time pol ynomial
+in logmiand log|Fh|using the quantum algorithm for the standard discrete logar ithm problem
+[31]) and checking whether Tαj
+h,i=Sh,i. If no such value αcan be found, we conclude that Kh,iis
+empty. Otherwise we take any such value αand compute kh,ias follows. Let us write the prime
+power decomposition of masm=pε1
+1···pεrrp′η1
+1···p′ηs
+sqδ1
+1···qδtt, where each prime pldividesmi
+forl∈{1,...,r}, each prime p′
+ldividesαbut notmiforl∈{1,...,s}, and each prime qldivides
+neitherminorαforl∈{1,...,t}. Thenthe integer
+kh,i=α+miqδ1
+1···qδttmodm
+is coprime with m(sinceαis coprime with miand then each prime pl,p′
+lorqldoes not divide
+kh,i), and hence is in Z∗
+m. From the choice of αand since any element in Th,ihas order mi, we
+conclude that kh,iisin theset Kh,i.
+5 Discrete Logarithm up toConjugacy
+5.1 Statement of theproblem
+Givenapositive integer randafinitefield F,remember that GL(r,F)denotes themultiplicative
+groupofinvertiblematricesofsize r×rwithentriesin F. Inthissectionweconsiderthefollowing
+problem. Here uisagain apositive integer which is aparameter of the proble m.
+DISCRETE LOG UP TO CONJUGACY
+INPUT: two lists of matrices (M(1)
+1,...,M(u)
+1)and(M(1)
+2,...,M(u)
+2)where, for each integer
+h∈{1,...,u},M(h)
+1andM(h)
+2are inGL(rh,Fh)for somepositive integer rhand some
+finite field Fh.
+OUTPUT: apositive integer kandumatricesM(h)∈GL(rh,Fh)such that
+M(h)·M(h)
+1=[M(h)
+2]k·M(h)for eachh∈{1,...,u}, if such elements exist.
+In the statement of the above problem, the notation [M(h)
+2]ksimply means M(h)
+2raised to the k-th
+power. Noticethatthecase u=1andr1=1isbasicallytheusualdiscrete logarithmproblem over
+themultiplicative group of the finitefield F1.
+Letm1andm2denote the smallest positive integers such that [M(h)
+1]m1=Iand[M(h)
+2]m2=Ifor
+allh∈{1,...,u}. Themainresult of this section isthe following theorem.
+Theorem 5.1. There exists a quantum algorithm that solves with high proba bility the prob-
+lemDISCRETE LOG UP TO CONJUGACY , and runs in time polynomial in u, log(m1+m2), and
+max1≤h≤u(rh+log|Fh|)
+145.2 Proof of Theorem 5.1
+Thequantumalgorithmsolvingtheproblem D ISCRETE LOG UP TO CONJUGACY followsfrom
+areduction totheproblem S ETDISCRETE LOGARITHM . Thekeyideaistorepresent eachmatrix
+by its set of elementary divisors. We will first introduce som e definitions and prove two lemmas
+before moving to the proof of Theorem 5.1. In this subsection we use the notations introduced in
+Subsection 2.3.
+LetMbe a matrix in GL(r,F), whereris a positive integer and Fis a finite field. The minimal
+polynomial as(x)ofMhas not in general all its roots in F, and, in order to define the elementary
+divisors of M, we need then to work on a field extension of Fcontaining all the roots of as(x).
+DenoteF=GF(q)whereqissomeprimepower. Itiswellknownthattherootsofanyirre ducible
+factor of degree dof a polynomial in F[x]are elements of the field extension GF(qd)ofF(see
+[24]for example). Thenthefieldextension GF(qd′)splits thepolynomial as(x), whered′denotes
+the least common multiple of the degrees of the irreducible f actors of as(x)overF. However, the
+valued′can be in general superpolynomial in r, and thus weneed to be more careful to obtain an
+algorithm with running time polynomial in rand log|F|. This is why we introduce the following
+definition (we also take in consideration the degrees of the a ssociated elementary divisors for
+technical reasons).
+Definition 5.1. Let M be a matrix in GL (r,F)where r is a positive integer and F=GF(q)is
+a finite field of prime power order q, and let d and ℓbe two positive integers. Suppose that
+{(x−λ1)ℓ,...,(x−λt)ℓ}is the subset of all elementary divisors of degree ℓof M such that each
+λiisanelement inGF (qd)but isnotinanypropersubfield ofGF (qd). Thenwedefine Σd,ℓ(M)as
+thesubset of GF (qd)with possible repetitions {λ1,...,λt}.
+Example . Define the two polynomials f1=(x2+x+1)andf2=(x2+x+1)2(x3+x+1)over
+GF(2), and the matrix M=diag(C1,C1,C2)whereC1(resp.C2) denotes the companion matrix
+off1(resp.f2). Notice that x2+x+1 andx3+x+1 are irreducible over GF(2). The matrix M
+has size 11×11, consists of 3 diagonal blocks of size 2 ×2, 2×2 and 7×7 respectively, and is
+actually already in rational normal form. In particular, it s invariant factors are (f1,f1,f2). Then
+the minimal polynomial of Misf2, which is split by GF(26). It can be checked that there exist
+twoelements α2∈GF(22)andα3∈GF(23)ofmultiplicative order respectively 3and7suchthat
+thepolynomial (x2+x+1)factorizes into (x−α2)(x−α2
+2)overGF(22)andthepolynomial (x3+
+x+1)factorizes into (x−α3)(x−α2
+3)(x−α4
+3)overGF(23). Then the set of elementary divisors
+ofMis{(x−α2),(x−α2),(x−α2
+2),(x−α2
+2),(x−α2)2,(x−α2
+2)2,(x−α3),(x−α2
+3),(x−α4
+3)}
+and the only sets Σd,ℓ(M)that are not empty are Σ2,1={α2,α2,α2
+2,α2
+2},Σ2,2={α2,α2
+2}and
+Σ3,1={α3,α2
+3,α4
+3}.
+Wewill need thefollowing result on Jordan matrices.
+Lemma 5.1. Letλbe a nonzero element in a finite field Kand c be a positive integer. Let k
+be a positive integer coprime with the multiplicative order of J(λ,c). Then the set of elementary
+divisors of the matrix [J(λ,c)]kis{(x−λk)c}.
+Proof.Let us write M=J(λ,c)and denote by pthe characteristic of K. The result is trivial if
+c=1so wesuppose that c≥2.
+Our proof is based on the simple fact that the k-th power of M is an upper triangular matrix
+withλkalong the main diagonal, kλk−1along the first superdiagonal, and possibly other nonzero
+entries in the other superdiagonals if c>2 (the values of these entries are easy to calculate, but
+15not relevant to this proof). Let mdenote the multiplicative order of M. Then, since Mm=Iand
+λ/\e}atio\slash=0, wehave mλm−1=0. Thenpdividesm.
+Letkbe apositive integer coprime with m. Thenkis necessary coprime with pfrom the above
+observation. Notice that amatrix in GL(c,K)has{(x−λk)c}as set of elementary divisors if and
+onlyif(x−λk)cisitsminimalpolynomial. Sincethecharacteristic polyno mial ofMkis(x−λk)c,
+theminimal polynomial of Mkdivides(x−λk)c. Wenowshow that (Mk−λkI)c−1/\e}atio\slash=0. Fromthe
+description of Mkgiven above, it is easy to show that (Mk−λkI)c−1is the matrix where the only
+nonzeroentryislocatedatthefirstrowandthe c-thcolumn. Thevalueofthisentryis (kλk−1)c−1.
+Sincekiscoprime with pandλ/\e}atio\slash=0, weconclude that (Mk−λkI)c−1/\e}atio\slash=0.
+Since two matrices are similar if and only if they have the sam e elementary divisors, Lemma
+5.1 shows that a Jordan matrix raised to a power coprime with i ts order is similar to itself. We
+now prove the following lemma (remember that, if Σ={x1,...,xt}is a subset of Fwith possible
+repetitions, wedenote by Σkthesubset of Fwithpossible repetitions {xk
+1,...,xk
+t}).
+Lemma 5.2. Let M1and M2be two matrices in GL (r,F), where r denotes a positive integer and
+Fdenotes a finite field. Let m be an integer such that Mm
+1=Mm
+2=I, and k be an integer in Z∗
+m.
+ThenM 1andMk
+2aresimilarinGL (r,F)ifandonlyif,forallpositive integers d and ℓ,theequality
+[Σd,ℓ(M2)]k=Σd,ℓ(M1)holds.
+Proof.LetKbe a field extension of Fsplitting the minimal polynomial of M2. Denote by (x−
+µ1)v1,...,(x−µs)vsthe elementary divisors of M2(where the µi’s are elements of Kthat may not
+be distinct). If kis coprime with m, then Lemma 5.1 implies (using the concept of the Jordan
+normal form) that theelementary divisors of Mk
+2are(x−µk
+1)v1,...,(x−µk
+s)vs. Sincetwomatrices
+are similar in GL(r,F)if and only if they have the same elementary divisors, the cla im follows
+from the fact that, if Kiis the smallest subfield of Kcontaining µi, thenKiis also the smallest
+subfield of Kcontaining µk
+i(sincekiscoprime withthe order of µi).
+Wenow present the proof of Theorem 5.1.
+Proof of Theorem 5.1. Remember that m1andm2denote the minimal positive integers such that
+[M(h)
+1]m1=Iand[M(h)
+2]m2=Ifor allh∈{1,...,u}. Notice that, if m1does not divide m2, then
+there is no solution to the problem D ISCRETE LOG UP TO CONJUGACY . Ifm1dividesm2but
+m1/\e}atio\slash=m2, then a solution (if it exists) can be found by replacing each matrixM(h)
+2by[M(h)
+2]m2/m1.
+Thus, without loss of generality, we suppose hereafter that m1=m2and denote by mthis value.
+Thenasolution kcanbe searched for in the set Z∗
+m.
+Let us fix an integer h∈{1,...,u}and suppose that Fh=GF(qh), whereqhis a some prime
+power. Wefirstcomputetheinvariant factorsover FhofM(h)
+1andM(h)
+2. Thiscanbedonein O(rh3)
+fieldoperations, usingforexamplethealgorithm byStorjoh ann [32]. Wethenfactor over Fhthese
+invariant factors using the Cantor-Zassenhaus algorithm [ 8], running intimepolynomial in rhand
+log|Fh|. Let usdenote by Dhthe set of degrees of the irreducible factors (over Fh) appearing inat
+least one of these invariant factors. Notice that obviously |D|≤2rhsince each M(h)
+1andM(h)
+2has
+atmostrhinvariant factors. Foreach d∈Dhand eachinteger ℓ∈{1,...,rh},wecompute thesets
+Σd,ℓ(M(h)
+1)andΣd,ℓ(M(h)
+2)defined in Definition 5.1 as follows: the irreducible factors of degree d
+of the invariant factors of M(h)
+1andM(h)
+2are factorized over GF(qd)using the Cantor-Zassenhaus
+algorithm [8],and the elementary factors of degree ℓarethen collected.
+16Lemma 5.2 implies that there exists a solution to the problem
+DISCRETE LOG UP TO CONJUGACY if and only if there exists some integer k∈Z∗
+msuch
+that[Σd,ℓ(M(h)
+2)]k=Σd,ℓ(M(h)
+1)for all integers h∈{1,...,u}, all integers d∈Dhand all integers
+ℓ∈{1,...,rh}. Such an integer k(if it exists) can then be found with high probability using t he
+quantum algorithm of Theorem 4.1in timepolynomial in u, logm,and max 1≤h≤u(rh+log|Fh|).
+Finally,ifsuchasolution kexists,then,foreach h∈{1,...,h},amatrix M(h)∈GL(rh,Fh)such
+thatM(h)M(h)
+1=[M(h)
+2]kM(h)canthenbecomputed forthisvalueof kintimepolynomial in rhand
+log|Fh|using efficient classical algorithms, for example the algor ithm by Storjohann [32].
+6 ProofofTheorem 1.2
+We first state some technical results by Le Gall [23] we use to p rove Theorem 1.2. We will
+first need the following result from [23] that shows necessar y and sufficient conditions for the
+isomorphism of twogroups inthe class S.
+Proposition 6.1 (Proposition 5.1 in [23]) .Let G and H be two groups in S. Let(A1,/a\}bracketle{ty1/a\}bracketri}ht)and
+(A2,/a\}bracketle{ty2/a\}bracketri}ht)be standard decompositions of G and H respectively and let ϕ1∈Aut(A1)(resp.ϕ2∈
+Aut(A2)) be the action by conjugation of y 1on A1(resp. of y 2on A2). The groups G and H are
+isomorphic if andonly if thefollowing threeconditions hol d: (i) A 1∼=A2; and(ii)|y1|=|y2|;and
+(iii) there exists a positive integer k and an isomorphism χ:A1→A2such that ϕ1=χ−1ϕk
+2χ,
+whereϕk
+2meansϕ2composed by itself k times.
+From now, we identify, for any prime p, the finite field of size pwithZp. The following
+proposition summarizes key elements used in the classical a lgorithm by Le Gall [23] that wewill
+need.
+Proposition 6.2 ([23]).Let A1and A2be two isomorphic abelian groups. Let (g1,...,gs)and
+(h1,...,hs)be bases of A 1and A2respectively. Suppose that A 1∼=(Zpf1
+1)r1×···×(Zpftt)rt,
+where each r iis a positive integer, and each p iis a prime but pfi
+i/\e}atio\slash=pfj
+jfor i/\e}atio\slash=j. Denote
+V=GL(r1,Zp1)×···×GL(rt,Zpt). Then there exists two homomorphisms Φ1:Aut(A1)→V
+andΦ2:Aut(A2)→Vsuch that, for any two automorphisms ζ1∈Aut(A1)andζ2∈Aut(A2)of
+order coprime with |A1|, thefollowing twoassertions are equivalent:
+(i) there exists an isomorphism χ:A1→A2such that ζ1=χ−1ζ2χ;
+(ii) there exists an element X ∈Vsuch that Φ1(ζ1)=X−1Φi(ζ2)X.
+Moreover,if,foreach j ∈{1,...,s},integersu ijandvijsuchthat ζ1(gj)=gu1j
+1···gusj
+sandζ2(hj)=
+hv1j
+1···hvsj
+sare known, then the following holds:
+(a) theimages Φ1(ζ1)andΦ2(ζ2)canbecomputed(classically) intimepolynomial in log|A1|;
+(b) givenanexplicit element X ∈Vsuchthat Φ1(ζ1)=X−1Φi(ζ2)X,anisomorphism χ:A1→
+A2such that ζ1=χ−1ζ2χcan be computed (classically) in timepolynomial in log|A1|.
+Wenow present our proof of Theorem 1.2.
+Proof of Theorem 1.2. Suppose that GandHare two groups in the class S. In order to test
+whether these two groups are isomorphic, we first run Procedu re DECOMPOSE onGandHand
+17obtain outputs (U1,y1)and(U2,y2)such that (/a\}bracketle{tU1/a\}bracketri}ht,/a\}bracketle{ty1/a\}bracketri}ht)and(/a\}bracketle{tU2/a\}bracketri}ht,/a\}bracketle{ty2/a\}bracketri}ht)are standard decompo-
+sitions of GandHrespectively with high probability (from Theorem 3.1). The running time of
+thisstepispolynomial inthelogarithms of |G|and|H|,fromTheorem 3.1. Denote A1=/a\}bracketle{tU1/a\}bracketri}htand
+A2=/a\}bracketle{tU2/a\}bracketri}ht. The orders of A1,A2,y1andy2are then computed using the quantum algorithms for
+Tasks(i)and(ii)inTheorem 2.1. Noticethat |G|=|A1|·|y1|and|H|=|A2|·|y2|. If|G|/\e}atio\slash=|H|,we
+concludethat GandHarenotisomorphic. Inthefollowing, wesuppose that |G|=|H|anddenote
+bynthis order.
+If|y1|/\e}atio\slash=|y2|we conclude that GandHare not isomorphic, from Proposition 6.1. Otherwise
+denote|y1|=|y2|=m. Then we compute a basis (g1,...,gs)ofA1and a basis (h1,...,hs′)ofA2
+using the quantum algorithm for Task (ii) in Theorem 2.1. Giv en these bases it is easy to check
+theisomorphism of A1andA2: thegroups A1andA2areisomorphic if andonlyif s=s′andthere
+exists a permutation σof{1,...,s}such that|gi|=|hσ(i)|for eachi∈{1,...,s}. IfA1/\e}atio\slash∼=A2we
+conclude that GandHarenot isomorphic, from Proposition 6.1.
+Now suppose that A1∼=A2∼=(Zpf1
+1)r1×···×(Zpftt)rt, where each piis a prime, but pfi
+i/\e}atio\slash=pfj
+j
+fori/\e}atio\slash=j. We want to decide whether the action by conjugation ϕ1∈Aut(A1)ofy1onA1and the
+action by conjugation ϕ2∈Aut(A2)ofy2onA2satisfy Condition (iii) in Proposition 6.1. Notice
+that, for each j∈{1,...,s}, we can compute (in time polynomial in log n) integers uijandvij
+such that ϕ1(gj) =y1gjy−1
+1=gu1j
+1···gusj
+sandϕ2(hj) =y2hjy−1
+2=hv1j
+1···hvsj
+susing the quantum
+algorithm for Task(iii) inTheorem2.1. FromProposition 6. 2,theimages Φ1(ϕ1)andΦ2(ϕ2)can
+then becomputed intimepolynomial in log n. Notice that [Φ1(ϕ1)]m=[Φ2(ϕ2)]m=I.
+Since the maps Φ2is a homomorphism, Proposition 6.2 implies that there exist s a posi-
+tive integer kand an isomorphism χ:A1→A2such that ϕ1=χ−1ϕk
+2χif and only if Φ1(ϕ1)
+and[Φ2(ϕ2)]kare conjugate in the group V=GL(r1,Zp1)×···×GL(rt,Zpt). If we denote
+Φ1(ϕ1)=(M(1)
+1,...,M(t)
+1)andΦ2(ϕ2)=(M(1)
+2,...,M(t)
+2), whereeach M(ℓ)
+1andeach M(ℓ)
+2arema-
+tricesinGL(rℓ,Zpℓ),thencheckingifthelaterconditionholdsbecomesaninsta nceoftheproblem
+DISCRETE LOG UP TO CONJUGACY , and can be decided using the algorithm of Theorem 5.1 in
+timepolynomial in t, logm,and max 1≤ℓ≤t(rℓ+logpℓ), i.e.,in timepolynomial inlog n.
+Iftheaboveinstance of D ISCRETE LOG UP TO CONJUGACY hasnosolution, weconclude that
+GandHare not isomorphic. Otherwise wetake one value ksuch that each Φ1(ϕ1)and[Φ2(ϕ2)]k
+are conjugate, along with an element X∈Vsuch that XΦ1(ϕ1) = [Φ2(ϕ2)]kX(such an element
+is obtained from the output of the algorithm of Theorem 5.1), and compute an isomorphism χ
+fromA1toA2such that ϕ1=χ−1ϕk
+2χusing the last part of Proposition 6.2. The map µ:G→H
+definedas µ(xyj
+1)=χ(x)ykj
+2foranyx∈A1andanyj∈{0,...,m−1}isthenanisomorphismfrom
+GtoH(a detailed proof of this statement can be found in the proof o f Proposition 6.1 included
+in[23]).
+Acknowledgments
+Theauthor isindebted toYoshifumi Inuiformanydiscussion s onsimilartopics. Healsothanks
+ErichKaltofen, Igor Shparlinski and Yuichi Yoshida for hel pful comments.
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+arXiv:1001.0609v2  [math.PR]  17 Aug 2010The evolution of the cover time
+Martin T. Barlow∗, Jian Ding†, Asaf Nachmias‡, Yuval Peres§
+October 30, 2018
+Abstract
+The cover time of a graph is a celebrated example of a parameter tha t is easy to ap-
+proximate using a randomized algorithm, but for which no constant f actor deterministic
+polynomial time approximation is known. A breakthrough due to Kahn , Kim, Lov´ asz and
+Vu [25] yielded a (loglog n)2polynomial time approximation. We refine the upper bound
+of [25], and show that the resulting bound is sharp and explicitly compu table in random
+graphs. Cooper and Frieze showed that the cover time of the large st component of the
+Erd˝ os-R´ enyi random graph G(n,c/n) in the supercritical regime with c >1 fixed, is asymp-
+totic toϕ(c)nlog2n, whereϕ(c)→1 asc↓1. However, our new bound implies that the
+cover time for the critical Erd˝ os-R´ enyi random graph G(n,1/n) has order n, and shows how
+the cover time evolves from the critical window to the supercritical phase. Our general esti-
+mate also yields the order of the cover time for a variety of other co ncrete graphs, including
+critical percolation clusters on the Hamming hypercube {0,1}n, on high-girth expanders,
+and on tori Zd
+nfor fixed large d. This approach also gives a simpler proof of a result of
+Aldous [2] that the cover time of a uniform labeled tree on kvertices is of order k3/2. For
+the graphs we consider, our results show that the blankettime, introduced by Winkler and
+Zuckerman [45], is within a constant factor of the cover time. Finally , we prove that for any
+connected graph, adding an edge can increase the cover time by at most a factor of 4.
+∗Department of mathematics, University of British Columbia , Research partially supported by NSERC
+(Canada) and the Peter Wall Institute of Advanced Studies. E mail: barlow@math.ubc.ca
+†Department of statistics, University of California at Berk eley, partially supported by Microsoft Research.
+Email: jding@berkeley.edu
+‡Department of Mathematics, Massachusetts Institute of Tec hnology. Email: asafnach@math.mit.edu
+§Microsoft Research. Email: peres@microsoft.com
+11 Introduction
+The cover time tcov(G) of a graph Gis the expected number of steps a simple random walk takes
+to visit every vertex of the graph G, starting from the worst possible vertex. It has been studie d
+extensively by computer scientists, due to its intrinsic ap peal and its applications to designing
+universal traversal sequences [4,8,9], testing graph conn ectivity [4,26], and protocol testing [36];
+see [1] for an introduction to cover times.
+Sophisticated methods to estimate the cover time have been d eveloped [18,19,25,35]. One
+of the most precise bounds was obtained by Kahn, Kim, Lov´ asz and Vu [25]. They gave poly-
+nomially computable upper and lower bounds that differ by a fac tor of order (loglog n)2. This
+breakthrough left several questions open:
+(i)Can the bounds in [25] be represented by an explicit formula f or concrete graphs of interest?
+(ii)For such graphs, can the (loglog n)2factor be removed?
+In this work we improve the upper bound from [25] and show the r esulting estimate is sharp
+up to a constant factor, and explicitly computable, for a lar ge variety of graphs, in particular
+random graphs.
+LetG= (V,E) be a simple graph and write Reff(x,y) andd(x,y) for the effective resistance
+andgraphdistancebetween twovertices x,y∈V, respectively. Seee.g. [33]or[31]fordefinitions
+and properties of effective resistance. It is known that Reff(x,y)≤d(x,y) and that Reff(·,·)
+forms a metric on G. Forx∈Vand a real number R >0 we write Beff(x,R) for the ball of
+radiusRin the resistance metric, that is,
+Beff(x,R) ={v∈G:Reff(x,v)≤R}.
+Theorem 1.1. LetG= (V,E)be a finite graph with diameter Rin the resistance metric. For
+i∈N, letAi=Ai(G)be a set of minimal size such that
+G⊂/uniondisplay
+v∈AiBeff/parenleftbig
+v,R
+2i/parenrightbig
+, (1.1)
+and write αi= 2−ilog|Ai|. Then there exists a universal constant C >0such that
+tcov≤C/parenleftbig/summationtextlog2logn
+i=1√αi/parenrightbig2R|E|. (1.2)
+The right hand side is approximable up to constant factors in polynomial time, see Remark
+2.2. Theorem 1.1 is a refinement on [25], in which it is shown th at
+max
+iαiR|E| ≤tcov(G)≤C(loglogn)2·max
+iαiR|E|. (1.3)
+The lower bound is a variant of Matthews’ estimate for cover t imes [35], and the upper bound
+is the main contribution of [25]. We refine the methods of [25] to deduce the stronger statement
+of Theorem 1.1 (clearly, (/summationtextlog2logn
+i=1√αi)2≤(log2logn)2maxiαi). This new bound turns out to
+be sharp in many concrete examples where we can show that max iαiand (/summationtextlog2logn
+i=1√αi)2are
+of the same order. Such examples are presented in Theorems 1. 2 and 1.3 below.
+Cooper and Frieze [11] studied the cover time of the largest c omponent of the Erd˝ os-R´ enyi
+[16] random graph model G(n,p), that is, the random graph obtained from the complete graph
+2Knby retaining each edge with probability pindependently. It is well known that if p=c
+n
+for some c >1, then the largest connected component, C1, is of size about xnwith probability
+tending to 1, where x=x(c) is the unique solution in (0 ,1) ofx= 1−e−cx. Cooper and Frieze
+[11] established the asymptotics for the cover time in this r egime,
+tcov(C1)∼ϕ(c)nlog2nwithϕ(c) =cx(2−x)
+4(cx−logc)
+with probability tending to 1 as n→ ∞.
+Sinceϕ(c) tends to 1 as c→1, one might be tempted to guess that tcov(C1) forG(n,1/n) is of
+ordernlog2n. However, it is known [39] that the maximal hitting time betw een two vertices in
+C1is typically of order n, so Matthews bound [35] shows that tcov(C1) is at most O(nlogn). In
+fact, inG(n,1/n) the largest component C1is roughly of size n2/3[7,17,32], and with probability
+uniformly bounded away from 0 it is a tree. Aldous [2] proved t hat a random tree on kvertices
+has cover time of order k3/2(see Theorem 3.2 for a precise statement and an alternative p roof).
+Combining these facts yields that tcov(C1) inG(n,1
+n) is of order nwith probability uniformly
+bounded away from 0. In the following theorem we show that thi s probability tends to 1, and
+moreover, we show how the order of the cover time continuousl y evolves from the critical regime
+c= 1 to the supercritical regime c >1.
+Theorem 1.2. Lettcov(C1)denote the cover time of largest component of G(n,p)and letλ∈R
+be fixed and ε(n)>0be a sequence such that ε(n)→0butn1/3ε(n)→ ∞. Then
+(a) Ifp=1−ε(n)
+n, then for any δ >0there exists B >0such that
+P/parenleftBig
+B−1ε−3log3/2(ε3n)≤tcov(C1)≤Bε−3log3/2(ε3n)/parenrightBig
+≥1−δ.
+(b) Ifp=1+λn−1/3
+n, then then for any δ >0there exists B >0such that
+P/parenleftBig
+B−1n≤tcov(C1)≤Bn/parenrightBig
+≥1−δ.
+(c) There exists a constant C >0such that if p=1+ε(n)
+n, then
+P/parenleftBig
+C−1nlog2(ε3n)≤tcov(C1)≤Cnlog2(ε3n)/parenrightBig
+→1.
+Theorem 1.1 also allows us to prove sharp bounds on cover time for critical percolation
+clusters, even when the underlying graph is notthe complete graph. Given a graph Gonn
+vertices and p∈[0,1], the random graph Gpis obtained from Gby retaining each edge with
+probability pindependently. In the special case of G=Kn, this yields the Erd˝ os-R´ enyi graph
+G(n,p). For a vertex v∈Gwe write C(v) for the connected component in Gpcontaining v, and
+denote by C1the largest connected component of Gp. We are interested in critical percolation
+in which |C1| ≈n2/3. This occurs in numerous underlying graphs G. A partial list of examples
+is:
+1. The complete graph on nvertices [7,17,32] with p=1+Θ(n−1/3)
+n,
+2. A random d-regular graph [38,41] with p=1+Θ(n−1/3)
+d−1,
+33. Expanders of high girth and degree d[37] with p=1+Θ(n−1/3)
+d−1,
+4. The Hamming hypercube {0,1}m[6] withpsatisfying Ep|C(v)|= Θ(n1/3),
+5. Discrete tori Zd
+mfor large but fixed dimension dwithp=pc(Zd) orpsatisfying Ep|C(v)|=
+Θ(n1/3) [6,21,22].
+In all the examples above it is known that for any δ >0 there exists B=B(δ)>0 such that
+Pp/parenleftbig
+B−1n2/3≤ |C1| ≤Bn2/3/parenrightbig
+≥1−δ.
+The following theorem is a generalization of part (b) of Theo rem 1.2, and states that in these
+casestcov(C1) has order n. This means that the cover time of the largest component has t he
+same order as the cover time of a random tree on the same number of vertices. We note that
+unlike the G(n,p) case, in examples 4 and 5, the probability that the largest c omponent is a
+tree tends to zero as the volume grows, so the Aldous estimate [2] does not apply.
+Theorem 1.3. In examples 1−5above, we have that for any δ >0there exists B=B(δ)>0
+such that
+Pp/parenleftbig
+B−1n≤tcov(C1)≤Bn/parenrightbig
+≥1−δ.
+In fact, in Section 3 we provide a general criterion for the co nclusion of Theorem 1.3 to hold,
+which applies to examples 1 −5, see Theorem 3.1.
+Remark. TheblankettimeBis the expected first time when the local times at all vertices
+are within a factor of 2 from each other (the local time at a ver texvis the number of visits to
+vdivided by the degree of v). This quantity was introduced by Winkler and Zuckerman [45 ]
+(we use the definition of [25]) who conjectured that B=O(tcov) for any graph. The bounds in
+Theorems 1 .1−1.3 also apply to Bin place of tcov. This will be clear from the proofs.
+Finally, it is natural to guess that adding edges to a graph ca n only decrease the cover time.
+However, this is not the case, as shown by the following examp le. LetK∗
+nbe the graph obtained
+fromKn(the complete graph on nvertices) by adding a new vertex vand connecting it to one
+vertex of Kn. The cover time of K∗
+nis easily seen to be n2. On the other hand, if we replaces
+KnbyHn, a bounded degree expander on nvertices, and construct H∗
+nby adding a new vertex
+vand connecting it to one vertex of Hn, then the cover time of H∗
+nis of order nlogn. SinceH∗
+n
+is a subgraph of K∗
+non the same n+ 1 vertices, we conclude that adding an edge to a graph
+may increase the cover time. The increase is at most by a const ant factor:
+Proposition 1.4. LetGbe a connected graph and let u,v∈Gbe two vertices. Let G+be the
+graph obtained from Gby adding the edge {u,v}(if an edge connecting these two vertices already
+exists, then we add a multiple edge, and if u=v, then we add a loop). Then we have
+tcov(G+)≤4tcov(G).
+2 Proof of Theorem 1.1
+LetStbe a simple random walk on G, and for an integer t≥0, define the local time Lv
+tof a
+vertexv∈Vby
+Lv
+t△=1
+dvt/summationdisplay
+k=01{Sk=v},for allv∈Vandt∈N, (2.1)
+4wheredvis the degree of vertex v. Furthermore, let τv
+k= min{t∈N:Lv
+t=k/dv}be the time
+of thek-th visit of the random walk to v. The following lemma of [25] implies that if the local
+time at a vertex uis large, then with high probability, the local time is also l arge at vertices v
+that are close to uin the resistance metric.
+Lemma 2.1. [25, Lemma 5.2] For allu,v∈V, numbers λ≥0andt∈Nwe have
+Pu/parenleftbig
+Lu
+τu
+t−Lv
+τu
+t≥λ/parenrightbig
+≤exp/parenleftbig
+−λ2
+4tReff(u,v)/parenrightbig
+.
+We use an idea of Kolmogorov [42, page 91]. For all i≥1 and for each u∈Ai, we can always
+selectv∈Ai−1such that Reff(u,v)≤2−(i−1)R(see (1.1)). Write v=h(u). Setα′
+i=αi∨2−i/2
+and define
+Ψ = 128(/summationtext∞
+i=1/radicalbig
+α′
+i)2,andβi=/radicalbig
+α′
+i
+2/summationtext∞
+i=1/radicalbig
+α′
+ifor alli∈N.
+Fori∈N, letti= (1−/summationtexti
+j=1βj)Ψ, and for u∈AidefineMi(u) to be the difference of the local
+times of vertices h(u) anduat timeτh(u)
+ti−1R, by
+Mi(u) =ti−1R−Lu
+τh(u)
+ti−1R.
+Lemma 2.1 then gives that
+P(Mi(u)≥βiΨR)≤exp/parenleftBig
+−(βiΨR)2
+4R2−(i−1)·ti−1R/parenrightBig
+≤e−2i+1α′
+i.
+DefineMi= max
+u∈AiMi(u). Recalling the definition of αiandα′
+i, we apply a union bound and get
+P(Mi≥βiΨR)≤ |Ai|e−2i+1α′
+i≤e−2iα′
+i.
+It follows that
+P/parenleftBig/uniontext
+i≥1{Mi≥βiΨR}/parenrightBig
+≤/summationdisplay
+i≥1e−2iα′
+i≤/summationdisplay
+i≥1e−2i/2≤2
+3. (2.2)
+Now, take v∈Vand write τcovfor the cover time of the random walk. Provided that the
+eventM△=/intersectiontext
+i≥1{Mi≤βiΨR}occurs, we have that Lu
+τv
+ΨR≥(1−/summationtexti
+j=1βi)ΨRfor allu∈Aiand
+henceLu
+τv
+ΨR≥ΨR/2 by the definition of βi. In particular, on the event Mevery vertex in the
+graph should have been visited at least once. Combined with ( 2.2), it follows that
+Pv(τcov≥τv
+ΨR)≤2
+3,
+and hence Evτcov≤3Evτv
+ΨR. The expected return time to vsatisfiesEvT+
+v=2|E|
+dvwhence
+Evτcov≤3ΨRdvEvT+
+v= 6ΨR|E|. (2.3)
+Since the above holds for all v∈V, we have tcov≤6ΨR|E|. Note that |Ai| ≤nfor all
+i∈Nand hence/summationtext
+i≥log2logn√αi=O(1). Observing also that αi≤α′
+i+ 2−i/2, we get Ψ ≤
+256((/summationtextlog2logn
+i=1√αi)2+16). It completes the proof of the theorem together with the fact that
+α1≥1
+2log2 (since |A1|has to be at least 2).
+5Remark 2.2. Note that the sum/summationtext
+i√αican be easily approximated up to constant. To
+see this, one can use greedy algorithm to find a maximal collec tion of centers ˜Aisuch that
+{Beff(v,2−(i+1)R) :v∈˜Ai}forms a collection of disjoint balls. Thus, |˜Ai−1| ≤ |Ai| ≤ |˜Ai|and
+1√
+2/summationdisplay
+i/radicalBig
+2−ilog|˜Ai| ≤/summationdisplay
+i√αi≤/summationdisplay
+i/radicalBig
+2−ilog|˜Ai|.
+3 Cover time of critical percolation clusters
+Weareinterested incritical percolationinwhich |C1| ≈n2/3. Thisoccursinnumerousunderlying
+graphsGas listed in the introduction (examples 1 −5). Recall the definition of Gp, and write
+dGp(x,y) for the length of the shortest path between xandyinGp, or∞if there is no such
+path. We call dtheintrinsic metric onGp. Define the random sets
+Bp(x,r;G) ={u:dGp(x,u)≤r}, ∂B p(x,r;G) ={u:dGp(x,u) =r},
+and the event Hp(x,r;G) =/braceleftBig
+∂Bp(x,r;G)/\e}atio\slash=∅/bracerightBig
+.Finally, define
+Γp(x,r;G) = sup
+G′⊂GP(Hp(x,r;G′)),
+wherePhere is the percolation probability measure over subgraphs ofG′. The reason for taking
+a supremum in the definition of Γ pis that the event Hp(x,r;G) isnotmonotone with respect
+to edge addition (indeed, adding an edge can potentially sho rten a shortest path and make
+∂Bp(x,r;G) empty even if it were not empty before). The quantity Γ pis called the intrinsic
+metricarm exponents and was introduced in [39], see Theorem 2 .1 of that paper for further
+details there.
+Theorem 3.1. LetG= (V,E)be a graph and let p∈[0,1]. Suppose that for some constants
+c1,c2>0and all vertices x∈Vthe following two conditions are satisfied:
+(i)E|Bp(x,r;G)| ≤c1r,(ii) Γp(x,r;G)≤c2/r.
+Then for any β,δ >0there exists B >0such that
+P/parenleftBig
+∃Cwith|C| ≥βn2/3andtcov(C)/\e}atio\slash∈[B−1n,Bn]/parenrightBig
+≤δ.
+Proof. The fact that there exists B >0 such that
+P/parenleftBig
+∃Cwith|C| ≥βn2/3andtcov(C)≤B−1n/parenrightBig
+≤δ/2,
+follows immediately from the corresponding lower bound on t he maximal hitting time, see part
+(c.2) of Theorem 2 .1 of [39] and Lemma 4 .1 in that paper. Also from [39] we have that for any
+β,δ′>0 there exists D=D(β,δ′)>0 such that
+P/parenleftBig
+|C(v)| ≥βn2/3and diam( C(v))/\e}atio\slash∈[D−1n1/3,Dn1/3]/parenrightBig
+≤δ′n−1/3. (3.1)
+To see this, combine (3.1) and (3.3) of [39]. Denote diam eff(C(v)) for the diameter of C(v)
+according to the resistance metric. We first show that with hi gh probability components of size
+6n2/3have diam effof order n1/3. Indeed, the upper bound follows immediately from Theorem
+2.1 of [39] and the fact that Reff(x,y)≤d(x,y). For the lower bound, we use Proposition 5.6 of
+[39], the Nash-Williams inequality and (3.1) to deduce that for large enough D=D(β,δ′)>0
+we have
+P/parenleftbig
+|C(v)| ≥βn2/3and diam eff(C(v))≤D−1n1/3/parenrightbig
+≤δ′n−1/3. (3.2)
+We now proceed to construct covering sets of Gon different scales. Fix an integer i≥0 and
+we define a sequence of radii {rj}j≤2D22iwhich have the following properties:
+(a)r0= 0,(b)(j−1/2)n1/3
+2D2i≤rj≤jn1/3
+2D2i,(c)E∂Bp(v,rj;G)≤4D2c12i.
+This is possible by condition (i) of the theorem, which impli es that for each j≤2D22i
+jn1/3/(2D2i)/summationdisplay
+ℓ=(j−1/2)n1/3/(2D2i)E∂Bp(v,ℓ;G)≤c1Dn1/3,
+and so there must exists ℓ∈[(j−1/2)n1/3/(2D2i),jn1/3/(2D2i)] such that rj=ℓsatisfies
+condition (c). Given such radii {rj}we say that a vertex u∈∂Bp(v,rj;G) isi-goodif there
+exist a path between uand∂Bp(v,rj+1;G) which does not go through Bp(v,rj;G). We now
+construct a sequence of sets {A′
+i}which will serve as a covering. Define
+A′
+i=/uniondisplay
+j≤2D22i/braceleftbig
+u∈∂Bp(v,rj;G) :uisi-good/bracerightbig
+.
+Observe that if diam( C(v))≤Dn1/3then we have that
+C(v)⊂/uniondisplay
+u∈A′
+iBp/parenleftbig
+u,2−iD−1n1/3;G/parenrightbig
+.
+Furthermore, if in addition R= diam eff(C(v))≥D−1n1/3, then we have that C(v)⊂/uniontext
+u∈A′
+iBp/parenleftbig
+u,2−iR;G/parenrightbig
+. Given these two events and the fact that Bp/parenleftbig
+u,r;G)⊂Beff(u,r;C(v)),
+we deduce that
+C(v)⊂/uniondisplay
+u∈A′
+iBeff/parenleftbig
+u,2−iR;C(v)/parenrightbig
+,
+and therefore |Ai|=|Ai(C(v))| ≤ |A′
+i|for alli∈N(see (1.1)). By (3.1) and (3.2), we get that
+P(|C(v)| ≥βn2/3,∃i∈N:|A′
+i|<|Ai|)≤2δ′n−1/3. (3.3)
+Now, by condition (ii) of our theorem and our construction of {rj}we get that
+E|A′
+i| ≤/summationdisplay
+j≤2D22iE∂Bp(v,rj;G)·4Dc22i
+n1/3≤16D3c1c222in−1/3.
+So we can choose a large integer m=m(c1,c2,D,δ′) such that
+∞/summationdisplay
+i=1P/parenleftbig
+|A′
+i| ≥em·2i/2/parenrightbig
+≤∞/summationdisplay
+i=116D3c1c222in−1/3
+em·2i/2≤δ′n−1/3. (3.4)
+7Recalling that (see Theorem 1.1) αi= 2−ilog|Ai|and combining the above estimate with (3.3),
+we obtain that
+P/parenleftbig
+|C(v)| ≥βn2/3,/summationtext∞
+i=1√αi≥4m/parenrightbig
+≤P/parenleftbig
+|C(v)| ≥βn2/3,∃i∈N:|A′
+i|<|Ai|/parenrightbig
++∞/summationdisplay
+i=1P/parenleftbig
+|A′
+i| ≥em·2i/2/parenrightbig
+≤3δ′n−1/3.(3.5)
+We say that C(v) isbadif|C(v)| ≥βn2/3and one of the following holds:
+•/summationtext∞
+i=1√αi≥4m, or
+•diameff(C(v))≥Dn1/3, or
+• |E(C(v))| ≥Dn2/3.
+By (3.5) and Theorem 2 .1 of [39] we learn that we can choose Dlarge enough so that the
+probability that C(v) is bad is at most 5 δ′n−1/3, whence EX≤5δ′n2/3. Note that if there exists
+vsuch that C(v) is bad, then X≥βn2/3. By Theorem 1.1 we learn that there exists some
+large constant B=B(D,m) such that if |C(v)| ≥βn2/3andtcov(C(v))≥Bn, thenC(v) is bad
+(takingB= 16Cm2D2, whereCis the constant of Theorem 1.1 suffices). Hence, by Markov’s
+inequality
+P/parenleftBig
+∃Cwith|C| ≥βn2/3andtcov(C)≥Bn/parenrightBig
+≤P(X≥βn2/3)≤5δ′/β,
+which concludes the proof of the theorem by setting δ′=δ/(10β). /square
+Proof of Theorem 1.3. We only need to show that the conditions of Theorem 3.1 holds i n
+examples 1 −5. Indeed, it is shown in [39] that the conditions hold for exa mples 1−3, and
+in [29] and [30] it is shown for examples 4 −5. In [21,22] it is shown for example 5 that at
+p=pc(Zd) the largest cluster size is of order n2/3. /square
+We will require the following result of Aldous [2]. For the re ader’s convenience we provide a
+simpler proof of this theorem based on Theorem 1.1.
+Theorem 3.2. LetTbe a Galton-Watson tree with progeny mean 1and variance σ2<∞.
+Then for any δ >0there exists A=A(δ,σ2)>0such that
+P/parenleftbig
+tcov(T)/\e}atio\slash∈[A−1k3/2,Ak3/2]/vextendsingle/vextendsingle|T| ∈[k,2k]/parenrightbig
+≤δ.
+Proof. This is very similar to the proof of Theorem 3.1. Firstly, we c laim that there exists
+D >0 such that
+P/parenleftbig
+diam(T)/\e}atio\slash∈[D−1k1/2,Dk1/2],|T| ∈[k,2k]/parenrightbig
+≤k−1/2δ/2.
+Indeed, it is a classical fact [27] that P(diam(T)≥Dk1/2) =O(D−1k−1/2). Furthermore, the
+expected number of particles in Tup to level D−1k1/2is precisely D−1k1/2, and the event
+{diam(T)≤D−1k1/2,|T| ≥k}implies that this quantity is at least k. Hence by Markov’s
+inequality we have that P(diam(T)≤D−1k1/2,|T| ≥k)≤D−1k−1/2.
+8Now, for each iwe define rj=j2−i−1D−1√
+kforj= 0,...,2i+1D2and define A′
+ito be the
+set of particles at level rjwhich survive up to level rj+1. As in the proof of Theorem 3.1, if
+diam(T)∈[D−1k1/2,Dk1/2], then
+T⊂/uniondisplay
+u∈A′
+iBeff/parenleftbig
+u,2−iR;T/parenrightbig
+,
+whereRis the diameter of Twith respect to the resistance metric. Now, for each jthe expected
+number of particles in level rjis precisely 1 and for each, the probability of surviving up t o
+levelrj+1is of order ( rj+1−rj)−1(see [27] again), hence E|A′
+i| ≤C22i+2D3k−1/2and the proof
+continues as in (3.4) to show using Theorem 1.1 that there exi stsAsuch that
+P/parenleftbig
+tcov(T)≥Ak3/2,|T| ∈[k,2k]/parenrightbig
+≤k−1/2δ/2.
+LetLbe the offspring random variable of T. We have that |T|is distributed as the first hitting
+time of 0 of a random walk starting 1 with increments distribu ted asL−1 (see exercise 5 .26 of
+[33]). We use this and Theorem 1a of chapter XII.7 in [20] to de duce that
+P(|T| ∈[k,2k]) = (1+ o(1))Ck1/2,
+for some constant C >0. This gives the required upper bound on the cover time. The c orre-
+sponding lower bound follows immediately from the lower bou nd on the maximal hitting time,
+which we obtain via the√
+klower bound on the diameter of Ttogether with commute time
+identity. /square
+Proof of part (a) and (b) of Theorem 1.2. Part (b) of the theorem follows immediately
+from Theorem 3.1, so we are only left to prove part (a). In this case it is known that the largest
+cluster is a uniform random tree of order ε−2log(ε3n) (see [23]). It is a classical fact (see chapter
+2.2 of [28]) that a uniform random tree of size kis distributed as a Poisson(1) Galton-Watson
+treeTconditioned on |T|=k. Hence the following statement concludes the proof: let Tbe a
+Poisson(1) Galton-Watson tree, then for any δ >0 there exists A >0 such that
+P/parenleftbig
+tcov(T)/\e}atio\slash∈[A−1k3/2,Ak3/2]/vextendsingle/vextendsingle|T|=k/parenrightbig
+≤δ. (3.6)
+Note that this assertion does not immediately follow from Th eorem 3.2. To fill in the gap, we
+will infer from a result Luczak and Winkler [34], that there e xists a coupling between a random
+treeTkof sizekand a random tree Tk+1of sizek+1 such that Tk⊂Tk+1. This together with
+Theorem 3.2 shows the the upper bound on the cover time of (3.6 ) and concludes the proof (the
+lower bound on the cover time is easier and follows, as in the r emark above, by the easy lower
+bound on the maximal hitting time).
+To see that such a coupling exists write T(d)
+kfor a Bin( d,1/d) Galton-Watson tree conditioned
+on being of size k. Theorem 4 .1 in [34] shows that there exists a coupling between T(d)
+kand
+T(d)
+k+1such that T(d)
+k⊂T(d)
+k+1. Now, for any fixed kwe may take d→ ∞and we get the required
+coupling between Poisson(1) Galton-Watson trees. This con cludes our coupling since the latter
+trees are uniform random trees. /square
+94 Cover time for mildly supercritical Erd˝ os-R´ enyi graph
+In this section, we prove Part (c) of Theorem 1.2, which incor porates the order of the cover
+time for the largest component of Erd˝ os-R´ enyi graph G(n,p) withp=1+ε
+n, whereε=o(1) and
+ε3n→ ∞. Our proof makes use of the following structure result of [12 ].
+Theorem 4.1. [12]LetC1be the largest component of G(n,p)forp=1+ε
+n, whereε3n→ ∞
+andε→0. Letµ <1denote the conjugate of 1+ε, that is, µe−µ= (1+ε)e−(1+ε). ThenC1is
+contiguous to the model ˜C1constructed in the following 3 steps:
+(a) LetΛ∼ N/parenleftbig
+1+ε−µ,1
+εn/parenrightbig
+and assign i.i.d. variables Du∼Poisson(Λ) (u∈[n]) to the
+vertices, conditioned that/summationtextDu1Du≥3is even. Let Nk= #{u:Du=k}andN=/summationtext
+k≥3Nk. Select a random graph KonNvertices, uniformly among all graphs with Nk
+vertices of degree kfork≥3.
+(b) Replace the edges of Kby paths of lengths i.i.d. Geom(1−µ).
+(c) Attach an independent Poisson(µ)-Galton-Watson tree (PGW tree in what follows) to each
+vertex.
+That is, P(˜C1∈ A)→0impliesP(C1∈ A)→0for any set of graphs A.
+By the above theorem, it suffices to analyze the cover time of ˜C1. In what follows, we will
+repeatedly use some known facts about ˜C1and one can see [12,13] for references.
+4.1 Lower bound
+We first show that w.h.p. there are ( ε3n)1/4attached trees, as in part (c) of the construction
+of˜C1, of height at least1
+2ε−1log(ε3n). To this end, note that the height Hof a PGW( µ) tree
+satisfies the following for some constant c >0 (see, e.g., [13, Lemma 4.2])
+P/parenleftbig
+H≥1
+2ε−1log(ε3n)/parenrightbig
+≥cε(ε3n)−1/2+o(1), (4.1)
+where we used the fact that µ= (1−(1+o(1))ε). It is an immediate consequence of parts (a)
+and (b) of the construction of ˜C1that w.h.p. there are (2 + o(1))ε2ni.i.d. attached PGW( µ)
+trees. Hence, by (4.1), we learn that with high probability t here are at least ( ε3n)1/4PGW trees
+of height at least1
+2ε−1log(ε3n). Now, take exactly one leaf in the bottom level from each of
+these trees and denote by Bthe set of these leaves. We will use the following lemma (see, e.g.,
+[44], and also see [33, Proposition 2.19]) to bound the hitti ng time between vertices in B.
+Lemma 4.2. Given a finite network with a vertex vand a subset of vertices Zsuch that v/\e}atio\slash∈Z.
+Letvol(·)be the voltage when a unit current flows from vtoZandvol(Z) = 0. Then we
+have that Ev[τZ] =/summationtext
+x∈Vc(x)vol(x), wherec(x) =/summationtext
+x∼yc(x,y)andc(x,y)is the conductance
+between(x,y).
+In our setting, c(x,y) = 1 if ( x,y) is an edge of ˜C1, and otherwise c(x,y) = 0. Let u,v∈B,
+and letT(v) be the attached PGW tree that contains v. It is clear that for all w/\e}atio\slash∈T(v) the
+effective resistance between wandvsatisfiesReff(w,v)≥(2ε)−1log(ε3n). Now, if a unit current
+flows from utovand the voltage at vis set to be 0, we can then deduce that the voltage at
+vertexwis at least (2 ε)−1log(ε3n), for all w/\e}atio\slash∈T(v). Note that w.h.p. simultaneously for all
+10v∈Bwe have |˜C1\T(v)|= (2+o(1))εn(see [12]) and we then assume this. Lemma 4.2 then
+yields that for all u,v∈B
+Euτv≥(2ε)−1log(ε3n)(2+o(1))εn= (1+o(1))nlog(ε3n).
+At this point, an application of the Matthews lower bound [35 ] (see also, e.g., [31]) stating that
+for any subset A⊂Gwe have tcov(G)≥log|A|minu,v∈AEuτv, completes the proof of the lower
+bound. /square
+4.2 Upper bound
+Inthissection weestablish theupperboundon thecover time . Inlight of Theorem1.1, it suffices
+to show that w.h.p. for ˜C1we have that |Ai| ≤(ε3n)2isimultaneously for all i≥1. LetRbe
+the diameter of ˜C1in resistance metric. As shown in [13], with high probabilit y the diameter
+in graph metric is (3 + o(1))ε−1log(ε3n) and also the two highest attached trees have height
+(1+o(1))ε−1log(ε3n) each. It implies that (2+ o(1))ε−1log(ε3n)≤R≤(3+o(1))ε−1log(ε3n)
+w.h.p., and we assume this in what follows.
+Fixi∈N, we now construct A′
+isuch that balls of radius 2−iRaround vertices in A′
+iform
+a covering of ˜C1. We first cover the 2-core Hof˜C1by balls of radius 2−(i+1)R. To this end,
+consider the disjoint balls of radius 2−(i+2)Rthat can be packed in H. Take such a maximal
+packing and denote by A′
+i,1the set of these centers. Since the packing is maximal, we hav e that
+H ⊆/uniondisplay
+v∈A′
+i,1Beff(v,2−(i+1)R).
+SinceReff(x,y)≤d(x,y), it follows that |Beff(v,2−(i+2)R)∩ H| ≥2−(i+2)Rfor allv∈A′
+i,1.
+Therefore, since the balls Beff(v,2−(i+2)R) forv∈A′
+i,1are disjoint, we conclude that |A′
+i,1| ≤
+4·2i|H|/R.
+We now turn to cover the attached trees. For a rooted tree T, letH(T) be the height of T.
+Forv∈T, denote by Tvthe subtree of Trooted at vthat contains all the descendants of v.
+Also, denote by Lkthe vertices in level k2−(i+1)RofT. Define
+FT△=∪∞
+k=1{v∈Lk:H(Tv)≥2−(i+1)R}.
+LetTbe the collection of attached PGW trees in ˜C1and letA′
+i,2=∪T∈TFT. Defining A′
+i=
+A′
+i,1∪A′
+i,2, we deduce from the definition that ˜C1⊆/uniontext
+v∈A′
+iBeff(v,2−iR). It remains to bound
+|A′
+i,2|. Using [13, Lemma 4.2] again, we obtain that for a PGW( µ) treeTand some absolute
+constant C,
+P(H(T)≥2−(i+1)R)≤/braceleftBigg
+Cε, if 2i≤log(ε3n),
+C
+2−(i+1)R,if 2i≥log(ε3n).(4.2)
+Also, it is immediate that E[|Lk|] =µk2−(i+1)R. Furthermore, by the Markov property, given
+|Lk|the set{Tv:v∈Lk}is distributed as |Lk|independent copies of T. By this and (4.2) we
+get that for some absolute constant C >0
+E[FT] =/summationdisplay
+k≥1E|{v∈Lk:H(Tv)≥2−(i+1)R}|=/summationdisplay
+k≥1E[|Lk|]P(H(Tv≥2−(i+1)R))
+≤/braceleftBigg/summationtext
+k≥1µk2−iR/2·Cε≤C2ε, if 2i≤log(ε3n),/summationtext
+k≥1µk2−(i+1)R·C
+2−(i+1)R≤C222i/Rif 2i≥log(ε3n).
+11Hence, we can always get E[FT]≤C2ε22i. Furthermore, it is known that |H|= (2+o(1))ε2n
+with high probability so we may assume this. By Markov’s ineq uality and the fact that |A′
+i,1| ≤
+4·2i|H|/R=o((ε3n)2i) we have that
+P(|A′
+i| ≥(ε3n)2i) =P(|A′
+i,2| ≥(ε3n)2i−|A′
+i,1|)≤E[|A′
+i,2|]
+(ε3n)2i−|A′
+i,1|=|H|E[FT]
+(1+o(1))(ε3n)2i
+≤(2+o(1))C2ε3n22i(ε3n)−2i≤o(1)C2(ε3n/8)−2(i−1).
+A simple union bound gives that with high probability |A′
+i| ≤(ε3n)2isimultaneously for all
+i≥1. Recalling the facts that |E(˜C1)|= (2+o(1))εnandR≤3+o(1)ε−1log(ε3n), we conclude
+the proof of the upper bound by an application of Theorem 1.1. /square
+5 Proof of Proposition 1.4
+We may assume that |E(G)| ≥2. Letπbe the stationary distribution of Gand let{S+
+t}t≥0be
+a random walk on G+starting from the initial distribution π(note that πisnotthe stationary
+distribution for G+). Letτ0=τ′
+0= 0 and for all i≥1 define
+τi△= min/braceleftbig
+t≥τ′
+i−1:{S+
+t,S+
+t+1}={u,v}/bracerightbig
+, Xi△=S+
+τi,andτ′
+i△= min{t > τi:S+
+t=Xi}.
+WriteTi={t:τi< t≤τ′
+i}and for all t∈Nfurther define
+Φ(t) = min{k:|[0,k]\∪∞
+i=1Ti|=t}.
+Now let St=S+
+Φ(t). We first claim that Stis a simple random walk on the graph G. In order
+to see that, one just need to note that Stis obtained from S+
+tby omitting all the excursions
+started with traveling through the edge ( u,v). Letτcovbe the first time when Stvisits every
+vertex of Gand it then remains to bound E[Φ(τcov)].
+To this end, it is more convenient to consider the first time τ∗
+covwhenStvisits every vertex of
+Gand returns to the starting point. We wish to bound the number of steps spent on the above
+defined excursions before τ∗
+cov. Define
+Lu(τ∗
+cov) =/vextendsingle/vextendsingle/braceleftbig
+t≤τ∗
+cov:St=u/bracerightbig/vextendsingle/vextendsingleandLv(τ∗
+cov) =/vextendsingle/vextendsingle/braceleftbig
+t≤τ∗
+cov:St=v/bracerightbig/vextendsingle/vextendsingle,
+Nu(τ∗
+cov) =/vextendsingle/vextendsingle/braceleftbig
+i:Ti⊆[0,Φ(τ∗
+cov)],Xi=u/bracerightbig/vextendsingle/vextendsingleandNv(τ∗
+cov) =/vextendsingle/vextendsingle/braceleftbig
+i:Ti⊆[0,Φ(τ∗
+cov)],Xi=v/bracerightbig/vextendsingle/vextendsingle.
+Note that every time when St=u, the corresponding random walk S+
+Φ(t)is also at uand has
+chance1
+du+1to travel to vand thus starts an excursion, and moreover, once started the number
+of excursions has law Geom(1 /(du+1)) independent of {St}. Therefore, we have
+Nu(τ∗
+cov) =Lu(τ∗
+cov)/summationdisplay
+i=1YiZi,
+where{(Yi,Zi)}are independent and Yi∼Ber(1/(du+1)) and Zi∼Geom(1/(du+1)). Thus,
+E[Nu(τ∗
+cov)] =1
+duE[Lu(τ∗
+cov)]. By [3, Chapter 2, Proposition 3], we know that E[Lu(τ∗
+cov)] =
+du
+2|E(G)|E[τ∗
+cov] and therefore E[Nu(τ∗
+cov)] =1
+2|E(G)|E[τ∗
+cov]. Suppose Xi=u, eachTiis distributed
+12as 1 +τ+
+uwhereτ+
+uis the hitting time of S+
+ttoustarted at v. Observing that {|Ti|}are
+independent of Nu(τ∗
+cov), we can then obtain that
+Exc(u)△=E[|∪i{Ti⊆[0,Φ(τ∗
+cov)] :Xi=u}|] =1
+2|E(G)|E[τ∗
+cov](1+Ev[τ+
+u]).
+In the same manner, we derive that
+Exc(v)△=E[|∪i{Ti⊆[0,Φ(τ∗
+cov)] :Xi=v}|] =1
+2|E(G)|E[τ∗
+cov](1+Eu[τ+
+v]).
+Note that Ev[τ+
+u] +Eu[τ+
+v] is the expected commute time between uandvand hence by
+commute identity [10], we have Ev[τ+
+u] +Eu[τ+
+v] = 2|E(G+)|R+(u,v), where R+(u,v) is the
+resistance between uandvinG+. SinceGis connected, we get R+(u,v)≤|E(G)|
+|E(G)|+1. Altogether,
+tcov(G+) =E[Φ(τcov)]≤tcov(G)+Exc(u)+Exc(v)≤3tcov(G)+2
+|E(G)|tcov(G)≤4tcov(G),
+where we used the inequality E[τ∗
+cov]≤2tcovand the assumption that |E(G)| ≥2. /square
+Remark 5.1. IfG+is obtained from a connected graph Gby adding kextra edges, a similar
+argument gives that
+tcov(G+)≤/parenleftbig
+2k+1+2k2
+|E|/parenrightbig
+tcov(G).
+6 A concluding remark
+The bound (1.2) is reminiscent of Dudley’s entropy bound for Gaussian process [15]. Motivated
+by this, Ding, Lee and Peres [14] show the link to Gaussian pro cesses is much tighter. In
+particular, Talagrand’s majorizing measures bound for Gau ssian processes (see [43]) can be
+used to estimate the cover time up to a multiplicative consta nt.
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+15
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+arXiv:1001.0610v1  [math.PR]  5 Jan 2010Conditional negative association for competing urns∗
+J. Kahn and M. Neiman
+Rutgers University and UCLA
+email: jkahn@math.rutgers.edu; neiman@math.ucla.edu
+Abstract
+Weproveconditionalnegativeassociationforrandomvariables xj=1{|σ−1(j)|≥tj}(j∈[n] :=
+{1,...,n}), where σ(1),...,σ(m) are i.i.d. from [ n]. (Theσ(i)’s are thought of as the locations
+of balls dropped independently into urns 1 ,...,naccording to some common distribution, so
+that, for some threshold tj,xjis the indicator of the event that at least tjballs land in urn
+j.) We mostly deal with the more general situation in which the σ(i)’s need not be identically
+distributed, proving results which imply conditional negative associa tionin the i.i.d. case. Some
+of the results—particularly Lemma 8 on graph orientations—are tho ught to be of independent
+interest.
+We also give a counterexample to a negative correlation conjecture of D. Welsh, a strong
+version of a (still open) conjecture of G. Farr.
+1 Introduction
+Competing urns referstotheexperimentinwhich mballsaredropped,randomlyandindependently,
+into urns 1 ,...,n. Formally, we have a random σ: [m]→[n] (where [ m] ={1,...,m}) with the
+σ(i)’s independent. We then take xjto be the indicator for occupation of urn jand are interested
+in the law, µ, of (x1,...,xn) (a measure on {0,1}n). In the traditional case where the balls are
+identical (i.e. the σ(i)’s are i.i.d.) we call µanurn measure , or, for emphasis, an ordinary urn
+measure. More generally, setting Bj=|σ−1(j)|, we may consider thresholds t1,...,tn, and let xj
+be the indicator of {Bj≥tj}; for i.i.d. balls, we then call the law of ( x1,...,xn) athreshold urn
+measure. When the balls are not required to be identical we sp eak ofgeneralized urn measures and
+generalized threshold urn measures .
+We are interested in correlation properties of these measur es, but before proceeding further
+need to briefly recall a few definitions. A fuller version of th e following discussion is given in [10],
+and further background and motivation may be found e.g. in [1 5].
+Recall that events A,Bin a probability space are positively correlated —we write A ↑ B—if
+Pr(AB)≥Pr(A)Pr(B), andnegatively correlated ( A ↓ B) if the reverse inequality holds.
+We will be interested in measures on finite product spaces Ω =/producttextn
+i=1Ωiwith each Ω ia chain
+(totally orderedset), oftensimply {0,1}. Weuse M(Ω), orsimply M, forthesetofprobabilitymea-
+sures on Ω, and MSforM({0,1}S). We will occasionally identify {0,1}Swith 2S({subsets of S}
+ordered by inclusion) in the usual way.
+Recall that an event A ⊆Ω isincreasing (really, nondecreasing) if x≥y∈ A ⇒ x∈ A
+(where Ω is endowed with the product order), and similarly fo rdecreasing . For real-valued random
+AMS 2000 subject classification: 60C05, 05A20
+Key words and phrases: competing urns, correlation inequal ities, conditional negative association, log-concavity
+* Supported by NSF grant DMS0701175.
+1variables X,Y, writeX↓Yif
+{X≥s} ↓ {Y≥t} ∀s,t∈ ℜ, (1)
+or, equivalently, if
+Ef(X)g(Y)≤Ef(X)Eg(Y) for all increasing f,g:ℜ → ℜ. (2)
+(N.B. this differs from the usage in [15]. Of course X↑Ymeans the reverse inequalities hold, but
+we don’t need this.)
+Sayi∈[n]affectsA ⊆Ω if there are η∈ Aandτ∈Ω\ Awithηj=τj∀j∝\e}atio\slash=i, and write
+A ⊥ Bif no coordinate affects both AandB. Thenµ∈ Misnegatively associated (or hasnegative
+association ; we use “NA” for either) if A ↓ Bwhenever A,Bare increasing and A ⊥ B. We say µ
+hasnegative correlations (oris NC) ifηi↓ηj(whereηis the random string) whenever i∝\e}atio\slash=j.
+We are primarily concerned with conditional negative association: µ∈ Misconditionally
+negatively associated (CNA) if any measure obtained from µby conditioning on the values of some
+of the variables is NA. (Throughout the paper we assume that a ny conditioning event we consider
+has positive probability.) Conditional negative correlation (CNC) for µis defined similarly.
+When Ω = {0,1}n, stronger properties are obtained by demanding NC (resp. NA ) for every
+measure W◦µ∈ Mof the form
+W◦µ(η)∝µ(η)/productdisplay
+Wηi
+i
+withW= (W1,...,W n)∈ ℜn
++. (Borrowing Ising terminology, one says that W◦µis obtained from
+µbyimposing an external field .) Thenµis said to be Rayleigh orNC+(resp.NA+), the reference
+in the former case being to Rayleigh’s monotonicity law for e lectric networks (see e.g. [4] or [3]).
+ThecompetingurnsmodelwasexploredinsomedetailbyDubha shiandRanjan[5]∗, whoproved
+inter alia that threshold urn measures are NA. Another proof of this is g iven in [15]. Actually the
+argument of [5], which proves the stronger statement that th e (law of the) r.v.’s
+ξij=1{σ(i)=j} (3)
+is NA, does not require identical balls. (The argument of [15 ] does not work for nonidentical balls.)
+The main purpose of the present note is to prove
+Theorem 1 Threshold urn measures are CNA.
+In contrast, as observed in [10], even ordinary urn measures need not be Rayleigh (but see the
+remark on R+in Section 5). We don’t know whether Theorem 1 extends to noni dentical balls
+(again see Section 5).
+Let us quickly say what Theorem 1 has to do with [10]. Followin g [15], we say that µ∈
+M({0,1}n) isultra-log-concave (ULC) if its rank sequence ,{ri:=µ(|η|=i)}n
+i=0(where|η|=/summationtextηi),
+has no internal zeros and the sequence {ri//parenleftbign
+i/parenrightbig}n
+i=0is log-concave. A set of four conjectures from
+[15] (see his Conjecture 4) states that each of CNC, CNA, NC+ a nd NA+ implies ULC; but, as
+shown in [1] and [10], even the weakest of these (NA+ ⇒ULC) is false. Theorem 1 provides a
+more natural counterexample to the stronger “CNA ⇒ULC,” since, as observed in [10] (disproving
+another conjecture from [15]), urn measures need not be ULC.
+∗They say “bins” rather than “urns.”
+2The proof of Theorem 1 gives something a little more general, as follows. Suppose that for each
+j∈[n] we are given a sequence 0 = a0(j)<···< akj(j) =m+1, and for σ: [m]→[n] set
+xj(σ) =tiffat(j)≤Bj< at+1(j). (4)
+Theorem 2 If theσ(i)’s are i.i.d. then the xj’s in(4)are CNA.
+Call the law of ( x1,...,xn) as in (4) a (generalized) interval urn measure .
+The paper is organized as follows. Section 2 reduces Theorem 2 to either of our two main
+inequalities, (10) and (14). Each of these is valid at the lev el of generalized interval urns; they are
+equivalent in the case of ordinary urns but not obviously so i n general (though the argument in
+Section 3 uses some interplay between the two). It is only in t he derivation of Theorem 2 from (10)
+that we need the σ(i)’s to be i.i.d.
+We give two quite different ways of getting at these main inequa lities. Theorem 4 in Section
+3 essentially restates (10) and (14) in induction-friendly form; the proof of the theorem given in
+this section is inspired by [5]. Section 4 takes a different app roach, based on a graph-theoretic
+observation, Lemma 8, that is thought to be of independent in terest. The lemma is used to:
+reprove (10); in combination with a result from [11] (Theore m 11 below), to prove a stronger,
+ultra-log-concavity version of (14); and to prove “log-sub modularity” for some classes of measures.
+Finally, Section 5 contains some discussion of the question of whether Theorem 1 extends to
+nonidentical balls, mentions a conjecture of G. Farr and a st ronger one of D. Welsh, and sketches
+a counterexample to the latter.
+Some notation. For a nonnegative vector
+γ= (γij:i∈[m],j∈[n]), (5)
+A⊆[m] andK⊆[n], the probability measure on KA(functions from AtoK)corresponding to γ
+is that given by
+Pr(σ)∝W(σ) :=/productdisplay
+i∈Aγi,σ(i). (6)
+Thus the r.v.’s σ(i) are independent; they are i.i.d. if γijdoes not depend on i, in which case
+we write simply γj. We also use PrL(L⊆[m]) for the measure on [ n]Lcorresponding to γ(so
+Pr = Pr[m]).
+2 Setting up
+Let the law of σ∈[n][m]be given by (6) and let x1,...,xnbe as in (4). Let I∪J∪Kbe a partition
+of [n] andtj∈ {0,...,k j−1}forj∈K, and set
+Q:={x(σ)≡tonK}(={atj(j)≤ |σ−1(j)|< atj+1(j)∀j∈K}), (7)
+X=|σ−1(I)|andY=|σ−1(J)|. The main point for the proof of Theorem 2 is
+X↓YgivenQ, (8)
+given which we finish easily:
+Proof of Theorem 2. With notation as above, let A,B ⊆Ω be increasing events determined by
+IandJ(more precisely, by the values of the variables xjindexed by IandJ) respectively. For
+3Theorem 2 we should show A ↓ BgivenQ. Define f,g:N→ ℜ(whereN={0,1,...}) by
+f(k) = Pr(A|X=k),g(l) = Pr(B|Y=l). A standard coupling argument shows that fandgare
+increasing, whence, according to (8),
+Pr(A∩B|Q) =E[f(X)g(Y)|Q]≤E[f(X)|Q]E[g(Y)|Q] = Pr(A|Q)Pr(B|Q)
+(where the first equality follows from conditional independ ence ofAandBgiven (X,Y)).
+We continue to condition on Qand write µkfor the law of Ygiven{X=k}; that is,
+µk(l) = Pr(Y=l|X=k). (9)
+We will actually prove
+µk+1(l+1)
+µk+1(l)≤µk(l+1)
+µk(l)(10)
+(whenever neither side is 0 /0, where we agree that x/0 =∞forx >0), which is a strengthening
+of (8) once we rule out some pathologies. We recall the standa rd
+Definition 3 C ⊆Nnisconvexifa,c∈ Canda≤b≤cimplyb∈ C.
+It will follow from Proposition 6 below that
+supp(Pr) := {(k,l) : Pr(X=k,Y=l)>0}is convex. (11)
+Given this, (10) implies that Yis stochastically decreasing in X—that is,
+µk+1(Y≥t)≤µk(Y≥t)∀k,t
+(the easy implication is essentially Proposition 1.2 of [15 ])—which in turn easily implies X↓Y.
+Let
+Z=|σ−1(I∪J)|. (12)
+Whenthe σ(i)’s arei.i.d., analternate way to specify XandYis: letZbeas in(12), X∼Bin(Z,α)
+andY=Z−X, whereα=γI/γI∪J(withγI=/summationtext
+i∈Iγi) and Bin( Z,α) is the binomial distribution
+with parameters Zandα.
+In general, for νthe law of an N-valued r.v. Zandα∈[0,1], letX=Xν,α∼Bin(Z,α),
+Y=Yν,α=Z−Xand, for lack of a better name, say νisbinomially negatively associated (BNA)
+ifX↓Yfor every α. Call a nonnegative sequence a= (ai)∞
+i=0strongly log-concave (SLC) if
+ia2
+i≥(i+1)ai−1ai+1∀i≥1 (13)
+(that is, ( i!ai)∞
+i=0is log-concave), and say ν∈ M(N) is SLC if the sequence ( ν(i))∞
+i=0is. A
+straightforward calculation shows that this is equivalent to saying that (10) holds for any α,X=
+Xν,αandY=Yν,α(andµkas in (9)): since
+µk(l) =ν(k+l)Pr(X=k|Z=k+l)
+Pr(X=k)=ν(k+l)/parenleftbigk+l
+k/parenrightbigαk(1−α)l
+Pr(X=k),
+we may rewrite (10) as
+ν(k+l+2)/parenleftbigk+l+2
+k+1/parenrightbigν(k+l)/parenleftbigk+l
+k/parenrightbig≤ν(k+l+1)/parenleftbigk+l+1
+k+1/parenrightbigν(k+l+1)/parenleftbigk+l+1
+k/parenrightbig,
+4which is SLC for ν. (Ifνis Poisson—that is, if (13) holds with equality—then XandYare
+independent Poisson r.v.’s and the inequalities (1) are equ alities.) Thus, in the i.i.d. case, (10) is
+equivalent to saying that Zas in (12) is SLC. The latter again turns out to be true at the le vel of
+generalized urns; that is, for any γas in (5), σ∈[n][m]with law given by (6), Qas in (7) and Zas
+in (12),
+the law of Zis SLC. (14)
+It’s also easy to see that absence of internal zeros in ( ν(i)) is equivalent to (11) for X=Xν,α,
+Y=Yν,α(which, again, is given by Proposition 6), so that (14) again implies (8) (and Theorem 2).
+It seems interesting that both (10) and (14) are valid for gen eralized urns, though the equivalence
+that holds for i.i.d. balls disappears in the more general se tting.
+As mentioned earlier, in Section 4 we will combine Lemma 8 wit h a result from [11] to obtain
+an improvement of (14):
+the law of Zis ultra-log-concave. (15)
+(Recall—see following Theorem 1—this means that the sequen ce{Pr(|Z|=i)//parenleftbigm
+i/parenrightbig}m
+i=0is log-
+concave without internal zeros.)
+3 First proof
+Let Pr be the measure on [ n][m]corresponding to some γ(see the end of Section 1), and for
+a,b∈Nn−1andk∈N, set
+p(k,a,b) = Pr(Bn=k|Bj∈[aj,bj]∀j∈[n−1]) (16)
+(recalling that Bj=|σ−1(j)|).
+Theorem 4 With notation as above,
+(a)p(k+1,a,b)
+p(k,a,b)is nonincreasing in (a,b), and
+(b)p(k+1,a,b)
+p(k,a,b)≤k
+k+1·p(k,a,b)
+p(k−1,a,b),
+where we say nothing about the case 0/0and agree that x/0 =∞whenx >0.
+See also Theorem 14 in Section 5 for a related result.
+As noted earlier, part (b) of Theorem 4 is just a reformulatio n of (14), while (a) is a mild
+generalization of (10) (which in fact—see (19)—quickly red uces to (10)). To see this, note that in
+(10) we may assume that each of I,Jis a singleton, say K= [n−2],I={n−1},J={n}—formally
+we could pass to
+γ′
+ij=
+
+γij ifj∈K= [n−2]/summationtext{γij:j∈I}ifj=n−1/summationtext{γij:j∈J}ifj=n
+—and similarly in (14) we may assume K= [n−1],I={n}andJ=∅. Then Theorem 4(b),
+which may also be stated
+for fixed a,b∈N[n−1]the sequence {p(k,a,b)}is SLC,
+5is (up to some name changes) the same as (14), while (10) is equ ivalent to
+p(k+1,a,b)/p(k,a,b)≥p(k+1,a′,b′)/p(k,a′,b′),
+wherean−1=bn−1=t,a′
+n−1=b′
+n−1=t+1 (for some t) andaj=a′
+j,bj=b′
+jforj∈[n−2].
+On the other hand, the inductive proof of Theorem 4 employs bo th the more general form of
+(a) and some interplay between the two parts.
+Before proving the theorem we note one further consequence a nd give the promised Proposition
+6. Forf,a∈Nn, letMf(a) ={σ∈[n][m]:|σ−1(j)| ∈[aj,aj+fj]∀j∈[n]}andMf(a) =
+Pr(Mf(a)). Though we won’t use the next result (but see the remark fol lowing Corollary 9), it
+seems natural and worth mentioning.
+Corollary 5 For each f∈Nn,M=Mfsatisfies the negative lattice condition:
+M(a)M(c)≥M(a∨c)M(a∧c)∀a,c∈Nn. (17)
+This is more or less immediate from Theorem 4 once we have the n ext little observation, which, as
+noted earlier, also gives (11) and absence of internal zeros in the law of Zin (12).
+Proposition 6 For any fandMfas above, the support of M=Mfis convex.
+Proof.This will follow easily from
+Claim.For any σ,τ∈[n][m]with Pr( σ),Pr(τ)>0 andi∈[n] with|σ−1(i)|>|τ−1(i)|, there are
+j∈[n] andρ∈[n][m]with Pr(ρ)>0,|σ−1(j)|<|τ−1(j)|and
+|ρ−1(k)|=
+
+|σ−1(i)|−1 ifk=i
+|σ−1(j)|+1 ifk=j
+|σ−1(k)|ifk∈[n]\{i,j}.
+This is a standard type of graph-theoretic observation: reg ardingσandτas edge sets of bipartite
+graphs on [ m]∪[n] in the natural way,†we need a path with edges alternately from σ\τandτ\σ
+that begins with a σ-edge at iand ends with a τ-edge at some jas above. (We then get ρby
+switching σandτon this path.) We omit the routine proof that such a path must e xist.
+To prove Proposition 6, we should show that for all distinct a,b,c∈Nnwitha≤b≤cand
+a,c∈supp(M), we also have b∈supp(M). Of course it suffices to show this when there is some
+i∈[n] withbi=ci−1 andbk=ckfor allk∝\e}atio\slash=i. Choose τ∈ M(a) :=Mf(a) andσ∈ M(c) with
+Pr(τ),Pr(σ)>0. We assume |σ−1(i)|=ci+fi, since otherwise σ∈ M(b) and we are finished.
+Lettingj,ρbe as in the claim (note |τ−1(i)|< ci+fi), we have
+|ρ−1(i)|=bi+fi
+and
+|ρ−1(j)|=|σ−1(j)|+1∈[cj+1,aj+fj]⊆[bj,bj+fj],
+whenceρ∈ M(b) andb∈supp(M).
+†We pretend [ m]∩[n] =∅.
+6Proof of Corollary 5. It is easy to see (and standard) that convexity of M(given by Proposition 6)
+implies that it’s enough to prove (17) when there are indices iandjwithai=ci−1,aj=cj+1,
+andak=ckfor allk∝\e}atio\slash=i,j. In this case—assuming, w.l.o.g., that i=n−1 andj=n—we set
+p1(k) = Pr(Bn=k|Bl∈[al,al+fl]∀l∈[n−1])
+and
+p2(k) = Pr(Bn=k|Bl∈[cl,cl+fl]∀l∈[n−1]).
+Then (17) is
+
+cn+fn+1/summationdisplay
+k=cn+1p1(k)
+
+cn+fn/summationdisplay
+k=cnp2(k)
+≥
+cn+fn+1/summationdisplay
+k=cn+1p2(k)
+
+cn+fn/summationdisplay
+k=cnp1(k)
+
+and follows immediately from
+p1(k)p2(l)≥p1(l)p2(k)whenever k≥l,
+which is a consequence of Theorem 4(a) (and Proposition 6).
+We now assume (as we may) that/summationtext
+jγij= 1 for each i. The proof of Theorem 4 resembles that
+of Theorem 33 in [5], and is based on
+Observation 7 For any i∈[n],k∈N, and event Qdetermined by (σ−1(j) :j∝\e}atio\slash=i),
+Pr(Bi=k+1,Q) =1
+k+1/summationdisplay
+l∈[m]γliPr[m]\{l}(Bi=k,Q).
+(Recall PrLwas defined at the end of Section 1.) We also use the trivial
+min
+iαi
+βi≤α1+···+αk
+β1+···+βk≤max
+iαi
+βi(18)
+(for allα1,...,α k,β1,...,β k≥0 withβ1+···+βk>0, where, again, x/0 :=∞whenx >0).
+Proof of Theorem 4. We proceed by induction on m, omitting the easy base cases with m= 1.
+For (a), it’s enough to show that the ratio in question does no t increase when we increase a single
+entry—w.l.o.g. the ( n−1)st—of one of a,b. Thus, by (18), it suffices to show that
+Pr(Bn=k+1|Bn−1=t,R)
+Pr(Bn=k|Bn−1=t,R)is nonincreasing in t,
+whereR={aj≤Bj≤bj∀j∈[n−2]}; and by Proposition 6, this will follow if we show
+Pr(Bn=k+1|Bn−1=t+1,R)
+Pr(Bn=k|Bn−1=t+1,R)≤Pr(Bn=k+1|Bn−1=t,R)
+Pr(Bn=k|Bn−1=t,R)(19)
+for alltfor which the probabilities appearing in (19) are positive. (This is the easy reduction of
+(a) to (10) mentioned earlier.)
+7By Observation 7 we may write the left side of (19) as
+/summationtext
+l∈[m]γl,n−1Pr[m]\{l}(Bn=k+1,Bn−1=t,R)
+/summationtext
+l∈[m]γl,n−1Pr[m]\{l}(Bn=k,Bn−1=t,R),
+which, by (18), is at most
+max
+l∈[m]Pr[m]\{l}(Bn=k+1|Bn−1=t,R)
+Pr[m]\{l}(Bn=k|Bn−1=t,R).
+Thus, setting Q={Bn−1=t}∧Rand assuming (w.l.o.g.) that the maximum occurs at l=m, we
+will have (19) if we show
+Pr(Bn=k+1|Q)
+Pr(Bn=k|Q)≥Pr[m−1](Bn=k+1|Q)
+Pr[m−1](Bn=k|Q). (20)
+Now
+Pr(Bn=k+1|Q)
+Pr(Bn=k|Q)=/summationtext
+j∈[n]Pr(σ(m) =j|Q)Pr(Bn=k+1|Q,σ(m) =j)
+/summationtext
+j∈[n]Pr(σ(m) =j|Q)Pr(Bn=k|Q,σ(m) =j),
+so that (20) will follow (again using (18)) from
+Pr(Bn=k+1|Q,σ(m) =j)
+Pr(Bn=k|Q,σ(m) =j)≥Pr[m−1](Bn=k+1|Q)
+Pr[m−1](Bn=k|Q)for allj∈[n] (21)
+(where, again, “for all j∈[n]” really includes only those for which Pr( Q,σ(m) =j)>0).
+There are three cases to consider. If j=n, the left side of (21) is
+Pr[m−1](Bn=k|Q)
+Pr[m−1](Bn=k−1|Q),
+which is at least the right side of (21) by (part (b) of) our ind uction hypothesis. If j=n−1, the
+left side of (21) is
+Pr[m−1](Bn=k+1|Bn−1=t−1,R)
+Pr[m−1](Bn=k|Bn−1=t−1,R),
+which is at least the right side of (21) by (part (a) of) the ind uction hypothesis. Finally, if j∝\e}atio\slash=
+n−1,n, the left side of (21) is
+Pr[m−1](Bn=k+1|Bn−1=t,R∗)
+Pr[m−1](Bn=k|Bn−1=t,R∗),
+whereR∗is obtained from Rby replacing the condition aj≤Bj≤bjby the condition aj−1≤
+Bj≤bj−1; again this is at least the right side of (21) by part (a) of th e induction hypothesis.
+We now turn to (b) and set Q={aj≤Bj≤bj∀j∈[n−1]}. Then we have, again using
+Observation 7 and (18),
+Pr(Bn=k+1|Q)
+Pr(Bn=k|Q)=k/summationtext
+l∈[m]γlnPr[m]\{l}(Bn=k,Q)
+(k+1)/summationtext
+l∈[m]γlnPr[m]\{l}(Bn=k−1,Q)
+≤max
+l∈[m]kPr[m]\{l}(Bn=k,Q)
+(k+1)Pr[m]\{l}(Bn=k−1,Q)
+w.l.o.g.=kPr[m−1](Bn=k,Q)
+(k+1)Pr[m−1](Bn=k−1,Q)
+8(notingthatwemayassume,byProposition6, thatPr( Bn=r|Q)ispositivefor r∈ {k−1,k,k+1});
+so we will be done if we can show
+Pr[m−1](Bn=k|Q)
+Pr[m−1](Bn=k−1|Q)≤Pr(Bn=k|Q)
+Pr(Bn=k−1|Q). (22)
+Proceeding as in the proof of part (a), we may rewrite
+Pr(Bn=k|Q)
+Pr(Bn=k−1|Q)=/summationtext
+j∈[n]Pr(σ(m) =j|Q)Pr(Bn=k|Q,σ(m) =j)
+/summationtext
+j∈[n]Pr(σ(m) =j|Q)Pr(Bn=k−1|Q,σ(m) =j);
+so for (22) it is enough to show that, for each j∈[n],
+Pr(Bn=k|Q,σ(m) =j)
+Pr(Bn=k−1|Q,σ(m) =j)≥Pr[m−1](Bn=k|Q)
+Pr[m−1](Bn=k−1|Q),
+which, as did (21), follows easily from our induction hypoth esis (here we only need to consider the
+two cases j=nandj∝\e}atio\slash=n).
+4 A graphical approach
+We begin here with a natural and seemingly new graph theoreti c statement which we regard as
+the main point of this section. Given a multigraph Gon vertex set Vanda,b∈NV, letO(a,b) =
+OG(a,b) be the set of orientations of Gfor which
+d+(x)≥axandd−(x)≥bxfor allx∈V
+andN(a,b) =NG(a,b) =|O(a,b)|. Hered+andd−are, as usual, out- and in-degrees. We will also
+usedxfor the degree of xinG. Note we regard a loop (at x, say) as having two orientations, each
+of which contributes 1 to each of d+(x) andd−(x).
+Lemma 8 Ifa,b,r,s∈NVsatisfy
+a≥r,sanda+b≥r+s
+(where the inequalities are with respect to the product orde r onNV), then
+N(a,b)≤N(r,s). (23)
+Of course the idea is that it’s harder to satisfy a set of deman ds that always requires large out-
+degrees than one for which these requirements are mixed. For the sake of comparison, let us also
+mention the specialization of Corollary 5 to the present sit uation:
+Corollary 9 Ifa+b=r+sthen
+N(a,b)N(r,s)≥N(a∨r,b∧s)N(a∧r,b∨s). (24)
+Proof.Interpret vertices of Gas urns and edges as balls, and assume that for each edge (ball )e
+we have γex= 1 or 0 according to whether xis or is not an end of e. Then (24) is just Corollary 5
+withf=d−a−b(=d−r−s), where d= (dx:x∈V(G)) is the vector of degrees.
+9Remark. It’s possibleto simplify theproof of Lemma8 usingCorollar y 9; butof course this depends
+on Theorem 4, so is really harder than the following direct pr oof. On the other hand, it’s not too
+hard to derive Theorem 4(a) from Lemma 8; see [14]. (And below we use Lemma 8 to prove (15),
+which is stronger than Theorem 4(b).)
+Proof of Lemma 8. We proceed by induction on ϕ(G,a,b) :=|E(G)|+/summationtext
+x∈V(dx−ax−bx), calling
+x∈Vsaturated ifax+bx=dx. SinceNis nonincreasing in each of its arguments, we may assume
+a+b=r+s(or we can increase rors).
+Suppose first that there is at least one saturated vertex, x. We may assume there are no loops
+atx, since otherwise (23) follows easily from the induction hyp othesis applied to the graph gotten
+fromGby deleting such loops. Let α=ax,β=bx,ρ=rx,σ=sx, and let X={e1,...,eα+β}be
+the set of edges incident with x.
+Consider a set πconsisting of βpairs{ei,ej} ⊆X, with the 2 βedges appearing in πdistinct,
+and, say, yithe vertex joined to xbyei(so theyi’s need not be distinct). Let G(π) be the graph
+with vertex set V\{x}and edge set E(G)\X∪{eij:{ei,ej} ∈π}, whereeijjoinsyiandyj. Let
+U(π) be the set of edges in Xnot belonging to pairs from π, andUz(π) the set of edges of U(π)
+incident to z.
+Defineaπ,bπ∈NV(G(π))by
+aπ
+z=azandbπ
+z= max{bz−|Uz(π)|,0}for allz∈V\{x}(=V(G(π))).
+For each πas above and T∈/parenleftbigU(π)
+ρ−β/parenrightbig(where/parenleftbigA
+k/parenrightbig={B⊆A:|B|=k}), define rπ,T,sπ,T∈NV(G(π))
+by
+rπ,T
+z= max{rz−|Uz(π)\T|,0}
+and
+sπ,T
+z= max{sz−|Uz(π)∩T|,0}
+for allz∈V\{x}.
+Eachσ∈ OG(π)(aπ,bπ) maps naturally to a (unique) ˆ σ∈ OG(a,b), namely: ˆ σagrees with σon
+E(G−x); orients all edges of U(π) away from x; and orients eifromyitoxandejfromxtoyj
+whenever σorientseijfromyitoyj(where, when yi=yj, we interpret one orientation of the loop
+eijasyi→yjand the other as yj→yi). Since each τ∈ OG(a,b) is in the range of this map for
+exactly/parenleftbigα
+β/parenrightbigβ! choice of π, we have
+N(a,b) =1/parenleftbigα
+β/parenrightbigβ!/summationdisplay
+πNG(π)(aπ,bπ). (25)
+Similarly,
+N(r,s) =1/parenleftbigρ
+β/parenrightbig/parenleftbigσ
+β/parenrightbigβ!/summationdisplay
+π/summationdisplay
+T∈(U(π)
+ρ−β)NG(π)(rπ,T,sπ,T). (26)
+Sinceaπ+bπ≥rπ,T+sπ,Tandaπ≥rπ,T,sπ,T, it follows from the induction hypothesis that
+NG(π)(aπ,bπ)≤NG(π)(rπ,T,sπ,T) for allπandT∈/parenleftbigU(π)
+ρ−β/parenrightbig. (27)
+10(Note that ϕ(G(π),aπ,bπ)< ϕ(G,a,b), since|E(G(π))|<|E(G)|and, for z∈V\{x},aπ
+z+bπ
+z≥
+az+bz−|Uz(π)|, while the degree of zinG(π) isdz−|Uz(π)|.) Combining (25), (26) and (27), we
+have
+N(r,s)≥1/parenleftbigρ
+β/parenrightbig/parenleftbigσ
+β/parenrightbigβ!/summationdisplay
+π/parenleftBigg
+α−β
+ρ−β/parenrightBigg
+NG(π)(aπ,bπ) =α!β!
+ρ!σ!N(a,b)≥N(a,b),
+where the last inequality follows from the assumptions α≥ρ,σandα+β=ρ+σ.
+So we may assume there are no saturated vertices. In this case we fixx∈Vwithax> bx. (Of
+course if there is no such vertex, then a=b=r=sand (23) is an equality.) For γ,δ∈Nlet
+N′(γ,δ) be the number of orientations of Gwith
+(d+
+y,d−
+y)≥/braceleftBigg
+(ay,by) ify∝\e}atio\slash=x
+(γ,δ) ify=x,
+and letN′′(γ,δ) be defined analogously with ( r,s) in place of ( a,b). Letα=ax,β=bx,ρ=rx,σ=
+sx, so that (23) is
+N′(α,β)≤N′′(ρ,σ). (28)
+By induction we have
+N′(γ,δ)≤N′′(η,ξ) whenever γ≥η,ξandγ+δ=η+ξ > α+β. (29)
+We apply this to the identity
+N′(α,β) =N′(α,β+1)+N′(dx−β,β). (30)
+Ifα > σ, then, by (29), the right side of (30) is at most
+N′′(ρ,σ+1)+N′′(dx−σ,σ) =N′′(ρ,σ).
+Ifα > ρ, then, again using (29), the right side of (30) is at most
+N′′(ρ+1,σ)+N′′(ρ,dx−ρ) =N′′(ρ,σ).
+(And, since α > β, we have at least one of α > σ,α > ρ.)
+The next result isolates (and generalizes) the main point in the derivation of (15) from Lemma
+8. We consider a hypergraph H=H1∪H2on a set Wof size 2l, where
+(i) the edges of H1are pairwise disjoint and
+(ii) the edges of H2are of size 2 and pairwise disjoint.
+LetSbe the set of vertices of Hnot covered by edges of H2, and|S|= 2t. Given α:H1→N,
+letNibe the number of partitions ( X,Y) ofWwith each of X,Ya vertex cover of H2, each of
+|X∩H|,|Y∩H|at leastαHfor each H∈ H1, and|X|=i.
+Lemma 10 In the above situation, Nl≥t+1
+tNl+1.
+11Proof.Fori∈Nandπa collection of t−1 disjoint 2-sets contained in S, letNi(π) be the set of
+partitions as above for which each of X,Yalso covers the edges of π, and set Ni=|Ni|. We assert
+that (for each π)
+Nl(π)≥2Nl+1(π). (31)
+This implies the proposition since (as is easily seen)
+Nl=1
+t·t!/summationdisplay
+πNl(π)
+and
+Nl+1=1/parenleftbigt+1
+2/parenrightbig(t−1)!/summationdisplay
+πNl+1(π).
+For the proof of (31) let x,ybe the two vertices of Wnot contained in members of H′
+2:=H2∪π.
+Noting that ( X,Y)∈ Nl+1(π) implies {x,y} ∈X, we may regard ( X,Y) as an orientation of H′
+2,
+where orienting {u,v}fromutovcorresponds to putting uinX(andvinY). The orientations
+corresponding to ( X,Y)’s from Nl+1are those for which, for each H∈ H1,
+d+(H)≥αH−|H∩{x,y}|andd−(H)≥αH,
+where, for the given orientation, d+(H) (resp.d−(H)) is the number of oriented edges whose tails
+(resp. heads) lie in H.
+If we let Gbe the multigraph gotten from ( H,π) by collapsing each H∈ H1to a single vertex
+(so for example, any {u,v} ∈ H′
+2contained in some H∈ H1becomes a loop in G), then the above
+discussion says that Nl+1(π) =NG(a,b) (see Lemma 8 for the notation), where
+az=αH−|H∩{x,y}|andbz=αH
+ifzis the vertex of Gcorresponding to H∈ H1, andaz=bz= 0 ifzis not of this type (i.e.
+z∈W\∪{H:H∈ H1}).
+A similar discussion shows that Nl(π) =NG(r,s)+NG(s,r), where
+rz=αH−1{x∈H}andsz=αH−1{y∈H}
+ifzis the vertex of Gcorresponding to H∈ H1, andrz=sz= 0 ifzis not of this type. (For
+example, NG(r,s) counts pairs ( X,Y) withx∈X(andy∈Y).)
+Finally, Lemma 8 gives NG(r,s),NG(s,r)≥NG(a,b), so we have (31). (Strictly speaking we
+may be applying Lemma 8 with some negative entries in b,rand/ors; but it’s easy to see that this
+slightly more general version follows from the lemma as stat ed.)
+As mentioned earlier, the proof of (15) also requires Theore m 11 below. (If we just wanted (14)
+then Lemma 10 alone would suffice.) For µ∈ Mm:=M(2[m]), set
+αi(µ) =/parenleftBigg
+m
+i/parenrightBigg−1/summationdisplay
+{µ(A)µ(¯A) :A⊆[m],|A|=i} (32)
+(where¯A= [m]\A). Sayµ∈ M2khas theantipodal pairs property (APP) if αk(µ)≥αk−1(µ), and
+µ∈ Mmhas theconditional antipodal pairs property (CAPP) if every measure obtained from µ
+by conditioning on the values of some m−2kvariables (for some k) has the APP (where we view
+conditioning on the values indexed by Tas producing a measure in M(2[m]\T).
+12Theorem 11 ([11]) A measure with the CAPP and no internal zeros in its rank sequen ce is ULC.
+Proof of (15). It’s again enough to show this when K= [n−1],I={n}andJ=∅. Setting
+U=σ−1(K) and letting µbe the law of U, we prove the equivalent
+µis ULC. (33)
+ForA⊆[m] andσ∈KA, sayσ∈Qif it satisfies the conditions in (7), which we now rewrite
+Sj≤ |σ−1(j)| ≤Tj∀j∈K, (34)
+whereSj=atj(j) andTj=atj+1(j)−1. (This extends the Qof (7), which was a subset of [ n][m].)
+Writeσ∼Aifσ∈KA∩Qandσ∼lifσ∼Afor some Aof sizel, and set T(A) =/producttext{γin:i∈
+[m]\A}. Thenµis given by
+µ(A)∝T(A)/summationdisplay
+{W(σ) :σ∼A}(A⊆[m]).
+By Theorem 11 we will have (33) if we show that µsatisfies the CAPP and its rank sequence
+has no internal zeros. The latter condition is given by Propo sition 6, applied with
+fj=/braceleftBigg
+Tj−Sjifj∈K
+0 if j=n
+(so that µ(|U|=k) =Mf(S1,...,S n−1,m−k).
+To show that µhas the CAPP, we should verify the APP for the conditional mea suresµX,Y∈
+MY\Xgiven by
+µX,Y(A)∝µ(A∪X) (A⊆Y\X),
+whereX⊆Y⊆[m] and|Y\X|= 2k(for some k). Fixing X,Yand letting A,Brun over subsets
+ofZ:=Y\X, this amounts to
+T(X)T(Y)/summationdisplay
+|A|=k/summationdisplay
+{W(σ)W(τ) :σ∼A∪X,τ∼(Z\A)∪X}
+≥k+1
+kT(X)T(Y)/summationdisplay
+|B|=k−1/summationdisplay
+{W(α)W(β) :α∼B∪X,β∼(Z\B)∪X}(35)
+Regard each of σ,τ,α,β in (35) as a bipartite graph on the vertex set [ m]∪Kin the natural way.
+Then for each pair ( σ,τ) appearing in (35) the multiset unionG=σ∪τis a bipartite multigraph
+with exactly 2( |X|+k) edges and
+dG(i) =
+
+2 ifi∈X
+1 ifi∈Y\X
+0 ifi∈[m]\Y
+(and similarly for pairs ( α,β)). We may thus rewrite (35) as
+/summationdisplay
+G/summationdisplay
+{W(σ)W(τ) :σ∪τ=G,σ∼ |X|+k}
+≥k+1
+k/summationdisplay
+G/summationdisplay
+{W(α)W(β) :α∪β=G,α∼ |X|+k−1},
+13and it is enough to show that for each fixed Gwe have the corresponding inequality for the inner
+sum, i.e.
+/summationdisplay
+{W(σ)W(τ) :σ∪τ=G,σ∼ |X|+k} ≥k+1
+k/summationdisplay
+{W(α)W(β) :α∪β=G,α∼ |X|+k−1}.(36)
+This has the advantage that the weights no longer play a role, since for σ,...,βas in (36),
+W(σ)W(τ) =W(α)W(β);
+so we will have (36) if we show
+Nk≥k+1
+kNk−1, (37)
+whereNi=Ni(G) is the number of partitions G=γ∪δwithγ∼ |X|+i.
+Now letH=H1∪H2be the hypergraph with vertex set W=E(G),
+H1={Hj:j∈K},
+whereHj={e∈W:j∈e}, and
+H2={{e,f}:e∝\e}atio\slash=f,eandfhave the same end in [ m]}.
+Defineα:H1→Nby
+αHj= max{Sj,dG(j)−Tj}
+(recallSj,Tjwere defined following (34)). We are then in the situation of L emma 10: a partition
+W=W1∪W2with|W1|=|X|+i(and|W2|=|Y| −i) as in the lemma (i.e. with ( W1,W2) in
+place of ( X,Y)) is the same thing as a partition G=γ∪δwithγ∼ |X|+i(andδ∼ |Y|−i), and
+thetin the lemma is equal to the present k; so we have (37).
+Remark. Log-concavity results being of some interest, we mention on e appealing specialization
+of (15). (See e.g. [16] or [2] for much more on log-concavity i n combinatorial settings.) For a
+bipartite graph G= (V∪K,E), define a G-mapto be a function f:A→KwithA⊆Vand
+(v,f(v))∈Efor allv∈A. Givenl,u∈NK(withlj≤uj), call aG-mapvalidif|f−1(j)| ∈[lj,uj]
+for allj∈K. Letsk=sk(G,l,u) be the number of valid G-mapsf:A→Kwith|A|=k.
+Theorem 12 For any G,l,uthe sequence (s0,...,s|V|)is ultra-log-concave.
+In the special case l≡0,u≡1,skbecomes Φ k= Φk(G), the number of matchings of size kinG.
+Heilman and Lieb [8, 9] and Kunz [12] (see also [13, Chapter 8] ) proved that for any (not necessarily
+bipartite) graph G, thematching generating polynomial
+p(x) =ν/summationdisplay
+k=0Φkxk
+(where, as usual, ν= max{k: Φk>0}is the matching number of G) has all real (negative) roots.
+This implies, by Newton’s inequalities (e.g. [7, Theorem 51 ]), that
+(Φ0,...,Φν) is ULC, (38)
+14which, if ν <|V|, is somewhat stronger than this case of Theorem 12. In contra st, for general l
+anduthe polynomial
+|V|/summationdisplay
+k=0skxk
+neednothaveallrealroots. (Forexample, let V={y,v,w},K={1},E(G) ={{y,1},{v,1},{w,1}},
+l1= 1,u1= 3.)
+Actually Theorem 11 can be used to show that for any G, (Φ0,...,Φτ) is ULC (where, as usual,
+τ=τ(G) is the vertex cover number), which in particular recovers ( 38) when Gis bipartite. As this
+doesn’t use Lemma 8, we won’t go into it here. It would be very i nteresting to see a combinatorial
+proof of (38) for general G.
+Before closing this section we point out one further consequ ence of Lemma 8, which seems to
+us interesting for its own sake. With notation as in the above proof of (15), set f(A) =/summationtext{W(σ) :
+σ∈KA∩Q}(A⊆[m]). We assert that fsatisfies the negative lattice condition:
+f(A∪B)f(A∩B)≤f(A)f(B)∀A,B⊆[m]. (39)
+While we don’t see how to get (14) (or (15)) from this in genera l, it’s not hard to see that it does
+imply (14) in case the σ(i)’s are i.i.d., so gives yet another proof Theorem 2. We omit t he details.
+Proof of (39). We may rewrite the inequality as
+/summationdisplay/summationdisplay
+{W(σ)W(τ) :σ∼A∪B,τ∼A∩B} ≤/summationdisplay/summationdisplay
+{W(α)W(β) :α∼A,β∼B}.(40)
+As before we regard σ,τ,α,β in (40) as bipartite graphs on [ m]∪K. For each pair ( σ,τ) appearing
+in (40), the (multiset) union G=σ∪τis a bipartite multigraph with
+dG(i) =
+
+2 ifi∈A∩B
+1 ifi∈A△B
+0 otherwise
+(and similarly for pairs ( α,β)), and it’s enough to show that, for each such G, (40) still holds if we
+restrict to pairs ( σ,τ) and (α,β) with
+σ∪τ=α∪β=G. (41)
+Again the weights ( W(σ) etc.) cancel and it’s enough to show
+N(A∪B,A∩B)≤N(A,B), (42)
+where, for C,D⊆[m],N(C,D) =NG(C,D) is the number of partitions E(G) =γ∪δwithγ∼C
+andδ∼D.
+Notice now that we are really counting partitions ˆ σ∪ˆτand ˆα∪ˆβof the edges of G′:=
+G[(A∩B)∪K], since for any σ,...,β (as in (40)) satisfying (41), any edge of Gwith an end in
+A\B(resp.B\A) must belong to σ∩α(resp.σ∩β).
+Forj∈KandC⊆[m] writedC(j) for the number of edges of GjoiningjtoC. In terms of
+ˆσ,...,ˆβthe requirement that σ,...,βsatisfy (34) becomes the condition that for each j∈K,
+Sj−dA∆B(j)≤ |ˆσ−1(j)| ≤Tj−dA∆B(j);Sj≤ |ˆτ−1(j)| ≤Tj;
+Sj−dA\B(j)≤ |ˆα−1(j)| ≤Tj−dA\B(j) andSj−dB\A(j)≤ |ˆβ−1(j)| ≤Tj−dB\A(j).(43)
+15Now let Hbe the multigraph on vertex set E(G′) with edge set {ex:x∈A∩B}, whereexjoins
+the two edges of G′containing x. We may identify a partition E(G′) =γ∪δwith the orientation
+ofHgotten by directing exfromatobwhenever a∈γandb∈δ, wherea,bare the edges on xin
+G′. The orientations corresponding to pairs (ˆ σ,ˆτ) as in (43) are then those satisfying
+Sj−dA∆B(j)≤d+(j)≤Tj−dA∆B(j) andSj≤d−(j)≤Tj∀j∈K
+while those corresponding to pairs (ˆ α,ˆβ) are those with
+Sj−dA\B(j)≤d+(j)≤Tj−dA\B(j) andSj−dB\A(j)≤d−(j)≤Tj−dB\A(j)∀j∈K.
+That the number of orientations of the first type is at most the number of the second type is then
+an instance of Lemma 8.
+5 Final remarks
+The most interesting question left open by the present work i s whether Theorem 1 (even without
+thresholds) extends to nonidentical balls; that is,
+Question 13 Are generalized urn measures (or generalized threshold or i nterval urn measures)
+CNA?
+As mentioned in the introduction, Dubhashi and Ranjan [5] sh owed NA for the ξij’s defined in (3),
+which immediately gives NA for generalized threshold urn me asures. That the weaker (than CNA)
+CNC, at least, does hold for generalized threshold (or, more generally, “interval”) urn measures is
+a special case of the following result; this is a somewhat mor e general version of Corollary 34 of [5],
+which implies CNC for generalized threshold urn measures. W e again take Pr to be the measure
+on [n][m]corresponding to some γ, and, for A ⊆2[m]anda,b∈Nn−1, setp(A,a,b) = Pr(σ−1(n)∈
+A|Bj∈[aj,bj]∀j∈[n−1]).
+Theorem 14 For any increasing A,p(A,a,b)is decreasing in (a,b).
+The proof is more or less the same as that of Theorem 4(a), so wi ll not be given here; see [14,
+Theorem 1.23]. (The proof of Corollary 34 in [5] is not quite c orrect, since it depends on the
+incorrect Proposition 24.)
+Thusonereasontobeinterested inwhethergeneralized urnm easuresareCNAis thatanegative
+answer would provide a counterexample to an important conje cture of Pemantle [15] stating that
+CNC implies CNA. (He also conjectures that the Rayleigh prop erty NC+ implies NA+.) Pursuing
+this a little further, say µ∈ M({0,1})nisR+if everyW◦µwithWi∈ {0} ∪[1,∞)∀iis NC.
+An easy simulation shows that CNC for the class of generalize d urn measures is the same as R+
+for this class. (Note this is without thresholds; it’s easy t o see that R+need not hold for (even
+ordinary) threshold urn measures.) So failure of CNA here wo uld in fact disprove
+Conjecture 15 R+implies CNA,
+a weakening of the first conjecture of Pemantle above.
+At this writing we can (e.g.) give a positive answer to Questi on 13 when each ball chooses
+from just two urns; this is of course quite special, but seems of some interest since it corresponds
+16to in- and out-degree statistics for a random orientation of a graph (where edges are oriented
+independently, but the two orientations of an edge may have d ifferent probabilities). Even this
+special case seems to require an interesting argument, but w e will not give this here as the paper
+seems long enough without it.
+As far as we can see, even the following very general statemen t could be true.
+Question 16 Suppose T0∪T1∪···∪Tsis a partition of [m]×[n], andar,br∈Nforr= 1,...,s.
+Is it true that the ξij’s in(3)are NA given
+{ξ(Tr)∈[ar,br]∀r∈[s]}
+(whereξ(T) =/summationtext
+(i,j)∈Tξij)?
+This would be a considerable strengthening of CNA for genera lized threshold urn measures.
+LetusalsojustmentiononepossibleapproachtoQuestion13 . Recall thatfor µ,ν∈ M({0,1}n),
+µstochastically dominates ν(written µ∝{ollowsequalν) ifµ(A)≥ν(A) for each increasing A ⊆ {0,1}n, and
+thatµhasthenormalized matching property if, withXchosenaccordingto µandξ=|X|,µ(·|ξ=k)
+is stochastically increasing in k(meaning, of course, that µ(·|ξ=k)∝{ollowsequalµ(·|ξ=l) whenever k > l).
+It is not too hard to show (this is somewhat like the derivatio n of CNA from CNC in [6]) that CNA
+for generalized interval urn measures would follow from a po sitive answer to
+Question 17 Is it true that for any σ∈[n][m]with law given by (6)andQas in(7), the law of
+σ−1(K)given Q has the normalized matching property?
+Finally we turn to the conjectures of Farr (unpublished circ a 2004; see [18]) and Welsh [18]
+mentioned at the end of Section 1. To put these in our framewor k, we add an urn Λ and assume
+Pr(σ(i) =j) =p∀i∈[m],j∈[n].
+(So Pr(σ(i) = Λ) = 1 −np.) LetI ⊆2[m]be decreasing and set Aj={σ−1(j)∈ I}and
+AJ=∩{Aj:j∈J}. Then Farr’s conjecture (somewhat rephrased) is
+Conjecture 18 If G is a graph on [m]andIis the collection of independent sets of G, then for
+any disjoint I,J,K⊆[n],AI↓ AJgivenAK.
+It’s not clear why this should require that Ibe of the type described, and Welsh’s conjecture was
+that the same conclusion holds for an arbitrary I. Here we sketch a counterexample to this stronger
+version. At present we don’t see how to extend to a counterexa mple to Conjecture 18, though we
+feel that this too is likely to be false.
+Example. Letn= 3 and p= 1/3 (so we don’t need Λ). Let M∪A∪B∪Cbe a partition of
+V:= [m] with|M|=s(large) and |A|=|B|=|C|=t=s+3. Let I1= 2V\M,
+I2={X⊆V:|X∩M|< .4|M|andXmeets at most two of A,B,C},
+andI=I1∪I2. Then, we assert,
+Pr(A{3})Pr(A{1,2,3})>Pr(A{1,3})Pr(A{2,3}),
+17which contradicts Welsh’s conjecture (with I={1},J={2}andK={3}). We omit the precise
+calculations; roughly, with α= (2/3)tandc= (3/2)3, we have (as t→ ∞)
+Pr(AL)∼
+
+(c+3)αif|L|= 1
+6α2if|L|= 2
+6α3if|L|= 3.
+Acknowledgment Parts of this work were carried out while the authors were vis iting the Isaac
+Newton Institute and while the first author was visiting MIT. The hospitality of both is gratefully
+acknowledged.
+References
+[1] J. Borcea, P. Br¨ and´ en and T.M. Liggett, Negative depen dence and the geometry of polynomi-
+als,J. Amer. Math. Soc. 22(2009), 521-567.
+[2] F. Brenti, Log-concave and unimodal sequences in algebr a, combinatorics, and geometry: an
+update,Contemporary Math. 178(1994), 71-89.
+[3] Y. Choe and D. Wagner, Rayleigh Matroids, Combin. Probab. Comput. 15(2006), 765-781
+(arXiv:math.CO/0307096v3).
+[4] P. Doyle and J.L. Snell, Random Walks and Electric Networks , Carus Mathematical Mono-
+graphs22, MAA, Washington DC, 1984.
+[5] D. Dubhashi and D. Ranjan: Balls and bins: a study in negat ive dependence, Random Struc-
+tures & Algorithms 13(1998), 99-124.
+[6] T. Feder and M. Mihail, Balanced matroids, pp. 26-38 in Proc. 24th STOC , ACM, 1992.
+[7] G.H. Hardy, J.E. Littlewood and G. P´ olya, Inequalities , 2nd edition, Cambridge University
+Press, Cambridge, 1952.
+[8] O.J. Heilman and E.H. Lieb, Monomers and dimers, Phys. Rev. Letters 24(1970), 1412-1414.
+[9] O.J. Heilman and E.H. Lieb, Theory of monomer-dimer syst ems,Comm. Math. Physics 25
+(1972), 190-232.
+[10] J. Kahn and M. Neiman, Negative correlation and log-con cavity,Random Structures & Algo-
+rithms, to appear.
+[11] J. Kahn and M. Neiman, A strong log-concavity property f or measures on Boolean algebras,
+submitted.
+[12] H. Kunz, Location of the zeros of the partition function for some classical lattice systems,
+Phys. Lett. (A) (1970), 311-312.
+[13] L. Lov´ asz and M.D. Plummer, Matching Theory , North Holland, Amsterdam, 1986.
+[14] M. Neiman, Ph.D. Thesis, Rutgers University, 2009.
+18[15] R. Pemantle, Towards a theory of negative dependence, J. Math. Phys. 41(2000), 1371-1390.
+[16] R.P. Stanley, Log-concave and unimodal sequences in al gebra, combinatorics, and geometry,
+pp. 500-535 in Graph theory and its applications: East and West (Jinan, 198 6), Ann. New
+York Acad. Sci. 576,
+[17] D. Wagner, Negatively correlated random variables and Mason’s conjecture for independent
+sets in matroids, Ann. Comb. 12(2008), 211-239.
+[18] D. Welsh, Harris’s inequality and its descendents, lec ture at Isaac Newton Inst., June 2008,
+http://www.newton.ac.uk/programmes/CSM/seminars/062 310005.pdf.
+19
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+arXiv:1001.0611v2  [math.DG]  29 Dec 2010FROBENIUS MANIFOLDS FROM PRINCIPAL CLASSICAL
+W-ALGEBRAS
+YASSIR IBRAHIM DINAR
+Abstract. We obtain polynomial Frobenius manifolds from classical
+W-algebras associated to principal nilpotent elements in si mple Lie al-
+gebras.
+Contents
+1. Introduction 1
+2. Preliminaries 3
+2.1. Frobenius manifolds and local bihamiltonian structur es 3
+2.2. Principal nilpotent element and opposite Cartan subal gebra 5
+3. Drinfeld-Sokolov reduction 9
+3.1. The nondegeneracy condition 12
+3.2. Differential relation 14
+4. Some results from Dirac reduction 15
+5. Polynomial Frobenius manifold 18
+5.1. Conclusions and remarks 19
+References 20
+1.Introduction
+Thisworkisacontinuationof[6]wherewebegantodevelopac onstruction
+of algebraic Frobeniusmanifolds from Drinfeld-Sokolov re duction to support
+a Dubrovin conjecture.
+A Frobenius manifold is a manifold Mwith the structure of Frobenius
+algebra on the tangent space Ttat any point t∈Mwith certain compati-
+bility conditions [11]. We say Mis semisimple or massive if Ttis semisim-
+ple for generic t. This structure locally corresponds to a potential satis-
+fying a system of partial differential equations known in topo logical field
+theory as the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. We
+sayMis algebraic if, in the flat coordinates, the potential is an a lgebraic
+function. Dubrovin conjecture is stated as follows: Semisi mple irreducible
+2000Mathematics Subject Classification. Primary 37K10; Secondary 35D45.
+Key words and phrases. Bihamiltonian geometry, Frobenius manifolds, classical W-
+algebras, Drinfeld-Sokolov reduction, Slodowy slice.
+12 YASSIR IBRAHIM DINAR
+algebraic Frobenius manifolds with positive degrees corre spond to quasi-
+Coxeter (primitive) conjugacy classes in finite Coxeter gro ups. We discussed
+in [6] how the examples of algebraic Frobenius manifolds con structed from
+Drinfeld-Sokolov reduction support this conjecture.
+Letebe aprincipal nilpotent element in a simple Lie algebra gover
+C. We fix, by using the Jacobson-Morozov theorem, a semisimple elementh
+and a nilpotent element fsuch that A={e,h,f}is ansl2-triple. Let κ+1
+be the Coxeter number of g. We prove the following
+Theorem 1.1. TheSlodowy slice
+(1.1) Q′:=e+keradf
+has a natural structure of polynomial Frobenius manifold of degreeκ−1
+κ+1.
+Let us recall some structures related to the principal nilpo tent element
+e. The element h∈ Adefines a Z-grading on gcalled the Dynkin grading
+given as follows
+(1.2) g=⊕i∈Zgi,gi={q∈g:ad h(q) =iq}.
+We fix below a certain nonzero element a∈g−2κ. It will follow from the
+work of Kostant [20] that y1=e+ais regular semisimple. The Cartan
+subalgebra h′= kerad y1is called the opposite Cartan subalgebra.
+Our main idea is to use the theory of local bihamiltonian stru cture on a
+loop space to construct the polynomial Frobenius manifold o nQ′. Recall
+that a bihamiltonian structure on a manifold Mis two compatible Pois-
+son brackets on M. It is well known that the dispersionless limit of a local
+bihamiltonian structureonthe loop space L(M) of afinitedimensional man-
+ifoldM(if it exists) always gives a bihamiltonian structureof hyd rodynamic
+type:
+(1.3) {ti(x),tj(y)}1,2=gij
+1,2(t(x))δ′(x−y)+Γij
+1,2;k(t(x))tk
+xδ(x−y),
+defined on the loop space L(M). This in turn gives a flat pencil of met-
+ricsgij
+1,2onMwhich under some assumptions corresponds to a Frobenius
+structure on M[12].
+We perform Drinfeld-Sokolov reduction [9] (see also [6] or [ 18]) using the
+representation theory of Aand the propertiesof h′to obtain a bihamiltonian
+structure on the affine loop space
+(1.4) Q=e+L(keradf).
+To this end we start by defining a bihamiltonian structure P1andP2in
+L(g). The Poisson structure P2is the standard Lie-Poisson structure and
+P1depends on the adjoint action of a. In the Drinfeld-Sokolov reduction
+the space Qwill be transversal to an action of the adjoint group of L(n) on
+a suitable affine subspace of L(g). Herenis the subalgebra
+(1.5) n:=/circleplusdisplay
+i≤−2giFROBENIUS MANIFOLDS AND W-ALGEBRAS 3
+The space of local functionals with densities in the ring Rof invariant differ-
+ential polynomials of this action is closed under P1andP2. This defines the
+Drinfeld-Sokolov bihamiltonian structure on Qsince the coordinates of Q
+can be interpreted as generators of the ring R. The second reduced Poisson
+structure on Qis called the classical W-algebra . We call it principal
+since it is related to the principal nilpotent element. We th en prove that the
+Drinfeld-Sokolov bihamiltonian structure admits a disper sionless limit and
+gives the promised polynomial Frobenius manifold.
+WementionthatfromtheworkofDubrovin[10]andHetrling[1 6]semisim-
+ple polynomial Frobenius manifolds with positive degrees a re already clas-
+sified. They correspond to Coxeter conjugacy classes in Coxe ter groups.
+Dubrovin constructed all these polynomial Frobenius manif olds on the orbit
+spaces of Coxeter groups using the results of [23]. There is a nother method
+to obtain the classical W-algebra associated to principal nilpotent elements
+known in the literature as Muira type transformation [9]. It was used in [14]
+(see also [7]) to prove that the dispersionless limit of the D rinfeld-Sokolov
+bihamiltonian structure gives the polynomial Frobenius ma nifold defined on
+the orbit space of the corresponding Weyl group [10]. The pro of depends
+also on the invariant theory of Coxeter groups. In the presen t work we give
+a new method to uniform the construction of polynomial Frobe nius mani-
+folds from Drinfeld-Sokolov reduction which depends only o n the theory of
+opposite Cartan subalgebras.
+2.Preliminaries
+2.1.Frobenius manifolds and local bihamiltonian structures. Start-
+ing we want to recall some definitions and review the construc tion of Frobe-
+nius manifolds from local bihamiltonian structure of hydro dynamics type.
+AFrobenius manifold is a manifold Mwith the structure of Frobenius
+algebra on the tangent space Ttat any point t∈Mwith certain com-
+patibility conditions [11]. This structure locally corres ponds to a potential
+F(t1,...,tn) satisfying the WDVV equations
+(2.1) ∂ti∂tj∂tkF(t)ηkp∂tp∂tq∂trF(t) =∂tr∂tj∂tkF(t)ηkp∂tp∂tq∂tiF(t)
+where (η−1)ij=∂tn∂ti∂tjF(t) is a constant matrix. Here we assume that the
+quasihomogeneity condition takes the form
+(2.2)n/summationdisplay
+i=1diti∂tiF(t) = (3−d)F(t)
+wheredn= 1. This condition defines the degrees diandthe charge dof
+the Frobenius structure on M. IfF(t) is an algebraic function we call Man
+algebraic Frobenius manifold .
+LetL(M) denote the loop space of M, i.e the space of smooth maps from
+the circle to M. A local Poisson bracket {.,.}1onL(M) can be written in4 YASSIR IBRAHIM DINAR
+the form [15]
+(2.3) {ui(x),uj(y)}1=∞/summationdisplay
+k=−1ǫk{ui(x),uj(y)}[k]
+1.
+Hereǫis just a parameter and
+(2.4) {ui(x),uj(y)}[k]
+1=k+1/summationdisplay
+s=0Ai,j
+k,sδ(k−s+1)(x−y),
+whereAi,j
+k,sare homogenous polynomials in ∂j
+xui(x) of degree s(we assign
+degreejto∂j
+xui(x))andδ(x−y) is the Dirac delta function defined by
+/integraldisplay
+S1f(y)δ(x−y)dy=f(x).
+The first terms can be written as follows
+{ui(x),uj(y)}[−1]
+1=Fij
+1(u(x))δ(x−y) (2.5)
+{ui(x),uj(y)}[0]
+1=gij
+1(u(x))δ′(x−y)+Γij
+1k(u(x))uk
+xδ(x−y) (2.6)
+Here the entries gij
+1(u),Fij
+1(u) and Γij
+1k(u) are smooth functions on the finite
+dimension space M. We note that, under the change of coordinates on M
+the matrices gij
+1(u),Fij
+1(u) change as a (2 ,0)-tensors.
+The matrix Fij
+1(u) defines a Poisson structure on M. IfFij
+1(u(x)) = 0
+and{ui(x),uj(y)}[0]
+1/ne}ationslash= 0 we say the Poisson bracket admits a disper-
+sionless limit . If the Poisson bracket admits a dispersionless limit then
+{ui(x),uj(y)}[0]
+1definesaPoissonbracketon L(M)knownas Poisson bracket
+of hydrodynamic type . By nondegenerate Poisson bracket of hydrody-
+namic type we mean those with the metric gij
+1is nondegenerate. In this case
+the matrix gij
+1(u) defines a contravariant flat metric on the cotangent space
+T∗Mand Γij
+1k(u) is its contravariant Levi-Civita connection [13].
+Assume there are two Poisson structures {.,.}2and{.,.}1onL(M) which
+form a bihamiltonian structure, i.e {.,.}λ:={.,.}2+λ{.,.}1is a Poisson
+structure on L(M) for every λ. Consider the notations for the leading terms
+of{.,.}1given above and write the leading terms of {.,.}2in the form
+{ui(x),uj(y)}[−1]
+2=Fij
+2(u(x))δ(x−y) (2.7)
+{ui(x),uj(y)}[0]
+2=gij
+2(u(x))δ′(x−y)+Γij
+2k(u(x))uk
+xδ(x−y) (2.8)
+Suppose that {.,.}1and{.,.}2admit a dispersionless limit as well as {.,.}λ
+for generic λ. In addition, assume the corresponding Poisson brackets of
+hydrodynamicstypearenondegenerate. Thenbydefinition gij
+1(u)andgij
+2(u)
+form what is called flat pencil of metrics [12], i.egij
+λ(u) :=gij
+2(u)+λgij
+1(u)
+defines a flat metric on T∗Mfor generic λand its Levi-Civita connection is
+given by Γij
+λk(u) = Γij
+2k(u)+λΓij
+1k(u).FROBENIUS MANIFOLDS AND W-ALGEBRAS 5
+Definition 2.1. A contravariant flat pencil of metrics on a manifold M
+defined by the matrices gij
+1andgij
+2is called quasihomogenous of degree
+dif there exists a function τonMsuch that the vector fields
+E:=∇2τ, Ei=gis
+2∂sτ (2.9)
+e:=∇1τ, ei=gis
+1∂sτ
+satisfy the following properties
+(1) [e,E] =e.
+(2)LE(,)2= (d−1)(,)2.
+(3)Le(,)2= (,)1.
+(4)Le(,)1= 0.
+Here for example LEdenote the Lie derivative along the vector field E
+and (,)1denote the metric defined by the matrix gij
+1. In addition, the
+quasihomogenous flat pencil of metrics is called regular if the (1,1)-tensor
+(2.10) Rj
+i=d−1
+2δj
+i+∇1iEj
+is nondegenerate on M.
+The connection between the theory of Frobenius manifolds an d flat pencil
+of metrics is encoded in the following theorem
+Theorem 2.2. [12]A contravariant quasihomogenous regular flat pencil of
+metrics of degree don a manifold Mdefines a Frobenius structure on Mof
+the same degree.
+It is well known that from a Frobenius manifold we always have a flat
+pencil of metrics but it does not necessary satisfy the regul arity condition
+(2.10). In the notations of (2.1) from a Frobenius structure onM, the flat
+pencil of metrics is found from the relations
+ηij=gij
+1 (2.11)
+gij
+2= (d−1+di+dj)ηiαηjβ∂α∂βF
+This flat pencil of metric is quasihomogenous of degree dwithτ=t1.
+Furthermore we have
+(2.12) E=/summationdisplay
+iditi∂ti, e=∂tn
+2.2.Principal nilpotent element and opposite Cartan subalgebra.
+We review some facts about principal nilpotent elements in s imple Lie al-
+gebra we need to perform the Drinfeld-Sokolov reduction. In particular, we
+recall the concept of the opposite Cartan subalgebra introd uced by Kostant
+which is the main ingredient in this work.
+LetgbeasimpleLiealgebra over Cof rankr. We fixaprincipal nilpotent
+elemente∈g. By definition a nilpotent element is called principal if ge:=
+keradehas dimension equals to r. Using the Jacobson-Morozov theorem we6 YASSIR IBRAHIM DINAR
+fix a semisimple element hand a nilpotent element fingsuch that {e,h,f}
+generate sl2subalgebra A ⊂g, i.e
+(2.13) [ h,e] = 2e,[h,f] =−2f,[e,f] =h.
+The element hdefine a Z-grading on gcalled the Dynkin grading given
+as follows
+(2.14) g=⊕i∈Zgi,gi={q∈g:ad h(q) =iq}.
+It is well known that gi= 0 ifiis odd and
+(2.15) b=⊕i≤0gi
+is a Borel subalgebra with
+(2.16) n=⊕i≤−2gi= [b,b]
+is a nilpotent subalgebra.
+We normalize the invariant bilinear from /an}b∇acketle{t.|./an}b∇acket∇i}htongsuch that /an}b∇acketle{te|f/an}b∇acket∇i}ht= 1
+and we denote the exponents of the Lie algebra gas follows
+(2.17) 1 = η1< η2≤η3...≤ηr−1< ηr.
+We will refer to the number ηrbyκ. Recall that κ+1 is the Coxeter number
+ofgand the exponents satisfy the relation
+(2.18) ηi+ηr−i+1=κ+1.
+We also recall that for all simple Lie algebras the exponents are different
+except for the Lie algebra of type D2nthe exponent n−1 appears twice.
+Consider the restriction of the adjoint representation of gtoA. Under
+this restriction gdecomposes to irreducible A-submodules
+(2.19) g=⊕Vi.
+with dim Vi= 2ηi+1 [17]. We normalize this decomposition by using the
+following proposition
+Proposition 2.3. There exists a decomposition of ginto a sum of ir-
+reducible A-submodules g=⊕r
+i=1Viin such a way that there is a basis
+Xi
+I,I=−ηi,−ηi+1,...,ηiin eachVi, i= 1,...,rsatisfying the following
+relations
+(2.20) Xi
+I=1
+(ηi+I)!adeηi+IXi
+−ηi, I=−ηi,−ηi+1,...,ηi.
+and
+(2.21) < Xi
+I,Xj
+J>=δi,jδI,−J(−1)ηi−I+1/parenleftbigg2ηi
+ηi−I/parenrightbigg
+.
+Furthermore
+adhXi
+I= 2IXi
+I. (2.22)
+adeXi
+I= (ηi+I+1)Xi
+I+1.
+adf Xi
+I= (ηi−I+1)Xi
+I−1.FROBENIUS MANIFOLDS AND W-ALGEBRAS 7
+Proof.The proof that one could compose the Lie algebra as irreducib leA-
+submodules satisfying (2.20) and (2.22) is standard and can be found in
+[17] or [20]. Let g=⊕r
+i=1Vibe such decomposition. It is easy to prove
+/an}b∇acketle{tVi|Vj/an}b∇acket∇i}ht= 0 in the case ηi/ne}ationslash=ηjby applying the step operators ad eand
+using the invariance of the bilinear form. Hence the proof is reduced to
+the case of irreducible A-submodules of the same dimension. But there is
+at most two irreducible submodules of the same dimension. As sumeVi1
+andVi2have the same dimension and denote the corresponding basis Xi1
+I
+andXi1
+J, respectively. Then one can prove by using the step operator ade
+that the subspaces Vi1andVi2are orthogonal if and only if /an}b∇acketle{tXi1
+0|Xi2
+0/an}b∇acket∇i}ht= 0.
+But it obvious that the restriction of the invariant bilinea r form to Xi1
+0and
+Xi2
+0is nondegenerate. Hence by applying the Gram-Schmidt proce dure we
+can assume that /an}b∇acketle{tXi1
+0|Xi2
+0/an}b∇acket∇i}ht= 0. Therefore, we can assume that the given
+decomposition satisfying /an}b∇acketle{tVi|Vj/an}b∇acket∇i}ht= 0 ifi/ne}ationslash=j. It remains to obtain the
+normalization (2.21). From the invariance of the bilinear f orm we have
+(2.23) /an}b∇acketle{th.Xi
+I|Xi
+J/an}b∇acket∇i}ht= (2I)/an}b∇acketle{tXi
+I|Xi
+J/an}b∇acket∇i}ht
+while
+(2.24) −/an}b∇acketle{tXi
+I|h.Xi
+J/an}b∇acket∇i}ht=−(2J)/an}b∇acketle{tXi
+I|Xi
+J/an}b∇acket∇i}ht
+Therefore /an}b∇acketle{tXi
+I|Xj
+J/an}b∇acket∇i}ht= 0 ifI+J/ne}ationslash= 0. We calculate using the step operator
+adewhereI≥0 the value
+/an}b∇acketle{tXi
+I|Xi
+−I/an}b∇acket∇i}ht=1
+(ηi−I)/an}b∇acketle{tXi
+I|e.Xi
+−I−1/an}b∇acket∇i}ht (2.25)
+=−1
+ηi−I/an}b∇acketle{te.Xi
+I|Xi
+−I−1/an}b∇acket∇i}ht
+=(−1)(ηi−I+1)
+ηi−I/an}b∇acketle{tXi
+I+1|Xi
+−I−1/an}b∇acket∇i}ht
+=(−1)ηi−I(ηi−I+1)(ηi−I+2)...2ηi
+(ηi−I)(ηi−I−1)...(1)/an}b∇acketle{tXi
+ηi|Xi
+−ηi/an}b∇acket∇i}ht
+= (−1)ηi−I/parenleftbigg2ηi
+ηi−I/parenrightbigg
+/an}b∇acketle{tXi
+ηi|Xi
+−ηi/an}b∇acket∇i}ht.
+The result follows by multiplying Xi
+Iby the value of −/an}b∇acketle{tXi
+ηi|Xi
+−ηi/an}b∇acket∇i}ht−1. We
+note that the formula (2.21) will give the same result when re placingIwith
+−I. This ends the proof.
+/square
+Note that the normalized basis for V1areX1
+1=−e, X1
+0=h, X1
+−1=f
+since it is isomorphic to Aas a vector subspace.
+It is easy to see that
+(2.26) ad e:gi→gi+28 YASSIR IBRAHIM DINAR
+is injective for i≤ −1 and surjective for i≥0. Hence the subalgebra
+gf:= kerad fhas a basis Xi
+−ηi, i= 1,...,rand
+(2.27) b=gf⊕ade(n).
+The affine space
+Q′=e+gf
+is called the Slodowy slice . It is transversal to the orbit of eunder the
+adjoint group action.
+We summarize Kostant results about the relation between the principal
+nilpotent element eand Coxeter conjugacy class in Weyl group of g.
+Theorem 2.4. [20]The element y1=e+Xr
+−2κis regular semisimple. De-
+noteh′the Cartan subalgebra containing y1, i.eh′:= kerady1and con-
+sider the adjoint group element wdefined by w:= expπi
+κ+1adh. Thenw
+acts onh′as a representative of the Coxeter conjugacy class in the Weyl
+group acting on h′. Furthermore, the element y1can be completed to a basis
+yi, i= 1,...,rforh′having the form
+yi=vi+ui, ui∈g2ηi, vi∈g2ηi−2(κ+1)
+and such that yiis an eigenvector of wwith eigenvalue expπiηi
+κ+1.
+Letadenote the element Xr
+−2κ. The element y1=e+ais called a cyclic
+element and the Cartan subalgebra h′= kerad y1is called the opposite
+Cartan subalgebra . We fix a basis yiforh′satisfying the theorem above.
+It is easy to see that ui,i= 1,...,rform a homogenous basis for ge. We
+assume the basis yiare normalized such that
+(2.28) ui=−Xi
+ηi.
+Form construction this normalization does not effect y1.
+Let us define the matrix of the invariant bilinear form on h′
+(2.29) Aij:=/an}b∇acketle{tyi|yj/an}b∇acket∇i}ht=−/an}b∇acketle{tXi
+ηi|vj/an}b∇acket∇i}ht−/an}b∇acketle{tvi|Xi
+ηj/an}b∇acket∇i}ht, i,j= 1,...,r.
+The following proposition summarize some useful propertie s we need in the
+following sections.
+Proposition 2.5. The matrix Aijis a nondegenerate and antidiagonal with
+respect to the exponents ηi, i.eAij= 0,ifηi+ηj/ne}ationslash=κ+1. Moreover, the
+commutators of aandXi
+ηisatisfy the relations
+(2.30)/an}b∇acketle{t[a,Xi
+ηi]|Xj
+ηj−1/an}b∇acket∇i}ht
+2ηj+/an}b∇acketle{t[a,Xj
+ηj]|Xi
+ηi−1/an}b∇acket∇i}ht
+2ηi=Aij
+for alli,j= 1,...,r.FROBENIUS MANIFOLDS AND W-ALGEBRAS 9
+Proof.The matrix Aijis nondegenerate since the restriction of the invariant
+bilinear form to a Cartan subalgebra is nondegenerate. The f act that it is
+anidiagonal with respect to the exponents follows from the i dentity
+(2.31) /an}b∇acketle{tyi|yj/an}b∇acket∇i}ht=/an}b∇acketle{twyi|wyj/an}b∇acket∇i}ht= exp(ηi+ηj)πi
+κ+1/an}b∇acketle{tyi|yj/an}b∇acket∇i}ht
+wherew:= expπi
+κ+1adh. For the second part of the proposition we note
+that the commutator of y1=e+aandyi=vi−Xi
+ηigives the relation
+(2.32) [ e,vi] = [a,Xi
+ηi], i= 1,...,r.
+Which in turn give the following equality for every i, j= 1,...,r
+/an}b∇acketle{t[a,Xi
+ηi]|Xj
+ηj−1/an}b∇acket∇i}ht=/an}b∇acketle{t[e,vi]|Xj
+ηj−1/an}b∇acket∇i}ht=−/an}b∇acketle{tvi|[e,Xj
+ηj−1]/an}b∇acket∇i}ht (2.33)
+=−2ηj/an}b∇acketle{tvi|Xj
+ηj/an}b∇acket∇i}ht
+but then
+(2.34)/an}b∇acketle{t[a,Xi
+ηi]|Xj
+ηj−1/an}b∇acket∇i}ht
+2ηj+/an}b∇acketle{t[a,Xj
+ηj]|Xi
+ηi−1/an}b∇acket∇i}ht
+2ηi=−/an}b∇acketle{tvi|Xj
+ηj/an}b∇acket∇i}ht−/an}b∇acketle{tvj|Xi
+ηi/an}b∇acket∇i}ht=Aij.
+/square
+3.Drinfeld-Sokolov reduction
+We will review the standard Drinfeld-Sokolov reduction ass ociated with
+the principal nilpotent element [9] (see also [6]).
+We introduce the following bilinear form on the loop algebra L(g):
+(3.1) ( u|v) =/integraldisplay
+S1/an}b∇acketle{tu(x)|v(x)/an}b∇acket∇i}htdx, u,v∈L(M),
+and we identify L(g) withL(g)∗by means of this bilinear form. For a
+functional FonL(g) we define the gradient δF(q) to be the unique element
+inL(g) such that
+(3.2)d
+dθF(q+θ˙s)|θ=0=/integraldisplay
+S1/an}b∇acketle{tδF|˙s/an}b∇acket∇i}htdxfor all ˙s∈L(g).
+Recall that we fixed an element a∈gsuch that y1=e+ais a cyclic
+element. Let us introduce a bihamiltonian structure on L(g) by means of
+Poisson tensors
+P2(v)(q(x)) =1
+ǫ[ǫ∂x+q(x),v(x)]. (3.3)
+P1(v)(q(x)) =1
+ǫ[a,v(x)].
+It is well known fact that these define a bihamiltonian struct ure onL(g)
+[21].
+We consider the gauge transformation of the adjoint group GofL(g)
+given by
+q(x)→expads(x)(∂x+q(x))−∂x (3.4)10 YASSIR IBRAHIM DINAR
+wheres(x), q(x)∈L(g). Following Drinfeld and Sokolov [9], we consider
+the restriction of this action to the adjoint group NofL(n).
+Proposition 3.1. ([6],[22]) The action of NonL(g)with Poisson tensor
+(3.5) Pλ:=P2+λP1
+is Hamiltonian for all λ. It admits a momentum map Jto be the projection
+J:L(g)→L(n+)
+wheren+is the image of nunder the killing map. Moreover, Jis Ad∗-
+equivariant.
+We take eas regular value of J. Then
+(3.6) S:=J−1(e) =L(b)+e,
+sincebis the orthogonal complement to n. It follows from the Dynking
+grading that the isotropy group of eisN.
+Recall that the space Qis defined as
+(3.7) Q:=e+L(gf).
+The following proposition identified S/Nwith the space Q. Which allows
+us to define the set Rof functionals on Qas functionals on Swhich have
+densities in the ring R.
+Proposition 3.2. [9]The space Qis a cross section for the action of Non
+S, i.e for any element q(x) +e∈Sthere is a unique element s(x)∈L(n)
+such that
+(3.8) z(x)+e= (expads(x))(∂x+q(x))−∂x∈Q.
+The entries of z(x)are generators of the ring Rof differential polynomials
+onSinvariant under the action of N.
+The Poisson pencil Pλ(3.3) is reduced on Qusing the following lemma.
+Lemma 3.3. [9]LetRbe the functionals on Qwith densities belongs to R.
+ThenRis a closed subalgebra with respect to the Poisson pencil Pλ.
+HenceQhas a bihamiltonian structure PQ
+1andPQ
+2fromP1andP2,
+respectively. The reduced Poisson structure PQ
+2is called a classical W-
+algebra. For a formal definition of classical W-algebras see [19]. We obtain
+thereducedbihamiltonian structurebyusinglemma3.3as fo llows. Wewrite
+the coordinates of Qas differential polynomials in the coordinates of Sby
+means of equation (3.8) and then apply the Leibnitz rule. For u,v∈Rthe
+Leibnitz rule have the following form
+(3.9) {u(x),v(y)}λ=∂u(x)
+∂(qI
+i)(m)∂m
+x/parenleftBig∂v(y)
+∂(qJ
+j)(n)∂n
+y/parenleftbig
+{qI
+i(x),qJ
+j(y)}λ/parenrightbig/parenrightBig
+The generators of the invariant ring Rwill have nice properties when
+we use the normalized basis we developed in last section. Let us begin byFROBENIUS MANIFOLDS AND W-ALGEBRAS 11
+writing the equation of gauge fixing (3.8) after introducing a parameter τ
+as follows
+q(x)+e=τr/summationdisplay
+i=1ηi/summationdisplay
+I=0qI
+iXi
+−I+e∈S
+z(x)+e=τr/summationdisplay
+i=1zi(x)Xi
+−ηi+e∈Q
+s(x) =τr/summationdisplay
+i=1ηi/summationdisplay
+I=1sI
+i(x)Xi
+−I∈L(n).
+Then equation (3.8) expands to
+r/summationdisplay
+i=1zi(x)Xi
+−ηi+r/summationdisplay
+i=1ηi/summationdisplay
+I=1(ηi−I+1)sI
+iXi
+−I+1=
+r/summationdisplay
+i=1ηi/summationdisplay
+I=0qI
+i(x)Xi
+−I−r/summationdisplay
+i=1ηi/summationdisplay
+I=1∂xsI
+i(x)Xi
+−I+O(τ).(3.10)
+It obvious that any invariant zi(x) has the form
+zi(x) =qηi
+i−∂xsηi
+i+O(τ) (3.11)
+=qηi
+i(x)−∂xqηi−1
+i+O(τ).
+That is, we obtained the linear term of each invariant zi(x). Furthermore,
+since/an}b∇acketle{te|f/an}b∇acket∇i}ht= 1 then z1(x) has the expression
+z1(x) =q1
+1(x)−∂xs1
+1+τ/an}b∇acketle{te|[s1
+i(x)Xi
+−1,q0
+iXi
+0]/an}b∇acket∇i}ht
++1
+2τ/an}b∇acketle{te|[s1
+i(x)Xi
+−1,[sI
+i(x)Xi
+−1,e]]/an}b∇acket∇i}ht.(3.12)
+Which is simplified by using the identity
+(3.13) [ s1
+i(x)Xi
+−1,[sI
+i(x)Xi
+−1,e]] =−[s1
+i(x)Xi
+−1,q0
+i(x)Xi
+0]
+and
+(3.14)
+/an}b∇acketle{te|[s1
+i(x)Xi
+−1,q0
+iXi
+0]/an}b∇acket∇i}ht=−/an}b∇acketle{t[s1
+i(x)Xi
+−1,e]|q0
+i(x)Xi
+0/an}b∇acket∇i}ht= (q0
+i(x))2/an}b∇acketle{tXi
+0|Xi
+0/an}b∇acket∇i}ht
+withs1
+1(x) =q0
+1(x) to the expression
+(3.15) z1(x) =q1
+1(x)−∂xq0
+1(x)+1
+2τ/summationdisplay
+i(q0
+i(x))2/an}b∇acketle{tXi
+0|Xi
+0/an}b∇acket∇i}ht
+The invariant z1(x) is called a Virasoro density and the expression above
+agree with [1].
+Our analysis will relay on the quasihomogeneity of the invar iantszi(x)
+in the coordinates of q(x)∈L(b) and their derivatives. This property is
+summarized in the following corollary12 YASSIR IBRAHIM DINAR
+Corollary 3.4. If we assign degree 2J+2l+2to∂l
+x(qJ
+i(x))thenzi(x)will
+be quasihomogenous of degree 2ηi+ 2. Furthermore, each invariant zi(x)
+depends linearly only on qηi
+i(x)and∂xqηi−1
+i(x). In particular, zi(x)with
+i < ndoes not depend on ∂l
+xqηrr(x)for any value l.
+Let us fix the following notations for the leading terms of the Drinfeld-
+Sokolov bihamiltonian structure on Q
+{zi(x),zj(y)}Q
+1=∞/summationdisplay
+k=−1ǫk{zi(x),zj(y)}[k]
+1 (3.16)
+{zi(x),zj(y)}Q
+2=∞/summationdisplay
+k=−1ǫk{zi(x),zj(y)}[k]
+2.
+where
+{zi(x),zj(y)}[−1]
+1=Fij
+1(z(x))δ(x−y) (3.17)
+{zi(x),zj(y)}[0]
+1=gij
+1(z(x))δ′(x−y)+Γij
+1k(z(x))zk
+xδ(x−y)
+{zi(x),zj(y)}[−1]
+2=Fij
+2(z(x))δ(x−y)
+{zi(x),zj(y)}[0]
+2=gij
+2(z(x))δ′(x−y)+Γij
+2k(z(x))zk
+xδ(x−y)
+3.1.The nondegeneracy condition. In this section we find the antidiag-
+onal entries of the matrix gij
+1with respect to the exponents of g, i.e the entry
+gij
+1withηi+ηj=κ+1. Our goal is to prove this matrix is nondegenerate.
+Let Ξi
+Idenote the value /an}b∇acketle{tXi
+I|Xi
+I/an}b∇acket∇i}htand we set
+[a,Xi
+I] =/summationdisplay
+j∆ij
+IXj
+I−ηr.
+By definition, for a functional Fong
+(3.18) δF(x) =/summationdisplay
+iηi/summationdisplay
+I=01
+Ξi
+IδF
+δqI
+i(x)Xi
+I
+and the Poisson brackets of two functionals IandFongreads
+(3.19) {I,F}1=/an}b∇acketle{tδI(x)|[a,δF(x)]/an}b∇acket∇i}ht=/summationdisplay
+iηi/summationdisplay
+I=0/summationdisplay
+j∆ij
+I
+Ξi
+IδI
+δqκ−I
+j(x)δF
+δqI
+i(x).
+Therefore, the Poisson brackets in coordinates have the for m
+(3.20) {qκ−I
+j(x),qI
+i(y)}1=∆ij
+I
+Ξi
+Iδ(x−y).FROBENIUS MANIFOLDS AND W-ALGEBRAS 13
+Recall that the Poisson bracket {v(x),u(y)}Q
+1of elements u, v∈Ris
+obtained by the Leibnitz rule which expands as
+{v(x),u(y)}Q
+1=/summationdisplay
+i,I;j/summationdisplay
+l,h∆ij
+I
+Ξi
+I∂v(x)
+∂(qκ−I
+j)(l)∂l
+x/parenleftBig∂u(y)
+∂(qI
+i)(h)∂h
+y(δ(x−y))/parenrightBig
+=/summationdisplay
+i,I;j/summationdisplay
+l,h,m,n(−1)h/parenleftbiggh
+m/parenrightbigg/parenleftbiggl
+n/parenrightbigg∆ij
+I
+Ξi
+I∂v(x)
+∂(qκ−I
+j)(l)/parenleftBig∂u(x)
+∂(qI
+i)(h)/parenrightBigm+n
+δh+l−m−n(x−y).
+Here we omitted the ranges of the indices since no confusion c an arise. Let
+A(v,u) denote the coefficient of δ′(x−y)
+(3.21) A(v,u) =/summationdisplay
+i,I,J/summationdisplay
+h,l(−1)h(l+h)∆ij
+I
+Ξi
+I∂v(x)
+∂(qκ−I
+j)(l)/parenleftBig∂u(x)
+∂(qI
+i)(h)/parenrightBigh+l−1
+Obviously, we obtain the entry gij
+1fromA(zi,zj).
+Lemma 3.5. Ifηi+ηj< κ+ 1thenA(zi,zj) = 0. In particular, the
+matrixgij
+1is lower antidiagonal with respect to the exponents of gand the
+antidiagonal entries are constants.
+Proof.We note that if v(x) andu(x) are in Rand quasihomogenous of
+degreeθandξ, respectively, then A(v,u) will be quasihomogenous of degree
+θ+ξ−(2κ+2)−4.
+The proof is complete by observing that the generators zi(x) of the ring R
+is quasihomogeneous of degree 2 ηi+2. /square
+Proposition 3.6. The matrix gij
+1is nondegenerate and its determinant is
+equal to the determinant of the matrix Aijdefined in (2.5).
+Proof.Fromthelastlemmaweneedonlytoconsidertheexpression A(zn,zm)
+withηn+ηm=κ+1. Here
+(3.22) A(zn,zm) =/summationdisplay
+i,I,J/summationdisplay
+h,l(−1)h(l+h)∆ij
+I
+Ξi
+I∂zn(x)
+∂(qκ−I
+j)(l)/parenleftBig∂zm(x)
+∂(qI
+i)(h)/parenrightBigh+l−1
+wherezmandznare quasihomogenous of degree 2 ηm+2 and 2 κ−2ηm+4,
+respectively. The expression∂zm(x)
+∂(qI
+i)(h)gives the constrains
+2I+2≤2ηm+2 (3.23)
+2κ−2I+2≤2κ−2ηm+4
+which implies
+ηm−1≤I≤ηm
+Therefore the only possible values for the index Iin the expression of
+A(zn,zm) that make sense are ηmandηm−1. Consider the partial sum-
+mation of A(zn,zm) whenI=ηm. The degree of zmyieldsh= 0 and that
+zmdepends linearly on qηm
+i. But then equation (3.11) implies iis fixed and
+equals to m. A similar argument on zn(x) we find that the indices landj14 YASSIR IBRAHIM DINAR
+are fixed and equal to 1 and n, respectively. But then the partial summation
+whenI=ηmgives the value
+∆mn
+ηm
+Ξmηm∂zn(x)
+∂(qκ−ηmn)(1)∂zm(x)
+∂(qηmm)(0)=−∆mn
+ηm
+Ξmηm.
+We now turn to the partial summation of A(zn,zm) whenI=ηm−1. The
+possible values for hare 1 and 0. When h= 0 we get zero since landh
+can only be zero. When h= 1 we get, similar to the above calculation, the
+value
+(−1)∆mn
+ηm−1
+Ξi
+I∂zn(x)
+∂(qκ−ηmn)(0)∂zm(x)
+∂(qηm−1
+m)(1)=∆mn
+ηm−1
+Ξm
+ηm−1.
+Hence we end with the expression
+A(zn,zm) =∆mn
+ηm−1
+Ξm
+ηm−1−∆mn
+ηm
+Ξmηm
+=/an}b∇acketle{t[a,Xn
+ηn]|Xm
+ηm−1/an}b∇acket∇i}ht
+2ηm+/an}b∇acketle{t[a,Xm
+ηm]|Xn
+ηn−1/an}b∇acket∇i}ht
+2ηn=Amn
+where we derive the last equality in proposition 2.5. Hence t he determinate
+ofgij
+1equals to the determinant of Amnwhich is nondegenerate. /square
+3.2.Differential relation. We want to observe a differential relation be-
+tween the first and the second Poisson brackets. This relatio n is a con-
+sequence of the fact that zr(x) is the only generator of the ring Rwhich
+depends explicitly on qκ
+r(x) and this dependence is linear.
+Proposition 3.7. The entries of matrices of the reduced bihamiltonian
+structure on Qsatisfy the relations
+∂zrFij
+2=Fij
+1 (3.24)
+∂zrgij
+2=gij
+1
+Proof.The fact that we calculate the reduced Poisson structure by u sing
+Leibnitz rule and zr(x) depends on qκ
+r(x) linearly, means that the invariant
+zr(x) will appear on the reduced Poisson bracket {zi(x),zj(y)}Q
+2only as a
+result of the following “brackets”
+(3.25) [ qκ−I
+j(x),qI
+i(y)] :=qκ
+r(x)∆ij
+I
+Ξi
+Iδ(x−y)
+which are the terms of the second Poisson bracket on L(g) depending ex-
+plicitly on qκ
+r(x). We expand the “brackets” [ zi(x),zj(y)] by imposing theFROBENIUS MANIFOLDS AND W-ALGEBRAS 15
+Leibnitz rule. We find the coefficient of δ(x−y) andδ′(x−y) are, respec-
+tively,
+B=/summationdisplay
+i,I,J/summationdisplay
+h,l(−1)h∆ij
+I
+Ξi
+Iqκ
+r(x)∂zi(x)
+∂(qκ−I
+j)(l)/parenleftBig∂zj(x)
+∂(qI
+i)(h)/parenrightBigh+l
+(3.26)
+D=/summationdisplay
+i,I,J/summationdisplay
+h,l(−1)h(l+h)∆ij
+I
+Ξi
+Iqκ
+r(x)∂zi(x)
+∂(qκ−I
+j)(l)/parenleftBig∂zj(x)
+∂(qI
+i)(h)/parenrightBigh+l−1
+Obviously, We have ∂zrFij
+2from∂qκrBand∂zrgij
+2from∂qκrD. But we see
+that∂qκrDis just the coefficient A(zi,zj) ofδ′(x−y) of{zi(x),zj(y)}Q
+1. This
+prove that
+∂zrgij
+2=gij
+1.
+A similar argument show that
+∂zrFij
+2=Fij
+1.
+/square
+4.Some results from Dirac reduction
+We recall that the Poisson bracket {.,.}Q
+2can be obtained by performing
+theDiracreductionof {.,.}2onQ. Wederivefromthissomefactsconcerning
+the dispersionless limit of the bihamiltonian structure on Q. Letndenote
+the dimension of g.
+LetξI, I= 1,...,nbe a total order of the basis Xi
+Isuch that
+(1) The first rare
+(4.1) X1
+−η1< X2
+−η2< ... < Xr
+−ηr
+(2) The matrix
+(4.2) /an}b∇acketle{tξI|ξJ/an}b∇acket∇i}ht, I,J= 1,...,n
+is antidiagonal.
+Letξ∗
+Idenote the dual basis of ξIunder/an}b∇acketle{t.|./an}b∇acket∇i}ht. Note that if ξI∈gµthen
+ξ∗
+I∈g−µ.
+We extend the coordinates on Qto allL(g) by setting
+(4.3) zI(b(x)) :=/an}b∇acketle{tb(x)−e|ξ∗
+I/an}b∇acket∇i}ht, I= 1,...,n.
+Let us fix the following notations for the structureconstant s and the bilinear
+form on g
+(4.4) [ ξ∗
+I,ξ∗
+J] :=/summationdisplay
+cIJ
+Kξ∗
+K,/tildewidegIJ=/an}b∇acketle{tξ∗
+I|ξ∗
+J/an}b∇acket∇i}ht.
+Now consider the following matrix differential operator
+(4.5) FIJ=ǫ/tildewidegIJ∂x+/tildewideFIJ.
+Here
+/tildewideFIJ=/summationdisplay
+K/parenleftbig
+cIJ
+KzK(x)/parenrightbig
+.16 YASSIR IBRAHIM DINAR
+Then the Poisson brackets of P2will have the form
+(4.6) {zI(x),zJ(y)}2=FIJ1
+ǫδ(x−y).
+Proposition 4.1. [1]The second Poisson bracket {.,.}Q
+2can be obtained by
+performing Dirac reduction of {.,.}2onQ.
+A consequence of this proposition is the following
+Proposition 4.2. [1]
+{z1(x),z1(y)}2=ǫδ′′′(x−y)+2z1(x)δ′(x−y)+z1
+xδ(x−y) (4.7)
+{z1(x),zi(y)}2= (ηi+1)zi(x)δ′(x−y)+ηizi
+xδ(x−y).
+Remark 4.3.The bihamiltonian reduction is a method introduced in [2] to
+reduced a bihamiltonian structure to a certain submanifold . We can use it
+to obtain a bihamiltonian structure from (3.3) associated t o the principal
+nilpotent element e[3]. The resulting bihamiltonian structure is defined on
+Q. We generalize the bihamiltonian reduction in [6] by imposi ng some con-
+ditions. The result is a bihamiltonian structure associate d to any nilpotent
+element in a simple Lie algebra. This generalization also si mplifies the bi-
+hamiltonian reduction given in [3]. The Drinfeld-Sokolov r eduction is also
+generalized to any nilpotent element in simple Lie algebra [ 19]. A similar
+result to proposition 4.1 for generalized Drinfeld-Sokolo v reduction was ob-
+tained in [1]. We used it in [8] to prove that the generalized D rinfeld-Sokolov
+reduction and the generalized bihamiltonian reduction for any nilpotent el-
+ement are the same. This in turn complete the comparison betw een the
+two reductions began by the work of Pedroni and Casati [3]. In [6] we
+also obtained proposition 4.2 by performing the generalize d bihamiltonian
+reduction.
+For the rest of this section we consider three types of indice s which have
+different ranges; capital letters I,J,K,... = 1,..,n, small letters i,j,k,... =
+1,....,rand Greek letters α,β,δ,... =r+1,...,n. Recall that the space Qis
+defined by zα= 0.
+We note that the matrix /tildewideFIJdefine the finite Lie-Poisson structure on g.
+It is well known that the symplectic subspaces of this struct ure are the orbit
+spaces of gunder the adjoint group action and we have rglobal Casimirs
+[21]. Since the Slodowy slice Q′=e+gfis transversal to the orbit of e, the
+minor matrix /tildewideFαβis nondegenerate. Let /tildewideFαβdenote its inverse.
+Proposition 4.4. [6]The Dirac formulas for the leading terms of {.,.}Q
+2is
+given by
+(4.8) Fij
+2= (/tildewideFij−/tildewideFiβ/tildewideFβα/tildewideFαj)
+(4.9) gij
+2=/tildewidegij−/tildewidegiβ/tildewideFβα/tildewideFαj+/tildewideFiβ/tildewideFβα/tildewidegαϕ/tildewideFϕγ/tildewideFγj−/tildewideFiβ/tildewideFβα/tildewidegαj.
+Now we are able to prove the followingFROBENIUS MANIFOLDS AND W-ALGEBRAS 17
+Proposition 4.5. The Drinfeld-Sokolov bihamiltonian structure on Qad-
+mits a dispersionless limit. The corresponding bihamiltoni an structure of
+hydrodynamic type gives a flat pencil of metrics on the Slodow y sliceQ′.
+Proof.We note that (4.8) is the formula of the Dirac reduction of the Lie-
+Poisson brackets of gto the finite space Q′. The fact that Slodowy slice
+is transversal to the orbit space of the nilpotent element an d this orbit has
+dimension n−ryieldFij
+2is trivial. From proposition 4.2 it follows that gij
+2is
+not trivial. This prove that the brackets {.,.}Q
+2admits a dispersionless limit.
+From propositions 3.6 and 3.7 it follows that {.,.}Q
+1admits a dispersionless
+limit and the matrix gij
+2is nondegenerate. Therefore, the two matrices gij
+1
+andgij
+2define a flat pencil of metrics on Q′. /square
+Now we want to study the quasihomogeneity of the entries of th e matrix
+gij
+2. We assign the degree µI+2 tozI(x) ifξ∗
+I∈gµI. These degrees agree
+with those given in corollary 3.4. We observe that degree zn−I+1equal to
+−µI+2 from our order of the basis, and an entry /tildewideFIJis quasihomogenous
+of degree µI+µJ+2 since [ gµI,gµJ]⊂gµI+µJ, .
+The following proposition proved in [5]
+Proposition 4.6. The matrix /tildewideFβαrestricted to Qis polynomial and the
+entry/tildewideFβαis quasihomogenous of degree −µβ−µα−2
+Proposition 4.7. The entry gij
+2is quasihomogenous of degree 2ηi+2ηj
+Proof.We will derive the quasihomogeneity from the expression (4. 9). We
+knowthatthematrix /tildewidegIJisconstant antidiagonal, i.e gIJ=CIδI
+n−J+1where
+CIare nonzero constants. In particular gij= 0. Now for a fixed iwe have
+/tildewidegiβ/tildewideFβα/tildewideFαj=Ci/tildewideFn−i+1,α/tildewideFαj.
+But then the left hand sight is quasihomogenous of degree
+µj+µα+2−µα−(−µi)−2 =µj+µi= 2ηi+2ηj.
+A similar argument show that /tildewideFiβ/tildewideFβα/tildewidegαjis quasihomogeneous of degree
+2ηi+2ηj. Let us consider
+/tildewideFiβ/tildewideFβα/tildewidegαϕ/tildewideFϕγ/tildewideFγj=/summationdisplay
+αCα/tildewideFiβ/tildewideFβα/tildewideFn−α+1,γ/tildewideFγj.
+Then any term in this summation will have the degree
+µi+µβ+2−µβ−µα−2−µn−α+1−µγ−2+µγ+µj+2 = 2ηi+2ηj
+This complete the proof. /square18 YASSIR IBRAHIM DINAR
+5.Polynomial Frobenius manifold
+Let us consider the finite dimension manifold Q′defined by the coordi-
+natesz1,...,zn. We will obtain a natural polynomial Frobenius structure on
+Q′.
+The proof of the following proposition depends only on the qu asihomo-
+geneity of the matrix gij
+1.
+Proposition 5.1. [10]There exist quasihomogenous polynomials coordi-
+nates of degree diin the form
+ti=zi+Ti(z1,...,zi−1)
+such that the matrix gij
+1(t)is constant antidiagonal.
+For the remainder of this section, wefix a coordinates ( t1,...,tn) satisfying
+the proposition above. The following proposition emphasis that under this
+change of coordinates some entries of the matrix gij
+2remain invariant.
+Proposition 5.2. The second metric gij
+2(t)and its Levi-Civita connection
+have the following entries
+(5.1) g1,n
+2(t) = (ηi+1)ti,Γ1j
+2k(t) =ηjδj
+k
+Proof.We know from proposition 4.2 that in the coordinates zithe matrix
+gij
+2(z) and its Levi-Civita connection have the following entries
+(5.2) g1,n
+2(z) = (ηi+1)zi,Γ1j
+2k(z) =ηjδj
+k
+LetE′denote the Euler vector field give by
+(5.3) E′=/summationdisplay
+i(ηi+1)zi∂zi.
+Then from the quasihomogeneity of tiwe have E′(ti) = (ηi+ 1)ti. The
+formulafor change of coordinates and thefact that t1=z1give thefollowing
+(5.4) g1j(t) =∂zat1∂zbtjgab
+2(z) =E′(tj) = (ηj+1)tj.
+For the contravariant Levi-Civita connection the change of coordinates has
+the following formula
+(5.5) Γij
+2k(t)dtk=/parenleftBig
+∂zati∂zc∂zbtjgab
+2(z)+∂zati∂zbtjΓab
+2c(z)/parenrightBig
+dzc.
+But then we get
+Γ1j
+2kdtk=/parenleftBig
+E′(∂zctj)+∂zbtjΓ1b
+2c/parenrightBig
+dzc(5.6)
+=/parenleftBig
+(ηj−ηc)∂zctj+ηc∂zctj/parenrightBig
+dzc=ηj∂zctjdzc=ηjdtj
+/square
+We arrive to our basic resultFROBENIUS MANIFOLDS AND W-ALGEBRAS 19
+Theorem 5.3. The flat pencil of metrics on the Slodowy slice Q′obtained
+from the dispersionless limit of Drinfeld-Sokolov bihamil tonian structure on
+Q(see proposition 4.5) is regular quasihomogenous of degreeκ−1
+κ+1.
+Proof.In the notations of definition 2.1 we take τ=1
+κ+1t1then
+E=gij
+2∂tjτ ∂ti=1
+κ+1/summationdisplay
+i(ηi+1)ti∂ti, (5.7)
+e=gij
+1∂tjτ ∂ti=∂tr.
+We see immediately that
+[e,E] =e
+The identity
+(5.8) Le(,)2= (,)1
+follows from and the fact that ∂tr=∂zrand proposition 3.7. The fact that
+(5.9) Le(,)1= 0.
+is a consequence from the quasihomogeneity of the matrix gij
+1(see lemma
+3.5). We also obtain from proposition 4.7
+(5.10) LE(,)2= (d−1)(,)2
+since
+(5.11) LE(,)2(dti,dtj) =E(gij
+2)−ηi+1
+κ+1gij
+2−ηj+1
+κ+1gij
+2=−2
+κ+1gij
+2.
+The (1,1)-tensor
+(5.12) Rj
+i=d−1
+2δj
+i+∇1iEj=ηi
+κ+1δj
+i.
+Hence it is nondegenerate. This complete the proof. /square
+Now we are ready to prove theorem 1.1.
+Proof.[Theorem 1.1] It follows from theorem 5.3 and 2.2 that Q′has a
+Frobenius structure of degreeκ−1
+κ+1from the dispersionless limit of Drinfeld-
+Sokolov bihamiltonian structure. This Frobenius structur e is polynomial
+since in the coordinates tithe potential Fis constructed from equations
+(2.11) and we know from proposition 4.6 that the matrix gij
+2is polynomial.
+/square
+5.1.Conclusions and remarks. The results of the present work can be
+generalized to some class of distinguished nilpotent eleme nts in simple Lie
+algebras. In particular, we notice that the existence of opp osite Cartan
+subalgebras is the main reason behind the examples of algebr aic Frobenius
+manifolds constructed in [6] which are associated to distin guished nilpotent
+elements in the Lie algebra of type F4. In [6] we discussed how these ex-
+amples support Dubrovin conjecture. Our goal is to develop a method to
+uniform the construction of all algebraic Frobenius manifo lds that could20 YASSIR IBRAHIM DINAR
+be obtained from distinguished nilpotent elements in simpl e Lie algebras
+by performing the generalized Drinfeld-Sokolov reduction . Similar treat-
+ment of the present work for algebraic Frobenius manifolds t hat could be
+obtained from subregular nilpotent elements in simple Lie a lgebras is now
+under preparation.
+Acknowledgments.
+ThisworkispartiallysupportedbytheEuropeanScienceFou ndationPro-
+gramme “Methods of Integrable Systems, Geometry, Applied M athematics”
+(MISGAM).
+References
+[1] Balog, J.; Feher, L.; O’Raifeartaigh, L.; Forgacs, P.; W ipf, A., Toda theory and
+W-algebra from a gauged WZNW point of view. Ann. Physics 203 , n o. 1, 76–136
+(1990).
+[2] Casati, Paolo; Magri, Franco; Pedroni, Marco, Bi-Hamil tonian manifolds and τ-
+function. Mathematical aspects of classical field theory, 2 13–234 (1992).
+[3] Casati, Paolo; Pedroni, Marco; Drinfeld-Sokolov reduc tion on a simple Lie algebra
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+[18] Feher, L.; O’Raifeartaigh, L.; Ruelle, P.; Tsutsui, I. ; Wipf, A. On Hamiltonian re-
+ductions of the Wess-Zumino-Novikov-Witten theories. Phy s. Rep. 222, no. 1 (1992).
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+Faculty of Mathematical Sciences, University of Khartoum, Sudan; and
+TheAbdus Salam InternationalCentre for TheoreticalPhysi cs (ICTP), Italy.
+Email: dinar@ictp.it.
+E-mail address :dinar@ictp.it
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+arXiv:1001.0612v3  [math.PR]  21 Feb 2011A Berry Esseen Theorem for the Lightbulb Process
+Larry Goldstein and Haimeng Zhang
+University of Southern California and Mississippi State University
+October 30, 2018
+Abstract
+In the so called lightbulb process, on days r= 1,...,n, out ofnlightbulbs, all initially off, exactly r
+bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice
+versa. With Xthenumberofbulbsonat theterminal time n, aneveninteger, and µ=n/2,σ2= Var(X),
+we have
+sup
+z∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingleP/parenleftbiggX−µ
+σ≤z/parenrightbigg
+−P(Z≤z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤n
+2σ2∆0+1.64n
+σ3+2
+σ
+whereZis a standard normal random variable, and
+∆0=1
+2√n+1
+2n+1
+3e−n/2forn≥6,
+yielding a bound of order O(n−1/2) asn→ ∞. A similar, though slightly larger bound holds for nodd.
+The results are shown using a version of Stein’s method for bo unded, monotone size bias couplings. The
+argument for even ndepends on the construction of a variable Xson the same space as Xthat has the
+X-size bias distribution, that is, that satisfies
+E[Xg(X)] =µE[g(Xs)] for all bounded continuous g,
+and for which there exists a B≥0, in this case B= 2, such that X≤Xs≤X+Balmost surely. The
+argument for nodd is similar to that for neven, but one first couples Xclosely to V, a symmetrized
+version of X, for which a size bias coupling of VtoVscan proceed as in the even case. In both the
+even and odd cases, the crucial calculation of the variance o f a conditional expectation requires detailed
+information on the spectral decomposition of the lightbulb chain.
+1 Introduction
+The problem we consider here arises from a study in the pharmaceut ical industry on the effects of dermal
+patches designed to activate targeted receptors. An active rec eptor will become inactive, and an inactive
+one active, if it receives a dose of medicine released from the dermal patch. Let the number of receptors, all
+initially inactive, be denoted by n. On each day of the study, some number of randomly selected rece ptors
+will each receive one dose of medicine, changing their statuses betw een the inactive and active states. We
+adopt the following, somewhat more colorful, though equivalent, ‘ligh tbulb process’ formulation from [7].
+Consider ntoggle switches, each connected to a lightbulb, all of which are initially off. Pressing the toggle
+switch connected to a bulb changes its status from off to on and vice versa. The problem of determining the
+properties of X, the number of light bulbs on at the end of day n, was first considered in [7] for the case
+where on each day r= 1,...,n,exactlyrof thenswitches are randomly pressed.
+More generally, consider the lightbulb process on nbulbs with some number kof stages, where sr∈
+{0,...,n}lightbulbs are toggled in stage r, forr= 1,...,k; we refer to the vector s= (s1,...,s k) recording
+0AMS 2010 subject classifications: Primary: 62E17, 60C05 Sec ondary: 60B15, 62P10.
+0Key words and phrases: Normal approximation, toggle switch , Stein’s method, size biasing.
+1the number of bulbs affected on each study day as the ‘switch patte rn.’ In order to consider quantities that
+depend on some subset of size bof thenbulbs, we define
+λn,b,s=b/summationdisplay
+t=0/parenleftbiggb
+t/parenrightbigg
+(−2)t(s)t
+(n)tandλn,b,s=k/productdisplay
+r=1λn,b,sr, (1)
+where(n)k=n(n−1)···(n−k+1)denotes the fallingfactorial, andthe empty product is 1. Gener alizingthe
+results in [7], writing Xsfor the number of bulbs on at the terminal time when applying the swit ch pattern
+s= (s1,...,s n), the martingale method in Proposition 4 of [11] shows that if the pro cess is initialized with
+all bulbs off, then
+EXs=n
+2(1−λn,1,s) and Var( Xs) =n
+4(1−λn,2,s)+n2
+4(λn,2,s−λ2
+n,1,s), (2)
+where, from (1),
+λn,1,s= 1−2s
+nandλn,2,s= 1−4s
+n+4s(s−1)
+n(n−1)fors= 1,...,n. (3)
+Lettingn= (1,...,n), we call the standard lightbulb process the one where s=n, and in this case
+we will write Xshort for Xn. In particular, (2) with s=nrecovers the mean µ=EXand variance of
+σ2= Var(X) as computed in [7]. Other results in [7] include recursions for determ ining the exact finite
+sample distribution of X. Though computational approximations to the distribution of X, including by the
+normal, were also considered in [7], the quality of such approximations , and the asymptotic normality of X
+was left open.
+Theorem 1.1 settles the matter of the asymptotic distribution of Xby providing a bound to the normal
+which holds for all finite n, and which tends to zero at the rate n−1/2asntends to infinity. We consider
+the case of even and odd nseparately. In the even case we directly couple the variable Xto one having the
+X-size bias distribution, as described later on in this section. In the ev en case (3) yields λn,1,n/2= 0, and
+therefore λn,1,n= 0, hence from (2) we obtain EX=n/2, and also that σ2= Var(X) is given by
+σ2=n
+4(1−λn,2,n)+n2
+4λn,2,n. (4)
+To state our result for nodd, for s= (s1,...,s k) let
+λn,b,r=/braceleftbigg1
+2(λn,b,m+λn,b,m+1)r∈ {m,m+1}
+λn,b,r otherwiseandλn,b,s=k/productdisplay
+j=1λn,b,sj, (5)
+that is,λn,b,sis obtained from λn,b,sby replacing any occurrences of λn,b,mandλn,b,m+1in the product (1)
+by their average. In the odd case, we proceed by first coupling Xto a more symmetric random variable V
+with mean and variance given respectively by
+µV=n
+2andσ2
+V=n
+4/parenleftbig
+1−λn,2,n/parenrightbig
++n2
+4λn,2,n. (6)
+Then, with Vin hand, we couple Vto a variable with the V-size bias distribution, and proceed as in the
+even case. In Theorem 1.1, and the remainder of the paper, Zdenotes a standard normal random variable.
+Theorem 1.1 LetXbe the number of bulbs on at the terminal time nin the standard lightbulb process.
+Then for all neven, with σ2as given in (4),
+sup
+z∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingleP/parenleftbiggX−n/2
+σ≤z/parenrightbigg
+−P(Z≤z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤n
+2σ2∆0+1.64n
+σ3+2
+σfor alln≥6, (7)
+2where
+∆0=1
+2√n+1
+2n+1
+3e−n/2,
+and for all n= 2m+1odd,
+sup
+z∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingleP/parenleftbiggX−n/2
+σV≤z/parenrightbigg
+−P(Z≤z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤n
+2σ2
+V∆1+1.64n
+σ3
+V+2
+σV/parenleftbigg
+1+1√
+2π/parenrightbigg
+for alln≥7,
+whereσ2
+Vis given in (6) and
+∆1=1√n+1
+2√
+2e−n/4. (8)
+In the even case, as λn,2,ndecays exponentially fast to zero, the variance σ2is of order nand the bound (7),
+therefore, of order 1 /√n; analogous remarks hold for the case where nis odd.
+We now more formally describe the lightbulb process on nbulbs with kstages. With n∈Nfixed and
+s= (s1,...,s k) withsr∈ {0,...,n}forr= 1,...,k, we will let Xs={Xrj:r= 0,1,...,k,j = 1,...,n}
+denote a collection ofBernoulli variables. The initial state ofthe bulb s is givendeterministically by {X0j,j=
+1,...,n}, which will be taken to be state zero, that is, all bulbs off, unless spe cifically stated otherwise; non
+zero initial conditions are considered only in the Appendix. For r∈ {1,...,k}the components of the ‘switch
+variables’ Xshave the interpretation that
+Xrj=/braceleftbigg1 if the status of bulb jis changed at stage r,
+0 otherwise.
+At stage r,srof thenbulbs are chosen uniformly to have their status changed, and the s tages are
+independent of each other. Hence, with e={erj}1≤r≤k,1≤j≤nan array of {0,1}valued variables, the
+distribution of Xsis given by
+P(Xs=e) =/braceleftigg/producttextk
+r=1/parenleftbign
+sr/parenrightbig−1if/summationtextn
+j=1erj=sr,r= 1,...,k
+0 otherwise.(9)
+Clearly, the vectorsofstage rswitch variables ( Xr1,...,X rn) are exchangeableand the marginaldistribution
+of the components Xrjare Bernoulli with success probability sr/n. In general, for j= 1,...,n, the variables
+Xj=/parenleftiggk/summationdisplay
+r=0Xrj/parenrightigg
+mod 2 and Xs=n/summationdisplay
+j=1Xj (10)
+are the indicator that bulb jis on at the terminal time k, and the total number of bulbs on at that time,
+respectively. For the standard lightbulb process we will write XandXforXnandXnrespectively.
+The lightbulb process, where the individual states of the nbulbs evolve according to the same marginal
+Markovchain, is a special case of a class of multivariate chains studie d in [11], known as composition Markov
+chains of multinomial type. As shown in [11], such chains admit explicit fu ll spectral decompositions, and in
+particular, the transition matrices for the stages of the lightbulb p rocess can be simultaneously diagonalized
+by a Hadamard matrix. These properties were put to use in [7] for t he calculation of the moments needed to
+compute the mean and variance of X. Here we put these same properties to somewhat more arduous wo rk,
+the calculation of moments of fourth order.
+That no higher order moments are required for the derivation of a fi nite sample bound holding for all
+nis one distinct advantage of the technique we apply here, Stein’s met hod for the normal distribution,
+brought to life in the seminal monograph [10]. By contrast, the meth od of moments requires the calculation
+and appropriate convergence of moments of all orders, and obta ins only convergence in distribution. Stein’s
+3method for the normal is based on the characterization of the nor mal distribution in [9], which states that
+Zis a standard normal variable if and only if
+E[Zg(Z)] =E[g′(Z)] (11)
+for all absolutely continuous functions gfor which these expectations exist. The idea behind Stein’s method
+is that if a mean zero, variance one random variable Wis close in distribution to Z, thenWwill satisfy (11)
+approximately. Hence, to gauge the proximity of WtoZfor a given test function h, one can evaluate the
+difference Eh(W)−Nh, whereNh=Eh(Z), by solving the Stein equation
+f′(w)−wf(w) =h(w)−Nh
+forfand evaluating E[f′(W)−Wf(W)]. A priori it may appear that an evaluation of E[f′(W)−Wf(W)]
+would be more difficult than that for Eh(W)−Nh. However, the former form may be handled through
+couplings.
+Here we consider size bias couplings to evaluate E[f′(W)−Wf(W)]. Given a nonnegative random
+variable Ywith positive finite mean µ=EY, we say Yshas theY-size bias distribution if P[Ys∈dy] =
+(y/µ)P[Y∈dy], or more formally, if
+E[Yg(Y)] =µE[g(Ys)] for all bounded continuous functions g. (12)
+The use of size bias couplings in Stein’s method was introduced in [1], whe re it was applied to derive
+bounds of order σ−1/2for the normal approximation to the number of local maxima Yof a random function
+on a graph, where σ2= Var(Y). In [5] the method was extended to multivariate normal approxima tions,
+and the rate was improved to σ−1, for the expectation of smooth functions of a vector Yrecording the
+number of edges with certain fixed degrees in a random graph. In [4] the method was used to give bounds in
+the Kolmogorov distance of order σ−1for various functions on graphs and permutations, and in [6] for tw o
+problems in the theory of coverage processes, with bounds of this same order. A more complete treatment
+of Stein’s method and its applications can be found in [3].
+Here we prove and apply Theorem 2.1 for bounded, monotone size bia s couplings, which requires that the
+random variable Yof interest, and a random variable Ys, having the Y-size bias distribution, be constructed
+on a common space such that for some nonnegative constant B,
+Y≤Ys≤Y+B
+with probability one. Loosely speaking, Theorem 2.1 says that given a ny such coupling of YandYson a
+common space, an upper bound on the Kolmogorov distance betwee n the distribution of Yand the normal
+can computed in terms of EY,Var(Y),Band the quantity
+∆ =/radicalbig
+Var(E(Ys−Y|Y)). (13)
+Theorem 2.1 is based on a concentration type inequality provided in Le mma 2.1
+For the standard lightbulb process, a size bias coupling of XtoXsis achieved in the even case by the
+construction, for each i= 1,...,n, of a collection Xifrom the given Xas follows. Recalling (10), where for
+s=nwe have k=n, ifXi= 1, that is, if bulb iis on at the terminal time, we set Xi=X. Otherwise, let
+Jbe uniformly chosen from all jfor which Xn/2,j= 1−Xn/2,iand letXibe the same as Xbut that the
+values of Xn/2,iandXn/2,Jare interchanged. Let Xibe the number of bulbs on at the terminal time when
+applying the switch variables Xi. Then, with Iuniformly chosen from 1 ,...,n, the variable Xs=XIhas
+theXsize bias distribution, essentially due to the fact, shown in Lemma 3.2, that
+L(Xi) =L(X|Xi= 1).
+Due to the parity issue, to handle the odd case when n= 2m+ 1, we first construct a coupling of X
+to a more symmetric variable V. The variable Vis constructed by randomizing stages mandm+1 in the
+4switch variables that yield X. In particular at stage mwe add an additional switch with probability 1 /2
+and, independently, at stage m+1 we remove an existing switch with probability 1 /2. A size bias coupling
+ofVtoVscan be achieved as in the even case, thus yielding a bound to the norm al forX. We remark that
+the size biased couplings developed here are used in [2] to show the dis tribution of X, in both the even and
+odd cases, obey concentration of measure inequalities.
+In Section 2 we present Theorem 2.1, which gives a bound to the norm al when a bounded, monotone size
+biased coupling can be constructed for a given X. Our coupling construction and the proof of the bound for
+the even case of the lightbulb process is given in Section 3.1. Symmetr ization, that is, the construction of V
+fromX, coupling constructions for V, and the proof of the bound for the odd case are given in Section 3.2 .
+Calculations of the bounds on the variance ∆ in (13) require estimate s onλn,b,sin (1). These estimates,
+based on the work of [11], yield the spectral decomposition of the un derlying transition matrices of the chain
+and are given in Section 4. Some of the more detailed calculations have been relegated to the Appendix.
+2 Bounded Monotone Couplings
+Theorem 2.1 for bounded monotone size bias couplings depends on th e following lemma, which is in some
+sense the size bias version of Lemma 2.1 of [8]. With Yhaving mean µand variance σ2, both finite and
+positive, with some slight abuse of notation in the definition of Ws, we set
+W=Y−µ
+σandWs=Ys−µ
+σ. (14)
+Lemma 2.1 LetYbe a nonnegative random variable with mean µand variance σ2, both finite and positive,
+and letYsbe given on the same space as Y, having the Y-size bias distribution, and satisfying Ys≥Ywith
+probability one. Then with WandWsgiven in (14), for any z∈Randa >0,
+µ
+σE(Ws−W)1{Ws−W≤a}1{z≤W≤z+a}≤a.
+Proof.For fixed z∈Rlet
+f(w) =
+
+−a w ≤z
+w−z−a z < w ≤z+2a
+a w > z +2a.
+Then, using |f(w)| ≤afor allw∈R, Var(W) = 1 and the Cauchy Schwarz inequality to obtain the first
+inequality, followed by definition (14) and the size bias relation (12), w e have
+a≥E(Wf(W))
+=1
+σE(Y−µ)f/parenleftbiggY−µ
+σ/parenrightbigg
+=µ
+σE(f(Ws)−f(W))
+=µ
+σE/integraldisplayWs−W
+0f′(W+t)dt
+≥µ
+σE/integraldisplayWs−W
+01{0≤t≤a}1{z≤W≤z+a}f′(W+t)dt,
+where in the final inequality we have used that Ws≥Wandf′(w)≥0 for all w∈R. Noting that
+f′(W+t) =1{z≤W+t≤z+2a}, and that 0 ≤t≤aandz≤W≤z+aimplyz≤W+t≤z+2a, we have
+1{0≤t≤a}1{z≤W≤z+a}f′(W+t) =1{0≤t≤a}1{z≤W≤z+a},
+5and therefore obtain
+a≥µ
+σE/integraldisplayWs−W
+01{0≤t≤a}1{z≤W≤z+a}dt
+=µ
+σE/parenleftbig
+min(a,Ws−W)1{z≤W≤z+a}/parenrightbig
+≥µ
+σE(Ws−W)1{Ws−W≤a}1{z≤W≤z+a},
+as claimed. /square
+Theorem 2.1 LetYbe a nonnegative random variable with mean µand variance σ2, both finite and positive,
+and letYsbe given on the same space as Y, with the Y-size bias distribution, satisfying Y≤Ys≤Y+B
+with probability one, for some positive constant B. Then with WandWsgiven in (14), we have
+sup
+z∈R|P(W≤z)−P(Z≤z)| ≤µ
+σ2∆+0.82δ2µ
+σ+δ,
+where
+∆ =/radicalbig
+Var(E(Ys−Y|Y))andδ=B/σ. (15)
+Proof.Forz∈Rarbitrary, let h(w) =1{w≤z}and letf(w) be the unique bounded solution to the Stein
+equation
+f′(w)−wf(w) =h(w)−Nh, (16)
+whereNh=Eh(Z). Substituting Winto (16) and using definition (14) and the size bias relation (12) yields
+E(h(W)−Nh)
+=E(f′(W)−Wf(W))
+=E/parenleftig
+f′(W)−µ
+σ(f(Ws)−f(W))/parenrightig
+=E/parenleftigg
+f′(W)(1−µ
+σ(Ws−W))−µ
+σ/integraldisplayWs−W
+0(f′(W+t)−f′(W))dt/parenrightigg
+. (17)
+As compiled in [3], we have the following bounds on the solution ffrom Lemma 2 in Chapter II of [10],
+0< f(w)<√
+2π/4 and |f′(w)| ≤1, (18)
+and also, as previously noted in [8], as a consequence of (18) and th e mean value theorem, we obtain
+|(w+t)f(w+t)−wf(w)| ≤(|w|+√
+2π/4)|t|. (19)
+Noting that EYs=EY2/µby (12), we find
+µ
+σE(Ws−W) =µ
+σ2/parenleftbiggEY2
+µ−µ/parenrightbigg
+= 1.
+Therefore, taking expectation by conditioning and then applying (1 8) and the Cauchy Schwarz inequality,
+we bound the first term in (17) as
+/vextendsingle/vextendsingle/vextendsingleE/braceleftig
+f′(W)E/parenleftig
+1−µ
+σ(Ws−W)|W/parenrightig/bracerightig/vextendsingle/vextendsingle/vextendsingle≤µ
+σ/radicalbig
+Var(E(Ws−W|W)) =µ
+σ2∆.
+6To bound the remaining term of (17), using (16), we have
+µ
+σ/integraldisplayWs−W
+0(f′(W+t)−f′(W))dt
+=µ
+σ/integraldisplayWs−W
+0[(W+t)f(W+t)−Wf(W)]dt+µ
+σ/integraldisplayWs−W
+0(1{W+t≤z}−1{W≤z})dt. (20)
+Applying (19) to the first term in (20), and using 0 ≤Ws−W≤δandEW2= 1, shows that the absolute
+value of the expectation of this term is bounded by
+µ
+σE/parenleftigg/integraldisplayWs−W
+0/parenleftigg
+|W|+√
+2π
+4/parenrightigg
+tdt/parenrightigg
+=µ
+2σE/parenleftigg
+(Ws−W)2(|W|+√
+2π
+4)/parenrightigg
+≤µ
+2σδ2(1+√
+2π
+4)≤0.82δ2µ
+σ.
+Taking the expectation of the absolute value of the second term in ( 20), we obtain
+µ
+σE/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayWs−W
+0(1{W+t≤z}−1{W≤z})dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=µ
+σE/parenleftigg/integraldisplayWs−W
+01{z−t<W≤z}dt/parenrightigg
+≤µ
+σE(Ws−W)1{z−δ<W≤z}
+again using that 0 ≤Ws−W≤δwith probability 1. Lemma 2.1 with a=δandzreplaced by z−δshows
+this term can be no more than δ. Sincez∈Rwas arbitrary the proof is complete. /square
+3 Normal approximation of X
+The next lemma shows that the size bias distribution of a sum may be ac hieved by taking certain mixtures.
+The result is a special case of Lemma 2.1 of [5], but we give a short dire ct proof to make the paper more
+self-contained.
+Lemma 3.1 Suppose Xis a sum of nontrivial exchangeable Bernoulli variables X1,...,X n, and that for
+i∈ {1,...,n}the variables Xi
+1,...,Xi
+nhave joint distribution
+L(Xi
+1,...,Xi
+n) =L(X1,...,X n|Xi= 1).
+Then
+Xi=n/summationdisplay
+j=1Xi
+j
+has theX-size bias distribution Xs, as does the mixture XIwhenIis a random index with values in
+{1,...,n}, independent of all other variables.
+Proof.Fori∈ {1,...,n}, we firstneed toshowthat Xisatisfies(12), that is, that E[X]E[g(Xi)] =E[Xg(X)]
+holds for a given bounded continuous g. Now, for such g
+E[Xg(X)] =n/summationdisplay
+j=1E[Xjg(X)] =n/summationdisplay
+j=1P[Xj= 1]E[g(X)|Xj= 1].
+As exchangeability implies that E[g(X)|Xj= 1] =E[g(X)|Xi= 1] for all j= 1,...,n, we have
+E[Xg(X)] =
+n/summationdisplay
+j=1P[Xj= 1]
+E[g(X)|Xi= 1] =E[X]E[g(Xi)],
+7proving the first claim. The second claim now follows from
+Eg(XI) =n/summationdisplay
+i=1E[g(XI),I=i] =n/summationdisplay
+i=1E[g(XI)|I=i]P(I=i) =n/summationdisplay
+i=1Eg(Xi)P(I=i)
+=n/summationdisplay
+i=1Eg(Xs)P(I=i) =Eg(Xs)n/summationdisplay
+i=1P(I=i) =Eg(Xs).
+/square
+3.1 Even case
+In this subsection we provide the proof of Theorem 1.1 for even n. We begin by describing a coupling of X,
+the total number of bulbs on at the terminal time nin the standard lightbulb process, to a variable Xswith
+theX-size bias distribution. Throughout, we let U(S) denote the uniform distribution over a finite set S.
+Theorem 3.1 Withn∈Neven, let the collection of switch variables X={Xrj:r,j= 1,...,n}andX
+satisfy (9) and (10), respectively, with s=n. For every i= 1,...,nletXibe given from Xas follows. If
+Xi= 1thenXi=X. Otherwise, with L(Ji|X) =U{j:Xn/2,j= 1−Xn/2,i}, letXi={Xi
+rj:r,j= 1,...,n}
+where
+Xi
+rj=
+
+Xrjr/\⌉}atio\slash=n/2
+Xn/2,jr=n/2,j/\⌉}atio\slash∈ {i,Ji}
+Xn/2,Jir=n/2,j=i
+Xn/2,ir=n/2,j=Ji,
+and letXi=/summationtextn
+j=1Xi
+jwhere
+Xi
+j=/parenleftiggn/summationdisplay
+r=1Xi
+rj/parenrightigg
+mod2.
+Then, with Iuniformly chosen from {1,...,n}and independent of all other variables, the mixture XI=Xs
+has theX-size bias distribution and satisfies
+Xs−X= 21{XI=0,XJI=0}andX≤Xs≤X+2. (21)
+We prove Theorem 3.1 making use of a preliminary lemma, and also of the fact that
+P(Xj= 0) =P(Xj= 1) =1
+2for allj= 1,...,n. (22)
+The equalities in (22) follow from EX=n/2, itself implied by (2) and that λ1,n,n= 0, as noted earlier.
+Lemma 3.2 For alli= 1,...,n, the collection of random variables Xiconstructed from Xas specified in
+Theorem 3.1 satisfies
+L(Xi) =L(X|Xi= 1).
+Proof.For a given i∈ {1,...,n}lete={erj:r,j= 1,...,n}witherj∈ {0,1}forr,j= 1,...,n. First note
+that since Xi=XwhenXi= 1 we have
+P(Xi=e) =P(Xi= 1)P(Xi=e|Xi= 1)+P(Xi= 0)P(Xi=e|Xi= 0)
+=P(Xi= 1)P(X=e|Xi= 1)+P(Xi= 0)P(Xi=e|Xi= 0),
+so the desired conclusion is equivalent to
+P(Xi=e|Xi= 0) =P(X=e|Xi= 1). (23)
+8As the construction of Xipreserves the number of switches in each stage rwe may assume/summationtext
+jerj=r
+for allr, as otherwise both sides of (23) are zero. If/summationtext
+reri= 0mod2 then the left hand side of (23) is zero
+sinceXi
+i= 1 by construction; similarly, the right hand side is zero as X=eimpliesXi= 0. Hence we need
+only verify (23) assuming
+n/summationdisplay
+j=1erj=rfor allr= 1,...,n, andn/summationdisplay
+r=1eri= 1mod2 . (24)
+Writing JforJifor simplicity, and letting ei,jdenote the array ewith coordinates en/2,ianden/2,j
+interchanged, by (22) we have
+P(Xi=e|Xi= 0) = 2 P(Xi=e,Xi= 0)
+= 2n/summationdisplay
+j=1P(Xi=e,Xi= 0,J=j)
+= 2n/summationdisplay
+j=1P(X=ei,j,Xi= 0,J=j).
+Note that when en/2,i=en/2,j, or equivalently, ei,j
+n/2,i=ei,j
+n/2,jthen
+P(X=ei,j,Xi= 0,J=j) =P(X=ei,j,J=j) (25)
+as both sides are zero, since on J=jwe haveXn/2,i/\⌉}atio\slash=Xn/2,j. Otherwise, en/2,i/\⌉}atio\slash=en/2,j, and by the second
+equality in (24),
+n/summationdisplay
+r=1ei,j
+ri=/summationdisplay
+r/n⌉}ationslash=n/2eri+ei,j
+n/2,i=/summationdisplay
+r/n⌉}ationslash=n/2eri+1−en/2,i=n/summationdisplay
+r=1eri+1 = 0,with equalities modulo 2,
+so (25) holds again. Hence
+P(Xi=e|Xi= 0)
+= 2n/summationdisplay
+j=1P(X=ei,j,J=j)
+= 2n/summationdisplay
+j=1P(J=j|X=ei,j)P(X=ei,j)
+= 2n/productdisplay
+s=1/parenleftbiggn
+s/parenrightbigg−1n/summationdisplay
+j=1P(J=j|X=ei,j) = 2n/productdisplay
+s=1/parenleftbiggn
+s/parenrightbigg−1
+,
+where in the second to last equality we have used that ei,j, ase, satisfies the first equality of (24), and
+the distribution of Xgiven by (9), and in the last equality that the sum of probabilities of th e conditional
+distribution of Jgiven an Xconfiguration that satisfies the first equality of (24) for r=n/2 must sum to
+one. Now, again using the second equality in (24),
+2n/productdisplay
+s=1/parenleftbiggn
+s/parenrightbigg−1
+= 2P(X=e) = 2P(X=e,Xi= 1) =P(X=e|Xi= 1),
+proving (23), and the lemma. /square
+We now present the proof of Theorem 3.1.
+9Proof.ThatXshas theX-size bias distribution follows from Lemmas 3.1 and 3.2. To prove the fir st equality
+in (21), note that if XI= 1 then XI=X, hence in this case Xs=X. Otherwise XI= 0 and the collection
+XIis constructed from Xby interchanging the stage n/2, unequal, switch variables Xn/2,IandXn/2,JI.
+IfXJI= 1 then after the interchange XI
+I= 1 and XI
+JI= 0, yielding Xs=X. IfXJI= 0 then after
+the interchange XI
+I= 1 and XI
+JI= 1, yielding Xs=X+ 2. The second claim in (21) is an immediate
+consequence of the first. /square
+The following lemma shows that for the case at hand the variance of t he conditional expectation term
+(15) in Theorem 2.1 may be expressed in terms of quantities of the fo rm
+g(l)
+α,β,s=P(X1=···=Xα+β= 0,Xl1=···=Xlα= 0,Xl,α+1=···=Xl,α+β= 1), (26)
+the probability that when applying the switch pattern s= (s1,...,s k) onnbulbs over kstages, bulbs
+numbered 1 though α+βterminate in the off position, and in some stage l∈ {1,...,k}, bulbs 1 through α
+receive switch variable 0, and bulbs numbered α+1 through α+βreceive switch variable 1. We keep the
+number of bulbs nimplicit in the notation, and also suppress swhens=n, writing more simply g(l)
+α,β.
+Using the spectral decomposition in Section 4 to handle the probabilit ies in (26), we now provide an
+upper bound to the term (15) when applying Theorem 2.1 for neven. For a given s= (s1,...,s k) and
+l∈ {1,...,k}let
+sl= (s1,...,s l−1,sl+1,...,s k), (27)
+the vector swith its lthcomponent deleted. For l/\⌉}atio\slash=jwe similarly let sl,jdenoteswith its lthandjth
+components deleted.
+Lemma 3.3 Letnbe even and XandXsgiven by Theorem 3.1. Then for n≥6,
+∆0≤∆0where∆0=/radicalbig
+Var(E(Xs−X|X))and∆0=1
+2√n+1
+2n+1
+3e−n/2.
+Proof.We apply the construction of Xs, and the conclusions, of Theorem 3.1. For notational simplicity let
+JI=J, so in particular, from (21) we have Xs−X= 21{XI=0,XJ=0}. Expanding the indicator over the
+possible values of IandJ, and then over the values of the switch variables Xn/2,iandXn/2,jyields
+1{XI=0,XJ=0}=n/summationdisplay
+i,j=11{Xi=0,Xj=0}1{I=i,J=j}
+=n/summationdisplay
+i,j=11{Xi=0,Xj=0,Xn/2,i=0}1{I=i,J=j}+n/summationdisplay
+i,j=11{Xi=0,Xj=0,Xn/2,i=1}1{I=i,J=j}
+=/summationdisplay
+i/n⌉}ationslash=j1{Xi=0,Xj=0,Xn/2,i=0,Xn/2,j=1}1{I=i,J=j}+/summationdisplay
+i/n⌉}ationslash=j1{Xi=0,Xj=0,Xn/2,i=1,Xn/2,j=0}1{I=i,J=j}
+= 2/summationdisplay
+i/n⌉}ationslash=j1{Xi=0,Xj=0,Xn/2,i=0,Xn/2,j=1}1{I=i,J=j},
+where the second to last equality holds almost surely, as the probab ility of the event {I=i,J=j}is zero
+whenever Xn/2,iandXn/2,jagree, and the last inequality holds as the final expression is the sum of two
+terms which can be seen to be equal by reversing the roles of iandj.
+To obtain a tractable bound on the required variance we apply the ine quality
+Var(E(Xs−X|X))≤Var(E(Xs−X|F)), (28)
+which holds when Fis anyσ-algebra with respect to which Xis measurable (see [5], for example). Here
+we letFbe theσ-algebra generated by X, the collection of all switch variables. The first indicator in the
+10final sum above, 1{Xi=0,Xj=0,Xn/2,i=0,Xn/2,j=1}, is measurable with respect to F. For the second indicator,
+conditioning on Fyields
+E/parenleftbig
+1{I=i,J=j}|F/parenrightbig
+=P(I=i,J=j|F) =2
+n21{Xn/2,i/n⌉}ationslash=Xn/2,j},
+as for any i, chosen with probability 1 /n, there are n/2 choices for jsatisfying the condition in the indicator.
+Hence, recalling from (21) that Xs−X= 21{XI=0,XJ=0}, we have
+E/parenleftbigg
+Xs−X/vextendsingle/vextendsingle/vextendsingle/vextendsingleF/parenrightbigg
+=UnwhereUn=4
+n2/summationdisplay
+i/n⌉}ationslash=j1{Xi=0,Xj=0,Xn/2,i=0,Xn/2,j=1}, (29)
+and ∆2
+0≤Var(Un) by (28).
+Taking the expectation of Unin (29), by the exchangeability of the ( n)2terms in the sum, applying
+Corollary 4.1 and recalling the notation defined in (27), we obtain
+EUn=4
+n2(n)2g(n/2)
+1,1=1
+4/parenleftbig
+1−λn,2,nn/2/parenrightbig
+. (30)
+Squaring (29) in order to obtain the second moment of Un, we obtain a sum over indices i1,i2,j1,j2with
+{i1,i2}∩{j1,j2}=∅, so|{i1,i2,j1,j2}| ∈ {2,3,4}, and we may write
+U2
+n=U2
+n,2+U2
+n,3+U2
+n,4where (31)
+U2
+n,p=16
+n4/summationdisplay
+|{ii,i2,j1,j2}|=p
+{i1,i2}∩{j1,j2}=∅1{Xi1=0,Xj1=0,Xn/2,i1=0,Xn/2,j1=1}1{Xi2=0,Xj2=0,Xn/2,i2=0,Xn/2,j2=1}.
+Beginning the calculation with the main term U2
+n,4, where all four indices are distinct, taking expectation
+using exchangeability, Corollary 4.1 yields
+EU2
+n,4=16
+n4(n)4g(n/2)
+2,2=/parenleftbiggn−2
+4n/parenrightbigg2/parenleftbig
+1−2λn,2,nn/2+λn,4,nn/2/parenrightbig
+. (32)
+With the inequalities over the summation in (31) in force, the event |{i1,i2,j1,j2}|= 3 can only occur
+when
+a)i1/\⌉}atio\slash=i2,j1=j2orb)i1=i2,j1/\⌉}atio\slash=j2.
+Now continuing to use Corollary 4.1 without further notice, case a) le ads to a contribution of
+16
+n4(n)3g(n/2)
+2,1=n−2
+4n2/parenleftbig
+1+λn,1,nn/2−λn,2,nn/2−λn,3,nn/2/parenrightbig
+,
+while in the same manner the contribution from case b) is
+16
+n4(n)3g(n/2)
+1,2=n−2
+4n2/parenleftbig
+1−λn,1,nn/2−λn,2,nn/2+λn,3,nn/2/parenrightbig
+.
+Totaling we find
+EU2
+n,3=n−2
+2n2/parenleftbig
+1−λn,2,nn/2/parenrightbig
+. (33)
+With the inequalities over the summation in (31) in force, the event |{i1,i2,j1,j2}|= 2 can only occur
+wheni1=i2andj1=j2. Hence,
+EU2
+n,2=16
+n4(n)2g(n/2)
+1,1=1
+n2/parenleftbig
+1−λn,2,nn/2/parenrightbig
+. (34)
+11Summing (32), (33) and (34) we obtain
+EU2
+n=/parenleftbiggn−2
+4n/parenrightbigg2/parenleftbig
+1−2λn,2,nn/2+λn,4,nn/2/parenrightbig
++1
+2n/parenleftbig
+1−λn,2,nn/2/parenrightbig
+.
+Subtracting the square of the first moment, given in (30), yields
+Var(Un) =1
+16/parenleftbigg
+1−2
+n/parenrightbigg2/parenleftbig
+1−2λn,2,nn/2+λn,4,nn/2/parenrightbig
++1
+2n/parenleftbig
+1−λn,2,nn/2/parenrightbig
+−1
+16/parenleftbig
+1−λn,2,nn/2/parenrightbig2
+=1
+16/parenleftig
+λn,4,nn/2−λ2
+n,2,nn/2/parenrightig
++1−n
+4n2/parenleftbig
+1−2λn,2,nn/2+λn,4,nn/2/parenrightbig
++1
+2n/parenleftbig
+1−λn,2,nn/2/parenrightbig
+=1
+4n+1
+4n2+1
+16/parenleftig
+λn,4,nn/2−λ2
+n,2,nn/2/parenrightig
+−1
+2n2λn,2,nn/2+1−n
+4n2λn,4,nn/2.
+Now applying Lemma 4.5 from Section 4, for n≥6 we obtain
+Var(Un)≤1
+4n+1
+4n2+1
+16/parenleftbigg1
+2e−n+e−2n/parenrightbigg
++1
+2n2e−n+n−1
+8n2e−n
+≤1
+4n+1
+4n2+e−n/parenleftbigg1
+16+1
+n2+1
+8n/parenrightbigg
+.
+The inequality√
+a+b+c≤√a+√
+b+√c, holding for all nonnegative a,bandc, now yields the claim of
+the lemma. /square
+With all ingredients at hand, we may now prove the bound for even n.
+Proof of Theorem 1.1, even case. The size biased coupling given in Theorem 3.1 satisfies the hypotheses of
+Theorem 2.1 with B= 2, by the second inequality in (21). Hence the result for the even c ase follows by
+applying Theorem 2.1 with µ=n/2,δ= 2/σand the bound ∆0on ∆0given in Lemma 3.3. /square
+3.2 Odd case
+Now we move to the case where n= 2m+1 is odd. Instead of directly forming a size biased coupling to X,
+we first couple Xclosely to a more symmetrical random variable Vfor which a coupling like the one in the
+even case may be applied. The variable Vis constructed by randomizing stages mandm+1. In particular
+at stage mwe add an additional switch with probability 1 /2 and, independently at stage m+1 we remove
+an existing switch with probability 1 /2.
+Formally, let X={Xrj:r,j= 1,...,n}be a collection of switch variables with distribution given by (9)
+withs=n, and let X=Xnbe given by (10) with k=n. Let
+L(Bm|X) =U{j:Xmj= 0}andL(Bm+1|X) =U{j:Xm+1,j= 1},
+and letCmandCm+1be symmetric Bernoulli variables, independent of Xand ofBmandBm+1. Now let a
+collection of switch variables V={Vrj,r,j= 1,...,n}be defined by
+Vrj=
+
+Xrj r/\⌉}atio\slash∈ {m,m+1}
+Xmj r=m,j/\⌉}atio\slash=Bm
+Cm r=m,j=Bm
+Xm+1,jr=m+1,j/\⌉}atio\slash=Bm+1
+Cm+1r=m+1,j=Bm+1,(35)
+and set
+V=n/summationdisplay
+j=1VjwhereVj=/parenleftiggn/summationdisplay
+r=1Vrj/parenrightigg
+mod 2. (36)
+12In other words, in all stages other than mandm+1 the switch variables that produce Vare the ones from
+the given collection X. In stage m, the switch variable for all bulbs but bulb Bm, chosen uniformly over all
+bulbs in that stage that were not toggled, are the ones given by X. The switch variable for Bmin stage
+m, however, is set to Cm, which takes the values 0 and 1 equally likely. Hence with probability 1 /2, one
+additional bulb in stage mis toggled. Similarly, in stage m+1, the switch variable of bulb Bm+1, uniformly
+selected from all the bulbs that were toggled in that stage, is no long er toggled with probability 1 /2. Since
+XandVdiffer in at most two switches, we have
+|X−V| ≤2. (37)
+We now show some basic facts about the distribution of the switch va riablesV, that as in even case
+EV=n/2, and verify the variance formula (6). In the following, let
+n(a,b) = (1,...,m−1,m+a,m+b,m+2,...,n) fora,b∈ {0,1}. (38)
+Lemma 3.4 The collections of variables {Vrj,j= 1,...,n}are mutually independent for r= 1,...,n. For
+r∈ {m,m+1},
+L(Vr1,...,V rn) =1
+2L(Xm1,...,X mn)+1
+2L(Xm+1,1,...,X m+1,n), (39)
+and furthermore, with Vjgiven by (36), we have P(Vj= 1) = 1 /2for allj= 1,...,n,EV=n/2and
+Var(V) =σ2
+Vas given in (6).
+Proof.The first claim follows by the independence of {Xrj,j= 1,...,n}forr= 1,...,n, and that {Vrj,j=
+1,...,n}={Xrj,j= 1,...,n}forr/\⌉}atio\slash∈ {m,m+1}, and otherwise {Vrj,j= 1,...,n}is given by randomizing
+{Xrj,j= 1,...,n}independently of the remaining stages, and of stage 2 m+1−r.
+Toprove(39),firstconsiderthecase r=m. Lete1,...,e n∈ {0,1}. Since(Vm1,...,V mn) = (Xm1,...,X mn)
+whenCm= 0, and since Cmis independent of ( Xm1,...,X mn), we have
+P(Vm1=e1,...,V mn=en)
+=1
+2P(Vm1=e1,...,V mn=en|Cm= 0)+1
+2P(Vm1=e1,...,V mn=en|Cm= 1)
+=1
+2P(Xm1=e1,...,X mn=en|Cm= 0)+1
+2P(Vm1=e1,...,V mn=en|Cm= 1)
+=1
+2P(Xm1=e1,...,X mn=en)+1
+2P(Vm1=e1,...,V mn=en|Cm= 1).
+Hence (39) is equivalent to
+P(Vm1=e1,...,V mn=en|Cm= 1) =P(Xm+1,1=e1,...,X m+1,n=en). (40)
+Now assuming/summationtext
+jej=m+1, as both sides of (40) are zero otherwise, using the independen ce ofCmfrom
+BmandXwe have
+P(Vm1=e1,...,V mn=en|Cm= 1)
+=n/summationdisplay
+j=1P(Vm1=e1,...,V mn=en|Bm=j,Cm= 1)P(Bm=j|Cm= 1)
+=/summationdisplay
+j:ej=1P(Xml=el,l/\⌉}atio\slash=j,Xmj= 0|Bm=j,Cm= 1)P(Bm=j)
+=/summationdisplay
+j:ej=1P(Xml=el,l/\⌉}atio\slash=j,Xmj= 0|Bm=j)P(Bm=j)
+=/summationdisplay
+j:ej=1P(Bm=j|Xml=el,l/\⌉}atio\slash=j,Xmj= 0)P(Xml=el,l/\⌉}atio\slash=j,Xmj= 0).
+13Now, since Bmis chosen uniformly from the n−mvariables taking the value 0 in stage m, recalling/summationtext
+jej=m+1, we have
+P(Vm1=e1,...,V mn=en|Cm= 1) =1
+n−m/summationdisplay
+j:ej=1P(Xml=el,l/\⌉}atio\slash=j,Xmj= 0)
+=m+1
+n−m/parenleftbiggn
+m/parenrightbigg−1
+=/parenleftbiggn
+m+1/parenrightbigg−1
+=P(Xm+1,1=e1,...,X m+1,n=en),
+proving (40). The argument for r=m+1 is essentially the same.
+Recalling the notation in (38), by (39) the distribution of Vis the equal mixture of L(Xs) over the four
+casess=n(a,b) for{a,b} ∈ {0,1}. Hence, from (2),
+EV=n
+8/summationdisplay
+r,t∈{m,m+1}/parenleftbig
+1−λn,1,n(a,b)/parenrightbig
+=n
+2,
+as
+1
+4/summationdisplay
+a,b∈{0,1}λn,1,n(a,b)=1
+4
+/productdisplay
+s/n⌉}ationslash∈{m,m+1}λn,1,s
+/parenleftbig
+λ2
+n,1,m+2λn,1,mλn,1,m+1+λ2
+n,1,m+1/parenrightbig
+=
+/productdisplay
+s/n⌉}ationslash∈{m,m+1}λn,1,s
+((λn,1,m+λn,1,m+1)/2)2= 0,
+since
+λn,1,m+λn,1,m+1=/parenleftbigg
+1−2m
+n/parenrightbigg
++/parenleftbigg
+1−2(m+1)
+n/parenrightbigg
+= 0.
+ClearlyEV=n/2 implies P(Vj= 1) = 1 /2
+Next, by the conditional variance formula and (2) we obtain
+Var(V) =1
+4/summationdisplay
+a,b∈{0,1}/parenleftbigg
+Var(Xn(a,b))+E/parenleftig
+Xn(a,b)−n
+2/parenrightig2/parenrightbigg
+=1
+4/summationdisplay
+a,b∈{0,1}/parenleftbiggn
+4(1−λn,2,n(a,b))+n2
+4λn,2,n(a,b)/parenrightbigg
+,
+and now applying (1 /4)/summationtext
+a,b∈{0,1}λn,1,n(a,b)=λn,1,n, and a similar identity for λn,2,n(a,b), yields the vari-
+anceσ2
+Vin (6). /square
+We now present a size bias coupling for V. As in the even case, the variable Vsis obtained by first
+constructing, for each i= 1,...,n, switch variables Vithat satisfy
+L(Vi) =L(V|Vi= 1). (41)
+For a given i= 1,...,n, to construct Vi, one first determines if Vi= 1. If so, set Vi=V. Otherwise, let M
+be a variable that chooses from the stages mandm+1 uniformly and independently of V. As in this case
+Vi= 0, one may achieve Vi
+i= 1 by changing the switch variable VMito 1−VMi. The coupling accomplishes
+this change in one of two possible ways.
+To introduce the first way, called a flip, we saya configuration eof binaryswitch variable values is feasible
+ifP(V=e)/\⌉}atio\slash= 0, that is, when
+n/summationdisplay
+j=1erj=rforr/\⌉}atio\slash∈ {m,m+1}andn/summationdisplay
+j=1erj∈ {m,m+1}forr∈ {m,m+1}.
+14If flipping VMito 1−VMiresults in feasible configuration, then the flip is made with probability 1 /(m+1).
+In other words, given e,r∈ {m,m+1}andi= 1,...,nleter,ibe the configuration with entries
+er,i
+sl=/braceleftbiggesls/\⌉}atio\slash=rorl/\⌉}atio\slash=i
+1−eris=randl=i.
+Defining VM,iin like manner, the distribution of Fi, the indicator that VMiis flipped, is given by
+P(Fi= 1|V,M) =1
+m+11(VM,iis feasible) . (42)
+In the flip is unsuccessful, that is, if Fi= 0, we perform an ‘interchange’ in stage M, much like the
+coupling in the even case. For a configuration e,r∈ {m,m+1}andi,j∈ {1,...,n}, let
+er,i,j
+sl=
+
+esls/\⌉}atio\slash=rorl/\⌉}atio\slash=i
+erls=r,l/\⌉}atio\slash∈ {i,j}
+eris=r,l=j
+erjs=r,l=i,
+that is,er,i,jis the configuration ewith the variables in the r,iandr,jpositions interchanged. Now let Ji
+be a random index with distribution given by
+L(Ji|V,M) =U{j:VMj/\⌉}atio\slash=VMi}. (43)
+Defining VM,i,jin like manner, when Fi= 0, we interchange VMiwithVM,Ji.
+Hence, overall the configuration Viis specified by
+Vi=
+
+V Vi= 1
+VM,iVi= 0,Fi= 1
+VM,i,JiVi= 0,Fi= 0.(44)
+The following theorem shows that Visatisfies (41).
+Before stating the theorem, we claim that if eis feasible, then for r∈ {m,m+1}, the configuration er,iis
+feasible if and only if/summationtext
+j/n⌉}ationslash=ierj=m. Botheander,iarefeasible if and only if/summationtext
+j/n⌉}ationslash=ierj+eri∈ {m,m+1}and/summationtext
+j/n⌉}ationslash=ierj+1−eri∈ {m,m+1}. Theclaimnowfollowsnotingthat {m−eri,m+1−eri}∩{m−1+eri,m+eri}=
+{m−1,m}∩{m,m+1}={m}. Summarizing, for a feasible configuration e,
+er,iis feasible if and only if/summationdisplay
+j/n⌉}ationslash=ierj=mthat is, if and only if eri=/summationtextn
+j=1erj−m. (45)
+In the following we denote FiandJibyFandJrespectively for simplicity.
+Theorem 3.2 LetVbe constructed from Xas in (35), let Mbe a random variable uniformly distributed
+over{m,m+1}and independent V, and for i∈ {1,...,n}, givenVandMletFandJhave distributions
+as specified in (42) and (43), respectively. Then Vigiven by (44) satisfies (41).
+Further, letting Vi=/summationtextn
+j=1Vi
+jwhere
+Vi
+j=/parenleftiggn/summationdisplay
+r=1Vi
+rj/parenrightigg
+mod2,
+whenIis uniformly chosen from {1,...,n}independent of all other variables, the mixture VI=Vshas the
+V-size bias distribution and satisfies
+Vs−V=1{VI=0,F=1}+21{VI=0,XJ=0,F=0}andV≤Vs≤V+2. (46)
+15Proof.As in the even case, since Vi=VwhenVi= 1 and P(Vi= 1) = 1 /2, the proof that VIhas the
+V-size bias distribution follows by verifying, for all i= 1,...,n, that
+P(Vi=e|Vi= 0) =P(V=e|Vi= 1) or equivalently P(Vi=e,Vi= 0) =P(V=e,Vi= 1) (47)
+holds for all feasible configurations esatisfying/summationtextn
+r=1eri= 1mod2.
+Recalling that M,FandJare the stage selected, the indicator that the bit in question in the Vconfig-
+uration is flipped, and the index with which an interchange is to be perf ormed, respectively, we find
+P(Vi=e,Vi= 0)
+=P(Vi=e,Vi= 0,F= 1)+P(Vi=e,Vi= 0,F= 0)
+=m+1/summationdisplay
+r=m
+P(Vi=e,Vi= 0,M=r,F= 1)+n/summationdisplay
+j=1P(Vi=e,Vi= 0,J=j,M=r,F= 0)
+
+=m+1/summationdisplay
+r=m
+P(V=er,i,Vi= 0,M=r,F= 1)+n/summationdisplay
+j=1P(V=er,i,j,Vi= 0,J=j,M=r,F= 0)
+
+=m+1/summationdisplay
+r=m
+P(V=er,i,M=r,F= 1)+n/summationdisplay
+j=1P(V=er,i,j,J=j,M=r,F= 0)
+ (48)
+where in the final equality we have used that/summationtextn
+r=1eri= 1mod2.
+Let/summationtextn
+j=1erj=m+b, wherenecessarily b∈ {0,1}. Forthe first term in the rthsummand of(48), by (45),
+(42), the independence of MfromVand that P(V=e) has the same value for any feasible configuration,
+we have
+P(V=er,i,M=r,F= 1)
+=P(F= 1|V=er,i,M=r)P(V=er,i,M=r) =1
+2P(V=e)1(eri=b)
+m+1. (49)
+Next, for the second term in the rthsummand of (48), we have
+n/summationdisplay
+j=1P(V=er,i,j,J=j,M=r,F= 0)
+=n/summationdisplay
+j=1P(J=j|F= 0,V=er,i,j,M=r)P(F= 0|V=er,i,j,M=r)P(V=er,i,j,M=r)
+=1
+2P(V=e)n/summationdisplay
+j=1P(J=j|F= 0,V=er,i,j,M=r)P(F= 0|V=er,i,j,M=r), (50)
+the final equality holding by again using that Mis uniformly chosen over {m,m+ 1}, independent of all
+other variables.
+To handle the summand terms in (50), note that the index Jchoosesuniformly overthe m+b(1−er,i,j
+ri)+
+(1−b)er,i,j
+riindices whose stage rswitch value is opposite to er,i,j
+ri, that is,
+P(J=j|F= 0,V=er,i,j,M=r) =1(er,i,j
+ri/\⌉}atio\slash=er,i,j
+rj)
+m+b(1−er,i,j
+ri)+(1−b)er,i,j
+ri=1(erj/\⌉}atio\slash=eri)
+m+beri+(1−b)(1−eri).(51)
+Again using er,i,j
+ri=erj, by (45) and (42),
+P(F= 0|V=er,i,j,M=r) =1(erj= 1−b)+m
+m+11(erj=b). (52)
+16Taking the product of (51) and (52) yields
+1(erj/\⌉}atio\slash=eri)
+m+beri+(1−b)(1−eri)/parenleftbigg
+1(erj= 1−b)+m
+m+11(erj=b)/parenrightbigg
+=1
+m+11(eri/\⌉}atio\slash=erj). (53)
+Hence (50) may be written as
+P(V=e)
+2(m+1)n/summationdisplay
+j=11(eri/\⌉}atio\slash=erj) =1
+2P(V=e)/parenleftbiggm
+m+11(eri=b)+1(eri= 1−b)/parenrightbigg
+.
+Adding this factor to (49) we find that the rthsummand of (48) equals
+1
+2P(V=e)1(eri=b)
+m+1+1
+2P(V=e)/parenleftbiggm
+m+11(eri=b)+1(eri= 1−b)/parenrightbigg
+=1
+2P(V=e).
+Hence, adding up the two summands r=mandr=m+1 in (48) yields (47), as
+P(Vi=e,Vi= 0) =P(V=e) =P(V=e,Vi= 1),
+where we have applied/summationtextn
+r=1eri= 1mod2 for the final equality.
+Lastly, to prove (46), first note that by construction Vs=VwhenVI= 1. When VI= 0 and F= 0
+thenVsequals either VorV+2, depending on whether XJtakes the value 1 or 0, respectively, as in the
+even case. When VI= 0 and F= 1 the status of bulb Iis changed from off to on and the status of all other
+bulbs remain unchanged, hence Vs−V= 1. /square
+With the coupling of VtoVsnow in hand, we prove a bound to the normal for V. We recall from
+Lemma 3.4 that EV=n/2 and Var( V) =σ2
+V, given in (6).
+Theorem 3.3 Ifn= 2m+1, an odd number, and Vis given by (36), then
+sup
+z∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingleP/parenleftbiggV−n/2
+σV≤z/parenrightbigg
+−P(Z≤z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤n
+2σ2
+V∆1+1.64n
+σ3
+V+2
+σVfor alln≥7,
+whereσ2
+Vand∆1are given in (6) and (8) respectively.
+Proof.We apply Theorem 2.1 to the coupling construction given in Theorem 3.2 . Asδ= 2/σVfrom (46),
+andEV=n/2 from Lemma 3.4, it remains only to show that/radicalbig
+Var(E(Vs−V|V))≤∆1for alln≥7.
+As in the even case, we obtain an upper bound by conditioning on the σ-algebra Fgenerated by V, with
+respect to which Vis measurable. Again for notational simplicity we drop the superscrip ts onFandJ
+unless clarity demands them.
+Taking conditional expectation in (46) yields
+E(Vs−V|F) =ζn+2ξnwhereζn=E/parenleftbig
+1{VI=0,F=1}|F/parenrightbig
+andξn=E/parenleftbig
+1{VI=0,VJ=0,F=0}|F/parenrightbig
+.(54)
+Letting
+A= Var(ζn) and B= Var(ξn),
+since|Cov(ζn,ξn)| ≤√
+ABby the Cauchy-Schwarz inequality yields, we have
+Var(E(Vs−V|F))≤A+4√
+AB+4B. (55)
+First computing a bound on A, expanding the indicator defining ζnin (54) over the possible values of M
+andIyields
+1{VI=0,F=1}=m+1/summationdisplay
+r=mn/summationdisplay
+i=11{M=r,I=i,Vi=0,F=1}.
+17Recall that conditional on V, configurations for which Vr,iare feasible, that is, by (45), those for which/summationtext
+j/n⌉}ationslash=iVrj=m, are flipped with probability 1 /(m+ 1). Now, since IandMare chosen uniformly and
+independentlyoverindices {1,...,n}and{m,m+1}respectively,takingconditionalexpectationwithrespect
+toFwe obtain
+ζn=1
+2n(m+1)m+1/summationdisplay
+r=mn/summationdisplay
+i=11
+
+Vi= 0,/summationdisplay
+j/n⌉}ationslash=iVrj=m
+
+. (56)
+To determine Eζnwe now demonstrate that the events {Vi= 0}and{/summationtext
+j/n⌉}ationslash=iVrj=m}are independent
+forr∈ {m,m+1}. Note that for each fixed i,
+P
+/summationdisplay
+j/n⌉}ationslash=iVrj=m
+=P
+Vi= 0,/summationdisplay
+j/n⌉}ationslash=iVrj=m
++P
+Vi= 1,/summationdisplay
+j/n⌉}ationslash=iVrj=m
+,
+so it suffices to prove that
+P
+Vi= 0,/summationdisplay
+j/n⌉}ationslash=iVrj=m
+=P
+Vi= 1,/summationdisplay
+j/n⌉}ationslash=iVrj=m
+ (57)
+sinceP(Vi= 0) = 1 /2, by Lemma 3.4. However, as the map that sends etoer,iis a bijection between the
+two events in (57), since all feasible configurations have equal pro bability the equality in (57) holds, and
+{Vi= 0}and{/summationtext
+j/n⌉}ationslash=iVrj=m}are therefore independent.
+Now, using that
+P
+/summationdisplay
+j/n⌉}ationslash=iVrj=m
+=P
+/summationdisplay
+j/n⌉}ationslash=iVrj=m,Cr= 0
++P
+/summationdisplay
+j/n⌉}ationslash=iVrj=m,Cr= 1
+
+=1
+2P
+/summationdisplay
+j/n⌉}ationslash=iVrj=m/vextendsingle/vextendsingle/vextendsingle/vextendsingleCr= 0
++1
+2P
+/summationdisplay
+j/n⌉}ationslash=iVrj=m/vextendsingle/vextendsingle/vextendsingle/vextendsingleCr= 1
+
+=1
+2P(Vri= 0|Cr= 0)+1
+2P(Vri= 1|Cr= 1)
+=1
+2/parenleftbiggm+1
+n+m+1
+n/parenrightbigg
+=m+1
+n, (58)
+and that P(Vi= 0) = 1 /2, taking expectation and using independence in (56) we have
+Eζn=1
+2n(m+1)m+1/summationdisplay
+r=mn/summationdisplay
+i=1m+1
+2n=1
+2n. (59)
+Turning to the second moment of ζn, squaring the right hand side of (56) we obtain
+ζ2
+n=1
+4n2(m+1)2/summationdisplay
+r,t∈{m,m+1}n/summationdisplay
+i=1n/summationdisplay
+l=11
+
+Vi= 0,Vl= 0,/summationdisplay
+j/n⌉}ationslash=iVrj=m,/summationdisplay
+j/n⌉}ationslash=lVtj=m
+
+.
+We decompose 4 n2(m+1)2ζ2
+ninto the following four terms:
+1. Forr=t,i=lwe obtain ζ2
+n,1=/summationtextm+1
+r=m/summationtextn
+i=11{Vi= 0,/summationtext
+j/n⌉}ationslash=iVrj=m}
+2. Forr=t,i/\⌉}atio\slash=lwe obtain ζ2
+n,2=/summationtextm+1
+r=m/summationtext
+i/n⌉}ationslash=l1{Vi= 0,Vl= 0,/summationtext
+j/n⌉}ationslash=iVrj=m,/summationtext
+j/n⌉}ationslash=lVrj=m}
+183. Forr/\⌉}atio\slash=t,i=lwe obtain ζ2
+n,3=/summationtext
+r/n⌉}ationslash=t/summationtextn
+i=11{Vi= 0,/summationtext
+j/n⌉}ationslash=iVrj=m,/summationtext
+j/n⌉}ationslash=iVtj=m}
+4. Forr/\⌉}atio\slash=t,i/\⌉}atio\slash=lwe obtain ζ2
+n,4=/summationtext
+r/n⌉}ationslash=t/summationtext
+i/n⌉}ationslash=l1{Vi= 0,Vl= 0,/summationtext
+j/n⌉}ationslash=iVrj=m,/summationtext
+j/n⌉}ationslash=lVtj=m}
+For the expectation of ζ2
+n,1, using independence as in the calculation of Eζnand (58), we obtain
+Eζ2
+n,1=m+1/summationdisplay
+r=mn/summationdisplay
+i=1P(Vi= 0,/summationdisplay
+j/n⌉}ationslash=iVrj=m) =m+1. (60)
+Turning to ζ2
+n,2we have
+{Vi= 0,Vl= 0,/summationdisplay
+j/n⌉}ationslash=iVrj=m,/summationdisplay
+j/n⌉}ationslash=lVrj=m}
+=/uniondisplay
+b∈{0,1}{Vi= 0,Vl= 0,/summationdisplay
+j/n⌉}ationslash∈{i,l}Vrj=m−b, Vri=b,Vrl=b},
+and hence may write
+ζ2
+n,2=/summationdisplay
+b∈{0,1}ζ2
+n,2,bwhereζ2
+n,2,b=m+1/summationdisplay
+r=m/summationdisplay
+i/n⌉}ationslash=l1
+
+Vi= 0,Vl= 0,/summationdisplay
+j/n⌉}ationslash∈{i,l}Vrj=m−b, Vri=b,Vrl=b
+
+.(61)
+Hence, to compute Eζ2
+n,2,bwe sum
+P
+Vi= 0,Vl= 0,/summationdisplay
+j/n⌉}ationslash∈{i,l}Vrj=m−b, Vri=b,Vrl=b
+
+=P
+Vi= 0,Vl= 0,/summationdisplay
+j/n⌉}ationslash∈{i,l}Vmj=m−b, Vmi=b,Vml=b
+
+=P
+Vi= 0,Vl= 0,n/summationdisplay
+j=1Vmj=m+b, Vmi=b,Vml=b
+
+=P(Vi= 0,Vl= 0,Vmi=b,Vml=b,Cm=b)
+=/summationdisplay
+a∈{0,1}P(Vi= 0,Vl= 0,Vmi=b,Vml=b,Cm=b,Cm+1=a)
+=1
+4/summationdisplay
+a∈{0,1}P(V1= 0,V2= 0,Vm1=b,Vm2=b|Cm=b,Cm+1=a),
+where in the second line we used the exchangeability of stages mandm+1 inV, and for the final equality
+thati/\⌉}atio\slash=l, and exchangeability again.
+Recalling definition (38), for a,b,∈ {0,1}easily we have that
+L(V|Cm=b,Cm+1=a) =L(Xn(b,a)),
+and so, applying definition (26), we obtain
+P(V1= 0,V2= 0,Vm1=b,Vm2=b|Cm=b,Cm+1=a) =g(m)
+2−2b,2b,n(b,a).
+Hence, recalling (61),
+Eζ2
+n,2=(n)2
+2/summationdisplay
+a,b∈{0,1}g(m)
+2−2b,2b,n(b,a). (62)
+19Now turning to Eζ2
+n,3, by again considering the bijection that maps etoer,iwe may conclude that
+P
+Vi= 0,/summationdisplay
+j/n⌉}ationslash=iVrj=m,/summationdisplay
+j/n⌉}ationslash=iVtj=m
+=P
+Vi= 1,/summationdisplay
+j/n⌉}ationslash=iVrj=m,/summationdisplay
+j/n⌉}ationslash=iVtj=m
+
+and hence, by the independence of the switch variables in stages r/\⌉}atio\slash=t, as provided by Lemma 3.4, we obtain
+(m+1)2
+n2=P
+/summationdisplay
+j/n⌉}ationslash=iVrj=m,/summationdisplay
+j/n⌉}ationslash=iVtj=m
+= 2P
+Vi= 0,/summationdisplay
+j/n⌉}ationslash=iVrj=m,/summationdisplay
+j/n⌉}ationslash=iVtj=m
+,
+and therefore
+Eζ2
+n,3=(m+1)2
+n. (63)
+Arguing similarly to compute ζ2
+n,4, the identity
+(m+1)2
+n2=P
+/summationdisplay
+j/n⌉}ationslash=iVrj=m,/summationdisplay
+j/n⌉}ationslash=lVtj=m
+= 4P
+Vi= 0,Vl= 0,/summationdisplay
+j/n⌉}ationslash=iVrj=m,/summationdisplay
+j/n⌉}ationslash=lVtj=m
+
+yields
+Eζ2
+n,4=(n)2(m+1)2
+2n2. (64)
+Combining (60), (62), (63), (64) and (59) yields
+A=E(ζ2
+n)−E(ζn)2
+=1
+4n2(m+1)2
+(m+1)+(n)2
+2/summationdisplay
+a,b∈{0,1}g(m)
+2−2b,2b,n(b,a)+(m+1)2
+n+(n)2(m+1)2
+2n2
+−1
+4n2.
+To obtain a bound on Afirst note that for any bwe have n(b,0)m=nm+1andn(b,1)m=nm, and that
+themthcomponent of n(b,a) ism+b. Hence when b= 0 Corollary 4.1 yields
+g(m)
+2,0,n(0,a)=/braceleftigg1
+4(1+2λn,1,nm+1+λn,2,nm+1)(m+1)2
+(n)2a= 0
+1
+4(1+2λn,1,nm+λn,2,nm)(m+1)2
+(n)2a= 1,
+and when b= 1 that
+g(m)
+0,2,n(1,a)=/braceleftigg1
+4(1−2λn,1,nm+1+λn,2,nm+1)(m+1)2
+(n)2a= 0
+1
+4(1−2λn,1,nm+λn,2,nm)(m+1)2
+(n)2a= 1.
+Summing these four terms, and using that ( λn,2,nm+λn,2,nm+1)/2 =λn,2,nm, as defined in (5), yields
+/summationdisplay
+a,b∈{0,1}g(m)
+2−2b,2b,n(b,a)=1
+4/parenleftbig
+4+2λn,2,nm+2λn,2,nm+1/parenrightbig(m+1)2
+(n)2= (1+λn,2,nm)(m+1)2
+(n)2.
+Hence,
+A
+=1
+4n2(m+1)2/parenleftbigg
+(m+1)+(n)2
+2/parenleftbigg(m+1)2
+(n)2+λn,2,nm(m+1)2
+(n)2/parenrightbigg
++(m+1)2
+n+(n)2
+2(m+1)2
+n2/parenrightbigg
+−1
+4n2
+=1
+4n2/parenleftbigg1
+m+1+m
+2(m+1)+1
+n+n−1
+2n−1/parenrightbigg
++m
+8n2(m+1)λn,2,nm
+=3m+2
+8n3(m+1)+m
+8n2(m+1)λn,2,nm.
+20Applying Lemma 4.5 we obtain
+A≤3m+2
+8n3(m+1)+e−n
+8n2≤3
+8n3+e−n
+8n2. (65)
+Next, expanding the indicator in (54) to obtain B= Var(ξn), and recalling that an interchange is
+performed when F= 0, we have
+1{VI=0,VJ=0,F=0}
+=m+1/summationdisplay
+r=mn/summationdisplay
+i,j=11{Vi=0,Vj=0}1{I=i,J=j,M=r,F=0}
+=m+1/summationdisplay
+r=mn/summationdisplay
+i,j=1/parenleftbig
+1{Vi=0,Vj=0,Vri=0}+1{Vi=0,Vj=0,Vri=1}/parenrightbig
+1{I=i,J=j,M=r,F=0}
+=m+1/summationdisplay
+r=m/summationdisplay
+i/n⌉}ationslash=j/parenleftbig
+1{Vi=0,Vj=0,Vri=0,Vrj=1}+1{Vi=0,Vj=0,Vri=1,Vrj=0}/parenrightbig
+1{I=i,J=j,M=r,F=0}
+= 2m+1/summationdisplay
+r=m/summationdisplay
+i/n⌉}ationslash=j1{Vi=0,Vj=0,Vri=0,Vrj=1}1{I=i,J=j,M=r,F=0},
+where the last inequality holds due to the symmetry between iandjin the penultimate sum, as in the even
+case.
+AsVisF-measurable, recalling the definition ξn=E(1{VI=0,VJ=0,F=0}|F) in (54), taking conditional
+expectation yields
+ξn= 2m+1/summationdisplay
+r=m/summationdisplay
+i/n⌉}ationslash=j1{Vi=0,Vj=0,Vri=0,Vrj=1}E(1{I=i,J=j,M=r,F=0}|F). (66)
+To compute the conditional expectation in the sum, using the indepe ndence of IfromMandV, andM
+fromV, we obtain
+E(1{I=i,J=j,M=r,F=0}|F)
+=P(J=j|M=r,I=i,F= 0,V)P(F= 0|I=i,M=r,V)P(I=i|M=r,V)P(M=r|V)
+=1
+2nP(J=j|M=r,I=i,F= 0,V)P(F= 0|I=i,M=r,V)
+=1
+2nP(Ji=j|M=r,Fi= 0,V)P(Fi= 0|M=r,V)
+=1
+2n1{Vri/\⌉}atio\slash=Vrj}
+m+1,
+the final equality since the product of (52) and (51) is given by (53) . Hence, as the first indicator in the sum
+in (66) specifies that Vri= 0,Vrj= 1, we obtain
+ξn=1
+n(m+1)m+1/summationdisplay
+r=m/summationdisplay
+i/n⌉}ationslash=j1{Vi=0,Vj=0,Vri=0,Vrj=1}. (67)
+Taking the expectation of ξn, by the exchangeability in Vof stages mandm+1, we have
+E(ξn) =2
+n(m+1)/summationdisplay
+i/n⌉}ationslash=jP(Vi= 0,Vj= 0,Vmi= 0,Vmj= 1). (68)
+21Exhausting over the possible values of Cm=bandCm+1=afora,b∈ {0,1}, we obtain
+P(Vi= 0,Vj= 0,Vmi= 0,Vmj= 1)
+=1
+4/summationdisplay
+a,b∈{0,1}P(Vi= 0,Vj= 0,Vmi= 0,Vmj= 1|Cm=b,Cm+1=a)
+=1
+4/summationdisplay
+a,b∈{0,1}g(m)
+1,1,n(b,a).
+Now, by (68), using Corollary 4.1,
+Eξn=n−1
+2(m+1)/summationdisplay
+a,b∈{0,1}g(m)
+1,1,n(b,a)=n−1
+2(m+1)/parenleftbig
+1−λn,2,nm/parenrightbigm(m+1)
+(n)2=m
+2n/parenleftbig
+1−λn,2,nm/parenrightbig
+.(69)
+To calculate Eξ2
+n, squaring ξnin (67) we obtain a sum over indices r,t∈ {m,m+1}andi1,i2,j1,j2with
+i1/\⌉}atio\slash=j1,i2/\⌉}atio\slash=j2, and therefore |{i1,i2,j1,j2}|=pforp∈ {2,3,4}. Hence we may write
+ξ2
+n=/summationdisplay
+p∈{1,2,3,4}ξ2
+n,pwhereξ2
+n,p=/summationdisplay
+q∈{1,2}ξ2
+n,p,qand
+ξ2
+n,p,q=1
+n2(m+1)2/summationdisplay
+|{r,t}|=q/summationdisplay
+|{ii,i2,j1,j2}|=p
+i1/n⌉}ationslash=j1,i2/n⌉}ationslash=j21{Vi1=0,Vj1=0,Vr,i1=0,Vr,j1=1}1{Vi2=0,Vj2=0,Vt,i2=0,Vt,j2=1}.
+First consider the case p= 4, that is, {i1,i2,j1,j2}|= 4. We begin with ξ2
+n,4,1, that is, when r=t. Using
+again that {Cr=b,Cn−r=a}partitions the space for a,b∈ {0,1}, we have
+P(Vi1= 0,Vi2= 0,Vj1= 0,Vj2= 0,Vr,i1= 0,Vr,i2= 0,Vr,j1= 1,Vr,j2= 1,Cr=b,Cn−r=a)
+=1
+4P(Vi1= 0,Vi2= 0,Vj1= 0,Vj2= 0,Vr,i1= 0,Vr,i2= 0,Vr,j1= 1,Vr,j2= 1|Cr=b,Cn−r=a)
+=1
+4P(V1= 0,V2= 0,V3= 0,V4= 0,Vr,1= 0,Vr,2= 0,Vr,3= 1,Vr,4= 1|Cr=b,Cn−r=a)
+=1
+4g(r)
+2,2,n(b,a).
+Summing over the four distinct indices i1,i2,j1,j2and applying Corollary 4.1 we obtain
+E(ξ2
+n,4,1) =(n)4
+4n2(m+1)2/summationdisplay
+a,b∈{0,1}m+1/summationdisplay
+r=mg(r)
+2,2,n(b,a)
+=2(n)4
+16n2(m+1)2/parenleftbig
+1−2λn,2,nm+λn,4,nm/parenrightbig(m)2(m+1)2
+(n)4
+=m2(m−1)
+8n2(m+1)/parenleftbig
+1−2λn,2,nm+λn,4,nm/parenrightbig
+. (70)
+In order to proceed further we need to consider functions such a s those in (26), but which specify
+switch variables in two stages, rather than in only one. In particular , fors= (s1,...,s k), distinct stages
+r,t∈ {1,...,k}and bulbs i1<···< iαandj1<···< jβin{1,...,n}, let
+g(r,t)
+[(i1,a1),...,(iα,aα);(j1,b1),...,(jβ,bβ)],s(71)
+=P(Xi1= 0,...,X iα= 0,Xj1= 0,...,X jβ= 0,Xr,i1=a1,...,X r,iα=aα,Xt,j1=b1,...,X t,jβ=bβ)
+whenXhas switch pattern s. We note that it is not required that {i1,...,iα} ∩ {j1,...,j β} /\⌉}atio\slash=∅. The
+functions in (71) give the probability that bulbs {i1,...,iα}∪{j1,...,j β}are off, and that the switch values
+for bulbs i1,...,iαandj1,...,j βassume the prescribed values in stages randt, respectively.
+22Now moving to ξ2
+n,4,2, considering first r=m,t=m+1 and arguing similarly as for the ξn,4,1,
+P(Vi1= 0,Vi2= 0,Vj1= 0,Vj2= 0,Vm,i1= 0,Vm+1,i2= 0,Vm,j1= 1,Vm+1,j2= 1,Cm=b,Cm+1=a)
+=1
+4P(V1= 0,V2= 0,V3= 0,V4= 0,Vm,1= 0,Vm,2= 1,Vm+1,3= 0,Vm+1,4= 1|Cm=b,Cm+1=a)
+=1
+4g(m,m+1)
+[(1,0),(2,1);(3,0),(4,1)],n(b,a),
+applying the definition (71). Summing over the four distinct indices, n oting that the case r=m+1,t=m
+contributes equally, using (103) from the Appendix we obtain
+E(ξ2
+n,4,2) =(n)4
+2n2(m+1)2/summationdisplay
+a,b∈{0,1}g(m,m+1)
+[(1,0),(2,1);(3,0),(4,1)],n(b,a)
+=(n)4
+2n2(m+1)2/parenleftbigg(m+1)3
+(n)4/parenrightbigg2/parenleftbig
+(2m−1)2+2(2m−1)λn,2,nm,m+1+λn,4,nm,m+1/parenrightbig
+.(72)
+Summing E(ξ2
+n,4,1) andE(ξ2
+n,4,2) from (70) and (72) respectively, and simplifying the latter using n= 2m+1
+yields
+E(ξ2
+n,4) =m2(m−1)
+8n2(m+1)/parenleftbig
+1−2λn,2,nm+λn,4,nm/parenrightbig
+(73)
++(m+1)2
+3
+2n2(m+1)2(n)4/parenleftbig
+(2m−1)2+2(2m−1)λn,2,nm,m+1+λn,4,nm,m+1/parenrightbig
+=m2(m−1)
+8n2(m+1)/parenleftbig
+1−2λn,2,nm+λn,4,nm/parenrightbig
++(m)2(2m−1)
+8n3+(m)2
+4n3λn,2,nm,m+1+(m)2
+2
+2n2(n)4λn,4,nm,m+1.
+Moving to p= 3 and the evaluation of E(ξ2
+n,3,q), under the constraints i1/\⌉}atio\slash=j1,i2/\⌉}atio\slash=j2, sums over indices
+that satisfy |{i1,i2,j1,j2}|= 3 can be partitioned into the following four cases:
+1.i1=i2,j1/\⌉}atio\slash=j2,2.i1/\⌉}atio\slash=i2,j1=j2,3.i1=j2,i2/\⌉}atio\slash=j1,4.i2=j1,i1/\⌉}atio\slash=j2,
+Subscripting by these subcases, here and similarly in what follows, we write
+ξ2
+n,3,q=ξ2
+n,3,q,1+ξ2
+n,3,q,2+ξ2
+n,3,q,3+ξ2
+n,3,q,4.
+Starting with q= 1, forEξ2
+n,3,1,1, we have
+P(Vi= 0,Vj1= 0,Vj2= 0,Vr,i= 0,Vr,j1= 1,Vr,j2= 1,Cr=b,Cn−r=a)
+=1
+4P(V1= 0,V2= 0,V3= 0,Vr,1= 0,Vr,2= 1,Vr,3= 1|Cr=b,Cn−r=a)
+=1
+4g(r)
+1,2,n(b,a),
+while for Eξ2
+n,3,1,2,
+P(Vi1= 0,Vi2= 0,Vj= 0,Vr,i1= 0,Vr,i2= 0,Vr,j= 1,Cr=b,Cn−r=a)
+=1
+4P(V1= 0,V2= 0,V3= 0,Vr,1= 0,Vr,2= 0,Vr,3= 1|Cr=b,Cn−r=a)
+=1
+4g(r)
+2,1,n(b,a).
+23Underq= 1, cases 3 and 4 have probability zero, so summing and applying Coro llary 4.1 yields
+E(ξ2
+n,3,1) =E/parenleftbig
+ξ2
+n,3,1,1+ξ2
+n,3,1,2/parenrightbig
+=(n)3
+4n2(m+1)2m+1/summationdisplay
+r=m/summationdisplay
+a,b∈{0,1}/parenleftig
+g(r)
+1,2,n(b,a)+g(r)
+2,1,n(b,a)/parenrightig
+=(n)3
+2n2(m+1)2/summationdisplay
+a,b∈{0,1}/parenleftig
+g(m)
+1,2,n(b,a)+g(m)
+2,1,n(b,a)/parenrightig
+=m(2m−1)
+4n2(m+1)/parenleftbig
+1−λn,2,nm/parenrightbig
+. (74)
+Moving to q= 2, forEξ2
+n,3,2,1we compute
+P(Vi= 0,Vj1= 0,Vj2= 0,Vr,i= 0,Vt,i= 0,Vr,j1= 1,Vt,j2= 1,Cr=b,Ct=a)
+=1
+4P(V1= 0,V2= 0,V3= 0,Vr,1= 0,Vr,2= 1,Vt,1= 0,Vt,3= 1|Cr=b,Ct=a)
+=1
+4g(r,t)
+[(1,0),(2,1);(1,0),(3,1)],n(b,a),
+while similarly, with factors of 1 /4 forEξ2
+n,3,2,2we have
+P(V1= 0,V2= 0,V3= 0,Vr,1= 1,Vr,2= 0,Vt,1= 1,Vt,3= 0|Cr=b,Ct=a) =g(r,t)
+[(1,1),(2,0);(1,1),(3,0)],n(b,a),
+forEξ2
+n,3,2,3we have
+P(V1= 0,V2= 0,V3= 0,Vr,1= 0,Vr,2= 1,Vt,1= 1,Vt,3= 0|Cr=b,Ct=a) =g(r,t)
+[(1,0),(2,1);(1,1),(3,0)],n(b,a),
+and forEξ2
+n,3,2,4we consider
+P(V1= 0,V2= 0,V3= 0,Vr,1= 1,Vr,2= 0,Vt,1= 0,Vt,3= 1|Cr=b,Ct=a) =1
+4g(r,t)
+[(1,1),(2,0);(1,0),(3,1)],n(b,a).
+Summing up these q= 2 case terms and applying (108) in the Appendix, we obtain
+E(ξ2
+n,3,2) =E(ξ2
+n,3,2,1+ξ2
+n,3,2,2+ξ2
+n,3,2,3+ξ2
+n,3,2,4)
+=(n)3
+4n2(m+1)2/summationdisplay
+a,b∈{0,1}/summationdisplay
+r/n⌉}ationslash=t/parenleftig
+g(r,t)
+[(1,0),(2,1);(1,0),(3,1)],n(b,a)+g(r,t)
+[(1,1),(2,0);(1,1),(3,0)],n(b,a)
++g(r,t)
+[(1,0),(2,1);(1,1),(3,0)],n(b,a)+g(r,t)
+[(1,1),(2,0);(1,0),(3,1)],n(b,a)/parenrightig
+=(n)3
+4n2(m+1)2/parenleftbigg4(m+1)2
+2(2m−1)2
+(n)2
+3/parenrightbigg
+=m(2m−1)
+2n3.(75)
+Now summing (75) and (74) we obtain
+Eξ2
+n,3=m(2m−1)
+4n2(m+1)/parenleftbig
+1−λn,2,nm/parenrightbig
++m(2m−1)
+2n3. (76)
+It remains to consider Eξ2
+n,2. Under the constraints i1/\⌉}atio\slash=j1,i2/\⌉}atio\slash=j2, sums over indices that satisfy
+|{i1,i2,j1,j2}|= 2 can be partitioned into the following two cases:
+1. i1=i2,j1=j2and 2. i1=j2,j1=i2.
+24Consider first q= 1. Under Case 1, by consideration of the term P(Vi= 0,Vj= 0,Vr,i= 0,Vr,j= 1,Cr=
+b,Ct=a) and the application of Corollary 4.1, we obtain
+Eξ2
+n,2,1,1=(n)2
+4n2(m+1)2m+1/summationdisplay
+r=m/summationdisplay
+a,b∈{0,1}g(r)
+1,1,n(b,a)=m
+2n2(m+1)/parenleftbig
+1−λn,2,nm/parenrightbig
+. (77)
+Case 2 has zero probability.
+Now letq= 2. For stages r/\⌉}atio\slash=t, under Case 1 we consider
+P(Vi= 0,Vj= 0,Vr,i= 0,Vr,j= 1,Vt,i= 0,Vt,j= 1,Cr=b,Ct=a) =1
+4g(r,t)
+[(1,0),(2,1);(1,0),(2,1)],n(b,a).
+Now, summing over the two ways one can achieve r/\⌉}atio\slash=t, which yield identical terms, and using (113) from
+the Appendix, we obtain
+Eξ2
+n,2,2,1=(n)2
+2n2(m+1)2/summationdisplay
+a,b∈{0,1}g(r,t)
+[(1,0),(2,1);(1,0),(2,1)],n(b,a)
+=(n)2
+2n2(m+1)2/parenleftbig
+1+2λn,1,nm,m+1+λn,2,nm,m+1/parenrightbig(m+1)2
+2
+(n)2
+2
+=m
+4n3(1+2λn,1,nm,m+1+λn,2,nm,m+1). (78)
+Under Case 2 we consider
+P(Vi= 0,Vj= 0,Vr,i= 0,Vr,j= 1,Vt,i= 1,Vt,j= 0,Cr=b,Ct=a) =1
+4g(r,t)
+[(1,0),(2,1);(1,1),(2,0)],n(b,a)
+and similarly arrive at
+Eξ2
+n,2,2,2=m
+4n3(1−2λn,1,nm,m+1+λn,2,nm,m+1). (79)
+Summing up the terms (77), (78) and (79) yields
+Eξ2
+n,2=m
+2n2(m+1)(1−λn,2,nm)+m
+2n3(1+λn,2,nm,m+1). (80)
+Finally, recalling
+B= Var(ξn) =E/parenleftbig
+ξ2
+n,4+ξ2
+n,3+ξ2
+n,2/parenrightbig
+−(Eξn)2,
+collecting terms (69), (73), (76), and (80) and simplifying we obtain
+B=m(8m+3)
+8n3+m(m2+m−1)
+4n2(m+1)λn,2,nm+(m+1)2
+4n3λn,2,nm,m+1
++m2(m−1)
+8n2(m+1)λn,4,nm+(m)2
+2
+2n2(n)4λn,4,nm,m+1−m2
+4n2λ2
+n,2,nm.
+Applying Lemma 4.5, we find for n≥7
+B≤m(8m+3)
+8n3+/parenleftbigg1
+16+1
+16n+1
+64+1
+64n2/parenrightbigg
+e−n≤m(8m+3)
+8n3+e−n
+11.
+25Now, using (65),
+√
+A·B
+≤/radicalbigg
+3
+8n3m(8m+3)
+8n3+/radicalbigg
+m(8m+3)
+8n3e−n
+8n2+/radicalbigg
+3
+8n3e−n
+11+/radicalbigg
+e−n
+8n2e−n
+11
+≤5m+1
+8n3+/radicalbigg
+1
+4ne−n
+8n2+/radicalbigg
+3
+8n3e−n
+11+/radicalbigg
+e−2n
+88n2
+=5m+1
+8n3+/radicalbigg
+1
+32n3e−n/2+/radicalbigg
+3
+88n3e−n/2+e−n
+8n
+≤5m+1
+8n3+e−n/2
+2n3/2+e−n
+8n.
+Hence, from (55), again using n≥7,
+Var(E(Vs−V|F))≤A+4B+4√
+AB
+≤/parenleftbigg3
+8n3+e−n
+8n2/parenrightbigg
++4/parenleftbiggm(8m+3)
+8n3+e−n
+11/parenrightbigg
++4/parenleftbigg5m+1
+8n3+e−n/2
+2n3/2+e−n
+8n/parenrightbigg
+≤1
+n+e−n/2
+8.
+Taking square root and using√
+a+b≤√a+√
+bnow yields the upper bound (8), thus completing the proof.
+/square
+We now provide a bound for the normal approximation of Xin the odd case. We remark that using V,
+fewer error terms, and therefore a smaller bound, results when s tandardizing Xas in Theorem 1.1, that is,
+not by its own mean and variance but by the (exponentially close) mea n and variance of the closely coupled
+V.
+Proof of Theorem 1.1: odd n.LettingW= (X−n/2)/σVandWV= (V−n/2)/σV, recalling |X−V| ≤2
+from (37), we have
+|W−WV|=|X−V|/σV≤2/σV. (81)
+With Φ(z) =P(Z≤z) and
+CV=n
+2σ2
+V∆1+1.64n
+σ3
+V+2
+σV,
+by (81) and Theorem 3.3 we obtain
+P(W≤z)−Φ(z)≤P(WV−2/σV≤z)−Φ(z)
+=P(WV≤z+2/σV)−Φ(z+2/σV)+Φ(z+2/σV)−Φ(z)
+≤CV+2/(σV√
+2π).
+As a corresponding lower bound can be similarly demonstrated, the c laim is shown. /square
+4 Spectral Decomposition
+In [11] the lightbulb chain was analyzed as a composition chain of multino mial type. Such chains in general
+are based on a d×dMarkov transition matrix Pthat describes the transition of a single particle in a
+system of nidentical particles, a subset of which is selected uniformly to underg o transition at each time
+step according to P.
+26In the case of the lightbulb chain there are d= 2 states and the transition matrix Pof a single bulb is
+given by
+P=/bracketleftbigg
+0 1
+1 0/bracketrightbigg
+,
+where we let e0= (1,0)′ande1= (0,1)′denote the 0 and 1 states of the bulb, for off and on, respectively.
+Withb∈ {0,1,...,n}letPn,b,sbe the 2b×2btransition matrix of a subset of size bof thentotal lightbulbs
+whensof thenbulbs are selected uniformly to be switched. Letting Pn,0,s= 1 for all nands, andI2the
+2×2 identity matrix, for n≥1 the matrix Pn,b,sis given recursively by
+Pn,b,s=s
+n(P⊗Pn−1,b−1,s−1)+(1−s
+n)(I2⊗Pn−1,b−1,s) forb∈ {1,...,n},
+as any particular bulb among the bin the subset considered is selected with probability s/nto undergo
+transition according to P, leaving the s−1 remaining switches to be distributed over the remaining b−1
+ofn−1 bulbs, and with probability 1 −s/nthe bulb is left unchanged, leaving all the sswitches to be
+distributed.
+The transition matrix Pis easily diagonalizable by the orthogonal matrix Tas
+P=T′ΓTwhere T=1√
+2/bracketleftbigg1 1
+−1 1/bracketrightbigg
+and Γ =/bracketleftbigg1 0
+0−1/bracketrightbigg
+, (82)
+hencePn,b,sis diagonalized by
+Pn,b,s=⊗bT′Γn,b,s⊗bT (83)
+where Γ n,0,s= 1, and is given the recursion
+Γn,b,s=s
+n(Γ⊗Γn−1,b−1,s−1)+(1−s
+n)(I2⊗Γn−1,b−1,s) forb∈ {1,...,n}. (84)
+The next result describes the diagonal matrices Γ n,b,smore explicitly in terms of a sequence of vectors abof
+length 2bfor allb≥1 defined through the recursion
+ab= (ab−1,ab−1+1b−1) forb≥2, witha1= (0,1), (85)
+where1b= (1,...,1) is of size 2b. For example,
+a1= (0,1),a2= (0,1,1,2) and a3= (0,1,1,2,1,2,2,3). (86)
+Lettinganbe thenthterm of the vector abfor anybsatisfying 2b≥nresults in a well defined sequence
+a1,a2,....
+Lemma 4.1 Forn∈ {0,1,...,},b,s∈ {0,...,n}, andλn,b,sgiven by (1), the matrix Γn,b,sin (83) satisfies
+Γn,b,s=diag(λn,a1,s,...,λ n,a2b,s).
+In particular, with 02b−1the vector of zeros of length 2b−1, forb≥1
+Γn,b,s=diag(λn,a1,s,...,λ n,a2b−1,s,02b−1)+diag(02b−1,λn,a1+1,s,...,λ n,a2b−1+1,s).
+For instance, from (86), for b= 2 we have
+Γn,2,s= diag(λn,0,s,λn,1,s,λn,1,s,λn,2,s),
+and forb= 3,
+Γn,3,s= diag(λn,0,s,λn,1,s,λn,1,s,λn,2,s,λn,1,s,λn,2,s,λn,2,s,λn,3,s).
+27Proof.Asa1= 0 we have Γ n,0,s= 1 =λn,0,s, so the lemma is true for b= 0. For the inductive step, assuming
+the lemma is true for b−1, by (84) and the definition of Γ from (82), it suffices to verify
+s
+nλn−1,a,s−1+(1−s
+n)λn−1,a,s=λn,a,sand
+−s
+nλn−1,a,s−1+(1−s
+n)λn−1,a,s=λn,a+1,s,
+for alla= 0,1,.... To prove the first equality, by (1) we have
+s
+nλn−1,a,s−1+(1−s
+n)λn−1,a,s=s
+na/summationdisplay
+t=0/parenleftbigga
+t/parenrightbigg
+(−2)t(s−1)t
+(n−1)t+(1−s
+n)a/summationdisplay
+t=0/parenleftbigga
+t/parenrightbigg
+(−2)t(s)t
+(n−1)t
+=a/summationdisplay
+t=0/parenleftbigga
+t/parenrightbigg
+(−2)t/parenleftbiggs
+n(s−1)t
+(n−1)t+/parenleftig
+1−s
+n/parenrightig(s)t
+(n−1)t/parenrightbigg
+=a/summationdisplay
+t=0/parenleftbigga
+t/parenrightbigg
+(−2)t/parenleftbigg(s)t+1+(n−s)(s)t
+(n)t+1/parenrightbigg
+=a/summationdisplay
+t=0/parenleftbigga
+t/parenrightbigg
+(−2)t/parenleftbigg(s)t(s−t+n−s)
+(n)t+1/parenrightbigg
+=a/summationdisplay
+t=0/parenleftbigga
+t/parenrightbigg
+(−2)t/parenleftbigg(s)t(n−t)
+(n)t+1/parenrightbigg
+=a/summationdisplay
+t=0/parenleftbigga
+t/parenrightbigg
+(−2)t(s)t
+(n)t
+=λn,a,s.
+The second equality can be shown in similar, though slightly more involve d, fashion. /square
+We note that [11] expresses these eigenvalues in terms of the hype rgeometric function.
+If thekstages of the process 1 ,...,kuse switches s= (s1,...,s k), then since the matrices Pn,b,s,s∈
+{0,1,...,n}are simultaneously diagonalizable by (83), the transition matrix Pn,b,sfor any subset of bbulbs
+can be diagonalized as
+Pn,b,s=k/productdisplay
+j=1Pn,b,sj=⊗bT′Γn,b,s⊗bT=⊗bT′diag(λn,a1,s,...,λ n,a2b,s)⊗bT, (87)
+whereλn,a,sis given in (1) and
+Γn,b,s=k/productdisplay
+j=1Γn,b,sj. (88)
+Ifπis a permutation of {1,...,k}letπ(s) = (sπ(1),...,s π(k)). As all matrices Γ n,b,sare diagonal, from (88)
+we have Γ n,b,s= Γn,b,π(s), and now from (87) we have the following result.
+Lemma 4.2 The distribution of the lightbulb chain is independent of th e order in which the switch variables
+sare applied, that is, for all permutations π,
+Pn,b,s=Pn,b,π(s).
+The following lemma helps us compute probabilities such as g(l)
+α,β,sin (26). For j∈ {0,1,...}let Ωb,jbe
+the 2b×2bdiagonal matrix in the variables xk,k∈ {0,1,...}given by
+Ωb,j= diag(xa1+j,...,x a2b+j),and set ub=⊗bTe⊗b
+0andwb=⊗bTe⊗b
+1, (89)
+28where we recall e0= (1,0)′ande1= (0,1)′. Note that for b= 1 we have
+u1=Te0=1√
+2(1,−1)′andw1=Te1=1√
+2(1,1)′. (90)
+Lemma 4.3 Lett∈ {0,1,...}andΩt= Ωt,0, and suppose that for some vector vt∈R2t, that
+v′
+tΩtut=1
+2tt/summationdisplay
+j=0/parenleftbiggt
+j/parenrightbigg
+a(j)xj (91)
+holds for t=b−1with some sequence a(j),j= 0,...,b−1. Then (91) holds for t=bwhen replacing vtby
+vb=u1⊗vtanda(j)by
+au(j) =/parenleftbiggb−j
+b/parenrightbigg
+a(j)+/parenleftbiggj
+b/parenrightbigg
+a(j−1), (92)
+and fort=bwhen replacing vtbyvb=w1⊗vtanda(j)by
+aw(j) =/parenleftbiggb−j
+b/parenrightbigg
+a(j)−/parenleftbiggj
+b/parenrightbigg
+a(j−1). (93)
+Proof.By (85) we may write
+Ωb=/bracketleftbigg
+Ωb−1,00
+0 Ω b−1,1/bracketrightbigg
+and by (90) we have
+ub=u1⊗ub−1=1√
+2(u′
+b−1,−u′
+b−1)′.
+Hence, when vb=u1⊗vb−1= (v′
+b−1,−v′
+b−1)′/√
+2, we obtain
+v′
+bΩbub=1
+2/parenleftbig
+v′
+b−1Ωb−1,0ub−1+v′
+b−1Ωb−1,1ub−1/parenrightbig
+=1
+2bb−1/summationdisplay
+j=0/parenleftbiggb−1
+j/parenrightbigg
+a(j)xj+1
+2bb−1/summationdisplay
+j=0/parenleftbiggb−1
+j/parenrightbigg
+a(j)xj+1
+=1
+2bb−1/summationdisplay
+j=0/parenleftbiggb−1
+j/parenrightbigg
+a(j)xj+1
+2bb/summationdisplay
+j=1/parenleftbiggb−1
+j−1/parenrightbigg
+a(j−1)xj
+=1
+2bb/summationdisplay
+j=0/parenleftbigg/parenleftbiggb−1
+j/parenrightbigg
+a(j)+/parenleftbiggb−1
+j−1/parenrightbigg
+a(j−1)/parenrightbigg
+xj
+=1
+2bb/summationdisplay
+j=0/parenleftbiggb
+j/parenrightbigg/parenleftbigg/parenleftbiggb−j
+b/parenrightbigg
+a(j)+/parenleftbiggj
+b/parenrightbigg
+a(j−1)/parenrightbigg
+xj
+=1
+2bb/summationdisplay
+j=0/parenleftbiggb
+j/parenrightbigg
+au(j)xj,
+as claimed. The proof is essentially the same, using (90), when vb=w1⊗vb−1= (v′
+b−1,v′
+b−1)′/√
+2. /square
+In Corollary 4.1 we express the functions g(l)
+α,β,sin (26) using the corresponding conditional probabilities
+fα,β,s=P(Xi= 0,i= 1,...,α+β|X0,i= 0,i= 1,...,α,X 0,i= 1,i=α+1,...,α+β) (94)
+that are the subject of the next lemma. As with g(l)
+α,β,s, we drop the dependence on swhens=n.
+29Lemma 4.4 For given α,β≥0, setting b=α+βthe probability fα,β,sin (94) is given by
+fα,β,s=1
+2bb/summationdisplay
+j=0/parenleftbiggb
+j/parenrightbigg
+aα,β(j)λn,j,s, (95)
+whereaα,0(j) = 1for allα≥0and
+aα,β(j) =/parenleftbiggb−j
+b/parenrightbigg
+aα,β−1(j)−/parenleftbiggj
+b/parenrightbigg
+aα,β−1(j−1)for allα≥0,β≥1. (96)
+Proof.Using exchangeability for the first equality, extracting the relevan t component of the k-step transition
+matrix and applying (87) we obtain
+fα,β,s=/parenleftbig
+(e′
+1)⊗β⊗(e′
+0)⊗α/parenrightbig
+Pn,b,se⊗b
+0=/parenleftbig
+(e′
+1)⊗β⊗(e′
+0)⊗α/parenrightbig
+⊗bT′Γn,b,s⊗bTe⊗b
+0=v′
+bΓn,b,sub,
+whereuαandwβare given as in (89) and vb=wβ⊗uα. Hence, with Ω b= Ωb,0as in (89), the result follows
+from
+v′
+bΩbub=1
+2bb/summationdisplay
+j=0/parenleftbiggb
+j/parenrightbigg
+aα,β(j)xj. (97)
+We first prove the case β= 0 by induction in α; note in this case vb=ub. Equality (97) holds with
+aα,0(j) = 1 for α= 0, as both sides equal x0in this case. Assuming that (97) holds for some α≥0 with
+aα,0(j) = 1, then (92) of Lemma 4.3 implies that (97) holds for α+1 and β= 0 with
+aα+1,0(j) =/parenleftbiggb−j
+b/parenrightbigg
+aα,0(j)+/parenleftbiggj
+b/parenrightbigg
+aα,0(j−1) = 1.
+Hence (97) holds for all α≥0 andβ= 0 with aα,0(j) = 1. Similarly, assuming now that (97) holds for
+aα,β−1(j) with nonnegative α,β−1, we have that (97) holds with aα,β(j) given by (96) by (93) of Lemma
+4.3, thus completing the induction. /square
+As our computations involve only moments up to fourth order, we hig hlight these particular special cases
+of Lemma 4.4 in the following Corollary.
+Corollary 4.1 Forα,β≥0, the probability g(l)
+α,β,sin (26) is given by
+g(l)
+α,β,s=fα,β,slpα,β(s,l)wherepα,β(s,l) =(n−sl)α(sl)β
+(n)α+β, (98)
+andfα,β,sis given by (95). For 0≤α+β≤4, the sequences in (95) specialize to
+a0,0(j) = 1
+a0,1(j) = (−1)janda1,0(j) = 1
+a0,2(j) = (−1)j, a1,1(j) = 1−janda2,0(j) = 1
+a0,3(j) = (−1)j, a1,2(j) = (−1)j(1−2j/3), a2,1= 1−2j/3anda3,0(j) = 1
+and
+a0,4(j) = (−1)j, a1,3(j) = (−1)j(1−j/2), a2,2= 1−j(4−j)/3, a3,1= 1−j/2anda4,0(j) = 1.
+30Proof.By Lemma 4.2, that is, the fact that the switch variables can be applie d in any order, conditioning
+on the values of the switch variables in stage lyields the same result as assuming these values as initial
+conditions in stage 0, and applying the switch pattern sl, that is, sskipping stage l. Hence the first claim
+in (98) follows, as the first factor is the probability of the given even t conditioned on the values in stage l,
+while the second factor is the probability of the conditioning event, a s
+P(Xl1=···=Xlα= 0,Xl,α+1=···=Xl,α+β= 1) =α−1/productdisplay
+i=0/parenleftbiggn−sl−i
+n−i/parenrightbiggβ−1/productdisplay
+i=0/parenleftbiggsl−i
+n−α−i/parenrightbigg
+=pα,β(s,l).
+The specific forms of the sequences aα,β(j) for 0≤α+β≤4 follow directly from the initial condition
+and recursion in Lemma 4.4. /square
+Applying Corollary 4.1, we obtain, for example, the formulas
+g(l)
+2,1,s=1
+8(1+λn,1,sl−λn,2,sl−λn,3,sl)(n−sl)2sl
+(n)3,
+and
+g(l)
+2,2,s=1
+16(1−2λn,2,sl+λn,4,sl)(sl)2(n−sl)2
+(n)4.
+Lastly we present the bounds on products of eigenvalues of the ch ain used to handle the variance term
+(15) when applying Theorem 1.1 to the lightbulb chain.
+Lemma 4.5 For all even n≥6, ands=nn/2, or for all odd n= 2m+1≥7ands∈ {nm,nm+1,nm,m+1},
+|λn,2,s| ≤e−nand|λn,4,s| ≤1
+2e−n.
+Moreover, with λn,b,sas given in (5), for all odd n= 2m+1≥7we have
+|λn,2,nm| ≤e−nand|λn,4,nm| ≤1
+2e−n.
+Proof.The following calculations slightly generalize the arguments of [7]. Let mbe given by n= 2m, and
+n= 2m+1, in the even and odd cases, respectively. For n≥2 consider the second degree polynomial (cf.
+(3))
+f2(x) = 1−4x
+n+4(x)2
+(n)2,0≤x≤n.
+It is simple to verify that f2(x) achieves its global minimum value of −1/(n−1) atn/2, and that f2(x) has
+exactly two roots, at ( n+√n)/2 and (n−√n)/2. Hence, as f2(x)≤0 for all xbetween these roots, and
+additionally, as ( x−1)/(n−1)≤x/nfor allx∈[0,n], we obtain the bound
+|f2(x)| ≤/braceleftigg
+1
+n−1forx∈[n−√n
+2,n+√n
+2]/parenleftbig
+1−2x
+n/parenrightbig2forx∈[n−√n
+2,n+√n
+2]c∩[0,n].(99)
+Forx∈Rlet⌊x⌋and⌈x⌉denote the greatest integer less than or equal to x, and the smallest integer
+greater than or equal to x. In both the even and odd cases let
+t=/braceleftbigg
+⌈n−√n
+2⌉,···,⌊n+√n
+2⌋/bracerightbigg
+\{m,m+1}
+so that
+|λn,2,nm,m+1|=
+⌊n−√n
+2⌋/productdisplay
+s=0|f2(s)|
+/parenleftigg/productdisplay
+s∈t|f2(s)|/parenrightigg
+n/productdisplay
+s=⌈n+√n
+2⌉|f2(s)|
+.
+31If either of the roots ( n−√n)/2 or (n+√n)/2 is an integer then equality holds as both expressions above
+are zero. Now assuming neither value is an integer, the product is ov er disjoint indices.
+Applying the bound (99), ⌊n/2−x⌋+⌈n/2+x⌉=nand 1−2(n−s)/n=−(1−2s/n) yields
+|λn,2,nm,m+1| ≤
+⌊n−√n
+2⌋/productdisplay
+s=0/parenleftbigg
+1−2s
+n/parenrightbigg2
+/parenleftigg/productdisplay
+s∈t1
+n−1/parenrightigg
+n/productdisplay
+s=⌈n+√n
+2⌉/parenleftbigg
+1−2s
+n/parenrightbigg2
+
+=
+⌊n−√n
+2⌋/productdisplay
+s=0/parenleftbigg
+1−2s
+n/parenrightbigg
+4/parenleftbigg1
+n−1/parenrightbigg|t|
+,
+where|t|is the cardinality of t.
+Using 1−x≤e−xand that ⌊x⌋ ≥x−1 on the first product, we obtain the bound
+|λn,2,nm,m+1| ≤/parenleftig
+e−2
+n(⌊n−√n
+2⌋)(⌊n−√n
+2⌋+1)/2/parenrightig4
+e−|t|log(n−1)≤e−(n−2√n−1+2/√n+|t|log(n−1)).
+To control |t|, note that as ⌈x⌉ ≤x+1, we have
+|t|=⌊n+√n
+2⌋−⌈n−√n
+2⌉−1≥√n−3.
+As log63 ≥4, the result follows for all n≥64 from
+−2√n−1+2/√n+|t|log(n−1)≥ −2√n−1+4(√n−3) = 2√n−13≥0,
+and by direct verification for all odd 7 ≤n≤63, thus proving the claimed inequality for λn,2,nm,m+1for all
+oddn≥7.
+By (99) we have |f2(x)| ≤1 forx∈[0,n], and hence
+|λn,2,nm|=/vextendsingle/vextendsinglef2(m+1)λn,2,nm,m+1/vextendsingle/vextendsingle≤/vextendsingle/vextendsingleλn,2,nm,m+1/vextendsingle/vextendsingle, (100)
+so the claimed bound holds for λn,2,nm, and likewise for λn,2,nm+1, for all odd n≥7. The claim for
+λn,2,nm= (λn,2,nm+λn,2,nm+1)/2 now follows. Direct verification of the claim for λn,2,nmfor all even
+6≤n≤62 now also completes the proof for all cases involving λn,2,s.
+Now turning to λn,4,s, for alln≥6 consider the fourth degree polynomial
+f4(x) = 1−8x
+n+24(x)2
+(n)2−32(x)3
+(n)3+16(x)4
+(n)4,0≤x≤n.
+It can be checked that the four roots of f4(x) are given by
+x1±=n±/radicalbig√
+2√
+3n2−9n+8+3n−4
+2andx2±=n±/radicalbig
+−√
+2√
+3n2−9n+8+3n−4
+2,
+and that additionally the three roots to the cubic equation f′
+4(x) = 0 occur at
+y1=n/2 and y2±= (n±√
+3n−4)/2.
+These roots satisfy
+0< x1−< y2−< x2−< y1< x2+< y2+< x1+< n.
+32To obtain a bound over the interval [ x1−,x1+], evaluating f4(x) at its critical values we obtain
+f4(y1) =3
+(n−1)(n−3)≤3
+(n−3)2and
+f4(y2±) =−2(3n2−9n+8)
+n(n−1)(n−2)(n−3)≥ −6
+(n−3)2.
+To bound f4(x) byf2
+2(x) in the remaining part of [0 ,n], write
+f2
+2(x)−f4(x) =16x(n−x)p(x)
+(n−1)2n2(n2−5n+6)
+where
+p(x) = (4n−6)x2+(6n−4n2)x+n3−2n2+n.
+The roots of the quadratic p(x) are given by
+z±=n
+2±1
+2/radicalbigg
+n(n−2)
+2n−3.
+As 5n2−15n+12≥0 for alln, we have (2 n−3)(3n−4)≥n(n−2), and therefore
+/parenleftig√
+2/radicalbig
+3n2−9n+8+3n−4/parenrightig
+(2n−3)≥n(n−2).
+Dividing by 2 n−3 and taking square roots demonstrates that
+x1−≤z−< z+< x1+.
+Hencep(x) is nonnegative on the complement of [ z−,z+], and we obtain
+|f4(x)| ≤/braceleftbigg6
+(n−3)2forx∈[x1−,x1+]
+f2
+2(x) forx/\⌉}atio\slash∈[x1−,x1+],x∈[0,n].(101)
+Now write for short
+C(n) =/radicalig√
+2/radicalbig
+3n2−9n+8+3n−4 so that x1−=n−C(n)
+2.
+Using (101), (99) and that |s−n/2|>√n/2 whenever s≤(n−C(n))/2, 1−x≤e−xforx≥0 and finally
+⌊x⌋ ≥x−1, we obtain
+⌊n−C(n)
+2⌋/productdisplay
+s=0|λn,4,s| ≤⌊n−C(n)
+2⌋/productdisplay
+s=0f2
+2(s)≤
+⌊n−C(n)
+2⌋/productdisplay
+s=0/parenleftbigg
+1−2s
+n/parenrightbigg
+4
+≤e−(n−2(C(n)+1)+C(n)2/n+2C(n)/n).(102)
+Now let
+u=/braceleftbigg
+⌈n−C(n)
+2⌉,···,⌊n+C(n)
+2⌋/bracerightbigg
+\{m,m+1}.
+Usingλn,4,s=λn,4,n−s, (101),|u| ≥C(n)−3 and (102), we have that
+|λn,4,nm,m+1|
+=⌊n−C(n)
+2⌋/productdisplay
+s=0|λn,4,s|/productdisplay
+s∈u|λn,4,s|n/productdisplay
+s=⌈n+C(n)
+2⌉|λn,4,s| ≤
+⌊n−C(n)
+2⌋/productdisplay
+s=0λ2
+n,4,s
+/parenleftbigg6
+(n−3)2/parenrightbiggC(n)−3
+≤e−2(n−2(C(n)+1)+C(n)2/n+2C(n)/n)+(C(n)−3)(log(6) −2log(n−3))
+=e−n·e−(n−4(C(n)+1)+2 C(n)2/n+4C(n)/n+(C(n)−3)(2log( n−3)−log(6))).
+33Since for n≥96 we have
+n−4(C(n)+1)≥log(2) and ( C(n)−3)(2log( n−3)−log(6))≥0,
+we conclude that |λn,4,nm,m+1| ≤e−n/2 for all such n≥96, and direct verification shows this same bound
+holdsforall7 ≤n≤95. Arguingasin (100), the claimedinequalitythereforeholdsfor λn,4,nmandλn,4,nm+1,
+and hence also for λn,4,nm= (λn,4,nm+λn,4,nm+1)/2 for all odd n≥7, as well as for λn,4,nn/2for all even
+n≥8. The proof is completed by directly verifying the bound for λn,4,nn/2forn= 6. /square
+Appendix
+We work out some of the detailed calculations in the proof of Theorem 3.3. First, we show
+/summationdisplay
+a,b∈{0,1}g(m,m+1)
+[(1,0),(2,1);(3,0),(4,1)],n(b,a)
+=/parenleftbigg(m+1)3
+(n)4/parenrightbigg2/parenleftbig
+(2m−1)2+2(2m−1)λn,2,nm,m+1+λn,4,nm,m+1/parenrightbig
+. (103)
+To handle this sum, first write
+/summationdisplay
+a,b∈{0,1}g(m,m+1)
+[(1,0),(2,1);(3,0),(4,1)],n(b,a)
+=/summationdisplay
+a,b∈{0,1}/summationdisplay
+e1,e2,e3,e4∈{0,1}P(Xi= 0,i= 1,...,4,Xm1= 0,Xm2= 1,Xm3=e1,Xm4=e2
+Xm+1,1=e3,Xm+1,2=e4,Xm+1,3= 0,Xm+1,4= 1).(104)
+We may write the probability in the sum as the conditional probability of {Xi= 0,i= 1,...,4}given
+{(Xm1,Xm2,Xm3,Xm4) = (0,1,e1,e2),(Xm+1,1,Xm+1,2,Xm+1,3,Xm+1,4) = (e3,e4,0,1)},multipliedbythe
+unconditional probability of this last event. By Lemma 4.2, that is, th at the switch variables may be applied
+in any order, conditioning the first event on the values of the switch variables for these four bulbs in stages m
+andm+1 is equivalent to conditioning on the combined event {(X01,X02,X03,X04) = (e3,e4+1,e1,e2+1)}
+in, say, an initial stage 0, and running the lightbulb process in the rem ainingn−2 stages, with, recalling
+our extension of the notation defined in (27), pattern nm,m+1= (1,...,m−1,m+2,...,n).
+For the conditional probability, by (94) we may write
+P(Xi= 0,i= 1,...,4|(X01,X02,X03,X04) = (e3,e4+1,e1,e2+1)) =fα,β,nm,m+1 (105)
+whereα+β= 4 and
+α=1(e3= 0)+1(e4+1 = 0)+ 1(e1= 0)+1(e2+1 = 0).
+Having conditioned on the switch variables in stages mandm+1, we note that this conditional probability
+does not depend on aorb.
+For the unconditional probability, we have
+P((Xm1,Xm2,Xm3,Xm4) = (0,1,e1,e2),(Xm+1,1,Xm+1,2,Xm+1,3,Xm+1,4) = (e3,e4,0,1))
+=/parenleftbigg(m+1−b)3−e1−e2(m+b)1+e1+e2
+(n)4/parenrightbigg/parenleftbigg(m+1−a)3−e3−e4(m+a)1+e3+e4
+(n)4/parenrightbigg
+,
+as the stages are independent, and the event requires 1+ e1+e2switches to be drawn from m+bin stage
+m, and 1+ e3+e4switches to be drawn from m+ain stagem+1. Hence, changing the order of summation
+34in (104), and recalling that the conditional probability fα,β,nm,m+1does not depend on a,b, we have
+/summationdisplay
+a,b∈{0,1}g(m,m+1)
+[(1,0),(2,1);(3,0),(4,1)],n(b,a)=4/summationdisplay
+α=0fα,β,nm,m+1wα,β (106)
+where
+wα,β=/summationdisplay
+Eα,β/summationdisplay
+{a,b}∈{0,1}/parenleftbigg(m+1−b)3−e1−e2(m+b)1+e1+e2
+(n)4/parenrightbigg/parenleftbigg(m+1−a)3−e3−e4(m+a)1+e3+e4
+(n)4/parenrightbigg
+,
+with
+Eα,β={(e3,e4,e1,e2) :1(e3= 0)+1(e4+1 = 0)+ 1(e1= 0)+1(e2+1 = 0) = α}.
+Letting
+pk(d) =/summationdisplay
+b∈{0,1}(m+1−b)k−d(m+b)d
+(n)k(107)
+we may write
+wα,β=/summationdisplay
+Eα,βp4(1+e1+e2)p4(1+e3+e4).
+Writing out the sets Eα,βrequired, we have that
+E4,0={(0,1,0,1)},
+E3,1={(0,1,0,0),(0,1,1,1),(0,0,0,1),(1,1,0,1)},
+E2,2={(0,0,0,0),(0,0,1,1),(0,1,1,0),(1,0,0,1),(1,1,0,0),(1,1,1,1)},
+E1,3={(0,0,1,0),(1,0,0,0),(1,0,1,1),(1,1,1,0)}andE0,4={(1,0,1,0)}.
+Aspk(d) =pk(k−d), and since Eβ,αis obtained from Eα,βby replacing each entry of each vector by its
+binary complement,
+wα,β=/summationdisplay
+Eα,βp4(3−e1−e2)p4(3−e3−e4) =/summationdisplay
+Eβ,αp4(1+e1+e2)p4(1+e3+e4) =wβ,α.
+Now we calculate the required weights. As the one element of E4,0satisfies ( e1+e2,e3+e4) = (1,1), we
+have
+w4,0=p4(2)2=/bracketleftbigg2(m)2(m+1)2
+(n)4/bracketrightbigg2
+.
+Next, as the four elements in E3,1yield the vectors (1 ,0),(1,2),(0,1),(2,1) for (e1+e2,e3+e4), we have
+w3,1= 2p4(2)p4(1)+2p4(2)p4(3) = 4p4(2)p4(1) = 8/bracketleftbigg(m)2(m+1)2
+(n)4/bracketrightbigg/bracketleftbigg(m+1)3m+(m)3(m+1)
+(n)4/bracketrightbigg
+.
+35Lastly, as the six vectors in E2,2yield (0,0),(0,2),(1,1),(1,1),(2,0),(0,0) for (e1+e2,e3+e4), we have
+w2,2= 2p4(1)2+2p4(1)p4(3)+2p4(2)2= 4p4(1)2+2p4(2)2
+= 4/bracketleftbigg(m+1)3m+(m)3(m+1)
+(n)4/bracketrightbigg2
++8/bracketleftbigg(m)2(m+1)2
+(n)4/bracketrightbigg2
+.
+Now using wα,β=wβ,α, applying Corollary 4.1 in (106) we obtain
+/summationdisplay
+a,b∈{0,1}g(m,m+1)
+[(1,0),(2,1);(3,0),(4,1)],n(b,a)
+=1
+2(1+6λn,2,nm,m+1+λn,4,nm,m+1)/parenleftbigg(m)2(m+1)2
+(n)4/parenrightbigg2
++(1−λn,4,nm,m+1)/parenleftbigg(m)2(m+1)2
+(n)4/parenrightbigg/parenleftbigg(m+1)3m+(m)3(m+1)
+(n)4/parenrightbigg
++1
+16(1−2λn,2,nm,m+1+λn,4,nm,m+1)/parenleftigg
+4/bracketleftbigg(m+1)3m+(m)3(m+1)
+(n)4/bracketrightbigg2
++8/bracketleftbigg(m)2(m+1)2
+(n)4/bracketrightbigg2/parenrightigg
+.
+Simplification yields equation (103).
+Next we demonstrate that
+/summationdisplay
+a,b∈{0,1}/summationdisplay
+r/n⌉}ationslash=t/parenleftig
+g(r,t)
+[(1,0),(2,1);(1,0),(3,1)],n(b,a)+g(r,t)
+[(1,1),(2,0);(1,1),(3,0)],n(b,a)
++g(r,t)
+[(1,0),(2,1);(1,1),(3,0)],n(b,a)+g(r,t)
+[(1,1),(2,0);(1,0),(3,1)],n(b,a)/parenrightig
+=4(m+1)2
+2(2m−1)2
+(n)2
+3. (108)
+Starting with the first term, indexed by [(1 ,0),(2,1);(1,0),(3,1)], arguing as for the case p= 4, with
+α+β= 3 as in (105) we obtain the conditioning event ( X01,X02,X03) = (0,e2+1,1+e1) and may write
+/summationdisplay
+a,b,∈{0,1}g(r,t)
+[(1,0),(2,1);(1,0),(3,1)],n(b,a)=3/summationdisplay
+α=0fα,β,nm,m+1wα,β, (109)
+where
+Eα,β={(e1,e2) : 1+1(e2+1 = 0)+ 1(e1+1 = 0) = α}={(e1,e2) : 1+1(e2= 1)+1(e1= 1) =α}
+and, with pk(d) as in (107),
+wα,β=/summationdisplay
+Eα,βp3(1+e1)p3(1+e2).
+Explicitly, the required sets are given by
+E3,0={(1,1)}E2,1={(0,1),(1,0)}, E1,2={(0,0)}andE0,3=∅,
+and calculating the weights, using p3(1) =p3(2) = (m+1)2(2m−1)/(n)3, we obtain
+w3,0=(m+1)2
+2(2m−1)2
+(n)2
+3, w2,1=2(m+1)2
+2(2m−1)2
+(n)2
+3andw1,2=(m+1)2
+2(2m−1)2
+(n)2
+3.
+36Hence, applying Corollary 4.1 in (109) yields
+/summationdisplay
+a,b,∈{0,1}g(r,t)
+[(1,0),(2,1);(1,0),(3,1)],n(b,a)(110)
+=(m+1)2
+2(2m−1)2
+(n)2
+3/parenleftbigg1
+8(1+3λn,1,nm,m+1+3λn,2,nm,m+1+λn,3,nm,m+1)
++1
+4(1+λn,1,nm,m+1−λn,2,nm,m+1−λn,3,nm,m+1)
++1
+8(1−λn,1,nm,m+1−λn,2,nm,m+1+λn,3,nm,m+1)/parenrightbigg
+.
+Similarly, for the term in (108) subscripted by [(1 ,0),(2,1);(1,1),(3,0)] the conditioning event becomes
+(X01,X02,X03) = (1,e2+1,e1), leading to the collection of sets
+E3,0=∅, E2,1={(1,0)}, E1,2={(0,0),(1,1)},andE0,3={(0,1)},
+that give rise to the same weights as for the terms in (110). Hence, this term contributes
+(m+1)2
+2(2m−1)2
+(n)2
+3/parenleftbigg1
+8(1+λn,1,nm,m+1−λn,2,nm,m+1−λn,3,nm,m+1) (111)
++1
+4(1−λn,1,nm,m+1−λn,2,nm,m+1+λn,3,nm,m+1)
++1
+8(1−3λn,1,nm,m+1+3λn,2,nm,m+1−λn,3,nm,m+1)/parenrightbigg
+,
+and adding to (110) results in
+(m+1)2
+2(2m−1)2
+(n)2
+3/parenleftbigg1
+4(1+3λn,2,nm,m+1)+1
+2(1−λn,2,nm,m+1)+1
+4(1−λn,2,nm,m+1)/parenrightbigg
+=(m+1)2
+2(2m−1)2
+(n)2
+3.(112)
+Fortheterminthesumsubscriptedby[(1 ,1),(2,0);(1,1),(3,0)]theconditioningeventis( X01,X02,X03) =
+(0,e1,e2), resulting in sets
+E3,0={(0,0)}, E2,1={(0,1),(1,0)}, E1,2={(1,1)}andE0,3=∅
+and a term, therefore, agreeing with (110). Lastly, for the term subscripted by [(1 ,1),(2,0);(1,0),(3,1)] the
+conditioning event is ( X01,X02,X03) = (1,e1,1+e2), the collection of sets becomes,
+E3,0=∅, E2,1={(0,1)}, E1,2={(0,0),(1,1)},andE0,3={(1,0)},
+and we thus obtain a term agreeing with (111). Hence these last two terms also contribute the factor (112),
+and summing over the cases ( r,t) = (m,m+1) and ( r,t) = (m+1,m) yields (108).
+Lastly, we show that for r,t∈ {m,m+1},r/\⌉}atio\slash=t,
+/summationdisplay
+a,b∈{0,1}g(r,t)
+[(1,0),(2,1);(1,0),(2,1)],n(b,a)=/parenleftbig
+1+2λn,1,nm,m+1+λn,2,nm,m+1/parenrightbig(m+1)2
+2
+(n)2
+2. (113)
+In this case the conditioning event is simply ( X01,X02) = (0,0), and selecting one on and one off switch in
+each of the stages randtyields the weight w2,0=p2(1)2= 4(m+ 1)2
+2/(n)2
+2; applying Corollary 4.1 now
+completes the verification of (113).
+37Bibliography
+1. Baldi, P., Rinott, Y. and Stein, C. (1989). A normal approximation for the number of local maxima of
+a random function on a graph, In Probability, Statistics and Mathematics, Papers in Honor o f Samuel
+Karlin.T. W. Anderson, K.B. Athreya and D. L. Iglehart eds., Academic Pre ss , 59-81.
+2. Ghosh, S. and Goldstein, L. (2009). Concentration of measure s via size biased couplings. Probability
+Theory and Related Fields, to appear
+3. Chen, L.H.Y., Goldstein, L., and Shao, Q.M. (2010). Normal Approx imation by Stein’s Method.
+Springer.
+4. Goldstein, L. (2005). Berry-Esseen bounds for combinatorial central limit theorems and pattern oc-
+currences, using zero and size biasing. J. Appl. Probab. 42, 661–683.
+5. Goldstein, L. and Rinott, Y. (1996). On multivariate normal appr oximations by Stein’s method and
+size bias couplings. J. Appl. Prob. 33, 1-17.
+6. Goldstein, L. and Penrose, M. (2009). Normal approximation fo r coverage models over binomial point
+processes. Ann. Appl. Prob .20, 696 - 721.
+7. Rao, C., Rao, M., and Zhang, H. (2007). One Bulb? Two Bulbs? How m any bulbs light up? A
+discrete probability problem involving dermal patches. Sanky¯a,69, 137-161.
+8. Shao, Q. M. and Su, Z. (2005). The Berry-Esseen bound for ch aracter ratios. Proceedings of the
+American Mathematical Society ,134, 2153-2159.
+9. Stein, C. (1981) Estimation of the mean of a multivariate normal d istribution. Ann. Statist. 9,
+1135-1151.
+10. Stein, C. (1986). Approximate Computation of Expectations. IMS, Hayward, California.
+11. Zhou, H. and Lange, K. (2009). Composition Markov chains of m ultinomial type. Advances in Applied
+Probability ,41, 270 - 291.
+38
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@@ -0,0 +1,270 @@
+arXiv:1001.0613v1  [hep-ph]  5 Jan 2010Sensitivity of the parameters measured in pp collisions
+on the gluon PDF.
+E. Cuautle,∗A. Ortiz,†and G. Pai´ c‡
+Instituto de Ciencias Nucleares, Universidad Nacional Aut ´ onoma de M´ exico
+Apartado Postal 70-543, M´ exico Distrito Federal 04510, M´ exico.
+(Dated: November 17, 2018)
+The sphericity distribution in minimum bias events obtaine d with PYTHIA event generator, in-
+corporating different parton distribution functions is pre sented. The results show that for minimum
+bias pp collisions the sphericity distribution for differen t parameters like the mean transverse mo-
+mentum and multiplicity exhibit strong sensitivity on the p arton distribution function used. The
+results indicate that early data at the LHC could be used in th is type of analysis to fix the gluon
+distribution function in the proton.
+PACS numbers: 24.85.+p, 12.38.-t
+I. INTRODUCTION
+The calculation of the production cross sections at
+hadron colliders relies upon a knowledge of the distribu-
+tion of the momentum fraction xof the partons (quarks
+and gluons) in a proton in the relevant kinematic range,
+and the corresponding fragmentation functions. Most
+probably, at LHC, we will be confronted with a copious
+production of jets what will severely limit the exclusive
+analysis of individual jets. It seems therefore interesting
+to separate the events in function of their multi-jetiness
+and to study the parameters of the collisions e.g.mean
+pt, mean multiplicity, etc., in function of different par-
+ton distribution functions (PDF’s) present on the mar-
+ket, which as quoted by the CTEQ group allows for an
+uncertainty of about 10-20% [1]. Obviously this uncer-
+taintywillreflectsinthe quantificationofthe kinematical
+processes. The PDF’s are of importance to analyze stan-
+dard model processes at LHC energies [2], and the usual
+leading order event generators use the PDF’s as an input
+to evaluate hard subprocess matrix elements. In those
+calculations the multiple parton interactions that make
+up the bulk of the underlying event, require extensive
+tuning which depends strongly on the details of the in-
+put parton distributions.
+The Event Shape Analysis (ESA) has been applied suc-
+cessfully to the restricted acceptance experiments at
+RHIC and ALICE to survey the possibilities of the
+method to analyze and isolate specific jet topology
+events [3]. In the present paper we are investigating the
+bulk properties of the event shape variables obtained in
+minimum bias pp events: the sensitivity obtained at dif-
+ferent collisions energies focussing to minimum bias col-
+lisions, corresponding to the minimum x-Bjorken (xB)
+achievable. The article is organized as follows: in the
+section II we give the relation for the sphericity and re-
+∗Electronic address: ecuautle@nucleares.unam.mx
+†Electronic address: antonio.ortiz@nucleares.unam.mx
+‡Electronic address: guypaic@nucleares.unam.mxcoil distributions; in the section III we discuss the parton
+distribution functions used and the way the simulations
+were done in the framework of minimum bias PYTHIA.
+The results are presented in section IV and finally, in
+section V the conclusion are drawn.
+II. EVENT SHAPE VARIABLES
+Event shape variables allow us to measure geometrical
+properties of QCD final state [4]. Those variables has
+been used to analyze dijet production in hadron-hadron
+collisions and to study the sensitivity of physical param-
+eters [4]. The shape variables at hadron colliders are
+defined over particles within the acceptance of the detec-
+tor. The thrust ( T) is defined in the transverse plane as
+follows:
+T≡max/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+− →nt/summationtext
+i|− →pt,i·− →nt|/summationtext
+i|− →pt,i|(1)
+where the sum runs over all particles in the final state
+withintheacceptance,− →pt,irepresentthetransversecom-
+ponent of the momenta of emitted particles and− →ntis
+the transverse vector that maximizes the ratio. In the
+literature it is more common to find the quantity 1 −T
+related to the transverse sphericity of the event. In the
+present work we do reference to this as sphericity. The
+range of sphericity is between 0 (for events with narrow
+back-to-backjets) and 1 −2/π(for events with a uniform
+momentum distribution).
+Another interesting observable is the recoil, defined as:
+R≡1/summationtext
+i|− →pt,i||/summationdisplay
+i− →pti|. (2)
+It is an indirect measurement for the radiation which re-
+mains undetected for acceptance effects (high inbalance:
+R→1, small inbalance: R→0).2
+TABLE I: The xBvalues for low ptproduction at different en-
+ergies. The evaluation was done at central rapidity, assumi ng
+pt= 500 MeV/c.
+√s x B Collider
+200 GeV ∼5×10−3RHIC
+900 GeV ∼10−3TEVATRON/LHC
+10 TeV ∼10−4LHC
+III. THE PARTON DISTRIBUTION
+FUNCTIONS
+The QCD factorization theorem provides us with a
+framework where distinct parameters can be indepen-
+dently studied in order to interpret the experimental re-
+sults. Prominent among them we find the parton distri-
+bution functions whichcorrespondto the momentum dis-
+tributions of partons inside a proton. The extraction of
+the parton distribution function from experimental data
+andtheoreticalapproachisacomplicatedtask[1,5]. The
+uncertaintiesgrowwith diminishing xBand aregenerally
+very much larger for gluons than for quarks. At the LHC
+(Large Hadron Collider) energies, the gluons will play
+an important role to understand the new physics. Some
+QCD analysis of parton distribution has been done [1].
+At this energies, one will be able to explore regions of
+smallxBas shown in Table I for minimum bias events.
+IV. RESULTS ON EVENT SHAPE VARIABLES
+AND PDF OF GLUONS
+In order to describe the recoil and sphericity variables
+and their behavior as function of the parton distribu-
+tion function of the gluon used, we generated a sample of
+PYTHIA 6.214 minimum bias events [6]. The generation
+incorporates some parameters proposed as ATLAS tune ,
+based onreference [7]. This tuning parameterswerefixed
+to describe the experimental minimum bias data of UA5
+and CDF experiments. The pseudorapidity and trans-
+verse momenta distribution were the first variables fitted
+in this tuning. Setting from these fits after that, predic-
+tion of the properties of minimum bias and underlying
+event at LHC energy were scaled.
+We have used this version of PYTHIA to implant two
+different parton distribution functions: ZEUS2005 [8]
+and CTEQ5L [9]. The latter PDF was extracted from
+a global analysis to several experimental data while the
+former is extracted only from HERA data. Plotting the
+ratio of the two gluon distribution as we have done in
+Fig. 1, we see that at different energies and Qvalues
+they differ substantially, the lowest values of xshowing
+the largest difference.
+In Fig. 2 we show in the top panel the mean multi-
+plicity variation with the sphericity variable obtained at
+central pseudorapidity. We observe an interesting depen-
+dence on the PDF used: the mean multiplicity at highx-510-410-310-210-110)2(x,Q
+CTEQ)/xf2(x,QZEUSxf
+0.40.50.60.70.80.911.11.21.31.4
+ = 200 GeV )s Q = 6.3 GeV  ( 
+ = 900 GeV )s Q = 20.0 GeV  ( 
+ = 10 TeV )s Q = 142.1 GeV  ( 
+FIG. 1: Ratio of the gluon contributions for the ZEUS and
+CTEQ5L parton distribution functions, the ratio is shown
+for different Q values corresponding to the mean Q values
+obtained from the simulations at 0.2, 0.9 and 10 TeV, respec-
+tively
+sphericities is much higher for the CTEQ5L PDF than
+for the ZEUS one. The phenomenon is the most accen-
+tuated in the case of the 10 TeV collisions where the xB
+value reached is also the smallest. The bottom part of
+Fig. 2 illustrates this fact, it shows the behavior of the
+ratio of the mean multiplicities for the three cases as a
+function of sphericity.
+The studies displayed here are within the reach of all
+the LHC experiments, though we have done the simula-
+tion for the case of a relatively small acceptance detec-
+tor which has the capability to detect low charged par-
+ticle momenta. We consider only primary tracks within
+|η| ≤0.9 andpt≥0.5 GeV/c defined according to refer-
+ence [10]. In this context they are the particles produced
+in the collision, including products of strong and electro-
+magnetic decays, but excluding feed-down products from
+strange weak decays and particles produced in secondary
+interactions.
+In Fig. 3 we show the ratio of the sphericity distri-
+butions obtained for the three energies. One observes
+that in all cases the difference between the two PDF’s
+increases with growing spherificity but much more so in
+the case of the 10 TeV simulations. This is due to the
+smallerxBinthatcase,wherethedifferencesbetweenthe
+two PDF’s are the largest. The strong slope in case of 10
+TeV data demonstrate the sensitivity of the approach.
+Next, the recoil variable is computed according to
+Eq. 2, the results are shown in Fig. 5 where a charac-
+teristic behavior of the ratio of the mean R values is ob-
+served for the 3 energies. Namely, The ZEUS PDF has a
+larger momentum inbalance than CTEQ5L. We explain
+the feature by the fact that, due to the smaller rise of the
+gluon PDF in ZEUS one is confronted with an emission
+of particles resulting from collisions of partons of differ-
+ent momenta, while in the case of the more peaky gluon3
+1-T0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4chmean N
+024681012141618202224
+200 GeV, CTEQ5L
+200 GeV, ZEUS
+900 GeV, CTEQ5L
+900 GeV, ZEUS
+10 TeV, CTEQ5L
+10 TeV, ZEUS
+x0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4CTEQch / mean NZEUSchmean N
+0.40.50.60.70.80.911.11.2
+ = 200 GeVs
+ = 900 GeVs
+ = 10 TeVs
+FIG. 2: top: Mean multiplicity in the pseudo rapidity range
+±0.9 obtained at the three energies and the two PDFs in
+function of the sphericity variable. Bottom: ratio of the me an
+multiplicities for the two PDF’s used for the three energies ,
+again in function of the sphericity variable.
+1-T0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Ratio
+00.20.40.60.811.21.4 = 200 GeVs
+ = 900 GeVs
+ = 10 TeVs
+FIG. 3: Ratio of sphericity distributions at different energ ies1-T0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4CTEQ)
+t,j p
+j∑/mean (
+ZEUS)
+t,i p
+i∑mean (
+0.40.50.60.70.80.911.11.21.31.4
+ = 200 GeVs
+ = 900 GeVs
+ = 10 TeVs
+FIG. 4: Ratio of mean total transverse momenta for the two
+PDFs versus sphericity calculated for 3 energies
+1-T0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4CTEQ/mean (Recoil)
+ZEUSmean (Recoil)
+0.50.60.70.80.911.11.21.31.41.5
+ = 200 GeVs
+ = 900 GeVs
+ = 10 TeVs
+FIG. 5: Ratio of the mean recoil values for two PDF’s versus
+sphericity calculated for 3 energies.
+distribution of CTEQ5L the chance of getting a collision
+of two partons of similar momentum is more probable,
+hence resulting in a larger probability that the resulting
+jets be contained within the acceptance.
+V. CONCLUSION
+The bulk sphericity distributions of several parameters
+exhibit marked dependence on the gluon PDF used for
+the computation. As the PDFs for low xBhave a larger
+uncertainty we believe that the present results point to
+a possibility to further refine gluon PDFs at low xB
+using the approach presented. In the course of this
+work we have also tried to check the dependence of the
+parameters on different fragmentation functions used.
+The results however do not demonstrate a similar depen-
+dence as for the PDFs. As one can expect, varying the
+fragmentation functions one affects slightly the absolute4
+value of the parameter dependence on sphericity but not
+the shape.
+In summary, the sphericity distributions of several pa-
+rameters are found to be highly sensitive on the details
+of the gluon PDF. This opens an interesting possibility
+to compare the existing gluon PDFs with the experimen-
+tal results using a small sample of minimum bias data
+in pp collisions. The Pb Pb collisions, we are currently
+studying [11] in the same framework will benefit verymuchfromthepresentanalysistodeterminetheexpected
+gluon saturation at low xB.
+Acknowledgments
+Support for this work has been received in part by
+DGAPA-UNAM under PAPIIT grants IN115808 and
+IN116508 as well as by the HELEN program.
+[1] Pavel, M. Nodolsky, et. al., Phys. Rev. D 78,013004
+(2008).
+[2] H-L. Lai, et. al., Parton Distribution for Event Genera-
+tors arXiv:0910.4183
+[3] A. Ayala et. al., Eur.Phys.J. C62535, (2009). A. Ortiz,
+G. Pai´ c, ALICE-INT-2009-015 (hep-ex/0912.0909).
+[4] A. Banfi et. al., JHEP 0408, 062 (2004).
+[5] W. K. Tung Perturbative QCD and the parton structure
+of the nucleon In *Shifman, M. (ed.): At the frontier of
+particle physics, Vol. 2, 887-971 (2001) .[6] T. Sj¨ ostrand, M. Bengtsson, Comput. Phys. Commun. 43
+(1987) 367.
+[7] A. Morales, et. al., Eur.Phys.J.C , 50:435-466, (2007).
+[8] (ZEUS Collaboration), Eur.Phys.J.C , 42-1:1-16, (2005).
+[9] H. L. Lai et. al., Eur.Phys.J.C , 12-3:375-392, (2000).
+[10] J. F. Grosse-Oetringhaus and C. Ekman, ALICE-INT-
+2007-005.
+[11] E. Cuautle, A. Ortiz, G. Pai´ c, work in progress
\ No newline at end of file
diff --git a/1001.0614.txt b/1001.0614.txt
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@@ -0,0 +1,528 @@
+arXiv:1001.0614v2  [cond-mat.stat-mech]  7 Jan 2010Correlation functions for the three state
+superintegrable chiral Potts spin chain of finite
+lengths
+Klaus Fabricius∗
+Physics Department, University of Wuppertal, 42097 Wuppertal, G ermany
+Barry M. McCoy†
+C.N.Yang Institute for Theoretical Physics,
+State University of New York, Stony Brook, NY 11794-3840
+October 27, 2018
+Abstract
+Wecomputethecorrelation functionsofthethreestatesupe rintegrable
+chiral Potts spin chain for chains of length 3,4,5. From thes e results
+we present conjectures for the form of the nearest neighbor c orrelation
+function.
+1 Introduction
+The free energy of the Ising model was first solved by Onsager [1] in 1944
+who invented a method of solution based on what is now known as “Ons ager’s
+algebra”. This algebra is generated from operators A0andA1which form a
+Hamiltonian
+H=A0+λA1 (1)
+where
+[A0,[A0,[A0,A1]]] =const[A0,A1] (2)
+and from A0andA1the full algebra is generated as [2]
+[Aj,Ak] = 4Gj−k
+[Gm,Al] = 2Al+m−2Al−m
+[Gj,Gk] = 0 (3)
+∗e-mail Fabricius@theorie.physik.uni-wuppertal.de
+†e-mail mccoy@max2.physics.sunysb.edu
+1For 41 years the Ising model was the only known model which satisfie d this al-
+gebra but in 1985von Gehlen and Rittenberg [3] made the remarkab le discovery
+that the Hamiltonian (1) with
+A0=−N/summationdisplay
+j=1N−1/summationdisplay
+r=1eiπ(2r−N)/(2N)
+sinπr/NZr
+jZ†r
+j+1 (4)
+A1=−N/summationdisplay
+j=1N−1/summationdisplay
+r=1eiπ(2r−N)/(2N)
+sinπr/NXr
+j (5)
+whereZjandXjare direct product matrices
+Zj=I⊗···⊗Z⊗···⊗I (6)
+Xj=I⊗···⊗X⊗···⊗I (7)
+Iis theN×Nidentity matrix, the N×Nmatrices ZandXare in the jth
+position in the product and have the matrix elements
+Zj,k=ωjδj,k (8)
+Xj,k=δj,k+1 (9)
+with
+ω=e2πi/N(10)
+is also a representation of Onsager’s algebra (3). The Hamiltonian (1 ) withA0
+andA1given by (4) and (5) is called the Nstate superintegrable chiral Potts
+spin chain. When N= 2 the Hamiltonian of the superintegrable chiral Potts
+spin chain reduces to the Hamiltonian studied by Onsager [1].
+It was discovered in [4] by means of explicit computations on chains of small
+length that the eigenvalues of the superintegrable chiral Potts Ha miltonian are
+all of the form
+E=A+Bλ+Nm/summationdisplay
+j=1±(1+λ2+ajλ)1/2(11)
+This form was proven to follow directly from Onsager’s algebra by [5, 6 , 7]. The
+parameters aj,A,Bin (11) have been computed by the method of functional
+equations [8, 9, 10, 11] where it is found, for λsuitably less than unity, that in
+the thermodynamic limit the ground state energy per site is
+e0(λ) = lim
+N→∞1
+NE0(λ;N) =−(1+λ)N−1/summationdisplay
+l=1F(−1
+2,l
+N;1;4λ
+(1+λ)2) (12)
+whereF(a,b;c;z) is the hypergeometric function.
+In this paper we extend these finite chain computations from the eig envalues
+of the Hamiltonian to the correlation functions in the ground state /an}bracketle{tZr
+0Z†r
+R/an}bracketri}ht.
+2There are several constraints these correlations must satisfy. One such con-
+straint is the obvious requirement that
+/an}bracketle{tZ(N−r)
+0Z†(N−r)
+1/an}bracketri}ht=/an}bracketle{tZr
+0Z†r
+1/an}bracketri}ht∗(13)
+Furthermore in thermodynamic limit N → ∞ the correlation is related to the
+order parameter by
+M2
+r= lim
+R→∞/an}bracketle{tZr
+0Z†r
+R/an}bracketri}ht (14)
+ForN= 2 the order parameter is the spontaneous magnetization of the I sing
+model
+M= (1−λ2)1/8(15)
+which was reported by Onsager [12] in 1948 and proven by Yang [13] in 1952.
+For theNstate superintegrable chiral Potts spin chain it was conjectured b y
+Albertini, McCoy, Perk and Tang [4] in 1988 and proven by Baxter [14 ] in 2005
+that
+Mr= (1−λ2)r(N−r)/2N2(16)
+The order parameter of the Ising model (15) may be computed [15] by use
+of Szeg˝ o’s theorem applied to the representation of the correlat ion function
+/an}bracketle{tZ0Z†
+R/an}bracketri}htas a determinant [16] which is derived using free fermion methods firs t
+invented by Kaufmann [17]. However, the chiral Potts order param eter (16) is
+computed by Baxter [14] by functional equation methods which do n ot extend
+to a computation of the correlation functions /an}bracketle{tZr
+0Z†r
+R/an}bracketri}ht
+Because the superintegrable chiral Potts model is a generalization of the
+Ising model with the same underlying Onsager algebra there must be structure
+of the Ising correlations which generalizes to superintegrable chira l Potts. How-
+ever, because the Ising correlations have been computed by mean s of free Fermi
+methods and not by use of Onsager algebra this structure remains unknown.
+Recently Au-Yang and Perk [18, 19, 20, 21] and Nishino and Deguch i [22]
+have initiated the study of the eigenvectors of the superintegrab le chiral Potts
+spin chain by use of the Onsager algebra. These important studies a re the
+necessary foundation for the computation of the correlation fun ctions from the
+point of view of the Onsager algebra.
+The purpose of this paper is to provide insight into the correlation fu nc-
+tions of the superintegrable chiral Potts spin chain by explicitly calcu lating the
+correlations /an}bracketle{tZr
+0Z†r
+R/an}bracketri}htfor the three state case N= 3 for chains of finite length
+N= 3,4,5 which extends to correlation functions the study of [4] of the gro und
+state energy E0(λ,N). For any value of Nthe nearest neighbor correlations
+satisfy a sum rule coming from the ground state energy E0(λ;N). This sum
+rule is presented and discussed in sec. 2. In sec. 3 we present the r esults of our
+finite chain computations and we conclude in sec. 4 with a discussion of the
+implications which the results for finite chains have for the correlatio ns in the
+thermodynamic limit.
+32 Sum rule
+There is an elementary result known as Feynman’s theorem that for any Hamil-
+tonian which depends on a parameter λthat
+∂
+∂λ/an}bracketle{tH(λ)/an}bracketri}ht=∂
+∂λE0(λ) (17)
+where/an}bracketle{tO/an}bracketri}htdenotes the expectation value of the operator Oin the ground state
+andE0(λ) denotes the ground state energy as a function of λ. Therefore it fol-
+lows from (17) that for the superintegrable chiral Potts Hamiltonia n (1),(4),(5)
+the expectation /an}bracketle{tXr
+0/an}bracketri}htsatisfies the sum rule
+−NN−1/summationdisplay
+r=1eiπ(2r−N)/(2N)
+sinπr/N/an}bracketle{tXr
+0/an}bracketri}ht=∂E0(λ;N)
+∂λ
+=B−Nm/summationdisplay
+j=1λ+aj/2
+(1+λ2+ajλ)1/2(18)
+and/an}bracketle{tZr
+0Z†r
+1/an}bracketri}htsatisfies
+−NN−1/summationdisplay
+r=1eiπ(2r−N)/(2N)
+sinπr/N/an}bracketle{tZr
+0Z†r
+1/an}bracketri}ht=E0(λ;N)−λ∂E0(λ;N)
+∂λ
+=A−Nm/summationdisplay
+j=11+λaj/2
+(1+λ2+ajλ)1/2(19)
+3 Correlations for N= 3andN= 3,4,5
+We will explicitly consider the three state case N= 3 where the Hamiltonian
+(1) is explicitly written as
+H=−N/summationdisplay
+j=1/parenleftbigg
+(1−i√
+3)(ZjZ†
+j+1+λXj)+(1+i√
+3)(Z2
+jZ†2
+j+1+λX2
+j)/parenrightbigg
+(20)
+and using (13) the sum rule (19) is
+(1−i√
+3)/an}bracketle{tZ0Z†
+1/an}bracketri}ht+(1+i√
+3)/an}bracketle{tZ0Z†
+1/an}bracketri}ht∗=−A
+N+N
+Nm/summationdisplay
+j=11+ajλ/2
+(λ2+1+ajλ)1/2(21)
+It is known from [10] that for small λthe ground state is in the sector P= 0
+andQ= 0 where Pis the momentum of the state and
+ei2πQ/3=N/productdisplay
+k=1Xk (22)
+4For the ground state
+A=B=−Pa,forN ≡ −Pamod 3Pa= 0,1,2 (23)
+and
+aj=2(1+t3
+j)
+1−t3
+j(24)
+andtjare the roots of the polynomial equation
+0 =t−Pa{(t−1)N(tω2−1)Nω−Pa
++(tω−1)N(tω2−1)N+(t−1)N(tω−1)NωPa} (25)
+withω=e2πi/3. From computations of Galois groups done on Maple we see
+forN ≤12 that (25) can be explicitly solved in terms of radicals only for
+N= 3,4,5,6,9,12
+The eigenspace for the states with P=Q= 0 are of dimension 5 for N= 3,
+dimension 8 for N= 4 and dimension 17 for N= 5. The correlations may
+now be computed in principle by computing the normalized eigenvector s in
+the subspace P=Q= 0 and then explicitly computing the matrix elements
+ofZ0Z†
+R. In practice the algebra is too formidable to do by hand and the
+computation must be computerized. When this is done and expressio ns are
+simplified by removing common factor the results are as follows
+3.1N= 3
+From (24) and (25) we find
+9a2−20 = 0 (26)
+and thus
+a1=−2√
+5
+3, a 2=2√
+5
+3(27)
+Thus defining
+xj= (λ2+1+ajλ)1/2(28)
+we find
+/an}bracketle{tZ0Z†
+1/an}bracketri}ht= (1−i√
+3)/parenleftbigg3
+40−3λ2−7
+40x1x2/parenrightbigg
++3
+8(1+i√
+3)/parenleftbigg1+a1λ/2
+x1+1+a2λ/2
+x2/parenrightbigg
+(29)
+From (29) we find
+(1−i√
+3)/an}bracketle{tZ0Z†/an}bracketri}ht=−i4√
+3/parenleftbigg3
+40−3λ2−7
+40x1x2/parenrightbigg
++1
+2/parenleftbigg1+a1λ/2
+x1+1+a2λ/2
+x2/parenrightbigg
+(30)
+5and hence the sum rule (19) is obviously satisfied. For λ→0 (29) is expanded
+as
+/an}bracketle{tZ0Z†
+1/an}bracketri}ht= 1−2
+9λ2−17−i3√
+3
+54λ4+O(λ6) (31)
+and forλ→ ±∞
+/an}bracketle{tZ0Z†
+1/an}bracketri}ht=1
+3(1+i√
+3)|λ|−1+1
+6(1−i√
+3)λ−2+O(|λ|−3) (32)
+3.2N= 4
+ForN= 4
+27a2+36a−20 = 0 (33)
+and thus there are again two roots ajwhich are now given by
+a1=−2
+3+4
+9√
+6, a 2=−2
+3−4
+9√
+6 (34)
+We define xjas before (28) and obtain for /an}bracketle{tZ0Z†
+1/an}bracketri}htthe result
+/an}bracketle{tZ0Z†
+1/an}bracketri}ht=1
+4−(1−i√
+3)2λ−3
+40x1x2
++(1+a1λ/2)3
+320x1[36−√
+6+i√
+2(3+2√
+6)] (35)
++(1+a2λ/2)3
+320x2[36+√
+6−i√
+2(3−2√
+6]
+=3
+16(1+i√
+3)+1
+16(1−i√
+3)−(1−i√
+3)2λ−3
+40x1x2
++(1+a1λ/2)3
+320x1][30(1+i√
+3)+(6−√
+6)(1−i√
+3)]
++(1+a2λ/2)3
+320x2[30(1+i√
+3)+(6+√
+6)(1−i√
+3)] (36)
+We note that the coefficients of 1+ i3−1/2and 1−i31/2are both real, that
+(1−i√
+3)/an}bracketle{tZ0Z†
+1/an}bracketri}ht
+=1
+4−i1
+4√
+3+i1√
+32λ−3
+10x1x2
++(1+a1λ/2)3
+320x1][40−i(6−√
+6)4√
+3]
++(1+a2λ/2)3
+320x2[40−i(6+√
+6)4√
+3] (37)
+6and thus the sum rule (19) is satisfied. For λ→0 (37) reduces to
+/an}bracketle{tZ0Z†
+1/an}bracketri}ht= 1−2
+9λ2+1+i√
+3
+162λ4+8√
+3i
+729λ5+O(λ6) (38)
+and forλ→ ±∞
+/an}bracketle{tZ0Z†
+1/an}bracketri}ht=1
+3(1+i√
+3)|λ|−1+1
+18(1−i√
+3)λ−2+4
+81(1−4i√
+3)|λ|−3+O(λ−4).(39)
+We also compute /an}bracketle{tZ0Z†
+2/an}bracketri}htand find
+/an}bracketle{tZ0Z†
+2/an}bracketri}ht=1
+4−2λ−3
+10x1x2
++(1+a1λ/2)3(6−√
+6)
+80x1+(1+a2λ/2)3(6+√
+6)
+80x2(40)
+For small λthis reduces to
+/an}bracketle{tZ0Z†
+2/an}bracketri}ht= 1−2
+9λ2−1
+81λ4+O(λ5) (41)
+and forλ→ ±∞|
+/an}bracketle{tZ0Z†
+2/an}bracketri}ht=2
+9λ−2+20
+81|λ|−3+84
+729λ−4+O(λ−5) (42)
+3.3N= 5
+ForN= 5 there are three roots ajthat satisfy
+81a3+54a2−228a−88 = 0 (43)
+which are given by
+a1=−2
+9−2
+9(W++W−) (44)
+a2=−2
+9−2
+9(ω2W++ωW−) (45)
+a3=−2
+9−2
+9(ωW++ω2W−) (46)
+where
+W±=1
+17(7±i√
+19)/parenleftbigg5
+2/parenrightbigg1/3
+w± (47)
+with
+w±= (311±i9√
+19)1/3(48)
+We define xjas before (28) and obtain for /an}bracketle{tZ0Z†
+1/an}bracketri}htthe result
+7/an}bracketle{tZ0Z†
+1/an}bracketri}ht=3
+40(1+i√
+3)+37
+40·19(1−i√
+3)+9
+40(1+i√
+3)3/summationdisplay
+j=11+ajλ/2
+xj
++1
+23·52·19(1−i√
+3)3/summationdisplay
+j=191+bj−27aj/2−λ(67/3+7bj/3−7·9aj/2)
+xj
++1
+23·52·19(1−i√
+3)/summationdisplay
+1≤j<k≤3Pj,k
+xjxk(49)
+where
+b1= 1+/parenleftbigg5
+2/parenrightbigg1/3
+(w++w−)
+b2= 1+/parenleftbigg5
+2/parenrightbigg1/3
+(ω2w++ωw−)
+b3= 1+/parenleftbigg5
+2/parenrightbigg1/3
+(ωw++ω2w−),
+(50)
+and
+Pj,k= 164+81
+2(aj+ak)+7(bj+bk)
++λ
+3[−56+49·9
+2(aj+ak)+2(bj+bk)]
++/parenleftbiggλ
+3/parenrightbigg2
+[−504−13·81(aj+ak)−9·13(bj+bk)]
+(51)
+It is easy to see that (49) satisfies the sum rule (19). When λ→0 (49)
+reduces to
+/an}bracketle{tZ0Z†
+1/an}bracketri}ht= 1−2
+9λ2+1
+162(−9+i√
+3)λ4+56
+729λ5+O(λ6) (52)
+and forλ→ ±∞
+/an}bracketle{tZ0Z†
+1/an}bracketri}ht=1
+3(1+i√
+3)|λ|−1+1
+18(1−i√
+3)λ−2+4
+81(1+i√
+3)|λ|−3+O(λ−4) (53)
+4 Discussion
+From the results (29),(36) and (49) for N= 3 we conjecture for arbitrary N
+that/an}bracketle{tZ0Z†
+1/an}bracketri}hthas the form
+/an}bracketle{tZ0Z†
+1/an}bracketri}ht=P0+/summationdisplay
+jPj(λ)
+xj+(1−i√
+3)/summationdisplay
+j<kPj,k(λ)
+xjxk(54)
+8whereP0is a constant, Pj(λ) is a polynomial linear in λandPj,k(λ) is a poly-
+nomial at most quadratic in λwith real coefficients. In the limit N → ∞ the
+correlation /an}bracketle{tZ0Z†
+1/an}bracketri}htwill be a double integral. For N= 2 this correlation is a
+single integral and thus we expect that for general Nthe correlation /an}bracketle{tZr
+0Zr†
+1/an}bracketri}ht
+will be an N−1 fold integral.
+We expect for arbitrary Nthat/an}bracketle{tZr
+0Zr†
+R/an}bracketri}htwill be a ( N−1)Rfold integral but
+the first evidence for this for N= 3 can only come N= 6.
+We also conjecture from the result (41) for /an}bracketle{tZ0Z†
+2/an}bracketri}htandN= 4 that for all
+evenNthe correlation for N= 3/an}bracketle{tZ0Z†
+N/2/an}bracketri}htis real. In the limit N → ∞ we
+must have
+lim
+N→∞/an}bracketle{tZr
+0Z†r
+N/2/an}bracketri}ht= (1−λ2)r(N−r)/N2(55)
+Acknowledgement
+The bulk of this computation was only possible using the program FORM
+written by Prof. J.A.M. Vermaseren.
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+arXiv:1001.0615v1  [q-fin.PR]  5 Jan 2010Adaptive Wave Models for Option Pricing Evolution:
+Nonlinear and Quantum Schr¨ odinger Approaches
+Vladimir G. Ivancevic
+Defence Science & Technology Organisation, Australia
+Abstract
+Adaptive wave model for financial option pricing is proposed, as a hig h-complexity
+alternative to the standard Black–Scholes model. The new option-p ricing model, repre-
+senting a controlled Brownian motion, includes two wave-type appro aches: nonlinear and
+quantum, both based on (adaptive form of) the Schr¨ odinger equ ation. The nonlinear ap-
+proach comes in two flavors: (i) for the case of constant volatility, it is defined by a single
+adaptive nonlinear Schr¨ odinger (NLS) equation, while for the case of stochastic volatil-
+ity, it is defined by an adaptive Manakov system of two coupled NLS eq uations. The
+linear quantum approach is defined in terms of de Broglie’s plane waves and free-particle
+Schr¨ odinger equation. In this approach, financial variables have quantum-mechanical
+interpretation and satisfy the Heisenberg-type uncertainty rela tions. Both models are
+capable of successful fitting of the Black–Scholes data, as well as defining Greeks.
+Keywords: Black–Scholes option pricing, adaptive nonlinear Schr¨ odinger equa tion,
+adaptive Manakov system, quantum-mechanical option pricing, ma rket-heat potential
+PACS:89.65.Gh, 05.45.Yv, 03.65.Ge
+11 Introduction
+Recall that the celebrated Black–Scholes partial differenti al equation (PDE) describes the
+time–evolution of the market value of a stock option [1, 2]. Formally, for a function u=u(t,s)
+defined on the domain 0 ≤s <∞,0≤t≤Tand describing the market value of a stock
+option with the stock (asset) price s, theBlack–Scholes PDE can be written (using the
+physicist notation: ∂zu=∂u/∂z) as a diffusion–type equation:
+∂tu=−1
+2(σs)2∂ssu−rs∂su+ru, (1)
+whereσ >0 is the standard deviation, or volatility ofs,ris the short–term prevailing
+continuously–compounded risk–free interest rate, and T >0 is the time to maturity of the
+stock option. In this formulation it is assumed that the underlying (typically the stock)
+follows a geometric Brownian motion with ‘drift’ µand volatility σ, given by the stochastic
+differential equation (SDE) [3]
+ds(t) =µs(t)dt+σs(t)dW(t), (2)
+whereWis the standard Wiener process. The Black-Scholes PDE (1) is usually derived from
+SDEs describing the geometric Brownian motion (2), with the stock-price solution given by:
+s(t) =s(0)e(µ−1
+2σ2)t+σW(t).
+In mathematical finance, derivation is usually performed us ing Itˆ o lemma [4] (assuming that
+the underlying asset obeys the Itˆ o SDE), while in physics it is performed using Stratonovich
+interpretation [5, 6] (assuming that the underlying asset o beys the Stratonovich SDE [8]).
+The Black-Sholes PDE (1) can be applied to a number of one-dim ensional models of
+interpretations of prices given to u, e.g., puts or calls, and to s, e.g., stocks or futures,
+dividends, etc. The most important examples are European ca ll and put options, defined by:
+uCall(s,t) =sN(d1)e−Tδ−kN(d2)e−rT, (3)
+uPut(s,t) =kN(−d2)e−rT−sN(−d1)e−Tδ, (4)
+N(λ) =1
+2/parenleftbigg
+1+erf/parenleftbiggλ√
+2/parenrightbigg/parenrightbigg
+,
+d1=ln/parenleftbigs
+k/parenrightbig
++T/parenleftBig
+r−δ+σ2
+2/parenrightBig
+σ√
+T,d2=ln/parenleftbigs
+k/parenrightbig
++T/parenleftBig
+r−δ−σ2
+2/parenrightBig
+σ√
+T,
+where erf(λ) is the (real-valued) error function, kdenotes the strike price and δrepresents the
+dividend yield. In addition, for each of the call and put opti ons, there are five Greeks (see,
+e.g. [9, 10]), or sensitivities, which are partial derivati ves of the option-price with respect
+2Figure 1: Fitting the Black–Scholes call option with β(w)-adaptive PDF of the shock-wave
+NLS-solution (10).
+to stock price (Delta), interest rate (Rho), volatility (Ve ga), elapsed time since entering into
+the option (Theta), and the second partial derivative of the option-price with respect to the
+stock price (Gamma).
+Using the standard Kolmogorov probability approach, instead of the market value of an
+option given by the Black–Scholes equation (1), we could con sider the corresponding prob-
+ability density function (PDF) given by the backward Fokker –Planck equation (see [6, 7]).
+Alternatively, we can obtain the same PDF (for the market val ue of a stock option), using the
+quantum–probability formalism [11, 12], as a solution to a time–dependent linear or nonlinear
+Schr¨ odinger equation for the evolution of the complex–valued wave ψ−function for which the
+absolute square, |ψ|2,is the PDF. The adaptive nonlinear Schr¨ odinger (NLS) equat ion was
+recently used in [10] as an approach to option price modellin g, as briefly reviewed in this
+section. The new model, philosophically founded on adaptiv e markets hypothesis [13, 14]
+and Elliott wave market theory [15, 16], as well as my own rece nt work on quantum congition
+[17, 18], describes adaptively controlled Brownian market behavior. This nonlinear approach
+to option price modelling is reviewed in the next section. It s important limiting case with
+low interest-rate reduces to the linear Schr¨ odinger equat ion. This linear approach to option
+price modelling is elaborated in the subsequent section.
+3Figure 2: Fitting the Black–Scholes put option with β(w)–adaptive PDF of the shock-wave
+NLSψ2(s,t) solution (10). Notice the kink near s= 100.
+2 Nonlinear adaptive wave model for general option pricing
+2.1 Adaptive NLS model
+The adaptive, wave–form, nonlinear and stochastic option– pricing model with stock price
+s,volatilityσand interest rate ris formally defined as a complex-valued, focusing (1+1)–
+NLS equation, defining the time-dependent option–price wave function ψ=ψ(s,t), whose
+absolute square |ψ(s,t)|2represents the probability density function (PDF) for the o ption
+price in terms of the stock price and time. In natural quantum units, this NLS equation
+reads:
+i∂tψ=−1
+2σ∂ssψ−β|ψ|2ψ,(i =√
+−1), (5)
+whereβ=β(r,w) denotes the adaptive market-heat potential (see [19]), so the termV(ψ) =
+−β|ψ|2represents the ψ−dependent potential field. In the simplest nonadaptive scen arioβ
+is equal to the interest rate r, while in the adaptive case it depends on the set of adjustabl e
+synaptic weights {wi
+j}as:
+β(r,w) =rn/summationdisplay
+i=1wi
+1erf/parenleftbiggwi
+2s
+wi
+3/parenrightbigg
+. (6)
+Physically, the NLS equation (5) describes a nonlinear wave (e.g. in Bose-Einstein conden-
+sates) defined by the complex-valued wave function ψ(s,t) of real space and time parameters.
+In the present context, the space-like variable sdenotes the stock (asset) price.
+4The NLS equation (5) has been exactly solved using the power s eries expansion method
+[20, 21] of Jacobi elliptic functions [22]. Consider the ψ−function describing a single plane
+wave, with the wave number kand circular frequency ω:
+ψ(s,t) =φ(ξ)ei(ks−ωt),withξ=s−σktandφ(ξ)∈R. (7)
+Its substitution into the NLS equation (5) gives the nonline ar oscillator ODE:
+φ′′(ξ)+[ω−1
+2σk2]φ(ξ)+βφ3(ξ) = 0. (8)
+We can seek a solution φ(ξ) for (8) as a linear function [21]
+φ(ξ) =a0+a1sn(ξ),
+where sn(s) = sn(s,m) are Jacobi elliptic sine functions with elliptic modulus m∈[0,1], such
+that sn(s,0) = sin(s) and sn(s,1) = tanh(s). The solution of (8) was calculated in [10] to be
+φ(ξ) =±m/radicalbigg−σ
+βsn(ξ),form∈[0,1]; and
+φ(ξ) =±/radicalbigg−σ
+βtanh(ξ),form= 1.
+This gives the exact periodic solution of (5) as [10]
+ψ1(s,t) =±m/radicalbigg−σ
+β(w)sn(s−σkt)ei[ks−1
+2σt(1+m2+k2)],form∈[0,1); (9)
+ψ2(s,t) =±/radicalbigg−σ
+β(w)tanh(s−σkt)ei[ks−1
+2σt(2+k2)], form= 1,(10)
+where (9) defines the general solution, while (10) defines the envelope shock-wave1(or, ‘dark
+soliton’) solution of the NLS equation (5).
+Alternatively, if we seek a solution φ(ξ) as a linear function of Jacobi elliptic cosine
+functions, such that cn( s,0) = cos(s) and cn(s,1) = sech(s),2
+φ(ξ) =a0+a1cn(ξ),
+1A shock wave is a type of fast-propagating nonlinear disturb ance that carries energy and can propagate
+through a medium (or, field). It is characterized by an abrupt , nearly discontinuous change in the character-
+istics of the medium. The energy of a shock wave dissipates re latively quickly with distance and its entropy
+increases. On the other hand, a soliton is a self-reinforcin g nonlinear solitary wave packet that maintains its
+shape while it travels at constant speed. It is caused by a can celation of nonlinear and dispersive effects in
+the medium (or, field).
+2A closely related solution of an anharmonic oscillator ODE:
+φ′′(s)+φ(s)+φ3(s) = 0
+5then we get [10]
+ψ3(s,t) =±m/radicalbiggσ
+β(w)cn(s−σkt)ei[ks−1
+2σt(1−2m2+k2)],form∈[0,1); (11)
+ψ4(s,t) =±/radicalbiggσ
+β(w)sech(s−σkt)ei[ks−1
+2σt(k2−1)], form= 1, (12)
+where (11) defines the general solution, while (12) defines th eenvelope solitary-wave (or,
+‘bright soliton’) solution of the NLS equation (5).
+In all four solution expressions (9), (10), (11) and (12), th e adaptive potential β(w) is
+yet to be calculated using either unsupervised Hebbian lear ning, or supervised Levenberg–
+Marquardt algorithm (see, e.g. [23, 24]). In this way, the NL S equation (5) becomes the
+quantum neural network (see [18]). Any kind of numerical analysis can be easily perf ormed
+using above closed-form solutions ψi(s,t) (i= 1,...,4) as initial conditions.
+The adaptive NLS–PDFs of the shock-wave type (10) has been us ed in [10] to fit the
+Black–Scholes call and put options (see Figures 1 and 2). Spe cifically, the adaptive heat
+potential (6) was combined with the spatial part of (10)
+φ(s) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalbiggσ
+βtanh(s−ktσ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2, (13)
+while parameter estimates where obtained using 100 iterati ons of the Levenberg–Marquardt
+algorithm.
+As can be seen from Figure (2) there is a kink near s= 100. This kink, which is a natural
+characteristic of the spatial shock-wave (13), can be smoot hed out (Figure 3) by taking the
+sum of the spatial parts of the shock-wave solution (10) and t he soliton solution (12) as:
+φ(s) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalbiggσ
+β[d1tanh(s−ktσ)+d2sech(s−ktσ)]/vextendsingle/vextendsingle/vextendsingle/vextendsingle2. (14)
+TheadaptiveNLS–basedGreeks(Delta, Rho,Vega, ThetaandG amma)havebeendefined
+in [10], as partial derivatives of the shock-wave solution ( 10).
+is given by
+φ(s) =/radicalbigg
+2m
+1−2mcn/parenleftBigg/radicalbigg
+1+2m
+1−2ms, m/parenrightBigg
+.
+6Figure 3: Smoothing out the kink in the put option fit, by combi ning the shock-wave solution
+with the soliton solution, as defined by (14).
+2.2 Adaptive Manakov system
+Next, for the purpose of including a controlled stochastic volatility3into the adaptive–NLS
+model (5), the full bidirectional quantum neural computati on model [18] for option-price
+forecasting has been formulated in [10] as a self-organized system of two coupled self-focusing
+NLSequations: onedefiningthe option–price wave function ψ=ψ(s,t)andtheotherdefining
+thevolatility wave function σ=σ(s,t):
+Volatility NLS : i ∂tσ=−1
+2∂ssσ−β(r,w)/parenleftbig
+|σ|2+|ψ|2/parenrightbig
+σ, (15)
+Option price NLS : i ∂tψ=−1
+2∂ssψ−β(r,w)/parenleftbig
+|σ|2+|ψ|2/parenrightbig
+ψ. (16)
+In this coupled model, the σ–NLS (15) governs the ( s,t)−evolution of stochastic volatil-
+ity, which plays the role of a nonlinear coefficient in (16); th eψ–NLS (16) defines the
+(s,t)−evolution of option price, which plays the role of a nonlinea r coefficient in (15). The
+purpose of this coupling is to generate a leverage effect , i.e. stock volatility is (negatively)
+correlated to stock returns4(see, e.g. [27]). This bidirectional associative memory effe ctively
+performs quantum neural computation [18], by giving a spati o-temporal and quantum gener-
+alization of Kosko’s BAM family of neural networks [28, 29]. In addition, the shock-wave and
+solitary-wave nature of the coupled NLS equations may descr ibe brain-like effects frequently
+3Controlled stochastic volatility here represents volatil ity evolving in a stochastic manner but within the
+controlled boundaries.
+4The hypothesis that financial leverage can explain the lever age effect was first discussed by F. Black [26].
+7occurring in financial markets: volatility/price propagat ion, reflection and collision of shock
+and solitary waves (see [30]).
+The coupled NLS-system (15)–(16), without an embedded w−learning (i.e., for constant
+β=r– the interest rate), actually defines the well-known Manakov system ,5proven by S.
+Manakov in 1973 [31] to be completely integrable, by the exis tence of infinite number of
+involutive integrals of motion. It admits ‘bright’ and ‘dar k’ soliton solutions. The simplest
+solution of (15)–(16), the so-called Manakov bright 2–soliton , has the form resembling that
+of the sech-solution (12) (see [34, 35, 36, 37, 38, 39, 40]), a nd is formally defined by:
+ψsol(s,t) = 2bcsech(2b(s+4at))e−2i(2a2t+as−2b2t), (17)
+whereψsol(s,t) =/parenleftbiggσ(s,t)
+ψ(s,t)/parenrightbigg
+,c= (c1,c2)Tis a unit vector such that |c1|2+|c2|2= 1. Real-
+valued parameters aandbare some simple functions of ( σ,β,k), which can be determined by
+the Levenberg–Marquardt algorithm. I have argued in [10] th at in some short-time financial
+situations, the adaptation effect on βcan be neglected, so our option-pricing model (15)–(16)
+can be reduced to the Manakov 2–soliton model (17), as depict ed and explained in Figure 4.
+Figure 4: Hypothetical market scenario including sample PD Fs for volatility |σ|2and|ψ|2of
+the Manakov 2–soliton (17). On the left, we observe the ( s,t)−evolution of stochastic volatil-
+ity: we have a collision of two volatility component-solito ns,S1(s,t) andS2(s,t), which join
+together into the resulting soliton S2(s,t), annihilating the S1(s,t) component in the process.
+On the right, we observe the ( s,t)−evolution of option price: we have a collision of two op-
+tion component-solitons, S1(s,t) andS2(s,t), which pass through each other without much
+change, except at the collision point. Due to symmetry of the Manakov system, volatility
+and option price can exchange their roles.
+5Manakov system has been used to describe the interaction bet ween wave packets in dispersive conservative
+media, and also the interaction between orthogonally polar ized components in nonlinear optical fibres (see,
+e.g. [32, 33] and references therein).
+83 Quantum wave model for low interest-rate option pricing
+In the case of a low interest-rate r≪1, we have β(r)≪1, soV(ψ)→0,and therefore
+equation (5) can be approximated by a quantum-like option wave packet. It is defined by
+a continuous superposition of de Broglie’s plane waves , ‘physically’ associated with a free
+quantumparticleofunitmass. Thislinearwavepacket, give nbythetime-dependentcomplex-
+valued wave function ψ=ψ(s,t), is a solution of the linear Schr¨ odinger equation with zero
+potential energy, Hamiltonian operator ˆHand volatility σplaying the role similar to the
+Planck constant. This equation can be written as:
+iσ∂tψ=ˆHψ, where ˆH=−σ2
+2∂ss. (18)
+Thus, we consider the ψ−function describing a single de Broglie’s plane wave, with t he
+wave number k, linear momentum p=σk,wavelength λk= 2π/k,angular frequency ωk=
+σk2/2,and oscillation period Tk= 2π/ωk= 4π/σk2. It is defined by (compare with [41, 42,
+12])
+ψk(s,t) =Aei(ks−ωkt)=Aei(ks−σk2
+2t)=Acos(ks−σk2
+2t)+Aisin(ks−σk2
+2t),(19)
+whereAis the amplitude of the wave, the angle ( ks−ωkt) = (ks−σk2
+2t) represents the phase
+of the wave ψkwith the phase velocity: vk=ωk/k=σk/2.
+The space-time wave function ψ(s,t) that satisfies the linear Schr¨ odinger equation (18)
+can be decomposed (using Fourier’s separation of variables ) into the spatial part φ(s) and
+the temporal part e−iωtas:
+ψ(s,t) =φ(s)e−iωt=φ(s)e−i
+σEt.
+Thespatial part, representing stationary (or,amplitude )wave function ,φ(s) =Aeiks,satisfies
+thelinear harmonic oscillator, which can be formulated in several equivalent forms:
+φ′′+k2φ= 0, φ′′+/parenleftBigp
+σ/parenrightBig2
+φ= 0, φ′′+/parenleftbiggωk
+vk/parenrightbigg2
+φ= 0, φ′′+2Ek
+σ2φ= 0.(20)
+Planck’s energy quantum of the option wave ψkis given by: Ek=σωk=1
+2(σk)2.
+From the plane-wave expressions (19) we have: ψk(s,t) =Aei
+σ(ps−Ekt)−for the wave
+going to the ‘right’ and ψk(s,t) =Ae−i
+σ(ps+Ekt)−for the wave going to the ‘left’.
+The general solution to (18) is formulated as a linear combin ation of de Broglie’s option
+waves (19), comprising the option wave-packet:
+ψ(s,t) =n/summationdisplay
+i=0ciψki(s,t),(withn∈N). (21)
+9Its absolute square, |ψ(s,t)|2,represents the probability density function at a time t.
+Thegroup velocity of an option wave-packet is given by: vg=dωk/dk.It is related to
+the phase velocity vkof a plane wave as: vg=vk−λkdvk/dλk.Closely related is the center
+of the option wave-packet (the point of maximum amplitude), given by:s=tdωk/dk.
+The following quantum-motivated assertions can be stated:
+1. Volatility σhas dimension of financial action , orenergy×time.
+2. Thetotal energy Eof an option wave-packet is (in the case of similar planewave s) given
+byPlanck’ssuperpositionoftheenergies Ekofnindividualwaves: E=nσωk=n
+2(σk)2,
+whereL=nσdenotes the angular momentum of the option wave-packet, representing
+the shift between its growth and decay, and vice versa.
+3. The average energy /an}bracketle{tE/an}bracketri}htof an option wave-packet is given by Boltzmann’s partition
+function:
+/an}bracketle{tE/an}bracketri}ht=/summationtext∞
+n=0nEke−nEk
+bT
+/summationtext∞
+n=0e−nEk
+bT=Ek
+eEk
+bT−1,
+wherebis the Boltzmann-like kinetic constant and Tis the market temperature.
+4. The energy form of the Schr¨ odinger equation (18) reads: Eψ= iσ∂tψ.
+5. The eigenvalue equation for the Hamiltonian operator ˆHis thestationary Schr¨ odinger
+equation:
+ˆHφ(s) =Eφ(s),orEφ(s) =−σ2
+2∂ssφ(s),
+which is just another form of the harmonic oscillator (20). I t has oscillatory solutions
+of the form:
+φE(s) =c1ei
+σ√2Eks+c2e−i
+σ√2Eks,
+calledenergy eigen-states with energies Ekand denoted by: ˆHφE(s) =EkφE(s).
+The Black–Scholes put and call options have been fitted with t he quantum PDFs (see
+Figures 5 and 6) given by the absolute square of (21) with n= 7 andn= 3, respectively.
+Using supervised Levenberg–Marquardt algorithm and Mathematica 7, the following coeffi-
+cients were obtained for the Black–Scholes put option:
+σ∗=−0.0031891, t∗=−0.0031891, k1= 2.62771, k2= 2.62777, k3= 2.65402,
+k4= 2.61118, k5= 2.64104, k6= 2.54737, k7= 2.62778, c1= 1.26632, c2= 1.26517,
+c3= 2.74379, c4= 1.35495, c5= 1.59586, c6= 0.263832, c7= 1.26779,
+10Figure 5: Fitting the Black–Scholes put option with the quan tum PDF given by the absolute
+square of (21) with n= 7.
+withσBS=−94.0705σ∗, tBS=−31.3568t∗.
+Using the same algorithm, the following coefficients were obt ained for the Black–Scholes call
+option:
+σ∗=−11.9245, t∗=−11.9245, k1= 0.851858, k2= 0.832409,
+k3= 0.872061, c1= 2.9004, c2= 2.72592, c3= 2.93291,
+withσBS−0.0251583σ∗, t=−0.00838609t∗.
+Now, given some initial option wave function, ψ(s,0) =ψ0(s),a solution to the initial-
+value problem for the linear Schr¨ odinger equation (18) is, in terms of the pair of Fourier
+transforms ( F,F−1),given by (see [42])
+ψ(s,t) =F−1/bracketleftbig
+e−iωtF(ψ0)/bracketrightbig
+=F−1/bracketleftbigg
+e−iσk2
+2tF(ψ0)/bracketrightbigg
+. (22)
+For example (see [42]), suppose we have an initial option wav e-function at time t= 0
+given by the complex-valued Gaussian function:
+ψ(s,0) = e−as2/2eiσks,
+whereais the width of the Gaussian, while pis the average momentum of the wave. Its
+11Figure 6: Fitting the Black–Scholes call option with the qua ntum PDF given by the absolute
+square of (21) with n= 3. Note that fit is good in the realistic stock region: s∈[75,140].
+Fourier transform, ˆψ0(k) =F[ψ(s,0)],is given by
+ˆψ0(k) =e−(k−p)2
+2a√a.
+The solution at time tof the initial value problem is given by
+ψ(s,t) =1√
+2πa/integraldisplay+∞
+−∞ei(ks−σk2
+2t)e−a(k−p)2
+2adk,
+which, after some algebra becomes
+ψ(s,t) =exp(−as2−2isp+ip2t
+2(1+iat))
+√1+iat,(withp=σk).
+As a simpler example,6if we have an initial option wave-function given by the real- valued
+6An example of a more general Gaussian wave-packet solution o f (18) is given by:
+ψ(s,t) =/radicalBigg/radicalbig
+a/π
+1+iatexp/parenleftBigg
+−1
+2a(s−s0)2−i
+2p2
+0t+ip0(s−s0)
+1+iat/parenrightBigg
+,
+wheres0,p0are initial stock-price and average momentum, while ais the width of the Gaussian. At time t= 0
+the ‘particle’ is at rest around s= 0, its average momentum p0= 0. The wave function spreads with time
+while its maximum decreases and stays put at the origin. At ti me−tthe wave packet is the complex-conjugate
+of the wave-packet at time t.
+12Gaussian function,
+ψ(s,0) =e−s2/2
+4√π,
+the solution of (18) is given by the complex-valued ψ−function,
+ψ(s,t) =exp(−s2
+2(1+it))
+4√π√1+it.
+From (22) it follows that a stationary option wave-packet is given by:
+φ(s) =1√
+2π/integraldisplay+∞
+−∞ei
+σksˆψ(k)dk,where ˆψ(k) =F[φ(s)].
+As|φ(s)|2is the stationary stock PDF, we can calculate the expectation values of the stock
+and the wave number of the whole option wave-packet, consist ing ofnmeasured plane waves,
+as:
+/an}bracketle{ts/an}bracketri}ht=/integraldisplay+∞
+−∞s|φ(s)|2dsand /an}bracketle{tk/an}bracketri}ht=/integraldisplay+∞
+−∞k|ˆψ(k)|2dk. (23)
+The recordings of nindividual option plane waves (19) will be scattered around the mean
+values (23). The width of the distribution of the recorded s−andk−values are uncertainties
+∆sand ∆k,respectively. They satisfy the Heisenberg-type uncertain ty relation:
+∆s∆k≥n
+2,
+which imply the similar relation for the total option energy and time:
+∆E∆t≥n
+2.
+Finally, Greeks for both put and call options are defined as th e following partial deriva-
+tives of the option ψ−function PDF:
+Delta =∂s|ψ(s,t)|2=
+2i/summationtextn
+j=1cjkjekj(is−1
+2iσkjt)Abs/bracketleftBig/summationtextn
+j=1cjekj(is−1
+2iσkjt)/bracketrightBig
+Abs′/bracketleftBig/summationtextn
+j=1cjekj(is−1
+2iσkjt)/bracketrightBig
+Vega =∂σ|ψ(s,t)|2=
+−it/summationtextn
+j=1cjkj2ekj(is−1
+2iσkjt)Abs/bracketleftBig/summationtextn
+j=1cjekj(is−1
+2iσkjt)/bracketrightBig
+Abs′/bracketleftBig/summationtextn
+j=1cjekj(is−1
+2iσkjt)/bracketrightBig
+Theta =∂t|ψ(s,t)|2=
+−iσ/summationtextn
+j=1cjkj2ekj(is−1
+2iσkjt)Abs/bracketleftBig/summationtextn
+j=1cjekj(is−1
+2iσkjt)/bracketrightBig
+Abs′/bracketleftBig/summationtextn
+j=1cjekj(is−1
+2iσkjt)/bracketrightBig
+13Gamma =∂ss|ψ(s,t)|2=
+−2/summationtextn
+j=1cjkj2ekj(is−1
+2iσkjt)Abs/bracketleftBig/summationtextn
+j=1cjekj(is−1
+2iσkjt)/bracketrightBig
+Abs′/bracketleftBig/summationtextn
+j=1cjekj(is−1
+2iσkjt)/bracketrightBig
+−/parenleftBig/summationtextn
+j=1cjkjekj(is−1
+2iσkjt)/parenrightBig
+2Abs′/bracketleftBig/summationtextn
+j=1cjekj(is−1
+2iσkjt)/bracketrightBig
+2−/parenleftBig/summationtextn
+j=1cjkjekj(is−1
+2iσkjt)/parenrightBig
+2Abs/bracketleftBig/summationtextn
+j=1cjekj(is−1
+2iσkjt)/bracketrightBig
+Abs′′/bracketleftBig/summationtextn
+j=1cjekj(is−1
+2iσkjt)/bracketrightBig
+,
+where Abs denotes the absolute value, while Abs′and Abs′′denote its first and second
+derivatives.
+4 Conclusion
+I have proposed an adaptive–wave alternative to the standar d Black-Scholes option pricing
+model. The new model, philosophically founded on adaptive m arkets hypothesis [13, 14]
+and Elliott wave market theory [15, 16], describes adaptive ly controlled Brownian market
+behavior. Two approaches have been proposed: (i) a nonlinea r one based on the adaptive
+NLS (solved by means of Jacobi elliptic functions) and the ad aptive Manakov system (of
+two coupled NLS equations); (ii) a linear quantum-mechanic al one based on the free-particle
+Schr¨ odinger equation and de Broglie’s plane waves. For the purpose of fitting the Black-
+Scholes data, the Levenberg-Marquardt algorithm was used.
+Thepresentedadaptive andquantumwave modelsarespatio-t emporal dynamical systems
+of much higher complexity [25] then the Black-Scholes model . This makes the new wave
+models harder to analyze, but at the same time, their immense variety is potentially much
+closer to the real financial market complexity, especially a t the time of financial crisis.
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+17
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+arXiv:1001.0616v1  [cond-mat.other]  5 Jan 2010Continuum Mechanics for Quantum Many-Body Systems:
+The Linear Response Regime
+Xianlong Gao
+Department of Physics, Zhejiang Normal University, Jinhua, Zhejiang Province, 321004, China
+Jianmin Tao
+Theoretical Division and Center for Nonlinear Studies,
+Los Alamos National Laboratory, Los Alamos, New Mexico 8754 5
+G. Vignale
+Department of Physics, University of Missouri-Columbia, C olumbia, Missouri 65211
+I. V. Tokatly
+IKERBASQUE, Basque Foundation for Science, E-48011, Bilba o, Spain
+ETSF Scientific Development Centre, Dpto. F´ ısica de Materi ales,
+Universidad del Pa´ ıs Vasco, Centro de F´ ısica de Materiale s CSIC-UPV/EHU-MPC,
+Av. Tolosa 72, E-20018 San Sebasti´ an, Spain and
+Moscow Institute of Electronic Technology, Zelenograd, 124 498 Russia
+(Dated: November 1, 2018)
+Wederiveaclosed equation ofmotion for thecurrentdensity of aninhomogeneous quantummany-
+body system under the assumption that the time-dependent wa ve function can be described as a
+geometric deformation of the ground-state wave function. B y describing the many-body system in
+terms of a single collective field we provide an alternative t o traditional approaches, which emphasize
+one-particle orbitals. We refer to our approach as continuu m mechanics for quantum many-body
+systems. In the linear response regime, the equation of moti on for the displacement field becomes
+a linear fourth-order integro-differential equation, whos e only inputs are the one-particle density
+matrixandthepaircorrelation functionoftheground-stat e. Thecomplexityofthisequationremains
+essentially unchanged as the number of particles increases . We show that our equation of motion is
+a hermitian eigenvalue problem, which admits a complete set of orthonormal eigenfunctions under a
+scalar product that involves the ground-state density. Fur ther, we show that the excitation energies
+derived from this approach satisfy a sum rule which guarante es the exactness of the integrated
+spectral strength. Our formulation becomes exact for syste ms consisting of a single particle, and for
+any many-body system in the high-frequency limit. The theor y is illustrated by explicit calculations
+for simple one- and two-particle systems.
+I. INTRODUCTION
+The dynamics of quantum many-particle systems
+poses a major challenge to computational physicists and
+chemists. In the study of ground-state properties one
+can rely on a variational principle, which enables a vari-
+ety of powerful statistical methods (in addition to exact
+diagonalization) such as the quantum variational Monte
+Carlo method and the diffusion Monte Carlo method.1
+In time-dependent situations, the absence of a practi-
+cal variational principle has greatly hindered the devel-
+opment of equally powerful methods. Yet it is hard to
+overestimate the importance of developing effective tech-
+niques to tackle the quantum dynamical problem. Such
+a technique could allow, for example, to follow in real
+time the evolution of chemical reactions, ionization and
+collision processes.
+One of the most successful computational meth-
+ods developed to date is the time-dependent density
+functional theory (TDDFT), or its more recent ver-
+sion – time-dependent current density functional theory
+(TDCDFT).2In this approach, the interacting electronicsystem is treated as a noninteracting electronic system
+subjected to an effective scalar potential (a vector poten-
+tial in TDCDFT) which is self-consistently determined
+by the electronic density (or by the current density).3,4
+Thus, one avoids the formidable problem of solving the
+time-dependent Schr¨ odinger equation for the many-body
+wave function. Even this simplified problem, however,
+is quite complex, since it involves the determination of
+Ntime-dependent single particle orbitals – one for each
+particle. Furthermore, there are features such as multi-
+particle excitations5and dispersion forces6that are very
+difficult to treat within the conventional approximation
+schemes.
+An alternative approach, which actually dates back
+to the early days of the quantum theory, attempts
+to calculate the collective variables of interest, density
+and current, without appealing to the underlying wave
+function.7–9This approach we call “quantum continuum
+mechanics” (QCM), because in analogy with classical
+theoriesofcontinuousmedia(elasticityandhydrodynam-
+ics), itattempts todescribethe quantum many-bodysys-
+tem without explicit reference to the individual particles2
+of which the system is constituted.10
+Thatsuchadescriptionispossibleisguaranteedbythe
+very same theorems that lie at the foundation ofTDDFT
+and TDCDFT.11,12Indeed, considerasystem ofparticles
+of massmdescribed by the time-dependent hamiltonian
+ˆH(t) =ˆH0+/integraldisplay
+drˆn(r)V1(r,t) (1)
+where
+ˆH0=ˆT+ˆW+ˆV0 (2)
+is the sum of kinetic energy ( ˆT), interaction potential
+energy ( ˆW), and the energy associated with an external
+staticpotentialV0(r),
+ˆV0=/integraldisplay
+drV0(r)ˆn(r), (3)
+where ˆn(r) is the particle density operator. V1(r,t) is an
+external time-dependent potential.
+The exact Heisenberg equations of motion for the den-
+sity and the current density operators, averaged over the
+quantum state, lead to equations of motion for the aver-
+age particle density n(r,t) and the average particle cur-
+rent density j(r,t):
+∂tn(r,t) =−∂µjµ(r,t) (4)
+and
+m∂tjµ(r,t) =−n(r,t)∂µ[V0(r)+V1(r,t)]
+−∂νPµν(r,t), (5)
+where∂tdenotes the partial derivative with respect to
+time and∂νis a short-hand for the derivative with re-
+spect to the cartesian component νof the position vector
+r. Here and in the following we adopt the convention
+that repeated indices are summed over. These equations
+simply express the local conservation of particle number
+(Eq. (4)) and momentum (Eq. (5)). The key quantity on
+the right hand side of Eq. (5) is the stress tensor Pµν(r,t)
+– a symmetric tensor which will be defined in the next
+section as the expectation value of a hermitian operator,
+and whose divergence with respect to one of the indices
+yields the force density arising from internal quantum-
+kinetic and interaction effects.
+Now the Runge-Gross theorem of TDDFT guarantees
+that the stress tensor, like every other observable of the
+system, is a functional of the current density and of the
+initial quantum state. Thus, Eq. (5) is in principle a
+closed equation of motion for j– the only missing piece
+of information being the explicitexpression for Pµνin
+terms of the current density.
+In recent years much effort has been devoted to con-
+structing an approximate QCM13–21and several applica-
+tionshaveappearedin the literature(seeRef. 22forsome
+representative examples). All approximation schemes so
+far have been based on the local density approximationand generalizations thereof. The objective of this pa-
+per is to derive and discuss an approximate expression
+forPµν(r,t), and, more importantly, for the associated
+force density −∂νPµν(r,t), as functionals of the current
+density and the initial state. We will do this in the lin-
+ear response regime , i.e. for systems that start from the
+ground-state of the static hamiltonian ˆH0and perform
+small-amplitude oscillations about it. The external po-
+tentialV1(r,t) will be treated as a small perturbation.
+In this regime the equations of motion (4) and (5) are
+conveniently expressed in terms of the displacement field
+u(r,t), defined by the relation
+j(r,t) =n0(r)∂tu(r,t), (6)
+wheren0(r) is the ground-state density. It is also conve-
+nient to write the density and the stress tensor as sums
+of a large ground-state component and a small time-
+dependent part, in the following manner
+n(r,t) =n0(r)+n1(r,t),
+Pµν(r,t) =Pµν,0(r)+Pµν,1(r,t), (7)
+where the equilibrium components, marked by the sub-
+script 0, satisfy the equilibrium condition
+n0(r)∂µV0(r)+∂νPµν,0(r) = 0. (8)
+Then the two equations (4) and (5) take the form
+n1(r,t) =−∂µ[n0(r)uµ(r,t)], (9)
+and
+mn0(r)∂2
+tuµ(r,t) =−n0(r)∂µV1(r,t)
+−n1(r,t)∂µV0(r)−∂νPµν,1(r,t). (10)
+Our task is to find an expression for the force density
+∂νPµν,1(r,t) as a linear functional of u(r,t). If this can
+be achieved, then the excitation energies of the system
+will be obtained from the frequenciesof the time-periodic
+solutions of Eq. (10) in the absence of external field (i.e.,
+withV1= 0).
+It iseasytoseethat the spatialdependence oftheseso-
+lutions will be proportional to the matrix element of the
+current density operator between the ground-state and
+the excited state in question. This is because, in a many-
+body system with stationary states |ψ0∝an}b∇acket∇i}ht,|ψ1∝an}b∇acket∇i}ht,...,|ψn∝an}b∇acket∇i}ht,...
+(|ψ0∝an}b∇acket∇i}htis the ground-state), and corresponding energies
+E0,E1,...,En..., the n-th linear excitation is described
+by the time-dependent state
+|ψ0∝an}b∇acket∇i}hte−iE0t+ε|ψn∝an}b∇acket∇i}hte−iEnt, (11)
+whereεis an arbitrarily small “mixing parameter”. The
+expectation value of the current density operator in this
+state is
+j(r,t) =ε∝an}b∇acketle{tψ0|ˆj(r)|ψn∝an}b∇acket∇i}hte−i(En−E0)t+c.c.(12)3
+Thus, in principle, almost all the excitation energies
+(En−E0) of the system can be obtained by Fourier-
+analyzing the displacement field – the only exception
+being those excitations that are not connected to the
+ground-state by a finite matrix element of the current-
+density operator.
+In this paper we will introduce an approximate expres-
+sion for the force density
+Fµ,1(r,t)≡ −n1(r,t)∂µV0(r)−∂νPµν,1(r,t),(13)
+which appearson the right hand side of Eq. (10), as a lin-
+earfunctional of u(r,t). The expressionwill be presented
+in terms of the functional
+E[u]≡ ∝an}b∇acketle{tψ0[u]|ˆH0|ψ0[u]∝an}b∇acket∇i}ht, (14)
+which is the energy of the distorted ground-state |ψ0[u]∝an}b∇acket∇i}ht,
+obtained from the undistorted ground-state |ψ0∝an}b∇acket∇i}htby vir-
+tually displacing the volume element located at rto a
+new position r+u(r,t). More precisely, we will show
+that the equation of motion for utakes the form:
+mn0(r)∂2
+tu(r,t) =−n0(r)∇V1(r,t)−δE2[u]
+δu(r,t),
+(15)
+whereE2[u] is the second order term in the expansion
+ofE[u] in powers of u. The functional E2[u] has an
+exact expression in terms of the one-particle density ma-
+trix and the pair correlationfunction of the ground-state ,
+which is a major simplification, since ground-state prop-
+erties, unlike time-dependent properties, are accessible
+to computation by a variety of numerical and analytical
+methods.
+Furthermore, we will show that the kinetic part of the
+force density functional δE2[u]/δuis local, in the sense
+that it depends only on a finite number of spatial deriva-
+tives (up to the fourth) of the displacement field at a
+given position. Thus, our equation of motion reduces to
+a fourth-order differential equation for uwhen interac-
+tion effects are neglected. The inclusion of interaction
+effects leads to the appearance of nonlocal contributions
+to the energy, and the equation of motion becomes a
+fourth-order integro-differential equationforthedisplace-
+ment field. However, the complexity of this equation re-
+mains essentially unchanged as the number of particles
+increases.
+Our equation of motion has two especially appealing
+features: (i) it is exact for one-electron systems at all fre-
+quencies and (ii) it can be physically justified for generic
+many-electron systems at high frequency or, more gener-
+ally, at all frequencies for which a collective description
+of the motion is plausible. Thus the range of frequencies
+for which our approximation makes sense is expected to
+be wider in strongly correlated systems than in weakly
+correlated ones.
+We discuss several qualitative features of our equation
+(uniform electron gas limit, harmonic potential theorem)and present its solution in simple one- and two-electron
+models, where the results can be checked against exact
+calculations. The results are encouraging. Although we
+are not able to resolve all the different excitation ener-
+gies of the models under study, we find that groups of ex-
+citation characterized by similar displacement fields are
+represented by a single mode of an average frequency,
+in such a way that the spectral strength of this mode
+equals the sum of the spectral strengths of all the excita-
+tions in the group. In this sense our approximation can
+be viewed asa(considerable)refinement and extensionof
+the traditional single-mode approximation for the homo-
+geneouselectrongastostronglyinhomogeneousquantum
+systems. In spite of the somewhat limited range of valid-
+ity of the present treatment (the linear response regime),
+we feel that this is an important first step in a direc-
+tion that might eventually lead to the construction of
+useful force density functionals for far-from-equilibrium
+processes.
+This paper is organized as follows. In Section II we
+presentacompletederivationofthelinearizedequationof
+motion for the displacement field. We begin by deriving
+a formally exact expression for the force density (Section
+II A), on which we perform the “elastic approximation”
+(Section II B). The expression for the force density in the
+elastic approximation is worked out in sections II C (ki-
+netic part) and II D (potential part). A simplified form
+of the equation of motion, valid for one-dimensional sys-
+tems, is presented in Section II E. Appendixes A through
+C provide supporting material for this part. In Section
+III we discuss the relation between quantum continuum
+mechanicsandtime-dependent currentdensityfunctional
+theory. In section IV we show how the linear equation
+of motion derived in Section II leads to an eigenvalue
+problem for the excitation energies. In Section IV A
+we demonstrate the hermiticity of this eigenvalue prob-
+lem and the positive-definiteness of the eigenvalues. In
+Section IV B we connect the eigenvalue problem to the
+high-frequency limit of the linear response theory. In
+Section IV C we prove that the first moment of the cur-
+rent excitation spectrum obtained from the solution of
+our eigenvalue problem is exact. Appendixes D and E
+contain supporting material for this part. In Section V
+we present a few simple applications of our theory for the
+excitations of (i) a homogeneous electron gas (Section V
+A) (ii) the linear harmonic oscillator and the hydrogen
+atom (Section V B), and (iii) a system of two-electrons
+in a one-dimensional parabolic potential interacting via
+a soft Coulomb potential. The analytic solution of the
+last model in the strong correlation regime is featured in
+Appendix F. Finally, Section VI contains our summary
+and a few speculations about future applications of the
+theory.4
+II. LINEARIZED EQUATION OF MOTION
+A. Derivation of the force density
+In this section we undertake the construction of an ap-
+proximate expression for the force density, Eq. (13), as a
+linear functional of u. The stress tensor Pµν(r,t), whose
+divergence determines the force density, is defined as the
+expectation value of the stress tensor operator ˆPµν(r) in
+the evolving quantum state |ψ(t)∝an}b∇acket∇i}ht:
+Pµν(r,t) =∝an}b∇acketle{tψ(t)|ˆPµν(r)|ψ(t)∝an}b∇acket∇i}ht. (16)
+An exact and unambiguous expression for the operator
+ˆPµν(r) in an arbitrary system of coordinates is obtained
+by considering the universal many-body Hamiltonian
+ˆHu=ˆT+ˆW (17)
+(external potential notincluded) in the presence of a
+“metric tensor field” gµν(r). As is well known23, the
+metric tensor gµν(r) allows us to express the length dsof
+an infinitesimal displacement from rtor+drin terms
+the corresponding increments of the coordinates drµ:
+ds2=gµν(r)drµdrν. (18)
+In ordinary Euclidean space and in cartesian coordi-
+natesgµν(r) =δµν, independent of position. In gen-
+eral,however,anon-Euclideanspaceischaracterizedbya
+position-dependent, symmetric gµν(r). A non-Euclidean
+metric can also be generated by a change of coordinates
+in an Euclidean space, as we will see shortly. As an im-
+portant technical point we also introduce the tensor gµν
+as theinverseofgµν, and we define gas thedeterminant
+ofgµν(sog−1is the determinant of gµν).
+The hamiltonian ˆHuundergoes the following changes
+in the presence of a non-trivial metrics. First, the lapla-
+cian operator ∂µ∂µfor the kinetic energy is replaced by
+1√g∂µ√ggµν∂ν. (19)
+Second, the Euclidean distance between two points
+(which controls the interaction energy) is replaced by the
+non-Euclideanlength ofthe shortestpath (geodesic) con-
+necting the two points. We denote by ˆHu[g] the Hamil-
+tonian in the presence of the metric field gµν. Then
+the stress tensor operator is defined as the first varia-
+tion ofˆHu[g] under an instantaneous virtual variation of
+the metric tensor gµν(r), i.e.,
+ˆPµν(r)≡2√gδˆHu[g]
+δgµν(r). (20)
+Thefirst-orderchangeintheHamiltonianduetoachange
+δgµνin the metric tensor is given by
+ˆHu[g]→ˆHu[g]+/integraldisplay
+dr/radicalbig
+g(r)
+2ˆPµν(r)δgµν(r).(21)Noticethatthestresstensoroperatordefinedinthisman-
+ner is itself a functional of the metrics. This definition
+is completely analogous to the standard definition of the
+current density operator as the derivative of the Hamil-
+tonian with respect to a vector potential. An explicit
+expression for ˆPµνin Euclidean metrics is reported for
+completeness in Appendix A (see also Refs 18, 19, 24
+and 25). We note that the definition of the quantum me-
+chanical stress tensor via the variational derivative with
+respect to the metric tensor has been also employed in
+Ref. 26.
+We will now focus on the calculation of Pµν,1– the
+correction to Pµνof first order in u. In order to ex-
+pressPµν,1(r,t) and its divergence as functionals of the
+displacement field we resort to Tokatly’s recent formu-
+lation of quantum dynamics in the co-moving reference
+frame.18–21The co-moving frame is an accelerated refer-
+ence frame which, at each point and each time, moves
+with the velocity of the volume element of the fluid at
+that point and that time, so that the density is constant
+and equal to the ground-state density, while the current
+density is zero. The time-dependent transformation from
+the laboratory frame (coordinates r) to the co-moving
+frame (coordinates ξ) is defined by the solution of the
+equation
+∂tr(t) =v(r,t),r(0) =ξ, (22)
+wherev(r,t) =j(r,t)
+n(r,t)is the velocity field. In the
+linear response regime the velocity is approximated as
+j(r,t)/n0(r), wheren0(r) is the ground-state density. In
+this regime we can write
+r(t) =ξ+u(ξ,t) (23)
+whereu(ξ,t) is the (small) displacement of a fluid ele-
+ment for its initial position ξ. Expressing ds2=dr·dr
+in terms of the new coordinates ξand making use of
+Eq. (18) we see that the metric tensor in the co-moving
+frame is given by
+gµν(ξ,t) =∂rα
+∂ξµ∂rα
+∂ξν. (24)
+From Eq. (23), to first order in the displacement field, we
+immediately get
+gµν=δµν+2uµν, (25)
+and
+gµν=δµν−2uµν, (26)
+where
+uµν≡1
+2(∂νuµ+∂µuν) (27)
+is thestrain tensor . Also to first order in uthe determi-
+nant of the metric tensor is easily seen to be
+g≃1+2∇·u, (28)5
+so that, for example, g−1/2≃1−∇·u. In view of these
+relations we will, in the rest of this paper, replace ˆHu[g]
+byˆHu[u], with the understanding that ucompletely de-
+termines the metrics. We also notice that, by virtue of
+Eq. (26), we have
+δˆHu[g]
+δgµν(r)=−1
+2δˆHu[u]
+δuµν(r). (29)
+The main reason for introducing the co-moving refer-
+ence frame is that in this frame we can make a simple
+approximation, which enormously simplifies the task of
+linearizing the stress tensor. This will be discussed in the
+next section. For the time being we proceed in a formally
+exact manner. To begin with, we observe that the gen-
+eral relation between the stress tensor in the lab frame
+and that in the co-moving frame is
+Pµν(r,t) = (∂µξα)(∂νξβ)˜Pαβ(ξ(r,t),t),(30)
+where
+˜Pµν(r,t) =−1/radicalbig
+g(r,t)∝an}b∇acketle{t˜ψ(t)|δˆHu[u]
+δuµν(r)|˜ψ(t)∝an}b∇acket∇i}ht,(31)
+where|˜ψ(t)∝an}b∇acket∇i}htis the quantum state in the co-moving
+frame.44
+After expanding the stress tensor in the co-moving
+frame to first order in the displacement field,
+˜Pµν(ξ,t) =Pµν,0(ξ)+˜Pµν,1(ξ,t),(32)
+it is easy to see that in the lab frame we have
+Pµν,1=˜Pµν,1−u·∇Pµν,0−(∂µuα)Pαν,0−(∂νuα)Pµα,0.
+(33)
+From this we get
+∂νPµν,1=∂ν[˜Pµν,1−(∂µuα)Pαν,0−(∂νuα)Pµα,0]
+−∂ν(u·∇Pµν,0), (34)
+and, after some algebra,
+n1∂µV0+∂νPµν,1=n0u·∇∂µV0
++∂ν[˜Pµν,1+(∇·u)Pµν,0−(∂µuα)Pαν,0−2uανPµα,0]
+(35)
+where we have made use of the equilibrium condition (8)
+and the definition (27) of the strain tensor.
+Itisconvenientatthispointtointroducethefirst-order
+stress force density
+Fµ,1≡ −∂ν[˜Pµν,1+(∇·u)Pµν,0−(∂µuα)Pαν,0−2uανPµα,0]
+(36)
+so that the equation of motion (10) takes the form
+m∂2
+tuµ(r,t)+u·∇∂µV0=Fµ,1(r,t)
+n0(r)−∂µV1(r,t).
+(37)Finally, it is possible to prove (see Appendix B) that
+the first order force density is exactly given by the ex-
+pression
+Fµ,1(r,t) =−∝an}b∇acketle{t˜ψ(t)|δˆHu[u]
+δuµ(r)|˜ψ(t)∝an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle
+1,(38)
+whereδˆHu[u]
+δuµ(r)is its functional derivative calculated with
+respect to a virtual(time-independent) variation of the
+displacement field. The vertical bar/vextendsingle/vextendsingle/vextendsingle
+1mandates that
+we keep only the first-order in upart of the bracketed
+expression.
+With the help of this identity we see that the equation
+of motion for the displacement field takes the form
+m∂2
+tuµ(r,t)+u·∇∂µV0=
+−1
+n0(r)∝an}b∇acketle{t˜ψ(t)|δˆHu[u]
+δuµ(r)|˜ψ(t)∝an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle
+1−∂µV1(r,t).
+(39)
+The same result could have been derived almost imme-
+diately by using the more sophisticated machinery of the
+generally covariant Lagrangian formalism introduced in
+Ref. 20. In fact, Eq. (39) is simply a linerized version of
+the equation of motion for an infinitesimal fluid element,
+Eq. (39) of Ref. 20.
+As a reality check, let us askourselveswhether the sys-
+tem can support excitations in which the displacement
+field is uniform in space: u(r,t) =u(t). Clearly in this
+case the strain vanishes and there is no change in met-
+rics, soδˆHu[u]/δuis null. As a result, after setting the
+external field V1= 0 we get the equation
+m∂2
+tuµ(t)+u(t)·∇∂µV0= 0 (40)
+which has the solution uµ(t)∝cos(ωt+φ) if and only if
+the potential is of the harmonic form V0(r) =1
+2mω2r2.
+This is just a statement of the harmonic potential theo-
+rem27according to which a many-body system in a har-
+monic potential performs a rigid simple harmonicmotion
+with frequency ωimposed by the curvature of the har-
+monic potential. We have now shown that the harmonic
+potential is the only potential with this property.
+B. The elastic approximation
+Equation (39) is formally exact, but it still contains
+the time-dependent state |˜ψ(t)∝an}b∇acket∇i}ht, which of course is not
+known. In spite of the simple behavior of the density
+(constant) and the current density (null), the evolution
+of the many-body wave function in the co-moving frame
+is far from trivial. Nevertheless, a simple and physically
+appealing approximation suggests itself. Namely, we as-
+sume that the wave function ˜ψis time-independent (just
+asthe density), and coincides with the ground-statewave6
+function ofthe laboratoryframe( ψ0)evaluatedat theco-
+ordinates ξof the co-moving frame:
+˜ψ(ξ1,...,ξN,t)≃ψ0(ξ1,...,ξN). (41)
+The physical idea behind this approximation is that
+the time evolution of the wave function in the laboratory
+frame can be approximated as a continuously evolving
+elastic deformation of the ground-state wave function.
+Such a deformation affects all the particles simultane-
+ously and instantaneously. The burden of describing the
+time-evolution of the system is entirely placed on the
+time-dependent geometry (i.e. the time-dependent re-
+lation between ξandr), while the wave function itself
+remains independent of time.
+What is lostin this approximation is the fact that in
+theactualtimeevolutionthesystemwillundergointernal
+relaxation in order to optimize the correlations between
+the particles. In other words, the probability of finding
+the particles in a certain configuration r1,...rNat timet
+isnotstrictly determined by the probability that those
+particleswereinitiallyintheconfiguration ξ1,...,ξNfrom
+whichr1,...rNevolve according to Eqs. (22) and (23).
+However, our approximation should always be valid at
+sufficiently high frequency, i.e., when the evolution of the
+geometry is very fast on the scale of the characteristic
+response times of the system.
+The equation of motion resulting from the elastic ap-
+proximation is also strictly valid (and therefore, not an
+approximation at all) for any one-particle system, be-
+cause in this case the wave function iscompletely deter-
+mined by the displacement field and there is no room
+for internal relaxation. Finally, our equation of motion
+is also strictly valid for non-interacting Bose systems in
+the ground-state(since these systems behave like a single
+particle), and for non-interacting Fermi systems consist-
+ing of at most two particles of opposite spins in the same
+orbital (since these behave like non-interacting Bosons).
+In all other cases – including the apparently simple case
+of a non-interacting many-fermion system – the appro-
+priateness of the elastic approximation must be assessed
+a posteriori and may depend on the objective of the cal-
+culation. In general, we can only say that the elastic
+approximation is expected to work better for collective
+(many-particle) excitations than for single particle ex-
+citations, and better for strongly correlated many-body
+systems(whichexhibit bosonicbehavior)thanforweakly
+correlated systems.
+It is important to appreciate the profound difference
+that exists between the present approximation and an-
+other common approximation which also entails an in-
+stantaneousresponseto atime-dependent field: the adia-
+batic approximation . In the adiabatic approximation one
+assumes that the system remains in the instantaneous
+ground-state of the hamiltonian ˆH(t) – an assumption
+that is justified only if the time evolution is slowon the
+scale of the characteristic response time of the system.
+This is exactly the opposite of the regime of validity of
+the present approximation. The geometrically distortedwave function ˜ψis not at all close to the instantaneous
+ground-stateof ˆH(t). Rather,itistheground-stateofthe
+“deformed hamiltonian” ˆH0[u] which is obtained from
+the initial-time Hamiltonian ˆH0by a coordinate trans-
+formation – indeed an elastic deformation.
+As anticipated in the foregoing discussion the elastic
+approximation paves the way for a relatively simple cal-
+culation of the complicated expression that appears in
+the second line of Eq. (39). Namely, thanks to the fact
+that˜ψ=ψ0is independent of the displacement field we
+can take the functional derivative of Eq. (38) aftertaking
+the average and we arrive at
+Fµ,1(r,t) =−δEu[u]
+δuµ(r)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+1, (42)
+where
+Eu[u]≡ ∝an}b∇acketle{tψ0|ˆHu[u]|ψ0∝an}b∇acket∇i}ht. (43)
+We further observe that
+u·∇∂µV0(r) =1
+n0(r)δV0[u]
+δuµ(r)/vextendsingle/vextendsingle/vextendsingle
+1, (44)
+where
+V0[u]≡/integraldisplay
+drV0(r+u(r))n0(r). (45)
+is the external potential energy of the distorted ground-
+state.
+Putting all together we arrive at the elegant result:
+m∂2
+tuµ(r,t) =−1
+n0(r)δE2[u]
+δuµ(r)−∂µV1(r,t).
+(46)
+whereE2[u] is the second-order term in the expansion of
+thetotal energy
+E[u]≡Eu[u]+V0[u] (47)
+of the distorted ground state. Equivalently, E[u] can be
+obtained as the expectation value of the original hamil-
+tonianˆH0in the distorted ground state
+ψ0[u](r1,...,rN) =ψ0(ξ1,...,ξN)N/productdisplay
+i=1g−1/4(ξi),(48)
+whereξi=ri−u(ri)andthe lastfactoronthe righthand
+side is for normalization. This proves that E[u]−E[0] is
+a positive definite quantity since E[0] is the ground-state
+energy of ˆH0whileE[u] is the expectation value of ˆH0
+in a state that is not the ground-state.7
+C. Calculation of δE2[u]/δu – kinetic part
+The evaluation of the distorted ground-state energy
+E2[u] is in principle straightforward if the exact one-
+particle and two-particle density matrices of the ground-
+state are known. In this section we focus on the con-
+struction of the kinetic contribution, which, as we will
+show, leads to a localequation of motion, which involvesonly a finite number of derivatives (up to the fourth) of
+the displacement field. For a calculation of the kinetic
+contribution to the elastic energy only the one-particle
+density matrix
+ρ(r,r′)≡ ∝an}b∇acketle{tψ0|ˆΨ†(r)ˆΨ(r′)|ψ0∝an}b∇acket∇i}ht (49)
+is needed.
+The kinetic energy of the distorted state is
+T[u] =1
+2m/integraldisplay
+dr√ggµν∂µ∂′
+ν[g−1/4(r)g−1/4(r′)ρ(r,r′)]r=r′, (50)
+which reduces to the kinetic energy of the ground-state when u= 0. Expanding the above expression to second order
+inuwe arrive, after some laborious algebra (see Appendix C for the der ivation) to the following expression
+T2[u] =/integraldisplay
+dr/braceleftbigg
+2Tµν,0/bracketleftbigg
+uµαuνα−1
+4(∂µuα)(∂νuα)/bracketrightbigg
++n0
+8m(∂µuνν)(∂µuνν)
++n0
+2m[(∂µuνα)(∂µuνα)−(∂µuνµ)(∂νuαα)]/bracerightBig
+, (51)
+where
+Tµν,0=1
+2m/parenleftbig
+∂µ∂′
+ν+∂ν∂′
+µ/parenrightbig
+ρ(r,r′)/vextendsingle/vextendsingle/vextendsingle
+r=r′−1
+4m∇2n0δµν (52)
+is the equilibrium stress tensor. Notice that T2[u] is a local functional of u, i.e. it presents no coupling between
+displacement fields at different positions. Taking the functional der ivative with respect to u(r) we arrive at the
+desired expression for the kinetic force density:
+−δT2[u]
+δuµ=∂α[2Tνµ,0uνα+Tνα,0∂µuν]−1
+4m∂ν∂µ(n0∂ν∇·u)
++1
+4m∂ν/braceleftbig
+2(∇2n0)uνµ+(∂νn0)∂µ∇·u+(∂µn0)∂ν∇·u−2∂µ[(∂αn0)uνα]/bracerightbig
+. (53)
+D. Calculation of δE2[u]/δu – potential part
+To calculate the potential energy functional W[u] we need the two-particle density matrix of the ground-state:
+ρ2(r,r′)≡ ∝an}b∇acketle{tψ0|ˆΨ†(r)Ψ†(r′)ˆΨ(r′)ˆΨ(r)|ψ0∝an}b∇acket∇i}ht. (54)
+For a system of electrons interacting via Coulomb interaction (char ge−e) we have
+W[u] =e2
+2/integraldisplay
+dr/integraldisplay
+dr′ρ2(r,r′)
+|r+u(r)−r′−u(r′)|. (55)
+Expanding to second order in uwe easily obtain
+W2[u] =−1
+2/integraldisplay
+dr/integraldisplay
+dr′[uµ(r)−uµ(r′)]Kµν(r,r′)[uν(r)−uν(r′)] (56)
+where
+Kµν(r,r′) =ρ2(r,r′)∂µ∂′
+νe2
+|r−r′|. (57)
+Finally, taking the functional derivative with respect to
+u(r) we get
+−δW2[u]
+δuµ(r)=/integraldisplay
+dr′Kµν(r,r′)[uν(r)−uν(r′)].(58)Thus, the inclusion of interactions transforms our equa-
+tion of motion into an integro-differential equation. No-8
+tice, however, that the interaction contribution vanishes
+ifu(r) is constant in space, as expected from the trans-
+lational invariance of the interaction.
+E. Equation of motion for one-dimensional systems
+The formulas presented in the preceding subsections
+simplify dramatically in one-dimensional systems, where
+the displacement field has only one component, u(x), the
+strain tensor reduces to the derivative of the displace-
+ment fieldu′(x), and the equilibrium kinetic stress tensor
+reduces to a scalar
+T0(x) =1
+m/bracketleftbigg
+∂x∂x′ρ(x,x′)/vextendsingle/vextendsingle
+x=x′−n′′
+0
+4/bracketrightbigg
+.(59)
+ThenthecombinationonthelastlineofEq.(53)vanishes
+and we are left with the simpler expression
+−δT2[u]
+δuµ= (3T0u′)′−1
+4m(n0u′′)′′(60)
+where the primes denote derivatives with respect to x.
+The complete equation of motion for one dimensional
+systems is thus
+mn0∂2
+tu=−n0uV′′
+0+(3T0u′)′−1
+4m(n0u′′)′′
++/integraldisplay
+dx′K(x,x′)[u(x)−u(x′)]−n0V′
+1,
+(61)
+whereK(x,x′) is given by the one-dimensional version of
+Eq. (57). We will make use of this form of the equation
+of motion in the model applications presented below.
+III. CURRENT-DENSITY FUNCTIONAL
+APPROACH
+Our discussion thus far has not relied on time-
+dependent current density functional theory, except on a
+very abstract level, i.e. as a basis for the statement that
+the stress tensor must be a functional of the current den-
+sity. The formulas presented in the last two subsections
+relied on the knowledge of the exact density matrices
+ρandρ2of the many-body ground-state – two quanti-
+ties that are amenable to treatment by powerful numer-
+ical techniques (e.g. the quantum Monte Carlo method)
+which have little in common with DFT. Before proceed-
+ing, we wish to clarify how the time-dependent CDFT
+can help us in more concrete ways when the exact ρand
+ρ2arenotknown, which is by far the most common case.
+One of the main ideas of TDCDFT is that the current
+and density evolutions of the interacting many-body sys-
+tem can be simulated in a non-interacting many-body
+system subject to an effective time-dependent vector po-
+tential which includes the Hartree electrostatic potentialand dynamical exchange-correlation (xc) effects. This
+non-interacting system is known as the “Kohn-Sham ref-
+erence system” and its ground-state density coincides
+with the exact ground-state density of the interacting
+system, i.e. n0(r). The potential that produces this ex-
+actground-statedensityin the Kohn-Shamreferencesys-
+tem is known as the static Kohn-Sham potential and is
+usually written as
+Vs,0(r) =V0(r)+VH,0(r)+Vxc,0(r),(62)
+whereVH,0andVxc,0are, respectively, theHartreepoten-
+tialandthexcpotentialofthe ground-state. Thesestatic
+potentials should not to be confused with the additional
+dynamical Hartree and xc potentials, which appear when
+the system is not in equilibrium.
+The idea is now to apply our continuum mechanics
+formulation directly to the Kohn-Sham reference sys-
+tem. There is a small technical problem in doing this,
+namely the time-dependent xc vector potential Axc(r,t)
+that acts on the Kohn-Sham system has in general a
+transverse component, which cannot be represented as
+the gradient of a scalar potential. Indeed, a complete
+representation of Arequires that we introduce both an
+electric field Excand a magnetic field Bxc. The inclusion
+of the xc magnetic field does not create any difficulties
+in principle (see Ref. 20), and leads to the appearance
+of a Lorentz-force term in the equation of motion for the
+current. But this Lorentz force term can be safely disre-
+garded in the linear response approximation, because it
+has the form j×Bwhich is of second order in the devi-
+ation from equilibrium. Thus we can take into account
+dynamical xc effects simply by adding the force −eExc
+to the driving force −n0∇V1on the right hand side of
+Eq. (15). All this considered, our equation of motion
+takes the form
+m∂2
+tu(r,t)+(u·∇)∇Vs,0=−1
+n0δTs2[u]
+δu(r)
+−∇[V1(r,t)+VH,1(r,t)]−eExc,1(r,t),(63)
+whereVH,1is the first-order term in the expansion of the
+time-dependent Hartree potential, and Exc,1is the first
+order term in the expansion of Excin powers of u(r).
+The non-interacting kinetic force density −δTs2[u]/δu(r)
+is given by Eq. (53) in which, however, the equilibrium
+kinetic stress tensor Tµν,0is replaced by the correspond-
+ing quantity for the Kohn-Sham reference system, i.e
+Ts
+µν,0=1
+2m/parenleftbig
+∂µ∂′
+ν+∂ν∂′
+µ/parenrightbig/summationdisplay
+ℓψ∗
+ℓ(r)ψℓ(r′)/vextendsingle/vextendsingle/vextendsingle
+r=r′
+−1
+4m∇2n0δµν (64)
+whereψℓ(r) are Kohn-Sham orbitals for the ground-state
+and the sum over ℓruns over the occupied orbitals.
+Assuming that the Kohn-Sham ground-state has been
+obtained by one of the available approximations for Vs,0,
+the remaining problem is to find a suitable approximate9
+expression for Exc,1. The “natural” approximation, in
+the presentcontext, would be the high-frequencyapprox-
+imation, which expresses Exc,1as the functional deriva-
+tive of the exchange-correlation energy functional with
+respect to u. In practice, since the latter is not known,
+one has to rely on more or less uncontrolled approxima-
+tions, such as the high-frequency limit of the local den-
+sity approximation proposed in Refs.19,28–30– see Eq.
+(117) of Ref. 19 (a general discussion of these approxi-
+mation can be found in Ref. 31). This approximation is
+local both in space and time and is obtained by applying
+an instantaneous geometric deformation to the homoge-
+neous electron gas. In the second respect it is perfectly
+consistent with our elastic approximation for the nonin-
+teracting kinetic force, but we must keep in mind that
+the latter is fully nonlocal.
+Unfortunately, the local deformation approximation
+for the xc potential suffers, like all electron-gas based
+approximations, from a serious defect: it fails to cancel
+the unphysical self-interaction that is contained in the
+Hartree term. This makes it unsuitable for the treat-
+ment of strongly correlated system, where the spurious
+self-interaction energy can be very large. More accu-
+rate approximations32do not suffer from this defect, but
+are more difficult to implement. Furthermore, such ap-
+proximations would do little (apart from fixing the self-
+interaction problem) to capture the physics of strongly
+correlated electrons. Alternatively, one could use func-
+tionals explicitly designed for electronic systems in the
+strong coupling limit, such as the ones developed by
+Seidl, Perdew and Kurth33and, more recently by Seidl,
+Gori-Giorgi, and Savin34.
+IV. THE CALCULATION OF EXCITATION
+ENERGIES
+A. The eigenvalue problem
+An immediate application of our equation of motion
+is the calculation of excitation energies.3,4To this end
+we turn off the external potential V1and consider the
+homogeneous equation
+mn0(r)∂2
+tu(r,t) =−δE2[u]
+δu(r,t). (65)
+Fourier-transforming with respect to time and carrying
+out the indicated expansionof the energy to second order
+inuwe get
+mω2n0(r)uµ(r,ω) =/integraldisplay
+dr′δ2E[u]
+δuµ(r)δuν(r′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+u=0uν(r′,ω).
+(66)
+Although this expression is not the most useful in prac-
+tice, it does bring forth some important features of the
+problem. First, because the kernel of the integral equa-
+tionisasymmetricsecondfunctionalderivative,weareinthe presence of an essentially hermitian eigenvalue prob-
+lem. More precisely, the problem is hermitian with re-
+spect to a scalar product with weight n0(r), i.e. defined
+as
+(f,g)≡/integraldisplay
+drn0(r)f(r)g(r), (67)
+wherefandgare arbitrary functions. This can be seen
+by rewriting Eq. (66) as an equation for ˜u≡/radicalbig
+n0(r)u(r)
+and noting that this equation has the form of a standard
+eigenvalue problem with a symmetric kernel:
+/integraldisplay
+dr′1/radicalbig
+mn0(r)δ2E[u]
+δuµ(r)δuν(r′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+u=01/radicalbig
+mn0(r′)˜uν(r′,ω)
+=ω2˜uµ(r,ω). (68)
+This means that all the eigenvalues will be real, and
+eigenfunctions ˜ucorresponding to different eigenvalues
+are orthogonal with respect to the ordinary scalar prod-
+uct. It follows that the original eigenfunctions uare or-
+thogonal with respect to the scalar product defined by
+Eq. (67). Second, the kernel of the integral equation is
+positive definite, because E[u] has an absolute minimum
+atu= 0, which corresponds to the ground-state energy.
+For this reason, it is guaranteed that all the eigenvalues
+are positive. The square roots of these eigenvalues are
+the approximate excitation energies of the system, start-
+ing from the ground-state. The eigenfunctions also have
+a simple interpretation as approximate matrix elements
+of the current density operator between the ground-state
+and the excited state under consideration, divided by the
+ground-state density. We will show this more clearly in
+the next section.
+B. Derivation from linear response theory
+Additional insight into the significance of the eigen-
+value problem for the excitation energies is obtained by
+deriving the equation of motion directly from the lin-
+ear response of the current density to an external vector
+potential in the high-frequency regime. To this end we
+write
+jµ(r,ω) =/integraldisplay
+dr′χµν(r,r′,ω)Aν,1(r′,ω),(69)
+whereχµν(r,r′,ω) is the current-current response func-
+tion, and notice that, at high frequency, this function has
+the well-known expansion35
+χµν(r,r′,ω) =n0(r)
+mδ(r−r′)δµν+Mµν(r,r′)
+m2ω2,(70)
+where the first term (diamagnetic term) is frequency-
+independent and
+Mµν(r,r′)≡ −m2∝an}b∇acketle{tψ0|[[ˆH0,ˆjµ(r)],ˆjν(r′)]|ψ0∝an}b∇acket∇i}ht,(71)10
+is the first spectral moment of the current-current re-
+sponse function. ([ ˆA,ˆB] denotes the commutator of
+two operators ˆAandˆB, andψ0is the undeformed
+ground-state wave function.) Now, replacing j(r,ω) =
+−iωn0(r)u(r,ω) andA1(r,ω) =∇V1(r,ω)
+iω, and solving
+for∇V1to leading order in 1 /ω2we obtain
+∂µV1=mω2uµ−1
+n0/integraldisplay
+dr′Mµν(r,r′)uν(r′).(72)
+This is equivalent to our equation of motion (46) if and
+only if
+Mµν(r,r′) =δ2E[u]
+δuµ(r)δuν(r′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+u=0.(73)
+To show that this is indeed the case we observe that the
+deformed ground-statewavefunction can be expanded as
+ψ0[u] =ψ0+ψ(1)
+0+ψ(2)
+0+... (74)
+whereψ0is the undeformed ground-state wave function
+andψ(1)
+0,ψ(2)
+0are corrections of first and second order
+inurespectively. The various corrections are not mu-
+tually independent. If ψ0[u] is normalized to a constant
+independent of u, then we must have
+∝an}b∇acketle{tψ0|ψ(2)
+0∝an}b∇acket∇i}ht+∝an}b∇acketle{tψ(2)
+0|ψ0∝an}b∇acket∇i}ht=−∝an}b∇acketle{tψ(1)
+0|ψ(1)
+0∝an}b∇acket∇i}ht.(75)
+Taking this into account it is easy to verify that the sec-
+ond order correction to the energy is
+E2[u] =∝an}b∇acketle{tψ(1)
+0|ˆH0−E0|ψ(1)
+0∝an}b∇acket∇i}ht, (76)
+whereE0is the ground-state energy. Finally, we observe
+that the first-order correction to the ground-state wave
+function is given by
+|ψ(1)
+0∝an}b∇acket∇i}ht=−im/integraldisplay
+drˆj(r)·u(r)|ψ0∝an}b∇acket∇i}ht (77)
+where we have used the fact that the momentum density
+operatormˆj(r) is the generator of a local translation of
+all the particles in an infinitesimal volume located at r.
+Thus, the operatoron the right hand side ofEq. (77) per-
+forms different translations by vectors u(r) at different
+points in space, i.e. precisely deforms the ground-state
+according to the displacement field u(r). Substituting
+the above expression for |ψ(1)
+0∝an}b∇acket∇i}htinto Eq. (76) for E2[u]
+one can easily verify that
+E2[u] =1
+2/integraldisplay
+dr/integraldisplay
+dr′uµ(r)Mµν(r,r′)uν(r′),(78)
+withMµν(r,r′) given by Eq. (71). This establishes the
+validity of Eq. (73).C. Eigenfunctions and sum rule
+The spectral representation of the kernel of our equa-
+tion of motion gives additional insight into the nature of
+our approximation and shows clearly where things can
+go wrong. Namely we can write
+Mµν(r,r′) =m2/summationdisplay
+nωn0{[jµ]0n(r)[jν]n0(r′)
++ [jν]0n(r′)[jµ]n0(r)}, (79)
+whereωn0are the exact excitation energies of the system
+from the ground-state (0) to the n-th excited state, and
+[jµ]n0(r)≡ ∝an}b∇acketle{tn|ˆjµ(r)|0∝an}b∇acket∇i}htarethematrixelementsofthecur-
+rent density operator between the corresponding states.
+An exact linear equation for the excitations would have
+to give±ωn0as excitation energies and n−1
+0(r)[jµ]0n(r)
+(=n−1
+0(r)[jµ]∗
+n0(r)) as the corresponding eigenfunctions.
+In general this will not be the case. However, for the
+special case of a one-particle system, in the absence of a
+magnetic field, we can showrathereasily that [ jµ]0n(r) =
+−[jµ]n0(r)andfurthermorethat n−1
+0(r)[jµ]0n(r)isindeed
+an eigenfunction of the operator n−1
+0(r)Mµν(r,r′) with
+eigenvaluemω2
+n0. This follows from the orthonormality
+relation
+2m√ωn0ωk0/integraldisplay
+dr[jν]0n(r)[jν]k0(r)
+n0(r)=δnk,(80)
+which is valid for one-particle systems (in the absence of
+a magnetic field) and is proved in Appendix D. Then the
+eigenvalues of the equation of motion (72) with V1= 0
+areω=±ωn0as they should be. This result is per-
+fectly consistent with our previous observation that the
+“elastic approximation” is not an approximation at all
+when it comes to one-electron system, due to the lack of
+retardation effects in such systems.
+In general, in a many-particle system the matrix ele-
+mentsofthecurrentdensityoperatorbetweentheground
+state and different excited states are not necessarily or-
+thogonal. Indeed, we will see that two completely dif-
+ferent excited states can produce the same eigenfunction
+for the displacement field, up to a proportionality con-
+stant. The reason why this can happen is that the ex-
+actequation of motion for the displacement field of a
+many-body system is not an eigenvalue problem (even
+though it is linear) due to the frequency dependence of
+the kernel. As a result, the normalization of the solu-
+tions becomes relevant: two “eigenfunctions” that differ
+by a mere proportionality constant can result in different
+excitation energies when the kernel of the linear equa-
+tion is itself energy-dependent. In such cases, the elastic
+approximation will fail to resolve the different excitation
+energies, replacing them by a single excitation energy at
+an “average” value.
+In spite of this shortcoming, an exact sum rule can
+be established, which relates the exact eigenvalues ωλ
+of the elastic eigenvalue problem to the exact excitation11
+energiesωn0. The sum rule reads:
+ω2
+λ=/summationdisplay
+nfλ
+nω2
+λ0 (81)
+where the “oscillator strengths”
+fλ
+n=2m/vextendsingle/vextendsingle/integraltext
+drj0n(r)·uλ(r)/vextendsingle/vextendsingle2
+ωn0, (82)
+and where uλ(r) is the solution of the elastic eigen-
+value problem normalized with respect to the scalar
+product (67), ωλis its eigenvalue, and the sum runs
+over all the exact eigenfunctions. Further, the oscillator
+strengths satisfy the sum rule
+/summationdisplay
+nfλ
+n= 1 (83)
+for allλ. The proof of these results is presented in Ap-
+pendix E.
+From this vantage point we see that the elastic ap-
+proximation is the extension of the well-known collective
+approximation40of the homogeneous electron gas to in-
+homogeneous systems. Each eigenvalue ωλof the elastic
+equation of motion is a weighted average of exact exci-
+tation energies, with a weight controlled by the overlap
+of the exact current matrix element with the eigenmode
+uλ(r). Sinceuλ(r) form an orthogonal basis in the space
+of displacements, we can say that ωλrepresents the av-
+erage energy of exact excitations in the “direction” λ.
+Thus, the full excitation spectrum is replaced by a set of
+spectral lines, one for each orthogonal direction in dis-
+placement space, and each one carrying the entire and
+exact spectral weight for that particular direction.
+V. MODEL APPLICATIONS
+For orientation we now examine the application of our
+theory to a few simple models.
+A. Homogeneous electron gas
+In a homogeneous electron gas the ground-state den-
+sityn0=nis independent of position. The equilibrium
+kinetic stress tensor has a constant value
+Tµν,0=2
+3nt(n)δµν, (84)
+wheret(n) is the kinetic energy per particle. The two
+particle density matrix ρ2(r,r′) is a function of |r−r′|.
+In such a homogeneous system the displacement eigen-
+functions are simply plane waves
+u(r,ω) =˜u(q,ω)ei(q·r−ωt)(85)characterized by a wave vector q. Of these there are
+two kinds: longitudinal, in which ˜uis parallel to q, and
+transverse, in which ˜uis perpendicular to q. The expres-
+sion (53) for the kinetic force density reduces to
+−δT2[u]
+δu=−2
+3nt(n)[2q(q·˜u)+q2˜u]+nq2
+4mq(q·˜u).
+(86)
+Theforcedensityfrompotentialenergy, Eq.(58), isgiven
+by
+−δW2[u]
+δuµ= [Kµν(0)−Kµν(q)]˜uν,(87)
+where
+Kµν(q) =−/integraldisplaydq′
+(2π)3ρ2(q−q′)q′
+µq′
+νv(q′),(88)
+wherev(q) = 4πe2/q2is the Fourier transform of the
+Coulomb potential in three dimensions, and ρ2(q) – the
+Fourier transform of ρ2(r−r′) – is related to the static
+structure factor S(q) in the following manner:35
+ρ2(q) =n[S(q)−1]. (89)
+Thus, we get
+−δW2[u]
+δu=−n/integraldisplay
+dq′[S(q−q′)−S(q′)]v(q′)q′[q′·˜u(q)].
+(90)
+Finally, in order to take into account the neutralizing
+background of positive charge (required for the stability
+of the electron gas), we add the external potential
+V0(r) =m
+2ω2
+p(r·ˆq)2, (91)
+whereω2
+p= 4πne2/mis the square of the plasmon fre-
+quency and ˆqis the unit vector in the direction of q.
+Notice that this potential is assumed to vary only in the
+direction of ˆq, because it is only in this direction that the
+displacement generatesboundarycharges: the system re-
+mains perfectly homogeneous in the direction perpendic-
+ular toq.45The corresponding force in the equation of
+motion (15) is
+u·∇∂µV0=mω2
+p(ˆq·u)ˆqµ. (92)12
+Putting everything together, the equation of motion takes the fo rm
+mω2˜u=2
+3t(n)/bracketleftbig
+2q(q·˜u)+q2˜u/bracketrightbig
++q2
+4mq(q·˜u)+mω2
+p(ˆq·˜u)ˆq+mω2
+p
+n/integraldisplaydq′
+(2π)3[S(q−q′)−S(q′)]ˆq′[ˆq′·˜u(q)],
+(93)
+This can be further decoupled into longitudinal and transverse com ponents denoted by ˜ uLand ˜uTrespectively. The
+corresponding eigenvalues are
+ω2
+L(q) =ω2
+p+2t(n)q2
+m+q4
+4m2+ω2
+p
+n/integraldisplaydq′
+(2π)3(ˆq·ˆq′)2[S(q−q′)−S(q′)], (94)
+and
+ω2
+T(q) =2
+3t(n)q2
+m+ω2
+p
+2n/integraldisplaydq′
+(2π)3(ˆq׈q′)2[S(q−q′)−S(q′)]. (95)
+Of course, this is exactly what one would have obtained
+by assuming that the spectrum of the current-current
+response function consist of a single δ-function peak at
+ωLorωTand requiring satisfaction of the first moment
+sum rule (see Ref. 35, Eq. 3.191 and Ref. 39).
+In Fig. (1) we plot the excitation spectrum of the ho-
+mogeneous electron gas calculated from Eqs. (94) and
+(95). We have used the static structure factor calculated
+in Ref. 36 by the quantum Monte Carlo method, and the
+kinetic energy has been computed from the parametrized
+correlation energy of Appendix B of Ref. 36, using the
+virial theorem. In the longitudinal channel, the exact
+spectrum is dominated at small qby the plasmon and
+at largeqby free particle excitations (energy q2/2m).
+There are also also electron-hole pair excitations at lower
+qandω, as well as multiple electron-hole pair excitations
+all over the plane. The elastic approximation replaces
+this complex spectrum by a single branch of longitudinal
+excitations which has the correct spectral moment. In
+particular, we get the correct dispersion of the plasmon
+at smallqand the correct free particle behavior at large
+q. In the transverse channel the plasmon and the high- q
+free particle excitations are absent. The exact transverse
+spectrum consists primarily of low-energy electron-hole
+pair excitations. The current vector, k+q/2, where k
+andk+qare the momenta of the hole and the elec-
+tron, respectively, is essentially perpendicular to qwhen
+kandk+qboth lie near the Fermi surface. On the
+other hand, high-energy excitations, with energy q2/2m
+are essentially longitudinal because the current vector is
+essentially parallel to qwhenqis much larger than the
+Fermi momentum. This is consistent with the fact that
+the frequency of our transverse collective mode ωT(q)
+grows linearly with qat largeq.B. Linear harmonic oscillator and hydrogen atom
+In order to demonstrate the exactness of the formu-
+lation for one-electron systems let us now consider the
+canonical examples of the one-dimensional harmonic os-
+cillator and the hydrogen atom.
+For a harmonic oscillator of natural frequency ω0, ex-
+ternal potential V0(x) =mω2
+0x2/2, and equilibrium den-
+sityn0(x) =e−x2/ℓ2
+√πℓ, whereℓ≡(mω0)−1/2, the equation
+of motion (61) reduces to
+1
+4u′′′′−xu′′′+(x2−2)u′′+3xu′−ω2−ω2
+0
+ω2
+0u= 0.(96)
+Solving the eigenvalue problem with the boundary con-
+dition ofn1/2
+0(x)u(x)→0 as|x| → ∞, we obtain the
+exact excitation spectra ωn=±nΩ, wheren= 1,2,...
+The corresponding eigenfunctions are
+un(x)∝Hn−1(x) (97)
+which are mutually orthogonal with respect to the scalar
+product (67). These are indeed proportional to the ma-
+trix elements of the current density operator between the
+ground-state and the n-th excited state.
+A similar calculation can be done for hydrogen-like
+atoms of atomic number Z. Focusing for simplicity on
+excitations of spherical symmetry, we introduce a radial
+displacementfield ur(r) whichdepends onlyonthe radial
+coordinate r. Thenursatisfies the equation
+1
+4u′′′′
+r−/parenleftbigg
+1−1
+r/parenrightbigg
+u′′′
+r+/parenleftbigg
+1−2
+r−1
+r2/parenrightbigg
+u′′
+r
++3
+r2u′
+r−/parenleftbigg2
+r3+ω2
+Z4/parenrightbigg
+ur= 0,(98)
+where the primes now denote derivatives with respect to
+r. Solving the eigenvalue problem with boundary con-
+ditionn1/2
+0(r)ur(r)→0 forr→ ∞yields the correct13
+ 0 10 20 30 40 50 60 70 80 90
+ 0  2  4  6  8  10  12 
+ 
+ 0 10 20 30 40 50 60 70 80 90
+ 0  2  4  6  8  10  12ω+(q)
+ω-(q)
+ω+(q)
+ω-(q)
+qqωL
+ωp(a)
+(b)
+ωTωL(q)
+ωT(q) 0.5 1 1.5 2 2.5 3
+ 0rs=1rs=5
+ 3  2.5  2  1.5  1  0.5  0 0 0.5 1 1.5 2 2.5
+rs=1rs=5
+rs=3 3
+ 0  0.5  1  1.5  2  2.5  3
+FIG. 1: (Color online) Longitudinal (a) and transverse (b)
+modes for a homogeneous electron gas at rs= 1,3,5. Wave
+vectorqis in units of kFand frequency is in units of 2 EF.
+The curveslabelled ω+(q)andω−(q)are theboundariesof the
+electron-hole continuum.35. Single particle excitations exist
+forω−(q)< ω < ω +(q), whereas multiparticle excitations
+are distributed all over the plane. The elastic approximati on
+replaces the exact spectrum by the two branches ωL(q) and
+ωT(q), which carry the entire spectral weight. Notice that
+thers-dependence is barely discernible on a large qscale, but
+becomes clearly visible at smaller qas shown in the insets.
+excitation energies ωn= (Z2/2)(1−1/n2) (n= 1,2,..).
+The corresponding eigenfunctions are given by Laguerre
+polynomials:
+un(r)∝L2
+n−2/parenleftbigg2r
+n/parenrightbigg
+. (99)
+C. Two-electron systems
+As a final example let us consider the case of two
+electrons repelling each other with the “soft” Coulomb
+potentiale2√
+(x1−x2)2+a2in a one-dimensional parabolic
+trap of natural frequency ω0(the cutoff a >0 serves
+to eliminate the pathological behavior of the interaction
+atx1=x2). This is a model that can be solved nu-
+merically thanks to the separation of center of mass andrelative variable, and analytically in the limit of strong
+correlation. The hamiltonian is
+ˆH0=P2
+4m+mω2
+0X2+p2
+m+m
+4ω2
+0x2+e2
+√
+x2+a2,(100)
+whereX=x1+x2
+2andP=p1+p2are, respectively,
+the coordinate and the momentum of the center of mass
+andx=x1−x2,p=p1−p2
+2are the coordinate and the
+momentum in the relative channel. Notice that for a
+fixed strength e2of the interaction we can go from the
+weakly correlated to the strongly correlated regime by
+varying the value of ω0:ω0→ ∞corresponds to the non-
+interacting limit, and ω0→0 to the strongly correlated
+limit.
+The ground-state wave function is
+Ψ0(x1,x2) =ψ0(X)φ0(x), (101)
+whereψ0(X) andφ0(x) are, respectively, the ground-
+state wave functions of the center of mass and of the
+relative hamiltonian. The general excited state is
+Ψnm(x1,x2) =ψn(X)φm(x) (102)
+whereψn(X) is the wave function of the n-th excited
+state of the center of mass hamiltonian and φm(x) is the
+wave function of the m-th excited state of the relative
+hamiltonian.
+The ground-state of the system is a spin singlet ( S=
+0) and for this reason in the following we consider only
+singletstates, whichareconnectedtotheground-stateby
+the current density operator. The relative wave function
+for such states is symmetric: φm(−x) =φm(x). This
+wave function has 2 mnodes, of which mwithx>0 and
+m(symmetricallyplaced)with x<0. Thecenterofmass
+wave function,
+ψn(X) =Hn/parenleftbiggX
+ℓcm/parenrightbigg
+e−X2/2ℓ2
+cm,(103)
+can be either symmetric or antisymmetric, depending on
+the parity of n, and hasnnodes. Here ℓcm=/radicalbig
+/planckover2pi1/2mω0.
+The ground-state has n= 0,m= 0 and all the other
+states are characterized by positive values of the integers
+nandm.
+In the non interacting limit ( ω0→ ∞, ore2→0) the
+relative wave function is
+φm(x) =H2m/parenleftbiggx
+ℓ0/parenrightbigg
+e−x2/2ℓ2
+r, (104)
+whereℓ0=/radicalbig
+2/planckover2pi1/mω0. The ground-state density is a
+gaussian centered at the origin. The excitation energies,
+expressed in units of ω0, are sums of the excitation ener-
+gies of two identical harmonic oscillators:
+lim
+ω0→∞Enm
+ω0=n+2m (105)14
+ 0 1 2 3 4 5 6
+ 0  2  4  6  8  10E/ω0
+ω0-1/2(1,0)(0,1)(2,0)(1,1)(3,0)(0,2)(2,1)(4,0)(1,2)(3,1)(5,0)
+FIG. 2: (Color online) Evolution of the excitation energies
+for two electrons in a one-dimensional harmonic trap. Solid
+lines denote exact excitation energies, labelled by ( n,m) as
+explained in the text. Solid dots indicate the calculated ei gen-
+values of the QCM equation of motion. Crosses on the right
+denote the strong-correlation limit of the eigenvalues, gi ven
+by Eq. (113).
+withn,mnon-negative integers. The degeneracy of the
+excited states is the number of integers less or equal n+
+2mwith the same parity as n+2m, i.e.
+Dnm= 1+/bracketleftbiggn+2m
+2/bracketrightbigg
+, (106)
+where [y] denotes the integer part of y. The displacement
+field associated with the ( n,m) excitation is
+unm(x)∝Hn+2m−1(x/λ0), (107)
+whereλ0=/radicalbig
+ℓ2cm+(ℓ0/2)2=/radicalbig
+/planckover2pi1/mω0. The parity is
+(−1)n+2m−1and the number of nodes in n+2m−1.
+The situation is quite different in the strongly corre-
+lated limit ( ω0→0, ore2→ ∞). The relative Hamilto-
+nian reduces to a harmonic oscillator of frequency ω0√
+3
+withequilibriumdistance x0= (2e2/mω2
+0)1/3(weassume
+a≪x0). The ground-state wave function for the rel-
+ative motion is a symmetric linear combination of two
+Gaussians of width ℓ∞= (2/planckover2pi1/√
+3mω0)1/2centered at
+x1−x2=±x0. The corresponding ground state den-
+sityn0(x) consists of two Gaussian peaks of the width
+λ∞=/radicalbig
+ℓ2cm+(ℓ∞/2)2= [/planckover2pi1(1 +√
+3)/2√
+3mω0]1/2cen-
+tered atx=±x0/2. The excitation spectrum has the
+form
+lim
+ω0→0Enm
+ω0=n+m√
+3, (108)and the degeneracyis completely removed. The displace-
+ment field is analytically found to be
+unm(x)∝Hn+m−1/parenleftbiggx−x0/2
+λ∞/parenrightbigg
+θ(x)
++ (−1)mHn+m−1/parenleftbiggx+x0/2
+λ∞/parenrightbigg
+θ(−x).(109)
+The parity is ( −1)n−1and the number of nodes is 2( n+
+m−1)+mod(n−1,2), where mod( n−1,2)≡n−1 (mod
+2).
+The evolution of the lowest-lying energy levels with
+given value of the pair n,mas a function of ω0is shown
+by the solid lines in Fig. (2). Some of the displacement
+fields of the low-lying excitations in the strongly corre-
+lated regime (Eq. (109)) are shown in Fig. (3).
+From these figureswe see that the displacement field of
+the (1,0) excitation, which corresponds to a rigid trans-
+lation of the center of mass, is uniform in space, while
+the displacement field of the (0 ,1) excitation, which cor-
+responds to the classical breathing mode, changes sign
+around the origin. The (1 ,0) and (0,1) excitations corre-
+spond to the classical phonon modes of a system of two
+localized particles. The remaining excitations are quan-
+tummechanicalincharacter,ascanbesurmisedfromthe
+fact that their displacement fields (in the strongly corre-
+lated regime) have significant variation over the regions
+where the density has peaks, i.e. the places where the
+particles would be classically localized. These modes de-
+scribe the dynamics of the wave function of the localized
+electrons.
+Looking at the figures we observe that, in the
+strongly correlated regime , there are groups of excited
+states (e.g. {(0,2),(2,0)};{(3,0),(1,2)};{(2,1),(0,3)};
+{(4,0),(0,2),(0,4)}, such that all the excited states
+within one group produce the same displacement field,
+up to a normalization constant. In general, states with a
+given value of n+mand the same parity of mhave the
+same displacement field, but different energies. Clearly,
+this is a feature of the exact solution that cannot be
+reproduced by any linear eigenvalue problem with a
+frequency-independent kernel.
+The phenomenon of different excited states produc-
+ing the same displacement field occurs also in the non-
+interacting limit: all the states with the same value of
+n+ 2m(e.g. (0,1) and (2,0)) have the same displace-
+ment field. But, in this case, the states with the same
+displacement field also have the same energy: therefore
+the noninteracting excitation energies can be accurately
+reproduced by a linear eigenvalue problem.
+Letusnowseewhatourelasticequationofmotion(61)
+predicts for this system. The kinetic part of the equilib-
+rium stress tensor T0(x) works out to be15
+-3-2-1 0 1 2 3
+-15 -10 -5  0  5  10  15un,m(x)
+x(0,1)
+(0,2), (2,0)
+(2,1), (0,3)
+-5-4-3-2-1 0 1 2 3 4
+-15 -10 -5  0  5  10  15un,m(x)
+x(1,0)
+(1,1)
+(3,0), (1,2)
+FIG. 3: (Color online) Top panel: the displacement field
+unm(x) for (n,m) = (0,1),(0,2) and (0,3) in the strong corre-
+lation limit. Bottom panel: the same for ( n,m) = (1,1),(1,2)
+and (2,1). The thin solid lines represent the density profile.
+The large value of the displacement field for x∼0 does not
+have a physical significance since the density is exponentia lly
+small in that region.
+T0(x) =2
+m/integraldisplay
+dy/braceleftBigg/bracketleftbigg
+φ′
+0(x−y)ψ0/parenleftbiggx+y
+2/parenrightbigg
++1
+2φ0(x−y)ψ′
+0/parenleftbiggx+y
+2/parenrightbigg/bracketrightbigg2
+−1
+4∂2
+x/bracketleftbigg
+φ2
+0(x−y)ψ2
+0/parenleftbiggx+y
+2/parenrightbigg/bracketrightbigg/bracerightBigg
+(110)
+whereψ0andφ0are the ground-state wave function in the center of mass and rela tive channel respectively. The
+interaction kernel K(x,x′) is given by
+K(x,x′) =−2(x−x′)2+a2
+[(x−x′)2+a2]5/2φ2
+0(x−x′)ψ2
+0/parenleftbiggx+x′
+2/parenrightbigg
+. (111)
+We now have all the input that is necessary to set up and solve the fo urth-order integro-differential equation (61).16
+In the limit of weak correlation ( ω0→ ∞) the eigen-
+values of the integro-differential equation coincide with
+the exact (degenerate) excitation energies. This is un-
+derstandable, since in this limit the two electrons are de-
+coupled and the excitation spectrum of the two-electron
+system coincides with that of a single electron starting
+from its own ground-state. This spectrum, as we have
+seen, is exactly reproduced by our equation of motion.
+Unfortunately, this nice feature ofthe present model can-
+not be extrapolated to general systems. If, for example,
+the system contains more than two electrons, then even
+in the non-interacting limit a generic excitation will en-
+tail the transition of a single electron from an occupied
+orbital that is notthe ground-state orbital to an unoccu-
+piedone. Suchanexcitationwill notbedescribedexactly
+by our method.
+In the limit of strong correlation our integro-
+differential equation can be solved analytically, as shown
+in Appendix F. The eigenfunctions can be classified as
+even (+) or odd (-) and are given by
+uk,±(x)∝Hk/parenleftbiggx−x0/2
+λ∞/parenrightbigg
+θ(x)
+±(−1)kHk/parenleftbiggx+x0/2
+λ∞/parenrightbigg
+θ(−x),(112)
+wherekisanon-negativeinteger. Thenumberofnodesis
+2kfor even eigenfunctions, 2 k+1 for odd eigenfunctions.
+The corresponding eigenvalues are given by
+lim
+ω0→0Ek,±
+ω0=/bracketleftBig
+2+3√
+3k+6k(k−1)(2−√
+3)
+∓(−1)k(2−√
+3)k/bracketrightBig1/2
+. (113)
+Notice that, within each symmetry sector (even or odd),
+the eigenvalues increase monotonically with increasing k.
+In Table I, fourth and fifth column, we present a de-
+tailed comparison between the exact excitation energies
+and the eigenvalues of our equation of motion in the
+strong correlation limit. For the sake of clarity, we list
+the excitations that produceeven displacement fields and
+those that produce odd displacement fields separately.
+The elastic equation of motion can also be solved numer-
+ically, and the results are in very good agreement with
+the analytical solution. This, and the fact that the sum
+rule (81) is satisfied with good accuracy, builds our con-
+fidence in the numerical solution.
+In Fig. (2) we present the numerical results for some of
+the lowest-lying excitations as a function of ω0. We can
+immediately see that the “non-degenerate” excitations,
+by which we mean the excitations (1 ,0) (0,1), and (1,1),
+which are uniquely associated to a given displacement
+field, are rather well reproduced by our calculation for
+all values of ω0. On the other hand, the “degenerate
+excitations”, which yield the same displacement field but
+havedifferentenergies, arereplacedbya singleexcitation
+of an average energy, in such a way that the total spec-
+tral strength of the group is preserved. Two examples ofEven modes
+(n,m)E0
+nmN0E∞
+nmE∞
+n+m−1,+N∞
+(1,0)1.001.0 1.0 0
+(1,1)3.022.732 2.732 2
+(3,0)3.023.03.942 4
+(1,2)5.044.464 4
+(3,1)5.044.732 5.220 6
+(1,3)7.066.196 6
+(5,0)5.045.06.486 8
+(3,2)7.066.464 8
+(1,4)9.087.928 8
+(5,1)7.066.732 7.755 10
+(3,3)9.088.196 10
+(1,5)7.069.660 10
+Odd modes
+(n,m)E0
+nmN0E∞
+nmE∞
+n+m−1,−N∞
+(0,1)2.011.732 1.732 1
+(2,0)2.012.02.632 3
+(0,2)4.033.464 3
+(2,1)4.033.732 3.960 5
+(0,3)6.055.196 5
+(4,0)4.034.05.217 7
+(2,2)6.055.464 7
+(0,4)8.076.928 7
+(4,1)6.055.732 6.487 9
+(2,3)8.077.196 9
+(0,5)10.098.660 9
+TABLE I: Exact excitation energies Enmof the two-electron
+model with hamiltonian specified in Eq. (100) in the non-
+interacting limit ( E0
+nm, second column) and in the strongly
+correlated limit ( E∞
+nm, fourth column). The eigenvalues of
+the QCM equations of motion ( E∞
+n+m−1,±) in the strongly
+correlated regime are listed in the fifth column. The top half
+of the table lists excitations with even displacement fields and
+the bottom half lists excitations with odd displacement fiel ds.
+The third and the last column on the right list the number
+of nodes in the displacement field in the non interacting limi t
+(N0) and in the strongly correlated limit ( N∞).
+this phenomenon are evident in Fig. (2): the (2 ,0) and
+(0,2) excitations, which in the strong correlation limit
+(ω0→0) have energies 2 ω0and 3.464ω0respectively,
+are replaced by a single excitation – the fourth one in
+Fig. (2) – which tends to the “average” energy 2 .632ω0.
+Similarly, the (3 ,0) and (1,2) excitations, which, in the
+ω0→0 limit tend to 3 ω0and 4.464ω0respectively are
+replaced by a single excitation – the fifth one in Fig. (2)
+– which tends to the “average”energy 3 .942ω0. The pat-
+tern recurs for more complex multiplets of excitations,
+involving three or more states with the same displace-
+ment field and different energies.
+We notice that the displacement field associated with,
+say, the (n,m) excited state has a number of nodes that
+generally growsfromn+ 2m−1 in the weak coupling
+limit to 2(n+m−1) (oddn) or 2(n+m−1)+1 (even n)
+in the strong coupling limit. This effect is particularly17
+ 0(b)
+ 0
+-4 -2  0  2  4 0(a)
+ 0
+-4 -2  0  2  4u11(x)
+u30(x)u20(x)
+u21(x)χ=100
+χ=  10
+χ=    1
+χ= 0.1
+FIG. 4: (Color online) Evolution of the displacement fields
+of excitations (1 ,1),(3,0) (even) and (2 ,0),(2,1) (odd) as a
+function of correlation strength χ=ω−1/2
+0, as shown in the
+top left panel. Notice the variation in the number of nodes as
+χincreases from the weakly correlated to the strongly corre-
+lated limit.
+pronounced for states of small m, and is absent in the
+n= 0 states. Fig. (4) shows the evolution of the dis-
+placement field for the even excitations (1 ,1) and (3,0)
+and for the odd excitations (2 ,0) and (2,1). We see that
+in the “non-degenerate”(1 ,1) state, the number of nodes
+staysconstantandequal to2 asonegoesfromthe weakly
+correlated to the strongly correlated regime. In the (3 ,0)
+state the number of nodes grows from 2 to 4 nodes, so
+that the displacement field of this state becomes pro-
+portional, in the strong-correlation limit, to that of the
+much higher in energy (0 ,3) state. The same behavior is
+observed in state (2 ,0), for which the number of nodes
+grows from 1 to 3, and in state (2 ,1), for which it grows
+from 3 to 5. By this mechanism, states of very different
+energy end up sharing the same displacement field (up
+to a proportionality constant) in the strong correlation
+limit.
+Our discussion has been limited to singlet states (sym-
+metric wave function in the relative channel). It would
+be easyto extend the calculation toinclude triplet states.
+To this end, we simply replace the density, kinetic energy
+density, and pair correlation function of the ground-state
+(a singlet) by the same quantities calculated from the
+ground state in the triplet ( S= 1) sector of the Hilbert
+space. The relative wave functions of these states are an-
+tisymmetric. The correct symmetry of the wave function
+is automatically taken into account through the ground-
+state properties, and does not further appear in the elas-
+tic equation of motion.VI. DISCUSSION AND SUMMARY
+The elastic approximation is, in a very precise
+sense, the extension of the well-known collective
+approximation39,40of the homogeneous electron gas to
+non-homogeneous electronic systems. In the case of two
+electronsinteracting by Coulombpotential in a harmonic
+trap, we have seen that the elastic approximation re-
+places groups of excitations characterized by the same
+displacement field by a single excitation that carries the
+oscillator strength of the whole group. In more com-
+plex systems, we do not expect to be able to identify
+small groups of excitations that share the same displace-
+ment field. All that can be said is that the displace-
+ments associated with different excitations will not be
+linearly independent. Each eigenfunction of the elastic
+equation of motion will overlap with many different ex-
+citations. However, the integrated spectral strength of
+the elastic eigenmodes will still add up to the correct
+value. For this reason, our approximation should be use-
+ful in dealing with collective effects which depend on the
+integrated strength of the excitation spectrum, such as
+the dipolar fluctuations that are responsible for van-der
+Waalsattraction.37,38Otherpossibleapplicationsinclude
+possible nonlocal refinements of the plasmon pole ap-
+proximation in GW theory41and studying the dynam-
+ics of strongly correlated systems, which are dominated
+by a collective response. As a byproduct we got an ex-
+plicit analytic representation of the exact xc kernel in
+the high-frequency (anti-adiabatic) limit.42This kernel
+should help us to study an importance of the space and
+time nonlocalities in the KS formulation of TD(C)DFT.
+It would be particularly interesting to try and interpo-
+latebetweentheadiabaticandanti-adiabaticextremesto
+construct a reasonable frequency-dependent functional.
+The elastic equation of motion derived in this paper
+relied on the knowledge of the exact density matrices
+ρ(1)andρ(2)of the ground-state. In many cases, these
+ground-state properties can be extracted from Quantum
+MonteCarlocalculations. Whenthiscannotbedone, one
+can still resortto density functional theory, i.e. apply the
+QCM formulation directly to the Kohn-Sham system, in
+which case we do not need the exact ground-state den-
+sity matrices, but only the ground-state KS orbitals and
+a reasonable approximation for the exchange-correlation
+field. While the standard KS method treats the nonin-
+teractingkinetic stresstensorexactly, our method should
+be computationally more agile, for large systems, since
+it does not involve time-dependent orbitals and/or the
+inversion of large linear response matrices.
+It remains a challenge to extend the present formal-
+ism to the nonlinear regime, as well as including external
+magnetic fields and spin-orbit interactions.18
+VII. ACKNOWLEDGEMENTS
+This work was supported by DOE grant DE-FG02-
+05ER46203 (GV) and DE-AC52-06NA25396 (JT) and
+by the IKERBASQUE Foundation. GX was supported
+by NSF of China under Grant No. 10704066 and
+10974181. IVT acknowledges funding by the Span-
+ish MEC (FIS2007-65702-C02-01), “Grupos Consolida-
+dosUPV/EHUdelGobiernoVasco”(IT-319-07),andthe
+European Community through e-I3 ETSF project (Con-
+tract Number 211956). GV gratefully acknowledges the
+kind hospitality of the ETSF in San Sebastian where this
+work was completed. We thank Dr. Stefano Pittalis for
+his help in calculating an plotting the curves shown in
+Fig. 1, and Dr. Paola Gori-Giorgi for kindly providing
+the code for calculating the structure factor of the elec-
+tron gas.
+Appendix A: Stress tensor operator
+From the evaluation of Eq. (20) at the Euclidean met-
+ricsgµν=δµνwe get18,19
+ˆPµν=ˆTµν+ˆWµν, (A1)
+where
+ˆTµν=1
+2m/braceleftbigg
+(∂µˆΨ†)(∂νˆΨ)+(∂νˆΨ†)(∂µˆΨ)−1
+2∇2ˆnδµν/bracerightbigg
+,
+(A2)
+and
+ˆWµν=−1
+2/integraldisplay
+dr′r′
+µr′
+ν
+r′∂w(r′)
+∂r′
+×/integraldisplay1
+0dλˆρ2(r+λr′,r−(1−λ)r′).(A3)
+HereˆΨ(r) is the field operator,
+ˆρ2(r,r′) =ˆΨ†(r)ˆΨ†(r′)ˆΨ(r′)ˆΨ(r) (A4)
+is the diagonal two-particle density operator, and w(r) is
+the interaction potential.
+Appendix B: Derivation of the force identity
+Eq. (38)
+In this appendix we derive an identity which is used
+in Sec. II A to identify the right hand sides of Eqs. (36)
+and (38). Namely, we consider a functional S[gµν] of the
+following metric tensor
+gµν(ξ,t) =∂rα
+∂ξµ∂rα
+∂ξν, rα(ξ) =ξα+uα(ξ) (B1)
+and prove that the following equality holds
+δS
+δrµ=−2∂
+∂ξα/parenleftbigg∂rµ
+∂ξβδS
+δgαβ/parenrightbigg
+≡∂
+∂ξα/parenleftbigg∂rµ
+∂ξβ√gPαβ/parenrightbigg
+(B2)Note that the identity relates the functional derivative
+ofSwith respect to the displacement to the functional
+derivativewith respect to the metric/deformationtensor,
+which physically means a connection of the force to the
+stress.
+To prove Eq. (B2) we consider a small variation of the
+functionrµ(ξ):rµ(ξ)∝ma√sto→rµ(ξ)+δrµ(ξ). The correspond-
+ing variation of the functional Stakes the form
+δS=/integraldisplay
+dξδS
+δrµδrµ(ξ) (B3)
+On other hand, the variation of rµ(ξ) induces the follow-
+ing variation of the metric tensor: gµν(ξ)∝ma√sto→gµν(ξ) +
+δgµν(ξ), where
+δgµν=∂δrα
+∂ξµ∂rα
+∂ξν+∂rα
+∂ξµ∂δrα
+∂ξν(B4)
+Hence the variation of Scan be also written as
+δS=/integraldisplay
+dξδS
+δgµνδgµν(ξ)
+=/integraldisplay
+dξδS
+δgµν/parenleftbigg∂δrα
+∂ξµ∂rα
+∂ξν+∂rα
+∂ξµ∂δrα
+∂ξν/parenrightbigg
+(B5)
+Performing the partial integration in the right hand side
+of Eq. (B5), and using the symmetry of the tensor gµν
+we reduce Eq. (B5) to the following form
+δS=−/integraldisplay
+dξ2∂
+∂ξα/parenleftbigg∂rµ
+∂ξβδS
+δgαβ/parenrightbigg
+δrµ(ξ) (B6)
+The direct comparison of Eqs. (B3) and (B6) proves the
+announced identity (B2).
+Finally, to make a connection to Eqs. (36) and (38) we
+setS[gµν] =ˆHu[gµν] and take the expectation value of
+Eq. (B2) in the state |˜ψ(t)∝an}b∇acket∇i}ht. Then, linearizing with re-
+spect the displacement u(ξ), we find that the right hand
+side of Eq. (B2) becomes identical to the right hand side
+of Eq. (36), while the left hand side of Eq. (B2) is exactly
+equal to the right hand side of Eq. (38).
+Appendix C: Derivation of Eq.(51)
+In this appendix we derive the linearized form of the
+kinetic energy Eqs. (51) for the instantaneously distorted
+ground state.
+We start with the general nonlinear expression for
+the kinetic energy T[u] in the elastic approximation [see
+Eq. (50)]
+T[u] =1
+2m/integraldisplay
+dξ√ggµν∂µ∂′
+ν[g−1/4(r)g−1/4(ξ′)ρ(ξ,ξ′)]ξ=ξ′,
+(C1)
+whereρ(ξ,ξ′) is the exact ground state one-particle den-
+sity matrix, and gµν(ξ) andg(ξ) are, respectively, the
+inverse and the determinant of the metric tensor gµν(ξ)19
+that is the functional of the displacement u(ξ), which is
+defined by Eq. (B1)
+Our aim is to expand the functional of Eq. (C1) to
+the second order in the displacement field, i. e., to the
+first non-vanishing contribution correspondingto the lin-
+earized theory.
+First we explicitly calculate the derivatives in the right
+handsideofEq.(C1) andset ξ′=ξ. AsaresultEq.(C1)
+reduces to the form
+T[u] =/integraldisplay
+dξgµν/braceleftBig
+Kµν+n0
+8m(∂µln√g)(∂νln√g)
+−1
+8m/bracketleftbig
+(∂µln√g)(∂νn0)+(∂νln√g)(∂µn0)/bracketrightbig/bracerightBig
+,(C2)
+wheren0(ξ) =ρ(ξ,ξ) is the ground state density, and
+Kµν(ξ) =1
+2m[∂µ∂′
+νρ(ξ,ξ′)]ξ=ξ′.(C3)
+Making use of the following representation for√g,
+√g= det/parenleftbigg∂rα
+∂ξβ/parenrightbigg
+,
+we can write the derivative of ln√gas follows:
+∂µln√g=∂µlndet/parenleftbigg∂rα
+∂ξβ/parenrightbigg
+=∂ξα
+∂rβ∂µ∂rβ
+∂ξα.(C4)It is now straightforward to expand the right hand side
+of Eq. (C4) to the second order in u:
+∂µln√g≈∂µ∂αuα−(∂βuα)∂µ∂αuβ.(C5)
+Next we consider the covariant tensor gµν(the inverse of
+gµν):
+gµν=/bracketleftbig
+δµν+∂µuν+∂νuµ+(∂µuα)(∂νuα)/bracketrightbig−1.(C6)
+Expanding the inverse matrix in Eq. (C6) to the second
+order and expressing the result in terms of the strain
+tensoruµνwe get
+gµν≈δµν−2uµν+4uµαuνα−(∂µuα)(∂νuα).(C7)
+Finally, we substitute Eqs. (C5) and (C7) into Eq. (C2)
+and keep terms up to the second order in u. The second
+order contribution to Ttakes the form
+T2=/integraldisplay
+dξ/braceleftBig
+Kµν[4uµαuνα−(∂µuα)(∂νuα)]+n0
+8m(∂µuαα)(∂µuνν)+∂νn0
+2muµν∂µuαα+∂νn0
+4m(∂µuα)∂ν(∂αuµ)/bracerightBig
+.(C8)
+The last term in Eq. (C8) can be identically represented as follows
+∂νn0
+4m(∂µuα)∂ν(∂αuµ) =−∇2n0
+8m[4uµαuµα−(∂µuα)(∂µuα)]−∂νn0
+2muµα∂νuµα. (C9)
+Note that the coefficient in front ofthe first term in Eq. (C9) and th e correspondingcoefficient in first term in Eq. (C8)
+are naturally combined into the kinetic stress tensor Tµν,0= 2Kµν−δµν∇2n0/4m. Hence, inserting Eq. (C9) into
+Eq. (C8) and integrating by parts terms proportional to ∂νn0/2mwe arrive at the following final representation for
+the linearized kinetic energy of the distorted state:
+T2=/integraldisplay
+dξ/braceleftBig1
+2Tµν,0[4uµαuνα−(∂µuα)(∂νuα)]+n0
+8m(∂µuαα)(∂µuνν)+n0
+4m[∂µuνα∂µuνα−∂µuνµ∂µuαα],(C10)
+which is identical to Eq. (51). It is worth noting a con-
+venient feature of this representation – the last term in
+Eq. (C10) vanishes in all 1D systems and for homoge-
+neous systems with n0=constin any number of dimen-
+sions.Appendix D: Proof of the orthonormality relation
+for single-particle transition currents
+In this appendix we prove Eq. (80) for one-particle
+systems in the absence of a magnetic field.
+The matrix elements of the current density operator is
+j0n(r) =∝an}b∇acketle{t0|ˆj(r)|n∝an}b∇acket∇i}ht=−i
+2m[ψ0∇ψn−ψn∇ψ0],(D1)
+whereψ0,...,ψnare orthonormal eigenfunctions of the20
+one-electron hamiltonian, which can be assumed to be
+real if there is no magnetic field. Now observe that
+j0n(r)/radicalbig
+n0(r)=−i
+2mψ0∇ψn−ψn∇ψ0
+ψ0=−i
+2mψ0∇/parenleftbiggψn
+ψ0/parenrightbigg
+,
+(D2)
+andjn0(r) =−j0n(r). Also, from the continuity equation
+we get
+∇·(ψ0∇ψn−ψn∇ψ0) =−2mωn0ψ0ψn.(D3)
+Combining these two equations we get
+∇·/parenleftbigg
+ψ2
+0∇/parenleftbiggψn
+ψ0/parenrightbigg/parenrightbigg
+=−2mωn0ψ0ψn.(D4)
+From this we see that
+/integraldisplay
+dr/parenleftBigg/radicalbigg
+2m
+ωn0j0n(r)/radicalbig
+n0(r)/parenrightBigg
+·/parenleftBigg/radicalbigg
+2m
+ωk0jk0(r)/radicalbig
+n0(r)/parenrightBigg
+=
+=1
+2m1√ωn0ωk0/integraldisplay
+dr/bracketleftbigg
+∇/parenleftbiggψn
+ψ0/parenrightbigg/bracketrightbigg
+·ψ2
+0(r)∇/parenleftbiggψk
+ψ0/parenrightbigg
+=−1
+2m1√ωn0ωk0/integraldisplay
+drψn
+ψ0∇·/bracketleftbigg
+ψ2
+0(r)∇/parenleftbiggψk
+ψ0/parenrightbigg/bracketrightbigg
+=/radicalbiggωk0
+ωn0/integraldisplay
+drψnψk
+=δnk (D5)
+which proves Eq. (80).
+Appendix E: Proof of the sum rules (81) - (83)
+Our starting point is the first moment sum rule for
+the current-current response function (or “third-moment
+sum rule” for the density-density response function),
+which states that
+−1
+π/integraldisplay∞
+0dωωℑmχµν(r,r′,ω) =/summationdisplay
+nωn0[jµ(r)]0n[jν(r′)]n0.
+(E1)
+On the other hand, the equation of motion (72) for
+the current density in the elastic approximation can be
+rewritten as
+/integraldisplay
+dr′/braceleftBig
+ω2δµνδ(r−r′)−˜Mµν(r,r′)/bracerightBig/radicalbiggm
+n0(r′)jν(r′)
+=ω2/radicalbigg
+n0(r)
+mA1,µ(r), (E2)
+whereA1,µ=∂µV1/(iω),jν=−iωn0uν, and
+˜Mµν(r,r′) =1/radicalbig
+mn0(r)Mµν(r,r′)1/radicalbig
+mn0(r′)
+=/radicalbiggm
+n0(r)/summationdisplay
+nωn0{[jµ(r)]0n[jν(r′)]l0
++ [jν(r′)]0n[jµ(r)]n0}/radicalbiggm
+n0(r′).(E3)is a hermitian positive definite operator, which admits
+a complete set of orthonormal eigenfunctions. Let us
+denote by ˜uλ(r) these eigenfunctions, and by ω2
+λtheir
+eigenvalues. The orthonormality relation reads
+/integraldisplay
+dr˜uλ(r)·˜uλ′(r) =δλλ′ (E4)
+and the completeness relation is
+/summationdisplay
+λ[˜uλ(r)]µ[˜uλ(r′)]ν=δ(r−r′)δµν.(E5)
+The kernel itself can be written as
+˜Mµν(r,r′) =/summationdisplay
+λω2
+λ[˜uλ(r)]µ[˜uλ(r′)]ν.(E6)
+The equation of motion for the current density can be
+rewritten as
+/summationdisplay
+λ/integraldisplay
+dr′(ω2−ω2
+λ)[˜uλ(r)]µ[˜uλ(r′)]ν/radicalbiggm
+n0(r′)jν(r′)
+=ω2/radicalbigg
+n0(r)
+mAµ(r), (E7)
+Its solution is obtained by projecting both sides of the
+equation along the eigenvector ˜uλ. We get
+/bracketleftbigg/radicalbiggm
+n0j/bracketrightbigg
+λ=/parenleftbigg
+1+ω2
+λ
+ω2−ω2
+λ/parenrightbigg/bracketleftbigg/radicalbiggn0
+mA/bracketrightbigg
+λ,(E8)
+where the subscript λdenotes projection along ˜uλ. From
+this we obtain
+jµ(r) =n0(r)
+mAµ(r)+/integraldisplay
+dr′/summationdisplay
+λω2
+λ
+ω2−ω2
+λ/radicalbigg
+n0(r)
+m[˜uλ(r)]µ
+[˜uλ(r′)]ν/radicalbigg
+n0(r′)
+mAν(r′). (E9)
+Hence, the current-current response function in the elas-
+tic approximation is
+χel
+µν(r,r′,ω) =n0(r)
+mδµνδ(r−r′)
++/summationdisplay
+λω2
+λ
+ω2−ω2
+λ/radicalbigg
+n0(r)
+m[˜uλ(r)]µ[˜uλ(r′)]ν/radicalbigg
+n0(r′)
+m.
+(E10)
+Evaluating the sum rule we obtain
+−1
+π/integraldisplay∞
+0dωωℑmχel
+µν(r,r′,ω) =
+1
+2/summationdisplay
+λω2
+λ/radicalbigg
+n0(r)
+m[˜uλ(r)]µ[˜uλ(r′)]ν/radicalbigg
+n0(r′)
+m=
+1
+2m/radicalbig
+n0(r)˜Mµν(r,r′)/radicalbig
+n0(r′) (E11)21
+On the other hand, the first moment sum rule (E1) can
+be rewritten as
+−1
+π/integraldisplay∞
+0dωωℑmχµν(r,r′,ω) =
+1
+2m/radicalbig
+n0(r)˜Mµν(r,r′)/radicalbig
+n0(r′).(E12)
+Comparing the last two equations we conclude that
+−1
+π/integraldisplay∞
+0dωωℑmχik(r,r′,ω) =
+−1
+π/integraldisplay∞
+0dωωℑmχel
+ik(r,r′,ω),(E13)
+i.e. the sum rule is satisfied in the elastic approximation.
+More pointedly, making use of Eq. (E3) the sum rule
+can be written in the form
+/summationdisplay
+λω2
+λ[˜uλ(r)]µ[˜uλ(r′)]ν= 2m/summationdisplay
+nωn0[jµ(r)]0n/radicalbig
+n0(r)[jν(r′)]n0/radicalbig
+n0(r′),
+(E14)
+from which it follows that
+ω2
+λ=/summationdisplay
+nω2
+n0fλ
+n, (E15)
+wherethe“oscillatorstrengths” fλ
+narepositivequantities
+defined as
+fλ
+n=2m|Fλ
+n|2
+ωn0(E16)
+with
+Fλ
+n≡/integraldisplay
+dr˜uλ(r)·[j(r)]0n/radicalbig
+n0(r)(E17)
+As a final step we prove that
+/summationdisplay
+lfλ
+n= 1 (E18)
+for allλ. This is of course nothing but the f-sum rule
+−1
+π/integraldisplay∞
+0dωℑmχµν(r,r′,ω)
+ω=n0(r)
+2mδµνδ(r−r′),(E19)which is manifestly satisfied by χel
+µνby virtue of the com-
+pleteness relation (E5) for ˜uλ(r). When applied to the
+exact response function the f-sum rule implies that
+/summationdisplay
+n[jµ(r)]0n[jν(r′)]n0
+ωn0=n0(r)
+2mδµνδ(r−r′).(E20)
+Then we see that
+/summationdisplay
+n2m|Fλ
+n|2
+ωn0= 2m/integraldisplay
+dr/integraldisplay
+dr′
+/summationdisplay
+n[jµ(r)]0n[jν(r′)]n0
+ωn[˜uλ(r)]µ/radicalbig
+n0(r)[˜uλ(r′)]ν/radicalbig
+n0(r′)
+=/integraldisplay
+dr/integraldisplay
+dr′n0(r)δµνδ(r−r′)[˜uλ(r)]µ/radicalbig
+n0(r)[˜uλ(r′)]ν/radicalbig
+n0(r′)
+= 1. (E21)
+QED.
+Appendix F: Analytic solution of the elastic
+eigenvalue problem in the strong-correlation regime
+In this appendix we present an asymptotically exact
+solution of the 1D continuum mechanics eigenvalue prob-
+lem for two particles confined by the harmonic potential
+V0=1
+2mω2
+0x2and interacting with a soft-Coulomb po-
+tential
+w(x−x′) =e2
+/radicalbig
+(x−x′)2+a2. (F1)
+At the exact many-body theory level the system is de-
+scribedby the Hamiltonian ofEq.(100). Within ourcon-
+tinuum mechanics the excitation energies are obtained
+from the solution of the following “elastic” eigenvalue
+problem for the displacement u(x,t) =u(x)e−iωt(see
+Eq. (61) in the main text)
+mω2n0u(x) =1
+4m∂2
+x[n0∂2
+xu(x)]−3∂x[T0∂xu(x)]+mω2
+0n0u(x)+2/integraldisplay
+dx′[∂2
+xw(x−x′)]Ψ2
+0(x,x′)[u(x)−u(x′)].(F2)
+Here Ψ 0(x,x′) is the ground state two-particle wave function, which in this case c oincides with the square root of
+the two-particle density matrix, n0(x) is the ground state density, and T0(x) is the ground state kinetic stress tensor
+defined by Eq. (59).
+Equation (F2) possesses an analytic solution in the limit of strong Cou lomb interaction e2m2ω0≫1 when the
+ground state wave function reduces to the following asymptotic fo rm
+Ψ0(x1,x2) =1
+ℓ∞√πe−(x1+x2)2
+2√
+3ℓ2∞/bracketleftbigg
+e−(x1−x2−x0)2
+2ℓ2∞+e−(x1−x2+x0)2
+2ℓ2∞/bracketrightbigg
+, (F3)22
+whereℓ∞= (2/planckover2pi1/√
+3mω0)1
+2, andx0= (2e2/mω2
+0)1
+3is the
+classicaldistancebetweenparticles, i.e., thedistancethat
+minimizes the classical energy of two charged particles in
+the 1D harmonic potential. The corresponding ground
+statedensitytakesthe formoftwowellseparated“blobs”
+n0(x) =1
+λ∞√π/bracketleftbigg
+e−(x−x0/2)2
+λ2∞+e−(x+x0/2)2
+λ2∞/bracketrightbigg
+,(F4)
+where the size λ∞of the density blobs located at x=
+±x0/2 is
+λ∞=/parenleftBigg
+/planckover2pi1(√
+3+1)
+2√
+3mω0/parenrightBigg1
+2
+. (F5)The kinetic stress tensor Eq. (59) for the ground state
+wave function (F3) becomes simply proportional to the
+density:
+T0(x) =√
+3+1
+4ω0n0(x). (F6)
+Another technical observation which simplifies calcu-
+lations in the strong interaction limit is that the cross-
+products of the two exponentials in the square brack-
+ets in (F3) are irrelevant for the expressions of the type
+Ψ0(x,x2)Ψ0(x′,x2) in the limit of x0√mω0≫1. For the
+two-particle density matrix entering the nonlocal term in
+(F2) this implies the following result
+2Ψ2
+0(x,x′) =2
+ℓ2∞πe−(x+x′)2
+√
+3ℓ2∞/bracketleftbigg
+e−(x−x′−x0)2
+ℓ2∞+e−(x−x′+x0)2
+ℓ2∞/bracketrightbigg
+. (F7)
+Simplification of the integral term in the equation of
+motion (F2) comes from the fact that in the limit of
+x0√mω0≫1the paircorrelationfunction (F7) ispeaked
+at|x−x′| ∼x0(thiskeepstheparticlesatadistanceclose
+to the classical value). Hence in the integral kernel the
+interaction factor can be approximated as
+∂2
+xw(x−x′)≈2e2
+|x−x′|3≈2e2
+x3
+0=mω2
+0.(F8)Substituting (F6) and (F8) into the (F2) and using the
+obvious identity
+n0(x) =/integraldisplay
+dx′2Ψ2
+0(x,x′),
+we reduce the equation of motion to the following form
+m/bracketleftbig
+ω2−2ω2
+0/bracketrightbig
+n0(x)u(x) =−mω2
+0/integraldisplay
+dx′2Ψ2
+0(x,x′)u(x′)+1
+4m∂2
+x[n0(x)∂2
+xu(x)]−3ω0√
+3+1
+4∂x[n0(x)∂xu(x)].(F9)
+Now the following observations are in order:
+(i) The integro-differential operator in (F9) (in fact in
+the original Eq. (F2)) is symmetric under inversion of x.
+Hence the solutions can be classified by parity u±(x) =
+±u±(−x). Therefore it is sufficient to consider Eq. (F9)
+only in the region of positive x;
+(ii) In (F9) for x >0 all local terms contain n0(x)
+which is a narrow Gaussian located at x∼x0/2. There-
+fore these terms are nonzero only around x0/2;
+(iii) The integralkernel Ψ2
+0(x,x′) forx>0is aproduct
+of two Gaussian peaks, one at x+x′∼0, and another at
+x−x′∼x0, which confines xto the region of the right
+density blob, x∼x0/2, andx′to the region of the left
+blob,x′∼ −x0/2.
+Therefore for positive xall terms in (F9) are nonzero
+only in the region x∼x0/2, while the integration re-
+gion in the nonlocal (interaction) term is confined by the
+Gaussianfactorsto x′∼ −x0/2. Notethatforthisreasonthe integral term will contribute with opposite signs to
+the equations of motion for the modes of opposite parity.
+To further simplify the eigenvalue problem in the
+strong coupling limit we proceed as follows. (i) Con-
+sidering Eq. (F9) in the region x>0 we make a shift of
+coordinates x→x+x0/2, andx′→x′−x0/2. After
+that because of the Gaussian factors the integration can
+be extended to the whole axis. This completely elimi-
+natesx0(i.e. the coupling constant) from the problem,
+as it should be in the strong coupling limit; (ii) Go to di-
+mensionless coordinates ξ=x/λ∞, andξ′=x′/λ∞; (iii)
+Divide everything by the ground state density n0(which
+is simply a Gaussian located at the origin after the above
+shift,n0(ξ) =e−ξ2/√π).
+The result of these three steps is the following dimen-
+sionless equation of motion23
+/parenleftbiggω2
+ω2
+0−2/parenrightbigg
+u±=3eξ2∂2[e−ξ2∂2u±]
+(√
+3+1)2−3√
+3
+2eξ2∂[e−ξ2∂u±]∓√
+3+1
+2√π31/4∞/integraldisplay
+−∞dξ′e−(√
+3+1)2
+4√
+3/parenleftBig
+ξ′+√
+3−1√
+3+1ξ/parenrightBig2
+u±(ξ′).(F10)
+To solve the eigenvalue problem (F10) we employ the following identities for Hermite polynomials43
+ex2∂x/parenleftBig
+e−x2∂xHk/parenrightBig
+=−2kHk, ex2∂2
+x/parenleftBig
+e−x2∂2
+xHk/parenrightBig
+= 4k(k−1)Hk, (F11)
+a+1
+2√aπ∞/integraldisplay
+−∞dx′e−(a+1)2
+4a(x′+a−1
+a+1x)2
+Hk(x′) = (−1)k/parenleftbigga−1
+a+1/parenrightbiggk
+Hk(x). (F12)
+From the identities (F11)and (F12) we seethat Hermite polymomials a rethe eigenfunctions of eachofthree terms in
+the integro-differentialoperator(both for odd and for even mod es) on the right hand side in (F10). The corresponding
+eigenvalues (one should apply the identity (F12) with a=√
+3) are
+/parenleftbiggω±
+k
+ω0/parenrightbigg2
+= 2+34k(k−1)
+(√
+3+1)2+3√
+3k∓(−1)k/parenleftBigg√
+3−1√
+3+1/parenrightBiggk
+, (F13)
+wherek= 0,1,2,...is the quantum number labeling the eigenmodes (for every nthere are two modes of opposite
+parity). Note that the second, third, and fourth terms on the rig ht hand side of Eq. (F13) are, respectively, the
+eigenvalues of the first, second, and the third terms on the right h and side in Eq. (F10). With a little algebra we
+simplify the eigenvalues of Eq. (F13) as follows
+ω±
+k=ω0/bracketleftBig
+2+3√
+3k+6k(k−1)(2−√
+3)∓(−1)k(2−√
+3)k/bracketrightBig1/2
+. (F14)
+In the physical units of length the eigenfunctions in the whole space take the form
+n0(x)u±
+k(x)∼e−(x−x0/2)2
+λ2∞Hk/parenleftbiggx−x0/2
+λ∞/parenrightbigg
+±(−1)ke−(x+x0/2)2
+λ2∞Hk/parenleftbiggx+x0/2
+λ∞/parenrightbigg
+. (F15)
+Finally, the displacement eigenmodes normalized by the condition
+/integraldisplay
+dxn0(x)u±
+k(x)u±
+l(x) =δkl (F16)
+can be written as follows
+u±
+k(x) =1√
+2k+1k!e−(x−x0/2)2
+λ2∞Hk/parenleftBig
+x−x0/2
+λ∞/parenrightBig
+±(−1)ke−(x+x0/2)2
+λ2∞Hk/parenleftBig
+x+x0/2
+λ∞/parenrightBig
+e−(x−x0/2)2
+λ2∞+e−(x+x0/2)2
+λ2∞. (F17)
+Equations (F14) and (F17) give the asymptotically exact
+solutions of the elastic eigenvalue problem in the limitof strong correlations. In Sec. IV C we have used this
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+44To be completely accurate, we point out that the hamilto-
+nianˆ˜H(t), which governs the time evolution of the quan-
+tum state |˜ψ(t)/angbracketrightin the co-moving reference frame, does
+not coincide with the instantaneously deformed hamilto-
+nianˆH[u]. The difference arises from the fact that the
+coordinate transformation to the co-moving frame is time-
+dependent, and this generates an additional vector poten-
+tial (also afunctional of u), which guarantees thevanishing
+ofthe currentdensityin theco-movingframe. Happily, this
+vector potential becomes irrelevant in the high-frequency
+limit, and therefore does not contribute to the elastic ap-
+proximation proposed in this paper.
+45Alternatively, we could set V0= 0 and include the q=
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+n[S(q)−1].
\ No newline at end of file
diff --git a/1001.0617.txt b/1001.0617.txt
new file mode 100644
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@@ -0,0 +1,663 @@
+arXiv:1001.0617v2  [astro-ph.CO]  15 Mar 2010Mon. Not.R. Astron. Soc. 000, 000–000 (0000) Printed 29 October 2018 (MN L ATEX style filev2.2)
+FractalDimensionas ameasureofthescaleofHomogeneity
+JaswantK. Yadav1,2, J.S. Bagla2andNishikanta Khandai3
+1Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun -gu, Seoul 130722, South Korea
+2Harish-Chandra Research Institute, Chhatnag Road,Jhusi, Allahabad 211019, INDIA
+3McWilliams Center for Cosmology, Carnegie Mellon Universi ty, Pittsburgh, PA15213, U.S.A.
+E-Mail: jaswant@kias.re.kr, jasjeet@hri.res.in, nkhand ai@andrew.cmu.edu
+29 October 2018
+ABSTRACT
+In the multi-fractal analysis of large scale matter distrib ution, the scale of transition to ho-
+mogeneityisdefinedasthescaleabovewhichthefractaldime nsion(Dq)ofunderlyingpoint
+distributionis equaltotheambientdimension(D)ofthespaceinwhichpointsaredis tributed.
+With finite sized weakly clustered distribution of tracers o btained from galaxy redshift sur-
+veysit isdifficult to achievethisequality.Recently Bagla et al. (2008) havedefinedthe scale
+of homogeneity to be the scale above which the deviation ( ∆Dq) of fractal dimension from
+theambientdimensionbecomessmallerthanthestatistical dispersionof ∆Dq,i.e.,σ∆Dq.In
+this paper we use the relation between the fractal dimension sand the correlation function to
+compute σ∆Dqfor any given model in the limit of weak clustering amplitude . We compare
+∆Dqandσ∆Dqfor theΛCDM model and discuss the implicationof this comparisonfor the
+expectedscaleofhomogeneityintheconcordantmodelofcos mology.Weestimatetheupper
+limit to the scale of homogeneity to be close to 260h−1Mpc for the ΛCDM model. Actual
+estimatesofthescaleofhomogeneityshouldbesmallerthan thisaswehaveconsideredonly
+statistical contribution to σ∆Dqand we have ignored cosmic variance and contributions due
+to survey geometry and the selection function. Errors arisi ng due to these factors enhance
+σ∆Dqand as∆Dqdecreases with increasing scale, we expect to measurea smaller scale of
+homogeneity. We find that as long as non linear correction to t he computation of ∆Dqare
+insignificant, scale of homogeneity does not change with epo ch. The scale of homogeneity
+dependsveryweaklyonthechoiceoftracerofthedensityfiel d.Thusthesuggesteddefinition
+ofthescale ofhomogeneityisfairlyrobust.
+Keywords: cosmology: theory,largescale structureoftheuniverse— m ethods:statistical
+1 INTRODUCTION
+One of the primary aims of galaxy redshift surveys is to deter mine
+the distribution of luminous matter in the Universe (Huchra et al.
+1983; Lillyetal. 1995; Shectmanet al. 1996; Huchra et al. 19 99;
+Yorket al. 2000; Colless etal. 2001; Giavaliscoet al. 2004;
+LeF` evre et al. 2005; Scovilleet al. 2007). These surveys ha ve re-
+vealed a large variety of structures starting from groups an d clus-
+ters of galaxies, extending to super-clusters and an interc onnected
+network of filaments which appears to extend across very larg e
+scales (Bond, Kofman, & Pogosyan 1996; Springel etal. 2005;
+Faucher-Gigu` ere, Lidz,& Hernquist 2008). We expect the ga laxy
+distributiontobehomogeneousonlargescales.Infact,the assump-
+tion of large scale homogeneity and isotropy of the universe is the
+basis of most cosmological models (Einstein 1917). In addit ion to
+determiningthelargescalestructures,theredshiftsurve ysofgalax-
+ies can also be used to verify whether the galaxy distributio n does
+indeed become homogeneous at some scale(Einasto&Gramann
+1993; Martinez et al. 1998; Yadavet al. 2005; SylosLabini et al.
+2009a,b). Fractal dimensions of the galaxy matter distribu tion can
+be used to test the conjecture of homogeneity. One advantage ofusing fractal dimensions over other analyses is that in the f ormer
+we don’t require the assumption of an average density in the p oint
+set(Jones et al.2004).
+When doing our analysis we often work with volume limited
+sub-samples extracted from the the full magnitude samples o f the
+galaxies. Thisis done inorder toavoidanexplicit use of the selec-
+tion function. The volume limited sub-samples constructed in this
+manner from a flux limited parent sample naturally have a much
+smallernumberofgalaxies.Thiswasfoundtobetoorestrict ivefor
+the earliest surveys and corrections for varying selection function
+were used explicitly in order to determine the scale of homog ene-
+ity(Bharadwaj et al.1999; Amendola & Palladino1999).Butw ith
+modern galaxyredshift surveys, thislimitationis less sev ere.
+Making a volume limited sub-sample requires assumption of
+acosmological modelandthismaybethought ofasanundesira ble
+feature of data analysis. However, if we directly use raw dat a and
+do not account for a redshift dependent selection function i n any
+way, it is obvious that the selection function will dominate in any
+large scale description of the distribution of galaxies. Th is is only
+c/circlecopyrt0000 RAS2Yadav, Baglaand Khandai
+to be expected as we see only brighter galaxies at larger dist ances
+ina fluxormagnitude limitedsample.
+For a given survey one computes the fractal dimension of the
+pointsetunderconsiderationandthescalebeyondwhichthe fractal
+dimension is equal tothe ambient dimension of the space inwh ich
+the particles are distributed is referred to as the scale of h omo-
+geneity of that distribution. Mathematically the fractal d imension
+is defined for an infinite set of points. Given that the observa tional
+samples are finite there is a need to understand the deviation s in
+the fractaldimensions arisingdue tothefinitenumber of poi nts.In
+practice we should identify a scale to be the scale of homogen eity
+where the rmserror in the fractal dimension is comparable to or
+greater than the deviation of the fractal dimension from the ambi-
+ent dimension (Bagla etal. 2008). It is not possible to disti nguish
+between a given point set and a homogeneous point set with the
+same number of points in the same volume beyond the scale of
+homogeneity .
+In all practical cases of interest the fractal dimension is n ever
+equal to the ambient integer dimension of the space. Bagla et al.
+(2008) have shown that these deviations in fractal dimensio n oc-
+cur mainly due to weak clustering present in the galaxy distr ibu-
+tion,withasmaller contributionarisingdue tothefinite nu mber of
+galaxies in the distribution. This work assumed the standar d cos-
+mological model in order to derive a relation between the fra ctal
+dimension on one hand, and, a combination of number density o f
+tracers of the density field and clustering on the other. It wa s as-
+sumedthattheclusteringisweakatscalesofinterest.Inth ispaper,
+we revisit the expected deviation of fractal dimension for a finite
+distribution of weakly clustered points and verify the rela tions de-
+rived in Bagla et al. (2008). For this purpose we have used a pa rti-
+cledistributionfromalargevolume N−Bodysimulations.Further,
+wegeneralisetherelationbetweenclusteringandfractald imension
+to compute the rmserror in determination of fractal dimensions
+and clustering from data. We then discuss the implications o f this
+for the expected scale of homogeneity in the standard cosmol ogi-
+cal model. We alsocomment on some recent determinations of t he
+scale of homogeneity from observations of galaxy distribut ion in
+redshift surveys.
+The plan of the paper is as follows. In section §2 we present
+a brief introduction to fractals and fractal dimensions. Su bsection
+§2.1 describes the fractal dimension for a weakly clustered d istri-
+bution of finite number of points. Section §3 discusses the calcu-
+lation of variance in ξand hence the variance in offset to fractal
+dimension. We present the results insection §4, and conclude with
+a summary ofthe paper in §5.
+2 FRACTALSANDFRACTALDIMENSIONS
+The name fractalwas introduced by Benoit B. Mandelbrot
+(Mandelbrot 1982) to characterise geometrical figures whic h may
+notbesmoothorregular.Oneofthedefinitionsofafractalis thatit
+isashapemadeofpartssimilartothewholeinsomeway.Itisu se-
+ful to regard a fractal as a set of points that has properties s uch as
+those listedbelow, rather than tolook for a more precise defi nition
+which will almost certainly exclude some interesting cases . A set
+Fis a fractalif itsatisfiesmost of the following(Falconer 20 03):
+(i)Fhas a fine structure, i.e.,detail onarbitrarilysmall scale s.
+(ii)Fis too irregular to be described in traditional geometrical
+language, both locallyandglobally.
+(iii)Fhassomeformofself-similarity,perhaps approximate or
+statistical.(iv) In most cases of interest Fis defined ina very simple way,
+perhaps recursively.
+Fractals are characterised using the so called fractal dime nsions.
+These can be defined in different ways, which do not necessari ly
+coincidewithoneanother.Therefore,animportantaspecto fstudy-
+ingafractalstructureisthechoiceofadefinitionforfract aldimen-
+sionthat best applies tothe case instudy.
+Fractaldimensionprovidesadescriptionofhowmuchspacea
+point set fills. It is also a measure of the prominence of the ir regu-
+laritiesof a point set when viewed at a given scale. We may reg ard
+the dimension as an index of complexity. We can, in principle , use
+the concept of fractaldimensions tocharacterise anypoint set.
+Thesimplestdefinitionofthefractaldimensionisthesocal led
+Box counting dimension . Here we place a number of mutually ex-
+clusiveboxesthatcovertheregioninspacecontainingthep ointset
+and count the number of boxes that contain some of the points o f
+thefractal.Thefractionofnon-emptyboxesclearlydepend s onthe
+size of boxes. Box counting dimension of a fractal distribut ion is
+defined in terms of non empty boxes N(r)of radius rrequired to
+cover the distribution. If
+N(r)∝rDb(1)
+we define Dbto be the box counting dimension
+(Barab´ asi & Stanley 1995). In general Dbis a function of
+scale. One of the difficulties with such a definition is that it does
+not depend on the number of particles inside the boxes and rat her
+depends only on the number of boxes. It provides very limited
+information about the degree of clumpiness of the distribut ion and
+ismoreofageometricalmeasure.Togetmoredetailedinform ation
+on clustering of the distribution we use the concept of corre lation
+dimension. We have chosen the correlation dimension, among
+various other definitions (see e.g. Borgani 1995; Mart´ ınez &Saar
+2002), due to its mathematical simplicity and the ease with w hich
+it can be adapted to calculations for a given point set. The fo rmal
+definition of correlation dimension demands that the number of
+particles in the distribution should be infinite (e.g. N→ ∞in
+equation 2). We use a working definition for a finite point set.
+Calculation of the correlation dimension requires the intr oduction
+of correlationintegral given by
+C2(r) =1
+NMM/summationdisplay
+i=1ni(r) (2)
+Hereweassumethatwehave Npointsinthedistribution. Misthe
+number of centres on which spheres of radius rhave been placed
+with the requirement that the entire sphere lies inside the d istribu-
+tionofpoints.Spheresarecentredonpointswithinthepoin tsetand
+itisclearthat M < N.Hereni(r)denotes thenumber ofparticles
+withina distance rfromithpoint:
+ni(r) =N/summationdisplay
+j=1Θ(r− |xi−xj|) (3)
+whereΘ(x)is the Heaviside function. If we consider the distribu-
+tion function for the number of points inside such spherical cells,
+wecan rewrite C2interms ofthe probability of finding nparticles
+inasphere of radius r.
+C2(r) =1
+NN/summationdisplay
+n=0nP(n;r,N) (4)
+c/circlecopyrt0000 RAS,MNRAS 000, 000–000Thescaleof homogeneity 3
+whereP(n;r,N)is the normalised probability of getting nout
+ofNpoints as neighbours inside a radius rof any of the points.
+The correlation dimension D2of the distribution of points can
+be defined via the power law scaling of correlation integral, i.e.,
+C2(r)∝rD2.
+D2(r) =∂logC2(r)
+∂logr(5)
+Since the scaling behaviour of C2can be different at different
+scales, we expect the correlation dimension to be a function of
+scale. For the special case of a homogeneous distribution we see
+thatthecorrelationdimensionequals the ambientdimensio n foran
+infiniteset of points.
+C2(r)definedinthemanner givenbyequation 4provides the
+averageofthedistributionfunctionof P(n;r,N).Inordertochar-
+acterise the distribution of points, we need information ab out all
+the moments of the distribution function. This leads us to th e gen-
+eralised dimension Dq, also known as Minkowski-Bouligand di-
+mension. This is a generalisation of the correlation dimens ionD2.
+The correlationintegral canbe generalised todefine Cq(r)as
+Cq(r) =1
+NN/summationdisplay
+n=0nq−1P(n)≡1
+N∝an}bracketle{tNq−1∝an}bracketri}htp (6)
+whichis usedtodefine the Minkowski-Bouligand dimension
+Dq(r) =1
+q−1dlogCq(r)
+dlogr(7)
+Weexpect thegeneralised dimensiontobe scaledependent in gen-
+eral.Ifthevalueof Dq(r)isindependentofboth qaswellas rthen
+the point distributionis calledamono-fractal.
+If we are dealing with a finite number of points in a finite
+volume then we can always define an average density. This al-
+lows us to relate the generalised correlation integral with correla-
+tion functions. For q >1, the generalised correlation integral and
+the Minkowski-Bouligand dimension can be related to a combi na-
+tion ofn-point correlation functions with the highest nequal toq
+and the smallest nequal to2. The contribution to Cqis dominated
+by regions of higher number density of points for q≫1whereas
+forq≪0the contribution is dominated by regions of very low
+number density. This clearly implies that the full spectrum of gen-
+eralised dimension gives us information about the entire di stribu-
+tion:regionscontainingclustersofpointsaswellasvoids thathave
+few points.
+This allows us to connect the concept of fractal dimensions
+with the statistical measures used to quantify the distribu tion of
+matter at large scales in the universe. In the situation wher e the
+galaxy distribution is homogeneous and isotropic on large s cales,
+we intuitively expect Dq≃D= 3independent of the value of
+q. Whereas at smaller scales we expect tosee a spectrum of valu es
+for the fractal dimension, all different from 3. It is of considerable
+interest tofind out the scale where we can consider the univer se to
+behomogeneous. Weaddressthisissueusingtheoreticalmod elsin
+this paper.
+In an earlier paper we have derived a leading order expres-
+sion for generalised correlation integral for homogeneous as well
+as weakly clustered distributions of points (Bagla et al. 20 08). We
+found that even for a homogeneous distribution of finite numb er
+of points the Minkowski-Bouligand dimension, Dq(r), is not ex-
+actly equal to the ambient dimension, D. The difference in these
+two quantities arises due to discreteness effects. Hence fo r a finite
+sample of points, the correct benchmark is not Dbutthe estimated
+value ofDqfor a homogeneous distribution of same number ofpoints in the same volume. An interesting aside is that the co rrec-
+tion due to a finite size sample always leads to a smaller value for
+Dq(r)thanD. As expected, this correction is small if the average
+number of points inspheres islarge,i.e., ¯N≫1.
+Bagla et al.(2008)havedemonstratedthatthegeneralexpre s-
+sion formthorder moment of a weakly clustered distribution of
+points is givenby:
+∝an}bracketle{tNm∝an}bracketri}htp=¯Nm/bracketleftbigg
+1+(m)(m−1)
+2¯N+m(m+1)
+2¯ξ
++O/parenleftbig¯ξ2/parenrightbig
++O/parenleftbigg¯ξ
+¯N/parenrightbigg
++O/parenleftBig1
+¯N2/parenrightBig/bracketrightbigg
+(8)
+where
+¯N=nV&¯ξ(r) =3
+r3r/integraldisplay
+0ξ(x)x2dx (9)
+isthe average number of particles ina randomly placedspher e and
+the volume averaged twopoint correlationfunction respect ively.
+Thegeneralisedcorrelationintegral(eq:6)forthisdistr ibution
+cannow be writtenas
+NCq(r) =∝an}bracketle{tNq−1∝an}bracketri}htp
+=¯Nq−1/bracketleftbigg
+1+(q−1)(q−2)
+2¯N+q(q−1)
+2¯ξ
++O/parenleftbig¯ξ2/parenrightbig
++O/parenleftbigg¯ξ
+¯N/parenrightbigg
++O/parenleftBig1
+¯N2/parenrightBig/bracketrightbigg
+(10)
+The third term on the right hand side in fact encapsulates the con-
+tribution of clustering. This differs from the last term in t he corre-
+sponding expressionforahomogeneous distributionasinth atcase
+the“clustering”isonlyduetocellsbeingcentredatpoints whereas
+inthiscasethelocationsofeverypairofpointshasaweakco rrela-
+tion. It is worth noting that the highest order term of order O/parenleftbig¯ξ2/parenrightbig
+has a factor O/parenleftbig
+q3/parenrightbig
+and hence can become important for suffi-
+cientlylarge q.Thismaybequantifiedbystatingthat q¯ξ≪1isthe
+more relevant smallparameter for thisexpansion. The Minko wski-
+Bouliganddimension for this system canbe expressed as:
+Dq(r)≃D/parenleftbigg
+1−(q−2)
+2¯N/parenrightbigg
++q
+2∂¯ξ
+∂logr
+=D/parenleftbigg
+1−(q−2)
+2¯N/parenrightbigg
++qr
+2∂¯ξ
+∂r
+=D/parenleftbigg
+1−(q−2)
+2¯N−q
+2/parenleftbig¯ξ(r)−ξ(r)/parenrightbig/parenrightbigg
+=D−(∆Dq)¯N−(∆Dq)clus
+∆Dq(r) =−(∆Dq)¯N−(∆Dq)clus(11)
+Fora weaklyclustered distributionwenote that
+•Forhierarchicalclustering,bothtermshave thesamesigna nd
+leadtoa smallervalue for Dqas compared to D.
+•Unless the correlation function has a feature at some scale,
+smaller correlation corresponds to a smaller correction to the
+Minkowski-Bouligand dimension.
+•If the correlation function has a feature then it is possible to
+have a small correction term (∆Dq)clusfor a relativelylarge ξ. In
+suchasituation,therelationbetween ξand(∆Dq)clusisnolonger
+one toone. Insuch acase (∆Dq)clusdoes not vary monotonically
+withscale.
+c/circlecopyrt0000 RAS,MNRAS 000, 000–0004Yadav, Baglaand Khandai
+Figure1. Comparison of ∆D4calculated usingequation 7(solidline), and
+estimated using equation 11(dashed line) for the large volu meN−Body
+simulation.
+The model described here has been validated with the
+multinomial-multi-fractalmodel (see e.g.Bagla et al.200 8).Inthe
+followingdiscussion,wevalidateourmodelwiththehelpof alarge
+volumeN-Bodysimulationinordertocheckitinthesettingw here
+we wishtouse it.
+2.1N-BodySimulations
+We use a the TreePM code for cosmological simulations (Bagla
+2002; Bagla & Ray 2003; Khandai & Bagla 2009) to simulate the
+distribution of particles. The simulations were run with th e set
+of cosmological parameters favoured by WMAP5 (Komatsu et al .
+2009)asthebestfitforthe ΛCDMclassofmodels: Ωnr= 0.2565,
+ΩΛ= 0.7435,ns= 0.963,σ8= 0.796,h= 0.719, and,
+Ωbh2= 0.02273. The simulations were done with 5123particles
+ina comoving cube of side 1024h−1Mpc.
+We computed the two point correlation function ξ(r)directly
+fromtheN−Bodysimulationoutputbyusingasubsampleofparti-
+cles(Peebles1980;Kaiser1984).Thevolumeaveragedcorre lation
+function¯ξ(r)follows from ξ(r)usingequation 9.
+Figure 1presents the comparison between ∆Dqestimateddi-
+rectly using equation 7, from the N−Body simulation, and, the
+values computed using equation 11. For the last curve we use t he
+valuesof ξ(r)and¯ξ(r)computedfromthesimulation.Wefindthat
+the two curves track each other and the differences are less t han
+10%at all scales larger than 60h−1Mpc. This is fairly impressive
+given that we have only taken the leading order contribution s into
+account. This validates the relation between the correlati on func-
+tions and the fractal dimensions in the limit of weak cluster ing.
+We find that as we go to larger scales, ∆Dqbecomes smaller but
+does notvanish.One maythenask,istherenoscalewheretheu ni-
+verse becomes homogeneous? The answer tothisquestion lies ina
+comparison of the offset ∆Dqwith the dispersion expected due to
+statisticalerrors, wediscuss thisindetail inthe followi ngsection.3 VARIANCEIN FRACTALDIMENSION
+In this section we use the relation between the two point corr ela-
+tionfunction,numberdensityofpointsandthefractaldime nsionto
+estimate the statistical error in determination of the frac tal dimen-
+sion. We have shown in Bagla etal. (2008) that for most tracer s
+of the large scale density field in the universe, the contribu tion of
+the finite number density of points is much smaller than the co n-
+tribution of clustering in terms of the deviation of fractal dimen-
+sion from the ambient dimension. In the following discussio n we
+assume that the contribution of a finite number density of poi nts
+can be ignored. With this assumption, we elevate the relatio n be-
+tween the fractal dimension and the two point correlation fu nction
+to the dispersion of the two quantities. The statistical err or in the
+Minkowski-Bouligand Dimension can then be estimated from t he
+statisticalerror inthe correlation function.
+Var{∆Dq} ≃Var{(∆Dq)clus} (12)
+Thisimplies that
+Var{∆Dq} ≃Var{Dq
+2/parenleftbig¯ξ(r)−ξ(r)/parenrightbig
+}
+=/parenleftBigDq
+2/parenrightBig2/parenleftbig
+Var{¯ξ(r)}+Var{ξ(r)}
++2Cov{¯ξ(r)ξ(r)}/parenrightbig
+(13)
+Wecanmakeuse ofthefactthat |Cov(x,y)| ≤σxσywherexand
+yare random variables. Weget:
+/vextendsingle/vextendsingleσ¯ξ(r)−σξ(r)/vextendsingle/vextendsingle≤2σ∆Dq
+Dq≤/vextendsingle/vextendsingleσ¯ξ(r)+σξ(r)/vextendsingle/vextendsingle (14)
+If we find that one of the σ¯ξ(r)orσξ(r)is much larger than the
+other thenwe get:
+σ∆Dq≃Dq
+2Max/parenleftbig
+σ¯ξ(r),σξ(r)/parenrightbig
+(15)
+The problem isthen reduced tothe evaluation of statistical error in
+the correlationfunction.
+Starting point in the analytical estimate of the statistica l error
+inξ(r)is the assumption that the variance in the power spectrum
+is that expected for Gaussian fluctuations with a shot-noise com-
+ponent arising from the finite number of objects used to trace the
+densityfield(Feldmanetal. 1994; Smith2009):
+σP(k) =/radicalbigg
+2
+V/parenleftBig
+P(k)+1
+¯n/parenrightBig
+, (16)
+whereVisthesimulationvolume,and ¯nisthemeandensityofthe
+objects considered (dark matter particles or halos). Angul o et al.
+(2008)foundgoodagreementbetweenthisexpressionandthe vari-
+ance inP(k)measured from numerical simulations. In order to
+develop a consistent approach, we use 1/¯n= 0in the following
+discussion.
+Thecovarianceofthetwo-pointcorrelationfunctionisdefi ned
+by(Bernstein1994;Cohn 2006;Smithetal. 2008):
+Cξ(r,r′)≡/angbracketleftbig
+(ξ(r)−¯ξ(r))(ξ(r′)−¯ξ(r′))/angbracketrightbig
+=/integraldisplay
+dkk2
+2π2j0(kr)j0(kr′)σ2
+P(k), (17)
+where the last term can be replaced by Eq. (16). The variance i n
+the correlation function is simply σ2
+ξ(r) =Cξ(r,r). Kazinet al.
+(2009) have tested this formula against the variance derive d from
+themockcatalogsforbothdimandbrightgalaxysamplesofSD SS,
+andfound thatat 50< r <100h−1Mpcthe variance isconsistent
+c/circlecopyrt0000 RAS,MNRAS 000, 000–000Thescaleof homogeneity 5
+withequation 17. The direct application of Eq.(17) leads to a sub-
+stantial over prediction of the variance, since it ignores t he effect
+of binning in pair separation which reduces the covariance i n the
+measurement (Cohn2006;Smithetal. 2008).
+An estimate of the correlation function in the ithpair separa-
+tionbinˆξicorresponds tothe shell averaged correlationfunction
+ˆξi=1
+Vi/integraldisplay
+Viξ(r)d3r, (18)
+whereViisthevolumeoftheshell.Thecovariance ofthisestimate
+isgivenbythefollowingexpression,e.g.,see(S´ anchez et al.2008)
+Cˆξ(i,j) =1
+ViVj/integraldisplay
+d3r/integraldisplay
+d3r′Cξ(r,r′)
+=/integraldisplay
+dkk2
+2π2¯j0(k,i)¯j0(k,j)σ2
+P(k), (19)
+where
+¯j0(k,i) =1
+Vi/integraldisplay
+Vij0(kr)d3r. (20)
+is the volume averaged Bessel function. So the variance in ¯ξis
+σ2
+¯ξ(r) =Cˆξ(i,i).Wefindthatatscalesofinterest σ¯ξ(r)≪σξ(r).
+From here, it is straightforward to compute the standard dev iation
+in∆Dqusing equation 15.
+4 SCALEOFHOMOGENEITY
+We can describe a distribution of points to be homogeneous if the
+standarddeviationof ∆Dqisgreaterthan ∆Dq.Notethatthisfor-
+mulation gives us a unique scale: above this scale it is not po ssible
+to distinguish between the given point distribution and a ho moge-
+neous distribution. In the cosmological context, we have by passed
+the details of contribution to the variance arising from the survey
+geometry, survey size, etc. Clearly, if we were to take these con-
+tributions into account, the error in determination of ∆Dqwill be
+larger and hence we will recover a smaller scale of homogenei ty
+(Sarkar etal.2009).Thischangeresultsnotfromanychange inthe
+realscaleofhomogeneitybutbecauseofthelimitationsofo bserva-
+tions.Inourview,the realscaleofhomogeneity isone wherethese
+limitations do not matter. In the limit where we ignore these addi-
+tionalsourcesoferrors,wehavethefollowinggeneralconc lusions:
+•As long as non-linear correction are not important, the scal e
+of homogeneity does not change withepoch.
+•In real space, the scale of homogeneity is independent of the
+tracerused,because thedeviation ∆Dqaswellasthedispersionin
+thisquantityscaleinthesamemanner withbias.Thisfollow sfrom
+the fact that bias in correlation function can be approximat ed by a
+constant number at a given epoch at sufficiently large scales (Bagla
+1998; Seljak2000).
+•Redshift space distortions introduce some bias dependence in
+the scale of homogeneity.
+•As long as our assumption of q¯ξ≪1is valid, the scale of
+homogeneity isthe same forall q.
+These points highlight the robust and unique nature of the sc ale of
+homogeneity definedinthe manner proposed inthis work.
+Wenowturntomorespecificconclusions inthecosmological
+settingbelow.Wehavecalculated σ∆Dqand∆Dqusingthepower
+spectrum of a large N-body simulation as well as using the lin -
+ear power spectrum obtained from WMAP5 parameters. We haveFigure2. Variation of ∆Dqand its predicted standard deviation with scale
+is shown in these plots. The top panel shows these for the ΛCDM simula-
+tiondescribedinthetext,themiddlepanelshowsthesameus ingthelinearly
+evolved powerspectrum andthelowerpanelagain showsthesa mequantity
+with data from simulations patched with the linearly evolve d power spec-
+trum at large scales. ∆Dq/(0.5qD)is shown using a solid curve in each
+panel, the dashed line shows the dispersion σ∆Dq/(0.5qD)and the dot-
+dashed line shows 2σ∆Dq/(0.5qD). The intersection of ∆Dq/(0.5qD)
+andσ∆Dq/(0.5qD)is the scale beyond which we cannot distinguish be-
+tween the ΛCDM modeland ahomogeneous distribution.
+c/circlecopyrt0000 RAS,MNRAS 000, 000–0006Yadav, Baglaand Khandai
+summarised these findings in Figure 2. The top panel in the fig-
+ure shows the deviation and the dispersion as estimated from an
+N-Body simulation. In this case the estimated scale of homog ene-
+ity is just above 320h−1Mpc. The corresponding calculation with
+the linear theory gives a scale of just over 280h−1Mpc. Part of
+the reason for this difference is that the simulation does no t con-
+tain perturbations at very large scales. If we patch the powe r spec-
+trumderivedfromsimulationswiththelinearlyextrapolat edpower
+spectrumoffluctuationsatthesescalesthenweget 260h−1Mpcas
+thescaleofhomogeneity, broadlyconsistentwiththevalue derived
+from linear theory. Curves for this last estimate are shown i n the
+lowestpanelofFigure2.Ifwecompare ∆Dqwith2σ∆Dqinstead,
+thenwegetscalesof 230,215and220h−1Mpc,respectively,inthe
+threecases.Asexpected, increasingtheerrorordispersio nleadsto
+a smaller scale of homogeneity. Once again, the answers deri ved
+using different approaches are broadly consistent with eac h other
+andeventhesmalldeviationscanbeunderstoodintermsofge neric
+aspects of non-linear evolution of perturbations (Bharadw aj 1996;
+Bagla & Padmanabhan 1997;Eisenstein, Seo,& White2007).
+5 SUMMARY
+In this paper we have verified the relation between the two poi nt
+correlation function and fractal dimensions for a simulate d dis-
+tribution of matter at very large scales. This relation indi cates
+(Bagla etal. 2008) that if we use a strongly biased tracer, su ch as
+LuminousRedGalaxiesorclustersofgalaxiesthenthedevia tionof
+thefractaldimensionfrom 3islarger.Thisisconsistentwithobser-
+vations (Einastoet al. 1997). It is also worthwhile to menti on that
+severalobservations havereporteddetectionofexcess clu steringat
+ascaleof 120h−1Mpc(Broadhurst etal.1990;Einastoetal.1997)
+anddistancesbetweenthelargestoverdensitieshavebeeno bserved
+toexceedthescaleofhomogeneityderivedfrommostobserva tions
+(Einasto2009).
+Inthis paper, wehave alsousedthe relationtoestimate stat is-
+tical uncertainty in determination of fractal dimensions. The scale
+of homogeneity isthentaken tobe the scale where thisuncert ainty
+is the same as the offset of fractal dimension from 3. We show ex-
+plicitly that the uncertainty scales with bias in the same ma nner as
+the offset of fractal dimension, thus the true scale of homog ene-
+ity is not sensitive to which objects are used as long as the su rvey
+volume is large enough to contain a sufficient number. The spi rit
+of this paper is to estimate the fractal dimensions for an ide al ob-
+servation, and not to worry about the limitations of current obser-
+vations. The connection between the fractal dimension and c orre-
+lation function allows us to make this leap in the limit of wea k
+clusteringthatisclearlyapplicable atlargescales.Appl ying thisto
+acosmological situationwiththemodel favouredbyWMAP-5o b-
+servations, we have estimatedthe scale ofhomogeneity tobe close
+to260h−1Mpc. Aswehaveignoredseveralsourcesofuncertainty
+thatare likelytobe present inmost observational data sets ,thises-
+timated scale of homogeneity is in some sense the upper limit of
+whatcanbe determinedasthe scaleofhomogeneity fromobser va-
+tions.Itiscomfortingtonotethatthescaleofhomogeneity ismuch
+smaller thanthe Hubble scale.
+Anattractivefeatureofthiswayofdefiningthescaleofhomo -
+geneity is that it can be defined self consistently within the setting
+of the cosmological model withdensity perturbations. Furt her,this
+scaleisindependentofepochandlargelyindependentofthe choice
+of tracer for the large scale density field in the universe. Th e scale
+of homogeneity is the same for the entire spectrum of fractal di-mensions, within the constraints of the underlying assumpt ion that
+q|ξ| ≪1,making this a unique scale inthe problem.
+A comparison with recent determination of the scale of ho-
+mogeneity from observations is pertinent. Observational a nalyses
+have shown that the scale of homogeneity may be as small as
+60−70h−1Mpc(Yadavet al.2005;Hogg etal.2005;Sarkar et al.
+2009).Thisismuchsmallerthanthescaleofhomogeneityweh ave
+found using our method. However, we have ignored the effect o f
+survey geometry and size, and hence in the analysis of any obs er-
+vational dataset there are additional contributions to σ∆Dq. Any
+increase in the value of σ∆Dqleads to a smaller scale of homo-
+geneity.Inthissenseourestimateisthe idealscaleofhomogeneity
+and may be treated as an upper limit. Note that the additional con-
+tributions to σ∆Dqare required to increase its value by more than
+anorder of magnitude above our determination for the scale o f ho-
+mogeneity to be as small as 70h−1Mpc. It should be possible to
+demonstrate consistency with our calculation through an ex plicit
+estimate of errors in observational survey. In the long term we ex-
+pectthatanincrease insurveydepthandsizewillgradually leadto
+loweringoferrorsfromothersourcesandweshouldobtainal arger
+scale of homogeneity.
+ACKNOWLEDGMENTS
+Computational work for this study was carried out at the clus -
+ter computing facility in the Harish-Chandra Research Inst itute
+(http://cluster.hri.res.in/). This research has made use of NASA’s
+Astrophysics Data System. The authors thank Prof. Jaan Eina sto
+for useful suggestions.
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+arXiv:1001.0618v1  [math.AG]  5 Jan 2010THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF
+MARI˜NO-VAFA FORMULA AND ITS APPLICATIONS
+SHENGMAO ZHU
+Abstract. By the same method introduced in [9], we calculate the Laplac e transform of the
+celebrated cut-and-join equation of Mari˜ no-Vafa formula discovered by C. Liu, K. Liu and J.
+Zhou [17]. Then, we study the applications of the polynomial identity (1) obtained in theorem
+1.1 of this paper. We show the proof Bouchard-Mari˜ no conjec ture for C3which was given by
+L. Chen [5] firstly. Subsequently, we will present how to obta in series Hodge integral identities
+from this polynomial identity (1). In particular, the main r esult in [9] is one of special case in
+such series of Hodge integral identities. At last, we give a e xplicit formula for the computation
+of Hodge integral /angbracketleftτbLλgλ1/angbracketrightgwherebL= (b1,...,bl).
+Contents
+1. Introduction 1
+2. The Laplace Transform of Mari˜ no-Vafa Formula 5
+3. The Laplace Transform of The Cut-and-join Equation of Mar i˜ no-Vafa Formula 6
+4. Application I: Proof of the Bouchard-Mari˜ no conjecture forC312
+4.1. Bouchard-Mari˜ no conjecture for C312
+4.2. Residue calculation 15
+4.3. Calculation in new variable v 17
+4.4. Proof of the Bouchard-Mari˜ no conjecture for C319
+5. Application II: Derivation of Some Hodge integral identi ties 23
+5.1. Preliminary calculations 23
+5.2. Expansion of τat∞ 27
+5.3. Expansion of τat 0 29
+5.4. Explicit expression for Hodge integral /an}bracketle{tτbLλgλ1/an}bracketri}htg 32
+6. Conclusion 33
+7. Appendix 34
+7.1. Appendix A 34
+7.2. Appendix B 35
+References 47
+1.Introduction
+In a recent paper [9], Eynard, Mulase and Safnukstated the La place transform of the cut-and-
+joinequationsatisfiedbythepartitionfunctionofHurwitz numbers. Theyobtainedapolynomial
+identity of linear Hodge integrals. As an application, they proved the Bouchard-Mari˜ no conjec-
+ture on Hurwitz numbers. Then, Mulase and Zhang [20] stated t hat this polynomial identity
+can also be used to derive the DVVequation and λg-integral with the same method introduced
+by [14].
+12 SHENGMAO ZHU
+In 2003, C. Liu, K. Liu and J. Zhou [17] proved the celebrated M ari˜ no-Vafa conjecture [19].
+The main step in their proof is to show the generating functio n of Hodge integral with triple λ-
+classesC(λ,p;τ) satisfies the cut-and-join equation. Combining the cut-an d-join restriction from
+combinatorial side formula [21], they finished the proof of M ari˜ no-Vafa formula. Moreover, the
+famous ELSV [7] formula for Hurwitz numbers is a large framin g limit of Mari˜ no-Vafa formula
+[18].
+With the above motivations, we calculate the Laplace transf orm of the cut-and-join equation
+for Mari˜ no-Vafa’s case in the first part of this paper. The ma in result is
+Theorem 1.1. Forg≥1andl≥1, we have the following equation:
+−(τ2+τ)l−2/summationdisplay
+bL≥0/parenleftbigg
+(l−1)(2τ+1)/an}bracketle{tτbLΓg(τ)/an}bracketri}htg+(τ2+τ)/an}bracketle{tτbLd
+dτΓg(τ)/an}bracketri}htg/parenrightbigg
+ˆΨbL(tL;τ) (1)
+−(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgl/summationdisplay
+i=1/parenleftbigg∂
+∂τˆΨbi(ti;τ)+1
+tiτ+1ˆΨbi+1(ti;τ)/parenrightbigg
+ˆΨbL\{i}(tL\{i};τ)
+=−(τ2+τ)l−2
+τ+1/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Γg(τ)/an}bracketri}htgˆΨbL\{i,j}(tL\{i,j};τ)
+·(tj−1)(t2
+iτ+ti)ˆΨa+1(ti;τ)−(ti−1)(t2
+jτ+tj)ˆΨa+1(tj;τ)
+ti−tj
++(τ2+τ)l−1
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0/parenleftig
+(τ2+τ)/an}bracketle{tτa1τa2τbL\{i}Γg−1(τ)/an}bracketri}htg−1
+−stable/summationdisplay
+g1+g2=g
+I/coproducttextJ=L\{i}/an}bracketle{tτa1τbIΓg1(τ)/an}bracketri}htg1/an}bracketle{tτa2τbJΓg2(τ)/an}bracketri}htg2
+2/productdisplay
+n=1ˆΨan+1(ti;τ)ˆΨbL\{i}(tL\{i};τ)
+WhereL={1,2,..,l}is an index set, and for any subset I⊂L, we denote
+tI= (ti)i∈I,bI={bi|i∈I},τbI=/productdisplay
+i∈Iτbi,ˆΨbI(tI,τ) =/productdisplay
+i∈IˆΨbi(ti,τ)
+andΓg(τ) = Λ∨
+g(1)Λ∨
+g(−τ−1)Λ∨
+g(τ),ˆΨn(t;τ) =/parenleftig
+(t2−t)(tτ+1)
+τ+1d
+dt/parenrightign/parenleftig
+t−1
+τ+1/parenrightig
+forn≥0. The last
+summation in the formula is taken over all partitions of g and disjoint subsets I/coproducttextJ=Lsubject
+to the stability condition 2g1−1+|I|>0and2g2−1+|J|>0.
+We remark that theorem 1.1 is equivalent to the symmetrized c ut-and-joint equation of
+Mari˜ no-Vafa formula obtained by L. Chen [4]. We will presen t this equivalence in appendix
+B.
+In [3], V. Bouchard and M. Mari˜ no proposed a conjecture for t he calculation of topological
+string amplitudes of the toric three fold C3based on the recent work [2].(We will call this
+conjectureasBouchard-Mari˜ noconjecturefor C3. TheBourchard-Mari˜ noconjectureforHurwitz
+number proved in [9] is the large framing limit of it). We calc ulate in section 4 that, the
+topological relation in Bouchard-Mari˜ no conjecture for C3is equivalent to the following identityTHE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 3
+of Hodge integral
+(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgdˆΨbL(tL;τ) (2)
+=−(τ2+τ)l−2l/summationdisplay
+i=2/summationdisplay
+a,bL\{1,i}/an}bracketle{tτaτbL\{1,i}Γg(τ)/an}bracketri}htgPa(t1,ti;τ)dt1dtidˆΨbL\{1,i}(tL\{1,i};τ)
++/summationdisplay
+a1,a2≥0
+bL\{1}(τ2+τ)l−1/parenleftig
+(τ2+τ)/an}bracketle{tτa1τa2τbL\{1}Γg−1(τ)/an}bracketri}htg−1
+−stable/summationdisplay
+g1+g2=g
+I/unionsqtextJ=L\{i}/an}bracketle{tτa1τbIΓg1(τ)/an}bracketri}htg1,|I|+1/an}bracketle{tτa2τbJΓg2(τ)/an}bracketri}htg2,|J|+1
+Pa1,a2(t1;τ)dt1dˆΨbL\{1}(tL\{1};τ)
+On the other hand, thanks to the Mari˜ no-Vafa formula, the to pological amplitudes of C3is
+completely determined by the relation (1) in theorem 1.1. In section 4, we will illustrate that
+the identity (2) is implied in (1). Therefore, the Bouchard- Mari˜ no conjecture for C3holds.
+Furthermore,weusetheresultoftheorem1.1toobtainsomeH odgeintegralidentities. Indeed,
+ˆΨb(t,τ) can be written as
+ˆΨb(t;τ) =b/summationdisplay
+k=0τk
+(τ+1)b+1Ψk
+b(t)
+and Ψk
+b(t), 0≤k≤bcould be computed from the recursion relation they defined on .
+At first, we consider the expansion of τat∞. Taking the highest level of formula (1), we get
+Corollary 1.2.
+/summationdisplay
+bL≥0/an}bracketle{tτbLΛ∨
+g(1)/an}bracketri}ht/parenleftigg
+(2g−2+l)ΨbL
+bL(tL)+l/summationdisplay
+i=1(t2
+i−ti)∂
+∂tiΨbi
+bi(ti)ΨbL\{i}
+bL\{i}(tL\{i})/parenrightigg
+=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Λ∨
+g(1)/an}bracketri}htgΨbL\{i,j}
+bL\{i,j}(tj−1)t2
+iΨa+1
+a+1(ti)−(ti−1)t2
+jΨa+1
+a+1(tj)
+ti−tj
++1
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0
+/an}bracketle{tτa1τa2τbL\{i}Λ∨
+g−1(1)/an}bracketri}htg−1+stable/summationdisplay
+g1+g2=g
+I∪J=L\{i}/an}bracketle{tτa1τbIΛ∨
+g1(1)/an}bracketri}htg1/an}bracketle{tτa2τbJΛ∨
+g2(1)/an}bracketri}htg2
+
+×Ψa1+1
+a1+1(ti)Ψa2+1
+a2+1(ti)ΨbL\{i}
+bL\{i}(tL\{i})
+We show in section 5 that Ψb
+b(t) is equal to ˆξb(t) which is defined by ˆξb(t) = ((t3−t2)d
+dt)b(t−1)
+in formula (1.2) in [9]. Hence, the main theorem 1.1 in [9] is a special case of formula (1) in this
+paper. Taking the sub-highest level of theorem 1.1, we also g et another Hodge integral identity
+corollary 5.2 in section 5. Unfortunately, this identity do es not contain any new information.
+Next, we consider the expansion of τatτ= 0. In this case, the lowest level of formula (1) is4 SHENGMAO ZHU
+Corollary 1.3.
+/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgΨ0
+bL(tL) =1
+l−1/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})
+·(tj−1)t2
+iΨ0
+a+1(ti)−(ti−1)t2
+jΨ0
+a+1(tj)
+ti−tj
+We can rederive the λg-integral [13] from Corollary 1.3.
+We also pick up the sub-lowest level of formula (1) and have
+Corollary 1.4.
+/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgl
+−(|bL|+1)Ψ0
+bL(tL)+/summationdisplay
+j=1Ψ1
+bj(tj)Ψ0
+bL\{j}(tL\{j})
+
++/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgl/summationdisplay
+i=1(t2
+i−ti)∂
+∂tiΨ0
+bi(ti)Ψ0
+bL\{i}(tL\{i})+/summationdisplay
+bL≥0/an}bracketle{tτbL3g−3/summationdisplay
+d=g−1Pd(λ)/an}bracketri}htglΨ0
+bL(tL)
+=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})[(tj−1)ti/parenleftbig
+Ψ1
+a+1(ti)+(ti−|bL\{i,j}|−a−3)Ψ0
+a+1(ti)/parenrightbig
+ti−tj
+−(ti−1)tj/parenleftbig
+Ψ1
+a+1(tj)+(tj−|bL\{i,j}|−a−3)Ψ0
+a+1(tj)/parenrightbig
+ti−tj]
++/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg/summationdisplay
+r/\e}atio\slash=i,jΨ1
+br(tr)Ψ0
+bL\{i,j}(tL\{i,j})(tj−1)tiΨ0
+a+1(ti)−(ti−1)tjΨ0
+a+1(tj)
+ti−tj
++/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}3g−3/summationdisplay
+d=g−1Pd(λ)/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})(tj−1)tiΨ0
+a+1(ti)−(ti−1)tjΨ0
+a+1(tj)
+ti−tj
++1
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0stable/summationdisplay
+g1+g2=g
+|I|∪|J|=L\{i}/an}bracketle{tτa1τbIλg1/an}bracketri}htg1/an}bracketle{tτa2τbJλg2/an}bracketri}htg2Ψ0
+a1+1(ti)Ψ0
+a2+1(ti)Ψ0
+bL\{i}(tL\{i})
+where/summationtext3g−3
+d=g−1Pd(λ) = Λ∨
+g(1)a1(λ)anda1(λ) =/summationtextg
+m=1mλg−mλg−(−1)gΛ∨
+g(−1)λg−1.
+The above identity contains Hodge integral of type /an}bracketle{tτbL/summationtext3g−3
+d=g−1Pd(λ)/an}bracketri}htg. By some direct
+calculations,
+Pg−1(λ) =λg−1,Pg(λ) =gλg,Pg+1(λ) =−λgλ1,···,P3g−3(λ) = (−1)g+1λgλg−1λg−2
+As an application of corollary 1.4, we get the following Hodg e integral recursion.
+Theorem 1.5. If/summationtextl
+i=1bi= 2g−4 +l, there exists a constant C(g,l,b1,..,bl)related to
+g,l,b1,..,bl, such that
+/an}bracketle{tτbLλgλ1/an}bracketri}htg=1
+l/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bj−1τbL\{i,j}λgλ1/an}bracketri}htg(bi+bj)!
+bi!bj!+C(g,l,b1,..,bl)
+whereC(g,l,b1,..,bl)is a very verbose combinatoric constant which is given at App endix B.THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 5
+The initial value /an}bracketle{tτ2g−3λgλ1/an}bracketri}htg=1
+12[g(2g−3)bg+b1bg−1] has been computed by Y. Li [15].
+Thus, by the recursion formula in theorem 1.5, we can compute out all the Hodge integral of
+type/an}bracketle{tτbLλgλ1/an}bracketri}htg. In fact, the Hodge integrals /an}bracketle{tτbLPd(λ)/an}bracketri}htgappeared in Corollary 1.4 could also
+be calculated via the same method. But the computation will b e more complicated.
+Acknowledgements. The author would like to thank his advisor Kefeng Liu’s encou rage-
+ment, valuable discussions, and bringing the paper [9] to hi s attention.
+2.The Laplace Transform of Mari ˜no-Vafa Formula
+At first, we introduce some notations followed by [17]. Let
+Λ∨
+g(t) =tg−λ1tg−1+···+(−1)gλg
+be the Chern polynomial of E∨, the dual of the Hodge bundle. For a partition µgiven by
+µ1≥µ2≥ ··· ≥µl(µ)>0,
+let|µ|=/summationtextl(µ)
+i=1µiand Γg(τ) = Λ∨
+g(1)Λ∨
+g(−τ−1)Λ∨
+g(τ), define
+Cg,µ(τ) =−√−1|µ|+l(µ)
+|Aut(µ)|[τ(τ+1)]l(µ)−1l(µ)/productdisplay
+i=1/producttextµi−1
+a=1(µiτ+a)
+(µi−1)!/integraldisplay
+Mg,l(µ)Γg(τ)
+/producttextl(µ)
+i=1(1−µiψi)
+The Deligne-Mumford stack Mg,lis defined as the moduli space of stable curves satisfying
+the stability condition 2 g−2+l>0. For the unstable cases ( g,l) = (0,1) and (0,2), we define/integraldisplay
+M0,11
+1−µψ=1
+µ2
+/integraldisplay
+M0,21
+(1−µ1ψ1)(1−µ2ψ2)=1
+µ1+µ2
+Thus
+C0,d(τ) =−√
+−1d+11
+τ/producttextd−1
+a=0(dτ+a)
+d!d−2
+C0,(µ1,µ2)(τ) =√−1µ1+µ2
+|Aut(µ1,µ2)|τ+1
+τ2/productdisplay
+i=1/producttextµi−1
+a=0(µiτ+a)
+µi!1
+µ1+µ2
+The Mari˜ no-Vafa formula proved in [17] gives a direct combi natorial formula associated to
+representation of symmetric groups for the generation func tion ofCg,µ(τ). But we don’t go
+further to this formula here.
+Through a direct calculation, we have
+Cg,µ(τ) =−√−1|µ|+l(µ)
+|Aut(µ)|[τ(τ+1)]l(µ)−1/summationdisplay
+bi≥0
+i=1,..,l(µ)/an}bracketle{tl(µ)/productdisplay
+i=1τbiΓg(τ)/an}bracketri}htgl(µ)/productdisplay
+i=11
+τ/producttextµi−1
+a=0(µiτ+a)
+µi!µbi
+i (3)
+LetCg(µ;τ) =|Aut(µ)|Cg,µ(τ), the Laplace transform of Cg,µ(τ) is defined by,
+Cg,l(w1,..,wl) =/summationdisplay
+µ∈Nl1
+√−1l+|µ|Cg(µ;τ)e−(µ1w1+···+µlwl)
+where we have let l=l(µ).6 SHENGMAO ZHU
+In order to simplify the above expression of laplace transfo rm, we introduce some new vari-
+ables. Consider the framed Lambert curve x=y(1−y)τ, and the coordinate change x=e−w.
+By Lagrange inversion theorem, L. Chen [4] got,
+y=/summationdisplay
+k≥1/producttextk−2
+a=0(kτ+a)
+k!xk
+Now we introduce the variable tby,
+t= 1+1+τ
+τ/summationdisplay
+k≥1/producttextk−1
+a=0(kτ+a)
+k!e−kw
+and
+ˆΨn(t;τ) =1
+τ/summationdisplay
+k≥1/producttextk−1
+a=0(kτ+a)
+k!kne−kw
+which is a polynomial with top degree 2 n+1 in variable tvia the recursion relation
+ˆΨn+1(t;τ) = (t2−t)tτ+1
+τ+1d
+dtˆΨn(t;τ)
+forn≥0 andˆΨ0(t;τ) =t−1
+τ+1.
+Now, the laplace transform of Cg,µ(τ) can be written in tvariable,
+ˆCg,l(t1,..,tl) =/summationdisplay
+µ∈Nl1
+√−1l+|µ|Cg(µ;τ)e−(µ1w1+···+µlwl)(4)
+=−(τ(τ+1))l−1/summationdisplay
+bi≥0
+i=1,..,l/an}bracketle{tl/productdisplay
+i=1τbiΓg(τ)/an}bracketri}htgl/productdisplay
+i=1ˆΨbi(ti;τ)
+3.The Laplace Transform of The Cut-and-join Equation of Mari ˜no-Vafa
+Formula
+Let us introduce formal variables p= (p1,..,pn,..), and define
+pµ=pµ1···pl(µ)
+for a partition µ. Define generating functions
+Cg,l(λ,p;τ) =/summationdisplay
+µ,l(µ)=lCg(µ;τ)
+|Aut(µ)|pµλ2g−2+l
+C(λ,p;τ) =/summationdisplay
+g≥0
+l≥1Cg,l(λ,p;τ)
+In 2003, C. Liu, K. Liu and J. Zhou [17] proved that C(λ,p;τ) satisfies the cut-and-join
+equation
+∂C
+∂τ=√−1
+2λ/summationdisplay
+i,j≥1((i+j)pipj∂C
+∂pi+j+ijpi+j∂2C
+∂pi∂pj+ijpi+j∂C
+∂pi∂C
+∂pj)
+For every choice of g≥1 and a partition µ, the coefficient of pµλ2g−2+l(µ)isTHE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 7
+∂
+∂τ/parenleftbiggCg(µ;τ)
+|Aut(µ)|/parenrightbigg
+=√
+−1/summationdisplay
+i<j(µi+µj)Cg(µ;τ)
+|Aut(µ(ˆi,ˆj))|(5)
++√−1
+2l/summationdisplay
+i=1/summationdisplay
+α+β=µiαβ
+Cg−1(µ(α,ˆi;τ))
+|Aut(µ(α,ˆi))|+/summationdisplay
+g1+g2=g
+ν1/coproducttextν2=µ(α,ˆi)Cg1(ν1;τ)
+|Aut(ν1|)Cg2(ν2;τ)
+|Aut(ν2)|
+
+Let us first calculate the Laplace transform of the cut-and-j oin equation for the l= 1 case
+which includes Proposition 3.3 in [9] as its special case.
+Proposition 3.1. Forg≥1, the Laplace transform of the cut-and-join equation for the l= 1
+case is:
+−/summationdisplay
+b≥0/an}bracketle{tτbd
+dτΓg(τ)/an}bracketri}htgˆΨb(t;τ)−/summationdisplay
+b≥0/an}bracketle{tτbΓg(τ)/an}bracketri}htg/parenleftbiggd
+dτˆΨb(t;τ)+1
+tτ+1ˆΨb+1(t;τ)/parenrightbigg
+=1
+2/summationdisplay
+a1,a2≥0
+(τ(τ+1))/an}bracketle{tτa1τa2Γg−1(τ)/an}bracketri}htg−1−/summationdisplay
+g1+g2=g
+g1>0
+g2>0/an}bracketle{tτa1Γg1(τ)/an}bracketri}htg1/an}bracketle{tτa2Γg2(τ)/an}bracketri}htg2
+ˆΨa1+1(t;τ)ˆΨa2+1(t;τ)
+Proof.The cut-and-join equation for l= 1 case is
+∂Cg(µ;τ)
+∂τ=√−1
+2/summationdisplay
+α+β=µαβ/parenleftigg
+Cg−1((α,β);τ)
+|Aut((α,β))|+/summationdisplay
+g1+g2=gCg1(α;τ)Cg2(β;τ)/parenrightigg
+(6)
+Then, the Laplace transform of the LHS of (6) is
+∂
+∂τˆCg,1(t;τ) =∂t
+∂τ∂
+∂tˆCg,1(t;τ)+∂
+∂τˆCg,1(t;τ) (7)
+The Laplace transform of the stable part of RHS of (6) is
+1
+2(τ(τ+1))/summationdisplay
+a1,a2≥0/an}bracketle{tτa1τa2Γg−1(τ)/an}bracketri}htg−1ˆΨa1+1(t;τ)ˆΨa2+1(t;τ)
+−1
+2/summationdisplay
+g1+g2=g
+g1>0
+g2>0/an}bracketle{tτa1Γg1(τ)/an}bracketri}htg1/an}bracketle{tτa2Γg2(τ)/an}bracketri}htg2ˆΨa1+1(t;τ)ˆΨa2+1(t;τ)
+The unstable term is C0(α;τ)Cg(β;τ)+Cg(α;τ)C0(β;τ). Because
+C0,d(τ) =−√
+−1d+11
+τ/producttextd−1
+a=0(dτ+a)
+d!1
+d2
+It’s Laplace transform ˆC0(w;τ) =/summationtext
+d≥1−1√−1d+1C0,d(τ)e−wsatisfies:
+d
+dwˆC0(w;τ) =t−1
+τ+1=y
+1−(τ+1)y8 SHENGMAO ZHU
+It is easy to calculated
+dw=(1−y)y
+1−(1+τ)yd
+dy, hence, ˆC0(w;τ) =−ln(1−y). Moreover, remember that
+the framed Lambert curve is y(1−y)τ=x, where y depends on τ. Taking derivation of τ, we
+have the identity,
+−ln(1−y) =∂y
+∂τ1−(τ+1)y
+y(1−y)
+Therefore, the unstable part of RHS of (6) is
+∂y
+∂τ1−(τ+1)y
+y(1−y)(t2−t)(tτ+1)
+τ+1∂
+∂tˆCg,1(t;τ) =∂y
+∂τt2(τ+1)∂
+∂tˆCg,1(t;τ) (8)
+where we have used t=1
+1−(τ+1)y. we also have
+∂t
+∂τ=t2y+∂y
+∂τt2(τ+1) =t2−t
+τ+1+∂y
+∂τt2(τ+1) (9)
+Hence, move the unstable part to left hand side, by (7),(8) an d (9), we get
+/parenleftbigg∂
+∂τ+t2−t
+(τ+1)∂
+∂t/parenrightbigg
+ˆCg,1(t;τ) =stable part
+which is just the Proposition 3.1. /square
+For the general case of l, we need to introduce two lemmas first.
+Lemma 3.2. When(g,l) = (0,2), we have the following Laplace transformation formula:
+ˆC0,2(w1,w2;τ) =−/summationdisplay
+α,β≥1τ+1
+τ1
+α+β/producttextα−1
+a=0(ατ+a)
+α!/producttextβ−1
+a=0(βτ+a)
+β!e−αw1e−βw2
+=−ln/parenleftbiggy1−y2
+x1−x2/parenrightbigg
+−τ(ln(1−y1)+ln(1−y2))
+Proof.By definition,
+ˆC0,2(w1,w2;τ) =/summationdisplay
+α,β≥11
+√−12+α+β|Aut(α,β)|ˆC0,(α,β)
+=−/summationdisplay
+α,β≥1τ+1
+τ1
+α+β/producttextα−1
+a=0(ατ+a)
+α!/producttextβ−1
+a=0(βτ+a)
+β!e−αw1e−βw2
+Thus/parenleftbiggd
+dw1+d
+dw2/parenrightbigg
+ˆC0,2(w1,w2;τ) =−τ(τ+1)ˆΨ0(t1;τ)ˆΨ0(t2;τ)
+=−τ(τ+1)y1y2
+(1−(τ+1)y1)(1−(τ+1)y2)
+Because,
+d
+dw=xd
+dx=(1−y)y
+1−(τ+1)yd
+dy
+Then, it is easy to get
+ˆC0,2(w1,w2;τ) =−ln/parenleftbiggy1−y2
+x1−x2/parenrightbigg
+−τ(ln(1−y1)+ln(1−y2))
+/squareTHE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 9
+Lemma 3.3.
+/summationdisplay
+α,β≥11
+τ/producttext(α+β)−1
+a=0((α+β)τ+a)
+(α+β)!(α+β)a+1e−αwie−βwj
+=xi
+xi−xjˆΨa+1(ti;τ)−xj
+xi−xjˆΨa+1(tj;τ)−ˆΨa+1(ti;τ)−ˆΨa+1(tj;τ)
+Proof.This calculation is the same as formula (3 .15) showed in paper [9].
+Letµ=α+βandν=β, then
+/summationdisplay
+α,β≥11
+τ/producttext(α+β)−1
+a=0((α+β)τ+a)
+(α+β)!(α+β)a+1e−αwie−βwj
+=/summationdisplay
+α,β≥01
+τ/producttext(α+β)−1
+a=0((α+β)τ+a)
+(α+β)!(α+β)a+1e−αwie−βwj
+−/summationdisplay
+α≥11
+τ/producttextα−1
+a=0(ατ+a)
+α!αa+1e−αwi−/summationdisplay
+β≥11
+τ/producttextβ−1
+a=0(βτ+a)
+β!βa+1e−βwj
+=/summationdisplay
+µ≥0µ/summationdisplay
+ν=01
+τ/producttextµ−1
+a=0(µτ+a)
+µ!µa+1e−(µ−ν)wie−νwj−ˆΨa+1(ti;τ)−ˆΨa+1(tj;τ)
+=/summationdisplay
+µ≥01
+τ/producttextµ−1
+a=0(µτ+a)
+µ!µa+1xµ+1
+i−xµ+1
+j
+xi−xj−ˆΨa+1(ti;τ)−ˆΨa+1(tj;τ)
+=xi
+xi−xjˆΨa+1(ti;τ)−xj
+xi−xjˆΨa+1(tj;τ)−ˆΨa+1(ti;τ)−ˆΨa+1(tj;τ)
+/square
+Now we give our main result in this paper.
+Theorem 3.4. Forg≥1andl≥1, we have the following equation:
+−(τ2+τ)l−2/summationdisplay
+bL≥0/parenleftbigg
+(l−1)(2τ+1)/an}bracketle{tτbLΓg(τ)/an}bracketri}htg+(τ2+τ)/an}bracketle{tτbLd
+dτΓg(τ)/an}bracketri}htg/parenrightbigg
+ˆΨbL(tL;τ) (10)
+−(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgl/summationdisplay
+i=1/parenleftbigg∂
+∂τˆΨbi(ti;τ)+1
+tiτ+1ˆΨbi+1(ti;τ)/parenrightbigg
+ˆΨbL\{i}(tL\{i};τ)
+=−(τ2+τ)l−2
+τ+1/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Γg(τ)/an}bracketri}htgˆΨbL\{i,j}(tL\{i,j};τ)
+·(tj−1)(t2
+iτ+ti)ˆΨa+1(ti;τ)−(ti−1)(t2
+jτ+tj)ˆΨa+1(tj;τ)
+ti−tj
++(τ2+τ)l−1
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0/parenleftig
+(τ2+τ)/an}bracketle{tτa1τa2τbL\{i}Γg−1(τ)/an}bracketri}htg−110 SHENGMAO ZHU
+−stable/summationdisplay
+g1+g2=g
+I/coproducttextJ=L\{i}/an}bracketle{tτa1τbIΓg1(τ)/an}bracketri}htg1/an}bracketle{tτa2τbJΓg2(τ)/an}bracketri}htg2
+2/productdisplay
+n=1ˆΨan+1(ti;τ)ˆΨbL\{i}(tL\{i};τ)
+Proof.The Laplace transform of LHS of equation (5) is
+/summationdisplay
+µ∈Nl1
+√−1l(µ)+|µ|∂
+∂τCg(µ;τ)e−(µ1w1+···+µlwl)(11)
+=∂
+∂τ
+/summationdisplay
+µ∈N1
+√−1l(µ)+|µ|Cg(µ;τ)e−(µ1w1+···+µlwl)
+
+=∂
+∂τˆCg,l(t1,..,tl;τ)
+=l/summationdisplay
+i=1∂ti
+∂τ∂
+∂tiˆCg,l(t1,..,tl;τ)+∂
+∂τˆCg,l(t1,..,tl;τ)
+The Laplace transform of stable geometry in the cut-term of R HS of (5) is
+√−1
+2/summationdisplay
+µ∈Nl1
+√−1l(µ)+|µ|l/summationdisplay
+i=1/summationdisplay
+α+β=µiαβ(Cg−1(µ(α,ˆi);τ) (12)
++/summationdisplay
+g1+g2=g
+ν1/coproducttextν2=µ(α,ˆi)
+2g1−2+|ν1|>0
+2g2−2+|ν2|>0Cg1(ν1;τ)Cg2(ν;τ))e−(µ1w1+···+µlwl)
+=(τ2+τ)l−1
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{k}≥0((τ2+τ)/an}bracketle{tτa1τa2τbL\{k}Γg−1(τ)/an}bracketri}htg−1
+−/summationdisplay
+g1+g2=g
+I/coproducttextJ={1,.,ˆi,.l}
+2g1−1+|I|>0
+2g2−1+|J|>0/an}bracketle{tτa1τbIΓg1(τ)/an}bracketri}htg1/an}bracketle{tτa2τbJΓg2(τ)/an}bracketri}htg2)2/productdisplay
+n=1ˆΨan+1(ti;τ)ˆΨbL\{i}(tL\{i};τ)
+The unstable geometry in the cut-term of RHS has two terms for l≥2.
+U1=√−1
+2l/summationdisplay
+i=1/summationdisplay
+α+β=µiαβ/parenleftbigg
+C0(α;τ)·Cg(µˆi,β;τ)
+|Aut(µˆi)|+Cg(µˆi,α;τ)
+|Aut(µˆi,α)|·C0(β;τ)/parenrightbigg
+U2=√−1
+2l/summationdisplay
+i=1/summationdisplay
+α+β=µiαβ/summationdisplay
+j/\e}atio\slash=i/parenleftigg
+C0(µj,α;τ)
+|Aut(µj,α)|·Cg(µˆi,ˆj,β;τ)
+|Aut(µˆi,ˆj)|+Cg(µˆi,ˆj,α;τ)
+|Aut(µˆi,ˆj,α)|·C0(β;τ)/parenrightigg
+As we have calculated in the proof of proposition 3.1, the Lap lace transform of U1is
+l/summationdisplay
+i=1∂yl
+∂τt2
+i(τ+1)∂
+∂tiˆCg,l(t1,..,tl;τ) (13)THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 11
+Moving the formula (13) to left hand side, it will cancel the fi rst term of formula (11), i.e.
+(11)−(13) =/parenleftigg
+∂
+∂τ+l/summationdisplay
+i=1t2
+i−ti
+τ+1∂
+∂ti/parenrightigg
+ˆCg,l(t1,..,tl;τ) (14)
+=−(τ2+τ)l−2/summationdisplay
+bL≥0/parenleftbigg
+(l−1)(2τ+1)/an}bracketle{tτbLΓg(τ)/an}bracketri}htg+(τ2+τ)/an}bracketle{tτbLd
+dτΓg(τ)/an}bracketri}htg/parenrightbigg
+ˆΨbL(tL;τ)
+−(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgl/summationdisplay
+i=1/parenleftbigg∂
+∂τˆΨbi(ti;τ)+1
+tiτ+1ˆΨbi+1(ti;τ)/parenrightbigg
+ˆΨbL\{i}(tL\{i};τ)
+The Laplace transform of U2equals to
+/summationdisplay
+µ∈Pl1
+√−1l+|µ|U2e−(µ1w1+···+µlwl)(15)
+=−l/summationdisplay
+i=1/summationdisplay
+j/\e}atio\slash=i/summationdisplay
+µj≥1
+α≥11
+√−12+µj+ααC0(µj,α;τ)e−µjwje−αwi
+×/summationdisplay
+β≥0
+µk≥0
+k/\e}atio\slash=i,j1
+√−1l+|µ|−α−1βCg(µˆi,ˆj,β;τ)e−βwie−/summationtext
+k/\e}atio\slash=i,jµkwk
+=l/summationdisplay
+i=1/summationdisplay
+j/\e}atio\slash=i
+∂
+∂wiˆC0,2(wi,wj;τ)(τ2+τ)l−2/summationdisplay
+a≥0
+bk≥0,k/\e}atio\slash=i,j/an}bracketle{tτa/productdisplay
+k/\e}atio\slash=i,jτbkΓg(τ)/an}bracketri}htgˆΨa+1(ti;τ)/productdisplay
+k/\e}atio\slash=i,jˆΨbk(tk;τ)
+
+=l/summationdisplay
+i=1/summationdisplay
+j/\e}atio\slash=i/parenleftbigg
+−yi
+yi−yj(1+τyj
+1−(τ+1)yi)+xi
+xi−xj/parenrightbigg
+·(τ2+τ)l−2/summationdisplay
+a≥0
+bk≥0,k/\e}atio\slash=i,j/an}bracketle{tτa/productdisplay
+k/\e}atio\slash=i,jτbkΓg(τ)/an}bracketri}htgˆΨa+1(ti;τ)/productdisplay
+k/\e}atio\slash=i,jˆΨbk(tk;τ)
+= (τ2+τ)l−2/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bk≥0,k/\e}atio\slash=i,j/an}bracketle{tτa/productdisplay
+k/\e}atio\slash=i,jτbkΓg(τ)/an}bracketri}htg/productdisplay
+k/\e}atio\slash=i,jˆΨbk(tk;τ)(xi
+xi−xjˆΨa+1(ti;τ)
+−xj
+xi−xjˆΨa+1(tj;τ)−yi
+yi−yj(1+τyj
+1−(τ+1)yi)ˆΨa+1(ti;τ)+yj
+yi−yj(1+τyi
+1−(τ+1)yj)ˆΨa+1(tj;τ))
+where we have used the lemma 3.3.
+At last, we needtocalculate theLaplace transformof jointe rm inRHS.It’s Laplace transform
+is
+√
+−1/summationdisplay
+µ∈Nl/summationdisplay
+i<j1
+√−1l(µ)+|µ|(µi+µj)Cg(µ(ˆi,ˆj);τ)e−µiwie−µjwje−/summationtext
+k/\e}atio\slash=i,jµkwk(16)12 SHENGMAO ZHU
+=−(τ2+τ)l−2/summationdisplay
+µ∈Nl/summationdisplay
+i<j/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Γg(τ)/an}bracketri}htg/productdisplay
+k/\e}atio\slash=i,j1
+τ/producttextµk−1
+a=0(µkτ+a)
+µk!µbk
+ke−/summationtext
+k/\e}atio\slash=i,jµkwk
+·1
+τ/producttextµi+µj−1
+a=0((µi+µj)τ+a)
+(µi+µj)!(µi+µj)a+1e−µiwie−µjwj
+=−(τ2+τ)l−2/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Γg(τ)/an}bracketri}htgˆΨbL\{i,j}(tL\{i,j};τ)/parenleftbiggxi
+xi−xjˆΨa+1(ti;τ)
+−xj
+xi−xjˆΨa+1(tj;τ)−ˆΨa+1(ti;τ)−ˆΨa+1(tj;τ)/parenrightbigg
+Combining (15) and (16), we get
+(10)+(11) = −(τ2+τ)l−2
+τ+1/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Γg(τ)/an}bracketri}htgˆΨbL\{i,j}(tL\{i,j};τ) (17)
+·(tj−1)(t2
+iτ+ti)ˆΨa+1(ti;τ)−(ti−1)(t2
+jτ+tj)ˆΨa+1(tj;τ)
+ti−tj
+Collecting the remainder equations (12),(14) and (17), we o btain the equation (10) of theorem
+3.4. /square
+4.Application I: Proof of the Bouchard-Mari ˜no conjecture for C3
+Motivated by Eynard and his collaborators’ series works on M atrix model [12, 10, 8, 11] ,
+Bourchard, Klemm, Mari˜ no and Pasquetti propose a new appro ach to compute the topological
+string amplitudes for local Calabi-Yau manifolds [2]. Base d on such proposal, Bouchard and
+Mari˜ no calculated the string amplitude for toric three fol dC3[3]. In this section, we will follow
+the work of [9] and [5] to illustrate that Bouchard-Mar˜ no’s approach is implied by theorem 1.1.
+Following the work of [9], in subsection 4.1, we illustrate B ourchard-Mari˜ no’s proposal of
+calculation the string amplitude of C3( which is also called Bouchard-Mari˜ no conjecture for C3
+). By some residue computations, we obtain a equivalent desc ription of Bourchard-Mar˜ no’s
+approach in subsection 4.2. The calculation in subsections 4.3 and 4.4 to prove the Bouchard-
+Mari˜ no conjecture for C3is firstly done by L. Chen [5]
+4.1.Bouchard-Mari˜ no conjecture for C3.Here we consider the spectral curve
+C:y(1−y)τ=x (18)
+and the variable t=1
+1−(τ+1)y.
+It is easy to see that, variables x,y,thave the relations that
+xd
+dx=(1−y)y
+1−(τ+1)yd
+dy=/parenleftbiggt(t−1)(tτ+1)
+τ+1/parenrightbiggd
+dt
+We defined in section 2 that,
+ˆΨn(t;τ) =/parenleftbigg/parenleftbiggt(t−1)(tτ+1)
+τ+1/parenrightbiggd
+dt/parenrightbiggnt−1
+τ+1
+forn≥0.
+In the following state, we need to introduce the differential,
+Ψn(t;τ) =dˆΨ(t;τ)THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 13
+which can be formulated in variables x,yas
+Ψn(y;τ) =dy1−(τ+1)y
+y(1−y)/parenleftbiggy(1−y)
+1−(τ+1)yd
+dy/parenrightbiggn+11
+(τ+1)(1−(τ+1)y)(19)
+Ψn(x;τ) =1
+τ/summationdisplay
+k≥1/producttextk−1
+a=0(kτ+a)
+k!kn+1xk−1dx (20)
+respectively.
+In [3], in order to formulate their conjecture, Bourchard an d Mari˜ no introduce
+Wg(x1,..,xl;τ) =/summationdisplay
+µ,l(µ)=lzµWg,µ(τ)1
+|Aut(µ)|/summationdisplay
+σ∈Sll/productdisplay
+i=1xµi−1
+σ(i)(21)
+whereWg,µ(τ) =−1√−1g+lCg,µ(τ).
+By (3),(19), (20) and (21), we have
+Wg(x1,..,xl;τ)dx1···dxl= (−1)g+l(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgΨbL(yL;τ)
+Definition 4.1. Let us call the symmetric polynomial differential form
+d⊗lˆCg,l(t1,..,tl;τ) =−(τ(τ+1))l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgΨbL(tL;τ)
+onClthe Mari˜ no-Vafa differential of type ( g,l).
+Thus
+Wg(x1,..,xl;τ)dx1···dxl= (−1)g+l−1d⊗lˆCg,l(t1,..,tl;τ)
+t=1
+1−(τ+1)y, it then follows that y=t−1
+(τ+1)t. Substituting to the spectral curve (18), we have
+t−1
+t/parenleftbigtτ+1
+t/parenrightbigτ1
+(τ+1)τ+1=x. Lets(t) be the solution of function equation
+t−1
+t/parenleftbiggtτ+1
+t/parenrightbiggτ
+=s(t)−1
+s(t)/parenleftbiggs(t)τ+1
+s(t)/parenrightbiggτ
+(22)
+From (22), we have
+ln/parenleftbiggtτ+1
+t/parenrightbigg
+−ln/parenleftbiggs(t)τ+1
+s(t)/parenrightbigg
+=1
+τ/parenleftbigg
+ln/parenleftbiggs(t)−1
+s(t)/parenrightbigg
+−ln/parenleftbiggt−1
+t/parenrightbigg/parenrightbigg
+(23)
+It is clear that the spectral curve C:y(1−y)τ=xhas only one ramification point of the
+x-projection, which we denote by ν. It is given by y(ν) =1
+τ+1. Then, we can find two points q
+andqon the curve such that x(q) =x(q) near the ramification point ν. Let us write
+y(q) =1
+τ+1−z, y(q) =1
+τ+1−P(z)
+with
+P(z) =−z+O(z2)
+From functional equation y(q)(1−y(q))τ=y(q)(1−y(q))τ, we can solve exact P(z) as a power
+series inz,
+P(z) =−z−2(−1+τ2)
+3τz2−4(−1+τ2)2
+9τ2z3−2(1+τ)3(−22+57τ−57τ2+22τ3)
+135τ3z4+···
+Note thatP(z) is an involution P(P(z)) =z.14 SHENGMAO ZHU
+In terms of the t-coordinate, the involution P(z) corresponds to s(t) in (22). We have,
+(24)
+
+t(q) =1
+1−(τ+1)y(q)=1
+τ+11
+z=t
+t(q) =1
+1−(τ+1)y(q)=1
+τ+11
+P(z)=s(t)
+It follows that
+(τ+1)dt
+(t2−t)(tτ+1)=(τ+1)ds(t)
+s(t)2(s(t)−1)=dx
+x
+Using the coordinate tof spectral curve C, the Bergman kernel is defined by
+B(t1,t2) =dt1dt2
+(t1−t2)
+Define a 1-form on Cby,
+dE(q,q;t2) =1
+2/integraldisplayq
+qB(·,t2) =1
+2/parenleftbigg1
+t1−t2−1
+s(t1)−t2/parenrightbigg
+dt2
+where the integral is taken with respect to the first variable ofB(t1,t2) along any path from q
+toq. The natural holomorphic symplectic form on C×Cis given by Ω = dln(1−y)∧ln(x).
+Again, let us introduce another 1-form on the curve Cby,
+ω(q,q) =/integraldisplayq
+qΩ(·,x) = (ln(1−y(q))−ln(1−y(q)))dx
+x
+=1
+τ(lny(q)−lny(q))dx
+x
+=1
+τ/parenleftbigg
+ln/parenleftbigg
+1−1
+t/parenrightbigg
+−ln/parenleftbigg
+1−1
+s(t)/parenrightbigg/parenrightbigg(τ+1)
+t2−tdt
+(tτ+1)
+The kernel operator is defined as the quotient
+K(t1,t2;τ) =dE(q,q,t2)
+ω(q,q)=τ
+2(τ+1)(s(t1)−t1)(t2
+1−t1)(t1τ+1)
+(t1−t2)(s(t1)−t2)(ln(1−1
+t1)−ln(1−1
+s(t1)))1
+dt1dt2
+It is clear that K(s(t1),t2;τ) =K(t1,t2;τ).
+Definition 4.2. The topological recursion formula is an inductive mechanis m of defining a
+symmetricl-formWg,l(t1,..,tl;τ) onClfor (g,l) subject to 2 g−2+l>0 by
+Wg,l+1(t0,tL;τ) =−1
+2πi/contintegraldisplay
+γ∞[K(t,t0;τ)(Wg−1,l+2(t,s(t),tL;τ)+
+l/summationdisplay
+i=1/parenleftbig
+Wg,l(t,tL\{i})B(s(t),ti)+Wg,l(s(t),tL\{i};τ)B(t,ti)/parenrightbig
++stable/summationdisplay
+g1+g2=g,I∪J=LWg1,|I|+1(t,tI;τ)Wg2,|J|+1(s(t),tJ;τ)
+
+
+wheretI= (ti)i∈Ifor a subset I⊂L, and the last sum is taken over all partitions of gand
+disjoint decompositions I/coproducttextJ=Lsubject to the stability condition 2 gi−1 +|I|>0 and
+2g2−1 +|J|>0. The integration is taken with respect to dton the contour γ∞, which is aTHE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 15
+positively oriented loop in the complex t-plane of large radius such that |t|>max(|t0|,s(t0)) for
+t∈γ∞.
+Remark 4.3. The original topological recursion formula for Wg,l(t1,..,tl;τ) was formulated by
+taken the residue at z= 0 [3]. But here, for simplicity, we have changed the coordin ate fromz
+totbyt=1
+(τ+1)z. Hence, we get the contour integral in definition 4.2.
+Now, we can formulate the Bouchard-Mari˜ no conjecture for C3.
+Conjecture 4.4. For everygandlsubject to the stability condition 2g−2+l>0, the topological
+recursion formula with the initial condition
+(25)
+
+W0,3(t1,t2,t3;τ) =d⊗3ˆC0,3(t1,t2,t3;τ) =−τ2
+(τ+1)dt1dt2dt3
+W1,1(t1;τ) =−dˆC1,1(t1;τ) =1
+24((1+τ+τ2)Ψ0(t1;τ)−τ(τ+1)Ψ1(t1;τ))
+gives the Mari˜ no-Vafa differential with a signature.
+Wg,l(t1,..,tl;τ) = (−1)g+l−1d⊗lˆCg,l(t1,..,tl;τ) (26)
+4.2.Residue calculation. In this subsection, we calculate the residues(contour inte grals) ap-
+pearing definition 4.2. In fact, we have to calculate the foll owing two types of residues:
+i)Ra,b(t;τ) =−1
+2πi/contintegraltext
+γ∞K(t′,t;τ)Ψa(t′;τ)Ψb(s(t′);τ)
+ii)Rn(t,ti;τ) =−1
+2πi/contintegraltext
+γ∞K(t′,t;τ)(Ψn(t′;τ)B(s(t′),ti)+Ψn(s(t′);τ)B(t′,ti))
+We need one lemma first,
+Lemma 4.5. In thez-coordinate, the kernel K(t′(z),t;τ)has the expansion,
+K=/parenleftbiggτ2
+2(τ+1)3z−1−(τ−1)
+2(τ+1)2+3τ2t2+2τ(1−τ)t−4τ
+6(τ+1)z
++/parenleftbiggτ−τ2
+6t2+τ2−2τ+1
+9t+(τ+1)
+18/parenrightbigg
+z2+···/parenrightbigg
+dt1
+dz
+In particular, the coefficients of zin above expansion are polynomials of t.
+Proof.Substituting t′=1
+(τ+1)zands(t′) =1
+(τ+1)s(z)toK(t′,t;τ) and calculate directly, we
+obtain lemma 4.5. /square
+Definition 4.6. For a laurent series/summationtext
+n∈Zantn, we denote
+/bracketleftigg/summationdisplay
+n∈Zantn/bracketrightigg
++=/summationdisplay
+n>0antn
+Now, we have two theorems for the calculation of Ra,b(t;τ) andRn(t,ti;τ).
+Theorem 4.7.
+Ra,b(t;τ) =−1
+2πi/contintegraldisplay
+γ∞K(t′,t;τ)Ψa(t′;τ)Ψb(s(t′);τ)
+=−τ
+2
+1
+ln/parenleftbig
+1−1
+t/parenrightbig
+−ln/parenleftig
+1−1
+s(t)/parenrightig/parenleftig
+Ψa(t;τ)ˆΨb+1(s(t);τ)+ˆΨa+1(s(t);τ)Ψb(t;τ)/parenrightig
+
++16 SHENGMAO ZHU
+Proof.We note that, in term of the original z-coordinate of [3], the residue Ra,b(t;τ) is the
+coefficient of z−1inK(t′,t;τ)Ψa(t′;τ)Ψb(s(t′);τ), after expanding it in the Laurent series in z.
+Ψn(t′;τ) is a polynomial in t′=1
+(τ+1)z, thus the contribution to the z−1term in the expression
+is a polynomial in tby lemma 4.5. Hence Ra,b(t;τ) is a polynomial in t.
+Let us write Ψ n(t′;τ) =fn(t′;τ)dt′. Letγǫbe a positively oriented loop, such that function
+(t′2−t′)(t′τ)+1
+ln/parenleftbig
+1−1
+t′/parenrightbig
+−ln/parenleftig
+1−1
+s(t′)/parenrightigs′(t′)fa(t′;τ)fb(s(t′);τ) (27)
+has no singularity on and outside loop γǫ. On the compact set γǫwe have a bound Mǫ,
+|(t′2−t′)(t′τ)+1
+ln/parenleftbig
+1−1
+t′/parenrightbig
+−ln/parenleftig
+1−1
+s(t′)/parenrightigs′(t′)fa(t′;τ)fb(s(t′);τ)|<Mǫ
+Choose|t|>>1. Then,
+|1
+2πi/contintegraldisplay
+γǫK(t′,t;τ)Ψ(t′;τ)Ψ(s(t′);τ)|
+=|1
+2πi/contintegraldisplay
+γǫτ
+2(τ+1)/parenleftbigg1
+t′−t−1
+s(t′)−t/parenrightbigg(t′2−t′)(t′τ+1)
+ln/parenleftbig
+1−1
+t′/parenrightbig
+−ln/parenleftig
+1−1
+s(t′)/parenrightigs′(t′)fa(t′;τ)fb(s(t′);τ)dt′|dt
+<τMǫ
+4π(τ+1)/contintegraldisplay
+γǫ|1
+t′−t−1
+s(t′)−t|dt′dt∼τMǫ
+4π(τ+1)|t|dt
+Thus, we have
+1
+2πi/contintegraldisplay
+γ∞K(t′,t;τ)Ψa(t′;τ)Ψb(s(t′);τ) =1
+2πi/contintegraldisplay
+γ∞−γǫK(t′,t;τ)Ψa(t′;τ)Ψb(s(t′);τ)+O(t−1)
+=/bracketleftbigg1
+2πi/contintegraldisplay
+γ∞−γǫK(t′,t;τ)Ψa(t′;τ)Ψb(s(t′);τ)/bracketrightbigg
++
+1
+2πi/contintegraldisplay
+γ∞−γǫK(t′,t;τ)Ψa(t′;τ)Ψb(s(t′);τ)
+=τ
+4(τ+1)πi/contintegraldisplay
+γ∞−γǫ/parenleftbigg1
+t′−t−1
+s(t′)−t/parenrightbigg(t′2−t′)(t′τ+1)
+ln/parenleftbig
+1−1
+t′/parenrightbig
+−ln/parenleftig
+1−1
+s(t′)/parenrightigs′(t′)fa(t′;τ)fb(s(t′);τ)dt′dt
+=τ
+2(τ+1)(t2−t)(tτ+1)
+ln/parenleftbig
+1−1
+t/parenrightbig
+−ln/parenleftig
+1−1
+s(t′)/parenrightigs′(t′)fa(t;τ)fb(s(t);τ)dt
++τ
+2(τ+1)(s(t)2−s(t))(s(t)τ+1)
+ln/parenleftbig
+1−1
+t/parenrightbig
+−ln/parenleftig
+1−1
+s(t)/parenrightigfa(s(t);τ)fb(t;τ)dt
+=τ
+2(τ+1)(t2−t)(tτ+1)
+ln/parenleftbig
+1−1
+t/parenrightbig
+−ln/parenleftig
+1−1
+s(t)/parenrightigs′(t)(fa(t;τ)fb(s(t);τ)+fa(s(t);τ)fb(t;τ))dt
+=τ
+21
+ln/parenleftbig
+1−1
+t/parenrightbig
+−ln/parenleftig
+1−1
+s(t)/parenrightig/parenleftig
+Ψa(t;τ)ˆΨb+1(s(t);τ)+Ψa+1(s(t);τ)ˆΨb(t;τ)/parenrightig
+/square
+Similarly, with the same proof as showed above, we haveTHE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 17
+Theorem 4.8.
+Rn(t,ti;τ) =−1
+2πi/contintegraldisplay
+γ∞K(t′,t;τ)/parenleftbig
+Ψn(t′;τ)B(s(t′),ti)+Ψn(s(t′);τ)B(t′;ti)/parenrightbig
+=−τ
+1
+ln/parenleftbig
+1−1
+t/parenrightbig
+−ln/parenleftig
+1−1
+s(t)/parenrightig/parenleftig
+ˆΨn+1(t;τ)B(s(t),ti)+ˆΨn+1(s(t);τ)B(t,ti)/parenrightig
+
++
+Let us define polynomials Pa,b(t;τ) andPn(t,ti;τ) by
+Pa,b(t;τ)dt=−τ(τ+1)
+2
+1
+ln/parenleftbig
+1−1
+t/parenrightbig
+−ln/parenleftig
+1−1
+s(t)/parenrightig/parenleftig
+ˆΨa+1(t;τ)ˆΨb+1(s(t);τ) (28)
++ˆΨa+1(s(t);τ)ˆΨb+1(t;τ)/parenrightigdt
+(t2−t)(tτ+1)/bracketrightbigg
++
+Pn(t,ti;τ)dtdti=−τ
+1
+ln/parenleftbig
+1−1
+t/parenrightbig
+−ln/parenleftig
+1−1
+s(t)/parenrightig/parenleftig
+ˆΨn+1(t;τ)B(s(t),ti)+ˆΨn+1(s(t);τ)B(t,ti)/parenrightig
+
++(29)
+With the above notations, the Bourchard-Mari˜ no conjectur e forC3is equivalent to
+(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgdˆΨbL(tL;τ)(30)
+=−(τ2+τ)l−2l/summationdisplay
+i=2/summationdisplay
+a,bL\{1,i}/an}bracketle{tτaτbL\{1,i}Γg(τ)/an}bracketri}htgPa(t1,ti;τ)dt1dtidˆΨbL\{1,i}(tL\{1,i};τ)
++/summationdisplay
+a1,a2≥0
+bL\{1}(τ2+τ)l−1/parenleftig
+(τ2+τ)/an}bracketle{tτa1τa2τbL\{1}Γg−1(τ)/an}bracketri}htg−1
+−stable/summationdisplay
+g1+g2=g
+I/unionsqtextJ=L\{i}/an}bracketle{tτa1τbIΓg1(τ)/an}bracketri}htg1,|I|+1/an}bracketle{tτa2τbJΓg2(τ)/an}bracketri}htg2,|J|+1
+Pa1,a2(t1;τ)dt1dˆΨbL\{1}(tL\{1};τ)
+4.3.Calculation in new variable v.Now, let us introduce a new variable vbyx=e−w=
+e−1
+2v2. Asy(1−y)τ=xandt=1
+1−(τ+1)y, we have
+/parenleftbiggt−1
+(τ+1)t/parenrightbigg/parenleftbigg
+1−t−1
+(τ+1)t/parenrightbiggτ
+=e−1
+2v2
+Thus
+v2=−2/parenleftbigg
+ln/parenleftbigg
+1−1
+t/parenrightbigg
++τln/parenleftbigg
+1+1
+τt/parenrightbigg
++τln(τ)−(τ+1)ln(τ+1)/parenrightbigg
+= 2
+/summationdisplay
+n≥21
+n/parenleftbigg
+(−1)n+1
+τn−1/parenrightbigg1
+tn+(τ+1)ln(τ+1)−τln(τ)
+18 SHENGMAO ZHU
+Solving the above functional equation,
+v=1
+t
+/radicalbigg
+τ+1
+τ+/summationdisplay
+k≥1ak(τ)1
+tk
+ (31)
+and
+−v=1
+s(t)
+/radicalbigg
+τ+1
+τ+/summationdisplay
+k≥1ak(τ)1
+s(t)k
+ (32)
+Taking the inverse of (31), we have
+1
+t=v
+/radicalbiggτ
+τ+1+/summationdisplay
+k≥1bk(τ)vk
+ (33)
+and
+1
+s(t)= (−v)
+/radicalbiggτ
+τ+1+/summationdisplay
+k≥1bk(τ)(−v)k
+ (34)
+Lemma 4.9. Let
+η−1(v;τ) =1
+2/parenleftig
+ˆΨ−1(t;τ)−ˆΨ−1(s(t);τ)/parenrightig
+=−1
+2/parenleftbigg
+ln/parenleftbiggtτ+1
+tτ/parenrightbigg
+−ln/parenleftbiggs(t)τ+1
+s(t)τ/parenrightbigg/parenrightbigg
+=1
+2τ/parenleftbigg
+ln/parenleftbigg
+1−1
+t/parenrightbigg
+−ln/parenleftbigg
+1−1
+s(t)/parenrightbigg/parenrightbigg
+and defineηn+1(v;τ) =−1
+vd
+dvηn(v;τ)whenn≥ −1, then
+ηn(v;τ) =1
+2/parenleftig
+ˆΨn(t;τ)−ˆΨn(s(t);τ)/parenrightig
+Moreover, there exists formal series Fn(w;τ)such thatηn(v;τ) =ˆΨn(t;τ)+Fn(w;τ).
+Proof.LetˆΨ−1(t;τ) =−ln/parenleftbigtτ+1
+tτ/parenrightbig
+be the solution of differential equation
+(tτ+1)(t2−t)
+τ+1d
+dtˆΨ−1(t;τ) =ˆΨ0(t;τ) =t−1
+τ+1
+then
+η−1(v;τ) =1
+2/parenleftig
+ˆΨ−1(t;τ)−ˆΨ−1(s(t);τ)/parenrightig
+=−1
+2/parenleftbigg
+ln/parenleftbiggtτ+1
+tτ/parenrightbigg
+−ln/parenleftbiggs(t)τ+1
+s(t)τ/parenrightbigg/parenrightbigg
+By definition,
+xd
+dx=−d
+dw=−1
+vd
+dv=(tτ+1)(t2−t)
+τ+1d
+dt=(s(t)τ+1)(s(t)2−s(t))
+τ+1d
+ds(t)(35)
+Therefore
+ηn(v;τ) =1
+2/parenleftig
+ˆΨn(t;τ)−ˆΨ(s(t);τ)/parenrightig
+Moreover,
+η−1(v;τ)−ˆΨ−1(t;τ) (36)THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 19
+=−1
+2/parenleftig
+ˆΨ−1(t;τ)+ˆΨ−1(s(t);τ)/parenrightig
+=1
+2/parenleftbigg
+ln/parenleftbigg
+1+1
+tτ/parenrightbigg
++ln/parenleftbigg
+1+1
+s(t)τ/parenrightbigg/parenrightbigg
+=1
+2
+/summationdisplay
+n≥1(−1)n−1
+n1
+τn/parenleftbigg1
+tn+1
+s(t)n/parenrightbigg
+
+By (33) and (34), we note that right hand side of (36) is a serie s of variable w=1
+2v2, we write
+it asF−1(w;τ). (36) is equivalent to
+η−1(v;τ) =ˆΨ−1(t;τ)+F−1(w;τ)
+Let us define
+Fn(w;τ) =/parenleftbigg
+−d
+dw/parenrightbiggn+1
+F−1(w;τ) (37)
+then, we have
+ηn(v;τ) =ˆΨn(t;τ)+Fn(w;τ)
+/square
+Corollary 4.10.
+Pa,b(t;τ)dt=−1
+2/parenleftbiggηa+1(v;τ)ηb+1(v;τ)
+η−1(v;τ)vdv|v=v(t)/parenrightbigg
++
+Proof.By (23) and (28), we have
+1
+21
+ln/parenleftbig
+1+1
+τt/parenrightbig
+−ln/parenleftig
+1+1
+τs(t)/parenrightig/parenleftig
+ˆΨa+1(t;τ)ˆΨb+1(s(t);τ)+ˆΨa+1(s(t);τ)ˆΨb+1(t;τ)/parenrightig(τ+1)dt
+(t2−t)(tτ+1)
+=1
+2η−1(v;τ)(τ+1)dt
+(t2−t)(tτ+1)/parenleftigg
+(ˆΨa+1(t;τ)−ˆΨa+1(s(t);τ))(ˆΨb+1(t;τ)−ˆΨb+1(s(t);τ))
+4
+−(ˆΨa+1(t;τ)+ˆΨa+1(s(t);τ))(ˆΨb+1(t;τ)+ˆΨb+1(s(t);τ))
+4/parenrightigg
+=−vdv
+2η−1(v;τ)(ηa+1(v;τ)ηb+1(v;τ)−Fa+1(w;τ)Fb+1(w;τ))
+Noticing that/parenleftbiggFa+1(w;τ)Fb+1(w;τ)vdv
+η−1(v;τ)|v=v(t)/parenrightbigg
++= 0
+we complete the proof. /square
+4.4.Proof of the Bouchard-Mari˜ no conjecture for C3.In the proof of the theorem 3.4,
+we know that LHS of (10) is equal to
+d
+dτˆCg,l(t1,t2,..,tl;τ)−l/summationdisplay
+i=1∂yi
+∂τt2
+i(τ+1)∂
+∂tiˆCg,l(t1,t2,..,tl;τ)
+=∂
+∂τˆCg,l(t1,t2,..,tl;τ)+l/summationdisplay
+i=1∂vi
+∂τ∂
+∂viˆCg,l(v1,v2,..vl;τ)−l/summationdisplay
+i=1∂yi
+∂τt2
+i(τ+1)∂
+∂tiˆCg,l(t1,t2,..tl;τ)20 SHENGMAO ZHU
+Recall the following relationship of x,y,t,v.
+x=e−1
+2v2,y(1−y)τ=x,t=1
+1−(τ+1)y
+Thus
+∂v1
+∂τ= 0,∂y
+∂τ=y(1−y)ln(1−y)
+y(τ+1)−1,(t2−t)(tτ+1)
+τ+1d
+dt=−1
+vd
+dv
+After some simple calculation,
+LHS of (10) =∂
+∂τˆCg,l(t1,t2,..,tl;τ)+l/summationdisplay
+i=1ln(1−yi)/parenleftbigg
+−1
+vid
+dvi/parenrightbigg
+ˆCg,l(t1,t2,..,tl;τ)
+The direct image of a function f(v) onCvia the projection π:C→Cis:
+π∗f=f(v)+f(−v)
+Thus,
+π∗(ˆCg,l(t1,t2,..,tl;τ)) =−(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htg/parenleftig
+ˆΨb1(t1;τ)+ˆΨb1(s(t1);τ)/parenrightig
+ˆΨb\{1}(tL\{1};τ)
+By the definition of Fb(w;τ),
+π∗/parenleftbigg∂
+∂τˆCg,l(t1,t2,..,tl;τ)/parenrightbigg
+=∂
+∂τ/parenleftig
+π∗ˆCg,l(t1,t2,..,tl;τ)/parenrightig
+=∂
+∂τ
+(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgFb1(w1;τ)ˆΨbL\{1}(tL\{1})
+
+is regular in w1. We will use O(w1) to denote some function regular in w1.
+Consider the direct sum
+π∗/parenleftiggl/summationdisplay
+i=1ln(1−yi)/parenleftbigg
+−1
+v1d
+dv1/parenrightbigg
+ˆCg,l(t1,t2,..,tl;τ)/parenrightigg
+wheni= 1,
+π∗/parenleftbigg
+ln(1−y1)/parenleftbigg
+−1
+v1d
+dv1/parenrightbigg
+ˆCg,l(t1,t2,..,tl;τ)/parenrightbigg
+=−(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htg/bracketleftbigg
+ln/parenleftbiggt1τ+1
+(τ+1)t1/parenrightbigg/parenleftbigg
+−1
+v1d
+dv1/parenrightbigg
+(ηb1(v1;τ)−Fb1(w1;τ))
++ln/parenleftbiggs(t1)τ+1
+(τ+1)s(t1)/parenrightbigg/parenleftbigg
+−1
+v1d
+dv1/parenrightbigg
+(ηb1(v1;τ)−Fb1(w1;τ))/bracketrightbigg
+ˆΨb\{1}(tb\{1})
+=−(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htg/bracketleftbigg/parenleftbigg
+ln/parenleftbiggt1τ+1
+(τ+1)t1/parenrightbigg
+−ln/parenleftbiggs(t1)τ+1
+(τ+1)s(t1)/parenrightbigg/parenrightbigg
+ηb1(v1;τ)
++/parenleftbigg
+ln/parenleftbiggt1τ+1
+(τ+1)t1/parenrightbigg
+−ln/parenleftbiggs(t1)τ+1
+(τ+1)s(t1)/parenrightbigg/parenrightbigg
+Fb1(w1;τ)/bracketrightbigg
+ˆΨb\{1}(tb\{1})
+= (τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htg2η−1(v1;τ)ηb1+1(v1;τ)ˆΨbL\{1}(tL\{1};τ)+O(w1)THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 21
+and it is easy to see that
+π∗/parenleftiggl/summationdisplay
+i=2ln(1−yi)/parenleftbigg
+−1
+v1d
+dv1/parenrightbigg
+ˆCg,l(t1,t2,..,tl;τ)/parenrightigg
+=O(w1)
+Thus
+π∗(LHS of (10)) = ( τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htg2η−1(v1;τ)ηb1+1(v1;τ)ˆΨbL\{1}(tL\{1};τ)+O(w1)
+(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgdˆΨbL(tL;τ)
+= (τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgdv1d
+dv1/parenleftig
+ˆΨb1(t1;τ)/parenrightig
+dˆΨbL\{1}(tL\{1};τ)
+= (−v1dv1)d2···dl
+(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htg/parenleftbigg
+−1
+v1d
+dv1/parenrightbigg
+(ηb1(v1;τ)−Fb1(w1;τ))
+ˆΨbL\{1}(tL\{1})
+=−v1dv1
+2η−1(v1;τ)d2···dl
+(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htg2η−1(v1;τ)(ηb1+1(v1;τ)−Fb1(w1;τ))
+ˆΨbL\{1}(tL\{1})
+=−v1dv1
+2η−1(v1;τ)d2···dl
+π∗(LHS of (10)) −2η−1(v1;τ)Fb1(w1;τ)(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgˆΨbL\{1}(tL\{1})
+−O(w1))
+It is important to note that
+/parenleftbigg−v1dv1O(w1)
+2η−1(v1;τ)|v1=v1(t1)/parenrightbigg
++= 0
+and
+/parenleftbig
+−v1dv1Fb1+1(w1;τ)|v1=v1(t1)/parenrightbig
++= 0
+Thus,
+(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgdˆΨbL(tL;τ) =/parenleftbigg
+−v1dv1
+2η−1(v1;τ)d2···dl(π∗(LHS of (10)))/parenrightbigg
+(38)
+Then we need to calculate/parenleftig
+−v1dv1
+2η−1(v1;τ)d2···dlπ∗(T1)|v1=v1(t1)/parenrightig
++
+Let us write T1=T11+T12, where
+T11=−(τ2+τ)l−2
+(τ+1)/summationdisplay
+2≤j≤l/summationdisplay
+a≥0
+bL\{1,j}≥0/an}bracketle{tτaτbL\{1,j}Γg(τ)/an}bracketri}htgˆΨbL\{1,j}(tL\{1,j};τ)
+(tj−1)(t2
+1τ+t1)ˆΨa+1(t1;τ)−(t1−1)(t2
+jτ+tj)ˆΨa+1(tj;τ)
+t1−tj
+T12=−(τ2+τ)l−2
+(τ+1)/summationdisplay
+2≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Γg(τ)/an}bracketri}htgˆΨbL\{i,j}(tL\{i,j};τ)22 SHENGMAO ZHU
+(tj−1)(t2
+iτ+ti)ˆΨa+1(ti;τ)−(ti−1)(t2
+jτ+tj)ˆΨa+1(tj;τ)
+ti−tj
+By the definition,
+/parenleftbigg−v1dv1
+2η−1(v1;τ)π∗T12|v1=v1(t1)/parenrightbigg
++= 0 (39)
+we have that
+−v1dv1
+2η−1(v1;τ)dj/parenleftigg
+(tj−1)(t2
+1τ+t1)ˆΨa+1(t1;τ)
+t1−tj/parenrightigg
+(40)
+=1
+2η−1(v1;τ)dj/parenleftbigg(τ+1)dt1
+(t2
+1−t1)(t1τ+1)(tj−1)(t2
+1τ+t1)
+t1−tjˆΨa+1(t1;τ)/parenrightbigg
+=τ+1
+2η−1(v1;τ)ˆΨa+1(t1;τ)dt1dtj
+(t1−tj)2
+=τ+1
+2η−1(v1;τ)ˆΨa+1(t1;τ)B(t1,tj;τ)
+Similarly,
+−v1dv1
+2η−1(v1;τ)dj/parenleftigg
+(tj−1)(s(t1)2τ+s(t1))ˆΨa+1(s(t1);τ)
+s(t1)−tj/parenrightigg
+(41)
+=τ+1
+2η−1(v1;τ)ˆΨa+1(s(t1);τ)B(s(t1),tj;τ)
+Moreover, by the expansion,
+t1−1
+t1−tj+s(t1)−1
+s(t1)−tj=/summationdisplay
+k≥0/parenleftigg
+tk
+j
+tk
+1+tk
+j
+tk+1
+1+tk
+j
+s(t1)k+tk
+j
+s(t1)k+1/parenrightigg
+we have
+/parenleftbigg−v1dv1
+2η−1(v1;τ)/parenleftbiggt1−1
+t1−tj+s(t1)−1
+s(t1)−tj/parenrightbigg
+|v1=v1(t1)/parenrightbigg
++= 0 (42)
+Therefore
+/parenleftbigg−v1dv1
+2η−1(v1;τ)d2···dlπ∗T1|v1=v1(t1)/parenrightbigg
++by (39) (43)
+=/parenleftbigg−v1dv1
+2η−1(v1;τ)d2···dlπ∗T11|v1=v1(t1)/parenrightbigg
++by (40), (41), (42)
+= (τ2+τ)l−2/summationdisplay
+2≤j≤l/summationdisplay
+a≥0
+bL\{1,j}/an}bracketle{tτaτbL\{1,j}Γg(τ)/an}bracketri}htgdˆΨbL\{1,j}(tL\{1,j};τ)
+/parenleftiggˆΨa+1(t1;τ)B(t1,tj)+ˆΨa+1(s(t1);τ)B(s(t1),tj)
+2η−1(v1;τ)|v1=v1(t1)/parenrightigg
++
+=−(τ2+τ)l−2/summationdisplay
+2≤j≤l/summationdisplay
+a≥0
+bL\{1,j}/an}bracketle{tτaτbL\{1,j}Γg(τ)/an}bracketri}htgdˆΨbL\{1,j}(tL\{1,j};τ)THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 23
+/parenleftiggˆΨa+1(t1;τ)B(s(t1),tj)+ˆΨa+1(s(t1);τ)B(t1,tj)
+2η−1(v1;τ)|v1=v1(t1)/parenrightigg
++
+=−(τ2+τ)l−2/summationdisplay
+2≤j≤l/summationdisplay
+a≥0
+bL\{1,j}/an}bracketle{tτaτbL\{1,j}Γg(τ)/an}bracketri}htgPa+1(t1,tj;τ)dt1dtjdˆΨbL\{1,j}(tL\{1,j};τ)
+Finally, we need to calculate/parenleftig
+−v1dv1
+2η−1(v1;τ)d2···dlπ∗(T2+T3)|v1=v1(t1)/parenrightig
++
+Because/parenleftbigg−v1dv1
+2η−1(v1;τ)π∗/parenleftig
+ˆΨa1+1(t1;τ)ˆΨa2+1(t1;τ)/parenrightig
+|v1=v1(t1)/parenrightbigg
++
+=/parenleftbigg−v1dv1
+2η−1(v1;τ)/parenleftig
+ˆΨa1+1(t1;τ)ˆΨa2+1(t1;τ)+ˆΨa1+1(s(t1);τ)ˆΨa2+1(s(t1);τ)/parenrightig
+|v1=v1(t1)/parenrightbigg
++
+=/parenleftbigg−v1dv1
+2η−1(v1;τ)(2ηa1+1(v1;τ)ηa2+1(v1;τ)+2Fa1+1(w1;τ)Fa2+1(w1;τ))|v1=v1(t1)/parenrightbigg
++
+= 2Pa1,a2(t1;τ)dt1
+In the last ”=”, we have used Corollary 4.10 and/parenleftigFa1+1(w1;τ)Fa2+1(w1;τ)v1dv1
+η−1(v1;τ)|v1=v1(t1)/parenrightig
++= 0.
+We have
+/parenleftbigg−v1dv1
+2η−1(v1;τ)d2···dlπ∗(T2+T3)|v1=v1(t1)/parenrightbigg
++= (τ2+τ)l−1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0/parenleftig
+(τ2+τ)/an}bracketle{tτa1τa2τbL\{i}Γg−1(τ)/an}bracketri}htg−1(44)
+−stable/summationdisplay
+g1+g2=g
+I/uniontextJ=L\{i}/an}bracketle{tτa1τbIΓg1(τ)/an}bracketri}htg1/an}bracketle{tτa2τbJΓg2(τ)/an}bracketri}htg2
+Pa1,a2(t1;τ)dt1dˆΨbL\{1}(tL\{1};τ)
+Hence, by equations (38), (43), (44) and (10), we have proved the identity (30), i.e the
+Bouchard-Marino conjecture for C3case.
+Remark 4.11. At the same time as L. Chen published his proof [5], J. Zhou als o proved the
+Bouchard-Mari˜ no conjecture for C3[22]. He formulated this new recursion relation of conjec-
+ture and equation (1) of theorem 1.1 in the coordinate v. Then, he regarded equation (1) as
+meromorphic functions in v1, taking the principal parts and only the even powers in v1would
+get the recursion relation for this conjecture. J. Zhou has a lso applied his method to formulate
+and prove this new recursion relation for topological verte x [23] based on the work [16].
+5.Application II: Derivation of Some Hodge integral identiti es
+In this section, we will use the main theorem 1.1 to obtain som e Hodge integral identities.
+5.1.Preliminary calculations. For convenience, let us recall the formula (1) in theorem 1.1
+at first: for g≥1 andl≥1, Then,
+LHS:=−(τ2+τ)l−2/summationdisplay
+bL≥0/parenleftbigg
+(l−1)(2τ+1)/an}bracketle{tτbLΓg(τ)/an}bracketri}htg+(τ2+τ)/an}bracketle{tτbLd
+dτΓg(τ)/an}bracketri}htg/parenrightbigg
+ˆΨbL(tL;τ)(45)24 SHENGMAO ZHU
+−(τ2+τ)l−1/summationdisplay
+bL≥0/an}bracketle{tτbLΓg(τ)/an}bracketri}htgl/summationdisplay
+i=1/parenleftbigg∂
+∂τˆΨbi(ti;τ)+1
+tiτ+1ˆΨbi+1(ti;τ)/parenrightbigg
+ˆΨbL\{i}(tL\{i};τ)
+=T1+T2+T3
+where
+T1:=−(τ2+τ)l−2
+τ+1/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Γg(τ)/an}bracketri}htgˆΨbL\{i,j}(tL\{i,j};τ)
+·(tj−1)(t2
+iτ+ti)ˆΨa+1(ti;τ)−(ti−1)(t2
+jτ+tj)ˆΨa+1(tj;τ)
+ti−tj
+T2:=(τ2+τ)l
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0/an}bracketle{tτa1τa2τbL\{i}Γg−1(τ)/an}bracketri}htg−12/productdisplay
+n=1ˆΨan+1(ti;τ)ˆΨbL\{i}(tL\{i};τ)
+T3:=−(τ2+τ)l−1
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}/summationdisplay
+g1+g2=g
+I/coproducttextJ=L\{i}
+2g1−1+|I|>0
+2g2−1+|J|>0/an}bracketle{tτa1τbIΓg1(τ)/an}bracketri}htg1/an}bracketle{tτa2τbJΓg2(τ)/an}bracketri}htg22/productdisplay
+n=1ˆΨan+1(ti;τ)ˆΨbL\{i}(tL\{i};τ)
+ˆΨn(t;τ) =/parenleftbigg(t2−t)(tτ+1)
+τ+1d
+dt/parenrightbiggn/parenleftbiggt−1
+τ+1/parenrightbigg
+,n≥0
+and
+Γg(τ) = Λ∨
+g(1)Λ∨
+g(−τ−1)Λ∨
+g(τ)
+Let us denote the τexpansion of ˆΨn(t;τ) and Γ g(τ) as follow,
+ˆΨb(t;τ) =b/summationdisplay
+k=0τk
+(τ+1)b+1Ψk
+b(t) (46)
+and
+Γg(τ) =2g/summationdisplay
+m=0Λ∨
+g(1)am(λ)τm(47)
+By the definition of Λ∨
+g(t) =/summationtextg
+j=0(−1)g−jλg−jtjand Mumford’s relation Λ∨
+g(t)Λ∨
+g(−t) =
+(−1)gt2g, we can compute out the coefficients am(λ) in (47). For example,
+a2g(λ) = (−1)g,a2g−1(λ) = (−1)gg,··· (48)
+a1(λ) =g/summationdisplay
+m=1mλg−mλg−(−1)gλg−1Λ∨
+g(−1)
+a0(λ) = (−1)gλgΛ∨
+g(−1)
+We also need to show the expansion form of (46) and how to calcu late the coefficients Ψk
+b(t).
+By definition ˆΨb+1(t,τ) = (t2−t)(yτ+1
+τ+1)d
+dyΨb(y,τ),ˆΨ0(t,τ) =t−1
+τ+1, we have the expansion formTHE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 25
+(46). Moreover, we have
+Ψk
+b+1(t) = (t3−t2)d
+dtΨk−1
+b(t)+(t2−t)d
+dtΨk
+b(t),k= 0,..,b+1 (49)
+Therefore, through the recursion formula (49), all the Ψk
+b(t) can be calculated.
+As an illustration, we calculate some cases which will be use d in the following discussion.
+Ψb+1
+b+1(t) = (t3−t2)d
+dtΨb
+b(t) (50)
+Ψb
+b+1(t) = (t3−t2)d
+dtΨb−1
+b(t)+(t2−t)d
+dtΨb
+b(t) (51)
+Ψ1
+b+1(t) = (t3−t2)d
+dtΨ0
+b(t)+(t2−t)d
+dtΨ1
+b(t) (52)
+Ψ0
+b+1(t) = (t2−t)d
+dtΨ0
+b(t) (53)
+It is clear that, the recursion relation for Ψb
+b(t) in (50) is just same for the definition of ˆξb(t)
+in [9], and they share the same initial. thus
+Ψb
+b(t) =ˆξb(t) (54)
+By (50), we write Ψb
+b(t) as
+Ψb
+b(t) =2b+1/summationdisplay
+i=b+1fb(b,i)ti(55)
+then all the fb(b,i) could be calculated from the recursion (50). But we can expl icitly write
+downfb(b,2b+1) = (2b−1)!!,fb(b,b+1) = (−1)bb! which are the coefficients used in the proof
+ofDVVequation and λgintegral respectively[20, 6, 14].
+We can also let
+Ψb−1
+b(t) =2b/summationdisplay
+i=bfb−1(b,i)ti(56)
+then by (51) we have
+fb(b+1,k) = (k−2)fb−1(b,k−2)−(k−1)fb−1(b,k−1)+(k−1)fb(b,k−1)−kfb(b,k)(57)
+Thus, allfb(b+1,k) can be calculated from (57). For example,
+fb(b+1,2b+2) = 2bfb−1(b,2b)+(2b+1)fb(b,2b+1) = 2bfb−1(b,2b)+(2b+1)!!
+Hence
+fb−1(b,2b) =b−1/summationdisplay
+k=0(2b−2)!!(2k+1)!!
+(2k)!!(58)
+Similarly,
+fb(b+1,b+1) =−bfb+1(b,b)−(b+1)fb(b,b+1) =−bfb(b,b)+(−1)b+1(b+1)!
+Thus
+fb−1(b,b) = (−1)b(b+1)!
+2(59)26 SHENGMAO ZHU
+On the other side, from (53), we can let
+Ψ0
+b(t) =b+1/summationdisplay
+i=1f0(b,i)yi
+Then
+f0(b+1,i) = (i−1)f0(b,i−1)−if0(b,i)
+Thus,
+f0(b,b+1) =b! (60)
+Then, from (52), we can let
+Ψ1
+b(t) =b+2/summationdisplay
+j=2f1(b,j)tj(61)
+and
+f1(b+1,k) = (k−2)f0(b,k−2)−(k−1)f0(b,k−1)+(k−1)f1(b,k−1)−kf1(b,k)
+Thus
+f1(b+1,b+3) = (b+1)f0(b,b+1)+(b+2)f1(b,b+2) (62)
+by initial value, f1(1,3) = 1 and f0(b,b+1) =b!, we get
+f1(b,b+2) = (b+1)!b+1/summationdisplay
+k=21
+k. (63)
+Now, let us substitute (46) and (47) to (45), we have
+LHS=−/summationdisplay
+bL≥0/summationdisplay
+0≤kL≤bL2g/summationdisplay
+m=0/an}bracketle{tτbLΛ∨
+g(1)am(λ)/an}bracketri}htgΨkL
+bL(tL) (64)
+×(l−2+m+|kL|−|bL|)τ|kL|+l+m−1+(l−1+m+|kL|)τ|kL|+l+m−2
+(τ+1)|bL|+2
+−/summationdisplay
+bL≥0/summationdisplay
+0≤kL≤bL2g/summationdisplay
+m=0/an}bracketle{tτbLΛ∨
+gam(λ)/an}bracketri}htgl/summationdisplay
+i=1(t2
+i−ti)∂
+∂tiΨki
+bi(ti)ΨbL\{i}
+bL\{i}(tL\{i})τ|bL|+m+l−2
+(τ+1)|bL|+2
+T1=−/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥02g/summationdisplay
+m=0/summationdisplay
+0≤k≤a+1
+0≤kL\{i,j}≤bL\{i,j}/an}bracketle{tτaτbL\{i,j}Λ∨
+g(1)am(λ)/an}bracketri}htgΨbL\{i,j}
+bL\{i,j}(tL\{i,j}) (65)
+×(tj−1)t2
+iΨk
+a+1(ti)−(ti−1)t2
+jΨk
+a+1(tj)
+ti−tjτ|kL\{i,j}|+k+l+m−1
+(τ+1)|bL\{i,j}|+a+3
+−/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥02g/summationdisplay
+m=0/summationdisplay
+0≤k≤a+1
+0≤kL\{i,j}≤bL\{i,j}/an}bracketle{tτaτbL\{i,j}Λ∨
+g(1)am(λ)/an}bracketri}htgΨbL\{i,j}
+bL\{i,j}(tL\{i,j})
+×(tj−1)tiΨk
+a+1(ti)−(ti−1)tjΨk
+a+1(tj)
+ti−tjτ|kL\{i,j}|+k+l+m−2
+(τ+1)|bL\{i,j}|+a+3THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 27
+T2=1
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥02(g−1)/summationdisplay
+m=0/summationdisplay
+0≤n1≤a1+1
+0≤n2≤a2+1
+0≤kL\{i}≤bL\{i}/an}bracketle{tτa1τa2τbL\{i,j}Λ∨
+g−1(1)am(λ)/an}bracketri}htg−1 (66)
+×Ψn1
+a1+1(ti)Ψn2
+a2+1(ti)ΨbL\{i}
+bL\{i}(tL\{i})τ|kL\{i}|+n1+n2+l+m
+(τ+1)a1+a2+|kL\{i}|+3
+T3=−1
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0/summationdisplay
+g1+g2=g
+I∪J=L\{i}
+2g1−1+|I|>0
+2g2−1+|J|>0/summationdisplay
+0≤m1≤2g1
+0≤m2≤2g2/summationdisplay
+0≤n1≤a1+1
+0≤n2≤a2+1
+0≤kL{i}≤bL\{i}/an}bracketle{tτa1τbIΛ∨
+g1(1)am1(λ)/an}bracketri}htg1 (67)
+×/an}bracketle{tτa2τbJΛ∨
+g2(1)am2(λ)/an}bracketri}htg2Ψn1
+a1+1(ti)Ψn2
+a2+1(ti)ΨkL\{i}
+bL\{i}(tL\{i})τ|kL\{i}|+n1+n2+l+m1+m2−1
+(τ+1)a1+a2+|kL\{i}|+4
+Where we have used the notation |bL|=/summationtextl
+i=1biin above formulas. We note that (64), (65),
+(66) and (67) are all the functions of τ. If we expand them as the series of τat different points,
+we will get some identities of Hodge integrals after recolle cting the coefficients of τat both sides
+of (64) = (65)+(66)+(67).
+5.2.Expansion of τat∞.Forb≥0, expanding at τ=∞,
+τa
+(τ+1)b=/summationdisplay
+h≥0/parenleftbigg−b
+h/parenrightbigg
+τa−b−i
+for example, the term containing τin (35),
+τ|kL|+l+m−1
+(τ+1)|bL|+2=/summationdisplay
+h≥0/parenleftbigg−|bL|−2
+h/parenrightbigg
+τl+m−3+|kL|−|bL|−h
+After this expansion, it is easy to see that the largest degre e ofτis 2g+l−3 in the formulas
+(64), (65), (66), (67). If F(t,τ)∈Q[t][[τ]], we will use the notation [ τk]F(t,τ) to mean the
+coefficient of τkinF(t,τ). Bya2g= (−1)gin (48) , after some calculations,
+[τ2g+l−3]LHS= (−1)g+1/summationdisplay
+bL≥0/an}bracketle{tτbLΛ∨
+g(1)/an}bracketri}ht/parenleftigg
+(2g−2+l)ΨbL
+bL(tL)+l/summationdisplay
+i=1(t2
+i−ti)∂
+∂tΨbi
+bi(ti)ΨbL\{i}
+bL\{i}(tL\{i})/parenrightigg(68)
+[τ2g+l−3]T1= (−1)g+1/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Λ∨
+g(1)/an}bracketri}htgΨbL\{i,j}
+bL\{i,j}(tL\{i,j}) (69)
+(tj−1)t2
+iΨa+1
+a+1(ti)−(ti−1)t2
+jΨa+1
+a+1(tj)
+ti−tj
+[τ2g+l−3]T2= (−1)g+11
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0/an}bracketle{tτa1τa2τbL\{i}Λ∨
+g−1(1)/an}bracketri}htg−1Ψa1+1
+a1+1(ti)Ψa2+1
+a2+1(ti)ΨbL\{i}
+bL\{i}(tL\{i})(70)28 SHENGMAO ZHU
+[τ2g+l−3]T3= (−1)g+11
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0/summationdisplay
+g1+g2=g
+I∪J=L\{i}/an}bracketle{tτa1τbIΛ∨
+g1(1)/an}bracketri}htg1/an}bracketle{tτa2τbJΛ∨
+g2(1)/an}bracketri}htg22/productdisplay
+n=1Ψan+1
+an+1(ti)ΨbL\{i}
+bL\{i}(tL\{i})(71)
+As showed in formula (54), Ψb
+b(t) =ˆξb(t), we have got the following corollary from (68) =
+(69)+(70)+(71),
+Corollary 5.1.
+/summationdisplay
+bL≥0/an}bracketle{tτbLΛ∨
+g(1)/an}bracketri}ht/parenleftigg
+(2g−2+l)ˆξbL(tL)+l/summationdisplay
+i=1(t2
+i−ti)∂
+∂tˆξbi(ti)ˆξbL\{i}(tL\{i})/parenrightigg(72)
+=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Λ∨
+g(1)/an}bracketri}htgˆξbL\{i,j}(tj−1)t2
+iˆξa+1(ti)−(ti−1)t2
+jˆξa+1(tj)
+ti−tj
++1
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0
+/an}bracketle{tτa1τa2τbL\{i}Λ∨
+g−1(1)/an}bracketri}htg−1+stable/summationdisplay
+g1+g2=g
+I∪J=L\{i}/an}bracketle{tτa1τbIΛ∨
+g1(1)/an}bracketri}htg1/an}bracketle{tτa2τbJΛ∨
+g2(1)/an}bracketri}htg2
+
+׈ξa1+1(ti)ˆξa2+1(ti)ˆξbL\{i}(tL\{i})
+which is just the main theorem 1.1 showed in paper [9]. Formul a (72) has been applied by
+many people to derive DVV equation [6, 20], λg-conjecture [14, 20] and λg−1Hodge integral
+recursion [24].
+Obviously, in this procedure, we can obtain a series of such c orollaries. For example, calcu-
+lating out the coefficients of τ2g+l−4bya2g= (−1)g,a2g−1(λ) = (−1)ggin (48) we have
+[τ2g+l−4]LHS(73)
+= (−1)g+1(2g−3+l)/summationdisplay
+bL≥0/an}bracketle{tτbΛ∨
+g(1)/an}bracketri}htg[(g−1−|bL|)ΨbL
+bL(tL)+l/summationdisplay
+i=1Ψbi−1
+bi(ti)ΨbL\{i}
+bL\{i}(tL\{i})]
++(−1)g+1/summationdisplay
+bL≥0/an}bracketle{tτbLΛ∨
+g(1)/an}bracketri}htg[l/summationdisplay
+i=1(t2
+i−ti)∂
+∂ti/parenleftig
+(g−|bL|−2)Ψbi
+bi(ti)+Ψbi−1
+bi(ti)/parenrightig
+(ΨbL\{i}
+bL\{i}(tL\{i})
++/summationdisplay
+j/\e}atio\slash=iΨbj−1
+bj(tj)ΨbL\{i,j}
+bL\{i,j}(tL\{i,j})]
+[τ2g+l−4]T1= (−1)g+1/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Λ∨
+g(1)/an}bracketri}htgΨbL\{i,j}
+bL\{i,j}(tL\{i,j}) (74)
+((tj−1)t2
+i[Ψa
+a+1(ti)+(g−a−|bL\{i,j}|−3+1
+ti)Ψa+1
+a+1(ti)]
+ti−tjTHE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 29
+−(ti−1)t2
+j[Ψa
+a+1(tj)+(g−a−|bL\{i,j}|−3+1
+tj)Ψa+1
+a+1(tj)]
+ti−tj)
+[τ2g+l−4]T2= (−1)g+11
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0/an}bracketle{tτa1τa2τbL\{i}Λ∨
+g−1(1)/an}bracketri}htg−1[(g−a1−a2−|bL\{i,j}|−4)(75)
+·Ψa1+1
+a1+1(ti)Ψa2+1
+a2+1(ti)+Ψa1+1
+a1(ti)Ψa2+1
+a2+1(ti)+Ψa1+1
+a1+1(ti)Ψa2
+a2+1(ti)]ΨbL\{i}
+bL\{i}(tL\{i})
++(−1)g+11
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0/an}bracketle{tτa1τa2τbL\{i}Λ∨
+g−1(1)/an}bracketri}htg−1Ψa1+1
+a1(ti)Ψa2+1
+a2+1(ti)/summationdisplay
+j/\e}atio\slash=iΨbj−1
+bj(tj)ΨbL\{i,j}
+bL\{i,j}(tL\{i,j})
+[τ2g+l−4]T3(76)
+= (−1)g+11
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0stable/summationdisplay
+g1+g2=g
+I∪J=L\{i}/an}bracketle{tτa1τbIΛ∨
+g1(1)/an}bracketri}htg1/an}bracketle{tτa2τbJΛ∨
+g2(1)/an}bracketri}htg2[(g−a1−a2−|bL\{i,j}|−4)
+·Ψa1+1
+a1+1(ti)Ψa2+1
+a2+1(ti)+Ψa1+1
+a1(ti)Ψa2+1
+a2+1(ti)+Ψa1+1
+a1+1(ti)Ψa2
+a2+1(ti)]ΨbL\{i}
+bL\{i}(tL\{i})
++(−1)g+11
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0stable/summationdisplay
+g1+g2=g
+I∪J=L\{i}/an}bracketle{tτa1τbIΛ∨
+g1(1)/an}bracketri}htg1
+/an}bracketle{tτa2τbJΛ∨
+g2(1)/an}bracketri}htg2Ψa1+1
+a1(ti)Ψa2+1
+a2+1(ti)/summationdisplay
+j/\e}atio\slash=iΨbj−1
+bj(tj)ΨbL\{i,j}
+bL\{i,j}(tL\{i,j})
+Thus, by (73) = (74)+(75)+(76), we get a Hodge integral ident ity. However, unfortunately,
+the only Hodge integral contained in this identity is /an}bracketle{tτbLΛ∨
+g(1)/an}bracketri}htgbecause only the terms a2g(λ) =
+(−1)ganda2g−1(λ) = (−1)ggwas involved in the above calculation which contains no more new
+information.
+In order to get the Hodge integral identity with more than one λ-class, we need to compare
+the coefficients of τ2g+l−5with the same procedure, we will arrive at a Hodge integral id entity
+which involves /an}bracketle{tτbLΛ∨
+g(1)/an}bracketri}htgand/an}bracketle{tτbLΛ∨
+g(1)λ1/an}bracketri}htg, becausea2g−2(λ) = (−1)g/parenleftig
+g(g−1)
+2−λ1/parenrightig
+.
+From such Hodge integral identity, we can calculate all the H odge integral of type /an}bracketle{tτbLλkλ1/an}bracketri}htg,
+k= 1,..,g. However, more terms will appear in the calculation of coeffic ients ofτ2g+l−5which
+make the computation very complicated. so far, we have not wr itten down the explicit formula
+to calculate the Hodge integral of type /an}bracketle{tτbLλkλ1/an}bracketri}htg. But when k=g, in the following subsection
+5.4, we will give an explicit formula for /an}bracketle{tτbLλgλ1/an}bracketri}htg.
+5.3.Expansion of τat 0.Now, let us consider the expansion ofτa
+(τ+1)batτ= 0, we have
+τa
+(τ+1)b=/summationdisplay
+i≥0/parenleftbigg−b
+i/parenrightbigg
+τa+i30 SHENGMAO ZHU
+After this expansion, it is easy to show that the lowest degre e ofτin (64) isl−2. In fact, by
+a0(λ) = (−1)gλgΛ∨
+g(−1) show in (48) and Λ∨
+g(1)Λ∨
+g(−1) = (−1)g, we have
+[τl−2]LHS=−(l−1)/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgΨ0
+bL(tL) (77)
+[τl−2]T1=−/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})(tj−1)t2
+iΨ0
+a+1(ti)−(ti−1)t2
+jΨ0
+a+1(tj)
+ti−tj(78)
+[τl−2]T2= [τl−2]T3= 0 (79)
+Thus, we get
+Corollary 5.2.
+/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgΨ0
+bL(tL) =1
+l−1/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j}) (80)
+·(tj−1)t2
+iΨ0
+a+1(ti)−(ti−1)t2
+jΨ0
+a+1(tj)
+ti−tj
+Here, we introduce two notations. Let g(x1,..,xl) =/summationtext
+ik≥0ai1i2···ilxi1
+1···xil
+l∈Q[x1,..,xl],
+Fd(g(x1,···,xl)) =/summationdisplay
+/summationtextl
+k=1ik=dai1i2···ilxi1
+1···xil
+l
+and
+[xj1
+1···xjl
+l]g(x1,···,xl) =aj1j2···jl.
+Now, let us use these two operator to corollary 5.2. By (60), w e have
+F2g−3+2l(LHSof (80)) = ( l−1)/summationdisplay
+bL≥0/an}bracketle{tτbLλ/an}bracketri}htgbL!ybL+1
+L
+F2g−3+2l(RHSof (80)) =/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL≥0/an}bracketle{tτbL\{i,j}λg/an}bracketri}htg(a+1)!bL\{i,j}!tbL\{i,j}+1
+L\{i,j}tjta+4
+i−tita+4
+j
+ti−tj
+Then,
+[tbL+1
+L]F2g−3+2l(LHSof (80)) = ( l−1)/an}bracketle{tτbLλ/an}bracketri}htgbL!
+[tbL+1
+L]F2g−3+2l(RHSof (80)) =/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bj−1τbL\{i,j}λg/an}bracketri}htg(bi+bj)!bL\{i,j}!
+Hence, we have
+/an}bracketle{tτbLλ/an}bracketri}htg=1
+l−1/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bj−1τbL\{i,j}λg/an}bracketri}htg(bi+bj)!
+bi!bj!THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 31
+Induction on l, we get the λgintegral,
+/an}bracketle{tτbLλ/an}bracketri}htg=/parenleftbigg2g−3+l
+bL/parenrightbigg
+cg
+Now, let us calculate the next degree of τ. Bya0(λ) = (−1)gλgΛ∨
+g(−1) showed in (48) and
+Λ∨
+g(1)Λ∨
+g(−1) = (−1)g, we have
+[τl−1]LHS=−/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgl
+−(|bL|+1)Ψ0
+bL(tL)+/summationdisplay
+j=1Ψ1
+bj(tj)Ψ0
+bL\{j}(tL\{j})
+(81)
+−/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgl/summationdisplay
+i=1(t2
+i−ti)∂
+∂tiΨ0
+bi(ti)Ψ0
+bL\{i}(tL\{i})−/summationdisplay
+bL≥0/an}bracketle{tτbLΛ∨
+g(1)a1(λ)/an}bracketri}htglΨ0
+bL(tL)
+[τl−1]T1(82)
+=−/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})[(tj−1)ti/parenleftbig
+Ψ1
+a+1(ti)+(ti−|bL\{i,j}|−a−3)Ψ0
+a+1(ti)/parenrightbig
+ti−tj
+−(ti−1)tj/parenleftbig
+Ψ1
+a+1(tj)+(tj−|bL\{i,j}|−a−3)Ψ0
+a+1(tj)/parenrightbig
+ti−tj]
+−/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg/summationdisplay
+r/\e}atio\slash=i,jΨ1
+br(tr)Ψ0
+bL\{i,j}(tL\{i,j})(tj−1)tiΨ0
+a+1(ti)−(ti−1)tjΨ0
+a+1(tj)
+ti−tj
+−/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}Λ∨
+g(1)a1(λ)/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})(tj−1)tiΨ0
+a+1(ti)−(ti−1)tjΨ0
+a+1(tj)
+ti−tj
+[τl−1]T2= 0 (83)
+[τl−1]T3=−1
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0stable/summationdisplay
+g1+g2=g
+|I|∪|J|=L\{i}/an}bracketle{tτa1τbIλg1/an}bracketri}htg1/an}bracketle{tτa2τbJλg2/an}bracketri}htg2Ψ0
+a1+1(ti)Ψ0
+a2+1(ti)Ψ0
+bL\{i}(tL\{i})(84)
+Recall formula (48), a1(λ) =/summationtextg
+m=1mλg−mλg−(−1)gΛ∨
+g(−1)λg−1and Mumford’s relation
+Λ∨
+g(1)Λ∨
+g(−1) = (−1)g. We can write
+Λ∨
+g(1)a1(λ) =3g−3/summationdisplay
+d=g−1Pd(λ) (85)
+wherePd(λ) is some combinatoric of λ-class with degree d. For example
+Pg−1(λ) =λg−1,Pg(λ) =gλg,Pg+1(λ) =−λgλ1,···,P3g−3(λ) = (−1)g+1λgλg−1λg−2 (86)
+By the above calculation, substituting (85) to the identity (81) = (82)+(84), we have32 SHENGMAO ZHU
+Corollary 5.3.
+/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgl
+−(|bL|+1)Ψ0
+bL(tL)+/summationdisplay
+j=1Ψ1
+bj(tj)Ψ0
+bL\{j}(tL\{j})
+(87)
++/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgl/summationdisplay
+i=1(t2
+i−ti)∂
+∂tiΨ0
+bi(ti)Ψ0
+bL\{i}(tL\{i})+/summationdisplay
+bL≥0/an}bracketle{tτbL3g−3/summationdisplay
+d=g−1Pd(λ)/an}bracketri}htglΨ0
+bL(tL)
+=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})[(tj−1)ti/parenleftbig
+Ψ1
+a+1(ti)+(ti−|bL\{i,j}|−a−3)Ψ0
+a+1(ti)/parenrightbig
+ti−tj
+−(ti−1)tj/parenleftbig
+Ψ1
+a+1(tj)+(tj−|bL\{i,j}|−a−3)Ψ0
+a+1(tj)/parenrightbig
+ti−tj]
++/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg/summationdisplay
+r/\e}atio\slash=i,jΨ1
+br(tr)Ψ0
+bL\{i,j}(tL\{i,j})(tj−1)tiΨ0
+a+1(ti)−(ti−1)tjΨ0
+a+1(tj)
+ti−tj
++/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}3g−3/summationdisplay
+d=g−1Pd(λ)/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})(tj−1)tiΨ0
+a+1(ti)−(ti−1)tjΨ0
+a+1(tj)
+ti−tj
++1
+2l/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bL\{i}≥0stable/summationdisplay
+g1+g2=g
+|I|∪|J|=L\{i}/an}bracketle{tτa1τbIλg1/an}bracketri}htg1/an}bracketle{tτa2τbJλg2/an}bracketri}htg2Ψ0
+a1+1(ti)Ψ0
+a2+1(ti)Ψ0
+bL\{i}(tL\{i})
+Since/an}bracketle{tτbLλg/an}bracketri}htg=/parenleftbig2g−3+l
+bL/parenrightbig
+cg, thus all the Hodge integral of type
+/an}bracketle{tτbL3g−3/summationdisplay
+d=g−1Pd(λ)/an}bracketri}htg (88)
+can be calculated from formula (87) in corollary 4.3.
+5.4.Explicit expression for Hodge integral /an}bracketle{tτbLλgλ1/an}bracketri}htg.Although corollary 4.3 says that
+all the Hodge integral (88) can be computed, it is not easy to w rite down their explicit formula.
+In this subsection, we will show how to obtain an explicit for mula for /an}bracketle{tτbLλgλ1/an}bracketri}htg.
+Both sides of (87) belong to Q[t1,..,tl]. We have known that /an}bracketle{tτbLλg/an}bracketri}htg=/parenleftbig2g−3+l
+bL/parenrightbig
+cg, and
+/an}bracketle{tτbLλg−1/an}bracketri}htgcan also be calculated easily from a recursion formula showe d in [24]. Thus, we have
+Theorem 5.4. If/summationtextl
+i=1bi= 2g−4 +l, there exists a constant C(g,l,b1,..,bl)related to
+g,l,b1,..,bl, such that
+/an}bracketle{tτbLλgλ1/an}bracketri}htg=1
+l/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bj−1τbL\{i,j}λgλ1/an}bracketri}htg(bi+bj)!
+bi!bj!+C(g,l,b1,..bl) (89)
+C(g,l,b1,..,bl)is a very complicated combinatoric constant which is given a t Appendix B.
+Proof.Taking all the terms with degree 2 g+2l−4 in formula (87). we have
+F2g+2l−4(LHSof (87)) = −l/summationdisplay
+bL≥0/an}bracketle{tτbLλgλ1/an}bracketri}htgbL!tbL+1
+L+G1(t1,..,tl) (90)THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 33
+F2g+2l−4(RHSof (87)) = −/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λgλ1/an}bracketri}htg (91)
+·bL\{i,j}!(a+1)!a+1/summationdisplay
+m=0tbL\{i,j}+1
+L\{i,j}ta+2−m
+itm+1
+j+G2(t1,..,tl)
+whereGi(t1,..,tl) =/summationtext
+|dL|=2g+2l−4Ci(g,l,d1,..,dl)tdL
+L∈Q[t1,..tl],Ci(g,l,d1,..,dl) is a combina-
+toric constant related to g,l,d1,..,dl. Thus, we have
+/summationdisplay
+bL≥0/an}bracketle{tτbLλgλ1/an}bracketri}htgbL!tbL+1
+L=1
+l/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λgλ1/an}bracketri}htg (92)
+·bL\{i,j}!(a+1)!a+1/summationdisplay
+m=0tbL\{i,j}+1
+L\{i,j}ta+2−m
+itm+1
+j+G(t1,..,tl)
+whereG(t1,..,tl) =/summationtext
+|dL|=2g+2l−4ˆC(g,l,b1,..,bl)tdL
+L=1
+l(G1(t1,..tl)−G2(t1,..,tl))
+After taking the coefficients of (92),
+/an}bracketle{tτbLλgλ1/an}bracketri}htg=1
+l/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bj−1τbL\{i,j}λgλ1/an}bracketri}htg(bi+bj)!
+bi!bj!+ˆC(g,l,b1+1,..,bl+1)
+bL!(93)
+LetC(g,l,b1,..,bl) =ˆC(g,l,b1+1,..,bl+1)
+bL!, thus, the theorem is proved. /square
+We know that the initial value /an}bracketle{tτ2g−3λgλ1/an}bracketri}htg=1
+12[g(2g−3)bg+b1bg−1] has been computed
+by Y. Li [15]. Therefore, from the recursion formula (62), we can compute out all the Hodge
+integral of type /an}bracketle{tτbLλgλ1/an}bracketri}htg.
+6.Conclusion
+Theoriginal cut-and-join equation of Mari˜ no-Vafa formul a is a complicated formula about the
+partitionwhichcanbechangedtoapolynomialidentity with thehelpoftheLaplacetransformin
+this paper or the symmetrization method in [4]. This polynom ial identity has some applications
+because it is manageable. We derived some corollaries about the Hodge integral identities. In
+fact, we have found an algorithm to calculate the Hodge integ ral appearing in Mari˜ no-Vafa
+formula.
+Step1, expanding the τat some special point, and collect the corresponding level o fτin
+theorem 1.1.
+Step2, taking certain degrees of tL, we will get the corresponding Hodge integral appearing
+in Mari˜ no-Vafa formula.
+In this paper, we only calculate four cases by using step1 and calculate out a new Hodge
+integral via Step2 from the result of last case after doing st ep1. We have not considered other
+cases here, because the computation will be more complicate d. We hope that this algorithm can
+be compiled to a computer program.
+We note that, the recursion formulas to calculate /an}bracketle{tτbLλg−1/an}bracketri}htg[24] and /an}bracketle{tτbLλgλ1/an}bracketri}htgin theorem
+1.5 have the similar recursion structure. For a given g, Hodge integral with l-point will reduce
+to 1-point after only ltimes recursions. Thus, it is a very effective recursion relat ion. In fact,
+many Hodge integral recursions will have this type structur e if they are calculated out by above
+algorithm. Thus it is interesting to consider the following combinatoric problem.
+LetQl(bL) =Ql(b1,b2,..,bl) be a symmetric function on ( b1,b2,..,bl)∈(Z+)lwhich is defined
+by the following recursion relation.34 SHENGMAO ZHU
+For giveng,k,
+Q1(b) =/braceleftbiggconstant, b = 3g−2−k, (94)
+0, others .
+Ql(bL) =
+
+/summationdisplay
+1≤i<j≤lQl−1(bi+bj−1,bL\{i,j})Al(bL)+Bl(bL)/summationtextl
+i=1bi= 3g−3+l−k, (95)
+0, others.
+WhereAl(bL) =Al(b1,..,bl) andBl(bL) =Bl(b1,..,bl) are some fixed functions defined on
+(b1,b2,..,bl)∈(Z+)l.
+It is interesting to study the properties of Ql(bL) defined above. We hope that this combina-
+toric structure of Hodge integral will be studied further in future.
+7.Appendix
+7.1.Appendix A. In this appendix, we will show how to derive our main result in theorem
+3.4 from the symmetrized cut-and-join equation of the Mari˜ no-Vafa formula [4].
+whenb≥0, Let
+Ψb(y;τ) = ((y2−y)(yτ+1
+τ+1)d
+dy)b(y−1
+τ+1) =b/summationdisplay
+k=0τk
+(τ+1)b+1ψk
+b(y)
+Then the symmetrized partition function of Mari˜ no-Vafa fo rmula can be written as
+Wg,n(y1,..,yn;τ) =−(τ(τ+1))n−1/summationdisplay
+bi≥0
+i=1,..,n/an}bracketle{tn/productdisplay
+i=1τbiΓg(τ)/an}bracketri}htgn/productdisplay
+i=1Ψbi(yi;τ) (96)
+The main result of [4], i.e Theorem 3 in that paper, says
+Theorem 7.1. The symmetrized generating series Wg,n(y1,..,yn;τ)is a polynomial in the yi
+variables of total degree 6g−6+3nand satisfies the symmetrized cut–and-join equation.
+(∂
+∂τ+n/summationdisplay
+l=1y2
+l−yl
+τ+1∂
+∂yl)Wg,n(y1,..,yn;τ) =T1+T2+T3
+where
+T1=symy
+1,1y1(y2−1)
+y1−y2(y1τ+1
+τ+1)Dτ
+y1Wg,n−1(y1,y3,..,yn;τ)
+T2=−1
+2n/summationdisplay
+l=1Dτ
+ylDτ
+yn+1Wg−1,n+1(y1,..,yn,yn+1;τ)|yn+1=yl
+T3=−1
+2/summationdisplay
+g1+g2=g
+n1+n2=n+1
+2g1−2+n1>0
+2g2−2+n2>0symy
+1,n1−1Dτ
+y1Wg1,n1(y1,..,yn1;τ)Dτ
+y1Wg2,n2(y1,yn1+1,..,yn;τ)
+We need to explain the notations appear in above theorem. Whe reDτ
+y= (y2−y)(yτ+1
+τ+1)∂
+∂y. For
+i,j≥0,i+j≤n, letsymx
+i,jbe the mapping, applied to a series in x1,..,xn, given by
+symx
+i,jf(x1,..,xn) =/summationdisplay
+R,S,Tf(xR,xS,xT)THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 35
+where the sum is over all ordered partitions (R,S,T)of{1,2,..,n},R={xr1,..,xri},S=
+{xs1,..,xsj},T={xt1,..,xtn−i−j}and(xR,xS,xT) = (xr1,..,xri,xs1,..,xsj,xt1,..,xtn−i−j),
+wherer1<···<ri,s1<···<sj, andt1<···<tn−i−j.
+Then, by equation (96), we have
+LHS=−(τ2+τ)n−2/summationdisplay
+bi≥0
+i=1,..,n/parenleftigg
+(n−1)(2τ+1)/an}bracketle{tn/productdisplay
+i=1τbiΓg(τ)/an}bracketri}htg+(τ2+τ)/an}bracketle{tn/productdisplay
+i=1τbid
+dτΓg(τ)/an}bracketri}htg/parenrightiggn/productdisplay
+i=1Ψbi(yi;τ)
+−(τ2+τ)n−1/summationdisplay
+bi≥0
+i=1,2,..,n/an}bracketle{tn/productdisplay
+i=1τbiΓg(τ)/an}bracketri}htgn/summationdisplay
+l=1(∂
+∂τΨbl(yl;τ)+1
+ylτ+1Ψbl+1(yl;τ))n/productdisplay
+k=1
+k/\e}atio\slash=lΨbk(yk;τ)
+T1=−(τ2+τ)n−2
+τ+1/summationdisplay
+1≤i<j≤n/summationdisplay
+a≥0
+bk≥0
+k/\e}atio\slash=i,j/an}bracketle{tτan/productdisplay
+k=1
+k/\e}atio\slash=i,jτbkΓg(τ)/an}bracketri}htgn/productdisplay
+k=1
+k/\e}atio\slash=i,jΨbk(yk;τ)
+×(yj−1)(y2
+iτ+yi)Ψa+1(yi;τ)−(yi−1)(y2
+jτ+yj)Ψa+1(yj;τ)
+yi−yj
+T2=1
+2(τ2+τ)nn/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bk≥0,k/\e}atio\slash=i/an}bracketle{tτa1τa2n/productdisplay
+k=1
+k/\e}atio\slash=iτbkΓg−1(τ)/an}bracketri}htg−1Ψa1+1(yi;τ)Ψa2+1(yi;τ)n/productdisplay
+k=1
+k/\e}atio\slash=iΨbk(yk;τ)
+T3=−(τ2+τ)n−1
+2n/summationdisplay
+i=1/summationdisplay
+a1≥0
+a2≥0
+bk≥0,k/\e}atio\slash=i/summationdisplay
+g1+g2=g
+I/coproducttextJ={1,.,ˆi,.n}
+2g1−1+|I|>0
+2g2−1+|J|>0/an}bracketle{tτa1/productdisplay
+k∈IτbkΓg1(τ)/an}bracketri}htg1
+×/an}bracketle{tτa2/productdisplay
+k∈JτbkΓg2(τ)/an}bracketri}htg2Ψa1(yi;τ)Ψa2(yi;τ)n/productdisplay
+k=1
+k/\e}atio\slash=iΨbi(yi;τ)
+Then, by theorem 5.1, we arrive our main results in section 3, i.e. theorem 3.4.
+7.2.Appendix B. In this appendix, we will calculate the constant C(g,l,b1,..,bl) which ap-
+pears in theorem 1.5. The computation is verbose and boring.
+Let us write the formula (77) of Corollary 5.3 as
+L1+L2+L3+L4+L5+L6+L7(97)
+=R1+R2+R3+R4+R5+R6+R7+R8+R9+R10+R11+R12+R13+R14+R15+R16
+where
+L1=/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgl(g−|bL|−1)Ψ0
+bL(tL)
+L2=/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgll/summationdisplay
+j=1Ψ1
+bj(tj)Ψ0
+bL\{j}(tL\{j})36 SHENGMAO ZHU
+L3=/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgl/summationdisplay
+j=1t2
+j∂
+∂tjΨ0
+bj(tj)Ψ0
+bL\{j}(tL\{j})
+L4=−/summationdisplay
+bL≥0/an}bracketle{tτbLλg/an}bracketri}htgl/summationdisplay
+j=1tj∂
+∂tjΨ0
+bj(tj)Ψ0
+bL\{j}(tL\{j})
+L5=−/summationdisplay
+bL≥0/an}bracketle{tτbLλg−1/an}bracketri}htglΨ0
+bL(tL)
+L6=−/summationdisplay
+bL≥0/an}bracketle{tτbLλgλ1/an}bracketri}htglΨ0
+bL(tL)
+L7=/summationdisplay
+bL≥0/an}bracketle{tτbL3g−3/summationdisplay
+d=g+2Pd(λ)/an}bracketri}htglΨ0
+bL(tL)
+R1=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})tjtiΨ1
+a+1(ti)−titjΨ1
+a+1(tj)
+ti−tj
+R2=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})−tiΨ1
+a+1(ti)+tjΨ1
+a+1(tj)
+ti−tj
+R3=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})tjt2
+iΨ0
+a+1(ti)−tit2
+jΨ0
+a+1(tj)
+ti−tj
+R4=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})−t2
+iΨ0
+a+1(ti)+t2
+jΨ0
+a+1(tj)
+ti−tj
+R5=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})(|bL\{i,j}|+a+3)−tjtiΨ0
+a+1(ti)+titjΨ0
+a+1(tj)
+ti−tj
+R6=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})(|bL\{i,j}|+a+3)tiΨ0
+a+1(ti)−tjΨ0
+a+1(tj)
+ti−tj
+R7=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg/summationdisplay
+r∈L\{i,j}Ψ1
+br(tr)Ψ0
+bL\{i,j,r}(tL\{i,j,r})tjtiΨ0
+a+1(ti)−titjΨ0
+a+1(tj)
+ti−tj
+R8=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg/summationdisplay
+r∈L\{i,j}Ψ1
+br(tr)Ψ0
+bL\{i,j,r}(tL\{i,j,r})−tiΨ0
+a+1(ti)+tjΨ0
+a+1(tj)
+ti−tjTHE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 37
+R9=−/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg−1/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})tjtiΨ0
+a+1(ti)−titjΨ0
+a+1(tj)
+ti−tj
+R10=−/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg−1/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})−tiΨ0
+a+1(ti)+tjΨ0
+a+1(tj)
+ti−tj
+R11=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htggΨ0
+bL\{i,j}(tL\{i,j})tjtiΨ0
+a+1(ti)−titjΨ0
+a+1(tj)
+ti−tj
+R12=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htggΨ0
+bL\{i,j}(tL\{i,j})−tiΨ0
+a+1(ti)+tjΨ0
+a+1(tj)
+ti−tj
+R13=−/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λgλ1/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})tjtiΨ0
+a+1(ti)−titjΨ0
+a+1(tj)
+ti−tj
+R14=−/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}λgλ1/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})−tiΨ0
+a+1(ti)+tjΨ0
+a+1(tj)
+ti−tj
+R15=/summationdisplay
+1≤i<j≤l/summationdisplay
+a≥0
+bL\{i,j}≥0/an}bracketle{tτaτbL\{i,j}3g−3/summationdisplay
+d=g+2Pd(λ)/an}bracketri}htgΨ0
+bL\{i,j}(tL\{i,j})(tj−1)tiΨ0
+a+1(ti)−(ti−1)tjΨ0
+a+1(tj)
+ti−tj
+R16=1
+2l/summationdisplay
+j=1/summationdisplay
+a1≥0
+a2≥0
+bL\{j}≥0stable/summationdisplay
+g1+g2=g
+I∪J=L\{j}/an}bracketle{tτa1τbIλg1/an}bracketri}htg1/an}bracketle{tτa2τbJλg2/an}bracketri}htg2Ψ0
+a1+1(tj)Ψ0
+a2+1(tj)Ψ0
+bL\{j}(tL\{j})
+Then,
+F2g+2l−4(L1) =/summationdisplay
+|bL|=2g−3+l/an}bracketle{tτbLλg/an}bracketri}htgl(2−g−l)l/summationdisplay
+j=1f0(bj,bj)tbj
+jf0(bL\{j},bL\{j}+1)tbL\{j}+1
+L\{j}(98)
+F2g+2l−4(L2) =/summationdisplay
+|bL|=2g−3+l/an}bracketle{tτbLλg/an}bracketri}htgl
+l/summationdisplay
+j=1f1(bj,bj+1)tbj+1
+j/summationdisplay
+k∈L\{j}f0(bk,bk)tbk
+k(99)
+f0(bL\{j,k},bL\{j,k}+1)tbL\{j,k}+1
+L\{j,k}+l/summationdisplay
+j=1f1(bj,bj)tbj
+jf0(bL\{j},bL\{j}+1)tbL\{j}+1
+L\{j}
++l/summationdisplay
+j=1f1(bj,bj+2)tbj+2
+j/summationdisplay
+k∈L\{j}f0(bk,bk−1)tbk−1
+kf0(bL\{j,k},bL\{j,k}+1)tbL\{j,k}+1
+L\{j,k}38 SHENGMAO ZHU
++l/summationdisplay
+j=1f1(bj,bj+2)tbj+2
+j/summationdisplay
+k1∈L\{j}f0(bk1,bk1)tbk1
+k1/summationdisplay
+k2∈L\{j,k1}f0(bk2,bk2)tbk2
+k2
+f0(bL\{j,k1,k2},bL\{j,k1,k2}+1)tbL\{j,k1,k2}+1
+L\{j,k1,k2}/parenrightig
+F2g+2l−4(L3) =/summationdisplay
+|bL|=2g−3+l/an}bracketle{tτbLλg/an}bracketri}htg
+l/summationdisplay
+j=1bjf0(bj,bj)tbj+1
+j/summationdisplay
+k∈L\{j}f0(bk,bk)tbk
+k(100)
+f0(bL\{j,k},bL\{j,k}+1)tbL\{j,k}+1
+L\{j,k}+l/summationdisplay
+j=1(bj−1)f0(bj,bj−1)tbj
+jf0(bL\{j},bL\{j}+1)tbL\{j}+1
+L\{j}
++l/summationdisplay
+j=1(bj+1)f0(bj,bj+1)tbj+2
+j/summationdisplay
+k∈L\{j}f0(bk,bk−1)tbk−1
+kf0(bL\{j,k},bL\{j,k}+1)tbL\{j,k}+1
+L\{j,k}
++l/summationdisplay
+j=1(bj+1)f0(bj,bj+1)tbj+2
+j/summationdisplay
+k1∈L\{j}f0(bk1,bk1)tbk1
+k1/summationdisplay
+k2∈L\{j,k1}f0(bk2,bk2)tbk2
+k2
+f0(bL\{j,k1,k2},bL\{j,k1,k2}+1)tbL\{j,k1,k2}+1
+L\{j,k1,k2}/parenrightig
+F2g+2l−4(L4) =−/summationdisplay
+|bL|=2g−3+l/an}bracketle{tτbLλg/an}bracketri}htg
+l/summationdisplay
+j=1bjf0(bj,bj)tbj
+jf0(bL\{j},bL\{j}+1)tbL\{j}+1
+L\{j}(101)
++l/summationdisplay
+j=1(bj+1)f0(bj,bj+1)tbj+1
+j/summationdisplay
+k∈L\{j}f0(bk,bk)tbk
+kf0(bL\{j,k},bL\{j,k}+1)tbL\{j,k}+1
+L\{j,k}
+
+F2g+2l−4(L5) =−/summationdisplay
+|bL|=2g−2+l/an}bracketle{tτbLλg−1/an}bracketri}htgl
+l/summationdisplay
+j=1f0(bj,bj−1)tbj−1
+jf0(bL\{j},bL\{j}+1)tbL\{j}+1
+L\{j}(102)
++l/summationdisplay
+j=1f0(bj,bj)tbj
+j/summationdisplay
+k∈L\{j}f0(bk,bk)tbk
+kf0(bL\{j,k},bL\{j,k}+1)tbL\{j,k}+1
+L\{j,k}
+
+F2g+2l−4(L6) =−/summationdisplay
+|bL|=2g−4+l/an}bracketle{tτbLλgλ1/an}bracketri}htglf0(bL,bL+1)tbL+1
+L(103)
+F2g+2l−4(L7) = 0 (104)
+F2g+2l−4(R1) =/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−4/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg/parenleftig
+f0(bL\{i,j},bL\{i,j}+1)tbL\{i,j}+1
+L\{i,j}(105)THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 39
+f1(a+1,a+1)a/summationdisplay
+k≥0tk+1
+ita+1−k
+j+/summationdisplay
+r∈L\{i,j}f0(br,br)tbrrf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}
+f1(a+1,a+2)a+1/summationdisplay
+k≥0tk+1
+ita+2−k
+j+/summationdisplay
+r∈L\{i,j}f0(br,br−1)tbr−1
+rf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}
+f1(a+1,a+3)a+2/summationdisplay
+k≥0tk+1
+ita+3−k
+j+/summationdisplay
+r1∈L∈L\{i,j}f0(br1,br1)tbr1r1/summationdisplay
+r2∈L\{i,j,r1}f0(br2,br2)tbr2r2
+f0(bL\{i,j,r1,r2},bL\{i,j,r1,r2}+1)tbL\{i,j,r1,r2}+1
+L\{i,j,r1,r2}f1(a+1,a+3)a+2/summationdisplay
+k≥0tk+1
+ita+3−k
+j
+
+F2g+2l−4(R2) =−/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−4/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg/parenleftig
+f0(bL\{i,j},bL\{i,j}+1)tbL\{i,j}+1
+L\{i,j}(106)
+f1(a+1,a+2)a+2/summationdisplay
+k≥0tk
+ita+2−k
+j+/summationdisplay
+r∈L\{i,j}f0(br,br)tbrrf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}
+f1(a+1,a+3)a+3/summationdisplay
+k≥0tk
+ita+3−k
+j
+
+F2g+2l−4(R3) =/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−4/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg/parenleftig
+f0(bL\{i,j},bL\{i,j}+1)tbL\{i,j}+1
+L\{i,j}(107)
+f0(a+1,a)a/summationdisplay
+k≥0tk+1
+ita+1−k
+j+/summationdisplay
+r∈L\{i,j}f0(br,br)tbrrf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}
+f0(a+1,a+1)a+1/summationdisplay
+k≥0tk+1
+ita+2−k
+j+/summationdisplay
+r∈L\{i,j}f0(br,br−1)tbr−1
+rf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}
+f0(a+1,a+2)a+2/summationdisplay
+k≥0tk+1
+ita+3−k
+j+/summationdisplay
+r1∈L∈L\{i,j}f0(br1,br1)tbr1r1/summationdisplay
+r2∈L\{i,j,r1}f0(br2,br2)tbr2r2
+f0(bL\{i,j,r1,r2},bL\{i,j,r1,r2}+1)tbL\{i,j,r1,r2}+1
+L\{i,j,r1,r2}f0(a+1,a+2)a+2/summationdisplay
+k≥0tk+1
+ita+3−k
+j
+
+F2g+2l−4(R4) =−/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−4/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg/parenleftig
+f0(bL\{i,j},bL\{i,j}+1)tbL\{i,j}+1
+L\{i,j}(108)40 SHENGMAO ZHU
+f0(a+1,a+1)a+2/summationdisplay
+k≥0tk
+ita+2−k
+j+/summationdisplay
+r∈L\{i,j}f0(br,br)tbrrf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}
+f0(a+1,a+2)a+3/summationdisplay
+k≥0tk
+ita+3−k
+j
+
+F2g+2l−4(R5) =−/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−4/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg/parenleftig
+(2g+l−1)f0(bL\{i,j},bL\{i,j}+1)tbL\{i,j}+1
+L\{i,j}(109)
+f0(a+1,a+1)a/summationdisplay
+k≥0tk+1
+ita+1−k
+j+(2g+l−1)/summationdisplay
+r∈L\{i,j}f0(br,br)tbrrf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}
+f0(a+1,a+2)a+1/summationdisplay
+k≥0tk+1
+ita+2−k
+j
+
+F2g+2l−4(R6) =/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−4/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg/parenleftig
+(2g+l−1)f0(bL\{i,j},bL\{i,j}+1)tbL\{i,j}+1
+L\{i,j}(110)
+f0(a+1,a+2)a+2/summationdisplay
+k≥0tk
+ita+2−k
+j
+
+F2g+2l−4(R7) =/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−4/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg
+/summationdisplay
+r∈L\{i,j}f1(br,br+2)tbr+2
+r(111)
+f0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}f0(a+1,a)a−1/summationdisplay
+k≥0tk+1
+ita−k
+j+/summationdisplay
+r∈L\{i,j}f1(br,br+2)tbr+2
+r
+/summationdisplay
+s∈L\{i,j,r}f0(bs,bs−1)tbs−1
+sf0(bL\{i,j,r,s},bL\{i,j,r,s}+1)tbL\{i,j,r,s}+1
+L\{i,j,r,s}f0(a+1,a+2)a+1/summationdisplay
+k≥0tk+1
+ita+2−k
+j
++/summationdisplay
+r∈L\{i,j}f1(br,br+2)tbr+2
+r/summationdisplay
+s1∈L\{i,j,r}f0(bs1,bs1)tbs1s1/summationdisplay
+s2∈L\{i,j,r1,s1}f0(bs2,bs2)tbs2s2
+f0(bL\{i,j,r,s1,s2},bL\{i,j,r,s1,s2}+1)tbL\{i,j,r,s1,s2}+1
+L\{i,j,r,s1,s2}f0(a+1,a+2)a+1/summationdisplay
+k≥0tk+1
+ita+2−k
+j+/summationdisplay
+r∈L\{i,j}f1(br,br)tbrr
+f0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}f0(a+1,a+2)a+1/summationdisplay
+k≥0tk+1
+ita+2−k
+j+/summationdisplay
+r1∈L\{i,j}f1(br1,br1+1)tbr1+1
+r1
+/summationdisplay
+r2∈L\{i,j,r1}f0(br2,br2)tbr2r2f0(bL\{i,j,r1,r2},bL\{i,j,r1,r2}+1)tbL\{i,j,r1,r2}+1
+L\{i,j,r1,r2}f0(a+1,a+2)THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 41
+a+1/summationdisplay
+k≥0tk+1
+ita+2−k
+j+/summationdisplay
+r∈{i,j}f1(br,br+1)tbr+1
+rf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}f0(a+1,a+1)
+a/summationdisplay
+k≥0tk+1
+ita+1−k
+j+/summationdisplay
+r1∈L\{i,j}f1(br1,br1+2)tbr1+2
+r1/summationdisplay
+r2∈L\{i,j,r1}f0(br2,br2)tbr2r2
+f0(bL\{i,j,r1,r2},bL\{i,j,r1,r2}+1)tbL\{i,j,r1,r2}+1
+L\{i,j,r1,r2}f0(a+1,a+1)a/summationdisplay
+k≥0tk+1
+ita+1−k
+j
+
+F2g+2l−4(R8) =−/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−4/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htg
+/summationdisplay
+r∈L\{i,j}f1(br,br+2)tbr+2
+r(112)
+f0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}f0(a+1,a+1)a+1/summationdisplay
+k≥0tk
+ita+1−k
+j+/summationdisplay
+r∈L\{i,j}f1(br,br+1)tbr+1
+r
+f0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}f0(a+1,a+2)a+2/summationdisplay
+k≥0tk
+ita+2−k
+j+/summationdisplay
+r∈L\{i,j}f1(br,br+2)tbr+2
+r
+/summationdisplay
+s∈L\{i,j,r}f0(bs,bs)tbssf0(bL\{i,j,r,s},bL\{i,j,r,s}+1)tbL\{i,j,r,s}+1
+L\{i,j,r,s}f0(a+1,a+2)a+2/summationdisplay
+k≥0tk
+ita+2−k
+j
+
+F2g+2l−4(R9) =−/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−3/an}bracketle{tτaτbL\{i,j}λg−1/an}bracketri}htg/parenleftig
+f0(bL\{i,j},bL\{i,j}+1)tbL\{i,j}+1
+L\{i,j}(113)
+f0(a+1,a)a−1/summationdisplay
+k≥0tk+1
+ita−k
+j+/summationdisplay
+r∈L\{i,j}f0(br,br)tbrrf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}
+f0(a+1,a+1)a/summationdisplay
+k≥0tk+1
+ita+1−k
+j+/summationdisplay
+r∈L\{i,j}f0(br,br−1)tbr−1
+rf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}
+f0(a+1,a+2)a+1/summationdisplay
+k≥0tk+1
+ita+2−k
+j+/summationdisplay
+r1∈L∈L\{i,j}f0(br1,br1)tbr1r1/summationdisplay
+r2∈L\{i,j,r1}f0(br2,br2)tbr2r2
+f0(bL\{i,j,r1,r2},bL\{i,j,r1,r2}+1)tbL\{i,j,r1,r2}+1
+L\{i,j,r1,r2}f0(a+1,a+2)a+1/summationdisplay
+k≥0tk+1
+ita+2−k
+j
+
+F2g+2l−4(R10) =/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−3/an}bracketle{tτaτbL\{i,j}λg−1/an}bracketri}htg/parenleftig
+f0(bL\{i,j},bL\{i,j}+1)tbL\{i,j}+1
+L\{i,j}(114)42 SHENGMAO ZHU
+f0(a+1,a+1)a+1/summationdisplay
+k≥0tk
+ita+1−k
+j+/summationdisplay
+r∈L\{i,j}f0(br,br)tbrrf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}
+f0(a+1,a+2)a+2/summationdisplay
+k≥0tk
+ita+2−k
+j
+
+F2g+2l−4(R11) =/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−4/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgg/parenleftig
+f0(bL\{i,j},bL\{i,j}+1)tbL\{i,j}+1
+L\{i,j}(115)
+f0(a+1,a+1)a/summationdisplay
+k≥0tk+1
+ita+1−k
+j+/summationdisplay
+r∈L\{i,j}f0(br,br)tbrrf0(bL\{i,j,r},bL\{i,j,r}+1)tbL\{i,j,r}+1
+L\{i,j,r}
+f0(a+1,a+2)a+1/summationdisplay
+k≥0tk
+ita+2−k
+j
+
+F2g+2l−4(R12) =−/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−4/an}bracketle{tτaτbL\{i,j}λg/an}bracketri}htgg/parenleftig
+f0(bL\{i,j},bL\{i,j}+1)tbL\{i,j}+1
+L\{i,j}(116)
+f0(a+1,a+2)a+2/summationdisplay
+k≥0tk
+ita+2−k
+j
+
+F2g+2l−4(R13) =−/summationdisplay
+1≤i<j≤l/summationdisplay
+a+|bL\{i,j}|=2g+l−5/an}bracketle{tτaτbL\{i,j}λgλ1/an}bracketri}htg/parenleftig
+f0(bL\{i,j},bL\{i,j}+1)tbL\{i,j}+1
+L\{i,j}(117)
+f0(a+1,a+2)a+1/summationdisplay
+k≥0tk+1
+ita+2−k
+j
+
+F2g+2l−4(R14) =F2g+2l−4(R15) = 0 (118)
+F2g+2l−4(R16) =1
+2l/summationdisplay
+j=1stable/summationdisplay
+g1+g2=g
+I∪J=L\{j}/summationdisplay
+a1+|bI|=2g1−2+|I|
+a2+|bJ|=2g2−2+|J|/an}bracketle{tτa1τbIλg1/an}bracketri}htg1/an}bracketle{tτa2τbJλg2/an}bracketri}htg2 (119)
+f0(a1+1,a1+2)f0(a2+1,a2+2)ta1+a2+4
+jf0(bL\{j},bL\{j}+1)tbL\{j}+1
+L\{j}
+[tbL+1
+L]F2g+2l−4(L1) =l/summationdisplay
+j=1/an}bracketle{tτbj+1τbL\{j}λg/an}bracketri}htgl(2−g−l)f0(bj+1,bj+1)f0(bL\{j},bL\{j}+1)(120)THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 43
+[tbL+1
+L]F2g+2l−4(L2) =l/summationdisplay
+j=1/summationdisplay
+k∈L\{j}/an}bracketle{tτbjτbk+1τbL\{j,k}λg/an}bracketri}htglf1(bj,bj+1)f0(bk+1,bk+1)(121)
+f0(bL\{j,k},bL\{j,k}+1)+l/summationdisplay
+j=1/an}bracketle{tτbj+1τbL\{j}λg/an}bracketri}htglf1(bj−1,bj+1)f0(bL\{j},bL\{j}+1)
++l/summationdisplay
+j=1/summationdisplay
+k∈L\{j}/an}bracketle{tτbj−1τbk+2τbL\{j,k}λg/an}bracketri}htglf1(bj−1,bj+1)f0(bk+2,bk+1)f0(bL\{j,k},bL\{j,k}+1)
++l/summationdisplay
+j=1/summationdisplay
+k1∈L\{j}/summationdisplay
+k2∈L\{j,k1}/an}bracketle{tτbj−1τbk1+1τbk2+1τbL\{j,k1,k2}λg/an}bracketri}htglf1(bj−1,bj+1)
+f0(bk1+1,bk1+1)f0(bk2+1,bk2+1)f0(bL\{j,k1,k2},bL\{j,k1,k2}+1)
+[tbL+1
+L]F2g+2l−4(L3) =l/summationdisplay
+j=1/summationdisplay
+k∈L\{j}/an}bracketle{tτbjτbk+1τbL\{j,k}λg/an}bracketri}htgbjf0(bj,bj)f0(bk+1,bk+1)(122)
+f0(bL\{j,k},bL\{j,k}+1)+l/summationdisplay
+j=1/an}bracketle{tτbj+1τbL\{j}λg/an}bracketri}htgbjf0(bj+1,bj)f0(bL\{j},bL\{j}+1)
++l/summationdisplay
+j=1/summationdisplay
+k∈L\{j}/an}bracketle{tτbj−1τbk+2τbL\{j,k}λg/an}bracketri}htgbjf0(bj−1,bj)f0(bk+2,bk+1)f0(bL\{j,k},bL\{j,k}+1)
++l/summationdisplay
+j=1/summationdisplay
+k1∈L\{j}/summationdisplay
+k2∈L\{j,k1}/an}bracketle{tτbj−1τbk1+1τbk2+1τbL\{j,k1,k2}λg/an}bracketri}htgbjf0(bj−1,bj)
+f0(bk1+1,bk1+1)f0(bk2+1,bk2+1)f0(bL\{j,k1,k2},bL\{j,k1,k2}+1)
+[tbL+1
+L]F2g+2l−4(L4) =−l/summationdisplay
+j=1/an}bracketle{tτbj+1τbL\{j}λg/an}bracketri}htg(bj+1)f0(bj+1,bj+1)f0(bL\{j},bL\{j}+1)(123)
+−l/summationdisplay
+j=1/summationdisplay
+k∈L\{j}/an}bracketle{tτbjτbk+1τbL\{j,k}λg/an}bracketri}htg(bj+1)f0(bj,bj+1)f0(bk+1,bk+1)f0(bL\{j,k},bL\{j,k}+1)
+[tbL+1
+L]F2g+2l−4(L5) =−l/summationdisplay
+j=1/an}bracketle{tτbj+2τbL\{j}λg−1/an}bracketri}htglf0(bj+2,bj+1)f0(bL\{j},bL\{j}+1)(124)
+−l/summationdisplay
+j=1/summationdisplay
+k∈L\{j}/an}bracketle{tτbj+1τbk+1τbL\{j,k}λg−1/an}bracketri}htglf0(bj+1,bj+1)f0(bk+1,bk+1)f0(bL\{j,k},bL\{j,k}+1)44 SHENGMAO ZHU
+[tbL+1
+L]F2g+2l−4(L6) =−/an}bracketle{tτbLλgλ1/an}bracketri}htglf0(bL,bL+1) (125)
+on the right hand side,
+[tbL+1
+L]F2g+2l−4(R1) =/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bjτbL\{i,j}λg/an}bracketri}htgf0(bL\{i,j},bL\{i,j}+1)f1(bi+bj+1,bi+bj+1)(126)
++/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bj−1τbr+1τbL\{i,j,r}λg/an}bracketri}htgf1(bi+bj,bi+bj+1)f0(bL\{i,j,r},bL\{i,j,r}+1)
+f0(br+1,br+1)+/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bj−2τbr+2τbL\{i,j,r}λg/an}bracketri}htgf1(bi+bj−1,bi+bj+1)f0(br+2,br+1)
+f0(bL\{i,j,r},bL\{i,j,r}+1)+/summationdisplay
+1≤i<j≤l/summationdisplay
+r1∈L\{i,j}/summationdisplay
+r2∈L\{i,j,r1}/an}bracketle{tτbi+bj−2τbr1+1τbr2+1τbL\{i,j,r1,r2}λg/an}bracketri}htg
+f0(br1+1,br1+1)f0(br2+1,br2+1)f0(bL\{i,j,r1,r2},bL\{i,j,r1,r2}+1)f1(bi+bj−1,bi+bj+1)
+[tbL+1
+L]F2g+2l−4(R2) =−/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bjτbL\{i,j}λg/an}bracketri}htgf0(bL\{i,j},bL\{i,j}+1)f1(bi+bj+1,bi+bj+2)(127)
+−/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bj−1τbr+1τbL\{i,j,r}λg/an}bracketri}htgf0(br+1,br+1)f0(bL\{i,j,r},bL\{i,j,r}+1)
+f1(bi+bj+1,bi+bj+3)
+[tbL+1
+L]F2g+2l−4(R3) =/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bjτbL\{i,j}λg/an}bracketri}htgf0(bL\{i,j},bL\{i,j}+1)f0(bi+bj+1,bi+bj)(128)
++/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bj−1τbr+1τbL\{i,j,r}λg/an}bracketri}htgf0(bi+bj,bi+bj)f0(bL\{i,j,r},bL\{i,j,r}+1)
+f0(br+1,br+1)+/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bj−2τbr+2τbL\{i,j,r}λg/an}bracketri}htgf0(bi+bj−1,bi+bj)f0(br+2,br+1)
+f0(bL\{i,j,r},bL\{i,j,r}+1)+/summationdisplay
+1≤i<j≤l/summationdisplay
+r1∈L\{i,j}/summationdisplay
+r2∈L\{i,j,r1}/an}bracketle{tτbi+bj−2τbr1+1τbr2+1τbL\{i,j,r1,r2}λg/an}bracketri}htg
+f0(br1+1,br1+1)f0(br2+1,br2+1)f0(bL\{i,j,r1,r2},bL\{i,j,r1,r2}+1)f0(bi+bj−1,bi+bj)
+[tbL+1
+L](F2g+2l−4(R4)+F2g+2l−4(R5)) =−/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bjτbL\{i,j}λg/an}bracketri}htg(2g+l)(129)
+f0(bL\{i,j},bL\{i,j}+1)f0(bi+bj+1,bi+bj+1)−/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bj−1τbr+1τbL\{i,j,r}λg/an}bracketri}htg
+(2g+l)f0(br+1,br+1)f0(bL\{i,j,r},bL\{i,j,r}+1)f0(bi+bj,bi+bj+1)THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 45
+[tbL+1
+L]F2g+2l−4(R6) =/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bjτbL\{i,j}λg/an}bracketri}htg(2g+l−1)f0(bL\{i,j},bL\{i,j}+1) (130)
+f0(bi+bj+1,bi+bj+2)
+[tbL+1
+L]F2g+2l−4(R7) =/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bj+1τbr−1τbL\{i,j,r}λg/an}bracketri}htgf1(br−1,br+1)f0(bL\{i,j,r},bL\{i,j,r}+1)(131)
+f0(bi+bj+2,bi+bj+1)+/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/summationdisplay
+s∈L\{i,j,r}/an}bracketle{tτbi+bj−1τbr−1τbs+2τbL\{i,j,r,s}λg/an}bracketri}htg
+f1(br−1,br+1)f0(bs+2,bs+1)f0(bL\{i,j,r,s},bL\{i,j,r,s}+1)f0(bi+bj,bi+bj+1)
++/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/summationdisplay
+s1∈L\{i,j,r}/summationdisplay
+s2∈L\{i,j,r,s1}/an}bracketle{tτbi+bj−1τbr−1τbs1+1τbs2+1τbL\{i,j,r,s1,s2}λg/an}bracketri}htgf1(br−1,br+1)
+f0(bs1+1,bs1+1)f0(bs2+1,bs2+1)f0(bL\{i,j,r,s1,s2},bL\{i,j,r,s1,s2}+1)f0(bi+bj,bi+bj+1)
++/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bj−1τbr+1τbL\{i,j,r}λg/an}bracketri}htgf1(br+1,br+1)f0(bL\{i,j,r},bL\{i,j,r}+1)
+f0(bi+bj,bi+bj+1)+/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/summationdisplay
+s∈L\{i,j,r}/an}bracketle{tτbi+bj−1τbrτbs+1τbL\{i,j,r,s}λg/an}bracketri}htgf1(br,br+1)
+f0(bs+1,bs+1)f0(bL\{i,j,r,s},bL\{i,j,r,s}+1)f0(bi+bj,bi+bj+1)+/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bjτbrτbL\{i,j,r}λg/an}bracketri}htg
+f1(br,br+1)f0(bL\{i,j,r},bL\{i,j,r}+1)f0(bi+bj+1,bi+bj+1)
++/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/summationdisplay
+s∈L\{i,j,r}/an}bracketle{tτbi+bjτbr−1τbs+1τbL\{i,j,r,s}λg/an}bracketri}htgf1(br−1,br+1)f0(bs+1,bs+1)
+f0(bL\{i,j,r,s},bL\{i,j,r,s}+1)f0(bi+bj+1,bi+bj+1)
+[tbL+1
+L]F2g+2l−4(R8) =−/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bj+1τbr−1τbL\{i,j,r}λg/an}bracketri}htgf1(br−1,br+1)(132)
+f0(bL\{i,j,r},bL\{i,j,r}+1)f0(bi+bj+2,bi+bj+2)−/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bjτbrτbL\{i,j,r}λg/an}bracketri}htg
+f1(br,br+1)f0(bL\{i,j,r},bL\{i,j,r}+1)f0(bi+bj+1,bi+bj+2)
+−/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/summationdisplay
+s∈L\{i,j,r}/an}bracketle{tτbi+bjτbr−1τbs+1τbL\{i,j,r,s}λg/an}bracketri}htgf1(br−1,br+1)f0(bs+1,bs+1)
+f0(bL\{i,j,r,s},bL\{i,j,r,s}+1)f0(bi+bj+1,bi+bj+2)
+[tbL+1
+L]F2g+2l−4(R9) =−/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bj+1τbL\{i,j}λg−1/an}bracketri}htgf0(bL\{i,j},bL\{i,j}+1)f0(bi+bj+2,bi+bj+1)(133)46 SHENGMAO ZHU
+−/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bjτbr+1τbL\{i,j,r}λg−1/an}bracketri}htgf0(br+1,br+1)f0(bL\{i,j,r},bL\{i,j,r}+1)
+f0(bi+bj+1,bi+bj+1)−/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/summationdisplay
+s∈L\{i,j,r}/an}bracketle{tτbi+bj−1τbr+1τbs+1τbL\{i,j,r,s}λg−1/an}bracketri}htg
+f0(br+1,br+1)f0(bs+1,bs+1)f0(bL\{i,j,r,s},bL\{i,j,r,s}+1)f0(bi+bj,bi+bj+1)
+−/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bj−1τbr+2τbL\{i,j,r}λg−1/an}bracketri}htgf0(br+2,br+1)f0(bL\{i,j,r},bL\{i,j,r}+1)
+f0(bi+bj,bi+bj+1)
+[tbL+1
+L]F2g+2l−4(R10) =/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bj+1τbL\{i,j}λg−1/an}bracketri}htgf0(bL\{i,j},bL\{i,j}+1)f0(bi+bj+2,bi+bj+2)(134)
++/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bjτbr+1τbL\{i,j,r}λg−1/an}bracketri}htgf0(br+1,br+1)f0(bL\{i,j,r},bL\{i,j,r}+1)
+f0(bi+bj+1,bi+bj+2)
+[tbL+1
+L]F2g+2l−4(R11) =/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bjτbL\{i,j}λg/an}bracketri}htggf0(bL\{i,j},bL\{i,j}+1)f0(bi+bj+1,bi+bj+1)(135)
++/summationdisplay
+1≤i<j≤l/summationdisplay
+r∈L\{i,j}/an}bracketle{tτbi+bj−1τbr+1τbL\{i,j,r}λg/an}bracketri}htggf0(br+1,br+1)f0(bL\{i,j,r},bL\{i,j,r}+1)
+f0(bi+bj,bi+bj+1)
+[tbL+1
+L]F2g+2l−4(R12) =−/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bjτbL\{i,j}λg/an}bracketri}htggf0(bL\{i,j},bL\{i,j}+1)f0(bi+bj+1,bi+bj+2)(136)
+[tbL+1
+L]F2g+2l−4(R13) =−/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bj−1τbL\{i,j}λgλ1/an}bracketri}htgf0(bL\{i,j},bL\{i,j}+1)f0(bi+bj,bi+bj+1)(137)
+[tbL+1
+L]F2g+2l−4(R16) =1
+2l/summationdisplay
+j=1/summationdisplay
+a1+a2=bj−3stable/summationdisplay
+g1+g2=g
+I∪J=L\{j}/an}bracketle{tτa1τbIλg1/an}bracketri}htg1/an}bracketle{tτa2τbJλg2/an}bracketri}htg2 (138)
+f0(a1+1,a1+2)f0(a2+1,a2+2)f0(bL\{j},bL\{j}+1)
+Therefore, from the equation (97), we get the identity
+(120)+(121)+(122) +(123)+(124)+(125) = (126)+(127)(139)
++(128)+(129) +(130) +(131)+(132)+(133)+(134)+(135) +13 6)+(137) +(138)
+We note that, three types of the Hodge integrals appear in ide ntity (139), /an}bracketle{tτbLλg/an}bracketri}htg,/an}bracketle{tτbLλg−1/an}bracketri}htg
+and/an}bracketle{tτbLλgλ1/an}bracketri}htg.THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION OF MARI ˜NO-VAFA FORMULA 47
+By now, we have known the closed formula for λg-integral [13] and a recursion formula for
+λg−1-integral [24].
+(140)/integraldisplay
+Mg,lψd1
+1...ψdl
+lλg=/parenleftbigg2g−3+l
+b1,..,bl/parenrightbigg
+cg,
+where/summationtextl
+i=1bi= 2g−3+l,b1,..,bl≥0 andcgis a constant that depends only on g.
+/an}bracketle{tτb1···τblλg−1/an}bracketri}htg=1
+l/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bj−1/productdisplay
+k/\e}atio\slash=i,jτbkλg−1/an}bracketri}htg(bi+bj)!
+bi!bj!+Cg,l(b1,..,bl) (141)
+whereCg,n(b1,..,bl) is a constant related to b1,..,blandg,l.
+On the other hand, the constants f0(b,k) andf1(b,k) are available by their definition and the
+recursions (52), (53). For example, f0(b,b+1) =b!,f1(b,b+2) = (b+1)!/summationtextb+1
+k=21
+k. Thus, solving
+the equation (139), we will get a formula for the computation of Hodge integral /an}bracketle{tτbLλgλ1/an}bracketri}htg.
+We observe that only the terms (125) and (137) contain the Hod ge integral of type /an}bracketle{tτbLλgλ1/an}bracketri}htg.
+Moving and combining the corresponding terms, we have
+/an}bracketle{tτbLλgλ1/an}bracketri}htg=1
+l/summationdisplay
+1≤i<j≤l/an}bracketle{tτbi+bj−1τbL\{i,j}λgλ1/an}bracketri}htg(bi+bj)!
+bi!bj!+C(g,l,b1,..,bl) (142)
+Where
+C(g,l,b1,..,bl) =−1
+l/producttextl
+i=1bi!((126)+(127)+(128) +(129)+(130)+(131)+(132) (143)
++(133)+(134) +(135)+(136)+(138) −(120)−(121)−(122)−(123)−(124))
+Formula (143) contains many terms, we hope to make a computer program to calculate
+C(g,l,b1,..,bl).
+References
+[1] G. Borot, B. Eynard, M. Mulase and B. Safnuk, A Matrix model for simple Hurwitz numbers, and topologicla
+recursion , math.AG/0906.1206.
+[2] V. Bouchard, A. Klemm, M. Mari˜ no, and S. Pasquetti, Remodeling the B-model , Commun. Math. Phys.
+287(2009), 117-178.
+[3] V. Bouchard and M. Mari˜ no, Hurwitz numbers, matrix models and enumerative geometry ,
+math.AG/0709.1458.
+[4] L. Chen, Symmetrized cut-join equation of the Mari˜ no-Vafa formula , Pacific Journal of Math., vol. 235,
+no.2, 2008, 201-212.
+[5] L. Chen, Bouchard-Klemm-Mari˜ no-Pasquetti Conjecture for C3, arXiv:0910.3739v1.
+[6] L. Chen, Y. Li and K. Liu, Localization, Hurwitz numbers and the Witten conjecture , math.AG/0609263.
+[7] T. Ekedahl, S. Lando, M. Shapiro, A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of
+curves, Invent. Math. 146(2001), no. 2, 297-327.
+[8] B. Eynard, M. Mari˜ no and N. Orantin Holomorphic anomaly and matrix models , arXiv:hep-th/0702110.
+[9] B. Eynard, M. Mulase and B. Safnuk, The Laplace transform of the cut-and-join equation and the B ouchard-
+Mari˜ no conjecture on Hurwitz numbers , math.AG/0907.5224v1
+[10] B. Eynard and N. Orantin Invariants of algebraic curves and topological expansion , arXiv:math-th/0702045.
+[11] B. Eynard and N. Orantin Algebraic methods in random matrices and enumerative geome try,
+arXiv:0811.3531.
+[12] B. Eynard Topological expansion for 1-hermitian matrix model correl ation functions , arXiv:hep-th/0407261.
+[13] C. Faber, and R. Pandharipande Hodge integrals, partition matrices, and the λgconjecture , Ann.Math.(2)
+157(2003),no.1,97-124.48 SHENGMAO ZHU
+[14] I.P. Goulden, D.M. Jackson and R. Vakil A short prooof of the λg-Conjecture without Gromov-Witten theory:
+Hurwitz theory and the moduli of curves , math.AG/0604297.
+[15] Y. Li Some results of the Mari˜ no-Vafa Formula , Math.Res.Lett. 13(2006),no.00,10001-10018.
+[16] J. Li, C.-C. Liu, K. Liu and J. Zhou, A mathematical theory of the topological vertex , math.AG/0408426.
+[17] C.-C. Liu, K. Liu, J. Zhou, A proof of a conjecture of Mari˜ no-Vafa on Hodge integrals , J. Differential Geom.
+65(2003).
+[18] C.-C. Liu, K. Liu, J. Zhou, Mari˜ no-Vafa formula and Hodge integral identities , arXiv:math.AG/0308015
+[19] M. Mari˜ no, C. Vafa Framed knots at large N , Orbifolds in mathematics and physics (Madison, WI, 2001).
+[20] M. Mulase, N. Zhang Polynomial recursion formula for linear Hodge integrals , arXiv:0908.2267
+[21] J. Zhou Hodge integrals, Hurwitz numbers, and symmetric groups , math.AG/0308024.
+[22] J. Zhou Local Mirror Symmetry for One-Legged Topological Vertex , arXiv:0910.4320.
+[23] J. Zhou Local Mirror Symmetry for the Topological Vertex , arXiv:0911.2343.
+[24] S.M. Zhu, Hodge integral with one λ-class.
+Center of Mathematical Sciences, Zhejiang University, Han gzhou, Zhejiang 310027, China
+E-mail address :zhushengmao@gmail.com
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+arXiv:1001.0619v4  [math.AG]  1 Jul 2011BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS
+SABIN CAUTIS AND JOEL KAMNITZER
+Abstract. We introduce the idea of a geometric categorical Lie algebra action on derived
+categories of coherent sheaves. The main result is that such an action induces an action of
+the braid group associated to the Lie algebra. The same proof shows that strong categorical
+actions in the sense of Khovanov-Lauda and Rouquier also lea d to braid group actions. As
+an example, we construct an action of Artin’s braid group on d erived categories of coherent
+sheaves on cotangent bundles to partial flag varieties.
+Contents
+1. Introduction 2
+Acknowledgements 3
+2. Definitions and Main Results 3
+2.1. Notation 3
+2.2. Geometric categorical gactions 4
+2.3. Role of the deformation 6
+2.4.k×-equivariance 8
+2.5. Example from resolutions of Kleinian singularities 9
+2.6. Equivalences via geometric categorical sl2actions 9
+2.7. Braid group action via geometric categorical gactions 10
+3. Example: cotangent bundles to flag varieties 10
+3.1. The categorical gaction 10
+3.2. The braid group action 11
+3.3. Proof of sl2conditions (i) - (vii) 12
+3.4. Proof of Serre relation (viii) 13
+3.5. Existence of deformations: conditions (x) and (xi) 15
+4. Preliminaries 18
+4.1. Some general notions 18
+4.2. Some basic sl2relations 19
+4.3. Spaces of maps 20
+4.4. Some basic sl3relations 21
+4.5. Induced maps 23
+5. Proof of Main Theorem 2.10 31
+5.1. Step 1: Calculation of Eij∗F(/angbracketleftλ,αi/angbracketright+s)
+i 32
+5.2. Step 2: Calculation of Eij∗F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s)
+i 32
+5.3. Step 3: Calculation of Eij∗Ti 32
+5.4. Step 4: Proof that Tij∗Ti∼=Ti∗Tj 36
+5.5. Step 5: Proof of braid relation 37
+6. Braiding via strong categorical g-actions 37
+References 39
+Date: October 23, 2018.
+12 SABIN CAUTIS AND JOEL KAMNITZER
+1.Introduction
+LetXbe a smooth complex variety. Often the derived category of coher ent sheaves on X,
+denoted D(X), possesses interesting autoequivalences, not coming from auto morphisms of X
+itself. For example, if D(X) contains a spherical object E, then Seidel-Thomas [ST] defined a
+spherical twist TE:D(X)→D(X) which is a non-trivial autoequivalence.
+The notion of twists in spherical objects has been generalized by va rious authors (Horja [Ho],
+Anno [A], and Rouquier [Ro1]) to twists in spherical functors (a relativ e version). In [CKL2],
+[CKL3], we (jointly with Anthony Licata and following ideas of Chuang-R ouquier [CR]) defined
+the notion of geometric categorical sl2actions as a generalization of the notion of spherical
+functors. We showed that geometric categorical sl2actions give rise to equivalences of derived
+categories of coherent sheaves.
+Often autoequivalences of D(X) can be organized into an action of a braid group. Seidel-
+Thomas[ST]showedthatgivenacollectionofsphericalobjectswhic hformatypeΓarrangement,
+the spherical twists generate an action of the braid group BΓ. An important example from [ST]
+of this situation concerned the case where Xis the resolution of a surface quotient singularity
+C2/Hand the spherical objects come from the exceptional P1s.
+Another example of a braid group action was given by Khovanov-Tho mas in [KT]. They
+showed that Bnacts onD(T⋆Fl(Cn)), the derived category of the cotangent bundle to the
+full flag variety, with the generators acting by spherical twists. O ur purpose in this paper is
+to introduce a new method of constructing braid group actions (ca lled geometric categorical
+gactions), where the generators act by the equivalences coming fr om geometric categorical
+sl2actions. Roughly speaking, spherical objects are a special case o f geometric categorical sl2
+actions and type Γ arrangements of spherical objects are a spec ial case of geometric categorical
+gactions.
+To explain our motivation for this notion, let us recall that our proof that a geometric cat-
+egorical sl2action gives an equivalence came in two parts. First in [CKL2], we showe d that
+a geometric categorical sl2action gives a strong categorical sl2action, a notion introduced by
+Chuang-Rouquier [CR]. We then showed in [CKL3] that a strong categ oricalsl2action gives
+an equivalence, using an explicit complex introduced by Chuang-Rouq uier [CR]. The reason
+for introducing the notion of geometric categorical sl2action, rather than working with strong
+categorical sl2actions, is that the axioms of the former are much easier to check in examples.
+The notion of strong categorical sl2action has been generalized by Rouquier [Ro2] and
+Khovanov-Lauda [KL1, KL2, KL3] to the notion of strong categor icalgaction. Hence it is
+natural to conjecture that a strong categorical gaction gives an action of the braid group of
+typeg(denoted Bg). Also it is natural to search for a notion of geometric categorical gaction
+which implies strong categorical gaction but which is easier to check in geometric examples.
+In this paper, we essentially accomplish these goals. More specifically , we define the notion of
+geometric categorical gaction, whenever gis a simply-laced Kac-Moody Lie algebra. We then
+prove that a geometric categorical gaction gives rise to an action of Bg(Theorem 2.10). We also
+show that a strong categorical gaction gives an action of Bg(Theorem 6.3). This essentially
+answers a conjecture of Rouquier [Ro2] (Rouquier has also recent ly proven his conjecture via a
+different method). However, we do not prove that a geometric cat egoricalgaction gives a strong
+categorical gaction, though we expect this to be the case (the proof should follo w along the
+same lines as [CKL2], where we established this result for g=sl2).
+Wegiveaquickexampleshowinghowresolutionsof C2/Hgivegeometriccategorical gactions.
+In greaterdetail in section 3, we discuss the morecomplicated exam ple ofa geometric categorical
+slnaction on cotangent bundles to n-step partial flag varieties. This generalizes the work ofBRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 3
+Khovanov-Thomas [KT] for T⋆Fl(Cn) and also our previous work [CKL2, CKL3] on cotangent
+bundles to Grassmannians.
+In a forthcoming paper with Anthony Licata [CKL4], we will construct geometric categorical
+gactions on Nakajima quiver varieties, generalizing the two examples in this paper. Using the
+main result of this paper, this will provide many more examples of braid group actions.
+There are also many interesting examples of strong categorical gactions, not involving coher-
+ent sheaves. For examples, Khovanov-Lauda[KL1, KL2, KL3] hav econsideredstrongcategorical
+slnactions on categories of modules over cohomology rings of partial fl ag varieties and Chuang-
+Rouquier [CR] have defined strong categorical /hatwideslpactions on categories of representations of the
+symmetric group in characteristic p. Our Theorem 6.3 can be applied to these situations to
+produce braid group actions.
+Acknowledgements. We would like to thank PierreBaumann, Chris Brav, Mikhail Khovanov,
+Aaron Lauda, Anthony Licata and Rapha¨ el Rouquier for helpful d iscussions. In the course of
+this work, S.C. was supported by NSF Grant 0801939 and J.K. by NSE RC. We would also like
+to thank MSRI for excellent working conditions and hospitality. We als o thank the referee for
+his very careful reading of this paper.
+2.Definitions and Main Results
+In this section we define the concept of a geometric categorical gaction, review the con-
+struction of equivalences from strong categorical sl2actions and state our main result (Theorem
+2.10).
+2.1.Notation. Fix a base field k, which is not assumed to be of characteristic 0, nor alge-
+braically closed.
+Let Γ be a graph without multiple edges or loops and with finite vertex s etI. In addition,
+fix the following data.
+(i) a free ZmoduleX(the weight lattice),
+(ii) fori∈Ian element αi∈X(simple roots),
+(iii) fori∈Ian element Λ i∈X(fundamental weight),
+(iv) a symmetric non-degenerate bilinear form /an}bracke⊔le{⊔·,·/an}bracke⊔ri}h⊔onX.
+These data should satisfy:
+(i) the set {αi}i∈Iis linearly independent,
+(ii)Ci,j=/an}bracke⊔le{⊔αi,αj/an}bracke⊔ri}h⊔(the Cartan matrix) so that /an}bracke⊔le{⊔αi,αi/an}bracke⊔ri}h⊔= 2 and for i/ne}a⊔ionslash=j,/an}bracke⊔le{⊔αi,αj/an}bracke⊔ri}h⊔=
+/an}bracke⊔le{⊔αj,αi/an}bracke⊔ri}h⊔ ∈ {0,−1}depending on whether or not i,j∈Iare joined by an edge,
+(iii)/an}bracke⊔le{⊔Λi,αj/an}bracke⊔ri}h⊔=δi,jfor alli,j∈I,
+(iv) dim X=|I|+corank( C), where Cis the Cartan matrix associated to Γ.
+Lethk=X⊗Zkand leth′
+k= span(Λ i)⊂hk.
+Associated to Γ, we have a Kac-Moody Lie algebra g(defined over C). LetBgdenote the
+braid group of type g. It has generators σifori∈Iand relations
+σiσjσi=σjσiσjifiandjare connected in Γ,
+σiσj=σjσiifiandjare not connected in Γ.
+Recallthat BgmapstotheWeylgroup Wgoftypegwhichhasthe samegeneratorsandrelations,
+except that the generators square to the identity.
+When we write H⋆(X) we will mean the cohomology with kcoefficients of Xas a variety over
+Cbut shifted so that it lies between degrees −dim(X) and dim( X). For example, H⋆(P1) =
+k[−1]⊕k[1]. By convention, H⋆(P−1) = 0.4 SABIN CAUTIS AND JOEL KAMNITZER
+2.2.Geometric categorical gactions. In [CKL2] we introduced the concept of a geometric
+categorical gaction when g=sl2. We now extend this definition to arbitrary simply-laced g.
+All varieties will be defined over k.
+2.2.1.Fourier-Mukai formalism. We briefly recall the formalism of Fourier-Mukai (FM) kernels
+(see [Hu], section 5.1, for more details). All the functors which follow are derived. Let X,Y
+be two smooth varieties. A FM kernel is any object P ∈D(X×Y) of the derived category of
+coherent sheaves on X×Ywhose support is proper over XandY. It defines the associated
+Fourier-Mukai transform
+ΦP:D(X)→D(Y)
+F /maps⊔o→π2∗(π∗
+1(F)⊗P)
+FM transforms have right and left adjoints which are themselves FM transforms. In particular,
+the right adjoint of Φ Pis the FM transform with respect to PR:=P∨⊗π∗
+2ωX[dim(X)]∈
+D(Y×X) (here we use the natural isomorphism X×Y∼− →Y×X). Similarly, the left adjoint of
+ΦPis the FM transform with respect to PL:=P∨⊗π∗
+1ωY[dim(Y)]. Here P∨denotes the dual
+ofP.
+We canexpresscomposition ofFM transformsin terms oftheir kern els. IfX,Y,Zarevarieties
+and Φ P:D(X)→D(Y),ΦQ:D(Y)→D(Z) are FM transforms, then Φ Q◦ΦPis a FM
+transform with respect to the kernel
+Q∗P:=π13∗(π∗
+12(P)⊗π∗
+23(Q))
+where∗is called the convolution product.
+2.2.2. A geometric categorical gactionconsists of the following data.
+(i) A collection of smooth varieties Y(λ) forλ∈X.
+(ii) Fourier-Mukai kernels
+E(r)
+i(λ)∈D(Y(λ)×Y(λ+rαi)) andF(r)
+i(λ)∈D(Y(λ+rαi)×Y(λ)).
+We will usually write just E(r)
+iandF(r)
+ito simplify notation whenever possible. When
+r= 1 we just write EiandFi.
+(iii) For each Y(λ) a flat deformation ˜Y(λ)→h′
+k(where the fibre over 0 ∈h′
+kis identified
+withY(λ)).
+Denote by ˜Yi(λ)→span(Λ i)⊂h′
+kthe restriction of ˜Y(λ) to span(Λ i) (this is a one parameter
+deformation of Y(λ)).
+Remark 2.1. In practice we only need the first order deformation of ˜Y(λ), but in geometric
+examples there exists a natural deformation over h′
+k. Replacing this deformation by the corre-
+sponding first order deformation does not change the results and arguments in the rest of the
+paper.
+Similarly, onecan replace h′
+kby someabstractsmooth baseofthe samedimension, span(Λ i)⊂
+h′
+kby one-dimensional subvarieties etc. But we use h′
+kto keep notation simpler and because in
+many examples the base is naturally isomorphic to h′
+k.
+On this data we impose the following conditions.
+(i) Each Hom space between two objects in D(Y(λ)) is finite dimensional. In particular,
+this means that End( OY(λ)) =k·I.
+(ii) AllE(r)
+is andF(r)
+is are sheaves(i.e. complexes supported in cohomologicaldegree ze ro).
+(iii)E(r)
+i(λ) andF(r)
+i(λ) are left and right adjoints of each other up to shift. More precise lyBRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 5
+(a)E(r)
+i(λ)R=F(r)
+i(λ)[r(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+r)]
+(b)E(r)
+i(λ)L=F(r)
+i(λ)[−r(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+r)].
+(iv) For each i∈I,
+H∗(Ei∗E(r)
+i)∼=E(r+1)
+i⊗kH⋆(Pr).
+(v) If/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≤0 then
+Fi(λ)∗Ei(λ)∼=Ei(λ−αi)∗Fi(λ−αi)⊕P
+whereH∗(P)∼=O∆⊗kH⋆(P−/angbracketleftλ,αi/angbracketright−1).
+Similarly, if /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≥0 then
+Ei(λ−αi)∗Fi(λ−αi)∼=Fi(λ)∗Ei(λ)⊕P′
+whereH∗(P′)∼=O∆⊗kH⋆(P/angbracketleftλ,αi/angbracketright−1).
+(vi) We have
+(1) H∗(i23∗Ei∗i12∗Ei)∼=E(2)
+i[−1]⊕E(2)
+i[2]
+wherei12andi23are the closed immersions
+i12:Y(λ)×Y(λ+αi)→Y(λ)טYi(λ+αi)
+i23:Y(λ+αi)×Y(λ+2αi)→˜Yi(λ+αi)×Y(λ+2αi).
+(vii) If/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≤0 andk≥1 then the image of supp( E(r)
+i(λ−rαi)) under the projection
+toY(λ) is not contained in the image of supp( E(r+k)
+i(λ−(r+k)αi)) also under the
+projection to Y(λ). Similarly, if /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≥0 andk≥1 then the image of supp( E(r)
+i(λ))
+inY(λ) is not contained in the image of supp( E(r+k)
+i(λ)).
+(viii) Ifi/ne}a⊔ionslash=j∈Iare joined by an edge in Γ then
+Ei∗Ej∗Ei∼=E(2)
+i∗Ej⊕Ej∗E(2)
+i
+while if they are not joined then Ei∗Ej∼=Ej∗Ei.
+(ix) Ifi/ne}a⊔ionslash=j∈IthenFj∗Ei∼=Ei∗Fj.
+(x) Fori∈Ithe sheaf Eideforms over α⊥
+ito some
+˜Ei∈D(˜Y(λ)|α⊥
+i×α⊥
+i˜Y(λ+αi)|α⊥
+i).
+(xi) Ifi/ne}a⊔ionslash=j∈Iare joined by an edge, by Lemma 4.5, there exists a unique non-zero map
+(up to multiple) Tij:Ei∗Ej[−1]→ Ej∗Eiwhose cone we denote
+Eij:= Cone/parenleftbigg
+Ei∗Ej[−1]Tij− − → Ej∗Ei/parenrightbigg
+∈D(Y(λ)×Y(λ+αi+αj)).
+ThenEijdeforms over B:= (αi+αj)⊥⊂h′
+kto some
+˜Eij∈D(˜Y(λ)|B×B˜Y(λ+αi+αj)|B).
+Remark 2.2. The conditions (i), (ii), (iii), (vii) are technical conditions. The cond itions (iv),
+(v), (viii), (ix)arecategoricalversionsoftherelationsintheusua lpresentationoftheKac-Moody
+Lie algebra g(except as in [CKL2], we only impose parts of (iv), (v) at the level of c ohomology
+which is much easier to check). The conditions (vi), (x) and (xi) relat e to the deformation.
+Notice that conditions (i) - (vii) are precisely equivalent to saying tha t{Y(λ+nαi)}n∈Z,
+together with EiandFiand deformations ˜Yi(λ+nαi) generate a geometric categorical sl2
+action. Relations (viii) - (xi) then describe how these various sl2actions are related.6 SABIN CAUTIS AND JOEL KAMNITZER
+One can compare the geometric definition above to the notion of a 2- representationof gin the
+sense of Rouquier [Ro2], which in turn is very similar to the notion of an action of Khovanov-
+Lauda’s2-category[KL1,KL2,KL3]. Inthesedefinitions, therear efunctors Ei,Fiaswellassome
+natural transformations X,Tbetween these functors. The additional data of our deformation s is
+perhaps equivalent to the additional deformation of these natura l transformations. In the case of
+g=sl2, we were able to make this connection precise (see [CKL2]). For gen eralg, it remains an
+open problem to show that a geometric categorical gaction gives an action of Khovanov-Lauda
+or Rouquier’s 2-category. In any case, we work here with the abov e definition since these axioms
+can be checked in examples (as in section 3).
+Remark 2.3. SinceEi,Fiare biadjoint (up to shift), the conditions (iv), (vi) and (viii) imme-
+diately imply the same conditions where all Eiare replaced by Fi.
+Theorem 2.4. A geometric categorical gaction gives a naive categorical gaction. By this we
+mean that the functors E(r)
+i:= ΦE(r)
+iandF(r)
+i:= ΦF(r)
+isatisfy the defining relations of g, up to
+isomorphism:
+Ei◦E(r)
+i∼=E(r)
+i◦E(r+1)
+i⊗CH⋆(Pr)(and similarly with Ereplaced by F),
+Fi◦Ei∼=Ei◦Fi⊕idY(λ)⊗CH⋆(P−/angbracketleftλ,αi/angbracketright−1)if/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≤0and
+Ei◦Fi∼=Fi◦Ei⊕idY(λ)⊗CH⋆(P/angbracketleftλ,αi/angbracketright+1)if/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≥0,
+Ei◦Ej◦Ei∼=E(2)
+i◦Ej⊕Ej◦E(2)
+iifi,jare joined by an edge and
+Ei◦Ej∼=Ej◦Eiif they are not (and similarly with Ereplaced by F),
+Fj◦Ei∼=Ei◦Fjifi/ne}a⊔ionslash=j.
+HenceEi,Fiacting on the Grothendieck groups {K(Y(λ))}gives a representation of U(g).
+Proof.Note that the first three statements differ from the conditions (iv ), (v) given in the
+definition, in that statements on the level of homology are turned in to direct sums. These facts
+are proven (with the help of the deformations) in [CKL2]. /square
+Finally, we define two more maps we will use repeatedly. The first map is
+E(r+1)
+iι− → Ei∗E(r)
+i[−r]∼=E(r)
+i∗Ei[−r] andF(r+1)
+iι− → Fi∗F(r)
+i[−r]∼=F(r)
+i∗Fi[−r]
+which includes into the lowest degree summand of the right hand side ( the isomorphisms above
+follow from Proposition 4.2). Notice that there is a unique such map (u p to multiple) because
+by Lemma 4.5 we have that Endk(E(r+1)
+i) is zero if k <0 and one-dimensional if k= 0 (similarly
+with Endk(F(r+1)
+i)).
+The second map is
+F(r)
+i∗E(r)
+i(λ)ε− → O∆[r(/an}bracke⊔le{⊔λ,αi+r/an}bracke⊔ri}h⊔)] andE(r)
+i∗F(r)
+i(λ)ε− → O∆[−r(/an}bracke⊔le{⊔λ,αi+r/an}bracke⊔ri}h⊔)]
+given by adjunction (by definition E(r)
+iandF(r)
+iare adjoint to each other up to shifts). This
+map is also uniquely defined (up to multiple).
+We say that a geometric categorical g-action is integrable if for every weight λandi∈Iwe
+haveY(λ+nαi) =∅forn≫0 orn≪0. From hereon we assume all actions are integrable.
+2.3.Role of the deformation.BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 7
+2.3.1.Generalities on deformations and obstructions. LetYbe a variety with a 1-parameter
+deformation ˜Y→A1. The obstruction to deforming A ∈D(Y) is given by a map c(A) :
+A[−1]→ A[1]. By definition, c(A)[1] is the connecting map coming from the exact triangle
+A[1]→t∗t∗A → A, wheretis the inclusion t:Y→˜Y(see [HT, appendix]). More generally,
+if˜Y→Vis a deformation over a k-vector space V, then for each v∈V, we get a 1-parameter
+deformation over span( v) and we write cv(A) :A[−1]→ A[1] for the obstruction map. Let us
+summarize the properties of these maps.
+Proposition 2.5. Let˜Y→Vbe a deformation over a k-vector space V.
+(i)There is a functorial distinguished triangle
+A[−1]cv(A)−−−→ A[1]→tv∗tv∗A → A,
+wheretvdenotes the inclusion of Yinto the fibre of ˜Yoverspan(v).
+(ii)cis linear in V, in the sense that cv+w(A) =cv(A)+cw(A).
+(iii)IfAdeforms over span(v)thencv(A) = 0.
+Proof.The functoriality follows from the definition of cvas a morphism of FM kernels. The
+linearity follows from the fact that cv(A) is the product of the Atiyah class of Awith the
+Kodaira-Spencer map V→H1(˜Y,T˜Y) which is linear (see page 3 of [HT]). /square
+We will also need some special properties of these obstruction maps related to products and
+FM kernels. Let Y1,Y2,Y3be three varieties, all with deformations ˜Y1,˜Y2,˜Y3over the same base
+V. Then the pairwise products Y1×Y2andY2×Y3admit deformations ˜Y1טY2,˜Y2טY3over
+V⊕V. Given v,w∈V, we will consider the 1-parameter deformation ˜Y1|span(w)×k˜Y2|span(v)
+and write tv,wfor the corresponding inclusion and cv,wfor the obstruction map (the reason for
+“switching” the order of v,wwill become clear in a moment).
+LetA12∈D(Y1×Y2) andA23∈D(Y2×Y3). Letv,w∈V. SinceA23∗(·) is a functor, we
+get a map A23∗A12[−1]Icv,w(A12)− −−−−−− → A 23∗A12[1].
+Proposition 2.6. With the above notation, the following holds for all v∈V.
+(i)We have an equality Ic0,v(A12) =c0,v(A12∗A23)
+(ii)There is distinguished triangle
+A23∗A12[−1]Icv,0(A12)− −−−−−− → A 23∗A12[1]→t0,v∗(A23)∗tv,0∗(A12)
+Proof.(i) follows from functoriality. (ii) follows from [CKL2, Lemma 4.1]. /square
+2.3.2.Deformations in geometric categorical gactions. In a geometric categorical gaction, we
+have a deformation ˜Y(λ)→h′
+k. In light of Proposition 2.5.(iii), condition (x) implies that
+cv,v(Ei) = 0 for v∈α⊥
+i. Similarly, condition (xi) implies that cv,v(Eij) = 0 for v∈(αi+αj)⊥.
+Let us now study cΛi,Λi(Ei) and in particular the map
+IcΛi,Λi(Ei) :Ei∗Ei[−1]→ Ei∗Ei[1]
+By Proposition 2.5.(ii), cΛi,Λi(Ei) =cΛi,0(Ei)+c0,Λi(Ei).
+First consider IcΛi,0(Ei) :Ei∗Ei[−1]→ Ei∗Ei[1]. We will examine this map on the level of
+cohomology. Proposition 2.6.(ii) and Condition (vi) imply that
+H∗(Cone(IcΛi,0(Ei)))∼=E(2)
+i[−1]⊕E(2)
+i[2].
+This means that on cohomology, IcΛi,0(Ei) induces an isomorphism
+E(2)
+i∼=H0(Ei∗Ei[−1])→ H0(Ei∗Ei[1])∼=E(2)
+i.8 SABIN CAUTIS AND JOEL KAMNITZER
+On the other hand, by Proposition 2.6.(i), Ic0,Λi(Ei) =c0,Λi(Ei∗Ei). Hence Ic0,Λi(Ei) is given
+by a diagonal matrix
+/parenleftbigg∗0
+0∗/parenrightbigg
+:/parenleftbigg
+E(2)
+i[−2]
+Ei/parenrightbigg
+→/parenleftBigg
+E(2)
+i
+E(2)
+i[2]/parenrightBigg
+and thus acts by 0 at the level of cohomology.
+Combining together these observations, we deduce that IcΛi,Λi(Ei) gives an isomorphism
+H0(Ei∗Ei[−1])→ H0(Ei∗Ei[1]).
+We can summarize the results of this section, with the following state ment.
+Proposition 2.7. Letv∈h′
+k.
+(i)If/an}bracke⊔le{⊔v,αi/an}bracke⊔ri}h⊔= 0, thencv,v(Ei) = 0.
+(ii)If/an}bracke⊔le{⊔v,αi/an}bracke⊔ri}h⊔ /ne}a⊔ionslash= 0, thencv,v(Ei)is non-degenerate in the sense that the resulting map
+H0(Icv,v(Ei)) :E(2)
+i∼=H0(Ei∗Ei[−1])→ H0(Ei∗Ei[1])∼=E(2)
+i
+is an isomorphism.
+Remark 2.8. For all practical purposes we do not actually need that Eideforms over all of α⊥
+i
+andEijdeforms over all of ( αi+αj)⊥. We just need them to deform over some one-dimensional
+subspaces. We use the definition above since it is satisfied in all examp les we know and, in our
+opinion, is more aesthetically pleasing.
+2.4.k×-equivariance. There is a slightly more general k×-equivariant version of a geometric
+categorical gaction. In this setting every variety Y(λ) and deformation ˜Y(λ) is equipped with
+ak×-action. The action on ˜Y(λ) is equivariant with respect to a weight 2 action on the base h′
+k.
+Subsequently, each D(Y(λ)) becomes the derived category of k×-equivariant coherent sheaves.
+This extra k×-structure givesus anothergradingon ourcategorieswhich we de note by{·}. More
+precisely, {k}is tensoring with the line bundle O{k}where if f∈ O(U) is a local function and
+t∈k×then viewed as a section f′∈ O{k}(U) we have t·f′=t−k(t·f).
+The conditions on the data are essentially the same. The adjoint con ditions become
+(i)E(r)
+i(λ)R=F(r)
+i(λ)[r(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+r)]{−r(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+r)}
+(ii)E(r)
+i(λ)L=F(r)
+i(λ)[−r(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+r)]{r(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+r)}.
+Condition (vi) on composition of deformed Es becomes
+H∗(i23∗Ei∗i12∗Ei)∼=E(2)
+i[−1]{1}⊕E(2)
+i[2]{−3}
+whileEijis defined as the cone of Ei∗Ej[−1]{1}→ Ej∗Ei. All the other conditions are the same
+once we replace H⋆(Pn) by the doubly graded version
+H⋆(Pn) :=k[−n]{n}⊕k[−n+2]{n−2}⊕···⊕ k[n]{−n}.
+In this setup, Theorem 2.4 shows that the functors EiandFiacting on the Grothendieck
+groups{Kk×(Y(λ))}gives us a representation of the quantum enveloping algebra Uq(g).
+All the results in this paper have natural k×-equivariant analogues. However, we will not
+workk×-equivariantly because keeping track of the extra {·}shifts would make the notation
+hard to read. One of the reasons to even consider this k×-equivariant setup is that it shows
+up naturally in various examples. The cotangent bundles of partial fl ag varieties considered in
+section 3 is one such example.BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 9
+2.5.Example from resolutions of Kleinian singularities. An instructive example of geo-
+metric categorical gaction comes from the minimal resolution of a Kleinian singularity. Let H
+denote a finite subgroup of SL2(C) and let π:Y→Cn/Hbe a minimal resolution. Recall that
+Hdetermines a finite type simply-laced Dynkin diagram Γ whose vertex s etIis in bijection
+with the components of the exceptional fibre π−1(0).
+From the work of Seidel-Thomas [ST], we know that each component Eiofπ−1(0) determines
+a spherical object Si=OEi(−1). These Siform a type Γ arrangement of spherical object and
+thus by the work of Seidel-Thomas give an action of the braid group BgonD(Y) (as usual, here
+gis the Lie algebra associated to Γ).
+Let us use the same data to construct a C×-equivariant geometric categorical gaction. Let
+Y(λ) be defined as follows. Let Y(0) =Y,Y(λ) = pt for λa root of g, andY(λ) =∅
+for all other λ. The action of C×onYis comes from the scaling action on Cn. We define
+Ei(0) :D(Y)→D(pt) using the kernel Si∈D(Y×pt) and similarly with Ei(−αi),Fi(0) and
+Fi(−αi) (all other Ei,Fiwe need to define are functors D(pt)→D(pt) which we take to be the
+identity). The deformation ˜YofYis the standard deformation (which may be constructed by
+thinking of Cn/Has a Slodowy slice or by deforming the polynomial defining the singularit y
+Cn/H).
+Let us check condition (viii) of the geometric categorical slnaction (all other conditions are
+immediate or follow along the same lines). Let i,jbe connected by an edge in Γ (so Ei,Ej
+intersect in a point). Then condition (viii) states that
+Ei(αj)∗Ej(0)∗Ei(−αi)∼=Ej(αi)∗E(2)
+i(−αi)⊕E(2)
+i(−αi+αj)∗Ej(−αi).
+NowY(−αi+αj) =∅whileE(2)
+i(−αi) =O∆pt=Ei(αj). So we see that this is equivalent to the
+fact that the composition
+D(pt)Ei(−αi)− −−−− → D(Y)Ej(0)− −− →D(pt)
+is the identity. Since the first functor is given by tensoring with the o bjectOEi(−1) and the
+second functor is Ext∗(OEj(−1)[−1],·), this condition corresponds to the fact that
+Extm(OEj(−1),OEi(−1)) = 0,
+unlessm= 1 in which case it is C.
+2.6.Equivalences via geometric categorical sl2actions. In [CKL2] we proved that a geo-
+metric categorical sl2action induces a strong categorical sl2action. In [CKL3] we showed that
+a strong categorical sl2action can be used to construct equivalences (using ideas of Chuan g-
+Rouquier [CR]). We briefly review this construction starting from a ca tegorical gaction.
+Given a geometric categorical gaction one can construct for each vertex i∈Ia geometric
+categorical sl2action. More precisely, we use as kernels E(r)
+iandF(r)
+iand use the one parameter
+deformation ˜Yi(λ). Consequently by the main result of [CKL2], we obtain a strong sl2action
+generated by the functors induced by the kernels EiandFi.
+Consider for each s≥0 and/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≥0 the kernel
+Ts
+i(λ) :=F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s)
+i(λ)[−s]∈D(Y(λ)×Y(λ−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔αi))
+(if/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≤0 then we consider instead the kernel E(−/angbracketleftλ,αi/angbracketright+s)
+i ∗ F(s)
+i). There exists a natural
+mapds
+i:Ts
+i(λ)→ Ts−1
+i(λ) given as the composition
+Ts
+i(λ)∼=F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s)
+i[−s]
+ιι− → F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗Fi[−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔−s+1]∗Ei∗E(s−1)
+i[−(s−1)][−s]
+IεI− − → F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗E(s−1)
+i[−s+1]∼=Ts−1
+i(λ).10 SABIN CAUTIS AND JOEL KAMNITZER
+Note that Ts
+i(λ) = 0 for s≫0 since we only deal with integrable representations. The main
+result of [CKL3] is the following.
+Theorem 2.9.
+··· → Ts
+i(λ)ds
+i− → Ts−1
+i(λ)ds−1
+i− −− →...d1
+i− → T0
+i(λ)
+is a complex of kernels which has a unique right convolution d enotedTi(λ). Moreover, the kernel
+Ti(λ)induces an equivalence of triangulated categories D(Y(λ))∼− →D(Y(λ−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔αi)).
+In the theorem above, by right convolution we mean an iterated con e starting from the right
+(see section 5.3).
+2.7.Braid group action via geometric categorical gactions. As noted in the previous
+section, each vertex i∈Iinduces a geometric categorical sl2action and subsequently an equiv-
+alenceTi(or, more precisely, a series of equivalences Ti(λ), one for each weight λ). The main
+result of this paper is to prove that these equivalences braid.
+Theorem 2.10. LetY(λ),...be a geometric categorical sl2action.
+Ifi,j∈Iare joined by an edge then the corresponding equivalences TiandTjsatisfy the braid
+relationTi∗ Tj∗ Ti∼=Tj∗Ti∗ Tj. Ifi,j∈Iare not joined by an edge then Ti∗ Tj∼=Tj∗Ti.
+Hence there is an action of the braid group of type gonD(⊔Y(λ))compatible with the action of
+the Weyl group on the weight lattice.
+Let us examine this action on the level of the Grothendieck groups ⊕K(Y(λ)). The above
+theorem provides us with an action of the braid group Bgon⊕K(Y(λ)). On the other hand,
+Theorem 2.4 provides us with an action of Uq(g) on⊕K(Y(λ)). These two structures are
+compatible via Lusztig’s quantum Weyl group map Bg→/hatwideUq(g). This follows from [CKL3]. (In
+the non-equivariant case, i.e. q= 1, then this is the same as the usual map Bg→/hatwideU(g)).
+Example 2.11. As a simple application of this theorem, we can consider the minimal res olution
+Yof the Kleinian singularity C2/H. In section 2.5, we explained that Y=Y(0) was the 0 weight
+space of a geometric categorical gaction. Hence by Theorem 2.10, we obtain an action of the
+braid group BgonD(Y). As mentioned earlier, such an action was previously studied by Seid el-
+Thomas [ST]. A more substantial application will be given in the next sec tion.
+3.Example: cotangent bundles to flag varieties
+Before we prove Theorem 2.10 we would like to illustrate a geometric ca tegorical slnaction
+on theC×-equivariant derived category of coherent sheaves on the cotan gent bundle to partial
+flag varieties (Theorem 3.1). We work C×-equivariantly in order to have condition (i) hold (if
+not, End( OY(λ))∼=H0(OY(λ)) will be infinite dimensional). Functors will always be considered
+in the derived sense (i.e. as functors between derived categories) .
+3.1.The categorical gaction. Fix integers n≤N. We consider the variety Fln(CN) of
+n-step flags in CN. This variety has many connected components, which are indexed b y the
+possible dimensions of the spaces in the flags. In particular, let
+C(n,N) :={λ= (λ1,...,λ n)∈NN:λ1+···+λn=N}.
+Forλ∈C(n,N), we can consider the variety of n-steps flags where the jumps are given by λ:
+Flλ(CN) :={0 =V0⊂V1⊂ ··· ⊂Vn=CN: dimVi/Vi−1=λi}.
+LetY(λ) =T⋆Flλ(CN) (ifλ/ne}a⊔ionslash∈C(n,N) we take Y(λ) =∅). These will be our varieties for
+the geometric categorical slnaction. We regard each λas a weight for slnvia the identification
+of the weight lattice of slnwith the quotient Zn/(1,···,1). For compatibility with [CKL4], weBRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 11
+choose the convention that the simple roots αiare equal to (0 ,...,0,−1,1,0,...,0) where the
+−1 is in position i.
+We will make use of the following description of the cotangent bundle t o the partial flag
+varieties.
+Y(λ) ={(X,V) :X∈End(CN),V∈Flλ(CN),XVi⊂Vi−1}
+This description immediately leads to the following deformations of Y(λ) overCn
+˜Y(λ) :={(X,V,x) :X∈End(CN),V∈Flλ(CN),x∈Cn,XVi⊂Vi,X|Vi/Vi−1=xi·id}.
+In more Lie-theoretic terms, Flλ(CN) is the variety of parabolic subalgebras pofglNof typeλ.
+Y(λ) is the variety of pairs ( X,p) whereX∈glN,pis a parabolic subalgebra of type λandX
+is in the nilradical of p. Finally ˜Y(λ) is the variety of triples ( X,p,x) whereXis inpand its
+image in the Levi of pis the central element x.
+We will restrict our deformation over the locus {(x1,...,x n)∈Cn:xn= 0}which we identify
+withh′, the Cartan for sln.
+We define an action of C×on˜Y(λ) byt·(X,V,x) = (t2X,V,t2x). Restricting to Y(λ) =
+T⋆Flλ(CN) this corresponds to a trivial action on the base and a scaling of the fibres.
+To construct the kernels E(r)
+i, we consider correspondences Wr
+i. More specifically, let λ,i,r
+be such that λ∈C(n,N) andλ+rαi∈C(n,N) (ieλi≥r). Then we define
+Wr
+i(λ) :={(X,V,V′) : (X,V)∈Y(λ),(X,V′)∈Y(λ+rαi),Vj=V′
+jforj/ne}a⊔ionslash=i,andV′
+i⊂Vi}
+From this correspondence we define the kernel
+E(r)
+i(λ) =OWr
+i(λ)⊗det(Vi+1/Vi)−r⊗det(V′
+i/Vi−1)r{r(λi−r)} ∈D(Y(λ)×Y(λ+rαi))
+where{·}denotes an equivariant shift and, abusing notation, Videnotes the vector bundle on
+Y(λ) whose fibre over ( X,V)∈Y(λ) is naturally identified with Vi. Similarly, we define the
+kernel
+F(r)
+i(λ) =OWr
+i(λ)⊗det(V′
+i/Vi)λi+1−λi+r{rλi+1} ∈D(Y(λ+rαi)×Y(λ)).
+Note that now we regard Wr
+i(λ) as a subvariety of Y(λ+rαi)×Y(λ) which means that Vi⊂V′
+i
+(we will continue to use this convention).
+Theorem 3.1. This datum defines a geometric categorical slnaction on D(T⋆Fln(CN)).
+By Theorem 2.4, this gives us a representation of Uq(sln) on
+K(T⋆Fln(CN)) =⊕λK(T⋆Flλ(CN)).
+Since
+dimK(T⋆Flλ(CN)) =/parenleftbigg
+N
+λ1···λn/parenrightbigg
+and since representations of Uq(sln) are determined by the dimensions of their weight spaces, we
+can identify this representation with V⊗N
+Λ1.
+3.2.The braid group action. As a corollaryof this theorem and the main result of this paper
+(Theorem 2.10), we obtain the following.
+Theorem 3.2. There is an action of the braid group Bnon the derived category of coherent
+sheaves on T⋆Fln(CN). This action is compatible with the action of Snon the set of connected
+components C(n,N).
+In particular, if N=dnfor some integer dand we choose λ= (d,...,d), then we obtain an
+action of the braid group of the derived category of coherent she aves on the connected variety
+T⋆Flλ(CN).12 SABIN CAUTIS AND JOEL KAMNITZER
+Example 3.3. Considerthecase n=N. LetT⋆(Fl(Cn))denotethecotangentbundletothefull
+flag variety. We have constructed an action of the braid group BnonD(T⋆(Fl(Cn))). Such an
+actionwaspreviouslyconstructedbyKhovanov-Thomas[KT]and byRiche[Ric], Bezrukavnikov-
+Mirkovic-Rumynin [BMR]. Their work served as motivation for this pape r. In this case, the
+generators of the braid group act by spherical twists (see [CKL3 , section 2.5]). This is the
+simplest case of our result, since in general the equivalences which g enerate the braid group
+action are given by more complicated complexes than spherical twist s.
+In [KT, Ric, BMR], the braid group action is extended to an affine braid g roup action. The
+extra generators for this extended action are given by tensoring with certain line bundles. One
+can similarly construct such an affine braid group action on D(T⋆(Fln(CN))) using line bundles.
+We discuss this in greater detail in [CKL4] where we build on the results from this paper to
+construct affine braid group actions on Nakajima quiver varieties.
+Even though the construction of each equivalence
+Ti:D(T⋆Flλ(CN))→D(T⋆Flλ−/angbracketleftλ,αi/angbracketrightαi(CN))
+via a categorical sl2action is indirect, the kernels one obtains are fairly concrete. More precisely,
+the kernel Tiwhich induces the functor Tiis always a sheaf supported on the variety
+Zi(λ) :={(X,V,V′) :X∈End(CN),V∈Flλ(CN),V′∈Flλ−/angbracketleftλ,αi/angbracketrightαi(CN)
+XVj⊂Vj−1,XV′
+j⊂V′
+j−1andVj=V′
+jifj/ne}a⊔ionslash=i}.
+In general Tiis not the structure sheaf of Zi(λ) but rather some rank one Cohen-Macaulay sheaf
+onZi(λ). In [C] we give a concrete description of this sheaf in the Grassman nian case ( n= 2
+case). A similar description of Tiis possible in general.
+In the rest of this section we prove Theorem 3.1.
+3.3.Proof of sl2conditions (i) - (vii). In [CKL2] (based on the computations in [CKL1])
+we proved that the sl2relations (i) - (vii) hold for cotangent bundles to Grassmannians (i.e . the
+casen= 2). The same proof with virtually no changes necessary applies to p rove these relations
+for anyn.
+As an example, we will check the adjunction relations (iii). We begin by computing some
+canonical bundles.
+Lemma 3.4. ωT⋆Flλ(CN)∼=OT⋆Flλ(CN){−2/summationtext
+i<jλiλj}
+Proof.The variety T⋆Flλ(CN) is symplectic (since it is a cotangent bundle) and the symplectic
+form has weight 2 for the C×action. Hence the dth wedge power of the symplectic form gives
+a non-vanishing section of the canonical bundle, where dis the dimension of Flλ(CN). Since
+d=/summationtext
+i<jλiλj, the result follows. /square
+Lemma 3.5. We have
+ωWr
+i(λ)∼=det(Vi+1/Vi)−rdet(Vi/V′
+i)−λi+λi+1+rdet(V′
+i/Vi−1)r{−2/summationdisplay
+i<jλiλj+2λi+1r}.
+Proof.Letµ:= (λ1,...,λ i−1,λi−r,r,λi+1,...,λ n) and let Tdenote the variety
+T:={(X,U) : (X,U)∈Y(µ),XUi+1⊂Ui−1}
+We can view Wr
+ias a subvariety of T, by setting Uj=Vjforj < i,Ui=V′
+i,Ui+1=Viand
+Uj=Vi−1forj > i+1. Moreover we can see that Wr
+iis carved out of Tas the vanishing locus
+ofX:Ui+2/Ui+1→Ui+1/Ui{2}. ThusWr
+iis cut out of Tby a section of the vector bundle
+(Ui+2/Ui+1)∨⊗Ui+1/Ui{2}.BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 13
+Similarly, Tis cut out of T⋆Flµ(CN) by a section of ( Ui+1/Ui)∨⊗Ui/Ui−1{2}. Thus
+ωWr
+i(λ)∼=det((Ui+2/Ui+1)∨⊗Ui+1/Ui{2})⊗det((Ui+1/Ui)∨⊗Ui/Ui−1{2})⊗ωT⋆Flµ(CN)|Wr
+i(λ).
+Here we use that if A⊂Bis cut out by a section of a vector bundle WthenωA=ωB⊗det(W).
+Combining all this with our previous calculation of ωT⋆Flµ(CN)we obtain the desired result.
+/square
+We will now give the proof of the first adjunction statement. The ot her proofs are similar.
+Corollary 3.6. E(r)
+i(λ)R∼=F(r)
+i(λ)[r(λi+1−λi)+r2]{−r(λi+1−λi)−r2}
+Proof.We have
+E(r)
+i(λ)R∼=E(r)
+i∨⊗π∗
+2ωY(λ)[dimY(λ)]
+∼=O∨
+Wr
+i⊗det(Vi+1/V′
+i)rdet(Vi/Vi−1)−r{−r(λi−r)}⊗π∗
+2ωY(λ)[dimY(λ)]
+∼=ωWr
+i(λ)ω∨
+Y(λ+rαi)×Y(λ)[−codimWr
+i(λ)]⊗det(Vi+1/V′
+i)r⊗det(Vi/Vi−1)−r
+⊗π∗
+2ωY(λ){−r(λi−r))}[dimY(λ)]
+∼=OWr
+i(λ)⊗det(Vi+1/V′
+i)−rdet(V′
+i/Vi)−λi+λi+1+rdet(Vi/Vi−1)r{−2/summationdisplay
+i<jλiλj+2λi+1r}
+⊗π∗
+1ω∨
+Y(λ+rαi)det(Vi+1/V′
+i)rdet(Vi/Vi−1)−r{−r(λi−r)}[dimY(λ)−codimWr
+i(λ)]
+∼=OWr
+i(λ)⊗det(V′
+i/Vi)−λi+λi+1+r[r(λi+1−λi)+r2]
+{2(/summationdisplay
+i<jλiλj+rλi−rλi+1−r2)+(−2/summationdisplay
+i<jλiλj+2λi+1r)−r(λi−r)}
+∼=F(r)
+i(λ)[r(λi+1−λi)+r2]{−r(λi+1−λi)−r2}
+where for the last isomorphism we use that F(r)
+i(λ) =OWr
+i(λ)⊗det(V′
+i/Vi)λi+1−λi+r{rλi+1}./square
+3.4.Proof of Serre relation (viii). Since we are in the Lie algebra sln, having i,j∈Ijoined
+by an edge is equivalent to j=i±1. So let us consider j=i+1 (the case j=i−1 is the same).
+We will show that
+Ei∗Ei+1∗Ei=E(2)
+i∗Ei+1⊕Ei+1∗E(2)
+i.
+Here is the outline of the proof. On the left hand side computing Ei∗Ei+1is straight-forward,
+meaningthat intersectionsareofthe expecteddimension andthe p ushforwardisone-to-one. The
+intersection when computing Ei∗ Ei+1∗ Eiis also of the expected dimension but contains two
+components AandB. Pushing forward by π13then gives us two terms (one for each component)
+which are equal to E(2)
+i∗Ei+1andEi+1∗E(2)
+i(these are also easy to compute).
+Lemma 3.7. E(2)
+i∗Ei+1andEi+1∗E(2)
+iare isomorphic to
+OWi(2)i+1⊗det(Vi+2/Vi+1)−1det(V′
+i+1/Vi)−1det(V′
+i/Vi−1)2{2λi+λi+1−5}
+OWi+1i(2)⊗det(Vi+2/Vi+1)−1det(Vi+1/Vi)−2det(V′
+i+1/V′
+i)det(V′
+i/Vi−1)2{2λi+λi+1−3}
+respectively where
+Wi(2)i+1:=/braceleftbig
+(X,V,V′) :V′
+i⊂Vi⊂V′
+i+1⊂Vi+1,andVj=V′
+jforj/ne}a⊔ionslash=i,i+1/bracerightbig
+Wi+1i(2):=/braceleftbig
+(X,V,V′) :V′
+i⊂Vi,V′
+i+1⊂Vi+1,XVi+1⊂V′
+i,andVj=V′
+jforj/ne}a⊔ionslash=i,i+1/bracerightbig
+insideY(λ)×Y(λ+2αi+αi+1).14 SABIN CAUTIS AND JOEL KAMNITZER
+Proof.Computing E(2)
+i∗ Ei+1is easy since the intersection π−1
+12(Wi+1)∩π−1
+23(W2
+i) is of the
+expected dimension and the map π13maps this intersection one-to-one onto its image
+π13(π−1
+12(Wi+1)∩π−1
+23(W2
+i))∼=Wi(2)i+1.
+So neither the tensor product nor the pushforward π13∗have lower or higher terms. Keeping
+track of the line bundles gives the result.
+Computing Ei+1∗E(2)
+iis very similar. /square
+Lemma 3.8. We have
+Ei∗Ei+1∼=OWii+1⊗det(Vi+2/Vi+1)∨det(V′
+i/Vi−1){λi+λi+1−2}
+Ei+1∗Ei∼=OWi+1i⊗det(Vi+2/Vi)∨det(V′
+i+1/Vi−1){λi+λi+1−1}
+where
+Wii+1:=/braceleftbig
+(X,V,V′) :V′
+i⊂Vi⊂V′
+i+1⊂Vi+1,andVj=V′
+jforj/ne}a⊔ionslash=i,i+1}
+Wi+1i:=/braceleftbig
+(X,V,V′) :V′
+i⊂Vi,V′
+i+1⊂Vi+1,XVi+1⊂V′
+i,andVj=V′
+jforj/ne}a⊔ionslash=i,i+1/bracerightbig
+insideY(λ+αi)×Y(λ+2αi+αi+1)andY(λ)×Y(λ+αi+αi+1)respectively.
+Proof.At the level of sets
+Wii+1∼=π13(π−1
+12(Wi+1)∩π−1
+23(Wi))
+where the intersection is transverse and the push forward is one- to-one. Hence in computing
+Ei∗Ei+1neitherthe tensorproduct northe pushforward π13∗havelowerorhigherterms. Keeping
+track of the line bundles gives the result. Computing Ei+1∗Eiis very similar. /square
+Notice that the varieties Wi(2)i+1,Wi+1i(2),Wii+1are all smooth. This is because each of them
+is a vector bundle over a iterated Grassmannian bundle. The fibre of these vector bundles is
+given by the Xdata and the base is given by the V,V′data.
+Now we can compute ( Ei∗Ei+1)∗Ei. We find that
+π−1
+12(Wi)∩π−1
+23(Wii+1) =/braceleftbig
+(X,V,V′,V′′) :V′′
+i⊂V′
+i⊂Vi,V′
+i⊂V′′
+i+1⊂Vi+1=V′
+i+1,
+andVj=V′
+j=V′′
+jifj/ne}a⊔ionslash=i,i+1/bracerightbig
+.
+This intersection is of the expected dimension but the push-forwar d underπ13is only generically
+one-to-one. This variety has two components which we denote by AandBwhich are defined by
+A:=/braceleftbig
+(X,V,V′,V′′) :V′′
+i⊂V′
+i⊂Vi⊂V′′
+i+1⊂Vi+1=V′
+i+1,andVj=V′
+j=V′′
+jifj/ne}a⊔ionslash=i,i+1/bracerightbig
+B:=/braceleftbig
+(X,V,V′,V′′) :V′′
+i⊂V′
+i⊂Vi,V′
+i⊂V′′
+i+1⊂Vi+1=V′
+i+1,XVi+1⊂V′′
+i,
+andVj=V′
+j=V′′
+jifj/ne}a⊔ionslash=i,i+1/bracerightbig
+.
+Thevarieties A,Baresmoothforthesamereasonsasexplainedabovefor Wi(2)i+1,Wi+1i(2),Wii+1.
+Keeping track of the line bundles shows that ( Ei∗Ei+1)∗Ei∼=π13∗/parenleftbig
+OA∪B⊗L) where
+L:= det(Vi+2/Vi)−1det(V′
+i/V′′
+i)det(V′′
+i/Vi−1)2{2λi+λi+1−3}.
+LetE:=A∩B. It is a divisorinside ofeach of A,B. Consider the standard short exact sequence
+0→ OA(−E)⊕OB(−E)→ OA∪B→ OE→0.
+Now,Eis cut out of Aby a section of Hom( Vi+1/V′′
+i+1,V′
+i/V′′
+i{2}), namely the map X:
+Vi+1/V′′
+i+1→V′
+i/V′′
+i{2}. Hence
+OA(−E) =OA⊗(Vi+1/V′′
+i+1)⊗(V′
+i/V′′
+i)∨{−2}.BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 15
+Similarly, Eis cut out of Bby a section of Hom( Vi/V′
+i,Vi+1/V′′
+i+1), namely the natural map
+Vi/V′
+i→Vi+1/V′′
+i+1induced by the inclusion Vi→Vi+1. Hence we see that
+OB(−E) =OB⊗(Vi/V′
+i)⊗(Vi+1/V′′
+i+1)∨.
+Putting all this together, we obtain a distinguished triangle
+(2) π13∗(OA⊗LA)⊕π13∗(OB⊗LB)→ Ei∗Ei+1∗Ei→π13∗(OE⊗L)
+where
+LA:= det(Vi+2/Vi)−1det(Vi+1/V′′
+i+1)det(V′′
+i/Vi−1)2{2λi+λi+1−5},
+LB:= det(Vi/V′′
+i)det(V′′
+i/Vi−1)2det(Vi+2/Vi)−1det(Vi+1/V′′
+i+1)−1{2λi+λi+1−3}.
+Now, we note that π13(A) =Wi(2)i+1. The map π13|A:A→Wi(2)i+1is generically one-
+to-one and LAis pulled back from Wi(2)i+1. SinceAandWi(2)i+1are both smooth we have
+π13∗(OA)∼=OWi(2)i+1and hence
+π13∗(OA⊗LA)∼=OWi(2)i+1⊗det(Vi+2/Vi+1)−1det(V′
+i+1/Vi)−1det(V′
+i/Vi−1)2{2λi+λi+1−5}
+∼=E(2)
+i∗Ei+1.
+A very similar argument shows that π13∗(OB⊗LB)∼=Ei+1∗E(2)
+i.
+Finally, we see that π13|Eis aP1bundle. Moreover Lrestricts to OP1(−1) on these fibres.
+Hence we conclude that π13∗(OE⊗L) = 0. So distinguished triangle (2) gives us an isomorphism
+E(2)
+i∗Ei+1⊕Ei+1∗E(2)
+i∼=Ei∗Ei+1∗Ei
+as desired.
+Remark 3.9. There is an interesting similarity between the proof of the braid relat ion in [KT]
+and the proof of the Serre relation above. In particular, the proo f of Proposition 4.6 of [KT]
+inspired our proof above. The geometry occuring in that proof is sim ilar to the geometry we
+consider here.
+Finally, the identity Ei∗ Ej∼=Ej∗ Eiwheni,j∈Iare not joined by an edge (i.e. when
+|i−j|>1) follows from a direct calculation of both sides (all intersections ar e of the expected
+dimension and push-forwards are one-to-one so this calculation is s traight-forward). The same
+argument works to show that Fj∗Ei∼=Ei∗Fjfor anyi,j∈I(condition (ix)).
+3.5.Existence of deformations: conditions (x) and (xi). We now explain why condition
+(xi) holds. Since we are in the Lie algebra sln, having i,j∈Ijoined by an edge is equivalent to
+j=i±1. So let us consider j=i+1 (the case j=i−1 is the same). We must show that the
+sheafEii+1= Cone(Tii+1) deforms over the subspace ( αi+αi+1)⊥.
+Here is an outline of the argument. Recall that Eii+1∼=Cone(Ei∗Ei+1[−1]Tii+1− −− → E i+1∗Ei).
+NowEi∗ Ei+1is a line bundle supported on Wii+1andEi+1∗ Eia line bundle supported on
+Wi+1i. We show that the connecting map Tii+1must be (up to tensoring by a line bundle) the
+connecting map in the standard triangle
+OWi+1i(−D)→ OUii+1→ OWii+1
+whereUii+1=Wi+1i∪Wii+1andD:=Wi+1i∩Wii+1. Thus we identify Eii+1with a line bundle
+supported on Uii+1and then write down an explicit deformation of it.16 SABIN CAUTIS AND JOEL KAMNITZER
+Lemma 3.10. We have
+Eii+1∼=OUii+1⊗det(Vi+2/V′
+i)∨det(Vi+1/Vi−1){λi+λi+1−1}
+where
+Uii+1:={(X,V,V′) :V′
+i⊂Vi,V′
+i+1⊂Vi+1,Vj=V′
+jifj/ne}a⊔ionslash=i,i+1} ⊂Y(λ)×Y(λ+αi+αj).
+Proof.Recall that by Lemma 3.8 the objects Ei∗Ei+1andEi+1∗EiinD(Y(λ)×Y(λ+αi+αi+1))
+are line bundles supported on smooth varieties Wii+1andWi+1i. Notice that the difference
+between the varieties Wii+1andWi+1iis that in the former we demand that Vi⊂V′
+i+1, while
+in the latter we demand that XVi+1⊂V′
+i.
+We claim that Uii+1=Wii+1∪Wi+1i. To see this, let ( X,V,V′)∈Uii+1. ThenV′
+i⊂Viand
+V′
+i⊂V′
+i+1. Sincedim Vi/V′
+i= 1,thisimpliesthateither Vi⊂V′
+i+1orV′
+i+1∩Vi=V′
+i. IfVi⊂V′
+i+1,
+then (X,V,V′)∈Wii+1. On the other hand, if V′
+i+1∩Vi=V′
+i, then the conditions XVi+1⊂Vi
+andXVi+2⊂V′
+i+1, forceXVi+1⊂Vi∩V′
+i+1=V′
+iand so we see that ( X,V,V′)∈Wi+1i.
+ThusUii+1=Wii+1∪Wi+1iand these are the two irreducible components of Uii+1. These
+two components intersect in a divisor Dand gives us a short exact sequence of sheaves
+0→ OWi+1i(−D)→ OUii+1→ OWii+1→0.
+Using the fact that Dis cut out of Wi+1iby a section of Hom( Vi/V′
+i,Vi+1/V′
+i+1), we see that
+OWi+1i(−D)∼=OWi+1i⊗(Vi+1/V′
+i+1)⊗(Vi/V′
+i)∨.
+Substitutingthisintothepreviousexactsequenceandrotating,w eobtainadistinguishedtriangle
+OWii+1[−1]→ OWi+1i⊗(Vi+1/V′
+i+1)(Vi/V′
+i)∨→ OUii+1.
+Tensoring with the line bundle det( Vi+2/Vi+1)∨det(V′
+i/Vi−1){λi+λi+1−1}, we obtain the
+distinguished triangle
+Ei∗Ei+1[−1]{1} → Ei+1∗Ei→ OUii+1⊗det(Vi+2/Vi+1)∨det(V′
+i/Vi−1){λi+λi+1−1}
+Moreover the first map in this distinguished triangle is non-zero. Sinc eTii+1is the unique
+such map (up to multiple) the first map must equal Tii+1up to multiple. The result follows. /square
+Now that we have identified Eii+1more explicitly we can write down a deformation ˜Eii+1over
+B:= (αi+αi+1)⊥= (0,...0,−1,0,1,0,...,0)⊥={(x1,...,x n−1) :xi=xi+2}.
+Define the variety
+˜Uii+1:={(X,V,V′,x) : (X,V,x)∈˜Y(λ),(X,V′,x)∈˜Y(λ+αi+αi+1),x∈(αi+αi+1)⊥,
+V′
+i⊂Vi,V′
+i+1⊂Vi+1,Vj=V′
+jforj/ne}a⊔ionslash=i,i+1}
+and consider
+˜Eii+1:=O˜Uii+1⊗det(Vi+2/V′
+i)∨det(Vi+1/Vi−1){λi+λi+1−1} ∈D(˜Y(λ)|B×B˜Y(λ+αi+αi+1)|B).
+Proposition 3.11. We have j∗˜Eii+1=Eii+1wherejis the inclusion of the central fibre Y(λ)×
+Y(λ+αi+αi+1)into˜Y(λ)|B×B˜Y(λ+αi+αi+1)|B.
+Proof.Since the line bundles on both sides agree, it suffices to show that j∗O˜Uii+1=OUii+1. To
+do this it suffices to show that ˜Uii+1is an irreducible variety of dimension dim Uii+1+ dimB
+and that the scheme theoretic central fibre of ˜Uii+1→Bis reduced. To do this, we will pass to
+local coordinates. To simplify our task of finding local coordinates w e will use an idea of Riche
+[Ric] and pass to a subvariety.BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 17
+Note that ˜Uii+1has an action of the group SLN. LetZdenote the subvariety of ˜Uii+1
+consisting of those points ( X,V,V′,x) where
+0⊂V1⊂ ··· ⊂Vi−1⊂V′
+i⊂V′
+i+1⊂Vi+1⊂Vi+2⊂ ··· ⊂Vn−1⊂CN
+is the standard partial flag. This means that V1= span(e1,...,e λ1), etc. The variety Zhas an
+action of the parabolic subgroup P⊂SLNwhich fixes the flag above. Note that as before, we
+have a map Z→B.
+The variety ˜Uii+1is obtained from Zby associated bundle construction, ˜Uii+1∼=Z×PSLN.
+Moreover, this is actually an isomorphism as varieties over B. Hence it suffices to prove that Z
+is irreducible of expected dimension and the central fibre over Bis reduced.
+We will show that Zcan be covered by open affine varieties Asuch that Acan be embedded
+intoCn−2+r(for some r) with the map to Bgiven by projection onto the first n−2 coordinates.
+Let us write the coordinates on Cn−2+rasx1,...,x i+1,xi+3,...,x n−1,z1,...,z r. We will show
+that under this embedding Ais given by the single equation xi−xi+1=z1z2. This proves the
+desired facts concerning the central fibre of Aand hence also for ˜Uii+1.
+To find this open affine variety A, note that Zhas a smooth surjective affine map Z→
+P(Vi+1/V′
+i), taking ( X,V,V′,x) toVi. Pickk,lsuch that
+V′
+i+1=V′
+i⊕span(ek,...,e l−1) andVi+1=V′
+i⊕span(ek,...,e l).
+InP(Vi+1/V′
+i) there is an open affine subspace consisting of those Viwhich are of the form
+(3) Vi=V′
+i⊕/an}bracke⊔le{⊔ek+ck+1ek+1+···+clel/an}bracke⊔ri}h⊔
+We letAdenote the preimage of this affine subspace in Z. So a point in Ais described by
+(ck+1,...,c l) and (x1,...,x n−1) and the matrix X. We will now describe equations for A. To
+do this, let us introduce the variety
+T={(X,Vi,x1,...,x n−1) :Vias above in (3) ,(X−xj)V′
+j⊂V′
+j−1for allj,andxi=xi+2}.
+HenceAis the closed subvariety of Tdefined by the equations ( X−xi)Vi⊂Vi−1and (X−
+xi+1)Vi+1⊂Vi. Since dim( Vi/V′
+i) = 1 and dim( Vi+1/V′
+i+1) = 1, we see that these equations are
+equivalent to
+(X−xi)(ek+ck+1ek+1+···+clel)∈Vi−1(X−xi+1)(el)∈Vi.
+Now, if ( X,Vi,x1,...,x n−1)∈T, thenXis upper triangular where the diagonal is broken up
+into blocks corresponding to the V′
+jwith each block a diagonal matrix with xjon the diagonal.
+HenceTisanaffinespacewithcoordinatesgivenbytheentriesinthematrices intheblocksabove
+the diagonals along with ( ck+1,...,c l) and (x1,...,x n−1). Now let ( X,Vi,x1,...,x n−1)∈T,
+and let us consider the square diagonal submatrix of Xcontaining matrix coefficients for the
+basis elements ek−r,...,e l, wherer=λi−1. This square submatrix has the form
+
+xiIrwkwk+1 ... w l
+xi+1 ak
+... ak+1
+......
+xi+1al−1
+xi
+
+whereIris ther×ridentity matrix. Here we mean that
+Xej=xi+1ej+wj+...fork≤j≤l−1
+Xel=xiel+al−1el−1+···+akek+wl+...18 SABIN CAUTIS AND JOEL KAMNITZER
+wherewj∈span(ek−r,...,e k−1) (sowjis a column matrix of height r) and...denotes terms
+inVi−1. Note also that el∈V′
+i+2so the (l,l) matrix entry is xi=xi+2.
+Now (X,Vi,x1,...,x n−1) lies inAif and only if ( X−xi)(ek+ck+1ek+1+···+clel)∈Vi−1
+and (X−xi+1)(el)∈Vi. These conditions translate into the equations
+ak+1=akck+1
+...
+al−1=akcl−1
+xi−xi+1=akcl
+wk=−(ck+1wk+1+···+clwl)
+Hence we can embed Ainto the subaffine space given by the x1,...,x i+1,xi+3,...,x n−1,ak, the
+cj, the entries in wj(j/ne}a⊔ionslash=k), and all the other free matrix entries. Inside of this affine space, A
+will be defined by the single equation xi−xi+1=akclas desired. /square
+Remark 3.12. It is interesting to notice that neither Wii+1norWi+1ideform over Bbut that
+Uii+1=Wii+1∪Wi+1idoes deform. In fact Wii+1andWi+1ionly deform over α⊥
+i∩α⊥
+i+1.
+The proof that Eideforms over α⊥
+i(condition (x)) is the same but easier since we already
+have an explicit description of Eias a line bundle supported on Wi.
+4.Preliminaries
+In this section we fix some further notation and prove various tech nical results about compo-
+sitions of functors EandFand about spaces of maps (natural transformations) between t hem.
+The reader can choose to skim this section on a first reading, using it as a reference.
+4.1.Some general notions.
+4.1.1.Idempotent completeness. LetCbe a graded additive category over kwhich is idempotent
+complete. Graded means that Chas a shift functor [1] which is an equivalence. Idempotent
+complete means that if e∈End(A) where e2=ethenA∼=A1⊕A2whereeacts by the
+identity on A1and by zero on A2. Notice that the derived category of coherent sheaves on any
+variety is idempotent complete. This is because the derived categor y of any abelian category is
+idempotent complete (see, for instance, Corollary 2.10 of [BS]). So a ll the categories we work
+with are idempotent complete.
+Suppose that (each graded piece of) the space of homs between t wo objects is finite dimen-
+sional (by condition (i) this is true in our setup). Then every object inChas a unique, up to
+isomorphism, direct sum decomposition into indecomposables (see se ction 2.2 of [Rin]). Assume,
+moreover, the following fact:
+for any non-zero object A∈ Cwe have A∼=A[k]⇒k= 0.
+Then ifA,B,C∈ C, we have the following cancellation laws:
+A⊕B∼=A⊕C⇒B∼=C (4)
+A⊗kV∼=B⊗kV⇒A∼=B (5)
+whereVis a graded kvector space. The first law above follows by uniqueness of direct su m
+decomposition. To see the second law, decompose AandBinto indecomposables as
+A∼=/circleplusdisplay
+i,jXaij
+i[j]B∼=/circleplusdisplay
+i,jXbij
+i[j]BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 19
+whereXiare indecomposable, aij,bij∈NandXi∼=Xi′[j] implies that i=i′. We must show
+thataij=bij. Now fix iand consider just the summands Xi[j]. By the uniqueness of the direct
+sum decomposition, we get
+⊕jXaij
+i[j]⊗kV∼=⊕jXbij
+i[j]⊗kV
+Now consider the Poincar´ e polynomials
+A(t) :=/summationdisplay
+jaijtj,B(t) :=/summationdisplay
+jbijtjandV(t) :=/summationdisplay
+jdim(Vj)tj
+whereVjis thejth graded piece of V. Then since Ai[j]/ne}a⊔ionslash∼=Ai[j′] forj/ne}a⊔ionslash=j′it follows that
+A(t)V(t) =B(t)V(t) which implies A(t) =B(t) and we are done.
+4.1.2.Bricks and ranks. A brick is an indecomposable object AinCsuch that End( A) =k·id.
+Suppose that Ais a brick and that X,Yare arbitrary objects of C. Letf:X→Ybe a
+morphism. fgives rise to a bilinear pairing Hom( A,X)×Hom(Y,A)→Hom(A,A) =k. We
+define the A-rank of fto be the rank of this bilinear pairing.
+We may also define A-rank as follows. Choose (non-canonical) direct sum decomposition s
+X=A⊗V⊕BandY=A⊗V′⊕B′whereV,V′arekvector spaces and B,B′do not contain
+Aas a direct summand. Then one of the matrix coefficients of fis a map A⊗V→A⊗V′,
+which (since Ais a brick) is equivalent to a linear map V→V′. TheA-rank of fequals the
+rank of this linear map. If the A-rank of fisk, then we will say that “ fgives an isomorphism
+onksummands isomorphic to A”.
+Note that the notion of brick makes no reference to the shift func tor inC. On the other hand,
+ifAis a brick, then A[n] is a brick for all n. Moreover, if f:X→Yis a morphism, we will say
+that “fgives an isomorphism on ksummands of the form A[·]” if the sum (over all n) of the
+A[n]-ranks of fisk.
+4.1.3.Gaussian elimination. Finally, we will repeatedly use the following cancellation Lemma
+which Bar-Natan [BN] calls “Gaussian elimination”.
+Lemma 4.1. LetX,Y,Z,W be four objects in a triangulated category. Let f=/parenleftbiggA B
+C D/parenrightbigg
+:
+X⊕Y→Z⊕Wbe a morphism. If Dis an isomorphism, then Cone(f)∼=Cone(A−BD−1C:
+X→Z).
+Proof.This is essentially Lemma 4.2 from [BN] (or Lemma 5.25 from [CK]). /square
+4.2.Some basic sl2relations. We begin by reviewing some of the relations which follow from
+the definition of a geometric categorical sl2action. These results are strictly about sl2actions
+and so they all follow from [CKL2].
+Proposition 4.2. We have the direct sum decomposition
+Ei∗E(r)
+i∼=E(r+1)
+i⊗kH⋆(Pr)∼=E(r)
+i∗Ei.
+More generally, we have
+E(r1)
+i∗E(r2)
+i∼=E(r1+r2)
+i⊗kH⋆(G(r1,r1+r2))
+whereG(r1,r1+r2)denotes the Grassmannian of r1-planes in Cr1+r2.
+Proof.This follows by Proposition 4.2 and Corollary 4.7 of [CKL2]. /square20 SABIN CAUTIS AND JOEL KAMNITZER
+Proposition 4.3. We have the direct sum decompositions:
+Fi(λ)∗Ei(λ)∼=Ei(λ−αi)∗Fi(λ−αi)⊕O∆⊗kH⋆(P−/angbracketleftλ,αi/angbracketright−1)if/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≤0
+Ei(λ−αi)∗Fi(λ−αi)∼=Fi(λ)∗Ei(λ)⊕O∆⊗kH⋆(P/angbracketleftλ,αi/angbracketright−1)if/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≥0.
+Proof.This follows by Proposition 4.5 of [CKL2]. /square
+Corollary 4.4. We have
+E(b)
+i∗F(a)
+i∼=/circleplusdisplay
+j≥0F(a−j)
+i∗E(b−j)
+i(λ)⊗kH⋆(G(j,/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔−a+b))if/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔−a+b≥0
+F(b)
+i∗E(a)
+i(λ)∼=/circleplusdisplay
+j≥0E(a−j)
+i∗F(b−j)
+i⊗kH⋆(G(j,−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+a−b))if/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔−a+b≤0
+where, by convention, E(l)
+i= 0 =F(l)
+iifl <0. Moreover, if i/ne}a⊔ionslash=jthen
+F(b)
+i∗E(a)
+j∼=E(a)
+j∗F(b)
+i.
+Proof.This first statement is a formal consequence of Proposition 4.3 and the cancellation
+relations. See [CKL3] Lemma 4.2 for a sketch of the proof. The seco nd commutation relation
+follows by cancellation from Proposition 4.2 and by repeatedly applying the fact that Fi∗Ej∼=
+Ej∗Fiifi/ne}a⊔ionslash=j. /square
+4.3.Spaces of maps. Next we have some results about maps between various combination s of
+Es.
+Lemma 4.5. Ifi,j∈Iare joined by an edge then
+Extk(E(b)
+i∗E(a)
+j,E(a)
+j∗E(b)
+i)∼=/braceleftbigg
+0ifk < ab
+kifk=ab(6)
+while
+Extk(E(b)
+i∗E(a)
+j,E(b)
+i∗E(a)
+j)∼=/braceleftbigg
+0ifk <0
+k·idifk= 0(7)
+for anya,b≥0(here we are assuming that E(b)
+i∗ E(a)
+j/ne}a⊔ionslash= 0). Thus E(b)
+i∗ E(a)
+jis a brick. The
+same results hold if we replace all Es byFs.
+We will denote the unique map (up to non-zero multiple) in (6) when k=abby
+T(b)(a)
+ij:E(b)
+i∗E(a)
+j[−ab]→ E(a)
+j∗E(b)
+i.
+Whena=b= 1 we omit the superscripts.
+Proof.The proof is by (decreasing) induction on /an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔and also a,b. The base case being a= 0
+orb= 0 and follows by Lemma 4.9 of [CKL2].
+Using adjunction and (ix), we have
+Extk(E(b)
+i(λ+aαj)∗E(a)
+j(λ),E(a)
+j(λ+bαi)∗E(b)
+i(λ))
+∼=Extk(F(a)
+j(λ+bαi)[−a(/an}bracke⊔le{⊔λ+bαi,αj/an}bracke⊔ri}h⊔+a)]∗E(b)
+i(λ+aαj)∗E(a)
+j(λ),E(b)
+i(λ))
+∼=Extk(E(b)
+i(λ)∗F(a)
+j(λ)∗E(a)
+j(λ)[−a(/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔+a−b)],E(b)
+i(λ))BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 21
+Now, let us suppose /an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔ ≤0 (if not then we rewrite the equation above by moving the left
+E(a)
+jto the right hand side using adjunction and proceeding in the same wa y). Then by Corollary
+4.4, we have
+F(a)
+j(λ)∗E(a)
+j(λ)∼=s=a/circleplusdisplay
+s=0E(a−s)
+j∗F(a−s)
+j⊗kH⋆(G(s,−/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔))
+so we need to understand
+Extk(E(b)
+i∗E(a−s)
+j∗F(a−s)
+j(λ−(a−s)αj)⊗kH⋆(G(s,−/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔)),E(b)
+i[a(/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔+a−b)]).
+ButF(a−s)
+j(λ−(a−s)αj)L∼=E(a−s)
+j(λ−(a−s)αj)[(a−s)(/an}bracke⊔le{⊔λ−(a−s)αj,αj/an}bracke⊔ri}h⊔+a−s)] so this
+equals
+Extk(E(b)
+i∗E(a−s)
+j⊗kH⋆(G(s,−/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔)),E(b)
+i∗E(a−s)
+j(λ−(a−s)αj)[(2a−s)(/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔+s)−ab]).
+NowH⋆(G(s,−/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔)) is supported in degrees ∗ ≤ −s(/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔+s). So we get summands of
+the form
+Extk(E(b)
+i∗E(a−s)
+j,E(b)
+i∗E(a−s)
+j(λ−(a−s)αj)[2(a−s)(/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔+s)−ab−∗])
+where∗ ≥0. Ifk < abthen 2(a−s)(/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔+s)−ab−∗+k <0 so we get zero (here we use
+that
+Ext<0(E(b)
+i∗E(a−s)
+j,E(b)
+i∗E(a−s)
+j(λ−(a−s)αj)) = 0
+bytheinductionhypothesis). If k=abthen2(a−s)(/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔+s)−∗isnon-negativepreciselywhen
+∗= 0 ands=aand we get only one such summand since H⋆(G(s,−/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔)) is one-dimensional
+in top degree.
+This completes half the induction argument (i.e. relation (7) implies (6) ). To prove the other
+half we repeat the analogous argument with
+Extk(E(b)
+i∗E(a)
+j,E(b)
+i∗E(a)
+j)
+to show that relation (7) holds assuming (6).
+Notice that to ensure the induction terminates we need the assump tion that the action is
+integrable. The corresponding result for Fs follows by taking adjoints. /square
+The following result shows that E(b)
+i∗E(a)
+jare bricks if iandjare not connected.
+Corollary 4.6. Ifi,j∈Iare not joined by an edge then
+Extk(E(b)
+i∗E(a)
+j,E(b)
+i∗E(a)
+j)∼=/braceleftbigg
+0ifk <0
+k·idifk= 0(8)
+and similarly if we replace all the Es byFs (here we are assuming that E(b)
+i∗E(a)
+j/ne}a⊔ionslash= 0).
+Proof.The proof is precisely the induction from Lemma 4.5. The main differenc e is that in the
+computation we replace /an}bracke⊔le{⊔αi,αj/an}bracke⊔ri}h⊔=−1 by/an}bracke⊔le{⊔αi,αj/an}bracke⊔ri}h⊔= 0. Also, the induction has only one part
+since now E(b)
+iandE(a)
+jcommute (because EiandEjcommute). /square
+4.4.Some basic sl3relations. We first generalize the relation Ei∗Ej∗Ei∼=E(2)
+i∗Ej⊕Ej∗E(2)
+i
+wheni,j∈Iare joined by an edge.
+Proposition 4.7. Ifi,j∈Iare joined by an edge then
+E(a)
+i∗Ej∗Ei∼=E(a+1)
+i∗Ej⊗kH⋆(Pa−1)⊕Ej∗E(a+1)
+i
+and similarly
+Ei∗Ej∗E(a)
+i∼=Ej∗E(a+1)
+i⊗kH⋆(Pa−1)⊕E(a+1)
+i∗Ej.22 SABIN CAUTIS AND JOEL KAMNITZER
+Proof.We prove the first relation by induction on a(the second relation follows similarly). The
+base case is a= 1 which is precisely one of the conditions of having a geometric categ oricalg
+action.
+Applying E(a)
+ito the left of the relation for a= 1 we get
+E(a)
+i∗Ei∗Ej∗Ei∼=E(a)
+i∗Ej∗E(2)
+i⊕E(a)
+i∗E(2)
+i∗Ej.
+Tensoring both sides with H⋆(P1) and using that E2
+i=E(2)
+i⊗kH⋆(P1) we get
+E(a)
+i∗Ei∗Ej∗Ei⊗kH⋆(P1)∼=E(a)
+i∗Ej∗E2
+i⊕E(a)
+i∗E2
+i∗Ej
+∼=/parenleftBig
+E(a+1)
+i∗Ej∗Ei⊗kH⋆(Pa−1)⊕Ej∗E(a+1)
+i∗Ei/parenrightBig
+⊕E(a)
+i∗E2
+i∗Ej
+where the second isomorphism follows by induction. Thus we get
+E(a+1)
+i∗Ej∗Ei⊗kH⋆(P1×Pa)∼=E(a+1)
+i∗Ej∗Ei⊗kH⋆(Pa−1)⊕Ej∗E(a+1)
+i∗Ei⊕E(a)
+i∗E2
+i∗Ej.
+By cancellation law (4), we obtain
+E(a+1)
+i∗Ej∗Ei⊗kH⋆(Pa+1)∼=Ej∗E(a+1)
+i∗Ei⊕E(a)
+i∗E2
+i∗Ej
+∼=Ej∗E(a+2)
+i⊗kH⋆(Pa+1)⊕E(a+2)
+i∗Ej⊗kH⋆(Pa×Pa+1).
+Using cancellation law (5) we get
+E(a+1)
+i∗Ej∗Ei∼=Ej∗E(a+2)
+i⊕E(a+2)
+i∗Ej⊗kH⋆(Pa)
+and the induction step is complete. /square
+Corollary 4.8. Ifi,j∈Iare joined by an edge then
+E(a)
+i∗Ej∗E(b)
+i∼=E(a+b)
+i∗Ej⊗kH⋆(G(b,a+b−1))⊕Ej∗E(a+b)
+i⊗kH⋆(G(a,a+b−1)).
+Proof.The proof is by induction on a. To compute E(a+1)
+i∗Ej∗E(b)
+ione looks at
+Ei∗E(a)
+i∗Ej∗E(b)
+i∼=E(a+1)
+i∗Ej∗E(b)
+i⊗kH⋆(Pa).
+By induction the left hand side is
+Ei∗/parenleftBig
+E(a+b)
+i∗Ej⊗kH⋆(G(b,a+b−1))⊕Ej∗E(a+b)
+i⊗kH⋆(G(a,a+b−1))/parenrightBig
+.
+Now we have
+Ei∗E(a+b)
+i∗Ej∼=E(a+b+1)
+i∗Ej⊗kH⋆(Pa+b)
+and by Proposition 4.7
+Ei∗Ej∗E(a+b)
+i∼=E(a+b+1)
+i∗Ej⊕Ej∗E(a+b+1)
+i⊗kH⋆(Pa+b−1)
+So by the cancellation law (5), the induction step comes down to prov ing that
+H⋆(G(b,a+b−1)×Pa+b)⊕H⋆(G(a,a+b−1))∼=H⋆(G(b,a+b)×Pa)
+and
+H⋆(G(a,a+b−1)×Pa+b−1)∼=H⋆(G(a+1,a+b)×Pa)
+which one can prove by standard techniques. /squareBRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 23
+4.5.Induced maps.
+Lemma 4.9. Ifi,j∈Iare connected by an edge then
+Eij∗Ei∼=Cone(E(2)
+i∗Ej[−1]T(2)(1)
+ij[1]
+− −−−−− → E j∗E(2)
+i[1])
+and
+Eij∗Ej∼=Cone(Ei∗E(2)
+j[−2]T(1)(2)
+ij− −−− → E(2)
+j∗Ei).
+In particular,
+Ei∗Ej∗Ei[−1]TijI− −− → E j∗Ei∗Ei
+induces an isomorphism on the Ej∗E(2)
+i[−1]summand while
+Ei∗Ej∗Ej[−1]TijI− −− → E j∗Ei∗Ej
+induces an isomorphism on the Ei∗E(2)
+jsummand. We also have the analogous results for Ei∗Eij
+andEj∗Eij.
+Proof.We deal with the case of Eij∗Eisince the other cases follow similarly.
+Now (Ei∗Ej[−1]Tij− − → Ej∗Ei)∗Eiinduces a map
+/parenleftbigg
+α0
+β γ/parenrightbigg
+:/parenleftBigg
+Ej∗E(2)
+i[−1]
+E(2)
+i∗Ej[−1]/parenrightBigg
+→/parenleftBigg
+Ej∗E(2)
+i[−1]
+Ej∗E(2)
+i[1]/parenrightBigg
+.
+We need to show that α/ne}a⊔ionslash= 0/ne}a⊔ionslash=γbecause then αis a non-zero multiple of the identity and by
+the cancellation Lemma 4.1 the cone is isomorphic to Cone( γ) whereγmust be T(2)(1)
+ij[1] (up to
+a multiple) by Lemma 4.5.
+Letv∈h′
+kbe a vector with /an}bracke⊔le{⊔v,αi/an}bracke⊔ri}h⊔= 1 and /an}bracke⊔le{⊔v,αj/an}bracke⊔ri}h⊔=−1.
+Denote by tthe natural inclusion
+t:Y(λ)×Y(λ+αi+αj)→˜Y(λ)|span(v)×span(v)˜Y(λ+αi+αj)|span(v).
+FromProposition2.5.(i), forall A ∈Y(λ)×Y(λ+αi+αj), wehavethefunctorialdistinguished
+triangle
+A[−1]cv,v(A)− −−−− → A [1]→t∗t∗A → A.
+Applyingthistothedistinguishedtriangle Ei∗Ej[−1]Tij− − → Ej∗Ei→ Eijweobtainthecommutative
+diagram
+(9) t∗t∗(Ei∗Ej[−1])t∗t∗Tij/d47/d47
+(adj)
+/d15/d15t∗t∗(Ej∗Ei)
+(adj)
+/d15/d15/d47/d47t∗t∗Eij
+(adj)
+/d15/d15
+Ei∗Ej[−1]
+cv,v(Ei∗Ej)
+/d15/d15Tij/d47/d47Ej∗Ei
+cv,v(Ej∗Ei)[1]
+/d15/d15/d47/d47Eij
+cv,v(Eij)
+/d15/d15
+Ei∗Ej[1]Tij[2]/d47/d47Ej∗Ei[2]g′/d47/d47Eij[2].
+As noted in section 2.3.2, because /an}bracke⊔le{⊔v,αi+αj/an}bracke⊔ri}h⊔= 0,cv,v(Eij) = 0.24 SABIN CAUTIS AND JOEL KAMNITZER
+Now apply ∗Eito the whole diagram to get
+(10) t∗t∗(Ei∗Ej[−1])∗Eit∗t∗TijI/d47/d47
+(adj)
+/d15/d15t∗t∗(Ej∗Ei)∗Ei
+(adj)
+/d15/d15/d47/d47t∗t∗Eij∗Ei
+(adj)
+/d15/d15
+Ei∗Ej∗Ei[−1]
+cv,v(Ei∗Ej)I
+/d15/d15TijI/d47/d47Ej∗Ei∗Ei
+cv,v(Ej∗Ei)I[1]
+/d15/d15/d47/d47Eij∗Ei
+0
+/d15/d15
+Ei∗Ej∗Ei[1]TijI[2]/d47/d47Ej∗Ei∗Ei[2]g′I/d47/d47Eij∗Ei[2].
+We now examine the map
+cv,v(Ej∗Ei)I[1] :Ej∗Ei∗Ei∼=Ej∗E(2)
+i[−1]⊕Ej∗E(2)
+i[1]→ Ej∗E(2)
+i[1]⊕Ej∗E(2)
+i[3]∼=Ej∗Ei∗Ei[2].
+We claim that this map is an isomorphism on the summand Ej∗E(2)
+i[1].
+To see this, note that cv,v(Ej∗ Ei)I[1] =cv,0(Ej∗ Ei)I[1] +c0,v(Ej∗ Ei)I[1] by Proposition
+2.5.(ii). Let us consider each of these terms.
+First, by Proposition 2.6.(i), cv,0(Ej∗Ei)I=cv,0(Ej∗Ei∗Ei). Hence this map is given by the
+diagonal matrix
+/parenleftBigg
+cv,0(Ej∗E(2)
+i)[−1] 0
+0 cv,0(Ej∗E(2)
+i)[1]/parenrightBigg
+:/parenleftBigg
+Ej∗E(2)
+i[−2]
+Ej∗E(2)
+i/parenrightBigg
+→/parenleftBigg
+Ej∗E(2)
+i
+Ej∗E(2)
+i[2]/parenrightBigg
+because the deformation is only along the left-hand factor. In par ticular, it induces the zero
+map between the summands Ej∗E(2)
+i.
+On the other hand, since /an}bracke⊔le{⊔v,αi/an}bracke⊔ri}h⊔= 1, we see by Proposition 2.7 that c0,v(Ej∗Ei)I=Ic0,v(Ei)I
+gives an isomorphism between the Ej∗E(2)
+isummands.
+Hence we conclude that cv,v(Ej∗Ei)I[1] is an isomorphism between the Ej∗E(2)
+i[1] summands.
+Finally, looking back at diagram (10), we see that ( g′I)◦cv,v(Ej∗Ei)I[1] = 0 so that cv,v(Ej∗
+Ei)I[1] must factor as
+Ej∗Ei∗Ei→ Ei∗Ej∗Ei[1]TijI[2]− −−− → E j∗Ei∗Ei[2].
+Sincecv,v(Ej∗Ei)I[1] induces an isomorphism on the summand Ej∗E(2)
+i[1] then so must TijI[2].
+ThusTijIinduces an isomorphism on the summand Ej∗E(2)
+i[−1]. This concludes the proof that
+α/ne}a⊔ionslash= 0.
+It remains to show γ/ne}a⊔ionslash= 0. To see this we consider the map
+Ei∗Ei∗Ej∗Ei[−1]ITijI− −− → E i∗Ej∗Ei∗Ei.
+We will examine this map in two ways, by associating in two different ways . In particular,
+we will obtain a contradiction by examining the number of summands of the form E(3)
+i∗Ej[·] on
+which this map induces an isomorphism.
+On the one hand, we consider the last three factors together and obtain a map
+Ei∗(Ej∗E(2)
+i[−1]⊕E(2)
+i∗Ej[−1]→ Ej∗E(2)
+i[−1]⊕Ej∗E(2)
+i[1])
+which is I/parenleftBig
+α0
+β γ/parenrightBig
+.
+Suppose γ= 0. Then this map induces an isomorphism on at most one summand of t he
+formE(3)
+i∗Ej[·]. This is because it can only induce an isomorphism on a summand coming f rom
+Ei∗Ej∗E(2)
+iand this summand contains one copy of E(3)
+i∗Ejby Proposition 4.7.BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 25
+On the other hand, we consider the first three factors together and obtain a map
+(E(2)
+i∗Ej⊕E(2)
+i∗Ej[−2]→ E(2)
+i∗Ej⊕Ej∗E(2)
+i)∗Ei
+We can apply the above reasoning to this map as well since we can write it as/parenleftBig
+α′0
+β′γ′/parenrightBig
+I. Now
+sinceα′/ne}a⊔ionslash= 0, we see that this map induces an isomorphism on at least two summa nds of the
+formE(3)
+i∗Ej[·] (sinceE(2)
+i∗Ej∗Eicontains two copies of E(3)
+i∗Ej). This gives a contradiction
+and means that γ/ne}a⊔ionslash= 0 (so we are done). /square
+Corollary 4.10. Ifi,j∈Iare connected by an edge then for s≥0
+Eij∗E(s)
+i∼=Cone(E(s+1)
+i∗Ej[−1]T(s+1)(1)
+ij[s]
+− −−−−−− → E j∗E(s+1)
+i[s]).
+In particular,
+Ei∗Ej∗E(s)
+i[−1]TijI− −− → E j∗Ei∗E(s)
+i
+induces an isomorphism on all summands of the form Ej∗E(s+1)
+i[·]on the left hand side. Similarly
+Eij∗E(s)
+j∼=Cone(Ei∗E(s+1)
+j[−s−1]T(1)(s+1)
+ij− −−−−− → E(s+1)
+j∗Ei).
+We also have the analogous results for E(s)
+i∗EijandE(s)
+j∗Eij.
+Remark 4.11. The proof only assumes the result when s= 1 (everything else is a formal
+consequence of the fact that E(a)
+i∗E(b)
+jare bricks).
+Proof.We prove only the first identity as the others follow similarly.
+Step 1. First we show by induction on sthat
+Eij∗E(s)
+i∼=Cone(E(s+1)
+i∗Ej[−1]g− → Ej∗E(s+1)
+i[s]) (11)
+for some map g. The base case s= 1 is covered in Lemma 4.9. Consider
+(Ei∗Ej[−1]Tij− − → Ej∗Ei)∗E(s+1)
+i
+which we can rewrite as
+(12) Ej∗E(s+2)
+i⊗kH⋆(Ps)[−1]⊕E(s+2)
+i∗Ej[−1]f1− → Ej∗E(s+2)
+i⊗kH⋆(Ps+1).
+Lettbe the number of summands of the form Ej∗E(s+2)
+i[·] on which f1induces an isomorphism.
+On the other hand we also have the map
+(13) ( Ei∗Ej[−1]Tij− − → Ej∗Ei)∗E(s)
+i∗Ei.
+SinceE(s)
+i∗Ei∼=E(s+1)
+i⊗H⋆(Ps) we see that (13) induces an isomorphism on t(s+1) summands
+of the form Ej∗E(s+2)
+i[·].
+Now we can rewrite (13) as
+/parenleftBig
+Ej∗E(s+1)
+i⊗kH⋆(Ps−1)[−1]⊕E(s+1)
+i∗Ej[−1]f2− → Ej∗E(s+1)
+i⊗kH⋆(Ps)/parenrightBig
+∗Ei.
+By induction, f2induces an isomorphism on ssummands of the form Ej∗E(s+1)
+i[·]. Now we can
+rewrite both sides as
+Ej∗E(s+2)
+i⊗kH⋆(Ps−1×Ps+1)[−1]⊕Ej∗E(s+2)
+i[−1]⊕E(s+2)
+i∗Ej⊗kH⋆(Ps)[−1]
+f3− → Ej∗E(s+2)
+i⊗kH⋆(Ps×Ps+1).26 SABIN CAUTIS AND JOEL KAMNITZER
+The map f3induces an isomorphism on either s(s+ 2) ors(s+2) +1 summands of the form
+Ej∗E(s+2)
+i[·] (we do not know a prioriif it induces an isomorphism on the middle summand on
+the left hand side).
+Combining with above, we see that t(s+1) =s(s+2) ort(s+1) =s(s+2)+1 for some t
+with 0≤t≤s+1. This forces t=s+1. Hence by Gaussian elimination,
+Eij∗E(s+1)
+i∼=Cone(E(s+2)
+i∗Ej[−1]g− → Ej∗E(s+2)
+i[s+1])
+for some map g. This completes the induction.
+Step 2. Next we show that the map gin (11) is non-zero since then by Lemma 4.5, g=
+T(s+1)(1)
+ij[s] (up to a non-zeromultiple). If g= 0 then applying ∗Eito (11) we get s+1summands
+Eij∗E(s+1)
+i∼=Cone(E(s+2)
+i∗Ej[−1]→ Ej∗E(s+2)
+i[s+1])
+on the left hand side and
+E(s+1)
+i∗Ej∗Ei⊕Ej∗E(s+1)
+i∗Ei[s]
+on the right hand side. But then the right side contains s+3 summands Ej∗E(s+2)
+iinstead of
+s+1 on the left side (contradiction). Thus g/ne}a⊔ionslash= 0. /square
+Corollary 4.12. Ifi,j∈Iare connected by an edge then the composition
+E(s+1)
+i∗EjιI− → Ei∗E(s)
+i[−s]∗EjIT(s)(1)
+ij− −−−− → E i∗Ej∗E(s)
+i
+is an isomorphism of E(s+1)
+i∗Ejonto the lone summand in
+Ei∗Ej∗E(s)
+i∼=E(s+1)
+i∗Ej⊕Ej∗E(s+1)
+i⊗kH⋆(Ps).
+Proof.SinceιIis an inclusion (into lowest cohomological degree) it suffices to show th at
+E(s+1)
+i∗Ej⊗kH⋆(Ps)[−s]∼=Ei∗E(s)
+i[−s]∗EjIT(s)(1)
+ij− −−−− → E i∗Ej∗E(s)
+i∼=E(s+1)
+i∗Ej⊕Ej∗E(s+1)
+i⊗kH⋆(Ps)
+is an isomorphism onto the one copy of E(s+1)
+i∗Ejon the right hand side.
+Now consider the map
+Ei∗Ej∗E(s−1)
+i[−1]TijI− −− → E j∗Ei∗E(s−1)
+i.
+By Corollary 4.10, on the corresponding summands of both sides, th is map restricts to
+E(s)
+i∗Ej[−1]T(s)(1)
+ij[s−1]−−−−−−−→ E j∗E(s)
+i[s−1].
+So applying Ei∗it suffices to show that
+(14) Ei∗Ei∗Ej∗E(s−1)
+i[−1]ITijI− −− → E i∗Ej∗Ei∗E(s−1)
+i
+is an isomorphism onto all copies of E(s+1)
+i∗Ejon the right hand side.
+Now, the right hand side of (14) is isomorphic to E(2)
+i∗ Ej∗ E(s−1)
+i⊕ Ej∗ E(2)
+i∗ E(s−1)
+iand
+(by Lemma 4.9) the map induces an isomorphism onto the first summan d. Since all copies of
+E(s+1)
+i∗Ejon the right hand side of (14) come from this first summand the map in (14) surjects
+onto all summands E(s+1)
+i∗Ejon the right side. /square
+Corollary 4.13. Ifi,j∈Iare joined by an edge then
+Extk(E(a)
+i∗Eij,E(a)
+i∗Eij)∼=/braceleftBigg
+0ifk <0
+k·idifk= 0.BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 27
+Here we are assuming that E(a)
+i∗Eij/ne}a⊔ionslash= 0.
+Proof.By Corollary 4.10 we have the following exact triangle
+E(a+1)
+i∗Ej[−a−1]T(a+1)(1)
+ij− −−−−− → E j∗E(a+1)
+i→ E(a)
+i∗Eij.
+Applying Hom( ·,E(a)
+i∗Eij) to this triangle we get
+··· →Extk+a(E(a+1)
+i∗Ej,E(a)
+i∗Eij)→Extk(E(a)
+i∗Eij,E(a)
+i∗Eij)→Extk(Ej∗E(a+1)
+i,E(a)
+i∗Eij)→...
+Now, applying Hom( Ej∗E(a+1)
+i,·) to the exact triangle and using Lemma 4.5 it is easy to see
+that Extk(Ej∗E(a+1)
+i,E(a)
+i∗Eij) = 0 ifk <0 and is one dimensional if k= 0. Similarly, applying
+Hom(E(a+1)
+i∗Ej,·) we get
+...→Extk+a(E(a+1)
+i∗Ej,Ej∗E(a+1)
+i)→Extk+a(E(a+1)
+i∗Ej,E(a)
+i∗Eij)
+→Extk(E(a+1)
+i∗Ej,E(a+1)
+i∗Ej)→Extk+a+1(E(a+1)
+i∗Ej,Ej∗E(a+1)
+i)→...
+By Lemma 4.5 this means Extk+a(E(a+1)
+i∗Ej,E(a)
+i∗Eij) = 0 ifk <0 and ifk= 0 we get
+0→Exta(E(a+1)
+i∗Ej,E(a)
+i∗Eij)→k·id→k·T(a+1)(1)
+ij→0
+where the third map is an isomorphism. Thus Extk+a(E(a+1)
+i∗Ej,E(a)
+i∗Eij) = 0 for any k≤0.
+Thus, ifk≤0, we get the exact sequence
+0→Extk(E(a)
+i∗Eij,E(a)
+i∗Eij)→Extk(Ej∗E(a+1)
+i,E(a)
+i∗Eij)
+and the result follows. /square
+Lemma 4.14. Ifi,j∈Iare connected by an edge and /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≥0then
+Eij∗Fi(λ)∼=Cone(Fi∗Ei∗Ej[−1]IT⊕εI− −−− → F i∗Ej∗Ei⊕Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1]).
+In particular,
+Ei∗Ej∗Fi[−1]TijI− −− → E j∗Ei∗Fi
+induces an isomorphism on every summand of the form Ej[·]on the left hand side.
+Remark 4.15. This result is a formal consequence of Lemma 4.9 and Corollary 4.4.
+Proof.First we consider the map
+(15) Ei∗Ej∗Ei∗Fi(λ−αi)[−1]TijII− −− → E j∗Ei∗Ei∗Fi.
+On the one hand, we can group the first three factors together t o obtain
+(16) ( Ej∗E(2)
+i[−1]⊕E(2)
+i∗Ej[−1]→ Ej∗E(2)
+i[−1]⊕Ej∗E(2)
+i[1])∗Fi(λ−αi)
+The map on the first summands is an isomorphism by Lemma 4.9. Using Co rollary 4.4 we have
+(17) Ej∗E(2)
+i[−1]∗Fi(λ−αi)∼=Ej∗Ei[−1]⊗kH⋆(P/angbracketleftλ,αi/angbracketright)⊕Fi∗Ej∗E(2)
+i[−1].
+So this induces an isomorphism between at least /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1 summands of the form Ej∗Ei[·].
+On the other hand, using Proposition 4.3 we have
+Ei∗Fi(λ−αi)∼=O∆⊗H⋆(P/angbracketleftλ,αi/angbracketright−1)⊕Fi∗Ei
+where, as before, H⋆(P−1) = 0 by convention. Hence we can rewrite map (15) as
+(18)Ei∗Ej⊗H⋆(P/angbracketleftλ,αi/angbracketright−1)[−1]⊕Ei∗Ej∗Fi∗Ei[−1]Tij⊕TijII−−−−−−→ E j∗Ei⊗H⋆(P/angbracketleftλ,αi/angbracketright−1)⊕Ej∗Ei∗Fi∗Ei28 SABIN CAUTIS AND JOEL KAMNITZER
+and then as
+Ei∗Ej⊗H⋆(P/angbracketleftλ,αi/angbracketright−1)[−1]⊕Ej∗Ei⊗H⋆(P/angbracketleftλ+αj,αi/angbracketright+1)[−1]⊕Fi∗Ei∗Ej∗Ei[−1]→
+Ej∗Ei⊗H⋆(P/angbracketleftλ,αi/angbracketright−1)⊕Ej∗Ei⊗H⋆(P/angbracketleftλ,αi/angbracketright+1)⊕Fi∗Ej∗Ei∗Ei.
+NowFi∗Ei∗Ej∗Eicontains no summand Ej∗Eisince
+Hom(Ej∗Ei,Fi(λ+αi+αj)∗Ei∗Ej∗Ei)
+∼=Hom(Ei[/an}bracke⊔le{⊔λ+αi+αj,αi/an}bracke⊔ri}h⊔+1]∗Ej∗Ei,Ei∗Ej∗Ei)
+∼=Hom(E(2)
+i∗Ej⊕Ej∗E(2)
+i,E(2)
+i∗Ej⊕Ej∗E(2)
+i[−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔−2])
+vanishes by using Lemma 4.5 and /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≥0. Hence (15) induces an isomorphism between at
+most/an}bracke⊔le{⊔λ+αj,αi/an}bracke⊔ri}h⊔+2 =/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1 summands of the form Ej∗Ei[·].
+Combining these two observations, we see that the map in (15) induc es an isomorphism
+between exactly /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+ 1 summands of the form Ej∗ Ei[·]. This means that the map TijII
+from (18) also induces an isomorphism on /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1 summands of the form Ej∗Ei[·].
+Now, let us consider
+(19) Ei∗Ej∗Fi(λ)[−1]TijI− −− → E j∗Ei∗Fi
+which we can rewrite as
+Ej⊗H⋆(P/angbracketleftλ,αi/angbracketright)[−1]⊕Fi∗Ei∗Ej[−1]→ Ej⊗H⋆(P/angbracketleftλ,αi/angbracketright+1)⊕Fi∗Ej∗Ei
+Since the map TijIIfrom (18) induces an isomorphism on /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+ 1 summands of the form
+Ej∗ Ei[·] the map from (19) must also induce an isomorphism on /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+ 1 summands of the
+formEj[·]. Hence we can apply Gaussian elimination to conclude that
+Eij∗Fi(λ)∼=Cone(Fi∗Ei∗Ej[−1]f1⊕f2− −−− → F i∗Ej∗Ei⊕Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1])
+for some maps f1,f2. Now
+Hom(Fi(λ+αi+αj)∗Ei∗Ej[−1],Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1])∼=Hom(Ei∗Ej[−1],Ei∗Ej[−1])∼=k
+which is spanned by the adjunction map εI. Similarly, we have
+Hom(Fi(λ+αi+αj)∗Ei∗Ej[−1],Fi∗Ej∗Ei)
+∼=Hom(Ei∗Ej[−1],Ei∗Fi∗Ej∗Ei[−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔−2])
+∼=Hom(Ei∗Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1],Fi(λ+2αi+αj)∗Ei∗Ej∗Ei⊕Ej∗Ei⊗kH⋆(P/angbracketleftλ+αi+αj,αi/angbracketright+1))
+∼=Hom(Ei∗Ei∗Ej[2/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+4],Ei∗Ej∗Ei)⊕Hom(Ei∗Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1],Ej∗Ei⊗kH⋆(P/angbracketleftλ,αi/angbracketright+2)).
+The first term above equals
+Hom(E(2)
+i∗Ej[−1]⊕E(2)
+i∗Ej[1],(E(2)
+i∗Ej⊕Ej∗E(2)
+i)[−2/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔−4])
+and thus vanishes by Lemma 4.5 since /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≥0. The second term is one-dimensional. Thus
+Hom(Fi∗Ei∗Ej[−1],Fi∗Ej∗Ei)∼=k (20)
+which is spanned by ITij. So it remains to show that f1andf2are non-zero.
+To show that f1/ne}a⊔ionslash= 0, we look again at the map (15). When we rewrite it as in (16), we kno w
+that the map on first summands is an isomorphism. By (17), these fir st summands contain a
+copy ofFi∗ Ej∗ E(2)
+i[−1]. On the other hand, when we rewrite (15) as in (18), we see that
+this copy of Fi∗ Ej∗ E(2)
+i[−1] is a direct summand of Fi∗ Ei∗ Ej∗ Ei[−1]. Thus the map on
+Fi∗Ei∗Ej∗Ei[−1] must be non-zero. However, this is precisely f1Iand hence f1/ne}a⊔ionslash= 0.BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 29
+To show f2/ne}a⊔ionslash= 0 we consider the map
+(21) Ei∗/parenleftbig
+Ei∗Ej[−1]Tij− − → Ej∗Ei/parenrightbig
+∗Fi(λ)
+On the one hand we can rewrite this as
+E(2)
+i∗Ej∗Fi[−2]⊕E(2)
+i∗Ej∗Fi→ Ej∗E(2)
+i∗Fi⊕E(2)
+i∗Ej∗Fi
+where the map is an isomorphism on the second summands. Since
+E(2)
+i∗Ej∗Fi(λ)∼=Ei∗Ej⊗kH⋆(P/angbracketleftλ,αi/angbracketright+1)⊕Fi∗E(2)
+i∗Ej,
+this means that (21) is an isomorphism on at least /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+2 summands of the form Ei∗Ej[·].
+On the other hand, as we showed above, (19) is an isomorphism on /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1 summands of the
+formEj[·]. Now, above we showed using Gaussian elimination that the map (19) c an be written
+as a direct sum of a map which is an isomorphism on /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1 summands of the form Ej[·] and
+the map f1⊕f2. But since (21) is obtained from (19) by applying Ei∗, this shows that
+Ei∗Fi∗Ei∗Ej[−1]If1⊕If2− −−−− → E i∗Fi∗Ej∗Ei⊕Ei∗Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1]
+must be an isomorphism on precisely one summand of the form Ei∗Ej[·]. This means that If2
+must induce an isomorphism on the right hand summand Ei∗Ej. In particular, f2/ne}a⊔ionslash= 0. /square
+Lemma 4.16. For any i/ne}a⊔ionslash=j∈Ithe composition F(s)
+i∗Ej∼=Ej∗F(s)
+iis brick.
+Proof.We have
+Extk(Ej(λ)∗F(s)
+i,Ej(λ)∗F(s)
+i)∼=Extk(F(s)
+i∗Ej(λ+sαi),Ej(λ)∗F(s)
+i)
+∼=Extk(Ej(λ)L∗F(s)
+i,F(s)
+i∗Ej(λ+sαi)L)
+∼=Extk(Fj∗F(s)
+i[−/an}bracke⊔le{⊔λ,αj/an}bracke⊔ri}h⊔−1],F(s)
+i∗Fj[−/an}bracke⊔le{⊔λ+sαi,αj/an}bracke⊔ri}h⊔−1])
+∼=Extk(Fj∗F(s)
+i,F(s)
+i∗Fj[−s/an}bracke⊔le{⊔αi,αj/an}bracke⊔ri}h⊔]).
+The result now follows from Lemma 4.5 if iandjare joined by an edge and from Corollary 4.6
+ifiandjare not joined. /square
+Corollary 4.17. Ifi,j∈Iare connected by an edge and /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s≥0then
+Eij∗F(s)
+i(λ)∼=Cone(F(s)
+i∗Ei∗Ej[−1]IT⊕IεI◦ιII− −−−−−−− → F(s)
+i∗Ej∗Ei⊕F(s−1)
+i∗Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s]).
+In particular,
+Ei∗Ej∗F(s)
+i[−1]TijI− −− → E j∗Ei∗F(s)
+i
+induces an isomorphism on every summand of the form F(s−1)
+i∗Ej[·]on the left hand side.
+Remark 4.18. This result is a formal consequence of Lemma 4.14 and Corollary 4.4.
+Proof.The proof is similar to that of Corollary 4.10.
+Step 1. First we prove by induction on sthat
+(22) Eij∗F(s)
+i∼=Cone(F(s)
+i∗Ei∗Ej[−1]g1⊕g2− −−− → F(s)
+i∗Ej∗Ei⊕F(s−1)
+i∗Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s])
+forsomemaps g1,g2. Thebasecase s= 1iscoveredbyLemma4.14. Nowsuppose /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s+1≥
+0 and consider
+(23) ( Ei∗Ej[−1]Tij− − → Ej∗Ei)∗F(s+1)
+i(λ).30 SABIN CAUTIS AND JOEL KAMNITZER
+We have
+Ei∗Ej∗F(s+1)
+i(λ)[−1]∼=F(s)
+i∗Ej⊗kH⋆(P/angbracketleftλ+αj,αi/angbracketright+s+1)[−1]⊕F(s+1)
+i∗Ei∗Ej[−1]
+Ej∗Ei∗F(s+1)
+i(λ)∼=F(s)
+i∗Ej⊗kH⋆(P/angbracketleftλ,αi/angbracketright+s+1)⊕F(s+1)
+i∗Ej∗Ei.
+We first want to show that the map induced in (23) is an isomorphism on all/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s+ 1
+summands F(s)
+i∗Ejon the left hand side. If /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s+1 = 0 we are done since there are no
+such summands.
+On the other hand, if /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s+1>0 then we also have the map
+(Ei∗Ej[−1]Tij− − → Ej∗Ei)∗Fi(λ)∗F(s)
+i(λ+αi)
+which induces a map
+/bracketleftBig
+Ej⊗kH⋆(P/angbracketleftλ+αj,αi/angbracketright+1)[−1]⊕Fi∗Ei∗Ej[−1]f− → Ej⊗kH⋆(P/angbracketleftλ,αi/angbracketright+1)⊕Fi∗Ej∗Ei/bracketrightBig
+∗F(s)
+i(λ+αi).
+By Lemma 4.14, the map fIinduces an isomorphism on all the /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+ 1 summands of the
+formEj∗F(s)
+i[·] on the left hand side and also induces
+/parenleftbigg
+Fi∗Ei∗Ej[−1]ITij− −− → F i∗Ej∗Ei/parenrightbigg
+∗F(s)
+i(λ+αi).
+By induction this induces an isomorphism on all the /an}bracke⊔le{⊔λ+αi+αj,αi/an}bracke⊔ri}h⊔+s+1 summands of the
+formFi∗F(s−1)
+i∗Ej[·] on the left hand side. So in total fIinduces an isomophism on
+(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+1)+s(/an}bracke⊔le{⊔λ+αi+αj,αi/an}bracke⊔ri}h⊔+s+1) = (s+1)(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s+1)
+summands of the form F(s)
+i∗Ej[·]. This shows that (23) induces an isomorphism on the /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+
+s+1 summands (which is what we wanted to show).
+Thus, by the cancellation Lemma 4.1,
+Eij∗F(s+1)
+i∼=Cone(F(s+1)
+i∗Ei∗Ej[−1]→ F(s+1)
+i∗Ej∗Ei⊕F(s)
+i∗Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s+1])
+which completes the induction.
+Step 2. Next, one can check
+Hom(F(s)
+i∗Ei∗Ej[−1],F(s)
+i∗Ej∗Ei)∼=k∼=Hom(F(s)
+i∗Ei∗Ej[−1],F(s−1)
+i∗Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s]).
+To do this one moves the factor F(s)
+ifrom the left side to the right side using adjunction,
+simplifies ( E(s)
+i∗F(s)
+i)∗Ej∗Eiand then uses adjunction again (just like in the computation used
+to prove (20)). This is a long but straight-forward calculation which we omit.
+Step 3. Finally we show that g1andg2are non-zero. This implies that g1must must be
+ITijandg2must be the composition
+F(s)
+i∗Ei∗Ej[−1]ιII− − → F(s−1)
+i∗Fi(λ+sαi+αj)[−s+1]∗Ei∗Ej[−1]
+IεI− − → F(s−1)
+i∗Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s]
+(up to a non-zero multiple). Recall that ιdenotes the unique inclusion of F(s)
+iinto the lowest
+degree summand of F(s−1)
+i∗Fi.
+To show g1/ne}a⊔ionslash= 0 we look at Eij∗F(s)
+i∗Fi. NowF(s)
+i∗Fi∼=F(s+1)
+i⊗kH⋆(Ps) and from the
+proof of Step 1,/parenleftbigg
+Ei∗Ej[−1]Tij− − → Ej∗Ei/parenrightbigg
+∗F(s+1)
+iinduces an isomorphism on all /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s+1
+summands of the form F(s)
+i∗Ej[·] on the left hand side. Thus
+(24) ( Ei∗Ej[−1]Tij− − → Ej∗Ei)∗F(s)
+i∗FiBRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 31
+induces an isomorphism on ( s+1)(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s+1) such summands.
+On the other hand, (24) induces an isomorphism on /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+ssummands F(s−1)
+i∗Ej∗Fior
+equivalently on s(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s) summands of the form F(s)
+i∗Ej[·] and what is left over is the map
+from equation (22):
+/parenleftBig
+F(s)
+i∗Ei∗Ej[−1]g1⊕g2− −−− → F(s)
+i∗Ej∗Ei⊕F(s−1)
+i∗Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s]/parenrightBig
+∗Fi.
+This means that the map above must induce an isomorphism on /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+2s+1 summands of
+the form F(s)
+i∗Ej[·]. This is impossible if g1= 0 since F(s−1)
+i∗Ej∗Fi∼=F(s)
+i∗Ej⊗kH⋆(Ps−1)
+contains only ssuch summands. Thus g1/ne}a⊔ionslash= 0.
+To show that g2/ne}a⊔ionslash= 0 we apply Ei∗to (23). On the one hand we get
+(25)/parenleftbigg
+Ei∗Ei∗Ej[−1]ITij− −− → E i∗Ej∗Ei/parenrightbigg
+∗F(s)
+i.
+By Lemma 4.9, this induces an isomorphism ( E(2)
+i∗Ej∼− → E(2)
+i∗Ej)∗F(s)
+iand the map
+/parenleftBigg
+E(2)
+i∗Ej[−2]T(1)(2)
+ij− −−− → E j∗E(2)
+i/parenrightBigg
+∗F(s)
+i.
+Now one can show, along the same lines as above, that the map T(1)(2)
+ijinduces an isomorphism
+on every summand of the form F(s−2)
+i∗Ej[·] on the left hand side. Thus the map in (25) induces
+an isomorphism on every summand of the form F(s−2)
+i∗Ejon the left hand side. On the other
+hand, ifg2= 0 then the map
+Ei∗/parenleftBig
+F(s)
+i∗Ei∗Ej[−1]g1⊕g2− −−− → F(s)
+i∗Ej∗Ei⊕F(s−1)
+i∗Ej[/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s]/parenrightBig
+cannot induce an isomorphism on all summands F(s−2)
+i∗Ejon the left hand side (contradiction).
+So we must have g2/ne}a⊔ionslash= 0. /square
+5.Proof of Main Theorem 2.10
+In this section we will assume that i,j∈Iare joined by an edge. For convenience we also
+assume that /an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≥0 and/an}bracke⊔le{⊔λ,αi+αj/an}bracke⊔ri}h⊔ ≥0, since the other cases are similar.
+The main idea of the proof is as follows. We will show that Ti∗Tj=Tij∗TiwhereTijis an
+equivalence coming from an sl2action generated by the kernel Eij. From a similar argument, we
+will also show that Ti∗Tj=Tj∗Tij. This immediately implies the braid relation. The kernel
+Eijshould be thought of as a root vector for the root αi+αj.
+Inordertoprovethat Ti∗Tj=Tij∗Ti, wewillcompute Eij∗Ti. Recallthat Tiistheconvolution
+of a complex where each term in the complex is of the form Ts
+i=F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s)
+i[−s]. So to
+compute Eij∗Tiwe first calculate Eij∗F(/angbracketleftλ,αi/angbracketright+s)
+i (Step 1) which follows directly from Corollary
+4.17. Next we calculate Eij∗ F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s)
+i(Step 2) which basically follows from Corollary
+4.10. This gives us a simplified expression for Eij∗Ts
+i.
+Next, in the most difficult step, we put all these terms together and simplify to come up with
+an expression for Eij∗Ti. We compare with a similarly simplified expression for Ti∗Ej(this is
+much easier to calculate) and conclude that Eij∗Ti∼=Ti∗Ej(Corollary 5.4). It then follows by
+formal arguments that Tij∗Ti∼=Ti∗Tj.32 SABIN CAUTIS AND JOEL KAMNITZER
+5.1.Step 1: Calculation of Eij∗F(/angbracketleftλ,αi/angbracketright+s)
+i .The first step is to compute
+Eij∗F(/angbracketleftλ,αi/angbracketright+s)
+i ∈D(Y(λ+sαi)×Y(λ−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔αi)).
+To simplify things we will abuse notation a little and write ds
+ifor any map obtained as the
+composition
+F(k)
+i∗E(s)
+iιι− → F(k−1)
+i∗Fi∗Ei∗E(s−1)
+iIεI− − → F(k−1)
+i∗E(s−1)
+i
+for anyk∈N(we omit the necessary shifts here to simplify notation).
+Proposition 5.1. Eij∗F(/angbracketleftλ,αi/angbracketright+s)
+i (λ−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔αi)is isomorphic to the cone of
+F(/angbracketleftλ,αi/angbracketright+s)
+i ∗Ei∗Ej[−1]ITij⊕d1
+iI− −−−−− → F(/angbracketleftλ,αi/angbracketright+s)
+i ∗Ej∗Ei⊕F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗Ej[s].
+Proof.This follows directly from Corollary 4.17 since
+/an}bracke⊔le{⊔λ−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔αi,αi/an}bracke⊔ri}h⊔+(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+s) =s≥0.
+/square
+5.2.Step 2: Calculation of Eij∗F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s)
+i.The second step is to compute
+Eij∗F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s)
+i(λ)∈D(Y(λ)×Y(λ−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔αi)).
+Proposition 5.2. Eij∗F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s)
+i(λ)is isomorphic to the cone of
+F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s+1)
+i∗Ej[−1]IT(s+1)(1)
+ij[s]⊕γs−−−−−−−−−−→ F(/angbracketleftλ,αi/angbracketright+s)
+i ∗Ej∗E(s+1)
+i[s]⊕Ej∗F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗E(s)
+i[s]
+whereγsis the composition
+F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s+1)
+i∗Ej[−1]ds+1
+iI− −−− → F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗E(s)
+i∗Ej
+IT(s)(1)
+ij[s]
+− −−−−− → F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗Ej∗E(s)
+i[s].
+Proof.This is a direct consequence of Proposition 5.1. Applying ∗E(s)
+ito the main expression
+in Proposition 5.1 we get the two maps
+(26) F(/angbracketleftλ,αi/angbracketright+s)
+i ∗Ei∗Ej[−1]∗E(s)
+iITijI− −− → F(/angbracketleftλ,αi/angbracketright+s)
+i ∗Ej∗Ei∗E(s)
+i
+(27) F(/angbracketleftλ,αi/angbracketright+s)
+i ∗Ei∗Ej[−1]∗E(s)
+id1
+iII− −− → F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗Ej[s]∗E(s)
+i.
+By Corollary 4.10, (26) induces an isomorphism on all summands of the formF(/angbracketleftλ,αi/angbracketright+s)
+i ∗Ej∗
+E(s+1)
+i[·] on the left hand side and cancelling out these terms leaves
+F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s+1)
+i∗Ej[−1]IT(s+1)(1)
+ij[s]
+− −−−−−−− → F(/angbracketleftλ,αi/angbracketright+s)
+i ∗Ej∗E(s+1)
+i[s].
+Now the map in (27) when restricted to the summand F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s+1)
+i∗Ej[−1] is by Corollary
+4.12 the composition
+F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s+1)
+i∗Ej[−1]IιI− − → F(/angbracketleftλ,αi/angbracketright+s)
+i ∗Ei∗E(s)
+i∗Ej[−s−1]
+d1
+iT(s)(1)
+ij− −−−−− → F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗Ej∗E(s)
+i[s].
+Up to multiple this is the same as the map γs(completing the proof). /square
+5.3.Step 3: Calculation of Eij∗Ti.BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 33
+5.3.1.Convolutions. First we recall the precise definition of a (right) convolution in a trian gu-
+lated category (see [GM] section IV, exercise 1).
+Let (A•,f•) =Anfn− →An−1→ ···f1− →A0be a sequence of objects and morphisms such that
+fi◦fi+1= 0. Such a sequence is called a complex. A (right) convolution of a com plex (A•,f•)
+is any object Bsuch that there exist
+(i) objects A0=B0,B1,...,B n−1,Bn=Band
+(ii) morphisms gi:Bi[−i]→Ai,hi:Ai→Bi−1[−(i−1)] (with h0=id)
+such that
+(28) Bi[−i]gi− →Aihi− →Bi−1[−(i−1)]
+is a distinguished triangle for each iandgi−1◦hi=fi. Such a collection of data is called a
+Postnikov system. Notice that in a Postnikov system we also have fi+1◦gi= (gi+1◦hi)◦gi= 0
+sincehi◦gi= 0.
+The convolution of a complex need not exist nor is it always unique. How ever, in the case of
+the complex
+...ds+1
+i− −− → Ts
+i(λ)ds
+i− → Ts−1
+i(λ)ds−1
+i− −− →...d1
+i− → T0
+i(λ)
+whereTs
+i(λ) =F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s)
+i(λ)[−s] we showed in [CKL3] that the right convolution exists,
+is unique and gives an object Ti(λ) which is invertible.
+5.3.2.Calculation. We denote the partial right convolution
+T≤s
+i(λ) := Conv/parenleftbigg
+Ts
+i(λ)ds
+i− → Ts−1
+i(λ)ds−1
+i− −− →...d1
+i− → T0
+i(λ)/parenrightbigg
+.
+Subsequently we have a standard exact triangle
+T≤s
+i→ T≤s+1
+iπ− → Ts+1
+i[s+1].
+Proposition 5.3. For any s≥ −1,Eij∗T≤s
+iis isomorphic to
+Cone/parenleftBigg
+T≤s+1
+i∗Ej[−1]IT(s+1)(1)
+ij[s]◦π− −−−−−−−−− → E j∗F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s+1)
+i[s]/parenrightBigg
+where the map above is the composition
+T≤s+1
+i∗Ej[−1]−→ Ts+1
+i∗Ej[s] =F(/angbracketleftλ+αj,αi/angbracketright+s+1)
+i ∗E(s+1)
+i∗Ej[−1]
+IT(s+1)(1)
+ij[s]
+− −−−−−−− → E j∗F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s+1)
+i[s].
+Proof.The proof is by induction on s. The base case is when s=−1 which follows since
+IT(0)(1)
+ij[−1] is an isomorphism and Eij∗T≤−1
+i= 0.
+Now we will prove the result for s+ 1 assuming it holds for s. The key is the following
+commutative diagram.34 SABIN CAUTIS AND JOEL KAMNITZER
+(29)
+F(/angbracketleftλ,αi/angbracketright+s+1)
+i ∗E(s+2)
+i∗Ej[−2]γs+1[−1]
+⊕IT(s+2)(1)
+ij[s]/d47/d47
+ds+2
+iI
+/d15/d15Ej∗F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s+1)
+i[s]
+⊕Ej∗F(/angbracketleftλ,αi/angbracketright+s+1)
+i ∗E(s+2)
+i[s]/d47/d47
+I⊕0
+/d15/d15Eij∗Ts+1
+i[s]
+f
+/d15/d15/d31/d31/d31
+T≤s+1
+i∗Ej[−1]
+/d15/d15IT(s+1)(1)
+ij[s]◦π/d47/d47Ej∗F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s+1)
+i[s]
+/d15/d15/d47/d47Eij∗T≤s
+i
+T≤s+2
+i∗Ej[−1]g/d47/d47/d95 /d95 /d95 /d95 /d95 /d95 /d95 /d95 /d95Ej∗F(/angbracketleftλ,αi/angbracketright+s+1)
+i ∗E(s+2)
+i[s+1]
+The top left square commutes because, by definition, γs+1[−1] = (IT(s+1)(1)
+ij[s])◦(ds+2
+iI). The
+first two rows and two columns are exact triangles – note that the s econd column is exact by
+the cancellation Lemma 4.1. The maps fandgare to be determined as explained below.
+In general, if one has a commutative square such as the upper left s quare in (29), then one
+can fill it with some maps f,gmaking all the squares commute and so that Cone( f)∼=Cone(g)
+(see Proposition 1.1.11 of [BBD]).
+Using the exact sequence T≤s
+i→ T≤s+1
+i→ Ts+1
+i[s+1] and Lemma 5.7 we find that
+Hom(T≤s+2
+i∗Ej[−1],Ej∗F(/angbracketleftλ,αi/angbracketright+s+1)
+i ∗E(s+2)
+i[s+1])
+∼=Hom(Ts+2
+i∗Ej[−1],Ej∗F(/angbracketleftλ,αi/angbracketright+s+1)
+i ∗E(s+2)
+i[s+1])
+is one dimensional and hence it must be spanned by IT(s+2)(1)
+ij[s+1]◦π. It is easy to see that
+g/ne}a⊔ionslash= 0 and thus g=IT(s+2)(1)
+ij[s+1]◦π(up to multiple).
+Similarly, by Lemma 5.7 we have
+Hom(Eij∗Ts+1
+i[s],Eij∗T≤s
+i)∼=Hom(Eij∗Ts+1
+i[s],Eij∗Ts
+i)∼=k.
+Sincef/ne}a⊔ionslash= 0 we find f=Ids+1
+i(up to multiple). Subsequently
+Eij∗T≤s+1
+i∼=Cone(Ids+1
+i)∼=Cone(f)∼=Cone(g)∼=Cone(IT(s+2)(1)
+ij[s+1]◦π)
+and the induction is complete. /square
+If we take s≫0 in Proposition 5.3 then we find that
+Corollary 5.4. Eij∗Ti∼=Ti∗Ej.
+The significance of this is that Eijis conjugate to Ej, namely Eij∼=Ti∗Ej∗T−1
+i. Thus, for
+anyk≥0 we can define
+E(k)
+ij(λ) :=Ti∗E(k)
+j(λ−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔αi)∗T−1
+iandF(k)
+ij(λ) :=Ti∗F(k)
+j(λ−/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔αi)∗T−1
+i.
+Corollary 5.5. Assuming both E(r)
+ijandF(r)
+ijare sheaves they generate a geometric categorical
+sl2action where the one parameter defomation of Y(λ)is the restriction of ˜Y(λ)to the subspace
+spanned by αi+αj. Moreover, we have
+E(r)
+ij∗Ti∼=Ti∗E(r)
+jandF(r)
+ij∗Ti∼=Ti∗F(r)
+j.
+Remark 5.6. In most examples one can verify directly that E(r)
+ijandF(r)
+ijare sheaves.BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 35
+Lemma 5.7. Ifk <0andt≥sork= 0andt > s+1
+Extk(Tt
+i∗Ej(λ),Ej∗F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗E(s)
+i(λ)) = 0andExtk(Eij∗Tt
+i,Eij∗Ts
+i) = 0.
+Whenk= 0andt=sort=s+1then both of these spaces are one-dimensional.
+Proof.Proposition 5.2 of [CKL3] claims that Extk(Tt
+i,Ts
+i) = 0 ift > s+1 andk≤0. However,
+what we showed there is a little stronger than that. One has
+Extk(Tt
+i,Ts
+i)∼=Extk(F(/angbracketleftλ+αj,αi/angbracketright+s)
+i L∗F(/angbracketleftλ+αj,αi/angbracketright+t)
+i ∗E(t)
+i[−t],E(s)
+i[−s])
+and one can repeatedly use Corollary 4.4 to write F(/angbracketleftλ+αj,αi/angbracketright+s)
+i L∗F(/angbracketleftλ+αj,αi/angbracketright+t)
+i ∗E(t)
+i[−t+s]
+as a direct sum/circleplusdisplay
+l≥0F(l)
+i∗E(s+l)
+i⊗kVl
+for some graded vector spaces Vl. Then one can rewrite/circleplustext
+l≥0Extk(F(l)
+i∗E(s+l)
+i⊗kVl,E(s)
+i) as
+/circleplusdisplay
+l≥0Extk(E(s+l)
+i⊗kVl,F(l)
+iR∗E(s)
+i)∼=/circleplusdisplay
+l≥0Extk(E(s+l)
+i,E(s+l)
+i⊗kV′
+l)
+for some (other) graded vector spaces V′
+l. Then the statement proven in Proposition 5.2 in
+[CKL3] is that each V′
+lis supported in degrees
+d≤ −(2l2+2l(−2b+t−s)+(t−s)2+2b(b−t+s)−1)
+whereb=/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+t−1. If we view this as a quadratic in lthen the discriminant simplifies to
+give−4(t−s)2+ 8 (as it happens it is independent of b). This is negative if t > s+1 which
+shows that V′
+lmust lie in negative degrees. If t=sort=s+1 then there is precisely one value
+oflfor which dis non-negative, namely l=b, and we show in [CKL3] that dim( V′
+b) = 1.
+Now consider
+Extk(Tt
+i∗Ej,Ej∗F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗E(s)
+i)∼=Extk(Tt
+i∗Ej,F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗Ej∗E(s)
+i)
+and suppose we are not in the case k= 0 and t=sort=s+1. Then, by the same argument
+as above, we get/circleplusdisplay
+l≥0Extk(F(l)
+i∗E(s+l)
+i∗Ej⊗kVl,Ej∗E(s)
+i[s]).
+By adjunction we can move the term F(l)
+ifrom the left side to the right side and try to simplify
+as before. But now, using Corollary 4.8, we get terms of the form
+E(l)
+i∗Ej∗E(s)
+i∼=E(l+s)
+i∗Ej⊗kH⋆(G(s,s+l−1))⊕Ej∗E(l+s)
+i⊗kH⋆(G(l,s+l−1))
+instead of terms of the form
+E(l)
+i∗E(s)
+i∗Ej∼=E(l+s)
+i∗Ej⊗kH⋆(G(s,s+l)).
+In the oldcasewe sawthat weend up with termsofthe formExtk(E(l+s)
+i∗Ej,E(l+s)
+i∗Ej[n]) where
+n <0. Since H⋆(G(s,s+l)) is supported in degrees −ls≤m≤lswhereas H⋆(G(s,s+l−1))
+in degrees −(l−1)s≤m≤(l−1)s(a difference of s) andH⋆(G(l,s+l−1)) in degrees
+−l(s−1)≤m≤l(s−1) (a difference of l) it must be that we end up with terms of the form
+Extk(E(l+s)
+i∗Ej,E(l+s)
+i∗Ej[s][n−s]) and Extk(E(l+s)
+i∗Ej,Ej∗E(l+s)
+i[s][n−l])
+wheren <0. These vanish by Lemma 4.5. If k= 0 and t=sort=s+ 1 then the right
+hand terms still vanish while the left hand terms also vanish except wh enl=bwhen it is one
+dimensional.36 SABIN CAUTIS AND JOEL KAMNITZER
+To deal with Extk(Eij∗Tt
+i,Eij∗Ts
+i) we take adjoints and work instead with Extk(Tt
+i∗Eij,Ts
+i∗
+Eij). Now the same argument as above leaves us with
+Extk(Tt
+i∗Eij,Ts
+i∗Eij)∼=⊕lExtk(E(s+l)
+i∗Eij,E(s+l)
+i∗Eij⊗V′
+l)
+whereV′
+lis supported in negative degrees unless k= 0 and t=sort=s+1 andl=bin which
+case it is one-dimensional in degree zero. The result now follows by Co rollary 4.13. /square
+5.4.Step 4: Proof that Tij∗Ti∼=Ti∗Tj.In analogy with Ts
+i(λ) we define
+Ts
+ij(λ) :=F(/angbracketleftλ,αi+αj/angbracketright+s)
+ij ∗E(s)
+ij(λ)[−s]∈D(Y(λ)×Y(λ−/an}bracke⊔le{⊔λ,αi+αj/an}bracke⊔ri}h⊔(αi+αj))).
+Notice that Ts
+ij∼=Ti∗Ts
+j∗T−1
+iso that
+Hom(Ts
+ij,Ts−1
+ij)∼=Hom(Ti∗Ts
+j∗T−1
+i,Ti∗Ts−1
+j∗T−1
+i)∼=Hom(Ts
+j,Ts−1
+j)∼=k
+where the last isomorphism follows from Lemma 5.9 (assuming Ts
+ij/ne}a⊔ionslash= 0 or equivalently Ts
+j/ne}a⊔ionslash= 0).
+We denote this map by ds
+ij(it is well defined only up to a non-zero multiple). One can also
+describe ds
+ijas before by the composition
+Ts
+ij(λ)∼=F(/angbracketleftλ,αi+αj/angbracketright+s)
+ij ∗E(s)
+ij[−s]
+ιι[−s]− −−− → F(/angbracketleftλ,αi+αj/angbracketright+s−1)
+ij ∗Fij[−/an}bracke⊔le{⊔λ,αi+αj/an}bracke⊔ri}h⊔−s+1]∗Eij∗E(s−1)
+ij[−(s−1)][−s]
+IεI− − → F(/angbracketleftλ,αi+αj/angbracketright+s−1)
+ij ∗E(s−1)
+ij[−s+1]∼=Ts−1
+ij(λ).
+Proposition 5.8. The complex
+··· → Ts
+ij(λ)ds
+ij− − → Ts−1
+ij(λ)ds−1
+ij− −− →...d1
+ij− − → T0
+ij(λ)
+has a unique convolution which we denote Tij(λ). Moreover,
+Tij∗Ti∼=Ti∗Tj.
+Proof.The key is the following commutative diagram
+... /d47/d47Ts
+ijds
+ij/d47/d47
+∼=
+/d15/d15Ts−1
+ij
+∼=
+/d15/d15/d47/d47...
+... /d47/d47Ti∗Ts
+j∗T−1
+iIds
+jI/d47/d47Ti∗Ts−1
+j∗T−1
+i/d47/d47...
+where one needs to choose appropriate multiples of the vertical iso morphisms. This is a conse-
+quence of the fact that Hom( Ts
+ij,Ti∗Ts−1
+j∗T−1
+i)∼=Hom(Ts
+ij,Ts−1
+ij)∼=kand that ds
+ij/ne}a⊔ionslash= 0/ne}a⊔ionslash=ds
+j.
+NowT•
+jhas aunique convolutionso Ti∗T•
+j∗T−1
+ihas a unique convolutionand hence sodoes T•
+ij.
+Finally, the commutativity ofthe diagramalsoimplies that the convolut ionsmust be isomorphic:
+i.e.Tij∼=Ti∗Tj∗T−1
+i. /square
+Lemma 5.9. IfTs
+i/ne}a⊔ionslash= 0ands≥1then the space of maps Hom(Ts
+i,Ts−1
+i)is one-dimensional.
+Proof.We must show that
+Hom(F(/angbracketleftλ,αi/angbracketright+s)
+i ∗E(s)
+i[−s],F(/angbracketleftλ,αi/angbracketright+s−1)
+i ∗E(s−1)
+i[−s+1])∼=k.
+This essentially done in [CKL3] proof of Proposition 5.2. There we show that
+Hom(Ts
+i[k−1],Ts−k
+i) = 0BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 37
+fork≥2 whereas we are interested in the left side when k= 1. The argument there shows
+that when k= 1 the left side is equal to a direct sum of terms Hom( E(a−j+s)
+i,E(a−j+s)
+i[l]) where
+a=λ+s−1,j= 0,...,aandl≤ −2(j−a)(j−a+1). Thus l <0 and these terms vanish
+unlessj=a. Ifj=athenl= 0 and the argument shows we get exactly one such term and then
+Hom(Ts
+i,Ts−1
+i)∼=Hom(E(s)
+i,E(s)
+i)∼=k
+where the second isomorphism follows by Lemma 4.5). /square
+5.5.Step 5: Proof of braid relation. Proposition 5.8 claims that Tij∗ Ti∼=Ti∗ Tjwhich
+follows from the fact that Eij∗Ti∼=Ti∗Ej.
+Now the same proof can be used to show that Tj∗ Tij∼=Ti∗ Tj. Namely, Step 1 is a
+consequence of the analogous version of Corollary 4.17 which compu tesF(s)
+i∗Eij(this in turn
+can be traced back to follow formally from Lemma 4.9 and Corollary 4.4 – see Remarks 4.18 and
+4.15). Then Step 2 and 3 follow formally (they also use Lemma 4.9 and th ere is some vanishing
+one needs to check which is a formal consequence of the Lie algebra relations). This shows that
+Tj∗Eij∼=Ei∗Tjand then Step 4 follows as before.
+Putting these two identities together we get the braid relation
+Ti∗Tj∗Ti∼=Tj∗Tij∗Ti∼=Tj∗Ti∗Tj.
+Finally, if i,j∈Iare not joined by an edge then any EiorFicommutes with any EjorFj.
+SinceTiis build out of Ei’s andFi’s andTjis built out of Ej’s andFj’s we get the commutativity
+relationTi∗Tj∼=Tj∗Ti. This concludes the proof of Theorem 2.10.
+6.Braiding via strong categorical g-actions
+Strong categorical g-actions have been defined by Khovanov and Lauda in [KL1, KL2, KL3 ]
+and independently by Rouquier in [Ro2]. Their definitions are very simila r though not identical.
+One should think of the geometric categorical g-action introduced here as a geometric analogue
+of their definition which is easier to check in practice.
+In [CKL2] we prove that when g=sl2a geometric g-action implies a strong g-action in the
+sense of Rouquier. There is good reason to believe the same is true f or arbitrary (simply-laced)
+Kac-Moody Lie algebras g. Nevertheless, in this paper we show that the braid relation follows
+directly from the geometric g-action.
+On the other hand, our proof of Theorem 2.10 works to show that a strongg-action gives a
+braid group action. In fact it seems that not all the axioms of a stro nggaction are needed to
+obtain the braid group action. We will now explain this, starting with a s implified version of
+Rouquier’s definition.
+A (simplified) strong categorical gaction consists of
+(i) For each weight λwe have a triangulated category D(λ).
+(ii) Exact functors E(r)
+i(λ) :D(λ)→ D(λ+rαi) andF(r)
+i(λ) :D(λ+rαi)→ D(λ).
+Remark 6.1. We actually need a little more than this. For each pair λ,λ′there should be
+a triangulated category D(λ,λ′) and an additive functor Φ : D(λ,λ′)→Hom(D(λ),D(λ′)),
+denoted E /maps⊔o→ΦE. We assume that Φ commutes with the cohomological shift [1]. We furt her
+assume that there is an associative monoidal structure ∗:D(λ,λ′)× D(λ′,λ′′)→ D(λ,λ′′)
+such that Φ intertwines this operation with the composition of funct ors. Moreover, if E →
+F → G → E [1] is a distinguished triangle in D(λ,λ′), then for any A∈ D(λ), we require that
+ΦE(A)→ΦF(A)→ΦG(A)→ΦE(A)[1] be a distinguished triangle in D(λ′).38 SABIN CAUTIS AND JOEL KAMNITZER
+Finally, we assume that D(λ,λ′) is idempotent complete and that the hom space between
+any two objects is finite dimensional. This way D(λ,λ′) satisfies the Krull-Schmidt property.
+Moreover, we assume that for any non-zero E ∈ D(λ,λ′) we have E∼=E[k]⇒k= 0.
+We then require the following relations:
+(i) For any weight λ, Hom(id D(λ),idD(λ)[l]) = 0 ifl <0 while End(id D(λ)) =k·id.
+(ii) (a) E(r)
+i(λ)R=F(r)
+i(λ)[r(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+r)]
+(b)E(r)
+i(λ)L=F(r)
+i(λ)[−r(/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔+r)].
+(iii)
+Ei◦E(r)
+i(λ)∼=E(r+1)
+i(λ)⊗kH⋆(Pr)∼=E(r)
+i◦Ei(λ)
+whileEi◦Ej∼=Ej◦Eiifi,j∈Iare not joined by an edge and
+Ei◦Ej◦Ei∼=E(2)
+i◦Ej⊕Ej◦E(2)
+i
+ifi,j∈Iare joined by an edge.
+(iv) If/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≤0 then
+Fi(λ)◦Ei(λ)∼=Ei(λ−αi)◦Fi(λ−αi)⊕id⊗kH⋆(P−/angbracketleftλ,αi/angbracketright−1)
+while if/an}bracke⊔le{⊔λ,αi/an}bracke⊔ri}h⊔ ≥0 then
+Fi(λ−αi)◦Ei(λ−αi)∼=Ei(λ)◦Fi(λ)⊕id⊗kH⋆(P/angbracketleftλ,αi/angbracketright−1).
+Ifi/ne}a⊔ionslash=j∈IthenFj◦Ej∼=Ei◦Fj.
+along with the following natural transformations (2-morphisms):
+(i)Xi:Ei(λ)[−1]→Ei(λ)[1] for each i∈Iand weight λ
+(ii)Tij:Ei◦Ej(λ)[/an}bracke⊔le{⊔αi,αj/an}bracke⊔ri}h⊔]→Ej◦Ei(λ) for any i,j∈Iand weight λ
+with relations
+(i) For each i∈ItheXi’s andTii’s satisfy the nil affine Hecke relations:
+(a)T2
+ii= 0
+(b) (ITii)◦(TiiI)◦(ITii) = (TiiI)◦(ITii)◦(TiiI) as endomorphisms of Ei◦Ei◦Ei.
+(c) (XiI)◦Tii−Tii◦(IXi) =I=−(IXi)◦Tii+Tii◦(XiI) as endomorphisms of
+Ei◦Ei.
+(ii) Ifi/ne}a⊔ionslash=j∈Iare joined by an edge then Tji◦Tij=XiI+IXjas endomorphisms of
+Ei◦Ej.
+This list of 2-morphisms and the relations appear in both the Khovano v-Lauda and Rouquier
+definitions and it turns out they suffice to prove the braid relations.
+Remark 6.2. Relation (i) above is the only finiteness condition we need. It follows fo rmally
+that the space of maps between any two compositions of E’s andF’s is finite. The reason we
+needed the stronger finiteness condition (i) in the definition of geom etric categorical actions
+(section 2.2) is because in that case we do not require that E(r)
+i∗EiandFi∗Eisplit as a direct
+sum (the condition is only at the level of cohomology). The argument that they split requires
+the cancellation property which in turn requires us to know that all m aps are finite dimensional.
+Theorem 6.3. A categorical strong g-action as defined above gives rise to equivalences Ti(i∈I)
+satisfying the braid relations.
+Proof.The fact that we have the nil affine Hecke relations means that for e achi∈Iwe have a
+strong categorical sl2action (in the sense of [CKL3]) so we can construct equivalences Ti. What
+remains is to show that they braid, which we do by running again throu gh the proof of Theorem
+2.10.BRAIDING VIA GEOMETRIC LIE ALGEBRA ACTIONS 39
+The more complicated relations among compositions of E’s andF’s (such as Propositions 4.2,
+4.3, 4.7 and Corollary 4.8) were all formal arguments which work in any abstract (idempotent
+complete) category. The same goesfor the calculationsof Hom-sp aces(Lemma 4.5and Corollary
+4.6).
+The only place where something more interesting happens is in the pro of of Lemma 4.9.
+Notice that the Tijfrom that Lemma and our Tijdefined above must be equal (up to non-zero
+scalars) since Hom( Ei◦Ej[−1],Ej◦Ei)∼=k(by Lemma 4.5).
+The whole proof of Lemma 4.9 comes down to showing that the map
+Ej◦E(2)
+i[−1]⊕E(2)
+i◦Ej[−1]∼=Ei◦Ej◦Ei[−1]TijI− −− →Ej◦Ei◦Ei∼=Ej◦E(2)
+i[−1]⊕Ej◦E(2)
+i[1]
+induces an isomorphism on the Ej◦E(2)
+i[−1] summand. In 4.9 we use the fact that Eijdeforms
+to˜Eijto show this. In the abstract setting we use instead the relation Tji◦Tij=XiI+IXj.
+More precisely, suppose the map does not induce an isomorphism. Th is means it must induce
+zero since End( Ej◦E(2)
+i) =k·id. But then, the composition
+Ei◦Ej◦Ei[−1]TijI− −− →Ej◦Ei◦EiITii−−→Ej◦Ei◦Ei[−2]
+must be zero. On the other hand, pre-composing with ( TjiI)◦(ITii) we get
+(ITii)◦(TijI)◦(TjiI)◦(ITii) = (ITii)(XjII+IXiI)◦(ITii)
+= (XjII)◦(ITii)2+(ITii)◦((ITii)◦(IIXi)+I)
+=ITii
+where we use T2
+ii= 0 twice in the last equality. This is non-zero (contradiction).
+This proves Lemma 4.9. Then Corollaries 4.10 and 4.12 follow by formal a rguments. Lemma
+4.14 is a formal consequence of Lemma 4.9 and Corollary 4.4 and Corolla ry 4.17 follows from
+Lemma 4.14 and Corollary 4.4.
+This brings us up to section 5. One can easily check that the argumen ts there are formal
+consequences of the Lie algebra relations and results from section 4. So the braid relation
+follows. /square
+Remark 6.4. In the setting of (non-categorified) quantum groups, there is a b raid group action
+onUq(g) (constructed by Lusztig) which is compatible with the braid group a ction on repre-
+sentations. Hence we would expect there should be a braid group ac tion on the 2-category of
+Rouquier/Khovanov-Lauda which is compatible with the above action of the braid group on the
+representations.
+From the proof of the main theorem in this paper, we would expect th at generators σiof this
+braid group action would obey the following two conditions
+σi(Ej) =Ti◦Ej◦T−1
+i= Cone(Ei◦Ej[−1]→Ej◦Ei) ifi,j∈Iare joined by an edge (30)
+σi(Ei) =Ti◦Ei◦T−1
+i=Fi(up to a shift) (31)
+In a forthcoming paper, Khovanov-Lauda will construct a braid gr oup action on the 2-
+categories from [KL1, KL2, KL3] which satisfies (30) and (31) abov e. Our braid group action
+will then be compatible with theirs.
+References
+[A] R. Anno, Spherical functors; math.CT/0711.4409 .
+[BBD] A.Beilinson, J.Bernstein and P.Deligne, Faisceaux p ervers.Analyseettopologie surlesespaces singuliers
+(I),Ast´ erisque 100(1982).
+[BMR] R. Bezrukavnikov, I. Mirkovic, and D. Rumynin, Singul ar localization and intertwining functors for
+semisimple Lie algebras in prime characteristic, Nagoya Math. J. 184(2006), 1–55.40 SABIN CAUTIS AND JOEL KAMNITZER
+[BN] D. Bar-Natan, Fast Khovanov homology computations, J. Knot Theory Ramifications ,16(2007) no. 3,
+243–255; math.GT/0606318 .
+[BS] P Balmer and M. Schlichting, Idempotent completion of t riangulated categories, J. Algebra ,236(2001),
+no. 2, 819–834.
+[C] S. Cautis, Equivalences and stratified flops, Compositio Math. (to appear); math.AG/0909.0817 .
+[CK] S. Cautis and J. Kamnitzer, Knot homology via derived ca tegories of coherent sheaves II, sl(m) case,
+Invent. Math. 174(2008), no. 1, 165–232. math.AG/0710.3216
+[CKL1] S. Cautis, J. Kamnitzer and A. Licata, Categorical ge ometric skew Howe duality, Invent. Math. 180
+(2010), no. 1, 111–159. math.AG/0902.1795 .
+[CKL2] S. Cautis, J. Kamnitzer and A. Licata, Coherent sheav es and categorical sl2actions, Duke Math. J. 154
+(2010), no. 1, 135–179; math.AG/0902.1796 .
+[CKL3] S. Cautis, J. Kamnitzer and A. Licata, Derived equiva lences for cotangent bundles of Grassmannians via
+categorical sl2actions, J. Reine Angew. Math. (to appear); math.AG/0902.1797 .
+[CKL4] S. Cautis, J. Kamnitzer and A. Licata, Coherent sheav es on quiver varieties and categorification;
+arXiv:1104.0352 .
+[CR] J. Chuang and R. Rouquier, Derived equivalences for sym metric groups and sl2-categorification, Ann. of
+Math.167(2008), no. 1, 245–298; math.RT/0407205 .
+[GM] S. Gelfand and Y. Manin, Methods of homological algebra, second edition , Springer-Verlag, 2003.
+[Ho] R.P. Horja, Derived Category Automorphisms from Mirro r Symmetry, Duke Math. J. 127(2005), 1–34;
+math.AG/0103231 .
+[Hu] D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry , Oxford University Press, 2006.
+[HT] D. Huybrechts and R. Thomas, P-objects and autoequivalences of derived categories, Math. Res. Lett.
+13(2006), no. 1, 87–98; math.AG/0507040 .
+[KL1] M. Khovanov and A. Lauda, A diagrammatic approach to ca tegorification of quantum groups I, Repre-
+sent. Theory 13(2009), 309–347; math.QA/0803.4121 .
+[KL2] M. Khovanov and A. Lauda, A diagrammatic approach to ca tegorification of quantum groups II, Trans.
+Amer. Math. Soc. 363(2011), 2685–2700; math.QA/0804.2080 .
+[KL3] M. Khovanov and A. Lauda, A diagrammatic approach to ca tegorification of quantum groups III, Quan-
+tum Topology 1, Issue 1 (2010), 1–92; math.QA/0807.3250 .
+[KT] M. Khovanov and R. Thomas, Braid cobordisms, triangula ted categories, and flag varieties, Homology,
+Homotopy, Appl. 9(2007), 19–94; math.QA/0609335 .
+[Ric] S. Riche, Geometric braid group action on derived cate gory of coherent sheaves, Represent. Theory 12
+(2008), 131–169.
+[Rin] C. M. Ringel, Tame Algebras and Integral Quadratic For ms,Lecture Notes in Mathematics ,1099(1984),
+Springer Berlin.
+[Ro1] R. Rouquier, Categorification of sl2and braid groups, Trends in representation theory of algebras and
+related topics (2006) Amer. Math. Soc., 137–167.
+[Ro2] R. Rouquier, 2-Kac-Moody algebras; math.RT/0812.5023 .
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+108, (2001), 37–108; math.AG/0001043 .
+E-mail address :scautis@math.columbia.edu
+Department of Mathematics, Columbia University, New York, NY
+E-mail address :jkamnitz@math.toronto.edu
+Department of Mathematics, University of Toronto, Toronto, ON Canada
\ No newline at end of file
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+arXiv:1001.0620v1  [cond-mat.str-el]  5 Jan 2010Metal-insulator transition in quarter-filled Hubbard mode l on triangular lattice and
+its implication for the physics of Na0.5CoO2
+Tao Li
+Department of Physics, Renmin University of China, Beijing 100872, P.R.China
+(Dated: November 18, 2018)
+The metal-insulator transition of the quarter-filled Hubba rd model on triangular lattice is studied
+at the mean field level. We find a quasi-one dimensional metall ic state with a collinear magnetic
+order competes closely with an insulating state with a non-c oplanar magnetic order for both signs
+of the hopping integral t. In the strong correlation regime( U/|t| ≫1), it is found that the metal-
+insulator transition of the system occurs in a two-step mann er. The quasi-one dimensional metallic
+state with collinear magnetic order is found to be stable in a n intermediate temperature region
+between the paramagnetic metallic phase and the non-coplan ar insulating phase. Possible relevance
+of these results to the physics of metal-insulator transiti on inNa0.5CoO2is discussed.
+PACS numbers:
+NaxCoO2is a remarkable transition metal oxide sys-
+tem in which strong electronic correlation and geomet-
+ric frustration interplay with each other. Unconven-
+tional superconductivity, novel magnetic and charge or-
+dered states and state with peculiar Curie-Weiss metal
+behavior are observed in this system at different carrier
+concentrations[1–8]. To date, most of these observations
+remains not well understood.
+The quarter-filled system, namely Na0.5CoO2is es-
+pecially interesting[3, 6, 8]. It divides the phase dia-
+gram of the NaxCoO2system into two parts of quali-
+tatively different nature. For x <0.5, the system ex-
+hibits conventional metallic behavior in the normal state
+and with water intercalation a superconducting state
+with unknown paring symmetry in a small doping region
+aroundx= 0.3[1]. While fore x >0.5, the system ex-
+hibits Curie-Weiss metal behavior at high temperature
+and a weak static magnetic order at low temperature
+with ferromagnetic intra-layer spin correlation[4, 5, 7].
+Right at x= 0.5, the observations are even more strik-
+ing. The system undergoes two phase transitions at 88K
+and 53K[3, 6]. At the 88K transition, the system de-
+velops a magnetic order with a ordering wave vector
+q=1
+2(/vectorb1+/vectorb2)[9]. However, the longitudinal resistivity
+of the system seems to be totally ignorant of the transi-
+tion. While at the 53K transition, the system enters the
+true insulating state as signaled by the rapid increase of
+the longitudinal resistivitybelow this temperature. How-
+ever, boththe peakpositionand theintensityofthemag-
+netic Bragg peak developed at 88K seems to be ignorant
+of the transition at 53K. The observation of two transi-
+tions and the apparent mutual independence of the mag-
+netic order and the charge transport have arose many
+interest in the field. However, this is still not the whole
+story. In fact, it is found that Hall resistivity begins to
+drop abruptly around the 88K transition which indicates
+that it is not a transition in the spin sector only[3].
+No coherent picture exists yet on the low tempera-
+ture phase transitions of the Na0.5CoO2system. It isgenerally anticipated that the ordering of the Na ions
+outside the CoO2plane may plays an important role in
+the low temperature physics of this system[3, 6, 11, 13].
+Charge ordering(induced by the Sodium order) in the
+CoO2plane is invoked in many works to explain the
+multiple phase transitions. In this paper, we suggest a
+possible picture for the multiple phase transitions of the
+Na0.5CoO2system as an intrinsic property of a quarter-
+filled stronglycorrelated system on the triangularlattice.
+We backup our reasoning with a mean field study of the
+metal-insulator transition of the quarter-filled Hubbard
+model on triangular lattice. We find this seemingly over-
+simplified model does exhibit two successive transitions
+at quarter filling in its way toward the low temperature
+insulating state. In our picture, the magnetic transition
+at 88K transform the system into an quasi-one dimen-
+sional metallic state through a spin ordering induced di-
+mensional reduction process. This quasi-onedimensional
+metal can then transforminto the true insulatingstate at
+53K in various possible ways. We thus predict that the
+intermediate phase between 53K and 88K to be a state
+with one dimensional transport property.
+FIG. 1: (a)The magnetic structure of the proposed state for
+the intermediate temperature region between 53K and 88K.
+Note the state has no charge order and all lattice sites are st ill
+uniformly quarter-filled. (b)The insulating state with a fo ur
+sublattice and non-coplanar magnetic structure predicted by
+the mean field theory of the quarter-filled Hubbard model
+on triangular lattice. The magnetic moments on the four
+sublattices point to the four corners of a perfect tetrahedr on
+as shown in the inset.
+The state we proposefor the intermediate temperature
+region between 53K and 88K is shown schematically in
+Figure 1a. In this state, the system self-organizes into a
+seriesofchainswithferromagneticorderonthem. Neigh-
+boring chains are antiferromagnetically correlated with
+each other. Unlike other proposals, there is no charge or-
+dering in this state and all lattice sites are still quarter-2
+filled uniformly and thus the ferromagnetic chains are
+metallic. Itisinterestingtonotethatthismetallicityisof
+one dimensional nature as any inter-chain electron hop-
+ping must cross the spin gap. Thus the spin ordering in
+suchaparticularpatternrendersthesystemaonedimen-
+sional metal and so we dub it as a spin ordering induced
+dimensional reduction process. Similar dimensional re-
+duction process also occurs in other condensed matter
+system with multiple low energy degree of freedoms. The
+most well known example is the orbital ordering induced
+dimensional reduction in ruthenium oxides[15].
+The magnetic structure of the state shown in Fig.1a is
+given by </vectorSi>=/vectormexp(iQ3·Ri) withQ3=1
+2(/vectorb1+/vectorb2).
+The states with Q1=1
+2/vectorb1andQ2=1
+2/vectorb2are degener-
+ate with it by lattice symmetry. All the three states
+are consistent with the neutron scattering measurement
+and the real system will contain domains of all three
+kinds. Since the ferromagnetic chains are still metallic,
+the longitudinal resistivity of the system is not expected
+to change dramatically across the 88K transition. How-
+ever, the transverse transport is totally different. As the
+system becomess quasi-one dimensional in charge trans-
+portstate, we expect the Hallresistivityto dropabruptly
+at the 88K transition, in accordance with the experimen-
+tal observation. Thus the state shown in Fig.1a has the
+correct characteristics to account for the phenomenology
+of the intermediate phase between 53K and 88K.
+With further lowering of temperature, the quasi-one
+dimensional intermediate state will eventually become a
+true insulator. It is important to note that the inter-
+mediate state has an enlarged unit cell with two lattice
+sites. Thus the system is half-filled below the 88K tran-
+sition. To arrive at the true insulating state, the sys-
+tem should lower its symmetry further to have a unit
+cell with at least four sites. Through mean field cal-
+culation on the Hubbard model on triangular lattice,
+we find a state with a four sublattice structure and
+non-coplanar spin configuration (see Fig.1b) that meet
+this requirement. This state has no charge order either
+and the magnetic structure can be written as </vectorSi>=
+/vectorm1exp(iQ1·Ri)+/vectorm2exp(iQ2·Ri)+/vectorm3exp(iQ3·Ri).
+Here the three vectors /vectorm1,/vectorm2,/vectorm3are of same length
+and are mutually orthogonal. The magnetic moments in
+the four sublattices points to the four corners of a per-
+fect tetrahedron. This state has the same set of magnetic
+Bragg peaks as the collinear state shown in Fig.1a if the
+three kinds of domains of the collinear state appear with
+the same probability. The mean field theory presented
+in this paper also indicates that the transition between
+collinear state and the four sublattice insulating state
+leaves the intensity of the magnetic Bragg peak almost
+intact. This is in accordance with the observation at the
+53K transition.
+It should be noted that both kinds of order shown in
+Fig.1 have already been predicted in a mean field studyof the quarter-filled Hubbard model on triangular lattice
+with anegative hopping integral[13]. In that work, both
+kinds of order develop as a result of the nesting property
+of the Fermi surface and exist for any non-zero interac-
+tion. However, the Fermi surface of the Na0.5CoO2is al-
+mostfeatureless. Aswillbeshownbelow, the Na0.5CoO2
+system should be described as a quarter-filled system
+with apositive hopping integral. Since the triangular
+lattice has no particle-hole symmetry, any magnetic or-
+der can develop only for interaction greater than certain
+critical value. Indeed, we find the magnetic structures
+shown in Fig.1 exist only in the strong correlationregime
+forNa0.5CoO2.
+We now present the results of the mean field study of
+the metal-insulator transition in the quarter-filled Hub-
+bard model on triangularlattice. The Hamiltonian of the
+model reads
+H =−t/summationdisplay
+<i,j>,σ(c†
+i,σcj,σ+h.c.)+U/summationdisplay
+ini,↑ni,↓.(1)
+It is generally believed that the NaxCoO2system re-
+sides in the strongly correlated regime withU
+t>>1[10].
+Since the triangular lattice has no particle-hole symme-
+try, the sign of the hopping integral tis important. For
+NaxCoO2system, it is negative as inferred from ARPES
+measurement[12]. In Na0.5CoO2, eachCosite hosts 1 .5
+electrons on average is thus a3
+4-filled system with a neg-
+ative hopping integral. After a particle-hole transforma-
+tion, it is equivalent to a quarter-filled system with a
+positive hopping integral. For purpose of comparison,
+we also consider the case of quarter-filled system with a
+negative hopping integral which is studied already in Ref
+12.
+From the weak coupling point of view, a quarter-filled
+system on triangular lattice with a positive hopping in-
+tegral has no speciality at all as compared to a general
+incommensurate filling. The Fermi surface has a almost
+perfect rounded shape. Thus, it is hard to guess what
+would happen in the strong correlationregime from weak
+coupling analysis. To the contrary, a quarter-filled sys-
+tem on triangular lattice with a negative hopping inte-
+gral is special in the sense that the chemical potential
+rests just on the Van Hove singularity of the density of
+state(µ=−2|t|at zero temperature). The Fermi surface
+is nested with nesting vectors Q1,Q2, andQ3. As the
+result of the nesting property of the Fermi surface, the
+system develops magnetic order at an arbitrarily small
+U. In Ref. 12, the collinear state shown in Fig. 1a
+and the non-coplanar state shown in Fig.1.b are found to
+be the most favorable choice for the magnetic structure.
+Among the two, the non-coplanar state is found to be
+slightly more favorable.
+To have an idea on what would happen in the strong
+correlation regime for the quarter-filled Hubbard system
+with a positive hopping integral, we have conducted a
+unrestricted mean field search at zero temperature on a3
+12×12 lattice. The search is done in the mean field
+space with the local density niand the three compo-
+nents of local spin density /vectorSias variational parameters.
+We have used both the conjugate gradient method and
+the simulated annealing method to perform the search
+and used more than 100 random initial configurations
+for the local density and local spin density. We find the
+search always converge to the two configurations shown
+in Fig.1(up to global spin rotations and point group op-
+erations)forlargeenough valueofU
+t. Thus, althoughthe
+quarter-filled system with positive hopping integral has
+no nesting property, it has the same magnetic ordering
+pattern in the strong correlation regime as the negative
+hopping system.
+The meanfield equationsforthe collinearstate isgiven
+by
+1 =2U
+N/summationdisplay
+k/bracketleftbigg(ξk−E+
+k)f(E+
+k)
+(ξk−E+
+k)2+(Um)2+(ξk−E−
+k)f(E−
+k)
+(ξk−E−
+k)2+(Um)2/bracketrightbigg
+,
+(2)
+inwhichthesumislimited inthemagneticBrillouinzone
+andξk=−2t(coskx+cosk y+cos(k x+ ky))−µis the
+dispersion on the triangular lattice. Here we have used
+the convention for the momentum that k= kx/vectorb1+kx/vectorb2.
+In this convention, the first Brillouin zone of the trian-
+gular lattice is given by (k x,ky)∈[−π,π]⊗[−π,π] and
+the magnetic Brillouin zone of the collinear state is given
+by [0,π]⊗[−π,π].E±
+k=ξk+ξk+Q3±√
+(ξk−ξk+Q3)2+(Um)2
+2
+is quasiparticle energy in the magnetic ordered state.
+In the non-coplanar insulating state, the magnetic
+Brillouin zone reduces further into the region given by
+[0,π]⊗[0,π]. The mean field equation for the non-
+coplanar state is given by
+m=1
+N/summationdisplay
+k,nWn
+kf(En
+k), (3)
+in which Wn
+k=1
+2√
+3/summationtext
+α,βφn∗
+α(k)Mα,βφn
+β(k) ,α,β,n=
+1···8 and the sum is limited in the magnetic Brillouin
+zone.φn
+α(k) is theα−thcomponent of the eigenvector
+for the mean field Hamiltonian with eigenvalue En
+k. The
+mean field Hamiltonian matrix is of the following form
+HMF=
+ξkI M 1M∗
+2M2
+M1ξk+Q1I M 2M∗
+2
+M∗
+2M2ξk+Q2I M 1
+M2M∗
+2M1ξk+Q3I
+,(4)
+here I is a 2 ×2 identity matrix, M 1=−Um
+3/parenleftbigg1√
+2√
+2−1/parenrightbigg
+,
+M2=−Um
+3/parenleftigg
+1−1+i√
+3√
+2
+−1−i√
+3√
+2−1/parenrightigg
+. The matrix Misgiven by
+M=−3
+Um
+0 M1M∗
+2M2
+M10 M2M∗
+2
+M∗
+2M20 M1
+M2M∗
+2M10
+.(5)
+FIG. 2: The order parameters of magnetic ordered states at
+zero temperature as functions of the coupling strength U/|t|
+for both signs of hopping integral. (a) t >0, (b)t <0.
+Fig.2 shows the order parameter mas a function of
+U/|t|at zero temperature for both signs of the hopping
+integral. For negative t, the magnetic order exist for
+all non-zero Uas a result of the nesting property of the
+system. While for positive t, the magnetic order exists
+only forU/|t|> Uc/|t|= 7.2. For both sign of t, the non-
+coplanar state is slightly more stable than the collinear
+state at zero temperature.
+Fig.3showsthe meanfield phasediagramatfinite tem-
+perature for both signs of the hopping integral. At fi-
+nite temperature, the gapless Fermion excitation in the
+collinear state will contribute more entropy than the
+gappedFermionexcitationinthenon-coplanarinsulating
+state. We find for sufficiently large U/|t|, the collinear
+state will become more stable than the non-coplanar
+state above a critical temperature at which a first order
+transition between the two occurs. In the phase diagram,
+the first order transition line between the collinear state
+and the non-coplanar state end at a critical end point on
+the magnetic-paramagnetic transition line. For positive
+t, the critical end point locates at U/|t|= 7.9, while for
+negative t, it locates at a smaller value of U/|t|= 5.5.
+Thus for both sign of the hopping integral, the collinear
+state is stable only in the strong correlation regime.
+It is interesting to note that the transition between the
+collinear state and non-coplanar state, though is of first
+order in nature, has almost no observable effect on the
+peak position and intensity of the magnetic Bragg peak.
+As we have explained above, the non-coplanar state has
+the same sets of magnetic Bragg peak as the collinear
+state, if the three domains of the latter as mixed with
+equal probability. In Fig.4, we plot the change of the
+magnitude of the order parameter at the collinear - non-
+coplanar phase transition as a function of U/|t|. We find
+the change to be always positive but its absolute value
+is always less than 1 .5% of the ordered moment for all
+value ofU/|t|and for both signs of t. Thus we conclude
+that the neutron scattering experiments are not sensitive
+to the collinear-non-coplanar transition.
+However, as the system opens a full gap abruptly on
+the quasiparticle spectrum at the collinear-non-coplanar
+transition, the longitudinal resistivity is expected to in-
+crease dramatically. Below the transition point, the sys-
+tem will enter the true insulating state. Thus in this sys-
+tem, the metal-insulator transition occurs in a two-step4
+manner. The quarter-filled system first breaks the lattice
+symmetry by a two-fold enlargement of the unit cell and
+entersahalf-filled quasi-onedimensionalmetallicstate at
+the paramagnetic-magnetic transition and then reduces
+its symmetry further by again a two-fold enlargement of
+the unit cell at the collinear-non-coplanar transition and
+enters a integer-filled insulating state. It is interesting
+to note that such a multi-step transition behavior is not
+limited to the quarter-filled system. At a more general
+filling fraction1
+qp, the system can first enter a1
+p(1
+q)-filled
+intermediate metallic phase by a q(p)-times enlargement
+of unit cell and then arrive at the integer-filled true in-
+sulating state by a second transition with a p(q)-times
+enlargement of the unit cell.
+FIG. 3: The finite temperature phase diagram of the model
+for both signs of the hopping integral.(a) t >0, (b)t <0.
+FIG. 4: The jump of the order parameter at the collinear-
+non-coplanar transition point as a function of U/|t|for both
+signs of the hopping integral.
+In the intermediate state between 53K and 88K, the
+collinear spin ordering induces dimensional reduction in
+the charge transport. In fact, all metal-insulator tran-
+sition can be viewed as a dimensional reduction process
+in the charge sector: an insulator can be viewed as a
+system in which the charge transport is quenched in all
+dimension. Such a dimensional reduction process in the
+charge sector can occur either directly, or like our exam-
+ple, in a multi-step manner in spatial dimension larger
+than 2. In the intermediate state , the charge trans-
+port is quenched in some dimension but remain active
+in other dimension. Such dimensional reduction in the
+bulk of the system can occur in condensed matter sys-
+tem with multiple low energy degree of freedoms. One
+famous example is the orbital ordering induced dimen-
+sional reduction in ruthenium oxides in which the elec-
+tronhoppingis blockedexceptin onedirection asaresult
+of the symmetry property of the electron orbital wave
+function. Another example is the stripe-like structure in
+some cuprates superconductors in which the charge mo-
+tion becomes effectively one dimensional as a result of
+microscopic phase separation[14].
+Finally, we discuss the relevance of our results to the
+physics of the Na0.5CoO2system. The quasi-one di-
+mensional metallic state with collinear magnetic order
+has the proper characteristics to account for the exper-imental observations between 53K and 88K. Thus it is
+not unreasonable to assign this state to the intermedi-
+ate temperature region. However, as our model is purely
+two-dimensional, additional three dimensional coupling
+should be invoked to explain the finite temperature ex-
+istence of the magnetic order in the experiment. Here
+we assume that such three dimensional coupling will not
+alter the essential physics discussed in this paper. Fur-
+thermore, as our treatment of the model is limited to the
+mean field level, it is important to know how fluctuation
+effect will change the results. We leave this important is-
+sueto future study. As to the insulatingstate below53K,
+we note that in real system, the perfect hexagonal sym-
+metry maybe broken by perturbation such as the sodium
+order potential, or spontaneously through the Peries in-
+stability process of the quasi-one dimensional intermedi-
+ate state, the real system have many different ways to
+become a true insulator. It is important to make sure
+that the non-coplanar state studied in this paper is sta-
+ble with respect to these perturbations before assigns it
+as the true ground state of the Na0.5CoO2system.
+T. Li is supported by NSFC Grant No. 10774187,
+National Basic Research Program of China No.
+2007CB925001 and Beijing Talent Program.
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+[9]/vectorb1=4π√
+3a(0,1) and/vectorb2=2π
+a(1,−1√
+3) are the two recip-
+rocal lattice vectors of the triangular lattice.
+[10] G. Baskaran, Phys. Rev. Lett 91, 097003 (2003).
+[11] S. Zhou and Z. Wang, Phys. Rev. Lett. 98,226402(2006).
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+(2008).
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+[15] S. Lee et al., Nature Materials 5, 471 (2006)./s40/s97/s41
+/s40/s98/s41/s48/s46/s48/s48/s46/s49/s48/s46/s50
+/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s109
+/s85/s47/s124/s116/s124
+/s32/s32/s109/s32/s99/s111/s108/s108/s105/s110/s101/s97/s114/s32/s115/s116/s97/s116/s101
+/s32/s110/s111/s110/s45/s99/s111/s112/s108/s97/s110/s97/s114/s32/s115/s116/s97/s116/s101/s40/s97/s41
+/s32/s99/s111/s108/s108/s105/s110/s101/s97/s114/s32/s115/s116/s97/s116/s101
+/s32/s110/s111/s110/s45/s99/s111/s112/s108/s97/s110/s97/s114/s32/s115/s116/s97/s116/s101
+/s32/s32/s40/s98/s41/s48/s49/s50/s51/s52/s53
+/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s49/s50/s51/s52/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s116/s41/s110/s111/s110/s45/s99/s111/s112/s108/s97/s110/s97/s114/s32/s115/s116/s97/s116/s101/s99/s111/s108/s108/s105/s110/s101/s97/s114/s32/s115/s116/s97/s116/s101
+/s32/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s116/s41/s112/s97/s114/s97/s109/s97/s103/s110/s101/s116/s105/s99/s32/s115/s116/s97/s116/s101
+/s85/s47/s124/s116/s124/s40/s97/s41
+/s110/s111/s110/s45/s99/s111/s112/s108/s97/s110/s97/s114/s32/s115/s116/s97/s116/s101/s99/s111/s108/s108/s105/s110/s101/s97/s114/s32/s115/s116/s97/s116/s101/s112/s97/s114/s97/s109/s97/s103/s110/s101/s116/s105/s99/s32/s115/s116/s97/s116/s101
+/s32
+/s32/s40/s98/s41/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s46/s48/s48/s48/s48/s48/s46/s48/s48/s48/s53/s48/s46/s48/s48/s49/s48/s48/s46/s48/s48/s49/s53/s48/s46/s48/s48/s50/s48
+/s32/s110/s111/s110/s45/s99/s111/s112/s108/s97/s110/s97/s114/s32/s115/s116/s97/s116/s101
+/s32/s99/s111/s108/s108/s105/s110/s101/s97/s114/s32/s115/s116/s97/s116/s101
+/s32/s32/s109
+/s85/s47/s124/s116/s124
\ No newline at end of file
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+arXiv:1001.0621v1  [cond-mat.other]  5 Jan 2010Hydrodynamic theory of supersolids: Variational principl e and effective Lagrangian
+C.-D. Yoo1and Alan T. Dorsey1
+1Department of Physics, University of Florida, P.O. Box 1184 40, Gainesville, FL 32611-8440
+(Dated: November 4, 2018)
+We develop an effective low-energy, long-wavelength theory of a bulk supersolid—a putative phase
+of matter with simultaneous crystallinity and Bose condens ation. Using conservation laws and gen-
+eral symmetry arguments we derive an effective action that co rrectly describes the coupling between
+the Bose condensation and the elasticity of the solid. We use our effective action to calculate the
+correlation and response functions for the supersolid, and we show that the onset of supersolidity
+produces peaks in the response function, corresponding to p ropagating second sound modes in the
+solid. Throughout our work we make connections to existing w ork on effective theories of superflu-
+ids and normal solids, and we underscore the importance of co nservation laws and symmetries in
+determining the number and character of the collective mode s.
+PACS numbers: 67.80.bd, 67.25.dg
+I. INTRODUCTION
+In1969AndreevandLifshitz proposedanovelphaseof
+matterinquantumBosecrystals, whereinaBoseconden-
+sate of point defects would coexist with the crystallinity
+of the solid.1This is perhaps the most conceptually clear
+picture of what we now call a “supersolid,” although sug-
+gestions of the coexistence (or non-coexistence) of Bose
+condensation and crystallinity can be traced to the ear-
+lier work of Penrose and Onsager2and Chester.3An-
+dreev and Lifshitz provided an elegant (albeit incom-
+plete) formulation of the hydrodynamics of supersolids,
+andpredictedpropagatingmodesanalogoustosecond(or
+fourth) sound in liquid4He. Their hydrodynamic formu-
+lation was further extended by Saslow4and by Liu.5Ex-
+perimental searchesforsignaturesofthe supersolidphase
+proved fruitless6until recently, when Kim and Chan ob-
+served rotational inertia anomalies in solid4He that they
+interpreted as evidence for supersolidity.7–9Their work
+fueled extensive searches for further evidence of this elu-
+sive phase of matter,10–17and there are now a number
+of extensive reviews of the experimental and theoretical
+progress in this area—see Refs. 18,19,20, and21.
+The presentworkis adetailed—and we believe novel—
+study of the hydrodynamics of bulk supersolids, of the
+type originally proposed by Andreev and Lifshitz. Our
+work uses conservation laws and general symmetry prin-
+ciples to derive an effective action for a supersolid. We
+rely extensively on a variational principle used to obtain
+the dynamic equations in various continuum systems:
+normal fluids,22–24superfluids,24–29normal solids,22,30
+liquid crystals,31trapped superfluid gases,32and rela-
+tivistic fluids.33This effective action is a powerful tool
+forcalculatingandelucidatingthecollectivemodesofthe
+supersolid phase, and one of our important new results is
+a calculation of the correlation and response functions in
+thesupersolidphase. Movingbeyondlinearizedhydrody-
+namics, the effective action can also be use to study the
+dynamics and interaction of topological defects—vortices
+and dislocations—in the supersolid, the topics of subse-
+quentpublications.34Weshouldalsostatewhatthisworkisnot—it is not an explanation of the recent experiments
+onpossiblesupersolidityin4He, asthe prevailingwisdom
+suggests that structural disorder plays a key role in most
+of the experiments, and our simplified model assumes an
+ordered solid.
+This work is organized as follows. In Sec. II we de-
+rive the supersolid hydrodynamics and the effective La-
+grangiandensityusingthevariationalprinciple. Weshow
+that the equation of motion are equivalent (up to a term
+nonlinearintheelasticstrain)tothehydrodynamicequa-
+tions of motion derived by Andreev and Lifshitz1. We
+also discuss the connection to the work of Saslow,4Liu,5
+and Son.35In Sec. III we use a quadratic version of the
+Lagrangian to investigate the linearized hydrodynamics
+of a supersolid. Finally, in Sec. IV the collective modes
+and the density-density correlation function of a model
+supersolid are calculated in detail. The Appendices pro-
+vide additional detail, as an aid to the reader.
+II. VARIATIONAL PRINCIPLE AND AN
+EFFECTIVE LAGRANGIAN OF SUPERSOLIDS
+We start with a Lagrangian density for a supersolid
+in the Eulerian description (in which all quantities are
+depicted at fixed position xand timet),
+LSS=1
+2ρsijvsivsj+1
+2(ρδij−ρsij)vnivnj
+−USS(ρ,ρsij,s,Rij), (1)
+whereρsijis the superfluid density tensor, ρis the total
+density,vsis the velocity of the super-components, vnis
+the velocity of the normal-components, sis the entropy
+density, and
+Rij≡∂iRj (2)
+is the deformation tensor, with Rthe coordinate affixed
+to material elements ( ∂i≡∂/∂xiand∂t≡∂/∂tin what
+follows). The first two terms in the Lagrangian density
+are the kinetic energy densities of the super-component2
+and the normal-component, respectively, and the third
+term is the internal energy density which is a function
+ofρ,ρsij,s, andRij. In contrast to a superfluid, Rij
+appears explicitly in USSfor a supersolid, a reflection
+of the solid’s broken translational symmetry. As shown
+in Appendix B, the internal energy density satisfies the
+thermodynamic relation
+dUSS=Tds+/bracketleftbigg
+µ+1
+2(vni−vsi)2/bracketrightbigg
+dρ−λikdRik
+−1
+2(vni−vsi)(vnj−vsj)dρsij, (3)
+whereµis the chemical potential per unit mass, and
+λijthe stress tensor. Given the Lagrangian density in
+Eq. (1), the action is
+SSS=/integraldisplay
+dt/integraldisplay
+d3xLSS. (4)
+The equations of motion for a supersolid are obtained
+from variations of SSSwith respect to the dynamical
+variables. However, as illustrated in Appendix A, the
+dynamical variables are not independent and one must
+insure that conservation laws and broken symmetries are
+incorporated in the action through the use of auxiliary
+fields (Lagrange multipliers). For a three dimensional
+supersolid there are five conserved quantities: the mass,
+the entropy and the three components of the momentum.
+Among these constraints we impose only the mass and
+entropy conservation laws, and show below that the mo-
+mentum conservation is the byproduct of the variational
+principle. Conservation of mass is expressed through the
+equation of continuity,
+∂tρ+∂iji= 0, (5)
+where the mass current jiis
+ji=ρsijvsj+(ρδij−ρsij)vnj. (6)
+The entropy conservation law is
+∂ts+∂i(svni) = 0, (7)
+in which only vnis involved because the entropy is
+transported by the normal component. Finally, we ac-
+count for the broken translational symmetry using Lin’s
+constraint ,24
+DnRi
+Dt= 0, (8)
+whereDn/Dt≡∂t+vni∂i. This constraint states that
+the Lagrangian coordinates (i.e., the initial positions of
+particles) do not change along the paths of the normal
+component. Indeed, Lin’s constraint was first introduced
+to generate vorticity in the Lagrangian description of an
+isentropic normal fluid.24We incorporate all of the con-
+straints, Eqs. ( 5)-(8), into the Lagrangiandensity Eq. ( 1)by using the Lagrange multipliers α,φ, andβi, with the
+result:
+LSS=1
+2ρsijvsivsj+1
+2(ρδij−ρsij)vnivnj
+−USS(ρ,ρsij,s,Rij)+α/bracketleftbigg
+∂ts+∂i(svni)/bracketrightbigg
++φ/braceleftigg
+∂tρ+∂i/bracketleftbigg
+ρsijvsj+(ρδij−ρsij)vnj/bracketrightbigg/bracerightigg
++βi/bracketleftbigg
+∂t(sRi)+∂j(sRivnj)/bracketrightbigg
+. (9)
+Note that in our formulationLin’s constraintis combined
+with the entropy conservation law.
+We are now in a position to derive the hydrodynamic
+equations of motion for supersolids. First of all, the vari-
+ation of the action with respect to vsiproduces
+vsi=∂iφ. (10)
+Therefore, the superfluid component of the velocity is a
+potential flow, as expected [rotational flow can be ob-
+tained by introducing another constraint; see Ref. 27].
+The remaining equations of motion are
+•δρ:
+1
+2v2
+n−∂USS
+∂ρ−∂tφ−vnivsi= 0, (11)
+•δρsij:
+∂USS
+∂ρsij=−1
+2(vsi−vni)(vsj−vnj),(12)
+•δs:
+Dnα
+Dt+RiDnβi
+Dt+∂USS
+∂s= 0, (13)
+•δvni:
+∂iα+Rj∂iβj=1
+s(ρδij−ρsij)(vnj−vsj),(14)
+•δRi:
+Dnβi
+Dt−1
+s∂j/parenleftbigg∂USS
+∂Rji/parenrightbigg
+= 0. (15)
+In the above equations of motion we have eliminated the
+gradientofφbyusingEq.( 10). Inadditiontothederived
+equations of motion, the variations with respect to the
+Lagrange multipliers reproduce the imposed constraints,
+Eqs. (5)-(8). Therefore, Eqs. ( 5)-(8), (10)-(15) are the
+hydrodynamic equations for supersolids.3
+In the following we demonstrate that the equations
+of motion derived above are equivalent to the non-
+dissipative supersolid hydrodynamics developed by An-
+dreev and Lifshitz,1Saslow,4and Liu.5First, the taking
+the gradient of Eq. ( 11) produces the Josephson equation
+∂tvsi=−∂iµ−1
+2∂ivs2, (16)
+where we have used the thermodynamic relation for
+∂Uss/∂ρgiven in Eq. ( 3). Second, we derive the momen-
+tum conservation equation; the following identity simpli-
+fies the derivation:
+D(a∂ib)
+Dt=∂ibDa
+Dt+a∂i/parenleftbiggDb
+Dt/parenrightbigg
+−a∂jb∂ivj,(17)
+whereD/Dt≡∂t+vi∂i. TakeDn/Dtof Eq. (14), and
+eliminate the Lagrange multipliers by using Eqs. ( 7), (8),
+(13)-(15). The result is
+Dn
+Dt/bracketleftbigg
+(ρδij−ρsij)(vnj−vsj)/bracketrightbigg
+=−s∂i/parenleftbigg∂USS
+∂s/parenrightbigg
+−∂iRj∂k/parenleftbigg∂USS
+∂Rkj/parenrightbigg
+−(ρδij−ρsij)(vnj−vsj)∂kvnk
+−(ρδjk−ρsjk)(vnk−vsk)∂ivnj.
+(18)
+Third, combine Eq. ( 18) with the thermodynamic rela-
+tion, Eq. ( 3), the continuity equation, Eq. ( 5), and the
+Josephson equation, Eq. ( 16). After some algebra, we
+obtain the momentum conservation law
+∂tji+∂jΠij= 0, (19)
+wherejiis the mass current given in Eq. ( 6), and Π ijis
+the (non-dissipative) stress tensor
+Πij=ρvsivsj+vsipj+vnjpi−Rikλjk
+−/bracketleftbigg
+ǫ−Ts−µρ−(vnj−vsj)pj/bracketrightbigg
+δij,(20)
+wherepi≡(ρδij−ρsij)(vnj−vsj) andǫsatisfies a ther-
+modynamic relation given by Eq. ( B3). Note that the
+Josephson equation, Eq. ( 16), and the momentum con-
+servation equation, Eq. ( 19), are Eqs. (9) and (12) of
+Andreev and Lifshitz1[Andreev and Lifshitz neglected
+nonlinear strain terms, effectively replacing Rikbyδik
+in the last term of Eq. ( 20) above]. Moreover, the mo-
+mentum conservationequation is equivalent to Eq. (4.16)
+of Saslow4whenvsis taken as a Galilean velocity, and
+Eq. (3.40) of Liu5in the case where the superthermal
+current vanishes.
+The Lagrangian density used to derive the hydrody-
+namics of supersolids, Eq. ( 9), can be recast into a more
+familiar and compact form by using the equations of mo-
+tion, as illustrated for an ideal fluid in Appendix A. To
+see this, we integrate the terms involving the Lagrange
+multipliers by parts (neglecting boundary terms), anduse Eqs. ( 10) and (13) to eliminate αandβi. We then
+obtain
+LSS=−ρ∂tφ−1
+2ρsij∂iφ∂jφ+1
+2(ρδij−ρsij)vnivnj
+−(ρδij−ρsij)vnj∂iφ−f(ρ,ρsij,T,Rij),(21)
+wheref≡USS−Tssatisfies the thermodynamic relation
+df=−sdT+/bracketleftbigg
+µ+1
+2(vni−vsi)2/bracketrightbigg
+dρ−λikdRik
+−1
+2(vni−vsi)(vnj−vsj)dρsij. (22)
+When cast in this form, we see that the coupling be-
+tween the superfluid and the normal fluid [the fourth
+terminEq.( 21)]is−(ρδij−ρsij)vnivsj; thisisa“current-
+current” interaction, where the coupling constant is the
+normal fluid density. This coupling coefficient is univer-
+sal–it is determined by conservation laws and Galilean
+invariance.
+As mentioned earlier, several other authors have re-
+cently proposed Lagrangian descriptions for supersolids.
+Son35used symmetry-based arguments to derive an ef-
+fective Lagrangian for a supersolid. To connect to Son’s
+results, we first invert Lin’s constraint, Eq. ( 8), to obtain
+vni=−R−1
+ji∂tRj, (23)
+whereR−1
+ji≡∂xi/∂RjandRijR−1
+jk=δik. Substituting
+this into Eq. ( 21), we obtain a Lagrangiansimilar in form
+to Eq. (23) of Son’s paper (however, our energy density f
+depends upon ρ,ρs, andTin addition to Rij). A differ-
+ent approach was used by Josserand et al.,36who applied
+a homogenization procedure to a nonlocal version of the
+Gross-Pitaevskiiequationtosystematicallyderivealong-
+wavelength Lagrangian for a supersolid. On the whole,
+our Eq. ( 21) agrees with their Eq. (4), once we identify
+theirρ(n) with our normal fluid density ρδij−ρsij. Fi-
+nally, Ye37proposed a phenomenological supersolid La-
+grangian; however, the Lagrangian in his Eq. (1) is not
+manifestly Galilean invariant, so that his coupling con-
+stantsaαβarearbitrary. His approachalsomissescertain
+nonlinear terms that are important when treating topo-
+logical defects.
+III. QUADRATIC LAGRANGIAN DENSITY
+AND THE LINEARIZED HYDRODYNAMICS OF
+SUPERSOLIDS
+In this section we discuss the propagation of collec-
+tive modes in supersolids by examining the response to
+small fluctuations away from equilibrium. The number
+of collective hydrodynamic modes of a system can be
+inferred by enumerating the system’s conservation laws
+and broken symmetries.38Since we are more interested
+in the effect of density or defect fluctuations than ther-
+mal fluctuations, for simplicity we ignore thermal fluctu-4
+ations in what follows. For a three dimensional super-
+solid there are conservation laws for mass, three compo-
+nents of momentum, and energy; however, by ignoring
+thermal fluctuations we can omit the energy conserva-
+tion law. In addition the conservation laws, there are
+three broken translational symmetries (due to the crys-
+tallinity) and one broken gauge symmetry (due to the
+Bose-Einstein condensation). Thus, a three dimensional
+supersolid without thermal fluctuations has eight con-
+servation laws and broken symmetries (nine, if conser-
+vation of energy is included). Correspondingly, there
+are eight hydrodynamic modes: two pairs of the ordi-
+nary transverse propagating modes, a pair of longitudi-
+nal first sound modes, and a pair of longitudinal second
+sound modes (note that a solution of the hydrodynamic
+equations with dispersion ω=±ckwould count as two
+modes–a pair of modes, with one propagating and a sec-
+ond counter-propagating). The appearance of the longi-
+tudinal second sound modes is one of the key signatures
+of a supersolid.
+Westartbyestablishingthenotationforthesmallfluc-
+tuationsawayfromequilibrium. Theequilibriumvalueof
+the densities will be denoted with a subscript of ‘0’, and
+the density fluctuations will be denoted by δρ, so that
+ρ=ρ0+δρandρsij=ρs0ij+δρsij. For the lattice fluc-
+tuations, let udenote the (small) deformation field away
+from the unstrained solid (i.e. the difference between the
+Lagrangian and the Eulerian coordinates):
+x=R+u. (24)
+Then the deformation tensor becomes
+Rij=δij−wij, (25)
+wherewij≡∂iuj. Finally, the velocity of the normal
+component is obtained by linearizing the inverted Lin’s
+constraint, Eq. ( 23), so that to lowest order in the strain
+field the normal velocity is the time derivative of the dis-
+placement field,
+vni=∂tui. (26)
+ExpandingtheLagrangiandensity,Eq.( 21),uptosecond
+order in the small quantities δρ,δρsij,wij,∂iφand∂tφ,
+we obtain
+Lquad
+SS=−ρ0∂tφ−λ0ijwij−µ0δρ−δρ∂tφ
+−1
+2ρ0(∂iφ)2−∂µ
+∂wij/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρδρwij−1
+2∂µ
+∂ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wij(δρ)2
++1
+2ρn0ij(∂tui−∂iφ)(∂tuj−∂jφ)
+−1
+2∂λij
+∂wlk/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρwijwlk, (27)
+whereρn0ij≡ρ0ij−ρs0ijand the thermodynamic re-
+lation forf, Eq. (22), is used. In the above expansion
+we have dropped constants which do not contribute tothe equations of motion, and have neglected terms pro-
+portional to ∂f/∂ρ sijbecause they are of higher order
+[see Eq. ( 12)]; consequently, the quadratic Lagrangian
+turns out to be independent of fluctuations of the su-
+perfluid density. However, we have kept the first two
+terms in Eq. ( 27); although they are total derivatives
+and would seem to be irrelevant to the equations of mo-
+tion, they are non-trivial for topological defects such as
+vortices or dislocations. In fact, one can show34that the
+first term produces the Magnus force on a vortex39and
+the second term generates the Peach-Koehler force on
+a dislocation.40,41We will defer the discussion of these
+effects to a subsequent publication.34
+Now we are in a position to study the hydrodynamic
+modes of a supersolid. The quadratic Lagrangian den-
+sity, Eq. ( 27), produces three linearized equations of mo-
+tions which are equivalent to the continuity equation,
+the Josephson equation and the momentum conserva-
+tion equation. Before proceeding further, it is useful to
+rewrite the quadratic Lagrangian density in terms of the
+defect density fluctuation by using one of the equations
+of motion. By varying the action with respect to δρwe
+obtain
+δρ=∂ρ
+∂wij/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+µwij+∂ρ
+∂µ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wijδµ, (28)
+where we have used the linearized Josephson equation
+(∂tφ=−µ0−δµ) and the identity
+∂x
+∂y/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+z=−∂z
+∂y/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+x∂x
+∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+y. (29)
+From Eq. ( 28) it is clear that the density fluctuation is
+an independent hydrodynamic variable–it is not slaved
+to the lattice deformation, as would be the case for a
+commensurate solid, where δρ=−ρ0∇·u. Indeed, in a
+real (incommensurate) crystal a density fluctuation can
+be produced by lattice deformations or by point defects
+(vacancies and interstitials). To highlight the role of de-
+fects we will introduce the defect density fluctuation δρ∆
+asour independent hydrodynamicvariable, instead of δρ.
+The local defect density is defined as the difference be-
+tween the density of vacancies, ρV, and the density of
+interstitials, ρI:
+ρ∆=ρI−ρV. (30)
+The minus sign is necessary so that the total defect den-
+sity is conserved–in the bulk of the crystal vacancies and
+interstitials are created and destroyed in pairs (ignoring
+surface effects). Then we have
+δµ=∂µ
+∂wij/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ∆wij+∂µ
+∂ρ∆/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wijδρ∆,(31)
+and from Eq. ( 28) we obtain
+δρ=∂ρ
+∂wij/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ∆wij+∂ρ
+∂ρ∆/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wijδρ∆,(32)5
+where we have used Eq. ( 29) and the identity
+∂x
+∂y/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+0=∂x
+∂y/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+z+∂x
+∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+y∂z
+∂y/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+0. (33)
+Equation ( 32) shows that a density fluctuation in an
+isothermal supersolid is caused either by a lattice defor-
+mation orby adefect density fluctuation δρ∆, just asin a
+normal solid.43–45Following Zippelius et al.(ZHN),44we
+canidentify( ∂ρ/∂w ij)ρ∆=−ρ0δijand(∂ρ/∂ρ ∆)wij= 1.
+We finally obtain
+δρ=−ρ0wii+δρ∆, (34)
+whichillustratestherolesofthelatticedeformation wii=
+∇ ·uand the defect density fluctuation in determining
+the total density fluctuation. We note in passing that in
+a higher order expansion of the Lagrangian density the
+terms proportional to the superfluid density fluctuation
+must also be retained in Eq. ( 34). This would resemble
+the “three-fluid” scenario proposed by Saslow46in which
+the lattice density and velocity are introduced for the
+third fluid component.
+We can now use Eq. ( 34) to rewrite the quadratic La-
+grangian density in terms of the defect density, with the
+result
+Lquad
+SS=ρ0wii∂tθ−ρn0ij∂tui∂jθ−1
+2ρ2
+0∂µ
+∂ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wijw2
+ii
+−δρ∆∂tθ−1
+2ρs0ij∂iθ∂jθ−∂µ
+∂wij/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρδρ∆wij
++ρ0∂µ
+∂ρ∆/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wijwiiδρ∆−1
+2∂µ
+∂ρ∆/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wijδρ∆2
+−1
+2∂λji
+∂wlk/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρwijwlk+ρ0∂µ
+∂wij/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρwijwkk
++1
+2ρn0ij∂tui∂tuj, (35)
+where we introduced θ=φ+µ0t. Next, we derive the
+linearized equations of motion from the Lagrangian den-
+sity. First, note that the variation with respect to δρ∆
+reproduces Eq. ( 31) because∂tθis−δµ. Second, taking
+the variation with respect to θproduces the linearized
+equation of continuity, expressed in terms of δρ∆:
+∂tδρ∆+∂ij∆
+i= 0, (36)
+where the defect current density is given by
+j∆
+i=ρs0ij(∂jθ−∂tuj). (37)
+We see that the defect current arises from counterflow
+between the superfluid velocity ∇θand the normal fluid
+velocity∂tu, and vanishes when ρs0ij= 0, in the normal
+state. In other words, ∂tδρ∆= 0 in the normal state,
+in agreement with the non-dissipative description of nor-
+mal solids44in which defect currents are only produced
+through diffusion (i.e., the defect current is dissipative inthe normal solid). The last equation of motion derived
+from the variation of uiis
+ρn0ij∂2
+tuj−/parenleftigg
+∂µ
+∂wji/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ−ρn0ij∂µ
+∂ρ∆/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wij/parenrightigg
+∂jδρ∆
+−/parenleftigg
+∂λji
+∂wlk/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ−ρn0ij∂µ
+∂wlk/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ∆−ρ0∂µ
+∂wij/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρδlk/parenrightigg
+∂jwlk= 0.
+(38)
+When the time derivative of Eq. ( 36) is combined with
+Eq. (31), we obtain
+∂2
+tδρ∆−ρs0ij∂µ
+∂ρ∆/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wij∂i∂jδρ∆
+−ρs0ij∂i∂2
+tuj−ρs0ij∂µ
+∂wlk/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ∆∂i∂jwlk= 0.(39)
+Our linearized equations of motion, Eqs. ( 38) and (39),
+are equivalent to Eq. (19) of Andreev and Lifshitz.1
+Inthe particularcaseinwhichthe latticesitesarefixed
+(sothatu= 0)werecoverfromEq.( 39)thefourthsound
+modesobtainedbyAndreevandLifshitz,1whichhavethe
+dispersion relation
+ω2=ρs0ij∂µ
+∂ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wijqiqj, (40)
+where we have used ( ∂/∂ρ∆)wij= (∂/∂ρ)wij. On the
+other hand, when there are no defect fluctuations ( δρ∆=
+0), Eqs. ( 38) and (39) are combined into
+ρ0∂2
+tui=∂λji
+∂wlk/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ∂jwlk−ρ0∂µ
+∂wlk/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ∆∂iwlk
+−ρ0∂µ
+∂wij/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ∂jwkk. (41)
+Amodeanalysisofthisequationwouldproducesixsound
+modes of an anisotropic normal solid. We see that with-
+out defects there are no additional sound modes, as ex-
+pected.
+IV. DENSITY-DENSITY CORRELATION
+FUNCTION OF A MODEL SUPERSOLID
+In this section we will calculate the density-density
+correlation function of a model supersolid, a measurable
+quantity in a light scattering experiment. However, be-
+foredelvingintothecalculationsforasupersolidlet’sfirst
+review what’s revealed by scattering light from a struc-
+tureless, normal fluid [for example, see Ref. 47]. The
+mode counting for the fluid is simple–there are five col-
+lective modes, due to conservation of mass, energy, and
+three components of momentum (in three dimensions).
+The five collective modes are a pair of transverse mo-
+mentum diffusion modes and three longitudinal modes:6
+a pair of propagating sound modes and a thermal diffu-
+sion mode. The density fluctuations important for light
+scatteringonlycoupleto the longitudinal modes, sothree
+modes are observed: the diffusion mode appears as the
+Rayleigh peak ω= 0 and the pair of sound modes as the
+Brillouin doublet at ω=±cq(with a sound speed c). In
+the absence of dissipation these peaks are δ-functions;
+dissipation turns each δ-function into a Lorentzian of
+widthDq2, withDbeing an attenuation coefficient.
+What happens in a superfluid? In addition to the five
+conserved densities that exist in a normal fluid there is
+a broken gauge symmetry, so from mode counting we
+conclude there are six collective modes. Two of these
+are transverse momentum diffusion modes (just as for
+the normal fluid), leaving fourlongitudinal modes for
+the superfluid: a pair of propagating first sound modes,
+and a new pair of propagating second sound modes. In
+essence, the central Rayleigh peak in the normal fluid
+splits into a new Brillouin doublet upon passing into the
+superfluid phase. This remarkable phenomenon has been
+observed in light scattering experiments on4He.48,49We
+show below that an analogous splitting occurs in a su-
+persolid, and should be observable in a light scattering
+experiment.
+A. Dynamics of supersolid without dissipation
+To facilitate the calculation of the density-density cor-
+relation function for a supersolid we’ll make two simpli-fying assumptions: the solid is isotropic, and two dimen-
+sional. The isotropy causes the transverse and longitu-
+dinal modes to neatly decouple; the two dimensionality
+results in only one pair of propagating transverse modes,
+rather than two pair. Since we’re interested in longi-
+tudinal fluctuations, the latter simplification is of little
+consequence to the main results of this section. With
+these assumptions, the thermodynamic relations are
+∂λji
+∂wlk/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ=˜λδjiδlk+ ˜µ(δilδjk+δikδjl),(42)
+∂µ
+∂wij/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ=γδij, (43)
+∂µ
+∂ρ∆/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wij=∂µ
+∂ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wij=1
+ρ2
+0χ, (44)
+whereχis the isothermal compressibility at constant
+strain,γis a phenomenological coupling constant be-
+tweenthestrainandthedensity, and ˜λand ˜µarethebare
+Lam´ e coefficients at constant density. Then in Fourier
+space the Lagrangian density, Eq. ( 35), reduces to
+LSS=1
+2/parenleftbigδρ∆(Q)θ(Q)uL(Q)/parenrightbig
+A
+δρ∆(−Q)
+θ(−Q)
+uL(−Q)
++1
+2/parenleftbig
+ρn0ω2
+n+ ˜µq2/parenrightbig
+uT(Q)uT(−Q), (45)
+whereωn=iω,Q= (q,ωn),uL= (q·u)/qwithq=|q|,uT=u−(uL/q)q, and
+A=
+1
+ρ2
+0χ−ωn −iq/parenleftig
+γ−1
+ρ0χ/parenrightig
+ωnq2ρs0 iωnqρs0
+iq/parenleftig
+γ−1
+ρ0χ/parenrightig
+iωnqρs0ρn0ω2
+n+q2/parenleftig
+λ+1
+χ−2ρoγ/parenrightig
+, (46)
+whereλ≡˜λ+2˜µ. The collective modes are determined
+from the determinant ∆ AofA:
+∆A=ρn0ω4
+n+/bracketleftbigg
+λ+ρn0/parenleftbigg1
+ρ0χ−2γ/parenrightbigg/bracketrightbigg
+q2ω2
+n
+−ρs0/parenleftbigg
+γ2−λ
+χρ2
+0/parenrightbigg
+q4. (47)Setting ∆ A= 0, we find the longitudinal first sound
+speedcLand second sound speed c2:
+c2
+L=λ
+2ρn0+1
+2ρ0χ−γ
++1
+2/radicaligg/parenleftbiggλ
+ρn0+1
+ρ0χ−2γ/parenrightbigg2
+−4ρs0
+ρn0/bracketleftbiggλ
+χρ2
+0−γ2/bracketrightbigg
+,
+(48)7
+c2
+2=λ
+2ρn0+1
+2ρ0χ−γ
+−1
+2/radicaligg/parenleftbiggλ
+ρn0+1
+ρ0χ−2γ/parenrightbigg2
+−4ρs0
+ρn0/bracketleftbiggλ
+χρ2
+0−γ2/bracketrightbigg
+.
+(49)
+In particular, when ρs0= 0 (normal solids), c2vanishes,
+and we only have
+c2
+NS= (˜λ+2˜µ+1/χ)/ρ0−2γ, (50)
+which agrees with the longitudinal sound speed obtained
+byZippelius et al.44onceweidentify ˜λ=λZHN+2γZHN+
+1/χandγ= (γZHN+1/χ)/ρ0. Moreover,when the Lam´ e
+coefficientsandthecouplingconstant γvanishwerecover
+the sound speed of a normal fluid. As discussed earlier,
+there is one pair of transverse sound modes with speed
+cT=/radicaligg
+˜µ
+ρn0. (51)
+Finally, we can calculatethe correlationfunctions from
+Eq. (45):
+/angb∇acketleftδρ∆(Q)δρ∆(−Q)/angb∇acket∇ight=ρs0q2ρ0ω2
+n+(λ−2ρ0γ+1/χ)q2
+∆A,
+(52)
+/angb∇acketleftδρ∆(Q)uL(−Q)/angb∇acket∇ight=iqρs0
+ρ0ρ0ω2
+n−(ρ0γ−1/χ)q2
+∆A,(53)
+and
+/angb∇acketleftuL(Q)uL(−Q)/angb∇acket∇ight=1
+ρ2
+0χρ2
+0χω2
+n+ρs0q2
+∆A.(54)
+Since the density fluctuation is related to the defect den-
+sity fluctuation and the strain tensor by Eq. ( 34), the
+density-density correlation function becomes
+/angb∇acketleftδρ(Q)δρ(−Q)/angb∇acket∇ight=A/parenleftbigg1
+iω−cLq−1
+iω+cLq/parenrightbigg
++B/parenleftbigg1
+iω−c2q−1
+iω+c2q/parenrightbigg
+,(55)
+where
+A=−qρ0ρn0c2
+L−ρs0λ
+2cLρn0(c2
+L−c2
+2), (56)
+B=−qρ0ρn0c2
+2−ρs0λ
+2c2ρn0(c2
+2−c2
+L). (57)
+Then, by performing the analytic continuation iωn=
+ω+iδ, the density-density response function can be ob-
+tained from the imaginary part of the density-density
+correlation function:
+χ′′
+ρρ(q,ω) =−πA/bracketleftbigg
+δ(ω−cLq)−δ(ω+cLq)/bracketrightbigg
+−πB/bracketleftbigg
+δ(ω−c2q)−δ(ω+c2q)/bracketrightbigg
+,(58)where we have used the identity
+1
+ω′−ω−iǫ=P1
+ω′−ω+iπδ(ω−ω′).(59)
+It is easy to show that the response function satisfies the
+thermodynamic sum rule (for the derivation of the static
+correlation function see Appendix B)
+/integraldisplay∞
+−∞dω
+πχ′′
+ρρ(q,ω)
+ω=ρ2
+0χλ
+λ−ρ2
+0γ2χ,(60)
+and the f-sum rule
+/integraldisplay∞
+−∞dω
+πωχ′′
+ρρ(q,ω) =ρ0q2. (61)
+B. Dynamics of supersolid with dissipation
+We continue our discussion of the density correlation
+and response functions by including dissipative terms in
+the equations of motion. As mentioned above, the dis-
+sipative terms will broaden the δ-function peaks in the
+density response function. In addition, as noted by Mar-
+tin et al.38, the dissipation is necessary to identify the
+“missing” defect diffusion mode in normal solids. The
+dissipative hydrodynamic equations of motion for a su-
+persolid were first obtained by Andreev and Lifshitz,1
+whousedstandardentropy-productionargumentstogen-
+erate the dissipative terms. For an isotropic supersolid
+(including the nonlinear term neglected by Andreev and
+Lifshitz) we have (with the new dissipative terms on the
+right hand side)
+∂tρ+∂iji= 0, (62)
+∂tji+∂jΠij=ζ∂i∂kvnk+η∂2vni
+−Σ∂i∂k/bracketleftbigg
+ρs(vnk−vsk)/bracketrightbigg
+,(63)
+∂tui−vni+vnk∂kui+ui∂kvnk= Γ∂kλki,(64)
+∂tvsi+∂i/parenleftbigg
+µ+1
+2vs2/parenrightbigg
+=−Λ∂i∂k/bracketleftbigg
+ρs(vnk−vsk)/bracketrightbigg
++Σ∂i∂kvnk, (65)
+wherej=ρnvn+ρsvsis the total mass current, Σ and Λ
+coefficients of viscosity, ζthe bulk viscosity coefficient, η
+the shear viscosity coefficient, Γ the diffusion coefficient
+for defects.
+We next linearize the dissipative hydrodynamic equa-
+tions by considering small fluctuations from the equilib-
+rium values. Writing δµandλijin terms of δρandδwij,
+δµ=1
+ρ2
+0χδρ+γwii, (66)8
+δλij=γδijδρ+˜λδijwkk+ ˜µ(wij+wji).(67)
+We replace δµandδλijinto Eqs. ( 62)-(65), and divide
+them into transverse and longitudinal parts. The equa-
+tions of motion for transverse parts are
+ρn0∂tvnT−˜µ∂2uT−η∂2vnT= 0,(68)
+where∂≡/radicalbig
+∂2
+i, and
+∂tuT−vnT−˜µΓ∂2uT= 0. (69)
+These equations support a propagating transverse sound
+mode with the transverse sound speed cT=/radicalbig
+˜µ/ρn0, as
+obtained in the previous section, and an attenuation con-
+stant Γ T=η+ρn0˜µΓ. Next, the longitudinal equations
+of motion are
+∂tδρ+ρs0∂vsL+ρn0∂vnL= 0, (70)
+ρn0∂tvnL+/parenleftbiggρn0
+ρ2
+0χ−γ/parenrightbigg
+δρ−/parenleftbigg
+λ−ρn0γ/parenrightbigg
+∂2uL
+−/parenleftbigg
+˜ζ−2ρs0σ−ρs2
+0Λ/parenrightbigg
+∂2vnL−ρs0σ∂2vsL= 0,(71)∂tuL−vnL−γΓ∂δρ−λΓ∂2uL= 0,(72)
+∂tvsL+1
+ρ2
+0χ∂δρ+γ∂2uL−σ∂2vnL−ρs0Λ∂2vsL= 0,(73)
+whereσ≡Σ−ρs0Λ, and˜ζ≡ζ+η. The Laplace-Fourier
+transform of Eqs. ( 70) - (73) yields
+C
+δρ(q,z)
+vnL(q,z)
+uL(q,z)
+vsL(q,z)
+=
+δρ(q)
+vnL(q)
+uL(q)
+vsL(q),
+(74)
+where
+C=
+−iz iqρ n0 0 iqρs
+iq/parenleftig
+1
+ρ2
+0χ−γ
+ρn0/parenrightig
+−iz+q21
+ρn0/parenleftig
+˜ζ−2ρs0σ−ρs2
+0Λ/parenrightig
+q21
+ρn0(λ−ρn0γ)q2ρs0
+ρn0σ
+−iqγΓ −1 −iz+q2λΓ 0
+iq1
+ρ2
+0χq2σ −q2γ−iz+q2ρs0Λ
+. (75)
+FromEq.( 75)wecalculatetwosoundspeeds cL, Eq.(48),
+andc2, Eq. (49), with two attenuation constants,
+DL=−1
+ρn0(c2
+L−c2
+2)/parenleftbig
+c2
+Ln1+n2/parenrightbig
+,(76)
+D2=1
+ρn0(c2
+L−c2
+2)/parenleftbig
+c2
+2n1+n2/parenrightbig
+, (77)
+where
+n1≡˜ζ−2ρs0σ+ρn0Γλ+ρs0(ρn0−ρs0)Λ,(78)
+n2≡1
+ρ2
+0χ/braceleftigg
+2ρ0ρs0(ρ0χγ−1)σ+ρ0ρn0(λ−ρ2
+0χγ2)Γ
++ρs0˜ζ+ρ0ρs0/bracketleftbigg
+ρn0−ρs0+ρ0χ(λ−2γρn0)/bracketrightbigg
+Λ/bracerightigg
+.
+(79)
+Now we can see that when ρs= 0 (a normal solid),
+the second sound modes disappear but there remainsthe defect diffusion mode with the diffusion constant
+D2= (λ−ρ2
+0χγ2)Γ/ρ0χc2
+L.
+We also calculate the density-density Kubo function
+from Eq. ( 75),50
+Kρρ(q,z) =χρρ(q)
+kBTiz3+bρρz2+dρρq2z+eρρq2
+Z
++χuLρ(q)
+kBTdρuLq2z+eρuLq2
+Z,(80)
+where the static susceptibilities χρρandχuLρin Eq. (80)
+are given in Appendix B, and
+Z≡(z2−c2
+Lq2+izq2DL)(z2−c2
+2q2+izq2D2),(81)
+bρρ=−Γγq2−ρs0(ρn0−ρs0)
+ρn0Λq2−˜ζ−2ρs0σ
+ρn0q2,(82)
+dρρ=−iλ
+ρn0+iγ, (83)
+eρρ=ρs0Λλ
+ρn0q2−ρs0Λγq2+ρs0σγ
+ρn0q2,(84)9
+dρuL= (λ−ρ0γ)q, (85)
+eρuL=iρs0/parenleftbigg
+Λ−σ
+ρn0/parenrightbigg
+λq3
++iρs0
+ρn0/bracketleftbigg
+2σρ0−˜ζ−ρ0Λ(ρ0−2ρs0)/bracketrightbigg
+γq3.(86)
+Then the susceptibility χ′′
+ρρ(q,ω) can be obtained from
+the real part of Eq. ( 80),43,50
+χ′′
+ρρ(q,ω)
+ω=−iq4c2
+LDLI1(q)
+(ω2−c2
+Lq2)2+(ωq2DL)2
+−iq4c2
+2D2I2(q)
+(ω2−c2
+2q2)2+(ωq2D2)2+(ω2−c2
+Lq2)I3(q)
+(ω2−c2
+Lq2)2+(ωq2DL)2
++(ω2−c2
+2q2)I4(q)
+(ω2−c2
+2q2)2+(ωq2D2)2,
+(87)
+whereI1(q),I2(q),I3(q), andI4(q) are given in Ap-
+pendix C.
+Equation( 87) isoneofthe centralresultsofthis paper,
+it’s worth exploringsome of its features and limits. First,
+one can show that Eq. ( 87) satisfies both the thermody-
+namic sum rule, Eq. ( 60), and f-sum rule, Eq. ( 61). Sec-
+ond, the first and second terms in Eq. ( 87) produce two
+Brillouin doublets centered at ω=±cLqandω=±c2q
+with widths DLq2andD2q2, respectively. The third and
+fourth terms in Eq. ( 87) are negligible near the Brillouin
+doublets, and in fact these terms vanish in the limit of
+zero dissipation. Therefore the non-dissipative density-
+density correlation function, Eq. ( 58), is obtained from
+the first two terms in the limit DL,D2→0. Finally, for
+normal solids ( ρs= 0), the second term in Eq. ( 87) van-
+ishes, and there is only one Brillouin doublet due to the
+longitudinal first sound modes. In this case the fourth
+term in Eq. ( 87) becomes the Rayleigh peak of the de-
+fect diffusion mode centered at ω= 0. Therefore, we see
+that in analogy with a superfluid,48the defect diffusion
+peak that exists in a normal solid will split into a Bril-
+louin doublet of second sound modes upon entering the
+supersolid phase .
+To get a sense of the size of this effect, let’s substi-
+tute some physically realistic numbers into the correla-
+tion function. Assuming ρs≪ρ0andγ= Λ = Σ = 0,
+we have
+I1(q) =−I2(q)+iαρ0
+c2
+NS
+=iρ0
+c2
+NS−2i(α−1)2
+α2ρs0
+c2
+NS+O/parenleftbiggρs2
+0
+ρ2
+0/parenrightbigg
+,(88)−0.06 −0.04 −0.02 0 0.02 0.04 0.060200400600800100012001400
+(D∆ / cNS2) ωNormalized Density−Density Correlation Function
+  
+normal solid
+ρs / ρ = 0.1
+FIG. 1: (Color online). Density-density correlation funct ions
+of a normal solid (black dashed line) and a supersolid (blue
+solid line). The supersolid fraction is assumed to be 10 %.
+I3(q) =−I4(q)
+=−(α−1)2
+αρ0
+c2
+NSD∆q2
++2q2α−1
+α2/bracketleftbigg(α−1)2(α−3)
+αD∆+Dl/bracketrightbiggρs0
+c2
+NS
++O/parenleftbiggρs2
+0
+ρ2
+0/parenrightbigg
+, (89)
+where the longitudinal sound speed of normal solid cNS
+is given in Eq. ( 50), the defect diffusion constant D∆≡
+Γ/χ, andα≡ρ0χc2
+NS. We show in Fig. 1the normalized
+density-density correlation functions of a normal solid
+and a supersolid. We have used the first sound speed
+cNS= 550 m/s, the density ρ= 0.19048 g/cm3, the
+isothermal compressibility χ= 0.29615×10−8cm s2/g
+for4He solid51,52at the molar volume 21 cm3/mole, the
+viscosity of4Hefluidof 2×10−5g/cm s, the typical wave
+number involved in light scattering q−1= 100 nm, and
+Γ = 8×10−11cm3s/g. In Fig. 2we show the splitting
+of the Rayleigh peak due to defect diffusion in a normal
+solid into a Brillouin doublet of second sound modes, for
+two values of the supersolid fraction.
+V. CONCLUSION
+Starting from general conservation laws and sym-
+metry principles we derived the effective action for a
+supersolid—a state of matter with simultaneous broken
+translational symmetry and Bose condensation. The re-
+sulting effective action in Eq. ( 21) is one of the two im-
+portantresultsofthiswork,andwillbefurtherdeveloped
+in subsequent work on vortex and dislocation dynamics
+in supersolids. In this work, however, we used the lin-10
+−0.01−0.008−0.006−0.004−0.002 00.0020.0040.0060.0080.010100200300400500600700
+(D∆ / cNS2) ωNormalized Density−Density Correlation Function
+  
+normal solid
+ρs / ρ = 0.01
+ρs / ρ = 0.02
+FIG. 2: (Color online). Splitting of the Rayleigh peak (blac k
+dashed line)due to the defect diffusion mode in the normal
+solid phase into the Brillouin doublet of the second sound
+modes in the supersolid phase. The red dash-dot line is for
+ρs/ρ= 1%, and the green solid line ρs/ρ= 2%.
+earized version of this action to calculate the second of
+ourimportantresults–thedensity-densitycorrelationand
+response functions for a model supersolid (with isotropic
+elastic properties), see Eq. ( 87). In complete analogy
+with a superfluid, we showed that the onset of superso-
+lidity causes the zero-frequency defect diffusion mode to
+split into propagatingsecond sound modes, and from our
+calculation we can determine the spectral weight in these
+modes as well as the weight in the “normal” longitudinal
+sound waves in a solid.
+Acknowledgments
+The authors would like to thank K. Dasbiswas, J.
+Dufty, P. Goldbart, D. Goswami, Y. Lee, M. Meisel, and
+J. Toner for helpful discussions and comments. A.T.D.
+would like to thank the Aspen Center for Physics, where
+part of this work was completed. This work was sup-
+ported by NSF Grant No. DMR-0705690.
+Appendix A: Variational principle for an ideal fluid
+and the Gross-Pitaevskii action
+Thisappendixdemonstratesthesimplicityofthevaria-
+tional principle for deriving the hydrodynamic equations
+of motion and the Lagrangian density for continuum sys-
+tems. Consider the simplest case of an ideal fluid (IF)
+which is irrotational, inviscid and incompressible. The
+Lagrangian density for the ideal fluid is
+LIF=1
+2ρv2−UIF(ρ), (A1)whereρis the mass density, vthe velocity field, and UIF
+the internal energy density which satisfies dUIF=µdρ.
+From the Lagrangian density we construct the action
+S=/integraltextdt/integraltextd3xLIF. The variational principle states that
+the equationsofmotion arederivedfrom variationsofthe
+action with respect to all the dynamical variables. The
+naive application of this principle to the ideal fluid leads
+to the trivial equation of motion v= 0. The difficulty
+is that the dynamical variables ρandvare not indepen-
+dent, but constrained by the conservation of mass,
+∂tρ+∂i(ρvi) = 0. (A2)
+Thisconstraint is incorporated into the Lagrangian den-
+sity by introducing a Lagrange multiplier φ:
+LIF=1
+2ρv2−UIF(ρ)+φ/bracketleftbigg
+∂tρ+∂i(ρvi)/bracketrightbigg
+.(A3)
+Then the equations of motion are obtained from varia-
+tions of the action S[ρ,v,φ] with respect to ρ,v, and
+φ:
+δS
+δρ=1
+2v2−∂UIF
+∂ρ−Dφ
+Dt= 0, (A4)
+δS
+δvi=ρvi−ρ∂iφ= 0, (A5)
+δS
+δφ=∂tρ+∂i(ρvi) = 0. (A6)
+From Eq. ( A5) we obtain the velocity field
+vi=∂iφ, (A7)
+which implies that there is no vorticity, as expected for
+an ideal fluid. We can derive the Euler equation from
+Eq. (A4) by taking its gradient, using Eq. ( A7), and then
+the Gibbs-Duhem relation to obtain
+ρDvi
+Dt=−∂iP, (A8)
+wherePis the pressure. The Lagrangian density may be
+cast into an equivalent form by substituting v=∇φinto
+Eq. (A3) and integrating by parts, with the result
+LIF=−ρ∂tφ−1
+2ρ(∂iφ)2−UIF(ρ).(A9)
+For comparison, consider a system of weakly interact-
+ing bosons with a condensate wave function ψ(r,t) that
+satisfies Gross-Pitaevskii equation42
+i/planckover2pi1∂ψ
+∂t=−/planckover2pi12
+2m∇2ψ+g|ψ|2ψ. (A10)
+This equation of motion can be derived from the La-
+grangian density
+LGP=i/planckover2pi1
+2[ψ∗∂tψ−ψ∂tψ∗]−/planckover2pi12
+2m(∇ψ∗)·(∇ψ)−g
+2(ψ∗ψ)2.
+(A11)11
+Takingψ=√nei˜φwith the number density n, the Gross-
+Pitaevskii Lagrangian density becomes
+LGP=i/planckover2pi1
+2∂tn−/planckover2pi1n∂t˜φ−/planckover2pi12
+8mn(∂in)2−/planckover2pi12
+2mn(∂i˜φ)2−g
+2n2.
+(A12)
+The first term contributes i/planckover2pi1N/2 to the action (with
+Nthe number of particles) and does not contribute to
+the dynamics. Identifying ρ=mnandφ= (/planckover2pi1/m)˜φ,
+we see that the Gross-Pitaevskii Lagrangian density,
+Eq. (A12), is identical to the ideal fluid Lagrangian den-
+sity, Eq. ( A9), with
+UIF(ρ) =/planckover2pi12
+2m2(∇√ρ)2+g
+2m2ρ2.(A13)
+Appendix B: Thermodynamic Relations and the
+Static Correlation Functions of Supersolids
+In this appendix we calculate the thermodynamic re-
+lation for the potential energy density in Eq. ( 1). Given
+the Lagrangian density, Eq. ( 1), the total energy density
+forasupersolidis definedasthe sumofthe kinetic energy
+densities and the internal energy density
+ESS=1
+2ρsijvsivsj+1
+2(ρδij−ρsij)vnivnj
++USS(ρ,ρsij,s,Rij). (B1)
+Following Andreev and Lifshitz1, this total energy den-
+sity can be related to the energy density ǫmeasured in
+the frame where the super-component is at rest as
+ESS=1
+2ρv2
+s+(ρδij−ρsij)(vnj−vsj)vsi+ǫ,(B2)
+whereǫhas a thermodynamic relation
+dǫ=Tds+µdρ−λikdRik
++(vni−vsi)d/bracketleftbig
+(ρδij−ρsij)(vnj−vsj)/bracketrightbig
+.(B3)
+We can obtain the thermodynamic relation for the total
+energyESSbydifferentiatingEq.( B2)andusingEq.( B3)
+fordǫ, with the result
+dESS=Tds−λikdRik−(vni−vsi)vnjdρsij
++/bracketleftbigg
+µ+1
+2(2vn2
+i−2vnivsi+vs2
+i)/bracketrightbigg
+dρ
++ρsijvsjdvsi+(ρδij−ρsij)vnidvnj.(B4)
+This thermodynamic relation agrees with Eq. (2.18)
+of Saslow4and Eq. (2.1) of Liu5after identifyingǫSaslow, Liu=ESS, andµSaslow, Liu=µ−vnivsi+
+vs2
+i/2. Then the differentiation of Eq. ( B1) and the use
+of Eq. (B4) give the thermodynamic relation for USS,
+Eq. (3).
+For a supersolid at rest we can expand the free energy
+FSS=ESS−TSup to the second order in the density
+fluctuations δρand the strains wij=∂iuj:
+FSS=1
+2∂µ
+∂ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+wij(δρ)2+∂µ
+∂wij/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρδρwij+1
+2∂λij
+∂wlk/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρwijwlk.
+(B5)
+Using Eqs. ( 42) - (44) for an isotropic supersolid, the free
+energy (in Fourier space) can be written as
+FSS=1
+2˜µq2u2
+T+1
+2/parenleftbigδρ(q)uL(q)/parenrightbig
+B/parenleftbigg
+δρ(−q)
+uL(−q)/parenrightbigg
+,
+(B6)
+where
+B=/parenleftbigg1
+ρ2
+0χ−iqγ
+iqγ q2λ/parenrightbigg
+. (B7)
+Then the static correlation functions can be easily read
+off from Eq. ( B6):
+χρρ(q) =β/angb∇acketleftδρ(q)δρ(−q)/angb∇acket∇ight=ρ2
+0χλ
+λ−ρ2
+0γ2χ,(B8)
+χuLρ(q) =β/angb∇acketleftuL(q)δρ(−q)/angb∇acket∇ight=iρ2
+0χγ
+q(λ−ρ2
+0γ2χ).(B9)
+Appendix C: Calculation of the density-density
+correlation function
+Each term in the Kubo function given in Eq. ( 80) can
+be separated into the first sound part and the second
+sound part by performing a partial fraction expansion,
+ajkz3+bjkz2+djkq2z+q2ejk
+(z2−c2
+Lq2+izDLq2)(z2−c2
+2q2+izD2q2)
+=˜Ajkz+˜Bjk
+z2−c2
+Lq2+izDLq2+˜Cjkz+˜Djk
+z2−c2
+2q2+izD2q2,(C1)
+wherej,k= (ρ,uL). Then, ˜Ajk,˜Bjk,˜Cjkand˜Djkcan
+be written in terms of ajk,bjk,djkandejkalong with
+the sound velocities ( cLandc2) and the attenuation co-
+efficients (DLandD2):
+˜Ajk=ajk/bracketleftbig
+c4
+L−c2
+2c2
+L+q2DL(DLc2
+2−D2c2
+L)/bracketrightbig
+(c2
+L−c2
+2)2+q2(DL−D2)(DLc2
+2−D2c2
+L)+ibjk(c2
+2DL−D2c2
+L)+djk(c2
+L−c2
+2)+iejk(DL−D2)
+(c2
+L−c2
+2)2+q2(DL−D2)(DLc2
+2−D2c2
+L),(C2)12
+˜Bjk=iajkc2
+Lq2(DLc2
+2−D2c2
+L)+bjkc2
+L(c2
+L−c2
+2)
+(c2
+L−c2
+2)2+q2(DL−D2)(DLc2
+2−D2c2
+L)+idjkc2
+Lq2(DL−D2)+ejk/bracketleftbig
+c2
+L−c2
+2+q2DL(D2−DL)/bracketrightbig
+(c2
+L−c2
+2)2+q2(DL−D2)(DLc2
+2−D2c2
+L).(C3)
+The coefficients ˜Cjkand˜Djkare the same as ˜Ajkand
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+I2(q) =χρρ(q)˜Cρρ+χuLρ(q)˜CρuL,(C5)I3(q) =χρρ(q)˜Bρρ+χuLρ(q)˜BρuL−iq2DLI1(q),(C6)
+I4(q) =χρρ(q)˜Dρρ+χuLρ(q)˜DρuL−iq2D2I2(q).(C7)
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\ No newline at end of file
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+arXiv:1001.0622v1  [math.CV]  5 Jan 2010In the name of Allah, the Beneficent, the Merciful
+A radius of absolute convergence for power series in many var iables
+Ural Bekbaev
+Turin Polytechnical University in Tashkent,
+INSPEM, Universiti Putra Malaysia.
+e-mail: bekbaev@science.upm.edu.my
+Abstract
+In this paper for power series in many (real or complex)varia bles a radius of (absolute) convergence is
+offered. This radius can be evaluated by a formula similar to C auchy-Hadamard formula and in one variable
+case they are same.
+Mathematics Subject Classification: 32A05, 15A60.
+Key words: multiindex, power series in many variables.
+In [1] a matrix representation of polynomial maps was offered and by the use of a new product of matrices the
+matrix representation of composition of polynomial maps wa s given. In this paper an application of this product to
+power series in many (real or complex)variables is presente d. Namely, by the use of this product power series in many
+variables are presented in the form of power series in one var iable. Then by the use of ρ-norm (defined in [1]) and the
+Cauchy-Hadamard type formula a radius of (absolute) conver gence for such series is introduced and investigated. In one
+variable case it is the same Cauchy-Hadamard formula for rad ius of convergence of power series in one variable.
+Here are some definitions and results related to the new produ ct introduced in [1].
+For a positive integer nletInstand for all row n-tuples with nonnegative integer entries with the followin g linear
+order:β= (β1,β2,...,βn)< α= (α1,α2,...,αn) if and only if |β|<|α|or|β|=|α|andβ1> α1or|β|=|α|,β1=α1
+andβ2> α2etcetera, where |α|stands for α1+α2+...+αn.
+It is clear that for α,β,γ∈Inone has α < βif and only if α+γ < β+γ. We write β≪αifβi≤αifor all
+i= 1,2,...,n,/parenleftBigg
+α
+β/parenrightBigg
+stands forα!
+β!(α−β)!,α! =α1!α2!...αn!.
+In future nandn′are assumed to be any fixed positive integers. Let Fstand for the field of real or complex numbers.
+For any nonnegative integer numbers p,p′letMn,n′(p,p′;F) =M(p,p′;F) stand for all ” p×p′” size matrices
+A= (Aα,α′)|α|=p,|α′|=p′(αpresents row, α′presents column and α∈In,α′∈In′) with entries from F. Over such kind
+matrices in addition to the ordinary sum and product of matri ces we consider the following ”product”/circledottext
+as well:
+Definition 1. IfA∈M(p,p′;F) andB∈M(q,q′;F) thenA/circledottext
+B=C∈M(p+q,p′+q′;F) that for any |α|=p+q,
+|α′|=p′+q′, whereα∈In,α′∈In′,
+Cα,α′=/summationdisplay
+β,β′/parenleftBigg
+α
+β/parenrightBigg
+Aβ,β′Bα−β,α′−β′
+, where the sum is taken over all β∈In,β′∈In′, for which |β|=p,|β′|=p′,β≪αandβ′≪α′.
+Let us agree that h(H,v,V) stands for any element of M(0,1;F) (respect. M(0,p;F),M(1,0;F),M(p,0;F) ,
+wherepmay be any nonnegative integer). We use Ekfor ”k×k” size ordinary unit matrix from Mn,n(k,k;R). For the
+sake of convenience it will be assumed that Aα,α′= 0 (α! =∞) whenever α /∈Inorα′/∈In′(respect. α /∈In).
+Proposition 1. For the above defined product the following are true.
+11.A/circledottext
+B=B/circledottext
+A.
+2. (A+B)/circledottext
+C=A/circledottext
+C+B/circledottext
+C.
+3. (A/circledottext
+B)/circledottext
+C=A/circledottext
+(B/circledottext
+C)
+4. (λA)/circledottext
+B=λ(A/circledottext
+B) for any λ∈F
+5.A/circledottext
+B= 0 if and only if A= 0 orB= 0.
+6.A(B/circledottext
+H) = (AB)/circledottext
+H
+7. (Ek/circledottext
+V)A=A/circledottext
+V
+In future A(m)means the mth power of the matrix Awith respect to the new product.
+Proposition 2. Ifh= (h1,h2,...,hn)∈M(0,1;F),v= (v1,v2,...,vn)∈M(1,0;F), then
+(h(m))0,α′=/parenleftBigg
+m
+α′/parenrightBigg
+hα′,(v(m))α,0=m!vα
+, wherehαstands for hα1
+1hα2
+2...hαnn
+In future let ρ≥1 be any fixed real number and ̺stand for the real number for which1
+ρ+1
+̺= 1. We consider the
+following ρ-norm of elements A∈M(p,p′;F):
+Definition 2.
+/bardblA/bardbl=/bardblA/bardblρ= (/summationdisplay
+α,α′|Aα,α′|ρ
+α!(p!p′!)ρ−1)1/ρ
+In the case of ρ=∞theρ-norm is defined by
+/bardblA/bardbl=/bardblA/bardbl∞=supα,α′|Aα,α′|
+p!p′!
+Theorem 1. 1. IfA,B∈Mat(p,p′;F) andλ∈Fthen
+a)/bardblA/bardbl= 0 if and only if A= 0,
+b)/bardblλA/bardbl=|λ|/bardblA/bardbl,
+c)/bardblA+B/bardbl ≤ /bardblA/bardbl+/bardblB/bardbl.
+2. For any nonnegative integer numbers p,p′,qandq′there is such a positive number λ(p,p′,q,q′) that for any
+A∈Mat(p,p′;F),B∈Mat(q,q′;F) the following inequality is valid:
+λ(p,p′,q,q′)/bardblA/bardbl/bardblB/bardbl ≤ /bardblA/circledotdisplay
+B/bardbl ≤ /bardblA/bardbl/bardblB/bardbl
+Proof.Here is a proof of part 2. First let us show the inequality /bardblA/circledottext
+B/bardbl ≤ /bardblA/bardbl/bardblB/bardbl.
+Due to the H¨ older inequality for A/circledottext
+B=Cone has
+|Cα,α′|=|/summationdisplay
+β≪α,β′≪α′/parenleftBigg
+α
+β/parenrightBigg
+Aβ,β′Bα−β,α′−β′| ≤α!/summationdisplay
+β≪α,β′≪α′|Aβ,β′Bα−β,α′−β′|
+(β!(α−β)!)1/ρ1
+(β!(α−β)!)1/̺≤
+α!(/summationdisplay
+β≪α,β′≪α′|Aβ,β′Bα−β,α′−β′|ρ
+β!(α−β)!)1/ρ(/summationdisplay
+β≪α,β′≪α′(1
+(β!(α−β)!)1/̺)̺)1/̺=
+α!(/summationdisplay
+β≪α,β′≪α′|Aβ,β′|ρ
+β!|Bα−β,α′−β′|ρ
+(α−β)!)1/ρ(/summationdisplay
+β≪α1
+β!(α−β)!/summationdisplay
+β′≪α′1)1/̺≤
+α!(/summationdisplay
+β≪α,β′≪α′|Aβ,β′|ρ
+β!|Bα−β,α′−β′|ρ
+(α−β)!)1/ρ(/parenleftBigg
+p+q
+p/parenrightBigg
+1
+α!/parenleftBigg
+p′+q′
+p′/parenrightBigg
+)1/̺
+2as far as according to Proposition 1 one has/summationtext
+β≪α1
+β!(α−β)!=/parenleftBigg
+p+q
+p/parenrightBigg
+1
+α!and/summationtext
+β′≪α′1≤/parenleftBigg
+p′+q′
+p′/parenrightBigg
+. Therefore
+/bardblC/bardbl= (/summationdisplay
+α,α′|Cα,α′|ρ
+α!((p+q)!(p′+q′)!)ρ−1)1/ρ≤
+(/summationdisplay
+α,α′1
+α!((p+q)!(p′+q′)!)ρ−1(α!)ρ/summationdisplay
+β≪α,β′≪α′|Aβ,β′|ρ
+β!|Bα−β,α′−β′|ρ
+(α−β)!(/parenleftBigg
+p+q
+p/parenrightBigg
+1
+α!/parenleftBigg
+p′+q′
+p′/parenrightBigg
+)ρ/̺)1/ρ=
+(/summationdisplay
+ββ′|Aβ,β′|ρ
+β!(p!p′!)ρ−1)1/ρ(/summationdisplay
+γγ′|Bγγ′|ρ
+γ!(q!q′!)ρ−1)1/ρ=/bardblA/bardbl/bardblB/bardbl
+due toρ/̺=ρ−1
+To show the inequality λ(p,p′,q,q′)/bardblA/bardbl/bardblB/bardbl ≤ /bardblA/circledottext
+B/bardbllet us consider
+X={(A,B) :A∈Mat(p,p′;F),/bardblA/bardbl= 1,B∈Mat(q,q′;F),/bardblB/bardbl= 1}
+, which is a compact set in the corresponding finite dimension al vector space, and the continuous map ( A,B)/mapsto→A/circledottext
+B.
+The image of X, with respect to this map, is a compact set which doesn’t cont ain zero vector due to Proposition 1.
+Letλ(p,p′,q,q′)>0 stand for the distance between zero vector and this image se t with respect to the corresponding ρ-
+norm. So λ(p,p′,q,q′)≤ /bardblA/circledottext
+B/bardblfor any ( A,B)∈Xand due to Proposition 1 one has
+λ(p,p′,q,q′)/bardblA/bardbl/bardblB/bardbl ≤ /bardblA/circledotdisplay
+B/bardbl
+for anyA∈Mat(p,p′;F),B∈Mat(q,q′;F)
+Remark. It would be nice if one could offer an expression for
+λ(p,p′,q,q′) = inf
+/bardblA(p,p′)/bardbl=/bardblB(q,q′)/bardbl=1/bardblA/circledotdisplay
+B/bardblρ
+in terms of n,n′,p,p′,q,q′andρ.
+With respect to the ordinary product of matrices a result sim ilar to/bardblA/circledottext
+B/bardbl ≤ /bardblA/bardbl/bardblB/bardblis not valid. But one can
+have the following result.
+Proposition 3. The following inequality
+/bardblA(p,q)B(q,q′)/bardbl ≤(q!)2−1/ρ(q′!)2/ρ−1/bardblA/bardbl/bardblB/bardbl̺
+is true.
+Proof.Indeed due to the H¨ older inequality one has
+/bardblA(p,q)B(q,q′)/bardblρ=/summationdisplay
+α,α′1
+α!(p!q′!)ρ−1|/summationdisplay
+βAα,βBβ,α′|ρ≤/summationdisplay
+α,α′1
+α!(p!q′!)ρ−1/summationdisplay
+β|Aα,β|ρ(/summationdisplay
+γ|Bγ,α′|̺)ρ/̺=
+/summationdisplay
+α,β|Aα,β|ρ
+α!(p!q!)ρ−1(/summationdisplay
+γ,α′|Bγ,α′|̺
+γ!(q!q′!)̺−1γ!)ρ/̺(q!)ρ(q′!)2−ρ≤ /bardblA/bardblρ/bardblB/bardblρ
+̺(q!)2ρ−1(q′!)2−ρ
+as far as γ!≤q!.
+In particular case the following estimation is also true.
+Proposition 4. For any nonnegative integer numbers m,k,q′andh∈Matn,n(0,1;F),
+A∈Matn,n′(m+k,q′;F) the following inequality
+/bardbl(h(m)
+m!/circledotdisplay
+Ek)A/bardbl ≤/parenleftBigg
+m+k
+k/parenrightBigg
+/bardblh/bardblm
+̺/bardblA/bardbl
+3is valid.
+Proof.Indeed
+/bardbl(h(m)
+m!/circledotdisplay
+Ek)A/bardblρ=/summationdisplay
+α,α′1
+α!(k!q′!)ρ−1|((h(m)
+m!/circledotdisplay
+Ek)A)α,α′|ρ=/summationdisplay
+α,α′1
+α!(k!q′!)ρ−1|/summationdisplay
+β(h(m)
+m!/circledotdisplay
+Ek)α,βAβ,α′|ρ=
+/summationdisplay
+α,α′1
+α!(k!q′!)ρ−1|/summationdisplay
+βhβ−α
+(β−α)!1/̺Aβ,α′
+(β−α)!1/ρ|ρ
+as far as
+(h(m)
+m!/circledotdisplay
+Ek)α,β=hβ−α
+(β−α)!
+Due to the H¨ older inequality
+(/summationdisplay
+β|hβ−α
+(β−α)!1/̺Aβ,α′
+(β−α)!1/ρ|)ρ≤/summationdisplay
+β|Aβ,α′|ρ
+(β−α)!(/summationdisplay
+β|h̺(β−α)|
+(β−α)!)ρ/̺=/summationdisplay
+β|Aβ,α′|ρ
+(β−α)!(/bardblh/bardblm̺
+̺
+m!)ρ/̺
+Therefore
+/bardbl(h(m)
+m!/circledotdisplay
+Ek)A/bardblρ≤/summationdisplay
+β,α′|Aβ,α′|ρ
+β!((m+k)!q′!)ρ−1/parenleftBigg
+m+k
+k/parenrightBiggρ−1/summationdisplay
+α/parenleftBigg
+β
+α/parenrightBigg
+/bardblh/bardblmρ
+̺=
+/summationdisplay
+β,α′|Aβ,α′|ρ
+β!((m+k)!q′!)ρ−1/parenleftBigg
+m+k
+k/parenrightBiggρ
+/bardblh/bardblmρ
+̺=/bardblA/bardblρ/parenleftBigg
+m+k
+k/parenrightBiggρ
+/bardblh/bardblmρ
+̺
+Corollary. Foranynonnegative integer number m,q′,A∈Matn,n′(m,q′;F)andhi∈Matn,n(0,1;F),i= 1,2,...,m
+the following inequality
+/bardblh1/circledottext
+h2/circledottext
+.../circledottext
+hm
+m!A/bardbl ≤ /bardblh1/bardbl̺/bardblh2/bardbl̺.../bardblhm/bardbl̺/bardblA/bardbl
+is valid.
+Fromnowletusassumethat x1,x2,...,xnarevariablesover F,q′isafixednonnegativeintegerand x= (x1,x2,...,xn)∈
+Mn,n(0,1;F[x]).
+Nowwearegoingtoconsideranapplicationofthenewproduct topowerseries/summationtext
+α∈Inxαaα, whereaα∈Mat(0,q′;F).
+To do it we represent the power series/summationtext
+α∈Inxαaαin the form/summationtext∞
+m=0x(m)
+m!A(m), where A(m)∈M(m,q′;F).
+It is well known that all ρ-norms define the same topology in Fn. Therefore one can speak about convergence of the
+above power series without refereing to any particular ρ-norm.
+Definition 3. A power series/summationtext∞
+m=0x(m)
+m!A(m) is said to be absolute convergent at h∈Fnif its each component is
+absolute convergent at h∈Fne.i. for each α′∈In′,|α′|=q′, the positive series/summationtext
+α∈In|hα
+α!A(m)α,α′|converges.
+Due to the inequality
+|h(m)
+m!A(m)| ≤ /bardblh/bardblm
+̺/bardblA(m)/bardbl
+(Proposition 4) the power series
+∞/summationdisplay
+m=0x(m)
+m!A(m) (1)
+absolutely converges whenever /bardblx/bardbl̺< R=1
+r, wherer=limm→∞/bardblA(m)/bardbl1
+mρ.
+Theorem 2. Power series (1) is absolute convergent at h∈Fnwhenever /bardblh/bardbl̺< Rand for any R1> Rnρ−1
+ρthere
+exists such h∈Fnthat/bardblh/bardbl̺=R1and power series (1) is not absolute convergent at h
+Proof.
+IfR=∞there is nothing to prove. Assume that 1 ≤ρ <∞,R <∞,R1> Rnρ−1
+ρand for any h∈Fnfor which
+/bardblh/bardbl̺=R1power series (1) is absolute convergent at h. Due to convergence of the series
+4/summationtext∞
+m=0/summationtext
+|α|=m|hα
+α!A(m)α,α′|for big enough mone has
+|hα
+α!A(m)α,α′| ≥ |hα
+α!A(m)α,α′|ρ≥ |hα|ρ|A(m)α,α′|ρ
+α!(m!q′!)ρ−1
+and therefore the series/summationdisplay
+|α′|=q′∞/summationdisplay
+m=0/summationdisplay
+|α|=m|hα|ρ|A(m)α,α′|ρ
+α!(m!)ρ−1
+should converge whenever h∈Fnand/bardblh/bardbl̺=R1. Consider this series for h= (R1n−1
+̺,R1n−1
+̺,...,R 1n−1
+̺)∈Fnfor
+which/bardblh/bardbl̺=R1.
+/summationdisplay
+|α′|=q′∞/summationdisplay
+m=0/summationdisplay
+|α|=m|hα|ρ|A(m)α,α′|ρ
+α!(m!q′!)ρ−1=∞/summationdisplay
+m=0(R1n−1
+̺)mρ/summationdisplay
+|α|=m,|α′|=q′|A(m)α,α′|ρ
+α!(m!q′!)ρ−1=∞/summationdisplay
+m=0(R1n−1
+̺)mρ/bardblA(m)/bardblρ
+But the radius of convergence of the ordinary number series/summationtext∞
+m=0tm/bardblA(m)/bardblρequalsRρand (R1n−1
+̺)ρ> Rρthis
+contradiction indicates that the Theorem is true in this cas e.
+Let us consider now the ρ=∞case. Assume that R1> Rnand power series (1) is absolute convergent at any
+h∈Fnfor which /bardblh/bardbl1=R1. In particular for any α′∈In′,|α′|=q′andh= (R1n−1
+̺,R1n−1
+̺,...,R 1n−1
+̺)∈Fn, for
+which/bardblh/bardbl1=R1, the series
+∞/summationdisplay
+m=0/summationdisplay
+|α|=m|hα
+α!A(m)α,α′|=∞/summationdisplay
+m=0(R1
+n)m/summationdisplay
+|α|=m|A(m)α,α′
+α!|
+is convergent. In this case the series
+∞/summationdisplay
+m=0(R1
+n)m/summationdisplay
+|α|=m,|α′|=q′|A(m)α,α′
+m!q′!|
+is convergent as well. Due to the equality/summationtext
+|α|=m1 =/parenleftBigg
+m+n−1
+n−1/parenrightBigg
+the following inequality
+/bardblA(m)/bardbl∞≤/summationdisplay
+|α|=m,|α′|=q′|A(m)α,α′
+m!q′!| ≤ /bardblA(m)/bardbl∞/parenleftBigg
+m+n−1
+n−1/parenrightBigg/parenleftBigg
+q′+n′−1
+n′−1/parenrightBigg
+is clear. It implies that the radius of convergence of the pow er series/summationtext∞
+m=0tm/summationtext
+|α|=m,|α′|=q′|A(m)α,α′
+m!q′!|is the same
+R. But in our caseR1
+n> Rand/summationtext∞
+m=0(R1
+n)m/summationtext
+|α|=m,|α′|=q′|A(m)α,α′
+m!q′!|converges. This contradiction indicates that the
+Theorem is true in this case as well.
+Due to this result if for power series (1) the radius of conver genceRis zero then in any neighborhood of zero one can
+find a point where (1) is not absolute convergent. This theore m indicates also a privileged position of 1-norm among all
+ρ-norms as far as in this case Rnρ−1
+ρ=Rand forρ >1 power series (1) has a solid layer
+{h∈Fn:R≤ /bardblh/bardbl̺≤Rnρ−1
+ρ}
+of indeterminacy. Investigation the behavior of power seri es (1)in this layer of indeterminacy could be interesting. I t is
+hoped that the emergence of a layer of indeterminacy is not a d efect of our approach.
+Question 1. Let/summationtext
+αaαxαbe any series for which/summationtext∞
+m=0Hm(x), where Hm(x) =/summationtext
+|α|=maαxα, is absolute
+convergent in some neighborhood of zero. Does it imply that t he original series/summationtext
+αaαxαis also absolute convergent in
+some neighborhood of zero as well?
+Question 2. Consider power series (1), for each nonnegative mmultilinear map
+(Fn)m−→Fn′: (h1,h2,...,hm)/mapsto−→(h1/circledotdisplay
+h2/circledotdisplay
+.../circledotdisplay
+hm)A(m)
+5and its norm /bardblA(m)/bardbl([2]) defined by
+/bardblA(m)/bardbl= sup
+/bardblh1/bardbl̺=/bardblh2/bardbl̺=...=/bardblhm/bardbl̺=1/bardbl(h1/circledotdisplay
+h2/circledotdisplay
+.../circledotdisplay
+hm)A(m)/bardblρ
+According to the above Corollary for each mthe inequality /bardblA(m)/bardbl ≤ /bardblA(m)/bardblρis true. Is it true that
+limm→∞/bardblA(m)/bardbl1
+m=limm→∞/bardblA(m)/bardbl1
+mρ
+References
+[1] U.Bekbaev. A matrix representation of composition of po lynomial maps. arXiv:math 0901.3179v3
+[2] H.Cartan. Calcul diff´ erntiel. Forms diff´ erentielles. Hermann. Paris, 1967.
+6
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+arXiv:1001.0624v1  [math.LO]  5 Jan 2010RANK FUNCTIONS AND PARTIAL STABILITY SPECTRA
+FOR TAME ABSTRACT ELEMENTARY CLASSES
+MICHAEL J. LIEBERMAN∗
+November 9, 2018
+Abstract
+We introduce a family of rank functions and related notions o f total transcendence
+for Galois types in abstract elementary classes. We focus, i n particular, on abstract
+elementary classes satisfying the condition know as tamene ss—currently regarded as a
+necessary condition for the development of a reasonable cla ssification theory—where the
+connections betweenstabilityandtotal transcendenceare mostevident. Asabyproduct,
+we obtain a partial upward stability transfer result for tam e abstract elementary classes
+stable in a cardinal λsatisfying λℵ0> λ, a substantial generalization of a result of
+[BaKuVa06].
+1INTRODUCTION
+For several decades, one of the overwhelming preoccupations of model theorists has been the
+pursuit of a classification theory for nonelementary classes, enco mpassing both the resolution
+of various categoricity conjectures generalizing Morley’s theorem and the development of
+something resembling stability theory in the non-first-order conte xt. The impetus to study
+such questions is as much practical as theoretical: many natural m athematical objects defy
+first-orderaxiomatization (e.g. Banach spaces, Artinian commuta tive rings, and the complex
+numbers with exponentiation), and are thus beyond the scope of t he results and methods
+of the classical theory. On a theoretical level, the study of nonele mentary classes offers
+an opportunity to obtain a deeper understanding of how classical s tability theory actually
+works—which aspects are purely general, and which are inextricably linked to the unique
+structure of first-order logic.
+The initial analysis of the model theory of more general logics (such asLκ,ω) was largely
+rooted in syntactic considerations, unfailingly identifying types with satisfiable sets of for-
+mulas, and relying on methods closely tailored to the peculiarities of th e logics in question.
+Abstract elementary classes (AECs) and their associated, funda mentally non-syntactic Ga-
+lois types (both notions due to Shelah) represent a broad, unifying framework that supports
+a moreuniform treatment of questionsconcerningcategoricityan d stability, transcending the
+focus on individual logics that limited the cope of earlier assays. Esse ntially the category-
+theoretic hulls of elementary classes—we discard syntax entirely, b ut retain the essential
+diagrammatic properties of elementary embeddings—AECs are suffic iently general to en-
+compass classes of models of sentences in L∞,ω,L(Q), andLω1,ω(Q), as well as homogeneous
+∗The contents of this paper are drawn from the author’s doctor al dissertation, completed at the University
+of Michigan under the supervision of Andreas Blass. The auth or also acknowledges useful conversations with
+John Baldwin and Alexei Kolesnikov.
+1classes, but also seemingly rich enough in structure to permit the de velopment of an inter-
+esting classification theory.
+Naturally, with the added generality comes added difficulty: for AECs satisfying a variety
+of conditions intended to guarantee reasonable behavior, two dec ades of work have typically
+yielded only partial categoricity and stability transfer results. In [J aSh08] and [Sh600], for
+example, Shelah has established a variety of broad structural res ults (including a version of
+his categoricityconjecture) for AECs with good or semi-good fram es; that is, AECs equipped
+with forking relations reminiscent of those associated with stable or superstable theories in
+the classical realm. In [GrVa06c], Grossberg and Vandieren obtain a large-scale categoricity
+transfer result for the considerably larger class of AECs in which th e Galois types satisfy the
+important locality condition known as tameness using a toolkit that inc ludes such classical
+standbys as indiscernibles and Morley sequences. With these same m ethods, augmented
+with the notion of splitting, Grossberg and VanDieren also establish, in [GrVa06b], a partial
+stability spectrum result for tame AECs.
+Most recently, Baldwin, Kueker and VanDieren use a splitting argume nt to prove a strik-
+ing upward stability transfer result for ℵ0-Galois-stable AECs (Theorem 2.1 in [BaKuVa06]):
+ifκis a cardinal of uncountable cofinality and the AEC is Galois stable in eve ry cardinal
+less thanκ, it must be Galois stable in κas well. We here introduce an entirely new set
+of vaguely classical methods: a family of rank functions for Galois ty pes, which bear a cer-
+tain resemblance to Morley rank, and use these to prove a generaliz ation of the result of
+[BaKuVa06]. In particular, we show that this kind of transfer is can b e triggered not merely
+byℵ0-Galois stability, but by stability in any cardinal λsatisfying the inequality λℵ0> λ,
+and significantly weaken the assumption on stability below κ.
+2PRELIMINARIES
+We begin with a very brief introduction to AECs, Galois types, and a fe w relevant properties
+thereof. Naturally this exposition will not be exhaustive; readers in terested in further details
+may wish to consult [Ba08] or [Gr02]. To begin:
+Definition 2.1. LetLbe a finitary signature (one-sorted, for simplicity). A class of L-
+structures equipped with a strong submodel relation, ( K,≺K), is anabstract elementary class
+(AEC)if bothKand≺Kare closed under isomorphism, and satisfy the following axioms:
+A0The relation ≺Kis a partial order.
+A1For allM,NinK, ifM≺KN, thenM⊆LN.
+A2(Unions of Chains) Let ( Mα|α<δ) be a continuous ≺K-increasing sequence.
+1./uniontext
+α<δMα∈ K.
+2. For allα<δ,Mα≺K/uniontext
+α<δMα.
+3. IfMα≺KMfor allα<δ, then/uniontext
+α<δMα≺KM.
+A3(Coherence) If M0,M1≺KMinK, andM0⊆LM1, thenM0≺KM1.
+A4(Downward L¨ owenheim-Skolem) There exists an infinite cardinal LS (K) with the prop-
+erty that for any M∈ Kand subset AofM, there exists M0∈ KwithA⊆M0≺KM
+and|M0| ≤ |A|+LS(K).
+2The prototypical example, of course, is the case in which Kis an elementary class—the
+class of models of a particular first-order theory T—and≺Kis the elementary submodel
+relation, in which case LS( K) is, naturally, ℵ0+|L(T)|.
+For any infinite cardinal λ, we denote by Kλthe subclass of Kconsisting of all models of
+cardinality λ(with the obvious interpretations for such notations as K≤λandK>λ). We say
+thatKisλ-categorical if Kλcontains only a single model up to isomorphism. For M,N∈ K,
+we say that a map f:M→Nis aK-embedding (or, more often, a strong embedding) if
+fis an injective homomorphism of L(K)-structures, and f[M]≺KN, that is;finduces an
+isomorphism of Monto a strong submodel of N. In that case, we write f:M ֒→KN. We
+also adopt the following notational shorthand: for any M∈ Kand any cardinal λ, we denote
+by Sub ≤λ(M) the set of all strong submodels N≺KMwith|N| ≤λ.
+Definition 2.2. LetKbe an AEC.
+1. We say that an AEC Khas thejoint embedding property (JEP) if for anyM1,M2∈ K,
+there is an M∈ Kthat admits strong embeddings of both M1andM2,fi:Mi֒→KM
+fori= 1,2.
+2. We say that an AEC Khas theamalgamation property (AP) if for anyM0∈ Kand
+strong embeddings f1:M0֒→KM1andf2:M0֒→KM2, there are strong embeddings
+g1:M1֒→KNandg2:M2֒→KNsuch thatg1◦f1=g2◦f2.
+Notice that both properties hold in elementary classes, as a conseq uence of the compact-
+ness of first-order logic. In this more general context, devised t o subsume classes of models
+in logics without any compactness to fall back on, both appear as ad ditional (and nontrivial)
+assumptions on the class. We will work exclusively with AECs that satis fy both the joint
+embedding and the amalgamation properties.
+It is not immediately clear what we might embrace as a suitable notion of type in AECs,
+given that we have dispensed with syntax, and removed ourselves t o a world of abstract
+embeddings and diagrams thereof. The best candidate—the Galois t ype—has its origins in
+the work of Shelah (first appearing in [Sh88] and [Sh394]). Although G alois types can be
+defined in very general AECs, we restrict our attention to those w ith amalgamation and joint
+embedding. In any AEC Kof this form, we may fix a monster model C∈ K, and consider
+all modelsM∈ Kas strong submodels of Cwith|M|<|C|. In this case, Galois types have
+a particularly simple characterization.
+Definition 2.3. LetM∈ K, anda∈C. TheGalois type of aoverM, denoted ga-tp( a/M),
+is the orbit of ainCunder Aut M(C), the group of automorphisms of Cthat fixM. We
+denote by ga-S( M) the set of all Galois types over M.
+In caseKis an elementary class with ≺Kas elementary submodel, the Galois types over
+Mcorrespond to the complete first-order types over M:
+ga-tp(a/M) = ga-tp(b/M) if and only if tp( a/M) = tp(b/M)
+In general, however, Galois types and syntactic types do not matc h up, even in cases when
+the logic underlying the AEC is clear (say, K= Mod(ψ), withψ∈Lω1,ω). We run through
+a series of basic definitions and notations:
+Definition 2.4. 1. We say that Kisλ-Galois stable if for every M∈ Kλ,|ga-S(M)|=λ.
+2. For any M,a∈C, andN≺KM, therestriction of ga-tp( a/M) toN, which we denote
+by ga-tp(a/M)↾N, is the orbit of aunder Aut N(C). This notion is well-defined: the
+restriction depends only on ga-tp( a/M), not onaitself.
+33. LetN≺KMandp∈ga-S(N). We say that Mrealizespif there is an element a∈M
+such that ga-tp( a/M)↾N=p. Equivalently, Mrealizespif the orbit in Ccorrespond-
+ing topmeetsM.
+4. We say that a model Misλ-Galois-saturated if for every N≺KMwith|N|< λand
+everyp∈ga-S(N),pis realized in M.
+Henceforth, the word “type” should be understood to mean “Galo is type,” unless other-
+wise indicated. Moreover, when there is no risk of confusion, we will o mit the word “Galois”
+altogether, speaking simply of types, λ-stability, and λ-saturation.
+As in first-order classification theory, one of the central preocc upations of those working
+with AECsisthe identificationof“dividinglines,” inthe senseofShelah: p ropertiesofclasses
+which, when present, guarantee nice structural properties, pr operties that disappear when
+onefindsoneselfonthe wrongsideofadividingline, onlytobereplaced bynewmisbehaviors;
+that is, “nonstructure.” Certain of these properties for AECs (e xcellence, as described in
+[GrKo05], and the existence of good or semi-good frames, as consid ered in [Sh600] and
+[JaSh08], respectively) echo classical dividing lines (simplicity, stability , and superstability),
+but we here concern ourselves primarily with a property that has no particularly natural
+classical analogue: tameness of Galois types. Roughly speaking, ta meness means that types
+are completely determined by their restrictions to small submodels, a condition slightly
+reminiscent of the locality properties of syntactic types. A possible intuition is the following:
+we may regard types over small models as playing a role analogous to f ormulas in first-
+order model theory, in which case tameness guarantees that typ es are determined by their
+constituent “formulas.” In this analogy, tameness of Galois types m ay in fact be most closely
+identified with completeness in the realm of syntactic types.
+We present two degrees of tameness:
+Definition 2.5. 1. We say that Kisχ-tameif for every M∈ K, ifpandp′are distinct
+types overM, then there is an N∈Sub≤χ(M), such that p↾N/ne}ationslash=p′↾N.
+2. We say that Kisweaklyχ-tameif for every saturated M∈ K, ifpandp′are distinct
+types overM, then there is an N∈Sub≤χ(M), such that p↾N/ne}ationslash=p′↾N.
+A few words about the significance of the assumption of tameness: although it holds
+automaticallyinelementaryclasses(aswellasin homogeneousclasse s,amongothersettings),
+it fails in some contexts—[BaKo07] and [BaSh07], for example, constr uct examples of non-
+tame AECs from Abelian groups. Moreover, it is independent from th e other properties
+one associates with well-behaved AECs. It is established in [BaSh07], f or example, that
+given an AEC with amalgamation, there is an AEC without amalgamation w ith precisely
+the same tameness spectrum. Its utility more than justifies the los s of generality involved
+in its assumption, though: it plays an indispensable role in establishing a ll of the existing
+upward categoricity transfer results (such as those of [GrVa06a ], [GrVa06c], and [Sh394]),
+the downward transfer result included in Chapter 15 of [Ba08], and n early all of the partial
+stability spectrum results of [BaKuVa06], [GrVa06b], and [Li09b], altho ugh weak tameness
+occasionally suffices in [BaKuVa06] and [Li09b]. At present, attempts to produce results of
+this nature for non-tame classes have not met with any success.
+We close with a brief topologicalaside (a more detailed treatment can be found in [Li09c])
+which, although it will not necessarily be emphasized in the sequel, pro vides essential moti-
+vation for the definitions that follow. Recall the identification that f ormed the basis of our
+intuition concerning tameness of Galois types—types over small sub models as “formulas”—
+and fix our notion of “small” as “of cardinality less than or equal to λ,” withλ≥LS(K).
+4Just as the Stone space topology on syntactic types is generated from a basis of clopen sets,
+each one consisting of all complete types containing a given formula, we here consider sets
+of Galois types extending given “formulas”; that is, for each model M∈ K, we consider all
+sets of the form
+Up,N={q∈ga-S(M) :q↾N=p}
+whereN∈Sub≤λ(M) andp∈ga-S(N). It is an easy exercise to show that the Up,Nform
+a basis for a topology on ga-S( M)—we denote the resulting space by Xλ
+M—and, moreover,
+that they are clopen sets, analogous to the first-order case. In fact, ifKis an elementary
+class (in which case Galois types may be identified with complete first-o rder types), we see
+that the topologies thus obtained refine the standard syntactic t opology.
+In fact, what we obtain is a spectrum of spaces Xλ
+Massociated with each M∈ K, with
+one space for each λ≥LS(K), and a tight correspondence between topological properties
+of the spaces in the various spectra and model-theoretic propert ies of the AEC (see [Li09c]
+for a complete account). In particular, χ-tameness of an AEC means precisely that any two
+distinct types over a model M∈ Kcan be distinguished by their restrictions to a submodel
+ofMof size at most χ; that is, they can be separated by basic open sets in each of the sp aces
+Xλ
+Mforλ≥χ. It should come as no surprise, then, that we have:
+Proposition 2.6. IfKisχ-tame,Xλ
+Mis Hausdorff for all M∈ Kandλ≥χ.
+Furthermore, in light of the fact that every Xλ
+Mhas a basis of clopens, the spaces in
+question will be totally disconnected—nearly Stone spaces. As it hap pens, though, tameness
+rules out not merely compactness, but even countable compactne ss. The chief complication
+results from the following result:
+Proposition 2.7. For anyM∈ Kandλ≥LS(K), the intersection of any λopen subsets
+ofXλ
+Mis open.
+Proof.It suffices to show that the intersection of λbasic open sets is open. To that end,
+let{Upi,Ni|i< λ}be a family of open subsets of Xλ
+M, and letq∈/intersectiontext
+i<λUpi,Ni. The union
+of theNiis of cardinality λso, by the Downward L¨ owenheim Skolem Property, there is a
+submodelN ֒→KMthat contains it. By coherence, Ni֒→KNfor alli. Consider the basic
+open neighborhood Uq↾N,N. It certainly contains q. For anyq′∈Uq↾N,N, moreover,
+q′↾Ni=q′↾N↾Ni=q↾N↾Ni=q↾Ni=pi
+Henceq′is contained in/intersectiontext
+i<λUpi,Niand, as this holds for any such q′,Uq↾N,N⊆/intersectiontext
+i<λUpi,Ni.
+As a result, if an AEC is χ-tame, the spaces Xλ
+Mforλ≥χexhibit an extraordinary
+degree of separation, as captured in our final topological results , Propositions 6.5 and 7.1 in
+[Li09c], respectively:
+Proposition 2.8. LetKbeχ-tame, and λ≥χ. Any setS⊆Xλ
+Mwith|S| ≤λis closed and
+discrete.
+Thus there will be a number of infinite sets with no accumulation point, ruling out
+countable compactness. We close with a closely-related character ization of what it means to
+be an accumulation point in Xλ
+M:
+Proposition 2.9. LetKbeχ-tame, and λ≥χ. A typeq∈Xλ
+Mis an accumulation point
+of a setS⊆Xλ
+Mif and only if for every neighborhood Uofq,|U∩S|>λ.
+53RANKS FOR GALOIS TYPES
+WehavejustseenthatifanAEC Kisχ-tame, thenforany λ≥χ(and, ofcourse, λ≥LS(K)),
+the criterion for a type q∈ga-S(M) to be an accumulation point of a family of types in Xλ
+M
+is quite stringent: every neighborhood must contain more than λtypes in the given family.
+This condition inspires the definition of the following Morley-like ranks:
+Definition 3.1. [RMλ] Assume Kisχ-tame for some χ≥LS(K). Forλ≥χ, we define
+RMλby the following induction: for any q∈ga-S(M) with|M| ≤λ,
+•RMλ[q]≥0.
+•RMλ[q]≥αfor limitαif RMλ[q]≥βfor allβ <α.
+•RMλ[q]≥α+1 if there exists a structure M′≻KMsuch thatqhas strictly more than
+λextensions to types q′overM′with the property that
+RMλ[q′↾N]≥αfor allN∈Sub≤λ(M′)
+For typesqoverMof arbitrary size, we define
+RMλ[q] = min{RMλ[q↾N] :N∈Sub≤λ(M)}.
+Before we proceed, a bit of motivation for this longwinded definition: in topologizing
+ga-S(M) asXλ
+M(and in developing our intuition with regard to tameness), we essent ially
+allowthe types oversubstructures of size at most λto play a role analogousto that ordinarily
+played by formulas. In defining the ranks RMλ, we first define the rank of these “formulas”
+and subsequently define the rank on types extending them as the m inimum of the ranks
+of their constituent “formulas,” just as in the definition of Morley ra nk in classical model
+theory.
+There is something to check here, though. The types over models o f cardinality at most
+λwhose ranks were defined by the inductive clause are also covered b y the second clause.
+We must ensure that there is no possibility of conflict between the tw o. Let RMλ
+∗denote the
+ranks assigned to such types using only the inductive clause.
+Lemma 3.2. The ranks RMλ
+∗are monotone: for any N,N′∈ K≤λ,N′≺KN, and any
+q∈ga-S(N), RMλ
+∗[q]≤RMλ
+∗[q↾N′].
+Proof:We show that for any ordinal α, RMλ
+∗[q]≥αimplies RMλ
+∗[q↾N′]≥α. We proceed
+by an easy induction on α. The zero case is trivial, by the first bullet point. The limit cases
+follow from the induction hypothesis. For the successor case, not ice that if RMλ
+∗[q]≥α+1,
+the extension M≻KNthat witnesses this fact is also an extension of N′and witnesses that
+RMλ
+∗[q↾N′]≥α+1.✷
+So we needn’t worry that the rank assigned to a type q∈ga-S(M) withM∈ K≤λunder
+the second clause (RMλ[q] = min{RMλ[q↾N] :N≺KM,|N| ≤λ}) will differ from the rank
+ofpunder the first clause. Hence the ranks RMλare well-defined. In fact, we can now restate
+the third bullet point above in a more compact form: for q∈ga-S(M) withM∈ K≤λ,
+•RMλ[q]≥α+1 if there exists a structure M′≻KMsuch thatqhas strictly more than
+λmany extensions to types q′overM′with RMλ[q′]≥α.
+We will invariably use this formulation in the sequel.
+We now consider the basic properties of the ranks RMλ, beginning with an extension of
+our partial monotonicity result, Lemma 3.2.
+6Proposition 3.3. [Monotonicity] If M≺KM′andq∈ga-S(M′), then RMλ[q]≤RMλ[q↾
+M].
+Proof:Trivial, from the second clause of the definition. ✷
+Bear in mind that “ pis a restriction of q” corresponds to the first-order “ q⊢p,” so this
+is indeed an appropriate analogue of the monotonicity result for the classical Morley rank.
+Proposition 3.4. [Invariance] Let fbe an automorphism of the monster model C, letq∈
+ga-S(M), and letM′=f[M]. Then the type f[q]∈ga-S(M′) satisfies RMλ[f[q]] =RMλ[q].
+Proof:We again proceed by induction, and once again the zero and limit cases are trivial.
+It remains to show that whenever RMλ[q]≥α+ 1, RMλ[f[q]]≥α+ 1, with the converse
+following by replaying the argument with f−1in place of f. As before, if RMλ[q]≥α+1,
+then eachN∈Sub≤λ(M) has an extension MNover which q↾Nhas more than λmany
+extensions of rank at least α. Notice that for any N′∈Sub≤λ(M′),N′=f[N] for some
+N∈Sub≤λ(M) and, moreover, that
+f[q]↾N′=f[q]↾f[N] =f[q↾N]
+The extensions of q↾NtoMNof rank at least αmap bijectively under the automorphism f
+to extensions of f[q]↾N′to types over f[MN] and, by the induction hypothesis, these image
+typesarealsoofrankatleast α. Asaresult,forany N′∈Sub≤λ(M′), RMλ[f[q]↾N′]≥α+1.
+Thus RMλ[q]≥α+1, as desired. ✷
+Trivially,
+Proposition 3.5. [≤λ-Local Character] For any q∈ga-S(M), there is an N∈Sub≤λ(M)
+with RMλ[q↾N] =RMλ[q].
+Slightly less trivially,
+Proposition 3.6. [Relations Between RMλ] Whenever λ≤µ, RMµ[q]≤RMλ[q].
+Proof:By induction, with the zero and limit cases still trivial. If RMµ[q]≥α+1, then for
+everyN∈Sub≤µ(M),q↾Nhas morethan µextensionsto types ofRMµ-rankat least αover
+someMN≻KN. In particular, there are more than λsuch extensions and, by the induction
+hypothesis, they are all of RMλ-rank at least α. Noticing that Sub ≤λ(M)⊆Sub≤µ(M), one
+sees that the same holds for all N∈Sub≤λ(M), meaning that, ultimately, RMλ[q]≥α+1.
+✷
+The ranks RMλ[−] also have the following attractive property: letting CBλ[q] denote the
+topological Cantor-Bendixson rank of a type q∈Xλ
+M, whereMis the domain of q, we have
+Proposition 3.7. For anyq(sayq∈ga-S(M)), CBλ[q]≤RMλ[q].
+Proof:We show by induction that CBλ[q]≥αimplies RMλ[q]≥α. The zero and limit
+cases are not interesting. If CBλ[q]≥α+1,qis an accumulation point of types of CBλ-rank
+at leastα. This means, by Proposition 2.9, that every basic open neighborhoo d ofqcontains
+more thanλsuch types, which is to say that each q↾NwithN∈Sub≤λ(M) has more than
+λmany extensions to types over Mof CBλ-rank at least α. By the induction hypothesis,
+these types are of RMλ-rank at least α, and for each N∈Sub≤λ(M) witness the fact that
+RMλ[q↾N]≥α+1. Naturally, it follows that RMλ[q]≥α+1.✷
+A quick fact about CBλ, incidentally, which we obtain in the usual way:
+Proposition 3.8. If everyq∈Xλ
+Mhas ordinal CBλ-rank, isolated points are dense in Xλ
+M.
+7In particular, then, if all of the types over a model M∈ Khave ordinal RMλ-rank,
+isolated points are dense in Xλ
+M.
+One of the great virtues of the classical Morley rank is that types h ave unique extensions
+of same Morley rank: if p∈S(A) has RM[p] =αandB⊇A, there is at most one type
+q∈S(B) that extends pand has RM[ q] =α. While we do not have a unique extension
+property for the ranks RMλ, we do have a very close approximation, Proposition 3.10 below.
+First, we notice:
+Lemma 3.9. LetM≺KM′,q∈ga-S(M) with an ordinal RMλ-rank. There are at most λ
+extensionsq′∈ga-S(M′) with RMλ[q′] =RMλ[q].
+Proof:Forthesakeofnotationalconvenience,sayRMλ[q] =α. LetS={q′∈ga-S(M′)|q′↾
+M=q,and RMλ[q′] =α}, and suppose that |S|>λ. For anyN∈Sub≤λ(M), eachq′is an
+extensionof q↾N, meaningthatRMλ[q↾N]≥α+1forallsuch N. ButthenRMλ[q]≥α+1.
+Contradiction. ✷
+It is worthnotinghere that we havemade nouse oftameness in the d iscussionabove—not
+in the definition of the ranks RMλ, nor in the proofs of the properties thereof—and have
+merely made use of the fact that λ≥LS(K). From now on, however, it will often be critically
+important that we have the ability to separate distinct types, leadin g us to restrict attention
+to those RMλwithλ≥χ, the tameness cardinal of the AEC at hand. We will be very
+explicit in making this assumption whenever it is necessary, beginning w ith the following
+essential property of the RMλ:
+Proposition 3.10. [Quasi-unique Extension] Let Kbeχ-tame with χ≥LS(K), and letλ≥
+χ. LetM≺KM′,q∈ga-S(M), and say that RMλ[q] =α. Given any rank αextensionq′ofq
+to a type over M′, there is an intermediate structure M′′,M≺KM′′≺KM′,|M′′| ≤ |M|+λ,
+and a rank αextensionp∈ga-S(M′′) ofqwithq′∈ga-S(M′) as its unique rank αextension.
+Proof:Again, letS={q′′∈ga-S(M′)|q′′↾M=q,and RMλ[q′′] =α}. From the lemma,
+|S| ≤λ, meaning that Sis discrete (by tameness and Corollary 3.35). Thus there is some
+N∈Sub≤λ(M′) with the property that q′↾NsatisfiesUq′↾N,N∩S={q′}. TakeM′′∈ K
+containingboth MandNandwith |M′′| ≤ |M|+|N|+LS(K)≤ |M|+λ, andsetp=q′↾M′′.
+Notice that, by monotonicity,
+α= RMλ[q′]≤RMλ[p]≤RMλ[q] =α
+so RMλ[p] =α. Naturally, q′is a rankαextension of p. Any such extension q′′ofpis also
+a rankαextension of q, hence inS, and also an extension of q′↾N. By choice of N, then,
+we must have q′′=q′.✷
+A possible gloss for this complicated-looking result: while a type p∈ga-S(M) of RMλ-
+rankαmay have many extensions of rank αover a model M′≻KM, we need only expand
+its domain ever so slightly (adding at most λelements of M′) to guarantee that the rank
+αextension is unique. As innocuous as this proposition may seem, it is th e linchpin of the
+analysis of stability in this framework, and will see extensive use in Sec tions 5 and 6. Of
+course, quasi-unique extension applies only to types with ordinal RMλ-rank. To take the
+fullest possible advantage of this property, then, we would be wise t o confine our attention
+to AECs where RMλis ordinal-valued on all types associated with the class.
+4STABILITY AND λ-TOTAL TRANSCENDENCE
+In classical model theory, we call a first-order theory totally tra nscendental if all types over
+subsets of the monster model are assigned ordinal values by the M orley rank. We now define
+8analogous notions for Galois types in AECs, one for each Morley-like r ank RMλ.
+Definition 4.1. We say that Kisλ-totally transcendental if for every M∈ Kandq∈
+ga-S(M), RMλ[q] is an ordinal.
+Proposition 4.2. IfKisλ-totally transcendental, it is µ-totally transcendental for all µ≥λ.
+Proof:SeeProposition3.6, theresultconcerningtherelationshipbetween thevariousMorley
+ranks.✷
+Putting this proposition together with Propositions 3.7 and 3.8, we ha ve:
+Proposition 4.3. IfKisλ-totally transcendental, then for any Mandµ≥λ, isolated
+points are dense in Xµ
+M.
+As a nice special case, if LS( K) =ℵ0, andKisℵ0-tame and ℵ0-totally transcendental,
+then isolated points are dense in each space Xℵ0
+M. This bears a certain resemblance to the
+classical result linking total transcendence to the density of isolat ed points in the spaces of
+syntactic types.
+As interesting as this result may be, the true power of the notion of λ-total transcendence
+lies in the leverage it provides in rather a different area: bounding the number of types over
+models. We will take advantage of this in Sections 5 and 6 below to prov e a variety of results
+related to Galois stability. Before we give any further thought to λ-totally transcendental
+AECs, though, it seems natural to inquire whether the notion is, in f act, a meaningful
+one; that is, whether one can actually find λ-totally transcendental AECs. One can, as the
+following proposition suggests:
+Theorem 4.4. IfKisχ-tame for some χ≥LS(K), and isλ-stable with λ≥χandλℵ0>λ,
+thenKisλ-totally transcendental.
+Proof:Suppose that RMλis unbounded; that is, suppose that there is a type pover some
+N≺KCwith RMλ[p] =∞. Indeed, we may assume that |N| ≤λ(if there is a type of rank
+∞over a larger structure, consider one of its restrictions). We pro ceed to construct a K-
+structure ¯Nof sizeλover which there are λℵ0many types, thereby establishing that Kis
+notλ-stable. We first need the following:
+Claim 4.5. For anyλ, there exists an ordinal αCsuch that for any qoverM≺KC, RMλ[q] =
+∞just in case RMλ[q]≥α.
+Proof:The number of all such types can be bounded in terms of |C|, meaning that the set
+of their ranks cannot be cofinal in the class of all ordinals. ✷(Claim)
+We construct ¯Nas follows:
+Step 1: Set p∅=p. Sincep∅satisfies RMλ[p∅] =∞, RMλ[p∅]> α. Hence there is an
+extensionM≻KNover which p∅has more than λmany extensions of rank at least αand
+thus of rank ∞. Take any λmany of them, say {qi∈ga-S(M)|i<λ}. By Proposition 2.8,
+this set is discrete in Xλ
+M, meaning that for each i<λ, there is an N′
+i∈Sub≤λ(M) with the
+property that qj↾N′
+i/ne}ationslash=qi↾N′
+iwheneverj/ne}ationslash=i. LetN∅be aK-substructure of Mcontaining
+N∪(/uniontext
+i<λNi),|N∅|=λ. For eachi<λ, definepi=qi↾N∅. Notice that:
+•RMλ[pi]≥RMλ[qi] =∞
+•We havepi↾N= (qi↾N∅)↾N=qi↾N=p∅.
+•Ifpi=pj, then
+qi↾Ni= (qi↾N∅)↾Ni=pi↾Ni=pj↾Ni= (qj↾N∅)↾Ni=qj↾Ni,
+9in which case we must have i=j. That is, the piare distinct.
+So we have a family {pi∈ga-S(N∅)|i<λ}of distinct extensions of p∅, all of rank ∞.
+Stepn+1: For each σ∈λn, we have a structure Nσof size at most λand a family of
+distinct types {pσi∈ga-S(Nσ)|i<λ}, all of rank ∞. For eachi<λ, apply the process of
+step 1 to each pσi. By this method, we obtain (for each i<λ) an extension Nσi≻KNσwith
+|Nσi|=λand a family {pσij∈ga-S(Nσi)|j <λ}of distinct extensions of pσi, all of rank ∞.
+Stepω: Notice that for each τ∈λω, the sequence N∅,Nτ↾1,...,N τ↾n,...is increasing
+and, moreover, ≺K-increasing (by coherence and the fact that everything embeds s trongly
+inC). It follows that Nτ:=/uniontext
+n<ωNτ↾nis inK. Notice also that the pτ↾(n+1)∈ga-S(Nτ↾n)
+forn<ωform an increasing sequence of types over the structures in this u nion. By (ω,∞)-
+compactness of AECs (see Theorem 11.1 in [Ba08]), there is a type pτ∈ga-S(Nτ) with the
+property that for all n<ω,pτ↾Nτ↾n=pτ↾(n+1). Let¯Nbe a structure of size λcontaining
+the union /uniondisplay
+σ∈λ<ωNσ
+(which is possible, since the union is of size at most λ·λ<ω=λ·λ=λ). For each τ∈λω,
+letqτbe an extension of pτto a type over ¯N. There are λℵ0many such types, all over ¯N—it
+remains only to show that they are distinct. To that end, suppose t hatτ,τ′∈λω,τ/ne}ationslash=τ′. It
+must be the case that for some σ∈λ<ω,τ=σi...andτ′=σj...withi/ne}ationslash=j. Then
+qτ↾Nσ= (qτ↾Nτ)↾Nσ=pτ↾Nσ=pσi
+Similarly, we have
+qτ′↾Nσ= (qτ′↾Nτ′)↾Nσ=pτ′↾Nσ=pσj
+Sincepσiandpσjaredistinct by construction, qτ/ne}ationslash=qτ′, and the types are distinct as claimed.
+By our assumption that λℵ0>λ, then,Kis not stable in λ.✷
+So the notion of λ-total transcendence is far from vacuous: totally transcenden tal AECs
+exist. Indeed, they exist in great abundance.
+5TRANSFER RESULTS
+As promised, we will now employ λ-total transcendence as a tool to analyze the proliferation
+of types over models in AECs. In fact, we introduce two methods of bounding the number
+of types over large models, the first of which is captured by the follo wing result:
+Theorem 5.1. LetKbeχ-tame for some χ≥LS(K), andλ-totally transcendental with
+λ≥χ+|L(K)|. Then for any structure Mwith|M|>λ,|ga-S(M)| ≤ |M|λ.
+Proof:Takeq∈ga-S(M), RMλ[q] =α. It must be the case that RMλ[q↾N] =αfor some
+N∈Sub≤λ(M) whence, by Proposition 3.10, there is an N′≻KNwith|N′| ≤λsuch that
+q↾N′has rankαand has a unique rank αextensionqoverM. Hence
+|{q∈ga-S(M)|RMλ[q] =α}| ≤ |{poverN∈Sub≤λ(M)|RMλ[p] =α}|
+and thus
+|ga-S(M)|=|/uniontext
+α{q∈ga-S(M)|RMλ[q] =α}|
+≤ |/uniontext
+α{poverN∈Sub≤λ(M)|RMλ[p] =α}|
+=|{poverN∈Sub≤λ(M)}|
+10Obviously, |N| ≤λfor anyN∈Sub≤λ(M), so by a general principle (see Proposition 2.10
+in [Li09b], for example) it must be the case that |ga-S(N)| ≤2λ. Hence
+|ga-S(M)| ≤2λ·|Sub≤λ(M)|
+≤2λ·|M|λ
+=|M|λ
+So we are done. ✷
+A major consequence of this theorem is:
+Corollary 5.2. IfKisχ-tame for some χ≥LS(K), andλ-totally transcendental with
+λ≥χ+|L(K)|, then for any µsatisfyingµλ=µ,Kisµ-stable.
+In light of Theorem 4.4, we may eliminate any mention of λ-total transcendence and
+translate the corollary above into a pure upward stability transfer result:
+Proposition 5.3. IfKisχ-tame for some χ≥LS(K), andλ-stable with λ≥χ+|L(K)|
+andλℵ0>λ, thenKis stable in every µwithµλ=µ.
+Proof:Under the assumptions on λ, stability in λimpliesλ-total transcendence of K(by
+the theorem invoked above). We are then in the situation of Corollar y 5.2, and the result
+follows. ✷
+It should be noted that this result is weaker than one found in [GrVa0 6b]—namely
+Corollary 6.4—obtained by a splitting argument, which dispenses with t he assumption that
+λℵ0> λ, inferring stability in µwithµλ=µfrom stability in any λlarger than the tame-
+ness cardinal. The potential advantages of our method become slig htly more evident if we
+reinvent Proposition 5.3 in such a way as to make clear the larger proj ect for which it should
+be the jumping off point:
+Theorem 5.4. LetKbeχ-tame for some χ≥LS(K), andλ-stable with λ≥χ+|L(K)|and
+λℵ0>λ. Then there exists κ≤λsuch that Kisµ-stable in every µsatisfyingµκ=µ.
+Recall that if Kis stable in such a λ, it isλ-totally transcendental, and we may take
+κ= min{ν|Kisν-totally transcendental }.
+For this to be an actual improvement on existing results, of course , one would need to be able
+to give a meaningful bound on κ, the cardinal at which total transcendence first occurs, that
+places it well below λ. Given that there is something of an analogy between this cardinal a nd
+the cardinal parameter of nonforking in elementary classes, ther e is some hope that this can
+be achieved, transforming the theorem above into a stability spect rum result reminiscent of
+those familiar from first-order model theory.
+There is a different tack to be taken, though. We introduce a secon d method for bounding
+the number of types over models of size larger than the cardinal of total transcendence:
+Theorem 5.5. LetKbeχ-tame for some χ≥LS(K), andλ-totally transcendental with
+λ≥χ. For anyM∈ Kwith cf(|M|)>λ,
+|ga-S(M)| ≤ |M|·sup{|ga-S(N)||N≺KM,|N|<|M|}.
+Proof:Take a filtration of M:
+M0≺KM1≺K...≺KMα≺K...≺KM
+11forα <|M|, with|Mα|<|M|for allαandM=/uniontext
+α<|M|Mα. Letq∈ga-S(M). By
+λ-total transcendence, RMλ[q] =βfor some ordinal βand, moreover, this is witnessed by
+a restriction to a small submodel of M. That is, there is a submodel N∈Sub≤λ(M) with
+RMλ[q↾N] =β. By Proposition 3.10, there is an intermediate extension N≺KN′≺KM
+with|N′|=λsuch thatq↾N′has unique rank βextensionqoverM. Since cf( |M|)> λ,
+N′⊆Mαfor someα(and, by coherence, N′≺KMα). Clearly, the type q↾Mαhasqas
+its only rank βextension over M. By a computation similar to that found in the proof of
+Proposition 5.1, then, we have
+|ga-S(M)| ≤ |M|·sup{|ga-S(Mα)||α<|M|}
+The inequality in the statement of the theorem is a trivial consequen ce.✷
+In essence, the theorem asserts that for models Mof cardinality µwith cf(µ)> λ,
+the equality |ga-S(M)|=µfails only if there is a submodel N≺KMwith|N|< µand
+|ga-S(N)|> µ. To ensure that this does not occur—that we have, in short, stab ility in
+µ—we need considerably less than full stability in the cardinals below µ. To be precise:
+Theorem 5.6. LetKbeχ-tame for some χ≥LS(K) andλ-totally transcendental with
+λ≥χ, and letµsatisfy cf(µ)>λ. If for every M∈ K<µ,|ga-S(M)| ≤µ,Kisµ-stable.
+Proof:For anyM∈ Kµ,
+|ga-S(M)| ≤ |M|·sup{|ga-S(N)||N≺KM,|N|<|M|} ≤ |M|·|M|=|M|.✷
+We may tighten the statement up a bit, once we notice the following:
+Lemma 5.7. If there is a set Sof cardinals cofinal in an interval [κ,µ)which has the
+property that for every M∈ Kwith|M| ∈S,|ga-S(M)| ≤µ, then|ga-S(M)| ≤µfor every
+M∈ K<µ.
+Proof:LetM∈ K<µ. If|M| ∈S, we are done. If not, take ν∈Swithν >|M|, and letM′
+be a strong extension of cardinality ν,M ֒→KM′֒→KC. By assumption, |ga-S(M′)| ≤µ.
+Every type over Mhas an extension to a type over M′, and, of course, distinct types must
+have distinct extensions. Hence
+|ga-S(M)| ≤ |ga-S(M′)| ≤µ.✷
+So Theorem 5.6 becomes
+Theorem 5.8. LetKbeχ-tame for some χ≥LS(K) andλ-totally transcendental with
+λ≥χ, and letµsatisfy cf(µ)>λ. If there is a set Sof cardinals cofinal in an interval [κ,µ)
+which has the property that for every M∈ Kwith|M| ∈S,|ga-S(M)| ≤µ,Kisµ-stable.
+The assumption on the number of types over models of cardinality les s thanµin the
+theorem above is very weak—it is certainly more than sufficient to ass ume that Kisν-stable
+for eachνin a cofinal set below µ. In particular, we have:
+Corollary 5.9. IfKisχ-tame for some χ≥LS(K) andλ-totally transcendental with λ≥χ,
+µsatisfies cf( µ)>λ, andKis stable on a set of cardinals cofinal in an interval [κ,µ), then
+Kisµ-stable.
+Using Theorem 4.4 again, we may recast the corollary as a stability tra nsfer result that
+makes no mention of total transcendence. Though the product o f this rewriting is a trivial
+consequence of the foregoing discussion (and, indeed, an extrem ely special case of Theo-
+rem 5.8), it nonetheless generalizes a state-of-the-art result: T heorem 2.1 in [BaKuVa06].
+First, the statement of the theorem:
+12Theorem 5.10. IfKisχ-tame for some χ≥LS(K) andλ-stable with λ≥χandλℵ0>λ,
+then for any µwith cf(µ)>λ, ifKis stable on a set of cardinals cofinal in an interval [κ,µ),
+Kisµ-stable.
+If we restrict our attention to the special case in which the AEC is ℵ0-stable, and assume
+more stability below µthan we need, strictly speaking, we have:
+Corollary 5.11. IfKhas LS(K)=ℵ0and isℵ0-tame and ℵ0-stable, then for any µwith
+cf(µ)>ℵ0, ifKisκ-stable for all κ<µ,Kisµ-stable.
+In other words: in an AEC satisfying the hypotheses of the corollar y, a run of stability
+will never come to an end at a cardinal µof uncountable cofinality; it will, in fact, include
+µ, thenµ+, thenµ++, and so on. This remarkable fact is the aforementioned result of
+[BaKuVa06] (which also appears as Theorem 11.11 in [Ba08]). What we h ave discerned by
+our method—and distilled in Theorem 5.10—is that it is not stability in ℵ0which is critical,
+but rather stabilityin anycardinal λsatisfyingλℵ0>λ. Moreover,it is not necessaryto have
+stability in every cardinality below the cardinal of interest, µ, but rather to have stability
+(or, indeed, somewhat less than stability) in a cofinal sequence of c ardinals below µ.
+6WEAKLY TAME AECS
+We now shift our frame of reference. Thus far we have limited ourse lves to AECs that
+are tame in some cardinal χ. The time has come to consider how the picture may differ
+in an AEC that is weakly χ-tame, rather than χ-tame. The most important difference is
+that the argument for Theorem 4.4 no longer works, meaning that w e can no longer infer
+total transcendence from stability. In a sense, then, we must tr eat total transcendence as
+a property in its own right, independent from the more conventiona l properties of AECs.
+On the other hand, the argument for Theorem 5.5 still goes throug h, provided the model in
+question is saturated. That is,
+Proposition 6.1. LetKbe weaklyχ-tame for some χ≥LS(K), andµ-totally transcendental
+withµ≥χ. IfM∈ Kis a saturated model with cf( |M|)>µ, then
+|ga-S(M)| ≤ |M|·sup{|ga-S(N)||N≺KM,|N|<|M|}
+We can use this to get bounds on the number of types over more gen eral (that is, not
+necessarily saturated) models, provided that there are enough s mall saturated extensions in
+K. In particular, if we can replace a model M∈ Kλwith a saturated extension M′∈ Kλ, we
+have
+|ga-S(M)| ≤ |ga-S(M′)|
+where the latter is governed by the bound in the proposition above. Given an adequate
+supply of such extensions, then, we can prove results similar to tho se found above, but
+now for weakly tame AECs. The existence of saturated extensions also has the following
+consequence:
+Lemma 6.2. If every model in Kλhas a saturated extension in Kλ, then for any M∈ K<λ,
+|ga-S(M)| ≤λ.
+Proof:TakeM∈ K<λ. LetM′be a strong extension of Mof cardinality λ. By assumption,
+M′has a saturated extension ¯Mof cardinality λ. This model, ¯M, realizes all types over
+13submodels of size less than λ, hence realizes all types over the original model, M. It follows
+that|ga-S(M)| ≤ |¯M|=λ.✷
+Notice that this is precisely the type-counting condition from which w e are able to infer
+stability using the bound in Proposition 6.1. Hence we have
+Theorem 6.3. LetKbe weaklyχ-tame for some χ≥LS(K), andµ-totally transcendental
+withµ≥χ. Suppose that λis a cardinal with cf (λ)> µ, and that every M∈ Kλhas a
+saturated extension M′∈ Kλ. ThenKisλ-stable.
+Proof:LetM∈ Kλ. By assumption, we may replace Mwith a saturated extension M′of
+cardinality λ, over which there can be at most λtypes, by Proposition 6.1 and Lemma 6.2.
+As noted above, |ga-S(M)| ≤ |ga-S(M′)|, and we are done. ✷
+As a simple special case:
+Corollary 6.4. Suppose Kis weaklyχ-tame for some χ≥LS(K) andµ-totally transcen-
+dental with µ≥χ, and suppose that λis a regular cardinal with λ>µ. IfKis stable on an
+interval[κ,λ),Kisλ-stable.
+Proof:LetM∈ Kbe of cardinality λ. By the usual union of chains argument, Mhas a
+saturatedextension M′∈ Kwhichisalsoofcardinality λ. UsingtheboundinProposition6.1,
+we get|ga-S(M′)|=λ, whenceλ=|M| ≤ |ga-S(M)| ≤λ, and thus |ga-S(M)|=λ. The
+result follows. ✷
+In particular,
+Corollary 6.5. Suppose Kis weaklyχ-tame for some χ≥LS(K),µ-totally transcendental
+withµ≥χ, andλ-stable for some λ≥µ. ThenKisλ+-stable.
+This is weaker than a result of [BaKuVa06], which infers λ+-stability from λ-stability in
+any AEC that is weakly tame in χ≤λ, with no additional assumptions. The methods of
+[BaKuVa06]—splitting, limit models—arebetter adapted to the task of provinglocal transfer
+results of this form in the context of weakly tame AECs. The machine ry of ranks and total
+transcendenceprovidesuswithleverageofadifferentsort,wells uitedtothetaskofproducing
+partial spectrum results of a more global nature.
+In particular, we saw in Theorem 6.3 that in a µ-totally transcendental AEC, even if
+merely weakly tame, we may infer stability in λgiven an adequate supply of saturated
+extensionsofcardinality λ. As it happens, the category-theoreticnotionofweak λ-stability—
+introduced in [Ro97], and analyzed in the context of AECs in [Li09a]—ens ures that this
+requirement is met. The implications of this fact for the Galois stability spectra of weakly
+tame AECs are examined in [Li09a].
+References
+[Ba08] John Baldwin. Categoricity in abstract elementary classes. A vailable at
+http://www.math.uic.edu/ ~jbaldwin/pub/AEClec.pdf , August 2008.
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+To appear in: Israel Journal of Math, November 2007.
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+for tame abstract elementary classes. Notre Dame Journal of Formal Logic , 47(2):291–
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+of Symbolic Logic. Available at http://www.math.uic.edu/ ~jbaldwin/model.html ,
+November 2007.
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+tame. Submitted. Available at http://www.math.cmu.edu/ ~rami/, September 2005.
+[GrVa06a] Rami Grossbergand Monica VanDieren. Categoricityfro m one successorcardinal
+in tame abstract elementary classes. Journal of Mathematical Logic , 6(2):181–201,2006.
+[GrVa06b] Rami Grossberg and Monica VanDieren. Galois-stability in t ame abstract ele-
+mentary classes. Journal of Mathematical Logic , 6(1):25–49, 2006.
+[GrVa06c] Rami Grossberg and Monica VanDieren. Shelah’s categor icity conjecture from a
+successor for tame abstract elementary classes. Journal of Symbolic Logic , 71(2):553–
+568, 2006.
+[JaSh08] Adi Jarden and Saharon Shelah. Good frames minus stabilit y. Available at
+http://shelah.logic.at/files/875.pdf , September 2008.
+[Li09a] Michael Lieberman. Category-theoretic aspects of abstr act elementary classes.
+Preprint, December 2009.
+[Li09b] Michael Lieberman. Topological and category-theoretic aspects of abstract el ementary
+classes. PhD thesis, University of Michigan, 2009.
+[Li09c] Michael Lieberman. A topology for Galois types in AECs. arXiv:0 906.3573v1. Sub-
+mitted, June 2009.
+[Ro97] Jiˇ r´ ı Rosick´ y. Accessible categories, saturation and ca tegoricity. Journal of Symbolic
+Logic, 62:891–901, 1997.
+[Sh600] Saharon Shelah. Categoricity in abstract elementary class es: going up inductive
+step. Preprint 600. Available at http://shelah.logic.at/short600.ht ml.
+[Sh88] Saharon Shelah. Classification theory for nonelementary cla sses, II. In Classification
+Theory, volume 1292 of Lecture Notes in Mathematics , pages 419–497. Springer-Verlag,
+Berlin, 1987.
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+15
\ No newline at end of file
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+arXiv:1001.0625v1  [cond-mat.mtrl-sci]  5 Jan 2010Spatial distribution functions of random packed granular s pheres obtained by direct
+particle imaging
+Andreea Panaitescu and Arshad Kudrolli
+Department of Physics, Clark University, Worcester, MA 016 10
+(Dated: (August 30, 2018))
+We measure the two-point density correlations and Voronoi c ell distributions of cyclically sheared
+granular spheres obtained with a fluorescence technique and compare them with random packing
+of frictionless spheres. We find that the radial distributio n function g(r) is captured by the Percus-
+Yevick equation for initial volume fraction φ= 0.59. However, small but systematic deviations are
+observed because of the splitting of the second peak as φis increased towards random close packing.
+The distribution of the Voronoi free volumes deviates from p ostulated Γ distributions, and the
+orientational order metric Q6shows disorder compared to numerical results reported for f rictionless
+spheres. Overall, these measures show significant similari ty of random packing of granular and
+frictionless spheres, but some systematic differences as we ll.
+PACS numbers: 45.70.Qj, 05.65.+b
+The packing of spheres is one of the enduring prob-
+lems in physics, and a basis to understand the structure
+and strength ofgranularmatter. Assuming dominanceof
+stericinteractions,densepackingofsteelsphereswasfirst
+used to understand structure of simple liquids with the
+radialdistributionfunction g(r)andtheorientationorder
+metricQ6[1]. However, experimental measurements at
+boundaries [2] and computer simulations in the bulk [3]
+have since shown that inter-particle friction can affect
+granular packing. Friction between particles changes the
+fundamental stability condition at contact from the fric-
+tionless case, causing a packing to be protocol dependent
+and the system to be out-of-equilibrium.
+The difficulty of accurately measuring significant num-
+ber of particle positions in the bulk away from the influ-
+ence of boundaries has also stymied progress. Recent ex-
+perimental studies [4, 5] have examined packing of gran-
+ular spheres and find that the associated free volume dis-
+tributions are described by a Γ distribution with two fit-
+ting parameters which were then given a thermodynamic
+interpretation [5]. These results are puzzling in light of
+earlier analytical work in one-dimension and simulations
+in two and three dimensions that show a Γ distribution
+with 3 fitting parameters is needed to describe a broad
+range of volume fraction for elastic particles [6].
+Here, we discuss new experiments with spherical gran-
+ular particles which enable us to directly determine sta-
+tistical measures to understand the effect of friction, test
+the effect of shear, and perform a rigorous comparison
+with frictionless hard sphere packing. Using a fluores-
+cence technique [4, 7, 8], we obtain the packing of glass
+spheres before and after application of cyclic shear, and
+compare with random packing of frictionless spheres. We
+find that the overall shape of g(r) for volume fraction
+φ∼0.6 is captured by the Percus-Yevick equation [9]
+which assumes random packing of spheres without angu-
+lar correlations. But, systematic deviations are observed
+because of the splitting of the second peak as φis in-
+creased toward random close packing, Q6shows partial
+hexagonal order, and the distribution of the Voronoi free
+FIG. 1. (a) Schematic diagram of the cyclic shear cell used in
+the experiments. (b) An image of the initial packing observe d
+in the central vertical slice of the shear cell after particl es
+are filled inside the cell. (c) Ordering grows near the top
+boundary which is free to move after 600 shear cycles but
+the packing in the bulk appears random. (d) The probability
+distribution function of the diameter of the glass beads. (e )
+The volume fraction φas a function of shear cycle number
+measured in the bulk (red/grey) evolves more slowly than
+in the entire cell (black) because of the ordering near the
+boundaries.
+volumes shows enhanced probabilities at higher values
+compared with Γ distributions postulated [6] for random
+packing of spheres.
+The experiments to measure structure of granular
+packing are performed using a shear cell shown schemat-
+ically in Fig. 1(a). The parallelepiped shaped cell con-
+sists of a rigid front, back, and bottom transparent glass
+boundary, andside boundaries that can be tilted through
+a prescribed angle θto cyclically shear and perturb the
+packing. Glass spheres with density ρg= 2.5×1032
+kg m−3and average diameter d= 1.034mm are gently
+added inside a shear cell filled with an interstitial liquid
+with the same refractive index as the glass spheres, den-
+sityρl= 1.0×103kg m−3, and viscosity ν= 2.2×10−2
+Pas. Then a flat plate is placed on the top, which is
+constrained to move only in the vertical and horizontal
+directionandnot allowedtorotateusingarigidsetoflin-
+ear guides. The initial volume fraction of the glass beads
+is measuredusingthe massofthe particlesadded and the
+volume of the cell occupied and found to be φ= 0.59. A
+normal stress σz= 0.4Pa is applied on the top boundary
+which is about five times the net gravitational stress due
+to the weight of the grains alone inside the cell, and is
+found to eliminate gradientsdue togravityin the system.
+A dye is added to the liquid and a thin slice of the
+cell is illuminated with a laser and a cylindrical lens [10].
+The resulting fluorescent light causes the particles to ap-
+pear dark against a bright background, and is imaged
+from an orthogonal direction with a resolution of 20 pix-
+els to a particle diameter using a 1000 ×1000 pixel 10-
+bit camera. A stack of images is recorded by linearly
+translating the plane of illumination. We then examine
+a 40d×5d×17.5dcentral region as in indicated by box
+in Fig. 1(b,c) to avoid any effect of the boundaries, and
+locate the absolute position of the spheres to within the
+slight polydispersity of the particles (see Fig. 1(d)).
+We impose quasi-static shear strain by varying θbe-
+tween±π/36 radians with a mean angular speed ω=
+8.0×10−3rad s−1, with a wait time of 50s while the stack
+of images is acquired every time the system returns to its
+original position, θ0= 0rad. The lubrication forces [11]
+due to liquid draining at contacts can be estimated to
+be 10−5lower than the confining forces, and the particle
+Reynolds number is ∼10−1. Therefore the particles can
+be assumed to be in contact during the entire experiment
+and the interstitial liquid does not have any impact on
+the observed structure.
+The mean packing volume fraction of the spheres in-
+side the entire cell is first obtained by measuring the po-
+sition of the top plate as a function of the shear cycles.
+The mean φis observed to increase logarithmically from
+the initial value by 5% (Fig. 1e) consistent with previous
+reports with a similar setup [12]. However, it is note-
+worthy that this is the total φinside the cell and can
+be different than φin the bulk because of influence of
+boundaries [10]. Examining the images corresponding to
+the initial state of the packing, before applying the shear
+deformation, N= 1 and after shearcycle N= 600shown
+in Fig. 1(b,c), we indeed note greater ordering near the
+top where the boundary shears the spheres and moves to
+accommodate changes in the total φ. The boundary be-
+tween ordered and disordered region appears sharp and
+moves downward as Nis increased further, similar to de-
+velopment of crystalline order inside a Couette shear cell
+upon extended shearing [7]. Therefore, we focus on the
+first 600 shear cycles where the particles inside the bulk
+in the observation window appear uniformly random and
+obtainφfrom the ratio of the particle volume and theφ = 0.605
+φ = 0.590
+FIG. 2. The radial distribution function g(r) plotted as func-
+tion of distance rnormalized by the mean diameter dfor
+initial packing φ= 0.59 and final packing φ= 0.605 obtained
+after shearing, (black), is compared withthetheoretical c alcu-
+lation (red/grey) obtained by using the Percus-Yevick equa -
+tion, and the measured polydispersity of the beads. The
+φ= 0.605 case is offset for clarity. Inset: The calculated
+pair distribution functions gijfor particles coarsened to three
+sizes (φ= 0.59). The thick red/grey curve represents the av-
+erage of the six distinct contributions gij(r) and is used for
+comparison with the experimental g(r).
+average Voronoi volume in the bulk. The Voronoi vol-
+ume is defined by points in the volume closest to that
+particle, and is calculated using algorithms written by
+Rycroft [13] and measured particle positions. As shown
+in Fig. 1(e), the evolution of φin the bulk is observed to
+be slower compared with φmeasured for the entire cell
+and is used in all subsequent discussion.
+To analyze the structure from the measured particle
+positions, we first discuss the radial distribution function
+g(r) which represents the probability that the center of
+a particle is found at a distance rfrom another particle.
+g(r) obtained from the experimental data is shown in
+Fig. 2. For the initial volume fraction φ= 0.59,g(r)
+shows a tall peak at r∼d, and a broad peak at r∼
+2d, but for the higher φobtained after cyclic shear, the
+second peak splits and a weak secondary peak occurs at
+r=√
+3d, corresponding to the next nearest neighbor in
+a face-centered-cubic (FCC) lattice.
+Now, the Percus-Yevickequation [9, 14] can be used to
+analytically find g(r) for randomly distributed particles
+for a given particle size, and volume fraction. Because of
+the slight polydispersity of the particles used in our ex-
+periments, we in fact have to compute the Percus-Yevick
+pairdistributionfunction forpolydispersesphericalpack-
+ingsgij(r), wheretheindices i,jrepresenttheprobability
+offinding a particle with diameter djat a distance rfrom
+a particle with diameter di.
+gPY(r) =1
+n(n−1)n/summationdisplay
+i=1
+j≥igij(r) (1)3
+(a)
+(b)
+FIG. 3. (a) The probability distribution of Q6measured for
+each particle using Voronoi neighbors are observed to de-
+scribed by Gaussian fits. (b) The mean Q6,localmeasured
+as a function of φ. The curve is a guide to the eye and shows
+thatQ6increase somewhat over the narrow φinvestigated.
+herenis the number of particle sizes in the system.
+Coarse graining the observed size distribution - shown in
+Fig. 1(d) - into three sizes d1= 0.98mm,d2= 1.02mm,
+d3= 1.07mm (using a greater ndoes not change the
+results significantly), we calculate the corresponding six
+distinctgij(r) terms, which are plotted in the Inset to
+Fig. 2 for φ= 0.59, and thus the computed gPY(r) for
+the polydisperse packing according to the Percus-Yevick
+approximation, which is plotted in Fig. 2. We observe
+that the amplitude and width of the primary peak and
+the broad features of the secondary peaks are in good
+agreement with the Percus-Yevick approximation. This
+comparison is especially noteworthy because there are
+no fitting parameters. At higher φ, the primary peak
+and the overall form is still captured by the gPY(r), but
+details such as the splitting of the second peak which
+can indicate hexagonal ordering is not captured because
+Percus-Yevick approximation assumes random angular
+orientation.
+To investigate orientational order in the packing, we
+use the orientation order metric
+Ql≡N/summationdisplay
+i=1/parenleftBigg
+4π
+l(l+1)m=l/summationdisplay
+m=−l|/angbracketleftYlm(Θ(r),Φ(r))/angbracketright|2/parenrightBigg1/2
+,(2)where,l= 6 to examine hexagonal order, Ylmare the
+spherical harmonics, Θ( r) and Φ(r) are the polar and az-
+imuthal angle, respectively, and ris the position vector
+from a particle to its neighbor [15]. We define particle
+neighborsasthosewhichshareaVoronoicellsurface[13].
+This removes any ambigity as is introduced when consid-
+ering only neighbors at contact due to roundoff errors
+in finding a particle center. In order to compare with
+packing of elastic particles, we first compute Q6,global
+by averaging Ylm(Θ(r),Φ(r)) over all the bonds of the
+packing, and find Q6,global= 0.27±0.02. If particles
+neighbors are uncorrelated, then Q6is small because it
+goes as square root of the total number of bonds [16].
+On the other hand, Q6for a FCC crystal with 12 neigh-
+bors is 0.5745. But even a slight perturbation due to
+roundoff errors introduces 2 extra neighbors and the cor-
+responding Q6is on average 0.454 [17]. Therefore, the
+observed distribution shows ordering but the degree of
+order appears lower compared with simulations of fric-
+tionless hard spheres [18], where Q6as high as 0.4 was
+reported for a frictionless hard spheres at comparable φ.
+In those studies which were performed with considerably
+smaller system size, particle inelasticity was observed to
+lowerQ6but not as significantly as in our experiments.
+To examine the local orientational order more closely,
+we plot the observed probability distribution of Q6for
+each particle in Fig. 3(a), and the mean of the distribu-
+tion/angbracketleftQ6,local/angbracketrightin Fig. 3(b). /angbracketleftQ6,local/angbracketrightis more sensitive
+thanQ6,globalto small crystalline regions within a pack-
+ing and allows us to avoid the possibility of destructive
+interference between different crystalline regions [19]. No
+significant enhancement of distribution is found at the
+values corresponding to FCC crystal, and the observed
+Q6distribution can be described rather well in fact by
+Gaussian fits. From these observations we conclude that
+while there is some local hexagonal order which increases
+slightly over the φinvestigated, no significant crystal-
+lites occurin this dense regimeapproachingrandomclose
+packing.
+A complementary method to examine the packing at
+the particle scale is using the free volume vfassoci-
+ated with each particle given by subtracting the min-
+imum Voronoi volume corresponding to close packing,
+vc=d3/√
+2 from the Voronoi volume. This statistical
+quantity has gained prominence because it may be used
+to define a new measure of entropy based on disorder
+in packing [6, 20], and may be amenable to thermody-
+namic interpretation [5]. It has been postulated based
+on analytical work in 1-dimensional systems, that vfdis-
+tribution of random packing of spheres can be described
+by a Γ distribution [6]:
+f(vf) =δα(m/δ2)
+Γ(m/δ2)vf(m/δ−1)e−αvfδ(3)
+with three fitting parameters m,δandαthat control dif-
+ferent parts of the distribution and were determined by
+numerical simulations with frictionless hard spheres [6].
+In Fig. 4, we plot vfnormalized by the mean free volume4
+FIG. 4. The probability distribution function of the free vo l-
+ume associated with a sphere vfnormalized by the mean free
+volume/angbracketleftvf/angbracketrightplotted for various φ. The smooth black curve
+is obtained after averaging over all the experimental data.
+Γ function corresponding to elastic frictionless spheres, the
+smooth red/grey curve, is shown for comparison, and is ob-
+served to deviate systematically at higher vf. Allowing the
+fitting parameters to float improves the fit, but systematic
+differences persist for vf>/angbracketleftvf/angbracketright(blue dashed curve).
+/angbracketleftvf/angbracketrightat thatφalong with the mean distribution obtained
+after averaging over all the measured φ. The errors asso-
+ciated with the slightly polydispersity and errors in find-
+ing particle centersis oforderofsymbol size. Further, we
+plot Eq. 3 using m= 15.5,δ= 1.3 reported in Ref. [6],
+and allowing αto float to obtain best fit. Clearly, sys-
+tematicdeviationsareobservedfromthefrictionlesscase,
+complementing the results for Q6.In order to check if a Γ-distribution can capture the
+experimentally observed free volume distributions, we
+tested both the three fitting parameter distribution, and
+thetwofittingparameterdistribution, whichcorresponds
+to setting δ= 1 in Eq. 3. The best fit obtained with
+m= 12.3,α= 24.5, andδ= 0.73 is also shown in Fig. 4.
+Even in this case we obtain enhanced probabilities for vf
+greater than the mean. Therefore, our distribution differ
+from the experimental distributions used to give a sim-
+plethermodynamicinterpretationofgranularpacking[5].
+While it is possible that such deviations arise because of
+the differences in preparationprotocol, we note no signif-
+icant differences in the distributions obtained before and
+after application of cyclic shear in our experiments.
+Inconclusion,wemeasuredpackingofgranularspheres
+andcomparedexperimentallyobtained two-pointdensity
+correlations and free volume distributions before and af-
+ter application of shear. We find that the radial distri-
+bution function is captured overall by the Percus-Yevick
+equation, which is important because it is fundamental
+to calculating the strength, heat conduction, and electro-
+magnetic wave scattering properties of a material. How-
+ever, angular correlation can be observed using the ori-
+entational order metric. In comparing with numerical
+simulations reported for frictionless sphere at compara-
+ble volume fractions, we find systematic differences in
+packing as measured by lower angular correlations, and
+deviation of free volume distributions from Γ distribu-
+tions postulated for frictionless hard spheres.
+ACKNOWLEDGMENTS
+We thank V. Kumaran, Ashish Orpe, and Michael
+Berhanu for many stimulating discussions, and Chris
+Rycroft for providing us Voro++ software library. This
+work was supported by the National Science Foundation
+under NSF Grant No CBET 0853943.
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+arXiv:1001.0626v3  [hep-ph]  4 Mar 2010Production of tidal-charged black holes at the Large Hadron
+Collider
+Douglas M. Gingrich∗
+Centre for Particle Physics, Department of Physics,
+University of Alberta, Edmonton, AB T6G 2G7 Canada
+(Dated: November 17, 2018)
+Abstract
+Tidal-charged black hole solutions localized on a three-br ane in the five-dimensional gravity
+scenario of Randall and Sundrum have been known for some time . The solutions have been used
+to study the decay, and growth, of black holes with initial ma ss of about 10 TeV. These studies are
+interesting in that certain black holes, if produced at the L arge Hadron Collider, could live long
+enough to leave the detectors. I examine the production of ti dal-charged black holes at the Large
+Hadron Collider and show that it is very unlikely that they wi ll be produced during the lifetime
+of the accelerator.
+PACS numbers: 04.50.Gh, 04.50.-h, 04.50.Kd, 12.60.-i
+∗Also at TRIUMF, Vancouver, BC V6T 2A3 Canada; gingrich@ualberta .ca
+1I. INTRODUCTION
+Models of low-scale gravity could allow particle physics experiments to study the strong-
+gravity regime [1–5]. One consequence of these models is the possibilit y of black hole pro-
+duction at the Large Hadron Collider (LHC) if the fundamental scale of gravity MDis about
+a TeV [6–9].
+Black hole solutions in higher dimensions have a complicated dependenc e on both the
+gravitational field of the brane and the geometry of the extra dime nsions. The models with
+large extra dimensions in a flat spacetime (ADD) [1, 2] are popular. On e reason for this
+interest is that if the geometrical scales of the extra dimensions ar e all large compared to
+1/MD, then there is a wide regime in which the geometry of the extra dimens ions plays
+no essential role. In ADD this means considering black holes with horiz on radius much
+smaller than the size of the extra dimensions. For large flat compact ified extra dimensions,
+the topology of the black hole can be assumed to be spherically symme tric in the ( n+3)-
+spatial dimensions and the boundary conditions from the compactifi cation can be neglected.
+This allows the simple Myers-Perry metric [10] to be used. This metric is also useful in
+the Randall-Sundrum scenario [3] if the warping scale or (AdS radius) is large compared to
+1/MD[9].
+In a regime in which the brane can not be neglected, the so called bran e world scenario,
+there are very few physical solutions to the higher-dimensional bla ck hole problem (see
+Ref. [11, 12] for a review). One of the earliest and most widely discus sed solutions is a black
+string solution [13], but this solution is far from physical. Actually, it is u nknown if physical
+black hole solutions exist at all in the Randall-Sundrum model.
+Aclass of Randall-Sundrumsolutions have been foundinwhich thesolu tion isa Reissner-
+N¨ ordstrommetricwiththeelectric chargereplaced byatidalchar ge[14]. Iftheeffectsofthis
+tidal-charge term are negligible, the solution becomes an effective fo ur-dimensional solution
+andtheeffectsoflow-scalegravityareunlikelytobeobserved. Ift heextratermissignificant,
+the effects of low-scale gravity may be observable if the fundament al scale of gravity is low
+enough.
+The decays of tidal-charged black holes in the contexts of the micro canonical picture
+have been studied [15–18]. The microcanonical corrections may be s ignificant when the
+object reaches the Planck size and the classical black hole descript ion fails (For the case of
+2the microcanonical treatment of ADD black hole decays at the LHC, also see Ref. [19].).
+The microcanonical corrections to the canonical decay treatmen t are larger in the Randall-
+Sundrum scenario for Planck-sized objects, and in certain cases t hey may live long enough
+to be considered quasi-stable. The possibility that black holes produ ced at the LHC might
+be quasi-stable has raised safety concerns (see for e.g. Ref [20]), which have been addressed
+in general [21–23], and inthe context of tidal charged black holes [16 –18]. I will have nothing
+new to add to the decay discussion.
+Based on recent phenomenological constraints on the tidal charg e [18], I estimate the
+production cross section at LHC energies and show that it is tiny ove r most of the allowed
+parameter space. The probability of producing these black holes du ring the lifetime of the
+LHC accelerator is vanishingly small. Hence, a scenario of high missing- energy signatures
+of black holes at the LHC, because of a long lifetime, is disfavoured.
+II. PRODUCTION CROSS SECTION
+A static solution for a mass Mlocalized on a three-brane in the five-dimensional gravity
+scenario of Randall-Sundrum is [14]
+−gtt= (grr)−1= 1−2M
+M2
+p1
+r+q
+M2
+51
+r2, (1)
+whereqis a dimensionless tidal charge, Mp= 1.2×1016TeV is the effective Planck scale on
+thebrane, and M5isthefundamental Planckscaleinthefive-dimensional bulk. Equatio n(1)
+is an exact solution of the effective Einstein equations on the brane a nd represents the
+induced metric on the brane in the strong-gravity regime. Attempt s to extend the metric
+off the brane have failed [11, 12] so it is hoped that the brane chara cteristics of the black
+hole dominate.
+The tidal term arises due the gravitational affects from the fifth d imension and the
+dimensionless parameter qcan be thought of as a tidal charge arising from the projection
+onto the brane of free gravitational field effects in the bulk. The pa rameterqis determined
+by both the brane source, i.e. the mass M, and any Coulomb part of the bulk Weyl tensor
+that survives when Mis set to zero [14]. It may be positive or negative. Ref. [14] argues
+that negative qis more natural. In this case, there will be one horizon that is outside the
+Schwarzschild horizon RS= 2M/M2
+p. The bulk effects tend to strengthen the gravitational
+3field.
+The tidal charge affects the geodesics and the gravitational pote ntial, and hence indirect
+limits may be placed on it by observations. The effects of the tidal ter m die off quickly
+with increasing r. At astrophysical distances, the weak-field effects are being pro bed and
+the correction term is much less than the Schwarzschild term in the e ffective potential.
+However, in the strong-gravity regime qcan be large, and the horizon radius would become
+RH∼√−q
+M5. (2)
+The tidal term will dominate the Schwarzschild term, which reflects t he fact that gravity
+becomes effectively five-dimensional at high energies. Current obs ervations place only weak
+limits on the tidal charge [14], but in principle significant tidal modificatio ns could arise in
+the strong-field regime. Thus it is useful to investigate the phenom enology of these black
+holes at the LHC.
+The tidal-charge parameter can only be deduced on dimensional gro unds. Ref. [14] sug-
+gests a value of q=−M/M5. Ref. [15] generalized this to q=−(Mp/M5)α(M/M5), whereα
+is a real parameter. For the tidal term to dominate, α≫ −4. Ref. [15–17] took as a typical
+choiceα≥0. A recent more general parameterization is given by [18]
+q=−/parenleftbiggMp
+M5/parenrightbiggα/parenleftbiggM
+M5/parenrightbiggβ
+, (3)
+whereβ >0 is a real parameter. Thus the original calculations used the specia l case of
+α= 0 and β= 1. For the tidal-charged black hole to be semiclassical, α>∼−2 [18].
+A feature of black hole production in particle collisions is that its cross section is essen-
+tially the horizon area of the forming black hole and grows with the cen tre of mass energy
+of the colliding particles as some power. In the ADD scenario, this cro ss section is rather
+large and given by
+σADD∼1
+M2
+D∼1 nb, (4)
+whereMD∼1 TeV is the fundamental Planck scale in Dspacetime dimensions. For the
+tidal-charged black hole case,
+σ∼−q
+M2
+5∼1
+M2
+5/parenleftbiggMp
+M5/parenrightbiggα/parenleftbiggM
+M5/parenrightbiggβ
+. (5)
+4For this cross section to be comparable to the ADD case, |q|must be of O(1). Recent
+calculations [15, 17] used α≥0 andβ= 1, which could give a huge cross section and exceed
+the ADD case. However, αcan be restricted from above by precision measurements of the
+deviation of Newton’s law [24] and thus α >0 may not be a valid choice.
+In estimating the upper bound on α, I follow a similar argument to Ref. [18]. The brane
+has a thickness L, which must be less than the length over which deviations from Newto n
+gravity are not observed: L<∼44µm = 2.2×1014TeV−1[24]. For the tidal-charge term
+to dominate over the Schwarzschild term, the horizon radius must b e less than the critical
+radiusrc, which can be defined to be the radius at which the Schwarzschild ter m equals the
+tidal term in the metric:
+rc=1
+2Mp/parenleftbiggMp
+M5/parenrightbigg3+α/parenleftbiggM
+M5/parenrightbiggβ−1
+. (6)
+In addition, this critical radius must be less than the brane thicknes s,rc≪L, else we would
+have already observed deviations in Newton’s law due to the tidal ter m. Thus, requiring
+rc≪Lgivesα<∼αc, where
+αc≡ln(2LMp)+(1−β)ln(M/M5)
+ln(Mp/M5)−3. (7)
+In low-scale gravity, black holes could be produced at the LHC in the r ange 1<∼M<∼
+14 TeV. The upper bound is given by the maximum LHC energy, which wo uld be very un-
+likely, and the lower bound is given by the lower experimental limit on M5. ForM5= 1 TeV,
+α<∼−1.09 forM=M5, independent of β, andα<∼−1.02 forM= 14 TeV, which occurs
+atβ= 0. For larger values of M5,α<∼−0.94 for the extreme case of M=M5= 14 TeV,
+which is highly unlikely. The LHC is a proton-proton collider and thus the probability of
+producing a parton-parton collision at the maximum LHC energy is van ishingly small. Be-
+cause of this, the upper bound on αcis likely to be less than −0.94 for realizable values of
+highM=M5. Ignoring factors of two in the metric or picking a different definition for the
+fundamental Planck scale M5changesαcby at most about 0.1, and in no case is αclarger
+than about −1.0.
+Assuming the largest allowed value of αis−1, gives the upper bound on the parton cross
+section of
+5σ<∼1
+M5Mp/parenleftbiggM
+M5/parenrightbiggβ
+<∼(M/TeV)β
+Mp/TeVTeV−2, (8)
+where in the last expression I have used M5= 1 TeV, which gives the maximum cross
+section. Because of Mpin the denominator, the cross section will be exceedingly low unless
+βis very large. In terms of numerical values
+σ<∼10−16+β∼10−10+βfb. (9)
+To calculate the proton-proton cross section, the parton cross section in Eq. (8) must be
+convoluted with the parton distribution functions of the proton, s ummed over all partons
+thatcouldformtheblack hole, andintegratedover allblackholemas ses. Iwillintegrateover
+thefullkinematically allowedrangealthoughtheapplicabilityofEq. (8) maybequestionable
+nearM∼M5. To ignore this region would significantly reduce the cross section. T he results
+are shown in Fig. 1 as a function of the βparameter for M5= 1 TeV.
+I assume the total integrated luminosity delivered by the LHC will be 1 ab−1. Thus the
+probability of producing a single tidal-charged black hole event during the lifetime of the
+accelerator is vanishingly small for reasonable values of β. Only for β>∼11 will black hole
+production become significant. We would expect βto be a few and large values of βare
+unnatural [18]. The functional form for qin Eq (3) can not be valid over all mass scales and
+some mass dependence of the parameters αandβis expected.
+TheπR2
+Hform for the cross section is usually considered a maximum cross sec tion.
+Possible form factors could increase the cross section by a factor of about three. There is
+far greater discussion on how the cross section is reduced from th e geometrical form. For
+example, initial-state beam radiation or radiation during a balding phas e could reduce the
+cross section by several orders of magnitude (see Ref. [25] for a review of the ADD case).
+The analysis has been confined to the brane and the bulk space has n ot been taken into
+account. Some approximations are involved in the terms that encod e the bulk behaviour
+on the brane. Assuming these simplifications do not alter the underly ing phenomenology, it
+appears very unlikely that tidal-charged black holes could be produc ed at the LHC.
+6β0 2 4 6 8 10 12 14Proton-Proton Cross Section [fb]
+-910-810-710-610-510-410-310-210-1101
+FIG. 1. Proton-proton cross section versus the βparameter for M5= 1 TeV.
+7ACKNOWLEDGMENTS
+This work was supported in part by the Natural Sciences and Engine ering Research
+Council of Canada.
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+9
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diff --git a/1001.0627.txt b/1001.0627.txt
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index 0000000000000000000000000000000000000000..8e22836182f1326b3873c5d9a27190e2bd2156d7
--- /dev/null
+++ b/1001.0627.txt
@@ -0,0 +1,987 @@
+The Labor Economics of Paid Crowdsourcing
+John J. Horton
+Harvard University
+383 Pforzheimer Mail Center
+56 Linnaean Street
+Cambridge, MA 02138
+john.joseph.horton@gmail.comLydia B. Chilton
+University of Washington
+AC101 Paul G. Allen Center, Box 352350
+185 Stevens Way
+Seattle, WA 98195-2350
+hmslydia@cs.washington.edu
+ABSTRACT
+Crowdsourcing is a form of \peer production" in which work
+traditionally performed by an employee is outsourced to an
+\undened, generally large group of people in the form of
+an open call." We present a model of workers supplying
+labor to paid crowdsourcing projects. We also introduce a
+novel method for estimating a worker's reservation wage |
+the smallest wage a worker is willing to accept for a task and
+the key parameter in our labor supply model. It shows that
+the reservation wages of a sample of workers from Ama-
+zon's Mechanical Turk (AMT) are approximately log nor-
+mally distributed, with a median wage of $1.38/hour. At
+the median wage, the point elasticity of extensive labor sup-
+ply is 0.43. We discuss how to use our calibrated model to
+make predictions in applied work. Two experimental tests
+of the model show that many workers respond rationally to
+oered incentives. However, a non-trivial fraction of sub-
+jects appear to set earnings targets. These \target earners"
+consider not just the oered wage|which is what the ra-
+tional model predicts|but also their proximity to earnings
+goals. Interestingly, a number of workers clearly prefer earn-
+ing total amounts evenly divisible by 5 , presumably because
+these amounts make good targets.
+Categories and Subject Descriptors
+J.4 [Social and Behavioral Sciences ]: Economics;; J.m
+[Computer Applications ]: Miscellaneous
+General Terms
+Design, Economics
+Keywords
+Crowdsourcing, Experimentation
+1. INTRODUCTION
+Crowdsourcing is a form of\peer production"that outsources
+work traditionally performed by an employee to an \unde-
+ned, generally large group of people in the form of an opencall"[2, 11]. Despite the successes and the perceived promise
+of crowdsourcing,1would-be users face a serious practical
+challenge: they need to attract a crowd. Crowdsourcing
+projects have used a variety of inducements, including en-
+tertainment [20], information [1, 13], the chance to be al-
+truistic and attention from others [12]. Until recently, it
+was extremely dicult to oer a crowd money, but with the
+advent of online labor markets like oDesk, Elance and Ama-
+zon's Mechanical Turk (AMT), buyers can now easily pay
+workers with cash [8].
+Compared to cash, non-monetary crowdsourcing incentives
+are limited in at least two ways. First, some non-monetary
+incentives depend on the nature of the task or the identity
+of the proposer, which limits their usefulness. For example,
+tasks such as classifying web pages, transcribing scanned
+documents and validating search results are ideal for crowd-
+sourcing, yet they are unlikely to attract volunteers because
+they are tedious and the private benets come only to the
+proposer. Second, even when appropriate non-monetary in-
+centives exist, they are hard to adjust. For example, assum-
+ing one could compute a \fun" labor supply elasticity, , it
+is not clear how we could make a task 10(1 =)% more fun
+in order to boost provision by 10%. With cash incentives,
+computing supply elasticities is straightforward and making
+price adjustments is easy.
+The possibility of cash payments raises a design question:
+how does one eectively and eciently employ monetary in-
+centives? To solve this problem, designers need two things:
+(1) a theoretical model that predicts how people will re-
+spond to dierent price/task scenarios and (2) data-driven
+renements of that model. The renements should allow the
+model to account for behavioral biases|such as those iden-
+tied by experimental economics|and the idiosyncrasies of
+particular applications. Borrowing an analogy from the economist
+and market designer Al Roth, just as builders of suspension
+bridges must be informed by both the elegant theory of me-
+chanics and the messy details of metallurgy and geology,
+builders of social systems must possess both a general the-
+ory of behavior and detailed contextual knowledge [17].
+An appropriate theoretical model for the labor supply as-
+pect of crowdsourcing design should tell the designer (1)
+how workers decide whether or not to participate in a crowd-
+sourcing project and (2) how workers decide the amount to
+1Examples include Wikipedia, Digg, Yelp, Yahoo! Answers
+and InnoCentive.arXiv:1001.0627v1  [cs.HC]  5 Jan 2010produce, conditional upon participating. In the language of
+economics, the designer must be able to predict the labor
+supply on both the (1) extensive and (2) intensive margins.
+These two aspects of labor supply have intrigued economists
+from Adam Smith onward, but because crowdsourcing\jobs"
+last seconds and pay pennies, results from conventional labor
+economics might not transfer readily (or at least not with-
+out modication). This caveat aside, labor economics oers
+a theoretical framework for understanding decision-making
+in paid crowdsourcing scenarios, and it potentially provides
+the predictive model that designers need.
+1.1 Overview
+There are several papers on the design of incentives in crowd-
+sourcing [5] and on obtaining work from AMT [18, 19], but
+our paper is most closely related to work by Watts and
+Mason [16], who also conducted labor supply experiments.
+They found that workers respond to prices in a way that is at
+least consistent with rational behavior. However, their nd-
+ings also suggest that the relationship between incentives
+and output is complex: higher pay rates did not improve
+work quality|a result the authors attribute to high wages
+aecting worker beliefs. They also found that the structure
+of incentives (i.e., whether workers faced a piece rate or a
+quota system) aected output.
+Motivated by the evidence from [16] that AMT workers re-
+spond to cash incentives, in Section 2.1 we develop a sim-
+ple rational model of crowdsourcing labor supply. We also
+present a novel method for estimating the reservation wage
+of each worker (Section 2.3) that makes use of a highly con-
+cave earnings function rather than a simple piece-rate.2The
+reservation wage is the minimum wage a worker is willing
+to accept as compensation in exchange for performing some
+task; it is the key parameter in models of labor supply.
+With subjects recruited from AMT, we test the predictions
+of the model and its underlying assumption in two sepa-
+rate experiments. Subjects performed simple tasks in which
+they chose how much output to produce. We test whether
+subjects adjusted output in response to task diculty (Ex-
+periment A, Section 4) and price (Experiment B, Section 5).
+We nd mixed evidence for the rational model: workers are
+clearly sensitive to price but insensitive to variations in the
+amount of time it takes to complete a task.
+Reservation wages are supposed to be xed and thus should
+be invariant to our experimental manipulations. Although
+we nd that the imputed wage distributions are quite simi-
+lar in Experiment A, large dierences appear in Experiment
+B. The cause seems to be that some workers workers are
+\target earners" who focus on reaching salient earnings tar-
+gets. This stands in sharp contrast to the rational model
+that predicts workers should only consider the oered wage.
+These ndings have important implications for the design of
+incentives and are discussed in Section 6.
+The rational model clearly misses important elements of re-
+ality, but there are no immediate replacements and it makes
+reasonable predictions in some cases. For these reasons, we
+2All of our data and experimental materials
+will be available on John Horton's webpage:
+http://www.people.fas.harvard.edu/~horton.demonstrate in Section 7 how the model can be used to
+predict labor supply for any price/task scenario, using cal-
+ibration data pooled from both experiments. We conclude
+with a discussion of the contributions and limitations of the
+paper and our thoughts on directions for future research.
+2. THEORY
+Every time-consuming activity generates an opportunity cost .
+The opportunity cost of doing A is the foregone net bene-
+ts one would have obtained from doing a next best option
+B. Researchers generally cannot observe the net benets of
+a person's options, but when they observe them doing A,
+they can infer that they nd doing A preferable to doing B,
+with all rewards and costs for the two tasks being taken into
+account. Applying this inference to work decisions yields
+a prediction: a person will work only when the net benets
+from working exceed the hypothetical net benets from their
+next best alternative, be it another job, leisure or a renewed
+job search. In labor models, the economic value of this \next
+best alternative" is characterized as a reservation wage.
+The reservation wage is dicult to estimate in practice for
+at least two reasons. First, all jobs oer a mixture of non-
+monetary benets and costs, or amenities and dis-amenities.
+For example, there are obvious non-monetary dierences be-
+tween working as a coal miner and working as an ice cream
+taste-tester. Second, observing someone working tells us
+only that their total benets exceed total costs, but not
+what those total cost actually are. Imagine a job oers a
+wagewand a stream of amenities, a, and a stream of dis-
+amenities,d. If the worker works for time t, they receive
+benets (w+a)tand bear costs dt. If the worker has a
+reservation wage, !, then observing someone working tells
+us only that w+ad!.
+To estimate !, we need to identify the worker's indier-
+ence point, i.e., the wwherew+ad=!. To push
+a worker down to his or her indierence point, we could
+continuously lower their wage by small amounts until they
+chose to quit. Assuming workers viewed this process san-
+guinely and their marginal costs were not increasing, their
+wage when they exit|i.e., when they are indierent between
+working and continuing|is their reservation wage for that
+task. This \decreasing wage" method is clearly impractical
+in traditional labor relationships, but [4] show that it can
+easily be done for small, piece-rate tasks when the process
+is explained up front and workers have little emotional in-
+vestment in their seconds-old \job." In our experiments, we
+capitalize on this feature of piece-rate work and and use a
+continuously decreasing payment function.
+2.1 Model of labor supply
+Because the payments in crowdsourcing contexts are so small,
+we do not expect a worker's marginal utility of wealth to
+change while working and thus we assume u(w) =w. Work-
+ers choose some positive, continuous quantity to produce,
+y0. They are paid P(y) for their choice of y. Let
+P0(y) =p(y). We assume that P(y) is strictly increasing,
+p()>0, and concave, p0()<0. We assume that it costs the
+workerC(y) to complete ytasks, with C0(y) =c(y). Theworker's maximization problem is:
+max
+yP(y)C(y)s:t: y0 (1)
+The rst-order condition is p(y) =c(y), i.e., the marginal
+benet equals marginal cost. An interior solution exists only
+whenB(y) =P(y)C(y) reaches a global maximum, which
+occurs when B(y) is concave.
+2.2 Cost curves and output
+In most applied contexts, P() will be linear (i.e., a constant
+piece-rate), so B(y) will be concave only when C(y) is
+strictly concave, i.e., marginal costs are increasing. If a task
+is very tiring, marginal costs are increasing ( c0(y)>0), but
+this is not necessarily the case: costs could be decreasing
+if workers get better with experience ( c0(y)<0), and costs
+can also be approximately linear ( c0(y) = 0) or even have
+dierent properties at dierent points (e.g., decreasing at
+rst, then linear).
+Consistent with our opportunity cost framework, we assume
+that the only costs of performing a task is the time it takes:
+C(y) =Ry
+0!t(x)dxwheret() is the marginal completion
+time and hence c(y) =t(y)!. The advantage of this as-
+sumption is that because the t(y)curve is observable, we
+can tell whether costs are increasing, decreasing or constant.
+In our experiments, we nd that completion times are essen-
+tially constant and we assume linear costs.3This linearity
+is unsurprising given the simplicity and brevity of our tasks.
+In practice P() is often linear, with a constant piece-rate 
+for each unit of output, giving a payment function P(y) =
+y. If costs are linear, there is no interior solution and work-
+ers are willing to produce either nothing when ! > =t or
+an innite amount when !<=t . The actual manifestation
+of this pheneomena is that worker's either produce nothing
+or produce all the way up to some cap set by employers.
+For example, in the experiments run by [16], workers could
+choose how many picture sorting tasks to perform, up to a
+cap of 100 and with each task paying a constant amount. In
+the experimenter's high-wage, low-diculty condition, mean
+output was over 90 tasks, and presumably many of the sub-
+jects hit the 100 task cap.
+2.3 Using the payment function to impute the
+reservation wage
+With linear costs and a linear payment function (constant
+piece-rate), an interior solution to the maximization problem
+does not exist. However, if we make P(y) strictly concave
+by having the piece-rate fall with greater output, then an in-
+terior solution at p(y) =!t0exists. A worker's estimated
+reservation wage is then estimated directly from their output
+choice: if a worker completes y
+i, then ^!i=p(y
+i)
+ti, where tiis
+the worker's average completion time. If costs were increas-
+ing, then instead of a point estimate of costs, ^t0, we use
+some parametric prediction, ^ti(y
+i) (recall that we observe
+completion times).
+3Excluding the very rst task, completion times rise by less
+than 1 millisecond per task. There is, however, a gap be-
+tween the rst task and second task, with the second task
+requiring about 1.5 fewer secondsFor ease of exposition, we modeled worker output as contin-
+uous. In most practical crowdsourcing applications, subjects
+make discrete output choices. Subjects chose some number
+of whole number of tasks to complet and P(y) =Ry
+0p(x)dx.
+The discrete analog to this function is P(y) =Py
+x=0p(x),
+hencep(y+ 1) =P(y+ 1)P(y). When output is discrete,
+measuring reservation wages is somewhat more complex, but
+we know that !ip(y
+i)=tiand!i> p(y
+i+ 1)=ti. If the
+piece-rate tasks are small and output is ne-grained, then:
+^!i(2ti)1[p(y
+i) +p(y
+i+ 1)]
+reasonably approximates the reservation wage.
+To use this method, it is important that workers are aware
+that the marginal payment is falling and will continue to fall.
+Otherwise, incorrect expectations that wages might increase
+eventually might cause them to continue working even when
+wages fall below their reservation wage.
+3. EXPERIMENTAL PRELIMINARIES
+Any reasonable labor supply model should predict that low-
+ering wages will reduce output. There are two ways to lower
+a person's wages: (a) increase the output they must produce
+in order to earn their previous wage or (b) lower their wage
+while keeping the required amount of work constant. As we
+will show, our model predicts that output will fall in either
+scenario. We then test these predictions in two experiments.
+In Experiment A, we test whther increasing the task di-
+culty reduces output, and in Experiment B, we test whether
+lowering wages reduces output. Both of our experiments
+had the same basic set-up: workers from AMT were asked
+to perform piece-rate tasks. Critically, they had complete
+freedom to choose how many pieces to complete. Before we
+discuss the experiments in depth, we rst provide necessary
+background information.
+3.1 Amazon’s Mechanical Turk
+Amazon's Mechanical Turk is an online labor market where
+workers can perform \Human Intelligence Tasks" (HIT) for
+\requesters." HITs vary, but most are small, simple tasks
+that are dicult for computers, but relatively easy for hu-
+mans. Common tasks include transcribing audio clips, clas-
+sifying and tagging images, reviewing documents and check-
+ing websites for pornographic content. When posting a HIT,
+a requester describes the task, sets a piece-rate payment,
+sets worker qualications, determines how long workers can
+work on the task, determines how many times they want
+each HIT performed and creates an interface for workers to
+use when working on the task.
+To become an AMT worker, a person must create an AMT
+account and provide Amazon with a bank account number.
+Workers are only allowed to have one account, and Amazon
+uses several technical and legal means to enforce this restric-
+tion. Workers can observe the collection of HITs and their
+attributes prior to starting work and can normally view a
+sample of the required work before starting. They are free
+to work on any task for which they are qualied, and they
+can begin work immediately after accepting a HIT. Once
+a worker completes a HIT, they must submit it for review.
+A requester then may review the work and decide whether
+or not to \approve" the HIT. If the HIT is approved, the
+worker is paid the piece-rate. The worker is also paid ifthe requester does not review and approve the work within
+a specied amount of time. Solely at their discretion, re-
+questers may \reject" work, in which case the worker is not
+paid.4Requesters may also elect to pay bonuses, which
+makes it easy to tailor payments to individual workers based
+on their performance within a nominally piece-rate HIT.
+3.2 Conduct of the experiments
+Experiment A and B were conducted in sequence, with ap-
+proximately 3 days between them. Of the 92 subjects who
+participated in A, 38 of them also participated in B, which
+had 198 subjects.
+Subjects were not informed that the task had an experimen-
+tal component. The HIT was simply posted on AMT like
+any other HIT, with the task title \User interface test." It
+is important to note that all subjects read identical instruc-
+tions before accepting the HIT and before observing any
+unique feature of their assigned treatment group. Thus we
+are condent that subjects did not select out of the experi-
+ment in response to the nature of their assigned treatment.
+In both experiments, all subjects accepting the HIT submit-
+ted a response, so no outcome data are missing.5
+3.3 Task and user interface
+For a crowdsourcing task, we had subjects click back and
+forth between two narrow vertical bars separated by some
+number of pixels. The \target" bar was green and the other
+bar was gray. The target bar alternated between the left bar
+and the right bar, with the rst target bar always begining
+on the left. If they missed a bar, the bar 
ashed red but
+workers could continue. Figure 1 represents the interface
+and the cursor movement needed to complete the task.
+We chose the clicking task because it is time consuming,
+requires the full attention of subjects and is not obviously
+fun. This task is also culturally neutral, easy to understand
+and easy to make harder (by narrowing the bars or spacing
+the bars farther apart). We organized work into blocks , with
+each block consisting of 10 back-and-forth clicks. A block
+is one unit of output (i.e., if a worker completes 3 blocks,
+y= 3). Subjects had 4 seconds to complete each click. If
+they did not perform the click within this time, the HIT
+ended, but across both experiments, no subject exceeded
+this cuto. If a subject missed more than 40% of the clicks
+in a block, the HIT ended. As with the time cuto, no
+subject came anywhere close to this limit. We capped total
+output at 200 blocks, but this proved unnecessary as the
+observed maximum output was only 58 blocks.
+After completing a block, subjects chose whether to quit
+or continue. If they had already completed y1 blocks,
+they were oered p(y) to perform an additional block, where,
+p(y) =P(y)P(y1). If they chose to quit, they performed
+no more tasks and earned P(y1). When making this exit
+decision, the interface showed then all the relevant rates and
+4[10] provide a model of the accept/reject decision in mar-
+kets like AMT and show under what conditions an \all re-
+ject" equilibrium does not occur.
+5A discussion of the causal inference issues raised by
+revealed-preference experiments conducted online can be
+found in [9].totals, as well as their error rate and their average per-block
+completion time in seconds. Subjects could rest as long as
+they liked before starting a new block.
+Figure 1: Crowdsourcing task. Subjects were asked
+to click between the two vertical rectangles. The
+\target" bar (shown on left here) was always colored
+green.
+3.4 Payment function for our experiments
+To create a concave payment function, each subsequent piece
+of work must pay less than the previous piece of work, but
+still oer positive payment. One payment function with this
+property is:
+P(y) =P
+1eky
+(2)
+Note that total earnings asymptotically approach Pasy
+increases. The parameter k0 can be thought of as deter-
+mining the \half-life" of the payment schedule. For example,
+if a givenkhas the property that P(10;k) =1
+2P, then the
+half-life of the schedule is 10 tasks. In all experiments, we
+used a half-life of 10; Table 1 shows samples of the total
+earnings and the marginal payment using Equation 2, given
+this half-life.6
+Table 1: Sample earnings and wages with 10 task
+half-life
+y P (y)=P p (y+ 1)=P
+1 0:07 0:0625
+5 0:29 0:0474
+25 0:82 0:0118
+One complication in our payment scheme is that it is im-
+possible to pay workers fractional cents. To circumvent this
+problem, we used a payment system where a worker was
+paid all of their whole-cent earnings for sure, and their frac-
+tional earnings stochastically. If a worker earned hwhole
+cents andffractions of a cent, with probability 1 f, we
+paid them hand with probability f, we paid them h+ 1.
+Payment was thus correct in expectation, since E[P(y)] =
+(1f)h+ (h+ 1)f=h+f. This procedure was explained
+to subjects before they joined the experiment.
+6For subjects producing only one unit of output, an ad-
+justment must be made when imputing reservation wages
+beacuseP(1) needs to incorporate the \show-up" fee.4. EXPERIMENT A: 
DIFFICULTY
+In Experiment A, subjects were randomly assigned to groups
+EASY and HARD . The only dierence between the two
+groups was that in EASY the vertical bars were 100 pix-
+els apart compared to 600 pixels apart in HARD . Of the
+92 subjects that participated, 42 were assigned to EASY
+and 38 were self-reported females. Subjects completed a to-
+tal of 18934 clicks. In both groups, subjects' earnings were
+determined using the same payment function, Equation 2,
+with the parameter values of P= 10 andk= 101log 1=2.
+With these parameter values, total earnings asymptotically
+approached 10 cents and the \half-life" was 10 blocks, i.e.,
+P(10) = 5.
+4.1 Model prediction
+Using the rst-order condition from Equation 1, p(y) =!t
+(for simplicity, we drop the notation) and treating optimal
+output as a function of t, we take the total derivative with
+respect totand gety0(t) =!=p0(y(t)). Because p0()<0,
+it follows that y0(t)<0. As expected, increasing the unit
+completion time reduces a worker's output.
+4.2 Results
+4.2.1 Effort
+Usurprisingly, subjects assigned to HARD needed more time
+to complete a block. Regressing average per block comple-
+tion time (in seconds), Ti, on the treatment indicator EASY i
+(with robust standard errors under each coecient)7we
+have:
+Ti=4:885|{z}
+1:25
EASY i+ 10:931|{z}
+1:2
+withR2= 0:13 and sample size N= 92. The treatment ef-
+fect is large and highly signicant: subjects in HARD took
+about 11 seconds to complete a block; subjects in EASY
+only needed about 6 seconds. If we examine between-click
+times instead of average block completion times, we see fur-
+ther evidence that the treatment was eective. In Figure 2,
+the distributions for both \hits" an \misses" are shown for
+each group. As expected, the \hit" distribution for HARD is
+right-shifted. Conrming our intuitions about the relation-
+ship between eort and quality, misses were associated with
+faster cursor movement.
+4.2.2 Output
+Even though subjects in HARD needed more time to com-
+plete each block, they had nearly the same pattern of output
+as subjects in EASY . Figure 3 shows output histograms for
+both groups. There is no discernible dierence in mean out-
+put, and although although more workers in HARD quit
+after performing just one block (12 vs. 7), this dierence is
+not statistically signicant.8Regressing output on a group
+indicator we have:
+yi=0:247|{z}
+3:76
EASY i+ 20:08|{z}
+2:72
+withR2= 5e05 and the sample size 92. Subjects in
+EASY had slightly less average output, though the eect
+7All standard errors in the paper are robust.
+8We regressed 1fyi= 1gonEASY i.
+Time between clicksdensity0.00.51.01.52.02.5
+0.00.51.01.52.02.5
+0.5 1.0 1.5 2.0 2.5EASY HARDoutcome
+hit
+missFigure 2: Distribution of time-between-clicks for
+\hits" and \misses" by treatment group. Distribu-
+tions are computed using a kernel density estimator.
+is imprecisely estimated. Transforming output with the log
+function and running the same regression we have:
+logyi= 0:136|{z}
+0:28
EASY i+ 2:298|{z}
+0:2
+withR2= 0:00256 and N= 92. Even though the coe-
+cient on EASY changes signs, both the regressions and the
+graphical analysis of Figure 3 lead to the same conclusion:
+despite the greater time required per-block for subjects in
+HARD , there was no discernible dierence in the patterns
+of output across groups.
+4.2.3 Reservation wages
+We imputed the reservation wage for each subject using the
+method explained in Section 2.3. The distribution of the
+log of estimates is plotted in Figure 4, which shows both
+the smooth kernel density estimate of the distributions as
+well as the the actual estimated values (displayed as tick
+marks along the horizontal axis). The top two panels contain
+the results of Experiment A. We can see that the imputed
+reservation wage distributions are quite similar, except that
+HARD has a fat left tail, implying that a cluster of workers
+inHARD exhibited output patterns consistent with very low
+reservation wages. However, this could be due to sampling
+variance. Indeed, in a regression of log reservation wages on
+a group indicator we have:
+log ^!i= 0:523|{z}
+0:37
EASY i+0:117|{z}
+0:26
+withR2= 0:02 and sample size N= 92. Assignment
+toEASY had a positive but statistically insignicant ef-
+fect. The coecients in this regression are more easily in-
+terpretable following transformation. They imply that theBlocks completedCount of subjects02468101214
+02468101214
+10 20 30 40 50EASY HARDFigure 3: Output by task diculty. The red vertical
+line indicates mean output in both groups, while the
+shaded band shows plus/minus the standard error in
+the estimated mean. Bins have unit length.
+geometric mean reservation wage was $0 :89/hour in HARD
+and $1:5/hour in EASY .
+4.3 Discussion
+The experimental results are theoretically ambiguous but
+are internally consistent. Given that workers in HARD took
+longer per task but had essentially the same output pattern
+compared to subjects in EASY , we would expect subjects in
+HARD to have relatively higher reservation wages, which is
+what we nd, albeit the eect only has a t-statistic of 1 :41.
+Perhaps the simplest explanation for the lack of treatment
+eects is that our treatment was not strong enough and that
+workers are not attuned to fairly small dierences in time.
+The quote by Lord Lionel Robbins about the \marginal util-
+ity of not bothering with marignal utility" seems particular
+apt, especially if worker's have a heuristic task-based (as
+opposed to time based) reservation rate.
+5. EXPERIMENT B: 
PRICE
+In Experiment B, 198 subjects were randomly assigned to
+groups HIGH and LOW . The task was identical in both
+groups, with the bars 100 pixels apart, but earnings asymp-
+totically approached either 10 cents (in LOW ) or 30 cents
+(inHIGH ). Because of the imprecise treatment eect esti-
+mates in Experiment A, we doubled the sample size. By
+chance, both groups had the same number of subjects. Of
+the 198 subjects, 72 were self-reported females. Subjects
+completed a total of 45710 clicks.
+5.1 Model prediction
+Consider an alternative payment function 
P(y), where
+is some positive number. The rst order condition for the
+log(wage)density0.000.050.100.150.200.250.30
+0.000.050.100.150.200.250.30
+0.000.050.100.150.200.250.30
+0.000.050.100.150.200.250.30
+−3 −2 −1 0 1 2 3EASY HARD HIGH LOWFigure 4: Imputed log reservation wage distribu-
+tions both experimental groups, in both experi-
+ments.
+Blocks completedCount of subjects05101520
+05101520
+10 20 30 40 50HIGH LOW
+Figure 5: Output by price group. The red vertical
+line indicates mean output in both groups, while the
+shaded band shows plus/minus the standard error in
+the estimated mean. Bins have unit length.original maximization problem is now 
p(y)t!= 0, and
+if we treat the optimal output yas a function of 
, the total
+derivative of the rst order condition with respect to 
is
+p(y) +
p0(y(
))y0(
) = 0 and thus:
+y0(
) =p(y)
+
p0(y(
))
+Because payment is strictly increasing, p()>0, and because
+the total payment function is cocave, p0()<0, it follows that
+y0(
)>0, i.e., output increases when payment is higher.
+5.2 Results
+5.2.1 Output
+Figure 5 shows the histogram of output in the two groups.
+Although LOW has more early quits, both groups have siz-
+able numbers of early quitters as well as some stalwarts that
+completed in excess of 50 blocks. Unlike in Experiment A,
+mean output in LOW is noticeably lower than in HIGH .
+Regressing output on a group indicator we have:
+yi=2:808|{z}
+2:89
LOW i+ 24:071|{z}
+2
+withR2= 0:0048 andN= 198. Subjects in LOW had
+lower mean output, but the eect is imprecisely estimated
+due to the bi-modality of output, which can be seen in the
+output histograms. Estimating the regression in logs rather
+than levels we have:
+logyi=0:303|{z}
+0:17
LOW i+ 2:712|{z}
+0:11
+withR2= 0:02 andN= 198. There is a large and signi-
+cant dierence in geometric means across the groups. From
+Figure 5, it appears that most of the eect is due to the
+relatively large number of subjects in LOW quitting after
+small amounts of output. We can conrm this graphical ob-
+servation by regressing an indicator for whether a subject
+completed fewer than 10 blocks on the treatment indicator:
+1fyi<10g= 0:152|{z}
+0:07
LOW i+ 0:273|{z}
+0:05
+withR2= 0:03 andN= 198. The choice of 10 blocks is
+somewhat arbitrary, but the eect is strong for all low cuto
+values, and a QQ-plot (not shown) makes the pattern even
+more apparent.
+5.2.2 Reservation Wages
+Unlike in Experiment A, group assignment strongly aected
+the imputed reservation wage. The bottom two panels of
+Figure 4 show that the mass of observations at the center
+of the distribution is left-shifted for LOW relative to HIGH .
+Both distributions also have a second mass of observations
+with very low reservation wages, but as with the main hump,
+theLOW observersations are shifted further left than the
+low-wage hump in HIGH . Regressing log wages on a group
+indicator we have:
+log ^!i=0:792|{z}
+0:22
LOW i+ 0:447|{z}
+0:14
+withR2= 0:06 andN= 198. Transforming the predic-
+tions into levels, the geometric mean reservation wage was
+$1:56/hour in HIGH and $0:71/hour in LOW .5.3 Discussion
+As per the model's prediction, lower pay reduces output, but
+the nding that being in LOW lowers a worker's reservation
+wage implies that the lower output in LOW is not as low as
+itshould be. Because the actual tasks are identical across
+groups, the dierence in imputed reservation wages cannot
+be due to uncontrolled-for dierences in amenities. There
+are several other possible explanations for the data, includ-
+ing worker error, increasing but non-time based marginal
+costs and target earning behavior; we believe target earning
+provides the most compelling explanation for our ndings.
+Worker error could explain our results: we would nd lower
+reservation wages in LOW if (1) subjects in LOW falsely
+believe that they are being paid more than they actually
+are, (2) subjects in HIGH falsely believe that they are be-
+ing paid less than they actually are or (3) some combination
+of (1) and (2). Liebman and Zeckhauser [15] argue that
+people systematically misinterpret schedules, usually by let-
+ting early marginal payments \bleed over" to aect percep-
+tions of future marginal payments. However, with our con-
+cave schedules, this bleed over would occur in both groups.
+While we have no evidence on this question, it seems likely
+that any bleed over eects would be greater in HIGH , which
+would lead to eects in the opposite direction of what we do
+nd. Furthermore, because subjects were informed of their
+precise marginal payment after each block, this bleed over
+explanation seems unlikely.
+Constant marginal costs are a key assuption in our reserva-
+tion wage estimation method. If marginal costs are rapidly
+increasing, then we would incorrectly infer that subjects in
+LOW had reservation wages too low: the additional output
+subjects in HIGH should produce because of their higher
+wages is moderated by the higher costs of high output. Al-
+though an increasing marginal cost explanation is possible,
+it seems unlikley, given that subjects could rest for as much
+time as they liked between blocks. Furthermore, even sub-
+jects producing lots of output spent less than 15 minutes
+total, including resting time (recall that per block time for
+the 100 pixel group was less than 7 seconds). Rapid in-
+creases in marginal costs over such a short duration of time
+seem rather unlikely.9
+An additional explanation for the results is that some work-
+ers may be target earners . Target earners try to obtain some
+self-imposed earnings goal rather than respond to the cur-
+rent oered wage. If at least some workers are target earn-
+ers, then the results are easy to explain: because their wages
+are lower, subjects in LOW must produce more output to
+acheive their earnings targets than they would have to pro-
+duce if they were instead assigned to HIGH .
+6. DEPARTURES FROM THE RATIONAL
+MODEL
+The dierences in the imputed reservation wage distribution|
+particularly those found in Experiment B|strongly suggest
+9One possibility we must consider with more complex work
+is that workers are likely to need more experience with a task
+before deciding whether they are good at it. In these cases,
+we might mistakenly view learning and exit as increasing
+marginal costs.that the rational model cannot explain the behavior of all
+workers. Our best conjecture is that workers create earnings
+targets that in
uence their output decisions. While having
+goals seems sensible and perhaps heuristically \rational" if
+the target provides motivation, earnings targets lead work-
+ers to commit a kind of sunk cost fallacy because, in the
+absence of income eects, past earnings are irrelevant to the
+decision they must make at the margin (i.e., whether the
+next bit of earnings is worth the the trouble of the next bit
+of work).
+To return to the suspension bridge building analogy from
+Section 1, we are now dealing with metallurgy instead of
+mechanics: target earners do not behave like rational work-
+ers in certain contexts, and this dierence will matter in
+applications. For example, in the absence of income eects,
+when wages are high, a target earner works less and a ra-
+tional worker works more. There is mixed evidence on the
+question of whether target earning occurs in \real life" [3, 6,
+7], but we nd fairly unambiguous evidence of target earning
+in several places in the results.
+The rst piece of evidence of target earning is that in ev-
+ery experiment, at least some subjects try to pursue the
+maximum earnings possible, despite the low wages associ-
+ated with this strategy. For rational workers to generate
+this pattern, the wage distribution would have to be highly
+bi-modal. A more plausible explanation is that workers try
+to earn the full amount possible (i.e., Pis their target) and
+only quit when they realize this goal is unattainable.10
+6.1 Preferences for “focal point” earnings
+More evidence of target earning can be see in the pattern of
+output. In Figure 6, we plot histograms of the output for
+HIGH , grouped in horizontal panels by the 
oor of earnings,
+bP(y)c. Subjects show a preference for working the mini-
+mum amount possible to earn some whole number of cents:
+when multiple output values yield the same number of whole
+cents, the far left bar of the histogram is the highest. The
+only exception is in the 29 cent panel, however, recall that
+inHIGH subjects' earnings asymptotically approached 30
+cents. Subjects quit once they realized that they could not
+break out of the 29 cent band. Although suggestive of tar-
+geting, the preference for whole cents could be symptomatic
+of extreme risk aversion (recall from Section 3.2 that pay-
+ment is stochastic due to the fractional cent problem), or
+workers could believe that we might cheat them on the frac-
+tional cents (i.e., ignore their fractional earnings) but that
+we would not be willing to withhold whole-cent earnings.
+Harder to reconcile with a rational model is another pat-
+tern seen in Figure 6: subjects show a preference for earn-
+ings amounts evenly divisible by 5. The smallest earnings
+amounts (e.g., 2, 3 and 5 cents) all get several subjects,
+but because these are people who quit very early, they pre-
+sumably do not have a target or would not need a target.
+However, we see clear output spikes at 15, 20 and 25 cents.
+10A more pedestrian explanation could be that some subjects
+believe that the payment is in dollars, not cents, despite
+the clear instructions that stated that payment would be in
+cents. One subject did email us insisting that payment was
+supposed to be in dollars|we sent him screen shots proving
+this was not the case.Assume that in the absence of\modulo 5"targeting, the pro-
+portion of subjects earning amounts divisible by 5 should be
+equal, in expectation, to the proportion of potentially real-
+izable amounts divisible by 5. Consider the set of possible
+whole-cent earnings:
+Pw=fbP(1)c;bP(2)c;:::bP(N)cg
+and the fraction of those elements of Pwthat are divisible
+by 5, which is given by:
+q=jPwj1X
+x2Pw1fxmod 5 = 0g
+wherej:jindicates the number of elements in the set. Let Y
+be the set of realized output choices for an experiment group
+and letnbe the number of observations, n=jYjand let
+sbe the actual number of observations in Ysuch that the
+whole-cent earnings were divisible by 5: s=P
+y2Y1fbP(y)c
+mod 5 = 0g
+Under our assumption of proportionally, we can compute
+the probability of observing ssuccesses out of ntrials when
+the per-trial probability of success is q. We had 33 successes
+out of 99 trials when the probability was only q= 0:22.
+The probability of observing this many successes or more
+by chance is only 0 :0027.
+7. CONCLUSION
+We nd some agreement with a simple rational model, as
+well as important anomolies; we nd fairly strong evidence
+that at least some worker's work to targets. Designers should
+consider this propensity when designing incentives schemes
+and give people natural targets that will increase output,
+though they should also consider that such schemes might
+seem manipulative and could backre (and potentially be
+unethical).
+In the following section, we demonstrate how our calibrated
+model can be used in applied work. While we only nd par-
+tial agreement between the model's predictions and reality,
+we beleive it still oers a useful approximation. We conclude
+by discussing how our paper ts into a larger research pro-
+gram and lay out some directions for future investigations..
+7.1 Using the calibrated model
+It is straightforward to use our calibrated model for pre-
+diction. Consider some crowdsourcing task that takes t0to
+complete and that will pay a piece-rate of p0. Workers whose
+reservation wage exceeds the oered wage, p0=t0, will accept
+the task. The fraction of workers producing at least one unit
+of output is equal to the probability that a given worker
+will nd participating attractive: Pr(p0!it0) =F
+p0
+t0
+whereFis the cumuliative density function of the estimated
+reservation wage distribution.
+The labor supply curve is S(w) =NsF(logw), whereNs
+is the number of workers in the population and w=p0=t0.
+The point elasticity of extensive labor supply is thus w=
+f(logw)=F(logw). To compute the intensive elasticity, we
+would have to know something about the change in marginal
+costs.11
+11If we learn that c(y) =!yt+ (y1)2, output on theWorker Output10 20 30 40 502
+3
+5
+7
+8
+10
+11
+12
+13
+15
+16
+17
+18
+19
+20
+21
+22
+23
+25
+26
+27
+28
+29Divisible 
+ by 5?
+FALSE
+TRUEFigure 6: Earnings in the HIGH Group. This plot shows the distribution of output, grouped by the whole
+digit of worker earnings (in cents). The vertical axis scale is not shown and is scaled dierently by groups.
+However, each horizontal white line indicates a gap of one.
+If we pool estimates from both experiments and assume that
+wages are log normally distributed, the distribution parame-
+ters are ^= 0:074 and ^= 1:634, with the reservation wage
+measured in dollars per hour. In Table 2, the 25th, 50th and
+75th percentiles of the pooled reservation wage distribution
+are shown, as well as the extensive labor supply elasticity
+computed at that point using the log normal approxima-
+tion. Consisent with a right-skewed distribution like the log
+normal, the arithmetic mean is considerably higher than the
+median wage.
+Table 2: Pooled reservation wage distribution prop-
+erties
+Wage ($/hour) Point Elasticity
+25th p. 0.321 0.81
+Median 1.384 0.43
+75th p. 2.876 0.28
+Mean 3.625 0.24
+It is important to remember the fairly strong assumptions
+underlying these predictions. First, the reservation wage
+distribution was estimated using a selected sample of AMT
+workers willing to participate in our experiment. Second,
+we are assuming constant marginal costs, and we are assum-
+ing that task amenities/dis-amenities for the new task are
+equivalent to those from the back-and-forth clicking task we
+used. Third, we are assuming that there is a xed reserva-
+intensive margin can be predicted: each worker produces
+y, such that p(y)!t+ 2y= 0.tion wage and that workers respond rationally to the oered
+wage, despite some evidence to the contrary.
+7.2 Future research
+Crowdsourcing is still a new development and many open
+questions remain. Perhaps the most obvious next empirical
+step is to collect data on the compensating dierentials asso-
+ciated with dierent kinds of crowdsourcing tasks. In other
+words, compared to a baseline task, how much more or less
+do workers have to be paid to generate the same amount of
+output? A related question is how costs change with output,
+i.e., does a task get much easier or much harder as the worker
+gains experience? Finally, it would be interesting to learn
+about the correlates of reservation wages. For example, do
+groups with lower opportunity costs (e.g., the unemployed,
+non-US citizens, etc.) have lower reservation wages?
+Perhaps the biggest limitation of the current model is that it
+only predicts the fraction of workers that will accept a task.
+It tells us nothing about how many workers will actually see
+a given oer on AMT. In our experience, the time since a
+HIT was posted aects uptake, presumably because \fresh"
+HITs are easier to nd when searching by date posted. Fur-
+thermore, AMT workers are not uniformly distributed across
+time zones and presumably the number of potential work-
+ers waxes and wanes over the day. It would be useful to
+learn more about search and uptake and it might shed light
+on the process of job search|a topic of great interest to
+economists.
+7.3 Main contributionWe view our paper as a step towards the theoretical frame-
+work for crowdsourcing/human computation advocated by
+[14]. Jain and Parkes argue (correctly in our view) that
+advances in crowdsourcing as a methodology will require
+broadly applicable, predictive models. They discuss game
+theory as a potential generalizing framework, but point out
+that crowdsourcing \games" are not so much about workers
+revealing private information (which would suggest a mech-
+anism design approach) as they are about getting workers
+to show up and exert the eort needed to accomplish a task.
+In a labor relationship, there is a \game" between work-
+ers and their employers: workers must choose how much
+(costly) eort to provide and employers|who prefer high
+eort|cannot directly observe this choice. This classic prin-
+cipal/agent problem arises in many real-world contexts, but
+we argue that in most crowdsourcing applications, it is a
+fairly easy problem to solve.12When eort is highly corre-
+lated with output and output is observable, simple mecha-
+nisms can eliminate moral hazard. By rejecting low quality
+work (or not hiring low quality workers in the future), buyers
+can easily make shirking a dominated strategy.
+Although the moral hazard problem is solvable, the problem
+of getting a sucient supply of labor persists. Once an em-
+ployer sets up a quality control mechanism (e.g., screening,
+ring, spot checks, etc.), the worker is essentially playing
+a game against nature. Individual workers will make la-
+bor supply decisions by comparing the costs and benets of
+working, and although workers must think rationally about
+their preferences, they do not have to think strategically.
+In short, we do not need a game theory of crowdsourcing,
+but rather a price theory of crowdsourcing. Our model and
+reservation wage estimation procedure provide the ground-
+work for such a theory.
+8. ACKNOWLEDGMENTS
+John Horton thanks the Berkman Center for Internet and
+Society and the NSF-IGERT Multidisciplinary Program in
+Inequality & Social Policy for generous nancial support.
+Thanks to Richard Zeckhauser for extremely useful com-
+ments and suggestions. All plots were made using the open
+source R package ggplot2 , developed by Hadley Wickham
+[21].
+9. REFERENCES
+[1] L. Adamic, J. Zhang, E. Bakshy, and M. Ackerman.
+Knowledge sharing and yahoo answers: everyone
+knows something. 2008.
+[2] Y. Benkler. The Wealth of Networks: How Social
+Production Transforms Markets and Freedom . Yale
+University Press, 2007.
+[3] C. Camerer, L. Babcock, G. Loewenstein, and
+R. Thaler. Labor supply of new york city cabdrivers:
+One day at a time. The Quarterly Journal of
+Economics , 112(2):407{441, 1997.
+[4] D. L. Chen and J. J. Horton. The wages of paycuts:
+12We qualify this statement because problems do arise when
+buyers do not know \what right looks like" and aggre-
+gating the inputs of other workers is not helpful for de-
+termining quality. For an example of this problem, see
+http://groups.csail.mit.edu/uid/deneme/?p=90.Evidence from a eld experiment. Working Paper ,
+2009.
+[5] D. DiPalantino and M. Vojnovic. Crowdsourcing and
+all-pay auctions. In Proceedings of the tenth ACM
+conference on Electronic commerce , pages 119{128.
+ACM, 2009.
+[6] H. S. Farber. Reference-dependent preferences and
+labor supply: The case of new york city taxi drivers.
+American Economic Review , 98(3):1069|1082, 2008.
+[7] E. Fehr and L. Goette. Do workers work more if wages
+are high? evidence from a randomized eld
+experiment. American Economic Review ,
+97(1):298{317, 2007.
+[8] B. Frei. Paid crowdsourcing: Current state & progress
+toward mainstream business use. Produced by
+Smartsheet.com , 2009.
+[9] J. J. Horton and R. J. Zeckhauser. The potential of
+online experiments. Working Paper , 2010.
+[10] J. J. Horton and X. Zhu. The allocation of decision
+rights in designed markets. Working Paper , 2010.
+[11] J. Howe. Crowdsourcing: Why the power of the crowd
+is driving the future of business . Three Rivers Press,
+2009.
+[12] B. A. Huberman, D. Romero, and F. Wu.
+Crowdsourcing, attention and productivity. Journal of
+Information Science (in press) , 2009.
+[13] S. Jain, Y. Chen, and D. Parkes. Designing incentives
+for online question and answer forums. In Proceedings
+of the tenth ACM conference on Electronic commerce ,
+pages 129{138. ACM, 2009.
+[14] S. Jain and D. C. Parkes. The role of game theory of
+human computation systems. In HCOMP 2009 . ACM,
+2009.
+[15] J. Liebman and R. Zeckhauser. Schmeduling. Working
+Paper , 2004.
+[16] W. A. Mason and D. J. Watts. Financial incentives
+and the `performance of crowds'. Knowledge Discovery
+and Data Mining- Human Computation
+(KDD-HCOMP) , 2009.
+[17] A. E. Roth. The economist as engineer: Game theory,
+experimentation, and computation as tools for design
+economics. Econometrica , 70:1341{1378, 2002.
+[18] V. S. Sheng, F. Provost, and P. G. Ipeirotis. Get
+another label? improving data quality and data
+mining using multiple, noisy labelers. Knowledge
+Discovery and Data Minding 2008 (KDD-2008) , 2008.
+[19] R. Snow, B. O'Connor, D. Jurafsky, and A. Y. Ng.
+Cheap and fast|but is it good? evaluating non-expert
+annotations for natural language tasks. Proceedings of
+the Conference on Empirical Methods in Natural
+Language Processing (EMNLP 2008) , 2008.
+[20] L. Von Ahn. Games with a purpose. IEEE Computer
+Magazine , 39(6):96{98, 2006.
+[21] H. Wickham. ggplot2: An implementation of the
+grammar of graphics. R package version 0.7, URL:
+http://CRAN. R-project. org/package= ggplot2 , 2008.
\ No newline at end of file
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+arXiv:1001.0628v2  [math.PR]  7 Apr 2010On the distribution of the Brownian motion process
+on its way to hitting zero
+Konstantin Borovkov1
+Dept. of Maths & Stats, University of Melbourne,
+Parkville 3010, Australia
+email:borovkov@unimelb.edu.au .
+Abstract
+We presentfunctional versionsof recent resultsontheuniv ariatedistributions
+of the process Vx,u=x+Wuτ(x),0≤u≤1, where W•is the standard Brownian
+motion process, x >0 andτ(x) = inf{t >0 :Wt=−x}.
+AMS 2000 Subject Classification: 60J65
+Key words: Brownian motion, hitting time, Brownian meander , Bessel bridge.
+Let{Wt}t≥0be the standard univariate Brownian motion process and, for x >0,
+Wx,t:=x+Wt, t≥0, τ(x) := inf{t >0 :Wx,t= 0}.
+As is well known, τ(x) is a proper random variable with density
+px(t) =xe−x2/2t
+√
+2πt3, t > 0, (1)
+so one can introduce
+Vx,u:=Wx,uτ(x),0≤u≤1.
+These random variables were studied in the recent paper [5], where it was shown
+(Theorem 1.1) that, for any fixed u∈(0,1), Vx,uhas density
+px,u(y) :=d
+dyP(Vx,u≤y)
+=4/radicalbig
+u(1−u)xy2
+π(uy2+(1−u)(y−x)2)(uy2+(1−u)(y+x)2), y > 0,(2)
+∼4x/radicalbig
+u(1−u)
+πy2asy→ ∞ (3)
+1Research supported by ARC Discovery Grant DP0880693.
+12
+(here and in what follows, a∼bmeans that a/b→1). Representation (2) implies, in
+particular, that, for any fixed u∈[0,1], one has
+Vx,ud=xV1,u. (4)
+Using a direct tedious calculation, it was also demonstrated in Section 3 of [5] that, for
+a fixedu∈(0,1), the density px,ucoincides with that of a “scaled Brownian excursion
+at the corresponding time, averaged over its length”. The mathem atical formulation
+of that result was given by formula (3.3) in [5] that can be rewritten a s follows. Let
+{RT
+x,t}t≤Tbe a three-dimensional Bessel bridge of length Tpinned at xat timet= 0
+and at 0 at time t=T, which is independent of our process Wx,•(and hence of τ(x)).
+Recall that one can represent the process as
+RT
+x,t=/vextenddouble/vextenddoubleW(3)
+x,t−tT−1W(3)
+x,T/vextenddouble/vextenddouble,0≤t≤T, (5)
+where
+W(3)
+x,t= (x,0,0)+W(3)
+t, t≥0, (6)
+W(3)
+•being a standard three-dimensional Brownian motion process and /bardbl · /bardblthe Eu-
+clideannormin R3. Theabove-mentionedformulafrom[5]isequivalenttotheassertio n
+that, for any fixed u∈[0,1],one has
+Vx,ud=Rτ(x)
+x,uτ(x). (7)
+Note that RT
+x,•is not exactly an excursion (an excursion returns to the same point
+where it started) but, rather, a time-reversed Brownian meande r (see e.g. p.63 in [2]),
+and that on the right-hand side of (7) it is averaged not over its leng th, but rather of
+that of an independent version of Wx,•on its way to hitting zero.
+Observe also that, due to the self-similarity of the Brownian motion p rocess, rep-
+resentation (5)–(6) implies that
+RT
+x,•Td=T1/2RT−1/2x,•
+(as processes in C[0,1]), where we put Rx,t:=R1
+x,t.
+The main aim of the present note is to give simple proofs to functional versions
+of (4) and (7) (that had “remained elusive”, as was noted in [5]).
+Theorem 1 For anyx >0,
+{Vx,u}u≤1d={xV1,u}u≤1. (8)
+Furthermore, there exists a regular version of the conditio nal distribution of V1,•in
+C[0,1]givenτ(1) =Tthat coincides with the law of T1/2RT−1/2,•, and therefore, if
+τd=τ(1)is independent of the Brownian motion process from represen tation(5)–(6),
+then one has
+{V1,u}u≤1d={τ1/2Rτ−1/2,u}u≤1. (9)3
+Before proceeding to prove Theorem 1, we will make the following obs ervation.
+That the asymptotic behaviour (3) holds is most natural in view of re presentation (9)
+and the form (1) of the density of τ(x), the latter implying that τ1/2has the density
+/radicalbigg
+2
+πy−2e−1/2y2, y > 0.
+Moreover, since Wx,•isamartingalewith Wx,0=x, fromtheoptionalsamplingtheorem
+one immediately obtains that, using notation
+ˆXt:= sup
+s≤tXs,ˇXt:= inf
+s≤tXs
+for the maximum and minimum processes of X•, one has
+P(ˆVx,1> y) =P/parenleftbigˆWx,τ(x)> y/parenrightbig
+=xy−1, y > x, (10)
+with the same asymptotic behaviour ∝y−2asy→ ∞for the density.
+Finally, denoting by qx,y,u(z) the density of a random variable following the condi-
+tional distribution L/parenleftbig
+y−1Vx,u|ˆVx,1> y/parenrightbig
+ofy−1Vx,ugivenˆVx,1> y(here and in what
+follows,L/parenleftbig
+X|C/parenrightbig
+denotes the conditional distribution of the random element Xin the
+respective measureable space given condition C), it is easily seen from (2), (3) and (10)
+that, for any fixed x,z >0 andu∈(0,1),
+qx,y,u(z)∼4/radicalbig
+u(1−u)
+πz2asy→ ∞.
+Proof of Theorem 1. First we observe that
+Wx,t=x(1+x−1Wt) =x/tildewiderW1,tx−2, t≥0, (11)
+where/tildewiderW1,•is a Brownian motion process starting at 1. All quantities related to t his
+process we will label with tilde. As τ(x) is the first time the LHS of (11) turns into
+zero, we see that /tildewideτ(1) =τ(x)x−2.Therefore
+Vx,u=Wx,uτ(x)=x/tildewiderW1,u/tildewideτ(1)=x/tildewideV1,u, u∈[0,1],
+which proves (8). So from now on, we can assume without loss of gen erality that x= 1.
+Next let, for some functions fj∈C[0,1] and numbers rj>0, j= 1,2,...,n,
+A:=/intersectiondisplay
+j≤n{f∈C[0,1] :/bardblf−fj/bardbl< rj}
+be a finite intersection of open balls in C[0,1] (/bardbl·/bardblstands for the uniform norm). For
+T,h,δ > 0, put
+AT:={f(•/T) :f∈A} ⊂C[0,T], ε(δ) := max
+j≤nωfj(δ),4
+whereωf(δ) := sup0≤s<t≤s+δ≤1|f(s)−f(t)|is the continuity modulus of the function f.
+Finally, we denoteby Aε(h/T)
+Ttheε(h/T)-neighbourhoodof AT(intheuniformtopology
+onC[0,T]) and introduce the event
+BT,h:=/braceleftbig
+{W1,t}t∈[0,T]∈Aε(h/T)
+T/bracerightbig
+.
+Now, using the Markov property and the well-know relations
+P/parenleftbigˇWT+h<0|WT=y/parenrightbig
+= 2Φ(yh−1/2),P/parenleftbigˇW1,T>0|W1,T=y/parenrightbig
+= 1−e−2y/T, y >0,
+whereΦ = 1−Φ,Φ being the standard normal distribution function, we have
+P/parenleftbig
+V1,•∈A, τ(1)∈(T,T+h)/parenrightbig
+≤P/parenleftbig
+BT,h, τ(1)∈(T,T+h)/parenrightbig
+=/integraldisplay∞
+0P/parenleftbig
+BT,h, τ(1)∈(T,T+h)|W1,T=y/parenrightbig
+P(W1,T∈dy)
+=/integraldisplay∞
+0P/parenleftbig
+BT,h,ˇW1,T>0,ˇW1,T+h<0|W1,T=y/parenrightbig
+P(W1,T∈dy)
+=/integraldisplay∞
+0P/parenleftbig
+BT,h,ˇW1,T>0|W1,T=y/parenrightbig
+P/parenleftBig
+min
+t∈[T,T+h]W1,t<0/vextendsingle/vextendsingle/vextendsingleW1,T=y/parenrightBig
+P(W1,T∈dy)
+=/integraldisplay∞
+0P/parenleftbig
+BT,h|ˇW1,T>0,W1,T=y/parenrightbig
+P/parenleftbigˇW1,T>0|W1,T=y/parenrightbig
+2Φ(yh−1/2)P(W1,T∈dy)
+= 2/integraldisplay∞
+0P/parenleftbig
+BT,h|ˇW1,T>0,W1,T=y/parenrightbig/parenleftbig
+1−e−2y/T/parenrightbig
+Φ(yh−1/2)P(W1,T∈dy)
+= (4+o(1))h1/2/integraldisplayh1/4
+0P/parenleftbig
+BT,h|ˇW1,T>0,W1,T=y/parenrightbig
+gT(yh−1/2)dy+o(h) (12)
+ash↓0,where
+gT(u) =1√
+2πuT−3/2e−1/(2T)Φ(u), u > 0,
+and we used Mills ratio to infer that/integraltext∞
+h1/4=o(h)
+Next we will show that the probability in the last integrand in (12) conv erges to
+the respective probability for the Brownian meander process as y↓0.
+Recall that the Brownian meander process {W⊕
+s}s≤1can be defined as follows (see
+e.g. [4] or p.64 in[2]): letting ζ:= sup{t≤1 :Wt= 0}bethe last zero ofthe Brownian
+motion in [0 ,1], we set
+W⊕
+s:= (1−ζ)−1/2/vextendsingle/vextendsingleWζ+(1−ζ)s/vextendsingle/vextendsingle,0≤s≤1.
+This is a continuous nonhomogeneous Markov process whose trans ition density can be
+found e.g. in [4] (relations (1.1) and (1.2)). It is known that the cond itional version
+of the process pinned at x >0 at time s= 1 coincides in distribution with the three-
+dimensional Bessel process starting at zero and also pinned at xat times= 1 (see e.g.
+p.64 in [2]), which can be written as
+L/parenleftbig
+{W⊕
+s}s≤1|W⊕
+1=x/parenrightbig
+=L/parenleftbig
+{/bardblW(3)
+s/bardbl}s≤1|/bardblW(3)
+1/bardbl=x/parenrightbig
+.5
+It is not hard to deduce from here, the spherical symmetry of the Brownian motion
+processW(3)
+•and representation (5)–(6) above that
+L/parenleftbig
+{W⊕
+s}s≤1|W⊕
+1=x/parenrightbig
+=L/parenleftbig
+{/bardblW(3)
+x,1−s−(1−s)W(3)
+x,1/bardbl}s≤1/parenrightbig
+=L/parenleftbig
+{Rx,1−s}s≤1/parenrightbig
+.(13)
+An alternative insightful interpretation of the Brownian meander is given by the
+fact that its distribution (in C[0,1]) coincides with the weak limit of conditional dis-
+tributions of W•conditioned to stay above −ε↑0 :
+L/parenleftbig
+{W⊕
+s}s≤1/parenrightbig
+= w-lim
+ε↓0L/parenleftbig
+{Ws}s≤1|ˇW1>−ε/parenrightbig
+(Theorem (2.1) in [4]; w-lim stands for the limit in weak topology). A cond itional
+version of a result of this type is used in the calculation displayed in (14 ) below.
+Now return to the probability in the integrand in the last line in (12) and recall the
+well-known property of Brownian bridges that conditioning a Brownia n motion on its
+arrival at a point y/ne}ationslash= 0 at time Tis equivalent to conditioning on its arrival to zero
+at that time and then adding the deterministic linear trend componen tyt/T. This
+implies that, for any ε≥ε(h/T),
+P/parenleftbig
+BT,h|ˇW1,T>0,W1,T=y/parenrightbig
+=P/parenleftbig
+{W1,t+ytT−1}t≤T∈Aε(h/T)
+T|W1,T= 0;W1,s>−ysT−1,s∈[0,T]/parenrightbig
+=P/parenleftbig
+{WT−t+ytT−1}t≤T∈Aε(h/T)
+T|WT= 1;Ws>−y(T−s)T−1,s∈[0,T]/parenrightbig
+=P/parenleftbig
+{T1/2W1−v+yv}v≤1∈Aε(h/T)|W1=T−1/2;Wv>−yT−1/2(1−v),v∈[0,1]/parenrightbig
+≤P/parenleftbig
+{T1/2W1−v+yv}v≤1∈Aε|W1=T−1/2;Wv>−yT−1/2(1−v),v∈[0,1]/parenrightbig
+→P/parenleftbig
+{T1/2W⊕
+1−v}v≤1∈Aε|W⊕
+1=T−1/2/parenrightbig
+=P/parenleftbig
+{T1/2RT−1/2,s}s≤1∈Aε/parenrightbig
+(14)
+asy↓0,where the second last relation follows from the weak convergence e stablished
+in Theorem 6 in [3] (as it is obvious that Aεhas null boundary w.r.t. the limiting
+distribution) and the last one follows from (13).
+Sinceε(h/T)→0 ash↓0, andAhas a null boundary under L/parenleftbig
+{T1/2RT−1/2,s}s≤1/parenrightbig
+,
+we conclude from (12) (changing there the variables: u=yh−1/2) that
+limsup
+h↓01
+hP/parenleftbig
+V1,•∈A, τ(1)∈(T,T+h)/parenrightbig
+≤limsup
+h↓04P/parenleftbig
+T1/2RT−1/2,•∈A/parenrightbig/integraldisplayh−1/4
+0gT(u)du
+=P/parenleftbig
+T1/2RT−1/2,•∈A/parenrightbig
+p1(T), (15)
+owing to/integraltext∞
+0uΦ(u)du=1
+4and (1).6
+As the same argument as employed in (14) and (15) will also work for t he comple-
+ment ofA, we obtain that
+P/parenleftbig
+V1,•∈A, τ(1)∈(T,T+h)/parenrightbig
+∼P/parenleftbig
+T1/2RT−1/2,•∈A/parenrightbig
+p1(T)hash↓0.
+This relation implies that, for any fixed 0 < T1< T2<∞,
+P/parenleftbig
+V1,•∈A, τ(1)∈(T1,T2)/parenrightbig
+=/integraldisplayT2
+T1P/parenleftbig
+T1/2RT−1/2,•∈A/parenrightbig
+p1(T)dT.
+Since intersections of finite collections of open balls form determining classes in sepa-
+rable spaces (see e.g. Section I.2 in [1]), this completes the proof of the theorem.
+Acknowledgment. The author is grateful to the referee whose valuable comments
+helped to improve the exposition of the paper.
+References
+[1] Patrick Billingsley. Convergence of Probability Measures. 2nd edn. Wiley, New
+York, 1999. MR1700749
+[2] Andrei N. Borodin and Paavo Salminen. Handbook of Brownian motion—facts and
+formulae. 2nd edn. Birkh¨ auser Verlag, Basel, 2002. MR1912205
+[3] Konstantin Borovkov and Anrew N. Downes. On boundary cross ing probabilities
+for diffusion processes. Stoch. Proc. Appl. 2009. DOI 10.1016/j.spa.2009.11.002
+[4] Richard T. Durrett, Donald L. Iglehart and Douglas R. Miller. Weak convergence
+to Brownian meander and Brownian excursion. Ann. Probab. 5:117–129, 1977.
+MR0436353
+[5] Fima C. Klebaner and Pavel Chigansky. Distribution of the Brownia n motion on
+its way to hitting zero. Electr. Comm. Probab. 13:641–648, 2008. MR2466191
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+ 1A New Algorithm for Multicommodity Flow 
+ 
+Dhananjay P. Mehendale 
+Sir Parashurambhau College,  Tilak Road, Pune 411030, 
+India 
+ 
+Abstract 
+ 
+We propose a new algorithm to obtai n max flow for the multicommodity 
+flow. This algorithm utilizes the max -flow min-cut theorem and the well 
+known labeling algorithm due to Ford an d Fulkerson [1]. We proceed as 
+follows: We select one source/si nk pair among the n distinguished 
+source/sink pairs at a time and treat the given multicommodity network as a 
+single commodity network for such chosen source/sink pair. Then applying standard labeling algorithm, separately for each sink/source pair, the feasible 
+flow which is max flow and the co rresponding minimum cut corresponding 
+to each source/sink pair is obtained. A r ecord is made of these cuts and the 
+paths flowing through the edges of these cuts. This record is  then utilized to 
+develop our algorithm to obtain max fl ow for multicommodity flow. In this 
+paper we have pinpointed the difficu lty behind not getti ng a max flow min 
+cut type theorem for multicommod ity flow and found out a remedy. 
+ 
+1. Introduction:  Multicommodity flow problem arises in a wide variety of 
+important applications. Many communicati ons, logistics, manufacturing, and 
+transportation problems can be formul ated as large multicommodity flow 
+problems. The max-flow min-cut theore m due to Ford and Fulkerson [1] is 
+applicable to transport networks containing one source and one sink, 
+represented by simple connected weight ed digraphs. However, if there are 
+several sources 
+ks ss ,,,21L  and several sinks kt tt ,,,21L  forming a 
+directed network and if there exists a restriction that the flow from the 
+specified source, is, should be sent to the specified sink, it, then this 
+problem is called the problem of multic ommodity flow. There is no result 
+similar to max-flow min-cut theorem for multicommodity flow problem in 
+general. This paper aims  at pinpointing the reason behind not getting a max 
+flow min cut type theorem for multicom modity flow and further a way to 
+surmount this difficulty.  2In multicommodity flow problems several distinct commodities are flowing 
+simultaneously through the given tran sport network formed by directed 
+edges connecting vertices of the netw ork and each commodity has its own 
+source and own sink. All flows share th e edge capacities and as in the case 
+of single commodity, having single sour ce and single sink, the sum of all 
+flows through an edge must not exceed the capacity of the edge. Also, the 
+conservation conditions, as in the ca se of single commodity case, are 
+observed at each intermediate node.   
+2. Preliminaries: For network with one source and one sink there exists 
+very well known algorithm due to Ford and Fulkerson to obtain max flow 
+[1]. This so called labeling algorithm star ts with a feasible flow as an input 
+and proceeds to find an  
+f- augmenting path P with positive leeway ε and  
+then increasing the flow by ε along forward edges of Pand decreasing the 
+flow by ε along backward edges of P produces a feasible flow /f with 
+value higher by ε, i.e. value(/f) = value( f) + ε. If this labeling algorithm 
+terminates at some iteration then it te rminates (only) when it does not find 
+anyf- augmenting path P.This termination (in fi nitely many iterations) 
+always takes place, i.e., starting with node set S = {s}, where element s is 
+source node, we reach at nodes, which get labels by the algorithm, and from 
+these labeled nodes we cannot furthe r label any nodes and the labeling 
+algorithm thus terminates producing a cut [ S,/S] where /S is nonempty 
+complement of S in the entire node set. In short, the labeling algorithm 
+always terminates after finitely many iterations by landing at a cut (or cutset) 
+with same value as this flow. By  weak duality theorem, namely, if fis a 
+feasible flow and [ S,T] is a source/sink cut, then value( f)≤ cap([ S,T]), 
+where cap stands for capac ity, the cut is a minimum cut and the flow is a 
+maximum flow, i.e. both are optim al (See 4.3, page 158, [2]).   
+ 
+3. Basic Preparation for Algorithm:  The input for this algorithm is the 
+given multicommodity network, a digraph with a nonnegative capacity, 
+)(ec, on each (directed) edge e and there are n distinguished source 
+vertices (or nodes) ns ss ,,,2 1L  and corresponding n distinguished sink 
+vertices (or nodes) nt tt ,,,21L  with a restriction that the flow from the 
+specified source, is, should be sent to the specified sink, it. We begin with  3describing certain steps to be carried  out on given multicommodity network 
+and we call this processing “Initialization”. 
+ 
+Step 1: We begin with treating this multicommodity network as a single 
+commodity network with a si ngle source vertex (or node) 1s and single sink 
+vertex (or node) 1t, and treat all other nodes as ordinary nodes as in the 
+sense of usual one sour ce and one sink network. 
+ 
+Step 2:  We apply the algorithm of Ford and Fulkerson to the network, which 
+is now treated as single source/sink networ k arrived at step 1, and (in finitely 
+many iterations) find ma x flow and min cut. 
+ 
+Remark 3.1:  We want to repeat these steps 1, 2 with all other source/sink 
+pairs (is,it), i = 2, 3, …, nand at each carrying out of these steps 1, 2 for 
+each source/sink pair we want to keep record of the max flow we get, the 
+min cut we get,  the paths that are flow ing this max flow we get, and finally 
+the edges through which these paths pa ss. We actually want to use in our 
+algorithm this information mentioned he re that is associated with each 
+source/sink pair. So, we proceed as follows: 
+ 
+Step 3:  At the end of step 2 we make a record of distinct paths which are 
+starting at source vertex 1s, then passing through some edge of min cut, and 
+terminating in sink vertex 1t. We assign a distinct color with each of these 
+distinct paths and further the color which is assigned to a path is also 
+assigned to each edge on that path. So, let {1
+1P,1
+2P,…, 1
+1rP} be the paths 
+starting at source vertex 1s, then passing respectively through edges 
+{1 1
+21
+11,,,re eeL } = min cut, and terminating in sink vertex 1t. Also, let 
+{1 1
+21
+11,,,ju j j e eeL } are the edges on paths 1
+jP, 1,,2,1 r jL= . We 
+associate distinct colors {1 1
+21
+11,,,rc ccL } with paths {1
+1P,1
+2P,…, 1
+1rP} 
+respectively and also associate same co lors with each edge on a path, thus, 
+color 1
+jc is associated with each  edge among the edges {1 1
+21
+11,,,ju j j e eeL } 
+on path 1
+jP.  
+  4Step 4:  For keeping this coloring record in a systematic and convenient way, 
+we then proceed, for the given multicommodity network N, with 
+construction of a bitableau, ca lled the Edge-Color-Bitableau, ECBT (N), to 
+keep the record of colors we have a ssociated with edges in step 3, in a 
+systematic way for using it later as follows: 
+                       Initially, we have (befor e assignment of any color) a left 
+tableau consisting of a column vector re presenting edge labels of edges in N 
+and a blank right tableau:  
+               
+
+
+
+
+=
+LMLMLLpj
+eeee
+N ECBT21
+)(
+ 
+ 
+After step 3, we have assigned colors  with certain edges that appeared on 
+certain path among the path s. We write down the colo rs assigned to an edge 
+in the same row in the right tableau in  which that edge label appears in the 
+left tableau. We carry out this pr ocedure for every edge on every path 1
+jP, 
+1,,2,1 r jL=  and thus partially build the in itially empty right tableau with 
+the corresponding color labels asso ciated with the edges involved. 
+ 
+Step 5:  We now go back to step 1. We  replace there source/sink pair ( 1s,1t) 
+by new source/sink pair ( 2s,2t) and proceed again with steps 1, 2, 3, 4 
+leading to new paths {2
+1P,2
+2P,…, 2
+2rP} with {2 2
+22
+12,,,re eeL } = min cut, 
+and {2 2
+22
+12,,,ju j j e eeL } are the edges on paths 2
+jP, 2,,2,1 r jL= . We 
+associate distinct colors {2 2
+22
+12,,,rc ccL } with paths {2
+1P,2
+2P,…, 2
+2rP} 
+respectively and also associate same co lors with each edge on a path, thus, 
+color 2
+jc is associated with each  edge among the edges {2 2
+22
+12,,,ju j j e eeL } 
+on path 2
+jP. We write down the colors assigne d to an edge in the same row  5in the right tableau in which that edge  label appears in the left tableau. We 
+carry out this procedure for every edge on every path 2
+jP, 2,,2,1 r jL=  
+and thus continue with further partially  building of the right tableau with the 
+corresponding color labels associated with the edges involved this time. 
+ 
+Step 6:  We now proceed with repeating step 5 for every other source/sink 
+pair (is,it), i = 3, 4, …, n, and thus, complete the construction of 
+ECBT (N).  
+ 
+Step 7: We now proceed, for the gi ven multicommodity network N, with 
+construction a bivector calle d the Edge-Capacity-Bivector, ECBV (N), one 
+more bitableau called th e Path-Recorder-Bitableau, PRBT (N), one more 
+bivector called Minimum- Edge-Capacity-Bivector, MECBV (N), and one 
+more bivector called Path-Cardinality-Bivector, PCBV (N), for using it later 
+to make decision about choosing paths to maximize the flow as follows: 
+ 
+
+
+
+
+=
+)()()()(
+)(2 21 1
+p pj j
+e Cap ee Cap ee Cap ee Cap e
+N ECBV
+MM
+ 
+ 
+, 
+
+
+
+
+=
+) ( )( )() ( )( )() ( )( )()( )( )(
+)(
+1 11 11
+21
+22
+21
+21
+211
+211
+21
+11
+11
+11
+11
+111
+111
+1
+2 21 1
+n
+urn
+run
+jrn
+jrn
+rn
+rn
+ri
+jui
+jui
+jji
+jji
+ji
+ji
+ju u j ju u j j
+nrn n n n n n nj j
+we we we Pwe we we Pwe we we Pwe we we P
+N PRBT
+L LML LML LL L
+  6 
+,  
+
+
+
+
+=
+)()()()(
+)(1
+21
+21
+11
+1
+n
+rn
+ri
+ji
+j
+n ne Cap Pe Cap Pe Cap Pe Cap P
+N MECBV
+MM
+ 
+and,  
+ 
+  
+
+
+
+
+=
+)()()()(
+)(1
+21
+21
+11
+1
+n
+rn
+ri
+ji
+j
+n nE Card PE Card PE Card PE Card P
+N PCBV
+MM
+ 
+ 
+a) The left vector of ECBV (N) is a column vector consisting of edge 
+labels of the edges of network N.  
+b) The right vector of ECBV (N) is a column vector consisting of 
+capacities of the correspondi ng edge labels written in the left vector in 
+the same row. 
+c) The left tableau of PRBT (N) is a column vector consisting of path 
+labels of the paths, i
+jP, ir j ,,2,1L=  and i = 3, 4, …, n, obtained in 
+succession for  all source/sink pair (is,it) of network N.  
+d) In the right tableau of PRBT (N) the vector of edge labels, along with 
+their capacities in bracket in front of them, are written in the same 
+order as they have occurred on these paths. 
+e) The left vector of MECBV (N) is same as left tableau of PRBT (N).  
+f) The right vector contains vector of minimum capacities, i.e. with 
+minimum capacity on the paths whose path labels are written in the  7same row in the left vector. It is clear that initially the value will be 
+the value of the capacity of the edge belonging to minimum cut on 
+that on the path. 
+g) The left vector of PCBV (N) is same as the left tableau of PRBT (N). 
+h) The right vector of PCBV (N) is a column vector consisting of count of 
+the distinct colors associated with  the edges on this path. Note that 
+these colors associated with e dges through their appearance on 
+different paths is depicted in ECBT (N). Some more explanation is as 
+follows: Let the directed edges present on pathi
+jPbe 
+{i
+jui
+ji
+jie ee ,,,2 1L }. Now look in those rows of right tableau of 
+ECBT (N) where there are edge labels i
+jui
+ji
+jie ee ,,,2 1L  present in the 
+left tableau of these rows and collect labels of all distinct (without 
+repetition) colors in a set, say, i
+jE. Let Card (i
+jE) denotes the 
+cardinality of this set of distinct co lor labels associated together with 
+the edges on the pathi
+jPand so these distinct color labels together 
+(collected in i
+jE) are essentially associ ated with the path i
+jP.  
+The basic preparation for the algorithm  is now complete and we are now 
+ready for the discussion of the actual steps of the algorithm.   
+ 
+4. Algorithm for Multicommodity Flow:  Let us note down first some 
+important observations: 
+1. We have associated a unique color w ith each path and also that same 
+color with each edge on this path.  
+2. Same edge can appear on more than one path and therefore more than 
+one color can get associated  with the same edge.  
+3. When we select some path, i
+jP, to pass the commodity, say, iM, of 
+quantity, say, )(i
+je Cap , where i
+je is the edge on the path 
+i
+jPbelonging to min cut obtained for pair (is,it), then we cannot use 
+the same edge to pass some other commodity kM if the same edge 
+appears as edge k
+leon other pathk
+lPon the min cut obtained for pair 
+(ks,kt).  
+4. The maximum possible flow on a path is equal to the minimum of the 
+capacities of the edges belonging to that  path, i.e. if the directed edges  8present on pathi
+jPare {i
+jui
+ji
+jie ee ,,,2 1L } and if edge 
+i
+je∈{i
+jui
+ji
+jie ee ,,,2 1L } has minimum capacity, or , is the one which 
+belongs to corresponding min cut, th en max flow that is possible on 
+this path = )(i
+je Cap .  
+5. When the quantity of commodity with value = )(i
+je Cap  
+corresponding to pathi
+jPis transported the “free (or, usable) capacity” 
+for the other edges on this path becomes | )(i
+jme Cap ─ )(i
+je Cap |, 
+iu m ,,1L= . 
+6. If more than one color gets associ ated with an edge due to its 
+appearance on more than one path th en we can use that edge with its 
+full capacity only for some one and one color among these associated 
+colors, i.e. for flowing only some one commodity.  
+7. If more than one color gets associ ated with an edge due to its 
+appearance on more than one path then we can use that edge with 
+partial capacity for multiple colors am ong these associated colors, i.e. 
+for transporting more than one commodity, by observing the capacity constraint for that edge.  
+8. If 
+)(i
+jE Card  = 1 for all i andj in the right vector of PCBV (N), i.e. if 
+the right vector of PCBV (N) is made up of ones, then it meant that 
+there is no sharing of edges while flowing the commodities through the network and we can achiev e max flow for each commodity 
+separately (as per max flow min cut theorem). Therefore, the total 
+flow, equal to sum of individual ma x flows, will be the max flow for 
+multicommodity network, N.  
+9. In general, 
+)(i
+jE Card  > 1 for many pairs of i andj. This meant same 
+edge appears on many paths. This e ssentially implies that many paths  
+try to share this same edge fo r transporting their associated 
+commodities for achieving their i ndividual max flow through max 
+flow min cut algorithm. 
+10.  “Max flow” possible for multicommodity flow is equal to f, where,  
+        ) ( )1( ) ( )1()(2 1)1( 1
+nn
+jij i
+ii C CCS CCS CS f ∩∩∩ −++∩ −+ =−
+<∑∑ L L  
+           and where, )(iCS  = sum of capacities of edges of min cut iC.  9 
+11.  The question that naturally arises  is: How to achieve this maximum 
+possible flow? Which path among the paths that passes through a 
+certain edge, common to them, should be  chosen to flow its associated 
+commodity and thus use the capacity of  this edge which is member of 
+these paths?  The simple and natura l answer is: Choose that path (or 
+that color), any one among the paths that pass through the edge which 
+is common to them, to flow the commodity, which in this process 
+eliminates minimum number of paths for further transport. 
+ 
+Steps of the Algorithm:  
+1. Carry out the process of “Initialization” on the given 
+mulicommodity network, i.e. ca rry out the process described 
+under the heading “Initialization” in the steps given there. 
+2. From PCBV (N) find out the path labelsi
+jPin the left vector for 
+which the values of )(i
+jE Card  in right vector are unity and 
+transport the commodities on these paths and mark these paths as 
+used for transport. Make the reco rd of total transport achieved in 
+this process. 
+3. From PCBV (N), among the unmarked ro ws after the first step, 
+find out the path labelsi
+jPin the left vector of PCBV (N) for which 
+the values of )(i
+jE Card  in right vector is minimum. Among such 
+rows select any row w ith path label say, i
+jP , to transport the 
+commodity associated with thus c hosen path with value equal to 
+value of minimum capacity  edge on that path, say, )(i
+je Cap , and 
+mark this path as used for trans port. Modify the record of total 
+transport achieved in this process.  Subtract this value of minimum 
+capacity from the original capacity of every edge that has occurred 
+on this path. Modify ECBV (N), PRBT (N), MECBV (N), PCBV (N). 
+If this modification produces any pa ths containing some edge with 
+zero (left out) capacity then discard those paths from further 
+selections, by marking such paths as discarded.  
+4. Continue step 3 till every path is e ither is marked as selected or 
+discarded. 
+5. Record total transport achieved in this process, carried out till end. 
+It will be the “max flow” for the given multicommodity network 
+under consideration.      10 
+     Example:  Consider following multicommodity network w ith two                  
+commodities, as depicted in FIG. 2.1.  The capacities are written near the 
+edges.  
+                       
+FIG. 2.1 
+ 
+This network has two source/sink pairs (11,ts) and (22,ts). 
+                      As per the “Initializati on” procedure, the paths forming 
+PRBT (N) are:      
+                                           5 
+                          1
+11P: 1 1 t s→ : Violet 
+                                           10       10        10 
+                          1
+12P:  1 1 t b a s →→→ : Red 
+                                           10        10        10 
+                           2
+11P:  2 1 2 t a s s →→→ : Green 
+                                           10        10        10 
+                           2
+12P:  2 1 2 t t b s →→→ : Yellow 
+ 
+where on the top of every edge the cap acity is depicted. Also, we have 
+written the colors applied to paths in itially on the right side of the path. 
+Using these paths and directed edge s and their associated colors and 
+capacities, We can build ECBT (N) and using it we can further build 
+ECBV (N), PRBT (N), MECBV (N), PCBV (N).  
+When we proceed as per algorithm we see that finally (from construction of 
+PCBV (N)): 
+ 
+¾ Count of colors associated with 1
+11P = 1 s1 
+t1 t2 
+s2 a 
+b 1010
+10
+1010
+1010
+5  11¾ Count of colors associated with 1
+12P = 3 
+¾ Count of colors associated with 2
+11P = 2 
+¾ Count of colors associated with 2
+12P = 2 
+Therefore, by proceeding as per algorithm:  
+ 
+1) We choose first path 1
+11P 
+2) We then choose path 2
+11P  
+3) We then choose path 2
+12P 
+4) We discard path 1
+12P 
+ 
+And we achieve max flow = 5 units + 10 units + 10 units = 25 units. Out of 
+these units, 5 units are of the first co mmodity and 20 units are of the second 
+commodity.  
+Theorem:  Algorithm for multicommodity flow produces max flow. 
+ Proof: The algorithm begins with applying max flow min cut theorem 
+individually and separately to each  source/sink pair of the given 
+multicommodity flow and produces separate  record of paths, min cuts, etc. 
+producing max flow for individual comm odities. When there is no sharing of 
+edges and thus no intersection of these paths then the sum of individual max 
+flows will be the max flow for multicommodity network. 
+             When th ere is sharing of edges by different paths the algorithm finds 
+out the optimal way of selecting pa ths among these paths for flowing the 
+commodities so that minimum numbers of paths get eliminated from the 
+totality of paths and thus causes mini mum possible reduction in the sum of 
+individual max flows, and thus, pr oduces the desired max flow for 
+multicommodity network under consideration. 
+    
+References 
+ 
+1. Ford L. R., and D. R. Fulkerso n, Flows in Networks, Princeton 
+University Press, Princeton, N. J., 1962. 
+2. West Douglas B., Introduction to Graph Theory, Prentice-Hall of 
+India, Pvt. Ltd., New Delhi 110001, 1999. 
\ No newline at end of file
diff --git a/1001.0630.txt b/1001.0630.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6c83274b9aea810ef14e54ffd2a2f5c200de4bc9
--- /dev/null
+++ b/1001.0630.txt
@@ -0,0 +1,247 @@
+The  Boundary  Conditions  Geometry  in  
+Lattice-Ising  model 
+ 
+You-gang Feng 
+ 
+Department of Basic Science, College of Scien ce, Gui Zhou University. Cai Jia Guan, Guiyang 
+550003, China 
+E-mail: ygfeng45@yahoo.com.cn 
+ 
+ 
+   We found that the differential topology of the lattice-system of Ising model 
+determines whether there can be the cont inuous phase transition , the geometric 
+topology of the space the latt ice-system is embedded in determines whether the 
+system can become ordered. If the system  becomes ordered it may not admit the 
+continuous phase transition. The spin-projection orie ntations are strongly influenced 
+by the geometric topology of the spac e the lattice-system is embedded in.  
+ 
+Keywords:  Ising model, periodic boundary, sp in-projection orientation, topology  
+PACS: 64.60. Cn, 05.50.+q, 02.40.Pc 
+ 
+1. Introduction 
+ 
+Physicists worked on Ising models for many years have obtained brilliant 
+achievements in the theory of  continuous phase transition. The model’s charm is that 
+the studying and researching will never be  at an end; some  new properties and 
+problems will be discovered and explored. Kadanoff pointed out, “The physical problem of critical phenomena reduces to the mathematical problem of enumerating 
+and describing the different universality class.”
+[]1 The late Chinese mathematician 
+Xing-sheng Chen, well-known for his great achievements in differential topology 
+around the world, said, “Physics ju st is geometry, and they ar e in one family.” In this 
+paper we will discuss how the difference be tween differential and geometric topology 
+influences on the magnetic order of the lattice system of Ising model. 
+   
+2. Applications 
+ 
+It used to be that the discussion about th e continuous phase transition of Ising model 
+is concentrated on the property of derivatives of order 2 of the system’s free energy at 
+ 1the critical temperature. If it is divergen t the system will have magnetic order below 
+the critical temperature, if it is not divergent the ordered stat e of system will not 
+appear. If we regard the thermodynamic proper ties of the system as a kind of manifold, 
+and the system’s free energy is a functi on defined in the coordinate system.  
+The property of derivatives of order 2 of the free energy reflects the differential  
+topology of the manifold. The phase transi tions of the systems with  the same 
+differential topology belong to one kind of pha se transitions—a phase transition of the 
+second kind. The order parameters are actually  the invariance linked to the differential 
+topology of such systems. C onsequently, the discussion about the phase transition of 
+the Ising model has a meaning in terms of  differential topology, although people did 
+not pay much attention to relate the pha se-transition property to the property of 
+differential topology, and were willing to cons ider it as a physical property only. As a 
+result of this, they were so indifferent to the geometric topological property of the 
+space in which the lattice system is embedded, that the selection of spin-projection orientations has never been mentioned. This aspect is dependent upon the geometric topology of the space. Consider ing the spin-projection orientations, the lattice gas will 
+not be equivalent to the Ising model because the lattice gas itself has not any 
+projecting orientation . 
+[]2
+  The property of differential topology and the property of geometry topology are 
+two distinct properties of topology: the form er determines whether there can exist the 
+continuous phase transition in a lattice system, and the la ter determines whether the 
+system can become ordered. Being a continuous phase transition, the system will 
+certainly become ordered, which was proved by physicists. But if the system behaves 
+ordered, it may not be in the state of c ontinuous phase transition for some systems. 
+This results from the difference between th e two topologies above mentioned. So it is 
+necessary to discuss the infl uence of the geometric topol ogy on the magnetic order of 
+the Ising model. We will discuss the probl ems in the following three respects . 
+First, the geometric topology of the em bedding space for the lattice system will 
+restrict the spin-projection orientations. A key point of the Onsager’s solution is his 
+periodic boundary conditions. Under the condi tions, as illustrated in FIG .1, the two  
+ 
+ 
+ 0 xy  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ FIG. 1 
+ 2 
+ 
+lattices in sites (- ) ,y∞  or (x,-  and (+)∞ ) ,y∞  or (x,+ )∞ have the same spin state. 
+The periodic boundary conditions change th e plane square-lattice system into a 
+torus-lattice system, which should be embedde d in 3-dimensional Euclidean space. If 
+a lattice system can become ordered, it will be in one of  the following two states: The 
+entire lattice system can be referred to as  a lattice with total spin. So, the space is 
+simply connected; or there is a non-vanishing vector field, of which the appearance is 
+dependent upon the geometric topology of th e lattice system.  The torus-lattice 
+system is not contractible because of its geometric topological structure [. FIG .2 
+shows that three simple closed curves on th e torus, and only one’s inner block of   ]3
+ 
+ 
+               A
+B
+C
+FIG.2 
+ 
+which, the curve B inner block, can become  contractible. The singularity of the 
+Onsager’s equation indicated that the sy stem certainly has the continuous phase 
+transition, which implied the system is ordered but contractible. We noticed that there 
+are only two spin states for the Ising m odel: spin-up and spin-down, and there are 
+only three important sp in-projection orientations in va rious projecting orientations: 
+the directions normal to the plane, parallel to  the x-axis and to the y-axis as illustrated 
+in FIG.1. In fact, the partiti on function of the system is not affected by the projecting 
+orientations because there is no the spin-pro jection orientation term in it. We found 
+that the spin-projection orientation norma l to the plane with the periodic boundary 
+conditions is forbidden. The reason is that if the system were ordered, the total spin-projection orientation of  the system should be normal to the torus surface, which 
+cannot shrink. Then its projecting orient ation is uncertain because the normal 
+orientations of the torus are different and divergent everywhere.  
+However, if the spin-projection orientations are parallel to the x-axis or to the 
+y-axis, the situation is completely differ ent. Let the total spin of the system be  
+below its critical temperature, the average lattice spin be , and ,  be 
+the total number of the lattices in the system, S
+s N S s/= N
+∞→N , and 0≠s , in accord with the 
+thermodynamic limit condition. When the toru s-lattice system becomes ordered, there 
+is a continuous non-vanishing vector field on the torus, as illustrated in FIG 3 each 
+ 
+ 
+ 3 
+ 
+ 
+ 
+  
+   
+ 
+ 
+ 
+ 
+ FIG . 3 
+ 
+ 
+little arrow represents an average lattice spin, which means that the topological 
+property of torus allows the system to be or dered. It is the geometric topology that not 
+only determines system is ordered, but also restricts the spin-projection orientations. 
+Onsager’s solution means that the torus syst em’s differential topology is equivalent to 
+the plane system’s differential topology, but the principle of topology tells us that their 
+geometry topology are different. Notwithstanding they cannot change into each other 
+automatically, (i.e., they are not one-to-one). 
+In fact, the free energy of the torus-lattice system is different from the free- energy 
+of the plane square-lattice system. The part ition function of the la tter is defined as 
+ 
+∑∑∑ − + − = − =
+ii iss s B B B T kH
+T kH
+T kHQ) 1() 2 (
+) exp( ) exp( ) exp(            ( 1 )  
+∑− =
+j ij is s J H
+,                        ,                              ( 2 )  
+where H is the system Hamiltonian,  is a spin-coupling constant ,  
+denotes the sum over all possibl e nearest neighbor lattices, ∑
+j i,J
+∑
+is does the sum over 
+all possible states of spins,  is the sum over all possible states of spins with the 
+periodic boundary conditions, and denotes the sum over all possible states of 
+spins which fail to keep the conditions. The first term on the right hand of Eq.(1) is expressed by ∑) 1 (
+is
+∑) 2 (
+is
+∑− =1
+1 ) exp(
+is BT kHQ                                                   ( 3 )  
+Obviously,  is the partition function of the lattice system with the periodic 1Q
+ 4boundary conditions, namely, the partition func tion of the torus-lattice system. The 
+free-energy of the plane square-lattice system and the free-energy of the torus-lattice 
+system are respectively given by 
+Q T k FBln−=                              ( 4 )  
+1 1 lnQ T k FB−=                             ( 5 )  
+Because of the exponential f unction property, we have 
+ 
+                   ,         ,                         ( 6 )  01〉Q1Q Q〉 0〉Q
+the difference between  and  is 1F F
+0 ) ln(
+11 〉 = − = ΔQQT k F F FB                        ( 7 )  
+Under the thermodynamic limit condition FΔ changes into infinitesimal but 
+non-zero, which makes the critical temperature of the torus-lattice system infinitely 
+close to the critical temper ature of the plane square-l attice system so that the 
+Onsager’s solution can be regarded as an exact solution of the plane square-lattice 
+system. The infinitesimal difference 0〉ΔF shows that the original state cannot 
+change into the state w ith the periodic boundary condi tions in the thermodynamic 
+equilibrium state, and the small offset of the free-energy makes the transformation of 
+the topological structure of the system so great that its fundamental group is 
+completely different from the original []3. The thermodynamic limit guarantees the 
+systems with and without the periodic bounda ry conditions have the same property of 
+differential topology so as to have the same property of the continuous phase 
+transition. This is not only seen in the plane- lattice system, but also will be seen in the 
+one-dimensional lattice system []2. Furthermore, if the spin states of all lattices at the 
+boundary are the same (which is one kind of the periodic boundary conditions) these 
+lattices themselves will gather up to one lattice to make the plane-lattice system 
+become a sphere-lattice system. The solution of  such a system will also be equivalent 
+to Onsager’s solution because of its peri odic boundary conditions, and there exists 
+certainly the continuous phase transition. Its geometric topology shows the sphere 
+surface is simply connected , which allows the system to be ordered. However, 
+there is no non-vanishing vector field on the sphere surface because of the “hairy ball 
+theorem”[, which is distinct from the torus-la ttice system. When the sphere-lattice 
+system becomes ordered its total spin-proje ction orientation will be along the sphere 
+radial, which is arbitrary. []3
+]3
+ 5  The second sense of the influence of the geometric properties is that the ordered 
+system is not equivalent to  its appearance of continuous phase transition all the time. 
+A one-dimensional Ising model can be cons tructed on each site, which coordinate 
+positions are  on which is laid one lattice spin with free degree 
+. Such a system can be referred as to a Ising model on the real line. Using the 
+periodic boundary conditions, it is proved th at the 1-dimensional system has not the 
+continuous phase transition. The periodic boundary conditi ons changed the real line 
+into a circle embedded in a plane, which means that the circle-lattice system also 
+cannot have the continuous phase transition li ke the real-line-lattice system because of 
+the the thermodynamic limit c ondition. However, the real -line–lattice system can 
+become ordered because of its simply connected,...., 2 , 2 , 1 , 1 , 0− + − + = x
+1=n
+[]3. In other words, the whole lattice 
+system can be looked as a lattice with th e total spin, but ther e is no non-vanishing 
+vector field. The circle is not simply  connected, and it allows the non-vanishing 
+vector field to exist, which direction is cl ockwise or anticlockwis e. Here, we noticed 
+that the 1-dimensional lattice system with or without th e periodic boundary conditions 
+have the same property of differential t opology but the geometric topology makes 
+them appear in different spin-projection orientations. The key point to cause the 
+system to become ordered is its geomet ric topology, not its di fferential topology. 
+While the free degree of spin changes from 1=n  into 〈+∞〈n1 , the chain–lattice 
+system can also become ordered w ithout the continuous phase transition . We 
+cannot say that all these phe nomena are physical abnormalities for all possible values 
+of . As a matter of fact, they just s how a common regular pattern, namely, 
+agreement with their geom etric topology and differential topology parameters []2
+n
+   Last, the spin-projection orientations  also depend on the dimensionality of the 
+embeding space for the lattice system. If the embedded space is 2-dimensional, the 
+spin-projection orientation is forbidden in  the plane’s normal direction for a plane 
+lattice system or the ci rcle-lattice system. The torus must be embedded in 
+3-dimensional space, and its non- vanishing vector field is al so 2-dimensional at least. 
+In the 1-dimensional space the spin-projec tion orientation normal to the real line 
+cannot exist. 
+ 
+3.Conclusion 
+ 
+  In summary, we found that the differ ential topology of the la ttice-system of the 
+Ising model determines whether there can be the continuous phase transition. The geometric topology of the space the lattice-system embedded in determines whether the system can become ordered. Also if the system behaves ordered it may not have a continuous phase transition. Thus the spin -projection orientations are strongly 
+influenced by the geometric topology of  the lattice-system embedding space in. 
+ 6References 
+[]1 L. P. Kadanoff, Statistical Physics: Static s, Dynamics and Renormalizeation 
+(World Scientific, Singapore,2000) p. 266 
+[]2 R. K. Pathria, Statistical Mechanics (Butterworth Heinemann, Oxford, 2001) p. 
+372-376, 319-321  
+[]3 M. A. Armstrong, Basic Topology (Springer-Verlag, Ne w York, 1983) p. 173, 198, 
+96 
+ 7
\ No newline at end of file
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@@ -0,0 +1,660 @@
+arXiv:1001.0631v2  [hep-th]  2 Jul 2010Design of a Cyclic Multiverse
+Yun-Song Piao
+College of Physical Sciences, Graduate School of Chinese Ac ademy of Sciences, Beijing 100049, China
+Recently, it has been noticed that the amplification of the am plitude of curvature perturbation
+cycle by cycle can lead to a cyclic multiverse scenario, in wh ich the number of universes increases
+cycle by cycle. However, this amplification will also inevit ably induce either the ultimate end of
+corresponding cycle, or the resulting spectrum of perturba tions inside corresponding universe is not
+scale invariant, which baffles theexistence ofobservable un iverses. In this paper, we propose adesign
+of a cyclic multiverse, in which the observable universe can emerges naturally. The significance of a
+long period of dark energy before the turnaround of each cycl e for this implementing is shown.
+PACS numbers:
+Recently, a cosmological cyclic scenario, in which the
+universe experiences the periodic sequence of contrac-
+tions and expansions [1], has been rewaked [2], and
+brought the distinct insights into the origin of observ-
+able universe. There has been lots of studies for cyclic
+or oscillating universe [3],[4],[5],[6],[7],[8],[9],[10],[11],[12],
+also [13] for a review. However, the global configuration
+of cyclic universe is actually more complex than imag-
+ined. In cyclic universe the amplitude of the curvature
+perturbation on super horizon scale is generally increas-
+ing during the contraction of each cycle, while is nearly
+constant during the expansion of each cycle. Thus the
+net result is that the amplitude of perturbation is ampli-
+fied, which occurs cycle by cycle,
+Recently, it has been argued that the amplification of
+the amplitude of curvature perturbation cycle by cycle
+can lead to a cyclic multiverse scenario [14]1, in which
+the universe proliferates, i.e. the number of universes
+increases cycle by cycle. There this was illustrated by
+including a contracting phase with w≃0 in each cycle of
+a cycle universe. In principle, since shortly after the end
+of each cycle the curvature perturbation on super hori-
+zon scale can be amplified to order one, the universe will
+be inevitably separated into lots of parts independent of
+oneanotheraftereachcycle, eachofwhichactuallycorre-
+sponds to a new universe and independently evolves up
+to succedent cycle, and then proliferates again2. This
+result in some sense incorporates the second law of ther-
+modynamics in such a fashion that the increase of total
+entropy in consecutive cycles is explained as or replaced
+1Here, the cyclic multiverse means that there are many indepe n-
+dent universes in each cycle or each spacelike slice. While i n
+usual cyclic universe there is only single universe in each c ycle or
+each spacelike slice, which can be regarded as a multiverse o nly
+when we count it along the time sequence.
+2In [15], an eternal expanding recurrent universe, in which hos-
+cillates periodically while aexpands all along, has been proposed
+phenomenologically, also latest [16], which can be impleme nted
+by appealing to a phantom component with w <−1. In this
+scenario, we live in one period of cycling, while the present ac-
+celeration with w <−1 is just a start of the phantom inflation
+[17],[18] for the next period of cycling. Similarly, a multi verse
+scenario might be obtained here, which will be explored.with the increase of the number of new universes, which
+has been discussed in [14] in detailed.
+The cyclic multiverse scenario might be significant for
+understanding the origin of observable universe. How-
+ever, generally the amplitude of perturbation modes that
+enter into the horizon during the expansion in previous
+cycle and then leave it during the contraction in cur-
+rent cycle is generally larger than that induced by the
+quantum fluctuation of background field in current cy-
+cle. This will inevitably lead to either the ultimate end of
+corresponding cycle, or the resulting spectrum of pertur-
+bations inside each universe is not scale invariant even if
+they leavethe horizonduringthe contractionwith w≃0,
+which baffles possible existence of observableuniverses in
+this scenario. We, in this paper, will discuss this problem
+in detail and then design a possible solution.
+We begin with an illustration of cyclic universe model
+andthe reviewofthe evolutionofperturbationin acyclic
+universe with the contraction with w≃0. There have
+been lots of studies of bounce cosmology [13]. Recently,
+the nonsingular bounce has been implemented in non-
+local higher derivative theories of gravity [19], which is
+ghostfree, while here that provided by us is only a simple
+example serving the purpose of illustration. We intro-
+duce a normal field ϕwith its potential M2ϕ2−Λ∗, and
+a fieldχwith negative ˙ χ2, which plays a crucial role in
+giving a nonsingular bounce [20]. Λ ∗is a small posi-
+tive constant. Thus the minimum of potential is neg-
+ative, which is responsible for the turnaround of cyclic
+universe. We show such an illustration of cyclic universe
+in Fig.1, in which M= 0.9, Λ∗= 10−10, the initial value
+104ϕ0= 5Mand 104χ0= 3Mare used and MP= 1 is
+set. It seems that this model suffers from the quantum
+instabilities and inconsistencies [19]. However, it can be
+expectedthatthiseffectivedescriptionisonlyanapproxi-
+mation of a fundamental theory with the UV completion
+below certain physical cutoff, while the appearance of
+ghost terms or the quantum instabilities is only an arte-
+fact of this approximation. In string theory, a slowly
+decaying D3 brane can be depicted by an open string
+tachyon mode, which is described within a open string
+field theory. This will bring a nonlocal polynomial inter-
+action. In this case, the behavior of open string tachyon
+can be effectively simulated by a ghost field, e.g.[21],[22],2
+10000 20000 30000 40000t1.21.41.61.82.02.2a
+FIG. 1: A simple model of cyclic universe
+as used here. However, the string theory is expected to
+be UV complete.
+We will regard the beginning of the contracting phase
+as the beginning of a cycle, in each cycle the universe will
+orderly experience the contraction, bounce, and expan-
+sion, and then arrive at the turnaround, which signals
+the end of a cycle. The motive equation for perturbation
+in the momentum space is
+u′′
+k+/parenleftbigg
+k2−z′′
+z/parenrightbigg
+uk= 0, (1)
+whereukis related to the curvature perturbation ζby
+uk≡zζk[23],[24], and the prime denotes the deriva-
+tive with respect to the conformal time η, andz=ϕ′/h,
+wherehistheHubble parameterand ϕisthebackground
+field. When w≃0,z∼η2. Thuswehavez′′
+z∼2
+η2, which
+is actually the same as that in inflation, in which a∼1
+η
+leadsz∼1
+ηand thusz′′
+z∼2
+η2. This gives the spectrum
+ns≃1 is scale invariant, as has been shown in [25, 26],
+also see [27],[28],[29] and earlier [30] for tensor pertur-
+bation. Thus the amplitude of curvature perturbation is
+given by
+P1/2
+ζ≃k3/2/vextendsingle/vextendsingle/vextendsingleuk
+z/vextendsingle/vextendsingle/vextendsingle≃he
+mp, (2)
+where we neglected the factor with order one, and he
+is determined by the energy scale ρeat the end time of
+contracting phase, he≃√ρe
+mp. This amplitude of pertur-
+bation spectrum is actually determined by the increasing
+mode of metric perturbation Φ during the contraction,
+which enters into the constant mode of ζor Φ after the
+bounce by k2orderofζ. The amplitude ofcurvature per-
+turbation during the contraction is increased, up to the
+end of contracting phase in corresponding cycle, while
+during the expansion it becomes constant on super hori-
+zon scale. Thus for a cycle the net result is the amplitude
+of curvature perturbation on super horizon scale is am-
+plified, which is inevitable here.
+We then consider the evolution of perturbation, which
+is generated in previous cycle, entering into current cy-ln1
+a|h|
+ln(t) t0a
+b
+c
+d
+i cycle j cycleperturbation
+modes
+expansioncontraction
+FIG. 2: In a cyclic universe, the sketch of ln(1
+a|h|) vs. ln( t).
+The adjacent cycles have been signed as iandjcycles, respec-
+tively. The blue lines denote the evolutions of perturbatio n
+modes, and the red regions denote the contracting phases in
+each cycle. The dashed line denotes the period of dark energy
+domination, and the following solid line denotes the evolut ion
+after it. t0is the turnaround epoch. In principle, at the
+bounce and turnaround points, h= 0, thus there is a diver-
+gence for1
+a|h|, which is not plotted here and also Fig.3 and
+4. However, the discussions are not affected by this neglect.
+cle. We regard adjacent cycles as iandjcycles for con-
+venience, respectively. The solution of metric perturba-
+tion Φ is Φ ≃ΦC+ ΦSh/a. When w≃0, during the
+contraction the constant mode Φ C∼√
+k3and the in-
+creased mode Φ S∼1/√
+k7. The solutions of ζis gener-
+allyζ≃ΦC+k2ΦSf(η) [26], where
+f(η) =/integraldisplaydη
+z2∼1
+η3∼h (3)
+forw≃0, since z∼η2andh∼1/η3. There is
+not bounce, but the contraction phase after t0. Thus
+ζis dominated by its increasing mode Φ S, i.e.ζ≃
+k2ΦSf(η)∼k−3/2h. ThusP1/2
+ζ≃k3/2|ζ| ∼his scale in-
+variant. In jcycle, sinceatthebeginningtime t0,h=h0,
+see Fig.2, and the initial value of perturbation is P1/2
+ζ(i),
+the amplitude of perturbation mode all along on super
+horizon scale, which is generated during the contraction
+inicycle and can not reenter into the horizon during the
+expansion of icycle, see ‘a’ mode in Fig.1, is given by
+P1/2
+ζ(ji)=P1/2
+ζ(i)/parenleftbigghj
+h0/parenrightbigg
+=P1/2
+ζ(i)e3Nj, (4)
+where the subscript iandjdenote the quantities in cor-
+responding cycles, respectively, and Nj=1
+3ln(hj
+h0) is the
+efolding number that the perturbation mode klasts after
+the beginning of contracting phase in jcycle.
+Theamplitudeofperturbationresponsibleforthelarge
+scale structure of observable universe is P1/2
+ζ∼10−5.
+Thus ifP1/2
+ζ(i)∼10−5is required in icycle, we can see
+whenNj≃5
+3ln10∼3,P1/2
+ζ(ji)∼1 injcycle, where3
+it should be noticed that when P1/2
+ζ(ji)approaches 1, the
+enhancement of nonlinear effect will make the required
+Njless. Thus this actually means that nearly at the
+beginning time of jcycle, the curvature perturbation on
+super horizon scale will have the amplitude be in order
+one. This will lead to the density perturbation
+δρ
+ρ∼ P1/2
+ζ(ji)∼1 (5)
+on corresponding super horizon scale. In this case, it is
+obviously impossible that the different regions of global
+universe will evolve synchronously, even if they are syn-
+chronous in icycle. This means that the global universe
+at the beginning time of jcycle will be inevitably sep-
+arated into many different parts, each of which actually
+corresponds to a new universe and will evolve indepen-
+dently ofoneanother, up tosuccedent cycle3. Therefore,
+it is evident that the global universe will proliferate af-
+ter each cycle. In this sense, a cyclic multiverse actually
+comes into being.
+However, inside each new universe, generally the am-
+plitude of primordial perturbation, see ‘b’ mode in Fig.1,
+is contributed by not only that of perturbations induced
+by the quantum fluctuation of background field in jcy-
+cle, but also that of perturbations that enter into the
+horizon during the expansion of icycle and then leave it
+during the contraction of jcycle4. In general, the latter
+amplitude can be larger, and dependent of the expansion
+behavior and the matter contents in icycle, its spectrum
+is also not scale invariant. This can be seen as follows.
+Whenk∗η≃1 during icycle,k∗mode just enters into
+the horizon. Hereafter, its evolution obeys Eq.(1), but
+the termz′′
+zis negligible, since k≫ah. In this case, uk
+isapproximatelyconstant, uptothetime whenthecorre-
+sponding mode leaves the horizon5, which occurs during
+the contraction in jcycle. Thus we have |ui∗|=|uj∗′|,
+where∗′is the time when the corresponding mode leaves
+the horizon in jcycle, which implies P1/2
+j∗′= (ai∗
+aj∗′)P1/2
+i∗.
+However, after this mode leaves the horizon in jcycle,
+its amplitude will increase like Eq.(4). Thus
+P1/2
+je=/parenleftbigghje
+hj∗′/parenrightbigg
+P1/2
+j∗′
+=/parenleftbiggkje
+kj∗′/parenrightbigg3/parenleftbiggai∗
+aj∗′/parenrightbigg
+P1/2
+i∗ (6)
+3It has been argued that if a region with super horizon scale ha s
+the density perturbation on corresponding scale larger tha n 1,
+such a region will correspond to a separated close universe [ 31].
+In principle, the initial conditions of each local universe can be
+obtained by rescaling the background of parent universe.
+4We thank R. Brandenberger for talking this point to us.
+5We neglected the effect of the transfer function on the pertur ba-
+tion spectrum after the corresponding perturbation mode en ters
+into the horizon, which is dependent of the matter content in cor-
+responding cycle. However, the behaviors of spectrum discu ssed
+here are not altered qualitativelyis obtained, where h∼(ah)3∼k3forw≃0 has been
+applied, and the subscript ‘ je’ denotes the end time of
+the contracting phase in jcycle. We can see that if
+P1/2
+i∗∼10−5,P1/2
+jewill be larger than that induced by
+the quantum fluctuation of background field in jcycle
+sincekj∗′≪kjeandai∗≃aj∗′, and also the spectrum is
+quiteredsince P1/2
+je∼k−3
+j∗′. Thisresultimpliesthatthere
+can hardly an observable universe after the bounce of j
+cycle. In addition, it can be noticed that the amplitudes
+of these modes may be also amplified up to orderone, i.e.
+after the contraction lasts some times, P1/2
+j∼1. In this
+case, the universe will be possibly split into smaller and
+smallerfragments,whichwillrendersthecycleultimately
+end [14], see also [32]. Thus in this sense the feasibility
+of such a cyclic multiverse scenario is questionable.
+The perturbation modes that enter into the horizon
+during the expansion of icycle and then leave it during
+the contraction of jcycle are baneful. Their amplitudes
+can hardly be suppressed by certain mechanism. Thus
+a reasonable solution is to push these baneful modes to
+larger scale, i.e. outside of observable universe in cor-
+responding cycle. This can be implemented by intro-
+ducing a period of accelerated expansion in each cycle.
+This period can be set before the turnaround of each
+cycle. In this case, there will be a period of dark en-
+ergy domination in corresponding cycle, hereafter the
+universe collapses and the next cycle begins. The per-
+turbation modes that enter into the horizon during the
+expansion of icycle will possibly leave it during the dark
+energy domination of icycle, but not during the con-
+traction of jcycle. Thus the amplitude of these modes
+will not increase, up to the turnaround. In this case,
+P1/2
+je= (hje
+h0)(ai∗
+a0)(a0
+aj∗′)P1/2
+j∗′, wheret0inFig.2isthe time
+when the period of dark energy domination begins, thus
+Eq.(6) is changed as
+P1/2
+je=/parenleftbiggkje
+kj0/parenrightbigg3/parenleftbiggk0
+ki∗/parenrightbigg3
+P1/2
+i∗, (7)
+wherea∼(ah)n
+n−1∼kn
+n−1has been applied, which
+gives that for n=2
+3,a∼1
+k2and forn≫1,a∼k.
+kje/kj0denotes the efolding number of primordial per-
+turbation contributed by the contraction in jcycle, i.e.
+Nj= ln(kje
+kj0). The larger it is, the larger the ampli-
+tude of perturbation is amplified during the contraction
+is. This point is essentially the same as that obtained in
+Eq.(4). While k0/ki∗is the ratio of the wavenumber of
+the mode that corresponds to the beginning time of dark
+energy domination to that of the given mode, which ac-
+tually corresponds to the efolding number that the dark
+energy phase lasts before the given mode leaves the hori-
+zon during the dark energy domination of icycle, and
+is an exponentially suppressed factor to the amplitude
+P1/2
+je. This result means that the period of dark energy
+domination not only helps to the continuance of cycling
+but also the emergence of observable universe.
+This design can be explained detailed in Fig.2. ‘a’4
+ln1
+a|h|
+ln(t) t0a
+b
+i cycle j cycleperturbation
+modes
+c
+dcontraction
+expansion
+FIG. 3: In a cyclic universe, the sketch of ln(1
+a|h|) vs. ln( t)
+for the case that there is a period of inflation before bounce i n
+jcycle. The blue lines denote the evolutions of perturbation
+modes, and the red regions denote the contracting phases in
+each cycle. The dashed line denotes the inflation, and the
+following solid line denotes the evolution after inflation.
+mode denotes the modes that leave the horizon in icycle
+but can not enter into the horizon during the expansion
+oficycle. These modes will destine to stay on super
+horizon scale all along up to jcycle. Their amplitudes
+will be inevitably amplified to order one around the be-
+ginning time of jcycle. This leads to that the universe
+is separated into lots of independent new universes, each
+of which will evolve independently up to the succedent
+cycle. ‘b’ mode denotes the modes that enters into the
+horizon during the expansion of icycle and then leaves
+it during the contraction of jcycle. These modes gen-
+erally have amplitudes larger than that induced only by
+the fluctuation of background field in jcycle, which thus
+is baneful for the existence of observable universe in j
+cycle. The dashed line denotes the period of dark en-
+ergy domination, which can push those baneful modes,
+such as ‘b’ mode, outside of observable universe. In this
+case, ‘d’ modewill providethe primordialperturbationof
+observable universe, which is induced only by the quan-
+tum fluctuation of background field in jcycle, and its
+amplitude is given by (2) and thus can be suitable for
+observations6.
+6The inflation might occur after the bounce in some cycles, see
+Fig.3, since the energy scale of bounce is generally high, it can be
+expected that after the bounce the field might land on a higher
+plain of effective potential. In this case, ‘b’ mode will be pu shed
+outside of observable universe by the inflation in jcycle, and the
+primordial perturbations suitable for observable univers e, such
+as ‘c’ and ‘d’ modes, can emerge during inflation, which are in -
+duced by the quantum fluctuation of inflaton field. The inflatio n
+after bounce has been originally studied in Refs. [33, 34], i n
+which the imprint of bounce on cosmic microwave background
+has been pointed out, see also for latest studies [35],[36]. Here
+for cyclic multiverse, the occurrence of inflation has simil ar effect
+asthat ofa period ofdark energy domination. However, both c an
+be different in the theoretical model building and observati onalr = 0
+evolution of 1
+a|h|k modebounce surfaceturnaround 
+    surface
+FIG. 4: The Penrose diagram of a cyclic multiverse. This
+is obtained by gluing the corresponding parts of diagrams
+of contracting universe, expanding universe and dS univers e,
+where the dashed lines denoting the turnaround surface and
+the bounce surface have been signed respectively. Before ea ch
+turnaround surface, the universe will experience the contr ac-
+tion, bounce, expansion successively and then enter into a
+period of dark energy domination. After this, the parent
+universe is separated into lots of parts independent of one
+another, each of which corresponds to a new universe and
+repeats above evolution. In principle, since the experienc e
+of each universe after the proliferation is generally differ ent,
+they will not be expected to be synchronous in cycling. This
+means that generally when we are in a period of dark energy
+domination, it is possible that there are many other univers es
+which are in the period of contraction or bounce or others.
+The causal patch diagram of cyclic multiverse can be
+plotted in Fig.4, which is obtained by gluing the corre-
+spondingpartsofdiagramsofcontractingphase, expand-
+ing phase and dS phase. This causal diagram is slightly
+similar to that of inflationary multiverse, e.g.[37]. The
+reason is that before the turnaround in each cycle the
+universe is in a dS phase. In some sense, such a pe-
+riod leads that the universe has an average positive en-
+ergy density for each cycle. In inflationary multiverse,
+the new universe is generated either in nucleated bubble,
+e.g. [38],[39], or in the region that the inflaton field is in
+stochasticwalking[40],[41],bothareinducedbyquantum
+effects7. Here, however, the mechanism resulting in mul-
+tiverse is the amplification of the amplitude of curvature
+perturbation cycle by cycle, which thus occurs in classi-
+cal sense, though it initially originates from the quantum
+fluctuation of field.
+We can have a concrete implement to this multiverse
+scenario, like in [5]. In this implement, the potential of
+signals, which might be interesting for studying.
+7However, in slow rollinflation, the multiverse can be also in duced
+by the classical rolling of inflaton along a web of branches of its
+effective potential [42].5
+scalar field has a nearly flat region, the value of whose
+energy density equals to that of cosmological constant
+observed, and a minimum with negative energy density.
+When the field is in the nearly flat region of potential, we
+can have a period of dark energy, and then it rolls down
+towardsits minimum with negativepotential energy den-
+sity, which leads to the collapse of universe, the contrac-
+tion with w≃0 can be obtained by the oscillation of
+field around its minimum. In this case, it is obvious that
+the evolution that the period of dark energy domination
+precedes the contraction with w≃0 can be obtained. In
+general, the period of dark energy will help to dilute the
+matter and radiation in icycle, which assures that in j
+cycle the energy density of matter and radiation from i
+cycle can not exceed that of field till the enough efold-
+ing number is obtained. In addition, the anisotropy and
+some baneful leftovers, which are menaces for cycling,
+e.g. discussions in [3], can be also diluted during this pe-
+riod of dark energy. Thus it is required that this period
+should be enough long. i.e. the corresponding potential
+should be enough flat.
+In different cycle of cyclic universe, the universe can
+be in different minima of a given potential in field space,
+or landscape8[5], in which these minima have negative
+potential energy density. Here, the generality is actu-
+ally straight. Thus in principle it can be argued that in
+a cyclic multiverse scenario, generally each of multiverse
+will have different minima and thus evolutions. However,
+in this case, the requirement for an enough period of ac-
+celerated expansion in icycle actually corresponds to a
+fine tunning for the initial condition of jcycle. Thus
+though in a cyclic universe driven by a given landscape,
+the number of universes having an enough long period of
+dark energy domination might be quite small, such uni-
+verses can certainly exist, which is significant not only
+for assuring the continuance of cycle, but also for the
+emergence of observable universe in which we might live.
+In general, in a string landscape both positive and neg-
+ative minima exist. However, unless all minima in the
+landscape are negative, the cycle ofuniverse will not con-
+tinue eternally, since if the universe enters into a positive
+minimum, it will stop cycling, and expand for ever, as
+illustrated earlier in [12].
+In cyclic universe [2], the effect of the increase of met-
+ric perturbation on global universe has been discussed
+in [43]. However, in this model it requires that on su-
+per horizon scale the increasing mode of metric per-
+turbation is inherited by the constant mode of curva-
+ture perturbation in leading order. Whether the cor-
+responding inheriting can occur remains controversial
+[44],[45],[46],[47],[48],[49],[50],[51]. However, if this dose
+not occur, the same mode is decayed during the expan-
+8In the low energy limit, the string landscape can be visualis ed as
+an effective potential in a given field space with multiple dim en-
+sions.sion of each cycle, thus even if it is increased during
+the contraction, the net result is still decayed in each
+cycle, since in each cycle the amount of the expansion
+of the universe is generally larger than that of the con-
+traction. Here, however, the increasing mode of metric
+perturbation is inherited by the constant mode of cur-
+vature perturbation in k2order, which certainly occurs
+for the bounce connecting the contracting and expanding
+phases, e.g. [29, 52] fortheoreticalandnumericalstudies.
+The metric perturbation after the bounce will be dom-
+inated by the same constant mode as that of curvature
+perturbation. Thus the net amplitude of metric pertur-
+bation is increased in each cycle. We has argued this
+effect will potentially lead to a cyclic multiverse [14]. In
+this paper, the problems baffling this multiverse scenario
+is discussed, and the possible solution is designed.
+In a class of oscillating universe, in which the bounce
+is implemented by quintom matter [8],[20],[53], the cyclic
+multiverse scenario might be naturally applied, since a
+period ofdarkenergycan be congenitallyincluded in this
+model. This study can be in order. In [54], a scenario
+that many small contractinguniverses can be spawned at
+the end of each cycle has been proposed, which is imple-
+mented by appealing to the brane world cosmology and
+the phantom dark energy. In the design given here, these
+additional appeals are needless, the mechanism resulting
+in the multiverse is the natural increase of perturbation
+on super horizon scale. The role of a period of dark en-
+ergy for cycling was also discussed in [55], in which the
+appearanceof new universeis obtained by the wormholes
+leaded by the accretion of phantom dark energy [56].
+In cyclic inflation [11],[12], the scale factor in consec-
+utive cycles increases by a constant factor. This, after
+some cycles, can give a net exponential growth of the
+scale factor, and thus can imitate inflation. Whether
+there can be a scale invariant spectrum in this model
+remains an open issue. However, combining it with the
+result obtained here, we might argue that a multiverse
+will come into being after cyclic inflation, some of which
+might have a subsequent period of slow roll inflation and
+thus correspond to our observable universe.
+In conclusion, in a cyclic universe, since the metric
+perturbation on super horizon scale is amplified cycle by
+cycle, after each cycle the universe will be inevitably sep-
+arated into many parts independent of one another, each
+of which corresponds to a new universe and evolves up to
+succedent cycle, and then is separated again. This mech-
+anism brings us a scenario of cyclic multiverse, in which
+the number of universes increases cycle by cycle. How-
+ever, in general, the amplitude of perturbation modes in
+previous cycle can be preserved and amplified in current
+cycle, which is generally larger than that induced by the
+quantum fluctuation of background field in current cycle.
+This baffles the possibility that this scenario is regarded
+as the origin of observable universe. We, in this paper,
+have provided a viable design of a cyclic multiverse, in
+which the observable universe can emerge naturally. The
+significance of a long period of dark energy before the6
+turnaround of each cycle for this implementing is shown.
+In this design, the causal diagram of cyclic multiverse
+likes that of inflationary multiverse. Thus the measure
+for the multiverse might be discussed similarly. Dark en-
+ergy is an important issue of current cosmology, which
+is being intensively explored. In some sense, this work
+might provide an alternative motivation for the existenceof dark energy.
+Acknowledgments We thank R. Brandenberger,
+Y.F. Cai, J. Zhang for helpful discussions. This
+work is supported in part by NSFC under Grant
+No:10775180, in part by the Scientific Research Fund
+of GUCAS(NO:055101BM03), in part by National Ba-
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+arXiv:1001.0632v1  [math.AP]  5 Jan 2010Global Existence for the Vlasov-Poisson System with
+Steady Spatial Asymptotics
+Stephen Pankavich
+Department of Mathematical Sciences
+Carnegie Mellon University
+Pittsburgh, PA 15213
+sdp@andrew.cmu.edu
+November 20, 2018
+MOS Classification : 35L60, 35Q99, 82C21, 82C22, 82D10.
+Keywords : Vlasov, kinetic theory, Cauchy problem, global existence, spatial decay.
+Abstract
+A collisionless plasma is modelled by the Vlasov-Poisson sy stem in three space dimen-
+sions. A fixed background of positive charge, dependant upon only velocity, is assumed.
+The situation in which mobile negative ions balance the posi tive charge as |x| → ∞is
+considered. Thus, the total positive charge and the total ne gative charge are both infinite.
+Smoothsolutions withappropriateasymptotic behaviorfor large|x|, whichwerepreviously
+shown to exist locally in time, are continued globally. This is done by showing that the
+charge density decays at least as fast as |x|−6. This paper also establishes decay estimates
+for the electrostatic field and its derivatives.
+INTRODUCTION
+LetF:R3→[0,∞),f0:R3×R3→[0,∞), andA: [0,∞)×R3→R3be given. We seek a
+solution, f: [0,∞)×R3×R3→[0,∞) satisfying
+1∂tf+v·∇xf−(E+A)·∇vf= 0,
+ρ(t,x) =/integraltext
+(F(v)−f(t,x,v))dv,
+E(t,x) =/integraltext
+ρ(t,y)x−y
+|x−y|3dy,
+f(0,x,v) =f0(x,v).
+
+(1)
+HereFdescribes a number density of positive ions which form a fixed backgr ound, and
+fdenotes the density of mobile negative ions in phase space. Notice th at iff0(x,v) =F(v)
+andA= 0, then f(t,x,v) =F(v) is a steady solution. Thus, we seek solutions for which
+f(t,x,v)→F(v) as|x| → ∞. It is important to notice that (1) is a representative problem,
+and that problems concerning multiple species of ions can be treated in a similar manner.
+Precise conditions which ensure local existence and conditions for c ontinuation were given
+in [18]. Also, a prioribounds on ρand a quantity related to the energy were obtained in [17].
+Therefore, we will work towards establishing global existence with t hese bounds and using sim-
+ilar assumptions. Other work on the infinite mass case has been done in [9] and [3].
+The case when F(v) = 0 and f→0 as|x| → ∞has been studied extensively. Smooth solu-
+tions were shown to exist globally in time in [14] and independently in [11]. I mportant results
+prior to global existence appear in [1], [5], [6], [7], and [10]. Also, the met hod used by [14] is
+refined in [8] and [16]. Global existence for the Vlasov-Poisson syste m in two dimensions was
+established in [12] and [19]. A complete discussion of the literature con cerning Vlasov-Poisson
+may be found in [4]. We also mention [2] and [15] since the problem treat ed in these papers is
+periodic in space, and thus the solution does not decay for large |x|.
+1 Section 1
+Letp∈(3,4) and denote
+R(x) =R(|x|) = (1+ |x|2)1
+2.
+We will use the following notation :
+/ba∇dblg/ba∇dblL∞(Rn)= sup
+z∈Rn|g(z)|
+and
+/ba∇dblρ/ba∇dblp:=/ba∇dblρ(x)Rp(x)/ba∇dblL∞(R3),
+2but never use Lp. We will write, for example, /ba∇dblρ(t)/ba∇dblpfor the/ba∇dbl·/ba∇dblpnorm ofx/mapsto→ρ(t,x).
+Following [17] and [18], we assume the following conditions hold for some C >0 and all
+t≥0,x∈R3, andv∈R3, unless otherwise stated :
+(I)F(v) =FR(|v|) is nonnegative and C2with
+F′′
+R(0)<0, (2)
+and there is W∈(0,∞) such that
+F′
+R(u)<0 foru∈(0,W)
+FR(u) = 0 for u≥W.
+
+(3)
+(II)f0isC1and nonnegative.
+(III)AisC1with
+|A(t,x)| ≤C(0)R−2(x), (4)
+|∂xiA(t,x)| ≤CR−3(x), (5)
+and
+∇x·A(t,x) = 0. (6)
+Finally, we assume there is a continuous function a: [0,T]→Rsuch that
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleA(t,x)−a(t)x
+|x|3/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤CR1−p(x).
+(IV)F−f0has compact support in v, and there is N >0 such that for |x|> N, we have
+|F(v)−f0(x,v)| ≤CR−6(x).
+Then, we have global existence:
+Theorem 1 Assuming conditions (I)-(IV)hold, there exists f∈ C1([0,∞)×R3×R3)that
+satisfies (1) with /ba∇dbl/integraltext
+(F−f)(t)dv/ba∇dblpbounded on t∈[0,T], for every T >0. Moreover, fis
+unique.
+3In addition, due to the previously known result of [13], stated here a s Theorem 2, we are
+able to conclude further decay of the charge density in Corollary 1 :
+Theorem 2 LetT >0andfbe theC1solution of (1) on [0,T]×R3×R3. Then, we have
+/ba∇dblρ(t)/ba∇dbl6≤Cp,t
+for anyt∈[0,T], whereCp,tdepends upon
+sup
+τ∈[0,t]/ba∇dblρ(τ)/ba∇dblp.
+Corollary 1 LetT >0andfbe theC1solution of (1) on [0,T]×R3×R3. Then, we have
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay
+(F−f)(t)dv/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+6≤C
+for anyt∈[0,T].
+To prove Theorem 1, we will use Theorems 1 and 3 of [18], which guaran tee local existence
+and continuation of the local solution so long as /ba∇dbl/integraltext
+(F−f)(t)/ba∇dblpis bounded for some p >3.
+Therefore, the following lemma will be all that is needed to complete th e proof of the theorem.
+Lemma 1 Assume conditions (I)-(IV)hold. Let fbe aC1solution of (1) on [0,T]×R3×R3.
+Then,
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay
+(F−f)(t)dv/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+p≤C
+for allt∈[0,T], whereCis determined by F,A,f0, andT.
+Define
+Qf(t) := sup{|v|:∃x∈R3,τ∈[0,t] such that f(τ,x,v)/ne}ationslash= 0} (7)
+and
+Qg(t) := max {W,Qf(t)}.
+In Section 2, we will bound E,∇E, and∇x,vf. Also, we will estimate the energy, and obtain
+bounds on Qf(t) and thus Qg(t). To better reveal the line of thought, the proofs of Lemmas 2
+through 4 are deferred to Section 3.
+4We will denote by “ C” a generic constant which changes from line to line and may depend
+uponf0,A,F, orT, but not on t,x, orv. When it is necessary to refer to a specific constant,
+we will use numeric superscripts to distinguish them. For example, C(0), as in (III), will always
+refer to the same numerical constant.
+To estimate Eand∇E, we will use
+Lemma 2 For anyq >0andb∈[0,5
+18)withb≤2
+q, we have
+|E(t,x)| ≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigb
+for anyt∈[0,T],|x| ≥1, whereCmay depend upon /ba∇dblρ(t)/ba∇dbl∞.
+and
+Lemma 3 For anyq >0anda∈[0,1)witha≤3
+q, we have
+|∇E(t,x)| ≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbiga
+for anyt∈[0,T],|x| ≥1, whereCmay depend upon /ba∇dblρ(t)/ba∇dbl∞and/ba∇dbl∇ρ(t)/ba∇dbl∞.
+Then, we will use the following lemma to bound the p-norm of ρand obtain decay of both
+Eand∇E:
+Lemma 4 For any q∈[0,54
+13),/ba∇dblρ(t)/ba∇dblqis bounded for t∈[0,T], whereCmay depend upon
+/ba∇dblρ(t)/ba∇dbl∞,/ba∇dbl∇ρ(t)/ba∇dbl∞, andQf(T).
+Thus, once we show /ba∇dblρ(t)/ba∇dbl∞and/ba∇dbl∇ρ(t)/ba∇dbl∞are bounded for t∈[0,T], andQf(T) is finite,
+we may use Lemma 4 to bound /ba∇dblρ(t)/ba∇dblpsincep∈(3,4). Then, we can prove Lemma 1, and thus
+Theorem 1.
+52 Section 2
+2.1 Characteristics
+Define the characteristics, X(s,t,x,v) andV(s,t,x,v), by
+∂X
+∂s(s,t,x,v) =V(s,t,x,v)
+∂V
+∂s(s,t,x,v) =−(E(s,X(s,t,x,v))+A(s,X(s,t,x,v)))
+X(t,t,x,v) =x
+V(t,t,x,v) =v.
+
+(8)
+Then, we have
+∂
+∂sf(s,X(s,t,x,v),V(s,t,x,v)) =∂tf+V·∇xf−(E+A)·∇vf= 0.
+Therefore, fis constant along characteristics, and
+f(t,x,v) =f(0,X(0,t,x,v),V(0,t,x,v)) =f0(X(0,t,x,v),V(0,t,x,v)). (9)
+Thus, we find by ( II) thatfis nonnegative.
+Defineg: [0,∞)×R3×R3→Rby
+g(t,x,v) :=F(v)−f(t,x,v).
+Then, we see that supx,v|g| ≤ /ba∇dblF/ba∇dblL∞+/ba∇dblf0/ba∇dblL∞<∞, and
+∂
+∂sg(s,X(s,t,x,v),V(s,t,x,v)) =∂tg+V·∇xg−(E+A)·∇vg
+=−∇vF(V(s))·(E+A)(s,X(s)).(10)
+Finally, for any s∈[0,T] withf(s,X(s),V(s))/ne}ationslash= 0, we have for t∈[0,T] andx,v∈R3,
+|X(s,t,x,v)−x|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt
+s˙X(τ,t,x,v)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤/integraldisplayt
+s|V(τ,t,x,v)|dτ
+≤TQg(T).
+6So, assuming we can bound Qg(T), we have for |x| ≥2TQg(T),v∈R3,t∈[0,T], ands∈[0,t],
+|X(s,t,x,v)| ≥ |x|−TQg(T)≥1
+2|x| (11)
+and
+|X(s,t,x,v)| ≤ |x|+TQg(T)≤3
+2|x|. (12)
+Unless it is necessary, we will omit writing the dependence of X(s) andV(s) ont,x, andv
+for the remainder of the paper.
+2.2 Bounds on the Field
+Much of the work that follows will rely on energy estimates found in [17 ]. In particular, we
+combine Lemma 3 and Theorem 4 from [17] to obtain an a priori bound o nρ.
+First, define σ: [0,∞)→Rby
+σ(h) =−/integraldisplaymin{h,F(0)}
+0/parenleftBig
+F−1(˜h)/parenrightBig2
+d˜h
+andS: [0,∞)×[0,∞)→[0,∞) by
+S(h,η) = (h−F(η))η2+σ(h)−σ(F(η)).
+Then, from [17], we have the following lemma:
+Lemma 5 Let
+k(t,x) =/integraldisplay
+S(f(t,x,v),|v|)dv.
+Assuming conditions (I)−(IV)hold, we have
+/integraldisplay
+|F(v)−f(t,x,v)|dv≤C(k(t,x)3
+5+k(t,x)1
+2) (13)
+and /integraldisplay
+k(t,x)dx≤C. (14)
+7It follows directly from (13) that
+|ρ(t,x)| ≤C(k(t,x)3
+5+k(t,x)1
+2) (15)
+Then, using Lemma 5 , we may bound the field.
+Assume/ba∇dblρ(t)/ba∇dblL∞(R6)≤Cfor allt∈[0,T]. Then, for t≥0,x∈R3, and any R >0
+|E(t,x)| ≤/integraltext|ρ(t,y)|
+|x−y|2dy
+≤/integraltext
+|x−y|<R/ba∇dblρ(t)/ba∇dblL∞|x−y|−2dy+/integraltext
+|x−y|>R|ρ(t,y)| |x−y|−2dy
+≤4π/ba∇dblρ(t)/ba∇dblL∞/integraltextR
+0dr+C/integraltext
+|x−y|>R(k1
+2(t,y)+k3
+5(t,y))|x−y|−2dy
+≤4π/ba∇dblρ(t)/ba∇dblL∞R+C(/integraltext
+k(t,y)dy)1
+2(/integraltext
+|x−y|>R|x−y|−4)1
+2
++C(/integraltext
+k(t,y)dy)3
+5(/integraltext
+|x−y|>R|x−y|−5)2
+5
+≤4π/ba∇dblρ(t)/ba∇dblL∞R+C(/integraltext∞
+Rr−2dr)1
+2+C(/integraltext∞
+Rr−3dr)2
+5
+≤C(/ba∇dblρ(t)/ba∇dblL∞R+R−1
+2+R−4
+5).
+
+(16)
+Obviously, we may choose R= 1 and deduce that |E(t,x)| ≤C. Suppose that the v-support
+ofgis bounded for bounded times. Then, we find
+|ρ(t,x)| ≤/integraldisplay
+|v|≤Qg(t)|g(t,x,v)|dv≤(/ba∇dblf0/ba∇dblL∞+/ba∇dblF/ba∇dblL∞)Q3
+g(t)≤CQ3
+g(t).
+Thus, we know for every t∈[0,T],
+/ba∇dblρ(t)/ba∇dblL∞≤CQ3
+g(t).
+Now, choose R=Q−5
+3g(t). IfR≥1, thenQg(t) is bounded. Otherwise, R≤1 and thus
+R−1
+2≤R−4
+5.
+8Then, we have
+|E(t,x)| ≤C(/ba∇dblρ(t)/ba∇dblL∞R+R−4
+5)
+≤C/parenleftBig
+Q3
+g(t)Q−5
+3g(t)+(Q−5
+3g(t))−4
+5/parenrightBig
+≤CQ4
+3g(t)
+=:C(1)Q4
+3g(t)
+
+(17)
+2.3 Bounds on Derivatives of Density and Field
+We proceed as in Section 4 .2.5 of [4] with one modification : we will not assume that ρ(t)∈
+L1(R3) for allt∈[0,T]. Instead, we find for any R >0, (letting r=|x−y|)
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
+4π/integraldisplay
+|y−x|≥Rρ(t,y)/parenleftbigg3(yk−xk)2
+r5−1
+r3/parenrightbigg
+dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1
+π/integraldisplay
+|y−x|≥R|ρ(t,y)|r−3dy
+≤C/integraldisplay
+|y−x|≥R/parenleftBig
+k1
+2(t,y)+k3
+5(t,y)/parenrightBig
+r−3dy
+≤C/parenleftbigg
+(/integraldisplay
+k(t,y)dy)1
+2(/integraldisplay
+|y−x|≥Rr−6dy)1
+2
++(/integraldisplay
+k(t,y)dy)3
+5(/integraldisplay
+|y−x|≥Rr−15
+2dy)2
+5/parenrightbigg
+≤C/parenleftBig
+R−3
+2+R−9
+5/parenrightBig
+.
+Thus, introducing this estimate into the result of [4], we have for an y 0< d≤R
+|∇xE(t,x)| ≤C(1+ln(R/d))/ba∇dblρ(t)/ba∇dblL∞+Cd/ba∇dbl∇xρ(t)/ba∇dblL∞+C(R−3
+2+R−9
+5).(18)
+Then, wemayfollowtheargumentinSection4 .2.6of[4]andfindaprioriboundson /ba∇dbl∇xE(t)/ba∇dblL∞
+and/ba∇dbl∇x,vf(t)/ba∇dblL∞as long as /ba∇dblρ(t)/ba∇dblL∞is bounded. This work will be included in Appendix A.
+2.4 Energy Bound
+First, notice from (3) that we may write W >0 as
+W= inf{η >0 :F(η) = 0}.
+9Recall the definitions of σ,S, andk, as well as, (14), which we may write as
+/integraldisplay /integraldisplay
+S(f(t,x,v),|v|)dv dx≤C
+for anyt∈[0,T].
+Letη≥2W. We find
+S(h,η) = (h−F(η))η2+σ(h)−σ(F(η))
+=hη2+σ(h)−σ(0)
+=hη2+σ(h)
+=hη2−/integraldisplaymin{h,F(0)}
+0(F−1(˜h))2d˜h
+≥hη2−min{h,F(0)}(F−1(0))2
+≥hη2−hW2
+=h(η2−W2).
+Then, since η≥2W, we have1
+4η2≥W2and
+h(η2−W2) =1
+2hη2+h(1
+2η2−W2)
+≥1
+2hη2.
+Thus, for any h≥0 andη≥2W,
+S(h,η)≥1
+2hη2
+and, finally, for P≥2W
+/integraldisplay /integraldisplay
+|v|>P|v|2f(t,x,v)dv dx≤2/integraldisplay /integraldisplay
+S(f(t,x,v),|v|)dv dx≤C. (19)
+2.5 The Good, the Bad, and the Ugly Revisited
+The method we will employ is very similar to that used in Section 4 .4 of [4]. The differences are
+due mainly to the lack of positivity in the charge density, changes in th e conserved energy, and
+the contribution of the applied field. We will assume throughout this s ection that Qg(t) is not
+already bounded and (17) applies.
+Define
+Q(t) :=/parenleftBig
+max{(2W)4
+3,C(0)}/parenrightBig15
+13+Qf(t).
+10Then, define C(2):= (C(0))−7
+13+C(1)so that we use (17) and ( III) to find
+|E(τ,x)+A(τ,x)| ≤C(1)Q4
+3(t)+C(0)R−2(x)
+≤C(1)Q4
+3(t)+/parenleftbig
+C(0)/parenrightbig−7
+13/parenleftbig
+C(0)/parenrightbig20
+13
+≤C(1)Q4
+3(t)+/parenleftbig
+C(0)/parenrightbig−7
+13Q4
+3(t)
+=C(2)Q4
+3(t)
+for anyτ∈[0,t].
+Now, let ( ˆX(t),ˆV(t)) be any fixed characteristic :
+d
+dtˆX=ˆV,d
+dtˆV=E(t,ˆX)
+for which
+f(t,ˆX(t),ˆV(t))/ne}ationslash= 0.
+For any 0 ≤∆≤t, we have
+/integraldisplayt
+t−∆|E(s,ˆX(s))|ds≤C/integraldisplayt
+t−∆/integraldisplay /integraldisplay|g(s,y,w)|
+|y−ˆX(s)|2dw dy ds (20)
+=C/integraldisplayt
+t−∆/integraldisplay /integraldisplay|g(s,X(s,t,x,v),V(s,t,x,v))|
+|X(s,t,x,v)−ˆX(s)|2dv dx ds. (21)
+LetP=Q13
+20(t),R >0, and ∆ =P
+4C(2)Q4
+3(t). From the definition of Q, notice that P≥2W.
+Let us partition the integral into IG,IB, andIU, whereIAis the integral in (21) over the set
+A, and the three sets are defined as :
+G:={(s,x,v) :t−∆< s < tand (|v|< Por|v−ˆV(t)|< P)},
+B:={(s,x,v) :t−∆< s < tand|v|> Pand|v−ˆV(t)|> P
+and|X(s,t,x,v)−ˆX(s)|< R},
+U:={(s,x,v) :t−∆< s < tand|v|> Pand|v−ˆV(t)|> P
+and|X(s,t,x,v)−ˆX(s)|> R}.
+We will use the invertibility of the characteristics as described in [4], so that when we set
+y=X(s,t,x,v)
+11w=V(s,t,x,v)
+we can invert using
+x=X(t,s,y,w)
+v=V(t,s,y,w).
+In particular, notice w=V(s,t,X(t,s,y,w),V(t,s,y,w)) and∂(y,w)
+∂(x,v)= 1 .
+To handle the integral over G, we must first deal with some preliminary inequalities :
+1. First notice that |V(s,t,x,v)−v| ≤/integraltextt
+s|E(τ,X(τ))+A(τ,X(τ))|dτ.
+Thus, for s∈[t−∆,t], we have
+|V(s,t,x,v)−v| ≤∆C(2)Q4
+3(t) =1
+4P.
+2. For|v|< P,
+|V(s,t,x,v)| ≤ |v|+1
+4P <2P.
+3. For|v−ˆV(t)|< P,
+|V(s,t,x,v)−ˆV(s)| ≤ |v−ˆV(t)|+|ˆV(t)−ˆV(s)|+|V(s,t,x,v)−v| ≤P+1
+4P+1
+4P <2P.
+4. For|v|> Pandτ∈[t−∆,t],
+|V(τ,t,x,v)| ≥ |v|−1
+4P
+>3
+4P
+≥3
+4·2W
+> W
+Now, let
+χG(s,x,v) :=/braceleftbigg1 (s,x,v)∈G
+0 else.
+12Then, we have
+IG=/integraldisplay /integraldisplay /integraldisplay
+G|g(s,X(s,t,x,v),V(s,t,x,v))|
+|X(s,t,x,v)−ˆX(s)|2dv dx ds
+=/integraldisplayt
+t−∆/integraldisplay /integraldisplayχG(s,x,v)|g(s,X(s,t,x,v),V(s,t,x,v))|
+|X(s,t,x,v)−ˆX(s)|2dv dx ds
+=/integraldisplayt
+t−∆/integraldisplay /integraldisplayχG(s,X(t,s,y,w),V(t,s,y,w))|g(s,y,w)|
+|y−ˆX(s)|2dw dy ds.
+IfχG/ne}ationslash= 0, then |V(t,s,y,w)|< Por|V(t,s,y,w)−ˆV(t)|< P. Then, byPreliminaryInequalities
+2 and 3, we have either |w|<2Por|w−ˆV(s)|<2P. Set
+˜ρ(s,y) :=/integraldisplay
+|g(s,y,w)|χG(s,X(t,s,y,w),V(t,s,y,w))dw.
+Then,/ba∇dbl˜ρ(s)/ba∇dblL∞≤CP3. Also, using (13), we know
+˜ρ(s,y)≤/integraldisplay
+|g(s,y,w)|dw≤C(k3
+5(s,y)+k1
+2(s,y)).
+Thus, we employ the method of (16) and (17) to find
+/integraldisplay˜ρ(s,y)
+|y−ˆX(s)|2dy≤CP4
+3
+and, finally, we have
+IG=/integraldisplayt
+t−∆/integraldisplay˜ρ(s,y)
+|y−ˆX(s)|2dy ds≤C∆P4
+3. (22)
+Estimating IB, we have
+IB=/integraldisplay /integraldisplay /integraldisplay
+B|g(s,X(s,t,x,v),V(s,t,x,v))|
+|X(s,t,x,v)−ˆX(s)|2dv dx ds
+≤/integraldisplayt
+t−∆/integraldisplay
+|y−ˆX(s)|<R/integraldisplay|g(s,y,w)|
+|y−ˆX(s)|2dw dy ds
+≤(/ba∇dblF/ba∇dblL∞+/ba∇dblf0/ba∇dblL∞)Q3(t)/integraldisplayt
+t−∆/integraldisplay
+|y−ˆX(s)|<R|y−ˆX(s)|−2dw ds
+≤CQ3(t)/integraldisplayt
+t−∆/integraldisplayR
+0dr ds,
+and thus
+13IB≤C∆Q3(t)R. (23)
+Finally, to estimate IU, we use Section 3 (specifically, line (15)) of [16] to find
+/integraldisplayt
+t−∆|X(s,t,x,v)−ˆX(s)|−2χU(s,x,v)ds≤C
+RP, (24)
+for (x,v) as inU. The proof of this result is quite long, so we shall include a sketch rat her than
+the entire proof. First, let
+Z(s) =X(s,t,x,v)−ˆX(s).
+Then, choose s0∈[t−∆,t] such that
+|Z(s0)| ≤ |Z(s)|
+for alls∈[t−∆,t]. It is shown in [16] that
+|Z(s)| ≥1
+4P|s−s0|. (25)
+Now, define
+Σ(r) :=
+
+1
+R20≤r≤R2
+1
+rr≥R2.
+Notice that Σ is non-negative, non-increasing and
+|Z(s)|−2χU(s,x,v)≤Σ(|Z(s)|2).
+Using these properties of Σ with (25), we find
+/integraldisplayt
+t−∆|Z(s)|−2χU(s,x,v)ds≤/integraldisplayt
+t−∆Σ(|Z(s)|2)ds
+≤/integraldisplayt
+t−∆Σ/parenleftbigg
+(1
+4P|s−s0|)2/parenrightbigg
+ds
+≤/integraldisplay
+Σ/parenleftbiggP2
+16τ2/parenrightbigg
+dτ
+=16
+PR.
+This shows (24).
+14Now, using (24) with (3), (9), (19), and preliminary inequality 4, we h ave
+IU=/integraldisplay /integraldisplay /integraldisplay
+U|g(s,X(s,t,x,v),V(s,t,x,v))|
+|X(s,t,x,v)−ˆX(s)|2dv dx ds
+=/integraldisplay /integraldisplay /integraldisplay
+U|F(V(s,t,x,v))−f(s,X(s,t,x,v),V(s,t,x,v))|
+|X(s,t,x,v)−ˆX(s)|2dv dx ds
+=/integraldisplay /integraldisplay /integraldisplay
+U|−f(t,x,v)|
+|X(s,t,x,v)−ˆX(s)|2dv dx ds
+=/integraldisplayt
+t−∆/integraldisplay
+|v|>P∩ |v−ˆV(t)|>P/integraldisplay
+|X(s,t,x,v)−ˆX(s)|>Rf(t,x,v)
+|X(s,t,x,v)−ˆX(s)|2dx dv ds
+=/integraldisplay
+|v|>P/integraldisplay
+f(t,x,v)/parenleftbigg/integraldisplayt
+t−∆|X(s,t,x,v)−ˆX(s)|−2χU(s,x,v)ds/parenrightbigg
+dx dv
+≤C
+RP/integraldisplay /integraldisplay
+|v|>Pf(t,x,v)dv dx
+≤C
+RP3/integraldisplay /integraldisplay
+|v|>P|v|2f(t,x,v)dv dx
+and so
+IU≤C
+RP3. (26)
+Finally, collecting the estimates (22), (23), and (26), we find,
+1
+∆/integraldisplayt
+t−∆|E(s,ˆX(s))|ds≤C/parenleftbigg
+P4
+3+RQ3(t)+1
+∆RP3/parenrightbigg
+.
+We take R=Q−32
+15(t) and then
+1
+∆/integraldisplayt
+t−∆|E(s,ˆX(s))|ds≤CQ13
+15(t).
+Using (III), we have
+1
+∆/integraltextt
+t−∆/parenleftBig
+|E(s,ˆX(s))+A(s,ˆX(s))|/parenrightBig
+ds≤CQ13
+15(t)+1
+∆/integraltextt
+t−∆C(0)R−2(ˆX(s))ds
+≤CQ13
+15(t)+1
+∆/integraltextt
+t−∆C(0)ds
+≤CQ13
+15(t)+/parenleftBig
+(C(0))15
+13/parenrightBig13
+15
+≤CQ13
+15(t)+Q13
+15(t)
+≤CQ13
+15(t).(27)
+15Finally, we use the argument in Section 4 .5 of [4] to bound the velocity support, since the power
+ofQ(t) is less than one. This work will be explored in Appendix B. Thus, for all t∈[0.T],
+Q(t)≤C, (28)
+andthis implies bounds on Qf(t) andQg(t) for allt∈[0,T]. Furthermore, if V(s,t,x,v)satisfies
+f0(X(0,t,x,v),V(0,t,x,v))/ne}ationslash= 0, then
+|V(s,t,x,v)| ≤C
+for anys∈[0,T], including V(t,t,x,v) =v.
+Notice then, the bound on the velocity support implies a priori bound s on/ba∇dblρ(t)/ba∇dblL∞,/ba∇dblE(t)/ba∇dblL∞,
+/ba∇dbl∇xE(t)/ba∇dblL∞, and/ba∇dbl∇x,vf(t)/ba∇dblL∞for allt∈[0,T].
+Now that /ba∇dblρ(t)/ba∇dbl∞≤Cand/ba∇dbl∇ρ(t)/ba∇dbl∞≤Cfor allt∈[0,T], andQf(T) is finite, we apply
+Lemma 4 since p >3, and find /ba∇dblρ(t)/ba∇dblp≤Cfor allt∈[0,T], and the proof of Theorem 1 is
+complete.
+3 Section 3
+To conclude the paper, this section contains the proofs of Lemmas 2 through 4.
+Proof(Lemma 2) :Letq,T >0 be given with /ba∇dblρ(t)/ba∇dbl∞≤Cfor allt∈[0,T]. Letb∈[0,5
+18)
+be given with b≤2
+q. Consider |x| ≥1 and define
+η:=/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigb
+and divide the field into the following pieces :
+|E(t,x)| ≤I+II+III.
+where
+I:=/integraldisplay
+|x−y|<η|ρ(t,y)| |x−y|−2dy,
+II:=/integraldisplay
+η<|x−y|<1
+2|x||ρ(t,y)| |x−y|−2dy,
+III:=/integraldisplay
+|x−y|>1
+2|x||ρ(t,y)| |x−y|−2dy.
+16Then, the first estimate satisfies
+I≤C/ba∇dblρ(t)/ba∇dbl∞η. (29)
+The second estimate satisfies, for any m∈[0,1
+3),
+II≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigm/integraldisplay
+η<|x−y|<1
+2|x||ρ(t,y)|1−m|x−y|−2dy
+≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigm/integraldisplay
+η<|x−y|<1
+2|x|/parenleftBig
+k3(1−m)
+5(t,y)|x−y|−2+k1−m
+2(t,y)|x−y|−2/parenrightBig
+dy
+≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigm/bracketleftBigg
+(/integraldisplay
+η<|x−y|<1
+2|x||x−y|−4
+1+mdy)1+m
+2
++ (/integraldisplay
+η<|x−y|<1
+2|x||x−y|−10
+3m+2dy)3m+2
+5/bracketrightBigg
+≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigm/bracketleftBig
+η−2+3(1+m
+2)+η−2+3(3m+2
+5)/bracketrightBig
+≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigm/braceleftBigg
+η3m−1
+2, η≥1
+η9m−4
+5, η≤1
+Then, for η≥1, we choose m=3b
+2+3b, and for η≤1, we choose m=9b
+5+9b. To guarantee
+convergence of the above integrals, we must have m <1
+3, and thus, b <5
+18. Thus, we find
+II≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigb. (30)
+Finally, for b≤2
+q,
+III≤ /ba∇dblρ(t)/ba∇dblb
+q(1
+2|x|)−qb/integraldisplay
+|x−y|>1
+2|x||ρ(t,y)|1−b|x−y|bq−2R−bq(y)dy
+≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigb/integraldisplay
+|x−y|>1
+2|x||ρ(t,y)|1−b(1
+4R(y))bq−2R−bq(y)dy
+≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigb/integraldisplay
+|x−y|>1
+2|x|(k1−b
+2(t,y)+k3
+5(1−b)(t,y))R−2(y)dy
+≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigb/bracketleftbigg
+(/integraldisplay
+R−4
+1+b(y)dy)1+b
+2+(/integraldisplay
+R−10
+3b+2(y)dy)3b+2
+5/bracketrightbigg
+≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigb
+for−4
+1+b<−3, which is satisfied since b <5
+18.
+17Combining the estimates for I,II, andIII, the lemma follows.
+Proof(Lemma 3) :Letq,T >0 be given with /ba∇dblρ(t)/ba∇dbl∞≤Cand/ba∇dbl∇ρ(t)/ba∇dbl∞≤Cfor all
+t∈[0,T]. Leta∈[0,1) be given with a≤3
+q. Consider |x| ≥1 and define
+η:=/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbiga.
+For anyi,k= 1,2,3, we have
+|∂xiEk(t,x)|=|/integraltext
+|x−y|<η∂yiρ(t,y)(x−y)k
+|x−y|3dy|+|/integraltext
+|x−y|=ηρ(t,y)(x−y)k
+|x−y|3(x−y)i
+|x−y|dSy|
++|/integraltext
+|x−y|>ηρ(t,y)∂yi((x−y)k
+|x−y|3)dy|
+=:I+II+III.
+Then,
+I≤/integraldisplay
+|x−y|<η/ba∇dbl∇xρ(t)/ba∇dbl∞|x−y|−2dy
+≤Cη.
+We estimate IIfor the large |x|and small |x|cases. For |x|>2η,
+II≤/integraldisplay
+|x−y|=η(/ba∇dblρ(t)/ba∇dblqR−q(y))a|ρ(t,y)|1−a|x−y|−2dSy
+≤ /ba∇dblρ(t)/ba∇dbla
+q/ba∇dblρ(t)/ba∇dbl1−a
+L∞η−2/integraldisplay
+|x−y|=ηR−aq(|x|−η)dSy
+≤C/ba∇dblρ(t)/ba∇dbla
+qη−2R−aq(1
+2|x|)η2
+≤C(/ba∇dblρ(t)/ba∇dblqR−q(x))a.
+For|x|<2η,
+II≤ /ba∇dblρ(t)/ba∇dbl∞η−2/integraldisplay
+|x−y|=ηdSy
+≤C
+≤C(2η|x|−1)
+≤Cη.
+18Then,
+III≤C/integraldisplay
+η<|x−y|<1
+2|x||ρ(t,y)| |x−y|−3dy+C/integraldisplay
+|x−y|>1
+2|x||ρ(t,y)| |x−y|−3dy
+=:A+B.
+Estimating A, we have for n∈[0,1),
+A≤ /ba∇dblρ(t)/ba∇dbln
+q/integraldisplay
+η<|x−y|<1
+2|x||ρ(t,y)|1−nR−nq(y)|x−y|−3dy
+≤C/ba∇dblρ(t)/ba∇dbln
+qR−nq(1
+2|x|)/integraldisplay
+η<|x−y|<1
+2|x|(k1−n
+2(t,y)+k3(1−n)
+5(t,y))|x−y|−3dy
+≤C(/ba∇dblρ(t)/ba∇dblqR−q(x))n/bracketleftBigg
+(/integraldisplay
+η<|x−y|<1
+2|x||x−y|−6
+1+ndy)1+n
+2
++ (/integraldisplay
+η<|x−y|<1
+2|x||x−y|−15
+3n+2)3n+2
+5/bracketrightBigg
+≤C(/ba∇dblρ(t)/ba∇dblqR−q(x))n/bracketleftbigg
+(/integraldisplay∞
+ηr−6
+1+n+2dr)1+n
+2+(/integraldisplay∞
+ηr−15
+3n+2+2dr)3n+2
+5/bracketrightbigg
+≤C(/ba∇dblρ(t)/ba∇dblqR−q(x))n(η3
+2(1+n)−3+η3
+5(3n+2)−3)
+≤C(/ba∇dblρ(t)/ba∇dblqR−q(x))n(η−3
+2(1−n)+η−9
+5(1−n))
+≤C(/ba∇dblρ(t)/ba∇dblqR−q(x))n/braceleftbiggη−9
+5(1−n), η≤1
+η−3
+2(1−n), η≥1
+Forη≤1, we may choose n=14a
+9a+5, and for η≥1, we may choose n=5a
+3a+2. Thus,
+A≤C(/ba∇dblρ(t)/ba∇dblqR−q(x))a.
+Finally, we estimate Band find
+B≤/integraldisplay
+|x−y|>1
+2|x||ρ(t,y)| |x−y|−3dy
+≤ /ba∇dblρ(t)/ba∇dbla
+q(1
+2|x|)−aq/integraldisplay
+|x−y|>1
+2|x||ρ(t,y)|1−a|x−y|aq−3R−aq(y)dy
+≤C(/ba∇dblρ(t)/ba∇dblqR−q(x))a/integraldisplay
+|x−y|>1
+2|x|(k1−a
+2(t,y)+k3
+5(1−a)(t,y))R−3(y)dy
+≤C(/ba∇dblρ(t)/ba∇dblqR−q(x))a/bracketleftbigg
+(/integraldisplay
+R−6
+1+a(y)dy)1+a
+2+(/integraldisplay
+R−15
+3a+2(y)dy)3a+2
+5/bracketrightbigg
+≤C(/ba∇dblρ(t)/ba∇dblqR−q(x))a
+19for−6
+1+a<−3, which is satisfied since a <1.
+Combining the estimates for I,II,A, andB, the lemma follows.
+Proof(Lemma 4) :LetT >0 andq∈[0,54
+13) be given with /ba∇dblρ(t)/ba∇dbl∞≤Cand/ba∇dbl∇ρ(t)/ba∇dbl∞≤C
+for allt∈[0,T] andQf(T)<∞. Lett∈[0,T] be given. For any D >0, taking |x| ≤D, we
+find
+|ρ(t,x)| ≤ /ba∇dblρ(t)/ba∇dbl∞
+≤ /ba∇dblρ(t)/ba∇dbl∞Rq(D)
+Rq(x)
+≤CR−q(x).(31)
+Now, define C(3):= max{1,8TQg(T)}and let|x| ≥C(3). Define for every t∈[0,T] and
+x∈R3,
+E(t,x) =E(t,x)+A(t,x).
+From the Vlasov equation,
+∂
+∂s(g(s,X(s),V(s))) =−E(s,X(s))·∇vF(V(s)),
+and thus
+g(t,x,v) =g(0,X(0),V(0))−/integraldisplayt
+0E(s,X(s))·∇vF(V(s))ds. (32)
+Thus, to estimate ρ, we must consider/integraltext
+E(s,X(s))· ∇vF(V(s))dv. Assume fis nonzero
+along (X(s),V(s)). Then,
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+E(s,X(s))·∇vF(V(s))dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+E(s,X(s))·(∇vF(V(s))−∇vF(v+/integraldisplayt
+sE(τ,x)dτ))dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle
++/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+(E(s,X(s))−E(s,x+(s−t)v))
+·∇vF(v+/integraldisplayt
+sE(τ,x)dτ)dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle
++/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+∇v·(F(v)E(s,x+(s−t)v))dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle
++/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+F(v)∇v·(E(s,x+(s−t)v))dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=:I+II+III+IV.
+20Using Lemmas 2 and 3, as well as (III), we find a∈[0,1) andb∈[0,5
+18) witha≤3
+q,b≤2
+q,
+anda+b= 1 such that
+|E(t,x)| ≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbigb
+and
+|∇E(t,x)| ≤C/parenleftbig
+/ba∇dblρ(t)/ba∇dblqR−q(x)/parenrightbiga.
+By the Mean Value Theorem, for τ∈[s,t] andi= 1,2,3, there exist ξi
+1on the line segment
+between X(τ) andxsuch that
+Ei(τ,X(τ))−Ei(τ,x) =∇xEi(τ,ξi
+1)·(X(τ)−x).
+Hence,
+I≤/integraldisplay
+|v|≤Qg(T)|E(s,X(s))| /ba∇dbl∇2F/ba∇dbl∞|V(s)−/parenleftbigg
+v+/integraldisplayt
+sE(τ,x)dτ/parenrightbigg
+|dv
+≤C/integraldisplay
+|v|≤Qg(T)/parenleftbig
+/ba∇dblρ(s)/ba∇dblqR−q(X(s))/parenrightbigb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt
+s(E(τ,X(τ))−E(τ,x))dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingledv
+≤C/integraldisplay
+|v|≤Qg(T)/parenleftbig
+/ba∇dblρ(s)/ba∇dblqR−q(X(s))/parenrightbigb/integraldisplayt
+ssup
+i|∇E(τ,ξi
+1)| |X(τ)−x|dτ dv
+≤C/integraldisplay
+|v|≤Qg(T)/parenleftbig
+/ba∇dblρ(s)/ba∇dblqR−q(X(s))/parenrightbigbTQg(T)/integraldisplayt
+ssup
+i/parenleftbig
+/ba∇dblρ(τ)/ba∇dblqR−q(ξi
+1)/parenrightbigadτ dv.
+Since we know, by (11), for any i= 1,2,3
+|ξi
+1| ≥ |X(τ)|−|X(τ)−x|
+≥1
+2|x|−TQg(T)
+≥1
+4|x|,
+we find
+I≤C/parenleftbigg
+/ba∇dblρ(s)/ba∇dblqR−q(1
+2|x|)/parenrightbiggb
+R−aq(1
+4|x|)/integraldisplayt
+s/ba∇dblρ(τ)/ba∇dbla
+qdτ
+≤CR−q(a+b)(x)/parenleftbigg
+/ba∇dblρ(s)/ba∇dblb
+q/integraldisplayt
+s/ba∇dblρ(τ)/ba∇dbla
+qdτ/parenrightbigg
+.
+21Similarly, using the above lemmas and the Mean Value Theorem, for any i= 1,2,3 there
+existξi
+2between X(τ) andx+(s−t)vsuch that
+II≤C3/summationdisplay
+i=1/integraldisplay
+|v|≤Qg(T)sup
+i|∇E(s,ξi
+2)| |X(s)−(x+(s−t)v)| /ba∇dbl∇F/ba∇dbl∞dv
+≤C3/summationdisplay
+i=1/integraldisplay
+|v|≤Qg(T)sup
+i/parenleftbig
+/ba∇dblρ(s)/ba∇dblqR−q(ξi
+2)/parenrightbiga|/integraldisplayt
+s/integraldisplayt
+τE(ι,X(ι))dι dτ|dv.
+Since we know, using (11), for any i= 1,2,3
+|ξi
+2| ≥ |X(s)|−|X(s)−(x+(s−t)v)|
+≥1
+2|x|−|/integraldisplayt
+s(V(τ)−v)dτ|
+≥1
+2|x|−2TQg(T)
+≥1
+4|x|,
+it follows that
+II≤C/integraldisplay
+|v|≤Qg(T)/parenleftbigg
+/ba∇dblρ(s)/ba∇dblqR−q(1
+4|x|)/parenrightbigga/integraldisplayt
+s/parenleftbig
+/ba∇dblρ(s)/ba∇dblqR−q(X(τ))/parenrightbigbdτ dv
+≤CR−q(a+b)(x)/parenleftbigg
+/ba∇dblρ(s)/ba∇dbla
+q/integraldisplayt
+s/ba∇dblρ(τ)/ba∇dblb
+qdτ/parenrightbigg
+.
+By the Divergence Theorem,
+III= 0,
+and finally,
+IV≤/integraldisplay
+|v|≤Qg(T)/ba∇dblF/ba∇dbl∞|∇x·E(s,x+(s−t)v)| |s−t|dv
+≤C/integraldisplay
+|v|≤Qg(T)|ρ(s,x+(s−t)v)|dv
+Collecting the estimates for I−IV, we have
+22|/integraltext
+E(s,X(s))·∇vF(V(s))dv| ≤C/parenleftBig
+R−q(a+b)(x)(/ba∇dblρ(s)/ba∇dblb
+q/integraltextt
+s/ba∇dblρ(τ)/ba∇dbla
+qdτ)
++R−q(a+b)(x)(/ba∇dblρ(s)/ba∇dbla
+q/integraltextt
+s/ba∇dblρ(τ)/ba∇dblb
+qdτ)
++/integraltext
+|v|≤Qg(T)|ρ(s,x+(s−t)v)|dv/parenrightBig
+.(33)
+Sincea+b= 1, we proceed from (32) and using (33), ( IV), and (28), we find
+|ρ(t,x)|=|/integraldisplay
+g(t,x,v)dv|
+≤/integraldisplay
+|v|≤Qg(T)|g(0,X(0),V(0))|dv+|/integraldisplayt
+0/integraldisplay
+|v|≤Qg(T)E(s,X(s))·∇vF(V(s))dv ds|
+≤CR−q(x)+C/integraldisplayt
+0/parenleftbigg
+R−q(a+b)(x)(/ba∇dblρ(s)/ba∇dblb
+q/integraldisplayt
+s/ba∇dblρ(τ)/ba∇dbla
+qdτ)
++R−q(a+b)(x)(/ba∇dblρ(s)/ba∇dbla
+q/integraldisplayt
+s/ba∇dblρ(τ)/ba∇dblb
+qdτ)+/integraldisplay
+|v|≤Qg(T)|ρ(s,x+(s−t)v)|dv/parenrightBigg
+ds
+≤CR−q(x)/parenleftbigg
+1+/integraldisplayt
+0/ba∇dblρ(s)/ba∇dblb
+q/integraldisplayt
+s/ba∇dblρ(τ)/ba∇dbla
+qdτds+/integraldisplayt
+0/ba∇dblρ(s)/ba∇dbla
+q/integraldisplayt
+s/ba∇dblρ(τ)/ba∇dblb
+qdτds/parenrightbigg
++C/integraldisplay
+|v|≤Qg(T)/integraldisplayt
+0|ρ(s,x+(s−t)v)|ds dv
+≤CR−q(x)/parenleftbigg
+1+(/integraldisplayt
+0/ba∇dblρ(s)/ba∇dblb
+qds) (/integraldisplayt
+0/ba∇dblρ(τ)/ba∇dbla
+qdτ)+(/integraldisplayt
+0/ba∇dblρ(s)/ba∇dbla
+qds) (/integraldisplayt
+0/ba∇dblρ(τ)/ba∇dblb
+qdτ)/parenrightbigg
++C/integraldisplay
+|v|≤Qg(T)/integraldisplayt
+0|ρ(s,x+(s−t)v)|ds dv.
+Then, applying H¨ older’s inequality twice,
+|ρ(t,x)| ≤CR−q(x)/parenleftBigg
+1+2T2/parenleftbigg/integraldisplayt
+0/ba∇dblρ(s)/ba∇dblqds/parenrightbigga+b
++Rq(x)/integraldisplay
+|v|≤Qg(T)/integraldisplayt
+0/ba∇dblρ(s)/ba∇dblqR−q(x+(s−t)v)ds dv/parenrightBigg
+≤CR−q(x)/parenleftbigg
+1+/integraldisplayt
+0/ba∇dblρ(s)/ba∇dblqds+Rq(x)R−q(7
+8|x|)Q3
+g(T)/integraldisplayt
+0/ba∇dblρ(s)/ba∇dblqds/parenrightbigg
+≤CR−q(x)/parenleftbigg
+1+/integraldisplayt
+0/ba∇dblρ(s)/ba∇dblqds/parenrightbigg
+.
+23Combining this with (31) (with D=C(3)) and multiplying by Rq(x), we have for all x,
+Rq(x)|ρ(t,x)| ≤C(1+/integraldisplayt
+0/ba∇dblρ(s)/ba∇dblqds).
+Since the right side of the inequality is independent of x, we take the supremum over all xto
+find,
+/ba∇dblρ(t)/ba∇dblq≤C(1+/integraldisplayt
+0/ba∇dblρ(s)/ba∇dblqds)
+and by Gronwall’s Inequality,
+/ba∇dblρ(t)/ba∇dblq≤C.
+Thus,/ba∇dblρ(t)/ba∇dblqis bounded for all t∈[0,T]. Notice, too, that the choice of a+b= 1 forces
+q <54
+13, and the proof of the lemma is complete.
+A Appendix A
+In this appendix, we will explore the argument used in Section 4 .2.6 of [4] to bound derivatives
+offandE.
+LetDbe anyxderivative. Then, using the Vlasov equation,
+∂t(Df)+v·∇x(Df)−E·∇v(Df) =DE·∇vf
+so thatd
+dsDf(s,X(s),V(s)) =DE·∇vf(s,X(s),V(s)).
+Hence,
+|Df(t,x,v)| ≤ |Df(0,X(0),V(0))|+/integraldisplayt
+0|DE·∇vf(s,X(s),V(s))|ds.
+Define
+|f(s)|1= sup
+x,v|∂xf(s,x,v)|+sup
+x,v|∂vf(t,x,v)|
+and
+|E(s)|1= sup
+x|∂xE(s,x)|.
+Then,
+|Df(t,x,v)| ≤C+/integraldisplayt
+0|E(s)|1|f(s)|1ds.
+24We see that ∂vfsatisfies an inequality of similar type because
+∂t(∂vf)+v·∇x(∂vf)−E·∇v(∂vf) =−∂vv·∇xf.
+It follows that
+|f(t)|1≤C+/integraldisplayt
+0(1+|E(s)|1)|f(s)|1ds.
+However, from (18), we can conclude (with d=/ba∇dbl∇xρ/ba∇dbl∞) that
+|E(s)|1≤C(1+ln∗|f(s)|1)
+where
+ln∗s=/braceleftbigg
+s 0≤s≤1
+1+lns s > 1.
+Therefore, for t≤T,
+|f(t)|1≤C/parenleftbigg
+1+/integraldisplayt
+0(1+ln∗|f(s)|1)|f(s)|1ds./parenrightbigg
+.
+It now follows by an application of Gronwall’s Iequality that
+|f(t)|1≤C
+and
+|E(t)|1≤C
+fort≤T. This concludes the argument to bound field derivatives and derivat ives of the density,
+and thus ends Appendix A.
+B Appendix B
+The bound on Q(t) is obtained as follows. From (8) and (27), we have
+|ˆV(t)| ≤ |ˆV(t−∆)|+/integraldisplayt
+t−∆|E(s,ˆX(s))|ds
+≤Q(t−∆)+C∆Q13
+15(t).
+Since the above constant is independent of the particular charact eristic, we find
+Q(t)≤Q(t−∆)+C∆Q13
+15(t)
+25for
+∆ = min/braceleftbigg
+t,1
+4C(2)Q−4
+3(t)·Q13
+20(t)/bracerightbigg
+.
+= min/braceleftbigg
+t,1
+4C(2)Q−41
+60(t)/bracerightbigg
+SinceQis non-decreasing, there exists T1such that
+∆ =/braceleftbiggt t ≤T1
+1
+4C(2)Q−41
+60(t)t≥T1.
+Taket0in the interval of existence. Without loss of generality, t0> T1. Let
+t1=t0−1
+4C(2)Q−41
+60(t0),
+ti+1=ti−1
+4C(2)Q−41
+60(ti) (i= 1,2,...)
+as long as ti> T1. Then
+ti−ti+1=1
+4C(2)Q−41
+60(ti)≥1
+4C(2)Q−41
+60(t0)
+which is a uniform lower bound on the length of each subinterval. So, t here is a first i, sayi=k,
+such that tk≤T1. Thus,tk≥0 and therefore
+(t0−t1)+(t1−t2)+···+(tk−1−tk)≥k·1
+4C(2)Q−41
+60(t0)
+which implies that
+Q−41
+60(t0)·k≤4C(2)t0.
+Now we have
+Q(t0) =Q(tk)+k−1/summationdisplay
+i=0[Q(ti)−Q(ti+1)]
+≤Q(tk)+Ck−1/summationdisplay
+i=0∆·Q13
+15(ti)
+≤Q(T1)+Ck−1/summationdisplay
+i=01
+4C(2)Q−41
+60(t0)·Q13
+15(ti)
+≤Q(T1)+C·kQ−41
+60(t0)·Q13
+15(t0)
+≤Ct0Q13
+15(t0).
+Therefore, Q(t0) is bounded, and the proof is complete. This ends Appendix B.
+26References
+[1] Batt, J. Global symmetric solutions of the initial-value problem of s tellar dynamics. J. Diff.
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+[12] Okabe, S.; Ukai, T. On classical solutions in the large in time of two- dimensional Vlasov’s
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+[13] Pankavich, S. Global existence for the Radial Vlasov-Possion S ystem with Steady Spatial
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+[14] Pfaffelmoser, K. Global classical solution of the Vlassov-Poisso n system in three dimensions
+for general initial data. J. Diff. Eq. 1992, 95(2):281-303.
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+28
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@@ -0,0 +1,334 @@
+arXiv:1001.0633v1  [astro-ph.EP]  5 Jan 2010Research inAstron. Astrophys. 2009Vol.9No.XX,000–000
+http://www.raa-journal.org http://www.iop.org/journa ls/raaResearch in
+Astronomy and
+Astrophysics
+Aggregatedust modelto study the polarization properties o fcomet
+C/1996 B2Hyakutake
+H.S.Das, A.Suklabaidya, S.Datta Majumder and A.K.Sen
+DepartmentofPhysics,Assam University,Silchar788011,I ndia;hsdas@iucaa.ernet.in
+Received [year] [month] [day]; accepted [year] [month] [da y]
+Abstract In ourpresentstudy,theobservedlinearpolarizationdata ofcometHyakutake
+are studied at wavelengths λ= 0.365µm,λ= 0.485µmand 0.684 µmthrough simula-
+tions using Ballistic Particle-Cluster Aggregate and Ball istic Cluster-Cluster Aggregate
+aggregates of 128 spherical monomers. We first investigated that the size parameter
+of the monomer, x∼1.56 – 1.70, turned out to be most suitable which provides the
+best fits to the observed dust scattering properties at three wavelengths λ= 0.365µm,
+0.485µmand 0.684 µm. Thus the effective radius of the aggregate (r) lies in the ra nge
+0.45µm≤r≤0.49µmatλ= 0.365µm;0.60µm≤r≤0.66µmatλ= 0.485µm
+and0.88µm≤r≤0.94µmatλ= 0.684µm. Now using superposition T-MATRIX
+code and the power-lawsize distribution, n(r)∼r−3, the best-fitting values of complex
+refractive indices are calculated which can best fit the obse rved polarization data at the
+above three wavelengths. The best-fitting complex refracti ve indices (n,k)are found to
+be (1.745,0.095) at λ= 0.365µm, (1.743,0.100) at λ= 0.485µmand (1.695, 0.100)
+atλ= 0.684µm.Therefractiveindicescomingoutfromthepresentanalysi scorrespond
+to mixture of both silicates and organics, which are in good a greement with the in situ
+measurementofcometsbydifferentspacecraft.
+Keywords: comets:general–dust,extinction–scattering–polarizat ion
+1 INTRODUCTION
+Thestudyofcometarypolarization,overvariousscatterin ganglesandwavelengths,givesvaluableinfor-
+mation about the nature of cometary dust. The numerical and e xperimentalsimulations of polarization
+data gives information about the physical propertiesof the cometary dust, which include size distribu-
+tion,shapeandcomplexrefractiveindices.Severalinvest igators(Kikuchietal.1987;Lamyetal.1987;
+Sen et al. 1991a,1991b;Chernovaet al. 1993;Xing & Hanner 19 97;Petrovaet al. 2004;Kimura et al.
+2006; Das et al. 2004; Kolokolova et al. 2007, Bertini et al. 2 007 etc.) have studied linear and circular
+polarizationmeasurementsofmanycomets.Thesestudieshe lpustounderstandthedustgrainnatureof
+comets.
+Comet Hyakutake (C/1996 B2) was the brightest comet appeare d in the sky in the year 1996. Its
+passage near the Earth was one of the closest cometary approa ches of the previous 200 years which
+passed within 0.1 AU of the Earth in March 1996. Comet Hyakuta ke was bright enough to make high
+precisionpolarimetricobservationsduringpre-periheli onphase.Observationsforthelinearpolarization
+of comet Hyakutake were carried out at three different wavel engths: 0.365 µm0.485µmand 0.684 µm
+bydifferentinvestigators(Joshiet al. 1997;Kiselev&Vel ichko1998andManset&Bastien 2000).
+Greenberg and Hage (1990) first suggested that cometary part icles are not spherical and porous.
+They originally proposed the presence of large numbers of porousgrains in the coma of comets to2 H. S.Das, A.Suklabaidya, S.DattaMajumder & A. K.Sen
+explain the spectral emission at 3.4 µmand 9.7µm. Dollfus (1989) discussed the results of laboratory
+experiments by microwave simulation and laser scattering o n various complex shapes with different
+porosities. The results of in situmeasurements carried out on the Giotto spacecraft at Comet H alley
+(Fulle et al. 2000) and the analysis of the infrared spectra o f Comet Hale-Bopp (Moreno et al. 2003)
+also agree with the model of aggregates. It is clear from rece nt modeling of optical (Xing & Hanner
+1997; Kimura 2001; Kimura et al. 2006; Petrova et al. 2004; Ti shkovetset al. 2004; Lasue et al. 2006;
+Kolokolova et al. 2007; Bertini et al. 2007; Levasseur-Rego urd et al. 2007, 2008; Das et al. 2008a;
+2008b etc.), thermal-infrared observations (Lisse et al. 1 998; Harker et al. 2002), laboratory studies
+(Wurm & Blum 1998; Gustafson & Kolokolova1999; Hadamcik et a l. 2002 etc.), and especially from
+the ‘Stardust’ returned samples ( H¨ orz et al. 2006), that co metary dust consists of irregular, mostly
+aggregatedparticles.
+Das & Sen (2006) studied the non-spherical dust grain charac teristics of Comet Levy 1990XX
+using the T-matrix theory. They foundthat compact prolate g rains as comparedto spherical grains can
+betterexplaintheobservedlinearpolarizationdata.Rece ntly,Dasetal.(2008a)haveagainanalyzedthe
+observedpolarizationdata of Comet Levy 1990XXand success fully reproducedthe polarizationcurve
+through simulations using aggregate dust model, where the fi t was still better. It has been found from
+their analysis that aggregate particles can produce a still better fit to the observed data as compared to
+compactprolategrains.Recently,usingaggregatedustmod el,Dasetal.(2008b)successfullyexplained
+thepolarizationcharacteristicsofcometHale-Boppat λ= 0.485µmand0.684 µm.Lasueetal.(2009)
+haveexplainedsuccessfullythepolarizationpropertieso fcometHale-BoppandHalleybyusingamodel
+of light scattering through a size distribution of aggregat es(spherical or spheroidal) mixed with single
+spheroidalparticles.
+Inthepresentwork,theaggregatedustmodelisproposedtos tudytheobservedpolarizationdataof
+CometHyakutakeat λ=0.365µm,0.485µmand0.684 µm.
+2 AGGREGATEMODELOFCOMETARY DUST:
+The aggregates are built by using ballistic aggregation pro cedure. Two types of aggregates are
+considered here- BPCA (Ballistic Particle-Cluster Aggreg ate) and BCCA (Ballistic Cluster-Cluster
+Aggregate). In actual case, the BPCA clusters are more compa ct than BCCA clusters (Mukai et al.
+1992). A systematic explanation on dust aggregate model is a lready discussed in our previous work
+(Dasetal.2008a).Laboratorydiagnosisofparticlecoagul ationinthesolarnebulasuggeststhatthepar-
+ticles grow underBCCA process. It is also foundthat the morp hologyof dust particles doesnot play a
+majorroleindeterminingtheshapeofpolarization(Kimura 2001;Kimuraetal.2003,2006;Kolokolova
+etal.2006;Lasue&Levasseur-Regourd2006;Bertinietal.2 007;Daset al.2008a,2008b).Thesizeof
+the individualmonomerin a cluster playsan importantrole i n scatteringcalculations. These havebeen
+confirmed by the results of previous work on dust aggregate mo del (Kimura et al. 2003; Kimura et al.
+2006; Petrova et al. 2004; Hadamcik et al. 2006; Bertini et al . 2007) and also from our previous work
+(Daset al. 2008a).
+3 COMPOSITION
+Theinsituobservationsofcomets,laboratoryanalysisofsamplesofI nterplanetaryDustParticles(IDP)
+and remote infrared spectroscopic study of comets give usef ul information about the composition of
+cometarydust. The in situmeasurementof impact-ionizationmass spectra of Comet Hal ley’sdust, has
+suggested that the dust consists of magnesium-rich silicat es, carbonaceousmaterials, and iron-bearing
+sulfides(Kissel etal. 1986;Jessbergeret al. 1988andJessb erger1999).Actually,thefirst evidencefor
+carbonaceousmaterial in comets comes from the study of Vega spacecraft data by Kissel et al. (1986).
+These materials are also known to be the major constituents o f IDPs (Brownlee et al. 1980). The stud-
+ies of comets and IDPs have shown the presence of amorphous an d crystalline silicate minerals (e.g.
+forsterite, enstatite) and organic materials (Hanner & Bra dley 2004). Laboratory studies have shown
+that majority of the collected IDPs fall into the spectral cl asses defined by their 10 µmfeature profil.Polarizationproperties of comet C/1996 B2Hyakutake 3
+These observed profiles indicate the presence of pyroxene,o livine and layer lattice silicates. This is in
+good agreement with results obtained from Giotto and Vega ma ss spectrometer observationsof Comet
+Halley(Lamyetal.1987).Theinfrared(IR)measurementofc ometshasalsoprovidedimportantinfor-
+mationonthesilicatecompositionsincometarydust.Thesp ectroscopicstudiesofsilicateshaveshown
+the predominance of both crystalline and amorphous silicat es consisting of pyroxene or olivine grains
+(Woodenetal.1999;Haywardetal.2000,Bockel´ ee-Morvane tal.2002etc.).Mg-richcrystalsarealso
+found within IDPs and are predicted by comparing the IR spect ral features of Comet Hale-Bopp with
+syntheticspectraobtainedfromlaboratorystudies(Hanne r1999;Woodenetal. 1999,2000).‘Stardust’
+sampleshavealsoconfirmedavarietyofolivineandpyroxene silicatesinComet81P/Wild2(Zolensky
+et al.2006).
+Levasseur-Regourd et al. (1996) studied a polarimetric dat a base of several comets and from the
+nature of phase angle ( α) dependence, they concluded that there is a clear evidence f or at least two
+classes of comets according to the values of polarization at α≈800−1000: comets with a high
+maximum in polarization, of about 25% for one group and small er than 15% for the other group. The
+two classes of cometsare distinct only for α >350. It has been also observedthat there is a verygood
+correlation between the existence of a high maximum in polar ization and a strong silicate emission
+feature (Levasseur-Regourd1999). The observed polarizat ion data of comet Hyakutake showed a high
+maximum in polarization.The polarizationat a given phase a ngle larger than 300most often increases
+linearlywithincreasingwavelengthinthevisibledomaina ndthisincreasebeingsteeperforlargerphase
+angles(Levasseur-Regourd& Hadamcik2003).Recent studie shaveprovidedusefulinformationabout
+the two groups of polarimetrically different comets (Kisel ev et al., 2001; Kiselev et al., 2004; Jewitt
+2004;Jockersetal., 2005).
+Ithasbeenalreadyfoundthatthesilicatecompositioncanb estreproducetheobservedpolarization
+dataofCometLevy1990XXandCometHale-Bopp(Daset al. 2008 a,b).
+4 NUMERICAL SIMULATIONS:
+ThescatteringcalculationsforBCCA &BPCA particleshaveb eendonebytheSuperpositionT-matrix
+code, which gives rigoroussolutions for ensembles of spher es (Mackowski & Mishchenko 1996). The
+observedlinearpolarizationdataof cometHyakutakeat λ=0.365µm,0.485µm&0.684µmaretaken
+fromJoshi etal. (1997),Kiselev&Velichko(1998),Manset& Bastien(2000).
+Thelinearpolarizationisgivenby
+P(θ) =−S21
+S11(1)
+For modeling comet Hyakutake, we will use a power-law size di stribution, n(r) =dn/dr∼r−3.
+For a particular type of aggregate with fixed N, the size distr ibution is just dn/da m∼a−3
+m. Thus the
+averagedpolarizationis(Shenet al. 2009):
+¯P=/integraltextamax
+aminp(am,θ)S11(am,θ)n(am)dam/integraltextamax
+aminS11(am,θ)n(am)dam(2)
+whereaminandamaxaretheminimumandmaximumvaluesofthemonomersize in ours ize distribu-
+tion.
+Theradiusofanaggregateparticlecanbedescribedbythera diusofasphereofequalvolumegiven
+byr=amN1/3, whereN is the numberof monomersin the aggregate.In the pre sent work,N =128is
+taken. The size parameter of the monomer is given by x=2πam
+λ. We first investigated that x∼1.56
+– 1.70 turned out to be most suitable which may provide the bes t qualitative fits to the observed dust
+scattering properties at three wavelengths λ= 0.365µm, 0.485µmand 0.684 µm. This correspond to
+0.090µm≤am≤0.098µmatλ= 0.365µm;0.120µm≤am≤0.131µmatλ= 0.485µmand
+0.174µm≤am≤0.186µmatλ= 0.684µm. Thus the effective radius of the aggregate (r) lies in4 H. S.Das, A.Suklabaidya, S.DattaMajumder & A. K.Sen
+the range 0.45µm≤r≤0.49µmatλ= 0.365µm;0.60µm≤r≤0.66µmatλ= 0.485µmand
+0.88µm≤r≤0.94µmatλ= 0.684µm.
+We start calculations considering the refractive indices f or amorphous pyroxene and amorphous
+olivine at λ=0.365µm,0.485µm&0.684µm. The refractive indices of the materials are calculated by
+linearly interpolating the data obtained from laboratory s tudies (Dorschner et al. 1995). Olivines and
+pyroxenes are described by Mg 2yFe2−2ySiO4, withy= 0.4, 0.5 and Mg yFe1−ySiO3, withy= 1.00,
+0.95,0.8,0.7,0.6,0.5and0.4.
+Ithasbeenalreadyinvestigatedthatthechoiceoftheabove valuesofrefractiveindicescan’tmatch
+the observed polarization data of Comet Levy 1990XX and Hale -Bopp (Das et al. 2008a,b).The same
+set of refractive indices is now chosen to fit the observed pol arization data of Comet Hyakutake. The
+calculations have been done for BCCA aggregates. But no such good fit has been observed using the
+above value of refractive indices. Next, the calculation ha s been repeated for carbonaceous materials,
+butnoneofthemcouldmatchtheobserveddatawell.
+Wenowuse χ2minimizationtechniquetoevaluatethebest-fittingvalues of(n,k),whichcanfit to
+theobservedpolarizationdata.Wehavealreadyusedthismi nimizationtechniquetofittheobservedlin-
+earpolarizationdataof CometLevy1990XXat λ= 0.485µmandComet Hale-Boppat λ= 0.485µm
+and0.684µm(Daset al.2008a,b),with aggregatemodelsofdust.
+Theerrorinthefittingprocedurecanbedefinedas
+χ2
+pol=J/summationdisplay
+i=1/vextendsingle/vextendsingle/vextendsinglePobs(θi,λ)−Pmodel(θi,λ)
+Ep(θi,λ)/vextendsingle/vextendsingle/vextendsingle2
+(3)
+Here,P obs(θi,λ)isthedegreeoflinearpolarizationobservedatscattering angleθi(i=1,2,....,J)and
+wavelength λ, Pmodel(θi,λ)is the polarization values obtained from model calculation s and E p(θi,λ)
+istheerrorin theobservedpolarizationatscatteringangl eθiandwavelength( λ).
+We now introduce a quantity χ2=χ2
+pol/J, where J is the number of data points. The values
+of(n,k)are varied over a large range simultaneously with amand we find for a particular value of
+(n,k),χ2becomes minimum. This particular value of (n,k)is our best fitted (n,k)value and the
+corresponding minimum value of χ2is denoted as χ2
+min. It is also observed that this technique of
+minimizationof χ2is quite unique.The value of χ2
+mingivesthe confidencelevel on our best fit values
+of(n,k)andalso inthe overallfittingprocedure.
+We need to fine-tune the free parameters (n,k)in the model to make the best fit to the observed
+linearpolarizationdataofCometHyakutake.Therealparto ftherefractiveindexisincreasedfrom n=
+1.4to 2.0with0.001steps, while theimaginarypart ofthere fractiveindexisincreasedfrom k= 0.001
+to 1.0 with 0.001 steps. The same range has been already used f or Comet Levy 1990XX and Comet
+Hale-Bopp(Dasetal. 2008a,b).
+NowweanalyzetheobserveddataofcometHyakutakeat0.365 µm.We calculate ¯Paveragedover
+the size distribution n(r)∼r−3withrmin= 0.45µmandrmax= 0.49µm(amin= 0.090µm,
+amax= 0.098µm).Thebestfittingrefractiveindexat0.365 µmisfoundtoben=1.745,k=0.095.The
+simulatedpolarizationcurveat 0.365 µmisshownin Fig.1.
+Wenowextendourcalculationfurthertofittheobservedpola rizationdataat λ=0.485µmand0.684
+µm. Here we also calculate ¯Paveraged over the size distribution n(r)∼r−3withrmin= 0.60µm
+andrmax= 0.66µm(amin= 0.120µm,amax= 0.131µm)atλ= 0.485µmandrmin= 0.88µmand
+rmax= 0.94µm(amin= 0.174µm,amax= 0.186µm) atλ= 0.684µm. The best fitting refractive
+indicesareobtainedfromthepresentanalysisarefoundtob e(1.743,0.100)at λ=0.485µmand(1.695,
+0.100)λ=0.684µm.ThesimulatedpolarizationcurveforcometHyakutakeat λ=0.485µmand0.684
+µmforBCCA aggregatesareshownin Fig.2andFig.3.
+5 DISCUSSION:
+The negative polarization behaviour of comet is one of the ma jor features observed in comets. Several
+cometsshow negativepolarizationbeyond1570scattering angle (Kikuchiet al., 1987;Chernovaet al.,Polarizationproperties of comet C/1996 B2Hyakutake 5
+-5 0 5 10 15 20 25
+ 60  80  100  120  140  160  180Polarization (in %)
+Scattering angleλ = 0.365 µm
+Fig.1Polarization values as observed at wavelength λ= 0.365µmfor comet Hyakutake by
+Joshi et al. (1997)and Kiselev & Velichko(1998).The solid c urverepresentsthe best-fitting
+polarization curve obtained for BCCA particles with 128 mon omers for a size distribution
+n(r)∼r−3for0.45µm≤r≤0.49µmatλ= 0.365µm.Heren= 1.745,k=0.095.
+1993; Ganesh et al., 1998 etc.). Interestingly, all comets s how very similar characteristics of negative
+polarization (minimum value of polarization ∼– 2%near 1700and inversion angle at 20 – 220).
+Comet Hyakutake was observed over a wide scattering angle ra nge (68.60– 143.10), but there was no
+observationrecordedbeyond143.10(Joshietal.,1997;Kiselev&Velichko1998andManset&Bast ien
+2000).In thepresentwork,it isinterestingtoobservethat theused dustaggregatemodelreproducethe
+negativepolarizationbehaviourbeyond1570.
+Thestrengthofthesilicatefeatureisdefinedastheratioof thefluxbetween10and11 µmtothatof
+the underlyingcontinuum(Lisse 2002;Sitko et al., 2004;Ko lokolovaet al., 2007).The silicate feature
+strength of Comet Hyakutake is >1.5(Lisse 2002) whereas the values for Comet Levy 1990XX and
+CometHale-Bopparegivenby1.8(Harkeretal.,1999)and2.1 6(Sitkoetal.,2004).CometHale-Bopp
+is an intrinsicallybrightcomet, with polarizationvalues muchhigherthan thoseof other comets.It has
+beenfoundthatCometHale-Boppshowsthehighestsilicatef eaturestrength.Thestrongsilicatefeature
+indicates high abundance of silicates in the dust. It can be s een that the refractive indices coming out
+from the present calculation is closed to the refractive ind ices of silicates and organics. Again the in
+situmeasurements of comet Halley (Lamy et al. 1987) and the ‘Star dust’ returned samples of comet
+Wild 2 (Zolensky et al., 2006) showed the presence of a mixtur e of silicates and organic refractory in
+cometary dust. Thus, our model calculations represent the m ore realistic type of grains which may be
+considered as a mixture of silicates and carbonaceous mater ials. It is to be noted that the presence of
+negative polarization in the backscatter domain has been co mmonly attributed to silicates or dirty ice
+grains(Kimuraetal., 2006).
+It hasbeeninvestigatedthat the aggregatedust modelcan we ll fit the observedpolarizationdata of
+cometHyakutakewhenthesizeparameterofthemonomer, x∼1.56–1.70.Thusthesizerangesofthe6 H. S.Das, A.Suklabaidya, S.DattaMajumder & A. K.Sen
+-5 0 5 10 15 20 25
+ 60  80  100  120  140  160  180Polarization (in %)
+Scattering angleλ = 0.485 µm
+Fig.2Polarization values as observed at wavelength λ= 0.485µmfor comet Hyakutake by
+Joshi et al. (1997)and Kiselev & Velichko(1998).The solid c urverepresentsthe best-fitting
+polarization curve obtained for BCCA particles with 128 mon omers for a size distribution
+n(r)∼r−3for0.60µm≤r≤0.66µmatλ= 0.485µm.Heren= 1.743,k=0.100.
+monomerdifferfor three wavelengthswhich is unlikely.The proposedmodel can be furtherdeveloped
+ifwetakeamixtureofcompactspheroidalgrainsandaggrega tesoverawidesizerangewhichLasueet
+al.(2009)usedintheirpaper.TheystudiedcometHalleyand cometHale-Boppusingamixtureoffluffy
+aggregatesandcompactsolidgrainsandsuccessfullyexpla inedtheobservedpolarizationcharacteristics
+of two comets. In a follow-up paper, we also plan to model come tary dust as a mixture of aggregates
+andcompactparticles.
+6 CONCLUSIONS
+1. The size parameter of the monomer, x∼1.56 – 1.70, turned out to be most suitable which pro-
+vides the best fits to the observed polarization data of comet Hyakutake at three wavelengths
+λ= 0.365µm, 0.485µmand 0.684 µm. This correspond to 0.090µm≤am≤0.098µmat
+λ= 0.365µm;0.120µm≤am≤0.131µmatλ= 0.485µmand0.174µm≤am≤0.186µmat
+λ= 0.684µm.
+2. Thebestfitrefractiveindicescomingoutfromthepresent analysisare n= 1.745andk= 0.095for
+N= 128at λ= 0.365µm;n= 1.743andk= 0.100forN = 128at λ= 0.485µmandn= 1.695
+andk= 0.100for N = 128 at λ= 0.684µm. These values resemble the mixture of silicates and
+carbonaceouscompounds.
+3. The negative polarization values have been successfully generated for θ >1570at three wave-
+lengths.
+4. We plan a follow-up paper where computationswill be made c onsidering a mixture of aggregates
+andcompactspheroidalparticlesoverawide sizerangeofth epartcles.Polarizationproperties of comet C/1996 B2Hyakutake 7
+-5 0 5 10 15 20 25 30
+ 60  80  100  120  140  160  180Polarization (in %)
+Scattering angleλ = 0.684 µm
+Fig.3Polarization values as observed at wavelength λ= 0.684µmfor comet Hyakutake by
+Joshietal.(1997),Kiselev&Velichko(1998)andManset&Ba stien(2000).Thesolidcurve
+representsthebest-fittingpolarizationcurveobtainedfo rBCCAparticleswith128monomers
+for a size distribution n(r)∼r−3for0.88µm≤r≤0.94µmatλ= 0.684µm. Heren=
+1.695,k= 0.100.
+ACKNOWLEDGMENTS
+The authors HSD and AKS acknowledge Inter University Centre for Astronomy and Astrophysics
+(IUCAA), Pune for its associateship programme. The authors acknowledge T. Mukai and Y. Okada
+for help on the execution of BPCA and BCCA codes. The authors a re thankful to D. Mackowski, K.
+Fuller,andM.Mishchenko,whomadetheirsuperpositionT-m atrixcodepubliclyavailable.Theauthors
+highly acknowledge the referee of the paper for his valuable suggestions and comments for which the
+qualityofthe paperhasbeenimproved.
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\ No newline at end of file
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+arXiv:1001.0634v1  [math.RA]  5 Jan 2010On low-dimensional filiform Leibniz algebras
+and their invariants
+1I. S. Rakhimov and2Munther A. Hassan
+1,2Institute for Mathematical Research (INSPEM)
+1Department of Mathematics, FS,
+UPM, 43400, Serdang, Selangor Darul Ehsan, Malaysia,
+1risamiddin@gmail.com2munther abd@yahoo.com
+Abstract
+The paper deals with the complete classification of a subclas s of complex filiform Leibniz algebras
+in dimensions 5 and 6. This subclass arises from the naturall y graded filiform Lie algebras. We give a
+complete list of algebras. In parametric families cases, th e corresponding orbit functions (invariants)
+are given. In discrete orbits case, we show a representative of the orbits.
+2000 Mathematics Subject Classification: Primary 17A32, 17A60, 1 7B30; Secondary 13A50
+Key words: filiform Leibniz algebra, classification, invariant.
+1 Introduction
+Leibniz algebraswere introduced by J. -L. Loday [5]. A skew-symme tricLeibniz algebra is a Lie alge-
+bra. The main motivation of Loday to introduce this class of algebras was the search of an “obstruction”
+to the periodicity in algebraic K−theory. Besides this purely algebraic motivation, some relationships
+with classical geometry, non-commutative geometry, and physics have been recently discovered. The
+present paper deals with the low-dimensional case of a subclass of fi liform Leibniz algebras.
+Theoutlineofthepaperisasfollows. Section1isanintroductiontoth esubclassofLeibnizalgebras
+that we are going to investigate. The main results of the paper cons isting of complete classification of
+a subclass of low dimensional filiform Leibniz algebras are in Section 2. H ere, for 5- and 6-dimensional
+cases, we give lists of isomorphism classes. For parametric family cas es, the corresponding invariant
+functions are presented. Since the proofs in 6-dimensional case a re similar to those in 5-dimensional case,
+the detailed proofs are given for dimension 5 only.
+Definition 1.1. An algebra L over a field K is called a Leibniz algebra, if its bi linear operation [·,·]
+satisfies the following Leibniz identity: [x,[y,z]] = [[x,y],z]−[[x,z],y], x,y,z ∈L.
+Onward, all algebras are assumed to be over the fields of complex nu mbersC.
+LetLbe a Leibniz algebra. We put:
+L1=L, Lk+1= [Lk,L], k≥1.
+Definition 1.2. A Leibniz algebra L is said to be nilpotent, if there exists s∈N,such that
+L1⊃L2⊃...⊃Ls={0}.
+Definition 1.3. A Leibniz algebra Lis said to be filiform, if dimLi=n−i,wheren=dimLand
+2≤i≤n.
+The set of n−dimensional filiform Leibniz algebras we denote by Leibn.
+The following theorem from [3] splits Leibn+1into three disjoint subset.
+Theorem 1.1. Any(n+1)−dimensional complex filiform Leibniz algebra admits a basis {e0,e1,...,en}
+called adapted, such that the table of multiplication of the algebra has one of the following forms, where
+non defined products are zero:
+FLeib n+1=
+
+[e0,e0] =e2,
+[ei,e0] =ei+1, 1≤i≤n−1,
+[e0,e1] =α3e3+α4e4+...+αn−1en−1+θen,
+[ej,e1] =α3ej+2+α4ej+3+...+αn+1−jen, 1≤j≤n−2,
+α3,α4,...,αn,θ∈C.2 On low-dimensional filiform Leibniz algebras and their invariants
+SLeibn+1=
+
+[e0,e0] =e2,
+[ei,e0] =ei+1, 2≤i≤n−1,
+[e0,e1] =β3e3+β4e4+...+βnen,
+[e1,e1] =γen,
+[ej,e1] =β3ej+2+β4ej+3+...+βn+1−jen, 2≤j≤n−2,
+β3,β4,...,βn,γ∈C.
+TLeib n+1=
+
+[ei,e0] =ei+1, 1≤i≤n−1,
+[e0,ei] =−ei+1, 2≤i≤n−1,
+[e0,e0] =b0,0en,
+[e0,e1] =−e2+b0,1en,
+[e1,e1] =b1,1en,
+[ei,ej] =a1
+i,jei+j+1+···+an−(i+j+1)
+i,j en−1+bi,jen,1≤i < j≤n−1,
+[ei,ej] =−[ej,ei], 1≤i < j≤n−1,
+[ei,en−i] =−[en−i,ei] = (−1)ibi,n−ien,
+whereak
+i,j,bi,j∈Candbi,n−i=bwhenever 1 ≤i≤n−1,andb= 0 for even n.
+According to this theorem, the isomorphism problem inside of each cla ss can be studied separately.
+Theclasses FLeib n,SLeib nin lowdimensionalcaseshavebeenconsideredin[7], [8]. Thegeneralme thods
+of classification for Leibnhas been given in [1], [2] and [6] (1). This paper deals with the classification
+problem of low-dimensional cases of TLeib n.
+Observe that the class of n−dimensional filiform Lie algebras is in TLeib n.
+Definition 1.4. Let{e0,e1,...,en}be an adapted basis of L∈TLeib n+1.Then a nonsingular linear
+transformation f:L→Lis said to be adapted, if the basis {f(e0),f(e1),...,f(en)}is adapted.
+The set of all adapted elements of GLn+1forms a subgroup and it is denoted by Gad.The following
+proposition specifies elements of Gad.
+Proposition 1.1. Letf∈Gad.Thenfcan be represented as follows:
+f(e0) =e′
+0=n/summationdisplay
+i=0Aiei,
+f(e1) =e′
+1=n/summationdisplay
+i=1Biei,
+f(ei) =e′
+i= [f(ei−1),f(e0)],2≤i≤n,
+A0,Ai,Bj,(i,j= 1,...,n) are complex numbers and A0B1(A0+A1b)/ne}ationslash= 0.
+Proof.
+Since a filiform Leibniz algebra is 2-generated (see Theorem 1.1), it is s ufficient to consider the
+adapted action of fon the generators e0,e1:
+f(e0) =e′
+0=n/summationdisplay
+i=0Aiei, f(e1) =e′
+1=n/summationdisplay
+i=0Biei.
+Thenf(ei) = [f(ei−1),f(e0)] =Ai−2
+0(A1B0−A0B1)ei+n/summationdisplay
+j=i+1(∗)ej,2≤i≤n.
+Note that, A0/ne}ationslash= 0,andA1B0−A0B1/ne}ationslash= 0,otherwise f(en) = 0.The condition A0B1(A0+A1b)/ne}ationslash= 0
+appears naturally, since fis not singular.
+1Combined version of [1], [2] and [6] will appear in Communications in Algebra.I. S. Rakhimov, Munther A. Hassan 3
+Let now consider [ f(e1),f(e2)] =B0(A1B0−A0B1)e3+n/summationdisplay
+j=4(∗)ej.Then for the basis
+{f(e0),f(e1),...,f(en)}to be adapted B0(A1B0−A0B1) must be zero. However, according to the obser-
+vation above, A1B0−A0B1/ne}ationslash= 0.Therefore, B0= 0.
+2 The description of TLeibn,n= 5,6.
+2.1 5-dimensional case
+In this section we deal with the class TLeib 5.By virtue of Theorem 1.1. we can represent TLeib 5as
+follows:
+TLeib 5=
+
+[ei,e0] =ei+1, 1≤i≤3,
+[e0,ei] =−ei+1, 2≤i≤3,
+[e0,e0] =b0,0e4,
+[e0,e1] =−e2+b0,1e4,
+[e1,e1] =b1,1e4,
+[e1,e2] =−[e2,e1] =b1,2e4,
+b0,0,b0,1,b1,1,b1,2∈C.
+Further, the elements of TLeib 5will be denoted by L(α) =L(b0,0,b0,1,b1,1,b1,2).
+Theorem 2.1. (Isomorphism criterion for TLeib 5)Two algebras L(α)andL(α′)fromTLeib 5are
+isomorphic, if and only if there exist complex numbers A0,A1andB1:A0B1/ne}ationslash= 0and the following
+conditions hold:
+b′
+0,0=A2
+0b0,0+A0A1b0,1+A2
+1b1,1
+A3
+0B1, (1)
+b′
+0,1=A0b0,1+2A1b1,1
+A3
+0, (2)
+b′
+1,1=B1b1,1
+A3
+0, (3)
+b′
+1,2=B1b1,2
+A2
+0. (4)
+Proof.Part “ if ”. Let L1andL2fromTLeib 5be isomorphic: f:L1∼=L2.We choose the corresponding
+adapted bases {e0,e1,e2,e3,e4}and{e′
+0,e′
+1,e′
+2,e′
+3,e′
+4}inL1andL2,respectively. Then, in these bases the
+algebraswillbe presentedas L(α)andL(α′),whereα= (b0,0,b0,1,b1,1,b1,2),andα′= (b′
+0,0,b′
+0,1,b′
+1,1,b′
+1,2).
+According to Proposition 1.1 one has:
+e′
+0=f(e0) =A0e0+A1e1+A2e2+A3e3+A4e4, (5)
+e′
+1=f(e1) =B1e1+B2e2+B3e3+B4e4.
+Then we get
+e′
+2=f(e2) = [f(e1),f(e0)] =A0B1e2+A0B2e3+(A0B3+A1B1b1,1+(A2B1−A1B2)b1,2)e4,
+e′
+3=f(e3) = [f(e2),f(e0)] =A2
+0B1e3+(A2
+0B2−A0A1B1b1,2)e4,
+e′
+4=f(e4) = [f(e3),f(e0)] =A3
+0B1e4.
+By using the table of multiplications one finds the relations between th e coefficients
+b0,0,b0,1,b1,1,b1,2andb′
+0,0,b′
+0,1,b′
+1,1,b′
+1,2.First, consider the equality [ f(e0),f(e0)] =b′
+0,0f(e4),we
+get equation (1) and from the equality [ f(e1),f(e0)] + [f(e0),f(e1)] =b′
+0,1f(e4) we have (2) ,and
+[f(e1),f(e1)] =b′
+1,1f(e4) gives (3) .Finally, the equality (4) comes out from [ f(e1),f(e2)] =b′
+1,2f(e4).4 On low-dimensional filiform Leibniz algebras and their invariants
+“Only if” part.
+Let the equalities (1)–(4) hold. Then, the base change (5) above is adapted and it transforms L(α)
+intoL(α′).Indeed,
+[e′
+0,e′
+0] =/bracketleftBigg4/summationdisplay
+i=0Aiei,4/summationdisplay
+i=0Aiei/bracketrightBigg
+=A2
+0[e0,e0]+A0A1[e0,e1]+A0A1[e1,e0]+A2
+1[e1,e1]
+=/parenleftbig
+A2
+0b0,0+A0A1b0,1+A2
+1b1,1/parenrightbig
+e4=b′
+0,0A3
+0B1e4=b′
+0,0e′
+4.
+[e′
+0,e′
+1] =/bracketleftBigg4/summationdisplay
+i=0Aiei,4/summationdisplay
+i=1Biei/bracketrightBigg
+=−(A0B1e2+A0B2e3+(A1B1b1,1+A2B1b1,2−A1B2b1,2+A0B3)e4)
++B1(b0,1A0+2A1B1,1)e4
+=−e′
+2+A3
+0B1b′
+0,1e4=−e′
+2+b′
+0,1e′
+4.
+By the same manner one can prove that [ e′
+1,e′
+1] =b′
+1,1e′
+4,[e′
+1,e′
+2] =b′
+1,2e′
+4and the other products are
+zero.
+For the simplification purpose, we establish the following notation: ∆ = 4b0,0b1,1−b2
+0,1.
+Now, we list the isomorphism classes of algebras from TLeib 5.
+Represent TLeib 5as a disjoint union of the following subsets: TLeib 5=9/uniondisplay
+i=1Ui
+5,where
+U1
+5={L(α)∈TLeib 5:b1,1/ne}ationslash= 0, b1,2/ne}ationslash= 0};
+U2
+5={L(α)∈TLeib 5:b1,1/ne}ationslash= 0, b1,2= 0,∆/ne}ationslash= 0};
+U3
+5={L(α)∈TLeib 5:b1,1/ne}ationslash= 0, b1,2= ∆ = 0};
+U4
+5={L(α)∈TLeib 5:b1,1= 0, b0,1/ne}ationslash= 0, b1,2/ne}ationslash= 0};
+U5
+5={L(α)∈TLeib 5:b1,1= 0, b0,1/ne}ationslash= 0, b1,2= 0};
+U6
+5={L(α)∈TLeib 5:b1,1=b0,1= 0, b0,0/ne}ationslash= 0, b1,2/ne}ationslash= 0};
+U7
+5={L(α)∈TLeib 5:b1,1=b0,1= 0, b0,0/ne}ationslash= 0, b1,2= 0};
+U8
+5={L(α)∈TLeib 5:b1,1=b0,1=b0,0= 0, b1,2/ne}ationslash= 0};
+U9
+5={L(α)∈TLeib 5:b1,1=b0,1=b0,0=b1,2= 0}.
+Proposition 2.1.
+1. Two algebras L(α)andL(α′)fromU1
+5are isomorphic, if and only if/parenleftBigg
+b′
+1,2
+b′
+1,1/parenrightBigg4
+∆′=/parenleftbiggb1,2
+b1,1/parenrightbigg4
+∆.
+2. For any λfromC,there exists L(α)∈U1
+5:/parenleftbiggb1,2
+b1,1/parenrightbigg4
+∆ =λ.
+Proof. 1. “ ⇒”. LetL(α) andL(α′) be isomorphic. Then, due to Theorem 2.1 it is easy to see
+that/parenleftBigg
+b′
+1,2
+b′
+1,1/parenrightBigg4
+∆′=/parenleftbiggb1,2
+b1,1/parenrightbigg4
+∆.
+“⇐”. Let the equality/parenleftBigg
+b′
+1,2
+b′
+1,1/parenrightBigg4
+∆′=/parenleftbiggb1,2
+b1,1/parenrightbigg4
+∆ hold. Consider the base change (5) above withI. S. Rakhimov, Munther A. Hassan 5
+A0=b1,1
+b1,2, A1=−b0,1
+2b1,2andB1=b2
+1,1
+b3
+1,2.This changing leads L(α) intoL/parenleftBigg/parenleftbiggb1,2
+b1,1/parenrightbigg4
+∆,0,1,1/parenrightBigg
+.
+An analogous base change transforms L(α′) intoL
+/parenleftBigg
+b′
+1,2
+b′
+1,1/parenrightBigg4
+∆′,0,1,1
+.
+Since/parenleftBigg
+b′
+1,2
+b′
+1,1/parenrightBigg4
+∆′=/parenleftbiggb1,2
+b1,1/parenrightbigg4
+∆,thenL(α) is isomorphic to L(α′).
+2. Obvious.
+Proposition 2.2. The subsets U2
+5, U3
+5, U4
+5, U5
+5, U6
+5, U7
+5, U8
+5andU9
+5are single orbits with repre-
+sentatives L(1,0,1,0), L(0,0,1,0), L(0,1,0,1), L(0,1,0,0), L(1,0,0,1), L(1,0,0,0), L(0,0,0,1)and
+L(0,0,0,0),respectively.
+Proof.To prove it, we give the appropriate values of A0,A1andB1in the base change (as for other
+Ai, Bj, i,j= 2,3,4 they are any, except where specified otherwise).
+ForU2
+5:
+e′
+0=A0e0+A1e1+A2e2+A3e3+A4e4,
+e′
+1=B1e1+B2e2+B3e3+B4e4,
+e′
+2=A0B1e2+A0B2e3+(A1B1b1,1+A0B3)e4,
+e′
+3=A2
+0B1e3+A2
+0B2e4,
+e′
+4=A03B1e4,
+whereA0=∆1
+4√
+2, A1=−b0,1∆1
+4
+2√
+2b1,1andB1=∆3
+4
+2√
+2b1,1.
+ForU3
+5:
+e′
+0=A0e0+A1e1+A2e2+A3e3+A4e4,
+e′
+1=B1e1+B2e2+B3e3+B4e4,
+e′
+2=A0B1e2+A0B2e3+(A1B1b1,1+A0B3)e4,
+e′
+3=A2
+0B1e3+A02B2e4,
+e′
+4=A03B1e4,
+whereA0∈C∗, A1=−A0b0,1
+2b1,1andB1=A3
+0
+b1,1.
+ForU4
+5:
+e′
+0=A0e0+A1e1+A2e2+A3e3+A4e4,
+e′
+1=B1e1+B2e2+B3e3+B4e4,
+e′
+2=A0B1e2+A0B2e3+(A0B3+(A2B1−A1B2)b1,2)e4,
+e′
+3=A2
+0B1e3+(A2
+0B2−A1A0B1b1,2)e4,
+e′
+4=A03B1e4,
+whereA2
+0=b0,1, A1=−A0b0,0
+b0,1andB1=A2
+0
+b1,2.
+ForU5
+5:
+e′
+0=A0e0+A1e1+A2e2+A3e3+A4e4,
+e′
+1=B1e1+B2e2+B3e3+B4e4,
+e′
+2=A0B1e2+A0B2e3+A0B3e4,
+e′
+3=A2
+0B1e3+A2
+0B2e4,
+e′
+4=A3
+0B1e4,
+whereA2
+0=b0,1, A1=−b0,0/radicalbig
+b0,1andB1∈C∗.
+ForU6
+5:
+e′
+0=A0e0+A1e1+A2e2+A3e3+A4e4,
+e′
+1=B1e1+B2e2+B3e3+B4e4,
+e′
+2=A0B1e2+A0B2e3+(A0B3+(A2B1−A1B2)b1,2)e4,6 On low-dimensional filiform Leibniz algebras and their invariants
+e′
+3=A2
+0B1e3+(A2
+0B2−A1A0B1b1,2)e4,
+e′
+4=A03B1e4,
+whereA3
+0=b0,0b1,2, A1∈CandB1=b2
+3
+0,0
+b1
+3
+1,2.
+ForU7
+5:
+e′
+0=A0e0+A1e1+A2e2+A3e3+A4e4,
+e′
+1=B1e1+B2e2+B3e3+B4e4,
+e′
+2=A0B1e2+A0B2e3+A0B3e4,
+e′
+3=A2
+0B1e3+A2
+0B2e4,
+e′
+4=A3
+0B1e4,
+whereA0∈C∗, A1∈CandB1=b0,0
+A0.
+ForU8
+5:
+e′
+0=A0e0+A1e1+A2e2+A3e3+A4e4,
+e′
+1=B1e1+B2e2+B3e3+B4e4,
+e′
+2=A0B1e2+A0B2e3+(A0B3+(A2B1−A1B2)b1,2)e4,
+e′
+3=A2
+0B1e3+(A2
+0B2−A1A0B1b1,2)e4,
+e′
+4=A03B1e4,
+whereA0∈C∗, A1∈CandB1=A2
+0.
+ForU9
+5:
+e′
+0=A0e0+A1e1+A2e2+A3e3+A4e4,
+e′
+1=B1e1+B2e2+B3e3+B4e4,
+e′
+2=A0B1e2+A0B2e3+A0B3e4,
+e′
+3=A2
+0B1e3+A2
+0B2e4,
+e′
+4=A3
+0B1e4,
+whereA0,B1∈C∗andA1∈C.
+2.2 6-dimensional case
+This section is devoted to the description of TLeib 6. This class can be represented by the following
+table of multiplication:
+TLeib 6=
+
+[ei,e0] =ei+1, 1≤i≤4,
+[e0,ei] =−ei+1, 2≤i≤4,
+[e0,e0] =b0,0e5,[e0,e1] =−e2+b0,1e5,[e1,e1] =b1,1e5,
+[e1,e2] =−[e2,e1] =b1,2e4+b1,3e5,
+[e1,e3] =−[e3,e1] =b1,2e5,
+[e1,e4] =−[e4,e1] =−[e2,e3] = [e3,e2] =−b2,3e5.
+The elements of TLeib 6will be denoted by L(α),whereα= (b0,1,b1,1,b1,2,b1,3,b2,3).
+Theorem 2.2. (Isomorphism criterion for TLeib 6)Two filiform Leibniz algebras L(α)andL(α′)
+fromTLeib 6are isomorphic, iff there exist A0,A1,B1,B2,B3∈C:such that A0B1(A0+A1b2,3)/ne}ationslash= 0
+and the following equalities hold:I. S. Rakhimov, Munther A. Hassan 7
+b′
+0,0=A2
+0b0,0+A0A1b0,1+A2
+1b1,1
+A3
+0B1(A0+A1b2,3),
+b′
+0,1=A0b0,1+2A1b1,1
+A3
+0(A0+A1b2,3),
+b′
+1,1=B1b1,1
+A3
+0(A0+A1b2,3), (6)
+b′
+1,2=B1b1,2
+A2
+0,
+b′
+1,3=2A0A1B2
+1b2
+1,2+A2
+0B2
+1b1,3+/parenleftbig
+A2
+0/parenleftbig
+−2B1B3+B2
+2/parenrightbig
++A2
+1B2
+1b2
+1,2/parenrightbig
+b2,3
+A4
+0B1(A0+A1b2,3),
+b′
+2,3=B1b2,3
+A0+A1b2,3.
+The proof is the similar to that of Theorem 2.1.
+Represent TLeib 6as a union of the following subsets:
+U1
+6={L(α)∈TLeib 6:b2,3/ne}ationslash= 0, b1,1/ne}ationslash= 0};
+U2
+6={L(α)∈TLeib 6:b2,3/ne}ationslash= 0, b1,1= 0, b0,1/ne}ationslash= 0};
+U3
+6={L(α)∈TLeib 6:b2,3/ne}ationslash= 0, b1,1=b0,1= 0, b1,2/ne}ationslash= 0, b0,0/ne}ationslash= 0};
+U4
+6={L(α)∈TLeib 6:b2,3/ne}ationslash= 0, b1,1=b0,1= 0, b1,2/ne}ationslash= 0, b0,0= 0};
+U5
+6={L(α)∈TLeib 6:b2,3/ne}ationslash= 0, b1,1=b0,1=b1,2= 0, b0,0/ne}ationslash= 0};
+U6
+6={L(α)∈TLeib 6:b2,3/ne}ationslash= 0, b1,1=b0,1=b1,2=b0,0= 0};
+U7
+6={L(α)∈TLeib 6:b2,3= 0, b1,2/ne}ationslash= 0, b1,1/ne}ationslash= 0};
+U8
+6={L(α)∈TLeib 6:b2,3= 0, b1,2/ne}ationslash= 0, b1,1= 0, b0,1/ne}ationslash= 0};
+U9
+6={L(α)∈TLeib 6:b2,3= 0, b1,2/ne}ationslash= 0, b1,1=b0,1= 0, b0,0/ne}ationslash= 0};
+U10
+6={L(α)∈TLeib 6:b2,3= 0, b1,2/ne}ationslash= 0, b1,1=b0,1=b0,0= 0};
+U11
+6={L(α)∈TLeib 6:b2,3=b1,2= 0, b1,1/ne}ationslash= 0, b1,3/ne}ationslash= 0};
+U12
+6={L(α)∈TLeib 6:b2,3=b1,2= 0, b1,1/ne}ationslash= 0, b1,3= 0,∆/ne}ationslash= 0};
+U13
+6={L(α)∈TLeib 6:b2,3=b1,2= 0, b1,1/ne}ationslash= 0, b1,3= ∆ = 0};
+U14
+6={L(α)∈TLeib 6:b2,3=b1,2=b1,1= 0, b0,1/ne}ationslash= 0, b1,3/ne}ationslash= 0};
+U15
+6={L(α)∈TLeib 6:b2,3=b1,2=b1,1= 0, b0,1/ne}ationslash= 0, b1,3= 0};
+U16
+6={L(α)∈TLeib 6:b2,3=b1,2=b1,1=b0,1= 0, b0,0/ne}ationslash= 0, b1,3/ne}ationslash= 0};
+U17
+6={L(α)∈TLeib 6:b2,3=b1,2=b1,1=b0,1= 0, b0,0/ne}ationslash= 0, b1,3= 0};
+U18
+6={L(α)∈TLeib 6:b2,3=b1,2=b1,1=b0,1=b0,0= 0, b1,3/ne}ationslash= 0};
+U19
+6={L(α)∈TLeib 6:b2,3=b1,2=b1,1=b0,1=b0,0=b1,3= 0}.
+Proposition 2.3.
+1. Two algebras L(α)andL(α′)fromU1
+6are isomorphic, if and only if
+/parenleftBigg
+b′
+2,3
+2b′
+1,1−b′
+0,1b′
+2,3/parenrightBigg2
+∆′=/parenleftbiggb2,3
+2b1,1−b0,1b2,3/parenrightbigg2
+∆and
+/parenleftbig
+2b′
+1,1−b′
+2,3b′
+0,1/parenrightbig3b′3
+1,2
+b′2
+2,3b′4
+1,1=(2b1,1−b2,3b0,1)3b3
+1,2
+b2
+2,3b4
+1,1.
+2. For any λ1,λ2∈C,there exists L(α)∈U1
+6:
+/parenleftbiggb2,3
+2b1,1−b0,1b2,3/parenrightbigg2
+∆ =λ1,(2b1,1−b2,3b0,1)3b3
+1,2
+b2
+2,3b4
+1,1=λ2.
+Then orbits from the set U1
+6can be parameterized as L(λ1,0,1,λ2,0,1), λ1,λ2∈C.8 On low-dimensional filiform Leibniz algebras and their invariants
+Proposition 2.4.
+1. Two algebras L(α)andL(α′)fromU2
+6are isomorphic, if and only if/parenleftbig
+b′
+0,1−b′
+2,3b′
+0,0/parenrightbig4b′3
+1,2
+b′3
+2,3b′5
+0,1=(b0,1−b2,3b0,0)4b3
+1,2
+b3
+2,3b5
+0,1.
+2. For any λ∈C,there exists L(α)∈U2
+6:(b0,1−b2,3b0,0)4b3
+1,2
+b3
+2,3b5
+0,1=λ.
+Therefore orbits from U2
+6can be parameterized as L(0,1,0,λ,0,1), λ∈C.
+Proposition 2.5.
+1. Two algebras L(α)andL(α′)fromU7
+6are isomorphic, if and only if
+4b′
+0,0b′4
+1,2−2b′
+1,3b′
+0,1b′2
+1,2+b′2
+1,3b′
+1,1
+b′
+1,2b′2
+1,1=4b0,0b4
+1,2−2b1,3b0,1b2
+1,2+b2
+1,3b1,1
+b1,2b2
+1,1and
+/parenleftbig
+b′
+0,1b′2
+1,2−b′
+1,3b′
+1,1/parenrightbig2
+b′
+1,2b′3
+1,1=/parenleftbig
+b0,1b2
+1,2−b1,3b1,1/parenrightbig2
+b1,2b3
+1,1.
+2. For any λ1,λ2∈C,there exists L(α)∈U7
+6:
+4b0,0b4
+1,2−2b1,3b0,1b2
+1,2+b2
+1,3b1,1
+b1,2b2
+1,1=λ1,/parenleftbig
+b0,1b2
+1,2−b1,3b1,1/parenrightbig2
+b1,2b3
+1,1=λ2.
+The orbits from U7
+6can be parameterized as L(λ1,λ2,1,1,0,0), λ1,λ2∈C.
+Proposition 2.6.
+1. Two algebras L(α)andL(α′)fromU8
+6are isomorphic, if and only if/parenleftbig
+2b′
+0,0b′2
+1,2−b′
+1,3b′
+0,1/parenrightbig3
+b′3
+1,2b′4
+0,1=/parenleftbig
+2b0,0b2
+1,2−b1,3b0,1/parenrightbig3
+b3
+1,2b4
+0,1.
+2. For any λ∈C,there exists L(α)∈U8
+6:/parenleftbig
+2b0,0b2
+1,2−b1,3b0,1/parenrightbig3
+b3
+1,2b4
+0,1=λ.
+The orbits from the set U8
+6can be parameterized as L(λ,1,0,1,0,0), λ∈C.
+Proposition 2.7.
+1. Two algebras L(α)andL(α′)fromU11
+6are isomorphic, if and only if/parenleftBigg
+b′
+1,3
+b′
+1,1/parenrightBigg6
+∆′=/parenleftbiggb1,3
+b1,1/parenrightbigg6
+∆.
+2. For any λ∈C,there exists L(α)∈U11
+6:/parenleftbiggb1,3
+b1,1/parenrightbigg6
+∆ =λ.
+The orbits from U11
+6can be parameterized as L(λ,0,1,0,1,0),λ∈C.
+Proposition 2.8.
+The subsets U3
+6, U4
+6, U5
+6, U6
+6, U9
+6, U10
+6, U12
+6, U13
+6, U14
+6, U15
+6, U16
+6, U17
+6, U18
+6andU19
+6are single or-
+bits with representatives L(1,0,0,1,0,1), L(0,0,0,1,0,1), L(1,0,0,0,0,1), L(0,0,0,0,0,1), L(1,0,0,1,0,0),
+L(0,0,0,1,0,0), L(1,0,1,0,0,0), L(0,0,1,0,0,0), L(0,1,0,0,1,0), L(0,1,0,0,0,0), L(1,0,0,0,1,0),
+L(1,0,0,0,0,0), L(0,0,0,0,1,0)andL(0,0,0,0,0,0),respectively.I. S. Rakhimov, Munther A. Hassan 9
+3 Conclusion
+1. InTLeib 5,we distinguished nine isomorphism classes (one parametric family and e ight concrete)
+of three dimensional Leibniz algebras and shown that they exhaust all possible cases.
+2. In the case of TLeib 6,there are 19 isomorphism classes (five parametric families and 14 con crete)
+and they exhaust all possible cases.
+Remark. It should be pointed out that the filiform Lie algebras case is covered byU8
+5, U9
+5and
+U4
+6, U6
+6, U10
+6, U18
+6, U19
+6in fiveand six dimensional cases, respectively. Therefore, the list o ffiliform
+Lie algebras in the paper agrees with the list given in [4].
+Acknowledgments. The first named author is grateful to Prof. B. A. Omirov for helpfu l dis-
+cussions. The research was supported by the research grant 06 -01-04-SF0122 MOSTI(Malaysia). The
+authorsalsoaregratefultotherefereesfortheircriticalread ingofthemanuscript, forvaluablesuggestions
+and comments in the original version of this paper.
+References
+[1] U. D. Bekbaev and I. S. Rakhimov, On classification of finite dimens ional complex filiform Leibniz
+algebras (part 1), (2006), http://front.math.ucdavis.edu/, ArX iv:math. RA/01612805.
+[2] U. D. Bekbaev and I. S. Rakhimov, On classification of finite dimens ional complex filiform Leibniz
+algebras (part 2), (2007), http://front.math.ucdavis.edu/, arX iv:0704.3885v1 [math.RA] .
+[3] J. R. G´ omez and B. A. Omirov, On classification of complex filiform L eibniz algebras, (2006),
+http://front.math.ucdavis.edu/, ArXiv:0612735v1 [math.RA].
+[4] J. R. G´ omez, A. Jimenez-Merchan and Y. Khakimdjanov, Low-d imensional filiform Lie algebras,
+Journal of Pure and Applied Algebra ,130(1998), 133–158.
+[5] J. -L. Loday, Une version non commutative d´ es alg´ ebras de Lie : les alg´ ebras de Leibniz, L’Ens.
+Math.,39(1993), 269–293.
+[6] B. A. Omirov and I. S. Rakhimov, On Lie-like complex filiform Leibniz alg ebras,Bulletin of the
+Australian Mathematical Society ,79(2009), 391-404.
+[7] I. S. Rakhimov and S. K. Said Husain, On isomorphism classes and inv ariants of low-dimensional
+Complex filiform Leibniz algebras (Part 1), http://front.math.ucdav is.edu/, arXiv:0710.0121
+v1.[math RA] (2007). Linear and Multilinear Algebra (to appear).
+[8] I. S. Rakhimov and S. K. Said Husain, On isomorphism classes and inv ariants of low-dimensional
+Complex filiform Leibniz algebras (Part 2), http://front.math.ucdav is.edu/, arXiv:0806.1803v1
+[math.RA] (2008). Linear and Multilinear Algebra (to appear).
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+arXiv:1001.0635v2  [physics.atm-clus]  10 Jan 2010On the Nature of Optical Excitations in Hydrogenated Alumin ium Cluster Al 4H6: A
+Theoretical Study
+Sridhar Sahu and Alok Shukla
+Department of Physics, Indian Institute of Technology, Bom bay, Powai, Mumbai 400076, INDIA∗
+In this paper, we present a theoretical investigation of pho toabsorption spectrum of the newly
+synthesized hydrogenated cluster of aluminium, Al 4H6. The calculations are performed within the
+wave-function-based semi-empirical method employing the complete neglect of differential overlap
+(CNDO) model, employing a large-scale configuration intera ction (CI) methodology, and our re-
+sults are found to be in very good agreement with the earlier o nes obtained from the ab initio
+time-dependent density functional theory (TDDFT). We care fully analyze the many-particle wave
+functions of various excited states up to 8 eV, and find that th ey are dominated by single-particle
+band to band excitations. This is in sharp contrast to bare al uminium clusters, in general, and Al 4,
+in particular, whose optical excitations are plasmonic in n ature. We attribute this difference to be
+a consequence of hydrogenation.
+PACS numbers: 36.40.Vz, 31.10.+z, 31.15.bu, 31.15.vq
+I. INTRODUCTION
+Because of the future importance of hydrogen as a
+fuel, now-a-days a considerable amount of research ac-
+tivity is taking place in the field of hydrogen storage
+materials[1, 2]. This is because hydrogen is an extremely
+combustible gas, therefore, the preferred way to store it
+is in physically or chemically adsorbed forms. It is from
+this point of view that a large amount of theoretical and
+experimental research is taking place in the field of metal-
+lic hydrides. Among all possibilities, hydrides of group
+III elements, boron and aluminium, which are called bo-
+ranes and alanes, respectively, are considered to be strong
+candidates for the purpose[3–5]. It is with this aim in
+mind, that Li et al.[6] recently synthesized and studied
+the properties of a new alane, Al 4H6. The same group fol-
+lowed up this work, by performing an experimental and
+theoretical study of the electronic structure and prop-
+erties of closo-alanes, Al 4H4, AlnHn+2,4≤n≤8[7].
+Besides their possible applications, alanes are of inter-
+est from a fundamental point of view as well. Boron
+and aluminium are in the same group of periodic table
+with three valence electrons each, yet their properties are
+very different. In the bulk form boron is a semiconduc-
+tor, while aluminium is a metal. As far as the hydride
+chemistry is concerned, boron exhibits a huge variety of
+boranes with n-vertex polyhedral structures[8, 9], while
+the number and types of alanes is much smaller. The pio-
+neering works of Lipscomb[10] which provide the explana-
+tions of the cage structures of boranes using the concepts
+of three-centered two-electron(3c-2e) bonding, ultimate ly
+resulted in an ingenious molecular orbital theory known
+as polyhedral skeletal electron pair theory (PSEPT) or
+simply Wade-Mingos rule[11–13]. On the other hand
+only a small number of hydrides of aluminium have
+∗Electronic address: sridhar@phy.iitb.ac.in, shukla@phy .iitb.ac.inbeen investigated theoretically and experimentally such
+as molecules AlH[14–18], AlH 3[19], Al 2H4[20] Al 2H6[19–
+24], of the general type Al nHn+2[25, 26], and cage-like
+clusters such as Al 13H−m[27]. Fu et al.[26] in a recent
+comparative study investigated theoretically Al nHn+2
+clusters, and their borane analogues. Grubisic et al.[7]
+have tried to arrive at a Wade-Mingos type of rule set
+for aluminium hydrides, so that their structures could be
+predicted in a way similar to boranes, though Martinez
+and Alonso[25] have recently raised doubts on this sim-
+ilarity. With this background in mind, the recent study
+of Al 4H6,[6] a material whose structure is consistent with
+Wade-Mingos rule, has again revived the analogy of alu-
+minium hydrides with boranes. In this work we present
+an extensive study of the electronic structure and op-
+tical properties of Al 4H6, with the aim of predicting its
+photoabsorption spectrum, and to understand the nature
+of its optical excitations, which can be used for optical
+characterization of this substance. For the purpose, we
+use our recently developed CNDO/INDO Hamiltonian-
+based semiempirical multi-reference singles-doubles con -
+figuration interaction (MRSDCI) methodology described
+elsewhere[28–30]. To benchmark our approach, we also
+apply this approach to AlH molecule and demonstrate
+good agreement with the experimental results. Martinez
+and Alonso[25] calculated the photoabsorption spectra of
+several alanes of the type Al nHn+2including Al 4H6, us-
+ing the ab initio time-dependent density functional the-
+ory (TDDFT), and our results are found to be in very
+good agreement with their results[25]. Upon analyzing
+the many-particle wave functions of the excited states
+corresponding to important peaks in the spectrum of
+Al4H6, we conclude that all the way up to 8 eV the
+states correspond to inter-band excitations, and not to
+plasmonic excitations common in bare Al clusters.
+The remainder of this paper is organized as follows. In
+section II we briefly describe the theoretical methodology
+employed for the present calculations. This is followed by
+the presentation and discussion of our results in section
+III. Finally, in section IV we present our conclusions.2
+II. THEORY
+For our study, we adopted a wave-function based
+electron-correlated approach employing the semi-
+empirical valence-electron CNDO/2 model Hamiltonian
+developed by Pople and coworkers[31–33]. The method-
+ology adopted in this work is discussed in detail in
+our earlier papers[28–30], therefore, we present only a
+brief description of it here. As compared to our earlier
+works on boron-based clusters[29, 30] where we had
+used the INDO model, the choice of CNDO/2 model
+here has to do with the fact that INDO parameters are
+not available for the third row atoms like aluminium
+in the original Pople approach[34]. Our calculations
+are initiated at the Hartree-Fock (HF) level, within
+the CNDO/2 model, using a computer program de-
+veloped recently by us[28]. The CNDO-HF molecular
+orbitals (MOs) thus obtained, are used to transform the
+Hamiltonian from the original atomic-orbital (AO) to
+the MO representation, which is subsequently used in
+the post-HF correlated calculations. The transformed
+CNDO/2 Hamiltonian matrix elements in the MO repre-
+sentation are supplied to the computer program package
+MELD[35], which is used to perform the correlated
+calculations using the multi-reference singles-doubles
+configuration-interaction (MRSDCI) approach. Using
+the ground- and excited-state wave functions obtained
+from the MRSDCI calculations, electric dipole matrix
+elements are computed and subsequently utilized to
+compute the linear absorption spectrum, under the
+electric-dipole approximation, assuming a Lorentzian
+line shape. By analyzing the excited states contributing
+to the peaks of the computed spectrum obtained from
+a given calculation, bigger MRSDCI calculations are
+performed with a larger number of reference states.
+This procedure is repeated until the computed spectrum
+converges within an acceptable tolerance. In the past,
+we have used such an iterative MRSDCI approach on a
+number of conjugated polymers to perform large-scale
+correlated calculations of their linear and nonlinear
+optical spectra[36].
+III. RESULTS AND DISCUSSION
+In this section we present and discuss our results on
+the electronic structure and optical properties computed
+using our CNDO-MRSDCI approach. However, before
+that, to benchmark our methodology we present and dis-
+cuss our CNDO-CI results on the simplest hydride of
+aluminium, namely AlH.
+A. AlH Molecule
+First we performed the geometry optimization of the
+AlH molecule at the CNDO-HF level, by calculating the
+total energy of the system for different bond lengths, andthen locating the minimum. Our optimized bond length
+was found to be 1.75 Å which is 0.1 Å larger than the
+reported experimental value of 1.65 Å[14, 15]. However,
+our aim here is to compute the optically-active excited
+states of the molecule, and as we will show later that this
+much of difference in the geometry leads to insignificant
+differences in their excitation energies.
+In the CNDO/2 model Al has nine Slater-type basis
+functions (1s, 3p, 5d), while H has only one basis func-
+tion, leading to ten basis functions in all. Being a va-
+lence electron approach, the total number of electrons
+being explicitly considered for AlH within the CNDO/2
+model is four, with Al contributing three electrons, and
+H one. Therefore, with four electrons, and ten basis
+functions full-CI (FCI) calculations for the system are
+feasible, because in the singlet subspace the total num-
+ber of all possible configuration state functions (CSFs) is
+only 825. Thus, the FCI results presented here are exact
+within the chosen model (CNDO/2), and any disagree-
+ments with the experiments will indicate the deficiency
+of the model rather than that of the CI expansion.
+The CNDO-HF calculations predict the Mulliken pop-
+ulation of Al(H) atoms to be +0.24(-0.24), implying
+a partial ionic character to this system, with Al be-
+ing the electron donor, fully consistent with previous
+works[23]. The absorption spectrum of AlH is well-known
+and has been studied extensively over the years both
+experimentally[16] and theoretically[15, 17, 18]. The
+ground state of the system is classified as X1Σ+while the
+dipole connected excited states are A1Π,C1Σ+,D1Σ+,
+andE1Π. The first excited state A1Πcorresponds to
+HOMO(H)→LUMO(L) transition which is 5σ→2πin
+nature. The reported experimental value of this transi-
+tion is 2.91 eV, with which our calculations are in good
+agreement with the computed values of 3.11 eV, and
+3.12 eV at the bond lengths of 1.75 Å (our optimized
+bond length) and 1.65 Å (experimental bond length)
+respectively. This also shows that small differences in
+geometry have negligible impact on the calculated exci-
+tation energy of the lowest excited state. The Cand
+DΣ-bands correspond to Rydberg-type excitations and,
+therefore, cannot be computed using CNDO approach,
+because they require the use of extremely diffused basis
+functions[15]. The E1Πband corresponds to |H−1→L/angbracketright
+(4σ→2π) excitation, and has been measured close to
+6.9 eV[16], while our calculation predicts its value to be
+7.1 eV, thus overestimating it by 0.2 eV. We note that
+quantitatively speaking, the disagreement between our
+results, and the experimental ones, both for the A1Π,
+andE1Πbands is 0.2 eV, which is quite satisfying, given
+the semiempirical nature of this approach. Having bench-
+marked the CNDO-CI approach for the case of the AlH
+molecule, next we present the results of our calculations
+for Al 4H6.3
+B. Al 4H6
+We first carried out the geometry optimization of
+Al4H6within a density functional theory (DFT) based
+approach, using Gaussian03 computational package[37]
+employing B3LYP functional, and 6-311+g(d) basis set.
+The optimized geometry is presented in Fig.1, and it is
+virtually identical to that reported by Li et al.[6] As noted
+by Li et al.[6] also, this structure is very similar to the
+computed structure of the borane, B 4H6[38] which still
+has not been synthesized. We used this DFT optimized
+geometry to perform the calculations of the photoabsorp-
+tion spectrum within our CNDO-CI approach, as well as
+theab initio TDDFT method.
+.
+Figure 1: Optimized geometry of Al 4H6obtained from
+theab initio DFT calculations (see text for details). Each
+bond length (in Å) has also been indicated.
+Before discussing our calculated spectra, we present
+some of the molecular orbitals (MOs) of the system close
+to the Fermi level. In Fig. 2 MOs obtained from the
+CNDO-HF calculations are presented. Although, here
+we have not presented the ab initio B3LYP MOs which
+were used to perform the TDDFT calculations, we note
+that among the occupied orbitals, H−3,H−1, and
+Hobtained from the CNDO-HF and the DFT calcula-
+tions were qualitatively very similar. Mulliken charges
+of various atoms indicate Al to be in slightly cationic
+state, while the H atoms carry small negative charges.
+Therefore, we conclude that the bonding in the system is
+largely covalent, with some polar character. The covalent
+nature of the bonding is also obvious from the MO plots,
+which show that there is significant amount of charge
+density in the region between the atoms.
+Next, we present and discuss the linear optical ab-
+sorption spectrum of the system computed using the
+CNDO-CI and the ab initio TDDFT methods. The cal-
+culations were performed using the geometry shown in
+Fig. 1, which as mentioned above, was obtained at the
+(a) HOMO-3
+ (b) HOMO-2
+ (c) HOMO-1
+(d) HOMO
+ (e) LUMO
+ (f) LUMO+1
+ (g) LUMO+2
+(h) LUMO+3
+Figure 2: (Color online) Molecular orbitals (iso plots) of
+Al4H6from HOMO-3 to LUMO+3, obtained from the
+CNDO-HF calculations.
+B3LYP/6-311+g(d) level of theory. For the CNDO-CI
+calculations the total number of valence electrons was 18
+and the total number of basis functions was 42, thus rul-
+ing out the FCI calculations. Therefore, as mentioned
+in section II, the MRSDCI method was adopted to com-
+pute correlated wave functions and energies of the ground
+and the optically active excited states. The calculations
+were performed without the use of point-group symme-
+try, therefore, the CI expansion became fairly large scale,
+with the largest CI calculation consisting of 1.3 millions
+CSFs. The CI matrix was iteratively diagonalized[35]
+to obtain 60 lowest roots which required several hours of
+CPU time. The dipole moments connecting these excited
+states to the ground state were used to compute the pho-
+toabsorption spectrum presented here. Before discussing
+our final spectrum, in Fig. 3 we present the convergence
+of our optical absorption results with respect to the size
+of the CI matrix, represented in this case by the num-
+ber of reference configurations (N ref) from which singly-
+and doubly- excited CSFs were generated to obtain the
+MRSDCI expansion. For example, in our largest MRS-
+DCI calculation we used N ref=22 which generated 1.3
+million total CSFs. The convergence of our calculations
+with respect to N refis obvious from Fig. 3, in which
+the spectra corresponding to N ref=14 and N ref=22 are
+virtually indistinguishable.
+In Fig. 4, we present our final linear optical absorp-
+tion spectrum of Al 4H6computed using the CNDO-CI
+method. Before discussing our spectrum in detail, we4
+0 5 10 15
+E (eV)050100150200Intensity (arb. units)Nref  =1
+Nref  =14
+Nref  =22
+Figure 3: (Color online) Convergence of the linear
+absorption spectrum of Al 4H6computed using the
+CNDO-MRSDCI method with respect to the number of
+reference states used in MRSDCI calculations. A line
+width of 0.1 eV was used to compute the spectra.
+would like to benchmark it against other calculations to
+ensure its correctness. Therefore, to ascertain the ac-
+curacy of our CNDO-CI calculations, we also performed
+ab initio TDDFT calculation of its lowest photoexcited
+state, at the B3LYP/6-311+g(d) level of theory using the
+Gaussian03 program package[37]. Our CNDO-CI value
+of 2.98 eV (peak I in Fig.4), compares excellently with the
+TDDFT value of 3.03 eV, which gives us confidence as to
+correctness of our calculation. Martinez and Alonso[25]
+used an ab initio TDDFT methodology to compute the
+photoabsorption spectrum of Al 4H6all the way up to
+12 eV ( cf.Fig. 7 of Ref.[25]). The first peak in the
+TDDFT spectrum of Martinez and Alonso[25] is at an
+energy slightly higher than 3 eV, and their highest peak
+occurs at an energy close to 9 eV, thus making their spec-
+trum slightly blue-shifted compared to ours. However,
+the qualitative nature of their spectrum[25] is quite sim-
+ilar to our spectrum discussed below.
+Next, we examine the nature of excited states corre-
+sponding to the peaks present in our calculated spec-
+trum ( cf.Fig. 4). The important peaks in the spectrum
+up to the excitation energy ≈8 eV have been labeled,
+and the many-particle wave functions of these excited
+states, along with their ground-state transition dipole
+moments, are presented in table I. The first peak in
+Fig. 4 occurs at 2.98 eV, and is a relatively weak peak
+corresponding to the across the gap H→Ltransition.
+The next peak (II) located at 3.28 eV is also similarly
+weak and corresponds to H−1→Lexcitation. When
+we examine the nature of the many-particle wave func-
+tions ( cf. Table I) of various excited states, we realize
+that: (a) the wave functions of all the states consist pre-
+dominantly of singly-exited CSFs, (b) all the states have012 3 4 5678 9 10 1112
+E (eV)050100150Intensity (arb. units)
+IIIIIIIVVVIVIIVIII
+Figure 4: Linear optical absorption spectrum of Al 4H6,
+computed using the CNDO-MRSDCI approach. Peaks
+with energies up to 8 eV have been labeled. A line
+width of 0.1 eV was used to compute the spectrum.
+one dominant configuration with magnitude of its coef-
+ficient always greater than 0.8, with some configuration
+mixing with the increasing excitation energy. The high-
+est peak in the spectrum is located at 8.05 eV with the
+dominant CSF being |H−1→L+7/angbracketright, with some con-
+tributions from |H→L+ 9/angbracketrightand|H−2→L+ 3/angbracketright.
+Koutecký and coworkers[39] have formulated a criterion
+according to which if the many particle wave function of
+a given excited state is dominated by one singly-excited
+configuration, it is classified as a normal inter-band ab-
+sorption. On the other hand, if the wave function ex-
+hibits strong mixing of several CSFs with coefficients of
+almost equal magnitude, it is considered to be a plas-
+monic collective excitation[39]. In metallic clusters suc h
+as Al 4H6, to know whether the excited states are plas-
+monic in nature or not, is always of interest. As per the
+Koutecký [39] criterion, the excited states correspond-
+ing to peaks I to VIII do not appear to be of plasmonic
+type. This is to be contrasted with the nature of optical
+absorption in the bare Al 4rhombus cluster which were
+found to be of plasmonic type by Deshpande et al.[40] in
+anab initio time-dependent local-density approximation
+(TDLDA) calculation. In their work, authors reported
+that low-intensity optical absorption in Al 4begins close
+to 1 eV, and increases in intensity with energy peaking
+around 8 eV. This suggests that in Al 4H6, it is the in-
+fluence of hydrogenation which increases the optical gap,
+as well as changes the nature of excited states from plas-
+monic to inter-band excitations.
+IV. CONCLUSIONS
+In conclusion, we have preseneted the linear optical ab-
+sorption spectrum of Al 4H6calculated using a wave func-
+tion based methodology employing the CNDO-CI ap-5
+Table I: Excitation energies and many-particle wave
+functions of excited states corresponding to the peaks in
+the CNDO-CI linear absorption spectrum of Al 4H6(cf.
+Fig. 4), along with the squares of their dipole coupling
+(µ2=/summationtext
+i|/angbracketleftf|di|G/angbracketright|2) to the ground state. |f/angbracketrightdenotes
+the excited state in question, |G/angbracketright, the ground state, and
+diis thei-th Cartesian component of the electric dipole
+operator. In the wave function column, the numbers in
+the parentheses are the CI coefficients of a given
+electronic configuration. Symbols H/Ldenote
+HOMO/LUMO orbitals.
+Peak Energy (eV)/angbracketleftbig
+µ2/angbracketrightbig
+Wave function
+I 2.98 0.08 |H→L/angbracketright(0.9425)
+II 3.28 0.09|H−1→L/angbracketright(0.9423)
+III 4.93 0.02|H−1→L+2/angbracketright(0.9226)
+|H→L+1/angbracketright(0.1531)
+IV 5.66 0.03|H→L+3/angbracketright(0.9313)
+V 6.24 0.07|H−1→L+3/angbracketright(0.9027)
+|H−1→L+4/angbracketright(0.1609)
+|H→L+4/angbracketright(0.1056)
+VI 6.93 0.11|H→L+5/angbracketright(0.9249)
+|H→L+4/angbracketright(0.1242)
+VII 7.31 0.12|H−3→L/angbracketright(0.8391)
+|H−1→L+5/angbracketright(0.3062)
+|H−2→L+2/angbracketright(0.1600)
+VIII 8.05 0.44|H−1→L+7/angbracketright(0.8590)
+|H→L+9/angbracketright(0.2880)
+|H−2→L+3/angbracketright(0.1812)proach, and compared it with that computed using the ab
+initio TDDFT approach. The photoabsorption spectra
+computed using the two approaches are generally in good
+agreement with each other, with the published TDDFT
+spectrum[25] being slightly blue-shifted compared to our
+work. Therefore, it will be of considerable interest to per-
+form optical absorption experiments on Al 4H6cluster, to
+determine the nature of its optical excitations, and shed
+further light on its electronic structure. Our results can
+then be used for optical characterization of this cluster.
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\ No newline at end of file
diff --git a/1001.0636.txt b/1001.0636.txt
new file mode 100644
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+arXiv:1001.0636v1  [math.AP]  5 Jan 2010Global Existence and Increased Spatial Decay for the Radial
+Vlasov-Poisson System with Steady Spatial Asymptotics
+Stephen Pankavich
+Department of Mathematics
+Indiana University
+Bloomington, IN 47401
+sdp@indiana.edu
+Abstract
+A collisionless plasma is modelled by the Vlasov-Poisson sy stem in three space dimensions. A fixed
+background of positive charge, which is independent of time and space, is assumed. The situation in
+which mobile negative ions balance the positive charge as |x| → ∞is considered. Hence the total positive
+charge, total negativecharge, andtotal energy are infinite . Smooth solutions with appropriate asymptotic
+behavior for large |x|, which were previously shown to exist locally in time, are co ntinued globally for
+spherically symmetric data. This is done by showing that the charge density decays at least as fast as
+|x|−4. Finally, an increased decay rate of |x|−6is shown in the general case without the assumption of
+spherical symmetry.
+INTRODUCTION
+LetF:R3→[0,∞),f0:R3×R3→[0,∞), andA: [0,∞)×R3→R3be given. We seek a solution,
+f: [0,∞)×R3×R3→[0,∞) satisfying
+∂tf+v·∇xf−(E+A)·∇vf= 0,
+ρ(t,x) =/integraltext
+(F(v)−f(t,x,v))dv,
+E(t,x) =/integraltextρ(t,y)x−y
+|x−y|3dy,
+f(0,x,v) =f0(x,v).
+
+(1)
+HereFdescribes a number density of positive ions which form a fixed backgr ound, and fdenotes the density
+of mobile negative ions in phase space. Notice that if f0(x,v) =F(v) andA= 0, then f(t,x,v) =F(v) is a
+steady solution. Thus, we seek solutions for which f(t,x,v)→F(v) as|x| → ∞. Precise conditions which
+ensure local existence were given in [17]. It is important to notice tha t (1) is a representative problem, and
+that problems concerning multiple species of ions can be treated in a s imilar manner.
+The paper will be divided into two main results. The first will be devoted to showing global existence
+of a smooth solution to (1) in the case of spherically symmetric data. This serves to continue the local
+existence result of [17]. The second, then, is devoted to achieving a n increased rate of spatial decay of the
+charge density without the spherical symmetry assumption. We will assume throughout that Fhas compact
+velocity support, so that decay in vof the background is not an issue. Thus, the main difficulty in showing
+global existence arises in showing ρdecays rapidly enough in |x|. To see this, consider the following heuristic
+1argument, discussed in [17]. Let r=|x|. Then, we typically expect the electric field, E, to decay like r−2
+for large r. If we let g=F−f, then
+∂tg+v·∇xg−(E+A)·∇vg=−(E+A)·∇vF. (2)
+Viewing ( E+A)· ∇vFas a source term for g, we can only conclude r−2decay for both gandρ. But, if
+ρreally decayed like r−2, the integral for Ecould certainly fail to decay like r−2. In fact, we cannot show
+thatgdecays faster than r−2, but due to cancellation in the vintegral, ρmust decay faster.
+The Vlasov-Poisson system has been studied extensively in the case whenF(v) = 0 and solutions decay
+to zero as |x| → ∞. Smooth solutions were shown to exist globally in time in [13] and indepe ndently in [11].
+Important results prior to global existence appear in [1], [6], [7], and [1 0]. Also, the method used by [13] is
+refined in [8] and [15]. Global existence for the Vlasov-Poisson syste m in two dimensions was established in
+[12] and [18]. A complete discussion of the literature concerning Vlaso v-Poisson may be found in [4]. We
+also mention [2] since the problem treated in said paper is periodic in spa ce, and thus the solution does not
+decay for large |x|. The works cited above make extensive use of the laws of conserva tion of charge and
+energy. However, in the problem considered here (and those of [3], [9], [16], and [17]), the charge and energy
+are infinite, and it is less clear how to use the conservation laws. Ther efore, the use of conservation laws
+will be an important issue, and we will utilize a lemma (stated here as Lem ma 2) from [16] to deal with it
+properly.
+Preliminaries
+We will use notation which follows [17]. For q >7+√
+33, let
+p= 4−8
+q
+and denote
+R(x) =R(|x|) = (1+ |x|2)1
+2.
+We will use the norms
+/ba∇dblg/ba∇dbl∞= sup
+z∈Rn|g(z)|
+/ba∇dblρ/ba∇dblp=/ba∇dblρRp(x)/ba∇dbl∞,
+/ba∇dblg/ba∇dblq=/ba∇dblg(1+|x|2+|v|q)/ba∇dblL∞(R6)
+and
+|||g|||=/ba∇dblg/ba∇dblq+/ba∇dbl∇g/ba∇dblq+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay
+g dv/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+p,
+but never use LporLqfor finite pandq. We will write, for example, /ba∇dblg(t)/ba∇dblqfor the/ba∇dbl · /ba∇dblqnorm of
+(x,v)/ma√sto→g(t,x,v). Notice that we may take qarbitrarily large since we take the initial data to be of compact
+support.
+Following [16] and [17], we assume the following conditions hold for some C >0 and all t≥0,x∈R3,
+andv∈R3, unless otherwise stated :(I)F(v) =FR(|v|) is nonnegative and C2with,
+F′′
+R(0)<0. (3)
+In addition, there is W∈(0,∞) such that
+F′
+R(u)<0 foru∈(0,W)
+FR(u) = 0 for u≥W.
+
+(4)
+(II)f0isC1with compact v-support, nonnegative, and satisfies the condition of spherical s ymmetry,
+f0(x,v) =fR(|x|,|v|,x·v) (5)
+which is equivalent to
+f0(x,v) =f0(Ux,Uv)
+for every rotation U.
+Also, there is N >0 such that for |x|> N, we have
+f0(x,v) =F(v).
+(III)AisC1and
+|A(t,x)|+|∂xA(t,x)| ≤CR−2(x)
+|∇x·A(t,x)| ≤CR−4(x).
+Furthermore, Asatisfies the condition of spherical symmetry,
+A(t,x) =α(t,|x|)x
+|x|2(6)
+which is equivalent to
+A(t,x) =UTA(t,Ux)
+for every rotation U.
+Finally, we assume that there is a continuous function a: [0,T]→Rsuch that for |x|> N
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleα(t,|x|)−a(t)
+|x|/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤R2−p(x).
+It should be noted that the assumptions (5) and (6), imply the sphe rical symmetry of f(t,x,v),g(t,x,v),
+andE(t,x) for all t∈[0,∞),x∈R3, andv∈R3. Thus, where necessary we will write g(t,x,v) =
+g(t,|x|,|v|,x·v) when using the spherical symmetry of g. Furthermore, the spherical symmetry of Eandf
+will be instrumental in the first results of the paper, while the last th eorem is dedicated to eliminating the
+symmetry assumptions to conclude similar decay rates. We wish to pr ove the following :
+Theorem 1 Assuming conditions (I),(II), and(III)hold, there exists f∈ C1([0,∞)×R3×R3)that
+satisfies (1) with |||(F−f)(t)|||bounded on t∈[0,T], for every T >0. Moreover, fis unique.In [17], both local existence and a criteria for continuation of a boun ded, unique solution of (1) are shown.
+Thus, to prove Theorem 1 we will need to establish the continuation c riteria for all T >0. Specifically,
+Theorems 2 and 3 of [17], when combined, state :
+Theorem 2 Assumeq >7+√
+33and conditions (I),(II)and(III)hold, without (5) and (6). Let fbe a
+C1solution of (1) on [0,T]×R3×R3withT >0. If/ba∇dbl/integraltext
+(F−f)(t)dv/ba∇dblpis bounded on [0,T], then we may
+uniquely extend the solution to [0,T+δ]for some δ >0with|||(F−f)(t)|||bounded on [0,T+δ].
+Therefore, to prove Theorem 1, we will find a solution to (1) on [0 ,T] for some T >0 using the local
+existence theorem (again, see [17]), that has been uniquely extend ed using Theorem 2. Since this may be
+done as long as the p-norm stays bounded, we will only need to prove the following lemma to establish
+Theorem 1.
+Lemma 1 Assume q >7+√
+33and conditions (I),(II)and(III)hold. Let fbe aC1solution of (1) on
+[0,T]×R3×R3. Then,
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay
+(F−f)(t)dv/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+p≤C
+for allt∈[0,T], whereCis determined by F,A,f0, andT.
+Althoughit isnot explicitly statedin [17], the proofpresentedtheres howsthat δin Theorem2is bounded
+away from zero as long as /ba∇dbl/integraltext(F−f)(t)dv/ba∇dblpis bounded. Using this observation, Theorem 1 follows from
+Lemma 1.
+Once Lemma 1 has been established, we will show increased decay of t he charge density, ρ, under slightly
+modified assumptions. Instead of ( III), we will assume the following conditions for some C >0 and all
+t≥0,x∈R3, andv∈R3:
+(IV)AisC1with
+|A(t,x)| ≤CR−2(x), (7)
+|∂xA(t,x)| ≤CR−3(x), (8)
+and
+|∇x·A(t,x)|= 0. (9)
+Notice that condition ( IV) does not involve spherical symmetry of A, and thus the result which follows
+requires no such assumption.
+Theorem 3 Assume conditions (I)and(II)hold, without (5), and condition (IV)holds. Let T >0andf
+be theC1solution of (1) on [0,T)×R3×R3with
+|||(F−f)(t)|||<∞
+for every t∈[0,T). Then, we have/ba∇dblρ(t)/ba∇dbl6≤Cp,t
+for anyt∈[0,T), whereCp,tdepends upon
+sup
+τ∈[0,t]/ba∇dblρ(τ)/ba∇dblp.
+In Section 1 we will prove Lemma 1, and thus Theorem 1. Then, Sectio n 2 will contain the proof of
+Theorem 3. We will denote by “ C” a generic constant which changes from line to line and may depend up on
+f0,A,F, orT, but not on t,x, orv. When it is necessary to refer to a generic constant which depends
+upon other variables, we will use variable subscripts to distinguish th em. For example, we will use Cp,tquite
+frequently in Section 3, and it will always denote dependence upon /ba∇dblρ(t)/ba∇dblpandt. When it is necessary to
+refer to a specific constant, we will use numeric superscripts to dis tinguish them. For example, C(1)will
+always refer to the same constant.
+Section 1
+Define the characteristics X(s,t,x,v) andV(s,t,x,v) by
+∂X
+∂s(s,t,x,v) =V(s,t,x,v)
+∂V
+∂s(s,t,x,v) =−(E(s,X(s,t,x,v))+A(s,X(s,t,x,v)))
+X(t,t,x,v) =x
+V(t,t,x,v) =v.
+
+(10)
+Then, we have
+d
+dsf(s,X(s,t,x,v),V(s,t,x,v)) =∂tf+V·∇xf−(E+A)·∇vf= 0.
+Therefore, fis constant along characteristics, and
+f(t,x,v) =f(0,X(0,t,x,v),V(0,t,x,v)) =f0(X(0,t,x,v),V(0,t,x,v)). (11)
+Thus, we find by ( II) thatfis nonnegative and supx,vf=/ba∇dblf0/ba∇dbl∞<∞. Unless necessary, we will omit
+writing the dependence of X(s) andV(s) ont,x, andvfor the remainder of the paper.
+In order to bound the electric field, and thus the velocity support, we must use Lemma 3 and Theorem
+4 from [16]. In particular, we state the following lemma without proof.
+Lemma 2 Assuming conditions (I)and(III)hold, without (6), there exists k: [0,T]×R3→[0,∞)such
+that
+|ρ(t,x)| ≤C(k(t,x)3
+5+k(t,x)1
+2)
+and /integraldisplay
+k(t,x)dx≤C.Furthermore, notice that, due to the radial symmetry, we may wr ite
+E(t,x) =m(t,|x|)
+|x|2x
+|x|(12)
+where the enclosed charge is given by
+m(t,r) :=/integraldisplay
+|y|<rρ(t,y)dy.
+Lemma 2 will be exactly what is needed to bound the electric field, E. We have for any x∈R
+/integraldisplay
+|y|≤|x|k(t,y)1
+2dy≤/parenleftBigg/integraldisplay
+|y|≤|x|k(t,y)dy/parenrightBigg1
+2/parenleftBigg/integraldisplay
+|y|≤|x|dy/parenrightBigg1
+2
+≤/parenleftBigg/integraldisplay
+|y|≤|x|k(t,y)dy/parenrightBigg1
+2/parenleftbig
+C|x|3/parenrightbig1
+2
+≤C|x|3
+2
+and
+/integraldisplay
+|y|≤|x|k(t,y)3
+5dy≤/parenleftBigg/integraldisplay
+|y|≤|x|k(t,y)dy/parenrightBigg3
+5/parenleftbig
+C|x|3/parenrightbig2
+5
+≤C|x|6
+5.
+Now, we use the bounds on kto estimate the enclosed charge.
+|m(t,|x|)| ≤/integraldisplay
+|y|≤|x||ρ(t,y)|dy
+≤C/integraldisplay
+|y|≤|x|(k(t,y)3
+5+k(t,y)1
+2)dy
+≤C(|x|3
+2+|x|6
+5).
+Thus,
+|E(t,x)| ≤C|m(t,|x|)||x|−2
+≤C|x|−2(|x|6
+5+|x|3
+2)
+≤C(|x|−4
+5+|x|−1
+2).
+So,
+|E(t,x)| ≤CG(|x|),
+and combining this with ( III),
+|E(t,x)+A(t,x)| ≤CG(|x|)
+where
+G(r) :=/braceleftbigg
+r−4
+5, r≤1
+r−1
+2, r≥1.Next, we use methods from [7] to estimate the velocity support. As sume
+f(t,x,v) =f(0,X(0,t,x,v),V(0,t,x,v))/ne}ationslash= 0
+so that, using the compact support in voff0, we have
+|˙X(0)|=|V(0)| ≤C.
+For anyǫ >0, define a:= 2+ǫ,b:=a
+a−1, and
+B:= (2/integraldisplay∞
+1G(|x|)adx)1
+a.
+Lett1,t2∈[0,T] witht1< t2, and assume ˙Xi≥0 on [t1,t2].
+We claim/integraldisplayXi(t2)
+Xi(t1)G(|x|)dx≤B(Xi(t2)−Xi(t1))1
+b+10. (13)
+To show (13), let
+G1(x) :=/braceleftbigg
+|x|−4
+5if|x| ≤1
+0 if |x|>1
+and
+G2(x) :=/braceleftbigg0 if |x| ≤1
+|x|−1
+2if|x|>1.
+Then,
+/integraldisplayXi(t2)
+Xi(t1)G(|x|)dx=/integraldisplayXi(t2)
+Xi(t1)G1(x)dx+/integraldisplayXi(t2)
+Xi(t1)G2(x)dx
+≤/integraldisplay1
+−1G1(x)dx+/parenleftBigg/integraldisplayXi(t2)
+Xi(t1)(G2(x))adx/parenrightBigg1
+a
+(Xi(t1)−Xi(t2))1
+b
+≤10+/parenleftbigg
+2/integraldisplay∞
+1(G2(x)/parenrightbigga
+dx)1
+a(Xi(t1)−Xi(t2))1
+b
+≤10+B(Xi(t1)−Xi(t2))1
+b.
+This establishes (13). Now, using this result and following [7], we have
+|˙Xi(t2)2−˙Xi(t1)2|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/integraldisplayt2
+t1˙Xi(s)¨Xi(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤C/integraldisplayt2
+t1G(|X(s)|)˙Xi(s)ds
+≤C/integraldisplayt2
+t1G(|Xi(s)|)˙Xi(s)ds
+=:C(0)/integraldisplayXi(t2)
+Xi(t1)G(|x|)dx
+≤C(0)B(Xi(t2)−Xi(t1))1
+b+10C(0)
+≤C(0)B(( sup
+τ∈[t1,t2]|˙Xi(τ)|)(t2−t1))1
+b+10C(0).Define
+W:= sup
+s∈[0,T]|˙Xi(s)|.
+Then,
+|˙Xi(t2)2−˙Xi(t1)2| ≤C(0)B(Wt2)1
+b+10C(0). (14)
+Note that this holds if ˙Xi≤0 on [t1,t2], as well. Let us consider t∈[0,T] and˙Xi(t)>0. Define
+¯t:= inf{τ≥0 :˙Xi(s)≥0,∀s∈[τ,t]}.
+If¯t= 0, then by (14)
+˙X2
+i(t)≤˙X2
+i(0)+C(0)B(Wt)1
+b+10C(0)
+≤˙X2
+i(0)+C(0)B(WT)1
+b+10C(0).
+If¯t >0, then ˙Xi(¯t) = 0 and by (14), we have
+˙X2
+i(t)≤˙X2
+i(¯t)+C(0)B(Wt)1
+b+10C(0)
+=C(0)B(WT)1
+b+10C(0)
+≤ |˙X2
+i(0)|+C(0)B(WT)1
+b+10C(0).
+We may repeat this process for ˙Xi(t)<0, so summing over iyields
+|˙X(t)|2≤ |˙X(0)|2+3C(0)B(WT)1
+b+30C(0).
+Since the right-hand side is independent of t, we take the supremum and find,
+W2≤3C(0)BT1
+bW1
+b+M
+whereM:=|˙X(0)|2+30C(0). If 3C(0)BT1
+bW1
+b≤M, then
+W≤√
+2M=:C1(T).
+IfM≤3C(0)BT1
+bW1
+b, then
+W≤(6C(0)B)b
+2b−1T1
+2b−1:=C2(T).
+So, combining the inequalities
+W≤max{C1(T),C2(T)}.
+Thus, all characteristics V(s,t,x,v), along which fis non-zero, are bounded for any s∈[0,T], including
+V(t,t,x,v) =v. Define
+Qt:= sup{|v|:∃x∈R3, τ∈[0,t]such that f(τ,x,v)/ne}ationslash= 0}.
+Notice that Qis an increasing function of time, so that we may write for every s∈[0,T]
+|V(s,t,x,v)| ≤Qs≤QT.
+Since the momentum is bounded, bounds on position follow. Note that
+|X(s)−x| ≤/integraldisplayt
+s|V(τ)|dτ
+≤QTT.So, for|x| ≥2QTT,
+|X(s)| ≥ |x|−QTT≥1
+2|x| (15)
+and
+|X(s)| ≤ |x|+QTT≤3
+2|x|. (16)
+Hence, we have control on Xcharacteristics.
+Now, to bound the charge, we must first estimate the correspond ing density. We use (2) to find
+d
+ds(g(s,X(s),V(s))) =−(E(s,X(s))+A(s,X(s)))·∇vF(V(s)).
+Thus,
+g(t,x,v) =g(0,X(0),V(0))−/integraldisplayt
+0(E(τ,X(τ))+A(τ,X(τ)))·∇vF(V(τ))dτ (17)
+and by (III), (12), and (15),
+|g(t,x,v)| ≤ |g0(X(0,t,x,v),V(0,t,x,v))|
++/integraltextt
+0(|E(τ,X(τ,t,x,v))|+|A(τ,X(τ,t,x,v))|)|∇vF(V(τ,t,x,v))|dτ
+≤C|X(0,t,x,v)|−2+C/integraltextt
+0|X(τ,t,x,v)|−2/ba∇dbl∇vF/ba∇dbl∞dτ
++C/integraltextt
+0|m(τ,X(τ,t,x,v))| |X(τ,t,x,v)|−2/ba∇dbl∇vF/ba∇dbl∞dτ
+≤C|x|−2/parenleftBig
+1+/integraltextt
+0|m(τ,X(τ,t,x,v))|dτ/parenrightBig(18)
+for|x| ≥2QTT.
+To show that gdecays like r−2, we must bound the enclosed charge m. For|x| ≤2QTT,
+|m(t,|x|)| ≤/integraldisplay
+|y|≤2QTT|ρ(t,y)|dy
+≤(2QTT)3/integraldisplay
+|v|≤QT/ba∇dblF−f0/ba∇dbl∞dv
+≤CNow, let r=|x|and define M(t) := supr|m(t,r)|. We know by (2), (18), and the Divergence Theorem,
+|m(t,r)| ≤ |m(0,r)|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt
+0/integraldisplay
+|y|≤r/integraldisplay
+|v|≤QT∂s(g(s,y,v))dv dy ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=|m(0,r)|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt
+0/integraldisplay
+|y|≤r/integraldisplay
+|v|≤QTv·∇yg(s,y,v)dv dy ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤ |m(0,r)|+/integraldisplayt
+0/integraldisplay
+|y|=r/integraldisplay
+|v|≤QT|v| |g(s,|y|,|v|,y·v)|dv dSyds
+≤M(0)+C/integraldisplayt
+0/integraldisplay
+|y|=r/integraldisplay
+|v|≤QT|v|/parenleftBigg
+1+/integraltexts
+0M(τ)dτ
+|y|2/parenrightBigg
+dv dSyds
+=M(0)+C/integraldisplayt
+0Q4
+T(1+/integraldisplays
+0M(τ)dτ)ds
+≤C+C/integraldisplayt
+0M(τ)dτ
+forr≥2QTT. Then, since m(t,r)≤Cforr≤2QTT, we find, for all r,
+m(t,r)≤C+C/integraldisplayt
+0M(τ)dτ
+and consequently
+M(t)≤C+C/integraldisplayt
+0M(τ)dτ.
+By Gronwall’s Inequality,
+M(t)≤(M(0)+CT)(1+Ctexp(Ct))≤(M(0)+CT)(1+CTexp(CT))≤C. (19)
+Notice, then, that this bound and (18) imply
+|g(t,x,v)| ≤C
+|x|2
+for|x| ≥2QTT.
+Proceeding in the standard way, we next estimate the gradient of t he electric field.
+∂xiEk=∂
+∂xi/parenleftBigg
+xk
+|x|3/integraldisplay
+|y|≤|x|ρ(t,y)dy/parenrightBigg
+=∂
+∂xi/parenleftbiggxk
+|x|3/parenrightbigg
+m(t,|x|)+xk
+|x|3∂
+∂xi(m(t,|x|)).
+Then, /vextendsingle/vextendsingle/vextendsingle/vextendsingle∂
+∂xi/parenleftbiggxk
+|x|3/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingleδik
+|x|3−3xixk
+|x|5/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C
+|x|3
+and, letting r=|x|,/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂
+∂xim(t,r)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsinglemr(t,r)/parenleftBigxi
+r/parenrightBig/vextendsingle/vextendsingle/vextendsingle≤C|ρ(t,r)|r2.Thus,
+|∂xiEk| ≤ |m(t,r)|C
+r3+C|ρ(t,r)|r2
+r2
+≤C
+r3+C|ρ(t,r)|.
+Since the best we can do a priori is |ρ(t,r)| ≤Cr−2, we can only conclude
+|∂xiEk| ≤C|x|−2. (20)
+We begin to estimate the spatial decay of ρby first estimating the large |x|behavior of the field integral
+obtained by integrating (17) in v. Assume |x| ≥8QTTandf(t,x,v) is nonzero. Define
+E(t,x) =E(t,x)+A(t,x).
+Then,
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+|v|≤QTE(s,X(s))·∇vF(V(s))dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+|v|≤QTE(s,X(s))·(∇vF(V(s))−∇vF(v))dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
++/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+|v|≤QT(E(s,X(s))−E(s,x+(s−t)v))·∇vF(v)dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
++/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+|v|≤QT∇v·(F(v)E(s,x+(s−t)v))dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
++/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+|v|≤QTF(v)∇v·(E(s,x+(s−t)v))dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=:I+II+III+IV.
+By the Mean Value Theorem, (12), (15), and (19),
+I≤/integraltext
+|v|≤QT|E(s,X(s))| /ba∇dbl∇2
+vF/ba∇dbl∞|V(s)−v|dv
+≤/integraltext
+|v|≤QT(C|X(s)|−2/ba∇dbl∇2
+vF/ba∇dbl∞|/integraltextt
+sE(τ,X(τ))dτ|)dv
+≤CQ3
+T|x|−2(t−s)|x|−2
+≤C|x|−4.(21)
+To estimate II, we again use the Mean Value Theorem and (20), so that there is ξxbetween X(s) and
+x+(s−t)vwith
+|Ei(s,X(s))−Ei(s,x+(s−t)v)|=|∇xEi(s,ξx)·(X(s)−x−(s−t)v)|
+≤C|ξx|−2|X(s)−x+(t−s)v|.
+Thus, we find
+II≤C/integraldisplay
+|v|≤QT|ξx|−2|X(s)−(x+(s−t)v)| /ba∇dbl∇vF/ba∇dbl∞dv
+≤C/integraldisplay
+|v|≤QT|ξx|−2/parenleftbigg/integraldisplayt
+s|V(τ)−v|dτ/parenrightbigg
+dv
+≤CT2Q3
+T|x|−2|ξx|−2,and by (15),
+|ξx| ≥ |X(s)|−|X(s)−(x+(s−t)v)|
+≥1
+2|x|−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt
+s(V(τ)−v)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≥1
+2|x|−2TQT
+≥1
+4|x|.
+Therefore,
+II≤C|x|−4. (22)
+By the Divergence Theorem, we find
+III= 0. (23)
+Then, evaluating IVyields
+IV≤/integraldisplay
+|v|≤QT/ba∇dblF/ba∇dbl∞|∇x·E(s,x+(s−t)v)| |s−t|dv
+≤CT/integraldisplay
+|v|≤QT|ρ(s,x+(s−t)v)+∇x·A(s,x+(s−t)v)|dv
+≤C/integraldisplay
+|v|≤QT|ρ(s,x+(s−t)v)|dv+CR−4(x+(s−t)v).
+Since
+|x+(s−t)v| ≥ |x|−TQT≥1
+2|x|,
+we have
+IV≤C/integraldisplay
+|v|≤QT|ρ(s,x+(s−t)v)|dv+C|x|−4. (24)
+Finally, collecting (21), (22), (23), (24), we have for |x| ≥8QTT,
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+|v|≤QTE(s,X(s))·∇vF(V(s))dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|x|−4+C/integraldisplay
+|v|≤QT|ρ(s,x+(s−t)v)|dv. (25)
+Now, we can bound /ba∇dblρ(t)/ba∇dbl4. Using the bound on the velocity support, we have
+/ba∇dblρ(t)/ba∇dbl∞= sup
+x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+|v|≤QT(F(v)−f(t,x,v))dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤4π
+3Q3
+T/ba∇dblF−f0/ba∇dbl∞
+≤C.
+Thus, for |x|<8QTT, we have
+|ρ(t,x)| ≤ /ba∇dblρ(t)/ba∇dbl∞/parenleftbigg8QTT
+|x|/parenrightbigg4
+≤C|x|−4.Now, recall (17) :
+g(t,x,v) =g(0,X(0,t,x,v),V(0,t,x,v))−/integraldisplayt
+0E(s,X(s,t,x,v))·∇vF(V(s,t,x,v))ds.
+We use this equation and (25) so that for |x| ≥8QTT,
+|ρ(t,x)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+|v|≤QTg(t,x,v)dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤/integraldisplay
+|v|≤QT/parenleftbigg
+|g0(X(0),V(0))|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt
+0E(s,X(s))·∇vF(V(s))ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg
+dv
+≤CQ3
+T|x|−4+/integraldisplayt
+0/parenleftBigg
+CQ3
+T|x|−4+C/integraldisplay
+|v|≤QT|ρ(s,x+(s−t)v)|dv/parenrightBigg
+ds
+≤C|x|−4+C/integraldisplayt
+0/integraldisplay
+|v|≤QT|ρ(s,x+(s−t)v)|dv ds.
+So,
+|x|4|ρ(t,x)| ≤C+C/integraldisplayt
+0/integraldisplay
+|v|≤QT|x|4|ρ(s,x+(s−t)v)|dv ds.
+Define
+P(t) := sup
+x(|x|4|ρ(t,x)|).
+Then, for all x∈R3,
+P(t)≤C+C/integraldisplay
+|v|≤QT/integraldisplayt
+0P(s)ds dv
+≤C+C/integraldisplayt
+0P(s)ds
+and using the Gronwall Inequality, we find
+P(t)≤C
+and thus |ρ(t,x)| ≤C|x|−4for allx∈R3. Finally, we may apply Theorem 2 with any q >7 +√
+33, and
+since/ba∇dblρ(t)/ba∇dbl∞is bounded, we conclude that |ρ(t,x)| ≤CR−4(x). Since this estimate is independent of T,
+we find
+sup
+t∈[0,T]/ba∇dblρ(t)/ba∇dblp≤C.
+This remains true for any T >0, so the proof of Lemma 1, and thus Theorem 1, is complete.
+Section 2
+As in Section 1, we will show that the charge density decays at a fast er rate than previously known. In
+the work that follows, we will use the framework of the previous sec tions and assume conditions ( I) and
+(II) from Section 1, without the spherical symmetry of f0. However, we will not take condition ( III) as an
+assumption, and instead, assume condition ( IV) holds for some C >0 and all t≥0,x∈R3, andv∈R3.
+Since we have made a change in the assumptions, we may not use resu lts from the previous sections, unless
+otherwise stated. Thus, this section can be viewed independently f rom the others, as we will rely more
+on results shown previously in [17]. Recall from the introduction that we will use Cp,tto denote a genericconstant which depends upon /ba∇dblρ(t)/ba∇dblpandt.
+To begin, we apply Theorems 1 and 2 of [17], finding a unique f∈C1([0,T]×R3×R3) which satisfies
+(1) and assume
+|||(F−f)(t)|||<∞ (26)
+for allt∈[0,T). To first bound the velocity support, we write, as before,
+g(t,x,v) :=F(v)−f(t,x,v)
+and
+E(t,x) :=E(t,x)+A(t,x).
+Using Lemma 1 of [17], we find for all x∈R3andt∈[0,T),
+/integraldisplayt
+0|E(τ,x)|dτ≤Csup
+τ∈[0,t]/ba∇dblρ(τ)/ba∇dblp.
+Then, for any s∈[0,t),
+|V(s)−V(0)| ≤/integraldisplays
+0|E(τ,X(τ,t,x,v))|dτ
+≤Cp,s.
+Assuming f/ne}ationslash= 0, we know f(t,x,v) =f0(X(0),V(0)), and by ( II) it must follow that
+|V(0,t,x,v)| ≤C.
+Thus,|V(s)| ≤Cp,sfor anys∈[0,t). Following previous notation let us write
+Qt:= sup{|v|:∃x∈R3,τ∈[0,t) such that f(τ,x,v)/ne}ationslash= 0}
+so that for all s∈[0,t)
+|V(s)| ≤Qt. (27)
+This bound on the velocity establishes some of the relations shown in p revious sections. Most importantly,
+(15) and (16) must hold for |x| ≥2Qtt.
+Next, we denote the v-derivatives of characteristics by
+Aij:=∂Xi
+∂vj(s,t,x,v)
+and
+Bij:=∂Vi
+∂vj(s,t,x,v).
+Then, using the characteristic equations of (28),
+∂A
+∂s(s) =B(s),
+∂B
+∂s(s) =∇xE(s,X(s))A(s),
+A|s=t= 0,
+B|s=t=I.
+
+(28)
+Thus,
+|A(s)| ≤/integraldisplayt
+s|B(τ)|dτand
+|B(s)| ≤1+/integraldisplayt
+s|∇xE(τ,X(τ))| |A(τ)|dτ
+which leads to
+|A(s)|+|B(s)| ≤1+/integraldisplayt
+s(|B(τ)|+|∇xE(τ,X(τ))| |A(τ)|)dτ. (29)
+Again, using Lemma 1 from [17], we find
+|∇xE(τ,X(τ))| ≤Cp,τR−3(X(τ)). (30)
+So, define
+H(s) := sup
+{(x,v):|x|>2Qtt}(|A(s,t,x,v)|+|B(s,t,x,v)|). (31)
+Then, for |x|>2Qtt, we use (15) to find
+R−3(X(τ))≤R−3(Qtt),
+and thus, from (29)
+H(s)≤1+/integraldisplayt
+smax{1,Cp,τR−3(Qtt)} H(τ)dτ.
+Using the Gronwall Inequality, we find
+H(s)≤Cp,t (32)
+fors∈[0,t], and v-derivatives of both characteristics are bounded for |x|>2Qtt.
+Summarizing (28), we may write
+∂2A
+ds2(s) =∇xE(s,X(s))A(s);A|s=t= 0;∂A
+∂s|s=t=I.
+So,
+A(s) = ( s−t)I+/integraldisplayt
+s/integraldisplayt
+τ∇xE(λ,X(λ))A(λ)dλdτ
+=: (s−t)I+γ1(s,t,x,v).
+Again using (28),
+∂B
+∂s(s) =∇xE(s,X(s))A(s)
+=∇xE(s,X(s))((s−t)I+γ1(s,t,x,v))
+=: (s−t)∇xE(s,X(s))+γ2(s,t,x,v).
+Thus,
+B(s) =I+γ3(s,t,x,v)
+where
+γ3(s,t,x,v) =−/integraldisplayt
+s((τ−t)∇xE(τ,X(τ))+γ2(τ,t,x,v))dτ.
+Now,
+∂
+∂s(detB) = det
+˙B11˙B12˙B13
+B21B22B23
+B31B32B33
++det
+B11B12B13
+˙B21˙B22˙B23
+B31B32B33
++det
+B11B12B13
+B21B22B23
+˙B31˙B32˙B33
+
+=:I+II+III.Estimating the first term yields
+I= det
+(s−t)∂E1
+∂x1+(γ2)11(s−t)∂E1
+∂x2+(γ2)12(s−t)∂E1
+∂x3+(γ2)13
+(γ3)21 1+(γ3)22 (γ3)23
+(γ3)31 (γ3)32 1+(γ3)33
+
+=: (s−t)∂E1
+∂x1+(γ2)11+σ1(s,t,x,v).
+Similarly,
+II=: (s−t)∂E2
+∂x2+(γ2)22+σ2(s,t,x,v)
+and
+III=: (s−t)∂E3
+∂x3+(γ2)33+σ3(s,t,x,v).
+So, we find
+∂
+∂s(detB) = (s−t)∇x·E(s,x)|x=X(s)+3/summationdisplay
+j=1((γ2)jj+σj)
+and since B|s=t=I,
+detB= 1−4π/integraldisplayt
+s(τ−t)ρ(τ,X(τ))dτ−/integraldisplayt
+s3/summationdisplay
+j=1((γ2)jj+σj)dτ.
+Let
+ǫ(s,t,x,v) := 4π/integraldisplayt
+s(τ−t)ρ(τ,X(τ))dτ+/integraldisplayt
+s3/summationdisplay
+j=1((γ2)jj+σj)dτ.
+Then, for |ǫ|<1,
+1
+detB=1
+1−ǫ
+= 1+ǫ+/summationtext∞
+n=2ǫn
+=: 1+4 π/integraltextt
+s(τ−t)ρ(τ,X(τ))dτ+η(s,t,x,v).(33)
+Now that the determinant has been written in a nicer form, we estima te the remaining terms. Let
+|x|>2Qtt. Using (15), (26), (30), (31), and (32), we find the following boun ds on the error terms for any
+i,j= 1,2,3:
+|(γ1)ij| ≤Cp,tT2R−3(x) sup
+s∈[0,t]|A(s,t,x,v)|
+≤Cp,tR−3(x),
+|(γ2)ij| ≤Cp,tR−6(x),
+and
+|(γ3)ij| ≤Cp,tT2R−3(x)+Cp,tR−6(x)
+≤Cp,tR−3(x).Then, using the estimates of the γterms, for any k= 1,2,3,
+|σk| ≤C(|∇xE|+|γ2|)(2|γ3|+2|γ3|2)+2(|∇xE|+|γ2|)(|γ3|+2|γ3|2)
+≤[Cp,tTR−3(x)+Cp,tR−6(x)][2Cp,tR−3(x)+2Cp,tR−6(x)]
++ 2[Cp,tTR−3(x)+Cp,tR−6][Cp,tR−3(x)+2Cp,tR−6(x)]
+≤Cp,tR−6(x).
+Now, we may bound ǫ.
+|ǫ| ≤4πT2sup
+τ∈[0,t]/ba∇dblρ(τ)/ba∇dblpR−p(x)+3Tsup
+τ∈[s,t](|γ2(τ)|+|σ(τ)|)
+≤Cp,tR−p(x)+Cp,tR−6(x)
+≤Cp,tR−p(x)
+=:C(1)R−p(x)
+<1
+2
+for|x|>(2C(1))1
+p. Thus/summationtext∞
+n=2ǫnconverges and ηis well-defined for large enough values of |x|. Finally, we
+have, for |x|>max{2Qtt,(2C(1))1
+p},
+|η|=/vextendsingle/vextendsingle/vextendsingle/integraltextt
+s/summationtext3
+j=1((γ2)jj+σj)dτ+/summationtext∞
+n=2ǫn/vextendsingle/vextendsingle/vextendsingle
+≤3Cp,tTR−6(x)+((C(1))2R−2p(x)+(C(1))3R−3p(x)+...)
+≤Cp,tR−6(x).(34)
+Using (27), we find for t∈[0,T),
+/ba∇dblρ(t)/ba∇dbl∞= sup
+x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+(F(v)−f(t,x,v))dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤4π
+3Q3
+t/ba∇dblF−f0/ba∇dbl∞
+≤Cp,t.
+Thus, if|x| ≤Dfor some D >0, we have
+|ρ(t,x)| ≤ /ba∇dblρ(t)/ba∇dbl∞R6(D)R−6(x)
+≤Cp,tR−6(x).(35)
+Now, denote C(2):= max{2Qtt,(2C(1))1
+p,2N}, and let |x|> C(2). Then, by (15) we have
+|X(0,t,x,v)| ≥1
+2|x|> N.Using (V), (11), (33), and (34) to estimate ρ(t,x) yields
+/integraldisplay
+f(t,x,v)dv=/integraldisplay
+f0(X(0,t,x,v),V(0,t,x,v))dv
+=/integraldisplay
+F(V(0,t,x,v))dv
+=/integraldisplay
+F(V(0,t,x,v))(detB(0))1
+detB(0)dv
+=/integraldisplay
+F(V(0,t,x,v))(1+4π/integraldisplayt
+0(τ−t)ρ(τ,X(τ,t,x,v))dτ+η(0,t,x,v))(det∂V
+∂v(0,t,x,v))dv
+=/integraldisplay
+F(w)dw+/integraldisplay
+F(V(0,t,x,v))η(0,t,x,v)(det∂V
+∂v(0,t,x,v))dv
++4π/integraldisplay
+F(V(0,t,x,v))/integraldisplayt
+0(τ−t)ρ(τ,X(τ,t,x,v))dτ(det∂V
+∂v(0,t,x,v))dv.
+Sinceρ(t,x) =/integraltext
+(F(v)−f(t,x,v))dv, we have
+ρ(t,x) = 4π/integraltext
+F(V(0,t,x,v))/integraltextt
+0(t−τ)ρ(τ,X(τ,t,x,v))dτ(det∂V
+∂v(0,t,x,v))dv
+−/integraltext
+F(V(0,t,x,v))η(0,t,x,v)(det∂V
+∂v(0,t,x,v))dv.(36)
+Now, let
+Ψ(t) :=/ba∇dblρ(t)/ba∇dbl6= sup
+x(|ρ(t,x)|R6(x)).
+In order to make the change of variables w=V(0,t,x,v) in the v-integral, we must first show that the
+mapping v→V(0,t,x,v) is bijective. For the moment, we will take this for granted, and con tinue with the
+estimate of ρ, delaying the proof of this fact until the very end. By (34), (36), and the work of Section 3 .6,
+we have for |x|> C(2),
+|ρ(t,x)| ≤4π/integraltext
+F(V(0,t,x,v))(/integraltextt
+0(t−τ)Ψ(τ)R−6(X(τ,t,x,v))dτ)(det∂V
+∂v(0,t,x,v))dv
++Cp,tR−6(x)/integraltext
+F(w)dw
+≤CR−6(x)/integraltext
+F(V(0,t,x,v))(det∂V
+∂v(0,t,x,v))dv/integraltextt
+0(t−τ)Ψ(τ)dτ
++Cp,tR−6(x)
+≤R−6(x)(C/integraltextt
+0(t−τ)Ψ(τ)dτ+Cp,t).(37)
+So, applying (35) with D=C(2)and combining with (37), we have for all x,
+|ρ(t,x)|R6(x)≤Cp,t+C/integraldisplayt
+0(t−τ)Ψ(τ)dτ
+and since the right side is independent of |x|,
+Ψ(t)≤Cp,t+C/integraldisplayt
+0(t−τ)Ψ(τ)dτ.
+By the Gronwall Inequality,
+Ψ(t)≤Cp,t.
+Therefore, for every t∈[0,T),
+/ba∇dblρ(t)/ba∇dbl6≤Cp,t
+and the proof is complete.Change of Variables
+In order to justify the change of variables used in line (37) of the pr evious section, we must first demonstrate
+that the mapping v→V(0,t,x,v) is bijective. This is done below.
+From (30), we know there is C(3)
+p,t>0 such that
+|∇E(t,x)| ≤C(3)
+p,t|x|−3
+for every x∈R3,t∈[0,T]. Also, using (32), we know there is C(4)
+p,t>0 such that for all s,t∈[0,T],
+x,v∈R3, with|x|>2TQ(T),/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂X
+∂v(s,t,x,v)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(4)
+p,t.
+Finally, using Lemma 1 of [17], we know there is C(5)
+p,t>0 such that for all x∈R3andt∈[0,T],
+/integraldisplayt
+0|E(τ,x)|dτ≤C(5)
+p,t.
+For any D >0, define CD:= max{8/parenleftBig
+D+C(5)
+p,tT/parenrightBig
+,2TQ(T),4(6C(4)
+p,tC(3)
+p,tT)1
+3}andB(0,D) :={v∈R3:
+|v| ≤D}. Injectivity on B(0,D) can now be shown in the following lemma.
+Lemma 3 For any D >0,|x|> CD, andt∈[0,T], the mapping v→V(0,t,x,v)is injective on B(0,D).
+Proof:LetD >0,|x|> CDandt∈[0,T] be given. Then, let v1,v2∈B(0,D) be given with
+V(0,t,x,v 1) =V(0,t,x,v 2).
+We have
+v1+/integraldisplayt
+0E(τ,X(τ,t,x,v 1))dτ=v2+/integraldisplayt
+0E(τ,X(τ,t,x,v 2))dτ.
+So, using the Mean Value Theorem, for any i= 1,2,3 there is ξi
+xbetween X(τ,t,x,v 1) andX(τ,t,x,v 2) and
+ξi
+vbetween v1andv2such that
+Ei(τ,X(τ,t,x,v 1))−Ei(τ,X(τ,t,x,v 2)) =∇xEi(τ,ξi
+x)·(X(τ,t,x,v 1)−X(τ,t,x,v 2))
+and
+Xi(τ,t,x,v 1)−Xi(τ,t,x,v 2) =∇vXi(τ,t,x,ξi
+v)·(v2−v1).
+Then, since
+|ξi
+x| ≥ |X(τ,t,x,v 1)|−|X(τ,t,x,v 1)−X(τ,t,x,v 2)|
+≥1
+2|x|−/integraldisplayt
+τ|V(ι,t,x,v 1)−V(ι,t,x,v 2)|dι
+≥1
+2|x|−/integraldisplayt
+τ(|V(ι,t,x,v 1)−v1|+|v1−v2|+|V(ι,t,x,v 2)−v2|)dι
+≥1
+2|x|−/parenleftbigg
+|v1−v2|+2/integraldisplayt
+τ/integraldisplayι
+0/ba∇dblE(λ)/ba∇dbl∞dλ dι/parenrightbigg
+≥1
+2|x|−2/parenleftBig
+D+C(5)
+p,tT/parenrightBig
+≥1
+4|x|,and
+|Xi(τ,t,x,v 1)−Xi(τ,t,x,v 2)| ≤C(4)
+p,t|v2−v1|,
+for anyi= 1,2,3, we find
+|Ei(τ,X(τ,t,x,v 1))−Ei(τ,X(τ,t,x,v 2))| ≤ |∇ xEi(τ,ξi
+x)| |X(τ,t,x,v 1)−X(τ,t,x,v 2)|
+≤/parenleftbigg
+sup
+iC(3)
+p,t|ξi
+x|−3/parenrightbigg/parenleftBig√
+3C(4)
+p,t|v1−v2|/parenrightBig
+≤√
+3C(3)
+p,tC(4)
+p,t/parenleftbigg1
+4|x|/parenrightbigg−3
+|v1−v2|.
+Therefore, we have
+|v1−v2| ≤/integraldisplayt
+0|E(τ,X(τ,t,x,v 1))−E(τ,X(τ,t,x,v 2))|dτ
+≤3C(4)
+p,t|v1−v2|C(3)
+p,t/integraldisplayt
+0/parenleftbigg1
+4|x|/parenrightbigg−3
+dτ.
+≤(3C(3)
+p,tC(4)
+p,tT(1
+4|x|)−3)|v1−v2|
+≤1
+2|v1−v2|.
+Thus,|v1−v2|= 0, which implies v1=v2, and injectivity follows.
+/square
+Now, let t∈[0,T] andx∈R3be given. Define
+S:={w:F(w)/ne}ationslash= 0}
+and
+V−1(S) :={v:F(V(0,t,x,v))/ne}ationslash= 0}.
+Using the compact support of F, we conclude that v∈V−1(S) implies
+|v| ≤W+/integraldisplayt
+0/ba∇dblE(τ)/ba∇dbl∞dτ≤Cp,t.
+Therefore, there is a D >0, such that V−1(S)⊂B(0,D). Thus, for |x|> CD, andt∈[0,T], the mapping
+v→V(0,t,x,v) is injective on V−1(S) and bijective from V−1(S) toS. Finally,
+/integraldisplay
+F(V(0,t,x,v))det/parenleftbigg∂V
+∂v/parenrightbigg
+dv=/integraldisplay
+V−1(S)F(V(0,t,x,v))det/parenleftbigg∂V
+∂v/parenrightbigg
+dv
+=/integraldisplay
+SF(w)dw
+=/integraldisplay
+F(w)dw.
+Thus, the change of variables is valid and the justification is complete .References
+[1] Batt, J. (1977). Global symmetric solutions of the initial-value pr oblem of stellar dynamics, J. Diff. Eq.
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+[15] Schaeffer, J. (1991). Global existence of smooth solutions to the Vlasov-Poisson system in three dimen-
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+Pure Appl. Math. 33:173-197.
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+arXiv:1001.0637v2  [hep-ph]  20 Oct 2010WIS/16/09-DEC-DPPA
+Inverted Sparticle Hierarchies from
+Natural Particle Hierarchies
+O. Aharony, L. Berdichevsky, M. Berkooz,
+Y. Hochberg and D. Robles-Llana1
+Department of Particle Physics and Astrophysics,
+Weizmann Institute of Science,
+Rehovot 76100, Israel
+ABSTRACT
+A possible resolution of the flavor puzzle is that the fermion mass hier archy
+can be dynamically generated through the coupling of the first two g eneration
+fieldstoastronglycoupledsector, whichisapproximately conforma llyinvariant
+and leads to large anomalous dimensions for the first two generation fields over
+a large range of energies. We investigate the possibility of using the s ame
+sector to also break supersymmetry. We show that this automatic ally gives
+an “inverted hierarchy” in which the first two generation squarks a nd sleptons
+aremuch heavier thanthe other superpartners. Implementing th is construction
+genericallyrequiressomefine-tuninginordertosatisfytheconstr aintsonflavor-
+changing neutral currents at the same time as solving the hierarch y problem.
+We show that this fine-tuning can be reduced to be milder than the pe rcent
+level by making some technically natural assumptions about the for m of the
+strongly coupled sector and its couplings to the standard model.
+1Emails: ofer.aharony, leon.berdichevsky, micha.berkooz, yonit.hoc hberg, daniel.robles@weizmann.ac.il1 Introduction
+The standard model (SM), when embedded into a theory valid at ene rgy scales higher than
+a TeV, contains two types of unnaturally small couplings: the mass- squared of the Higgs
+field (the hierarchy problem), and many of the Yukawa couplings and mixing anglesin both
+the quark and lepton sectors (the flavor puzzle). Both issues can be addressed in various
+ways, andsomeofthemostelegant solutionsinvolve addingtotheth eoryadditionalsectors
+that decouple at some high energy scale.
+A natural solution to the hierarchy problem is supersymmetry (SUS Y), which to avoid
+fine-tuning must be broken softly at energies not much higher than the weak scale (see
+[1] for a review). This breaking cannot happen within the simplest sup ersymmetric gener-
+alization of the SM itself, and so an additional sector breaking super symmetry is usually
+required, as well as some mechanism for coupling the two sectors su ch that SUSY breaking
+is transmitted to the SM.
+The flavor puzzle can be addressed by introducing approximate hor izontal flavor sym-
+metrieswhichsuppress thesmall Yukawacouplings[2,3,4]. Another waytosolvetheflavor
+puzzle is to consider a strongly coupled and approximately conforma l sector [5, 6, 7, 8].
+The quarks and leptons of the first two generations can obtain larg e anomalous dimensions
+through direct couplings to this “conformal field theory (CFT) sec tor”.2If the conformal
+regime persists for a large range of energy scales, renormalization group (RG) evolution
+can lead to realistic hierarchical Yukawa couplings at low energies, st arting from an un-
+structured (anarchical) UV theory. The resulting Yukawa struct ures are similar to those
+obtained using horizontal symmetries.
+In principle the above solutions can be combined by making the “flavor CFT sector”
+supersymmetric, and some implications of this were discussed in [7, 9].3However, it is
+natural to wonder whether it may be possible to use the flavor CFT s ector also for super-
+symmetry breaking. This could allow for the construction of more ec onomical models with
+fewer additional sectors that have not yet been observed. Trad itionally models with less
+unobserved sectors have been preferred, following Occam’s razo r, though in the context of
+a complete theory of nature like string theory it is not clear if generic vacua, agreeing with
+all observations made so far, contain a small or a large number of se ctors. In this paper we
+will simply assume that the same sector plays both roles described ab ove – it generates the
+flavor hierarchy through large anomalous dimensions and it also brea ks supersymmetry –
+2We will refer to the sector responsible for generating the flavor hie rarchy as the CFT sector, even
+though its dynamics is only approximately conformal in some range of energies.
+3As discussed in [7, 9], if one assumes that the SUSY breaking is transm itted to the flavor CFT sector
+above the scale where this sector is superconformal, the RG flow of the CFT sector naturally suppresses
+also SUSY-breaking flavor-violating terms, so it helps in solving the su persymmetric flavor problem.
+1and we will analyze the phenomenological implications of this assumptio n.
+Our assumption, however, cannot be taken at face-value since it im mediately leads to
+a severe tension between an appropriate suppression of flavor ch anging neutral currents
+(FCNCs) and naturalness of the Higgs sector. On the one hand, th e dynamical scale of our
+CFT sector determines the scale of SUSY breaking, and this cannot be too large without
+introducing fine-tuning of the Higgs mass (for instance, the super symmetric partner of
+the top quark should not be much heavier than a TeV or so [10, 11, 12 ]). On the other
+hand, and unlike models with flavor-universal SUSY breaking, in our m odels the first two
+generations must directly couple to theCFT sector in aflavor non-u niversal way inorder to
+generate the flavor hierarchy. Generic non-universal flavor dyn amics leads to large FCNCs
+unless it decouples fromthe standard model fields at a rather heav y scale; the experimental
+constraints on FCNCs place lower bounds on the scale of flavor-viola ting new physics that
+are of the order of tens of thousands of TeVs [13, 14]. In general, an even more serious
+problem wouldarise fromthe experimental boundsonCPviolation[13, 14]. Inouranalysis
+in most of this paper we consider the bounds from CP-conserving pr ocesses; in the final
+section we discuss the implications of the stronger bounds from CP v iolation.
+The above fine-tuning problem does not have to be faced in its most s evere form,
+since in our models there is always some built-in hierarchy between the scale of the stop
+mass and the scale of flavor-violating effects involving the first two g enerations, to which
+the strongest FCNC bounds apply. The superpartners of the firs t two generations couple
+directly to the CFT sector and obtain masses of the order of the SU SY breaking scale.
+The existence of these direct couplings implies that the CFT sector m ust have a symmetry
+group (which is a global symmetry group from the point of view of the CFT sector) that
+contains the SM gauge group as a subgroup. As we do not allow direct couplings for them
+(whichwoulddestroytheflavor hierarchy), thethirdgenerations calars, gauginosandHiggs
+sector fields will obtain SUSY breaking masses mostly through gauge -mediated couplings
+to the CFT sector [15, 16]. These masses will then be suppressed by an extra small
+factor from SM gauge couplings. Such inverted hierarchy models [17 , 18, 19], where the
+first two generation scalars are relatively heavy, have been consid ered before as a possible
+resolution of the supersymmetric flavor problem, from a variety of different perspectives
+[20, 21, 22, 23, 24, 25, 26, 27]. In our models this feature is an auto matic outcome of our
+assumptions.
+Unfortunately, the natural inverted hierarchy obtained from th e above considerations
+is still not large enough to reduce the fine-tuning to acceptable leve ls. One reason for this
+is that our CFT sector generally contains many fields charged under the SM gauge group,
+so that in the language of gauge mediation we have a large effective nu mber of messengers
+[28], reducing the suppression of the gauge-mediated superpartn er masses compared to
+2the directly-mediated masses. Another issue to address is a gener ic problem in inverted
+hierarchy models, where the two-loop RG flow often leads to negativ e masses-squared for
+some of the third generation sfermions, which must be avoided [29, 3 0, 31].
+We are thus led to make several additional assumptions. We first re quire that the
+CFT sector does not directly couple the first generation and the se cond generation fields
+(“separable CFT sectors”). This can be accomplished either by hav ing a “decomposable
+CFT sector”, in which the fields coupled to the first generation do no t couple to the
+second and vice versa (implying a factorization of the CFT sector ga uge group), or by
+introducing some horizontal-type symmetry which suppresses cou plings between the first
+two generations. The assumption of separable CFT sectors plays t wo important roles:
+•It helps to suppress FCNCs. Naively FCNCs are completely absent in s uch a setup,
+butonemustkeepinmindthatthedecouplingbetweenthegeneratio nsdiscussedhere
+is in the interaction basis, so in the mass basis some flavor-non-diago nal couplings
+suppressed by powers of the mixing angles between the first two ge nerations are
+always present. (For quarks, these are typically of order the Cab ibbo angle ∼0.22
+or smaller.) Thus, separability helps alleviate the fine-tuning problem, but does not
+solve it completely.
+•Ithelpstosolvethe“gracefulexit”probleminmodelsofdynamically- generatedflavor
+hierarchies as in [6, 7]. It was emphasized there that the dynamics re sponsible for the
+breaking of the conformal symmetry and the superconformal U(1)Rsymmetry of the
+CFT sector should not generate a large mixing in the kinetic terms of t he first two
+generation fields, since this would wash out the hierarchy generate d between their
+Yukawa couplings in the conformal range of energies. The assumpt ion of separable
+CFTsectorsautomaticallysolvesthisproblem, asitimpliesthatthere isnoadditional
+source of mixing between the first two generation fields.
+Even if the CFT sector is separable, we are still left with a rather larg e amount of fine-
+tuning, so we then consider two additional ways to reduce this in our models. First, we
+assume the scale of SUSY breaking√
+Fto be much smaller than the scale Mat which the
+SM fields decouple from the CFT sector. This hierarchy can be achiev ed – as in traditional
+models of gaugemediation [32, 33] –if the fields inthe CFT sector coup ling to thestandard
+model fields have a mass which is larger thanthat of other fields in the CFT sector involved
+in the SUSY breaking, or by some other mechanism which suppresses the SUSY breaking
+scale [34]. This allows us to raise the scale Mat which generic flavor-violating operators
+are generated, though the fact that in these models the first two generation sfermions
+are much lighter than the scale Mimplies that one must separately worry about flavor-
+violating effects involving sfermion loops [35, 36]. Second, we assume t hat only some of
+3the first two generation fields directly couple to the CFT sector, wh ile others do not; in
+particular we consider models where only the left-handed superfield s couple directly to the
+CFT sector, or only the right-handed superfields, or only the supe rfields which are in the
+10representation of SU(5) (in the language of SU(5) grand unified (GUT) models). Such
+partial couplings are natural in brane realizations of the standard model. We are then able
+to suppress the strongest flavor-violating effects coming from ch irality mixing operators in
+the down sector, and still solve the flavor puzzle by generating larg e anomalous dimensions
+for those fields which do couple directly to the CFT sector.
+After making all the assumptions mentioned above, we find that only a small fine-
+tuning (milder than the percent level) is necessary in order to avoid F CNCs and solve the
+hierarchy problem, taking all couplings and correlation functions of the CFT sector to be
+otherwise generic. While the CFT sector must obey many requiremen ts, all of these are
+technically natural and could arise in the context of some complete t heory of high-energy
+physics. Finding an explicit model satisfying all these conditions is com plicated, and we
+do not consider it here, but there is no apparent obstruction to su ch constructions.4
+The outline of this paper is the following. We begin in Section 2 by reviewin g how
+flavor hierarchies can be generated by a CFT, as well as the experim ental constraints on
+FCNCs. In Section 3 we present a general discussion of our models in which all of the
+first two generation fields are coupled to the CFT sector, and we int roduce the notion of
+a separable CFT sector. We briefly consider one-scale models, F∼M2, and then move to
+a more detailed analysis of two-scale models, F≪M2. In Section 4 we discuss partially
+coupledscenarios, whichresult inlessfine-tuning, andpresent som eexamplesofthespectra
+of superpartners arising from models of this type. We end in Section 5 with a summary of
+our results and conclusions.
+2 Review
+In this section we review the two main tools of our constructions. Se ction 2.1 reviews the
+Nelson-Strassler mechanism, namely how a CFT sector can explain th e intergenerational
+mass hierarchy. In Section 2.2 we outline the main experimental FCNC bounds derived
+fromgeneraldimension-sixflavor-violatingoperatorsandfromdia gramswithsquark-gluino
+boxes. These will be basic constraints on all of our models.
+4See [37] for recent examples of superconformal models with dynam ical SUSY breaking, and [38] in the
+context of metastable SUSY breaking.
+42.1 CFTs and flavor hierarchies
+2.1.1 The Nelson-Strassler Mechanism
+In [6], Nelson and Strassler considered a mechanism that dynamically g enerates the IR
+hierarchical structure of the minimal supersymmetric standard m odel (MSSM) Yukawa
+couplings starting from a flavor anarchical UV theory.5In these models, short-distance
+Yukawa couplings are arbitrary order one matrices, and the patte rn of masses and mixing
+angles at low energies arises as the result of renormalization group fl ow.
+The basic idea is the following. Unlike in asymptotically free theories in th e weak cou-
+pling regime, where dimensionless couplings have logarithmic running, in a nearly scale-
+invariant theory such couplings can have power law running. Furthe rmore, if the theory is
+not weakly coupled, the anomalous dimensions of such couplings can b e significant. Fol-
+lowing an initial idea put forward in [5], the Nelson-Strassler mechanism is the application
+of this idea to the Yukawa couplings. When light quarks and leptons ha ve direct couplings
+to astrongly coupled conformal sector, they can acquire substa ntial anomalous dimensions.
+As a result, a large Yukawa hierarchy can be generated if the confo rmal regime persists
+over a wide enough range of energy scales, and if the anomalous dime nsions of these fields
+are appreciable.
+We will assume below that the fixed point theory, near which the RG flo w occurs, has
+N= 1 SUSY. In this case the anomalous dimensions, and hence the IR Yu kawa couplings,
+are determined by the U(1)Rcharges of the fixed point theory [39]. This U(1)Ris approx-
+imate and may be accidental from the UV point of view. The resulting m asses and mixing
+angles resemble those predicted in models with horizontal Abelian sym metries, but they
+differ from these models in that the R-symmetry which sets the char ges arises dynamically
+and accidentally. Furthermore, the characteristic ratio between different Yukawas is not
+set by a vacuum expectation value (VEV) of a field, but by the numbe r of energy decades
+spent near the fixed point.
+To be more concrete and to set the notation, we consider an N= 1 supersymmetric
+theory with gauge group ˜G, charged matter Q, neutral matter Xand a superpotential
+W(X,Q), and we will assume that it flows to a conformal theory in the infrar ed. Here Q
+is not to be confused with the SM quarks, which are CFT sector single ts and are included
+in theXfields.
+By unitarity all gauge-invariant operators (except the identity op erator) have scaling
+dimensions dim[ O(Q,X)]≥1 [40], with a strict inequality when Ois not a free field. As a
+consequence, Qmay have a negative anomalous dimension but X, being gauge-invariant,
+5We assumeforsimplicity that the SUSYgeneralizationofthe standar dmodel containsonlythe MSSM.
+Our considerations are easily generalized when additional fields are p resent.
+5always has anomalous dimension γX= 2[dim[ X]−1]≥0. Consider now superpotential
+terms of the form ck,OXkO(Q), whereO(Q) is a non-trivial ˜G-invariant operatorbuilt from
+the charged fields Q, which therefore has dim[ O(Q)]>1. At or near the superconformal
+fixed point, most ofthese terms areirrelevant since dim[ XkO(Q)]> k+1, andthus if k≥2
+the coupling ck,Owill flow to zero. Conversely, relevant superpotential terms can d rive the
+theory to a fixed point in which fields of type Xpossess positive anomalous dimensions.
+We will be interested in terms of the form XO(Q) (for dim[ O]<2), which can exist at a
+non-trivial fixed point when 0 < γX<2.
+If the superconformal sector has a global symmetry G– in which the standard model
+gauge group GSM≡SU(3)c×SU(2)L×U(1)Ycan be embedded – one can consider the
+following scenario. The effective theory has gauge group ˜G׈G, with˜Gstrongly coupled
+and an appropriate subgroup ˆGofG, which contains GSM, weakly gauged. As far as ˜G
+is concerned the SM quarks and leptons Li,Rj(i,j= 1,2) of the first two generations are
+of typeXabove – they are charged under Gbut neutral under ˜G, and they can directly
+couple in the superpotential to charged matter of ˜Gby couplings of the form W=XO.
+In this case, since they are X-type fields, the conformal dynamics associated to ˜Gmay
+cause them to develop large positive anomalous dimensions. These will in turn cause their
+standard model Yukawa couplings W=yijLiHRj(where we canonically normalize the
+fields) to run as6
+yij(µ) =yij(µ0)/parenleftbiggµ
+µ0/parenrightbigg1
+2[γ(Li)+γ(Rj)]
+. (2.1)
+We can now suppose that all Yukawa couplings are unstructured an d of order one at
+some high scale, say the Planck or string scale M0. If the gauge couplings of ˜Gand the
+direct couplings become conformal near some scale µ0=M>and remain so until some
+lower scale µ=M<, withǫ≡M</M>≪1, then the standard model Yukawa couplings
+will run down to
+yij(M<)∼/parenleftbiggM<
+M>/parenrightbigg1
+2(γLi+γRj)
+=ǫ1
+2(γLi+γRj)≡ǫLiǫRj. (2.2)
+We assume here (this will later be slightly modified) that the CFT secto r has a mass gap of
+orderM<, so that below this scale we are left only with the weakly coupled MSSM fi elds,
+with no remaining mixing to the CFT sector.
+The above factorized texture of Yukawa couplings, with each ferm ion coming with its
+own suppression factor, leads to the following structure of mixing a ngles [2, 3, 4, 6]
+(Vq
+L)ij∼ǫLi
+ǫLj∼(Vq
+L)ji,(Vq
+R)ij∼ǫRi
+ǫRj∼(Vq
+R)ji(i < j), q=u,d,ℓ, (2.3)
+6This is the running during the range of energies where the CFT secto r is conformal. In addition, (2.1)
+will obtain order one corrections from the entrance to and exit fro m this regime.
+6and fermion mass relationsmi
+mj∼ǫLiǫRi
+ǫLjǫRj, (2.4)
+where we assume without loss of generality γL1≥γL2,γR1≥γR2. Combining the two
+above equations in the case of the quarks one obtains
+(Vq
+L,R)ii∼1,(Vq
+L)ij∼ |Vij|,(Vq
+R)ij∼mqi/mqj
+|Vij|(i < j), q=u,d, (2.5)
+whereVijis the CKM matrix V=Vu
+LVd†
+L.
+It is essential for our purposes that the measured Yukawa hierar chy can then be ex-
+plained by coupling only the first two generations to the CFT. To this e nd, suppose a CFT
+has a spectrum of chiral operators O1,2with distinct scaling dimensions d1,2related to their
+R-charges r1,2asd1,2=3
+2r1,2, and with r1< r2<4/3. If some linear combination of MSSM
+superfields couples to O1and this coupling flows to a non-trivial fixed point, then at the
+fixed point this linear combination (which we can call the first generat ion and denote by
+Φ1) will have rΦ1= 2−r1anddΦ1= 3−d1. If some other linear combination of MSSM su-
+perfields couples to O2, then this other linear combination will contain a component along
+the Φ1direction in flavor space whose coupling will be irrelevant at the fixed p oint, and
+a component orthogonal to Φ 1which we can denote by Φ 2(and which will be the second
+generation). Assuming that the O2Φ2coupling also flows to a fixed point7, we will have at
+that fixed point rΦ2= 2−r2anddΦ2= 3−d2. By construction there are no R-symmetry
+violating operators of the form O1Φ2. The R-symmetry violating operators O2Φ1are irrel-
+evant, so after the RG flow they will be suppressed by factors of o rderǫ1
+2(γ(Φ1)−γ(Φ2)). One
+can then start with a generic gauge-invariant K¨ ahler potential an d superpotential in the
+UV, and the conformal dynamics will guarantee that in the effective low-energy theory at
+the scale M<the K¨ ahler potential and superpotential are approximately flavo r diagonal in
+the CFT basis [6], and exhibit a hierarchy in the Yukawa couplings.
+2.1.2 Some further assumptions and requirements
+After the conformal dynamics has done its job, some relevant def ormation must drive the
+theory away from the fixed point. For example, the escape from th e conformal window can
+be the result of small masses for some CFT sector particles, or of c onfinement of the CFT
+sector gauge group. It is important, however, that threshold eff ects atM<do not spoil the
+hierarchy of the Yukawa couplings built during the conformal regime . In particular, flavor
+7We assume that this new fixed point is stable, namely that it does not h ave any relevant operators
+that can appear in the K¨ ahler potential. In our examples we will typic ally know that the original CFT
+before coupling to the MSSM fields is stable, and it seems reasonable t o assume that this remains true
+after this coupling, but we do not prove this.
+7off-diagonal wavefunction renormalizations Zijshould be small. This is the concept of a
+graceful exit [6, 7]. In our constructions this will be guaranteed th rough the notion of sep-
+arable CFT sectors. In such scenarios, described in Section 3.1.2 be low, intergenerational
+couplings are absent in the interaction (CFT) basis, up to terms simila r to the previously
+discussed O2Φ1. Such a term can induce off-diagonal wavefunction renormalization s of
+orderǫ1/ǫ2(whereǫ1andǫ2are typical suppression factors for the operators of the first
+two generations), which are small enough such that the Yukawa hie rarchy in the canonical
+basis is not offset. Note that this implies that in our models we expect t o have flavor
+violating terms at least of order ǫ1/ǫ2.
+We should also worry about proton decay. As emphasized in [6, 7], if th e CFT sector
+contains couplings violating baryon number, the graceful exit must take place above a
+scaleM<∼1015GeV in order to appropriately suppress proton decay from dimensio n
+six operators. Dimension five operators pose additional constrain ts. It is argued there,
+that even assuming that these terms are not generated by the CF T, they can already be
+present at Mpl(R-paritycan beimposed to forbiddimension four couplings, but it do es not
+preclude the presence of dimension five operators). In that case the natural suppression
+factor 1/Mplhas to be strengthened by small coefficients in the range (10−6−10−7) [6, 41].
+In many of the scenarios considered by Nelson-Strassler (at small tanβ) this is achieved
+by the same suppression factors ǫiin (2.2) responsible for the Yukawa hierarchy. We will
+assume in our constructions that the CFT sector preserves bary on number and that the
+appropriate suppression of dimension five operators at Mplproceeds along the same lines
+as just discussed. This allows for low M<and no proton decay.
+To illustrate part of the above discussion, we now review two example s presented in [6].
+Wenotethatnoneofthesemodelsfullysatisfiestherequirements d iscussed abovethatneed
+to be imposed on our CFT sectors. Apart from the fact that they d o not break SUSY,
+the first example is separable but baryon-violating, whereas the se cond respects baryon
+number but is not separable. In this paper we are interested in a qua litative discussion
+on general classes of possible models, and are not concerned with c oncrete model building.
+We assume that a model satisfying all our requirements can be built.
+2.1.3 An SU(3)3example
+We first present anexample which is separable but can induce proton decay. As mentioned,
+the scale M<must then be above ∼1015GeV in order to suppress baryon number violating
+dimension six operators in the effective K¨ ahler potential.
+InthismodelthefullsymmetrygroupoftheCFTsectoris SU(3)3×Z3×SU(5)×SU(4).
+The group SU(5)×SU(4) is a gauge symmetry of the CFT sector under which all MSSM
+fields are neutral. These groups become strongly coupled at scales Λ5and Λ 4respectively,
+8SU(3)3SU(4)SU(5)dimension
+271,272,273(3,¯3,1)+(1,3,¯3)+(¯3,1,3)115
+3,4
+3,1
+27H,27′
+H(3,¯3,1)+(1,3,¯3)+(¯3,1,3)11 1,1
+¯27H,¯27′
+H(¯3,3,1)+(1,¯3,3)+(3,1,¯3)11 1,1
+Q (3,1,1)+(1,3,1)+(1,1,3)415
+6
+¯Q (¯3,1,1)+(1,¯3,1)+(1,1,¯3)¯415
+6
+Q′(3,1,1)+(1,3,1)+(1,1,3)152
+3
+¯Q′(¯3,1,1)+(1,¯3,1)+(1,1,¯3)1¯52
+3
+Table 1: Quantum numbers and scaling dimensions at the fixed point of the chiral super-
+fields in the SU(3)3model.
+and the couplings g5andg4eventually reach a fixed point. SU(3)3is a global symmetry of
+the CFT sector. After weakly gauging an appropriate subgroup, t he firstSU(3) factor is
+identified with the color gauge group, the electroweak SU(2)Lgroup resides in the second
+SU(3), and hypercharge is embedded in the second and third SU(3)’s.SU(3)3can be
+seen as a subgroup of the GUT group E6[42], and the MSSM quarks and leptons are
+contained in three copies of 27 ≡(3,¯3,1) + (¯3,1,3)+ (1,3,¯3). In addition to the (grand
+unified generalization of the) usual chiral superfield content of th e MSSM, this example
+contains chiral superfields Q(¯Q) andQ′(¯Q′) which transform in the fundamental (anti-
+fundamental) representations of SU(4) and SU(5), respectively. Including appropriate
+Higgs multiplets, the field content of the model is summarized in Table 1 .
+Apart from gauge and Yukawa interactions this model contains the following gauge and
+Z3-invariant superpotential:
+W=/summationdisplay
+i=1,2λi¯QQ27i+λ′¯Q′Q′271. (2.6)
+Here we used flavor rotations between the three generations to e nsure that only the first
+generation couples to ¯Q′Q′and only the first two to ¯QQ, as described above; the third
+generation then does not couple directly to the CFT sector. With th is convention the
+MSSM fields acquire the anomalous dimensions listed in Table 1, as we will n ow describe.
+Going down in energies, the dynamics of this theory can be analyzed in terms of scales
+Λ5and Λ 4(we take Λ 5>Λ4). TheSU(5) group has nine charged flavors, which drive the
+theory to a Seiberg conformal fixed point (since3
+2Nc< Nf<3Nc) [43]. At this fixed point
+the dimensions of gauge-invariant chiral operators are related to their R-charges as d=3
+2r
+through the superconformal algebra of the fixed point theory. T he R-charges can in turn
+be read from the microscopic theory and result in d¯Q′=dQ′=3
+2Nf−Nc
+Nf=2
+3. Therefore
+9the fields ¯Q′andQ′acquire negative anomalous dimensions of γ¯Q′,Q′=−2/3 at the fixed
+point, making the coupling λ′relevant and driving the theory to a new fixed point where
+the last term in the superpotential (2.6) is marginal. Assuming the ex istence of this fixed
+point, and given the anomalous dimensions of ¯Q′,Q′, the field 27 1must acquire a positive
+anomalous dimension of γ271= 4/3. This in turn causes λ1and the coupling of the 27 1to
+the Higgs field to become irrelevant and highly suppressed at low ener gies.
+As we continue flowing, the SU(4) theory becomes strongly coupled at the scale Λ 4,
+and its dynamics is also superconformal below this energy scale. In a similar fashion to the
+primed fields, ¯QandQacquire negative anomalous dimensions of γ¯Q,Q=−1/3, causing
+the coupling λ2to become relevant, while λ1remains irrelevant. The coupling λ2drives
+the theory to a fixed point where the λ2term in the superpotential (2.6) is marginal. The
+anomalous dimension of the 27 2field at the fixed point is γ272= 2/3, causing its Yukawa
+coupling to the Higgs fields to be suppressed at low energies.
+The theory can naturally exit the conformal regime, for example, t hrough confinement
+of an additional strong group Hwith no extra matter charged under it. If we have non-
+renormalizable couplings, generated at some high scale M∗≥M>, of the form
+/integraldisplay
+d2θ1
+M2∗tr(W2
+α)/parenleftbig¯QQ+¯Q′Q′/parenrightbig
+(2.7)
+whereWαare the gauge superfields of H, then this gives small masses to the Q,¯Q,Q′,¯Q′
+fields after condensation of the Hgauginos contained in W2
+α, and leads SU(5)×SU(4) to
+a confining phase.8
+Since the anomalous dimension of 27 1is twice that of 27 2, this model makes the generic
+prediction that ǫ1∼ǫ2
+2(assuming for simplicity that Λ 5∼Λ4and that both strong groups
+exit the conformal window around the same scale). However, since in this model suppres-
+sion factors are universal within a generation, it predicts mu/mt∼md/mb, a prediction
+that is off by around two orders of magnitude. For more details on th is model, see [6].
+2.1.4 A “10-centered” example
+In this model the CFT sector does not induce proton decay and the exit scale M<could
+in principle be as low as 10TeV. The gauge groups and matter content are listed in Table
+2, classifying fields according to their representations in SU(5) GUTs.
+The usual MSSM superfields are T1,2,3,¯F1,2,3,¯H,H. The gauge-invariant superpotential
+of the model is schematically given by
+W=T1QL+/summationdisplay
+i=1,2TiQM+A5+(JK)(JK)+A3LM+(MJ)(MK).(2.8)
+8The term (2.7) is slightly enhanced by the negative anomalous dimensio n of the quark bilinears, but
+it can still easily give small masses.
+10SU(5)GUTSp(8)Sp(8)′dimension
+T1,2,3 10 1142
+25,69
+50,1
+¯F1,2,3,¯H ¯5 11 1
+H 5 11 1
+Q 10 8187
+100
+A 1 27 13
+5
+J,K,L,M 1 813
+4,3
+4,3
+4,9
+20
+¯Q′10 18confined
+R,S 1 18confined
+Table 2: Quantum numbers and scaling dimensions of chiral superfield s in the10-centered
+model.
+Note that in this model only the standard model particles in the 10representation of
+SU(5) couple directly to the CFT sector. Relevant interactions such a sA3andA4must be
+forbidden by discrete symmetries, or be initially very small for some r eason. In this model
+theSp(8) group flows to a fixed point where the above superpotential is m arginal, while
+theSp(8)′group confines at low energies.
+From the anomalous dimensions of the fields T1andT2one can see that this model
+makes the prediction ǫ101=ǫ34/19
+102which is in quite good agreement with predictions from
+SU(5) models (see [6]).
+2.2 Flavor Changing Neutral Currents
+Generic models of new flavor physics introduce flavor violating terms beyond the SM. Such
+terms are highly constrained by experimental FCNC measurements . The most stringent
+constraints in our models will arise from either the K0−K0system [35, 36, 13] or the
+D0−D0system [44, 45]. Lepton number violating processes will generically giv e much
+milder constraints, so we confine this general discussion to ∆ S= 2 and ∆ C= 2 processes.
+The effective Hamiltonians for ∆ F= 2 processes at the scale of new flavor physics Λ NP
+can be written as [36]
+H∆S=2
+Eff=1
+Λ2
+NP/parenleftBigg5/summationdisplay
+I=1zK
+IQsd
+I+3/summationdisplay
+I=1˜zK
+I˜Qsd
+I/parenrightBigg
+,
+H∆C=2
+Eff=1
+Λ2
+NP/parenleftBigg5/summationdisplay
+I=1zD
+IQcu
+I+3/summationdisplay
+I=1˜zD
+I˜Qcu
+I/parenrightBigg
+,(2.9)
+11where the 4-fermi operators are
+Qqiqj
+1= ¯qα
+Ljγµqα
+Li¯qβ
+Ljγµqβ
+Li,Qqiqj
+2= ¯qα
+Rjqα
+Li¯qβ
+Rjqβ
+Li,Qqiqj
+3= ¯qα
+Rjqβ
+Li¯qβ
+Rjqα
+Li,
+Qqiqj
+4= ¯qα
+Rjqα
+Li¯qβ
+Ljqβ
+Ri, Qqiqj
+5= ¯qα
+Rjqβ
+Li¯qβ
+Ljqα
+Ri,(2.10)
+and where ˜Q1,2,3are obtained from Q1,2,3throughL↔R. The dimensionless coefficients
+zK,D
+I,˜zK,D
+Iencode the details of the new physics generating the above operat ors at Λ NP.
+In order to compare to experiment, the above effective Hamiltonian s have to be evolved
+from the scale of new physics Λ NPdown to the mKandmDscales. Taking into account
+QCD running and operator mixing one obtains (we do not write the tilde d part of the
+effective Hamiltonian, but it is implicit in our discussion)
+/angbracketleftK0|H∆S=2
+Eff|K0/angbracketrightI=1
+Λ2
+NP5/summationdisplay
+j=15/summationdisplay
+r=1/parenleftBig
+b(r,I)(K)
+j+ηc(r,I)(K)
+j/parenrightBig
+ηa(K)
+jzK
+I/angbracketleftK0|Qsd
+r|K0/angbracketright,
+/angbracketleftD0|H∆C=2
+Eff|D0/angbracketrightI=1
+Λ2
+NP5/summationdisplay
+j=15/summationdisplay
+r=1/parenleftBig
+b(r,I)(D)
+j+ηc(r,I)(D)
+j/parenrightBig
+ηa(D)
+jzD
+I/angbracketleftD0|Qcu
+r|D0/angbracketright,(2.11)
+whereη≡α3(ΛNP)/α3(mt), and the other relevant inputs can be found in [46] for the
+K0−K0system and in [13] for the D0−D0system. Throughout this paper we only use
+the bounds coming from CP-conserving processes, see the discus sion in the final section.
+Imposing that the new physics contribution is not larger than the me asured values of
+∆mK= 2Re/angbracketleftK0|H∆S=2
+eff|K0/angbracketright ≃3.483·10−12MeV and ∆ mD= 2Re/angbracketleftD0|H∆C=2
+eff|D0/angbracketright ≃
+1.89·10−11MeV [45, 47], the following constraints are obtained:
+ΛNP∼>/radicalBig
+|zK
+1|1.0×103TeV,ΛNP∼>/radicalBig
+|zK
+2|7.3×103TeV,ΛNP∼>/radicalBig
+|zK
+3|4.1×103TeV,
+ΛNP∼>/radicalBig
+|zK
+4|17×103TeV,ΛNP∼>/radicalBig
+|zK
+5|10×103TeV,
+(2.12)
+and
+ΛNP∼>/radicalBig
+|zD
+1|1.2×103TeV,ΛNP∼>/radicalBig
+|zD
+2|3.1×103TeV,ΛNP∼>/radicalBig
+|zD
+3|1.6×103TeV,
+ΛNP∼>/radicalBig
+|zD
+4|6.2×103TeV,ΛNP∼>/radicalBig
+|zD
+5|3.5×103TeV.
+(2.13)
+The above bounds are obtained after running α3up to Λ NPassuming that only SM fields
+contribute to the beta-function up to that scale. In our models we will always have ad-
+ditional fields below the scale Λ NP, which will change the running contribution to these
+bounds, but since the bounds are up to unknown order one coefficie nts in any case, we
+ignore this issue here. For a strongly coupled theory with generic fla vor structure one
+12expects|zK,D
+I|,|˜zK,D
+I| ∼1 (barring accidental cancelations), and the above inequalities
+directly translate into lower bounds on Λ NP.
+In supersymmetric extensions of the SM with perturbative quark- squark-gluino interac-
+tions, the operators (2.10) will also be generated through box diag rams with squark-gluino
+exchange [35, 36], and in this case the coefficients zK,D
+I,˜zK,D
+Iwill be suppressed by SM
+gauge couplings. Neglecting ˜ qL−˜qRmixing, they can be written as
+zK,D
+1=−α2
+3
+216/parenleftBig
+24xf6(x)+66˜f6(x)/parenrightBig
+(δd,u
+12)2
+LL,
+˜zK,D
+1=−α2
+3
+216/parenleftBig
+24xf6(x)+66˜f6(x)/parenrightBig
+(δd,u
+12)2
+RR,
+zK,D
+4=−α2
+3
+216/parenleftBig
+504xf6(x)−72˜f6(x)/parenrightBig
+(δd,u
+12)LL(δd,u
+12)RR,
+zK,D
+5=−α2
+3
+216/parenleftBig
+24xf6(x)+120˜f6(x)/parenrightBig
+(δd,u
+12)LL(δd,u
+12)RR,(2.14)
+wheref6(x) and˜f6(x) are kinematical functions whose expressions can be found in [36]
+andxis the ratio of the gluino mass squared to the squark mass squared, x= ˜m2
+˜g/˜m2
+˜q.
+Flavor violation is encoded in the mass insertion [48] parameters
+(δq
+12)NN≡(Kq
+21)N(Kq∗
+22)N(˜m2
+qN2−˜m2
+qN1)/˜m2
+˜q,(q=u,d, N =L,R) (2.15)
+In the above expression the average squark mass is taken to be ˜ m˜q≡(˜mq1+ ˜mq2)/2 [49]
+andKq
+N=Vq
+N˜Vq†
+N(N=L,R), where Vq
+Nand˜Vq
+Nare hermitian matrices that diagonalize
+the quark and squark mass matrices (again, neglecting ˜ qL−˜qRmixing), respectively, as
+diag(mq1,mq2,mq3) =Vq
+LMqVq†
+R,diag(˜m2
+qN1,˜m2
+qN2,˜m2
+qN3) =˜Vq
+N˜M2
+qNN˜Vq†
+N.(2.16)
+With this nomenclature, non-degenerate masses correspond to ( ˜m2
+qN2−˜m2
+qN1)∼˜m2
+˜qand
+theamountofalignment canbeestimatedfromthesizeofthemixinga ngles(Kq
+21)N(Kq∗
+22)N.
+Generic squark mass matrices for the first two generations with no degeneracy and no
+alignment lead to ( δq
+12)NN∼1. To assess the amount of flavor violation introduced by
+SUSY breaking in our subsequent (two-scale) models, we will use α3(1 TeV) ≃0.089 in
+(2.14), set Λ NP∼˜m˜qin (2.11) with zK,D
+I,˜zK,D
+Igiven by (2.14) and require that the total
+sum of the contributions proportional to a single δdoes not exceed the experimental values
+of ∆mK,D.
+3 Fully coupled models
+In this paper we wish to explore the possibility that the same dynamics generating the
+weak-scale flavor hierarchy is also responsible for SUSY breaking, a nd to work out the
+13qualitative form of the resulting spectrum. Our basic framework is a n implementation of
+the Nelson-Strassler scenario as reviewed in Section 2.1, leading to a suppression of the
+Yukawa couplings through coupling of the first two generations to a strongly coupled CFT
+sector as in (2.2), but with SUSY dynamically broken at a low scale by th e CFT sector
+itself. Thisimpliesthatthefirsttwogenerationsuperfieldsfeelstr ongdirectSUSYbreaking
+effects; note that this is very different from the high-scale SUSY br eaking discussed in
+[7, 9].9In our scenario the standard model gauge group GSMnecessarily couples (weakly)
+also to the CFT sector, so gauge-mediated soft terms naturally als o arise.
+The above couplings are summarized in the following interaction Lagra ngian:
+Lint=/parenleftbigg/integraldisplay
+d2θ(λ1O1Φ1+λ2O2Φ2+W(O1,O2))+h.c./parenrightbigg
++3/summationdisplay
+A=12gA/integraldisplay
+d4θJAVA,(3.1)
+where the chiral operators O1,2have different and definite R-charges, Φ 1,2denote generic
+first two generation MSSM matter superfields, the couplings λ1,2∼1 and we disregard
+irrelevantcouplingsfollowingthediscussion inSection2.1. VAareMSSMvector superfields
+andJAare the CFT sector global symmetry current superfields for GSM[16]. Note that,
+in contrast to models of general gauge mediation, taking αA→0 in our extended scenario
+does not fully decouple the MSSM from the CFT sector, since the dire ct couplings λiare
+dynamically set by the superconformal theory.
+We define Mas the mass of the degrees of freedom of the CFT sector which cou ple
+directly to the first two generations. We assume Mto be close to the bottom of the
+conformal window M∼M<, although this assumption is easily relaxed. The scale Mmay
+or may not be equal to the scale of the mass gap for all CFT sector fi elds. We distinguish
+models according to the number of scales involved:
+•One-scale models : models in which the SUSY breaking scale is also given by M.
+•Two-scale models : modelswhere theSUSY breaking scale Fisparametrically smaller
+than the scale of decoupling and conformal symmetry breaking, F≪M2.
+In the class of models that we are discussing, the Nelson-Strassler mechanism necessi-
+tates that the CFT sector and SUSY breaking are not flavor blind. T his is simply because
+the first two generations couple differently to the CFT sector. The main qualitative con-
+straint on our models therefore comes from the tension between r equiring small FCNCs,
+which pushes the mass of the CFT sector and the first two generat ion sfermions upwards,
+9In these works SUSY breaking occurs at a high scale well above the b ottom of the conformal window,
+and so the dynamics of the conformal regime leads to the suppress ion of many soft terms. There must
+then be a long enough RG flow so that gauginos can drive the soft mas ses back up to acceptable values,
+and so the scale M<is compelled to be high.
+14andrequiring small fine-tuningforelectroweak symmetry breaking , pushing themassscales
+of the theory downwards.
+Our main result will be a class of models which are technically natural an d – despite
+direct couplings between the first two generations and the strong ly coupled sector – accom-
+modate the FCNC constraints without too much fine-tuning. The be st models do so as a
+result of being separable rather than non-separable, two-scale m odels rather than one-scale
+models, and partially coupled rather than fully coupled.
+In this section we consider fully coupled models and their limitations. In Section 3.1 we
+discuss thedynamicsassociatedwiththescale M, inparticulartheconstraintscomingfrom
+four-Fermi terms generated at the threshold scale M. Through constraints on FCNCs we
+obtain a lower bound on Massuming the theory is generic (non-separable). We then show
+how a technically natural structural requirement on the theory, which we call separability,
+reduces the bound on M. Section 3.2 discusses the soft terms. In Section 3.3 we address
+one-scale models, where the same scale Msets the soft masses in the theory, leading to
+large fine-tuning. In Section 3.4 we discuss two-scale models, in which the mass of the first
+two generation sfermions is reduced compared to one-scale models . Although reducing the
+fine-tuning, this also introduces new sources of FCNCs which arise d ue to the running of
+the first two generation sfermions in loops [35, 36]. Therefore the fi ne-tuning issue is not
+fully resolved, but it is brought to a much lower level than the single sc ale case.
+3.1 Flavor restrictions on M
+3.1.1 Non-separable vs. separable models
+We are interested in low-scale SUSY breaking, leading to a spectrum c ompatible with a
+light enough Higgs without fine-tuning. However, generically, exiting the conformal and
+SUSY-preserving phase through some strong coupling dynamics ca n be accompanied by
+troublesome flavor-violating wave-function renormalizations (the graceful exit problem)
+and four-Fermi operators. If generic four-Fermi terms of the first two generations are
+generated at M∼M<, then the bound on Mis the largest of the bounds in Section 2.2,
+M∼>17·103TeV. This leads to a very heavy sparticle spectrum, resulting in large
+fine-tuning in the Higgs sector. A concrete mechanism suppressing such effects must be
+provided.
+A possible resolution is the suppression of direct flavor-mixing terms involving the first
+twogenerationsintheCFTbasis. Onewaytoachievethisisbyusingho rizontalsymmetries
+under which the CFT sector fields are charged. Another way to acc omplish this is by
+assuming a “decomposable” CFT sector. In such a setup, the CFT s ector decomposes into
+two separate sectors, each of which couples to a specific linear com bination of the three
+15MSSM generations. In particular, the gauge group ˜Gdecomposes into the product of two
+groups, and the fields charged under one couple to those charged under the other only
+via SM interactions. In both constructions terms similar to O2Φ1can still exist but, as
+discussed in Section 2.1, they are negligible and do not affect the Yuka wa structure. We
+will henceforth omit these flavor mixing, R-symmetry violating terms from the discussion;
+their effect on the flavor structure of the soft terms will be at mos t comparable to the
+effects we discuss. We shall refer to CFT sectors which are either d ecomposable or have
+horizontal symmetries as “separable”, and restrict ourselves to such scenarios from now
+on.
+Our separable CFT sectors give diagonal wave-function renormaliz ations in the inter-
+action basis, which realize the graceful exit in a natural way. Never theless, even in the case
+of separable models there can still be dangerous FCNC effects. The se come about from
+four-Fermi operators which appear flavor-diagonal in the intera ction basis, but are iden-
+tified as flavor-changing when switching to the mass basis. (These e ffects are generically
+much larger than the couplings induced by the SM interactions betwe en the two sectors.)
+Constraints from such operators, suppressed by powers of Mas well as mixing angles, are
+still in tension with low SUSY-breaking masses, as discussed in the follo wing subsections.
+3.1.2 Separable CFT sectors
+We now discuss the constraints on separable CFT sectors. In the f ollowing we focus on
+decomposable CFT sectors; the case of horizontal symmetries giv es similar results.
+We couple the first two generations of the MSSM to a decomposable s uperconformal
+sectorHwith group GטG1טG2. The gauge subgroups ˜G1and˜G2become strongly
+coupled at scales Λ 1and Λ 2, respectively. The interaction Lagrangian (3.1) becomes
+Lint=/parenleftbigg/integraldisplay
+d2θ(λ1O1Φ1+λ2O2Φ2+W1(O1)+W2(O2))+h.c./parenrightbigg
++3/summationdisplay
+A=12gA/integraldisplay
+d4θJAVA,
+(3.2)
+where now the fundamental fields forming the composite operator sO1,O2are not charged
+under˜G2,˜G1, respectively, and there are no direct couplings between fields cha rged under
+˜G1and˜G2. We collectively denote the CFT sector fields charged under ˜G1,2byH1,2. An
+example ofthistype ofmodels(althoughnotSUSYbreaking itself) ist heexamplereviewed
+in Section 2.1.3 [6].
+In these models each part of the CFT sector is perturbed away fro m the fixed point
+by separate relevant deformations, and no new intergenerationa l couplings are introduced.
+We then naturally obtain a flavor diagonal exit from the conformal r egime. SUSY breaking
+can occur in either sector separately, or in both. We will assume SUS Y is broken in both
+16sectors, andforsimplicitywetakethemassscalesof H1,2tobeofsimilarorderofmagnitude
+M.
+The leading non-renormalizable interactions of the SM fields are given by dimension
+six four-Fermi operators as in (2.9). Schematically,
+L4−fermi=1
+Λ2
+NP/parenleftBigg5/summationdisplay
+I=1zIQqkql
+I+3/summationdisplay
+I=1˜zI˜Qqkql
+I/parenrightBigg
+, (3.3)
+andtheconstants zIand ˜zIdependonthedetailsofthestrongdynamics. Operatorsinvolv-
+ing the first two generations are suppressed by Λ NP∼M. In the absence of any accidental
+cancelations in the coefficients, the first two generations’ terms h avezI,˜zI∼ O(1), while
+terms involving the third generation are further suppressed by po wers of SM gauge cou-
+plings. Since in separable models the two parts of the CFT sector talk to each other only
+via SM gauge interactions, in the CFT basis all first two generation fo ur-Fermi opera-
+tors are flavor-diagonal at leading order. Dangerous flavor violat ing terms are generated,
+however, when rotating to the mass basis (2.16):
+¯ˆqMiˆqNi¯ˆqPiˆqSi⊃/parenleftbig
+¯qM1qN2¯qP1qS2/parenrightbig/parenleftBig
+VM
+1iVN†
+i2VP
+1iVS†
+i2/parenrightBig
++/parenleftbig
+¯qM2qN1¯qP2qS1/parenrightbig/parenleftBig
+VM
+2iVN†
+i1VP
+2iVS†
+i1/parenrightBig
+,
+(3.4)
+where in the above q=u,d;i= 1,2;M,N,P,S =R,L; the hatted states on the left-hand
+side denote quarks in the CFT basis, while the unhatted states on th e right-hand side
+denote quarks in the mass basis.
+A bound on Mcan now be obtained by combining (3.4) with (2.5), (2.10), (2.12) and
+(2.13). We find that the strongest constraint comes from Qsd
+4in theK0−K0system,
+which picks up a factor of/radicalbig
+md/ms∼0.22 compared to the general CFT sector estimate
+M∼>17·103TeV, namely (up to order one numbers coming from the coefficients zIand
+˜zIin (3.3))
+M∼>3.7·103TeV. (3.5)
+In models with partial couplings to the CFT sector the FCNC constra ints can be
+relaxed. Such models will be discussed in Section 4.
+3.2 The soft terms in our models
+In our models there are two mechanisms for transmitting SUSY brea king to the standard
+model fields. The first two generation superfields couple directly to the CFT sector and
+obtain soft terms directly through these strong couplings. All oth er fields of the standard
+model couple to the CFT sector only indirectly, mostly through their interactions with the
+standard model gauge fields. These couple directly, though weakly , to the CFT sector,
+since the SM gauge group GSMis a subgroup of the CFT sector global symmetry group.
+17There are also very small contributions from the Yukawa couplings o f these fields to the
+first two generations, which we will ignore here. The mechanism for g enerating these soft
+terms is thus gauge mediation through coupling to a strongly coupled theory.
+In a theory of gauge mediation, the leading order (in standard mode l gauge couplings)
+soft terms may be expressed in terms of correlators of the GSMcurrents in the CFT sector
+theory [16]. Since our CFT sector is strongly coupled, we assume tha t these correlation
+functions are all of order one, up to an overall factor of N(A)
+effwhich captures the effective
+number of degrees of freedom in the CFT sector that are charged under the A’th factor in
+the standard model gauge group and participate in SUSY breaking. In a weakly coupled
+CFT sector this would simply be the number of messenger fields, while in strongly coupled
+theories it does not have to take integer values. We denote by µSthe scale at which the
+SUSY breaking dynamics is integrated out, and by Λ Sthe effective SUSY breaking scale
+transmitted to the MSSM (in one-scale models this is just the scale M, but we will later
+analyze models where this scale is different). Our assumptions then im ply that at the
+scaleµSthe gaugino masses, sfermion masses-squared (for multiplets whic h do not couple
+directly to the CFT sector) and A-terms are
+MA=N(A)
+effαA
+4πΛS,˜m2
+f∼3/summationdisplay
+A=1N(A)
+eff/parenleftBigαA
+4π/parenrightBig2
+Λ2
+S, Au,d,ℓ
+ij∼yu,d,ℓ
+ij3/summationdisplay
+A=1N(A)
+eff/parenleftBigαA
+4π/parenrightBig2
+ΛS,
+(3.6)
+wherei,jare generation indices, yu,d,ℓare Yukawa couplings, and the sum over Ashould
+include only the SM gauge groups that the specific fields in question ar e charged under.
+Note that we define N(A)
+effin terms of the corresponding gaugino mass.
+In the equations above we ignored possible contributions from a D-t erm of the hyper-
+charge current coming fromthe CFT sector [50, 51]. In gaugemedia tion this D-term comes
+from a vacuum expectation value for the lowest component of the U(1)Ycurrent super-
+field,JY[16], and at tree-level it gives additional possible contributions to th e sfermion
+masses going as ˜ m2
+f=g2
+1Yf/angbracketleftJY/angbracketright, whereYfis the hypercharge of the sfermions. Naively
+we expect /angbracketleftJY/angbracketright ∼ ±N(1)
+effΛ2
+S/(4π)2, yielding contributions to sfermion masses of order
+˜m2
+f∼ ±YfN(1)
+effα1
+4πΛ2
+S. Such terms are larger than the other contributions to third gene ra-
+tion sfermion masses, and since they always take both positive and n egative values, they
+are problematic.
+Thus, we will always assume that U(1)Yis embedded in a non-Abelian group in the
+global symmetry Gof the CFT sector. This means that at leading order we actually have
+/angbracketleftJY/angbracketright= 0, and the dominant contribution to /angbracketleftJY/angbracketrightcomes from the leading effects breaking
+this non-Abelian symmetry, which we expect to be one-loop effects o f orderα3/4π[51] (for
+instance, if it is a GUT symmetry). The D-term contributions to the s fermion masses are
+then expected to be of order ˜ m2
+f∼ ±YfN(1)
+effα1α3
+(4π)2Λ2
+S, which is already of the same order as
+18the other gauge-mediated contributions (3.6) described above. W e will analyze the effect
+of these terms in each scenario separately. In two-scale models, a dditional contributions
+to the D-term from the first two generation sfermions can arise via one-loop RG evolution,
+the effect of which will be discussed in Section 3.4.
+Having outlined some generalities of the soft terms structure, we m ove on to discuss in
+detail separable one-scale and two-scale models.
+3.3 Separable one-scale models
+In these models the scale of SUSY breaking is the same as that of con formal symmetry
+breaking, and therefore all dimensionful quantities are given in ter ms of powers of M.
+Specifically, the soft terms for the first two generations are domin antly given by
+(˜M2
+u,d,ℓ)NNii∼M2, Au,d,ℓ
+ii∼yu,d,ℓ
+iiM, i= 1,2, N=R,L, (3.7)
+and off-diagonal terms are not generated at leading order due to s eparability of the CFT
+sector. Since their mass is of order M, the first two generation sfermions can be viewed as
+part of the CFT sector and are integrated out when writing the effe ctive action below M.
+The effective Lagrangian below the scale Mthus takes the form:
+Leff=/summationdisplay
+i=1,2iqiDµγµ¯qi+/integraldisplay
+d4θΦ3eVΦ†
+3+/integraldisplay
+d2θW˜q1,2=0
+MSSM+L˜q1,2=0
+soft+L≥4,(3.8)
+whereqiare the first two generation fermions, ˜ qiare their superpartners, and the MSSM
+superpotential W˜q1,2=0
+MSSMand the soft Lagrangian L˜q1,2=0
+softdo not include the first and second
+generation sfermions. As we are discussing separable models, wave -function renormaliza-
+tions below Mare diagonal. The soft Lagrangian is given by
+−L˜q1,2=0
+soft=1
+2/parenleftbig
+MAλAλA+c.c./parenrightbig
++/parenleftBig
+(˜M2
+q)NN33˜q∗
+N3˜qN3+Aq
+33Hu,d˜qL3˜qR3+c.c./parenrightBig
++/bracketleftbig
+m2
+HuH∗
+uHu+m2
+HdH∗
+dHd+(BµHuHd+c.c.)/bracketrightbig(3.9)
+whereN=R,Landq=u,d,ℓ. The gaugino masses MA, third generation soft masses
+(˜M2
+q)33, soft trilinear terms Aq
+33and soft Higgs terms are all dominantly generated through
+gauge mediation (3.6) with an effective SUSY breaking scale Λ S∼M.
+Substituting the lowest value of Mcompatible with the four-Fermi operator bound
+(3.5) into the general formulas (3.6) and (3.7), one finds that the s pectrum of such one-
+scale models is very heavy. For instance, the gluino mass is of order 2 2N(3)
+effTeV at the
+scaleM, and grows as it evolves down to the weak scale. Clearly, a spectrum of such
+massive gauginos and third generation sparticles is unacceptable sin ce it implies severe
+fine-tuning of the weak scale [10, 11, 12]. For this reason we discard further discussion of
+these one-scale models, and turn to models involving two scales.
+193.4 Separable two-scale models
+We now study models in which the SUSY breaking scale Fis parameterically suppressed,
+F≪M2. For simplicity, we discuss models in which SUSY breaking occurs in both H1
+andH2, assuming F1∼F2∼F.
+3.4.1 Soft terms
+The soft terms for the first two generation sfermions can be dete rmined to leading order
+inF/M2by dimensional analysis and by requiring these leading contributions t o vanish
+in the supersymmetric limit F→0, as well as in the limit of no direct coupling M→ ∞.
+Chirality-preserving diagonal soft masses are dominantly given by
+(˜M2
+u,d,ℓ)NNii∼/parenleftbiggF
+M/parenrightbigg2
+, i= 1,2, N=R,L. (3.10)
+Off-diagonal soft masses-squared for the first two generations are not generated from direct
+couplings, and are zero at leading order in gauge mediation. The first two generation
+diagonal A-terms are given by
+Au,d,ℓ
+ii∼yu,d,ℓ
+iiF
+M, i= 1,2, (3.11)
+while off-diagonal first two generation A-terms are much smaller (see below). In this
+scenario the first two generation sfermions are much lighter than t he CFT sector fields, so
+the parametric suppression of the SUSY breaking order paramete rF/M2≪1 can account
+for lighter masses ˜ m1,2even for a large CFT sector scale M.
+The third generation and gaugino soft masses are generated thro ugh gauge mediation,
+as are additional soft trilinear couplings. The precise definition of Λ S, the effective scale
+of SUSY breaking transmitted to the visible sector, in terms of FandMdepends on the
+specific class of models under consideration, as we will discuss short ly. At this point, the
+only requirement is that Λ S→0 asF→0. For example, in weakly coupled messenger
+models of gauge mediation Λ Scorresponds to F/M, withMthe messenger scale and F
+the vacuum energy squared. In terms of this scale Λ S, we then have at the scale µS:
+MA=N(A)
+effαA
+4πΛS,
+(˜M2
+u,d,ℓ)NN33∼N(A)
+eff/parenleftBigαA
+4π/parenrightBig2
+Λ2
+S±Yu,d,ℓNN(1)
+effα1α3
+(4π)2Λ2
+S, N=R,L,
+Au,d,ℓ
+ij∼yu,d,ℓ
+ijN(A)
+eff/parenleftBigαA
+4π/parenrightBig2
+ΛS, i/negationslash=jori=j= 3, i,j= 1,2,3,(3.12)
+where off-diagonal gauge-mediated soft masses-squared are ne gligible,Yu,d,ℓL,Rdenotes the
+hypercharge of the appropriate sfermion, and in the above we tak e the largest contribution
+when several SM gauge factors are possible.
+20In these models we need to consider the RG contribution of the first two generation
+sparticles to the D-term [19]. The soft masses-squared in (3.10) ar e more accurately given
+by
+(˜M2
+u,d,ℓ)NNii∼/parenleftbiggF
+M/parenrightbigg2
+±α3
+4π/parenleftbiggF
+M/parenrightbigg2
+, i= 1,2, N=R,L. (3.13)
+The coefficient of the first term is assumed to be invariant under G. The second term
+encodes the leading violation of the Gsymmetry by the standard model. In the simplest
+case that Gis a unified version of GSM, the strongest such effect is proportional to α3.
+Generally, the masses (3.13) contribute to the D-term of U(1)Yand to the soft masses of
+lighter particles. For some other light sfermion of hypercharge Y, the D-term contributes
+Yα1
+4πTr(Yf˜m2
+f) (3.14)
+to its soft mass-squared beta function. Depending on the sign of Y, some of the third
+generation scalar masses may then be driven tachyonic. Recall, how ever, that we assume
+thatU(1)Yis embedded in a non-Abelian factor in the global symmetry group Gof the
+CFT sector which includes the first two generations (for example an SU(5) GUT). As a
+result, the first contribution from the heavy first two generation masses in (3.13) yields
+a vanishing hypercharge trace. Since the (say) GUT symmetry is br oken by SM gauge
+couplings, thesecond termin (3.13)will induce a non-zero D-term, le ading tocontributions
+to light sfermion mass-squared beta functions of order
+Yα1
+4πTrYf˜m2
+f∼Yα1α3
+(4π)2/parenleftbiggF
+M/parenrightbigg2
+. (3.15)
+In the scenarios considered in the rest of the paper the integral o f (3.15) is always smaller
+than the first contribution to the sfermion masses-squared in (3.1 2), and will henceforth
+be neglected. The second term in the sfermion masses-squared (3 .12) can be negative and
+therefore potentially dangerous, so it may require mild tuning. We will discuss the effect
+of this D-term in further detail for partially coupled models in Section 4.
+Operators of dimension d≥4 can also be generated. Since dimension four sfermion
+operators can only come from SUSY breaking, they are parametric ally suppressed via
+F/M2≪1, and are thus not a concern. In dimension five fermion-sfermion o perators the
+SUSY preserving effects dominate over the SUSY breaking ones. Th e four-Fermi operators
+of dimension six are dominated by the SUSY preserving effects, and a re down by 1 /M2.
+They dictate the bound (3.5) on M.
+Equation (3.10) presents the pleasant feature of two-scale mode ls: the first two gen-
+eration soft terms and the higher-dimensional operator coefficien ts can simultaneously be
+suppressed by taking an enhanced value of Mfor fixed F, relaxing the contributions to
+FCNCs while lowering the scale of the scalar masses.
+We consider the following two classes of models:
+21Class (a)
+In these models the CFT sector is assumed to have a mass gap of ord erM, and all of
+the CFT sector physics is integrated out at the scale M∼µS. In the effective theory
+obtained at scale Msupersymmetry is dynamically broken, with an accidentally small
+SUSY breaking parameter F≪M2. As a result Λ S∼F/M. In the limit M→ ∞the
+CFT sector decouples and SUSY is not broken.
+Class (b)
+In these models the physics of the CFT sector coupling to the first t wo generations is
+integrated out at the scale M(namely, the CFT sector fields that couple directly to the
+first two generations have a mass of order M). The rest of the CFT sector is integrated
+out at a lower scale√
+F≪M, at which SUSY is then naturally broken in the effective
+theory. Here the mass gap in the CFT sector is the same as the SUSY breaking F-term.
+As a result µS∼ΛS∼√
+F, while the soft terms for the first two generations are still given
+by (3.10). In this class of models the limit M→ ∞corresponds to a gauge mediation
+scenario with SUSY breaking scale√
+F.
+Since in models of Class (b) part of the CFT sector dynamics is integra ted out at M,
+we expect the SUSY breaking dynamics in this class to yield smaller value s ofNeffthan
+in Class (a). Given the direct couplings and the MSSM quantum number s of the operators
+O1,2we expect Neffto be at least twice the MSSM contribution (in two generations) for
+Class (a) models. In Class (b) models we can have lower Neff, though requiring dynamical
+SUSY breaking in the CFT sector typically means that it cannot be sma ller than ∼5.
+3.4.2 Further FCNC bounds on scales
+As described above, in both classes of two-scale models generic fou r-Fermi operators gen-
+erated by the superconformal dynamics are suppressed by 1 /M2, and bound Maccording
+to (3.5). The non-zero soft terms involving the first two generatio ns are set by F/Mand
+are of the form (3.10), and chirality mixing terms are negligible. At lead ing order, sfermion
+mass-squared matrices for the first two generations are then alr eady diagonalized in the
+CFT basis at the scale µS.
+In these models the sfermions are much lighter than Mand so additional FCNC con-
+straints are present. The most stringent bound on F/Mis obtained from the K0−K0
+FCNC box diagrams involving squarks and gluino exchange. This proce ss constrains the
+first two generation (down sector) masses at the weak scale, obt ained by RG evolution
+(RGE) from the boundary conditions (3.10) at µSdown tomZ. We can neglect RG effects
+22on the first two generation mass-squared matrices due to the sho rt range of scales involved,
+and use (3.10) at the weak scale as well. In light of the above, in the sq uark sector we have
+(˜Vq
+L,R)ij∼δij, i,j,= 1,2, q=u,d. (3.16)
+The quark sector diagonalizing matrices have the structure (2.5), and so the mixing ma-
+trices in (2.15) are of the form
+(Kq
+L)ij∼(Vq
+L)ij∼ |Vij|,(Kq
+R)ij∼(Vq
+R)ij∼mqi/mqj
+|Vij|(i < j), i,j= 1,2, q=u,d.
+(3.17)
+Note that ( Kd
+L)12∼ |V12| ∼(Kd
+R)12∼md/ms
+|V12|∼0.22 are all of order a Cabibbo factor. In
+terms of the δd
+12parameters of Section 2.2, the separable CFT sector scenario with no de-
+generacy then corresponds to taking all δd
+12of order a Cabibbo factor ∼0.22. Additionally,
+(3.10) and (3.12) imply that the ratio x≡m2
+˜g/˜m2
+˜qappearing in (2.14) is small yet depends
+onN(3)
+eff. In models of Class (a), this ratio is independent of F/M2and is given by
+x(a)=/parenleftBigα3
+4πN(3)
+eff/parenrightBig2
+(3.18)
+while in models of Class (b) we have
+x(b)=/parenleftBigα3
+4πN(3)
+eff/parenrightBig2M2
+F. (3.19)
+Themoststringentboundscomefromthemixed( δd
+12)LL(δd
+12)RRtermsin(2.14). Wepresent
+plots of these bounds as a function of N(3)
+efffor Class (a) and Class (b) models in Figure 1.
+Representative bounds are10
+Class (a) :F
+M∼>81 TeV ( N(3)
+eff= 27),
+Class (b) :F
+M∼>87 TeV ( N(3)
+eff= 5).(3.20)
+The values for N(3)
+effare chosen in agreement with the two-loop effect [29, 30] that will
+be discussed in Section 4. Flavor changing chirality-mixing terms ( δd
+12)LRgive much less
+stringent constraints on F/Mdue to the relative smallness of the A-terms (3.11).
+Given some amount of degeneracy in the first two generation down s ector, the bound is
+weakened and lower values of F/Mare accessible. For example, allowing additional ∼0.5
+degeneracy the boundsin bothclasses areweakened to F/M∼>41TeV and F/M∼>43 TeV
+respectively, for the same values of N(3)
+effas in (3.20).
+10Recall that all the bounds we write are up to unknown order one con stants coming from correlators
+in the CFT sector.
+230 20 40 60 80 1005000060000700008000090000
+NeffF/Slash1M GeV
+0 5 10 155000060000700008000090000
+NeffF/Slash1M GeV
+Figure 1: The relevant K0−K0system bounds on F/Mas a function of N(3)
+efffor fully
+coupled two-scale models with a separable CFT sector ( M= 3.7·103TeV). The bounds
+depicted here come from ( δd
+12)LL(δd
+12)RRin Class (a) (left) and Class (b) (right) models.
+Using the bounds on MandF/Mgiven in (3.5) and (3.20), we compute a bound for√
+Frelevant for models of Class (b), yielding
+√
+F∼>570 TeV. (3.21)
+Itisclear from(3.10), (3.12), (3.20)and(3.21)thatinthetwo-sca lemodelsdescribed above
+the generic sfermion soft mass spectrum at the scale µSis still given by an inverted hier-
+archy, provided Neffis not too large. In such models, avoiding third generation tachyons
+fromheavy firsttwo generationtwo-loopcontributions[29,30]pu tsalower boundoninitial
+third generation masses, which in our models translates into a lower b ound on N(A)
+eff. This
+two-loop effect will be discussed in greater detail in Section 4. In the current context, this
+effect along with the initial soft terms (3.12) imply unnaturally large init ial soft masses for
+the third generation sfermions and the gauginos. For instance, glu ino masses at the scale
+µSare of order ∼12 TeV in Class (a), and ∼18 TeV in Class (b). These fully coupled
+separable models thus still require large fine-tuning in the electrowe ak sector.
+244 Partially coupled models
+In the models analyzed in the previous section all MSSM fields of the fir st two generations
+coupled directly to the CFT sector. This resulted in relatively high sca lesMandF/M,
+which led to large fine-tuning of the weak scale mZ. In this section we explore models with
+partial couplings, in which the strongest constraints on the scales can be relaxed and the
+fine-tuning can be alleviated.
+When fully coupling all first two generation fields to the CFT sector, t he main sources
+of flavor violation are operators made out of both left- and right-h anded fields in the
+down sector. We can thus allow for a lower scale Mby coupling only some of these fields
+(in particular only some of the down quarks) directly to the CFT sect or. Here we will
+consider several such models – models in which only the left-handed ( LH) fields in the
+first two generations couple directly to the CFT sector, similar mode ls with right-handed
+(RH) direct couplings, and a 10-centered model, in which only the 10’s ofSU(5) couple to
+the CFT sector, thus coupling both up sector chiralities but suppre ssing mixed chirality-
+operators in the down sector. We consider here only two-scale mod els, of both Class (a)
+and (b).
+A few comments concerning D-terms arein order. When coupling only a single chirality
+to the CFT sector, unification is lost. In order to suppress the tre e-level gauge mediation
+andone-loopbeta-functionD-termsasintheGUTcase(see(3.12) and(3.15),respectively),
+oneisforcedtoembed U(1)Yinsomenon-Abeliangroupunderwhichleftandrightchirality
+superfields transformseparately [19, 51]. In thiscase one shouldr equire that theparameter
+ζmeasuring the magnitude of the breaking of the non-Abelian symmet ry should be such
+that the one-loop gauge-mediated masses-squared
+˜m2
+f∼3/summationdisplay
+A=1N(A)
+eff/parenleftBigαA
+4π/parenrightBig2
+Λ2
+S±YfN(1)
+effα1ζ
+(4π)2Λ2
+S (4.1)
+do not lead to tachyonic sfermions. As discussed in Section 3.2, the D -term contribution
+to the one-loop beta function for the sfermion masses-squared ( 3.15) (where α3is replaced
+by a general ζ) is negligible in comparison to the bare D-term and can be neglected fr om
+subsequent discussion.11
+We will analyze the effect of the remaining gauge mediation D-term initia l masses-
+squared in (4.1) for the third generation sfermions case by case.
+11We implicitly assume here that ζis not larger (at the scale M) than GUT breaking parameters in the
+standard model such as α3, and that N(1)
+effis not much larger than the other Neff’s.
+254.1 The two-loop effect
+As in fully coupled models, all the constructions discussed here will ex hibit a (partial)
+inverted hierarchy in the sfermion sector. Inverted hierarchy mo dels are known to be
+susceptible to two-loop effects in the RGE, where the heavy first tw o generations can
+render light third-generation sparticles tachyonic [29, 30].
+Positive physical masses-squared then require heavy initial soft m asses for the third
+generation, such that a fair amount of fine-tuning may be necessa ry to stabilize the weak
+scale. This concern, however, will turn out to be unjustified – some fine-tuning is needed,
+but for a reasonable range of scales and parameters it can be milder than, say, the percent
+level.
+The beta function for the third generation (and all non-directly co upled) sfermion
+masses-squared up to two-loops can be written as [52]
+d
+dt˜m2
+f=1
+16π2β(1)
+˜m2
+f+1
+(16π2)2β(2)
+˜m2
+f, t≡log(µ/µ0). (4.2)
+In our analysis we will neglect Yukawa couplings and A-terms throughout in the RGE,
+as well as gaugino contributions at the two-loop level. With these app roximations the
+beta-functions can be written as [52]
+β(1)
+˜m2
+f≃ −8/summationdisplay
+Ag2
+ACA(Rf)|MA|2,
+β(2)
+˜m2
+f≃+4/summationdisplay
+Ag2
+ACA(Rf)σA+12
+5g2
+1YfS′,(4.3)
+where the sum over Aruns over the three SM gauge groups, CA(Rf) denotes the quadratic
+Casimir invariant of the representation Rfof the gauge group A,Yfis the hypercharge of
+the sfermion f, and (ignoring small contributions from soft Higgs masses) one has
+σ1≃1
+5g2
+1Tr/parenleftBig
+˜M2
+QL+3˜M2
+LL+8˜M2
+uR+2˜M2
+dR+6˜M2
+eR/parenrightBig
+,
+σ2≃g2
+2Tr/parenleftBig
+3˜M2
+QL+˜M2
+LL/parenrightBig
+,
+σ3≃g2
+3Tr/parenleftBig
+2˜M2
+QL+˜M2
+uR+˜M2
+dR/parenrightBig
+,(4.4)
+and
+S′≃8
+3g2
+3Tr/parenleftBig
+˜M2
+QL−2˜M2
+uR+˜M2
+dR/parenrightBig
++3
+2g2
+2Tr/parenleftBig
+˜M2
+QL−˜M2
+LL/parenrightBig
++g2
+1Tr/parenleftbigg
+−3
+10˜M2
+LL+1
+30˜M2
+QL−16
+15˜M2
+uR+2
+15˜M2
+dR+6
+5˜M2
+eR/parenrightbigg
+.(4.5)
+In the above equations we write the sfermion mass-squared matric es in the standard SU(2)
+notation. In all the subsequent estimates made in this section, (4.2 ) is integrated in the
+26leading log approximation for the appropriate light sparticles, where the two-loop contri-
+bution runs down to the approximate decoupling scale for the heavy first two generation
+scalars∼F/M, and the one-loop term flows down to a scale µ0∼1 TeV. Neglecting
+chirality-mixing, we then impose that for the lightest sparticle
+˜m2
+f(µ0)≃˜m2
+f(µS)+1
+16π2β(1)
+˜m2
+f(µS)log/parenleftbiggµS
+µ0/parenrightbigg
++1
+(16π2)2β(2)
+˜m2
+f(µS)log/parenleftbiggµS
+F/M/parenrightbigg
+∼>0.(4.6)
+TakingN(A)
+eff≡Nfor simplicity in all initial sfermion and gaugino masses, a rough lower
+bound on Nmay then be found in each scenario, affecting the initial soft terms a nd thus
+the physical spectrum. A full analysis will of course introduce corr ections to the obtained
+bounds, but the order of magnitude of Nis captured in this approximation. The sample
+spectra thatwepresent inSection4.3usethefulltwo-loopRGevolu tion, andareconsistent
+with this estimate.
+4.2 Models
+The ground is now set to explore some partially coupled models. We beg in with chiral
+models in which only the MSSM fermion-sfermion fields of one chirality ar e coupled to the
+superconformal sector. We then address an example of non-chir al partial couplings – a
+unified10-centered scenario.
+4.2.1 Chiral models
+We assume that only the left-handed (right-handed) MSSM fields ar e coupled to the CFT
+sector. Therefore, in the quark and lepton sectors, there are o nly left-handed (right-
+handed) suppression factors generated by the superconforma l dynamics, and the factoriz-
+able Yukawa structure of (2.2) contains a single ǫL(ǫR) factor. These models can still lead
+to acceptable flavor structures, but producing appropriate sup pression for given anoma-
+lous dimensions requires increasing the ratio M>/M<. There are, however, no a priori
+constraints on this ratio.12To be more concrete, in chiral left-handed models, the sup-
+pression pattern predicts that the mixing angles are of order the m ass ratios, and so some
+additional suppression is needed in order to be in full agreement with measurement. Right-
+handed models suffer from a difficulty to parametrically produce the C KM mixing angles.
+When discussing right-handed models in the following, we assume a Cab ibbo factor has
+been generated in estimating the right-handed mixing matrices (2.5) ; the bounds on the
+scales obtained in this way are conservative.
+12We need torequirethat M>isbelow the Planckscale, but this iseasyto satisfy. Additional const raints
+from Landau poles will be discussed in the next section.
+27A comment is in order. When only right-handed superfields are couple d to the CFT
+sector, the minimal and more natural assumption is that the CFT se ctor is uncharged
+underSU(2)L⊂GSM. In such a scenario the Wino gauge-mediated mass will not be
+generated at the high scale µS, leading to unacceptably light neutralinos and charginos at
+the weak scale. Therefore, from here on when discussing chiral rig ht-handed couplings we
+assume that the CFT sector is charged under SU(2)L. This also leaves open the possibility
+of directly coupling the Higgs sector to the CFT sector.
+Among the four-Fermi operators (3.3) bounding M, only the chiral operators Q1(˜Q1)
+are now generated at M. In light of (2.5), the bound (3.5) in separable models can then
+be relaxed here to
+LH couplings : M∼>264 TeV (from the D0−D0system),
+RH couplings : M∼>220 TeV (from the K0−K0system).(4.7)
+In the sfermion sector, soft mass-squared terms for the left-h anded (right-handed) first
+two generation sfermions are given by direct coupling as in (3.10)
+(˜M2
+u,d,ℓ)LL(RR)ii∼/parenleftbiggF
+M/parenrightbigg2
+, i= 1,2. (4.8)
+All other soft terms are gauge-mediated similarly to (3.12):
+MA=N(A)
+effαA
+4πΛS,
+(˜M2
+u,d,ℓ)LL,RR 33∼N(A)
+eff/parenleftBigαA
+4π/parenrightBig2
+Λ2
+S±N(1)
+effYu,d,ℓL,Rα1ζ
+(4π)2Λ2
+S,
+(˜M2
+u,d,ℓ)RR(LL)ii∼N(A)
+eff/parenleftBigαA
+4π/parenrightBig2
+Λ2
+S±N(1)
+effYu,d,ℓR(L)α1ζ
+(4π)2Λ2
+S, i= 1,2,
+Au,d,ℓ
+ij∼yu,d,ℓ
+ijN(A)
+eff/parenleftBigαA
+4π/parenrightBig2
+ΛS, i,j= 1,2,3.(4.9)
+Avoiding initial tachyonic right-handed sleptons when the sign of the D-term is negative
+requires ζ∼α3/5, implying a mild tuning of the D-term generated by the CFT sector
+in this case. In Class (b) models we can achieve suppression of the D- term naturally by
+assuming a messenger parity [51] symmetry (taking JY→ −JY) below the scale M. In
+such a case /angbracketleftJY/angbracketright1−loop= 0 [16, 51] and so
+/angbracketleftJY/angbracketright=O/parenleftbiggζ2
+16π2/parenrightbigg
+, (4.10)
+i.e, the D-term mass-squared acquires an extra factor of ζ/(4π).
+In the chiral coupling scenarios discussed here N(A)
+effcould be smaller than in the case of
+direct coupling of both left- and right-handed fields to the superco nformal sector, schemat-
+ically by a factor of a half.
+280 20 40 60 80 1002000300040005000600070008000
+NeffF/Slash1M GeV
+0 2 4 6 8 102000300040005000600070008000
+NeffF/Slash1M GeV
+Figure 2: Relevant D0−D0system (dashed) and K0−K0system (bold) bounds on F/M
+as a function of N(3)
+efffor chiral left-handed models ( M= 264TeV). The left graph refers
+to Class (a) models and the right graph refers to Class (b) models.
+In chiral left-handed models the most stringent bounds on the mas ses of the heavy
+first two generations, F/M, are set by processes involving the ( δu
+12)LLterm inD0−D0
+box diagrams. The ( δd
+12)LLterm inK0−K0box diagrams gives comparable yet milder
+constraints. Since the right-handedsoft masses-squared areg iven by gaugemediation there
+is near degeneracy in the first two generations of right-handed sq uarks, and so ( δu,d
+12)RR≈0
+for left-handed models.
+Ontheotherhand, inchiralright-handedmodels, thestrongestc onstraintsontheheavy
+scaleF/Mare obtained from processes involving ( δd
+12)RRterms in K0−K0box diagrams.
+Processes involving ( δu
+12)RRterms in D0−D0box diagrams are comparatively suppressed
+by the relative smallness of the right mixing angles in the up sector (se e (2.5)). Since
+the left-handed soft masses-squared are given by gauge mediatio n there is near degeneracy
+in the first two generations of left-handed squarks, and so ( δu,d
+12)LL≈0 for right-handed
+models.
+Plots of the bounds on F/Mas a function of N(3)
+effare given in Figure 2 for chiral
+left-handed models, and in Figure 3 for chiral right-handed models.
+290 20 40 60 80 10020003000400050006000
+NeffF/Slash1M GeV
+0 2 4 6 8 1020003000400050006000
+NeffF/Slash1M GeV
+Figure 3: Relevant K0−K0system bounds on F/Mas a function of N(3)
+efffor chiral
+right-handed models ( M= 220TeV). The left graph refers to Class (a) models and the
+right graph refers to Class (b) models.
+Representative bounds in chiral left-handed coupled models are
+LH Class (a) :F
+M∼>5 TeV ( N(3)
+eff= 62),
+LH Class (b) :F
+M∼>6 TeV ( N(3)
+eff= 5),(4.11)
+while in chiral right-handed coupled models they are
+RH Class (a) :F
+M∼>6 TeV ( N(3)
+eff= 15),
+RH Class (b) :F
+M∼>5 TeV ( N(3)
+eff= 5).(4.12)
+Combining the bounds on MandF/Mof (4.7), (4.11) and (4.12) we obtain for models of
+Class (b):
+LH couplings :√
+F∼>40 TeV,
+RH couplings :√
+F∼>33 TeV.(4.13)
+As explained in Section 4.1, two-loop contributions to third generatio n masses put
+a lower bound on N≡N(A)
+eff. For concreteness we obtain the bounds for the point in
+parameter space where all order-one numbers in (4.8) are equal t o one. The analysis
+30for other points in parameter space is qualitatively the same. Using ( 4.8) and (4.9) for
+left-handed models, at this point in parameter space one has
+S′≃/bracketleftbigg16
+3g2
+3−8
+15g2
+1/bracketrightbigg/parenleftbiggF
+M/parenrightbigg2
+,
+σ1≃8
+5g2
+1/parenleftbiggF
+M/parenrightbigg2
+, σ2≃8g2
+2/parenleftbiggF
+M/parenrightbigg2
+, σ3≃4g2
+3/parenleftbiggF
+M/parenrightbigg2
+.(4.14)
+For Class (a) the strongest bound on Ncomes from the right-handed charged sleptons, for
+which (4.6) reads
+˜m2
+eRi(TeV)∼α1Λ2
+S
+(4π)2/bracketleftBigg
+α1N+12
+5/parenleftbigg
+−16
+3α3−16
+15α1/parenrightbigg/parenleftbiggF/M
+ΛS/parenrightbigg2
+log/parenleftbiggµS
+F/M/parenrightbigg
++24
+5α2
+1
+4πN2log/parenleftBigµS
+TeV/parenrightBig/bracketrightbigg
+∼>0, i= 1,2,3.(4.15)
+For Class (b) the strongest bound comes from ˜ uRi, although other sparticles give compa-
+rable bounds.13By solving the relevant inequalities we obtain the bounds:
+LH Class (a) : N∼>62 for Λ S∼F/M= 5 TeV, µS∼M= 264 TeV ,
+LH Class (b) : N∼>2 for Λ S∼µS∼√
+F= 40 TeV ( F/M= 6 TeV) .(4.16)
+The analysis is analogous for right-handed models. For Class (a) mod els all sparticles,
+except right-handed charged sleptons which do not become tachy onic, give similar bounds,
+while in Class (b) models all sparticles typically give comparable constra ints. We obtain:
+RH Class (a) : N∼>15 for Λ S∼F/M= 6 TeV, µS∼M= 220 TeV ,
+RH Class (b) : N∼>2 for Λ S∼µS∼√
+F= 33 TeV ( F/M= 5 TeV) .(4.17)
+The values of N(3)
+effin (4.11) and (4.12) are chosen in agreement with the above.
+4.2.2 10-centered models
+As an alternative scenario, one could contemplate evading the stro ngestK0−K0mixing
+bounds by disallowing non-chiral couplings in the down sector alone. I nSU(5)-based GUT
+models, where the ¯5contains LL,dRand the10contains QL,uRandeR, coupling only the
+101,2to the CFT sector can accomplish this, while still generating the Yuka wa hierarchies
+in the up, down and lepton sectors and guaranteeing a vanishing hyp ercharge D-term at
+13Here and in the following, since we neglect Yukawa couplings in the runn ing, all generations of the
+non-directly coupled fields have similar RG evolution. Including the effe cts of Yukawas, the strongest
+constraints will come specifically from third generation sparticles wit hin the relevant sector.
+31leading order. In this scenario, suppression factors for both chir alities are generated in
+the up quark sector. In the down quark and lepton sectors only fie lds of one chirality, left
+and right respectively, acquire large anomalous dimensions. A detaile d discussion of this
+fermion flavor structure can be found in [6].
+In this setup the strongest bound on Mcomes from Q2of (3.3) in the D0−D0system
+[13] and reads
+M∼>682 TeV. (4.18)
+First two generation up squarks feel direct SUSY breaking, as do le ft-handed down squarks
+and right-handed sleptons, while all other sparticles have gauge me diated soft masses and
+masses-squared. Diagonal first two generation soft trilinear cou plings in the up sector have
+direct SUSY breaking in them and similarly to (3.7) are proportional to the appropriate
+Yukawa couplings and are not suppressed by SM gauge factors. All otherA-terms are
+dominantly given by gauge mediation. The expressions in (4.8) and (4.9 ) are appropriately
+modified.
+As in chiral models, avoiding initial tachyonic right-handed sleptons r equires a tuning
+of order 1 /5 in the D-term for one of the possible signs of its contribution to the soft
+masses-squared. In Class (b) models, this tuning is not necessary if we assume messenger
+parity below the scale M.
+Thestrongest constraints ontheheavy scalearenowtypically set byprocesses involving
+(δu
+12)LLterms in D0−D0box diagrams. At relatively high N(3)
+eff, processes involving
+(δu
+12)LL(δu
+12)RRbecome dominant. Plots of the bounds on F/Mas a function of N(3)
+effare
+given in Figure 4.
+Representative bounds can be taken to be
+Class (a) :F
+M∼>7 TeV ( N(3)
+eff= 28),
+Class (b) :F
+M∼>7 TeV ( N(3)
+eff= 5),(4.19)
+and the bound on the effective scale of SUSY breaking for models of C lass (b) is then
+√
+F∼>70 TeV. (4.20)
+Two-loop contributions then impose the following bounds on N≡N(A)
+effat, say, the point
+in parameter space where all order one numbers in (4.8) are equal t o one:
+Class (a) : N∼>28 for Λ S∼F/M= 7 TeV, µS∼M= 682 TeV ,
+Class (b) : N∼>1 for Λ S∼µS∼√
+F= 70 TeV ( F/M= 7 TeV) .(4.21)
+Atthispointinparameterspace, Class(a)boundscometypicallyfro mleft-handedsleptons,
+whereas in Class (b) all light sparticles (except right-handed charg ed sleptons, which do
+not become tachyonic) give comparable bounds.
+320 20 40 60 80 100300040005000600070008000
+NeffF/Slash1M GeV
+0 2 4 6 810 122000300040005000600070008000
+NeffF/Slash1M GeV
+Figure 4: Relevant D0−D0system bounds on F/Mas a function of N(3)
+efffrom (δu
+12)LL
+(dashed) and ( δu
+12)LL(δu
+12)RR(dotted) processes, for 10-centered models ( M= 682TeV).
+The left graph refers to Class (a) models and the right graph refer s to Class (b) models.
+4.3 Spectra
+We now present some sample spectra for the various partially couple d models discussed
+above. The results are obtained using the program SuSpect [53], sampling the parameter
+space using various effective messenger numbers, specific choices of order one coefficients
+coming from the unknown correlation functions in the CFT sector, a nd various values of
+tanβ. For simplicity, all CFT sector N(A)
+efffactors are taken to be equal to Nand we always
+set the scale Mto its lower bound. In Class (a) models, at the classical level, the sca leM
+can be taken to be much higher than its lower bound, as long as F/Mis kept fixed, but
+this will modify the RG effects (4.6). Additionally, we set the gauge-me diated D-term to
+zero, since the qualitative behavior of the spectra is not modified.
+In the simulations presented in Tables 4, 5, 6, 7 and 8 at the end of th e paper, the
+O(1) coefficients coming from the CFT sector in direct-mediated soft terms (for instance,
+multiplying the first two generation sfermion masses-squared in (4.8 )) have all been taken
+to be one for simplicity. This implies that all the heavy sparticles are de generate, which
+is certainly not the case generically in our models. (Models with separa ble CFT sectors
+involving two different scales of SUSY breaking also seem perfectly via ble, though we do
+not discuss them here.) Otherwise, the spectra we present are ty pical. In Table 3 we
+show the inputs for the simulations and references to the corresp onding tables containing
+33Model Class ΛSF/M µSNTableNLSP mNLSP Fine-
+[TeV][TeV][TeV] [GeV] tuning
+Left-handed (b)456.5 45104˜τ1215 99
+Right-handed (a) 99220205 ˜ν120 16
+Right-handed (b)35 53556˜τ1153 30
+10-centered (b)70 77057˜τ1280 106
+Right-handed (a) 99220158˜χ0168 16
+Table 3: Inputs and some results of simulations of partially coupled mo dels. In the above
+we include references to the corresponding tables containing the lo w-energy spectra. For
+each simulation, the identity of the NLSP, its mass and the amount of fine-tuning for
+tanβ= 10 are presented.
+representative sparticles of the low-energy spectra. In the tab les, the mass scale of the
+heavy sparticles is of order F/M. For completeness, we also include in Table 3 the identity
+of the NLSP, its mass and the amount of fine-tuning (defined below) in each type and class
+of viable model when tan β= 10. A few examples of our full spectra are depicted in Figure
+5.
+All spectra obtained in these partially coupled models exhibit inverted hierarchies. The
+sectors in which the first two generation sfermions are heavy differ amongst the models
+depending on the coupling scenario. Fields that are not directly coup led to the CFT
+sector present a mass pattern similar to models of general gauge m ediation. In our setups,
+the gluino typically interpolates between the heavy and light scales. A s in models of
+gauge mediation, the LSP is always the gravitino. The NLSP can vary b etween sneutrino,
+stau, neutralino and chargino identities (see Tables 4 through 8 for samples of this, and
+[54, 55, 56] for recent NLSP parameter space discussions in gener al gauge mediation).
+Collider studies involving NLSPs of stau, sneutrino, neutralino and ch argino character can
+be found in [15] (and references therein) and in [57, 58, 59].
+In the simulations, the values of µandBµare determined in order to reproduce the
+correct pattern of electroweak symmetry breaking (namely, the correct VEVs for the two
+Higgs fields). A rough order of magnitude estimate of the fine-tunin g of the weak scale in
+these models is then given by ∼2µ2/m2
+Z. In chiral left-handed models and 10-centered
+modelsµisO(600−800) GeV and so these models typically present fine-tuning at the 1%
+level, as expected from the relatively large gluino and stop masses. C hiral right-handed
+couplings have lower values of µ,O(300) GeV, and so these models can present fine-
+tunings milder than 1%. A more accurate quantification of the fine-t uning with respect
+to a parameter λican be given by the Barbieri-Giudice parameter ∆( m2
+Z;λi) [10]. In this
+language, ∆( m2
+Z;λi)∼<100 corresponds to the appropriate fine tuning being milder than
+34the percent level. In the tables, ∆ denotes the strongest fine-tu ning between µ2andBµ,
+calculated by SuSpect [53], which in all cases considered is ∆( m2
+Z;µ2). Fine-tunings with
+respect to other parameters, e.g.the scale Λ S, are expected in our models to be at most
+comparable to the ones presented [60].
+Note that in the spectra in Table 3 left-handed and 10-centered Class (a) models are
+not presented. This is because these models require large values of N(A)
+eff, and so run
+into Landau poles. In fact, many (if not all) of our models exhibit Land au poles for
+the standard model gauge couplings below the GUT scale, and many o f the models do not
+exhibit gaugecoupling unification. We view our models aseffective theo ries validbelow the
+scaleM>, so we do not worry about UV completions above this scale (except f or needing to
+suppress baryon-violating operators by a higher scale Mpl). Models without large numbers
+of degrees of freedom in the CFT sector charged under the stand ard model group can be
+safe from Landau poles within the conformal window. However, our models that do have
+large numbers of degrees of freedom in the CFT sector charged un der the standard model
+group, say N(A)
+eff∼>20, could have Landau poles already below the scale M>, and then these
+models are not really valid as effective field theories (at least not as an alyzed above). This
+means that some of our Class (a) models are not really self-consiste nt.
+To derive a rough estimate of the upper bound on N(A)
+eff, we impose that the conformal
+window is such that a 10−5hierarchy in the up sector can be generated, and demand
+that the strong coupling α3does not blow up in this window. Combining this, we find14
+the order of magnitude constraint −3 +1
+2Nadd∼<7γ, whereNaddstands for additional
+non-MSSM degrees of freedom charged under SU(3) in the conformal window, and γis
+the sum of the relevant anomalous dimensions in the up sector, γ=1
+2(γQ+γ¯u). Under
+the definition of N(A)
+effthrough the gaugino masses (3.6) (relating it in weakly coupled
+theories to, say, thenumber ofpairsof superfields inthefundame ntal andanti-fundamental
+representations), we obtain for Class (a) models N(3)
+eff,(a)≈1
+2(Nadd+ 8) for fully coupled
+models (where the second term here stems from the fact that our definition of N(A)
+eff,(a)
+contains the first two generations of the MSSM, directly coupled to the CFT), yielding
+N(3)
+eff,(a)∼<7γ+ 7. Similarly, in chiral models this gives N(3)
+eff,(a)∼<7γ+ 5, and in 10-
+centered models N(3)
+eff,(a)∼<7γ+ 6. In Class (b) models, the relation between Naddand
+N(A)
+effis less direct, since N(A)
+effonly contains the fields that survive to the lower scale√
+F;
+clearlyN(A)
+eff,(b)<1
+2Nadd. In all our Class (b) models N(A)
+eff,(b)is small and consistent with
+this, andit seems that it should always be possible to add few enough d egrees of freedom at
+14We use here equations for the beta functions that ignore the anom alous dimensions; we know that
+some of our fields always have positive anomalous dimensions and othe rs negative, and we expect some
+overall correction coming from this issue, but we ignore it here since our bounds are up to order one
+numbers anyway.
+35Figure 5: The full sparticle spectrum for the models described in Tab les 4, 5 and 7, for
+tanβ= 10.
+the higher scale to be compatible with the bound on Naddabove. In the above simulations
+we have presented only models that can be consistent with these bo unds. This issue seems
+to favor Class (b) models, although some Class (a) models can also be consistent.
+Notethat our superconformal sector hasan(accidental) U(1)Rsymmetry thatis broken
+at the exit from the conformal window. If this breaking is spontane ous, one may worry
+that we would have a light R-axion [61] as the corresponding Nambu-G oldstone boson.
+However, the R-symmetry in our models is violated by Yukawa coupling s and by terms like
+O2Φ1, and despite the small coefficient of these terms, this is enough to r aise the R-axion
+mass to a level where it does not pose any problems for phenomenolo gy or for cosmology.
+In any case, we assume that the R-symmetry is broken at the exit b y a large amount, so
+that it does not affect the spectrum of soft masses; it may be inter esting to also investigate
+36models where this breaking is small.
+5 Discussion and conclusions
+The Nelson-Strassler mechanism is an elegant means of generating t he pattern of the
+Yukawa couplings ex-nihilio – a sector with a strongly coupled fixed poin t with a finite
+basin of attraction naturally suppresses the Yukawa couplings via R G effects. In this
+paper we explored the extension of the Nelson-Strassler mechanis m to a CFT sector which
+not only determines the Yukawa couplings, but also triggers SUSY br eaking in the MSSM.
+A coarse look suggests that “here be dragons” – using this CFT sec tor to set the scale of
+SUSY breaking implies that its scale cannot be very high, and then, sin ce the CFT sector
+is flavor-dependent (as much as conceivably possible), the Nelson- Strassler mechanism can
+run into problems with FCNC constraints if precautions are not take n when SUSY is
+broken.
+In this work we analyzed these models in detail, focusing on the const raints from
+FCNCs, as well as on other problems that are typical in inverted hier archy models. An
+inverted hierarchy is inevitable in our case, where the strongly coup led SUSY breaking
+CFT sector couples directly to (some of) the first two generation fi elds, giving rise to large
+first two generationssfermionmasses, whereas thethirdgenera tionmasses arisefromgauge
+mediation. We show that despite the apparent difficulties one can con struct models with
+a viable level of fine-tuning.
+In order to achieve low fine-tuning, we had to make several assump tions about the CFT
+sector:
+1. The CFT sector conserves baryon number.
+2. The standard model U(1)Yis embedded in a non-Abelian group which is a subgroup
+of the global symmetry group of the CFT sector dynamics. This is ce rtainly true in
+GUT models (such as SU(5) models, on which our 10-centered model is based), but
+it can be implemented in other cases as well.
+3. In order to suppress flavor-violating effects, including those co ming from mixings at
+the exit from the conformal window (accomplishing a graceful exit) , we require two
+features of the CFT sector, both compatible with the Nelson-Stra ssler construction:
+(a) The CFT sector is separable (for instance by coupling each of th e first two
+generations to a separate CFT sector), meaning that the CFT sec tor does not
+directly produce any flavor changing in the interaction basis.
+37(b) The models are partially coupled – only a subset of the fields in the fi rst two
+generations couple directly to the CFT sector.
+4. The CFT sector has two scales, namely the scale of SUSY breaking√
+Fis much
+smaller than the scale Mwhere the first two generations decouple from the CFT
+sector.
+In this paper we did not attempt to construct a model that obeys a ll of these require-
+ments (as well as a Nelson-Strassler mechanism and dynamical SUSY breaking). It seems
+that such a model should be quite complicated, but we do not see any obstruction inprinci-
+ple to the construction of such models (see [37, 38] for related, th ougha priorinot directly
+applicable, examples in this direction). Our discussion assumed a stro ngly coupled CFT
+sector, but otherwise it is completely general; in particular it applies t o models [62] where
+this sector has a weakly coupled description in higher dimensions. In C lass (b) models
+it is possible that the CFT sector below the scale M, and in particular at the scale of
+SUSY breaking, could become weakly coupled, but we do not discuss t his possibility here.
+As discussed above, our models generally exhibit Landau poles below t he GUT scale, but
+suitable UV completions can perhaps be found, e.g.via Seiberg duality.
+We believe that even without building explicit models, our construction s should have
+variousdistinctive featuresthatcouldperhapsbetestedattheL HC.Itwouldbeinteresting
+to perform a general analysis of the phenomenology of these mode ls, but this is beyond
+the scope of the present paper. Let us just mention here the gen eral form of the spectrum
+that the assumptions above imply. We always have a partial inverted hierarchy for the
+sparticles, where the masses of the scalar components of the sup erfields coupled directly
+to the CFT sector are around 10 TeV. The gluinos are lighter, with a m ass of a few TeV,
+and the rest of the superpartners are all around 2 TeV or below. T he NLSP can be a
+stau, sneutrino, neutralino or chargino, as in general gauge media tion. In comparison to
+gauge mediation, our spectra differ by the heaviness of some of the first two generation
+fields. The partial inverted hierarchy that we presented here cou ld be even more partial
+in separable models with only one SUSY-breaking component, but we d o not analyze this
+case here.
+In comparison to other models with an inverted hierarchy, here the light sparticles have
+a mass spectrum dictated specifically by gauge mediation. Additionally , in our partially-
+coupled models only some chiralities exhibit an inverted hierarchy (see [63] for an example
+of a previous model with related characteristics). We also have rat her heavy gluinos.
+Recently, a suggestion for joining a different model of flavor physic s with supersymmetry
+breaking [24] was realized in [26, 27]; there are many similarities of thes e models to what
+we find here, but also some differences. Note in particular that in our models the difference
+between thethree generations ofthestandard model is generat edpurely dynamically, while
+38in the models of [24, 26, 27] one must put in by hand that only the third generation fields
+areelementary athighenergies. Recent LHC-orientedphenomeno logical studiesofinverted
+hierarchy models have appeared in [64].
+In this paper we have thus far ignored the bounds on new physics co ming from CP-
+violating processes. Even if the hidden sector conserves CP, this is not really justified,
+since the order one CP-violating phase in the CKM matrix generically me ans that the
+CP-violating flavor-changing processes in our model will be of the sa me order as the CP-
+conserving ones.15Thus, in the general case the bounds used for CP-conserving pro cesses
+would be replaced by the bounds including CP-violating processes. Th e fine-tuning re-
+quired in our models would then be enhanced by a factor of order 25, depending on the
+precise model, leading to unacceptably large fine-tuning. The level o f fine-tuning can be
+reduced (with or without the CP-violating operators) by making som e additional assump-
+tions about the structure of our models. For instance, one could a ssume that at high
+energies there is an SU(3)×SU(3) global symmetry that guarantees that the up-Yukawa
+couplings (or the down-Yukawa couplings) are proportional to the identity matrix (this
+symmetry is then broken by the couplings to the hidden sector). Or , one could assume
+that there is some sort of alignment that guarantees that the up- Yukawa couplings (or
+the down-Yukawa couplings) are diagonal in the same basis as the co uplings to the hidden
+sector (this can naturally arise in extra dimensional scenarios, whe re different fields are lo-
+cated at different positions in the extra dimensions). In both cases , most of the discussion
+in our paper is not modified, but flavor-changing processes involving either up or down
+quarks are suppressed. Since in most of our models one of the two t ypes of processes gives
+much stronger bounds than the other, this allows us to reduce the amount of fine-tuning
+by a factor of 8 or so. Note that in all cases the bounds coming from flavor-conserving
+CP-violating processes, such as electric dipole moments [65, 66] (se e alsoe.g.[67]), are less
+constraining or at most comparable. We leave a detailed discussion of the implications of
+these additional assumptions, and of the construction of such mo dels, to future work.
+Finally, we should emphasize that in this paper we did not attempt to ad dress the µ/Bµ
+problem, which is generic in models of gauge mediation. Presumably, re cent solutions to
+this problem (such as [68]) can be applied to our models as well. Note tha t in the natural
+scenario in which the Higgs fields do not couple directly to the CFT sect or,Bµ= 0 at
+leading order at the scale µS, and it was recently claimed [56] that such a scenario is not
+impossible if tan βis large enough. We leave a detailed analysis of the Higgs sector and its
+couplings to future work.
+15We thank P. Paradisi and Y. Nir for discussions on this issue.
+39Acknowledgements
+We would like to thank Oram Gedalia, Vadim Kaplunovsky, Lorenzo Mann elli, Gilad
+Perez and Tomer Volansky for many useful conversations, and Sh amit Kachru, Zohar
+Komargodski and Yossi Nir for fruitful discussions and comments on a draft of this paper.
+This work was supported in part by the Israel-U.S. Binational Scienc e Foundation, by a
+center of excellence supported by the Israel Science Foundation (grant number 1468/06),
+by a grant (DIP H52) of the German Israel Project Cooperation, and by the Minerva
+foundation with funding from the Federal German Ministry for Educ ation and Research.
+tanβmhm˜χ0
+1m˜χ+
+1m˜t1m˜b1m˜τ1m˜u1m˜d1m˜e1M3µ∆
+310875476621502274215228122752164446761218
+101196316362165227221522812275217444562999
+Table 4: Example ofa spectrum for Class (b) modelswith N= 10, when coupling onlyleft-
+handedfirsttwogenerationfieldstoadecomposableCFTsector. W epresent representative
+sparticles of the low energy spectrum. The appropriate heavy spa rticle masses are of order
+F/M. The results are obtained using F/M= 6.5 TeV,µS= ΛS= 45 TeV for two different
+values of tan β, choosing all unknown coefficients coming from the CFT sector to eq ual
+one. The entries for ˜ u1,˜d1,˜e1refer to ˜c1,˜s1,˜µ1as well. ∆ is the fine-tuning of m2
+Zwith
+respect to µ2. All dimensionful quantities are given in GeV.
+tanβmhm˜χ0
+1m˜χ+
+1m˜t1m˜b1m˜τ1m˜u1m˜d1m˜e1m˜νM3µ∆
+398218277567697141722726142123153329534
+10111206238580696140722727143120153324616
+Table 5: The same for Class (a) models with N= 20, when coupling only the right-
+handed first two generation fields to a decomposable CFT sector, u singF/M= 9 TeV,
+µS= 220 TeV and Λ S= 9 TeV.
+40tanβmhm˜χ0
+1m˜χ+
+1m˜t1m˜b1m˜τ1m˜u1m˜d1m˜e1m˜νM3µ∆
+31013143941018109915411291131277268186040864
+101143003361029109715311291132277266186034230
+Table 6: The same for Class (b) models with N= 5, when coupling only the right-
+handed first two generations to a decomposable CFT sector, using F/M= 5 TeV and
+µS= ΛS= 35 TeV.
+tanβmhm˜χ0
+1m˜χ+
+1m˜t1m˜b1m˜τ1m˜d1m˜e1m˜νM3µ∆
+31076647831943211228121135585533538782231
+101196296551959211028021135585523538650106
+Table 7: The same for Class (b) models with N= 5 in the 10-centered coupling scenario
+with a decomposable CFT sector, using F/M= 7 TeV and µS= ΛS= 70 TeV.
+tanβmhm˜χ0
+1m˜χ+
+1m˜t1m˜b1m˜τ1m˜u1m˜d1m˜e1m˜νM3µ∆
+397170267522666190702706191178118130737
+10110168234538664188703707192175118125216
+Table 8: Example of a spectrum for Class (a) models with N= 15, in the chiral right-
+handed coupling scenario with a decomposable CFT sector. The resu lts are obtained using
+F/M= 9 TeV, µS= 220 TeV and Λ S= 9 TeV, choosing the O(1) numbers equal to 1 for
+direct-mediated soft terms and 3 for gauge-mediated ones.
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+arXiv:1001.0638v4  [math.DS]  28 May 2012The Annals of Probability
+2012, Vol. 40, No. 3, 1357–1374
+DOI:10.1214/11-AOP671
+c/circlecopyrtInstitute of Mathematical Statistics , 2012
+MAHARAM EXTENSION AND STATIONARY STABLE
+PROCESSES
+By Emmanuel Roy
+Universit´ e Paris
+We give a second look at stationary stable processes by inter -
+preting the self-similar property at the level of the L´ evy m easure as
+characteristic of a Maharam system. This allows us to derive struc-
+tural results and their ergodic consequences.
+1. Introduction. In a fundamental paper [ 9], Rosi´ nski revealed the hid-
+den structure of stationary symmetric α-stable (SαS) processes. Namely, he
+proved that, following Hardin [ 5], through what is called a minimal spectral
+representation , such a process is driven by a nonsingular dynamical system.
+Such a result was proved to classify those processes accordi ng to their
+ergodicpropertiessuchasvariouskindsofmixing. In[ 13], weusedadifferent
+approach as we considered the whole family of stationary infi nitely divisible
+processes without Gaussian part (called IDp processes ). The key tool there
+was the L´ evy measure system of the process, which was measur e-preserving
+and not just merely nonsingular. So far, in the stable case, t he connection
+between the L´ evy measure and the nonsingular system was not clear. This
+is the purpose of this paper, to fill the gap and go beyond both a pproaches.
+Indeed, we will prove that L´ evy measure systems of α-stable processes
+have the form of a so-called Maharam system. This observatio n has some
+interesting consequences as it allows us to derive very quic kly minimal spec-
+tral representations in the SαScase, to reinforce factorization results, and
+to refine ergodic classification.
+Let us explain very loosely the mathematical features of sta ble distribu-
+tions we will beusing. Observe that stable distributions ar e characterized by
+a self-similar property which is obvious when observing the corresponding
+L´ evy process:
+IfXtisanα-stableL´ evy process,then b−1/αXbthasthesamedistribution.
+Received January 2010; revised March 2011.
+AMS 2000 subject classifications. Primary 60G52, 60G10, 37A40; secondary 37A50.
+Key words and phrases. Stable stationary processes, Maharam system, ergodic prop er-
+ties.
+This is an electronic reprint of the original article published by the
+Institute of Mathematical Statistics inThe Annals of Probability ,
+2012, Vol. 40, No. 3, 1357–1374 . This reprint differs from the original in
+pagination and typographic detail.
+12 E. ROY
+However, if not obvious or useful, this property is also pres ent for any
+α-stable object but takes another form. The common feature is to be found
+in the L´ evy measure:
+Loosely speaking, if {Xt}t∈Sis anα-stable process indexed by a set S,
+then for any positive number c, the image of the L´ evy measure Qby the
+mapRc:={xt}t∈S/mapsto→{cxt}t∈Sisc−αQ.
+This property of the L´ evy measure is characteristic of α-stable processes
+and can be translated into an ergodic theoretic statement:
+The measurable nonsingular flow {Rc}c∈R+is dissipative and the multi-
+plicative coefficient c−αhas an outstanding importance in this matter, since
+it reveals the structure of a Maharam transformation. The im portance is
+even greater when there is more invariance involved (statio naryα-stable
+processes, etc ....), as in the present paper.
+The paper is organized as follows. In Section 2we recall what a spectral
+representation is, and in Section 3, we give the necessary background in
+nonsingular ergodic theory. Maharam systems are introduce d in Section 4,
+and the link with L´ evy measures of stable processes, togeth er with spectral
+representations is exposed in Section 5. Section 6is a refinement of the
+structure of stable processes. We deduce from the preceding results some
+ergodic properties in Section 7.
+2. Spectral representation. We warn the reader that we will, most of
+the time, omit the implicit “ µ-a.e.” or “modulo null sets” throughout the
+document.
+A family of functions {ft}t∈T⊂Lα(Ω,F,µ) where (Ω ,F,µ) is aσ-finite
+Lebesguespaceissaidtobeaspectralrepresentationof SαSprocess{Xt}t∈T
+if
+{Xt}t∈T=/braceleftbigg/integraldisplay
+Ωft(ω)M(dω)/bracerightbigg
+t∈T
+holds in distribution, Mbeing an independently scattered SαS-random
+measure on (Ω ,F) with intensity measure µ.
+We will say that aspectral representation is properif Supp{ft,t∈T}=Ω.
+Of course we obtain a proper representation from a general on e by removing
+the complement of Supp {ft,t∈T}.
+To express that a representation contains the strict minimu m to define
+the process, the notion of minimality has been introduced (H ardin [5]):
+A spectral representation is said to be {ft}t∈T⊂Lα(Ω,F,µ)minimal if
+it is proper and σ(ft
+fs1{fs/ne}ationslash=0},s,t∈T)=F.
+Hardin proved in [ 5] the existence of minimal representations for SαS
+processes.
+In the stationary case ( T=RorZ), Rosi´ nski has explained the form of
+the spectral representation:MAHARAM EXTENSION AND STATIONARY STABLE PROCESSES 3
+Theorem 1 (Rosi´ nski). Let{ft}t∈T⊂Lα(Ω,F,µ)be a minimal repre-
+sentation of a stationary SαS-process; then there exists nonsingular flow
+{φt}t∈Ton(Ω,F,µ)and a cocycle {at}t∈Tfor this flow with values in
+{−1,1}(or in|z|=1in the complex case) such that, for each t∈T,
+ft=at/braceleftbiggdµ◦φt
+dµ/bracerightbigg1/α
+(f0◦φt).
+3. Some terminology. A quadruplet (Ω ,F,µ,T) is called a dynamical
+systemor shortly a systemifTis anonsingular automorphism , that is, a bi-
+jective bi-measurable map such that T∗µ∼µ. IfT∗(µ)=µ, then (Ω ,F,µ,T)
+is ameasure-preserving (abr.m.p.) dynamical system.
+A system (Ω 2,F2,µ2,T2) is said to be a nonsingular (resp. measure pre-
+serving) factor of the system (Ω 1,F1,µ1,T1) if there exists a measurable
+nonsingular (resp. measure preserving) homomorphism between them, that
+is, a measurable map Φ from Ω 1to Ω2such that Φ T1=T2Φ and Φ∗µ1∼µ2
+(resp. Φ∗µ1=µ2). If Φ is invertible and bi-measurable it is called a nonsin-
+gular (resp. measure preserving) isomorphism , and the system is said to be
+nonsingular (resp. measure preserving) isomorphic .
+3.1.Krieger types. Consideranonsingulardynamicalsystem(Ω ,F,µ,T).
+A setA∈Fsuch that µ(A)>0 is said to be periodic of period nifTiA,
+0≤i≤n−1, are disjoint and TnA=Aandwandering ifTiA,i∈Zare
+disjoint. A set is exhaustive if/uniontext
+k∈ZTkA=Ω. A system is conservative if
+there is no wandering set and dissipative if there is an exhaustive wandering
+set.
+(Ω,F,µ,T) is said to be of Krieger type :
+•Inif there exists an exhaustive set of period n;
+•I∞if it is dissipative;
+•II1if there is no periodic set and exists an equivalent finite T-invariant
+measure;
+•II∞if is is conservative with an equivalent infinite T-invariant continuous
+measure but no absolutely continuous finite T-invariant measure;
+•III if there is no absolutely continuous T-invariant measure.
+4. Maharam transformation.
+Definition 2. An m.p. dynamical system is said to be Maharam if
+it is isomorphic to (Ω ×R,F ⊗B,µ⊗esds,/tildewideT), where Tis a nonsingular
+automorphism of (Ω ,F,µ), and/tildewideTis defined by
+/tildewideT(ω,s):=/parenleftbigg
+T(ω),s−lndT−1
+∗µ
+dµ(ω)/parenrightbigg
+.4 E. ROY
+Observe that the dissipative flow {τt}t∈Rdefined by τt:=(ω,s)/mapsto→(ω,s−t)
+commutes with /tildewideT.
+Note that we have chosen the usual additive representation b ut we could
+(andeventually will!)usethefollowingmultiplicative re presentationofaMa-
+haram system. Take 0 <α<2. We can represent (Ω ×R,F⊗B,µ⊗esds,/tildewideT)
+by the system (Ω ×R∗
++,F ⊗B+,µ⊗1
+s1+αds,/tildewideTα), where/tildewideTαis defined by
+/tildewideTα(ω,s):=/parenleftbigg
+T(ω),s/parenleftbiggdT−1
+∗µ
+dµ(ω)/parenrightbigg1/α/parenrightbigg
+.
+Theisomorphismisprovidedbythemap( ω,s)/mapsto→(ω,(2−α)−1/(2−α)e(2−α)s).
+Observe that, under this isomorphism, {τt}t∈Ris changed into {Set/α}t∈R∗
++,
+whereStis the multiplication by ton the second coordinate.
+In [2], the authors proved the following characterization, as a s traightfor-
+ward application of Krengel’s representation of dissipati ve transformations:
+Theorem 3. A system (X,A,ν,γ)is Maharam if and only if there exists
+a measurable flow {Zt}t∈Rcommuting with γsuch that (Zt)∗ν=etν.{Zt}t∈R
+corresponds to {τt}t∈Runder the isomorphism with the Maharam system
+under the additive representation.
+In the original theorem they assumed ergodicity of γto prove that the
+resulting nonsingular transformation Tin the above representation was ac-
+tually living on a nonatomic measure space (Ω ,F,µ). The ergodicity as-
+sumption is therefore not necessary in the way we present thi s theorem.
+We end this section with a very natural lemma which is part of f olklore.
+We omit the proof.
+Lemma 4. Consider two Maharam systems (Ω1×R∗
++,F1⊗B,µ1⊗1
+s1+αds,
+/tildewiderT1)and(Ω2×R∗
++,F2⊗ B,µ2⊗1
+s1+αds,/tildewiderT2), and denote by {St}t∈R∗
++and
+{Zt}t∈R∗
++their respective multiplicative flows. Assume there exists a (measure-
+preserving) factor map (resp. isomorphism) Φbetween the two systems such
+that, for all t∈R∗
++,StΦ=ZtΦ. ThenΦinduces a nonsingular factor map
+(resp. isomorphism) φbetween(Ω1,F1,µ1,T1)and(Ω2,F2,µ2,T1).
+Remark5. ObservealsothattheMaharamsystemsassociatedto(Ω ,F,
+µ1,T) and (Ω ,F,µ2,T) whereµ1∼µ2are isomorphic.
+4.1.Refinements of type III(see [3]).Since the flow {St}t∈Rcommutes
+with/tildewideT, it acts nonsingularly on the space ( Z,ν) of ergodic components of /tildewideT
+and is called the associated flow ofT. This flow is ergodic whenever Tis
+ergodic, and its form allows us to classify ergodic type III s ystems:
+•Tis of type III λ, 0< λ <1, if the associated flow is the periodic flow
+x/mapsto→x+tmod(−logλ);MAHARAM EXTENSION AND STATIONARY STABLE PROCESSES 5
+•Tis of type III 0, if the associated flow is free;
+•Tis of type III 1, if the associated flow is the trivial flow on a singleton.
+In particular /tildewideTis ergodic if and only if Tis of type III 1.
+5. L´ evy measure as Maharam system and spectral representat ions.
+5.1.L´ evy measure of α-stable processes. For simplicity we will only con-
+sider discrete time stationary processes.
+Let usrecall, following [ 8] (seealso[ 13]), that theL´ evy measureof station-
+ary IDp process Xof distribution Pis the shift-invariant σ-finite measure
+onRZ,Q, such that Q(0RZ)=0,/integraltext
+RZ(x2
+0∧1)Q(d{xn}n∈Z)<∞and
+E/bracketleftBigg
+exp/parenleftBigg
+in2/summationdisplay
+k=n1akXk/parenrightBigg/bracketrightBigg
+=exp/bracketleftBigg/integraldisplay
+RZ/parenleftBigg
+exp/parenleftBigg
+in2/summationdisplay
+k=n1akxk/parenrightBigg
+−1−in2/summationdisplay
+k=n1akc(xk)/parenrightBigg
+Q(d{xn}n∈Z)/bracketrightBigg
+for any choice of −∞<n1≤n2<+∞,{ak}n1≤n2∈Rn2−n1.
+cis defined by:
+c(x)=−1 ifx<−1;
+c(x)=xif−1≤x≤1;
+c(x)=1 if x>1.
+The system ( RZ,B⊗Z,Q,S) whereSis the shift on RZis called the L´ evy
+measure system associated to the process X.
+Theα-stable stationary processes, 0 <α<2, are (see Chapter 3 in [ 17])
+completely characterized as those IDp processes such that t heir L´ evy mea-
+sure satisfies
+(Rt)∗Q=t−αQ (5.1)
+for any positive t,Rtbeing the multiplication by t, that is,
+{xn}n∈Z/mapsto→{txn}n∈Z.
+We also recall the fundamental result of Maruyama that allow s to rep-
+resent any IDp process with L´ evy measure Qas a stochastic integral with
+respect to a Poisson measure with intensity Q.
+Theorem 6 (Maruyama representation [ 8]).LetPbe the distribution of
+a stationary IDp process with L´ evy measure Qand((RZ)∗,(B⊗Z)∗,Q∗,S∗)
+the Poisson measure over the L´ evy measure system (RZ,B⊗Z,Q,S). SetX0
+as{xn}n∈Z/mapsto→x0and define, on (RZ)∗, the stochastic integral I(X0)as the6 E. ROY
+limit in probability, as ntends to infinity, of the random variables
+ν/mapsto→/integraldisplay
+|X0|>1/nX0dν−/integraldisplay
+|X0|>1/nc(X0)dQ.
+Then the process {I(X0)◦Sn
+∗}n∈Zhas distribution P.
+5.2.L´ evy measure as Maharam system.
+Theorem 7. Let(RZ,B⊗Z,Q,S)be the L´ evy measure system of an α-
+stable stationary process. Then there exists a probability s pace(Ω,F,µ),
+a nonsingular transformation T, a function f∈Lα(µ)such that, if Mde-
+notes the map (ω,t)/mapsto→tf(ω), then the map Θ:=(ω,t)/mapsto→{M◦/tildewideTn
+α(ω,t)}n∈Z
+yields an isomorphism of the Maharam system (Ω×R+,F⊗B+,µ⊗1
+s1+αds,
+/tildewideTα)with(RZ,B⊗Z,Q,S).
+Proof. First observe that Theorem 3can be applied to ( RZ,B⊗Z,Q,S)
+since the measurable and (obviously) dissipative flow {Ret/α}t∈Rsatisfies
+the hypothesis, thanks to equation ( 5.1). Therefore, there exists an isomor-
+phism Ψ between the Maharam system (Ω ×R+,F ⊗B+,µ⊗1
+s1+αds,/tildewideTα)
+and (RZ,B⊗Z,Q,S) for an appropriate nonsingular system (Ω ,F,µ,T). Set
+f:=Ψ(ω,1)0[i.e., Ψ(ω,1)0is the 0th coordinate of the sequence Ψ( ω,1)],
+and let us check that f∈Lα(µ). Indeed, as Qis a L´ evy measure, we have/integraldisplay
+RZx2
+0∧1Q(d{xn}n∈Z)<∞,
+but since Ψ is an isomorphism and Ψ( ω,t)=Ψ◦St(ω,1) =Rt◦Ψ(ω,1)=
+tΨ(ω,1), we have/integraldisplay
+RZx2
+0∧1Q(d{xn}n∈Z)
+=/integraldisplay
+Ω/integraldisplay
+R+Ψ(ω,t)2
+0∧11
+tα+1dtµ(dω),
+/integraldisplay
+Ω/integraldisplay
+R+(t2Ψ(ω,1)2
+0)∧11
+tα+1dtµ(dω)
+=/parenleftbigg/integraldisplay
+R+z2∧11
+zα+1dz/parenrightbigg/integraldisplay
+Ω|Ψ(ω,1)0|αµ(dω)
+after the change of variable z:=t|Ψ(ω,1)0|. Therefore/integraltext
+Ω|Ψ(ω,1)0|αµ(dω)<
+∞./square
+In the symmetric case we can make the theorem more precise:
+Theorem 8. Let(RZ,B⊗Z,Q,S)be the L´ evy measure system of a sym-
+metricα-stable stationary process. Then there exists a probability spaceMAHARAM EXTENSION AND STATIONARY STABLE PROCESSES 7
+(X,A,ν), a nonsingular transformation R, a function f∈Lα(ν)and a mea-
+surable map ξ:X→{−1,1}such that, if Mdenotes the map (x,t)/mapsto→tf(x),
+then the map (x,t)/mapsto→ {M◦Rαn(x,t)}n∈Zyields an isomorphism between
+(X×R∗,A⊗B,ν⊗1
+|s|1+αds,Tα)with(RZ,B⊗Z,Q,S),Rαbeing defined by
+(x,t)/mapsto→(Rx,ξ(x)t(dR−1
+∗µ
+dµ(x))1/α).
+Proof. Start by applying Theorem 7to the L´ evy measure.
+Observe that the symmetry involves the presence of a measure preserving
+involution I, namely I{xn}n∈Z={−xn}n∈Z.Ialso preserves the L´ evy mea-
+sure of the process. Observe also that Icommutes with the shift and with
+the flow Rt. Therefore/tildewideI:=Θ−1IΘ is a measure preserving automorphism
+of (Ω×R∗
++,F ⊗B,µ⊗1
+s1+αds,/tildewideT),and we can apply Lemma 4to deduce
+that/tildewideIinduces a nonsingular involution φon (Ω,F,µ,T). It is standard that
+such transformation admits an equivalent finite invariant m easure, so, up to
+another measure preserving isomorphism, we can assume that φpreserves
+the probability measure µ.
+Using Rohklin’s structure theorem, the compact factor asso ciated to the
+compact group {Id,φ}tells us that we can represent (Ω ,F,µ,T) as (X×
+{−1,1},A⊗P{− 1,1},ν⊗m,Rξ), where Ris a nonsingular automorphism
+of (X,A,ν),mis the uniform measure on ( {−1,1},P{−1,1}),ξa cocycle
+fromXto{−1,1}andRξ:=(x,ε)/mapsto→(Rx,ξ(x)ε).
+It is now clear that ( X× {−1,1} ×R∗
++,(A ⊗ P{− 1,1})⊗ B,ν⊗m⊗
+1
+s1+αds,/tildewiderSξ) is isomorphic to ( X×R∗,A⊗B,ν⊗1
+|s|1+αds,Rα) thanks to the
+mapping ( x,ε,t)/mapsto→(x,21/αεt) andRα:=(x,t)/mapsto→(Rx,ξ(x)(dR−1
+∗µ
+dµ(x))1/αt).
+/square
+5.3.Spectral representation. It is now very easy to derive spectral rep-
+resentations from the above results. In particular, if ( RZ,B⊗Z,Q,S) is the
+L´ evy measure system of an SαSstationary process, under the notation of
+Theorem 8, (X,A,ν) together with the function f∈Lα(ν), the cocycle φ
+and the nonsingular automorphism Tyields a spectral representation of the
+process. Indeed, by building the Poisson measure over ( X×R∗,A⊗B,ν⊗
+1
+|s|1+αds,Rα) and by applying to it, fandξ(Theorem 3.12.2, page 156
+in [16]), we recover the SαSprocess with L´ evy measure Q, which proves
+the validity of the spectral representation. The minimalit y can be obtained
+without difficulty thanks to Proposition 2.2 in [ 10].
+5.4.Maharam systems as L´ evy measure. We can ask if whether a Ma-
+haram system (Ω ×R+,F ⊗B+,µ⊗1
+s1+αds,/tildewideTα) can be coded into a L´ evy
+measure system of a stable process. We can answer this questi on affirma-
+tively in the only interesting case, that is, when the Mahara m system has no
+finiteabsolutely continuous /tildewideTα-invariant measure,thatis,whentheresulting
+L´ evy measure system leads to an ergodic stable process.8 E. ROY
+Recall that a Maharam system (Ω ×R+,F ⊗B+,µ⊗1
+s1+αds,/tildewideTα) has no
+finiteabsolutely continuous /tildewideTα-invariant measureifandonlyifthenonsingu-
+lar system (Ω ,F,µ,T) has the same property. But then a famous theorem
+of Krengel [ 7] shows that such a system possesses a 2-generator, that is,
+there exists a measurable function f:Ω→{0,1},µ{f=1}<∞, such that
+σ{f◦Tn,n∈Z}=F.
+To be more precise, this means that, up to isomorphism, (Ω ,F,µ,T) can
+be represented as ( {0,1}Z,B({0,1}Z),ν,S) for an appropriate measure ν,
+whereSis the shift transformation. Then the Maharam system can be r ep-
+resented as ( {0,1}Z×R∗
++,B({0,1}Z)⊗B,ν⊗1
+s1+αds,/tildewideSα). But if ϕis the
+map ({xn}n∈Z,t)/mapsto→ {txn}n∈ZandQ=ϕ∗(ν⊗1
+s1+αds), we obtain a L´ evy
+measure system ( RZ,B⊗Z,Q,S) of anα-stable system, as {xn}n∈Z/mapsto→{x0}
+is inLα(µ) (see the proof of Theorem 7). Moreover, the sequence {yn}n∈Z
+takes only two values, 0 or sup {yn}n∈ZQ-almost everywhere ( {yn}n∈Zcan
+not be identically zero with positive measure as such a const ant sequence
+forms a shift-invariant set of finite measure); therefore, ϕ−1exists and is
+defined by {yn}n∈Z/mapsto→(sup{yn}n∈Z,{yn
+sup{yn}n∈Z}n∈Z).
+({0,1}Z×R∗
++,B({0,1}Z)⊗B,ν⊗1
+s1+αds,/tildewideSα)isisomorphicto( RZ,B⊗Z,Q,S).
+6. Refinements of the representation. Ergodic stationary processes are
+building blocks of stationary processes; prime numbers are the building
+blocks of integers; factors are building blocks of Von Neuma nn algebras etc.
+What are the building blocks of stationary infinitely divisi ble processes? Let
+us get more precise.
+Given a stationary ID process X, what are the solutions to the equation
+(in distribution)
+X=X1+X2,
+whereX1andX2are independent stationary ID processes. Of course, if Q
+is the L´ evy measure of X, then taking X1with L´ evy measure c1QandX2
+with L´ evy measure c2Qwithc1+c2=1 gives a solution. If these are the
+only solutions, we said in [ 12] thatXispure, meaning that is impossible to
+reduceXto “simpler” pieces. It was then very easy to show that Xis pure
+if and only if its L´ evy measure is ergodic.
+Proposition 9. A stationary IDp process Xis pure if and only if its
+L´ evy measure Qis ergodic.
+Proof. Assume Q is not ergodic. There exists a partition of RZinto
+two shift invariant sets AandBboth of positive measure. Therefore, Q|A
+andQ|Bcan be taken as L´ evy measures of two stationary IDp processe sXA
+andXBand taking them independently leads to
+X=XA+XB
+in distribution, as Q=Q|A+Q|B.MAHARAM EXTENSION AND STATIONARY STABLE PROCESSES 9
+In the converse, assume Qis ergodic, and supposethere exist independent
+stationary IDp processes X1andX2with L´ evy measure Q1andQ2such
+that
+X=X1+X2
+holds in distribution. As Q=Q1+Q2we getQ1≪Q. But as Qis ergodic,
+this in turns implies that there exists c >0 such that Q1=cQand thus
+Q2=(1−c)Q./square
+In this section, we will try to comment the above equation acc ording to
+the Krieger type of the associated nonsingular transformat ion. A descrip-
+tion of the interesting class of those stable processes driv en by nonsingular
+transformations of type III 0is unknown.
+6.1.The type III1case, pure stable processes. It was an open question
+whether there exist pure stable processes. It can now be solv ed thanks to
+the Maharam structure of the L´ evy measure: an α-stable process is pure if
+and only if the underlying nonsingular system is of type III 1.
+Theexistence of purestableprocesses (guaranteed bytheco mments made
+in Section 5.4) is reassuring as it validates the specific study of stable pr o-
+cesses.
+6.2.The type IIIλcase,0<λ<1.In this section we derive the form of
+thoseα-stable processes associated with an ergodic, type III λnonsingular
+automorphism, 0 <λ<1.
+6.2.1.Semi-stable stationary processes. An infinitely divisible probabil-
+ity measure µonRdis called α-semi-stable withspanbif its Fourier trans-
+form satisfies
+ˆµ(z)bα= ˆµ(bz)ei/an}bracketle{tc,z/an}bracketri}ht
+for some c∈Rd.
+By extension, an α-semi-stable process process is a process whose finite-
+dimensional distributions are α-semi-stable. Using once again results of
+Chapter 3 in [ 17], one gets the following characterization of α-semi-stable
+stationary processes:
+A shift-invariant L´ evy measure Qon (RZ,B⊗Z) is the L´ evy measure of an
+α-semi-stable stationary process of span b>0 if and only if it satisfies
+(Rb)∗Q=b−αQ,
+whereRbis the multiplication by b
+{xn}∈Z/mapsto→{bxn}∈Z.
+Of course by iterating Rb, we easily observe that ( Rbn)∗Q=b−nαQfor all
+n∈Z.10 E. ROY
+6.2.2.Discrete Maharam extension. Assume(Ω ,F,µ,T)isanonsingular
+system such that there exists λ >0 so thatdT−1
+∗µ
+dµ∈ {λn,n∈Z}µ-almost
+everywhere. We can form its discrete Maharam extension , that is, the m.p.
+system (Ω ×Z,F ⊗ B,µ⊗λndn,/tildewideT) where λndnstands for the measure/summationtext
+n∈Zλnδnon (Z,B) and/tildewideTis defined by
+/tildewideT(ω,n)=/parenleftbigg
+Tω,n−logλdT−1
+∗µ
+dµ(ω)/parenrightbigg
+.
+6.2.3.Ergodic decomposition of Maharam extension of type IIIλtransfor-
+mations. Let (Ω,F,µ,T) be an ergodic type III λsystem. Up to a change
+of measure we can assume that the Radon–Nykodim derivative t ake its val-
+ues in the group {λn,n∈Z}wherer(T) ={0,λn,n∈Z,+∞}is the ratio
+set ofT(see [6]). Therefore, the discrete Maharam extension (Ω ×Z,F ⊗
+B,βµ⊗λndn,/tildewideT), where β:=/integraltext−lnλ
+0e−sds, exists. Now form the product
+system/parenleftbigg
+Ω×Z×[0,−lnλ[,F ⊗B⊗B ([0,−lnλ[),βµ⊗λndn⊗e−s
+βds,/tildewideT×Id/parenrightbigg
+.
+The dissipative nonsingular flow St:(ω,n,s)/mapsto→(ω,n+⌊s−t
+−lnλ⌋,s−t+
+lnλ⌊s−t
+−lnλ⌋) satisfies St◦/tildewideT×Id=/tildewideT×Id◦Stand (St)∗µ⊗λndn⊗e−sds=
+e−tµ⊗λndn⊗e−sds, and it is very easy to see that ( Z×[0,−lnλ[,B ⊗
+B([0,−lnλ[),λndn⊗e−sds)isjustareparametrizationof( R,B,esds),thanks
+to the mapping ( n,s)/mapsto→−nlnλ−s.
+Therefore (Ω ×Z×[0,−lnλ[,F⊗B⊗B ([0,−lnλ[),µ⊗λndn⊗e−sds,/tildewideT×
+Id) can be seen as the Maharam extension of (Ω ,F,µ,T).
+It remains to prove the ergodicity of (Ω ×Z,F ⊗B,βµ⊗λndn,/tildewideT). This
+follows, for example, from Corollary 5.4 in [ 18], as the ratio set is precisely
+the set of essential values of the Radon–Nykodim cocycle.
+We then obtain the ergodic decomposition of the Maharam exte nsion: it
+is the discrete Maharam extension (Ω ×Z,F⊗B,βµ⊗λndn,/tildewideT) randomized
+by the measuree−s
+βds on [0,−lnλ[.
+6.2.4.Application to stable processes. Let (RZ,B⊗Z,Q,S) be the L´ evy
+measure system of an α-stable process driven by an ergodic type III λsys-
+tem (Ω,F,µ,T), and let f∈Lα(µ) be given as in Theorem 7. Letb >1
+so thatb−α=λ, we need to obtain a multiplicative version of the above
+structure adapted to our parameters. Up to a change of measur e we can
+assume that (dT−1
+∗µ
+dµ)1/α∈ {bn,n∈Z}µ-almost everywhere. Consider the
+discrete Maharam extension (Ω ×Gb,F ⊗B,βµ⊗mb,/tildewideT) (in a multiplica-
+tive representation) where β=/integraltextb
+11
+s1+αds,mbis the measure/summationtext
+g∈Gbg−αδg
+onGb:={bn,n∈Z}and/tildewideT:=(ω,g)/mapsto→(Tω,g(dT−1
+∗µ
+dµ(ω))1/α). FormMAHARAM EXTENSION AND STATIONARY STABLE PROCESSES 11
+thesystem ( RZ,B⊗Z,Qr,S) asafactorof(Ω ×Gb,F⊗B,βµ⊗mb,/tildewideT)given
+bythemapping ϕ:=(ω,g)/mapsto→{M◦/tildewideTn(ω,g)}n∈ZwhereM(ω,g)=gf(ω) and
+Qr=ϕ∗(βµ⊗mb).
+Now, as above, we recover the Maharam extension of (Ω ,F,µ,T) by con-
+sidering (Ω ×Gb×[1,b[,F⊗B⊗B ([1,b[),βµ⊗mb⊗1
+βs1+αds,/tildewideT×Id). As the
+system(Gb×[1,b[,B⊗B([1,b[),mb⊗1
+s1+αds)isisomorphicto( R∗
++,B,1
+s1+αds)
+thanks to ( g,t)/mapsto→gt, we obtain ( RZ,B⊗Z,Q,S) by applying the map
+({xn}n∈Z,t)/mapsto→{txn}n∈Zto (RZ×[1,b[,B⊗Z⊗B([1,b[),Qr⊗1
+βs1+αds,S×Id).
+At last, we can check that Qris a L´ evy measure, and indeed we know that/integraldisplay
+RZ(x2
+0∧1)Q(d{xn}n∈Z)<+∞,
+but/integraldisplay
+RZ(x2
+0∧1)Q(d{xn}n∈Z)=/integraldisplayb
+1/parenleftbigg/integraldisplay
+RZ((sx0)2∧1)Qr(d{xn}n∈Z)/parenrightbigg1
+βs1+αds;
+therefore, for some 1 ≤s<b,/integraltext
+RZ((sx0)2∧1)Qr(d{xn}n∈Z)<+∞,and this
+is enough to prove that Qris a L´ evy measure.
+(RZ,B⊗Z,Qr,S) is the L´ evy measure system of an α-semi-stable station-
+ary process with span b. Heuristically, if Xhas L´ evy measure Q,Xcan
+be thought of as the continuous sum of independent processes Yt, 1≤t<b
+weighted by the probability measure1
+βs1+αdswhere1
+tYthas L´ evy mea-
+sureQr. More formally, if ((Ω ×Gb×[1,b[)∗,(F ⊗ B ⊗ B ([1,b[))∗,(βµ⊗
+mb⊗1
+βs1+αds)∗,(/tildewideT×Id)∗) denotes the Poisson suspension over (Ω ×Gb×
+[1,b[,F ⊗B ⊗B ([1,b[),βµ⊗mb⊗1
+βs1+αds,/tildewideT×Id), then, if Idenotes the
+stochastic integral as in Theorem 6,X:={I{M1}◦(/tildewideT×Id)n
+∗}n∈Zhas L´ evy
+measure QandY:={I{M2}◦(/tildewideT×Id)n
+∗}n∈ZwhereM1(ω,g,s) =sgf(ω)
+andM2(ω,g,s)=gf(ω).
+We therefore observe that Xis entirely determined by a pure α-semi-
+stable stationary process with span b,Y. It is very easy to see that XandY
+share the same type of mixing.
+6.2.5.Examples. Itisnotdifficulttoexhibitexamplesof stableprocesses
+of the kind described above, as the structure detailed allow s us to build
+such processes. We can, for example, consider the systems Tp,1
+2< p <1
+introduced in [ 4]. We will follow the presentation given in ([ 1], page 104).
+Let Ω be the group of dyadic integers, let τacts by translation by 1 on Ω
+andfor1
+2<p<1, letµpbeaprobability measureon Ωdefinedoncylindersby
+µp([ε1,...,εn])=n/productdisplay
+k=1p(εk),
+wherep(0)=1−pandp(1)=p.12 E. ROY
+If we set1−p
+p=λ, we get
+dτ−1
+∗µp
+dµp=λφ,
+whereφ(x) = min{n∈N,xn= 0}−2. It is proved in [ 4] that the discrete
+Maharam extension (Ω ⊗Z,F ⊗B,µ⊗λndn,/tildewideτ) is ergodic.
+We can form a new system, which will be the L´ evy measure syste m of
+a stationary semi-stable process with span λα, thanks to the following map:
+f:(ω,n)/mapsto→λαn/summationdisplay
+i≥1ωi2−i.
+The L´ evy measure Qris the image of µ⊗λndnby the map ( ω,n)/mapsto→
+{f◦/tildewideτk(ω,n)}k∈Z.
+By randomizing this L´ evy measure as explained above, we obt ain the
+L´ evy measure Qof a stationary α-stable process; that is, Qis the image
+measure of Qr⊗1
+βs1+αdsby the map
+({xn}n∈Z,t)/mapsto→{txn}n∈Z.
+To obtain a realization of these two processes as stochastic integrals over
+Poisson suspensions,we can proceed as explained at theend o f thepreceding
+section.
+Anticipating the next sections, we derive the ergodic prope rties of these
+processes:
+τbeing of type III λ, the Maharam extension is of type II ∞which means
+that the L´ evy measure system of the corresponding stationa ryα-stable pro-
+cess (with L´ evy measure Q) is of type II ∞. Therefore the associated Poisson
+suspension is weakly mixing. As stochastic integrals with r espect to this
+Poisson suspension, both processes (with L´ evy measures QandQr) are
+weakly mixing.
+Thanks to (Lemma 1.2.10, page 30 in [ 1]),τis rigid for the sequence
+{2n}n∈N. Therefore, by Theorem 18(or with a slight adaptation for the
+semi-stable case), both processes are also rigid for the sam e sequence.
+6.3.The type IandIIcases.This case is easy to deal with as we can
+assume the associated ergodic nonsingular system is actual ly measure pre-
+serving, that is, the L´ evy measure system ( RZ,B⊗Z,Q,S) is isomorphic to
+(Ω×R∗
++,F ⊗B,µ⊗1
+s1+αds,/tildewideT) whereTpreserves µand/tildewideTacts asT×Id,
+that is,/tildewideT(ω,t) = (Tω,t). Considering f∈Lα(µ) furnished by Theorem 7,
+(Ω×R∗
++,F ⊗B,µ⊗1
+s1+αds,/tildewideT) is isomorphic to ( RZ×R∗
++,B⊗Z⊗B,Qs⊗
+1
+s1+αds,S×Id) through the map ( ω,t)/mapsto→({f◦Tn(ω)}n∈Z,t) and
+/integraldisplay
+RZ(x2
+0∧1)Q(d{xn}n∈Z)=/integraldisplay
+Ω/integraldisplay
+R+((tf(ω))2∧1)1
+tα+1dtµ(dω)MAHARAM EXTENSION AND STATIONARY STABLE PROCESSES 13
+=/integraldisplay
+R+/parenleftbigg/integraldisplay
+RZ((tx0)2∧1)Qs(d{xn}n∈Z)/parenrightbigg1
+tα+1dt
+<+∞.
+For the same reason as above, Qsis a L´ evy measure. We draw the same
+conclusions as in the preceding section, taking into accoun t that the weight
+is now the infinite measure1
+tα+1dtonR∗
++, andQscan be any L´ evy measure
+(of a stationary IDp process).
+7. Ergodicproperties. SomeergodicpropertiesofgeneralIDpstationary
+processes have been given in terms of ergodic properties of t he L´ evy measure
+system in [ 13]. For an α-stable stationary processes, it is more interesting to
+give them in terms of the associated nonsingular system (Ω ,F,µ,T). This
+work has been undertaken in the symmetric ( SαS) case in a series of papers
+(see, in particular, [ 11] and [15]).
+We have a new tool to deal with this problem:
+As the L´ evy measure of an α-stable stationary processes can now be seen
+as the Maharam extension (Ω ×R∗
++,F ⊗ B,µ⊗1
+s1+αds,/tildewideT) of the system
+(Ω,F,µ,T), it suffices to connect ergodic properties of Tand/tildewideT, and then
+apply the general results relating ergodic properties of a s tationary IDp
+process with respect to those of its L´ evy measure system.
+Observe that linking ergodic properties of Tand/tildewideTis a general problem
+in nonsingular ergodic theory which is of great interest.
+We will illustrate this in the following sections dealing wi th mixing, K-
+property and rigidity, the last two having been neglected in theα-stable
+literature.
+7.1.Mixing. First recall that if Sis a nonsingular transformation of
+a measure space ( X,A,m), it induces a unitary operator USonL2(m) by
+USf(x)=/radicalBigg
+dS−1∗µ
+dµ(x)f◦S(x).
+We first give a general result:
+Proposition 10. The Maharam system (Ω×R∗
++,F⊗B+,µ⊗1
+s1+αds,/tildewideTα)
+is of zero type if and only if for all f∈L2(µ),/an}⌊ra⌋ketle{tUn
+Tf,f/an}⌊ra⌋ketri}htL2(µ)→0asntends
+to infinity.
+Theproofcan beextracted from [ 11] but follows also from theobservation
+thattheconditionsbelowareequivalent (weassumethat µisaprobability):
+(1) for all f∈L2(µ),/an}⌊ra⌋ketle{tUn
+Tf,f/an}⌊ra⌋ketri}htL2(µ)→0 asntends to infinity;
+(2)|logdT−n
+∗µ
+dµ|→∞in probability;14 E. ROY
+(3)m(Aε∩/tildewideTn
+αAε)→0 for all 0 <ε<1 wherem=µ⊗1
+s1+αdsandAε:=
+Ω×[ε,1
+ε];
+(4)/tildewideTαis of zero type.
+Combining this result with the characterization of the L´ ev y measure system
+as a Maharam system and the mixing criteria found in [ 13], we obtain the
+following theorem, already known in the SαS-case (see [ 11]):
+Theorem 11. A stationary stable process (RZ,B⊗Z,P,S)with associ-
+ated system (Ω,F,µ,T)is mixing if and only if for all f∈L2(µ),/an}⌊ra⌋ketle{tUn
+Tf,
+f/an}⌊ra⌋ketri}htL2(µ)→0asntends to infinity.
+In the forthcoming sections, we are interested in less-know n ergodic prop-
+erties (Kproperty and rigidity) that have been neglected in the α-stable
+literature.
+7.2.Kproperty.
+Definition 12 (see[19]). Aconservativenonsingularsystem(Ω ,F,µ,T)
+is aK-system if there exists a sub- σ-algebra G ⊂ Fsuch that T−1G ⊂ G,
+T−nG ↓{Ω,∅},TnG ↑Fanddµ
+dT∗µisG-measurable.
+AK-system is always ergodic (see [ 19]) .
+Definition 13. A measure-preserving system ( X,A,m,S) is remotely
+infinite if there exists a sub- σ-algebra C ⊂Asuch that T−1C ⊂C,SnC ↑A
+and/intersectiontext
+n≥1S−nCcontains zero or infinite measure sets only.
+Proposition 14. If(Ω,F,µ,T)is aK-system which is not of type II1,
+then its Maharam extension (Ω×R∗
++,F ⊗B+,µ⊗1
+s1+αds,/tildewideTα)is remotely
+infinite.
+Proof. LetGbe as in Definition 12. Observe that, asdµ
+dT∗µisG-measur-
+able,G ⊗B+is/tildewideTα-invariant, that is, /tildewideT−1
+αG ⊗ B +⊂G ⊗ B +. Indeed, take g
+G-measurable and fB+-measurable, and we get
+g⊗f(/tildewideTα(ω,s))=/parenleftbigg
+g(Tω),s/parenleftbiggdT−1
+∗µ
+dµ(ω)/parenrightbigg1/α/parenrightbigg
+=/parenleftbigg
+g(Tω),s/parenleftbiggdµ
+dT∗µ(Tω)/parenrightbigg1/α/parenrightbigg
+.
+We aregoing toshowthat P:=/intersectiontext
+n∈N/tildewideT−n
+αG⊗B+only contains sets of zero
+or infinite measure. Observe that, as Stcommutes with /tildewideTαand preservesMAHARAM EXTENSION AND STATIONARY STABLE PROCESSES 15
+G ⊗ B+for allt >0, thenS−1
+t(/tildewideT−n
+αG ⊗ B+)⊂/tildewideT−n
+αG ⊗ B+and therefore
+S−1
+tP ⊂Pfor allt>0. Now consider the measurable union, say K, ofP-
+measurable sets of finite and positive measure. It is a /tildewideTα-invariant set and
+aSt-invariant set as well. Recall that the nonsingular action o f the flow St
+on the ergodic components of (Ω ×R∗
++,F ⊗B+,µ⊗1
+s1+αds,/tildewideTα) is ergodic;
+therefore, if K/ne}ationslash=∅, thenK=Ω×R∗
++mod.ν⊗1
+s1+αds.
+AssumeK=Ω×R∗
++. This implies that the measure µ⊗1
+s1+αdsisσ-finite
+onP, and therefore Pis a factor of (Ω ×R∗
++,F ⊗B+,µ⊗1
+s1+αds,/tildewideTα). Now
+consider the quotient space (Ω ×R∗
++)/upslopePthat we can endow, with a slight
+abuse of notation with the σ-algebra P. Letρbe the image measure of
+µ⊗1
+s1+αdsby the projection map π. On ((Ω ×R∗
++)/upslopeP,P,ρ)/tildewideTαand the
+dissipative flow Stinduce a transformation Uand a dissipative flow Ztthat
+satisfy
+π◦/tildewideTα=U◦π,π◦St=U◦πandU◦Zt=Zt◦U.
+Of course, thanks to Theorem 3, ((Ω×R∗
++)/upslopeP,P,ρ,U) is a Maharam
+system; therefore, we can represent it as ( Y×R∗
++,K⊗B+,σ⊗1
+s1+αds,/tildewideLα)
+for a nonsingular system ( Y,K,σ,L). Applying Lemma 4,πinduces a non-
+singular factor map Γ from (Ω ,G,µ,T) to (Y,K,σ,L), which means that
+there exists an R-invariant σ-algebra Z ⊂ Gsuch that Γ−1K=Z. But
+we can observe, that for all n >0, the factor /tildewideT−n
+αG ⊗B+corresponds to
+a Maharam system that corresponds to the factor T−nGof (Ω,G,µ,T).
+Therefore, for all n>0,Z ⊂T−nG, that is, Z ⊂/intersectiontext
+n∈NT−nG={Ω,∅}. This
+means that K={Y,∅}, or, in other words, that ( Y,K,σ,L) is the trivial
+(one-point) system. ( Y×R∗
++,K ⊗B+,σ⊗1
+s1+αds,/tildewideLα) then possesses lots
+of invariant sets of positive finite measure, for example A:=Y×[1,2]. But
+π−1(A) is in turn a positive and finite measure invariant set for the system
+(Ω×R∗
++,F⊗B+,µ⊗1
+s1+αds,/tildewideTα), and the existence of such set is impossible
+in a Maharam extension of an ergodic system which does not pos ses a finite
+T-invariant probability measure ν≪µ. We can conclude that K=∅.
+To prove that (Ω ×R∗
++,F ⊗B +,µ⊗1
+s1+αds,/tildewideTα) is remotely infinite, it
+remains to show that/logicalortext
+n∈Z/tildewideT−n
+αG⊗B+=F ⊗B+. We only sketch the proof
+which consists of verifying that the operation of taking nat ural extension
+and Maharam extension commute.
+Of course, we have/logicalortext
+n∈Z/tildewideT−n
+αG⊗B+⊂F ⊗B +. It is not difficult to check
+that/logicalortext
+n∈Z/tildewideT−n
+αG ⊗B+corresponds to a Maharam system that comes from
+aσ-algebra H ⊂ F. But we also have G ⊂ Hand asT−1H=H, we get/logicalortext
+n∈ZT−nG ⊂H. By assumption,/logicalortext
+n∈ZT−nG=F, and we deduce H=F
+which implies/logicalortext
+n∈Z/tildewideT−n
+αG ⊗B+=F ⊗B+./square
+As before we deduce the following result for α-stable stationary processes:16 E. ROY
+Theorem 15. Let(RZ,B⊗Z,P,S)be a stationary stable process with
+associated system (Ω,F,µ,T). If(Ω,F,µ,T)isKand not of type II1, then
+(RZ,B⊗Z,P,S)isK.
+Proof. From Proposition 14, we know that the L´ evy measuresystem of
+thestableprocessis remotely infinite.ThecorrespondingP oisson suspension
+isKby a result from [ 14]. By applying Maruyama’s representation Theorem
+(Theorem 6), we recover the stable process as a factor of the suspension ,
+which therefore inherits the Kproperty. /square
+Recall that in the probability preserving context, Kis strictly stronger
+than mixing. In [ 11], to produce examples of mixing α-stable stationary pro-
+cesses that were not based on dissipative nonsingular syste ms, the authors
+considered indeed null recurrent Markov chains as base syst ems. These sys-
+tems are well-known examples of K-systems; therefore Theorem 15shows
+that the associated α-stable stationary processes are not just merely mixing
+but are indeed K.
+7.3.Rigidity. We recall that a system (Ω ,F,µ,T) is rigid if there ex-
+ists an increasing sequence nksuch that Tnk→Id in the group of nonsin-
+gular automorphism on (Ω ,F,µ) [the convergence being equivalent to the
+weak convergence in L2(µ) of the associated unitary operators UTnk:f/mapsto→/radicalbigg
+dT−nk∗µ
+dµf◦Tnkto the identity]. Observe that in the finite measure case,
+rigidity does not imply ergodicity but prevents mixing.
+Proposition 16. The Maharam system (Ω×R∗
++,F⊗B+,µ⊗1
+s1+αds,/tildewideTα)
+is rigid for the sequence nkif and only (Ω,F,µ,T)is rigid for the se-
+quencenk.
+Proof. First observethat themap T/mapsto→/tildewideTαis acontinuous group homo-
+morphism from the group of nonsingular automorphism of (Ω ,F,µ) to the
+group of measure preservingautomorphism of (Ω ×R∗
++,F⊗B+,µ⊗1
+s1+αds).
+AsTnk→Id, then/tildewideTnkα→Id; therefore/tildewideTnkαis rigid for the sequence nk.
+Conversely, if /tildewideTαis rigid for the same sequence, then, as
+/an}⌊ra⌋ketle{tUnk
+/tildewideTαf⊗g,f⊗g/an}⌊ra⌋ketri}htL2(µ⊗1/(s1+α)ds)≤/⌊ard⌊lg/⌊ard⌊l2
+2/an}⌊ra⌋ketle{tUnk
+Tf,f/an}⌊ra⌋ketri}htL2(µ)≤/⌊ard⌊lg/⌊ard⌊l2
+2/⌊ard⌊lf/⌊ard⌊l2
+2
+and/an}⌊ra⌋ketle{tUnk
+/tildewideTαf⊗g,f⊗g/an}⌊ra⌋ketri}htL2(µ⊗1/(s1+α)ds)→/⌊ard⌊lg/⌊ard⌊l2
+2/⌊ard⌊lf/⌊ard⌊l2
+2, we get /an}⌊ra⌋ketle{tUnk
+Tf,f/an}⌊ra⌋ketri}htL2(µ)→
+/⌊ard⌊lf/⌊ard⌊l2
+2; thusTis rigid. /square
+We need the following general result:MAHARAM EXTENSION AND STATIONARY STABLE PROCESSES 17
+Proposition 17. A stationary IDp stationary process (RZ,B⊗Z,P,S)is
+rigid for the sequence nkif and only if its L´ evy measure system (RZ,B⊗Z,Q,S)
+is rigid for the sequence nk.
+Proof. Consider X:={Xn}n∈ZwhereXn:={xk}k∈Z/mapsto→xnonRZand
+let/an}⌊ra⌋ketle{ta,X/an}⌊ra⌋ketri}htbe a finite linear combination of the coordinates. exp i/an}⌊ra⌋ketle{ta,X/an}⌊ra⌋ketri}ht −
+E[expi/an}⌊ra⌋ketle{ta,X/an}⌊ra⌋ketri}ht] is a centered square integrable vector under Pwhose spectral
+measure (under P) isλa:=|E[expi/an}⌊ra⌋ketle{ta,X/an}⌊ra⌋ketri}ht]|2/summationtext∞
+k=11
+k!σ∗k
+awhereσais the spec-
+tral measure of exp i/an}⌊ra⌋ketle{ta,X/an}⌊ra⌋ketri}ht−1 underQ(see [13]). Therefore/hatwiderσa(nk)→/hatwiderσa(0)
+if and only if /hatwiderλa(nk)→/hatwiderλa(0). This implies that exp i/an}⌊ra⌋ketle{ta,X/an}⌊ra⌋ketri}ht−E[expi/an}⌊ra⌋ketle{ta,X/an}⌊ra⌋ketri}ht]
+is a rigid vector for nkunderPif and only if exp i/an}⌊ra⌋ketle{ta,X/an}⌊ra⌋ketri}ht−1 is a rigid vector
+fornkunderQ. Observe now that the smallest σ-algebra generated by vec-
+tors of the kind exp i/an}⌊ra⌋ketle{ta,X/an}⌊ra⌋ketri}ht−E[expi/an}⌊ra⌋ketle{ta,X/an}⌊ra⌋ketri}ht] underPisB⊗Z, and the same
+is true with vectors of the kind exp i/an}⌊ra⌋ketle{ta,X/an}⌊ra⌋ketri}ht−1 underQ. As in any dynam-
+ical system if there exists a rigid vector for the sequence nk, there exists
+a nontrivial factor which is rigid for the sequence nk, we get the announced
+result./square
+Theorem 18. A stationary stable process (RZ,B⊗Z,P,S)with associ-
+ated system (Ω,F,µ,T)is rigid for the sequence nkif and only (Ω,F,µ,T)
+is rigid for the sequence nk.
+Proof. This is the combination of the last two results. /square
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+France
+E-mail: roy@math.univ-paris13.fr
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+arXiv:1001.0639v1  [cs.DS]  5 Jan 2010Optimal Exploration of Terrains with Obstacles
+Jurek Czyzowicz∗†David Ilcinkas‡Arnaud Labourel∗§Andrzej Pelc∗¶
+September 18, 2018
+Abstract
+A mobile robot represented by a point moving in the plane has to explor e an unknown
+terrain with obstacles. Both the terrain and the obstacles are mod eled as arbitrary polygons.
+We consider two scenarios: the unlimited vision , when the robot situated at a point pof the
+terrain explores (sees) all points qof the terrain for which the segment pqbelongs to the terrain,
+and the limited vision , when we require additionally that the distance between pandqbe at
+most 1. All points of the terrain (except obstacles) have to be exp lored and the performance of
+an exploration algorithm is measured by the length of the trajector y of the robot.
+For unlimited vision we show an exploration algorithm with complexity O(P+D√
+k), where
+Pis the total perimeter of the terrain (including perimeters of obsta cles),Dis the diameter of
+the convex hull of the terrain, and kis the number of obstacles. We do not assume knowledge
+of these parameters. We also prove a matching lower bound showing that the above complexity
+is optimal, even if the terrain is known to the robot. For limited vision we show exploration
+algorithms with complexity O(P+A+√
+Ak), where Ais the area of the terrain (excluding
+obstacles). Our algorithms work either for arbitrary terrains, if o ne of the parameters Aorkis
+known, or for c-fat terrains, where cis any constant (unknown to the robot) and no additional
+knowledge is assumed. (A terrain Twith obstacles is c-fat ifR/r≤c, whereRis the radius of
+the smallest disc containing Tandris the radius of the largest disc contained in T.) We also
+prove a matching lower bound Ω( P+A+√
+Ak) on the complexity of exploration for limited
+vision, even if the terrain is known to the robot.
+keywords: mobile robot, exploration, polygon, obstacle.
+∗D´ epartement d’informatique, Universit´ e du Qu´ ebec en Ou taouais, Gatineau, Qu´ ebec J8X 3X7, Canada. E-mails:
+jurek@uqo.ca ,labourel.arnaud@gmail.com ,pelc@uqo.ca
+†Partially supported by NSERC discovery grant.
+‡CNRS, Universit´ e de Bordeaux (LaBRI), 33405 Talence, Fran ce. E-mail: david.ilcinkas@labri.fr
+§This work was done during this author’s stay at the Universit ´ e du Qu´ ebec en Outaouais as a postdoctoral fellow.
+¶Partially supported by NSERC discovery grant and by the Rese arch Chair in Distributed Computing at the
+Universit´ e du Qu´ ebec en Outaouais.1 Introduction
+The background and the problem. Exploring unknown terrains by mobile robots has impor-
+tant applications when the environment is dangerous or of di fficult access for humans. Such is
+the situation when operating in nuclear plants or cleaning t oxic wastes, as well as in the case of
+underwater or extra-terrestrial operations. In many cases a robot must inspect an unknown terrain
+and come back to its starting point. Due to energy and cost sav ing requirements, the length of the
+robot’s trajectory should be minimized.
+We model the exploration problem as follows. The terrain is r epresented by an arbitrary polygon
+P0with pairwise disjoint polygonal obstacles P1,...,Pk, included in P0, i.e., the terrain is T=
+P0\(P1∪··· ∪P k). We assume that borders of all polygons Pibelong to the terrain. The robot
+is modeled as a point moving along a polygonal line inside the terrain. It should be noted that
+the restriction to polygons is only to simplify the descript ion, and all our results hold in the more
+general case where polygons are replaced by bounded subsets of the plane homeotopic with a disc
+(i.e., connected and without holes) and regular enough to ha ve well-defined area and boundary
+length. Every point of the trajectory of the robot is called visited . We consider two scenarios:
+theunlimited vision , when the robot visiting a point pof the terrain Texplores (sees) all points
+qfor which the segment pqis entirely contained in T, and the limited vision , when we require
+additionally that the distance between pandqbe at most 1. In both cases the task is to explore
+all points of the terrain T. The cost of an exploration algorithm is measured by the leng th of the
+trajectory of the robot, which should be as small as possible . We assume that the robot does not
+know the terrain before starting the exploration, but it has unbounded memory and can record the
+portion of the terrain seen so far and the already visited por tion of its trajectory.
+Our results. For unlimited vision we show an exploration algorithm with c omplexity O(P+D√
+k),
+wherePis the total perimeter of the terrain (including perimeters of obstacles), Dis the diameter
+of the convex hull of the terrain, and kis the number of obstacles. We do not assume knowledge
+of these parameters. We also prove a matching lower bound for exploration of some terrains (even
+if the terrain is known to the robot), showing that the above c omplexity is worst-case optimal.
+For limited vision we show exploration algorithms with comp lexityO(P+A+√
+Ak), where Ais
+the area of the terrain∗. Our algorithms work either for arbitrary terrains, if one o f the parameters
+Aorkis known, or for c-fat terrains, where cis any constant larger than 1 (unknown to the robot)
+and no additional knowledge is assumed. (A terrain Tisc-fat ifR/r≤c, whereRis the radius of
+the smallest disc containing Tandris the radius of the largest disc contained in T.) We also prove
+a matching lower bound Ω( P+A+√
+Ak) on the complexity of exploration, even if the terrain is
+known to the robot.
+The main open problem resulting from our research is whether exploration with asymptotically
+optimal cost O(P+A+√
+Ak) can be performed in arbitrary terrains without anya priori knowledge.
+Related work. Exploration of unknown environments by mobile robots was ex tensively studied
+both for the unlimited and for the limited vision. Most of the research in this domain concerns the
+competitive framework, where the trajectory of the robot no t knowing the environment is compared
+to that of the optimal exploration algorithm having full kno wledge.
+One of the most important works for unlimited vision is [8]. T he authors gave a 2-competitive
+∗Since parameters D,P,Aare positive reals that may be arbitrarily small, it is impor tant to stress that complexity
+O(P+A+√
+Ak) means that the trajectory of the robot is at most c(P+A+√
+Ak), for some constant candsufficiently
+largevalues of PandA. Similarly for O(P+D√
+k). This permits to include, e.g., additive constants in the c omplexity,
+in spite of arbitrarily small parameter values.
+1algorithm for rectilinear polygon exploration without obs tacles. The case of non-rectilinear polygons
+(without obstacles) was also studied in [7, 12] and a competi tive algorithm was given in this case.
+For polygonal environments with an arbitrary number of poly gonal obstacles, it was shown in [8]
+that no competitive strategy exists, even if all obstacles a re parallelograms. Later, this result was
+improved in [1] by giving a lower bound in Ω(√
+k) for the competitive ratio of any on-line algorithm
+exploring a polygon with kobstacles. This bound remains true even for rectangular obs tacles. On
+the other hand, there exists an algorithm with competitive r atio inO(k) [7].
+Exploration of polygons by a robot with limited vision has be en studied, e.g., in [9, 10, 11, 13, 14, 15].
+In [9] the authors described an on-line algorithm with compe titive ratio 1 + 3(Π S/A), where Π is a
+quantity depending on the perimeter of the polygon, Sis the area seen by the robot, and Ais the
+area of the polygon. The exploration in [9, 10] fails on a cert ain type of polygons, such as those
+with narrow corridors. In [11], the authors consider explor ation in discrete steps. The robot can
+only explore the environment when it is motionless, and the c ost of the exploration algorithm is
+measured by the number of stops during the exploration. In [1 3, 14], the complexity of exploration
+is measured by the trajectory length, but only terrains comp osed of identical squares are considered.
+In [15] the author studied off-line exploration of the boundar y of a terrain with limited vision.
+An experimental approach was used in [2] to show the performa nce of a greedy heuristic for ex-
+ploration in which the robot always moves to the frontier bet ween explored and unexplored area.
+Practical exploration of the environment by an actual robot was studied, e.g., in [6, 18]. In [18], a
+technique is described to deal with obstacles that are not in the plane of the sensor. In [6] landmarks
+are used during exploration to construct the skeleton of the environment.
+Navigation is a closely related task which consists in findin g a path between two given points in
+a terrain with unknown obstacles. Navigation in a n×nsquare containing rectangular obstacles
+aligned with sides of the square was considered in [3, 4, 5, 17 ]. It was shown in [3] that the navigation
+from a corner to the center of a room can be performed with a com petitive ratio O(logn), only using
+tactile information (i.e., the robot modeled as a point sees an obstacle only when it touches it). No
+deterministic algorithm can achieve better competitive ra tio, even with unlimited vision [3]. For
+navigation between any pair of points, there is a determinis tic algorithm achieving a competitive
+ratio ofO(√n) [5]. No deterministic algorithm can achieve a better compe titive ratio [17]. However,
+there is a randomized approach performing navigation with a competitive ratio of O(n4
+9logn) [4].
+Navigation with little information was considered in [19]. In this model, the robot cannot perform
+localization nor measure any distances or angles. Neverthe less, the robot is able to learn the
+critical information contained in the classical shortest- path roadmap and perform locally optimal
+navigation.
+2 Unlimited vision
+LetSbe a smallest square in which the terrain Tis included. Our algorithm constructs a quadtree
+decomposition ofS. A quadtree is a rooted tree with each non-terminal node havi ng four children.
+Each node of the quadtree corresponds to a square. The childr en of any non-terminal node v
+correspond to four identical squares obtained by partition ing the square of vusing its horizontal
+and vertical symmetry axes. This implies that the squares of the terminal nodes form a partition
+of the root†. More precisely,
+†In order to have an exact partition we assume that each square of the quadtree partition contains its East and
+South edges but not its West and North edges.
+21.{S}is a quadtree decomposition of S
+2. If{S1,S2,...,S j}is a quadtree decomposition of S, then
+{S1,S2,...,S i−1,Si1,Si2,Si3,Si4,Si+1,...,S j}, whereSi1,Si2,Si3,Si4form a partition of Si
+using its vertical and horizontal symmetry axes, is a quadtr ee decomposition of S
+The trajectory of the robot exploring Twill be composed of parts which will follow the boundaries of
+Pi, for 0≤i≤k, and of straight-line segments, called approaching segments , joining the boundaries
+ofPi, 0≤i≤k. Obviously, the end points of an approaching segment must be visible from each
+other. The quadtree decomposition will be dynamically cons tructed in a top-down manner during
+the exploration of T. At each moment of the exploration we consider the set QSof all squares of
+the current quadtree and the set QTof squares being the terminal nodes of the current quadtree.
+We will also construct dynamically a bijection f:{P0,P1,...,Pk} −→ Q S\QT.
+When a robot moves along the boundary of some polygon Pi, it may be in one of two possible
+modes: the recognition mode - when it goes around the entire boundary of a polygon without
+any deviation, or in the exploration mode - when, while moving around the boundary, it tries to
+detect (and approach) new obstacles. When the decision to ap proach a new obstacle is made at
+some point rof the boundary of Pithe robot moves along an approaching segment to reach the
+obstacle, processes it by a recursive call, and (usually muc h later), returning from the recursive
+call, it moves again along this segment in the opposite direc tion in order to return to point rand to
+continue the exploration of Pi. However, some newly detected obstacles may not be immediat ely
+approached. We say that, when the robot is in position r, an obstacle Pjisapproachable , if there
+exists a point q∈ Pj, belonging to a square St∈ QTof diameter D(St) such that |rq| ≤2D(St).
+It is important to state that if exactly one obstacle becomes approachable at moment t, then it is
+approached immediately and if more than one obstacle become approachable at a moment t, then
+one of them (chosen arbitrarily) is approached immediately and the others are approached later,
+possibly from different points of the trajectory. Each time a n ew obstacle is visited by the robot
+(i.e., all the points of its boundary are visited in the recog nition mode) the terminal square of the
+current quadtree containing the first visited point of the ne w obstacle is partitioned. This square
+is then associated to this obstacle by function f.
+The trajectory of the robot is composed of three types of sect ions:recognition sections ,exploration
+sections andapproaching sections . The boundary of each polygon will be traversed twice: first t ime
+contiguously during a recognition section and second time t hrough exploration sections, which may
+be interrupted several times in order to approach and visit n ewly detected obstacles. We say that
+an obstacle is completely explored , if each point on the boundary of this obstacle has been trave rsed
+by an exploration section. We will prove that the sum of the le ngths of the approaching sections is
+O(D√
+k).
+Algorithm ExpTrav (polygon R, starting point r∗on the boundary of R)
+1 Make a recognition traversal of the boundary of R
+2 Partition square St∈QTof the quadtree containing r∗into four identical squares
+3f(R) :=St
+4repeat
+5 Traverse the boundary of Runtil, for the current position r, there exists a visible point q
+of a new obstacle Qbelonging to square St∈ QT, such that |rq| ≤2D(St)
+6 Traverse the segment rq
+7 ExpTrav( Q,q)
+38 Traverse the segment qr
+9untilRis completely explored
+Before the initial call of ExpTrav , the robot reaches a position r0at the boundary of the polygon
+P0. This is done as follows. At its initial position v, the robot chooses an arbitrary half-line α
+which it follows as far as possible. When it hits the boundary of a polygon P, it traverses the
+entire boundary of P. Then, it computes the point uwhich is the farthest point from vinP ∩α.
+It goes around Puntil reaching uagain and progresses on α, if possible. If this is impossible, the
+robot recognizes that it went around the boundary of P0and it is positioned on this boundary. It
+initialises the quadtree decomposition to a smallest squar eScontaining P0. This square is of size
+O(D(P0)). The length of the above walk is less than 3 P.
+Lemma 2.1 Algorithm ExpTrav visits all boundary points of all obstacles of the terrain T.
+Proof: Note that Algorithm ExpTrav always terminates. Indeed, since there is a finite number of
+obstacles, there is a finite number of calls of ExpTrav and steps 5-8 are executed a finite number of
+times. Moreover, since each obstacle has a finite boundary, s tep 1 and the repeat loop are always
+completed.
+Consider the quadtree decomposition QofSarising at the completion of the algorithm. Suppose,
+for contradiction, that there exists a point on the boundary of an obstacle which was never visited.
+Letpbe a point among all unvisited boundary points for which the t erminal square Sjbelonging
+to the quadtree Qhas the smallest possible diameter. Consider the square Sm, the parent of Sjin
+Q.Smwas partitioned in step 2 of some call of ExpTrav as a result of detecting some obstacle P′
+intersecting Sm. Consider the segment qp, whereq∈Smbelongs to the boundary of P′. Since both
+pointspandqbelong to the boundary of Tandqwas visited while pwas not, there exists a pair
+of points q′andp′, both belonging to the boundary of Tand to the segment qp, such that q′was
+visited,p′was not and p′is visible from q′. Such a pair exists because at the end of the exploration
+the boundary of each polygon is either entirely visited or no t at all. Consider the quadtree at the
+moment twhen the robot visited point q′, and its terminal square Sicontaining point p′. Clearly,
+D(Si)≥D(Sj), because Sjis a square with the smallest diameter containing unvisited boundary
+points. Hence |q′p′| ≤ |qp| ≤D(Sm) = 2D(Sj)≤2D(Si) andp′was approachable from q′at time
+t, a contradiction. /square
+Lemma 2.2 Function fis a bijection from {P0,P1,...,Pk}toQS\QT, whereQSandQTcor-
+respond to the final quadtree decomposition produced by Algo rithmExpTrav.
+Proof: WhenExpTrav is called for the first time, the robot is on the boundary of P0and the
+quadtree has exactly one non-terminal node - its root S, andf(P0) =S. By induction, each time
+a new obstacle Qis approached in step 6 of a call of ExpTrav ,f(Q) is set to some Stintersecting
+QandStbecomes a nonterminal node of the quadtree in step 2. Hence ea ch square corresponding
+to a non-terminal node of the quadtree is an image of a different polygon Pi, 0≤i≤k. /square
+Lemma 2.3 For any quadtree T, rooted at a square of diameter Dand having xnon-terminal
+nodes, the sum σ(T)of diameters of these nodes is at most 2D√x.
+Proof: The diameter of a square at depth tof the quadtree isD
+2t. We prove first that among
+all quadtrees with xnon-terminal nodes, σ(T) is maximized for the quadtree having all terminal
+4nodes of at most two consecutive depths. Suppose, to the cont rary, that there exists a quadtree T
+maximizing σ(T) having terminal nodes of two different depths t1andt2witht1< t2−1. Letpbe
+a terminal node of Tof depth t1andqbe a non-terminal node of depth t2−1. LetT′be the tree
+obtained from Tby detaching from Tnodep∈Tand the subtree of Trooted at qand exchanging
+their places. T′is again a quadtree with xnon-terminal nodes, having one less non-terminal node of
+deptht2−1 and one extra non-terminal node of depth t1. Henceσ(T′)≥σ(T)−D
+2t2−1+D
+2t1> σ(T),
+which contradicts the maximality of σ(T).
+Therefore it is sufficient to consider only quadtrees having t erminal nodes of at most two consecutive
+depthstandt+ 1. Suppose that there are yterminal nodes of depth t+ 1, 0≤y≤4t+1. Then
+the number xof non-terminal nodes equals4t−1
+3+y
+4and
+σ(T) =y
+4D
+2t+t−1/summationdisplay
+i=04iD
+2i
+=D/parenleftBig
+2t−1 +y
+2t+2/parenrightBig
+We need to prove that σ(T)≤2D√x, i.e. that
+D/parenleftBig
+2t−1 +y
+2t+2/parenrightBig
+≤2D/radicalbigg
+4t−1
+3+y
+4
+Hence it is sufficient to show that
+/parenleftBig
+2t−1 +y
+2t+2/parenrightBig2
+≤4/parenleftbigg4t−1
+3+y
+4/parenrightbigg
+i.e., that
+0≤y/parenleftbigg
+1 +1
+2t+1−1
+2−y
+4t+2/parenrightbigg
++/parenleftbigg
+4·4t−1
+3−22t−1 + 2t+1/parenrightbigg
+The second term is clearly positive for t≥0 and the first term is also positive since 0 ≤y≤4t+1.
+We conclude that σ(T)≤2D√x. /square
+Theorem 2.1 Algorithm ExpTrav explores the terrain Tof perimeter Pand convex hull diameter
+Dwithkobstacles in time O(P+D√
+k).
+Proof: Take an arbitrary point pinsideTand a ray outgoing from pin an arbitrary direction.
+This ray reaches the boundary of Tat some point q. Since, by Lemma 2.1 point qwas visited by
+the robot, pwas visible from qduring the robot’s traversal, and hence pwas explored.
+To prove the complexity of the algorithm, observe that the ro bot traverses twice the boundary of
+each polygon of T, once during its recognition in step 1 and the second time dur ing the iterations
+of step 5. Hence the sum of lengths of the recognition and expl oration sections is 2 P. The only
+other portions of the trajectory are produced in steps 6 and 8 , when the obstacles are approached
+and returned from. According to the condition from step 5, an approaching segment is traversed
+in step 6 only if its length is shorter than twice the diameter of the associated square. If k= 0
+then the sum of lengths of all approaching segments is 0, due t o the fact that exploration starts
+at the external boundary of the terrain. In this case the leng th of the trajectory is at most 5 P.
+Hence we may assume that k > 0. By Lemma 2.2 each obstacle is associated with a different
+5non-terminal node of the quadtree and the number xof non-terminal nodes of the quadtree equals
+k+ 1. Hence the sum of lengths of all approaching segments is at most 2σ(T). By Lemma 2.3 we
+haveσ(T)≤2D√x= 2D√
+k+ 1, hence the sum of lengths of approaching segments is at mos t
+2σ(T)≤4D√
+k+ 1≤4D√
+2k≤6D√
+k. Each segment is traversed twice, so the total length of
+this part of the trajectory is at most 12 D√
+k. It follows that the total length of the trajectory is
+at most 5 P+ 12D√
+k. /square
+Theorem 2.2 Any algorithm for a robot with unlimited visibility, explor ing polygonal terrains
+withkobstacles, having total perimeter Pand the convex hull diameter D, produces trajectories in
+Ω(P+D√
+k)in some terrains, even if the terrain is known to the robot.
+Proof: In order to prove the lower bound, we show two families of terr ains: one for which P∈Θ(D)
+(Pcannot be smaller), Dandkare unbounded and still the exploration cost is Ω( D√
+k), and the
+other in which Pis unbounded, Dis arbitrarily small, k= 0 and still the exploration cost is Ω( P).
+Consider the terrain from Figure 1(a) where kidentical tiny obstacles are distributed evenly at the√
+k×√
+kgrid positions inside a square of diameter D. The distance between obstacles is at least
+D√
+2
+2(√
+k+1)−ǫwhereǫ >0 may be as small as necessary by choosing obstacles sufficient ly small. The
+obstacles are such that to explore the small area inside the c onvex hull of the obstacle the robot
+must enter this convex hull. Since each such area must be expl ored, the trajectory of the robot
+must be of size at least ( k−1)/parenleftBig
+D√
+2
+2(√
+k+1)−ǫ/parenrightBig
+, which is clearly in Ω( D√
+k). Note that the perimeter
+Pis in Θ(D).
+The terrain from Figure 1(b) is a polygon of arbitrarily smal l diameter (without obstacles), whose
+exploration requires a trajectory of size Ω( P), where Pis unbounded. Indeed, each ”corridor” must
+be traversed almost completely to explore points at its end. Hence the two families of polygons
+from Figure 1 lead to the Ω( P+D√
+k) lower bound. /square
+(a) (b)
+Figure 1: Lower bound for unlimited visiblity
+63 Limited vision
+In this section we assume that the vision of the robot has rang e 1. The following algorithm is at
+the root of all our positive results on exploration with limi ted vision. The idea of the algorithm
+is to partition the environment into small parts called cells (of diameter at most 1) and to visit
+them using a depth-first traversal. The local exploration of cells can be performed using Algorithm
+ExpTrav , since the vision inside each cell is not limited by the range 1 of the vision of the robot.
+The main novelty of our exploration algorithm is that the rob ot completely explores anyterrain.
+This should be contrasted with previous algorithms with lim ited visibility, e.g. [9, 10, 13, 14] in
+which only a particular class of terrains with obstacles is e xplored, e.g., terrains without narrow
+corridors or terrains composed of complete identical squar es. This can be done at cost O(A). Our
+lower bound shows that exploration complexity of arbitrary terrains depends on the perimeter and
+the number of obstacles as well. The complete exploration of arbitrary terrains achieved by our
+algorithm significantly complicates both the exploration p rocess and its analysis.
+Algorithm LimExpTrav (LET , for short)
+INPUT: A point sinside the terrain Tand a positive real F≤√
+2/2.
+OUTPUT: An exploration trajectory of T, starting and ending at s.
+Tile the area with squares of side F, such that sis on the boundary of a square. The connected
+regions obtained as intersections of Twith each tile are called cells. For each tile S, maintain a
+quadtree decomposition QSinitially set to {S}. Then, arbitrarily choose one of the cells containing
+sto be the starting cell Cand callExpCell (C,s).
+Procedure ExpCell (current cell C, starting point r∗∈C)
+1 Record Cas visited
+2ExpTrav (C,r∗) using the quadtree decomposition QS, whereSis the tile containing C
+3repeat
+4 Traverse the boundary of Cuntil the current position rbelongs to an unvisited cell U
+5ExpCell (U,r)
+(ifris in several unvisited cells, choose arbitrarily the first c ell to be processed)
+6until the boundary of Cis completely traversed
+It is worth to note that, at the beginning of the exploration o f the first cell belonging to a tile
+S, the quadtree of this tile is set to a single node. However, at the beginning of explorations of
+subsequent cells belonging to S, the quadtree of Smay be different. So the top-down construction
+of this quadtree may be spread over the exploration of many ce lls which will be visited at different
+points in time.
+Consider a tile Tand a cell C⊆T. LetACbe the area of C,RCbe the length of the part of the
+boundary of Cissued from the boundary of T, andPCbe the length of the part of the boundary
+ofCissued from the boundary of T.
+Lemma 3.1 There is a positive constant c, such that RC≤c(AC/F+PC), for any cell C.
+Proof: We consider three cases:
+Case 1: PC< F/ 2 andAC< F2/2
+In this case, we will show that there is a positive constant csuch that RC≤c·PC. We call a
+borderline a maximum connected part of the boundary of Tinside the tile Tdelimiting the cell
+C. LetL={L1,...,L l}be the set of borderlines of C. There are two types of borderlines: the
+7linking borderlines that link two points of the boundary of Tand theclosed borderlines that are
+closed polygonal lines inside S. A borderline Lseparates the tile Sinto two connected regions, the
+inside region denoted by IL, i.e., the region containing C, and the outside region, denoted by OL.
+If the area of ILis smaller than that of OL, thenLis asmall-inside borderline, otherwise Lis a
+large-inside borderline. We denote by MLthe region among ILandOLwhich has the smaller area.
+Notice that the two endpoints of a linking borderline can either be on the same side of Sor on
+two adjacent sides. Indeed, any borderline linking two poin ts on opposite sides of Swould have a
+length at least F, a contradiction with the inequality PC< F/ 2. IfLis a linking borderline with
+both endpoints xandyon the same side of S, then the length of segment xyis smaller than |L|.
+Hence, the perimeter of MLis smaller than 2 · |L|. IfLis a linking borderline with endpoints x
+andyon two sides that intersect at a vertex v, then the lengths of segments vxandvyare both
+less than |L|. Therefore the perimeter of MLis smaller than 3 ·|L|. IfLis a closed borderline, then
+the perimeter of MLis exactly L. Hence, the perimeter of MLis always less than 3 ·|L|. Moreover,
+the area of MLis less than (3 |L|)2/4πby the isoperimetric inequality [16].
+Now we are ready to show that at least one borderline in Lis a small-inside borderline. Suppose,
+for contradiction that, for all i= 1,...,l , the borderline Liis a large-inside borderline. We have
+C=S\/uniontextl
+i=1MLi. It follows that:
+AC=Area (S)−l/summationdisplay
+i=1Area (MLi)
+≥F2−l/summationdisplay
+i=1(3|Li|)2
+4πsince (3|Li|)2/4π≥Area (MLi) for alli= 1,...,l
+≥F2−9P2
+C
+4πsinceP2
+C=/parenleftBiggl/summationdisplay
+i=1|Li|/parenrightBigg2
+≥l/summationdisplay
+i=1|Li|2
+>F2
+2sinceF/2> PC.
+We obtain AC> F2/2, a contradiction. This shows that there exists a small-ins ide borderline
+L∈ L. We have C⊆MLand thus RC<3|L| ≤3PC.
+Case 2: PC< F/ 2 andAC≥F2/2
+We have:
+RC≤4F≤8
+FACsinceF≤2AC/F.
+Case 3: PC≥F/2
+We have:
+RC≤4F≤8PCsinceF≤2PC.
+In all cases, we have RC≤8(AC/F+PC). /square
+The following is the key lemma for all upper bounds proved in t his section. Let S={T1,T2,...,T n}
+be the set of tiles with non-empty intersection with TandC={C1,C2,···,Cm}be the set of cells
+that are intersections of tiles from SwithT. For each T∈ S, letkTbe the number of obstacles of
+Tentirely contained in T.
+Lemma 3.2 For any F≤√
+2/2, Algorithm LETexplores the terrain Tof areaAand perimeter
+P, using a trajectory of length O(P+A/F +F/summationtextn
+i=1/radicalbig
+kTi).
+8Proof: First, we show that Algorithm LET explores the terrain T. Consider the graph Gwhose
+vertex set is Cand edges are the pairs {C,C′}such that CandC′have a common point at
+their boundaries. The graph Gis connected, since Tis connected. Note that for any cell C
+and point ron the boundary of C,ExpTrav (C,r) and thus ExpCell (C,r) starts and ends on r.
+Therefore, Algorithm LET performs a depth first traversal of graph G, since during the execution
+ofExpCell (C,... ), procedure ExpCell (U,···) is called for each unvisited cell Uadjacent to C.
+Hence,ExpCell (C,... ) is called for each cell C∈ C, sinceGis connected. During the execution of
+ExpCell (C,r),Cis completely explored by ExpTrav (C,r) by the same argument as in the proof of
+Lemma 2.1, since the convex hull diameter of Cis less than one.
+It remains to show that the length of the LET trajectory is O(P+A/F +F/summationtextn
+i=1/radicalbig
+kTi). For each
+j= 1,...,m , the part of the LET trajectory inside the cell Cjis produced by the execution of
+ExpCell (Cj,...). In step 2 of ExpCell (Cj,...), the robot executes ExpTrav withD=√
+2Fand
+P=PCj+RCj. The sum of lengths of recognition and exploration sections of the trajectory in Cj
+is at most 2( PCj+RCj). The sum of lengths of approaching sections of the trajecto ry inTiis at
+most 6√
+2F/radicalbig
+kTiand each approaching section is traversed twice (cf. proof o f Theorem 2.1). In
+step 3 ofExpCell (Cj,...), the robot only makes the tour of the cell Cj, hence the distance traveled
+by the robot is at most PCj+RCj. It follows that:
+|LET| ≤ 3m/summationdisplay
+j=1(PCj+RCj) + 12√
+2Fn/summationdisplay
+i=1/radicalbig
+kTi
+≤3m/summationdisplay
+i=1((1 +c)PCj+cACj/F) + 12√
+2Fn/summationdisplay
+i=1/radicalbig
+kTiby Lemma 3.1
+≤3(c+ 1)P+ 3cA/F + 12√
+2Fn/summationdisplay
+i=1/radicalbig
+kTi.
+/square
+In view of Lemma 3.2, exploration of a particular class of ter rains can be done at a cost which will
+be later proved optimal.
+Theorem 3.1 Letc >1be any constant. Exploration of a c-fat terrain of area A, perimeter P
+and with kobstacles can be performed using a trajectory of length O(P+A+√
+Ak)(without any
+a priori knowledge).
+Proof: The robot executes Algorithm LET withF=√
+2/2. By Lemma 3.2, the total cost is
+O(P+A+/summationtextn
+i=1/radicalbig
+kTi). Recall that nis the number of tiles that have non-empty intersection with
+the terrain. We have/summationtextn
+i=1/radicalbig
+kTi≤/summationtextn
+i=1/radicalBig
+k
+n=√
+nk. Hence, it remains to show that n=O(A) to
+prove that the cost is O(P+A+√
+Ak). By definition of a c-fat terrain, there is a disk D1of radius r
+included in the terrain and a disk D2of radius Rthat contains the terrain, such thatR
+r≤c. There
+are Θ(r2) tiles entirely included in D1and hence in the terrain. So, we have A= Ω(r2). Θ(R2)
+tiles are sufficient to cover D2and hence the terrain. So n=O(R2). Hence, we obtain n=O(A)
+in view of R≤cr. /square
+Consider any terrain Tof areaA, perimeter Pand with kobstacles. We now turn attention to
+the exploration problem if some knowledge about the terrain is available a priori. Notice that if
+Aandkare known before the exploration, Lemma 3.2 implies that Alg orithmLET executed for
+F= min{/radicalbig
+A/k,√
+2/2}explores anyterrain at cost O(A+P+√
+Ak). (Indeed, if F=/radicalbig
+A/k
+9thenA/F =√
+AkandkF=√
+Ak, whileF=√
+2/2 implies A/F = Θ(A) andkF=O(A).) This
+cost will be later proved optimal. It turns out that a much mor e subtle use of Algorithm LET
+can guarantee the same complexity assuming only knowledge o fAork. We present two different
+algorithms depending on which value, Aork, is known to the robot. Both algorithms rely on the
+same idea. The robot executes Algorithm LET with some initial value of Funtil either the terrain
+is completely explored, or a certain stopping condition, de pending on the algorithm, is satisfied.
+This execution constitutes the first stage of the two algorit hms. If exploration was interrupted
+because of the stopping condition, then the robot proceeds t o a new stage by executing Algorithm
+LET with a new value of F. Values of Fdecrease in the first algorithm and increase in the second
+one. The exploration terminates at the stage when the terrai n becomes completely explored, while
+the stopping condition is never satisfied.
+In each stage the robot is oblivious of the previous stages, e xcept for the computation of the new
+value of Fthat depends on the previous stage. This means that in each st age exploration is
+done “from scratch”, without recording what was explored in previous stages. In order to test the
+stopping condition in a given stage, the robot maintains the following three values: the sum A∗of
+areas of explored cells, updated after the execution of ExpTrav in each cell; the length P∗of the
+boundary traversed by the robot, continuously updated when the robot moves along a boundary
+for the first time (i.e., in the recognition mode); and the num berk∗of obstacles approached by the
+robot, updated when an obstacle is approached. The values of A∗,P∗andk∗at the end of the i-th
+stage are denoted by Ai,Piandki, respectively. Let Fibe the value of Fused by Algorithm LET
+in thei-th stage. Now, we are ready to describe the stopping conditi ons and the values Fiin both
+algorithms.
+Algorithm LETA, forAknown before exploration
+The value of Fused in Algorithm LET for the first stage is F1=√
+2/2. The value of F
+for subsequent stages is given by Fi+1=A
+kiFi. The stopping condition is {k∗Fi≥2A/Fiand
+k∗Fi≥P∗+ 1}.
+Algorithm LETk, forkknown before exploration
+The value of Fused in Algorithm LET for the first stage is F1=1
+k+√
+2. The value of F
+for subsequent stages is given by Fi+1= min/braceleftBig
+Ai
+kFi,√
+2
+2/bracerightBig
+. The stopping condition is {A∗/Fi≥
+2kFiandA∗/Fi≥P∗+ 1 and Fi<√
+2/2}.
+Consider a moment tduring the execution of Algorithm LET . LetCtbe the set of cells recorded as
+visited by Algorithm LET at moment t, and let Otbe the set of obstacles approached by the robot
+until time t. For each C∈ Ct, letBCbe the length of the intersection of the exterior boundary of
+cellCwith the boundary of the terrain. For each O∈ Ot, let|O|be the perimeter of obstacle O
+and letkt=|Ot|. The following proposition is proved similarly as Lemma 3.2 .
+Proposition 3.1 There is a positive constant dsuch that the length of the trajectory of the robot
+until any time t, during the execution of Algorithm LET, is at most d(/summationtext
+C∈Ct(BC+AC/F) + (kt+
+1)·F+/summationtext
+O∈Ot|O|).
+Proof: LetUbe the current cell at moment t, which means that the instance of procedure ExpCell
+executed at moment twas called with Uas its first parameter. Let O′
+tbe the set of obstacles in
+Otthat are not included in Uand letC′
+t=Ct\ {U}. All the obstacles included in C∈ C′
+twere
+approached and visited by the robot. Hence, we have/summationtext
+C∈C′
+tPC=/summationtext
+C∈C′
+tBC+/summationtext
+O∈O′
+t|O|. By
+10the same arguments as in the proof of Lemma 3.2, it follows tha t the total length of the trajectory
+of the robot in the cells in C′
+tis at most d′(/summationtext
+C∈C′
+t(PC+AC/F) +F|O′
+t|) =d′(/summationtext
+C∈C′
+t(BC+
+AC/F) +/summationtext
+O∈O′
+t|O|+F|O′
+t|) for some positive constant d′. The length of the trajectory of the
+robot in Uis at most 3 GU+ 2/summationtext
+O∈Ot\O′
+t|O|+|Ot\O′
+t|·F, whereGUis the length of the exterior
+boundary of cell U. Hence, the length of the trajectory of the robot until time tis at most
+d(/summationtext
+C∈Ct(BC+AC/F)+(kt+1)·F+/summationtext
+O∈Ot|O|) for some positive constant d, sinceGU≤4F+BU.
+/square
+The following lemma establishes the complexity of explorat ion if either the area of the terrain or
+the number of obstacles is known a priori.
+Lemma 3.3 Algorithm LETA(resp.LETk) explores a terrain Tof areaA, perimeter Pand with k
+obstacles, using a trajectory of length O(P+A+√
+Ak), ifA(resp.k) is known before exploration.
+Proof:
+Part 1: complexity of Algorithm LETA
+First, we show that the algorithm eventually terminates and completely explores T. Remark that
+for each i >1,Fi=A
+ki−1Fi−1≤Fi−1
+2sinceki−1Fi−1≥2A
+Fi−1. SinceF1=√
+2/2, for each i≥1,
+we have Fi≤√
+2
+2i. The algorithm eventually terminates, since there exists m∈Nsuch that
+kFm<2A/F1<2A/Fmand the stopping condition is never satisfied in this case. In the last stage,
+Algorithm LET performs complete exploration of the terrain by Lemma 3.2, s ince the value of F
+used for exploration is at most√
+2/2.
+LetDibe the distance traveled by the robot during the i-th stage and let nbe the number of stages.
+By Proposition 3.1, if stage iends at moment tithenDi≤d(/summationtext
+C∈Cti(BC+AC/Fi)+(|Oti|+1)·Fi+/summationtext
+O∈Oti|O|) for each i≥1. Since the algorithm can only interrupt stage i < n when approaching
+an obstacle, we have Pi=/summationtext
+C∈CtiBC+/summationtext
+O∈Oti|O|. We obtain that Di≤d(Pi+A/Fi+(ki+1)Fi)
+for each i≥1.
+Ifn= 1, then the stopping condition is never satisfied and kF1≤max{2A/F1,P+ 1}. The total
+cost is at most d(P+A/F1+ (k+ 1)F1) =O(P+A), sinceF1=√
+2/2. On the other hand, if
+n > 1, then for each isuch that 1 ≤i < n , we have Di≤d(Pi+A/Fi+ (ki+ 1)Fi)≤3dkiFi.
+Indeed, we have kiFi≥A/FiandkiFi≥Pi+Fi, since the stopping condition is satisfied at
+the end of the i-th stage. Moreover, we have1
+2kiFi≥A/Fi=ki−1Fi−1, for all 1 < i < n . It
+follows that/summationtextn−1
+i=1Di≤6dkn−1Fn−1. In order to show that the total cost is O(P+A+√
+Ak),
+it is sufficient to show that P+A/Fn+kFn=O(P+A+√
+Ak), sincekn−1Fn−1=A/Fnand
+Dn≤d(Pn+A/Fn+ (k+ 1)Fn).
+Take the moment tn−1when the ( n−1)-th stage was interrupted, i.e., when both inequalities o f
+the stopping condition started to be satisfied. Consider the inequality which was not satisfied just
+before moment tn−1. If this is the first of the two inequalities, then at time tn−1the Algorithm
+LETAmust have increased the value of k∗, hence just before moment tn−1we had (kn−1−1)Fn−1<
+2AFn−1. Similarly, if the second inequality was not satisfied just b efore moment tn−1, then we had
+(kn−1−1)Fn−1< Pn−1+ 1.
+Case 1: (kn−1−1)Fn−1<2A/Fn−1
+11We have:
+kn−1Fn−1≤2A
+Fn−1+Fn−1
+A
+Fn≤2A
+Fn−1+ 1 since Fn−1=A
+kn−1FnandFn−1≤1
+A
+Fn≤√
+2/radicalbig
+Akn−1+ 1 since kn−1Fn−1≥2A
+Fn−1and thus/parenleftbiggA
+Fn−1/parenrightbigg2
+≤1
+2kn−1Fn−1A
+Fn−1
+A
+Fn≤2√
+Ak+ 1 since kn−1≤k
+Since the stopping condition in the last stage is not satisfie d, we have kFn≤max{2A/Fn,P+
+1}. We obtain P+A/Fn+kFn=O(P+A/Fn). Hence, we have P+A/Fn+kFn=O(P+√
+Ak).
+Case 2: (kn−1−1)Fn−1< Pn−1+ 1
+We have:
+kn−1Fn−1≤Pn−1+Fn−1+ 1
+A
+Fn≤Pn−1+Fn−1+ 1 since kn−1Fn−1=A
+Fn
+A
+Fn≤P+ 2 since Fn−1≤1 andPn−1≤P
+Since the stopping condition in the last stage is not satisfie d, we have kFn≤max{2A/Fn,P+
+1}, as before. We obtain P+A/Fn+kFn=O(P+A/Fn) =O(P).
+Part 2: complexity of Algorithm LETk
+The proof is similar to that concerning Algorithm LETA. The main difference follows from the
+additional clause Fi<√
+2/2 in the stopping condition. This clause was not necessary in Algorithm
+LETAbecause, as opposed to the present case, sides of tiles were d ecreasing. First, we show that
+the algorithm eventually terminates and completely explor esT. Remark that for each i >1, we
+haveFi= min{Ai−1/kFi−1,√
+2/2} ≥ min{2Fi−1,√
+2/2}, sinceAi−1/Fi−1≥2kFi−1. Hence, the
+algorithm eventually terminates. Indeed, even if the first t wo inequalities remain true, the third
+must become false at some point. Notice that F1≤√
+2/2 sincek≥0, and for all i >1 we have
+Fi≤√
+2/2. In the last stage, Algorithm LET performs complete exploration of the terrain by
+Lemma 3.2, since the value of Fused for the exploration is at most√
+2/2.
+LetDibe the distance traveled by the robot during the i-th stage and let nbe the number of
+stages. By Proposition 3.1, if stage iends at moment ti, thenDi≤d(/summationtext
+C∈Cti(BC+AC/Fi) +
+(|Oti|+ 1)·Fi+/summationtext
+O∈Oti|O|) for each i≥1. Since the algorithm can only stop when completing
+the exploration of a cell, we have Pi=/summationtext
+C∈CtiBC+/summationtext
+O∈Oti|O|andAi=/summationtext
+C∈CtiAC. We obtain
+thatDi≤d(Pi+Ai/Fi+ (k+ 1)Fi) for each i≥1.
+Ifn= 1, then the stopping condition is never satisfied and either A/F1≤max{2kF1,P+ 1}or
+F1=√
+2/2. In the first case, the total cost is at most d(P+A/F1+ (k+ 1)F1) =O(P) since
+kF1≤1. In the second case, we have k= 0 and the total cost is at most d(P+A/F1+ (k+
+1)F1) =O(P+A). On the other hand, if n > 1 then for each isuch that 1 ≤i < n , we have
+Di≤d(Pi+Ai/Fi+ (k+ 1)Fi)≤3dAi/Fi. Indeed, we haveAi
+Fi≥kFiandAi
+Fi≥Pi+Fi, since the
+12stopping condition is satisfied at the end of the i-th stage. Moreover, we haveAi
+2Fi≥kFi=Ai−1
+Fi−1
+for all 1< i < n . It follows that/summationtextn−1
+i=1Di≤6dAn−1
+Fn−1.
+Notice that the third inequality of the stopping condition i s always satisfied during the ( n−1)-
+stage. Take the moment tn−1when the ( n−1)-th stage was interrupted, i.e., when the two first
+inequalities of the stopping condition started to be satisfi ed. Consider the inequality which was not
+satisfied just before moment tn−1. If this is the first of the two inequalities, then at time tn−1the
+Algorithm LETkmust have increased the value of A∗by at most F2
+n−1since each cell is included in a
+square of size Fn−1. Hence just before moment tn−1we hadAn−1−F2
+n−1
+Fn−1<2AFn−1. Similarly, if the
+second inequality was not satisfied just before moment tn−1, then we hadAn−1−F2
+n−1
+Fn−1< Pn−1+ 1.
+Case 1:An−1−F2
+n−1
+Fn−1<2AFn−1
+Notice that k≥1, since otherwise F1=√
+2/2 and the stopping condition would never be
+satisfied in the first stage. We have:
+An−1
+Fn−1≤2kFn−1+Fn−1
+kFn≤2kFn−1+ 1 since Fn−1≤An−1
+kFnandFn−1≤1
+kFn≤√
+2/radicalbig
+An−1k+ 1 sinceAn−1
+Fn−1≥2kFn−1and thus ( kFn−1)2≤1
+2An−1
+Fn−1kFn−1
+kFn≤2√
+Ak+ 1 since An−1≤A
+Since the stopping condition in the last stage is not satisfie d, we have either A/Fn≤
+max{2kFn,P+ 1}and thus A/Fn=O(P+kFn), orFn=√
+2/2 and thus A/Fn=O(A). We
+obtain that Dn≤d(P+A/Fn+ (k+ 1)Fn) =O(P+√
+Ak+A). From the previous sequence
+of inequalities, we also have/summationtextn−1
+i=1Di≤6d(An−1/Fn−1) =O(√
+Ak). Hence, the total cost is
+O(P+√
+Ak+A).
+Case 2:An−1−F2
+n−1
+Fn−1< Pn−1+ 1
+We have:
+An−1
+Fn−1≤Pn−1+Fn−1+ 1
+kFn≤Pn−1+Fn−1+ 1 since kFn≤An−1
+Fn−1
+kFn≤P+ 2 since Fn−1≤1 andPn−1≤P
+As before, we have A/Fn=O(P+kFn) orA/Fn=O(A). We obtain that Dn≤d(P+
+A/Fn+ (k+ 1)Fn) =O(P+A). We have/summationtextn−1
+i=1Di≤6d(An−1/Fn−1) =O(P). Hence, the
+total cost is O(P+A).
+/square
+The following theorem shows that the lengths of trajectorie s in Lemma 3.3 and in Theorem 3.1 are
+asymptotically optimal.
+13Theorem 3.2 Any algorithm for a robot with limited visibility, explorin g polygonal terrains of
+areaA, perimeter Pand with kobstacles, produces trajectories of length Ω(P+A+√
+Ak)in some
+terrains, even if the terrain is known to the robot.
+Proof: In order to prove our lower bound we present three families of terrains. A terrain in the
+first family (cf. Fig. 1(a)) is a square with identical obstac les of diameter ǫlocated on a√
+k×√
+k
+grid. The side of the square is√
+A+x, wherexis the negligible total area of all kobstacles and
+the total perimeter of all obstacles is 1. By the same argumen ts as in the proof of Theorem 2.2,
+any exploration trajectory must be of length at least ( k−1)/parenleftBig√A+x√
+k+1−ǫ/parenrightBig
+, which is in Ω(√
+Ak). At
+the same time we have P= Θ(√
+A) (Pcannot be smaller). A terrain in the second family (cf.
+Fig. 1(b)) is a polygon of arbitrarily small area (without ob stacles), whose exploration requires a
+trajectory of size Ω( P). A terrain in the third family is the empty square of side√
+A. Now we have
+P= Θ(√
+A) andk= 0. When the robot traverses a distance d, it explores a new area of at most
+2d. So, the robot has to travel a distance Ω( A) to explore such a terrain. These three families of
+terrains justify our lower bound. /square
+The examples from the above proof can be adjusted so that all c onsidered terrains are c-fat, for
+any fixed constant c >1. Lemma 3.3 and Theorem 3.2 imply
+Theorem 3.3 Consider terrains of area A, perimeter Pand with kobstacles. If either Aork
+is known before the exploration, then the exploration of any such terrain can be performed using a
+trajectory of length Θ(P+A+√
+Ak), which is asymptotically optimal.
+Notice that in order to explore a terrain at cost O(P+A+√
+Ak), it is enough to know the parameter
+Aorkup to a multiplicative constant, rather than the exact value . This can be proved by a carefull
+modification of the proof of Lemma 3.3. For the sake of clarity , we stated and proved the weaker
+version of Lemma 3.3, with knowledge of the exact value.
+Suppose now that no a priori knowledge of any parameters of th e terrain is available. We iterate
+Algorithm LETAorLETkforA(resp. k) equal 1 ,2,4,8,...interrupting the iteration and doubling
+the parameter as soon as the explored area (resp. the number o f obstacles seen) exceeds the current
+parameter value. The algorithm stops when the entire terrai n is explored (which happens at the
+first probe exceeding the actual unknown value of A, resp.k). We get an exploration algorithm
+using a trajectory of length O((P+A+√
+Ak) logA), resp.O((P+A+√
+Ak) logk). By interleaving
+the two procedures we get the minimum of the two costs. Thus we have the following corollary.
+Corollary 3.1 Consider terrains of area A, perimeter Pand with kobstacles. Exploration of any
+such terrain can be performed without any a priori knowledge at cost differing from the worst-case
+optimal cost with full knowledge only by a factor O(min{logA,logk}).
+References
+[1] S. Albers and K. Kursawe and S. Schuierer, Exploring unkn own environments with obstacles,
+Algorithmica 32 (2002), 123–143.
+[2] T. Bandyopadhyay, Z. Liu, M.H Ang, M.H, W.K.G Seah, Visib ility-based exploration in un-
+known environment containing structured obstacles, Advan ced Robotics (2005), 484-491.
+14[3] E. Bar-Eli, P. Berman, A. Fiat and R. Yan, On-line navigat ion in a room, Journal of Algorithms
+17 (1994), 319-341.
+[4] P. Berman, A. Blum, A. Fiat, H. Karloff, A. Rosen and M. Saks, Randomized robot navigation
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+[6] N. Cuperlier, M. Quoy, C. Giovanangelli, Navigation and planning in an unknown environment
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+[7] X. Deng, T. Kameda and C. H. Papadimitriou, How to learn an unknown environment, Proc.
+32nd Symp. on Foundations of Computer Science (FOCS 1991), 2 98–303.
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+robot, Computational Geometry: Theory and Applications (2 003), 24(3):197-224.
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+visibility - exploring unknown polygonal environments, Ro botics & Automation Magazine 15
+(2008), 67-76.
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+Comput. 31 (2001), 577–600.
+[13] C. Icking, T. Kamphans, R. Klein and E. Langetepe. Explo ring an unknown cellular envi-
+ronment. In Abstracts of the 16th European Workshop on Compu tational Geometry, pages
+140-143, 2000.
+[14] A. Kolenderska, A. Kosowski, M. Ma/suppress lafiejski and P. ˙Zyli´ nski. An Improved Strategy for Ex-
+ploring a Grid Polygon, SIROCCO 2009.
+[15] S. Ntafos, Watchman routes under limited visibility, C omput. Geom. Theory Appl. 1 (1992),
+149–170.
+[16] R. Osserman. The isoperimetric inequality. Bull. Amer. Math. Soc. , 84:1182–1238, 1978.
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+(1991), 127–150.
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+ronment without sensing distances, IEEE Transactions on Ro botics 23 (2007), 506-518.
+15
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+arXiv:1001.0640v2  [math.AG]  6 Oct 2010Contemporary Mathematics
+Two exact sequences for lattice cohomology
+Andr´ as N´ emethi
+Dedicated to Henri Moscovici on his 60th birthday.
+Abstract. This article is a continuation of [ N08], where the lattice cohomology of connected
+negative definite plumbing graphs was introduced. Here we co nsider the more general situation of
+non–degenerate plumbing graphs, and we establish two exact sequences for their lattice cohomol-
+ogy. The first is the analogue of the surgery exact triangle pr oved by Ozsv´ ath and Szab´ o for the
+Heegaard–Floer invariant HF+; for the lattice cohomology over Z2–coefficients it was proved by
+J. Greene in [ Gr08]. Here we prove it over Z, and we supplement it by some additional properties
+valid for negative definite graphs. The second exact sequenc e is an adapted version which does
+not mix the classes of the characteristic elements ( spinc–structures); it was partially motivated by
+the surgery formula for the Seiberg–Witten invariant obtai ned in [BN10]. For this, we define the
+relative lattice cohomology and we determine its Euler characteristic in terms of Seiber g–Witten
+invariants.
+1. Introduction
+The lattice cohomology {Hq(Γ)}q≥0was introduced in [ N08]. In its original version, it was associated
+with anyconnected negative definiteplumbinggraph Γ, or, eq uivalently, with any oriented3–manifold, which
+might appear as the link of a local complex normal surface sin gularity. Lattice cohomology (together with
+the graded roots) plays a crucial role in the comparison of th e analytic and topological invariants of surface
+singularities, cf. [ N05, N07, N08 ], see also [ BN10, NN02, N10 ] for relations with the Seiberg–Witten
+invariants of the link. Additionally, the lattice cohomolo gy (conjecturally) offers a combinatorial description
+for the Heegaard–Floer homology of Ozsv´ ath and Szab´ o (for this theory see [ OSz03, OSz04, OSz04b ] and
+thelonglistofarticles ofOzsv´ athandSzab´ o). Indeed, in [N08]theauthorconjecturedthat ⊕q even/odd Hq(Γ)
+is isomorphic as a graded Z[U]–module with HF+
+even/odd(−M(Γ)), where M(Γ) is the plumbed 3–manifold
+associated with Γ. (Recall that at this moment there is no com binatorial definition/characterization of
+HF+.)
+For rational and ‘almost rational’ graphs this corresponde nce was established in [ OSz03, N05 ] (see also
+[NR10] for a different situation and [ N08] for related results). A possible machinery which might hel p to
+prove the general conjecture is based on the surgery exact se quences. They are established for the Heegaard–
+Floer theory in the work of Ozsv´ ath and Szab´ o. Our goal is to prove the analogous exact sequences for the
+lattice cohomology. In fact, independently of the above con jecture and correspondence, the proof of such
+exact sequences is of major importance, and they are fundame ntal in the computation and in finding the
+main properties of the lattice cohomologies.
+The formal, combinatorial definition of the lattice cohomol ogy permits to extend its definition to ar-
+bitrary graphs (plumbed 3–manifolds), the connectedness a nd negative definiteness assumptions can be
+dropped. Nevertheless, in the proof of the exact sequences, we will deal only with non–degenerate graphs
+(they are those graphs whose associated intersection form h as non–zero determinant).
+2000Mathematics Subject Classification. Primary. 14B05, 14J17, 32S25, 57M27, 57R57. Secondary. 14E 15,
+32S45, 57M25.
+Key words and phrases. plumbed manifolds, plumbing graphs, 3-manifolds, lattice cohomology, surface singular-
+ities, rational singularities, Seiberg-Witten invariant , Heegaard-Floer homology.
+The author is partially supported by OTKA Grant K67928.
+c/circlecop†rt0000 (copyright holder)
+12 Andr´ as N´ emethi
+More precisely, for any graph Γ and fixed vertex j0, we consider the graphs Γ \j0and Γ+
+j0, where the
+first one is obtained from Γ by deleting the vertex j0and adjacent edges, while the second one is obtained
+from Γ by replacing the decoration ej0of the vertex j0byej0+1. We will assume that all these graphs are
+non–degenerate. Then Theorem 5.5.3 establishes the follow ing long exact sequence.
+Theorem A. Assume that the graphs Γ+
+j0,ΓandΓ\j0are non–degenerate. Then
+··· −→Hq(Γ+
+j0)Aq
+−→Hq(Γ)Bq
+−→Hq(Γ\j0)Cq
+−→Hq+1(Γ+
+j0)−→ ···
+is an exact sequence of Z[U]–modules.
+The first 3 terms of the exact sequence (i.e. the H0–part) were already used in [ OSz03] (see also [ N05]),
+and the existence of the long exact sequence was already prov ed overZ2–coefficients in [ Gr08]. Here we
+establish its validity over Z. In the proof we not only find the correct sign–modifications, but we also replace
+some key arguments. (Nevertheless, the proof follows the ma in steps of [ Gr08].)
+For negative definite graphs (i.e. when Γ+
+j0, hence Γ and Γ \j0too are negative definite), the above exact
+sequence has some important additional properties. By gene ral theory (cf. [ N08]), if Γ is negative definite
+thenH0(Γ) contains a canonical submodule Tand one has a direct sum decomposition H0=T⊕H0
+red.
+Tis the analogue of the image of HF∞inHF+in the Heegaard–Floer theory. On the other hand, in
+Heegaard–Floer theory by a result of Ozsv´ ath and Szab´ o, th e operator C0restricted on T(Γ\j0) is zero
+(this follows from the fact that the corresponding cobordis m connecting Γ \j0and Γ+
+j0is coming from a
+non–negative definite surgery). The analogue of this result is Theorem 6.1.2:
+Theorem B. Assume that Γ+
+j0is negative definite. Consider the exact sequence
+0−→H0(Γ+
+j0)A0
+−→H0(Γ)B0
+−→H0(Γ\j0)C0
+−→H1(Γ+
+j0)−→ ···
+and the canonical submodule T(Γ\j0)ofH0(Γ\j0). Then the restriction C0|T(Γ\j0)is zero.
+Using the above results one proves for a graph Γ with at most nbad vertices (for the definition, see 6.2)
+the vanishing Hq
+red(Γ) = 0 for any q≥n(whereHq
+red=Hqforq>0).
+H∗(Γ) has a natural direct sum decomposition indexed by the set of the characteristic element classes
+(in the case of the Heegaard–Floer, or Seiberg–Witten theor y, they correspond to the spinc–structures of
+M(Γ)). Namely, H∗(Γ) =⊕[k]H∗(Γ,[k]). In the exact sequence of Theorem A the operators mix these
+classes. Theorem 7.2.2 provides an exact sequence which con nects the lattice cohomologies of Γ and Γ \j0
+with fixed (un–mixed) characteristic element classes. More precisely, for any characteristic element kof Γ,
+we define a Z[U]–module {Hq
+rel(k)}q≥0, therelative lattice cohomology associated with (Γ ,j0,k). It has finite
+Z–rank and it fits in the following exact sequence:
+Theorem C. Assume that ΓandΓ\j0are non–degenerate. One has a long exact sequence of Z[U]–modules:
+··· −→Hq
+rel(k)Aq
+rel−→Hq(Γ,[k])Bq
+rel−→Hq(Γ\j0,[R(k)])Cq
+rel−→Hq+1
+rel(k)−→ ···
+whereR(k)is the restriction of k(and the operators also depend on the choice of the represent ativek).
+The existence of such a long exact sequence is predicted and m otivated by the surgery formula for
+the Seiberg–Witten invariant established in [ BN10]: the results of section 7 resonate perfectly with the
+corresponding statements of Seiberg–Witten theory. This a llows us to compute the Euler characteristic
+of the relative lattice cohomology in terms of the Seiberg–W itten invariants associated with M(Γ) and
+M(Γ\j0).
+2. Notations and preliminaries
+2.1. Notations. First we will introduce the needed notations regarding plum bing graphs and we recall
+the definition of the lattice cohomology from [N08]. Since in [loc.cit.] the graphs were connected and
+negative definite (as the ‘normal’ plumbing representations of isolated comp lex surface singularity links), at
+the beginning we will start with these assumptions .
+Let Γ be such a plumbing graph with vertices Jand edges E; we set|J|=s. and we fix an order on J.
+Γ can also be codified in the lattice L, the free Z–module generated by {Ej}j∈Jand the ‘intersection form’
+{(Ei,Ej)}i,j, where (Ei,Ej) fori/\e}atio\slash=jis 1 or 0 corresponding to the fact that ( i,j) is an edge or not; and
+(Ei,Ei) is the decoration of the vertex i, usually denoted by ei. (The graph is negative definite if this form is
+so.) The graph Γ may have cycles, but we will assume that all th e genus decorations are zero (i.e. we plumb
+S1–bundles over S2). The associated plumbed 3–manifold M(Γ) is not necessarily a rational homologyTwo exact sequences for lattice cohomology 3
+sphere, this happens exactly when the graph is a tree. Let L′be the dual lattice {l′∈L⊗Q: (l′,L)⊂Z};
+it is generated by {E∗
+j}j, where (E∗
+j,Ei) =−δij(thenegative of the Kronecker–delta). Moreover,
+Char:={k∈L′:χk(l) :=−1
+2(k+l,l)∈Zfor alll∈L}
+denotes the set of characteristic elements of L(or Γ).
+As usual (following Ozsv´ ath and Szab´ o), T+
+0denotes the Z[U]–module Z[U,U−1]/UZ[U] with grading
+deg(U−d) = 2d(d≥0). More generally, for any r∈Qone defines T+
+r, the same module as T+
+0, but graded
+(byQ) in such a way that the d+r–homogeneous elements of T+
+rare isomorphic with the d–homogeneous
+elements of T+
+0. (E.g., for m∈Z,T+
+2m=Z[U,U−1]/U−m+1Z[U].)
+2.2. The lattice cohomology associated with k∈Char[N08].L⊗R=Zs⊗ZRhas a natural
+cellular decomposition into cubes. The set of zero–dimensi onal cubes is provided by the lattice points L.
+Anyl∈Land subset I⊂ Jof cardinality qdefines aq–dimensional cube, which has its vertices in the
+lattice points ( l+/summationtext
+j∈I′Ej)I′, whereI′runs over all subsets of I. On each such cube we fix an orientation.
+This can be determined, e.g., by the order ( Ej1,...,E jq), wherej1<···<jq, of the involved base elements
+{Ej}j∈I. The set of oriented q–dimensional cubes defined in this way is denoted by Qq(0≤q≤s).
+LetCqbe the free Z–module generated by oriented cubes /squareq∈ Qq. Clearly, for each /squareq∈ Qq, the
+oriented boundary ∂/squareqhas the form/summationtext
+kεk/squarek
+q−1for someεk∈ {−1,+1}. Here, in this sum, we write only
+those (q−1)–cubes which appear with non–zero coefficient. One sees tha t∂◦∂= 0, but, obviously, the
+homology of the chain complex ( C∗,∂) is trivial: it is just the homology of Rs. In order to get a more
+interesting (co)homology, one needs to consider a weight functions w:Qq→Z(0≤q≤s). In the present
+case this will be defined, for each k∈Charfixed, by
+w(/squareq) := max {χk(l) :lis a vertex of /squareq}.
+Once the weight function is defined, one considers Fq, the set of morphisms Hom Z(Cq,T+
+0) with finite
+support on Qq. Notice that Fqis, in fact, a Z[U]–module by ( p∗φ)(/squareq) :=p(φ(/squareq)) (φ∈ Fq,p∈Z[U]).
+Moreover, Fqhas a2Z–grading:φ∈ Fqis homogeneous ofdegree 2 d∈Zif for each /squareq∈ Qqwithφ(/squareq)/\e}atio\slash= 0,
+φ(/squareq) is a homogeneous element of T+
+0of degree 2d−2·w(/squareq).
+Next,onedefines δ:Fq→ Fq+1. Forthis, fix φ∈ Fqandweshowhow δ(φ)actsonacube /squareq+1∈ Qq+1.
+First write ∂/squareq+1=/summationtext
+kεk/squarek
+q, then set
+(2.2.1) ( δ(φ))(/squareq+1) :=/summationdisplay
+kεkUw(/squareq+1)−w(/squarek
+q)φ(/squarek
+q).
+One verifies that δ◦δ= 0, i.e. ( F∗,δ) is a cochain complex (with δhomogeneous of degree zero); its
+homology is denoted by {Hq(Γ,k)}q≥0.
+(F∗,δw) has a natural augmentation too. Indeed, set mk:= min l∈L{χk(l)}. Then one defines the
+Z[U]–linear map ǫ:T+
+2mk−→ F0such thatǫ(U−mk−s)(l) is the class of U−mk+χk(l)−sinT+
+0for any integer
+s≥0. Thenǫis injective and homogeneous of degree zero, δ◦ǫ= 0. The homology of the augmented
+cochain complex
+0−→ T+
+2mkǫ−→ F0δ−→ F1δ−→...
+is called the reduced lattice cohomology H∗
+red(Γ,k). For any q≥0, both HqandHq
+redadmit an induced
+gradedZ[U]–module structure and Hq=Hq
+redforq>0.
+Note thatǫprovides a canonical embedding of T+
+2mkintoH0. Moreover, one has a graded Z[U]–module
+isomorphism H0=T+
+2mk⊕H0
+red, andH∗
+redhas finite Z–rank.
+Remark 2.2.2. For the definition of the lattice cohomology for more general weight functions and graphs
+with non–zero genera, see [ N08].
+2.3. Reinterpretation of the lattice cohomology. Ifk′=k+2lfor somel∈LthenH∗(Γ,k) and
+H∗(Γ,k′) are isomorphic up to a degree–shift, cf. [ N08, (3.3)]. In fact, all the modules {H∗(Γ,k)}kcan be
+packed into only one object, more in the spirit of [ OSz03]. This was used in [ Gr08] too.
+In this way, for any fixed k∈Char,Lis identified with the sublattice k+ 2L∈Char, and with the
+notationl′:=k+2lone hasχk(l) =−1
+8(l′,l′)+1
+8(k,k). In particular, up to a shift in degree, for each fixed
+k,l/mapsto→χk(l) andl′/mapsto→ −1
+8(l′,l′) define the same weight–function (on the cubes, see below), h ence the same
+cohomology. In fact, we will modify even this weight functio n bys/8 (in this way the blowing up will induce
+a degree preserving isomorphism, cf. §3), and set:
+(2.3.1) w:Char→Q, w(k) :=−k2+s
+8(s=|J|).4 Andr´ as N´ emethi
+Theq–cubes/squareq∈ Qqare associated with pairs ( k,I)∈Char× P(J),|I|=q, (hereP(J) denotes the
+power set of J), and have the form {k+2/summationtext
+j∈I′Ej)I′, whereI′runs over all subsets of I. The weights are
+defined by
+(2.3.2) w(/squareq) =w((k,I)) = max
+I′⊂I/braceleftbig
+w(k+2/summationdisplay
+j∈I′Ej)/bracerightbig
+.
+Moreover, Fqare elements of Hom Z(Qq,T+
+0) with finite support. Similarly as above, Fqis aZ[U]–module
+with aQ–grading:φ∈ Fqis homogeneous of degree rif for each /squareq∈ Qqwithφ(/squareq)/\e}atio\slash= 0,φ(/squareq) is a
+homogeneous element of T+
+0of degreer−2·w(/squareq).
+It is convenient to consider the module of (infinitely suppor ted) homological cycles too: let Fqbe the
+direct product of Z≥0× Qqcopies of Z(considered already in [ OSz03] forq= 0). We write the pair
+(m,/square) asUm/square.Fqbecomes a Z[U]–module by U(Um/square) =Um+1/square. Clearly Fq= Hom Z[U](Fq,T+
+0), i.e.
+φ(U/square) =Uφ(/square) for anyφ.
+δ:Fq→ Fq+1is defined as in (2.2.1) using the new weight–function, or by δ(φ)(/square) =φ(∂(/square)), where
+for/square= (k,I) = (k,{j1,...,j q}) one has:
+(2.3.3) ∂(k,I) =q/summationdisplay
+l=1(−1)l/parenleftbig
+Uw(k,I)−w(k,I\jl)(k,I\jl)−Uw(k,I)−w(k+2Ejl,I\jl)(k+2Ejl,I\jl)/parenrightbig
+.
+Thecohomology of( F∗,δ)isdenotedby H∗(Γ). Sincetheverticesofacubebelongtothesameclass Char/2L
+(where a class has the form [ k] ={k+2l}l∈L⊂Char),H∗(Γ) has a natural direct sum decomposition
+H∗(Γ) =⊕[k]∈Char/2LH∗(Γ,[k]).
+In fact, if [k1] = [k2] thenw(k1)−w(k2)∈Z.
+Since Γ is negative definite, for each class [ k]∈Char/2Lone has a well–defined rational number
+d[k] :=−max
+k∈[k]k2+s
+4= 2·min
+k∈[k]w(k).
+Then (cf. 2.2) one has a direct sum decomposition:
+(2.3.4) H0(Γ,[k]) =T+
+d[k]⊕H0
+red(Γ,[k]).
+Sometimes we write T:=⊕[k]T+
+d[k]for the canonical submodule of H0, which satisfies H0=T⊕H0
+red.
+Following [ N05, N08, N10 ] we define the Euler characteristic of H∗(Γ,[k]) by
+(2.3.5) eu(H∗(Γ,[k])) :=−d[k]/2+/summationdisplay
+q≥0(−1)qrankZHq
+red(Γ,[k]).
+Remark 2.3.6. For anyφ∈ Fqandℓ≥0 setU−ℓ∗φ∈ Fqdefined as follows: if φ(/square) =/summationtext
+m≥0am,/squareU−m,
+then (U−ℓ∗φ)(/square) =/summationtext
+m≥0am,/squareU−m−ℓ. Notice that U(U−1∗φ) =φ(but, in general, U−1∗(Uφ)/\e}atio\slash=φ).
+For any /square∈ Qqlet/square∨be that element of Fqwhich sends /squarein 1∈ T+
+0and any other element into
+zero. Then any element of Fqis a finite Z–linear combination of elements of type U−ℓ∗/square∨.
+2.4. Generalizations. Graphs which are not negative definit e.Since the definition of the lattice
+cohomology is purely algebraic/combinatorial, its definit ion can be considered for any graph, or even for any
+latticeLwith fixed base elements {Ej}j. Nevertheless, in this note we assume that the graphs are non –
+degenerate. Of course, doing this generalization, some of t he properties of the lattice cohomology associated
+with connected negative definite graphs will not survive. He re we wish to point out some of the differences.
+First, we notice that dropping the connectedness of the grap h basically has no effect (this fact already
+was used in the main inductive proofs of [ OSz03] and [N05, (8.3)]). Dropping the negative definiteness is
+more serious. In order to explain this, we will introduce the following spaces: for any fixed class [ k] and any
+real number rwe defineSras the union of all cubes /square∈ Qqwith all vertices in [ k] andw(/square)≤r.
+If Γ is negative definite then Sris compact for any r. In particular, the weight function takes its
+minimum on it, and for each class [ k] =Char/2La decomposition H0(Γ,[k]) =T+
+d([k])⊕H0
+red(Γ,[k]) can be
+defined. This property will not survive for general graphs. I n fact, if the components of Srare not compact,
+then we even might have the vanishing H0(Γ) = 0 (and this can happen simultaneously with the non–
+vanishing of H1), see e.g. (2.4.1). Also, in [ N08, Theorem 3.1.12] the lattice cohomology is recovered from
+the simplicial cohomology of the spaces Sr. In the general case, the adapted version of that theorem (wi th
+similar proof), valid for any lattice, states that the same f ormula is valid once if we replace the cohomology
+groupsHq(Sr,Z) by the cohomology groups with compact support Hq
+c(Sr,Z). Namely:
+H∗(Γ,[k])≃/circleplusdisplay
+r∈w(k)+ZH∗
+c(Sr,Z).Two exact sequences for lattice cohomology 5
+We wish toemphasize, that replacing thecohomology with coh omology with compact support is a conceptual
+modification: the two groups have different functorial prope rties.
+In fact, exactly this second point of view is determinative i n the definition of the lattice cohomology
+(and in the definition of the maps in the surgery formula treat ed next): basically we mimic the infinitely
+supported homology and the (dual) cohomology with compact s upport, and the corresponding functors
+associated with them.
+Since the surgery exact sequence considered in the next sect ion is valid for any non–degenerate lattice,
+it is very convenient to extend the theory to arbitrary graph s (or, at least for non–degenerate ones) in order
+to have a larger flexibility for computations. Nevertheless , we have to face the following crucial problem. An
+oriented plumbed 3–manifold M(Γ) has many different plumbing representations Γ. They are c onnected by
+the moves of the (oriented) plumbing calculus. For negative definite graphs the only moves are the blowups
+(and their inverses) by (rational) ( −1)-vertices. In [ N08, (3.4)] it is proved that the lattice cohomology is
+stable with respect to these moves, see also §3 here. On the other hand, if we enlarge our plumbing graphs,
+this stability condition will not survive: the same 3–manif old can be represented by many different lattices
+with rather different lattice cohomologies.
+More precisely, for negative definite graphs one conjecture s (cf. [N08]) that from the lattice cohomology
+one can recover (in a combinatorial way) the Heegaard–Floer homology of Ozsv´ ath–Szab´ o. In particular,
+the lattice cohomology carries a geometric meaning dependi ng only on M(Γ). This geometric meaning is
+lost in the context of general graphs (or, at least, it is not s o direct).
+Example 2.4.1. S3can be represented by a graph with one vertex, which has decor ation−1. Computing
+the lattice cohomology of this graph we get Hq= 0 forq/\e}atio\slash= 0 and H0=H0
+red=T+
+0. (This is the Heegaard–
+Floer homology HF+(S3).) On the other hand, it is instructive to compute the lattic e cohomology of
+another graph, which has one vertex, but now with decoration +1. Its lattice cohomology is Hq= 0 for
+q/\e}atio\slash= 1 and H1=T+
+−1/2. This graph also represent S3, and the two graphs can be connected by a sequence
+of non–empty graphs and ±1–blow ups and downs. In particular, we conclude that the lat tice cohomology
+is not stable with respect to blowing up/down (+1)–vertices .
+Remark 2.4.2. Replacing negative definite graphs with arbitrary non–dege nerate ones we can also adopt
+the definition of the weight function (2.3.1) by modifying in to
+(2.4.3) w†:Char→Q, w †(k) :=−k2+3σ+2s
+8(σ= signature of ( ,), s=|J|),
+a more common expression associated with (4–manifold inter section) forms which are not negative definite.
+In this note we will not do this, but the interested reader pre ferring this expression might shift all the weights
+below by 3( σ+s)/8.
+3. Blowing up Γ
+3.1.Since in the main construction we will need some of the operat ors induced by blowing down, we
+will make explicit the involved morphisms.
+The next discussion provides a new proof of the stability of t he lattice cohomology in the case when we
+blow up a vertex (for the old proof see [ N08, (3.4)]).
+Starting from now, the graph is neither necessarily connect ed, nor necessarily negative definite. Never-
+theless, we will assume that the intersection form is non–de generate.
+We assume that Γ′is obtained from Γ by ‘blowing up the vertex j0’. More precisely, Γ′denotes a graph
+with one more vertex and one more edge: we glue to the vertex j0by the new edge the new vertex Enew
+with decoration −1 (and genus 0), while the decoration of Ej0is modified from ej0intoej0−1, and we keep
+all the other decorations. We will use the notations L(Γ), L(Γ′), L′(Γ), L′(Γ′). Letw′:Char(Γ′)→Q
+be the weight function of Γ′defined similarly as in (2.3.1). (We may use the following con vention for the
+ordering of the indices: if j/\e}atio\slash=j0, thenj <j0<jnew.) The following facts can be verified:
+3.1.1.Consider the maps π∗:L(Γ′)→L(Γ) defined by π∗(/summationtextxjEj+xnewEnew) =/summationtextxjEj, andπ∗:
+L(Γ)→L(Γ′) defined by π∗(/summationtextxjEj) =/summationtextxjEj+xj0Enew. Then (π∗x,x′)Γ′= (x,π∗x′)Γ. This shows that
+(π∗x,π∗y)Γ′= (x,y)Γand (π∗x,Enew)Γ′= 0 for any x,y∈L(Γ). Bothπ∗andπ∗extend over L⊗Qand
+L′.
+3.1.2.Set the (nonlinear) map: c:L′(Γ)→L′(Γ′),c(l′) :=π∗(l′)+Enew. Thenc(Char(Γ))⊂Char(Γ′)
+andcinduces an isomorphism between the orbit spaces Char(Γ)/2L(Γ)≃Char(Γ′)/2L(Γ′). Moreover, for
+anyk∈Char(Γ) andk′:=c(k) one hasw′(k′) =w(k).
+3.1.3.π∗(Char(Γ′))⊂Char(Γ). For any k′∈Char(Γ′) writek:=π∗(k′). Thenk′=π∗(k)+aEnewfor
+some odd integer a, andw′(k′)−w(k) =a2−1
+8∈Z≥0.6 Andr´ as N´ emethi
+The mapscandπ∗can be extended to the level of cubes and complexes as follows .
+3.1.4.For anyq≥0 and/square= (k′,I)∈ Qq(Γ′) one defines π∗((k′,I)) := (π∗(k′),I)∈ Qq(Γ), provided that
+jnew/\e}atio\slash∈I. By (3.1.3) one has wΓ′(/square)−wΓ(π∗(/square))≥0. This defines a homological morphism πh
+∗:Fq(Γ′)→
+Fq(Γ) by
+(3.1.5) πh
+∗(/square) =/braceleftbigg
+UwΓ′(/square)−wΓ(π∗(/square))π∗(/square) ifjnew/\e}atio\slash∈I,
+0 else .
+Using (2.3.3) one verifies that πh
+∗◦∂=∂◦πh
+∗, henceπh
+∗is morphism of homological complexes. In particular,
+its dualπc
+∗:Fq(Γ)→ Fq(Γ′), defined by πc
+∗(φ) =φ◦πh
+∗, satisfiesπc
+∗◦δ=δ◦πc
+∗, hence it is a morphism of
+complexes as well.
+3.1.6.Before we extend cto the level of complexes, notice that for any k∈Char(Γ)
+c(k+2Ej) =/braceleftbiggc(k)+2Ej ifj/\e}atio\slash=j0,
+c(k)+2Ej+2Enewifj=j0.
+In particular, the pair c(k) andc(k+2Ej0) does not form a 1–cube in Γ′. Hence, the application ( k,I)/mapsto→
+(c(k),I), sending the vertex k+ 2/summationtext
+j∈I′Ejintoc(k) + 2/summationtext
+j∈I′Ej,does not commute with the boundary
+operator. The ‘right’ operator ch:Fq(Γ)→ Fq(Γ′) is defined by
+(3.1.7)ch((k,I)) =/braceleftbigg(c(k),I) ifj0/\e}atio\slash∈I,
+(c(k),I)+Uw(k,I)−w(k+2Ej0,I0)(c(k)+2Ej0,I0∪jnew) ifI=I0∪j0,j0/\e}atio\slash∈I0.
+In fact, by (3.1.8) below, w(k+2Ej0,I0) above can be replaced by w′(c(k)+2Ej0,I0), or even by w′(c(k)+
+2Ej0,I0∪jnew). By (3.1.2) and a computation one gets (where in the third li neI′
+0is any subset of Jwith
+j0/\e}atio\slash∈I′
+0):
+(3.1.8)w((k,I)) =w′((c(k),I)),
+w((k+2Ej0,I0)) =w′((c(k)+2Ej0,I0)),
+w′(c(k)+2/summationtext
+j∈I′
+0Ej+2Ej0) =w′(c(k)+2/summationtext
+j∈I′
+0Ej+2Ej0+2Enew).
+These and a (longer) computations shows that chcommutes with the boundary operator ∂.
+3.1.9.Usingπ∗◦c, the definitions and the first equation of (3.1.8) one gets πh
+∗◦ch=idF∗(Γ). On the other
+hand,ch◦πh
+∗is not the identity, but it is homotopic to idF∗(Γ′). Indeed, we define the homotopy operator
+K:F∗(Γ′)→ F∗+1(Γ′) as follows. Write any kascπ∗(k)+2aEnewfor somea∈Z. Then define K((k,I))
+as 0 if either jnew∈Iora= 0. Otherwise take for K((k,I)):
+sign(a)·/summationdisplay
+Uw(k,I)−w(cπ∗(k)+2lEnew,I∪jnew)(cπ∗(k)+2lEnew,I∪jnew),
+where the summation is over l∈ {0,1,...,a−1}ifa>0 andl∈ {a,...,−1}ifa<0. (The exponents are
+non–negative because of (3.1.3).) Then, again by a computat ion,∂◦K−K◦∂=id−ch◦πh
+∗.
+In particular, πh
+∗andchinduce (degree preserving) isomorphisms of the correspond ing lattice cohomolo-
+gies. In the sequel the operator πh
+∗will be crucial.
+4. Comparing ΓandΓ\j0
+4.1. Notations, remarks. We consider a non–degenerate graph Γ as in 3.1, and we fix one of its
+verticesj0∈ J. The new graph Γ \j0is obtained from Γ by deleting the vertex j0and all its adgacent
+edges. We will denote by L(Γ),L′(Γ),L(Γ\j0),L′(Γ\j0) the corresponding lattices.
+The operator i:L(Γ\j0)→L(Γ),i(Ej,Γ\j0) =Ej,ΓidentifiesL(Γ\j0) with a sublattice of L(Γ). The
+dual operator (restriction) is R:L′(Γ)→L′(Γ\j0),R(E∗
+j,Γ) =E∗
+j,Γ\j0forj/\e}atio\slash=j0andR(E∗
+j0,Γ) = 0. It
+satisfies (iQ(x),y)Γ= (x,R(y))Γ\j0for anyx∈L′(Γ\j0) andy∈L′(Γ). (Here, E∗
+j,Γrespectively E∗
+j,Γ\j0are
+the usual dual generators of L′considered in Γ, respectively in Γ \j0.)
+Recall that l′=/summationtext
+jajE∗
+j,Γis characteristic if and only if aj≡ej(mod2). In particular, Rsends charac-
+teristic elements into characteristic elements. On the oth er hand,iQnot necessarily preserves characteristic
+elements.
+AlthoughR(Char(Γ))⊂Char(Γ\j0), this operator cannot be extended to the level of cubes. Ind eed,
+notice that
+(4.1.1) R(Ej0) =−/summationdisplay
+(j,j0)∈EΓE∗
+j,Γ\j0,
+where the sum runs over the adjacent vertices of j0in Γ. In particular, the endpoints of the 1–cube ( k,j0)
+are sent into R(k) andR(k)+2R(Ej0), which cannot be expressed as combination of 1–cubes in Γ \j0. InTwo exact sequences for lattice cohomology 7
+the next subsection we will consider another operator, whic h can be extended to the level of cubes (and
+which, in fact, operates in a different direction than R, cf. 2.4).
+4.2. TheB–operator. Considerb:L′(Γ\j0)→L′(Γ) defined by
+/summationdisplay
+jajE∗
+j,Γ\j0/mapsto→/summationdisplay
+jajE∗
+j,Γ.
+Clearly, ifk∈Char(Γ\j0) thenb(k)+aj0E∗
+j0,Γ∈Char(Γ) for any aj0withaj0≡ej0(mod2). In order to
+see how it operates on cubes, notice that in Γ \j0for anyj∈ J(Γ\j0) one has
+Ej,Γ\j0=−ejE∗
+j,Γ\j0−/summationdisplay
+(i,j)∈EΓ\j0E∗
+i,Γ\j0,
+and a similar relation holds in Γ too. Therefore, one gets tha t
+(4.2.1) b(Ej) =Ej+(Ej0,Ej)Γ·E∗
+j0,Γ.
+Therefore,
+(4.2.2) b(l′) =iQ(l′)+(Ej0,iQ(l′))Γ·E∗
+j0,Γ.
+In particular, bis a ‘small’ modification of iQ, but this modification is enough to extend it to the level of
+cubes. Although the vertices of ( k,I) are not sent to the vertices of a cube (provided that Icontains elements
+adjacent to j0in Γ), cf. (4.2.1), nevertheless if we consider all the shift s in the direction of the ‘error’ of
+(4.2.1) we get a well–defined operator
+Bq:Fq(Γ\j0)→ Fq(Γ)
+given by
+(4.2.3) Bq((k,I)) :=/summationdisplay
+aj0≡ej0(mod2)(b(k)+aj0E∗
+j0,Γ,I).
+(Here we keep the notation ‘ B’, since this notation was used in similar contexts, as in [ OSz03, N05, Gr08 ].)
+Bqinduces a morphism Bq:Fq(Γ)→ Fq(Γ\j0) via (Bqφ)(/square) =φ(Bq/square). A straightforward (slightly
+long) computation shows that Bqcommutes with the boundary operator ∂, henceB∗◦δ=δ◦B∗too. In
+particular, one gets a well–defined morphism of Z[U]–modules B∗:H∗(Γ)→H∗(Γ\j0).
+4.3. The sign–modified B–operator. In the exact sequences considered in the next section we will
+need to modify the B–operator by a sign (compare with the end of §2 of [OSz03], where the case q= 0 is
+discussed). The definition depends on some choices.
+For eachL–orbit [k] :=k+ 2L(Γ\j0)⊂Char(Γ\j0) we will fix a representative r[k]∈[k]. Hence,
+for any characteristic element k, with orbit [ k],k−r[k]∈2L(Γ\j0), hence (k−r[k],E∗
+j,Γ\j0)∈2Zfor any
+j/\e}atio\slash=j0. In particular, ( k−r[k],R(Ej0))∈2ZforR(Ej0) defined in (4.1.1). Then the modified operator
+Bq:Fq(Γ\j0)→ Fq(Γ)
+is given by
+(4.3.1) Bq((k,I)) :=/summationdisplay
+aj0≡ej0(mod2)(−1)n(k,aj0)/2·(b(k)+aj0E∗
+j0,Γ,I),
+where
+n(k,aj0) := (k−r[k],R(Ej0))+aj0+ej0.
+Then, again, B∗commutes with the boundary operator, hence B∗:Fq(Γ)→ Fq(Γ\j0) defined by
+(Bqφ)(/square) =φ(Bq/square) commutes with δ, hence it also defines a Z[U]–modules morphism B∗:H∗(Γ)→
+H∗(Γ\j0).
+Remark 4.3.2. The morphisms B∗andB∗donotpreserve the gradings neitherthe orbitsChar/2L(i.e.
+they do not split in direct sum with respect to these orbits).8 Andr´ as N´ emethi
+5. The surgery exact sequence
+5.1. Notations. Let Γ be a non–degenerate graph as above with s=|J| ≥2. We fix a vertex j0∈ J.
+Associated with j0we consider two other graphs, namely Γ \j0and Γ+
+j0. The second one is obtained from
+Γ by modifying the decoration ej0of the vertex j0intoej0+1. In order to stay in our category of objects,
+we will assume that both graphs Γ \j0and Γ+
+j0are non–degenerate.
+Our goal is to establish a long exact sequence connecting the lattice cohomologies of these three graphs,
+similar which is valid for the Heegaard–Floer cohomologies provided by surgeries. As usual, in order to
+define a long exact sequence, we need first to determine a short exact sequence of complexes.
+The graphs Γ and Γ \j0will be connected by the B–operator, cf. 4.3. We define the ‘ A’–operator
+connecting Γ and Γ+
+j0as follows. Let Γbbe the graph obtained from Γ by attaching a new vertex jnew(via
+a single new edge) to j0, such that the decoration of the new vertex is −1. Then, for the pair Γ, Γb(since
+Γ = Γb\jnew), we can apply theconstruction andresults of §4. Inparticular, we get morphisms of complexes:
+B∗:F∗(Γ)→ F∗(Γb) andB∗:F∗(Γb)→ F∗(Γ). Next, since Γ+
+j0obtained from Γbby blowing down the
+new vertex jnew, by§3 we get morphisms of complexes: πh
+∗:F∗(Γb)→ F∗(Γ+
+j0) andπc
+∗:F∗(Γ+
+j0)→ F∗(Γb).
+By composition we get the A–operators:
+A∗:=πh
+∗◦B∗:F∗(Γ)→ F∗(Γ+
+j0),andA∗:=B∗◦πc
+∗:F∗(Γ+
+j0)→ F∗(Γ).
+In particular, we can consider the short sequence of complex es
+(5.1.1) 0 → F∗(Γ+
+j0)A∗
+−→ F∗(Γ)B∗
+−→ F∗(Γ\j0)→0.
+Theorem 5.1.2. The short sequence of complexes (5.1.1) is exact.
+The proof is given in several steps.
+5.2. The injectivity of Aq.Take an arbitrary non–zero φ∈ Fq(Γ+
+j0). In order to prove that Aqφ/\e}atio\slash= 0,
+we need to find f∈ Fq(Γ) such that φ(Aq(f))/\e}atio\slash= 0. LetNbe the smallest non–negative integer such that
+UN+1φ= 0. Then replacing φbyUNφwe may assume that Uφ= 0. In particular, in f(or inAq(f)) any
+term whose coefficient has the form Un(n>0) is irrelevant.
+Sinceφ/\e}atio\slash= 0, there exists ( ¯k,I)∈ Qq(Γ+
+j0) withφ((¯k,I))/\e}atio\slash= 0. Write
+(5.2.1) ¯k=/summationdisplay
+j∈JajE∗
+j,Γ+
+j0+E∗
+j0,Γ+
+j0,
+withaj−ejeven for all j∈ J. Sinceφis finitely supported, we may assume that
+(5.2.2) aj0is minimal with the property φ((¯k,I))/\e}atio\slash= 0.
+We wish to construct f∈ Fq(Γ) such that φ(Aq(f)−(¯k,I)) = 0. For the weight functions in Γband Γ+
+j0we
+will use the notations wbrespectively w+.
+First, set
+(5.2.3) k:=/summationdisplay
+jajE∗
+j,Γ∈Char(Γ).
+Then
+B0(k) =/summationdisplay
+a≡1kb
+a,wherekb
+a=/summationdisplay
+jajE∗
+j,Γb+aE∗
+new.
+Set also
+ka:=π∗(kb
+a) =/summationdisplay
+jajE∗
+j,Γ+
+j0+aE∗
+j0,Γ+
+j0.
+Notice that k1=¯k. Sinceπ∗ka=kb
+a−aEnew, by (3.1.3)
+(5.2.4) wb(kb
+a) =w+(ka)+(a2−1)/8.
+Moreover,B((k,I)) =/summationtext
+a≡1(kb
+a,I) andπ∗((kb
+a,I)) = (ka,I).
+For anyI′⊂IwriteEI′=/summationtext
+j∈I′Ej. LetδI′be 1 ifj0∈I′, and zero otherwise. Since
+(5.2.5) wb(kb
+a+2EI′)−wb(kb
+a) =1
+2/summationdisplay
+j∈I′aj−1
+2(EI′,EI′)Γb,
+and
+(5.2.6) w+(ka+2EI′)−w+(ka) =1
+2/summationdisplay
+j∈I′aj+1
+2δI′a−1
+2(EI′,EI′)Γ+
+j0,Two exact sequences for lattice cohomology 9
+we get
+(5.2.7) wb(kb
+a+2EI′)−w+(ka+2EI′) =a2−1
+8−δI′·a−1
+2.
+Assume that I/\e}atio\slash∋j0. Then, by (5.2.7), we get that wb(kb
+a,I)−w+(ka,I) = (a2−1)/8, hence
+(5.2.8) Aq((k,I)) =/summationdisplay
+a≡1,a<0Ua2−1
+8(ka,I)+(¯k,I)+/summationdisplay
+a≡1,a≥3Ua2−1
+8(ka,I).
+Both sums are killed by φ, the first one by (5.2.2), the second one by Uφ= 0, henceφ(Aq((k,I))−(¯k,I)) = 0.
+Next, assume that I∋j0. In that case, again by (5.2.7), we get that wb(kb
+a,I)−w+(ka,I)>0 whenever
+a/\e}atio\slash∈ {−1,1,3}. For the other three special values we have the following fac ts. It is convenient to set for each
+k(represented as in (5.2.3)) the integer:
+(5.2.9) M(k) := max
+I′{/summationdisplay
+j∈I′aj−(EI′,EI′)Γb}.
+•Fora= 1 one has wb(kb
+a,I)−w+(ka,I) = 0 always.
+•Ifa= 3, then the right hand side of (5.2.7) is 1 −δI′, hencewb(kb
+a,I)−w+(ka,I) = 0 if and only if
+maxI′{wb(kb
+a+2EI′)}can be realized by some I′withI′∋j0. By (5.2.5) this is equivalent to
+(5.2.10) M(k) can be realized by some I′withI′∋j0.
+•Ifa=−1, then the right hand side of (5.2.7) is δI′, hencewb(kb
+a,I)−w+(ka,I) = 0 if and only if
+(5.2.11) M(k) can be realized by some I′withI′/\e}atio\slash∋j0.
+Assume that in the case I∋j0fora= 3 one has wb(kb
+a,I)−w+(ka,I)>0. Then by an identical
+argument as in the case I/\e}atio\slash∋j0we get that φ(Aq((k,I))−(¯k,I)) = 0.
+Hence, the remaining final case is when there exists at least o neI′⊂Iwhich contains j0and satisfies
+(5.2.10). Then
+Aq((k,I) = (¯k,I)+(¯k+2E∗
+j0,Γ+
+j0,I) (mod Uand ker(φ)).
+Now, we will consider k+2E∗
+j0,Γinstead ofk. Via the operator Aqit provides the same cubes as k(where
+the index set will have a shift a/mapsto→a−2) but with different Um–coefficients. Notice that by (5.2.10) and
+(5.2.11),M(k+2E∗
+j0,Γ) can be realized only by subsets I′withI′∋j0, hence (5.2.11) will fail. Therefore,
+Aq((k+2E∗
+j0,Γ,I) = (¯k+2E∗
+j0,Γ+
+j0,I)+(¯k+4E∗
+j0,Γ+
+j0,I) (mod U).
+More generally, for any positive integer ℓ, by the same argument:
+Aq((k+2ℓE∗
+j0,Γ,I) = (¯k+2ℓE∗
+j0,Γ+
+j0,I)+(¯k+(2ℓ+2)E∗
+j0,Γ+
+j0,I) (mod U).
+Sinceφis finitely supported, φ((2ℓ+2)E∗
+j0,Γ+
+j0,I)) = 0 for some ℓ≥0, let us consider the minimal such ℓ.
+ThenAq, moduloUand ker(φ), restricted on the relevant finite dimensional spaces look s as an (ℓ+1)×(ℓ+1)
+upper triangular matrix whose diagonal is the identity matr ix. Since this is invertible over Z, the result
+follows: some linear combinations of the elements Aq((k+ 2tE∗
+j0,Γ,I), 0≤t≤ℓ, is (¯k,I) moduloUand
+ker(φ).
+5.3. The surjectivity of Bq.Weprovidethesameargumentas[ Gr08]: Foranyfixed a≡ej0(mod2),
+Bq/parenleftbig
+U−ℓ∗(b(k)+aE∗
+j0,Γ,I)∨/parenrightbig
+=±U−ℓ∗(k,I)∨.
+Since the collection of U−ℓ∗(k,I)∨generate Fq(Γ\j0) (overZ), the surjectivity follows.
+5.4.Bq◦Aq=0.Take an arbitrary ( k,I)∈ Qq(Γ\j0). Then, by (5.2.7), one has:
+(Aq◦Bq)((k,I)) =/summationdisplay
+a≡1/summationdisplay
+c≡ej0(−1)N(k)+c+ej0
+2·Ua2−1
+8/parenleftbig
+π∗b(k)+(a+c)E∗
+j0,Γ+
+j0,I/parenrightbig
+.
+Those pairs ( a,c) for which a2−1 anda+care fixed hit the same element of Fq. Writea= 2i+1. Then
+the two solutions of ( a2−1)/8 =i(i+1)/2 =tsatisfiesi1+i2=−1. Since the corresponding two cvalues
+satisfies 2i1+c1= 2i2+c2, one gets that ( c2−c1)/2 is odd. Hence the terms cancel each other two by two.10 Andr´ as N´ emethi
+5.5.kerBq⊂imAq.Set kerUm:={φ∈ Fq(Γ) :Umφ= 0}. Notice that the inclusion
+(5.5.1) ker Um∩kerBq⊂imAq
+form= 1 (together with (5.4)) implies by induction its validity f or anym(cf. [Gr08]). Indeed, assume
+that the inclusion is true for m−1 and setφ∈kerUm∩kerBq. ThenUφ=Aq(ψ) for someψ. Moreover,
+˜φ:=φ−Aq(U−1∗ψ)∈kerU. On the other hand, by (5.4), ˜φ∈kerBqtoo. Therefore, by (5.5.1) applied
+form= 1, we get ˜φ∈imAq, henceφ∈imAqtoo.
+Notice that ∪m(kerUm∩kerBq) = kerBq, hence this would prove ker Bq⊂imAqtoo.
+Next, we show (5.5.1) for m= 1. First notice that ker U∩kerBqis generated by elements of type
+(k,I)∨+(k+2E∗
+j0,Γ,I)∨whereI/\e}atio\slash∋j0, and (k,I)∨whereI∋j0.
+In the first case (i.e. I/\e}atio\slash∋j0), let us write kas in (5.2.3), and set ¯kas in (5.2.1). Then by (5.2.7) and
+(5.2.8) one has
+Aq((k,I)) = (¯k−2E∗
+j0,Γ+
+j0,I)+(¯k,I) (mod U),
+Aq((k+2E∗
+j0,Γ,I)) = (¯k,I)+(¯k+2E∗
+j0,Γ+
+j0,I) (mod U),
+and (¯k,I) does not appear in any other term with non–zero coefficient ( modU). HenceAq(¯k,I)∨= (k,I)∨+
+(k+2E∗
+j0,Γ,I)∨.
+Next, we fix an element of type ( k,I)∨withI∋j0. It belongs to the collection {(k(i),I)}a∈Z, where
+k(i) =k+2iE∗
+j0,Γ, which will be treated simultaneously via the discussions o f (5.2). Write kas in (5.2.3)
+and set¯kvia (5.2.1); it is also convenient to write ¯k(i) :=¯k+ 2iE∗
+j0,Γ+
+j0. Notice that k(i) has coefficients
+{{aj}j/\egatio\slash=j0,aj0+2i}. Therefore, for i≪0 (5.2.10) will fail and (5.2.11) is satisfied. Let i0−1 be maximal
+when (5.2.10) fails. Then for i=i0both conditions are satisfied, and for i>i0only (5.2.10). Therefore,
+(5.5.2)Aq((k(i),I)) = (¯k(i−1),I)+(¯k(i),I) ( mod U) ifi<i0,
+Aq((k(i0),I)) = (¯k(i0−1),I)+(¯k(i0),I)+(¯k(i0+1),I) (mod U)
+Aq((k(i),I)) = (¯k(i),I)+(¯k(i+1),I) ( mod U) ifi>i0.
+This reads as
+Aq((¯k(i),I)∨) =
+
+(k(i),I)∨+(k(i+1),I)∨ifi<i0
+(k(i),I)∨ifi=i0
+(k(i),I)∨+(k(i−1),I)∨ifi>i0
+Taking finite linear combination we get that any ( k(i),I)∨is in the image of Aq. This ends the proof of
+Theorem (5.1.2). As a corollary we get:
+Theorem 5.5.3. Assume that the graphs Γ+
+j0,ΓandΓ\j0are non–degenerate. Then
+··· −→Hq(Γ+
+j0)Aq
+−→Hq(Γ)Bq
+−→Hq(Γ\j0)Cq
+−→Hq+1(Γ+
+j0)−→ ···
+is an exact sequence of Z[U]–modules.
+6. The exact sequence for negative definite graphs
+6.1. Preliminaries. For any graph Γ we write det(Γ) for the determinant of the negative of the
+intersection form associated with Γ. If Γ is negative definit e then det(Γ) is obviously positive.
+If Γ is negative definite then Γ \j0is automatically so for any j0. Nevertheless, this is not the case for
+Γ+
+j0(although, if Γ+
+j0is negative definite then Γ is so too).
+Lemma 6.1.1. Assume that Γis negative definite. Then Γ+
+j0is negative definite if and only if det(Γ)>
+det(Γ\j0). If these conditions are satisfied then E∗
+j0,Γ/\e}atio\slash∈L(Γ)and(E∗
+j0,Γ)2/\e}atio\slash∈Z.
+Proof. Γ+
+j0isnegativedefiniteifandonlyifdet(Γ+
+j0)ispositive(providedthatΓ \j0isnegativedefinite).
+But det(Γ+
+j0) = det(Γ) −det(Γ\j0). The last statement follows from −(E∗
+j0,Γ)2= det(Γ\j0)/det(Γ), which
+cannot be an integer. /square
+In the sequel we assume that all the graphs are negative defini te, but not necessarily connected. The
+next theorem is an addendum of Theorem 5.5.3. Since its proof is rather long, it will be published elsewhere.
+Theorem 6.1.2. Assume that Γ+
+j0is negative definite. Consider the exact sequence
+0−→H0(Γ+
+j0)A0
+−→H0(Γ)B0
+−→H0(Γ\j0)C0
+−→H1(Γ+
+j0)−→ ···
+and the canonical submodule T(Γ\j0)ofH0(Γ\j0). ThenT(Γ\j0)⊂imB0; or, equivalently, the restriction
+C0|T(Γ\j0)is zero.Two exact sequences for lattice cohomology 11
+6.2. Graphs with nbad vertices. We say that a negative definite connected graph is ‘rational’ if it
+is the plumbing representation of a the link of a rational sin gularity (or, the resolution graph of a rational
+singularity). They were characterized combinatorially by Artin, for more details see [ N99, N05 ].
+We fix an integer n≥0. We say that a negative definite graph has at most n‘bad’ vertices if we can find
+nvertices{jk}1≤k≤n, such that replacing their decorations ejkby some more negative integers e′
+jk≤ej0we
+get a graph whose all connected components are rational. (No tice that this is a generalization of the notion
+of ‘bad’ vertices of [ OSz03]. A graph with at most one bad vertex is called ‘almost ration al’ in [N05, N08 ].
+Any ‘star–shaped’ graph, i.e. normal form of a Seifert manif old, has at most one bed vertex, namely the
+‘central’ vertex.)
+Theorem 6.2.1. IfΓhas at most nbad vertices then Hq
+red(Γ) = 0forq≥n.
+This is a generalization of [ N08, (4.3.3)], where it is proved for n= 1 (compare also with the vanishing
+theorems of [ OSz03, N05 ]).
+Proof. We run induction over n. Ifn= 0, then all the components of Γ are rational. By [ N05],
+their reduced lattice cohomology is vanishing. This fact re mains true for more components too, since the
+cohomology of a tensor product of two acyclic complexes is ac yclic.
+Assume now that the statement is true for n−1 and take Γ with nbad vertices. Let jbe one of them.
+Let Γ j(−ℓ) be the graph obtained from Γ by replacing the decoration ejbyej−ℓ(ℓ≥0). Then consider
+the long exact sequence (5.5.3) associated with Γ j(−ℓ), Γj(−ℓ−1) and Γ \j0, for allℓ≥0. Then, by the
+inductive step, we get that Hq(Γ) =Hq(Γj(−ℓ)) for allℓandq≥n. (Here, in the case n= 1, Theorem 6.1.2
+is also used.) Since for ℓ≫0 the graph Γ j(−ℓ) has onlyn−1 bad vertices, all these modules vanish. /square
+In fact, the above statement can be improved as follows.
+Theorem 6.2.2. Assume that Γhas at most n≥2bad vertices {jk}1≤k≤nsuch that Γ\j1has at most
+(n−2)bad vertices. Then Hq
+red(Γ) = 0forq≥n−1.
+Proof. The proof is same as above, if one eliminates first the vertex j1. /square
+See [N05, (8.2)(5.b)] for a graph Γ with 2 bad vertices {j1,j2}such that Γ \j1has only rational
+components.
+7. The ‘relative’ surgery exact sequence
+7.1. Preliminaries. The motivation for the next exact sequence is two–folded. Fi rst, the exact se-
+quence (5.5.3) mixes the classes of the characteristic elem ents. (Note that the surgery exact sequence valid
+for the Heegaard–Floer theory — which is one of out models for the theory — does the same.) These classes,
+in topological language, correspond to the spinc–structures of the corresponding plumbed 3–manifolds. It
+would be desirable to have a surgery exact sequence which do n ot mix them, and allows inductively the
+computation of each H∗(Γ,[k]) for each [ k] independently.
+The second motivation is the main result of [ BN10]. This is a surgery formula for the Seiberg–Witten
+invariant of negative definite plumbed 3–manifolds, it comp ares these invariants for Γ and Γ \j0for fixed
+(non–mixed) spinc–structures. The third term in the main formula of [ BN10] comes from a ‘topological’
+Poincar´ e series associated with the plumbing graph, and it s nature is rather different than the other two
+terms.
+Here our goal is to determine an exact sequence connecting H∗(Γ,[¯k]) andH∗(Γ\j0,[R(¯k)]) (whereR(¯k)
+is the restriction of ¯k, see 4.1) with the newly defined third term, playing the role o f the relative cohomology.
+Its relationship with the Poincar´ e series used in [ BN10] will also be treated.
+7.2. The ‘relative’ complex and cohomology. We consider a non–degenerate graph Γ and j0one
+of its vertices.
+We fix [k]∈Char(Γ\j0)/2L(Γ\j0) and a characteristic element km∈[k] withw(km) = min k∈[k]w(k).
+Furthermore, we fix a0satisfyinga0≡ej0(mod 2). Then ka0:=i(km)+/parenleftbig
+(i(km),Ej0)+a0/parenrightbig
+E∗
+j0∈Char(Γ).
+In fact, for any k′∈[k] one gets that i(k′)+/parenleftbig
+(i(km),Ej0)+a0/parenrightbig
+E∗
+j0∈Char(Γ) and it is an element of [ ka0].
+For simplicity we write r0for (i(km),Ej0)+a0.
+We define
+B0,rel:F0(Γ\j0,[k])→ F0(Γ,[ka0])
+by
+(7.2.1) B0,rel(k′) =i(k′)+/parenleftbig
+(i(km),Ej0)+a0/parenrightbig
+E∗
+j0=i(k′)+r0E∗
+j0.12 Andr´ as N´ emethi
+This extends to the level of complexes B∗,rel: (F∗(Γ\j0,[k]),∂)→(F∗(Γ,[ka0]),∂) byB∗,rel((k,I)) =
+(B0,rel(k),I). Its dualB∗
+rel: (F∗(Γ,[ka0]),δ)→(F∗(Γ\j0,[k]),δ) is defined by B∗
+rel(φ) =φ◦B∗,rel.
+By a similar argument as in (5.3) we get that
+B∗
+rel:F∗(Γ,[ka0])→ F∗(Γ\j0,[k]) is surjective .
+We define the ‘relative’ complex F∗
+rel=F∗
+rel(Γ,j0,[k],a0]) via ker(B∗
+rel). Let the cohomology of the complex
+(F∗
+rel,δ) beH∗
+rel=H∗
+rel(Γ,j0,[k],a0). It is a graded Z[U]–module. We refer to it as the relative lattice
+cohomology .
+Note that both F∗
+relandB∗
+reldepend on the choice of the representative ka0of [ka0] and are not
+invariants merely of the classes [ ka0] and [k].
+Theorem 7.2.2. One has the short exact sequence of complexes:
+0−→ F∗
+relA∗
+rel−→ F∗(Γ,[ka0])B∗
+rel−→ F∗(Γ\j0,[k])−→0,
+which provides a long exact sequence of Z[U]–modules:
+··· −→Hq
+relAq
+rel−→Hq(Γ,[ka0])Bq
+rel−→Hq(Γ\j0,[k])Cq
+rel−→Hq+1
+rel−→ ···
+(Again, the relative cohomology modules and the operators i n the above exact sequence depend on the choice
+ofka0, and not only on its class [ka0].)
+In the case when Γ is negative definite, the restriction of B0
+reltoT+
+d[ka0]has its image in T+
+d[k].
+Proposition 7.2.3. Assume that Γ(but not necessarily Γ+
+j0) is negative definite. Then
+B0
+rel:T+
+d[ka0]→ T+
+d[k]is onto.
+In particular, H∗
+relhas finite rank over Z. Moreover, one has an exact sequence of finite Z–modules:
+0−→H0
+rel−→H0
+red(Γ,[ka0])⊕Zn−→H0
+red(Γ\j0,[k])−→H1
+rel−→H1
+red(Γ,[ka0])−→H1
+red(Γ\j0,[k])···
+wheren:=wΓ(i(km)+r0E∗
+j0)−min{wΓ|[ka0]} ∈Z≥0.
+Proof. First note that
+wΓ(i(k′)+r0E∗
+j0) =wΓ\j0(k′)−1+r2
+0(E∗
+j0)2
+8.
+This applied for k′=kmprovides
+(7.2.4) n+d[ka0]
+2=d[k]
+2−1+r2
+0(E∗
+j0)2
+8.
+For anyl≥0 and¯k∈[ka0] set¯φl∈ F0(Γ,[ka0]) defined by
+¯φl(¯k) =U−l−d[ka0]/2+wL(¯k)(¯k∈[ka0]).
+For different l≥0 they generate T+
+d[ka0]⊂H0(Γ,[ka0]). Similarly, set {φl}l≥0inF0(Γ\j0,[k]), where
+φl(k′) =U−l−d[k]/2+w(k′)(k′∈[k]).
+They generate T+
+d[k]. From the above identities and from the definition of B0
+relone gets for any l≥0:
+B0
+rel(¯φn+l) =φl.
+Hence the restriction B0
+rel:T+
+d[ka0]→ T+
+d[k]is onto and the Z–rank of its kernel is n. /square
+The reader is invited to recall the definition of the Euler cha racteristic of the lattice cohomology from
+(2.3.5). We define the Euler characteristic of the relative lattice cohomology by
+eu(H∗
+rel) :=/summationdisplay
+q≥0(−1)qrankZHq
+rel.
+Then the exact sequence of Proposition 7.2.3 and equation (7 .2.4) provide
+Corollary 7.2.5. With the notation r0:= (i(km),Ej0)+a0one has
+eu(H∗
+rel(Γ,j0,[k],a0)) =eu(H∗(Γ,[ka0]))−eu(H∗(Γ\j0,[k]))−1+r2
+0(E∗
+j0)2
+8.
+Fix anyl∈L(Γ\j0), thenka0+2l=i(km+2l)+r0E∗
+j0, hence we also getTwo exact sequences for lattice cohomology 13
+Corollary 7.2.6. For anyl∈L(Γ\j0)one has
+eu(H∗
+rel(Γ,j0,[km],a0)) =eu(H∗(Γ,[ka0]))−(ka0+2l)2
+Γ+|J|
+8
+−eu(H∗(Γ\j0,[km]))+(km+2l)2
+Γ\j0+|J \j0|
+8.(7.2.7)
+7.3. Reinterpretation. Above, we started with an element kmof the class [ k], and we constructed
+one of its extensions ka0. Sinceka0+i(2l) =i(km+ 2l) +r0E∗
+j0, we havekm+ 2l=R(ka0+ 2l) for any
+l∈L(Γ\j0).
+This procedure can be inverted. Indeed, let us fix any ¯k∈Char(Γ) (which plays the role of ka0+i(2l)).
+Then setR(¯k) and define r0by the identity r0E∗
+j0=¯k−iR(¯k). Finally, define
+B0,rel:F0(Γ\j0,[R(¯k)])→ F0(Γ,[¯k]) byB0,rel(k′) =i(k′)+r0E∗
+j0,
+whose kernel is the relative complex F∗
+relwith cohomology H∗
+rel(Γ,j0,¯k). Then (7.2.7) reads as
+eu(H∗
+rel(Γ,j0,¯k)) =eu(H∗(Γ,[¯k]))−(¯k)2
+Γ+|J|
+8
+−eu(H∗(Γ\j0,[R(¯k)]))+(R(¯k))2
+Γ\j0+|J \j0|
+8.(7.3.1)
+The identity (7.3.1) depends essentially on the choice of th e choice of ¯k∈Char(Γ). In fact, even if we fix
+the class [ ¯k], the choice of the representative ¯kfrom the class [ ¯k] provides essentially different identities of
+type (7.3.1): not only the terms eu(H∗
+rel(Γ,j0,¯k)), (¯k)2
+Γand (R(¯k))2
+Γ\j0depend on the choice of ¯k, but even
+the class [R(¯k)].
+7.4. The connection with the topological Poincar´ e series. LetKΓ∈L′denote the canonical
+characteristic element of Γ defined by the adjunction formulae (KΓ+Ej,Ej)+2 = 0 for all j∈ J. Similarly,
+one defines KΓ\j0∈Char(Γ\j0). Note that Char(Γ) =K+2L′(Γ) andKΓ\j0=R(KΓ). The next result
+computeseu(H∗
+rel(Γ,j0,K+2l′)) in terms of l′via the coefficients of a series associated with Γ.
+Consider the multi-variable Taylor expansion Z(t) =/summationtextpl′tl′at the origin of
+(7.4.1)/productdisplay
+j∈J(1−tE∗
+j)δj−2,
+where for any l′=/summationtext
+jljEj∈L′we write tl′=/producttext
+jtlj
+j, andδjis the valency of j. This lives in Z[[L′]], the
+submodule of formal power series Z[[t±1/d]] in variables {t±1/d
+j}j, whered= det(Γ). The series Z(t) was
+used in several articles studying invariants of surface sin gularities, see [ CDG04, CDG08, CHR04, N08b,
+N08c, N10 ] for different aspects.
+For any series S(t)∈Z[[L′]],S(t) =/summationtext
+l′cl′tl′, we have the natural decomposition
+S=/summationdisplay
+h∈L′/LSh,whereSh:=/summationdisplay
+l′: [l′]=hcl′tl′.
+In particular, for any fixed class [ l′]∈L′/L, one can consider the component Z[l′](t) ofZ(t). In fact, see
+e.g. [N08b, (3.1.20)],
+(7.4.2) Z[l′](t) =1
+d/summationdisplay
+ρ∈(L′/L)/hatwideρ([l′])−1·/productdisplay
+j∈J(1−ρ([E∗
+j])tE∗
+j)δj−2,
+where (L′/L)/hatwideis the Pontjagin dual of L′/L.
+Furthermore, once the vertex j0of Γ is fixed, for any class [ l′]∈L′/Lwe set
+H[l′],j0(t) :=Z[l′](t)/vextendsingle/vextendsingle
+tj0=td
+tj=1forj/\egatio\slash=j0∈Z[[t]].
+LetS(t) =/summationtext
+i≥0citibe a formal power series. Suppose that for some positive inte gerp, the expression/summationtextpn−1
+i=0ciis a polynomial Pp(n) in the variable n. Then the constant term of Pp(n) is independent of p. We
+call this constant term the periodic constant ofSand denote it by pc( S) (cf. [NO09]).
+Proposition 7.4.3. Fix the vertex j0ofΓand write H[l′],j0(t)as/summationtext
+i≥0citi.
+(a) Ifl′=/summationtext
+jajE∗
+j=/summationtext
+jl′
+jEj∈L′(Γ)with allajsufficiently large then
+/summationdisplay
+i<dl′
+j0ci=eu(H∗
+rel(Γ,j0,K+2l′)).14 Andr´ as N´ emethi
+(b) Take ¯l′=/summationtext
+j¯l′
+jEj∈L′(Γ)with¯l′
+j0∈[0,1). Then
+pc(H[¯l′],j0) =eu(H∗
+rel(Γ,j0,K+2¯l′)).
+Proof. Use (7.3.1) from above and the identities (3.2.7) and (3.2.1 3) from [ N10]. /square
+7.5. The connection with the Seiberg–Witten invariants. LetM(Γ) be the oriented plumbed
+3–manifold associated with Γ, and −M(Γ) the same 3–manifold with opposite orientation. In is kno wn that
+thespinc–structures of M(Γ) (and of −M(Γ) too) can be identified with Char/2L, see e.g. [ N05, N08 ].
+Letsw(M(Γ),[¯k]) be the Seiberg–Witten invariant of M(Γ) associated with the spinc–structure [ ¯k]. Then,
+by Theorem B of the Introduction of [ N10] for a negative definite graph Γ one has
+(7.5.1) eu(H∗(Γ,[¯k])) =sw(−M(Γ),[¯k]).
+Hence the above statements can be reinterpreted in terms of S eiberg–Witten invariants as well.
+Example 7.5.2. Let Γ be the next graph, where all the vertices have decoratio n−2 except the j0vertex
+which has −3.
+s s s s s s s s
+sj0
+det(Γ) = det(Γ \j0) = 1, hence for both graphs we have only one class.
+Moreover, Γ \j0is rational (an E8–graph), hence min wΓ\j0= 0,H∗
+red(Γ\j0) = 0 andeu(H∗(Γ\j0)) = 0.
+On the other hand, Γ is minimally elliptic, min wΓ= 0,H∗
+red(Γ) =H0
+red(Γ) =Z(0), the rank one
+Z–module concentrated at degree zero. Hence eu(H∗(Γ)) = 1.
+By the long exact sequence, we get Hq
+rel(Γ,j0,¯k) = 0 for any ¯kandq>0.
+It is easy to see that KΓ=−E∗
+j0, and (E∗
+j0)2=−1. Therefore, if ¯k=K+2l′, andr0:=−(¯k,E∗
+j0) and
+l′
+j0:=−(l′,E∗
+j0), thenr0= 2l′
+j0−1. By (7.2.5) or (7.3.1)
+rankH0
+rel(Γ,j0,¯k) =eu(H∗(Γ,j0,¯k)) =r2
+0+7
+8= 1+l′
+j0(l′
+j0−1)
+2.
+By a computation one obtains
+H[0],j0(t) =1−t6
+(1−t3)(1−t2)(1−t)=1−t+t2
+(1−t)2= 1+t+2t2+3t3+4t4+···.
+Then
+/summationdisplay
+i<l′
+j0ci= 1+1+2+3+ ···+(l′
+j0−1) = 1+l′
+j0(l′
+j0−1)
+2,
+hence (7.4.3)(a) follows. In order to exemplify part (b), we have to take ¯l′with¯l′
+j0= 0, hence in this case
+eu(H∗(Γ,j0,¯k)) = 1. But one also has
+pc1−t+t2
+(1−t)2= 1.
+Example 7.5.3. Assume that Γ is a star–shaped graph with central vertex j0and we fix ¯k=KΓ.
+Since all the connected components of Γ \j0are strings (i.e. rational graphs), for them (cf. [ N08])
+eu(H∗(Γ\j0,[KΓ\j0]))−(KΓ\j0)2+|J \j0|
+8= 0.
+Moreover, Hq(Γ,[K]) =Hq
+rel(Γ,j0,[K]) = 0 forq>0, and
+rankH0
+rel(Γ,j0,[K]) =eu(H∗(Γ,[K])−K2+|J|
+8= rankH0(Γ,[K])−minwΓ−K2+|J|
+8
+= rankH0(Γ,[K])−minχK.(7.5.4)
+This equals the periodic constant of H[0],j0(t) by [N08, BN10 ]. Moreover, if ( X,o) is a weighted homoge-
+neous normal surface singularity with minimal good resolut ion graph Γ, then its geometric genus pgequals
+the last term of (7.5.4), cf. e.g. [ NN04, N05 ]. Hencepg(X,o) = rankH0
+rel(Γ,j0,[K]).Two exact sequences for lattice cohomology 15
+References
+[BN10] Braun, G. and N´ emethi, A., Surgery formula for Seiberg–Witten invariants of negative definite plumbed
+3-manifolds, Journal f¨ ur die Reine und Angewandte Mathematik 638(2010), 189–208.
+[CDG04] Campillo, A., Delgado, F. and Gusein-Zade, S. M., Poincar´ e series of a rational surface singularity, Invent.
+Math.155(1) (2004), 41–53.
+[CDG08] Guse˘ ın-Zade, S. M., Delgado, F. and Campillo, A., Universal abelian covers of rational surface singularitie s
+and multi-index filtrations, Funk. Anal. i Prilozhen. 42(2) (2008), 3–10.
+[CHR04] Cutkosky, S. D., Herzog, J. and Reguera, A., Poincar´ e series of resolutions of surface singularities , Trans.
+Amer. Math. Soc. 356(5) (2004), 1833–1874.
+[Gr08] Greene, J., A surgery triangle for lattice cohomology, arXiv:0810.0862 .
+[N99] N´ emethi, A., Five lectures on normal surface singularities, lectures delivered at the Summer School in Low
+dimensional topology Budapest, Hungary, 1998; Bolyai Society Math. Studies 8(1999), 269-351.
+[N05] N´ emethi, A., On the Ozsv´ ath-Szab´ o invariant of negative definite plumb ed 3-manifolds, Geometry and Topology
+9(2005), 991-1042.
+[N07] N´ emethi, A., Graded roots and singularities, inSingularities in geometry and topology , World Sci. Publ.,
+Hackensack, NJ, 2007, 394–463.
+[N08] N´ emethi, A., Lattice Cohomology of Normal Surface Singulariites, Publ. of RIMS, Kyoto University, 44(2)
+(2008), 507-543.
+[N08b] N´ emethi, A., Poincar´ e series associated with surface singularities , inSingularities I: Algebraic and Analytic
+Aspects, Contemporary Math. 474(2008), 271–299.
+[N08c] N´ emethi, A., The cohomology of line bundles of splice-quotient singular ities, arXiv:0810.4129.
+[N10] N´ emethi, A., The Seiberg–Witten invariants of negative definite plumbed 3–manifolds,
+Alg.Geo.arXiv:1003.1254, to appear in the Journal of EMS.
+[NN02] N´ emethi, A. and Nicolaescu, L.I., Seiberg-Witten invariants and surface singularities, Geometry and Topol-
+ogy,6(2002), 269–328.
+[NN04] N´ emethi, A. and Nicolaescu, L.I., Seiberg-Witten invariants and surface singularities II (s ingularities with
+goodC∗-action), Journal of London Math. Soc. 69(2) (2004), 593–607.
+[NO09] N´ emethi, A. and Okuma, T., On the Casson Invariant Conjecture of Neumann-Wahl, Journal of Algebraic
+Geometry 18(2009), 135–149.
+[NR10] N´ emethi, A. and Rom´ an, F., The lattice cohomology of S3
+−d(K), manuscript in preparation.
+[OSz03] Ozsv´ ath, P.S. and Szab´ o, Z., On the Floer homology of plumbed three-manifolds, Geometry and Topology 7
+(2003), 185–224.
+[OSz04] Ozsv´ ath, P.S. and Szab´ o, Z., Holomorphic disks and topological invariants for closed th ree-manifolds, Ann.
+of Math. (2) 159(3) (2004), 1027–1158.
+[OSz04b] Ozsv´ ath, P.S. and Szab´ o, Z., Holomorphic triangle invariants and the topology of symple ctic four-manifolds,
+Duke Math. J. 121(1) (2004), 1–34.
+[R04] Rustamov, R., A surgery formula for renormalized Euler characteristic of Heegaard Floer homology,
+math.GT/0409294.
+A. R´enyi Institute of Mathematics, 1053 Budapest, Re ´altanoda u. 13-15, Hungary.
+E-mail address :nemethi@renyi.hu
\ No newline at end of file
diff --git a/1001.0641.txt b/1001.0641.txt
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--- /dev/null
+++ b/1001.0641.txt
@@ -0,0 +1,741 @@
+arXiv:1001.0641v1  [cs.LO]  5 Jan 2010Least and Greatest Fixpoints
+in Game Semantics
+PierreClairambault
+pierre.clairambault@pps.jussieu.fr
+PPS — Universit´ e Paris 7
+Abstract
+We show how solutions to many recursive arena equations can b e com-
+puted in a natural way by allowing loops in arenas. We then equ ip arenas
+with winning functions and total winning strategies. We pre sent two nat-
+ural winning conditions compatible with the loop construct ion which re-
+spectively provide initial algebras and terminal coalgebr as for a large class
+of continuous functors. Finally, we introduce an intuition istic sequent cal-
+culus, extended with syntactic constructions for least and greatest fixed
+points, and prove it has a sound and (in a certain weak sense) c omplete
+interpretation in our game model.
+1 Introduction
+The idea to model logic by game-theoretic tools can be traced back t o the work
+of Lorenzen [ 21]. The idea is to interpret a formula by a game between two
+players O and P, O trying to refute the formula and P trying to prove it. The
+formulaAis then valid if P has a winning strategy on the interpretation of A.
+Later, Joyal remarked [ 18] that it is possible to compose strategies in Conway
+games [8] in an associative way, thus giving rise to the first category of game s
+and strategies. This, along with parallel developments in Linear Logic and
+Geometry of Interaction, led to the more recent construction of compositional
+game models for a large variety of logics [ 3,23,9] and programming languages
+[17,4,22,5].
+We aim here to use these tools to model an intuitionistic logic with induct ion
+and coinduction. Inductive/coinductive definitions in syntax have b een defined
+and studied in a large varietyof settings, such as linear logic [ 6],λ-calculus [ 1] or
+Martin-L¨ of’s type theory [ 10]. Motivations are multiple, but generally amount
+to increasing the expressive power of a language without paying the price of
+exponential modalities (as in [ 6]) or impredicativity (as in [ 1] or [10]). However,
+less work has been carried out when it comes to the semantics of suc h con-
+structions. Of course we have the famous order-theoretic Knas ter-Tarski fixed
+point theorem [ 25], the nice categorical theory due to Freyd [ 12], set-theoretic
+1models [10] (for the strictly positive fragment) or PER-models [ 20], but it seems
+they have been ignored by the current trend for intensional mode ls (i.e.games
+semantics, GoI ...). We fix this issue here, showing that (co)induc tion admits
+a nice game-theoretic model which arises naturally if one enriches Mc Cusker’s
+[22] work on recursive types with winning functions inspired by parity ga mes
+[24].
+In Section 2, we first recall the basic definitions of the Hyland-Ong- Nickau
+setting of game semantics. Then we sketch McCusker’s interpreta tion of re-
+cursive types, and show how most of these recursive types can be modelled by
+means of loops in the arenas. For this purpose, we define a class of f unctors
+calledopen functors , including in particular all the endofunctors built out of
+the basic type constructors. We also present a mechanism of winnin g functions
+inspired by [ 16], allowing us to build a category Gamof games and total win-
+ning strategies. In section 3, we present µLJ, the intuitionistic sequent calculus
+with least and greatest fixpoints that we aim to model. We briefly discu ss its
+proof-theoretic properties, then present its semantic counter part: we show how
+to build initial algebras and terminal coalgebras to most positive open func-
+tors. Finally, we use this semantic account of (co)induction to give a sound and
+(weakly) complete interpretation of µLJinGam.
+2 Arena Games
+2.1 Arenas and Plays
+We recall briefly the now usual definitions of arena games, introduc ed in [17].
+Moredetailedaccountscanbe found in [ 22,14]. We areinterestedin gameswith
+two participants: Opponent (O, the environment ) and Player (P, the program).
+Possibleplaysaregeneratedbydirectedgraphscalled arenas, whicharesemantic
+versionsof typesorformulas . Hence, aplayisasequenceof movesoftheambient
+arena, each of them being annotated by a pointerto an earlier move — these
+pointers being required to comply with the structure of the arena. Formally, an
+arenais a structure A= (MA,λA,⊢A) where:
+•MAis a set of moves,
+•λA:MA→{O,P}×{Q,A}is alabelling function indicating whether a
+move is an Opponent or Player move, and whether it is a question (Q) o r
+an answer (A). We write λOP
+Afor the projection of λAto{O,P}andλQA
+A
+for its projection on {Q,A}.λAwill denote λAwhere the{O,P}part has
+been reversed.
+•⊢Ais a relation between MA+{⋆}toMA, calledenabling , satisfying:
+–⋆⊢m=⇒λA(m) =OQ;
+–m⊢An∧λQA
+A(n) =A=⇒λQA
+A(m) =Q;
+–m⊢An∧m/\e}atio\slash=⋆=⇒λOP
+A(m)/\e}atio\slash=λOP
+A(n).
+2In other terms, an arenais a directed bipartite graph, with a set of distinguished
+initialmoves (msuch that ⋆⊢Am) and a distinguished set of answers (m
+such that λQA
+A=A) such that no answer points to another answer. We now
+define plays as justified sequences overA: these are sequences sof moves
+ofA, each non-initial move minsbeing equipped with a pointer to an earlier
+movenins, satisfying n⊢Am. In other words, a justified sequence soverAis
+such that each reversed pointer chain sφ(0)←sφ(1)←...←sφ(n)is a path on
+A, viewed as a directed bipartite graph.
+The role of pointers is to allow reopenings in plays. Indeed, a path on Amay
+be (slightly naively) understood as a linear play on A, and a justified sequence
+as an interleaving of paths, with possible duplications of some of them . This
+intuition is made precise in [ 15]. When writing justified sequences, we will often
+omit the justification information if this does not cause any ambiguity .⊑will
+denote the prefix ordering on justified sequences. If sis a justified sequence on
+A,|s|will denote its length.
+Given a justified sequence sonA, it has two subsequences of particular
+interest: the P-view and O-view. The view for P (resp. O) may be und erstood
+as the subsequence of the play where P (resp. O) only sees his own d uplications.
+In a P-view, O never points more than once to a given P-move, thus h e must
+always point to the previous move. Concretely, P-views correspon d to branches
+of B¨ ohm trees [ 17]. Practically, the P-view /rightanglenws/rightangleneofsis computed by forgetting
+everything under Opponent’s pointers, in the following recursive wa y:
+•/rightanglenwsm/rightanglene=/rightanglenws/rightanglenemifλOP
+A(m) =P;
+•/rightanglenwsm/rightanglene=mif⋆⊢Amandmhas no justification pointer;
+•/rightanglenws1ms2n/rightanglene=/rightanglenws/rightanglenemnifλOP
+A(n) =Oandnpoints to m.
+The O-view/rightanglesws/rightangleseofsis defined dually. Note that in some cases — in fact if s
+does not satisfies the visibility condition introduced below — /rightanglenws/rightangleneand/rightanglesws/rightanglesemay
+not be correct justified sequences, since some moves may have po inted to erased
+parts of the play. However, we will restrict to plays where this does not happen.
+Thelegal sequences overA, denoted byLA, are the justified sequences son
+Asatisfying the following conditions:
+•Alternation. Iftmn⊑s, thenλOP
+A(m)/\e}atio\slash=λOP
+A(n);
+•Bracketing. A question qisanswered byaifais an answer and a
+points to q. A question qisopeninsif it has not yet been answered.
+We require that each answer points to the pending question, i.e.the last
+open question.
+•Visibility. Iftm⊑sandmisnot initial, then if λOP
+A(m) =Pthe justifier
+ofmappears in/rightanglenwt/rightanglene, otherwise its justifier appears in /rightangleswt/rightanglese.
+32.2 The cartesian closed category of Innocent strategies
+Astrategy σonAis a prefix-closed set of even-length legal plays on A. A
+strategyis deterministic ifonlyOpponentbranches, i.e.∀smn,smn′∈σ, n=
+n′. Of course, if Arepresents a type (or formula), there are often many more
+strategies on Athan programs (or proofs) on this type. To address this issue
+we need innocence . An innocent strategy is a strategy σsuch that
+sab∈σ∧t∈σ∧ta∈LA∧/rightanglenwsa/rightanglene=/rightanglenwta/rightanglene=⇒tab∈σ
+We now recall how arenas and innocent strategies organize themse lves into
+a cartesian closed category. First, we build the product A×Bof two arenas
+AandB:
+MA×B=MA+MB
+λA×B= [λA,λB]
+⊢A×B=⊢A+⊢B
+We mention the empty arena I= (∅,∅,∅), which will be terminal for the
+category of arenas and innocent strategies. We mention as well th e arena⊥=
+(•,• /mapsto→OQ,(⋆,•)) with only one initial move, which will be a weak initial
+object. We define the arrowA⇒Bas follows:
+MA⇒B=MA+MB
+λA⇒B= [λA,λB]
+m⊢A⇒Bn⇔
+
+m/\e}atio\slash=⋆∧m⊢An
+m/\e}atio\slash=⋆∧m⊢Bn
+⋆⊢Bm∧⋆⊢An
+m=⋆∧⋆⊢Bn
+We define composition of strategies by the usual parallel interactio n plus
+hiding mechanism. If A,BandCare arenas, we define the set of interactions
+I(A,B,C) as the set of justified sequences uoverA,BandCsuch that u↾A,B∈
+LA⇒B,u↾B,C∈LB⇒Candu↾A,C∈LA⇒C. Then, if σ:A⇒Bandτ:B⇒C,
+we define parallel interaction:
+σ||τ={u∈I(A,B,C)|u↾A,B∈σ∧u↾B,C∈τ}
+Composition is then defined as σ;τ={u↾A,C|u∈σ||τ}. It is associative and
+preserves innocence (a proof of these facts can be found in [ 17] or [14]). We also
+define the identity on Aas the copycat strategy (see [ 22] or [14] for a definition)
+onA⇒A. Thus, there is a category Innwhich has arenas as objects and
+innocent strategies on A⇒Bas morphisms from AtoB. In fact, this category
+is cartesianclosed, the cartesianstructure given by the arenapr oduct aboveand
+the exponential closure given by the arrow construction. This cat egory is also
+equipped with a weak coproduct A+B[22], which is constructed as follows:
+MA+B=MA+MB+{q,L,R}
+4λA+B= [λA,λB,q/mapsto→OQ,L/mapsto→PA,R/mapsto→PA]
+m⊢A+Bn⇔
+
+m,n∈MA∧m⊢An
+m,n∈MB∧m⊢Bn
+m=⋆∧n=q
+(m=q∧n=L)∨(m=q∧n=R)
+(m=L∧⋆⊢An)∨(m=R∧⋆⊢Bn)
+2.3 Recursive types and Loops
+Let us recall briefly the interpretation of recursive types in game s emantics, due
+to McCusker [ 22]. Following [ 22], we first define an ordering /triangleleftequalon arenas as
+follows. For two arenas AandB,A/triangleleftequalBiff
+MA⊆MB
+λA=λB↾MA
+⊢A=⊢B∩(MA+{⋆}×MA)
+This defines a (large) dcpo, with least element Iand directed sups given by
+the componentwise union. If F:Inn→Innis a functor which is continuous
+with respect to/triangleleftequal, we can find an arena Dsuch that D=F(D) in the usual
+way by setting D=/unionsqtext∞
+n=0Fn(I). McCusker showed [ 22] that when the functors
+areclosed(i.e.their action can be internalized as a morphism ( A⇒B)→
+(FA⇒FB)), and when they preserve inclusion and projection morphisms
+(i.e.partial copycat strategies) corresponding to /triangleleftequal, this construction defines
+minimal invariants [12]. Note that the crucial cases of these constructions are
+the functors built out of the product, sum and function space con structions.
+We give now a concrete and new (up to the author’s knowledge) desc ription
+of a large class of continuous functors, that we call open functors . These
+includeallthefunctorsbuiltoutofthebasicconstructions,andallo warereading
+of recursive types, leading to the model of (co)induction.
+2.3.1 Open arenas.
+LetTbe a countable set of names. An open arena is an arena Awith dis-
+tinguished question moves called holes, each of them labelled by an element of
+T. We denote by/squareXthe holes annotated by X∈T. We will sometimes write
+/squareP
+Xto denote a hole of Player polarity, or /squareO
+Xto denote a hole of Opponent
+polarity. If Ahas holes labelled by X1,...,X n, we denote it by A[X1,...,X n].
+By abuse of notation, the corresponding open functor we are goin g to build will
+be also denoted by A[X1,...,X n] : (Inn×Innop)n→Inn.
+2.3.2 Image of arenas.
+IfA[X1,...,X n] is an open arena and B1,...,B n,B′
+1,...,B′
+nare arenas (possi-
+bly open as well), we build a new arena A(B1,B′
+1,...,B n,B′
+n) by replacing each
+5occurrence of/squareP
+XibyBiand each occurrence of /squareO
+XibyB′
+i. More formally:
+MA(B1,B′
+1,...,Bn,B′n)= (MA\{/squareX1,...,/squareXn})+n/summationdisplay
+i=1(MBi+MB′
+i)
+λA(B1,B′
+1,...,Bn,B′n)= [λA,λB1,λB′
+1,...,λ Bn,λB′n]
+m⊢A(B1,B′
+1,...,Bn,B′n)p⇔
+
+m⊢A/squareP
+Xi∧⋆⊢Bip
+m⊢A/squareO
+Xi∧⋆⊢B′
+ip
+⋆⊢Bim∧/squareP
+Xi⊢Ap
+⋆⊢B′
+im∧/squareO
+Xi⊢Ap
+m⊢Bip
+m⊢B′
+ip
+m⊢Ap
+Note that in this definition, we assimilate all the moves sharing the sam e hole
+label/squareXiand with the same polarity. This helps to clarify notations, and is
+justified by the fact that we never need to distinguish moves with th e same hole
+label, apart from when they have different polarity.
+2.3.3 Image of strategies.
+IfAisanarena, wewill, byabuseofnotation, denoteby IAboththe setofinitial
+moves of Aand the subarena of Awith only these moves. Let A[X1,...,X n] be
+an open arena, B′
+1,B1,...,B′
+n,BnandC′
+1,C1,...,C′
+n,Cnbe arenas. Consider
+the application ξdefined on moves as follows:
+ξ(x) =/braceleftbigg/squareXiifx∈/uniontext
+i∈{1,...,n}(IB′
+i∪IBi∪IC′
+i∪ICi)
+xotherwise
+and then extended recursively to an application ξ∗on legal plays as follows:
+ξ∗(sa) =/braceleftbiggξ∗(s)ifais a non-initial move of Bi,B′
+i,CiorC′
+i
+ξ∗(s)ξ(a)otherwise
+ξ∗erases moves in the inner parts of B′
+i,Bi,C′
+i,Ciand agglomerates all the
+initialmovesbacktotheholes. Thiswaywewillbeabletocomparether esulting
+play with the identity on A[X1,...,X n]. Now, if σi:Bi→Ciandτi:C′
+i→B′
+i
+are strategies, we can now define the action of open functors on t hem by stating:
+s∈A(σ1,τ1,...,σ n,τn)⇔
+
+∀i∈{1,...,n}, s↾Bi⇒Ci∈σi
+∀i∈{1,...,n}, s↾C′
+i⇒B′
+i∈τi
+ξ∗(s)∈idA[X1,...,Xn]
+Proposition 1. For any A[X1,...,X n], this defines a functor A[X1,...,X n] :
+(Inn×Innop)n→Inn, which is monotone and continuous with respect to /triangleleftequal.
+6Proof sketch. Preservation of identities and composition are rather direct. A
+little care is needed to show that the resulting strategy is innocent: this relies
+on two facts: First, for each Player move the three definition case s are mutually
+exclusive. Second, a P-view of s∈A(σ1,τ1,...,σ n,τn) is (essentially) an initial
+copycat appended with a P-view of one of σiorτi, hence the P-view of s
+determines uniquely the P-view presented to one of σi,τioridA[X1,...,Xn].
+Example. Consider the open arena A[X] =/squareX⇒/squareX. For any arena B,
+we have A(B) =B⇒Band for any σ:B1→C1andτ:C2→B2, we have
+A(σ,τ) =τ⇒σ: (B2⇒B1)→(C2⇒C1), the strategy which precomposes
+its argument by τand postcomposes it by σ.
+2.3.4 Loops for recursive types.
+Since these open functors are monotone and continuous with resp ect to/triangleleftequal, solu-
+tions to their corresponding recursive equations can be obtained b y computing
+the infinite expansion of arenas ( i.e.infinite iteration of the open functors).
+However, for a large subclass of the open functors, this solution c an be ex-
+pressed in a simple way by replacing holes with a loop up to the initial move s.
+Suppose A[X1,...,X n] is an open functor, and iis such that/squareXiappears only
+in non-initial, positivepositions in A. Then wedefine an arena µXi.Aasfollows:
+MµXi.A= (MA\/squareXi)
+λµXi.A=λA↾MµXi.A
+m⊢µXi.An⇔/braceleftbiggm⊢An
+m⊢A/squareXi∧⋆⊢An
+A simple argument ensures that the obtained arena is isomorphic to t he one
+obtained by iteration of the functor. For this issue we take inspirat ion from
+Laurent [ 19] and prove a theorem stating that two arenas are isomorphic in the
+categorical sense if and only if their set of paths are isomorphic. A pathinA
+is a sequence of moves a1,...,a nsuch that for all i∈{1,...,n−1}we have
+ai⊢Aai+1. Apath isomorphism between AandBis a bijection φbetween
+the set of paths of Aand the set of paths on Bsuch that for any non-empty
+pathponA,φ(ip(p)) =ip(φ(p)) (where ip(p) denotes the immediate prefix of
+p). We have then the theorem:
+Theorem 1. LetAandBbe two arenas. They are categorically isomorphic if
+and only if there is a path isomorphism between their respect ive sets of paths.
+Now, it isclearbyconstructionthat, if A[X]is anopen functorsuchthat /squareX
+appearsonlyinnon-initialpositivepositionsin A, thesetofpathsof/unionsqtext∞
+n=0An(I)
+and ofµX.Aare isomorphic. Therefore µX.Ais solution of the recursive equa-
+tionX=A(X), and when A[X] is closed and preserves inclusions and projec-
+tions,µX.Adefines as well a minimal invariant for A[X]. But in fact, we have
+the following fact:
+7Proposition 2. IfA[X]is an open functor, then it is closed and preserves
+inclusions and projections. Hence µX.Ais a minimal invariant for A[X].
+This interpretation of recursive types as loops preserves finitene ss of the
+arena, and as we shall see, allows to easily express the winning condit ions nec-
+essary to model induction and coinduction.
+2.4 Winning and Totality
+Atotal strategy onAis a strategy σ:Asuch that for all s∈σ, if there is a
+such that sa∈LA, then there is bsuch that sab∈σ. In other words, σhas a
+response to any legal Opponent move. This is crucial to interpret lo gic because
+the interpretation of proofs in game semantics always gives total s trategies: this
+isacounterpartinsemanticstothecuteliminationpropertyinsynta x. Tomodel
+induction and coinduction in logic, we must therefore restrict to tot al strategies.
+However, it is well-known that the class of total strategies is not clo sed under
+composition, because an infinite chattering can occur in the hidden p art of the
+interaction. This is analogousto the fact that in λ-calculus, the class of strongly
+normalizing terms is not closed under application: δ=λx.xxis a normal form,
+however δδis certainly not normalizable. This problem is discussed in [ 2,16]
+andmorerecentlyin[ 7]. Wetakeherethesolutionof[ 16], and equiparenaswith
+winning functions: for every infinite play we choose a loser, hence re stricting to
+winning strategies has the effect of blocking infinite chattering.
+The definition of legal plays extends smoothly to infinite plays. Let Lω
+A
+denote the set of infinite legal plays over A. Ifs∈Lω
+A, we say that s∈σ
+when for all s⊏s,s∈σ. We writeLA=LA+Lω
+A. Agamewill be a pair
+A= (A,GA) whereAis an arena, and GAis a function from infinite threads
+onA(i.e.infinite legal plays with exactly one initial move) to {W,L}. The
+winning function GAextends naturally to potentially finite threads by setting,
+for each finite s:
+GA(s) =/braceleftbiggWif|s|is even ;
+Lotherwise.
+Finally,GAextends to legal plays by saying that GA(s) =WiffGA(t) =Wfor
+every thread tofs. By abuse of notation, we keep the same notation for this
+extended function. The constructions on arenas presented in se ction2.2extend
+to constructions on games as follows:
+•GA×B(s) = [GA,GB] (indeed, a thread on A×Bis either a thread on Aor
+a thread on B) ;
+•GA+B(s) =Wiff all threads of s↾Aare winning forGAand all threads of
+s↾Bare winning forGB.
+•GA⇒B(s) =Wiff if all threads of s↾Aare winning forGA, thenGB(s↾B) =
+W.
+It is straightforward to check that these constructions commut e with the
+extension of winning functions from infinite threads to potentially infi nite legal
+8plays. We now define winning strategies σ:Aas innocent strategies σ:A
+such that for all s∈σ,GA(s) =W. Now, the following proposition is satisfied:
+Proposition 3. Letσ:A⇒Bandτ:B⇒Cbe two total winning strategies.
+Thenσ;τis total winning.
+Proof sketch. Ifσ;τis not total, there must be infinite sin their parallel inter-
+actionσ||τ, suchthat s↾A,Cisfinite. Byswitching, we havein fact |s↾A|evenand
+|s↾C|odd. ThusGA(s↾A) =WandGC(s↾C) =L. We reason then by disjunction
+of cases. EitherGB(s↾B) =Win which caseGB⇒C(s↾B,C) =Landτcannot
+be winning, orGB(s↾B) =Lin which caseGA⇒B(s↾A,B) =Landσcannot be
+winning. Therefore σ;τis total.
+σ;τmust be winning as well. Suppose there is s∈σ;τsuch thatGA⇒C(s) =
+L. By definition of GA⇒C, this means that GA(s↾A) =WandGC(s↾C) =L. By
+definition of composition, there is u∈σ||τsuch that s=u↾A,C. But whatever
+the value ofGB(u↾B) is, one of σorτis losing. Therefore σ;τis winning.
+It is clear from the definitions that all plays in the identity are winning. It
+is also clear that all the structural morphisms of the cartesian clos ed structure
+ofInnare winning (they are essentially copycat strategies), thus this de fines a
+cartesian closed category Gamof games and innocent total winning strategies.
+3 Fixpoints
+3.1µLJ: an intuitionistic sequent calculus with fixpoints
+3.1.1 Formulas.
+S::=S⇒T|S∨T|S∧T|µX.T|νX.T|X|⊤|⊥
+A formula Fisvalidif for any subformula of Fof the form µX.F′,
+(1)Xappears only positively in F′,
+(2)Xdoes not appear at the root of F′(i.e.Xappears at least under a ∨
+or a⇒in the abstract syntax tree of F′).
+(2)corresponds to the restriction to arenas where loops allow to expr ess recur-
+sive types, whereas (1)is the usual positivity condition. We could of course
+hack the definition to get rid of these restrictions, but we choose n ot to obfus-
+cate the treatment for an extra generality which is neither often c onsidered in
+the literature, nor useful in practical examples of (co)induction.
+3.1.2 Derivation rules.
+We present the rules with the usual dichotomy.
+Identity group
+ax
+A⊢AΓ⊢A∆,A⊢B
+Cut
+Γ,∆⊢B
+9Structural group
+Γ,A,A⊢B
+C
+Γ,A⊢BΓ⊢B
+W
+Γ,A⊢BΓ,A,B,∆⊢C
+γ
+Γ,B,A,∆⊢C
+Logical group
+Γ,A⊢B
+⇒rΓ⊢A⇒BΓ⊢A∆,B⊢C
+⇒lΓ,∆,A⇒B⊢C⊥lΓ,⊥⊢A⊤rΓ⊢⊤
+Γ⊢AΓ⊢B
+∧rΓ⊢A∧BΓ,A⊢C← −∧lΓ,A∧B⊢CΓ,B⊢C− →∧lΓ,A∧B⊢C
+Γ⊢A← −∨rΓ⊢A∨BΓ⊢B− →∨rΓ⊢A∨BΓ,A⊢C∆,B⊢C
+∨lΓ,∆,A∨B⊢C
+Fixpoints
+Γ⊢T[µX.T/X ]
+µrΓ⊢µX.TT[A/X]⊢A
+µlµX.T⊢AT[νX.T/X ]⊢B
+νlνX.T⊢BA⊢T[A/X]
+νrA⊢νX.T
+Note that the µl,νlandνrrules are not relative to any context. In fact,
+the general rules with a context Γ at the left of the sequent are de rivable from
+these ones (even if, for µlandνr, the construction of the derivation requires
+an induction on T), and we stick with the present ones to clarify the game
+model. Cut elimination on the ⇒,∧,∨fragment is the same as usual. For the
+reduction of µandν, we need an additional rule to handle the unfolding of
+formulas. For this purpose, we add a new rule [ T] for each type Twith free
+variables. This method can already be found in [ 1] for strictly positive functors:
+no type variable appears on the left of an implication. From now on, T[A/X]
+will be abbreviated T(A). This notation implies that, unless otherwise stated,
+Xwill be the variable name for which Tis viewed as a functor. In the following
+rules,Xappears only positively in Tand only negatively in N:
+Functors
+A⊢B
+[T]
+T(A)⊢T(B)A⊢B
+[N]
+N(B)⊢N(A)
+The dynamic behaviour of this rule is to locally perform the unfolding. W e give
+someof the reduction rules. These areoftwo kinds: the rules for t he elimination
+of [T], and the cut elimination rules. Here are the main cases:
+10π
+A⊢B
+[T](X/negationslash∈FV(T))
+T⊢T/squiggleright ax
+T⊢Tπ
+A⊢B
+[X]
+A⊢B/squigglerightπ
+A⊢B
+π
+A⊢B
+[N⇒T]
+N(A)⇒T(A)⊢N(B)⇒T(B)/squigglerightπ
+A⊢B
+[N]
+N(B)⊢N(A)π
+A⊢B
+[T]
+T(A)⊢T(B)
+⇒l
+N(A)⇒T(A),N(B)⊢T(B)
+⇒r
+N(A)⇒T(A)⊢N(B)⇒T(B)
+π
+A⊢B
+[µY.T]
+µY.T(A)⊢µY.T(B)/squigglerightπ
+A⊢B
+[T[µY.T(B)/Y]]
+T(A)[µY.T(B)/Y]⊢T(B)[µY.T(B)/Y]
+µr
+T(A)[µY.T(B)/Y]⊢µY.T(B)
+µl
+µY.T(A)⊢µY.T(B)
+We omit the rule for ν, which is dual, and for ∧and∨, which are simple
+pairing and case manipulations. Note also that most of these cases h ave a
+counterpart where Tis replaced by negative N, which has the sole effect of π
+being a proof of B⊢Ainstead of A⊢Bin the expansion rules. With that, we
+can express the cut elimination rule for fixpoints:
+π1
+Γ⊢T[µX.T/X ]
+µr
+Γ⊢µX.Tπ2
+T[A/X]⊢A
+µl
+µX.T⊢A
+Cut
+Γ⊢A/squiggleright
+π1
+Γ⊢T[µX.T/X ]π2
+T[A/X]⊢A
+µl
+µX.T⊢A
+[T]
+T[µX.T/X ]⊢T[A/X]
+Cut
+Γ⊢T[A/X]π2
+T[A/X]⊢A
+Cut
+Γ⊢A
+We skip once again the rule for ν, which is dual to µ. We choose consciously
+not to recall the usual cut elimination rules nor the associated comm utation
+rules, since they are not central to our goals. µLJ, as presented above, does
+not formally eliminate cuts since there is no rule to reduce the following (and
+11its dual with ν):π1
+T(A)⊢A
+µlµX.T⊢Aπ2
+Γ,A⊢B
+Cut
+Γ,µX.T⊢B
+This cannot be reduced without some prior unfolding of the µX.Ton the left.
+This issue is often solved [ 6] by replacing the rule for µpresented here above by
+the following:
+T(A)⊢AΓ,A⊢B
+µ′
+Γ,µX.T⊢B
+With the corresponding reduction rule, and analogously for ν. We choose here
+not to do this, first because our game model will prove consistency without
+the need to prove cut elimination, and second because we want to pr eserve the
+proximitywiththe categoricalstructureofinitialalgebras/termin alcoalgebras.
+3.2 The games model
+We present the game model for fixpoints. We wish to model a proof s ystem,
+therefore we need our strategies to be total. The base arenas of the interpreta-
+tion offixpoints will be the arenaswith loopspresentedin section 2.3.4, towhich
+we will adjoin a winning function. While the base arenas will be the same f or
+greatest and least fixpoints, they will be distinguished by the winning function:
+intuitively, Player loses if a play grows infinite in a least fixpoint (inductiv e)
+game, and Opponent loses if this happens in a greatest fixpoint (coin ductive)
+game. The winning functions we are going to present are strongly infl uenced
+by Santocanale’s work on games for µ-lattices [ 24]. Awin open functor is
+a functor T: (Gam×Gamop)n→Gamsuch that there is an open functor
+T[X1,...,X n] such that for all games A1,...,A2nof base arenas A1,...,A 2n,
+the base arena of T(A1,...,A2n) isT(A1,...,A n). In other terms, it is the
+natural lifting of open functors to the category of games. By abu se of notation,
+we denote this by T[X1,...,X n], andT[X1,...,X n] will denote its underlying
+open functor.
+3.2.1 Least fixed point.
+LetT[X1,...,X n] be a win open functor such that /squareX1appears only positively
+and at depth higher than 0 in T[X1,...,X n]. Then we define a new win open
+functorµX1.T[X2,...,X n] as follows:
+•Its base arena is µX1.T[X2,...,X n] ;
+•IfA3,...,A2n∈Gam,GµX1.T(A3,...,A2n)(s) =Wiff
+–There is N∈Nsuch that no path of stakes the external loop more
+thatNtimes, and ;
+12–sis winning in the subgame inside the loop, or more formally:
+GT(I,I,A3,...,A2n)(s↾T(I,I,A3,...,A2n)) =W.
+3.2.2 Greatest fixed point.
+Dually, if the same conditions are satisfied, we define the win open fun ctor
+νX1.T[X1,...,X n] as follows:
+•Its base arena is µX1.T[X2,...,X n] ;
+•IfA3,...,A2n∈Gam,GνX1.T(A3,...,A2n)(s) =Wiff
+–For anyN∈N, there is a path of scrossing the external loop more
+thanNtimes, or ;
+–sis winning in the subgame inside the loop, or more formally:
+GT(I,I,A3,...,A2n)(s↾T(I,I,A3,...,A2n)) =W.
+It is straightforward to check that these are still functors, and in particular
+win open functors. There is one particular case that is worth noticin g: ifT[X]
+has only one hole which appears only in positive position and at depth gr eater
+than 0, then µX.Tis a constant functor, i.e.a game. Moreover, theorem 1
+implies that it is isomorphic in InntoT(µX.T). It is straightforward to check
+that this isomorphism iT:T(µX.T)→µX.Tis winning (it is nothing but the
+identity strategy), which shows that they are in fact isomorphic in Gam. Then,
+one can prove the following theorem:
+Theorem 2. IfT[X]has only one hole which appears only in positive position
+and at depth greater than 0, then the pair (µX.T,iT)defines an initial algebra
+forT[X]and(νX.T,i−1
+T)defines a terminal coalgebra forT[X].
+Proof.We give the proof for initial alebras, the second part being dual. Let
+(A,σ) another algebra of T[X]. We need to show that there is a unique σ†:
+µX.T⇒Bsuch that
+T(µX.T)T(σ†)/d47/d47
+iT/d15/d15T(B)
+σ/d15/d15
+µX.T
+σ†/d47/d47B
+commutes. The idea is to iterate σ:
+...T3(σ)/d47/d47T3(B)T2(σ)/d47/d47T2(B)T(σ)/d47/d47T(B)σ/d47/d47B
+and somehow to take the limit. In fact we can give a direct definition of σ†:
+σ(1)=σ
+σ(n+1)=Tn(σ);σ(n)
+σ†={s∈LµX.T⇒B|∃n∈N∗, s∈σ(n)}
+13This defines an innocent strategy, since when restricted to plays o fµX.T, these
+strategies agree on their common domain. This strategy is winning. I ndeed,
+take an infinite play s∈σ†. Suppose s↾µX.Tis winning. By definition of GµX.T,
+thismeansthatthereis N∈Nsuchthatnopathof s↾µX.Ttakestheexternalloop
+more than Ntimes. Thus, s∈LTn(I)⇒B. But this implies that s∈σ(n), and
+σ(n)is a composition of winning strategies thus winning, therefore sis winning.
+Moreover, σ†is the unique innocent strategy making the diagram commute:
+suppose there is another fmaking this square commute. Since T(µX.T) and
+µX.Thavethesamesetofpaths, iTisinfacttheidentity, thuswehave T(f);σ=
+f. By applying Tand post-composing by σ, we get:
+T2(f);T(σ);σ=T(f);σ=f
+And by iterating this process, we get for all n∈N:
+Tn+1(f);Tn(σ);...;T(σ);σ=f
+Thus:
+Tn+1(f);σ(n)=f
+Now take s∈f, and let nbe the length of the longest path in s. SinceT[X]
+has no hole at the root, no path of length ncan reach BinTn+1(B), thus
+s∈σ(n),therefore s∈σ†. Thesamereasoningalsoworksfortheotherinclusion.
+Likewise, if σ:B→T(B), we build a unique σ‡:B→νX.Tmaking the
+coalgebra diagram commute.
+3.3 Interpretation of µLJ
+3.3.1 Interpretation of Formulas.
+As expected, we give the interpretation of valid formulas.
+/llbracket⊤/rrbracket=I /llbracketA⇒B/rrbracket=/llbracketA/rrbracket⇒/llbracketB/rrbracket
+/llbracket⊥/rrbracket=⊥ /llbracketX/rrbracket=/squareX
+/llbracketA∨B/rrbracket=/llbracketA/rrbracket+/llbracketB/rrbracket /llbracket µX.T/rrbracket=µX./llbracketT/rrbracket
+/llbracketA∧B/rrbracket=/llbracketA/rrbracket×/llbracketB/rrbracket /llbracket νX.T/rrbracket=νX./llbracketT/rrbracket
+3.3.2 Interpretation of Proofs.
+As usual, the interpretation of a proof πof a sequent A1,...,A n⊢Bwill be a
+morphism /llbracketπ/rrbracket:/llbracketA1/rrbracket×...×/llbracketAn/rrbracket−→/llbracketB/rrbracket. The interpretation is computed by
+induction on the proof tree. The interpretation of the rules of LJ is standard
+and its correctness follows from the cartesian closed structure o fGam. Here
+are the interpretations for the fixpoint and functor rules:
+/Hugellbrackettop
+/Hugellbracketex/Hugellbracketbotπ
+Γ⊢T[µX.T/X ]
+µr
+Γ⊢µX.T/Hugerrbrackettop
+/Hugerrbracketex/Hugerrbracketbot=/llbracketπ/rrbracket;i/llbracketT/rrbracket/Hugellbrackettop
+/Hugellbracketex/Hugellbracketbotπ
+T[A/X]⊢A
+µl
+µX.T⊢A/Hugerrbrackettop
+/Hugerrbracketex/Hugerrbracketbot=/llbracketπ/rrbracket†
+14/Hugellbrackettop
+/Hugellbracketex/Hugellbracketbotπ
+T[νX.T/X ]⊢B
+νl
+νX.T⊢B/Hugerrbrackettop
+/Hugerrbracketex/Hugerrbracketbot=i−1
+/llbracketT/rrbracket;/llbracketπ/rrbracket/Hugellbrackettop
+/Hugellbracketex/Hugellbracketbotπ
+A⊢T[A/X]
+νr
+A⊢νX.T/Hugerrbrackettop
+/Hugerrbracketex/Hugerrbracketbot=/llbracketπ/rrbracket‡
+/Hugellbrackettop
+/Hugellbracketex/Hugellbracketbotπ
+A⊢B
+[T]
+T(A)⊢T(B)/Hugerrbrackettop
+/Hugerrbracketex/Hugerrbracketbot=/llbracketT/rrbracket(/llbracketπ/rrbracket)
+We do not give the details of the proof that this defines an invariant o f
+reduction. The main technical point is the validity of the interpretat ion of the
+functor rule; more precisely when the functor is a (least or greate st) fixpoint.
+Given that, we get the following theorem.
+Theorem 3. Ifπ/squigglerightπ′, then /llbracketπ/rrbracket=/llbracketπ′/rrbracket.
+In particular, this proves the following theorem which is certainly wor th
+noticing, because µLJhas large expressive power. In particular, it contains
+G¨ odel’s system T [ 13].
+Theorem 4. µLJis consistent: there is no proof of ⊥.
+Proof.There is no total strategy on the game ⊥.
+3.3.3 Completeness.
+When it comes to completeness, we run into the issue that the total winning
+innocent strategiesare not necessarilyfinite, hence the usualde finability process
+does not terminate. Nonetheless, we get a definability theorem in an infinitary
+version of µLJ. Whether a more precise completeness theorem is possible is
+a subtle point. First, we would need to restrict to an adequate subc lass of
+the recursive total winning strategies (for example, the Ackerma nn function is
+definable in µLJ). Then again, the problem to find a proof whose interpretation
+isexactlythe original strategy would be highly non-trivial: if σ:µX.T⇒A, we
+have toguessan invariant B, a proof π1ofT(B)⊢Band a proof π2ofB⊢A
+such that /llbracketπ1/rrbracket†;/llbracketπ2/rrbracket=σ. Perhaps it would be more feasible to look for a proof
+whose interpretation is observationally equivalent to the original strategy, which
+would be very similar to the universality result in [ 17].
+4 Conclusion and Future Work
+We have successfully constructed a games model of a propositiona l intuition-
+istic sequent calculus µLJwith inductive and coinductive types. It is striking
+that the adequate winning conditions on legal plays to model (co)ind uction
+are almost identical to those used in parity games to model least and greatest
+fixpoints, to the extent that the restriction of our winning conditio n to paths
+coincides exactlywith the winning condition used in [ 24]. It would be worthwile
+to investigate this connection further: given a game viewed as a bipa rtite graph
+15along with winning conditions for infinite plays, under which assumption s can
+these winning conditions be canonically lifted to the set of legal plays o n this
+graph, viewed as an arena? Results in this direction might prove usef ul, since
+they would allow to import many game-theoretic results into game sem antics,
+and thus programming languages.
+This work is part of a larger project to provide game-theoretic mod els to
+total programming languages with dependent types, such as COQ o r Agda.
+In these settings, (co)induction is crucial, since they deliberately la ck general
+recursion. We believe that in the appropriate games setting, we can push the
+present results further and model Dybjer’s Inductive-Recursiv e[11] definitions.
+4.0.4 Acknowledgements.
+We would like to thank Russ Harmer, Stephane Gimenez and David Baeld e for
+stimulating discussions, and the anonymous referees for useful c omments and
+suggestions.
+References
+[1] A. Abel and T. Altenkirch. A predicative strong normalisation pro of for a
+lambda-calculus with interleaving inductive types. In TYPES, 1991.
+[2] S. Abramsky. Semantics of interaction: an introduction to game semantics.
+Semantics and Logics of Computation , pages 1–31, 1996.
+[3] S. Abramsky and R. Jagadeesan. Games and full completeness f or multi-
+plicative linear logic. J. Symb. Log. , 59(2):543–574, 1994.
+[4] S. Abramsky, R. Jagadeesan, and P. Malacaria. Full Abstractio n forPCF.
+Info. & Comp , 2000.
+[5] S. Abramsky, H. Kohei, and G. McCusker. A fully abstract game s emantics
+for general references. In LICS, pages 334–344, 1998.
+[6] D. Baelde and D. Miller. Least and greatest fixed points in linear logic . In
+LPAR, pages 92–106, 2007.
+[7] P. Clairambault and R. Harmer. Totality in Arena Games. Submitted. ,
+2008.
+[8] J.H. Conway. On Numbers and Games . AK Peters, Ltd., 2001.
+[9] J. De Lataillade. Second-ordertype isomorphismsthroughgame semantics.
+Ann. Pure Appl. Logic , 151(2-3):115–150, 2008.
+[10] P. Dybjer. Inductive sets and families in Martin-L¨ ofs Type The ory and
+their set-theoretic semantics: An inversion principle for Martin-L¨ ofs type
+theory.Logical Frameworks , 14:59–79, 1991.
+16[11] P. Dybjer. A general formulation of simultaneous inductive-re cursive defi-
+nitions in type theory. J. Symb. Log. , 65(2):525–549, 2000.
+[12] P. Freyd. Algebraically complete categories. In Proc. 1990 Como Category
+Theory Conference , volume 1488, pages 95–104. Springer, 1990.
+[13] K. Godel. ¨Uber eine bisher noch nicht bentzte Erweiterung des finiten
+Standpunktes. Dialectica , 1958.
+[14] R. Harmer. Innocent game semantics. Lecture notes , 2004.
+[15] R. Harmer, J.M.E. Hyland, and P.-A. Melli` es. Categorical combin atorics
+for innocent strategies. In LICS, pages 379–388, 2007.
+[16] J.M.E. Hyland. Game semantics. Semantics and Logics of Computation ,
+1996.
+[17] J.M.E. Hyland and C.H.L. Ong. On full abstraction for PCF:I,II, and
+III.Inf. Comput. , 163(2):285–408, 2000.
+[18] A. Joyal. Remarques sur la th´ eorie des jeux ` a deux personne s.Gaz. Sc.
+Math. Qu. , 1977.
+[19] O. Laurent. Classical isomorphisms of types. Mathematical Structures in
+Computer Science , 15(5):969–1004, 2005.
+[20] R. Loader. Equational theories for inductive types. Ann. Pure Appl. Logic ,
+84(2):175–217, 1997.
+[21] P. Lorenzen. Logik und Agon. Atti Congr. Internat. di Filosofia , 1960.
+[22] G. McCusker. Games and full abstraction for FPC.Inf. Comput. , 160(1-
+2):1–61, 2000.
+[23] P.-A. Melli` es. Asynchronous games 4: A fully complete model of proposi-
+tional linear logic. In LICS, pages 386–395, 2005.
+[24] L. Santocanale. Free µ-lattices. J. Pure Appl. Algebra , 168(2-3):227–264,
+2002.
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+cific Journal of Mathematics , 5(2):285–309, 1955.
+17
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+PREPARATION OF PAPERS FOR THE INTERNATIONAL JOURNAL ON EMERGING TECHNOLOGIES IN LEARNING  
+ Contextual Mobile Learning Strongly Related to 
+Industrial Activities: Principles and Case Study 
+doi:10.3991/ijxx.vxix.xxx  (Please do not delete this line) 
+Bertrand DAVID, Chuantao YIN and René CHALON 
+LIESP Laboratory, Ecole Centrale de Lyon, Ecully, France 
+ 
+ 
+ 
+ 
+Abstract —M-learning (mobile learning) can take various 
+forms. We are interested in contextualized M-learning, i.e. 
+the training related to the si tuation physically or logically 
+localized. Contextualization a nd pervasivity are important 
+aspects of our approach. We propose in particular 
+MOCOCO principles (Mobility - COntextualisation - 
+COoperation) using IMERA platform (Mobile Interaction 
+in the Augmented Real Environment) covering our 
+university campus in which we prototype and test our approach. We are studying various mobile learning contexts 
+related to professional activities, in order to master 
+appliances (Installation, Use, Breakdown diagnostic and Repairing). Contextualization, traceability and checking of 
+execution of prescribed operations are based mainly on the 
+use of RFID labels. Investigatio n of the appropriate training 
+methods for this kind of learning  situation, applying mainly 
+a constructivist approach known as "Just-in-time learning", 
+"learning by doing", "learnin g and doing", constitutes an 
+important topic of this project. From an organizational 
+point of view we are in perfect symbiosis with EPSS - 
+Electronic Performance Support System [12] and our objective is to integrate learning  in professional activities in 
+three ways: 1/ before work i.e. to learn about coming 
+actions, 2/ after work i.e. to  learn about past actions to 
+understand what happened and accumulate experience, 3/ 
+during work i.e. to master the problem just-in-time We are 
+also studying an appropriate re lationship between the just-
+in-time M-learning approach and preliminary training 
+performed in a serious game approach based on typical action scenarios. 
+Index Terms —mobile learning, contextualization, RFID, 
+augmented reality, wearable computer. 
+I. INTRODUCTION  
+As mobile technologies are becoming more and more 
+widespread in people’s daily life, there has been a huge 
+increase in studies and experiments regarding the use of 
+mobile technologies for learning and training in professional and industrial situations. More and more 
+people are starting to benefit from various mobile learning 
+technologies, but in the meanwhile arguments for the mobile learning definition have never stopped evolving 
+over the years. 
+Compared to e-learning, we identified mobile learning 
+cartography [1] in relation with ubiquitous computing, 
+mobile computing and wearable computing, a mobile 
+learning cartography as shown in Fig. 1. 
+Four essential characteristics  are proposed to describe 
+mobile learning situations [3]: devices, mobility, context, 
+and location. Several categories of mobile learning can be identified according to the variation of these 
+characteristics. 
+We consider it very important to separate M-learning 
+into two categories in relation with the context. Either the 
+learning activity is totally independent from the actor’s location and the context in which he/ she is evolving, 
+taking into account only the opportunity to use mobile 
+devices to learn (in public transportation, waiting for the bus, etc.) or on the contrary, th e learning activity relates to 
+the location (physical, geographical or logical) of the actor 
+and the context in which he/ she is evolving. We are mainly concerned with this second category of mobile 
+learning, which is also called contextual mobile learning 
+or context-aware mobile learning. 
+The learning context in mobile learning is a very 
+important aspect, which can be described as “any 
+information that can be used to characterize the situation 
+of learning entities that is considered relevant to the interactions between a learne r and an application” [4]. 
+Three categories are considered : computing context, user 
+context, and physical cont ext [5]. Thanks to the 
+development of wearable computers which are 
+complemented by accessorial sensors, such as GPS 
+receivers, RFID readers [6], cameras, etc, and software 
+sensors, such as network congestion manager, web log 
+analyzer, user behavior analy zer etc, the learning context 
+such as learners’ location, activity, network connectivity, 
+learning situation, etc, can be captured to improve the 
+learning activities.  
+ 
+Wearable
+Comp.Mobile
+Comp.Ubiquitous
+Computing
+ 
+Figure 1.  Mobile learning cartography 
+ PREPARATION OF PAPERS FOR THE INTERNATIONAL JOURNAL ON EMERGING TECHNOLOGIES IN LEARNING  
+ On the other hand, Mixed Reality [7] better known as 
+Augmented Reality, appeared in 1993 [8], is becoming 
+increasingly essential. This technology attempts to merge 
+the physical and numerical worlds in order to facilitate the 
+user’s task with special devices and particular interaction techniques. A lot of hardware materials (like see-through 
+goggles) and software applica tions were developed, which 
+are very helpful for the m obile learning research in 
+specific situations.  
+Study of learning methods is another important aspect 
+of mobile learning. New and more appropriate methods 
+are needed for the great re volution of mobile learning 
+compared to the traditional classroom methods. We 
+consider that these mobile learning methods should take 
+into account the specific cond itions in which learning 
+occurs: limited time, particular working conditions and 
+operational result orientation. Main learning method 
+characteristics are such as problem-based learning [2], 
+case-based learning [9], scenar io-based learning [10], and 
+just-in-time learning.  
+Several existing e-learning projects provide us with 
+much useful inspiration for designing new mobile learning systems. What we are intere sted in is applying mobile 
+learning theories to industrial learning and training 
+activities. Our contextual mobile learning system is part of a larger project called Help  M e  t o  D o  ( H M T D )  [ 1 1 ] ,  
+which aims at assisting different users in domestic, public 
+or professional mastering of appliances by wearable computer. The aim of this paper is to discuss new learning 
+methods to be used in and for industry in relation with 
+web-based knowledge access a nd digital era evolutions 
+and their integration in M-learning MOCOCO 
+environment to allow the user to learn just-in-time use, 
+diagnosis and repair procedures. 
+In the following sections, we present a discussion of 
+learning in and for industry. Next, augmentations of 
+human, environment and appliance in order to facilitate coordination, cooperation and conversation are described, 
+as well as global process and learning unit production. We 
+finish by a case study and a conclusion 
+II. L
+EARNING IN & FOR INDUSTRY  
+Learning of industrial professionals is an important 
+challenge which can accommodate  different solutions in 
+respect with different professional situations. Without identifying all these situations, we can point out two, 
+which are extreme and thus re presentative: before starting 
+professional work, mainly during university studies and during professional, in the field, life. University studies 
+form an appropriate period in which to assimilate 
+important theories and generic methodological approaches. In the field, thes e aspects are more difficult to 
+acquire. On the contrary, practical, precise behaviors, 
+operations and gestures are difficult to acquire at university and easy in the field. It is important to take into 
+account these two kinds of knowledge to be assimilated.  
+As an old Chinese proverb says “give a fish to a man 
+and he can eat one day, give him a fishing rod and he can eat all his life”. This prove rb is a motto (device) for 
+university studies, allowing students to understand generic 
+theories and approaches on wh ich they can build their life. 
+But to cook a particular fish is also important. 
+Unfortunately, university is not an appropriate place to learn cooking. Industry and in the field situations are more 
+appropriate workpl aces for this kind of learning.  
+Of course, optimum conditions for learning theories and 
+main approaches through typical case studies are 
+combined at university, before starting to work, and 
+precise command of special methods, tools and gestures is naturally mastered during in the field practices. However, 
+appropriate conditions for continual learning are not 
+always available. It seems important to create appropriate context, to acquire just-in-time, missing knowledge. 
+For industrial firms, it could be appropriate to separate 
+blended learning situations into alternative periods with 
+theory and approach acquisition in a university context and their application and adaptation to precise industrial 
+situations during associated industrial periods.  
+It is also possible to create individual conditions for this 
+learning by e-learning, either out of working time (in the evening) or during work ing time (if allowed).  
+In the scope of this confer ence and our orientations, we 
+are mainly interested in ICT enhanced learning. E-
+learning is the generic term for this field of studies as described in the introduction. As mentioned, we are 
+specifically interested in Cont extual Mobile Learning, the 
+aim of which is to learn, ju st-in-time in the workplace, 
+precise behaviors, actions and situations related to 
+maintenance, diagnosis and repair of industrial appliances. 
+For this kind of learning we identified three approach alternatives, but also the following complementary and 
+cooperative approaches: 
+y Formal, firm-specific learning organization in 
+direct relation with the firm’s operational information, 
+y Web-based and forum-base d access to world-wide 
+accessible knowledge, 
+y Coupling between off-line learning of industrial 
+practices by serious games and concrete working 
+practical situations.  
+A. Formal Approach EPSS 
+EPSS: Electronic Performance Support System is an 
+appropriate answer to the proble m of industrial in the field 
+learning, which is formal, firm-specific, learning 
+organization in direct relation with the firm’s operational information [12]. An EPSS is a web-based content 
+management system for  
+y Providing storage and delivery of plant reference 
+materials including: 
+— Training documents 
+— Operating procedures 
+— Historical maintenance information 
+y Improving plant performance through knowledge 
+transfer 
+y Storing knowledge captured from seasoned 
+employees 
+y Providing procedure management 
+By using an EPSS, employees will receive the basic and 
+additional support they need during their everyday tasks. The EPSS may be used as a training tool for new staff 
+members or as a quick reference library for experienced 
+personnel. The EPSS provides the user with  information 
+including information on plant systems and delivers plant 
+procedures to the people that  need them. The combination PREPARATION OF PAPERS FOR THE INTERNATIONAL JOURNAL ON EMERGING TECHNOLOGIES IN LEARNING  
+ of digital photos, video, animation and sound is a 
+powerful tool when attempting to stimulate a learner. All of these can be incorp orated into the EPSS. 
+The aim of the EPSS is to  improve human performance 
+by providing staff with the tools they need to perform 
+their jobs in a safe, efficient manner, which in turn, improves plant performance. 
+As presented later in this paper our approach is clearly 
+compatible with this EPSS approach. However, it seems 
+important to mention other more open-ended approaches. 
+B. Open-ended approaches 
+ In their paper [13] “A Theory of Learning for the 
+Mobile Age”, Mike Sharples and his colleagues propose a new way to learn, based mainly on conversation. They 
+propose a theory of learning for a mobile society in which 
+new forms of educational activity take place based mainly on informal and workplace lear ning that is fundamentally 
+mobile. This new learning is characterized as 
+personalized, learner centere d, situated, collaborative, 
+ubiquitous and lifelong as related to new technologies 
+characterized as personal , user centered, mobile, 
+networked, ubiquitous and durable. In this approach that considers learning as conversation, teacher and learner 
+jointly develop understanding through dialogue. In this 
+way the teacher can progressive ly totally disappear and be 
+replaced by the conversation between “partners” in a 
+conversation web–based forum.  These partners can be 
+either humans or also machin es, mainly in the form of a 
+web accessible Knowledge base. 
+This practice is already observable in software 
+development teams, where members of different and 
+maybe rival teams can mutually help each other via a web-based forum to solve recurrent technological 
+problems. 
+Another interesting point of view is expressed by Don 
+Tapscott in his book “Grown up digital: How the net Generation is changing your world” [14]. He mentions 
+new learner behavior as well as the behavior of workers, 
+and also further interesting observations and expectations in relation with this new digital world. 
+C. Mobile learning and Serious Game coupling  
+Another interesting approach for industrial learning is 
+the coupling between off-line learning of industrial 
+practices by serious games and concrete working 
+situations in the field. In this way it is possible to put into perspective the scenarios studied during the game and the 
+concrete case on which the technician (learner) is 
+working. With this connection he/ she can refresh the knowledge acquired during the game and re-contextualize 
+it in the current precise situation. He/she can thus combine 
+theories, methods and practices and thoroughly understand all interactions. 
+In our approach we take into account these aspects and 
+we propose a contextualized mobile cooperative learning 
+for industrial situations. In the following paragraphs we describe first different augmentations, as the way to create 
+a continuum between the real and virtual (digital) world 
+for contextualized workin g conditions for a human 
+augmented technician (hum an) evolving in a real 
+augmented environment and working on, observing, 
+studying, diagnosing and repairing real augmented appliances. 
+ 
+Figure 2.  Goggles with integrated screen, See-Through goggles, 
+TabletPC, RFID reader and Data glove 
+III. AUGMENTATIONS  
+A. Augmented human 
+In order to create appropriate MOCOCO (mobile, 
+contextual and cooperativ e) working and learning 
+conditions, we provide the user (worker – learner) with a 
+wearable computer and asso ciated input/output devices 
+(Fig. 2). Different users (actors) evolving in different 
+working scenarios receive di fferent configurations of 
+wearable computers and associ ated peripherals. Several 
+requirements can guide this choice: light and small hand 
+free equipment, or heavier equipment but with better 
+visualization capacities or better interaction performance …. For example, to present appropriate information to the 
+user, screen integrating goggles or see-through goggles 
+are carefully considered and chosen in order to create 
+suitable Augmented Reality info rmation allowing the user 
+to concentrate on the repair work and associated learning. 
+Elaboration of this configuration is based on a study of user’s tasks, and their characterization by requirements 
+concerning graphic informa tion complexity (textual, 
+graphic diagrams or precise blueprints, etc.), interaction complexity (writing, observation, manipulation) and 
+working conditions (seating, standing, hands availability, 
+etc.) and so on. A precise selection process based on a selection space allows comparis on of different interaction 
+modes and system implementations, with their typical 
+supporting devices organized onto axes and classified for each axis by one of their mo st relevant characteristics 
+[15]. 
+B. Augmented environment 
+Our user (worker – learner) is evolving in a real 
+augmented environment, which is able to provide him (by 
+his/ her wearable computer  and peripherals) with 
+appropriate information mainly related to the context. In 
+this way he/ she can decide what is relevant to do. The 
+delivering of appropriate context information allows him/ her to decide what to do. This is a crucial aspect in mobile 
+learning system design the aim of which is to deliver the 
+right learning content into the right learning context.  
+In our industrial scenarios, we mainly use RFID and 
+other associated technologies to collect contextual 
+information. Radio-frequency id entification (RFID) is an 
+automatic identification method, relying on storing and remotely retrieving data using devices called RFID tags 
+and RFID readers. The RFID tag can be applied to or 
+incorporated into a product,  animal, or person for the 
+purpose of identification using radio-waves and can be 
+read from several centimeters to meters away and beyond 
+the line of sight of the read er. In our scenarios, each 
+appliance registered in the manufacturer’s database has a PREPARATION OF PAPERS FOR THE INTERNATIONAL JOURNAL ON EMERGING TECHNOLOGIES IN LEARNING  
+ RFID tag, which indicates its own product serial number. 
+The user equipped with an RFID reader on his/ her wearable computer is able to obtain immediately all the 
+relative information and operation history about the 
+appliance from the server. In this way, concerned learning 
+units can receive precise c ontext information which is 
+used in the learning process to master or repair the 
+appliance. We also explore the use of RFID tags and 
+readers as part of the user interface and in relation with 
+hand and body based interactions.  
+1) RFID use 
+In mobile context-aware applications, easy access to 
+information describing and characterizing this context is important. Grounded communicating objects such as 
+RFID tags can be used to deliver static information to the 
+user’s wearable computer RFID reader. This information can be used to access more precise and complete 
+information which can be obt ained from the EPSS system 
+accessed by a wireless network. However, in situations in 
+which wireless network is not systematically available and 
+this information is mandatory, it seems important to be 
+able to store these data in-situ. 
+2) In-situ storage 
+In situ storage is possible using RFID tags. These tags 
+can be updated by the same reader, which is usually able 
+not only to read but also to write information to the RFID 
+tag. Storage capacity is ex tended significantly allowing 
+storage not only of access identifiers, but of real 
+information. 
+3) Traceability  
+In maintenance and repair of specific equipment 
+traceability is a major requireme nt. The aim is to register 
+all operations performed in order to be able to show these 
+operations later, in order to verify their application or to 
+understand or explain specific situations (behavior or 
+accident). For this reason, th e operations, tools used and 
+product parts concerned can be collected and stored in a RFID tag or a central databa se (EPSS), tools and product 
+parts are ”tagged” by RFID ta gs and the user has a RFID 
+reader in his wearable computer.  
+4) Operation prescription 
+Unlike traceability, execution of  prescribed operations 
+and respect of the operation sequence (workflow) can also 
+be managed and controlled by RFID. To guarantee safety 
+and reliability of the repairing process, it is possible to control respect of procedures . Control can concern actor 
+identification, in order to de termine his/ her accreditation 
+level or the process, i.e. appropriate execution of the 
+prescribed operation (sequence, tools and product parts). 
+C. Augmented appliance 
+In order to respond fully to HMTD problems, it is 
+necessary to use wearable computers for different activities of diagnosis, maintenance, repair and associated 
+M-learning. These computers must be correlated with 
+technologies of the appliance on which the actor wants to act. Three forms of relations hip between the wearable 
+computer and the appliance can  be identified (Fig. 3a, 3b 
+and 3c): 
+y When the appliance do es not propose any 
+connection with the wearab le computer, the user 
+has to establish this connection indirectly by 
+observing the appliance relating to the wearable computer observed situation and asking it to suggest an appropriate ope ration on the appliance 
+(Fig. 3a); 
+y When the appliance is able  to receive orders via, 
+for example, an infra-red interface, it is possible to 
+establish a unilateral contact from the wearable 
+computer towards the app liance, to give it an 
+order. The other method, which provides the 
+wearable computer with the information observed 
+on the appliance is replaced by user  assistance (Fig. 3b); 
+y When the appliance is able to establish with the 
+wearable computer two-directional 
+communication, it is possible to substitute the original interface of the appliance for one 
+proposed by the wearable computer interface. In 
+this case this new interf ace can be totally adapted 
+to user requirements (Fig. 3c). 
+D. IMERA platform 
+The IMERA platform designed to create an 
+environment for experimentations consists of a main 
+workplace and three auxiliary  distant workspaces. The 
+main working area is a CAE (Computer Augmented Environment) in which different mobile actors evolve. 
+This CAE is an area of variable size covered by a WiFi 
+network or another wireless network such as an adhoc network. The area is able to  receive RFID tags, either 
+freely set or integrated into  real objects (augmented 
+objects). The actors move freel y in this area with their 
+wearable computers, each eq uipped with a WiFi card and 
+an RFID reader. These wearab le computers are connected 
+to the network and are able to  collect contextual data via 
+RFID technology. The WiFi network allows actors to be 
+both connected with one another and with central systems 
+(database servers, EPSS, etc. ) so they can communicate 
+and access large amounts of data . Independently from this 
+working area, three separate distant management and 
+observation workplaces comple te this platform [11]. 
+Appliance Wearable 
+Com puter 
+b 
+Appliance Wearable Com
+puter 
+c Wearable 
+Com puter Appliance 
+a 
+ 
+Figure 3.  Appliance – Wearable computer communication relations PREPARATION OF PAPERS FOR THE INTERNATIONAL JOURNAL ON EMERGING TECHNOLOGIES IN LEARNING  
+ The IMERA platform can be used in a number of 
+collaborative situations, managing educational, industrial, cultural and sporting events. For each application, it is 
+important to identify the actors and their tasks with the 
+data to be collected and manipulated. We thereby determine the technologies to use on the main working 
+area and the most appropriate interaction devices for each 
+actor. As we mentioned in the previous section, with the support of WiFi access points to internet, the user also has 
+the possibility to collaborate with other actors located 
+elsewhere. Communication can assume a variety of forms: voice, text message, drawing or gesture and so on. During 
+the repair process of an ap pliance the technician can 
+sometimes have problems checking out the reasons for dysfunction. He can ask for help and be guided in his/ her 
+activity by his/ her technical supervisor. Other experts, 
+anyone who can be located anywhere in the world can 
+contribute to this collaborative problem solving. 
+IV. G
+LOBAL PROCESS AND ORGANIZATION  
+A. Learning unit production  
+Production of specific learning units for an appliance is 
+schematized in Fig. 4 and is carried out by analysis of 
+manufacturer documentation. For this step, an existing IMAT (Integrating Manuals and Training) project [16] 
+provided us with inspiration in indexing and management 
+of learning fragments. This, however, was initially designed for fixed expert users and did not consider 
+mobile users and contextualization possibilities. 
+Information provided is not limited to the textual 
+description of the tasks to be carried out (and thus learning), but can also contain diagrams, photographs, 
+figures and models of elements (parts making up the 
+equipment, its front face, its control panel, etc.). The goal is to build a computer supported model that is as complete 
+as possible in order to be able to produce suitable learning 
+units. The original source of documentation can assume a 
+variety of forms, such as paper manuals, .doc, .pdf, video, 
+voice, etc. Paper manuals are scanned and converted to electronic versions with OCR (Optical Character 
+Recognition) technologies; other documents are also converted to uniform formats, which become learning 
+fragments and finally learning units.  
+These learning units are se gmented according to their 
+usability, taking into account actors’ experience characteristics i.e. beginner’s actions, basic tasks, 
+advanced tasks, expert tasks for mastering the use of 
+appliances, as well as main tasks related to dysfunction identification, diagnosis process, repairing, dismounting 
+the parts and reassembling, etc.  The degree of specificity 
+or generality of the explanation and later classification (indexing) is defined in order to be able to retrieve these 
+units easily in the future. For specific industrial 
+knowledge firm-protected units, they are stored in the EPSS, while for general know ledge units an open-ended 
+web-based Knowledge base is used. All these units are 
+expressed in XML using the standard SCORM and LOM to allow adaptation.  
+B. IMERA working principles 
+During the working period, actors evolve in the IMERA 
+workspace equipped with thei r wearable computers and 
+associated peripherals, and work (use, master, diagnose 
+and repair) on appliances (Fig. 5). The control engine of the actor’s wearable computer assists him/ her with their 
+activities and helps him/ her by providing appropriate 
+working and learning units, in relation with the chosen method (working, just-in-time learning, learning and 
+doing). It plays the role of a manager able to link concrete 
+working situations with database stored learning units and is also responsible for adapting this information to 
+visualization and interacti on devices attached to the 
+wearable computer.  
+Contextualization is provided by use of different 
+markers, mainly RFID tags. In this way precise 
+information on the appliance can be transmitted to the 
+learning system. Actors (user, repairer) also use wearable 
+computers to communicate w ith the system, between 
+actors and, if possible,  with the appliance. 
+ 
+Provider 
+(constructor)  
+documentation of 
+appliance Structuring in leaning units:
+• Beginner action, 
+• Basic tasks 
+• Advanced tasks 
+• Expert tasks 
+• Dysfunction 
+identification  
+• Diagnosis 
+• Repairing 
+• Taking to pieces and 
+reassembling Study of dysfunction identification  
+Diagnosis approaches 
+ 
+Tree of weaknesses 
+Learning of problem situations 
+Analysis and 
+modeling of tasks Database of 
+learning units   
+EPSS or Web-based KB  
+ 
+Figure 4.  Learning unit production process PREPARATION OF PAPERS FOR THE INTERNATIONAL JOURNAL ON EMERGING TECHNOLOGIES IN LEARNING  
+   
+EPSS or Web-based KB 
+IMERA platform Wearable computer Learning methods: 
+• Just in time learning 
+• Learning by doing 
+• Learning and doing 
+Appliance CommunicationDatabase of 
+learning units   
+Control 
+engine Interaction 
+Patterns 
+Wearable computer 
+Wearable computer  
+ 
+Figure 5.  Learning units use on IMERA platform 
+ 
+V. CASE STUDY  
+A. Description of the case study 
+To explain the use of our MOCOCO system, we 
+decided to present a commonly understandable problem, assistance with co mputer maintenance, and, 
+more precisely, replacement of a hard disk of a desktop 
+computer. We show how to assist a young technician who is carrying out, and learning about, replacement of 
+a hard disk. Based on the contextual mobile learning 
+principles described in the previous section, we propose a contextualized solution, using mobile devices (Tablet 
+PC or PDA) and augmented reality accessories (semi-
+transparent glasses) to pr ovide a just-in-time learning 
+solution when the user needs a guide to do this task. 
+Through the contextualization process, the user 
+immediately obtains brief information on the computer 
+to maintain and a list of all possible operating guides on his/ her screen. This information helps him/ her to 
+initiate the maintenance work. 
+In the scenario of changi ng the hard disk of a 
+computer, work is made up of several steps such as: 1. 
+Remove the screws on the case. 2. Remove the case. 3. 
+Pull out the power connector…14. Fit on the case. If the user is not clear on one or more steps, he/ she can 
+trigger the just-in-time learning process and learn how to carry out the actions. 
+He/ she can view the global list of actions or ask for 
+more precise action by an explanation of each step 
+activity. He/ she can watch the sequence of actions, listen to the voice guidance, etc… and perform his task 
+step-by-step (Fig.6.). 
+If the user is equipped with augmented reality semi-
+transparent glasses, the sy stem can show him/ her 
+guided actions superimposed on real objects (Fig. 7). 
+If the user still experiences difficulties during work 
+after individual learning, a collaborative interaction can 
+launch communication with other remote experts via the network, to guide the user by appropriate methods 
+(voice, text message, real gestures, etc.). 
+During this learning process, the user can not only 
+finish the job by following the guide, but also try to 
+understand the whole sequence of actions. Moreover, 
+some additional information is displayed for the user to learn more about the task. Fo r example, when the user 
+removes the hard disk, he/she is informed on the screen 
+of all the characteristics c oncerning the disk (Fig.6). 
+ 
+ 
+Figure 6.  Changing hard disk manipulation process 
+ PREPARATION OF PAPERS FOR THE INTERNATIONAL JOURNAL ON EMERGING TECHNOLOGIES IN LEARNING  
+  
+ 
+ 
+Figure 7.  Semi transparent see-through augmented reality HMD 
+B. Learning and applying which relationships? 
+We can propose three ways in which learning can be 
+integrated in professional activities: 
+y Before work i.e. to learn about coming actions  
+y After work i.e. to learn about past actions to 
+understand what happened and accumulate experience 
+y During work i.e. to ma ster the problem just-in-
+time. 
+In our case study we can offer learners better mastery of 
+different hard disk transfer rates, interfaces, multi device 
+connections and so on, as explained in the appendix. Of 
+course learners can either apply the method before, after 
+or during their work and learn in identical manner, or apply only without in-depth understanding of what they 
+have done. 
+VI. C
+ONCLUSIONS  
+M-learning (mobile learning) can take various forms. 
+We are interested by contextu alized M-learning, i.e. the 
+training related to the situation physically or logically localized. The contextualiz ation and pervasivity are 
+important aspects of our approach. We propose in 
+particular MOCOCO pr inciples (Mobility - 
+COntextualisation - COoperation) using IMERA platform 
+(Mobile Interaction in the Augmented Real Environment) 
+covering our university campus in which we are prototyping and testing our approach. From an 
+organizational point of view we are in perfect symbiosis 
+with EPSS - Electronic Perform ance Support System [12] 
+and our objective is to integrate learning in professional 
+activities in three ways: 1/ before work i.e. to learn about 
+coming actions, 2/ after work i.e. to learn about past actions to understand what happened and accumulate 
+experience, 3/ during work i.e. to master the problem just-
+in-time We are also studying an appropriate relationship between the just-in-time M-learning approach and 
+preliminary training performed in a serious game 
+approach based on typical action scenarios. 
+Different experimentations and evaluations of this 
+approach and platform have  been carried out in our 
+laboratory, mainly with students as actors. The results are 
+very positive and many useful suggestions were given. We are open to other applications to validate our approach. 
+We are also studying an appropriate relationship 
+between the just-in-time M-learning approach and 
+preliminary training performed in a serious game approach based on a typical action scenario. 
+R
+EFERENCES  [1] David B., Yin C., Chalon R., Contextual Mobile Learning for 
+Repairing Industrial Machines: System Architecture and Development Process, ICELW 2008, June 12-13, New York, NY, USA 
+[2] O’Malley, C., “Mobile Learning Guidelines for Learning in a 
+Mobile Environment”, UoN, http://www.mobilearn.org, 2003. 
+[3] Meyer C., et al, “Caractérisation de Situations de M-Learning ”, 
+TICE 2006, 25-27 October 2006, Toulouse. 
+[4] Yuan-Kai Wang, “Context Awareness and Adaptation in Mobile 
+Learning”, Proceedings of the 2nd IE EE International Workshop 
+on Wireless and Mobile Technologies in Education (WMTE’04) 
+[5] Schilit W.N., “System Architectur e for Context-aware Mobile 
+Computing”, Ph.D thesis, Columbia University, May, 1995 
+[6] Srivasta L., “Ubiquitous Network Societies: The Case of Radio 
+Frequency Identification”, Bac kground paper of ITU Workshop 
+on Ubiquitous Network Societies, Geneva, 2005. 
+[7] Milgram P., et al., “Merging Real and Virtual Worlds”, 
+Proceedings of IMAGINA’ 95, Monte-Carlo, 1995. 
+[8] Wellner P., et al. “Computer Au gmented Environments: Back to 
+the Real World”. Special Issue of Communications of the ACM, vol. 36. 1993. 
+[9] Kolodner, J.L., et al., “Case-Based Learning Aids”. Handbook of 
+Research for Educational Communications and Technology, 2nd Ed. Mahwah, NJ: Lawrence Erlbaum Associates, pp. 829 – 861, 2004. 
+[10] Lave, J., “Cognition in Practice: Mind, mathematics, and culture 
+in everyday life”. Cambridge Univ ersity Press, Cambridge, Jean 
+Lave, UK, 1988. 
+[11] David B., Chalon R., “IMERA: Experimenta tion Platform for 
+Computer Augmented Environment for Mobile Actors”, 3rd IEEE International Conference on Wireless and Mobile Computing, Networking and Communications , WiMob 2007, October 8-10, 
+2007, Crowne Plaza Hotel, White Plains, New York, USA. 
+[12] Finley E., Bollinger R., What is EPSS (Electronic Performance 
+Support System) and How Do Two Utilities Use It?, WUTAB 2003, Oklahoma City. 
+[13] Sharples M., Tyalor J., Vavoula G.  A Theory of Learning for the 
+Mobile Age, in R. Andrews and C. Haythornthwaite (eds.) The 
+Sage Handbook of Elearning Research, London, Sage, pp. 221-247. 
+[14] Tapscott D., Grown Up Digital: How the Net  Generation is 
+Changing Your World, Mc Graw Hill 2009. 
+[15] Masserey G., et al., “Démarche d’Aide au Choix de Dispositifs 
+pour l'Ordinateur Porté”, Ac tes du colloque ERGO’IA 2006  
+L’humain comme facteur de performance des systèmes complexes , octobre 2006, Biarritz. 
+[16] De Hoog R. et al. “Reusing Tec hnical Manuals for Instruction: 
+Creating Instructional Material with the Tools of the IMAT Project”. Workshop on Integrating Technical and Training Documentation, 2002 
+VII. APPENDIX  
+We show the nature of information which can be useful 
+for the actor in charge of ha rd disk replacement. We list a 
+set of knowledge items growing understanding of 
+connection technologies. This information is coming from 
+in Wikipedia : http://en.wikipedia.org/ .  
+ 
+ 
+(a) IDE   (b) SCSI  
+        (c) FC        (d) SATA  
+Figure 8.  Main connection technologies PREPARATION OF PAPERS FOR THE INTERNATIONAL JOURNAL ON EMERGING TECHNOLOGIES IN LEARNING  
+ A. Data transfer rate 
+In 2008, a typical 7200rpm desktop hard drive has a 
+sustained "disk-to-buffer" data transfer rate of about 70 
+megabytes per second. This rate depends on the track location, so it will be highest for data on the outer tracks 
+(where there are more data s ectors) and lower toward the 
+inner tracks (where there are fewer data sectors); and is generally somewhat higher for 10,000rpm drives. A 
+current widely-used standard for the "buffer-to-computer" 
+interface is 3.0 Gbit/s SATA, which can send about 300 
+megabyte/s. from the buffer to the computer, and thus is 
+still comfortably ahead of today's disk-to-buffer transfer 
+rates. Data transfer rate (read/write) can be measured by writing a large file to disk using special file generator 
+tools, then reading back the file. Transfer rate can be 
+influenced by the fragmentation of the drive and the layout of the files. For example, a single file of 10GB will 
+be read significant faster  than 1000 files of 10MB. 
+B. Interfaces 
+Word serial interfaces: connect to a host bus adapter 
+with two cables, one for data/control and one for power. 
+The earliest versions of these interfaces typically had a 16 
+bit parallel data transfer to/f rom the drive and there are 8 
+and 32 bit variants. Modern versions have serial data 
+transfer. The word nature of  data transfer makes the 
+design of a host bus adapter significantly simpler than that of the precursor HDD controller: 
+y IDE or ATA, P-ATA 
+y EIDE 
+y SCSI 
+Bit serial interfaces: connect to a host bus adapter with 
+two cables, one for data/control and one for power: 
+y Fibre Channel (FC), 
+y Serial ATA (SATA). 
+y Serial Attached SCSI (SAS) 
+C. Integrated Drive Electronics (IDE) 
+Later renamed to ATA, and then later to P-ATA 
+("parallel ATA", to distinguish it from the new Serial 
+ATA). The original name reflected the innovative integration of HDD controlle r with HDD itself, which was 
+not found in earlier disks. Moving the HDD controller 
+from the interface card to th e disk drive helped to 
+standardize interfaces, and to reduce the cost and 
+complexity. The 40 pin IDE/ATA connection of PATA 
+transfers 16 bits of data at a time on the data cable. The data cable was originally 40 conductors, but later higher 
+speed requirements for data transfer to and from the hard 
+drive led to an "ultra DMA" mode, known as UDMA. Progressively faster versions of this standard ultimately 
+added the requirement for an 80 conductor variant of the 
+same cable; where half of the conductors provide grounding necessary for enhanced high-speed signal 
+quality by reducing cross ta lk. The interface for 80 
+conductors only has 39 pins, the missing pin acting as a key to prevent incorrect inser tion of the connector to an 
+incompatible socket, a common cause of disk and 
+controller damage. 
+D. Multiple devices on a cable: 
+If two devices attach to a single cable, one must be 
+designated as device 0 (comm only referred to as master) and the other as device 1 (slave). This distinction is 
+necessary to allow both drives to share the cable without conflict. The mode that a drive must use is often set by a 
+jumper setting on the drive itself, which must be manually 
+set to master or slave. If there is a single device on a cable, it should be configured as master. However, some hard 
+drives have a special setting called single for this 
+configuration (Western Digital, in particular). Also, depending on the hardware and software available, a 
+single drive on a cable can work reliably even though 
+configured as the slave drive (this configuration is most 
+often seen when a CD ROM has a channel to itself). 
+E. EIDE  
+It was an unofficial update (by Western Digital) to the 
+original IDE standard, with the key improvement being 
+the use of direct memory access (DMA) to transfer data 
+between the disk and the computer without the 
+involvement of the CPU, an improvement later adopted by 
+the official ATA standards. By  directly transferring data 
+between memory and disk, DM A eliminates the need for 
+the CPU and operating system to copy byte per byte. And 
+can therefore process other ta sks while the data transfer 
+occurs. 
+F. Small Computer System Interface (SCSI) 
+It was an early competitor of ESDI. SCSI disks were 
+standard on servers, workstations, Commodore Amiga and 
+Apple Macintosh computers through the mid-90s, by which time most models had been transitioned to IDE 
+(and later, SATA) family disks. Only in 2005 did the 
+capacity of SCSI disks fall behind IDE disk technology, though the highest-performance disks are still available in 
+SCSI and Fiber Channel only. The length limitations of 
+the data cable allows for external SCSI devices. Originally SCSI data cables used single ended data transmission, but 
+server class SCSI could use differential transmission, 
+either low voltage differential (LVD) or high voltage 
+differential (HVD). 
+G. Fibre Channel (FC) 
+It is a successor to parallel SCSI interface on enterprise 
+market. It is a serial protocol. In disk drives usually the 
+Fiber Channel Arbitrated Loop (FC-AL) connection 
+topology is used. FC has much broader usage than mere disk interfaces, it is the co rnerstone of storage area 
+networks (SANs). Recently othe r protocols for this field, 
+like iSCSI and ATA over Ethernet have been developed as well. Confusingly, drives usually use copper twisted-pair 
+cables for Fiber Channel, not fiber optics. The latter are 
+traditionally reserved for larger  devices, such as servers or 
+disk array controllers. 
+H. Serial ATA (SATA) 
+The SATA data cable has one data pair for differential 
+transmission of data to the device, and one pair for 
+differential receiving from the device, just like EIA-422. 
+That requires that data be transmitted serially.  
+I. Topology: 
+SATA uses a point-to- point architecture. The 
+connection between the controller and the storage device 
+is direct. Modern PC systems usually have a SATA controller on the motherboard, or installed in a PCI or PCI 
+Express slot. Some SATA controllers have multiple PREPARATION OF PAPERS FOR THE INTERNATIONAL JOURNAL ON EMERGING TECHNOLOGIES IN LEARNING  
+ SATA ports and can be connected to multiple storage 
+devices. There are also port expanders or multipliers which allow multiple storage devices to be connected to a 
+single SATA controller port. 
+J. Serial Attached SCSI (SAS  
+The SAS is a new generation serial communication 
+protocol for devices designed to allow for much higher 
+speed data transfers and is compatible with SATA. SAS uses a mechanically identical data and power connector to 
+standard 3.5" SATA1/SATA2 HDDs, and many server-
+oriented SAS RAID controlle rs are also capable of 
+addressing SATA hard drives. SAS uses serial 
+communication instead of the parallel method found in 
+traditional SCSI devices but still uses SCSI commands.  
+A
+UTHORS  
+Bertrand DAVID, Chuantao YIN and René 
+CHALON are with the LIESP Laboratory, Ecole Centrale 
+de Lyon, 36, Avenue Guy de Collongue, 69134 Ecully Cedex, France. (e-mail: {Bertrand.David, Chuantao.Yin, 
+Rene.Chalon}@ec-lyon.fr).  
+Manuscript received 18 July 2009  Published as submitted by the authors.  
+ 
\ No newline at end of file
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+Response to I. I. Mazin’s correspondence on “Electronic correlations in the iron 
+pnictides”  
+ M. M. Qazilbash
+1, J. J. Hamlin1,2, R. E. Baumbach1,2, Lijun Zhang3, D. J. Singh3,  
+M. B. Maple1,2 and D. N. Basov1. 
+ 
+1. Department of Physics, University of Californi a, San Diego, La Jolla, California 92093, USA  
+2. Institute for Pure and Applied Physical Sciences, University of California, San Diego, La Jolla, 
+California 92093, USA  
+3. Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,  
+Tenne ssee 37831, USA  
+ 
+ 
+There is accumulating evidence for substantial renormalization of the iron d- bands on a 
+several tenth of eV scale in the iron pnictides, starting with photoemission experiments on LaFePO [1]. Recent direct measurements of the Fermi surface of LaFePO using 
+quantum oscillations support this, finding mass renormalizations of ~2 depending on the 
+particular Fermi surface sheet compared with band structure calculati ons [2]. We 
+compared optical spectra and electronic structure calculations and di scovered a 
+renormalization  of the Drude weight in LaFePO consistent with the mass renormalization 
+found in the above experiments [3]. Specifically, we showed that the experimental Drude weight (K
+exp) is substantially suppressed relative to  the bare band st ructure value ( Kband) 
+and we attributed this phenomenon to correlation effects of electronic origin [3].  
+ 
+Recently, Mazin has criticized our  conclusion arguing that band shifts of +100 meV for 
+the electron bands and -100 meV for the hole bands would reduce  the needed mass 
+renormalization by ~2/3, leaving only  the remaining  ~1/3 to be explained by electronic 
+correlations [4].  In this response, we point out that the effect discussed by Mazin is 
+considerably smaller than he claims and, in particular, is too small to change our 
+conclusions. In addition, the minor  band shifts may well be a consequence of the 
+electronic correlations that we find.  
+ 
+Mazin bases his criticism on the measurements by Coldea and co -workers [2]. Actually, 
+Coldea and co -workers note that shifts will improve the agreement between the density 
+functional bands and their quantum oscillation measurements, but the shifts  are 
+substantially  smaller  than those assumed by Mazin, and even then some of the shifts may be related to sample stoichiometry . None of these  shifts  on any Fermi surface is  as large 
+as 100 meV  quoted by Mazin. A noteworthy  effect is the removal of  the heavy 3D hole  
+sheet. However, because of its heavy mass, this sheet contributes very little to the Drude 
+weight  that we register in our experiments . The main contribution (~65%) to the Drude 
+weight  according to our calculations  in fact comes from the electron sections.  This 
+dominance of the electron sheets is im portant because the overall effect  of band shifts on 
+the electron contrib ution are smaller than the effect of similar shifts on the hole 
+contribution. There are two electron  sections, an  inner section and an outer se ction, which 
+according to Coldea and co -workers  can be brought into optimum agreement with  their 
+experiment using shifts of 83 and 30 meV accompanied by a mass  renormalization . The 
+optimum shift of the hole bands of  -53 meV  is also much smaller  in magnitude than  -100 
+meV  invoked by Mazin. The actual change in the electron Fermi surface volume is small  
+(~15%), as may be seen from Fig s. 3c and 3d of Ref. 2.  This contradicts the claim of 
+Mazin, who asserts that the main contribution to the discrepancy in the optical Drude 
+spectral weight (proportional to n/ mopt) is due to a change in the effective Fermi surface 
+volume ( n) and not the effective optical mass ( mopt).  
+ 
+Finally, we note that although Mazin invokes the well known density functional band gap 
+error in simple insulators  such as Ge as the mechanism for band shifts , there is no reason 
+to suppose that the same physi cs is operative in a partially filled metallic d-band system. 
+For example, t here are strong shifts  in electronic structure relative to density functional 
+calculations  in correlated oxides such as NiO, and these are primarily associated with 
+electron correl ations (Mott physics in that case ).  
+ To summarize, Mazin’s arguments do not invalidate our conclusions because  (1) the 
+band shifts seen in experiment are significantly smaller than what he proposed, leaving a substantial renormalization of the Drude weig ht to be explained by correlations of 
+electronic origin , (2) the observed band shifts may also reflect electronic correlations  
+themselves  and (3) many experiments are converging on the fact that there are substantial 
+band renormalizations  on the tenths of eV scale consistent with our report . These include 
+optical measurements on EuFe
+2As2 as compared with density functional calculations by Moon, Mazin and co- workers [5]. Moreover, our conclusion about the importance of 
+electronic correlations in the iron pni ctides was also based on the significant 
+renormalization of the Drude weight ( Kexp/Kband ~ 0.3)  in paramagnetic BaFe 2As2 [3]. A 
+more recent infrared work on LaFeAsO also reported a factor -of-3 renormalization of the 
+Drude weight [6]. As for the detailed nature of the electronic correlation effects, this 
+remains an open question [3] . 
+ 
+ 
+[1] D. H. Lu et al. , Nature 455, 81 (2008).  
+[2] A. I. Coldea et al. , Phys. Rev. Lett. 101, 216402 (2008). 
+[3] M. M. Qazilbash et al. , Nature Physics 5, 647 (2009) . 
+[4] I. I. Mazin, arXiv:0910.4117 (2009).  
+[5] S. J. Moon et al. , arXiv:0909.3352 (2009).  
+[6] Z. G. Chen et al. , arXiv:0910.1318 (2009) . 
\ No newline at end of file
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@@ -0,0 +1,1295 @@
+arXiv:1001.0644v2  [astro-ph.CO]  6 May 2010Mon. Not.R. Astron. Soc. 000, 000–000 (0000) Printed 1November 2018 (MN L ATEX style filev2.2)
+Searchingfordarkmatterin X-rays:howto checkthedarkmat ter
+originof aspectralfeature
+Alexey Boyarsky1,2, Oleg Ruchayskiy1,Dmytro Iakubovskyi2,
+Matthew G. Walker3,Signe Riemer–Sørensen4, Steen H. Hansen4
+1Ecole Polytechnique F´ ed´ erale deLausanne, FSB/ITP/LPPC ,BSPCH-1015, Lausanne, Switzerland
+2Bogolyubov Institute for Theoretical Physics, Metrologic hna str.,14-b, Kiev 03680, Ukraine
+3Institute of Astronomy, University ofCambridge, UK
+4DarkCosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej30,DK-2100 Copenhagen, Den mark.⋆
+ABSTRACT
+A signal from decaying dark matter (DM) can be unambiguously distinguished from spec-
+tral features of astrophysical or instrumental origin by st udying its spatial distribution. We
+demonstratethis approachby examiningthe recent claim of L oewenstein&Kusenko (2009)
+regarding the possible DM origin of the 2.5 keV line in Chandr a observations of the Milky
+Way satellite known as Willman 1. Our conservative strategy is to adopt, among reasonable
+mass estimates derived here and in the literature, a relativ ely large dark mass for Willman 1
+andrelativelysmalldarkmassesforthecomparisonobjects .Inlightofthelargeuncertaintyin
+theactualdarkmattercontentofWillman1,thisstrategypr ovidesminimumexclusionlimits
+ontheDMoriginofthereportedsignal.Weanalyzearchivalo bservationsby XMM-Newton of
+M31andFornaxdwarfspheroidalgalaxy(dSph)and Chandra observationsofSculptordSph.
+By performinga conservativeanalysisofX-rayspectra,we s howthe absenceofa DM decay
+line with parameters consistent with those of Loewenstein& Kusenko (2009). For M31, the
+observationsoftheregionsbetween10and20kpcfromthecen ter,wheretheuncertaintiesin
+theDMdistributionareminimal,makeastrongexclusionatt helevelabove 10σ.TheFornax
+dSphprovidesa ∼3.3σexclusioninsteadofa predicted 4σdetection,andtheSculptordSph
+providesa 3σexclusioninsteadofapredicted 2.5σdetection.Theobservationsofthecentral
+regionofM31(1-3kpcoff-center)areinconsistentwithhav ingaDMdecaylineatmorethan
+20σifonetakesthemostconservativeamongthebestphysically motivatedmodels.Themin-
+imalestimatefortheamountofDMinthecentral40kpcofM31i sprovidedbythemodelof
+Corbellietal. (2010), assuming the stellar disk’s mass to l ight ratio ∼8and almost constant
+DM density within a core of 28 kpc. Even in this case one gets an exclusion at 5.7σfrom
+central region of M31, whereasmodeling allprocessed data from M31 and Fornax produces
+morethan 14σexclusion.Therefore,despitepossiblesystematicuncert ainties,weexcludethe
+possibilitythatthespectralfeatureat ∼2.5keVfoundinLoewenstein&Kusenko(2009)isa
+DM decayline. We conclude,however,that the searchfor DM de cayline, althoughdemand-
+ing prolonged (up to 1 Msec) observations of well-studied dS phs, M31 outskirts and other
+similar objects, is rather promising, as the nature of a poss ible signal can be checked. An
+(expected) non-observation of a DM decay signal in the plann ed observations of Willman 1
+shouldnotdiscouragefurtherdedicatedobservations.
+1 INTRODUCTION
+Modern astrophysical and cosmological data strongly indic ate that
+a significant amount of matter inthe Universe exists in the fo rm of
+dark matter . The nature of dark matter (DM) remains completely
+unknown and its existence presents one of the major challeng es to
+modern physics. It is commonly believed that the dark matter is
+made of particles.1However, the Standard Model of elementary
+particlesfailstoprovide aDMcandidate. Therefore thehyp othesis
+ofaparticlenatureofDMimpliesextensionoftheStandardM odel.
+1Although more exotic possibilities (such as, for example, p rimordial
+black holes, Carr (2005)) exist.Weakly interacting massive particles (WIMPs) are probably
+the most popular class of DM candidates. They appear in many
+extensions of the Standard Model (see e.g. Bertone etal. 200 5;
+Hooper 2009). These stable particles interact with the SM se c-
+tor with roughly electroweak strength (Lee & Weinberg 1977) . To
+provide a correct dark matter abundance, WIMPs should have
+mass in the GeV range. A significant experimental effort is de -
+voted to the detection of the interaction of WIMPs in the Gala xy’s
+DM halo with laboratory nucleons ( direct detection experiments ,
+see e.g. Baudis (2007)) or to finding their annihilation sign al in
+space (indirect detection experiments , see e.g. Carret al. (2006);
+Bergstrom (2008);Hooper & Baltz(2008)).
+Another large class of DM candidates are superweakly inter-
+c/circlecopyrt0000 RAS2Boyarskyet al.
+acting particles (so called super-WIMPs ), i.e. particles, whose in-
+teraction strength with the SM particles is much more feeble than
+the weak one. Super-WIMPs appear in many extensions of the
+Standard Model: extensions of the SM by right-handed neutri nos
+(Dodelson & Widrow 1994; Asaka etal. 2005; Lattanzi& Valle
+2007), supersymmetric theories (Takayama & Yamaguchi 2000 ;
+Buchmuller et al. 2007; Fenget al. 2004, 2003), models with
+extra dimensions (Fenget al. 2003) and string-motivated mo d-
+els (Conlon& Quevedo 2007). The feeble interaction strengt h
+makes the laboratory detection of super-WIMPs challenging (see
+e.g. Bezrukov &Shaposhnikov (2007)). On the other hand, man y
+super-WIMP particles possess a 2-body radiative decay channel:
+DM→γ+ν,γ+γ, producing a photon with energy Eγ=
+MDM/2. One can therefore search for the presence of such a
+monochromatic line with energy in the spectra of DM-contain ing
+objects (Abazajianet al. 2001; Dolgov & Hansen 2002). Altho ugh
+the lifetime of any realistic decaying DM is much longer than the
+lifetime of the Universe (see e.g. Boyarsky &Ruchayskiy (20 08)),
+the huge amount of potentially decaying DM particles in a typ ical
+halo means a potentially detectable decay signal in the spec tra of
+DM-dominated objects. Searchingfor such a line provides a m ajor
+wayofdetection of super-WIMP DM particles .
+The strategy of the search for decaying dark matter signal
+drasticallydiffersfrom itsannihilationcounterpart. In deed, the de-
+caysignal is proportional to the DM column density – integral of a
+DM distribution ρDMalong the line of sight S=/integraltext
+l.o.s.ρDM(r)dr
+as opposed to/integraltext
+l.o.s.ρ2
+DM(r)drin the case for annihilating DM.
+The DM column density varies slowly with the mass of DM ob-
+jects (Boyarsky et al. 2006b, 2009a) and as a result a vast var iety
+of astrophysical objects of different nature would produce a com-
+parable decay signal (Boyarsky etal. 2006b; Bertone et al. 2 007;
+Boyarsky et al. 2009a). Therefore (a)without sacrificing the ex-
+pected signal one has a freedom of choosing observational ta rgets,
+avoidingcomplicatedastrophysicalbackgrounds; (b)ifacandidate
+line is found, its surface brightness profile may be measured (as it
+does not decay quickly away from the centers of the objects) a nd
+distinguished from astrophysical lines that usually decay in out-
+skirtsof galaxies andclusters. Moreover, any tentative de tectionin
+one object would imply a signal of certain (comparable) sign al-to-
+noiseratiofromanumberofotherobjects.Thiscanbechecke dand
+thesignalcaneitherbeunambiguously confirmedtobethatof DM
+decay origin or ruled out. This allows to distinguish the dec aying
+DMlinefromanypossibleastrophysicalbackground andther efore
+makes astrophysical search for the decaying DM another type of a
+direct detection experiment .
+In this paper we demonstrate the power of this approach.
+We check the recent conjecture by Loewenstein & Kusenko
+(Loewenstein& Kusenko 2009, LK09in what follows), who re-
+ported that a spectral feature at 2.51±0.07keV with the flux
+(3.53±1.95)×10−6photonscm−2s−1(all errors at 68%
+confidence level) in the spectrum of the Milky Way satellite
+known as Willman 1 (Willmanetal. 2005) may be interpreted
+as a DM decay line. According to LK09, the line with such
+parameters is marginally consistent the restrictions on de caying
+dark matter from some objects (see e.g. Loewenstein et al. (2 009);
+Riemer-Sørensen& Hansen(2009)ortherestrictionsfromth e7ks
+observation of Ursa Minor dwarf spheroidal in Boyarsky et al .
+(2007b)). The results of other works (e.g. Watsonetal. 2006 ;
+Abazajian etal. 2007; Boyarsky et al. 2008a) are inconsiste nt at
+3σlevel with having 100%of decaying DM with the best-fit pa-
+rameters of LK09. However, it is hard to exclude completely n ew
+unknown systematic uncertainties in the DM mass estimates a ndtherefore in expected signal, from any given object. To this end, in
+this work we explicitly check for a presence of a line with par am-
+eters, specified above, by using archival observations by XMM-
+NewtonandChandra of several objects, where comparable or
+stronger signalwas expected. Wecompare the signalsfrom se veral
+objects and show that the spatial (angular) behavior of the s pectral
+feature is inconsistent with its DM origin so strongly, that system-
+atic uncertainties cannot affect this conclusion. We also d iscuss a
+possible origin ofthe spectral feature ofLK09.
+2 CHOICE OFOBSERVATIONALTARGETS
+Theparticlefluxofthedecayingdarkmatter(inphotons cm−2×s))
+isgiven by
+FDM=Γ
+4πmDMMfov
+DM
+D2
+L=Γ
+4πmDMΩfovS, (1)
+wheremDMis the mass of the dark matter particle, Γis the decay
+ratetophotons(dependingontheparticularmodelofdecayi ngdark
+matter),Mfov
+DMisthedarkmattermasswithintheinstrument’sfield-
+of-view (FoV) ΩfovandDLis the luminous distance towards the
+object.2
+Willman 1 signal. In LK09 the signal was extracted from
+the central 5′of the Willman 1 observation. Assuming the
+DM origin of this signal, the observed flux (3.53±1.95)×
+10−6photonscm−2s−1corresponds to FW1= (4.50±2.5)×
+10−8photonscm−2s−1arcmin−2flux per unit solid angle. The
+mass within the FoV of the observation of Willman 1 was esti-
+matedinLK09tobe Mfov
+W1= 2×106M⊙,usingthebest-fitvalues
+from Strigarietal. (2008). Taking the luminous distance to the ob-
+ject to be DL= 38kpc , the average DM column density of Will-
+man 1 is SW1≃208.5M⊙pc−2(see however discussion in the
+Section 5 below). To check the presence of a DM decay line, we
+should lookfor targets withcomparable signal.
+The signal-to-noise ratio for a weak line observed against
+a featureless continuum is given by (see e.g. discussion
+inBoyarsky etal. (2007a)):
+(S/N)∝ S/radicalbig
+texpAeffΩfov∆E (2)
+whereSis the average DM column density within an instrument’s
+field-of-view Ωfov,texpistheexposuretime, Aeffisaneffectivearea
+of the detector and ∆Eis the spectral resolution (notice that the
+lineinquestionhasamuchsmallerintrinsicwidththanthes pectral
+resolution of ChandraandXMM-Newton ). For the purpose of our
+estimate we consider the effective areas of XMM-Newton MOS
+cameras and Chandra ACIS-I (as well as their spectral resolution)
+to be approximately equal, and effective area of PN camera is 2
+timesbigger.3
+Galactic contribution. The contribution of the Milky Way’s
+DM halo along the line of sight should also be taken into accou nt,
+whenestimatingtheDMdecaysignal.4Usingapseudo-isothermal
+profile
+ρiso(r) =ρc
+1+r2/r2c, (3)
+2Eq.(1) isvalid for all objects whosesize is much smaller tha nDL.
+3Seee.g.http://xmm.esa.int/external/xmm_user_support/docume ntation/uhb/node32.html
+andhttp://cxc.harvard.edu/ccr/proceedings/07_proc/pres entations/drake/ .
+4Notice, that both in LK09 and in this work only instrumental ( particle)
+background has been subtracted fromthe observed diffuse sp ectra.
+c/circlecopyrt0000 RAS,MNRAS 000, 000–000Searchingfordarkmatterin X-rays 3
+ 0 100 200 300 400 500 600
+ 90  100  110  120  130SDM [MSun/pc2]
+φ [deg]Willman 1, LK09 M31on, M31B
+M31on, CorbelliM31off, M31B
+FornaxSculptor
+Figure 1. Comparison of DM column densities of the Milky Way (red
+solid line) with the density profiles of M31, Fornax and Sculp tor dSphs,
+discussed in the text. The DM column density estimate for Wil lman 1 is
+based on the “optimistic” DM density profile from Strigari et al. (2008),
+used in LK09 (see Section 5.1 for discussion). The angle φmarks the di-
+rection off Galactic center (for an object with galactic coo rdinates(l,b)
+cosφ= coslcosb).
+with parameters, adopted in Boyarsky et al. (2006b, 2007b) ( rc=
+4 kpc,ρc= 33.5×106M⊙/kpc3,r⊙= 8 kpc ), the cor-
+responding Milky Way column density in the direction of Will -
+man 1 (120.7◦off Galactic center) is 73.9M⊙pc−2. As demon-
+strated in Boyarsky et al. (2008b), the value of DM column den -
+sity computed with this DM density profile coincides with the
+best-fit Navarro-Frenk-White (Navarroetal. 1997, NFW) profiles
+of Klypinet al.(2002)and Battagliaet al.(2005) withinfew %for
+the off-center angle φ/greaterorsimilar90◦. Other dark matter distributions pro-
+duce systematically larger values of dark matter column den sity.
+For example, the best-fit pseudo-isothermal profile of Kerin s et al.
+(2001)wouldcorresponds toanadditionalMilkyWaycontrib ution
+∼119M⊙pc−2in the direction of Willman 1. The comparison
+between the DM column densities of several objects with that of
+MilkyWayintheirdirections isshown inFig.1.
+3 XMM-NEWTONDATA ANALYSIS
+For each XMM-Newton observation observation, we extracted
+bothEPICMOSandPNspectra.5Weselecteventswith PATTERN
+<= 12for MOS and PATTERN == 0 for PN camera and use
+FLAG == 0 . We used SAStaskedetect chainto exclude
+point sources, detected with the likelihood values above 10 (about
+4σ). For EPIC PN camera, bright strips with out-of-time (OOT)
+events were also filtered out, following the standard proced ure.6
+We imposed stringent flare screening criteria. To identify “ good
+time intervals” we performed a light-curve cleaning, by ana -
+lyzing temporal variability in the hard ( E >10 keV) X-ray
+band (as described e.g. in Read& Ponman 2003; Nevalainen et a l.
+2005; Carter &Read 2007). The resulting light curves were in -
+spected “by eye” and additional screening of soft proton flar es
+was performed by employing a method similar to that describe d
+in e.g. Leccardi &Molendi (2008); Kuntz & Snowden (2008) (i. e.
+by rejecting periods when the count rate deviated from a cons tant
+5WeuseSASversion9.0.0andXSPEC12.6.0 .
+6Asdescribedforexampleat http://xmm.esac.esa.int/sas/current/documentation/t hreads/EPIC_OoT.shtml .bymorethan 2σ).Theextractedspectrawerecheckedforthepres-
+enceofresidualsoftprotoncontaminantsbycomparingcoun trates
+inandout of FoV(DeLuca & Molendi 2004;Leccardi & Molendi
+2008).7In all spectra that we extracted the ratio of count rates in
+and out of FoV did not exceed 1.13 which indicates thorough so ft
+protonflare cleaning (c.f.Leccardi & Molendi 2008).
+Following the procedure, described in Carter &Read (2007)
+we extracted the filter-wheel closed background8from the same
+regionassignal(indetectorcoordinates). Thebackground wasfur-
+therrenormalizedbasedonthe E >10keVbandcountratesofthe
+extracted spectra. The resulting source and background spe ctra are
+grouped by atleast 50counts per bin using FTOOL(Irby, B.2008)
+commandgrppha.
+The extracted spectra with subtracted instrumental back-
+ground were fitted by the powerlaw model ofXSPECin the en-
+ergy range 2.1–8.0 keV data for MOS and 2.1–7.2 keV for the
+PN camera. This choice of the baseline model is justified by the
+fact that for all considered objects we expect no intrinsic X -ray
+emission above 2 keV and therefore the observed signal is dom -
+inated by the extragalactic diffuse X-ray background (XRB) with
+thepowerlawslope Γ = 1.41±0.06andthenormalization (9.8±
+1.0)×10−7photonscm−2s−1keV−1arcmin−2at 1 keV (90%
+CL errors) (see e.g. Lumbet al. 2002; De Luca & Molendi 2004;
+Nevalainen et al.2005;Hickox & Markevitch2006;Carter & Re ad
+2007; Moretti etal. 2009). Galactic contribution (both in a bsorp-
+tion and emission) is negligible at these energies and we do n ot
+take it into account in what follows. For all considered obse rva-
+tionspowerlaw model provided a very good fit, fully consistent
+withthe above measurements of XRB. The powerlaw index was
+fixed to be the same for all cameras (MOS1, MOS2, PN), observ-
+ingthesamespatialregion,whilethenormalizationwasall owedto
+vary independently to account for the possible off-axis cal ibration
+uncertainties between the different cameras, see e.g. Mate os et al.
+2009; Carteret al. 2010. The best-fit values of the powerlaw in-
+dexand normalizations arepresented inTable 1.
+To check for the presence of the LK09 line in the spectra, we
+added a narrow ( ∆E/E∼10−3) Gaussian line ( XSPECmodel
+gaussian )tothebaseline powerlaw model.AssumingtheDM
+originofthisline,wefixitsnormalizationatthelevel FW1,derived
+inLK09,multipliedbytheratiooftotal(includingMilkyWa ycon-
+tribution)DMcolumndensities andinstrument’s field-of-v iew. We
+varythe positionof the lineinthe interval corresponding t othe3σ
+interval of LK09: 2.30 – 2.72 keV. Eq. (2) allows to estimate t he
+expected significance of the line.
+When searching for weak lines, one should also take into ac-
+count uncertainties arising from the inaccuracies in the ca libra-
+tion of the detector response and gain. This can lead to syste m-
+aticresidualscaused bycalibrationinaccuracies, atthel evel∼5%
+of the model flux (some of them having edge-like or even line-
+like shapes) (see e.g. Kirschetal. 2002, 2004) as well as dis cus-
+sion in Boyarsky etal. (2008c) for similar uncertainties in Chan-
+dra. To account for these uncertainties we perform the above pro -
+cedure with the 5%of the model flux added as a systematic error
+(usingXSPECcommandsystematic ).
+7Weusethescript http://xmm2.esac.esa.int/external/xmm_sw_cal/backgr ound/Fin_over_Fout DeLuca &Molendi
+(2004)provided by the XMM-Newton EPIC Background working group.
+8Availableat http://xmm2.esac.esa.int/external/xmm_sw_cal/backgr ound/filter_closed/index.shtml
+c/circlecopyrt0000 RAS,MNRAS 000, 000–0004Boyarskyet al.
+ObsID PLindex PLnorm, 10−4ph.cm−2s−1keV−1)χ2/dof
+at1 keV (MOS1/MOS2 /PN)
+M31onobservations:
+0109270101 1.33 5.71 /5.80 /7.88 1328/1363
+0112570101 ... 5.37 /6.12 /6.86 ...
+0112570401 ... 6.13 /6.66 /6.37 ...
+M31off observation:
+0402560301 1.46 6.28 /6.12 /9.97 895/997
+M31out observations:
+0511380101 1.53 2.75 /2.88 /3.89 875/857
+0109270401 1.47 5.02 /4.41 /5.22 606/605
+0505760401 1.27 1.91 /2.04 /2.47 548/528
+0505760501 1.28 2.25 /1.85 /3.03 533/506
+0402561301 1.57 2.94 /3.10 /3.95 489/515
+0402561401 1.47 4.33 /3.58 /4.47 751/730
+0402560801 1.55 5.22 /4.49 /5.83 902/868
+0402561501 1.44 2.70 /4.29 /6.67 814/839
+0109270301 1.39 4.29 /4.58 /3.75 413/436
+Fornax observation:
+0302500101 1.52 4.46 /4.02 /3.78 938/981
+Table 1. Best-fit values of powerlaw index and normalization for each XMM-Newton observation analyzed in this paper. The systematic uncerta inty is
+included (see text). In M31onregion,powerlaw index for different observations is chosen to bethe same,be cause they point to thesamespatial region.
+4 ANDROMEDAGALAXY
+4.1 Dark matter contentof M31
+The Andromeda galaxy is the closest spiral galaxy to the
+Milky Way. Its dark matter content has been extensively stud -
+ied over the years (see e.g. Kerins etal. 2001; Klypin etal. 2 002;
+Widrow &Dubinski 2005;Geehan etal. 2006; Tempelet al.2007 ;
+Cheminet al.2009;Corbelli etal.2010,andreferencesther ein)for
+an incomplete list of recent works. The total dynamical mass (out
+to∼40kpc) can be determined from the rotation curve measured
+fromHIkinematics.Themajoruncertaintyindeterminationofdark
+matter content is then related to a separation of contributi ons of
+baryonic (stellar bulge and especially, extended stellar d isk) and
+dark components to the total mass.9The baryonic mass is often
+obtained from the (deprojected) surface brightness profile (optical
+or infrared), assuming certain mass-to-light ratio for the luminous
+matter inthe bulge andthe diskof a galaxy.
+Boyarsky etal. (2008a) ( B08in what follows) analyzed dark
+matter distributions of M31, existing in the literature at t hat time
+in order to provide the most conservative estimate of the exp ected
+DM decay signal. The DM column density Sin more than 10
+9Someworksalsotakeintoaccountapresenceofsupermassive blackhole
+in the center of a galaxy (see e.g Widrow &Dubinski 2005) and a dditional
+contributionofagaseousdisk(c.f.Chemin etal.2009;Corb elli et al.2010).
+Their relative contributions to the mass at distances of int erest turn out to
+benegligible and wedo not discuss them in whatfollows.models from the works (Kerins etal. 2001; Klypin etal. 2002;
+Widrow& Dubinski 2005;Geehan et al.2006;Tempel etal.2007 )
+turns out to be consistent within a factor of ∼2for off-center dis-
+tances greater than ∼1kpc (see gray lines in Fig.2). The most
+conservative estimate of dark matter column density was pro vided
+by one of the models of Widrow& Dubinski (2005) (calledin tha t
+workM31B), marked as redsolidline on Fig.2.
+Recently two new H Isurveys of the disk of M31 were per-
+formed (Cheminet al. 2009; Corbelli et al. 2010) and new data
+on mass distribution of M31 became available.10In Cheminet al.
+(2009) modeling of H Irotation curve between ∼0.3kpc and
+38 kpc was performed. Cheminet al. (2009) used R-bandphoto-
+metric information (Walterbos & Kennicutt 1987, 1988) to de ter-
+mine the relative contribution of the stellar disk and the bu lge.
+Based on this information and taking into account foregroun d
+and internal extinction/reddening effects (see section 8. 2.2 in
+Cheminet al. 2009, for details) they determined mass-to-li ght ra-
+tio, using stellar population synthesis models (Bellet al. 2003)
+Υdisk≃1.7Υ⊙(in agreement with those of Widrow& Dubinski
+(2005); Geehan et al. (2006)). Aiming to reproduce the data i n the
+inner several kpc, they explored different disk-bulge deco mposi-
+tions with two values of the bulge’s mass-to-light ratio Υbulge≃
+0.8Υ⊙andΥbulge≃2.2Υ⊙(all mass-to-light ratios are in solar
+units). Cheminet al. (2009) analyzed several dark matter de nsity
+10We thank A. Kusenko and M. Loewenstein in drawing our attenti on to
+these works (Kusenko & Loewenstein 2010).
+c/circlecopyrt0000 RAS,MNRAS 000, 000–000Searchingfordarkmatterin X-rays 5
+4x1014x103
+1x1021x103
+ 2 4 6 8 10 12 14 16 18 20 10 20 30 40 50 60 70 80DM column density [MSun/pc2]
+Off-center distance [kpc]Off-center distance [arcmin]
+Widrow Dubinski (2005), M31B
+Chemin et al. (2009), ISO
+Corbelli et al. (2009), rB = 28 kpc
+Maximum disk, Kerins et al.'00
+Figure 2. Dark matter column density of M31 as a function of off-
+center distance. The gray lines represent models from (Keri ns etal. 2001;
+Klypin et al. 2002; Kerins 2004; Widrow &Dubinski 2005; Geeh an et al.
+2006; Tempel et al. 2007) analyzed in B08. The red solid line r epresents
+the most conservative model M31B of Widrow & Dubinski (2005) . The
+model of Chemin et al. (2009) (red dashed line), the maximum d isk model
+of Kerins (2004) (blue dashe double-dotted line) and the “mi nimal” model
+of Corbelli et al. (2010) (red dotted line) (see text) are sho wn for compari-
+son.
+profiles(NFW,Einasto,coredprofile).Accordingtotheir6b est-fit
+models (3 DM density profiles times two choices of mass-to-li ght
+ratio) the DM column density in the Andromeda galaxy is highe r
+thanthe one, adapted inB08(model M31B) everywhere but insi de
+the inner 1kpc (see reddashed line onFig.2).
+Corbelli et al.(2010)usedcircularvelocitydata,inferre dfrom
+the HIrotation curves, also extending out to ∼37 kpc. The
+authors exclude from the analysis the inner 8 kpc, noticing t he
+presence of structures in the inner region (such as a bar), as soci-
+atedwithnon-circularmotions. Corbelliet al.(2010)also useopti-
+cal data of Walterbos & Kennicutt (1988) and determine an upp er
+and lower bounds on the disk mass-to-light ratio using the sa me
+models (Bell& de Jong 2001; Bellet al. 2003) as Cheminet al.
+(2009). They obtain possible range of values of the B-bandmass-
+to-light ratio 2.5Υ⊙≤Υdisk≤8Υ⊙. These values do not take
+intoaccountcorrectionsfortheinternalextinction.11Corbelliet al.
+(2010) fit the values of Υdiskrather than fixing it to a theoreti-
+cally preferred value. The mass-to-light ratio of the disk o f a spi-
+ral galaxy is known to be poorly constrained in such a procedu re
+sincethecontributionsofthediskandDMhaloaresimilar(s eee.g.
+discussions in Widrow &Dubinski (2005); Cheminet al. (2009 )).
+Corbelli etal. (2010) analyzed variety of DM models and mass -
+to-light ratios, providing good fit to the data. The mass mode l
+that maximizes the contribution of the stellar disk ( Υdisk= 8)
+has the core of the Burkert profile (Burkert 1995) rB= 28 kpc
+(if one imposes additional constraint on the total mass of M3 1
+within several hundred kpc). The DM column density remains i n
+this model remains essentially flat from the distance ∼rBin-
+ward(seeFig.2).Notice,thatthe“maximumdisk”fittingofK erins
+11The correction for foreground extinction was taken into acc ount
+in Corbelli et al. (2010), using the results of Seigar et al. ( 2008). The uni-
+formdisk extinction was notapplied (which explains high va lues ofΥdisk).
+The authors of Corbelli etal. (2010) had chosen not to includ e any uni-
+form extinction corrections due to the presence of a gradien t of B-R index
+(E.Corbelli, private communication).ObsID Cleaned exposure [ks] FoV [ arcmin2]
+(MOS1/MOS2 /PN) (MOS1 /MOS2 /PN)
+0109270101 16.8 /16.7 /15.3 335.4/336.0 /283.6
+0112570101 39.8 /40.0 /36.0 332.9/333.1 /285.9
+0112570401 29.8 /29.9 /23.5 335.6/336.1 /289.5
+Table 2. Cleaned exposures and FoV after the removal of point sources
+and OOT events (calculated using BACKSCAL keyword) of three M31on
+observations.
+(2004) (blue dashed double-dotted line on Fig.2) has higher DM
+content. Therefore in this work we will adopt the Burkert DM
+model of Corbelli et al. (2010) as a minimalpossible amount of
+dark matter, consistent with the rotation curve data on M31. The
+correspondingDMcolumndensityisfactor2–3lowerthanthe one
+given by the previously adapted model M31B in the inner 10kpc.
+Inwhat follows we will provide the restrictions for both M31 B (as
+themost conservative among physically motivated models of DM
+distributioninM31) and Burkert model of Corbelli etal. (20 10).
+4.2 M31central part
+Three observations of the central part of M31 (obsIDs
+0112570401 ,0109270101 and0112570101 ) were used in
+B08 to search for a dark matter decay signal. After the remova l of
+brightpointsources,thediffusespectrumwasextractedfr omaring
+with inner and outer radii 5′and13′, centered on M3112(see B08
+for details, where this region was referred to as ring5-13 ). We
+call these three observations collectively M31onin what follows.
+The baseline powerlaw model has the total χ21328 for 1363
+d.o.f.(reduced χ2= 0.974).
+Let us estimate the improvement of the signal-to-noise ra-
+tio(2),assumingtheDMoriginofLK09spectralfeature.The aver-
+ageDMcolumndensityinthemodelM31BofWidrow& Dubinski
+(2005)) is SM31on= 606M⊙pc−2. The dark mattercolumn den-
+sityfrom the Milky Wayhalo inthis directionis 74.1M⊙pc−2.
+The ratio of texp×Ωfov×Aeffof all observations of M31on
+(Table 2) and the LK09 observation is 12.9. Thus one expects the
+followingimprovement of the S/Nratio:
+(S/N)M31on
+(S/N)W1=606+74 .1
+208.5+73.9√
+12.90≈8.65,(4)
+i.e. the∼2.5σsignal of LK09 should become a prominent fea-
+ture (formally about ∼21.6σabove the background) for the
+M31onobservations. As described in Section 3 we add to the
+baseline powerlaw spectrum a narrow Gaussian line with the
+normalization fixed at FM31on=606+74 .1
+208.5+73.9FW1≈1.08×
+10−7photons cm−2s−1arcmin−2. The quality of fit becomes
+significantly worse (see Fig. 3). The increase of the total χ2due
+to the adding of a line is equal to (23.2)2,(22.9)2and(22.2)2for
+1σ,2σand3σintervals withrespect tothe central respectively.
+In addition to the M31onobservations, we processed an
+XMM-Newton observation 0402560301 , positioned ≈22′off-
+center M31 (RA = 00h40m47.64s, DEC = +41d18m46.3s) –
+M31off observation, see Table 3. We collected the spectra from
+12We adopt the distance to M31 DL= 784 kpc (Stanek &Garnavich
+1998)(at this distance 1′corresponds to 0.23kpc).
+c/circlecopyrt0000 RAS,MNRAS 000, 000–0006Boyarskyet al.
+ObsID Cleaned exposure [ks] FoV [ arcmin2]
+(MOS1 /MOS2/PN) (MOS1 /MOS2 /PN)
+0402560301 41.9 /42.2 /35.2 405.6/ 495.4 /433.3
+Table 3.Cleaned exposures and FoV (calculated using BACKSCAL key-
+word) of the observation M31off (obsID0402560301 ). The significant
+difference in FoVs between MOS1 and MOS2 cameras is due to the loss
+CCD6 in MOS1camera.
+the central 13′circle. Several point sources were manually ex-
+cluded from the source spectra. The rest of data reduction is de-
+scribed in the Section 4.2. The fit by the powerlaw model is
+excellent, the total χ2equals to 895 for 997 d.o.f. (the reduced
+χ2= 0.898). Using the DM estimate based on the model M31B
+we find that the average DM column density for the M31off
+SM31off= 388.6M⊙pc−2plus the Milky Way halo contribution
+74.3M⊙pc−2. The estimate of the line significance is similar to
+the previous Sectionandgives
+(S/N)M31off
+(S/N)W1=388.6+74.3
+208.5+73.9√
+8.76≈4.85 (5)
+and therefore, under the assumption of DM nature of the fea-
+ture of LK09, one would expect ∼12.1σdetection. After that,
+we add a narrow line with the normalization FM31off≈7.37×
+10−8photons cm−2s−1arcmin−2and perform the procedure,
+described inSection3.The observation M31off rules outthe DM
+decay line origin of the LK09 feature with high significance ( see
+Fig. 4): the increase of χ2due to the addition of this line has the
+minimum value ∆χ2≃(10.4)2at 2.44 keV, which is within 1σ
+interval of the quoted central value of the LK09feature.
+We also perform the analysis, adding a line, whose flux is
+determined according to the DM density estimates based on th e
+model of Corbelliet al. (2010), that we consider to be a minim al
+DM model for M31 (as discussed in 4.1). For this model the col-
+umn density in the central 1–3 kpcdecreases by a factor ∼3.4as
+comparedwiththevalue,basedonM31B.Kusenko & Loewenstei n
+(2010) claimed that in this case the LK09 line becomes consis tent
+with theM31onobservations. However, our analysis shows that
+the totalχ2increases, when adding the corresponding line to the
+model, by (5.7)2for3σvariation of the position of the line. The
+corresponding increase of total χ2forM31off region is (2.0)2.
+Combining M31onandM31off observations, one obtains (6.2)2
+increase of the total χ2.
+We conclude therefore that despite the uncertainties in DM
+modeling, the analysis of diffuse emission from the central part of
+M31 (1–8kpcoffthe center), as measured by XMM-Newton , dis-
+favors the hypothesis that the spectral feature, observed i n Will-
+man 1, is due to decaying dark matter. Nevertheless, to stren gthen
+this conclusion further we analyzed available XMM-Newton ob-
+servations of M31 in the region 10–20 kpcoff-center, where the
+uncertainties in the mass modeling of M31 reduce significant ly
+as compared with the central 5−8 kpc(c.f. Cheminet al. 2009;
+Corbelli etal.2010,see alsoSection4.1,inparticular Fig .2).
+4.3 M31off-center (10-20 kpc)
+We selected 9 observations with MOS cleaned exposure greate r
+than 20 ks (Table 4). The total exposure of these observation s is
+about 300 ks. Based on the statistics of these observations o ne
+would expect a detection of the signal of LK09 with the signif -
+icance12.1σ(for M31B) and 11.2σ(for Corbelli et al. (2010)).ObsID Cleaned exposure [ks] FoV [ arcmin2]
+(MOS1 /MOS2 /PN) (MOS1 /MOS2/PN)
+0511380101 44.3/ 44.5/37.6 356.2 /400.4 /333.3
+0109270401 38.5/ 38.4/33.5 387.8 /384.1 /366.7
+0505760401 26.0/ 26.0/21.7 330.5 /374.1 /325.0
+0505760501 23.8/ 23.8/19.8 344.8 /395.2 /313.2
+0402561301 22.7/ 22.7/20.1 367.2 /419.5 /382.6
+0402561401 39.3/ 39.3/33.7 349.9 /408.3 /343.6
+0402560801 42.6/ 42.6/36.6 373.6 /435.8 /393.7
+0402561501 38.5/ 38.5/34.0 369.3 /421.0 /407.5
+0109270301 24.4/ 24.6/22.2 443.4 /439.6 /405.0
+Table4.Cleaned exposures and FoVafter theremoval ofpointsources and
+OOTevents (calculated usingBACKSCALkeyword) ofnineM31o bserva-
+tions off-set by morethan 10 kpcfrom the center ( M31out observations).
+The fit to the baseline model is very good, giving total χ2equal to
+5931 for 5883 d.o.f. (the reduced χ2= 1.008). However, adding
+the properly rescaled line significantly reduced the fit qual ity, in-
+creasingthetotal χ2by(12.0)2/(10.7)2/(10.7)2whenvaryingline
+position within 1σ,2σand3σintervals (using M31B model) and
+by(11.7)2,(10.7)2and(10.6)2for1σ,2σand3σintervals(using
+the model of Corbelli etal. 2010, which gives DM column densi ty
+inM31out about160−180M⊙pc−2(includingMilkyWaycon-
+tribution),see Fig.2).
+4.4 Combinedexclusionfrom M31
+Finally, performing a combined fit to all 13 observations of M 31
+(M31,M31off,M31out) we obtain the exclusion of more than
+26σ(using the DM model M31B of Widrow & Dubinski (2005))
+and more than 13σfor the minimal DM model of Corbelli et al.
+(2010). As described in Section 3, in deriving these results we
+allowed the normalization of the baseline powerlaw model to
+varyindependently foreachcamera, observingthe samespat ialre-
+gion, added additional 5% of the model flux as a systematic unc er-
+tainty and allowed the position of the narrow line to vary wit hin
+2.3−2.72keVinterval.
+5 DWARFSPHEROIDALGALAXIES
+Since Willman 1 is purported to be a dwarf spheroidal (dSph)
+galaxy (but see section 5.1), as the next step in examining th e hy-
+pothesis of LK09 we compare estimates of the dark mass of Will -
+man 1 to dark masses estimated for other dSph satellites of th e
+Milky Way. Specifically we consider two of the (optically) br ight-
+est dSphs, Fornax and Sculptor, for which comparable X-ray d ata
+exist. To characterize the dark matter halos of dSphs, we ado pt the
+general DMhalo model of Walker etal. (2009a), withdensity p ro-
+filegiven by
+ρ(r) =ρs/parenleftbiggr
+rs/parenrightbigg−γ/bracketleftbigg
+1+/parenleftbiggr
+rs/parenrightbiggα/bracketrightbiggγ−3
+α
+, (6)
+where the parameter αcontrols the sharpness of the transition
+from inner slope limr→0dln(ρ)/dln(r)∝ −γto outer slope
+c/circlecopyrt0000 RAS,MNRAS 000, 000–000Searchingfordarkmatterin X-rays 7
+Figure3. Spectra ofthree XMM-Newton observations ofM31 ( M31on,Sec. 4.2).TheGaussian line, obtained byproper scaling of theresult of LK09isalso
+shown. The top row is for MOS1, the middle row – for MOS2, and th e bottom row for PN cameras. The spectra in the left column are for the observation
+0109270101 ,the middle column – 0112570101 and the right column – for 0112570401 .The error bars include 5%of the model flux as an additional
+systematic error. For all these spectra combined together, theχ2increases by at least22.22, when adding a narrow Gaussian line in any position between
+2.3keVand2.72 keV(seetext).
+limr→∞dln(ρ)/dln(r)∝ −3. This model includes as special
+cases both cored(α= 1,γ= 0) and NFW (Navarroetal. 1997)
+(α=γ= 1)profiles.AssumingthatdSphstarsaresphericallydis-
+tributedasmasslesstestparticlestracingtheDMpotentia l,theDM
+densityrelatestoobservables—theprojectedstellardens ityandve-
+locity dispersion profiles—via the Jeans Equation (see Equa tion
+3 of Walkeret al. (2009a)). Walker etal. (2009a) show that wh ile
+the data for a given dSph do not uniquely specify any of the hal o
+parameters individually, the bulk mass enclosed within the opti-
+cal radius is generally well constrained, subject to the val idity of
+the assumptions of spherical symmetry and dynamic equilibr ium
+and negligible contamination of stellar velocity samples f rom un-
+resolved binary-orbital motions.
+5.1 Willman1
+Despite some indications that Willman 1 is a dark-matter dom -
+inated dSph galaxy (Martinetal. 2007; Strigarietal. 2008) , it
+shouldbenotedthatthegalacticstatusofWillman1remains uncer-
+tain.Claims that Willman1is a bona fide galaxy—as opposed toastarclusterdevoidofdarkmatter—stemprimarilyfromtwoc onsid-
+erations. First,under simple dynamical models, the stella r velocity
+dispersion ( 4.3±2.5km s−1; Martinetal. (2007)) of Willman 1
+implies a large mass-to-light ratio ( M/LV∼700M⊙/LV,⊙), in-
+dicative of a dominant dark-matter component like those tha tchar-
+acterizeotherdSphgalaxies.Second,theinitialspectros copicstudy
+byMartin etal. (2007) suggests that Willman1 stars have a me tal-
+licity range −2.0≤[Fe/H]≤ −1.0, significantly broader than
+is observed in typical star clusters. However, both argumen ts are
+vulnerable to scrutiny. For example, if the measured veloci ty dis-
+persion of Willman 1 receives a significant contribution fro m un-
+resolved binary stars and/or tidal heating, then the inferr ed mass
+maybe significantlyoverestimated.Demonstrations thatbi naryor-
+bital motions contribute negligibly to dSph velocity dispe rsions
+have thus far been limited to intrinsically “hotter” system s, with
+velocity dispersions σV∼10km s−1(Olszewski etal. 1995;
+Hargreaves etal. 1996). Furthermore, given Willman 1’s low lu-
+minosity, kinematic samples must include faint stars close to the
+main-sequence turnoff(seeFigure8ofMartinet al.(2007)) .These
+stars are physically smaller than the bright red giants obse rved in
+c/circlecopyrt0000 RAS,MNRAS 000, 000–0008Boyarskyet al.
+Figure 4. The spectra of off-center M31 observation 0402560301 (M31off, Sec.4.3). The decaying dark matter signal, obtained by pro per scaling of the
+resultofLK09isalsoshown.Theerrorbarsinclude 5%ofthemodelfluxasanadditional systematic error.Fitting t hesespectratogether excludes theproperly
+scaled line of LK09,atthe level of at least 10.4σ(seetext).
+brighter dSphs, and thus admit tighter, faster binary orbit s. There-
+fore, the degree to which binary motions may inflate the esti-
+matedmassofWillman1remainsunclear.Finally,follow-up spec-
+troscopy by Siegelet al. (2008) suggests that the relativel y high-
+metallicity stars observed by Martinetal. (2007) are in fac t fore-
+ground contaminants contributed by the Milky Way along the l ine
+of sight to Willman 1. Removal of these stars from the sample
+would erase the large metallicity spread reported by Martin et al.
+(2008), and the remaining stars would have a narrow metallic ity
+distribution consistent with that of a metal-poor globular clus-
+ter(Siegelet al. 2008). Nevertheless, we proceed under the as-
+sumption that Willman 1 is indeed a dSph galaxy for which the
+measured stellar velocity dispersion provides a clean esti mate of
+the dark matter content. This assumption gives conservativ e esti-
+mates of the expected DM decay signal in other objects (inclu ding
+M31), since any overestimate of the DM column density in Will -
+man 1 will result in an underestimate of the relative S/N expe cted
+in other objects (see Eq. (2)). For Willman 1 we adopt the line -of-
+sight velocity data for 14 member stars observed by Martinet al.
+(2007). These stars, which lie a mean distance of 30 pcin projec-
+tion from the center of Willman 1, have a velocity dispersion of
+σV= 4.3±2.5km s−1. To specify the stellar surface density,
+we adopt a Plummer profile, I(R) =I0[1 +R2/r2
+half]−2, with
+parameters rhalf= 25 pc andI0= 0.5L⊙pc−2as measured by
+Martinet al.(2008).Ofcourse thesingle point inthe empiri calve-
+locitydispersion“profile”relatesnoinformationaboutth eshapeofthe profile, and these data therefore provide only weak const raints
+on the dark matter content of Willman 1. The solid black line i n
+the bottom-left panel of Figure 5 displays the median spheri cally-
+enclosed mass profile for the general halo model of Equation 6 ,
+with dotted lines enclosing the region corresponding to the central
+68%of posterior parameter space from the Markov-Chain Monte
+Carlo analysis (see Walker et al. (2009a) for a detailed desc ription
+of the MCMC procedure). The small published kinematic data s et
+(14stars)forWillman1allows foronlya single bininthe vel ocity
+dispersion profile (see the upper right panel in Fig. 5 and com pare
+ite.g. withFornaxdSph(theleftpanel)). Thisresultingco nstraints
+on the mass profile of Willman 1 are therefore very poor. The al -
+lowed masses span several orders of magnitude at all radii of in-
+terestand are consistent with negligible dark matter content—the
+lower bound ( 68%confidence interval) on the mass within 200pc
+(includingall ofthe Willman1) is aslow as 100M⊙. Strigariet al.
+(2008)obtainedforWillman1 M(<100pc) = 1 .3+1.5
+−0.8×106M⊙
+with the DM column density for the best-fit values of parame-
+ters being SW1∼200M⊙pc−2. The discrepancy with results of
+Strigariet al. (2008) may arise from two possible sources. F irst,
+Strigariet al. (2008) use an independent, still unpublishe d kine-
+matic data set (B. Willman et al., in preparation) for Willma n 1.
+Although the global velocity dispersion is statistically e quivalent
+to that which we measure from Martinet al. (2008) sample, the
+data set used by Strigarietal. (2008) is larger, containing 47 Will-
+man 1 members. Second, Strigariet al. (2008) adopt more stri n-
+c/circlecopyrt0000 RAS,MNRAS 000, 000–000Searchingfordarkmatterin X-rays 9
+Figure 5. DM mass modeling in Fornax and Willman 1. Top panels display e mpirical velocity dispersion profiles as calculated by Walk er etal. (2009a) for
+Fornax, Sculptor and from the Willman 1 data of Martin et al. ( 2007) (for Willman 1, the small published sample of 14 stars a llows for only a single bin).
+Overplotted are the median profiles obtained from the MCMC an alysis described by Walker etal. (2009a), using the general dark matter halo model (6), as
+wellasthebest-fitting NFWandcoredhalomodels.Bottompan elsindicate thecorresponding spherically-enclosed mass profiles.Inthebottompanels,dashed
+lines enclose thecentral 68%ofaccepted general models (forWillman 1,thelower bound fa lls outside theplotting window). Vertical dotted lines ind icate the
+outer radius of the field of view of the X-ray observations ( 14′for Fornax, 7.5′for Sculptor and 5′for Willman 1).
+gent priors motivated by cosmological N-body simulations, under
+whichρsandrsarestronglycorrelated(Bullock& Johnston2007;
+Diemand etal. 2007). We have used more general models here in
+order to help illustrate the uncertainties in the mass model ing of
+Willman 1. However, in the interest of obtaining the most con ser-
+vative estimates for the signal we expect to see from other ob jects
+if the LK09 detection is real, we shall continue to adopt one o f
+the largest available estimates of the dark matter column de nsity
+in Willman 1—namely, the value of SW1= 208.5M⊙pc−2that
+results from the modeling of Strigariet al.(2008).
+5.2 FornaxandSculptordSphs
+For Fornax and Sculptor, two of the brightest and most well-
+studied dSph satellites of the Milky Way, the availability o f large
+kinematic data sets of ∼2600and∼1400members, respec-
+tively(Walker etal. 2009b), allows us toplace relativelyt ight con-
+straints on the dark-matter column densities. Figure 5 disp lays the
+empirical velocity dispersion profiles, as well as the fits we obtain
+from the general (Eq.6),NFWand cored halomodels. Adopting a
+Fornax distance of ∼138kpc (Mateo 1998), we integrate the pro-Fornax Sculptor
+ρs[M⊙pc−3]rsρs[M⊙pc−3]rs
+NFW 0.1 500 0.04 920
+Core 0.57 280 0.48 350
+Table 5.Best-fit NFW ( α=γ= 1in Eq.(6)) and core profiles ( α= 1,
+γ= 0) for Fornax and Sculptor dSphs
+jected density profile out to a radius of 560 pc, which corresponds
+to the14′region (extraction region of XMM-Newton observation,
+see below) and is similar to the half-light radius of Fornax. For
+Fornax, the best-fitting density profile parameters are give n in Ta-
+ble 5.13For NFW density profile ( Vmax= 17.0kms−1) gives
+13Note that the fitting parameters ρsandrsare typically degenerate in
+the mass modeling of dSphs (Strigari etal. 2006; Penarrubia et al. 2008).
+Thus neither parameter is constrained uniquely. The best fit s considered
+here generally represent a “family” of models that follow a ρs,rsrelation
+consistent with the data, but which all tend to have the same b ulk mass
+enclosed within the optical radius (Pe˜ narrubia etal. 2009 ).
+c/circlecopyrt0000 RAS,MNRAS 000, 000–00010Boyarskyetal.
+SForn,NFW= 55.2M⊙pc−2and the best-fitting cored profile
+(Vmax= 17.4 kms−1) gives a slightly lower value SForn=
+54.4M⊙pc−2that we adopt for subsequent analysis.
+Adopting a Sculptor distance of 79 kpc (Mateo 1998), we
+integrate the projected density profile over a square of 1752pc2,
+which corresponds to the 7.6′2region (extraction region of Chan-
+draobservations on the ACIS-S3 chip, see below). The best-
+fit parameters for Sculptor are given in Table 5. The best-fitt ing
+cored profile ( Vmax= 19.8 kms−1) gives a DM column den-
+sitySScul,core= 147M⊙pc−2and the best-fitting NFW profile
+(Vmax= 19.4kms−1), givesSScul,NFW= 140.3M⊙pc−2and
+thatweadoptforsubsequent analysis.Noticethatusingthe param-
+etersofthe“universalDMdensityprofile”ofWalker etal.(2 009a),
+one would get only slightly higher values. In any case, the DM
+column densities of Fornax and Sculptor are much more tightl y
+constrained than that of Willman 1. The Fornax column densit y is
+smallerthantheexpectedcolumndensityofMilkyWayhaloin the
+direction of Fornax, which we estimate to be 83.2M⊙pc−2. The
+estimatedDMcolumndensityoftheMilkyWayinthe direction of
+Sculptor is 95.7M⊙pc−2(see Section2).
+5.3XMM-Newton observation of FornaxdSph
+Fornax is one of the most deeply observed dSphs in X-ray wave-
+lengths. We have processed the 100 ks XMM-Newton observation
+(ObsID0302500101 ) of the central part of Fornax dSph. The
+data analysis is described in the Section 3. In each camera, w e ex-
+tract signal from the 14′circle, centered on the Fornax dSph. The
+cleaned exposure and corresponding fields of view (calculat ed us-
+ingthe BACKSCALkeyword) for all three cameras (after subtr ac-
+tionofCCDgaps,OOTstringsinPNcamera andCCD6inMOS1
+camera) are shown in Table 6. Based on the DM estimates, pre-
+sented in the Section 5.2 we expect the following improvemen t in
+S/Nratio
+(S/N)Forn
+(S/N)W1=54.4+83.2
+208.5+73.9√
+12.12≈1.70 (7)
+i.e. we expect a ∼4.2σsignal from Fornax, assuming DM decay
+line originof the LK09feature.
+The fit to the baseline model is very good, χ2= 955
+for 983 d.o.f. (reduced χ2= 0.972). Note that the param-
+eters are consistent with those of extragalactic diffuse X- ray
+background as it should be for the dwarf spheroidal galaxy
+where we do not expect any diffuse emission at energies above
+2keV(see also Boyarsky et al. 2006b, 2007b; Jeltema & Profumo
+2008; Loewensteinet al.2009;Riemer-Sørensen& Hansen 200 9).
+After adding a narrow line with the normalization FForn≈
+2.19×10−8photonscm−2s−1arcmin−2, theχ2increases, the
+minimal increase is equal to (5.1)2,(4.2)2and(3.3σ)2for a posi-
+tion of the line within 1σ,2σand3σintervals from LK09 feature,
+respectively (after adding 5%of model flux as a systematic error).
+We see that adding a properly scaled Gaussian line worsens χ2by
+at least∼3.3σ(instead of expected improvement of the quality of
+fit by a factor of about 16if the LK09 feature were the DM decay
+line),see Fig.6for details.
+The combination of all XMM-Newton observations used in
+this work ( M31on,M31off,M31out andFornax) provide a
+minimum of 26.7σexclusion at 2.35 keV (which falls into the 3σ
+energy range for the LK09 spectral features), after adding a 5%
+systematic error. Within 1σand2σenergy intervals, the minimal
+increase of χ2is29.02and27.32,correspondingly. Therefore, one
+can exclude the DM origin of the LK09 spectral feature by 26σ.XMM-Newton Cleaned exposure [ks] FoV [ arcmin2]
+ObsID (MOS1 /MOS2 /PN) (MOS1/ MOS2/PN)
+0302500101 53.8/53.9 /48.2 459.1 /548.5 /424.9
+Table 6. Cleaned exposures of the XMM-Newton observation of Fornax
+dSph
+This exclusion is produced by using the most conservative DM
+model M31B of Widrow & Dubinski (2005). Using the minimal
+DMmodelofCorbelliet al.(2010),onecanexcludetheLK09fe a-
+ture atthe level more than 14σ.
+5.4Chandra observations of theSculptordSph
+5.4.1 The dataand datapreparation
+For the Sculptor dSph there is one observation of 50ks(ObsId
+9555) and 21 observations of 5−6ks(ObsID4698-4718 ),see
+Table 7 for details. In all the observations, the centre of Sc ulptor
+is on the ACIS-S3chip, and we have restricted the analysis to this
+chip.
+Throughout the entire analysis we used CIAO 4.1
+withCALDB 4.1 (Fruscione et al. 2006) and XSPEC 12.4
+(Arnaudet al. 2009). All the data were observed using the
+VFAINT telemetry mode and thus required reprocessing prior
+to any data analysis in order to take advantage of the improve d
+background event rejection based on 5×5pixel islands instead of
+the normal 3×3pixel.
+Weusedthewaveletalgorithm wavdetect inCIAOtofindand
+exclude point sources for each observation. The removed are as are
+very small compared to the field of view, and were consequentl y
+neglected in the following analysis. Furthermore the obser vations
+werelightcurvecleanedusing lcclean.Theobservations 4705and
+4712wereflaredandleftoutinthesubsequentanalysis.Thenum-
+berofpointsourcesremovedandtheexposuretimeafterligh tcurve
+cleaning are given in Tab. 7. The variation in the number of po int
+sources is mainly due to slightly different observed areas. The ob-
+servations are allcenteredonthe samespot, butthere isava riation
+in detector roll angles. Assuming spherical symmetry of Scu lptor,
+thishas noimportance for the further analysis.
+5.4.2 The spectra
+Thespectrawereextractedfromasquareregionof 7.6′×7.6′cov-
+ering most of the ACIS-S3chip but excluding the edges where t he
+calibration is less precise ( Chandra X-rayCentre 2009). For later
+fittingtheywhere binned toat least 15counts per bin.
+The spectra of the short observations ( 4698-4718 , except
+4705,4712)werecombinedintooneobservation,seeFig.7.This
+is justified because they fulfill the following criteria: The y are ob-
+servations of the same (non-variable) object and thus have s imilar
+count rates, they are extracted from the same region of the sa me
+chip, their rmf’s(response matrixfunctions) are extremel ysimilar.
+The spectra were added using the FTOOLmathpha(Irby, B. 2008)
+withthe uncertainties propagated as iftheywere pure Gauss ian.
+c/circlecopyrt0000 RAS,MNRAS 000, 000–000SearchingfordarkmatterinX-rays 11
+Figure 6. The spectra of the XMM-Newton observation 0302500101 , Sec. 5.2. The error bars include 5%of the model flux as an additional systematic
+error.Fitting thesespectratogether excludes theproperl y scaled LK09lineatthelevel ofatleast 2σ(3.3σifonerestricts theposition ofthelinetotheinterval
+2.30 – 2.72keV) instead of expected improving of the quality of fitby about 4σ(if theline wereof the DM origin).
+5.4.3 Background subtraction
+We subtracted only the particle background as observed with the
+ACIS detector stowed14. Around 2005 there was a change in the
+spectral shape of ACIS-S3 and consequently new particle bac k-
+ground files were produced (Markevitch, M. 2009). We used the
+new particle background files (2005-2009) for both spectra. The
+particle background was normalised in the 10−12 keVinter-
+val where any continuous emission from Sculptor is negligib le
+(Markevitch et al. 2003). For 9555the effect of changing the nor-
+malization interval to e. g. 4−6 keVis less than 3%. For the
+combined observations there is a larger excess at low energi es,
+which makes it more sensitive to the choice of normalization in-
+terval and consequently we chose the conservative approach with
+the10−12keVinterval.Thelargerexcessmightcomefrompoint
+sources, which are unresolved in the short exposures, but re solved
+14The files can be downloaded from
+http://cxc.harvard.edu/contrib/maxim/acisbg/and thus removed in the long exposure. The excess is mainly at
+energies below 2keVandare irrelevant tothe present analysis.
+The background subtracted spectra are shown in Fig. 7 with
+the combined observations in black and ObsID 9555in a lighter
+(green) color. Also shown is the best fit to a power law model an d
+the Gaussian expected according to the signal proposed by LK 09
+whichis further discussed below.
+5.4.4 Constraints
+The total exposure of Sculptor is 162.1 ks which is 1.63times
+longer than the Willman 1 observation. Sculptor is observed with
+the ACIS-S3 chip, which at 2.5 keVhas an effective area that is
+1.23timeslarger thanthat ofACIS-IwhichWillman1is observed
+with. The Sculptor spectra are extracted from a (7.6′)2square,
+whichis0.74ofthe5′circleforwhichtheWillman1spectrawere
+extracted.
+(S/N)Sc
+(S/N)W1=140.3+95.7
+208.5+73.9√
+1.63×1.23×0.74 = 1.02(8)
+c/circlecopyrt0000 RAS,MNRAS 000, 000–00012Boyarskyetal.
+00.010.020.03normalized counts s−1 keV−1
+5−202∆S χ2
+Energy (keV)
+Figure 7. The background subtracted Chandra spectra of Sculptor dSph with the short combined observatio ns in black and ObsID 9555in a lighter (green)
+color. Also shown the best fit power law model fitted to both obs ervations simultaneously (index fixed to 1.46) and a Gaussian corresponding to the expected
+DM line emission assuggested by LK09.Thedip in the residual s is avisual indication of the absence of any line signal at ∼2.5keV.
+ObsID Obsdate ExposureaPoint sourcesb
+4698 2004 Apr 26 6060 s 9
+4699 2004 May 7 6208 s 6
+4700 2004 May 17 6104 s 8
+4701 2004 May 30 6069 s 8
+4702 2004 Jun 12 5883 s 7
+4703 2004 Jun 27 5880 s 8
+4704 2004 Jul12 5915 s 9
+4705 2004 Jul24 — 6
+4706 2004 Aug 4 6075 s 8
+4707 2004 Aug 17 5883 s 8
+4708 2004 Aug 31 5880 s 8
+4709 2004 Sep 16 6091 s 10
+4710 2004 Oct 1 5883 s 8
+4711 2004 Oct 11 5727 s 9
+4712 2004 Oct 24 — 7
+4713 2004 Nov 5 6073 s 11
+4714 2004 Nov 20 5649 s 13
+4715 2004 Dec 5 5286 s 10
+4716 2004 Dec 19 6016 s 8
+4717 2004 Dec 29 6073 s 11
+4718 2005 Jan 10 6060 s 10
+shortobs combined — 113169 s —
+9555 2008 Sep 12 48935 s 21
+Allobs — 162104 s —
+Table 7.The analyzed Chandra observations of Sculptor Dwarf.aThe ex-
+posure times after lightcurve cleaning using lcclean.bThe number point
+sources found by wavdetect and removed.i.e.weexpect asignificance of 2.5σincaseofa DMsignal similar
+tothe one of LK09.
+Fitting apowerlaw to the two spectra simultaneously over
+theinterval 2.1−10keVgivesanindexof 1.54+0.35
+−0.18andanormali-
+sationof(1.22+0.74
+−0.29)×10−6photonskeV−1cm−2arcmin−2.The
+fitisexcellent with χ= 1135for1078 d.o.f.(reduced χ2=1.05).
+The ratio (Mfov
+DM/D2
+L)Scu= 1138M⊙/kpc2for Sculptor
+which is only 0.605 of the same ratio for Willman 1 (including
+the MilkyWaycontributioninbothcases) givinga fluxnormal iza-
+tion of2.03×10−6photonscm−2s−1. We add this Gaussian to
+the power law model varying the position over the energy inte rval
+2.1-2.8 keV. The quality of the fit worsen(instead of expected im-
+provement) with the lowest increase in χ2being 1141 at 2.3keV.
+At2.5 keVtheincreasedvalue is1144whichallowsustoexclude
+thepossibilityofa 2.5keVfeaturetobeadecaylineatthelevelof√1144−1135 = 3 σ.
+6 DISCUSSION
+This work provides more than 80times improvement of statistics
+of observations, as compared to LK09 (factor ∼5times larger to-
+tal exposure time and factor ∼4improvement in both the effec-
+tive area and the field-of-view). Even under the most conserv ative
+assumption about the DM column density such an improvement
+should have led to an about 14σdetection of the DM decay line.
+In our analysis no significant lines in the position, predict ed by
+LK09 were found. However, in the XMM-Newton observations
+that we processed there are several spectral features in the range
+2−3keV(seeTable8).Thisisnotsurprising,asthe XMM-Newton
+c/circlecopyrt0000 RAS,MNRAS 000, 000–000SearchingfordarkmatterinX-rays 13
+Observations Powerlaw index, PLnormalization Linepositi on Significance Line flux Line flux
+best-fit value best-fit value, [keV] best-fit value 3σupperbound
+10−6ph./(cm2sarcmin2keV) ph ./(cm2sarcmin2) ph./(cm2sarcmin2)
+M31on 1.32 1.66 /1.82 /2.39 2.55 1.7σ 7.35×10−92.09×10−8
+M31off 1.42 1.51 /1.19 /2.22 2.47 2.3σ 6.69×10−93.10×10−8
+M31out ... .../.../... 2.52 1.6σ 3.76×10−91.16×10−8
+Fornax 1.37 0.79 /0.59 /0.70 2.19 2.6σ 9.63×10−92.11×10−8
+Table 8. Parameters of the best-fit powerlaw components (assuming the powerlaw+gauss model) and of the maximally allowed flux in the narrow
+Gaussian line in the interval 2.1−2.8keV. Theslight difference between these values and those of Table 1 is due to the presence of gaussian component
+and the proportionality of powerlaw normalization to field-of-view solid angle (for M31on); for each observation, the corresponding parameters coin cide
+within 90% confidence range. The values of best-fit powerlaw parameters for M31out are not shown because they are different for different obser vations
+(seeTable 1).
+gold-coated mirrors have Au absorption edge at ∼2.21 keV
+(Kirschetal. 2002, 2004) and therefore their effective are as pos-
+sess several prominent features in the energy range 2−3 keV.
+According to XMM-Newton calibrationreport (Kirschet al.2002,
+2004), there is about 5% uncertainty at the modeling of the ef fec-
+tive area of both MOS and PN cameras at these energies. These
+uncertainties in the effective area can lead to artificialsp ectral fea-
+tures due to the interaction of the satellite with cosmic ray s.15In
+particular,thesolarprotons withenergiesoffewhundreds keV can
+be interpreted as X-ray photons (so called soft proton flares ). The
+interactionefficiencyofsolarprotons withtheinstrument isknown
+tobe totallydifferent from that of real photons (Kuntz & Sno wden
+2008). Inparticular, their fluxis not affected by the Auedge of the
+XMM-Newton mirrors. According to Kuntz & Snowden (2008),
+the spectrum of soft proton flares for EPIC cameras is well de-
+scribed by the unfolded (i.e. assuming that the instrument’s re-
+sponse is energy-independent) broken powerlaw model with t he
+break energy around ∼3.2keV,bknpow/b inXSPEC v.11 .
+Therefore, modeling the spectrum that is significantly cont ami-
+natedwithsoft protonflares ina standardway (i.e.byusing folded
+powerlaw model) will produce artificial residual excess at ener-
+gies of sharpdecreases of the instrument’s effective area.
+In our data analysis we performed both the standard
+flare screening (Read&Ponman 2003; Nevalainen et al. 2005;
+Carter& Read 2007) that uses the inspection of high-energy ( 10–
+15 keV) light curve and an additional soft proton flare cleani ng.
+As found by Pradas & Kerp (2005), it is possible to screen out
+the remaining flares, e.g. by the visual inspection of the cle aned
+lightcurve at low and intermediate energies. To provide the ad-
+ditional cleaning after the rejection of proton flares at hig h ener-
+gies, we followed the procedure of De Luca & Molendi (2004);
+Kuntz & Snowden (2008), leaving only the time intervals, whe re
+the totalcount rate differs from itsmeanvalue byless than 2σ.
+To test the possible instrumental origin of the features, di s-
+cussed in this Section, we first performed only a high-energy light
+curve cleaning (Read& Ponman 2003; Nevalainen et al. 2005;
+Carter& Read 2007) and determined the maximally allowed flux
+in a narrow line in the energy interval of interest. After tha t we
+performed an additional soft proton cleaning. This procedu re im-
+proved the quality of fit. The significance contours of the res ulting
+“line”(forM31out region)areshownontheleftpanelinFig.8.In
+deriving these limits we allowed both the parameters of the b ack-
+ground, the position of the line and its total normalization to vary
+(the latter from negative values). Finally, we added an unfo lded
+15Seee.g.http://www.star.le.ac.uk/ ˜amr30/BG/BGTable.html .broken powerlaw component bknpow/b to model a contribution
+of remaining soft proton flares (see e.g. Kuntz & Snowden 2008 ).
+The break energy was fixed as 3.2 keV so no degeneracy with the
+parameters of the gaussian is expected. The bknpow/b pow-
+erlaw indices at low and high energies were fixed to be the same
+fordifferentcamerasobservingthesamespatialregion.fu rtherim-
+provedthequalityoffitandmadethelinedetectiontotallyi nsignif-
+icant(see rightFig.8fordetails).Ouranalysis clearlysu ggests the
+instrumentaloriginoftheline-likefeaturesattheenergy range2.1–
+2.8keV.
+As a final note, we should emphasize that the Chandra ACIS
+instrument used in LK09 has the Ir absorption edges near 2.11
+and2.55keV(seeMirabal & Nieto2010foradditionaldiscuss ion).
+They also report negative result of searching for the LK09 fe ature,
+inagreement withour findings.
+7 CONCLUSION
+In this work we demonstrated that the DM decay line can
+be unambiguously distinguished from spectral features of o ther
+origin (astrophysical or instrumental) by studying its spa -
+cial distribution. Many DM-containing objects would pro-
+vide a comparable DM decay signal (Boyarsky etal. 2006b;
+Bertone et al. 2007; Boyarsky et al. 2009a) which makes such a
+study possible (cf. Boyarsky et al. 2009b; den Herder et al. 2 009;
+Riemer-Sørensen&Hansen 2009). If an interesting candidat e line
+is found in an object with an unusually high column density, t he
+differences (always within an order of magnitude) in column den-
+sitieswithother objectscanbe compensated bylonger expos ure or
+bigger grasp of observing instruments.
+To illustrate the power of this strategy, we have applied thi s
+approach to the recent result of LK09 who put forward a hypoth -
+esis about a decaying DM origin of a ∼2.51 keVspectral fea-
+ture in Chandra observations of the Milky Way satellite know n as
+Willman1.Although the parameters ofdecaying DM,correspo nd-
+ing to such an interpretation, would lie in the region, alrea dy ex-
+cluded at least 3σlevel by several existing works (Watsonet al.
+2006; Abazajianet al. 2007; Boyarsky etal. 2008a) we perfor med
+a new dedicated analysis of several archival observations o f M31,
+Fornaxand SculptordSphs.
+The exclusions provided by M31 are extremely strong (at the
+level of10−26σ, depending on the observed region). The 10σ
+bound is obtained using the off-center ( 10−20kpc) observations
+and a very conservative model of DM distribution with a heavy
+stellar disc and a very large ( ∼28 kpc) DM core (the model
+ofCorbelli et al.(2010),seeSection4.1).Suchamodelleav esDM
+c/circlecopyrt0000 RAS,MNRAS 000, 000–00014Boyarskyetal.
+2.3 2.35 2.4 2.45 2.5 2.55 2.6 2.65 2.70 10−82×10−8Line flux (photons/(cm2 sec arcmin2))
+Line energy (keV)Confidence contours: Chi−Squared
++xmin = 5.926310e+03; Levels =  5.928610e+03 5.930920e+03 5.935520e+03
+2.3 2.35 2.4 2.45 2.5 2.55 2.6 2.65 2.70 10−82×10−8Line flux (photons/(cm2 sec arcmin2))
+Line energy (keV)Confidence contours: Chi−Squared
+xmin = 5.868254e+03; Levels =  5.870554e+03 5.872864e+03 5.877464e+03
+Figure 8. Left:∆χ2= 2.3(dashed-dotted), 4.61(dashed) and 9.21(solid) contours (corresponding to 68%, 90% and 99% confiden ce intervals) for a thin
+gaussian lineallowed bythejointfitof M31out spectrausedinthiswork.Theresulting linenormalization s areshownforanaveraged DMcolumndensity
+in theM31out region (about 170M⊙pc−2, including the Milky Way contribution). The rescaled LK09 l ine is marked with a cross x; its normalization is
+therefore 4.50×10−8×170
+208.5+73.9= 2.71×10−8photons cm−2s arcmin−2.Itis clearly seen that the hypothesis of DM origin for aLK09 line-like
+feature is excluded atextremely high significance, corresp onding to morethan ≈10σ(seetext). Right:Thesameas inthe leftfigurebutwith an addition of a
+bknpow/b componenttomodelthecontribution oftheremaining softpr oton contamination (seetext). Thebreak energy isfixedas3. 2keVsonodegeneracy
+with the parameters of the gaussian is expected. The bknpow/b powerlaw indices at low and high energies are fixed to bethe sa mefor different cameras
+observing the samespatial region. Theline significance dro ps below 1σin this case.
+column density almost unchanged from about ∼30 kpcinward.
+Combiningall XMM-Newton observationsofM31atthedistances
+1–20 kpcoff-center provides more than 13σbound assuming the
+model by Corbelli etal. (2010) and the 26σbound with the model
+M31B of Widrow & Dubinski (2005) (with more physically moti-
+vated values of the mass of the stellar diskof M31).
+Fornax dSph provides an exclusion between 3.3σan5σ(de-
+pending on how far from its nominal value 2.51±0.07keV the
+position of the line is allowed to shift) instead of ∼4σdetection
+expectedifthespectralfeatureofLK09wasoftheDMorigin. The
+bound of 3.3σcorresponds to the interval 2.3−2.72keV and the
+5σexclusion corresponds to 2.44−2.58keV interval.
+To summarize, by comparing the strength and position of the
+featurefoundbyLoewenstein&Kusenkowithobservationsof sev-
+eral DM-dominated objects (M31, Fornax and Sculptor dSphs) we
+foundthatthehypothesisofdarkmatteroriginofLK09isexc luded
+withthecombinedsignificanceexceeding 14σevenunderthemost
+conservative assumptions. This is possible because of the l arge in-
+crease of the statistics in our observations as compared wit h the
+observation used inLK09.
+To change this conclusion, a number of extreme (and not oth-
+erwise motivated) systematic errors would need to be presen t in
+determination of DM content of M31, Fornax and Sculptor dSph .
+The DM origin of the spectral feature of LK09 would remain con -
+sistent with existing archival data only if the DM amount in M 31,
+Fornax and Sculptor are strongly overestimated and/or the m ass of
+Willman 1 is grossly underestimated even compared to the bes t-fit
+model of Strigarietal.(2008).Itshould be stressedthat in our cal-
+culations of the expected signal in other objects, we have ad opted
+for Willman 1 the DM column density that results from the mass
+modelingofStrigarietal.(2008).Ourownmodelingindicat esthat
+the mass of Willman 1 is highly uncertain. Even supposing tha t
+the measured velocity dispersion of Willman 1 provides a cle an
+estimate of the mass, the general models of Walker et al. (200 9a)
+allow the mass of Willman 1 to vary over several orders of magn i-
+tude withinthe region of interest. Furthermore, plausible scenariossuch as tidal heating and/or contamination from unresolved bina-
+ries in the kinematic sample would cause the Willman 1 mass to
+be over-estimated. We conclude that the column density deri ved
+from the modeling of Strigariet al. (2008) is approximately an up-
+per limit. In contrast, the large kinematic samples of Forna x and
+Sculptor provide tighter mass constraints under the assump tions
+of our modeling, and given the larger inherent velocity disp ersion,
+themodelingassumptionsaremoresecureforFornaxandScul ptor
+thantheyareforWillman1.Asaresult,weexpectthatsignifi cance
+of our exclusion of dark matter origin of LK09 feature to be ve ry
+conservative.
+Small line-like features (with the significance ∼2σ) are
+present in the XMM-Newton spectrum at energies between 2 and
+3 keV (see Fig. 8, left and Table 8 for details). Their most pro b-
+able cause is the interaction of the satellite with soft prot ons (see
+discussion in Section 6) which can lead to an appearance of ar tifi-
+cial spectral features due to the uncertainties in the effec tive area
+at these energies (c.f. Kirschet al. 2002, 2004). Indeed, th e signif-
+icance of the detection of these features drops below 1σwhen the
+residualcontaminationwithsoftprotonsismodeledbyanun folded
+broken powerlaw (see previous Sectionfor details).
+The search for decaying DM is a well-motivated task, having
+crucial importance not only for cosmology and astrophysics , but
+also for particle physics. X-ray astronomy has a significant poten-
+tialinthisrespect(Abazajian2009;den Herder et al.2009) .Dueto
+the small amount of baryons present in dwarf spheroidal gala xies,
+the uncertainties in DM determinations are smaller (as comp ared
+to e.g. spiral galaxies) and DM modeling provides tight boun ds
+on the amount of DM. However, the above-mentioned uncertain -
+ties make Willman 1 (or other ultra-faint dwarfs) to be bad ob -
+servational targets from the point of view of searching for d ecay-
+ing DM. On the contrary, the “classical” dSphs provide excel lent
+observational targets (Boyarsky et al. 2006b). A signal in F ornax,
+Sculptor (or other “classical” dSphs such as Draco or Ursa Mi nor
+not considered here because of the absence of archival obser va-
+tions for them) would therefore have the potential toprovid e much
+c/circlecopyrt0000 RAS,MNRAS 000, 000–000SearchingfordarkmatterinX-rays 15
+moreinformationabout thenatureofDMthanwouldasignalfr om
+Willman1. Toimprove significantly the best available bound s (see
+e.g. Boyarsky et al. 2009b) and to probe the most interesting re-
+gion of parameter space, an extended (300-500 ks) observati ons
+of the well-studied dSphs are needed. However as we are searc h-
+ing for the weak diffuse signal, a special care should be take n in
+choosing the instruments and observational times in such a w ay
+thatminimizepossibleflarecontamination(seee.g.Boyars ky et al.
+2007b). Drastic improvement in the decaying DM search is pos -
+sible with a new generation of spectrometers, having large fi eld
+of view and energy resolution close to several eV (Boyarsky e t al.
+2007a;Piroet al.2009;den Herder etal. 2009; Abazajian200 9).
+Acknowledgments.
+We are grateful to L. Chemin, E. Corbelli, M. Markevitch,
+S.Molendi, J.Nevalainen, M.Shaposhnikov formanyuseful c om-
+ments and discussions. DI is also grateful to Scientific and E duca-
+tionalCentreoftheBogolyubovInstituteforTheoreticalP hysicsin
+Kiev,Ukraine,andespeciallytoV.Shadura,forcreatingwo nderful
+atmosphere for young Ukrainian scientists. The work of AB an d
+OR was supported in part by the Swiss National Science Founda -
+tion.TheworkofDIissupportedinpartfromtheSCOPESproje ct
+No. IZ73Z0 128040 of Swiss National Science Foundation, the
+Cosmomicrophysics programme and the Program of Fundamenta l
+Research of the Physics and Astronomy Division of the Nation al
+Academy of Sciences of Ukraine. The Dark Cosmology Centre is
+funded by theDanish National ResearchFoundation.
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+arXiv:1001.0645v4  [math.AG]  13 Sep 2012A going down theorem for Grothendieck Chow motives
+Charles De Clercq∗
+Abstract. LetXbe a geometrically split, geometrically irreducible varie ty over a
+fieldFsatisfying Rost nilpotence principle. Consider a field exte nsionE/Fand a finite
+fieldF. We provide in this note a motivic tool giving sufficient condi tions for so-called
+outer motives of direct summands of the Chow motive of XEwith coefficients in Fto be
+lifted to the base field. This going down result has been used S . Garibaldi, V. Petrov and
+N. Semenov to give a complete classification of the motivic de compositions of projective
+homogeneous varieties of inner type E6and to answer a conjecture of Rost and Springer.
+Introduction
+Throughout this note Fwill be the base field and by an F-variety we will mean a smooth,
+projective scheme over F. Given an F-varietyX, we denote by Ch(X)the Chow group
+CH(X)⊗ZFof cycles on Xmodulo rational equivalence with coefficients in a finite field F.
+We write Ch(X)for the colimit of all Ch(XK), whereKruns through all field extensions
+K/Fand ifXis integral we denote by F(X)its function field.
+For any field extension L/F, an element lying in the image of the natural morphism of
+Ch(XL)−→Ch(X)is called L-rational. The image of any correspondence α∈Ch(XL)
+under the canonical morphism Ch(XL)−→Ch(X)is denoted by α. AnF-varietyX
+is geometrically split if the Grothendieck Chow motive of XF=X×Spec(F)Spec(F)with
+coefficients in Fis isomorphic to a finite direct sum of Tate motives, for an alg ebraic closure
+F/F. The variety Xsatisfies the Rost nilpotence principle with coefficients in Fif for
+any field extensions L/E/F the kernel of the restriction map resL/E: End(M(XE))−→
+End(M(XL))consists of nilpotents.
+As shown in [1], any projective homogeneous F-variety under the action of a semisimple
+affine algebraic group is geometrically split and satisfies th e Rost nilpotence principle. It
+follows by [2, Corollary 35] (see also [7, Corollary 2.6]) th at the Grothendieck Chow motive
+of these varieties with coefficients in Fdecomposes in an essentially unique way as a direct
+sum of indecomposable motives. The study of these decomposi tions have already shown
+to be very fruitful (see [6], [7], [11]).
+∗Address : Université Paris VI, 4 place Jussieu, 75252 Paris CEDEX 5.
+Keywords : Chow groups, Grothendieck motives, upper motives, project ive homogeneous varieties.
+Email : declercq@math.jussieu.fr
+1The notion of upper motives, previously defined by Vishik in the context of quadr ics
+in [11], was further developed by Karpenko in [7] to describe the indecomposable motives
+lying in the motivic decomposition of projective homogeneo us varieties. If Xis a ho-
+mogeneous F-variety, E/Fa field extension and the upper motive of M(XE)is a direct
+summand of another motive ME, [11, Theorem 4.15] and [6, Proposition 4.6] give suffi-
+cient conditions for the upper motive of Xto be a direct summand of M. The purpose
+of the present note is to push these ideas further. We define th e notions of upper,lower
+andouter direct summands of a direct summand Nof the motive of a geometrically split
+F-variety. We then show some lifting property of outer summan ds ofNEto the base field
+with the following result.
+Theorem 1. LetNbe a direct summand of the motive (with coefficients in F) of a geo-
+metrically split, geometrically integral F-varietyXsatisfying the Rost nilpotence principle
+with coefficients in FandMa twisted direct summand of the motive of another F-variety
+Y. Assume that there is a field extension E/Fsuch that
+1. every E(X)-rational cycle in Ch(X×Y)isF(X)-rational;
+2. the motive NEhas an indecomposable outer direct summand which is also a di rect
+summand of the motive ME.
+Then the motive Nhas an outer direct summand which is also a direct summand of M.
+Theorem 1 allows one to descend outer motives of direct summa nds projective ho-
+mogeneous varieties which appear on some field extension E/Fof the base field. This
+generalizes [6, Proposition 4.6], one of the key ingredient s in the proof of [6, Theorem
+1.1], replacing the whole motive of a variety Xby a direct summand. To replace Xby an
+arbitrary direct summand, one needs to construct explicitl y the rational cycles to get an
+outer summand defined over F, and thus theorem 1 gives a new proof of [6, Proposition
+4.6]. Note that assumption 1 of theorem 1 holds if the field ext ensionE(X)/F(X)is
+unirational, i.e. if there is a field extension L/E(X)such that L/F(X)is purely tran-
+scendental.
+The following particular case of theorem 1 was used by Gariba ldi, Petrov and Semenov
+in [5] to both determine all the motivic decompositions of ho mogeneous F-varieties of inner
+typeE6and prove a conjecture of Rost and Springer in [5].
+Corollary. ([5, Proposition 3.2]) Let XandYbe two projective homogeneous F-varieties
+for a semisimple affine algebraic group, and let MandNbe direct summands of the
+motives of YandXrespectively with coefficients in F. Assume that NF(Y)is an inde-
+composable direct summand of MF(Y)andYhas anF(X)-point. Then Nis a direct
+summand of M.
+Proof. SettingE=F(Y), the field extension E(X)/F(X)is purely transcendental, hence
+assumption 1)of theorem 1 holds.
+21 Grothendieck Chow motives
+Our main reference for the construction of the category of Gr othendieck Chow motives
+overFwith coefficients in Fis [3,§63-65].
+LetXandYbe twoF-varieties and X=/coproducttextn
+k=1Xkbe the decomposition of Xas
+a disjoint union of irreducible components with respective dimension d1,...,dn. For any
+integerithe group of correspondences between XandYof degree iwith coefficients in F
+is defined by Corri(X,Y) =/coproducttextn
+k=1Chdk+i(Xk×Y).We now consider the category C(F;F)
+whose objects are pairs X[i], whereXis anF-variety and iis an integer. Morphisms
+are defined in terms of correspondences by HomC(F;F)(X[i],Y[j]) = Corr i−j(X,Y). For
+any correspondences f:X[i]/squigglerightY[j]andg:Y[j]/squigglerightZ[k]inMor(C(F;F))the composite
+g◦f:X[i]/squigglerightZ[k]is defined by
+g◦f=/parenleftbigXpZ
+Y/parenrightbig
+∗/parenleftbig
+(X×YpZ)∗(f)·(pY×Z
+X)∗(g)/parenrightbig
+(∗)
+whereUpW
+V:U×V×W→U×Wis the natural projection.
+The category C(F;F)is preadditive and its additive completion CR(F;F)is the cat-
+egory of correspondences over Fwith coefficients in F, which has a structure of tensor
+additive category given by X[i]⊗Y[j] = (X×Y)[i+j]. The category CM(F;F)of
+Grothendieck Chow motives with coefficients in Fis the pseudo-abelian envelope of the
+category CR(F;F). Its objects are couples (X,π), whereXis an object of the category
+CR(F;Λ), andπ∈End(X)is a projector (i.e. π◦π=π). Morphisms are given by
+HomCM(F;F)((X,π),(Y,ρ)) =ρ◦HomCR(F;F)(X,Y)◦πand the objects of CM(F;F)are
+called motives . For any F-varietyXthe motives (X[i],ΓidX)(whereΓidXis the graph
+of the identity of X) will be denoted X[i]andX[0]is the motive of X. The motives
+F[i] = Spec( F)[i]are the Tate motives .
+Lemma 1. Let(X,π)be a direct summand of the motive of an F-varietyX. A motive M
+is a direct summand of (X,π)if and only if Mis isomorphic to (X,ρ), for some projector
+ρsatisfying π◦ρ◦π=ρ.
+Proof. SinceEnd((X,π)) =π◦Chdim(X)(X×X)◦π, any projector ρinEnd((X,π))
+satisfiesπ◦ρ◦π=ρ.
+Definition 1. LetM∈CM(F;F)be a motive and ian integer. The i-dimensional Chow
+groupChi(M)ofMis defined by HomCM(F;F)(F[i],M). Thei-codimensional Chow group
+Chi(M)ofMis defined by HomCM(F;F)(M,F[i]).
+For any field extension E/Fand any correspondence α:X[i]/squigglerightY[j]the pull-back of
+αalong the natural morphism (X×Y)E→X×Ywill be denoted αE. IfN= (X,π)[i]
+is a twisted motivic direct summand of X, the motive (XE,πE)[i]will be denoted NE.
+Finally the category CM(F;F)is endowed with a duality functor. If XandYare two
+F-varieties and α∈Ch(X×Y)is a correspondence, the image of αunder the exchange
+isomorphism X×Y→Y×Xis denotedtα. The duality functor is the additive functor †:
+CM(F;Λ)op−→CM(F;Λ)determined by the formula M(X)[i]†=M(X)[−dim(X)−i]
+and such that for any correspondence α:X[i]/squigglerightY[j],α†=tα.
+32 Direct summands of geometrically split F-varieties
+Throughout this section we consider a geometrically split F-varietyXandE/Fa splitting
+field ofX. By [8, Proposition 1.5] the pairing
+Ψ :Ch(XE)×Ch(XE)−→ F
+(α,β) /mapsto−→deg(α·β)
+is non degenerate hence gives rise to an isomorphism of F-modules between Ch(XE)and
+its dual space HomF(Ch(XE),F)given by α/mapsto→Ψ(α,·). The dual basis of a homogeneous
+basis(xk)n
+k=1ofCh(XE)with respect of Ψis the basis (x∗
+k)n
+k=1ofCh(XE)such that for
+any1≤i,j≤n,Ψ(xi,x∗
+j) =δij, whereδijis the Kronecker symbol. By definition of the
+composition ( ∗) inCM(F;F), ify(resp.y′) lies inCh(Y)(resp.Ch(Y′)) for two other
+F-varieties YandY′and if(i,j)are two integers, the composition of the correspondences
+xi×y∈Ch(XE×Y)andy′×x∗
+j∈Ch(Y′×XE)is given by
+(xi×y)◦(y′×x∗
+j) =δij(y′×y)∈Ch(Y′×Y). (1)
+Note that the Kunneth decomposition holds in Ch(XE×Y)andCh(Y′×XE)in the
+view of [3, Proposition 64.3], since XEis split, and thus the cycles of Ch(XE×Y)and
+Ch(Y′×XE)may always be written that way.
+Upper, lower and outer motives. Letπ∈Chdim(X)(X×X)be a non-zero projector
+andN=(X,π)the associated summand of the motive of X. The baseofNis the set
+B(N) ={i∈Z,Chi(NE)is not trivial }. The bottom ofN(denoted by b(N)) is the least
+integer of B(N)and the topofN(denoted by t(N)) is the greatest integer of B(N). We
+now introduce the notion of upper and lower direct summands o fN, previously introduced
+by Vishik in the context of the motives of quadrics in [11, Defi nition 4.6].
+Definition 2. LetNbe a direct summand of the twisted motive of a geometrically s plit
+F-variety and Ma motivic direct summand of N. We say that
+1.Misupper inNifb(M) =b(N);
+2.Mislower inNift(M) =t(N);
+3.Misouter inNifMis both lower and upper in N.
+Remark 1. Keeping the same F-varietyXand any direct summand N= (X,π), consider
+a homogeneous basis (xk)n
+k=1ofCh(XE)and its dual basis (x∗
+k)n
+k=1. The base, bottom and
+top ofNcan be easily determined by the decomposition
+πE=n/summationdisplay
+i,j=1πi,j(xi×x∗
+j)
+noticing that B(N) ={dim(xi), πi,j/ne}ationslash= 0for some j}.
+4Lemma 2. LetNbe a motivic direct summand of a geometrically split F-variety and
+Ma direct summand of N. ThenMis lower in N(resp. upper in N) if and only if the
+dual motive M†is upper in N†(resp.M†is lower in N†).
+Proof. For any motive Oand for any integer i,Chi(O†) = Ch −i(O). It follows that
+b(O†) =−t(O)andt(O†) =−b(O).
+The Krull-Schmidt property. LetCbe a pseudo-abelian category and Cbe the
+set of the isomorphism classes of objects of C. We say that the category Csatisfies the
+Krull-Schmidt property if the monoid (C,⊕)is free. The Krull-Schmidt property holds
+for the motives of geometrically split F-varieties satisfying the Rost nilpotence principle
+inCM(F;F)by [7, Corollary 2.6].
+Proof of the main result. In order to prove theorem 1, we will need the following
+lemma, which will allow us to construct explicitly the ratio nal cycles lifting outer motives
+to the base field.
+Lemma 3. LetNbe a motivic direct summand of a geometrically split, geomet rically
+irreducible F-varietyXsatisfying the Rost nilpotence principle and Ma twisted direct
+summand of an F-varietyY. Assume the existence of a field extension E/Fsuch that
+1. anyE(X)-rational cycle in Ch(X×Y)isF(X)-rational;
+2. there are two correspondences α:NE/squigglerightMEandβ:ME/squigglerightNEsuch that β◦αis
+a projector and (XE,β◦α)is a lower direct summand of NE.
+Then there are two correspondences γ:N/squigglerightMandδ:ME/squigglerightNEsuch that (XE,δ◦γE)
+is a direct summand of NEwhich contains all lower indecomposable direct summands of
+(XE,β◦α). Furthermore if βisF-rational, then δis alsoF-rational.
+Proof. We may assume by lemma 1 that M=(Y,ρ)[i]andN=(X,π). We construct ex-
+plicitly the two correspondences γandδ. SinceE(X)is a field extension of E,αisE(X)-
+rational, hence F(X)-rational by assumption 1. LetL/Fbe a field extension. Then the
+morphism Spec(F(XL))−→XLinduces a pull-back morphism ε∗: Ch(X×Y×X)−→
+Ch((X×Y)F(X))which maps F-rational cycles onto F(X)-rational cycles by [3, Corollary
+57.11], and so there is a cycle α1∈Ch(X×Y×X)such that ε∗(α1) =α. Sinceε∗maps
+any homogeneous cycle/summationtext
+ixi×yi×1to/summationtext
+ixi×yiand vanishes on homogeneous cycles
+whose codimension on the third factor is strictly positive, we haveα1=α×1+···where
+”···”is a linear combination of homogeneous cycles in Ch(X×Y×X)with strictly
+positive codimension on the third factor.
+We now look at α1as a correspondence X/squigglerightY×Xand consider the cycle α2=α1◦π.
+By formula 1 we have
+α2= (α×1)◦π+···,
+where”···”is a linear combination of homogeneous cycles in Ch(X×Y×X)with di-
+mension at most t(N)on the first factor (since these terms come from the first facto rs
+5ofπ) and strictly positive codimension on the third factor (sin ce these terms come from
+the third factors of α1−α×1). Finally considering the pull-back of the morphism
+∆ :X×Y→X×Y×Xinduced by the diagonal embedding Xand setting α3= ∆∗(α2),
+we have
+α3=α◦π+···
+where”···”stands for a linear combination of homogeneous cycles in Ch(X×Y)with
+dimension strictly lesser than t(N)on the first factor.
+Composing with π◦βon the left and πon the right, we get that
+π◦β◦α3◦π=π◦β◦α◦π+ξ
+whereξis a linear combination of homogeneous cycles of strictly le sser dimension than
+t(N)on the first factor since they come from the first factors of α3◦π−α◦π. The
+correspondence β◦αdefines a direct summand of the motive NEand thus by lemma 1
+π◦β◦α3◦π=β◦α+ξ.
+By formula 1, β◦α◦ξ,ξ◦ξandξ◦β◦αare linear combinations of homogeneous cycles
+of dimension strictly lesser than t(N)on the first factor. Repeating the same procedure
+and since k◦his a projector, we see that for any integer n
+/parenleftbig
+π◦β◦α3◦π/parenrightbign=β◦α+··· (2)
+where”···”is a linear combination of homogeneous cycles in Ch(X×X)with dimen-
+sion on the first factor strictly lesser than t(N). Since the direct summand (X,β◦
+α)is lower, all these correspondences are non-zero and by [7, C orollary 2.2] an ap-
+propriate power (πE◦β◦(α3)E◦πE)◦n0is a projector. If we set γ=ρ◦α3◦πand
+δ=(πE◦β◦(α3)E◦πE)◦n0−1◦πE◦β, we see that δisF-rational if βisF-rational. The
+correspondence δ◦γEis a projector which defines a direct summand of NEby lemma 1.
+Consider the decomposition β◦α=/summationtexts
+i,j=1pij(xi×x∗
+j)ofβ◦αwith respect to a basis
+(xi)s
+i=1ofCh(X). By formula 2, the decomposition of δ◦γin(xi×x∗
+j)s
+i,j=1has a non-zero
+coefficient for any couple (i,j)such that pijis non zero and dim(xi) =t((XE,β◦α)). The
+Krull-Schmidt property and remark 1 then imply that any lowe r indecomposable direct
+summand of (XE,β◦α)is a direct summand of (X,δ◦γE).
+We now show how we can derive the proof of theorem 1 from the rat ional cycles
+constructed in lemma 3. To lift the outer motive to the base fie ldF, we apply lemma 3
+and the duality functor twice in order produce two correspon dences which are defined on
+the base field.
+Proof of Theorem 1. LetO=(XE,κ)be an outer indecomposable direct summand of NE
+which is also a direct summand of ME. We prove theorem 1 by applying lemma 3 once,
+then the duality functor and finally lemma 3 another time to ge t all our correspondences
+defined over the base field F.
+6SinceOis a direct summand of ME, there are two correspondences α:NE/squigglerightME
+andβ:ME/squigglerightNEsuch that β◦α=κ. Moreover Ois lower in NE, so lemma 3 justifies
+the existence of two other correspondences α′:N/squigglerightMandβ′:ME/squigglerightNEsuch that
+O2= (XE,β′◦α′
+E)is a direct summand of NE, and the motive O2is outer in NEsince it
+contains O. The dual motive O†
+2= (XE,tα′
+E◦tβ′)[−dim(X)]is therefore outer in N†
+Eby
+lemma 2 and is a direct summand of the dual motive M†
+E. Twisting these three motives
+bydim(X), we can apply lemma 3 again. The correspondencetα′isF-rational, so lemma
+3 gives two correspondences γ:N†/squigglerightM†andδ:M†/squigglerightN†such that the motive
+(XE,δE◦γE)is both an outer direct summand of N†(since it contains the dual motive
+O†) and a direct summand of M†. Transposing again, the motive (X,tγ◦tδ)is an outer
+direct summand of Nand a direct summand of M.
+3 Motivic decompositions for groups of inner type E6
+The purpose of this section is to discuss the complete classi fication of the motivic decom-
+positions of projective homogeneous varieties of inner typ eE6, which is achieved in [5].
+LetGbe an algebraic group of inner type E6andXa projective G-homogeneous variety.
+We choose the following numbering of the Dynkin diagram G.
+s
+s s s s s
+1 3 4 5 62
+The results of [9] show that in the case where Xisgenerically split , any indecomposable
+summand of the Fp-motive of Xis isomorphic to a shift of the upper motive Rp(G)of
+the variety of Borel subgroups of G. Furthermore, the structure of the motives Rp(G)is
+determined in [9] in terms of the so-called J-invariant modulo pofG.
+TheJ-invariant was first introduced by Vishik in [12] in the conte xt of quadratic
+forms. Petrov, Semenov and Zainoulline define in [9] the noti on ofJ-invariant modulo
+pof an arbitrary semisimple algebraic group G, denoted by Jp(G), which is an r-tuple
+of integers (j1,...,jr)given by the rational cycles in Ch(G). By [9, Table 4.13], the J-
+invariant modulo 3of a semisimple adjoint algebraic group of inner type E6is(j1,j2),
+with0≤j1≤2and0≤j2≤1.
+Another invariant attached to Gis the Tits index, which consists of the data of the
+Dynkin diagram of Gwith some vertices being circled. The complete classificati on of the
+Tits indices of type E6, provided in [10], is as follows :
+r ❡
+r r r r r❡ ❡ ❡ ❡ ❡r ❡
+r r r r r ❡r
+r r r r r❡ ❡r
+r r r r r
+LetΘbe a subset of the vertices of the Dynkin diagram of GandXΘa projective
+G-homogeneous variety of type Θ. By [9, Table 3.6] and a case by case anylisis of the
+above Tits indices, the variety XΘis either split or generically split if p/ne}ationslash= 3,Θ/ne}ationslash={2},{4}
+7or{2,4}and ifj1= 0. Finally, the results of [1] and [5, §8] imply that the unders tanding
+of the motivic decompositions of X{2},X{4}, andX{2,4}is reduced to the study of the
+upper motive of X2inCM(F;F3), which is denoted by Mj1,j2.
+Using theorem 1, Garibaldi, Petrov and Semenov provide some restrictions on the
+Poincaré polynomial of the upper motive of X, ifXis an anisotropic projective homoge-
+neous variety satisfying some technical assumptions (see [ 5, Proposition 7.6]). Assuming
+thatJ3(G)=(1,0), they observe that although the variety X{2}satisfies all those techni-
+cal assumptions, the Poincaré polynomial of its upper motiv e does not match with the
+conclusion of [5, Proposition 7.6]. In particular X{2}has a0-cycle of degree coprime to 3,
+andM1,0is the Tate motive F3.
+Furthermore, the authors deduce from the fact that M1,0is the Tate motive that the J-
+invariant modulo 3ofGcannot be (2,0)(see [5, Corollary 8.10]). Indeed, if J3(G)=(2,0)
+andSB(3,A)is the Severi-Brauer variety of right ideals of reduced dime nsion3in the
+Tits algebra of G, thenJ3(GF(SB(3,A)))=(1,0). In particular, X2has a zero-cycle of degree
+coprime to 3over the function field of SB(3,A). It follows that the upper motive of X2
+would be isomorphic to the upper motive of SB(3,A), and thus the canonical 3-dimensions
+ofX2andSB(3,A)would be equal, a contradiction.
+The authors use similar techniques to provide isotropy crit eria for projective homoge-
+neous varieties. They consider several varieties which sat isfy the technical assumptions of
+[5, Proposition 7.6] without fulfilling its conclusion, and thus have a zero cycle of degree 3
+(or a rational point). These examples include projective ho mogeneous varieties for orthog-
+onal group with application to the isotropy of varieties of t ypeE7and varieties of type E8
+(see [5, Lemma 10.15,10.21]). They also produce with these t echniques isotropy criteria
+for projective homogeneous varieties in terms of the Rost in variant (see [5, Proposition
+10.18] for type E7and [5, Propositions 10.22] for type E8).
+Acknowledgements : I am grateful to N. Karpenko for raising this question and for
+his suggestions. I also would like to thank N. Semenov for the very useful conversations.
+References
+[1]Chernousov, V., Gille, S., Merkurjev, A. Motivic decomposition of isotropic projective homo-
+geneous varieties , Duke Math. J., 126(1) :137-159, 2005.
+[2]Chernousov, V., Merkurjev, A. Motivic decomposition of projective homogeneous varieties and
+the Krull-Schmidt theorem , Transformation Groups 11, 371-386, 2006.
+[3]Elman, R., Karpenko, N., Merkurjev, A. The Algebraic and Geometric Theory of Quadratic
+Forms , AMS Colloquium Publications, Vol. 56, 2008.
+[4]Fulton, W. Intersection theory , Springer-Verlag, 1998.
+[5]Garibaldi, S., Petrov, V., Semenov, N. Shells of twisted flag varieties and non-decomposibility
+of the Rost invariant , preprint (2010), available on the webpage of the authors.
+[6]Karpenko, N. Hyperbolicity of orthogonal involutions , Doc. Math. Extra Volume: Andrei A. Suslin’s
+Sixtieth Birthday (2010), 371–392, with an appendix of J-.P . Tignol
+8[7]Karpenko, N. Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties ,
+to appear in J. Reine Angew. Math.
+[8]Merkurjev, A. R-equivalence on three-dimensional tori and zero-cycles , Algebra and Number The-
+ory 2, 69-89, 2008.
+[9]Petrov, V., Semenov, N. Zainoulline, K. J-invariant of linear algebraic groups , Ann. Sci. Éc.
+Norm. Sup. 41, 1023-1053, 2008.
+[10]Tits, J. ,Classification of algebraic semisimple groups , Algebraic Groups and Discontinuous Sub-
+groups, Amer. Math. Soc., Providence, RI, 1966.
+[11]Vishik, A. Motives of quadrics with applications to the theory of quadrat ic forms , Lect. Notes in
+Math. 1835, Proceedings of the Summer School "Geometric Met hods in the Algebraic Theory of
+Quadratic Forms, Lens 2000", 25-101, 2004.
+[12]Vishik, A. On the Chow groups of Quadratic Grassmannians , Documenta Mathematica 10, 111-130,
+2005.
+9
\ No newline at end of file
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+arXiv:1001.0646v2  [hep-ph]  30 Mar 2010EPJ manuscript No.
+(will be inserted by the editor)
+Primakoff effect in η-photoproduction off protons
+A. Sibirtsev1,2,3, J. Haidenbauer3,4, S. Krewald3,4and U.-G. Meißner1,3,4
+1Helmholtz-Institut f¨ ur Strahlen- und Kernphysik (Theori e) und Bethe Center for Theoretical Physics, Universit¨ at B onn,
+D-53115 Bonn, Germany
+2Excited Baryon Analysis Center (EBAC), Thomas Jefferson Nat ional Accelerator Facility, Newport News, Virginia 23606,
+USA
+3Institut f¨ ur Kernphysik and J¨ ulich Center for Hadron Phys ics, Forschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany
+4Institute for Advanced Simulation, Forschungszentrum J¨ u lich, D-52425 J¨ ulich, Germany
+Received: date / Revised version: date
+Abstract. We analyse data on forward η-meson photoproduction off a proton target and extract the η→γγ
+decay width utilizing the Primakoff effect. The hadronic ampl itude that enters into our analysis is strongly
+constrained because it is fixed from a global fit to available γp→pηdata for differential cross sections
+and polarizations. We compare our results with present info rmation on the two-photon η-decay from the
+literature. We provide predictions for future PrimEx exper iments at Jefferson Laboratory in order to
+motivate further studies.
+PACS.11.55.Jy Regge formalism – 13.60.Le Meson production – 13.6 0.-r Photon and charged-lepton
+interactions with hadrons
+The decay of the light pseudoscalar mesons into two
+photonsis relatedtosymmetry breakingthroughthe axial
+vector anomaly and reveals one of the fundamental prop-
+erties of QCD [1,2,3,4,5,6,7,8]. The Adler-Bell-Jackiw
+(ABJ) anomaly [1,2] allows to determine the decay con-
+stant of those pseudoscalar mesons from the two-photon
+decay width (invoking a smooth extrapolation from the
+chiral limit to the physical light quark masses). While this
+is indeed feasible for the π0-meson, the situation is more
+complicated for the η-meson since the extrapolation from
+theη-masstozeroandthedominanceofthe ABJanomaly
+is debatable. Moreover, for the ηandη′mesons the decay
+constants [9] alone do not determine the two-photon de-
+cay widths uniquely. It is necessary to know in addition
+the singlet-octet mixing angle.
+In an extension of chiral perturbation theory including
+theη′meson – from now on called the Extended Effective
+Theory (EET) – the evaluation of the η→γγdecay width
+requiresapropermixingofthe SU(3)pseudoscalarsinglet
+and octet states so that [10]
+Γ(η→γγ) =α2m3
+η
+96π3/bracketleftBigg
+cosθP
+f8η−√
+8sinθP
+f0η/bracketrightBigg2
+.(1)
+Here,f0
+ηandf8
+ηare the singlet and octet decay constants,
+θPis the mixing angle, and αis the fine structure con-
+stant. To estimate the η→γγradiative decay width we
+take from Ref. [11] the following set of SU(3) param-
+eters:f0
+η=(1.17±0.03)fπ0,f8
+η=(1.26±0.04)fπ0andθP=
+−(21.20±1.60). The pion decay constant has the valuefπ0=130±5 MeV and the η-meson mass is given by the
+PDG [12] as mη=547.853±0.024 MeV. With these param-
+eters the η→γγdecay width can be estimated to be
+0.39≤Γ(η→γγ)≤0.52 keV. (2)
+Alternatively, assuming that f0
+η=f8
+η=fπ0one can, in prin-
+ciple, extract the SU(3) mixing angle from the η→γγde-
+cay width [10].
+Experimentallythe η→γγdecaywidthwasdetermined
+through the QED process e+e−→e+e−ηand also from
+measurementsofthe Primakoffeffect with nucleartargets.
+The results from these two classes of experiments are in
+conflict [12]. While all of the QEDresults [13,14,15,16,17,
+18] are in line with Eq. (2) within the experimental error,
+the Primakoff measurements [19,20] agree with the EET
+only within three times the experimental error bars, say.
+It was argued [21] that most of the uncertainties in the
+evaluation from the Primakoff effect are due to the nu-
+clear response. Recently it was shown [22] that the η→γγ
+results might depend significantly on contributions from
+incoherent η-meson photoproduction off nuclei.
+The Primakoff measurements [23] on a proton target
+are not included in the list of the PDG [12,21], although
+the quality of these data, collected at DESY, is compa-
+rable or even better than the results obtained with nu-
+clear targets [19,20]. These data were considered [24] as a
+strongmotivation for further η→γγPrimakoffstudies [25]
+at JLab, however, so far they were not used explicitely for
+an extraction of the two-photon decay width.2 A. Sibirtsev et al.: Primakoff effect in η-photoproduction off protons
+Fig. 1. Differential cross section for the reaction γp→pηas
+a function of the four-momentum transfer squared at differen t
+photon energies Eγor invariant collision energies√s. Data are
+from ELSA-Bonn[29] (filled triangles), the DaresburyElect ron
+Synchrotron [30] (open circles), Cornell [26] (inverse tri angles),
+DESY [23] (open squares), SLAC [31] (open triangles), and
+CLAS at JLab [32] (filled squares). The lines are the results o f
+our model calculation.
+In the present paper we utilize the data [23,26] avail-
+able for forward η-meson production in the γp→pηreac-
+tion at photon energies of 4 GeV and 6 GeV in order
+to extract the η→γγdecay width. To isolate the Pri-
+makoff effect from the hadronic contribution we use the
+Regge amplitudes established in our previous study [27] of
+theγp→pπ0reaction. The free parameters of the helicity
+amplitudes, namely coupling constants and form factors,
+were fixed by a global fit to η-photoproduction data. In
+the present study we compare our results to some γp→pη
+data on differential cross sections and polarization avail-
+able at high energies. A more thorough comparison to ex-
+perimental results and predictions at photon energies be-
+low 3 GeV, together with more detailed information on
+the used model parameters, will be given elsewhere [28].
+Fig. 1 shows γp→pηdifferential cross sections at pho-
+ton energies from 3 GeV to 5.5 GeV. The different mea-
+surements available at the same energy are in good agree-
+mentwitheachother,consideringtheiruncertainties,apart
+from the recent CLAS results [32]. Actually, there are sys-
+tematic discrepancies between the ELSA-Bonn data pub-
+lished in 2005 [29], in 2008 [33] and in 2009 [34], and the
+2009 data from CLAS [32]. A detailed discussion of these
+discrepancies will be given in Ref. [28].
+Differential cross sections for η-photoproduction at en-
+ergies from 6 to 9 GeV are displayed in Fig. 2. The data
+from the different measurements are described rather well
+by our model calculation.
+Data for beam and target asymmetries are presented
+in Fig. 3 as a function of the four-momentum transferFig. 2. Differential cross section for the reaction γp→pηas
+a function of the four-momentum transfer squared at differen t
+photon energies Eγor invariant collision energies√s. Data are
+from Cornell [26] (inverse triangles), DESY [23] (open squa res)
+and SLAC[31] (open triangles). The lines are the results of o ur
+model calculation.
+squared. Those measurements [30,35] were performed at
+the Daresbury Laboratory. It is known [30] that polariza-
+tion data constitute a rather crucial test for models based
+on Regge phenomenology. Indeed, all model calculations
+[36,37] available at the time when the experiments were
+published failed to reproduce the data on the beam asym-
+metry. The most recent analysis [22] of the γp→pηreac-
+tion is based on a Regge model proposed in 1970 [23] and
+considersonlydifferentialcrosssectionsatphotonenergies
+above 4 GeV, but no polarization data. Furthermore, the
+ρ-trajectoryused in Refs. [22,23] is very different from the
+onedeterminedinourglobalanalysis[38]ofthe π−p→π0n
+reaction and from total cross sections for various other re-
+actions [39]. The lines in Fig. 3 are our results which are
+Fig. 3. The beam (filled squares) and target (open squares)
+asymmetries in photoproduction of η-mesons from protons
+measured [30,35] at the Daresbury Laboratory. The lines are
+the results of our model calculation.A. Sibirtsev et al.: Primakoff effect in η-photoproduction off protons 3
+clearly in good agreement with the data on beam and tar-
+get asymmetries.
+Obviously our Regge model reproduces the ηphoto-
+production data on differential cross sections and polar-
+izations availableat photon energiesabove ≃3 GeV rather
+well. We take that asconfirmation for the hadronicpart of
+the reaction amplitude to be reliably determined so that
+it can be used with confidence in the evaluation of the
+Primakoff effect.
+The Primakoff effect [40] is due to the one-photon ex-
+change (OPE) contribution to neutral meson photopro-
+duction. The OPE amplitude FPis proportional to the
+two photon decay width of the η-meson and given by
+FP=8mp
+t/radicalBigg
+πΓ(η→γγ)
+m3ηFD(t)Fηγγ∗(t),(3)
+wherempandmηare the proton and η-meson masses,
+respectively, and Γis theη→γγdecay width. FDand
+Fηγγ∗are form factors at the ppγ∗andηγγ∗vertices, re-
+spectively ( γ∗signifies the virtual (exchanged) photon).
+ForFD(t), the Dirac form factor of the proton, we adopt
+the parameterization given in Ref. [41],
+FD(t) =4m2
+p−2.8t
+4m2p−t1
+(1−t/t0)2(4)
+witht0=0.71 GeV2, which is derived under the assump-
+tions that the Diracform factor FDofthe neutron and the
+isoscalar Pauli form factor vanish and that a dipole form
+is satisfactory for GM≈µGE(t), cf. [41]. There are slight
+deviationsfromthedipoleformintheregion −t<0.5GeV2
+we are concerned with here, cf. for example Ref. [42], but
+these are negligible for the present study. For the form
+factorFηγγ∗(t) we use the parameterization given by the
+CLEO Collaboration in Ref. [43]:
+|Fηγγ∗(t)|2=1
+(4πα)264πΓ(η→γγ)
+m3η1
+(1−t/Λ2)2.(5)
+Afittotheirdata[43]yieldedthevalue Λ=0.774GeVthat
+is close to the vector dominance model and to the predic-
+tion for the soft nonperturbative region given in Ref. [44].
+Forthe Primakoffamplitude asgivenin Eq.(3) we haveto
+renormalize this form factor so that Fηγγ∗(0)=1. The am-
+plitude ofEq. (3) should be added to the Regge amplitude
+F1, cf. Refs. [27,28].
+Since the OPE contributes essentially only at very
+small|t|it is sensible to consider the differential cross sec-
+tion as a function of the center-of-mass (cm) angle and
+not of the four-momentum transfer squared. This is done
+in Fig. 4 where we include data from DESY at energies of
+4 GeV and 6 GeV [23] (open squares) and from Cornell at
+4 GeV [26] (inverse triangles). The dashed lines indicate
+our calculations without the OPE amplitude. It is clear
+that at angles below 10◦the data are underestimated,
+which is a direct indication for the required additional
+contribution due to the Primakoff amplitude, Eq. (3).
+Let us now fit the data, taking the η→γγradiative
+decay width as free parameter. Unfortunately, there isFig. 4. Differential cross section for the reaction γp→pηas
+a function of the angle ϑin the cm system for the photon
+energies Eγ=4 and 6 GeV. The data are from DESY [23] (open
+squares)andCornell [26](inversetriangles). Thesolid (d ashed)
+lines show the results of our Regge calculation with (withou t)
+inclusion of the OPE contribution. The shaded band indicate s
+the uncertainty of our results due to the hadronic amplitude .
+no information with regard to the angular resolution of
+the measurements [23,26]. Only the intervals of the four-
+momentumsquaredwheretheDESYresultswereobtained
+are known and those are indicated in Fig. 4 as an uncer-
+tainty in the cm angle ϑ. Fortunately, below ϑ≤15◦the
+cross-section data exhibit practically no angular depen-
+dence so that these uncertainties have no significant influ-
+ence on our solution. We obtain for the decay width the
+valuesΓ(η→γγ)=0.86±0.11 keV at the photon energy of
+4 GeV and Γ(η→γγ)=0.70±0.09 keV at Eγ=6 GeV. The
+fit includes all data points shown in Fig. 4. A somewhat
+unpleasant finding is that for both photon energies the fit
+results in a rather low χ2, namely χ2/data point ≃0.3. For
+a statistically uncorrelated set of data points one would
+expect a value around 1. The shaded band in Fig. 4 indi-
+cates the uncertainty in the employed hadronic amplitude
+[28].
+Keepinginmindpossibleambiguitiesofourresultsdue
+to the unknown resolution we now compare the η→γγra-
+diative decay width evaluated here with the world data.
+Fig. 5 contains results from different measurements avail-
+ablein the literature.Specifically, circlesindicatedata [13,
+14,15,16,17,18] determined from the e+e−→e+e−ηreac-
+tion, while the squares are results [19,20] obtained with
+the Primakoff effect from measurements on nuclear tar-
+gets. A detailed discussion of the various measurements is
+given in Ref. [21].
+RecentlytheresultsfromtheCornellmeasurement[19],
+indicated by G in Fig. 5, were re-analyzed [22] assuming
+differentcontributionsfromincoherent η-mesonphotopro-
+duction from nuclei which led to an η→γγradiative de-
+cay width of 0.476 keV. This illustrated that the analysis
+of the data [19] depends significantly on the incoherent
+background at angles below ≃20in the laboratory frame4 A. Sibirtsev et al.: Primakoff effect in η-photoproduction off protons
+Fig. 5.Theη→γγradiative decay width from different anal-
+yses. The results are taken from: A [13], B [14], C [15], D [16] ,
+E [17], F [18], G [19], H [20]. Our own results obtained from
+a fit of the γp→ηpdifferential cross sections [26,23] at the
+photon energies of 4 and 6 GeV, are denoted by I and J, re-
+spectively. Circles are data obtained from the e+e−→e+e−η
+reaction, while squares indicate results obtained by Prima koff-
+effect measurements. The shaded box, indicated as PDG, is an
+averaged result [12]. The box indicated as EET shows the limi t
+given by Eq. (2). The lines indicate data distribution funct ions
+as explained in the text.
+and on the model applied. Clearly the data obtained [26,
+23] with a proton target are clean with respect to such
+background contributions to the Primakoff effect so that
+our evaluation here should be more reliable. Therefore, we
+agree with the conclusion of Laget [24] that future mea-
+surements with a proton target could be quite promising.
+Ourresultsobtainedforphotonenergiesof4and6GeV
+are indicated in Fig. 5 as I and J, respectively1. We see
+that there is a partial conflict between these results and
+data obtained from the e+e−→e+e−ηmeasurements. The
+shaded band in Fig. 5 illustrates the averaged result from
+the PDG [12], based on the measurements A to D. The
+lines show data distribution functions that were obtained
+in the following way [12]: To each measurement shown in
+Fig. 5 a Gaussian distribution is assigned with a central
+value, and a dispersion given by the error bar and the
+integral area proportional to the inverse error bar. The
+dotted line in Fig. 5 represents the sum of the Gaussians
+for the measurements A to D. The dashed line indicates
+the sum of the measurements A to H. The solid line is the
+sum obtained with all data, including our results I and J.
+From this data distribution analysis we conclude that
+even when taking into account all measurements and con-
+sidering the correction proposed [22] for G, it is still diffi-
+culttoinferthatthe η→γγradiativedecaywidthisknown
+nowwith high accuracy,say5%,asgivenby the PDG [12].
+1We remark that the effects of the form factors in Eq. (3) is
+small, setting e.g. Fηγγ∗(t) to one leads to a few percent shift
+in the central value for the width, well within the uncertain ty
+induced from the hadronic amplitudes.Fig. 6.Differential cross section for the reaction γp→pηas a
+function of the angle ϑin the cm system at the photon en-
+ergyEγ=11 GeV. The solid (dashed) lines show the results
+of our Regge calculations with (without) inclusion of the on e-
+photon-exchange contribution. The shaded band indicates t he
+uncertainty of our prediction due to the hadronic amplitude .
+Note that the recent averaged value of Γ(η→γγ) of the
+PDG was obtained by neglecting the e+e−→e+e−ηmea-
+surements [17,18] indicated as E and F in Fig. 5. In addi-
+tion, the results G and H obtained by the Primakoff mea-
+surement were always neglected by the PDG [12] as rec-
+ommended in Ref. [45] without any solid argumentation,
+solely on the basis that these results are in disagreement
+with other measurements.
+In Fig. 6 we display the differential cross section at
+forward angles for η-meson photoproduction at the pho-
+ton energy 11 GeV. This energy was suggested by the
+PrimEx Collaboration for a future experiment in Hall D
+atJLab[25].Herethedashedlineisthehadroniccontribu-
+tion alone, while the solid line accounts for the sum of the
+OPE and the hadronic amplitude. The shaded band indi-
+catestheuncertaintyofourprediction dueto the hadronic
+amplitude. Note that for this prediction we used the stan-
+dard value Γ(η→γγ)=0.51 keV [12]. Obviously also at
+higher energies the signal from the Primakoff effect is
+much larger than the uncertainty in the hadronic ampli-
+tude, even up to angles around ϑ≈5◦, Thus, we believe
+that it is rather promising to perform further high preci-
+sionmeasurementsofthetwo-photondecayofthe η-meson
+utilizing the Primakoff effect as proposed by the PrimEx
+Collaboration at JLab [25].
+This work is partially supported by the Helmholtz Associ-
+ation through funds provided to the virtual institute “Spin
+and strong QCD” (VH-VI-231), by the EU Integrated Infras-
+tructure Initiative HadronPhysics2 Project (WP4 QCDnet)
+and by DFG (SFB/TR 16, “Subnuclear Structure of Matter”).
+This work was also supported in part by U.S. DOE Contract
+No. DE-AC05-06OR23177, under which Jefferson Science As-
+sociates, LLC, operates Jefferson Lab. A.S. acknowledges su p-
+port by the JLab grant SURA-06-C0452 and the COSY FFE
+grant No. 41760632 (COSY-085).A. Sibirtsev et al.: Primakoff effect in η-photoproduction off protons 5
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\ No newline at end of file
diff --git a/1001.0647.txt b/1001.0647.txt
new file mode 100644
index 0000000000000000000000000000000000000000..56c6822a5d79176c91f596d9fee8250d1f8faaab
--- /dev/null
+++ b/1001.0647.txt
@@ -0,0 +1,766 @@
+A comparison of methods to determine neuronal
+phase-response curves
+Ben Torben-Nielsen, Marylka Uusisaari, Klaus M. Stiefel
+Abstract
+The phase-response curve (PRC) is an important tool to determine the excitabil-
+ity type of single neurons which reveals consequences for their synchronizing prop-
+erties. We review ve methods to compute the PRC from both model data and
+experimental data and compare the numerically obtained results from each method.
+The main dierence between the methods lies in the reliability which is in
uenced by
+the 
uctuations in the spiking data and the number of spikes available for analysis.
+We discuss the signicance of our results and provide guidelines to choose the best
+method based on the available data.
+1 Introduction
+The phase-response curve (PRC) of a regularly ring neuron quanties the shift in the
+next spike time as a function of the timing of a small perturbation delivered to that neu-
+ron. The PRC is an important measure for several reasons. First, the ability of neurons
+to synchronize in excitatory coupled pairs, chains or networks can be predicted from the
+PRC: type-I PRCs (purely positive, all excitatory perturbations lead to an acceleration
+of spiking) do not allow synchronization while type-II PRCs (biphasic, acceleration or
+delay of spiking depending on the phase of the perturbation) allow synchronization with
+excitatory connections and short delays [6]. Furthermore, the PRC is informative about
+the type of bifurcation leading from rest to spiking [8], thus constraining quantitative
+models of the neuron under investigation. Also, the PRC is correlated with the type of
+excitability of a neuron [7, 9].
+More precisely, a regular ring neuron can be seen as a stable oscillator with period
+T and only the phase to describe its state. T results from the characteristic angular
+velocity!of the oscillator, thusd
+dt=!. In the absence of inputs a regularly ring
+neuron res exactly when =kT(withkan integer and T corresponding the the average
+ISI,dISI). Now suppose an input to that neuron with a small amplitude at phase ,
+(). Then, the in
uence of this perturbation on the next spike time is described by
+d
+dt=!+ ()Z() whereZ() is the PRC. In other words, the time required to reach
+the next spike deviates from Taccording to the perturbation and the PRC. Since the
+exact spike times, perturbations ( ) and!(i.e.,2
+T) are known, we can estimate the
+PRCZ() from these data.
+1arXiv:1001.0647v3  [q-bio.QM]  26 Mar 2010Several methods have been proposed to compute the PRC from experimental or
+modeled data. For basic neuronal models, the PRC can be directly computed from the
+underlying dierential equations by the adjoint method [1], but for all other cases the
+PRC has to be determined numerically (for complex models) or experimentally (for real
+neurons). In this work, we review ve methods for determining PRCs and compare their
+performance on data sets containing modeled data and experimental data. We identify
+pitfalls in estimating the PRC, lay out guidelines for approximating the PRC, and assess
+the reliability of the resulting PRCs.
+2 Methods
+Here we concisely outline ve methods to estimate PRCs, describe the data sets used
+in the comparison, and describe how we will compare the outcomes. In addition, we
+describe the direct method which is used to benchmark the performance of the other
+ve methods. In the direct method, the PRC is constructed by injecting excitatory
+pulses at dierent phases of the inter-spike interval and measuring the resulting phase
+shift of the next spike.The PRC is produced directly by plotting the phase of the pulse
+on the x-axis and the resulting phase shift on the y-axis. In a noise-free case, such as
+a deterministic simulation, a ne-grained PRC can be generated by injecting pulses at
+many dierent phases.
+2.1 Five methods to determine a PRC
+In the case of experimental data or stochastic simulations, the data points resulting from
+applying the direct method will be jittered, and it is necessary to either t a curve to these
+points [3, 17] or bin them [12, 14]. We include one such method in our review of methods
+(Galan's method, see below). Due to the often quite signicant jitter, it is required to
+measure the spike time shift in hundreds of ISIs at randomized phases. Furthermore,
+it is necessary to intersperse them with inter-spike intervals without perturbing pulses
+to avoid entrainment of spiking and to have an unperturbed baseline to compare them
+to. Thus, large amounts of data are necessary to determine the PRC with the direct
+method.
+To alleviate this problem, novel methods have been proposed that use predictions of
+how spike times will be altered by incoming pulses, and, methods that use continuous
+
uctuation signals to obtain a more robust PRC measurement based on less spikes. The
+ve reviewed methods consist of one variation of the direct method (Galan's method),
+two methods that use spike-time predictions to reconstruct the PRC (the modied-
+Izhikevich method and the STEP method), and two methods that derive the PRC from
+the incoming continuous 
uctuating signal (the STA and WSTA method). The dierent
+methods are outlined below and illustrated in Figure 1.
+22S SMn
+'tn
+M(delay)(advance)'M = T-tMi  Direct method / Galan’s fitting
+'phase
+'tFFFn
+T
+2S
+02S
+0M't1't2
+MMn'tn
+tn t2 t1
+2S
+02S
+0TT T TT
+...Perturbation data
+FFFFFt1 t2 t3t4t5  T = t ...Continuous fluctuation data
+averageT
+'t1
+'t2
+'t3
+'t4
+'t5Trescale apply weight
+* ' 
+* ' 
+* ' 
+* ' 
+* ' 'ti = T - t i 'i = tiT - ti
+F
+F
+F
+F
+FF’
+F’
+F’
+F’
+F’
+S
+M
+SZ(M) = ‹'i * F’i›WSTA
+tmax
+F
+F
+F
+F
+F
+integratePRC = ∫
+0Tmax
+- STA
+6F
+nTmax
+STA =STAF
+F
+F
+F
+F
+T
+'t1
+'t2
+'t3't4
+'t5
+rescaleS 
+zc(M)MMMn
+E(M)
+F’
+F’
+F’
+F’
+F’
+**
+***
+****
+* ****
+*S
+SEi (Mj)= √('ti(Mj) - ti)2 
+MMM'ti(Mj) = F
+i’(Mj)* zc(Mj)~
+~STEP
+...
+zc(M)
+S'tM = FM* zc(M)~
+EM = √(t - tM)2 ~MMMnFFFn
+*
+**E(M)
+S MMMn...
+...update t  with 'tM ~~
+minimize
+E(M)update
+zc(M)predict spike times
+compute errorIzhikevich - modified
+minimize
+E(M)update
+zc(M)predict spike times
+compute errorFigure 1: Schematic overview of the reviewed methods. Two methods use pulsed-
+perturbation data and three methods use continuous 
uctuation data. More details
+in the main text and Table 1.
+32.1.1 Galan's method
+Galan's method [3] uses pulses as perturbations (see the top panel of Figure 1), and
+ts the PRC to the spike time shifts as a function of the phase of the perturbation.
+This is one of the methods which is an extension of the direct method for noisy data
+[14, 17, 12]. The PRC Z() is substituted by a truncated Fourier series, i.e., Z()
+nP
+0asin(n) +bcos(n) and the parameters describing the curve ( aandb) are optimized
+using the Euclidean distance between the data points and the curve as an error signal.
+The resulting curve is the best approximation of the PRC (only constrained by the length
+of the expansion of the Fourier series).
+2.1.2 Modied-Izhikevich method
+Izhikevich proposed an inverse solution to compute the PRC in [8, chapter10]. The
+method relies on predicting the next spike time and minimizing the error between the
+predicted spike time and the true next spike time. The prediction is based on the sum
+of the phase shifts that a small perturbation (part of a continuous 
uctuation) would
+cause. However, the proposed method does not converge to a correct solution after a
+reasonable number of tting rounds (e.g., in about 2-3 hours of computation while other
+methods converge within minutes). Therefore, we modied the method and used it with
+less complex perturbation data. In the modied-Izhikevich method, one pulse x(t) is
+injected per phase and the next spike is predicted according to a candidate PRC, i.e.,
+~st+1=x()zc() in which zc() is the candidate PRC. Then, the candidate PRC is
+optimized to match the spiking data by computing an error signal proportional to the
+dierence between the predicted next spike time and the actual next spike time, e.g.,
+Err=p
+(sn+1~sn+1).
+2.1.3 Spike-triggered average method (STA)
+In the spike triggered average method [2] the the spike-triggered average is computed
+from continuous low amplitude current 
uctuations (see Figure 1). Then, this spike-
+triggered average is numerically integrated to produce the PRC. The connection between
+the integral of the spike-triggered average and the PRC is proven for regularly ring
+neurons and small perturbations in [2]. Formally, this method uses the fact that with
+STA(s)=hx(snsn1)i(which is on the interval [0 ;ISImax] withISImaxbeing the
+largest ISI of s), the relationship PRCISImax

+0STA (s) holds. In contrast to both
+Galan's method and the moded-Izhikevich method, only a single pass over the complete
+noise signal and the voltage trace is required because there is no optimization step. From
+this single pass, the spike-triggered average is computed and subsequently integrated.
+42.1.4 Weighted spike triggered average method (WSTA)
+The weighted spike triggered average method devised in [10] is an extension of the STA
+method and also integrates the continuous low amplitude current 
uctuations to derive
+the PRC. However, in the WSTA method, the 
uctuations in between the dierent spike
+times are normalized to the average ISI (dISI) of all spikes in s(esicISI
+it, with instan-
+taneous ISI i). Then, as Ota et al. [10] prove, with an appropriate weighting function
+the weighted sum of these normalized stretches of current 
uctuations constitutes the
+PRC (for regularly ring neurons). The weighing function is =(cISIi)
+i. Therefore,
+PRCWSTA (es)hx(esi)i.
+2.1.5 Standardized error prediction method (STEP)
+The STEP method [16] is an extension of the modied-Izhikevich method to work with
+continuous 
uctuation data (in a dierent way than originally proposed by Izhikevich).
+In this method, instead of using a prediction error averaged over all ISIs and all phases,
+the temporal information in the error is preserved by binning the errors of all ISIs
+independently per phase.
+The 
uctuations are binned equidistantly ondISI(i.e., normalized) and are treated
+as if they were independent, i.e., for each phase bin of each ISI, the predicted next spike
+time is computed and the mismatch with the true next spike time is used to optimize the
+parameters of a curve representing the PRC. More precisely, all inter-spike intervals are
+normalized to [0 ;2] and discretized into N bins. For each bin, an independent prediction
+is made about the next spike time: ~ si;j=!+[x(binj)zc()], wherecorresponds to phase
+ofbinj. Subsequently, one obtains a 2-dimensional array with the the dimensions given
+by N bins and M spikes. Finally, a least-squares tting algorithm is used to minimize
+the 2-D prediction error array and obtain a PRC that predicts the recorded spike time
+shifts [16].
+Table 1 summarizes the ve implemented methods (plus the direct method) and how
+they relate to each other. In addition to the reviewed methods, there are several other
+published methods which we omitted because they proved impractical (with respect to
+experimental demands), e.g., the MAP-estimation algorithm [11] and the post-stimulus
+time histogram method [5].
+No optimization Optimization
+Perturbation data(Direct method) Galan's method
+Izhikevich-derived method
+Continuous 
uctuation dataSTA STEP
+WSTA
+Table 1: Comparison of the implemented methods in terms of the required data and the
+requirement to optimize the outcome.
+52.2 Data sets
+We tested the ve methods (and the direct method) with three dierent data sets when-
+ever possible. The rst two data sets contain model data while the third set contains
+experimental data. We used the single-compartmental model as developed by Golomb
+and Amitai [4] and modied by [14] to generate the data. This model uses a Hodgkin-
+Huxley-type formalism to model neural spiking behavior:
+CMdV
+dt=m3hgNa(VENa)ngKDR (VEK)sgKs(VEK)gleak(VEleak)Iinj
+anddx
+dt=(V) (xx1(V)), whereVis the membrane potential, gxthe maximum
+conductance for ion xandExthe reversal potential for ion x. The parameter values can
+be found in [4, 14]. By turning the adaptation current on or o, this model switches
+between type-II or type-I excitability [14], respectively.
+The rst data set contains noise-free model data in which a single compartmental
+model neuron (see below) is perturbed at dierent phases. This set contains 128 pulses
+evenly spaced over [0 ;2]. The second data set contains modeled data from the same
+single-compartmental model but with an additionally injected 
uctuating current. The
+
uctuations are generated through a stationary Orstein-Uhlenbeck process around a
+given mean value and parametrized by the reversion rate ( g= 0:1) and 4 dierent
+volatility levels ( D= 1e4;5e4;1e5;5e5). The advantage of the noisy modeled data
+is that the excitability type is known with certainty because small perturbations do
+not change the PRC type [8]. The injected 
uctuating current and the resulting spike
+trains are illustrated in Figure 2. The dierent noise levels result in four groups of data
+each containing approximately 950 spikes. The noise and the resulting spike trains are
+illustrated in Figure 2. The last data set contains experimental data recorded from a layer
+3/4 pyramidal cell of the mouse visual cortex with the whole cell patch-clamp technique
+in vitro. Standard patch-clamp techniques as in [13] were used. Membrane potential
+voltage data and the injected 
uctuations were digitized at 40 kHz, and two levels of
+current (= 50 and 100 pA) were injected as 
uctuations. The 
uctuations consisted of
+white-noise low-pass ltered at 200 Hz. In both the model data and experimental data,
+the 
uctuations are on top of a step current ( Is) which is required to get the model/cell
+into a regime of regular ring
+2.3 Quantitative comparison
+To investigate which method produces the most reliable result, we compared the dierent
+methods with the calibrated PRC resulting from the direct method. The dierence
+resulting from the ve implemented methods with the directly observed PRC can be
+quantied by examining the Euclidean distance (by taking the mean-squared error, MSE)
+and correlation (with the Pearson correlation) between the obtained PRCs.
+It is important to note that Galan's method and the modied-Izhikevich method
+cannot be used with continuous 
uctuating data because they are designed to work
+solely with pulsed perturbation data. However, the methods intended for continuous
+6NL = 1
+NL = 2
+NL = 3
+NL = 4
+1 s100 mV
+50 pAFigure 2: Noise and the resulting spike trains as generated by our model neurons. The
+four noise levels have increasing amplitudes around the same mean and are generated
+by an Ornstein-Uhlenbeck process. The noise has a profound in
uence on the regularity
+of the spikes.
+
uctuation data (e.g., STA, WSTA and STEP) can be used to compute the PRC from
+both perturbation data and 
uctuation data because the former is a simplied case of
+the latter: instead of a continuous stream of 
uctuations only a single 
uctuation per
+period is injected. Hence, we can compare the PRCs resulting from the ve implemented
+method with the directly observed PRC on the perturbation data, but only the STA,
+WSTA and STEP on the more complex continuous 
uctuation data (i.e., noisy model
+data set and experimental data set).
+To compare the methods with each other, we normalize the PRCs in a post-processing
+step. All but the STA method are dened (along the x-axis) over [0 ;2]. We scale the
+time of the STA produced PRC to the same interval. Along the y-axis we normalize the
+PRC so that the (positive) peaks are equal to 1. Since most researchers are primarily
+interested in the type of the PRC (type-I vs. type-II), and because the normalization
+does not aect qualitative features of the PRC such as the slope and the positive and
+negative areas, the normalization of the results is valid.
+72.4 Implementation details
+We implemented the dierent methods in Python in combination with the Numpy/Scipy
+and Matplotlib1. In three methods (Galan's method, modied-Izhikevich and STEP)
+a curve is optimized to t the data. Any smooth curve such as a polynomial or a
+Fourier series can be used for this purpose. To be consistent with the implementation
+in previously published studies [3, 8] we use the third expansion of the Fourier series
+(n=3) in the remainder of this manuscript. Larger expansion would provide better ts
+in some cases (when there is a steep slope in the PRC) but it would also be more
+prone to overtting and hence the third expansion seems suitable. Moreover, Galan's
+method and (the original) Izhikevich method do not prescribe a particular optimization
+algorithm although Galan uses least-squares optimization. We follow his work and also
+use least-squares optimization in Galan's method, the modied-Izhikevich method and
+STEP method.
+For the WSTA method, the authors suggest to t a polynomial to the raw outcome
+of their algorithm because this raw output is noisy with a smaller number of spikes. For
+the sake of clarity we show the raw outcome to illustrate the true capabilities of this
+method.
+All methods require conguration of the estimated inter-spike interval (gISI2). In our
+implementation of the dierent methods, thegISI can be given as an argument to the
+algorithm or automatically computed. The automatic computation straightforwardly
+takes the mean and, therefore, works only for highly regular ring neurons. In addition
+we exclude ISIs that do not satisfy 0 :1gISIgISI2gISIbecause larger spread of
+ISIs generally causes the methods to fail (remember that the PRC is a characteristic of
+regularly ring neurons.).
+3 Results
+3.1 Performance on noise-free model perturbation data
+Figure 3 illustrates the PRCs as computed by all implemented methods for noise-free
+model data with both type-I and type-II parameters. Table 2 quanties the dierence
+between each PRC and the directly observed PRC by means of the MSE and Pear-
+son correlation. For brevity, the modied-Izhikevich method is labeled as `IzhiLQ' and
+Galan's method is referred to as `GalanLQ'; in both cases the LQ sux indicates the use
+of least-squares optimization. The PRCs resulting from the direct method is plotted as
+a dashed line and serves to calibrate the results. It can easily be veried from Figure 3
+that the ve methods compare qualitatively; it can be veried from Table 2 that they
+are also quantitatively similar.
+1The code is developed for scientic use and can be obtained from
+http://www.irp.oist.jp/tenu/btn/Tools.html
+2In theory, the average inter-spike interval dISIis required. However, in most cases when ring is not
+regular and the ISI histogram mildly skewed, we have to estimate the `average' inter-spike interval gISI.
+In the remainder of this manuscript, dISIandgISIare used as synonyms.
+8The best results using this type of data are obtained by the methods designed to work
+with pulse-perturbation data only, namely Galan's method and the modied-Izhikevich
+method. The results of these two methods are better in terms of the MSE and the
+Pearson correlation compared to the results of the methods intended for continuous
+
uctuating data. Moreover, the quantitative analysis shows equal results of the modied-
+Izhikevich and Galan's method on type-I data. This result is due to the simplicity of
+the curve to t and the low dimension of the search space (i.e., 3 values for aandb
+of the Fourier series). On type-II data, the modied-Izhikevich method performs best.
+From methods intended for continuous 
uctuating data, the STEP method has the best
+performance on both the type-I and type-II model data set as the MSE and Pearson
+correlation indicate closest resemblance to the directly obtained PRC.
+0π 2 π
+Phase (radians)-101Normalized Δ phasePRC comparison. Type-I model, w/o noise
+STEP
+STA
+WSTA
+IzhiLQ
+GalanLQ
+Direct
+0π 2 π
+Phase (radians)-101Normalized Δ phasePRC comparison. Type-II model, w/o noise
+STEP
+STA
+WSTA
+IzhiLQ
+GalanLQ
+Direct
+Figure 3: Comparison of all PRC estimation methods on noise-free model data. The
+dotted black line is the directly observed PRC. The PRCs from all ve methods agree
+on the PRC type, have a similar shape, and resemble the directly determined PRC.
+3.2 Performance on noisy model data
+The noisy model data are obtained by continuous 
uctuating current injection. Here
+we compare the three methods designed to analyze such data. We used four dierent
+noise levels for the comparison. Each noise level has the same mean and only diers
+in the variance around the mean3. Figure 4 illustrates the PRCs produced by the
+STA, WSTA and STEP method at the four levels of 
uctuations. From this gure it
+is clear that for low noise levels all methods agree qualitatively on the PRC type as all
+methods correctly indicate type-II PRC in the model. However, for the two higher noise
+levels, the WSTA method results in a (mostly) nonnegative, type-I, PRC while the two
+other methods correctly assess the neuron as type-II. Hence, we can say that the STA
+3The reversion rate in the Orstein-Uhlenbeck noise was kept constant while the volatility was in-
+creased.
+9Method Type-I Type-II
+MSE Pearson MSE Pearson
+STA 0.691571 199.188386 0.833662 218.912789
+STEP 0.951353 67.878615 0.939346 130.261712
+WSTA 0.959762 76.051482 0.760755 235.848697
+IzhiLQ 0.961458 66.999838 0.991354 54.368821
+GalanLQ 0.961458 66.999838 0.988336 59.415021
+Table 2: Quantitative performance of the dierent methods to estimate the PRC. The
+means-squared error (MSE) and the Pearson correlation with respect to the directly
+observed PRC are used for the quantication. For both type-I and type-II data, the
+modied-Izhikevich method performs best both in terms of MSE and Pearson correlation.
+On the type-I data set, the modied-Izhikevich and Galan's method obtain the same
+performance (and hence perform equal) due to the low degree of freedom (6 parameters
+to be optimized for the Fourier series). Of the methods designed to work with continuous
+data, the STEP method performs the best.
+and STEP method cope better with high amplitude 
uctuations. Table 3 quanties the
+dierence between the computed PRCs and the directly observed PRC. In contrast to
+the qualitative observation that STA and STEP perform better with 
uctuation, the
+WSTA method has the highest resemblance to the directly observed PRC in terms of
+MSE and Pearson correlation (except at the highest 
uctuation level where the STEP
+method obtains the best Pearson correlation). We explain this observation as follows:
+with higher amplitude 
uctuations, the regularity of the spikes decreases. As a result,
+the PRC as computed from this less regularly ring data is dierent than the PRC from
+the noise-free data [15, 17]. However, the PRC is a quantitative measurement of regular
+ring neurons and hence, the true PRC of a neuron is the PRC measured from spikes in
+the regular ring regime. Therefore, we compare the PRCs always to the PRC directly
+observed in the noise-free case although the shape of the PRC at higher 
uctuation levels
+might look dierent. The PRC produced by the WSTA method has more resemblance to
+the directly observed PRC although is does provide a wrong categorization of the PRC
+type at higher amplitudes. For low-amplitude 
uctuations we conclude that the WSTA
+method produces the most reliable PRC, but for higher amplitude-
uctuations (NL=3,4)
+the STEP and especially the STA method produce more reliable results (Figure 4)).
+The reliability of the STA method has to be inferred from a visual inspection of
+the produced PRC and knowledge of the noise-free, directly observed PRC because the
+quantitative analysis always indicates low similarity between the PRC produced by the
+STA method and the noise-free, directly observed PRC. This phenomenon is due to the
+fact that the STA method is computed on the interval [0 ;ISImax] and only afterwards
+scaled to [0 ;gISI]. As a consequence, the STA is always relatively 
at at the beginning
+of the normalized interval and the PRC is shifted to later phases. Even though the
+interpretation of the PRC is beyond the scope of this review, the shift to higher phases
+10does not matter for the reliability as the shift is consistent and proportional to increasing
+irregularity of the spike times.
+0π 2 π
+Phase-101Normalized Δ phasePRC comparison, NL=1
+STA
+WSTA
+STEP
+0π 2 π
+Phase-101Normalized Δ phasePRC comparison, NL=2
+STA
+WSTA
+STEP
+0π 2 π
+Phase-101Normalized Δ phasePRC comparison, NL=3
+STA
+WSTA
+STEP
+0π 2 π
+Phase-101Normalized Δ phasePRC comparison, NL=4
+STA
+WSTA
+STEP
+Figure 4: Comparison of PRC methods on noisy model data. The four panels illustrate
+the resulting PRCs at dierent noise levels (NL=1,2,3,4).
+3.3 Performance on experimental data
+The most interesting test case is the comparison of dierent PRC methods applied to
+real, experimental data. We compare the STA, WSTA and STEP method on data from
+layer 2/3 neurons. Figure 5 illustrates the results with two dierent noise levels. With
+650 spikes and considerable spread of ISIs, the WSTA method produces a noisy outcome
+while the STA and STEP result in smooth PRCs (as they are optimized Fourier series of
+small expansion). The two noise levels produce PRCs that very similar; all three methods
+indicate a type-II PRC although at both noise levels the WSTA produced PRC is rather
+noisy. In the case of experimental data, there is no calibrated data to test against and
+11Method NL=1 NL=2 NL=3 NL=4
+MSE PC MSE PC MSE PC MSE PC
+STA -0.089941 514 -0.252694 562 -0.191698 541 0.279241 410
+STEP 0.751355 343 0.702422 544 0.671676 336 0.551901 387
+WSTA 0.847633 212 0.776861 282 0.815145 318 0.788307 459
+Table 3: Quantitative performance of the dierent methods to estimate the PRC using
+noisy, continuously 
uctuating data. The WSTA performs the best on most noise levels
+in terms of both the MSE and Pearson correlation (PC) when compared to the directly
+observed PRC. An interpretation of these results is in the main text.
+hence we only look at the Pearson correlations between the dierent methods to assess
+the level of agreement between the dierent methods (Table 4). For the lower noise level,
+the Pearson correlation between the three methods is always higher than 0.85, indicating
+good correspondence between the three methods. Also, for the higher noise level, the
+correlations stay above 0.73 which still indicates good agreement between the dierent
+methods. The STEP method has the highest Pearson correlation with the other two
+outcomes and can therefore be seen as a sort of `average' of the other two methods.
+Additionally, the high resemblance between the three methods corroborates that these
+PRCs are reliable.
+0π 2 π
+Phase-101Normalized Δ phasePRC comparison, NL=1
+STA
+WSTA
+STEP
+0π 2 π
+Phase-101Normalized Δ phasePRC comparison, NL=2
+STA
+WSTA
+STEP
+Figure 5: Comparison of dierent PRC methods on experimental data. Two noise levels
+were tested and the three methods show a fair agreement on the type (type-II) of the
+PRC curve.
+3.4 Reliability and parameter sensitivity
+Here we address the reliability of the various methods for estimating the PRC. Moreover,
+we investigate how the reliability is aected by parameter sensitivity in the algorithm.
+12STA STEP WSTA
+Noise 1
+STA 1.000000 0.872780 0.852972
+STEP 1.000000 0.887956
+WSTA 1.000000
+Noise 2
+STA 1.000000 0.794771 0.737648
+STEP 1.000000 0.837448
+WSTA 1.000000
+Table 4: Agreement between dierent PRCs on the experimental data. Shown are the
+Pearson correlation between the PRCs generated by three dierent methods. All show
+strong correlation indicating similar trends in the three curves.
+The reliability of the estimated PRC depends strongly on the number of spikes avail-
+able for analysis. The data used to obtain the PRCs in Figures 3, 4 and 5 contain a
+reasonable number of spikes4, and the results indicate that all three methods are { to a
+certain extent { capable of producing reliable PRCs even under the presence of higher-
+amplitude 
uctuations and more diverse ISIs. However, the number of spikes available
+for analysis can be limited in experimental data because the experimental protocol is
+often not exclusively used to gather data to determine a PRC but for other scientic
+goals. Therefore, we examined the performance of the STA, WSTA and STEP method
+with less spikes, namely 50, 100 and 500 spikes. Figure 6 illustrates these results. The
+columns illustrate the PRC with (from left to right) 50, 100 and 500 spikes (the spikes
+are the rst 50, 100 and 500 of the available spikes in the data set). The PRCs from
+the top row use spikes from the continuous 
uctuating data set at the lowest 
uctuation
+level. The PRCs from the bottom row use experimental data at the second 
uctuation
+level. For the model data, the WSTA and STEP methods give a fair result when using
+as little as 50 spikes while the STA method produces no usable PRC5. With 100 spikes
+and more, all three methods produce a reliable PRC on this set of model data with little
+variance in the ISIs. For the experimental data, a dierent view emerges. For both 50
+and 100 spikes, the WSTA output is useless because of the high noise. Moreover, the
+STEP method wrongly classies the PRC as type-I with only 50 spikes; a negative part
+in the STEP-produced PRC using 100 or 500 spikes correctly indicates type-II PRC.
+With 500 spikes, all three method provide a reliable PRC. In contrast to the PRCs from
+the modeled data, the STA method produces fair PRC with as little as 50 spikes. This
+result demonstrates that the produced PRCs are in
uenced by regularity of the data
+and the number of spikes, rather than claiming superiority of any method of the other.
+4`Reasonable' is here used to denote the number of spikes from experimental data (at least 650) or
+higher.
+5If the spike-triggered average is purely positive, the STA-produced PRC will steeply go negative as
+illustrated in the top left panel of Figure 6.
+13With a dierent combination of spike times (i.e., not the initial 50, 100, or 500) the
+results look slightly dierent (not shown).
+0π 2 π
+Phase-101Normalized Δ phaseModel NL=1, 50 spikes
+STA
+WSTA
+STEP
+0π 2 π
+Phase-101Normalized Δ phaseModel NL=1, 100 spikes
+STA
+WSTA
+STEP
+0π 2 π
+Phase-101Normalized Δ phaseModel NL=1, 500 spikes
+STA
+WSTA
+STEP
+0π 2 π
+Phase-0.501Normalized Δ phaseExperimental NL=2, 50 spikes
+STA
+WSTA
+STEP
+0π 2 π
+Phase-0.501Normalized Δ phaseExperimental NL=2, 100 spikes
+STA
+WSTA
+STEP
+0π 2 π
+Phase-0.501Normalized Δ phaseExperimental NL=2, 500 spikes
+STA
+WSTA
+STEP
+Figure 6: The in
uence of the number of spikes on the reliability of the produced PRCs.
+The top and bottom row represent noisy modeled data and experimental data, respec-
+tively. We tested 50, 100 and 500 spikes and observe that the PRC type is correctly
+assessed by the STEP and STA method for data sets containing more than 100 spikes.
+The reliability increases for higher number of spikes but with 500 spikes the STA and
+STEP methods seem to converge on both modeled data and experimental data while
+the WSTA method is still very noisy on the experimental data set with 500 spikes.
+The results presented in this paper demonstrate that all methods are capable of
+producing reliable PRC on pulsed-perturbation data. Moreover, provided a sucient
+number of spikes are available, the STA, WSTA and STEP methods also work well
+with continuous 
uctuating data. This result might be, however, misleading since the
+reliability of the tested methods is sensitive to algorithm parameters such as the a priori
+estimated ISI (gISI) and the estimated injected step current Is(on top of which the
+
uctuations are modulated). These parameters can be set manually or computed by the
+algorithm itself. Below we describe the eects of these two parameters on the dierent
+methods to estimate a PRC.
+We observed that the STA method is highly sensitive to a correct estimation of the
+current step, i.e., Isthe injected current to make the neuron re regularly. Figure 7 (top
+left) illustrates this eect. With a very small deviation of the estimated mean from the
+real injected DC the outcome becomes unstable. In most cases, this eect will be evident
+to the researcher: when the amplitude of the spike-triggered average (in the STA method)
+is close to zero, minor osets in the estimated DC may shift the spike-triggered average
+(to either all positive or all negative), and, in turn, shift the PRC resulting in a wrong
+14indication of the PRC type. The top left panel in gure 6 also clearly demonstrates this
+eect: the PRC drops steeply (which is accentuated by the normalizing of the positive
+peak to 1). However, when the spike-triggered average is further away from zero, or
+when the spike-triggered average crosses zero, a faulty PRC is hard to observe because
+the resulting PRC will resemble a PRC but will not be representative of the data. Hence,
+a few dierent settings for the estimated mean (for instance, the calculated mean from
+the injected signal in simulations, or, straightforwardly the injected current from the
+experimental setup) should be used and the resulting PRCs should be compared to
+PRCs produced by the other methods. In addition, we observed that the PRC produced
+by the STA is dicult to interpret for two reasons. First the PRC is always shifted on the
+x-axis towards later phases because it is computed on the interval [0 ;ISImax] and later
+normalized to [0 ;dISI]. Second, the y-axis provides the integral of the spike-triggered
+average; it is not obvious how this relates to the exact delay or advance in spike times.
+Unfortunately, this second diculty also arises for the WSTA method where the y-axis
+is dened by the non-symmetric (see below), weighted sum of the input 
uctuations.
+The STEP method has a straightforward interpretation of the y-axis as it stands for the
+phase shift.
+We observed that the WSTA method is highly sensitive to the estimated ISI (see
+Figure 7, top right). In eect, thegISIshifts the resulting PRC along the y-axis. This
+observation can be explained as follows: spikes with relatively short ISIs compared to
+thegISI(i) are stretched to thegISIand become straight lines with little variation, and
+(ii) receive a high weight in the weighted sum (Figure 7, bottom left). These two eects
+combined lead to shorter ISIs pulling the PRC upwards. On the other hand, relatively
+long ISIs compared to thegISImake the PRC resulting from the WSTA method noisy
+because they are compressed to t on the normalized interval; during this compression
+smooth 
uctuations become steep and are added up to the PRC. For highly regularly
+ring neurons, the estimated ISI (gISI) is simply the mean of the ISIs (dISI). However,
+in the more realistic cases with more variation and a skewed ISI histogram, it becomes
+less clear what the `best' a priori estimatedgISI should be: mean, median, or mode
+of the ISIs? In type-II cases this estimation might cause unreliable results because the
+crossing point (i.e., the point where the curve changes sign) can be shifted along the
+x-axis and the ratio of positive to negative surface will clearly be modied, even up to
+the point that the negative part may became insignicant. In addition, gure 7 (bottom
+right) illustrates the accumulated eect of dierent settings for thegISI and the step
+currentIs. One problem is that all of the PRCs produced by the WSTA method shown
+in that panel resemble what a researcher might anticipate: some PRC estimates are of
+type-II while others are of type-I. The dierence is caused by changing two conguration
+options in the algorithm.
+In general, the fact that one can congure two settings (gISIandIs) opens a possi-
+bility for a bias in the resulting PRC: as we just illustrated a PRC can be easily `tuned'
+under the presence of considerable 
uctuations to a particular PRC type by changing
+gISI andIs. Therefore, one might tune the settings in a way as to prove a particular
+PRC type. Moreover, even without predispositions about the PRC type, all of the PRC
+150 50 100 150 200 250
+Phase (ms)1.51.00.50.00.51.01.5Δ phase (raw)STA dependence on Is
+Is = 105%
+Is = 101%
+Is = 100%
+Is = 99%
+Is = 95%
+0 ISI 2xISI
+inter s p ike interval6420246weight
+DelayAdvanceWTSA: asymmetric weight assignment
+weight
+ISI x 0.66
+I S Ix2
+0 20 40 60 80 100 120
+Phase (ms)0.00150.00100.00050.00000.00050.00100.0015Δ phase (raw)WSTA dependence on /tildewidestISI and Is
+set Is/tildewidestISI = 105%
+set Is/tildewidestISI = 100%
+set Is/tildewidestISI = 95%
+man Is/tildewidestISI = 105%
+man Is/tildewidestISI = 100%
+man Is/tildewidestISI = 95%Figure 7: In
uence of conguration settings in dierent PRC methods. All PRCs are
+generated from noisy modeled data with NL=2. Top left: the STA method is highly
+sensitive to errors in the estimated current step ( Is). Top right: the WSTA method
+is sensitive to estimates of thegISI. Bottom left: the WSTA weighing function is not
+symmetric and non symmetric distributions of ISIs will lead to upward and downward
+drifts of the PRC. Bottom right: dierent outcomes of the WSTA method depend on
+a combination of the estimateddISIand the step current which can be either manually
+specied as the predetermined mean (man) or computed from the input 
uctuations
+(set).
+16methods (and especially the methods that optimize a smooth curve) can produce PRCs
+that appear realistic without being representative of the data.
+4 Discussion
+We reviewed ve dierent methods to determine the PRC from experimental or modeled
+data. Two methods are variations of the direct method and require the perturbation-
+stimulation protocol to gather data. The three other methods use continuous 
uctuation
+data, which requires the continuous injection of (a step current and) a 
uctuation into
+a regularly ring neuron. We found that on noise-free modeled perturbation data, all
+methods worked well and showed little dierence. Moreover, the two methods requiring
+pulse perturbation data produced the best results, but this comes at the cost of a spe-
+cialized stimulation protocol and the requirement of a large number of spikes in order to
+cover all the phases. In contrast, the methods that can use the continuous 
uctuation
+data use all available data eciently as random 
uctuations are by denition delivered
+at every phase and thus require less spikes. For instance, one study [3] uses 7000 (highly
+regular) spikes while we show that the continuous 
uctuation methods provide reliable
+results after a few hundreds spikes (e.g., 500). Hence, for experimental situations where
+little time is available it is best to use the continuous 
uctuation protocol. When an ex-
+periment is done solely for the purpose of determining a PRC, the perturbation protocol
+should be used.
+However, we also demonstrated that the estimated PRC not only depends on the
+method employed, but also on the settings of the method ( IsandgISI) and the regularity
+of the spikes (as altered by the amplitude of the 
uctuations). Moreover, we demon-
+strated that the dierent techniques might generate PRC curves that appear plausible
+but are not representative for the data. A `panel of experts' strategy can be applied to
+enhance the reliability: one can run all the dierent methods (that are appropriate for
+the given data set) with slightly dierent congurations. A stable PRC for the widest
+range of settings can be considered the most correct. And, pitfalls such as upward or
+downward shifts in the PRC can be detected by trying several settings. W also suggest
+researchers dealing with PRC to inspect carefully the spiking data and obtain good es-
+timates for thegISIand the DC step current before running the analysis and using any
+of the PRC estimation methods.
+Interpretation of the PRCs is beyond the scope of this review. Brie
y, however,
+dierent criteria are used to classify PRC curves into type-I curves and type-II curves.
+For instance, the ratio between the negative amplitude and the positive amplitude [15]
+or the ratio between the positive and negative surface [17] have been proposed as PRC
+categorization criteria. Moreover, some reports suggest that the exact shape of the PRC,
+skewness, zero-crossings and other features contain information about the underlying
+system, e.g., [5, 15]. These features can only be reliably interpreted after obtaining a
+PRC in a reliable manner following guidelines and avoiding potential pitfalls as outlined
+above.
+We hope that this review of methods for determining the PRC will motivate re-
+17searchers to determine this measure of spiking behavior for the neural cell types they
+investigate. The PRC is a measure with considerable predictive power for the behavior
+of a single neuron in a network [1]. Given knowledge of a neuron's PRC and its synap-
+tic connections (excitatory/inhibitory, waveform duration, and delay), it is possible to
+analytically determine its participation in population activities such as synchronization,
+asynchrony and beating. Specically, pairs, chains or networks of neurons with a type
+I PRC will synchronize when coupled by fast inhibitory synapses. For type II neurons,
+synchrony ensues with excitatory synapses. Strictly speaking, these predictions only
+hold for homogeneous networks of regularly spiking neurons perturbed by small synap-
+tic potentials. Nevertheless, they allow for insights about network activity emerging from
+single neuron properties in many activity regimes. Thus, reliable knowledge of PRCs
+from more neural cell types will aid the understanding of how single neurons contribute
+to brain function.
+Acknowledgments
+The authors thank Drs. Yasuhiro Tsubo and Stijn Vanderlooy for fruitful discussions.
+References
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uc-
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+1672, 1993 May.
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+changes phase response curve shape and type in cortical pyramidal neurons. PLoS
+One, 3(12):e3947, 2008.
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+in interneurons of rat somatosensory cortex. Biophys J , 92(2):683{95, Jan 2007.
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+the phase-response curves of neurons based on minimizing spike-time prediction
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+19
\ No newline at end of file
diff --git a/1001.0648.txt b/1001.0648.txt
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--- /dev/null
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@@ -0,0 +1,1336 @@
+arXiv:1001.0648v2  [astro-ph.SR]  23 Feb 2010Exploring massloss,low-Z accretion, andconvective overs hootin solar
+modelsto mitigatethe solarabundance problem
+JoyceAnn Guzikand KatieMussack
+XTD-2,MS T-086,LosAlamosNationalLaboratory,LosAlamos ,New Mexico,87545
+joy@lanl.gov
+ABSTRACT
+Solar models using the new lower abundancesof Asplund et al. (2005, 2009) or Ca ffau et al. (2008,
+2009)do not agree as well with helioseismic inferencesas mo dels that use the higher Grevesse & Noels
+(1993) or Grevesse & Sauval (1998) abundances. Adopting the new abundances leads to models with
+soundspeeddiscrepanciesofupto1.4%belowthebaseofthec onvectionzone(comparedtodiscrepancies
+of less than 0.4% with the old abundances),a convectionzone that is too shallow, and a convectionzone
+helium abundance that is too low. Here we review briefly recen t attempts to restore agreement, and
+we evaluate three changes to the models: early mass loss, acc retion of low-Z material, and convective
+overshoot. Onegoaloftheseattemptsistoexploremodelsth atcouldpreservethestructureintheinterior
+obtained with the old abundances while accommodating the ne w abundances at the surface. Although
+the mass-losing and accretion models show some improvement in agreement with seismic constraints, a
+satisfactory resolution to the solar abundance problem rem ains to be found. In addition, we perform a
+preliminaryanalysis of modelswith the Ca ffau et al. (2008,2009)abundancesthat shows that the sound
+speed discrepancy is reduced to only about 0.6% at the convec tionzone base, compared to 1.4% for the
+Asplund et al. (2005) abundancesand 0.4% for the Grevesse & N oels (1993) abundances. Furthermore,
+includingmass loss in modelswith the Ca ffau et al. (2008,2009)abundancesmay improvesoundspeed
+agreementandhelpresolvethesolarlithiumproblem.
+Subject headings: Sun: abundances –Sun: evolution –Sun: helioseismology –Su n: interior –Sun: oscillations
+1. Introduction
+1.1. The SolarAbundance Problem
+Before 2005, we believed we knew very well how
+tomodeltheevolutionandinteriorstructureofthesun.
+Evolvedsolarmodelswiththelatestinputphysics(in-
+cludingdiffusiveheliumandelementsettlingandwith-
+out tachocline mixing) reproduced the sound speed
+profile determined from seismic inversions to within
+0.4%, as well as the seismically-inferred convection
+zone depth and convection zone helium abundance.
+Solar interior modelers had little impetus to progress
+beyond one-dimensional spherical models of the sun,
+with perhaps the exception of introducing some ad-
+ditional mixing below the convection zone (hereafter
+CZ) to deplete surface lithium and reduce the small
+remaining sound-speed discrepancy at the CZ base.
+For the Grevesse&Noels (1993, hereafter GN93)or Grevesse& Sauval (1998, hereafter GS98) abun-
+dances, the simplest ‘spherical sun’ assumptions ap-
+peared nearly adequate for solar modeling. These in-
+clude one-dimensional zoning (concentric shells in
+hydrostaticequilibrium),initial homogeneouscompo-
+sition, negligible mass loss or accretion, neglecting
+rotation and magnetic fields, simple surface bound-
+ary conditions, mixing-length theory of convection
+(e.g., B¨ ohm-Vitense (1958)), and no additional mix-
+ing or structural changes from convective overshoot,
+shearfromdifferentialrotation,meridionalcirculation,
+waves,oroscillations.
+However, new analyses of solar spectral lines re-
+vise downward the abundances of elements heavier
+thanhydrogenandhelium,particularlytheabundances
+of carbon, nitrogen, and oxygen that contribute to the
+opacityjust below the CZ. See Table 1 for a summary
+ofsomeofthemajorabundancerevisionsoverthelast
+1Table 1: Mass fractions of metals in the present-day photosp here, Z, and ratio of metals to hydrogen mass fraction,
+Z/X,evaluatedoverthe last twentyyears.
+Year Source Z Z /X
+1989 Anders&Grevesse(1989) 0 .0201 0.0274
+1993 Grevesse&Noels(1993) 0 .0179 0.0244
+1998 Grevesse&Sauval(1998) 0 .0170 0.0231
+2005 Asplundet al.(2005) 0 .0122 0.0165
+2009 Asplundet al.(2009) 0 .0134 0.0181
+2009 Ludwiget al.(2009) 0 .0154 0.0209
+aValues for Zare inferred from Z /Xassuming Y=0.248.
+bUncertainties are<10%.
+twentyyears.
+Compared to the older GS98 and GN93 abun-
+dances, the Asplundet al. (2005, hereafter AGS05)
+abundance of C is lower by 35%, N by 27.5%, O by
+48%, and Ne by 74%. The abundances of elements
+from Na to Ca are lower by 12 to 25%, and Fe is de-
+creased by 12%. For the GS98 abundances, the ratio
+oftheelementtohydrogenmassfractionZ /X=0.023,
+and the heavy element abundance Z ∼0.018, while,
+forthenewabundances,Z /X=0.0165,andZ∼0.0122.
+Models evolved with the AGS05 abundance mixture
+give worse agreement with helioseismic constraints;
+the sound-speed discrepancy is 1.4% below the CZ
+base,andtheCZdepthisshallowandCZheliumabun-
+dance is low compared to those derived from seismic
+inversions.
+Recently, Asplundet al. (2009, hereafter AGSS09)
+re-evaluated the spectroscopic abundances, carefully
+considering the atomic input data and selection of
+spectral lines and using improved radiative trans-
+fer and opacities. They revised the heavy element
+abundance to Z/X=0.0181 and Z=0.0134. This
+slight increase over the AGS05 values yields some
+improvement in agreement with seismic constraints
+(see Serenellietal. 2009), although the higher abun-
+dances of GN93 or GS98 still provide the best agree-
+ment. In addition, the Cosmological Impact of the
+FIrst STars Team and its collaborators used the 3D
+model atmosphere code CO5BOLD to perform an
+independent investigation of the solar abundances
+(Caffauet al.2008a,b,2009;Ludwiget al.2009,here-
+after CO5BOLD). They derived heavy element abun-
+dances of Z/X=0.0209 and Z=0.0154, in between
+theAGSS09andGN93values.Spectroscopic determinations measure the pho-
+tospheric abundances. The continuous convective
+overshoot into the photosphere should leave the
+photosphere with the same abundances as the con-
+vection zone. In addition, convective timescales
+are much shorter than element di ffusion and evo-
+lution timescales, making the convection zone well
+mixedandhomogeneous. Thereforethespectroscopic
+abundances should be indicative of the abundances
+throughout the convection zone. However, we are re-
+luctant to dismiss the recent abundance re-analyses
+because of the many improvementsin the physicsand
+models included, namely 3D dynamical atmosphere
+models,non-localthermodynamicequilibriumcorrec-
+tions for important elements, and updated atomic and
+molecular data. Line profile shapes now agree nearly
+perfectlywith observations. Also, it is impressivethat
+abundances derived from several di fferent atomic and
+molecular lines for the same element now are consis-
+tent.
+For this paper, we will focus on the AGS05 abun-
+dancesasthemostextremeexampleofthelowerabun-
+dance determinations. However, we will devote sec-
+tion 7 to a preliminary exploration of the CO5BOLD
+abundances. We first review the results of helioseis-
+mic tests using the old and new abundances and on-
+goingattemptsto resolvethese discrepancies(Section
+1). We discuss in detail the followingthree mitigation
+attempts: an early mass-loss phase in solar evolution
+(Section 3), accretion of low-Z material early in the
+sun’s lifetime (Section 4), and extending the CZ be-
+lowthedepthinferredbyhelioseismology(Section5).
+We thenexamineincludingsomeoftheseadjustments
+in models with the higher metallicity of CO5BOLD
+2(Section 7). The models used to explore all of these
+changesaredescribedin Section2.
+1.2. Helioseismologyand solarmodels
+Helio- and asteroseismology have turned out to be
+excellent tools to test the physics of stellar models.
+Basu &Antia (2004); Bahcall& Pinsonneault(2004);
+Turck-Chi` ezeet al. (2004) authored some of the first
+papers to examine the e ffects of the new lower abun-
+dances on solar models, demonstrating that the new
+abundances lead to greater discrepancy with seismic
+inferences. This is demonstrated in Table 2 which
+compares our calibrated evolution models using the
+GN93andAGS05mixtures(detailsofourmodelscan
+be foundin Section 2). For the AGS05 model,the CZ
+heliumabundanceYislow(0.22730)andtheCZbase
+(RCZB=0.72944R⊙) is shallow comparedto the seis-
+mically inferredCZ Y abundanceof 0.248 ±0.003and
+CZbaseradiusof0.713 ±0.001R⊙fromBasu & Antia
+(2004).
+Figure1showsthedi fferencesbetweeninferredand
+calculated sound speed for calibrated evolved mod-
+els using the old and new abundances. The inferred
+sound speed is from Basuet al. (2000). The uncer-
+tainties in sound speed inversions are much smaller
+than the differences between these curves, at most a
+few widths of the plotting line. Figure 2 shows the
+observed minus calculated frequency di fferences for
+modes of angular degree ℓ=0, 2, 10, and 20 that
+propagate into the solar interior below the convection
+zone. Thecalculatedfrequencieswerecomputedusing
+thePesnell(1990)non-adiabaticstellarpulsationcode.
+TheobservationaldataarefromBiSON(Chaplinetal.
+2007), LowL (Schou&Tomczyk 1996), or GOLF
+(Garciaetal. 2001). The observational uncertainties
+forthemodesarelessthan0.1 µHz,muchsmallerthan
+themodeldiscrepancies. TheO-Ctrendsforthemodel
+with the Fergusonet al. (2005) low-temperatureopac-
+ities are flatter for higher frequencies that are more
+sensitive to the solar surface. These newer opacities
+include an improved treatment of grains, finer wave-
+lengthspacing,andadditionalmolecularlines,andare
+higherby12%uptoafactorofthree. Asillustratedby
+bothoftheseplots,thediscrepancywiththenewabun-
+dancesismuchlargerthanwiththeoldabundances.
+Helioseismology has also been used to investigate
+heavyelementabundances. Lin&D¨ appen(2005)find
+that a reduction in carbon abundance, in the direc-
+tion of the AGS05 abundances, can improve sound-0.00500.0050.010.015
+0 0.2 0.4 0.6 0.8 1GN93
+AGS05c
+s Difference
+(Sun - Model)/Sun
+R (R
+sun)CZ base
+Fig. 1.— Difference between inferred and calculated sound
+speeds with error bars for models with the GN93 and AGS05 abun -
+dances. The sound speed inversion is from Basu et al. (2000).
+The seismically inferred convection zone base at R =0.713 R⊙
+(Basu &Antia 2004) isshown with thevertical line.
+-20246810
+500 1000 1500 2000 2500 3000 3500GN93 with 2005 low-T opacities
+AGS05
+GN93 with 1995 opacitiesObserved minus Calculated
+Frequency ( µHz)
+Calculated Frequency ( µHz)
+Fig. 2.— Observed minus calculated versus calculated frequency
+for our models with the GN93 mixture (triangles or circles), and
+the AGS05 mixture (diamonds) for modes of degree ℓ=0, 2, 10,
+and 20. The models producing the triangle and diamond points use
+the newer Ferguson et al. (2005) low-temperature opacities , while
+the model with the circle points uses 1995 Alexander & Fergu-
+son opacities. The calculated frequencies were computed us ing the
+Pesnell (1990) non-adiabatic stellar pulsation code, and t he data are
+from Chaplin et al. (2007); Schou &Tomczyk (1996); Garcia et al.
+(2001).
+3speed inference. However, Linet al. (2007) show that
+lower Z increases the discrepancy with adiabatic in-
+dex inversions; Antia &Basu (2006) use the ioniza-
+tion signature in the sound speed derivative to infer
+ZCZ=0.0172±0.002. These results favor the old,
+higher abundances. In addition, Chaplinet al. (2007)
+use small frequency separations between low-degree
+modes that are sensitive to the core structure to con-
+strainthecoreZabundance. TheyfindZ core=0.0187-
+0.0239. Zaatrietal. (2007) find that the mean low-
+degree frequency spacings of a model using AGS05
+abundances are incompatible with those determined
+from the GOLF measurementsof Lazreketal. (2007)
+and Gellyet al. (2002). Similarly, Basu et al. (2007)
+find that models constructed with low metallicity are
+incompatiblewiththesmallfrequencyspacingandfre-
+quency separation ratios calculated from BiSON data
+(Chaplinet al. 1996). These results do not rule out
+accretion, enhanced di ffusion, or other options that
+can retain high core Z. However, they do disfavor the
+prospect that the new, lower abundanceswere present
+initiallythroughoutthesun.
+1.3. Attempts to restore agreement through solar
+modelmodifications
+Herewebrieflyreviewrecentattemptstoadjustso-
+lar models in order to mitigate the discrepancy with
+seismic constraints for the new abundances. For a
+moredetailedreviewofpreviousattempts,see,forex-
+ample, Basu &Antia (2008). In the following sec-
+tions, we discuss increased opacities below the CZ
+(11−30%); increased neon abundance ( ×∼4); in-
+creased abundances (within uncertainty limits, or us-
+ingalternativedeterminations);enhanceddi ffusiveset-
+tling rates (×1.5 or more); accretion of lower-Z ma-
+terial early in the sun’s lifetime; structure modifica-
+tion below the CZ base due to radiative damping of
+gravity waves; tachocline mixing (also used with old
+abundances); convectiveovershoot; and combinations
+oftheabove. Theconclusionhasbeenthatitisdi fficult
+to match simultaneously the new Z /X and helioseis-
+micconstraintsforCZdepth,soundspeedanddensity
+profiles, and CZ helium abundance by applying these
+changes.
+1.4. Opacityincreases
+Heavy-element abundances primarily a ffect solar
+structurethroughtheir e ffect on opacity,which a ffects
+the structure of the radiative zone and the locationof the CZ base. The structure of most of the con-
+vection zone is essentially independent of the opac-
+ity. Christensen-Dalsgaardet al. (2009) determined
+the change in opacity required to restore the sound-
+speed agreement of a solar model using the AGS05
+abundances to the level of success originally attained
+with the GN93 abundances. They find that opacity
+would need to be increased ranging from about 30%
+belowtheCZ toa fewpercentinthecore.
+Although improvements to the solar model can
+be made by ad hocopacity increases, there is lit-
+tle justification for such large enhancements. The
+presently available opacity tables from three sepa-
+rate projects (OPAL, OP, and LANL T-4) for con-
+ditions below the CZ di ffer by only a few percent
+(Neuforge-Verheeckeet al.2001; Badnelletal. 2005),
+making it difficult to justify such large opacity en-
+hancements. Using the Los Alamos National Lab-
+oratory T-4 opacity library data (Mageeet al. 1995;
+Hakelet al. 2006), Guziketal. (2009) find that,to ob-
+tain a 30% opacity increase with the new abundances,
+the contribution from oxygen alone would need to in-
+crease by a factor of two to three. Alternatively, the
+ironabsorptioncontributionwouldneedtoincreaseby
+a factorofthree.
+Including additional elements has a negligible ef-
+fect on Rosseland mean opacities for solar interior
+conditions. The Lawrence LivermoreOPAL opacities
+for the AGS05 mixture included 17 of the most abun-
+dant elements. With the LANL T-4 opacity library,
+Guziketal. (2009) find that including in the mixture
+all of the elements up to atomic number Z =30 in-
+creases the mixture opacity by only 0.2% for solar in-
+terior conditions. Includingadditionalelements in the
+mixturefrom30<Z<93,an83-elementmixture,fur-
+ther increases the mixture opacity by less than 0.1%.
+An as-yet unidentified error in calculationsof the line
+wingsforK-shelltransitionsin O,C, N, andNe at en-
+ergies around 800 eV, the peak of the weighting of
+the Rosseland mean opacity for the temperature con-
+ditionsat the CZ base,couldprovidesomeincreasein
+the calculated Rosseland mean opacities (Guziketal.
+2009).
+Inordertoshedlightontheissue,Turck-Chi` ezeet al.
+(2009) suggest experimental investigations of opac-
+ity coefficients for the radiative zones of solar-like
+stars. The cominglargelaser facilities (LIL +PETAL,
+OMEGA EP, FIREX II, LMJ, NIF) have the potential
+toattainsufficientlyhightemperaturesandhighdensi-
+tiesat LTEalongwiththeprecisediagnosticsrequired
+4for stellar opacity measurements. Experiments to in-
+vestigate the opacities relevant for stellar interiors are
+being conducted at Sandia’s Z facility by Bailey etal.
+(2009).
+1.5. Neonandotherelementabundanceincreases
+For a while, it was thought that an increase in the
+solarneonabundanceprovidedthemostplausibleres-
+olution to this problem. Neon is not measured in the
+photosphere due to a lack of suitable spectral lines.
+Instead, its abundance is determined relative to oxy-
+gen using lines formed in the solar corona, XUV and
+gamma ray spectroscopy of quiet and active regions,
+and solar wind particle collections. AGS05 adopt a
+Ne/O abundance ratio of 0.15, and apply this ratio
+to the photospheric oxygen abundance to derive the
+neon abundance. The neon abundance has been re-
+vised downwardby74% fromthe GS98value,for the
+mostpartduetotheoxygenabundancereduction.
+Several groups explored solar models with en-
+hanced neon. Some improvement in agreement was
+shown for Ne increases ranging from 0.5-0.67 dex
+(Antia&Basu2005;Bahcallet al.2005a;Turck-Chi` ezeetal .
+2005; Zaatriet al. 2007; Delahaye&Pinsonneault
+2006). However, Linet al. (2007) find that increas-
+ing Ne alone actually increases the discrepancy in the
+adiabatic exponent in the region 0.75-0.9 R ⊙. They
+find that the discrepancy is reduced if only C, N, and
+O abundancesareincreased.
+MoremodestNe enhancements,combinedwith in-
+creasesinthe otherelementabundancesof ∼0.05dex,
+atthelimitoftheAGS05uncertainties,havealsobeen
+considered. The best model of Bahcall etal. (2005a)
+has Ne enhanced by 0.45 dex (2.8 ×), Ar by 0.4 dex,
+andC,N,andOby0.05dex. Thismodelproducesrea-
+sonablygoodagreementwiththeinferredsoundspeed
+and density profile, and has CZ base radius 0.715 R ⊙
+andacceptableCZY =0.2439. Ofcourseanyincrease
+inabundancesfromtheAGS05valuewill mitigatethe
+problembyincreasingopacitiesbelowtheCZ.Forex-
+amples, see Zaatrietal. (2007) or Turck-Chi` ezeetal.
+(2004).
+1.6. Enhanced di ffusion
+Severalgroups,e.g.,Basu &Antia(2004),Montalb´ anetal.
+(2004), Guziket al. (2005, hereafter GWC05), and
+Yang& Bi (2007) considered the e ffects of enhanced
+diffusion. At first this idea might seem promising,be-
+cause the solar interior could have higher abundancesthat give good sound speed agreement, while the CZ
+elementscouldbedepletedtotheAGS05photospheric
+values. Inpractice,therequireddi ffusionincreasesare
+quite large (factors of 1.5 to 2 on absolute rates), and
+enhanced diffusion also depletes the CZ Y abundance
+towellbelowtheseismicdeterminationandleavesthe
+CZ tooshallow.
+GWC05 investigate the enhancement of thermal
+diffusion for elements and He by di fferent amounts.
+A modelwithresistance coe fficients×1/4forC, N,O,
+Ne,andMgand×2/3forHeshowssomeimprovement
+in the sound-speed discrepancy, but the CZ depth is
+still a little shallow (0.718 R ⊙), and the CZ Y is still
+a little low (0.227). Moreover,there is no justification
+for these ad hocchanges in thermal di ffusion coeffi-
+cients.Thediffusioncoefficientsthemselvesshouldnot
+beinerrorbysucha largefactor.
+1.7. Gravitywavesand dynamicale ffects
+Arnett,Meakin,& Young(2006,privatecommuni-
+cation)havebeeninvestigating,followingPress(1981)
+and Press&Rybicki (1981), the e ffects of gravity
+waves excited and launched inward at the CZ base.
+The radiative damping of these waves as they travel
+inwarddepositsenergyandchangesthesolarstructure
+in the same way as would an opacity enhancement.
+The amount of damping and distance that the waves
+propagate depend on the initial amplitudes and the
+degree of the mode, with low-frequency, high-degree
+waves damped more heavily, after traveling a shorter
+distance. Theexpectedwavespectrumandamplitudes
+still needto beworkedout,but couldremoveas much
+as one third of the sound-speeddiscrepancy. More re-
+cently, Arnettetal. (2009) have re-examined convec-
+tion at the surface and sub-surface layers of the sun,
+proposinga way to eliminate astronomical calibration
+from stellar convection theory. By choosing charac-
+teristic lengths that are determined by the flow, they
+eliminatetheneedforthefreeparameterstraditionally
+used in mixing length theory. They show that some
+of the discrepancy between the new abundances and
+helioseismicinferencesmayresultfromtheneglectof
+hydrodynamicprocessesinthestandardsolar model.
+1.8. Combinationsofe ffects
+In addition to single changes to solar models,
+several groups considered combinations of changes,
+such as diffusion, opacity, and abundance enhance-
+ments. See, for example, Basu &Antia (2004),
+5Montalb´ anet al. (2004),andBahcall etal. (2006).
+Although the above modifications to the input
+physicshaveachievedsomesuccessinrestoringagree-
+ment between seismic constraintsand modelsthat use
+the new abundances, agreement is not fully restored
+andthereislittlephysicaljustificationfortheproposed
+changes. In this paper, we discuss the motivation for
+and the results of three additional attempts to restore
+agreement: anearlymasslossphase,accretionoflow-
+Zmaterial,andconvectiveovershoot.
+2. Solarevolutionmodels
+Solarmodelsrequireinputdataforopacitiessuchas
+OPAL(Iglesias&Rogers1996)orOP(Seaton&Badnell
+2004) supplemented by low-temperature opacities
+(e.g., Fergusonet al. 2005); equation of state such
+as OPAL (Rogerset al. 1996), MHD (D¨ appenetal.
+1988), or CEFF (Christensen-Dalsgaard& D¨ appen
+1992);nuclearreactionrates(e.g.,Anguloetal.1999);
+and diffusive element settling (e.g., Burgers 1969;
+Coxet al.1989; Thoulet al.1994).
+The solar models produced by the Los Alamos
+group shown hereare evolved from the pre-main se-
+quenceusinganupdatedversionoftheone-dimensional
+evolution codes described in Iben (1963, 1965a,b).
+TheevolutioncodeusestheSIREFFEOS(seeGuzik&Swenson
+1997), Burgers’ di ffusion treatment as implemented
+by Cox etal. (1989), the nuclear reaction rates from
+Anguloet al. (1999) with a correction to the14N rate
+from Formicolaetal. (2004), and the OPAL opac-
+ities (Iglesias& Rogers 1996) supplemented by the
+Fergusonet al. (2005) or Alexander & Ferguson (pri-
+vatecommunication,1995)low-temperatureopacities.
+The Fergusonet al. (2005) low-temperature opacities
+include an improved treatment of grains, finer wave-
+length spacing, and additionalmolecularlines and are
+higher than the Alexander & Ferguson (1995) low-
+temperature opacities by 12% up to a factor of three.
+As discussed in Section 1.2, the observed minus cal-
+culated frequency trends for a model with the newer
+low-T opacities are flatter for higher frequencies that
+aresensitivetothe solarsurface,asseenin Figure2.
+Themodelsarecalibratedtothepresentsolarradius
+6.9599×1010cm (Allen1973), luminosity3.846 ×1033
+erg/g(Willsonet al.1986),mass1.989 ×1033g(Cohen&Taylor
+1986), age 4.54±0.04Gyr (Guentheretal. 1992), and
+adoptedphotosphericZ /Xratio. Forevolutionmodels,
+the initial helium abundance Y, initial element mass
+fraction Z, and mixing length to pressure-scale-heightratioαareadjustedsothatthefinalluminosity,radius,
+andsurfaceZ/Xmatchtheobservationalconstraintsto
+within uncertainties. See Guziket al. (2005) for ref-
+erences and a description of the physics used in the
+evolutionandpulsationcodesandmodels.
+3. Mass loss
+3.1. Motivationandmethod
+Willsonet al. (1987) explored the possibility that
+significant mass loss could occur during the early
+part of the main-sequence for ∼1-2.5 M⊙stars. They
+considered mass-loss rates ranging from 10−9to 10−8
+M⊙/yrthatwouldremoveasubstantialfractionofmass
+from a star before it evolves o ffthe main sequence.
+Mass loss in these stars, possibly including the early
+sun, would be driven by pulsation, which provides
+the necessary mechanical energy flux, and facilitated
+by rapid rotation. The mass-loss rate would diminish
+upon the development of a surface convection zone,
+which channels mechanical energy away from pulsa-
+tion,andofmagneticfieldswhichprovideangularmo-
+mentum transfer and rotational braking. Guziketal.
+(1987) showed that mass-losing solar models have
+steeper molecular weight gradients, shorter main-
+sequence lifetimes, higher8B neutrino fluxes, deeper
+surface convection, higher surface3He abundances,
+and earlier, more pronounced dredge-up of CN cycle
+processed material in the post-main-sequence phase
+compared to the standard model. In addition, mass-
+losing models predict complete destruction of pro-
+tosolar Li and Be, requiring a mechanism, such as
+production in spallation reactions or flares, for partial
+replenishmentto theobservedsurfaceabundances.
+Such mass loss in other stars could potentially ex-
+plain blue stragglers and the earlier-than-predicted
+dredge-up of carbon and nitrogen in solar-mass stars
+ascending the first red giant branch (see Guzik 1988).
+However, there are also drawbacks. If the sun re-
+mains at too high a mass for too long, all surface Li
+in the subsequent solar model at present age is de-
+stroyed, too much surface3He is produced, and dis-
+crepancies with the inferred sound speed arise (Guzik
+1988). In addition, Guzik&Cox (1995) compare ob-
+served and calculated p-mode oscillation frequencies
+to test the structure of solar models including early
+main-sequence mass loss. They show that extreme
+solar mass loss has a significant e ffect on solar struc-
+ture and can be ruled out by the p-mode oscillation
+frequencies.
+6Whilemoreextremeearlymain-sequencemassloss
+hasnot been observed,smaller mass lossin the sun of
+about 0.1 M⊙looks promising to solve a number of
+problems. The advantagesofan earlymassloss phase
+in solar evolution include: solving the faint early sun
+problem, explaining early liquid water on Mars, early
+innersolarsystembombardment,andsolarlithiumde-
+struction.
+Models with early mass loss using the older,
+higher element abundances were explored previously
+by Guziketal. (1987), Swenson&Faulkner (1992),
+Guzik&Cox (1995), and Sackmann& Boothroyd
+(2003). In addition, Minton&Malhotra (2007) re-
+cently assessed consequences for Earth climate and
+solar system formation. Here we re-investigate so-
+lar mass loss in light of the new abundances. In our
+models, we use the mass-loss treatment implemented
+by Brunish (1981) in the Iben (1963, 1965a,b) code.
+We evolve two models with initial masses 1.3 and
+1.15 M⊙, having exponentially-decaying mass-loss
+rates with an e-folding time of 0.45 Gyr. Following
+Guziketal. (1987), we adopt this exponential mass-
+loss prescription because it is simple and decreases
+smoothly with time. This is a physically plausible de-
+scription,asthemass-lossrateshouldbehighestwhen
+a rotatingstar arrivesonthe main sequencewithin the
+instability strip, where pulsation and rotation can fa-
+cilitate mass loss. The mass loss shouldthen decrease
+as the star moves out of the instability strip, ceases
+to pulsate, spins down, and develops a surface con-
+vection zone. The present solar mass-loss rate is 2
+×10−14M⊙/yr(e.g.,Feldmanetal.1977),toosmallto
+affect the sun’s evolution. The initial mass-loss rates
+for the two models are 6.55 and 3.38 ×10−10M⊙/yr,
+respectively.
+3.2. Results
+Table2summarizestheinitialYandmixing-length
+parameterneededtocalibrateeachmodelandthefinal
+CZYandCZbaseradius. Theseismically-inferredCZ
+YabundanceandCZbaseradiusare0.248 ±0.003and
+0.713±0.001 R⊙, respectively (Basu& Antia 2004).
+Figure 3 shows the luminosity versus time for these
+models, as well as for two constant, one solar-mass
+calibrated models. Figure 4 shows the inferred mi-
+nus calculated sound speed for these models. For the
+models with the AGS05 abundances, the sound speed
+agreementisconsiderablyimprovedbyincludingearly
+mass loss. For the model with initial mass 1.3 M ⊙
+sound-speedagreementisalmostrestoredneartheCZ0.511.522.5
+0 1 2 3 4GN93
+AGS05
+M
+0 = 1.3 M
+sun
+M
+0 = 1.15 M
+sunLuminosity (L
+sun)
+Time (Gyr)
+Fig. 3.— Luminosity versus time for standard one solar-mass
+models using the GN93 and AGS05 abundances and for two mass-
+losingmodelsusingtheAGS05abundanceswithinitialmass1 .3and
+1.15 M⊙. Mass-loss rates are exponentially decaying with e-foldin g
+time 0.45 Gyr.
+-0.00500.0050.010.015
+0 0.2 0.4 0.6 0.8 1GN93
+AGS05
+M
+0 = 1.3 M
+sun
+M
+0 = 1.15 M
+sunc
+s Difference
+(Sun - Model)/Sun
+R (R
+sun)
+Fig. 4.— Inferred minus calculated sound speed di fferences
+for calibrated standard one solar-mass models using the GN9 3 and
+AGS05 abundances and for models with AGS05 abundances and
+initial mass1.3 and 1.15 M ⊙including early mass loss.
+base, but the agreement is not as good in the more H-
+depleted core. Unfortunately, while the model with
+initial mass 1.15 M ⊙has a little better sound speed
+agreement in the central 0.1 R ⊙, the improvement is
+70246810
+500 1000 1500 2000 2500 3000 3500GN93
+AGS05
+M
+0 = 1.3 M
+sun
+M
+0 = 1.15 M
+sunObserved minus Calculated
+Frequency ( µHz)
+Calculated Frequency ( µHz)
+Fig. 5.— Observed minus calculated versus calculated frequen-
+cies for calibrated standard one solar mass models using the GN93
+and AGS05 abundances, and for models with AGS05 abundances
+with initial mass 1.3 and 1.15 M ⊙including early mass loss. Fre-
+quencies compared are for modes of angular degrees ℓ=0, 2, 10,
+and 20. The data are from Chaplin et al. (2007), Schou &Tomczy k
+(1996), and Garcia et al. (2001). The calculated frequencie s were
+computed using the Pesnell (1990) non-adiabatic stellar pu lsation
+code.
+not as pronounced for the region below the CZ. Fig-
+ure 5 showsthe observedminuscalculatedversuscal-
+culatednonadiabaticfrequenciesformodesofangular
+degreesℓ=0,2,10,and20thatpropagateintotheso-
+larinteriorbelowtheconvectionzone. Includingmass
+loss improves agreement, but models with old abun-
+dancesandnomasslossstill givethebestagreement.
+The mass-losing models described here would de-
+stroy all of the observed surface lithium. Lithium
+is destroyed relatively rapidly in the solar interior at
+temperatures≥2.8 million K. For standard models,
+on the main sequence the surface layers are never
+mixed to high enough temperatures to deplete Li by
+theobservedfactorof150fromtheinitialsolarsystem
+abundance (Asplundet al. 2009), and additional mix-
+ingmechanismsmustbeinvoked. However,withmass
+loss,layersthatarenowatthesurfacewereinitiallyin
+theinteriorattemperatureshighenoughtoquicklyde-
+stroy Li. Figure 6 shows the temperature experienced
+by the surface layer throughout the evolution of each
+model. Duringthemass-losingphase,thehightemper-
+atures that depleted the lithium in the current surface
+layerwereexperiencedwhenthematerialthatisatthe
+surface of the now 1 M ⊙sun was deeper, before the23456
+0 1 2 3 4GN93
+AGS05
+M
+0 = 1.3 M
+sun
+M
+0 = 1.15 M
+sunTemperature (106 K)
+Time (Gyr)
+Fig. 6.— Temperature experienced by the present-day solar sur-
+face layer as a function of time for the mass-losing and stand ard
+models. For the mass-losing phase, the lithium-destroying temper-
+atures are attained because the layer that is now at the surfa ce once
+resided deeper inside the sun. In the post-mass-loss phase, the rele-
+vant temperatures are attained by envelope convection whic h mixes
+surface layers downward, exposing the surface material to t he tem-
+perature at the CZ base. 2.8 million K is the temperature requ ired
+for relatively rapid Lidestruction.
+previous surface layers were lost. After the mass-loss
+phase, the temperature that a ffects the surface layer is
+that of the CZ base, since the material currentlyat the
+surfaceiscontinuallymixedthroughtheCZ.
+The mass-losing models also produce more3He
+at the surface as the now-surface layers were once
+processed at higher interior temperatures where3He
+buildsuptohigherequilibriumvalues. Forthe1.3M ⊙
+initialmassmodel,thesurface3Hemassfractionisen-
+hancedfromitsinitialvalueof5.0 ×10−5to9.0×10−5,
+while for the 1.15 M ⊙initial mass model, the surface
+3He mass fraction is only slightly enhanced from its
+initial value of 5.0 ×10−5to 5.1×10−5. The final
+3He/4Heabundanceratiosforthe1.3and1.15M ⊙ini-
+tial massmodelsare3.9 ×10−4and2.2×10−4,respec-
+tively,enhancedfromaninitial valueof2.0 ×10−4.
+4. Accretionoflow-Zmaterial
+4.1. Motivationandmethod
+Asawaytokeepthesolarinteriormorelikemodels
+obtained using higher abundances, Guziket al. (2004,
+hereafterGWC04)andGWC05 proposedaccretionof
+8materialdepletedinheavierelementsearlyinthesun’s
+lifetime. In this scenario, the pre-main sequence sun
+would have∼98% of its present mass and a higher Z
+with a mixture similar to the GN93 or GS98 abun-
+dances. After the sun begins core hydrogen burn-
+ing and is no longer fully convective, the remaining
+∼2% of material accreted would have lower Z provid-
+ing a convection-zone abundance similar to the cur-
+rent photospheric abundances of AGS05 or AGSS09.
+Possible justifications for this scenario are discussed
+by Nordlund (2009) and Mel´ endezetal. (2009). One
+plausibleexplanationisthatplanetformationremoves
+some high-Z elements from the solar nebula, leaving
+lower-Z material to be accreted after the sun is no
+longerfullyconvective.
+Winnicket al. (2002) explored the accretion of
+metal-rich material in models with the older GS98
+abundances. They show that some solar models with
+enhanced metallicity in the convection zone might be
+viable as small perturbations to the standard GS98
+model. Haxton& Serenelli (2008) discuss the ques-
+tionof accretionof metal-depletedgasontothesun as
+a motivation for future experiments to measure CN-
+cycle neutrinos. The flux of these neutrinos should
+dependnearlylinearlyontheinitialcoreabundanceof
+C and N. A successful measurement of the CN-cycle
+solar neutrino flux would therefore place constraints
+onpossibleaccretionbydeterminingthemetallicityof
+thesolarcore.
+Totestthepossibilityoflow-Zaccretionintheearly
+sun, a model is evolved starting with Z=0.0197 on
+the zero-age main sequence, and material is progres-
+sively added to reduce the CZ Z by 0 .001 in each of
+six steps of about six million years. After each step,
+the model is given time to equilibrate to a new shal-
+lowerCZdepth,leavingbehindthehigher-Zcomposi-
+tiongradient(Guzik2006). Thefinalaccretionepisode
+leavestheCZwith Z=0.0137. After36millionyears
+oflow-Zaccretion,themodelisevolvednormally(in-
+cludingdiffusivesettling)andcalibratedasusualtothe
+observedluminosity,radius,andAGS05 Z/Xvalue.
+4.2. Results
+Figure 7 shows the heavy element abundance
+throughoutthe sun in the accretion model, and Figure
+8showstheYabundance. Asintended,theabundance
+in the interior is similar to the GN93 model while the
+CZ abundance matches the new, lower Z from photo-
+sphericobservations.0.0120.0140.0160.0180.020.022
+0 0.2 0.4 0.6 0.8 1GN93
+AccretionZ
+R (R
+sun)
+Fig. 7.— HeavyelementabundanceprofilesfortheGN93mixture
+model and the low-Z accretion model.
+0.240.2480.2560.2640.2720.280.2880.296
+0 0.2 0.4 0.6 0.8 1GN93
+AccretionY
+R (R
+sun)
+Fig. 8.— Heliumabundanceprofiles fortheGN93mixturemodel
+and thelow-Z accretion model.
+Figure9showstherelativesound-speeddi fferences
+for models with the GN93 mixture, the AGS05 mix-
+ture, andlow-Z accretion. The accretionmodel shows
+improvementinsound-speedagreementinthe interior
+where Z is similar to the GN93 mixture. However,
+discrepancy remains near the CZ base. Compared to
+theAGS05model,theaccretionmodelhasalessshal-
+low CZ base radiusof 0.7235R ⊙and a nearlyaccept-
+9-0.00500.0050.010.015
+0 0.2 0.4 0.6 0.8 1GN93
+AGS05
+Accretionc
+s Difference
+(Sun - Model)/Sun
+R (R
+sun)
+Fig. 9.— Relative difference between inferred and calculated
+sound speeds for models with the GN93 and AGS05 abundances
+and with low-Z accretion.
+able CZ Y abundance of 0 .2407. Figure 10 shows
+the observed minus calculated versus calculated non-
+adiabatic frequencies for modes of angular degrees ℓ
+=0,2,10,and20that propagateintothesolar interior
+below the CZ. Including accretion in a model using
+thenewabundancesimprovesagreementwiththisfre-
+quencydata.
+Castroet al. (2007) also approximatedan accretion
+model by instantaneouslydecreasingthe Z abundance
+gradient in the CZ in an early main-sequence model
+(age 74 My). They do not find an improvementin the
+CZ depth, as we did for our model, but find about the
+same CZ Y abundance, 0 .240, and improved sound-
+speedagreementbelow0.5R ⊙.
+5. Convectiveovershoot
+5.1. Motivationandmethod
+It is possible that the CZ depth predicted using
+standardmixing-lengththeoryis tooshallow andcon-
+vective motions extend the nearly adiabatically strat-
+ified part of the CZ to the depth inferred seismically.
+Rempel (2004) presents a semi-analytical model of
+overshoot. This approach facilitates the understand-
+ing of the relation between numerical simulationsand
+classical overshoot theories in terms of physical pa-
+rameters. Montalb´ anet al. (2006) developed solar
+models that include convective overshoot. By adopt-0246810
+500 1000 1500 2000 2500 3000 3500GN93
+AGS05
+AccretionObserved minus Calculated
+Frequency ( µHz)
+Calculated Frequency ( µHz)
+Fig. 10.— Observed minus calculated frequency versus calcu-
+lated ferquency of GN93, AGS05, and accretion models for de-
+greeℓ=0, 2, 10, and 20 modes. The calculated frequencies were
+computed using the Pesnell (1990) non-adiabatic stellar pu lsation
+code. The data are from Chaplin et al. (2007), Schou &Tomczyk
+(1996), and Garcia etal. (2001). The accretion model shows i m-
+proved, though notacceptable, O-C agreement.
+ing an overshoot parameter of the order of 0.15 times
+the pressure scale height and increasing the opacity
+by∼7% (within the uncertainty limits of the abun-
+dances), they were able to reproduce the seismically
+inferred CZ base and Y CZ. However, large sound-
+speed discrepancies remain in the radiative region of
+theirmodel.
+To explore the possibility of convective overshoot,
+weevolvemodelswith AGS05abundancesbutextend
+the CZ that follows the adiabatic gradient to a depth
+that optimizes agreement with the sound speed inver-
+sions. We hoped that a deeper CZ would also inhibit
+diffusionandkeepthe CZY abundancehigher.
+5.2. Results
+Figure 11 shows the relative sound-speed di ffer-
+ences for models with the GN93 mixture, the AGS05
+mixture, and the first convective overshooting model.
+For the first overshoot model, the CZ depth is 0.704
+R⊙, deeperthaninferredseismically. Thesoundspeed
+agreement is improved only within the CZ but not
+muchbelowit. ThedeeperCZdoesnotinhibitYdi ffu-
+sionaswehadhoped. Inthesecondmodel,weextend
+the CZ even deeper, to 0.64 R ⊙. Figure 12 shows the
+relativesound-speeddi fferenceforthesecondconvec-
+10-0.00500.0050.010.015
+0 0.2 0.4 0.6 0.8 1GN93
+AGS05
+Overshootc
+s Difference
+(Sun - Model)/Sun
+R (R
+sun)
+Fig. 11.— Relative difference between inferred and calculated
+soundspeedsformodelswiththeGN93andAGS05abundancesan d
+the first convective overshoot model that extends the adiaba tically
+stratified layer to 0.704 R ⊙.
+tive overshootingmodelandstandardmodelswith the
+GN93 and AGS05 mixtures. The sound speed gradi-
+entatthebaseofthisadiabaticallystratifiedCZclearly
+does not agree with the seismically inferred one, and
+sound speed agreement in the central 0.2 R ⊙is much
+worse than in any of the other models. We therefore
+omit the second overshoot model from further analy-
+sis. Figure 13 shows the observed minus calculated
+frequency for the first overshoot model compared to
+theGN93andAGS05models. TheAGS05modeland
+the overshoot model where the core Z is low do not
+agree as well with the data as the GN93 model does.
+It appears that overshooting alone is not a solution to
+thisproblem.
+6. Summary ofmitigationattempts
+Table 2 summarizesthe CZ Y, CZ base radius, and
+photosphereZ/Xforthemodelsexaminedhere. Figure
+14 shows the relative sound-speed di fferences of our
+models with the GN93 mixture, the AGS05 mixture,
+and models with mass-loss, accretion, and convective
+overshoot. Theobservedminuscalculatedfrequencies
+of these models are shown in Figure 15. As seen in
+Table 2 and Figures 14 and 15, the AGS05 model and
+overshootmodel do not agree well with the data. The
+GN93model,the1.3M ⊙masslossmodel,andtheac-
+cretionmodelshowbetteragreement,thoughnomodel-0.04-0.03-0.02-0.0100.010.02
+0 0.2 0.4 0.6 0.8 1GN93
+AGS05
+Overshoot 2c
+s Difference
+(Sun - Model)/Sun
+R (R
+sun)
+Fig. 12.— Relative difference between inferred and calculated
+soundspeedsformodelswiththeGN93andAGS05abundancesan d
+thesecondconvectiveovershootmodelthatextendstheadia batically
+stratified layer to 0.64 R ⊙.
+0246810
+500 1000 1500 2000 2500 3000 3500GN93
+AGS05
+OvershootObserved minus Calculated
+Frequency ( µHz)
+Calculated Frequency ( µHz)
+Fig. 13.— Observed minus calculated frequency versus cal-
+culated frequency for models with the GN93 and AGS05 abun-
+dances and for the overshoot model for degree ℓ=0, 2, 10, and 20
+modes. Thecalculated frequencies werecomputedusingtheP esnell
+(1990) non-adiabatic stellar pulsation code, and the data a re from
+Chaplin et al. (1998), Schou &Tomczyk (1996), and Garcia eta l.
+(2001).
+matchesthedataperfectly.
+Figure16showsthesmallfrequencyseparationdif-
+ferences of the Guzik et al. models minus the solar-
+11Table2:Initialmassandsurfaceabundances,mixinglength parameter,andfinalabundancesandCZbaseforoursolar
+models.
+Model GN93 AGS05 ML1 ML2 Accretion Overshoot C1 C2 C3
+Mo/M⊙1.00 1.00 1.30 1.15 0.98 1.00 1.00 1.00 1.10
+Yo 0.26930 0.25700 0.24659 0.25279 0.26927 0.25699 0.26370 0. 27780 0.27530
+Zo 0.01970 0.01350 0.01351 0.01351 0.01973 0.01351 0.01740 0. 01700 0.01700
+α 2.0379 2.0004 2.0571 2.0104 1.8958 1.9962 1.9918 2.0635 2.0 652
+Z/X 0.0240 0.0163 0.0178 0.0171 0.0162 0.0164 0.0209 0.0208 0. 0219
+YCZ 0.2412 0.2273 0.2388 0.2349 0.2402 0.2292 0.2349 0.2473 0.2 551
+RCZB/R⊙0.7125 0.7294 0.7217 0.7264 0.7241 0.7038 0.7186 0.7190 0.7 181
+aSeismically inferred values from Basu &Antia (2004): Y CZ=0.2485±0.0035, R CZB/R⊙=0.713±0.001.
+bModels ML 1 and ML 2 are the AGS05 models including mass loss. M odels C1, C2, and C3 are the CO5BOLD models with GN93 opacities,
+AGS05 opacities, and AGS05 opacities including mass loss.
+-0.00500.0050.010.015
+0 0.2 0.4 0.6 0.8 1GN93
+AGS05
+M
+0 = 1.3 M
+sun
+M
+0 = 1.15 M
+sun
+Accretion
+Overshootc
+s Difference
+(Sun - Model)/Sun
+R (R
+sun)
+Fig. 14.— Relative difference between inferred and calculated
+sound speeds for models with the GN93 and AGS05 abundances
+and models with mass-loss,accretion, and convective overs hoot.
+cycle corrected frequency di fferences from the Bi-
+SONgroup(Chaplinet al.2007)for ℓ=0and2modes,
+which are sensitive to the structure of the core. This
+plot illustrates that including low-Z accretion in the
+model retains the core structure of the GN93 model.
+The overshoot model and AGS05 model do not agree
+aswellwiththedataasthemodelswithhighercoreZ.
+The mass losing model with the best sound speed and
+observedminuscalculatedfrequencyagreement(M 0=
+1.3 M⊙) shows worse agreementthan any of the other0246810
+500 1000 1500 2000 2500 3000 3500GN93
+AGS05
+M
+0 = 1.3 M
+sun
+M
+0 = 1.15 M
+sun
+Accretion
+OvershootObserved minus Calculated
+Frequency ( µHz)
+Calculated Frequency ( µHz)
+Fig. 15.— Observed minus calculated frequency versus calcu-
+lated frequency for degree ℓ=0, 2, 10, and 20 modes in GN93 and
+AGS05 models and in models with mass-loss, accretion, and co n-
+vective overshoot. The calculated frequencies were comput ed us-
+ing the Pesnell (1990) non-adiabatic stellar pulsation cod e. The
+data are from Chaplin et al. (2007), Schou & Tomczyk (1996), a nd
+Garcia etal. (2001).
+models. However,mass loss does change the value of
+thesmallfrequencyseparationintherightdirectionto
+correct for the discrepancy seen with the new abun-
+dances. Perhaps the over-compensation indicates that
+this model has too much mass loss; a model with a
+smallerinitialmasswouldreducethedisagreement,as
+seenwith theM 0=1.15M⊙model.
+12-0.6-0.4-0.200.20.4
+9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25GN93
+AGS05M
+0=1.3 M
+sun
+M
+0=1.15 M
+sunAccretion
+OvershootCalculated - Observed
+small frequency separations
+between l = 0 and  l = 2
+Radial order for l = 0
+Fig. 16.— Differencebetweencalculated andobservedsmallsep-
+arations forℓ=0 and 2 modes for the Guzik et al. models. The data
+are fromChaplin et al. (2007).
+Mass-losing models can improve seismic agree-
+ment for the new abundances, but they do not fully
+restore agreement. In addition, the destruction of too
+much Li and the production of too much surface3He
+make the two models considered here unlikely. A
+smaller amount of mass loss that leads to destruction
+of some, but not all, of the initial Li could provide
+a plausible partial mitigation of the solar abundance
+problem.
+The accretion model allows for a solar interior that
+is similar to models developed with the higher abun-
+dances and therefore agrees nicely with seismic in-
+ferences in the central 0.5 R ⊙of the sun. The model
+stillshowspooragreementneartheCZbase. Thevery
+steep Z abundance gradient developed at the CZ base
+seen in Figure 7 (see Guzik 2006) might have a de-
+tectable signature in the seismic frequencies. Basu
+(1997) finds that inversions appear to rule out such
+steepcompositiongradientsatthe CZ base.
+The convective overshoot model with a CZ depth
+of 0.704R⊙improvessound speed agreement slightly
+within the CZ but not much below it. Extending the
+CZ depth to 0.64 R ⊙results in even worse agreement.
+In addition, Y diffusion is not inhibited by the deeper
+CZ, aswehadhoped.7. CO5BOLD abundances
+At the suggestion of both the referee and a col-
+league P. Bonifacio, we also include here a prelimi-
+naryexplorationof solarmodelsusing theCO5BOLD
+abundances with a Z /X of 0.0209 and Z of 0.0154,
+intermediate between the AGS05 and AGSS09 abun-
+dances,but lowerthantheGN93abundances.
+Because the abundances of only 12 elements have
+beenre-evaluatedatthistimebytheCO5BOLDgroup
+(Ludwigetal. 2009), it is a little premature to create
+new opacity tables for the CO5BOLD mixture, which
+can, in principle be done using the Lawrence Liver-
+moreOPAL web requestat http: //opalopacity.llnl.gov.
+In addition, we do not have low-temperatureopacities
+for a mixture representative of the CO5BOLD abun-
+dances available to us. Therefore, here we decided
+to calibrate standard models to the Z /X of CO5BOLD
+using opacity tables based on the GN93 and AGS05
+mixtures. We observe that the O /Fe (oxygen to iron)
+mass fraction ratio of CO5BOLD is 4.98, intermedi-
+ate between the O/Fe mass ratio of 6.56 for the GN93
+mixture and 4.65 for the AGS05 mixture, so our two
+standard models should bracket results that use the
+CO5BOLD mixture in the opacity tables. We did up-
+date abundances to the CO5BOLD values for our in-
+line equation of state calculation and for tracking the
+diffusionofthemajorelements.
+The initial Y, Z, and mixing length to pressure-
+scale-heightratiosneededtocalibratetotheCO5BOLD
+Z/X using either opacity set are listed in Table 2; Fig-
+ures 17, 18, and 19 show the results for sound-speed
+differences, small separations between calculated ℓ=0
+andℓ=2 modes, and observed minus calculated fre-
+quenciesforℓ=0,2,10,and20. Thesoundspeeddis-
+crepancyis reducedto only about0.6%at the convec-
+tion zone base for the CO5BOLD abundances, com-
+pared to 1.4% for the AGS05 abundances and 0.4%
+forthe GN93abundances. Theresultsforeither opac-
+ity set are identicalabove0.6R ⊙, but differbelowthis
+where oxygen is the main opacity contributor, and in
+the core where iron is the main opacity contributor,as
+expected. ThemodelusingtheAGS05opacitymixture
+requires a higher Y abundance to compensate for the
+relatively higher Fe opacity contribution in the core.
+The small separations and observed minus calculated
+frequenciesare slightly higher than foundfora model
+calibratedtotheGN93abundancesonaverage,butnot
+ashighasforamodelcalibratedtotheAGS05Z /X.
+Asalsosurmisedbythereferee,sincetheCO5BOLD
+13sound speed differences are closer to observed, im-
+provement could be obtained with a smaller amount
+of mass loss. Here we have calculated an addi-
+tional mass-loss model with initial mass 1.1 M ⊙and
+an initial mass-loss rate of 2.25 x 10−10M⊙/yr, ex-
+ponentially decaying with e-folding time 0.45 Gyr.
+This model was calibrated to the CO5BOLD Z/X us-
+ing the AGS05 opacities that have O /Fe abundance
+closer to that of the CO5BOLD O/Fe. Previous work
+(e.g. Swenson&Faulkner 1992; Guzik& Cox 1995;
+Sackmann&Boothroyd 2003) indiciated that an ini-
+tial mass of 1.1 M ⊙or less and a relativelyshort mass
+loss phase (less than 0.2-0.5 Gyr) could deplete the
+lithiumtothepresent-dayobservedvaluesfrominitial
+solar-system abundance without completely destroy-
+ing the lithium or building up too much3He. The
+sound speed agreement (Figure 17) shows consider-
+able improvement with this smaller amount of mass
+loss; however, the agreement for the solar core is not
+as good as for the non-mass losing models, as can be
+seenmoreclearlyinthesmallseparations(Figure18).
+Thismass-losingmodelrestoresthelevelofagreement
+attained with the GN93 abundances for the observed
+minuscalculatedfrequencies(Figure19).
+Figure20showstheluminosityversustimeforthis
+model,aswellasforthestandardsolarmodelsevolved
+with GN93 and AGS05 abundances. Figure 21 shows
+the effective temperature experienced by the surface
+layer throughoutthe evolution of the mass-loss model
+comparedto that experiencedby the standardmodels.
+During the mass-losing phase, the high temperatures
+that depleted the lithium in the current surface layer
+were experienced when the material that is at the sur-
+faceofthenow1M ⊙sunwasdeeper,beforetheprevi-
+oussurfacelayerswerelost. Afterthemass-lossphase,
+the temperature that a ffects the surface layer is that of
+theCZ base,sincethematerialcurrentlyatthesurface
+is continually mixed through the CZ. We see that in-
+cludingmasslossintheCO5BOLDmodelexposesthe
+surface layers to high enough temperatures to deplete
+Li early in the evolution (2.8 million K is the temper-
+ature requiredfor relativelyrapid Li destruction). It is
+not clear that the advantages of mass loss (e.g. Li de-
+pletion and better sound speed agreement in the outer
+80% of the solar radius) can be retained while at the
+sametimenotcreatingadiscrepancyintheinner20%.-0.00500.0050.010.015
+0 0.2 0.4 0.6 0.8 1GN93
+AGS05
+COBOLD with GN93
+COBOLD with AGS05
+COBOLD with mass lossc
+s Difference
+(Sun - Model)/Sun
+R (R
+sun)
+Fig. 17.— Relative difference between inferred and calculated
+sound speeds for models using the CO5BOLD abundances created
+with either theGN93orAGS05 opacities and amodelcreated us ing
+theCO5BOLDabundances with theAGS05 opacities and including
+massloss.
+-0.3-0.2-0.100.10.20.30.4
+9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25GN93
+AGS05
+COBOLD with GN93COBOLD with AGS05
+COBOLD with mass lossCalculated - Observed
+small frequency separations
+between l = 0 and l = 2
+Radial order for l=0
+Fig. 18.— Differencebetweencalculated andobservedsmallsep-
+arations forℓ=0 and 2 modes for the Guzik et al. models using the
+CO5BOLD abundances with either the GN93 or AGS05 opacities
+and with the AGS05 opacities and mass loss. The data are from
+Chaplin et al. (2007).
+8. Conclusionsand futurework
+In spite of the seismic evidence in favor of the old
+abundances,the new abundancescannotbe easily dis-
+missed. Theimprovementsinthephysicsoftheatmo-
+140246810
+500 1000 1500 2000 2500 3000 3500GN93
+AGS05
+COBOLDwith GN93
+COBOLD with AGS05
+COBOLD mass lossObserved minus Calculated
+Frequency ( µHz)
+Calculated Frequency ( µHz)
+Fig. 19.— Observed minus calculated frequency versus calcu-
+lated frequency for degree ℓ=0, 2, 10, and 20 modes in models
+using the CO5BOLD abundances with either the GN93 or AGS05
+opacities and with the AGS05 opacities and mass loss. The cal -
+culated frequencies were computed using the Pesnell (1990) non-
+adiabatic stellar pulsation code. The data are from Chaplin etal.
+(2007),Schou &Tomczyk (1996), and Garcia etal. (2001).
+0.60.70.80.911.1
+0 1 2 3 4GN93
+AGS05
+COBOLD mass lossLuminosity (L
+sun)
+Time (Gyr)
+Fig. 20.— Luminosity versus time for standard one solar-mass
+models using the GN93 and AGS05 abundances and for a mass-
+losing model using the CO5BOLD abundances with the AGS05
+opacities withinitialmass1.1M ⊙. Mass-lossratesareexponentially
+decaying with e-folding time 0.45 Gyr.
+spheric models used to determine the abundances, the
+successachievedinline-profilematching,andtheself-22.533.54
+0 1 2 3 4GN93
+AGS05
+COBOLD with mass lossTemperature (106 K)
+Time (Gyr)
+Fig. 21.— Temperature experienced bythepresent-day solar sur-
+face layer as a function of time for the standard models and th e
+mass-losing model with CO5BOLD abundances and AGS05 opaci-
+ties. Forthemass-losing phase, thelithium-destroying te mperatures
+areattained because thelayer thatis nowatthesurface once resided
+deeper inside thesun. In thepost-mass-loss phase, the rele vant tem-
+peratures are attained by envelope convection which mixes s urface
+layersdownward,exposingthesurfacematerialtothetempe ratureat
+the CZbase. 2.8million Kis the temperature required for rel atively
+rapid Lidestruction.
+consistency of the abundance determinations provide
+great credibility to the new lower abundances. How-
+ever,solarmodelsdevelopedwiththenewabundances
+remain discrepant with helioseismic constraints, even
+with a variety of (often unjustified) changes to the in-
+put physics. Adjustments to the evolution of solar
+models, such as the early mass loss and low-Z accre-
+tion discussed here, show some promise but do not
+fully restore agreement. Any single adjustment to so-
+larmodelsdoesnotfullyresolvetheproblem. Combi-
+nationsofchangesmightprovidebetteragreementbut
+seem contrived. A resolution to the solar abundance
+problem(orthesolarmodelproblem)remainselusive.
+In the future, a more comprehensiveexplorationof
+parameter space, including opacity variations, di ffer-
+entmass-lossoraccretionprescriptions,di ffusion,and
+perhaps even combinations of these e ffects could be
+useful. In particular, models with AGS05 abundances
+and a smaller amount of mass loss than exploredhere
+may provide a way to retain the core structure with-
+out completely destroying Li or creating too much
+3He build-up. Further examination of the CO5BOLD
+15modelswithrevisedabundancesforeveryelementand
+opacitytables(includinglow-Topacities)adjustedfor
+a newmixturewouldalso beenlightening.
+The authors thank David Arnett, Alfio Bonanno,
+PiercarloBonifacio,SarbaniBasu,JoergenChristensen-
+Dalsgaard, Wick Haxton, Ross Rosenwald, Sylvaine
+Turck-Chi` eze, and our anonymousreferee for helpful
+discussions. WethankScottWatsonforcodeimprove-
+mentandearlierversionsofthe AGSmodels.
+REFERENCES
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+arXiv:1001.0649v1  [cs.SC]  5 Jan 2010A complete algorithm to find exact minimal polynomial
+by approximations
+Xiaolin Qina,b,c, Yong Fenga,∗, Jingwei Chena,b,c, Jingzhong Zhanga,b
+aLab. of Computer Reasoning and Trustworthy Comput., Univer sity of Electronic
+Science and Technology of China, Chengdu 610054, China
+bLab. for Automated Reasoning and Programming, Chengdu Inst itute of Computer
+Applications, CAS, Chengdu 610041, China
+cGraduate School of the Chinese Academy of Sciences, Beijing 100049, China
+Abstract
+We present a complete algorithm for finding an exact minimal polynomia l
+from its approximate value by using an improved parameterized integ er re-
+lation construction method. Our result is superior to the existence of error
+controlling on obtaining an exact rational number from its approxima tion.
+The algorithm is applicable for finding exact minimal polynomial of an al-
+gebraic number by its approximate root. This also enables us to prov ide
+an efficient method of converting the rational approximation repre sentation
+to the minimal polynomial representation, and devise a simple algorith m to
+factor multivariate polynomials with rational coefficients.
+Compared with the subsistent methods, our method combines adva ntage
+of high efficiency in numerical computation, and exact, stable result s in sym-
+bolic computation. we also discuss some applications to some transce ndental
+numbers by approximations. Moreover, the Digitsof our algorithm is far less
+than the LLL-lattice basis reduction technique in theory. In this pa per, we
+completely implement how to obtain exact results by numerical appro ximate
+computations.
+✩This research was partially supported by the National Key Basic Res earch Project
+of China (2004CB318003), the Knowledge Innovation Program of t he Chinese Academy
+of Sciences (KJCX2-YW-S02), and the National Natural Science F oundation of China
+(10771205).
+∗Corresponding author.
+Email addresses: qinxl@casit.ac.cn (Xiaolin Qin), yongfeng@casit.ac.cn (Yong
+Feng )
+Preprint submitted to Elsevier November 6, 2018Keywords: algebraic number, numerical approximate computation,
+symbolic-numerical computation, integer relation algorithm, minimal
+polynomial
+1. Introduction
+Symbolic computations are principally exact and stable. However, th ey
+have the disadvantage of intermediate expression swell. Numerical approxi-
+mate computations can solve large and complex problems fast, wher eas only
+give approximate results. The growing demand for speed, accurac y and relia-
+bility in mathematical computing has accelerated the process of blur ring the
+distinction between two areasof research that were previously qu ite separate.
+Therefore, algorithms that combine ideas from symbolic and numeric com-
+putations have been of increasing interest in the recent two decad es. Sym-
+bolic computations are for sake of speed by intermediate use of floa ting-point
+arithmetic. The work reported in [22, 10, 15, 11, 9] studied the re covery of
+approximate value from numerical intermediate results. A somewha t related
+topic is algorithms that obtain the exact factorization of an exact in put poly-
+nomial by use of floating point arithmetic in a practically efficient techn ique
+[6]. In the meantime, symbolic methods are applied in the field of numeric al
+computations for ill-conditioned problems [8, 5, 23]. The main goal of h ybrid
+symbolic-numeric computation is to extend the domain of efficiently so lvable
+problems. However, there is a gap between approximate computat ions and
+exact results[25].
+We consider the following question: Suppose we are given an approxi-
+mate root of an unknown polynomial with integral coefficients and a b ound
+on the degree and size of the coefficients of the polynomial. Is it poss ible to
+infer the polynomial and its exact root? The question was raised by M anuel
+BluminTheoretical Cryptography, andJingzhong ZhanginAutomat edRea-
+soning, respectively. Kannan et alanswered the question in [17]. However,
+their technique is based on the Lenstra-Lenstra-Lovasz(LLL) la ttice reduc-
+tion algorithm, which is quite unstable in numerical computations. In [ 16],
+Justet alpresented an algorithm for finding an integer relation on nreal
+numbers using the LLL-lattice basis reduction technique, which nee ded the
+high precision. The function MinimalPolynomial inmaple, which finds min-
+imal polynomial for an approximate root, was implemented using the s ame
+technique.
+2In this paper, we present a new algorithm for finding exact minimal po ly-
+nomial and reconstructing the exact root by approximate value. O ur algo-
+rithm is based on the improved parameterized integer relation const ruction
+algorithm PSLQ( τ), whose stability admits an efficient implementation with
+lower run times on average than the existing algorithms, and can be u sed to
+prove that relation bounds obtained from computer runs using it is n umer-
+ically accurate. The other function identifyinmaple, which finds a closed
+form for a decimal approximation of a number, was implemented using the
+integer relation construction algorithm. However, the choice of Digitsof ap-
+proximate value is fairly arbitrary[4]. In contrast, we fully analyze nu merical
+behavior of an approximate to exact value and give how many Digitsof ap-
+proximate value, which can be obtained exact results. The work is re gard as
+a further research in [26]. We solve the problem, which can be descr ibed as
+follows:
+Given an approximate value ˜ αat arbitrary accuracy of an unknown alge-
+braic number, and we also know the upper bound degree nof the algebraic
+number and an upper bound Nof its height on minimal polynomial in ad-
+vance. The problem will be solved in two steps: First, we discuss how m uch
+the error εshould be, so that we can reconstruct the algebraic number α
+from its approximation ˜ αwhen it holds that |α−˜α|< ε. Of course, εis a
+function in nandN. Second, we give an algorithm to compute the minimal
+polynomial of the algebraic number.
+Based on our method, we are able to extend our results with the sam e
+techniques to transcendental numbers of the form sin−1(α), log(α), etc.,
+whereαis algebraic. we also propose a simple polynomial-time algorithm
+to factor multivariate polynomials with rational coefficients, and pro vide a
+natural, efficient technique to the minimal polynomial representatio n. The
+basic idea is from [17]. However, we have greatly improved their efficien cy.
+The rest of this paper is organized as follows. Section 2 illustrates th e
+improved parameterized integer relation construction algorithm. S ection 3
+discusses how to reconstruct minimal polynomial and some applicatio ns to
+some transcendental numbers by approximations. Section 4 gives some ex-
+perimental results. The final section concludes this paper.
+This paper is the final journal version of [21], which contains essent ially
+the entire contents of this paper.
+32. Preliminaries
+In this section, we first give some notations, and a brief introductio n on
+integer relation problems. Then an improved parameterized integer relation
+construction algorithm is also reviewed.
+2.1. Notations
+Throughout this paper, Zdenotes the set of the integers, Qthe set of the
+rationals, Rthe set of the reals, O(Rn) the corresponding system of ordinary
+integers, U(n−1,R) the group of unitary matrices over R,GL(n,O(R)) the
+group of unimodular matrices with entries in the reals, coliB the i-th column
+of the matrix B. The ring of polynomials with integral coefficients will be
+denoted Z[X]. Thecontentof a polynomial p(X) inZ[X] is the greatest
+common divisor of its coefficients. A polynomial in Z[X] isprimitive if its
+content is 1. A polynomial p(X) has degree difp(X) =/summationtextd
+i=0piXiwith
+pd/negationslash= 0. We write deg(p) =d. Thelength|p|ofp(X) =/summationtextd
+i=0piXiis the
+Euclidean length of the vector ( p0,p1,···,pd); theheight|p|∞ofp(X) is the
+L∞-norm of the vector( p0,p1,···,pd), so|p|∞= max 0≤i≤d|pi|. Analgebraic
+number is a root of a polynomial with integral coefficients. The minimal
+polynomial of an algebraic number αis the irreducible polynomial in Z[X]
+satisfied by α. The minimal polynomial is unique up to units in Z. The
+degreeandheightof an algebraic number are the degree and height of its
+minimal polynomial, respectively.
+2.2. Integer relation algorithm
+There exists an integer relation amongst the numbers x1,x2,···,xnif
+there are integers a1,a2,···,an, not all zero, such that/summationtextn
+i=1aixi= 0. For
+the vector x= [x1,x2,···,xn]T, the nonzero vector a= [a1,a2,···,an]∈Zn
+is an integer relation for xifa·x= 0.
+In order to introduce the integer relation algorithm, we recall some useful
+definitions and theorems[14, 3]:
+Definition 2.1. (Mx) Assume x= [x1,x2,···,xn]T∈Rnhas norm |x|=1.
+Definex⊥tobetheset ofallvectorsin Rnorthogonalto x. LetO(Rn)∩x⊥be
+the discrete lattice of integral relations for x. Define Mx>0 to be the
+smallest norm of any relation for xin this lattice.
+Definition 2.2. (Hx) Assume x= [x1,x2,···,xn]T∈Rnhas norm |x|=1.
+Furthermore, suppose that no coordinate entry of xis zero, i.e., xj/negationslash= 0 for
+41≤j≤n(otherwise xhas an immediate and obvious integral relation). For
+1≤j≤ndefine the partial sums
+s2
+j=/summationdisplay
+j≤k≤nx2
+k.
+Given such a unit vector x, define the n×(n−1) lower trapezoidal matrix
+Hx= (hi,j) by
+hi,j=
+
+0 if 1 ≤i < j≤n−1,
+si+1/si if 1≤i=j≤n−1,
+−xixj/(sjsj+1) if 1≤j < i≤n.
+Note that hi,jis scale invariant.
+Definition 2.3. (Modified Hermite Reduction) Let H be a lower trapezoidal
+matrix, with hi,j= 0 ifj > iandhj,j/negationslash= 0. Set D=In, define the matrix
+D= (di,j)∈GL(n,O(R)) recursively as follows: For i from 2 to n, and for j
+from i-1 to 1(step -1), set q=nint(hi,j/hj,j); then for k from 1 to j replace
+hi,kbyhi,k−qhj,k, and for k from 1 to nreplacedi,kbydi,k−qdj,k, where the
+function nintdenotes a nearest integer function, e.g., nint(t) =⌊t+1/2⌋.
+Theorem 2.4. Letx/negationslash= 0∈Rn. Suppose that for any relation mofxand
+for any matrix A ∈GL(n,O(R))there exists a unitary matrix Q ∈U(n-1)
+such that H=AHxQis lower trapezoidal and all of the diagonal elements of
+H satisfy hj,j/negationslash= 0. Then
+1
+max1≤j≤n−1|hj,j|= min
+1≤j≤n−11
+|hj,j|≤ |m|.
+Proof.See Theorem 1 of [14].
+Remark2.5.The inequality of Theorem 2.4 offers an increasing lower bound
+on the size of any possible relation. Theorem 2.4 can be used with any
+algorithm that produces GL(n,O(R)) matrices. Any GL(n,O(R)) matrix A
+whatsoever can be put into Theorem 2.4.
+Theorem 2.6. Assume real numbers, n≥2,τ >1,γ >/radicalbig
+4/3, and that
+0/negationslash=x∈RnhasO(R)integer relations. Let Mxbe the least norm of relations
+forx. Then PSLQ( τ) will find some integer relation for xin no more than
+/parenleftbiggn
+2/parenrightbigglog(γn−1Mx)
+logτ
+iterations.
+5Proof.See Theorem 2 of [14].
+Theorem 2.7. LetMxbe the smallest possible norm of any relation for x.
+Letmbe any relation found by PSLQ( τ). For all γ >/radicalbig
+4/3for real vectors
+|m| ≤γn−2Mx.
+Proof.See Theorem 3 of [14].
+Remark2.8.For n=2, Theorem 2.7 proves that any relation 0 /negationslash=m∈O(R2)
+found has norm |m|=Mx. In other words, PSLQ(τ) finds a shortest rela-
+tion. For real numbers this corresponds to the case of the Euclide an algo-
+rithm.
+Based on these theorems as above, and if there exists a known err or
+controlling ε, then an algorithm for obtaining the integer relation can be
+designed as follows:
+Algorithm 2.9. Parameterized Integer Relation Construction
+Input: a vector x, the upper bound Non the height of minimal polynomial,
+and an error ε >0;
+Output: an integer relation m.
+Step 1: Set i:= 1,m:=0,τ >2/√
+3, and unitize the vector xto
+¯x;
+Step 2: Set H¯xby definition 2.2;
+Step 3: Producematrix D∈GL(n,O(R))using modifiedHermite
+Reduction by definition 2.3;
+Step 4: Set ¯x:=¯x·D−1,H:=D·H,A:=D·A,B:=B·D−1,
+case 1: if ¯xj= 0, then m:=coljB;
+case 2: if hi,i< ε, thenm:=coln−1B;
+Step 5: if 0 <|m|∞≤N, then goto Step 12;
+if|m|∞> N, there is no such an integer relation, algorithm
+terminating.
+Step 6:i:=i+1;
+Step 7: Choose an integer r, such that τr|hr,r| ≥τj|hj,j|, for all
+1≤j≤n−1;
+6Step 8: Define α:=hr,r,β:=hr+1,r,λ:=hr+1,r+1,σ:=/radicalbig
+β2+λ2;
+Step 9: Change hrtohr+1, and define the permutation matrix R;
+Step 10: Set ¯x:=¯x·R,H:=R·H,A:=R·A,B:=B·R, if
+i=n-1, then goto Step 4;
+Step 11: Define Q:= (qi,j)∈U(n−1,R),H:=H·Q, goto Step
+4;
+Step 12: return m.
+By algorithm 2.9, we can find the integer relation U(n−1,R) of the
+vectorx= (1,˜α,˜α2,···,˜αn). So, we get a nonzero polynomial of degree n,
+which denotes G(x) for the rest of this paper, i.e.,
+G(x) =m·(1,x,x2,···,xn)T. (2.1)
+Our main task is to show that polynomial (2.1) is uniquely determined un der
+assumptions, and discuss the controlling error εin algorithm 2.9 in the next
+section.
+3. Reconstructing minimal polynomial from its approximati on
+In this section, we will solve such a problem: For a given floating numbe r
+˜α, which is an approximation of unknown algebraic number α, how do we
+obtain the exact value? At first, we state some lemmas as follows:
+Lemma 3.1. Letfbe a nonzero polynomial in Z[x]of degree n. Ifε=
+max1≤i≤n|αi−˜αi|1, where˜αifor1≤i≤nare the rational approximations
+to the powers αiof an algebraic number α, and˜α0= 1, then
+|f(α)−f(˜α)| ≤ε·n·|f|∞. (3.1)
+Proof.Clear.
+Lemma 3.2. Lethandgbe two nonzero polynomials in Z[x]of degree nand
+m, respectively, and let α∈Rbe a zero of hwith|α| ≤1. Ifhis irreducible
+andg(α)/negationslash= 0, then
+|g(α)| ≥n−1·|h|−m·|g|1−n. (3.2)
+1εis defined by the same way for the rest of this paper.
+7Proof.See Proposition(1.6) of [17]. If |α|>1, a simple transform of it
+does.
+Corollary 3.3. Lethandgbe two nonzero polynomials in Z[x]of degrees
+nandm, respectively, and let α∈Rbe a zero of hwith|α| ≤1. Ifhis
+irreducible and g(α)/negationslash= 0, then
+|g(α)| ≥n−1·(n+1)−m
+2·(m+1)1−n
+2·|h|−m
+∞·|g|1−n
+∞.(3.3)
+Proof.First notice that |f|2≤(n+1)·|f|2
+∞holds for any polynomial fof
+degree at most n >0, so|f| ≤√n+1·|f|∞. From Lemma 3.2, we get
+|g(α)| ≥n−1·(n+1)−m
+2·(m+1)1−n
+2·|h|−m
+∞·|g|1−n
+∞.
+This proves Corollary 3.3.
+Theorem 3.4. Let˜αbe an approximate value to an unknown algebraic num-
+berαwith degree n >0,Nbe the upper bound on the height of minimal
+polynomial of α. For any G(x)inZ[x]with degree n, if
+|G(˜α)|< n−1·(n+1)−n+1
+2·|G|−n
+∞·N1−n−n·ε·|G|∞,
+then
+|G(α)|< n−1·(n+1)−n+1
+2·|G|−n
+∞·N1−n.
+Proof.Letα∈Rbe with|α| ≤1. From Lemma 3.1, we notice that |G(α)−
+G(˜α)| ≤ε·n·|G|∞, and
+|G(α)|−|G(˜α)| ≤ |G(α)−G(˜α)|,
+so,
+|G(α)| ≤ |G(˜α)|+n·ε·|G|∞. (3.4)
+From the assumption of the theorem, since
+|G(˜α)|< n−1·(n+1)−n+1
+2·|G|−n
+∞·N1−n−n·ε·|G|∞,(3.5)
+combined with (3.4), we have proved Theorem 3.4.
+8Corollary 3.5. Let˜αbe an approximate value to an unknown algebraic
+numberαwith degree n >0,Nbe the upper bound on the height of minimal
+polynomial of α. For any G(x)inZ[x]with degree n, if|G(α)|< n−1·(n+
+1)−n+1
+2·|G|−n
+∞·N1−n, then
+G(α) = 0. (3.6)
+The primitive part of polynomial G(x)is the minimal polynomial of algebraic
+numberα.
+Proof.(Proof by Contradiction) Let α∈Rbe with|α| ≤1. According to
+Lemma 3.2, suppose on the contrary that G(α)/negationslash= 0, then
+|G(α)| ≥n−1·(n+1)−n+1
+2·|G|−n
+∞·N1−n.
+From the assumption of the corollary, we have
+|G(α)|< n−1·(n+1)−n+1
+2·|G|−n
+∞·N1−n.
+However, it leads to a contradiction. So, G(α) = 0.
+LetG(x) =/summationtextn
+i=0aixi, which is constructed by the parameterized integer
+relation construction algorithm from the vector x = (1 ,α,α2,···,αn). Since
+algebraic number αwith degree n >0, according to the definition of minimal
+polynomial, then the primitive polynomial of G(x), denoted by pp(G(x)).
+Hencepp(G(x)) is just irreducible and equal to g(x). Of course, it is unique.
+This proves Corollary 3.5.
+3.1. Obtaining minimal polynomial by approximation
+Ifαis a real number, then by definition αis algebraic exactly, for some
+n, the vector
+(1,α,α2,···,αn) (3.7)
+has an integer relation. The integral coefficients polynomial of lowes t degree,
+whose root an algebraic number αis, is determined uniquely up to a con-
+stant multiple; it is called the minimal polynomial forα. Integer relation
+algorithm can be employed to search for minimal polynomial in a straigh t-
+forward way by simply feeding them the vector (3.7) as their input. L et ˜α
+be an approximate value belonging to an unknown algebraic number α, con-
+sidering the vector v= (1,˜α,˜α2,···,˜αn), how to obtain the exact minimal
+polynomial from its approximate value? We have the same technique a nswer
+to the question from the following theorem.
+9Theorem 3.6. Let˜αbe an approximate value belonging to an unknown al-
+gebraic number αof degree n >0. If
+ε=|α−˜α|<1/(n2(n+1)n−1
+2N2n), (3.8)
+whereNis the upper bound on the height of its minimal polynomial, th en
+G(α) = 0, and the primitive part of G(x)is its minimal polynomial.
+Proof.Letα∈Rbe with|α| ≤1. From Theorem 3.4 and Corollary 3.5, it
+is obvious that
+G(α) = 0,
+if and only if
+|G(α)|< n−1·(n+1)−n+1
+2·|G|−n
+∞·N1−n. (3.9)
+Under the assumption of the theorem, we get the upper bound of d egreen
+and an approximate value ˜ αbelonging to an unknown algebraic number α.
+For substituting the approximate value ˜ αinG(x), denoted by G(˜α), there
+are two cases:
+Case 1:G(˜α)/negationslash= 0,|G(˜α)|>0. We have the inequality (3.5)holds, i.e.,
+0< n−1·(n+1)−n+1
+2·|G|−n
+∞·N1−n−n·ε·|G|∞.(3.10)
+Clearly, the inequality (3.10) satisfies from the condition (3.8). This p roves
+the Case 1.
+Case 2:G(˜α) = 0. From Lemma 3.1, we have |G(α)−G(˜α)|< n·ε·|G|∞,
+hence|G(α)|< n·ε·|G|∞. In order to satisfy condition (3.9), we only need
+the following inequality holds,
+n·ε·|G|∞< n−1·(n+1)−n+1
+2·|G|−n
+∞·N1−n. (3.11)
+From theorem 2.7, and algorithm 2.9 in Step 5, |G|∞is not more than N.
+Hence we replace |G|∞byN. So the correctness of the inequality (3.11)
+follows from (3.8). This proves Theorem 3.6.
+It is easiest to appreciate the theorem by seeing how it justifies the fol-
+lowing algorithm for obtaining minimal polynomials from its approximation :
+Algorithm 3.7. Reconstructing Minimal Polynomial
+Input: a floating number (˜ α,n,N) belonging to an unknown algebraic num-
+berα, i.e., satisfying (3.8).
+Output: g(x), the minimal polynomial of α.
+10Step 1: Construct the vector v;
+Step 2: Compute εsatisfying (3.8);
+Step 3: Call algorithm 2.9 to find an integer relation wforv;
+Step 4: Obtain w(x) the corresponding polynomial;
+Step 5: Evaluate the primitive part of w(x) tog(x);
+Step 6: return g(x).
+Theorem3.8. Algorithm3.7 workscorrectlyas specifiedanduses O( n(logn+
+logN)) binary bit operations, where nandNare the degree and height of its
+minimal polynomial, respectively.
+Proof.Correctness follows from Theorem 3.6. The cost of the algorithm is
+O(n(logn+logN)) binary bit operations obviously.
+3.2. Some applications
+In this subsection, we discuss some applications to the practicalities . The
+method of obtaining exact minimal polynomial from an approximate ro ot
+can be extended to the set of complex numbers and many application s in
+computer algebra and science.
+This yields a simple factorization algorithm for multivariate polynomials
+with rational coefficients: We can reduce a multivariate polynomial to a
+bivariate polynomial using the Hilbert irreducibility theorem, the basic idea
+was described in [9], and then convert a bivariate polynomial to a univa riate
+polynomial by substituting a transcendental number in [24] or an a lgebraic
+numberofhighdegreeforonevariatein[7]. Afterthissubstitution wecanget
+an approximate root of the univariate polynomial and use our algorit hm to
+find the irreducible polynomial satisfied by the exact root, which mus t then
+be a factor of the given polynomial. It can find the bivariate polynomia l’s
+factors, from which the factors of the original multivariate polyno mial can
+be recovered using Hensel lifting. This is repeated until all the fact ors are
+found.
+The other yields an efficient method of converting the rational appr ox-
+imation representation to the minimal polynomial representation of an al-
+gebraic number. The traditional representation of algebraic numb ers is by
+their minimal polynomials [1, 2, 13, 19]. We now propose an efficient meth od
+11to the minimal polynomial representation, which only needs an appro ximate
+value, degree and height of its minimal polynomial, i.e., an ordered triple
+<˜α,n,N > instead of an algebraic number α, where ˜αis its approximate
+value, and nandNare the degree and height of its minimal polynomial,
+respectively, denoted by < α >=<˜α,n,N > . It is not hard to see that the
+computations in the representation can be changed to computatio ns in the
+other without loss of efficiency, the rational approximation method is closer
+to the intuitive notion of computation.
+we also discuss some applications to some transcendental numbers by
+using an improved parameterized integer relation construction met hod. The
+formofthesetranscendental numbersissin−1(α),cos−1(α), log(α)etc., where
+αis an algebraic number. Moreover, a large number of results were fo und
+by using integer relation detection algorithm in the course of resear ch on
+multiple sums and quantum field theory in [12].
+Suppose βis the principle value of sin−1(α) for some unknown α, which
+is, however, known to the algebraic of degree and height at most nandN,
+respectively. We consider inferring the minimal polynomial of αfrom an
+approximation ˜βtoβin the deterministic polynomial time. We show that if
+|β−˜β|is at most ε= 1/(n2(n+1)n−1
+2N2n), this can be done. The specific
+technique is similar with the method in [17].
+Thus, in polynomial time we can compute from ˜βan approximation ˜ αto
+an unknown algebraic number αsuch that |α−˜α| ≤ε, withεas above. Now
+Theorem (3.6) guarantees that we can find the minimal polynomial of αin
+polynomial time.
+4. Experimental results
+Our algorithmshave beenimplemented in Maple. The following examples
+run in the platform of Maple 12 under Windows and PIV2.53GB, 512MB o f
+mainmemory. Table1proposesthe Digitsofapproximatevaluestocompare
+our method against the LLL-lattice basis reduction algorithm.
+InTable1, wepresentmanyexamplestocompareournewmethodag ainst
+the LLL-lattice basis reduction algorithm. For each example, we con struct
+the irreducible polynomial with random integral coefficients in the ran ge
+−20≤c≤20. Here nandNdenote the degree of algebraic number and the
+height of its corresponding minimal polynomial respectively; wherea sDLLL
+andDPSLQare relative Digitsto obtain the exact minimal polynomial in
+theory,EPSLQis in our optimal experimental results respectively.
+12Ex.nNDLLLDPSLQEPSLQ
+14131612 8
+27173625 11
+310155736 16
+41519102 59 25
+5239145 77 29
+62719240109 55
+73015277118 57
+83411327126 67
+94015435161 82
+104517532189 110
+115013620200 123
+12100132033 427 381
+Table 1: Comparison between our algorithm and LLL-lattice basis red uction technique
+From Table 1, we observe that the Digitsof our algorithm is far less than
+the LLL-lattice basis reduction technique in theory. However, the Digitsof
+our algorithm is more than that of the optimal experimental results . So, in
+the further work we would like to consider improving the error contr olling.
+Thefollowingfirsttwoexamplesilluminatehowtoobtainanexactquadr atic
+algebraic number and minimal polynomial. Example 3 uses a simple exam-
+ple to test our algorithm for factoring primitive polynomials with integr al
+coefficients.
+Example 4.1. Letαbe an unknown quadratic algebraic number. We only
+know an upper bound of height on its minimal polynomial N= 47. Ac-
+cording to theorem 3.6, compute quadratic algebraic number αas follows:
+First obtain control error ε= 1/(12∗√
+3∗N4) = 1/(1807729447692 ∗√
+3)≈
+1.0×10−8. And then assume that we use some numerical method to get an
+approximation ˜ α= 11.937253933, such that |α−˜α|< ε. Calling algorithm
+3.7 yields as follows: Its minimal polynomial is g(x) =x2−8∗x−47. So,
+we can obtain the corresponding quadratic algebraic number α= 4+3√
+7.
+Remark 4.2.The function identifyin maple 12 needs Digits:=13, whereas
+our algorithm only needs 9 digits.
+Example 4.3. Let a known floating number ˜ αbelonging to some algebraic
+numberαof degree n= 4, where ˜ α= 3.14626436994198, we also know an
+13upper bound of height on its minimal polynomial N= 10. According to
+theorem 3.6, we can get the error ε= 1/(n2(n+ 1)n−1
+2N2n) = 1/(42∗57
+2∗
+108)≈2.2×10−12. Calling algorithm 3.7, if only the floating number ˜ α, such
+that|α−˜α|< ε, then we can get its minimal polynomial g(x) =x4−10∗
+x2+1. So, the exact algebraic number αis able to be denoted by < α >=<
+3.14626436994198 ,4,10>, i.e.,<√
+2+√
+3>=<3.14626436994198 ,4,10>.
+Remark4.4.In the example 4.3, we only propose a simple example to repre-
+senttheexactalgebraicnumberbyapproximatemethod. Inthefu rtherwork,
+we would like to discuss the efficient arithmetic algorithms for summatio n,
+multiplication and inverse of the rational approximation representa tion.
+Example 4.5. This example is an application in factoring primitive polyno-
+mials over integral coefficients. For convenience and space-saving purposes,
+we choose a very simple polynomial as follows:
+p= 3x9−9x8+3x7+6x5−27x4+21x3+30x2−21x+3
+We want to factor the polynomial pvia reconstruction of minimal polyno-
+mials over the integers. First, we transform pto a primitive polynomial as
+follows:
+p=x9−3x8+x7+2x5−9x4+7x3+10x2−7x+1,
+We see the upper bound of coefficients on polynomial pis 10, which has
+relation with an upper bound of coefficients of the factors on the pr imitive
+polynomial pby Landau-Mignotte bound [20]. Taking N= 5,n= 2 yields
+ε= 1/(22∗(2+1)2−1
+2∗54) = 1/(7500∗√
+3)≈8.0×10−5. Then we compute
+the approximate root on x. With Maple we get via [fsolve( p= 0,x)]:
+S= [2.618033989 ,1.250523220 ,−.9223475138 ,.3819660113 ,.2192284350] .
+According to theorem 3.6, let ˜ α= 2.618033989 be an approximate value
+belonging to some quadratic algebraic number α, calling algorithm 3.7 yields
+as follows:
+p1=x2−3∗x+1.
+And then we use the polynomial division to get
+p2=x7+2∗x3−3∗x2−4∗x+1.
+Based on the Eisenstein’s Criterion [18], the p2is irreducible in Z[X]. So,
+thep1andp2are the factors of primitive polynomial p.
+145. Conclusions
+In this paper, we propose a new method for obtaining exact results by
+numerical approximate computations. The key technique of our me thod is
+based on an improved parameterized integer relation construction algorithm,
+which is able to find an exact relation by the accuracy control. The er rorεin
+formula (3.8) is an exponential function in degree and height of its min imal
+polynomial. The result is superior to the existence of error controllin g on
+obtaining an exact rational number from its approximation in [26]. Usin g
+our algorithm, we have succeed in factoring multivariate polynomials w ith
+rationalcoefficients andproviding anefficient methodofconverting theratio-
+nal approximation representation to the minimal polynomial repres entation.
+Ourmethodcanbeappliedinmanyaspects, such asproving inequality state-
+ments and equality statements, and computing resultants, etc.. T hus we can
+take fully advantage of approximate methods to solve larger scale s ymbolic
+computation problems. Furthermore, our basic idea can be genera lized easily
+to complex algebraic numbers.
+References
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+real numbers as functions. In Turner. D., editor, Research Topics in
+Functional Programming, Addison-Wesley, (1990), pp. 43-64.
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+2002 Internat. Symp. Comput. ISSAC’02 , ACM press, pp. 37-45.
+[10] Corless, R.M., Giesbrecht, M. W., et al. Numerical implicitization of
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+SAC’01, ACM press, pp. 85-92.
+[12] David H. B.. Integer Relation Detection. Computing in Science and En-
+gineering, 2, 1(2000), pp. 24-28.
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+dations of Computer Science, (1989), pp. 314-320.
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+16[18] Lang, S. Algebra, 3rd edition, Springer-Verlag, New York, 2002.
+[19] Loos, B. Computing in Algebraic Extensions, Computer Algebra, (Ed.
+by B. Buchberger, et al), Springer-Verlag, (1982), pp. 173-187 .
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+28, 128(1974): 1153-1157.
+[21] Qin, X. L., Feng, Y., Chen J. W., et al. Finding Exact Minimal Polyno-
+mialbyApproximations. InProc.2009Internat.WorkshoponSym bolic-
+Numeric Comput. ACM press,125-131,2009.
+[22] Sasaki, T., Suzuki, M., et al. Approximate factorization of multiva riate
+polynomials and absolute irreducibility testing. Japan J. Indust. Appl.
+Math.8 (1991), 357-375.
+[23] Sch¨onhage, A.Quasi-gcdcomputations. J.Complexity .1(1985): 118-137.
+[24] Van Der Hulst, M.-P. and Lenstra, A. K. Factorization of polyno mials
+by transcendental evaluation. EUROCAL’85 , (1985), pp. 138-145.
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+tween algebraic equation and its application. Proc. IWMN’92, Internat.
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+17
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+arXiv:1001.0650v1  [astro-ph.SR]  5 Jan 2010Astronomy& Astrophysics manuscriptno.orion˙0110 c∝circlecopyrtESO 2018
+September16,2018
+Disk andoutflow signatures inOrion-KL: The power of
+high-resolution thermal infraredspectroscopy⋆
+H. Beuther1,H.Linz1, A.Bik1,M.Goto1, and Th.Henning1
+Max-Planck-Institute forAstronomy, K¨ onigstuhl 17, 6911 7 Heidelberg, Germany
+e-mail:name@mpia.de
+Preprintonline version: September 16, 2018
+ABSTRACT
+Context. The Orion-KL region contains the closest examples of high-m ass accretion disk candidates. Studying their properties i s an
+essential stepinstudying high-mass starformation.
+Aims.Resolving at high spatial and spectral resolution the molec ular line emission in the immediate environment of the excit ing
+sources toinfer the physical properties of the associated g as.
+Methods. WeusedtheCRIREShigh-resolutionspectrographmountedon theVLTtostudythero-vibrational12CO/13CO,thePfundβ
+and H 2emission between 4.59 and 4.72 µm wavelengths toward the BN object, the disk candidate sourc e n, and a proposed dust
+densityenhancement IRC3.
+Results.We detected CO absorption and emission features toward all t hree targets. Toward the BN object, the data partly confirm
+the results obtained more than 25 years ago by Scoville et al. , however, we also identify several new features. While the b lue-shifted
+absorption is likely due to outflowing gas, toward the BN obje ct we detect CO in emission extending in diameter to ∼3300AU
+with a velocity structure close to the vlsr. Although at the observational spectral resolution limit, the13CO line width of that feature
+increases with energy levels, consistent with a disk origin . If one attributes the extended CO emission also toa disk ori gin, its extent
+is consistent with other massive disk candidates in the lite rature. For source n, we also find the blue-shifted CO absorpt ion likely
+from an outflow. However, it also exhibits a narrower range of redshifted CO absorption and adjacent weak CO emission, con sistent
+with infalling motions. We do not spatially resolve the emis sion for source n. For both sources we conduct a Boltzmann ana lysis
+of the13CO absorption features and find temperatures between 100 and 160K, and H 2column densities of the order a few times
+1023cm−2. The observational signatures from IRC3 are very di fferent with only weak absorption against a much weaker contin uum
+source. However, the CO emission is extended and shows wedge -like position velocity signatures consistent with jet-en trainment of
+molecular gas, potentially associatedwiththe Orion-KL ou tflow system. We alsopresent and discuss the Pfund βand H2emissionin
+the region.
+Conclusions. This analysis toward the closest high-mass disk candidates outlines the power of high spectral and spatial resolution
+mid-infrared spectroscopy to study the gas properties clos e to young massive stars. We will extend qualitatively simil ar studies to
+larger samples of high-mass young stellar objects to constr ain the physical properties of the dense innermost gas struc tures in more
+detail alsoina statisticalsense.
+Key words. Stars:formation –Stars:early-type –Accretion, accretio ndisks –Techniques: spectroscopic –ISM: jets and outflows –
+Stars:individual: Orion-BN, Orionsource n
+1. Introduction
+Understandingthe physical structure of massive accretion disks
+is one of the main unsolved problems in high-mass star for-
+mation. Although indirect, the main line of arguments for ac -
+cretion disks stems from massive molecular outflow obser-
+vations that identify collimated and energetic outflows fro m
+high-mass young stellar objects (YSOs, e.g., Henninget al.
+2000;Beutheret al.2002b;Zhanget al.2005;Arceet al.2007 ).
+Collimated jet-like outflow structures are usually attribu ted
+to massive accretion disks and magneto-centrifugal accele r-
+ation. Recent 2D and 3D magneto-hydrodynamical simula-
+tions of massive collapsing gas cores also result in the for-
+mation of massive accretion disks (Yorke&Sonnhalter, 2002 ;
+Krumholzet al.,2007,2009).However,itisstillunclearwh ether
+such massive disks are similar to their low-mass counterpar ts,
+hence dominated by the central YSO and in Keplerian rotation ,
+⋆Based onobservations of the ESOprogram 380.C-0380(A).or whether they are maybe self-gravitating non-Keplerian e nti-
+ties.
+While studies at (sub)mm wavelengths are a powerful tool
+to mainly study the cold gas and dust components on spatial
+scales of the order 1000AU (e.g., Cesaroniet al. 2007), such
+observations are not that well suited to investigate the inn er
+and warmer components of massive rotating structures. In co n-
+trast to that, mid-infrared spectral lines, e.g., ro-vibra tionally
+excited CO emission lines, can trace these warm gas compo-
+nents.However,thespectraland /orspatialresolutionwasmostly
+lackingbecauseabsorptionfeatureswere dominatingand he nce
+prohibiting the detection of the accretion disks in emissio n.
+Several recent studies have further demonstrated the power of
+high-spectraland high-spatialresolution CO mid-infrare dspec-
+troscopy for disks around low-mass young stellar objects an d
+HerbigAestars (e.g.,Gotoetal. 2006; Pontoppidanet al.20 08;
+vanderPlaset al. 2009).
+Toachievethehighestangularandspectralresolutionposs i-
+ble, we observedsome of the closest massive disk candidates in2 H.Beuther etal.:Diskand outflow signatures inOrion-KL
+Orion at a distance of 414pc (Mentenet al., 2007) – Orion-BN
+(the Becklin-Neugebauer Object), source n and IRC3 – in the
+COv=1−0transitionsaround4.65 µmwiththe CRyogenichigh
+resolutionInfraRedEchelleSpectrograph (CRIRES,K¨ auflet al.
+2004)attheVLT.Bothobjectsarewell detectedatmid-infra red
+wavelengthsexhibitingvariouskindsofdisk-signatures.
+The Becklin-Neugebauer Object: Since its detection in
+the 1960s, the BN object is one of the archetypical high-mass
+YSOs (Becklin&Neugebauer, 1967; Henninget al., 1990).
+Scovilleetal. (1983) observed the source in several freque ncy
+settings between 2 and 5 µm and detected molecular emission
+fromseveralCOisotopologues(fundamentalandovertoneem is-
+sion), and they inferred that BN exhibits an outflow /wind as
+well as a highly confined region of molecular gas at high den-
+sitiesandtemperaturesof ∼3500K.Theestimatedluminosityof
+the BN object is 1 −2×104L⊙corresponding to a B0.5 main
+sequence star (Scovilleet al., 1983). More recently, Jiang et al.
+(2005) observed BN in polarized near-infrared emission, an d
+they also identified signatures caused by an embedded accre-
+tion disk. The BN object has a high velocity along the line of
+sight of∼21kms−1comparedwith the cloudvelocityof around
+5kms−1(Scovilleet al., 1993). This is consistent with the mea-
+sured high proper motions of that object (e.g., Plambecket a l.
+1995). Whether the BN object is expelled from the Trapezium
+system or during a disintegration of a bound system once con-
+taining source I, source n and the BN object itself is still a m at-
+ter of debate (e.g., Tan 2004; G´ omezet al. 2005; Zapataet al .
+2009).
+Source n: Based on a bipolar radio morphology and H 2O
+maser association, Menten& Reid (1995) suggested that this
+sourcemaybeoneofthedrivingsourcesofthepowerfulmolec -
+ularoutflowswithinOrion-KL.Extendedmid-infraredemiss ion
+wasobservedperpendiculartothe outflowaxis(Greenhillet al.,
+2004; Shupingetal., 2004), and Luhman (2000) detected CO
+overtone emission. Both features are interpreted as likely be-
+ing due to an accretion disk. Source n is believed to be in an
+evolutionary younger stage than the BN object, and the lumi-
+nosity is estimated to be lower as well, of the order 2000L ⊙
+(Greenhillet al., 2004).
+IRC3:The source n observations serendipitously covered
+the extended infrared source IRC3 (e.g., Dougadoset al. 199 3)
+which we present here as well. At 3.6 µm wavelengths, IRC3 is
+elongatedin the northeast-southwestdirection(Dougados et al.,
+1993) and shows highly polarized near- to mid-infrared emis -
+sion (Minchinetal., 1991). The observations are consisten t
+with IRC3 being a dust density enhancementreprocessing lig ht
+from another source, potentially IRC2 (Downeset al., 1981;
+Minchinet al., 1991; Dougadoset al., 1993).
+Figure1givesanoverviewoftheregionmarkingthesources
+discussed in the paper as well as the slit orientations (see a lso
+section 2). The nominal absolute positions for the three sou rces
+arelisted inTable1.
+Table 1.Sourcepositions(fromDougadosetal. 1993)
+Source R.A. Dec.
+(J2000.0) (J2000.0)
+BNobject 05h35m14.12s -05d22m22.9s
+Source n 05h35m14.35s -05d22m32.9s
+IRC3 05h35m13.90s -05d22m30.0s
+Fig.1.Overview image of the region. The grey-scale Ksband
+image is taken from UKIDSS DR5 (Lawrenceet al., 2007), the
+white stellar peak positions are due to saturation. The slit s and
+importantsourcesaremarked.The crossdenotesthe positio nof
+IRc2.
+2. Observations
+We obtained high-resolution spectra between 4.6 and 4.7 µm
+with CRIRES (K¨ auflet al., 2004)) mounted on UT1 at the VLT
+on Paranal, Chile. Two grating settings were selected (12 /-1/n,
+λref=4662.1and12/-1/i,λref=4676.1)to observethespectralin-
+terval covering the12COv=1−0 [P(1)–P(5)/R(0)–R(8)]lines
+withoutgaps.
+One slit covered the BN object with a position angle of 126
+degrees east of north, whereas the second slit included sour ce
+n and IRC3 with a position angle of 110 degrees east of north.
+ForthestrongsourceBN, only40secs(DIT =2secs, NDIT=10)
+wererequired.Forthesecondobjectwiththeweakersources we
+had a total on-slit integrationtime of 20 minutes(DIT =10secs,
+NDIT=2). For the BN observationa slit width of 0 .2” was used
+while for the observation of source n a slit with of 0 .4” was se-
+lected which correspond to spectral resolving powers λ/∆λof
+100,000 and 50,000, respectively. The non-AO mode was ap-
+pliedsincenonaturalguidestarisavailableinthecloseen viron-
+ment. The infraredseeing measuredfrom the spectra was 0.35′′
+during the BN observation and 0.45′′for the source n observa-
+tion. The nod-throw of all the observations was set to 10′′. To
+correctforthetelluricabsorptionlines,attachedtoever yscience
+observationa telluric standard star (HR 1666 with spectral type
+A3III)wasobserved.
+Asextendedemissionin theCO lineswaspresentinthe ob-
+servations,we correctedthe framesfor distortionin order to get
+awavelengthsolutionvalidforthewholechip.Firstly,the chips
+areslightlyrotatedwithrespecttotheslit(rangingfrom0 .05de-
+greeforchip3to0.45degreeinthecaseofchip4).Wemeasure d
+thepositionofthebrightestobjectasfunctionofwaveleng thand
+calculate from the displacement in position the rotation an gle.
+Secondly,after the rotation angle hasbeen corrected,the c urva-
+ture of the slit is corrected by measuring the position (cent ralH.Beuther etal.:Diskand outflow signatures inOrion-KL 3
+wavelength) of a sky-line as function of the spatial coordin ate.
+This could be described with a 2nd degree polynomial. Using
+the IDL routines polywarp and poly 2d, the distortion was cor-
+rected.
+After the distortion correction, the raw files are processed
+by the ESO CRIRES pipeline (version 1.10.1) in combination
+with the Gasgano software. The data are dark subtracted, flat
+field corrected as well as corrected for non-linearity.The w ave-
+length calibration is done using the telluric emission line s in
+combination with a HITRAN model spectrum (Rothmanet al.,
+2005). A cross-correlation of the spectra with the HITRAN
+spectra showed that the wavelength accuracy of the spectra i s
+±0.5kms−1.
+Two absorption lines are present in the standard star: the
+Pfundβand the Humphreys ǫline. Due to the strong telluric
+absorption, these lines are not trivial to remove. Therefor e, we
+first reduced the spectrum without correcting for the intrin sic
+absorption lines of the standard star. In the final reduced sp ec-
+trum these lines become eminent as emission lines free from
+contamination by the atmosphere. The Pfund βline is also seen
+in our science object. However, the Humphreys ǫline is not
+present in the science spectrum of BN before division by the
+standard star. Therefore, it can be used to correct for the ab -
+sorption lines of the standard star. We used a high resolutio n
+Kurucz model spectrum from an A0V star (http: //kurucz.har
+vard.edu/stars.html) and scaled and shifted the model spectrum
+such that the Humphreys ǫprofile would fit that of the observed
+Humphreysǫprofile in the standard star. Assuming that the
+Pfundβline scales in the same way as the Humphreys ǫline
+does, we divided with the model spectrum to remove the line
+contamination.
+Thespectrawerecorrectedfortheearthvelocitytothe local
+standard of rest using rvcorrect in IRAF. The velocity correc-
+tionsappliedforthetwoobservingdateswere-3.8kms−1forthe
+BN data (observed on 21st October 2007) and +43.2kms−1for
+the source n/IRC3 observations (taken on 21st February 2008).
+The velocity relative to the local standard of rest vlsrof Orion
+varies between 2.5 and 9kms−1(e.g., Comitoet al. 2005), and
+weadopttheapproximatevalueof +5kms−1.
+3. Results
+Towardallthreesourceswedetectedthewholesuiteof12COv=
+1−0linespresentinthespectralwindow,the13COv=1−0lines
+fromR(6) toR(13)that werenotblendedbythe12COv=1−0
+lines,aswell asthe Pfund βline.Figure2presentsthecomplete
+spectrumtowardtheBNobject,andTable2givesanoverviewo f
+thecoveredlines,theirwavelengths λandthelower-levelenergy
+state of the transitions ( Elower/k). In total, this setup covers a
+broadrangeofenergylevelsextendingupto504K.
+3.1. The BNobject
+3.1.1. COabsorption andemission
+Figure 3 presents a zoom compilation of12CO data from the
+R(1) to R(8) line covering lower energy levels between 5.5 an d
+199K. If one ignores the telluric line feature at ∼3kms−1two
+broadabsorptionfeaturescanbeidentifiedatapproximatel y−14
+and+8kms−1. Since all12CO lines are saturated, their peak
+absorption velocities are unreliable, and we refer to the13CO
+data (Fig. 4). The peak velocity of the blue-shifted compone nt
+is∼ −15kms−1extending from∼ −28 to∼ −8kms−1. The
+second absorption feature has its peak at ∼8kms−1, close toTable 2.Observedlines
+Line λElower/k
+µm K
+12CO
+R(9) 4.5876 249
+R(8) 4.5950 199
+R(7) 4.6024 155
+R(6) 4.6090 116
+R(5) 4.6176 83
+R(4) 4.6254 55
+R(3) 4.6333 33
+R(2) 4.6412 17
+R(1) 4.6493 5.5
+R(0) 4.6575 0
+P(1) 4.6742 5.5
+P(2) 4.6826 17
+P(3) 4.6912 33
+P(4) 4.7000 55
+P(5) 4.7088 83
+13CO
+R(13) 4.6641 504
+R(12) 4.6741 432
+R(11) 4.6782 365
+R(10) 4.6853 304
+R(9) 4.6926 249
+R(8) 4.7000 199
+R(7) 4.7075 155
+R(6) 4.7150 116
+Pfundβ4.6538 157954a
+H20−0S9 4.6947 7198a
+The temperatures listed for the Pfund βand H 20−0S9 lines do not
+indicate gas temperatures since these lines are not excited by collisions
+within thermal gas, but rather by ionizing photons and withi n shocked
+regions, respectively.
+Fig.3.The12CO R(8) to R(1) lines from top to bottom toward
+theBNobject.Telluricand vlsrvelocitiesaremarked.Thefeature
+between0 and8 kms−1isatelluricartifact.
+thevlsrof the cloud, and extends from ∼ −8 to∼18kms−1.
+Furthermore, red-shifted from the absorption we clearly id en-
+tify a CO emission peaking in12CO and13CO at∼20kms−1
+andextendingfrom ∼15to∼30kms−1.Theoverallextentofthe
+12CO absorption and emission is from ∼−30 to∼+30kms−1.
+It should be noted that while the vlsrof the different cloud com-
+ponents for the Orion-KL region vary between approximately
+3 and 9kms−1, Scovilleet al. (1983) inferred that the corre-4 H.Beuther etal.:Diskand outflow signatures inOrion-KL
+Fig.2.CRIRES observations of the R and P CO line series around 4.65 µm. The strong absorption features are the CO lines, and
+theemissionlineat ∼4.654µmisPfundβ.
+spondingvelocityoftheBNobjectissignificantlylargerar ound
+21kms−1(consistent with the di fferent BN ejection scenarios,
+e.g., Tan 2004; G´ omezetal. 2005). While the absorption fea -
+tures stem from the warmer protostellar envelopes and the su r-
+roundingcloud with usual temperaturesof the order 100K (se e
+alsoBoltzmannanalysisbelow),thero-vibrationallinesi nemis-
+sion can be caused by di fferent processes. For example, fluo-
+rescence via UV photons or resonance scattering from strong
+infrared fields can excite these lines without significantly heat-
+ingthegas(e.g.,Blake& Boogert2004;Ryde&Sch¨ oier2001) .
+Alternatively, the ro-vibrational lines could be caused by hot-
+ter gas components (see discussion in section 4). We note tha t
+the critical densities of these lines are of order1013cm−3which
+practically implies that extended gas componentscan be har dly
+responsibleforthe emission.
+Although the blue-shifted part of the spectrum with respect
+tothevlsrofthemolecularcloudisslightlybroaderthanthered-
+shiftedpart, neverthelesswe clearlyidentifyred-shifte dabsorp-
+tionaswell.Tofirstorder,theblue-shiftedgasseeninabso rption
+can be identified with outflowing gas from the region, whereas
+thered-shiftedfeaturesbelongtogasinfallinginthedire ctionof
+thecentralsource.Theoutflowinggaswithamaximumvelocit y
+relativetothe vlsrof∼35kms−1isconsistentwithoutflowwings
+often observed at mm wavelengths from young massive star-
+forming regions (e.g., Beutheretal. 2002b). It should be no ted
+thattheevenbroaderoutflowwingsobservedatmmwavelengthtowardOrion-KLexceeding ±50kms−1(e.g.,Chernin&Wright
+1996) are likely not related to the BN object but rather to one
+or more sources about 10′′south-east of BN (source I, source n
+and/orSMA1,e.g.,Greenhillet al.2004;Jianget al.2005;Bally
+2008; Beuther& Nissen 2008).
+How do these general features compare with the data pub-
+lishedbyScovilleet al.(1983)whichwereobserveddurings ev-
+eralobservingrunsbetweenDecember1977andFebruary1981 ,
+hence about 30 years priorto our observations.The general C O
+line structure with two strong absorption features plus one red-
+shifted emission peak are largely the same. Also the overall ex-
+tent of the emission is quite similar. Compared to our measur ed
+values of∼−15,∼+9 and∼20kms−1for the three compo-
+nents, Scovilleetal. (1983) report for the corresponding f ea-
+tures velocities of −18,+9 and+20kms−1. While two veloci-
+tiesagreewell,themostblue-shiftedabsorptionpeakappe arsto
+have shifted a little bit between the two observations. Howe ver,
+giventhattheirspectralresolutionwasmorethanafactor2 lower
+thanthatofthenewCRIRESdata(7versus3kms−1),werefrain
+from further interpretation of this. Furthermore, Scovill eet al.
+(1983) identify two more absorption features, one at −3kms−1
+and one at+30kms−1. Regarding the−3kms−1component we
+are not able to infer any changes because that features lies v ery
+close to the telluric emission which obscures any reliable s ig-
+nature there. However, Fig. 3 shows that we do not detect any
+additional absorption feature blue-shifted from the 26.6k ms−1H.Beuther etal.:Diskand outflow signatures inOrion-KL 5
+Fig.4.13CO opticaldepthstowardtheBN object.
+emission. Therefore, the +30kms−1absorption dip reported by
+Scovilleetal. (1983) waseithera transientfeatureornots ignif-
+icantwithrespectto thesignal-to-noiseratio.
+Since the12CO data are so strongly saturated, for the fol-
+lowing analysis we use the corresponding13CO data covering
+theR(6)toR(13)transitions(Fig.2).Sincetheabsorption depth
+I/I0isrelatedto the opticaldepth τviaI
+I0=e−τwe can directly
+estimatetheopticaldepthofthe13CO absorptionlinesshownin
+Fig. 4 if the lines are spectrally resolved.Since the full wi dth at
+zerointensity(FWZI)isoforder20kms−1(seeabove),thiscri-
+terionis fulfilledwith ourspectralresolution λ/∆λ=105which
+corresponds to a velocity resolution of 3kms−1. Except of the
+lowest13COR(6)linecomponentat +9kms−1allotherobserved
+13CO absorption features have optical depths below 1. This al-
+lows us to estimate rotational temperaturesvia Boltzmann p lots
+from the equivalent line width following the approach outli ned
+in Scovilleetal. (1983) also adopting their finite optical d epth
+corrections.Usingtheirequation(A8),thecolumndensity Nlof
+thelower-levelenergystate is
+Nl=1.13×1012FA
+flucm−2
+withF,Aandfuldenoting the optical depth correction fol-
+lowingtheappendixinScovilleet al.(1983),theequivalen tline
+width (in units of cm−1) and absorption oscillator strength, re-
+spectively. Because the absorption feature at ∼9kms−1is af-
+fected by the telluric line, in the following we only work wit h
+thedatafromthe∼−15kms−1line.Theparametersandthecal-
+culated column densities are listed in Table 3. In a Boltzman n
+distribution,thetotalcolumndensity Ntot(13CO)isrelatedtothe
+temperatureandthelowerstate columndensityvia:
+ln/parenleftBiggNl
+gl/parenrightBigg
+=ln/parenleftBiggNtot(13CO)
+Q(T)/parenrightBigg
+−1
+TrotElower
+k
+withglthestatisticalweightofthelower-leveltransitionand
+Q(T)thepartitionfunctionat thegiventemperature.
+Figure 5 presents the corresponding Boltzmann plot where
+the lower-state column density Nldivided by the statistical
+weightgJisplottedagainstthelower-stateenergy Elowerdivided
+byk. A linearfit to thedata givestherotationaltemperature TrotTable 3.13CO valuesforBoltzmannplots
+Elower/k A f ulF g JNl(13CO)
+K km/s×10−6×1015cm−2
+BN@-15kms−1
+R(6) 116 3.70 6.0 1.18 13 6.0
+R(7) 155 3.16 5.9 1.14 15 5.0
+R(9) 249 1.90 5.9 1.08 19 2.9
+R(10) 304 1.36 5.8 1.06 21 2.0
+R(11) 365 0.69 5.8 1.04 23 1.0
+R(12) 432 0.51 5.8 1.02 25 0.7
+R(13) 504 0.27 5.8 1.02 27 0.4
+source n @-7kms−1comp.
+R(7) 155 4.1 5.9 1.14 15 6.5
+R(9) 249 2.7 5.9 1.08 19 4.1
+R(10) 304 2.0 5.8 1.08 21 3.1
+R(12) 432 1.3 5.8 1.06 25 2.0
+R(13) 504 1.5 5.8 1.05 27 2.2
+source n @+5kms−1comp.
+R(7) 155 4.6 5.9 1.23 15 7.8
+R(9) 249 2.7 5.9 1.16 19 4.4
+R(10) 304 1.8 5.8 1.12 21 2.8
+R(12) 432 0.6 5.8 1.04 25 0.9
+R(13) 504 0.1 5.8 1.00 27 0.1
+The Table entries areexplained inthe maintext.
+and the column density of13CO divided by the partition func-
+tionQ(T). For the−15kms−1 13CO componentthe fitted Trotis
+∼112±20K. Scovilleet al. (1983) calculatedthe rotationtem-
+perature for the more blue-shifted absorption component fr om
+the12COv=2−0transitions,andtheyfind ∼150Kthere,which
+is approximately consistent with our new determination. Th e
+other fitted parameter is Ntot(13CO)/Q(T)≈1.35×1015cm−2.
+For 2-atomic molecules like13CO the partition function can be
+approximated by Q(T=112K)∼kT/(hB)∼42 (where Bis
+the rotation constant), and we get a13CO column density of
+∼5.7×1016cm−2. Using furthermore the12CO to13CO iso-
+topologicratio of 69(e.g.,She fferet al. 2007), we derivea total
+COcolumndensityofthe −15kms−1componenttowardOrion-
+BN of≈3.9×1018cm−2. These column density estimates are
+inexcellentagreementwith theresultsderivedbyScoville et al.
+(1983)fromthe12COv=2−0lines.6 H.Beuther etal.:Diskand outflow signatures inOrion-KL
+Fig.5.Boltzmann plot for the13CO v=1-0 lines toward the
+BNobject.Thex-axisshowsthelower-levelenergiesofthet ran-
+sitions and the y-axis presents the natural logarithm of the cor-
+respondingcolumndensitiesdividedbytheirstatisticalw eights.
+How do these values compare to other observations? The
+BN object was detected at 1.3mm wavelength by Blake et al.
+(1996) at a 0.15Jy level with a spatial resolution of 1 .5′′×
+1.0′′. Following Plambecket al. (1995), about 50mJy of that
+flux can be attributed to circumstellar dust emission. Assum ing
+optically thin dust emission, an average dust temperature o f
+100K, a dust opacity index βof 2 and a standard gas-to-dust
+ratio of 100, we can calculate the total H 2column density
+(Hildebrand,1983;Ossenkopf&Henning,1994;Henninget al .,
+1995;Beutheretal.,2002a,2005a;Draineet al.,2007).The de-
+rivedH 2columndensityandmasswithintheir1 .5′′×1.0′′synte-
+sizedbeamarethen ∼1.3×1024cm−2and0.1M⊙,respectively.
+Forcomparison,usingastandardCO-to-H 2ratioof8×10−5,the
+H2columndensityestimatedherefromtheCO IRspectroscopy
+data is∼4.9×1022cm−2. While the mm continuum data trace
+allvelocitycomponentsalongthelineofsight,thenear-in frared
+data trace selected velocity components. Judging from Figu res
+3 and 4, where we see more than 1 velocity component, the to-
+tal columndensitytracedbytheCO dataismorethanafactor2
+higher.Furthermore,theCOabsorptiononlytracesthegasi nthe
+foregroundofthenear-infraredsourcereducingthetraced gasby
+anotherfactor2. While bothapproachesare a ffectedbysystem-
+atics – e.g., the mm derived column densities can be wrong by
+a factor 5 depending on the assumptions on the dust propertie s
+and temperatures – other reasons are more important for some
+of the differences. In particular, the mm continuum emission is
+sensitive to the cold and warm dust emission whereas the near -
+infraredabsorptionof the13CO transitionsuseable forouranal-
+ysistracesmainlythewarmergas.Therefore,thesedataind icate
+thata largefractionofthegasisat relativelylowtemperat ures.
+3.1.2. Potentialsignaturesfromthe BNdisk?
+Whilemostoftheobservedemissionisspatiallyunresolved ,we
+findweakextended12COemissiontowardtheBNobject.Figure
+6 presents a position-velocity diagram along the slit direc tion
+(PA of 126oeast of north, see sec. 2) which corresponds to the
+proposeddiskorientation(Jianget al.,2005).We findCO emi s-
+sionaroundtherestvelocityoftheBNobjectof ∼21kms−1ex-
+tending approximately ±4′′in both directions. Since we do not
+identify any strong velocity dispersion, this extended emi ssion
+is unlikely to be due to an outflow, but it may potentially comeFig.6.Positionvelocitydiagramofthe12CO(R0)linealongthe
+slit direction (PA of 126oeast of north, see sec. 2) plotted in
+logarithmic intensity scale. Around o ffset 0 the figure is domi-
+nated by the continuum emission, but outside ∼±2′′the emis-
+sion stems from the CO line. Positive o ffsets go in northwest
+direction.
+fromtheproposeddisk(Jiangetal.,2005).Themeasuredext ent
+of∼8′′wouldthencorrespondatthe givendistanceofOrionof
+414pc to an approximatedisk diameter of ∼3300AU or a disk
+radius of 1650AU. While such disk size would be comparably
+largewithrespecttotypicallow-massdisks(e.g.,several reviews
+in Reipurthetal. 2007), it is consistent with measured size s of
+rotatingstructuresinotherhigh-massstar-formingregio ns(e.g.,
+Schreyeretal. 2002; Cesaroniet al. 2005; Beltr´ an etal. 20 06;
+Beuther&Walsh 2008).
+While we do not identify extended emission is the rarer
+13CO isotopologue, the CO &13CO emission feature around
+20kms−1exhibits an additional interesting feature in the13CO
+data (Fig. 4): While the emission feature is at the edge of the
+bandpass for the13CO R(6) line, it remains undetected for the
+R(7) line and only comes up for the lines greater R(9). This
+emission feature is also visible in all12CO lines (R(1) to R(8))
+shown in Figure 3, but it does not exhibit any significant shap e
+change there. Likely, the12CO emission is optically thick and
+traces only an outer envelope that does not show big variatio ns
+with excitation temperature. In contrast to that, the measu red
+Gaussian line width of that component for the more optically
+thin13CO R(9), R(10), R(11), R(12) and R(13) lines are 4.3,
+5.3,6.5,6.4and6.6kms−1,respectively.Althougherrorsonthe
+line widthare difficulttoquantifybecausethe emissionis at the
+flankofthestrongabsorptionfeature,andfurthermoreours pec-
+tral resolution is only 3kms−1, the data are indicative of a line
+width increases with excitation temperature of the line tha t ap-
+pears to saturate for the highest detectable transitions R( 11) to
+R(13). While for a centrifugally supported disk with a centr alH.Beuther etal.:Diskand outflow signatures inOrion-KL 7
+Fig.7.HydrogenrecombinationPfund βline towardthe BN ob-
+ject. The histogram shows the data, and the three other
+lines present a two-component Gaussian fit to the data. Both
+Gaussiansareshownseparatelyaswellasthecombinedfit.Th e
+FWHM∆vof both components is given in the figure. The big
+dots reproduce the best fit to the Br αline (their Figure 7 re-
+scaledtoournormalizedPfund βspectrum)foranopticallythin
+outflowwith avelocitylaw v∝r−2/3byScovilleetal. (1983).
+mass of∼10M⊙the circular velocity at 3300AU is relatively
+small of order∼1.6kms−1, the measuredline width increase is
+consistentwitharotatingdiskstructurebecausetheinner region
+with larger rotation velocities should have higher tempera tures
+causedbythe excitingcentralstar. One hasto keepin mindth at
+for purely thermal excitation, rotation would not cause suc h an
+effect because the upper levels – all in the 3000K regime – do
+notexhibitbigrelativeexcitationdi fferences.Hence,theydonot
+trace significantlydi fferent regionsof a disk then. In contrast to
+that, for lines that are excited by UV fluorescence such an ef-
+fect is possible because the excitation mechanism via an ele c-
+tronic excited state tends to preserve the level population of the
+v=0 rotational states (e.g., Brittain et al. 2009). Therefore , in
+thiscasetheR(9)toR(13)linesaresensitivetogastempera tures
+in a relatively broader range between 150 and 500K (Table 2).
+Anexactreproductionofthelinewidthincreasewouldrequi rea
+detaileddiskmodel,includingitsdensityandtemperature struc-
+ture. Since we cannot constrain these parameters well from o ur
+data, this is beyond the scope of this paper. Although the lin e
+widthincreaseisbelowournominalspectralresolutionele ment,
+qualitativelythe observationsare consistent with a rotat ing disk
+structure.
+3.1.3. The Pfundβlinetowardthe BNobjectand H 2
+emissionnearbyBN
+Figure 7 presents a zoom into the Pfund βline. The line shape
+consists of two components, one central Gaussian component
+and broad line wings. It is possible to fit the whole profile rel a-
+tivelywell witha two-componentGaussianfit wherethecentr al
+componenthas a FWHM ∆v=39.3kms−1and the broad com-
+ponenthasaFWHMof ∆v=175.5kms−1.Thefullwidthdown
+tozerointensityisapproximately340kms−1(between-170and
+170kms−1).Closetothepeakoftheprofileataround25kms−1,
+thespectrumexhibitsasmalldipwhichislikelyanartifact from
+thetelluriccorrections(seesection2).Consideringthis ,thepeak
+ofthecentralGaussianfitat ∼16.5kms−1isstill consistentwith
+the velocity derived for the BN object by Scovilleet al. (198 3)
+of∼21kms−1.
+Figure 7 also presents as thick dots the best fit obtained by
+Scovilleetal. (1983) for the Br αline (their Figure 7 re-scaledFig.8.PositionvelocitydigramoftheH 2emission∼4.3′′south-
+east oftheBN-objectassociatedwith theinfraredsourceIR C15
+(Dougadosetal., 1993).
+to our normalized spectrum). Their model consists of a super -
+sonic, optically thin decelerating outflow with a velocity l aw of
+v∝r−2/3. It is remarkable how well the shape of their Br αline
+obtained∼30years ago corresponds to the shape of the newly
+observed Pfundβline. Hence, these Pfund βdata are also con-
+sistent withtheiroutflowmodel.
+Furthermore, approximately 4′′south-east of the BN object
+we detect H 2emission from the H 20−0S9 transition with
+Elower/k=7198K. This H 2emission feature is spatially associ-
+ated with the correspondingH 2knot in near-infraredH 2images
+(e.g.,Nissen etal. 2007)aswell aswiththemid-infraredso urce
+IRc15 reported by Shupinget al. (2004). Figure 8 shows a po-
+sition velocity diagram of this feature. While we do not reso lve
+anyspatialstructureoftheH 2knot,itshowsaverybroadveloc-
+ityextentgoingtoblue-shiftedvelocitiesof ∼−80kms−1,inex-
+cess ofthe CO absorptionfeaturesmeasuredtowardthe BN ob-
+ject. Since we are mainly interested in the BN object, source
+n and IRC3, we refrain from further analysis of this o ffset H2
+emission.
+3.2. Source n
+3.2.1. COabsorption andemission
+Due to the different observing dates, for source n (and IRC3
+in the following section) the telluric lines are almost enti rely
+shifted out of the CO spectrum. Figures 9 and 10 present the
+corresponding12CO R-series lines and the detected13CO lines
+with their associated optical depths. In comparisonto the p revi-
+ousBNdata,forsourcenweonlyidentifyonebroadabsorptio n
+featureinthe12COdatawhichisdominatedbyblue-shiftedout-
+flowing gas. It is interesting to note that the most blue-shif ted
+absorption does not have a Gaussian shape but rather a more
+extended wing-like structure. In the framework of accelera ted
+winds with distance from the driving star or disk, such a spec -
+tral behavior would be expected. These accelerated winds in -
+crease in velocity with distance from the star. Simultaneou sly,
+with increasing distance from the driving source the densit y of
+the surrounding gas and dust envelope also decreases, lower ing
+the corresponding absorption depth at those velocities. He nce,
+in this picture, higher-velocity gas should show shallower ab-
+sorptionfeaturesinthespectra(e.g.,Lamers&Cassinelli 1999).
+Furthermore,wealsoidentifyared-shiftedemissioncompo nent
+at∼+15kms−1.However,comparedtotheBNobject,wherethe
+emission feature can be identified in all transitions, for so urce8 H.Beuther etal.:Diskand outflow signatures inOrion-KL
+Fig.9.The12CO R(8) to R(1) lines from top to bottom to-
+wardsource-n.Telluricand vlsrvelocitiesaremarked.Thelinear
+slopesanddipsaround-40kms−1associatedwiththeR(8),R(6)
+andR(4)lines(forthelatterextendingfrom ∼-5to∼-65kms−1)
+areartifactsduetothe telluriccorrections.
+n it is more prominent in the higher excited lines. The total
+width of the CO absorption and emission is ∼90kms−1rang-
+ing from approximately ∼-65 to∼+25kms−1(broader than for
+BN).Theblue-shiftedendislesswelldeterminedbecauseof tel-
+luric line contamination. Nevertheless the blue-shifted o utflow
+part of the spectrum extends ∼70kms−1from the assumed vlsr
+of+5kms−1. This value exceeds that measured toward BN by
+about20kms−1.Althoughattheedgeofthetelluriclinecontam-
+ination,we tentativelyidentifyan additionaldiscrete ab sorption
+featureat∼-35kms−1.
+Since the12CO line is again always saturated (Fig. 9), for
+the additional analysis we use the corresponding13CO data
+(Fig. 10). Because of line-blending and telluric emission, we
+only measure five13CO lines for source n. The broad, saturated
+12CO absorption feature is clearly resolved into two compo-
+nentswithapproximatevelocitiesat ∼-7kms−1and∼+5kms−1.
+While the∼+5kms−1shows the deeper absorption features for
+the lower energy R-lines, it is interesting to note that the s ingle
+absorption peak observed for the13CO R(13) line is associated
+with the∼-7kms−1component, indicating hotter gas at more
+blue-shifted velocities. Following the approachoutlined for BN
+in section 3.1.1, the derived optical depth of13CO is∼0.6 at its
+highest.
+Since the telluric lines are far away in velocity space (see
+Fig. 9), for source n we can conductthe Boltzmannanalysis fo r
+both absorption components separately. The measured equiv a-
+lent widths Afor both componentsare listed in Table 31. Again
+calculating the13CO v=1 column densities (Table 3) and pro-
+ducing Boltzmann plots (Fig. 11), we find that for both veloc-
+ity components fits to all 5 data points are less good than in
+the case of the BN object. This is likely due to the fact that fit -
+ting 2 Gaussians to the broad R(13) line, which does not show
+two well separated absorption features anymore, may overes ti-
+mate the contribution of the ∼-7kms−1component and hence
+underestimatethecontributionfromthe ∼+5kms−1component.
+Therefore, we also fit only the 4 lower-energy transitions be -
+tween R(7) and R(12) resulting in better fits (Fig. 11). As ex-
+1The equivalent width Ais measured by simultaneous Gaussian fits
+tobothabsorption features.pected from the di fferent behavior of the two absorption com-
+ponentswith increasing energylevels, the derived rotatio n tem-
+peratures using the 4 lower energy lines of the ∼-7kms−1and
+∼+5kms−1are∼163±20K and∼103±10K, respectively. As a
+comparison, rotational temperatures measured at submm wav e-
+length from CH 3OH emission lines toward source n are around
+200K (Beutheret al., 2005b). Considering that these measur e-
+mentsare conductedwith di fferentmoleculesandverydi fferent
+observational techniques, the overall range of similar tem pera-
+tures derived at mid-infrared and submm wavelengths is reas -
+suring for the complementarity of such multi-wavelengths o b-
+servations.
+Fitting only the 4 lower-energy lines, the column den-
+sity of13CO divided by the partition function Q(T) results in
+Ntot(13CO)/Q(T)≈1.0×1015cm−2andNtot(13CO)/Q(T)≈
+2.5×1015cm−2for the∼-7kms−1and∼+5kms−1compo-
+nents, respectively. Approximating again the partition fu nction
+byQ(T)∼kt/(hB) (section 3.1.1) we have Q(163K)≈62 and
+Q(103K)≈39,andusingthe12COto13COisotopologicratioof
+69(e.g.,Shefferetal.2007),thederivedtotalCOcolumndensi-
+tiestowardsourcenare ≈4.3×1018cm−2and≈6.7×1018cm−2,
+of the same orderas for the BN object.With the CO-to-H 2con-
+versionfactorof8×10−5,thecorrespondingH 2columndensities
+are5.3×1022cm−2and8.4×1022cm−2.Addingthesetwocolumn
+densityvalues,thetotalH 2columndensitytracedtowardsource
+n by this near-infrared observations is ∼1.4×1023cm−2, more
+than an order of magnitude below the H 2column densities de-
+rivedwiththeSubmillimeterArrayat865 µm(∼6×1024cm−2,
+Beutheretal. 2004). Similar to the BN-case discussed in sec -
+tion3.1.1,thisshowsthatthenear-infrareddatamainlytr acethe
+warmer gaswhereasthe (sub)mmcontinuumobservationstrac e
+warmandcoldgascomponents.
+3.2.2. The Pfundβline
+The Pfundβline is also detected toward source n (Fig. 12), and
+we can fit a Gaussian to the recombinationline with FWHM of
+∆v≈24.3kms−1, a width down to 0 intensity of ≈70kms−1
+(from∼ −35 to∼+35kms−1) and a central velocity of
+∼0kms−1.Thelineiscoveredby2CRIRESchips,andwhilethe
+generalPfundβlineshapeisthesameforbothchips,thecentral
+dipat∼0kms−1cannotindependentlybereproduced.Therefore,
+thedipislikelyonlyanartifactduetoinsu fficientsignaltonoise.
+The thermal line width of a 104K hydrogen gas is
+∼21.4kms−1.Convolvingthatwiththe6kms−1spectralresolu-
+tion, the observable thermal line width should be ∼22.2kms−1.
+Therefore, the measured line width does not exceed much the
+thermallinewidthofanH iiregion.Hence,thePfund βemission
+fromsourcendoesnotexhibitstrongsignaturesfromawindb ut
+isratherconsistentwithathermalH iiregion.Wealsocannotex-
+cludethatthePfund βemissiontowardsourceniscontaminated
+bymorebroadlydistributedOrionnebulaemission.
+3.3. IRC3
+Asoutlinedin theIntroduction,in contrasttotheBN object and
+source n, the source IRC3 is unlikely to be a YSO, and it is
+not prominent in any typical hot core tracer (e.g., Blake et a l.
+1996;Beutheret al.2005b).Thesourceratherresemblesamo re
+extended dust density enhancement that reprocesses light f rom
+othersources, potentiallyIRC2. Figure13 presentsa part o f the
+2D slit spectrum clearlyshowingthe extendednatureof the C O
+emission toward IRC3 in contrast to the point-like continuu mH.Beuther etal.:Diskand outflow signatures inOrion-KL 9
+Fig.10.13CO optical depths toward source-n. The absorption feature a t∼+25kms−1of the R(7) line is likely an artifact due to
+telluriclines.
+Fig.13.Original data from the slit covering source n and IRC3. Sourc e n is the bright continuum source at o ffset∼4′′whereas
+IRC3 exhibitstheextendedCO emissionaroundo ffset∼10′′.
+structure from source n. The continuum emission from IRC3 is
+veryweakcomparedtothat fromsourcen.
+Figure 14 shows the CO lines extracted as an average spec-
+trum of length 5.16′′toward IRC3. While we again see a broad
+absorption feature against the weaker background extendin g
+from∼ −5kms−1out to the telluric contamination at about
+−40kms−1, IRC3 shows a broad and strong emission feature
+peakingat∼2kms−1.Since thisemission isextendedoversev-
+eral arcseconds, Figure 15 presents the position velocity d ia-
+grams of selected12CO and all13CO lines. In contrast to the
+BN object where the extended CO emission is just around the
+rest velocity of the star (Fig. 6), here we see a clear trend of in-
+creasing velocity with increasing distance from the contin uum
+peak (the so-called Hubble law of outflows). Furthermore, in
+particular the12CO position velocity diagrams show a twofold
+structurewithonevelocityincreasetovalues >10kms−1atoff-
+sets of∼ −1.8′′and another increase to similar velocities at
+∼ −3.3′′offset from the main continuum peak. To guide the
+eye, these wedge-like structures are sketched in the top-ri ght
+panelofFigure15).Suchmultiplewedgepositionvelocitys truc-
+tures are what jet-bow-shock entrainment models for molecu -
+lar outflows predict (e.g., Arceet al. 2007). Furthermore, s imi-
+lar to source n, the CO absorption shows also for IRC3 the ex-
+tended wing-like features toward the most blue-shifted abs orp-
+tion. Similar to the pv-diagrams, where we see acceleration of
+thegaswithdistancefromthesource,thesewing-likeabsor ptioncan also be interpreted in the framework of accelerated wind s
+(see also section 3.2.1 or Lamers&Cassinelli 1999). In sum-
+mary, IRC3 is not only a dust density enhancement scattering
+the light from another source, it may also be part of the fa-
+mous outflow emanating from the Orion-KL region. Based on
+these data, we cannot clearly define the driving source of the
+CO outflow feature. While the continuum peak of IRC3 could
+be driving an outflow, the CO emission may also be part of the
+large-scale outflow where di fferent driving sources have been
+proposed for in the literature, e.g., source I, SMA1, or the d is-
+integration of a bound system once containing sources I, n an d
+the BN-object (e.g., Menten&Reid 1995; G´ omezet al. 2005;
+Beuther&Nissen 2008; Zapataet al. 2009). While the positio n
+velocitystructureofthe13CO lineslargelyresemblesthatofthe
+12CO lines (Fig. 15), it is interesting to note that the stronge st
+13CO feature is at higher velocities and relatively distant fr om
+the continuum peak (o ffset∼−2.8′′and velocity∼+7kms−1.
+This difference is likely caused by the varying optical depth of
+the12COand13COtransitions.Sincethevelocityoftheemission
+feature is very close to the vlsrof the cloud (typically between
+5 and 9kms−1, e.g., Comitoetal. 2005), self-absorption of the
+mostabundant12COisotopologuemayveilthisfeature,whereas
+the less abundant13CO can be utilised to penetratemoredeeply
+intothecloudandhencedetectthe emissionbetter.10 H.Beuther etal.:Diskand outflow signatures inOrion-KL
+Fig.15.Position velocitycuts along the slit axis for IRC3 with a PA o f 110 degreeseast of north.The top row shows diagramsfor
+selected12CO lines, and the bottom row presents the13CO data. For13CO the scaling is adapted to highlight the weaker extended
+emissionincontrasttothecontinuumat o ffset0′′. Thewedge-likeoutflowstructuresare sketchedinthe top-r ightpanel.
+4. Discussionand conclusion
+High spectral resolution mid-infrared observations allow us to
+infer several important characteristics of some of the majo r
+sources within the Orion-KL region.The fundamentalCO line s
+are largely dominated by absorption from the individual YSO
+envelopes and their surrounding gas cloud. These absorptio n
+features are blue- and red-shifted with respect to the vlsrof the
+molecularcloudindicatingthatoutflowingandinflowinggas are
+simultaneously present toward our target sources. However , we
+also identify interesting emission features. Together the y con-
+firm the youth of the sources where infall and likely accretio n
+arestill ongoing.
+BN object: For the BN object, our data confirm, at double
+the spectral resolution, several of the assessments conduc ted
+already by Scovilleet al. (1983). However, also discrepanc ies
+arise. For example, an absorption feature reported previou sly
+around+30kms−1is not found in the new data, indicating ei-
+ther transient components or poor signal-to-noise in the ol der
+data. From a Boltzmann analysis, the rotational temperatur e is
+around 112K, and we derive CO column densities of several
+times 1018cm−2, well in agreement with the older results based
+on12COv=2-0byScovilleetal. (1983). Using standardCO-to-
+H2conversion factors, these column densities are about an or-
+der of magnitude below H 2column density estimates based on
+mm continuum emission. As discussed in section 3.1.1, while
+systematics may account for some of this discrepancy,the ma in
+differenceisthatthemmcontinuumemissionissensitivetocold
+and warm dust, whereasthe near-infraredabsorptionlines t race
+onlythe warmgascomponents.We also identify extended CO emission, likely emanating
+from the close environment of BN. While the absorption stems
+fromwarmgaswithtemperaturesoftheorder100K,gettingth e
+ro-vibrationally lines in emission implies already much hi gher
+temperaturesfor the correspondinggascomponents.The vel oc-
+ity range of the CO emission between 15 and 30kms−1en-
+compasses the previously inferred velocity of the BN object
+of∼21kms−1. Following Scovilleet al. (1983), who covered
+moreJ-transitions(fundamentalandovertoneemission)th anwe
+do, the rotational temperature of the compact emitting gas i s
+around 600K, significantly higher than the above derived rot a-
+tional temperature of the absorbing envelope gas. As outlin ed
+in section 3.1.1, high critical densities are required to pr oduce
+this line emission by pure thermal collisional excitation, and
+other processes like UV fluorescence or resonance scatterin g
+frominfraredemissionmayalsocontributetotheemissionl ines.
+Scovilleetal.(1983)inferanapproximatesizeforthatemi ssion
+of∼20AU, whereas we now resolve the emission extending to
+∼±4′′,correspondingtoadiameterof ∼3300AU.Thisdiscrep-
+ancycanbeexplainedbythelargedynamicalrangebetweenth e
+compact emission close to the source itself and the compara-
+bly very weak more extended emission. The small size found
+by Scovilleet al. (1983) can be attributed to the strong comp act
+emission which also emits in the overtone bands, whereas our
+moresensitivenewdataalsodetecttheweakerextendedfeat ures
+duetothehighdynamicrangeavailablewithCRIRES.Whileth e
+absolute ratio of the peak emission (CO plus continuum) to th e
+extended CO emission is ∼210, the ratio of the continuum sub-
+tracted CO emission toward BN compared to the extended CO
+isstill70.Scovilleet al.(1983)werenotsensitiveenough tode-
+tect this faint extended emission compared to the strong cen tralH.Beuther etal.:Diskand outflow signatures inOrion-KL 11
+Fig.11.Boltzmann plots for the13CO v=1-0 lines toward
+source-n for the∼-7kms−1and the∼+5kms−1components in
+the upper and lower panels, respectively. The x-axis shows t he
+lower-levelenergiesofthetransitionsandthey-axispres entsthe
+naturallogarithmofthecorrespondingcolumndensitiesdi vided
+bytheirstatisticalweights.Inbothpanels,thedashedlin eshows
+a fit to all five data points, whereas the full line presents a fit to
+onlythe fourlowertransitions.
+Fig.12.HydrogenrecombinationPfund βline ofsourcen.
+source. Although at the spectral resolution limit of our obs er-
+vations, the measured line width increase of the13CO emission
+featurewith increasingexcitationtemperatureis consist entwith
+a disk origin of the CO emission. Therefore, these parameter s
+make the CO emitting gas potentially to be associated with a
+disk around the BN object (see also Jianget al. 2005). Furthe r
+supportingthe disk-outflowinterpretation for the BN objec t areFig.14.The12CO R(8) to R(1) lines (except of the R(4) line
+fromtoptobottomtowardIRC3.Sincetheemissionisextende d,
+thisspectrumisanaverageover5 .16′′alongtheslit.Telluricand
+vlsrvelocitiesaremarked.
+thePfundβhydrogenrecombinationlinedatawhichshowbroad
+high-velocitylinewingsconsistentwiththedecelerating outflow
+scenarioproposedbyScovilleet al.(1983).
+Source n: While bright sources like the BN object were al-
+ready feasible to be observed a while ago (e.g., Scovilleet a l.
+1983) weaker sources like source n or IRC3 allowed reason-
+able high-spectral-resolutionspectroscopy only with the advent
+ofrecentinstrumentslikeCRIRESontheVLT.Thegeneralpic -
+ture for source n is relatively similar. A single broad absor ption
+feature extending approximately to −65kms−1traces mainly
+themolecularoutflowwhereasred-shiftedemissionlikelys tems
+from an inner infalling and accreting envelope /disk. This pic-
+tureisconsistentwith disk /outflowproposalsforthissourcede-
+duced from cm and mid-infrared wavelength imaging projects
+(e.g.,Menten&Reid 1995; Greenhilletal. 2004; Shupinget a l.
+2004). The clearly resolved double-peaked13CO structure al-
+lows to conduct the Boltzmann analysis for both components.
+We find that the colder component has higher H 2column den-
+sities (∼103K &∼8×1022cm−2) compared with the second
+warmer component( ∼163K &∼5×1022cm−2). As discussed
+above for the BN object, di fferences between the infrared CO
+derived column densities and those estimated from mm wave-
+lengthsobservationsmayarisebecausebothtracersaresen sitive
+to gas(anddust)at di fferenttemperatures.Incontrastto theBN
+object, the Pfundβemission from source n is consistent with a
+thermal Hiiregion without a strong wind component to the line
+shape.
+IRC3:The observational signatures from the dust density en-
+hancement IRC3 are very di fferent compared to the two pre-
+viously discussed sources. The continuum from IRC3 is much
+weaker, nevertheless we detect CO absorption between ∼ −5
+and∼−40kms−1.However,moreimportantly,weclearlydetect
+extended CO and13CO emission over scales of ∼4′′. This ex-
+tendedCOemissionshowsamultiplewedge-likevelocitystr uc-
+ture consistent with jet-entrainment models of molecular o ut-
+flows. Hence IRC3 may be not merely a dust density enhance-12 H.Beuther etal.:Diskand outflow signatures inOrion-KL
+ment, but it may be part of the famous outflow from the Orion-
+KLregion.
+Limitationsand future: One shortcomingof the data is that for
+our primary targets, the BN object and source n, we could not
+spatially resolve the inner region of the emission as done in
+thelower-masscasepresentedinGotoet al.(2006).Thereas ons
+may be different for the two sources. Source n is probably still
+too young and too deeply embedded so that the envelope over-
+whelms any emission from the embedded disk itself. The BN
+object has the advantage that its velocity of rest is o ffset from
+that of the cloud by more than 10km−1, and hence absorption
+couldbelessofaproblemforsuchkindofsource.However,BN
+is likely significantly more evolved, and the continuum-to- line
+ratio is so high that we cannot reasonably filter out the conti n-
+uum emission. Hence, we cannot well study the inner region of
+the proposed disk. Nevertheless, observing higher J-trans itions
+aswellastheCOovertoneemissionwithtodayshighersensit iv-
+itycomparedtotheScovilleet al.(1983)observationswill likely
+constraintheproposeddisk structureinmoredetail.
+How to proceed now if one wants to do similar-type fun-
+damental CO line studies of disks in high-mass star formatio n?
+On the one hand,it is important to not select too youngsource s
+because their envelopes will likely almost always “destroy ”
+the emission signatures. There may exist exceptions where o ne
+views straight through the outflow cavity face-on toward the
+disk.Ontheotherhand,formoreevolvedregions,thecontin uum
+emission can be very strong or maybe the remaining disk size
+can be reduced again making the spatial resolution a problem .
+Therefore, in addition to very careful target selections, a dding
+AOtoachievethebestspatialresolutionwillbeacrucialel ement
+forsuchkindofstudiesinthecomingyears.Furthermore,in par-
+ticularforthemostlysaturated12COlines,itwillbeimportantto
+extend the spectral coverage to also observe higher excited CO
+lines, that will likely not saturate anymore,as well as CO ov er-
+tone emission. This will allow us to better assess the hotter gas
+components and hence to conduct a more detailed analysis of
+the12CO data themselves. On longer time-scales, the ELT with
+itsproposedmid-infraredinstrumentMETISpromisesorder sof
+magnitude progress in this field based on its superior sensit iv-
+ity and spatial resolution. With this instrument,we will be truly
+capable to resolve the gas signatures of accretion disks aro und
+(high-mass)YSOs.
+Acknowledgements. We like to thank a lot the anonymous referee as well as
+theEditorMalcolm Walmsley forthorough reviews which help ed improving the
+paper. H.B. acknowledges financial support by the Emmy-Noet her-Program of
+the Deutsche Forschungsgemeinschaft (DFG, grant BE2578).
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+arXiv:1001.0652v1  [nucl-th]  5 Jan 2010Quantum Monte-Carlo method applied to Non-Markovian barri er transmission
+Guillaume Hupin1and Denis Lacroix1
+1GANIL, Bd Henri Becquerel, BP 55027, 14076 Caen Cedex 5, Fran ce
+In nuclear fusion and fission, fluctuation and dissipation ar ise due to the coupling of collective
+degrees offreedom with internal excitations. Close tothe b arrier, both quantum, statistical and non-
+Markovian effects are expected to be important. In this work, a new approach based on quantum
+Monte-Carlo addressing this problem is presented. The exac t dynamics of a system coupled to
+an environment is replaced by a set of stochastic evolutions of the system density. The quantum
+Monte-Carlo methodis applied tosystems with quadraticpot entials. Inall range oftemperature and
+coupling, the stochastic method matches the exact evolutio n showing that non-Markovian effects
+can be simulated accurately. A comparison with other theori es like Nakajima-Zwanzig or Time-
+ConvolutionLess ones shows that only the latter can be compe titive if the expansion in terms of
+coupling constant is made at least to fourth order. A systema tic study of the inverted parabola
+case is made at different temperatures and coupling constant s. The asymptotic passing probability
+is estimated in different approaches including the Markovia n limit. Large differences with the exact
+result are seen in the latter case or when only second order in the coupling strength is considered
+as it is generally assumed in nuclear transport models. On op posite, if fourth order in the coupling
+or quantum Monte-Carlo method is used, a perfect agreement i s obtained.
+PACS numbers: 24.60-k,25.70.Jj,05.60.Gg
+Keywords: open quantum systems, Monte-Carlo methods, non- Markovian effect.
+I. INTRODUCTION
+To understand nuclear reactions, the dynamics of nu-
+clei is often replaced by few selected collective degrees of
+freedom expected to contain important information on
+the dynamic. This is for instance the case in fusion reac-
+tions wherethe relativedistance and/ormassasymmetry
+is retained [1, 2]. Another example is provided by the fis-
+sion process which is often treated as a trajectory in an
+energy landscape function on different deformation pa-
+rameters [3, 4]. Although the evolution is projected onto
+few variables, other internal degrees of freedom may play
+an important role to understand the onset of dissipa-
+tion or fluctuation phenomena [5]. To treat these effects,
+the relevant degrees of freedom should be regarded as an
+Open QuantumSystemcoupled toanenvironmentwhich
+simulates the internal dynamics.
+To include dissipation in collective space, two impor-
+tant simplifications are often made. First, most of cur-
+rent models treating fusion/fission neglect quantum ef-
+fects and consider a classical treatment [6–9]. Such an
+approximation is however expected to be valid only if
+the internal excitation is high and therefore is not ex-
+pected to hold close to or below the Coulomb barrier. As
+it is discussed in ref. [10], a proper treatment of quan-
+tum and decoherence effects might be crucial in this re-
+gion. Second, when the time scale associated to collec-
+tive dynamics cannot be dissociated from the one of the
+environment, ”non-Markovian” (also called ”memory”)
+effect should also be properly treated [11, 12]. Large ef-
+fort is now devoted to account for both quantal and non-
+Markovian effects in nuclear reactions [13–18] and more
+generally in open quantum systems [19].
+Recently, the description of open quantum systems
+by stochastic methods has received much attention [19–21]. In the Markovian limit, several methods have been
+proposed to treat fluctuation and dissipation starting
+from a quantum master equation on the system density
+[19, 20, 22–28]. These methods have been extended also
+totreatnon-MarkovianeffectslikeinQuantumStateDif-
+fusion (QSD) [29–32] or quantum Monte-Carlo (QMC)
+methods [33]. Several groups have shown that these
+effects could eventually be simulated using Feynman-
+Vernon influence functional [21, 34] or directly stochastic
+master equations [35, 36].
+In this work, we apply the stochastic formulation pro-
+posed in ref. [36] to the case of quadratic potentials
+coupled to a heat-bath, the so-called Caldeira-Leggett
+model [37]. The case of inverted potential is the first
+steptowardsrealisticsituationslikefusionorfission. The
+aim of the present work is threefold. First, to introduce
+the new QMC method and apply it for potentials with
+barriers similar to those appearing in fusion/fission pro-
+cesses. Second, we show that the exact quantum Monte-
+Carlo method can be rather accurate to treat the dis-
+sipative dynamics of a quantum system. Last, we also
+present a comparison of this theory with other meth-
+odsbasedonprojection, namelyNakajima-Zwangig(NZ)
+and Time-ConvolutionLess (TCL) [38–41], which are ac-
+tually widely used to treat non-Markovian effects. Doing
+so, we show that only TCL up to at least fourth orders
+in the coupling constant can provide a competitive the-
+ory. The paper is organized as follows. In section II, the
+ingredients and properties of the quantum Monte-Carlo
+approach are discussed and the link with functional inte-
+gral is precised. In section III, the method is first illus-
+trated to the case of parabolic potential. Then, the pass-
+ing probabilities are estimated for the inverted parabola
+case.2
+II. QUANTUM MONTE-CARLO METHOD
+Our starting point is a system (S) interacting with a
+surrounding environment (E). We assume here that the
+total system (S+E) is described by the Hamiltonian
+H=HS+HE+V. (1)
+HS(resp.HE) acts on the system (resp. env.) only
+whileVinduces a coupling between the two sub-systems.
+Startingfromaninitialtotaldensity D(0), thedynamical
+evolutionisgivenbytheLiouvillevon-Neumannequation
+on the density:
+i/planckover2pi1dD(t)
+dt= [H,D(t)]. (2)
+In many physical situations, the total number of degrees
+offreedomto followin time preventsfromsolvingexactly
+this equation. One of the leitmotiv of Open Quantum
+System (OQS) theory is to find accurate approximations
+forthesystemevolutionwithoutfollowingexplicitlyirrel-
+evant degrees of freedom associated to the environment
+and therefore reduces the complexity of the initial prob-
+lem. Conventional strategy to treat dissipation and fluc-
+tuation in an open quantum system is to reduce the in-
+formation to the system density only ρS(t) = Tr E(D(t))
+while accounting approximately for environment effect.
+Here, we use a different strategy, the dynamics of the
+total system is first replaced by a set of stochastic evo-
+lutions where the total density remains separable along
+each path, i.e. D=ρS(t)⊗ρE(t). Then, the stochastic
+evolution of the environment is projected onto the rele-
+vant degrees of freedom to obtain a closed equation for
+the system density. It is shown that the new approach
+provides a proper treatment of dissipation and fluctua-
+tion for a system coupled to a surrounding heat-bath.
+A. Quantum Monte-Carlo formulation of Open
+Quantum Systems
+Recently, new stochastic formulations [34, 35, 42, 43]
+have been developed to study the system+environment
+problem that avoid evaluation of non-local memory ker-
+nels although non-Markovian effects are accounted for
+exactly (see also [35, 36, 42, 44–47]). One example of
+such a theory based on quantum Monte-Carlo method is
+presented here.
+Hereafter, it is assumed that the coupling is separable:
+V=Q⊗BwhereQandBact on the system and en-
+vironment respectively. For simplicity, initial separable
+density is considered, i.e. D(0) =ρS(0)⊗ρE(0). We
+want to replace the evolution of the total density (Eq.
+(2)) by an ensemble of stochastic evolutions of both the
+system and environment such that:
+
+
+dρS=dt
+i/planckover2pi1[HS,ρS]+dξSQρS+dλSρSQ
+dρE=dt
+i/planckover2pi1[HE,ρE]+dξEBρE+dλEρEB(3)wheredξS/EanddλS/Edenote Markovian Gaussian
+stochastic variables with zero means and where we use
+Ito convention of stochastic calculus [48]. In the follow-
+ing, we assume in addition that
+dξSdλE=dλSdξE= 0, (4)
+where the overline denotes the stochastic average. Along
+each path, the total density remains separable, i.e.
+D(t) =ρS(t)⊗ρE(t). Starting from such a density, at
+timet+dt, the average evolution deduced from Eq. (3)
+is given by
+dD(t) =dt
+i/planckover2pi1[hS+hE,D(t)]
++dξSdξE(Q⊗B)D(t)+dλSdλED(t)(Q⊗B).
+Therefore, under the condition
+dξSdξE=dt
+i/planckover2pi1,dλSdλE=−dt
+i/planckover2pi1, (5)
+the average evolution over the separable densities that
+evolve according to Eq. (3) identify with the exact Li-
+ouville von Neumann equation of motion (Eq. (2)). The
+possibility to use simple Gaussian noises to incorporate
+the environment effect might appear surprising. Indeed,
+noises used in standard approaches for Open Quantum
+Systems generally reflect properties of the environment.
+It should be howeverkept in mind that such environment
+dependent noises appear once the environment dynam-
+ics has been projected out on the system density evo-
+lution. Anticipating the discussion of section IIB, once
+such a projection has been made, the Gaussian noises in-
+troduced here, transform into new random variables that
+explicitly depend on the environment properties.
+The discussion above for one time step can then be
+iterated to show that the exact dynamics of a sys-
+tem+environment could be replaced by an average over
+an ensemble of stochastic evolutions of separable densi-
+ties [34–36]. To be really useful, mainly two difficulties
+should be overcome (i) in general, the environment cor-
+responds to a large number of degrees of freedom that
+could not be followed explicitly in time. (ii) the numeri-
+cal implementation of such a theory is possible only if the
+statistical errors do not growth too fast during the time-
+evolution. This statistical errors are directly connected
+to the number of trajectories necessary to accurately de-
+scribe the physical process. Difficulty (i) is solved in the
+next section by projecting out the effect of the environ-
+ment on the system leading to a closed equation for the
+system density only. Let us first concentrate on statis-
+tical errors. At any time, a measure of the statistical
+fluctuation around the average trajectory is given by
+Λstat=Tr/braceleftBig/parenleftBig
+D†(t)−D†(t)/parenrightBig/parenleftBig
+D(t)−D(t)/parenrightBig/bracerightBig
+=Tr/braceleftBig
+D†D(t)/bracerightBig
+−Tr/braceleftBig
+D(t)2/bracerightBig
+. (6)3
+Starting from Eq. (3), the evolution of Λ statover a small
+time step reads
+dΛstat=2dt
+/planckover2pi1/braceleftBig
+/an}bracketle{tQ2/an}bracketri}htS+/an}bracketle{tB2/an}bracketri}htE/bracerightBig
+, (7)
+where/angbracketleftbig
+Q2/angbracketrightbig
+S≡TrS(Q2ρS(t)) and/angbracketleftbig
+B2/angbracketrightbig
+E≡
+TrE(B2ρE(t)). Statistical errors associated with Eq. (3)
+have been estimated numerically and turn out to grow
+very fast in time [46]. As a consequence, the stochastic
+process in the present form is useless to simulate physical
+situationsand methodsto reducestatisticalerrorsshould
+be used.
+To do so, it is worth to note that the stochastic equa-
+tion of motion is not unique. Indeed, any stochastic pro-
+cess of the form:
+
+
+dρS=dt
+i/planckover2pi1[HS+Q∆E,ρS]
++dξS(Q−∆S)ρS+dλSρS(Q−∆S)
+dρE=dt
+i/planckover2pi1[HE+B∆S,ρE]
++dξE(B−∆E)ρE+dλEρE(B−∆E),(8)
+where ∆ S(t) and ∆ E(t) are time-dependent parameters
+leads to the same average evolution. These stochastic
+equations also provide a reformulation of the initial sys-
+tem+environment problem. Indeed, we have
+dρS⊗ρE+ρS⊗dρE=dt
+i/planckover2pi1[HS+Q∆E,ρS⊗ρE]
++dt
+i/planckover2pi1[HE+B∆S,ρS⊗ρE]
+dρS⊗dρE=dt
+i/planckover2pi1[(Q−∆S)⊗(B−∆E),ρS⊗ρE].
+Therefore, terms appearing in the deterministic part are
+exactly compensated by equivalent terms coming from
+the average over the noise. Accordingly, the evolution of
+the average density identifies with the exact equation of
+motion (2).
+Up tonow, the flexibilityhasbeen essentiallyexploited
+by using [35, 36]
+∆E(t) =/an}bracketle{tB(t)/an}bracketri}htE,∆S(t) =/an}bracketle{tQ(t)/an}bracketri}htS.(9)
+This choice is justified by the fact that it directly appears
+when the Ehrenfest theorem is applied to separable to-
+tal density for system or environment observables. By
+modifying the stochastic evolution, part of the couplingis already contained in the deterministic evolution. Ac-
+cordingly, we do expect that the amount of coupling to
+be treated by the noise is significantly reduced as well
+as the statistical errors. In the latter case, statistical
+fluctuations are given by:
+dΛstat=2dt
+/planckover2pi1/braceleftBig
+(/an}bracketle{tQ2/an}bracketri}htS−/an}bracketle{tQ/an}bracketri}ht2
+S)+(/an}bracketle{tB2/an}bracketri}htS−/an}bracketle{tB/an}bracketri}ht2
+S)/bracerightBig
+,(10)
+and are always smaller than the original ones (7). As
+shown numerically in ref. [35], introduction (9) signifi-
+cantlyreduces statisticalfluctuations andopens new per-
+spectives for the application of the present framework.
+The modified stochastic theory has other advantages.
+For instance, the traces of densities are constant and
+remain equal to their initial values, i.e. dTr(ρS/E) =
+0. This greatly simplifies expectation values of system
+and/or environment observables. Indeed, denoting by X
+a system operator, along a trajectory, we have
+/an}bracketle{tX/an}bracketri}ht= TrE(XD(t)) = Tr(ρE(t))TrS(XρS(t)).(11)
+For stochastic processes with varying trace of densi-
+ties, the observable evolution will contain terms com-
+ing from dTr(ρS/E) and cross terms coming from
+dTr(ρS/E)dTr(XρS(t)). In the case considered here, we
+simply have
+d/an}bracketle{tX/an}bracketri}ht= TrE(ρE(t))dTrS(XρS(t)).(12)
+The QMC theory with centered noise overcomes the dif-
+ficulty (ii) but does not help for (i) since the environment
+degreesof freedom should still be followed in time. In the
+next section, we show how irrelevant degrees of freedom
+can be projected out to obtain a closed stochastic master
+equation for the system only.
+B. Reduced system density evolution and link with
+influence-functional theory
+The stochastic formulation suffers a priori from the
+same difficulty as the total dynamics: the environment
+is in general rather complex and has a large number of
+degrees of freedom which can hardly be followed in time.
+In Eq. (8), the influence of the environment on the sys-
+tem only enters through /an}bracketle{tB(t)/an}bracketri}htE. Therefore, instead of
+following the full environment density evolution, one can
+concentrate on this observable only. As shown in ref.
+[43], the second equation in Eqs. (8) can be integrated
+in time to give:
+/an}bracketle{tB(t)/an}bracketri}htE= TrE(BI(t−t0)ρE(t0))−1
+/planckover2pi1/integraldisplayt
+0D(t,s)/an}bracketle{tQ(s)/an}bracketri}htSds−/integraldisplayt
+0D(t,s)duE(s)+/integraldisplayt
+0D1(t,s)dvE(s),(13)4
+whereBIdenotes the operator Bwritten in the interaction picture while DandD1are the memory function given
+by
+D(t,s)≡i/an}bracketle{t[B(t),B(t−s)]/an}bracketri}htE,andD1(t,s)≡ /an}bracketle{t{B(t),B(t−s)}+/an}bracketri}htE−2/an}bracketle{tB(t)/an}bracketri}htE/an}bracketle{tB(t−s)/an}bracketri}htE.(14)
+A new set of stochastic variables dvS/EandduS/Ehave been introduced through dξS/E=dvS/E−iduS/Eand
+dλS/E=dvS/E+iduS/E, and verify
+duSduE=dvSdvE=dt
+2/planckover2pi1,duSdvE=dvSduE= 0. (15)
+Reporting the evolution of /an}bracketle{tB(t)/an}bracketri}htEinto the evolution of ρS, a closed stochastic equation of motion for the system
+density is obtained:
+dρS=dt
+i/planckover2pi1[HS,ρS]+dt[Q,ρS]/integraldisplayt
+0dsD(t−s)/an}bracketle{tQ(s)/an}bracketri}htS+dξ(t)[Q,ρS]+dη(t){Q−/an}bracketle{tQ/an}bracketri}htS,ρS} (16)
+with
+dξ(t) =dt/integraldisplayt
+0D1(t−s)dvE(s)−dt/integraldisplayt
+0D(t−s)duE(s)−idvS(t), dη(t) =duS(t). (17)
+By integrating out the evolution of the environment, a
+new stochastic term is found that depends not only on
+the noise at time tbut also on its full history through
+the time integrals. Using second moments given by Eqs.
+(15) leads to:
+dη(t)dη(t′) = 0,
+dξ(t)dη(t′) =−dt
+2/planckover2pi1Θ(t−t′)D(t−t′),
+dξ(t)dξ(t′) =−idt
+2/planckover2pi1D1(|t−t′|),
+where Θ( t−t′) = 1 if t > t′and 0 elsewhere. Inter-
+estingly enough, the stochastic equation given by (16)
+identifies with the stochastic master equation obtained
+in ref. [21] using a completely different method based on
+the Feynman-Vernon influence functional theory [49]. It
+should however be kept in mind that different strategies
+to design the stochastic equation (see discussion in sec-
+tion IIA) would have given a different stochastic master
+equation.
+Despite the apparentcomplexityofEq. (16), the QMC
+approach has been recently applied with success to the
+spin-boson model coupled to a heat bath of oscillators
+[43]. In particular, the introduction of (9) seems to cure
+the numerical difficulties that have been encountered in
+this model [46]. By projecting the environment effect
+onto the system density evolution we do not need any-
+more to follow the environment density and we expect
+that a rather limited number of trajectories will be suffi-
+cient to accurately simulate the onset of dissipation and
+fluctuation in an open quantum system. Eq. (16) is
+the equation that is solved in practice. It should be
+noted that the present stochastic process differs signif-
+icantly from conventional approaches. Indeed, accord-
+ing to the noise properties, system densities are non-hermitian along a stochastic path. As a consequence, ex-
+pectation value of observables are complex loosing their
+physical meaning before averaging over the stochastic
+path. Nevertheless, we illustrate in the following that
+the new theory can be a very powerful tool.
+III. APPLICATION
+The Caldeira-Leggett (CL) model [37] corresponds to
+a single harmonic oscillatorcoupled to an environment of
+harmonic oscillators initially at thermal equilibrium, i.e.
+HS=Hc+P2
+2M+ε1
+2Mω2
+0Q2, (18)
+HE=/summationdisplay
+n/parenleftbiggp2
+n
+2mn+1
+2mnω2
+nx2
+n/parenrightbigg
+(19)
+andB≡ −/summationtext
+nκnxn[19]. Here, Hc=Q2/summationtext
+nκ2
+n
+2mnω2nis
+the counter-termthat insuresthat the physical frequency
+isω0. In the following, εis either +1 (harmonic case) or
+−1 (inverted parabola case). Such a model can be solved
+exactly.
+As shown in ref. [43], the two functions DandD1
+defined by Eqs. (14) and estimated along the stochastic
+trajectories identify with the standard times correlation
+functions:
+D(τ) = 2/planckover2pi1/integraldisplay+∞
+0dωJ(ω)sin(ωτ), (20)
+D1(τ) = 2/planckover2pi1/integraldisplay+∞
+0dωJ(ω)coth(/planckover2pi1ω/2kBT)cos(ωτ),(21)
+whereJ(ω)≡/summationdisplay
+nκ2
+n
+2mnωnδ(ω−ωn) denotes the spectral5
+density characteristic of the environment [19, 50]. In the
+following, a Drude spectral density [43]
+J(ω) =2Mη
+πωΩ2
+ω2+Ω2, (22)
+is considered where Mis the nucleon mass.
+A. Quantum Monte-Carlo method for parabolic
+potentials
+As a first illustration, the harmonic case ( ε= +1) is
+considered. This case has been already studied in ref.
+[51] using the stochastic method proposed in ref. [21]. In
+the CL model, starting from an initial Gaussian density,
+the system density remains gaussian along the stochastic
+path. Therefore, the stochastic evolution of the system
+densityreducestothefirstandsecondmomentsevolution
+of/an}bracketle{tP/an}bracketri}htand/an}bracketle{tQ/an}bracketri}htgiven by [36]:
+
+
+d/an}bracketle{tQ/an}bracketri}ht=/angbracketleftP/angbracketright
+Mdt+2duSσQQ
+d/an}bracketle{tP/an}bracketri}ht=−Mω2
+0/an}bracketle{tQ/an}bracketri}htdt−dt/an}bracketle{tB/an}bracketri}ht+2duSσPQ−/planckover2pi1dvS
+dσQQ= 2dt
+MσPQ
+dσPP=−2Mω2
+0dtσPQ
+dσPQ=dt
+MσPP−Mω2
+0σQQdt(23)
+These equations illustrate the differences between the
+new exact reformulation and standard methods to treat
+dissipation. Generally, the noise enters into the evolu-
+tion of/an}bracketle{tP/an}bracketri}htonly and affects directly the second moment.
+Here, we see that second moments identify with the un-
+perturbed oneswhile the randomforcesenterin both /an}bracketle{tQ/an}bracketri}ht
+and/an}bracketle{tP/an}bracketri}ht. In addition, the noise is complex, which im-
+plies that observables make excursions into the complex
+plane. This stems from the specific noise used to design
+the exact formulation that leads to non-Hermitian den-
+sities along paths. Part of the conceptual difficulties in
+understanding the physical meaning of observable evolu-
+tions can be overcome by noting that if ρS(t) belongs to
+the set of trajectories, by symmetry ρ†
+S(t) will also be-
+longs to the set. By grouping these two trajectories to
+estimate observables, real quantities are deduced.
+The exact evolution is obtained by averaging over dif-
+ferent trajectories. For second moments, this leads to
+ΣQQ≡/an}bracketle{tQ2/an}bracketri}ht−/an}bracketle{tQ/an}bracketri}ht2=σQQ+/an}bracketle{tQ/an}bracketri}ht2−/an}bracketle{tQ/an}bracketri}ht2(24)
+where/an}bracketle{tX/an}bracketri}htdenotes the statistical average of quantum
+expectation values /an}bracketle{tX/an}bracketri}ht. It is a particularity of the
+CL model that total fluctuation is recovered by simply
+adding up quantum and statistical fluctuations.
+An example of Σ QQ(t) evolution obtained using Eq.
+(24) (red filled circles) is compared to the exact result
+(solid line) in figure 1. As an indication, the evolution of 1 2 3 4 5
+0 3 6ΣQQ  γ2
+ωo t  1 2 3 4 5
+0 3 6ΣQQ  γ2
+ωo t 
+FIG. 1: (Color online) Evolution of Σ QQ(filled circles) ob-
+tained by averaging over 105trajectories. This evolution is
+compared to the exact result (solid line) and to the quan-
+tum fluctuation σQQevolution (dotted line). Parameters of
+the simulation are kBT=/planckover2pi1ω0,/planckover2pi1Ω = 5/planckover2pi1ω0,η= 0,5/planckover2pi1ω0and
+/planckover2pi1ω0= 14MeV. The factor γdefined as γ2=/planckover2pi1/(2Mω0)
+equals here γ= 1.216 fm−1.
+quantum fluctuation σQQ, which is identical for all tra-
+jectories, is also displayed (dotted line). Note that, Eq.
+(16) is already exact for the evolution of first moments
+/an}bracketle{tP/an}bracketri}htand/an}bracketle{tQ/an}bracketri}ht, even if the noise is omitted. However, it
+completely fails to account for fluctuation. While the
+quantum evolution does not present any damping, the
+average evolution closely follows the exact solution. The
+harmonic oscillations in σQQare due to the fact that the
+width of the initial density differs from the width of the
+coherent state associated to the considered oscillator, i.e.
+σQQ(0)/ne}ationslash=/planckover2pi1/(2Mω0). This is at variance with the simu-
+lation made in ref. [51].
+The accuracy of the quantum Monte-Carlo theory has
+beensystematicallyinvestigatedforvarioustemperatures
+and coupling strengths. In all cases, averaged evolutions
+could almost not be distinguished from the exact evo-
+lution. This is illustrated in figure 2 where Σ QQ(left),
+ΣPP(middle) and Σ PQ(right) are displayed as a func-
+tion of time and compared to exact solutions for various
+temperatures. Figure 2, clearly shows that the stochas-
+tic method properly includes all non-Markovian effects.
+In particular at low temperature, typically kBT </planckover2pi1ω0,
+and medium coupling constant η, large memory effect is
+expected.
+B. Application of NZ and TCL
+Conjointly to the benchmark of quantum Monte-Carlo
+approaches, we also tested projection method either
+based on the Nakajima-Zwanzig [19, 38, 39, 52] or Time
+ConvolutionLess [19, 40, 41, 52] formalisms. Both theo-
+ries provide a priori exact re-formulations of the initial
+problem and lead to a closed master equation for the sys-
+tem density. However,they differcompletely inthe strat-
+egy and equation of motion used to incorporate memory6
+0.10.20.3
+ 0 3 6ωo t 0.10.20.3
+ 0 3 6ωo t 0.10.20.3ΣPP  −h2 / γ2
+0.10.20.3ΣPP  −h2 / γ20.20.61 
+0.20.61 
+1.422.6
+036 
+ωo t 1.422.6
+036 
+ωo t 1.534.5ΣQQ  γ2
+1.534.5ΣQQ  γ271421 
+71421 
+-0.200.2
+036 
+ωo t -0.200.2
+036 
+ωo t -0.300.3ΣPQ  −h
+-0.300.3ΣPQ  −h012 
+012 
+FIG. 2: (Color online) Evolution of Σ PP(left), Σ QQ(middle)
+and Σ PQ(right) obtained with 105trajectories are displayed
+with red filled circles as a function of time and systematical ly
+compared with the exact evolution (solid line). kBT= 5/planckover2pi1ω0,
+/planckover2pi1ω0and 0,1/planckover2pi1ω0are respectively shown from top to bottom.
+In all cases, η= 0,5/planckover2pi1ω0,/planckover2pi1Ω = 5/planckover2pi1ω0and/planckover2pi1ω0= 14MeV.
+effects. In the NZ case, the evolution of the system den-
+sity at time tdepends on its full history (i.e. on ρS(s)
+for alls≤t). In the TCL case, the master equation is
+local in time and non-Markovian effects are treated in
+time-dependent transport coefficients. To illustrate the
+differences between our new QMC method and TCL, we
+remind below the corresponding local master equation,
+forV=Q⊗B.
+/planckover2pi1d
+dtρS(t) =−i[HS,ρS(t)]−i
+2∆(t)[Q,{Q,ρS(t)}]
+−2iλ(t)[Q,{P,ρS(t)}]−DPP(t)
+/planckover2pi1[Q,[Q,ρS(t)]]
++2DPQ(t)
+/planckover2pi1[Q,[P,ρS(t)]]. (25)
+∆(t),λ(t),DPQ(t) andDPP(t) are time-dependent
+transport coefficients that contain memory effects. Simi-
+larly to the QMC case, the solution of the master Eq.
+(25) is equivalent to follow first and second momentsgiven by:
+
+
+d/an}bracketle{tQ/an}bracketri}ht
+dt=/an}bracketle{tP/an}bracketri}ht
+M
+d/an}bracketle{tP/an}bracketri}ht
+dt=−Mω2
+p(t)/an}bracketle{tQ/an}bracketri}ht−2λ(t)/an}bracketle{tP/an}bracketri}ht
+dΣPP
+dt=−2Mω2
+p(t) ΣPQ−4λ(t)ΣPP+2DPP(t)
+dΣQQ
+dt= 2ΣPQ
+M
+dΣPQ
+dt=−Mω2
+p(t) ΣQQ−2λ(t)ΣPQ
++ΣPP
+M+2DPQ(t),
+withω2
+p(t) =ω2
+0+ ∆(t). In practice, the exact NZ or
+TCLtheorycannotbeexactlysolvedandanexpansionin
+powersofthecouplingconstantismade. Inthefollowing,
+NZ2 (or TCL2) will refer to the expansion up to second
+order while NZ4 (or TCL4) will refer to the expansion is
+made up to fourth order. By neglecting higher orders in
+the coupling in NZ2 (resp. TCL2) or NZ4 (resp. TCL4),
+both theory are not exact anymore. In the following, the
+efficiency of each method is systematically discussed.
+In Figure 3, the evolution of Σ PPfor different cut-off
+frequencies /planckover2pi1Ω and coupling strengths ηare compared
+to the exact evolution (solid line). Explicit forms of the
+equation of motion in the NZ and TCL case can be re-
+spectivelyfoundinref. [19,38,39,52]and[19,40,41,52].
+Severalimportantremarkscouldbedrawnfromthiscom-
+parison: (i) In all cases, when the coupling strength is
+considered up to second order, NZ2 (open triangles) pro-
+vides a better approximation than TCL2 (open squares).
+This might indeed be expected since NZ2 and TCL2 are
+respectively equivalent to the Born and Redfield master
+equationand the formercontainsa priorilessapproxima-
+tions than the latter. (ii) While the TCL4 (filled squares)
+leadstoaclearimprovementcomparedtotheTCL2, NZ4
+(filled triangles) is in general worse than NZ2. This is a
+known difficulty of NZ approach and was one of the mo-
+tivation for the introduction of TCL method (see discus-
+sioninref. [19,40,41]). Thisstemsfromthefactthatthe
+order in perturbation in NZ cannot be identified. For in-
+stance, NZ2 (resp. NZ4) contains orders in coupling con-
+stant greaterthan 2 (resp. 4). As a result, the NZ theory
+doesnot lead tobetter results when the ”apparent”order
+in the coupling increases. The TCL method essentially
+cures this pathology and precise orders in the coupling
+can be selected. (iii) Rather large deviations between
+the exact and TCL2 are observed for different cut-off fre-
+quencies and coupling strengths. This issue is important
+since several theories have been recently developed along
+the line of TCL2 to include memory effects in fusion and
+fission reactions [14, 15, 17]. Note that, the accuracy of
+TCL2 depends on different parameters used in the spec-
+tral density, in particular of the parameter /planckover2pi1Ω. Here,7
+we have used a value of the cut-off frequency between 10
+and 20 MeV, which gives realistic dissipation and fluc-
+tuation for the fusion or fission mechanism[14, 15, 18].
+Our study clearly points out that a proper treatment of
+memory requires to include higher order effects. (iv) Fi-
+nally, in all cases, TCL4 could not be distinguished from
+the exact result. As we will see, the efficiency of TCL4
+is similar for the inverted parabola.
+0.20.30.4
+ 0 3 6ΣPP −h2/γ2
+ωo t0.20.30.4
+ 0 3 6ΣPP −h2/γ2
+ωo t0.20.40.6 ΣPP −h2/γ2
+0.20.40.6 ΣPP −h2/γ2
+0.150.20.25
+036ΣPP −h2/γ2
+ωo t0.150.20.25
+036ΣPP −h2/γ2
+ωo t0.30.60.9ΣPP −h2/γ2
+0.30.60.9ΣPP −h2/γ2
+FIG. 3: (Color online) Evolution of Σ PPfor different approx-
+imations: NZ2 (open triangles), NZ4 (filled triangles), TCL 2
+(open squares) and TCL4 (filled squares). The exact evo-
+lution is displayed with solid line. In all cases, /planckover2pi1ω0= 14
+MeV, and kBT=/planckover2pi1ω0are used. The left side, corre-
+sponds to different cut-off frequencies: /planckover2pi1Ω = 20/planckover2pi1ω0(top) and
+/planckover2pi1Ω = 5/planckover2pi1ω0(bottom). In both cases, η= 0,5/planckover2pi1ω0. In the right
+side,/planckover2pi1Ω = 10/planckover2pi1ω0and different coupling strengths are used:
+η=/planckover2pi1ω0(top) and η= 0.1/planckover2pi1ω0(bottom).
+Since NZ method is not competitive, only the quan-
+tum Monte-Carlo and TCL methods are considered in
+the following application.
+C. Quantum Monte-Carlo method applied to
+inverted oscillators
+Severalapproacheshavebeen recentlydevelopedtode-
+scribe fusion and fission reactions [6, 14, 15, 17, 18, 53].
+In these mechanisms, few collective degrees of freedom
+couple to a sea of internal excitations while passing an
+inverted barrier. At very low energy, both quantum and
+non-Markovian effects are expected to play a significant
+role. Most of the theory currently used start from quan-
+tum master equations deduced from TCL2. The quan-
+tum Monte-Carlo method offers a practical alternative
+which has similarities with path integrals theory. Path
+integrals are known to provide a possible framework to
+includedissipationwhilepassingbarriers(seeforinstance
+[54]). However, due to their complexity, only few appli-
+cations have been made so far [2, 55]. We compare here
+the different approaches for inverted potential ( ε=−1).1. Initial conditions, trajectories and mean evolution
+Initially, we consider a Gaussian density with quantum
+widthσQQ(0) = 0.16 fm2andσPQ(0) = 0 MeV.fm/c
+and positioned on one side of the potential (here taken
+arbitrarily at /an}bracketle{tQ(0)/an}bracketri}ht=Q0>0 while the barrier height is
+is located at 0 fm and is by convention taken as VB= 0
+MeV). The initial kinetic energy, denoted EK(0) is set by
+boosting the density with an initial momentum /an}bracketle{tP(0)/an}bracketri}ht=
+P0<0.
+Contrary to the classical theory of Brownian motion,
+the notion of trajectories is not so easy to tackle in the
+present Monte-Carlo framework. First, observables are
+complex. As mentioned in section IIIA, this difficulty
+canbeovercomedbygroupingtrajectoriesbypairswhich
+is equivalent to replace expectation of observables by
+their real parts. Second, it should be kept in mind that
+thepresenttheoryisapurelyquantumtheorywhereden-
+sities associated to wave-packets are evolved. Therefore,
+each trajectory should be interpreted in the statistical
+sense of quantum mechanics and contains many classical
+paths. Nevertheless, to visualize the trajectory we define
+the following energies:
+E(t) =P(t)2
+2m−1
+2mω2
+0Q(t)2(26)
+whereQ(t) andP(t) denote the real part of /an}bracketle{tQ(t)/an}bracketri}htand
+/an}bracketle{tP(t)/an}bracketri}htalong the trajectory. An illustration of two trajec-
+tories, one passing the barrier and one reflected is shown
+in figure 4. As illustrated in the following, it is conve-
+nient to group trajectories according to the quantity ∆ E
+defined by
+∆E=E(0)−VB (27)
+which is nothing but the difference between total initial
+energy and barrier high. Both trajectories shown in fig-
+ure 4 correspond to ∆ E= 0 MeV.
+It is tempting to group trajectories into those passing
+the barrier and those reflected by the potential to get
+information on the passing probability or passing time,
+however, it should be kept in mind that the present the-
+ory is fully quantal. Since each trajectories are associ-
+ated with densities with quantum widths, both trajecto-
+ries presented in Fig. 4 contribute to the transmission
+probability.
+The accuracy of different methods is illustrated in fig-
+ure 5 where evolutions of /an}bracketle{tQ/an}bracketri}ht,/an}bracketle{tP/an}bracketri}ht, ΣQQand Σ PPare
+shown as a function of time. Values of parameters re-
+tained for this figure are typical values generally taken in
+thenuclearcontext[16]. Inallcases,includingTCL2, sec-
+ond moments are well reproduced. However, only TCL4
+and the stochastic simulation provides a correct descrip-
+tion of first moments. Calculations are shown here for
+∆E= 0 MeV. TCL2 provides a better and better ap-
+proximation when ∆ Eincreases while the disagreement
+increases below the barrier. This will be further illus-
+trated below.8
+0
+0.5
+Re(<Q>)-2-1012
+E (MeV)
+0 0.5
+Re(<Q>)-2-1012
+E (MeV)
+0 0.5
+Re(<Q>)-2-1012
+E (MeV)
+0 0.5
+Re(<Q>)-2-1012
+E (MeV)
+FIG. 4: (Color online) Evolution of E(t) as a function of Q(t)
+for twotrajectories (darkand brightlines) with ∆ E= 0MeV,
+kBT= 1MeVandη= 0.003MeV. The black arrow indicates
+the initial position of the trajectories while the potentia l is
+also shown with bold line as a reference.
+Different coupling parameters, cut-off frequencies and
+temperatures have been investigated showing that both
+TCL4 and quantum Monte-Carlo are very accurate the-
+ories leading always to results on top of the exact ones.
+It should be noted that the number of trajectories used
+in the stochastic approach to get small statistical errors
+is rather small (around 105) which is quite encouraging
+for future applications.
+0150300
+ 0 10 20 30 40〈 P 〉 −h /γ
+ωo t0150300
+ 0 10 20 30 40〈 P 〉 −h /γ
+ωo t024
+ 0 10 20 30 40〈 Q 〉 γ
+ωo t024
+ 0 10 20 30 40〈 Q 〉 γ
+ωo t090000180000
+     ΣPP  −h2 / γ2
+090000180000
+     ΣPP  −h2 / γ2
+01020
+     ΣQQ  γ2
+01020
+     ΣQQ  γ2
+FIG.5: (Color online)Evolutionof /angbracketleftQ/angbracketright,/angbracketleftP/angbracketright, ΣQQandΣ PPas
+a function of time obtained with quantum Monte-Carlo (filled
+circles), TCL2 (open squares) and TCL4 (filled squares). The
+exact evolution is also displayed with a solid line. Param-
+eters of the simulations are VB= 4MeV,/planckover2pi1ω0= 1MeV,
+η= 0.03MeV,/planckover2pi1Ω = 15/planckover2pi1ω,kBT= 1MeV, ∆E= 0 MeV
+andγ= 0.402 fm−1. 105trajectories have been used for the
+quantum Monte-Carlo case.D. Transmission probability
+The accuracy of the method used to incorporate non-
+Markovian effects directly affects the predicting power of
+the theory. This aspect is illustrated here with the pass-
+ing probabilities. Such a probability is a crucial ingredi-
+ent in particular for models dedicated to the formation of
+very heavy elements [7, 14, 15, 17, 53, 56, 57] and should
+be precisely estimated.
+The asymptotic passing probability is usually defined
+as:
+P(+∞) = lim
+t→+∞1
+2erfc/parenleftBigg
+−|q(t)|/radicalbig
+2σqq(t)/parenrightBigg
+.(28)
+whereq(t) andσqq(t) denote the expectation value and
+second moment of Qdeduced from the considered the-
+ory. In the quantum Monte-Carlo case, these quantities
+identify with /an}bracketle{tQ(t)/an}bracketri}htand Σ QQ(t) respectively. To quan-
+tify the precision of each theory, we have systematically
+investigated the difference between the estimated pass-
+ing probability and the exact one using the parameter
+∆P/P:
+∆P
+P≡P(+∞)−Pex(+∞)
+Pex(+∞)(29)
+whereP(+∞) andPex(+∞) denote the results of the
+specific calculation considered and the exact one respec-
+tively. Figure6presentstheevolutionof∆ P/Pasafunc-
+tion of ∆ Eobtained for different coupling strengths and
+temperatures for the quantum Monte-Carlo (filled cir-
+cles), TCL2 (open squares), TCL4 (filled squares) cases.
+Inthisfigure,the Markovianapproximationisalsoshown
+by open crosses.
+Once again, the TCL4 and the quantum Monte Carlo
+methods are in perfect agreement with the exact solu-
+tion for any input parameters. Small differences some-
+times observed between the stochastic approach and the
+exact value come from the limited number of trajectories
+used to obtain figure 6. Well above the barrier (here at
+least two times), TCL2 convergestowards the exact case.
+However, at low ∆ E, it turns out to be a rather poor ap-
+proximation. The difference seen in the TCL2 case can
+directly be traced back to the discrepancy already ob-
+served in figure 5 and further confirms the difficulty of
+treatingnon-Markovianeffectsbelowthe barrier. We can
+see that at lowest energy considered here, the error could
+be as large as 20% in the weak coupling case and more
+than 100%in the strongcouplinglimit. It is worthfinally
+to mention that below the barrier, the Markovian limit
+gives an even better result that the TCL2 case.
+IV. SUMMARY AND DISCUSSION
+In this work, the quantum Monte-Carlo approach re-
+cently proposed in ref. [43] to incorporate exactly non-
+Markovian effects is introduce and applied to the case of
+harmonic potentials coupled to a heat-bath.9
+00.1
+-2 0 2 4∆P /P
+∆E (MeV)00.1
+-2 0 2 4∆P /P
+∆E (MeV)00.1∆P /P
+00.1∆P /P00.1∆P /P
+00.1∆P /P
+01
+-2024∆P /P
+∆E (MeV)01
+-2024∆P /P
+∆E (MeV)01∆P /P
+01∆P /P01∆P /P
+01∆P /P
+FIG. 6: (Color online) Evolution of ∆ P/Pas a function of
+∆Ecalculated using quantum Monte-Carlo (filled circles),
+TCL2 (open squares), TCL4 (filled squares) and Markovian
+approximation (open crosses) for different coupling consta nt:
+η= 0.003MeV(left) and η= 0.03MeV(right). In both
+cases,T= 5/planckover2pi1ω,/planckover2pi1ωand 0,1/planckover2pi1ωare shown from top to bottom.
+For both non-inverted and inverted potentials, the
+new technique is rather effective to reproduce the exact
+evolution with a rather limited numbers of trajectories.Other methods have also been benchmarked. The TCL2
+method, which is now widely used in nuclear physics
+to estimate passing probabilities, turns out to deviate
+significantly from the expected result especially below
+the barrier and even in the weak coupling regime. To
+properly treat the dynamics of barrier transmission,
+higher orders in the coupling strength should be incorpo-
+rated. TCL4 gives very good agreement with the exact
+evolution in all cases considered here. The conclusion of
+our present work is that both quantum Monte-Carlo ap-
+proachandTCL4couldbe consideredasgoodcandidates
+to include memory effects for situations of interest in
+nuclear physics. Henceforth, the TCL2 method which is
+widely used nowadays should be replaced by TCL4. The
+possibility to use stochastic formulation that are exact
+in average open new perspectives to describe a system
+coupled to a complex environment. The application to
+harmonic potential gives interesting insight into such a
+theory. Application to more general potential turns out
+to be less straightforward with the appearance of spikes
+which have been already observed in several formalism
+where non linear stochastic equations appear [28]. To
+make these theories more versatile, new methods like
+the one proposed recently in ref. [58] could be used.
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\ No newline at end of file
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+arXiv:1001.0653v1  [q-bio.BM]  5 Jan 2010The effects of mismatches on hybridization in DNA
+microarrays: determination of nearest neighbor
+parameters
+J. Hooyberghsa,b,∗, P. Van Hummelenc, and E. Carlona,
+aInstitute for Theoretical Physics, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001
+Leuven, Belgium
+bFlemish Institute for Technological Research (VITO), Boereta ng 200, B-2400 Mol, Belgium
+cMicroArray Facility, VIB, Herestraat 49, B-3000 Leuven, Bel gium
+Abstract
+Quantifying interactions in DNA microarrays is of central importance for a bette r
+understanding of their functioning. Hybridization thermodynamics for nucleic acid
+strands in aqueous solution can be described by the so-called nearest-neighbor mo del,
+which estimates the hybridization free energy of a given sequence as a sum of dinu-
+cleotide terms. Compared with its solution counterparts, hybridization in DNA mi-
+croarrays may be hindered due to the presence of a solid surface and of a high density
+of DNA strands. We present here a study aimed at the determination of hybridiza-
+tion free energies in DNA microarrays. Experiments are performed on custom Agilen t
+slides. The solution contains a single oligonucleotide. The microarray contains spots
+with a perfect matching complementary sequence and other spots with one or tw o mis-
+matches: in total 1006 different probe spots, each replicated 15 times per m icroarray.
+The free energy parameters are directly fitted from microarray data. The experiments
+demonstrate a clear correlation between hybridization free energies in the microarray
+and in solution. The experiments are fully consistent with the Langmuir model at
+low intensities, but show a clear deviation at intermediate (non-saturating) intensi-
+ties. These results provide new interesting insights for the quantification of molecularinteractions in DNA microarrays.
+Key words: DNA hybridization, DNA microarrays, Nearest-neighbor mod el
+Introduction
+DNA microarrays are widely used in the current research in molecular biology [1]. Such
+devices have several important applications [2] as for instance in ge ne expression profiling,
+in the detection of Single Nucleotide Polymorphisms, in the analysis of c opy number vari-
+ations and of target sequences for transcription factors. Seve ral different platforms, either
+commercial or home made, are currently available. They differ by the details of fabrications
+(via spotting or in situ growth), the length of the sequences (oligon ucleotides or long PCR
+fragments) and the chemistry of fixation. What all DNA microarray s have in common is the
+basic underlying reaction of hybridization between a nucleic acid stra nd in solution and a
+complementary strand linked covalently at a solid surface. Hybridiza tion is characterized by
+a (sequence dependent) free energy difference ∆ Gwhich measures the binding affinity for
+the two strands to form a duplex.
+In the past decades a large number of papers were dedicated to th e investigation of static
+and dynamic properties of the hybridization between nucleic acid str ands which are both
+floating in an aqueous solution (see [3] and references therein). Ne arest neighbor models
+provide resonable approximation of ∆ Gfor strands hybridizing in solution [4,5]. In these
+models ∆ Gis calculated as a sum of “stacking” parameters associated to dinuc leotides [3].
+The nearest neighbor model isknown to berather accurate atleas t forhybridization between
+complementary strands. The case of single internal mismatches [6] as well as the dependence
+∗Corresponding author.
+Email addresses: jef.hooyberghs@vito.be (J. Hooyberghs), paul.vanhummelen@vib.be (P.
+Van Hummelen), enrico.carlon@fys.kuleuven.be (E. Carlon).
+2of ∆Gon other parameters as the monovalent salt concentration [7] wer e also considered.
+Therehasbeensomediscussionintheliteratureabouttherelations hipbetweenhybridization
+in solution and hybridization in DNA microarrays. In early studies of ge l pad microarrays
+[8] a linear relationship between microarray hybridization free energ ies (∆Gµarray) and the
+corresponding free energies in solution (∆ Gsol) was found. Recently [9] a similar relationship
+wasobservedonself-spottedcodelinkactivatedslides.Otherstu diesonAffymetrixGenechips
+[10,11] report very weak correlation between ∆ Gµarrayand ∆Gsol. In some papers [12,13]
+however the same Affymetrix data could be fitted resonably well with a linearly rescaled
+∆Gsol. Also some recent measurements of thermodynamic parameters u sing a temperature
+dependent surface plasmon resonance [14] seem to suggest a dec reased ∆ Gµarraycompared to
+∆Gsol. Clearly, as also some recent literature points out [6,15,16], more sys tematic physico-
+chemicalstudiesarerequiredforabetterunderstandingofhybr idizationinDNAmicroarrays.
+A precise quantification of ∆ Gis important. Through a better understanding of molecular
+interactions between hybridizing strands it would be possible to turn microarrays into more
+precise toolsfor largescale genomic analysis. Forinstance onecould estimate gene expression
+levels or detect mutations through an analysis based on thermodyn amics instead of using
+empirical statistical methods.
+This paper is dedicated to the investigation of the applicability of the n earest neighbor
+model to describe hybridization reactions in DNA microarrays, with a focus on sequences
+that contain isolated mismatches. Experimental results involving th e hybridization of one
+sequence in solution with a large set of different sequences on a micro array will be presented.
+The stacking free energy parameters will be determined from the a nalysis of the behavior of
+the experimental fluorescent intensities measured from different spots of the microarray. We
+will be interested in the correlation between free energies resulting from these parameters
+and the equivalent quantities calculated from experimental stackin g free energy parameters
+of nucleic acid melting in aqueous solution. The analysis of the experime ntal data clearly
+3Table 1
+The oligos used as target in the four different hybridization e xpriments. The oligos were bought
+from Eurogentec in duplicate obtained from independent syn thesis cycles.
+Name Sequence Labeling
+Target1 5’ GTTTTCGAAGATTGGGTGGCACTGTTGTAA 3’ 20-mer poly A + Cy3 on 3’
+Target2 5’ CAGGGCCTCGTTATCAATGGAGTAGGTTTC 3’ 20-mer poly A + Cy3 on 3’
+Target3 5’ CTTTGTCGAGCTGGTATTTGGAGAACACGT 3’ 20-mer poly A + Cy3 on 3’
+Target4 5’ GCTTCTCCTTAATGTCACGCACGATTTCCC 3’ 20-mer poly A + Cy3 on 3’
+reveals agooddegreeofcorrelation. However, amuch betteragr eement withthermodynamic
+models is found if the thermodynamic parameters are directly fitted from the experimental
+microarray data. In addition to this tight agreement with theory, a regime is found where
+the data are cleary deviating from the Langmuir behavior.
+This paper is organized as follows. Materials and Methods discusses t he experimental setup,
+the thermodynamic model of hybridization and the fitting procedur e. In Results and Dis-
+cussion the experimental results are presented and a comparison between free energies fitted
+from the microarray data and their solution counterparts is done. The final part of the paper
+is dedicated to a general discussion in which some open issues are high lighted.
+Materials and Methods
+The design of the experiment
+For the present study several hybridization experiments were pe rformed, each with a single
+oligonucleotide sequence (referred to as the target in this paper) in solution at different
+4Table 2
+Design of probeset: probe sequences covalently linked at th e microarray surface contained up to
+two mismatches following the scheme shown in this table. In t otal there are 1006 different probe
+sequences, replicated 15 times in the custom 8 ×15K custom Agilent slide.
+Nr of probes type of mismatch location of mismatch
+1 perfect match —
+60 single mismatch (all 3 permutations) site 6 to 25
+945 double mismatch (all 9 permutations) site 6 to 25, separa ted by min. 5 sites
+concentrations. Four different targets were used in the experime nts, and their sequences are
+given in Table 1. The sequences contain a 30-mer hybridizing stretch followed by a 20-mer
+poly(A) spacer and a Cy3 label at the 3’ end of the sequence. Each target oligo was bought
+in duplicate, in order to check the quality of the target synthesis. I n the rest of the paper we
+will refer to the two duplicated oligos as a and b.
+The sequences printed at the microarray surfaces and referred to here as the probes, were
+chosen to contain up to two mismatches, following the scheme shown in Table 2. Mismatches
+were inserted from nucleotides 6 to 25 along the 30-mer sequences in order to avoid terminal
+regions. In the probes with two mismatches these were separated by at least 5 nucleotides.
+Given the nucleotide of the target strand there are three differen t possible mismatching
+nucleotides and 20 available positions, hence in total 60 single mismatc h sequences. A similar
+countingfordoublemismatchesyields945differentsequences (see Table2).Thetotalnumber
+of probe sequences, including the perfect matching one, is 1006.
+For each experiment one target and one 8x15K custom Agilent slide w as used. This slide
+consists of eight identical microarrays and each of these can cont ain up to more than 15
+thousand spots. The 1006 probe sequences were spotted in the c ustom array 15 times: in
+5Table 3
+Thetarget condition permicroarray: concentration, oligo synthesis aor b, fragmentation f if applied
+.
+Microarray Experiment/target 1 Experiment/target 2 Exper iment/target 3 Experiment/target 4
+1 10000 pM, a, f 10000 pM, a, f 10000 pM, a 1000 pM, a
+2 7500 pM, a, f 5000 pM, a, f 5000 pM, a 500 pM,a
+3 5000 pM, a, f 1000 pM, a, f 1000 pM, a 100 pM, a
+4 2500 pM, a, f 50 pM, a, f 50 pM, a 50 pM, a
+5 1000 pM, a, f 10000 pM, b, f 10000 pM, b 1000 pM, b
+6 500 pM, a, f 5000 pM, b, f 5000 pM, b 500 pM, b
+7 100 pM, a, f 1000 pM, b, f 1000 pM, b 100 pM, b
+8 50 pM, a, f 50 pM, b, f 50 pM, b 50 pM, b
+12 repicates a 30-mer poly(A) was added on the 3’ side (surface sid e), in order to asses
+the effect of a sequence spacer. Three replicates contained no po ly(A) spacer. The eight
+microarrays of one slide have to be hybridized during the same exper iment, but a different
+target solution can be used. In the experiments the target conce ntrations ranged from 50
+to 10,000 pM (picomolar) according to the scheme given in Table 3. In exper iment 1 only
+target a was used, while in the experiments 2, 3 and 4 both replicated targets (a and b) were
+used. Finally, in experiment 1 and 2 a fragmentation of the target wa s performed before
+hybridization (see section on hybridization protocol for details).
+The four 30-mer target sequences were selected from fragment s of human genes having a
+6GC content ranging from 43% to 50%. A criterion for selecting the ta rget sequences was
+the requirement that the probes constructed following the schem e in Table 2 would yield a
+roughly flat histogram of mismatch types, so that all mismatches ar e approximately equally
+present in the experiments.
+Hybridization protocol and scanning
+For the experiments we used the commercially available Agilent platfor m and followed a
+standard protocol with Agilent products, as described below. (Th e target oligonucleotides
+were OliGold /circleRfrom Eurogentec, Seraing, Belgium). Hybridization mixtures conta ined one
+target oligonucleotide with a 3’ Cy3 endlabeling diluted in nuclease-fre e water to the final
+concentration together with 5 µL 10x blocking agent and 25 µL 2x GEx hybridization buffer
+HI-RPM. Unfortunately Agilent Techologies does not disclose the pr ecise composition of the
+hybridization buffer in the content of salt and other chemicals. In ex periment 1 and 2 the
+addition of the hybridization buffer was proceeded by a fragmentat ion step, 1 µL fragmen-
+tation buffer was added followed by an incubation of 30 min at 60◦C. This fragmentation
+buffer is customarily used in Agilent hybridization platforms and produ ces targets sequences
+of reduced length in order to speed up the hybridization reaction. T oo long sequences, as
+obtained from biological extracts, e.g. from reverse transcriptio n of mRNA samples, have a
+reduced hybridization efficiency due to steric hindrance. By compar ing experiments with and
+without fragmentation we found that the fragmentation step has little effect on the results.
+(More information can be found in the online supplementary material.) The hybridization
+mixture was centrifuged at 13000 rpm for 1 min and each microarray of the 8x15K custom
+Agilent slides was loaded with 40 µL. The hybridization occurred in an Agilent oven at 65◦C
+for 17 hours with rotor setting 10 and the washing was performed a ccording to the manu-
+facturer’s instructions. The arrays were scanned on an Agilent sc anner (G2565BA) at 5 µm
+resolution, high+low laser intensity and further processed using Ag ilent Feature Extraction
+7Software (GE1 v5 95 Feb07) which performs automatic gridding, int ensity measurement,
+background subtraction and quality checks.
+Thermodynamic Model
+In the Langmuir model the dynamics of hybridization is described by a rate equation for θ,
+the fraction of hybridized probes from a spot as follows
+dθ
+dt=ck1(1−θ)−k−1θ (1)
+wherecis the target concentration and k1andk−1are the attachment and detachment
+rates. The equilibrium value for θcan be obtained from the condition dθeq/dt= 0. Using
+the link between the rates and equilibrium constants, i.e. k1/k−1=e−∆G/RT, with ∆Gthe
+hybridization free energy, Rthe gas constant and Tthe temperature one finds
+θeq=c e−∆G/RT
+1+c e−∆G/RT(2)
+which is the so-called Langmuir isotherm. To link this isotherm to the me asured quanti-
+ties one assumes that the fraction of hybridized probes is linearly re lated to the measured
+fluorescent intensity measured from a spot, which yields
+I=Ac e−∆G/RT
+1+c e−∆G/RT(3)
+HereIis the background-subtracted intensity, where the background subtraction, as ex-
+plained above is done by Agilent Feature Extraction software. In th e rest of the paper we
+will no longer explicitly state that the intensities are background sub tracted, and will sim-
+ply refer to them as intensities. Ais a constant which is an overall scale factor. Far from
+chemical saturation, i.e. when only a small fraction of surface sequ ences is hybridized (i.e.
+8c e−∆G/RT≪1) one can neglect the denominator in Eq. (2) to get:
+I≈Ac e−∆G/RT(4)
+Inthenearestneighbormodelthehybridizationfreeenergyofpe rfectcomplementary strands
+is approximated as a sum of dinucleotide terms. For instance:
+∆G/parenleftbiggATCCT
+TAGGA/parenrightbigg
+= ∆G/parenleftbiggAT
+TA/parenrightbigg
++∆G/parenleftbiggTC
+AG/parenrightbigg
++∆G/parenleftbiggCC
+GG/parenrightbigg
++∆G/parenleftbiggCT
+GA/parenrightbigg
++∆Ginit(5)
+where∆Ginitisaninitiationparameter.Sincewewillonlyconsider differencesof∆ Gbetween
+a perfect matching hybridization and a hybridization with one or multip le mismatches (see
+Eq. (7)), this initiation parameter will not contribute and it is omitted in the rest of the
+paper. For DNA/DNA hybrids, symmetries reduce the number of ind ependent parameters
+to 10 [3]. The nearest neighbor model can be extended to include sing le internal mismatches;
+as an example we consider the free energy of a stretch with an inter nal mismatch of CT type
+∆G/parenleftbiggATCCT
+TATGA/parenrightbigg
+= ∆G/parenleftbiggAT
+TA/parenrightbigg
++∆G/parenleftbiggTC
+AT/parenrightbigg
++∆G/parenleftbiggGT
+CC/parenrightbigg
++∆G/parenleftbiggCT
+GA/parenrightbigg
+(6)
+The mismatching nucleotides are underlined and for notational reas ons the mismatch is
+always put in the second part of the dinucleotide (which requires the use of symmetry like
+here in dinucleotide term three) . There are 12 types of mismatches and 4 types of flanking
+nucleotide pairs, hence in total there are 48 mismatch parameters of dinucleotide type.
+There are several possible ways of extracting the 48+10 dinucleot ide parameters from the
+experimental data. One can either fit the full Langmuir isotherm (E q. (2)) , or for experi-
+ments at sufficiently low concentrations one could consider the limiting case of Eq. (4) . In
+addition, the parameters could beextracted either fromanexper iment at fixed concentration
+c, by comparing the intensities of different probe sequences, or fro m experiments at different
+concentrations by analyzing the intensities of identical probe sequ ences over a concentration
+9range. As argued in the next sections the optimal strategy in our e xperimental setup is to
+fit Eq. (4) at a fixed low concentration (the supplementary online ma terial discusses other
+strategies).
+Equation (4) contains the constant Awhich is an overall scale factor relating the hybridiza-
+tion probability to the actual measured fluorescence intensity. Th is quantity may fluctuate
+from experiment to experiment. For instance, the optical scannin g influences A, as this is
+proportional to the laser intensity used. Also hybridizations in differ ent slides might occur
+at slightly varying conditions and there can be small differences in the manifacturing of the
+slides. In the rest of this paper we will focus on relative intensities an d relative free energies,
+i.e. for each microarray we will use the perfect match of that microa rray as a point of ref-
+erence. We denote the logarithmic ratios of the intensities with the p erfect match intensity
+as
+yi= lnIi−lnIPM=−∆G−∆GPM
+RT≡ −∆∆G
+RT(7)
+for which the exact value of Ais irrelevant and we only need to consider the relative free
+energy differences ∆∆ G(which is for each probe a positive number). In ∆∆ Gof a duplex,
+only dinucleotide parameters which are flanking a mismatch remain, th e other parameters
+cancel out in the subtraction. E.g. from Eq. (5) and (6) one gets
+∆∆G/parenleftbiggATCCT
+TATGA/parenrightbigg
+= ∆G/parenleftbiggTC
+AT/parenrightbigg
++∆G/parenleftbiggGT
+CC/parenrightbigg
+−∆G/parenleftbiggTA
+AT/parenrightbigg
+−∆G/parenleftbiggAC
+TG/parenrightbigg
+(8)
+Inthis equation the lower strandrefers to thetarget sequence in solution, which is fixed. The
+upper strand is that of the probe sequence attached to the solid s urface. Hence, the ∆∆ G
+of a duplex contains two mismatch dinucleotide parameters and two m atching dinucleotide
+parameters per mismatch. This holds for sequences that contain m ore mismatches as long
+as the nearest neighbor model is valid, which we assume in our setup s ince mismatches are
+10separated at least by five base pairs. The model can now be written as
+yi=58/summationdisplay
+α=1Xiα∆Gα
+RT(9)
+whereαis the index running over the 58 possible dinucleotide parameters and Xis a fre-
+quency matrix, whose elements Xiαare the number of times the dinucleotide parameter α
+enters in ∆∆ Gof probe sequence i. With a simple extension of matrices and vectors one can
+rewrite the problem as
+/vector y=X/vector ω (10)
+where we have defined ωα= ∆Gα/RT. Having written the problem in Eq. (10) as a linear
+one, we can now apply the standard approach to find the optimal va lues of the parameters.
+The procedure consists in minimizing S= (/vector y−X/vector ω)2, which amounts to solving the following
+linear equation
+XT(/vector y−X/vector ω) = 0 (11)
+whereXTis the transpose of X.
+Degeneracies of /vector ω
+To obtain /vector ωfrom Eq. (11) one has to invert the 58 ×58 matrix XTX. In the case that XTX
+is not invertible one applies a singular value decomposition [18]. In the pr esent case the
+matrix is not invertible. Zero eigenvalues of the matrix XTXcome from reparametrizations
+that leave the physically accessible parameters ∆∆ Ginvariant. It is known, indeed, that
+the dinucleotide mismatch parameters are not uniquely determined [1 7,18], as these param-
+eters are entering in the expression for the total ∆ Gin pairs (see Eq. (6)). For instance, a
+11reparametrization of the type:
+∆G′/parenleftbiggx C
+x′T/parenrightbigg
+= ∆G/parenleftbiggx C
+x′T/parenrightbigg
++ε∆G′/parenleftBiggy T
+y′C/parenrightBigg
+= ∆G/parenleftBiggy T
+y′C/parenrightBigg
+−ε (12)
+for every pair of complementary nucleotides x,x′andy,y′leaves the total ∆ Ginvariant, as it
+can be verified directly from Eq. (6). Similar reparametrizations are possible for mismatches
+of type AG, AC and TG. Next to these there are three more invarian ces which involve a
+reparametrization of both mismatch and matching dinucleotide para meters. Hence one has
+at least 7 zero eigenvalues in XTX. A more detailed discussion of degeneracies of XTXcan
+be found in the supplementary online material.
+Results and Discussion
+Control of the quality of the experiments
+Asacontrolofthereproducibility oftheresult weconsider theinte nsities correlationbetween
+analogous spots in replicated experiments. The replicated hybridiza tions were carried out on
+two microarrays of the same slide, with two identical but separately synthesized and labeled
+target oligos, at the same manually prepared concentration in solut ion, see Table 3. Figure 1
+is an example thereof. It shows correlation plots between two replic ated hybridizations. Two
+plots are shown, one with the full 15K intensities (left) and one in whic h the median of the
+intensities of the 15 replicated spots are taken (right). In the for mer some data spreading is
+observed, which is greatly reduced when the median over 15 replicat ed spots is taken. Note
+that the experimental data do not align perfectly on the diagonal o f the graph, this may be
+attributed to the manual preparation of the solutions or to differe nces in the oligos (synthesis
+or labeling). Data from different microarrays are aligned on a line of slo pe equal to one in the
+log-log plots of Fig. 1, which implies a linear relationship between the inte nsities. In general,
+replicates show a strong correlation between median intensities, wh ich is an indication of
+12a good reproducibility of the results. We included in this median the pro bes with and
+without poly(A) spacer. No significant difference was found in the int ensities from spots
+with poly(A) and without poly(A) spacer. From this point on, the med ian intensity of 15
+replicates is always used and simply referred to as the intensity of a p robe, and because of
+the good reproducibility we will only discuss the data produced by hyb ridizations with oligo
+synthesis a (see Table 3) .
+Data analysis with ∆Gsol
+Next, we consider the relation between the intensities and the corr esponding ∆ Gsolfor hy-
+bridizations insolutionwithoneortwo mismatches. Inthecase oftwo mismatches ∆ Gsolwas
+calculated as the sum of nearest neighbor parameters for individua l mismatches, assuming
+that the presence of two mismatches does not involve additional te rms in the free energy,
+i.e. they do not interact. In the experiment the minimal distance bet ween two mismatches
+is 5 nucleotides, which is considered sufficient, in first approximation t o support the non-
+interaction assumption. In the calculation of ∆ Gfrom the tabulated values of ∆ Hand ∆S
+the temperature was set to the experimental value T= 65◦C.
+Figure 2 (a) shows plots of the intensities vs. ∆∆ Gsolas taken from the nearest neighbor
+model with the existing tabulated values for hybridization in solution ( see Ref. [6] and
+references therein). ∆∆ Gsolis obtained by subtracting from all free energies that of the PM
+sequence, which istaken asareference. Asa consequence, fort hePM intensities ∆∆ Gsol= 0.
+Each plot in Fig. 2 contains 1006 data points obtained from the median value of the 15
+replicated spots on each array.
+As it is well-known fromseveral studies of melting/hybridization inaqu eous solution(see e.g.
+[7]), the hybridization free energy ∆ Gsoldepends on the buffer conditions, and in particular
+of the ionic strenght of the solution. Particularly studied was the eff ect of salt concentration
+13(NaCl), which is usually assumed to be independent of sequence, but to be dependent on
+oligonucleotide length. Melting experiments in solution are consistent with the following
+dependence on Na ions concentrations [7]
+∆Gsol= ∆Gsol(1M[Na+])−aNln[Na+] (13)
+where ∆Gsol(1M[Na+]) is measured at 1M NaCl, Nis the number of phosphates in the
+sequence and aa constant. To our knowledge, the salt effect on sequences with int ernal
+mismatches has not been investigated yet, as measurements were done at 1M NaCl [6].
+However, salt has mostly an effect on interactions with the negative ly charged phosphate
+molecules. It is hence plausible to expect the same type of correctio n as Eq. (13) also for
+sequences carrying mismatches. If that is the case, the salt depe ndence cancels out from
+∆∆Gsol, which is the quantity we are interested in. In the rest of the paper , we will set the
+value at 1M NaCl in ∆ Gsol.
+Figure 2(a) shows the data for Experiment 1 at three different con centrations, from bottom
+to top of 50, 500 and 5000 pM. When plotted as functions of ∆∆ Gsolthe data points tend
+to cluster along single monotonic curves. This already suggests a fa ir degree of correlation
+between ∆ Gsoland ∆Gµarray. The experiment at 5000 pM shows a pronounced saturation
+of the intensities, as expected from the Langmuir model (Eq. 2). S ufficiently far from
+saturation one expects a linear relationship between the logarithm o f the intensity and ∆ G,
+as given by Eq. (4). Figure 2 shows that the low concentration data at low intensities follow
+approximately a straight with the slope 1 /RTexpected from equilibrium thermodynamics
+atT= 65◦C, which is the experimental temperature.
+However, the global behavior of the three concentrations is at od ds with the Langmuir
+model, which predicts that Intensity vs. free energy plots for diffe rent concentrations should
+saturate at a common intensity value A, as indicated in Fig. 2(b). Although one may expect
+14some variations on Afrom experiment to experiment, the data of Figure 2(a) are hard t o
+reconcilewiththeLangmuirmodel.Weconcludethatthehybridization datadeviatefromthe
+full Langmuir model of Eq. (2), but they are in rather good agreem ent with its limiting low
+intensities behavior Eq. (4). Inorder toobtainestimates ofthefr eeenergies ∆∆ Gµarraysfrom
+microarray data we will use then Eq. (4) and restrict ourselves to t he lower concentration
+data. The analysis of the higher concentration regime is presented in the supplementary
+online material.
+Fitting the free energy parameters
+To fit the 58 parameters of the nearest neighbor model we use the lowest concentration
+data, i.e. 50 pM. Hereto we applied the algebraic procedure explained in Materials and
+Methods, which fits the logarithm of the ratios I/IPMand which assumes that the data can
+be described by Eq. (4). For low concentrations this assumption is e xpected to be correct
+for the lower intensities but not for the highest intensities, which de viate from the Langmuir
+isotherm as shown in Fig. 2. This poses a problem for the fitting proce dure since it was
+designed with the perfect match intensity IPMas a reference (Eq. (7)). One may think to
+circumvent this problem by restricing the fit to low intensities, for ins tance only to probes
+with two mismatches and rewrite Eq. (7) using as reference not IPM, but for instance one of
+the intensities of a probe with two internal mismatches. This proced ure turns out to be of
+little practical use for our purposes which is to estimate the free en ergy difference between
+perfect matching sequences and sequences with one or multiple mism atches and for which
+the PM reference value is necessary (a more detailed discussion is in t he online material).
+From the analysis of plots of Intensity vs. ∆∆ Gsol(Figure 2) one finds that the PM inten-
+sity is systematically lower than that predicted by (Eq. (4)), which is the straight line in
+Figure 2(a). Hence, the relative intensities I/IPMof the probes that contain mismatches
+are systematically higher than those predicted by Eq. (4). Conseq uently, a direct fit of the
+15experimental data to Eq. (7) underestimates the effect of a misma tch, which will result in
+free energy penalties that are too small. The result of the fit is show n in Figure 3. One can
+notice that the ∆∆ Grange is indeed smaller than the one from hybridization in solution
+(Figure 2). Moreover, the underestimation of ∆∆ Gis more severe for probes with two mis-
+matches than for those with only one, since ∆∆ Gis a sum of contributions per mismatch.
+This produces a discontinuity of thecurve fromdouble tosingle misma tches. Theappearance
+of this discontinuity is another evidence of the fact that Eq. (4) is n ot valid in the full range
+of intensities.
+In order to solve this problem, one would need to fit the data with a mo re general model
+I(c,∆G)thatincorporatestheobserved deviationsfromEq.(4). Asmen tionedabove,andas
+shown explicitely fromthe data analysis in the supplementary online ma terial, the deviations
+cannot be described within the general Langmuir model (Eq. (2)). At present, it is not yet
+clear which alternative model to use for I(c,∆G). Moreover, the choice of this model may
+considerably influence the fitted nearest neighbor parameters. A safer compromize is to start
+from the observation that Eq. (4) is followed by the large majority o f the low concentration
+data points in Fig. 2. Hence a fit to the low concentration limit of the La ngmuir model seems
+reasonable. Unfortunately, one of the points deviating from Eq. ( 4) is the PM intensity,
+which is used as reference measure. In order to calibrate the fit co rrectly one should reweight
+the reference PM intensity. We therefore fit the data against Eq. (7) using instead of the
+actual PM intensity as a reference, a rescaled value I∗
+PM=αIPM, which is the value the
+PM intensity would have if the data would agree with Eq. (4) in the whole intensity range.
+We estimate αfrom the crossing of the 50pM fitting line in Fig. 2(a) with the ∆∆ G= 0
+axis. This estimate is α= 30. The effects of a change in αon the fitting parameters will be
+discussed below.
+Figures 4(a-d) show the result of the fit to Eq. (7), using α= 30. In the main frames each
+experiment is fitted independently. In the insets the free energy p arameters are obtained
+16from a simultaneous fit of all 50pM experiments. The latter data pro duce more accurate
+parameters, as they come from using 4 independent experiments ( the 4 experiments at 50
+pM, oligo synthesis a, in Table 3), hence the 58 parameters are obta ined on sampling over
+1006×4 data points. Both the free energy range and the continuity of th e curves in Fig. 4
+are now as expected. The data show very little spreading in comparis on with the curves
+in Figure 2(a). A quantification of the spreading for a monotonic cur ve can be assessed
+by the Spearman’s rank correlation coefficient, which for all four ex periments is very close
+to 1. This is an indicator of the reliability of the the nearest neighbor fi tted parameters.
+The ratio of data points over tuning parameters is large 4024/58, w hich ought to yield a
+reliable fit. Moreover, although the data are fitted to a linear model, all four experiments
+show a clear deviation for the highest intensities. This is an indication a gainst overfitting,
+which would result in a fully linear curve with erroneous fitting paramet ers. Therefore we
+conclude that the deviations from the Langmuir isotherm observed in all four experiments is
+a robust feature of the system and that the resulting free energ y parameters are physically
+meaningful. We also verified that the free energy parameters obta ined from the fit are quite
+stable whether one fits the whole set of experimental data, or whe ther the fit is restricted to
+the lowest intensity scales (e.g. I/I∗
+PM≤5·10−3) where all data clearly follow Eq. (4). This is
+because the large majority of experimental points in Fig. 4 are locat ed in the lowest intensity
+scales, anyhow. Hence, this additional data filtering has little effect s on the parameters.
+Table 4 shows the free energy parameters ∆∆ Gµarrayas obtained from the above fitting
+procedure. Because of the degeneracies mentioned above (see e .g. Eq. (12) and Ref. [18])
+the dinucleotide parameters are not uniquely determined. Triplet pa rameters are however
+unique, and these are given in the Table. The parameters are the ∆∆ Gdefined, for instance,
+as:
+∆∆G/parenleftbiggACG
+TTC/parenrightbigg
+= ∆G/parenleftbiggACG
+TTC/parenrightbigg
+−∆G/parenleftbiggAAG
+TTC/parenrightbigg
+(14)
+17wheretheupperstrandis5’-3’oriented.Thelower strandistheinv arianttargetsequence, the
+upper strand are the probe sequences. Hence the ∆∆ Gparameters are measured subtracting
+the reference perfect match probe. Note that because of this s ubtraction one has
+∆∆G/parenleftbiggACG
+TTC/parenrightbigg
+/negationslash= ∆∆G/parenleftbiggCTT
+GCA/parenrightbigg
+(15)
+as the reference PM sequence is different in the two cases.
+Using standard linear regression tools we estimated the error bar o n the parameters of Table
+4 to be equal to 0 .2. In order to compare with existing published data [6] we present in T able
+5 the ∆∆ Gsolfor triplets following the same notation as in Table 4. As mentioned bef ore
+the data in solution are at T= 65◦Cand 1M [Na+]. Figure 5 shows a plot of the two free
+energies ∆∆ Gµarrayvs. ∆∆Gsol. A clear quantitative correlation between the two is observed.
+The Pearson correlation coefficient is 0 .839. In comparing the two sets we note that the 16
+mismatches of CC appear to be the most deviating in the two cases.
+As discussed above, the fit was done with a rescaled PM intensity, us ing a factor α= 30. We
+have repeated the analysis for other values of α. Varying αcauses a global shift of the data
+in Table 4 by an αdependent constant. This shift does not affect the slope or corre lation of
+the data in Fig. 5. By using α= 50 we found a positive shift of 0 .17, while setting α= 20
+produces a shift of −0.14. These two values of αare our estimate of the largest range of
+variability for this parameters. In general the procedure of rewe ighting the PM intensity
+withαintroduces a global error ±0.2 affecting all parameters in Table 4.
+Concluding remarks
+During the past decades a considerable amount of research was de voted to the quantification
+of interactions among hybridizing nucleic acid strands in aqueous solu tion. This lead to a
+parametrization, via the nearest-neighbor model, of the contribu tion to the total free energy
+18in terms of dinucleotide pairs for perfect matching DNA/DNA [7], RNA/ RNA [19] and
+DNA/RNA [20] duplexes, but also for strands with an internal mismat ch [6]. This large
+amount of data is currently used in various applications as for instan ce for calculation of
+DNA melting temperatures or for RNA secondary structure predic tions. As it has been
+widely recognized [6,15,16] a similar effort for quantifying interactions in DNA microarrays
+is very important. This effort will lead to a better understanding of m olecular interactions
+in DNA microarrays and ultimately on their functioning.
+A precise quantification of interactions brings some challenges. Firs t of all many different
+microarrayplatformsexist, theydiffer bythelengthofprobesequ ences andthewaythese are
+covalently linked to the solid surface. It is not unlikely that interactio ns between hybridizing
+strands are of slightly different nature in these different platforms . Hence, one should be
+careful for instance to generalize the results of this work to, say , Affymetrix GeneChips.
+In addition, in order to measure accurately interaction parameter s, one needs a careful ex-
+perimental setup in which possible competing reactions, as hybridiza tion between partially
+complementary strands in solution, are absent. In the case of the present work this was
+achieved by choosing a single sequence in solution hybridizing with perf ect matching probe
+sequences with one or two internal mismatches. It is difficult to direc tly fit the free energy
+parameters from complex biological experiments where the hybridiz ing solution contain a
+large number of interacting sequences. This may explain why in some c ases poor correlations
+between ∆ Gsoland ∆Gµarraywas reported [10,11]. One of the advantages of the experimen-
+tal setup chosen in this work is that one can obtain in principle all para meters in a single
+experiment, as all hybridization reactions with one or two mismatche s occur in “parallel”
+on a single array. However, a drawback is that in this setup one can d etermine only the free
+energy and not the contribution of enthalpy and entropy separat ely, which would allow to
+extend the parameters to other temperatures. It would be cert ainly interesting to extend the
+analysis to other platforms and hybridization conditions.
+19In the present work we focused on the determination of ∆∆ Gwhich is the free energy
+difference between a perfect matching hybridization and an hybridiz ation where the probe
+sequences hasoneormoreinternal mismatches. Quantifying thee ffect ofinternal mismatches
+is important for a better understanding of cross-hybridization eff ect, which is the unintended
+binding of non-perfectly complementary sequences to a given prob e. Moreover, this under-
+standing could have some practical consequences for optimal pro be design. An advantage
+of the parameter ∆∆ Gis that it is insensitive to the free energy initiation parameter (Eq.
+(5)) and the scaling factor A(Eq. (2), Eq. (4)) and that it is expected to be less sensitive
+to buffer conditions as ionic salt etc ...The determination of the per fect match parameters
+∆Gis also possible in principle from microarray experiment but it requires s ampling perfect
+match hybridizations from a large number of target sequences. Th is requires a different and
+more complex experimental design.
+The present work on custom Agilent arrays shows that there is a st rong correlation, also on
+the quantitative scale, between ∆∆ Gsoland ∆∆Gµarray. This correlation is shown in Fig. 5
+with explicit free energy values given in Tables 4 and 5. A fit of the inter action parameters
+from microarray data shows a much better agreement of the data with the thermodynamic
+models (compare Fig. 2 with Fig. 3). However, in absence of dedicate d experiments for the
+determination of interaction free energies on a DNA microarray, th e results of this work
+suggest that one could use as approximations for them the corres ponding hybridization free
+energies in solution. Recent work [9,15] has addressed the issue of t he correlation between
+∆Gsoland ∆Gµarray. Ref. [9] considered oligonucleotide microarrays on Codelink activat ed
+slides carrying one, two or three mismatches. The data plotted as a function of ∆ Gsolshowed
+a good agreement with the Langmuir model, implying a fair correlation b etween ∆ Gsoland
+∆Gµarray. However the number of data points was insufficient to perform a dir ect fit of the
+thermodynamic parameters fromthemicroarray data. Interest ingly, thelowest concentration
+data in Ref. [9] seem to indicate the existence of deviations from the Langmuir model similar
+20to those observed in Fig. 2(a). Fish et al. [15] performed a series of experiments on oligo
+sequences in solutions hybridizing to perfect match and to sequenc e carrying one to multiple
+mismatches. Their analysis included tandem mismatches, i.e. mismatch es on neighboring
+sequence sites (in our case the minimal distance between mismatche s is five nucleotides).
+An overall correlation between ∆ Gsoland microarray intensities was observed, implying a
+correlation between ∆ Gsoland ∆Gµarray. In these experiments ∆ Gsolwas measured directly
+from experiments in solution and did not rely on the nearest-neighbo r model parameters.
+As a correlation between ∆ Gsoland ∆Gµarrayhas by now been observed in several different
+microarray platforms, it is fair to expect that such a correlation is a general feature of
+microarrays. However, an accurate determination of nearest-n eighbor parameters in other
+platforms would be very useful for a better quantification of this c orrelation.
+An interesting issue is the deviation from the low concentration limit of the Langmuir model
+(Eq. (4)).Thesedeviationscannotbeexplained bythefullmodelo fEq.(2).Thereareseveral
+underlying approximations in the Langmuir model, as for instance hyb ridization is always
+considered two state. The model also assumes that hybridizing str ands, apart from forming
+a duplex, do not further interact with other strands at the surfa ce. Moreover, Eqs. (2) and
+(4) apply to a system in thermal equilibrium. More investigations are n ecessary for a better
+understanding on the deviation from the Langmuir model found in th is study. These will
+involve further experiments in different external conditions, e.g. d ifferent temperatures or
+salt concentrations as well as theoretical analysis, which are left f or some future work.
+It is interesting to remark that the deviation from the Langmuir mod el “enhances” the cross-
+hybridization problem because there is a smaller effect on intensity fo r a given free energy
+penalty (smaller slope in Figure 4). As an example, a mismatch with ∆∆ G= 2.5 kcal/mol
+(a typical value from Table 4) corresponds to a I/IPMratio of≈0.02 in the regime governed
+by the Langmuir model, compared to ≈0.2 in the deviating regime. This implies that in the
+deviating regime a significant fraction of the amount of target bindin g to a PM probe binds
+21to a probe carrying one internal mismatch.
+Although the origin of these deviations are not known it is remarkable that the data appear
+to follow approximately two straight lines separated by a sharp kink ( Fig. 4). Although
+extensions of the Langmuir model in the context of DNA microarray s have been discussed
+(see e.g. [21]) we are unaware of isotherms which could have a shape a s shown in Fig. 4. The
+presence of a second straigth line in the log plot implies that inthis rang e the data still follow
+the thermodynamic model of Eq. (4) but with a different “effective” temperature than the
+experimental one. A linear regression to the data yields Teff≈850K, which is higher than
+the experimental temperature. It is interesting to point out that recent analysis [12,13] of
+Affymetrix GeneChipdatauseLangmuir modelwith∆ Gsolrescaled tohighereffective higher
+temperatures. A better understanding of the regime governed b y an effective temperature
+may provide new insights on this issue.
+Acknowledgements
+We thank Kizi Coeck and Karen Hollanders for help with the experimen ts. We aknowledge
+financial support from the Fonds voor Wetenschappelijk Onderzo ek (FWO) under grant
+G.03111.08.
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+24Table 4. Free energy differences ∆∆ Gunique parameters obtained from fitting microarray data to E q. (7). The data refer to triplets with
+central mismatching nucleotides and flanking matching nucl eotides. The convention is that the numbers correspond for s ay, a mismatch
+AGT
+TTAto a free energy difference ∆ G/parenleftBig
+AGT
+TTA/parenrightBig
+−∆G/parenleftBig
+AAT
+TTA/parenrightBig
+. The upper strand has orientation 5’ - 3’. The error bar on the parameters is
+0.2.
+❍❍❍❍❍❍❍XY
+A C G T A C G T A C G T A C G T
+A
+XAY
+X′AY′2.2 2.0 2.4 2.2
+XAY
+X′CY′3.0 2.8 3.0 3.0
+XAY
+X′GY′2.5 1.8 2.5 2.2
+XCY
+X′AY′2.4 2.2 2.4 2.5
+C 2.3 2.1 2.5 2.4 3.0 2.8 3.0 3.0 2.5 1.7 2.5 2.1 2.4 2.2 2.4 2.5
+G 1.9 1.8 2.2 2.0 2.7 2.5 2.7 2.7 2.4 1.6 2.4 2.0 2.0 1.8 2.1 2.1
+T 2.2 2.1 2.5 2.3 3.1 2.9 3.1 3.1 2.4 1.7 2.5 2.1 2.4 2.2 2.4 2.5
+A
+XCY
+X′CY′3.9 3.4 3.4 4.0
+XCY
+X′TY′2.5 2.4 2.4 2.8
+XGY
+X′AY′1.5 1.3 1.7 1.7
+XGY
+X′GY′2.4 1.8 2.3 1.9
+C 3.4 3.0 2.9 3.5 2.4 2.3 2.3 2.7 1.7 1.6 1.9 1.9 2.7 2.1 2.6 2.2
+G 3.1 2.7 2.7 3.2 2.5 2.5 2.5 2.8 1.1 0.9 1.3 1.3 2.5 1.9 2.4 2.0
+T 3.8 3.4 3.3 3.9 2.5 2.5 2.4 2.8 1.7 1.6 2.0 2.0 2.8 2.2 2.7 2.3
+A
+XGY
+X′TY′2.0 1.8 1.9 1.9
+XTY
+X′CY′3.5 3.6 3.1 3.2
+XTY
+X′GY′2.2 2.2 2.0 2.4
+XTY
+X′TY′2.3 2.4 2.0 2.2
+C 1.6 1.4 1.5 1.5 3.2 3.3 2.8 3.0 2.3 2.3 2.1 2.5 2.1 2.2 1.7 2.0
+G 1.8 1.7 1.8 1.7 3.1 3.2 2.8 2.9 2.4 2.4 2.2 2.6 2.4 2.5 2.1 2.4
+T 1.6 1.4 1.6 1.5 3.2 3.3 2.9 3.0 2.3 2.3 2.1 2.5 2.2 2.3 1.9 2.1
+25Table 5. Data as in Table 4 using the nearest neighbor paramet ers obtained from melting experiments in solution (see [6] a nd references
+therein). The data are at T= 65◦C and at 1 M [Na+].
+❍❍❍❍❍❍❍XY
+A C G T A C G T A C G T A C G T
+A
+XAY
+X′AY′1.3 2.0 2.3 2.0
+XAY
+X′CY′2.9 3.6 3.5 2.6
+XAY
+X′GY′2.3 1.7 2.9 1.8
+XCY
+X′AY′1.4 1.8 2.1 1.8
+C 1.6 2.3 2.6 2.2 3.5 4.2 4.1 3.2 2.7 2.0 3.2 2.1 2.2 2.6 3.0 2.6
+G 1.6 2.3 2.6 2.3 3.1 3.8 3.7 2.9 2.6 2.0 3.2 2.1 2.1 2.5 2.9 2.6
+T 1.1 1.8 2.1 1.8 3.0 3.7 3.6 2.7 2.5 1.8 3.0 1.9 1.9 2.3 2.6 2.3
+A
+XCY
+X′CY′3.4 4.3 4.4 4.5
+XCY
+X′TY′2.2 2.5 2.3 2.2
+XGY
+X′AY′0.8 0.8 1.3 1.0
+XGY
+X′GY′2.1 1.6 2.6 1.7
+C 3.6 4.5 4.5 4.7 2.7 3.0 2.8 2.7 1.6 1.6 2.1 1.7 2.8 2.2 3.2 2.4
+G 3.1 4.0 4.1 4.2 2.3 2.6 2.4 2.4 0.7 0.7 1.1 0.8 2.1 1.5 2.6 1.7
+T 2.6 3.5 3.6 3.7 2.1 2.4 2.2 2.2 1.4 1.3 1.8 1.5 2.3 1.7 2.8 1.9
+A
+XGY
+X′TY′2.0 1.7 1.7 1.7
+XTY
+X′CY′3.3 3.6 3.6 3.1
+XTY
+X′GY′2.4 2.2 2.4 2.5
+XTY
+X′TY′2.5 2.8 2.4 2.6
+C 1.6 1.3 1.3 1.3 3.6 4.0 4.0 3.4 2.5 2.2 2.4 2.6 2.3 2.6 2.2 2.4
+G 1.4 1.1 1.1 1.1 3.6 3.9 3.9 3.4 2.6 2.4 2.6 2.7 2.4 2.8 2.4 2.6
+T 1.4 1.1 1.1 1.1 3.2 3.6 3.6 3.0 2.5 2.2 2.4 2.6 1.8 2.2 1.7 2.0
+26100101102103104
+I   (Exp. 3-4)100101102103104I  (Exp. 3-8)
+100101102103104
+I  (Exp. 3-4)100101102103104I  (Exp. 3-8)Median intensities All intensities
+FIGURE 1
+Fig. 1. Correlation plots for intensities in two replicated experiments at 50 pM for oligo 3a (x-axis)
+and oligo 3b (y-axis); these are the experiments 3-4 and 3-8 i n Table 2. The left plot shows the
+total intensities and the right plot concerns only the media n intensities taken for the 15 replicated
+spots. The dashed line has slope equal to one.
+27-8 -6 -4 -2 0
+-∆∆Gsol (kcal/mol)100101102103104105IExperiment 1
+50 pM
+FIGURE 2A500 pM5000 pM
+PM
+1MM
+2MM(a)
+-∆∆G (arb. units)I (arb. units)
+FIGURE 2Bslope 1/RTA
+(b)
+Fig. 2. (a) Plot of the intensities as functions of ∆∆ Gsol, the difference of hybridization free energy
+with respect to the perfect match free energies, from neares t neighbor free energies obtained from
+melting experiments in solution. With this choice of parame ters the perfect match is located
+at∆∆G= 0.The different plots correspond to concentrations of 50, 500 an d 5000 pM (from
+bottom to top). The lines drawn have slopes corresponding to 1/RT, withT= 65◦C= 338Kthe
+experimental temperature.(b) Behavior of threeconcentra tion dataas predictedfromtheLangmuir
+model (Eq. (2)).
+28-6 -5 -4 -3 -2 -1 0
+-∆∆Gµarray (kcal/mol)10-410-310-210-1100I/IPMExperiment 1
+c = 50 pM
+2MM1MMPM
+FIGURE 3A
+Fig. 3. Ratios of Intensities and perfect match intensities vs. ∆∆Gµarray, therelative hybridization
+free energy between two strands as obtained from a fit to Eq. (7 ). Three distinct groups of points
+are indicated: PM for perfect match, 1MM for probes with a sin gle internal mismatch and 2MM
+for probes with two mismatches. The dashed line in is drawn as a guide to the eye.
+29-8 -6 -4 -2 0
+-∆∆Gµarray (kcal/mol)10-510-410-310-210-1I/IPM*
+-8 -6 -4 -2 010-410-2Experiment 1
+c = 50 pM
+FIGURE 4ATexp
+Teff
+-8 -6 -4 -2 0
+-∆∆Gµarray (kcal/mol)10-510-410-310-210-1I/IPM*
+-8 -6 -4 -2 010-410-2Experiment 2
+c = 50 pM
+FIGURE 4B
+-8 -6 -4 -2 0
+-∆∆Gµarray (kcal/mol)10-510-410-310-210-1I/IPM*
+-8 -6 -4 -2 010-410-2Experiment 3
+c = 50 pM
+FIGURE 4C-8 -6 -4 -2 0
+-∆∆Gµarray (kcal/mol)10-510-410-310-210-1I/IPM*
+-8 -6 -4 -2 010-410-2Experiment 4
+c = 50 pM
+FIGURE 4D
+Fig. 4. Plot of I/I∗
+PMwhereI∗
+PM=αIPM(where we took α= 30 as explained in the text) as
+function of the nearest neighbor fitted ∆∆ Gµmarray. The alignment of the intensities onto single
+monotonic curves is a proof of the good quality of the fits. In t he main frame the four different
+experiments where fitted separately. The insets show the dat e from intensities of each experiments,
+but the fit was done globally on all experiments at 50 pM. As a me asurement of the goodness
+of the fits the Spearman’s rank correlation coefficient was use d. This coefficient is for the main
+frames plots (a-d): 0.9860 0.9911 0.9866 and 0.9867. For the four plots in the insets the correlation
+coefficients are: 0.9732 0.9705 0.9748 and 0.9699. The two str aight lines in the first main frame
+correspond to slopes 1 /RTwhere we took Texp= 65◦C = 338Kfor the experimental temperature
+andTeff= 850K.
+30012 3 4 5
+∆∆Gsol012345∆∆GµarrayCC mismatchesAll Experiments
+FIGURE 5
+Fig. 5. Comparison of data in Table 4 and 5: the free energy diffe rences between a perfect matching
+hybridization and hybridization with an internal mismatch as obtained from data from Ref. [6]
+(∆∆Gsol) and from a fit of the microarray data (∆∆ Gµarray). The results show a good quantitative
+correlation between the two quantities: the Pearson correl ation coefficient is 0.839
+31
\ No newline at end of file
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@@ -0,0 +1,2231 @@
+arXiv:1001.0654v1  [math.DG]  5 Jan 2010REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES
+RUNG-TZUNG HUANG
+Abstract. LetEbe a flat complex vector bundle over a closed oriented odd dime nsional manifold M
+endowed with a flat connection ∇. The refined analytic torsion for ( M,E) was defined and studied by
+Braverman and Kappeler. Recently Mathai and Wu defined and st udied the analytic torsion for the
+twisted de Rham complex with an odd degree closed differentia l formH, other than one form, as a flux
+and with coefficients in E. In this paper we generalize the construction of the refined a nalytic torsion
+to the twisted de Rham complex. We show that the refined analyt ic torsion of the twisted de Rham
+complex is independent of the choice of the Riemannian metri c onMand the Hermitian metric on E.
+We also show that the twisted refined analytic torsion is inva riant (under a natural identification) if His
+deformed within its cohomology class. We prove a duality the orem, establishing a relationship between
+the twisted refined analytic torsion corresponding to a flat c onnection and its dual. We also define the
+twisted analogue of the Ray-Singer metric and calculate the twisted Ray-Singer metric of the twisted
+refined analytic torsion. In particular we show that in case t hat the Hermtitian connection is flat, the
+twisted refined analytic torsion is an element with the twist ed Ray-Singer norm one.
+1.Introduction
+LetEbe a flat complex vector bundle over a closed oriented odd dimensiona l manifold Mendowed
+with a flat connection ∇. Braverman and Kappeler [4, 5, 6, 7, 8] defined and studied the re fined analytic
+torsion for ( M,E), which can be viewed as a refinement of the Ray-Singer torsion [19] and an analytic
+analogue of the Farber-Turaev torsion, [11, 12, 23, 24]. It was sh own that the refined analytic torsion is
+closely related with the Farber-Turaev torsion, [4, 5, 8, 15].
+In [17, 18] Mathai and Wu generalize the classical construction of t he Ray-Singer torsion to the twisted
+de Rham complex with an odd degree closed differential form H, other than one form, as a flux and with
+coefficients in E. The twisted de Rham complex is the Z2-graded complex (Ω•(M,E),∇H), where
+(Ω•(M,E) is the space of differential forms with coefficients in Eand∇H:=∇+H∧·. Its cohomology
+H•(M,E,H) is called the twisted de Rham cohomology. Mathai and Wu [17] define d the analytic
+torsion of the twisted de Rham complex τ(M,E,H)∈Det/parenleftbig
+H•(M,E,H)/parenrightbig
+as a ratio of ζ-regularized
+determinants of partial Laplacians, multiplied by the ratio of volume e lements of the cohomology groups.
+They showed that when dim Mis odd,τ(M,E,H) is independent of the choice of the Riemannian metric
+onMand the Hermitian metric on E. They also showed that the torsion τ(M,E,H) is invariant (under
+a natural identification) if His deformed within its cohomology class and discussed its connection w ith
+the generalized geometry [14].
+In this paper we define the refined analytic torsion for the twisted d e Rham complex ρan(∇H)∈
+Det/parenleftbig
+H•(M,E,H)/parenrightbig
+. We show that the twisted refined analytic torsion ρan(∇H) is independent of the
+choice of the Riemannian metric on Mand the Hermitian metric on E. We then show that the torsion
+ρan(∇H) is invariant (under a natural identification) if His deformed within its cohomology class. We
+2010Mathematics Subject Classification. Primary: 58J52.
+Key words and phrases. determinant, analytic torsion, twisted de Rham complex.
+12 RUNG-TZUNG HUANG
+also establish a duality theorem, establishing a relationship between t he twisted refined analytic torsion
+corresponding to a flat connection and its dual, which is a twisted ana logue of Theorem 10.3 of [5]. In the
+end we define the twisted analogue of the Ray-Singer metric and the n calculate the twisted Ray-Singer
+norm of the twisted refined analytic torsion. In particular we show t hat in case of flat Hermtitian metric,
+the twisted refined analytic torsion is a canonical choice of an elemen t with the twisted Ray-Singer norm
+one.
+The paper is organized as follows. In Section 2, we review some stand ard materials about determinant
+lines of a Z2-graded finite dimensional complex. Then we define and calculate the refined torsion of
+theZ2-graded finite dimensional complex with a chirality operator. In Sect ion 3, we define the graded
+determinant of the twisted version of the odd signature operator of a flat vector bundle Eover a closed
+oriented odd dimensional manifold M. We use this graded determinant to define a canonical element
+ρHof the determinant line of the twisted de Rham cohomology of the vec tor bundle E. We study the
+relationship between this graded determinant and the η-invariant of the twisted odd signature operator.
+In Section 4, we first study the metric dependence of the canonica l element ρHand then use this element
+to construct the refined analytic torsion twisted by the flux form H. We show that the twisted refined
+analytic torsion is independent of the metric gMand the representative Hin the cohomology class [ H].
+In Section 5, we first review the concept of the dual of a complex an d construct a natural isomorphism
+between the determinant lines of a Z2-graded complex and its dual. We then establish a relationship
+between the twisted refined analytic torsion corresponding to a fla t connection and that of its dual. In
+Section 6, we first define the twisted Ray-Singer metric and then ca lculate the twisted Ray-Singer norm
+of the twisted refined analytic torsion.
+Throughout this paper, the bar over an integer means taking the v alue modulo 2.
+Acknowledgement. The author would like to thank Maxim Braverman for suggesting this p roblem.
+2.The refined torsion of a Z2-graded finite dimensional complex with a chirality
+operator
+In this section we first review some standard materials about deter minant lines of a Z2-graded finite
+dimensional complex. Then we define and calculatethe refined torsio noftheZ2-gradedfinite dimensional
+complex with a chirality operator. The contents are Z2-graded analogues of Section 2, Section 4 and
+Section 5 of [5]. Throughout this section kis a field of characteristic zero.
+2.1.The determinant line of a Z2-graded finite dimensional complex. Given a k-vector space
+Vof dimension n, the determinant line of Vis the line Det( V) :=∧nV, where∧nVdenotes the n-th
+exterior power of V. By definition, we set Det(0) := k. Further, we denote by Det( V)−1the dual line of
+Det(V). Let
+0−→C0∂0−→C1∂1−→ ···∂m−1−→Cm∂m−→0 (2.1)
+be an odd length, i.e. m= 2r−1 being a positive odd integer, cochain complex of finite dimensional
+k-vector spaces. Set
+C¯0=Ceven=r−1/circleplusdisplay
+i=0C2i, C¯1=Codd=r−1/circleplusdisplay
+i=0C2i+1.
+Let
+(C•,d) :···d¯1−→C¯0d¯0−→C¯1d¯1−→C¯0d¯0−→ ··· (2.2)REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 3
+be aZ2-graded cochain complex of finite dimensional k-vector spaces. For example, we can choose
+d¯k=/summationtext
+i,i=kmod2∂i. Denote by H¯k(d¯k),(k= 0,1) its cohomology. Set
+Det(C•) := Det/parenleftbig
+C¯0/parenrightbig
+⊗Det/parenleftbig
+C¯1/parenrightbig−1,Det(H•(d)) := Det/parenleftbig
+H¯0(d¯0)/parenrightbig
+⊗Det/parenleftbig
+H¯1(d¯1)/parenrightbig−1.(2.3)
+2.2.The fusion isomorphisms. (cf. [5, Subsection 2.3]) For two finite dimensional k-vector spaces V
+andW, we denote by µV,Wthe canonical fusion isomorphism,
+µV,W: Det(V)⊗Det(W)→Det(V⊕W). (2.4)
+Forv∈Det(V),w∈Det(W), we have
+µV,W(v⊗w) = (−1)dimV·dimWµW,V(w⊗v). (2.5)
+By a slight abuse of notation, denote by µ−1
+V,Wthe transpose of the inverse of µV,W.
+Similarly, if V1,···,Vrare finite dimensional k-vector spaces, we define an isomorphism
+µV1,···,Vr: Det(V1)⊗···⊗Det(Vr)→Det(V1⊕···⊕Vr). (2.6)
+2.3.The isomorphism between the determinant lines of a Z2-graded complex and its coho-
+mology. Fork= 0,1, fix a direct sum decomposition
+C¯k=B¯k⊕H¯k⊕A¯k, (2.7)
+such that B¯k⊕H¯k= (Kerd¯k)∩C¯kandB¯k=dk+1/parenleftbig
+Ck+1/parenrightbig
+=dk+1/parenleftbig
+Ak+1/parenrightbig
+. Then H¯kis naturally
+isomorphic to the cohomology H¯k(d¯k) andd¯kdefines an isomorphism d¯k:A¯k→Bk+1.
+Fixc¯k∈Det(C¯k) anda¯k∈Det(A¯k). Letd¯k(a¯k)∈Det(Bk+1) denote the image of a¯kunder the map
+Det(A¯k)→Det(Bk+1) induced by the isomorphism d¯k:A¯k→Bk+1. Then there is a unique element
+h¯k∈Det(H¯k) such that
+c¯k=µB¯k,H¯k,A¯k/parenleftbig
+dk+1(ak+1)⊗h¯k⊗a¯k/parenrightbig
+, (2.8)
+whereµB¯k,H¯k,A¯kis the fusion isomorphism, cf. (2.6), see also [5, Subsection 2.3].
+Define the canonical isomorphism
+φC•=φ(C•,d): Det(C•)−→Det(H•(d)), (2.9)
+by the formula
+φC•:c¯0⊗c−1
+¯1/mapsto→(−1)N(C•)h¯0⊗h−1
+¯1, (2.10)
+where
+N(C•) :=1
+2/summationdisplay
+k=0,1dimA¯k·/parenleftbig
+dimA¯k+(−1)k+1/parenrightbig
+. (2.11)
+2.4.The fusion isomorphism for Z2-graded complexes. LetC•=C¯0⊕C¯1and/tildewideC•=/tildewideC¯0⊕/tildewideC¯1be
+finite dimensional Z2-gradedk-vector spaces. The fusion isomorphism
+µC•,/tildewideC•: Det(C•)⊗Det(/tildewideC•)→Det(C•⊕/tildewideC•),
+is defined by the formula
+µC•,/tildewideC•:= (−1)M(C•,/tildewideC•)µC¯0,/tildewideC¯0⊗µ−1
+C¯1,/tildewideC¯1, (2.12)
+where
+M(C•,/tildewideC•) := dim C¯1·dim/tildewideC¯0. (2.13)
+The following lemma is a Z2-graded analogue of [5, Lemma 2.7] and [12, Lemma 2.4]. The proof is a
+slight modification of the proof of [5, Lemma 2.7].4 RUNG-TZUNG HUANG
+Lemma 2.1. Let(C•,d)and(/tildewideC•,/tildewided)beZ2-graded complexes with finite dimensional k-vector spaces.
+Further, assume that the Euler characteristics χ(C•) =χ(/tildewideC•) = 0. Then the following diagram com-
+mutes:
+Det(C•)⊗Det(/tildewideC•)φC•⊗φ/tildewideC•−−−−−− → Det/parenleftbig
+H•(d)/parenrightbig
+⊗Det/parenleftbig
+H•(/tildewided)/parenrightbig
+µC•,/tildewideC•/arrowbt/arrowbtµH•(d),H•(/tildewided)
+Det(C•⊕/tildewideC•)φC•⊕/tildewideC•− −−−− → Det/parenleftbig
+H•(d⊕/tildewided)/parenrightbig∼=Det/parenleftbig
+H•(d)⊕H•(/tildewided)/parenrightbig(2.14)
+Proof.Proceed similar procedures as the proof of Lemma 2.7 of [5], cf. [5, P . 152-153], we conclude that
+to prove the commutativity of the diagram (2.14) it remains to show t hat, mod 2,
+N(C•⊕/tildewideC•)+N(C•)+N(/tildewideC•)+M(C•,/tildewideC•)+M(H•,/tildewideH•) (2.15)
+≡/summationdisplay
+k=0,1/parenleftbig
+dimA¯k·dim/tildewideAk+1+dimH¯k·dim/tildewideAk+1+dimA¯k·dim/tildewideH¯k/parenrightbig
+.
+Using the identity
+(x+y)(x+y+(−1)j)
+2−x(x+(−1)j)
+2−y(y+(−1)j)
+2=xy, (2.16)
+wherex,y∈C,j∈Z≥0, we have
+N(C•⊕/tildewideC•)−N(C•)−N(/tildewideC•) =/summationdisplay
+k=0,1dimA¯k·dim/tildewideA¯k. (2.17)
+By (2.7) and the equalities dim Ak+1= dimB¯k, dim/tildewideAk+1= dim/tildewideB¯k, we have
+dimC¯k= dimA¯k+dimAk+1+dimH¯k,dim/tildewideC¯k= dim/tildewideA¯k+dim/tildewideAk+1+dim/tildewideH¯k.(2.18)
+By (2.13), (2.18) and a straightforward computation, we obtain, m odulo 2,
+M(C•,/tildewideC•)+M(H•,/tildewideH•)+/summationtext
+k=0,1/parenleftbig
+dimA¯k·dim/tildewideAk+1+dimH¯k·dim/tildewideAk+1+dimA¯k·dim/tildewideH¯k/parenrightbig
+=/summationtext
+k=0,1dimA¯k·dim/tildewideA¯k+dimA¯1·(dim/tildewideH¯0+dim/tildewideH¯1)+(dim H¯0+dimH¯1)·dim/tildewideA¯1.
+(2.19)
+By (2.17), (2.19) and the assumption that the Euler characteristic of the complex ( C•,d) (resp. ( /tildewideC•,/tildewided))
+is zero, i.e./summationtext
+k=0,1dimH¯k≡0(mod 2) (resp./summationtext
+k=0,1dim/tildewideH¯k≡0(mod 2)), we obtain the equality
+(2.15). /square
+2.5.The refined torsion of a finite dimensional Z2-graded complex with a chirality operator.
+Let (C•,d) be aZ2-graded complex defined as (2.2). A chirality operator is an involution Γ : C•→C•
+such that Γ( C¯k) =Ck+1,k= 0,1. Forc¯k∈Det(C¯k), we denote by Γ c¯k∈Det(Ck+1) the image of c¯k
+under the isomorphism Det( C¯k)→Det(Ck+1) induced by Γ.
+Fix a nonzero element c¯0∈Det(C¯0) and consider the element
+cΓ:= (−1)R(C•)·c¯0⊗(Γc¯0)−1∈Det(C•), (2.20)
+where
+R(C•) :=1
+2dimC¯0·(dimC¯0+1). (2.21)
+The element defined in (2.20) is a Z2-graded analogue of the Z-graded one as defined in [5, (4-1)], by
+Braverman-Kappeler, and is chosen to fit the Z2-graded setting.REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 5
+Definition 2.2. Therefined torsion of the pair (C•,Γ)is the element
+ρΓ=ρC•,Γ:= ΦC•(cΓ), (2.22)
+whereΦC•is the canonical map defined by (2.9).
+The following is the Z2-graded analogue of Lemma 4.7 of [5].
+Lemma 2.3. Let(C•,d)and(/tildewideC•,/tildewided)beZ2-graded complexes defined as (2.2)and letΓ :C•→C•,
+/tildewideΓ :/tildewideC•→/tildewideC•be chirality operators. Then /hatwideΓ := Γ⊕/tildewideΓ :C•⊕/tildewideC•→C•⊕/tildewideC•is a chirality operator on the
+direct sum complex (C•⊕/tildewideC•,d⊕/tildewided)and
+ρ/hatwideΓ=µH•(d),H•(/tildewided)/parenleftbig
+ρΓ⊗ρ/tildewideΓ/parenrightbig
+. (2.23)
+Proof.Clearly,/hatwideΓ2= 1 and /hatwideΓ(C¯k⊕/tildewideC¯k) =Ck+1⊕/tildewideCk+1. Hence, /hatwideΓ is a chirality operator. By Lemma 2.1,
+to prove (2.23) it is enough to show that
+c/hatwideΓ=µC•,/hatwideC•/parenleftbig
+cΓ⊗c/tildewideΓ/parenrightbig
+. (2.24)
+Fix nonzero elements c¯0∈Det(C¯0),/tildewidec¯0∈Det(/tildewideC¯0) and set /hatwidec¯0=µC¯0,/tildewideC¯0(c¯0⊗/tildewidec¯0). We denote the operators
+induced by Γ and /tildewideΓ on Det( C•) and Det( /tildewideC•) by the same letters. Thus,
+/hatwideΓ/hatwidec¯0= (Γ⊕/tildewideΓ)◦µC¯0,/tildewideC¯0(c¯0⊗/tildewidec¯0) =µC¯1,/tildewideC¯1(Γc¯0⊗/tildewideΓ/tildewidec¯0).
+Hence, it follows from (2.12) and (2.20) that
+µC•,/tildewideC•(cΓ⊗c/tildewideΓ) = (−1)M(C•,/tildewideC•)+R(C•)+R(/tildewideC•)·/hatwidec¯0⊗(/hatwideΓ/hatwidec¯0)−1
+= (−1)M(C•,/tildewideC•)+R(C•)+R(/tildewideC•)−R(C•⊕/tildewideC•)·c/hatwideΓ. (2.25)
+Using the identity (2.16), we obtain from (2.21)
+R(C•)+R(/tildewideC•)−R(C•⊕/tildewideC•) = dimC¯0·dim/tildewideC¯0. (2.26)
+Using the isomorphism Γ : C¯0→C¯1, one sees that dim C¯0= dimC¯1. Combining this fact with (2.13)
+and (2.26), we conclude that
+M(C•,/tildewideC•)+R(C•)+R(/tildewideC•)−R(C•⊕/tildewideC•)≡0 mod 2 . (2.27)
+The identity (2.24) follows from (2.25) and (2.27). /square
+2.6.Dependence of the Z2-graded refined torsion on the chirality operator. Suppose that
+Γt,t∈R, is a smooth family of chirality operators on the Z2-graded complex ( C•,d). Let˙Γt:C¯k→
+Ck+1,k= 0,1, denote the derivative of Γ twith respect to t. Then, for k= 0,1, the composition ˙Γt◦Γt
+mapsC¯kinto itself. Define the supertrace Trs(˙Γt◦Γt) of˙Γt◦Γtby the formula
+Trs(˙Γt◦Γt) := Tr( ˙Γt◦Γt|C¯0)−Tr(˙Γt◦Γt|C¯1). (2.28)
+The following proposition is the Z2-graded analogue of Proposition 4.9 of [5]. We modify the proof of
+Proposition 4.9 of [5] slightly to fit our setting.
+Proposition 2.4. Let(C•,d)be aZ2-graded complex of finite dimensional k-vector spaces and let Γt:
+C¯k→Ck+1,t∈R, be a smooth family of chirality operators on C•. Then the following equality holds
+d
+dtρΓt=1
+2Trs(˙Γt◦Γt)·ρΓt. (2.29)6 RUNG-TZUNG HUANG
+Proof.Let Γt,¯0denote the restriction of Γ ttoC¯0. We denoted the map Det( C¯0)→Det(C¯1) induced by
+Γtby the same symbol Γ tabove. To avoid confusion we denote this map by ΓDet
+t,¯0in the proof.
+Fort0∈R, we have Γ t,¯0= Γt,¯0◦Γt0,¯1Γt0,¯0. Hence,
+d
+dt/vextendsingle/vextendsingle/vextendsingle
+t=t0ΓDet
+t,¯0=d
+dt/vextendsingle/vextendsingle/vextendsingle
+t=t0/bracketleftBig
+Det(Γt,¯0◦Γt0,¯1)ΓDet
+t0,¯0/bracketrightBig
+= Tr(˙Γt0,¯0◦Γt0,¯1)ΓDet
+t0,¯0,
+where for the latter equality we used the fact that for any smooth family of operators At:C¯1→C¯1,
+one hasd
+dtDet(At) = Tr( ˙AtA−1
+t)·Det(At) and that Γ−1
+t0,¯0= Γt0,¯1. Hence, for any nonzero element
+c¯0∈Det(C¯0), we have
+d
+dt(ΓDet
+t,¯0(c¯0))±=±Tr(˙Γt,¯0◦Γt,¯1)·(ΓDet
+t,¯0)±. (2.30)
+By (2.30) and the definition (2.20) of cΓ, we obtain
+d
+dtcΓt=−Tr(˙Γt,¯0◦Γt,¯1)·cΓt. (2.31)
+Since Γt,¯0◦Γt,¯1= 1, we have
+0 =d
+dtTr(Γt,¯0◦Γt,¯1) = Tr(˙Γt,¯0◦Γt,¯1)+Tr(Γ t,¯0◦˙Γt,¯1).
+Hence,
+Tr(˙Γt,¯0◦Γt,¯1) =−Tr(˙Γt,¯1◦Γt,¯0). (2.32)
+Combining (2.31) with (2.32), we obtain (2.29). /square
+2.7.The signature operator. We now introduce the Z2-graded analogue of the Z-graded finite dimen-
+sional odd signature operator of [5, Section 5]. The signature operators Bk,k= 0,1 are defined by the
+formula
+B¯k:= Γd¯k+dk+1Γ. (2.33)
+Define
+C¯k
++:= Ker(dk+1◦Γ)∩C¯k= Γ(Ker dk+1∩Ck+1), C¯k
+−:= Kerd¯k∩C¯k. (2.34)
+LetB±
+¯kdenote the restriction of B¯ktoC¯k
+±. Then one has
+ImB+
+¯k⊆Im(Γ◦d¯k|C¯k)⊆Γ(Kerdk+1|Ck+1)⊆C¯k
++; (2.35)
+ImB−
+¯k⊆Im(dk+1◦Γ|C¯k)⊆Im(dk+1|Ck+1)⊆C¯k
+−. (2.36)
+Hence,
+B+
+¯k= Γ◦d¯k:C¯k
++→C¯k
++,B−
+¯k=dk+1◦Γ :C¯k
+−→C¯k
+−.
+Note thar B¯k= Γ◦Bk+1◦Γ.
+The following lemma is the Z2-graded analogue of [5, Lemma 5.2]. The proof is a verbatim repetition
+of the proof of [5, Lemma 5.2], we skip the proof.
+Lemma 2.5. Suppose that the signature operators B¯k,k= 0,1are bijective. Then the complex (C•,d)is
+acyclic and
+C¯k=C¯k
++⊕C¯k
+−. (2.37)REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 7
+2.8.Calculation of the refined torsion in case Bis bijective. In this subsection we compute the
+Z2-graded refined torsion in the case that B¯k,k= 0,1 are bijective. Assume that the signature operators
+B¯k,k= 0,1 are bijective. Then, by Lemma 2.5, the complex ( C•,d) is acyclic. Note that Γ B−
+¯0Γ =B+
+¯1.
+Hence Det( B−
+¯0) = Det(B+
+¯1). Then we have the following definition.
+Definition 2.6. Thegraded determinant of the signature operator B¯0is defined by the formula
+Detgr(B¯0) := Det( B+
+¯0)/Det(−B−
+¯0) = Det(B+
+¯0)/Det(−B+
+¯1). (2.38)
+The following proposition is the Z2-graded analogue of [5, Proposition 5.6]. We modify the proof of
+[5, Proposition 5.6] slightly to fit our setting.
+Proposition 2.7. Suppose that the signature operators B¯k,k= 0,1are invertible and, hence, the complex
+(C•,d)is acyclic. Then
+ρΓ= Det gr(B¯0). (2.39)
+Proof.We choose the decomposition (2.7) to be C¯k=C¯k
+−⊕C¯k
++and define elements c¯kas follows. Fix a
+nonzero element a¯k∈Det(C¯k
++) and set
+c¯k=µC¯k
+−,C¯k
++(Γak+1⊗a¯k),
+whereµC¯k
+−,C¯k
++is the fusion isomorphism, cf. (2.4), see also [5, (2-5)]. Note that, b y (2.5),
+Γc¯k=µCk+1
++,Ck+1
+−(ak+1⊗Γa¯k)
+= (−1)dimCk+1
++·dimCk+1
+−·µCk+1
+−,Ck+1
++(Γa¯k⊗ak+1)
+= (−1)dimCk+1
++·dimCk+1
+−·ck+1. (2.40)
+Thus, from (2.20), we obtain
+cΓ= (−1)R(C•)·c¯0⊗(Γc¯0)−1
+= (−1)R(C•)+dimC¯1
++·dimC¯1
+−·c¯0⊗c−1
+¯1. (2.41)
+Hence, by (2.22) and (2.10), to compute ρΓwe need to compute the elements h¯k∈Det(H¯k)∼=k.
+IfLis a complex line and x,y∈Lwithy/\e}atio\slash= 0, we denote by [ x:y]∈kthe unique number such that
+x= [x:y]y. Then
+h¯k= [c¯k:µC¯k
+−,C¯k
++(dk+1ak+1⊗a¯k)]
+= [µC¯k
+−,C¯k
++(Γak+1⊗a¯k) :µC¯k
+−,C¯k
++(dk+1ak+1⊗a¯k)]
+= [Γak+1:dk+1ak+1]
+= [ak+1: Γdk+1ak+1]
+= Det(Γdk+1)−1. (2.42)
+Combining (2.10) and (2.42), we obtain
+ΦC•(c¯0⊗c−1
+¯1) = (−1)N(C•)·Det(Γd0)·Det(Γd1)−1
+= (−1)N(C•)+dimC¯1
++·Det(Γd0)·Det(−Γd1)−1. (2.43)
+Combining (2.22), (2.38), (2.41) and (2.43), we have
+ρΓ= ΦC•(cΓ) = (−1)R(C•)+dimC¯1
++·dimC¯1
+−+N(C•)+dimC¯1
++·Detgr(B¯0).8 RUNG-TZUNG HUANG
+Hence, we are remaining to show that
+F(C•) :=R(C•)+dimC¯1
++·dimC¯0
+++N(C•)+dimC¯1
++≡0 mod 2 , (2.44)
+here we use the fact that Γ B+
+¯0Γ =B−
+¯1. By the fact that Γ B+
+¯1Γ =B−
+¯0and (2.21), we have
+R(C•) =1
+2(dimC¯0
+++dimC¯0
+−)·(dimC¯0
+++dimC¯0
+−+1)
+=1
+2(dimC¯0
+++dimC¯1
++)·(dimC¯0
+++dimC¯1
+++1)
+=1
+2(dimC¯0
++)2+1
+2(dimC¯1
++)2+dimC¯0
++·dimC¯1
+++1
+2dimC¯0
+++1
+2dimC¯1
++.(2.45)
+Recall that, (2.11),
+N(C•) =1
+2dimC¯0
++(C¯0
+++1)+1
+2dimC¯1
++(C¯1
++−1)
+=1
+2(dimC¯0
++)2+1
+2dimC¯0
+++1
+2(dimC¯1
++)2−1
+2dimC¯1
++. (2.46)
+Combining (2.44), (2.45) and (2.46), we have
+F(C•) = (dim C¯0
++)2+(dimC¯1
++)2+2dimC¯0
++·dimC¯1
+++dimC¯0
+++dimC¯1
++. (2.47)
+By (2.47) and the fact that for any x∈Z,x(x+1)≡0(mod2), we obtain
+F(C•) = 0.
+/square
+2.9.Calculation of the refined torsion in case Bis not bijective. In this subsection we compute
+theZ2-graded refined torsion in the case that B¯k,k= 0,1 are not bijective. Note that the operator B2¯k
+mapsC¯kinto itself. For an arbitrary interval I, denote by C¯k
+I⊂C¯kthe linear span of the generalized
+eigenvectors of the restriction of B2¯ktoC¯k, corresponding to eigenvalue λwithλ∈ I. Since both Γ and
+d¯kcommute with B¯k(and, hence, with B2
+¯k), Γ(C¯k
+I)⊂Ck+1
+Iandd¯k(C¯k
+I)⊂Ck+1
+I. Hence, we obtain a
+subcomplex C•
+IofC•and the restriction Γ Iof Γ toC•
+Iis a chirality operator for C•
+I. We denote by
+H•
+I(d) the cohomology of the complex ( C•
+I,dI). Denote by d¯k,IandB¯k,Ithe restrictions of d¯kandB¯k
+toC¯k
+I. ThenB¯k,I= ΓId¯k,I+dk+1,IΓI.
+Lemma 2.8. If0/∈ I, then the complex (C•
+I,dI)is acyclic.
+Proof.If, fork= 0,1,x∈Kerd¯k,I, thenB2¯k,Ix= (dk+1Γ)2x∈Imdk+1,I⊂Kerdk,I. Since the operators
+B2
+¯k,I:C¯k
+I→C¯k
+I,k= 0,1 are invertible, we conclude that Ker dk,I= Imdk+1,I. /square
+For each λ≥0,C•=C•
+[0,λ]⊕C•
+(λ,∞)andH•(d) = 0 whereas H•
+[0,λ](d)∼=H•(d). Hence there are
+canonical isomorphisms
+Φλ: Det(H•
+(λ,∞)(d))→C,Ψλ: Det(H•
+[0,λ](d))→Det(H•(d)).
+In the sequel, we will write tfor Φλ(t)∈C.
+The following proposition is Z2-graded analogue of [5, Proposition 5.10].REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 9
+Proposition 2.9. Let(C•,d)be aZ2-graded complex of finite dimensional k-vector spaces and let Γbe
+a chirality operator on C•. Then, for each λ≥0,
+ρΓ= Det gr(B¯0
+(λ,∞))·ρΓ[0,λ],
+where we view ρΓ[0,λ]as an element of Det(H•(d))via the canonical isomorphism Ψλ: Det(H•
+[0,λ](d))→
+Det(H•(d)).
+Proof.Recall the natural isomorphism
+Det(H¯k
+[0,λ](d)⊗H¯k
+(λ,∞)(d))∼=Det(H¯k
+[0,λ](d)⊕H¯k
+(λ,∞)(d)) = Det( H¯k(d)) (2.48)
+From Definition 2.2, Proposition 2.7 and (2.48), we obtain the result. /square
+3.Graded determinant of the twisted odd signature operator
+In this section we define the graded determinant of the odd signatu re operator, cf. [1, 13], twisted by
+a flux form H, of a flat vector bundle Eover a closed oriented odd dimensional manifold M. We use this
+graded determinant to define an element of the determinant line of t he twisted de Rham cohomology of
+the vector bundle E. We also study the relationship between this graded determinant an d theη-invariant
+of the twisted odd signature operator.
+3.1.The twisted odd signature operator. LetMbe a closed oriented smooth manifold of odd
+dimension m= 2r−1 and let Ebe a complex vector bundle over Mendowed with a flat connection
+∇. We denote by Ωp(M,E) the space of p-forms with values in the flat bundle E, i.e., Ωp(M,E) =
+Γ(∧p(T∗M)R⊗E) and by
+∇: Ω•(M,E)→Ω•+1(M,E)
+the covariant differential induced by the flat connection on E. Fix a Riemannian metric gMonMand
+let⋆: Ω•(M,E)→Ωm−•(M,E) denote the Hodge ⋆-operator. We choose a Hermitian metric hEso that
+together with the Riemannian metric gMwe can define a scalar product <·,·>Mon Ω•(M,E). Define
+the chirality operator Γ = Γ( gM) : Ω•(M,E)→Ω•(M,E) by the formula, cf. [5, (7-1)],
+Γω:=ir(−1)q(q+1)
+2⋆ω, ω ∈Ωq(M,E), (3.1)
+wherergiven as above by r=m+1
+2. The numerical factor in (3.1) has been chosen so that Γ2= Id, cf.
+Proposition 3.58 of [3].
+Assume that His an odd degree closed differential form on M. Let Ω¯0(M,E) := Ωeven(M,E),
+Ω¯1(M,E) := Ωodd(M,E) and∇H:=∇+H∧·. We assume that Hdoes not contain a 1-form component,
+which can be absorbed in the flat connection ∇.
+Definition 3.1. The twisted odd signature operator is the operator
+BH=B(∇H,gM) := Γ∇H+∇HΓ : Ω•(M,E)→Ω•(M,E). (3.2)
+We denote by BH¯kthe restriction of BHto the space Ω¯k(M,E),k= 0,1.10 RUNG-TZUNG HUANG
+3.2.ζ-function and ζ-regularized determinant. In this subsection we briefly recall some definitions
+ofζ-regularized determinants of non self-adjoint elliptic operators. S ee [5, Section 6] for more details.
+LetD:C∞(M,E)→C∞(M,E) be an elliptic differential operator of order n≥1. Assume that θis an
+Agmon angle, cf. for example, Definition 6.3 of [5]. Let Π : L2(M,E)→L2(M,E) denote the spectral
+projection of Dcorresponding to all nonzero eigenvalues of D. Theζ-function ζθ(s,D) ofDis defined as
+follows
+ζθ(s,D) = TrΠD−s
+θ,Res >dimM
+n. (3.3)
+It was shown by Seeley [21] (See also [22]) that ζθ(s,D) has a meromorphic extension to the whole
+complex plane and that 0 is a regular value of ζθ(s,D).
+Definition 3.2. Theζ-regularized determinant of Dis defined by the formula
+Det′
+θ(D) := exp/parenleftBig
+−d
+ds/vextendsingle/vextendsingle/vextendsingle
+s=0ζθ(s,D)/parenrightBig
+.
+We denote by
+LDet′
+θ(D) =−d
+ds/vextendsingle/vextendsingle/vextendsingle
+s=0ζθ(s,D).
+LetQbe a 0-th order pseudo-differential projection, ie. a 0-th order p seudo-differential operator
+satisfying Q2=Q. We set
+ζθ(s,Q,D) = TrQΠD−s
+θ,Res >dimM
+n. (3.4)
+The function ζθ(s,Q,D) also has a meromorphic extension to the whole complex plane and, by Wodzicki,
+[25, Section 7], it is regular at 0.
+Definition 3.3. Suppose that Qis a0-th order pseudo-differential projection commuting with D. Then
+V:= ImQisDinvariant subspace of C∞(M,E). Theζ-regularized determinant of the restriction D|V
+ofDtoVis defined by the formula
+Det′
+θ(D|V) :=eLDet′
+θ(D|V),
+where
+LDet′
+θ(D|V) =−d
+ds/vextendsingle/vextendsingle/vextendsingle
+s=0ζθ(s,Q,D). (3.5)
+Remark 3.4.The prime in Det′
+θand LDet′
+θindicates that we ignore the zero eigenvalues of the operator
+in the definition of the regularized determinant. If the operator is in vertible we usually omit the prime
+and write Det θand LDet θinstead.
+3.3.The graded determinant of the twisted odd signature operato r.Note that for each k= 0,1,
+the operator ( BH)2maps Ω¯k(M,E) to itself. Suppose that Iis an interval of the form [0 ,λ],(λ,µ],or
+(λ,∞) (µ > λ≥0). Denote by Π (BH)2,Ithe spectral projection of ( BH)2corresponding to the set of
+eigenvalues, whose absolute values lie in I. set
+Ω•
+I(M,E) := Π (BH)2,I(Ω•(M,E))⊂Ω•(M,E).
+If the interval Iis bounded, then, cf. Section 6.10 of [5], the space Ω•
+I(M,E) is finite dimensional.
+For each k= 0,1, set
+Ω¯k
++,I(M,E) := Ker( ∇HΓ)∩Ω¯k
+I(M,E) =/parenleftbig
+Γ(Ker∇H)/parenrightbig
+∩Ω¯k
+I(M,E);
+Ω¯k
+−,I(M,E) := Ker(Γ ∇H)∩Ω¯k
+I(M,E) = Ker∇H∩Ω¯k
+I(M,E).(3.6)REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 11
+Then
+Ω¯k
+I(M,E) = Ω¯k
++,I(M,E)⊕Ω¯k
+−,I(M,E) if 0 /∈ I. (3.7)
+We consider the decomposition (3.7) as a grading of the space Ω¯k
+I(M,E), and refer to Ω¯k
++,I(M,E) and
+Ω¯k
+−,I(M,E) as the positive and negative subspaces of Ω¯k
+I(M,E). Denote by BH
+IandBH¯k,Ithe restrictions
+ofBHto the subspaces Ω•
+I(M,E) and Ω¯k
+I(M,E) respectively. Then BH
+¯k,Imaps Ω¯k
+±,I(M,E) to itself. Let
+BH,±
+¯k,Idenote the restriction of BH
+¯k,Ito the subspace Ω¯k
+±,I(M,E). Clearly, the operator BH,±
+¯k,Iare bijective
+whenever 0 /∈ I. Note that Γ BH,−
+¯0,IΓ =BH,+
+¯1,I. Hence Det θ(BH,−
+¯0,I) = Det θ(BH,+
+¯1,I). Then we have the
+following definition.
+Definition 3.5. Suppose that 0/∈ I. The graded determinant of the operator BH,I
+¯0is defined by
+Detgr,θ(BH
+¯0,I) :=Detθ(BH,+
+¯0,I)
+Detθ(−BH,−
+¯0,I)=Detθ(BH,+
+¯0,I)
+Detθ(−BH,+
+¯1,I)∈C\{0}, (3.8)
+whereDetθdenotes the ζ-regularized determinant associated to the Agmon angle θ∈(−π,0), cf. for
+example, Section 6 of [5].
+We define, cf. (3.5),
+LDetgr,θ(BH
+¯0,I) = LDet θ(BH,+
+¯0,I)−LDetθ(−BH,−
+¯0,I) = LDet θ(BH,+
+¯0,I)−LDetθ(−BH,+
+¯1,I).(3.9)
+It follows from formula (6-17) of [5] that (3.8) is independent of the choice of θ∈(−π,0).
+3.4.The canonical element of the determinant line. It is not difficult to check that ( ∇H)2= 0.
+Clearly,∇H: Ω¯k(M,E)→Ωk+1(M,E) and Γ : Ω¯k(M,E)→Ωk+1(M,E). Hence we can consider the
+following twisted de Rham complex with chirality operator Γ:
+/parenleftbig
+Ω•(M,E),∇H/parenrightbig
+:···∇H
+−→Ω¯0(M,E)∇H
+−→Ω¯1(M,E)∇H
+−→Ω¯0(M,E)∇H
+−→ ···. (3.10)
+We define the twisted de Rham cohomology groups of (Ω•(M,E),∇H) as
+H¯k(M,E,H)≡H¯k(∇H) :=Ker(∇H: Ω¯k(M,E)→Ωk+1(M,E))
+Im(∇H: Ωk+1(M,E)→Ω¯k(M,E)), k= 0,1.
+The groups H¯k(M,E,H),k= 0,1 are independent of the choice of the Riemannian metric on Mor the
+Hermitian metric on E. Suppose that His repalced by H′=H−dBfor some B∈Ω¯0(M), there is an
+isomorphism εB:=eB∧·: Ω•(M,E)→Ω•(M,E) satisfying
+εB◦∇H=∇H′◦εB.
+Therefore εBinduces an isomorphism on the twisted de Rham cohomology, also deno te byεB,
+εB:H•(M,E,H)→H•(M,E,H′). (3.11)
+Denote by ( ∇H¯k)∗the adjoint of ∇H¯kwith respect to the scalar product <·,·>M. Then the Laplacians
+∆¯k= ∆H
+¯k:= (∇H
+¯k)∗∇H
+¯k+∇H
+k+1(∇H
+k+1)∗, k= 0,1
+are elliptic operators and therefore the complex (3.10) is elliptic. By H odge theory, we have the isomor-
+phism Ker∆ ¯k∼=H¯k(M,E,H),k= 0,1. For more details of the twisted de Rham cohomology, cf. for
+example [17].12 RUNG-TZUNG HUANG
+Since∇Hcommuteswith BH, the subspaceΩ•
+I(M,E) is asubcomplex ofthe twisted de Rham complex
+(Ω•(M,E),∇H). Clearly, for each λ≥0, the complex Ω•
+(λ,∞)(M,E) is acyclic. Since
+Ω•(M,E) = Ω•
+[0,λ](M,E)⊕Ω•
+(λ,∞)(M,E), (3.12)
+the cohomology H•
+[0,λ](∇H)≡H•
+[0,λ](M,E,H) of the complex/parenleftbig
+Ω•(M,E),∇H/parenrightbig
+is naturally isomorphic
+to the cohomology H•(M,E,H). Let Γ Idenote the restriction of Γ to Ω•
+I(M,E). For each λ≥0, let
+ρΓ[0,λ]=ρΓ[0,λ](∇H,gM)∈Det/parenleftbig
+H•
+[0,λ](M,E,H)/parenrightbig
+(3.13)
+denote the refined torsion of the twisted finite dimensional complex/parenleftbig
+Ω•
+[0,λ](M,E),∇H/parenrightbig
+corresponding to
+the chirality operator Γ [0,λ], cf. Definition 2.2. We view ρΓ[0,λ]as an element of Det/parenleftbig
+H•(M,E,H)/parenrightbig
+via
+the canonical isomorphism between H•(M,E,H) andH•
+[0,λ](M,E,H).
+Proposition 3.6. Assume that θ∈(−π,0)is an Agmon angle for the operator BH
+¯0. Then the element
+ρH=ρ(∇H,gM) := Det gr,θ(BH
+¯0,(λ,∞))·ρΓ[0,λ]∈Det/parenleftbig
+H•(M,E,H)/parenrightbig
+(3.14)
+is independent of the choice λ≥0. Further, ρHis independent of the choice of the Agmon angle
+θ∈(−π,0)ofBH
+¯0.
+Proof.Clearly, for 0 ≤λ≤µ, we have
+Detgr(BH
+¯0,(λ,∞)) = Det gr(BH
+¯0,(λ,µ])·Detgr(BH
+¯0,(µ,∞)). (3.15)
+From Proposition 2.9, (3.15) and (6-17) of [5], we obtain the result. /square
+3.5.Theη-invariant. In this subsection we recall the definition of the η-invariant of a non-self-adjoint
+elliptic operator D, cf. [13], [5, Subsection 6.15].
+Definition 3.7. LetD:C∞(M,E)→C∞(M,E)be an elliptic differential operator of order n≥1
+whose leading symbol is self-adjoint with respect to some gi ven Hermitian metric on E. Assume that θis
+an Agmon angle for D, cf. Definition 6.3 of [5]. LetΠ>(resp.Π<) be a pseudo-differential projection
+whose image contains the span of all generalized eigenvecto rs ofDcorresponding to eigenvalues λwith
+Re>0(resp. with Re<0) and whose kernel contains the span of all generalized eigen vectors of D
+corresponding to eigenvalues λwithRe≤0(resp. with Re≥0). We define the η-function of Dby the
+formula
+ηθ(s,D) =ζθ(s,Π>,D)−ζθ(s,Π<,−D).
+Note that, by the abovedefinition, the purely imaginaryeigenvalues ofDdo not contribute to ηθ(s,D).
+It was shown by Gilkey, [13], that ηθ(s,D) has a meromorphic extension to the whole complex plane
+Cwith isolated simple poles, and that it is regular at 0. Moreover, the nu mberηθ(0,D) is independent
+of the Agmon angle θ.
+Since the leading symbol of Dis self-adjoint, the angles ±π/2 are principal angles for Dcf. [5,
+Definition 6.2]. Hence, there are at most finitely many eigenvalues of Don the imaginary axis. Let
+m+(D) (resp.m−(D)) denote the number of eigenvalues of D, counted with their algebraicmultiplicities,
+on the positive (resp. negative) part of the imaginary axis. Let m0(D) denote the algebraic multiplicity
+of 0 as an eigenvalue of D.
+Definition 3.8. Theη-invariant η(D)ofDis defined by the formula
+η(D) =ηθ(0,D)+m+(D)−m−(D)+m0(D)
+2. (3.16)REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 13
+Sinceηθ(0,D) is indenpendent of the choice of the Agmon angle θforD, cf. [13], so is η(D). Note
+that the defintion of η(D) is slightly different from the one proposed by Gilkey in [13]. See [5, Rem ark
+2.5].
+Denote by η(∇H) =η(BH
+¯0) theη-invariant of the restriction BH
+¯0of the twisted odd signature operator
+BHto Ω¯0(M,E).
+3.6.Relationship with the η-invariant. In this subsection we study the relationship between (3.8)
+and theη-invariant of BH
+¯0,(λ,∞).
+To simplify the notation set
+ηλ(∇H) :=η(BH
+¯0,(λ,∞)) (3.17)
+and
+ξλ=ξλ(∇H,gM,θ) =1
+2/parenleftBig
+LDet2θ/parenleftbig
+(BH,+
+¯0,(λ,∞))2/parenrightbig
+−LDet2θ/parenleftbig
+(BH,+
+¯1,(λ,∞))2/parenrightbig/parenrightBig
+. (3.18)
+LetP±
+¯k,I,k= 0,1 be the orthogonal projection onto the closure of the subspace Ω¯k
+±,I(M,E). Set
+d∓
+¯k,λ:= rank(Id −P±
+¯k,[0,λ]) = dimΩ¯k
+∓,I(M,E), k= 0,1. (3.19)
+IfI ⊂Rwe denote by LIthe solid angle
+LI={ρeiθ: 0< ρ <∞,θ∈ I}.
+Proposition 3.9. Let∇be a flat connection on a vector bundle Eover a closed Riemannian manifold
+(M,gM)of odd dimension m= 2r−1andHis an odd-degree closed differential form, other than one
+form, on M. Assume θ∈(−π/2,0)is an Agmon angle for the twisted odd signature operator BH
+¯0,(λ,∞)
+such that there are no eigenvalues of BHin the solid angles L(−π/2,θ]andL(π/2,θ+π]. Then, for every
+λ≥0, cf.(3.9),
+LDetgr,θ(BH
+¯0,(λ,∞)) =ξλ−iπηλ(∇H)−iπ
+2/summationdisplay
+k=0,1(−1)kd−
+¯k,λ. (3.20)
+Proof.From Definition 3.8 of the η-invariant it follows that
+η(−BH,+
+¯1,(λ,∞)) =−η(BH,+
+¯1,(λ,∞)). (3.21)
+By the fact that Γ BH,+
+¯1,(λ,∞)Γ =BH,−
+¯0,(λ,∞), we have
+η(BH,+
+¯1,(λ,∞)) =η(BH,−
+¯0,(λ,∞)). (3.22)
+Combining (3.17), (3.21) with (3.22), we have
+η(BH,+
+¯0,(λ,∞))−η(−BH,+
+¯1,(λ,∞)) =η(BH,+
+¯0,(λ,∞))+η(BH,−
+¯0,(λ,∞)) =ηλ(∇H). (3.23)
+By [4, (4.34)], for k= 0,1, we have
+LDetθ(±BH,+
+¯k,I) =1
+2LDet2θ/parenleftbig
+(BH,+
+¯k,(λ,∞))2/parenrightbig
+−iπ/parenleftBig
+η(±BH,+
+¯k,I)−ζ2θ(0,/parenleftbig
+BH,+
+¯k,(λ,∞)/parenrightbig2)
+2/parenrightBig
+(3.24)
+and, by [5, (6-6)] and (3.19), we have
+ζ2θ(0,/parenleftbig
+BH,+
+¯0,(λ,∞)/parenrightbig2)−ζ2θ(0,/parenleftbig
+BH,+
+¯1,(λ,∞)/parenrightbig2) =−/summationdisplay
+k=0,1(−1)kd−
+¯k,λ. (3.25)
+Combining (3.18), (3.23), (3.24) with (3.25), we obtain the result. /square14 RUNG-TZUNG HUANG
+4.Metric anomaly and the definition of the refined analytic to rsion twisted by a flux
+form
+In this section we study the metric dependence of the element ρH=ρ(∇H,gM) defined in (3.14).
+We then use this element to construct the twisted refined analytic torsion , which is a canonical element
+of the determinant line Det( H•(M,E,H)). We also show that the twisted refined analytic torsion is
+independent of the metric gMand the representative Hin the cohomology class [ H].
+4.1.Relationship between ρH(t)and the η-invariant. Suppose that gM
+t,t∈R, is a smooth family
+of Riemannian metrics on M. Let
+ρH(t) =ρ(∇H,gM
+t)∈Det/parenleftbig
+H•(M,E,H)/parenrightbig
+be the canonical element defined in (3.14).
+Let Γtdenote the chirality operator corresponding to the metric gM
+t, cf. (3.1), and let BH(t) =
+B(∇H,gM
+t) denote the twisted odd signature operator corresponding to th e Riemannian metric gM
+tand
+letBH,+(t) denote the restriction of BH(t) =B(∇H,gM
+t) to Ω•
++(M,E).
+Fixt0∈Rand choose λ≥0 such that there are no eigenvalues of ( BH(t0))2of absolute value λ.
+Further, assume that λis big enough so that the real parts of eigenvalues of ( BH
+(λ,∞)(t0))2are all greater
+than 0. Then there exists δ >0 small enough such that the same holds for the spectrum of ( BH(t))2for
+t∈(t0−δ,t0+δ). In particular, d−
+¯k,λ, cf. (3.19), is independent of t∈(t0−δ,t0+δ). Set
+ηλ(∇H,t) :=η(BH
+¯0,(λ,∞)(t)), ξλ(t,θ) =ξλ(∇H,gM
+t,θ).
+By definition (3.14),
+ρH(t) = Det gr,θ(BH
+¯0,(λ,∞)(t))·ρΓt,[0,λ].
+Assume that θ0∈(π/2,0) is an Agmon angle for BH(t0) such that there are no eigenvalues of BH(t0)
+inL(−π/2,θ0]andL(−π/2,θ0+π). Choose δ >0 so that for every t∈(t0−δ,t0+δ) bothθ0andθ0+π
+are Agmon angles of BH(t0). Fort/\e}atio\slash=t0, it might happen that there are eigenvalues of BH
+¯k,(λ,∞)(t) in
+L(−π/2,θ0]andL(−π/2,θ0+π). Hence, (3.20) is not necessarily true. However, from the indepen dence of the
+Agmon angle of the ζ-function, cf. [5, (6-16)], and (3.18), we conclude that for every angleθ∈(−π/2,0),
+so thatθandθ+πare Agmon angles for BH
+¯k,(λ,∞)(t),
+ξλ(t,θ)≡ξλ(t,θ0) modπi.
+Hence, from (3.20), we obtain
+ρH(t) =±eξλ(t,θ0)·e−iπηλ(∇H,t)·eiπ
+2/summationtext
+k=0,1(−1)kd−
+¯k,λ·ρΓt,[0,λ]. (4.1)
+Lemma 4.1. Under the above assumptions, the product
+eξλ(t,θ0)·ρΓt,[0,λ]∈Det/parenleftbig
+H•/parenleftbig
+M,E,H/parenrightbig/parenrightbig
+is independent of t∈(t0−δ,t0+δ).REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 15
+Proof.Let Γtdenote the chirality operator corresponding to the metric gM
+t. Following the Z-graded case,
+cf. [19], we set
+f(s,t) =/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0us−1Tr/bracketleftBig
+exp/parenleftbig
+−u(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+du
+= Γ(s)/summationdisplay
+k=0,1(−1)kζ/parenleftBig
+s,(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightBig
+(4.2)
+Using the fact that
+Γt(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)Γt= (∇HΓt)2/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E),
+we also have
+f(s,t) =−/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0us−1Tr/bracketleftBig
+exp/parenleftbig
+−u(∇HΓt)2/vextendsingle/vextendsingle
+Ω¯k
+−,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+du
+=−Γ(s)/summationdisplay
+k=0,1(−1)kζ/parenleftBig
+s,(∇HΓt)2/vextendsingle/vextendsingle
+Ω¯k
+−,(λ,∞)(M,E)/parenrightBig
+(4.3)
+We denote by ˙Γtwith respect to the parameter t. Then
+d
+dt(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)=˙ΓtΓt(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)
++(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)˙ΓtΓt(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)(4.4)
+where we used that Γ2
+t= 1. Similarly, we have
+d
+dt(∇HΓt)2/vextendsingle/vextendsingle
+Ω¯k
+−,(λ,∞)(M,E)= (∇HΓt)2Γt˙Γt/vextendsingle/vextendsingle
+Ω¯k
+−,(λ,∞)(M,E)+(∇HΓt)Γt˙Γt(∇HΓt)/vextendsingle/vextendsingle
+Ω¯k
+−,(λ,∞)(M,E).(4.5)
+IfAis of trace class and Bis a bounded operator, it is well known that Tr( AB) = Tr(BA). Hence, by
+this fact and the semi-group property of the heat operator, we h ave
+Tr/bracketleftBig
+(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)˙ΓtΓt(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+=Tr/bracketleftBig
+exp/parenleftbig
+−u
+2(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig
+(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)
+·˙ΓtΓt(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u
+2(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+=Tr/bracketleftBig
+˙ΓtΓt(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u
+2(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig
+·exp/parenleftbig
+−u
+2(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig
+(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/bracketrightBig
+=Tr/bracketleftBig
+˙ΓtΓt(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig
+(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/bracketrightBig
+=Tr/bracketleftBig
+˙ΓtΓt(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+, (4.6)
+here in the last equality we used the fact that
+(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig
+(Γt∇H)/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)
+=(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig
+.16 RUNG-TZUNG HUANG
+By (4.4), (4.2) and (4.6), we have
+d
+dtf(s,t) =/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0us−1Tr/bracketleftBig
+−2u˙ΓtΓt(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+du.
+(4.7)
+Simlarly, we have
+d
+dtf(s,t) =−/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0us−1Tr/bracketleftBig
+−2uΓt˙Γt(∇HΓt)2/vextendsingle/vextendsingle
+Ω¯k
+−,(λ,∞)(M,E)exp/parenleftbig
+−u(∇HΓt)2/vextendsingle/vextendsingle
+Ω¯k
+−,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+du.
+(4.8)
+By (4.7), (4.8) and the fact that ˙ΓtΓt=−Γt˙Γt, we conclude that
+d
+dtf(s,t) =−/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0usTr/bracketleftBig
+˙ΓtΓt(BH¯k,(λ,∞)(t))2exp/parenleftbig
+−u(BH¯k,(λ,∞)(t))2/parenrightbig/bracketrightBig
+du
+=/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0usd
+duTr/bracketleftBig
+˙ΓtΓtexp/parenleftbig
+−u(BH
+¯k,(λ,∞)(t))2/parenrightbig/bracketrightBig
+du
+=−s/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0us−1Tr/bracketleftBig
+˙ΓtΓtexp/parenleftbig
+−u(BH
+¯k,(λ,∞)(t))2/parenrightbig/bracketrightBig
+du, (4.9)
+where we used the integration by parts for the last equality. Since ( BH(t))2is an elliptic differential
+operator, the dimension of Ω•
+[0,λ](M,E) is finite. Let ε/\e}atio\slash= 0 be a small enough real number so that
+(BH(t))2+εis bijective and 2 θ0is an Agmon angle for ( BH(t))2+ε. Then we can rewrite (4.9) as
+d
+dtf(s,t) =−s/summationdisplay
+k=0,1(−1)k/integraldisplay1
+0us−1Tr/bracketleftBig
+˙ΓtΓt/parenleftbig
+exp/parenleftbig
+−u(BH
+¯k(t))2+ε/parenrightbig/bracketrightBig
+du
+−s/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+1us−1Tr/bracketleftBig
+˙ΓtΓt/parenleftbig
+exp/parenleftbig
+−u(BH
+¯k(t))2+ε/parenrightbig/bracketrightBig
+du
++s/summationdisplay
+k=0,1(−1)k/integraldisplay1
+0us−1Tr/bracketleftBig
+˙ΓtΓt/parenleftbig
+exp/parenleftbig
+−u(BH
+¯k,[0,λ](t))2/parenrightbig/bracketrightBig
+du
++s/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+1us−1Tr/bracketleftBig
+˙ΓtΓt/parenleftbig
+exp/parenleftbig
+−u(BH
+¯k,[0,λ](t))2/parenrightbig/bracketrightBig
+du. (4.10)
+Since˙ΓtΓtis a local quantitity and the dimension of the manifold Mis odd, the asymptotic expansion
+asu↓0 for Tr/bracketleftBig
+˙ΓtΓt/parenleftbig
+exp/parenleftbig
+−u(BH(t))2+ε/parenrightbig/bracketrightBig
+does not contain a constant term. Therefore the integrals
+of the first term on the right hand side of (4.10) do not have poles at s= 0. On the other hand, because
+of exponential decay of Tr/bracketleftBig
+˙ΓtΓt/parenleftbig
+exp/parenleftbig
+−u(BH(t))2+ε/parenrightbig/bracketrightBig
+and Tr/bracketleftBig
+˙ΓtΓt/parenleftbig
+exp/parenleftbig
+−u(BH
+[0,λ](t))2/parenrightbig/bracketrightBig
+for large u,
+the integrals of the second term and the fourth term on the right h and side of (4.10) are entire functions
+ins. Hence we have
+d
+dt/vextendsingle/vextendsingle/vextendsingle
+s=0f(s,t) =/parenleftBig
+s/summationdisplay
+k=0,1(−1)k/integraldisplay1
+0us−1Tr/bracketleftBig
+˙ΓtΓt/vextendsingle/vextendsingle/vextendsingle
+Ω¯k
+[0,λ](M,E)/bracketrightBig
+du/parenrightBig/vextendsingle/vextendsingle/vextendsingle
+s=0
+=/summationdisplay
+k=0,1(−1)kTr/bracketleftBig
+˙ΓtΓt/vextendsingle/vextendsingle/vextendsingle
+Ω¯k
+[0,λ](M,E)/bracketrightBig
+(4.11)REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 17
+and, by (4.2),
+d
+dt/vextendsingle/vextendsingle/vextendsingle
+s=0sf(s,t) =d
+dt/vextendsingle/vextendsingle/vextendsingle
+s=0/summationdisplay
+k=0,1(−1)kζ/parenleftBig
+s,(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightBig
+= 0. (4.12)
+By (3.18) and (4.2), we know that
+ξλ(t,θ0) =−1
+2lims→0/bracketleftBig
+f(s,t)−1
+s/summationdisplay
+k=0,1(−1)ζ(0,(Γt∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E))/bracketrightBig
+. (4.13)
+Combining (4.11), (4.12) with (4.13), we obtain
+d
+dt/vextendsingle/vextendsingle/vextendsingle
+t=0ξλ(t,θ0) =−1
+2/summationdisplay
+k=0,1(−1)kTr/bracketleftBig
+˙ΓtΓt/vextendsingle/vextendsingle/vextendsingle
+Ω¯k
+[0,λ](M,E)/bracketrightBig
+(4.14)
+Combining (2.29) with (4.14), we obtain
+d
+dt(eξλ(t,θ0)·ρΓt,[0,λ]) = 0.
+/square
+We need the following lemma, which is a slight modification of the result in S ubsection 9.3 of [5].
+Lemma 4.2. For any t1,t2∈(t0−δ,t0+δ), we have
+ηλ(∇H,t1)−ηλ(∇H,t2)≡η(BH
+¯0(t1))−η(BH
+¯0(t2)),modZ.
+LetBH
+trivial=B¯0(∇H
+trivial,gM) : Ω¯0(M,E)→Ω¯0(M,E) denote the even part of twisted odd signature
+operator corresponding to the metric gMand the trivial line bundle over Mendowed with the trivial
+connection ∇trivial. Put
+ηtrivial:=1
+2η(0,BH
+trivial).
+We now need to study the dependence of η(BH
+¯0) on the Riemannian metric gM. This was essentially
+done in [1] and [13].
+Lemma 4.3. The function η(BH
+¯0(t))−rank(E)ηtrivial(t)is, modulo Z, independent of t∈(t0−δ,t0+δ).
+4.2.Removing the metric anomaly and the definition of the twisted refined analytic torsion.
+The following theorem is the main theorem of this subsection.
+Theorem 4.4. LetMbe an odd dimensional oriented closed Riemannian manifold. Let(E,∇,hE)be
+a flat complex vector bundle over MandHis a closed differential form on Mof odd degree, other than
+one form. Then the element
+ρ(∇H,gM)·eiπ(rank(E))ηtrivial∈Det/parenleftbig
+H•(M,E,H)/parenrightbig
+, (4.15)
+where∇H:=∇+H∧·andρ(∇H,gM)∈Det/parenleftbig
+H•(M,E,H)/parenrightbig
+is defined in (3.14), is independent of gM.
+Proof.Consider a smooth family gM
+t,t∈Rof Riemannian metrics on M. From (4.1), we obtain for
+t∈(t0−δ,t0+δ)
+ρH(t)·eiπ(rank(E))ηtrivial=±eξλ(t,θ0)·e−iπηλ(∇H,t)·eiπ
+2/summationtext
+k=0,1(−1)kd¯k,λ·ρΓt,[0,λ]·eiπ(rank(E))ηtrivial.(4.16)
+Combining (4.14), (4.16) with Lemma 4.3 we conclude that for any t1,t2∈(t0−δ,t0+δ)
+ρH(t1)·eiπ(rank(E))ηtrivial=±ρH(t2)·eiπ(rank(E))ηtrivial.
+Since the function ρH(t)·eiπ(rank(E))ηtrivialis continuous and nonzero, the sign in the right hand side of
+the equality must be positive. This proves the statement. /square18 RUNG-TZUNG HUANG
+Definition 4.5. LetMbe an odd dimensional oriented closed Riemannian manifold. Let(E,∇,hE)be
+a flat complex vector bundle over MandHis a closed differential form on Mof odd degree, other than
+one form. The twisted refined analytic torsion ρan(∇H)is the element of Det/parenleftbig
+H•(M,E,H)/parenrightbig
+defined by
+(4.15).
+4.3.Variation of refined analytic torsion with respect to the flux in a cohomology class.
+Suppose that the (real) flux form His deformed smoothly along a one-parameter family with parameter
+v∈Rin such a way that the cohomology class [ H]∈H¯1(M,R) is fixed. Thend
+dvH=−dBfor some
+formB∈Ω¯0(M) that depends smoothly on v. Letβ=B∧·. Fixv0∈Rand choose λ >0 such that
+there are no eigenvalues of ( BH)2(v0) of absolute value λ. Further, assume that λis big enough so that
+the real parts of eigenvalues of ( BH
+(λ,∞)(v0))2are all greater than 0. Then there exists δ >0 small enough
+such that the same holds for the spectrum of ( BH(v))2forv∈(v0−δ,v0+δ). For simplicity, we often
+omit the parameter vin the notations of operators in the following discussion.
+We have the following two lemmas, see also Lemma 3.5 and lemma 3.7 of [17 ].
+Lemma 4.6. Under the above assumptions, we have
+d
+dvξλ=/summationdisplay
+k=0,1(−1)kTr(β/vextendsingle/vextendsingle
+Ω¯k
+[0,λ](M,E)).
+Proof.As in the proof of Lemma 4.1, we set
+f(s,v) =/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0us−1Tr/bracketleftBig
+exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+du. (4.17)
+We note that B, henceβ, is real. By (4.17) and the fact that
+d
+dv∇H= [β,∇H],
+we have
+d
+dvf(s,v)
+=/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0us−1Tr/bracketleftBig
+−u/parenleftbig
+Γ[β,∇H]Γ∇H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+du
++/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0us−1Tr/bracketleftBig
+−u/parenleftbig
+Γ∇HΓ[β,∇H]/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+du.
+(4.18)
+Using the fact that Γ2= 1, we have
+Tr/bracketleftBig/parenleftbig
+Γβ∇HΓ∇H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+= Tr/bracketleftBig/parenleftbig
+ΓβΓ/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)·/parenleftbig
+Γ∇H/parenrightbig2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+.(4.19)REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 19
+By using the trace property, Γ2= 1, and the semi-group property of the heat operator, we have
+Tr/bracketleftBig/parenleftbig
+Γ∇HβΓ∇H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+= Tr/bracketleftBig/parenleftbig
+Γ∇HΓ·ΓβΓ∇H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−1
+2u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig
+·exp/parenleftbig
+−1
+2u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+= Tr/bracketleftBig
+exp/parenleftbig
+−1
+2u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig
+·/parenleftbig
+Γ∇HΓ·ΓβΓ∇H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)
+·exp/parenleftbig
+−1
+2u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+= Tr/bracketleftBig/parenleftbig
+ΓβΓ∇H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−1
+2u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig
+·exp/parenleftbig
+−1
+2u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig
+·/parenleftbig
+Γ∇HΓ/parenrightbig/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)/bracketrightBig
+= Tr/bracketleftBig/parenleftbig
+ΓβΓ/parenrightbig/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)·/parenleftbig
+∇HΓ/parenrightbig2/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)exp/parenleftbig
+−u(∇HΓ)2/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+.(4.20)
+For the last equality of (4.20), we used the fact that
+∇H/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/parenleftbig
+Γ∇HΓ/parenrightbig/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)
+=/parenleftbig
+∇HΓ/parenrightbig2/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)exp/parenleftbig
+−u(∇HΓ)2/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)/parenrightbig
+.
+By combining (4.19) with (4.20), we obtain
+/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0usTr/bracketleftBig/parenleftbig
+Γ[β,∇H]Γ∇H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+du
+=/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0us/parenleftBig
+Tr/bracketleftBig/parenleftbig
+ΓβΓ/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)·/parenleftbig
+Γ∇H/parenrightbig2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+−Tr/bracketleftBig/parenleftbig
+ΓβΓ/parenrightbig/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)·/parenleftbig
+∇HΓ/parenrightbig2/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)exp/parenleftbig
+−u(∇HΓ)2/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)/parenrightbig/bracketrightBig/parenrightBig
+du
+=/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0usTr/bracketleftBig/parenleftbig
+ΓβΓ/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
+(λ,∞)(M,E)·(BH
+(λ,∞))2exp/parenleftbig
+−u(BH
+(λ,∞))2/bracketrightBig
+du. (4.21)
+Similarly, we have
+Tr/bracketleftBig/parenleftbig
+Γ∇HΓ∇Hβ/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+= Tr/bracketleftBig
+β/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)·/parenleftbig
+Γ∇H/parenrightbig2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+(4.22)
+and
+Tr/bracketleftBig/parenleftbig
+Γ∇HΓβ∇H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+= Tr/bracketleftBig
+β/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)·/parenleftbig
+∇HΓ/parenrightbig2/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)exp/parenleftbig
+−u(∇HΓ)2/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+.(4.23)20 RUNG-TZUNG HUANG
+By combining (4.22) with (4.23), we have
+/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0usTr/bracketleftBig/parenleftbig
+Γ∇HΓ[β,∇H]/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+du
+=/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0us/parenleftBig
+Tr/bracketleftBig
+β/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)·/parenleftbig
+Γ∇H/parenrightbig2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)exp/parenleftbig
+−u(Γ∇H)2/vextendsingle/vextendsingle
+Ω¯k
++,(λ,∞)(M,E)/parenrightbig/bracketrightBig
+−Tr/bracketleftBig
+β/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)·/parenleftbig
+∇HΓ/parenrightbig2/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)exp/parenleftbig
+−u(∇HΓ)2/vextendsingle/vextendsingle
+Ωk+1
+−,(λ,∞)(M,E)/parenrightbig/bracketrightBig/parenrightBig
+du
+=/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0usTr/bracketleftBig
+β/vextendsingle/vextendsingle
+Ω¯k
+(λ,∞)(M,E)·(BH
+(λ,∞))2exp/parenleftbig
+−u(BH
+(λ,∞))2/bracketrightBig
+du. (4.24)
+Combining (4.21), (4.24), with (4.18), we obtain
+d
+dvf(s,v) =−/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0usTr/bracketleftBig/parenleftbig
+ΓβΓ/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
+(λ,∞)(M,E)·(BH
+(λ,∞))2exp/parenleftbig
+−u(BH
+(λ,∞))2/bracketrightBig
+du
+−/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0usTr/bracketleftBig
+β/vextendsingle/vextendsingle
+Ω¯k
+(λ,∞)(M,E)·(BH
+(λ,∞))2exp/parenleftbig
+−u(BH
+(λ,∞))2/parenrightbig/bracketrightBig
+du
+=−2/summationdisplay
+k=0,1(−1)k/integraldisplay∞
+0usTr/bracketleftBig
+β/vextendsingle/vextendsingle
+Ω¯k
+(λ,∞)(M,E)·(BH
+(λ,∞))2exp/parenleftbig
+−u(BH
+(λ,∞))2/parenrightbig/bracketrightBig
+du,
+where for the latter equality we used the fact that Tr( β/vextendsingle/vextendsingle
+Ω¯k
+[0,λ](M,E)) = Tr(Γ βΓ/vextendsingle/vextendsingle
+Ω¯k
+[0,λ](M,E)). The rest is
+similar to the proof of Lemma 4.1. /square
+Lemma 4.7. Under the same assumptions, along any one parameter deforma tion ofHthat fixes the
+cohomology class [H], the element can be chosen so that
+d
+dvρΓ[0,λ]=−/summationdisplay
+k=0,1(−1)kTr(β/vextendsingle/vextendsingle
+Ω¯k
+[0,λ](M,E))ρΓ[0,λ],
+where we identify Det/parenleftbig
+H•(M,E,H)/parenrightbig
+along the deformation using (3.11).
+Proof.In order to compare the elements ρΓ[0,λ]∈Det/parenleftbig
+H•(M,E,H)/parenrightbig
+at different values of v. We choose
+a reference point, say v= 0, and let H(0),ρ(0)
+Γ[0,λ]be the values of H,ρΓ[0,λ], respectively, at v= 0. By
+(3.11), we have the isomorphism
+Det(εB) : Det/parenleftbig
+H•(M,E,H(0))/parenrightbig
+→Det/parenleftbig
+H•(M,E,H)/parenrightbig
+.
+SinceεB=eβon Ω•(M,E), we have, for k= 0,1,
+d
+dv(Det(εB))−1ρΓ[0,λ]=−/summationdisplay
+k=0,1(−1)kTr(β/vextendsingle/vextendsingle
+Ω¯k
+[0,λ](M,E))(Det(εB))−1ρΓ[0,λ].
+The result follows. /square
+The argument of the following lemma is similar to the argument of Lemma 9.4 of [5].
+Lemma 4.8. For any v1,v2∈(v0−δ,v0+δ), we have
+ηλ(∇H,v1)−ηλ(∇H,v2)≡η(BH
+¯0(v1))−η(BH
+¯0(v2)),modZ.REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 21
+Again we need to study the dependence of η(BH
+¯0) on the parameter v. As Lemma 4.3, this was
+essentially done in [1] and [13, P. 52].
+Lemma 4.9. The function η(BH
+¯0(v))−rank(E)ηtrivial(v)is, modulo Z, independent of v∈R.
+Now we have the main theorem of this subsection.
+Theorem 4.10. LetMbe an odd dimensional oriented closed Riemannian manifold. Let(E,∇,hE)be
+a flat complex vector bundle over M. Suppose that HandH′are closed differential forms on Mof odd
+degrees representing the same deRham cohomology class, and letBbe an even form so that H′=H−dB.
+Then the refined analytic torsion ρan(∇H′) = Det(εB)(ρan(∇H)).
+Proof.Again we choose a reference point, say v= 0, and let H(0),ρ(0)
+Γ[0,λ]be the values of H,ρΓ[0,λ],
+respectively, at v= 0. By (3.11), we have the isomorphism
+Det(εB) : Det/parenleftbig
+H•(M,E,H(0))/parenrightbig
+→Det/parenleftbig
+H•(M,E,H)/parenrightbig
+.
+Recall that εB=eβon Ω•(M,E). By combining Lemma 4.6, Lemma 4.7 with Lemma 4.9, we con-
+clude that (Det( εB))−1ρan(∇H) is, up to sign, invariant along the deformation. Since the function
+(Det(εB))−1ρan(∇H) is continuous and nonzero, the sign in the right hand side of the equ ality must be
+positive. This proves the statement. /square
+As pointed out by Mathai and Wu, [17], that when His a 3-form on M, the deformation of the
+Riemannian metric gMand that of the flux Hwithin its cohomology class can be interpreted as a
+deformation of generalized metrics on Mand the analytic torsion should be defined for generalized
+metrics so that the deformations of gMandHare unified. Similarly, the the refined analytic torsion
+should be also defined for generalized metrics so that the deformat ions ofgMandHare unified.
+5.A duality theorem for the twisted refined analytic torsion
+In this section wefirst reviewthe concept ofthe dual ofa complexa nd construct anaturalisomorphism
+between the determinant lines of a Z2-graded complex and its dual. We then show that this isomorphism
+is compatible with the canonical isomorphism (2.9). Finally we establish a relationship between the
+twisted refined analytic torsion corresponding to a flat connection and that of its dual. The contents of
+this section are Z2-graded analogues of Sections 3 and 10 of [5]. Throughout this sect ion,kis a field of
+characteristic zero endowed with an involutive automorphism τ:k→k.The main examples are k=C
+withτbeing the complex conjugation and k=Rwithτbeing the identity map.
+5.1.TheZ2-graded τ-dual space. IfV,Warek-vector spaces, a map f:V→Wis said to be
+τ−linearif
+f(x1v1+x2v2) =τ(x1)v1+τ(x2)v2,for anyv1,v2∈V,x1,x2∈k.
+The linear space V∗=V∗τof allτ-linear maps V→kis called the τ−dual space to V . There are
+naturalτ-linear isomorphisms, cf. [5, Subsection 3.1],
+αV: Det(V∗)→Det(V)−1, βV: Det(V)→Det(V∗)−1. (5.1)
+Then for any v∈Det(V), we have, cf. [5, (3-8)],
+/parenleftbig
+α−1
+V(v−1)/parenrightbig−1= (−1)dimVβV(v), (5.2)22 RUNG-TZUNG HUANG
+LetVandWbek-vector spaces, then, for any v∈Det(V),w∈Det(W), we have, cf. [5, (3-9)],
+/parenleftbig
+µV,W(v⊗w)/parenrightbig−1=αV⊕W◦µV∗,W∗/parenleftbig
+α−1
+V(v−1)⊗α−1
+W(w−1)/parenrightbig
+. (5.3)
+LetT:V→Wbe ak-linear map. The τ−adjoint of T is the linear map
+T∗:W∗→V∗
+such that
+(T∗w∗)(v) =w∗(Tv),for allv∈V,w∗∈W∗.
+IfTis bijective, then, for any nonzero v∈Det(V), we have, cf. [5, (3-11)],
+T∗α−1
+W/parenleftbig
+(Tv)−1/parenrightbig
+=α−1
+V(v−1). (5.4)
+LetV0,V1,···,Vmbe finite dimensional k-vector space, where m= 2r−1 is an odd integer. Denote
+byV¯0=/circleplustextr−1
+i=0V2iandV¯1=/circleplustextr−1
+i=0V2i+1. LetV•=V¯0⊕V¯1be a finite dimensional Z2-gradedk-vector
+space. We define the Z2−graded(τ−)dual space /hatwideV=/hatwideV¯0⊕/hatwideV¯1by
+/hatwideV¯k:= (Vk+1)∗, k= 0,1.
+Then (5.1) induces a τ-linear isomorphism
+αV•: Det(V•)→Det(/hatwideV•), (5.5)
+defined by
+αV•(v¯0⊗(v¯1)−1) = (−1)M(V•)·α−1
+V¯1(v−1
+¯1)⊗αV¯0(v¯0), (5.6)
+wherev¯k∈Det(V¯k),k= 0,1 and, cf. (2.13),
+M(V•) =M(V•,V•) = dimV¯0·dimV¯1.
+5.2.The dual complex of a Z2-graded complex. Consider the Z2-graded complex (2.2) of finite
+dimensional k-vector spaces. The dual complex of the Z2-graded complex (2.2) is the complex
+(/hatwideC•,d∗) :···d∗
+−→/hatwideC¯0d∗
+−→/hatwideC¯1d∗
+−→/hatwideC¯0d∗
+−→ ···, (5.7)
+where/hatwideC¯k= (Ck+1)∗andd∗is theτ-adjoint of d. Then the cohomology H¯k(d∗) of/hatwideC•is natural
+isomorphic to the τ-dual space to Hk+1(d)(k= 0,1). Hence by (5.5), we obtain τ-linear isomorphisms
+αC•: Det(C•)→Det(/hatwideC•),
+αH•(d): Det(H•(d))→Det(H•(d∗)).(5.8)
+The following lemma is the Z2-graded analogue of Lemma 3.6 of [5]. The proof is similar to the proof
+of Lemma 3.6 of [5]. We skip the proof.
+Lemma 5.1. Let(C•,d)be aZ2-graded complex of finite dimensional k-vector spaces, defined as (2.2).
+Further, assume that the Euler characteristics χ(C•) =χ(/hatwideC•) = 0. Then the following diagramm com-
+mutes:
+Det(C•)φC•− −−− → Det(H•(d))
+αC•/arrowbt/arrowbtαH•(d)
+Det(/hatwideC•)φ/hatwideC•− −−− →Det/parenleftbig
+H•(d∗)/parenrightbig(5.9)
+whereφC•andφ/hatwideC•are defined as in (2.10).REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 23
+5.3.The refined torsion of the Z2-graded dual complex. Suppose now that kis endowed with an
+involutive endomorphism τ. Let/hatwideC•be theτ-dual complex of Cand letαC•: Det(C•)→Det(/hatwideC•) denote
+theτ-isomorphism defined in (5.8). Let Γ∗be theτ-adjoint of Γ. Then Γ∗is a chirality operator for the
+complex /hatwideC•.
+The following lemma is the Z2-graded analogue of Lemma 4.11 of [5].
+Lemma 5.2. In the situation described above,
+ρΓ∗=αH•(d)(ρΓ). (5.10)
+Proof.Fixc¯0∈Det(C¯0) and set
+/hatwidec¯0=α−1
+C¯0/parenleftbig
+(Γc¯0)−1/parenrightbig
+∈Det(/hatwideC¯0). (5.11)
+Then, by (5.4),
+Γ∗/hatwidec¯0=α−1
+C¯0(c−1
+¯0)∈Det(/hatwideC¯0). (5.12)
+Using (5.2), we obtain from (5.11) and (5.12), that
+βC¯0(c¯0) = (−1)dimC¯0·(Γ∗/hatwidec¯0)−1, βC¯1(Γc¯0) = (−1)dimC¯0·/hatwidec−1
+¯0. (5.13)
+Combining (5.6), (2.20), (5.11), (5.12) and (5.13), we have
+αC•(cΓ) = (−1)M(C•)+dimC¯0·cΓ∗. (5.14)
+By (2.22), ρΓ=φC•(cΓ). Therefore, from Lemma 5.1, we obtain
+αH•(d)(ρΓ) =φ/hatwideC•◦αC•(cΓ) = (−1)M(C•)+dimC¯0·ρΓ∗. (5.15)
+By (2.13) and the fact dim C¯0= dimC¯1, we get
+M(C•)+dimC¯0= dimC¯1·dimC¯0+dimC¯0= dimC¯0·(dimC¯0+1)≡0,mod 2.(5.16)
+Combining (5.15) with (5.16), we obtain (5.10). /square
+5.4.The duality theorem. Suppose that Mis a closed oriented manifold of odd dimension m= 2r−1.
+LetE→Mbe a complex vector bundle over Mand let∇be a flat connection on E. Fix a Hermitian
+metrichEonE. Denote by ∇′the connection on Edual to the connection ∇, cf. [5, Subsection 10.1].
+We denote by E′the flat bundle ( E,∇′), referring to E′as the dual of the flat vector bundle E. Using the
+construction of Section 5.1, with τ:C→Cbe the complex conjugation, we have the canonical anti-linear
+isomorphism
+α: Det/parenleftbig
+H•(M,E,H)/parenrightbig
+→Det/parenleftbig
+H•(M,E′,H)/parenrightbig
+. (5.17)
+The following theorem is the main result of this section and is the twiste d analogue of Theorem 10.3
+of [5].
+Theorem 5.3. LetE→Mbe a complex vector bundle over a closed oriented odd dimensi onal manifold
+Mand let∇be a flat connection on E. Denote by ∇′the connection dual to ∇with respect to a Hermitian
+metrichEonEand letHbe a odd-degree cloed form, other than one form, on M. Then
+α(ρan(∇H)) =ρan(∇′H)·e2πi/parenleftbig
+¯η(∇H,gM)−(rankE)ηtrivial(gM)/parenrightbig
+, (5.18)
+whereαis the anti-linear isomorphism (5.17)andgMis any Riemannian metric on M.
+The rest of this section is concerned about the proof of Thoerem 5 .3.24 RUNG-TZUNG HUANG
+5.5.A choice of λ.Assume that no eigenvalue of BH
+(λ,∞)lies in the solid angles L[−θ−π,θ]andL[−θ,θ+π],
+cf. [5, Subsection 10.4 and 10.5], then it follows that no eigenvalue of ( BH
+(λ,∞))2lies in the solid angles
+L[−2θ,2θ+2π].
+LetB′Hdenote the twisted odd signature operator associated to the con nection∇′, the odd-degree
+formHand the Riemannian metric gM. One can check that
+(∇H)∗= Γ∇′HΓ and (∇′H)∗= Γ∇HΓ. (5.19)
+Using (3.2), (5.19) and the equality Γ∗= Γ, one can see that the adjoint B′HofBHsatisfies
+(BH)∗=B′H. (5.20)
+The choice of the angle θguarantees that ±2θare Agmon angles for the operator (Γ ∇′H)2=/parenleftbig
+(Γ∇H)2/parenrightbig∗.
+In particular, for each λ≥0, the number ξλ(∇′H,gM,θ) can be defined by the formula (3.18), with the
+same angle θand with ∇Hreplaced by ∇′Heverywhere.
+The following lemma is twisted analogue of Lemma 10.6 of [5] and the proo f is similar to the proof of
+Lemma 10.6 of [5]. We skip the proof.
+Lemma 5.4. Letθbe as above and let λ≥0be big enough so that the operator BH
+(λ,∞)does not have
+purely imaginary eigenvalues, cf. [5, Subsection 10.5] . Then
+ξλ(∇′H,gM,θ) =¯ξλ(∇H,gM,θ),
+and
+ηλ(∇′H) = ¯ηλ(∇H), (5.21)
+where¯zdenotes the complex conjugate of the number z∈C.
+5.6.Small eigenvalues of BHandB′H.We define Ω¯k
+±,I(M,E′),Ω¯k
+±(M,E′), and Ω¯k(M,E′) in similar
+ways as Ω¯k
+±,I(M,E),Ω¯k
+±(M,E) and Ω¯k(M,E), respectively. As (3.19), for k= 0,1, set
+d±
+¯k,λ= dimΩ¯k
+±,[0,λ](M,E), d′±
+¯k,λ= dimΩ¯k
+±,[0,λ](M,E′). (5.22)
+From the fact that Γ BH,±
+¯k,(λ,∞)Γ =BH,∓
+k+1,(λ,∞), we conclude that
+Γ/parenleftBig
+Ω¯k
+±,[0,λ](M,E)/parenrightBig
+=/parenleftBig
+Ωk+1
+∓,[0,λ](M,E)/parenrightBig
+, k= 0,1. (5.23)
+Therefore
+d±
+¯k,λ=d∓
+k+1,λ, k= 0,1.
+Hence/summationdisplay
+k=0,1(−1)kd−
+¯k,λ=d−
+¯0,λ−d−
+¯1,λ≡d−
+¯0,λ+d+
+¯0,λ= dimΩ¯0
+[0,λ](M,E),mod 2Z. (5.24)
+From (5.20) and (5.23), we obtain
+dimΩ¯0
+[0,λ](M,E) = dimΩ¯1
+[0,λ](M,E) = dimΩ¯0
+[0,λ](M,E′) = dimΩ¯1
+[0,λ](M,E′).(5.25)
+Hence by (5.24) and (5.25), we have
+/summationdisplay
+k=0,1(−1)kd′−
+¯k,λ= dimΩ¯0
+[0,λ](M,E),mod 2Z. (5.26)
+By the definition (3.16), we obtain
+2η(BH
+¯0,[0,λ])≡dimΩ¯0
+[0,λ](M,E),mod 2Z. (5.27)REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 25
+From (3.17), (5.26) and (5.27), we obtain, modulo 2 Z,
+2η(BH
+¯0(∇H)) = 2η(BH
+¯0,(λ,∞)(∇H))+2η(BH
+¯0,[0,λ])
+≡2ηλ(∇H)+/summationdisplay
+k=0,1d−
+¯k,λ. (5.28)
+Similarly,
+2η(BH
+¯0(∇′H))≡2ηλ(∇′H)+/summationdisplay
+k=0,1d−
+¯k,λ,mod 2Z. (5.29)
+5.7.Proof of Theorem 5.3. Letρ′
+Γ[0,λ]be the twisted refined torsion of the complex Ω•
+[0,λ](M,E′)
+associated to the restriction of Γ to Ω•
+[0,λ](M,E′).
+By Lemma 5.2, (5.17) and the fact that Γ∗= Γ, cf. [3, Proposition 3.58], we obtain
+ρ′
+Γ[0,λ]=α(ρΓ[0,λ]) (5.30)
+From (3.14), (3.20) and Definition 4.5, we obtain
+ρan(∇H) =ρΓ[0,λ]·exp/parenleftBig
+ξλ(∇H,gM,θ)−iπηλ(∇H)−iπ
+2/summationdisplay
+k=0,1(−1)kd−
+¯k,λ+iπ(rank(E))ηtrivial/parenrightBig
+.(5.31)
+Sinceαis an anti-linear isomorphism, α(ρan·z) =α(ρan)·¯zfor anyz∈C. Hence, from (5.30) and (5.31),
+we get
+α/parenleftbig
+ρan(∇H)/parenrightbig
+=ρ′
+Γ[0,λ]·exp/parenleftBig
+¯ξλ(∇H,gM,θ)+iπ¯ηλ(∇H)+iπ
+2/summationdisplay
+k=0,1(−1)kd−
+¯k,λ−iπ(rank(E))ηtrivial/parenrightBig
+.(5.32)
+Using Lemma 5.4 and the analogue of (5.31) for ρan(∇′H), we obtain from (5.32)
+α/parenleftbig
+ρan(∇H)/parenrightbig
+=ρan(∇′H)·exp/parenleftBig
+2iπ¯ηλ(∇H)+iπ/summationdisplay
+k=0,1(−1)kd−
+¯k,λ−iπ(rank(E))ηtrivial/parenrightBig
+.(5.33)
+From (5.33) and (5.29), we obtain (5.18).
+6.Comparison with the twisted analytic torsion
+In this section we first define the twisted Ray-Singer metric /ba∇dbl·/ba∇dblRS
+Det(H•(M,E,H))and then calculate the
+twisted Ray-Singer norm /ba∇dblρan(∇H)/ba∇dblRS
+Det(H•(M,E,H))of the twisted refined analytic torsion. In particular,
+we show that, if ∇is a Hermitian metric, then /ba∇dblρan(∇H)/ba∇dblRS
+Det(H•(M,E,H))= 1.
+6.1.The twisted analytic torsion. LetE→Mbe a complex vector bundle over a cloed oriented
+manifold Mof odd dimension m= 2r−1. Let∇be a flat connection on EandHbe a odd degree
+closed form, other than one form, on M. Fix a Riemannian metric gMonMand a Hermitian metric hE
+onE. Let∇H∗denote the adjoint of ∇H:=∇+H∧·with respect to the scalar product <·,·>Mon
+Ω•(M,E) defined by hEand the Riemannian metric gM.
+Now let
+∆H=∇H∗∇H+∇H∇H∗
+be the Laplacian twisted by the form H. We denote by ∆H
+¯kthe restriction of ∆Hto Ω¯k(M,E),k= 0,1.
+Assume that Iis an interval of the form [0 ,λ],(λ,µ],(λ,∞)(µ≥λ≥0) and let Π∆H
+¯k,Ibe the spectral
+projection of ∆H¯kcorresponding to I, cf. Subsection 3.3. Set
+ˇΩ¯k
+I(M,E) := Π∆H
+¯k,I/parenleftbig
+Ω•(M,E)/parenrightbig
+⊂Ω•(M,E).26 RUNG-TZUNG HUANG
+Let ∆H,I
+kdenote the restriction of ∆H
+¯ktoˇΩ¯k
+I(M,E) and define
+TRS
+I=TRS
+I(∇H) := exp/parenleftBig1
+2/summationdisplay
+k=0,1(−1)k+1LDet′
+−π/parenleftbig
+∇H∗∇H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
+I(M,E)/parenrightBig
+. (6.1)
+It is not difficult to check that, for any non-negative, real number sµ≥λ≥0,
+TRS
+(λ,∞)=TRS
+(λ,µ]·TRS
+[µ,∞). (6.2)
+Note that if η¯kis the unit volume element of H¯k(M,E,H),k= 0,1, then
+τ(M,E,H) :=/parenleftbig
+TRS
+(0,∞)/parenrightbig−1·η¯0⊗η−1
+¯1∈Det/parenleftbig
+H•(M,E,H)/parenrightbig
+(6.3)
+is thetwisted analytic torsion , introduced by V. Mathai and S. Wu in [17].
+For each λ >0, the cohomology of the finite dimensional complex/parenleftbigˇΩ•
+[0,λ](M,E),∇H/parenrightbig
+is naturally
+isomorphic to H•(M,E,H). Identifying these two cohomology spaces, we then obtain from ( 2.10) an
+isomorphism
+φλ=φˇΩ•
+[0,λ](M,E): Det/parenleftbigˇΩ•
+[0,λ](M,E)/parenrightbig
+→Det/parenleftbig
+H•(M,E,H)/parenrightbig
+. (6.4)
+The scalar product <·,·>onˇΩ•
+[0,λ](M,E)⊂Ω•(M,E) defined by gMandhEinduces a metric
+/ba∇dbl·/ba∇dblDet/parenleftbig
+ˇΩ•
+[0,λ](M,E)/parenrightbigon the determinant line Det/parenleftbigˇΩ•
+[0,λ](M,E)/parenrightbig
+. Let/ba∇dbl·/ba∇dblλdenote the metric on the
+determinant line Det/parenleftbig
+H•(M,E,H)/parenrightbig
+such that the isomorphism (6.4) is an isometry. Then, for c∈
+Det/parenleftbigˇΩ•
+[0,λ](M,E)/parenrightbig
+, we have
+/ba∇dblc/ba∇dblDet/parenleftbig
+ˇΩ•
+[0,λ](M,E)/parenrightbig=/ba∇dblφλ(c)/ba∇dblλ. (6.5)
+Using the Hodge theory, we have the canonical identification
+H¯k(M,E,H)∼=Ker∆H
+¯k, k= 0,1.
+By their inclusion in Ω¯k(M,E), the space of twisted harmonicforms Ker∆H¯kinherits a metric. We denote
+by|·|Det/parenleftbig
+H•(M,E,H)/parenrightbigthe corresponding metric on Det/parenleftbig
+H•(M,E,H)/parenrightbig
+. By definition
+/ba∇dbl·/ba∇dblDet/parenleftbig
+ˇΩ•
+{0}(M,E)/parenrightbig=|·|Det/parenleftbig
+H•(M,E,H)/parenrightbig. (6.6)
+The following is twisted analogue of [2, Proposition 1.5]. See also [20] f or the contact version.
+Proposition 6.1.
+/ba∇dbl·/ba∇dblλ=|·|Det/parenleftbig
+H•(M,E,H)/parenrightbig·TRS
+(0,λ]. (6.7)
+Proof.Fork= 0,1, fixc′
+¯k∈Det/parenleftbigˇΩ¯k
+{0}(M,E)/parenrightbig∼=Det/parenleftbig
+H¯k(M,E,H)/parenrightbig
+andc′′
+¯k∈Det/parenleftbigˇΩ¯k
+(0,λ](M,E)/parenrightbig
+. Then,
+using the natural isomorphism
+Det/parenleftbigˇΩ¯k
+{0}(M,E)/parenrightbig
+⊗Det/parenleftbigˇΩ¯k
+(0,λ](M,E)/parenrightbig
+∼=Det/parenleftbigˇΩ¯k
+{0}(M,E)⊕ˇΩ¯k
+(0,λ](M,E)/parenrightbig
+= Det/parenleftbigˇΩ¯k
+[0,λ](M,E)/parenrightbig
+,
+we can regard the tensor product c¯k:=c′¯k⊗c′′¯kas an element of Det/parenleftbigˇΩ¯k
+[0,λ](M,E)/parenrightbig
+. Denote by
+ˇΩ¯k,−
+I(M,E) = Ker∇H∩ˇΩ¯k
+I(M,E),ˇΩ¯k,+
+I(M,E) = Ker∇H∗∩ˇΩ¯k
+I(M,E).REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 27
+Since the complexes ˇΩ¯k
+(0,λ](M,E),k= 0,1, are acyclic, it is not difficult to see that each ˇΩ¯k
+(0,λ](M,E)
+splits orthogonally into
+ˇΩ¯k
+(0,λ](M,E) =ˇΩ¯k,−
+(0,λ](M,E)⊕ˇΩ¯k,+
+(0,λ](M,E). (6.8)
+Takea¯k∈Det/parenleftbigˇΩ¯k,+
+(0,λ](M,E)/parenrightbig
+so thatc′′
+¯k=∇H(ak+1)∧a¯k. Denote by c=c′⊗c′′, wherec′=c′
+¯0⊗c′
+¯1−1
+andc′′=c′′
+¯0⊗c′′−1
+¯1. Then
+/ba∇dblc/ba∇dblDet/parenleftbig
+ˇΩ•
+[0,λ](M,E)/parenrightbig=/ba∇dblc′/ba∇dblDet/parenleftbig
+ˇΩ•
+{0}(M,E)/parenrightbig×/ba∇dblc′′/ba∇dblDet/parenleftbig
+ˇΩ¯k
+(0,λ](M,E)/parenrightbig
+=/ba∇dblc′/ba∇dblDet/parenleftbig
+ˇΩ•
+{0}(M,E)/parenrightbig×/ba∇dbla¯0/ba∇dblDet/parenleftbig
+ˇΩ¯0,+
+(0,λ](M,E)/parenrightbig×/ba∇dbl∇H(a¯1)/ba∇dblDet/parenleftbig
+ˇΩ¯0,−
+(0,λ](M,E)/parenrightbig
+×/parenleftBig
+/ba∇dbla¯1/ba∇dblDet/parenleftbig
+ˇΩ¯1,+
+(0,λ](M,E)/parenrightbig/parenrightBig−1
+×/parenleftBig
+/ba∇dbl∇H(a¯0)/ba∇dblDet/parenleftbig
+ˇΩ¯1,−
+(0,λ](M,E)/parenrightbig/parenrightBig−1
+. (6.9)
+where/ba∇dbl · /ba∇dblVdenotes the naturally induced norm on the subspace V. The space ˇΩ¯k
+(0,λ](M,E) splits
+orthogonally into eigenspaces.
+ˇΩ¯k
+(0,λ](M,E) =⊕ν≤λˇΩ¯k
+{ν}(M,E)
+Givenν∈(0,λ], we choose an orthogonal basis ( v1,···,vnk) of each eigenspace ˇΩ¯k,+
+{ν}(M,E) and choose
+the element a¯k,ν=v1∧···∧vnk∈Det/parenleftbig
+H¯k,+
+{ν}(M,E,H)/parenrightbig
+, wherenk= dimˇΩ¯k,+
+{ν}(M,E). Then
+/ba∇dbl∇H(a¯k,ν)/ba∇dblDet/parenleftbig
+ˇΩk+1,−
+{ν}(M,E)/parenrightbig=/ba∇dbl∇Hv1∧···∧∇Hvnk/ba∇dblDet/parenleftbig
+ˇΩk+1,−
+{ν}(M,E)/parenrightbig
+=/ba∇dbl∇Hv1/ba∇dblDet/parenleftbig
+ˇΩk+1,−
+{ν}(M,E)/parenrightbig×···×/ba∇dbl∇Hvnk/ba∇dblDet/parenleftbig
+ˇΩk+1,−
+{ν}(M,E)/parenrightbig
+=νnk
+2/ba∇dblv1/ba∇dblDet/parenleftbig
+ˇΩ¯k,+
+{ν}(M,E)/parenrightbig×···×/ba∇dbl vnk/ba∇dblDet/parenleftbig
+ˇΩ¯k,+
+{ν}(M,E)/parenrightbig
+=νnk
+2/ba∇dbla¯k,ν/ba∇dblDet/parenleftbig
+ˇΩ¯k,+
+{ν}(M,E)/parenrightbig. (6.10)
+By combining (6.9) with (6.10), we obtain
+/ba∇dblc/ba∇dblDet/parenleftbig
+ˇΩ•
+[0,λ](M,E)/parenrightbig=/ba∇dblc′/ba∇dblDet/parenleftbig
+ˇΩ•
+{0}(M,E)/parenrightbig×/productdisplay
+k=0,1/parenleftbig
+Det−π/parenleftbig
+∇H∇H∗/parenrightbig/vextendsingle/vextendsingle
+Ω¯k
+(0,λ](M,E)/parenrightbig(−1)k+1/2.(6.11)
+By (6.1), (6.5), (6.6) and (6.11), we obtain the result. /square
+By (6.7), we have, for 0 ≤λ≤µ,
+/ba∇dbl·/ba∇dblµ=/ba∇dbl·/ba∇dblλ·TRS
+(λ,µ]. (6.12)
+Thetwisted Ray-Singer metric on Det/parenleftbig
+H•(M,E,H)/parenrightbig
+is defined by the formula
+/ba∇dbl·/ba∇dblRS
+Det/parenleftbig
+H•(M,E,H)/parenrightbig:=/ba∇dbl·/ba∇dblλ·TRS
+(λ,∞), λ≥0. (6.13)
+It follows immediately from (6.2) and (6.12) that /ba∇dbl · /ba∇dblDet/parenleftbig
+H•(M,E,H)/parenrightbigis independent of the choice of
+λ≥0. Note that for λ= 0, by (6.3) and (6.13), we have
+/ba∇dblτ(M,E,H)/ba∇dblRS
+Det/parenleftbig
+H•(M,E,H)/parenrightbig= 1.
+Theorem 6.2. LetEbe a complex vector bundle over a closed oriented odd dimensi onal manifold M
+and let∇be a flat connection on E. Further, let Hbe a odd-degree closed form on Mand denote by
+∇H:=∇+H∧·. Then
+/ba∇dblρan(∇H)/ba∇dblRS
+Det/parenleftbig
+H•(M,E,H)/parenrightbig=eπImη(∇H,gM), (6.14)28 RUNG-TZUNG HUANG
+where
+η(∇H,gM) =η/parenleftbig
+B¯0(∇H,gM)/parenrightbig
+.
+In particular, if ∇is a Hermitian connection, then /ba∇dblρan(∇H)/ba∇dblRS
+Det/parenleftbig
+H•(M,E,H)/parenrightbig= 1.
+The rest of this section is concerned with the proof of Theorem 6.2.
+6.2.Comparison between the twisted Ray-Singer metrics associa ted to a connection and to
+its dual. We assume that θ∈(−π/2,0) is any Agmon angle for the twisted odd signature operator BH
+such that no eigenvalue of BH
+(λ,∞)lies in the solid angles L[−θ−π,−π/2],L(−π/2,θ],L[−θ,π/2)andL(π/2,θ+π],
+cf. [5, Subsection 11.4]
+As in Subsection 5.4, let ∇′be the connection dual to ∇with respect to the Hermitian metric hEand
+letE′denote the flat bundle ( E,∇′). LetHbe an odd degree closed form, other than one form, on M
+and denote by ∇H:=∇+H∧·. Let
+∆′H= (∇′H)∗∇′H+∇′H(∇′H)∗,
+denote the twisted Laplacian of the connection ∇′twisted by the form H. For any λ≥0, we denote by
+ˇΩ•
+[0,λ](M,E′)⊂Ω•(M,E′)
+the image of the spectral projection Π ∆′H,[0,λ], cf. Subsection 3.3. As in Subsection 6.1, we use the scalar
+product induced by gMandhEonˇΩ•
+[0,λ](M,E′) to construct a metric /ba∇dbl·/ba∇dbl′
+λon Det/parenleftbig
+H•(M,E′,H)/parenrightbig
+and
+we define the twisted Ray-Singer metric on Det/parenleftbig
+H•(M,E′,H)/parenrightbig
+by the formula
+/ba∇dbl·/ba∇dblRS
+Det/parenleftbig
+H•(M,E′,H)/parenrightbig:=/ba∇dbl·/ba∇dbl′
+λ·TRS
+(λ,∞)(∇′H), λ≥0. (6.15)
+As the untwisted case, cf. [5, Subsection 11.6], we have the followin g identification,
+ˇΩ•
+[0,λ](M,E′)∼=ˇΩm−•
+[0,λ](M,E)∗,
+which preserves the scalar products induced by gMandhEonˇΩ•
+[0,λ](M,E′) andˇΩm−•
+[0,λ](M,E)∗. Hence,
+the anti-linear isomorphism α, cf. (5.17), is an isometry with respect to the metrics /ba∇dbl·/ba∇dblλand/ba∇dbl·/ba∇dbl′
+λ. In
+particular,
+/ba∇dblρan(∇H)/ba∇dblλ=/ba∇dblα(ρan(∇H))/ba∇dbl′
+λ.
+It follows from (5.18) that
+/ba∇dblρan(∇H)/ba∇dblλ=/ba∇dblρan(∇′H)/ba∇dbl′
+λ·e2πImη(∇H,gM). (6.16)
+We need the following lemma. For untwisted case, cf. for example [4, L emma 8.8].
+Lemma 6.3.
+TRS
+(λ,∞)(∇′H) =TRS
+(λ,∞)(∇H). (6.17)
+Proof.From (5.19), we have
+(∇H)∗∇H= Γ∇′HΓ∇H= Γ(∇′HΓ∇HΓ)Γ = Γ∇′H(∇′H)∗Γ. (6.18)
+As in (6.8), we have
+ˇΩ¯k
+(λ,∞)(M,E) =ˇΩ¯k,−
+(λ,∞)(M,E)⊕ˇΩ¯k,+
+(λ,∞)(M,E). (6.19)
+The operator ∇HmapsˇΩ¯k,+
+(λ,∞)(M,E) isomorphically onto ˇΩk+1,−
+(λ,∞)(M,E) and
+∇H/vextendsingle/vextendsingleˇΩ¯k,+
+(λ,∞)(M,E)/parenleftbig
+(∇H)∗∇H/parenrightbig/vextendsingle/vextendsingleˇΩ¯k,+
+(λ,∞)(M,E)=/parenleftbig
+∇H(∇H)∗/vextendsingle/vextendsingleˇΩk+1,−
+(λ,∞)(M,E)/parenrightbig
+·∇H/vextendsingle/vextendsingleˇΩ¯k,+
+(λ,∞)(M,E).(6.20)REFINED ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES 29
+Hence, by (6.20), we have
+LDet′
+−π/parenleftbig
+(∇′H)∗∇′H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k,+
+(λ,∞)(M,E)= LDet′
+−π/parenleftbig
+∇′H(∇′H)∗/parenrightbig/vextendsingle/vextendsingle
+Ωk+1,−
+(λ,∞)(M,E). (6.21)
+From (6.18), we obtain
+logTRS
+(λ,∞)(∇H)
+=1
+2/summationdisplay
+k=0,1(−1)k+1LDet′
+−π/parenleftbig
+(∇H)∗∇H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k,+
+(λ,∞)(M,E)
+=1
+2/summationdisplay
+k=0,1(−1)k+1LDet′
+−π/parenleftbig
+Γ∇′H(∇′H)∗Γ/parenrightbig/vextendsingle/vextendsingle
+Ω¯k,+
+(λ,∞)(M,E)
+=1
+2/summationdisplay
+k=0,1(−1)k+1LDet′
+−π/parenleftbig
+∇′H(∇′H)∗/parenrightbig/vextendsingle/vextendsingle
+Ωm−k,−
+(λ,∞)(M,E)
+=1
+2/summationdisplay
+k=0,1(−1)kLDet′
+−π/parenleftbig
+∇′H(∇′H)∗/parenrightbig/vextendsingle/vextendsingle
+Ω¯k,−
+(λ,∞)(M,E)
+=1
+2/summationdisplay
+k=0,1(−1)k+1LDet′
+−π/parenleftbig
+(∇′H)∗∇′H/parenrightbig/vextendsingle/vextendsingle
+Ω¯k,+
+(λ,∞)(M,E),
+where we use (6.21) for the last equality. This proves the lemma. /square
+Hence, from (6.13), (6.15), (6.16) and (6.17), we conclude that
+/ba∇dblρan(∇H)/ba∇dblRS
+Det/parenleftbig
+H•(M,E,H)/parenrightbig=/ba∇dblρan(∇′H)/ba∇dblRS
+Det/parenleftbig
+H•(M,E′,H)/parenrightbig·e2πImη(∇H,gM). (6.22)
+6.3.Direct sum of a connection and its dual. Let
+/tildewide∇=/parenleftbigg∇0
+0∇′/parenrightbigg
+denote the flat connection on E⊕Eobtained as a direct sum of the connections ∇and∇′. Denote by
+/tildewide∇H:=/parenleftbigg∇H0
+0∇′H/parenrightbigg
+.
+Then a discussion similar to [5, Subsection 11.7], where the untwisted c ase was treated, one easily obtains
+that,
+ρan(/tildewide∇H) =µH•(M,E,H),H•(M,E′,H)/parenleftbig
+ρan(∇H)⊗ρan(∇′H)/parenrightbig
+and
+/ba∇dblρan(/tildewide∇H)/ba∇dblRS
+Det/parenleftbig
+H•(M,E⊕E′,H/parenrightbig=/ba∇dblρan(∇H)/ba∇dblRS
+Det/parenleftbig
+H•(M,E,H/parenrightbig·/ba∇dblρan(∇′H)/ba∇dblRS
+Det/parenleftbig
+H•(M,E′,H/parenrightbig.
+Combining this later equality with (6.22), we get
+/ba∇dblρan(/tildewide∇H)/ba∇dblRS
+Det/parenleftbig
+H•(M,E⊕E′,H/parenrightbig=/parenleftbig
+/ba∇dblρan(∇H)/ba∇dblRS
+Det/parenleftbig
+H•(M,E,H/parenrightbig/parenrightbig2·e−2πImη(∇H,gM).
+Hence, (6.14) is equivalent to the equality
+/ba∇dblρan(/tildewide∇H)/ba∇dblRS
+Det/parenleftbig
+H•(M,E⊕E′,H/parenrightbig= 1. (6.23)
+By a slight modification of the deformation argument in [5, Section 11 , P.205-211], where the untwisted
+case was treated, we can obtain (6.23). Hence, we finish the proof of Theorem 6.2.30 RUNG-TZUNG HUANG
+References
+[1] M.F.Atiyah, V..K.Patodi, and I.M.Singer, Spectral asymmetry and Riemannian geometry II,Math.Proc.Cambridge
+Philos. Soc. 78(1975), 405-432.
+[2] J.-M. Bismut, H. Gillet, C. Soul’e, Analytic torsion and holomorphic determinant bundles I. Bo tt-Chern forms and
+analytic torsion , Commun. Math. Phys. 115, 49-78 (1988).
+[3] N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operators , Grundlehren Text Editions, Springer-Verlag,
+Berlin, 2004, Corrected reprint of the 1992 original.
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+[5] M. Braverman, T. Kappeler, Refined Analytic Torsion as an Element of the Determinant Lin e, Geom. Topol. 11
+(2007), 139-213.
+[6] M. Braverman, T. Kappeler, Ray-Singer type theorem for the refined analytic torsion , J. Funct. Anal. 243(2007)
+232-256.
+[7] M. Braverman, T. Kappeler, Comparison of the refined analytic and the Burghelea-Haller torsions, Ann. Inst. Fourier
+(Grenoble) 57(2007) 2361-2387.
+[8] M. Braverman, T. Kappeler, A canonical quadratic form on the determinant line of a flat ve ctor bundle , Int. Math.
+Res. Not. (2008) Art. ID rnn030, 21pp.
+[9] J-M. Bismut and W. Zhang, An extension of a theorem by Cheeger and M¨ uller , Ast´ erique , 205, SMF, Paris 1992.
+[10] S. E. Cappell, E. Y. Miller, Complex valued analytic torsion for flat bundles and for holo morphic bundles with (1,1)
+connections, arXiv:math.DG/0810.4833v1 .
+[11] M. Farber, V. Turaev, Absolute torsion , Tel Aviv Topology Conference: Rothenberg Festschrift (19 98), Contemp.
+Math., vol. 231, Amer. Math. Soc., Providence, RI, 1999, pp. 73–85.
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+topology-global differential geometry, Teubner-Texte Mat h. vol.70, Teubner, Leipzig, 1984, 49-87.
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+[16] W. M¨ uller, Analytic torsion and R-torsion for unimodular representat ions,J. Amer. Math. Soc. 61993, 721-753.
+[17] V. Mathai, S. Wu, Analytic torsion for twisted de Rham complexes, arXiv:math.DG/0810.4204v3 .
+[18] V. Mathai, S. Wu, Twisted analytic torsion ,arxiv:math.DG/0912.2184v1
+[19] D. B. Ray, I. M. Singer, R-torsion and the Laplacians on Riemannian manifolds , Adv. in Math. 7(1971), 145-210.
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+Ill., 1966)”, Amer. Math. Soc., Providence, R.I. 1967, 288-307.
+[22] M. A. Shubin, Pseudodifferential operators and spectral, Second edition, Springer, Berlin 2001.
+[23] V. G. Turaev, Reidemeister torsion in knot theory , Russian Math. Survey 41(1986), 119-182.
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+Institute of Mathematics, Academia Sinica, Nankang 11529, T aipei, Taiwan
+E-mail address :rthuang@math.sinica.edu.tw
\ No newline at end of file
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+arXiv:1001.0655v2  [astro-ph.GA]  10 Jan 2010Mon. Not. R. Astron. Soc. 000, 1–11 (2009) Printed 19 November 2018 (MN L ATEX style file v2.2)
+Interstellar extinction and polarization — A spheroidal
+dust grain approach perspective
+H.K. Das1⋆, N.V. Voshchinnikov2and V.B. Il’in2,3
+1IUCAA, Post Bag 4, Ganeshkhind, Pune 411 007, India
+2Sobolev Astronomical Institute, St. Petersburg Universit y, St. Petersburg 198504, Russia
+3Main (Pulkovo) Astronomical Observatory, St. Petersburg 1 96140, Russia
+Received 2009 ...
+ABSTRACT
+We extend and investigate the spheroidal model of interstellar dus t grains used to
+simultaneously interpret the observed interstellar extinction and p olarization curves.
+We compare our model with similar models recently suggestedby othe r authors, study
+its properties and apply it to fit the normalized extinction A(λ)/AVand the polarizing
+efficiency P(λ)/A(λ) measured in the near IR to far UV region for several stars seen
+through one large cloud. We conclude that the model parameter Ω b eing the angle
+between the line of sight and the magnetic field direction can be more o r less reliably
+determined from comparison of the theory and observations. This opens a way to
+study the spatial structure of interstellar magnetic fields by using multi-wavelength
+photometric and polarimetric observations.
+Key words: ISM: clouds – polarization – dust, extinction.
+1 INTRODUCTION
+Modelling of the wavelength dependencies of interstellar e x-
+tinctionA(λ) and polarization P(λ) along with analysis of
+infrared bands is the main source of information about the
+properties of interstellar grains (Draine 2003; Whittet 20 03;
+Voshchinnikov 2004).
+Extinction data were often used to constrain
+the size distribution of carbonaceous and silicate
+bare grains (e.g. Mathis, Rumpl & Nordsieck 1977;
+Zubko, Kre/suppress lowski & Wegner 1996; Weingartner & Draine
+2001). These data were also modeled with composite
+particles (Vaidya et al. 2001; Voshchinnikov et al. 2006; Ia t` ı
+et al. 2008). In all these studies the grains were assumed to
+be spherical, i.e. their shape was not involved.
+To get information about the grain shapes as well
+as the ambient magnetic fields one needs to consider in-
+terstellar polarization. Interpretation of its wavelengt h de-
+pendence includes calculations of the extinction cross sec -
+tions of rotating partially aligned non-spherical particl es. In
+early studies (e.g. Hong & Greenberg 1980; Voshchinnikov
+1989) an unphysical model of infinitely long cylinders
+was applied1. More realistic is the spheroidal model of
+⋆E-mail: hkdas@iucaa.ernet.in (HKD); nvv@astro.spbu.ru
+(NVV); ilin55@yandex.ru (VBI)
+1Li & Greenberg (1997)also considered finite cylinders formo d-
+elling interstellar extinction and polarization.grains with the shape of these axisymmetric particles be-
+ing characterized by the only parameter — the ratio of
+the major to minor semi-axis a/b. The optical properties
+of spheroids have long been of interest (e.g., Martin 1978;
+Rogers & Martin 1979; Onaka 1980; Draine & Lee 1984;
+Voshchinnikov & Farafonov 1993; Kim & Martin 1995). The
+growth of such grains in the interstellar medium (ISM) has
+been discussed by Stark, Potts & Diver (2006), their absorp-
+tion and scattering properties (in the Rayleigh limit) by
+Min et al. (2006) and mineralogy by Li, Zhao & Li (2007).
+Over the past decade spheroids have found wide
+applications in studies of optics of cometary dust —
+Das & Sen (2006), Moreno et al. (2007), circumstellar dust
+— Wolf et al. (2002), interplanetary dust — Lasue et al.
+(2007). Greenberg & Li (1996) used prolate spheroids and
+Lee & Draine (1985), Hildebrand & Dragovan (1995) oblate
+ones to interpret the silicate and ice polarization feature s of
+the BN object. Draine & Allaf-Akbari (2006) predicted ha-
+los due to X-ray scattering by oblate particles, detection o f
+which would serve as a test for dust models.
+The spheroidal model of cosmic dust grains is particu-
+larly promising for interpretation of the interstellar pol ariza-
+tion and extinction data. A detailed review of early works
+on the subject can be found in Voshchinnikov (2004), so
+we consider recent applications of the model. Gupta et al.
+(2005) well reproduced the average Galactic extinction cur ve
+in the region 0 .3−9.5µm−1with silicate and graphite
+oblate spheroidal ( a/b= 1.33) particles rotating in a plane
+c/circlecopyrt2009 RAS2Das et al.
+and having the size distribution n(rV)∝r−3.5
+V withrV=
+0.005−0.25µm whererVis the radius of a volume equiva-
+lent sphere. They found that the shape of grains did not af-
+fect the abundance problem, but essentially changed the po-
+larization efficiency factors at different particle orientat ion
+angles. It should be noted that interpretation of the extinc -
+tion data alone is usually made within the spherical model of
+dust grains (see for details the recent review of Draine 2009 ).
+Draine & Allaf-Akbari (2006) and Draine & Fraisse (2009)
+fitted both the average Galactic extinction and polarizatio n
+curves in the range 0 .4−9.5µm−1applying picket-fence ori-
+ented silicate and graphite oblate spheroids with the aspec t
+ratio in the interval a/b= 1−2 with specific size distribu-
+tions. The grain alignment degree was size dependent and
+showed a sharp increase at rV∼0.1µm. Voshchinnikov
+& Das (2008; hereafter VD08) studied the wavelength de-
+pendence of the ratio of the linear polarization degree to
+extinction (so called polarizing efficiency) from the ultrav io-
+let to near-infrared. They found that the wavelength depen-
+dence ofP(λ)/A(λ) was mainly determined by the particle
+composition and size whereas the absolute values of this ra-
+tio depended on the particle shape, degree and direction of
+alignment.
+In this paper we investigate further advantages of the
+spheroidal approach to interstellar grains. In contrast to
+other works we calculate the optical properties of spheroid s
+of all sizes exactly even at far UV wavelengths, consider
+ensembles of particles with various degree and direction of
+alignment, compare different approaches to extinction and
+polarization curve modelling, treat the observational dat a in
+a new way (the wavelength dependence of the polarizing ef-
+ficiency is fitted instead of the polarization curve), apply t he
+theory to observational data for concrete stars, and discus s
+a possibility to determine the direction of interstellar ma g-
+netic fields from the data. In Section 2 we describe details of
+our model and compare it with the models used earlier. In
+Section 3 the results obtained from our model are compared
+with observations of seven stars for which the line of sight
+intersects one main cloud, keeping in view that “mean ex-
+tinction curves as well as mean total-to-selective extinct ion
+ratios are possibly averaged over several interstellar clo uds
+of possibly different physical parameters” (Kre/suppress lowski 198 9).
+Section 4 contains our conclusions.
+2 THEORY
+Our modelling is based on the simple standard assumptions
+concerning dust grain materials, size distribution, shape and
+alignment. Homogeneous spheroids considered are charac-
+terized by their type (prolate or oblate), the aspect ratio a/b
+and a size parameter. As the latter we choose the radius rV
+of a sphere whose volume is equal to that of a non-spherical
+particle, i.e. r3
+V=ab2for prolate spheroids and r3
+V=a2b
+for oblate ones.
+2.1 Dust grain materials
+Interstellar dust mainly consists of five heavy elements: C, O,
+Mg, Si and Fe, which are locked in carbonaceous and silicate
+particles (Jones 2000; Draine 2009). We consider sub-micro ngrains from amorphous carbon (AC1) and amorphous sili-
+cate (astronomical silicate, astrosil). In order to reprod uce
+the 2175 ˚A absorption feature small graphite spheres with
+radiusrgra= 0.02µm are also involved2. The optical con-
+stants of the materials are taken from Rouleau & Martin
+(1991) and Laor & Draine (1993).
+2.2 Size distribution
+Interpretation of the interstellar extinction is usually a imed
+at reconstruction of the dust grain size distribution consi s-
+tent with the cosmic abundances and some other constraints
+(see discussion in Voshchinnikov 2004).
+We choose a power-law size distribution
+n(rV)∝r−q
+V, (1)
+which was derived by Mathis, Rumpl & Nordsieck (1977)
+(see also Draine & Lee 1984) from minimization of χ2-
+statistic. The distribution has three parameters: the lowe r
+(rV,min) and upper ( rV,max) limits and the power index
+q. It was widely used in modelling of interstellar extinc-
+tion and radiative transfer in dusty objects. The average
+Galactic extinction curve can be reproduced with the so-
+called standard MRN mixture — an ensemble of carbona-
+ceous (graphite) and silicate spheres (in nearly equal pro-
+portions) with the parameters: q= 3.5,rV,min≈0.005µm
+andrV,max≈0.25µm.
+2.3 Cross section averaging
+Let us consider non-polarized light passing through a cloud
+that contains three populations of grains: rotating sphero ids
+from amorphous silicate (Si), amorphous carbon (C) and
+graphite (gra) spheres. For a line of sight, extinction (in
+stellar magnitudes) and linear polarization (in percentag e)
+produced by the cloud can be written as
+A(λ) = 1.086Nd∝angbracketleftCext∝angbracketrightλ, P (λ) =Nd∝angbracketleftCpol∝angbracketrightλ100%.(2)
+Here,Nd=NC+NSi+Ngrais the total dust grain column
+density, ∝angbracketleftCext∝angbracketrightλand∝angbracketleftCpol∝angbracketrightλare the extinction and polar-
+ization cross sections, respectively, averaged over the gr ain
+populations, i.e.
+∝angbracketleftCext∝angbracketrightλ=KCCext,C(λ)+KSiCext,Si(λ)+KgraCext,gra(λ),(3)
+∝angbracketleftCpol∝angbracketrightλ=KCCpol,C(λ) +KSiCpol,Si(λ), (4)
+whereCext,gra(λ) =πr2
+graQext,gra(λ) andKi=Ni/Nd(i≡
+C,Si) is the relative column density of the i-th population.
+It is evident that
+KC+KSi+Kgra= 1.
+The values of Cext,pol,iare obtained by averaging of the
+cross sections over the size distribution and grain orienta -
+tions (Hong & Greenberg 1980; Voshchinnikov 1989)
+Cext,i(λ) =/parenleftbigg2
+π/parenrightbigg2/integraldisplayrV,max,i
+rV,min,i/integraldisplayπ/2
+0/integraldisplayπ/2
+0/integraldisplayπ/2
+0πr2
+V,iQext,i(λ)
+×ni(rV)f(ξ,β)dϕdωdβdr V, (5)
+2Note that graphite is not the only material considered as the
+carrier of the the 2175 ˚A absorption feature (see, e.g., Li et al.
+2008)
+c/circlecopyrt2009 RAS, MNRAS 000, 1–11Interstellar extinction and polarization 3
+XZ
+YBJ
+N
+kβω
+ϕ
+ψα
+ΦO2O1Ω
+θ
+Figure 1. Geometrical configuration of a spinning and wobbling
+prolate spheroidal grain. The major (symmetry) axis of the p ar-
+ticle O 1O2is situated in the spinning plane NO 1O2which is
+perpendicular to the the angular momentum J. The direction o f
+light propagation k is parallel to the Z-axis and makes the angle
+αwith the particle symmetry axis.
+Cpol,i(λ) =2
+π2/integraldisplayrV,max,i
+rV,min,i/integraldisplayπ/2
+0/integraldisplayπ
+0/integraldisplayπ/2
+0πr2
+V,iQpol,i(λ)
+×ni(rV)f(ξ,β) cos 2ψdϕdωdβdr V. (6)
+Here,βis the precession-cone angle for the angular mo-
+mentum J which spins around the direction of the mag-
+netic field B, ϕthe spin angle, ωthe precession angle
+(see Fig. 1). The quantities Qext= (QTM
+ext+QTE
+ext)/2 and
+Qpol= (QTM
+ext−QTE
+ext)/2 are the extinction and polariza-
+tion efficiency factors for non-polarized incident radiatio n
+(Voshchinnikov & Farafonov 1993),
+cos 2ψ=/braceleftBigg
+cos 2Φ/bracketleftBig
+2cos2ϕcos2θ
+1−cos2ϕsin2θ−1/bracketrightBig
+,prolate
+cos 2Φ ,oblate,(7)
+and
+cosθ= cos Ω cos β+ sin Ω sinβcosω, (8)
+cos 2Φ =/bracketleftbigg2 sin2βsin2ω
+sin2θ−1/bracketrightbigg
+, (9)
+cosα=/braceleftbiggsinθcosϕ , prolate/radicalbig
+1−sin2θcos2ϕ , oblate.(10)
+Here, Ω is the angle between the line of sight and the mag-
+netic field direction (0◦/lessorequalslantΩ/lessorequalslant90◦). The value Ω = 90◦
+corresponds to the case when the particle rotation plane con -
+tains the light propagation vector k, which gives the max-
+imum degree of linear polarization. For Ω = 0◦, the light
+falls perpendicular to the particle rotation plane and from
+symmetry reasons the net degree of polarization produced
+is zero.
+The efficiency factors were calculated by using a so-
+lution to the light scattering problem for a homogeneousspheroid obtained by the method of separation of vari-
+ables in spheroidal coordinates (Voshchinnikov & Farafono v
+1993) and a modern approach to computations of spheroidal
+wave functions (Voshchinnikov & Farafonov 2003). In con-
+trast to all works where the spheroidal model was recently
+applied (Gupta et al. 2005; Draine 2009; Draine & Fraisse
+2009) we computed the optical properties of the particles of
+all sizes exactly even in the far UV spectral region.
+2.4 Grain alignment
+Interstellar non-spherical grains should be partially ali gned.
+The “picket fence” or perfect rotating alignment often
+used in modelling of polarization (e.g. Kim & Martin 1995;
+Draine & Fraisse 2009) is a crude approximation giving a
+too high polarizing efficiency in comparison with the empir-
+ical upper limit (e.g., Greenberg 1978)
+Pmax/AV<∼3%/mag.
+Although the theory of grain alignment is actively dis-
+cussed (Lazarian 2007) it still has little practical signifi -
+cance. Therefore, we consider the classical alignment mech a-
+nism — so-called imperfect Davis–Greenstein (IDG) orienta -
+tion (Davis & Greenstein 1951). The IDG mechanism is de-
+scribed by the function f(ξ,β) depending on the alignment
+parameterξand the precession angle β(Hong & Greenberg
+1980; Voshchinnikov 1989)
+f(ξ,β) =ξsinβ
+(ξ2cos2β+ sin2β)3/2. (11)
+The parameter ξdepends on the particle size rV, the imagi-
+nary part of the grain magnetic susceptibility χ′′=κωd/Td,
+whereωdis the angular velocity of a grain and κ=
+2.5×10−12(Davis & Greenstein 1951), gas density nH, the
+strength of magnetic field B, and temperatures of dust Td
+and gasTg. By introducing a parameter
+δ0= 8.23×1023κB2
+nHT1/2
+gTd, µm, (12)
+one can get
+ξ2=rV+δ0(Td/Tg)
+rV+δ0. (13)
+We assume that carbon and silicate grains have similar
+alignment but different size distributions. Thus, our model
+has 10 main parameters: rV,min,rV,maxandqfor carbon and
+silicate grains, relative density for carbon ( KC) and silicate
+(KSi) grains, degree ( δ0) and direction (Ω) of grain align-
+ment.
+2.5 Comparison with other models
+There were just a few approaches to simultaneous mod-
+elling of the wavelength dependencies of interstellar ex-
+tinction and linear polarization. In most papers where an
+imperfect alignment of grains was involved (along with a
+power-law size distribution) the authors derived that part i-
+cles withrV<∼0.1µm should be poorly (randomly) oriented
+while those with rV>∼0.1µm should be nearly perfectly
+aligned (e.g., Kim & Martin 1995; Draine & Fraisse 2009).
+Some physical grounds for such alignment of interstellar
+c/circlecopyrt2009 RAS, MNRAS 000, 1–114Das et al.
+grains come, e.g., from Mathis (1986), Goodman & Whittet
+(1995), Martin (1995).
+As the actual composition of carbonaceous grains in the
+ISM is still unknown (Draine & Fraisse 2009), in interstel-
+lar dust modelling one often considered large (sub-micron)
+graphite grains instead of amorphous carbon ones. As po-
+larization produced by such graphite spheroids may be to a
+certain extent peculiar, the carbonaceous grains were ofte n
+assumed to be either spherical or randomly aligned (e.g.,
+Mathis 1986; Draine & Fraisse 2009).
+To compare these approaches with our one, we have
+made some calculations for three additional models :
+model A1 – imperfectly DG-aligned silicate and graphite
+spheroids;
+model A2 – imperfectly DG-aligned silicate spheroids
+andrandomly oriented amorphous carbon spheroids;
+model A3 – silicate and amorphous carbon spheroids
+withrandomly oriented small particle and perfectly DG-
+aligned large particle.
+The grain size distributions and the population of small
+graphite particles are not varied, i.e. they are the same as
+in our basic model described in Sects. 2.1–2.4.
+When considering graphite spheroids in the A1 model,
+we use the standard 2/3–1/3 approximation which gives rea-
+sonably accurate results (Draine & Malhotra 1993). The ori-
+entation function for amorphous carbon grains in the A2
+model isf(β) = sinβ. In the A3 model this function is size
+dependent
+f(β,rV) =/braceleftbiggsinβforrV/lessorequalslantrV,cut
+δ(β) forrV>rV,cut, (14)
+whereδ(β) is the Dirac delta function, rV,cuta cut-off pa-
+rameter.
+So, we assume that all particles whose volume is smaller
+than that of a sphere with the radius rV,cutare randomly
+oriented, while other particles have perfect rotation alig n-
+ment, i.e. their angular moment vectors are parallel to the
+magnetic field direction. In our calculations we choose rV,cut
+to be 0.11µm (cf. Draine & Fraisse 2009).
+Some results obtained with the models considered are
+presented in Fig. 2, the fitting procedure used is described
+in the next Section. We see that all models can give very
+similar extinction curves that well fit the observational da ta.
+Only the A1 model with large graphite particles provides the
+curve that differs a bit from others. Certainly, the extincti on
+data are not enough to constrain the models.
+The model dependencies of polarizing efficiency shown
+in the right panel of the figure vary in form. In particular,
+the model with randomly oriented carbonaceous particles
+produces too small polarization in the visual and IR parts of
+spectrum. Apparently, in this case a better fit can be reached
+if one takes a more complex size distribution function or
+(and) increases the alignment efficiency of silicate grains.
+However, this complicates the model, therefore, we prefer
+our model having a small number of the parameters and
+simple assumptions.3 COMPARISON WITH OBSERVATIONAL
+DATA
+So far, most efforts in modelling of the optics of interstel-
+lar dust were directed at fitting of the average Galactic
+extinction curve (often with some additional constraints)
+by making use of spherical particles. Interpretation of the
+curves observed towards individual objects was very sel-
+dom, and that of the extinction and polarization curves was
+done only for a few objects. Il’in (1987) applied the model
+of infinitely long coated cylinders with the imperfect DG-
+alignment to the interpretation of the wavelength depen-
+dence of interstellar extinction, linear and circular pola riza-
+tion for the star HD 204827 in the visual and IR parts of
+spectrum. Kim & Martin (1994) used infinitely long cylin-
+ders with the perfect DG-alignment (Ω = 90◦) to inter-
+pret the data for HD 25443 and HD 197770. Infinitely long
+cylinders with a silicate core and organic refractory mantl e
+were chosen by Li & Greenberg (1998) for the interpretation
+of the extinction and polarization curves for HD 210121.
+So, Voshchinnikov & Das (2008) were first who utilized the
+spheroidal model to consider both the extinction and polar-
+ization wavelength dependencies observed for two stars.
+3.1 Sample of objects
+For application of our model we chose stars seen through
+a single cloud with the extinction and polarization curves
+known in a wide wavelength region. Initially, preference
+was given to five stars (HD 24263, HD 62542, HD 99264,
+HD 147933 and HD 197770) with measured UV polarization
+(Anderson et al. 1996). These stars are not peculiar and not
+very distant. Later, two stars (HD 147165 and HD 179406)
+for which the polarization data in the visible were obtained
+by us was added to our sample.
+In this paper we discuss extinction and polarization of
+six stars (HD 24263, HD 99264, HD 147165, HD 147933,
+HD 179406, and HD 197770). Two objects (HD 62542 and
+HD 147933) were considered in our previous paper (VD08).
+Almost all the stars are not very far away from the galac-
+tic plane (Table 1) and are in dusty environment. One
+can assume that these stars are seen through single dust
+clouds as they have weak rotation angle of linear polariza-
+tion (Anderson et al. 1996) and narrow symmetric sodium
+D1and D 2lines (Zubko, Kre/suppress lowski & Wegner 1996). A few
+comments on the stars should be made.
+HD 24263 is a double star (the secondary is a B9 star) seen
+through a large cloud AG6 located between the Local Bubble
+and Orion–Eridanus Super-bubble at distance ∼150 pc and
+observed in NaI D and CaII lines (Genova & Beckman 2003;
+Welsh et al. 2005). One also detects the molecules CH, CH+
+and CN in the line of site (Penprase et al. 1990).
+HD 99264 is located in the region of the Scorpios–
+Centaurs OB association and the Chameleon–Musca dark
+clouds (Platais et al. 1998; Corradi, Franko & Knude 2004).
+Consideration of high-resolution NaI D profiles and
+uvbyβ photometry for over 60 nearby B stars allows
+Corradi, Franko & Knude (2004) to conclude that interstel-
+lar gas and dust in this region are distributed in two ex-
+tended sheet-like structures. The nearby feature located a t
+the distance <60 pc provides the colour excess E(b−y)≈
+c/circlecopyrt2009 RAS, MNRAS 000, 1–11Interstellar extinction and polarization 5
+0 2 4 6 800.511.522.5
+HD147933
+0 2 4 6 80123
+Figure 2. Normalized extinction (left panel) and polarizing efficienc y (right panel) for our basic model and three additional mode ls with
+graphite spheroids (A1); randomly oriented amorphous carb on spheroids (A2); small randomly oriented silicate and amo rphous carbon
+spheroids (A3) (see the text for mode details). Oblate spher oids with a/b= 3 and Ω = 33◦, for other parameters see the last line in
+Table 3. Observational data for HD 147933.
+0.05 while the second feature at 120–150 pc has E(b−y)≈
+0.2.
+HD 147165 (σSco) is a beta Cephei type pulsating vari-
+able and a spectroscopic binary with the secondary being a
+B1V star (North et al. 2007). It is a member of the Upper
+Scorpius subgroup within the Sco OB2 association related
+with reflection nebula in the Ophiuchus cloud complex re-
+gion (Mamajek 2008).
+HD 147933 is a component of the well-studied binary star
+ρOph AB. High-resolution KI, NaI, CaII observations
+show very complex profiles of atomic lines in its spectrum
+(Snow et al. 2008). The star is associated with a strong re-
+flection nebula and a significant amount of dust (and gas)
+is assumed to lie behind it.
+HD 179406 (20 Aql) is a variable star behind a translucent
+cloud (Hanson et al. 1992; Dobashi et. al. 2005). Though
+the absorption and emission lines have at least 3 compo-
+nents, the dominant one is seen in most of the atomic and
+all molecular lines.
+HD 197770 is an evolved spectroscopic eclipsing binary star
+with both components being B2III stars (Clayton 1996). The
+object is close to Cyg OB7 and Cep OB2 and probably lies
+on the edge of a star formation region including molecular
+clouds L1036 and L1049. It is associated with non-stellar
+IRAS sources (see for more details Gordon et al. 1998).
+So, the clouds observed in the selected lines of sight look
+to be rather different and hence their dust grain ensemble
+parameters may also differ.
+3.2 Observational data
+Basic information on the selected stars is presented in Ta-
+ble 1. The distance D, colour excess E(B−V), ratio of
+total extinction to the selective one RVand extinction
+AVfor all stars except HD 24263 and 197770 are from
+Fitzpatrick & Massa (2007), with the data for these twostars being from Valencic, Clayton & Gordon (2004). The
+MK spectral types were taken from the catalogue of stellar
+spectral classifications (Skiff 2009).
+The photometric data in the standard visual – near in-
+frared bands were found in the general catalogue of photo-
+metric data (Mermilliod et al. 1997) and recent papers re-
+ferred in the Simbad database (see Table 2). When the JHK
+data were unavailable, the JHK Svalues from the 2MASS
+catalogue (Cutri et al. 2003) were used. When the RI data
+were absent, we applied with caution the data from the
+USNO B1 catalogue (Monet et al. 2003). To derive the in-
+terstellar extinction in the line of sight the intrinsic col ours
+from Straizys (1992) were utilized. The UV extinction
+parameters were taken from Valencic, Clayton & Gordon
+(2004) and Fitzpatrick & Massa (2007).
+Uncertainties of UV extinction are mainly a result
+of errors in RVandAVvalues. Note that typical rel-
+ative errors of these quantites are 10–15% (see, e.g.,
+Valencic, Clayton & Gordon 2004). An additional source of
+relatively large errors of extinction values in the visual a nd
+near-IR regions is uncertainty of the intrinsic colors (cf. , e.g.,
+Bessell & Brett 1988; Straizys 1992; Ducati et al. 2001). So,
+relative errors of extinction for individual objects are ra ther
+large (see, e.g., Fig. 2).
+Table 2 also contains references to the papers with
+polarimetric data. Additional polarimetric observations of
+HD 147165 and HD 179406 were performed by one of the
+authors (HKD) in 2007 at a 2m telescope of the Girawali
+Observatory (Pune, India). The IUCAA Faint Object Spec-
+trograph and Camera (IFOSC) was used in the polarimet-
+ric mode. The polarization parameters were measured in 10
+bands in the wavelength region about 0.37 – 0.60 µm and a
+standard package was applied for data analysis.
+Note that for two stars (HD 62542 and HD 197770),
+far-UV extinction curves were derived from IUE and FUSE
+observations by Sofia et al. (2005). For two stars (HD 147933
+and HD 197770), the important polarization data in the
+c/circlecopyrt2009 RAS, MNRAS 000, 1–116Das et al.
+Table 1. Target stars.
+Star l b D(pc) Sp RVAV
+HD 24263 182.1 -34.9 171 B5V 3.44 0.72
+HD 62542 255.9 –9.2 396 B5V 2.74 0.99
+HD 99264 296.3 –10.5 266 B2.5V 3.15 0.85
+HD 147165 351.3 +17.0 137 B1III 3.60 1.48
+HD 147933 353.7 +17.7 118 B2.5V 4.41 2.07
+HD 179406 28.2 –8.3 227 B3V 2.88 0.89
+HD 197770 93.9 +9.0 943 B2V 2.77 1.61
+Table 2. Photometric and polarimetric data sources.
+Star PhotometryaPolarimetryb
+HD 24263 V04 (UV); M97 (UBV); M03 (RI); L05 (JHKL) A96 (UV & vis –nearIR)
+HD 62542 S05 (farUV); F07 (UV); M97 (BV); M03 (RI); C03 (JHK S) A96 (UV & vis–nearIR)
+HD 99264 F07 (UV); M97 (UBVRI); C03 (JHK S) A96 (UV & vis); S69 (UBV)
+HD 147165 V04 (UV); W02 (UBVRIJHKLM) this work (vis); C66 (vi s–nearIR)
+HD 147933 F07 (UV); M97 (UBVRI); C03 (JHK S) A96 (UV–nearIR)
+HD 179406 V04 (UV); W02 (UBVJHKL); M03 (RI) this work (vis)
+HD 197770 S05 (farUV); V04 (UV); M97 (UBV); M03 (RI); C03 (JHK S) A96 (UV–nearIR)
+aM97 is for Mermilliod et al. (1997), W02 for Wegner (2002), C0 3 for Cutri et al. (2003), M03 for Monet et al.
+(2003), V04 for Valencic, Clayton & Gordon (2004), L05 for La rson & Whittet (2005), S05 for Sofia et al. (2005),
+F07 for Fitzpatrick & Massa (2007).
+bC66 is for Coyne & Gehrels (1966), S69 for Serkowski & Roberts on (1969), A96 for Anderson et al. (1996).
+J, H and K bands are also available (Wilking et al. 1980;
+Wilking, Lebofsky & Rieki 1982).
+3.3 Fitting procedure and parameter variations
+A general problem of dust optics modelling is in obtaining
+the absolute values of the model parameters. As it has been
+noted many times (e.g., Greenberg 1978), similar extinctio n
+occurs when a product of the typical particle size ∝angbracketleftr∝angbracketrightand
+the particle refractive index does not change, i.e.
+∝angbracketleftr∝angbracketright|m−1| ≈const.
+The average Galactic extinction curve was successfully
+reproduced using quite different mixtures of particles
+with exponential (Hong & Greenberg 1980), power-law
+(Mathis, Rumpl & Nordsieck 1977) and more complicated
+grain size distributions (Mathis 1996; Weingartner & Drain e
+2001). Effects of variations of the distribution parameters
+(e.g.,rV,min,rV,max,qfor a power-law distribution) on
+extinction are well studied for spheres (see, for example,
+Voshchinnikov & Il’in 1993).
+Joint modelling of interstellar extinction and polariza-
+tion adds at least one parameter — particle shape — to the
+model. In the case of spheroids the particle shape is defined
+by the spheroid’s aspect ratio a/b. Besides this, we can con-
+sider prolate (“cigars”) and oblate (“pancakes”) spheroid s.
+If such particles are imperfectly oriented, the alignment p a-
+rameters (in our case, δ0and Ω) very weakly affect extinc-
+tion. Note that the normalized extinction Aλ/AVessentially
+changes in the UV region when a/bvaries (see Fig. 3 in
+VD08).In order to analyze importance of the model param-
+eters we reproduced the observational data available for
+HD 147933 with different particles. Some results are shown
+in Figs. 2 and 3 (see also Fig. 8 in VD08), the parameters
+of the fits are given in Table 3. Note that in VD08 the fit-
+ting was simply “chi-by-eye” while in this paper we apply
+minimization of χ2.
+The fitting approach used included two steps: i) finding
+of the size distribution parameters and the relative column
+densities Kifrom approximation of the extinction data using
+particles of given shape, and ii) a more accurate determina-
+tion of the alignment parameters δ0and Ω from fitting of the
+wavelength dependence of polarizing efficiency P(λ)/A(λ).
+It is interesting that by varying the parameters rV,min,
+rV,maxandqand the relative density of carbon ( KC) and sil-
+icate (KSi) grains one can reproduce the observational data
+with prolate as well as oblate particles of different aspect r a-
+tios. Note that after fixing the particle type (prolate/obla te)
+anda/bone can estimate only the values of KC,KSi,rV,min
+andqas the extinction and polarization curves is little af-
+fected by variation of rV,max.
+The alignment degree for three models presented in Ta-
+ble 3 is found to be slightly larger than the standard inter-
+stellar value ( δ0≈0.2µm, Voshchinnikov 1989). This can
+be a result of enhanced magnetic fields in the interstellar
+cloud near ρOph with which HD 147933 is related. The
+most exciting conclusion of our modelling is a close coinci-
+dence of the magnetic field directions — the values of Ω for
+various fits differ less than 10◦. We suggest that this model
+parameter can be more or less reliably determined from such
+modelling.
+Possibly, a better coincidence of the theory with the ob-
+c/circlecopyrt2009 RAS, MNRAS 000, 1–11Interstellar extinction and polarization 7
+Table 3. Modelling results for HD 147933.
+amorphous carbon astrosil graphite shape; χ2
+KCrV,minarV,maxqKSirV,minrV,maxqKgra a/b δ 0(µm) Ω (deg) A/AVP/A
+0.24 0.03 0.15 1.5 0.73 0.08 0.25 1.5 0.03 pro; 5 0.5 35 1.04 24. 6
+0.54 0.10 0.20 2.0 0.40 0.01 0.25 1.3 0.06 pro; 3 0.3 42 0.19 8.2
+0.50 0.10 0.25 2.0 0.47 0.07 0.20 2.0 0.03 obl; 3 0.3 33 0.19 2.2
+arV,minandrV,maxare inµm
+0 2 4 6 800.511.522.5
+HD147933 
+0 2 4 6 80123
+Figure 3. Comparison of the normalized extinction (left panel) and th e polarizing efficiency (right panel) observed for HD 147933 w ith
+results of modelling. The solid curves show the case of prola te (a/b= 3) spheroids having a power-law size distribution. Other m odel
+parameters are given in Table 3.
+servations could be obtained if we consider different align-
+ment parameters for silicate and the carbonaceous grains.
+Certainly, consideration of non-aligned carbonaceous gra ins
+cannot be ruled out ( Li & Greenberg 2002; Chiar et al.
+2005; see, however, Fig. 2).
+3.4 Fitting results and discussion
+The normalized extinction A(λ)/AVand the polarizing effi-
+ciencyP(λ)/A(λ) were calculated with the theory of Sect. 2
+and compared with the observational data available for the
+stars mentioned in Sect. 3.1. The results of this compari-
+son are plotted in Figs. 3–4 (see also Fig. 9 in VD08 for
+HD 62542) with the parameters of the fits being presented
+in Table 4. We start with comments on the individual ob-
+jects and then make general conclusions.
+HD 62542 . The UV extinction curve has a very broad and
+weak 2175 ˚A bump and a steep far-UV rise (Snow & McCall
+2006). Our model suggests that the major part of grains are
+small carbon particles with a very narrow size distribution .
+HD 99264 . The extinction curve for it is characterized by
+RVof about 3 (Fitzpatrick & Massa 2007), the UV extinc-
+tion curve is a bit “shallow” but not anomalous (Carnochan
+1986). The polarization in an UV region and the UBV bands
+is well described by Serkowski curve with λmax= 0.55µm
+(Anderson et al. 1996; Martin, Clayton & Wolff 1999). So,we have the case close to the average Galactic curves, but
+certainly there are not enough polarization data for sound
+comparison.
+HD 147165 andHD 179406 . Extinction in these lines of
+sight along with element abundances were interpreted by
+Zubko, Kre/suppress lowski & Wegner (1996, 1998) using rather wide
+size distributions of grains. These authors applied spheri cal
+particles of different materials as well as inhomogeneous pa r-
+ticles, which makes impossible a direct comparison of their
+results with those obtained in our work.
+HD 197770 . For this star the polarizing efficiency is max-
+imal: it reaches P/A = 3.7 at the J band ( λ= 1.25µm).
+The polarization is enhanced in the region of the UV bump
+(super-Serkowski behaviour) but strong UV bump in extinc-
+tion results in no structure for P/A curve (see insert in
+Fig. 4). Our model shows the largest fraction of graphite in
+the grain mixture for this star.
+Thus, we have fitted the observational data using pro-
+late or oblate spheroids with a/b= 2 or 3 and intermedi-
+ate values of the alignment parameter δ0= 0.3−0.5µm.
+Contributions of carbonaceous and silicate grains to extin c-
+tion and polarization are nearly equal excluding the case of
+HD 62542 for which extinction is very peculiar. A contribu-
+tion of small graphite particles to extinction usually is le ss
+than 10%. The parameters of the size distributions for sili-
+cate and amorphous carbon grains differ from the standard
+c/circlecopyrt2009 RAS, MNRAS 000, 1–118Das et al.
+0 2 4 6 801234HD24263
+0 2 4 6 800.511.522.5
+0 2 4 6 801234
+HD99264
+0 2 4 6 801234
+0 2 4 6 80123
+HD147165 
+0 2 4 6 80123
+Figure 4. Comparison of observed and modelled normalized extinction (left panel) and polarizing efficiency (right panel) for five s tars.
+The solid curves show results of calculations. The model par ameters are given in Table 4.
+c/circlecopyrt2009 RAS, MNRAS 000, 1–11Interstellar extinction and polarization 9
+Table 4. Modelling results.
+amorphous carbon astrosil graphite shape;
+Star KCrV,minarV,maxqKSirV,minrV,maxqKgra a/b δ 0(µm) Ω (deg)
+HD 24263 0.48 0.08 0.40 3.5 0.47 0.04 0.15 2.1 0.05 pro; 2 0.3 60
+HD 62542 0.85 0.07 0.10 2.7 0.10 0.005 0.25 2.7 0.05 pro; 2 0.5 4 0
+HD 99264 0.44 0.09 0.30 6.0 0.50 0.04 0.18 2.0 0.06 pro; 3 0.5 50
+HD 147165 0.49 0.06 0.30 2.2 0.44 0.04 0.20 2.2 0.07 pro; 2 0.5 5 4
+HD 147933 0.50 0.10 0.25 2.0 0.47 0.07 0.20 2.0 0.03 obl; 3 0.3 3 3
+HD 179406 0.45 0.10 0.40 6.0 0.47 0.04 0.18 2.5 0.08 pro; 3 0.5 4 9
+HD 197770 0.43 0.05 0.20 2.8 0.46 0.01 0.17 1.3 0.11 pro; 3 0.5 4 2
+arV,minandrV,maxare inµm
+MRN mixture (see Sect. 2.2). As usual, very small particles
+withrV,min<∼0.01µm are absent in the grain ensembles. Be-
+sides this, in many cases the slope of the n(rV) dependence
+is less steep than that for the MRN mixture (i.e. q<3.5).
+The inclination of the magnetic field direction relative
+to the line of sight determined for all target stars lies in
+the interval Ω ≈30 – 60◦. These values of Ω are derived
+rather accurately because of a strong dependence of P/A on
+Ω (see Figs. 4 and 6 in VD08). The magnetic fields seem
+to be connected with local dust clouds in the directions to
+the selected stars. At the same time, the obtained values
+of Ω do not contradict the general pattern of the galactic
+magnetic field in the solar neighbourhood as found from
+pulsar measurements (Vall´ ee 2004, 2008).
+It should be noted that one usually assumed that small
+grains were badly oriented as the interstellar polarizatio n de-
+gree decreases with a growing λ−1in the UV region (Whittet
+2003). However, for all the objects we have fitted the ob-
+servational data by using the simple size and orientation
+distributions with small particles being rather well align ed.
+To see that one should consider Eqs. (11)–(13) and keep in
+mind thatδ0is about 0.4 and the temperature ratio about
+0.1. ForrV∼0.06µm, this gives ξ∼0.5 whenξ= 0 cor-
+responds to the perfect alignment while ξ= 1 means the
+random orientation.
+4 CONCLUSIONS
+We have extended the spheroidal model of interstellar dust
+by considering imperfectly dynamically aligned amorphous
+carbon and silicate prolate and oblate spheroids with a
+power-law size distribution and a small fraction of graphit e
+spheres involved to explain the 2175 ˚A bump. Our model be-
+ing as simple as possible was applied to fit the the normalized
+extinctionA(λ)/AVand the polarizing efficiency P(λ)/A(λ)
+observed in the near IR to far UV region.
+In contrast to earlier works we exactly calculated the
+optical properties of spheroids, for the first time utilized
+their imperfect Davis–Greenstein alignment and applied th e
+spheroidal model to fit the data obtained for individual star s
+instead of the commonly used average Galactic curves.
+A comparison of our model with those recently sug-
+gested by other authors has demonstrated that the extinc-
+tion data are well fitted by all the models while only our
+model allows one to satisfactorily fit the polarization datawith simple size and orientation distributions. Contrary t o
+other works we find that the data can be well fitted with
+aligned carbonaceous and small silicate non-spherical gra ins.
+Investigation of our model has shown that the only pa-
+rameter that can be determined more or less accurately from
+observational data fitting is the direction of magnetic field
+relative to the line of sight. As the projection of the mag-
+netic field on the sky plane is given by the positional angle of
+polarization ϑ, multi-wavelength extinction and polarization
+observations can be used for study of the spatial structure
+of interstellar magnetic fields.
+Variations of the interstellar extinction and polariza-
+tion curves for the considered stars can be interpreted by
+differences in the relative fractions of carbon and silicate
+grains and their size distributions in the clouds crossed by
+the lines of sight with the alignment parameter having in-
+termediate values δ0= 0.3−0.5µm (for the particle aspect
+ratioa/b= 2−3). We assume that ambiguous results could
+be ruled out by considering dust phase abundances.
+ACKNOWLEDGMENTS
+One of the authors (HKD) is deeply indebted to the IU-
+CAA’s instrumentation and observatory team for their sup-
+port and cooperation extended during the course of this
+work. The work was partly supported by the grants RFBR
+07-02-00831, NTP 2.1.1/665 and NSh 1318.2008.2.
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\ No newline at end of file
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+ 
+A SECURITY PRICE VOLATILE TRADING CONDITIONING MODEL 
+ 
+Leilei Shi*1, Yiwen Wang2, Ding Chen3, Liyan Han2, Yan Piao, and Chengling Gou4
+ 
+1Complex System Research Group, Department of Modern Physics 
+University of Science and Technology of China 
+2Department of Fina nce, Beijing University of Aeronautics and Astronautics 
+3Harvest Fund Management Co. Ltd. 
+4Department of Physics, Beijing Univer sity of Aeronautic s and Astronautics 
+ 
+ 
+This Draft is on February 8, 2010 
+(Comments welcome) 
+ 
+ 
+Abstract 
+ 
+We develop a theoretical trading conditioning model subject to price volatility and return 
+information in terms of market psychological behavior, based on analytical transaction 
+volume-price probability wave distributions in wh ich we use transaction volume probability to 
+describe price volatility uncertainty and intensity. Applying the model to high frequent data test in China stock market, we have main findings as follows: 1) there is, in general, significant positive correlation between the rate of mean return and that of change in trading conditioning intensity; 2) it lacks significance in spite of positive correlation in two time intervals right before and just after bubble crashes; and 3) it shows, particularly, significant negative correlation in a time interval when SSE Composite Index is rising during bull market. Our model and findings can test both disposition effect and herd behavior simultaneously, and explain excessive trading (volume) and other anomalies in stock market.  
+ 
+Key words: behavioral finance, transaction volume-pri ce probability wave, price volatility, trading conditioning, 
+disposition effect, herd behavior, excessive trading, econophysics 
+JEL Classification: G12, G11, G10, C16 
+ 
+ 
+  
+ 
+*Corresponding author. E-mail address: Shileilei8@yahoo.com.cn  or leilei.shi@hotmail.com , working in an 
+international securities company in China. 
+The author appreciates va luable discussion with Huaiyu Wang, Ya ping Wang, Chunsheng Zhou, Jing Liu, and 
+Famin Liu etc. 
+ 
+ 
+ 
+ 1
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+1. INTRODUCTION 
+ 
+  It has been long that literature in finance focuses much more attention on price and return and 
+less on trading volume, even completely ignoring it. In the past 10 years, however, there is a changing trend that academics have increasing minds on the information contained in trading volume. In neoclassical finance framework, Lo and Wang (2006) explored the link between the dynamic properties of volume, together with price, and the economic fundamentals, and underlined the general point that trading volume and price should be an integral part of any analysis of asset market. In newly emerging behavioral finance, we have associated trading volume with investors’ emotion, belief, and pref erence. Behavioral finan ce is the study of the 
+influence of psychology on the behavior of financial practitioners and the subsequent effect on markets (Sewell, 2008). It helps explain why and how markets might be inefficient. Lee and 
+Swaminathan (2000) showed that past trading volume provides an important link between “momentum” and “value” strategies and these findings help to reconcile intermediate-horizon “underreaction” and long-horizon “overreaction” effects. Benos (1998) and Odean (1998) hypothesized that overconfidence produces excessive trading in stock market. Odean (1999) explained why those actively trade in financial markets to be more overconfident than the general population by three reasons: selection bias, survi vorship bias, and unrealistic belief, and tested 
+overconfident trading hypothesis by investigat ing whether the trading profits of discount 
+brokerage customers are sufficient to cover th eir trading costs. Barber and Odean (2000) 
+documented further evidence that active trading results poor performance and is hazardous to 
+wealth. Such irrational behavior can be explained only by overconfidence. Graham et al. (2009) found that investors who feel competent trade mo re, and thus explained that overconfidence leads 
+to higher trading frequency. Tested the trading volume predictions of formal overconfidence models, Statman et al. (2006) found that share turnover is positively related to lagged returns in both market-wide and individual security for many months. They are interpreted as the evidence of investor overconfidence and the disposition ef fect. Barber et al. (2009) demonstrated that 
+psychological biases that lead investors to systematically buy stocks with strong recent 
+performance, to refrain from selling stocks with a loss, and to be net buyers of stocks with unusually high trading volume, likely contribute to the evidence that the trading of individuals is highly correlated and persistent. Grinblatt and Keloharju (2001) evidenced that past return, reference price effect, tax-loss selling, and the size of the holding period capital gain or loss etc. affect trading. Overco nfident investors and sensation seeking investors trade more frequently 
+(Grinblatt and Keloharju, 2009). Hong and Stein (2007) found that trading volume appears to be an indicator of sentiment. In other words, when prices look to be high relative to fundamental value, disagreement on price is strong, and volu me is abnormally high. Thus, they proposed a 
+disagreement volume model which allows speaking directly to a joint behavior between stock price and trading volume.  
+  Whatever excessive trading (volume) hypothe ses are overconfidence, sensation seeking, and 
+disagreement, they all converge to  a point that trading volume reflects the intensity of investors’ 
+emotion, belief, and preference. 
+  Pavlov (1904), a Russian physiologist and Nobel laureate in physiology or medicine in 1904, proposed conditioned reflex (classical conditioning) when studied physiological reflex in animal, 
+ 
+ 2
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+using saliva volume to measure conditioned reflex intensity in dog. Conditioned reflex is a 
+physiological response to expect that an unconditioned stimulus will follow whenever a conditioned stimulus is present. 
+Thorndike (1913) is a pioneer in operant conditioning study. Skinner (1938) invented a 
+conditioning chamber, called as a Skinner box, to study operant conditioning. It was a finding that a rat settles into a smooth pattern of frequent ba r pressing, after it gets  foods (reinforcement) 
+several times by doing this. Today, psychologists define an operant reinforcement as any event that follows a response and increases its probability. 
+  Like classical conditioning, operant learning is based on information and expectancies. Operant conditioning is a behavioral response with expectancy that if a certain operation is made, it will be followed by a certain consequence (reinforcement) at certain times. In a viewpoint, operant conditioning is that a discriminative stimulus se ts the occasion for operant behavior, which is 
+followed by a consequence (Pierce and Cheney, 20 04). Or it occurs in the presence of certain 
+stimuli and is always followed by certain conseq uences (Irons and Buskist, 2008). Skinner called 
+this relationship a three-term contingency (Dragoi, 1997). In operant conditioning, reinforcement is used to alter the frequency of responses. It produces very high operant response rate and 
+tremendous resistance to extinction that reinfo rcement follows the uncertain number of operant 
+times (variable ratio) and time interval (variable interval) (Coon, 2007). 
+  Intra-cranial stimulation (ICS) that involves dire ct stimulation of “pleasure centers” in brain is 
+one of the most unusual and powerful reinforcement (Olds and Fobes, 1981). Some rats press a bar thousands of times per hour to obtain stimulation in experiment, ignoring food, water, and sex in favor of the bar pressing. 
+  Coon (2007) classified operant reinforcement into three categories: primary reinforcement, secondary reinforcement, and feedback. Primary rein forcement is natural, or unlearned. They are 
+usually rooted in biology and produce comfort, end discomfort, or fill an immediate physical need. Money, praise, approval, affection, and simila r rewards, all serve as learned or secondary 
+reinforcement. Secondary reinfo rcement that can be exchanged for primary reinforcement gain 
+their value more directly. Printed money obviously has little or no value of its own, neither eaten nor drunk. However, it can be exchanged for foods and services, and perhaps is the most important source of economic reinforcement or conditioned reinforcement (Pierce an d Cheney, 2004). For 
+example, chimpanzees were taught to work for toke ns in research (Cowles,  1937). Feedback is a 
+process that an operator makes an (input) ad justment on his behavior after receiving the 
+consequence of his response (output). 
+  Soros (1995) studied reflex theory in a perspective of philosophy at beginning, and gradually found that it was correlated with stock price behavior. In his opinion, it is the best platform to study reflex in stock market. He has descriptive explanation for a feedback process, the third reinforcement in Coons’ classifications, that investors participate in trading, cause price volatility, make a judgment from price information, and then decide trading again to further influence price. 
+  Studied joint behavior for security transaction volume distribution over a trading price range on every trading day, Shi (2006) derived a time-independent transaction volume-price probability wave equation and got two sets of analytical transaction volume distribution eigenfunctions over a trading price range in which transaction volume pr obability describes price volatility uncertainty 
+and intensity using normative econophysics methodology. Based on this, we develop a theoretical trading conditioning model subject to price volatility and return information in terms of market 
+ 
+ 3
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+psychological behavior, in which we use transaction volume probability to describe trading 
+conditioning intensity. By studying correlation between the rate of return and that of change in trading conditioning intensity, we test both disposition effect and herd behavior simultaneously, trace back to analy ze investors’ psychological behavior wi th price volatility when they make 
+trading decision, and attempt to provide a quantitative analysis and test model for behavioral finance. 
+  Although there are many literatures on the relation between return and volume, we have not yet found study on the relation between the rate of return and that of change in transaction volume (probability), and a successful result that incorporates normative analytical transaction volume-price probability wave behavior in econophysics with descriptive cognitive reflex and conditioning behavior in physiology and psychology—it is difficulty for most of non-Expected Utility models in trying to achieve both normative and descriptive goals that they end up doing an unsatisfactory job at both (Barberis and Thaler, 2003). Unlike transaction volume, the rate of change in transaction volume (probability) could be positive, negative, or zero. Instead of testing disposition effect and herd behavior separately using event study and psychological experiment previously, moreover, we use high frequency data to test them simultaneously, and explain excessive trading and other anomalies in a unified theory. 
+  The remainder of the paper is organized as follows. Section 2 will briefly introduce analytical transaction volume-price probability wave distributions and find a way to measure trading conditioning intensity subject to price volatility and return information; in section 3, we first study stationary equilibrium, determine the equilibrium price, and demonstrate the early finding that stationary equilibrium is prevailing in stock ma rket (Shi, 2006) using Huaxia SSE 50ETF every 
+trading high frequent data. Then, we empirically test correlations among the rate of mean return, the rate of change in trading conditioning intensity, and that of change in the amount of transaction in any two consecutive trading days. Section 4 is analyses and discussions on empirical results. They are: 1) Stationary equilibr ium exists widely as a consequence of interaction and coherence in 
+stock market; 2) General speaking, there are significant both disposition effect and herd behavior; 3) Cognitive and trading behavior changes with price volatility and environment; 4) Excessive trading explanation in trading conditioning theory; and 5) Potential application. Final are summaries and main conclusions. 
+ 
+2. TRANSACTION VOLUME DISTRIBITION AND 
+TRADING CONDITIONING INTENSITY 
+ 
+2.1 Transaction V olume-price Probability Wave Distribution 
+ 
+  Shi (2006) documented that stationary equilibrium exists widely in stock market by studying 
+transaction volume-price behavior, using econophysics method. Transaction volume v is 
+accumulative trading volume at a price in a given ti me interval. It shows a kurtosis near the price 
+mean value over a trading price range on every trading day regardless of stock price, price 
+volatility path and direction1. A stationary equilibrium price is the price to which transaction 
+ 
+ 4                                                        
+1 Shi (2006) fitted and tested 618 volume distributions over a trading price range. Although there are different in 
+price, price volatility path and direction, they show the same characteristic that accumulative trading volume exhibits kurtosis near the price mean value. 
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+volume kurtosis is corresponding. In addition, while trading price is volatile upward and 
+downward to a stationary equilibrium price constantly, the equilibrium price jumps from time to time
+1. 
+  There are many factors that could influenc e stock price and its volatility, for example, 
+management in listing company, macro-economy, news announcement, and psychological behavior etc. These factors impact on price change  more and less if and only if there is trading. 
+Shi (2006) did not consider them temporarily in his previous study. 
+  There is no price volatility at all if there is no trading in stock market. But if there is trading, we 
+can not draw a conclusion that there is price ch ange. Trading is a necessary condition for price 
+volatility. In addition, the amount of transaction c onstrains price volatility and transaction volume. 
+Studied how the amount of transaction constrains price volatility and transaction volume, and 
+how the volume distributes over a trading price range in stationary equilibrium in stock market, Shi (2006) proposed a relationship hypothesis among the constraint of transaction amount, the 
+transaction volume distribution of the amount, and the price volatility of the amount (price reversal to a stationary equilibrium price), i.e. transaction amount constraint hypothesis: 
+() 02
+= + + −p WVvp Et,           ( 1 )  
+where p is price, its dimension is [currency unit][share]-1 ; t v vt /=  is transaction momentum 
+and transaction volume in a given time interval , its dimension is [share][time]t -1;  
+is the amount of transaction at a correspondent price 2/t v p E⋅ =
+p in a given time interval , its dimension 
+is[currency][time]t
+ -2;  is impulse and momentum force in a direction deviating from a 
+stationary equilibrium pric e at a corresponding price 2/t v vtt=
+p in a given time interval , its dimension 
+is[share][time]t
+ -2,  is the volume probability of transaction amount at a price V v pt/2⋅ p in a 
+given time interval , its dimension is [currency unit][time]t -2; In addition, 
+()()()p p A p p A p W− ≈ − =0          ( 2 )  
+is the price volatility and linear reversal of transaction amount to a stationary equilibrium price at 
+a price p in a given time interval , its dimension is [currency unit][time]t -2, too;  is a 
+stationary equilibrium price and 0p
+p is price mean value; ()ttv V v A/ 1−=  is the magnitude of 
+a supply-demand restoring force or stationary equilibrium restoring force , its dimension is 
+[share][time] -2. 
+According to stock trading regulation, transaction priority is given by “p rice first and time first”. 
+If current trading price is , then next trading priority is given to minimize price volatility with cp
+                                                        
+ 
+ 51 Unlike Poisson-diffusion jump (Ait-Sahalia, 2004) and Levy jump (Li et al., 2008), stationary equilibrium price 
+jump has its clear mechanism. For ex ample, if there is sudden increasi ng buying volume to break stationary 
+equilibrium, trading price will be volatile upward and downw ard from a stationary equilibrium price to another by 
+its jump. We can measur e its jump behavior in terms of volume dist ribution over a trading price range (Shi, 2006).  
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+respect to it. It states that actual trading price path is chosen by transaction amount constraint 
+hypothesis, equation (1), functional 
+() ( )⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ − =dpd
+dpdpVBE W p Fψ ψψ ψ ψ** ,2
+     ( 3 )  
+to minimize its wave function ()pψ  with respect to price variati ons. Its mathematical expression 
+is 
+()∫=0 , dp p Fψ δ .            ( 4 )  
+  From equation (1) and (4), we have transaction volume-price probability wave equation 
+()[] 022 2
+= − +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ψψ ψp W Edpd
+dpdpVB.        ( 5 )  
+Substituting equation (2) into equation (5) with its natural boundary conditions 
+() 0 0=ψ , ()∞ <0pψ , and ()0→∞+ψ , 
+we get two sets of analytical transaction volum e distribution eigenfunctions over a trading price 
+range. One is that when there is coherence (the sum of momentum force  and restoring force 
+ is equal to an eigenvalue constant over a trading price range), we have ttv
+A
+() ( )[]0 0 p p J C pm m m − =ω ψ ,   ( )L, 2 , 1 , 0=m    (6) 
+where mω satisfy 
+tt tt m vVvA v= − =2ω ,  ()0>mω  ( )L, 2 , 1 , 0=m    (7) 
+a set of positive eigenvalue constants, ()[ ]0 0 p p Jm−ω  are a set of zero-order Bessel 
+eigenfunctions,  are normalized dimensionless constants; the other is that when restoring 
+force are a set of constants over a trading price range, we get the other set of multi-order 
+eigenfunctions as mC
+mA
+() ( )0 2 , 1 ,0p p A n F e C pmp p A
+m mm− − ⋅ =− −ψ , ( )L, 2 , 1 , 0 ,=m n  (8) 
+where 0 .2 1> =+= constnEAm
+m ,      ( )L, 2 , 1 , 0 ,=m n  (9) 
+( )0 2 , 1 , p p A n Fm− −  are a set of  order confluent hypergeometric eigenfunctions or the 
+first Kummer’s eigenfunctions. n
+The absolute of functions (6) and (8) ()pmψ  is transaction volume density and probability at 
+price p in a given time interval (see figure 1 and figure 2, in which x-coordinate is price, 
+ 
+ 6
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+y-coordinate is transaction volume distribution, and origin is a stationary equilibrium price),  
+is normalized constant. mC
+ 
+ 
+ 7Figure 1: Transaction te of function (6) 
+      
+    (b) The first-order 
+    
+    (d) the tenth-order 
+Figure 2: Transa nction (8) 
+ 
+ There are several different characteristics be tween figure 1 and figure 2. First, the volume 
+2.2 Eigenvalue and Probability Wave 
+  
+ volume distribution over a trading price range in the absolu
+ 
+ 
+(a) Zero-order     
+ 
+ 
+(c) The second-order 
+ction volume distribution over a trading price range in the absolute of fu
+ 
+distribution in figure 1 is an attenuated wave with price departing from stationary equilibrium 
+price, while it is a uniform wave with price exce pt for zero-order distribution which is exponent in 
+figure 2. The higher the order is in the volume distribution function, the better uniformly the volume distributes over a price range in figure 2. Second, it is an open distribution in figure 1, which can describe a pulse trading behavior far away from a stationary equilibrium price, for example, profit and loss transfer tr ading, whereas it is a closed distribution in figure 2, which can 
+describe random and uniform behavior. Third, the magnitude of eigenvalue in figure 2 is about two orders of magnitude larger than that in figure 1 except for exponent distribution. However, both distributions can fit exponent distribution very well in common.  -600 -400 -200 200 400 6000.10.20.30.40.5
+-40 -20 20 400.20.40.60.81-15 -10 -5 5 10 150.20.40.60.81
+-4 -2 2 40.20.40.60.811
+0.8
+0.6
+0.4
+-20 -10 100.2
+20
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+8 An eigenvalue is different from a parameter in a distribution function when we use it to describe 
+. The 
+) Classical wave       (b) Probability wave 
+          uncertainty in events. In a great majority of scenarios, we usually us e a distribution function with a 
+parameter to analyze statistic and uncertain behavior in events if we know their distribution in a 
+certain extent, but do not understand its mechanism.  In this approximate analysis, we do not clear 
+what exact information is contained in the parameter. An eigenvalue is observable and measurable. Thus, it is possible for us to test distribution eigenfunction validity quantitatively and understand its mechanism. In transaction volume-price probability  wave distribution, for example, two sets of 
+eigenvalues represent two observables. One is calculated by equation (7). It is the consequence of 
+a kind of coherence that the sum of momentum force 
+v and supply-demand restoring force is 
+equal to this eigenvalue constant over a trading price range other is calculated by equation (9). 
+It is the consequence that supply-demand restoring and reversal force is equal to the eigenvalue constant over a trading price range. It can describe exponent distribution and uniform random distribution. 
+ tt
+(a
+ 
+  
+() ) cos(0 t y t yω=         () ()[]0 0 p p J C pm m m − =ω ψ  
+Figure 3: Classical wave  and probability  wave 
+ 
+robability wave is a kind of wave in which we use volume distribution and probability rather 
+thP
+an the amplitude of wave to describe wave intensity. For example, we use transaction volume 
+probability rather than the amplitude of price volatility to describe price volatility intensity in a 
+transaction volume-price probability wave. Now, let us contrast and compare probability wave with classical wave. First, x-coordinate is price (in stock market) or distance (in quantum mechanics) and y-coordinate is accumulative volume probability in a probability wave, while x-coordinate is time and y-coordinate is amplitude in a classical wave. Second, we use volume distribution and probability to describe wave intensity, whereas we use the amplitude of wave to describe wave intensity. Third, intensity is equal to and larger than zero in probability wave, while it can be negative in classical wave; Forth, intensity displays wave change with an independent variable in a probability wave. Large amplitude is not equal to strong intensity in a probability wave. In a classical wave, the larger the amplitude is, the stronger the intensity does be in wave. Fifth, probability wave is the consequence of coherence in group. It can not exist independently. Classical wave can exist independently. Sixth, stri ctly speaking, we have not yet found that there 
+is a periodicity and time cycle in a probability wave. It is that there is uncertainty in time prediction. In classical wave, we can measure spee d in wave. Thus, there is periodicity and time 
+cycle. However, there are common in both. There is coherence in both probability wave and classical wave. And there is repeated change reference to an equilibrium point (see figure 3).  1
+19.6 19.8 20.2 20.40.20.40.60.8
+0.5
+0.25
+2 4 6 8 10 12 14
+-0.25
+-0.5
+-0.75
+-10.751
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+ 
+ 92.3 Price V olatile Trading Conditioning and Its Intensity 
+ 
+ Let us first consider a rigid trading system in which there is no any subjective cognition change 
+ of transaction to  buy, to break stationary equilibrium, and to 
+that could influence investors’ 
+ people have physiological demand for clothes, foods, and services in life. 
+Th
+x (classical conditioning) that traders produce the same 
+ph
+n in price volatile trading conditioning. It has several  
+in buying and selling behavior. The transaction volume is the same at any the same time interval 
+even if trading price is volatile.    If there is incremental amount
+cause stationary equilibrium price jump, then , there is one-to-one correspondence between 
+incremental amount of transaction and the equi librium price jump because total transaction 
+volume is the same in two time intervals before and after jump.    In a real financial market, there are a grea t many factors 
+subjective cognition, supply-demand balance, and tr ansaction volume. Theref ore, it is a key to 
+find a certain relation among incremental amount of transaction, transaction volume change, and stationary equilibrium price jump how we combine market subjective trading behavior and transaction volume change into transaction volume-price probability wave distribution function. It is also a key to develop a theoretical trading conditioning model subject to price volatility and return information.  
+It is objective that
+is is physiological response and unconditioned reflex. Printed money obviously has little or no 
+value of its own, neither eaten nor drunk. In commodity exchange economy, however, when we associate money, income, and return with the necessities of life and services tightly through exchange, we have been conditioned by them (conditioned reinforcement) and will produce the same physiological response to them, i.e. conditio ned reflex. Pierce and Cheney (2004) wrote that 
+money can be exchanged for foods and services, and perhaps is the most important source of economic and conditioned reinforcement. 
+In stock market, it is a conditioned refle
+ysiological response when they are stimulated by price volatility and return information as they 
+do for the necessities of life and services, and it is  a trading conditioning that practitioners trade 
+and expect return in the future after they analyze, judge and have decision making in the presence of price volatility and return information.   The reinforcement is money and retur
+characteristics as follows: First, it does no t loss reinforcement value as quickly as primary 
+reinforcement does because physiological demand is satisfied. Second, profit is a positive reinforcement whereas loss is a negative reinforcem ent. People trade and buy if they expect price 
+rising. On the other hand, they trade and sell if they expect price dropping. There are tradings no matter whether price rises or drops in expectancy. Third, because we have not yet found time cycle in volume-price probability wave, return occurs after stock holding in an uncertain time interval. This makes tremendous resistance to extinction in trading conditioning, similar to variable interval (VI) schedules in operant conditioning experiment (Coon, 2007). Forth, the trading consequence (profit and loss) that produces feedback to traders could influence their emotion and judgment, 
+and enhance or change their expectancy in return to make trading. For examples, loss stop selling 
+investors are likely to buy stock again because they  change their expectancy in return, stimulated 
+by price volatility and return. Just buying traders may sell their stock immediately because they find market movement in an opposite direction and change their previous expectancy. Therefore, 
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+ 10ers’ three-term contingency and operant conditioning (Irons and Buskist, 2008), 
+w
+ 
+Figure 4: Three-term continge ncy in trading conditioning 
+ 
+ There are two independent variables, price and transaction volume, in stock market. The 
+t 
+g conditionin g behavior, security price volatility and return 
+robability wave distribution function, we use transaction volume cognitive response in price volatile trading conditio ning includes both expe ctancy and expectancy 
+change in return. 
+Based on Skinn
+e define price volatile trading conditioning in stock market as participants trade (operant) and 
+expect return (profit a nd loss) in the future (consequence) after they analyze, judge, and have 
+decision making (cognition) in the presence of price volatility and return information 
+(discriminative stimuli). The trading consequence (profit and loss) that produces feedback to 
+influences their emotion and judgment, adjust their expectancy in return, and make trading (see figure 4).  
+Discriminative 
+Stimulus Operant Reinforcement 
+(Positive or Negative) 
+Price V olatility and 
+Return Information Expectancy in Return 
+(Profit or Loss) Trading 
+Consequence Feedback 
+ 
+amount of transaction is a constrain condition on them. According to classical mechanics (Greenwood, 1977), the degree of freedom is equal to that the number of independent variables 
+minus the number of constrain conditions. The degree of freedom is one in this trading system.    Osborne (1977), a pioneer in econophysics, found that price as a function of volume does no
+exist empirically, and then explained why volume is th e function of price, and this is not invertible. 
+McCauley (2000) proved that price as a function of volume does not exist mathematically. Therefore, Shi (2006) chose price as an inde pendent variable and transaction volume as a 
+dependent variable in stock market.    According to afore-developed tradin
+information stimulate practitioners to trade with  expectancy on its return in the future. The 
+subjective trading behavior (cognition) and accumulative trading volume (transaction volume) change synchronously. We use the change in transaction volume distribution (probability) over a trading price range to present the change in subjective trading behavior (cognition) subject to price 
+volatility and return information (We use transaction volume rather than the amount of transaction because the later is constraint).   In transaction volume-price p
+probability instead of the amplitude of price volatility to describe price volatility intensity. The 
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+larger the transaction volume probability is at a pr ice, the higher the trading intensity is (Unlike a 
+classic wave in that the larger the amplitude is, the stronger the intensity does be). The higher the 
+trading intensity is, the higher the trading frequency1. Therefore, expectancy  on return and trading 
+conditioning intensity are higher (The higher the transaction volume is, the higher volume traders buy. There are more expectancy on  price rising. On the other side, the higher the transaction 
+volume is, the higher volume traders sell. There ar e more expectancy on price dropping), and vice 
+versa. Price volatile intensity is approximately equal to trading conditioning intensity subject to price volatility and return information. Theref ore, we can use transaction volume probability 
+(distribution) to describe trading conditioning intensity and frequency (reference to figure 5).   
+Price V olatility Intensity 
+ 
+ 11 
+Figure 5: Price volatilit y intensity is approxim ately equal to trading conditioning intensity 
+ 
+  Now, we can use transaction volume probability to describe trading conditioning intensity. In 
+the same way, when we use stationary equilibriu m jump to represent price volatility mean return 
+in two consecutive trading days, we can use the rate of change in total transaction volume to 
+describe the rate of change in trading conditioning intensity subject to the rate of mean return and 
+its information in the two days. Obviously, the rate of change in trading conditioning intensity 
+could be positive, negative, and zero. In this way, we can study correlation between the rate of 
+mean return and that of change in trading conditioning intensity and trace back to analyze 
+investors’ physiological response and psychological behavior when they trade in decision making, 
+using high frequency data. 
+                                                        
+1According to the law of large number in probability and st atistics, if total transaction volume is much greater than 
+the volume of every trading, then, transaction volume probability is approximately equal to trading frequency. In 
+our study, total daily trading volume is about 360,000,000 shares in average. We can use transaction volume 
+probability to represent trading frequency . The higher the transaction volume is  at a price, the higher trading 
+frequency. Although there is difference in  information contained in a single larg e trading volume equivalent to total 
+volume of several small trades, transaction volume probability is still approximately equal to trading frequency. 
+The abnormal volume distribution disturbe d by large volume trading reveals that  stationary equili brium is easily 
+broken by capital advantage speculators. We will study it in the future. Trading Frequency 
+Trading Conditioning Frequency Price V olatility Frequency
+Trading Conditioning Intensity 
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+123. EMPIRCAL TESTS 
+ 
+In this section, we will fit and test accumulative ing volume distribution over a trading price  trad
+range by using the absolute of zero-order Besse l eigenfunction regression model. From this, we 
+can get a stationary equilibrium price on every trading day and calculate its jump and mean return 
+in any two consecutive trading days. For those that show the significance fitted by the model, we 
+can determine a stationary equilibrium from fitting figures directly. Otherwise, we substitute price 
+mean value for it. In this way, we can further study the correlations among the rate of mean return, 
+the rate of change in trading conditioning intensity, and that of change in the amount of 
+transaction. 
+ 
+3.1 Data 
+ 
+We use every trading high frequent data in Huaxia SSE 50ETF from April 2, 2007 to April 10, 
+2009, in which there are nearly 740 days and 495 trading days in total, i.e., the total number of 
+volume distributions over a trading price range is 495. The data is from HF2 database, provided by 
+Harvest Fund Management Co., Ltd.
+We process the data in two steps . First, we reserved two places of decimals in price by 
+rounding-off method and added volume at a corresponding price ( original data reserves three 
+places of decimals .). Second, transaction volume at a price is divided by total transaction volume 
+over a trading price range. Thus, we acquire transa ction volume probability (distribution) over a 
+trading price range in every trading day.   
+3.2 Distribution Fitting, Significance Test, and Stationary Equilibrium Price 
+ 
+In a st ationary equilibrium state, theoretical tr ansaction volume-price distribution function is  
+()[] ()
+ 
+ 0 0 m m m p p J C p− =ω ψ ,    ( )L2 , 1 , 0=m   （10） 
+mC, mω where and 0p are a normalized constant, an eigenvalue constant and a stationary 
+equilibri  p , respe vely, and determined by a nonlinear regression model as follow: 
+  ()um rice cti
+()[]i i m m i m 0 0 p p J C pε ω ψ + − = ,  ( )n i , 3 , 2 , 1L=   （11） 
+where  is the number of prices over a tr ading price range in a trading day; n  iε is random error 
+() ()[ ]0 0 p p J Ci m m−ω2, 0σN ; ()i mpψ  and subject to  are observed and theoretical 
+transaction v e over a nge, respectively. We fit the olume probability at a pric trading price ra
+volume distribution by Levenberg-Marquardt nonlinear least square method , get the numerical 
+values of mC, mω 0 and p, and find theoretical distribution. Origin 6.0 Professional software is 
+friendly use n ng (se igure 6 (a)). d i  fitti e f
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+We use F statistic to test significance. The coefficient of determination 2R is as follow: 
+  TSSRSS TSS
+TSSESSR−= =2,           （12） 
+where 
+ 
+ 13()2
+1ˆ∑
+=− =n
+iiY Y ESS ()2
+1ˆ∑
+=− =n
+ii iY Y RSS ()2
+1∑
+=− =n
+iiY Y , , and TSS  are the explained 
+sum of squares, the residual sum of squares, and total sum of squares, respectively. And, 
+  ()1 //
+− −=k n RSSk ESSF            （13） 
+where  nand k is sample size and the number of explanatory variables, respectively. If 
+05 . 0F F>  or  
+  ()105 . 005 . 0 2 2
+− − + ⋅⋅= >k n F kF kR Rcrit ,         （14） 
+the regression model (11) holds true at 95% significant level.
+Our test results show that 380 out of 495 distributions show significance from April 2, 2007 to 
+April 10, 2009 (76.77%). The remainders (23.23%) lack significance.
+There are two notable  characteristics among significance lacking distributions. First, the number 
+of trading prices is few or the sample size is too small in price. It is partly credited by previous 
+data process, in which we reserved two places of de cimals in price by rounding-off three places in 
+original data. The volume distribution characteristics can not be displayed.    
+To solve the problem, we add 0.005 in three places of decimals in price and subdivided volume at 
+corresponding prices. As a result, 28 volume distributions show significance, modeled by absolute 
+zero-order Bessel function. Thus, there are tota l 408 (380+28) volume distributions, around 
+82.42%, that show significance. Their stationary equilibrium prices can be determined by fitting 
+results directly.  
+Second, the remaining volume distributions (87) show at least two of kurtosis over a trading price 
+range. It is explained when abrupt change in supply and demand quantity, for example, continuous 
+large buying volume, breaks up original stationary equilibrium, price volatility is going to be 
+adjusted to a new equilibrium price. The stati onary equilibrium price appears a step and jump 
+change. After this, trading price is volatile around  the new stationary equilibrium price. In this 
+case, the volume distribution function is the linear superposition of function (10), that is, 
+() () [ ] ∑ − =n n n m m m p p J C p0 , 0ω ψ ,      ()L2 , 1=n  （15） 
+where n is the number of stationary equilibrium prices. We fit them with a two stationary 
+equilibrium price regression model as follow: 
+[   () ()]   ()L2 , 1=i  （16） i n n i n m m i i m p p J C p ε ω ψ + − =∑∑=2 , 1 , 0 , 0
+where . 2=n
+We test significance ( , here 2
+22
+2 critR R> 2=k ). Of 87 distributions, 59 distributions (11.92% 
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+in total) show significance at 95% level (see figure 6 (b)). 
+ 
+ 
+ 142.82 2.84 2.86 2.88 2.90 2.92 2.94 2.96 2.98 3.000.000.020.040.060.080.100.120.140.160.180.20Data: Data1_B
+Model: probawave   
+Chi^2 =  0.00083
+R^2 =  0.69725
+  
+P1 0.14967 ±0.01613
+P2 79.97463 ±2.96019
+P3 2.93479 ±0.00227volume probability
+price(yuan)510050
+2007-6-1
+(a) A typical sample is fitted by function (10)  
+（ ） 50 . 0 697 . 02 2= > =critR R2.28 2.29 2.30 2.31 2.320.000.020.040.060.080.100.120.140.160.18Data: Data1_B
+Model: probwave2.1 
+  
+Chi^2 =  0.00086R^2 =  0.75699
+  
+P1 0.13664 ±0.01516P2 263.84993 ±17.65125P3 2.29316 ±0.00202
+P4 2.3124 ±0.00261volume probability
+price(yuan)510050
+2008-7-23
+(b) A typical sample is fitted by function (16) 
+（ ） 575 . 0 757 . 02
+22
+2 = > =critR R
+2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.210.000.020.040.060.080.100.120.14Data: Data1_B
+Model: kummer1 
+  
+Chi^2 =  0.00093
+R^2 =  0.53632
+  
+P1 0.14756 ±0.01969
+P2 4223.7671 ±1236.31859
+P3 2.17427 ±0.00215volume probability
+price(yuan)3.42 3.44 3.46 3.48 3.50 3.52 3.54 3.560.000.010.020.030.040.050.060.07510050
+2008-2-1volume probability
+price(yuan)510050
+2008-7-17
+(c) A typical sample is fitted by function  (d) One of five insignificant samples 
+（ ） 232 . 0 536 . 02 2= > =critR R
+Figure 6: Distributions fitted by transaction volume-price probability wave model and significance test 1
+ 
+For the rest of 28 transaction volume distributions, we fit them by the second set of distribution 
+functions as follow: 
+() ( )0 2 , 1 ,0p p A n F e C pmp p A
+m mm− − ⋅ =− −ψ  .    （17） 
+In convenience, we choose , it is 1=n
+  () ( )0 2 , 1 , 10p p A F e C pmp p A
+m mm− − ⋅ =− −ψ  
+    0 2 10p p A e Cmp p A
+mm− − ⋅ =− − .      （18） 
+The result is that 23 distributions (about 4.65% in  total) show significance at 95% level by this 
+                                                        
+1 In figure, P1, P2, and P3  are a normalized constant, an eigenvalue, and a stationary equilibrium price, 
+respectively.  P4 is a stationary equilibrium price, too. 
+ 
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+regression model (see figure 6 (c)). The last 5 dist ributions lack significance (see figure 6 (d)).  
+In figure 6 shows some typical test results fitted by transaction volume-price distribution 
+functions (10), (16), and (18), respectively. 
+ 
+3.3 Correlation and Significant Test 
+ 
+408 transaction volume distributions (about 82.42% in total samples) show significance, tested by 
+the absolute of zero-order Bessel eigenfunction regression model. We can decide their stationary 
+equilibrium prices from fitting figures directly. For the rest samples, we substitute price mean 
+values for them. In this way, we can figure out price volatility mean return by stationary 
+equilibrium price jump in any two consecutive trading days. Thus, we can study the correlation 
+among the rate of mean return, the rate of change in trading conditioning intensity, and that of 
+change in the amount of transaction. Here, the rate  of change in trading conditioning intensity is 
+approximately equal to that in total transaction volume in any two consecutive trading days.  
+  Given correlation coefficient Y X,r as 
+()
+Y XY XY Xrσ σ, cov
+,= ,                （19） 
+where (
+ 
+ 15Xσ and Yσare the standard deviations of variable X and Y, )Y X, covis covariance. 
+We use t-statistic to test significance. If we have 
+0 :0=ρH  and 0 :1≠ρH , 
+then,  ()()2 / 12− −−=
+n rrtρ,            （20） 
+where r and  anre correlation coefficient and sample size, respectively. For 05 . 0=α , if 
+(22 / 05 . 0− = >n t t tcrit ), then, original hypothesis is rejected. Correlation coefficient is significant 
+not equal to zero at 95% level. 
+  In our test, we use Eviews6.0. We subdivide our data into 5 groups in terms of time, financial 
+crisis in the United States, and bubble crash in China. The first group is dated from April 2, 2007 
+to June 29, 2007, the first half before bubble crash in China. The second group is dated from July 
+2, 2007 to October 31, 2007, the second half before bubble crash in China. The third group is 
+dated from November 1, 2007 to April 40, 2008, the first half after bubble crash in China. The 
+forth group is dated from May 5, 2008 to October 31, 2008, the second half after bubble crash in 
+China. And latest one is dated from November 3, 2008 to April 10, 2009, the reversal time after a 
+year time deep dropping (reference to Table).  
+  There is a major advantage when we subdivide data into 5 time intervals. We can find how 
+market psychological behavior change with time and environment and quire why. 
+  By empiric results, we have main findings as follows: generally speaking, first, there are 
+significant positive correlations between the rate of mean return and that of change in both trading 
+conditioning intensity and the amount of transaction; second, correlation coefficient varies in 5 
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+ 
+Electronic copy available at: http://arxiv.org/abs/1001.0656   16subdivided periods. For examples, (a) they lack significance in spite of positive correlations in two 
+time intervals right before and just after bubble crashes; (b) it shows positive significance in the second half after bubble crash; (c) the positive correlation coefficient is 0.4766, the highest during reversal period after a year time deep dropping; (d ) particularly, there exists significant negative 
+correlation between the rate of mean return and that of change in trading conditioning intensity when SEE Composite Index is rising during bull market; third, the correlation coefficient between the rate of mean return and that of change in amount of transaction is always several points (0.03~0.08) higher than that between the rate of mean return and that of change in trading conditioning intensity. We will discuss them in next section.  
+ 17Notes: (1) Correlation 1 is the correlation between the rate of me an return and that of change in trading conditioning intensit y; correlation 2 is the correlation between the rate of mean return 
+and that of change in the amount of transaction; Correlation 3 is  the correlation between the rate of change in trading conditi oning intensity and that in th e amount of transaction; (2) 
+Differences are correlation coefficient 2 minus 1; (3)  is critt ()22 05 . 0Table: Correlation and Significance Test 
+ Time 
+(yr. mo. d.) Number of 
+DistributionsSSE 
+Composite IndexCorrelation 1 Correlation 2 Differences Correlation 3 
+A  
+2007.4.2—2009.4.10 
+  
+494 
+  
+3252.59—2444.230.1391 
+(t=3.115> =1.960) critt0.1970  
+(t=4.458> =1.960) critt 
+0.0579 0.9975 
+(t=316.3> =1.960) critt
+B  
+2007.4.2—2007.6.29 
+  
+59 
+  
+3252.59—3820.70-0.2567  
+(t=2.006> =2.001) critt-0.2120  
+(t=1.638< =2.001) critt 
+0.0447 0.9986  
+(t=142.0> =2.001) critt
+C  
+2007.7.2—2007.10.30 
+  
+83 
+  
+3836.29—5954.770.0729  
+(t=0.6583< =1.990) critt0.1053  
+(t=0.9529< =1.990) critt 
+0.0324 0.9993  
+(t=241.2 > =1.990) critt
+D  
+2007.11.1—2008.4.30 
+  
+122 
+  
+5914.28—3693.110.1026  
+(t=1.130< =1.980) critt0.1714  
+(t=1.906< =1.980) critt 
+0.0688 0.9968  
+(t=137.0> =1.980) critt
+E  
+2008.5.5—2008.10.31 
+  
+123 
+  
+3761.01—1728.790.1963  
+(t=2.202> =1.980) critt0.2706  
+(t=3.091> =1.980) critt 
+0.0743 0.9958  
+(t=119.2> =1.980) critt
+F  
+2008.11.3—2009.4.10 
+  
+107 
+  
+1719.77—2444.230.4766  
+(t=5.556> =1.983) critt0.5203  
+(t=6.243> =1.983) critt 
+0.0437 0.9981  
+(t=166.5 > =1.983) critt
+−n t ; If , then, the correlation coefficient is significantly not equal to zero; On the other hand, we can not 
+reject original hypothesis that the correlation coefficient is equa l to zero; (4) the lack of significance is printed in bold a nd red; (5) SSE Composite Index is measured by closing point. critt t> 
+ 
+4. ANALYSES AND DISCUSSIONS ON EMPIRICAL RESULTS 
+ 
+4.1 Stationary Equilibrium and the Equilibrium Price 
+ 
+Shi (2006) decomposed price volatility behavior in to two parts. First, trading price is volatile 
+upward and downward to a stationary equilibrium price constantly. It is quantitatively described 
+by a transaction volume-price probability wave equation (5), in which a stationary equilibrium 
+price is the price that transaction volume kurtosis corresponds to. Second, a stationary equilibrium 
+price jumps from time to time. It is explained that a stationary equilibrium is easily broken and its 
+restoring force is weak in stock market. If there is abrupt change in supply and demand quantity, 
+for example, by a large buying volume, then trading price is going to adjust to a new stationary 
+equilibrium price that jumps. After this, it is volatile upward and downward to new equilibrium 
+price. The jump is used to represent for mean return.
+We use every trading high frequent data in Huaxia SSE 50ETF from April 2, 2007 to April 10, 
+2009, in which there are nearly 740 days and 495 trading days in total, i.e., the number of volume 
+distribution over a price range is 495. We firstly apply transaction volume-price probability wave 
+distribution regression model, equation (11), to fitting them, and test significance. 408 
+distributions or 82.42% in total show significance. For those distributions that show significance, 
+we get stationary equilibrium prices directly from fitting results. For those that do not show 
+significance, the volume distributions display two and more than two of kurtosis over trading price 
+range in a day. It is that a stationary equilibrium price jumps on trading days. We substitute price 
+mean value for stationary equilibrium price. Thus, we can get price volatility mean return in any 
+two consecutive trading days by stationary equilibrium price jump. 
+Our empirical results further demonstrate the early finding that there is stationary equilibrium and 
+a stationary equilibrium price jumps from time to time in stock market (Shi, 2006). 
+The ratio of the number of abnormal distributions to total number of distributions is 23.23% from 
+2007 to 2009 in this paper and 5.66% in 2003 (Shi, 2006). By comparison, we can draw a 
+conclusion that it is much more stable and less volatile in Shanghai stock market in 2003. It is fact 
+that there was 26.17% maximum volatility in 2003, whereas there was 140.96%, 231.71%, and 
+88.60% from 2007 to 2009, respectively. The larger the ratio is, the more volatile and unstable 
+market is. Thus, the ratio can be used as a stability index in stock market.
+ 
+4.2 Test Disposition Effect and Herd Behavior 
+ 
+  F r o m  e m p i r i c a l  t e s t  r e s u l t s  i n  S e c t i o n  3 ,  w e  find that there is significant positive correlation 
+between the rate of mean return and that of change in trading conditioning intensity in general 
+from April, 2007 to April, 2009 (reference to line A in Table). In order to understand the result and 
+its implication in trading behavior, let us clarify disposition effect and herd behavior in stock market at first. 
+  Shefrin and Statman (1985) identified what they  termed the “disposition effect” that investors 
+have a desire to realize gains by selling stocks that have appreciated, but to delay the realization of losses. Odean (1998) tested and demonstrated the disposition effect among investors by analyzing trading records for 10,000 accounts at a large discount brokerage house. Weber and Camerer 
+(1998) designed experiments to test disposition effect validity and explained it by prospect theory (Tversky and Kahneman, 1992). Grinblatt and Keloharju (2001) found evidence that investors are reluctant to realize losses using a unique data set in Finnish stock market. Disposition effect is abnormal behavior in selling stock. 
+ 
+ 18  There are innumerable social and economic situations in which we are influenced in our 
+decision making by what others around us are doing (Banerjee, 1992). Herding has been 
+theoretically linked to many economic activities. It is often said to occur when many people take the same action (Graham, 1999). There are many herd behaviors (Hirshleifer and Teoh, 2009), for example, Nofsinger and Sias (1999) studied herding and feedback trading by institutional and 
+individual investors. Lux (1995) formalized herd behavior to explain the emergence of bubbles. 
+Trading conditioning herd is that we are participants with others when expecting return and onlookers with others when expecting loss subject to price volatility and return information in trading market. Herd is abnormal behavior in buying stock. 
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+  If, on selling side, the larger the positive mean  return is, the more shar e volume are sold, and the 
+larger the negative mean return is, the less share volume are sold, then, it is disposition effect. If, 
+on buying side, the larger the positive mean retu rn is, the more share volume are bought, and the 
+larger the negative mean return is, the less share volume are bought, then, it is trading conditioning herd behavior.  
+Therefore, it is true that ther e are significant disposition effect and trading conditioning herd 
+behavior simultaneously if there is a significant positive correlation between the rate of mean 
+return and that of change in trading conditioning intensity. The magnitude of correlation 
+coefficient indicates intensity in the behavior. Our empirical results show that there is significant positive correlation between the rate of mean return and that of change in trading conditioning intensity in general. Therefore, there are signif icant disposition effect and trading conditioning 
+herd behavior in stock market in general. It is physiological response and conditioned behavior for return in human beings when people stimulated by price volatility and return information trade in expectancy for return.  
+ 
+4.3 Cognitive and Trading Behavior 
+ 
+  We subdivide our two year high frequency data into five time intervals based on financial crises in the world and bubble crash in China in 2007. There is a major advantage for this because we can study cognitive and trading behavior change with price volatility, time, and environment.    
+  The correlation coefficients are 0.0729 and 0.1026 right before and just after bubble crashes, respectively. The positive correlation lacks significance. Disposition effect and herd behavior lack significance. It is explained that disagreement on price between bounded rational and irrational traders increases uncertainty in last bubble ridi ng (see line C and D in Table). If we can further 
+demonstrated that there is a relation between bu bble crash and the lack of significance in the 
+correlation, we might time and predict bubble crash at a certain extent. 
+  It is more interesting that  the correlation coefficient is 0.1963 when SH Index continues 
+dropping in second half time interval after bu bble crashes. The posi tive correlation shows 
+significance. The disposition effect and herd behavior are significance. In comparison with behavior just after bubble crash, we further demonstrate that disposition effect and herd behavior are stronger in a more loss situation (please compar e the correlation coefficient in line E with line 
+D in Table).  
+  The correlation coefficient is 0.4766, the largest among 5 subdivided time intervals, and shows 
+significance. It is the most notable disposition effect and herd behavior (see line F in Table). This 
+indicates two facts. First, investors have been conditioned by drop momentum after a year succession of large drop from top 6124.04 points to bottom 1664.04 points in SSE Composite Index. Disposition effect to realize gains shows the most significance in short term. Second, it is a 
+necessary condition for reversal after there is pr ice momentum drop in market that we enhance 
+market confidence and expectancy in return, keep incremental amount of capital into market 
+continuously, and buy a large volume of stock shar es that have appreciated and sold by strong 
+disposition effect in short term. 
+  Now, we will focus on a special case that ther e is a negative correlation, -0.2567, between the 
+rate of mean return and that of change in trading conditioning intensity. It shows significance (see line B in Table). 
+  SSE Composite Index increases almost continuously from 998.23 points at bottom in June, 2005 to 3183.98 points in March, 2007, the beginning of sampling. When investors are 
+conditioned by sustaining positive return, they are reluctant to sell shares with price rising. It is an 
+“anti-disposition effect” (see line B in Table). 
+We explain variation in the correlation coefficient by that Investors change  their expectancy in 
+return and psychological behavior in stock market. 
+  Finally, the correlation coefficient between the rate of mean return and that of change in amount of transaction is always several points (0.03~0.08) higher than that between the rate of mean return and that of change in trading conditioning intensity. Money plays more important role in mean return than psychological behavior and expectancy do in stock market. It demonstrates a 
+dictum in Wall Street: cash is king.  
+ 
+ 
+ 19
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+4.4 Trading Conditioning and Excessive Trading 
+ 
+  We extend three-term contingency operant conditioning to a six-term trading conditioning loop 
+as follows: price volatility and return information (discriminative stimuli) cognition, judgment, 
+and decision (decision making) trading and no trading (operant) expectancy on return 
+(profit and loss) in th e future (consequence) consequence emotion (new discriminative stimuli) 
+ adjustment in expectancy on return (feedback). Please see figures 4 and 7.  →
+→ →
+→
+→
+Let us go back to reconsider  intra-cranial stimulation (ICS) experiment in introduction. 
+Assumed that the rat is rational in bar pressing for foods and irrational for pleasures, we might 
+find a clue why investors behave irrational by excessive trading ignoring wealth loss and 
+maximum utility. 
+Pierce and Cheney (2004) wrote that money can be exchanged for foods and services, and 
+perhaps is the most important source of economic and conditioned reinforcement. Money and 
+return is positive reinforcement in price volatile trading conditioning in stock market. It has 
+several characteristics as follows:  First, it does not loss reinforcement value as quickly as primary 
+reinforcement does because physiological demand is satisfied. Second, profit is a positive reinforcement whereas loss is a ne gative reinforcement. People trade and buy if they expect price 
+rising. On the other hand, they trade and sell if they expect price dropping. There is trading no 
+matter whether price in expectancy rises or drops.  Trading frequency is higher. Third, because we 
+have not yet found time cycle in volume-price probability wave, profit occurs after stock holding in an uncertain time interval. This makes tremendous resistance to extinction in trading conditioning, similar to variable interval (VI) sc hedules in operant conditioning experiment (Coon, 
+2007). It can increase trading probability and frequency if people not leave market, for example, loss stopping traders are likely to buy stock again, stimulated by price volatility and positive return (herd behavior). Forth, the trading consequence (profit and loss) that produces feedback to traders could influence their emotion and judgment, and enha nce or change their expectancy in return to 
+make trading. Expectancy change also produces high frequency trading. For examples, just buying traders may sell their stock immediately because they find market movement in an opposite direction. Investors who have realized gains by selling stock may become overconfidence and buy stock again. Trading will not stop. In conclusion, we find that trading conditioning produces 
+excessive trading (volume) in stock market.  
+ 
+ 
+ 20
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+Price V olatility in Stock Market 
+(The Source of Conditioned Stimulus) 
+ 
+ 
+ 
+ 
+Conditioned Cognitive Reflex (“Physiological Response”) 
+ 
+ 
+ 
+ 
+ No, No Trading      Process Information   Feedback Adjustment 
+Make Judgment and Trading Decision (“Preference”)  
+Trading Behavior Constrained by Capita l, Number of Shares, and Regulation 
+(“Limits” to Trading Behavior) 
+ 
+No Impact on Price  
+ 
+ 
+ 
+Yes, Trading Conditioning Behavior (Operant) 
+ 
+ 
+ 
+ 
+Price V olatility Caused by Independent Traders in Decision Making 
+Price V olatility Path Constrained by “Price First and Time First” 
+ 
+ 
+ 
+ 
+Market Coherence Behavior 
+Transaction V olume Distribution over a Trading Price Range Is Governed by 
+Transaction V olume-Price Probability Wave Equation 
+Stationary Equilibrium Price Jump Explains Abnormal Distribution 
+Trading Conditioning Intensity ≈Price V olatility and Mean Return Intensity 
+Trace Back to Psychological Behavior in Trading Decision Making 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Consequence in Return Influences Emotion 
+Pride or Regret, Happy or Rejected, Greedy or Terrified 
+New Conditioned Cognitive Reflex 
+(Persistent Reinforcement Establishes “Beliefs”) 
+ 
+Figure 7: Excessive trading mechan ism in a trading conditioning loop 
+ 
+ 21
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+  
+4.5 Finance Theory in Interdisciplinary Fields 
+ 
+  Shiller (2006) summarized the history of financial theory over the last half century in terms of 
+two distinct revolutions. The first was the revolution by neoclassical finance that began around 1960s, and the second was the revolution by behavioral finance which began in the 1980s. Unlike neoclassical finance that assumed investors’ rationality and maximum utility in decision making, 
+behavioral finance attempts to establish a link among neoclassical finance, psychology, and 
+decision making science, study how investors make systematic biases in cognition, judgment, and decision, and explain anomalies in financial market from psychological behavior (Tversky and Kahneman, 1973). Today, we have made great progress on how preferences, beliefs, and limits to 
+arbitrage influence financial market, respectively. Of all non-expected utility (EU) theories, 
+prospect theory (Kahneman and Tversky,1979)(Tversky and Kahneman, 1992) may be the most promising (Barberis and Thaler, 2003). 
+It is necessary for newly emerging behavioral fina nce to have better explanations that we need 
+develop a unified hypothesis, methodology, and theory about financial economics while 
+incorporating salient aspect of human behavior so that both rational and irrational behaviors are 
+covered in extreme conditions (Dong, 2009). 
+  Barberis and Thaler (2003) concluded that behavioral models typically capture something about investors’ beliefs, or their preference, or the limits to arbitrage, but not all three. In addition, there 
+are obviously competing behavioral explanat ions for some of the empirical facts. 
+  Shiller (2006) continued his viewpoint that behavioral finance is not wholly different from neoclassical finance. Perhaps the best way to describe the difference is that behavioral finance is more eclectic, more willing to learn from other social sciences and less concerned about elegance 
+of models and more with the evidence that they describe actual human behavior. 
+  We attempt to use econophysics methodology to link physiology, cognitive psychology, 
+neuroeconomics (Camerer et al., 2005), probability  and statistics, economic finance, behavioral 
+finance, and econophysics together to study financial market, a multi-interdisciplinary science (see 
+figure 7).    
+ 
+4.6 Possible Applications 
+ 
+There are many possible applications in our res earch. First, we find that trading conditioning 
+produces excessive trading (volume) in stock market. This can help us to explain why and how 
+personality, e.g. overconfidence, sensation seekin g, and disagreement, produces excessive trading 
+and high volume from the viewpoints in cognitive psychology, neuroeconomics (Hsu et al., 2005), and medicine. For example, it supports disagreement excessive trading hypothesis (Hong and 
+Stein, 2007) in a certain extent because of positive correlation between the rate of mean return and 
+the rate of change in trading conditioning intensity in general. Second, we use high frequency data to test disposition effect and herd behavior simultaneously by positive correlation between the rate of return and that of change in trading conditioning intensity. It provides a new quantitative 
+analysis and test model for behavioral finance. Thir d, if we can further demonstrated that there is a 
+relation between bubble crash and the lack of sign ificance in correlation between the rate of mean 
+return and that of change in trading conditioning intensity, we might time and predict bubble crash 
+at a certain extent. In this way, we can improve investment strategy, avoid large risk, and capture 
+good opportunity in investment. Forth, policy makers may use reinforcement to manage irrational 
+behavior and reduce risk in financial market (Xiao and Houser, 2005). Fifth, it is the best 
+laboratory to do research on conditioning in physiology, psychology, and medicine field.  
+ 
+5. SUMMARIES AND MAIN CONCLUSIONS 
+ 
+  In a real financial market, there are a grea t many factors that could influence investors’ 
+subjective cognition and transactio n volume. Therefore, it is a key to develop a theoretical trading 
+conditioning model subject to price volatility and return information how we incorporate market 
+subjective trading behavior into transaction volume-price probability wave distribution function. 
+Based on classical conditioning and Skinners’ three-term contingency operant conditioning, we 
+ 
+ 22
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+define price volatile trading conditioning in stock market as participants trade (operant) and expect 
+return in the future (consequence) after they analyze, judge, and have decision making (cognition) in the presence of price volatility and return information (discriminative stimuli). The trading 
+consequence that produces feed back (feedback) to influences their emotion  and judgment, 
+adjusts their expectancy in return (emotion), and makes trading. 
+We develop a theoretical trading conditioning model subject to price volatility and return 
+information in terms of market psychological behavior, based on normative analytical transaction volume-price probability wave distribution function in econophysics. We use transaction volume 
+probability to represent trading conditioning intensity in the model.  
+  In the same way, when we use stationary equilibrium price jump to represent price volatility 
+mean return in any two consecutive trading days, we can use the rate of change in total transaction 
+volume to describe the rate of change in trading conditioning intensity subject to the rate of mean 
+return and its information in the two days. In this way, we can use high frequency data to study 
+correlation between the rate of mean return and that of change in trading conditioning intensity,  to 
+analyze psychological behavior change when participants make trading decision in stock market, and to provide a quantitative analysis method and test model for behavioral finance. 
+  In empiric test, we use every trading high frequency data in SSE Huaxia 50ETF from April 2, 
+2007 to April 10, 2009. We study stationary equilibrium at first in stock market and further evidence the early finding that st ationary equilibrium exists widely and transaction volume-price 
+probability wave distribution holds true. Second, we get many good and useful results by studying 
+the correlation between the rate of mean return and that of change in trading conditioning intensity 
+using high frequency data. They are, for examples, (a) we test and explain disposition effect and herd behavior using significant positive correlation. General speaking, there are significant disposition effect and herd behavior simultaneously. We explain them by trading conditioning 
+theory. It is physiological response and return conditioned behavior in human beings when people 
+stimulated by price volatility and return information trade in expectancy for return; (b) we find anti-disposition effect and explain it by sustainable positive return (conditioned reinforcement) rewarded to buying and holding behavior in bull ma rket; (c) it is a necessary condition for reversal 
+after there is price momentum drop in market th at we enhance market co nfidence and expectancy 
+in return, keep incremental amount of capital into market continuously, and buy a large volume of stock shares that have appreciated  and sold by strong disposition effect in short term; (d) it lacks 
+significance in spite of positive correlation in two time intervals right before and just after bubble crashes. Probably, it will help us to time bubble crash in stock market; (e) the magnitude of 
+correlation coefficient between the rate of mean return and that of change in the amount of 
+transaction is always several percent points (0.03~0.08) higher than that between the rate of mean return and that of change in trading conditioning intensity. We can draw an inference that money plays more important role in mean return than psychological behavior and expectancy do in stock 
+market. Third, trading conditioning produces excessive trading. 
+ 
+ 
+ 23
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+References: 
+ 
+Ait-Sahalia, Yacine (2004): “Dis entangling Diffusion from Jump,” Journal of Financial Economics, 74, 487-528. 
+Benos, Alexandros V . (1998): “Aggressiveness and Survival of Overconfident Traders,” Journal of Financial 
+Markets , 1, 353-383 
+Barber, Brad M., AND Terrance Odean (2000): “Trading Is Hazardous to Your Wealth: The Common Stock 
+Investment Performance,” Journal of Finance , 55, 773-806. 
+Barber, Brad M., Terrance Odean, AND Ning Zhu (2009): “Systematic Noise,” Journal of Financial Markets , 12, 
+547-569. 
+Barberis, Nicholas, AND Richard Thaler (2003) : “A Survey of Behavioral Finance,” Handbook of the Economics 
+of Finance (Edited by G .M. Constantinides, M. Harris and R. Stulz) , Elsevier Science B.V ., 1051-1121. 
+Camerer, Colin F., George Loewenst ein, AND Drazen Prelec (2005): “Neu rosciences: How neuroscience can 
+inform economics?” Journal of Economic Literature, XLIII, 9-64. 
+Coon, Dennis, AND John O. Mitterer (2007): Introduction to  Psychology—Gateways to Mind and Behavior (11th 
+Edition), Brooks/Cole Publishing Company. 
+Cowles, J.T. (1937): “Food-tokens as Incentive for Learning by Chimpanzees,” Comparative Psychology 
+Monographs , 14 (5, Whole no. 71). 
+Dong, Zhiyong (2009): Behavioral Financ e, Peking University Press, 303-304.  
+Dragoi, V . (1997) : “A Dynamic Theory of Acquisition and Extincti on in Operant Learning,” Neural Networks, 10, 
+201-229. 
+Graham, John R. (1999): “Herding among Investment Newsletters: Theory and Evidence,” Journal of Finance , 54, 
+237-268.  
+Graham, John R., Campbell R. Harvey, AND Hai Huang (2009 ): “Investor Competence, Trading Frequency, and 
+Home Bias,” Management Science , 55, 1094-1106. 
+Greenwood, D.T. (1977): Classical Dynamics (2nd edition), Prentice-Hall, Englewood Cliffs, NJ. 
+Grinblatt, Mark, AND Matti Keloharju ( 2001): “What Makes Investors Trade?” Journal of Finance , 56, 589-616. 
+Grinblatt, Mark, AND Matti Keloharju (2009): “Sensation Seeking, Overconfidence, and Trading Activity,” 
+Journal of Finance , 64, 549-578. 
+Hirshleifer, David AND Siew Hong Teoh (2009): “Thought  and Behavior Contagion in Capital Markets,” 
+Handbook of Financial Markets: Dynamics and E volution (edited by Hens and Schenk-Hoppe), 
+North-Holland/Elsevier.  
+Hong, Harrison, AND Jeremy C. Stein (200 7): “Disagreement and Stock Market,” Journal of Economic 
+Perspectives , 21, 109-128. 
+Hsu, Ming, Meghana Bhatt, Ralph Adolphs, Daniel Tranel, AND Colin F. Camerer (2005): “Neural Systems 
+Responding to Degrees of Uncertai nty in Human Decision-Making,” Science , 310 (5754), 1680-1683. 
+Irons, Jessica G., and William Buskist (2008): “Operant Conditioning,” 21st Century 
+Psychology—A Reference Handbook (ed. by Davis and Buskist), 329-339.  
+Kahneman, Daniel, AND Amos Tversky (1979): “Prospect  Theory: An Analysis of Decision under Risk,” 
+Econometrica , 47, 263-292. 
+Lee, Charles M.C., AND Bhaskara n Swaminathan (2000): “Price Momentum and Trading V olume,” Journal of 
+Finance , 55, 2017-2069. 
+Li, Haitao, Martin T. Wells, AND Ci ndy L. Yu (2008): “A Bayesian Analysis of Return Dynamics with Levy 
+Jumps,” The Review of Financial Studies , 21, No.5, 2345-2378. 
+Lo, Andrew W., AND Jiang Wang (2006): “Trading volume: Impl ications of an Intertemporal Capital Asset Pricing 
+Model,”  Journal of Finance , 61, 2805-2840. 
+Lux, Thomas (1995): “Herd Beha vior, Bubbles and Crashes,” The Economic Journal , 105 (July), 881-896. 
+McCauley J. L. (2000): “The Futility of Utility:  How Market Dynamics ma rginalize Adam Smith,” Physica A , 285, 
+506-538. 
+Nofsinger, John R. AND Richard W. Sias (1999): “Herding and Feedback Trading by Institutional and Individual Investors,” Journal of Finance , 54 (6), 2263-2295. 
+Odean, Terrance (1998a): “V olume, V olatility, Price,  and Profit When All Traders Are Above Average,” Journal of 
+Finance , 53, 1887-1934. 
+Odean, Terrance (1998b): “Are Investors Reluctant to Realize Their Losses?” Journal of Finance , 53, 1775-1798. 
+Odean, Terrance (1999): “Do Investors Trade Too Much?” The American Economic Review, 89, 1279-1298. 
+Olds, M.E. AND J.L. Fobes (1981): “T he Central Basis of Motivation: In tracranial Self-stimulation Studies,” 
+Annual Review of Psychology , 32, 523-574. 
+Osborne, M. F. M. (1977): The Stock Market and Financ e from a Physicist’s Viewpoint, Grossga, Mineapolis. 
+Pavlov, Ivan (1904): “Physiology of Digestion,” Nobel Lectures—Physiology  or Medicine 1901-1921, Elsevier 
+Publishing Company, Amsterdam, 1967.  
+(available at http://nobelprize.org/nobel_pr izes/medicine/laureates/1904/pavlov-lecture.html  ) 
+Pierce, W.D., AND C.D. Cheney (2004): Behavior Analysis and Learning (3rd.), Mahwah, NJ: Erlbaum. 
+Sewell, Martin (2008): “Behaviora l Finance,” working paper.  
+Shefrin, Hersh, AND Meir Statman (1985) : “The Disposition to Sell Winners To o Early and Ride Losers to Long: 
+Theory and Evidence,” Journal of Finance , 40, 777-790.  
+  24
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+Shi, Leilei (2006): “Does Security Transaction V olu me-price Behavior resemb le a Probability Wave?” Physica A , 
+366, 419-436. 
+Shiller， Robert J. (2006): “Tools for Financial Innovation: Ne oclassical versus Be havioral Finance,” The 
+Financial Review , 41, 1-8. 
+Skinner, B.F. (1938): The Behavior of Organisms: An Experimental Analysis, New Yo rk: Appleton-Century-Crofts. 
+Soros, George (1995): The Alchemy of Finance—Reading the Mind of the Market, International Publishing Co.. 
+Statman, Meir, Steven Thorley, AND Keith V orkink ( 2006): “Investor Overconfidence and Trading V olume,” The 
+Review of Financial Studies , 19, 1531-1565. 
+Thorndike, E.L. (1913): Educational Psycholog y (V ol. 2), New York: Teachers College. 
+Tversky, Amos, AND Daniel Kahneman ( 1973): “Availability: A Heuristic for Judging Frequency and Probability,” 
+Cognitive Psychology , 5, 207-232. 
+Tversky, Amos, AND Daniel Kahneman (1992): “Advances in Prospect Theory: Cumulative Representation of 
+Uncertainty,” Journal of Risk and Uncertainty,” 5, 297-323.  
+Weber, Martin, AND Colin F. Camerer (1998): “The disposition effect in se curities trading: an experimental 
+analysis,” Journal of Economic Behavior and Organization , 33, Issue 2, 167-184. 
+Xiao, Erte, AND Daniel Houser (2005): “Emotion Expression in Human Punish ment Behavior,” Proceedings of 
+the National Academy of Sciences of the Un ited States of Amer ica, 102 (20), 7398-7401.  
+ 
+ 
+ 
+ 
+FOLLOWED IS CHINESE VERSION 
+ 
+ 25
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+证券价格波动的交易性条件反射模型  
+ 
+石磊磊*1、王毅文2、陈定3、韩立岩2、朴燕、苟成玲4
+ 
+1中国科学技术大学近代物理系复杂系统研究组 
+2北京航空航天大学金融系 
+3嘉实基金管理公司 
+4北京航空航天大学物理系 
+ 
+ 
+（2010年2月8日修改稿） 
+（欢迎讨论） 
+ 
+摘要 
+ 
+基于解析的成交量价概率波分布函数，用成交量概率描述价格波动的不确定性和强度，
+我们根据市场群体的心理行为， 构造了一个关于价格波动和收益信息的交易性条件反射的理
+论模型。应用此模型对中国股市高频数据检验，我们主要有以下发现：1 ）总体来说平均收
+益率与交易性条件反射强度变化率之间存在着显著的正相关；2 ）在泡沫破裂前、后两个时
+期，它们之间的正相关缺乏显著性；3 ）比较特殊的是上证指数在牛市上升的一段时期内，
+它们之间存在着显著的负相关。 我们的模型和实证能够同时检验股市中的处置效应和羊群行为，并且解释过度交易等市场异象。 
+ 
+关键词：行为金融、成交量价概率波、价格波动、交易性条件反射、处置效应、羊群行为、过度交易、经
+济物理 
+JEL分类：G12 ，G11，G10，C16 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+*联系作者：在某著名证券公司工作，电子信箱 Shileilei8@yahoo.com.cn  或leilei.shi@hotmail.com   
+作者感谢与王怀玉、王亚平、周春生、刘劲和刘发民等非常有意义的讨论。 
+ 
+ 
+ 26
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+A SECURITY PRICE VOLATILE TRADING CONDITIONING MODEL 
+ 
+Leilei Shi*1, Yiwen Wang2, Ding Chen3, Liyan Han2, Yan Piao, and Chengling Gou4
+ 
+1Complex System Research Group, Department of Modern Physics 
+University of Science and Technology of China 
+2Department of Fina nce, Beijing University of Aeronautics and Astronautics 
+3Harvest Fund Management Co. Ltd. 
+4Department of Physics, Beijing Univer sity of Aeronautic s and Astronautics 
+ 
+ 
+This Draft is on February 8, 2010 
+(Comments welcome) 
+ 
+ 
+Abstract 
+ 
+We develop a theoretical trading conditioning model subject to price volatility and return 
+information in terms of market psychological behavior, based on analytical transaction 
+volume-price probability wave distributions in which we use transaction volume probability to 
+describe price volatility uncertainty and intensity. Applying the model to high frequent data test in China stock market, we have main findings as follows: 1) there is, in general, significant positive correlation between the rate of mean return and that  of change in trading conditioning intensity; 2) 
+it lacks significance in spite of positive correlation in two time intervals right before and just after bubble crashes; and 3) it shows, particularly, significant negative correlation in a time interval when SSE Composite Index is rising during bull market. Our Model and findings can test both disposition effect and herd behavior simultaneously, and explain excessive trading (volume) and 
+other anomalies in stock market.  
+ 
+Key words: behavioral finance, transaction volume-price probability wave, price volatility, trading conditioning, 
+disposition effect, herd behavior, excessive trading, econophysics 
+JEL Classification: G12, G11, G10, C16 
+ 
+ 
+ 
+ 
+*Corresponding author. E-mail address: Shileilei8@yahoo.com.cn  or leilei.shi@hotmail.com , working in an 
+international securities company in China. 
+The author appreciates valuable discussion with Huai yu Wang, Yaping Wang, Chunsheng Zhou, Jing Liu, and 
+Famin Liu etc. 
+  27
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+一、引言  
+ 
+ 长期以来，人们都十分关注价格波动行为的研究，而忽略了对交易量的认识。近 10年
+来这种现象有所改变， 人们开始逐步重视和研究在金融交易市场中交易量所包含的内容和信
+息。在新古典金融理论的框架内， Lo和Wang（2006）试图通过建立价格和交易量与基本面
+之间某种联系来强调应该把量价作为一个整体对资本市场进行分析。 在新兴发展的行为金融学范畴内，我们开始将交易量与投资者的情绪、信念和偏好联系在一起。 Sewell（2008）认
+为：行为金融学是研究心理学对金融市场参与者行为的影响并且由此产生的市场效应，它有助于解释市场为什么以及怎样无效的。Lee 和Swaminathan （2000）发现以前的成交量在趋
+势投资策略和价值投资策略之间提供了重要的连接， 它有益于调和价格在中期反应不足而在长期反应过度的效应；Benos （1998）和 Odean（1998）认为过于自信（overconfidence ）导
+致市场的过度交易（excessive trading） ； Odean（1999）例举三个理由说明过于自信的投资者
+产生过度的交易量：职业选择偏差、生存能力偏差以及对预期收益的不现实理念，并通过对客户交易收益和交易成本的研究来检验过于自信导致过度交易的假说；Barber 和Odean
+（2000）进一步的实证发现活跃的交易行为会降低收益率，这种非理性行为只能用过于自信来解释；Graham 等（2009）则从那些自认为比别人更具竞争力的群体交易更加频繁来解释
+过于自信导致过度交易假说。Statman 等（2006）用过于自信模型对交易量的预测性进行检
+验， 发现个股的换手率与滞后回报率以及与滞后市场回报率之间持续数月的关系分别显示了处置效应和过于自信。 Barber等（2009）用个人投资者系统地买入近期表现好的股票而不卖
+亏损的股票（使得净交易量非常大）的心理偏差来解释个体对价格表现的交易行为存在着高度持续的相关性。Grinblatt 和Keloharju （2001）通过实证发现过去的回报率、参照价格效
+应、避税卖出和持有期间盈亏的大小都是影响交易的因素，而过于自信和寻求兴奋感（sensation seeking ）的人交易都更加频繁和活跃（Grinblatt 和Keloharju ，2009） 。Hong 和
+Stein（2007）认为交易量似乎是反应投资者观点的指标，价格偏离基本价值越高，交易者对
+价格的分歧就越大，交易量就表现出异常地大，由此，提出了一个可以将股价和交易量统筹考虑的意见分歧交易量模型（disagreement model ） 。 
+ 无论是过于自信交易量假说， 还是寻求兴奋感或意见分歧交易量假说都认为交易量的大
+小反应了投资者情绪、信念和偏好的强弱程度。 
+1904年诺贝尔生理与医学奖获得者、俄国生理学家巴甫洛夫（Pavlov， 1904）在研究动
+物生理反应和反射时用狗的唾液量来表示反射的强度，提出了经典性条件反射。经典性条件反射是当条件反射刺激出现时预期非条件反射刺激物的一种生理反应。 
+ Thorndike （1913）最早研究了操作性条件反射。Skinner （1938）为了研究大鼠操作性
+条件反射专门设计了一个装置， “斯金纳箱” （Skinner box ） 。实验发现：经过几次操作获取
+食物（强化物）之后，大鼠便形成了稳定的频率按压杠杆获取食物的行为模式。现在，心理学家把操作性强化物定义为 “伴随一个反应发生并能增加这个反应再次发生的可能性事件” 。  
+与经典条件反射一样，操作性学习也是建立在信息和期望基础之上的。操作性条件反射
+是在特定时间通过特定操作预期得到特定结果 （强化物） 的行为反应。 Pierce和Cheney（2004）
+把操作性条件反射归纳成“由特定刺激营造的时机从事操作，进而产生某种后果（强化物） ”
+的行为反应。或者说在特定的刺激出现时发生操作性条件反射行为，并伴随着特定的结果
+(Irons and Buskist, 2008) 。Skinner将这种关系称之为三项相互联系、可能发生的事件
+（three-term contingency ） （Dragoi ，1997） 。在操作性条件反射中，强化物可被用于改变反应
+频率：强化物出现的不确定性（例如不确定反应次数后出现强化物的“可变比率模式”和不确定时间间隔后出现强化物的“不定时间间隔模式” ）使得条件反射的频率很高、消退的抑制力很强（库恩，2007 ） 。 
+ 颅内刺激（intracranial stimulation, ICS ）能够直接激活大脑中的快乐中枢，有着最强、
+最特殊的强化作用（Olds 和Fobes，1981） 。实验发现许多大鼠为获得愉快感每小时能按压
+杠杆几千次，远远大于为获取食物按压的频率（相差几个数量级） ，全然不顾对食物、水和性的需求！ 
+库恩（Coon ，2007）将操作性强化物分成三类：一级强化物、二级强化物和反馈。一
+级强化物是自然形成的和非习得性的，具有生理基础，能产生舒适感和消除不适感，或能满
+ 
+ 28
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+足即时的生理需求。金钱、赞扬、赞同、情感以及其他的奖赏都可以成为强化物，即习得的
+二级强化物。一些二级强化物具有“代币物”的功能，可用以交换一级强化物，因此就有了更直接的价值。例如，钱币本身没有太大的价值，既不能充饥也不能解渴，但是，我们可以用它换取商品和服务，也许是最重要的经济强化物（Pierce 和Cheney，2004） 。研究者曾通
+过实验教黑猩猩为获得代币物而工作（Cowles ，1937） 。反馈是指人们对操作反应结果信息
+的接收和处理之后再影响操作的一个过程。 
+索罗斯（Soros ，1995）最初从抽象的哲学思考出发研究反射理论，随后逐步察觉到它
+与股票价格行为的相关性。他认为股票市场可以成为研究反射现象的最佳起点。索罗斯定性描述的反射是指人们参与交易导致价格波动，通过重新对交易价格的认知、思考和判断再参与交易的反馈过程，属库恩（Coon，2007）分类中的第三类操作性强化物。 
+在研究每日证券成交量在价格波动区间分布整体行为时，Shi （2006）用规范的经济物
+理方法推导出一个解析的成交量价波动方程并且得到两组解析的成交量在价格波动区间的分布函数，用成交量概率描述价格波动的不确定性和强度。基于此，我们根据市场群体的心理行为用成交量概率描述价格波动的交易性条件反射强度， 构造了一个关于价格波动和收益信息的交易性条件反射的理论模型。 通过该模型研究平均收益率与交易性条件反射强度变化率的相关性，我们同时检验了股市中的处置效应和羊群行为，由此追溯和分析市场群体在做交易决策时因价格波动和信息环境不同而发生的心理行为的变化，解释过度交易（量）等市场异象，试图为行为金融提供一个新的分析和检验模型。 
+尽管在过去有许多关于回报与交易量的研究，也有一些交易量行为的研究，但是据笔者
+所知还没有回报率与成交量（概率）变化率关系的研究，没有将经济物理学中严谨规范的解析量价概率波分布与生理学和心理学中解释性的认知条件反射行为成功地结合在一起进行研究——绝大多数试图同时实现规范和解释性目标的非期望效用理论模型的困难之处是它们最终在这两个方面都做的不好（Barberis 和Thaler，2003） 。与成交量不同，成交量变化率
+或成交量概率变化可以是正的、也可以是负的、或者是零。与过去采用事件研究法和心理学实验方法不同，我们用高频数据同时检验处置效应和羊群行为，并且解释过度交易行为等市场异象。 
+ 以下文章的结构是： 第二部分是成交量价概率波分布函数简介并且找到了一种关于价格
+波动和收益信息的交易性条件反射强度的计量方法；第三部分是以华夏上证 50ETF每笔交
+易的高频数据为例，我们首先研究定态均衡、确定定态均衡价格、进一步佐证了在股票市场中普遍存在着定态均衡状态（Shi ，2006） ；之后，我们实证检验了前后交易日之间平均收益
+率、交易性条件反射强度变化率和成交金额变化率三者之间的相关性；第四部分是实证分析和讨论，这包括 1）市场群体相互作用和干涉结果出现的定态均衡；2 ）总体来说，市场同
+时具有显著的处置效应和羊群行为；3 ）人们的认知和交易行为随价格波动和环境不同发生
+的变化； 4）用交易性条件反射理论解释过度交易； 5）应用前景等等；最后是总结和主要结
+论。 
+ 
+二、成交量价概率波分布函数及交易性条件反射强度 
+ 
+2.1 成交量价概率波分布函数 
+ 
+ 在用经济物理的方法研究股票成交量价波动行为时，Shi （2006）观察到在股票交易市
+场中普遍存在着定态均衡现象。股票每日在价格波动区间的成交量（累计交易量）分布在价格加权平均值附近出现峰化，这种现象与股票价格高低、价格波动路径以及涨跌没有关系
+1。
+我们将成交量峰值所对应的价格定义为该日的定态均衡价格。另外，价格总是围绕定态均衡价格上下波动的同时，定态均衡价格表现出跳跃（jump）的变化
+2。 
+ 
+ 29                                                        
+1 Shi（2006）对 618个成交量价分布进行拟合和显著性检验。这些样本价格不同、价格波动的路径和方向
+也不同，但是都表现出在成交量加权价格平均值附近出现峰化现象。 
+2 与Poisson-diffusion （Ait-Sahalia ，2004）和 Levy（Li，Wells & Yu, 2008）等跳跃不同，定态均衡价格有清
+晰的跳跃机制。例如，当有大量资金进入市场买入股票，供需关系突然发生大的变化、打破平衡时，供需
+关系就会通过价格调节作用寻找新的价格平衡点，定态均衡价格随之出现不连续的跳跃；通过成交量在价
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 影响股票价格波动的因素有许多，例如，公司经营状况、宏观经济因素、政策因素、心
+理行为因素等等。它们对价格波动的影响有大有小。但是，各种因素只有通过交易传导才能
+造成价格的波动，是间接的作用。因此，Shi （2006）在前期建立价格围绕定态均衡价格波
+动模型时暂时没有考虑这些因素。 
+ 没有交易就没有价格的波动，价格波动的同时一定存在交易，但是有交易不一定就有价
+格的波动，交易是价格波动的必要条件。除此之外，成交金额约束着价格的波动和成交量的大小。在某一价格处的成交量 等于在此价格处的累计交易量。在研究定态均衡条件下成交
+金额是如何定量约束量价行为以及成交量是怎样在价格波动区间分布时，Shi （2006）提出
+了成交金额流动约束量、 成交金额流动分布量和成交金额流动波动量 （定态均衡价格回归量）
+之间相互关系假说，即成交金额约束关系假说： v
+() 02
+= + + −p WVvp Et          （1 ） 
+其中 p是价格，其量纲是[ 货币单位][ 股]-1 ； t v vt /=是在时间间隔 内，在价格 t p处的成
+交量（累计交易量） 、动量（ momentum ）和趋势，其量纲是[ 股][时间] -1； 是在价格V v/ p处
+的成交量概率，它等于在价格 p处的成交量 除以总成交量 V； 是在时间间隔
+内，在价格v2/t v p E⋅ =
+t p处的成交金额（累计交易金额） ，即成交金额流动约束量，其量纲是[ 货币单
+位][时间] -2； 是在时间间隔 内， 在价格2/t v vtt= t p处的冲量和偏离定态均衡价格的趋势力，
+其量纲是[ 股][时间] -2； 是在时间间隔 内，在价格V v pt/2⋅ t p处成交金额流动分布量，其
+量纲是[货币单位][ 时间] -2；另外， 
+()()()p p A p p A p W− ≈ − =0          （2 ） 
+是在时间间隔 内，在价格 t p处成交金额流动回归量，其量纲也是[ 货币单位][ 时间] -2； 代
+表在时间间隔 t内的定态均衡价格，0p
+p是价格加权平均值， ()ttv V v A/ 1−= 代表定态均衡
+回归力和供需回归力的大小，其量纲是[ 股][时间 ] -2。 
+ 根据股票交易“价格优先，时间优先”规则，我们有交易价格最小波动定律：股票交易
+价格波动的路径要求成交金额约束假说方程（1 ）的泛函 
+() ( )⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ − =dpd
+dpdpVBE W p Fψ ψψ ψ ψ** ,2
+     （3 ） 
+在各种可能的价格变化中，选择对当前价格变化最小，即对描述价格波动的函数 ()pψ取极
+值；其数学表达式是 
+()∫=0 , dp p Fψ δ 。           （4 ） 
+ 由方程（1 ）和方程（4 ） ，我们得到量价波动方程 
+()[] 022 2
+= − +⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ ψψ ψp W Edpd
+dpdpVB       （5 ） 
+将方程（2 ）代入方程（5 ） ，并利用自然边界条件 
+() 0 0=ψ 、()∞ <0pψ 、及()0→∞+ψ ， 
+我们得到两组解析解；其一，当偏离定态均衡价格的趋势力 与供需平衡回归力ttv A之和在
+价格波动区间等于本征值常数、发生相互干涉行为时，我们得到 
+() ( )[]0 0 p p J C pm m m − =ω ψ ，   ( )L, 2 , 1 , 0=m ，   （6） 
+其中 
+tt tt m vVvA v= − =2ω ， ()0>mω  ( )L, 2 , 1 , 0=m    （7） 
+                                                                                                                                                               
+ 
+ 30格波动区间的分布，我们能够观察和测量其跳跃行为（Shi ，2006） 。  
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+mω是大于零的本征值常数， ()[0 0 p p Jm]−ω 是带本征值的零阶贝塞尔（Bessel ）函数，
+是一无量纲的常数；其二，当供需平衡回归力在价格波动区间等于本征值常数时，我们得到
+另外一组多阶本征值函数 mC
+() ( )0 2 , 1 ,0p p A n F e C pmp p A
+m mm− − ⋅ =− −ψ ，( )L, 2 , 1 , 0 ,=m n  （8） 
+其中 02 1> =+= 常数nEAm
+m ,      ( )L, 2 , 1 , 0 ,=m n  （9） 
+( )0 2 , 1 , p p A n Fm− − 是一个 阶的合流超几何函数或第一类 Kummer 函数。 n
+波函数（6 ）和（ 8）的含义是它的绝对值 ()pmψ代表在价格 p处的成交量概率（参见
+图1和图 2， 其中横坐标可以代表价格， 纵坐标代表成交量分布， 原点是定态均衡价格） ，
+是归一化常数。 mC
+图1和图 2的分布函数有几点不同：第一，图 1的分布是随着价格不断偏离定态均衡价
+格表现出波浪式衰减的，而图 2除了零阶是指数分布之外，其分布是一种波浪式均匀的，阶
+数越大，均匀性越显著；第二，图 1的分布是开放的，能够描述股市中突发的、远离均衡价
+格的脉冲式交易行为，例如，利益输入或输出交易行为，而图 2的分布是封闭的，能够描述
+随机均匀分布行为；第三，除了零阶分布，图 2分布的本征值常数一般比图 1的大一、二个
+数量级。但是，图 1和图 2分布的共同点是它们都能够很好地描述指数分布。 
+ 
+ 
+图1：成交量概率在价格波动区间的零阶贝塞尔函数的绝对值分布函数 -20 -10 10 200.20.40.60.81
+ 
+         
+（a）零阶           （b）一阶 -4 -2 2 40.20.40.60.81
+-15 -10 -5 5 10 150.20.40.60.81
+ 
+ 
+ 31         
+（c）二阶        （d）十阶 -40 -20 20 41
+图2：成交量概率在价格波动区间的多阶函数的绝对值分布函数 
+ 
+2.2 本征值和概率波 
+ -600 -400 -200 200 400 6000.10.20.30.40.5
+0.8
+0.6
+0.4
+0.2
+0
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+ 32本征值又称为特征值。在描述事件分布的函数中，本征值与参数是不同的。在绝大多数
+情况下，当我们观察到某种事件的分布现象时，由于不清楚导致事件分布的机制，往往用一
+个带有参数的分布函数描述这种统计行为。在这种近似的描述中，我们并不知道该参数所包
+含的内容和实际意义。本征值是可观察和可测量的量，明确了产生分布的机制，因此能够对其正确与否进行定量检验。例如，在成交量价概率波分布函数中，本征值包含两种意义：在
+第一组解中，本征值描述的是偏离定态均衡价格的趋势力
+ttv与供需平衡回归力 A之和在价
+格波动区间等于本征值常数、发生相互干涉行为的结果，用方程（7 ）计量；在第二组解中，
+本征值描述的是供需平衡回归力在价格波动区间等于本征值常数、 出现指数和随机均匀分布
+的现象，用方程（9 ）计量。 
+所谓概率波是用数量分布 的概率描述波动的强弱程度，例如在成交量价概率波行为中，
+我们
+，都有周而复始的重复变化。 
+     用成交量分布和概率而不是价格波动的幅度描述价格波动的强度。现在，让我们来比较
+概率波和经典波的不同点和相同点（参见图 3） 。首先，概率波的横坐标是价格或距离，纵
+坐标是数量概率，而经典波的横坐标是时间，纵坐标是振幅；第二，概率波是用数量分布概率的大小描述波动的强弱和变化，而经典波是用振幅的大小描述波动的强弱和变化；第三，概率波的强度大于或等于零，而经典波的强度可以是负的；第四，相对于平衡位置，概率波波动的强度随着偏离平衡位置呈现出波浪式的变化（除了零阶分布函数） ，波动的幅度越大，不等于波动的强度也越大，而在经典波中波动的幅度越大，波动的强度就越大；第五，概率波描述群体之间的相互作用，不能独立存在，而经典波可以独立存在；第六，目前我们还没有发现概率波在时间上存在着严格的周期性，即预测重复的时间存在着不确定性，而经典波有传播速度，可以计量，因此存在着时间周期。 
+概率波和经典波的共性就是都会出现干涉现象
+ 
+  
+（a）经典波() ) cos(0 t y t yω=     （b）概率波 () ()[]0 0 pp J C pm m m − =ω ψ  
+ 
+.3 交易性条件反射的引入 
+我们首先考虑没有人们主观变化、没有影响买卖交易行为的刚性交易体系。当交易价格
+知、供需平衡和成
+交量
+生理反应是一种非条件反
+射。
+当人们看到价格波动和收益（盈利或亏损）时产生的对物质必需品和
+服务图3：经典波与概率波的比较 
+2
+ 
+ 
+波动时，在任意相同的时间间隔内总成交量是相同的。如果有增量扰动资金进入市场买入股
+票、打破供需平衡、导致定态均衡价格跳跃，那么增量资金与定态均衡价格跳跃幅度之间存在着一一对应的关系（因为跳跃前后定态均衡的总成交量是相等的） 。  
+在真实的金融交易市场中，有许许多多的因素都会影响人们的主观认
+的变化。 如何将市场群体的主观交易行为和成交量变化纳入到解析的成交量价概率波分
+布函数中，将是认识和解决成交量、价格波动与增量扰动资金之间相互关系的关键，也是建立关于价格波动和收益信息的交易性条件反射理论模型的关键。 
+人类对衣、食、住、行和服务的生理需求是客观的，对它们的
+印币既不能吃，也无法解渴，其本身几乎是没有任何价值。在商品经济中，当我们将货
+币、收入和收益（率）通过交换与人类赖以生存的物质必需品和服务紧密结合起来时，我们就已经被它们条件化，就会对货币、收入和收益（率）形成同样的生理反应，即经典性条件反射。 Pierce和Cheney（2004）认为在现代商品经济中钱可用于交换商品和服务，也许是最
+重要的经济强化物。 
+在股票交易市场，
+需求的生理刺激就是一种经典性条件反射， 而根据价格波动和收益信息进行分析、 处理、1
+19.6 19.8 20.2 20.40.20.40.60.81
+2 4 6 8 10 12 14
+-1-0.75-0.5-0.250.250.50.75
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+ 33易性条件反射的强化物是钱和收益率，它有以下几个特点。第一，它不会像一级强化
+物那
+互联系、可能发生的事件（three-term contingency ）和操作性条
+件反
+ 
+图4：在交易性条件反射中三项相互联系、可能发生的事件 
+ 
+在股票交易体系中，独立变量有两个，价格和成交量，成交金额是它们的约束量。根据
+经典
+场群体对该
+证券
+的幅
+                                                       判断和决策从事买卖交易行为，并预期未来收益（盈利或亏损）的反应是一种操作性条件反
+射。 
+交
+样因生理需求已经得到满足而失去强化作用；第二，收益对应与正强化物，亏损对应与
+负强化物。 预期股票价格上涨会促使人们买入持有， 预期股票价格下跌会促使人们卖出观望。
+无论是预期价格上涨还是下跌都会出现交易性条件反射，进行交易。第三，由于量价概率波行为没有确定的时间周期，类似与操作性条件反射中部分强化的不定时间间隔模式（库恩，2007） ，持有股票时间有长有短之后才出现收益，这使得交易性条件反射消退的抑制力很强；
+第四，损益结果（盈利、亏损或持平）反馈给交易者会影响人们的情绪（emotion ）和判断
+能力，会改变人们对未来收益的预期而导致交易。例如，投资者止损出来后很容易因为看到价格上涨导致收益预期变化再次买入股票， 而刚买入股票之后也会因为收益预期变化或已经
+获得微利又很快卖掉。因此，在交易性条件反射的认知过程中，除了预期收益和亏损外，还
+包括反馈导致的预期变化。 
+根据 Skinner关于三项相
+射的定义（Irons and Buskist ，2008） ，我们定义在股票市场中的交易性条件反射是一种
+受价格波动和收益信息 （discriminative stimuli ） 影响， 通过分析、 处理、 判断和决策 （cognition ）
+进行交易（ operant），并预期未来收益（盈利、亏损或持平） （consequence ）的一种操作性
+条件反射； 其结果 （收益、 亏损或持平） 反馈 （feedback ） 给交易者会影响他们的情绪 （emotion ）
+和判断能力，调整他们的收益预期并进行交易活动（参见图 4） 。 
+ 
+可识别刺激 操作 强化物（正负强化） 
+交易 预期收益（盈利和亏损）  价格波动和收益信息 
+结果反馈（调整收益预期）
+力学原理（周衍柏，1985） ，我们知道交易体系的独立自由度的数目等于 1（独立变量
+数2减去约束方程数 1） 。经济物理学先驱 Osborne（1977）发现价格作为供需量的被解释变
+量（因变量）在实证中是不存在的， McCauley （2000）从数学上证明了 Osborne的这一发现。
+因此，Shi （2006）选择价格是股票交易体系中的自变量，成交量是因变量。 
+根据刚刚建立的交易性条件反射行为， 证券的价格波动和收益信息会产生市
+未来收益的预期而进行交易。这种主观认知的交易行为和成交量是同步变化的。我们用
+成交量分布和概率随价格波动的变化（注：不是成交金额的变化，因为成交金额是约束量）表示市场群体受价格波动和收益信息影响，导致交易行为、供需关系和主观认知的变化。 
+在成交量价概率波分布函数中，我们用成交量概率描述价格波动的强度（不是价格波动
+度） 。在某一价格处，成交量概率越大，波动的强度就越大（注：与经典波不同，在概
+率波中波动的强度越大不等于波动的幅度就越大，反之亦然） ，波动的频率就越高1，交易频
+ 
+1根据概率统计学的大数定律，当总成交量远远大于单笔成交量时，成交量的概率近似等于频率。因为我们
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+率也就越高，对未来价格波动、收益和亏损的预期就越高（成交量越大，买量越大，表明市
+场对未来价格上涨的心理预期越强；成交量越大，卖量越大，表明市场对未来价格下跌的心理预期越强） ，交易性条件反射的强度也就越大，反之亦然。因此，价格波动的强度近似等于关于价格波动和收益信息的交易性条件反射的强度。 我们可以用成交量概率近似地描述交易性条件反射的强度和频率（参见图 5） 。 
+ 
+价格波动的强度
+价格波动的频率
+交易频率 
+交易性条件反射频率 
+ 
+ 34 
+图5：价格波动的强度（用成交量概率表示）近似等于交易性条件反射的强度 
+ 
+现在，我们用成交量概率表示交易性条件反射的强度。同样，当我们用定态均衡价格跳
+跃的
+三、数据分析及相关性检验 
+ 
+我们以华夏上证 50ETF（510050）每笔交易的高频数据为例，用零阶贝塞尔绝对值分布
+回归
+3.1 据 
+们采用嘉实基金管理公司提供的嘉实高频 HF2数据库中每笔交易的高频数据。样本
+区间
+我们分两步对原始数据进行了预处理。首先，在原始数据中价格保留了小数点后的三位交易性条件反射的强度 
+幅度率表示价格波动的平均收益率时， 我们也能够用定态均衡价格跳跃前后成交量的变
+化率来描述市场群体对该平均收益率的交易性条件反射强度的变化率。显然，交易性条件反
+射强度的变化率可以是正的、负的或是零。这样，我们就能够通过使用高频数据定量分析平均收益率与交易性条件反射强度变化率之间的相关性， 由此研究他们在交易决策时的生理反应和心理行为。 
+ 
+模型对每个交易日的成交量在价格波动区间的分布进行拟合并检验其显著性。 对于通过
+检验的，我们能够从拟合数据中直接得到该交易日的定态均衡价格。对其他交易日，我们用价格加权平均值近似表示该日的定态均衡价格。由此，我们能够计算出前后交易日之间定态均衡价格跳跃的幅度（平均收益率） ，研究平均收益率、交易性条件反射强度变化率以及成交金额变化率之间的相关性。 
+ 
+数
+ 
+我
+为2007年4月2日至 2009年4月10日。近 740天，共 495个交易日，即 495个量价
+分布。 
+                                                                                                                                                               
+在下一部分使用样本期间的日平均成交量很大，为 3.6亿股，所以在某一价格处成交量概率可近似等于交
+易频率。这样，在某一价格的成交量越大，其概率就越大，频率就越大。对于一笔大单提升的交易量，即
+使诸多小单对应同样的交易量，虽然两者对应的信息可能不一样，但是成交量概率依然可以近似地描述交
+易的频率，由此出现的量价分布“异常”现象说明市场资金优势的扰动很容易破坏定态均衡、操纵价格，
+我们会进一步研究。 
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+ 35采取四舍五入的原则保留到两位数， 同时将相应的成交量相加， 得到新的量价数据。
+接着
+ 数， 我们
+我们将每一价格处的成交量（累计交易量）除以当天总成交量得到在该价格处的成交量
+概率。这样，我们就得到了每个交易日的成交量概率在价格波动区间的实际分布。 
+ 
+3.2 成交量价分布拟合、总体显著性检验以及定态均衡价格确定 
+ 我们已知在定态均衡状态下，量价分布的理论函数是 
+() ()[]0 0 p p J C pm m m − =ω ψ ，   ( )L2 , 1 , 0=m   （10） 
+、mCmω 其中 和 分别是归一化常数、本征值常数和定态均衡价格。
+回归模型 0p 它们在单变量非线性
+()[] ()i i m m i m=0 0 p p J C pε ω ψ + −  ( )n i , 3 , 2 , 1L=   （11）   
+是服从()2, 0σN 中是三个待 间内的价 定常系数， n是样本在交易价格区 格数，iε 分布的随
+机误差项， ()i mpψ是在价格波动区间内任一处的成交量概率观察值， ()[]0 i m
+合，确定其中 待定常系数并且得到成交量概率分布的理论结果。
+使用 Origin6.0 软件（参见图 6（a） ） 。  
+对总体显著性检验，我们采用 F检验的方法。可决系数(coefficient of determination) 0 p p J Cm−ω
+是理论值。我们采用 Levenberg-Marquardt 非线性最小二乘法对每一个量价分布样本进行拟
+的三个 在拟合分析中，我们
+  TSS TSSR = =          （12） RSS TSS ESS−2，
+()2
+1ˆ∑
+=− =n
+iiY Y ESS 和(explained sum of squares) ，()2
+ˆ∑− =n
+Y Y RSS是 其中 是回归平方
+=ii i残
+差平方和(residual sum of squares) ，1
+()2
+∑− =n
+Y Y是总离差平方和
+统计量 1=ii TSS (total sum of squares) 。 
+() 1 //
+− −=k n RSSk ESSF              （13） 
+分别指样本个数和解 中， n和 释变量个数（独立自由度数） 。设定显著性水平 05 . 0= k α ，
+当 或 
+ 05 . 0F F>
+ () 105 . 005 . 0 2 2
+− − + ⋅⋅>F k=k n F kR Rcrit ，        （14） 
+即2R大于 ）在 95%的显著性水平下总体显著成立（在 临界值 时，则回归模型（11 这里
+） 。 
+检验结果显示自
+通过显著性检验，即 R R> 个，约 23.23%的样本缺少显著性。 
+成交量价 是因为我们
+在前期数据处理时只保留价格数据中小数点后的两位数，导致一些分布信息无法体现出来。
+供求平衡关系发生较大的变化，其成交量价行为通过价格的调节，使得交易2
+critR
+1=k
+2007年4月2日至 2009年4月10日的 495个交易日中有 380个样本
+2 2
+crit
+我们对缺少显著性的样本进行仔细的观测，发现这些样本有两个特征：第一，相当一部
+分未通过显著性检验的 分布，其成交价格太少，单日的样本数不足。这。其余 115
+为此，我们在小数点后第三位采用“ 二舍三入和七舍八入” 的原则增加一个数值 0.005，相应
+的成交量也叠加到对应的交易价格中去，得到较明显的特征分布，并对它们进行拟合和显著性检验。结果有 28天的分布样本通过零阶贝塞尔绝对值分布回归模型的检验。这样，我们
+总共有 408（380+28）个分布样本，约占总数的 82.42%，通过零阶贝塞尔绝对值分布回归
+模型的检验。 
+第二个特征是成交量在交易价格区间内出现两个或两个以上的峰值。 这是因为这些样本
+在交易日当天的
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+ 36价格从围绕某一定态均衡价格上下波动调整成围绕另外一个或几个定态均衡价格上下波动。
+定态均衡价格在当日出现不连续的跳跃变化。在这种情况下，其成交量价分布函数是（10）
+式的线性叠加，即 
+() () []∑ − =m p p J C pω ψ ，    n n n m m 0 , 0 ()L2 , 1=n  （15） 
+其中 是定态均衡价格的数目。我们采用具有两个定态均衡价格的量价分布回归模型
+  n  
+() () [ ]()L2 , 1=i  i n n i n m m i i m p J C p ω ψ=∑∑ pε+ −  =2 , 1 , 0 , 0 （ ）
+87个分布进行拟合和2 2critR R>显著性检验（这16  
+对剩余的 里2 22=k） 。结果显示 本，
+约占总数的 11.92%在95%的水平下显著成立（参见图 6（b） ） 。       
+28分布样 概率波
+函数的表达式如下  59个样
+对还没有通过显著性检验的 本，我们采用量价 分布的第二组解——含
+合流超几何函数（第一类 Kummer 函数）的量价分布回归模型来拟合。该
+() ( )0 2 , 1 ,0p p A n F e C pmp p A
+m mm− − ⋅ =− −ψ  。    （17） 
+由于这是一个多阶的分布函数， 比较特殊和复杂， 直接拟 时，
+采用一阶函数的展开式 
+ 合比较困难， 我们取定 1=n
+ () ( )0 m
+ 2 , 1 , 10p p A F e C pp p A
+m mm− − ⋅ =− −ψ  
+0 2 10p p A e Cmp p A
+mm− − ⋅ =− −      （18） 
+28个样本中有 23个（约占总分布数的 平下
+6（c） ） 。最后 5个分布样本呈现 4 日市   
+进行拟合和检验。结果显示 4.65%）在 95%的水
+显著成立（参见图 个或 4个以上的峰值，这表明当
+场非常不稳定（见图 6（d） ）。图 7是分别为采用函数（10 ）、函数（ 16）和函数（18 ）拟合
+并检验其显著性的典型样本。 
+  
+2.28 2.29 2.30 2.31 2.320.000.020.040.060.080.100.120.140.160.18Data: Data1_B
+Model: probwave2.1 
+  
+Chi^2 =  0.00086
+R^2 =  0.75699  
+P1 0.13664 ±0.01516
+P2 263.84993 ±17.65125P3 2.29316 ±0.00202P4 2.3124 ±0.00261volume probability
+price(yuan)5100502008-7-23
+2.82 2.84 2.86 2.88 2.90 2.92 2.94 2.96 2.98 3.000.000.020.040.060.080.100.120.140.160.180.20Data: Data1_B
+Model: probawave 
+  
+Chi^2 =  0.00083R^2 =  0.69725
+  
+P1 0.14967 ±0.01613
+P2 79.97463 ±2.96019
+P3 2.93479 ±0.00227volume probability
+price(yuan)510050
+2007-6-1
+（a）采用函数（10）拟合的 2007-6-1 日数据 
+（ ） 50. 0 697 . 02 2= > =critR R（b）采用函数（16）拟合的 2008-7-23 日数据 
+（ 575 . 0 757 . 02
+22
+2 = > =critR R ） 
+2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.210.000.020.040.060.080.100.120.14Data: Data1_B
+Model: kummer1 
+  Chi^2 =  0.00093
+R^2 =  0.53632
+  
+P1 0.14756 ±0.01969
+P2 4223.7671 ±1236.31859
+P3 2.17427 ±0.00215volume probability
+price(yuan)510050
+2008-7-17
+ 
+（c）采用函数（18）拟合的 2008-7-17 日数据 
+（ ） 232 . 0 536 . 02 2= > =critR R3.42 3.44 3.46 3.48 3.50 3.52 3.54 3.560.000.010.020.030.040.050.060.07volume probability
+price(yuan)510050
+2008-2-1
+（d）5个缺乏显著性的样本之一  
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+ 
+Electronic copy available at: http://arxiv.org/abs/1001.0656   37图6：用成交量价概率波回归模型拟合成交量在价格波动区间的分布并检验其显著性1
+ 
+3.3 平均收益率与交易性条件反射强度变化率的相关性分析 
+ 
+对于 408个通过零阶贝塞尔绝对值分布回归模型检验的样本（约占总数的 82.42%），我
+们从拟合数据中直接得到该交易日的定态均衡价格。对于其他 87个交易日，我们用价格加
+权平均值近似表示该日的定态均衡价格。由此，我们能够计算出任意前后交易日之间定态均
+衡价格跳跃的幅度（平均收益率） ，研究平均收益率、交易性条件反射强度变化率以及成交金额变化率之间的相关性。 在这里交易性条件反射强度变化率近似等于前后交易日总成交量变化率。 
+已知相关系数 
+()
+Y XY Xσ σ,= ，           
+XY X, covr （19） 
+其中σ和Yσ分别是变量 X和Y的标准差， ()Y X, cov是协方差， r是相关系数。Y X, 我们
+设 
+0 :0采用t分布对相关系数进行显著性检验。建立原假设和备择假
+=ρH ； 0 :1≠ρH  
+计算统计量  
+()()2 / 12− −−=
+n rrtρ
+，          （20） 
+其中 r、n分别为样本相关系数和样本容量。取显著性水平为 05 . 0=α ，当
+= >t t t()2−n时，拒绝原假设，相关系数在该显著性水平下显著不为 0。 
+软件，以美国次贷金融危机和上证综合指数走势为背景，将华夏上
+证50ETF（510050 ）整个样 个子
+样本：2007 年4月2日（上证综合指数 3252.59点）至 2007年6月29日（上证综合指数
+70为泡沫破裂前市场上升前期； 年 7月2 合指
+年 2007年11月
+3693.11点）为泡沫
+破裂后市场下跌前期；2008 年5月5日（上 合指数 37 01点）至 2008年10月31日
+数1728.79点）为泡沫破裂后市场下跌后期；2008 年11月3日（上证综合指
+数1719.77点）至 2009年4月 。 
+这样细分有一个优点：我们 化。 
+从实证结果（参见表中的检验 看到： ）总体来说，平均收益率与交易性条
+强 变 率和成交金额变 本期间的
+个不同阶段，平均收益 在泡沫
+破裂
+                                                        （上证综合指3820.点） 2007 日（上证综 数 3836.29点）至
+2007 10月31日（上证综合指数 5954.77点）为泡沫破裂前市场上升后期；
+1日（上证综合指数 5914.28点）至 2008年4月30日（上证综合指数2 / 05 . 0 crit
+我们使用 Eviews6.0
+本数据从 2007年4月2日至 2009年4月10日又分成为五
+证综 61.
+10日（上证综合指数 2444.23点）为市场反转上升初期
+能够看出不同时期市场群体的心理行为随环境发生的变
+数据） ，我们 1
+件反射 度 化 化率之间都呈现出显著的正相关；2 ）在整个样 5
+率与交易性条件反射强度变化率之间的相关性明显不同: （a）
+前、后两个时期，它们之间的正相关缺乏显著性； (b) 在泡沫破裂后的后半时期，它们
+具有显著的正相关性；(c) 泡沫破裂后市场出现反转上升的初期，其正相关系数最高，是
+0.4766；(d) 比较特殊的是在牛市上升的一段时期内，即在泡沫破裂前市场上升的前期，平
+均收益率与交易性条件反射强度变化率之间存在着显著的负相关，相关系数是-0.2567 ；3）
+成交金额变化率与平均收益率之间的相关系数比交易性条件反射强度变化率与平均收益率之间的相关系数总是要高出几个百分点。我们将在下一部分详细讨论实证结果。 
+ 
+ 1 在图 6中，拟合结果中的P1 、P2和P3分别代表归一化常数、本征值常数和定态均衡价格，在图 6（b）
+中，P3和P4分别代表跳跃前后的定态均衡价格。  
+ 
+ 
+Electronic copy available at: http://arxiv.org/abs/1001.0656   38表：相关系数和显著性检验 
+ 时期 样本 
+数量 上证综合指数
+（1A0001） 收益率与条件反射变化率 收益率与成交额变化率 差值 条件反射 变化率与成交额变化率 
+A  
+2007.4.2—2009.4.10  
+494  
+3252.59—2444.23 0.1391 
+（t=3.115> ） 0.1970  
+（t=4.458> ）  
+0.0579 
+（t=critt=1.960critt=1.9600.9975 
+316.3>critt=1.960） 
+B  
+2007.4.2—2007.6.29  
+59  
+3252.59—3820.70 -0.2567  
+（t=2.006> ） -0.2120  
+（t=1.638< ）  
+0.0447 
+（t=
+  critt=2.001critt=2.0010.9986  
+142.0>critt=2.001） 
+C  
+2007.7.2—2007.10.30  
+83  
+3836.29—5954.77 0.0729  
+t=0.6583< ）0.1053  
+（t=0.9529< ）  
+0.0324 
+（t=
+  （critt=1.990critt=1.9900.9993  
+241.2 >critt=1.990） 
+D  
+2007.11.1—2008.4.30  
+122  
+5914.28—3693.110.1026  
+（t=1.130< ） 0.1714  
+（t=1.906< ）  
+0.0688 
+（t=
+   
+critt=1.980critt=1.9800.9968  
+137.0>critt=1.980） 
+E  
+2008.5.5—2008.10.31  
+123  
+3761.01—1728.79 0.1963  
+（t=2.202> ） 0.2706  
+（t=3.091> ）  
+0.0743 
+（t=
+  critt=1.980critt=1.9800.9958  
+119.2>critt=1.980） 
+0.9981  
+166.5 >critt=1.983) F  
+2008.11.3—2009.4.10  
+107  
+1719.77—2444.23 0.4766  
+（t=5.556> ） 0.5203  
+（t=6.243> ）  
+0.0437 
+(t=
+  critt=1.983critt=1.983
+标注： 1 ） critt指()22−n，  
+  2 ） 差值是平均收益率与成交金额变化率之间的相关系数减平均收益率与交易性条件反射强度变化率之间的相关系数； 
+2） 红色黑体标出的是检验结果的相关性缺乏显著性； 
+3） 上证综合指数以当日收盘值计量。 05 . 0tcritt t>说明相关系数显著不为零；相反，不能拒绝相关系数为零的原假设； 
+ 
+ 39四、实证分析和讨论 
+ 
+4.1 定态均衡和定态均衡价格 
+ 
+S 2006） 价 行 分 成 。 一 价 绕某个定态均衡价
+格上下 动 种定 为可以 解析的 量价概 方程 ）定量描述，其中定态均
+衡价格 成 的 易 格 二 定态均衡价格表现出不连续的跳跃（jump）变
+化。在定态均衡条件下，定 均衡回 较弱， 均衡 被打破变化， 如有增 动资金 入市场 买入股 ，打破原有定态均衡，那么，供需关系就会通过价格的调节作用寻找新的定态均衡点，定态均衡价格出现不连续的跳跃。此后价格又开始围绕新的定态均衡价格上下波动。定态均衡价格跳跃的幅度表示价格波动的平均收益率。 
+我们对华夏上证 50E近740共49交易日，即 495个成交量价分布分析，用
+零阶贝塞尔绝对值分布函数对它们逐一拟合和显著性检验， 结果是通过显著性检验的样本数量是 408个，占总数的 82.42%，这表明在股票市场中普遍存在着定态均衡状态。对于通过
+显著性检验的分布，我们从拟合数据中就能够得到交易日当天的定态均衡价格。著性检验的成交量价分布“异常”现 对于
+这部分交易日， 用 交 加 近 代 其 态 价 。 ，后交易日定态均衡价格跳跃的幅度和价格波动的平均收益率。
+我们的实证进一步佐证了 S 2）前期发现：在股票市场中普遍存在着定态均衡状
+态，而 态均衡 会出现 连续的 变化。
+我们从本次“异常”样本数量比例（17.58 与Shi 06）前期的比例（ 6%）比
+较，就能够推断 03年的上海证券交易市场比 07~200 的市场平稳。事实上，上证指
+数在 2003年最大振幅是 26.17%，而在 2007、 2008及2009年的最大振幅分别是 0.96%、
+2 .71 88.60% 因此，我们能够用这种比值表示市场稳定性或波动性指标。比值越大，
+市场的波动性也就越大，越不稳定。
+ 
+4检验处置效应和羊群行为的新方法 
+ 
+前一部分的实证结果表明：从 20年4月 009年 整个样本期间，平均收益率与
+交易性条件反射变化率之间的相关系数是 0.139具有显 A栏） 。
+为了理 它所包 行为内 ，我们 解一下 市中的 效应和 群行 。
+Shefrin和Statman（1985）最早提出处置效应：投资者在处置处于不同赢利状态股票时，
+存在急于兑现收益、回避兑现亏损的倾向。之后， Odea 1998） 究了 00
+交易记录后充分证实了处置效应， er和C erer（1）通过心理学实验来检验并且用
+Kah 和Tversky（19 1992） 的前景理论来解释处置效应， Gr tt 和 loh u 001）
+用芬兰股市中 34卖 据检验 处置效 处置效 现的 股票 象。 
+为 存在于 的社 动中（ erjee， 2） ，理论上讲与许多经济活动有
+关系（Graham ， ） 。当模仿他人导致许多人采取同样的行为时，
+在股票市场中存在着许多类型的羊群行为（Hirshleifer 和Teoh，2001） ，例如机构和个人的
+羊群行为（No ger 和 ，1999 Lux (1 ) 曾提出传染模型描述投资者的羊群行为。
+我们定义交易性条件反射的羊群行为是受价格波动和收益信息影响，
+亏损的观望交易行为。羊群行为表现的是买股票时的行为异象。 
+对于卖方 果平均 率正值 大卖量 ，并且 果平均 率负值越
+大）卖量越少 么，这是处置效应。对于买 如果平 收益率 越大，买众跟风的行为越明显，并且如果平 益率负 越大（ 越多） 量越少，避亏损的气氛越浓，这是交易性条件反射羊群行为。 因此， 平均收益率与交易性条件反射强度变化率之间显著的正相关特性表明市场同时具有显著的处置效应和交易性条件反射羊群行为， 其正相关系数的大小也能够用来衡量处置效应和羊群行为的强弱。 我们的实证表明市场总体平均收益率与交易性条件反射变化率之间具有显hi（
+波是
+例将
+峰值
+量扰格波
+态行对应动的
+交
+态
+进为
+用价解
+；第
+归力
+大量两个
+成交
+，部分
+定态
+票第
+率波，
+容易格始
+（5终围
+。当供需关系发生较大，这交量
+TF 5个 天，
+没有通过显
+5.6
+14
+ 
+0个帐户的
+arj
+行为
+大（亏损越
+量越
+大家观望回象是在交易日当天定态均衡价格跳跃的行为特征。
+均值
+006
+跳跃我们
+价格
+20成量
+不权平
+hi（似替
+ 
+%）
+20定均衡
+（20
+9年格
+ 这样我们能够得到前
+为
+10
+Ke
+时的定
+31
+.2 %和
+解。
+含的 
+07
+先了到2
+1，
+股4月
+著的正相关性（参见表中处置
+容 羊
+n（
+998
+应表
+199在研
+inbla
+是卖
+我们说羊群行为出现了。Web
+了
+会活am
+应。
+Banneman
+羊群行79，
+单数
+人类（2
+异
+多2930
+普遍
+1999
+fsin Sias 995 ） 。
+预期收益的从众和规避
+，如
+，那收益 越
+均收越多
+方，
+值如
+均
+亏损收益
+正值
+，买，从
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+ 40著的正相关性，具有显著的处置效应和羊群行为。它们是人类受价格波动和收益信息刺激、
+获得收益和规避亏损从事交易而表现出来的一些生理反应和人性被收益条件化的行为。 
+.3 认知心理和交易行为的变化 
+ 
+如果我们今后的研究能够进一步证明这种缺乏显著性相关特征与股市价格泡沫破裂有
+一定
+年大幅下跌，股指从最高的
+6124
+人们收益预期，保持持续增量资金入市接
+盘、
+的情况，泡沫破裂前市场上升前期收益率与交易性条
+反射强度的变化率之间存在较强的负相关性，相关系数是 -0.2567，具有显著的负相关性
+上证综合指数从 2005年6月最低的 998.23点开始几乎是一路上涨到 2007年3月30日
+（样
+缩，而价
+格下
+ 
+） 损益结果影响情绪
+（新
+进行的操作性条件反射的行为是理性为
+ 
+4
+我们以美国次贷金融危机和我国股市泡沫破裂为背景，将整个样本时期细分 5个阶段。
+这样做有一个优点， 我们能够研究市场群体随价格波动和环境不同展现出的认知心理和交易
+行为的变化。 
+在泡沫破裂前的最后疯狂上涨和泡沫破裂初期，其相关系数分别是 0.0729和0.1026。
+它们的正相关都缺乏显著性，处置效应和羊群行为不显著。在这两个阶段，有限理性和非理性之间对价格认识的分歧增大，他们之间的博弈产生了很大的不确定性（参见表中 C和D
+栏） 。
+的相互联系，那么，我们就在一定程度上找到了一种预测和把握泡沫破裂的时机！ 有趣的是在泡沫破裂后的后期，持续下跌和超跌期间，相关系数是 0.1963，具有显著的
+正相关性，处置效应和羊群行为显著增大，换句话说下跌越多，亏损越大，卖出越少，处置效应越显著；同时下跌越多，买入越少，观望气氛越浓，羊群行为更显著（比较表中 D栏
+与E栏的相关系数）。 
+在反转上涨初期，相关系数最大，是 0.4766，具有显著的正相关性，处置效应和羊群行
+为最显著（参见表中 F栏）。这说明两个问题：第一，经历一
+.04点跌到 1664.93点后，人们已经形成了熊市下跌的趋势思维，短期获利了结的处置
+效应非常显著；第二，当下跌趋势形成之后，提高
+克服大量短期赢利兑现的处置效应是改变市场继续下跌并出现反转的必要条件。 总体来说， 在股票交易市场中普遍存在着处置效应和羊群行为， 而且我们发现下跌越多，
+亏损越大，平均收益率与交易性条件反射强度变化率之间的正相关系数就越大，处置效应和羊群行为就越显著。 
+现在，我们来关注实证中非常特殊
+件
+（参见表中 B栏）。  
+本选取区间起始日期）的 3183.98点。当持有股票与正收益率挣钱（条件反射刺激物）
+不断结合，持有股票的行为转化为正收益率挣钱效应（正强化）时，一种新的条件反射也就形成了。人们表现出惜售和持有股票的心态。价格上升，卖的数量减少，成交量萎
+跌，买的数量增加，成交量放大，出现“反处置效应” 。 最后， 我们还注意到平均收益率与成交金额变化率之间的相关性比它与交易性条件反射
+强度变化率之间的相关性总是高出几个点（0.03~0.08） 。由此，我们可以推断在股票交易市场，资金对平均收益率的作用大于人们的心理行为和收益预期的作用，印证了华尔街的一句至理名言：现金为王。 
+ 
+4.4 交易性条件反射循环和过度交易 
+ 我们将传统的操作性条件反射的三项相互联系、可能发生的事件进行扩展和补充，归纳
+出交易性条件反射的六个环节：价格波动和收益信息（可识别刺激）
+→认知、判断和决策
+（决策）→交易或不交易（操作行为） →预期收益或亏损（结果 →
+的可识别刺激） →调整收益预期（反馈） 。我们将这种周而复始的过程称之为交易性条
+件反射行为的六项重复过程（参见图 4和图 7） 。 
+ 现在让我们回过头来再仔细思考一下引言中提到的颅内刺激（ intra-cranial stimulation, 
+ICS）实验。如果我们做一个类比，大鼠为获取食物而
+的，为获取愉快感而进行的操作性条件反射的过度行为是非理性的，那么，我们就能够类推
+人们过度交易而忽视了资产增值或效用最大化的非理性行为的原因。 
+在现代商品经济中，钱、收入和收益可用于交换商品和服务，也许是最重要的经济强化
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+物（ Pierce和Cheney, 2004 ） 。在股票市场中，交易性条件反射的强化物是钱和收益率，它不
+会像一级强化物那样因生理需求已经得到满足而失去强化作用；第二，量价概率波行为没有
+ 
+ 41定的时间周期， 其收益率的不确定性增加了交易性条件反射的频率。 收益对应与正强化物，
+格上涨会促使人们买入持有，预期股票价格下跌会促使人
+卖出观望。因此，无论是预期价格上涨还是下跌都会促使人们交易，提高交易的频率，导
+致过
+很容易因为看到价格上涨导致收益预期变化再次买入股票（羊群行为） ，增
+加交
+4.5 
+与
+新古
+并且
+发展
+需要找到统一的理论将理性与非理性这两个极端的情况统一起来，以
+增强
+2003）认为行为金融学模型有代表性地扑捉到投资信念（beliefs ） 、
+偏好确
+亏损对应与负强化物。预期股票价
+们
+度交易。第三，类似与操作性条件反射中部分强化的不定时间间隔模式（库恩， 2007） ，
+持有股票时间有长有短之后才出现收益，这使得交易性条件反射消退的抑制力很强，例如投
+资者止损出来后
+易频率；第四，收益（盈利和亏损）结果会影响人们的情绪（emotion ）和判断能力，
+并且再反馈到交易者、改变人们的收益预期。例如，投资者赢利卖出后会因为自信增强而再次参与买进股票，而刚买入股票之后也会因为收益预期变化或已经获得微利（处置效应）又
+急于卖掉，交易很难停止，出现过度交易。我们的结论是：交易性条件反射导致过度交易！  
+ 
+多学科交叉的金融理论体系 
+ 
+ Shiller （2006）把过去半个世纪的金融理论史概括成两次革命。第一次起源于上个世纪
+六、七十年代的新古典金融学革命和第二次起源于上个世纪八十年代的行为金融学革命。
+典金融学理论要求投资者是完全理性和决策效用最大化不同， 行为金融学把新古典经济
+金融学理论与心理学和决策科学联系起来，研究投资者是如何产生系统的认知偏差、判断偏
+差和决策偏差， 并从心理学的角度解释金融市场上的异常现象 （Tversky 和Kahneman， 1973） 。
+目前，行为金融学分别在研究人们偏好、情绪和信念、以及有限套利三方面对金融市场的影响已经取得了许多卓有成效的成果，其中由 Kahneman 和Tversky（1979 ，1992）提出
+起来的描述性前景理论影响最深远。 
+ 然而， 新兴发展中的行为金融学需要将心理学同经济学更好地结合起来构建比较统一的
+假设前提和研究范式，
+其解释能力。而目前的研究还局限在基于投资者的认知偏差和前景理论，但在各自的目
+标函数上引入了五花八门的行为特征，用以解释某一特定现象（董志勇，2009 ） 。 
+ Barberis 和Thaler（
+（preferences ）或有限套利（the limits of arbitrage ）三个方面中的某一特征，而不能涵
+盖全部三个方面，而且对同一个实证结果又有明显不同的行为解释。  Shiller （2006）认为行为金融学不是完全不同与新古典金融学。也许描述两者不同的最
+好方式是行为金融学更加兼收并蓄、 更加愿意从其他社会科学吸取营养而较少关心模型是否完美、更加注重描述真实的人类行为现象。 
+我们试图通过经济物理学的方法，将生理学、认知心理学、神经经济学（Camerer 等，
+2005） 、医学、概率统计数学、经济金融学、行为金融学和经济物理学联系起来，在一个整体框架内系统全面地研究涉及多交叉学科的金融交易市场的行为和变化（参见图 7） 。 
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+42形成判断并做出交易决策（ “偏好” ） 
+及交易规则约束交易行为（ “限制” ） 
+ 
+ 
+分布特征类似一种概率波  
+股票市场的价格波动 
+（经典性条件反射刺激源） 
+ 
+ 
+ 
+ 
+认知条件反射（生理反应） 
+   
+ 
+  否定，不交易    对信息进行加工处理   反馈（调整收益预期）  
+资金、股数以
+   
+对价格波动没影响  
+ 
+ 
+肯定，交易性条件反射行为 
+ 
+  
+ 
+众多独立个体的买卖决策导致供求关系的不断变化和价格波动 
+“价格优先和时间优先”的交易规则约束价格波动的路径 
+ 
+   
+ 
+众多独立个体行为产生群体的市场效应 
+总成交量在价格波动区间的
+用量价概率波方程描述这种行为特征 
+定态均衡价格跳跃模型解释量价分布的“异常”现象 
+交易性条件反射强度 ≈价格波动的强度 
+追溯市场群体的判断决策和心理行为 
+  
+  
+ 
+ 
+市场结果影响人们（无论交易还是不交易者）的情绪 
+产生自豪或后悔、快乐或沮丧、甚至贪婪或恐惧等 
+新的认知条件反射（生理反应） 
+（不断强化的条件反射会形成“信念” ） 
+ 
+图7：用成交量概率定量描述价格波动的不确定性和市场群体交易性条件反射的强度 
+（交易性条件反射导致过度交易） 
+ 
+ 
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+ 434.6 应用前景 
+ 
+，2005）和医学角度解释为什么不同性格的
+人会有不同的交易频率（交易量） ，并且检验 于自信、寻找兴奋感或意见分歧导致过度交
+易的行为金融学假说，例如，当价格上涨成交量增加的市场效应在一定程度上支持意见分歧导致过度交易假说（Hong 和Stein，2007） ；第二，通过对平均收益率与交易性条件反射强
+度变化率的相关性分析，我们用高频数据同时检验了市场群体的处置效应和羊群行为。交易性条件反射理论模型为行为金融学假说和模型提供了一个新的定量分析和检验方法；第三，
+缺乏显著性的特征与股市价 我们就在一定程度上找到
+预测和把握泡沫破裂 策略，规避和防范
+大的金融风险和扑捉好的投资机会；第四，有助于监管部门通过正负强化（reinforcement ）
+金融市场 （Xiao 和Houser，2005） ， 降低大的金融风险和危机爆发的可能性；
+第五 很好的交易性条件反射实验室，有助于今后开展一些生理
+学、认知心理学、神经经济学和医学方面的研究工作。 
+ 
+ 
+五、总结和主要结论 
+ 
+在真实的金融交易市场中，有许许多多的因素都会影响人们的主观认知和成交量的变
+化。 如何将市场群体的主观交易行为和成交量变化纳入到经济物理学中规范的解析量价概率
+波分布函数中，键，也是建立关于价格波动和收益信息的交易性条件反射理论模型的关键。
+我们从经典条件反射和操作性条件反射出发，进一步提炼交易性条件反射行为的定义：
+在股票市场中，交易性条件反射是一种受价格波动和收益信息影响，通过分析、处理、判断和决策进行交易，并预期未来收益的一种操作性条件反射；交易性条件反射结果反馈给交易者会影响他们的情绪和判断能力，调整他们的收益预期并进行交易活动。 
+基于解析的量价概率波分布函数——用成交量概率描述价格波动的不确定性和强度、 用
+定态均衡价格跳跃的幅度 描价格波动和成交量变
+化的约束量，我们用成交量的变化率来计量交易性条件反射强度的变化率，构造了一个关于价格波动和收益信息的交易
+同理，当我们用定态均衡价格跳跃的幅度率表示前后交易日价格波动的平均收益率时，
+我们也能够用这两天成交 率的交易性条件反射强
+度的变化率。这样，我们能 交易性条件反射强度变化
+率之间的相关性， 追溯和分析市场群体在做交易决策时受价格波动和市场环境不同发生的心
+ 
+通过采用 2007年4月到 2009年4月我国股市中的华夏上证 50ETF每笔交易的高频数
+实证检验，我们首先来研究股票市场中的定态均衡，进一步佐证了 Shi（2006）前期的发
+——在股票交易市场中，定态均衡状态普遍存在，量价概率波分布函数是有效的；第二，
+我们得到许多有
+意义的结果。这些包 了市场群体的处置
+效应和羊群行为；总体来说， 羊群行为。我们用交易性条
+件反射理论来解释这些市 和收益信息刺激、为获
+得收益和规避亏损从事买卖交易而表现出来的一些生理反应和人性被收益条件化的行为； 2）
+发现了市 反射解
+释这种市场异象；3 ）当下跌趋 ，保持持续增量资金入市接
+盘、克服大量短期赢利兑现的处置效应是改变市场继续下跌并出现反转的必要条件；4 ）在
+泡沫破裂前后的两个时期，它们的正相关都缺乏显著性，这也许可以帮助我们把握泡沫破裂 
+通过量价概率波分布函数， 我们根据市场的心理行为用成交量概率定量描述了交易性条
+件反射的强度并且归纳出包含六项环节的交易性条件反射循环，用它解释过度交易行为，这有助于今后从认知心理学、神经经济学（Hsu
+等
+过
+如果我们今后的研究能够进一步证明平均收益率与交易性条件反射强度变化率之间相关性
+格泡沫破裂有一定的相互联系，那么，
+的时机！通过加深对市场的认识，我们能够优化投资
+措施和政策调控
+，证券市场为我们提供了一个
+将是定量解决成交量、价格波动的平均收益率与增量资金之间相互关系的关
+ 
+描述价格波动的平均收益、 并且用成交金额
+性条件反射的理论模型。 
+量的变化率来描述市场群体对该平均收益
+够用高频数据，通过研究平均收益率与
+理行为的变化，试图为行为金融提供一个量化分析的研究方法和检验模型。
+据
+现通过高频数据研究平均收益率与交易性条件反射强度变化率之间的相关性，
+括：1）我们用它们之间的正相关性同时检验和解释
+市场同时具有显著性的处置效应和
+场群体行为的异象：它们是人类受价格波动
+场中的反处置效应， 并且用牛市中买入持有行为不断获得收益的正强化条件
+势形成之后，提高人们收益预期
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
+ 44时机；5）成交金额变化率与收益率之间的相关系数比交易性条件反射强度变化率与它之
+是要高出几个百分点。我们由此能够推断：在股票交易市场，资金对收益率
+的作用大于人们的心理行为和损益预期的作用；第三，交易性条件反射导致过度交易。 的
+间的相关系数总
+Electronic copy available at: http://arxiv.org/abs/1001.0656    
+ 
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\ No newline at end of file
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+arXiv:1001.0657v1  [astro-ph.EP]  5 Jan 2010Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 15 November 2018 (MN L ATEX style file v2.2)
+Eccentricity pumping of a planet on an inclined orbit by a
+disc
+Caroline Terquem1,2⋆and Aikel Ajmia1
+1Institut d’Astrophysique de Paris, UPMC Univ Paris 06, CNRS , UMR7095, 98 bis bd Arago, F-75014, Paris, France
+2Institut Universitaire de France
+15 November 2018
+ABSTRACT
+In this paper, we show that the eccentricity of a planet on an inclined orbit with
+respect to a disc can be pumped up to high values by the gravitationa l potential of
+the disc, even when the orbit of the planet crosses the disc plane. T his process is an
+extension of the Kozai effect. If the orbit of the planet is well inside the disc inner
+cavity, the process is formally identical to the classical Kozai effec t. If the planet’s
+orbit crosses the disc but most of the disc mass is beyond the orbit, the eccentricity
+of the planet grows when the initial angle between the orbit and the d isc is larger
+than some critical value which may be significantly smaller than the clas sical value of
+39 degrees. Both the eccentricity and the inclination angle then var y periodically with
+time. When the period of the oscillations of the eccentricity is smaller t han the disc
+lifetime, the planet may be left on an eccentric orbit as the disc dissipa tes.
+Key words: celestial mechanics — planetary systems — planetary systems: for ma-
+tion — planetary systems: protoplanetary discs — planets and sate llites: general
+1 INTRODUCTION
+Amongthe240extrasolar planetsthathavebeendetectedso
+far with a semi–major axis larger than 0.1 astronomical unit
+(au), about 100 have an eccentricity e >0.3. Five of them
+even have e >0.8. Such large eccentricities, which cannot
+be the result of disc–planet interaction (Papaloizou et al.
+2001), are probably produced by planet–planet interaction s,
+either through scattering or secular perturbation (see For d
+& Rasio 2008 and references therein), that occur after the
+disc dissipates (Juric & Tremaine 2008, Chatterjee et al.
+2008, Ford & Rasio 2008).
+Here, we show that high eccentricities can be pumped
+by the disc if the orbit of the planet is inclined with respect
+tothe disc. The process involvedis an extension of theKozai
+mechanism, in which a planet is perturbed by a distant com-
+panion on an inclined orbit (Kozai 1962). While the Kozai
+effect has always been studied for the case in which the com-
+panion is far away from the planet, the process investigated
+here is shown to be efficient even if the orbit of the planet
+crosses the disc. The classical Kozai effect has of course bee n
+very well studied. Here, we show that some significant differ-
+ences occur when the classical scenario is extended to apply
+to a disc.
+In section 2 we review the Kozai effect, and show that
+the same behaviour is expected whether the planet is per-
+⋆E-mail: caroline.terquem@iap.frturbed by a distant companion or by a ring of material or-
+biting far away. In section 3, we present the results of nu-
+merical simulations of the interaction between a planet on
+an inclined orbit and a disc. We show that, provided most of
+the mass in the disc is beyond the orbit, and the initial in-
+clination is larger than some critical value, the gravitati onal
+potential from the disc causes the eccentricity and the in-
+clination of the planet’s orbit to oscillate with time. This
+may occur even if the orbit crosses the disc. In section 4 we
+summarise our findings, and discuss under which conditions
+this mechanism could operate. The important result is that
+a planet on an inclined orbit with respect to the disc and
+located in or within the planet formation region may have
+its eccentricity pumped up to high values by the interaction
+with the disc. This is of astronomical interest, since incli na-
+tions are beginning to be measured for extrasolar planets.
+2 REVIEW OF THE KOZAI EFFECT AND
+EXTENSION TO A DISC
+We consider a planet of mass Mporbiting around a star of
+massM⋆which is itself surrounded by a ring of material
+of mass Mdisc. The ring is in the equatorial plane of the
+star whereas the orbit of the planet is inclined with respect
+to this plane. The motion of the planet is dominated by
+the star, so that its orbit is an ellipse slightly perturbed b y
+the gravitational potential of the ring. We study the secula r
+c/circlecopyrt0000 RAS2Terquem and Ajmia
+perturbation of the orbit due to the ring. We denote by
+(X,Y,Z) the Cartesian coordinate system centred on the
+star and ( r,ϕ,θ) the associated spherical coordinates. The
+ring is in the ( X,Y)–plane between the radii RiandRo>
+Ri. We suppose that the angular momentum of the disc is
+large compared to that of the planet’s orbit so that the effect
+of the planet on the disc is negligible: the disc does not
+precess and its orientation is invariable. The gravitation al
+potential exerted by the ring at the location of the planet
+is:
+Φ =−G/integraldisplayRo
+RiΣ(r)rdr/integraldisplay2π
+0dα
+(r2+r2p−2rrpcosαsinθp)1/2,(1)
+where the subscript prefers to the planet and Σ( r) is the
+mass density in the ring. We assume:
+Σ(r) = Σ 0/parenleftBigr
+Ro/parenrightBig−n
+, (2)
+where:
+Σ0=(−n+2)Mdisc
+2(1−η−n+2)πR2o, (3)
+withη≡Ri/Ro. We suppose that Ri≫rp, so that the
+square root in equation (1) can be expanded in rp/rand
+integrated to give:
+Φ =−−n+2
+1−η−n+2GMdisc
+Ro/bracketleftbigg
+1−η1−n
+1−n+ (4)
+−1+η−1−n
+1+nr2
+p
+2R2o/parenleftBig
+−1+3
+2sin2θp/parenrightBig/bracketrightbigg
+. (5)
+In the classical Kozai effect, the planet is perturbed by
+a distant companion of mass M. If we assume the orbit of
+this outer companion is circular of radius R≫rpand lies
+in the (X,Y)–plane, then the potential averaged over time
+it exerts at the location ( rp,θp) is:
+ΦKozai=−GM
+R/bracketleftbigg
+1+r2
+p
+2R2/parenleftBig
+−1+3
+2sin2θp/parenrightBig/bracketrightbigg
+. (6)
+Because rpandθpappears in exactly the same way in
+Φ and Φ Kozai, the secular perturbation on the inner planet,
+obtained by averaging over the mean anomaly of its orbit,
+is the same in both cases to within an overall multiplicative
+factor. The results obtained for the classical Kozai effect c an
+therefore be extended to the case of the disc. In particular,
+the perturbation due to the disc makes the eccentricity e
+of the planet to oscillate with time if the initial inclinati on
+angleI0between the orbit of the planet and the plane of the
+disc is larger than a critical angle Icgiven by:
+cos2Ic=3
+5. (7)
+The maximum value reached by the eccentricity is then (In-
+nanen et al. 1997):
+emax=/parenleftBig
+1−5
+3cos2I0/parenrightBig1/2
+, (8)
+and the time tevolit takes to reach emaxstarting from e0is
+(Innanen et al. 1997):
+tevol
+τ= 0.42/parenleftBig
+sin2I0−2
+5/parenrightBig−1/2
+ln/parenleftBigemax
+e0/parenrightBig
+, (9)with the time τdefined as:
+τ=(1+n)(1−η−n+2)
+(−n+2)(−1+η−n−1)R3
+oM⋆
+a3MdiscT
+2π, (10)
+whereTis the orbital period of the planet and ais its semi–
+major axis. Note that the function/parenleftbig
+sin2I0−2/5/parenrightbig−1/2de-
+creases very sharply from infinity to ∼3 asI0increases
+fromIcto about 45◦and then decreases by about 50% as
+I0continues to increase up to 90◦.
+TheZ–component of the angular momentum of the or-
+bit,Lz∝√
+1−e2cosI, where Iis the inclination angle
+between the orbit and the plane of the disc, is constant.
+Therefore Ialso oscillates with time and is out of phase
+withe.
+3 NUMERICAL SIMULATIONS
+We consider a star of mass M⋆= 1M⊙surroundedbya disc
+ofmassMdiscandaplanetofmass Mpwhoseorbitisinclined
+with respect to the disc. The planet interacts gravitationa lly
+with the star and the disc but we take Mp≪Mdiscso that
+it has no effect on the disc. To study the evolution of the
+system, we use the N–body code described in Papaloizou &
+Terquem (2001) in which we have added the gravitational
+force exerted by the disc onto the planet.
+The equation of motion for the planet is:
+d2r
+dt2=−GM⋆r
+|r|3−∇Φ−GMpr
+|r|3+Γt,r, (11)
+whereris the position vector of the planet and Φ is the
+gravitational potential of the disc given by equation (1) wi th
+RiandRobeing the inner and outer radii of the disc. The
+third term on the right–hand side is the acceleration of the
+coordinate system based on the central star. Tides raised
+by the star in the planet and relativistic effects are include d
+through Γt,r, but they are unimportant here as the planet
+does not approach the star closely. Equation (11) is inte-
+grated using the Bulirsch–Stoer method and the integrals
+involved in ∇Φ are calculated with the Romberg method
+(Press et al. 1993). In most runs, the integration conserves
+the total energy of the planet and LZwithin 1 to 2%.
+The planet is set on a circular orbit at the distance rp
+from the star. The initial inclination angle of the orbit wit h
+respect to the disc is I0. In the simulations reported here we
+have taken n= 1/2 in equation (2). The functional form of
+Σ is shallower than what is usually used for discs, but that
+has no significant effect on the argument we develop here.
+We firstcompare thenumerical results with the analysis
+summarised in section 2 by setting up a case with Ri≫rp.
+In figure 1 we display the evolution of eandIforMp=
+10−3M⊙,rp= 1 au, Mdisc= 10−2M⊙,Ro= 100 au,
+Ri= 50 au and I0= 42◦.3. For this run, LZis conserved
+within 2% but the energy of the planet is conserved only
+within 10%. We are here in the conditions of the analysis
+of section 2 with η= 0.5. From equation (8), we expect
+emax= 0.3, which is a bit smaller than the value of 0.41
+found in the simulation. Also the minimum value of Ishould
+beIc= 39◦.2 and is observed to be 36◦.5. Note that since
+the energy varies by about 10% in this run, we do not expect
+exact agreement between the numerical and the analytical
+results. We observe that the time it takes to reach emaxfrom
+c/circlecopyrt0000 RAS, MNRAS 000, 000–000Eccentricity pumping of a planet on an inclined orbit by a dis c3
+theinitialconditionsis2 .8×107years,whichagrees wellwith
+tevolgivenbyequation(9)providedwetake e0≃2×10−2.As
+mentioned above, tevolbecomes very long when I0is smaller
+than 45◦. As the disc lifetime is only a few Myr, ewould
+not have time to reach the maximum value in this case, if
+starting from a very small value.
+Figure 2 shows the evolution of eandIforMp=
+10−3M⊙,rp= 20 au, Mdisc= 10−2M⊙,Ro= 100 au,
+Ri= 1 au and I0= 47◦.7 (case A). We see that eoscillates
+between emin= 10−2andemax= 0.7, whereas Ioscillates
+between Imin= 20◦.1 andImax=I0. The values of emin,
+emaxandImindiffer from those calculated in the analysis in
+section 2, but this is expected as the condition rp≪Ri, that
+was used in the analysis, is not valid here. However, since
+most of the mass in the disc is in the outer parts, beyond
+the planet’s orbit, the behaviour we get here is similar to
+that described in the analysis. The period of the oscillatio ns
+isTosc= 2.2×105years.
+For comparison, we have run the classical Kozai case,
+where the disc is replaced by a planet located on a circular
+orbit in the ( X,Y)–plane. This perturbing planet is at a dis-
+tanceRfrom the central star and has a mass M. We take
+Mto be the same as the value of Mdiscabove, and consider
+R= 50 and 100 au. The evolution of eandIfor the inner
+planet in that case is shown in Figure 3. Equation (8) gives
+emax= 0.5, which is in very good agreement with the values
+of 0.5 and 0.55 obtained from the numerical simulations for
+R= 100 and 50 au, respectively. The time it takes to reach
+emaxfromemin, which is Tosc/2, is givenbyequation(9)with
+e0being replaced by eminandτ= [R3M⋆/(a3M)]T/(2π).
+We gettevol= 5.5×105and 8×104years for R= 100 au
+(emin= 0.03) and 50 au ( emin= 0.02), respectively, which is
+in excellent agreement with the values seen on Figure 3. The
+minimum angle reached when R= 50 au is about 31◦, much
+larger than the value obtained when the disc is present. Of
+course this value would decrease if the perturbing planet
+were moved closer to the inner planet, but then the oscilla-
+tions would not be regular anymore, as can already be seen
+in the case R= 50 au.
+Figure 4 shows the evolution of eandIfor the same
+values of MdiscandI0as in case A but with Mp= 4×
+10−3M⊙,rp= 1 au,Ro= 50 au and Ri= 0.5 au (case B).
+Here we have emin= 5×10−2,emax= 0.72,Imin= 16◦.4,
+Imax=I0andTosc= 2.0×105years, comparable to the
+value found in the previous case.
+On dimensional grounds, Toscis expected to be pro-
+portional to τgiven by equation (10). We have run case A
+with different values of Mdiscand have checked that Tosc∝
+1/Mdisc. The values between which eandIoscillate though
+do not depend on Mdisc, as expected from the analysis.
+We have run case A with different values of rpranging
+from 2 to 50 au. We observe that for rproughly below 10 au,
+Toscdecreases when rpincreases, as expected from the ex-
+pression of τ. For larger values of rpthough, Toscdoes not
+vary much with rp, and is ∼2–3×105years. For rp= 10
+and 20 au, the extreme values of eandIare roughly the
+same if the calculations are started from the same I0and
+e0, but for rp= 50 au the amplitude of the oscillations is
+very small. In that case, most of the disc mass if not beyond
+the planet’s orbit, so that the Kozai effect disappears.
+Finally, we have checked the effect of varying I0in
+case A. We have found that there is a critical value of I0,Ic∼30◦, below which eccentricity growth was not observed.
+However, the fact that Imin= 20◦.1 in case A suggests that
+Icmay actually be smaller. Since the time it takes for eto
+grow from very small values when I0is close to Icis very
+long (see equation [9]), the simulations may not have been
+run long enough for a growth to be observed. As I0increases
+fromIcto 90◦,emaxgrows from 0 to 1. Toscis not observed
+to vary significantly with I0, although the time it takes for
+the eccentricity to grow from its initial value to emaxdoes
+depend on I0.
+The simulations reported here suggest that when the
+planet’s orbit crosses the disc, eccentricity growth occur s
+for significantly smaller initial inclination angles than i n the
+classical Kozai effect.
+4 DISCUSSION AND CONCLUSION
+We have shown that, when a planet’s orbit is inclined with
+respect to a disc, the gravitational perturbation due to the
+discresultsintheeccentricityandtheinclination oftheo rbit
+oscillating if: (i) most of the disc mass is beyond the planet ’s
+orbit, (ii) the initial inclination angle I0is larger than some
+critical value Ic, which may be significantly smaller than in
+the classical Kozai effect. In the simulations we have per-
+formed, in which Σ ∝r−1/2andMdisc≪Mp(so that the
+effect of the planet on the disc is negligible), oscillations oc-
+cur as long as the initial planet’s distance to the star, rp,
+is smaller than about half the disc radius Ro. We expect
+a smaller critical distance when Σ decreases more rapidly
+with radius. Note that Icshould be independent of the func-
+tional form of Σ as long as most of the disc mass is beyond
+the planet’s orbit. The amplitude of the oscillations depen ds
+only on I0. It is small for I0≃Icand becomes large when
+I0is increased. In particular, emax→1 asI0→90◦. The
+oscillations of eandIare 180◦out of phase. Their period
+Tosc∝1/Mdisc. When rp≪Ro,Toscdecreases as rpin-
+creases. For larger values of rp,Toscdoes not vary much
+with this parameter. In the simulations we have performed,
+Tosc∼105years. As this is much shorter than the disc life-
+time, there is a nonzero probability that the planet is left
+on a highly eccentric orbit as the disc dissipates.
+Note that the process discussed in this paper is not
+a mere trivial extension of the Kozai effect. Indeed, in the
+classical Kozai effect, the periodic behaviour of eis obtained
+because the dependence of the perturbing gravitational po-
+tential on the planet’s argument of pericentre ωis through
+a term proportional to cos2 ω. When the orbit of the planet
+crosses the disc, the gravitational potential has a very dif -
+ferent form, and the behaviour of eis much more difficult to
+predict.
+The effect discussed here could be inhibited if other
+processes induced a precession of the planet’s orbit on
+timescales shorter than Tosc. That may happenif other plan-
+ets are present in the system, or because of dissipative tida l
+torques exerted by the disc, which have been ignored here
+(Lubow & Ogilvie 2001). In the latter case, the Kozai effect
+would then work only for planets orbiting inside the disc
+inner cavity.
+When the planet crosses the disc, loss of energy and
+angular momentum circularises the orbit and aligns it with
+thedisc planeonatimescale ∼TMp/(Σ(rp)R2
+p),where Rpis
+c/circlecopyrt0000 RAS, MNRAS 000, 000–0004Terquem and Ajmia
+the planet radius (Syer et al. 1991, Ivanovet al. 1999), whic h
+can be smaller than Toscforrplarger than a few au. Taking
+this process into account together with the Kozai effect may
+lead to equilibrium values of eandI. This will be studied in
+a forthcoming paper. Note that the usual type II migration
+mechanism that applies to planets orbiting in discs would
+not be relevant here, as it happens when the planet orbits
+in a gap that is locked in the disc evolution. In the coplanar
+case, as the disc spirals toward the central star, it carries
+along the gap and the planet. When the planet is on an
+inclined orbit, it may still open up a gap but it is not locked
+in it, so that it is not being pushed in as the disc spirals in.
+The mechanism described here relies on the planet be-
+ing on an inclined orbit, which could happen as a result of:
+(i) dynamical relaxation of a population of planets formed
+through fragmentation of a protostellar envelope around a
+star surrounded by a disc (Papaloizou & Terquem 2001);
+(ii) mean motion resonances (Thommes & Lissauer 2003,
+also Yu & Tremaine 2001) and (iii) gravitational interac-
+tions between embryos during the planet formation stage
+(Levison et al. 1998, Cresswell & Nelson 2008). In all cases,
+the process that makes the orbits inclined also makes them
+eccentric. According to the results presented here, the dis c
+could pump the eccentricities up to even larger values.
+Measures of the projected angle between the axes of the
+planet’s orbit and the stellar rotation, using the Rossiter –
+McLaughlin effect, are becoming available. So far, only the
+system XO–3, which has a ∼12 Jupiter masses planet on a
+3.19daysorbitwith e= 0.26, hasbeenshowntohaveaspin–
+orbit misalignment of at least 37◦.3 (H´ ebrard et al. 2008,
+Winn et al. 2009). Misalignment has also been reported for
+the system HD 80606, which has a ∼4 Jupiter mass planet
+on a 111.44 days orbit with e= 0.93 (Moutou et al. 2009,
+Pont et al. 2009). As HD80606 is a component of a binary
+system, the classical Kozai effect could be responsible for
+the misalignment in this system.
+We thank J. Papaloizou and S. Balbus for very useful
+comments which have greatly improved the original version
+of this paper. C.T. thanks the MPIA in Heidelberg, where
+part of this paper was written, for hospitality and support.
+REFERENCES
+Chatterjee, S., Ford, E. B., Matsumura, S.,& Rasio, F. A. 200 8,
+ApJ, 686, 580
+Cresswell, P., & Nelson, R. P. 2008, A&A, 482, 677
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+AJ, 113, 1915
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+& Ida, S. 2000, in Protostars and Planets IV, eds V. Man-
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+p. 1111
+Lubow, S. H., & Ogilvie, G. I. 2001, ApJ, 560, 997
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+B. P. 1993, Numerical Recipes in FORTRAN (CUP)
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+c/circlecopyrt0000 RAS, MNRAS 000, 000–000Eccentricity pumping of a planet on an inclined orbit by a dis c5
+0373839404142
+time (years)00.10.20.30.4
+Figure 1. Eccentricity e(solid line ) and inclination angle I(in degrees, dotted line ) versus time (in years) for Mp= 10−3M⊙,rp= 1 au,
+Mdisc= 10−2M⊙,Ro= 100 au, Ri= 50 au and I0= 42◦.3.
+c/circlecopyrt0000 RAS, MNRAS 000, 000–0006Terquem and Ajmia
+0202530354045
+time (years)00.10.20.30.40.50.60.7
+Figure 2. Same as figure 1 but for rp= 20 au, Ri= 1 au and I0= 47◦.7.
+c/circlecopyrt0000 RAS, MNRAS 000, 000–000Eccentricity pumping of a planet on an inclined orbit by a dis c7
+04045
+time (years)00.10.20.30.40.5
+030354045
+time (years)00.10.20.30.40.50.6
+Figure 3. Classical Kozai effect: Same as figure 2 but with the disc being replaced by a planet of mass M= 10−2M⊙and located at a
+distance R= 100 au ( upper plot ) and 50 au ( lower plot ) from the star.
+c/circlecopyrt0000 RAS, MNRAS 000, 000–0008Terquem and Ajmia
+0202530354045
+time (years)00.10.20.30.40.50.60.7
+Figure 4. Same as figure 2 but for Mp= 4×10−3M⊙,rp= 1 au, Ro= 50 au, Ri= 0.5 au.
+c/circlecopyrt0000 RAS, MNRAS 000, 000–000
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+arXiv:1001.0658v2  [cond-mat.mtrl-sci]  23 Jun 2010Fluctuations and correlations during the shear flow of elast ic
+particles near the jamming transition
+Claus Heussinger, Pinaki Chaudhuri and Jean-Louis Barrat∗
+November 1, 2018
+Abstract
+We present a numerical study of the flow of an as-
+sembly of frictionless, soft discs at zero tempera-
+ture, in the vicinity of and slightly above the jam-
+ming density. We find that some of the flow prop-
+erties, such as the fluctuations in the number of
+contacts or the shear modulus, display a critical
+like behaviour that is governed by the proximity to
+the jamming point. Dynamical correlations dur-
+ing a quasistatic deformation, however, are non
+critical and dominated by system size. At finite
+strain rates, these dynamical correlations acquire
+a finite, strain-rate dependent amplitude, that de-
+creases when approaching the jamming point from
+above.
+Manuscript submitted to the themed Issue
+on granular and jammed materials of Soft
+Matter
+1 Introduction and back-
+ground
+The properties of granular materials, in particu-
+lar sand, are a constant source of fascination for
+children and adults alike, and are intrinsically re-
+lated tothe abilityofsuchsystems toexist in either
+solid or fluid states, under very similar conditions.
+For the scientist this fascination may arise through
+the apparent contradiction between the rigidity of
+the individual grains and the fragility of the as-
+sembly as a whole. This means that, for exam-
+ple, small changes in the loading conditions (such
+∗Universit´ e Lyon 1, Laboratoire de Physique de la
+Mati` ere Condens´ ee et Nanostructures, CNRS, UMR 5586
+Domaine Scientifique de la Doua F-69622 Villeurbanne
+Cedex, Franceas changing the inclination angle of the support)
+can lead to large scale structural rearrangements
+(“avalanches”) or even to the complete fluidization
+of the material. A few years ago, Liu and Nagel1
+have suggested a “phase-diagram” for this type of
+solid-liquid transition (“jamming”). At zero tem-
+perature the axes relate to the ways an unjamming
+transition can be triggered, either by increasingthe
+external driving (e.g. shear stress) or by decreasing
+the density of the material. The present study will
+probe the vicinity of this ”unjamming line” using
+quasistatic and finite strain rate simulations of a
+model granular system.
+If one applies a shear stress, which is small and
+below a certain threshold (“yield-stress”), the ma-
+terialwillrespondasanelasticsolid. Increasingthe
+stress above the yield-stress, the particles will un-
+jam and start to flow. This flow behavior is called
+“plastic flow” as the material will not revert to its
+original shape when the stress is removed.
+In the following we will assume that the sys-
+tem canflowatarbitrarysmallstrain-rateswithout
+showing flow-localization. That this is possible is
+by no means guaranteed, as in some instances co-
+existence of flowing and jammed states is observed
+.2We have not observed such a persistent strain
+localization in the simulations that are presented
+here.
+Plastic flow is observed in a large number of
+glassy materials, that are a priori very different
+fromthe athermalgranularsystems closeto point J
+(see below) studied in this paper. However, all ma-
+terials displaya ratheruniversalbehavior, that was
+illustrated already in early studies on the plastic-
+flow of metallic glasses.3,4These studies have given
+indications that in the flowing phase the main plas-
+tic activity is spatially localized to so called shear
+transformation zones.5,6These zones are non per-
+1sistent, localized in space, and presumably consist
+of a few atoms that undergo the irreversible re-
+arrangements responsible for the observed plastic
+flow. Recently, this plasticity has been further an-
+alyzedinsimulationswithafocusonthequasistatic
+dynamics at small strain rates, close to the flow ar-
+rest.7–12With these studies it was possible to trace
+back the origin of plastic activity to the softening
+of a vibrational mode and the vanishing of the as-
+sociated frequency.7In real space, this softening
+is associated with the formation of distinct, local-
+ized zones where the plastic failure is nucleated.8
+In turn, this can trigger the failure of nearby zones,
+such that avalanches of plastic activity form that
+may “propagate” through the entire system.
+It has been argued that the macroscopic extent
+of these avalanches is a signature of the quasistatic
+dynamics, which gives the system enough time to
+propagate the failure throughout the system. Be-
+yond the quasistatic regime, i.e. farther away from
+the jammed state, the size of these events is ex-
+pected to be finite. Thus, one naturally finds an
+increasing length-scale that is connected with the
+flow arrest upon reducing the stress towards the
+threshold value.13–15
+Without external drive, an (un)jamming transi-
+tion can occur for decreasing particle volume frac-
+tion below a critical value, φc. This special point,
+which is only present in systems with purely repul-
+sive steric interactions, has been given the name
+“point J”.16,17At this point the average number
+of particle contacts jumps from a finite value z0to
+zero just below the transition. The value of z0is
+given by Maxwell’s estimate for the rigidity transi-
+tion18,19and signals the fact that at point J each
+particle has just enough contacts for a rigid/solid
+state to exist. This marginally rigid state is called
+“isostatic”. Compressing the system above its iso-
+static state a number of non-trivial scaling prop-
+erties emerge.16,20As the volume fraction is in-
+creased, additional contacts are generated accord-
+ing toδz∼δφ1/2. The shear modululs scales as
+G∼p/δzand vanishes at the transition (unlike
+the bulk modulus).16This scaling is seen to be a
+consequence of the non-affine deformation response
+of the system,21with particles preferring to rotate
+around rather than to press into each other.22As-
+sociated with the breakdown of rigidity at point J
+is the length-scale, l⋆∼δz−1,23,24which quantifies
+the size overwhich additional contacts stabilize themarginally rigid isostatic state.
+In this article we present results from quasistatic
+and small strain-rate flow simulations of a two-
+dimensional system in the vicinity of point J. To-
+gether with the linear elastic shear modulus, at
+point J also the yield-stress σyvanishes.25–30Thus,
+point J is connected with a transition from plastic-
+flow behavior ( φ > φ c,σy>0) to normal fluid
+flow (φ < φ c,σy= 0), with either Newtonian26
+or Bagnold rheology27,29at small strain-rates. In
+consequence, both (un)jamming mechanisms as de-
+scribed above are present at the same time: the
+flow arrest, as experienced by lowering the stress
+towards threshold, is combined with the vanishing
+of the threshold itself.
+In this study we want to address two questions:
+In how far do the general plastic flow properties
+carry overto this situation of small or, indeed, van-
+ishing yield-stress? Is the vicinity to point J and its
+isostatic state at all relevant for the flow properties
+? It will be shown that while the stress fluctuations
+reflect the critical properties at point J, dynamical
+correlations are typical those observed in the flow
+of elasto-plastic solids.
+We will approach these questions starting with
+the quasistatic-flow regime. The advantage of qua-
+sistatic simulations is to provide a clean way of ac-
+cessing the transition region between elastic, solid-
+like behavior and the onset of flow. In the qua-
+sistatic regime flow is generated by a succession of
+(force-)equilibratedsolidstates. Thus, onecancon-
+nect a liquid-like flow with the ensemble of solid
+states that are visited along the trajectory through
+phase-space.
+In Section 3.1, we study the instantaneousstatis-
+tical properties of the configurations generated by
+this flow trajectory at zero strain rate, and show
+that they displaylargefluctuations in severalquan-
+tities, that are associated with the proximity to the
+jamming point.
+In Section 3.2, we follow the analysis of recent
+experiments31and use a ”four point correlation”
+tool to define a dynamical correlation length that
+characterizes the extension of the dynamical het-
+erogeneities observed in the flow process. This dy-
+namical length scale is shown to scaleas the system
+size in the zero strain rate limit, independently of
+the distance to point J. The heterogeneity in the
+system is maximal for strains that correspond to
+the typical duration between the plastic avalanches
+2described above.
+We complement this analysis with preliminary
+results from dissipative molecular-dynamics sim-
+ulations that access strain-rates above the qua-
+sistatic regime. This allows us to assess the im-
+portance of dynamic effects in limiting access to
+certain regions of the landscape. Indeed, the re-
+sults at larger strain rate are system size indepen-
+dent, and reveala surprisinggrowthofthe strength
+of the heterogeneities with increasing packing frac-
+tion away from φc.
+2 Simulations
+Our system consists of Nsoft spherical particles
+with harmonic contact interactions
+E(r) =k(r−rc)2. (1)
+Two particles, having radii riandrj, only interact,
+whentheyare“incontact”, i.e. whentheirdistance
+ris less than the interaction diameter rc=ri+rj.
+This system has been studied in several contexts,
+for example in.16,26,32The mixture consists of two
+types of particles (50 : 50) with radii r1= 0.5d
+andr2= 0.7din two-dimensions. Three different
+system-sizes have been simulated with N= 900,
+1600 and 2500 particles, respectively. The unit
+of length is the diameter, d, of the smaller par-
+ticle, the unit of energy is kd2, where kis the
+spring constant of the interaction potential. We
+usequasistaticshearsimulation, andcomparesome
+of the results with those obtained from dissipative
+molecular-dynamics simulations at zero tempera-
+ture.
+Quasistatic simulations consists of successively
+applying small steps of shear followed by a mini-
+mization of the total potential energy. The shear
+is implemented with Lee-Edwards boundary con-
+ditions with an elementary strain step of ∆ γ=
+5·10−5. After each change in boundary condi-
+tions the particles are moved affinely to define the
+starting configuration for the minimization, which
+isperformedusingconjugategradienttechniques.33
+The minimization is stopped when the nearest en-
+ergy minimum is found. Thus, as the energy land-
+scape evolves under shear the system always re-
+mains at a local energy minimum, characterized by
+a potential energy, a pressure pand a shear stress
+σ.The molecular dynamics simulations were per-
+formed by integrating Newton’s equations of mo-
+tion with elastic forces as deduced from Eq. (1)
+and dissipative forces
+/vectorFij=−b[(/vector vi−/vector vj)·ˆrij]ˆrij, (2)
+proportional to the velocity differences along the
+direction ˆ rijthat connects the particle pair. The
+damping coefficient is chosen to be b= 1. Rough
+boundaries are used during the shear, the bound-
+aries being built by freezing some particles at the
+extreme ends in the y-direction, from a quenched
+liquid configuration at a given φ. The system is
+sheared by driving one of the walls at a fixed ve-
+locity in the xdirection, using periodic boundary
+conditions in this direction.
+For all system sizes, the distance between the
+top and bottom boundaries is 52 .8dand each of
+the boundaries has a thickness of 4 .2d. The system
+size is changed by modifying the length of the box
+in thex-direction.
+3 Results
+3.1 Quasistatic simulations
+As is readily apparent from Fig. 1, a typical feature
+of quasistatic stress-strain relations is the interplay
+of “elastic branches” and “plastic events”. During
+elastic branches stress grows linearly with strain
+and the response is reversible. In plastic events the
+stress drops rapidly and energy is dissipated.
+In setting the elementary strain step, ∆ γ, care
+mustbe takento properlyresolvetheseevents. Too
+large strain steps would make the simulations miss
+certain relaxation events. We chose a strain step
+small enough, such that most minimization steps
+do not involve any plastic relaxations. In conse-
+quence, the elastic branches are well resolved, each
+consisting of many individual strain steps.
+The succession of elastic and plastic events de-
+fines the flow of the material just above its yield-
+stressσy(φ). The value of the yield-stress depends
+on volume-fraction and nominally vanishes at φc
+(see Fig. 2). For finite systems, however, finite-size
+effects dominate close to φcsuch that one cannot
+observe a clear vanishing of σy. Rather, as Fig. 1
+shows, one enters an intermittent regime, i.e. a fi-
+nite interval in volume-fraction in which the stress-
+3 01 10-42 10-43 10-4
+ 1.51 1.55 1.59 1.63 1.67σ
+γ
+ 02 10-54 10-56 10-5
+ 1.51 1.55 1.59 1.63 1.67σ
+γ
+Figure 1: Stress-strain relation for two different
+volume-fractions, φ= 0.847 (top) and φ=φc=
+0.8433 (bottom). At volume-fractions close to φc
+thesignalisintermittentshowinglongquietregions
+where the system flows without the building up of
+stress.
+signal shows a coexistence between jammed and
+“freely-flowing” states. This is evidence of a distri-
+bution of jamming thresholds, P(φc), which sharp-
+ens with increasing the system-size.16,28A finite-
+size scaling analysis of this distribution allows one
+to extract the critical volume-fraction. To this end
+we count the number of jamming events that lead
+from the freely-flowing state to the jammed state
+and back1. We find a maximum number of events
+at a certain φc(L), which can be extrapolated to
+L=∞to define the critical volume fraction of
+our simulation. The value we find, φc= 0.8433 is
+slightly higher than what has been obtained pre-
+viously, however, evidence is mounting that φcis
+non-universal35and depends on the details of the
+ensemble preparation. Scaling properties in the
+vicinity of a jamming threshold, on the other hand,
+appear to be universal.35
+In Fig. 2 we display the yield-stress σy, as a func-
+tionofvolume-fraction, δφ=φ−φc, andpressure p.
+Finite-size effects are particularly strong when us-
+ingφascontrolvariable. Intheintermittentregime
+the average stress levels off to a system-size depen-
+dent value.
+1For a closer illustration of a jamming event see the sup-
+porting material to our previous paper3410-510-410-310-2
+10-410-310-210-1σy
+δφ~x1.05N = 2500
+1600
+900
+400
+10-610-510-410-310-2
+10-510-410-310-210-1σy
+p~x0.95N = 2500
+1600
+900
+400
+Figure 2: Average yield-stress as function of
+volume-fraction (left) and pressure (right). The
+yield-stress is determined as an average over stress-
+values just before plastic events occur, i.e at the
+top end of each elastic branch.
+Much better scaling behavior can be obtained,
+when using pressure as control variable, as this is
+characterized by the same finite-size effects as the
+shear-stress. In the following we will therefore use
+pressure as control variable. Be aware, however,
+that we do notrun pressure-controlledsimulations,
+as for example Peyneau and Roux25but use the
+average pressure, /angbracketleftp/angbracketright(φ), only to plot our simula-
+tion results. The value σ/p≈0.1 obtained from
+Fig. 2 is consistent with these pressure-controlled
+simulations. On the other hand, the scaling with
+volume-fraction, p∼δφ1.1, is slightly strongerthan
+in linear elasticity at zero stress,16,20where the
+pressure simply scales as δφ. In view of the strong
+finite-size effects, the scaling with volume-fraction
+should, however, be taken with care.
+In the following we show results from five dif-
+ferent volume-fractions, φ= 0.846, 0.848, 0.85,
+0.86 andφ= 0.9, which are all above φcand
+outsidethe intermittent regime (i.e. no freely-
+flowing zero-stress states occur). For each volume-
+fraction we study three different system sizes with
+N= 900,1600 and 2500 particles.
+3.1.1 Elastic properties
+As reviewed in the introduction a hallmark of the
+elasticity of solids in the vicinity of point J is the
+scaling of the linear elastic shear modulus g∼
+p1/2. Similarly, the number of inter-particle con-
+tacts scale as z=z0+Ap1/2.
+We have analyzed the elastic branches in the
+steady-state flow to find (Figs. 3 and 4) that the
+same scaling properties characterize the average
+nonlinear elastic modulus gavg, which we define as
+4the local slope of the stress-strain curve, and also
+the associated contact numbers zavg. If we take
+these scaling properties as a signature of the criti-
+cality of point J, we can conclude that for the range
+of volume-fractions considered we are in the “criti-
+cal regime”.
+10-210-1
+10-310-210-1g
+p~x0.5N = 2500
+N = 1600
+N = 900
+10-310-210-1
+-9-6-3 0 3 6∆gP(g)
+(g-gavg)/∆g
+Figure 3: (Top) Average nonlinear elastic shear
+modulus gavgas function of pressure p. (Bottom)
+Probability distribution P(g) centered around av-
+erage value gavgand rescaled width according to
+∆g=p0.25/N0.5. Black solid line is a Gaussian
+pdf.
+As additional characterizationof the ensemble of
+elasticstateswereporttheprobabilitydistributions
+of shear moduli and contact numbers, respectively.
+Maybe surprisingly all the obtained distributions
+have approximately the same shape and can be su-
+perimposed on a single master curve. To achieve
+this we center each distribution around the average
+value and rescale the width with a factor pαNβ.
+By looking carefully at the individual distribu-
+tions we do observe a slight trend towards the de-
+velopment of non-Gaussian tails close to φc. While
+non-Gaussiandistributions areto be expected close10-1100
+10-310-210-1z-z0
+p~x0.5N = 2500
+N = 1600
+N = 900
+ 0.01 0.1 1
+-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6∆zP(z)
+(z-zavg)/∆z
+Figure 4: (Top) Average contact number zas func-
+tion of pressure p. The values for z0≡z(p= 0) are
+determined from a best fit. They are smaller than
+zc= 4 as to the presence of rattlers, which have
+not been accounted for. (Bottom) Probability dis-
+tribution P(z) centered around average value zavg
+and rescaled width according to ∆ z=p−0.35/N0.5
+to critical points,36the effect is quite small and all
+distributions have a well developed Gaussian core.
+The pronounced small- gtail ofP(g) is due to shear
+moduli thatextend downtozero. Similartailshave
+been observed in8and related to a softening of the
+response upon approach towards plastic instabil-
+ities. Indeed, we found that manually suppress-
+ing states close to plastic events, the weight in the
+small-gtail is reduced.
+For the width of the g-distribution we obtain
+∆g=p0.25/N0.5. Thus, the absolute width of
+the distribution decreaseswith decreasingpressure,
+while the relative width, ∆ g/gavgdiverges at point
+J. For the contact numbers, on the other hand,
+we find a divergence of the absolute width itself,
+∆z=p−0.35/N0.5. These enhanced fluctuations
+certainly support the view of δzas an order pa-
+5rameter for a continuous jamming transition. The
+quantity ∆ z2Nwould then be analogous to a sus-
+ceptibility, χ∼p−γ, diverging with an exponent
+γ= 0.7.
+OurresultsaredifferentthanthoseofHenkesand
+Chakraborty,37wherefluctuationsof zarefoundto
+be independent of pressure, ∆ z∼p0. Note, how-
+ever, the subtle difference in ensemble. These au-
+thorsstudyapressure-ensemble,inwhichstatesare
+generated by quenching randomparticle configura-
+tions to the local minimum of the potential energy
+landscape(similarto the procedurein Ref.16). One
+mayviewthesestatesastheinherentstructuresofa
+high temperature liquid. Our ensemble then corre-
+sponds to the inherent structures of a driven glassy
+material. We fix volume-fraction and sample only
+states that are connected by the trajectory of the
+system in phase-space. The ensemble therefore re-
+flects the dynamics of the system and the region of
+phase-space where it is guided to.
+3.1.2 Yield properties
+We now go beyond the properties of the elastic
+states and discuss aspects related to their failure
+during the plastic events. As indicated in the in-
+troduction, plastic events can be viewed as bifurca-
+tionsin energy-landscape. Alocalenergyminimum
+vanishes and the system has to search for a new
+minimum at lower energy and stress. In quasistatic
+dynamics this process is instantaneous. The asso-
+ciated stress-drop is therefore visible as a vertical
+line in the stress-strain relation (Fig. 1).
+In the following we characterize the amount of
+dissipation during a plastic event by the associated
+stress-drop, ∆ σ. The frequency of plastic events is
+discussed in terms of the length of elastic branches,
+∆γavg.
+Just like the probability distributions within the
+elastic states, the functions P(∆σ) andP(∆γavg)
+(Figs. 5 and 6) are universal and can be rescaled
+on a single master-curve. Here, it is sufficient to
+use the first moment of the distribution, i.e. the
+ensemble-averaged stress drops and elastic-branch
+lengths, respectively. As the logarithmic scale in
+the inset in Fig. 5 shows, the collapse for the stress-
+drop distribution is quite good for large as well
+as for small stress drops. The black line repre-
+sents a fit of the form ∆ σ−1exp(−∆σ/σL) with
+the stress-scale σL≈5∆σavg. The intermediate10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg10-410-2100
+ 0 5 10 15 20 25 30∆σavgP(∆σ)
+∆σ/∆σavg0.01 1~x-1exp(-x/x0)
+10-510-4
+10-310-2∆σavg
+pN = 900
+N = 1600
+N = 2500
+Figure 5: (Top) Distribution of stress-drops nor-
+malizedwith averagevalues∆ σavg. Inset showsthe
+same figure in a log-log representation. (Bottom)
+Scaling of ∆ σavgwith pressure p.
+power-law behaviour P∼∆σ−1reflects the lack of
+scale related to a typical event size. The only rele-
+vant scale is the exponential cut-off at σL. Tewari
+et al.38have reported an exponent of −0.7 in the
+energy-drop distribution at finite-strain rates. The
+simulated systems are somewhat smaller, however.
+Kablaet al.39have found an exponent of −1.5 in
+a vertex model for foams, in agreement with renor-
+malization group arguments.40
+The exponential tail has been observed in sev-
+eral different studies8,11,11,41,42in two and in three
+spatialdimensions. Tsamados et al.11havefurther-
+more related this feature of the stress-drop distri-
+bution to the diversity of local flow-defects causing
+the plastic event.
+A similar universality has been observed by Mal-
+oney and Lemaˆ ıtre.8Their simulations are con-
+ducted with three different interaction potentials
+but without changing the density, which is set to
+high values far away from the rigidity transition.
+610-310-210-1100
+ 0 1 2 3 4 5∆γavgP(∆γ)
+∆γ/∆γavg
+4⋅10-410-32⋅10-3
+10-310-2∆γavg
+pN = 900
+N = 1600
+N = 2500
+Figure 6: (Top) Distribution of elastic branch
+lengthnormalizedwithaveragevalues∆ γavg. (Bot-
+tom) Scaling of ∆ γavgwith pressure p. The ratio
+g= ∆σavg/∆γavg, which defines a shear-modulus,
+is consistent with the scaling in Fig. 3.
+The authors have argued for a universal value of
+the “flow-strain” σy/gof a few percent. Appar-
+ently, this can only be true far away from φc. As
+the yield-stress vanishes faster than the shear mod-
+ulus, one finds a ratio σy/g∼δφ1/2that vanishes
+at point J. Thus, particle configurations at the on-
+set of jamming are highly fragile and susceptible to
+even minute changes in the boundary conditions.
+The average stress-drop as well as the average
+length of elastic branches change with pressure and
+system-size as displayed in Figs. 5 and 62. As a
+function of pressure one observes an increase but
+with a slope that depends on system-size. The av-
+2Note, that these values also depend on the elementary
+strain step used in the simulations. For any finite step-size
+there will be some small plastic events, that cannot be re-
+solved but only lead to an apparent reduction of the stress
+increase. This will artificially increase the length of elas tic
+branches and decrease the weight of the small ∆ σ-tail of the
+stress-drop distribution.erage stress-drops increase somewhat slower with
+pressure than the yield-stress3. The relative stress
+fluctuations∆ σavg/σyarethusslightlyenhancedat
+small pressures close to φc. The same trend is also
+visible in the total stress fluctuations as calculated
+by/angbracketleftbig
+(σ−/angbracketleftσ/angbracketright)2/angbracketrightbig
+.
+The overall scale of both, ∆ σavgand ∆γavg, de-
+creases with system-size to give a smooth stress-
+strain relation in the thermodynamic limit. Pre-
+vious studies8,10,11have observed a scaling of the
+stress-drops with N−1/2. This includes8a sys-
+tem of harmonically interacting particles, as stud-
+ied here, but at a rather high pressure. In gen-
+eral, we observe a weaker dependence on system-
+size, with an effective exponent that increases with
+pressure. Our data is consistent, however, with the
+value of 1 /2 being the relevant high-pressure limit.
+3.2 Dynamical correlations
+Let us now turn to the dynamics of the system. In
+particular we want to characterize dynamic corre-
+lations in the motion of particles. While at volume-
+fractions above φcthe isostaticity length-scale l⋆is
+clearly finite4, there is nevertheless a large dynam-
+ical length-scale related to the flow arrest. This
+has, for example, been evidenced in a system of
+Lennard-Jones particles with dissipative dynam-
+ics.15We will show below that a similar length-
+scale occurs in our system of purely repulsively in-
+teracting particles, independent of the distance to
+φc.
+Let us start by presenting the results from the
+quasistatic simulations. To define a dynamical cor-
+relation length we study heterogeneities in the par-
+ticle mobilities. To this end we use the overlap-
+function43,44
+/angbracketleftQ(γ,a)/angbracketright=/angbracketleftBigg
+1
+NN/summationdisplay
+i=1exp/bracketleftbigg
+−uina(γ)2
+2a2/bracketrightbigg/angbracketrightBigg
+,(3)
+of particles undergoing nonaffine displacements
+uinaduring a strain interval of γ5. Particles mov-
+ing farther than the distance a(“mobile”), have
+3The effective exponents range from 0 .8 to 0.9 as com-
+pared to an exponent 0 .95 for the yield-stress
+4If we assume for the isostaticity length l⋆= 1/δz, we
+would have values l⋆≈5 atφ= 0.846 and l⋆≈1 atφ= 0.9
+(z-values taken from Fig. 4).
+5To calculate the overlap function we use the non-affine
+displacements in the gradient direction (irrespective of t he
+distance to the wall).
+7Q≈0, while those that stay within this distance
+(“immobile”) have Q≈1.
+As a function of strain γ, the average over-
+lap/angbracketleftQ/angbracketrightwill decay, when particle displacements
+unaare comparable to the probing length-scale a.
+The overlap function is similar to the intermedi-
+ate scattering function with wave-vector q∼1/a.
+Thus,asets the probing length-scale. The decay
+ofQ(γ,a) then gives an associated structural re-
+laxation strain, γ⋆(a), on which particle positions
+decorrelate.
+In the following we are interested in the dynami-
+cal heterogeneity of Qand the fluctuations around
+its average value
+χ4(a,γ) =N/parenleftBig/angbracketleftbig
+Q(γ,a)2/angbracketrightbig
+−/angbracketleftQa(γ,a)/angbracketright2/parenrightBig
+,(4)
+which defines the (self-part of the) dynamical sus-
+ceptibility χ4. This is displayed in Fig. 7 as a func-
+tion of both strain γand probing length-scale a.
+For each γit has a well defined peak (at a⋆(γ)) that
+parallels the decay of the overlap function /angbracketleftQ/angbracketright.31
+ 0 50 100 150 200
+10-410-310-210-1χ4
+a
+Figure7: Dynamicalsusceptibility χ4asfunctionof
+probing length-scale afor various different strains
+γ= (5,20,50,200,500,2000)·10−5(from left to
+right) and φ= 0.9. The maxima of the curves
+(black circles) define the amplitude h(γ).
+The strength of the correlations are encoded in
+the peak-height, h(γ)≡χ4(a⋆(γ),γ) (black circles
+in Fig. 7)). As χ4can be written as the integral
+over a correlation function, it is connected to the
+correlation volume, or to the number of correlated
+particles. Assuming that this volume forms a com-
+pact region in space45,46we can relate the am-
+plitude of χ4to a dynamic correlation length via
+ξ2(γ) =h(γ). 0.02 0.04 0.06 0.08
+10-1100101102h/N
+γ/∆γavg
+Figure 8: Amplitude h(γ) as taken from the qua-
+sistatic simulations. Data for different volume-
+fractions and system-sizes. The axes are nor-
+malized according to the scaling form h(γ) =
+N˜h(γ/∆γavg), with ∆ γavgtaken from Fig 6.
+Followingthe maxima in Fig. 7 from left to right,
+one sees that the amplitude first increases and then
+quickly drops to small values. This implies that
+thereisafinitestrain γ, atwhich χ4presentsan ab-
+solutemaximum. Toextractthismaximumweplot
+in Fig. 8 the amplitude h(γ) for various volume-
+fractions φand system-sizes N.
+There are two surprising features in this plot.
+First, by rescaling the strain-axis with the aver-
+age length of elastic branches, ∆ γavg(see Fig. 6)
+we find reasonable scaling collapse for all studied
+volume-fractions and system-sizes. This implies
+thatcooperativity,asmeasuredbytheamplitudeof
+χ4and the length of elastic branches are intimately
+related. The frequency of plastic events sets the
+strain-scale for dynamical heterogeneities. As the
+lengthoftheelasticbranchesdecreaseswith system
+size, the absolute maximum shifts towards smaller
+strains, with h(γ) becoming effectively a decreasing
+function of γin the thermodynamic limit.
+The second surprising feature in Fig. 8 is the
+system-size dependence of h. It turns out that
+h/Nrather than hitself is independent of system-
+size, indicating a finite variance of the distribu-
+tion ofQvalues in the thermodynamic limit (see
+Eq. (4)). Assuming the connection with the corre-
+lation length to hold, ξ2∼h, this implies a cor-
+relation length that is proportional to the length
+of the simulation box, ξ≈0.3L, independent of
+volume-fraction and distance to point J. This illus-
+trates the fact that quasistatic dynamics is inher-
+8 0 50 100 150 200 250
+1e-05 1e-04 1e-03 1e-02 1e-01 1e+00χ4
+γqs
+1e-06
+1e-05
+3e-05
+5e-05
+1e-04
+1e-03
+Figure 9: χ4(γ) for different strain-rates ˙ γand
+a= 0.01. comparison with quasistatic (‘qs’) sim-
+ulations. Note the different boundary conditions
+used. MD simulations are with walls, while qua-
+sistatic simulations have periodic boundary condi-
+tions. This may explain the difference in the am-
+plitude.
+ently dominated by system-size effects, as already
+shown in previous works .14,15,28,47Note, that this
+system-size dependence is of different origin than
+the finite-size effects present within the above men-
+tioned intermittent regime, which occurs close to
+φc. The intermittency can be avoided by staying
+away from φc. In contrast, the system-size de-
+pendence encountered here is quite independent of
+volume-fraction, but rather a generic feature of the
+quasistatic regime, as we show now.
+Tothisendlet usturntothemolecular-dynamics
+simulations. We will show that the dependence
+on system-size indeed reflects the saturation of a
+length-scale that is finite for larger strain-rates and
+increases towards the quasistatic regime6. As
+Fig. 9 shows, the amplitude of χ4increases when
+reducingthe strain-rate( a= 0.01,φ= 0.9)andap-
+proachesthe quasistaticlimit forsmall strain-rates.
+Also the strain γm, at which χ4is maximal is very
+well reproduced in the dynamic simulation.
+Fig. 10 demonstrates a saturation of the ampli-
+tudeχm≡χ4(γm) at small strain-rates indicating
+that the quasistatic regime is entered. Comparing
+with the quasistatic simulation, we find somewhat
+smaller values for the amplitude. It should be re-
+membered, however, that different boundary con-
+6A more detailed account of these simulations will be
+presented in: P. Chaudhuri and L. Bocquet, in preparation
+(2010). 1 2 4 10
+1e-061e-051e-041e-03χm1/2
+γ.N=2500
+N=1600
+N=900 1 2 4
+0.84 0.89 0.94 0.99 χm1/2
+φγ.=1e-04
+Figure 10: (Left) Peak-height χm=χ4(γm) as
+determined from Fig. 9. The saturation at small
+strain-ratesis an indication of the quasistatic limit,
+in which χm∼N. In contrast, at high strain-
+rates no significant dependence on system-size is
+observed. (Right) Peak-height χmas function of
+volume-fraction.
+ditions have been used. The rough walls used in
+the molecular dynamics simulations are likely re-
+sponsible for the reduction of the peak-height as
+compared to the quasistatic simulations (which are
+performed with periodic boundary conditions).
+The presence of the quasistatic regime is also ev-
+idenced by the fact that the amplitude χmwithin
+the plateau depends on system-size, just as in the
+quasistatic simulations. Outside this regime, on
+the other hand, no significant N-dependence is ob-
+served. In effect this means that the quasistatic
+regime shrinks with increasing system-size. The
+strainrate ˙ γqs(N) that describes the crossover to
+the quasistatic regime decreases with N. This is
+in line with Refs.,14,15where a power-law depen-
+denceγqs∼1/Nis reported. From our data we
+cannot make any definitive statement about this
+dependence.
+Theresultsarefurthermoreconsistentwiththose
+of Onoet al.48The lowest strain-rate accessible in
+the latter study was ˙ γ= 0.0001. At this strain-
+rate the correlation length was observed to be on
+the order of 3 in agreement with our data.
+We also probed the volume-fraction dependence,
+by performing runs at φ= 0.85,0.87,0.9,0.95 and
+1. The resulting amplitude of χ4is given in Fig. 10.
+Interestingly we observe a mild increase in the am-
+9plitude with volume-fraction, signalling enhanced
+correlations awayfromφc.
+This is not due to the special choice of the pa-
+rameter a= 0.01. We have found the same trend
+when fixing γand viewing χ4as function of aas
+for the quasistaticsimulationsin Fig. 7. Finally, we
+have also calculated the absolute maximum ofχ4,
+viewed as function of both aandγ. In all cases, the
+amplitude increases with volume-fraction. Both,
+the increase of the length-scale with lowering the
+strain-rate and the increase with volume-fraction
+are consistent with the recently proposed elasto-
+plastic model of Bocquet et al.13
+Given the trend in Fig. 10, one may specu-
+late abouta vanishingdynamicalcorrelationlength
+(taken at constant strain-rate ˙ γ= 10−4outside the
+quasistatic regime), as φcis approached. Such a
+behavior is indeed compatible with our data and
+has recently been observed in the rheology of a
+concentrated emulsion confined in gaps of differ-
+ent thickness.49Here we interpret this surprising
+feature in the following way: at a given packing
+fraction, dynamical correlations increase with de-
+creasing ˙ γ, and saturate at a N-dependent value in
+the quasistatic regime. This cross-over to the qua-
+sistatic regime does not only depend on system-size
+but also on packing fraction: ˙ γqs(N,φ).
+Forφcloser to φcthe energy landscape becomes
+increasingly flat. Particle relaxations take longer50
+and smaller strain-rates are needed to allow for full
+relaxation into the local energy minimum. Thus,
+smaller strain-rates are needed to reach the qua-
+sistatic behavior and ˙ γqsdecreases towards φc.
+Hence, reducing φtowardsφcat a fixed strain rate
+is ”equivalent” to increasing the strain rate relative
+to ˙γqs, and results in a decrease of dynamical corre-
+lations. In contrast, corrrelations taken at a strain
+rate ˙γ= ˙γqs(φ), are independent of φand given by
+their quasi-static values, as displayed in Fig. 8.
+4 Discussion and Conclusion
+We have discussed the small strain-rate elasto-
+plastic flow of an athermal model system of soft
+harmonic spheres. In particular, we were inter-
+ested in the flow properties at and above a critical
+volume-fraction (point J), at which the yield-stress
+of the material vanishes. This regime combines the
+more traditional elasto-plastic flow of solids abovetheir yield-stresswith the breakdownofthe rigidity
+of the solid state at point J.
+Wefoundthatthisbreakdownisvisibleintheen-
+semble of states visited during a flow simulation in
+a similar way as in the linear elasticity of the solid.
+For example (Fig. 4), we showed that the average
+number of particle contacts scale with the square-
+root of pressure, just as in linear elasticity. In con-
+trast, the fluctuations around this average value
+showadistinctbehaviorthathasnotbeenobserved
+previously. We showed that the contact-number
+fluctuations actually diverge upon approaching the
+critical volume-fraction from above, making the
+contact number an ideal candidate for an order
+parameter of a continuous jamming transition as
+observed under steady shear. The relative fluctua-
+tions of the shear modulus and those of the shear
+stress also diverge in the same limit. Going beyond
+the characterization of the average elastic proper-
+ties we have studied the statistics of plastic events
+(Figs. 5 and 6). It seems that all distributions have
+universalscalingformsreminiscentofstandardcrit-
+ical phenomena.
+From all these results, it would be tempting to
+say that it is the energy landscape as a whole
+that becomes critical at point J. Isostatic elastic-
+ity would then be just one aspect of this criticality,
+another one could be the intermediate power-law
+tail in the stress-drop distribution. This critical as-
+pect is also illustrated by the intermittency in the
+stressresponseof finite-size systems (see Fig.1) and
+by the growth of an isostatic correlation length in
+thequasistaticresponsewhenpointJisapproached
+from below.28
+At strain-rates above the quasistatic regime, the
+dynamics limits access to certain regions of the en-
+ergy landscape. While the dynamics is still highly
+correlated, the dynamical correlation length, as
+measured by the amplitude of the four-point sus-
+ceptibility χ4, remains finite and actually decreases
+with lowering the volume-fraction towards φc.
+In the quasistatic regime we have shown that χ4
+reflects, in two ways, the interplay of elastic load-
+ing and plastic energy release (Fig. 8). First, the
+typicalstrain-scaleofheterogeneityissetbythefre-
+quency of plastic events. Second, the amplitude of
+χ4scaleswithsystem-size,whichhighlightsthefact
+that the quasistatic, plastic flow regime is, in fact,
+a finite-size dominated regime with a correlation
+length that is limited by system size. This behavior
+10should be contrasted with the one observed below
+φc, where a large but finite correlation length has
+been identified, which is governed by the approach
+to point J.51
+Upon increasing the strain-rate we have shown
+that the correlation length starts to decrease out-
+side the finite-size scaling regime (Fig. 10). Olsson
+and Teitel26infer from their flow simulations that
+shear-stressshould be viewed as a “relevantpertur-
+bation” to point J, such that a different fixed-point
+and indeed different physics is relevant for the flow
+behaviouratfinite stress. Ourfindingssupport this
+picture for the dynamical correlations, which ap-
+pear to behave similarly to those observed in mod-
+els of elasto-plastic flow13,14,52or in low temper-
+ature glasses:10,15correlations increase upon low-
+ering the strain-rate and saturate at a system-size
+dependent value in the quasistatic regime.
+The flow behaviour in the vicinity of point J is
+therefore influenced by a complex combination of
+two critical behaviour. Large stress fluctuations
+(relativeto the yield stress), orgeometricalchanges
+(number of neighbours) reflect the enhanced sen-
+sitivity of the material to small changes in exter-
+nal conditions at point J, and are specific proper-
+ties of the energy landscape at this point. On the
+other hand, dynamical correlations above point J
+are dominated by the system size, and build up
+progressively as the strain rate is decreased, as in
+any elasto-plastic system, and are not particularly
+sensitive to the proximity of point J.
+Acknowledgments The authors acknowledge
+fruitful discussions with Ludovic Berthier, Lyd´ eric
+Bocquet, Erwin Frey; Craig Maloney and Michel
+Tsamados, as well as thank the von-Humboldt
+Feodor-Lynen, the Marie-Curie Eurosim and the
+ANR Syscom program for financial support.
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+arXiv:1001.0659v2  [math-ph]  16 Feb 2010On the Veldkamp Space of GQ(4,2)
+Metod Saniga
+Astronomical Institute, Slovak Academy of Sciences
+SK-05960 Tatransk´ a Lomnica, Slovak Republic
+(msaniga@astro.sk)
+(16 February 2010)
+Abstract
+TheVeldkampspace,inthesenseofBuekenhoutandCohen, ofthe generalizedquadrangleGQ(4,2)
+is shown not to be a (partial) linear space by simply giving several exam ples of Veldkamp lines
+(V-lines) having two or even three Veldkamp points (V-points) in com mon. Alongside the ordinary
+V-lines of size five, one also finds V-lines of cardinality three and two. There, however, exists a
+subspace of the Veldkamp space isomorphic to PG(3,4) having 45 pe rps and 40 plane ovoids as its
+85 V-points, with its 357 V-lines being of four distinct types. A V-line o f the first type consists of
+five perps on a common line (altogether 27 of them), the second typ e features three perps and two
+ovoids sharing a tricentric triad (240 members), whilst the third and fourth type each comprises
+a perp and four ovoids in the rosette centered at the (common) ce nter of the perp (90). It is also
+pointed out that 160non-planeovoids(tripods) fallinto twodistinc t orbits— ofsizes40and 120—
+with respect to the stabilizer group of a copy of GQ(2,2); a tripod o f the first/second orbit sharing
+with the GQ(2,2) a tricentric/unicentric triad, respectively. Finally , three remarkable subconfig-
+urations of V-lines represented by fans of ovoids through a fixed o void are examined in some detail.
+Keywords: GQ(4,2) — Geometric Hyperplane — Veldkamp Space — PG(3,4)
+1 Introduction
+Generalized quadrangles of types GQ(2 ,t), witht= 1, 2, and 4, have recently been found to play
+a prominent role in quantum information and black hole physics; the fir st type for grasping the
+geometrical nature of the so-called Mermin squares [1, 2], the seco nd for underlying commutation
+properties between the elements of two-qubit Pauli group [1, 2, 3], and the third one for fully
+encoding the E6(6)symmetric entropy formula describing black holes and black strings in D= 5
+[4]. Whereas GQ(2,2) is isomorphic to its point-line dual, this is not the c ase with the remaining
+two geometries; the dual of GQ(2,1) being GQ(1,2), that of GQ(2 ,4) GQ(4,2) [5]. These two
+duals, strangely, did not appear in the above-mentioned physical c ontexts. It is, therefore, natural
+to ask why this is so. We shall try to shed light on this matter by invokin g the concept of the
+Veldkamp space of a point line incidence structure [6]. The Veldkamp sp ace of GQ(2,4) was shown
+to bealinearspaceisomorphictoPG(5,2)[7]. Here, weshalldemonst ratethat the Veldkamp space
+of GQ(4,2), due to the existence of two different kinds of ovoids in G Q(4,2), is not even a partial
+linear space, though it contains a (linear) subspace isomorphic to PG (3,4) in which GQ(4,2) lives
+as a non-degenerate Hermitian variety.
+2 Rudiments of the Theory of Finite Generalized Quadran-
+gles and Veldkamp Spaces
+To make our exposition as self-contained as possible, and for the re ader’s convenience as well, we
+will first gather all essential information about finite generalized qu adrangles [5], then introduce
+the concept of a geometric hyperplane [8] and, finally, that of the Veldkamp space of a point-line
+incidence geometry [6].
+Afinite generalized quadrangle oforder( s,t), usuallydenoted GQ( s,t), isan incidencestructure
+S= (P,B,I), where PandBare disjoint (non-empty) sets of objects, called respectively poin ts
+and lines, and where I is a symmetric point-line incidence relation satisf ying the following axioms
+[5]: (i) each point is incident with 1+ tlines (t≥1) and two distinct points are incident with at
+most one line; (ii) each line is incident with 1+ spoints (s≥1) and two distinct lines are incident
+with at most one point; and (iii) if xis a point and Lis a line not incident with x, then there
+1exists a unique pair ( y,M)∈P×Bfor which xIMIyIL; from these axioms it readily follows that
+|P|= (s+1)(st+1) and |B|= (t+1)(st+1). It is obvious that there exists a point-line duality
+with respect to which each of the axioms is self-dual. Interchanging points and lines in Sthus
+yields a generalized quadrangle SDof order ( t,s), called the dual of S. Ifs=t,Sis said to have
+orders. The generalized quadrangle of order ( s,1) is called a grid and that of order (1 ,t) a dual
+grid. A generalized quadrangle with both s >1 andt >1 is called thick. Every finite generalized
+quadrangle is obviously a partiallinear space, that is the point-line incidence structure where a)
+any line has at least two points and b) any two points are on at most on e line.
+Given two points xandyofSone writes x∼yand says that xandyare collinear if there
+exists a line LofSincident with both. For any x∈Pdefine the perp of xasx⊥={y∈P|y∼x}
+and note that x∈x⊥, being its center; obviously, |x⊥|= 1+s+st. Given an arbitrary subset Aof
+P, theperpofA,A⊥, is defined as A⊥=/intersectiontext{x⊥|x∈A}andA⊥⊥:= (A⊥)⊥; in particular, if xand
+yare two non-collinear points, then {x,y}⊥⊥is called a hyperbolic line (through them). A triple
+of pairwise non-collinear points of Sis called a triad; given any triad T, a point of T⊥is called its
+center and we say that Tis acentric, centric or unicentric according as |T⊥|is, respectively, zero,
+non-zero or one. An ovoid of a generalized quadrangle Sis a set of points of Ssuch that each line
+ofSis incident with exactly one point of the set; hence, each ovoid conta insst+1 points. The
+dual concept is that of spread; this is a set of lines such that every point ofSis on a unique line of
+the spread. A rosetteof ovoids is a set of ovoids through a given point xofSpartitioning the set
+of points non-collinear with x. Afanof ovoids is a set of ovoids partitioning the whole point set
+ofS; ifShas order ( s, t) then every rosette contains sovoids and every fan features s+1 ovoids.
+Ageometric hyperplane Hof a point-line geometry Γ( P,B) is a proper subset of Psuch that
+each line of Γ meets Hin one or all points [8]. For Γ = GQ( s,t), it is well known that His one
+of the following three kinds [5]: (i) the perp of a point x,x⊥; (ii) a (full) subquadrangle of order
+(s,t′),t′< t; and (iii) an ovoid.
+Finally, we shall introduce the notion of the Veldkamp space of a point-line incidence geometry
+Γ(P,B),V(Γ) [6].V(Γ)isthe spaceinwhich(i) apointisageometrichyperplaneofΓand(ii) aline
+is the collection H1H2of all geometric hyperplanes Hof Γ such that H1∩H2=H1∩H=H2∩H
+orH=Hi(i= 1,2), where H1andH2are distinct points of V(Γ).
+3 Basic Properties of GQ(4,2)
+The unique generalized quadrangle GQ(4,2), associated with the cla ssical group PGU 4(2), can be
+represented by 45 points and 27 lines of a non-degenerate Hermitia n surface H(3,4) in PG(3,4),
+the three-dimensional projective space over GF(4) [5, 9, 10]. Ev ery line has five points and there
+are three lines through everypoint. This quadranglefeatures bot h unicentric and tricentrictriads,1
+and has no spreads. There are 16 tricentric triads through each p oint; hence, their total number
+is 45×16/3 = 240.Its geometric hyperplanes are (45) perps of points and (200) ovo ids, because it
+has no subquadrangles of type GQ(4,1) [5].
+Obviously, perps correspond to the cuts of H(3,4) by its 45 tangent planes. As first shown by
+Brouwer and Wilbrink [9], ovoids fall into two distinct orbits of sizes 40 a nd 160. The ovoids of
+the first orbit are called planeovoids, as each of them represents a section of H(3,4) by one of the
+40 non-tangent planes. The ovoids of the second orbit are referr ed to as tripods, each comprising
+nine isotropic points on three hyperbolic lines {x,y}⊥⊥,{x,z}⊥⊥and{x,w}⊥⊥, where{x,y,z,w}
+is a basis of non-isotropic points; in other words, every tripod can b e viewed as a unique union of
+three tricentric triads. Given a plane ovoid Pand any two distinct points x,y∈ P, it isalways
+true that {x,y}⊥⊥⊆ P. Hence, {P\{x,y}⊥⊥}∪{x,y}⊥is again an ovoid, and all the tripods can
+be obtained in this manner from plane ovoids. GQ(4,2) contains both fans and rosettes of ovoids
+[9]. There are altogether 520 fans, each made of a plane ovoid and fo ur tripods and falling into
+two orbits of sizes 480 and 40, and 26 rosettes on a given point; two of them feature four plane
+ovoids, the remaining ones consist of four tripods each.
+Apart from perps and ovoids, GQ(4,2) is endowed with one more kind of distinguished subge-
+ometry — that isomorphic to the unique generalized quadrangle GQ(2 ,2); this is, however, not a
+geometric hyperplane. There are altogether 36 distinct copies of G Q(2,2) living inside GQ(4,2),
+and with any of them an ovoid is found to share a triad. For a plane ovo id this triad is always
+tricentric. For tripods, however, it can be either tricentric (40 of them) or unicentric (120 of
+them); in what follows we shall occasionally refer to the former/latt er as tri-tripods/uni-tripods,
+respectively. There exists a remarkable partitioning of the point-s et of GQ(4,2) in terms of three
+1GQ(4,2) is also endowed with acentric triads, but these are o f no relevance for our subsequent reasoning.
+2Figure 1: A diagrammatical model of the structure of GQ(4,2) who se points are illustrated by
+bullets and lines by straight segments, arcs of ellipses and/or parab olas, and two circles (for more
+details, see the text). Note a particular copy of GQ(2,2) (black), its complement (red) being
+nothing but famous Schl¨ afli’s double-six of lines.
+GQ(2,1)s and three GQ(1,2)s such that one of the latter group fo rms with each of the former
+group a GQ(2,2). Another noteworthy property is the existence of pairs of plane ovoids and/or
+tri-tripods on the common (tricentric) triad whose symmetric differ ence is a disjoint union of two
+GQ(1,2)s.
+All the above-mentioned properties can be ascertained — some rea dily, some requiring a bit
+of work — from a diagrammatical illustration of GQ(4,2) depicted in Fig ure 1. In this picture,
+of a form showing an automorphism of order five, all the 45 points of GQ(4,2) are represented by
+bullets, whereas its 27 lines have as many as four distinct represent ations: two are represented by
+(concentric) circles, five by arcs of parabolas touching the inner c ircle, another five by parabolas
+touching the outer circle, five by arcs of ellipses, and, finally, 10 by s traight line-segments. (Note
+that there aremany intersectionsof segmentsand arcsthat do n ot stand for any point ofGQ(4,2).)
+A copy of GQ(2,2) is also highlighted (black bullets, all line-segments a nd all arcs of ellipses).
+4 Distinguished Features of the Veldkamp Space of GQ(4,2)
+4.1 Linear Subspace Isomorphic to PG(3,4)
+In PG(3,4) a point and a plane are duals of each other. On the other hand, both a perp and a
+plane ovoid are associated each with a unique plane of PG(3,4). Henc e, disregarding tripods for
+the moment, we find a subspace of the Veldkamp space of GQ(4,2) t hat is isomorphic to PG(3,4).
+The 85 V-points of this subspace are 45 perps and 40 planar ovoids, and the 357 V-lines split into
+four distinct types as shown in Figure 2. A V-line of the first type (1- st row in Figure 2) consists
+of five perps on a common pentad of collinear points i.e. on a common lin e; clearly, there are 27
+V-lines of this type as each line leads to a unique V-line. A second-type V-line (2-nd row) features
+three perps and two ovoids sharing a tricentric triad; since each su ch triad defines a unique V-
+line, there are altogether 240 V-lines of this type. Third-/fourth- type V-lines (3-rd/4-th row) each
+comprises a perp and four ovoids in the rosette centered at the pe rp’s center (the only common
+point); their total number thus amounts to 2 ×45 = 90.
+4.2 Examples of V-lines on Three and Two Common Points
+In order to show that V(GQ(4,2)) is not a (partial) linear space it suffices to find two (or mor e)
+V-lines sharing two (or more) V-points. From the previous subsect ion it should already be fairly
+obvious that it is the existence of tripods that prevents V(GQ(4,2)) from being a linear space.
+3Figure 2: Representatives of the four different types of V-lines fo rming the linear subspace of
+V(GQ(4,2) isomorphic to PG(3,4). The encircled bullets represent t he points shared by all the
+five V-points forming a given V-line.
+To this end, let us have again a look at a type-two V-line of the PG(3,4 )-subspace, reproduced
+once again in Figure 3, top row. It turns out that we can get a new ty pe of V-line by replacing its
+two plane ovoids by two particular tripods, as shown in Figure 3, bott om row. Hence, we have an
+example of two distinct V-lines having three V-points (the three per ps) in common.
+Figure 3: An instance of two distinct V-lines having three V-points in c ommon. We note in passing
+that the tripods are tri-tripods with respect to the selected copy of GQ(2,2).
+The next couple of examples feature three (Figure 4) and four (Fig ure 5) V-lines through two
+common V-points. In the first case, the V-lines consist each of a pe rp and four tripods in the
+rosettecentered at the perp’s center; the perp and the first tripod are t he common V-points. With
+respect to the selected GQ(2,2), the first two tripods in each V-lin e are tri-tripods, the other two
+being uni-tripods. In the second case, each of the four V-lines rep resents a fanof ovoids, the first
+ovoid being planar, second a tri-tripod, and the remaining three uni- tripods.
+4Figure4: AnexampleofthreedifferentV-linesontwocommonV-point s(aperpanda(tri-)tripod).
+Figure 5: An example of four different V-lines sharing two V-points (a plane ovoid and a (tri-
+)tripod).
+4.3 Examples of V-lines of Size Three and Two
+That the structure of V(GQ(4,2)) is much more complex and intricate than that of any (par tial)
+linear space is, alongsidethe above-introducedexamples, also illustr ated by the existence of V-lines
+of cardinalityless than five. Thus, we found many V-lines ofsizethre e, like the one shownin Figure
+6. This V-line consists of a perp and two (uni-)tripods on a common un icentric triad; since two
+perps, obviously, cannot share a unicentric triad and a given perp c ontains 48 such triads, we find
+5altogether 45 ×48 = 2160 V-lines of this particular kind. Finally, there are also a large n umber of
+V-lines of size two; the one depicted in Figure 7 is composed of a plane o void and a tripod having
+six points in common.
+Figure 6: An example of V-line of size three.
+Figure 7: An example of V-line of size two.
+4.4 V-lines as Fans of Ovoids Through a Given Ovoid
+We shall finish the paper by having a closer look at very impressive sub configurations of V-lines
+represented by fans of ovoids. One first recalls (Sec.3) that a fa n of ovoids of GQ(4,2) is a pentad
+of (pairwise disjoint) ovoids partitioning the point set. Every fan co mprises — as already shown
+in Figure 5 — a plane ovoid, a tri-tripod and three uni-tripods; in the fig ures below these are
+represented by red-, green- and yellow-coloured circles, respec tively.
+Given aplaneovoid, there are no plane ovoids, 10 tri-tripods and 30 uni-tripods d isjoint from
+it. Figure 8 (“spider”) depicts the configuration of 13 fans of ovoid s through a plane ovoid (P). We
+see a remarkable pattern. One tri-tripod (4) has a special standin g as there are four distinct fans
+passing through it. Similarly, three out of 30 uni-tripods (namely 19, 25 and 28) have a particular
+footing as there are four distinct fans through each of them. Lea ving the distinguished fan {P, 4,
+19, 25, 28 }aside, the remaining twelve fans form four classes of cardinality thr ee each. In each
+class the fans have two ovoids in common (namely P and 4, P and 19, P a nd 25, and P and 28);
+whereas in the first class the two ovoids are both plane ovoids, in the remaining three they are of
+two distinct kinds. Hence, considering the multiplicities and characte r of mutual intersection of
+fans, the totality of the latter is split into five subsets in a [3 + 3 + 3] + ( 3) + 1 fashion; the “1”
+corresponds, naturally, to the distinguished fan {P, 4, 19, 25, 28 }.
+Given atri-tripod, there are 10 plane ovoids, no tri-tripods and 21 uni-tripod s disjoint from it.
+Figure 9 (“bee”) depicts the configuration of 13 fans of ovoids thr ough a tri-tripod (X). We again
+see a remarkable pattern. One plane ovoid (5) has a special standin g as there are four distinct fans
+passing through it. Similarly, three out of 21 uni-tripods (11, 13 and 20) have a particular footing
+as there are seven distinct fans through each of them. The split of the whole family of fans again
+shows a [3 + 3 + 3] + (3) + 1 pattern; the “1” here symbolizes the distin guished fan {5, X, 11,
+13, 20}. This case differs from the previous one in two crucial aspects. Firs t, the three size-three
+classes here are such that their fans have threeovoids in common (namely X, 11 and 13; X, 11 and
+20; and X, 13 and 20), whilst the fans of the fourth class share only a couple of ovoids (X and 5).
+Second, one finds three pairs of ovoids, namely X and 11, X and 13, a nd X and 20, such that there
+go as many as sevendistinct fans through each of them.
+Given auni-tripod, there are 10 plane ovoids, 7 tri-tripods and 14 uni-tripods disjoint from it.
+Figure 10 (“frog”) illustrates the configuration of 13 fans throug h a uni-tripod (O). We again see
+a remarkable pattern. One plane ovoid (8) has a special standing as there pass four different fans
+through it. Similarly, one tri-tripod (4) and two uni-tripods (2 and 10 ) have also a special footing,
+as each of them is shared by seven distinct fans. One sees here, ho wever, a slightly different [3 + 3]
+67 18 2111 26
+2310 20
+292222
+2 1542427
+1861330
+10
+6
+9 531
+P
+17319714 12 28 2 8 25 9 2242516
+Figure 8: A diagrammatic illustration of the qualitative relation betwee n the 13 fans of ovoids
+sharing a plane ovoid (P). Here and in the following two figures as well, e ach ovoid is represented
+by a numbered circle — the particulars of the numbering being complet ely irrelevant for our
+purpose — and every fan comprises five different circles (one red, o ne green and three yellow ones,
+their order being also irrelevant) joined by broken line-segments an d/or arcs of big circles (e.g.,
+{P, 1, 28, 13, 30 },{P, 2, 28, 12, 14 },{P, 3, 28, 6, 18 }, etc.).
+19
+18
+2
+817
+616211034
+17123
+26911
+51320
+X
+4 7 9
+14 10 151 5 8
+Figure 9: The same as in the previous figure for a tri-tripod (X). Not e a “tighter coupling” (i.e.,
+higher-multiplicity intersections) between the fans within each of th e three classes and between the
+classes themselves when compared with the plane ovoid case.
+70
+6
+9
+4 3105410 29
+5
+4 631761
+217118712 14 135238
+Figure 10: The same as in the previous two figures for a uni-tripod (O ).
++ (3) + {3}+ 1 factorization; the “1” stands for the distinguished fan {8, 4, O, 2, 10 }. Moreover,
+whilst omitting the fixed/central ovoid in the previous two cases leav es us with only two kinds of
+ovoids, this case yields all the three types.
+What is the nature of the three distinguished fans? This is quite easy to spot. The nine points
+of aplaneovoid can always be partitioned into three pairwise disjoint tricentric triads. Since a
+plane ovoid contains 12 tricentric triads, there exist fourdifferent ways such a partitioning can be
+done. Given any of them, the nine centers of the triads generate a (unique) tripod; the plane ovoid
+and its four associated tripods form a distinguished fan of ovoids. I n all the three above-discussed
+cases, the distinguished fan is the only one sharing with eachof the remaining twelve at least one
+additional ovoid apart from the common one, and thus fully deserve s its name.
+5 Conclusion
+We have furnished several examples showing that the Veldkamp spa ce of GQ(4,2) is not a (partial)
+linear space. This is in a sharp contrast with the case of the dual, GQ( 2,4), whose Veldkamp space
+is a linear space, being isomorphic to PG(5,2) [7]. We surmise that this f act, among other things,
+might also contain an important clue why it is GQ(2,4), not GQ(4,2), w hich is relevant for a
+particular black-hole/qubit correspondence [4].
+Acknowledgements
+This work was conceived within the framework of the Cooperation Gr oup “Finite Projective Ring
+Geometries: An Intriguing Emerging Link Between Quantum Informa tion Theory, Black-Hole
+Physics, and Chemistry of Coupling” at the Center for Interdisciplin ary Research (ZiF), University
+of Bielefeld, Germany, being completed with the support of the VEGA grant agency, projects
+Nos. 2/0092/09, 2/0098/10 and 2/7012/27. I am extremely grat eful to Prof. Hans Havlicek
+(Vienna University of Technology) for providing me with an easy-to- use software for generating
+the first seven figures. I also thank Richard Green (University of C olorado,Boulder) for motivating
+combinatorial remarks and Petr Pracna (J. Heyrovsk´ y Institut e of Physical Chemistry, Prague)
+for providing me with the electronic versions of the last three figure s.
+8References
+[1] Planat M, and Saniga M. On the Pauli graphs ofN-qudits. Quant In f Comput 2008; 8:127–146
+(arXiv:quant-ph/0701211).
+[2] Saniga M, Planat M, and Pracna P. Projective ring line encompassin g two-qubits. Theoret
+Math Phys 2008;155:905–913 (arXiv:quant-ph/0611063).
+[3] Havlicek H, Odehnal B, and Saniga M. Factor-group-generated polar spaces and (multi-
+)qudits. SIGMA 2009;5:096 (arXiv:0903.5418).
+[4] L´ evay P, Saniga M, Vrana P, and Pracna P. Black hole entropy an d finite geometry. Phys Rev
+D 2009;79:084036 (arXiv:0903.0541).
+[5] Payne SE, and Thas JA. Finite Generalized Quadrangles. Pitman: B oston – London – Mel-
+bourne; 1984; see also Thas K. Symmetry in Finite Generalized Quadr angles. Birkh¨ auser:
+Basel; 2004.
+[6] Buekenhout F, and Cohen AM. Diagram Geometry (Chapter 10.2) . Springer: Berlin – Hei-
+delberg – New York – London – Paris – Tokyo – Hong Kong – Barcelona – Budapest; 2009.
+Preprints of separate chapters can be also found at http://www.w in.tue.nl/ /tildewideamc/buek/.
+[7] Saniga M, Green RM, L´ evay P, Pracna P, and Vrana P. The Veldka mp space of GQ(2,4). Int
+J Geom Methods Mod Phys, submitted (arXiv:0903.0715).
+[8] Ronan MA. Embeddings and hyperplanes of discrete geometries. European J Combin
+1987;8:179–185.
+[9] Brouwer AE, and Wilbrink HA. Ovoids and fans in the generalized qua drangle Q(4,2). Geom
+Dedicata 1990;36:121–124.
+[10] Hirschfeld JWP. On the history of generalized quadrangles. Bull Belg Math Soc 1994;3:417–
+421.
+9
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@@ -0,0 +1,398 @@
+arXiv:1001.0660v1  [cond-mat.mes-hall]  5 Jan 2010Are non-equilibrium Bose-Einstein condensates superfluid ?
+Michiel Wouters1and Iacopo Carusotto2
+1Institute of Theoretical Physics, Ecole Polytechnique F´ e d´ erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
+2CNR INFM-BEC Center and Dipartimento di Fisica, Universit` a di Trento, I-38123 Povo, Italy
+We theoretically study the superfluidity properties of a non -equilibrium Bose-Einstein condensate
+of exciton-polaritons in a semiconductor microcavity. The dynamics of the condensate is described
+at mean-field level in terms of a modified Complex Ginzburg Lan dau equation. A generalized
+Landau criterion is formulated which estimates the onset of the drag force on a small moving defect.
+Metastability of supercurrents in multiply connected geom etries persists up to higher flow speeds.
+PACS numbers: 03.75.Kk, 05.70.Ln, 71.36.+c.
+Superfluidity is among the most remarkable conse-
+quences of macroscopic quantum coherence in condensed
+matter systems and manifests itself in a number of fasci-
+nating effects [1, 2]. A unified description of these phe-
+nomenaisobtainedintheframeworkoftheso-calledtwo-
+fluid hydrodynamics, in which the macroscopic conden-
+satewavefunctionaddsupto thestandardhydrodynamic
+variables [3]. The phenomenon of macroscopic coherence
+isnotrestrictedtosystemsat(orcloseto)thermodynam-
+ical equilibrium such as liquid Helium, ultracold atomic
+gases, or superconducting materials, but has been ob-
+served also in systems far from thermodynamical equi-
+librium, whose state is determined by a dynamical bal-
+ance of driving and losses. Most remarkable examples
+are lasers and, more recently, Bose-Einstein condensates
+of magnons in magnetic solids [4] and exciton-polaritons
+in semiconductor microcavities [5, 6]. In particular, the
+issue of superfluidity in this latter system has attracted
+a significant interest from both the theoretical [7, 8] and
+experimental [9–11] points of view.
+Recent experiments with resonantly pumped polari-
+ton condensates [9] have demonstrated superfluidity as a
+dramatic reduction in the intensity of resonant Rayleigh
+scattering as originally predicted in [7]. The situation is
+less clear in the case of non-resonant [5] or parametrical
+(OPO) [10] pumping schemes: recent experiments [10]
+have observed propagation of polariton bullets without
+apparent friction, which is in contrast with the predic-
+tions ofa na¨ ıveLandaucriterionbasedonthe elementary
+excitation spectrum predicted in [12]. Another aspect
+of superfluidity, namely metastability of supercurrent in
+multiply-connected geometries was investigated theoret-
+ically in [8] and experimentally confirmed in [11]. The
+present paper reports a comprehensive theoretical inves-
+tigation ofthe meaningof superfluidity for polaritoncon-
+densates under a non-resonant pumping. Emphasis will
+be given to the novel features that originate from their
+non-equilibrium character.
+At the mean-field level, the condensate dynamics can
+be described in terms of the so-called Gross-Pitaevskii
+equation (GPE) [1], which was recently generalized to
+non-equilibrium condensates by including the effect ofpumping and losses [13, 14]. This mean-field descrip-
+tion has been able to explain a number of experimen-
+tal observations on polariton condensates, e.g. the ring-
+shaped momentum distribution of spatially narrow con-
+densates [6, 15], the synchronization/desynchronization
+transition [16], the spontaneous appearance of vor-
+tices [17]. Nonetheless, the implicit assumption that the
+pumping mechanism is not frequency-selective can lead
+to unphysical predictions, e.g. that in a spatially ho-
+mogeneous or ring-like geometry condensation is equally
+likely to occur in any momentum state. Kinetic calcu-
+lations [18] have pointed out the significant energy de-
+pendence of the polariton-polariton scattering processes
+that are responsible for replenishing the condensate. In-
+cludingthis featureasanenergy-dependentamplification
+mechanism turned out to be crucial in order to extract
+physicallymeaningfulpredictionsforthecondensatefluc-
+tuations in the Wigner Monte Carlo simulations of [19].
+A simplest generalization of the GPE to include
+frequency-dependent pumping has the form:
+idψ
+dt=/braceleftbigg
+−/planckover2pi1
+2m∇2+Vext
++i
+2/bracketleftbigg
+P/parenleftbigg
+1−i
+ΩKd
+dt/parenrightbigg
+−r|ψ|2−γ/bracketrightbigg
++g|ψ|2/bracerightbigg
+ψ.(1)
+The energy zero has been set for convenience at the bot-
+tom of the lower polariton branch; the efficiency of am-
+plification (proportional to the pumping strength P) de-
+creases to zero a frequency interval Ω Kabove it. As-
+suming a linear form of the frequency dependence of the
+amplification, the generalized GPE (1) maintains a tem-
+porally local form. The others terms describing gain sat-
+uration (r), losses (γ), polariton mass ( m), polariton-
+polariton interactions ( g), external potential ( Vext) have
+the same meaning as in previous works [13, 14].
+We first consider the evolution of the system starting
+from an initial state with no condensate ψ= 0. If the
+strength of pumping is enough to overcome losses P >γ,
+theψ= 0 state is dynamically unstable against the cre-
+ation of a finite condensate amplitude in all the low-
+momentum modes for which /planckover2pi1k2/2m <ΩK(1−γ/P).
+The rate of this instability is maximum at k= 0 and2
+decreases for increasing k. This fact is in agreement with
+the experimental observation of condensation naturally
+occurring around k= 0 as soon as the sample is suffi-
+ciently large and free from disorder [5].
+In spite of this natural preference, condensation can be
+forced to occur in finite momentum state by seeding the
+system with a short resonant pulse at the desired kcas
+proposed and numerically assessed in Ref. [8]. A related
+configuration was experimentally demonstrated for the
+OPO pump scheme in [11]. The effect of the frequency-
+dependent pumping is then visible in the state equation
+relating the pumping strength Pto the condensate den-
+sitync=|ψ|2in the stationary state:
+|ψ|2=/bracketleftbigg
+1−1
+ΩK/parenleftbigg/planckover2pi1k2
+c
+2m+g|ψ|2/parenrightbigg/bracketrightbiggP
+r.(2)
+As a consequence of the frequency dependent pumping,
+the interaction-induced blue-shift of the condensate fre-
+quencyωc=k2
+c/2m+g|ψ|2is responsible for a slower
+increase of density with pumping strength P.
+-505  -10  -5  0  5  10Re[ωBog(k)]
+-505-10-50510
+-505-10-50510
+-505
+q / kγ  -1  0Im[ωBog(k)]
+-505
+q / kγ-10
+-505
+q / kγ-10
+-5 0 5-0.2-0.100.1(a) (c) (e)
+(b) (d) (f)
+FIG. 1: Elementary excitation spectrum of a moving, spa-
+tially homogeneous, non-equilibrium condensate as descri bed
+by the linearized CGLE around the steady-state solution.
+Parameters: kc/kγ= 4,r/γ= 1,g/γ= 1, Ω K/γ= 50,
+ncg/γ= 0.1 (c,d) and ncg/γ= 1 (e,f). Inset: magnified
+view of (d). Panels (a,b): same as panels (e,f) in the limit
+of a frequency-independent pumping Ω K=∞. Momenta are
+measured in units of kγ=/radicalbig
+mγ//planckover2pi1.
+The dynamical stability of a moving condensate is to
+be assessed by linearization of the CGLE around the
+stationary solution. Stability requires that the imagi-
+nary part of the frequency is negative for all Bogoliubov
+modes. Examples of the Bogoliubov dispersion ωBog(q)
+for moving condensates are shown in Fig.1. In the left-
+most panels (a,b), the limit Ω K→ ∞of a negligible
+frequency-dependence of pumping is considered. Apart
+for the global Doppler tilting of the real part due to the
+finite condensate velocity, the plotted dispersion fully re-
+covers the diffusive character first discussed in [12–14]: aflat dispersion around the condensate wavevector and an
+imaginary part quadratically growing in q. In this limit,
+stable condensates exist for any value of kc.
+The effect of a frequency-selective pumping is ad-
+dressedin the otherpanels. The central panels(c,d) refer
+to the case of a pumping intensity just above the thresh-
+old. In this regime, the condensate density is very small
+and the real part of the Bogoliubov mode frequency re-
+duces to the single-particle one. On the other hand, the
+characteristic damping rate of density fluctuations (the
+gappedmodeat q= 0)andofthehigh-momentummodes
+issuppressedbyacriticalslowingdownphenomenon[13].
+As a result, the low energy modes around q≃ −kcbe-
+come dynamically unstable. Eventually, this mechanism
+leads to the disappearence of the original condensate and
+the formation of another -stable- condensate in a lower
+momentum state.
+FIG. 2: Growth rate (in units of γ) of maximally unstable
+mode as a function of condensate momentum kcand density
+nc. The dashed line indicates the boundary of the stability
+region. Same system parameters as in Fig.1(c-f).
+The right panels (e,f) refer to the case of a stronger
+pumping well above the threshold. In this case, the
+dampingrateofallmodesotherthantheGoldstoneoneis
+comparable to the polariton lifetime γ. In this case, the
+frequency-dependence of pumping is not able to desta-
+bilize the moving condensate. The stability domain as
+a function of the momentum kcand the density ncof
+the condensate is summarized in Fig.2: as the density
+is increased, stable condensates survive up to larger mo-
+menta.
+It is of crucial importance to note that this instability
+mechanism has no direct counterpart in standard equi-
+librium superfluids and is effective even in the absence
+of any defect potential. Of course, the present theory is
+directly applicable only to infinite, homogeneous conden-
+satesorinmultiplyconnectedring-shapedgeometries. In
+finite systems, assessing the stability of moving conden-
+sates would be made harder by the flow of condensate
+polaritons outside the pump regions [15].
+In the presence of weak defects, the frictionless flow of
+equilibrium condensates is limited to flow speeds below3
+the Landau critical velocity vc= min k/braceleftbig
+Re[ωo
+Bog(k)]/k/bracerightbig
+,
+whereωo
+Bog(k) is the real part of the Bogoliubov dis-
+persion for a condensate at rest. For dilute systems, vc
+coincides with the sound velocity cs=/radicalbig
+gnc/m[1]. As
+a consequence of the diffusive character of the Goldstone
+mode, a na¨ ıve application of this definition to the Bo-
+goliubov dispersion of non-equilibrium condensates an-
+ticipated in [12–14] gives a vanishing prediction for the
+critical velocity, vc= 0: Bogoliubov waves are emitted in
+a moving condensate hitting a (weak) defect at any value
+of the condensate speed.
+−20020−20−1001020
+x(µm)y(µm)(a)
+−20020−20−1001020
+x(µm)y(µm)(b)
+−20020−20−1001020
+x(µm)y(µm)(c)
+  
+0.980.9911.011.02
+FIG. 3: Density perturbation created in a moving condensate
+by a stationary weak defect for three different values of the
+condensate velocity v/cs= 1.5,1,0.4 across the critical value
+for superfluidity. Parameters: ncg/γ=ncr/γ= 1, Ω K/γ=
+50.
+A complete picture of non-equilibrium condensates re-
+quires taking into account the non-trivial dynamics of
+the imaginary part of the Bogoliubov dispersion. Nu-
+merical plots of the density perturbation induced by a
+single weak stationary defect in a (dynamically stable)
+moving condensate can be calculated from the general-
+izedGPE(1)inthepresenceofasuitabledefectpotential
+Vext(r) and are shown in Fig.3 for different values of the
+condensate velocity. In contrast to the predictions of the
+na¨ ıve Landau criterion stated above, the induced per-
+turbation closely resembles the one induced in an equi-
+librium condensate described by the standard GPE [22]:
+at high speeds [panel (a)], the defect creates a series of
+parabolic fringes that propagate away from the defect;
+at low speeds [panel (c)], the propagating fringes are re-
+placed by a localized perturbation in the vicinity of the
+defect. Remarkably, the characteristic speed at which
+the fringes disappear is of the order of the equilibrium
+sound speed cs=/radicalbig
+gnc/m, but does not correspond to
+any noticeable feature in the real part of the Bogoliubov
+dispersion of Fig.1(a,c,e).
+As it was pointed out in a different context in [20], the
+emission pattern by a localized monochromatic source is
+better understood in terms of the (complex) wavevector
+of the emitted wave at the given (real) frequency. As one
+can see in Fig. 4 (b), the wavevector ˜kof the Bogoliubov
+waveemitted at zero frequency by the static defect starts
+having a finite real part only after a branching point:
+extended oscillations in the density are observed as soon
+as the real part of ˜kexceeds the imaginary part.0 1 200.20.40.60.8(a)
+v/csF/Fγ
+0 1 2−4−2024Re[k/kγ]
+v/cs(b)
+0 1 2−3−2−10123Im[k/kγ]
+v/cs(c)
+FIG. 4: Panel (a): Force exerted on a weak stationary defect
+by a moving condensate as a function of the condensate veloc-
+ity for different values of rnc/γ= 0.1,1,2 (blue solid, green
+dashed, red dotted lines), in units of Fγ=/planckover2pi12k3
+γ/m. For com-
+parison, the same force on an equilibrium condensate is show n
+as a thin solid black line. Panel (b,c): Real and imaginary
+parts of the complex wavevector ˜kof the Bogoliubov emission
+in the upstream direction as a function of the condensate ve-
+locity, for rnc/γ=gnc/γ= 1. The other parameters are the
+same as in Fig. 3.
+This behavior is reflected in the drag force Fexerted
+by the moving condensate onto the defect as a function
+of the condensate velocity [Fig.4(a)]. In terms of the
+perturbed density profile n(r), the drag force is given by
+F=−/integraltext
+d2rn(r)∇rVdef(r) [21]. As a consequence of the
+finite lifetime of the Bogoliubov modes, the drag force
+has a non-vanishing value at all v. In addition, the drag
+force shows a marked threshold at a velocity value that
+closely corresponds to the onset of fringes in the density
+profile: the lower the value of the ncr/γparameter, the
+sharper the threshold. On the other hand, the role of Ω K
+on this effect is minor. Even though no experiment has
+so far investigated the superfluid properties of polariton
+condensatesundernon-resonantpumping, itislikelythat
+this generalized form of the Landau criterion provides at
+least a partial explaination of the recent experimental
+observation of superfluidity in a OPO regime [10].
+Supersonic flows in equilibrium condensates in ring-
+shaped geometries are generally strongly sensitive to the
+presence of defects: as soon as nodes appear in the con-
+densate wavefunction, the topological stability of the su-
+percurrent state is broken, which leads to a rapid slow
+down of the condensate motion. The finite damping
+rate of excitations in polariton condensates introduces
+a substantial modification to this picture: the perturba-
+tion created by each defect is not able to propagate on
+long distances, but rather remains localized in space on
+a length scale inversely proportional to Im[ ˜k]. As a re-
+sult, we expect it is much harder to break the topological
+stability of the supercurrents.
+This physical picture is confirmed by numerical simu-
+lations of the time evolution under the generalized GPE
+(1) starting from an initial condition with a supersonic
+flow. Periodic boundary conditions are assumed. For a
+low defect density, the supercurrent state is maintained
+for very long times with no sign of decay. The char-
+acteristic Cerenkov-like density patterns in the vicinity4
+kγ xkγ y(a)
+−20 020−20−1001020
+kx/kγky/kγ(b)
+−202−2−1012kγ xkγ y(c)
+−20 020−20−1001020
+kx/kγky/kγ(d)
+−202−2−1012
+FIG. 5: Real (linear color scale) and momentum space (loga-
+rithmic color scale) densities of a non-equilibrium conden sate
+after a temporal evolution of γt= 300 for an initial momen-
+tumk/kγ= 2 and two different densities of defects. The dots
+in the real space panels indicate the position of the defects .
+The square in the momentum space panels indicates the ini-
+tial momentum of the condensate. Parameters are the same
+as in Fig. 4, except for Ω K/γ= 10.
+of each defect are spatially separated and do not inter-
+fere [Fig.5(a)]. The momentum distribution in k-space
+[Fig.5(b)] shows the condensate peak right at the initial
+momentum state and a much fainter resonant Rayleigh
+scattering ring [7]. The situation is different for a higher
+densityofdefects. Inthiscase, therealspacedensityper-
+turbations created by the different defects substantially
+overlap with each other [Fig.5(c)]. As a result, interfer-
+ence effects are more likely to create nodes in the con-
+densate wavefunction and therefore to trigger dissipation
+of the supercurrent. The occurrence of such a process is
+apparent in Fig.5(d) as a much reduced late-time value
+of the condensate momentum.
+This fact illustrates another important difference with
+respect to standard, equilibrium condensates: in that
+case, the dissipation of a supercurrent leads to a signifi-
+cant heating and reduction of the condensate fraction. In
+the present case, the condensate turns out to be almost
+rigidly transferred to a lower momentum state; the (co-
+herent) momentum broadening that is visible in Fig.5(d)
+is an effect of the defects. Remarkably, the frequency de-
+pendence of pumping in the generalized CGLE (1) has a
+crucialrolein concentratingthe population in low-energymodes and preventing the momentum distribution from
+spreading all over the k-space.
+In conclusion, we have investigated how the non-
+equilibrium nature of exciton-polariton condensates af-
+fects their superfluidity properties. A mean-field model
+based on a generalized Gross-Pitaevskii equation includ-
+ing a frequency-dependent pumping is developed. Dif-
+ferent aspects of superfluidity have been considered. A
+non-equilibrium version of the Landau critical speed for
+the onset of drag force is formulated and metastability
+of supercurrents is shown to persist even at speeds well
+above the critical speed.
+We are indebted to Vincenzo Savona, Cristiano Ciuti
+andDavideSarchiforcontinuousenlighteningexchanges.
+Stimulating discussions with J. Keeling, C. Menotti, F.
+Piazza, D. Sanvitto, A. Smerzi are acknowledged.
+[1] L.P. Pitaevskii and S. Stringari, Bose-Einstein Conden-
+sation, Clarendon Press Oxford (2003).
+[2] A. J. Leggett, Rev. Mod. Phys. 71, S318 (1999).
+[3] K. Huang, Statistical Mechanics (New York, John Wiley
+& Sons, 1963).
+[4] O. Demokritov, et al., Nature 443, 430 (2006).
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+Kasprzak et al., Nature 443, 409 (2006).
+[6] M. Richard et al., Phys. Rev. Lett. 94, 187401 (2005).
+[7] I.Carusotto, C.Ciuti, Phys. Rev. Lett. 93, 166401 (2004).
+[8] M. Wouters and V. Savona, arXiv:0904.2966.
+[9] A. Amo, et al.Nature Phys. 5, 805 (2009).
+[10] A. Amo, et al., Nature 457, 291 (2009).
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+arXiv:1001.0662v1  [astro-ph.SR]  5 Jan 2010Astron. Nachr./ AN 0000,No.00,1–13(0000)/ DOIpleaseset DOI!
+Finding themostvariablestarsintheOrionBeltwiththeAll Sky Auto-
+matedSurvey
+Jos´eA. Caballero⋆,M. Cornide, andE. de Castro
+Departamento de Astrof´ ısica y Ciencias de la Atm´ osfera, F acultad de F´ ısica, Universidad Complutense de Madrid, E-
+28040 Madrid, Spain
+Received05 Sep2009, accepted 04 Jan2010
+Publishedonline later
+Key words stars: variables: general — stars: pre-main sequence — star s: oscillations — astronomical data bases: mis-
+cellaneous —Galaxy: open clusters and associations: Ori OB 1b
+We look for high-amplitude variable young stars in the open c lusters and associations of the Orion Belt. We use public
+data from the ASAS-3 Photometric V-band Catalogue of the All Sky Automated Survey, infrared ph otometry from the
+2MASSand IRAScatalogues,propermotions,andtheAladinskyatlastoobta inalistofthemostvariablestarsinasurvey
+areaof side5degcentredonthe bright starAlnilam( ǫOri)inthe centre ofthe OrionBelt.Weidentify32highly-va riable
+stars, of which 16 had not been reported to vary before. They a re mostly variable young stars and candidates (16) and
+background giants (8), but there are also fieldcataclysmic v ariables, contact binaries, and eclipsing binary candidat es. Of
+the young stars, which typically are active Herbig Ae/Be and T Tauri stars with H αemission and infrared flux excess,
+we discover four new variables and confirm the variability st atus of another two. Some of them belong tothe well known
+σOrionis cluster.Besides, sixof the eight giants are new var iables, andthree are new periodic variables.
+c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim
+1 Introduction
+TheOriOB1bsubgroupassociation,mostlycoincidentwith
+theasterismoftheOrionBelt,containsarichpopulationof
+very young stars (e.g. Haro & Moreno 1953; Walker 1969;
+Warren & Hesser 1978; Brown et al. 1994; Sterzik et al.
+1995;deZeeuwetal.1999;Brice˜ noetal.2001).Theyounges t
+regions, including Alnitak ( ζOri), which illuminates the
+Flame Nebula (NGC 2024), and σOri, which illuminates
+the Horsehead Nebula (Barnard 33 and IC 434), are to the
+east of the Belt and have ages of 1 to 5Ma. Here lies the
+σOrionis cluster, one of the most suitable sites for dis-
+covering and characterising substellar objects (B´ ejar et al.
+1999; Zapatero Osorio et al. 2002; Sherry et al. 2004; Ca-
+balleroetal.2007).ThepopulationofstarssurroundingAl -
+nilam (ǫOri) and Mintaka ( δOri), to the west of the Belt,
+are older (5–10Ma) and spread over a wider area than the
+much denser σOrionis cluster (Caballero & Solano 2008).
+Surroundingthe Belt, there is anotherdispersed populatio n
+of 10–20Ma-old stars, mostly associated to the Ori OB1a
+subgroup association (Jeffries et al. 2006; Caballero 2007 )
+and its agglomerates, such as the 25 Orionis group or the
+ηOrionisoverdensity(Brice˜ noetal.2007;Caballero&Di-
+nis2008).
+Hαand X-ray emission and infraredexcess are the pri-
+mary features for the identification of very young stars as
+those in the Orion Belt. Optical and near-infrared variabil -
+ity, cosmic lithiumabundances,surroundingjets, or forbi d-
+⋆Correspondingauthor:e-mail: caballero@astrax.fis.ucm.esden emission lines are other signpostsused to confirmvery
+young ages in stars. Photometric variability in young stars
+is mostly due to dark (cool)and bright (hot)spots and flare
+activity, which are related to the strengthened magnetic ac -
+tivy due to fast rotation,andinteractionswith circumstel lar
+discs, such as intense, variable accretionepisodesor occu l-
+tations(e.g.classicalworks:Joy1945;Walker1956;Herbi g
+1977 – theoretical works: Koenigl 1991; Collier Comeron
+& Campbell 1993; Shu et al. 1994 – modern observational
+works: Bouvier et al. 1993; Herbst et al. 1994; Th´ e et al.
+1994;Hartmann&Kenyon1996;Hamiltonetal.2001;Eiroa
+et al.2002).
+TheOrionNebulaClusterintheOriOB1dsubgroupas-
+sociation has been a traditional target for studyingthe pho -
+tometric variability of young stars (Jones & Walker 1988;
+Carpenteretal.2001;Herbstetal.2002;Stassunetal.2006 ;
+Parihar et al. 2009), but the surveys for variability in the
+OrionBeltarerelativelyscarce.Themostthoroughvariabi l-
+ity surveys in the Ori OB1 b association have been aimed
+at the low-mass pre-main sequence stars and brown dwarfs
+(Bailer-Jones & Mundt 2001; Zapatero Osorio et al. 2003;
+Caballeroetal.2004,2006;Scholz&Eisl¨ offel2004,2005;
+Brice˜ no et al. 2005; Scholz et al. 2009),while the brightes t
+variable stars were first recognised as early as the begin-
+ning of the 20th century and have not received quite atten-
+tion. Only a few (multiple) young stars of important astro-
+physical interest had been extensively monitored, such as
+VV Ori (Miller Barr 1904; Struve & Luyten 1949; Terrell
+et al. 2007), σOri E (Walborn & Hesser 1976; Landstreet
+& Borra 1978), or Mintaka AE–D (Stebbins 1915; Koch
+c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim2 Caballero etal.:The most variable ASASstars inthe OrionBe lt
+8282.58383.58484.58585.58686.5−2.5−2−1.5−1−0.500.51
+α (J2000)δ (J2000)
+Fig.1Spatiallocationofthe10029ASASsourcesinthe
+studiedareawithlightcurveswithmorethan40datapoints,
+markedwith(blue)smalldots.Northisup,eastistotheleft .
+The 100 ASAS sources with possible variable light curves,
+the threeOrionBelt supergiantstars(Alnitak,Alnilam,an d
+Mintaka, from east to west), and σOri are marked with
+(red) filled circles, big stars, and a small star, respective ly.
+Note the blank strip where ASAS did not survey (between
+Alnitak and σOrionis’ declinations) and the overlapping
+between different ASAS pointings (at about the Mintaka’s
+declination).
+& Hrivnak 1981). Many of the intermediate-mass T Tauri
+stars in the Orion Belt with variability denominations (e.g .
+BG Ori, V505 Ori) were compiled by Fedorovich (1960)
+and have never been monitored again. After a century of
+photometric searches in the region, there are still serendi p-
+itous discoveries of bright, UX Orionis-, A-type, variable
+stars(Caballeroetal. 2008).
+Our aim is to find the most variable stars in the Orion
+Belt, giving especial emphasis to the young Herbig Ae/Be
+(HAeBe) and T Tauri stars. To accomplish that, we have
+usedtheASAS-3Photometric V-bandCatalogueoftheAll
+SkyAutomatedSurvey(ASAS)1,whichis“alowcostproject
+dedicated to constant photometric monitoring of the whole
+availablesky”(Pojma´ nski1997,2002).
+2 Analysis
+First, we compiled 63276 ASAS light curves of objects in
+a square of side 5.0deg centred on Alnilam; see Fig. 1. We
+used the default parameters N >4 data points and r <
+15arcsec ( Nis the number of data points within a circle
+of radius rcentred on the coordinatesof an ASAS source).
+If not for certain blank strip, the surveyed area would have
+been 25deg2. To maximise the scientific return and min-
+imise possible further troubles with light curves of poor
+quality, we discarded all ASAS light curves with N≤40
+data points. It left 10029 light curves (16%) with N >40
+1http://www.astrouw.edu.pl/asas/.7 8 910 11 12 13 14 1500.10.20.30.40.50.6
+V0 ASAS−3 [mag]σ(V0) ASAS−3 [mag]
+Fig.2Diagram showing σ(V0)vs.V0. ASAS sources
+above the dashed line are variable star candidates in this
+work. X Ori is out of limits ( V0∼12.7mag, σ(V0)∼
+1.4mag).
+for next phases of the analysis. The ASAS data coveredal-
+mostsix yearsduringthe first decadeofthe21stcentury.
+Of the five available apertures (0, 1, 2, 3, 4), the “0”
+aperturephotometryseemedto providebetter data(i.e. les s
+dispersion and number of grade “C” and “D” data points
+– probably useless data or without photometric value at all
+duetobadpixels,frame-edgeandbrightstar/planetproxim -
+ity,orexcessivefaintness).AccordingtoPojma´ nski(200 2),
+the five apertures correspond to diameter sizes of the aper-
+ture photometry from 2 to 6pixels; the “0” aperture corre-
+spondstothenarrowerphotometry(Pojma´ nski,priv.comm. ).
+Fromthispointon,weused onlythe Johnson Vmagnitude
+withtheASAS “0”aperturephotometry,whichweindicate
+with thesymbol V0.
+Fig. 2 illustrates the selection of variable stars. It is a
+σ(V0)vs.V0diagram in which the variable star candidates
+have largervaluesof standarddeviation σ(V0)with respect
+to non-variable stars of the same average magnitude V0.
+ASAS stars brighterthan V0≈8.5mag are affected bysat-
+uration in the detector and were not considered. The lower
+envelopeat the non-saturated,brightest end of the cloud of
+data points in the diagram indicates the best photometric
+accuracy of the system, which is of about 0.02mag. The
+obvious rise of the lower envelope at faint magnitudes is a
+combinationofthestandarddeviations(root-mean-square s)
+from Poisson noise in the targets and from sky noise in
+the photometric aperture (e.g. Irwin et al. 2007). The line
+used as the selection criterion is the lower envelopeshifte d
+tolargerσ(V0)sbymagnitude-dependentamounts(roughly
+0.06magat V0∼8.5–11.5mag,0.12magat V0∼12.5mag,
+and0.18magat V0∼13.5mag).Thischoicewasrelatively
+arbitrary: more or less restrictive selection criteria wou ld
+have given significantly less or more variable star candi-
+dates, respectively.Thus, the used selection criterion wa s a
+compromisetominimisethenumberoffalsepositives(vari-
+ablestarcandidatesthatactuallydonotvary)andmaximise
+c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim www.an-journal.orgAstron. Nachr./ AN(0000) 3
+Table 1 Identifiedvariablesin theOrionBelt.
+No.α2MASSδ2MASSV0σ(V0)δV0N⋆
+obsSpectral Name Class Variable
+(J2000) (J2000) [mag] [mag] [mag] type
+01 0528 59.57 –03 3352.3 13.621 0.412 0.055 190 ... V1159 Ori D warf nova Known
+03 0531 01.46 –03 2324.3 12.492 0.341 0.056 294 ... IRAS05285 –0325 Giant New
+06 0533 22.45 –03 0955.7 11.185 0.085 0.032 207 ... No.06 Unkn own New
+07 0537 48.79 –03 0748.7 10.780 0.101 0.021 286 ... IRAS05353 –0309 Giant New
+11 0532 09.94 –02 4946.8 11.581 0.524 0.032 191 F5–8Vpe RYOri HAeBe Known
+12 0538 33.68 –02 4414.2 12.885 0.368 0.054 187 K4e TXOri TTau ri Known
+13 0538 25.87 –02 4351.2 13.059 0.356 0.054 142 K3e TYOri TTau ri Known
+14 0543 54.26 –02 4335.4 11.628 0.135 0.032 222 ... 2M054354– 0243.6 Contact binary Known
+15 0539 39.99 –02 4309.7 13.246 0.246 0.054 197 ... RWOri TTau ri Known
+18 0538 48.04 –02 2714.2 13.340 0.295 0.053 224 K7e+M: Mayrit 528005 AB TTauri New
+21 0536 20.91 –02 1057.6 12.607 0.192 0.056 252 F3e PQOri HAeB e Known
+29 0537 38.67 –01 4616.6 12.752 1.441 0.054 179 M8–9III: XOri Giant Known
+30 0537 56.59 –01 4050.3 10.734 0.250 0.021 273 ... IRAS05354 –0142 Giant New
+33 0530 02.89 –01 3005.3 12.786 0.263 0.056 176 ... No.33 Unkn own New
+36 0534 25.82 –01 2106.5 12.667 0.325 0.055 170 ... V469Ori TT auri Known
+41 0526 53.53 –01 0902.3 13.478 0.480 0.057 144 ... KisoA–090 3 135 TTauri? New
+43 0537 59.04 –01 0552.8 13.554 0.371 0.054 200 ... V993Ori TT auri? Known
+46 0531 26.41 –00 5833.1 10.454 0.076 0.021 192 F0 HD290509 Un known New
+48 0539 45.65 –00 5550.9 10.934 0.075 0.021 203 ... GSC04767– 00071 TTauri? Known
+51 0536 24.29 –00 4212.0 13.399 0.388 0.056 227 K3e PUOri TTau ri Known
+53 0534 48.91 –00 3716.7 13.384 0.298 0.056 178 ... V472Ori TT auri? Known
+54 0527 24.67 –00 3511.2 11.031 0.134 0.033 184 ... IRAS05248 –0037 Giant New
+55 0535 43.27 –00 3436.7 12.411 0.278 0.057 246 K4e StHA48 TTa uri New
+60 0526 34.28 –00 1927.2 12.353 0.137 0.056 181 ... No.60 Gian t New
+65 0536 28.55 +00 0445.7 11.987 0.131 0.056 257 ... No.65 Unkn own New
+66 0543 29.25 +00 0458.9 11.459 0.513 0.032 202 F0? GTOri Gian t Known
+70 0544 18.80 +00 0840.4 8.906 0.061 0.019 204 A7IIIe V351Ori HAeBe Known
+72 0546 11.86 +00 3225.9 13.390 0.439 0.053 167 ... VSSVI–32 U nknown New
+73 0541 59.81 +00 3527.1 12.698 0.435 0.053 185 ... No.73 Unkn own New
+74 0536 42.64 +00 3834.3 11.010 0.101 0.033 254 A2 HD290625 HA eBe New
+75 0533 47.85 +00 5536.3 10.433 0.166 0.021 181 ... IRAS05312 +0053 Giant New
+78 0528 17.85 +01 1006.1 11.219 0.165 0.033 173 F8–K0e StHA40 TTauri Known
+the number of actual variable stars, but always keeping a
+maneagablenumberofobjectstobefollowedup.
+There were 100 ASAS light curves above the selection
+criterion in Fig. 2. We were able to identify the 2MASS
+(Skrutskie et al. 2006) and USNO-B1 (Monet et al. 2003)
+possible stellar counterparts of 99 of them. The unidenfied
+ASASsourcewas054200–0154.7( V0=13.167mag, σ(V0)
+= 0.245mag, N obs= 49), which is actually a bright spot in
+NGC2024.Forthevisualidentification,weusedtheAladin
+interactiveskyatlas(Bonnareletal.2000).Theactualnum -
+ber of stars with light curves was 78, because there were
+21 variable star candidates with two corresponding light
+curves.Theseparateddatasets(withidenticalassociated co-
+ordinates and magnitudes within error bars) correspond to
+thesamestarindifferentobservedfields(Pojma´ nski2002) .
+WerevisitedtheASASdataforthe78starsbyretrieving
+alldatapointswithqualitygrades“A”and“B”inacircleof
+radiusr= 15arcsec centredon the stars, and mergingthem
+intoa uniquelightcurve.“A” and“B”qualitygrades,which
+are tabulated by the ASAS catalogue and derived from the
+average photometricquality of the frame for each aperture,correspond to the best and mean data, respectively2. Be-
+cause of the large ASAS pixel size (of about 15 µm) and
+typicalfullwidthathalfmaximum,ofnolessthan1.4pixels
+(Pojma´ nski2002),wediscardedallvisualbinariesofroug hly
+the same brightnessseparated by less than about 20arcsec.
+Besides, we also discarded relatively faint variable candi -
+datesatlessthan1arcmintoverybrightstars,includingAl -
+nilam,Mintaka,and σOri,whichareseenbythenakedeye
+and lead to a high sky background. After these removals,
+wekept32variablestarswhoselightcurveswerelikelynot
+contaminatedby other stars. The 32 light curvesare shown
+in Figs.A1to A4.
+The identification number, 2MASS coordinates, aver-
+ageV0magnitude,standard deviation σ(V0), mean error of
+magnitude δV0, and numberof finally usedlight curvedata
+points,N⋆
+obs, of the 32 identified variable stars are listed
+in Table 1. The ratios between the standard deviations and
+mean errors vary between ∼2.5 and 27; the expected ratio
+2The quality grades “A” and “B” of a new ASAS frame depends on
+theaverage dispersion between ‘old’ and ‘new’ magnitudes f or starsin the
+range 8–11mag (Pojma´ nski, priv. comm.).
+www.an-journal.org c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim4 Caballero etal.:The most variable ASASstars inthe OrionBe lt
+10110210310400.10.20.30.40.50.60.70.80.91Fraction of stars
+Power spectrum maximum [mag2]
+Fig.3Cumulative distribution function of power spec-
+trum maxima of the light curves of the 32 identified vari-
+ables in the OrionBelt. Stars to the rightof the dashed ver-
+tical line at powerspectrumof 500mag2are classified here
+asprobableperiodicvariablestars.
+for a non-variable star should be lower than 2 (e.g. Bailer-
+Jones&Mundt2001).InTable1,wealsoprovidethespec-
+traltypefromtheliterature(ifithaseverbeengiven),nam e,
+variability class (see Section 3), and if the variabilityst atus
+wasknownornot.Forthosestarsthathaveneverbeenmen-
+tioned in the literature, we use the nomenclature “No. X”,
+whereXistheidentificationnumber.
+Foratypicalnumberoflightcurvedatapointsof N⋆
+obs∼
+200 and a temporal coverage of almost six years, one may
+derivethattherewasanASASvisitevery ∼11dforeachof
+the32stars.Actually,thetypicaltimeintervalbetweenco n-
+secutive ASAS visits was shorter because there were long
+non-observinggapswithout data. The mediantime interval
+betweenconsecutiveASASvisitswas2.0–3.0d,depending
+on the star. Approximately 15–20, 20–33, 65–85, and 85–
+95%ofthetimeintervalswereshorterthan1,2,5,and10d,
+respectively,andonlyinfourcases,coincidentwiththefo ur
+Australsummers,thetimeintervalswerelongerthan100d.
+We performed a rutinary time-series analysis on the 32
+lightcurvesbyapplyingtheScargle(1982)procedure.Fig. 3
+shows the cumulativedistributionfunctionof the measured
+maximaofpowerspectrum.Roughly85%ofthestarshave
+power spectrum maxima lower than 500mag2, where there
+seems to be an inflection point in the cumulative distribu-
+tion function. The other 15% (6) of the stars with larger
+powerspectrummaximaaregiveninTable2andwereclas-
+sified by us as probableperiodic variable stars. We provide
+periods for two previously known periodic variable stars
+(V1159 Ori and X Ori) consistent with those in the liter-
+ature, and found probable periods for two previously un-
+knwon variable giant stars (with IRASdenomination), but
+failedtofindanyreliableperiodictyforRYOriandGTOri.
+Thesix ofthemwill bestudiedin detailinSection3.
+We performeda compilationof additional infrareddata
+by collecting JHKsmagnitudes from 2MASS and IRASTable2 Identifiedvariableswithpowerspectrummaxima
+largerthan500mag2.
+No. Power PName Remarks
+[mag2] [d]
+01 767.19 44.345 V1159 Ori P= 44.2–46.8d
+03 1785.4 78.226 IRAS05285–0325 New periodic
+11 632.39 600.38 RY Ori Noclear peak
+29 8779.7 416.05 X Ori P= 424.2±1.8d
+30 913.68 373.31 IRAS05354–0142 New periodic
+66 1671.2 920.32 GT Ori Pulsating
+−2 0 2 4 6 8 10 1200.20.40.60.811.21.41.61.82
+V0−J [mag]J−Ks [mag]
+Fig.4Optical-infraredcolour-colourdiagram( J−Ksvs.
+V0−J). Filled (red) stars: new variable young stars; filled
+(red) circles: previously known variable young stars; fille d
+(red) squares: variable young star candidates; open (blue)
+circles: variablegiants; open(blue)up-triangle:dwarfn ova
+(V1159 Ori); open (blue) down-triangle: contact binary
+(2M054354–0243.6); open (blue) squares: other new vari-
+ablestars.
+average non-colour corrected flux densities with qualities
+“3”or“2”(highandmoderatequalities,respectively).Whi le
+allthe32starshaveanear-infraredcounterpartinthe2MAS S
+catalogue, only 12 stars were identified in the IRAScata-
+logues (e.g. IRAScatalogue of point sources, version 2.0;
+IPAC 1986). Furthermore, only one star, the A7IIIe Her-
+big Ae/Be star V351 Ori, has high/moderate quality mea-
+surements at the four IRASpassbands at 12, 25, 60, and
+100µm.A colour-colourdiagramcombiningASASoptical
+and2MASSnear-infraredmagnitudesisshowninFig.4.
+Besides,wealsocollectedpropermotionsfromTycho-2
+(Høgetal.2000),PPMX(R¨ oseretal.2008),andUSNO-B1
+for the 32 stars. Given that only a few of them have appre-
+ciable proper motions, we do not list them, but are men-
+tionedin Section3whenappropriated.
+Finally, only 13 stars have variable star designations,
+and there were suspects of variability for other three stars .
+Besides, an importantfractionof the targetshave also been
+tabulated as variable stars of the Southern Hemisphere by
+Pojma´ nski (2002), to which our work complements. Next,
+c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim www.an-journal.orgAstron. Nachr./ AN(0000) 5
+Table 3 Infraredphotometryofidentifiedvariables.
+No. J H K s F12 F25 F60 F100
+[mag] [mag] [mag] [Jy] [Jy] [Jy] [Jy]
+01 13.817 ±0.027 13.781 ±0.046 13.675 ±0.050 ... ... ... ...
+03 6.097 ±0.019 5.083 ±0.027 4.639 ±0.021 1.08 ±0.08 0.39 ±0.04 <3 <20
+06 8.295 ±0.034 7.552 ±0.024 7.368 ±0.026 ... ... ... ...
+07 5.590 ±0.021 4.665 ±0.076 4.222 ±0.033 1.09 ±0.07 0.47 ±0.08 <0.9 <30
+11 9.444 ±0.023 8.885 ±0.055 8.277 ±0.031 0.75 ±0.09 0.71 ±0.09 <2 <20
+12 10.131 ±0.026 9.280 ±0.024 8.666 ±0.024 0.38 ±0.04 0.50 ±0.07 <0.7 <40
+13 10.445 ±0.027 9.726 ±0.024 9.311 ±0.028 ... ... ... ...
+14 10.240 ±0.026 9.943 ±0.026 9.816 ±0.021 ... ... ... ...
+15 10.647 ±0.027 9.920 ±0.023 9.530 ±0.019 ... ... ... ...
+18 10.156 ±0.023 9.463 ±0.026 9.187 ±0.019 ... ... ... ...
+21 11.303 ±0.022 11.065 ±0.028 10.979 ±0.023 ... ... ... ...
+29 2.322 ±0.292 1.379 ±0.306 0.861 ±0.346 96 ±4 49 ±7 6.0 ±0.9 <20
+30 5.152 ±0.018 4.608 ±0.076 3.972 ±0.320 2.0 ±0.2 0.63 ±0.07 <2 <40
+33 11.340 ±0.024 10.878 ±0.025 10.756 ±0.021 ... ... ... ...
+36 10.655 ±0.023 9.647 ±0.030 8.759 ±0.021 ... ... ... ...
+41 11.547 ±0.027 10.652 ±0.027 9.898 ±0.019 ... ... ... ...
+43 10.846 ±0.027 9.955 ±0.024 9.355 ±0.027 ... ... ... ...
+46 9.457 ±0.023 9.247 ±0.023 9.133 ±0.021 ... ... ... ...
+48 8.589 ±0.023 7.933 ±0.055 7.797 ±0.027 ... ... ... ...
+51 10.536 ±0.023 9.551 ±0.033 8.764 ±0.019 <0.4 0.38 ±0.05 <3 <20
+53 10.860 ±0.022 10.043 ±0.026 9.538 ±0.019 ... ... ... ...
+54 5.206 ±0.017 4.225 ±0.202 3.911 ±0.258 1.48 ±0.09 0.49 ±0.04 <5 1.7 ±0.3
+55 10.243 ±0.021 9.509 ±0.033 9.061 ±0.019 ... ... ... ...
+60 7.508 ±0.023 6.596 ±0.029 6.280 ±0.021 ... ... ... ...
+65 10.762 ±0.024 10.447 ±0.025 10.054 ±0.019 ... ... ... ...
+66 9.802 ±0.027 9.043 ±0.025 8.219 ±0.026 0.87 ±0.13 1.02 ±0.17 <5 <40
+70 7.950 ±0.020 7.504 ±0.040 6.846 ±0.026 1.16 ±0.07 4.1 ±0.2 26 ±3 21 ±3
+72 10.936 ±0.024 10.046 ±0.023 9.448 ±0.019 ... ... ... ...
+73 10.448 ±0.023 9.818 ±0.023 9.351 ±0.019 ... ... ... ...
+74 10.459 ±0.024 10.342 ±0.024 10.317 ±0.023 ... ... ... ...
+75 6.304 ±0.026 5.375 ±0.027 5.091 ±0.020 0.47 ±0.03 <0.2 <0.4 <20
+78 9.422 ±0.023 8.825 ±0.040 8.366 ±0.018 0.41 ±0.06 0.67 ±0.06 0.47 ±0.05 <1.1
+weclassifythe32variablestarsbasedonspectroscopic,as -
+trometric,andphotometriccriteria.
+3 Identified variablestars
+3.1 HAeBe andTTauri variablestars
+3.1.1 Previouslyknown variableyoungstarsand
+candidates
+This class is mostly made of probable accreting T Tauri
+stars with intense H αemission. Of the nine H αemitter
+stars,threebelongtothe σOrioniscluster(TXOri,TYOri,
+RW Ori), three have been associated to the Alnilam and
+Mintakayoungstarpopulations(V469Ori,PUOri,V472Ori),
+andthreehavenotbeenlinkedtoanyyoungregioninpartic-
+ularbutarebrightstarsattheboundarybetweenHAeBeand
+TTauristars(RYOri,V351Ori,StHa40–twoofthemmay
+haveedge-ondiscs). Thetruenatureofthe otherthreenon-
+Hαemitter stars (PQ Ori, V993 Ori, GSC 04767–00071)
+should be investigated with further spectroscopic analyse s.Besides,sixofthestarsinthisclassdisplayX-rayemissio n,
+andothersix,infraredfluxexcess.Onlythethree σOrionis
+members are known to simultaneously display H αand X-
+rayemissionsandinfraredfluxexcess.
+RY Ori (No. 11). It is a well investigated,emission-line
+staroftheOrionpopulationattheboundarybetweenHAeBe
+andTTauristars(e.g.Haro&Moreno1953;Herbig&Bell
+1988; Yudin & Evans 1998; Eiroa et al. 2001; Mora et al.
+2001;Hern´ andezetal.2004;Acke&vandenAncker2006).
+Its photometric variability was detected very early (Picke r-
+ing1904).AccordingtoBibo&Th´ e(1991),itisaUXOri-
+type star with a colour reversal in the colour-magnitudedi-
+agram, indicative of an edge-on disc. The disc of RY Ori
+isalsodetectedbecauseofitslong-time-knowninfraredex -
+cess(Cohen1973;Glass &Penston1974;Weintraub1990;
+Weaver & Gordon 1992). Its photometric variability in the
+ASAS lightcurve is obvious, with small mean photometric
+errorswithrespecttothepeak-to-peakamplitudefrom10.8
+to 13.0mag ( δV0= 0.032mag). RY Ori is one of the six
+identified variable stars with power spectra maxima larger
+www.an-journal.org c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim6 Caballero etal.:The most variable ASASstars inthe OrionBe lt
+10−11001011021031040100200300400500600700
+Period [d]Power [mag2]
+00.20.40.60.811.21.41.61.8210.5
+11
+11.5
+12
+12.5
+13
+13.5
+PhaseV0 [mag]
+Fig.5Periodogram of the ASAS light curve ( top) and
+phase-folded light curve of RY Ori to the period P=
+600.38d( bottom).
+than 500mag2(Table 2 – actually, it is the star with the
+lowestpowerspectrummaximum).Theperiodogramofthe
+ASAS lightcurve,displayedinFig.5,showsnoclearpeak,
+with a numberof power maxima increasingtowardslonger
+periods. The phase-folded light curve at the period of the
+heighestpeak,at P∼600d,doesnotshowanyevidenceof
+real periodicity.Actually, from the full-lengthligh curv e in
+Fig.A1,thetimescaleoftheirregularvariabilityofRYOri
+isoftheorderofafewdays,justlikemanyconfirmedyoung
+starsin thiswork.
+TXOri(Mayrit521199,No.12)andTYOri(Mayrit489196,
+No.13). Theyaretwoclose( ρ∼40arcsec)pre-mainse-
+quencestarsinthe σOrionisclusterthathaveappearedina
+numberofyoungandemission-linestar catalogues(Herbig
+& Kameswara Rao 1972; Cohen & Kuhi 1979; Herbig &
+Bell 1988; Ducourant et al. 2005). As in the case of many
+variablestarsinthiswork,theywerediscoveredasvariabl e
+stars by Pickering (1904). Both TX Ori and TY Ori have
+been subject of comprehensive studies and data compila-
+tions byOliveira & vanLoon(2004)and Caballero(2008).
+Their light curves are quite similar, with minima and max-ima at about V0∼12.5 and 14.0mag. Remarkably, the
+TX Ori light curve displayed an apparent dimming for he-
+liocentricJulian day3HJD>3500.
+RW Ori (Mayrit 931117, No. 15). It is another mem-
+ber in the σOrionis cluster, but it is much less known than
+thestarsabove.ItsphotometricvariabilityandH αemission
+werefirstdetectedbyPickering(1906)andHaro&Moreno
+(1953), respectively. Afterwards, it has been only investi -
+gatedbyFranciosinietal.(2006–with XMM-Newton ,Her-
+n´ andezetal.(2007–with Spitzer),andCaballero(2008).Its
+ASAS light curve varied between V0∼12.8 and 14.0mag
+(theoutlierdatapointisprobablyunreliable).
+PQOri(No.21). Inthediscoverypaper,McKnelly(1949)
+classified the star as a rapid, irregular variable with an ob-
+served range of variability of 12.5–13.5mag (photographic
+magnitude). Although its light curve displayed many max-
+ima and minima, he was unable to determine “any sem-
+blance of periodicity”. Herbig & Kameswara Rao (1975),
+Cohen & Kuhi (1979), and Hern´ andez et al. (2004) mea-
+suredF0,F5:,andF3.0 ±1.5spectraltypesandobservedno
+emission lines. However, while the first authours classified
+it as a star in the Orion population,Hern´ andezet al. (2004)
+did not find significant differences from F3 main-sequence
+stars nor near infrared excess, and categorized it as an ob-
+ject with uncertain evolutionary status. The variability p at-
+terns in the ASAS light curve of PQ Ori, although ranging
+betweenV0∼12.3and 13.3mag, do not resemble those of
+otheryoungstarsinthiswork,andits J−Kscolourisquite
+blue with respect to other early F stars in Ori OB1 b. As a
+result,itsfield dwarfnatureseemstousmoreplausible.
+V469Ori(No.36). Itisanemission-linestarwithX-ray
+emission (Haro & Moreno1953; Wiramihardjaet al. 1989;
+Nakanoetal.1999).AccordingtoCaballero&Solano(2008),
+the star is a member in the young Ori OB1 b stellar popu-
+lationsurroundingAlnilam,andisproablyassociatedtoth e
+[OS98]29J, [OS98]29H, and[OS98]29K remnantmolec-
+ularclouds(Ogura&Sugitani1998).Basedonits V0−Ks
+colour and the empirical scale of effective temperatures of
+main-sequence stars of Alonso et al. (1996), and extrapo-
+lating it to the pre-mainsequence, V469 Ori is likely a late
+GeorearlyKestar.ItsASASlightcurvelookslikethoseof
+otherTTauristars.
+V993Ori(No.43). Itisalong-time-knownvariablestar
+discovered by Luyten (1932). Caballero & Solano (2008)
+classified it as a probable member of the Ori OB1 b young
+stellar population surrounding Alnilam. It has a very red
+colourJ−Ks= 1.26±0.04mag for its brightness ( H=
+3The values of heliocentric Julian day used in the Figures are those
+tabulated bytheASAScatalogue. Actually, theyarenotheli ocentric Julian
+days HJD, but the HJD minus the constant 2450000.0 (do not con fuse
+with the constant used to transform the Julian day into the mo dified Julian
+day MJD,which is 2400000.5).
+c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim www.an-journal.orgAstron. Nachr./ AN(0000) 7
+9.82±0.02mag), which may indicate the presence of a cir-
+cumstellar disc or, alternatively, that it is a background g i-
+ant star. It may be the optical/near infrared counterpart of
+IRAS Z05354–0107,asourceinthe IRASFaintSourceRe-
+ject Catalog(Moshir1992).Thereisnospectroscopyavail-
+able to confirm or discard its possible T Tauri status. The
+largepeak-to-peakamplitudeofitslightcurveof V0∼13.0
+to fainter than 14.5mag,althoughshedslight on the photo-
+metricvariabilityofV993Ori, isnotconclusive.
+GSC04767–00071(No.48). Schirmeretal.(2009)clas-
+sifieditasachromosphericallyactivestarsuspectedofvar i-
+abilityinthe ROTSE-1 database.ThestarGSC04767–00071
+isprobablyassociatedtotheX-raysource1RXSJ053944.7–
+005612(Vogesetal.1999)andhasaverylowPPMXproper
+motion,of( µαcosδ,µδ)=(–6±2,–9±2)masa−1.Theop-
+tical and near-infraredphotometrysupportsmembership in
+Ori OB1 b. It may be a very young solar analog. The am-
+plitudeofphotometricvariabilityofitsASASlightcurve, of
+V0∼10.8to11.1mag,is relativelysmall.
+PU Ori (No.51). It is a pre-mainsequencestar near the
+Alnilam supergiant with H αin strong two-lobe emission
+(Haro & Moreno 1953; Herbig & Kameswara Rao 1972;
+Cohen & Kuhi 1979; Wiramihardja et al. 1989),photomet-
+ric variability(Fedorovich1960;Brice˜ no et al. 2005),mi d-
+infraredfluxexcessatthe IRASpassbands(Weintraub1990;
+Weaver & Jones1992),andforbiddenemission lines ([O I]
+λ6300.3˚A; Hirth et al. 1997). It has a very red colour of
+J−Ks=1.77±0.03mag(Caballero&Solano2008).From
+our light curve, PU Ori is an irregular variable with mini-
+mumandmaximummagnitudes V0∼12.5and14.5mag.
+V472 Ori (No. 53). It is an H αemitter star (Haro &
+Moreno 1953; Wiramihardja et al. 1989) of known photo-
+metricvariability(Fedorovich1960;Cieslinski etal.199 7).
+Itmaybetheoptical/nearinfraredcounterpartofIRASZ053 22–
+0039. V472 Ori is a probable member in the Mintaka clus-
+ter (Caballero & Solano 2008). Its ASAS light curve has a
+peak-to-magnitudelargerthan1mag.
+V351Ori(No.70). ItisaHAeBeA7IIIevariablestarof
+δScuti type discovered by Hoffmeister (1934). It has been
+investigatedindetail,especiallyafterKovalchuk(1984) ob-
+served a flare-like event on its light curve at the UBVR
+bands (Wouterloot & Walmsley 1986; Sterken et al. 1993;
+Rosenbush 1995; Marconi et al. 2000; Balona et al. 2002;
+Vieria et al. 2003). It has been proposed that a temporar-
+ily strong accretion of matter onto the star or extinction by
+circumstellardustcloudstakesplaceto explaintheremark -
+ableV351Orichangeofbehaviourfrom“thatofa[HAeBe]
+star with strong photometricvariations[...] to that of an a l-
+most non-variable star” (van den Ancker et al. 1996). The
+main period of pulsations of V351 Ori is only of the order
+of 0.06d with a peak-to-peak amplitude of about 0.1mag(van den Ancker et al. 1998; Marconi et al. 2001; Ripepi
+et al. 2003). The pre-main sequence nature of the star is
+supportedbythepresenceofH αinemission,apronounced
+inverseP Cygni profile,S Iλ1296˚A and O Iλ1304˚A lines
+inemission,andinfraredexcess(Valentietal.2000andref -
+erencesabove).With V0≈8.91magand20–30Jyatthe60
+and 100µmIRASpassbands, it is by far the brightest star
+in our sample at all wavelengths, but also the less variable
+one. However, the amplitude of variability, between V0∼
+8.80 and 9.15mag, is slightly larger than previously mea-
+sured.Itmaybeduetoourlong(ASAS)temporalcoverage,
+probablylongerthananyotherpreviousmonitoring.
+StHa 40 (No. 78). It was first listed in the H α-emission
+starcataloguesofMacConnell(1982)andStephenson(1986)
+and followed-up afterwards by Downes & Keyes (1988),
+Torres et al. (1995),Gregorio-Hetem& Hetem (2002),and
+Maheswar et al. (2003).StHa 40is a T Tauristar with vari-
+able spectral type (from F8, through G2 and G5, to K0),
+variable H αemission (from pEW(H α) = –14.5 ˚A to a faint
+absorption),lithiumabsorption(frompEW(Li I) =+0.13to
++0.33˚A),andradialvelocity(indicativeofpossiblespectral
+binarity). Besides, Gregorio-Hetem & Hetem (2002) asso-
+ciated StHa 40 to a ROSATX-ray source and derived ba-
+sic stellar and circumstellar parameters. Maheswar et al.
+(2003) estimated a magnitude difference ∆V∼1.6mag
+between two optical spectra taken almost three years apart
+and after comparison with spectro-photometric standards.
+TheHαlinewasinemission[absorption]whenthestarwas
+brighter,V∼10.9mag [fainter, V∼12.5mag]. However,
+while the brightest Maheswar et al. (2003)’s estimation for
+Vagreesverywellwithourlightcurve(fromwerewemea-
+sureV0,bright≈11.0mag), none of the 173 StHa 40 data
+pointscoveringalmostsixyearswerefainterthan V0,faint≈
+11.8mag.
+3.1.2 Previouslyunknown variableyoungstars
+Of the fourstars in thisclass, onlytwo (Mayrit528005AB
+andStHa48)haveuncontrovertiblesignpostsofyouth.The
+othertwostars(KisoA–0903135andHD290625)havenot
+beenwellinvestigatedyetandmaynotbelongto theyoung
+OrionBelt stellarpopulation.
+Mayrit 528005 AB (No. 18). It is a hierarchical triple
+systeminthe σOrioniscluster,madeofaclosebinary( ρA−B=
+0.40±0.08arcsec)resolvedwithadaptiveopticsat ρAB−C∼
+7.6arcsectothelow-massstarMayrit530005(SOriJ053847. 5–
+022711; Caballero 2005, 2009). The binary primary has a
+cosmic lithium abundance (Zapatero Osorio et al. 2002),
+double-peak H α(Caballero 2006), and strong X-ray emis-
+sions (Wolk 1996; Fraciosini et al. 2006; Caballero et al.
+2009),andanspectralenergydistributiontypicalofclass II
+objects(Hern´ andezetal.2007).TheASASlightcurveshows
+variability between V0∼13 and 14mag, and an apparent
+seriesofbrighteningsatHJD ∼3000.
+www.an-journal.org c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim8 Caballero etal.:The most variable ASASstars inthe OrionBe lt
+Kiso A–0903 135 (No. 41). It is an emission-line star
+only referenced once by Kogure et al. (1989), who mea-
+sured a weak H αemission intensity and placed it in the
+Orion region.Itsoptical (DENIS;Epchteinet al. 1997)and
+near-infrared(2MASS) photometryand null propermotion
+(within errorbars; USNO-B1) are consistent with member-
+ship in the Ori OB1 b association. With σV0= 0.480mag,
+it is one of the most variable stars in the study. While its
+brightest magnitude is V0∼12.8mag, the faintest one is
+V0∼15.0magorfainter.
+StHa 48 (No. 55). It is a K4-type T Tauri star with H α
+in emission and Li Iin absorption (Stephenson 1986; Ma-
+heswar et al. 2003; Caballero & Solano 2008). The star,
+whichislocatedinthesurroundingsofAlnilam,hasawide
+amplitudeofvariability,withmaximumandminimumbright-
+eness atV0≈11.9 and 13.3mag, respectively, and time
+scalesofvariationofa fewdays.
+HD290625(No.74). ItisanA2starclassifiedasaprob-
+able member in the young stellar population of Ori OB1 b
+(Guetter1979;Nesterovetal.1995).TheASASlightcurve
+displayed variations between V0≈10.8 and 11.3mag, ap-
+proximately.The Tycho-2and 2MASS photometryand the
+lowpropermotion,oflessthan4masa−1(Tycho-2,PPMX,
+USNO-B1),areconsistentwiththespectraltypeandalong
+heliocentricdistance.Besides,lifesofA2starsarerelat ively
+short, which confirm the HD 290625 youth. However, the
+starisacoupleofmagnitudesfainterthanexpectedfor“nor -
+mal” A2 Orion stars at d∼0.4kpc. Instead of resorting to
+another young population in the far background of Orion,
+of which there is no evidence, we proposethat HD 290625
+is a variable HAeBe star with a scattering edge-on disc in
+the Orion complex, just like UX Ori or the σOrionis star
+StHa 50 (Mayrit 459340 – Th´ e et al. 1994; Caballero et al.
+2008) or, alternatively, a bright, blue, subdwarf (there ha ve
+been previous detections of such objects in the area: Ca-
+ballero&Solano2007;Venneset al.2007).
+3.2 Variablebackgroundgiants
+Unfortunately,becauseoftherelativeresemblancebetwee n
+the spectral energy distributions of distant giant stars wi th
+extendedcoolenvelopesandofyoungstarswith developed
+discs (due to the common strong infrared flux excess), gi-
+antstarssometimescontaminatelistsofmembercandidates
+in star-forming regions. For example, the bright (in the in-
+frared)giantstarIRAS05358–0238wasonceclassifiedasa
+ClassIobjectinthe σOrioniscluster(Oliveira&vanLoon
+2004;Hern´ andezetal.2007;Caballero2008).Sincethesur -
+veyed area in this work is rich in star-forming regions (in-
+cludingσOrionis),we hadtobe carefulwith disentangling
+bothpre-mainsequenceandgiantstarpopulations.
+Of the eight identified variable backgroundgiants, only
+one (No. 60) has no IRASexcesses or were not detected
+with the Spatial Infrared Imaging Telescope onboard the10−11001011021031040100020003000400050006000700080009000
+Period [d]Power [mag2]
+00.20.40.60.811.21.41.61.829
+10
+11
+12
+13
+14
+15
+PhaseV0 [mag]
+Fig.6Same as Fig. 5, but for X Ori and a period P=
+416.05d.
+Midcourse Space Experiment (8.3–21.3 µm; Kraemer et al.
+2003).Therewerealsoonlythreestarspreviouslyclassifie d
+asgiants:XOri,GTOri,andIRAS05354–0142(forwhich
+variability had not ever been proposed). Therefore, in this
+work we present six new variable giants. Besides, we mea-
+sureperiodsofphotometricvariability(withmaximaofthe
+powerspectrumabove500mag2)between78and920dfor
+four of the eight giants, while only one, X Ori, had been
+reportedbeforeto displayperiodicity.
+3.2.1 Known backgroundgiants
+X Ori (No. 29). This giant or superginat star was the
+reddest object in the Alnilam-Mintaka area studied by Ca-
+ballero & Solano (2008), but is located at about twice the
+heliocentricdistance to the Ori OB1 association (Jura et al .
+1993).ItisanM8–9-typeMiraCeti variablefoundbyWolf
+(1904)with P=424.75±1.77d(Templetonetal.2005)and
+silicate dust emission (Sloan & Price 1998; Speck et al.
+2000). In particular, the 9.7 and 18 µm silicate dust fea-
+ture originatesfromthe circumstellarenvelopesofoxygen -
+richasymptoticgiantbranchstars(Kwoketal.1997).From
+our periodogram(Fig. 6), there are two clear peaks at P≈
+c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim www.an-journal.orgAstron. Nachr./ AN(0000) 9
+10−1100101102103104020040060080010001200140016001800
+Period [d]Power [mag2]
+00.20.40.60.811.21.41.61.8210.5
+11
+11.5
+12
+12.5
+13
+13.5
+PhaseV0 [mag]
+Fig.7Same as Fig. 5, but for GT Ori and a period P=
+920.32d.
+416d,consistentwithliteraturevaluesassumingareasoba le
+errorof2%, anditsharmonicat P≈832d.Thereisalso a
+spurious(systematic) peakat P= 1d. The triangularshape
+of minimaandmaximainthe phase-foldedlightcurvemay
+be useful for modelling the astrophysical properties of the
+Mira star.
+GTOri(No.66). Itisasemi-regularpulsatingstar,orig-
+inally proposed to be a likely Mira star with variations be-
+tween11.5and12.5magbyHoffmeister(1934).Afterwards,
+it has been monitored by several authors (e.g. Kurochkin
+1949; Braune 2001). The F0 spectral type determined by
+Nesterov et al. (1995) is in contradiction with the red op-
+tical and near-infrared colours, the detection of GT Ori by
+IRASatthe12and25 µm,anditslowpropermotion,ofless
+than 3masa−1. The photometric and astrometric informa-
+tion and the strong periodogrampeakat P∼920d and the
+phase-folded light curve in Fig. 7 are more consistent with
+the semi-regularpulsatingstatusanda probableK-type,gi -
+ant nature of GT Ori. The large amplitude of variability of
+the star, of almost 3mag ( V0≈10.6–13.5mag), and the
+overlapping of different (shorter) periods in the light cur ve10−110010110210310401002003004005006007008009001000
+Period [d]Power [mag2]
+00.20.40.60.811.21.41.61.8210.2
+10.4
+10.6
+10.8
+11
+11.2
+11.4
+PhaseV0 [mag]
+Fig.8Same as Fig. 5, but for IRAS 05354–0142 and a
+periodP=373.31d.
+areremarkable,aswellasitsabnormallyblue V0−Jcolour
+foritsred J−KsandIRASfluxexcessemission.
+3.2.2 New variablegiantcandidates
+IRAS05354–0142(No.30). WithVT−Ks=7.1±0.3mag,
+it had the reddest optical-infraredcolour amongthe ∼1500
+Tycho-2/2MASS stars investigated by Caballero & Solano
+(2008). According to them, “the closeness of IRAS 0534–
+0142 to the Ori I–2 globule may explain part of its red-
+dening, but not all. It might be an S-type or a C-type gi-
+ant with a very late spectral type and very low effective
+temperature”. From our ASAS data, IRAS 05354–0142 is
+the brightest star in Table 2 ( V0≈10.2–11.2mag) and has
+the fourth largest power spectrum. Besides, it is one the
+four most clearly variable stars in this study, together wit h
+XOri,RYOri,andGTOri.Thelightcurvepowerspectrum
+peaks atP= 373.31d, but there are also secondary max-
+ima at shorter periods (Fig. 8). The multi-pattern, phase-
+folded light curve indicates that a harmonic study is nec-
+essary to disentangle the different oscillation modes of th e
+giant. At about 3arcmin to the north lies the irregular vari-
+ablestarV1299Ori,foundbyKaiser(1992).Itwastheonly
+www.an-journal.org c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim10 Caballero etal.:The most variable ASASstars inthe OrionBe lt
+10−1100101102103104020040060080010001200140016001800
+Period [d]Power [mag2]
+00.20.40.60.811.21.41.61.8211.5
+12
+12.5
+13
+13.5
+14
+PhaseV0 [mag]
+Fig.9Same as Fig. 5, but for IRAS 05285–0325 and a
+periodP=78.226d.
+SIMBADobjectunidentifiedbyCaballero&Solano(2008)
+in the Alnilam/Mintaka region. Tabulated coordinates are
+likelyincorrect.IRAS05354–0142andV1299Orimightbe
+the same star, but the variability type and amplitude given
+by Kaiser (1992) are inconsistent with the data presented
+here.
+IRAS 05285–0325 (No. 03), IRAS 05353–0309 (No. 07),
+IRAS05248–0037(No.54),andIRAS05312+0053(No.75).
+Theyhad beenonlyreportedbyKraemeret al. (2003)as
+“infrared” sourcesand have “high quality” flux densities at
+theIRAS12µm(and25 µm)pass-bandsandpropermotion
+absolute valuesof a few milliarcsecondsperyear. Of them,
+only one had a power spectra peak larger than 500mag2
+(Table 2). The periodogram of IRAS 05285–0325 shows a
+forest of peaks associated to the different pulsation modes
+of the giant. The main mode,which is quite clear in Fig. 9,
+correspondsto aperiod P=78.3±0.1d.
+No. 60. It is reported here for the first time and has not
+a measured IRASfluxdensity.However,itsveryredoptical
+and near-infrared colours, which puts the star in the giant10−11001011021031040100200300400500600700800
+Period [d]Power [mag2]
+00.20.40.60.811.21.41.61.8212.5
+13
+13.5
+14
+14.5
+15
+PhaseV0 [mag]
+Fig.10 SameasFig.5,butforV1159Oriandaperiod P
+= 44.345d.
+locus in Fig. 4 ( V0−J∼4.8mag), and very low proper
+motionsmakeNo.60tobea fairgiantcandidate.
+3.3 Miscellanea
+In this class, we include variable objects that are neither
+young stars nor background giants. Of the eight objects in
+this class, only two (the dwarf nova cataclysmic variable
+V1159Oriandthecontactbinary2M054354–0243.6)were
+knowntovary.Atleastthreeoftheothersixstars,includin g
+thenearbyearlyF-typestarHD290509,mightbeeclipsing
+orinteractingbinaries.
+V1159 Ori (No. 01). It is a well-knowndwarf nova cat-
+aclysmic variable of ER UMa (or RZ LMi) type, which is
+a subgroup of SU UMa-type dwarf novae4with short re-
+currence time of super-outburst ( τ≤45d). Their super-
+4A dwarf nova is a close binary star system in which one of the co m-
+ponents is a white dwarf, which accretes matter from its comp anion. In
+contrast to classical novae, outbursts and super-outburst s in dwarf novae
+result from instabilities in the accretion disc. SU UMa dwar f novae are
+characterised bytwodistincttypesofoutburstsintimesca lesofdays(more
+frequent) and weeks (less frequent and with larger amplitud e).
+c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim www.an-journal.orgAstron. Nachr./ AN(0000) 11
+cycles are stable both in length and outburst pattern (Os-
+aki 1996; Patterson 1998; Downes et al. 2001). V1159 Ori
+was discovered by Wolf & Wolf (1906), who gave it the
+name“Var.,Orionis36.1906”,andfirst investigatedbyKip-
+penhahn (1953) and Jablonski & Cieslinski (1992). After-
+wards,itsveryshortoutburstcycle,superhumps,orbitalp e-
+riods, and high energy (X-ray, ultraviolet) emission have
+been studied by a number of authors (Nogami et al. 1995;
+Robertson et al. 1995; Patterson et al. 1995; Thorstensen
+et al. 1997; Szkody et al. 1999). We can add little to the
+knowledge on V1159 Ori except for its supercycle period.
+In our periodogram (Fig. 10), we detect a clear peak (and
+its harmonics)at P= 44.3±0.1d,whichisagreementwith,
+or may improve or complement, previous determinations
+at 44.2–46.8d (Robertson et al. 1995; Richter 1995; Kato
+2001;Pitts etal. 2002).
+2M054354–0243.6([GGM2006]12390250,No. 14). It
+is a contact binary discovered by Gettel et al. (2006), who
+measured an heliocentric distance of d= 288±5pc. Given
+the appreciable proper motion tabulated in the USNO-B1
+and PPMX catalogues, µ= 46masa−1, one can measure a
+transversal velocityof about 63kms−1, which is typical of
+Galactic disc stars. In our periodogram (not shown here),
+there are two faint peaks at 0.29 and 2.3d, but nothing at
+the 0.438118d measured by Gettel et al. (2006). Besides,
+the phase-folded light curve of 2M054354–0243.6 at this
+perioddoesnotshowanyclear periodicbehaviour.
+No. 33 and No. 65. They are reported here for the first
+time.Thestarsdisplayopticalandnear-infraredcolourst yp-
+ical of early K dwarfs and have no infrared excess. The
+lightcurvemagnitudesvariedbetween V0≈12.4and13.4–
+13.6mag (No. 33) and V0≈11.7 and 12.4mag (No. 65),
+withmostofthedatapointsconcentratedtowardsthebright -
+est limits. This fact may be an indication that the stars are
+possiblyneweclispingorinteractingbinaries,butthepos si-
+bilityofbeingstarswithmagneticspotsshouldnotberuled
+out.Thattheperiodograms(notshownhere)donotdisplay
+anystrongpeakputsalimitonthemaximumperiods,which
+are possiblyshorterthana fewdays.
+HD 290509 (No. 46). It is an anodyne F0-type Tycho-2
+starwithonlyoneentryinSIMBAD(Nesterovetal.1995).
+TheappreciablepropermotiontabulatedbyPPMX,of(30.6 ±1.3,
+–8.8±1.3)masa−1, indicates that it is a field star and not
+a young star in Orion. Its optical and red colours match
+those of dwarfs of the same spectral type. The ASAS light
+curve of HD 290509 resembles that of the star No. 33, but
+with a shorter range of variation, between V0≈10.35 and
+10.60mag. It might be another short-period eclipsing bi-
+nary.
+VSSVI–32(No.72). Itisanunpolarisedstarintheback-
+ground of the Messier 78 star-forming complex, only re-
+ported by Vrba et al. (1976). Its very red colours, of V0−J∼2.5mag and J−Ks= 1.49±0.03mag, may not be
+intrinsic, but due to the high extinction in the area. Its var i-
+abilityclassisunknowntous.
+No. 06, No. 73. They are also reported here for the first
+time and none of them have a measured IRASflux den-
+sity. They have very low proper motions and are the red-
+dest(inJ−Ks)starsinthissectionexceptforVSSVI–32,
+but do not seem to be red enough to be classified as giant
+star candidates. Their light curves are quite different, wi th
+No. 06’s one varying only between 11.05 and 11.45mag,
+and No. 73’s one between 12.0 and 14.0mag. Neither the
+powerspectrumanalysis northe light curveshapesprovide
+anyclearindicationofthe natureofthetwostars.
+4 Summary
+With the aim of discovering and confirming variable Her-
+big Ae/Be and T Tauri stars among the young stellar pop-
+ulation of the Orion Belt, we have analysed over 60000
+light curves from the ASAS-3 Photometric V-band Cata-
+logueoftheAllSkyAutomatedSurveyinasquaredareaof
+side5degcentredonthesupergiantstarAlnilam( ǫOri,the
+middlestaroftheOrionBelt).
+After consecutive filterings, visual inspections with the
+virtual observatorytool Aladin, and cross-matcheswith in -
+fraredandastrometriccatalogues,we kept32 variablestar s
+with average magnitudes V0= 8.5–13.5mag for a detailed
+analysis and data compilation. They are the most variable
+stars in the Orion Belt and complement the results in Po-
+jma´ nski (2002). Based on spectroscopic, photometric, and
+astrometric information from catalogues and the literatur e,
+we classifiedthe 32starsinto:
+–previously known variable young stars and candidates
+(12stars).Thelistincludeswell-investigatedHerbigAe/ Be
+andTTauristarssurroundingtheearly-typestars σOri,
+Alnilam,andMintakaintheOriOB1bassociation.We
+confirmthe variablestatusof two activestars suspected
+ofvariability(GSC04767–00071andStHa40)andmea-
+sureapeakofthepowerspectrumlargerthan500mag2
+inthelightcurveoftheemission-line,UXOri-typevari-
+able, disc-host star RY Ori. However, the correspond-
+ing period of photometric variability seems unreliable.
+Thus,the12starsappearto beirregularvariables;
+–previouslyunknownvariableyoungstars(4stars).They
+are two T Tauri stars with Li Iin absorption and H αin
+emission(Mayrit528005ABandSHa48)andtwostars
+earlierclassifiedasprobablemembersinthe OriOB1 b
+association. Besides, one of them, Mayrit 528005 AB,
+displaysinfraredfluxexcessandstrongX-rayemission,
+and is a close binary within a hierarchical triple system
+in theσOrionis cluster. Again, we do not find periodic
+trendsamongtheseyoungstars;
+–known background giants (2 stars). The two giants (or
+supergiants)areX Ori, forwhichourperioddetermina-
+tion is consistent with those provided in the literature,
+www.an-journal.org c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim12 Caballero etal.:The most variable ASASstars inthe OrionBe lt
+and GT Ori, for which we firstly provide a period of
+photometricvariabilityatabout920danddetermineab-
+normaloptical-nearinfraredcolours;
+–new variablegiant candidates(6 stars). Of them, we re-
+markthenew,extremelyred,periodicvariablesIRAS05354–
+0142 (P∼370d – with a lot of shorter periods) and
+IRAS 05285–0325( P∼78d);
+–miscellaneaobjects(8stars).Thisclasscontainsthecat-
+aclysmic variable V1159 Ori, for which we measure a
+periodP= 44.3±0.1d, in agreement with previous de-
+terminations,thecontactbinary2M054354–0243.6,and
+six poorlyinvestigated,newvariablestars.
+Some of these targets, including variable active Her-
+big Ae/Be and T Tauri starsin the Ori OB1 association and
+background giants, merit next spectroscopic analyses and
+photometricmonitoring.
+Acknowledgements. We thank M. Paegert for his helpful referee
+report and careful reading of the manuscript, and G. Pojma´ n ski
+for his kind and prompt answer to our inquiries. JAC formerly
+was an “investigador Juan de la Cierva” at the Universidad Co m-
+plutense deMadridandcurrentlyisan“investigadorRam´ on yCa-
+jal” at the Centro de Astrobiolog´ ıa (CSIC-INTA). This rese arch
+has made use of the SIMBAD, operated at Centre de Donn´ ees as-
+tronomiques de Strasbourg, France, and the NASA’s Astrophy sics
+Data System. Financial support was provided by the Universi dad
+Complutense de Madrid, the Comunidad Aut´ onoma de Madrid,
+the Spanish Ministerio Educaci´ on y Ciencia, and the Europe an
+Social Fund under grants: AyA2005-04286, AyA2005-24102-E ,
+AyA2008-06423-C03-03, AyA2008-00695, PRICITS-0505/ESP -
+0237, and CSD2006-0070.
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+1500 2000 2500 3000 3500 400012.5
+13
+13.5
+14
+14.5
+15
+HJDV1159 OriV0 [mag]
+1500 2000 2500 3000 3500 400011.5
+12
+12.5
+13
+13.5
+14
+HJDIRAS 05285−0325V0 [mag]
+1500 2000 2500 3000 3500 400011
+11.05
+11.1
+11.15
+11.2
+11.25
+11.3
+11.35
+11.4
+11.45
+11.5
+HJD#06V0 [mag]
+1500 2000 2500 3000 3500 400010.5
+10.6
+10.7
+10.8
+10.9
+11
+11.1
+11.2
+HJDIRAS 05353−0309V0 [mag]
+1500 2000 2500 3000 3500 400010.5
+11
+11.5
+12
+12.5
+13
+13.5
+HJDRY OriV0 [mag]
+1500 2000 2500 3000 3500 400012
+12.5
+13
+13.5
+14
+14.5
+HJDTX OriV0 [mag]
+1500 2000 2500 3000 3500 400012
+12.5
+13
+13.5
+14
+14.5
+HJDTY OriV0 [mag]
+1500 2000 2500 3000 3500 400011.3
+11.4
+11.5
+11.6
+11.7
+11.8
+11.9
+12
+HJDV0 [mag]2M054354−0243.6
+Fig.A1 ASAS (V0)lightcurvesoftheidentifiedvariablesNo.01to14(labele d).
+c/circlecopyrt0000WILEY-VCH VerlagGmbH&Co.KGaA, Weinheim www.an-journal.orgAstron. Nachr./ AN(0000) 15
+1500 2000 2500 3000 3500 400012.2
+12.4
+12.6
+12.8
+13
+13.2
+13.4
+13.6
+13.8
+14
+14.2
+HJDRW OriV0 [mag]
+1500 2000 2500 3000 3500 400012
+12.5
+13
+13.5
+14
+14.5
+HJDV0 [mag]Mayrit 528005 AB
+1500 2000 2500 3000 3500 400012.2
+12.4
+12.6
+12.8
+13
+13.2
+13.4
+13.6
+HJDPQ OriV0 [mag]
+1500 2000 2500 3000 3500 40009
+10
+11
+12
+13
+14
+15
+HJDX OriV0 [mag]
+1500 2000 2500 3000 3500 400010.2
+10.4
+10.6
+10.8
+11
+11.2
+11.4
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+arXiv:1001.0663v2  [cond-mat.dis-nn]  28 May 2010Self-organized chaos through polyhomeostatic optimizati on
+D. Markovic and Claudius Gros
+Institute for Theoretical Physics, Johann Wolfgang Goethe University, Frankfurt am Main, Germany
+(Dated: June 19, 2018)
+The goal of polyhomeostatic control is to achieve a certain t arget distribution of behaviors, in
+contrast to homeostatic regulation which aims at stabilizi ng a steady-state dynamical state. We con-
+sider polyhomeostasis for individual and networks of firing -rate neurons, adapting to achieve target
+distributions of firing rates maximizing information entro py. We show that any finite polyhomeo-
+static adaption rate destroys all attractors in Hopfield-li ke network setups, leading to intermittently
+bursting behavior and self-organized chaos. The importanc e of polyhomeostasis to adapting behav-
+ior in general is discussed.
+Introduction. – Homeostatic regulation plays a central
+role in all living, as well as in many technical applica-
+tions. Biological parameters, like the blood sugar level,
+the heart beating frequency or the average firing rates of
+neurons need to be maintained within certain ranges in
+order to guarantee survival. The same holds in the tech-
+nical regime for the rotation speed of engines and the
+velocity of airplanes, to give a few examples.
+Homeostasic control in the brain goes beyond the reg-
+ulation of scalar variables like the concentration of pro-
+teins and ions, involving the functional stability of neu-
+ral activity both on the individual as well as on a net-
+work level [1–3]. We use here the term ‘polyhomeosta-
+sis’ for self-regulating processes aimed at stabilizing a
+certain target distribution of dynamical behaviors. Poly-
+homeostasisisanimportantconceptusedhithertomostly
+implicitly and not yet well studied from the viewpoint
+of dynamical system theory. Polyhomeostasis is present
+whenever the goalof the autonomouscontrolis the stabi-
+lization of a non-trivial distribution of dynamical states,
+polyhomeostatic control hence generalizes the concept of
+homeostasis. The behavior of animals on intermediate
+time scales, to give an example, may be regarded as
+polyhomeostatic, aiming at optimizing a distribution of
+qualitatively different rewards, like food, water and pro-
+tection; animals are not just trying to maximize a single
+scalarrewardquantity. Aconceptlooselyrelatedtopoly-
+homeostasis is homeokinesis, proposed in the context of
+closed-loop motion learning [4], having the aim to sta-
+bilize non-trivial but steady-state movements of animals
+and robots.
+Here we study generic properties of dynamical systems
+governed by polyhomeostatic self-regulation using a pre-
+viouslyproposedmodel[5,6]forregulatingthefiring-rate
+distribution of individual neurons based on information-
+theoretical principles. We show that polyhomeostatic
+regulation,aimingatstabilizingaspecifictargetdistribu-
+tion of neural activities gives rise to non-trivial dynami-
+calstateswhenrecurrentinteractionsareintroduced. We
+find, in particular, that the introduction of polyhomeo-
+static controlto attractornetworksleads to a destruction
+ofallattractorsresultingforlargenetworks,asafunction
+of the average firing rate, in either intermittent burstingbehavior or self-organized chaos, with both states being
+globally attracting in their respective phase spaces.
+Firing-rate distributions. – We considera discrete-time,
+rate encoding artificial neuron with input x∈[−∞,∞],
+outputy∈[0,1] and a transfer function g(z),
+y(t+1) =g/parenleftbig
+a(t)x(t)+b(t)/parenrightbig
+, g(z) =1
+e−z+1.(1)
+The gain a(t) and the threshold −b(t)/a(t) in (1) are
+slow variables, their time evolution being determined by
+polyhomeostatic considerations.
+Information is encoded in the brain through the firing
+states of neurons and it is therefore plausible to postu-
+late [5], that polyhomeostatic adaption for the internal
+parameters a(t) andb(t) leads to a distribution p(y) for
+the firing rate striving to encode as much information
+as possible given the functional form (1) of the transfer
+function g(z). The normalized exponential distribution
+pλ(y) =λe−λy
+1−e−λ, µ=1
+λeλ−1−λ
+eλ−1,(2)
+maximizes the Shannon entropy [7], viz the information
+content, on the interval y∈[0,1], for a given expecta-
+tion value µ. A measure for the closeness of the two
+probability distributions p(y) andpλ(y) is given by the
+Kullback-Leibler divergence [7]
+Dλ(a,b) =/integraldisplay
+p(y)log/parenleftbiggp(y)
+pλ(y)/parenrightbigg
+dy , (3)
+which is, through (1), a function of the internal parame-
+tersaandb. The Kullback-Leibler divergence is strictly
+positive and vanishes only when the two distributions are
+identical. By minimizing Dλ(a,b) with respect to aand
+bone obtains [6] the stochastic gradient rules
+∆a=ǫa/parenleftBig
+1/a+x∆˜b/parenrightBig
+∆b=ǫb∆˜b,∆˜b= 1−(2+λ)y+λy2(4)
+which have been called ‘intrinsic plasticities’ [1]. The re-
+spective learning rates ǫaandǫbare assumed to be small,
+viz the time evolution of the internal parameters aandb
+is slow compared to the evolution of both xandy. For2
+0.2 0.4 0.6 0.8 1 µ 4812
+a
+stable fixpointsy
+xa, b
+FIG. 1: (color online) Left: As a function of the average
+firing rate µ, the region of stability (shaded area) of the fix-
+point (y∗,b∗), see Eq. (7) of the one-neuron network in the
+quasistatic limit. Right: A self-coupling neuron with inte rnal
+parameters a(t) andb(t).
+any externally given time series x(t) for the input, the
+adaption rules (4) will lead to a distribution of the out-
+put firing rates y(t) as close as possible, given the spec-
+ification (1) for the transfer function, to an exponential
+with given mean µ.
+Attractor relics. – In deriving the stochastic gradient
+rules (4) it has been assumed that the input x(t) is sta-
+tistically independent of the output y(t) (but not vice
+versa). This is not any more the case when a set of poly-
+homeostatically adapting neurons are mutually intercon-
+nected, forming a recurrent network, Here we will show
+that networks based on polyhomeostatic principles will
+generically show spontaneous and continuously ongoing
+activity.
+In a first step we analyze systematically the smallest
+possible network, viz the single-site loop, obtained by
+feeding the output back to the input, see Fig. 1, a setup
+which has been termed in a different context ‘autapse’
+[8]. We use the balanced substitution x→y−1/2 in
+Eqs. (1) and (4), the complete set of evolution rules for
+the dynamical variables y(t), a(t) andb(t) is then
+y(t+1) = g/parenleftbig
+a(t)[y(t)−1/2]+b(t)/parenrightbig
+b(t+1) = b(t)+ǫb∆˜b(t) (5)
+a(t+1) = a(t)+ǫa/parenleftBig
+1/a(t)+[y(t)−1/2]∆˜b(t)/parenrightBig
+with
+∆˜b(t) = 1−(2+λ)y(t+1)+λy2(t+1).(6)
+Note, that ∆ ˜b(t) in (6) depends on y(t+ 1), and not
+ony(t), as one can easily verify when going through the
+derivation of the rules (4) for the intrinsic plasticity. The
+evolution equations (5) are invariant under y↔(1−y),
+b↔(−b),a↔aandλ↔(−λ), the later corresponding
+to the interchange of µ↔(1−µ).
+We first consider the quasistatic limit ǫa≪ǫb, viz
+a(t)≃ais approximatively constant. The fixpoint
+(y∗,b∗) in the ( y,b) plane is then determined by
+0 =λ(y∗)2−(2+λ)y∗+1
+b∗=g−1(y∗)−a[y∗−1/2]
+= log(y∗/(1−y∗))−a[y∗−1/2](7)1000 2000 3000t0123y(t), b(t), a(t)-4a(t)-4
+b(t)
+y(t)
+FIG. 2: (color online) The time dependence of y(t) (red stars)
+b(t) (solid line) and of a(t)−4 (dashed line) for the balanced
+one-site problem (5). ǫa=ǫb=ǫ= 0.01 andµ= 0.28; the
+horizontal dotted lines are guides to the eye. The gain a(t) is
+initially small and the system relaxes, since ǫ≪1, fast to a
+fixpoint of y=g(a[y−1/2]+b). Oncea(t) surpasses a certain
+threshold, compare Fig. 1, the fixpoint becomes unstable and
+the system starts to spike spontaneously.
+A straightforward linear stability analysis shows, that
+the fixpoint ( y∗,b∗) remains stable for small gains aand
+becomes unstable for large gains, see Fig. 1. We now
+go beyond the quasistatic approximation and consider
+a small but finite polyhomeostatic adaption rate ǫafor
+the gain. Starting with a small gain awe see, compare
+Eq. (5), that the gain necessarily grows until it hits the
+boundary towards instability; for small ∆ ˜bthe growths
+of the gain is a(t)∼√
+t.
+In other words, a finite adaption rate ǫafor the gain
+turns the fixpoint attractor ( y∗,b∗) into an attractor
+relic and the resulting dynamics becomes non trivial and
+autonomously self-sustained. This route towards au-
+tonomous neural dynamics is actually an instance of a
+more general principle. Starting with an attractor dy-
+namical system, viz with a system governed by a finite
+number of stable fixpoint, one can quite generally turn
+these fixpoints into unstable attractor ruins by coupling
+locally to slow degree of freedoms [10, 11], in our case the
+slow local variables are a(t) andb(t). Interestingly, there
+are no saddlepoints present in attractor relic networks in
+general, and in (5) in particular, and hence no unstable
+heteroclines, as in a proposed alternative route to tran-
+sient state dynamics via heteroclinic switching. [12, 13]
+In Fig. 2 the time evolution is illustrated for λ= 3.017,
+which corresponds to µ= 0.28, see Eq. (2), and ǫa=
+ǫb=ǫ= 0.01. The system remains in the quasistation-
+ary initial regime until the gain asurpasses a certain
+threshold. The initial quasistationary fixpoint becomes
+therefore unstable via the same mechanism discussed an-
+alytically above, see Eq. (7), for the regime ǫa→0, com-
+pare Fig. 1. The output activity y(t) oscillates fast be-
+tween two transient fixpoints of y=g(a[y−1/2] +b),
+having a high and a low value respectively. This spiking
+behavior of the neural activity is driven by spontaneous
+oscillations in the threshold −b(t)/a(t), shifting the in-
+tersection of g(a[y−1/2]+b) withyforth and back.3
+0 1000 2000 3000t-1012y(t), a(t) - 4, b(t) + 2a(t)-4
+y(t)
+b(t)+2
+FIG. 3: (color online) The time dependence of y(t) (red stars)
+b(t)+2 (solid line) and of a(t)−4 (dashed line) for the one-
+site problem with inhibitory self-coupling x→1/2−y,ǫa=
+ǫb=ǫ= 0.01 andµ= 0.28. A Hopf-bifurcation occurs for the
+outputy(t) when the initial quasistationary fixpoint becomes
+unstable, giving place to a new fixpoint of period two.
+Evaluating the local Lyapunov exponents we find that
+thetrajectoryisstableagainstperturbationsforthetran-
+sient states close to one of the transient fixpoints of
+y=g(a[y−1/2]+b) and sensitive to perturbations and
+external modulation during the fast transition periods,
+an instantiation of the general notion of transient state
+dynamics [10, 11].
+Inhibitory self-coupling. – So far we discussed, see (5)
+and Fig. 2, a neuron having its output y(t) coupled back
+excitatorily to its input via x→y−1/2. The dynam-
+ics changes qualitatively for an inhibitory self-coupling
+x→1/2−y, see Fig. 3. There is now only a single
+intersection of g(a[−y+1/2]+b) withy. This intersec-
+tion corresponds to a stable fixpoint for small gains a. A
+Hopf-bifurcation [14] occurs when a(t) exceeds a certain
+threshold and a new fixpoint of period two becomes sta-
+ble. The coordinates of this fixpoint of period two slowly
+drift, compare Fig. 3, due to the residual changes in a(t)
+andb(t). Interestingly, the dynamics remains non-trivial,
+as a consequence of the continuous adaption, even in the
+case of inhibitory self-coupling.
+Self organized chaos. – We have studied numerically
+fully connected networks of i= 1,..,Npolyhomeostat-
+ically adapting neurons (4), coupled via
+xi(t) =/summationtext
+j/negationslash=iwijyj(t). (8)
+The synaptic strengths are wij=±1/√
+N−1, with in-
+hibition/excitation drawn randomly with equal proba-
+bility. The adaption rates are ǫa=ǫb= 0.01. We con-
+sider homogeneous networks where all neurons have the
+identical µfor the target output distributions (2), with
+µ= 0.15,0.28 andµ= 0.5.
+The activity patterns presented in the inset of Fig. 4
+are chaoticfor the µ= 0.28networkand laminarwith in-
+termittent bursting for the network with µ= 0.15. This
+behavior is typical for a wide range of network geome-
+tries, distributions of the synaptic weights and for all ini-
+tial conditions sampled. The respective Kullback-Leibler
+divergences Dλdecrease with time passing, as shown in50000 55000 60000
+t00.51y(t)49000 49500 5000000.51y(t)
+0 1 2t/1050123<Dλ(t)>N A
+B
+FIG. 4: (color online) Mean Kullback-Leibler divergence Dλ,
+forN= 500 networks, of the real and target output dis-
+tributions as a function of number of iterations t. The solid
+black, dashed-dottedredandthedashedbluelines correspo nd
+respectively to target firing rates µ= 0.15,µ= 0.28, and
+µ= 0.5. Inset: Output activity of two randomly chosen neu-
+ronsfor targetaveragefiringrates µ= 0.28(top)and µ= 0.15
+(bottom).
+Fig. 4, due to the ongoing adaption process (4). Dλbe-
+comes very small, though still finite, for long times in the
+self-organized chaotic regime ( µ= 0.28,0.5), remaining
+substantially larger in the intermittent bursting phase
+(µ= 0.15).
+We have evaluated the global Lyapunov exponent, [15]
+finding that two initial close orbits diverge until their
+distance in phase space corresponds, within the available
+phase space, to the distance of two randomly selected
+states, the tellmark sign of deterministic chaos. [14] The
+corresponding global Lyapunov exponent is about 5%
+per time-step for µ= 0.5 and initially increases with
+the decrease of µ, until the intermittent-bursting regime
+emerges, after which the global Lyapunov exponent de-
+clines with the decrease of µ.
+The system enters the chaotic regime, in close anal-
+ogy to the one-side problem discussed previously, when-
+ever the adaptive dynamics (4) has pushed the individ-
+ual gains ai(t), of a sufficient number of neurons, above
+their respective critical values. Hence, the chaotic state
+is self-organized. Chaotic dynamics is also observed in
+the non-adapting limit, with ǫa,ǫb→0, whenever the
+static values of aiare above the critical value, in agree-
+ment with the results of a large- Nmean field analysis
+of an analogous continuous time Hopfield network. [16]
+Subcritical static ailead on the other side to regular dy-
+namics controlled by point attractors.
+In Fig. 5 we present the distribution of the output ac-
+tivities for networks with target mean firing rates µ=
+0.28 andµ= 0.5. Shown are in both cases the firing rate
+distributions p(y) of the two neurons having respectively
+the largest and the smallest Kullback-Leibler divergence
+(3) with respect to the target exponential distributions
+(2). Also shown in Fig. 5 are the distributions of the
+network-averaged output activities.4
+0 0.5 100.010.020.030.04p(y)0 100.020.040.06A
+0 0.5 1y00.010.02p(y)0 100.020.040.06 B
+FIG. 5: (color online) Output distributions of the two neu-
+rons with highest (blue diamonds) and lowest (red circles)
+Kullback-Leibler divergence (3) compared to the mean out-
+put distribution (dashed green line) and the target exponen -
+tial output distribution (full black line). For a network wi th
+N= 500 neurons and a target output mean firing rate (A)
+µ= 0.28 and (B) µ= 0.5 (identical for all neurons). Insets:
+For the same parameters the mean output distribution of the
+single self-coupled neuron, see Fig. 1.
+ThedatapresentedinFig.5shows,thatthepolyhome-
+ostatically self-organized state of the network results in
+firing-rate distributions close to the target distribution.
+This result is quite remarkable. The polyhomeostatic
+adaption rules (4) are local, viz every neuron adapts its
+internal parameters aiandbiindependently on its own.
+Intermittency. – Interestingly, the neural activity pre-
+sented in the inset of Fig. 4 for µ= 0.15 shows inter-
+mittent or bursting behavior. In the quiet laminar peri-
+ods the neural activity y(t) is close to (but slightly be-
+low) the target mean of 0 .15, as one would expect for a
+homeostaticallyregulatedsystemandthelocalLyapunov
+exponent is negative. The target distribution of firing
+rates (2) contains however also a small but finite weight
+for large values for the output y(t), which the system
+achieves through intermittent bursts of activity. In the
+laminar/bursting periods the gain a(t) is below/above
+threshold, acting such as a control parameter [18].
+Both the intermittent and the chaotic regime are
+chaotic in terms of the global Lyapunov being positive in
+both regimes. A qualitative difference can be observed
+when considering the subspace of activities ( y1,...,y N).
+Evaluating the global Lyapunov exponent in this sub-
+space, viz the relative time evolution (∆ y1,...,∆yN) of
+differencesinactivities, wefindthisrestrictedglobalLya-
+punov to be negative in the intermittent and positive in
+the chaotic regime. Details on the evolution from inter-mittency to chaos as a function of µand system sizes will
+be given elsewhere.
+Conclusions. –It is well known [17], that individual and
+networks of spiking neurons may show bursting behav-
+ior. Here we showed, that networks of rate encoding
+neurons are intermittently bursting for low average fir-
+ing rates, when polyhomeostatically adapting, entering a
+fully chaotic regime for larger average activity level. Au-
+tonomous and polyhomeostatic self-control of dynamical
+systems may hence lead quite in general to non-trivial
+dynamical states and novel phenomena.
+Polyhomeostaticoptimizationis ofrelevanceina range
+of fields. Here we have discussed it in the framework of
+dynamical system theory. Alternative examples are re-
+source allocation problems, for which a given limited re-
+source, like time, needs to be allocated to a set of uses,
+with the target distribution being the percentages of re-
+source allocated to the respective uses. These ramifica-
+tions of polyhomeostatic optimization will be discussed
+in further studies.
+References
+[1] G.G. Turrigiano, Trends Neurosci 22, 221 (1999).
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+(2006).
+[3] E. Marder, and J.M. Goaillard, Nature Reviews Neuro-
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+[4] R. Der, U. Steinmetz, and F. Pasemann, in Computa-
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+[6] J. Triesch, in Proceedings of ICANN 2005 , W. Duch et
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+[8] H.S. Seung, D.D. Lee, B.Y. Reis, and D.W. Tank, J.
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+[9] J.H. Steil, Neural Networks 20, 353 (2007).
+[10] C. Gros, New Journal of Physics 9, 109 (2007).
+[11] C. Gros, Cognitive Computation 1, 77 (2009).
+[12] M. Rabinovich et al, Phys. Rev. Lett. 87, 68102 (2001).
+[13] C. Kirst, andM.Timme, Phys.Rev.E 78, 065201(2008).
+[14] C. Gros, Complex and Adaptive Dynamical Systems, A
+Primer, Springer (2008).
+[15] R. Ding, and J. Li, Phys. Lett. A 364, 396 (2007).
+[16] H. Sompolinsky, A. Crisanti, and H.H. Sommers, Phys.
+Rev. Lett. 61259 (1988).
+[17] M.A. Arbib, The handbook of brain theory and neural
+networks , MIT Press (2002).
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+70, 279 (1993).
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+arXiv:1001.0664v2  [hep-ph]  26 May 2010Preprint typeset in JHEP style - HYPER VERSION
+Hybrid inflation with moduli stabilization and low scale
+supersymmetry breaking
+Sander Mooij and Marieke Postma
+NIKHEF, Science Park 105, 1098 XG Amsterdam, The Netherland s.
+smooij@nikhef.nl, mpostma@nikhef.nl
+Abstract: We study the supergravity hybrid inflation model of Ref. [1] i n the presence of a
+modulus field. The η-problem is solved by a shift symmetry for the inflaton, which protects
+the inflaton mass even in the presence of the modulus field. Infl ation is (nearly) unaffected
+by moduli stabilization, provided the scale of supersymmet ry breaking in the post-inflation
+vacuum is small. Therefore the model has the nice phenomenol ogy that it combines low scale
+supersymmetry breaking with high scale (grand unification s cale) inflation.Contents
+1. Introduction 1
+2. The model: SUGRA hybrid inflation 3
+3. Adding the modulus sector 5
+3.1 General approach 6
+3.2 Discussion 9
+4. Inflation with a KL modulus sector 10
+4.1 Inflation 10
+4.2 Numerical analysis 12
+5. Conclusions 14
+A. Coleman-Weinberg potential 15
+1. Introduction
+Inflation, a short period of rapid expansion in the very early universe, is by now regarded
+as a fundamental part of standard cosmology, as it solves the horizon problem, the flatness
+problem and the monopole problem in one go. Furthermore the m echanism of hybrid in-
+flation [2], inflation that naturally comes to an end when a fiel d different from the inflaton
+field becomes tachyonic, has been shown to have a natural embe dding in the framework of
+supersymmetry. All couplings can be chosen to be of order one (no naturalness problem)
+and the field direction corresponding to the inflaton can easi ly be flat enough for slow-roll
+inflation (no η-problem). When moving on to local supersymmetry (supergra vity) [3], the
+η-problem may be circumvented by invoking a shift symmetry in the K¨ ahler potential, as was
+first proposed in [4, 5].
+It is well known that inflation is a UV sensitive theory; indee d, this is the root of the
+η-problem [6, 7]. In the language of effective field theory this c an be readily seen: higher
+order operators that are suppressed by some large cutoff scal e can nevertheless give a large
+and dominant contribution to the inflaton mass, and thus to th eη-parameter. To fully study
+inflation it is therefore imperative to consider a UV complet ion of the theory. In this paperwe
+consider embedding inflation in a higher dimensional Planck -scale theory, for example string
+theory. After dimensional reduction, the 4D effective field th eory will still carry traces of its
+– 1 –higher dimensional origin in the form of moduli fields, light scalar fields which parametrize
+the shapes and sizes of the compactified extra dimensional ma nifold. For definiteness, we
+will follow the seminal work of KKLT [8] and assume that all mo duli can be fixed at some
+high scale by fluxes, except for the volume modulus which is to be stabilized at a lower scale
+by non-perturbative effects. The dynamics of the the volume mo dulus thus enters the low
+energy effective field theory, and inflation should be studied i n conjunction with modulus
+stabilization.
+There are two ways to deal with the moduli fields in the context of inflation. The first is
+to make themodulipart of theinflaton dynamics. Thisis for ex ample donein racetrack [9, 10]
+and K¨ ahler [11, 12] inflation models, where a modulus field is identified with the inflaton field
+itself. Another approach, the one we follow in this paper, is to decouple the physics of moduli
+stabilization from the inflationary physics as much as possi ble. Our set-up is as follows: we
+have a hybrid inflation sector and a (volume) modulus stabili zation sector, which are coupled
+onlygravitationally as dictated by theSUGRA action. Event hough gravitational interactions
+are usually thought of as being weak, they are generically st rong enough to ruin inflation —
+inflation is UV sensitive. It has indeed been shown that “stan dard” SUSY hybrid inflation
+[13] cannot be combined with a KKLT-like modulus sector [14, 15] (but see [16] for a possible
+resolution). Instead we will consider a modified model of hyb rid inflation [1].
+We want to extend the hybrid inflation model of Ref. [1] with a m odulus sector. In
+our set-up the η-problem is solved by a shift symmetry for the inflaton, and in addition the
+property that the inflationary superpotential and its first d erivative w.r.t. the inflaton field
+vanishes during inflation. Note in this respect that a shift s ymmetry alone is not enough,
+as it is broken explicitly by the superpotential. This is wha t goes wrong in standard hybrid
+inflation. However, the η-problem is not the only potential difficulty. Making sure tha t the
+modulus field remains stabilized during inflation, implies t hat the scales appearing in the
+SUSY breaking modulus sector are large, resulting in a large gravitino mass. Even though
+the inflaton direction is protected, large soft corrections may destabilize the inflationary
+trajectory.
+In this paper we describe an explicit way to stabilize the mod ulus sector without running
+into the aforementioned troubles. The trick is to constrain the modulus sector in such a way
+that its gravitino mass is much smaller than the other scales in the problem. Although this
+presents some amount of tuning, the result is a phenomenolog ically favored scenario with low
+scale SUSY breaking and high scale inflation. An explicit mod ulus sector that does the job
+is the model developed by Kallosh & Linde [17, 18] (we will ref er to this as the KL model).
+Many SUGRA or string-derived models of inflation predict a la rge gravitino mass. In
+models based on a generic KKLT potential the gravitino mass h as to be larger than the
+Hubble constant during inflation m3/2/greaterorsimilarH∗[17, 18], whereas in models with a large volume
+compactification the bound is even stronger m3/2
+3/2/greaterorsimilarH∗[19]. It has proven hard to avoid this
+bound. The KL moduli potential decouples the SUSY breaking s cale from the modulus mass,
+at the cost of tuning, thereby invalidating the bound. Altho ugh it is not automatic that a
+– 2 –KL-based inflation scenario with low scale SUSY breaking can be constructed [14, 15, 20],
+successful models have been found [21], and the model discus sed in this paper is another
+example. Otherapproachestoobtainalight gravitinocanbe foundinRefs.[16,19,22,23,24].
+Thispaperisorganizedasfollows. Insection2wefirstbriefl ydescribetheshift-symmetric
+super gravitational model of hybrid inflation introduced in Ref. [1]. Then, in section 3,
+we explain why combining it with a generic KKLT-type modulus sector does not work: it
+is impossible to find a suitable inflationary trajectory stab le in field space. In the fourth
+section we show that a constrained modulus sector, of which K L is an explicit example, saves
+inflation. We discuss the inflationary observables, and show some numerical results. The
+paper concludes with a discussion of our results.
+2. The model: SUGRA hybrid inflation
+We briefly describe the supergravitational shift-symmetri c model of hybrid inflation, that
+we want to extend by including a moduli sector in the next sect ions. For a more detailed
+introduction we refer to the original paper: Ref. [1].
+The model is defined by its superpotential Winf
+Winf=κS/parenleftbig
+H2−M2/parenrightbig
++λ
+ΛN2H2, (2.1)
+and K¨ ahler potential Kinf
+Kinf=|H|2+|S|2+1
+2(N+N∗)2+κH
+Λ2|H|4+κS
+Λ2|S|4+κN
+4Λ2(N+N∗)4
++κSH
+Λ2|S|2|H|2+κSN
+2Λ2|S|2(N+N∗)2+κHN
+2Λ2|H|2(N+N∗)2+... . (2.2)
+where the ellipses denote higher order terms, and Λ is some cu toff scale. The superfield H
+plays the role of waterfall field responsible for ending infla tion. The superfield Sis the so-
+called driving field, as its F-term provides the energy density that drives inflation. Fin ally,
+the imaginary part of Nis the slowly rolling inflaton field. It is hoped that Ncan be
+identified with the right-handed sneutrino superfield, and Hwith the grand unified Higgs
+field that breaks B-L, thus providing an embedding of the model in a grand unified th eory
+[25]. The K¨ ahler potential is invariant under a shift of N→N+iµ; this is the before
+mentioned shift symmetry pivotal for keeping the inflaton di rection flat. The superfields can
+be decomposed in real and imaginary components: H= (hr+ihi)/√
+2,S= (sr+isi)/√
+2
+andN= (nr+ini)/√
+2.
+Inflation In this model inflation takes place as the field ni, the imaginary part of N, slowly
+rollsdowntoacritical value nc
+i, whiletheotherfieldsareintheirminimum( hr,hi,sr,si,nr) =
+(0,0,0,0,0). Let us first check the stability of this minimum and then br iefly explain how
+inflation comes about.
+– 3 –During inflation the F-term scalar potential1
+VF= eK/bracketleftBig
+DiWKi¯jD¯j¯W−3|W|2/bracketrightBig
+, (2.3)
+withDiW=Wi+KiWis constant:
+Vtree=κ2M4, (2.4)
+and drives inflation with H2
+inf=Vtree/3. All non-inflationary fields are at an extremum of
+the potential independent of the value of ni. Hence the inflationary valley is a classically
+and quantum mechanically stable trajectory provided all ma sses squared are positive, and
+the mass exceeds the Hubble scale during inflation. The masse s during inflation are
+{m2
+hr,m2
+hi}=/braceleftbig
+M2κ2/parenleftbig
+M2(1−κSH)−2/parenrightbig
++λ2n4
+i, M2κ2/parenleftbig
+M2(1−κSH)+2/parenrightbig
++λ2n4
+i/bracerightbig
+,
+{m2
+sr,m2
+si}={−4M4κ2κS,−4M4κ2κS},
+{m2
+nr,m2
+ni}={2M4κ2/parenleftbig
+1−κSN/parenrightbig
+,0}. (2.5)
+Here, and from now on, we set Λ = 1. We see that hrbecomes tachyonic when nidrops
+below the critical value ( n2
+i)c≈√
+2κM/λ. This will mark the end of inflation. The sr,si
+andnrdirections are stable as long as κSN<5
+6andκS<−1
+12. This is one of the reasons
+for including the higher order terms in the K¨ ahler potentia l (2.2). (The other is that the
+inflationary observables depend on κSH[1]; taking κSH=O(10) the spectral index can be
+brought closer to the WMAP central value.)
+Thanks to the shift symmetry, niitself does not acquire any mass, independent of the
+higher order terms in the K¨ ahler potential; at tree level it is a flat direction in field space. The
+slow roll parameter η=V′′/Vis small, with prime denoting derivative w.r.t. the canonic ally
+normalized inflation field, and there is no η-problem. This is in contrast with “standard”
+SUSY hybrid inflation [13] where higher order terms lift the fl atness of the potential, and thus
+must be tuned [3]. The reason for this marked difference is that in our model Winfvanishes
+during inflation (as well as many first and second derivatives ofWinf), thereby killing all
+possible inflaton mass terms. In standard SUSY hybrid inflati on on the other hand Winf∝ne}ationslash= 0,
+and theη-problem resurfaces despite the shift symmetry. It is this r emarkable property of the
+inflaton superpotential that led the authors of [1] to sugges t that the model can be combined
+with a modulus sector. In the next sections we will take a clos er look at this claim.
+The inflaton potential is generated by the 1-loop Coleman-We inberg potential [26], from
+the mass splitting between fermions and bosons. Only the wat erfall fields have inflaton-
+dependent mass terms and contribute to the inflaton potentia l. Writing the mass of the
+waterfall fields and their fermionic superpartners in the fo rmm2
+hr,i=µ2(x2+y2±1) and
+˜m2
+hr,i=µ2x2with
+µ2= 2κ2M2, x=λ2n4
+i
+2κ2M2, y=M2
+2(1−κSH), (2.6)
+1We set the reduced Plank mass mp= (8πGN)−1/2= 1.
+– 4 –then the loop potential is given by (A.4) in appendix A. We now have the effective potential
+Vinf=Vtree+Vloop(ni). (2.7)
+Theni-direction in field space, flat at tree-level, gets slightly l ifted at the one-loop level. With
+a suitable choice of parameters this effective potential can g enerate inflation. The inflaton
+fieldnislowly rolls down until it reaches the critical value nc
+iwhere inflation ends. We note
+that fory2<0, or equivalently κSH>1, the CW-potential has a maximum at nmax
+igiven in
+(A.6). This introduces a constraint on the initial field valu e of the inflaton field which has to
+be smaller than nmax
+i, to make sure that the inflaton rolls towards the “right” mini mum. On
+the other hand for y2<0, the loop potential steadily increases with ni, and there is no such
+problem.
+With the potential (2.7) one can calculate the inflaton value (ni)∗at horizon-exit, 60 e-
+folds before the end of inflation, when observable scales lea ve the horizon. Here the slow-roll
+parameters ǫ,ηandξ2can be evaluated, and consequently the power spectrum PR, the scalar
+spectral index nS, the tensor-to-scalar ratio r, and the running of the scalar spectral index
+dnS/dlogk.
+After inflation When the inflation field nireaches its critical value nc
+i, the waterfall field
+hrbecomes tachyonic. Inflation ends with a phase transition du ring which the waterfall field
+obtains a non-zero vev. The post-inflationary vacuum field va lues are{hr,hi,sr,si,nr,ni}=
+{±√
+2M,0,0,0,0,0}, andV= 0 corresponding to zero cosmological constant.
+Numerical results In Ref. [1] it is shown that in the parameter space
+/parenleftbigg
+κ=O(10−1),M=O(10−3),λ=O(10−1),κSH=O(1−10)/parenrightbigg
+(2.8)
+many solutions can be found that satisfy the WMAP 1 σrange for the power spectrum P1
+2
+R=
+(5.0±0.1)×10−5and scalar spectral index ns= 0.960+0.014
+−0.013[27]. The tensor to scalar ratio
+rtypically becomes of order (10−5) which easily satisfies the WMAP bound r <0.2. The
+model fails on the prediction of dns/dlogk: it typically predicts a value of order (10−4) while
+WMAP measured −0.0032+0.021
+−0.020. As the accuracy of this measurement is rather low, this
+does not seem to be a serious problem.
+3. Adding the modulus sector
+The inflaton model described in the previous section is an effec tive theory, arising as a low-
+energy effective description of an underlying Planck scale th eory. If the UV completion is an
+extra dimensional theory, we expect moduli fields to appear i n the 4D effective action. The
+moduli fields parametrize the sizes and shapes of the extra di mensions. In case the vacuum
+manifold is degenerate, the moduli correspond to massless m odes appearing in the low energy
+effective four-dimensional theory.
+– 5 –For definiteness we concentrate in this paper on a KKLT type mo duli sector [8], arising
+from compactifications in type IIB string theory. KKLT showe d that all complex structure
+moduli (shape moduli) can be stabilized by fluxes. In the simp lest case there is only one
+K¨ ahler modulus (size modulus) left, the volume modulus, wh ich appears in the 4D effective
+theory. This modulus, in turn, is stabilized by invoking non -perturbative effects, coming from
+either gaugino condensation or instantons. Finally, to arr ive at a zero cosmological constant,
+a non-supersymmetric uplifting term is added, generated by an anti-D3 brane located at the
+bottom of a throat in the compactification manifold.
+To combine the inflaton and modulus sector we simply add their respective K¨ ahler- and
+superpotentials2:
+W=Winf+Wmod, K=Kinf+Kmod (3.1)
+with
+Kmod=−3ln(T+¯T). (3.2)
+For themoment we ignore corrections to this tree level K¨ ahl er potential. Thesewill betreated
+in section 4.1.
+3.1 General approach
+The function Wmod(T) generically contains a constant term W0arising from integrating out
+the stabilized moduli and a non-perturbative potential tha t is to stabilize T. In this section
+we will work with a generic function Wmod(T) and see what restrictions on this function we
+get to make inflation work in this moduli-extended framework . We choose a KKLT uplifting
+potential Vup=c/(T+¯T)2, withca constant tuned to solve the cosmological constant
+problem. However, its specific form is not so important for ou r discussion.
+As before the modulus fields can be decomposed in real and imag inary parts: T=σ+iα.
+Choosing the phases in the superpotential judiciously, we c an setα= 0 to zero consistently.
+We define σ=σ0at the minimum of the F-term modulus potential in the absence of the
+inflaton sector, i.e.
+∂σVF
+mod/vextendsingle/vextendsingle
+σ=σ0= 0⇔DTW/vextendsingle/vextendsingle
+σ=σ0= 0. (3.3)
+Due to the uplift term and the presence of the inflaton sector, σis displaced from its F-term
+minimum both during and after inflation. If the displacement is minimal the inflationary
+trajectory and the post-inflation minimum are only slightly affected as well, and the moduli
+sector may be combined with inflation. In this case σ≈σ0andDTW≈0 are still good
+approximations. In the rest of this section we discuss the ge neral conditions the moduli
+sector has to satisfy for this to be the case, followed — in the next section — by an explicit
+example. There are many pitfalls. When a modulus sector is in cluded, the η-problem may
+reappear, the vacuum after inflation and/or the inflationary trajectory may be destabilized,
+and the corrections to the waterfall fields may hamper a succe ssful exit to inflation.
+2The K¨ ahler potential does not have to be separable in modulu s and inflaton field, e.g. Kinfcan appear
+inside the log. We checked that its exact form does not affect o ur qualitative results.
+– 6 –η-problem We have seen that in the absence of a moduli sector the tree lev el inflaton mass
+is zero, as a consequence of the shift symmetry and the fact th atWinf= 0. Due to the shift
+symmetry the K¨ ahler potential is independent of ni, and thus any mass for nimust come
+from the second derivative of the term in square brackets in ( 2.3). The fact that the modulus
+superpotential is non-zero does not change the results, the inflaton potential is still flat at tree
+level. All terms in m2
+niproportional to Wmodor its derivatives are multiplied by ( Winf)nini,
+which is zero during inflation. As the η-problem is usually the main obstacle to embedding
+inflation in a supergravity theory, this is no small feat.
+Stability of the vacuum Consider the vacuum after inflation. We suppose the post-
+inflationary minimum to occur at {hr,hi,sr,si,nr,ni,σ,α}={±√
+2M,0,0,0,0,0,σ0,0}. For
+the post-inflation scalar potential we find
+Vvac=c
+(4σ)2+VF
+mod+f(M2), VF
+mod=−3W2
+mod+4
+3σ2(DTWmod)2
+(2σ)3,(3.4)
+whereeach term in f(M2) is either proportional to DTWor toW(withDTW=−3W/(2σ)+
+WT). For parameters that keep the modulus stabilized during in flation (discussed below), the
+M-dependent corrections f(M2) to the modulus potential after inflation are small, and do
+not destabilize the potential minimum.
+The first derivatives of the scalar potential with respect to the eight real fields, evaluated
+at the postulated potential minimum after inflation, are man ifestly zero or involve again small
+functions of M2proportional to DTWorWindicating that the minimum of some of the fields
+is slightly displaced. One of the displaced fields is the modu lus field, which is shifted from its
+F-term potential minimum σ0due to the presence of the uplift term. This shift is typicall y
+small.
+Second derivatives again involve many functions of DTW,Wandc. The vacuum mass
+of the field nris most seriously affected by moduli corrections, and runs the risk of going
+tachyonic:
+m2
+nR/vextendsingle/vextendsingle
+vac=4(DTWmod)2σ2−3W2
+mod+O(M2)
+12σ3. (3.5)
+Indeed, for DTWmod≈0 the mass is tachyonic unless Wmod/lessorsimilarMis sufficiently small, and
+theO(M2) terms dominate.
+Stability during inflation The tentative inflationary trajectory is
+{hr,hi,sr,si,nr,ni,σ,α}={0,0,0,0,0,ni,σ0,0}. (3.6)
+We have to check whether this is still an extremum when the mod ulus potential is turned
+on. As before niis the slowly rolling inflaton field. The potential during infl ation along this
+trajectory is then
+Vinf=c
+(4σ)2+VF
+mod+κ2M4
+(2σ)3, (3.7)
+– 7 –withVF
+moddefined in (3.4). If σ≈σ0the first two terms in the above expression nearly
+cancel, and the last term is as before the energy density driv ing inflation. However, this
+energy density is now modulus dependent. If this term is too l arge, the displacement in σis
+large, or worse, the barrier separating in the potential dis appears and σrolls off to infinity.
+The fields are all at an extremum for the inflationary trajecto ry, except for the modulus
+and thesrfield. The non-vanishing first derivatives are
+∂srVinf=κM2((DTWmod)σ+Wmod)
+2√
+2σ3, (3.8)
+∂σVinf=−3(3κ2M4+4cσ)+8σ2(DTWmod)(−2DTWmod−3Wmod
+σ+σW′′
+mod)
+24σ4,
+where primes denote derivatives with respect to σ. We see indeed that during inflation the
+minimum of the σ-field does not occur at exactly σ=σ0: now it is both the uplift and the
+inflationary energy density that shifts the minimum away. In addition, the field sris not
+minimized at sr= 0. For DTWmod≈0, the first derivative is proportional to Wmodand is
+typically large. This can have dramatic consequences, as we will see. The matrix of second
+derivatives evaluated at the inflationary minimum is not dia gonal anymore, as Vsrσdoes not
+vanish. This coupling between srandσcould already be foreseen from (3.8). We also find a
+similar coupling between siandα, but as they both have their minimum at zero this coupling
+will not have any significant consequences.
+Just as in the case without moduli fields ( cf. the discussion below (2.5)), we need some
+tuning of the κ-parameters in the inflationary K¨ ahler potential to mainta in positive definite
+masses squared. Since expressions are long, we only explici tly give the mass of the field hr,
+that can be compared to (2.5)
+Vinf
+hrhr=1
+(2σ0)3/bracketleftbigg
+κ2M2(/parenleftbig
+M2(1−κSH)−2/parenrightbig
++λ2n4
+i−2W2
+mod+2σλn2
+iDTWmod
++Wmod/parenleftbig
+λn2
+i−4σDTWmod/parenrightbig
++4
+3σDTWmod/parenleftbig
+σDTWmod+3/parenrightbig/bracketrightbigg
+.(3.9)
+Once again, for DTWmod≈0, the corrections — in this case to the waterfall masses — sca le
+withWmodand are potentially large.
+Waterfall mechanism and CW-loop The above expression becomes much more com-
+plicated when we take the displacement of srinto account, see (3.8). If we allow srto be
+non-zero we find among many other terms
+δm2
+hr,i=κ2s2
+r
+2σ3+... (3.10)
+Thisindicates that theshiftin srcan doalot of harmtoourmodel. Oncethewaterfall masses
+get dominated by terms like (3.10), the Coleman-Weinberg lo op potential changes drastically
+and inflation is no longer possible. Therefore, we absolutel y need the displacement in srto
+be small.
+– 8 –3.2 Discussion
+As discussed, for a generic superpotential ( DTWmod)≈0 as this minimizes the F-term su-
+perpotential (3.3), and corrections to the inflaton potenti al scale with Wmod(which is the
+only scale in the moduli sector). Wmodshould be large enough to assure the modulus remains
+stabilized during inflation, yet small enough to ensure that the vacuum and inflationary tra-
+jectory is not destabilized. This does not seem to be easy. An d indeed, for a KKLT modulus
+sector, which is of the above described generic form, this is impossible. In the original KKLT
+paper [8] the non-perturbative potential is a single expone nt, and the superpotential is
+Wmod=−W0+Ae−aT. (3.11)
+where the sign in front of W0is chosen such that the potential is minimized by α= 0. The
+minimum of Vvacoccurs for DTWmod≈0 andWmod∼W0. Let us go through all modulus
+corrections for this specific choice of superpotential.
+For the nr-direction to be stable in the vacuum after inflation, see (3. 5), we have to
+demand W0/lessorsimilarκM2. From the perspective of the modulus field, the inflationary e nergy
+densityκ2M4acts as an additional upliftterm (3.7). Ifthis term is toola rge,σis destabilized.
+To avoid this one needs κ2M4/(2σ)3/lessorsimilarVupor
+κM2/lessorsimilarW0 (3.12)
+It follows that stabilizing the modulus during inflation plu s stabilizing the vacuum are both
+possible only for a very limited range of parameters: W0≈κM2. But what kills the KKLT
+model are the corrections it gives to the waterfall fields. As anticipated from (3.8) it follows
+that both srandσare displaced considerably during inflation. Numerically w e findsr∼
+O(10−1−10−2) (wherewe used V1/4
+inf∼MGUT). Thecorrespondingcorrection to the waterfall
+field (3.10) is enormous, hampering a graceful exit to inflati on.
+How to salvage inflation? Taking a look at the modulus correct ions (3.5, 3.7, 3.8, 3.10),
+we see they all vanish in the limit that both
+DTWmod/vextendsingle/vextendsingle
+σ=σ0≈0 & Wmod/vextendsingle/vextendsingle
+σ=σ0≈0. (3.13)
+The first condition is assured by minimizing of the F-term pot ential (3.3), but the second
+constitutes an extra constraint on the modulus potential wh ich can be satisfied by tuning the
+parameters in the superpotential. Such a tuning is not possi ble for the one-exponent KKLT
+model. Kallosh & Linde (KL) constructed a modulus sector wit h two exponents, with the
+parameters carefully tuned, such that (3.13) is satisfied [1 7, 18]. We will discuss this model in
+detail in the next section. The only constraint left is then ( 3.12), assuring that the modulus
+remains fixed during inflation.
+The fine-tuning required to set Wmod≈0 is the same tuning that creates a hierarchy
+between the gravitino and modulus mass with m3/2≪mT. SinceH∗/greaterorsimilarmT(from (3.12)),
+this tuning allows to have low scale SUSY breaking with high s cale inflation — something
+– 9 –that seems impossible in non-fine tuned models. This was the m otivation behind the KL
+model. Note that since in hybrid inflation Vinf∼M4
+GUT, without this tuning, it is impossible
+to get the phenomenologically favored TeV scale SUSY breaki ng.
+Finally we would like to contrast the results with standard S USY hybrid inflation [13,
+14, 15]. In the standard case, the η-problem reappears once a modulus sector is included;
+the reason is that in these models the inflaton superpotentia l is non-zero Winf∝ne}ationslash= 0, and
+many terms mixing the modulus and inflaton sector appear in VF. In addition, the waterfall
+masses get large corrections, just as we found above. Althou gh each of these problems can be
+solved separately by a fine-tuned condition on the modulus po tential, they cannot be solved
+simultaneously. Since in our case, the η-problem has dropped off the list, inflation can be
+rescued by a single tuning.
+4. Inflation with a KL modulus sector
+As discussed in the previous section, hybrid inflation may be combined with a modulus sector
+provided the latter satisfies (3.13). In this section we work out the details, focusing on the KL
+modulus sector introduced by Kallosh & Linde in [17, 18]. Aug menting the KKLT potential
+by anadditional non-perturbativeexponential factor, it i spossible(by tuningtheparameters)
+to construct a SUSY Minkowski minimum with DTW=WT= 0. The superpotential is
+Wmod=−W0+Ae−aT−Be−bT(4.1)
+withW0andσ0:
+W0=w0≡A/parenleftbiggbB
+aA/parenrightbigga/(a−b)
+−B/parenleftbiggbB
+aA/parenrightbiggb/(a−b)
+, σ 0= ¯σ0≡1
+a−bln/parenleftbiggaA
+bB/parenrightbigg
+.(4.2)
+So, at the cost of fixing W0and introducing another exponent in the non-perturbative p oten-
+tial, we now explicitly have DTW=WT= 0, and thus VF
+mod= 0, in thevacuum after inflation
+{hr,hi,sr,si,nr,ni,σ,α}={±√
+2M,0,0,0,0,0,σ0,0}. No uplift term is needed, and SUSY
+is unbroken.
+We can get a small but non-zero gravitino mass by perturbing t he SUSY Minkowski
+solution
+W0=w0+ǫw. (4.3)
+As long as the perturbation is small enough DTWmod≈0,Wmod≈ǫwand (3.13) is still
+satisfied. We will determine below how small ǫwhas to be. With this perturbation the
+minimum of the F-term potential, located at σ0= ¯σ0+O(ǫw), is SUSY AdS, and a small
+upliftVup≈3W2
+0/(2σ0)3is needed to get zero cosmological constant. SUSY is broken i n the
+process. In this set-up there is a large hierarchy between th e gravitino m3/2= eK|W| ∝ǫw
+and modulus mass mσ∝√Vσσ∝W0.
+4.1 Inflation
+Let us see how inflation works for the hybrid inflation model de scribed in section 2 combined
+with a KL modulus sector.
+– 10 –Stability of the vacuum Since the function f(M2)∼ǫ2in (3.4) the inflaton corrections
+to the modulus minimum after inflation are small. Likewise th e modulus correction to the
+inflaton sector are small. The mass of nrin (3.5) is manifestly positive definite in the vacuum.
+We checked numerically the stability of the vacuum.
+Stability during inflation From (3.7) we see that during inflation we now have
+Vinf≈κ2M4+O(ǫ2
+w)
+(2σ0)3. (4.4)
+The inflationary trajectory is slightly shifted from the ten tative inflationary trajectory (3.6),
+as the first derivatives Viwithi={sr,σ}are non-zero (3.8). Expanding in small ǫwthis shift
+is
+δσ=−9κ2M4/parenleftbig
+3−4κS/parenrightbig
+4a2b2κSW2
+0+ǫw3(1−2κS)
+4abκsW0σ0,
+δsr=−9κM2
+8√
+2abW0κSσ2
+0−ǫw1√
+2κκSM2. (4.5)
+The shift due to the inflation sector, which is the ǫwindependent part, is small, and harmless
+for inflation. The corrections due to the modulus sector scal e withǫwand can be larger
+depending on the size of ǫw. The mass matrix is nearly diagonal. Except for the sr-field, the
+masses for all the inflaton fields are as before (2.5), up to an o verall scaling by (2 σ0)3, and
+up to order δm2
+i=O(ǫ2
+w/(2σ0)3) corrections. From the masses of srandnrone can deduce
+constraints on κSNandκS, just as we did before around 2.5:
+m2
+nr=2κ2M4(1−κSN)
+(2σ0)3, m2
+sr=κ2M4(3−4κS)
+(2σ0)3(4.6)
+lead to the constraints κSN<5
+6andκS<2
+3. It follows that for ǫw/greaterorsimilarκM2the moduli
+corrections dominate, and one of the masses, depending on th e choice of κiparameters in the
+K¨ ahler potential (2.2), may go tachyonic, thereby destroy ing inflation.
+Waterfall mechanism and CW-loop A stronger bound on the value of ǫwmay be ob-
+tained by looking at the waterfall masses. Writing the masse s of the bosonic waterfall fields
+and their superpartners in the form m2=µ2(x2+y2±1) and ˜m2=µ2x2, we find
+µ2=2κ2M2
+(2σ0)3+O(ǫw), x2=λ2n4
+i
+2κ2M2+O(ǫw), y2=M2
+2(1−κSH)−ǫ2
+wκSHλ2n4
+i
+4κ2κ2
+SM2(4.7)
+where the dominant moduli correction for our purposes is the O(ǫ2
+w) correction in y2. Al-
+thoughy2≪x2, and is an unimportant contribution to the waterfall field ma ss, it is relevant
+for the 1-loop potential. As explained in the appendix A, the reason is that the dominant
+terms cancel between the bosons and the fermions in the Colem an-Weinberg potential (A.1).
+Indeed, even in the absence of moduli corrections the term ∝M2(1−κSH) in the boson mass
+– 11 –causes the loop potential to develop a maximum for κSH>1 (A.6). If the ǫ2
+w-correction in
+y2dominates, the loop potential steepens for large ni. In the case of κSH>0, this results in
+the maximum shifting to smaller values of ni, until at some point it becomes impossible to
+get 60 e-folds of inflation. In the opposite limit κSH<0 it results in a larger spectral index,
+in contradiction with observations. Either way, inflation i s ruined if the moduli corrections
+get too large. Using (4.7) this gives the bound
+ǫw/lessorsimilar0.1−0.01κM2(4.8)
+where the exact value depends on the κivalues, and the precise parameters. This estimate is
+confirmed by our numerical calculation.
+Corrections to modulus K¨ ahler potential What remains to address is an analysis of
+possible corrections to the tree-level K¨ ahler moduli pote ntial introduced in (3.2). Denoting
+α′[28] and gs[29, 30] corrections by θ1andθ2respectively, we replace (3.2) by
+Kmod=−2ln/bracketleftBig/parenleftbig
+T+¯T/parenrightbig3/2+θ1/bracketrightBig
++θ2/parenleftbig
+T+¯T/parenrightbig3/2. (4.9)
+In the natural limit θ1</parenleftbig
+T+¯T/parenrightbig3/2we can expand the logarithm to arrive at
+Kmod=−3ln/parenleftbig
+T+¯T/parenrightbig
++−2θ1+θ2/parenleftbig
+T+¯T/parenrightbig3/2≡ −3ln/parenleftbig
+T+¯T/parenrightbig
++θ
+/parenleftbig
+T+¯T/parenrightbig3/2,(4.10)
+where we have defined θ≡ −2θ1+θ2.
+To check analytically whether these corrections might spoi l inflation, we calculate the induced
+displacement in the minimum of the srfield. Before we have seen that large srcontributions
+ruin the waterfall mechanism and CW loop effect, so this seems t he right quantity to consider.
+In leading order in M2we find that these contributions are naturally small:
+δsr=δsr|θ=0/parenleftBigg
+1+θ
+/parenleftbig
+T+¯T/parenrightbig3/2/parenrightBigg
+. (4.11)
+This already suggests that K¨ ahler corrections leave the mo del unaffected. Indeed, this is
+confirmed by our numerical analysis. As long as θ </parenleftbig
+T+¯T/parenrightbig3/2all results of the previous
+sections apply: there is no danger for the K¨ ahler correctio ns to spoil the model.
+4.2 Numerical analysis
+Adding a modulus sector to inflation, the F-term potential an d thus all masses squared
+are rescaled by a factor eK= (2σ)−3. We can absorb this factor in the parameters of the
+superpotential via
+¯κ=κ
+(2σ0)3/2,¯λ=λ
+(2σ0)3/2,¯A=A
+(2σ0)3/2,¯B=B
+(2σ0)3/2,¯W0=W0
+(2σ0)3/2.(4.12)
+– 12 –0.00.10.20.30.40.50.6/Minus0.50.00.51.01.52.02.5
+n/ImagiV/Minusloop
+0.00.10.20.30.40.50.601234567
+n/ImagiV/Minusloop
+Figure 1: The Coleman-Weinberg potential, rescaled by a factor 1015, as a function of niforǫw=
+{0,10−7,10−5}corresponding to the solid, short dashed and dashed lines respect ively. On the left
+are the results for κSH= 1, on the right for κSH=−1.
+It is the barred quantities that give the effective couplings b etween the fields, and that can be
+measured (in principle) in experiments. The rescaling allo ws to easily compare the parameter
+space for hybrid inflation without moduli as described in Sec tion 2 and discussed in detail
+in Ref. [1], with the set-up where a modulus potential is incl uded. If in the former case the
+model gives the right predictions for the density perturbat ions for a given set of parameters,
+for example {κ= 0.14,M= 0.003,...}, the same observational results are obtained in the
+setup up with a modulus field if we choose the same numerical va lues for the barred quantities
+{¯κ= 0.14,M= 0.003,...}. This correspondence works up to O(ǫw) corrections. We checked
+numerically that with the above identification we get the sam e parameter space for successful
+inflation, e.g. including the same κSNdependence, as found in Ref. [1].
+Consider an explicit numerical example. For the inflaton sec tor we choose parameter
+values
+¯κ= 0.14, M= 0.003,¯λ= 0.1, κSH= 1, (4.13)
+and all other κiequal to −1. As discussed in section 2 this assures stability of the infl ationary
+trajectory. For the modulus sector we take
+¯A= 1,¯B= 1.03, a=2π
+100, a=2π
+99. (4.14)
+which gives W0= 0.276+ǫwandσ0= 62.41. The exact parameter values in (4.14) are not
+so important, what matters is the resultant value for W0and to some lesser extent σ0.
+As anticipated, as we increase ǫwwe see that the moduli corrections first appear in
+the loop potential. For ǫw={0,10−7,10−5}we getδσ={2.06,2.39,5.33} ×10−5and
+δsr={3.26,3.66,7.28} ×10−4. These values match our estimates (4.5). Although the
+increase in δsrseems quite moderate, the effects are nevertheless visible in the loop potential,
+where the maximum is shifting to increasingly small ni-values. This is plotted in Fig.1. For
+ǫ= 10−5the loop potential gets modified in such a way that the slow-ro ll trajectory is not
+large enough to accommodate 60 e-folds of inflation. Fig. 1 al so shows the equivalent results
+– 13 –/Minus10/Minus9/Minus8/Minus7/Minus60.750.800.850.900.951.00
+log/LParen1Εw/RParen1nS
+/Minus10/Minus9/Minus8/Minus7/Minus60.000000.000050.000100.000150.00020
+log/LParen1Εw/RParen1PR
+Figure 2: The spectral index ns(left) and power spectrum PR(right) as a function of ǫwfor the
+parameters mentioned in the text.
+forκSH=−1 and for the rest the same parameters; now the potential stee pens to fast for
+ǫw>10−5pushing the spectral index to values larger than one.
+Fig. 2 shows the spectral index and power spectrum as a functi on ofǫw. For small ǫw
+the results are identical to those found in the model without a modulus. As ǫwapproaches
+its critical value (4.8) the results for the spectral index a nd power spectrum change rapidly,
+and inflation breaks down abruptly.
+5. Conclusions
+In this paper we combined the hybrid inflation model of [1] wit h a KKLT like modulus sector.
+The inflaton mass is protected by a shift symmetry, and remain s massless (at tree level) even
+in the presence of the modulus sector. This is in sharp contra st with standard SUGRA hybrid
+inflation.
+The vacuum after inflation and the inflationary trajectory ar e corrected by the modulus
+sector. These corrections are under control and do not disru pt inflation provided the modulus
+sector satisfies the constraint DTWmod≈Wmod≈0. The first condition is automatic in the
+minimumofthepotential, thesecondconditionscanbesatis fiedbyfine-tuningtheparameters
+in the potential. This is the same fine-tuning needed to get a h ierarchy between the gravitino
+and modulus mass, and which allows for low scale SUSY breakin g yet high scale inflation. As
+explicit examples, the original KKLT modulus stabilizatio n scheme [8] does not satisfy the
+above condition, whereas the fine-tuned model of Kallosh & Li nde [17, 18] does.
+Why inflation works for a modulus sector with small scale SUSY breaking can be easily
+understood by considering the relevant scales in the system .
+1. The modulus mass mT∝W0which sets the height of the barrier in the modulus
+potential. It has to be larger than the inflationary scale for the modulus to be stabilized
+during inflation; this implies the condition (3.12).
+– 14 –2. The energy density during inflation Vinf= ¯κ2M4, which determines the size of the
+density perturbations. To get the observed amplitude we find V1/4
+inf∼MGUTis of the
+order of the grand unified scale.
+3. The vacuum gravitino mass m3/2= eK/2|W| ∝ǫW, which sets the scale of the mod-
+uli corrections to the inflationary potential. It cannot be t oo large, the bound (4.8)
+translates in a bound on the gravitino mass m3/2/lessorsimilar109−1010GeV/σ3/2
+0.
+In summary, we find that it is possible to extend the, in itself already very promising,
+model of supersymmetric hybrid inflation proposed in Ref. [1 ] with a moduli sector. It is
+absolutely necessary to have a modulussector that does not b reak SUSY too badly. Therefore
+we need to tune the parameters in the superpotential. As a bon us, however, we find that our
+extended model can accommodate TeV-scale SUSY breaking.
+Acknowledgments
+The authors are supported by a VIDI grant from the Dutch Scien ce Organization (NWO).
+A. Coleman-Weinberg potential
+The Coleman-Weinberg potential is [26]
+VCW=1
+64π2/summationdisplay
+i(−1)Fm4
+iln/parenleftbiggm2
+i
+Λ2/parenrightbigg
+(A.1)
+where the sum is over all masses, with F= 1 for bosons and F=−1 for fermions, and Λ
+is the cutoff scale. Only the ni-dependent mass terms lift the inflaton potential, in our cas e
+these are the waterfall field masses and their fermionic part ners. They can be written in the
+form
+m2
+hr,i=µ(x2+y2±1),˜m2
+hr,i=µ2x2(A.2)
+withµ,x,ygiven for inflation without and with a modulus field respectiv ely by (2.6,4.7). The
+waterfall field hrbecomes tachyonic and inflation ends for xc= 1.
+Even though y2≪x2is clearly subdominant in the expression for the mass terms ( A.2),
+they are important for the shape of the loop potential. This i s because the dominant contri-
+butions of the boson mass cancels with that of the fermion mas s in (A.1). The loop potential
+becomes
+VCW=/parenleftbiggµ4
+32π2/parenrightbigg /bracketleftbigg
+2(1+x2y2+y4)ln/parenleftbiggx2µ2
+Q2/parenrightbigg
+(A.3)
++ (x2+y2+1)2ln/parenleftbigg
+1+y2+1
+x2/parenrightbigg
++(x2+y2−1)2ln/parenleftbigg
+1+y2−1
+x2/parenrightbigg/bracketrightbigg
+,
+withQthe renormalization scale which we fix to Q=µ= ˜mhr,i|x=1. For negative values
+y2<0 the potential develops a maximum at large x. Inflation has to take place on the left
+– 15 –of the maximum, for the inflaton field to roll towards the “righ t” minimum. This also means
+that if the maximum is to close to the critical value, it is imp ossible to get 60 e-folds of
+inflation. To see the maximum appearing, we can take the large xlimit of the potential
+lim
+x→∞/parenleftbigg32π2
+µ4/parenrightbigg
+VCW= 3+4ln( x)+2y2x2/parenleftbig
+1+4ln(x)/parenrightbig
+. (A.4)
+The slope of the potential at large xgets a positive contribution from the y0-term, and a
+positive or negative contribution from the y2correction; if the latter is negative, the potential
+has a maximum. The slope is
+lim
+x→∞∂xVCW=4
+x+4xy2(3+4log( x)), (A.5)
+which vanishes for
+x2
+max=−(y2(3+4ln( xmax))−1(A.6)
+which is only a solution for y2<0. Numerically we find for κSH=O(1) thatxmax= 50−100
+in the absence of moduli corrections (i.e. using (2.6)).
+References
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diff --git a/1001.0665.txt b/1001.0665.txt
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@@ -0,0 +1,669 @@
+arXiv:1001.0665v1  [cond-mat.supr-con]  5 Jan 2010NaAlSi: a self-doped semimetallic superconductor
+with free electrons and covalent holes
+H. B. Rhee, S. Banerjee, E. R. Ylvisaker, and W. E. Pickett
+Department of Physics, University of California, Davis, CA 95616
+(Dated: November 12, 2018)
+The layered ternary spconductor NaAlSi, possessing the iron-pnictide “111” crys tal structure,
+superconducts at 7 K. Using density functional methods, we s how that this compound is an intrinsic
+(self-doped)low-carrier-density semimetal with anumber of unusualfeatures. CovalentAl-Sivalence
+bands provide the holes, and free-electron-like Al 3 sbands, which propagate in the channel between
+the neighboring Si layers, dip just below the Fermi level to c reate the electron carriers. The Fermi
+level(andtherefore thesuperconductingcarriers) lies in anarrowand sharppeakwithin apseudogap
+in the density of states. The small peak arises from valence b ands which are nearly of pure Si,
+quasi-two-dimensional, flat, and coupled to Al conduction b ands. Isostructural NaAlGe, which is
+not superconducting above 1.6 K, has almost exactly the same band structure except for one missing
+piece of small Fermi surface. Certain deformation potentia ls induced by Si and Na displacements
+along the c-axis are calculated and discussed. It seems likely that the mechanism of pairing is related
+to that of several other lightly doped two-dimensional nonm agnetic semiconductors (TiNCl, ZrNCl,
+HfNCl), which is not well understood but apparently not of ph onon origin.
+PACS numbers:
+I. INTRODUCTION
+The discovery of new superconductors in unex-
+pected materials brings the potential to understand
+something deeper, or perhaps something different,
+abouttheunderlyingpropertiesthatfavorsupercon-
+ducting pairing. The discovery of high-temperature
+superconductivity (to 56 K so far) in Fe-pnictides1
+is a recent spectacular example, and is also an ex-
+ample of close relationships between magnetism and
+superconductivity, though the connections are still
+far from clear.
+New superconductors where little or no mag-
+netic effects are present are also arising, and these
+clearly involve different physics from the cuprate
+or Fe-pnictide high-temperature superconductors.
+Electron-dopedMNCl, whereM=ZrorHf, becomes
+superconducting immediately upon undergoing the
+insulator-to-metal transition,2–5which, in the case
+of M = Hf, is higher than 25 K. The similarly lay-
+ered, electron-doped, ionic insulator TiNCl super-
+conducts up to 16 K. Magnetic behavior in these
+materials is at most subtle, amounting to an en-
+hancement in Pauli susceptibility near the metal-to-
+insulator transition.6
+In this paper we address the ternary silicide
+NaAlSi (space group P4/nmm ,Z= 2), another
+ionic and layered material that shows unexpected
+superconductivity, and does so in its native (with-
+out doping or pressure) stoichiometric state, at 7
+K.7NaAlSi introduces new interest from several
+viewpoints. First, it is an spelectron supercon-
+ductor, with a high Tcfor such materials at am-
+bient pressure. Pb is an spsuperconductor withcomparable Tc(7.2 K) but with simple metallic
+bonding and heavy atoms, making it very differ-
+ent. Doped Si8and doped diamond11superconduct
+in the same range, and are of course very different
+classes of materials. A more relevant example is the
+pseudo-ternary compound Ba 1−xKxBiO3(BKBO),
+which undergoes an insulator-to-metal transition for
+x≈0.4, beyond which its Tcsurpasses 30 K.[9,10]
+Second, the Al-Si layered substructure is like that
+oftheFeAs layerinthe Fe-pnictidesuperconductors,
+raising the possibility of some connections between
+their electronic structures. In fact, NaAlSi has the
+same structure as the Fe-pnictide “111”compounds,
+with Al being tetrahedrally coordinated by Si (anal-
+ogous to Fe being tetrahedrally coordinated by As).
+In spite of their structural similarities, these com-
+pounds have major differences; for example, the Fe
+pnictides are 3 delectron systems with magnetism,
+while NaAlSi is an spelectron system without mag-
+netism.
+Third,NaAlSiistheisovalentsister(onerowdown
+in the periodic table for each atom) of LiBC. LiBC
+itself is (in a sense) isovalent and also isostructural
+toMgB 2; howeverduetotheB-Calternationaround
+the hexagonin the honeycomb-structurelayer,LiBC
+is insulating rather than conducting. When hole-
+doped while retaining the same structure, Li 1−xBC
+has stronger electron-phonon coupling than does
+MgB2.12While NaAlSi has a substantially different
+structure than LiBC, its isovalence and its combina-
+tion of covalence with some ionic character is shared
+with LiBC.
+Yet another closely related compound is CaAlSi,
+whose two different stacking polymorphs and par-ent structure superconduct in the 5–8 K range.13,14
+Linear-response and frozen-mode calculations indi-
+cate electron-phonon coupling is the likely mech-
+anism; in particular, an ultra-soft phonon mode
+appears and is suggested to play a role in the
+superconductivity.15–20It is curious that in this
+compound, where divalent Ca (comparing it with
+NaAlSi) contributes one additional electron into the
+Si-Alspbands, the preferred structure is that of
+AlB2(i.e., MgB 2) withsp2planar bonding15,16,18
+rather than the more sp3-like bonding in NaAlSi.
+Electronic structure calculations show that CaAlSi
+has one electron in the conduction band above a
+bonding-antibonding band separation at the NaAlSi
+band-filling level, a situation which would not ap-
+pear to be particularly favorable for sp2bonding.
+In this paper we analyze first-principles elec-
+tronic structure calculations that revealthat NaAlSi
+is a naturally self-doped semimetal, with doping
+occurring—thus charge transfer occurring—between
+covalent bands within the Al-Si substructure, and
+two-dimensional(2D)free-electron-likebandswithin
+the Al layer. The resulting small Fermi surfaces
+(FSs) are unusual, complicated by the small but
+seemingly important interlayer coupling along the
+crystalline c-axis.
+II. COMPUTATIONAL METHODS
+First-principles, local density approximation
+(LDA) calculations were carried out using the full-
+potential local-orbital (FPLO) scheme.21Ak-point
+mesh of20 ×20×12wasused, and the Perdew-Wang
+FIG. 1: (Color online.) Crystal structure24of NaAlSi.
+Four Si atoms tetrahedrally surround an Al atom, and
+these Al-Si networks sandwich the Na atoms. The unit
+cell is outlined in black.92 approximation22was applied for the exchange-
+correlation potential. The experimental lattice con-
+stants obtained by Kuroiwa et al.7(a= 4.119˚A
+andc= 7.632˚A) and internal coordinates pub-
+lished by Westerhaus and Schuster23(zNa= 0.622,
+zSi= 0.223) were used in our calculations.
+III. ELECTRONIC STRUCTURE
+A. Discussion of the band structure
+The calculated band structure of NaAlSi is shown
+in Fig. 2. As expected, the Na ion gives up its elec-
+tron to the Si-Al-Si trilayer (see Fig. 1), which may
+have some ionic character, though it is not easy to
+quantify(the valence bandshavemuchmoreSi char-
+acter than Al character, seemingly more than sug-
+gested by their number of valence electrons). There
+are several readily identifiable classes of bands. Two
+primarily Si 3 sbands, with a small amount of Al 3 s
+character, are centered 9 eV below the Fermi level
+εFand have a width of 2.5 eV. Above them there is
+a six-band complex of Al-Si s-pbands (heavily Si)
+that are very nearly filled, the band maximum only
+slightly overlapping εF.
+AboveεFlie non-bonding and antibonding bands,
+and the Na sbands. Among these there are a pair of
+distinctive bands, which can be identified most eas-
+ily by their Al scharacter in the top panel of Fig. 2.
+These bands are nearly free-electron-like with large
+dispersions, and cross many other bands with little
+mixing. Along Γ-M they are degenerate and easily
+identifiableinFig. 2, astheydisperseupthroughthe
+Fermi level to nearly 10 eV at the M point. Along Γ-
+X, and similarly at the top of the zone Z-R, they are
+distinct: one againdispersesupwardrapidly, cutting
+through many other bands, also to nearly 10 eV at
+X; the other disperses much more weakly to X, with
+a bandwidth of about 2 eV. Their Al scharacter
+and nearly vanishing Si character identify these as
+free-electron states, in which electrons move down
+channels of Al atoms separately in xandydirec-
+tions. (Note that their lack of kzdispersion identi-
+fies them as planar bands.) There is some coupling
+to the states in a parallel channel of Al atoms, giv-
+ing rise to the 2 eV transverse dispersion. These
+bands lie 0.5 eV below εFat Γ and contain elec-
+trons. Without interference with other bands near
+the Fermi level and supposing them to be isotropic
+in the plane (but see below), such FSs might include
+3–4% of the area of the zone, which would equate
+to an intrinsic electron doping for two bands, both
+spins of around 0.12–0.16 carriers per unit cell, and
+the concentration of hole carriers would be equal.
+The anisotropy, discussed below, makes the actual
+2carrier concentration much lower.
+The small overlap of valence and conduction
+bands results in semimetallic character and small
+Fermi surfaces. The valence bands are quite
+anisotropic. Looking at the valence bands along
+Γ-X, one might try to characterize them as one
+“heavy-hole” and one “light-hole” band, degener-
+ate at Γ, with the band maximum lying 0.13 eV
+aboveεF. However, the heavy hole band is actually
+FIG. 2: (Color online.) Band structure, with projected
+fatbands, of NaAlSi. Top panel: the Al 3 scharacter of
+bands is indicated by broadening. A doubly degenerate
+pair of broad bands is evident along the Γ-M direction.
+Middle panel: Al 3 pcharacter is weak below 2–3 eV.
+Bottom panel: Si 3 s(black) bands below −8 eV, and Si
+3pin the valence bands and lower conduction bands.almost perfectly flat for the first third of the Γ-X
+line, before dispersing downward across εFand far-
+ther below. Due to this flatness, the band cannot
+be characterized by an effective mass. The conduc-
+tionbandscontributethepairoflightelectronbands
+described above. In addition, one conduction band
+dips slightly below the Fermi level along Γ-M near
+M.
+The fatbands representation in Fig. 2 that reveals
+the dominant band character shows that the Si 3 px,
+3py, and Al 3 sorbitals dominate the valence states
+nearεF. As anticipated from consideration of the
+layeredstructure as mentioned earlier, the electronic
+structure is quasi-2D, with generally small disper-
+sion along kznearεF. However, the small kzdis-
+persion of one band is important in determining the
+geometry of the FSs, as discussed in more detail be-
+low.
+FIG. 3: (Color online.) Total and partial (atom- and
+orbital-projected) DOSs of NaAlSi. Top panel: Total
+and atom-projected DOS in a 20 eV-wide region, show-
+ing the pseudogap centered at the Fermi level (the zero
+of energy) punctuatedbythe curiously narrow and sharp
+peak at the Fermi level. Middle panel: expanded view of
+the peak, and the variation of the DOS near the Fermi
+level, separated into Si sandpcontributions. Lower
+panel: the Al sandpcharacter; the scharacter “turns
+on” just below the Fermi level.
+3B. Density of states
+Fig. 3 shows the total, partial, and projected den-
+sities of states (DOS) of NaAlSi. The Na contri-
+bution near the Fermi level is negligible and thus
+not shown. Except for a strong dip (“pseudogap”)
+near the Fermi level and a less severe dip in the
+−3 to−4 eV range, the DOS hovers around 3
+states/eV throughout both valence and conduction
+bands. Within the pseudogap encompassing the
+Fermi energy, there is an anomalous sharp and nar-
+row peak with εFlying on its upper slope, as noted
+previously by Kuroiwa et al.7The value of N(εF)
+is 1.1 states/eV. We discuss below the FSs of both
+hole (Si) and electron (Al) character.
+It seems clear that the transport properties and
+low-energy properties (which have not yet been re-
+ported), and in particular the superconductivity of
+NaAlSi, are intimately associated with this sharp
+and narrow peak in the DOS, which includes the
+Fermi level. The projected DOS shows the flat
+bandsthat giveriseto this peakareverystronglySi-
+derived. There is Al 3 scharacter that turns on just
+belowεF, but it is relatively small compared to the
+Si character at εF, and its magnitude remains low
+and nearly constant through the peak. There is Al
+3pcharacter of the same magnitude in the vicinity
+of the Fermi level. The top edge of the peak coin-
+cides with the flat band along Γ-X at 0.13 eV. The
+width of the peak, about 0.35 eV, must be due to
+dispersionandanticrossingsthat aremostlynotvisi-
+ble alongsymmetry directionsand arisefrom mixing
+away from symmetry lines of the valence and con-
+duction bands.
+Nonetheless, the slope of the DOS at εFis rather
+steep, andthismaygiverisetohighthermopowerfor
+the material. The standardlow-temperaturelimit of
+thermopower (the Seebeck coefficient tensor) S(T)
+in semiclassical Bloch-Boltzmann theory is
+S(T)→ −π2kB
+3edlnσ(E)
+dE/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+εFkBT. (1)
+The conductivity tensor σ(E) can be written in
+terms of the average velocity ( /vector v(E)) product, DOS,
+and scattering time τ(E) over the constant energy
+(E) surface:
+σ(E) = 4πe2∝angbracketleft/vector v(E)/vector v(E)∝angbracketrightN(E)τ(E).(2)
+The thermopower thus picks up contributions from
+the energy variation of three quantities: the dyadic
+product ∝angbracketleft/vector v/vector v∝angbracketright,N(E), andτ(E). Often the energy
+dependence of τis neglected, out of lack of detailed
+knowledge, though it also can be argued to follow
+roughly 1 /τ(E)∝N(E) for elastic scattering. Theenergy dependence of v2(E) also counteracts the en-
+ergy dependence of N(E). Nevertheless it is ob-
+served that materials with large slope in N(E) fre-
+quently have large thermopower. For NaAlSi we cal-
+culatedlnN(E)/dE|εF=−4.0 eV−1. This value
+can be compared with other materials that have
+fine structure near the Fermi level: TiBe 2, where
+dlnN(E)/dE|εF= 10–12 eV−1andN(εF) also is
+much larger;25and MgCNi 3with its very impres-
+sive peak very near εF, for which dlnN(E)/dE|εF
+∼-15-20/eV−1.26
+The energy derivative of diagonal elements of σ
+that occurs in Eq. 1 can also be expressed as
+1
+σdσ
+dE=dlnτ(E)
+dE
++1
+2π2M−1(E)
+v2(E), (3)
+whereM−1(E) is (a diagonal element of) the in-
+verse mass tensor (second derivative of εk) averaged
+over the constant energy surface. This form makes
+it clear that the expressions should, for any quan-
+titative estimate, be generalized to two-band form,
+since the valence and conduction bands have differ-
+ing signs of their effective masses, and the scattering
+time—and its energy variation—is likely to be very
+different for Si-derived covalent valence bands and
+Al-derivedfree-electronconductionbands. Measure-
+ment of the thermopower, and a quantitative theo-
+retical treatment, would be very useful in extend-
+ing the understanding of the transport properties of
+NaAlSi.
+C. Unusual Fermi surfaces
+Fig. 4 depicts the calculated FSs. In spite of the
+generally 2D band structure, the small kzdispersion
+of bands at εFmake some of the FSs surprisingly
+three-dimensional. Electron pockets and hole pock-
+ets coexist in the Brillouin zone, with electron and
+hole concentrations necessarily being equal.
+Hole surfaces. Four hole “fan-blade” surfaces lie
+oriented in the xz- andyz-planes. At the center,
+extending from Γ half way to Z, lies a long and nar-
+row surface with square cross section. The top view
+allows the origin of these surfaces to be understood.
+The cross sections in the xy-plane are of two ellipses
+that are very anisotropic (in the xy-plane) and at
+right angels to each other. Each corresponds to a
+dispersion that is weak in one direction (the long
+major axis) and strong in the other (minor axis).
+These bands would intersect, but in fact are inter-
+sected by the electron band that cuts a squarishhole
+(rotated by 45◦), within which the elongated hole
+surface inside re-emerges.
+4Electron surfaces. In the bottom panels of Fig.
+4, the squarish electron surface (with kzvariation
+and resulting holes, shown in the lower two panels
+of Fig. 4) that cuts the aforementioned hole sur-
+face is pictured, and substantiates the discussion
+provided just above. In addition, there are simple
+electron ellipsoids centered along the Γ-M lines. It
+is curious that in a band structure that is for the
+most part strongly 2D, all the FSs have a rather
+definite three-dimensional character. Although the
+bands show little dispersion along Γ-Z, the bands
+just above the the Fermi level are quite different de-
+pending on whether kz= 0 orkz=π. In particular,
+the lowest band along R-A is rather flat, but the
+lowest conduction band along X-M has a dispersion
+of nearly 2 eV. Similar comparisons can be made for
+the bands along Γ-X and Z-R. The kzdispersion is
+not nearly as strong near kx=ky= 0, which is clear
+from both the band structure and the FS.
+Short discussion. It was noted in the Introduc-
+tion that the NaAlSi structure is the same as the
+Fe-pnictide “111”structure. Moreover,in both com-
+pounds, the relevantbandsinvolveonlythe (Si-Al-Si
+or As-Fe-As) trilayer. The top view of the fan-blade
+surfaces have characteristics in common with those
+ofsomeoftheFepnictides,27,28allofwhichhavethis
+same trilayer. The similarity is that the top view of
+the fan blades (if one ignores the diamond-shaped
+cutout at the intersection, centered at Γ) appears
+to show intersecting FSs, neither of which has the
+square symmetry of the lattice.
+FIG. 4: (Color online.) Views from the xy-plane (left)
+and top (right) of the FSs of NaAlSi, centered at Γ. The
+blue (dark) surfaces enclose holes and the yellow (light)
+surfaces enclose electrons.Such occurrence of intersecting FSs, each with
+lower symmetry than the lattice, has been analyzed
+for LaFeAsO (a “1111” compound) by Yaresko et
+al.29A symmetry of the Fe 2As2(also Al 2Si2) sub-
+structure is a non-primitive translation connecting
+Fe atoms (respectively, Al atoms) followed by z-
+reflection. This operation leads to symmetries that
+allowkz= 0 bands to be unfolded into a larger
+Brillouin zone (that is, a “smaller unit cell” having
+only one Fe atom) which unfolds the band structure
+and the intersecting FSs. The NaAlSi FSs appear
+to have this similar crossing (albeit interrupted by
+the free-electron bands), and the highly anisotropic
+dispersion is due to distinct (but symmetry-related)
+hopping along each of the crystal axes. In this re-
+spect NaAlSi may clarify the electronic structure of
+the pnictides: by analogy, there are separate bands
+that disperse more strongly along the (1 ,1) direc-
+tion or the (1 ,−1) direction, and give rise to the
+intersecting, symmetry-related surfaces. In NaAlSi
+the bands are much more anisotropic in the plane
+(approaching one-dimensional), making such char-
+acter much clearer. A difference that complicates
+the analogy is that in the pnictides the bands near
+εFare derived from the Fe atoms, which comprise
+the center layer of the trilayer, whereas in NaAlSi
+the bands under discussion are derived from the Si
+atoms, which comprise the two outer layers.
+D. Wannier functions
+Pictured in Fig. 5 are symmetry-projected Wan-
+nier functions (WFs) projected onto Si 3 porbitals.
+The extension of the WFs shows considerable in-
+volvement from nearby Al and Si atoms, and in ad-
+dition have some density extending into the Na lay-
+ers. The pxWF consists of an atomic pxfunction,
+with its density shifted downward by the bonding
+contribution of Al sp3hybrid orbitals. Beyond the
+pxlobes the nearest Si atoms form a bonding lobe
+that connects to the “small” side of the Al sp3func-
+tion. The large pxlobes and the extra contribution
+from nearby Si atoms are responsible for the largest
+hoppingamplitudesshowninTableI, althoughthere
+is some phase cancellation between the pxlobe and
+the lobe lying beyond the nodal surface.
+ThepzWF has one lobe extended well into the
+Na layer; this is responsible for the largest hoppings
+alongb∗in Table I, and they create the large disper-
+sion in the pzbands seen in Fig. 5(c). Again, the Al
+atomscontributewithan sphybridorbital, although
+it appearsto be more sp2-like than sp3-like. Thereis
+also a “ring” structure below the Al layer, where an
+sphybrid orbital from the Si atoms forms a bonding
+combination, but it is antibonding with the pzfunc-
+5tion on the central Si. The largest contribution to
+near-neighborhopping in the Al-Si plane between pz
+andpxorpzis most likely due to this ring structure,
+as theplobes are confined to the inside of a square
+ofnear-neighborAl atoms, which areonly edgeshar-
+ing with the nearest Si atoms along bvectors. This
+is the likely reason that all the hoppings along bare
+approximately of the same magnitude. The disper-
+sion which creates the FSs along Γ-Z (seen in Fig.
+4) is composed only of the pxandpyWFs. This
+is not caused by the large hoppings, but by smaller
+hoppings along b∗between pxandpyWFs. With-
+out these small hoppings, the band just above εFis
+dispersionless along Γ-Z.
+IV. RESPONSE TO CHANGES
+A. Electron-ion coupling
+A deformation potential Dis the shift in an en-
+ergybandwith respecttosublatticeatomicdisplace-
+ment. One can freeze in phonon modes to calculate
+deformation potentials, which at the FS are directly
+connected to electron-phonon matrix elements.30
+Moving the Si atoms in the zdirection by ±1%
+of the experimental parameter, such that the tetra-
+hedra surrounding the Al atoms stretch or flatten
+(while remaining centered on Al), gives an average
+FIG. 5: (Color online.) Isosurface of the WFs for (a)
+Si 3pxand (b) Si 3 pz. Na atoms are large and yellow
+(light) colored, Si atoms are small and blue (dark) col-
+ored. The two colors of the isosurface represent different
+signs. (c) The tight-binding fatbands band structure de-
+scribed in the text for the WFs, compared to the DFT
+band structure (black lines).pxpypz
+apx761 60
+py-62 60
+pz60 60 -40
+2apx128 27
+bpx361 300 360
+py300 361 360
+pz360 360 360
+b∗px12 5 50
+py5 12 50
+pz50 50 185
+TABLE I: Selected hopping integrals in meV for the Si
+3pWFs along the vectors a= (a,0,0) (hopping within a
+Si layer), b= (a/2,a/2,d) (hopping across an Al layer),
+andb∗= (a/2,a/2,c−d) (hopping across a Na layer). d
+is the distance in the zdirection between Si atoms above
+and below Al planes.
+FIG. 6: (Color online.) Comparison of band structures
+nearεFfor different zSivalues.
+deformation potential of ∼0.8 eV/˚A over five band
+positions near the Fermi level. The largest shift is
+for the ellipsoidal electron pockets, with D ∼1.2
+eV/˚A. These ellipsoids disappear when the Si atoms
+are displaced toward the Al plane (see Fig. 6).
+Flattening the Na bilayer, so as to remove the
+buckling of the Na atoms, requires a (very large)
+12%changeinthe zcomponentoftheNaatoms. We
+chose such a large displacement because we do not
+expect a substantial deformation potential for Na
+movement. Even this large displacement does not
+alter very much the valence bands, as expected, and
+the hole FSs remain virtually unchanged. The con-
+ductionbandsattheFermilevelhowevershiftappre-
+ciably, resulting in a modulation of the electron el-
+lipsoids along (1,1) near M. In addition, the acciden-
+tal four-band near-degeneracy that is 0.5 eV above
+εFat M splits the two separate doubly-degenerate
+states, opening up a gap of ∼0.7 eV, which is equiv-
+alent to a deformation potential of ∼0.8 eV/˚A (but
+6the bands are not at εF).
+B. Magnetic susceptibility
+The magnetic spin susceptibility χis given by
+χ=∂M
+∂H= (∂2E/∂M2)−1.
+Fixed-spin-moment calculations were conducted to
+produce an energy-vs.-moment E(M) curve, result-
+ing in a susceptibility of χ= 2.93µ2
+BeV−1, or
+3.76×10−6emu/mol. Thisistheexchange-enhanced
+susceptibility in the Stoner theory, and can be writ-
+ten as
+χ=Sχ0≡χ0
+1−IN(εF),
+whereS≡χ/χ0is the Stoner enhancement fac-
+tor,Iis the Stoner parameter, and the bare Pauli
+susceptibility χ0is equal to µ2
+BN(εF). Janak has
+shown how to calculate Iwithin density functional
+theory,31which must be equivalent (within a minor
+approximation he used) to our approach of using
+fixed-spin-moment calculations.
+With our calculated values we obtain S= 2.75,
+which translates to I= 0.60 eV. Some interpreta-
+tion of this value of Stoner Ishould be noted. First,
+εFfalls where most of the states are Si-derived, so
+for simplicity we neglect Al (and Na, which is ion-
+ized). Second, wenotethat therearetwoSiatomsin
+the primitive cell. Thus to get an “atomic value” of
+ISi, we should use a value of N(εF)/2 per Si atom.
+The result then is very roughly ISi∼1 eV. This
+value can be compared to the atomic value for Al
+(next to Si in the periodic table) in the elemental
+metal, which is IAl= 0.6 eV.31It seems, therefore,
+that Si in NaAlSi is considerably more “magneti-
+callyinclined”thanisAlin aluminum. However,the
+Stoner enhancement overall is not large, indicating
+relatively modest magnetic enhancement and rather
+conventional magnitude of magnetic interactions.
+C. Comparison to NaAlGe
+Isostructural and isovalent NaAlGe is not super-
+conducting (above 1.6 K, at least), so it should
+be instructive to compare its electronic structure
+to that of NaAlSi. Using its experimental lattice
+parameters,23we have calculated the band structure
+of NaAlGe, and compared it with that of NaAlSi on
+a rather fine scale in Fig. 7. The band structures are
+very similar, the one difference being that the band
+alongΓ-MnearMdoesnotcross εFin NaAlGe. The
+free-electron band is also identical.Supposing the tiny bit of FS along Γ-M cannot
+account for the difference in superconducting behav-
+iors, the factors relevant for electron-phonon cou-
+pling will be the difference in mass (Ge is more than
+twice as heavy as Si) and the difference in elec-
+tronic character, which can affect force constants
+and electron-phonon matrix elements. A real pos-
+sibility is that the pairing mechanism is electronic
+rather than phononic. In three dimensions purely
+electronic pairing mechanisms have been attracting
+seriousstudy(byShamandcollaborators32,33forex-
+ample), but 2D semimetals introduce new features
+that deserve detailed study.
+Another possibility is that these pockets are im-
+portant, and that superconductivity arises from an
+enhancement of electron-phonon coupling in these
+tiny electron pockets from low frequency electronic
+response, either interband transitions or plasma os-
+cillations, or both. A model in which the elec-
+tronic response of a 2D electronic superlattice plays
+a central role in the mechanism has been previously
+studied34,35using a model of parallel conducting
+sheets separated by a dielectric spacer. This model
+may be useful as a starting point for understanding
+NaAlSi.
+V. DISCUSSION
+The classes of materials that contain relatively
+high temperature superconductors36continues to
+expand. Superconductors derived from doped 2D
+semiconductors pose many of the most interesting
+issues in superconductivity today. The cuprates and
+the Fe-pnictides (and -chalcogenides) are strongly
+magnetic, and comprise one end of the spectrum
+(though they are themselves quite different). On
+FIG. 7: (Color online.) Blowup of the band structures
+of NaAlSi and NaAlGe near εF.
+7the other end lie those with little, perhaps negligi-
+ble, magnetism: electron-doped ZrNCl and HfNCl,
+and electron-doped TiNCl. There are several other,
+lower-Tcsystems, whose behavior seems different
+still (hydrated Na xCoO2, Li1−xNbO2, and several
+transition-metal disulfides and diselenides).
+A commonfeature ofmost ofthese systems is that
+they are 2D and have a small, but not tiny, con-
+centration of charge carriers, often in the range of
+0.05–0.15carriers per unit cell. These materials also
+have ionic character. NaAlSi differs in that it has sp
+carriers—the others have carriers in dbands—and
+is self-doped, a compensated semimetal. We suggest
+that a useful view of NaAlSi is that it be regardedas
+arising from an underlying ionic semiconductor, but
+that it has a small negative gap rather than a true
+gap. Without the overlapofthe valence and conduc-tion bands, it would be a 2D, ionic, and somewhat
+covalent semiconductor like the aforementioned ni-
+tridochloridecompounds, whichsuperconduct in the
+15–25 K range. Comparing the characteristics of
+these two classes of superconductors should further
+the understanding of 2D superconductivity.
+VI. ACKNOWLEDGMENTS
+This work was supported by DOE grant DE-
+FG02-04ER46111, the Strategic Sciences Aca-
+demic Alliance Program under grant DE-FG03-
+03NA00071, and by DOE SciDAC Grant No. DE-
+FC02-06ER25794.
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+arXiv:1001.0666v1  [physics.ins-det]  5 Jan 2010Radiation hardnessof CMSpixelbarrelmodules
+T.Rohe∗,a, A.Beanb,W. Erdmanna,H.-C. K¨ astlia,S. Khalatyanc, B. Meiera,V. Radiccib,J. Sibillea,b
+aPaulScherrer Institut, Villigen and W¨ urenlingen, Switze rland
+bUniversity ofKansas, Laurence KS, USA
+cUniversity of Illinois at Chicago, USA
+Abstract
+Pixel detectors are used in the innermost part of the multi pu rpose experiments at LHC and are therefore exposed to the hig hest
+fluences of ionising radiation, which in this part of the dete ctors consists mainly of charged pions. The radiation hardn ess of all
+detector components has thoroughly been tested up to the flue nces expected at the LHC. In case of an LHC upgrade, the fluence
+will bemuchhigheranditisnotyetclearhowlongthepresent pixelmoduleswill stayoperativeinsuchaharshenvironmen t. The
+aim ofthisstudywastoestablish suchalimit asabenchmarkf orotherpossibledetectorconceptsconsideredforthe upgr ade.
+Asthesensorsandthereadoutchiparethepartsmostsensiti veto radiationdamage,samplesconsistingofa smallpixels ensor
+bump-bondedtoaCMS-readoutchip(PSI46V2.1)havebeenirr adiatedwithpositive200MeVpionsatPSIupto6 ×1014neq/cm2
+andwith21GeV protonsatCERN upto 5 ×1015neq/cm2.
+After irradiation the response of the system to beta particl es from a90Sr source was measured to characterise the charge
+collection efficiency of the sensor. Radiation induced changes in the reado ut chip were also measured. The results show that
+the present pixel modules can be expected to be still operati onal after a fluence of 2 .8×1015neq/cm2. Samples irradiated up to
+5×1015neq/cm2still see the beta particles. However, further tests are nee ded to confirm whether a stable operation with high
+particledetectione fficiencyispossibleaftersuchahighfluence.
+Key words: LHC,superLHC, CMS,tracking,pixel,silicon,radiationha rdness
+PACS:07.89,29.40.Gx,29.40.Wk,61.80.-x
+1. Introduction
+Hybridsiliconpixeldetectorsareusedintheinnermostpar t
+of the multi purpose experiments at LHC at distances between
+4 and 12cm from the beam [1, 2]. They are therefore exposed
+to the highest fluences of ionising radiation, which in this p art
+of the detectors consists mainly of charged pions. The radia -
+tion hardness of all detector components has thoroughly bee n
+tested up to the fluences expected at the LHC [3]. A replace-
+mentofthepresentCMSpixeldetectorbyanimproved4-layer
+system is already planned for 2014 [4]. As a two stage lumi-
+nosity upgrade of the LHC is foreseen shortly after [5], it ha s
+tobeassuredthatthenewsystemissuitableforpeakluminos i-
+tiesabove1034cm−2s−1intermsofradiationhardnessanddata
+transfer. TheshorttimescaleallowsonlyforverylimitedR &D
+and it has to be tested if the present detector concept can be
+adjusted to the new specifications. This paper reports on the
+radiationhardnessofboththesensorandthereadoutchip,w ith
+the aim to estimate their fluence limits and their lifetime in an
+upgradedLHC.
+2. CMS Modules
+The pixel detector of CMS is a so called hybridpixel de-
+tector. This means that the sensor part, which actually dete cts
+∗Correponding author
+Emailaddress: tilman.rohe@cern.ch (T.Rohe)the charged particles, and the readout electronics are on se pa-
+rate substrates. Each sensing element (pixel) is connected to
+its preamplifier located on the readout chip by a small indium
+bowl,the so-calledbump. The powerand I /Opadsofthe read-
+out chip are then wire bonded to a high density interconnect
+(HDI)whichprovidestheconnectiontothe outside.
+The two parts most sensitive to radiation damage are the
+sensor and the readout chip. Therefore this study is limited
+to those components. In order to further simplify the irradi a-
+tion procedure only small modulesconsisting of one ROC and
+a small sensor covering the area of this ROC were used. Such
+samplesareabout1 ×1cm2largewhichisaboutthebeamspot
+size in the irradiation sites chosen. So, scanning during ir radi-
+ation could be avoided. However,in contrast to strip detect ors,
+where the strip lengthis important,small pixel devicescon tain
+allrelevantfeaturesofafullmoduleandtheresultsobtain edin
+thisstudyare fullyvalid.
+2.1. TheSensor
+The sensors for the CMS pixel barrel are made of silicon
+and follow the so called “n-in-n” approach. The collection o f
+electronsisadvantageousbecauseoftheirhighermobility com-
+pared to holes. It causes a larger Lorentz drift of the signal
+chargeswhichleadsto chargesharingbetweenbetweenneigh -
+boringpixelsandsoimprovesthespatialresolution. Furth erthe
+highermobilityofelectronsmakesthemlesspronetotrappi ng,
+Preprint submitted to Elsevier November 15,2018which is essential after high fluences of charged particles. Af-
+ter irradiation induced space charge sign inversion, the hi ghest
+electric field in the sensor is located close to the n-electro des
+usedto collectthecharge,whichisalso ofadvantage.
+The choice of n-substrate requires a double sided sensor
+process,meaningthat bothsidesofthe sensorneedphotolit ho-
+graphic steps. This leads to higher costs compared to single
+sidedp-in-n(orn-in-p)sensors. However,thedoublesided sen-
+sors have a guard ring scheme where all sensor edges are at a
+groundpotentialandthisgreatlysimplifiedthedesignofde tec-
+tormodules. Then-sideisolationwasimplementedthrought he
+so-calledmoderatedp-spraytechnique[6]withapunchthro ugh
+biasinggrid.
+Thesensorsamplesweretakenfromwafersofthemainpro-
+duction run for the CMS pixel barrel which were processed on
+approximately 285 µm thick n-doped DOFZ silicon according
+to the recommendationof the ROSE-collaboration[7]. The re -
+sistance of the material priorto irradiationwas 3 .7kΩcm lead-
+ingto aninitialfulldepletionvoltageof Vfd≈55V.
+2.2. The ReadoutChip
+The CMS pixel readout chip (called PSI46V2.1) described
+in detail in [8] has been fabricated in a commercial five-meta l
+layer 0.25µm process available via CERN. The chip contains
+about 1.3 million transistors on an area of 7 .9×9.8mm2. It
+is divided into three functional blocks: the array of 80 ×52
+pixel unit cells of 100 ×150µm2size organised in 26 double
+columns, the double column perifery, and a control and a chip
+periferycontaininga supplyandcontrolblock.
+The radiation hardness of the 0 .25µm technology is well
+knownifcertainlayoutrecommendationsarefollowed[9]. T his
+concernstheleakagecurrents,whicharestronglysurpress edby
+a “closedgate”geometryofthe NMOStransistorsandchannel
+stop rings. The shift of the transistor thresholdsis small d ue to
+the small thicknessof the gate oxides in such processes. Oth er
+radiationinducede ffectslikethereductionofthesurfacemobil-
+ity andthe minoritycarrierlife time remain. Theformermig ht
+effect the transconductance of the transistors and by this the
+speed ofthe chipwhile the latter influencesthe bipolardevi ces
+like diodes.
+3. Sample Preparation
+Thereadoutchipstakenareidenticaltothoseusedforbuild -
+ing the CMS pixel detector. The small sensors were produced
+on the same wafers as those for full modules and identical to
+those apart from the size. Both parts were joined using the
+bump bonding process used for the module production for the
+CMS pixel barrel [10]. As this procedure includes processin g
+steps at temperaturesabove200◦C, it was done beforeirradia-
+tion. This means that the sensors and readout chips had to be
+irradiated at the same time. This way, a realistic picture of the
+situationinCMSafterafewyearsofrunningcouldbeobtaine d.
+The sandwiches of sensor and readout chip were irradiated
+at the PSI-PiE1-beam line [11] with positive pions of momen-
+tum 200MeV/c to fluences up to 6 ×1014neq/cm2and with
+26GeV/cprotonsat CERN-PS [12]upto5 ×1015neq/cm2.All irradiated samples were kept in a commercial freezer
+at−18◦C after irradiation. However, the pion irradiated ones
+were accidentallywarmed up to roomtemperaturefor a period
+ofa fewweeks.
+4. SensorPerformance
+The aim of this study was to determine the signal obtained
+from minimumionizingparticles(m.i.p.) which is the limit ing
+factorfortheuse ofsuchtypeofsensors.
+4.1. MeasurementProcedure
+For reasons of simplicity, a90Sr source has been used for
+inducingthesignalinthesensor. The β-spectrumofthedaugh-
+ter decay of90Y has an endpoint energyof about 2.3MeV and
+therefore contains particles which approximate a m.i.p. we ll.
+However,therearealsoalotoflowenergyparticleswhichle ad
+toan energydepositionin thesensorwhichis muchhigher.
+ThesamplesweremountedonapairofwatercooledPeltier
+elementsandkeptbelow −10◦C.Thesourcewasplacedinside
+the box a few millimetres above the sample. For acquiring the
+data a so-called random trigger was used. In this mode of op-
+eration an arbitrary cycle of the clock is stretched by a larg e
+factor, and a trigger is sent in a way that all the hits within t his
+clockcyclearereadout. Thestretchfactorwasaadjustedsu ch
+that about 80% of the triggers showed hits. As this method
+does not allow for hit detection e fficiency measurements, the
+upgrade of the setup with an independentscintillator trigg er is
+currentlyunderway.
+The testing and calibration procedure was similar to what
+is used for the qualification of CMS pixel barrel modules[13] .
+Thethresholdtowhichthepixelswereadjustedwas4000e−for
+allsamplesuptoafluenceof1 .2×1015neq/cm2. Forthehigher
+fluencethe thresholdwasloweredtoa valuebetween2000and
+3000e−. After,calibrationdatawastakenusingthe90Srsource.
+Thesensorbiaswasvariedfrom50Vupto1100Vforsomeof
+the most irradiated samples. The temperature was kept stabl e
+within0.2◦Cduringthe biasscan.
+Thedatawasanalysedo ffline. First,allanalogsignalswere
+convertedintoan absolutechargevalue[14]. Thena mask was
+generated to exclude noisy and unbonded pixels from further
+analysis. Finally, all clusters not contiguous with a maske d
+pixel were reconstructed. The charge of the clusters was his -
+togrammed separately for each cluster size. The spectra wer e
+fittoaLandaufunctionconvolutedwithaGaussian. Thecharg e
+valuequotedisthe mostprobablevalue(MPV)fromthisfit.
+The radiation of the90Sr source contains a sizable fraction
+of low energy electrons which might be stopped in the sensor
+and cause a much higher signal than a minimum ionising par-
+ticle. A part of the secondary electrons, however, travels i n
+parallel to the sensor surface resulting in large clusters. Fig-
+ure1showsthepulseheightspectrumofasampleirradiatedt o
+2.8×1015neq/cm2ata biasvoltageof800V fordi fferentclus-
+ter sizes. The large fraction of 2-clusters and the occurren ce
+of clusters larger than 2 pixels cannot be explained by simpl e
+chargesharingandclearlyindicatedthepresenceoflowene rgy
+2Signal Height [a.u.]0 100 200 300 400 500 600 700 800Number of Entries
+310410
+All Clusters
+1 Pixel
+2 Pixels
+3 Pixels
+4 Pixels
+> 4 Pixels
+Figure 1: Pulseheight distribution ofasampleirradiated t o 2.8×1015neq/cm2
+at a bias voltage of 800V in arbitrary units (1 unit correspon ds to about 65
+electrons)
+Voltage [V]0 200 400 600 800 1000]-Signal [ke
+0510152025 Not irradiated
+, Pions2/cmeqn1410×4.3
+, Pions2/cmeqn1410×6.2
+, Protons2/cmeqn1410×6.1
+, Protons2/cmeqn1510×1.1
+, Protons2/cmeqn1510×2.8
+, Protons2/cmeqn1510×5.1
+Figure 2: Most probable signal from single pixel clusters as a function of the
+sensor bias. Each line represents one sample
+electrons from the source. The same is the case for the strong
+dependenceoftheMPVontheclustersize. Inordernottobias
+the dataonlythespectraofsinglehit clustershavebeenuse d.
+4.2. Results
+Figure 2 shows the valuesof the MPV for all the measured
+samples and all biases. For the unirradiated samples the ste ep
+rise of the signalat the full depletionvoltageof Vdepl≈55Vis
+visible. Thesignalplateauisreachedveryfast.
+Thesamplesirradiatedtofluencesinthe1014neq/cm2-range
+alsoshowasaturationofthesignalaboveroughly300V. How-
+ever, the variation of the saturated signal for the same irra dia-
+tion fluence is of the same order of magnitudeas the reduction
+ofthesignalfrom4 .3to6.2×1014neq/cm2. Adistinctionofthe
+irradiationfluenceofthesamplesfromtheheightoftheplat eau
+is not possible. This strong variation cannot be explained b y
+differences in the sensor thickness and is probably caused by
+the imperfectionof the pulse heightcalibration. It relies onthe
+assumptionthattheinjectionmechanismfortestpulsesise qual]2/cmeq n14 [10Φ0 10 20 30 40 50]-Signal [ke
+0510152025Not irradiated, 250V
+Pions, 600V
+Protons, 600V
+Protons, 800V
+Protons, 1000V
+Figure 3: Mostprobable signal as afunction of the irradiati on fluence
+forallreadoutchips,whichisnotthecase. Variationsofth ein-
+jectioncapacitorarelargerthan15%.
+Thedifferencesintheirradiationfluencearebetterreflected
+in the “low” voltage part of the curves, wherethe voltage fro m
+which the signal starts to rise reflects the radiation-induc edin-
+creaseofthespacecharge.
+For the samples irradiated to fluences above 1015neq/cm2,
+no saturation of the signal with increasing bias is visible. It
+is remarkable that even after a fluence of 2 .8×1015neq/cm2a
+charge of more than 10000 electrons can be collected if a bias
+voltageover800Vis applied.
+The samples irradiated to 5 .1×1015neq/cm2, can clearly
+detect the particles. However, the signal obtained is well b e-
+low5000electronswhichistoolittleforreliableoperatio nasa
+trackingdevice. The data were taken at a temperatureof abou t
+−15◦C and therefore the leakage current was above 50 µA for
+the about1cm2largedevice. Thisdid notallow measurements
+above800V.
+To display the development of the signal height as a func-
+tion of the fluence, the charge at 600V was measured for each
+sample (250V for the unirradiated ones) and plotted in Fig. 3 .
+Inaddition,valuesfor800Vand1000Vareshownforthehigh-
+est fluence. Apart from the large fluctuations the reduction o f
+the charge with fluence is visible. Further it becomes obviou s
+that it pays to go to very high bias voltages if the fluences ex-
+ceed1015neq/cm2.
+Thechargevaluesinthisstudyarelowerthanthosereported
+elsewhere [15, 16]. The authorsin these studies suggested t hat
+some kind of avalanche multiplication was the reason for the
+higherthanexpectedsignals. Thesensorsusedinthisstudy are
+designeddifferently. The layoutof the pixel’s n+-implantswas
+optimizedtoavoidpeaksoftheelectricfield. Thereforethe IV-
+curves of the sensors tested do not show evidence for an onset
+ofmicrodischarge.
+5. PerformanceoftheReadoutChip
+In general it can be stated that all of the ROCs, even at
+the highest fluence, worked well and could be operated using
+3DAC setting [a.u.]0 2 4 6 8 10 12 14 16 [V]digV
+1.61.71.81.922.12.22.32.42.52.6
+unirradiated
+, Pionseqn14 10×3.2
+, Pionseqn14 10×4.2
+, Pionseqn14 10×6.2
+, Protonseqn14 10×6.1
+, Protonseqn15 10×1.1
+, Protonseqn15 10×2.8
+, Protonseqn15 10×5.1
+Figure 4: Output voltage of the ROC-DAC ,, Vdig” for two unirradiated chips
+and irradiated samples up to afluence of 5 .1×1015neq/cm2
+the standard calibration software. For the ROC with a fluence
+above2×1015neq/cm2the feedbackcircuitofthepreamplifier
+and shaper had to be adjusted, and the sampling point of the
+ADC,readingthechipdatahadtobeshiftedforafewnanosec-
+onds.
+The parameters of the readout chip which are subject to
+changeduetoirradiationarenotdirectlyaccessibleinthe read-
+out chips. For an extraction of the transistor parameters a s pe-
+cial test chip would be necessary. However, there are some
+chip parameters which give, indirectly, hints on the radiat ion
+inducedchangesin theROC.
+One very sensitive block in the periphery is the band gap
+reference which is used as a reference for all internal volta ges
+set. OneofthesesvoltagesisstabilizedusinganexternalS MD-
+capacitor and can therefore be measured using a probe. The
+bandgapreferenceisinprincipleaforwardbiaseddiodewho se
+reference voltage might be dependent on the minority carrie r
+lifetime. Measuringthedigitalvoltage Vdigasafunctionofthe
+adjusted DAC setting is therefore a test of the stability of t he
+band gap reference. The result of this measurement is shown
+in Fig. 4. The output of the band gap reference shows a shift
+of about10% alreadyat thelowest fluenceandthen stayscon-
+stant. Thisisahintofasurfacee ffectwhichusuallysaturatesat
+low doses. The saturation of the voltage at DAC values above
+10 is due to the fact the supply voltage of 2.5V cannot be ex-
+ceeded.
+Another parameter, which might be prone to radiation in-
+ducedchanges, is the rise time of the preamplifier. Thiscaus es
+the so called time walk and either needs to be compensated by
+an increase of the analog current or results in a higher in-ti me
+threshold.
+The pulse shape of the PSI46V2.1 cannot be measured di-
+rectly, but an indirect method gives a hint on its leading edg e.
+Itisbasedoncalibrationpulseswhichcanbeinjectedintoe ach
+pixel. With a certain DACsetting, called calibrationdelay ,the
+timeofthispulsecanbeshiftedwithrespecttotheleadinge dge
+of the clock cycle whose signals will be read out. If this DAC
+is scanned while the height of the calibration signal and theTime [ns]0 10 20 30 40 50 60]-Amplitude [100 e
+406080100120140160180200220240
+Unirradiated
+, Pionseqn14 10×6.2
+Figure 5: Pulse shape of an injected test signal measured in o ne pixel of three
+unirradiated ROCs and two ROCs irradiated to 6 .2×1014neq/cm2
+threshold stays constant, an e fficiency curve will be obtained.
+If the pulse is injected too early it will cross the threshold in
+the clock cycle before the readout and will not be seen. If
+it is too late it will be assigned to the next clock cycle. The
+plateau of full efficiency in-between the two regions is exactly
+one clock cycle wide i.e. 25ns. If this scan is repeated with a
+lower threshold and the same calibration pulse height, the e f-
+ficiency plateau will be slightly shifted due to the fact that the
+thresholdwill be reached slightly earlier. If the whole spa ce of
+thethresholdandthecalibrationdelayisscanned,thetimi ngof
+theleadingedgeofthepulseshapecanbereconstructed.
+Theresultofsuchareconstructionisshowninfig.5forone
+pixeleachinthreeunirradiatedROCsandtwoROCsirradiate d
+to 6.2×1014neq/cm2. The analog current was the same in all
+themeasurements. Therisetimeofthepulseisabout35nsand ,
+mostremarkable,itdoesnotchangewithirradiation,atlea stnot
+uptothelevelinvestigated.
+6. Conclusion
+Sandwiches of a CMS pixel readout chip and a small sin-
+gle chip sensor have been irradiated up to flucences of 5 .1×
+1015neq/cm2. Test with a90Sr source showed a signal above
+10000 electrons per m.i.p. can be achived at bias voltage of
+or below 600V up to a fluence of about 1015neq/cm2. The
+samples irradiated up to 2 .8×1015neq/cm2, however, needed
+a bias voltage of 1000V to reach a signal of 10000 electrons.
+This indicates the suitabiliy of these devices for an upgrad ed
+LHC up to rather small radii. The samples which obtained a
+fluence of 5.1×1015neq/cm2could only be measured up to a
+bias voltage of 800V due to limitations of the cooling. They
+still showed signals, but their amplitude seems too low for a n
+eficient tracking. Measurements at higher bias voltages are a
+subject of further studies, as well as e fficiency measurements
+usinganindependentscintillatortrigger.
+Thereadoutchip performedwell evenafter the highest flu-
+ence. Itcouldbeoperatedwiththestandardcalibrationmet hod
+and only very few settings, like the feedback resistor of the
+4preamplifier and shaper, needed to be adjusted. A degradatio n
+of the amplifier’s rise time could not be detected by this stud y.
+The band gap reference of the chip drifted by about 10% al-
+readyafterlowestfluence,butthisshiftsaturated. Furthe rstud-
+iesto characterizetheirradiatedreadoutchipsare underw ay.
+Acknowledgement
+The pion irradiation at PSI would not have been possible
+without the beam line support by D. Renker and K. Deiters,
+PSI, the logistics providedby M. Glaser, CERN, and the great
+effort of C. Betancourt and M. Gerling, UC Santa Cruz (both
+were supportedbya financialcontributionofRD50andPSI).
+Theprotonirradiationwascarriedoutat theCERN irradia-
+tionfacility. TheauthorswouldliketothankM.Glaserandt he
+CERN teamforthe outstandingservice.
+Thanks are also due to N. Krzyzanowski and E. Stachura,
+UICforthemeasurementsoftheirradiatedROCs.
+The work of A. Bean and V. Radicci is supported by the
+PIRE grantOISE-0730173oftheUS-NSF.
+The sensors were produced by CiS GmbH in Erfurt, Ger-
+many.
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+[14] S. Dambach, CMS pixel module readout optimization and s tudy of the
+B0lifetime in the semileptonic decay mode, Ph.D. thesis, ETH Z ¨ urich,
+Switzerland, ETHDiss.No.18203 (2009).
+[15] A.Affolder, P.Allport, G. Casse, Title notyet known, these proce edings.
+[16] I. Mandi` c, V. Cindro, G. Kramberger, M. Miku˘ z, Observ ation of full
+charge collection e fficiency in p+n strip detectors irradiated up to 3 ×
+1016neq/cm2, presented at the RESMDD 2008 in Florence, to be pub-
+lished in NIM A.
+5
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+arXiv:1001.0667v1  [math.OA]  5 Jan 2010ASYMPTOTIC PROPERTIES OF RANDOM
+MATRICES AND PSEUDOMATRICES
+ROMUALD LENCZEWSKI
+Abstract. We study the asymptotics of sums of matricially free random variable s,
+called random pseudomatrices, and we compare it with that of rando m matrices with
+block-identical variances. For objects of both types we find the lim it joint distribu-
+tions of blocks and give their Hilbert space realizations, using operat ors called ‘matri-
+cially free Gaussian operators’. In particular, if the variance matric es are symmetric,
+the asymptotics of symmetric blocks of random pseudomatrices ag rees with that of
+symmetric random blocks. We also show that blocks of random pseud omatrices are
+‘asymptotically matricially free’ whereas the corresponding symmet ric random blocks
+are ‘asymptotically symmetrically matricially free’, where symmetric ma tricial free-
+ness is obtained from matricial freeness by an operation of symmet rization. Finally,
+we show that row blocks of square, lower-block-triangular and bloc k-diagonal pseu-
+domatrices are asymptotically free, monotone independent and bo olean independent,
+respectively.
+1.Introduction and main results
+We have recently shown that the Hilbert space construction of the free product of
+states onC∗-algebras given by Voiculescu [8] can be generalized to a framework w hich
+exhibits some matricial features [5].
+We considered arrays of noncommutative probability spaces, for w hich we defined the
+matricially free product of states . Thedefinitionofthisproductisbasedonreplacingthe
+family of canonical unital *-representations of C∗-algebras on a free product of Hilbert
+spaces by a diagonal-containing array of non-unital *-represent ations ofC∗-algebras on
+the matricially free product of Hilbert spaces. The crucial point is th at products of
+these representations imitate products of matrices rather than free products, although
+the main features of the latter are still present. We studied the as sociated concepts of
+noncommutative independence called matricial freeness and related to it strong matri-
+cial freeness which can be viewed as a scalar-type generalization of freeness und erlying
+other fundamental notions of noncommutative independence (mo notone and boolean)
+and some of their generalizations (conditional freeness and condit ional monotone inde-
+pendence).
+More importantly, matricial freeness is closely related to random ma trices. Their
+significance for free probability was discovered by Voiculescu [10], wh o showed that
+Gaussian random matrices with mutually independent entries are asy mptotically free.
+2000 Mathematics Subject Classification : 46L53, 46L54, 15A52
+Key words and phrases: free probability, freeness, matricial freeness, symmetric matric ial freeness,
+random pseudomatrix, random matrix, matricially free Fock space
+This work is partially supported by MNiSW research grant No N N201 36 4436
+12 R. LENCZEWSKI
+This connection was later generalized by Dykema [2] to non-Gaussian random matrices.
+Of course, the first indication that free probability might be related to random matrices
+was the central limit theorem for free random variables since its limit la w was the
+semicirle law obtained in the classical work of Wigner [13] as the limit distr ibution of
+certain self-adjoint random matrices with independent entries.
+The main objects of interest in our study of matricial freeness and strong matricial
+freeness arediagonal-containing arrays of (in general, non-unita l) subalgebras of a given
+unital algebra A,
+(Ai,j)(i,j)∈Jwith ∆ ⊆J⊆I×Iand ∆ = {(j,j) :j∈I},
+equipped with an array of states ( ϕi,j) onA, with respect to which notions of noncom-
+mutative independence are defined. Here, the states ϕi,jand their kernels play a very
+similar role to that of one distinguished state ϕand its kernel in free probability. Apart
+from square arrays, of particular interest are lower- (upper-) t riangular arrays related
+to monotone (anti-monotone) independence introduced by Murak i [6]. In all arrays, the
+roles of diagonal and off-diagonal entries are quite different, which is a characteristic
+feature of this theory reminding the random matrix theory.
+Let (Xi,j(n)) be ann-dimensional square array (or its diagonal-containing subarray)
+of self-adjoint random variables in a unital *-algebra A(n) which is matricially free with
+respect to the array ( ϕi,j(n)) defined in terms of a family ( ϕj(n))1≤j≤n. The sums
+S(n) =/summationdisplay
+(i,j)∈J(n)Xi,j(n),
+where (J(n)) is an appropriate sequence of sets, called random pseudomatrices , remind
+random matrices, whereas convex linear combinations
+ψ(n) =1
+nn/summationdisplay
+j=1ϕj(n),
+wheren∈N, replace the states τ(n) =E⊗tr(n), classical expectation tensored with
+normalized traces, under which distributions of random matrices ar e computed.
+Under suitable assumptions on the distributions of the variables Xi,j(n) in the states
+ϕi,j(n), respectively, wecannowstudytheasymptoticdistributions ofs ums correspond-
+ing to blocks of random pseudomatrices in various states. These blo cks are defined as
+sums of the form
+Sp,q(n) =/summationdisplay
+(i,j)∈Np×NqXi,j(n)
+where [n] :={1,2,...,n}=N1∪N2∪...∪Nris a partition into disjoint subsets whose
+sizes grow proportionately to n(it is convenient to think of intervals). The dependence
+oftheNj’sonnissupressed inthenotation. Inparticular, we assume thatthevar iances
+vi,j(n) :=ϕi,j(n)(X2
+i,j(n)) are identical within blocks.
+This leads to various asymptotic properties of sums of matricially fre e and strongly
+matricially free random variables. Let ϕi,j(n) be defined according to
+ϕj,j(n) =ϕ(n) andϕi,j(n) =ϕj(n) fori/\e}a⊔io\slash=j,ASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 3
+whereϕ(n) is a distinguished state on A(n) and (ϕj(n))1≤j≤nis a family of additional
+states called conditions associated with ϕ(n) for anyn∈N. Then, as n→ ∞, we
+obtain the following properties:
+(i) if the variances of square arrays are identical within blocks and r ows, then the
+sums
+Ap(n) :=n/summationdisplay
+q=1Sp,q(n),where 1 ≤p≤r,
+are asymptotically free with respect to ϕ(n),
+(ii) if the variances of block-lower-triangular arrays are identical w ithin blocks and
+rows, then the sums
+Bp(n) :=p/summationdisplay
+q=1Sp,q(n),where 1 ≤p≤r,
+are asymptotically monotone independent with respect to ϕ(n),
+(iii) if the variances of block-diagonal arrays are identical within bloc ks, then the
+sumsCp(n) :=Sp,p(n), where 1 ≤p≤r, are asymptotically boolean indepen-
+dent with respect to ϕ(n).
+Let us add that non-asymptotic analogs of (i)-(ii) for finite sums of stronglymatricially
+free random variables were proved in [5] and that (iii) also holds for fin ite sums of
+both matricially free and strongly matricially free random variables. R ecall that the
+‘strongly matricially free product of states’, on which the definition o f strong matricial
+freeness is based, is obtained from the matricially free product by r estriction. However,
+it is worth to remark that the difference between blocks of matricially free and strongly
+matricially free random variables disappears asymptotically.
+The scheme of matricial freeness and its symmetrized version, in wh ich ordered pairs
+are replaced by (non-ordered) sets consisting of one or two eleme nts, called symmet-
+ric matricial freeness , is also used for arrays ( ϕi,j(n)) defined in terms of a family
+(ϕj(n))1≤j≤nof states on A(n) according to
+ϕi,j(n) =ϕj(n) for anyi,j,
+and anyn∈N. Here, we do not a priori assume that the states ϕj(n) are conditions as-
+sociated with a distinguished state. In the study of asymptotics, o f importance become
+arrays defined by ‘normalized partial traces’
+ψq(n) =1
+nq/summationdisplay
+j∈Nqϕj(n)
+according to ψp,q(n) =ψq(n) for anyp,q∈[r] andn∈N, wherenqis the cardinality
+ofNq. Then, asn→ ∞, we obtain the following properties:
+(iv) if the variances are identical within blocks, then the array ( Sp,q(n)) is asymp-
+totically matricially free with respect to ( ψp,q(n)),4 R. LENCZEWSKI
+(v) if the variances are identical within blocks and form symmetric ma trices, then
+the array (Zp,q(n)) of symmetric blocks given by
+Zp,q(n) :=/summationdisplay
+(i,j)∈Np,qXi,j(n)
+whereNp,q= (Np×Nq)∪(Nq×Np), is asymptotically symmetrically matricially
+free with respect to ( ψp,q(n)),
+(vi) if the variances are identical within blocks of symmetric random m atrices, then
+the array (Tp,q(n)) of symmetric random blocks is asymptotically symmetrically
+matricially free with respect to ( τp,q(n)),
+where the array ( τp,q(n)) is defined by the family of partial traces τq=E⊗trq(n) and
+trq(n) denotes the normalized trace over the vectors indexed by Nq. Here, by symmetric
+random blocks we understand symmetric blocks of symmetric rando m matrices in the
+approach of Voiculescu [10] and Dykema [2].
+Concerning a non-asymptotic analog of (iv), we show in this paper th at blocks
+(Sp,q(n)) are matricially free for any finite nunder the stronger assumption that the
+variables have block-identical distributions. In turn, properties ( v) and (vi) do not seem
+to have their non-asymptotic analogs. Nevertheless, they are th e reason why we call
+the sumsS(n) random pseudomatrices. On the other hand, let us also remark th at
+we do not have an analog of (iv) for random matrices. Therefore, in formally, random
+pseudomatrices can be viewed as objects which remind random matr ices for large nif
+we consider blocks with symmetric variances, but exhibit different fe atures if the vari-
+ances are not symmetric, which leads to triangular arrays and relat ions to monotone
+independence.
+In that connection we would like to mention the result of Shlyakhtenk o [7] who estab-
+lished a connection between a class of symmetric random matrices ca lled random band
+matrices and an operator-type generalization of freeness called f reeness with amalga-
+mation. Our result (vi) gives a connection between another class of symmetric random
+matricescalledsymmetricrandomblockswithblock-identicaldistribu tionsandascalar-
+type generalization of freeness called symmetric matricial freenes s. At present, it is not
+clear to us whether there is a relation between these two approach es.
+In our previous work, we expressed the limit distributions of random pseudomatrices
+with respect to ϕ(n)’s andψ(n)’s in terms of certain functions on the class of colored
+non-crossing pair partitions, as well as in terms of some ‘continued m ultifractions’ ]5].
+Both these realizations showed that the limit laws can be viewed as mat ricial general-
+izations of the semicircle laws. Moreover, these distributions turne d out to be related
+to s-free additive convolutions and the associated notion of freen ess with subordination,
+or s-freeness [3], concepts motivated by the results of Voiculescu [11] and Biane [1] on
+analytic subordination in free probability. Using s-freeness, we also studied s-free and
+free multiplicative convolutions, which allowed us to establish relations between these
+convolutions (as well as some other multiplicative convolutions) and c ertain classes of
+walks on appropriately defined products of graphs [4].
+In this paper, we give Hilbert-space realizations of the limit distributio ns of random
+pseudomatrices and their blocks under ϕ(n),ψk(n) andψ(n), where the underlyingASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 5
+Hilbert space in all considered cases is the matricially free product
+(F,ξ) =∗M
+i,j(Fp,q)
+of ther-dimensional array ( Fp,q) of Fock spaces, where the diagonal and off-diagonal
+Fockspaces are, respectively, freeandbooleanFockspaces ove r one-dimensional Hilbert
+spaces. Using realizations on Fandoperatorswhich play the role ofGaussian operators
+in our approach, as well as their ‘symmetrizations’, we prove (i)-(vi) .
+In Section 2, we present the combinatorics of colored non-crossin g pair-partitions.
+In Section 3 we recall the basic notions and facts concerning matric ially free random
+variables. In Section 4 we introduce matricially free Gaussian operat ors onF. Using
+them, we find F-realizations of the limit distributions for random pseudomatrices in
+Section 5. In Section 6 we prove that blocks of (strongly) matricially free arrays of
+random variables with block-identical distributions are matricially fre e with respect to
+a suitably defined array of states. In turn, asymptotic matricial f reeness of blocks of
+randompseudomatrices is proved inSection 7, where we also find F-realizations of their
+limit joint distributions. In Section 8 we introduce the notion of ‘symme tric matricial
+freeness’. In Section 9 we find the limit joint distributions of symmetr ic random blocks
+and their F-realizations and we show that symmetric random blocks are asympt otically
+symmetrically matricially free. In Section 10 we obtain results on asym ptotic freeness
+and asymptotic monotone independence of rows of pseudomatrice s.
+2.Combinatorics
+The combinatorics of our model is based on the class of colored non-crossing pair
+partitions , to which we assign certain products of matrix elements whose indice s depend
+on the colorings of their blocks.
+For a given non-crossing pair partition π, we denote by B(π),L(π) andR(π) the
+sets of its blocks, their left and right legs, respectively. If πi={l(i),r(i)}andπj=
+{l(j),r(j)}are blocks of πwith left legs l(i) andl(j) and right legs r(i) andr(j),
+respectively, then πiisinnerwith respect to πjifl(j)<l(i)<r(i)<r(j). In that case
+πjisouterwith respect to πi. It is the nearestouter block of πiif there is no block
+πk={l(k),r(k)}such thatl(j)< l(k)< l(i)< r(i)< r(k)< r(j). Since the nearest
+outer block, if it exists, is unique, we can write in this case πj=o(πi),l(j) =o(l(i))
+andr(j) =o(r(i)). Ifπidoes not have an outer block, it is called a covering block. It
+is convenient to extend each partition π∈ NC2
+mto the partition /hatwideπobtained from πby
+adding one block, say π0={0,m+1}, called the imaginary block .
+LetFr(π) be the set of all mappings f:B(π)→[r] calledcolorings of the blocks of π
+by the set [r] :={1,2,...,r}. Then the pair ( π,f) plays the role of a colored partition.
+Its blocks will be denoted
+B(π,f) ={(π1,f),(π2,f),...,(πk,f)},
+where we use pairs ( πi,f) since to each block πiwe shall assign entries of a matrix
+which depend on the colors of both πiando(πi) for anyi∈[k]. If the imaginary block
+is used, it is convenient to assume that it is also colored by a number fr om the set [ r].
+In Fig. 1 we give an example with the notions defined above.6 R. LENCZEWSKI
+1 2 3 4 5 6i klimaginary block jblocksπ1={1,6},π2={2,3},π3={4,5}
+legsL(π)={1,2,4},R(π)={3,5,6}
+coloring f(π1)=l, f(π2)=i, f(π3)=k
+nearest outer blocks o(π2)=π1,o(π3)=π2
+Figure 1. A colored non-crossing partition
+Definition 2.1. Let (π,f) be a colored non-crossing partition with blocks as above,
+wheref∈Fr(π) and letB∈Mr(R) be given. For any 0 ≤j≤rwe define
+bj(π,f) =bj(π1,f)bj(π2,f)...bj(πk,f)
+where the functions bj:B(π,f)→Rare given by the following rules:
+(1)bj(πi,f) =bp,qiff(πi) =pandf(o(πi)) =q, where 0 ≤j≤r,
+(2)bj(πi,f) =bp,jiff(πi) =pandπidoes not have outer blocks, where 1 ≤j≤r,
+(3)b0(πi,f) =bp,piff(πi) =pandπidoes not have outer blocks.
+The indexj∈[r] inbj(π,f) can be interpreted as the color of the imaginary block
+(ifj= 0, then the imaginary block is not needed). Finally, NC2
+m=∅formodd and
+thus we shall understand in this case that the summation over π∈ NC2
+mgives zero.
+When we sum these products over all possible colorings, we obtain nu mbers
+bj(π) =/summationdisplay
+f∈Fr(π)bj(π,f)
+for any 0 ≤j≤r, which are used to express the limit laws of random pseudomatrices.
+Example 2.1. If we consider the partition πgiven by the diagram in Fig.1, we obtain
+b0(π) =/summationdisplay
+i,k,lbi,lbk,lbl,landbj(π) =/summationdisplay
+i,k,lbi,lbk,lbl,j
+wherej∈[r] and all indices run over [ r], which corresponds to all possible colorings
+fromFr(π) and is adequate for a square array B∈Mr(R). For other arrays, we obtain
+the same formulas except that range of the summation is smaller.
+Entries of the matrix Bwill be related to variances of matricially free random vari-
+ables obtained when computing their mixed moments. Therefore, it is natural to begin
+with an array ( ai,j) of random variables and assign a variable to each leg of the con-
+sidered partition. If the array is square, we assume that i,j∈[r], whereas in other
+cases the pairs ( i,j) belong to a proper subset J⊂[r]×[r] which includes the diagonal.
+To a given non-crossing partition π∈ NC2
+m, wheremis even, we can now associate a
+product of these variables in which indices can be interpreted as colo rs taken from the
+set [r]. These indices are related to each other in a natural way which refe rs to the way
+matricially free random variables can be multiplied to give a non-trivial c ontribution
+to the limit laws. The definition given below is based on this relation.
+Definition 2.2. We will say that the partition π∈ NC2
+misadaptedto a tuple of
+numbers (p1,q1,...,p m,qm) from the set [ r] ifASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 7
+(1) (pi,qi) = (pj,qj) whenever {i,j}is a block of π,
+(2)qk=po(k)wheneverk∈ R(π).
+The set of such partitions will be denoted NC2
+m(p1,q1,...,p m,qm).
+Ifπ∈ NC2
+m(p1,q1,...,p m,qm), then the tuple ( p1,q1,...,p m,qm) defines a unique
+coloring of both πand/hatwideπin which the block containing k∈ L(π) is colored by pkfor
+anykand the imaginary block is colored by qm.
+Example 2.2. Consider the partition πshown in Fig.1 and let p1,q1,...,p 6,q6∈[6]
+be given. Conditions (1)-(2) of Definition 2.2 lead to equations p1=p6,p2=p3,p4=p5
+andq2=q3=p1,q4=q5=p1,q1=q6, which gives four independent colors. Setting
+p1=l,p2=i,p4=kandq6=j, we obtain the coloring of Fig.1. Note in this context
+thatp1,p2,p4may be interpreted as colors of the left legs of πandq6as the color of
+the imaginary block. These equations lead to a product of variables o f the form
+ap1,q6ap2,p1ap2,p1ap4,p1ap4,p1ap1,q6,
+where each variable is taken from an array ( ap,q) of matricially free random variables
+and is associated with one inner-outer pair of blocks of /hatwideπ.
+We close this section with the symmetrized version of Definition 2.2 whic h will be
+needed in a combinatorial formula for the limit joint distribution of sym metric random
+blocks. The symmetrization is obtained by replacing ordered pairs by subsets.
+Definition 2.3. We will say thatthe partition π∈ NC2
+misadaptedto a tupleof subsets
+({p1,q1},...,{pm,qm}) of the set [ r] if
+(1){pi,qi}={pj,qj}whenever {i,j}is a block of π,
+(2){pk,qk}∩{po(k),qo(k)} /\e}a⊔io\slash=∅wheneverk∈ R(π).
+The set of such partitions will be denoted NC2
+m({p1,q1},...,{pm,qm}).
+Ifπ∈ NC2
+m({p1,q1},...,{pm,qm}), the tuple ( {p1,q1},...,{pm,qm}) defines color-
+ings of both πand/hatwideπ, calledadmissible , in which each block containing k∈ R(π) is
+colored bypkorqkand the imaginary block is colored by pmorqm.
+Example 2.3. Consider againthepartitionshowninFig.1andlet p1,q1,...,p 6,q6∈[6]
+be given. Conditions (1)-(2) of Definition 2.3 take the form
+(1){p1,q1}={p6,q6},{p2,q2}={p3,q3}and{p4,q4}={p5,q5},
+(2){p3,q3}∩{p1,q1} /\e}a⊔io\slash=∅and{p5,q5}∩{p1,q1} /\e}a⊔io\slash=∅.
+For instance, they are satisfied if ( p1,q1) = (q6,p6), (p2,q2) = (q3,p3), (p4,q4) = (q5,p5),
+withqi=pi+1for anyi= 1,...,5, which corresponds to the product of variables of the
+form
+ap1,p6ap6,p3ap3,p6ap6,p5ap5,p6ap6,p1,
+where we have four independent indices p1,p3,p5,p6, among which p1colors the imag-
+inary block and p3,p5,p6color the blocks of π(they are associated with the right legs
+ofπ). If we replace some indices from the set {p1,p3,p5,p6}by the corresponding qi’s,
+we obtain other admissible colorings and the associated products of variables.8 R. LENCZEWSKI
+3.Matricially free random variables
+Let us recall the definition of matricially free random variables as well as the main
+results of [5], where we refer the reader for details.
+LetAbe a unital algebra with an array ( Ai,j) of not necessarily unital subalgebras
+ofAand let (ϕi,j) be a family of states on A. Here, by a state on Awe understand a
+normalizedlinear functional. Further, we assumethat each Ai,jhasaninternal unit 1i,j,
+for which it holds that a1i,j= 1i,ja=afor anya∈ Ai,j, and that the unital subalgebra
+IofAgenerated by all internal units is commutative. If Ais a unital *-algebra, then,
+in addition, we require that the considered functionals are positive, the subalgebras are
+*-subalgebras and the internal units are projections.
+However, the definitions given below are slightly more general than t hose in [5].
+Namely, in contrast to the formulation given there, we do not assum e any particular
+form of the considered array ( ϕi,j). Instead, we will distinguish the diagonal states as
+those of the form ϕj,j, wherej∈I, but we will not assume that they all coincide. This
+will enable us to use the concept of matricial freeness for a wider cla ss of arrays, which
+turns out convenient in the formulation of our results.
+We shall use the subsets of ( I×I)mof the form
+Λm={((i1,i2),(i2,i3),...,(im,im+1)) : (i1,i2)/\e}a⊔io\slash= (i2,i3)/\e}a⊔io\slash=.../\e}a⊔io\slash= (im,im+1)},
+wherem∈N, with their union denoted Λ =/uniontext∞
+m=1Λm.
+Definition 3.1. We say that (1 i,j) is amatricially free array of units associated with
+(Ai,j) and (ϕi,j) if for any diagonal state ϕit holds that
+(1)ϕ(u1au2) =ϕ(u1)ϕ(a)ϕ(u2) for anya∈ Aandu1,u2∈ I,
+(2) ifak∈ Aik,jk∩Kerϕik,jk, where 1<k≤m, then
+ϕ(a1i1,j1a2...am) =/braceleftbiggϕ(aa2...an) if ((i1,j1),...,(im,jm))∈Λ
+0 otherwise.
+wherea∈ Ais arbitrary and ( i1,j1)/\e}a⊔io\slash=.../\e}a⊔io\slash= (im,jm).
+Definition 3.2. We say that ( Ai,j) ismatricially free with respect to ( ϕi,j) if
+(1) for any ak∈Kerϕik,jk∩Aik,jk, wherek∈[m] and (i1,j1)/\e}a⊔io\slash=.../\e}a⊔io\slash= (im,jm), and
+for any diagonal state ϕ, it holds that
+ϕ(a1a2...am) = 0
+(2) (1 i,j) is a matricially free array of units associated with ( Ai,j) and (ϕi,j).
+Definition 3.3. The array of variables ( ai,j) in a unital algebra (*-algebra) Awill be
+calledmatricially free (*-matricially free ) with respect to ( ϕi,j) if there exists an array
+of elements (projections) (1 i,j) which is a matricially free array of units associated with
+Aand (ϕi,j) and such that the array of algebras (*-algebras) generated by ai,jand 1i,j,
+respectively, is matricially free with respect to ( ϕi,j).
+Slightly less general was the setting given in [5], where we assumed tha t all diagonal
+states coincide with some distinguished state ϕand the off-diagonals states agree withASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 9
+a family of additional states ( ϕj)j∈I, calledconditions associated with ϕ, which are
+defined by
+ϕj(a) =ϕ(cjabj)
+for somecj,bj∈ Aj,j∩Kerϕsuch thatϕ(cjbj) = 1 (ifAis a *-algebra, cj=b∗
+j). In this
+setting, it is natural to assume that ϕandϕj’s are normalized according to
+ϕ(1i,j) =δi,jandϕj(1i,k) =δj,k
+for anyi,j,k. However, in this paper we will also use other arrays, for instance s uch in
+which all states in the j-th column agree with some ϕj, where (ϕj)j∈Iis a given family
+of states on Awhich are not a priori assumed to be conditions associated with some
+distinguished state. This motivates the following definition.
+Definition 3.4. Letϕandϕj,j∈I, be states on Aand let (ϕi,j), where (i,j)∈Jand
+∆⊆J⊆I×I, be an array of states on A. Ifϕj,j=ϕandϕi,j=ϕjfor any (i,j)∈J,
+we will say ( ϕi,j) isdefined byϕand the family (ϕj)j∈I. In turn, if ϕi,j=ϕjfor any
+(i,j)∈J, we will say that ( ϕi,j) isdefined by the family (ϕj)j∈I.
+In this context, let us remark that a Hilbert space setting, similar to that in [5], can
+be given for a family ( ϕj)j∈Iof product states on the unital *-algebra
+A:=⊔i,jAi,j
+which are not defined as conditions associated with some distinguishe d state on A.
+Here,⊔i,jAi,jstands for the free product of C∗-algebras without identification of units
+which is assumed to contain the empty word playing the role of the alge bra unit.
+In this ‘tracial’ framework, if ( Hi,j,πi,j,ξi,j) is the array of GNS triples associated
+with an array ( Ai,j,ϕi,j) of noncommutative probability spaces, the underlying product
+Hilbert spaces are of the form
+Hj=
+Cξj,j⊕∞/circleplusdisplay
+m=1/circleplusdisplay
+(i1,i2)/ne}ationslash=.../ne}ationslash=(im−1,j)/ne}ationslash=(j,j)H0
+i1,i2⊗...⊗H0
+j,j
+
+wherej∈IandH0
+i,jis the orthocomplement of Cξi,jinHi,jfor eachi,j, and their
+direct sum is
+H=/circleplusdisplay
+j∈IHj,
+with the canonical inner product used in all direct sums. The space Hreplaces the
+matricially free product of Hilbert spaces ∗M
+i,j(Hi,j,ξi,j) used in [5].
+Nevertheless, Hcan be embedded in a larger matricially free product of Hilbert
+spaces, namely ( H′,ξ) =∗M
+i,j(H′
+i,j,ξ′
+i,j), where
+H′
+i,j=/braceleftbiggHi,jifi/\e}a⊔io\slash=j
+Hj,j⊕Cξ′
+j,jifi=jandξ′
+i,j=/braceleftbiggξi,jifi/\e}a⊔io\slash=j
+ξ′
+j,jifi=j
+andξ′
+j,jis an additional unit vector for each j∈I. In order to define appropriate
+product states on Aassociated with vectors ξj,j, we first trivially extend each cyclic
+representation πj,j:Aj,j→B(Hj,j) to a non-cyclic representation π′
+j,j:Aj,j→B(H′
+j,j),
+and we keep the off-diagonal representations unchanged, namely π′
+i,j=πi,jfori/\e}a⊔io\slash=j.10 R. LENCZEWSKI
+Then, we take the free product λ′=∗M
+i,jπ′
+i,jand observe that His left invariant by λ′(a)
+for anya∈ A. This allows us to define the unital *-representation
+λ:A →B(H) asλ=λ′|H
+which then leads to product states ϕjonAdefined by
+ϕj=ηj◦λ,
+whereηjis the vector state on B(H) associated with ξj,j, wherej∈I. An equivalent
+formulation in terms of partial isometries leading to the product sta tesϕjcan also be
+given.
+Thepropositiongivenbelowprovides themainmotivationforallowingam oregeneral
+class of arrays in the definition of matricial freeness.
+Proposition 3.1. The array of C∗-algebras (Ai,j)viewed as *-subalgebras of A=
+⊔i,jAi,jis matricially free with respect to the array (ψi,j)of states on Adefined by the
+family(ϕj)j∈Iintroduced above.
+Proof.In fact, it can be seen that an analog of [5, Proposition 2.3] holds for the states
+ϕj:=ηj◦λdefined above. However, in this case it suffices to take ak∈ Aik,jk, where
+k∈[n] and (i1,j1)/\e}a⊔io\slash=.../\e}a⊔io\slash= (in,jn) (thus, we do not assume that ( i1,j1)/\e}a⊔io\slash= (j,j)/\e}a⊔io\slash=
+(in,jn)). Ifak∈Kerϕik,jkfork∈[n], then
+ϕj(a1a2...an) = 0
+for eachj. Moreover, if ar= 1ir,jrandam∈Kerϕim,jmforr<m≤n, then
+ϕj(a1...an) =/braceleftbiggϕj(a1...ar−1ar+1...an) if ((ir,jr),...,(in,jn))∈Λ
+0 otherwise.
+Finally, for any a∈ A,u1,u2∈ Iandi,j,k∈I, it holds that
+ϕj(u1au2) =ϕj(u1)ϕj(a)ϕj(u2) andϕj(1i,k) =δj,k.
+All these factsimply that ( Ai,j) is matricially freewith respect to thearray ( ψi,j), where
+ψi,j=ϕjfor anyi,j. /squaresolid
+Remark 3.1. Inthiscontext, let usobserve thatif( Ai,j)ismatriciallyfreewithrespect
+to the array defined by ϕand the associated conditions ( ϕj)j∈I, then (Ai,j) is not, in
+general, matricially free with respect to the array defined by ( ϕj)j∈Isince condition 1
+of Definition 3.2 does not need to hold if ϕis replaced by ϕjand we take a1,am∈ Aj,j.
+Let us assume now that for any natural nwe have an n-dimensional square array
+of self-adjoint variables ( Xi,j(n)) in a unital *-algebra A(n) equipped with an array
+of states (ϕi,j(n)) defined by a family of states ( ϕj(n))1≤j≤n. Blocks are defined by
+partitioning the set {1,2,...,n}into disjoint non-empty subsets,
+[n] =N1∪N2∪...∪Nr
+wherer∈N, with dependence on nsupressed in the notation, whose sizes increase as
+n→ ∞so thatnj/n→dj, wherenjis the cardinality of Njfor anyj∈[r]. Then the
+numbersdjform a diagonal matrix Dof trace one called the dimension matrix .
+Concerning the distributions of the considered arrays, we assume thatASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 11
+(A1) (Xi,j(n)) is matricially free with respect to ( ϕi,j(n)) for anyn∈N,
+(A2) the variables have zero expectations,
+ϕi,j(n)(Xi,j(n)) = 0
+for alli,j∈[n] andn∈N,
+(A3) their variances are block-identical and are of order 1 /n, namely
+ϕi,j(n)(X2
+i,j(n)) =up,q
+n
+for anyi∈Np,j∈Nq, where each up,qis a non-negative real number,
+(A4) their moments are uniformly bounded, i.e. ∀m∃Mm≥0 such that
+|ϕi,j(n)(Xm
+i,j(n))| ≤Mm
+nm/2
+for alli,j∈[n] andn∈N.
+In particular, if the distributions of the Xi,j(n)’s in the states ϕi,j(n) areblock-identical,
+assumptions (A3)-(A4) are satisfied.
+Following [5], where we studied limit distributions of random pseudomatr ices under
+ψ(n), we consider now normalized partial traces
+ψk(n) =1
+nk/summationdisplay
+j∈Nkϕj(n)
+wherek∈[r] andn∈N. Easy modifications of the proofs given in [5, Lemma 6.1] and
+[5, Theorem 6.1] lead to combinatorial formulas for the limit distributio ns under partial
+traces given above.
+Theorem 3.1. [5]Under assumptions (A1)-(A4), the limit distributions of ra ndom
+pseudomatrices under partial traces have the form
+lim
+n→∞ψk(n)(Sm(n)) =/summationdisplay
+π∈NC2
+mbk(π)
+wherek∈[r],m∈NandB=DU, withDbeing the dimension matrix. Consequently,
+ψ(n)(Sm(n))converges to/summationtext
+π∈NC2
+mb(π)asn→ ∞, whereb(π) =/summationtextr
+k=1dkbk(π).
+The above result refers to the ‘tracial’ framework which reminds th e limit theorem
+for random matrices and is of main interest to us in this work. Howeve r, we have also
+shown in [5] that a similar result holds for the ‘standard’ framework w hich reminds the
+central limit theorem and involves the distributions of random pseud omatrices in the
+distinguished states ϕ(n). This can be phrased as follows.
+Theorem 3.2. [5]Under assumptions (A1)-(A4) for arrays of states (ϕi,j(n))defined
+by a distinguished state ϕ(n)and the associated conditions (ϕj(n))1≤j≤nfor eachn,
+lim
+n→∞ϕ(n)(Sm(n)) =/summationdisplay
+π∈NC2
+mb0(π)
+wherem∈NandB=DU.12 R. LENCZEWSKI
+In the sequel we will derive Hilbert space realizations of the limit joint d istributions
+of random pseudomatrices S(n) and their blocks under ϕ(n),ψk(n) andψ(n). Interest-
+ingly enough, all these realizations are given on the same type of Hilbe rt space which
+is a matricially free product of Fock spaces.
+4.Matricially free Gaussian operators
+In this Section we shall introduce self-adjoint operators which play the role of matri-
+cially free Gaussian operators living in the matricially free product of F ock spaces.
+RecallthatbythebooleanandfreeFockspacesovertheHilbertsp aceH, respectively,
+we understand the direct sums
+F0(H) =Cξ⊕HandF(H) =Cξ⊕∞/circleplusdisplay
+m=1H⊗m,
+whereξis a unit vector, endowed with the canonical inner products. We sha ll use them
+to define an array of Fock spaces and their matricially free product .
+Definition 4.1. Let (Hi,j,ξi,j) be an array of Hilbert spaces with distinguished unit
+vectors. By the matricially free product of (Hi,j,ξi,j) we understand the pair ( H,ξ),
+where
+H=Cξ⊕∞/circleplusdisplay
+m=1/circleplusdisplay
+(i1,i2)/ne}ationslash=.../ne}ationslash=(im,im)H0
+i1,i2⊗H0
+i2,i3⊗...⊗H0
+im,im,
+withH0
+i,j=Hi,j⊖Cξi,jandξbeing a unit vector, with the canonical inner product.
+We denote it ( H,ξ) =∗M
+i,j(Hi,j,ξi,j).
+We already know that the matricially free Fock space, which is a matric ially free
+analog of the free Fock space, is the matricially free product of an a rray of free Fock
+spaces [5]. Nevertheless, in order to find Hilbert space realizations o f the limit laws
+of Theorems 3.1 and 3.2, it suffices to take a matricially free product o f an array of
+Fock spaces, in which free Fock spaces are put only on the diagonal and the boolean
+Fock spaces elsewhere. Clearly, we obtain a truncation of the matr icially free Fock
+space in this fashion. This structure is related to the difference bet ween diagonal and
+off-diagonal random variables and is related to the difference betwe en multiplication of
+diagonal and off-diagonal blocks of usual matrices.
+Definition 4.2. By thematricially free-boolean Fock space over the array /hatwideH= (Hi,j)
+we shall understand the matricially free product
+(F,ξ) =∗M
+i,j(Fi,j,ξi,j),whereFi,j=/braceleftbiggF(Hj,j) ifi=j
+F0(Hi,j) ifi/\e}a⊔io\slash=j
+andξi,jdenotes the distinguished unit vector in Fi,j.
+Remark 4.1. If we have a square array /hatwideH= (Hi,j), where Hi,j∼=Hifor anyi,j∈I
+and (Hi) is a family of Hilbert spaces, then we have a natural isomorphism
+F∼=F(/circleplusdisplay
+jHj)ASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 13
+since each tensor product
+(H⊗(n1−1)
+i1,i1⊗Hi1,i2⊗H⊗(n2−1)
+i2,i2)⊗...⊗(H⊗(nm−1−1)
+im−1,im−1⊗Him−1,im⊗H⊗nm
+im,im)
+is isomorphic to
+H⊗n1
+i1⊗H⊗n2
+i2...⊗H⊗nm
+im
+for anyi1/\e}a⊔io\slash=i2/\e}a⊔io\slash=.../\e}a⊔io\slash=imandm,n1,n2,...,n m∈N. Thus, Fis in this case also
+isomorphic to the strongly matricially free Fock space R(/hatwideH) introduced in [5].
+Example 4.1. A simple example of a matricially free-boolean Fock space is the matri-
+cially free product of the two-dimensional array of the form
+(Fi,j) =/parenleftbiggl2(G1,1)l2(G1,2)
+l2(G2,1)l2(G2,2)/parenrightbigg
+,
+whereGj,j=FS(gj,j), the free semigroup on one generator gj,j, wherej∈ {1,2}, and
+Gi,j=Z2with the generator denoted gi,jfori/\e}a⊔io\slash=j. ThenF=l2(G), whereGis the
+‘matricially free product of semigroups’ Gi,j, by which we understand the subset of their
+free product ∗i,jGi,jgiven by the union
+G=∞/uniondisplay
+n=0G(n)
+of disjoint subsets, where G(n)consists of words of type g1g2...gnwithgkbeing
+an element of G0
+ik,jk, whereG0
+i,j:=Gi,j\ {ǫi,j}, whereǫi,jis the unit in Gi,j, with
+((i1,j1),...,(in,jn))∈Λ andin=jn. For instance,
+G(0)={e},
+G(1)={gk
+1,1, gk
+2,2:k∈N},
+G(2)={g2,1gk
+1,1, g1,2gk
+2,2:k∈N},
+G(3)={gk
+2,2g2,1gm
+1,1, gk
+1,1g1,2gm
+2,2, g1,2g2,1gm
+1,1, g2,1g1,2gm
+2,2:k,m∈N},
+etc., withtheremainingsubsetsconsistingofwordsbuiltfrom‘matric iallyfreeproducts’
+of powers of the generators, where the diagonal generators ad mit all natural powers,
+whereas the off-diagonal ones admit only the powers equal to one.
+Definition 4.3. LetA= (αi,j) be a diagonal-containing array of positive real numbers
+and let ( Hi,j) = (Cei,j) be the associated array of Hilbert spaces. By the matricially
+free creation operators associated with Awe understand operators of the form
+ςi,j=αi,jτ∗ℓ(ei,j)τ,
+whereτ:F → F(/circleplustext
+i,jHi,j) is the canonical embedding and the ℓ(ei,j)’s denote the
+canonical free creation operators. By the matricially free annihilation operators and the
+matricially free Gaussian operators we understand their adjoints ς∗
+i,jand sums denoted
+ζi,j=ςi,j+ς∗
+i,j, respectively.
+We shall assume now that Ais a diagonal-containing subarray of a finite square array
+and that it is indexed by the set J(thus, ∆ ⊆J⊆I×I). Moreover, it is convenient
+to assume that Ais the square root of another array Btaken entrywise, i.e.
+αi,j=/radicalbig
+bi,jwrittenA=√
+B14 R. LENCZEWSKI
+whereBcan be assumed to be a subarray of a square array. The dependen ce onAof
+the operators defined above is supressed in our notations.
+We shall prove below that the *-algebras Ai,j, each generated by ςi,jand suitably
+defined unit 1 i,j, respectively, are matricially free with respect to a suitably defined
+array of states on B(F). Namely, 1 i,jis defined as the projection onto the subspace of
+Fonto which the *-algebra generated by the creation operator ςi,jacts non-trivially.
+To be more precise, let us introduce projections si,jandri,jfor any (i,j)∈Jwhich give
+an orthogonal decomposition of 1 i,j, i.e. 1 i,j=ri,j+si,jandri,jsi,j= 1. Namely, si,jis
+the canonical projection onto the subspace of Fspanned by tensors which begin with
+ei,jfor any (i,j)∈J. In turn,ri,jis the canonical projection onto the subspace of F
+spanned bytensors which beginwith ej,kforsomekifi/\e}a⊔io\slash=j, whereasrj,jisthecanonical
+projection onto the subspace spanned by the vacuum vector and tensors which begin
+withej,kfork/\e}a⊔io\slash=j, where (j,k)∈J.
+Two types of transformations on the considered operators will be performed: trunca-
+tions and symmetrizations. Here, we shall introduce truncated matricially free creation
+operators andtruncated units by
+℘i,j=ςi,jPandti,j= 1i,jP
+respectively, where Pis the canonical projection onto F ⊖CΩ. The truncated ma-
+tricially free annihilation operators andtruncated matricially free Gaussian operators ,
+respectively, will be denoted by ℘∗
+i,jandωi,j=℘i,j+℘∗
+i,jfor anyi,j. For uniformity,
+one can put Pon both sides of the creation, annihilation and unit operators.
+Finally, in the case of finite dimensional arrays, which will be considere d from now
+on, we use the same symbol to denote the sum of operators in a give n array, like
+ζ=/summationdisplay
+(i,j)∈Jζi,jandω=/summationdisplay
+(i,j)∈Jωi,j,
+called the Gaussian pseudomatrix and the truncated Gaussian pseudomatrix , respec-
+tively. All adjoints are denoted in the usual way. Note that all thes e operators are
+bounded since all considered sums are finite.
+Proposition 4.1. Let(ϕi,j)be the array of states on B(F)defined by the vacuum state
+ϕand the family (ψj)j∈I, whereψjis the vector state associated with ej,jfor anyj∈I.
+Then
+(1)theϕ-distribution of ζj,jis the semicircle law of radius 2αj,jfor anyj,
+(2)theψj-distribution of ζi,jis the Bernoulli law concentrated at ±√
+2αi,jfor any
+i/\e}a⊔io\slash=j,
+(3)the array (ζi,j)is matricially free with respect to (ϕi,j).
+Proof.Thefirsttwoclaimsfolloweasilyfromthedefinitionsoftheoperators involved.
+Next, instead of proving the third claim, we will prove a slightly more ge neral result
+that the array ( Ai,j), where Ai,j=C/a\}bracke⊔le{⊔ςi,j,ς∗
+i,j,1i,j/a\}bracke⊔ri}h⊔for any (i,j)∈J, is matricially free
+with respect to ( ϕi,j). Essentially, we proceed as inthe free case [9], but we have slightly
+more complicated relations between the operators involved (see als o Proposition 4.2 of
+[5]). In particular, one has to treat diagonal and off-diagonal suba lgebras separately.ASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 15
+Let us consider first the off-diagonal case, when the creation and annihilation operators
+satisfy relations
+ς∗
+i,jςi,j=bi,jri,jandς2
+i,j= 0, ς∗2
+i,j= 0
+for anyi/\e}a⊔io\slash=j. In that case we have additional relations
+ri,jςi,j= 0, ςi,jri,j=ςi,j, si,jςi,j=ςi,j, ςi,jsi,j= 0.
+Using all these relations and their adjoints, as well as equations
+ψj(ri,j) = 1, ψj(si,j) = 0
+we deduce that an arbitrary noncommutative polynomial from Ai,j∩Ker(ψj), where
+i/\e}a⊔io\slash=j, is spanned by ςi,j, ς∗
+i,j,andsi,j. In the diagonal case, the situation is similar
+to that in the free case since each pair of diagonal creation and ann ihilation operators
+satisfies the relation
+ς∗
+j,jςj,j=bj,j1j,jwithϕ(1j,j) = 1,
+and therefore any polynomial from Aj,j∩Ker(ϕ) is spanned by 1 j,jandςp
+j,jς∗q
+j,j, where
+p+q >0, for anyj. Moreover, si,jςk,l= 0 andςk,lsi,j=δl,iςk,lfor any (i,j)/\e}a⊔io\slash= (k,l).
+Hence, to prove matricial freeness of the array ( Ai,j) with respect to ( ϕi,j), it suffices
+to show that
+ϕ(ςp1
+i1,j1ς∗q1
+i1,j1...ςpm
+im,jmς∗qm
+im,jm) = 0
+for suitable powers p1,q1,...,p m,qmthat depend on whether the corresponding ope-
+ratorsarediagonalornot,since ϕ(si1,j1...sim,jm) = 0foranyoff-diagonalpairs( i1,j1)/\e}a⊔io\slash=
+.../\e}a⊔io\slash= (im,jm). At this point we can use the same inductive argument as in the free
+case [9], which gives the above ‘freeness condition’. Moreover, the d efinition of each 1 i,j
+shows that it is the projection onto the subspace onto which the *- algebra generated
+byςi,jacts non-trivially, which implies that the array (1 i,j) is the matricially free array
+of units. /squaresolid
+NotethatinProposition4.1each ψjisaconditionassociatedwith ϕsincethereexists
+bj∈ Aj,j∩Kerϕ, namelybj=ςj,j/αj,j, such that ψj(a) =ϕ(b∗
+jabj) for anya∈B(F).
+However, a similar result is obtained for arrays of states defined by the family ( ψj)j∈I,
+whereψjis now the vector state on B(F ⊖CΩ) associated with ej,jfor anyj∈I.
+Proposition 4.2. Let(ψi,j)be the array of states on B(F ⊖CΩ)defined by the family
+(ψj)j∈I, whereψjis the vector state associated with ej,jfor anyj. Then
+(1)theψj-distribution of ωj,jis the semicircle law of radius 2αj,jfor anyj,
+(2)theψj-distribution of ωi,jis the Bernoulli law concentrated at ±√
+2αi,jfor any
+i/\e}a⊔io\slash=j,
+(3)the array (ωi,j)is matricially free with respect to (ψi,j).
+Proof.The proof is similar to that of Proposition 4.1. /squaresolid16 R. LENCZEWSKI
+5.Fock-space realizations of limit distributions
+Using the matricially free Gaussian operators and their truncations , we will find
+realizations on Fof the limit distributions of Theorems 3.1-3.2.
+To eachπ∈ NC2
+mformeven we assign natural products of creation and annihilation
+operators. First, to each π∈ NC2
+mwe assign the sequence ǫ(π) = (ǫ1,ǫ2,...,ǫ m), where
+ǫj= 1 whenever j∈ R(π) andǫk=∗wheneverk∈ L(π). Then, to each π∈ NC2
+mwe
+assign products of creation and annihilation operators
+ς(π) =ςǫ1ςǫ2...ςǫmand℘j(π) =℘ǫ1℘ǫ2...℘ǫm
+for anyj∈[r], where (ǫ1,ǫ2,...,ǫ m) =ǫ(π).
+Thereis anicerelationbetween theexpectations ofthese product sandnumbers bj(π)
+defined in Section 2. In the case of b0(π) we will use the expectations in the vacuum
+stateϕand for the remaining j’s we takeψj’s associated with vectors ej,j. Finally, to
+obtainb(π), we will use the convex linear combination of states of the form
+ψ=r/summationdisplay
+j=1djψj
+where numbers d1,d2,...,d rare taken from the dimension matrix D.
+We are ready to give Fock-space realizations of the limit distributions of random
+pseudomatrices in terms of the distributions of Gaussian and trunc ated Gaussian pseu-
+domatrices.
+Lemma 5.1. For the limit distributions of Theorem 3.2, it holds that
+/summationdisplay
+π∈NC2
+mb0(π) =ϕ(ζm)
+for anym∈N, whereζis the Gaussian pseudomatrix.
+Proof.Ifmisodd, bothsidesofthefirstequationareclearly zero. Therefore , suppose
+thatmis even. We have
+ϕ(ζm) =/summationdisplay
+ǫ1,ǫ2,...,ǫm∈{1,∗}ϕ(ςǫ1ςǫ2...ςǫm) =/summationdisplay
+π∈NC2
+mϕ(ς(π))
+where we used the fact that products of creation and annihilation o perators corre-
+sponding to sequences ( ǫ1,ǫ2,...,ǫ m) which are not associated with non-crossing pair
+partitions give zero contribution (this follows from the definition of t heςǫ
+i,j). The as-
+sertion of the lemma follows then from the combinatorial formula
+b0(π) =ϕ(ς(π))
+for eachπ∈ NC2
+m. To prove it, first observe that
+ϕ(ςǫ1
+p1,q1...ςǫm
+pm,qm) = 0 unless ( ǫ1,...,ǫ m) =ǫ(π)
+for someπ∈ NC2
+m. Now, ifπ∈ NC2
+m(p1,q1,...,p m,qm) for somep1,q1,...,p m,qm∈[r]
+and (ǫ1,...,ǫ m) =ǫ(π), then we claim that
+ϕ(ςǫ1
+p1,q1...ςǫm
+pm,qm) =b0(π,f)ASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 17
+wherefis the coloring of πdefined by pk,k∈ L(π). This claim can be proved
+by induction. Suppose that the last annihilation operator in this mixed moment is
+indexed by k, i.e.ǫk=∗. Then{k,k+ 1}must be a block which does not have any
+inner blocks. The corresponding product of operators is of the fo rmς∗
+ik,jkςik,jk. Two
+cases are possible:
+(1) Ifk=m−1 andpk=qk, thenς∗
+pk,qkςpk,qkacts on Ω and gives bpk,qkΩ.
+(2) Ifk<m−1 andqk+1=pk+2, thenς∗
+pk,qkςpk,qkacts on some simple tensor hand
+givesbpk,qkh. Here,qkcolors the nearest outer block of {k,k+1}.
+If we repeat this procedure for the product of operators corre sponding to the partition
+π′obtained from πby removing block {k,k+1}, we obtain the product of bp,q’s which
+appears in the combinatorial formula for b0(π,f) after a finite number of steps. If
+we fixπand sum over all p1,q1,...,p m,qmto whichπis adapted (Definition 2.2), we
+obtain in fact the summation over pk,k∈ L(π), equivalent to the summation over
+Fr(π), which proves the formula for b0(π) since mixed moments corresponding to those
+p1,q1,...,p m,qmto whichπis not adapted are equal to zero. This completes the proof.
+/squaresolid
+Lemma 5.2. For the limit distributions of Theorem 3.1, it holds that
+/summationdisplay
+π∈NC2
+mbk(π) =ψk(ωm)
+fork∈[r]andm∈Nand hence/summationtext
+π∈NC2
+mb(π) =ψ(ωm), whereωis the truncated
+Gaussian pseudomatrix.
+Proof.The proof is similar to that of Lemma 5.1 and is based on the analogous
+combinatorial formula
+bk(π) =ψk(℘(π))
+for any 1 ≤k≤randπ∈ NC2
+m, wheremis even and positive. The proof of that for-
+mulareducestoshowingthatif( ǫ1,...,ǫ m) =ǫ(π)forsomeπ∈ NC2
+m(p1,q1,...,p m,qm)
+andk=qm, then
+ψk(℘ǫ1
+p1,q1...℘ǫm
+pm,qm) =bk(π,f)
+wherefis the coloring of πdefined by pk,k∈ L(π), withkcoloring the imaginary
+block. As compared with the proof for ϕ, instead of acting on Ω, we need to act on
+ek,k. However, thanks to the projection PontoF⊖CΩ, which appears in the definition
+of the℘ǫ
+k,k, eachek,kplays the role of the vacuum vector with respect to the ℘ǫ
+i,kfor
+anyi,k, including the case when i=k, since℘∗
+k,kek,k= 0. Therefore, the arguments
+are similar to those for ϕ, except that ℘i,kek,kis non-zero for arbitrary i. In terms
+of diagrams, this means that each covering block of πcontributes bi,kfor various i’s.
+Finally, when we use the definition of ψand sum over k∈[r], we obtain for each k
+the extra factor dkwhich appears in the combinatorial formula for b(π). Consequently,
+b(π) =ψ(℘(π)) for any such π, which then leads to the formula for ψ(ωn). /squaresolid
+Example 5.1. Consider themoment correspondingtothepartition π∈ NC2
+4consisting
+of two blocks: {1,4}and{2,3}. LetA=√
+B∈M2(R) be the square root taken
+entrywise, where B=DUand we set α1,1=α,α1,2=β,α2,1=γ,α2,2=δ. Then
+b0(π) =α4+α2γ2+δ2β2+δ4.18 R. LENCZEWSKI
+On the other hand,
+ς2Ω =α2(e1,1⊗e1,1)+αγ(e2,1⊗e1,1)
++βδ(e1,2⊗e2,2)+δ2(e2,2⊗e2,2)
+and thus the ‘symmetric’ action of ( ς∗)2gives exactly b0(π)Ω.
+Example 5.2. For the same partition πas in the above example we obtain
+b(π) =d1(α4+α2γ2+β2γ2+γ2δ2)
++d2(α2β2+β2γ2+β2δ2+δ4)
+On the other hand,
+ψ(℘(π)) =d1ψ1(℘(π))+d2ψ2(℘(π))
+where℘(π) =℘∗℘∗℘℘, and we get
+℘2e1,1=γ(δ(e2,2⊗e2,1⊗e1,1)+β(e1,2⊗e2,1⊗e1,1))
++α(α(e1,1⊗e1,1⊗e1,1)+γ(e2,1⊗e1,1⊗e1,1))
+which, in view of the ‘symmetric’ action of the adjoint, gives
+d1ψ1(℘∗℘∗℘℘) =d1(γ2δ2+β2γ2+α2γ2+α4).
+A similar expression is obtained for d2ψ2(℘∗℘∗℘℘) withαinterchanged with δandβ
+interchanged with γ. The sum of both expressions agrees with b(π).
+6.Matricial freeness of blocks
+We show in this section that blocks of finite-dimensional arrays of ma tricially free
+random variables are matricially free with respect to an appropriate ly defined array
+of states. This is the analog of the property of free random variab les which says that
+families of sums of free random variables are free.
+Definition 6.1. Suppose that the n-dimensional array ( Xi,j) of variables from a unital
+algebraAis matricially free with respect to ( ϕi,j) and let [n] =N1∪N2∪...∪Nrbe
+a partition of [ n] ino disjoint non-empty subsets. The sums of the form
+Sp,q=/summationdisplay
+(i,j)∈Np×NqXi,j
+wherep,q∈[r], will be called blocksof the pseudomatrix S=/summationtext
+i,jXi,j. We will
+say that the array ( Xi,j) hasblock-identical distributions with respect to ( ϕi,j) if the
+ϕi,j-distributions of Xi,jare the same for all ( i,j)∈Np×Nqand fixedp,q∈[r].
+We need to define the associated array of block units which satisfy t he conditions of
+Definition 3.1. In order to construct them on the level of noncommu tative probability
+spaces, let us first return to the more intuitive framework of the m atricially free product
+ofrepresentationsofthearray( Ai,j)ofunitalC∗-algebrasonthematriciallyfreeproduct
+of Hilbert spaces
+(H,ξ) =∗M
+i,j(Hi,j,ξi,j)ASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 19
+and then carry over the corresponding definition to the algebraic f ramework of noncom-
+mutative probability spaces. We assume that each Ai,jis equipped with an internal
+unit 1 i,jand a state ϕi,j. It is the matricially free product of Hilbert spaces on which
+one defines the canonical *-representations of ( Ai,j,ϕi,j). Namely, let ( Hi,j,πi,j,ξi,j) be
+the associated GNS triples, so that ϕi,j(a) =/a\}bracke⊔le{⊔πi,j(a)ξi,j,ξi,j/a\}bracke⊔ri}h⊔for anya∈ Ai,j. Byλi,j
+we denote the canonical *-representation of ( Ai,j,ϕi,j) on (H,ξ). Using these represen-
+tations and appropriate partial isometries, we have defined in [5] th eir matricially free
+productλ=∗M
+i,jπi,jwhich maps ⊔i,jAi,jintoB(H).
+In this connection note that λdoes not send the units 1 i,jonto the unit of B(H). In
+fact,λ(1i,j) =ri,j+si,j, where
+ri,j= projection onto H(i,j)
+si,j= projection onto K(i,j)
+whereK(i,j) =H0
+i,j⊗ H(i,j) andH(i,j) denotes the subspace of Honto which the
+left free action of λ(Ai,j) is non-trivial. By the left free action of λ(Ai,j) we understand
+the left action onto the subspace spanned by simple tensors which d o not begin with
+vectors from H0
+i,jand, in addition, by ξifi=j. Clearly,ri,j⊥si,jfor any fixed i,jand
+thus their sum is the canonical projection onto the subspace of Honto which λ(Ai,j)
+acts non-trivially.
+In the propositions given below we state basic properties of the pro jections involved
+that are needed for the construction of block units. Note that th e case of card( I) = 1
+is trivial from the point of view of matricially free product structure s and that is why
+it is not treated. By C[ai,i∈I] we denote the unital algebra of polynomials in the
+commuting indeterminates ai,i∈I. Recall that Istands for the commutative unital
+algebra generated by the units λ(1i,j)≡1i,j.
+Proposition 6.1. Ifcard(I)>1, thenI=C[ri,j,si,j,pξ:i,j∈I], wherepξis the
+canonical projection onto Cξ.
+Proof.Observe that we have relations
+1i,i1j,j=pξ,1j,j1j,k=sj,kand 1 j,j1k,j=sj,j
+wheneveri/\e}a⊔io\slash=j/\e}a⊔io\slash=k, which implies that sj,k,pξ∈C[1i,j:i,j∈I] for anyj,k, and since
+rj,k= 1j,k−sj,k, it holds that C[ri,j,si,j,pξ:i,j∈I]⊆C[1i,j:i,j∈I]. The reverse
+inclusion is obviously true, hence the proof is completed. /squaresolid
+Proposition 6.2. Ifcard(I)>1andJ1,J2⊆Iare identical or disjoint, then the
+algebraC[1i,j:i∈J1,j∈J2]contains the canonical projection 1J1,J2onto the subspace
+ofHonto which the algebra generated by {λ(ai,j) :ai,j∈ Ai,j, i∈J1,j∈J2}acts
+non-trivially.
+Proof.We want to construct a projection 1J1,J2onto the subspace of Hthat would
+be suitable for the left action of the λ(ai,j), wherei∈J1andj∈J2. IfJ1∩J2=∅,
+then this subspace is the orthogonal direct sum
+HJ1,J2=/circleplusdisplay
+i∈J1,j∈J2K(i,j)⊕/circleplusdisplay
+j∈J2(K(j,j)⊕H(j,j)⊖Cξ)20 R. LENCZEWSKI
+and the associated canonical projection is
+1J1,J2=/summationdisplay
+i∈J1,j∈J2si,j+/summationdisplay
+j∈J2(sj,j+rj,j−pξ)
+=/summationdisplay
+i∈J1,j∈J2(1i,j−1j,j+1i,i1j,j)+/summationdisplay
+j∈J2(1j,j−1j,j1k,k)
+wherek∈J1is arbitrary. In turn, if J1=J2=J, then we obtain the orthogonal direct
+sum
+HJ,J=Cξ⊕/circleplusdisplay
+j∈J(K(j,j)⊕H(j,j)⊖Cξ)
+and the corresponding canonical projection takes the form
+1J,J=/summationdisplay
+j∈J(sj,j+rj,j−pξ)+pξ
+=/summationdisplay
+j∈J(1j,j−1j,j1k,k)+1k,k1l,l,
+where indices k,lare such that k/\e}a⊔io\slash=land otherwise are arbitrary elements of J. We
+have used the fact that pξ= 1i,i1j,jfor anyi/\e}a⊔io\slash=j. This completes the proof. /squaresolid
+The proof of the above proposition enables us to construct block u nits which are
+internal units in the commutative algebras C[Sp,q,1p,q], wherep,q∈[r], each generated
+by a block and the corresponding block unit. Namely, we set
+1p,q:=1Np,Nq
+for anyp,q∈[r], where the right hand side is defined in terms of 1 i,j’s in exactly the
+same way as in the proof of Proposition 6.2.
+It will also be useful to introduce the following terminology. Namely, if
+ak∈ Aik,jk∩Ker(ϕik,jk) for 1≤k≤nand ((i1,j1),...,(in,jn))∈Λ,
+wewill saythattheproduct a1a2...anisinthematricially free kernel form withrespect
+to (ϕi,j).
+Theorem 6.1. Let(ϕi,j)be the arrayof states on Adefined by the family (ϕj)1≤j≤n, and
+let(ψp,q)be defined by the family of associated normalized partial tra ces(ψq)1≤q≤r. If
+(Xi,j)is matricially free (strongly matricially free) and has blo ck-identical distributions
+with respect to (ϕi,j), then(Sp,q)is matricially free with respect to (ψp,q).
+Proof.We will assume that ( Xi,j) is matricially free since the proof for strong matri-
+cialfreenessisanalogous. Clearly, theunitalalgebrageneratedb y(1p,q)iscommutative.
+We claim that the array ( 1p,q) is a matricially free array of units associated with ( Sp,q)
+and (ψp,q). The proof of condition (1) of Definition 3.1 for ( ψp,q) follows easily from the
+same condition for ( ϕi,j). To prove condition (2), we need to evaluate mixed momeints
+of type
+ϕj(w1p,qw1w2...wm)
+wherewk∈Ker(ψpk,qk) is a polynomial in Spk,qkfor eachk∈[m] andj∈[n] and the
+productw1w2...wmis in the matricially free kernel form with respect to ( ψp,q). WeASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 21
+shall reduce the computations to moments of type ϕj(w1i,jv1v2...vh), wherev1v2...vh
+is in the (strongly) matricially free kernel form with respect to ( ϕi,j). For that purpose,
+let us express each power of Spk,qkwhich appears in wkin terms of variables which are
+in the kernels of the ϕi,j. This procedure, described in more detail below, is applied to
+wm,wm−1...,w1(in that order).
+Letw(X) =/summationtexts
+r=0crXrbe an arbitrary polynomial and let Sp,q=/summationtext
+(i,j)∈Np×NqXi,j.
+We decompose each positive power of Xi,jwhich appears in w(Sp,q) as
+Xn
+i,j= (Xn
+i,j)0+ϕi,j(Xn
+i,j)1i,j,
+wherei∈Np,j∈Nq. Using the fact that (1 i,j) is a matricially free array of units, we
+observe that in the computations of the mixed moments of the given type we can use,
+without loss of generality, polynomials of the form
+w(Sp,q) =s/summationdisplay
+r=1/summationdisplay
+i1,...,ir+1/summationdisplay
+n1,...,nrcn1,...,nr
+i1,...,ir,ir+1(X(n1)
+i1,i2)0...(X(nr)
+ir,ir+1)0+cp,q/summationdisplay
+(i,j)∈Np×Nq1i,j,
+with summations over i1,...,i r+1andn1,...,n rrun over some finite sets of natural
+numbers, with n1+...+nr≤deg(w), since the remaining terms will give zero con-
+tribution to the considered moment if the product of variables stan ding to the right of
+this polynomial is in the matricially free kernel form with respect to ( ϕi,j), which is the
+case if we carry out our computations going from the right to the lef t.
+Note that if p/\e}a⊔io\slash=q, thens= 1, but if p=q, thens≤deg(w). Let us also point out
+that thanks to our assumption that the array ( Xi,j) has block-identical distributions
+with respect to ( ϕi,j), the same constant cp,qstands by each 1 i,jfor anyi∈Np,j∈Nq,
+i.e. it does not depend on i,j. Moreover,
+ψp,q(/summationdisplay
+(i,j)∈Np×Nq1i,j) =1
+nq/summationdisplay
+k∈Nqϕk(/summationdisplay
+(i,j)∈Np×Nq1i,j) =1
+nq/summationdisplay
+i,jϕj(1i,j) =np
+for anyp,q, whereas the first sum in the above expression for w(Sp,q) belongs to
+Ker(ψp,q). These arguments lead us to the conclusion that in the computatio ns of
+mixed moments oftheconsidered typewe cantakeeach polynomial wktobeoftheform
+w(Spk,qk) given above, with cpk,qk= 0 for each k∈[m] since the product w1w2...wmis
+assumed to be in the matricially free kernel form with respect to ( ϕi,j).
+Therefore, each moment of type ϕj(w1p,qw1w2...wm) is a sum of moments of type
+ϕj(w1p,qv1v2...vh), where the product v1v2...vhis in the matricially free kernel form
+with respect to ( ϕi,j). Similarly, each moment of type ϕj(ww1w2...wm) is a corre-
+sponding sum of moments of type ϕj(wv1v2...vh) since the reduction described above
+does not depend on what stands before the product w1w2...wm. It remains to observe
+that under each ϕj, the block unit 1p,qacts as the projection onto the linear span of
+productsv1v2...vjwhich are in the matricially free kernel form with respect to ( ϕi,j)
+and begin with v1∈ Ai1,j1, wherei1∈Nq. The proof of that fact follows from the
+definition of 1p,qexpressing it in terms of units 1 i,kwhich, under ϕj, act as projections
+onto the linear span of v1v2...vmwhich are in the matricially free kernel form with
+respect to ( ϕi,j) and begin with v1∈ Ai1,j1, wherei1=k. Of course, we use here the
+fact that (p,q)/\e}a⊔io\slash= (p1,q1). This completes the proof of condition (2) of Definition 3.1.22 R. LENCZEWSKI
+Moreover, using the normalization conditions for 1 i,j’s, we obtain the normalization
+conditions
+ψr(1p,q) =1
+nr/summationdisplay
+j∈Nrϕj(1p,q) =δr,q
+which completes the proof that ( 1p,q) is a matricially free array of units.
+Finally, the proof of condition (1) of Definition 3.2 is similar to that of co ndition
+(2) of Definition 3.1 presented above and is based on reducing the co mputations of the
+mixed moments ϕj(w1w2...wn), wherew1w2...wnis in thematricially free kernel form
+with respect to ( ψp,q), to mixed moments of type ϕj(v1v2...vh), wherev1v2...vhis in
+the matricially free kernel form with respect to ( ϕi,j). This completes the proof. /squaresolid
+7.Asymptotic matricial freeness of blocks
+In this section we study the asymptotic joint distributions of blocks of random pseu-
+domatrices. Inparticular, weshowthattheyare‘asymptoticallyma triciallyfree’, which
+is a notion analogous to asymptotic freeness. This generalizes the r esults of Section 6,
+where we proved matricial freeness of blocks in the case when the a rray of matricially
+free variables has block-identical distributions.
+ForthatpurposewewillusearealizationontheFockspace F. Havingestablishedthe
+realization of the limit laws of random pseudomatrices S(n) under normalized partial
+traces in Lemma 5.1, it is natural to expect that it can be carried ove r to the level of
+blocksofS(n) of the form
+Sp,q(n) =/summationdisplay
+(i,j)∈Np×NqXi,j(n)
+for anyn, where we require that the arrays ( Xi,j(n)) of self-adjoint random variables
+satisfy the assumptions of Theorem 3.1.
+Before we proceed with examining the limit joint distribution of these b locks, we
+define the notion of asymptotic matricial freeness, following the an alogous notion of
+asymptotic freeness. By the block units we shall understand units 1p,q(n) defined for
+eachnin terms of 1 i,j(n)’s in exactly the same way as in Section 6. In addition to the
+statesϕ(n) andψ(n) on the algebras A(n), we will also use normalized partial traces
+ψq(n), whereq∈[r] andn∈N, defined as before in the case of fixed n. Informally,
+the statesψq(n) will play the role of conditions which converge to the conditions ψqof
+Section 4 as n→ ∞.
+Definition 7.1. Let (ηp,q(n)) be anr-dimensional array of functionals on the algebra of
+noncommutative polynomials C/a\}bracke⊔le{⊔Sp,q(n),1p,q(n),p,q∈[r]/a\}bracke⊔ri}h⊔. We will say that ( Sp,q(n)) is
+asymptotically matricially free with respect to ( ηp,q(n)) if these functionals have point-
+wise limits ( ηp,q) asn→ ∞with respect to which the limit array is matricially free.
+Theorem 7.1. Under the assumptions of Theorem 3.1, the joint ψq(n)-distributions of
+blocks and block units converge to the joint ψq-distributions of the truncated matricially
+free Gaussian operators and the truncated units, respectiv ely, asn→ ∞.
+Proof.This result is a refinement of Theorem 3.1. The combinatorial argume nts
+referring to non-crossing partitions used in [5, Lemma 6.1] can be re peated except thatASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 23
+since we now take products of the Sp,q(n)’s instead of powers of S(n), all indices run
+only over some of the subsets N1,N2,...,N rof the set [ n]. Moreover, it suffices to
+consider the case of meven, saym= 2s, since ifmis odd, both sides are zero, by
+standard arguments. We will first show that
+lim
+n→∞ψq(n)(Sp1,q1(n)...Spm,qm(n)) =ψq(ωp1,q1...ωpm,qm)
+for anyq/\e}a⊔io\slash= 0. Thus, for given qandp1,q1,...,p m,qm, in order to compute the left-hand
+side of the above equation, it suffices to take into account only the m ixed moments
+ϕj(n)(Xi1,j1(n)...Xim,jm(n))
+in which (ik,jk)∈Npk×Nqkfor anyk∈[m] andj∈Nq.
+However, in the limit n→ ∞, further reductions takes place. As in the case of ψ(n)-
+andϕ(n)-distributions of random pseudomatrices investigated in [5, Lemma 6.1], the
+sum of mixed moments of matricially free random variables in the state sϕj(n) which
+correspond to a partition which is not a non-crossing pair partition is O(1/√n).
+Moreover, it suffices to take into account the mixed moments of the above type
+which are associated with π∈ NC2
+m(p1,q1,...,p m,qm) forq=qmsince the remaining
+moments vanish. However, these moments are the same for all ( i1,j1,...,i m,jm), for
+whichπ∈ NC2
+m(i1,j1,...,i m,jm) andjm=jsince the variances of ( Xi,j(n)) are block-
+identical, namely vi,j(n) =up,q/nwhenever ( i,j)∈Np×Nq. Thus we can use the
+matrix elements of U= (up,q) to compute our mixed moments and express them as
+ϕj(n)(Xi1,j1(n)...Xim,jm(n)) =uq(π,f)
+ns,
+wherefis the unique coloring of π∈ NC2
+m(p1,q1,...,p m,qm) defined by indices pk,
+wherek∈ L(π), withqcoloring the imaginary block. Since the coloring fis uniquely
+determined by p1,q1,...,p m,qm, we will use the simplified notation u∗
+q(π) =uq(π,f).
+For suchπit then remains to enumerate the tuples ( i1,j1,...,i m,jm), for which
+π∈ NC2
+m(i1,j1,...,i m,jm). Sincethecoloringof πdefinedbythesetuplesisdetermined
+by independent indices associated with pl(1),...,p l(s), where
+L(π) ={l(1),l(2),...,l(s)},
+and byq=qm, this enumeration boils down to enumerating the indices il(1),...,i l(s)
+andj=jm. An appropriate inductive argument for this fact can be easily prov ided.
+Instead of giving a formal proof, we refer the reader to Example 2 .2 and to a similar
+argument given in the context of random matrices (see the proof o f Theorem 9.1).
+Therefore, for given π, the cardinality of the corresponding set of equal mixed moments
+is of the same order as
+Θn(π) =nqmnpl(1)...npl(s)=O(ns+1)
+asn→ ∞, whereqm=q. In fact, the exact cardinality may be slightly smaller
+than Θ n(π) since all independent indices il, wherel∈ L(π), must be different in this
+computationto give anon-zero contribution tothelimit. Intheform ula forΘ n(π) given
+above this is not the case since the mapping L(π)→[r] given byk→p(k) may not be
+injective. Nevertheless, we can substitute Θ n(π) for the cardinality of the considered24 R. LENCZEWSKI
+set when taking the limit n→ ∞since the difference between these two cardinalities
+isO(ns).
+Therefore, from each considered partition we obtain the contribu tion
+lim
+n→∞u∗
+q(π)Θn(π)
+nqns=u∗
+q(π)dpl(1)...dpl(s)
+where the division by nqcomes from the normalization of the partial trace ψq(n).
+Collecting contributions associated with all π∈ NC2
+m(p1,q1,...,p m,qm), we obtain
+lim
+n→∞ψq(n)(Sp1,q1(n)...Spm,qm(n)) =/summationdisplay
+π∈NC2
+m(p1,q1,...,pm,qm)b∗
+q(π)
+whereb∗
+q(π) =b∗
+q(π,f) corresponds to the unique coloring fofπdefined by the tuple
+(p1,q1,...,p m,qm) and to the matrix B=DU. Finally, the proof that the expression
+on the right-hand side is equal to ψq(ωp1,q1...ωpm,qm) is similar to that of Lemma 5.1,
+which completes the proof. /squaresolid
+Corollary 7.1. Under the assumptions of Theorem 7.1, the joint ψ(n)-distributions of
+blocks and block units converge to the joint ψ-distribution of the truncated matricially
+free Gaussian operators and truncated units, respectively , asn→ ∞.
+Proof.If we replace ψq(n) byψ(n) at the end of the proof of Theorem 7.1, we need
+to sum over q∈[r] the corresponding right-hand sides with weights dq, respectively,
+which result from normalizations of partial traces. Therefore, in o rder to obtain a
+combinatorial expression for the corresponding mixed moment, it s uffices to replace
+b∗
+q(π) in the above formula by b∗(π) =/summationtext
+qdqb∗
+q(π). This gives ψ(ωp1,q1...ωpm,qm), which
+proves our assertion. /squaresolid
+Corollary 7.2. Under the assumptions of Theorem 7.1, the array (Sp,q(n))is asymp-
+totically matricially free with respect to the array of stat es(ψp,q(n))onA(n)defined by
+the family of normalized partial traces (ψq(n))1≤q≤rasn→ ∞.
+Proof.It follows from Theorem 7.1 that the ψq(n)-distribution of ( Sp,q(n)) converges
+to theψq-distribution of ( ωp,q) for anyp,q. This, Theorem 7.1 and Proposition 4.2 give
+our assertion. /squaresolid
+Theorem 7.2. Under assumptions (A1)-(A4), the joint ϕ(n)-distributions of blocks
+and block units converge to the joint ϕ-distribution of the matricially free Gaussian
+operators and the associated units, respectively, as n→ ∞.
+Proof.The proof is similar to that of Theorem 7.1 and is based on the combinat orial
+arguments of type used in [5, Lemma 6.1] which lead to [5, Lemma 6.2]. /squaresolid
+Corollary 7.3. Under assumptions (A1)-(A4), the array (Sp,q(n))is asymptotically
+matricially free with respect to the array (ϕp,q(n))of states on A(n)defined byϕ(n)
+and the family of normalized partial traces (ψq(n))1≤q≤rasn→ ∞.
+Proof.The assertion follows from Theorem 7.2 and Proposition 4.1. /squaresolidASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 25
+8.Symmetric matricial freeness
+In order to compare the asymptotics of blocks of random pseudom atrices with that
+of symmetric random blocks, we shall now introduce a symmetric ana logue of matricial
+freeness.
+Roughly speaking, in this concept one replaces ordered pairs by two -element sets
+and assumes that the array ( Ai,j) of subalgebras of a unital algebra Acontains the
+diagonal and is symmetric. Moreover, each algebra Ai,jcontains an internal unit 1 i,j
+which agrees with 1 j,ifor any (i,j)∈J. ByIwe denote the unital algebra generated
+by the internal units and we assume that it is commutative. By ( ϕi,j) we denote an
+array of states on A.
+Instead of sets Λ m, we shall use their symmetric counterparts, namely subsets of Im
+of the form
+Πm={({i1,i2},{i2,i3},...,{im,im+1}) :{i1,i2} /\e}a⊔io\slash={i2,i3} /\e}a⊔io\slash=.../\e}a⊔io\slash={im,im+1}}
+wherem∈N, with their union denoted Π =/uniontext∞
+m=1Πm. The main difference between
+‘symmetric matricial freeness’ and matricial freeness is that in all d efinitions we have
+to use Π instead of Λ.
+Definition 8.1. We say that the array (1 i,j) is asymmetrically matricially free array
+of unitsassociated with ( Ai,j) and (ϕi,j) if for any diagonal state ϕit holds that
+(1)ϕ(u1au2) =ϕ(u1)ϕ(a)ϕ(u2) for anya∈ Aandu1,u2∈ I,
+(2) ifak∈ Aik,jk∩Kerϕik,jk, where 1<k≤m, then
+ϕ(a1i1,j1a2...am) =/braceleftbiggϕ(aa2...am) if ({i1,j1},...,{im,jm})∈Π
+0 otherwise.
+wherea∈ Ais arbitrary and {i1,j1} /\e}a⊔io\slash=.../\e}a⊔io\slash={im,jm}.
+Definition 8.2. We say that a symmetric array ( Ai,j) issymmetrically matricially free
+with respect to ( ϕi,j) if
+(1) for any ak∈Kerϕik,jk∩Aik,jk, wherek∈[m] and{i1,j1} /\e}a⊔io\slash=.../\e}a⊔io\slash={im,jm}, and
+for any diagonal state ϕit holds that
+ϕ(a1a2...am) = 0
+(2) (1 i,j) is a symmetrically matricially free array of units associated with ( Ai,j) and
+(ϕi,j).
+The array of variables ( ai,j) in a unital algebra Awill be called symmetrically matri-
+cially free with respect to ( ϕi,j) if there exists a symmetrically matricially free array of
+units (1 i,j) inAsuch that the array of algebras ( Ai,j), each generated by ai,j+aj,iand
+1i,j, respectively, is symmetrically matricially free with respect to ( ϕi,j). The definition
+of *-matricially free arrays of variables is similar to that of *-matricia lly free arrays.
+We shall need the symmetrized operators on the matricially free-bo olean Fock space
+Fof Section 4, like the /hatwideωi,j’s defined as
+/hatwideωi,j=/braceleftbiggωj,jifi=j
+ωi,j+ωj,iifi/\e}a⊔io\slash=j.26 R. LENCZEWSKI
+In a similar way we define /hatwideςi,j,/hatwideζi,j,/hatwide℘i,j, etc. All these arrays are associated with matrix
+A= (αi,j), which is supressed in the notation. In turn, the symmetrized unit s onF
+andF ⊖CΩ, are of the form
+/hatwide1i,j= 1i,j+1j,i−1i,j1j,iand/hatwideti,j=/hatwide1i,jP,
+respectively, for any i,j, where (1 i,j) is the array of canonical units on FandPstands
+for the projection onto F⊖CΩ. More explicitly, /hatwide1i,jis the projection onto the subspace
+ofFspanned by tensors which begin with ei,korej,kfor somek, and, in addition, by
+Ω ifi=j.
+Proposition 8.1. If the matrix A= (αi,j)is symmetric, then
+(1)theψj-distribution of /hatwideωi,jis the semicircle law of radius 2αi,jfor any(i,j),
+(2)the array (/hatwideωi,j)is symmetrically matricially free with respect to (ψi,j).
+Proof.Ifi=j, then the first claim is similar to (1) of Proposition 4.1 since the action
+ofωj,jontoej,jis similar to the action of ζj,jonto the vacuum vector. If i/\e}a⊔io\slash=j, then for
+even and positive m, it holds that
+ψj(/hatwideωm
+i,j) =/summationdisplay
+ǫ1...,ǫm∈{1,∗}/summationdisplay
+(i1,j1),...,(im,jm)∈{(i,j),(j,i)}ψj(ℓǫ1(ei1,j1)...ℓǫm(eim,jm))
+and there is a bijection between NC2
+mand products of the form ℓǫ1(ei1,j1)...ℓǫm(eim,jm)
+whose action onto ej,jis non-trivial. This bijection is obtained as follows. If π∈ NC2
+m
+is given and {l,r}is a block of π, wherel<r, thenǫl=∗,ǫr= 1 and (il,jl) = (ir,jr)
+with
+(ir,jr) =/braceleftbigg(jo(r),io(r)) if{l,r}has an outer block
+(i,j) otherwise
+whereo(r) is the right leg of the nearest outer block of {l,r}. It can be seen that
+this mapping is onto since in order to get a non-trivial action on ej,jof the considered
+product of operators, the action of each ℓ(ei,j) (corresponding to the right leg of some
+block) must befollowed by theactionof ℓ(ej,i) (corresponding to theright leg of another
+block) orℓ∗(ei,j) (corresponding to the left leg of the same block). In particular, a cting
+first withℓ(ei,j) on a vector from Fand then with ℓ(ei,j) orℓ∗(ej,i) gives zero. This
+completes the proof of (1) since the even moments of the semicircle law of radius 2 αi,j
+are given by Mm=αm
+i,jcm/2, wherecm/2=|NC2
+m|,m∈2N, are Catalan numbers. The
+proof of (2) is similar to that of (2) of Proposition 4.1. We show the slig htly more
+general result that the array /hatwideBi,j=C/a\}bracke⊔le{⊔/hatwide℘i,j,/hatwide℘∗
+i,j,/hatwideti,j/a\}bracke⊔ri}h⊔is symmetrically matricially free
+with respect to ( ψi,j), where we use relations
+/hatwide℘∗
+i,j/hatwide℘i,j=bi,j/hatwideti,j,
+and normalization ψj(/hatwideti,j) = 1 for any ( i,j)∈J. The details are left to the reader. /squaresolid
+For a given partition [ n] =N1∪N2∪...∪Nrof the form considered before, it is
+useful to introduce the notation
+Np,q= (Np×Nq)∪(Nq×Np)ASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 27
+for sets of pairs which label ‘symmetric blocks’ of random pseudoma trices. If we let
+n→ ∞and assume that blocks grow as before, we can obtain the asympto tic behavior
+of these symmetric blocks.
+Theorem 8.1. Under the assumptions of Theorem 7.1, the array of symmetric blocks
+given by
+Zp,q(n) :=/summationdisplay
+(i,j)∈Np,qXi,j(n)
+is asymptotically symmetrically matricially free with res pect to(ψp,q(n))asn→ ∞.
+Proof.Observing that
+Zp,q(n) =/braceleftbiggSp,p(n) ifp=q
+Sp,q(n)+Sq,p(n) otherwise,
+it suffices to use Theorem 7.1 and Proposition 8.1 to prove the assert ion. /squaresolid
+9.Symmetric random blocks
+We show in this section that the tracial asymptotics of random pseu domatrices is the
+same as the asymptotics of complex Gaussian random matrices and t hat this similarity
+can be carried over to the level of their symmetric random blocks.
+The context for the study of random matrices originated by Voicule scu [10] is the
+following. Let µbe a probability measure on some measurable space without atoms an d
+letL=/intersectiontext
+1≤p<∞Lp(µ) be endowed with the state expectation Egiven by integration
+with respect to µ. The *-algebra of n×nrandom matrices is Mn(L) =L⊗Mn(C) with
+the stateτ(n) =E⊗tr(n), where tr( n) is the normalized trace.
+In order to compare the asymptotics of the blocks of random matr ices with that
+of random pseudomatrices, we partition each set [ n] into disjoint non-empty intervals
+as in the case of pseudomatrices and we set again D= diag(d1,d2,...,d r) to be the
+associated diagonal dimension matrix.
+By acomplex Gaussian random matrix we understand a matrix ( Yi,j(n))1≤i,j≤n, in
+whichYi,j(n) =Yj,i(n) for anyi,j,nand
+{ReYi,j(n)|1≤i≤j≤n}∪{ImYi,j(n)|1≤i≤j≤n}
+is an independent set of Gaussian random variables. Its submatrice s of the form
+Tp,q(n) =/summationdisplay
+(i,j)∈Np,qYi,j(n)⊗ei,j(n)
+will be called symmetric random blocks , wherep,q∈[r], and{ei,j(n)|1≤i≤j≤n}is
+a system of matrix units.
+We will study the asymptotics of symmetric random blocks under som e natural as-
+sumptions. Our setting is very similar to that in [10,12] except that we will assume
+that the variances of |Yi,j(n)|, wherei,j∈[n] andnis fixed, are block-identical rather
+than identical. The case of non-Gaussian random matrices [2] and th e corresponding
+symmetric blocks can be treated in a similar way.
+In order to find a Hilbert space realization of the limit joint distribution of symmetric
+random blocks under τ(n), we will use the symmetrized truncated Gaussian operators28 R. LENCZEWSKI
+/hatwideωp,qonFand the state ψ=/summationtext
+jdjψjonB(F), where the dependence of /hatwideωp,q’s on the
+matrixB=DUis supressed in the notation.
+Theorem 9.1. Let(Yi,j(n))1≤i,j≤nbe a complexGaussian randommatrix for each n∈N
+such that
+(1)E(Yi,j(n)) = 0for anyi,j,n,
+(2)E(|Yi,j(n)|2) =up,q/nfor anyi∈Npandj∈Nqand anyn,
+whereU:= (up,q)∈Mr(R). Then it holds that
+lim
+n→∞τ(n)(Tp1,q1(n)...Tpm,qm(n)) =ψ(/hatwideωp1,q1.../hatwideωpm,qm)
+for anyp1,q1,...,p m,qm∈[r], where/hatwideωp,q’s are associated with B:=DU, whereDis
+the dimension matrix.
+Proof.OurproofreferstotheoriginalproofofVoiculescu [10,12], which isf ollowedby
+some combinatorial arguments referring tonon-crossing pair par titions(inour approach
+the variances may vary). Let τq(n) =E⊗trq, where tr q(A) = 1/nq/summationtext
+j∈NqAj,jis the
+normalized partial trace of A∈Mn(C) corresponding to q∈[r]. We have
+τq(n)(Tp1,q1(n)...Tpm,qm(n))
+=/summationdisplay
+(i1,j1)∈Np1,q1,...,(im,jm)∈Npm,qmE(Yi1,j1(n)...Yim,jm(n)) trq(ei1,j1...eim,jm).
+Since all terms are zero for modd, throughout the rest of the proof we assume that
+m= 2sfor somes∈N.
+An individual term of this sum is then non-zero only if
+j1=i2,j2=i3,...,j m=i1∈Nq
+and there exists a bijection γ: [m]→[m] such that
+γ2= id, γ(k)/\e}a⊔io\slash=kandiγ(k)=jk, jγ(k)=ik.
+Each such term is O(n−s−1) asn→ ∞. For convenience, we restrict our attention to
+the case when
+(i1,j1)∈Np1×Nq1,...,(im,jm)∈Npm×Nqm,
+since the remaining cases can be treated in a similar way, with some pi’s interchanged
+with the corresponding qi’s.
+The number of non-zero terms of the considered type is/summationtext
+γΘn(γ), whereγruns over
+the set of permutations of [ m] such that
+γ2= id, γ(k)/\e}a⊔io\slash=kandpk=qγ(k), qk=pγ(k)
+for anyk∈[m]. Ifq=p1, we obtain
+Θn(γ) = card{(i1,...,i m)∈Np1×...×Npm:ik=iγ(k)+1, ik+1=iγ(k)},
+where addition is modulo m. Denote by π(γ) the pair-partition of [ m] defned by γ.
+Ifπ(γ) is a crossing pair-partition, then Θ n(γ)≤O(ns), as in the free case, which
+will make the corresponding mixed moments disappear as n→ ∞since we still need
+to multiply these cardinalities by the corresponding expectations, w hich areO(1/ns),
+and divide them by nqdue to the normalization of the partial trace.ASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 29
+inner blocks inner blocks inner blocks ...in(2) in(4)im(2)
+in(2r)
+m(1)n(1) n(2)n(3) n(4) n(2r−1) n(2r)m(2)
+Figure 2. Inductive step involving subpartitions of π(γ).
+In turn, ifπ(γ)∈ NC2
+m\NC2
+m({p1,q1},...,{pm,qm}) orq/\e}a⊔io\slash=p1, then again Θ n(γ)≤
+O(ns). On the other hand, if π(γ)∈ NC2
+m({p1,q1},...,{pm,qm}) andq=p1, then the
+number of independent indices which enter in the computation of Θ n(γ) iss+1. We
+claim that for such γwe have
+Θn(γ) = card{(i1,ir(1),...,i r(s))∈Np1×Npr(1)×...×Npr(s)},
+where
+R(π(γ)) ={r(1),r(2),...,r(s)}.
+In fact, it is not difficult to show that the conditions which define Θ n(γ) can be reduced
+tos+1 independent indices i1,ir(1),...,i r(s)which can be interpreted as independent
+colors assuming arbitrary values from the corresponding intervals Np1,Npr(1),...,N pr(s),
+respectively, with index i1coloring the imaginary block.
+The proof of this fact essentially reduces to the inductive step invo lving subpartitions
+ofπ(γ) of the form shown in Fig.2, where the covering block can also be inter preted as
+the imaginary block associated with π(γ). We want to show that the conditions which
+define Θ n(γ) reduce all indices involved here to those associated with the right le gs of
+the blocks of the subpartition shown above. In particular, on the le vel of blocks of
+depth one, we have pairings
+γ(n(1)) =n(2),γ(n(3)) =n(4),...,γ(n(2r−1)) =n(2r)
+which, in view of the condition ik=iγ(k)+1, lead to the equation
+in(1)=in(3)=...=in(2r−1)=im(2)
+which says that the indices associated with the left legs of blocks of d epth one are equal
+to the index associated with the right leg of its covering block. No bloc ks of depth
+greater than one lead to new conditions involving im(2)and for that reason the index
+im(2)can be used to color the covering block of the considered subpartit ion. The same
+pattern is repeated for blocks of arbitrary depth which are cover ed by the same block.
+The same holds for blocks of zero depth, where the role of the cove ring block in the
+above reasoning is played by the imaginary block (in that case the imag inary block is
+colored byi1).
+This argument can be viewed as a proof of the inductive step of our c laim that the
+conditions involving the indices i1,i2,...,i m, used to define the numbers Θ n(γ) for the
+‘right’γreduce tos+1 independent indices associated with the right legs of the blocks
+ofπ. If the covering block is interpreted as the imaginary block, we obta in the starting30 R. LENCZEWSKI
+case of the induction. This proves our formula for Θ n(γ), from which we easily get
+Θn(γ) =np1npr(1)...npr(s).
+Let us remark here that not all tuples ( i1,ir(1),...,i r(s)) which contribute to Θ n(γ),
+but only those which are pairwise different, are actually used in our co mputation of
+the asymptotic joint distribution. If any two (or more) of these ind ices coincide, the
+corresponding contribution from the associated expectations is O(n−1) and therefore
+becomes irrelevant in the limit.
+For those independent indices which are pairwise different and corre spond to parti-
+tionsπ(γ)∈ NC2
+m({p1,q1},...,{pm,qm}), the corresponding expectation of Gaussian
+random variables E(Yi1,j1(n)...Yim,jm(n)) is the product of
+E(Yik,jk(n)Yiγ(k),jγ(k)(n)) =vik,jk(n)
+whereγ(k)∈ R(π(γ)), withiγ(k)=jkandjγ(k)=ik. Now, since ik=io(k), the above
+expectationisequal to vi,j(n), wherei=io(k)andj=iγ(k). Inotherwords, jisthecolor
+assigned to block {k,γ(k)}since we have shown before that we color the blocks with
+indices which correspond to their right legs and iis the color assigned to its covering
+block. This includes the case of blocks which do not have an outer bloc k, then the role
+of the latter is played by the conditional block colored by i1=jm.
+Now, using our assumption on the variance matrices, vi,j(n) =vj,i(n) =uq,p/nwhen-
+everi∈Np,j∈Nq. Therefore, taking into account all pairings and using the definition
+of symboluj(π,f) of Section 2, we obtain
+E(Yi1,j1(n)...Yim,jm(n)) =up1(π,f)
+ns
+for the relevant Gaussian expectations of variables associated wit hπ(γ), wheref∈
+Fr(π(γ)) is the coloring of the associated π(γ) defined by indices pr(1),pr(2),...,p r(s)
+assigned to the blocks of π(γ), with color p1assigned to the imaginary block.
+Collecting expectations corresponding to all π∈ NC2
+mand taking into account that
+lim
+n→∞Θn(γ)
+ns=dp1dpr(1)...dpr(s),
+we obtain
+lim
+n→∞τq(n)(Tp1,q1(n)...Tpm,qm(n)) =/summationdisplay
+π∈NC2
+m({p1,q1},...{pm,qm})/hatwidebq(π)
+where/hatwidebq(π) =/summationtext
+fbq(π,f) is the sum over all admissible colorings of πand eachbq(π,f)
+coresponds to the matrix B=DUand to the color qof the imaginary block.
+If we replace τq(n) byτ(n) in the above expression, we just need to sum over q∈[r]
+the corresponding right-hand sides with weights dq, respectively, which result from
+normalizations of partial traces. Therefore, in order to obtain th e required expression
+for the corresponding mixed moment, we need to replace /hatwidebq(π) by/hatwideb(π) =/summationtext
+qdq/hatwidebq(π),
+which completes the proof of the combinatorial formula.
+It remains to show that the mixed moment of the symmetrized Gauss ian operators
+in the state ψgives the apropriate sum of combinatorial expressions obtained ab ove.ASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 31
+This, inturn, issimilar totheproofofLemma 5.1. Therefore, our pro ofiscompleted. /squaresolid
+Remark 9.1. In this context, let us remark that the array of symmetric random blocks
+(Tp,q(n)) is not symmetrically matricially free with respect to ( τp,q(n)) for finite n.
+In order to see this, it suffices to compute some simple examples of mix ed moments
+of theTp,q(n)’s associated with crossing partitions. For instance, if n= 3 and the
+blocks are one-dimensional, then τ1(T1,2T2,3T3,1T1,2T2,3T3,1)/\e}a⊔io\slash= 0, where τ1=τ1(3) and
+Tp,q=Tp,q(3) for any 1 ≤p,q≤3, which shows that condition (1) of Definition 8.2 is
+not satisfied.
+However, in view of Proposition 8.1 and Theorem 9.1, we can expect th at symmetric
+random blocks are symmetrically matricially free as n→ ∞. The notion of asymptotic
+symmetric matricial freeness is analogous to that of asymptotic ma tricial freeness (see
+Definition 7.1). It remains to define appropriate arrays of block unit s. Thus, let
+/hatwide1p,q(n) =/summationdisplay
+j∈Np∪Nq1⊗ej,j(n)
+for anyp,q∈[r] and eachn∈N. It is easy to see that /hatwide1p,q(n) is an internal unit in the
+algebra generated by Tp,q(n) and/hatwide1p,q(n) for any given p,q∈[r] andn∈N.
+Theorem 9.2. Under the assumptions of Theorem 9.1, let (τp,q(n))be ther-dimensio-
+nal array of states on the algebra A(n) =L⊗Mn(C)defined by the normalized partial
+tracesτq(n), whereq∈[r], for anyn∈N. Then the array of symmetric random blocks
+(Tp,q(n))is asymptotically symmetrically matricially free with res pect to(τp,q(n)).
+Proof.In view of Proposition 8.1, it suffices to show that the τm(n)-distributions
+of blocksTp,q(n) and units /hatwide1p,q(n) converge to the ψm-distribution of the symmetrized
+Gaussians /hatwideωp,qand summetrized units /hatwide1p,qasn→ ∞for any given m. It suffices to
+verify that /hatwide1p,q(n) leaves invariant any vector of the form
+w1(Tp1,q1(n))...wk(Tpk,qk(n))em
+where the product of polynomials w1,...,w kis in the symmetrically matricially free
+kernel formwithrespect to thearray( τp,q(n)) and{p,q}∩{p1,q1} /\e}a⊔io\slash=∅andkills all other
+vectors. Thisisquiteeasytoobservesinceanyvector ofthistype isalinearcombination
+of the form/summationtext
+i∈Npαiai⊗ei+/summationtext
+j∈Nqβjbj⊗ej, whereai,bj∈Landαi,βj∈Cfor any
+i,j, and thus it is left invariant by /hatwide1p,q(n), where (ej) is the canonical basis in Cn./squaresolid
+10.Asymptotic freeness and asymptotic monotone independence
+If we make additional assumptions on the variance matrices ( vi,j(n)) of matricially
+free arrays ( Xi,j(n)), we obtain asymptotic freeness which refers to the rows of ran dom
+pseudomatrices. The proposition given below can also be viewed as an operatorial
+version of [5, Proposition 8.2].32 R. LENCZEWSKI
+Proposition 10.1. If the arrays (Xi,j(n))of Theorem 3.2 are square and have identical
+variances within blocks and rows, then the sums
+Sp(n) :=r/summationdisplay
+q=1Sp,q(n), where 1≤p≤r,
+are asymptotically free with respect to both ϕ(n)andψ(n)asn→ ∞.
+Proof.In view of Theorems 7.1-7.2, it suffices to show that the variables ζ1,...,ζ r
+are free with respect to ϕ, and thatω1,...,ω rare free with respect to ψ, where
+ζp=r/summationdisplay
+q=1ζp,qandωp=r/summationdisplay
+q=1ωp,q,
+with the notations of Section 4. If {e1,e2,...,e r}is an orthonormal set of vectors, we
+have the natural isomorphism
+κ:F → F(/circleplusdisplay
+qCeq)
+of Remark 4.1 and then, since αp,q=αp,pfor anyp,q∈[r], we have
+ζp=αp,pκ∗ω(ep)κ
+whereω(ep) =ℓ(ep)+ℓ∗(ep) is the canonical free Gaussian operator on F(/circleplustext
+qCeq) for
+anyp. Let us remark that ωpandω(ep) denote different operators in our notation.
+Sinceω(e1),...,ω(er) are free with respect to the vacuum state on B(F(/circleplustext
+qCeq)), the
+operatorsζ1,...,ζ rare free with respect to the vacuum state on B(F). Similarly,
+ωp=αp,pγ∗ω(ep)γ
+for anyp∈[r], whereγ=P◦κandPis the canonical projection from F(/circleplustext
+qCeq) onto
+the orthocomplement of CΩ. Now, since Pω(e1)P,...,Pω (er)Pare free with respect
+toψq(.) =/a\}bracke⊔le{⊔.eq,eq/a\}bracke⊔ri}h⊔for anyq, the operators ω1,...,ω rare free with respect to ψqfor
+anyq. Moreover, the ψp-distribution of ωpdoes not depend on psince the variances are
+assumed to be identical in each row. Therefore, ω1,...,ω rare free with respect to ψ.
+This completes the proof. /squaresolid
+Analogous results hold for block-triangular arrays and lead to mono tone indepen-
+dence (lower-block-triangular arrays) and anti-monotone indepe ndence (upper-block-
+triangular arrays). One has to remember that order is important f or these notions of
+independence and therefore in that case we use sequences of var iables instead of fam-
+ilies. We formulate the result only for lower-triangular arrays since t he case of upper-
+triangular arrays is completely analogous. This result can be viewed a s an operatorial
+version of [5, Proposition 8.3].
+Proposition 10.2. If the arrays (Xi,j(n))of Theorem 3.2 are lower-block-triangular
+and have identical variances within blocks and rows, then th e sums
+Sp(n) :=p/summationdisplay
+q=1Sp,q(n), where 1≤p≤r,ASYMPTOTIC PROPERTIES OF RANDOM MATRICES AND PSEUDOMATRIC ES 33
+are asymptotically monotone independent with respect to ϕ(n)asn→ ∞.
+Proof.Letζp=/summationtextp
+q=1ζp,q, wherep∈[r]. In view of Theorem 7.1, it suffices to show
+that the operators ζ1,...,ζ rare monotone independent with respect to ϕ. LetF1be
+the subspace of F(/circleplustext
+qCeq) of the form
+F1=Cξ⊕r/circleplusdisplay
+m=1/circleplusdisplay
+p1>...>p m/circleplusdisplay
+n1,...,nm∈NH⊗n1
+p1⊗...⊗H⊗nm
+pm
+whereHq=Ceqfor anyq∈[r] and{e1,e2,...,e r}is an orthonormal basis in some
+Hilbert space and denote by Q:F(/circleplustext
+qCeq)→ F1the corresponding canonical projec-
+tion. Observe that we have
+ζp=αp,pβ∗ω(ep)β
+whereβ=Q◦κandκis the same as in the proof of Proposition 10.1. Since the op-
+eratorsQω(e1)Q,...,Qω (er)Qare monotone independent with respect to the vacuum
+state on the free Fock space, the operators ζ1,...,ζ rare monotone independent with
+respect toϕ. This completes the proof. /squaresolid
+Finally, we obtain asymptotic booleanindependence of blocks of block -diagonal pseu-
+domatrices.
+Proposition 10.3. If the arrays (Xi,j(n))of Theorem 3.2 are block-diagonal and have
+identical variances within blocks, then the sums Sp,p(n), where1≤p≤r, are asymp-
+totically boolean independent with respect to ϕ(n)asn→ ∞.
+Proof.This is a simple consequence of Theorem 7.2 and the fact that the fam ily
+(ζp,p)1≤p≤ris boolean independent with respect to the vacuum state ϕonF./squaresolid
+References
+[1] Ph. Biane, Processes with free increments, Math. Z. 227(1998), 143-174.
+[2] K. Dykema, On certain free product factors via an extended ma trix model, J. Funct. Anal. 112
+(1993), 31-60.
+[3] R. Lenczewski, Decompositions of the free additive convolution, J. Funct. Anal. 246(2007), 330-
+365.
+[4] R. Lenczewski, Operators related to subordination for free mu ltiplicative convolutions, Indiana
+Univ. Math. J. ,57(2008), 1055-1103.
+[5] R. Lenczewski, Matricially free random variables, arXiv:0812.0488 v1 [math.OA], 2008.
+[6] N. Muraki, Monotonic independence, monotonic central limit theo rem and monotonic law ofsmall
+numbers, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4(2001), 39-58.
+[7] D. Shlyakhtenko, Random Gaussian band matrices and freeness with amalgamation, Int. Math.
+Res. Notices 20(1996), 1013-1025.
+[8] D. Voiculescu, Symmetries of some reduced free product C∗-algebras, Operator Algebras and
+Their Connections with Topology and Ergodic Theory, Lecture Note s in Mathematics, Vol. 1132,
+Springer Verlag, 1985, pp. 556-588.
+[9] D. Voiculescu, Lectures on free probability theory, Lectures on probability theory and statistics
+(Saint-Flour, 1998), 279-349, Lecture Notes in Math. 1738, Spr inger, Berlin, 2000.
+[10] D. Voiculescu, Limit laws for random matrices and free products ,Invent. Math. 104(1991), 201-
+220.
+[11] D. Voiculescu, The analogues of entropy and of Fisher’s informa tion measure in free probability
+theory, I, Commun. Math. Phys. 155(1993), 71-92.34 R. LENCZEWSKI
+[12] D. Voiculescu , K. Dykema, A. Nica, Free random variables , CRM Monograph Series, No.1,
+A.M.S., Providence, 1992.
+[13] E. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. Math. 67(1958),
+325-327.
+Romuald Lenczewski,
+Instytut Matematyki i Informatyki, Politechnika Wroc/suppress law ska,
+Wybrze ˙ze Wyspia ´nskiego 27, 50-370 Wroc/suppress law, Poland
+E-mail address :Romuald.Lenczewski@pwr.wroc.pl
\ No newline at end of file
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+arXiv:1001.0668v5  [math.GT]  8 Sep 2015THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL
+CHARTS
+ANKE D. POHL
+Abstract. It is well-known that reduced smooth orbifolds and proper
+effective foliation Lie groupoids form equivalent categori es. However, for
+certain recent lines of research, equivalence of categorie s is not sufficient.
+We propose a notion of maps between reduced smooth orbifolds and a
+definition of a category in terms of marked proper effective ét ale Lie
+groupoids such that the arising category of orbifolds is iso morphic (not
+only equivalent) to this groupoid category.
+1.Introduction
+Given a reduced orbifold (in local charts) and an orbifold at las represent-
+ing its orbifold structure it is well known how to construct ( in an explicit
+way) a proper effective foliation groupoid (orbifold groupo id) from these data
+(see, e.g., Haefliger [ 4] or the book by Moerdijk and Mrčun [ 8]). Over the
+years various authors (in particular, Moerdijk [ 7], Pronk [ 10]) used this link
+to provide a definition of a category of orbifolds by proposin g a definition of
+a category of orbifold groupoids, either as a 2-category or a s a bicategory of
+fractions. Lerman [ 5] provides a very good discussion of these approaches.
+These approaches have in common that the morphisms in the orb ifold cate-
+gory are only given implicitly.
+While the notion of isomorphisms between orbifolds is clear , that of more
+general maps allows variations. Various propositions for o rbifold maps in
+local charts have been made serving different purposes, e.g. , by Borzellino
+and Brunsden [ 2] or the Chen-Ruan good maps [ 3]. Unfortunately, neither of
+these definitions model the morphisms coming from the groupo id category.
+We remark that Lupercio and Uribe [ 6], even though widely believed, do not
+prove that Chen-Ruan good maps correspond to groupoid homom orphisms
+(a counterexample to the characterization of groupoid homo morphisms via
+good maps and thus to [ 6, Proposition 5.1.7] is provided by the unique map
+between any orbifold with at least two charts in any orbifold atlas and the
+0-dimensional connected reduced orbifold).
+All these proposed groupoid categories are just equivalent , not isomorphic,
+to the orbifold category. This is caused by the fact that the c onstruction
+mentioned above assigns the same groupoid to different (but i somorphic)
+orbifolds, and conversely various (Morita equivalent) gro upoids to the same
+orbifold. Moerdijk and Pronk [ 9] show that isomorphism classes of orbifolds
+correspond to Morita equivalence classes of orbifold group oids. For many
+2010Mathematics Subject Classification. Primary: 57R18, 22A22, Secondary: 58H05.
+Key words and phrases. reduced orbifolds, orbifold maps, groupoids, groupoid
+homomorphisms.
+12 ANKE POHL
+investigations about orbifolds, an equivalence of categor ies suffices to trans-
+late the problem to groupoids. However, one cannot investig ate e.g. the
+diffeomorphism group of an orbifold using any of the groupoid categories.
+In this article we provide a correct characterization of gro upoid homomor-
+phisms in local charts. We use the arising maps to define a geom etrically
+motivated notion of orbifold maps as certain equivalence cl asses. This al-
+lows us to define a geometrically natural orbifold category ( with orbifolds as
+objects). Characterizing these orbifold maps in terms of gr oupoid homomor-
+phisms enables us to define a category in terms of marked prope r effective
+étale Lie groupoids (which is not the classical one) which is isomorphic to
+the orbifold category.
+The results in this article are used by Schmeding [ 13] to show that the
+diffeomorphism group of a paracompact reduced orbifold can b e endowed
+with the structure of an (infinite-dimensional) Lie group. M oreover, this Lie
+group is even Ck-regular for any k∈N0∪{∞} anda fortiori regular in the
+sense of Milnor; we refer to [ 13] for details.
+For convenience we briefly comment on the outline of the paper . We
+start by reviewing the necessary background material on orb ifolds, groupoids,
+pseudogroups, and the well-known construction of a groupoi d from an orb-
+ifold and an orbifold atlas representing its orbifold struc ture. Groupoids
+which arise in this way will be called atlas groupoids . To overcome the prob-
+lem that different orbifolds are identified with the same atla s groupoid we
+introduce, in Section 3, a certain marking of atlas groupoids. It consists in
+attaching to an atlas groupoid a certain topological space a nd a certain ho-
+momorphism between its orbit space and the topological spac e. The general
+concept of marking already appeared in [ 7]. There, however, the relation
+between a marking of a groupoid and an orbifold atlas (in loca l charts) is
+not discussed. The specific marking of an atlas groupoid intr oduced here al-
+lows us to recover the orbifold. There is a natural notion of h omomorphisms
+between marked atlas groupoids. In Section 4we characterize these homo-
+morphisms in local charts. On the orbifold side, this charac terization involves
+the choice of representatives of the orbifold structures, n amely those orbifold
+atlases which were used to construct the marked atlas groupo ids. Hence, at
+this point we get a notion of orbifold map with fixed represent atives of orb-
+ifold structures, which we will call charted orbifold maps . In Section 5we
+introduce a natural definition of composition of charted orb ifold maps and a
+geometrically motivated definition of the identity morphis m (a certain class
+of charted orbifold maps), which allows us to establish a nat ural equivalence
+relation on the class of charted orbifold maps. An orbifold m ap (which does
+not depend on the choice of orbifold atlases) is then an equiv alence class of
+charted orbifold maps. The leading idea for this equivalenc e relation is geo-
+metric: we consider charted orbifold maps as equivalent if a nd only if they
+induce the same charted orbifold map on common refinements of the orbifold
+atlases. Moreover, using the same idea, we define the composi tion of orbifold
+maps. In this way, we construct a category of reduced orbifol ds. Finally, in
+Section 6, we characterize orbifolds as certain equivalence classes of marked
+atlas groupoids, and orbifold maps as equivalence classes o f homomorphismsTHE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 3
+of marked atlas groupoids. These equivalence relations are natural adapta-
+tions of the classical Morita equivalence. In this way, ther e arises a category
+of marked atlas groupoids which is isomorphic to the orbifol d category. As
+an additional benefit the isomorphism functor is constructi ve. As a final
+step, in Section 7, we show that the category of marked atlas groupoids is
+isomorphic to an analogously defined category of marked prop er effective
+étale Lie groupoids (by embedding).
+Even though the category of marked proper effective étale Lie groupoids
+constructed in this article not the classical one, all the cl assical groupoid
+(2- or bi-) categories can be recovered by considering eleme nts in an equiv-
+alence class as stand-alone entities and the equivalence-p roviding maps as
+2-morphisms, and by forgetting the marking. The ultimate or bifold cate-
+gory constructed here is the one which is satisfying from a ge ometric point
+of view (c.f. the results in [ 13]) and also allows to adapt without any diffi-
+culties all results on groupoids that are invariant under Mo rita equivalence
+(e.g. the behavior of orbifold vector bundles under pullbac k).
+We expect that all these constructions can be performed in a s imilar way
+for orbifolds over other fields, e.g., complex ones, and for o rbifolds with ad-
+ditional structures such as Riemannian ones.
+Acknowledgments: This paper emerged from a workshop on orbifolds in
+2007 which took place in Paderborn in the framework of the Int ernational
+Research Training Group 1133 “Geometry and Analysis of Symm etries”. The
+author is very grateful to the participants of this workshop for their abid-
+ing interest and enlightening conversations. In particula r, she likes to thank
+Joachim Hilgert, Alexander Schmeding and an anonymous refe ree for very
+careful reading of the manuscript. Moreover, she wishes to t hank Dieter
+Mayer and the Institut für Theoretische Physik in Clausthal for the warm
+hospitality where part of this work was conducted. The autho r was par-
+tially supported by the International Research Training Gr oup 1133 “Geom-
+etry and Analysis of Symmetries”, the Sonderforschungsber eich/Transregio
+45 “Periods, moduli spaces and arithmetic of algebraic vari eties”, the Max-
+Planck-Institut für Mathematik in Bonn, the SNF (200021-127 145) and the
+Volkswagen Foundation.
+Notation and conventions: We useN0=N∪{0}to denote the set of non-
+negative integers. If not stated otherwise, every manifold is assumed to be
+real, paracompact, Hausdorff and smooth ( C∞). We also consider second-
+countable manifolds, and it will always be indicated whethe r we require a
+given manifold to be just paracompact or even second-counta ble. IfMis a
+manifold, then Diff(M)denotes the group of diffeomorphisms of M. IfGis
+a subgroup of Diff(M), then
+G\M:={Gm|m∈M}
+denotes the space of G-orbits endowed with the final topology. If A1,A2,B
+are sets (manifolds) and f1:A1→B,f2:A2→Bare maps (submersions),
+then we denote the fibered product of f1andf2byA1f1×f2A2and identify4 ANKE POHL
+it with the set (manifold)
+A1f1×f2A2={(a1,a2)∈A1×A2|f1(a1) =f2(a2)}.
+Finally, we say that a family V={Vi|i∈I}isindexed by IifI→ V,
+i/ma√s⊔o→Vi, is a bijection.
+2.Reduced orbifolds, groupoids, and pseudogroups
+In this section we recall the necessary background on reduce d orbifolds and
+groupoids. The notion of (reduced) orbifolds goes back to at least Satake
+[11,12], and has been refined since then, see e.g., [ 4,14,8,1]. Here, we use
+one of these more modern definitions of reduced orbifolds, an d we consider
+different flavors for the manifolds involved. We recall that t hroughout any
+manifold is required to be real, smooth, Hausdorff and paraco mpact.
+2.1.Reduced orbifolds. LetQbe a topological space, and n∈N0. A
+reduced orbifold chart of dimension nonQis a triple (V,G,ϕ)whereVis a
+connectedn–manifold without boundary, Gis a finite subgroup of Diff(V),
+andϕ:V→Qis a map with open image ϕ(V)that induces a homeomor-
+phism from G\Vtoϕ(V). In this case, (V,G,ϕ)is said to uniformize ϕ(V).
+One might argue that orbifold charts should better be called orbifold
+parametrizations. However, orbifold charts is the more est ablished notion
+and we prefer to stick to it. The orbifold charts are called re duced (or effec-
+tive) to indicate that the action of Gis effective.
+Two reduced orbifold charts (V,G,ϕ),(W,H,ψ)onQare called compatible
+if for each pair (x,y)∈V×Wwithϕ(x) =ψ(y)there are open connected
+neighborhoods /tildewideVofxand/tildewiderWofyand a diffeomorphism h:/tildewideV→/tildewiderWwith
+ψ◦h=ϕ|/tildewideV. The map his called a change of charts . The neighborhoods /tildewideV
+and/tildewiderWand the diffeomorphism hcan always be chosen in such a way that
+h(x) =y. Moreover /tildewideVmay assumed to be open G-stable. This means that
+/tildewideVis open and connected, and for each g∈Gwe either have g/tildewideV=/tildewideVor
+g/tildewideV∩/tildewideV=∅. In this case, /tildewiderWis openH-stable by [ 8, Proposition 2.12(i)].
+Areduced orbifold atlas of dimension nonQis a collection of pairwise
+compatible reduced orbifold charts
+V:={(Vi,Gi,ϕi)|i∈I}
+of dimension nonQsuch that/uniontext
+i∈Iϕi(Vi) =Q. Two reduced orbifold atlases
+areequivalent if their union is a reduced orbifold atlas. A reduced orbifold
+structure of dimension nonQis an equivalence class of reduced orbifold at-
+lases of dimension nonQ. Areduced (paracompact resp. second-countable)
+orbifold of dimension nis a pair (Q,U)whereQis a (paracompact resp.
+second-countable) Hausdorff space and Uis a reduced orbifold structure of
+dimensionnonQ. We note that since the manifolds Vin the orbifold
+charts(V,G,ϕ)are assumed to be connected and finite-dimensional, para-
+compactness of Vis equivalent to second-countability of V. However, since
+the topological space Qis not required to be connected, paracompactness
+does not necessarily imply second-countability of Q.
+LetUbe a reduced orbifold structure on Q. Each reduced orbifold atlas
+VinUis called a representative ofUor areduced orbifold atlas of (Q,U).THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 5
+Since we are considering reduced orbifolds only, we omit the term “re-
+duced” from now on. Moreover, when implicitly understood we will omit the
+terms “paracompact” and “second-countable”.
+Let(V,G,ϕ),(W,H,ψ)be orbifold charts of identical dimension on the
+topological space Q. Then an (open) embedding µ: (V,G,ϕ)→(W,H,ψ)
+between these two orbifold charts is an (open) embedding µ:V→Wbe-
+tween manifolds which satisfies ψ◦µ=ϕ. We remark that any embedding
+is automatically open by the Theorem on the Invariance of Dom ain. If, in
+addition,µis a diffeomorphism between VandW, thenµis called an iso-
+morphism from(V,G,ϕ)to(W,H,ψ). Suppose that Sis an openG–stable
+subset ofVand letGS:={g∈G|gS=S}denote the isotropy group of
+S. Then(S,GS,ϕ|S)is an orbifold chart on Qas well, the restriction of
+(V,G,ϕ)toS.
+Remark 2.1.Suppose that µ: (V,G,ϕ)→(W,H,ψ)is an embedding. By [ 8,
+Proposition 2.12(i)], µ(V)is an open H–stable subset of W, and moreover
+that there is a unique group isomorphism µ:G→Hµ(V)for whichµ(gx) =
+µ(g)µ(x)forg∈G,x∈V.
+In the following example we provide two orbifolds with the sa me under-
+lying topological space. These orbifolds are particularly simple since both
+orbifold structures have one-chart-representatives. Des pite their simplicity
+they serve as motivating examples for several definitions in this paper.
+Example 2.2. LetQ:= [0,1)be endowed with the induced topology of R.
+The map
+f:Q→Q, x/ma√s⊔o→x2
+is a homeomorphism. Further the map pr: (−1,1)→[0,1),x/ma√s⊔o→ |x|, induces
+a homeomorphism {±id}\(−1,1)→Q. Then
+V1:=/parenleftbig
+(−1,1),{±id},pr/parenrightbig
+andV2:=/parenleftbig
+(−1,1),{±id},f◦pr/parenrightbig
+are two orbifold charts on Q. They are easily seen to be non-compatible.
+LetU1be the orbifold structure on Qgenerated by V1, andU2be the one
+generated by V2.
+2.2.Groupoids and homomorphisms. A groupoid is a small category
+in which each morphism is an isomorphism. In the context of or bifolds
+this concept is most commonly expressed (equivalently) in t erms of sets and
+maps. The morphisms are then called arrows.
+AgroupoidGis a tupleG= (G0,G1,s,t,m,u,i )consisting of the set
+G0ofobjects , or the baseofG, the setG1ofarrows , and five structure
+maps, namely the source map s:G1→G0, thetarget map t:G1→G0, the
+multiplication orcomposition m:G1s×tG1→G1, theunit mapu:G0→G1,
+and the inversioni:G1→G1which satisfy that
+(i) for all (g,f)∈G1s×tG1we haves(m(g,f)) =s(f)andt(m(g,f)) =
+t(g),
+(ii) for all (h,g),(g,f)∈G1s×tG1we havem(h,m(g,f)) =m(m(h,g),f),
+(iii) for all x∈G0we haves(u(x)) =x=t(u(x)),
+(iv) for allx∈G0and all(u(x),f),(g,u(x))∈G1s×tG1it followsm(u(x),f) =
+fandm(g,u(x)) =g,6 ANKE POHL
+(v) for allg∈G1we haves(i(g)) =t(g)andt(i(g)) =s(g), andm(g,i(g)) =
+u(t(g))andm(i(g),g) =u(s(g)).
+We often use the notations m(g,f) =gf,u(x) = 1x,i(g) =g−1, and
+g:x→yorgx→yfor an arrow g∈G1withs(g) =x,t(g) =y. Moreover,
+G(x,y)denotes the set of arrows from xtoy.
+Classically, the base space of a Lie groupoid is required to b e a second-
+countable manifold. However, parts of the theory of Lie grou poids stay valid
+if the base manifold is just paracompact. Therefore, as with orbifolds, we
+consider two flavors of Lie groupoids here.
+A(paracompact resp. second-countable) Lie groupoid is a groupoid Gfor
+whichG0is a paracompact resp. second-countable manifold, G1is a smooth
+(possibly non-Hausdorff, possible non-second-countable) manifold, the struc-
+ture mapss,t:G1→G0are smooth submersions (hence G1s×tG1, the do-
+main ofm, is a smooth, possibly non-Hausdorff manifold), and the stru cture
+mapsm,uandiare smooth. A Lie groupoid is called étaleif its source and
+target map are local diffeomorphisms. It is called proper ifG1is Hausdorff
+and the map (s,t):G1→G0×G0is proper. An étale Lie groupoid Gis
+called effective if the map
+g/ma√s⊔o→germs(g)(t◦(s|U)−1),
+whereg∈G1andUis an open neighborhood of ginG1such thatt|Uand
+s|Uare injective, is injective.
+LetGandHbe groupoids. A homomorphism fromGtoHis a functor
+ϕ:G→H, i.e., it is a tuple ϕ= (ϕ0,ϕ1)of mapsϕ0:G0→H0and
+ϕ1:G1→H1which commute with all structure maps. If GandHare Lie
+groupoids, then ϕis a homomorphism between them if it is a homomorphism
+of the abstract groupoids with the additional requirement t hatϕ0andϕ1be
+smooth maps.
+LetGbe a groupoid. The orbitofx∈G0is the set
+Gx:=t(s−1(x)) =/braceleftBig
+y∈G0/vextendsingle/vextendsingle/vextendsingle∃g∈G1:xg→y/bracerightBig
+.
+Two elements x,y∈G0are called equivalent ,x∼y, if they are in the same
+orbit. The quotient space |G|:=G0/∼is called the orbit space ofG. The
+canonical quotient map G0→ |G|is denoted by prorprG, and[x]:= pr(x)
+forx∈G0.
+2.3.Pseudogroups and groupoids. We recall how to construct a Lie
+groupoid from an orbifold and a representative of its orbifo ld structure. This
+construction is well known, see e.g. the book by Moerdjik and Mrčun [ 8]. We
+provide it here for the convenience of the reader and to intro duce the nota-
+tions we will use later on. It is a two-step process in which on e first assigns
+a pseudogroup to the orbifold, which depends on the represen tative of the
+orbifold structure. Then one constructs an étale Lie groupo id from the pseu-
+dogroup. For reasons of generality and clarity we start with the second step.
+Definition 2.3. LetMbe a manifold. A transition onMis a diffeomor-
+phismf:U→VwhereU,Vare open subsets of M. In particular, the empty
+map∅ → ∅ is a transition on M. The product of two transitions f:U→V,THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 7
+g:U′→V′is the transition
+f◦g:g−1(U∩V′)→f(U∩V′), x/ma√s⊔o→f(g(x)).
+Theinverse offis the inverse of fas a function. If f:U→Vis a transi-
+tion, we use domfto denote its domain andcodfto denote its codomain .
+Further, ifx∈domf, thengermxfdenotes the germ of fatx.
+LetA(M)be the set of all transitions on M. Apseudogroup onMis
+a subsetPofA(M)which is closed under multiplication and inversion. A
+pseudogroup Pis called fullifidU∈Pfor each open subset UofM. It is
+said to be complete if it is full and satisfies the following gluing property:
+Whenever there is a transition f∈ A(M)and an open covering (Ui)i∈Iof
+domfsuch thatf|Ui∈Pfor alli∈I, thenf∈P.
+We now recall how to construct an étale Lie groupoid from a ful l pseu-
+dogroup.
+Construction 2.4. LetMbe a manifold and Pa full pseudogroup on M.
+Theassociated groupoid Γ:= Γ(P)is given by
+Γ0:=M,Γ1:={germxf|f∈P, x∈domf},
+and
+Γ(x,y):={germxf|f∈P, x∈domf, f(x) =y}.
+Forf∈PdefineUf:={germxf|x∈domf}. The topology and differential
+structure of Γ1is given by the germ topology and germ differential structure ,
+that is, for each f∈Pthe bijection
+ϕf:/braceleftbigg
+Uf→domf
+germxf/ma√s⊔o→x
+is required to be a diffeomorphism. The structure maps (s,t,m,u,i )ofΓ
+are the obvious ones, namely
+s(germxf):=x
+t(germxf):=f(x)
+m(germf(x)g,germxf):= germx(g◦f)
+u(x):= germxidUfor an open neighborhood Uofx
+i(germxf):= germf(x)f−1.
+Obviously, Γ(P)is an étale Lie groupoid.
+Special Case 2.5. Let(Q,U)be an orbifold, and let
+V={(Vi,Gi,πi)|i∈I}
+be a representative of Uindexed by I. We define
+V:=/coproductdisplay
+i∈IViandπ:=/coproductdisplay
+i∈Iπi.
+Then
+Ψ(V):=/braceleftbig
+ftransition on V/vextendsingle/vextendsingleπ◦f=π|domf/bracerightbig
+.
+is a complete pseudogroup on V. The associated groupoid
+Γ(V):= Γ(Ψ(V))8 ANKE POHL
+is the proper effective étale Lie groupoid we shall associate toQandV. Note
+that this groupoid depends on the choice of the representati ve of the orbifold
+structure UofQ. A groupoid which arises in this way we call atlas groupoid .
+If(Q,U)is a second-countable orbifold, then we can choose Vto be count-
+able and the associated atlas groupoid has a second-countab le base. When-
+ever dealing with second-countable orbifolds in the follow ing, we will assume
+that the chosen representative of the orbifold structure is countable.
+Example 2.6. Recall the orbifolds (Q,Ui) (i= 1,2)from Example 2.2, and
+consider the representative Vi:={Vi}ofUi. Proposition 2.12 in [ 8] implies
+that
+Ψ(Vi) =/braceleftbig
+g|U:U→g(U)/vextendsingle/vextendsingleU⊆(−1,1)open,g∈ {±id}/bracerightbig
+.
+In both cases the associated groupoid Γ:= Γ(Vi)is
+Γ0= (−1,1)
+Γ(x,y) =
+
+/braceleftbig
+germ0id,germ0(−id)/bracerightbig
+ifx= 0 =y/braceleftbig
+germxid/bracerightbig
+ifx=y/\e}a⊔io\slash= 0/braceleftbig
+germx(−id)/bracerightbig
+ifx=−y/\e}a⊔io\slash= 0
+∅ otherwise.
+3.Marked Lie groupoids and their homomorphisms
+In Example 2.6we have seen that it may happen that the same atlas
+groupoid is associated to two different orbifolds. The reaso n for this is that
+in the definition of the pseudogroup which is needed for the co nstruction
+of the atlas groupoid one loses information about the projec tion mapsϕof
+the orbifold charts (V,G,ϕ). To be able to distinguish atlas groupoids con-
+structed from different orbifolds, we mark the groupoids wit h a topological
+space and a homeomorphism. It will turn out that this marking suffices to
+identify the orbifold one started with.
+Amarked Lie groupoid is a triple (G,α,X)consisting of a Lie groupoid
+G, a topological space X, and a homeomorphism α:|G| →X.
+For atlas groupoids there exists a specific marking which is c rucial for the
+isomorphism between the orbifold category and the groupoid category. It is
+stated in the following lemma of which we omit its straightfo rward proof.
+Lemma 3.1. Let(Q,U)be an orbifold and
+V={(Vi,Gi,πi)|i∈I}
+a representative of Uindexed byI. SetV:=/coproducttext
+i∈IViandπ:=/coproducttext
+i∈Iπi:V→
+Q. Then the map
+α:/braceleftbigg
+|Γ(V)| →Q
+[x]/ma√s⊔o→π(x)
+is a homeomorphism.
+Let(Q,U)be an orbifold. To each (countable if (Q,U)is second-countable)
+orbifold atlas VofQwe assign the marked atlas groupoid (Γ(V),αV,Q)with
+αVbeing the homeomorphism from Lemma 3.1. We often only write Γ(V)
+to refer to this marked groupoid.THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 9
+Example 3.2. Recall from Example 2.6the orbifolds (Q,Ui)fori= 1,2,
+their respective orbifold atlases Vi, and the associated groupoids Γ = Γ(Vi).
+The orbit of x∈Γ0is{x,−x}. Hence the homeomorphism associated to
+(Q,Ui)isαVi:|Γ| →Qgiven byαV1([x]) =|x|resp.αV2([x]) =x2. Thus,
+the associated marked groupoids (Γ,αV1,Q)and(Γ,αV2,Q)are different.
+Proposition 3.3. Let(Q,U)and(Q′,U′)be orbifolds. Suppose that Vis a
+representative of U, andV′a representative of U′. If the associated marked
+atlas groupoids (Γ(V),αV,Q)and(Γ(V′),αV′,Q′)are equal, then the orbifolds
+(Q,U)and(Q′,U′)are equal. More precisely, we even have V=V′.
+Proof. Clearly,Q=Q′. Suppose that
+V={(Vi,Gi,πi)|i∈I}andV′={(V′
+j,G′
+j,π′
+j)|j∈J},
+indexed by Iresp. byJ. FromΓ(V) = Γ(V′)it follows that
+/coproductdisplay
+i∈IVi= Γ(V)0= Γ(V′)0=/coproductdisplay
+j∈JV′
+j.
+Since each Viand eachV′
+jis connected, there is a bijection between Iand
+J. We may assume I=JandVi=V′
+ifor alli∈I. Letx∈Vi. Then
+πi(x) =αV([x]) =αV′([x]) =π′
+i(x). Therefore πi=π′
+ifor alli∈I. Thus
+Ψ(V) = Ψ(V′). Moreover
+Gi={f∈Ψ(V)|domf=Vi= codf}=G′
+i.
+To show that the actions GiandG′
+ionViare equal, let g∈Gi. For each
+x∈Viwe haveπ′
+i(g(x)) =π′
+i(x). This shows that g(x)∈G′
+ixfor eachx∈Vi.
+By [8, Lemma 2.11] there exists a unique element g′∈G′
+isuch thatg=g′.
+From this it follows that Gi=G′
+ias acting groups. Thus, V=V′./square
+Lemma 3.4. Let(G,α,X)and(H,β,Y)be marked Lie groupoids and sup-
+pose thatϕ= (ϕ0,ϕ1):G→His a homomorphism of Lie groupoids. Then
+ϕinduces a unique map ψsuch that the diagram
+G0prG/d47/d47
+ϕ0
+/d15/d15|G|α/d47/d47X
+ψ
+/d15/d15
+H0prH/d47/d47|H|β/d47/d47Y
+commutes. Moreover, ψis continuous.
+Proof. The mapϕinduces a unique map |ϕ|:|G| → |H|such that |ϕ|◦prG=
+prH◦ϕ0, which is continuous. Then ψ=β◦|ϕ|◦α−1. /square
+4.Groupoid homomorphisms in local charts
+In this section we characterize homomorphisms between mark ed atlas
+groupoids on the orbifold side, i.e., in terms of local chart s. We proceed
+in a two-step process. First we define a concept which we call r epresenta-
+tives of orbifold maps. Each representative of an orbifold m ap gives rise to
+exactly one homomorphism between the associated marked atl as groupoids.
+Since, in general, each groupoid homomorphism corresponds to several such
+representatives, we then impose an equivalence relation on the class of all rep-
+resentatives for fixed orbifold atlases. The equivalence cl asses, called charted10 ANKE POHL
+orbifold maps, turn out to be in bijection with the homomorph isms between
+the marked atlas groupoids. The constructions in this secti on are subject to
+a fixed choice of representatives of the orbifold structures . In the following
+sections we will use this construction as a basic building bl ock for a notion of
+maps (or morphisms) between orbifolds which is independent of the chosen
+representatives.
+Throughout this section let (Q,U),(Q′,U′)denote two orbifolds.
+Definition 4.1. Letf:Q→Q′be a continuous map, and suppose that
+(V,G,π)∈ U,(V′,G′,π′)∈ U′are orbifold charts. A local lift offwith
+respect to (V,G,π)and(V′,G′,π′)is a smooth map ˜f:V→V′such that
+π′◦˜f=f◦π. In this case, we call ˜falocal lift of fatqfor eachq∈π(V).
+Recall the pseudogroup A(M)from Definition 2.3.
+Definition 4.2. LetMbe a manifold and Aa pseudogroup on Mwhich
+satisfies the gluing property from Definition 2.3and is closed under restric-
+tions. The latter means that if f∈AandU⊆domfis open, then the map
+f|U:U→f(U)is inA. Suppose that Bis a subset of A(M). ThenAis
+said to be generated byBifB⊆Aand for each f∈Aand eachx∈domf
+there exists some g∈Bwithx∈domgand an open set U⊆domf∩domg
+such thatx∈Uandf|U=g|U. In this case we say that BgeneratesA.
+Definition 4.3. LetMbe a manifold. A subset PofA(M)is called a
+quasi-pseudogroup onMif it satisfies the following two properties:
+(i) Iff∈Pandx∈domf, then there exists an open set Uwithx∈
+U⊆domfandg∈Psuch that there exists an open set Vwith
+f(x)∈V⊆domgand
+/parenleftbig
+f|U/parenrightbig−1=g|V.
+(ii) Iff,g∈Pandx∈f−1(codf∩domg), then there exists h∈Pwith
+x∈domhsuch that we find an open set Uwith
+x∈U⊆f−1(codf∩domg)∩domhandg◦f|U=h|U.
+A quasi-pseudogroup is designed to work with the germs of its elements.
+Therefore identities (like inversion and composition) of e lements in quasi-
+pseudogroups are only required to be satisfied locally, wher eas for (ordinary)
+pseudogroups these identities have to be valid globally. On e easily proves
+that each quasi-pseudogroup generates a unique pseudogrou p which satis-
+fies the gluing property from Definition 2.3and is closed under restrictions.
+Conversely, each generating set for such a pseudogroup is ne cessarily a quasi-
+pseudogroup.
+In the following definition of a representative of an orbifol d map, the
+underlying continuous map fis the only entity which is stable under change
+of orbifold atlases or, in other words, under the choice of lo cal lifts. The pair
+(P,ν)should be considered as one entity. It serves as a transport o f changes
+of charts from one orbifold to another. Here we ask for a quasi -pseudogroup
+Pinstead of working with all of Ψ(V)(recallΨ(V)from Special Case 2.5)
+for two reasons. In general, Pis much smaller than Ψ(V). Sometimes it
+may even be finite. In Example 4.6below we see that for some orbifolds, P
+may consist of only two elements. Moreover, if the orbifold i s a connectedTHE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 11
+manifold,Pcan always be chosen to be {id}. The other reason is that it
+is much easier to construct a quasi-pseudogroup Pand a compatible map ν
+from a given groupoid homomorphism than a map νdefined on all of Ψ(V).
+Examples 4.5and4.6below show that the objects requested in the fol-
+lowing definition need not exist nor, if they exist, are uniqu ely determined.
+Definition 4.4. Arepresentative of an orbifold map from(Q,U)to(Q′,U′)
+is a tuple
+ˆf:= (f,{˜fi}i∈I,P,ν)
+where
+(R1)f:Q→Q′is a continuous map,
+(R2) for each i∈I, the map ˜fiis a local lift of fw.r.t. some orbifold charts
+(Vi,Gi,πi)∈ U,(V′
+i,G′
+i,π′
+i)∈ U′such that
+/uniondisplay
+i∈Iπi(Vi) =Q
+and(Vi,Gi,πi)/\e}a⊔io\slash= (Vj,Gj,πj)fori,j∈I,i/\e}a⊔io\slash=j,
+(R3)Pis a quasi-pseudogroup which consists of changes of charts o f the
+orbifold atlas
+V:={(Vi,Gi,πi)|i∈I}
+of(Q,U)and generates Ψ(V).
+(R4) Letψ:=/coproducttext
+i∈I˜fiand letV′be a representative of U′which contains
+{(V′
+i,G′
+i,π′
+i)|i∈I}.
+Thenν:P→Ψ(V′)is a map which assigns to each λ∈Pan embed-
+ding
+ν(λ): (W′,H′,χ′)→(V′,G′,ϕ′)
+between some orbifold charts in V′such that
+(a)ψ◦λ=ν(λ)◦ψ|domλ,
+(b) for allλ,µ∈Pand allx∈domλ∩domµwithgermxλ= germxµ,
+we have
+germψ(x)ν(λ) = germψ(x)ν(µ),
+(c) for allλ,µ∈P, for allx∈λ−1(codλ∩domµ)we have
+germψ(λ(x))ν(µ)·germψ(x)ν(λ) = germψ(x)ν(h),
+wherehis an element of Pwithx∈domhsuch that there is an
+open setUwith
+x∈U⊆λ−1(codλ∩domµ)∩domh
+andµ◦λ|U=h|U,
+(d) for allλ∈Pand allx∈domλsuch that there exists an open set
+Uwithx∈U⊆domλandλ|U= idUwe have
+germψ(x)ν(λ) = germψ(x)idU′
+withU′:=/coproducttext
+i∈IV′
+i.12 ANKE POHL
+The orbifold atlas Vis called the domain atlas of the representative ˆf, and
+the set
+{(V′
+i,G′
+i,π′
+i)|i∈I}
+is called the range family ofˆf. The latter set is not necessarily indexed by
+I.
+Condition (R 4c) is in fact independent of the choice of h. The technical
+(and easily satisfied) condition in (R 2) that each two orbifold charts in Vbe
+distinct is required because we use Ias an index set for Vin (R3) and other
+places.
+Example 4.5below shows that the continuous map fin (R1) cannot be
+chosen arbitrarily. It is not even sufficient for fto be any homeomorphism.
+Example 4.5. Recall the orbifold (Q,U1)from Example 2.2. The map
+f:Q→Q, f(x) =√x,
+is a homeomorphism on Q. We show that fhas no local lift at 0. Each
+orbifold chart in U1that uniformizes a neighborhood of 0is isomorphic to
+an orbifold chart of the form (I,{±idI},pr)whereI= (−a,a)for some
+0< a <1. Seeking a contradiction assume that ˜fis a local lift of fat0
+with domain I= (−a,a). For each x∈I, necessarily ˜f(x)∈/braceleftbig
+±/radicalbig
+|x|/bracerightbig
+.
+Since˜fis required to be continuous, there remain four possible can didates
+for˜f, namely
+˜f1(x) =/radicalbig
+|x|, ˜f2=−˜f1,
+˜f3(x) =/braceleftBigg√x x ≥0
+−√−x x≤0,˜f4=−˜f3.
+But none of these is differentiable in x= 0, hence there is no local lift of f
+at0.
+The following example shows that the pair (P,ν)is not uniquely deter-
+mined by the choice of the family of local lifts.
+Example 4.6. Recall the orbifold (Q,U1)and the representative V1={V1}
+ofU1from Example 2.2. The map f:Q→Q,q/ma√s⊔o→0, is clearly continuous
+and has the local lift
+˜f:/braceleftbigg
+(−1,1)→(−1,1)
+x/ma√s⊔o→0
+with respect to V1andV1. Consider the quasi-pseudogroup P={±id(−1,1)}
+on(−1,1). Proposition 2.12 in [ 8] implies that Pgenerates Ψ(V1). The
+triple(f,˜f,P)can be completed in the following two different ways to rep-
+resentatives of orbifold maps on (Q,U1):
+(a)ν1(±id(−1,1)):= id(−1,1),
+(b)ν2(id(−1,1)):= id(−1,1),ν2(−id(−1,1)):=−id(−1,1).
+We will see in Example 4.8below that (f,˜f,P,ν1)and(f,˜f,P,ν2)give rise
+to different groupoid homomorphisms.THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 13
+Proposition 4.7. Letˆf= (f,{˜fi}i∈I,P,ν)be a representative of an orbifold
+map from (Q,U)to(Q′,U′). Suppose that V={(Vi,Gi,πi)|i∈I}, is the
+domain atlas of ˆf, which is an orbifold atlas of (Q,U)indexed byI. LetV′be
+an orbifold atlas of (Q′,U′)which contains the range family/braceleftbig
+(V′
+i,G′
+i,π′
+i)/vextendsingle/vextendsinglei∈
+I/bracerightbig
+. Define the map ϕ0: Γ(V)0→Γ(V′)0by
+ϕ0:=/coproductdisplay
+i∈I˜fi.
+Suppose that ϕ1: Γ(V)1→Γ(V′)1is determined by
+ϕ1(germxλ):= germϕ0(x)ν(λ)
+for allλ∈P,x∈domλ. Then
+ϕ= (ϕ0,ϕ1): Γ(V)→Γ(V′)
+is a homomorphism. Moreover, αV′◦|ϕ|=f◦αV.
+Proof. Obviously, ϕ0is smooth. To show that ϕ1is a well-defined map on
+all ofΓ(V)1, letg∈Ψ(V)andx∈domg. Then there exists λ∈Psuch that
+x∈domλand
+g|U=λ|U
+for some open subset U⊆domg∩domλwithx∈U. Hence germxg=
+germxλ. Thus
+ϕ1(germxg) =ϕ1(germxλ) = germϕ0(x)ν(λ).
+If there isµ∈Psuch thatx∈domµandg|W=µ|Wfor some open subset
+Wofdomg∩domµwithx∈W, thengermxµ= germxλ. By (R 4b),
+germϕ0(x)ν(µ) = germϕ0(x)ν(λ)and thus
+ϕ1(germxµ) =ϕ1(germxλ).
+This shows that ϕ1is indeed well-defined on all of Γ(V)1. The properties
+(R4a), (R4c) and (R 4d) yield that ϕcommutes with the structure maps.
+It remains to show that ϕ1is smooth. For this, let germxλ∈Γ(V)1with
+λ∈P. The definition of νshows that ϕ1maps
+U:={germyλ|y∈domλ}
+to
+U′:={germzν(λ)|z∈domν(λ)}.
+Now the diagram
+U
+s
+/d15/d15ϕ1/d47/d47U′
+s
+/d15/d15germyλ✤ /d47/d47
+❴
+/d15/d15germϕ0(y)ν(λ)❴
+/d15/d15
+domλϕ0/d47/d47domν(λ) y✤ /d47/d47ϕ0(y)
+commutes, the vertical maps (restriction of source maps) ar e diffeomorphisms
+andϕ0is smooth, so ϕ1is smooth. Finally, suppose x∈Vi. Then
+(αV′◦|ϕ|)/parenleftbig
+[x]/parenrightbig
+=αV′/parenleftbig
+[ϕ0(x)]/parenrightbig
+=π′
+i(˜fi(x)) =f(πi(x)) = (f◦αV)/parenleftbig
+[x]/parenrightbig
+.
+This completes the proof. /square14 ANKE POHL
+Example 4.8below shows that the choice of (P,ν)in Definition 4.4is not
+unique, and more elaborated examples would immediately sho w that not
+each pair (P,ν)automatically satisfies the compatibility conditions in R 4a).
+However, for some important types of maps between orbifolds (e.g. isomor-
+phisms) the choice of (P,ν)is canonical and the compatibility conditions are
+also automatic, which is the essence of Proposition 5.5below.
+Example 4.8. Recall the setting of Example 4.6and the associated groupoid
+Γ:= Γ(V1)from Example 2.6. The homomorphism ϕ= (ϕ0,ϕ1)ofΓinduced
+by(f,˜f,P,ν1)isϕ0=˜fand
+ϕ1(germx(±id(−1,1))) = germ0id(−1,1).
+The homomorphism ψ= (ψ0,ψ1): Γ→Γinduced by (f,˜f,P,ν2)isψ0=˜f
+and
+ψ1(germxid(−1,1)) = germ0id(−1,1),
+ψ1(germx(−id(−1,1))) = germ0(−id(−1,1)).
+The following proposition is the converse to Proposition 4.7. Its proof is
+constructive. In Section 6we will use this construction to define the functor
+between the category of orbifolds and that of marked atlas gr oupoids.
+Proposition 4.9. LetVbe a representative of U,V′a representative of U′,
+and
+ϕ= (ϕ0,ϕ1): Γ(V)→Γ(V′)
+a homomorphism. Then ϕinduces a representative of an orbifold map
+(f,{˜fi}i∈I,P,ν)
+with domain atlas V, range family contained in V′, and
+˜fi=ϕ0|Vi
+for alli∈I. Moreover, we have f=αV′◦|ϕ|◦α−1
+V.
+Proof. We start by showing that for each h∈Ψ(V)and eachx∈domh
+there exist an element g∈Ψ(V′)and an open neighborhood Uofx(which
+may depend on g) withU⊆domhsuch that for each y∈Uwe have
+(1) ϕ1(germyh) = germϕ0(y)g.
+So, leth∈Ψ(V)andx∈domh. By the definitions of Γ(V)1andϕ1, there
+existsg∈Ψ(V′)such that
+ϕ1(germxh) = germϕ0(x)g.
+Sinceϕ1is continuous, the preimage of the germϕ0(x)g–neighborhood
+U′
+g={germzg|z∈domg}
+is a neighborhood of germxh. Hence there exists an open neighborhood U
+ofxwithU⊆domhsuch that
+Uh|U={germyh|y∈U} ⊆ϕ−1
+1/parenleftbig
+U′
+g/parenrightbig
+.
+Thus, for all y∈Uwe have ( 1) as claimed. We remark that each two possible
+choices forgcoincide on some neighborhood of ϕ0(x).THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 15
+For eachh∈Ψ(V)and eachx∈domhwe now choose a pair (g,U)
+whereg∈Ψ(V′)is an embedding between some orbifold charts in U′and
+Uis an open neighborhood of xsuch thath|Uis a change of charts of
+V. LetP(h,x):= (g,U). We adjust choices such that for h1,h2∈Ψ(V)
+andx1∈domh1,x2∈domh2the chosen pairs P(h1,x1) = (g1,U1)resp.
+P(h2,x2) = (g2,U2)are either equal or U1/\e}a⊔io\slash=U2. LetPdenote the family of
+the changes of charts we have chosen in this way:
+P={h|U:U→h(U)|h∈Ψ(V), x∈domh, P(h,x) = (g,U)}.
+By construction, Pis a quasi-pseudogroup which generates Ψ(V). We define
+the mapν:P→Ψ(V′)by
+ν(λ):=g
+wheregis the unique element in Ψ(V′)attached to λ∈Pby our choices.
+Forλ∈Pandx∈domλwe clearly have
+(2) ϕ1(germxλ) = germϕ0(x)ν(λ).
+Properties (R 4) are easily checked using the compatibility of ϕwith the
+structure maps.
+It remains to show that the image of ϕ0|Viis contained in V′
+jfor some
+orbifold chart (V′
+j,G′
+j,π′
+j)∈ V′. SinceViis connected, the image ϕ0(Vi)is
+connected as well. The connected components of Γ(V′)0are exactly the sets
+W′with(W′,G′,ϕ′)∈ V. From this the claim follows. /square
+Proposition 4.9guarantees that each homomorphism
+ϕ= (ϕ0,ϕ1): Γ(V)→Γ(V′)
+induces a representative of an orbifold map (f,{˜fi}i∈I,P,ν)with domain
+atlasV, range family contained in V′,˜fi=ϕ0|Vi, andf=αV′◦ |ϕ| ◦α−1
+V.
+For the pair (P,ν), Proposition 4.9allows (in general) a whole bunch of
+choices. On the other hand, different representatives of an o rbifold map
+may induce the same groupoid homomorphism. In view of Propos ition4.7
+and the proof of Proposition 4.9, the relevant information stored by the pair
+(P,ν)are the germs of the elements in Pand the via νassociated germs of
+elements in Ψ(V′). This observation is the motivation for the equivalence
+relation in the following definition.
+Definition 4.10. Let
+ˆf:= (f,{˜fi}i∈I,P1,ν1)andˆg:= (g,{˜gi}i∈I,P2,ν2)
+be two representatives of orbifold maps with the same domain atlasVrep-
+resenting the orbifold structure UonQand both range families being con-
+tained in the orbifold atlas V′of(Q′,U′). Setψ:=/coproducttext
+i∈I˜fi. We say that ˆf
+isequivalent toˆgiff=g,˜fi= ˜gifor alli∈I, and
+germψ(x)ν1(λ1) = germψ(x)ν2(λ2)
+for allλ1∈P1,λ2∈P2,x∈domλ1∩domλ2withgermxλ1= germxλ2.
+This defines an equivalence relation. The equivalence class ofˆfwill be
+denoted by [ˆf]or
+(f,{˜fi}i∈I,[P1,ν1]),16 ANKE POHL
+or evenˆfif it is clear that we refer to the equivalence class. It is cal led an
+orbifold map with domain atlas Vand range atlas V′, in short orbifold map
+with(V,V′)or, if the specific orbifold atlases are not important, a charted
+orbifold map . The set of all orbifold maps with (V,V′)is denoted Orb(V,V′).
+For convenience we often denote an element ˆh∈Orb(V,V′)by
+Vˆh−→ V′.
+Remark 4.11.Instead of using the pair [P,ν]in the definition of charted
+orbifold maps one could also define a variant where [P,ν]is replaced by a
+mapping
+θ: Germ(Ψ( V))→Germ(Ψ( V′)),
+which maps germs of transitions to germs such that the equiva lence condi-
+tions in (R 4a) are satisfied by θ. This would stress the actual “germ nature”
+of[P,ν]and might simplify the compositions of charted orbifold map s as de-
+fined in Construction 5.9below and support the intuition for its definition.
+Anyhow, for the reasonings in this manuscript (and other con crete situa-
+tions) it is more convenient to work with the actual transiti ons and to have
+the possibility to choose “small” quasi-pseudogroups P, for which reason we
+decided to use the variant with the pair [P,ν].
+Propositions 4.7and4.9yield the following statement of which we omit
+the proof.
+Proposition 4.12. LetVbe a representative of U, andV′a representative
+ofU′. Then the set Orb(V,V′)of all orbifold maps with (V,V′)and the set
+Hom(Γ(V),Γ(V′))of all homomorphisms from Γ(V)toΓ(V′)are in bijection.
+More precisely, the construction in Proposition 4.7induces a bijection
+F1: Orb(V,V′)→Hom(Γ(V),Γ(V′)),
+and the construction in Proposition 4.9defines a bijection
+F2: Hom(Γ( V),Γ(V′))→Orb(V,V′),
+which is inverse to F1.
+5.The category of reduced orbifolds
+To define an orbifold category where the objects are orbifold s and the
+morphisms are equivalence classes of charted orbifold maps we have to answer
+the following questions:
+(i) When shall two charted orbifold maps be considered as equ al? In other
+words, what shall be the equivalence relation?
+(ii) What shall be the identity morphism of an orbifold?
+(iii) How does one compose ϕ∈Orb(V,V′)andψ∈Orb(V′,V′′)?
+(iv) What is the composition in the category?
+The leading idea is that charted orbifold maps are equivalen t if and only
+if they induce the same charted orbifold map on common refinem ents of
+the orbifold atlases. Therefore, we will introduce the noti on of an induced
+charted orbifold map.
+It turns out that answers to the questions ( ii) and ( iii) naturally extend
+to answers of ( i) and ( iv), and that the arising category has a counterpartTHE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 17
+in terms of marked atlas groupoids and homomorphisms. We sta rt with the
+definition of the identity morphism of an orbifold. This defin ition is based
+on the idea that the identity morphism of (Q,U)shall be represented by a
+collection of local lifts of idQwhich locally induce idSon some orbifold charts,
+and that each such collection which satisfies (R 2) shall be a representative.
+5.1.The identity morphism.
+Definition and Remark 5.1. Let(Q,U)and(Q′,U′)be orbifolds and let
+f:Q→Q′be a continuous map. Suppose that ˜fis a local lift of fw.r.t. the
+orbifold charts (V,G,π)∈ Uand(V′,G′,π′)∈ U′. Further suppose that
+λ: (W,K,χ)→(V,G,π)andµ: (W′,K′,χ′)→(V′,G′,π′)
+are embeddings between orbifold charts in Uresp. inU′such that
+˜f(λ(W))⊆µ(W′).
+Then the map
+˜g:=µ−1◦˜f◦λ:W→W′
+is a local lift of fw.r.t.(W,K,χ)and(W′,K′,χ′). We say that ˜finduces
+the local lift ˜gw.r.t.λandµ, and we call ˜gtheinduced lift of fw.r.t.˜f,λ
+andµ.
+V˜f/d47/d47V′
+λ(W)˜f|λ(W)/d47/d47/d63 /d31/d79/d79
+µ(W′)/d31 /d63/d79/d79
+Wλ/d69/d69☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛˜g/d47/d47/d60/d60λ/d60/d60 /d60/d60②②②②②②②②
+W′µ/d89/d89✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹/d99/d99µ/d99/d99 /d99/d99●●●●●●●●
+Suppose that ˜fis a local lift of the identity idQfor some orbifold (Q,U).
+Proposition 5.3below shows that ˜finduces the identity on sufficiently small
+orbifold charts. This means that locally ˜fis related to the identity itself
+via embeddings. In particular, ˜fis a local diffeomorphism. For its proof
+we need the following lemma, which is easily shown and crucia lly depends
+on the finiteness of Gand the Hausdorff property of M. We refer to [ 13,
+Lemma B.3] for the details of the construction of the G-stable neighborhood
+S.
+Lemma 5.2. LetMbe a manifold, Ga finite subgroup of Diff(M), and
+x∈M. There exist arbitrary small open G–stable neighborhoods Sofx.
+Moreover, one can choose Sso small that GS=Gx, the isotropy group of x.
+Proposition 5.3. Let(Q,U)be an orbifold and suppose that ˜fis a local lift
+ofidQw.r.t.(V,G,π),(V′,G′,π′)∈ U. For eachv∈Vthere exist a restric-
+tion(S,GS,π|S)of(V,G,π)withv∈Sand a restriction (S′,(G′)S′,π′|S′)of
+(V′,G′,π′)such that ˜f|Sis an isomorphism from (S,GS,π|S)to the orbifold
+chart(S′,(G′)S′,π′|S′). In particular, ˜f|Sinduces the identity idSw.r.t.idS
+and/parenleftBig
+˜f|S/parenrightBig−1
+.18 ANKE POHL
+Proof. Letv∈Vand setv′:=˜f(v). Thenπ(v) =π′(v′). By compatibility
+of orbifold charts there exist a restriction (W,H,χ)of(V,G,π)withv∈
+Wand an open embedding λ: (W,H,χ)→(V′,G′,π′)such thatλ(v) =
+v′. Lemma 5.2yields an open H–stable neighborhood SofvwithS⊆
+˜f−1(λ(W))∩W. Let
+˜g:=λ−1◦˜f|S:S→W
+denote the induced lift of idQ. Sinceχ◦˜g=χ, [8, Lemma 2.11] shows the
+existence of a unique h∈Hsuch that ˜g=h|S. Thus˜f|S=λ◦h|S:S→
+λ(h(S))is a diffeomorphism. In turn
+˜f|S: (S,HS,χ|S)→(˜f(S),G′
+˜f(S),π′|˜f(S))
+is an isomorphism of orbifold charts. /square
+Not each local lift of the identity is a global diffeomorphism , as the fol-
+lowing example shows.
+Example 5.4. LetQbe the open annulus in R2with inner radius 1and
+outer radius 2centered at the origin, i.e.,
+Q={w∈C|1<|w|<2}.
+The mapα:Q→C×R,
+α(w):=/parenleftbiggw2
+|w|2,|w|−1/parenrightbigg
+mapsQonto the cylinder Z:=S1×(0,1). Note that α(Q)coversZtwice.
+Then the map β:Z→C,
+β(z,s):=2
+2−sz
+is the linear projection of Zfrom the point (0,2)∈C×Rto the complex
+plane. The composed map ˜f=β◦α:Q→C,
+˜f(w):=2w2
+(3−|w|)|w|2
+is smooth (where we consider C=R2as a2-dimensional real manifold) and
+mapsQontoQ. Further it induces a homeomorphism between Q/{±id}
+andQ. Hence, if we endow Qwith the orbifold atlas
+/braceleftbig/parenleftbig
+Q,{±id},˜f/parenrightbig
+,/parenleftbig
+Q,{id},id/parenrightbig/bracerightbig
+,
+then˜fis a local lift of idQw.r.t.(Q,{±id},˜f)and(Q,{id},id)but not a
+global diffeomorphism.
+Proposition 5.5. Let(Q,U)be an orbifold and {˜fi}i∈Ia family of local
+lifts ofidQwhich satisfies (R 2). Then there exists a pair (P,ν)such that
+(idQ,{˜fi}i∈I,P,ν)is a representative of an orbifold map on (Q,U). The
+pair(P,ν)is unique up to equivalence of representatives of orbifold m aps.
+Proof. This follows immediately from Proposition 5.3in combination with
+(R4a). /squareTHE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 19
+Proposition 5.6. LetQbe a topological space and suppose that UandU′
+are orbifold structures on Q. Let
+ˆf=/parenleftbig
+f,{˜fi}i∈I,[P,ν]/parenrightbig
+be a charted orbifold map for which f= idQ, the domain atlas Vis a rep-
+resentative of U, the range family V′, which here is an orbifold atlas, is a
+representative of U′, and for each i∈I, the map ˜fiis a local diffeomorphism.
+ThenU=U′.
+Proof. Let(Vi,Gi,πi)∈ V,(V′
+j,G′
+j,π′
+j)∈ V′andx∈Vi,y∈V′
+jsuch that
+πi(x) =π′
+j(y). Since˜fi:Vi→V′
+iis a local diffeomorphism, there are open
+neighborhoods UofxinViandU′of˜fi(x)inV′
+isuch that ˜fi|U:U→U′is
+a diffeomorphism. We have
+π′
+i/parenleftbig˜fi(x)/parenrightbig
+=πi(x) =π′
+j(y).
+Therefore there exist open neighborhoods W′
+1of˜fi(x)inU′andW′
+2ofyin
+V′
+jand a diffeomorphism h:W′
+1→W′
+2satisfyingπ′
+j◦h=π′
+i. Shrinking U
+shows that (Vi,Gi,πi)and(V′
+j,G′
+j,π′
+j)are compatible. Thus U=U′./square
+The following example shows that the requirement in Proposi tion5.6that
+the local lifts be local diffeomorphisms is essential.
+Example 5.7. Recall the orbifolds (Q,Ui),i= 1,2, from Example 2.2, the
+representatives V1:={V1}andV2:={V2}ofU1resp.U2, and setg(x):=x2
+forx∈(−1,1). Thengis a lift of idQw.r.t.V2andV1. Further let
+P:={±id(−1,1)}andν(±id(−1,1)):= id(−1,1).
+Then(idQ,{g},[P,ν])is an orbifold map with (V2,V1)from(Q,U2)to(Q,U1),
+butU1/\e}a⊔io\slash=U2.
+Motivated by Propositions 5.5and5.6we make the following definition.
+Definition 5.8. Let(Q,U)be an orbifold and let ˆf= (f,{˜fi}i∈I,[P,ν])be
+a charted orbifold map whose domain atlas is a representativ e ofU. If and
+only iff= idQand˜fiis a local diffeomorphism for each i∈I, we call ˆfa
+lift of the identity id(Q,U)or arepresentative ofid(Q,U). The set of all lifts of
+id(Q,U)is the identity morphism id(Q,U)of(Q,U).
+5.2.Composition of charted orbifold maps.
+Construction 5.9. Let(Q,U),(Q′,U′)and(Q′′,U′′)be orbifolds, and
+V:={(Vi,Gi,πi)|i∈I},V′:={(V′
+j,G′
+j,π′
+j)|j∈J}
+resp.V′′be representatives for U,U′resp.U′′, whereVresp.V′are indexed
+byIresp.J. Suppose that
+ˆf= (f,{˜fi}i∈I,[Pf,νf])∈Orb(V,V′)
+and
+ˆg= (g,{˜gj}j∈J,[Pg,νg])∈Orb(V′,V′′)20 ANKE POHL
+are charted orbifold maps and that α:I→Jis the unique map such that
+for eachi∈I,˜fiis a local lift of fw.r.t.(Vi,Gi,πi)and(V′
+α(i),G′
+α(i),π′
+α(i)).
+The composition
+ˆg◦ˆf:=ˆh= (h,{˜hi}i∈I,[Ph,νh])∈Orb(V,V′′)
+is given by h:=g◦fand˜hi:= ˜gα(i)◦˜fifor alli∈I. To construct a
+representative (Ph,νh)of[Ph,νh]we fix representatives (Pf,νf)and(Pg,νg)
+of[Pf,νf]and[Pg,νg], respectively. The leading idea to define (Ph,νh)is
+to takePh=Pfandνh=νg◦νf. But since νf(λ)is not necessarily in Pg
+forλ∈Pf, the composition νg◦νfmight be ill-defined. In the following we
+refine this idea.
+Letµ∈Pfand suppose that domµ⊆Viandcodµ⊆Vjfor the orbifold
+charts(Vi,Gi,πi)and(Vj,Gj,πj)inV. By (R 4a)
+˜fj◦µ=νf(µ)◦˜fi|domµ,
+whereνf(µ)∈Ψ(U′). By possibly shrinking domains, we may assume that
+νf(µ)∈Ψ(V′). Forx∈domµwe setyx:=˜fi(x), which is an element of
+domνf(µ). Hence we find (and fix a choice) ξµ,x∈Pgwithyx∈domξµ,x
+and an open set U′
+µ,x⊆domξµ,x∩domνf(µ)such thatyx∈U′
+µ,xand
+ξµ,x|U′µ,x=νf(µ)|U′µ,x.
+Then we find (and fix) an open set Uµ,x⊆domµwithx∈Uµ,xsuch that
+˜fi(Uµ,x)⊆U′
+µ,x. By adjusting choices we achieve that for µ1,µ2∈Pfand
+x1∈domµ1,x2∈domµ2we either have
+(3) µ1|Uµ1,x1/\e}a⊔io\slash=µ2|Uµ2,x2orξµ1,x1=ξµ2,x2.
+Define
+Ph:=/braceleftbig
+µ|Uµ,x/vextendsingle/vextendsingleµ∈Pf, x∈domµ/bracerightbig
+,
+which obviously is a quasi-pseudogroup generating Ψ(V), and set
+νh/parenleftbig
+µ|Uµ,x/parenrightbig:=νg(ξµ,x)
+forµ|Uµ,x∈Ph. Property ( 3) yields that νhis a well-defined map from
+PhtoΨ(U′′). One easily sees that νhsatisfies (R 4a) - (R 4d), and that the
+equivalence class of (Ph,νh)does not depend on the choices we made for the
+construction of Phandνh.
+Remark 5.10.The construction of the composition of two charted orbifold
+maps immediately implies that the maps F1andF2(see Proposition 4.12)
+are both functorial.
+The following lemma provides the definition of induced chart ed orbifold
+map and shows its relation to lifts of the identity.
+Lemma and Definition 5.11. Let(Q,U)and(Q′,U′)be orbifolds. Further
+let
+V={(Vi,Gi,πi)|i∈I}be a representative of U, indexed by I,
+V′={(V′
+l,G′
+l,π′
+l)|l∈L}be a representative of U′, indexed by L,
+ˆf=/parenleftbig
+f,{˜fi}i∈I,[Pf,νf]/parenrightbig
+∈Orb(V,V′),THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 21
+and letβ:I→Lbe the unique map such that for each i∈I,˜fiis a local lift
+offw.r.t.(Vi,Gi,πi)and(V′
+β(i),G′
+β(i),π′
+β(i)). Suppose that we have
+•a representative W={(Wj,Hj,ψj)|j∈J}ofU, indexed by J,
+•a subset {(W′
+j,H′
+j,ψ′
+j)|j∈J}ofU′, indexed by J(not necessarily
+an orbifold atlas),
+•a mapα:J→I,
+•for eachj∈J, an embedding
+λj:/parenleftbig
+Wj,Hj,ψj/parenrightbig
+→/parenleftbig
+Vα(j),Gα(j),πα(j)/parenrightbig
+,
+and an embedding
+µj:/parenleftbig
+W′
+j,H′
+j,ψ′
+j/parenrightbig
+→/parenleftbig
+V′
+β(α(j)),G′
+β(α(j)),π′
+β(α(j))/parenrightbig
+such that
+˜fα(j)/parenleftbig
+λj(Wj)/parenrightbig
+⊆µj(W′
+j).
+For eachj∈Jset
+˜hj:=µ−1
+j◦˜fα(j)◦λj:Wj→W′
+j.
+Then
+(i)ε:=/parenleftbig
+idQ,{λj}j∈J,[Pε,νε]/parenrightbig
+∈Orb(W,V)(with[Pε,νε]provided by
+Proposition 5.5) is a lift of id(Q,U).
+(ii)The set{(W′
+j,H′
+j,ψ′
+j)|j∈J}and the family {µj}j∈Jcan be extended
+to a representative
+W′=/braceleftbig
+(W′
+k,H′
+k,ψ′
+k)/vextendsingle/vextendsinglek∈K/bracerightbig
+ofU′and a family of embeddings {µk}k∈Ksuch that
+ε′:=/parenleftbig
+idQ′,{µk}k∈K,[Pε′,νε′]/parenrightbig
+∈Orb(W′,V′)
+(with[Pε′,νε′]provided by Proposition 5.5) is a lift of the identity
+id(Q′,U′).
+(iii)There is a uniquely determined equivalence class [Ph,νh]such that
+ˆh:= (f,{˜hj}j∈J,[Ph,νh])∈Orb(W,W′)
+and such that the diagram
+Vˆf/d47/d47V′
+Wε/d63/d63⑦⑦⑦⑦⑦⑦⑦⑦ˆh/d47/d47W′ε′/d97/d97❇❇❇❇❇❇❇❇
+commutes.
+We say that ˆhisinduced byˆf.
+Proof. (i) is clear by Proposition 5.3and5.5. To show that ( ii) holds we
+construct one possible extension: Let
+y∈Q′\/uniondisplay
+j∈Jψ′
+j(W′
+j).
+Then there is a chart (V′,G′,π′)∈ V′such thaty∈π′(V′). Extend the set
+{(W′
+j,H′
+j,ψ′
+j)|j∈J}22 ANKE POHL
+with(V′,G′,π′)and the family {µj}j∈JwithidV′. If this is done iteratively,
+one finally gets an orbifold atlas of Q′as wanted. Then Proposition 5.3
+and5.5yield the remaining claim of ( ii). The following considerations are
+independent of the specific choices of extensions. Concerni ng (iii) we remark
+that each ˜hjis obviously a local lift of f. Fix a representative (Pf,νf)of
+[Pf,νf]. In the following we construct a pair (Ph,νh)for which ˆhis an
+orbifold map and the diagram in ( iii) commutes. It will be clear from the
+construction that the equivalence class [Ph,νh]is independent of the choice of
+(Pf,νf)and uniquely determined by the requirement of the commutati vity
+of the diagram. Let γ∈Ψ(W)andx∈domγ. Possibly shrinking the
+domain ofγ, we may assume that domγ⊆Wjandcodγ⊆Wkfor some
+j,k∈J. In the following we further shrink the domain of γto be able to
+defineνhas a composition of νfwith elements of {µj}j∈J. Lety:=λj(x).
+Since
+˜γ:=λk◦γ◦(λj|domγ)−1:λj(domγ)→λk(codγ)
+is an element of Ψ(V), we findβγ∈Pfsuch thaty∈domβγandgermyβγ=
+germy˜γ. Then
+z:=˜fα(j)(y)∈domνf(βγ)∩µj(W′
+j).
+Since
+νf(βγ)(z) =˜fα(k)(βγ(y))∈µk(W′
+k),
+the set
+U′:= domνf(βγ)∩µj(W′
+j)∩νf(βγ)−1(µk(W′
+k))
+is an open neighborhood of z. Define
+U1:={w∈domβγ∩λj(domγ)|germwβγ= germw˜γ},
+which is an open neighborhood of y. Then also
+U:=U1∩˜f−1
+α(j)(U′)
+is an open neighborhood of y. We fix an open neighborhood Uγ,xofxin
+λ−1
+j(U). Further we suppose that for γ1,γ2∈Ψ(W),x1∈domγ1,x2∈
+domγ2, we either have
+(4) γ1|Uγ1,x1/\e}a⊔io\slash=γ2|Uγ2,x2orνf(βγ1) =νf(βγ2).
+Then we define
+Ph:=/braceleftbig
+γ|Uγ,x/vextendsingle/vextendsingleγ∈Ψ(W), x∈domγ/bracerightbig
+and set
+νh/parenleftbig
+γ|Uγ,x/parenrightbig:=µ−1
+k◦νf(βγ)◦µj
+forγ|Uγ,x∈Phwithx∈Wjandγ(x)∈Wk(j,k∈J). The map νh:Ph→
+Ψ(W′)is well-defined by ( 4). One easily checks that (Ph,νh)satisfies all
+requirements of ( iii). /square
+We consider two charted orbifold maps as equivalent if they i nduce the
+same charted orbifold map on common refinements of the orbifo ld atlases.
+The following definition provides a precise specification of this idea.THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 23
+Definition 5.12. Let(Q,U)and(Q′,U′)be orbifolds. Further let V1,V2
+be representatives of U, andV′
+1,V′
+2be representatives of U′. Suppose that
+ˆf1∈Orb(V1,V′
+1)andˆf2∈Orb(V2,V′
+2). We call ˆf1andˆf2equivalent
+(ˆf1∼ˆf2) if there are a representative WofU, a representative W′of
+U′,ε1∈Orb(W,V1),ε2∈Orb(W,V2)lifts ofid(Q,U),ε′
+1∈Orb(W′,V′
+1),
+ε′
+2∈Orb(W′,V′
+2)lifts ofid(Q′,U′), and a map ˆh∈Orb(W,W′)such that the
+diagram
+V1ˆf1/d47/d47V′
+1
+Wε1/d62/d62⑦⑦⑦⑦⑦⑦⑦⑦ˆh/d47/d47
+ε2/d32/d32❅❅❅❅❅❅❅❅ W′ε′
+1/d96/d96❆❆❆❆❆❆❆
+ε′
+2 /d126/d126⑥⑥⑥⑥⑥⑥⑥⑥
+V2ˆf2/d47/d47V′
+2
+commutes.
+Proposition 5.15below shows that ∼is indeed an equivalence relation.
+For its proof we need the following two lemmas. The first lemma discusses
+how local lifts which belong to the same charted orbifold map are related to
+each other. The second lemma shows that two charted orbifold maps which
+are induced from the same charted orbifold map induce the sam e charted
+orbifold map on common refinements of orbifold atlases. This means that ∼
+satisfies the so-called diamond property.
+Lemma 5.13. Let(Q,U)and(Q′,U′)be orbifolds and let
+ˆf:= (f,{˜fi}i∈I,[P,ν])∈Orb(V,V′)
+be a charted orbifold map where Vis a representative of UandV′one ofU′.
+Suppose that we have orbifold charts (Va,Ga,πa),(Vb,Gb,πb)∈ Vand points
+xa∈Va,xb∈Vbsuch thatπa(xa) =πb(xb). Then there are arbitrarily small
+orbifold charts (W,K,χ)∈ U,(W′,K′,χ′)∈ U′and embeddings
+λ: (W,K,χ)→(Va,Ga,πa)
+λ′: (W′,K′,χ′)→(V′
+a,G′
+a,π′
+a)
+µ: (W,K,χ)→(Vb,Gb,πb)
+µ′: (W′,K′,χ′)→(V′
+b,G′
+b,π′
+b)
+withxa∈λ(W)andxb∈µ(W)such that the induced lift ˜goffw.r.t.˜fa,λ,λ′
+coincides with the one induced by ˜fb,µ,µ′. In other words, the diagram
+Va˜fa/d47/d47V′
+a
+Wλ/d62/d62⑦⑦⑦⑦⑦⑦⑦⑦˜g/d47/d47
+µ/d32/d32❅❅❅❅❅❅❅❅ W′λ′/d96/d96❆❆❆❆❆❆❆
+µ′/d126/d126⑥⑥⑥⑥⑥⑥⑥⑥
+Vb˜fb/d47/d47V′
+b24 ANKE POHL
+commutes.
+Proof. By compatibility of orbifold charts we find an arbitrarily sma ll re-
+striction (W,K,χ)of(Va,Ga,πa)withxa∈Wand an embedding
+µ: (W,K,χ)→(Vb,Gb,πb)
+such thatµ(xa) =xb. Thenµ:W→µ(W)is an element of Ψ(V). Fix a
+representative (P,ν)of[P,ν]. Hence there is γ∈Pwithxa∈domγand an
+open neighborhood Uofxasuch thatU⊆domγ∩Wand
+µ|U=γ|U.
+W.l.o.g.,γ=µ. Property (R 4a) yields that
+ν(µ)◦˜fa|W=˜fb◦µ.
+By shrinking the domain of ν(µ), we can achieve that codν(µ)⊆V′
+band
+still˜fa(W)⊆domν(µ) =:W′. Withµ′:=ν(µ)it follows
+˜fb(µ(W)) =µ′(˜fa(W))⊆µ′(W′)
+and further
+˜fa|W= (µ′)−1◦˜fb◦µ.
+This proves the claim. /square
+Lemma 5.14. Let(Q,U)and(Q′,U′)be orbifolds, Va representative of U,
+andV′one ofU′. Further let ˆf∈Orb(V,V′). Suppose that ˆh∈Orb(W1,W′
+1)
+andˆg∈Orb(W2,W′
+2)are both induced by ˆf. Then we find a representative
+WofUand charted orbifold maps ε1∈Orb(W,W1),ε2∈Orb(W,W2)
+which are lifts of id(Q,U), and a representative W′ofU′and charted orbifold
+mapsε′
+1∈Orb(W′,W′
+1),ε′
+2∈Orb(W′,W′
+2)which are lifts of id(Q′,U′), and
+a charted orbifold map ˆk∈Orb(W,W′)such that the diagram
+W1ˆh/d47/d47W′
+1
+Wε1/d62/d62⑤⑤⑤⑤⑤⑤⑤⑤ˆk/d47/d47
+ε2/d32/d32❇❇❇❇❇❇❇❇ W′ε′
+1/d97/d97❈❈❈❈❈❈❈
+ε′
+2 /d125/d125④④④④④④④④
+W2ˆg/d47/d47W′
+2
+commutes. If the orbifolds are second-countable, we can cho oseW,W′to be
+countable.
+Proof. Suppose that ˆf= (f,{˜fa}a∈A,[Pf,νf]),ˆh= (f,{˜hi}i∈I,[Ph,νh])and
+ˆg= (f,{˜gj}j∈J,[Pg,νg]). Let
+W1:={(W1,i,H1,i,ψ1,i)|i∈I},indexed by I,
+W′
+1:={(W′
+1,k,H′
+1,k,ψ′
+1,k)|k∈K},indexed by K,
+W2:={(W2,j,H2,j,ψ2,j)|j∈J},indexed by J,
+W′
+2:={(W′
+2,l,H′
+2,l,ψ′
+2,l)|l∈L},indexed by L,
+and letα1:I→Kresp.α2:J→Lbe the map such that for each i∈I,
+˜hiis a local lift of fw.r.t.(W1,i,H1,i,ψ1,i)and(W′
+1,α1(i),H′
+1,α1(i),ψ′
+1,α1(i))THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 25
+resp. for each j∈J,˜gjis a local lift of fw.r.t.(W2,j,H2,j,ψ2,j)and
+(W′
+2,α2(j),H′
+2,α2(j),ψ′
+2,α2(j)). Further let
+δ1= (idQ,{λ1,i}i∈I,[R1,σ1])∈Orb(W1,V),
+δ2= (idQ,{λ2,j}j∈J,[R2,σ2])∈Orb(W2,V)
+be lifts of id(Q,U)and
+δ′
+1= (idQ′,{µ1,k}k∈K,[R′
+1,σ′
+1])∈Orb(W′
+1,V′),
+δ′
+2= (idQ′,{µ2,l}l∈L,[R′
+2,σ′
+2])∈Orb(W′
+2,V′)
+be lifts of id(Q′,U′)such that ˆf◦δ1=δ′
+1◦ˆhandˆf◦δ2= ˆg◦δ′
+2. W.l.o.g.
+we assume that all λ1,i,µ1,k,λ2,jandµ2,lare embeddings. We will use
+Lemma 5.11to show the existence of ˆk. More precisely, we attach to each
+q∈Qan orbifold chart (Wq,Hq,ψq)∈ Uwithq∈ψq(Wq)and an orbifold
+chart(W′
+q,H′
+q,ψ′
+q)∈ U′withf(q)∈ψ′
+q(W′
+q). We consider orbifold charts
+defined for distinct qto be distinct. In this way, we get a representative
+(5) W:={(Wq,Hq,ψq)|q∈Q}
+ofUwhich is indexed by Q, and a subset {(W′
+q,H′
+q,ψ′
+q)|q∈Q}ofU′,
+indexed byQas well. Moreover, we will find maps β1:Q→Iandβ2:Q→J
+and embeddings
+ξ1,q:/parenleftbig
+Wq,Hq,ψq/parenrightbig
+→/parenleftbig
+W1,β1(q),H1,β1(q),ψ1,β1(q)/parenrightbig
+ξ2,q:/parenleftbig
+Wq,Hq,ψq/parenrightbig
+→/parenleftbig
+W2,β2(q),H2,β2(q),ψ2,β2(q)/parenrightbig
+χ1,q:/parenleftbig
+W′
+q,H′
+q,ψ′
+q/parenrightbig
+→/parenleftbig
+W′
+1,α1(β1(q)),H′
+1,α1(β1(q)),ψ′
+1,α1(β1(q))/parenrightbig
+χ2,q:/parenleftbig
+W′
+q,H′
+q,ψ′
+q/parenrightbig
+→/parenleftbig
+W′
+2,α2(β2(q)),H′
+2,α2(β2(q)),ψ′
+2,α2(β2(q))/parenrightbig
+such that for each q∈Qthe local lift ˜kqoffinduced by ˜hβ1(q),ξ1,qandχ1,q
+coincides with the one induced by ˜gβ2(q),ξ2,qandχ2,q. Then Lemma 5.11
+shows that ˆhresp.ˆginduces a charted orbifold map (f,{˜kq}q∈Q,[P1,ν1])
+resp.(f,{˜kq}q∈Q,[P2,ν2]). It then remains to show that we can choose all
+the embeddings ξ1,q,ξ2,q,χ1,q,χ2,qin a way that [P1,ν1]equals[P2,ν2].
+Letq∈Q. We fixi∈Isuch thatq∈ψ1,i(W1,i)and we pick w1∈W1,i
+withq=ψ1,i(w1). We setβ1(q):=i. Further we fix j∈Jsuch that
+q∈ψ2,j(W2,j), and pick an element w2∈W2,jwithq=ψ2,j(w2). We set
+β2(q):=j. By Lemma 5.13we find orbifold charts (Wq,Hq,ψq)∈ Uwith
+q∈ψq(Wq), sayq=ψq(wq), and(W′
+q,H′
+q,ψ′
+q)∈ U′withf(q)∈ψ′
+q(W′
+q)and
+embeddings ξ1,q,ξ2,q,χ1,q,χ2,qwithw1=ξ1,q(wq),w2=ξ2,q(wq), and a26 ANKE POHL
+local lift ˜kqoffsuch that the diagram
+λ1,β1(q)/parenleftbig
+W1,β1(q)/parenrightbig˜fβ1(q)/d47/d47µ1,α1(β1(q))/parenleftbig
+W′
+1,α1(β1(q))/parenrightbig
+W1,β1(q)λ1,β1(q)/d79/d79
+˜hβ1(q)/d47/d47W′
+1,α1(β1(q))µ1,α1(β1(q))/d79/d79
+Wqξ1,q/d56/d56qqqqqqqqqqqq
+ξ2,q/d38/d38▼▼▼▼▼▼▼▼▼▼▼▼▼˜kq/d47/d47W′
+qχ1,q/d104/d104PPPPPPPPPPPPP
+χ2,q/d118/d118♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥
+W2,β2(q)˜gβ2(q)/d47/d47
+λ2,β2(q)
+/d15/d15W′
+2,α2(β2(q))
+µ2,α2(β2(q))
+/d15/d15
+λ2,β2(q)/parenleftbig
+W2,β2(q)/parenrightbig˜fβ2(q)/d47/d47µ2,α2(β2(q))/parenleftbig
+W′
+2,α2(β2(q))/parenrightbig
+commutes. We may assume that ξ1,q= idandχ1,q= id. Now
+η:=λ2,β2(q)◦ξ2,q◦λ−1
+1,β1(q):λ1,β1(q)(Wq)→λ2,β2(q)/parenleftbig
+ξ2,q(Wq)/parenrightbig
+is an element of Ψ(V)withy:=λ1,β1(q)(ξ1,q(wq))in its domain. We pick
+a representative (Pf,νf)of[Pf,νf]. Then there is an element γ∈Pfwith
+y∈domγand an open neighborhood Uofysuch thatU⊆domγ∩domη
+andη|U=γ|U. By (R 4a),
+νf(γ)◦˜fβ1(q)|U=˜fβ2(q)◦γ|U=˜fβ2(q)◦η|U.
+The map
+µ:=µ2,α2(β2(q))◦χ2,q◦µ−1
+1,α1(β1(q)):µ1,α1(β1(q))(W′
+q)→µ2,α2(β2(q))/parenleftbig
+χ2,q(W′
+q)/parenrightbig
+is a diffeomorphism as well. Further there exists an open neig hborhoodV
+ofysuch that
+˜fβ2(q)◦η|V=µ◦˜fβ1(q)|V.
+Hence
+νf(γ)◦˜fβ1(q)=µ◦˜fβ1(q)
+on some neighborhood of y. Therefore, after possibly shrinking Wq, we can
+redefineW′
+q,χ2,qand˜kqsuch that
+(6) χ2,q=µ−1
+2,α2(β2(q))◦νf(γ)◦µ1,α1(β1(q))|W′q.
+We remark that this redefinition might be quite serious if ˜fβ1(q)and hence
+˜hβ1(q),˜gβ2(q)and˜fβ2(q)are highly non-injective. But since these maps all
+behave in the same way, we may perform the changes without run ning into
+problems. Let Wbe defined by ( 5). Lemma 5.11, more precisely its proof,
+shows that ˆhresp.ˆginduces the orbifold maps
+ˆk1= (f,{˜kq}q∈Q,[P1,ν1])resp.ˆk2= (f,{˜kq}q∈Q,[P2,ν2])
+with(W,W′), whereW′is a representative of U′which contains the set
+{(W′
+q,H′
+q,ψ′
+q)|q∈Q}THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 27
+(the proof of Lemma 5.11shows that we can indeed have the same W′for
+ˆk1andˆk2).
+It remains to show that [P1,ν1] = [P2,ν2]. Recall from Lemma 5.11that
+[P1,ν1]is uniquely determined by ˆh,{ξ1,q}q∈Qand{χ1,q}q∈Q, and analo-
+gously for [P2,ν2]. Alternatively, we may consider ˆk1andˆk2to be induced
+byˆf. Thus, [P1,ν1]is uniquely determined by ˆf,{λ1,β1(q)◦ξ1,q}q∈Qand
+{µ1,α1(β1(q))◦χ1,q}q∈Q, and[P2,ν2]is uniquely determined by ˆf,{λ2,β2(q)◦
+ξ2,q}q∈Qand{µ2,α2(β2(q))◦χ2,q}q∈Q. We fix a representative (Pf,νf)of
+[Pf,νf]. Letγbe a change of charts in Ψ(W)andx∈domγ. Suppose
+domγ⊆Wpandcodγ⊆Wq. Using the same arguments and notation as
+in the proof of Lemma 5.11(without discussing the necessary shrinking of
+domains, since we are only interested in equality in a neighb orhood ofx) we
+have
+βh=λ1,β1(q)◦γ◦λ−1
+1,β1(p),
+βg=λ2,β2(q)◦ξ2,q◦γ◦ξ−1
+2,p◦λ−1
+2,β2(p),
+ν1(γ) =µ−1
+1,α1(β1(q))◦νf(βh)◦µ1,α1(β1(p)),
+ν2(γ) =χ−1
+2,q◦µ−1
+2,α2(β2(q))◦νf(βg)◦µ2,α2(β2(p))◦χ2,p.
+Hence
+βg=λ2,β2(q)◦ξ2,q◦λ−1
+1,β1(q)◦βh◦λ1,β1(p)◦ξ−1
+2,p◦λ−1
+2,β2(p).
+Definition ( 6) shows that
+νf(λ2,β2(q)◦ξ2,q◦ξ−1
+1,q◦λ−1
+1,β1(q)) =µ2,α2(β2(q))◦χ2,q◦µ−1
+1,α1(β1(q)).
+Then
+ν2(γ) =µ−1
+1,α1(β1(q))◦νf(βh)◦µ1,α1(β1(p))=ν1(γ).
+Hence the induced equivalence classes [P1,ν1]and[P2,ν2]indeed coincide.
+The liftε1ofid(Q,U)is given by the family {ξ1,q}q∈Q, the liftε2by{ξ2,q}q∈Q,
+the liftε′
+1ofid(Q′,U′)is any extension of {χ1,q}q∈Q, and the lift ε′
+2is any
+extension of {χ2,q}q∈Q.
+If the orbifolds are second-countable, we find countable sub families of W
+andW′which are themselves orbifold atlases and restricted to whi ch the
+orbifold map ˆkstill satisfies the statement of the lemma. Alternatively, w e
+could have started the construction with an appropriate cou ntable subset of
+Q. /square
+The following proposition now follows immediately.
+Proposition 5.15. The relation ∼from Definition 5.12is an equivalence
+relation.
+The equivalence class of a charted orbifold map ˆfwith respect to the
+equivalence from Definition 5.12is denoted by [ˆf]. It will always be clear from
+context whether ˆfis a charted orbifold map and [ˆf]denotes an equivalence
+class of charted orbifold maps, or ˆfis a representative of an orbifold map
+and[ˆf]denotes an equivalence class of representatives, that is, a charted
+orbifold map (cf. Definition 4.10).28 ANKE POHL
+5.3.The orbifold category. Now we can define the category of reduced
+orbifolds.
+Definition 5.16. The category Orbof reduced (paracompact resp. second-
+countable) orbifolds is defined as follows: In both cases, th e class of objects is
+the class of orbifolds. For two paracompact orbifolds (Q,U)and(Q′,U′), the
+morphisms ( orbifold maps ) from(Q,U)to(Q′,U′)are the equivalence classes
+[ˆf]of all charted orbifold maps ˆf∈Orb(V,V′)whereVis any representative
+ofU, andV′is any representative of U′, that is
+Morph/parenleftbig
+(Q,U),(Q′,U′)/parenrightbig:=/braceleftbigg/bracketleftbigˆf/bracketrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆf∈Orb(V,V′),Vrepr. ofU,
+V′representative of U′/bracerightbigg
+.
+For second-countable orbifolds we restrict to countable re presentatives of the
+orbifold structures. We now describe the composition in Orb. For this let
+[ˆf]∈Morph/parenleftbig
+(Q,U),(Q′,U′)/parenrightbig
+and[ˆg]∈Morph/parenleftbig
+(Q′,U′),(Q′′,U′′)/parenrightbig
+be orbifold maps. Choose representatives ˆf∈Orb(V,V′)of[ˆf]andˆg∈
+Orb(W′,W′′)of[ˆg]. Then find (countable, if the orbifolds are second-
+countable) representatives K,K′,K′′ofU,U′,U′′, resp., and lifts of identity
+ε∈Orb(K,V),ε′
+1∈Orb(K′,V′),ε′
+2∈Orb(K′,W′),ε′′∈Orb(K′′,W′′)and
+two charted orbifold maps ˆh∈Orb(K,K′),ˆk∈Orb(K′,K′′)such that the
+diagram
+Vˆf/d47/d47V′W′ˆg/d47/d47W′′
+Kε/d63/d63⑧⑧⑧⑧⑧⑧⑧⑧ ˆh/d47/d47K′ε′
+1/d96/d96❆❆❆❆❆❆❆❆ε′
+2/d61/d61⑤⑤⑤⑤⑤⑤⑤⑤ˆk/d47/d47K′′ε′′/d97/d97❉❉❉❉❉❉❉❉
+commutes. The composition of [ˆg]and[ˆf]is defined to be
+[ˆg]◦[ˆf]:= [ˆk◦ˆh].
+The following lemma shows that this composition is always po ssible, Propo-
+sition 5.18below that it is well-defined.
+Lemma 5.17. Let(Q,U),(Q′,U′)and(Q′′,U′′)be orbifolds. Further let
+Vbe a representative of U,V′andW′be representatives of U′, andW′′a
+representative of U′′. Suppose that ˆf∈Orb(V,V′)andˆg∈Orb(W′,W′′).
+Then there exist representatives KofU,K′ofU′,K′′ofU′′, lifts of the
+respective identities ε∈Orb(K,V),η1∈Orb(K′,V′),η2∈Orb(K′,W′),δ∈
+Orb(K′′,W′′), and charted orbifold maps ˆh∈Orb(K,K′),ˆk∈Orb(K′,K′′)
+such that the diagram
+Vˆf/d47/d47V′W′ˆg/d47/d47W′′
+Kˆh/d47/d47ε/d63/d63⑧⑧⑧⑧⑧⑧⑧⑧
+K′ˆk/d47/d47η1/d96/d96❆❆❆❆❆❆❆❆η2/d61/d61⑤⑤⑤⑤⑤⑤⑤⑤
+K′′δ/d97/d97❉❉❉❉❉❉❉❉
+commutes. For second-countable orbifolds we can choose K,K′andK′′to be
+countable.THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 29
+Proof. Letˆf=/parenleftbig
+f,{˜fi}i∈I,[Pf,νf]/parenrightbig
+andˆg=/parenleftbig
+g,{˜gj}j∈J,[Pg,νg]/parenrightbig
+. Suppose
+that
+V={(Vi,Gi,πi)|i∈I},indexed by I,
+V′={(V′
+c,G′
+c,π′
+c)|c∈C},indexed by C,
+W′={(W′
+j,H′
+j,ψ′
+j)|j∈J},indexed by J,
+W′′={(W′′
+d,H′′
+d,ψ′′
+d)|d∈D},indexed by D.
+Letτ:I→Cbe the map such that for each i∈I,˜fiis a local lift of fw.r.t.
+(Vi,Gi,πi)and(V′
+τ(i),G′
+τ(i),π′
+τ(i)), andν:J→Dthe map such that for each
+j∈J,˜gjis a local lift of gw.r.t.(W′
+j,H′
+j,ψ′
+j)and(W′′
+ν(j),H′′
+ν(j),ψ′′
+ν(j)). By
+Lemma 5.11it suffices to find
+•a representative K={(Ka,La,χa)|a∈A}ofU, indexed by A,
+•a representative K′={(K′
+b,L′
+b,χ′
+b)|b∈B}ofU′, indexed by B,
+•a subset {(K′′
+b,L′′
+b,χ′′
+b)|b∈B}ofU′′, indexed by (the same set) B,
+•a mapα:A→I,
+•an injective map β:A→B,
+•for eacha∈A, an embedding
+λa: (Ka,La,χa)→(Vα(a),Gα(a),πα(a))
+and an embedding
+µa: (K′
+β(a),L′
+β(a),χ′
+β(a))→(V′
+τ(α(a)),G′
+τ(α(a)),π′
+τ(α(a)))
+such that
+˜fα(a)/parenleftbig
+λa(Ka)/parenrightbig
+⊆µa(K′
+β(a)),
+•a mapγ:B→J,
+•for eachb∈B, an embedding
+̺b: (K′
+b,L′
+b,χ′
+b)→(W′
+γ(b),H′
+γ(b),ψ′
+γ(b))
+and an embedding
+σb: (K′′
+b,L′′
+b,χ′′
+b)→(W′′
+ν(γ(b)),H′′
+ν(γ(b)),ψ′′
+ν(γ(b)))
+such that
+˜gγ(b)/parenleftbig
+̺b(K′
+b)/parenrightbig
+⊆σb(K′′
+b).
+Letq∈Qand setr:=f(q). We fixi∈Iandj∈Jsuch thatq∈
+πi(Vi)andr∈ψ′
+j(W′
+j). Further we choose v′∈V′
+τ(i)andw′∈W′
+jsuch
+thatπ′
+τ(i)(v′) =r=ψ′
+j(w′). By compatibility of orbifold charts we find a
+restriction (K′
+q,L′
+q,χ′
+q)of(V′
+τ(i),G′
+τ(i),π′
+τ(i))withv′∈K′
+qand an embedding
+̺q: (K′
+q,L′
+q,χ′
+q)→(W′
+j,H′
+j,ψ′
+j).
+Since˜fiis continuous, there is a restriction (Kq,Lq,χq)of(Vi,Gi,πi)such
+thatq∈χq(Kq)and˜fi(Kq)⊆K′
+q. We set
+(K′′
+q,L′′
+q,χ′′
+q):= (W′′
+j,H′′
+j,ψ′′
+j).30 ANKE POHL
+We consider orbifold charts constructed for distinct qto be distinct. Then
+we set
+A:=Q, α (q):=i, λq:= id, µq:= id,
+B:=Q⊔Q′\f(Q), β(q):=q, γ(q):=j, σq:= id forq∈Q.
+Forq′∈Q′\f(Q)we fixj∈Jwithq′∈ψ′
+j(W′
+j)and setγ(q′):=j. Further
+we set
+(K′
+q′,L′
+q′,χ′
+q′):= (W′
+j,H′
+j,ψ′
+j)and(K′′
+q′,L′′
+q′,χ′′
+q′):= (W′′
+j,H′′
+j,ψ′′
+j).
+Again we consider orbifold charts build for distinct q′to be distinct and to
+be distinct from all those defined for some q∈Q, and we define ̺q′:= id
+andσq′:= id. Then all requirements are satisfied. If the orbifolds are
+second-countable, then we see as in Lemma 5.14that we may restrict this
+construction to countably many points of QandQ′resp. countable sets A
+andBto achieve that K,K′andK′′are countable. /square
+Proposition 5.18. The composition in Orbis well-defined.
+Proof. We use the notation from the definition of the composition. We have
+to show that the composition of [ˆf]and[ˆg]neither depends on the choice of
+the induced orbifold maps ˆhandˆknor on the choice of the representatives
+of[ˆf]and[ˆg]. To prove independence of the choice of ˆhandˆk, suppose
+that we have two pairs (ˆhj,ˆkj)of induced orbifold maps ˆhj∈Orb(Kj,K′
+j),
+ˆkj∈Orb(K′
+j,K′′
+j)(j= 1,2) such that the diagram
+K1ˆh1/d47/d47
+/d31/d31❄❄❄❄❄❄❄❄K′
+1ˆk1/d47/d47
+/d126/d126⑦⑦⑦⑦⑦⑦⑦
+/d32/d32❇❇❇❇❇❇❇K′′
+1
+/d125/d125④④④④④④④
+Vˆf/d47/d47V′W′ˆg/d47/d47W′′
+K2ˆh2/d47/d47/d63/d63⑧⑧⑧⑧⑧⑧⑧⑧
+K′
+2ˆk2/d47/d47/d96/d96❅❅❅❅❅❅❅❅/d62/d62⑤⑤⑤⑤⑤⑤⑤⑤
+K′′
+2/d97/d97❈❈❈❈❈❈❈❈
+commutes. The non-horizontal maps are lifts of identity. Le mma5.14shows
+the existence of representatives HofU,H′,I′ofU′,I′′ofU′′, and charted
+orbifold maps ˆh3∈Orb(H,H′),ˆk3∈Orb(I′,I′′), and appropriate lifts of
+identity such that the diagrams
+K1ˆh1/d47/d47K′
+1 K′
+1ˆk1/d47/d47K′′
+1
+Hˆh3/d47/d47/d62/d62⑦⑦⑦⑦⑦⑦⑦⑦
+/d32/d32❅❅❅❅❅❅❅❅ H′/d96/d96❆❆❆❆❆❆❆
+/d126/d126⑥⑥⑥⑥⑥⑥⑥⑥I′ˆk3/d47/d47/d62/d62⑦⑦⑦⑦⑦⑦⑦
+/d32/d32❅❅❅❅❅❅❅❅ I′′/d96/d96❆❆❆❆❆❆❆
+/d126/d126⑥⑥⑥⑥⑥⑥⑥⑥
+K2ˆh2/d47/d47K′
+2 K′
+2ˆk2/d47/d47K′′
+2THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 31
+commute. By Lemma 5.17we find representatives K,K′,K′′ofU,U′,U′′,
+resp., charted orbifold maps ˆh∈Orb(K,K′),ˆk∈Orb(K′,K′′), and appro-
+priate lifts of identity such that
+Hˆh3/d47/d47H′I′ˆk3/d47/d47I′′
+Kˆh/d47/d47/d63/d63⑧⑧⑧⑧⑧⑧⑧⑧
+K′ˆk/d47/d47/d62/d62⑥⑥⑥⑥⑥⑥⑥⑥/d96/d96❇❇❇❇❇❇❇❇
+K′′/d97/d97❇❇❇❇❇❇❇❇
+commutes. Hence, altogether we have the commutative diagra m
+(7) K1ˆh1/d47/d47K′
+1ˆk1/d47/d47K′′
+1
+Kˆh/d47/d47/d79/d79
+/d15/d15K′ˆk/d47/d47/d79/d79
+/d15/d15K′′/d79/d79
+/d15/d15
+K2ˆh2/d47/d47K′
+2ˆk2/d47/d47K′′
+2
+which shows that ˆk1◦ˆh1andˆk2◦ˆh2are equivalent (see Remark 5.19below
+for some more details).
+For the proof of the independence of the choices of the repres entatives of [ˆf]
+and[ˆg], letˆf1∈Orb(V1,V′
+1),ˆf2∈Orb(V2,V′
+2)be representatives of [ˆf], and
+ˆg1∈Orb(W′
+1,W′′
+1),ˆg2∈Orb(W′
+2,W′′
+2)be representatives of [ˆg]. Further, for
+j= 1,2, letˆhj∈Orb(Kj,K′
+j)be induced by ˆfj, andˆkj∈Orb(K′
+j,K′′
+j)be
+induced by ˆgj. Sinceˆf1andˆf2are equivalent, we find representatives V,V′
+ofU,U′, resp., a charted orbifold map ˆf∈Orb(V,V′)and appropriate lifts
+of identities, and analogously for ˆg1andˆg2, such that the diagrams
+V1ˆf1/d47/d47V′
+1 W′
+1ˆg1/d47/d47W′′
+1
+Vˆf/d47/d47/d63/d63⑧⑧⑧⑧⑧⑧⑧⑧
+/d31/d31❄❄❄❄❄❄❄❄ V′/d95/d95❅❅❅❅❅❅❅
+/d127/d127⑦⑦⑦⑦⑦⑦⑦⑦W′ˆg/d47/d47/d61/d61④④④④④④④
+/d33/d33❈❈❈❈❈❈❈❈ W′′/d97/d97❉❉❉❉❉❉❉
+/d125/d125③③③③③③③③
+V2ˆf2/d47/d47V′
+2 W′
+2ˆg2/d47/d47W′′
+2
+commute. Lemma 5.17yields the existence of ˆh∈Orb(K,K′)andˆk∈
+Orb(K′,K′′)and appropriate lifts of identities such that
+Vˆf/d47/d47V′W′ˆg/d47/d47W′′
+Kˆh/d47/d47/d63/d63⑧⑧⑧⑧⑧⑧⑧⑧
+K′ˆk/d47/d47/d96/d96❆❆❆❆❆❆❆❆/d61/d61⑤⑤⑤⑤⑤⑤⑤⑤
+K′′/d97/d97❉❉❉❉❉❉❉❉
+commutes. Since ˆhis induced by ˆf1and byˆf2, and likewise, ˆkis induced
+byˆg1and byˆg2, we conclude as above that ˆk1◦ˆh1andˆk2◦ˆh2are both
+equivalent to ˆk◦ˆh. This yields that the composition map is well-defined. /square32 ANKE POHL
+Remark 5.19.We elaborate on a subtle issue in the proof of Proposition 5.18.
+By the reasoning in the proof, the diagram ( 7)a priori is of the form
+K1ˆh1/d47/d47K′
+1 K′
+1ˆk1/d47/d47K′′
+1
+Hˆh3/d47/d47/d79/d79
+H′/d79/d79
+I′ˆk3/d47/d47/d79/d79
+I′′/d79/d79
+Kˆh/d47/d47/d79/d79
+/d15/d15K′ˆk/d47/d47/d62/d62⑤⑤⑤⑤⑤⑤⑤⑤
+/d32/d32❇❇❇❇❇❇❇❇/d96/d96❇❇❇❇❇❇❇❇
+/d126/d126⑤⑤⑤⑤⑤⑤⑤⑤K′′/d79/d79
+/d15/d15
+Hˆh3/d47/d47
+/d15/d15H′
+/d15/d15I′ˆk3/d47/d47
+/d15/d15I′′
+/d15/d15
+K2ˆh2/d47/d47K′
+2 K′
+2ˆk2/d47/d47K′′
+2
+and we implicitly claim that the two maps from K′toK′
+1as well as from K′
+toK′
+2are identical. This might lead to the suspicion that the comp osition
+of charted orbifold maps as defined above is not well-defined.
+K
+/d15/d15h/d47/d47K′⊆K′
+1
+ε1/d122/d122✉✉✉✉✉✉✉✉✉
+ε2/d36/d36■■■■■■■■■■k/d47/d47K′′
+/d15/d15
+Hh3=f|H/d47/d47
+/d15/d15H′
+ε−1
+1
+/d36/d36■■■■■■■■■■ I′
+ε−1
+2
+/d122/d122✉✉✉✉✉✉✉✉✉✉k3=g|I′/d47/d47I′′
+/d15/d15
+K1h1/d47/d47
+/d15/d15K′
+1k1/d47/d47
+ε1/d122/d122✉✉✉✉✉✉✉✉✉✉
+ε2/d36/d36■■■■■■■■■■K′′
+1
+/d15/d15
+Vf/d47/d47V′W′g/d47/d47W′′
+K2/d79/d79
+h2/d47/d47K′
+2η1/d100/d100■■■■■■■■■■η2/d58/d58✉✉✉✉✉✉✉✉✉✉k2/d47/d47K′′
+2/d79/d79
+H/d79/d79
+h3=f|H/d47/d47H′η−1
+1/d58/d58✉✉✉✉✉✉✉✉✉✉
+I′η−1
+2/d100/d100■■■■■■■■■■k3=g|I′/d47/d47I′′/d79/d79
+K/d79/d79
+h/d47/d47K′ξ−1◦η1/d100/d100■■■■■■■■■■ξ−1◦η2/d58/d58ttttttttttk/d47/d47K′′/d79/d79
+Figure 1. Maps in local charts
+However, since ˆh,ˆh1,ˆh2are all induced by ˆf, andˆk,ˆk1,ˆk2are all induced
+byˆg, the arising maps K′→ K′
+1are indeed identical as well as the arising
+mapsK′→ K′
+2. The diagram in Figure 1shows the maps between the local
+charts, which proves the latter statement as well as that dia gram 7is indeed
+commutative. The notation in this diagram is intuitive, e.g .Kis an orbifoldTHE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 33
+chart inK, all greek letters refer to diffeomorphisms. To improve read ibility
+we may assume without loss of generality that K′⊆K′
+1,K′
+1⊆V′∩W′and
+K′
+2⊆V′∩W′. The latter two choices implicitly mean that we assume that
+the local diffeomorphism between V′andW′is the identity, which we indeed
+may. Otherwise we would just need to add the same local diffeom orphism
+to several maps in the diagram. The local diffeomorphism betw eenK′and
+K′
+2is denoted by ξ(note that we are not allowed to suppose that ξ= idin
+general). Note that the top and bottom maps in the diagram are identical.
+We end this section with a discussion of the equivalence clas s represented
+by a lift of identity. The following proposition shows that i t is precisely the
+class of all lifts of identity of the considered orbifold. Th is justifies the notion
+“identity morphism” in Definition 5.8.
+Proposition 5.20. Let(Q,U)be an orbifold and εa lift ofid(Q,U). Then
+the equivalence class [ε]ofεconsists precisely of all lifts of id(Q,U).
+Proof. Letε1∈Orb(V1,W1)andε2∈Orb(V2,W2)be two lifts of id(Q,U).
+Propositions 5.3and5.5imply that there is a representative VofUsuch
+thatε1andε2both induce the orbifold map
+/hatwideidQ:= (idQ,{idVi}i∈I,[R,σ])
+with(V,V). Thus, each two lifts of id(Q,U)are equivalent.
+Letˆfbe a charted orbifold map which is equivalent to ε. W.l.o.g. we may
+assume that ε=/hatwideidQ. To fix notation let
+V={(Vi,Gi,πi)|i∈I},indexed by I,
+K1={(K1,a,L1,a,χ1,a)|a∈A},indexed by A,
+K2={(K2,b,L2,b,χ2,b)|b∈B},indexed by B,
+W1={(W1,j,H1,j,ψ1,j)|j∈J},indexed by J,
+W2={(W2,k,H2,k,ψ2,k)|k∈K},indexed by K,
+be representatives of U. Let
+ˆf=/parenleftbig
+f,{˜fj}j∈J,[Pf,νf]/parenrightbig
+∈Orb(W1,W2).
+Suppose that
+ˆg=/parenleftbig
+g,{˜ga}a∈A,[Pg,νg]/parenrightbig
+∈Orb(K1,K2)
+is a charted orbifold map and
+ε1=/parenleftbig
+idQ,{λ1,a}a∈A,[P1,ν1]/parenrightbig
+∈Orb(K1,V)
+ε2=/parenleftbig
+idQ,{λ2,a}a∈A,[P2,ν2]/parenrightbig
+∈Orb(K1,W1)
+δ1=/parenleftbig
+idQ,{µ1,b}b∈B,[R1,σ1]/parenrightbig
+∈Orb(K2,V)
+δ2=/parenleftbig
+idQ,{µ2,b}b∈B,[R2,σ2]/parenrightbig
+∈Orb(K2,W2)34 ANKE POHL
+are lifts of id(Q,U)such that the diagram (which shows that ˆfand/hatwideidQare
+equivalent)
+V/hatwideidQ/d47/d47V
+K1ε1/d61/d61④④④④④④④④
+ε2/d33/d33❈❈❈❈❈❈❈❈ˆg/d47/d47K2δ1/d97/d97❈❈❈❈❈❈❈❈
+δ2 /d125/d125④④④④④④④④
+W1ˆf/d47/d47W2
+commutes. Clearly, g= idQand hencef= idQ. Letα:A→I,β:A→J,
+γ:A→B,δ:B→I,η:B→Kandζ:J→Kbe the induced maps on
+the index sets as, e.g., in Construction 5.9. For eacha∈A, we have
+idVα(a)◦λ1,a=µ1,γ(a)◦˜ga.
+SinceidVα(a),λ1,aandµ1,γ(a)are local diffeomorphisms, so is ˜ga. Now
+˜fβ(a)◦λ2,a=µ2,γ(a)◦˜ga
+for eacha∈A. Hence ˜fβ(a)is a local diffeomorphism. Lemma 5.13implies
+that˜fjis a local diffeomorphism for each j∈J. Therefore, ˆfis a lift of
+id(Q,U). /square
+6.The orbifold category in terms of marked atlas groupoids
+Proposition 4.12and Remark 5.10show that charted orbifold maps and
+their composition correspond to homomorphisms between mar ked atlas grou-
+poids and their composition. By characterizing lifts of iden tity and equiva-
+lence of charted orbifold maps in terms of marked atlas group oids and their
+homomorphisms, we construct a category for marked atlas gro upoids which
+is isomorphic to the one of reduced orbifolds. To that end we fi rst show that
+lifts of identity correspond to unit weak equivalences, a no tion we define
+below. Throughout this section let pr1denote the projection onto the first
+component.
+A homomorphism ϕ= (ϕ0,ϕ1):G→Hbetween Lie groupoids is called
+aweak equivalence if
+(i) the map
+t◦pr1:H1s×ϕ0G0→H0
+is a surjective submersion, and
+(ii) the diagram
+G1ϕ1/d47/d47
+(s,t)
+/d15/d15H1
+(s,t)
+/d15/d15
+G0×G0ϕ0×ϕ0/d47/d47H0×H0
+is a fibered product.
+Two Lie groupoids G,Hare called Morita equivalent if there is a Lie groupoid
+Kand weak equivalences
+G Kϕ/d111/d111ψ/d47/d47H.THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 35
+Definition 6.1. Let(G1,α1,X1)and(G2,α2,X2)be marked atlas groupoids.
+A homomorphism
+ϕ= (ϕ0,ϕ1): (G1,α1,X1)→(G2,α2,X2)
+is called a unit weak equivalence ifϕ:G1→G2is a weak equivalence and
+α2◦|ϕ| ◦α−1
+1= idX1. Necessarily we have X1=X2=:X. Aunit Morita
+equivalence between (G1,α1,X)and(G2,α2,X)is a pair (ψ1,ψ2)of unit
+weak equivalences
+ψj: (G,α,X)→(Gj,αj,X)
+where(G,α,X)is some marked atlas groupoid. If such a unit Morita equiv-
+alence exists, then the marked atlas groupoids (G1,α1,X)and(G2,α2,X)
+are called unit Morita equivalent .
+In contrast to Morita equivalence of Lie groupoids, unit Mor ita equivalence
+of marked atlas groupoids requires the third (marked) Lie gr oupoid to be an
+atlas groupoid. In Proposition 6.3below we will show that unit Morita
+equivalence of marked atlas groupoids is indeed an equivale nce relation.
+The following proposition identifies lifts of identity with unit weak equiv-
+alences.
+Proposition 6.2. LetUandU′be orbifold structures on the topological space
+Q. Further let Vresp.W′be a representative of Uresp. ofU′.
+(i)Suppose that U=U′. Ifˆf∈Orb(V,W′)is a lift of id(Q,U), thenF1(ˆf)
+is a unit weak equivalence.
+(ii)Letε∈Hom(Γ(V),Γ(W′))be a unit weak equivalence. Then U=U′,
+andF2(ε)is a lift of id(Q,U).
+Proof. Let
+V={(Vi,Gi,πi)|i∈I}resp.W′={(W′
+j,H′
+j,ψ′
+j)|j∈J},
+indexed by Iresp. byJ, and letG:= Γ(V)andH:= Γ(W′). We will first
+prove ( i). Suppose ˆf= (idQ,{˜fi}i∈I,[P,ν]). By Proposition 4.7it suffices to
+show thatε= (ε0,ε1):=F1(ˆf)is a weak equivalence. We first show that
+t◦pr1:/braceleftbigg
+H1s×ε0G0→H0
+(h,x)/ma√s⊔o→t(h)
+is a submersion. Let (h,x)∈H1s×ε0G0. Recall from Proposition 5.3that
+ε0is a local diffeomorphism, and from Special Case 2.5thatGandHare
+étale groupoids. Choose open neighborhoods UxofxinG0andUhofhin
+H1such thatε0|Uxands|Uhare embeddings with s(Uh) =ε0(Ux). Then
+Uhs×ε0Uxis open inH1s×ε0G0. Further
+Uhs×ε0Ux={(k,y)∈Uh×Ux|s(k) =ε0(y)}
+=/braceleftbig/parenleftbig
+k,ε−1
+0(s(k))/parenrightbig/vextendsingle/vextendsinglek∈Uh/bracerightbig
+.
+Therefore,
+pr1:Uhs×ε0Ux→Uh
+is a diffeomorphism. Since tis a local diffeomorphism, t◦pr1is a submersion.36 ANKE POHL
+Now we prove that t◦pr1is surjective. Let y∈H0, sayy∈W′
+j, and set
+ψ′
+j(y) =:q∈Q. Then there is an orbifold chart (Vi,Gi,πi)∈ Vsuch that
+q∈πi(Vi), sayq=πi(x).
+Vi˜fi/d47/d47
+πi/d30/d30❃❃❃❃❃❃❃❃W′
+i
+ψ′
+i /d127/d127⑦⑦⑦⑦⑦⑦⑦⑦
+Q
+Setz:=˜fi(x), henceψ′
+i(z) =q=ψ′
+j(y). Hence, there are a restriction
+(S′,K′,χ′)of(W′
+i,H′
+i,ψ′
+i)withz∈S′and an embedding
+λ: (S′,K′,χ′)→(W′
+j,H′
+j,ψ′
+j)
+such thatλ(z) =y. Thenλ∈Ψ(W′)and(germzλ,x)∈H1s×ε0G0with
+t◦pr1(germzλ,x) =t(germzλ) =y.
+This means that t◦pr1is surjective.
+Set
+K:= (G0×G0)(ε0,ε0)×(s,t)H1.
+It remains to show that the map
+β:/braceleftbigg
+G1→ K
+germxg/ma√s⊔o→(x,g(x),ε1(germxg))
+is a diffeomorphism. Note that β= (s,t,ε1). Let(x,y,germε0(x)h)be in
+K, hencegermε0(x)h:ε0(x)→ε0(y). By the definition of H1there are open
+neighborhoods U′
+1ofε0(x)andU′
+2ofε0(y)inW′:=/coproducttext
+j∈JW′
+jsuch that
+h:U′
+1→U′
+2is an element of Ψ(W′). Sinceε0is a local diffeomorphism,
+there are open neighborhoods U1ofxandU2ofyinV:=/coproducttext
+i∈IVisuch that
+ε0|Ukis an embedding with ε0(Uk)⊆U′
+k(k= 1,2). After shrinking U′
+kwe
+can assume that ε0(Uk) =U′
+k. Letγk:=ε0|Uk. Then
+g:=γ−1
+2◦h◦γ1:U1→U2
+is a diffeomorphism, hence g∈Ψ(V). By Proposition 5.5(or rather its
+proof), we have ν(g) =hand hence (see (R 4a) and recall Proposition 4.7)
+ε1(germxg) = germε0(x)h.
+Finally, we see
+β(germxg) = (x,g(x),ε1(germxg)) = (x,y,germε0(x)h).
+Thereforeβis surjective. Since germxgdoes not depend on the choice of Uk
+andU′
+k, the mapβis also injective. Finally, we will show that βis a local
+diffeomorphism. Since sandtare local diffeomorphisms, we only need to
+prove thatε1is one as well. Let germxf∈G1. Choose an open neighborhood
+Uofxsuch thatU⊆domfandε0|U:U→ε0(U)is a diffeomorphism. By
+the germ topology, the set
+˜U:={germyf|y∈U}
+is open inG1, and the set
+˜V:={germzν(f)|z∈ε0(U)}THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 37
+is open inH1. Further the diagrams
+˜Uε1/d47/d47
+/d15/d15˜V
+/d15/d15germyf✤ε1/d47/d47
+❴
+/d15/d15germε0(y)ν(f)❴
+/d15/d15
+Uε0/d47/d47ε0(U) y✤
+ε0/d47/d47ε0(y)
+commute. Since the vertical arrows are diffeomorphisms by de finition,
+ε1|˜U:˜U→˜V
+is a diffeomorphism as well. This completes the proof of ( i).
+We will now prove ( ii). Proposition 3.3shows that the orbifold atlases V
+andW′are determined completely by the marked atlas groupoids Γ(V)and
+Γ(W′), resp. Hence we can apply Proposition 4.9, which shows that F2(ε)is
+well-defined. Suppose that
+F2(ε) =/parenleftbig
+f,{˜fi}i∈I,[P,ν]).
+Proposition 4.9yieldsf= idQ. By [ 8, Exercises 5.16(4)] ε0is a local dif-
+feomorphism. Thus, Proposition 4.9implies that each ˜fiis a local diffeo-
+morphism. The domain atlas of F2(ε)isV, its range family is W′. From
+Proposition 5.6it follows that U=U′. By Definition 5.8F2(ε)is a lift of
+id(Q,U). /square
+The combination of Propositions 3.3,6.2and Remark 5.10now allows to
+identity each step in the construction of the category of red uced orbifolds
+and each intermediate object in terms of marked atlas groupo ids. For an
+orbifold(Q,U)we define
+Γ(Q,U):=/braceleftbig/parenleftbig
+Γ(V),αV,Q/parenrightbig/vextendsingle/vextendsingleVis a (countable) representative of U/bracerightbig
+.
+We recall from Special Case 2.5the convention that for second-countable
+orbifolds we always restrict here to countable representat ives of the orbifold
+structure. For non-second-countable, we also consider non -countable repre-
+sentatives.
+Then Proposition 5.20gives rise to the following proposition.
+Proposition 6.3. Unit Morita equivalence of marked atlas groupoids is an
+equivalence relation. Further, if Vis a (countable) representative of the orb-
+ifold structure Uof the orbifold (Q,U), then the unit Morita equivalence class
+of(Γ(V),αV,Q)isΓ(Q,U).
+Equivalence of charted orbifold maps translates to marked a tlas groupoids
+as follows.
+Definition 6.4. Let(G1,α1,X),(G2,α2,X), as well as (H1,β1,Y)and
+(H2,β2,Y)be marked atlas groupoids. For j= 1,2let
+ψj: (Gj,αj,X)→(Hj,βj,Y)
+be a homomorphism of marked Lie groupoids. We call ψ1andψ2unit Morita
+equivalent if there exist marked atlas groupoids (G,α,X)and(H,β,Y),
+a homomorphism χ: (G,α,X)→(H,β,Y), and unit weak equivalences38 ANKE POHL
+εj: (G,α,X)→(Gj,αj,X),δj: (H,β,Y)→(Hj,βj,Y)such that the dia-
+gram
+(G1,α1,X)ψ1/d47/d47(H1,β1,Y)
+(G,α,X)ε1/d55/d55♣♣♣♣♣♣♣♣♣♣♣
+ε2/d39/d39◆◆◆◆◆◆◆◆◆◆◆χ/d47/d47(H,β,Y)δ1/d103/d103◆◆◆◆◆◆◆◆◆◆◆
+δ2 /d120/d120♣♣♣♣♣♣♣♣♣♣♣
+(G2,α2,X)ψ2/d47/d47(H2,β2,Y)
+commutes.
+Proposition 5.15in terms of atlas groupoids means the following.
+Proposition 6.5. Unit Morita equivalence of homomorphisms between marked
+atlas groupoids is an equivalence relation.
+We define the category Agrof marked atlas groupoids as follows: Its class
+of objects consists of all Γ(Q,U). The morphisms from Γ(Q,U)toΓ(Q′,U′)
+are the unit Morita equivalence classes [ϕ]of homomorphisms ϕ: (G,α,Q)→
+(G′,α′,Q′)where(G,α,Q)is any representative of Γ(Q,U)and(G′,α′,Q′)
+is any representative of Γ(Q′,U′).
+The composition of two morphisms [ϕ]∈Morph/parenleftbig
+Γ(Q,U),Γ(Q′,U′)/parenrightbig
+and
+[ψ]∈Morph/parenleftbig
+Γ(Q′,U′),Γ(Q′′,U′′)/parenrightbig
+is defined as follows: Choose representa-
+tives
+ϕ: (G,α,Q)→(G′,α′,Q′)of[ϕ]
+and
+ψ: (H′,β′,Q′)→(H′′,β′′,Q′′)of[ψ].
+Then find representatives (K,γ,Q),(K′,γ′,Q′),(K′′,γ′′,Q′′)of the classes
+Γ(Q,U),Γ(Q′,U′),Γ(Q′′,U′′), resp., and unit Morita equivalences
+ε: (K,γ,Q)→(G,α,Q),
+ε′
+1: (K′,γ′,Q′)→(G′,α′,Q′),
+ε′
+2: (K′,γ′,Q′)→(H′,β′,Q′),
+ε′′: (K′′,γ′′,Q′′)→(H′′,β′′,Q′′),
+and homomorphisms of marked Lie groupoids
+χ: (K,γ,Q)→(K′,γ′,Q′),
+κ: (K′,γ′,Q′)→(K′′,γ′′,Q′′)
+such that the diagram
+(G,α,Q)ϕ/d47/d47(G′,α′,Q′) ( H′,β′,Q′)ψ/d47/d47(H′′,β′′,Q′′)
+(K,γ,Q)ε/d79/d79
+χ/d47/d47(K′,γ′,Q′)ε′
+1/d102/d102▼▼▼▼▼▼▼▼▼▼ε′
+2/d56/d56qqqqqqqqqqκ/d47/d47(K′′,γ′′,Q′′)ε′′/d79/d79
+commutes. Then the composition of [ϕ]and[ψ]is defined as
+[ψ]◦[ϕ]:= [κ◦χ].THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 39
+Invoking Lemmas 5.11,5.17and Proposition 5.18we deduce the following
+proposition.
+Proposition 6.6. The composition in Agris well-defined.
+We define an assignment Ffrom the orbifold category Orbto the category
+of marked atlas groupoids Agras follows. On the level of objects, Fmaps
+the orbifold (Q,U)toΓ(Q,U). Suppose that [ˆf]is a morphism from the
+orbifold(Q,U)to the orbifold (Q′,U′). ThenFmaps[ˆf]to the morphism
+[F1(ˆf)]fromΓ(Q,U)toΓ(Q′,U′).
+Theorem 6.7. The assignment Fis a covariant functor from OrbtoAgr.
+Even more, Fis an isomorphism of categories. The functor Fand its inverse
+are constructive.
+In the following example we show that the representatives of orbifold maps
+from Example 4.6define different orbifold maps. In this example we use
+G(x,y)to denote the set of arrows gof the groupoid Gwiths(g) =xand
+t(g) =y.
+Example 6.8. Recall the representatives of orbifold maps
+ˆf1= (f,/tildewidef,P,ν1)andˆf2= (f,/tildewidef,P,ν2)
+from Example 4.6and4.8. We claim that ˆf1andˆf2are representatives
+of different orbifold maps. Assume for contradiction that ˆf1andˆf2define
+the same orbifold map on (Q,U1). This means that the groupoid homomor-
+phismsϕandψfrom Example 4.8are Morita equivalent. Hence there exist
+marked atlas groupoids KandH, unit weak equivalences
+α= (α0,α1):K→Γ, γ = (γ0,γ1):H→Γ,
+β= (β0,β1):K→Γ, δ = (δ0,δ1):H→Γ,
+and a homomorphism χ= (χ0,χ1):K→Hsuch that the diagram
+Γϕ/d47/d47Γ
+Kα/d63/d63⑦⑦⑦⑦⑦⑦⑦
+β/d31/d31❅❅❅❅❅❅❅❅χ/d47/d47Hγ/d95/d95❅❅❅❅❅❅❅
+δ/d127/d127⑦⑦⑦⑦⑦⑦⑦⑦
+Γψ/d47/d47Γ
+commutes. Since αis a (unit) weak equivalence, there exists x∈Kand
+g∈Γwiths(g) =α0(x)andt(g) = 0 . Necessarily, g∈ {germ0(±id)},
+and henceα0(x) = 0. In turn,α1induces a bijection between K(x,x)and
+Γ(0,0). ThusK(x,x)consists of two elements, say K(x,x) ={k1,k2}. Let
+x′:=χ0(x). Then
+0 =ϕ0(α0(x)) =γ0(x′).
+This shows that γ1induces a bijection between H(x′,x′)andΓ(0,0). For
+j= 1,2we have
+γ1(χ1(kj)) =ϕ1(α1(kj)) = germ0id,40 ANKE POHL
+which implies that χ1(k1) =χ1(k2). Furtherβ1induces a bijection between
+K(x,x)andΓ(β0(x),β0(x)). Henceβ0(x) = 0, and thus
+ψ1(β1(k1))/\e}a⊔io\slash=ψ1(β1(k2)).
+But this contradicts to
+ψ1(β1(k1)) =δ1(χ1(k1)) =δ1(χ1(k2)) =ψ1(β1(k2)).
+In turn,ϕandψare not Morita equivalent.
+7.Extension to marked proper effective étale Lie groupoids
+LetPgrdenote the category of marked proper effective étale Lie grou poids
+which is defined analogously to the category Agrof marked atlas groupoids
+with the only difference that any marked atlas groupoid may be a marked
+proper effective étale Lie groupoid. Note that the objects in Pgrare not the
+single marked proper effective étale Lie groupoids themselv es but the unit
+Morita equivalence classes of these.
+In this final section we show that these two categories are iso morphic
+via the canonical embedding. As before we consider two differ ent flavors
+of categories distinguished by whether the involved topolo gical spaces are
+required to be just paracompact or even second-countable. T hese differences
+do not cause any distinctions in statements or proofs and the refore will be
+implicit.
+Theorem 7.1. The embedding functor Agr→Pgris an isomorphism.
+Proof. We prove that we can lift any given homomorphism (weak equiva -
+lence) between two proper effective étale Lie groupoids to a h omomorphism
+(weak equivalence) between weakly equivalent atlas groupo ids:
+G /d47/d47G′
+Γ(U)/d61/d61④④④④
+/d47/d47❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴Γ(U′)./d98/d98❋❋❋❋❋
+LetGbe a proper effective étale Lie groupoid. By [ 8, Corollary 5.31] there
+exists an orbifold atlas Uon the base space G0ofGsuch that Γ(U)is
+weakly equivalent to G. More precisely, let (Ui)i∈Ibe a covering of G0by
+open connected subsets such that for each i∈I, there exists a point x∈Ui
+such that
+Gi:=Gx:={g∈G1|gx−→x}=G1|Ui
+andUiisGi-stable, and such that the action groupoid Gi⋉Uiis isomorphic
+toG|Ui. Then the family
+U:={(Ui,Gi,pr)|i∈I}
+is an orbifold atlas on G0. We identify Uiwith{i} ×Ui. LetH:= Γ(U).
+Then the weak equivalence ϕ:H→Gis given by
+ϕ0:H0→G0,(i,x)/ma√s⊔o→x
+and
+ϕ1:H1→G1,k
+(i,x)−→(j,y)/ma√s⊔o→kx−→y.THE CATEGORY OF REDUCED ORBIFOLDS IN LOCAL CHARTS 41
+Now letG,G′be two proper effective étale Lie groupoids and χ:G→G′be
+a homomorphism (weak equivalence). Choose orbifold atlase s (in the way as
+above)
+U:={(Ui,Gi,pr)|i∈I},U′:={(U′
+j,G′
+j,pr)|j∈J}
+onG0resp.G′
+0such that there exists a map β:I→Jsuch that
+χ0(Ui)⊆U′
+β(i)
+for alli∈I(this is possible). Set H:= Γ(U),H′:= Γ(U′)and define
+α:H→H′via
+α0:H0→H′
+0,(i,x)/ma√s⊔o→(β(i),χ0(x))
+α1:H1→H′
+1,k
+(i,x)−→(j,y)/ma√s⊔o→χ1(k)
+(β(i),χ0(x))−→(β(j),χ0(y)).
+One easily checks that αis a homomorphism (weak equivalence).
+With this lifting property we can bring back all equivalence relations be-
+tween marked proper effective étale Lie groupoids and betwee n maps between
+these groupoids to statements purely between marked atlas g roupoids and
+maps between those. This proves the theorem. /square
+References
+1. A. Adem, J. Leida, and Y. Ruan, Orbifolds and stringy topology , Cambridge Tracts in
+Mathematics, vol. 171, Cambridge University Press, Cambri dge, 2007.
+2. J. Borzellino and V. Brunsden, A manifold structure for the group of orbifold diffeo-
+morphisms of a smooth orbifold , J. Lie Theory 18(2008), no. 4, 979–1007.
+3. W. Chen and Y. Ruan, Orbifold Gromov-Witten theory , Orbifolds in mathematics
+and physics (Madison, WI, 2001), Contemp. Math., vol. 310, A mer. Math. Soc., Prov-
+idence, RI, 2002, pp. 25–85.
+4. A. Haefliger, Groupoïdes d’holonomie et classifiants , Astérisque (1984), no. 116, 70–97.
+5. E. Lerman, Orbifolds as stacks? , Enseign. Math. (2) 56(2010), no. 3-4, 315–363.
+6. E. Lupercio and B. Uribe, Gerbes over orbifolds and twisted K-theory , Comm. Math.
+Phys.245(2004), no. 3, 449–489.
+7. I. Moerdijk, Orbifolds as groupoids: an introduction , Orbifolds in mathematics and
+physics (Madison, WI, 2001), Contemp. Math., vol. 310, Amer . Math. Soc., Provi-
+dence, RI, 2002, pp. 205–222.
+8. I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids , Cambridge
+Studies in Advanced Mathematics, vol. 91, Cambridge Univer sity Press, Cambridge,
+2003.
+9. I. Moerdijk and D. Pronk, Orbifolds, sheaves and groupoids , K-Theory 12(1997),
+no. 1, 3–21.
+10. D. Pronk, Etendues and stacks as bicategories of fractions , Compositio Math. 102
+(1996), no. 3, 243–303.
+11. I. Satake, On a generalization of the notion of manifold , Proc. Nat. Acad. Sci. U.S.A.
+42(1956), 359–363.
+12. ,The Gauss-Bonnet theorem for V-manifolds , J. Math. Soc. Japan 9(1957),
+464–492.
+13. A. Schmeding, The diffeomorphism group of a non-compact orbifold. , Diss. Math. 507
+(2015), 179.
+14. W. Thurston, The geometry and topology of three-manifolds , available at
+www.msri.org/publications/books/gt3m/.
+Mathematisches Institut, Georg-August-Universität Gött ingen, Bunsen-
+str. 3-5, 37073 Göttingen
+E-mail address :anke.pohl@math.uni-goettingen.de
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+arXiv:1001.0669v2  [cond-mat.soft]  8 Jun 2010Model for Density Waves in Gravity-Driven Granular Flow in N arrow Pipes
+Simen˚A. Ellingsen
+Department of Energy and Process Engineering, Norwegian Un iversity of Science and Technology, N-7491 Trondheim, Norw ay
+Knut S. Gjerden, Morten Grøva, and Alex Hansen
+Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
+(Dated: November 15, 2018)
+A gravity-driven flow of grains through a narrow pipe in vacuu m is studied by means of a one-
+dimensional model with two coefficients of restitution. Nume rical simulations show clearly how
+density waves form when a strikingly simple criterion is ful filled: that dissipation due to collisions
+between the grains and the walls of the pipe is greater per col lision than that which stems from
+collisions between particles. Counterintuitively, the hi ghest flow rate is observed when the number
+of grains per density wave grows large. We find strong indicat ion that the number of grains per
+density wave always approaches a constant as the particle nu mber tends to infinity, and that collapse
+to a single wave, which was often observed also in previous si mulations, occurs because the number
+of grains is insufficient for multiple wave formation.
+PACS numbers: 45.70.Mg, 83.10.Rs, 47.11.Mn, 47.57.Gc
+I. INTRODUCTION
+Transport of dry granular media through pipes and
+channels is a problem of fundamental interest which is of
+considerable industrial importance [1], and it has been
+studied intensely both experimentally and theoretically
+in recent decades [2–5]. Granular media behave radically
+different from both liquids and solids [3] and in granular
+pipe flow driveneither bygravityorpressurizedgas, non-
+linear dynamical phenomena such as clogging (density
+waves) are observed, but not yet well understood.
+We consider dry grains falling inside a pipe. In this
+system, encountered industrially, e.g., in emptying of si-
+los and transportation of sand or powder, there are three
+main mechanisms of interaction: (a) collisions between
+grains, (b) collisionsbetween grainsandwalls, and(c) in-
+teraction of the grains with air in the system. Assuming
+the pipe is narrow, we approach this problem by means
+of a simple one-dimensional model with periodic bound-
+ary conditions in which collisions are modelled by means
+of two coefficients of restitution µandν, corresponding
+to the collisions of mechanisms (a) and (b), respectively.
+We demonstrate that this is sufficient to observe the for-
+mation of density waves.
+Granular media has frequently been investigated by
+means of numerical simulations. Features such as clus-
+tering through dissipative collisions [6, 7] and inelastic
+collapse [8, 9] are among those reported. Various driv-
+ing mechanisms have been considered, which can roughly
+be divided into two categories: vibrating walls [10–12]
+and interior heating [13–18]. The case of an emptying
+hopper has been studied, see e.g. [19, 20], as has the de-
+compaction transient associated with the release of an
+initially dense packing [21].
+The particular case of gravity-driven granular flow
+through channels and pipes has been studied for some
+time, both experimentally and theoretically. In previ-
+ous experimental investigations [22–28] the importanceof the presence of air in the flow has been emphasized,
+and the formation of density waves in falling sand has
+been explained with primary reference to the air-grain
+interaction. Although clearly of importance in real sys-
+tems, we demonstrate that the introduction of air in the
+model is not necessary to observe density waves. Our
+simulation suggests that such flocculent behaviour, while
+certainly influenced by the presence of air, can occur also
+in evacuated pipes, provided dissipation from collisions
+between grains and walls is faster than dissipation from
+grain-grain collisions.
+Previoussimulationsofgranularpipe flowhavealsore-
+ported density waves [6, 29–34]. Several mechanisms for
+velocity dissipation through inelastic collisions, damping
+and static friction have been employed, see, e.g., [35].
+Earlypapers[6,29]wereunabletostudysufficientlylarge
+numbers of grains and collsions to reach a state indepen-
+dent of initial conditions. The model used by Lee [30]
+was a 2+1-dimensional time-driven molecular dynamics
+(MD) simulation, with at least eight significant param-
+eters. In the model of Peng and Herrmann [31, 32], a
+2+1 dimensional lattice gas automaton is used in which
+variouseventsareassignedprobalisticcollisionrules. An-
+other model was proposed by Liss, Conway and Glasser
+[33]. Again, this is a 2+1 dimensional model with more
+complicated collision rules than ours. In their simulation
+the grain-wall coefficient of restitution was found to be
+of little importance. In all these models stable density
+waves were seen.
+Our model is one-dimensional and much simpler, yet
+reproduces similar qualitative phenomena, dictated es-
+sentially by two coefficients of restitution whose inter-
+pretation is physically transparent.
+We present our model and the results of our simula-
+tions and demonstrate that the model has two clearly
+distinct regimes in the parameter plane, one in which
+density waves form (flocculent regime) and one where no
+such are present (gaseous regime). Average number of2
+FIG. 1: Spatiotemporal plots of grain trajectories. (a) Gas eous regime µ= 0.8,ν= 0.3. (b) Split velocity transition point
+µ=ν= 0.55. (c) and (d): Different multi-wave patterns, ( µ,ν) = (0.55,0.65) and (0 .30,0.85), respectively. (e) and (f):
+Collapse to a single wave; ( µ,ν) = (0.70,0.95) and (0 .15,0.85), respectively. Further details in text.
+grains per density wave and flow rate as functions of the
+two coefficients of restitution are studied.
+II. THE MODEL
+The pipe of our model is sufficiently narrow to pre-
+vent grainsfrom passing each other, and sufficiently wide
+to allow them to fall approximately freely between col-
+lisions. As a consequence, grain-wall collisions are trig-
+gered by, and occur immediately subsequent to, grain-
+grain collisions. The pipe is therefore assumed to be too
+narrowfor archingeffects to play a role. We thus propose
+thatthesystemisessentiallyone-dimensionalinbehavior
+and may be captured qualitatively by a one-dimensional
+model where collisions with walls are incorporated into
+the grain-grain collision rules themselves.
+The set-up is the following. Let Ngrains move in one
+dimension so that grain ihas position xiand velocity vi.
+We use periodic boundary conditions, with grains falling
+beneath x= 0 reinserted at x=L, whereLis the length
+of the pipe. The grains are accelerated by a constant
+gravitygtowards x= 0. In the dilute limit dN≪L,
+wheredis the diameter of a single grain, we find that
+the behaviour is independent of d, and we set d= 0 for
+simplicity.
+As mentioned, the collision rules employed are based
+on the idea that the falling grains lose energy due to two
+different types of collisions, with the walls and with each
+other. Grain-grain collisions are modelled by reducingthe relative velocity of the colliding grains through a co-
+efficient of restitution µ∈[0,1]. Thus, v+
+R,ab=−µv−
+R,ab
+wherevR,ab=va−vb, grain ‘a’ is directly above grain
+‘b’, and we use superscripts ‘ −’ and ‘+’ to denote times
+just before and after a collision, respectively.
+Grain-wall collisions tend to decrease the average ve-
+locity of the system, vav=N−1/summationtextN
+i=1vi, which would
+otherwisedivergeintime. Instead, vaveventuallyreaches
+and fluctuates around a constant value at which dissi-
+pated energy is in equilibrium with energy gained from
+the gravitational field. We model the interaction with
+the walls at each collision by a coefficient of restitution
+ν∈[0,1] which dissipates grain energy by reducing the
+center of mass (CM) velocity of the two colliding grains
+according to v+
+CM,ab=νv−
+CM,ab, where the CM velocity
+of the pair is vCM,ab=1
+2(va+vb). These rules combine
+to
+v+
+a=1
+2(ν−µ)v−
+a+1
+2(ν+µ)v−
+b(1a)
+v+
+b=1
+2(ν+µ)v−
+a+1
+2(ν−µ)v−
+b. (1b)
+Note that introducing a restitution of vCM, momentum
+is transfered for each collision from the grains to the (in-
+finitely massive) pipe, and the momentum of the grains
+alone is not conserved. This causes the flow to approach
+a constant flow rate as observed in experiments.
+While a collision changes the velocities of just two
+grains, the relative and CM velocities of three pairsof
+grains are affected: the colliding pair plus the neighbor-
+ing pairs above and below. In particular, a collision in-
+creasesthe relativevelocityofthe pair abovethe collision3
+inproportiontothelostCMvelocityofthe collidingpair.
+Thus, relative velocities are constantly regenerated, and
+the phenomenon of inelastic collapse [8, 9] is avoided for
+µ >0 andν <1.
+III. SIMULATION AND RESULTS.
+We use event-driven molecular dynamics for our sim-
+ulations. Each grain is given an initial velocity which
+is assigned randomly according to various initialization
+schemes. The grains are also given equidistant starting
+positions between 0 and L.
+Simulations are run for different grain numbers Nat
+constant grain density ρ=N/L. After a ‘thermaliza-
+tion’ time the system reaches a steady state where the
+flow rate of the whole system fluctuates about a mean
+value. Initializing the simulations with different initial
+conditions yields the same dynamic steady state, charac-
+terized by average flow rate, collision frequency and the
+average number of density waves.
+In Fig. 1 we plot grain trajectories in space and time.
+In theseplots wehaveused, in arbitraryunits, L= 1,g=
+0.01 andN= 100. In these simulations initial velocities
+between 0 and 1 were drawn randomly from a uniform
+distribution.
+We observe two distinct regimes in the µ,νplane.
+Whenν < µ, dense regions tend to spread out. In this
+‘gaseous’ regime, shown in Fig. 1a, no steady density
+waves are observed. Whenever ν > µ, density waves are
+observed, and the transition from gaseous to flocculent
+behavior at µ=νis sudden.
+For the special case µ=νseen in Fig. 1b, the eqs. (1)
+simplify to v+
+a=µv−
+bandv+
+b=µv−
+a. Now the veloc-
+ity of grain ‘a’ after a collision is only a function of the
+velocity of grain ‘b’ before the collision, and vice versa .
+After ‘thermalization’ we observe that the grain veloci-
+ties are organized about two values: a lower velocity for
+the grains whose last collision was with the grain below
+it and a higher velocity for those which last collided with
+the grain above it.
+While the qualitative behaviour in the gaseous regime,
+µ > ν, is insensitive to the values of µandν, the situa-
+tion in the flocculent regime is quite different. As exem-
+plified for the parameter pairs in Fig. 1, the wave pat-
+terns vary strongly in this half of the parameter plane.
+Both the magnitude and velocity of the density waves
+depend sensitively on the values of µandν. Moreover,
+the qualitative picture varies from a few large and sta-
+ble waves with approximately constant velocity to many
+small and volatile waves which emerge, merge and dis-
+solve and vary greatly in velocity even for a single set of
+parameters. Two examples are shown in Fig. 1c and d,
+but a multitude of different multiwave patterns may be
+observed.
+Beyondrescalingthetimeaxis,thevaluesof Landgdo
+not affect the wave patterns, so long as the grain number
+Nis large compared to the average number of grains per100 1000 500 200 50 1001000
+50 500
+200
+Particle number, NParticles per wave, N w
+  
+μ=0.25, ν=0.40
+μ=0.80; ν=0.95
+μ=0.70; ν=0.90
+N=Nw 
+100 1000 50 500 2000.10.5
+0.2
+0.05
+0.02
+Particle number, NAverage velocity v av 
+  (a)
+(b)
+FIG. 2: (Color online) (a) Numberof grains per waves and (b)
+average grain velocity as functions of particle number Nfor
+three sets of parameters (same in both panels). The dashed
+line in the above panel is N=Nw, i.e., a single density wave
+is present.
+density wave. For Nbelow some µ,ν-dependent thresh-
+old, the grains will gather in one or a few stable or
+metastable waves, as exemplified in panels (e) and (f)
+of Fig. 1.
+WhenNis increased beyond this threshold, however,
+we find that the number of density waves increases lin-
+early,while thenumberofgrainsperdensitywaveremain
+constant. Average flow rate and the average velocity of
+density waves also remain independent of Nin the limit
+of largeN. This represents the asymptotic limit of our
+system, wherethere is nodependence ondensity ρ, which
+we illustrate for three pairs of coefficients in Fig. 2. Due
+to limitations of computer time we have not investigated
+this limit when Nwgrows beyond ≈400, which happens
+in the limit of either small µ, orν≈1, but we conjec-
+ture that such an asymptotic regime always exists when
+µandνare both on the open interval /angbracketleft0,1/angbracketright. Single-wave
+pictures such as panels (e) and (f) of Fig. 1, seen also
+in various other simulations [30–33], thus appear in our
+model simply because there are not enough particles in
+the system to form more than one wave [36].
+In Fig. 3a we estimate the number of grains per wave
+in the flocculent regime, Nw, after ‘thermalization’ (cal-
+culated as Ndivided by the number of density waves) in
+the asymptotic (large N) regime. In practice we choose
+Nlargeenoughthat the observedvalue of Nwis constant
+with increasing N. We have used Nranging from 500 to
+4000 in different areas of the plotted region. Nwvaries4
+  
+0.2 0.4 0.6 0.80.20.30.40.50.60.70.80.9
+−2 −1.8−1.6−1.4−1.2−1 −0.8
+νμa
+fe
+dcbνμLog10 (N w)
+  
+0.2 0.4 0.6 0.80.20.30.40.50.60.70.80.9
+11.21.41.61.822.22.42.62.8
+a
+fe
+dcb
+Log10 (v av )(a)
+(b)
+FIG.3: (Color online)(a): Plotoflog NwwhereNwisnumber
+of grains per density wave as a function of µandνin the
+asymptotic limit. (b): Plot of average flow rate as a function
+ofµandνin the asymptotic limit. In both panels, encircled
+letters refer to parameter pairs in Fig. 1.
+nonmonotonouslyand spans severalordersofmagnitude.
+Indeed, when µdecreases below 0 .2 orνapproaches 1,
+bothNwand ‘thermalization’ time diverge rapidly, mak-
+ing calculation in this region expensive.
+In Fig. 3b the average flow rate is given for all values
+ofµandν. In the gaseous regime we see a monotonous
+variation supporting our observation that there is no sig-
+nificant parameter dependency on the structures formed
+in this region. Within the flocculent regime the pattern
+is similar to that observed for the number of grains per
+density wave.
+Counterintuitively, flow rate is highest in areas of the
+µ,νplane in which the density waves are large, and con-
+versely, smaller waves correspond to a low flow rate.
+Since wave velocity is always lower than the flow rate,
+each grain will approach a density wave from above, col-
+lide its way through it, and fall into a low-density area
+beneath it once more before meeting another wave. Al-0.01 110 −110 010 110 210 310 410 5
+Frequency Power  
+0.1μ=0.30
+ν=0.85
+5
+FIG. 4: Frequency spectrum of the wave pattern of Fig. 1d.
+The graph is the average over the power spectra of 200 sub-
+samples of a long time series of ≈107data points.
+thoughlargewavesmovemoreslowlythansmallerwaves,
+we find that this is more than compensated by the pres-
+ence of large low density areas between waves in which
+the grains can fall freely. Hence the total flow rate is
+governed by the length of these acceleration stretches
+whereas we find no clear connection between flow rate
+and wave velocity.
+Wehaveanalyzedthepowerspectrumofatypicalwave
+pattern (that of Fig. 1d) shown in Fig. 4. The analysis
+was performed by recording density in a region of length
+L/10 as a function of time. With sampling frequency
+10 per time unit, about 107data points were recorded.
+The series was then divided into subsamples which were
+Fourier transformed individually, and the average of the
+powerforallsub-sampleswerethen taken(differentnum-
+ber of sub-samples were used for comparison; 200 sub-
+samples are used in figure 4). The spectrum is peaked at
+a few frequencies, demonstrating how the wave pattern
+has a number of preferred wave-front velocities. This is
+not obvious from studying spatiotemporal diagrams such
+asFig.1d. Inthehighfrequencyregimethepowerdiesoff
+in a manner consistent with a power law with exponent
+-2, consistent with the wave-fronts undergoing Brownian
+fluctuations.
+IV. CONCLUSIONS.
+We have studied granular pipe flow by means of a one-
+dimensional two-parameter model. The two parameters
+are the coefficients of restitution for collisions between
+grains and collisions between grains and walls. The very
+simple collision rules are contrasted by the amount of
+structure exhibited by the model, and formation of den-
+sity waves varying greatly in magnitude and qualitative
+behavior is observed. We find a criterion for the forma-
+tion of density waves: that the dissipation from collisions
+with walls be greater than that from grain-grain colli-
+sion. Under this criterion our model predicts that den-
+sity waves can form also in the absence of any interstitial5
+gas.
+Contrary to intuition, the flow rate is largest when
+density waves are large, slow and far between. This indi-
+catesthatinsomecircumstances,theflowrateingravity-
+drivengranularpipe flowcanbe increasedbysofteningor
+roughening the pipe walls. For example, with soft grainsdescribed by µ= 0.3, a ’rough’ pipe with ν= 0.5 gives
+a flow rate two to three times faster than a ’smoother’
+pipe for which ν= 0.7.
+We have benefited from discussions with J. P. Hulin
+and the Granular Media Group at FAST laboratory, Or-
+say.
+[1] R. Jackson, The dynamics of fluidized particles (Cam-
+bridge Univ. Press, 2000).
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+[20] G. H. Risto and H. J. Herrmann, Phys. Rev. E 50, R5
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+Rev. Lett. 80, 2833 (1998).
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+Phys. Fluids 15, 3358 (2003).
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+Phys. Rev. E 59, 778 (1999).
+[26] S. Horikawa, T. Isoda, T. Nakayama, A. Nakahara, and
+M. Matsushita, Physica A 233, 699 (1996).
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+E53, 4345 (1996).
+[28] Y. Bertho, F. Giorgiutti-Dauphin´ e, T. Raafat, E. J.
+Hinch, H. J. Herrmann, and J. P. Hulin, J. Fluid Mech.
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\ No newline at end of file
diff --git a/1001.0670.txt b/1001.0670.txt
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,536 @@
+arXiv:1001.0670v3  [nucl-th]  25 May 2010Baryon Stopping in Heavy-Ion Collisions at Elab=2–160 GeV/nucleon
+Yu.B. Ivanov1,2,∗
+1GSI Helmholtzzentrum f¨ ur Schwerionenforschung GmbH, D-6 4291 Darmstadt, Germany
+2Kurchatov Institute, Moscow RU-123182, Russia
+It is argued that the experimentally observed baryon stoppi ng may indicate (within the present
+experimental uncertainties) a non-monotonous behaviour a s a function of the incident energy of
+colliding nuclei. This can be quantified by a midrapidity red uced curvature of the net-proton
+rapidity spectrum. The above non-monotonous behaviour rev eals itself as a “zig-zag” irregularity
+in the excitation function of this curvature. The three-flui d dynamic calculations with a hadronic
+equation of state (EoS) fail to reproduce this irregularity . At the same time, the same calculations
+with an EoS involving a first-order phase transition into the quark-gluon phase do reproduce this
+“zig-zag” behaviour, however only qualitatively.
+PACS numbers: 24.10.Nz, 25.75.-q
+Keywords: relativistic heavy-ion collisions, baryon stop ping, hydrodynamics, phase transition
+I. INTRODUCTION
+A degree of stopping of colliding nuclei is one of the
+basic characteristics of the collision dynamics, which de-
+termines a part of the incident energy of colliding nuclei
+deposited into produced fireball and hence into produc-
+tion of secondary particles. The deposited energy in its
+turn determines the nature (hadronic or quark-gluonic)
+of the produced fireball and thereby its subsequent evo-
+lution. Therefore, a proper reproduction of the baryon
+stopping is of prime importance for theoretical under-
+standing of the dynamics of the nuclear collisions.
+A direct measure of the baryon stopping is the net-
+baryon rapidity distribution. However, since experimen-
+talinformationonneutronsisunavailable,wehavetorely
+on proton data. Presently there exist extensive experi-
+mental data on proton (or net-proton) rapidity spectra
+at AGS [1–4] and SPS [5–9] energies. These data were
+analyzed within various models [10–18] The most exten-
+sive analysis has been done in [14, 17]. Since that time
+new data at SPS energies have appeared [7–9]. There-
+fore, it is appropriate to repeat this analysis of already
+extended data set. In the present Letter it is done within
+the framework of the model of the three-fluid dynamics
+(3FD) [17].
+II. ANALYSIS OF EXPERIMENTAL DATA
+Available data on the proton (at AGS energies) and
+net-proton (at SPS energies) rapidity distributions from
+central heavy-ion collisions are presented in Fig. 1. Only
+the midrapidity region is displayed in Fig. 1, since it is
+of prime interest in the present consideration. The data
+at 10AGeV are repeated in the right panel of Fig. 1
+in order to keep the reference spectrum shape for the
+∗e-mail: Y.Ivanov@gsi.decomparison. The data are plotted as functions of a “di-
+mensionless” rapidity ( y−ycm)/ycm, whereycmis the
+center-of-mass rapidity of colliding nuclei. In particular,
+this is the reason why the experimental distributions are
+multiplied ycm. This representation is chosen in order to
+make different distributions of approximately the same
+width and the same height. This is convenient for com-
+parison of shapes of these distributions. To make this
+comparison more quantitative, the data are fitted by a
+simple formula
+dN
+dy=a(exp{−(1/ws)cosh(y−ycm−ys)}
++ exp{−(1/ws)cosh(y−ycm+ys)}) (1)
+wherea,ysandwsare parameters of the fit. The form
+(1) is a sum of two thermal sources shifted by ±ysfrom
+the midrapidity. The width wsof the sources can be
+interpreted as ws= (temperature)/(transverse mass), if
+we assume that collective velocities in the sources have
+no spread with respect to the source rapidities ±ys. The
+parameters of the two sources are identical (up to the
+sign ofys) because we consider only collisions of identical
+nuclei. Results of these fits are demonstrated in Fig.
+1. Energy dependence of parameters ysandwsdeduced
+from these fits revels no significant irregularities: they
+monotonously rise with the energy.
+The above fit has been done by the least-squares
+method. Data were fitted in the rapidity range |y−
+ycm|/ycm<0.7. The choice of this range is dictated
+by the data. As a rule, the data are available in this
+rapidity range, sometimes the data range is even more
+narrow (40 A, 80AGeV and new data at 158 AGeV [9]).
+We put the above restriction in order to treat different
+data in approximately the same rapidity range. Notice
+that the rapidity range should not be too wide in order
+to exclude contribution of cold spectators.
+We met problems with fitting the data at 80 AGeV [8]
+and the new data at 158 AGeV [9]. These data do not
+go beyond the side maxima in the rapidity distributions.
+The fit within such a narrow region results in the source
+rapidities ysvery close (at 80 AGeV) or even exceeding2
+-0.4 0 0.4 
+(y-ycm)/ycmSPSAGS
+-0.4 0 0.4 
+(y-ycm)/ycm60 80 100 120 ycm dN/dyAu(2A GeV) 5%, E895
+Au(4A GeV) 5%, E895
+Au(6A GeV) 5%, E917
+Au(8A GeV) 5%, E917
+Au(10A GeV) 5%, E866
+Au(10A GeV) 5%, E917
+Pb(20A GeV) 7%, NA49
+Pb(30A GeV) 7%, NA49
+Pb(40A GeV) 7%, NA49
+Pb(80A GeV) 7%, NA49
+Pb(158A GeV) 5%, NA49
+FIG. 1: Rapidity spectra of protons (for AGS energies) and
+net-protons ( p−¯p) (for SPS energies) from central collisions
+of Au+Au (AGS) and Pb+Pb (SPS). Experimental data are
+from collaborations E802 [1], E877 [2], E917 [3], E866 [4], a nd
+NA49 [5–9]. The percentage shows the fraction of the total
+reaction cross section, corresponding to experimental sel ec-
+tion of central events. Solid lines connecting points repre sent
+the two-source fits by Eq. (1). The dashed line is the fit to
+old data on Pb(158 AGeV)+Pb [5], these data themselves are
+not displayed.
+(at 158AGeV)ycmand a huge width ws. As a result, the
+normalization of the net-proton rapidity distributions, as
+calculated with fit (1), turns out to be 330 (at 80 AGeV)
+and 400 (at 158 AGeV), which are considerably larger
+than the total proton number in colliding nuclei (=164).
+To avoid this problem, we performed a biased fit of these
+data. An additional condition restricted the total nor-
+malization of distribution (1) to be less than the total
+proton number in colliding nuclei (=164). This biased fit
+isthereasonwhythecurvefittedtothenewdataat158 A
+GeV does not perfectly hit the experimental points. In
+particular, because of this problem we keep the old data
+at 158AGeV [5] in the analysis. We also use old data
+at 40AGeV, corresponding to centrality 7% [8], instead
+of recently published new data at higher (5%) centrality
+[9], since the data at the neighboring energies of 20 A,
+30Aand 80AGeV are known only at centrality 7% [8].
+Similarity of conditions, at which the data were taken,
+prevents excitation functions, which are of prime interest
+here, from revealing artificial irregularities.
+Inspectingevolutionofthespectrumshapewiththein-
+cident energy rise, we observe an irregularity. Beginning4 8 12 16 20 
+s1/2 [GeV]04812 Cy hadr. EoS
+ 2-ph. EoS
+ exp. fit
+FIG. 2: Midrapidity reduced curvature of the (net)proton
+rapidity spectrum as a function of the center-of-mass energ y
+of colliding nuclei as deduced from experimental data and
+predicted by3FD calculations with hadronic EoS (hadr. EoS)
+[19] and a EoS involving a first-order phase transition into
+the quark-gluon phase (2-ph. EoS) [20]. The thin dashed-
+dotted line demonstrates the effect of the 2-ph. EoS without
+changing the friction in the quark-gluon phase.
+from the lowest AGS energy to the top one the shape of
+the spectrum evolves from convex to slightly concave at
+10AGeV.However,at20 AGeVtheshapeagainbecomes
+distinctly convex. With the further energy rise the shape
+again transforms from the convex form to a highly con-
+cave one. In order to quantify this trend, we introduce
+a reduced curvature of the spectrum in the midrapidity
+defined as follows
+Cy≡/parenleftbigg
+y3
+cmd3N
+dy3/parenrightbigg
+y=ycm//parenleftbigg
+ycmdN
+dy/parenrightbigg
+y=ycm
+= (ycm/ws)2/parenleftbig
+sinh2ys−wscoshys/parenrightbig
+.(2)
+This curvature is defined with respect to the “di-
+mensionless” rapidity ( y−ycm)/ycm. The factor
+1/(ycmdN/dy)y=ycmis introduced in order to get rid of
+overall normalization of the spectrum, i.e. of the apa-
+rameter in terms of fit (1). The second part of Eq. (2)
+presents this curvature in terms of parameters of fit (1).
+Values of the curvature Cydeduced from fit (1) to ex-
+perimental data are displayed in Fig. 2. To evaluate er-
+rorsofthese deduced values, we estimated the errorspro-
+duced by the least-squares method, as well as performed
+fits in different the rapidity ranges: |y−ycm|/ycm<0.5
+and|y−ycm|/ycm<0.9, where it is appropriate, and also
+fits of the data at 80 AGeV [8] and the new data at 158 A3
+GeV [9] with different bias on the overall normalization
+of the distributions: Nprot.≤208 (i.e., half of the net-
+nucleons can be participant protons) and Nprot.≤128
+(which is the hydrodynamic normalization of the dis-
+tribution). The error bars present largest uncertainties
+among mentioned above. The lower point at s1/2= 17.3
+GeV corresponds to the new data at 158 AGeV. Its up-
+per error, as well as that of 80 A-GeV point, results from
+the uncertainty of the normalization. The irregularity
+observed in Fig. 1 is distinctly seen here as a “zig-zag”
+irregularity in the energy dependence of Cy.
+It is somewhat suspicious that the “zig-zag” irregu-
+larity happens at the border between the AGS and SPS
+energies. Itcouldimplythatthisirregularityresultsfrom
+different ways ofselecting central events in AGS and SPS
+experiments. However, there are indirect evidences of a
+physical (rather than methodical) nature of this irregu-
+larity. The difference between Cyvalues in two different
+experiments at 10 AGeV can be taken as an estimate of
+the methodical uncertainty. The difference between Cy
+values at 10 AGeV and 20 AGeV is two to three times
+larger than this methodical uncertainty. Moreover, we
+could expect that Cyat 20AGeV would be larger than
+that at 10 AGeV because the incident energy is higher
+and centrality selection at 20 AGeV is less restrictive
+(7%) than at 10 AGeV (5%). Contrary to these expec-
+tations the Cyat 20AGeV is smaller than that at 10 A
+GeV. There should be a physical reason for that. Ex-
+citation functions of other quantities [21] deduced from
+the same AGS and SPS data do not reveal any misfit
+at the border between the AGS and SPS domains. The
+latter suggests that the AGS and SPS data were taken
+at similar physical conditions. However, new data taken
+at the same acceptance and the same centrality selection
+in this energy range are highly desirable to clarify this
+problem. Hopefully such data will come from new ac-
+celerators FAIR at GSI and NICA at Dubna, as well as
+from the low-energy-scan program at RHIC.
+III. THREE-FLUID MODEL SIMULATIONS
+Figure 2 also contains Cydeduced from results of 3FD
+simulationswithahadronicequationofstate(hadr. EoS)
+[19] and a EoS involving a first-order phase transition
+into the quark-gluon phase (2-ph. EoS) [20]. To obtain
+ysandws, the 3FD spectra were also fitted by the form
+(1). For central (5%) Au+Au collisions at AGS energies
+we performed our calculations taking a fixed impact pa-
+rameter b= 2 fm; for the central (5%) Pb+Pb reaction
+atElab= 158AGeV,b= 2.4 fm which is the experimen-
+tal estimate for this centrality [22]; for other central (7%)
+Pb+Pb collisions at 20 A-80AGeV,b= 3 fm.
+The 3FD model with the hadronic EoS reasonably re-
+produces a great body of experimental data in a wide
+energy range from AGS to SPS, see Ref. [17, 23–25]. De-
+scription of the rapidity distributions with the hadronic
+EoSis reported in Refs. [17, 18]. The reproductionofthe4 8 12 16 20 
+s1/2 [GeV]70 80 90 100 110 120 ycm (dN/dy)cmNA49
+E895, E917, E866
+hadr. EoS
+hadr. EoS w/o coal.
+2-ph. EoS
+FIG. 3: Midrapidity value of the net-proton rapidity spec-
+trum as a function of the center-of-mass energy of colliding
+nuclei. Experimental data are confronted to predictions of
+the 3FD model with the hadronic EoS (hadr. EoS) [19] and
+the EoS with a first-order phase transition (2-ph. EoS) [20].
+The thin long-dashed line corresponds to the hadr.-EoS cal-
+culation without fragment production.
+distributionsisquitegoodatthe AGSenergiesandatthe
+top SPS energies. At 40 AGeV the description is still sat-
+isfactory. However, at 20 Aand 30AGeV the hadr.-EoS
+predictions completely disagree with the data, cf. [18].
+At 20AGeV instead of a bump at the midrapidity the
+hadronic scenario predicts a quite pronounced dip. The
+problemswiththe descriptionofthelow-energySPSdata
+are clearly seen from Fig. 2 and also from Fig. 3, where
+midrapidity values of the rapidity distributions (multi-
+plied by the center-of-mass rapidity) are presented. In
+Ref. [18] it was demonstrated that the problem with the
+low-energy SPS data can be solved by considerable soft-
+ening the hadronic EoS. This softening may indicate an
+onset of the phase transition into the quark-gluon phase.
+Notice that a maximum in ycm(dN/dy)cmats1/2= 4.7
+GeVhappens only becausethe light fragment production
+becomes negligible above this energy. The 3FD calcula-
+tion without coalescence (i.e. without the fragment pro-
+duction)revealsamonotonousdecreaseof ycm(dN/dy)cm
+beginning from s1/2= 2.7 GeV, i.e. from the lowest en-
+ergy considered here.
+The 3FD simulations have been also done with a EoS
+involving a first-order phase transition into the quark-
+gluon phase (2-ph. EoS) [20]. In 2-ph. EoS the Gibbs
+construction was used for the mixed phase. These calcu-
+lations well reproduce the AGS data up to the energy of4
+1 10 100 
+s1/2 [GeV]020 40 60 80 (dN/dy)cmhadr. EoS  net-p
+2-ph. EoS  net-p
+E895,E917,E866
+NA49 net-p
+NA49 p
+STAR net-p
+STAR p
+FIG. 4: The same as in Fig. 3 but in conventional repre-
+sentation (without multiplying by ycm) and in a wider energy
+range including RHIC data on Au+Au collisions at 5% cen-
+trality [26]. Proton data [7, 26] are also displayed.
+6AGeV, where the purely hadronic scenario is realized.
+The data at the top SPS energy are also reproduced,
+which is achieved by a proper tune of the inter-fluid fric-
+tion in the quark-gluon phase. Quality of the reproduc-
+tion of above data is approximately the same as that
+with the hadronic EoS, as it is, e.g., seen from Figs. 2
+and from Fig. 3. However, at top AGS and lower SPS
+energies(8 A-80AGeV), wherethe mixed phase turnsout
+to be really important, the 2-ph. EoS completely fails.
+The fact that the 2-ph.-EoS line perfectly hits 20 A-40A-
+GeV experimental points in Fig. 3 is just a coincidence,
+shapes of the distributions are completely wrong, as seen
+from Figs. 2. This failure cannot be cured by variations
+of neither the friction nor the freeze-out criterion.
+However, the Cycurvature energy dependence in the
+first-order-transition scenario manifests qualitatively the
+same “zig-zag” irregularity (Fig. 2), as that in the
+data fit, while the hadronic scenario produces purely
+monotonous behaviour. This “zig-zag” irregularity of
+the first-order-transition scenario is also reflected in the
+midrapidity values of the (net)proton rapidity spectrum
+(Fig. 3). As for the experimental data, it is still diffi-
+cult to judge if the “zig-zag” anomaly in the midrapid-
+ity values is statistically significant. In the conventional
+representation of the data (Fig. 4) without multiplying
+byycm, the irregularity of the ( dN/dy)cmdata is hardly
+visible. However, the conventional representation clearly
+demonstrates the overall trend of the data: the midra-
+pidity net-proton yield gradually decreases with the in-cident energy, while the proton one stays approximately
+constant above the top SPS energy. Below the top SPS
+energy the proton and net-proton yields practically coin-
+cide. Model computations above the top SPS energy are
+at present not feasible because of high memory consump-
+tion required by the code (see discussion in Ref. [17]).
+Allabovediscussionconcernsonlycentralnuclearcolli-
+sions. Experimentaldataonmidcentralcollisionsismuch
+less complete. The model calculations for midcentral col-
+lisions (b≈6 fm) reveal the same quantitative behaviour
+of the excitation functions of Cyand (dN/dy)cmboth for
+hadr. EoS and 2-ph. EoS.
+The baryon stopping depends on a character of inter-
+actions (e.g., cross sections) of the matter constituents.
+If during the interpenetration stage of colliding nuclei a
+phasetransformation1ofthe hadronicmatter into quark-
+gluonic one happens, one can expect a change of the
+stopping power of the matter at this time span. This
+is a natural consequence of a change of the constituent
+content of the matter because hadron-hadron cross sec-
+tions differ from quark-quark, quark-gluon, etc. ones.
+This can naturally result in a non-monotonousbehaviour
+of the shape of the (net)proton rapidity-spectrum at an
+incident energy, where onset of the phase transition oc-
+curs. Of course, the first-order transition does not hap-
+penabruptly. Within theGibbs constructionthe fraction
+of the quark-gluon phase is gradually increasing, as well
+asweightsofthe correspondingcrosssections. Therefore,
+a non-monotonous behaviour will show up only if the dif-
+ference in cross sections in the hadronic and quark-gluon
+phases is large enough to override the above gradual in-
+crease of the fraction of the new phase. In fact, this is
+the case in the 3FD calculation with the phase transi-
+tion (2-ph. EoS). The friction in the quark-gluon phase
+was tuned to reproduce the data at the top SPS energy.
+Naturally, it does not continuously match the friction in
+the hadronic phase. In terms of parton-parton cross sec-
+tions, these cross sections in the quark-gluon phase turn
+out to be approximately twice as large as those in the
+hadronic phase2. In the quark-gluon phase these cross
+sections are compatible with those used, e.g., in a multi-
+phase transport model [27] and a parton cascade model
+[28].
+Notice that the proton rapidity distribution at 158 A
+GeV is well described within the color-glass-condensate
+framework based on small-coupling QCD [29]. This
+mechanism drastically differs from that of hadronic stop-
+ping. Therefore, it is not surprising that the 3FD model
+requires very different (from the hadronic one) phe-
+nomenological friction at the 158 A-GeV energy to repro-
+duce the data.
+1The term “phase transition” is deliberately avoided, since it usu-
+ally implies thermal equilibrium.
+2In the hadronic phase this parton cross section corresponds to
+the proton-proton one on the assumption of naive valence qua rk
+counting.5
+0 0.5 1 1.5 2
+nB[fm-3]01234ε−mNnB [GeV/fm3]2-ph. EoS
+ 4A GeV
+10A GeV
+20A GeV
+ε(T=0)-mNnB
+inaccessible
+regionmixed 
+phasehadr. EoS
+4A GeV
+10A GeV
+20A GeV
+FIG. 5: Dynamical trajectories of the matter in the central
+box of the colliding nuclei (4fm ×4fm×γcm4fm), where γcmis
+the Lorentz factor associated with the initial nuclear moti on
+in the c.m. frame, for central ( b= 0) collisions of Au+Au
+at 4Aand 10AGeV energies and Pb+Pb at 20 AGeV. The
+trajectories are plotted in terms of baryon density ( nB) and
+the energy density minus nBmultiplied by the nucleon mass
+(ε−mNnB). Only expansion stages of the evolution are dis-
+played for two EoS’s. Symbols on the trajectories indicate
+the time rate of the evolution: time span between marks is 1
+fm/c.
+However, if even the same friction is used in both
+phases, the calculated (with 2-ph. EoS) reduced cur-
+vature still reveals a “zig-zag” behaviour but with con-
+siderably smaller amplitude (see the thin dashed-dotted
+line in Fig. 2). This happens because the EoS in a gener-
+alized sense of this term, i.e. viewed as a partition of the
+total energy between kinetic and potential parts, also af-
+fects the stopping power. The friction is proportional to
+the relative velocity of the counter-streaming nuclei [17].
+Therefore, it is more efficient when the kinetic-energy
+part of the total energy is higher, i.e. when the EoS is
+softer. This effect of the softening was demonstrated in
+Ref. [18]. It wasshownthat applicationofasoft, butstill
+hadronic EoS changes the rapidity distributions, making
+them closer to the data at low SPS energies. This is pre-
+cisely what the phase transition does: it makes the EoS
+essentially softer in the mixed-phase region. The latter
+naturally results in a non-monotonous evolution of the
+proton rapidity spectra with the energy rise.
+Figure5demonstratesthattheonsetofthephasetran-
+sition in the calculations indeed happens at top-AGS–low-SPS energies, where the “zig-zag” irregularity takes
+place. Similarly to that it has been done in Ref. [30],
+the figure displays dynamical trajectories of the matter
+in the central box placed around the origin r= (0,0,0)
+in the frame of equal velocities of colliding nuclei: |x| ≤2
+fm,|y| ≤2 fm and |z| ≤γcm2 fm, where γcmis Lorentz
+factor associated with the initial nuclear motion in the
+c.m. frame. Initially, the colliding nuclei are placed sym-
+metrically with respect to the origin r= (0,0,0),zis
+the direction of the beam. The ε-nBrepresentation is
+chosen because these densities are dynamical quantities
+and, therefore, are suitable to compare calculations with
+different EoS’s. Subtraction of the mNnBterm is taken
+for the sake of suitable representation of the plot. Only
+expansion stages of the evolution are displayed, where
+the matter in the box is already thermally equilibrated.
+The size of the box was chosen to be large enough that
+the amount of matter in it can be representative to con-
+clude on the onset of the phase transition and to be
+small enough to consider the matter in it as a homo-
+geneous medium. Nevertheless, the matter in the box
+still amounts to a minor part of the total matter of col-
+liding nuclei. Therefore, only the minor part of the total
+matter undergoes the phase transition at 10 AGeV en-
+ergy. As seen, the trajectories for two different EoS’s are
+very similar at AGS energies and start to differ at SPS
+energies because of the effect of the phase transition.
+IV. CONCLUSIONS
+In conclusion, it is argued that the experimentally ob-
+served baryon stopping may indicate (within the present
+experimentaluncertainties)anon-monotonousbehaviour
+as a function of the incident energy of colliding nuclei.
+This reveals itself in a “zig-zag” irregularity in the exci-
+tation function of a midrapidity reduced curvature of the
+(net)proton rapidity spectrum. Notice that the energy
+location of this anomaly coincides with the previously
+observed anomalies for other hadron-production proper-
+tiesatthe lowSPSenergies[21, 31]. The 3FDcalculation
+with the hadronicEoS fails to reproduce this irregularity.
+At the same time, the same calculation with the EoS in-
+volvingafirst-orderphasetransitionintothequark-gluon
+phase (within the Gibbs construction) [20] reproduces
+this“zig-zag”behaviour,howeveronlyqualitatively. Pre-
+liminarysimulationswith the EoSofRef. [32], alsobased
+onthefirst-orderphasetransitionbutwithintheMaxwell
+construction, show the same qualitative trend. It is ar-
+gued that the non-monotonous behaviour of the baryon
+stopping is a natural consequence of a phase transition.
+The question why these calculations do not qualitatively
+reproduce the “zig-zag” irregularity deserves special dis-
+cussion elsewhere. It is very probable that either the
+Gibbs and Maxwell constructions are inappropriate for
+the fast dynamics of the heavy-ion collisions [33, 34] or
+the phase transition is not of the first order.
+Fruitful discussions with B. Friman, M. Gazdzicki, J.6
+Knoll, P. Senger, H. Str¨ obele, V.D. Toneev and D.N.
+Voskresensky are gratefully acknowledged. I am grateful
+to A.S.Khvorostukhin, V.V.Skokov, and V.D.Toneev for
+providing me with the tabulated 2-ph. EoS. I am grate-
+ful to members of the NA49 Collaboration for providing
+me with the experimental data in the digital form. This
+work was supported in part by the Deutsche Bundesmin-isterium f¨ ur Bildung und Forschung(BMBF project RUS
+08/038), the Deutsche Forschungsgemeinschaft (DFG
+projects 436 RUS 113/558/0-3 and WA 431/8-1), the
+Russian Foundation for Basic Research (RFBR grant 09-
+02-91331 NNIO a), and the Russian Ministry of Science
+and Education (grant NS-7235.2010.2).
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\ No newline at end of file
diff --git a/1001.0671.txt b/1001.0671.txt
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+arXiv:1001.0671v1  [cond-mat.supr-con]  5 Jan 2010Surface losses and self-pumping effects in a long
+Josephson junction — a semi-analytical approach
+Marek Jaworski
+Institute of Physics, Polish Academy of Sciences, Al. Lotni k´ ow 32/46, 02-668 Warszawa, Poland
+The flux-flow dynamics in a long Josephson junction is studied both analytically and numerically.
+A realistic model of the junction is considered by taking int o account a nonuniform current distri-
+bution, surface losses and self-pumping effects. An approxi mate analytical solution of the modified
+sine-Gordon equation is derived in the form of a unidirectio nal dense fluxon train accompanied
+by two oppositely directed plasma waves. Next, some macrosc opic time-averaged quantities are
+calculated making possible to evaluate the current-voltag e characteristic of the junction. The re-
+sults obtained by the present method are compared with direc t numerical simulations both for the
+current-voltage characteristics and for the loss factor mo dulated spatially due to the self-pumping.
+The comparison shows very good agreement for typical juncti on parameters but indicates also some
+limitations of the method.
+PACS numbers: 74.50.+r, 05.45.Yv, 85.25.Cp
+I. INTRODUCTION
+In recent years the flux-flow (FF) dynamics in a long
+Josephson junction has attracted considerable attention
+in view of possible applications in superconducting mm-
+wave electronics1–4. The FF mode appears in a junction
+immersed in a sufficiently large external magnetic field
+and can be described briefly as a unidirectional viscous
+flow of a dense train of fluxons (magnetic flux quanta).
+Recently, the FF oscillators have found various appli-
+cations,e.g. inmm-waveintegratedreceivers4–7, however
+some important parameters, such as the radiation line-
+width are still not satisfactory. Therefore, there is still
+a need for more adequate description of a real Joseph-
+son junction operating in the FF mode, by taking into
+account some additional factors which may affect the
+current-voltage ( I-V) characteristic of the junction and
+change its working conditions.
+In the majority of papers dealing with long Josephson
+junctions (see e.g. Refs. 8–14) a simplified version of the
+the sine-Gordon (sG) equation is considered, usually ne-
+glecting surface losses and assuming uniform distribution
+of the bias current density. Only recently, a few papers
+have been published15–18taking into account a nonuni-
+form bias current distribution and its influence on the
+junction behavior. Moreover, in Refs. 16,17 a realistic
+model of the FF oscillator has been investigated both ex-
+perimentally and numerically, including general bound-
+ary conditions, nonuniform bias current profile, surface
+losses as well as self-pumping effects19related to addi-
+tional tunneling of quasiparticles due to the Josephson
+radiation.
+Theaimofthe presentpaperistopresentananalytical
+approach to the modified sG equation, which takes into
+account:(i) nonuniform current distribution, (ii) surface
+losses, and (iii) spatial modulation of the loss factor re-
+sulting from the self-pumping effect. Contrary to Ref. 16
+we assume standard open-circuit boundary conditions to
+make the influence of various effects more pronounced.Nevertheless, the present analysis can be easily extended
+to include also more general boundary conditions.
+In the particular case of uniform current distribution,
+the present method makes it possible to obtain fully an-
+alytical closed-form expressions describing both the su-
+perconducting phase within the junction and the I-V
+characteristic. Such a solution can be regarded as the
+first-order approximation which appears sufficiently ac-
+curate for some moderate junction parameters. How-
+ever, in the general case, particularly for very long and
+weakly damped junctions, such an analytical approxi-
+mation is only the first step in an iterative procedure,
+which has to be performed numerically. Thus, in spite
+of analytical expressions describing the superconducting
+phase within the junction, the present approachhas been
+named “semi-analytical”.
+The paper is organized as follows. In Sec. II we for-
+mulate the problem, i.e. we present a generalized sG
+equation subject to open-circuit boundary conditions at
+the junction ends. Approximate analytical solutions to
+the sG equation are discussed in Sec. III. We start with
+a linearized (small-amplitude) solution and apply appro-
+priate boundary conditions. Next, some large-amplitude
+corrections are introduced, we discuss also possible self-
+pumping effects and their influence on the I-Vcharacter-
+istic. In Sec. IV analytical results are compared with di-
+rect numerical simulations. In particular, we discuss the
+influence of surface losses both for junctions of moderate
+length and for more realisticstructures, such as verylong
+junctions with small damping and strongly nonuniform
+currentdistribution. Sec.Vcontainsconcludingremarks,
+weindicatealsopossibleextensionsofthemethodbytak-
+ing into account more general boundary conditions.2
+II. FORMULATION OF THE PROBLEM
+Fluxon dynamics in a long Josephson junction is usu-
+ally described by the following modified sG equation8–12:
+φxx−φtt−αφt= sinφ−γ, (1)
+whereφdenote the quantum phase difference across the
+barrier,αis the loss factor, and γis the bias current
+density. The spatial coordinate xhas been normalized
+to the Josephson penetration depth λJand the time co-
+ordinatetto the inverse plasma frequency ω−1
+0, where
+λJ= (¯h/2µ0edjc)1/2,ω0= (2ejc/¯hC)1/2,jcisthecritical
+current density and Cdenotes the junction capacitance
+per unit area.
+However, to describe more adequately a real physical
+situation, one can consider a more general form16,17:
+φxx−φtt−α(x)φt+βφxxt= sinφ−γ(x),(2)
+whereβdenotes the surface loss parameter and we as-
+sume both αandγto bex-dependent, taking into ac-
+count both a spatial modulation of the loss factor due
+to the self-pumping and a nonuniform distribution of the
+current density along the junction.
+For the overlapgeometry one can neglect the self-fields
+and assume simple open-circuit boundary conditions9:
+φx(±L/2)+βφxt(±L/2) =h, (3)
+whereh=Hext/jcλJ,Hextdenotes the external mag-
+netic field, and L≫1 is the normalized junction length.
+Following Refs. 11–14,18 we look for an approximate
+solution in a form of a dense fluxon train traveling on
+a rotating background. Thus, a linearized solution of
+Eq. (2) can be written as
+φ=φ0+ψ, (4)
+whereφ0=θ(x) +Ωtis the background term (linear in
+time) andψdenotes a quasi-linear term (usually small)
+representing the motion of fluxons and plasma waves
+within the junction.
+Substituting Eq. (4) into Eq. (2) we find
+θxx+ψxx−ψtt−αΩ−αψt+βψxxt= sin(φ0+ψ)−γ.(5)
+We are interested in a steady-state, time periodic so-
+lution, thus the background frequency Ω should be equal
+to the fundamental frequency of the oscillating term ψ.
+Both experimental data and numerical simulations show
+that the output signal of a real FF oscillator is nearly
+sinusoidal1,15. Thus, it is reasonable to assume the time-
+dependence of the term ψto be harmonic in Ω t, and
+neglect any higher harmonics generated by the nonlin-
+ear term sin( φ0+ψ). Consequently, Eq. (5) can be split
+into a time-independent part and a part oscillating with
+frequency Ω, while the boundary conditions (3) can be
+written separately for θ(x) andψ(x,t):
+θx(±L/2) =h, (6)
+ψx(±L/2)+βψxt(±L/2) = 0. (7)III. APPROXIMATE ANALYTICAL SOLUTION
+Themethod forsolvingEq.(5) with the boundarycon-
+ditions(6)and(7)issimilartothatreportedrecently(see
+Ref. 18 for details). Thus, below we discuss briefly only
+the main points of the analysis.
+A. Time-independent equation
+It can be easily shown that for ψoscillating with fre-
+quency Ω, the nonlinearterm sin( φ0+ψ) contributes also
+to the time-independent part of Eq. (5), and such a con-
+tribution yields, in fact, the first-order approximation of
+theI-Vcharacteristic of the junction. Thus, the time-
+independent equation can be written as:
+θxx=α(x)Ω+S(x)−γ(x), (8)
+where
+S(x) =/angbracketleftsin(φ0+ψ)/angbracketrightT= (1/T)/integraldisplayT
+0sin(φ0+ψ)dt,(9)
+andT= 2π/Ω.
+Solving Eq. (8) for θ(x) we find
+θ=/integraldisplayx
+−L/2/bracketleftBigg/integraldisplayη
+−L/2[α(ξ)Ω+S(ξ)−γ(ξ)]dξ/bracketrightBigg
+dη+C1x+C2.
+(10)
+Without loss of generality C2can be set to zero while
+from the boundary condition (6) we find C1=hand
+γ0=α0Ω+1
+L/integraldisplayL/2
+−L/2S(x)dx, (11)
+whereγ0,α0denote the averaged current density and
+loss factor, respectively:
+γ0=1
+L/integraldisplayL/2
+−L/2γ(x)dx, α 0=1
+L/integraldisplayL/2
+−L/2α(x)dx.(12)
+It follows from the relation (4) that the time-averaged
+value ofφt, (being proportional to the constant voltage)
+isequaltothefrequencyΩ. Thus, theexpression(11)can
+be regarded as the current-voltage ( I-V) characteristic
+of the Josephson junction. One can see that the total
+current density γ0consists of a linear (Ohmic) part and a
+nonlinear contribution related to the Josephson current.
+B. Time-dependent equation
+Forψsufficiently small we can write
+sin(φ0+ψ)≃sinφ0+ψcosφ0. (13)3
+Noting that the term ψcosφ0gives no contribution of
+frequency Ω and collecting all the time-dependent terms
+in Eq. (5) we find
+ψxx−ψtt−α(x)ψt+βψxxt≃sinφ0.(14)
+Since all the terms in Eq. (14) oscillate with the
+same frequency Ω, it is convenient to use a complex
+notation14,18
+ψ(x,t) = Im/bracketleftBig
+/hatwideψ(x)eiΩt/bracketrightBig
+,sinφ0= Im/bracketleftBig
+ei(θ+Ωt)/bracketrightBig
+,
+(15)
+and rewrite Eq. (14) as an ordinary differential equation
+(1+iΩβ)/hatwideψxx+(Ω2−iα(x)Ω)/hatwideψ=eiθ,(16)
+or
+/hatwideψxx+δ2/hatwideψ=eiθ/P, (17)
+whereP= 1+iΩβ,δ2= (Ω2−iα(x)Ω)/P.
+The general approximate solution of Eq. (17) can be
+written in a WKB-like form20
+/hatwideψ(x) =1
+2iP√
+δ/bracketleftBig
+eif(x)F−(x)−e−if(x)F+(x)/bracketrightBig
++Aeif(x)
+√
+δ+Be−if(x)
+√
+δ, (18)
+where
+F±(x) =/integraldisplayx
+−L/2ei(θ(ξ)±f(ξ))dξ√
+δ, f(x) =/integraldisplayx
+−L/2δ(ξ)dξ.
+(19)
+The term in square brackets in Eq. (18) corresponds
+to a dense fluxon train moving unidirectionally along the
+junction, while the last two terms describe two plasma
+waves propagating in opposite directions.
+Using the complex formalism the boundary condition
+(7) can be rewritten as
+/hatwideψx(±L/2)+iΩβ/hatwideψx(±L/2) = 0 = ⇒/hatwideψx(±L/2) = 0.
+(20)
+Thus, using Eqs. (18) and (20) one can determine the
+integration constants AandBto be
+A=B=eiδ0LF−(L/2)+e−iδ0LF+(L/2)
+4Psinδ0L,(21)
+where
+δ0=1
+L/integraldisplayL/2
+−L/2δ(x)dx. (22)
+It should be mentioned here that the solutions for θ
+and/hatwideψ(Eqs. (10) and (18), respectively) are not given
+explicitly but rather form a system of coupled equations.
+Fortunately, this systemcanbeeasilysolvedbyamethod
+of consecutive iterations. Indeed, starting with S(x) = 0
+one can solve Eq. (10) for θand next solve Eq. (18) for/hatwideψ. Substituting θand/hatwideψinto Eq. (9) we obtain a new
+approximation for S(x) and consequently new approxi-
+mations for θand/hatwideψ.
+It is clear from Eq. (11) that S(x) = 0 corresponds
+simply to the Ohmic line γ0=α0Ω, while consecutive
+iterations yield a sequence of approximations for S(x)
+describing the time-independent contribution from the
+Josephson current.
+The rate of convergence of the iterative process de-
+pends strongly on the junction parameters. As shown
+in the next Section, for moderate values of L,h, andα0,
+the first iteration yields satisfactoryresults. However, for
+practical FF oscillators which are usually based on very
+long junctions, up to a few thousands of iterations are
+needed to obtain a self-consistent solution.
+C. Analytical approximation
+It is interesting that for the simplest, but widely
+used model of uniform bias current distribution ( γ(x) =
+const), all the integrationscan be performed analytically,
+leading to a compact and fully analytical expression for
+theI-Vcharacteristic.
+Indeed, assuming the small-amplitude limit ( /hatwideψ≪1)
+and ignoring self-pumping effects ( α(x) = const), one
+can calculate the first-order approximation for Eq. (11).
+Followingthe steps outlined abovewe find θ(x) =hxand
+consequently
+/hatwideψ(x) =−eihx
+P(h2−δ2)+hsin[(h+δ)L/2]
+P(h2−δ2)δsinδLeiδx
++hsin[(h−δ)L/2]
+P(h2−δ2)δsinδLe−iδx(23)
+and
+γ0=α0Ω+Im/bracketleftbigg
+−1
+2P(h2−δ2)+h2(cosδL−coshL)
+LP(h2−δ2)2δsinδL/bracketrightbigg
+,
+(24)
+wherePandδhave been defined in Eq. (17).
+When the surface losses are neglected ( β= 0) the
+above simple expression is reduced to the solution de-
+rived earlier14. Moreover, it can be shown that for β= 0
+the expression (24) is also equivalent to apparently dif-
+ferent solutions derived independently in Refs. 15,21 in a
+form of infinite series expansions.
+D. Large-amplitude corrections
+As follows from Eq. (11), the evaluation of S(x) is cru-
+cial for the determination of I-Vcharacteristic. So far
+we haveassumed ψ≪1 and used an approximation(13).
+However, for ψlarger we should consider the exact rela-
+tion
+S(x) =/angbracketleftsinφ0cosψ/angbracketrightT+/angbracketleftcosφ0sinψ/angbracketrightT.(25)4
+Sinceψis an oscillatory function, one can use well-
+known relations22involving Bessel function. As shown
+in Ref. 18, the leading terms for cos ψand sinψare given
+by
+cosψ≃J0(|/hatwideψ|),sinψ≃2J1(|/hatwideψ|)
+|/hatwideψ|ψ,(26)
+whereJ0,J1denote the Bessel functions of order 0 and
+1, respectively.
+Using Eq. (26) we can find the time-independent con-
+tributionS(x) to be18
+S(x) =J1(|/hatwideψ|)
+|/hatwideψ|Im/bracketleftBig
+/hatwideψe−iθ/bracketrightBig
+. (27)
+Accordingly, the right-hand sides of Eqs. (14) and (16)
+should be replaced by sin φ0cosψandJ0(|/hatwideψ|)eiθ, respec-
+tively.
+E. Self-pumping effects
+According to Refs. 16,17, the total current density due
+to the quasi-particle tunneling is given by
+γeff=∞/summationdisplay
+−∞J2
+n(evac/¯hω)γdc(vdc+n¯hω/e),(28)
+whereJnis the Bessel functions of order n,γdcdenotes
+the unpumped I-Vdependence, and the total voltage
+applied to the junction can be separated into a constant
+(vdc) part and and an oscillatory part of amplitude vac
+and frequency ω.
+Using the Josephson relation23vdc= ¯hω/2eand com-
+ing back to our dimensionless notation we find vdc= Ω,
+vac=|/hatwideψ|Ω and
+γeff=∞/summationdisplay
+−∞J2
+n/parenleftBig
+|/hatwideψ|/2/parenrightBig
+γdc[Ω(1+2n)].(29)
+For small arguments z=|/hatwideψ|/2≪1 the infinite ex-
+pansion (29) can be truncated to include only quadratic
+terms inz:
+γeff=γdc(Ω)+z2
+4[γdc(−Ω)−2γdc(Ω)+γdc(3Ω)].(30)
+As an unpumped I-Vcharacteristic we can take the
+nonlinear resistive model24
+γdc(Ω) =α0Ω/braceleftbigg
+b(Ω/Ωg)n
+[(Ω/Ωg)n+1]+1/bracerightbigg
+,(31)
+wheren≫1, Ωgdenotes the normalized gap voltage and
+b=Rj/Rnis the ratio of normal-state resistances below
+and above Ω g.Forn→ ∞the highly nonlinear continuous depen-
+dence (31) tends to a simple discontinuous linear form
+which will be used in further calculations:
+γdc=/braceleftbigg
+α0Ω
+α0Ω(b+1)forΩ<Ωg,
+Ω>Ωg.(32)
+It is clear that γeffisx-dependent due to the self-
+pumping via /hatwideψ(x), thus we can define an effective damp-
+ing factor αeff(x) =γeff(x)/Ω and compute it self-
+consistently by starting with αeff=α0, calculating /hatwideψ,
+substituting into Eq. (29), evaluating a new approxima-
+tion forαeff(x), and so on.
+IV. RESULTS AND DISCUSSION
+In this section we compare analytical results derived
+above with numerical simulations obtained by the finite-
+difference implicit scheme25. TheI-Vcharacteristic is
+given by Eq. (11), while S(x) follows from the self-
+consistent solutions of Eqs. (10) and (18). To illustrate
+the influence of surface losses we consider first the sim-
+plest caseγ(x) = const where the I-Vdependence can
+be expressed in a closed form (24).
+Fig. 1 shows the central part of the I-Vcharacteristic
+for a junction of moderate length L= 5. The remain-
+ing parameters ( α0= 0.2,h= 3) are similar to those
+assumed in Ref. 15. Open circles denote the results of
+numerical simulations for a discrete set of γ0points. The
+dashed line denotes the analytical solution (24) while the
+solid line follows from the self-consistent solutions (10)
+and (18) obtained by consecutive iterations for θand/hatwideψ.
+Fig. 1(a) shows the case β= 0 (no surface losses) and in
+Fig. 1(b) we assume β= 0.01. Similar results for h= 5
+are presented in Fig. 2(a),(b).
+One can see that the I-Vdependence departs from the
+Ohmic line γ0=α0Ω in the region of Ω ≃h, forming the
+main FF step modulated by a series of Fiske steps with
+voltage spacing ∆Ω ≃π/L. Analytical approximations
+are continuous and consist of a series of resonances while
+numerical results show typical hysteretic behavior and
+we observe only the segments of positive slope.
+It is clear that the presence of even very small surface
+losses changes significantly the I-Vdependence. Gen-
+erally, the steps (resonances) become smaller and this
+effect is more pronounced for larger values of the exter-
+nal fieldh. Such a result can be easily explained if we
+recall that the main FF step and accompanying Fiske
+steps are visible in the region Ω ≃hand the surface loss
+factor enters the formalism via P= 1 +iΩβ. In other
+words, for higher external magnetic field the influence
+of surface losses is stronger and the I-Vcharacteristic
+becomes more smooth. The influence of surface losses
+is clearly visible in experimental results16,17, where the
+Fiske steps gradually disappear as the external magnetic
+field is getting stronger.
+Comparing analytical and numerical results shown in
+Figs. 1 and 2 one can see that the fully analytical solu-5
+/s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48
+/s32/s32/s48/s40/s98/s41/s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48
+/s32/s32/s48/s40/s97/s41
+FIG. 1: (Color online) Current-voltage characteristic cal cu-
+lated analytically (solid line) and numerically (open circ les)
+forγ(x) = const, L= 5,h= 3,α0= 0.2: (a)β= 0, (b)
+β= 0.01. The dashed line shows the first-order approxima-
+tion (24).
+tion (dashed line) is fairly accurate. On the other hand,
+the self-consistent solution (solid line) reproduces very
+accurately all the details of the numerical solution.
+As the next example, let us consider a more realistic
+case of a very long junction with small damping. Follow-
+ing Ref. 17 we choose L= 40,α0= 0.033,β= 0.035,
+and a slightly asymmetric current profile γ(x) depicted
+in Fig. 3. The bias electrode ( x1≤x≤x2) is shorter
+than the total junction length. Consequently, the current
+distribution is assumed parabolic for x1≤x≤x2, and
+exponential in the unbiased tails ( x≤x1,x≥x2).
+Contrary to the previous example, now we cannot use
+the analytical approximation (24). First, the current dis-
+tribution is x-dependent, what means that θ(x) departs
+from a simple linear dependence θ(x) =hx, and rele-
+vant integrals cannot be calculated analytically. Second,
+it appears (even for a uniform current profile) that the
+first-order approximation is not accurate enough for very
+long junctions, and consecutive iterations are necessary
+to obtain a self-consistent solution./s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48/s48/s46/s56/s48/s46/s57/s49/s46/s48/s49/s46/s49/s49/s46/s50/s49/s46/s51
+/s32/s32/s48/s40/s98/s41/s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48/s48/s46/s56/s48/s46/s57/s49/s46/s48/s49/s46/s49/s49/s46/s50/s49/s46/s51
+/s32/s32/s48/s40/s97/s41
+FIG. 2: (Color online) the same as in Fig. 1 but for h= 5.
+Fig.4showsthe I-Vcharacteristicscalculatedforjunc-
+tion parameters specified above and for two values of
+the external magnetic field h= 2.5 andh= 3.5. As
+before, open circles correspond to numerical simulations
+while the solid line represents a self-consistent solution.
+One can see that the agreement between numerical and
+analytical results is rather poor for h= 2.5. To ex-
+plain this discrepancy we should recall the main assump-
+tion (see Sec. II) that Eq. (5) can be separated into a
+time-independent part and a part sinusoidal in Ω twhile
+neglecting higher-order harmonics. The Fourier analy-
+sis shows, however, that the anharmonic contribution
+toψ(t) is rather large for h= 2.5. E.g. for γ0= 0.3
+we find the content of the second harmonic to be about
+35%. Physically, it means that the fluxon train is not
+sufficiently dense for h= 2.5 and its time-dependence
+although periodic is not strictly sinusoidal.
+Contrarytothecase h= 2.5,forh= 3.5weobserveex-
+cellentagreementbetweennumericalandanalyticaldata.
+Now theI-Vcharacteristicis smooth and the Fiske steps
+disappeared completely as a combined result of surface
+losses and self-pumping effects. The Fourier analysis of
+ψ(t) showsnowthat the anharmonicitydiminishes rather
+quickly with increasing hand forh= 3.5 we observe only6
+/s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56
+/s120
+/s50
+/s32/s32/s120
+/s120/s120
+/s49
+FIG. 3: Normalized bias current distribution similar to tha t
+assumed in Ref. 17. L= 40,x1=−9,x2= 5.5.
+/s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56 /s50/s46/s48 /s50/s46/s50 /s50/s46/s52 /s50/s46/s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53
+/s32/s32/s48/s104 /s61/s50/s46/s53
+/s104 /s61/s51/s46/s53
+/s98
+FIG. 4: (Color online) Current-voltage characteristic cal cu-
+lated analytically (solid line) and numerically (open circ les)
+for two values of the external magnetic field: h= 2.5 and
+h= 3.5. Thecurrentprofileis shown inFig. 3andtheremain-
+ing junction parameters are: L= 40,α0= 0.033,β= 0.035.
+about 10% of the second harmonic.
+Another interesting feature which is visible for h= 3.5
+is an additional small step at Ω b= 1.9. Such a step ap-
+pears in experimental results17,26and can be attributed
+to the self-pumping effect described in Subsection III.E.
+As shown in Ref. 26, the position of the current step fol-
+lows from a simple relation Ω b= Ωg/3, where Ω gdenotes
+the gap voltage. Assuming typical junction parameters26
+we find a normalized dimensionless gap voltageΩ g= 5.7,
+hence Ω b= 1.9.
+It is interesting that the value of Ω bcan also be
+deduced directly form Eq. (30). Indeed, noting that
+γdc(−Ω) =−γdc(Ω) one can see the quadratic term in
+Eq. (30) to vanish, provided γdc(Ω) is a linear function.
+However, for a nonlinearity starting at Ω = Ω gwe find/s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50
+/s61/s48/s46/s51/s32
+/s32/s101/s102/s102/s40/s120 /s41
+/s120/s48/s61/s48/s46/s49
+FIG. 5: (Color online) Spatial distribution of αeff(x) calcu-
+lated analytically (solid line) and numerically (dotted li ne)
+forh= 3.5 and two values of the averaged current density
+γ0= 0.1 andγ0= 0.3. The remaining junction parameters
+are the same as in Fig. 4.
+γdc(3Ω)/negationslash= 3γdc(Ω), thus a nonzero contribution to γeff
+appears at 3Ω = Ω g, giving rise to a step at Ω b= Ωg/3.
+As shown in Subsection III.E, an effective loss factor
+αeff(x) is spatially modulated due to the self-pumping,
+and its average value becomes significantly larger than
+α0. Fig. 5 shows the spatial distribution of αeffplotted
+forγ0= 0.1 andγ0= 0.3. The junction parameters are
+assumed as before. i.e. L= 40,h= 3.5,α0= 0.033,
+β= 0.035. Now the dotted lines correspond to numeri-
+cal simulations and the solid lines denote self-consistent
+(analytical) solutions. One can see that the agreement is
+excellent, the self-consistent solution following closely all
+the details of the numerical simulations, taken here as a
+reference.
+It is clear that αeffgenerally grows with increasing
+value ofγ0. E.g. for γ0= 0.3 we obtain an averaged
+value ofαeff(x) about three times larger than the “un-
+pumped” value α0. Such an effect together with surface
+losses makes the I-Vcurve smooth, damping effectively
+the Fiske steps.
+V. CONCLUSIONS
+In this paper a semi-analytical approach has been sug-
+gested, making possible to solve a modified sG equa-
+tion (2) with both surface losses and self-pumping effects
+taken into account. The solution, as given by Eqs. (4),
+(10), and (18), consists of a rotating background and a
+unidirectional fluxon train accompanied by two plasma
+waves traveling in opposite directions with the velocity
+close to the critical value Ω /δ0≃ ±1. Having obtained
+an analytical solution to Eq. (2) one can determine some
+macroscopic, directly measurable quantities, such as the
+constant bias current and constant voltage, making pos-7
+sible to calculate the I-Vcharacteristic (11) of the junc-
+tion.
+For the uniform current density distribution ( γ(x) =
+const) it has been shown that the relevant expressionsfor
+θ(x),/hatwideψ(x), and consequently the I-Vdependence can be
+obtained in a closed fully analytical form (24). As fol-
+lows from Figs. 1 and 2, such an approximation is fairly
+accurate for moderate junction parameters. In general,
+however, practical FF oscillators are based on very long
+junctions with strongly nonuniform current profile. In
+such a case an analytical approximation (24) appears in-
+sufficient and one should look for a self-consistent solu-
+tion of Eqs. (10) and (18) which can be obtained by an
+iterative procedure.
+As shown in Figs. 1,2,4 and 5, the self-consistent so-
+lutions and numerical simulations are generally in very
+good agreement. The only exception, where one can ob-
+serve qualitative rather than quantitative agreement is
+theI-VcurveshowninFig.4for h= 2.5. Asexplainedin
+the previousSection, the fluxon trainis not dense enough
+forh= 2.5 and consequently the time-dependent part ofthe solution is periodic but not strictly sinusoidal, vio-
+lating the main assumption of the present approach.
+Finally, it should be mentioned that the method pre-
+sented here can be easily extended to include general,
+more realistic boundary conditions discussed in Refs.
+16,17,27. In the present paper, however, we intention-
+ally assumed standard open-circuit boundary conditions
+(3), where the plasma waves could interfere after com-
+plete reflection at the boundaries, giving rise to clearly
+visibleFiskesteps. Thisway,wewereabletoseparatethe
+influence of surface losses and self-pumping from that of
+boundary conditions, which (e.g. for a resistive load) can
+also affect the I-Vcharacteristic and make the physical
+mechanism discussed here less clear.
+Acknowledgments
+Financial support from the Institute of Physics, Polish
+Academy od Sciences is gratefully acknowledged.
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+435, 112 (2006).
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+(1985)
+20P. M. Morse and H. Feshbach, Methods of Theoretical
+Physics(McGraw-Hill, New York, 1953).
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+(1999).
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\ No newline at end of file
diff --git a/1001.0672.txt b/1001.0672.txt
new file mode 100644
index 0000000000000000000000000000000000000000..694ac4cb506d12368d396d77b0c52bd1a73a21c5
--- /dev/null
+++ b/1001.0672.txt
@@ -0,0 +1,219 @@
+Ferroelectricity 
+of Néel-type magnetic domain walls 
+ 
+A.P. Pyatakov*, D.A. Sechin, A.V. Nikolaev, E.P. Nikolaeva, A.S. Logginov 
+Physics Department, M.V. Lomonosov Moscow State University, Leninskie gory, 
+Moscow, 119991, Russia 
+ 
+*) E-mail: pyatakov@physics.msu.ru 
+ 
+The chirality-dependent magnetoelectric properties of Néel-type domain walls in iron garnet films is 
+observed. The electrically driven magnetic domain wall motion changes the direction to the opposite with the 
+reversal of electric polarity of the probe and with the chirality switching of the domain wall from clockwise to 
+counterclockwise. This proves that the origin of the electric field induced micromagnetic structure 
+transformation is inhomogeneous magnetoelectric interaction.  
+ 
+1. Introduction 
+ 
+In recent years there has been the remarkable progress in understanding of the coupling 
+mechanisms between magnetic and electric subsystems in multiferroics [1-11]. A lot of excitement was caused by the discovery of so called spiral multiferroics  in which ferroelectricity is induced by spatially 
+modulated spin structure since magnetoelectric coupling expected to be particularly strong in these media [12-17]. The chirality of the spin spiral determines th e direction of polarization, as was proved 
+experimentally by reversal the chirality with electric field in orthorhombic manganites RMnO
+3 (R=Dy, Tb) 
+[18; 19] and MnWO 4 [20]. The specific type of ferroelectric domains generated by chiral structures was 
+observed in [21,22]. This magnetically induced ferroelectricity is caused by the special type of 
+magnetoelectric interaction, the inhomogeneous  one, that is described by P iMj∇kMn coupling term, where P 
+and M, are electric and magnetic order parameters, respectively [23-26].   
+Micromagnetic structures like domain walls and magnetic vortexes can be also considered as a 
+source of ferroelectricity due to the modulation of magnetic order parameter that occurs in them. Noteworthy that the term “inhomogeneous magnetoelectric interaction” was originally used by V.G. Bariakhtar [25] in relation to the domain wall in magnet. In subsequent works [27] the magnetoelectric 
+properties of antiferromagnetic domain were discu ssed. This topic recently was revitalized by I.E. 
+Dzyaloshinskii [28] that proposed the nucleation of domain walls with electric field and its motion in the 
+gradient of electric field. It should be noted that ferroelectricity developed from micromagnetic structure is 
+universal phenomena and can appear in every magnetic insulator even in centrocymmetric one [28]. Quite surprisingly the experimental proof for electric properties of magnetic domain walls is still lacking. The 
+indirect evidence can be seen in the enhancement of elect romagnetooptical effect (i.e. electric field induced 
+Faraday rotation [29]) in the vicinity of domain wall observed in YIG films [30,31].  
+Recently the motion of domain walls in the gradient electric field provided by a tip electrode was 
+observed in rare earth iron garnet films [32-34]. Although the basic features of the effect agree with the 
+properties of inhomogeneous magnetoelectric interaction P
+iMj∇kMn (i.e. the oddness with respect to the 
+electric polarity and evenness with respect to the magnetic polarity of the domain), another mechanism of 
+the effect is possible such as the electrically induced local variation of magnetic anisotropy [30,31] that forces the domain walls to be attracted or repelled by the tip. Thus the origin of the effect have remained 
+unclear since the key feature of inhomogeneous magnetoelectric interaction, namely, the dependence on the 
+domain wall chirality has not been shown.  
+In this paper the magnetoelectric properties of Néel-type domain walls with controlled chirality are 
+studied in iron garnet film. The electrically induced domain wall displacement that depends on the chirality of the walls shows unambiguously that the origin of the effect is inhomogeneous magnetoelectric 
+interaction.   
+2. Theory 
+The inhomogeneous magnetoelectric effect corresponds to the following contribution to the free 
+energy of the crystal:  
+l k j ijkl ME M MP F ∇⋅⋅⋅=iγ ,    (1) 
+where  M=M(r) is magnetization distribution, P is electric polarization, ∇ is vector differential operator, 
+ijklγ is the tensor of inhomogeneous magnetoelectric interaction that is determined by the symmetry of the 
+crystal.  
+The inhomogeneous magnetoelectric contributions (1) for the bulk crystal of iron garnets with cubic 
+symmetry takes the following high symmetry form [26,13]:  
+()() ( )M M M MP ∇⋅−⋅∇⋅⋅⋅=γMEF . (2) 
+Taking into account that operator ∇can be represented in terms of magnetic spiral wave vector k as x∂∂k , 
+where x is the coordinate along the axis parallel to k the following form for (2) can be obtained:  
+[]()ΩkP MMkP ×⋅=⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎥⎦⎤
+⎢⎣⎡
+⎥⎦⎤
+⎢⎣⎡×∂∂×⋅=2MxFME γ γ    (3) 
+where Ω is the spin rotation axis. Thus in accordance with the rule formulated in [13,6], the electric 
+polarization of the spatially modulated structure can be found as a vector product of k and Ω: 
+[] kΩEP × =∂∂−=2MF
+eMEγχ    (4) 
+where χe is electric susceptibility.  
+ 
+a) 
+  b)
+  
+c)  
+   d)
+  
+ 
+Fig. 1. Spatially modulated magnetic structures in media a) a spin cycloid structure b) a spin helicoid structure  
+c) a Néel domain wall d) a Bloch domain wall. 
+ 
+It follows from (4) that the magnetically induced polarization is nonvanishing for cycloidal 
+modulation (fig.1 a), i.e. the spiral plane is parallel to the modulation wave vector and should be zero for 
+helicoidal modulation in which k||Ω (fig. 1 b). One can see that domain wall of Néel type can be considered 
+as a part of the spin cycloid (fig.1 c) and the Bloch -type domain wall as a part of helicoid (fig. 1d). It 
+follows also from (4) that upon changing the chirality of the cycloid or Néel domain wall ( Ω Ö -Ω) the 
+electric polarization of the spatially modulated structure reverses ( P Ö - P).  In the bulk of the magnetic media the domain walls of Bloch type correspond to the state with the 
+lowest energy since magnetic charges at the boundary are zero (div M=0).  However in the magnetic field 
+perpendicular to the domain wall plane the Bloch domain wall transforms into Néel type one [35]. In the following experimental part we report the magnetoelectric properties of domain wall in the Néel-type state with definite chirality induced by magnetic field. The e ffect of electric polarization reversal upon chirality 
+switching is demonstrated. 
+ 
+3. Experiment  
+Iron garnet films are well known magnetooptical materials, thus one can visualize domains and 
+domain walls, for example with Faraday effect [36, 37]. The experimental scheme is shown in Figure 2. To 
+induce the Néel-type state of the domain wall the in-plane magnetic field produced by magnetic coils was 
+applied perpendicular to the domain wall. To probe th e electrical properties of the domain wall the tip 
+electrode was used. It was made of copper wire that wa s sharpened by etching in ferric chloride to curvature 
+radius of ~2 μm at one end. Such probe allowed us to obtain an electric field with strength of up to 7 
+MV/cm near the point of contact by supplying voltage up to 1500 V. Strong electrostatic field did not cause 
+dielectric breakdown because it decreased rapidly with the distance from the probe and near the grounding 
+electrode (i.e. metal foil attached to substrate) the strength of the field did not exceed 100 V/cm. 
+In our experiment we used (BiLu) 3(FeGa) 5(O) 12 films grown by liquid-phase epitaxy on (210) 
+Gd 3Ga5O12 substrate (for details see [38]). The parameters  of the samples are listed in Table 1. Samples 
+with (210) substrate orientation were chosen because the anisotropy in them is strong enough to retain the 
+orientation of the domain wall along [ 120] axis even in the presence of magnetic field normal to the 
+domain wall plane, while in highly symmetrical (111) films the domain walls align along the field thus 
+making the induction of Néel-state impossible. 
+ 
+Table 1: Parameters of the samples under study. Symbol h stands for sample thickness, MS is the saturation 
+magnetization, and p is a period of domain structure [38]. 
+а)
+       b)
+           
+Fig. 2. a)  Schematic representation of the geometry of the experiment and the configurations of the magnetic field, 
+electric field and magnetization. The electric field is formed in the dielectric medium of the sample between the tip (1) 
+and the copper foil (2), which plays the role of the grounding electrode, (3) is iron garnet film, (4) is the GGG substrate, (5) is optical system, (6) are magnetic coils.   
+b) The top view magnetooptical image of the domain structure in iron garnet film: dark and bright gray stripes are 
+domain with upward and downward direction of magnetization, respectively. Vertical black lines are domain wall images, dark area at the bottom is the image of electrode tip.  4. Results  
+In experiment we registered magnetic structure in the initial state (fig. 2 b) and after applying 
+external electric and magnetic fields (fig. 3).  
+The application of in-plane magnetic field of several tens Oe without electric field did not lead to 
+significant changes of the micromagnetic structure while application electric voltage of 500÷1500V between the tip and the ground electrode caused the evident displacement of the domain wall. Its direction depends on both the polarity of the field and the magnetic field direction (fig. 3). As one can see on fig. 3 the change in magnetic field direction causes reversal of the sign of wall displacement. It is important to note simultaneous movement of the wall that is nearest to the probe and of the next wall to the right: while one is attracted the other is repulsed and vice versa. 
+It worth mentioning that the effect of the domain wall displacement was observed in the samples 
+listed in Table 1 even in the absence of magnetic field (for details, see [33]), however its value was much smaller and the positive voltage always caused the attraction of the domain wall nearest to the tip, while negative voltage always caused the repulsion irrespective of the position of the tip.  
+ 
+ 
+  
+  a )       b )   
+ 
+  
+c)      d)  
+Fig. 3. The transformation of the micromagnetic structure in the magnetic and electric fields applied simultaneously:  
+a) the positive potential of the tip, and LTR direction of magnetic field b) negative potential at the tip, LTR direction 
+of magnetic field; c) the positive potential of the tip, and RTL direction of magnetic field d) the negative potential of the tip, and RTL magnetic field. The sample 1 from the table is used.  
+ 
+Summarizing the experimental results we can point out th e following characteristic features of the electric 
+field driven domain wall motion:  
+ 
+(i) Switching the electric polarity changes the sign of the effect (ii) Neighboring walls shows opposite direction of the displacement for the same polarity of electric field (iii) Switching the magnetic polarity of the in -plane field changes the sign of the effect  
+ H H
+H H5. Discussion 
+ 
+The electric field driven domain wall motion can be explained by ferroelectricity of the walls in 
+accordance with the predictions [25,14,28]. However one can not rule out another scenario according to which the electric field induces variation of magnetic anisotropy [30,31]. In the immediate vicinity of the contact point the local anisotropy can favor (unfavor) the in-plane direction of the magnetization, resulting, respectively, in the attraction (repulsion) of the domain wall. Thus the discriminative feature of spin spiral induced ferroelectricity is the polarization reversal upon the switching chirality of the spiral. In the following we will show that properties of the effect (i-iii) are explained in terms of inhomogenous magnetoelectric interaction (2) and magnetically induced electric polarization (4). 
+ 
+  
+a )       b )  
+Fig. 4 The schematic representation of micromagnetic structures corresponding to the experimental images of 
+Fig. 3: (a) corresponds to the a, b of Fig.3  (b) corresponds to the c, d of Fig.3. For illustrative purposes the size of 
+domain wall is exaggerated. 
+ 
+Let us consider the stripe domain structure in magnetic film (fig.4). If we choose the modulation 
+vector k directed left to right than the left boundary of the central domain is characterized by the clockwise 
+direction of magnetization rotation while neighboring domain boundary has the opposite chirality (fig 4 a). 
+The corresponding electrical charges due to inhomoge neous magnetoelectric interaction (formula 4) are 
+shown at the domain wall.  
+When the in-plane magnetic field is reversed the chirality and hence the electric polarity of the 
+charges change the sign (fig 4 b) in accordance with the feature (iii).  
+It is easily seen from the picture (4 b) t hat neighboring walls have opposite chiralities: 
+magnetization rotates clockwise and counterclockwise.  The chirality determines the direction of electric 
+polarization in the wall so the neighboring walls should posses opposite electric polarities in accordance 
+with feature (ii). The later makes us rule out the effects related to the local anisotropy changes as in the single crystal the anisotropy is the same for all points of the sample and should not depend on the domain wall chirality.  
+The results agree with the equation (4) and the results of the theoretical work [27] in which the 
+antiferromagnetic domain wall in electric and magnet ic field applied simultaneously are considered.  
+Finally some  comments should be made about the effect of domain wall displacement that was 
+observed in the same samples even in the absence of magnetic field [32-34]. As this magnetic field induces Néel-type state of domain wall the question arises how the electric field alone can induce attraction or 
+repulsion of the domain wall. We relate it with some “spontaneous” Néel component of domain wall that arises due to the fact that direction of magnetization in the domain is not exactly parallel to the domain wall plane. It has the projection on the direction of modulation vector thus making nonzero terms in (2) 
+() () 0 ;0 ≠∇⋅≠⋅∇⋅ M M M M  (for details see [33]). 
+Authors are grateful to A.K. Zvezdin for the interest to the work and valuable discussions.   
+ 
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\ No newline at end of file
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+arXiv:1001.0673v2  [nucl-th]  1 Feb 2010Breaking and restoring symmetries within the
+nuclear energy density functional method
+T. Duguet1,2and J. Sadoudi1
+1CEA, Centre de Saclay, IRFU/Service de Physique Nucl´ eaire , F-91191
+Gif-sur-Yvette, France
+2National Superconducting Cyclotron Laboratory and Depart ment of Physics
+and Astronomy, Michigan State University, East Lansing, MI 48824, USA
+E-mail:thomas.duguet@cea.fr ,jeremy.sadoudi@cea.fr
+Abstract. We review the notion of symmetry breaking and restoration wi thin
+the frame of nuclear energy density functional methods. We f ocus on key
+differences between wave-function- and energy-functional -based methods. In
+particular, we point to difficulties to formulate the restora tion of symmetries
+within the energy functional framework. The problems tackl ed recently in
+connection with particle-number restoration serve as a bas eline to the present
+discussion. Reaching out to angular-momentum restoration , we identify an exact
+mathematical property of the energy density ELM(/vectorR) that could be used to
+constrain energy density functional kernels. Consequentl y, we suggest possible
+routes towards a better formulation of symmetry restoratio ns within energy
+density functional methods.
+1. Introduction
+1.1. Spontaneous symmetry breaking
+Symmetries are essential features of classical and quantal syst ems. For the latter in
+particular, symmetriescharacterizethe energeticsofthe syste m and providetransition
+matrix elements of operators with specific selection rules. In nuclea r systems for
+example, electromagnetic and electro-weak decays display patter ns associated with
+such selection rules.
+On the other hand, certain emergent phenomena relate to the spo ntaneous
+breaking of those symmetries [1]. In the thermodynamic limit, i.e. when the number
+of particles Nand the volume Vof the system go to infinity such that N/Vremains
+constant, a state with lower symmetry than the Hamiltonian can be r igorouslyused as
+an effectiveground-stateofthe system. Sucha stateisa linearsu perposition ofnearly-
+degenerate eigenstates, i.e. it is a wave-packet. In finite systems however, quantum
+fluctuations make such a wave-packet to relax into the symmetry- conserving ground-
+stateandcannotbeignored; i.e. theconceptofspontaneoussym metrybreakingisonly
+an intermediate description of the system that arises within certain approximations
+and symmetries must eventually be restored. Still, it makes physical sense to go
+through such an intermediate description as pseudo spontaneous ly-brokensymmetries
+(i) relate to specific features of the inter-particle interactions, ( ii) characterize internal
+correlations and (ii) leave clear fingerprints in the observed excitat ion spectrum of the
+system.Symmetries within EDF methods 2
+Figure 1. (Color online) Schematic representation of the different st rategies
+followed by many-body methods regarding the treatment of sy mmetries, e.g. in
+Density Functional Theory and nuclear Energy Density Funct ional approaches.
+In atomic nuclei, several symmetries, if allowed, tend to break spon taneously in
+approximate descriptions based on the mean-field concept. The mo st important ones
+relate to the invariance of the nuclear Hamiltonian Hunder spatial translations and
+rotationsas well asto the gaugeinvarianceassociatedwith particle -numbersymmetry.
+As described in Tab. 1, the spontaneous breaking of these three s ymmetries relates
+to the short-rangeand dominant quadrupole-quadrupole terms o f the nucleon-nucleon
+interaction as well as to its strong attraction in the L= 0 partial-wave of relative
+motion. In particular, the strong attraction in the L= 0 partial-wave in particular
+generates a S-wave di-neutron (di-proton) virtual state at almost zero scatt ering
+energy that is the precursor of neutron (proton) Cooper pairs a nd superfluidity in
+the nuclear medium. Even though such symmetries must be eventua lly enforced,
+their underlying breaking impacts the low-lying spectroscopy of finit e nuclei through
+the presence of surface vibrational excitations, rotational ban ds and a gap in the
+individual excitations of even-even nuclei, respectively [2]. Parity an d time-reversal
+are other good symmetries of Hthat can be spontaneously broken, while isospin
+symmetry is only approximate in the first place.
+1.2. Wave-function-based methods
+As schematically shown in Fig. 1, quantum many-body methods separ ate into
+two categories as for the way symmetries are dealt with, i.e. (i) meth ods enforcing
+symmetries throughout and (ii) those that explicitly single out the int ermediate
+breaking of symmetries. Although hybrid approaches that allow the breaking of some
+symmetries while enforcing the others can be set up, the present w ork focuses on
+those that strongly rely on the concept of symmetry breaking, i.e. methods whose
+philosophy, apart for computational constraints, is to allow all sym metries to break
+Table 1. Links between the spontaneous breaking of translational, r otational
+and particle-number symmetries and features of the nuclear force, correlations in
+the internal motion of nucleons and patterns in the excitati on spectrum.
+Invariance VNNInternal correlations Excitation patterns
+Spatial translation Short range Spatial localization Surface vibrations
+Gauge rotation S-wave attraction Pairing Energy gap
+Spatial rotation Quad.-quad. component Angular localization Rotational bandsSymmetries within EDF methods 3
+Figure 2. (Color online) Schematic view of the energy as a function of t he phase
+and magnitude of the order parameter qof a spontaneously broken symmetry.
+spontaneouslyapriori. The breakingofeachsymmetryismonitored by the magnitude
+and the phase of an order parameter q, such that the (approximate) energy is
+independentofthephaseasschematicallyshowninFig.2. Thiscorre spondstothefact
+thataspontaneoussymmetrybreakingisaccompaniedbythepres enceofazero-energy
+Goldstonemode. Ofcourse, thatacertainsymmetrydoesbreaks pontaneouslyusually
+depends on the number of elementary constituents of the system under consideration.
+For example, while translational symmetry (strongly) breaks in all n uclei, particle-
+number symmetry tend to (weakly) break in all but doubly-magic nuc lei whereas
+rotational symmetry remains unbroken if either the neutron numb er or the proton
+number is ”magic” ‡. Figure 3 displays the correlation energy incorporated in240Pu
+and120Sn ground-states energy through the spontaneous breaking of rotational and
+particle-number symmetries, respectively. Such symmetry break ings may account for
+up to 20 MeV correlation energy out of about 2 GeV binding energy, i.e . for about
+2%, which is much largerthan the targeted accuracyon nuclear mas ses. Incorporating
+such correlation energies through symmetry-conservingapproa ches, e.g. configuration
+interaction methods, would necessitate tremendous computation al efforts in such
+heavy open-shell nuclei.
+As already stated, methods authorizing the breaking of symmetrie s at a certain
+level of approximation must eventually restore them in a second sta ge. In wave-
+functions based methods, the symmetry breakingstep, e.g. the s ymmetry unrestricted
+Hartree-Fock-Bogoliubovapproximation, relies on minimizing the ave ragevalue of the
+Hamiltonian for a trial wave-function that does not carry good qua ntum numbers, i.e.
+which mixes irreducible representations of the symmetry group of in terest. Restoring
+symmetries amounts to using an enriched trial wave-function that does carry good
+quantum numbers. In terms of the schematic ”mexican-hat” pictu re of Fig. 2, this
+‡The fact that the neutron or proton number is magic is not know n a priori but is based on a
+posteriori observations and experimental facts. In partic ular, the fact that traditional magic numbers,
+i.e.N,Z= 2,8,20,28,50,82,126, remain as one goes to very isospin-asymmetric nuclei is the subject
+of intense on-going experimental and theoretical investig ations [3].Symmetries within EDF methods 4
+Figure 3. (Color online) Energy gain from spontaneous symmetry break ing and
+symmetry restoration as a function of the magnitude of the or der parameter q.
+Left: breaking and restoration of rotational symmetry in th e ground state of
+240Pu as a function of the axial quadrupole moment of the single- nucleon density
+distribution (adapted from Ref. [4]). Right: breaking and r estoration of neutron-
+number symmetry in the ground state of120Sn as a function of the norm of the
+anomalous pair density (adapted from Ref. [5]).
+corresponds to incorporating zero-energyfluctuations associa ted with the phase of the
+order parameter §. One typical approach is to project out from the symmetry-brea king
+trial state the component that belongs to the intended irreducible representation [2].
+Figure 3 shows that doing so for rotational and particle-number sy mmetries adds a
+few MeV correlation energy to the ground-state binding energy of heavy nuclei. This
+is still significant compared to the few hundreds keV targeted accu racy on nuclear
+masses. As shown in Fig. 1, a variant consists of performing the pro jection only
+approximately such that the calculation boils down to the minimization o f acorrected
+energy expressed in terms the original symmetry-breaking state . Typical examples
+are Lipkin [6, 7] of Kamlah approximate projection methods [8, 9]. While it is likely
+that the strongly broken translational symmetry can be safely tr eated through such
+approximate projection methods /ba∇dbl, it is still unclear whether the same is true for
+weakly broken symmetries such as particle number symmetry or rot ational symmetry
+in transitional nuclei.
+1.3. The nuclear energy density functional method
+Wave-function-based projection methods and their variants are well formulated
+quantum mechanically [2]. The goal of the present paper is to discuss their Energy
+Density Functional (EDF) counterparts [11] which have been empirically adapted from
+the former to deal quantitatively with properties of nuclei. The nuc lear EDF method
+doesrelyheavilyonthe conceptofspontaneoussymmetrybreakin gand(approximate)
+restoration. In that sense, it is intrinsically a two-step approach. The so-called single-
+reference EDF (SR-EDF) method has been originally adapted from t he symmetry-
+unrestricted Hartree-Fock-Bogoliubov method by using a density-dependent effective
+Hamilton”operator”[12]. Lateron,theapproximateenergywasfo rmulateddirectlyas
+§Although it is not the focus of the present work, the restorat ion of symmetries must be
+complemented with the inclusion of collective quantum corr elations associated with the fluctuations of
+themagnitude of the order parameter. This can be done through the so-calle d Generator Coordinate
+Method (or its energy density functional counterpart) that is formally identical to projection methods
+discussed here.
+/bardblSuch a statement is to be taken with a grain of salt for rather l ight nuclei [10].Symmetries within EDF methods 5
+a possibly richer functional of one-body density matrices compute d from a symmetry-
+breaking product-state of reference. The second step, carrie d out through the multi-
+reference(MR)extensionoftheSR-EDFhasbeenadaptedfromt heprojectedHartree-
+Fock-Bogoliubov method. Such a step necessitates a prescription to extend the
+SR energy functional ¶associated to a single auxiliary state of reference, i.e. a
+diagonal energy kernel, to the non-diagonal energy kernel asso ciated with a pair
+of reference states (see Sec. 4.2). Constraints based on physic al requirements have
+been worked out that limit the number of possible prescriptions to do so [13]. Still,
+pathologies [14, 15, 16] of MR-EDF calculations have been recently id entified and
+corresponding cures [17, 18, 19] have been proposed. Besides th e actual successes
+of nuclear EDF calculations [11], the work of Refs. [17, 18, 19, 20] de monstrates that
+nuclearSR-andMR-EDFmethods mustbefurtherconstrainedtob ecomesatisfactory
+many-body approaches to finite Fermi systems. The first goal of the present work will
+be to reformulate, focusing on group-theory considerations, co ncerns about MR-EDF
+calculations that have been dealt with in Refs. [17, 18, 19, 20]. Our se cond objective
+will be to provide a new mathematical property that could be used in t he case of
+angular-momentum restoration to constrain the form of basic EDF kernels at play.
+Given the efforts needed to better formulate EDF methods, one ma y question
+the necessity to stick to such approaches rather than to use a wa ve-function-based
+approach that strictly computes the energy from a (state-/den sity-independent)
+Hamiltonian, e.g. through many-body perturbation theory. Althou gh the breaking
+(up toafewtensofMeV) andthe restoration(up toafewMeV)ofs ymmetriesbringin
+both types of methods essential correlations that vary rapidly wit h nucleon numbers,
+incorporating the bulk of correlations+(hundreds of MeV) requires involved wave-
+function-based calculations [22] that are still impractical for heav y open-shell nuclei.
+The power of the EDF approach is to parameterize bulk correlations under the form
+of a functional of the one-body density (matrices) such that sys tematic calculations
+of heavy nuclei are tractable. The success of the overall approa ch, based on the
+resummation of bulk correlations into the EDF kernel and the furth er breaking and
+restoration of symmetries, relies on the empirical decoupling of the different categories
+of correlations at play, i.e. on the different scales that characteriz e them (see Tab. 2),
+and on the fact that quickly varying correlations with the filling of nuc lear shells are
+explicitly accounted for through symmetry breaking and restorat ion∗.
+Table 2. Schematic classification of correlation energies as they na turally appear
+in nuclear EDF methods. The quantity Avaldenotes the number of valence
+nucleons while Gdegcharacterizes the degeneracy of the valence major shell.
+Correlation energy Treatment Scale Vary with
+Bulk Summed into EDF kernel ∼8AMeVA
+Static collective Finite order parameter q/lessorsimilar25 MeV Aval,Gdeg
+Dynamical collective Fluctuations of q /lessorsimilar5 MeV Aval,Gdeg
+¶The density-dependence of the effective Hamilton operator i n more standard formulations.
++We take as a loose definition of bulk correlations the correla tion energy computed beyond a genuine
+Hartree-Fock approximation in terms of the vacuum (low-mom entum [21]) nuclear Hamiltonian for
+nuclei that do not break any symmetry besides translational invariance, i.e. doubly-magic nuclei.
+∗Here we have in mind to add the fluctuations of the magnitude of the order parameter.Symmetries within EDF methods 6
+1.4. Density functional theory
+As theaimofthe presentpaperis toraisequestionsabout thetrea tmentofsymmetries
+within the nuclear EDF method, let us make a few relevant statement s about Density
+Functional Theory (DFT) [23, 24, 25] that provides a formal fram ework to obtain the
+ground-state energy and one-body density of electronic many-b ody systems. It has
+become customary in nuclear physics to assimilate the SR-EDF metho d, eventually
+including corrections a laLipkin or Kamlah, with DFT, i.e. to state that the
+Hohenberg-Kohn theorem underlays nuclear SR-EDF calculations [2 6, 27, 28, 29, 30].
+This is a misconception asdistinct strategiesactually support both m ethods. Whereas
+the SR-EDF method minimizes the energy with respect to a symmetry -breaking trial
+density(ies), DFT reliesonan energyfunctional whoseminimum must be reachedfora
+one-body density that possesses allsymmetries of the actual ground-state density, i.e.
+that displays fingerprints of the symmetry quantum numbers of th e underlying exact
+ground-state [31]. As a matter of fact, generating a symmetry-b reaking solution is
+problematic in DFT, as it lies outside the frame of the Hohenberg-Koh n theorem, and
+isusuallyreferredtoasthe symmetry dilemma . Toby-passthesymmetrydilemmaand
+graspkinematicalcorrelationsassociatedwithgoodsymmetries,s everalreformulations
+of DFT have been proposed over the years, e.g. see Refs. [32, 33], some of which are
+actually close in spirit to the nuclear MR-EDF method [32].
+Recent efforts within the nuclear community have been devoted to f ormulating
+a Hohenberg-Kohn-like theorem in terms of the internal density, i.e . the matter
+distribution relative to the center of mass of the self-bound syste m [34, 35]. Together
+with an appropriate Kohn-Sham scheme [35], it allows one to reinterpr et the SR-
+EDF method as a functional of the internal density rather than as a functional
+of a translational-symmetry-breaking density. This constitutes a n interesting route
+whose ultimate consequence would be to remove entirely the notion o f breaking and
+restoration of symmetries from the EDF approach and make the SR formulation a
+complete many-body method, at least in principle. To reach such a po int though, the
+work of Refs. [34, 35] must be extended, at least, to rotational a nd particle-number
+symmetries, knowing that translational symmetry was somewhat t he easy case to deal
+with given the explicit decoupling of internal and center of mass motio ns.
+1.5. Outline
+The paper is organized as follows. Section 2 provides the minimal set o f group
+theory considerations that are needed to go through the remainin g of the paper.
+Section 3 discusses symmetry breaking and restoration within a wav e-function based
+method while Sec. 4 provides the analog within the EDF context. In Se c. 4,
+we focus on difficulties encountered to formulate symmetry restor ations within the
+energy functional approach. Finally, Sec. 5 briefly discusses a mat hematical property
+associated with angular-momentum conservation that could used t o formulate new
+constraints on off-diagonal energy functional kernels at play in MR -EDF calculations.
+2. Symmetry group
+Let us consider an arbitrary continuous compact group G ≡ {R(g)}parameterized by
+rreal parameters g≡ {gi;i= 1,...,r}and whose transformations leave Hinvariant.Symmetries within EDF methods 7
+We denote by vGthe volume of the group
+vG≡/integraldisplay
+Gdm(g), (1)
+wherem(g) is the invariant measure on G. Having in mind to deal more specifically
+with particle number and rotational symmetries, we further consid erGto be a Lie
+group, although this is not mandatory. We thus introduce the set o f infinitesimal
+generators C={Ci;i= 1,...,r}that make up the Lie algebra and in terms of which
+anytransformationofthegroupcanbe expressed. The Casimirof the groupbuilt from
+the infinitesimal generators and a non-degenerate invariant bilinea r form is denoted
+by Λ. We also denote by R(g) (C) a unitary representation of R(g) (C) on the Fock
+space of quantum mechanics and by Sλ
+ab(g)≡ /an}b∇acketle{tΘλa|R(g)|Θλb/an}b∇acket∇i}htthe matrix elements of
+the unitary irreducible representation labeled by λ. Noticing that Sλ
+ab(0) =δabfor all
+λ, the action of two successive transformations and the unitarity o f the representation
+can be both read off the following identity
+/summationdisplay
+cSλ∗
+ca(g′)Sλ
+cb(g) =/summationdisplay
+cSλ
+ac(−g′)Sλ
+cb(g) =Sλ
+ab(g−g′), (2)
+where−gandg−g′symbolically denote the parametersoftransformations R−1(g) and
+R−1(g′)R(g), respectively. States |Θλa/an}b∇acket∇i}htspan the irreducible representation λwhose
+degree is dλ. They are eigenstates of the Casimir Λ and of a chosen generator Ci
+Λ|Θλa/an}b∇acket∇i}ht=l(λ)|Θλa/an}b∇acket∇i}ht, (3)
+Ci|Θλa/an}b∇acket∇i}ht=g(a)|Θλa/an}b∇acket∇i}ht, (4)
+where eigenvalues l(λ) andg(a) are functions of labels λanda, respectively, with a
+running over dλvalues. A so-called irreducible tensor operator Tλ
+aand a state |Θλa/an}b∇acket∇i}ht
+transform according to
+R(g)Tλ
+aR(g)−1=/summationdisplay
+bTλ
+bSλ
+ba(g), (5)
+R(g)|Θλa/an}b∇acket∇i}ht=/summationdisplay
+b|Θλb/an}b∇acket∇i}htSλ
+ba(g). (6)
+Thediscussionbelowisconductedfortheenergy,i.e. forascalarop eratorHbelonging
+to the trivial irreducible representation λ= 0 characterizedby S0
+ba(g) =δab. However,
+such a discussion can be extended to any irreducible tensor operat or [36].
+For nuclear structure, two groups are of particular importance a s discussed in
+the introduction, i.e. SO(3) for rotations in the three-dimensional space and U(1) for
+rotations in the gauge space associated with particle number. The g roup of spatial
+translations is also essential but corresponds to a symmetry that is strongly broken
+in all nuclei and that does not need to be exactly restored in practic e. Consequently,
+Tab. 3gatheruseful elements that characterize U(1) andSO(3) suchthat the formulae
+given in the paper for a generic compact Lie group can be easily adapt ed to either of
+them.
+3. Wave-function-based method
+The present section describes what we denote as a wave-function -based method where
+energy kernels are explicitly and strictly computed as matrix elements of a Hamilton
+operator that does notdepend on the wave-function it is used with, e.g. His not an
+effective operator depending on the density of the system.Symmetries within EDF methods 8
+Table 3. Characteristics of SO(3) and U(1) relevant to the present study. The
+gauge angle of U(1) isϕ∈[0,2π] whereas Euler angles parameterizing SO(3)
+are Ω≡(α,β,γ)∈[0,4π]×[0,π]×[0,2π]. The one-dimensional irreducible
+representations of U(1) are labeled by m∈Zwhereas the (2 J+1)-dimensional
+ones of SO(3) are labeled by 2 J∈Nand are given by the so-called Wigner
+functions DJ
+MM′(Ω), where (2 M,2M′)∈Z2with−2J≤2M,2M′≤+2J. For
+U(1), one has l(N) =N2, whereas for SO(3), with the choice Ci≡Jz, one has
+l(J) =/planckover2pi12J(J+1) and g(M) =M/planckover2pi1.
+G g dm (g)vG{C}ΛCi R(g) Sλ
+ab(g)dλ
+U(1) ϕ dϕ 2π N N2- eiNϕeimϕ1
+SO(3)α,β,γ sinβdαdβdγ 16π2/vectorJ J2Jze−iαˆJze−iβˆJye−iγˆJzDJ
+MM′(Ω) 2J+1
+3.1. Symmetry breaking
+In the present discussion, states |Θλa/an}b∇acket∇i}htdefined above can be eigenstates of H, which
+correspondsto handlingexactsolutionsofthe many-bodyproblem . In generalthough,
+we will only consider states that are approximations to those exact eigenstates of H.
+According to our introductory discussion, a normalized, symmetry -breaking state |Θ/an}b∇acket∇i}ht
+takes the form
+|Θ/an}b∇acket∇i}ht=/summationdisplay
+λacλa|Θλa/an}b∇acket∇i}ht, (7)
+where/summationtext
+λa|cλa|2= 1. Using either Eq. 5 or Eqs. 6 and 7, one can easily prove that
+the average energy
+E≡/an}b∇acketle{tΘ|H|Θ/an}b∇acket∇i}ht
+/an}b∇acketle{tΘ|Θ/an}b∇acket∇i}ht, (8)
+is a scalar under all transformations of G. However, such an energy cannot be labeled
+by good quantum numbers ( λ,a), which is the fingerprint of the symmetry-breaking
+character of the many-body state |Θ/an}b∇acket∇i}ht.
+3.2. Symmetry restoration
+One extends the definition of the symmetry breaking energy E(Eq. 8) to the so-called
+energykernelE[g′,g] through
+E[g′,g]≡/an}b∇acketle{tΘ|R−1(g′)HR(g)|Θ/an}b∇acket∇i}ht
+/an}b∇acketle{tΘ|R−1(g′)R(g)|Θ/an}b∇acket∇i}ht, (9)
+where the norm overlap kernel is N[g′,g]≡ /an}b∇acketle{tΘ|R−1(g′)R(g)|Θ/an}b∇acket∇i}ht. Expanding |Θ/an}b∇acket∇i}ht
+according to Eq. 7 and using Eq. 2, one obtains
+E[g′,g]N[g′,g] =/summationdisplay
+λabc∗
+λacλbEλSλ
+ab(g−g′), (10)
+N[g′,g] =/summationdisplay
+λabc∗
+λacλbSλ
+ab(g−g′), (11)
+where the symmetry-restoredenergies /an}b∇acketle{tΘλa|H|Θλ′a′/an}b∇acket∇i}ht ≡Eλδλλ′δaa′provide the eigen-
+spectrum of Hif one is working with its exact eigenstates. Expressions 9 and 11
+correspond to the double expansion over the volume of the group
+F[g′,g]≡/summationdisplay
+λλ′/summationdisplay
+aba′b′Fλλ′
+aba′b′Sλ
+ab(g′)Sλ
+a′b′(g), (12)Symmetries within EDF methods 9
+applied to functions F[g′,g] =f[g−g′] that in fact only depend on the difference of
+the two arguments. In such a case, the double expansion reduces to a single expansion
+whose coefficients fλ
+abareeλ
+ab≡c∗
+λacλbEλandnλ
+ab≡c∗
+λacλbfor the functions of
+intereste[g−g′]n[g−g′] andn[g−g′], respectively. Given that coefficients fλ
+abtransform
+specifically under any member of G♯, the ratio of two such objects, e.g. Eλ, transforms
+as a scalar.
+Starting from E[g′,g] andN[g′,g], and given the orthogonality relationship
+/integraldisplay
+Gdm(g)Sλ∗
+ab(g)Sλ′
+a′b′(g) =vG
+dλδλλ′δaa′δbb′, (13)
+one can perform the integration
+/parenleftbiggdλ
+vG/parenrightbigg2/integraldisplay /integraldisplay
+Gdm(g′)dm(g)Sλ
+ca(g′)Sλ∗
+cb(g)E[g′,g]N[g′,g] =eλ
+ab,(14)
+to extract the energy Eλassociated with states |Θλa/an}b∇acket∇i}htspanning the irreducible
+representation 놆. Such a symmetry restoration stage is denoted as a multi-reference
+method in the sense that, while the energy computed through Eq. 8 involves a
+single reference state |Θ/an}b∇acket∇i}ht, the extraction of Eλinvolves the set of references states
+|Θ(g)/an}b∇acket∇i}ht ≡R(g)|Θ/an}b∇acket∇i}htobtained from |Θ/an}b∇acket∇i}htthrough all transformations of G.
+3.3. Transfer operator
+The above presentation of the symmetry restoration will be partic ularly suited to
+making the connection with the MR-EDF approach discussed in Sec. 4 . Still, it is
+worth noting that Eq. 14 is usually obtained within the wave-function -based approach
+by first introducing the transfer operator [2]
+Pλ
+ab≡dλ
+vG/integraldisplay
+Gdm(g)Sλ∗
+ab(g)R(g), (15)
+to extract the state with good symmetries
+|Θλa/an}b∇acket∇i}ht=1
+cλbPλ
+ab|Θ/an}b∇acket∇i}ht, (16)
+from which Eq. 14 can be recovered by taking the average value of H.
+3.4. Particular case of interest
+Let us consider the particular case where the symmetry-breaking state|Θ/an}b∇acket∇i}htis taken
+as aproduct state|Φ/an}b∇acket∇i}htof the Bogoliubov type. In this case, the method at
+play corresponds to symmetry-unrestricted and symmetry-pro jected Hartree-Fock-
+Bogoliubov approximations [2]. In such a situation, one can use the st andard Wick
+theorem[37]toexpressthesymmetry-breakingenergy E(Eq.8)asaspecificfunctional
+E[ρ,κ,κ∗] of the one-body density matrices computed from the symmetry b reaking
+state and given, in an arbitrary single-particle basis, by
+ρij≡/an}b∇acketle{tΦ|a†
+jai|Φ/an}b∇acket∇i}ht
+/an}b∇acketle{tΦ|Φ/an}b∇acket∇i}ht, κij≡/an}b∇acketle{tΦ|ajai|Φ/an}b∇acket∇i}ht
+/an}b∇acketle{tΦ|Φ/an}b∇acket∇i}ht, κ∗
+ij≡/an}b∇acketle{tΦ|a†
+ia†
+j|Φ/an}b∇acket∇i}ht
+/an}b∇acketle{tΦ|Φ/an}b∇acket∇i}ht.(17)
+♯The corresponding law is easily obtained from the transform ation of Sλ
+ab(g).
+††The fact that E[g′,g] andN[g′,g] only depend on the difference g−g′can be exploited to extract
+eλ
+abthrough a single integral rather than through a double integ ral as in Eq. 14. The reason why we
+keep explicitly two integrals here will only become clear in the last section of the paper (see Sec. 5.1).Symmetries within EDF methods 10
+Theexplicitformof E[ρ,κ,κ∗]dependsonthespecificHamiltonian H, e.g. onwhether
+it is local or non local in space and on whether it contains a three-bod y force, a four-
+body force...
+As for the symmetry restoration stage, the energy kernel at pla y (Eq. 11)
+can be computed in this case using the generalized Wick theorem [38] such that
+E[g′,g] =E[ρg′g,κg′g,κgg′∗], i.e. the transition energy kernel is expressed through
+the same functional as the symmetry-breakingenergy except th at diagonal symmetry-
+breaking one-body density matrices are replaced by transition one-body density
+matrices involving the two transformed product states |Φ(g)/an}b∇acket∇i}htand|Φ(g′)/an}b∇acket∇i}ht, i.e.
+ρg′g
+ij≡/an}b∇acketle{tΦ(g′)|a†
+jai|Φ(g)/an}b∇acket∇i}ht
+/an}b∇acketle{tΦ(g′)|Φ(g)/an}b∇acket∇i}ht, κg′g
+ij≡/an}b∇acketle{tΦ(g′)|ajai|Φ(g)/an}b∇acket∇i}ht
+/an}b∇acketle{tΦ(g′)|Φ(g)/an}b∇acket∇i}ht, κgg′∗
+ij≡/an}b∇acketle{tΦ(g′)|a†
+ia†
+j|Φ(g)/an}b∇acket∇i}ht
+/an}b∇acketle{tΦ(g′)|Φ(g)/an}b∇acket∇i}ht.(18)
+3.5. Actual implementation
+In practice, states |Θλa/an}b∇acket∇i}htonly provide approximations to exact eigenstates. This is for
+instance the case when starting from a symmetry-breaking produ ct state as discussed
+in Sec. 3.4. When dealing with a non abelian group, one must actually con sider
+an arbitrary linear combination of states spanning a given irreducible representation
+such that mixing coefficients are determined through the minimization of the resulting
+energy. This corresponds to considering that the link between the symmetry-restored
+states of interest and the symmetry-breaking one is in fact given b y
+|Θλa/an}b∇acket∇i}ht ≡/summationdisplay
+bgλbPλ
+ab|Θ/an}b∇acket∇i}ht, (19)
+rather than by Eq. 16, and to determining the {gλb}through the minimization of
+Eλ≡/an}b∇acketle{tΘλa|H|Θλa/an}b∇acket∇i}ht
+/an}b∇acketle{tΘλa|Θλa/an}b∇acket∇i}ht=/summationtext
+bb′gλb∗gλb′eλ
+bb′/summationtext
+bb′gλb∗gλb′nλ
+bb′, (20)
+witheλ
+bb′defined by Eq. 14 and nλ
+bb′given by a similar equation for the kernel N[g′,g].
+Such a minimization leads to solving a Hill-Wheeler-Griffin equation [39, 40 ].
+4. EDF-based method
+We are now interested in the nuclear EDF formalism within which energy kernels are
+formulated as functionals of the one-body (transition) densities d efined by Eqs. 17
+and 18. Although extensions can be envisioned, the standard EDF m ethod is
+based on the use of simple symmetry-breaking product states of t he Bogoliubov
+type. This choice stems from the historical construction of the nu clear EDF
+method by analogy with symmetry-unrestricted and symmetry-pr ojected Hartree-
+Fock-Bogoliubov approximations (Sec. 3.4) [2].
+4.1. SR-EDF step
+The SR-EDF method [11] relies on computing the analog to the symmet ry-breaking
+average energy E(Eq. 8) as an a priori generalfunctional E[ρ,κ,κ∗]. As opposed
+to what was considered in Sec. 3.4, the symmetry-breaking energy E[ρ,κ,κ∗] isnot
+computed from the average value of a genuine operator H. As a result, specific
+constraints must be imposed onto the functional form of E[ρ,κ,κ∗] for it to be a scalar
+under all transformations of G; i.e. under transforming |Φ/an}b∇acket∇i}htand the densities ρ,κ,Symmetries within EDF methods 11
+κ∗constructed from it. For a quasi-local functional of the Skyrme t ype, we refer
+the reader to Refs. [41, 42, 43] for the formulation of such const raints. For actual
+parameterizations of the nuclear EDF, we refer the reader to Ref . [11].
+4.2. MR-EDF step
+The MR-EDF step is built from the SR-EDF by analogy to the wave-fun ction-based
+method formulated in terms of product states (Sec. 3.4). As a res ult, the analog of the
+energy kernel E[g′,g] (Eq. 11) is naturally introduced over the volume of Gthrough
+the definition E[g′,g]≡ E[ρg′g,κg′g,κgg′∗], where E[ρ,κ,κ∗] is the SR-EDF kernel.
+FromE[g′,g], one extracts
+ελ
+ab≡/parenleftbiggdλ
+vG/parenrightbigg2/integraldisplay
+G/integraldisplay
+Gdm(g′)dm(g)Sλ
+ca(g′)Sλ∗
+cb(g)E[g′,g]N[g′,g],(21)
+by analogyto Eq.14. Whereasin the wave-function-basedmethod one could explicitly
+demonstrate the identity eλ
+ab=c∗
+λacλbEλ, this is not the case in the EDF approach
+where there no possibility in general to perform the equivalent to th e derivation that
+led to Eq. 9. Equation 21 corresponds simply to the application of exp ansion 12 to
+thefunction E[g′,g] over the irreducible representations of the group, without any
+reference to many-body states with good quantum numbers. As a matter of fact, and
+contrarily to what is often stated [11], ελ
+abisnotcomputed from a projected state in
+the MR-EDF method, i.e. the transfer operator Pλ
+abcannotbe factorized explicitly in
+Eq. 21. However, one can implicitly relate the MR-EDF energy Eλto the projected
+state|Φλa/an}b∇acket∇i}htobtained from |Φ/an}b∇acket∇i}htas in Eq. 19. With this in mind, it is natural and
+customary [11, 17] to definethe symmetry-restored energy Eλfromελ
+abthrough the
+analog of Eq. 20, i.e.
+Eλ≡/summationdisplay
+bb′gλb∗gλb′ελ
+bb′//summationdisplay
+bb′gλb∗gλb′nλ
+bb′, (22)
+where the {gλb}are determined through the minimization of Eλ.
+4.3. Puzzling questions
+We have clarified in previous sections that SR- and MR-EDF methods h ave been
+empiricallyconstructedbyanalogyto symmetry-unrestrictedand symmetry-projected
+Hartree-Fock-Bogoliubov approximations. The key difference with the latter is that
+the energy kernels at play in the EDF method are notdefined as matrix elements of
+a genuine operator between wave-functions. For the rest, expr essions utilized in both
+approaches, in particular regarding the extraction of the symmet ry-restored energy
+(Eqs. 14 and 20 versus Eqs. 21 and 22), look totally alike. Still, puzzlin g questions
+remain to be raised.
+As mentioned in Sec. 4.1, one must require at the symmetry-breakin g level
+that the SR energy E[ρ,κ,κ∗] is a scalar under all transformations of G. Such a
+requirement has led to formulating a set of constraints on the func tional form of
+E[ρ,κ,κ∗] [41, 42, 43]. The next question one may ask is the following: are thos e
+constraintsimposedon the energykernel E[ρ,κ,κ∗] atthe SR levelsufficient tomaking
+the MR-EDF method described in Sec. 4.2well defined, in particular fr om a symmetry
+standpoint? In particular, one may wonder whether the fact that the energy kernel
+E[g′,g], which is the key ingredient to the MR-EDF calculation, is not compute d as
+the matrix element of a (genuine) operator makes the method ill-defi ned in any way?Symmetries within EDF methods 12
+As a matter of fact, a set of physical constraints to be imposed on E[g′,g] have
+already been worked out [13]. The facts (i) that the MR energy shou ld be real, (ii)
+that the SR-EDF should be recovered from the Kamlah expansion an d (iii) that the
+RandomPhaseApproximationbasedonthe SR-EDF E[ρ,κ,κ∗] shouldbe recoveredas
+a limit of the MR-EDF calculation [44], has helped limiting the energy kerne lE[g′,g]
+to depend on transition densities only, e.g. E[g′,g]≡ E[ρg′g,κg′g,κgg′∗].
+The aim of the present contribution is to elaborate further on the q uestion raised
+above and to discuss a path that could be followed to constrain more tightly MR-EDF
+calculations. References [16, 17, 18, 19] have already provided imp ortant elements in
+the case of U(1), i.e. for particle-number restoration (PNR). Let us recall the main
+outcome of those studies prior to formulating the problem to be add ressed.
+4.4. Lessons learnt from particle-number restoration
+Equation 11 applied to U(1) provides the Fourier decomposition ε[ϕ]n[ϕ] =/summationtext
+N∈Zc2
+NENeiNϕof the periodic function ε[ϕ]n[ϕ] over [0,2π]. From a mathematical
+standpoint, the sum runs a priori over all irreducible representat ions of the group, i.e.
+over both positive and negative integers N. From a physics point of view though, the
+labelNdenotes the particle number of the physical system. Consequent ly, the sum
+should actually only run over positive integers, i.e. one should find c2
+NEN= 0 and
+EN= 0forN≤0. Inthewave-function-basedmethod, sucharesultisindeedobt ained
+from the fact that ENis computed as the average value of Hin|ΦN/an}b∇acket∇i}ht, the latter being
+zero [18] for N≤0. In the EDF context, however, it was demonstrated [18, 19] th at
+Fourier components ENare a priori different from zero for N≤0, i.e. one usually
+obtains a non-zero symmetry-restored energy for negative par ticle numbers! This
+problem was shown [18] to be related to unphysical mathematical p roperties of E[ϕ].
+Applying the regularization method proposed in Ref. [17], the cancela tion of non-
+physical Fourier components was recovered [18]. At the same time , components EN
+forN >0 were modified by up to 1 MeV, which is of the same order as the root- mean-
+square error on mass residuals reached by the best available partic le-number-restored
+EDF mass fits [45]. This demonstrates the practical need of constr aining further
+MR-EDF calculations in order to produce fully reliable results.
+5. Towards new constraints?
+The example discussed above is particularly enlightening given that cle ar-cut physical
+arguments can be used to argue that certain coefficients in the Fou rier expansion
+ε[ϕ]n[ϕ] should be strictly zero, although they are not if one does not pay p articular
+attention to it. Recovering such physical features removes at th e same time non-
+physical contaminations from other coefficients of the expansion [1 8]. This proves
+that the MR-EDF method, as performed so far, faces the danger to be ill-defined and
+that new constraints on the energy kernel E[g′,g] must be workedout in order to make
+the method physically sound. The regularization method proposed in Ref. [17] that
+restores the validity of PNR can only be applied if the underlying EDF E[ρ,κ,κ∗]
+depends on integer powers of the density matrices [19], which is an ex ample of such a
+constraint.
+For an arbitrary symmetry group, the situation might not be as tra nsparent as
+forU(1). Indeed, it is unlikely in general that certain coefficients of the e xpansion of
+E[g′,g]N[g′,g] over irreducible representations of the group are zero based on physicalSymmetries within EDF methods 13
+arguments. The challenge we face can be formulated in the following w ay: although
+expansion 12 that underlines the MR-EDF method is sound from a gro up-theorypoint
+ofview, mathematicalpropertiesdeducedfromawave-function- basedmethod mustbe
+workedoutandimposedon E[g′,g]tomake ελ
+abextractedfromEq.21physicallysound.
+The rest of the present contribution is dedicated to briefly introdu cing an example of
+such a property in the case of SO(3), i.e. for angular momentum restoration, that
+could be used to constrain E[Ω′,Ω]. Details of such an analysis will be reported
+elsewhere [36].
+5.1. Mathematical property associated with angular-momen tum conservation
+We omit spin and isospin for simplicity and consider the rotationally-inva riantnuclear
+Hamiltonian H=T+Vin which the central two-nucleon interaction
+V≡1
+2/integraldisplay /integraldisplay
+d/vector r1d/vector r2v(|/vector r1−/vector r2|)a†
+/vector r1a†
+/vector r2a/vector r2a/vector r1, (23)
+is local, i.e. non-antisymmetrized matrix elements are defined as /an}b∇acketle{t1 :/vector r1;2 :/vector r2|V|1 :
+/vector r3;2 :/vector r4/an}b∇acket∇i}ht ≡v(|/vector r1−/vector r2|)δ(/vector r1−/vector r3)δ(/vector r2−/vector r4), and in which three-nucleon and higher
+many-body forces are disregarded for simplicity. None of the conc lusions drawn below
+would be modified by the inclusion of many-body forces or by using a no n-local two-
+nucleon interaction. Operator a†
+/vector r(a/vector r) creates (annihilates) a nucleon at position
+/vector r. Considering an eigenstate |ΘLM/an}b∇acket∇i}htof/vectorL2andLz, as well as using center of mass
+/vectorR≡(/vector r1+/vector r2)/2 and relative coordinates /vector r≡/vector r1−/vector r2, the potential energy reads as
+VL≡/an}b∇acketle{tΘLM|V|ΘLM/an}b∇acket∇i}ht
+/an}b∇acketle{tΘLM|ΘLM/an}b∇acket∇i}ht=1
+2/integraldisplay
+d/vectorR/integraldisplay
+d/vector r v(r)ρ[2]LMLM
+/vectorR/vector r(24)
+≡/integraldisplay
+d/vectorR VLM(/vectorR), (25)
+which defines a potential energy densityVLM(/vectorR) in terms of the two-body density
+matrixρ[2]LMLM
+/vectorR/vector r≡ /an}b∇acketle{tΘLM|a†
+/vector r2a†
+/vector r1a/vector r1a/vector r2|ΘLM/an}b∇acket∇i}ht//an}b∇acketle{tΘLM|ΘLM/an}b∇acket∇i}ht. After tedious but
+straightforward calculations, one can demonstrate that [36]
+VLM(/vectorR) =2L/summationdisplay
+L′=0CLM
+LML′0υ[2]
+LL′(R)Y0
+L′(ˆR), (26)
+where the Clebsch-Gordan coefficient CLM
+LML′0carries the dependence on MwhileYm
+l
+denotes spherical harmonics. The weight υ[2]
+LL′(R) depends on the norm of /vectorRonly
+and is related to a reduced matrix element of the two-body density m atrix operator
+recoupled to a total angular momentum L′. The remarkable mathematical property
+identified through Eq. 26 is that the scalar potential energy VLis obtained from
+an intermediate energy density VLM(/vectorR) whose dependence on the orientation of /vectorR
+is tightly constrained by the angular-momentum quantum number of the underlying
+many-body state |ΘLM/an}b∇acket∇i}ht, i.e. its expansion over spherical harmonics is limited to
+L′≤2L. Such a result is unchanged when adding the kinetic energy (density ) to the
+potential energy (density) such that we restrict ourselves to th e latter for simplicity.
+Of course, the energy eventually extracts the coefficient of the lo west harmonic, i.e.
+VL=√
+4π/integraltext
+dRυ[2]
+L0(R).Symmetries within EDF methods 14
+5.2. Wave-function-based symmetry-restoration method
+Since property 26 is general, it can also be obtained within the frame o f the wave-
+function-based symmetry-restoration method presented in Sec . 3. Omitting again the
+kinetic energy for simplicity and using Eqs. 2-7-11-23, the potentia l energy kernel
+reads
+V[Ω′,Ω]N[Ω′,Ω] =1
+2/integraldisplay
+d/vectorRd/vector r V(r)/an}b∇acketle{tΘ|R†(Ω′) ˆρ[2]
+/vectorR/vector rR(Ω)|Θ/an}b∇acket∇i}ht (27)
+=1
+2/summationdisplay
+{L,M}c∗
+L1N1cL2N2DL1†
+N1M1(Ω′)DL2
+M2N2(Ω)/integraldisplay
+d/vectorRd/vector r V(r)ρ[2]L1M1L2M2
+/vectorR/vector r,
+where{L,M}denotes a sum over the six angular-momentum quantum numbers
+appearing in the formula. Applying Eq. 14 to the above expression (E q. 27) provides,
+thanks to the orthogonality property 13, the result
+VL=(2L+1)2
+(8π2)2/integraldisplay
+dΩ′dΩDL
+KM(Ω′)
+cLKDL∗
+KM(Ω)
+c∗
+LKV[Ω′,Ω]N[Ω′,Ω]
+=1
+2/integraldisplay
+d/vectorRd/vector r V(r)ρ[2]LMLM
+/vectorR/vector r, (28)
+so that Eqs. 25 and 26 are recovered. To obtain such a result it is ma ndatory to
+use the double-integral formulation of Eq. 14 rather than the mor e standard single-
+integral formulation that takes advantage, from the outset, of the fact that V[Ω′,Ω]
+andN[Ω′,Ω] only depend on the difference Ω −Ω′. We thus insist on using the double-
+integral formulation in the present discussion.
+5.3. EDF-based symmetry-restoration method
+Let us now come back to the EDF method described in Sec. 4. The poin t is to
+underline the fact that property 26 cannot be derived a priori give n that the potential
+energy part of the kernel E[Ω′,Ω] isnotexplicitly related to the two-body density
+matrix in this case. Taking a quasi-local Skyrme EDF as an example, alt hough this
+can be easily adapted to non-local EDF of the Gogny type, the ener gy kernel takes
+the form
+E[Ω′,Ω] =E[ρΩ′Ω,κΩ′Ω,κΩΩ′∗]
+≡/integraldisplay
+d/vectorRE(ρΩ′Ω(/vectorR),τΩ′Ω(/vectorR),/vectorjΩ′Ω(/vectorR),...), (29)
+whereE(ρΩ′Ω(/vectorR),τΩ′Ω(/vectorR),/vectorjΩΩ′(/vectorR)) is a general function of a set of one-body local
+transition densities built from the transition density matrices, e.g.
+ρΩ′Ω(/vectorR)≡/summationdisplay
+ijϕ∗
+j(/vectorR)ϕi(/vectorR)ρΩ′Ω
+ij, (30)
+τΩ′Ω(/vectorR)≡/summationdisplay
+ij/bracketleftbig/vector∇ϕ∗
+j(/vectorR)/bracketrightbig
+·/bracketleftbig/vector∇ϕi(/vectorR)/bracketrightbig
+ρΩ′Ω
+ij, (31)
+/vectorjΩ′Ω(/vectorR)≡ −i
+2/summationdisplay
+ij/braceleftBig
+ϕ∗
+j(/vectorR)/bracketleftbig/vector∇ϕi(/vectorR)/bracketrightbig
+−/bracketleftbig/vector∇ϕ∗
+j(/vectorR)/bracketrightbig
+ϕi(/vectorR)/bracerightBig
+ρΩ′Ω
+ij,(32)
+...Symmetries within EDF methods 15
+such that constraints imposed at the SR level [41, 42, 43] are fulfille d (see Secs. 4.1
+and 4.3). Given such an EDF, there is no reason a priori that the ene rgy density
+ELM(/vectorR) extracted from Eqs. 21 and 22 displays property 26; i.e. the angu lar
+dependence of ELM(/vectorR) is likely to display harmonics Y0
+L′(ˆR) withL′>2L. One
+might argue that it is not an issue considering that the symmetry-re stored energy EL
+eventually relates to the harmonic Y0
+0(ˆR) only. However, a formalism that provides
+ELM(/vectorR) with a spurious angular content will certainly also provide the coeffic ient
+EL0(R) of the lowest harmonic with unphysical contributions. To state it d ifferently,
+it is likely that constraining the MR-EDF kernels E[ρΩ′Ω,κΩ′Ω,κΩΩ′∗] to produce an
+energy density ELM(/vectorR) that fulfils the mathematical property 26 will impact at the
+same time the value of the weight EL0(R), and thus the value of EL. To some extent,
+thisissimilartothesituationencounteredwith U(1)whererestoringthemathematical
+property that Fourier coefficients ENwithN≤0 should be strictly zero impacted the
+value of all other Fourier coefficients [18].
+6. Conclusions
+The present contribution reviews the notion of symmetry breaking and restoration
+within the frame of nuclear energy density functional (EDF) metho ds. Multi-reference
+(MR) EDF calculations are nowadays routinely applied with the aim of inc luding
+long-range correlations associated with large-amplitude collective m otions that are
+difficult to incorporate in a more traditional single-reference (SR), i.e. ”mean-field”,
+EDF formalism [11].
+The framework for MR-EDF calculations was originally set-up by analo gy with
+projection techniques and the Generator Coordinate Method (GC M), which are
+rigorously formulated only within a Hamiltonian/wave-function-base d formalism [2].
+In the present work, we elaborate on key differences between wav e-function- and
+energy-functional-based methods. In particular, we point to diffic ulties encountered
+to formulate symmetry restoration within the energy functional a pproach. The
+analysis performed in Ref. [18] to tackle problems encountered in Re fs. [14, 15, 16] for
+particle number restoration serves as a baseline. Reaching out to a ngular-momentum
+restoration, we identify in a wave-function-based framework a ma thematical property
+of the energy density ELM(/vectorR) associated with angular momentum conservation that
+could be used to constrain EDF kernels. Consequently, possible fut ure routes to
+better formulate symmetry restorations in EDF-based methods c ould encompass the
+following points.
+(i) The fingerprints left on the energy density ELM(/vectorR) by angular momentum
+conservation in a wave-function-based method could be exploited t o constrain
+the functional form of the basic EDF E[ρ,κ,κ∗].
+(ii) The regularization method proposed in Ref. [17] to deal with spec ific spurious
+features of MR-EDF calculations should be investigated as to what im pact it has
+on properties of the energy density ELM(/vectorR) in the case of angular momentum
+restoration.
+(iii) Similar mathematical properties extracted from a wave-functio n-based method
+could be worked out for other symmetry groups of interest and us ed to constrain
+EDF kernels.
+Efforts in those directions are currently being made [36].Symmetries within EDF methods 16
+Acknowledgments
+The authors wish to thank D. Lacroix for proofreading the manusc ript and for useful
+discussions. T. D. wishes to thank M. Bender, P.-H. Heenen and D. L acroix for long
+term collaborations on matters related to the subject of the pres ent paper.
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+arXiv:1001.0674v2  [quant-ph]  20 Oct 2010The 3-dimensional cube is the only periodic,
+connected cubic graph with perfect state transfer
+Simone Severini∗
+Department of Physics and Astronomy, University College Lo ndon, WC1E 6BT London, United Kingdom
+There is perfect state transfer between two vertices of a graph, if a single excitation can tr avel
+with fidelity one between the corresponding sites of a spin sy stem modeled by the graph. When the
+excitation is back at the initial site, for all sites at the sa me time, the graph is said to be periodic.
+A graph is cubicif each of its vertices has a neighbourhood of size exactly th ree. We prove that
+the 3-dimensional cube is the only periodic, connected cubi c graph with perfect state transfer. We
+conjecture that this is also the only connected cubic graph w ith perfect state transfer.
+I. INTRODUCTION
+A. State transfer
+LetG= (V,E) be a graph with set of vertices V(G) ={1,2,...,n}and set of edges E(G)⊆V(G)×V(G)−{{i,i}:
+i∈V(G)}. TheorderofGis the number of its vertices.
+Let us consider a system of nspin-1/2 quantum particles with unit XYcouplings. The space assigned to the entire
+system is/parenleftbig
+C2/parenrightbig⊗n. Each particle is attached to a vertex of G. The coupling between two particles is nonzero if and
+only if the particles are attached to adjacent vertices. The couplin gs are then specified by the adjacency matrix of the
+graph. The ij-th entry of the adjacency matrix ofGis [A(G)]i,j= 1 if{i,j} ∈E(G) and [A(G)]ij= 0 if{i,j}/∈E(G).
+Weshallworkwith theXYmodel. Let XiandYibethePaulioperatorsactingonthe i-thparticle. TheHamiltonian
+governing the dynamics of the spin system can be written as HXY(G) = 2−1/summationtextn
+i/negationslash=j=1[A(G)]i,j(XiXj+YiYj). Let
+{|1/an}bracketri}ht ≡e1,|2/an}bracketri}ht,...,|n/an}bracketri}ht}be the standard basis of the space Cn. A vector |i/an}bracketri}htindicates the presence of an excitation at
+vertexionly. With respect to the standard basis, the ij-th entry of the Hamiltonian acting on Cnis [HXY(G)]i,j=
+2[A(G)]i,j. It follows that the Schr¨ odinger evolution of the excitation is prac tically induced by a unitary matrix of
+the formUG(t) =e−iA(G)t, wheret∈R+. We obtain a probability distribution supported by V(G) by performing
+a projective measurement on the state |ψt/an}bracketri}ht=UG(t)|ψ0/an}bracketri}ht. For regular graphs, the “practically” has a much larger
+extension, given that, with constant couplings, any kind of interac tion has an Hamiltonian proportional to A(G).
+Given two vertices i,j∈V(G), thefidelityat timetbetweeniandjis the function fG(i,j;t) =|/an}bracketle{tj|UG(t)|i/an}bracketri}ht|. We
+say that there is perfect state transfer (for short, PST) between the particles iandjat timetiffG(i,j;t) = 1 [13].
+We say that Gisperiodic, with period t, iffG(i,i;t) = 1 [17]. Sometime in the physics literature periodic graphs are
+said to afford perfect revival (see,e.g., [7]).
+Even if the topic is not directly discussed here, it is worth noticing tha t in theXYZmodel, the Hamiltonian
+restricted to Cnis proportional to the Laplacian matrix of G(see,e.g., [10]). This fact alone is sufficient to distinguish
+different approaches for the two models, when the graphs conside red have generic degree sequences. In this work, we
+consider regular graphs only. The result obtained is then also valid fo r theXYZmodel.
+The concept of PST has been introduced in [9] and [13]. The recent pa pers [6] and [3], even if not reviews, point
+out a good number of references embracing the more mathematica l aspects around the notion.
+B. Diameter
+A graphH= (W,F) is asubgraph ofGifW(H)⊆V(G) andF(H)⊆E(G). A subgraph H= (W,F) is an
+induced subgraph ofGifHis a subgraph of Gand, for every two vertices i,j∈V(H),{i,j} ∈E(H) if and only if
+{i,j} ∈E(G).
+Thedegreeof a vertex iis the number of edges incident with i. Apathof lengthlfromvertexitovertexj(if
+there is one) is an induced subgraph with lvertices and l−1 edges, such that iandjhave degree one and all other
+vertices in the path have degree two. A graph is said to be connected if every two vertices are in a path.
+∗Electronic address: simoseve@gmail.com2
+LetPi,j(G) be the set of all paths with end-vertices iandj. Thelengthof a path with end-vertices iandjis
+denoted by l(i,j). The (geodesic) distance between two vertices iandjis defined as d(i,j) = min Pi,j(G)l(i,j). The
+diameter of a connected graph Gis defined as dia( G) = max i,j∈V(G)d(i,j). Informally, the diameter is the longest of
+the shortest paths. Two vertices i,j∈V(G) are said to be antipodal ifd(i,j) = dia(G).
+C. Order/distance problems for state transfer
+There is a large and growing literature concerned with the mathemat ics of state transfer on spin systems. Some
+effort towards a classification may be seen as driven by three recur rent, but essentially unstated problems:
+•General graphs: GivenD∈N, find the graph G= (V,E) with the smallest possible number of vertices nD=
+|V(G)|such thatd(i,j) =DandfG(i,j;t) = 1 for some t∈R+. The set of these graphs is denoted by GD.
+•Fixed degree graphs: GivenD,∆∈N, find the graph G= (V,E) with the smallest possible number of vertices
+nD,∆=|V(G)|such that (i)the maximum degree of Gis ∆,(ii)d(i,j) =DandfG(i,j;t) = 1 for some t∈R+.
+The set of these graphs is denoted by GD,∆.
+•Regular graphs: GivenD,k∈N, find the graph G= (V,E) with the smallest possible number of vertices
+nD,k=|V(G)|such that (i)Gisk-regular, (ii)d(i,j) =DandfG(i,j;t) = 1 for some t∈R+. The set of these
+graphs is denoted by GR
+D,k. A graph is k-regularif all of its vertices have degree k.
+The requirement “smallest possible number” could be replaced with “la rgest possible number” to state the specular
+versions of the problems.
+D. Examples
+•D= 1: LetP2= ({1,2},{{1,2}}) be the path of length one. Then [ UP2(t)]1,2=−isin(t) and
+max
+t∈R+(|[UP2(t)]1,2|) =fP2(1,2;π/2) = 1.
+Hence,P2∈ G1.
+•D= 2: LetP3= ({1,2,3},{{1,2},{2,3}}) be the path of length two. Then [ UP3(t)]1,3=−sin/parenleftbig
+t/√
+2/parenrightbig2and
+max
+t∈R+(|[UP3(t)]1,3|) =fP3/parenleftBig
+1,3;π/√
+2/parenrightBig
+= 1.
+Hence,P3∈ G2.
+In both cases, PST is between antipodal vertices.
+•D= 3: LetP4= ({1,2,3,4},{{1,2},{2,3},{3,4}}) be the path of length three. Let a:=t/2. Then [UP4(t)]1,4=
+i√
+5sin/parenleftbig
+a+a√
+5/parenrightbig
+/10−isin/parenleftbig
+a+a√
+5/parenrightbig
+/2−i√
+5sin/parenleftbig
+a−a√
+5/parenrightbig
+/10−isin/parenleftbig
+a−a√
+5/parenrightbig
+/2 and
+max
+t∈R+(|[UP4(t)]1,4|) =fP4/parenleftBig
+1,3;2π/√
+5/parenrightBig
+= sin/parenleftBig
+π/√
+5/parenrightBig
+≈0.986.
+Hence,P4/∈ G3.
+E. Cartesian products
+TheCartesian product G=G1×G2= (V,E) of two graphs G1= (V1,E1) andG2= (V2,E2) has set of vertices
+V(G) =V(G1)×V(G2) and{{i,j},{k,l}} ∈E(G) if(i)i=kand{j,l} ∈E(G2) or(ii)j=land{i,k} ∈E(G1).
+Two facts are important: dia( G) = dia(G1)+ dia(G2); ifG1andG2arek-regular and l-regular graphs, respectively,
+thenGis (k+l)-regular.3
+Thek-dimensional cube is the graph P×k
+2(in Fig. 1,P×3
+2). We have the following four cases:
+[UP×k
+2(t)]1,2k=
+
+−sin(t)k,ifk= 2l,lodd;
+sin(t)k,ifk= 2l,leven;
+isin(t)k,ifk= 2l+1,lodd;
+−isin(t)k,ifk= 2l+1,leven.
+In all these cases,
+max
+t∈R+/parenleftBig/vextendsingle/vextendsingle/vextendsingle[UP×k
+2(π/2)]1,2k/vextendsingle/vextendsingle/vextendsingle/parenrightBig
+=fP×k
+2/parenleftbig
+1,2k;π/2/parenrightbig
+= 1.
+ForP×k
+2, we have/vextendsingle/vextendsingleV(P×k
+2)/vextendsingle/vextendsingle= 2kand dia(P×k
+2) =k·dia(P2) = 2log2/vextendsingle/vextendsingleV(P×k
+2)/vextendsingle/vextendsingle=k. Thus, for the k-regular graphs
+inGR
+m,m,nm,m≤2m, wherem=D,k.
+Thek-dimensional generalization of the 3 ×3 grid is denoted by P×k
+3(in Fig. 1, the grid P×3
+3). There are four
+types of matrix entries for the graph P×k
+3:
+[UP×k
+3(t)]1,3k=
+
+−sin/parenleftbig
+t/√
+2/parenrightbig2k,ifk= 2l,lodd;
+sin/parenleftbig
+t/√
+2/parenrightbig2k,ifk= 2l,leven;
+isin/parenleftbig
+t/√
+2/parenrightbig2k,ifk= 2l+1,lodd;
+−isin/parenleftbig
+t/√
+2/parenrightbig2k,ifk= 2l+1,leven.
+Then,
+max
+t∈R+/parenleftBig/vextendsingle/vextendsingle/vextendsingle[UP×k
+3/parenleftBig
+π/√
+2/parenrightBig
+]1,3k/vextendsingle/vextendsingle/vextendsingle/parenrightBig
+=fP×3
+3/parenleftBig
+1,3k;π/√
+2/parenrightBig
+= 1.
+ForP×k
+3, we have/vextendsingle/vextendsingleV(P×k
+3)/vextendsingle/vextendsingle= 3kand dia(P×k
+3) =k·dia(P3) = 2log3/vextendsingle/vextendsingleV(P×k
+3)/vextendsingle/vextendsingle= 2k.
+ForP×k
+2andP×k
+3PST is between antipodal vertices. The parameters related to PST between two vertices iand
+jin these graphs are given in the following tables [13]:
+P×k
+2k/vextendsingle/vextendsingleV(P×k
+2)/vextendsingle/vextendsingled(i,j)
+24 2
+38 3
+416 4
+532 5;P×k
+3k/vextendsingle/vextendsingleV(P×k
+3)/vextendsingle/vextendsingled(i,j)
+29 4
+327 6
+481 8
+5243 10.
+Notice that P×k
+3is nonregular for every k.
+A square matrix Mof sizenconsisting of unimodular entries |Mi,j|= 1 is called a Hadamard matrix ifHH†=nI,
+whereIis the identity matrix and†denotes the Hermitian transpose. In a complex Hadamard matrix ,Mi,j∈C[26].
+The matrix
+UP×k
+2/parenleftBigπ
+4/parenrightBig
+=1√
+2/bracketleftBigg
+1−i
+−i1/bracketrightBigg⊗k
+is a complex Hadamard matrix.
+F. Statement of the results
+We prove that the 3-dimensional cube is the only periodic, connecte d cubic graph with PST. Equivalently,
+Theorem. The 3-dimensional cube, P×3
+2, is the only periodic, connected cubic (3-regular) graph Gwith two different
+verticesiandjsuch thatfG(i,j;t) = 1, for some t∈R+.
+The statement is verified directly in the next section. Conclusions fo llow. The proof is based on two known results:
+a periodic, connected regular graph is integral; there are only thirt een connected cubic integral graphs. The proof4
+123
+4
+5
+6
+78123
+4
+5
+6
+78123
+4
+5
+6
+7
+89
+FIG. 1:(L): The graph W. There is PST between vertices 1 and 8. The spectrum of Wis not integral. (C): The 3-dimensional
+cube. There is PST between vertices 1 and 8. These vertices ar e antipodal and at distance 3. (R): The graph P×2
+3. Note that
+the graph has vertices of degree two and three. There is PST be tween the vertices 1 and 9. These vertices are antipodal and
+at distance four.
+is technically easy, but tedious, because it goes through a seemingly unavoidable case by case analysis. Nonetheless,
+establishingtheresultisalsoanexcuseforafurtherstepintoasys tematicalexplorationofperiodicquantumdynamics.
+The proof is interspersed with extra information. The broader aim w ould be to take a picture of periodic quantum
+dynamics on cubic graphs, even if here we do not state any further general result, beyond a crude analytic compilation
+of matrix entries.
+It is an open problem to prove that P×3
+2is the only connected cubic graph with PST.
+Conjecture. GR
+D,3={P×3
+2}ifD= 3 and GR
+D,3=∅, otherwise.
+Periodicity is not necessary for PST. There are examples of regular graphs that are not periodic but have PST [23].
+II. PROOF OF THE THEOREM
+A. Integral graphs
+Thespectrum of a graphGis the multiset {λ[m1]
+1(G),λ[m2]
+2(G),...,λ[mr]
+r(G)}of the eigenvalues of A(G). The index
+[mi] inλ[mi]
+i(G) denotes the multiplicity of the eigenvalue λi(G). For example, the spectrum of the complete graph
+on four vertices, K4, is{3,−1[3]}. A graph is said to be integralif its eigenvalues are integers. There is basically one
+survey on this area [5] . See also [15] for the relevant terminology a nd [1] for a nontrivial upper bounds on the total
+number of integral graphs with nvertices. There are a few general results that establish a relation between PST and
+integral graphs [17]. The next statement is directly useful to our p urposes:
+P0.A connected regular graph is periodic if and only if it is integral (Corolla ry 2.3 [17]).
+The converse is not necessarily true. This fact can be observed in t he examples discussed below. Moreover, integer
+eigenvalues have a weaker role in nonregular graphs. Let us conside r, for instance, a graph Wwith eight vertices
+{1,2,...,8}and set of edges E(W) ={{1,i},{j,8}: 2≤i,j≤7;}(see Fig. 1). The unitary governing the dynamics
+inWis defined by
+[UW(t)]i,j=
+
+−sin/parenleftbig√
+3t/parenrightbig2,ifi= 1 andj= 8 orviz.;
+−isin/parenleftbig
+2√
+3t/parenrightbig
+/2√
+3, if{i,j} ∈E(W);
+cos/parenleftbig√
+3t/parenrightbig2, Ifi=j= 1,8;/parenleftbig
+5+cos/parenleftbig
+2√
+3t/parenrightbig/parenrightbig
+/6, ifi=j/ne}ationslash= 1,8;
+−sin/parenleftbig√
+3t/parenrightbig2, otherwise.
+There is PST between vertices 1 and 8, because [ UW/parenleftbig
+π/2√
+3/parenrightbig
+]1,8=−sin(3π/2)2=−1. At the same time, the 6 ×6
+submatrix of UW/parenleftbig
+π/2√
+3/parenrightbig
+, excluding the first/last row/column, is 2 /3 in the diagonal and −1/3 off-diagonal. The
+spectrum of Wis{±2√
+3,0[6]}. Thus,Wis not an integral graph.
+We shall be interested in a special family of graphs. A cubicgraph is a 3-regular graph. All cubic integral graphs
+have been classified and explicitly constructed [11, 24]. In particular ,5
+P1.There are exactly thirteen connected cubic integral graphs.
+On the light of the statements P0 and P1, a proof of the theorem ca n be obtained via a case by case analysis.
+As we have seen, and this is an already known fact, the 3-dimensiona l cube,P×3
+2, has PST between its antipodal
+vertices. These are vertices at distance three. We shall verify th at none of the remaining twelve connected, cubic
+integral graphs affords PST. All graphs considered in this section a re periodic.
+B. The complete graph K4, the complete bipartite graph K3,3, and two connected copies of K2,3
+Acomplete graph onnvertices is a graph Kn= ({1,...,n},E), whereE(Kn) ={{i,j}: 1≤i,j≤n}. Theij-entries
+ofUK4are given by
+[UK4(t)]i,j=/braceleftBigg/parenleftBig
+3cos(t)+cos(3t)+4isin(t)3/parenrightBig
+/4,ifi=j;
+cos(t)sin(t)(−icos(t)−sin(t)),ifi/ne}ationslash=j.
+Then max t∈R+([UK4(t)]i,i) =fUK4(i,i;π/2) = 1, for 1 ≤i≤4; for every pair of vertices iandj,
+maxt∈R+([UK4(t)]i,j) =fUK4(i,j;π/4) = 1/2. Note that
+[UK4(π/4)]i,j=1√
+2/braceleftBigg
+(1+i)/2,ifi=j;
+−(1+i)/2,ifi/ne}ationslash=j.
+gives a complex Hadamard matrix.
+A graphG= (V=V1∪V2,E) isbipartite if each vertex in V1is adjacent to vertices in V2only and viz. Acomplete
+bipartite graph is a bipartite graph Kq,p= (V=V1∪V2,E) such that |V1|=p,|V2|=qandE={{i,j}:i∈V1and
+j∈V2}. The graph K3,3is on six vertices; V(K3,3) ={A,B}, with|A|=|B|= 3. By definition, {i,j} ∈E(K3,3) if
+and only if i∈Aandj∈B. The spectrum of K3,3is{±3,0[4]}and dia(K3,3) = 2. The ij-entries ofUK3,3are as
+follows:
+[UK3,3(t)]i,j=
+
+(2+cos(3t))/3, i =j;
+(−1+cos(3t))/3,{i,j}/∈E(K3,3);
+−isin(3t)/3,{i,j} ∈E(K3,3).
+Hence, max t∈R+/parenleftbig/vextendsingle/vextendsingle[UK3,3(t)]i,i/vextendsingle/vextendsingle/parenrightbig
+=fUK3,3(i,i;2π/3) = 1, for 1 ≤i≤6. For the off-diagonal entries, we need
+to distinguish two cases: (i)maxt∈R+/parenleftbig/vextendsingle/vextendsingle[UK3,3(t)]i,j/vextendsingle/vextendsingle/parenrightbig
+=fUK3,3(i,j;π/3) = 2/3, ifi,j∈Aori,j∈B;(ii)
+maxt∈R+/parenleftbig
+[/vextendsingle/vextendsingleUK3,3(t)]i,j/vextendsingle/vextendsingle/parenrightbig
+=fUK3,3(i,j;π/6) = 1/3, ifi∈Aandj∈B. Whent=π/2,
+[UK3,3(π/2)]i,j=
+
+2/3, i =j;
+−1/3,{i,j}/∈E(K3,3);
+i/3,{i,j} ∈E(K3,3).
+LetDK2,3be the graph on ten vertices obtained from two disjoint copies of K2,3, sayK1
+2,3andK2
+2,3, by adding
+three edges between the vertices of degree two in K1
+2,3andK2
+2,3. The spectrum of DK2,3is{±3,±2,±1[2],0[2]}and
+dia(DK2,3) = 3. The structure of UDK3,3consists of various kind of entries:
+UDK2,3(t) =a1a2a3a3a3a4a4a4a5a5
+a2a1a3a3a3a4a4a4a5a5
+a3a3a6a7a7a8a9a9a4a4
+a3a3a7a6a7a9a8a9a4a4
+a3a3a7a7a6a9a9a8a4a4
+a4a4a8a9a9a6a7a7a3a3
+a4a4a9a8a9a7a6a7a3a3
+a4a4a9a9a8a7a7a6a3a3
+a5a5a4a4a4a3a3a3a1a2
+a5a5a4a4a4a3a3a3a2a1,6
+where
+a1= (5+3cos(2 t)+2cos(3t))/10,
+a2= (−5+3cos(2t)+2cos(3t))/10,
+a3=−i(sin(2t)+sin(3t))/5,
+a4= (−cos(2t)+cos(3t))/5,
+a5=i(3sin(2t)−2sin(3t))/10,
+a6= (10cos(t)+2cos(2t)+3cos(3t))/15,
+a7= (−5cos(t)+2cos(2t)+3cos(3t))/15,
+a8=−i(10sin(t)−2sin(2t)+3sin(3t))/15,
+a9=i(10sin(t)+2sin(2t)−3sin(3t))/15.
+By considering a1anda6, we can see that max t∈R+/parenleftbig/vextendsingle/vextendsingle[UDK2,3(t)]i,i/vextendsingle/vextendsingle/parenrightbig
+=fDK2,3(i,i;2π) = 1, for every i. However,
+max1≤j≤9;j/negationslash=1,6maxt∈R+(|aj|) =/parenleftbig
+5−5/parenleftbig
+−1−√
+5/parenrightbig
+/4/parenrightbig
+/10≈0.9. The maximum is attained by a2fort= 2π/5. The
+segments in the matrix UDK3,3(t) help visualizing its structure and these do not have a mathematical meaning. We
+shall make a consistent use of this graphic tool also in the next case s. The graphs K3,3andDK2,3are in Fig. (2).
+C. The graphs C3+K2andC6+K2
+The (Cartesian )sumG=G1+G2= (V,E) of two graphs G1= (V1,E1) andG2= (V2,E2) has set of vertices
+V(G) =V(G1)×V(G2) and{{i,j},{k,l}} ∈E(G) if(i){i,k} ∈E(G1) or(ii){j,l} ∈E(G2) [22]. We denote by
+Cnthen-cycle:V(Cn) ={1,2,...,n}and{i,(i+1)modn} ∈E(Cn).
+The graph C3+K2has two cycles of length three and it can be drawn as a prism with trian gular basis. It is
+the undirected version the Cayley digraph of the dihedral group D6generated by the standard set. Its spectrum is
+{3,1,0[2],−2[2]}. Given the symmetry, the unitary matrix has a neat structure:
+UC3+K2(t) =a1a2a2a3a4a4
+a2a1a2a4a3a4
+a2a2a1a4a4a3
+a3a4a4a1a2a2
+a4a3a4a2a1a2
+a4a4a3a2a2a1,
+with
+a1=/parenleftbig
+2+e−it+2e2it+e−3it/parenrightbig
+/6,
+a2=−ie−it/2(sin(t/2)+sin(5t/2))/3,
+a3=/parenleftbig
+2−e−it−2e2it+e−3it/parenrightbig
+/6,
+a4=/parenleftbig
+−1−e−it+e2it+e−3it/parenrightbig
+/6.
+From this, max 2≤j≤4maxt∈R+(|aj|)≈0.9, forj= 3 andt=√
+3+π/17.
+The graph C6+K2is on twelve vertices. It has two cycles of length six and it can be draw n as a prism with
+hexagonal basis. In fact, in analogy with C3+K2, it is the undirected version the Cayley digraph of the dihedral
+groupD12generated by the standard set. As we have done for the other ca ses, we explicitly write down the unitary
+matrix. Let
+A=a1a2a3a4a3a2
+a2a1a2a3a4a3
+a3a2a1a2a3a4
+a4a3a2a1a2a3
+a3a4a3a2a1a2
+a2a3a4a3a2a1andB=a5a6a7a8a7a6
+a6a5a6a7a8a7
+a7a6a5a6a7a8
+a8a7a6a5a6a7
+a7a8a7a6a5a6
+a6a7a8a7a6a5.
+Then
+UC6+K2(t) =/bracketleftBigg
+A B
+B A/bracketrightBigg
+,7
+12 3
+4
+5 612
+3
+45
+6
+78
+9
+101
+234
+56
+123 4
+5
+6789 10
+11
+12
+FIG. 2: From the left: The complete bipartite graph K3,3. The graph obtained by connecting together two copies of K2,3. The
+graphC3+K2. The graph C6+K2. Although periodic, none of these graphs has PST.
+and
+a1= (2+cos( t)+2cos(2t)+cos(3t))/6,
+a2=−i(sin(t)+sin(2t)+sin(3t))/6,
+a3= (−1+cos(t)−cos(2t)+cos(3t))/6,
+a4=i(sin(t)−2sin(2t)+sin(3t))/6,
+a5=i(sin(t)−2sin(2t)−sin(3t))/6,
+a6=−(1+2cos(t))sin(t)2/3,
+a7=−i(sin(t)+sin(2t)−sin(3t))/6,
+a8= 16/parenleftBig
+cos(t/2)2sin(t/2)4/parenrightBig
+/3.
+Forj/ne}ationslash= 1,
+max
+t∈R+(|aj|)
+
+≈21/50, j= 2 andt≈π/6;
+= 2/3, j= 3 andt=π;
+≈29/50, j= 4 andt= 77/20
+≈9/20, j= 5 andt= 19/10;
+≈1/2, j= 6 andt≈1.21;
+≈9/25, j= 7 andt≈1.35;
+≈64/81, j= 8 andt≈1.9.
+The graphs C3+K2, andC6+K2are represented in Fig. (2).
+D. The Petersen graph, a graph on ten vertices, and L(S(K4))
+The Petersen graph, P, illustrated in Fig. (3), is one of the best-studied single objects in th e graph-theoretic
+literature. The Petersen graph has two cycles of length five. It is v ertex-transitive but not a Cayley graph, with
+spectrum {3,1[4],−2[4]}and dia(P) = 3. It is strongly regular with parameters (10 ,3,0,1). The symmetry appears
+also inUP(t), which reflects faithfully the structure of P:
+UP(t) =a1a2a3a3a2a2a3a3a3a3
+a2a1a2a3a3a3a2a3a3a3
+a3a2a1a2a3a3a3a2a3a3
+a3a3a2a1a2a3a3a3a2a3
+a2a3a3a2a1a3a3a3a3a2
+a2a3a3a3a3a1a3a2a2a3
+a3a2a3a3a3a3a1a3a2a2
+a3a3a2a3a3a2a3a1a3a2
+a3a3a3a2a3a2a2a3a1a3
+a3a3a3a3a2a3a2a2a3a1,8
+where
+a1=e−3it/parenleftbig
+1+5e2it+4e5it/parenrightbig
+/10,
+a2=e−3it/parenleftbig
+3+5e2it−8e5it/parenrightbig
+/30,
+a3=e−3it/parenleftbig
+3−5e2it+2e5it/parenrightbig
+/30.
+Thus, max j=2,3maxt∈R+(|aj|) = 8/15, forj= 2 andt=π. More generally, [ UP(π)]i,j=−1/5 ifi=j; 2/15 if
+{i,j}/∈E(P) and−8/15, otherwise.
+There is another cubic integral graph on ten vertices, obtained by replacing with triangles two nonadjacent vertices
+ofK3,3. Denoted by Z, it has spectrum {3,2,1[3],−1[2],−2[3]}. Fig. (3) contains a drawing. The unitary matrix
+obtained from Zis
+UZ(t) =a1a2a2a2a3a3a3a3a3a3
+a2a1a3a3a3a3a2a3a2a3
+a2a3a1a3a2a2a3a3a3a3
+a2a3a3a1a3a3a3a2a3a2
+a3a3a2a3a5a4a6a6a7a7
+a3a3a2a3a4a5a7a7a6a6
+a3a2a3a3a6a7a5a6a4a7
+a3a3a3a2a6a7a6a5a7a4
+a3a2a3a3a7a6a4a7a5a6
+a3a3a3a2a7a6a7a4a6a5.
+The entries are
+a1=e−3it/parenleftbig
+1+5e2it+4e5it/parenrightbig
+/10,
+a2=e−3it/parenleftbig
+3+5e2it−8e5it/parenrightbig
+/30,
+a3=e−3it/parenleftbig
+3−5e2it+25it/parenrightbig
+/30,
+a4=e−3it/parenleftbig
+eit−1/parenrightbig2/parenleftbig
+3+eit+4e2it+7e3it/parenrightbig
+/30,
+a5=e−3it/parenleftbig
+3+5eit+5e2it+10e4it+7e5it/parenrightbig
+/30,
+a6=e−3it/parenleftbig
+3+5eit−5e4it−3e5it/parenrightbig
+/30,
+a7=e−3it/parenleftbig
+3−5eit+5e4it−35it/parenrightbig
+/30.
+We then obtain max 1≤j≤9;j/negationslash=1,5maxt∈R+(|aj|)≈0.85, forj= 4 andt≈π−5/6. Note that a1,a2, anda3give the
+same dynamics as the Petersen graph.
+The graphX=L(S(K4)) is constructed by replacing eachvertex of K4with a triangle (see Fig. (3)). The triangles
+are then connected by independent edges. The notation indicates the line graph of the K4subdivision. Its spectrum
+is{3,±2[3],0[2],−1[3]}and dia(X) = 3. The unitary has some symmetry:
+UX(t) =a1a2a2a4a5a6a4a3a4a4a6a5
+a2a1a2a3a4a4a5a4a6a4a5a6
+a2a2a1a4a6a5a4a4a5a3a4a4
+a4a3a4a1a2a2a4a5a6a5a4a6
+a5a4a6a2a1a2a3a4a4a6a4a5
+a6a4a5a2a2a1a4a6a5a4a3a4
+a4a5a6a4a3a4a1a2a2a6a5a4
+a3a4a4a5a4a6a2a1a2a5a6a4
+a4a6a5a6a4a5a2a2a1a4a4a3
+a4a4a3a5a6a4a6a5a4a1a2a2
+a6a5a4a4a4a3a5a6a4a2a1a2
+a5a6a4a6a5a4a4a4a3a2a2a1,9
+123 4
+5
+67
+89
+10123 4
+5
+6
+7
+8 9101
+23
+456789
+101112
+FIG. 3: (L): The Petersen graph. (C): The graph Zon ten vertices. (R): The graph L(S(K4)) on twelve vertices. These
+graphs are periodic without PST.
+with
+a1=/parenleftbig
+2+3eit+3e−2it+3e2it+e−3it/parenrightbig
+/12,
+a2=−e−3it/parenleftbig
+−2−5eit+2e3it+2e4it+3e5it/parenrightbig
+/24,
+a3=e−3it/parenleftbig
+1+eit+2e3it−e4it−e5it/parenrightbig
+/12,
+a4=e−3it/parenleftbig
+eit−1/parenrightbig2/parenleftbig
+2+3eit+4e2it+3e3it/parenrightbig
+/24,
+a5=−e−3it/parenleftbig
+eit−1/parenrightbig3/parenleftbig
+2+3eit+3e2it/parenrightbig
+/24,
+a6=−e−3it/parenleftbig
+eit−1/parenrightbig3/parenleftbig
+eit+1/parenrightbig
+/12.
+Forj/ne}ationslash= 1,
+max
+t∈R+(|aj|)
+
+≈1/2, j= 2 andt≈π/4;
+≈1/2, j= 3 andt≈π/4;
+≈1/4, j= 4 andt≈6/5;
+= 2/3, j= 5 andt=π;
+=√
+3/4, j= 6 andt= 2π/3.
+E. The Desargues graph and its cospectral mate
+The bipartite double cover of the Petersen graph is called Desargues graph . There are many different notations for
+this graph. We adopt H5,2. The Desargues graph is on twenty vertices and its spectrum is/braceleftbig
+±3,±2[4],±1[5]/bracerightbig
+. The
+graphH5,2hasacospectral mate , whichwewilldenoteby H′
+5,2. Thisisanonisomorphicgraphwiththesamespectrum.
+Two graphs GandHare said to isomorphic if there is a permutation matrix Qsuch thatQA(G)QT=A(H). It is
+clear that two isomorphic graphs have the same spectrum. The con verse is not necessarily true. Indeed, H5,2and
+H′
+5,2are a counterexample. The graphs H5,2andH′
+5,2are drawn in Fig. (4). Let us define the arrays
+A=a1a2a2a2a3a2a2a2a3a3
+a2a1a2a2a2a3a3a2a2a3
+a2a2a1a3a2a2a2a3a2a3
+a2a2a3a1a2a2a3a2a3a2
+a3a2a2a2a1a2a3a3a2a2
+a2a3a2a2a2a1a2a3a3a2
+a2a3a2a3a3a2a1a2a2a2
+a2a2a3a2a3a3a2a1a2a2
+a3a2a2a3a2a3a2a2a1a2
+a3a3a3a2a2a2a2a2a2a1,10
+1234567
+8
+9
+10
+11
+12
+13
+14
+1516171819201
+2
+620
+317
+412
+515
+10
+7816
+91914
+1113 18
+FIG. 4:(L):The graph H′
+5,2.(R):The Desargues graph H5,2. This is cospectral with H′
+5,2.
+B=a4a4a4a8a8a8a8a8a8a7
+a4a8a8a4a8a4a8a8a7a8
+a4a8a8a8a7a8a4a4a8a8
+a5a4a6a4a8a6a6a4a8a8
+a6a4a5a8a4a6a6a8a4a8
+a8a8a4a8a4a4a8a7a8a8
+a8a8a4a7a8a8a4a8a4a8
+a6a8a6a4a4a5a6a8a8a4
+a8a7a8a8a8a4a4a8a8a4
+a6a8a6a8a8a6a5a4a4a4.
+Let/tildewiderMbe the matrix obtained by permuting the lines (rows and columns) of a square matrix Msuch that line iin
+Mis linen−j+1 in/tildewiderM. One can observe that
+UH′
+5,2(t) =/bracketleftBigg
+A/tildewideB
+B/tildewideA/bracketrightBigg
+,
+where
+a1=e−3it/parenleftbig
+1+4eit+5e2it+5e4it+4e5it+e6it/parenrightbig
+/20,
+a2=e−3it/parenleftbig
+3+2eit−5e2it−5e4it+2e5it+3e6it/parenrightbig
+/60,
+a3=e−3it/parenleftbig
+eit−1/parenrightbig4/parenleftbig
+3+4eit+3e2it/parenrightbig
+/60,
+a4=e−3it/parenleftbig
+3+8eit+5e2it−5e4it−8e5it−3e6it/parenrightbig
+/60,
+a5=−e−3it/parenleftbig
+eit−1/parenrightbig3/parenleftbig
+1+4eit+4e2it+e3it/parenrightbig
+/20,
+a6=−e−3it/parenleftbig
+eit−1/parenrightbig3/parenleftbig
+3+2eit+2e2it+3e3it/parenrightbig
+/60,
+a7=−e−3it/parenleftbig
+eit−1/parenrightbig5/parenleftbig
+1+eit/parenrightbig
+/20,
+a8=−e−3it/parenleftbig
+eit−1/parenrightbig3/parenleftbig
+3+7eit+7e2it+3e3it/parenrightbig
+/60.
+From these functions, we can see that max 1≤j≤8;j/negationslash=1maxt∈R+(|aj|)≈0.83, forj= 7 andt≈575/250. When t=π,
+the probability amplitude is supported by all vertices in a class of the b ipartition; in other words, the matrix UH′
+5,2(π)
+is block-diagonalwith two10 ×10blockscorrespondingto the classes. The functions in the above equationscompletely
+specify the dynamics in H5,2, sinceH′
+5,2andH′
+5,2are cospectral. One can verify that the matrix UH5,2(t) is obtained
+by rearranging the entries of UH′
+5,2(t).
+F. The Nauru graph and the Tutte-Coxeter graph
+TheNauru graph ,N24, is the only cubic symmetric (arc-transitive) graph on 24 vertices. It is bipartite, with
+spectrum/braceleftbig
+±3,±2[6],±1[3],0[4]/bracerightbig
+and dia(N24) = 4. A brief parenthesis: the Foster census of cubic symmetric gr aphs11
+highlights that a large portion of periodic cubic graph is symmetric [14]. Clearly, the two sets do not coincide. There
+are exactly seven different kind of entries in UN24(t). This can be seen as a 2 ×2 block matrix. The blocks (1 ,1) and
+(2,1) are below. The other blocks are just their rearrangements:
+a1a2a2a2a3a2a3a4a4a3a2a2
+a2a1a2a2a2a3a4a2a3a2a3a4
+a2a2a1a3a4a2a2a3a2a2a4a3
+a2a2a3a1a2a4a2a3a2a4a2a3
+a3a2a4a2a1a2a3a2a2a3a4a2
+a2a3a2a4a2a1a2a2a3a4a3a2
+a3a4a2a2a3a2a1a2a2a3a2a4
+a4a2a3a3a2a2a2a1a4a2a2a3
+a4a3a2a2a2a3a2a4a1a2a3a2
+a3a2a2a4a3a4a3a2a2a1a2a2
+a2a3a4a2a4a3a2a2a3a2a1a2
+a2a4a3a3a2a2a4a3a2a2a2a1
+and
+a5a5a6a5a6a7a7a7a7a7a6a7
+a5a6a5a7a7a5a6a7a7a7a7a6
+a6a7a7a7a5a5a7a6a6a7a7a5
+a7a5a7a6a5a6a7a5a7a6a7a7
+a7a7a6a7a6a5a5a5a7a7a6a7
+a7a7a5a6a7a6a5a7a5a6a7a7
+a6a5a5a7a7a7a7a6a6a5a7a7
+a7a7a6a7a6a7a7a7a5a5a6a5
+a7a5a5a7a7a7a6a5a7a5a5a6
+a5a6a5a6a7a6a7a7a7a6a5a5
+a6a7a5a5a7a7a5a6a6a7a5a7
+a7a7a5a5a5a7a6a7a5a7a7a6,
+where
+a1= (2+3cos( t)+6cos(2t)+cos(3t))/12,
+a2=−/parenleftBig
+(1+2cos(t))sin(t)2/parenrightBig
+/6,
+a3= 8/parenleftBig
+cos(t/2)2sin(t/2)4/parenrightBig
+/3,
+a4= 2/parenleftBig
+(1+2cos(t))sin(t/2)4/parenrightBig
+/3,
+a5=−i(sin(t)+4sin(2t)+sin(3t))/12,
+a6=isin(t)3/3,
+a7=−i(sin(t)−2sin(2t)+sin(3t))/12
+are all the the different entries of the unitary matrix. From these f unctions, we can write
+max
+t∈R+(|aj|) =
+
+= 1, j= 1 andt= 2π;
+≈1/2, j= 2 andt≈π/4;
+≈1/4, j= 3 andt≈6/5;
+= 2/3, j= 4 andt=π;
+≈9/20, j= 5 andt≈π/4;
+= 1/3, j= 6 andt=π/2;
+≈3/10, j= 7 andt≈3π/4.
+TheTutte-Coxeter graph ,TC, is illustrated in Fig. (5). This is the largest cubic graph giving a periodic dynamics.
+Its eigenvalues are/braceleftbig
+±3,±2[9],0[10]/bracerightbig
+. Let us describe how to write down the unitary matrix UTC. In Fig. (5) we12
+have ordered the vertices anticlockwise. Each even vertex is conn ected to odd vertices, and viz.As we have seen in
+previous examples, TCis bipartite and its unitary has a block structure:
+UTC=/bracketleftBigg
+A B
+B C/bracketrightBigg
+.
+Theij-th entry of the block Bis
+[B]i,j=/braceleftBigg
+−i(6sin(2t)+sin(3t))/15,{i,j} ∈E(TC);
+i(3sin(2t)−2sin(3t))/30,{i,j}/∈E(TC).
+The indexiruns over the odd numbers 1 ,3,...,29;jover the even ones, 2 ,4,...,30. For any chosen t, none of the
+entries ofBcan have unit absolute value. The blocks AandChave the same entries but rearranged:
+a1a2a3a3a2a3a2a2a3a2a3a3a3a3a2
+a2a1a2a3a3a2a3a3a3a2a3a2a2a3a3
+a3a2a1a2a3a2a3a2a3a3a3a3a3a2a2
+a3a3a2a1a2a3a3a2a3a2a2a3a2a3a3
+a2a3a3a2a1a2a3a3a2a3a3a3a2a3a2
+a3a2a2a3a2a1a2a3a2a3a2a3a3a3a3
+a2a3a3a3a3a2a1a2a3a3a2a3a2a3a3
+a2a3a2a2a3a3a2a1a2a3a3a2a3a2a3
+a3a3a3a3a2a2a3a2a1a2a3a2a3a3a3
+a2a2a3a2a3a3a3a3a2a1a2a3a3a2a3
+a3a3a3a2a3a2a2a3a3a2a1a2a3a2a2
+a3a2a3a3a3a3a3a2a2a3a2a1a2a3a2
+a3a2a3a2a2a3a2a3a3a3a3a2a1a3a3
+a3a3a2a3a3a3a2a3a2a2a3a3a2a1a2
+a2a3a2a3a2a3a3a3a3a3a2a2a3a2a1,
+where
+a1= (5+9cos(2 t)+cos(3t))/15,
+a2= (−5+3cos(2t)+cos(3t))/30,
+a3= 2/parenleftBig
+(7+8cos(t))sin(t/2)4/parenrightBig
+/15,
+and
+max
+t∈R+(|aj|)
+
+= 1, j= 1 andt= 2π;
+≈3/10, j= 2 andt≈1/4;
+≈13/50, j= 3 andt≈91/50.
+This last equation concludes the proof of the theorem. Notice that UTC(π) is 2×2 block diagonal: the diagonal
+entries are −13/15; the off-diagonal ones 2 /15. The two blocks are Grover matrices. When t=π/2, all entries in the
+off-diagonal blocks are equal to i/15.
+III. CONCLUSIONS
+We have shown that the 3-dimensional cube is the only periodic, conn ected cubic graph with PST. The proof is
+based on known results of graph theory and basic definitions of qua ntum dynamics on graphs. Our proof goes through
+the relevant cases, which can be interpreted as a systematical ex ploration of periodic quantum dynamics on cubic
+graphs. A necessary and sufficient condition for PST would give a sho rter, more elegant proof. However, any method
+for the same task requires the different cases. There is a small plus in the approach adopted here: we have written
+down explicitly the unitary matrices that specify the dynamics. The m atrices are potentially useful in further work.
+The well-known mathematical scenario of state transfer togethe r with various facts observed in this paper suggest
+some reflections. These may be worth a mention.13
+12345678
+9
+10
+11
+12
+13
+14
+15
+16
+17
+1819202122232412345678910
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+222324252627282930
+FIG. 5:(L): The Nauru graph. (R): The Tutte-Coxeter graph. This is the largest periodic cubi c graph. The Nauru graph and
+the Tutte-Coxeter graph do not have PST.
+A. Extremality
+The literature on interconnection networks for telecommunication s, parallel computers and distributed systems
+contain many optimization scenarios dealing with order/degree/diam eter [16]. The most famous one is perhaps the
+degree/diameter problem [20]: given natural numbers Dand ∆, find the largest possible number of vertices n∆,Din a
+graph of maximum degree ∆ and diameter ≤D. The graphs achieving the upper bound are known as Moore graphs .
+Interestingly, in the setting of PST, the original problem was someh ow the opposite one [13]. In fact, the main point
+does not seek to maximize the number of vertices without compromis ing reachability, but to perform long distance
+communication, with a minimum of physical resources. The number of particles equals the order of the graph and it
+is therefore a resource. The order/distance problems for state transfer are then a new scenario, where graphs with
+extremal properties can be defined via some parameter associate d to a dynamical process, instead of a topological
+condition. The landscape becomes even richer, if we consider more t han a single excitation, and lift the analysis to
+the graph powers defined in [4]. These considerations remind that t he problem of identifying extremal graphs with
+respect to state transfer is still a mostly unexplored area of rese arch.
+B. A notion of persistency
+As a generalization of the rule inducing a continuous-time random walk , the operator UG(t) =e−iA(G)thas been
+subject of intense study (see [21] and the references therein) . In analogy with random walks, given the importance
+of these objects for constructing distributions, a principal direc tion has pointed uniform sampling. Additionally, still
+from the quantitative side, it is useful to deal with problems related to single vertices, or pairs of vertices, more than
+the entire graph. For example, an instance of relevant parameter s is a quantum version of the hitting time. We would
+like to propose a notion that intends to quantify how the fidelity for s tate transfer between two vertices is constant
+in an interval, including small fluctuations. In the previous section, w e have seen that half of the diagonal entries
+ofUC6+K2(t) have the form a1= (2+cos( t)+2cos(2t)+cos(3t))/6. Whent∈ {π/2,3π/2},a1= 0. Roughly
+betweenπ/2 and 3π/2 there is a plateau: the value of the function is about 1 /3; it is exactly 1 /3, fort=π. Starting
+the process from some vertex i, if at time t=πwe sample from the distribution, we are going to obtain iwith
+probability 1 /9. This probability is not spicked, but fairly stable around π. Given a graph G, theǫ-persistency of
+a pair of vertices i,j∈V(G) is the length of the longest interval T⊂[0,2π] such that there exists a value kfor
+whichk−ǫ <|[UG(t)]i,j|< k+ǫ, withǫ≥0, for every t∈T. By looking at all vertices, one could define the
+maximum persistency or the average persistency, if extending the definition in the obvious ways. A process with
+higher persistency requires less clock precision for sampling. A first sight reason for giving attention to persistency
+could arise from a connection with energy transfer problems [12]. Le t us add an additional remark. Decoherence
+has been shown to behave as a natural smoothing mechanism on pro bability amplitudes [19]. Because of this fact,
+does decoherence increase persistency? With a similar scope but a o n a different line, is the presence of decoherence
+compatible with PST at all?14
+C. Discrete probability transfer
+The evolution governed by the Hamiltonian HXY(G) is driven by the exponentially smaller UG(t), when taking
+into account a single excitation only. While A(G) faithfully represents G, the matrix UG(t) does not preserve its
+topological structure. In other words, [ UG(t)]i,j/ne}ationslash= 0 does not imply [ A(G)]i,j= 1. A discrete evolution on Gcould be
+defined (in some cases [25]) by a unitary WG, with [WG]i,j/ne}ationslash= 0 if and only if [ A(G)]i,j= 1. Such a process does not
+describe the transfer of a single excitation, but it only allows to crea te a probability distribution supported by V(G).
+The process is discrete and it follows the iteration WGWt−1
+G/ma√sto−→Wt
+G. If there is t∈Nsuch that |/an}bracketle{tj|Wt
+G|i/an}bracketri}ht|= 1 then
+we haveperfect probability transfer from vertex ito vertexj. Unless we make use of some kind of lifting[2] (e.g., the
+introduction of extra degrees of freedom), this is the closest ana logue to the continuous object UG(t), even if this one
+does not always exist. For example, we can construct on K4the unitary below:
+WK4=
+0−1 1 1
+1 0−1 1
+1−1 0−1
+1 1 1 0
+.
+However, it is simple to observe that each power of WK4has one of the two zero-patterns
+
+0∗ ∗ ∗
+∗0∗ ∗
+∗ ∗0∗
+∗ ∗ ∗0
+or
+∗0∗ ∗
+0∗ ∗ ∗
+∗ ∗ ∗0
+∗ ∗0∗
+,
+where∗denotes a generic nonzero entry. This is sufficient to show that WK4does not give perfect probability transfer
+inK4. In more complicated situations, the zero-patterns of matrix pow ers are not immediately available to imply
+a general statement. To verify that a graph Gdoes not enjoy the property, we should study the spectra of unit ary
+matriceswiththesamezero-patternof WG. Sincetheprobleminvolvesbothspectra,zero-pattern,andano ptimization
+procedure, its flavour reminds of the matrix analysis questions app roached in [8] or various parametrizations coming
+from graph matrices [18]. In our context, the use of semidefinite pr ogramming techniques does not seem immediately
+useful, because the matrices are not stochastic, but in fact unita ry.
+Acknowledgments. I am supported by a Newton International Fellowship. I am gratefu l to Matthew Russell for
+finding an important error in a previous version of the paper and an a nonymous referee for valuable comments. This
+paper is dedicated to Anthony Sudbery in the occasion of his retirem ent.
+References
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+pp. 547-552.
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+Practice of Computer Science, LNCS ,4910(2008).
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+biology (Balatonlelle, 1996), Bolyai Soc. Math. Stud ,7, Janos Bolya math. Soc. Budapest (1999), 29-85.
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+arXiv:quant-ph/0512154v2
\ No newline at end of file
diff --git a/1001.0675.txt b/1001.0675.txt
new file mode 100644
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+++ b/1001.0675.txt
@@ -0,0 +1,514 @@
+arXiv:1001.0675v1  [math-ph]  5 Jan 2010Summation ofdivergentseries: Order-dependentmapping
+JeanZinn-Justin
+CEA,IRFUand Institut de Physique Th´ eorique, Centre deSac lay, F91191 Gif-sur-Yvette Cedex, France
+Abstract
+Summationmethodsplayaveryimportantroleinquantumfield theorybecauseallperturbationseriesaredivergent
+andthe expansionparameteris not always small. A numberof m ethodshave beentried in this context,most notably
+Pad´ e approximants, Borel–Pad´ e summation, Borel transfo rmation with mapping, which we briefly describe and one
+on which we concentrate here, Order-Dependent Mapping (ODM ). We recall the basis of the method, for a class of
+series we give intuitive argumentsto explainits convergen ceand illustrate its propertiesby several simple examples .
+Sincethemethodwasproposed,somerigorousconvergencepr oofsweregiven. Themethodhasalsofoundanumber
+ofapplicationsandwe shalllist afew.
+Keywords: Divergentseries; Summationmethods;Boreltransformatio n;Quantummechanics.
+1. The initialmotivation: Perturbativequantumfield theor y
+In quantum field theory, the main analytic calculation tool i s the perturbative expansion. As an illustration, we
+considerthe importantexampleof the φ4fieldtheory[24]. In thestatistical formulation,oneconsi dersthe Euclidean
+(orimaginarytime)action S, localfunctionalofthefield φ(x),x∈Rd,
+S(φ)=/integraldisplay
+ddx1
+2d/summationdisplay
+µ=1/bracketleftBig
+∂µφ(x)/bracketrightBig2+1
+2rφ2(x)+g
+4!φ4(x), (1)
+whererandgaretwoparameters. Tothisactionisassociatedafunctiona lmeasure e−S(φ)/Z,whereZisthepartition
+functiongivenbythefieldintegral
+Z=/integraldisplay
+[dφ]e−S(φ). (2)
+Thelimit d=0correspondstoa simpleintegral.
+Thecase d=1 correspondsto thequantumquarticanharmonicoscillator .
+Dimensions d>1correspondtoquantumfieldtheoryandtheexpression(1)is thensomewhatsymbolicsincethe
+theoryhastobemodifiedat shortdistancetoregularizeUV di vergencesandrenormalizedtocancelthem.
+In particular, the dimensions d=2,3 are especially relevant to classical statistical physics and the theory of
+phase transitions. Finally, d=4 is relevant to the theory of fundamental interactions at th e microscopic scale. The
+correspondingrelativistic quantumfieldtheoryispartoft heso-calledHiggsmechanism.
+Forthefieldtheory(1),theperturbativeexpansionamounts toanexpansioninpowersofthepositiveparameter g.
+Ford>1,thedifficultyofevaluatingthesuccessiveperturbativetermsincr easesveryrapidly. Moreover,questions
+like regularization and renormalization arise. Therefore , the calculation of renormalization group functions in the
+d=3 (φ2)2fieldtheory uporderg7[1]isa remarkableachievement.
+Emailaddress: jean.zinn-justin@cea.fr (Jean Zinn-Justin)
+Preprintsubmitted to Applied Numerical Mathematics September 1,20181.1. Largeorderbehaviourof perturbativeseries
+In theφ4field theory (1), g=0 corresponds to a singularity since the integral (2) is not d efined for g<0.
+Theperturbative series is divergent. Ford<4, the large order behaviour can be inferred from a steepest d escent
+calculationofthefieldintegral(2)[19][7]. Forthequarti canharmonicoscillator( d=1)theresultwasderivedearlier
+fromtheSchrdingerequation[3]. Foranyphysicalobservab lef, theresultshavethegeneralstructure
+fk∝
+k→∞(−1)kkbakk!, (3)
+whereadepends only on dandbis a half-integer that depends on the observable. The coe fficientA=1/ahas the
+value
+d=0 :A=3/2, (4)
+d=1 :A=8, (5)
+d=2 :A=35.10268957367896(1) (Zinn-Justin) , (6)
+d=3 :A=113.38350781527714(1) (Zinn-Justin) . (7)
+Ford=4,tothecontributioncomingfromthesteepestdescentcalc ulation,acontributionduetothelargemomentum
+singularitiesofFeynmandiagramshasin generaltobeadded .
+Finally,noticethatfor d<4,Borelsummabilityhasbeenproved.
+Similar results can be obtained for a number of quantum field t heories. When the formal expansion parameter
+is Planck’s constant, a divergence of the form (3) is in gener al found (except for some fermion theories), but the
+parameter amaybecomplex. Foran earlyreview,see [23].
+It follows from the large order behaviour analysis that, whe n the expansion parameter is not small, a summation
+oftheperturbativeexpansionis indispensable.
+1.2. Seriessummation
+In the study of the fundamentalinteractionsat the microsco picscale, it was realizedthat in the case of the strong
+nuclear force, unlike QED, the expansion parameter was larg e and, therefore, perturbation theory useless, leading
+manyphysicistseventorejectquantumfield theoryasaframe workto describesuchphenomena.
+Before the largeorderbehaviourwas evenknown,in [5] it was proposed,instead, to sum the perturbativeexpan-
+sion, using Pad´ e approximants and the idea was applied to a p henomenologicalmodel, the φ4field theory in d=4
+dimensions. Sinceonlytwoorthreetermscouldbecalculate d,thepossibleconvergenceofthePad´ esummationcould
+not be checked very well. However, the results obtained in th is way made much better physical sense than those of
+plainperturbationtheory. Fora reviewsee [22].
+In the seventies, one outstanding problem for which summati on methods was required, is the determination of
+critical exponentsand other critical quantitiesin the the ory of second order phase transitions. Following Wilson, fo r
+a whole class of physical systems, these quantities can be ob tained from the (φ2)2field theory in d=3 dimensions.
+One verifies immediately that the expansion parameter, the r enormalized interaction gr, is of order 1 and a series
+summationis required(wedonotdiscussherethe ε=4−dexpansion,buttheproblemisanalogous).
+To deal with the practical problem of series summation, a met hod was proposed based on Borel–Pad´ e approxi-
+mants[1]. With the knowledgeof the large orderbehaviour,a moreefficient method couldbe developed,combining
+a Borel transformation(actually Borel–Leroy)and a confor malmapping[17], [12], which we briefly present in next
+section. However, another method based only on the analytic properties of the series, the order-dependentmapping
+wasalsoinvestigated,whichwedescribeinmoredetailin se ction3(ageneralreferenceis[24]).
+2. Boreltransformationandconformalmapping
+The valuesof critical exponentsin a large class of continuo us(or second order)phase transitions can be inferred
+from so-called renormalization group (RG) functions of the (φ2)2quantum field theory. One important function is
+2Table1:
+SeriessummedbythemethodbasedonBoreltransformationan dmappingforthezero ˜g∗oftheRGβ(g)functionand
+theexponents γandνintheφ4
+3fieldtheory.
+k 2 3 4 5 6 7
+˜g∗1.8774 1.5135 1.4149 1.4107 1.4103 1.4105
+ν 0.6338 0.6328 0.62966 0.6302 0.6302 0.6302
+γ 1.2257 1.2370 1.2386 1.2398 1.2398 1.2398
+the RGβ-function whose zeros determine the RG fixed points. For exam ple, in the case of the φ4theory in d=3
+dimensions,Nickel [1]hascalculated
+˜β(˜g)=−˜g+˜g2−308
+729˜g3+0.3510695977˜ g4−0.3765268283˜ g5+0.49554751˜ g6
+−0.749689˜g7+O/parenleftBig
+˜g8/parenrightBig
+, (8)
+where ˜g=3gr/(16π) andgris the so-called renormalized interaction, related to the p arameter that appears in the
+action(1) by gr=g+O(g2) and
+β(gr)=16π
+3˜β(˜g)=−1
+dlng/dgr. (9)
+Theperturbativeexpansionis divergent (equation(3)). Forthe β-functioninthreedimensions,
+˜β(˜g)=/summationdisplay
+k˜βk˜gk,
+thelargeorderbehaviour,impliedbytheestimate(7),isgi venby
+˜βk∝
+k→∞(−a)kk7/2k!
+witha=0.147774232 ....
+Tocharacterizethelargedistancepropertiesofstatistic al systemsatthe phasetransition,onemustfirst determine
+thenon-trivialzero ˜ g∗oftheβ-functionandthencalculatevariousphysicalquantitiesl ikecriticalexponentsfor ˜ g=˜g∗.
+Onediscoversthat ˜ g∗is a numberof order1 and,thus, a numericaldeterminationfr omthe series 8 clearlyrequiresa
+summationofthe series.
+Inthreedimensions,theperturbativeexpansionisprovedt obeBorelsummable . Itisthusnaturaltointroducethe
+Borel–Laplacetransformation(here,Borel–Leroy):
+Bσ(g)=/summationdisplay
+kβk
+Γ(k+σ+1)gk,
+whereσisa freeparameter. Then,formallyin thesense ofpowerseri es
+β(g)=/integraldisplay+∞
+0tσe−tBσ(gt)dt.
+The function Bσ(g) is analytic in a circle of radius 1 /a. The series is said Borel summable if, in addition, Bσ(g) is
+analyticin aneighbourhoodoftherealpositivesemi-axisa ndtheintegralconverges.
+The series defines the function in a circle. It is thus necessa ry to perform an analytic continuation. In practice,
+with a small number of terms, the continuation requires a dom ain of analyticity larger than rigorously established.
+3Le Guillou and Zinn-Justin [17] have assumed maximal analyt icity,i.e., analyticity in a cut-plane. The continuation
+has then be obtained by a conformal mapping of the cut-plane onto a circle . Finally, variousmodificationshas been
+introducedto optimizethesummationmethod(fordetailsse e[17]).
+Further optimization of the summation technique and the add itional seven-loop contributions have led to new
+estimatesofcritical exponents[12]. Someresultsaredisp layedintable 1.
+3. Order-dependentmapping
+Theorder-dependentmapping(ODM)summationmethod[20] is basedonsomeknowledgeoftheanalyticprop-
+ertiesof the functionthat is expanded. It appliesboth to co nvergentanddivergentseries, althoughit is mainlyuseful
+inthelatter case.
+3.1. Thegeneralmethod
+Letf(z) beananalyticfunctionthathastheTaylorseriesexpansio n
+f(z)=/summationdisplay
+ℓ=0fℓzℓ.
+(the=signhastounderstoodinthesense ofseriesexpansion.)
+WhentheTaylorserieshasafiniteradiusofconvergence,toc ontinuethefunctioninthewholedomainofanalyt-
+icity,onecanmapthedomainontoa circle,whilepreserving theorigin.
+Divergent series: the intuitive idea. In a case of a divergen t series, one adds to the domain of analyticity a disk
+|z|<r of variable radius r and applies a similar mapping. Of course, the transformedse ries is still divergent. Then,
+onerecallstheempiricalrulethat,foradivergentseries, oneisinstructedtotruncatetheseriesatthetermofminima l
+modulus, the last term giving an order of magnitude of the err or. By adjusting the radius rorder by order, one can
+managetoset theminimumalwaysjust at thelast calculatedt erm.
+∝ma√sto−→−r −1
+Figure 1: Mapping z∝ma√sto→λ: example of afunction analytic in acut-plane.
+In what follows, we consider only functions analytic in a sec tor (as in the example of figure 1) and mappings
+z∝ma√sto→λoftheform
+z=ρζ(λ), ζ(λ)=λ+O/parenleftBig
+λ2/parenrightBig
+,
+whereζ(λ)isan explicitanalyticfunctionand ρanadjustableparameter .
+Althoughthetransformedseriesisstill divergentat ρfixed,we shallverifyonafewexamplesthat,byadjusting ρ
+orderbyorder (here,we limitourselvesto Borelsummableexamples) onecanconstructaconvergentalgorithm.
+Afterthetransformation, fisgivenbyaTaylorseriesin λoftheform
+f/parenleftbigz(λ)/parenrightbig=/summationdisplay
+k=0Pk(ρ)λk,
+4wherethecoefficientsPk(ρ)arepolynomialsofdegree kinρ. Sincetheresultisformallyindependentoftheparameter
+ρ,theparametercanbechosenfreely.
+Thek-th approximant f(k)(z) is constructed in the following way: one truncates the expa nsion at order kand
+choosesρas to cancel the last term. Since Pk(ρ) haskroots(real or complex),one choosesfor ρthe largest possible
+root(inmodulus) ρkforwhich P′
+k(ρ) issmall. Thisleadstoa sequenceofapproximants
+f(k)(z)=k/summationdisplay
+ℓ=0Pℓ(ρk)λℓ(ρk,z) with Pk(ρk)=0.
+Inthecaseofconvergentseries, it isexpectedthat ρkhasa non-vanishinglimit for k→∞. Bycontrast,fordivergent
+seriesit isexpectedthat ρkgoestozeroforlarge kas
+ρk=O/parenleftBig
+f−1/k
+k/parenrightBig
+.
+Theintuitiveideahereis that ρkcorrespondstoa ‘local’radiusofconvergence.
+Sinceρkgoestozero,thefunction ζ(λ)mustdivergeforafinitevalueof λ. Below,wechoose λ=1byconvention.
+Remark.Inthecaseofrealfunctions,whentherelevantzerosarecom plexitisoftenconvenienttochooseminima
+ofthepolynomials Pk,whichsatisfy
+P′
+k(ρk)=0,
+choosing, in general, the largest zero for which Pkis small. Other mixed criteria involving a combination of Pkand
+P′
+kcan also be used. Indeed, the approximant is not very sensitive to the precise value of ρk, within errors . Finally,
+Pk+1(ρk) givesanorderofmagnitudeofthe error.
+3.2. Functionsanalyticinacut-plane: Heuristic converge nceanalysis
+Although some rigorousconvergenceresults have been obtai ned [10], there are not optimal. Therefore, we give
+here heuristic but quantitative arguments that show the nat ure of the convergence of the ODM method. Following
+[20], to simplify we consider a real function analytic in a cu t-plane with a cut along the real negative axis (figure 1)
+anda Cauchyrepresentationoftheform
+E(g)=1
+π/integraldisplay0−
+dg′∆(g′)
+g′−g,
+butthegeneralizationissimple. Moreover,we assumethat
+∆(g)∝
+g→0−gbeA/g,A>0. (10)
+Thefunction E(g)canbeexpandedinpowersof g:
+E(g)=/summationdisplay
+kEkgkwithEk=1
+π/integraldisplay0−dg
+gk+1∆(g).
+Theassumption(10) thenimpliesalargeorderbehaviour
+Ek∝
+k→∞(−A)−kΓ(k−b)∼(−A)−kk−b−1k!,
+exactlyoftheformdisplayedinsection1.1. We introduceth e mapping
+g=ρλ
+(1−λ)α, α>1. (11)
+TheCauchyrepresentationthencanbewrittenas
+E/parenleftbigg(λ)/parenrightbig=1
+π/integraldisplay0−
+dλ′∆/parenleftbigg(λ′)/parenrightbig
+λ−λ′+R(λ),
+5whereR(λ)isa sumofcontributionsfromcutsatfinite distancefromth eorigin. We expand
+E/parenleftbigg(λ)/parenrightbig=/summationdisplay
+kPk(ρ)[λ(g)]k(12)
+with
+Pk(ρ)=1
+π/integraldisplay0−
+dλ∆/parenleftbigg(λ)/parenrightbigλ−k−1+finitedistancecontributions. (13)
+Fork→∞,thefactorλ−kfavourssmallvaluesof λbutfortoosmallvaluesof λtheexponentialdecayof ∆/parenleftbigg(λ)takes
+over. Thus, Pk(ρk) can be evaluatedbythe steepest descentmethod. With the An satz that at the saddlepoint λ<0 is
+independentof kand
+ρk∼R/k,R>0,
+which implies g(λ)→0,∆(g) can be replaced by its asymptotic form (10) for g→0−. At leading order, the saddle
+pointequationreducesto
+d
+dλ/parenleftbiggA
+Rλ(1−λ)α−ln|λ|/parenrightbigg
+=0. (14)
+Inwhatfollows,we set R/A=µ, since thisisthe onlyparameter. Theequationcanberewrit tenas
+µ+1
+λ(1−λ)α−1/parenleftbig(α−1)λ+1/parenrightbig=0.
+If the mapping (11) does not cancel all singularities (and th is excludes the case of the integral of section 4), then
+Pk(ρk) cannotdecreaseexponentiallywith k. Thisimpliesanotherequation
+1
+λ(1−λ)α−µln|λ|=0. (15)
+This is indeed the region where the contribution coming from the cut at the origin and from the other finite distance
+singularitiesarecomparableandwherethezerosof Pk(ρ) canlie.
+Returningtothe expansion(12), at gfixed,fromthebehaviourof ρkwe infer
+1−λ∼(R/kg)1/α⇒λk∼e−k1−1/α(R/g)1/α.
+Inagenericsituation,wethenexpect Pk(ρk)tobehavelike
+Pk(ρk)=O(eCk1−1/α)
+andthedomainofconvergencedependsonthesignoftheconst antC. ForC>0,thedomainofconvergenceis
+|g|<RC−α[cos(Argg/α)]α.
+Forα >2, this domain extends beyond the first Riemann sheet and requi res analyticity of the function E (g)in the
+correspondingdomain.
+ForC<0,thedomainofconvergenceisthe unionofthe sector |Argg|<πα/2andthedomain
+|g|>R|C|−α[−cos(Argg/α)]α.
+Againforα>2,thisdomainextendsbeyondthefirst Riemannsheet.
+3.3. Examples
+Forα=3/2,combiningequations(14)and(15),onefinds
+µ=4.031233504 , λ=−0.2429640300 .
+6Table2:
+α 3/2 2 5/2 3 4
+µ 4.031233504 4.466846120 4.895690188 5.3168634291 6.1359656420
+−λ0.2429640300 0.2136524524 0.1896450439 0.1699396648 0.14003129119
+Forα=2,equation(14) becomes
+λ2−µλ−1=0⇒λ=1
+2(µ−/radicalBig
+µ2+4).
+Forµ=3.017759126 ...onerecoverstheexponentialrate ofconvergence(16).
+Inthecase ofadditionalsingularities,withthe additiona lequation(15),oneobtains
+µ=4.466846120 ..., λ=−0.2136524524 ....
+Togivea fewotherexamples,againcombiningequations(14) and(15)onefindstheresultsdisplayedintable2.
+4. Application: The simple integral d=0
+Forr=1,theintegral(2)inthe case d=0reducesto thesimpleintegral
+Z(g)=1√
+2π/integraldisplay
+dxe−x2/2−gx4/4!,
+andtheconvergenceoftheODMmethodcanbestudiedanalytic ally.
+4.1. Theoptimalmapping
+Analyticpropertiessuggestthat theoptimalmappingisgiv enbysetting
+g=ρλ
+(1−λ)2andZ(g)=(1−λ)1/2f(λ).
+Then,fhasanexpansionoftheform
+f(λ)=/summationdisplay
+kPk(ρ)[λ(g)]k.
+Convergencecanbestudiedanalytically. First, fhastherepresentation
+f(λ)=1√
+2π/integraldisplay
+dse−s2/2+λ(s2/2−ρs4/24)=Pk(ρ)[λ(g)]k
+with
+Pk(ρ)=1
+k!1√
+2π/integraldisplay
+dse−s2/2(s2/2−ρs4/24)k.
+Setting
+s2/2=kt, ρ=R/k,
+onecanrewritetheexpressionas
+Pk(ρ)=kk+1/2
+k!/radicalbigg
+1
+π/integraldisplaydt√te−kt(t−Rt2/6)k.
+7Table3:
+ODM for the integral d =0for g→ ∞: Z(g)∼g−1/4(1/2)∗241/4∗√π/Γ(3/4). We define [Z(g)g1/4]exact−
+[Z(g)g1/4]ODM=δ
+k 5 10 15 20 25 30
+1/ρ 1.131726 2.35036 3.34050 4.5594 5.5495 6.8614
+−δ 5.7∗10−32.5∗10−53.7∗10−62.2∗10−83.4∗10−95.1∗10−11
+ln|δ|−5.1578−10.5921−12.5008−17.5923−19.4855−23.6818
+k 35 40 45 50 55 60
+1/ρ 7.7586 8.9778 9.9678 11.1869 12.1769 13.3958
+−δ3.5∗10−122.5∗10−143.9∗10−152.9∗10−174.5∗10−183.4∗10−20
+ln|δ|−26.3535−31.2859−33.1625−38.0643−39.9364−44.8208
+Fork→∞,theintegralcanbeevaluatedbythesteepest descentmetho d. Thesaddlepointequationis
+Rt2/6−(1+R/3)t+1=0
+and,thus,
+t=3
+R/parenleftBig
+1+R/3±/radicalbig
+1+R2/9/parenrightBig
+.
+Forkodd,thezerocorrespondsto acancellationbetweenthetwos addlepoints. Thisyieldstheequation
+e2√
+R2+9/R=√
+R2+9+R√
+R2+9−R⇔e√
+R2+9/R=1
+3/parenleftbigg/radicalbig
+R2+9+R/parenrightbigg
+.
+Asexpected,onefinds
+ρk∼R
+kwithR=4.526638689 ...(R/A=3.017759126 ...).
+Theminimumisgivenbyoneofthesaddlepoints
+Pk(ρ)∝e−3k/R=(0.5154353381 ...)k. (16)
+Atgfixed,λconvergesto1. Moreprecisely,
+λ=1−/radicalbig
+R/kg+O(1/k)⇒λk∼e−√
+Rk/g.
+The approximantsconverge geometrically on the entire Riem ann surface , a situation possible only because the func-
+tionZ(g)hasnoothersingularityat finitedistance.
+Numerical verifications. With about 60 terms, the slope is found to be 1 /kρk≈0.2209 in agreement with the
+prediction1 /R=0.2209(once even–oddorder oscillations are taken into accou nt). The logarithmof the error has a
+slope0.696/0.685tobecomparedwith theprediction3 /R=0.66(seetable 3).
+4.2. Analternativemapping
+Anothermappingthat alsoregularizesthepointat infinityi s
+g=ρλ
+(1−λ)4.
+8Table4:
+ODMfortheintegrald =0forg=5: we define [Z(g)]exact−[Z(g)]ODM=δ
+k 5 10 15 20 25 30
+1/ρ 0.5918 1.0297 1.5627 2.0779 2.5865 3.1376
+|δ|1.1∗10−33.7∗10−51.7∗10−69.2∗10−71.1∗10−73.5∗10−9
+ln|δ|−6.7454−10.2069−13.2837−13.8898−16.0103−19.4614
+k 35 40 45 50 55 60
+1/ρ 3.6167 4.1877 4.6557 5.3021 5.6959 6.2458
+−δ3.4∗10−98.0∗10−101.0∗10−102.9∗10−111.2∗10−112.4∗10−12
+ln|δ|−19.4706−20.9453−22.9796−24.2450−25.0907−26.7433
+Numericalresultsfor g=5 aredisplayedintable4. Finally,for k=65,70,
+1/ρk=6.7479,7.5724,ln|δ|=−30.3657,−28.9505
+Ontheaveragebetween k=5andk=70,
+ρk∼R/kwithR=9.75.
+Theresultsdisplayedintable2 leadtotheprediction
+R=9.2039.
+Theerrorisabout
+δ=−1.59k3/4
+g1/4
+and1.59hastobecomparedwith theexpectedasymptoticvalue1 .74if oneassumesconvergenceforall g>0.
+5. The quarticanharmonicoscillator: d=1
+Forr=1,thepathintegralcorrespondstothe quantumHamiltonian
+H=1
+2p2+1
+2x2+g
+4!x4.
+Theeigenvalues EofHaregivenbythesolutionofthetime-independentSchrdinge requation
+−1
+2ψ′′(x)+/parenleftbigg
+1
+2x2+g
+4!x4/parenrightbigg
+ψ(x)=Eψ(x),
+whereψ(x) isasquare-integrablefunction.
+As an example, we consider the perturbativeexpansion of the lowest eigenvalue, the groundstate energy. Varia-
+tionalargumentsandscalingsuggestthemapping
+g=ρλ
+(1−λ)3/2,E=E
+(1−λ)1/2. (17)
+Then,
+E=/summationdisplay
+kPk(ρ)[λ(g)]k.
+9Largeorderbehaviour(section1.1)anda steepestdescente valuation(table2)leadtothe prediction
+ρk∼R/kwithR=µA=32.25....
+Then,λconvergesto1as
+λ=1−/parenleftBiggR
+kg/parenrightBigg2/3
++O(k−4/3)⇒λk∼e−R2/3k1/3/g2/3withR2/3=10.131....
+Anunbiasedfit ofthenumericaldata for k≤60yieldsresultswithin10%ofthepredictedvalues. Finall y,afit ofthe
+relativeerrorfor g→∞yields[20]
+Pk+1(ρk)∝e−9.6k1/3.
+Therelativeerrorat order kisthusoforder e−k1/3(9.6+R2/3g−2/3). Onefindsconvergencefor
+0.95+|g|−2/3cos/parenleftBig2
+3Argg/parenrightBig
+>0.
+Thecorrespondingdomaincontainsa sectionofthe first Riem annsheetandextendsto thesecondRiemannsheetfor
+|g|largeenough.
+6.φ4field theoryin d=3dimensions
+In[20],theODMmethodhasbeenappliedonfunctionsofthein itialparameter goftheaction(1)rathertherenor-
+malizedparameter grintroducedin section2. Thenthe pointof physicalinterest isg→∞, whichcorrespondsto the
+zero ˜g∗oftheβ-function(8). DuetoUVdivergences,aneededregularizati onandrenormalization,scalingarguments
+are no longer applicable to determine an appropriate mappin g. The relation (9) between initial and renormalized
+parameter shows that, for g→∞, physical observables have an expansion in powers of g−ω, where the exponent
+ω=˜β′(˜g∗). Thisthensuggeststhe mapping
+g=ρλ
+(1−λ)1/ω, (18)
+but the difficulty is that ωhas to be inferred from the series (8) itself. The results obt ained in this way [20] are
+consistentwith thoseobtainedin[12]( ω=0.80(1)fromBoreltransformationandmapping),butempirica lerrorsare
+moredifficultto assess. Also theexpectedrateofconvergenceisofor dere−const.k1−ω=e−const.k0.2,whichis ratherslow
+(see table 5). See reference[20] for details. Finally,the i nformationabout the largeorderbehaviourcannoteasily be
+incorporated.
+Table5:
+Seriesfortheexponent ωsummedbyODM inthe φ4
+3fieldtheorywith ωin=0.79anddωcal./dωin=−0.6.
+k 2 3 4 5 6
+ωk 0.552 0.754 0.711 0.767 0.759
+Here, to illustrate the flexibility of the method, we work dir ectly with functions of ˜ g. We also take into account
+thecovarianceofthe β-functionsundera changeofparametrization:
+β1(g1)=dg1
+dg2β2(g2). (19)
+This transformation law is such that the derivative of the β-function at a zero (a fixed point), which is a physical
+observable,remainsunchanged.
+10Equation(19) suggestsamappingoftheform
+˜g=ρ/parenleftBigg1
+(1−λ)α−1/parenrightBigg
+,
+with a suitable choice of the parameter α, since unlike a mapping of the form (11), it introduces no new singularity.
+We thusset
+βλ(λ)=(1−λ)α+1
+αρ˜β/parenleftbig˜g(λ)/parenrightbig.
+A few trials, withouttryingto optimize,suggest thevalue α=3/2and thisis the valuewe haveadopted. Theresults
+forthe zero ˜ g∗, for comparisonwith the methodoutlinedin section 2, andth e exponentω=β′
+λ(λ∗) are givenin table
+6. Theorderofmagnitudeoftheerrorscanonlybeestimatedb ythesensitivitytotheprecisechoiceoftheparameter
+ρ(zero of last term or its derivative, for example). The indic ations are∆˜g∗≈∆ω≈0.006. In the case of complex
+zeros,we havegivenonlytherealpartinthe tables.
+Table6:
+Resultsfor the zero ˜g∗of theβ-functionand the exponent ωsummed by ODM in the φ4
+3field theory. With the method
+ofsection2([12], onefinds ˜g∗=1.411±0.004andω=0.799±0.011.
+k 3 4 5 6 7
+˜g∗1.09871 1.39330 1.41771 1.41737 1.41744
+ω 1 0.7984 0.7804 0.7806 0.7807
+Othercriticalexponentsare obtainedfromthe twoRG functi ons
+γ−1(˜g)=1−1
+6˜g+1
+27˜g2−0.0230696212˜ g3+0.0198868202˜ g4−0.0224595215˜ g5
++0.0303679053˜ g6−0.046877951 g7+O(g8),
+η(˜g)=0.0109739368˜ g2+0.0009142222˜ g3+0.0017962228 g4−0.0006537035˜ g5
++0.0012749100˜ g6−0.001697694˜ g7+O(g8),
+by setting ˜ g=˜g∗. We have also summed the series for γ−1(˜g) andη(˜g)/˜g2, and independently the series for the
+functionν−1(˜g) although it is related to the γandηbyγ(˜g)=ν(˜g)(2−η(˜g)). A verification of this relation after
+summationgivesanindicationabouttheerrors. Theresults , displayedintable 7, canbe comparedwith theresultsof
+[12]:
+γ=1.2396±0.0013, ν=0.6304±0.0013, η=0.0335±0.0025.
+Onenoticesareasonableconsistency.
+Table7:
+Critical exponents γ=γ(˜g∗),ν=ν(˜g∗)andη=η(˜g∗)for˜g∗=1.411.
+k 3 4 5 6 7
+γ 1.23717 1.23486 1.23845 1.23820 1.23923
+ν 0.62521 0.62486 0.62746 0.62771 0.62865
+η 0.0290 0.0289 0.0297 0.0306
+117. Convergenceproofsandphysicsapplications
+Since the ODM summationmethodhasbeenproposed,convergen cehasbeenprovedin specific cases[10],how-
+ever,to ourknowledge,systematicmathematicalinvestiga tionsarestill lackingandwouldbemostwelcome.
+Noticethatin[8]aspecialcaseoftheODMmethodhasalsobee nproposed. Moreover,methodsknownas linear
+orscaleddeltaexpansion ,oroptimizedperturbationtheory,introducedlater,ofte nreducetospecialcasesoftheODM
+method(see, forexample,[9],[4]).
+The ODM method has also found a numberof useful applications in physics (sometimesunder di fferent names).
+In[18],theproblemofthehydrogenatominstrongmagneticfi eldshasbeenconsidered. Thesummationoftheweak
+field series expansion has led to precise determinations of t he ground state energy for very strong fields. In [13],
+the location of the Bender–Wu singularities of the quartic a nharmonic oscillator has been determined numerically
+(for the anharmonic oscillator see also [2]). In [11], the eq uation of state of physical systems belonging to the Ising
+modeluniversalityclasshasbeendeterminednumericallyb yacombinationofBoreltransformationandmappingand
+the ODM summation method. Anotherapplication has been the B ose–Einstein condensationproblem [21], [14] (for
+similarproblemssee also[6]). Finally,let usalsomention theapplicationtothe Gross–Neveumodel[15],[16].
+References
+[1] G.A. Baker, B.G. Nickel, M.S. Green, D.I. Meiron, Ising- Model Critical Indices in Three Dimensions from the Callan- Symanzik Equation,
+Phys.Rev. Lett. 36 (1976) 1351-1354.
+[2] B. Bellet, P. Garcia, A. Neveu, Convergent sequences of p erturbative approximations for the anharmonic oscillator I, Int. J. Mod. Phys. A 11
+(1996) 5587-5606.
+[3] C.M.Bender, T.T.Wu,Anharmonic Oscillator. II. AStudy of Perturbation Theory in Large Order, D 7(1973) 1620-1636.
+[4] C. M. Bender, A. Duncan, H. F. Jones, Convergence of the op timizedδexpansion for the connected vacuum amplitude: Zero dimensi ons,
+Phys.Rev. D49 (1994) 4219-4225.
+[5] D. Bessis, M. Pusteria, Unitary Pad´ e approximants in st rong coupling field theory and application to the calculatio n of theρ- andf0-meson
+Regge trajectories, IlNuovo Cimento A(1965-1970) 54 (1968 ) 243-294.
+[6] E.Braaten, E.Radescu,Convergence oftheLinear δExpansionintheCritical O(N)FieldTheory,Phys.Rev.Lett.89(2002)271602[4pages].
+[7] E.Br´ ezin, J.C.LeGuillou, J.Zinn-Justin, Perturbati on theory atlarge order. I.The ϕNinteraction, Phys. Rev. D15 (1977) 1544-1557.
+[8] W.E.Caswell, Accurate energy levels fortheanharmonic oscillator andasummableseries forthedouble-well potent ial in perturbation theory,
+Ann.Phys.123 (1979) 153-184.
+[9] A.Duncan, H. F.Jones, Convergence proof for optimized δexpansion: Anharmonic oscillator, Phys. Rev. D47 (1993) 25 60-2572.
+[10] RGuida, KKonishi,HSuzuki,Improved Convergence Proo foftheDeltaExpansion andOrderDependent Mappings,Ann.P hys.249(1996)
+109-145.
+[11] R.Guida, J.Zinn-Justin, 3DIsing Model: TheScaling Eq uation ofState, Nucl. Phys.B 489 (1997) 626-652.
+[12] R.Guida, J.Zinn-Justin, Critical exponents of the N-vector model, J.Phys.A 31 (1998) 8103-8121.
+[13] H.Kleinert, W.Janke, Convergence behavior of variati onal perturbation expansion —A method for locating Bender- Wu singularities, Phys.
+Lett. A206 (1995) 283-289.
+[14] J.L.Kneur, M.B.Pinto, R.O.Ramos,Asymptotically imp roved convergence ofoptimized perturbation theory in theB ose-Einstein condensa-
+tion problem, Phys.Rev. A 68 (2003) 043615 [30 pages].
+[15] J. L. Kneur, M. B. Pinto, R. O. Ramos Critical and tricrit ical points for the massless 2D Gross–Neveu model beyond lar geN, Phys. Rev. D
+74 (2006) 125020 [18 pages].
+[16] J. L. Kneur, M. B. Pinto, R. O. Ramos, E. Staudt, Emergenc e of tricritical point and liquid-gas phase in the massless 2 +1 dimensional
+Gross–Neveu model, Phys.Rev. D76 (2007) 045020 [19 pages].
+[17] J.-C.LeGuillou, J.Zinn-Justin, Critical exponents f rom field theory, Phys.Rev. B 21 (1980) 3976-3998.
+[18] J.-C.LeGuillou, J.Zinn-Justin, Thehydrogen atom in s trong magnetic fields: summation oftheweak field series expa nsion, Ann.Phys.147
+(1983) 57-84.
+[19] L.N.Lipatov, Divergence ofthe perturbation-theory s eries and pseudoparticles, JETPLett. 25 (1977) 104-107.
+[20] R.Seznec, J.Zinn-Justin, Summation of divergent seri es by order dependent mappings: Application to the anharmon ic oscillator and critical
+exponents in field theory, J.Math. Phys.20 (1979) 1398-1408 .
+[21] F. de Souza Cruz, M.B. Pinto, R.O. Ramos, P. Sena, Higher -order evaluation of the critical temperature for interact ing homogeneous dilute
+Bosegases, Phys.Rev. A 65 (2002) 053613 [24 pages].
+[22] J.Zinn-Justin, Strong interactions dynamics with Pad ´ eapproximants, Phys.Rept. 1 (1971) 55-102.
+[23] J.Zinn-Justin, Perturbation series at large orders in quantum mechanics and field theories: application to the pro blem of resummation, Phys.
+Rept. 70 (1981) 109-167.
+[24] J.Zinn-Justin, Quantum Field Theory and Critical Phen omena: fourth edition, Oxford University Press,Oxford, 20 02.
+12
\ No newline at end of file
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--- /dev/null
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@@ -0,0 +1,109 @@
+arXiv:1001.0676v1  [astro-ph.HE]  5 Jan 2010Spectral/timing evolution of black-hole binaries
+TomasoM. Belloni
+INAF– OsservatorioAstronomicodi Brera,ViaE. Bianchi46, I-23807,Merate,Italy
+Abstract. I briefly outline the state-paradigm that has emerged from th e study of black-hole
+binarieswithRossiXTE.Thisisthestartingpointofanumbe rofstudiesthataddresstheconnection
+betweenaccretionand jet ejection andthe physicalnatureo f thehard spectralcomponentsin these
+systems.
+Keywords: Accretionandaccretiondisks– blackholebinaries
+PACS:97.10.Gz,04.25.dg
+STATESANDSTATE-TRANSITIONS
+AlmostfourteenyearsintothelifeoftheRossiXTEmission( atthetimeofwriting),the
+existingdatabaseofoutburstsoftransientblack-holebin ariesisveryrichandconstitutes
+a unique laboratory to understand accretion onto black hole s. The dense coverage and
+high-flexibility of RossiXTE are indeed unique and not likel y to be matched by new
+missions in the near future. A large amount of theoretical an d observational work has
+focused on the hard and soft states, in particular regarding the energy spectrum (see
+e.g. Gilfanov 2009). In the hard state (LHS), the energy spec trum shows a clear high-
+energy cutoff around ∼50–100 keV, while in the soft state (HSS) a cutoff was not
+observed up to ∼1 MeV (Grove et al. 1998). These spectra are interpreted as th e result
+of Comptonization from a combination of thermal and non-the rmal electrons (see e.g.
+Ibragimov et al. 2005). Different theoretical models diffe r in the details of the physical
+locationand combinationofthesecomponents.
+However,amoredynamicviewofthetimeevolutionoftheoutb urstsoftheseobjects
+is now possible. From this, it is possible to identify a small number of states, which in
+additiontothehardandsoftones,includeintermediatesta tes.Thesearerelativelyshort-
+livedandmarktransitionsbetweenhardandsoftandvice-ve rsa.Theirimportanceliesin
+anumberofaspects.First,thestudyofthetransitionbetwe entwoverydifferentstatesas
+thehardandsoftstate,whicharecharacterizedbyqualitat ivelydifferentenergyspectra,
+providesthe best opportunityto establishtheir nature. In simplewords: ifthe transition
+is between a thermal or hybrid Comptonization spectrum and a non-thermal spectrum,
+what happens in between? Second,it has been shown that some o f these transitions
+correspondtochangesintheinfrared/radioemission(seee .g.Homanetal.2005;Coriat
+et al. 2009) or to te ejection of relativistic jets (Fender, B elloni & Gallo 2004; Fender,
+Homan&Belloni2009;Fender 2009;Gallo2009).
+The basic tools used for RossiXTE data for approaching the ev olution of the X-
+ray emission of transient black hole binaries in outburst ar e the Hardness-Intensity
+Diagram (HID), where the total count rate is plotted as a func tion of hardness and
+the Hardness-Rms Diagram (HRD), where the fractional rms, i ntegrated over a broadFIGURE 1. Top panel: sketch of the main regions of the Hardness-Intens ity Diagram, corresponding
+todifferentsourcesstatesasidentifiedthroughvariabili typarameters.Thepathis anexampleofoutburst
+evolution. The black segment of the path correspondsto the i nterval covered by GX 339–4 in 2006 (see
+Motta,Homan&Belloni2009)andFig.2. Forthe definitionofQ POline,see text.
+rangeoffrequencies,isplottedversushardness.Bothdiag ramsareproducedwithX-ray
+photonsintheapprocimate3–20keV range. Youcan seesevera lexamplesinBelloni et
+al. (2005), Homan & Belloni (2005), Fender, Homan & Belloni ( 2009), Belloni (2009),
+Dunn et al. (2009). A schematic picture with the typical path of an outburst and the
+states and state transitions is shown in Fig. 1. The four stat es are Low/Hard (LHS),
+Hard-Intermediate (HIMS), Soft-Intermediate (SIMS) and H igh-Soft (HSS) (adapted
+fromBelloni2009).Theverticallinesmarkthestatetransi tions.
+State transitionsare sharp: althoughobviouslythe 3–20 ke V energy spectrum cannot
+be much different when crossing the vertical lines, detaile d analysis of the properties
+of fast aperiodic variability shows that indeed marked chan ges take place there (see
+Belloni 2009). Therefore, the positionof thetransition li nes is not at all arbitrary, but is0.10.20.30.40.50.60.70.80.940100300
+HardnessEcutoff (keV)
+FIGURE 2. High-energycutoff as a function of spectral hardnessfor th e 2006 data of GX 339–4 (see
+Motta et al. 2009). The lines mark state transitions (thick: LHS-HIMS; dashed: HIMS-SIMS, the QPO-
+line;dotted:HIMS-SIMS).
+fixedbyobservations.Particularlyimportantisthetransi tionbetweenHIMSandSIMS,
+markedasQPOlineinFig.1asitinvolvestheappearanceofap articularQuasi-Periodic
+OscillationassociatedwiththeSIMS(seeBelloni2009;Cas ella,Belloni&Stella2005).
+CONNECTION TOJETEJECTION
+Inthepastfewyears,thankstoanincreasedobservationalr adio/X-rayactivity,apicture
+of the connection between accretion and ejection is also eme rging (see Fender 2009;
+Gallo 2009). In particular, it is now clear that while the LHS is associated to compact
+radio jets and the HSS to quenched radio emission, the major j et ejections take place
+duringintermediatestates.Fender, Belloni&Gallo(2004) identifiedtheQPO-linewith
+thepositioncorrespondingto theejectionoffast jets(the “Jet-line”). However,recently
+Fender, Homan & Belloni (2009) found that this association i s not perfect nor causal:
+thetwo linesarecrossed withina coupleofweeks fromeach ot her,in eithersequence.
+HIGH-ENERGY COMPONENTS
+Although across transitions the energy spectrum does not ch ange appreciably, once
+energies above 20 keV are considered the picture changes. In particular, recent results
+on GX 339–4 have shown that the high-energy cutoff of the hard spectral componentchangesinnon-monotonicwayacrossthetransition(Mottae tal.2009).Figure2shows
+the evolution of this parameter as a function of hardness, ov er the path sketched in
+black in Fig. 1. in the LHS, the high-energy cutoff decreases from 120 to 60 keV
+as the sources brightens (and softens), then the trend is rev ersed and in the HIMS
+there is a marked increase back to ∼100 keV. After the QPO-line, the cutoff energy
+is high, possibly not detected significantly (see also Motta et al., this volume). This
+evolution is rather complex. Although the LHS behaviour is s imple to understand,
+as the simultaneous increased soft emission cools the elect rons responsible for the
+Comptonization. However, the trend reversal is still puzzl ing. The same behaviour was
+shownby othersources (seee.g. Joinetet al.2008;Mottaet a l. 2009).
+CONCLUSIONS
+Wenowhaveaclearphenomenologicalpictureoftheevolutio nofblack-holetransients,
+which can be compared to that of neutron-star binaries and ac tive galactic nuclei (see
+Belloni 2009). It is clear that state-transitions hold the k ey to a deeper understanding.
+Although they are transient and difficult to observe, the dat a already existing show that
+thespectral evolutioniscomplexandstillneeds tobeunder stood.
+REFERENCES
+1. T.M.Belloni,“Statesandtransitionsinblack-holebina ries,”inTheJetParadigm-FromMicroquasars
+toQuasars,Lect. NotesPhys.794 ,editedbyT.Belloni,Springer,2009,inpress(arXiv:0909 .2474)
+2. T..M.Belloni,J. Homan,P.Casella, et al., A&A,2005,440,207
+3. P.Casella, T.M.,Belloni,L. Stella, ApJ,2005,629.403
+4. M.Coriat,S., Corbel,M.M. Buxton,etal., MNRAS,2009,400,123
+5. R. Dunn,R. P.Fender,E.Koerding,et al., MNRAS,2009,inpress(arXiv:0912.0142)
+6. R. P. Fender, “’Disc-jet’ coupling in black hole X-ray bin aries and active galactic nuclei,” in The Jet
+Paradigm - From Microquasars to Quasars, Lect. Notes Phys. 7 94, edited by T. Belloni, Springer,
+2009,in press(arXiv:0909.2572)
+7. R. P.Fender,T..M.Belloni,E.Gallo, MNRAS,2004,355,1105
+8. R. P.Fender,J. Homan,T..M.Belloni, MNRAS,2009,396,1370
+9. E. Gallo, “Radio emission and jets from microquasars,”in The Jet Paradigm - From Microquasars to
+Quasars,Lect.NotesPhys.794 ,editedbyT.Belloni,Springer,2009,in press(arXiv:0909 .2585)
+10. M. Gilfanov, “X-ray emission from black-hole binaries, ” inThe Jet Paradigm - From Microquasars
+toQuasars,Lect. NotesPhys.794 ,editedbyT.Belloni,Springer,2009,inpress(arXiv:0909 .2567)
+11. J. E.Grove,W. N.Johnson,R. A.Kroeger,et al., ApJ,1998,500,899
+12. J. Homan,T..M.Belloni, Ap&SS,2005,300,107
+13. J. Homan,M.Buxton,S. Markoff,etal., ApJ,2005,624,295
+14. A.A. Ibragimov,J. Poutanen,M.Gilfanov,et al., MNRAS,2005,362,1435
+15. A.Joinet,E.Kalemci,F. Senziani, ApJ,2008,679,655
+16. S.Motta,T..M. Belloni,J.Homan, MNRAS,2004,inpress(arXiv:0908.2451)
\ No newline at end of file
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@@ -0,0 +1,138 @@
+SPATIAL AND FREQUENCY DEPENDENCIES OF LOCAL PHOTORESPONSE OF 
+HTS STRIP-LINE RESONATOR IN THE REGIME OF TWO-TONE MICROWAVE 
+INTERMODULATION EXCITATION  
+ 
+ 
+Alexander P. Zhuravel 
+B. Verkin Institute for Low temperature Physics & Engineering, NAS of Ukraine 
+Address: 47 Lenin Avenue, Kharkov, 61103, Ukraine 
+Tel.: +38(057)341-0907, Fax: +38(057)340-3370, E-mail: zhuravel@ilt.kharkov.ua  
+Steven M. Anlage 
+Center for Nanophysics and Advanced Materials, Department of Physics, University of Maryland 
+Address: College Park, MD 20742-4111 USA 
+Tel.: (301) 405-7321, Fax: (301) 405-3779, E-mail:anlage@umd.edu 
+Alexey V. Ustinov 
+Physikalisches Institut, Karlsruhe  Institute of Technology (KIT) 
+Address: Wolfgang-Gaede-Str. 1, D-76131, Karlsruhe, Germany 
+Tel.: +49-721-608-3455, Fax: +49-721-608-6103, E-mail: ustinov@physik.uni-karlsruhe.de 
+  
+High-T
+c superconducting (HTS) films have been widely a pplied in passive microwave electronics including 
+high-Q resonators for mobile telephony. One of the crucial problems in applying such HTS devices remains their nonlinear (NL) response with respect to pumping power. This manifests itself as global  intermodulation product 
+distortion (IMD) when two or more high-frequency (HF) signals are NL mixed in the resonant frequency band. A predominately local  origin of NL sources through global  NL media motivates the development of spatially 
+resolved methods to identify the dominant mechanisms of NL response. We have previously shown [1] that by using different imaging contrast modes of low-temperat ure (LT) laser scanning microscopy (LSM) one can solve 
+this problem with great success. Additionally, a new paradigm of HF frequency-dependent singularity of LSM 
+images was developed to spatial describe (i) local changes in surface resistance 
+δRs(x, y) and (ii) HF current 
+density distribution JHF(x, y) as a function of resonator ( x, y) surface coordinates by using a method of spatially-
+resolved complex impedance partition [2]. Here, we pr opose a new phenomenological approach to spatially-
+resolved research of NL microwave properties of operating thin-film superconducting resonators. The approach is based on frequency and spatial uniqueness of LSM images that can be extracted from a set of 2-D patterns representing x-y distribution of the LSM photoresponse, PR
+f(x, y), at fixed third-order IMD frequencies 2 f1-f2 and 
+2f2-f1 as a result of two-tone resonator microw ave excitation at equidistant frequencies f1 and f2 relative to the 
+fundamental resonance, f0. We show that similar to what is described in [2] manipulation of LSM images at 2 f1-f2 
+and 2 f2-f1 can be used to present NL components of IMD LSM PR(x, y) in terms of its independent spatial 
+variations of ( i) inductive PR IMD,X (x, y) and ( ii) resistive PR IMD,R(x, y) contributions. In particular the realistic 
+spatial coherence between such IMD components and JHF(x, y) flow, so crucial for determining the NL properties 
+of the device, have not been investigated. 
+The tested device was a half wave YBa 2Cu3O7-δ (YBCO) microstrip (with thickness d = 1 μm) resonator on 
+a 500 μm LaAlO 3 (εr=24.2) substrate (LAO) designed for operation at f0=1.85 GHz with Q~1230 at T=78 K. The 
+resonator was connected to the IMD scheme of a tw o-tone excitation centered with 1 MHz spacing around f0. For 
+LSM imaging, the device was x-y scanned by a 4 μm diameter thermal probe produced by a 1.1 μm diameter 
+focused laser beam that is amplitude modulated with a frequency of about 100 kHz on the sample surface. Position of the LSM raster was centered not far from the peak of the standing wave current pattern near the YBCO strip edge to investigate an area with maximum J
+HF(x, y) gradient that is the most suitable for the 
+following spatially-resolved analysis. All LSM images were acquired at T=78 K and at Pf1=Pf2=9 dBm using 
+step-by-step scanning of the sample by the laser probe with equal distance between steps of about 0.4 μm along 
+both x and y direction. The linear components of HF LSM PR f(x, y) were measures at f1 and f2 to make spatial 
+separation between inductive PR X(x, y) and dissipative PR R(x, y) contribution as explained by [2]. The derived 
+PR X(x, y) was used to present LSM image of JHF(x, y) calibrated in real units by using a procedure described in 
+[1]. Changes in δRs(x,y) were estimated as PR R(x,y)/PR X(x,y) considering that PR R(x, y)~ J2
+HF(x, y)*δRs(x, y) and 
+PR R(x, y)~ J2
+HF(x, y). 
+Additionally, two LSM PR 2f1-f2(x, y) and PR 2f2-f1(x, y) images were measured in the same area of the 
+resonator at combinational frequencies 2 f1-f2 and 2 f2-f1 of IMD harmonics. The spectral dependence of IMD 
+photoresponse was estimated by using an expression for global IMD power PIMD=Q4(P2
+f1Pf2)(1/P2
+c)[1+( ΔR/QΔL)2] that has been developed in [3]. Here, Pc=2πfL0I2
+IMD/ΔL is the 
+characteristic power, L0 and R0 are the linear parts,  ΔL and ΔR are the nonlinearity coefficients in changes of the 
+inductance  L=L0+ΔL(I/IIMD)2 and resistance  R=R0+ΔR(I/IIMD)2 per unit length of the microstrip related to the case 
+when jIMD is on order of the critical current density Jc. We considered the frequency dependence of thermally 
+induced modulation in PIMD in terms of the temperature derivative δPIMD(f)= (dPIMD/dT)δT, where  separate 
+contributions from PR IMD,X (x, y) and PR IMD,R(x, y) can be deduced through partial derivatives related to f0 and 
+1/Q, correspondingly: 
+ 0
+0() (1 / )
+( 1 / ) () ( ) ()IMD IMD IMD
+IMDf PP P QPRTQ T f T∂ ∂∂ ∂ ∂∝= +∂∂ ∂ ∂ ∂ (1) 
+One can make sure that at frequencies very close to the fundamental resonance ( f ~ f 0) dissipative and 
+inductive components of LSM PR may be presented as: 
+3
+, 422
+0(1 / )
+(1 / ) 4( / 1) (1/ )IMD
+IMD RPQPRQ ff Q∂−=∝∂ ⎡ ⎤ −+⎣ ⎦   (2a) 
+2
+0
+, 42200(/ 1 ) (1 / )
+() 4( / 1) (1/ )IMD
+IMD Xff P QPRff ff Q− ∂=∝∂ ⎡ ⎤ −+⎣ ⎦  (2b) 
+As evident from Eq. (2a), the spectrum of PR IMD,R(f) is approximated well by a symmetric (relative to f0) 
+Lorentzian-like curve of the same (negative) sign having equal magnitudes PR IMD,R(2f1-f2)= PR IMD,R(2f2-f1) at 
+equidistant IMD frequencies 2 f1-f2 and 2 f2-f1 around f0.  In contrast, the curve of PR IMD,X (f) dependence (see Eq. 
+(2b)) crosses “0” at f = f0 and has opposite signs i.e. PR IMD,X (2f1-f2)= (-) PR IMD,X (2f2-f1) there. By virtue such an f-
+singularity of PR IMD(f), one can present LSM images of NL media as independent distributions of inductive and 
+resistive NL sources by using the following transformations: 
+12 21
+,(2 ) (2 )(, )2IMD IMD
+IMD XPRf f P Rf fPR x y−− −=  (3 а) 
+12 21
+,(2 ) (2 )(, )2IMD IMD
+IMD RPRf f P Rf fPR x y−+ −=  (3b) 
+For the experimental verification of these statements, the same 40x20 μm2 area of the resonator was 
+visualized in different imaging modes of LSM contrast. In Fig. 1. 
+ 
+ 
+Fig.1. Set of LSM images that were analyzed and 
+discussed in the text. Fig.2. (a) Sketch of the sample cross section along x-
+scan and (b) profiles of LSM response there. 
+ 
+In Fig. 1(a) we have pictured an optical reflectiv ity LSM (false-color) map of the scanned area where the 
+surface quality of the YBCO film as well as the twinned structure of the LAO substrate is clearly visible. The 
+YBCO strip edge has a chamfered profile producing a darker YBCO reflectivity image within a t = 4 μm strip. 
+Top (P1 in Fig. 2(a)) and bottom (P2 in Fig. 2(a)) edge s of the chamfer are placed along two dashed white lines 
+in Fig. 2(b) where an LSM image of JHF(x, y) is shown. The brightest area there coincides with maximum amplitude of JHF(x, y) ~ 109A/m2 while the darkest ones correspond to zero current density. Spatial coherence 
+between (i) JHF(x, y), nonlinear (ii) inductive PR IMD,X (x, y) and (iii) resistive PR IMD,R(x, y) components of LSM 
+PR as well as (iv) δRs(x,y) is evident as one can see through comparison of all images presented in Fig. 1(b-e).  
+Fig. 2(b) shows typical profiles of different IMD and HF components of LSM PR(x, y = 4) together with 
+their exponential fit that were measured along a single x-scan (see dashed line numbered as ‘4’ in Fig. 4(a)) at fixed y(=4 lines)-cross-section. Similar profiles were used in Fig. 3 for log-log plotting of both local PR
+IMD,R and 
+PR IMD,X  responses as a function of local circulating HF power  that can be approximated by the linear term of 
+inductive LSM PR. As seen all four cross-sections (4, 18, 32 and 100 from Fig. 4(a)) of the resonator demonstrate a similar tendency for a linear increasing of PR
+IMD,X  and square-low growth of PR IMD,R with respect 
+to PR HF,X that is linked directly to IMD components through the same position of the laser probe. Also, the 
+dominant contribution of PR IMD,X  to the total IMD PR is seen at small JHF(x, y) closer to YBCO center. However 
+at peak values of HF current density near the HTS film edge those amplitudes of NL response are almost equal. 
+We believe that IMD LSM PR originates from a small laser-probe induced modulation δPIMD(x, y). of 
+intermodulation power. In this case, PR IMD,X (x, y) and PR IMD,R(x, y) components show local NL properties of the 
+microwave device in terms of corresponding local derivatives δPIMD,X (x, y) and δPIMD,R(x, y). Spatial integration 
+(see Fig. 4(b)) shows that below Jc(x, y) (see, for example data in Fig. 3(c) below P1), the power of real 
+(dissipative) NL sources follows a classical cubic dependence PIMD,R ~ P3
+IN on input HF power. There were no 
+changes in local values of the surface resistance found at all JRF(x, y) < Jc(x, y). Surprisingly, local sources of 
+imaginary (inductive) IMD response show almost a square-law power dependence (see Fig. 4(b)) even below J
+c(x, y) except very close to JRF(x, y) ~ 0. This contradicts previously published data considering Jc(x, y) as the 
+correct scale for the lower limit of NLs. 
+ 
+Fig.3. Log-log plots of IMD components of LSM photoresponse as a function of linear inductive one.  The line cuts in parts (a) – (d) correspond to those shown in Fig. 4. Fig.4. (a) LSM image of PR
+IMD,R(x, y) 
+showing line-scans analyzed in Fig. 3 and (b) restored P
+IMD(P) dependencies. 
+ 
+The authors acknowledge the support of the Fundamental Researches State Fund of Ukraine  and German 
+International Bureau of the Federal Ministry of Education and Research (BMBF) under grant project UKR08/011 and a NASU program on “nanostructures, materials and technologies”. 
+References 
+1. A.P. Zhuravel, A.G. Sivakov, O.G. Turutanov, A.N. Omelyanchouk, S. M. Anlage, A. Lukashenko, A.V. 
+ Ustinov, D. Abraimov, “Laser scanning microscopy of HTS films and devices (Review Article)”, Low  Temp. Phys., Vol. 32, Issue 6, , pp. 592-607, 2006. 
+2. A.P. Zhuravel, S.M. Anlage, A.V. Ustinov, “Measurement of local reactive and resistive photoresponse of a  superconducting microwave device”, Appl. Phys. Lett ., Vol. 88, p. 212503, 2006. 
+3. T. Dahm,  D J. Scalapino, B.A. Willemsen, “Phenomenological Theory of Intermodulation in HTS Resonators  and Filters”, Journal of Superconductivity, Vol. 12, No. 2,  pp. 339-342, 1999.  
\ No newline at end of file
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+arXiv:1001.0678v1  [nucl-th]  5 Jan 2010REVIEW ARTICLE
+TheR-matrix Theory
+P Descouvemont
+Physique Nucl´ eaire Th´ eorique et Physique Math´ ematique , C.P. 229,
+Universit´ e Libre de Bruxelles (ULB), B 1050 Brussels, Belg ium
+E-mail:pdesc@ulb.ac.be
+D Baye
+Physique Quantique, C.P. 165/82,
+Physique Nucl´ eaire Th´ eorique et Physique Math´ ematique , C.P. 229,
+Universit´ e Libre de Bruxelles (ULB), B 1050 Brussels, Belg ium
+E-mail:dbaye@ulb.ac.be
+Abstract. The different facets of the R-matrix method are presented pedagogically
+in a general framework. Two variants have been developed ove r the years: ( i) The
+”calculable” R-matrix method is a calculational tool to derive scattering properties
+from the Schr¨ odinger equation in a large variety of physica l problems. It was developed
+ratherindependentlyinatomicandnuclearphysicswithtoo littlemutualinfluence. ( ii)
+The ”phenomenological” R-matrix method is a technique to parametrize various types
+of cross sections. It was mainly (or uniquely) used in nuclea r physics. Both directions
+are explained by starting from the simple problem of scatter ing by a potential. They
+are illustrated by simple examples in nuclear and atomic phy sics. In addition to
+elastic scattering, the R-matrix formalism is applied to transfer and radiative-cap ture
+reactions. We also present more recent and more ambitious ap plications of the theory
+in nuclear physics.
+Submitted to: Rep. Prog. Phys.TheR-matrix Theory 2
+1. Introduction
+1.1. Principle
+TheR-matrix theory is a powerful tool of quantum physics, introd uced by Wigner
+and Eisenbud [1, 2, 3] where they simplified an original idea o f Kapur and Peierls [4].
+The advantage of their simplification is that the Rmatrix only involves real energy-
+independent parameters. Initially the theory was aimed at d escribing resonances in
+nuclear reactions. However even the very first developments a lso contained the principle
+of a technique for solving coupled-channel Schr¨ odinger eq uations in the continuum.
+At present, the main aim of the R-matrix theory is to describe scattering states
+resulting from the interaction of particles or systems of pa rticles, which can be nucleons,
+nuclei, electrons, atoms, molecules. Its principle relies on a division of the configuration
+space into two regions: the internal and external regions. T he boundary between
+these regions is defined by a parameter known as the channel ra dius. This radius is
+chosen large enough so that, in the external region, the diffe rent parts of the studied
+system interact only through known long-range forces and an tisymmetrization effects
+can be neglected. The scattering wave function is approxima ted there by its asymptotic
+expression which is known except for some coefficients relate d to the scattering matrix.
+In the internal region, the system is considered as confined. Its eigenstates thus form a
+discretebasiswhichcanbecalculated. Ascatteringwavefu nctionatanarbitraryenergy
+is expanded in the internal region over these square-integr able eigenstates. Then, the
+Rmatrix, which is the inverse of the logarithmic derivative o f the wave function at
+the boundary, can be calculated. A matching with the solutio n in the external region
+provides the scattering matrix. This method can also provid e the bound states of the
+system. In this case, the external solution behaves as a decr easing exponential. Since
+the exponential decrease depends on the unknown binding ene rgy, an iteration is then
+necessary.
+TheR-matrixtheorywasdevelopedintotwodifferentdirectionsw ithlittleexchange
+between these variants. Many of its practitioners often ign ore the progresses about the
+other aspect of this double-faced method.
+As already mentioned, the original goal was to provide an effici ent theory for the
+treatment of nuclear resonances [3, 5]. From information on bound states and low-
+energy resonances, it soon became clear that the R-matrix theory offers an efficient way
+for accurately parametrizing not only resonances but also t he non-resonant part of low-
+energy cross sections with a small number of parameters [5]. An important advantage is
+that most of these parameters have a physical meaning. This fi rst variant of the method
+is still very important and much employed, in particular to p arametrize the low-energy
+cross sections relevant in nuclear astrophysics. This vers ion of theR-matrix theory will
+be called hereafter the phenomenological Rmatrix. Its properties are reviewed in [5, 6].
+The other aspect of the R-matrix theory is that it can provide a simple and elegant
+way for solving the Schr¨ odinger equation. It is especially competitive in coupled-channel
+problems with large numbers of open channels where direct nu merical integrations mayTheR-matrix Theory 3
+become unstable. An additional advantage is that narrow reso nances which can escape
+a purely numerical treatment are easily studied. This other facet of the R-matrix theory
+has been mostly developed in atomic physics although we shal l see that it can also be
+very useful for nuclear-physics applications. This varian t will be called hereafter the
+calculable or computational Rmatrix. Its properties are reviewed in [7, 8, 9, 10].
+A very comprehensive review of the phenomenological R-matrix method has been
+given in 1958 by Lane and Thomas [5]. Their article contains m ost of the important
+aspects of the phenomenological applications of the Rmatrix to nuclear physics. Many
+of their results can also be useful for the calculable Rmatrix. However, in 1957, just
+before that review appeared in print, an important improvem ent of the method was
+published which is therefore not used in their review. Bloch introduced a singular
+operator defined on the boundary between the two regions, now known as the Bloch
+operator, which allows a more elegant and compact presentat ion of the method [11].
+The main interest of the Bloch operator is that its use led to e xtensions of the method
+to more general treatments of the resolution of the Schr¨ odi nger equation in the internal
+region and opened the way to accurate methods of resolution i n atomic and nuclear
+physics. Several reviews on the computational Rmatrix have been published in the
+context of nuclear physics [7] and of atomic physics [8, 12, 1 0]. Reference [9] deals with
+both aspects. An update is nevertheless timely.
+1.2. The phenomenological Rmatrix
+The phenomenological Rmatrix and most of its applications were already exhaustive ly
+described fifty years ago in [5]. Among these applications, le t us mention a detailed
+studyofresonancesandanextensionofthemethodtothedesc riptionofelectromagnetic
+processes. As far as we know, all these applications have been made in nuclear physics,
+i.e. for the scattering of neutrons on nuclei or of nuclei on n uclei with the presence of
+a repulsive Coulomb barrier. Nevertheless, using the method still revealed a number of
+difficulties. In a series of papers, Barker and collaborators provided practical solutions
+to the determination of the R-matrix parameters from experimental data [13, 14, 15]
+and applied this framework to the spectroscopy and reaction s of light nuclei [16, 17].
+They also explained non-intuitive effects such as the Thomas -Ehrman shift [17], ghosts
+of resonances [18] and extended the method to further proces ses such as radiative-
+capture reactions [19, 20, 21] and delayed βdecay [22]. The approach developed by
+Barker and collaborators has become a standard tool for the a nalysis of low-energy
+radiative-capture reactions useful in astrophysics. Rece nt progresses in the adjustment
+ofR-matrix parameters have been performed in [23, 24].
+TheRmatrixallowsparametrizingvariousphysicalprocessesan ditsdetermination
+providescollisionmatricesandcrosssections. Foreachse tofgoodquantumnumbers,i.e.
+total angular momentum and parity, the dimension of the phen omenological Rmatrix
+is equal to the number of channels relevant to the physical pr operties. When a single
+channel is considered, the Rmatrix for a partial wave with orbital momentum landTheR-matrix Theory 4
+total angular momentum Jis a function of the energy Eparametrized by the formula
+RlJ(E) =N/summationdisplay
+n=1γ2
+nlJ
+EnlJ−E. (1.1)
+Inprinciple,thisfunctionpossessesaninfinityofpolesat therealenergies EnlJbutonlya
+limitednumber Nofsuchpolesaffectthelow-energycrosssections. Thelowes tpolesare
+closely related to bound states at negative energies or to na rrow resonances at positive
+energies. Nevertheless, the poles and the energies of physic al states are slightly different.
+Because of this shift, the determination of these parameter s from data requires some
+skill. The real parameters γnlJare known as the reduced width amplitudes because their
+square is a crucial factor of the width of non-overlapping re sonances. More precisely, we
+shall see in section 3.6 that the width is given as Γ = 2 γ2
+nlJPlwherePlis the penetration
+factor which includes most of the effects of transmission thr ough the Coulomb barrier.
+This factor depends on energy and the width thus also depends on energy.
+A serious drawback of the phenomenological Rmatrix is that the poles and widths
+depend on the choice of channel radius, i.e. on a rather arbit rary value. This aspect
+of theRmatrix has been criticized by a number of authors and has led t o further
+developments of the competing phenomenological Kmatrix [25]. The Kmatrix, which
+provides an alternative formulation of the collision matri x, is also expanded in a series
+involvinganinfinityofpoles. Thisapproachisbasedonadel icatetreatmentofCoulomb
+functions. Inspiteofthefactthatthe Kmatrixdoesnotcontainanarbitraryparameter
+such as the channel radius, its parametrization is more diffic ult because its parameters
+may have a less direct physical interpretation.
+1.3. The calculable Rmatrix
+The aim of the calculable Rmatrix is to provide an efficient way of solving the
+Schr¨ odinger equation both at positive and negative energi es. It was proposed in 1965
+by Haglund and Robson and applied to a two-channel problem inv olving square-well
+potentials [26]. An expansion over a finite basis was introduc ed by Buttle [27]. He
+performed the first realistic application on12C + n scattering [27]. He also proposed
+a correction to the truncation of the Rmatrix to a finite number of poles, that is now
+named after him. A more serious problem is a discontinuity of the derivative of the
+wave function at the boundary between the regions that occur s with the traditional
+choice of basis states inspired by the original ideas in [3]. Various solutions to the lack
+of matching at the boundary have been suggested (see [9] for a review). This apparent
+problem has attracted a lot of attention even long after an effi cient technique where it
+does not occur was introduced [28, 29]. By dropping an unnece ssary condition as we
+will describe, the R-matrix method can be very accurate without matching proble ms
+and without need for a Buttle correction.
+Some users of the phenomenological Rmatrix consider the channel radius as a
+parameter which must be optimized when fitting the data. Even if this dependence on a
+parameter without strong physical meaning is weak, this is a drawback that would notTheR-matrix Theory 5
+beacceptablewhenaimingataccuratelysolvetheSchr¨ odin gerequation. Henceacrucial
+test of the results of the calculable Rmatrix is an almost perfect independence with
+respect to the choice of channel radius. This test provides a measure of the accuracy of
+the calculations.
+In spite of its introduction for nuclear-physics problems [ 26, 27], this approach
+was first extensively developed to study electron (or positr on) collisions on atoms and
+molecules [8, 12, 10]. It allows describing the excitation a nd ionization of these systems.
+Photoionization, i.e. collisions with a photon leading to t he single or double ionization
+of the atom, is also a well-studied application [30, 31].
+Important and difficult aspects of these atomic-physics prob lems are the non-
+locality of the interaction due to electron exchanges and th e long-range nature of the
+interactions, due to the polarization interactions. The no n-locality is well treated in the
+R-matrix approach. The long range of the force implies that th e asymptotic behaviour
+of the solution is only reached for very large values of the in terparticle distance. To
+avoid using a very large channel radius, propagation method s have been introduced
+[32]. They involve an intermediate region where the interac tion can be simplified, for
+example with an asymptotic expansion.
+Electron scattering on heavy atoms has required the introdu ction of relativistic
+corrections. The extension of the Rmatrix to the Dirac equation has been introduced
+as early as in 1948 [33] but its validity remained controvers ial during a long time.
+Relativistic corrections were thus first derived from the Br eit-Pauli equation. The
+validity of the Dirac extension is now well established and e laborate relativistic codes
+have been developed. This aspect will not be covered here (se e [34] for a recent review).
+In nuclear physics, the computational Rmatrix is much less used although it
+should be very useful in large coupled-channel calculation s. It has been much applied
+in microscopic cluster calculations in which the difficult an tisymmetrization is taken
+into account in the internal region only [35, 36]. The versat ility of the Rmatrix found
+interesting applications in processes where bound and scat tering states are mixed, such
+as radiative capture or delayed βdecay [37, 38]. The application of the R-matrix
+method to coupled-channel calculations has been simplified by its combination with
+the Lagrange-mesh method which avoids calculating matrix e lements of the potentials
+[39, 40, 41]. Recently this approach has been extended to non -local interactions [42].
+Other approaches to the same problem present a number of simi larities. The
+variational K-matrix method [43] has a very similar spirit. The Kohn varia tional
+principle for the logarithmic derivative is equivalent to t he calculable Rmatrix [44].
+Practical implementations of the Gamow-state method [45] a re also exactly equivalent
+toR-matrix calculations [46].
+1.4. Outline
+Inthisreview,wepresentbothcalculableandphenomenolog icalversionsofthe R-matrix
+theory. Very few papers deal with both aspects simultaneous ly [47, 48]. Since manyTheR-matrix Theory 6
+excellent reviews already exist, we try to make an introduct ory presentation, illustrated
+with simple numerical examples. We also show the parallel ev olutions of the method in
+atomic and nuclear physics and try to shed light on some commo n misunderstandings or
+controversies about the R-matrix methods. The numerical examples that we display are
+tailored to allow a motivated reader to test his/her underst anding by reproducing them
+with limited effort. Because of our background as nuclear phy sicists, most examples
+(but not all) correspond to nuclear applications. Finally, we also review state-of-the-art
+calculations in nuclear physics where the Rmatrix proves useful.
+Contrary to tradition, we start with the calculable Rmatrix on a finite basis,
+which provides a convenient numerical approach. Taking the limit for an infinite
+complete basis will introduce the theoretical Rmatrix which leads after truncation
+to the phenomenological approximation (1.1). For the sake o f simplicity, we detail
+potential scattering in the single-channel case so avoidin g the unpedagogical definitions
+of channel wave functions. We only provide the main steps for the multichannel case.
+The bibliography about the Rmatrix is enormous and can not be fully covered
+here. We have tried to quote papers that we think significant o r useful for further
+bibliographic research.
+In section 2, we introduce the necessary basics of scatterin g theory with radiative
+capture as a more elaborate application. The calculable Rmatrix is presented in section
+3 and relatively simple numerical applications in section 4 . The phenomenological R
+matrix and and its applications are presented in section 5. R ecent elaborate calculations
+in nuclear physics are reviewed in section 6. Concluding rem arks are made in section 7.TheR-matrix Theory 7
+2. Summary of scattering theory
+2.1. Coulomb scattering
+Consider the collision of two particles with respective mas sesm1andm2and charges
+Z1eandZ2eat a positive energy Ein the centre-of-mass frame. The wavenumber is
+defined as
+k=/radicalBig
+2µE/¯h, (2.1)
+whereµ=m1m2/(m1+m2) is the reduced mass.
+Let us start with some definitions about pure Coulomb scatter ing. In this case, a
+Bohr radius can be defined as
+aB=¯h2
+µ|Z1Z2|e2. (2.2)
+A useful parameter is the dimensionless Sommerfeld paramet er
+η=Z1Z2e2
+¯hv=sgn(Z1Z2)
+aBk(2.3)
+wherev= ¯hk/µis the relative velocity. Parameter ηmeasures the importance of
+Coulomb effects at a given energy. The neutral case is recover ed withη= 0.
+For a central potential, a wave function can be factorized in spherical coordinates
+r= (r,Ω) asψ(r) =r−1ul(r)Ym
+l(Ω). The spherical harmonics Ym
+l(Ω) depend on the
+orbital and magnetic quantum numbers landm, and on the angles Ω = ( θ,ϕ). They
+are defined according to the convention of Condon and Shortle y. The radial Schr¨ odinger
+equation for the Coulomb problem in partial wave lthen reads
+/parenleftBiggd2
+dr2−l(l+1)
+r2−2kη
+r+k2/parenrightBigg
+ul(r) = 0. (2.4)
+Its solutions are combinations of the regular and irregular Coulomb functions Fl(η,kr)
+andGl(η,kr) [49]. The regular function vanishes at the origin and is nor malized at
+infinity according to
+Fl(η,x)−→x→∞sin(x−1
+2lπ−ηln2x+σl), (2.5)
+where appears the Coulomb phase shift
+σl= argΓ(l+1+iη) (2.6)
+involving the Euler function Γ. The irregular function Gl(η,x) is unbound at the origin
+(except for η=l= 0) and is fixed by its asymptotic behaviour
+Gl(η,x)−→x→∞cos(x−1
+2lπ−ηln2x+σl). (2.7)
+Also very useful are the conjugate functions
+Il(η,x) =Gl−iFl, O l(η,x) =Gl+iFl, (2.8)
+which behave asymptotically like incoming and outgoing wav es, respectively,
+Il(η,x)−→x→∞e−i(x−1
+2lπ−ηln2x+σl), O l(η,x)−→x→∞ei(x−1
+2lπ−ηln2x+σl).(2.9)TheR-matrix Theory 8
+Notice that we do not follow here the same phase convention as L ane and Thomas [5]. In
+the neutral case, one has Fl(0,x) =xjl(x) andGl(0,x) =xnl(x) wherejlandnl=−yl
+are spherical Bessel functions [49].
+In some applications, solutions of (2.4) are also needed at n egative energies. The
+real solution decreasing at infinity is the Whittaker functi onW−ηB,l+1
+2(2κr) [49]. It
+depends on the wave number κ=√−2µE/¯hand on the Sommerfeld parameter
+ηB= sgn(Z1Z2)/aBκof the bound state. Whittaker functions behave asymptotica lly as
+W−ηB,l+1
+2(x)−→x→∞x−ηBe−x/2. (2.10)
+They are singular at the origin.
+A bounded solution of the three-dimensional Schr¨ odinger e quation at a positive
+energy with the same Coulomb potential is given by [50]
+ψ+
+C(r) = (2π)−3/2e−πη/2Γ(1+iη)eikz
+1F1(−iη,1,ik(r−z)) (2.11)
+where 1F1is the confluent hypergeometric function [49]. This wave fun ction has the
+asymptotic behaviour of an outgoing scattering state
+ψ+
+C(r)−→
+|r−z|→∞(2π)−3/2/parenleftBigg
+ei[kz+ηlnk(r−z)]+fC(Ω)ei(kr−ηln2kr)
+r/parenrightBigg
+.(2.12)
+The coefficient of the second term is the Coulomb scattering am plitude,
+fC(Ω) =−η
+2ksin21
+2θe2i(σ0−ηlnsin1
+2θ). (2.13)
+The square of the modulus of fCprovides the Rutherford cross section. Function ψ+
+C
+can be expanded in partial waves as
+ψ+
+C(r) = (2π)−3/2(kr)−1∞/summationdisplay
+l=0(2l+1)ileiσlPl(cosθ)Fl(η,kr), (2.14)
+wherePlis a Legendre polynomial [49].
+2.2. Scattering by a potential
+Consider a potential Vtending to the Coulomb potential faster than r−2,
+V(r)−→r→∞Z1Z2e2
+r+o/parenleftbigg1
+r2/parenrightbigg
+. (2.15)
+The radial Schr¨ odinger equation in partial wave lreads
+/parenleftBiggd2
+dr2−l(l+1)
+r2−2µV(r)
+¯h2+k2/parenrightBigg
+ul(r) = 0 (2.16)
+with the condition at the origin
+ul(0) = 0. (2.17)
+A real solution at positive energy Ebehaves asymptotically as
+ul(r)−→r→∞cosδlFl(η,kr)+sinδlGl(η,kr), (2.18)TheR-matrix Theory 9
+up to a normalization factor. The important physical quanti ty is the phase shift δl. For
+later use, it is however more convenient to write the solutio n as
+ul(r)−→r→∞Cl[Il(η,kr)−UlOl(η,kr)], (2.19)
+whereClcan be chosen in various ways. For example, (2.18) is recover ed with
+Cl=iexp(−iδl)/2 and the normalization of the ulwith respect to δ(k−k′) is obtained
+withCl=iexp(−iδl)/√
+2π. The collision or scattering ‘matrix’ Ulis given by
+Ul=e2iδl. (2.20)
+With the different partial solutions, one can construct an ou tgoing stationary
+solution
+Ψ+(r) = (2π)−3/2(2kr)−1∞/summationdisplay
+l=0(2l+1)il+1eiσlPl(cosθ)C−1
+lul(r) (2.21)
+behaving asymptotically as a Coulomb wave (2.11) propagati ng in thezdirection plus
+an outgoing spherical wave
+Ψ+(r)−→r→∞ψ+
+C(r)+(2π)−3/2f(Ω)ei(kr−ηln2kr)
+r. (2.22)
+The coefficient of the second term in this asymptotic expressi on determines the
+additional scattering amplitude
+f(Ω) =1
+2ik∞/summationdisplay
+l=0(2l+1)e2iσl(Ul−1)Pl(cosθ). (2.23)
+From (2.23), one obtains the elastic cross section
+dσ
+dΩ=|fC(Ω)+f(Ω)|2(2.24)
+also involving the Coulomb amplitude (2.13). The scatterin g wave function (2.21) is
+useful in various types of reactions. We illustrate it below with radiative capture.
+2.3. Collisions in a many-body system
+AnN-body system is described with the microscopic Hamiltonian
+H=T+V=N/summationdisplay
+i=1Ti+N/summationdisplay
+i>j=1Vij (2.25)
+where for simplicity we only display two-body forces. This Ha miltonian is invariant
+under rotations, translations and reflections. We do not dis play nor discuss the removal
+of the centre of mass.
+At positive excitation energies, several channels may be op en. In each channel, the
+particles are divided into various groups. Each such divisi on is known as a partition.
+A given partition is denoted as α. For simplicity, we only consider here channels where
+the particles form only two subsystems: N=N(1)
+α+N(2)
+α. Both subsystems of partition
+αare described with an internal Hamiltonian H(i)
+α(i= 1,2) which has the form (2.25)
+withNreplaced by N(i)
+α.TheR-matrix Theory 10
+A channelcis defined by specifying in addition the energy of each subsys tem, i.e.
+a certain eigenvalue for each internal Hamiltonian. The exac t or approximate energies
+E(i)
+cand wave functions φ(i)
+care related by
+E(i)
+c=/angbracketleftφ(i)
+c|H(i)
+c|φ(i)
+c/angbracketright. (2.26)
+Here and in what follows, subscript chas a variable symbolic meaning depending on
+the considered quantity. In H(i)
+c≡H(i)
+α, it represents the set of internal coordinates of
+subsystem i. InE(i)
+c, it represents a set of quantum numbers. In φ(i)
+c, it means both.
+Whenφ(i)
+cis not an exact eigenfunction of H(i)
+c, (2.26) remains valid from a variational
+perspective.
+To each partition αof the system into two subsystems may correspond several
+channels differing by their internal states. Each channel chas a threshold energy
+Ec=E(1)
+c+E(2)
+c (2.27)
+defined with respect to some common reference energy. A chann el is open or closed
+according to whether Ecis smaller or larger than the total energy Eof the system. In
+each open channel, one can define a wave number kc=/radicalBig
+2µc(E−Ec)/¯hand a relative
+velocityvc= ¯hkc/µc,µcbeing the reduced mass of partition α. In each closed channel,
+one can define a wave number κc=/radicalBig
+2µc(Ec−E)/¯h.
+For each partition α, the relative coordinate rc≡rαis the difference between
+the centre-of-mass coordinates of the subsystems. The eige nstates of each subsystem
+i= 1,2 are characterized by their energy E(i)
+cand by their good quantum numbers, i.e.
+the total angular momentum Ii(usually called spin) and its projection Mi. Under the
+timereversaloperator K[50], theyareassumedtotransformaccordingtotheconvent ion
+K|JM/angbracketright= (−1)J−M|J−M/angbracketright. (2.28)
+A channel state is an eigenstate of the total angular momentu m of the full system
+resulting from the coupling of both internal states φ(1)
+cI1M1andφ(2)
+cI2M2with a spherical
+harmonics depending on the angles Ω cdefining the orientation of the relative coordinate
+rc. More precisely, a channel state is represented as
+|c/angbracketright=ilc/bracketleftBig
+[φ(1)
+cI1⊗φ(2)
+cI2]Ic⊗Ylc(Ωc)/bracketrightBigJMπ, (2.29)
+whereIcis the channel spin resulting from the coupling of I1andI2,lcis the orbital
+momentum of the relative motion in channel c,Jis the total angular momentum
+quantum number of the many-body system, Mis its projection, and πis the total
+parity. Thanks to the phase factor ilc, the channel states transform under time reversal
+according to (2.28). For simplicity, we do not display the pa rity quantum numbers πc1
+andπc2of the subsystems. They are related to the total parity by
+π=πc1πc2(−1)lc. (2.30)
+Channel states are assumed orthogonal to each other and norm ed,
+/angbracketleftc|c′/angbracketright=δcc′. (2.31)TheR-matrix Theory 11
+The orthogonality is not obvious for different partitions αandα′. In that case, it is
+only true asymptotically.
+Since Hamiltonian His invariant under rotation and reflection, J,Mandπare
+good quantum numbers. A partial wave of the total wave functi on of the system at
+energyEcan be written as
+ΨJMπ
+(c0)=/summationdisplay
+cA|c/angbracketrightr−1
+cuc(c0)(rc), (2.32)
+where indices c0andcmay represent either all quantum numbers appearing (or
+understood) in the right-hand side of (2.29) or a relevant su bset of them. Projector
+Aperforms any antisymmetrization due to the indistinguisha bility of some identical
+particles. These can be electrons in atomic physics or nucle ons in nuclear physics
+within the isospin formalism. The subscript ( c0) recalls the entrance channel as
+explained below. This wave function is an approximate eigen state of the full many-
+body Hamiltonian
+HΨJMπ
+(c0)=EΨJMπ
+(c0). (2.33)
+To complete the definition of ΨJMπ
+(c0), one must specify its asymptotic behaviour.
+We shall describe the asymptotic behaviour in terms of the co llision or scattering
+matrixwhichisusuallydenotedas UintheR-matrixcontext. Thismatrixisinteresting
+physically because it is directly related to cross sections but it leads to complex radial
+wave functions. Calculations only involving real radial wa ve functions are also possible
+(for real potentials). They are dominantly used in atomic ph ysics [51] but are also
+encountered in nuclear physics [9]. The relation between bo th approaches is summarized
+in appendix A.
+The asymptotic behaviour in open channels generalizing (2. 19) is given by
+uc(c0)(rc)−→rc→∞Cc0v−1/2
+c[δcc0Ic(kcrc)−Ucc0Oc(kcrc)]. (2.34)
+In (2.34),IcandOcare defined as in (2.8) and Cc0is arbitrary (see appendix A). The
+asymptotic form is chosen in such a way that incoming flux only occurs in the entrance
+channelc0. Taking all possible entrance channels c0into account, the coefficients Ucc0of
+theoutgoingwavesformthecollisionmatrix U. Itisdefinedforeachangularmomentum
+Jand parity π. Its dimension is given by the number of open channels at ener gyE.
+Closed channels may also appear in expansion (2.32) but the a symptotic behaviour of
+the corresponding radial functions is exponentially decre asing according to (2.10).
+For real potentials, thanks to the introduction of coefficien tsv−1/2
+c, current
+conservation imposes that the collision matrix is unitary,
+UU†=U†U=1. (2.35)
+Because of time-reversal invariance, it is also symmetric,
+U=UT(2.36)
+whereTmeanstransposition. Thispropertyimposesthephas eilcinthedefinition(2.27)
+of the channel states. As shown by Huby [52], this factor is miss ing in some importantTheR-matrix Theory 12
+references [3, 53] and some phases must be corrected accordi ngly. This correction is
+not necessary if the symmetry property (2.36) is never used. A matrix with properties
+(2.35) and (2.36) can be diagonalized with an orthogonal (re al) matrixS,
+SUST=e2iδ, (2.37)
+whereδis a diagonal matrix whose elements are the eigenphases δn.
+Introducing (2.32) in the Schr¨ odinger equation (2.33) and projecting over a channel
+wave function |c/angbracketrightleads to the coupled equations
+[Tc+Vc(r)+Ec−E]uc(c0)(r)+/summationdisplay
+c′/integraldisplay∞
+0Wcc′(r,r′)uc′(c0)(r′)dr′= 0.(2.38)
+They involve the kinetic-energy operators
+Tc=−¯h2
+2µc/parenleftBiggd2
+dr2−lc(lc+1)
+r2/parenrightBigg
+. (2.39)
+The local or direct potentials are defined by
+Vc(rc) =/angbracketleftc|V|c/angbracketright (2.40)
+whereVis the total potential appearing in the many-body Hamiltonia n (2.25) and the
+integration is performed over the internal coordinates of t he subsystems appearing in
+channelc. The non-local potentials
+Wcc′(r,r′) =/angbracketleftc|δ(rc−r)VAδ(rc′−r′)|c′/angbracketright−Vc(r)δcc′δ(r−r′) (2.41)
+occur because of antisymmetrization and/or because severa l partitions are taken into
+account.
+Mathematically, system (2.38) can be written as a function o f a single coordinate
+r(which becomes r′in the integrals of the non-local terms). Wave functions how ever
+depend on several relative coordinates rcwhen several partitions are taken into account.
+Channelscdiffereitherbythenatureofthesubsystemsorbytheirlevel ofexcitation. To
+simplify the presentation, we now consider that a single par tition is taken into account
+or that all coordinates rcare approximated by a single one. Interactions may still be
+non local but this does not raise major problems as long as the non-local terms are
+short-ranged.
+The colliding systems have initial orientations specified b y the spin projections M1
+andM2in the entrance channel now denoted as c. One is looking for a solution of the
+Schr¨ odinger equation with the asymptotic behaviour
+Ψ+
+(cM1M2)(r)−→r→∞ψ+
+C(r)φ(1)
+cI1M1φ(2)
+cI2M2
++(2π)−3/2/summationdisplay
+c′M′
+1M′
+2ei(kc′r−ηc′ln2kc′r)
+rf(cM1M2)
+c′M′
+1M′
+2(Ω)φ(1)
+c′I′
+1M′
+1φ(2)
+c′I′
+2M′
+2. (2.42)
+Several scattering amplitudes f(cM1M2)
+c′M′
+1M′
+2appear. The partial wave functions (2.32) read
+with|c/angbracketrightrepresenting |(αI1I2)IlJM/angbracketright,
+ΨJMπ
+(c)(r) =/summationdisplay
+c′|c′/angbracketrightr−1uc′(c)(r). (2.43)TheR-matrix Theory 13
+A stationary scattering wave function is constructed as
+Ψ+
+(cM1M2)(r) =i(2π)−3/2π1/2k−1/summationdisplay
+Jπ/summationdisplay
+IlC−1
+c(2l+1)1/2eiσl
+×(I1I2M1M2|IM)(IlM0|JM)ΨJMπ
+(c)(r) (2.44)
+withM=M1+M2. From the outgoing waves of the asymptotic form of (2.44), on e
+deduces the scattering amplitudes
+f(cM1M2)
+c′M′
+1M′
+2(Ω) =i√π
+k/summationdisplay
+Jπ/summationdisplay
+Il/summationdisplay
+I′l′(2l+1)1/2ei(σl+σl′)(I1I2M1M2|IM)
+×(IlM0|JM)(I′
+1I′
+2M′
+1M′
+2|I′M′)(I′l′M′M−M′|JM)
+×(δc′cδI′Iδl′l−UJπ
+c′I′l′,cIl)YM−M′
+l′(Ω) (2.45)
+whereM′=M′
+1+M′
+2. The elastic cross section averaged over initial orientati ons and
+summed over final orientations reads
+dσel.
+dΩ=1
+(2I1+1)(2I2+1)
+×/summationdisplay
+M1M2/summationdisplay
+M′
+1M′
+2|fC(Ω)δM′
+1M1δM′
+2M2+f(cM1M2)
+cM′
+1M′
+2(Ω)|2(2.46)
+wherefCis defined in (2.13). Inelastic or reaction cross sections ar e given by
+dσc→c′
+dΩ=1
+(2I1+1)(2I2+1)/summationdisplay
+M1M2/summationdisplay
+M′
+1M′
+2|f(cM1M2)
+c′M′
+1M′
+2(Ω)|2(2.47)
+forc′/negationslash=c. Thesummationscanbeperformedanalytically. Alongbutsi mplecalculation
+provides [53]
+dσc→c′
+dΩ=π
+k21
+(2I1+1)(2I2+1)/summationdisplay
+λBλ(E)Pλ(cosθ), (2.48)
+where the anisotropy coefficients Bλ(E) are given by
+Bλ(E) =1
+4π/summationdisplay
+Jπ/summationdisplay
+IlL/summationdisplay
+J′π′/summationdisplay
+I′l′L′(−1)I−I′ei(σl+σl′−σL−σL′)Z(lJLJ′,Iλ)
+×Z(l′JL′J′,I′λ)UJπ
+c′I′l′,cIl(E)UJ′π′∗
+c′I′L′,cIL(E). (2.49)
+The real coefficients Zdefined in [53] are modified here for consistency with the
+symmetry property (2.36) of the collision matrix as [52]
+Z(lJLJ′,Iλ) = (−1)J+J′[(2λ+1)(2J+1)(2J′+1)(2l+1)(2L+1)]1/2
+×/parenleftBigg
+l L λ
+0 0 0/parenrightBigg/braceleftBigg
+l L λ
+J′J I/bracerightBigg
+. (2.50)
+They verify the symmetry relation
+Z(lJLJ′,Iλ) =Z(LJ′lJ,Iλ). (2.51)
+FromB0, one deduces the integrated inelastic or reaction cross sec tions
+σc→c′=π
+k21
+(2I1+1)(2I2+1)/summationdisplay
+Jπ(2J+1)/summationdisplay
+Il/summationdisplay
+I′l′|UJπ
+c′I′l′,cIl(E)|2.(2.52)TheR-matrix Theory 14
+For the elastic cross section, the summation over the orient ations can also be performed
+analytically [53] and provides three contributions: nucle ar and Coulomb cross sections,
+as well as an interference term. However, in practical applic ations, it turns out that
+definition (2.46) is more direct to use.
+2.4. Radiative capture
+In nuclear physics, radiative capture is an important proce ss because of its astrophysical
+applications [54]. In this reaction, the two colliding nucl ei fuse into a nucleus with
+massmwith the emission of a photon. In stars, this process often oc curs at very low
+scattering energies and requires that the reaction has a pos itive threshold energy Q=
+(m1+m2−m)c2. The analog reaction in atomic physics is electron capture. However,
+muchmoreattentionispaidinthatfieldtothereversedproce ss,namelyphotoionization.
+Here we shall proceed with radiative capture in a nuclear phys ics context but the
+formulas can be easily adapted to photodissociation or to ph otoionization by using
+the detailed balance.
+Radiative capture is an electromagnetic transition from a s cattering state to a
+bound state. The electromagnetic aspects of this process ca n be studied at first order
+of perturbation theory [5], with an outgoing scattering sta te Ψ+
+(cM1M2)as initial state
+at positive energy Eand a bound state in partial wave Jfπfas final state at negative
+energyEf. The final energy Efis equal to −Q+ExwhereExis the excitation energy
+of the final level with respect to the ground state. The radiat ive capture cross section
+to this state is then given by
+σJfπf(E) =64π4
+¯hv1
+(2I1+1)(2I2+1)/summationdisplay
+σλk2λ+1
+γ
+[(2λ+1)!!]2λ+1
+λ
+×/summationdisplay
+M1M2Mfµ|/angbracketleftΨJfMfπf|Mσλ
+µ|Ψ+
+(cM1M2)(E)/angbracketright|2(2.53)
+where the symbols σλlabel the electric (E λ) and magnetic (M λ) multipoles and the
+corresponding multipole operators are denoted as Mσλ
+µ(µ=−λ,+λ) [55]. The photon
+wave number is given by
+kγ= (E−Ef)/¯hc. (2.54)
+In practice, the sum over σλcan usually be restricted to the dominant electric multipol e
+(E1, or E2 when E1 is forbidden) because kγis small with respect to the inverse of the
+nucleus dimension. Below the Coulomb barrier, the cross sec tion strongly depends on
+energy. To reduce the energy dependence, it is often convert ed into the astrophysical
+S-factor, defined as
+SJfπf(E) =Eexp(2πη)σJfπf(E), (2.55)
+whereηis the Sommerfeld parameter.
+Let us restrict the scattering wave function to a single chan nel. The spins I1and
+I2are then fixed and only Iandlare needed to specify the entrance channel. By usingTheR-matrix Theory 15
+expansion (2.44) in this particular case, the cross section (2.53) can be written as
+σJfπf(E) =/summationdisplay
+σλσσλ
+Jfπf(E), (2.56)
+where the partial cross section of multipolarity σλreads, in analogy with (2.52),
+σσλ
+Jfπf(E) =π
+k21
+(2I1+1)(2I2+1)/summationdisplay
+Jπ(2J+1)/summationdisplay
+Il/vextendsingle/vextendsingle/vextendsingle˜Uσλ
+Il(E,Jπ→Jfπf)/vextendsingle/vextendsingle/vextendsingle2. (2.57)
+In (2.57), ˜Uσλ
+Ilis dimensionless and proportional to a matrix element of the
+electromagnetic operator between the final state and the ini tial partial wave. From
+(2.44) and (2.53), it is given by
+˜Uσλ
+Il(E,Jπ→Jfπf) =/parenleftbigg2Jf+1
+2J+1/parenrightbigg1/2/parenleftBigg8π(λ+1)k2λ+1
+γ
+¯hvλ(2λ+1)!!2/parenrightBigg1/2
+×1
+CIl/angbracketleftΨJfπf||Mσλ||ΨJπ
+(Il)(E)/angbracketright, (2.58)
+where the reduced matrix element is defined by
+/angbracketleftΨJfMfπf|Mσλ
+µ|ΨJMπ
+(Il)/angbracketright= (JλMµ|JfMf)/angbracketleftΨJfπf||Mσλ||ΨJπ
+(Il)/angbracketright.(2.59)
+For a number of reactions involving light nuclei, capture ma inly occurs at distances
+where the wave functions of the colliding nuclei overlap wea kly. This situation can
+be described by a simple model where the internal structure o f the colliding nuclei is
+neglected and the physics of the process is modeled by a local potentialVdepending
+on the distance rbetween the centres of mass of the nuclei. In this case, the as ymptotic
+form of the initial state is described by (2.42) where the int ernal states φ(1)
+cI1M1andφ(2)
+cI2M2
+reduce to spinors |I1M1/angbracketrightand|I2M2/angbracketright.
+The electric operators MEλ
+µare given in this simple model by
+MEλ
+µ=eZ(Eλ)
+effrλYµ
+λ(Ω) (2.60)
+whereZ(Eλ)
+effis the effective charge
+Z(Eλ)
+eff=Z1/parenleftbigg
+−m2
+m/parenrightbiggλ
++Z2/parenleftbiggm1
+m/parenrightbiggλ
+. (2.61)
+The radiative-capture cross section to a final bound state wi th angular momentum Jf
+can be calculated in this model. The initial scattering stat e with quantum numbers JM
+is defined by (2.44) where (2.43) is replaced by
+ΨJM(r) =/summationdisplay
+Ilil|(I1I2)IlJM/angbracketrightr−1uJ
+Il(r) (2.62)
+and radial functions are normalized according to (2.34). Th e normed final bound state
+with quantum numbers JfMfis assumed to be approximated by expression (2.62) with
+the normalization
+/summationdisplay
+Iflf/integraldisplay∞
+0/bracketleftBig
+uJf
+Iflf(r)/bracketrightBig2dr= 1. (2.63)TheR-matrix Theory 16
+The reduced matrix element reads
+/angbracketleftΨJfπf||MEλ||ΨJπ
+(Il)(E)/angbracketright=eZ(Eλ)
+eff[4π(2Jf+1)]−1/2
+×/summationdisplay
+Iflfli(−1)If−JZ(lfJfliJ,Ifλ)/integraldisplay∞
+0uJf
+Iflf(r)rλuJ
+Ifli(Il)(r)dr. (2.64)
+In practice, (2.64) must often be corrected empirically by m ultiplicative factors called
+spectroscopic factors to take an approximate account of the internal structure of the
+nuclei [56].TheR-matrix Theory 17
+3. The calculable Rmatrix
+3.1. Introduction
+The two variants of the Rmatrix method mainly differ by their types of applications.
+In the calculable Rmatrix, the aim is to accurately solve a given Schr¨ odinger e quation
+mostly in the continuum, i.e. for positive energies. In the p henomenological Rmatrix,
+the goal is to parametrize scattering data; it is thus essent ial to know the analytical
+form of the Rmatrix. Of course, both variants have much in common and it is a matter
+of taste to start with one or the other. Historically, in nucle ar physics, the emphasis
+has first been put on the phenomenological variant. Converse ly, most applications in
+atomic physics are related to the calculable Rmatrix. Here we will start with a general
+formalism leading to the calculable version and then deduce the properties allowing the
+phenomenological use.
+First, we restrict ourselves to a single channel for an arbit rary partial wave. This
+assumption is often valid, and allows simple notations. We t hus attempt to find
+approximatesolutionsoftheSchr¨ odingerequationforthe relativemotionoftwoparticles
+with reduced mass µinteracting via a central potential V. At large relative distances
+r, the interaction reduces to the Coulomb interaction VC.
+After separation of the angular part, the radial Schr¨ odinge r equation (2.16) for
+partial wave lcan be written as
+(Hl−E)ul= 0. (3.1)
+In this expression, the radial Hamiltonian Hlis defined as
+Hl=Tl+V(r), (3.2)
+whereTlisgivenby(2.39). Weareinterestedinboundedsolutions ul(r)of(3.1)verifying
+condition (2.17) at the origin. Bound-state solutions at ne gative energies are square
+integrable over (0 ,∞) and can be normed. Scattering solutions at positive energi es are
+assumed to be normalized according to (2.19) with the scatte ring matrix Uldefined
+in (2.20). We will essentially deal with real potentials for which the phase shifts are
+real and the scattering matrix is unitary. The generalizati on to complex potentials is
+straightforward.
+3.2. Definition and calculation of Rmatrix
+In theR-matrix method, the configuration space is divided at the cha nnel radius ainto
+an internal region and an external region. The channel radiu s is chosen large enough
+so thatVcan be approximated by VCin the external region at the required accuracy.
+This means that the channel radius can in principle always be increased although often
+at a cost of computational time. At each energy E, the wave function is defined by
+different expressions in these regions. In the external regi on, the wave function ul(r) isTheR-matrix Theory 18
+approximated by the exact asymptotic expression (2.19) ‡,
+uext
+l(r) =Cl[Il(kr)−UlOl(kr)]. (3.3)
+In the internal region, the wave function uint
+l(r) is expanded over some finite basis
+involvingNlinearly independent functions ϕjas
+uint
+l(r) =N/summationdisplay
+j=1cjϕj(r). (3.4)
+The functions ϕjmust vanish at the origin but are not necessarily orthogonal . In
+contrast with some traditional presentations of the R-matrix theory [8], we do not
+assume that they satisfy specific boundary conditions at r=a. Various choices are
+possible, as exemplified in section 4. The internal and exter nal pieces of the radial
+functions will be connected at the boundary aby the continuity of the wave function ul
+and of its first derivative.
+TheRmatrix at energy Eis defined through §
+ul(a) =Rl(E)[au′
+l(a)−Bul(a)]. (3.5)
+A dimensionless boundary parameter Bis included for later convenience. Its choice
+will be discussed later. The inverse of the Rmatrix is thus the difference between the
+logarithmic derivative of the radial wave function at the bo undary between both regions,
+and the boundary parameter B. This matrix has dimension 1 in a single-channel case
+andisjustafunctionofenergy. Inmultichannelproblems,t hedimensionofthe Rmatrix
+is equal to the number of channels (see section 3.10). The pri nciple of the method relies
+on the facts that the Rmatrix can be calculated from properties of the Hamiltonian i n
+the internal region and that its knowledge allows determini ng the scattering matrix in
+the external region.
+The operator Hlis not Hermitian over the internal region (0 ,a). This property is
+not convenient for practical resolutions of the Schr¨ oding er equation. This problem is
+elegantly solved with the help of the surface operator intro duced by Bloch [11]
+L(B) =¯h2
+2µδ(r−a)/parenleftBiggd
+dr−B
+r/parenrightBigg
+. (3.6)
+The operator Hl+L(B) is Hermitian over (0 ,a) whenBis real [11]. Moreover it has a
+fully discrete spectrum as the self-adjoint problem is defin ed over a finite interval.
+The Schr¨ odinger equation in the internal region is approxi mated by the
+inhomogeneous Bloch-Schr¨ odinger equation
+(Hl+L(B)−E)uint
+l=L(B)uext
+l, (3.7)
+where the external solution is used in the right-hand member . The mathematical
+problem is complemented with the continuity condition
+uint
+l(a) =uext
+l(a). (3.8)
+‡From now on, the Sommerfeld parameter ηis implied.
+§As defined here, the Rmatrix is dimensionless. In some works, it has the dimension of a length and
+differs from the present definition by a factor a. The definition (3.21) of the reduced width amplitudes
+must then be modified accordingly.TheR-matrix Theory 19
+Until now, the approximation only consists in using in the rig ht-hand side of (3.7) the
+asymptotic form (2.19) which is known except for the value of the scattering matrix Ul.
+The main advantage of the R-matrix method is that an expansion in square-integrable
+functions can now be used in the internal region.
+Because of the Dirac function in the Bloch operator, (3.7) an d (3.8) are equivalent
+to the Schr¨ odinger equation (3.1) restricted to the interv al (0,a) supplemented by the
+continuity condition at r=a[11],
+uint
+l′(a) =uext
+l′(a) (3.9)
+for anyB. Hence, beyond making Hl+L(B) Hermitian, the Bloch operator enforces
+the continuity of the derivative of the wave function. The im portance of this aspect
+of the Bloch operator has often been underestimated in the li terature. Condition (3.9)
+needs not be imposed to the basis functions ϕjsince the Bloch operator will impose it
+to the physical solution ul. For historical reasons, a lot of confusion about the Rmatrix
+arose from the misunderstanding of this property as we shall see in section 3.5.
+Formally, the inhomogeneous Bloch-Schr¨ odinger equation (3.7) can be solved with
+the Green function defined by
+(Hl+L(B)−E)Gl(r,r′) =δ(r−r′) (3.10)
+andGl(0,r) = 0. The solution reads
+uint
+l(r) =/integraldisplaya
+0Gl(r,r′)L(B)uext
+l(r′)dr′. (3.11)
+With (3.6) and (3.5), the Rmatrix is thus given by
+Rl(E) =¯h2
+2µaGl(a,a). (3.12)
+Thecalculable R-matrixmethodconsistsinsolvingtheBloch-Schr¨ odinger equationwith
+an approximate Green function expanded over a finite basis.
+To obtain a practical expression for (3.12), expansion (3.4 ) is introduced in (3.7)
+and the resulting equation is projected on ϕi(r), giving
+N/summationdisplay
+j=1Cij(E,B)cj=¯h2
+2µaϕi(a)/parenleftBig
+auext
+l′(a)−Buext
+l(a)/parenrightBig
+. (3.13)
+The elements of the symmetric matrix Care defined as
+Cij(E,B) =/angbracketleftϕi|Tl+L(B)+V−E|ϕj/angbracketright. (3.14)
+Dirac brackets correspond here to one-dimensional integra ls over the variable rfrom 0
+toa. Because of the Bloch operator, the right-hand side of (3.13 ) only involves values
+atr=a.
+Coefficients cjare obtained by solving system (3.13). Introducing them in ( 3.4) at
+r=aand comparing with (3.5) provides the calculable Rmatrix
+Rl(E,B) =¯h2
+2µaN/summationdisplay
+i,j=1ϕi(a)(C−1)ijϕj(a). (3.15)TheR-matrix Theory 20
+This expression is nothing but a finite-basis approximation of (3.12).
+The wave function in the internal region is given by
+uint
+l(r) =¯h2
+2µaRl(E,B)uext
+l(a)N/summationdisplay
+j=1ϕj(r)N/summationdisplay
+i=1(C−1)ijϕi(a). (3.16)
+We shall see in section 3.4 that it does not depend on B.
+3.3. Properties of the Rmatrix
+Temporarily, in this section, the basis functions ϕi(r) are assumed to be orthonormal.
+The matrix of overlaps /angbracketleftϕi|ϕj/angbracketrightis thus the unit matrix. Let us consider the eigenvalues
+Enland the corresponding normalized eigenvectors vnlof matrix C(0,B),
+C(0,B)vnl=Enlvnl (3.17)
+with the orthonormality property
+vT
+nlvn′l=δnn′. (3.18)
+With the spectral decomposition
+[C(E,B)]−1=N/summationdisplay
+n=1vnlvT
+nl
+Enl−E, (3.19)
+theRfunction (3.15) becomes
+Rl(E,B) =N/summationdisplay
+n=1γ2
+nl
+Enl−E(3.20)
+with
+γnl=/parenleftBigg¯h2
+2µa/parenrightBigg1/2
+φnl(a) (3.21)
+and
+φnl(r) =N/summationdisplay
+i=1vnl,iϕi(r), (3.22)
+wherevnl,iis theith component of vnl. Theγnlare known as the reduced width
+amplitudes and their squares γ2
+nlas the reduced widths [5]. Their interpretation
+is simple. They are proportional to the value at the channel r adius of variational
+approximations φnlof the eigenfunctions of the Hermitian operator Hl+L(B). Those
+corresponding to the lowest energies thus represent approx imate eigenfunctions of the
+physical problem confined over the interval (0 ,a) with logarithmic derivative Batr=a.
+Expression (3.20) looks familiar to practitioners of the R-matrix theory. It is
+however obtained here with a finite basis. The traditional ex pression for the Rmatrix
+is obtained when Ntends towards infinity in a complete basis as
+Rl(E,B) =∞/summationdisplay
+n=1γ2
+nl
+Enl−E. (3.23)
+Theenergies Enlarenowtheexacteigenvaluesoftheoperator Hl+L(B)andthereduced
+width amplitudes γnlare related to the values at r=aof its exact eigenfunctions.TheR-matrix Theory 21
+TheRmatrix is a real function when VandBare real. It has an infinity of real
+simple poles, bounded from below. Its derivative is always p ositive at regular points.
+It is a meromorphic function of the energy when the energy is c onsidered as a complex
+variable [5]. All residues are negative and given by minus the reduced widths γ2
+nl.
+3.4. Scattering matrix and phase shifts
+Since theRmatrix is known, the external function (3.3) can be introduc ed in relation
+(3.8) to determine the scattering matrix for the lth partial wave as
+Ul=e2iφl1−(L∗
+l−B)Rl(E,B)
+1−(Ll−B)Rl(E,B). (3.24)
+In this expression,
+Ll=kaO′
+l(ka)
+Ol(ka)(3.25)
+is the dimensionless logarithmic derivative of Olat the channel radius, L∗
+lis the
+conjugate of Ll, and
+φl= argIl(ka) =−arctan[Fl(ka)/Gl(ka)] (3.26)
+is the hard-sphere phase shift. Note that the same notation φlin [5] represents the
+opposite of the hard-sphere phase shift.
+Expression(3.24)hasthestrikingpropertythatitdoesnot dependontheboundary
+parameterB, independently of the size of the basis. Indeed, with the mat rix relation
+(B4) in Appendix B, one deduces from (3.15) and (3.14)
+1
+Rl(E,0)=1
+Rl(E,B)+B (3.27)
+for anyB, real or complex. Expression (3.27) means that the logarith mic derivative of
+the internal wave function at the boundary is independent of B. Introducing relation
+(3.27) into (3.24) shows that any Bvalue leads to the same scattering matrix as for
+B= 0. Equation (3.27) is well known in R-matrix theory (see equation (IV.2.5a) of [5]).
+It is also valid for the phenomenological Rmatrix with a finite number of poles [14].
+However its validity for the approximation (3.24) for any bas is size [29, 57] is sometimes
+overlooked.
+Like the scattering matrix Uland the external wave function uext
+l(r), the internal
+function (3.16) does not depend on the choice for B. Indeed, with the help of relation
+(B3), one easily shows that, for any B, it is equal to the similar expression where Bis
+replaced by zero.
+Forabetterphysicalinterpretationoftheresultswhichis importantinapplications,
+it is convenient to introduce some definitions. To this end, Llis separated into its real
+and imaginary parts as
+Ll=Sl+iPl. (3.28)TheR-matrix Theory 22
+The real part Sland imaginary part PlofLlare called the shift and penetration factors,
+respectively. They depend on energy and on the channel radiu s. The penetration factor
+can be written with the Wronskian relation IlO′
+l−I′
+lOl= 2ias
+Pl(E) =ka
+|Ol(ka)|2=ka
+Fl(ka)2+Gl(ka)2. (3.29)
+It is always positive and increasing [5]. The shift factor re ads
+Sl(E) =Pl(E)[Fl(ka)F′
+l(ka)+Gl(ka)G′
+l(ka)]. (3.30)
+It is always negative for η≥0 [5]. Although we do not know a proof that Slis always
+increasing in the same case, we could not find numerically a co unterexample. As shown
+below, none of these properties is valid in the attractive ca se.
+00.511.5
+0 1 2Pl
+l=3l=2l=1l=0
+-5-4-3-2-10
+0 1 2Sl
+2 2 ( /2 )E aµ
l=4l=3l=2l=1
+Figure 1. Penetration factors Pl(E) (upper panel) and shift factors Sl(E) (lower
+panel) in the neutral case ( η= 0) as a function of Ein units of ¯ h2/2µa2.
+In the neutral case ( η= 0), the penetration factors have simple analytical
+expressions such as
+P0(E) =ka, P 1(E) =(ka)3
+1+(ka)2, ... (3.31)
+They do not vary very fast with energy (see Fig. 1). This figure is universal, i.e.,
+independent of the collision. Notice that the derivative of P0with respect to energy is
+infinite at the origin. This property leads to the special beh aviour of neutron scattering
+in theswave. Penetration factors decrease with the orbital moment umlas expected
+from the occurrence of an increasing centrifugal barrier. T he shift factors read
+S0(E) = 0, S 1(E) =−1
+1+(ka)2, ... (3.32)TheR-matrix Theory 23
+They vary smoothly with energy, starting from the integer va lues−l(see Fig. 1). This
+weak energy dependence is the origin of the Thomas approxima tion [5] where the shift
+factor is assumed to vary linearly in a limited energy range.
+0 1 2Pl10-2
+10-4
+10-6
+10-8
+10-101
+l=3l=2l=1l=0
+l=4
+-5-4-3-2-10
+0 1 2Sl
+2 2 ( /2 )E aµ l=4l=3l=2l=1l=0
+Figure 2. Penetration factors Pl(E) (upper panel) and shift factors Sl(E) (lower
+panel) in the repulsive charged case for a=aBas a function of Ein units of ¯ h2/2µa2.
+The penetration factors are very different according to whet her both particles are
+charged or not. In the repulsive charged case, the energy dep endence of the penetration
+factors is much stronger (notice the logarithmic scale in Fi g. 2). Here we have to choose
+the strength of the Coulomb interaction. Figure 2 correspon ds to a channel radius a
+equal to the Bohr radius aB. The strong dependence at low energies is due to the
+difficulty of penetrating a Coulomb barrier when the scatteri ng energy becomes much
+smaller than the Coulomb barrier. Beyond l= 1, increasing from ltol+ 1 decreases
+the penetration factors by more than an order of magnitude. I n contrast, shift factors
+are much more similar in all cases. In the repulsive charged c ase, the shift factors are
+quite similar to those of the neutral case, except for l= 0 (see Fig. 2).
+In the attractive charged case, the energy dependence of the penetration factors is
+displayed in Fig. 3 for a=aB. It is rather similar to the neutral case except at low
+energies where it starts from a finite value at energy zero. Sh ift factors displayed in
+Fig. 3 resemble other cases. Notice however that S0is positive, and decreasing.
+With definition (3.28), the collision matrix (3.24) becomes
+e2iδl=e2iφl1−(Sl−B)Rl+iPlRl
+1−(Sl−B)Rl−iPlRl. (3.33)TheR-matrix Theory 24
+00.511.522.5
+0 1 2Pl
+l=3l=2l=1l=0
+-4-3-2-101
+0 1 2Sl
+2 2 ( /2 )E aµ l=4l=3l=2l=1l=0
+Figure 3. Penetration factors Pl(E) (upper panel) and shift factors Sl(E) (lower
+panel) in the attractive charged case for a=aBas a function of Ein units of ¯ h2/2µa2.
+One obtains an explicit expression for the phase shift,
+δl=φl+arctanPlRl
+1−(Sl−B)Rl. (3.34)
+The internal wave function (3.16) can be rewritten as
+uint
+l(r) =¯h2
+µaei(δl−1
+2π)Cl(kaPl)1/2
+|1−(Ll−B)Rl|N/summationdisplay
+j=1ϕj(r)N/summationdisplay
+i=1(C−1)ijϕi(a), (3.35)
+which explicitly shows the phase and modulus of uint
+l(up to a global sign). For an
+orthonormal basis, this expression can be rewritten using ( 3.19) and (3.22) as
+uint
+l(r) =¯h2
+µaei(δl−1
+2π)Cl(kaPl)1/2
+|1−(Ll−B)Rl|N/summationdisplay
+n=1φnl(r)φnl(a)
+Enl−E. (3.36)
+3.5. On the basis and boundary parameter choices
+Considerable confusion exists in the literature about the p roperties that basis states
+ϕishould have. Improper choices have led to the introduction o f corrections and to
+attempts to use the boundary parameter Bto correct drawbacks of the basis. However
+we have just seen that the results are independent of the choi ce ofB. It is thus
+worthwhile to devote this section to a clarification of this i ssue that has sometimes led
+to an undeserved reputation of poor convergence for the calc ulableR-matrix method.
+Intheirseminalpaper, WignerandEisenbudwantedtoprovid eaphenomenological
+description of resonances [3]. They did not intend to propos e a technique of resolution.TheR-matrix Theory 25
+To reach their goal they assume that the basis functions all s atisfy (forB= 0) the
+boundary conditions ϕj(0) = 0 and
+aϕ′
+j(a)−Bϕj(a) = 0. (3.37)
+This procedure leads to Rmatrix (3.23). When used as a technique of resolution, the
+finite-basis Rmatrix (3.15) or (3.20) obtained with this procedure does no t converge
+uniformly. The reason is simple. The first derivative of the w ave function (3.16) suffers
+from a discontinuity at r=a[58]. The limit of uint
+l′whenrtends towards ato the left
+is equal touext
+l′(a) but not to uint
+l′(a),
+lim
+r→a−uint
+l′(r) =uext
+l′(a)/negationslash=uint
+l′(a). (3.38)
+Forexample,if B= 0,ϕ′
+j(a)vanishesforall jvaluesandonereadilyseesthat uint
+l′(a) = 0
+at all energies. This property has unfavourable consequenc es on the convergence of
+numerical methods when the basis is truncated since the loga rithmic derivative of the
+external solution depends on the phase shift (and thus on ene rgy) and can not be
+matched with the internal solution (see Figures 5 and 8 in sec tion 4). Buttle [27] has
+proposed a correction to the R-matrix truncation. His idea is to replace the truncated
+part by an analytical approximation, i.e., in practice, by t he same expression for the
+zero potential. Although this correction improves the phase shifts, it does not really
+solve the problem because it does not improve the wave functi ons.
+This problem received a solution with the works of Lane and Ro bson [28, 29, 59].
+Their method was successfully applied in nuclear physics wh ere traditional basis
+functions do not satisfy (3.37) and, on the contrary, displa y a variety of behaviours at
+the channel radius. With oscillator basis functions, accur ate results for neutron-nucleus
+scattering could be obtained [59, 60]. At the same time, a mic roscopic extension of the
+Rmatrix using a fully antisymmetrized two-centre harmonic- oscillator model provided
+accurate phase shifts for collisions between light nuclei w ith few basis states [35, 36] (see
+section 6.2). The success of these calculations relies on th e fact that the Bloch operator
+makes condition (3.37) unnecessary. Since the results do no t depend on B, the choice
+B= 0 was used. A general though economical method for solving c oupled-channel
+problems is described in [41].
+The negative role of condition (3.37) remained long unnotic ed in atomic physics
+where in many cases basis states were imposed to satisfy (3.3 7) [61, 8, 51, 62, 63].
+In the literature, the basis functions are often assumed to b e solutions of a Sturm-
+Liouvilleproblemsatisfying(3.37). Asreviewedin[9], var ioussolutionstothepurported
+convergence problems of the calculable Rmatrix have been proposed, such as using two
+different sizes for the internal region. Another type of solut ion proposed in [64] requires
+a basis with a boundary condition depending on the eigenvalu eEnl. In 1983, Greene
+[65, 66] realized in the context of atomic physics that, in pl ace of the traditional choice
+(3.37) of a common boundary condition to all basis functions atr=a, a variety of
+values for the logarithmic derivatives of the basis functio ns should be far more efficient
+for accurate calculations. He also proposed to optimize the b oundary parameter BTheR-matrix Theory 26
+to have a better connection between the internal and externa l logarithmic derivatives.
+With his variational principle [66] applied to potential sc attering,Bis given at each
+energyEby the generalized eigenvalue problem
+C(E,B)c= 0 (3.39)
+in the present notation. One readily sees [67] that the only e igenvalueBis 1/R(E)
+which, as the logarithmic derivative of the internal wave fu nction, is obviously optimal.
+With this choice, the basis functions allow a perfect matchi ng but at the cost of a new
+calculation of the basis at each energy. As we have shown with ( 3.27), this complication
+is unnecessary since all physical results are independent o fB.
+The first accurate calculations in atomic physics with a basi s that fully ignores
+condition (3.37) can be found in [68, 39, 69, 70]. New developm ents making use of B
+splines now avoid this condition [71, 72, 73]. So let us empha size that the calculable
+Rmatrix does converge accurately when the basis functions ar e a well chosen part of
+a complete set displaying a variety of logarithmic derivati ves at the channel radius a.
+The reason of its accuracy is that the Bloch operator imposes a good matching at the
+boundary [40]. The choice of a boundary parameter Bis irrelevant and the Buttle
+correction is not necessary because the good matching of the internal and external wave
+functions allows an accurate determination of the phase shi ft. Practice has shown that
+this simple solution allows a much smoother connection with the external wave function
+[40, 41]. In opposition to the traditional presentation, th ere is thus no need for a special
+assumption about the behaviour of the basis functions at the boundary.
+3.6. Resonances
+Resonances can be studied in various ways. Each of them may be useful, either in
+calculable applications (section 4.8) or in phenomenologi cal applications (section 5).
+In a first approach, the boundary parameter can be chosen as [1 1, 29]
+B=Ll. (3.40)
+It thus depends on energy. This complex value leads to a compl ex function Rl(E,Ll)
+which is not an Rmatrix in the strict sense since Bis not a real constant. It is the
+function introduced by Kapur and Peierls [4]. Nevertheless i t is also given by expression
+(3.15). Equation (3.24) then takes the simpler form [29, 46]
+Ul=e2iφl[1+(Ll−L∗
+l)Rl(E,Ll)]. (3.41)
+This expression is also valid for complex kvalues ifφlis defined as the phase of
+Il(ka). Since (3.41) has no denominator, a direct relation appear s between a pole of
+the scattering matrix, i.e. a resonance energy, and a pole of the complex Rmatrix.
+This relation is however valid only when Llis calculated at the resonance energy. This
+means that the scattering and Rmatrices have only one common pole and only at
+specific energies. Determining S-matrix poles in this way thus requires some iterative
+procedure. The choice (3.40) for the boundary parameter can be used to analyze theTheR-matrix Theory 27
+mathematical nature of resonances in the complex plane in co upled-channel cases [74].
+However,thesameresultscanalsobeinterpretedwiththetra ditionalRmatrixinvolving
+only real parameters [75]. Let us return to real energies for the other approaches. We
+chooseB= 0 to simplify the presentation.
+Another definition of a resonance energy ERis that it corresponds to the value π/2
+of the resonant part δl−φlof the phase shift. From (3.34), it is therefore defined by
+the equation
+1−Sl(ER)Rl(ER) = 0. (3.42)
+In general this equation must be solved numerically. To defin e the resonance width, let
+us consider the collision matrix (3.33) for energies close t oER. A Taylor expansion of
+Sl(E)Rl(E) forE≈ERprovides the Breit-Wigner approximation
+UBW
+l(E)≈e2iφlER−E+iΓ(E)/2
+ER−E−iΓ(E)/2. (3.43)
+In this expression the (energy-dependent) width of the reso nance is given by
+Γ(E) =2Pl(E)Rl(E)
+[d(SlRl)/dE]E=ER. (3.44)
+Because of the shift of ERwith respect to a pole (see below), Rl(E) can be supposed to
+vary slowly near a narrow resonance (see section 5). The tota l width then reads
+Γ(E) = 2γ2Pl(E) =Pl(E)
+Pl(ER)Γ(ER), (3.45)
+where Γ(ER) is the width at the resonance energy. The reduced width γ2defined by
+(3.45) is given by
+γ2=Rl(ER)/[d(SlRl)/dE]E=ER. (3.46)
+Let us mention that (3.42) may have solutions which do not cor respond to physical
+states. The width of a physical resonance should be small eno ugh to make its lifetime
+longer that the typical collision time.
+In another way of studying a resonance, let us consider an ene rgy very close to a
+poleEnlof theRmatrix. If all terms with n′/negationslash=ncan be neglected, the Rmatrix is
+approximated as Rl(E,0)≈γ2
+nl/(Enl−E). A simple calculation provides
+δl≈φl+arctanγ2
+nlPl(E)
+Enl−γ2
+nlSl(E)−E. (3.47)
+This expression resembles the Breit-Wigner form of the phas e shift
+δBW
+l≈φl+arctan1
+2Γ(E)
+ER−E. (3.48)
+By comparison, one defines the resonance energy
+ER=Enl−γ2
+nlSl(ER) (3.49)
+and the formal width
+Γ(E) = 2γ2
+nlPl(E). (3.50)TheR-matrix Theory 28
+The resonance energy is defined by an implicit equation which can be solved
+approximately (see section 5.2). The width is an energy-dep endent quantity whose
+asymmetric shape depends on the behaviour of Pl. Its relation with a measured width
+is also discussed in section 5.2.
+With (3.21), (3.49) and (3.50), the internal wave function ( 3.36) can be
+approximated at the vicinity of a resonance by
+uint
+l(r)≈ei(δl−1
+2π)Cl/bracketleftBigg¯hvΓ
+(ER−E)2+(Γ/2)2/bracketrightBigg1/2
+φnl(r). (3.51)
+It is thus proportional to an approximate eigenfunction (3. 22) ofHl+L(B) with a
+proportionalityfactorexhibitingtheusualLorentzianen ergydependenceofaresonance.
+Equation (3.51) is at the basis of the so-called bound-state approximations where the
+resonance is described by a square-integrable wave functio n [76].
+3.7. Bound states
+TheR-matrix formalism can be extended to bound states ( EB<0) [37]. In that case
+the external wave function uext
+lis given by
+uext
+l(r) =ClWl(2κBr), (3.52)
+whereWl(x) is a shorthand notation for the Whittaker function (2.10) a nd whereκB
+andηBare the wave number and Sommerfeld parameter, respectively , of the bound
+state. In (3.52), Clis the asymptotic normalization constant (ANC) which determi nes
+the amplitude of the wave function at large distances. This q uantity plays an important
+role in some nuclear reactions of astrophysical interest [7 7] (see section 5.7).
+To determine EB, a convenient choice for the boundary parameter Bin (3.6) is
+B=Ll(EB) =Sl(EB) = 2κBaW′
+l(2κBa)
+Wl(2κBa), (3.53)
+because it suppresses the right-hand side of the Bloch-Schr ¨ odinger equation (3.7). Since
+the wave function is real, Llis real and identical to the shift factor. The penetration
+factorPlvanishes. The internal wave function is expanded over a basi s, as in (3.4).
+Projecting the Bloch-Schr¨ odinger equation (3.7) over a ba sis function ϕiprovides for
+i= 1,N,
+N/summationdisplay
+j=1/angbracketleftϕi|Tl+L(Ll)+V−EB|ϕj/angbracketrightcj= 0. (3.54)
+This system of equations is similar to a standard eigenvalue problem, but parameter Ll
+depends on energy EB. In practice one starts from Ll= 0 and iterates until energy EB
+has converged. At convergence, the cjcan be calculated by solving the system. If uint
+l
+is normed according to (3.18), the square of the norm of the wa ve function is given by
+Nl= 1+(Cl)2/integraldisplay∞
+a(Wl(2κBr))2dr. (3.55)TheR-matrix Theory 29
+This expression can be rewritten [5, 17] as
+Nl= 1+γ2
+nl/bracketleftBiggdSl
+dE/bracketrightBigg
+E=EB(3.56)
+whereγnlis the reduced width amplitude of the bound state.
+The internal wave function is given by (3.16) multiplied by N−1/2
+l,
+uint
+l(r) =N−1/2
+lN/summationdisplay
+j=1vnl,jϕj(r). (3.57)
+where coefficients vnl,jcorrespond for an orthogonal basis to the eigenvector vnlof
+C(0,Ll) at energy Enl=EBin (3.17). Using (3.22), it can be rewritten as
+uint
+l(r) =N−1/2
+lφnl(r). (3.58)
+From (3.8) and (3.52), the ANC is given by
+Cl=N−1/2
+lφnl(a)/Wl(2κBa) (3.59)
+or with (3.21) by [78]
+Cl= (2µa/¯h2Nl)1/2γnl/Wl(2κBa). (3.60)
+It should be independent of radius a. This relation between the ANC and a reduced
+width amplitude which corresponds to a vanishing width can b e useful for the
+phenomenological Rmatrix. A similar formalism can be applied to resonances [79 ],
+and provides widths as well as energies of resonances.
+3.8. Capture cross sections
+The determination of capture cross sections requires the ca lculation of matrix elements
+of the electromagnetic multipole operators Mσλ
+µbetween an initial scattering state and
+a final bound state. One can take into account the division of t he configuration space
+in the general case [37] but the principle of the calculation can be explained more easily
+in the simple potential model using equations (2.57), (2.58 ) and (2.64).
+According to the R-matrix framework, the radial matrix element between an ini tial
+scattering wave function ui(r)≡uJi
+Ifli(Il)(r) and a final bound-state wave function
+uf(r)≡uJf
+Iflf(r) appearing in (2.64) can be written for an electric multipol e as
+/integraldisplay∞
+0ufrλuidr=/integraldisplaya
+0uint
+frλuint
+idr+/integraldisplay∞
+auext
+frλuext
+idr. (3.61)
+The internal matrix element is given by
+/integraldisplaya
+0uint
+frλuint
+idr=/summationdisplay
+k,k′cf,k′ci,k/integraldisplaya
+0ϕk′rλϕkdr, (3.62)
+where coefficients cf,k′andci,kare related to the final and initial wave functions,
+respectively. Notice that the bases could be different for bot h states. In the external
+region, we have
+/integraldisplay∞
+auext
+frλuext
+idr=CiCf/integraldisplay∞
+aWlf(2κBr)rλ(Ili(kr)−UJi
+Ifli,IlOli(kr))dr. (3.63)TheR-matrix Theory 30
+As for elastic scattering, the total matrix element (3.61) sh ould not depend on the
+channel radius a, whereas each contribution does depend on a. For transitions to weakly
+bound states, the external term is dominant [76] since the Wh ittaker function slowly
+decreases at large distances. Neglecting completely the int ernal contribution leads to
+the “external-capture”model [80]. A typical example is the7Be(p,γ)8B reaction where
+the ground state is bound by 137 keV only [77]. On the contrary , resonant reactions, or
+transitions to deeply bound states provide a dominant inter nal term.
+3.9. Propagation methods
+For long-range potentials, the channel radius may become ve ry large. This may induce a
+prohibitivesizeforthe R-matrixbasisandleadtolongcomputationtimes. Thisprobl em
+was first met in atomic physics because of the long tail of pola rization potentials and
+led to the development of propagation methods [32, 81, 82]. T his situation can also
+occur in nuclear physics, for example, in three-body system s [83] or in coupled-channel
+calculations [84].
+Different methods have been proposed to address this problem . The basic idea is,
+either to propagate the wave function or the R-matrix over several intervals on which
+the basis size remains reasonable, or to determine “distort ed” Coulomb functions valid
+at distances shorter than the channel radius. We present her e a propagation method
+directly derived for the R-matrix (see for example [82]).
+The idea of propagation methods [32] is to divide the interna l region (0,a) inNs
+subregions ( aα−1,aα) forα= 1,Ns, witha0= 0 andaNs=a. The intermediate radii
+aαcan be chosen equidistant, but this is not mandatory. The wid th of the intervals
+and the basis size in each interval can also depend on the numb er of oscillations of the
+wave function [81, 82]. With small intervals, the R-matrix bases remain limited, but
+the number Nsof repetitions of the calculation may be large. Approximatio ns of the
+potentialcanoftenbeemployedinsomeintervals. Insomeca ses, thesizeoftheintervals
+is chosen small enough so that the potential may be considere d as constant which allows
+an analytical propagation [32].
+We briefly present here the principle of the propagation tech nique for potential
+scattering with a basis in each interval. We refer to referen ces [81, 82] for a multichannel
+extension. A Bloch operator Lαis now defined at each boundary aαas
+Lα=¯h2
+2µδ(r−aα)d
+dr, α= 1,Ns. (3.64)
+The Bloch-Schr¨ odinger equation is replaced by a set of equa tions
+(Hl+Lα−Lα−1−E)ul,α= (Lα−Lα−1)ul,α, α= 1,Ns(3.65)
+withL0= 0. The boundary conditions are ul,1(0) = 0 and
+ul,α(aα) =ul,α+1(aα), α= 1,Ns (3.66)
+withul,Ns+1≡uext
+l. In each interval, the wave function is expanded over a set of NαTheR-matrix Theory 31
+basis functions
+ul,α(r) =Nα/summationdisplay
+j=1cα
+jϕα
+j(r). (3.67)
+Equations (3.65) can be solved with approximate Green funct ions as in section 3.2. By
+projecting (3.65) on one of the basis functions, the solutio ns can be approximated as
+ul,α(r) =/summationdisplay
+jj′(C−1
+α)jj′/angbracketleftϕα
+j′|Lα−Lα−1|ul,α/angbracketrightϕα
+j(r), (3.68)
+where the symmetric matrix Cαis defined in each interval by
+Cα,ii′=/angbracketleftϕα
+i|Hl+Lα−Lα−1−E|ϕα
+i′/angbracketright. (3.69)
+The Dirac notation represents an integration limited to the range (aα−1,aα).
+For each interval, (3.68) can be used to determine a relation between values of the
+wave function and its first derivative at aα−1andaα,
+ul,α(aα−1) =Rα
+10u′
+l,α(aα)−Rα
+11u′
+l,α(aα−1), (3.70)
+ul,α(aα) =Rα
+00u′
+l,α(aα)−Rα
+01u′
+l,α(aα−1), (3.71)
+where various values of the approximate Green functions Rα
+ββ′are defined as
+Rα
+ββ′=¯h2
+2µ/summationdisplay
+jj′ϕα
+j(aα−β)(C−1
+α)jj′ϕα
+j′(aα−β′) (3.72)
+withββ′= 0,1.
+AnRmatrix can be defined at each boundary with an extension of (3. 5) as
+ul,α(aα) =aαR(aα)u′
+l,α(aα), α= 1,Ns. (3.73)
+Equations (3.70), (3.71) and (3.73) provide relationships betweenRmatrices at aα−1
+andaα(α= 2,Ns),
+aα−1R(aα−1) =−Rα
+11+Rα
+10[Rα
+00−aαR(aα)]−1Rα
+01, (3.74)
+aαR(aα) =Rα
+00−Rα
+01[Rα
+11+aα−1R(aα−1)]−1Rα
+10. (3.75)
+The latter equation provides an outwards propagation [ R(aα) fromR(aα−1)] and the
+former provides a backwards propagation [ R(aα−1) fromR(aα)]. This technique is
+quite efficient in multichannel calculations. Numerically th e main part of the R-matrix
+calculation arises in the inversion of matrices Cα[see (3.72)]. It may save computer
+time to diagonalize them when many energies are needed. The s ize of these matrices is
+given by the number of basis functions times the number of cha nnels. If the calculation
+involves many channels, reducing the number of basis functi ons may lead to a significant
+reduction of the computer times.
+The collision matrix is obtained from R(aNs) with (3.24). The external wave
+functionuext
+lis thus known. By starting from the last interval, the wave fu nction is
+determined by its coefficients in each interval. If the Lα−1terms are simplified in (3.65),
+one obtains the system
+/summationdisplay
+j′/bracketleftBig
+Cα,jj′+/angbracketleftϕα
+j|Lα−1|ϕα
+j′/angbracketright/bracketrightBig
+cα
+j′=/angbracketleftϕα
+j|Lα|ul,α/angbracketright (3.76)TheR-matrix Theory 32
+where (3.73) can be used to eliminate ul,α′(aα) in the right-hand side. The matrix in
+this system can be inverted with (B2). Starting with α=Nsand going backwards one
+obtains the coefficients in all intervals.
+Let us briefly discuss other techniques dealing with long-ra nge potentials. In [83],
+in the framework of three-body continuum states, we propaga te the wave functions from
+aα−1toaαby using the Numerov algorithm. This approach avoids the choi ce of a basis,
+but requires longer computer times. It is also difficult to app ly to non-local potentials.
+TheLight-Walkerpropagationconsistsinapproximatingth epotentialbyaconstant
+in small enough intervals [32]. In [84], this method is impro ved by considering linear
+approximations of the potential. In the method suggested by Gailitis [85], the Coulomb
+wave functions are modified by 1 /rexpansions, which can be used at short distances
+(see also [86, 82]).
+3.10. Extension to multichannel collisions
+Until now the presentation was, for the sake of clarity and sim plicity, limited to single-
+channel calculations which also neglect the spins of the col liding particles. This does not
+affect the general properties of the R-matrix theory. However many problems require a
+multichannel approach.
+In a many-body problem, the Schr¨ odinger equation (2.33) in volves Hamiltonian
+(2.25). The total wave function ΨJMπof the system, with total angular momentum J
+and parity π, is expanded over a set of channel functions (2.29), denoted as|c/angbracketright. If we
+assume that a single relative coordinate appears in the prob lem or that we approximate
+all relative coordinates by a single one, the wave function i s given by (2.43). The
+Schr¨ odinger equation is replaced by a set of differential eq uations
+/summationdisplay
+c′/bracketleftBig
+(Tc+Ec−E)δcc′+Vcc′/bracketrightBig
+uc′= 0, (3.77)
+where, as before, Tcincludes the centrifugal term. This system is a particular c ase or a
+local approximation of (2.38).
+A typical example is given by coupled-channel calculations where channel functions
+aredefinedby(2.29). Thissituationoccurs, forexample, in coupled-channelcalculations
+with a discretized continuum (CDCC) [87], see section 6.3. An other example
+corresponds to three-body scattering, where the channel fu nctions|c/angbracketrightcontain various
+quantum numbers defined in the hyperspherical formalism [88 ], see section 6.4.
+In all cases the problem is first to determine the potentials Vcc′and then to solve
+(3.77) for positive energies. Here we essentially deal with t he second step. We thus
+assume that the potentials are known and that they present th e asymptotic behaviour
+Vcc′−→r→∞Z1cZ2ce2
+rδcc′. (3.78)
+In these conditions the asymptotic form of the radial wave fu nctionsuc(r) is given byTheR-matrix Theory 33
+(2.34) and the radial wave functions in the external region a t energyEare defined as
+uext
+c(c0)(r) =
+
+v−1/2
+c/parenleftBig
+Ic(kcr)δcc0−Ucc0Oc(kcr)/parenrightBig
+forEc<E
+Acc0W−ηc,l+1
+2(2κcr) for Ec>E,(3.79)
+wherec0is the entrance channel ( Ec0<E).
+The Bloch operator (3.6) is defined in the multichannel forma lism as
+L=/summationdisplay
+c|c/angbracketrightLc/angbracketleftc|,Lc=¯h2
+2µcδ(r−a)/parenleftBiggd
+dr−Bc
+r/parenrightBigg
+, (3.80)
+where coefficients Bcare chosen as zero or as in (3.53) for open and closed channels ,
+respectively. Notice that these coefficients then depend on en ergy for closed channels.
+This choice may be inefficient in some variants of the Rmatrix [59] but is convenient in
+(3.88) and (3.89) below. The Bloch-Schr¨ odinger equation i s given by
+/summationdisplay
+c′/bracketleftBig
+(Tc+Lc+Ec−E)δcc′+Vcc′/bracketrightBig
+uint
+c′(r) =Lcuext
+c. (3.81)
+The internal parts of the radial wave functions are expanded over a basis ϕj(r),
+uint
+c(r) =N/summationdisplay
+j=1ccjϕj(r). (3.82)
+The determination of the Rmatrix and of the collision matrix Uare direct extensions
+of the formalism developed in sections 3.2 and 3.4.
+TheRmatrix is defined as
+uc(a) =/summationdisplay
+c′(µc/µc′)1/2Rcc′[au′
+c′(a)−Bc′uc′(a)]. (3.83)
+MatrixRis symmetric, with elements given by
+Rcc′(E) =¯h2
+2√µcµc′aN/summationdisplay
+i,i′=1ϕi(a)(C−1)ci,c′i′ϕi′(a), (3.84)
+where
+Cci,c′i′=/angbracketleftϕi|Tc+Lc+Ec−E|ϕi′/angbracketrightδcc′+/angbracketleftϕi|Vcc′|ϕi′/angbracketright. (3.85)
+Like in section 3.3, the spectral decomposition of the symme tric matrix Cfor an
+orthonormal basis provides the canonical form of the multic hannelRmatrix as
+Rcc′(E) =/summationdisplay
+nγncγnc′
+En−E(3.86)
+where the real poles Enare the eigenvalues of Cand the reduced-width amplitude of
+poleEnin channelcis expressed as a function of the components of the correspon ding
+normed eigenvector vnas
+γnc=/parenleftBigg¯h2
+2µca/parenrightBigg1/2N/summationdisplay
+i=1vn,ciϕi(a). (3.87)
+Thenumberoftermsin thesum (3.86) is given bytheproductof thenumberofchannels
+byN.TheR-matrix Theory 34
+The collision matrix is obtained with
+U=Z−1Z∗, (3.88)
+where an element of matrix Zreads
+Zcc′= (kc′a)−1/2/bracketleftBig
+Oc(kca)δcc′−kc′aRcc′O′
+c′(kc′a)/bracketrightBig
+. (3.89)
+For complex potentials as encountered in the optical model, expression (3.88) must be
+modified into
+U=Z−1
+OZI, (3.90)
+whereZO=ZandZI/negationslash=Z∗
+Ois given by a similar expression with outgoing functions
+replaced by incoming ones. Matrix Uis then not unitary.
+The dimension of the Rmatrix is equal to the number of channels included in
+the calculation; it does not depend on energy. On the contrar y the dimension of the
+collision matrix Uis given by the number of openchannels and may vary with energy.
+In (3.89), only open channels contribute thanks to the choic e of boundary parameters
+Bc. TheRmatrix can be modified by eliminating the closed channels. Wh en not all
+channels are open, let us denote open channels by cand closed channels by ¯ c. Equation
+(3.84) remains valid with matrix Creplaced by a smaller open-channel matrix Cowith
+elements [89, 5, 41]
+Co
+ci,c′i′=Cci,c′i′−/summationdisplay
+¯cj,¯c′j′/angbracketleftϕi|Vc¯c|ϕj/angbracketright(¯C−1)¯cj,¯c′j′/angbracketleftϕj′|V¯c′c′|ϕi′/angbracketright, (3.91)
+where¯Cis the restriction of the full matrix to closed channels and ¯ c, ¯c′are the
+corresponding indices.
+The internal components of the wave function are given for bo th open and closed
+channels by
+uint
+c(c0)(r) =/summationdisplay
+c′¯h2kc′
+2µc′a/bracketleftBig
+I′
+c′(kc′a)δc′c0−Uc′c0O′
+c′(kc′a)/bracketrightBigN/summationdisplay
+i,i′=1ϕi(r)(C−1)ci,c′i′ϕi′(a),(3.92)
+where the sum over c′runs over open channels only. The coefficients of the external
+part of the wave function for closed channels read
+A¯cc0= [W−η¯c,l+1
+2(2κ¯ca)]−1/summationdisplay
+c′(µckc′/¯h)1/2aR¯cc′/bracketleftBig
+I′
+c′(kc′a)δc′c0−Uc′c0O′
+c′(kc′a)/bracketrightBig
+,(3.93)
+where the sum over c′also runs over open channels only.TheR-matrix Theory 35
+4. Applications of the calculable Rmatrix
+4.1. Conditions of the calculations
+In this section we apply the calculable R-matrix method to the scattering by a potential.
+Our goal is not to fit experimental data, but to illustrate the method under different
+conditions, and with different types of basis wave functions . In this approach, both
+colliding particles are assumed to be structureless, and to interact through a potential
+[see (3.2)]. In general, this potential V(r) involves a local term U(r) and a non-local
+termW(r,r′) such that
+V(r)ul(r) =U(r)ul(r)+/integraldisplay∞
+0W(r,r′)ul(r′)dr′. (4.1)
+In the following, unless specified otherwise, only the local term is included. In nuclear
+physics applications, the local potential U(r) usually contains a nuclear and a Coulomb
+contributions, denoted as VN(r) andVC(r), respectively. The potential may be l
+dependent, by adapting its parameters to the lvalue. The Schr¨ odinger equation
+associated with potential (4.1) can be solved exactly (at th e computer precision) with
+the Numerov method [90, 91, 92]. The exact phase shifts and wav e functions will be of
+course compared with the R-matrix calculations.
+In nuclear-physics applications, the reduced mass is expre ssed in terms of the
+nucleon mass as
+µ=A1A2
+A1+A2mN, (4.2)
+whereA1andA2are the nucleon numbers. The calculations are performed wit h
+¯h2/2mN= 20.736 MeV.fm2ande2= 1.44 MeV.fm.
+Tocoverabroadvarietyofapplications, wehavefirstselect edthreetypicalsystems:
+(i) The12C+p system ( l= 0) which presents a narrow resonance (Γ = 37 keV) at
+E= 0.42 MeV.
+(ii) Theα+αsystem (l= 4) which presents a broad resonance (Γ = 3 .5 MeV) near
+E= 11.3 MeV.
+(iii) Theα+3He system ( l= 0) which is non resonant.
+Thesecasesaretypicalexamplesofnuclear-physicsapplic ations, andwillbetreated
+by different types of basis functions. Here, we only consider r eal potentials. This is
+consistent with the low-energy regime where the R-matrix method is well adapted. The
+extension to complex potentials is however trivial.
+Then, the calculation of e−−H phase shifts will provide a typical example of an
+atomic-physics application with a non-local potential. At omic units will be used.
+4.2. Basis functions
+We consider different families of basis functions ϕi(r) [i= 1,...,N, see (3.4)] vanishing
+atr= 0, commonly used in the literature. The corresponding matr ix elements are given
+in Appendix C.TheR-matrix Theory 36
+(i)Sine functions
+The basis functions are given by
+ϕi(r) = sinπr
+a(i−1/2). (4.3)
+This choice seems natural since it can simulate the oscillat ing behaviour of the wave
+function near the channel radius. However we will show that th ese basis functions
+are not suitable for accurate calculations since the deriva tive satisfies (3.37) (see
+section 3.5), i.e. it vanishes at r=afor eachivalue,
+ϕ′
+i(a) = 0. (4.4)
+Thispropertymeansthatanyfinitecombinationofthesebasi sfunctionswillpresent
+a zero derivative at r=a. As discussed before, the matching between the internal
+and external solutions is therefore expected to be poor, and theR-matrix phase
+shifts rather inaccurate.
+(ii)Gaussian functions
+The basis functions have a Gaussian dependence [43] with diff erent size parameters
+ϕi(r) =rl+1exp(−(r/bi)2), (4.5)
+where thebiare chosen as a geometric progression
+bi=b1xi−1
+0, (4.6)
+and are therefore determined by a set of 2 parameters ( b1,x0). Gaussian-type basis
+functions (4.5) allow an analytical calculation of the over lap and kinetic-energy
+matrix elements. For Gaussian potentials, the matrix eleme nts are analytical as
+well.
+(iii)Lagrange functions
+The Lagrange functions [42] are defined in the (0 ,a) interval as
+ϕi(r) = (−1)N+i/parenleftbiggr
+axi/parenrightbiggn/radicalBig
+axi(1−xi)PN(2r/a−1)
+r−axi, (4.7)
+wherePNis the Legendre polynomial of order N, andxiare the zeros of
+PN(2xi−1) = 0. (4.8)
+In (4.7), the factor ( r/axi)nis aimed at regularizing the basis function at the origin.
+For two-body calculations defined in the (0 ,a) range, we use n= 1, which ensures
+that the wave function vanishes at the origin.
+The basis functions satisfy the Lagrange conditions
+ϕi(axj) = (aλi)−1/2δij, (4.9)
+whereλiis the weight of the Gauss-Legendre quadrature correspondi ng to the (0 ,1)
+interval. This basis is exactly equivalent to the Legendre b asis defined by
+ϕi(r) =rPi−1(2r/a−1). (4.10)TheR-matrix Theory 37
+However, if the matrix elements with basis functions (4.7) ar e computed at
+the Gauss approximation of order N, consistent with the Nmesh points, their
+calculation is strongly simplified. At this approximation, the overlap is given by
+/angbracketleftϕi|ϕj/angbracketright=/integraldisplaya
+0ϕi(r)ϕj(r)dr≈δij, (4.11)
+and the local-potential matrix is diagonal with elements ju st given by the value of
+the potential at the mesh points,
+/angbracketleftϕi|U|ϕj/angbracketright=/integraldisplaya
+0ϕi(r)U(r)ϕj(r)dr≈U(axi)δij. (4.12)
+This calculation is easily extended to non-local potential sW(r,r′)
+/angbracketleftϕi|W|ϕj/angbracketright ≈a/radicalBig
+λiλjW(axi,axj). (4.13)
+These matrix elements do not need any evaluation of integral . The kinetic energy
+term is given by analytical expressions [93] (see Appendix C) . This method has
+been shown to be quite efficient and accurate in various fields f or bound states
+[93] as well as for scattering states [42, 83]. In the present applications, the Gauss
+approximation will be used systematically, except in the e−−H scattering, where
+this approximation will be also tested by a Simpson quadratu re for the matrix
+elements.
+4.3. Application to a narrow resonance:12C+p
+The12C+p system is known to present a narrow resonance ( Jπ= 1/2+,ER= 0.42
+MeV, Γ = 37 keV [94]) at low energies. As mentioned before, our ai m is not to fit these
+properties accurately, but to compare the exact solutions o f the Schr¨ odinger equation
+(3.1) with the R-matrix approach. For the nuclear and Coulomb potentials we choose
+VN(r) =−73.8exp(−(r/2.70)2),
+VC(r) = 6e2/r. (4.14)
+In what follows, the units in the nuclear potentials are fm an d MeV for lengths and
+energies, respectively. This potential reproduces the res onant behaviour of the sphase
+shifts at 0.42 MeV. To simulate the Pauli principle, a nucleus -nucleus potential may
+contain additional (unphysical) bound states [95, 96]. The se Pauli forbidden states
+show up in microscopic calculations, where the internal str ucture of the colliding nuclei
+istakenintoaccount, buttheireffectcanbepartlysimulate dinnon-microscopictheories
+by additional bound states in the potential (see [95, 96] for details). For the12C+p
+system, the l= 0 potential contains one forbidden state.
+InFig.4, wepresenttheexactphaseshiftsand R-matrixcalculationswithLagrange
+(a) and Sine (b) functions. The channel radius is chosen as a= 8 fm, where the
+nuclear interaction is negligible. Fig. 4 (a) shows that wit h the Lagrange functions, the
+convergenceisreachedwith N≥10. Weshowanexamplewith N= 7,where Nisaimed
+at being not large enough. Fig. 4 (b) illustrates the Sine fun ctions, poorly adapted to a
+good matching since the left derivative of the wave function atr=ais zero [see (4.4)].TheR-matrix Theory 38
+060120180
+(a)N=7N=10Lagrange, a=8 fm
+060120180
+0 0.5 1 1.5 2
+E (MeV)(c)a=5a=6a=7060120180
+(b)Sine, a=8 fm
+N=20
+N=15
+N=10
+N=7exactδ (deg.)
+Figure 4.12C+pR-matrix phase shifts (in degrees) for different bases and con ditions
+(l= 0). (a) Lagrange functions at a= 8 fm (the exact results are superimposed to the
+N= 10 curve), (b) Sine functions at a= 8 fm, (c) Convergence as a function of afor
+N= 15; the Lagrange and Gaussian results are superimposed ( a= 7 fm corresponds
+to the exact results).
+EvenN= 20 is far from the exact calculation. In Fig. 4 (c), we presen t the convergence
+as a function of the channel radius with the Lagrange functio ns (N= 15 is fixed).
+Results obtained with Gaussian functions (with b1= 1.4 fm,x0= 0.6) are identical at
+the scale of the figure. As expected, a= 5 fm is too small ( |VN(a)/VC(a)|= 1.4). To
+obtain a good stability, radii larger than 6 fm should be used .
+The matching problem is illustrated in Fig. 5, where we show t he wave function
+atE= 2 MeV, with a= 8 fm,N= 15. With the Lagrange functions, the matching
+between internal and external wave functions is quite smoot h. For Sine functions, the
+matching is poor, which has a direct impact on the phase shift s.
+The accuracy of the different basis functions is illustrated in Table 1 where we
+compare various calculations of the phase shifts. The relat ive difference between leftTheR-matrix Theory 39
+-1.2-0.8-0.400.40.8
+r (fm)u0(r) (fm-1/2)Lagrange
+Sine2 8 6 4
+Figure 5.12C+pl= 0 wave functions with Lagrange and Sine functions at E= 2
+MeV (a= 8 fm,N= 15). Solid lines represent the internal wave function, and dotted
+lines the external parts. The Lagrange wave function is supe rimposed to the exact
+result.
+Table 1.12C+pl= 0 phase shifts (in degrees) and matching parameters (4.15) for
+different bases ( a= 8 fm).
+phase shift matching parameter
+E(MeV) exact N= 7N= 10N= 15N= 7N= 10N= 15
+Lagrange
+0.5 154.66 112.90 154.94 154.59 5.96 0.65 0.01
+1.0 147.48 144.22 147.55 147.48 3.68 0.28 0.00
+1.5 133.30 311.02 133.35 133.30 1.67 0.20 0.00
+2.0 121.18 299.30 121.23 121.18 1.21 0.16 0.00
+Gaussian
+0.5 154.66 179.36 154.53 154.53 5.61 0.02 0.01
+1.0 147.48 146.52 147.47 147.47 0.04 0.01 0.01
+1.5 133.30 130.28 133.29 133.29 1.06 0.00 0.01
+2.0 121.18 112.15 121.18 121.18 1.55 0.01 0.01
+Sine
+0.5 154.66 2.03 3.74 6.68 0.12 0.66 1.54
+1.0 147.48 28.90 66.57 109.98 1.30 4.66 29.35
+1.5 133.30 63.84 94.53 113.28 11.46 6.97 3.53
+2.0 121.18 71.76 92.24 105.34 5.73 3.14 2.38
+and right derivatives provides the matching parameter ǫdefined as
+ǫ=|uext
+l′(a)−uint
+l′(a)|
+[uext
+l′(a)+uint
+l′(a)]/2. (4.15)TheR-matrix Theory 40
+This comparison is done for the different bases and at four typ ical energies. When ǫ
+is small, the matching is obviously accurate. However, in som e specific cases where
+uext
+l′(a) +uint
+l′(a) is small, this parameter may be large, although the phase sh ift is
+fairly good. This parameter should therefore be considered as indicative only. Table 1
+confirms that Lagrange and Gaussian functions represent a co nvenient basis, whereas
+Sine functions do not provide satisfactory results. Note tha t the use of Gaussian
+functions implies two additional parameters ( b1,x0) to define the basis. This choice has
+to be optimized for the different conditions and is therefore less direct than Lagrange
+functions.
+4.4. Application to a broad resonance: α+α
+Theα+αsystem presents a well known rotational band based on the 0+ground state.
+Thel= 0 narrow resonance corresponds to the unstable ground stat e of8Be. In order
+to illustrate the R-matrix formalism applied to a broad resonance, we consider thel= 4
+partial wave. Theexperimental energy andwidth ofthereson anceareER= 11.35±0.15
+MeVandΓ ≈3.5MeV[97]. However, forbroadresonancesthesepropertiesma ydepend
+on the definition used (see section 3.6). A comparison betwee n experiment and theory
+cannot be done with a high precision.
+Theα+αpotential of Buck et al.[98] has a simple Gaussian form, is l-independent,
+and reproduces the l= 0,2,4 experimental α+αphase shifts up to about 20 MeV. It
+is defined by
+VN(r) =−122.6225exp( −(r/2.132)2),
+VC(r) = 4e2erf(r/1.33)/r. (4.16)
+Fig. 6 shows the phase shifts obtained with the Lagrange and S ine functions, with
+a= 8 fm. With Lagrange functions, the R-matrix perfectly reproduces the exact phase
+shifts with N≥10. With Sine functions, the convergence is, as expected, mu ch slower.
+Table 2 gives the phase shifts and ǫvalues under different conditions and at different
+energies. Again, the importance of the matching on the phase s hifts is obvious.
+4.5. Application to a non-resonant system: α+3He
+Forl= 0, the experimental α+3He phase shifts are well reproduced by the potential of
+Buck and Merchant [99],
+VN(r) =−66.10exp(−(r/2.52)2),
+VC(r) = 4e2(3−(r/rC)2)/2rCforr≤rC,
+= 4e2/rforr≥rC, (4.17)
+withrC= 3.095 fm. In Fig. 7, we use the Lagrange functions for two radii: a= 8 fm,
+anda= 5 fm. For a= 8 fm, the convergence is reached as soon as N≥10. On the
+contrary the choice a= 5 fm is not consistent with one of the R-matrix requirements:TheR-matrix Theory 41
+060120180
+0 5 10 15 20exactN=7Lagrange
+060120180
+0 5 10 15 20Sine
+E (MeV)N=15
+exact
+N=7δ (deg.)
+Figure 6. α+α l= 4 phase shifts with the Lagrange and Sine functions for diffe rent
+Nvalues. For the Lagrange functions N≥10 is indistinguishable from the exact
+result.
+Table 2. α+α l= 4 phase shifts (in degrees) and matching parameters for Lag range
+and Sine bases ( a= 8 fm).
+phase shift matching parameter
+E(MeV) exact N= 7N= 10N= 15N= 7N= 10N= 15
+Lagrange
+5 0.79 0.81 0.80 0.80 0.05 0.02 0.00
+10 24.46 26.19 24.58 24.48 0.42 0.12 0.00
+15 125.88 137.90 126.00 125.88 0.31 0.18 0.01
+20 141.55 151.24 141.53 141.54 1.26 5.67 0.13
+Sine
+5 0.79 0.53 0.62 0.68 0.66 0.65 0.65
+10 24.46 15.55 17.74 19.84 2.23 2.20 2.17
+15 125.88 109.09 116.76 119.87 2.16 2.09 2.06
+20 141.55 139.82 141.24 141.37 3.27 5.84 6.21
+the nuclear contribution is not negligible. In that case, it is impossible to get a good
+convergence at all energies.
+In Fig. 8, we compare the wave functions at E= 8 MeV with the Sine and LagrangeTheR-matrix Theory 42
+-240-180-120-600
+0 5 10 15 20
+N=7N=10
+Lagrange, a=8 fm
+-180-120-600
+0 5 10 15 20
+N=7N=15
+Lagrange, a=5 fm
+-240-180-120-600
+0 5 10 15 20
+Sine, a=8 fmN=7N=15N=10
+E (MeV)δ (deg.)
+Figure 7. α+3HeR-matrix phase shifts with Lagrange ( a= 8 fm and a= 5 fm) and
+Sine (a= 8 fm) functions for l= 0. The thick lines represent the exact phase shifts.
+functions. In the former case, the matching between the inte rnal and external wave
+functions is poor, which has an impact on the accuracy of the p hase shifts. Conversely,
+theR-matrix wave function with the Lagrange basis is indistingu ishable from the exact
+wave function.
+Table 3 gives the phase shifts and matching coefficients under different conditions.
+Fora= 8 fm,N≥10 provides reasonable phase shifts. If aincreases (we choose here
+a= 10 fm), the R-matrix approximation is of course still valid, but the numb er of basis
+functions must be increased. This is expected as the interna l wave function must be
+reproduced in a wider range. In agreement with Fig. 8, the Sin e functions provide a
+poor approximation of the phase shifts.TheR-matrix Theory 43
+-1.5-1.0-0.50.00.51.01.5
+0 2 4 6 8 10 12Sine, a=8 fm, N=15
+r (fm)
+sineexactu0(r) (fm-1/2)
+Figure 8. α+3Hel= 0 wave functions with Sine functions at E= 8 MeV ( a= 8
+fm,N= 15). Solid lines represent the internal wave function, and the dotted line the
+external part with Sine functions. The Lagrange wave functi on is superimposed to the
+exact result.
+Table 3. α+3Hel= 0 phase shifts (in degrees) and matching parameters for diff erent
+bases.
+phase shift matching parameter
+E(MeV) exact N= 10N= 15N= 10N= 15
+Lagrange,a= 8 fm
+5 -63.97 -65.02 -63.96 0.16 0.01
+10 -101.2 -103.1 -101.2 0.89 0.22
+15 -125.9 -128.9 -125.9 0.13 0.01
+20 -144.2 -148.5 -144.2 0.14 0.02
+Lagrange,a= 10 fm
+5 -63.97 -70.25 -63.97 0.30 0.04
+10 -101.2 -109.5 -101.3 0.17 0.02
+15 -125.9 -134.5 -126.0 5.19 0.07
+20 -144.2 -149.4 -144.3 0.28 0.02
+Sine,a= 8 fm
+5 -63.97 -88.74 -81.18 2.66 2.47
+10 -101.2 -127.0 -117.2 1.16 0.81
+15 -125.9 -159.4 -146.5 2.66 2.40
+20 -144.2 -183.7 -165.4 1.95 1.82
+4.6. Application to a deep potential
+As mentioned before, the potential may contain additional (u nphysical) bound states to
+simulatethePauliprinciple. Intheapplicationsconsider edsofar,thepotentialsinvolved
+a small number of forbidden states. The situation is differen t in heavy-ion reactions,
+where the number mlof forbidden states may be rather large. In such conditions t he
+internal wave function presents several oscillations whic h must be reproduced by theTheR-matrix Theory 44
+basis functions to provide accurate phase shifts.
+This problem is illustrated here with the13C+12C system (l= 0,ml= 12). We take
+the deep Gaussian potential of [100] (depth of −273.7 MeV and range of 3.06 fm) for
+the nuclear interaction. The Coulomb potential has a point- sphere shape [see (4.17)],
+with a radius of 8.2 fm. The R-matrix phase shifts are determined with the Lagrange
+basis, and compared in Fig. 9 (upper panel) with the exact pha se shifts. The channel
+radius isa= 10 fm. It is clear that small Nvalues do not reproduce the phase shifts.
+Values larger than N≈35 are necessary to obtain accurate results.
+We show in Fig. 9 (lower panel) the wave function for N= 20,30,40 atE= 10
+MeV. Clearly, N= 20 is not able to give a good description of the oscillations of the
+internal wave function. The situation is of course improved withN= 30, but only
+N= 40 is superimposed to the exact wave function. This problem is common to all
+potentials presenting many bound states.
+-1.5-1-0.500.511.5
+0 2 4 6 8 10 12N=20N=30
+N=20-240-180-120-60060120180
+0 5 10 15 20
+E (MeV)δ (deg.)
+N=15N=30N=20N=25
+N=35
+r (fm)u0(r) (fm-1/2)
+Figure 9. Upper panel:13C+12Cl= 0 phase shifts with the Lagrange functions
+(a= 10 fm). Lower panel: corresponding wave functions at E= 10 MeV. The exact
+wave function (superimposed on the N= 40 result) is shown as a thick line.
+4.7. Application to a non-local potential: e−−H scattering in the static-exchange
+approximation
+The electron + hydrogen-atom scattering has been studied wi th various methods [51].
+In the simplest version, referred to as the static-exchange approximation [101, 102], theTheR-matrix Theory 45
+problem is limited to a single-channel calculation, but inv olves a non-local potential.
+The local term U(r) reads
+U(r) =−E0/parenleftbigga0
+r+1/parenrightbigg
+exp(−2r/a0). (4.18)
+We use here the traditional atomic units, involving the Bohr radiusa0and the Hartree
+energyE0=e2/a0
+The non-local contribution is defined as
+W(r,r′) = (−1)S+14rr′
+a3
+0exp/parenleftBigg
+−r+r′
+a0/parenrightBigg/parenleftBigg
+E−E0(1+a0
+r>)/parenrightBigg
+(4.19)
+whereEis the total energy, Sis the total spin of the electrons ( S= 0 or 1), and
+r>= max(r,r′). The 1/r>singularity is responsible for numerical problems [103], a nd
+deserves special attention in the literature.
+This example is used to illustrate the applicability of the L agrange functions for
+non-local potentials (notice that the singlet phase shifts have been already determined
+in theR-matrix framework [51]). Here the Gauss approximation assoc iated with the
+Lagrange basis is expected to present convergence problems , owing to the singularity
+mentioned above. This problem will be addressed by performi ng, in parallel with the
+consistent Gauss approximation of the potential matrix ele ments [see (4.12) and (4.13)],
+a “traditional” calculation using the Simpson method (with typically 1000 integration
+points) for the numerical quadrature.
+Results are shown in Tables 4 and 5 for S= 0 andS= 1, respectively, as a
+function of the wave number k. For the Lagrange-mesh method, we select a channel
+radiusa= 14a0, and consider various basis sizes. Comparing with the alter native
+method (Simpson integration) which is less sensitive to sin gularities in the potential,
+we immediately see that a reasonable accuracy (typically 0. 02 rad.) can be achieved
+with small Nvalues at the Gauss approximation. However, when Nincreases the
+convergence is rather slow. If we do not use the Gauss approxi mation for the potential
+matrix elements, at least 5 digits are exact with N= 20 and for a given avalue.
+The sensitivity with respect to the channel radius is about 1 0−4rad. when going from
+a= 14a0toa= 16a0. For comparison we give in Table 4 the results of Burke et al.
+[51], obtained for S= 0, up tok= 0.7a−1
+0.
+Similar conclusions can be drawn for the S= 1 phase shifts, but the Gauss
+approximation is slightly more accurate because the antisy mmetry of the spatial part
+of the wave function decreases the effect of the 1 /r>singularity. The S= 0 andS= 1
+phase shifts are also in good agreement with those of Apagyi et al.[103] who use the
+Schwinger variational method [104], much more sensitive to numerical problems than
+theR-matrix method.
+4.8. Discussion of resonances
+In this section we determine the resonance energies and widt hs of the12C+p (l= 0) and
+α+α(l= 4) systems, as an illustration of section 3.6. In Fig. 10, we show the functionsTheR-matrix Theory 46
+Table 4. R-matrix phase shifts (in rad.) for e−−H scattering ( S= 0). The potential
+matrix elements are computed, either with the associated Ga uss approximation (4.12)
+and (4.13), or with the Simpson quadrature. The results are c ompared with those of
+Burkeet al.[51].
+Gauss approximation, a= 14a0 N= 20, Simpson [51]
+k(a−1
+0)N= 20N= 40N= 60N= 80N= 100a= 14a0a= 16a0
+0.12.3790 2.3917 2.3940 2.3949 2.3952 2.3959 2.3958 2.3960
+0.21.8509 1.8654 1.8681 1.8691 1.8695 1.8703 1.8702 1.8704
+0.31.4902 1.5036 1.5061 1.5070 1.5074 1.5081 1.5081 1.5081
+0.41.2228 1.2352 1.2375 1.2384 1.2388 1.2394 1.2395 1.2391
+0.51.0155 1.0274 1.0296 1.0304 1.0308 1.0314 1.0315 1.0309
+0.60.8532 0.8650 0.8673 0.8681 0.8684 0.8691 0.8690 0.8685
+0.70.7281 0.7401 0.7424 0.7432 0.7435 0.7442 0.7441 0.7438
+0.80.6347 0.6470 0.6494 0.6502 0.6506 0.6513 0.6512
+0.90.5685 0.5813 0.5837 0.5846 0.5850 0.5857 0.5857
+1.00.5251 0.5384 0.5409 0.5417 0.5422 0.5429 0.5429
+Table 5. See caption to Table 4 for S= 1.
+Gauss approximation, a= 14a0 N= 20, Simpson
+k(a−1
+0)N= 20N= 40N= 60N= 80N= 100a= 14a0a= 16a0
+0.12.9080 2.9077 2.9077 2.9077 2.9077 2.9075 2.9076
+0.22.6799 2.6794 2.6793 2.6792 2.6792 2.6791 2.6792
+0.32.4622 2.4614 2.4612 2.4612 2.4611 2.4611 2.4611
+0.42.2588 2.2576 2.2574 2.2574 2.2573 2.2572 2.2573
+0.52.0721 2.0706 2.0703 2.0702 2.0702 2.0701 2.0701
+0.61.9032 1.9013 1.9009 1.9008 1.9007 1.9006 1.9006
+0.71.7521 1.7497 1.7492 1.7490 1.7490 1.7488 1.7488
+0.81.6180 1.6150 1.6145 1.6143 1.6142 1.6140 1.6141
+0.91.4998 1.4963 1.4957 1.4954 1.4953 1.4951 1.4952
+1.01.3959 1.3919 1.3911 1.3909 1.3908 1.3905 1.3905
+R(E) and 1/S(E) for both systems. The resonances energies are calculated f rom the
+roots of equation (3.42), i.e. from the crossing points of bo th curves. The numerical
+values, as well as the widths (3.45) are given in Table 6.
+The12C+p (l= 0) state is a typical example of a narrow resonance. The phas e
+shifts near 0.42 MeV are well described by the Breit-Wigner a pproximation. Although
+theRmatrices are rather different for a= 8 fm and a= 10 fm (see Fig. 10), the energy
+and width are weakly sensitive to the channel radius (see Tab le 6). This result is typical
+of narrow and isolated resonances.
+The situation is different for the α+α(l= 4) system, which is typical of a broadTheR-matrix Theory 47
+-20-15-10-50510
+0 5 10 15 20 25
+E (MeV)R(E), 1/S(E)α+α, =4
+1/S(E)-10-50510
+0 0.5 1 1.5 2
+E (MeV)R(E), 1/S(E)12C+p, =0
+1/S(E)
+Figure 10. R-matrixRl(E) and inverse of the shift function Sl(E) for the12C+p
+(l= 0) and α+α(l= 4) collisions. Solid lines correspond to a= 8 fm and dotted
+lines toa= 10 fm. The circles show the roots of (3.42).
+resonance (around 12.5 MeV, see Fig. 6). The resonance proper ties are rather sensitive
+to the channel radius, whereas the phase shifts are almost id entical. Table 6 shows that
+the second eigenvalue might be related to the 4+resonance. However the width is not
+significantly smaller than the energy difference with neighb ouring states. In that case,
+a single-level approximation is not well adapted to reprodu ce the phase shifts in a wide
+energy range.
+Table 6. Resonance energies and widths (in MeV) of the12C+p (l= 0) and
+α+α(l= 4) systems. The Lagrange basis is used with N= 15.
+a= 8 fm a= 10 fm
+Eigenvalue ERΓERΓ
+12C+p (l= 0)
+1 0.418 0.0375 0.417 0.0369
+α+α(l= 4)
+1 6.92 5.14 4.67 3.68
+2 12.3 2.47 10.2 2.74
+3 19.8 7.08 14.0 3.33TheR-matrix Theory 48
+4.9. Application to bound states
+TheRmatrix is applied to the13N ground state ( Jπ= 1/2−,l= 1) described by a
+12C+p potential model. In order to simulate the experimental b inding energy ( −1.94
+MeV),thedepthoftheGaussianpotentialusedinsection4.3f orl= 0hasbeenmodified
+to−55.3 MeV.
+The calculation is performed with the Lagrange basis, and th e results are presented
+in Table 7. With a= 8 fm,N= 7 does not provide the exact energy. This situation
+is of course improved by increasing N. ForN= 15, we have 5 exact digits, and the
+matching parameter is close to 0. In each case, 4 iterations a re sufficient in system (3.54)
+to provide an energy stable by better than 0.1 keV. The ANC value Cl[see (3.52)] is
+also very stable.
+As in section 4.3 for the phase shifts, the choice a= 5 fm is too small to make the
+nuclear interaction negligible. Although the matching para meter is quite acceptable, the
+convergence is slower and the final result does not converge t o the exact energy. This
+means that a small matching parameter is not sufficient to ensu re that the calculation
+is accurate. Testing the stability against the channel radi us is a more severe test.
+Table 7. Energy of13N ground state (in MeV) with a Lagrange basis and different a
+andNvalues. The exact binding energy and ANC are −1.942 MeV and 2.063 fm−1/2,
+respectively.
+Iteration a= 8,N= 7a= 8,N= 10a= 8,N= 15a= 5,N= 10
+1 −2.012 −2.052 −2.053 −3.236
+2 −1.897 −1.940 −1.941 −1.763
+3 −1.899 −1.941 −1.942 −1.894
+4 −1.899 −1.941 −1.942 −1.881
+5 −1.899 −1.941 −1.942 −1.882
+6 −1.882
+Cl(fm−1/2) 2.060 2 .071 2 .072 2 .034
+ǫ 9.1 0.19 0.005 0.002
+4.10. Application to a multichannel problem: α+d
+Here we use the α+deuteron potential of Dubovichenko [105] to investigate t heR-
+matrix formalism in a multichannel problem. This nuclear po tentialVN(r) contains
+centralVc(r) and tensor Vt(r) forces, defined as
+VN(r) =Vc(r)+Vt(r)S12
+Vc(r) =V0exp(−αr2)
+Vt(r) =V1exp(−βr2), (4.20)
+where
+S12=6
+r2(S·r)2−2S2(4.21)TheR-matrix Theory 49
+is the usual tensor operator. The Coulomb potential is the ba re potential. With
+V0=−91.979 MeV,V1=−25.0 MeV,α= 0.2 fm−2,β= 1.12 fm−2, this potential
+reproduces most of the6Li ground-state properties, in particular the binding ener gy,
+the quadrupole moment, and the d-wave admixture amplitude. The spin and parity of
+the deuteron being 1+, non-natural parity states (i.e. π= (−1)J+1) involve two lvalues:
+l=|J−1|andl=J+1. The system to be solved (3.77) involves the potentials (w e
+use the notation of section 3.10)
+V11=Vc+VC−2(J−1)Vt/(2J+1)
+V12=V21= 6/radicalBig
+J(J+1)Vt/(2J+1)
+V22=Vc+VC−2(J+2)Vt/(2J+1) (4.22)
+As in previous sections the R-matrix phase shifts are computed in various
+conditions, and compared to the “exact” results obtained by the Numerov algorithm.
+TheR-matrix calculations are performed with Lagrange function s. To compare with
+experiment [106], the collision matrix is diagonalized [se e (2.37)] and parametrized as
+U=/parenleftBigg
+cosǫ−sinǫ
+sinǫcosǫ/parenrightBigg/parenleftBigg
+exp(2iδ1) 0
+0 exp(2 iδ2)/parenrightBigg/parenleftBigg
+cosǫsinǫ
+−sinǫcosǫ/parenrightBigg
+, (4.23)
+whereδ1andδ2are the (real) eigenphases, and ǫis the mixing angle (removing the
+tensor force provides ǫ= 0). These values are presented in Fig. 11 for Jπ= 1+and
+compared with the exact solutions (also shown in Figs. 4 and 5 of [105]). We choose
+a= 8 fm and show the result for N= 7. With this small number of basis functions,
+slight deviations can be observed. As soon as N >7, theR-matrix phase shifts cannot
+be distinguished from the exact values.
+Other applications of the R-matrix theory to multichannel problems can be found
+in [41].
+4.11. Application to propagation methods
+To illustrate the propagation of the Rmatrix, we present here a numerical example,
+with theα+αsystem. We choose a= 80 fm, which is of course unnecessarily large for
+this system, but typical of long range potentials, where pro pagation methods should be
+used. The goal of this example is just to provide a numerical i llustration of the method.
+TheRmatrix at 80 fm has been computed in 3 ways: (i) exactly with th e Numerov
+algorithm, (ii) with 100 basis functions without propagati on (NS= 1), (iii) with 100
+basis functions split in several intervals. The R-matrix basis functions are the Lagrange
+functions, but any other choice would provide similar concl usions. Matrix elements over
+an interval ( a1,a2) are given in Appendix C [see (C14)-(C18)].
+Table 8 gives the l= 0R-matrix at a= 80 fm for typical energies. Without
+propagation ( NS= 1), we reproduce at least 4 significant digits. This is still true for
+NS= 2, but the computer time is reduced by more than a factor of tw o. Essentially the
+difference is that, in the former case we have to invert one 100 ×100 matrix, whereas
+the latter calculation requires two inversions of 50 ×50 matrices. The computer time isTheR-matrix Theory 50
+060120180
+0 2 4 6 8 10
+E (MeV)eigen phase (deg.)
+-120-90-60-30030
+0 2 4 6 8 10
+E (MeV)ε (deg.)
+Figure 11. α+d eigenphases (upper panel) and mixing angles (lower panel ) for
+Jπ= 1+. The solid lines represent the exact calculation and the dot ted lines
+correspond to the R-matrix calculation with a= 8 fm, N= 7. The experimental
+data are taken from [106].
+still reduced with NS= 4 (four inversions of 25 ×25 matrices). The precision is however
+reduced since the first interval (0 ,20 fm) only contains 25 basis functions. Increasing
+this number to 35 (and hence the full basis size to 110 functio ns) provides the exact
+R-matrix (with at least 4 significant digits) with similar com puter times.
+Table 8. R-matrix for the α+αcollision ( l= 0) and for different NSvalues
+(N= 100,a= 80 fm). For each interval the number of basis functions is 10 0/NS.
+E(MeV) exact NS= 1 NS= 2 NS= 4
+5 2 .648×10−22.648×10−22.648×10−22.643×10−2
+10 −3.729×10−1−3.729×10−1−3.730×10−1−3.957×10−1
+15 1 .195×10−31.195×10−31.195×10−31.157×10−3
+20 −1.246×10−2−1.246×10−2−1.246×10−2−1.262×10−2
+time (ms) 1.2 0.50 0.25
+4.12. Application to capture reactions:12C(p,γ)13N
+The12C(p,γ)13N reaction is the first reaction of the CNO cycle. It represents an ideal
+candidate for an R-matrix treatment since the 1 /2+(l= 0) resonance at ER= 0.42
+MeV determines the S-factor (2.55) in a wide energy range. As in previous sections , ourTheR-matrix Theory 51
+goal here is not to find the best fit to the data (see [17]). Rathe r, we want to illustrate
+the different contributions to the capture matrix elements ( 3.61), and to discuss various
+approximations. Our procedure is as follows:
+(i) The initial states are determined as in section 4.3.
+(ii) Asourgoalistoillustratethedifferentcontributionst othecapturematrixelements,
+the bound-state wave function (and the corresponding ANC) is c omputed exactly
+with the Numerov algorithm [90]. This avoids the optimizatio n of the basis for the
+bound state.
+(iii) The basis functions for the initial state are chosen as Lagrange functions ( N= 15),
+and we determine the internal and external contributions fr om (3.62) and (3.63),
+respectively.
+The matrix elements are computed as in section 3.8, but the R-matrix expansion of
+the final state is replaced by the exact wave function. Table 9 gives, at typical energies,
+the exact values of the matrix elements (i.e. with scatterin g wave functions obtained
+from the Numerov method), as well as their values in the R-matrix theory with a= 8
+fm anda= 10 fm. Large values must be used to ensure that the nuclear in teraction is
+negligible. Several comments can be made:
+(i) As expected the matrix element presents a maximum at E= 0.42 MeV, as the
+initial potential has been fitted to provide a resonance at th is energy.
+(ii) Since the external scattering wave function (3.63) inv olves the phase shift, the
+external contribution also presents a resonant behaviour.
+(iii) Each term in (3.61) depends on the channel radius. Thei r sum, however, should be
+insensitive to its choice, if the conditions of the calculat ion are properly defined.
+The relative difference is maximum near the resonance but is a lways less than 1%.
+For the sake of completeness we show in Fig. 12 the12C(p,γ)13NS-factor compared
+to the available data sets [107, 108]. As usual in the potentia l model, a spectroscopic
+factorSshould be introduced for a more realistic comparison with th e data. The
+spectroscopic factor scales the total cross section. Value s lower than unity mean that
+the final ground state is more complicated than a simple12C+p structure. With S= 1
+the theoretical S-factor is larger than the data. A reasonable agreement can b e obtained
+withS= 0.45, although the high energy part is slightly overestimated by the model.TheR-matrix Theory 52
+Table 9.12C(p,γ)13N matrix elements [see (3.62) and (3.63)] with Lagrange func tions
+(N= 15).
+exact R-matrix
+E(MeV) internal external total internal external total
+a= 8 fm
+0.20 220.78 76.56 297.34 219.24 76.51 295.74
+0.40 2034.4 414.38 2448.8 2002.1 410.58 2412.7
+0.60 247.93 19.26 267.18 246.76 19.32 266.08
+0.80 116.02 −3.83 112.19 115.35 −3.81 111.54
+1.00 74.60 −9.69 64.92 74.14 −9.68 64.46
+a= 10 fm
+0.20 248.93 48.41 297.34 247.97 48.42 296.40
+0.40 2240.6 208.22 2448.8 2228.7 208.03 2436.7
+0.60 266.55 0.63 267.18 265.39 0.64 266.03
+0.80 121.69 −9.49 112.19 121.13 −9.49 111.64
+1.00 76.28 −11.37 64.92 75.92 −11.36 64.56
+0 0.2 0.4 0.6 0.8 1
+E (MeV)S (MeV-b)
+10.45100
+10-1
+10-2
+10-3
+10-4
+Figure 12.12C(p,γ)13NS-factors with different spectroscopic factors S. The
+experimental data are from [107] (open circles) and [108] (c losed circles).TheR-matrix Theory 53
+5. The phenomenological Rmatrix
+5.1. Introduction
+The goal of the phenomenological R-matrix method is to use a parametrization based
+on expression (3.23) of the Rmatrix or its multichannel generalization (3.86) with a
+finite number of poles. The properties of these poles are adju sted to some data, in place
+of being derived from some Hamiltonian, as in the calculable a pproach. We present here
+various applications in nuclear physics. In particular, th is technique is very successful
+in nuclear astrophysics [109], where the main issue is to fit c ross-section data, and to
+extrapolate them down to stellar energies at which direct me asurements are in general
+impossible (see, for example, [21]). Another recent applica tion is the analysis of low-
+energy scattering data in experiments involving radioacti ve beams. Those experiments
+usually probe the nuclear structure at low level density, we ll adapted to the R-matrix
+formalism (see, for example references [110, 111, 112] for8B+p,11C+p and18Ne+p,
+respectively). The method is of course not limited to elasti c scattering, but can be
+extended to inelastic [113] as well as to transfer [114] reac tions.
+One of the main drawbacks of the phenomenological R-matrix formalism is that,
+though the pole energies and reduced widths are associated w ith physical properties,
+they cannot be directly compared with experiment. In partic ular, theR-matrix
+parameters significantly depend on the channel radius. This is in contrast with the
+so-called “observed” data, such as resonance energies and w idths directly fitted to
+experimental data which are by definition independent of suc h a radius. However, these
+“observed” values of the energy and width of a resonance depe nd on the assumptions
+made about the theoretical description of the resonance (se e section 3.6 for a part of the
+possible definitions). Within the phenomenological R-matrix approach, the “observed”
+quantities have a rather precise definition from which the R-matrix parameters can be
+determined. For an isolated resonance in some partial wave l, the observed energy ER
+and width Γ Rare obtained in the single-channel approach by fitting the el astic cross
+section (2.46) where the phase shift is parametrized by the g eneralized Breit-Wigner
+expression (3.48),
+δBW
+l≈φl+arctan1
+2ΓR
+ER−E. (5.1)
+This expression may be simplified by neglecting φlor made more realistic by multiplying
+ΓRbyPl(E)/Pl(ER). Notice that, except for the simplest Breit-Wigner approxi mation,
+these expressions slightly depend on athroughφlorPl. The “observed” reduced width
+γ2
+obsis then extracted from Γ Rwith the relation similar to (3.50),
+ΓR= 2γ2
+obsPl(ER). (5.2)
+The corresponding “formal” parameters, i.e. the pole locat ionEnland the reduced
+widthγ2
+nlcan then be deduced for a given channel radius abut their determination is
+not immediate because of the shift factor Sland its energy dependence. The problem is
+more complicated if several resonances or several channels must be taken into account.TheR-matrix Theory 54
+In the next sections, we present different methods to link the “formal”R-matrix
+parameters with the “observed” values. The simplest and mos t common case deals
+with a single isolated resonance, described by one pole (sec tion 5.2). However some
+applications require to include several poles in the same pa rtial wave or, in other words,
+to consider interference effects between these poles (secti on 5.3). Other applications
+require to take several channels into account (section 5.4) . As a first application,
+we consider the12C+p elastic scattering from 0.3 MeV to 1.8 MeV, where accurate
+data exist for many years (section 5.5). Then more recent mul tichannel application
+to1H(18Ne,p)18Ne(g.s.) and1H(18Ne,p’)18Ne*(2+, 1.887 MeV) will be shown as an
+example of radioactive-beam experiments (section 5.6). Fi nally, theR-matrix method
+will be applied to radiative capture reactions, with12C(p,γ)13N as a typical example
+(section 5.7).
+5.2. Single-pole approximation of elastic scattering
+In section 3.6, we have presented a general procedure to defin e the resonance energy
+and width from the R-matrix expression. Let us particularize (3.23) to a single pole
+with energy E1and reduced width γ2
+1(we drop the index lfor the sake of clarity); the
+Rmatrix is then given by
+Rl(E) =γ2
+1
+E1−E. (5.3)
+This approximation is frequently used at low energy, where s ingle isolated resonances
+are present. The R-matrix phase shift associated with (5.3) is given by (3.47) . To go
+further, let us use the Thomas approximation [5], which cons ists in a linearization of
+the shift function Sl(E). Near the pole energy E1, it reads
+Sl(E)≈Sl(E1)+(E−E1)S′
+l(E1), (5.4)
+whereS′
+lis the derivative of the shift factor with respect to energy ( which appears
+both in the wave number kand in the Sommerfeld parameter η). The validity of this
+approximation is supported by Figs. 1, 2 and 3, where it is cle ar that, in a limited energy
+range, the linearization of the shift function is quite appr opriate.
+Comparing expression (3.47) of the phase shift at the Thomas approximation with
+the Breit-Wigner expression (5.1), one derives the observe d properties ( ER,γobs) from
+the formal parameters ( E1,γ1) of an isolated pole. The observed energy ERreads
+ER=E1−γ2
+1Sl(E1)
+1+γ2
+1S′
+l(E1), (5.5)
+where the shift between both energies depends on E1. This shift also depends on a. It
+is in general non-negligible, unless γ2
+1is very small. The observed reduced width reads
+γ2
+obs=γ2
+1
+1+γ2
+1S′
+l(E1). (5.6)
+It also depends on E1.TheR-matrix Theory 55
+In practice, however, the reversed relationships are neede d. Indeed, in many cases,
+observed values ( ER,γobs) are known from experiments (or from other theoretical work s)
+and one wants to derive the corresponding R-matrix parameters ( E1,γ1). The inverses
+of (5.5) and (5.6) are obtained by linearizing the shift fact orSl(E) aroundERas
+γ2
+1=γ2
+obs
+1−γ2
+obsS′
+l(ER), (5.7)
+E1=ER+γ2
+1Sl(ER). (5.8)
+These results are well known [5].
+In the literature, the reduced width is frequently expresse d in units of the Wigner
+limit [89]
+γ2
+W=3¯h2
+2µa2, (5.9)
+which provides the dimensionless reduced widths
+θ2
+obs=γ2
+obs/γ2
+W,
+θ2
+1=γ2
+1/γ2
+W. (5.10)
+Notice that Lane and Thomas [5] define the dimensionless reduc ed width as θ2=
+γ2
+obsµa2/¯h2without an explicit reference to the Wigner limit. In genera l, a value of
+θ2not far from unity indicates the occurrence of a cluster stru cture, i.e. the colliding
+nuclei partly conserve their identity within the resonance . Conversely, a plausible guess
+onθ2(possibly inspired by a model) can be used to estimate the wid th of a resonance.
+This concept of dimensionless reduced width, as defined by (5 .10), was first used by
+various authors [115, 116, 117] in the interpretation of low -energy scattering data.
+The difference between observed and formal parameters is ill ustrated by numerical
+applications in Table 10 for the narrow 1 /2+resonance in12C+p (ER= 0.42 MeV,
+ΓR= 32 keV) and for the broad 1−resonance in12C+α(ER= 2.42 MeV, Γ R= 420
+keV). The formal parameters ( E1,γ1) are calculated for various channel radii and are
+then used to determine the phase shifts shown in Fig. 13. As usu al in applications of
+the phenomenological R-matrix method, the channel radii are somewhat smaller than
+in the “calculable” variant.
+From Table 10, it is clear that the dimensionless reduced wid thsθ2
+obsare less
+dependent on the channel radius than γ2
+obs. For both systems, and in particular for
+12C+α, theθ2
+obsvalues are rather large. These states can therefore be consi dered as
+cluster states. As expected, the formal parameters (5.7) are strongly dependent on the
+channel radius. However, the corresponding phase shifts in F ig. 13 are very close to
+each other, in particular for the narrow resonance in12C+p. For the broad resonance in
+12C+α, the phase shift is rather stable near the resonance energy, but more significant
+differences appear at higher energies. In such a case, the val idity of the Breit-Wigner
+approximation is more limited. In other words, the single-p ole approximation (5.3)
+should be replaced by an Rmatrix containing several terms.TheR-matrix Theory 56
+Table 10. R-matrix parameters (5.7) and (5.8) for resonances in12C+p and12C+α
+(in MeV). The observed reduced widths are obtained from (5.2 ).
+12C+p (Jπ= 1/2+,l= 0,ER= 0.42 MeV, Γ R= 32 keV)
+a= 4 fma= 5 fma= 6 fm a= 7 fm
+γ2
+obs1.089 0.592 0.353 0.227
+θ2
+obs0.258 0.220 0.189 0.165
+γ2
+13.083 1.157 0.569 0.323
+E1−2.152−0.614−0.110 0.113
+12C+α(Jπ= 1−,l= 1,ER= 2.42 MeV, Γ R= 420 keV)
+a= 5 fma= 6 fm a= 7 fm
+γ2
+obs 0.574 0.277 0.165
+θ2
+obs 0.6920 0.481 0.389
+γ2
+1 1.172 0.374 0.191
+E1 0.491 1.921 2.219
+060120180
+0 1 2 3 4 5δ (deg.)a=7a=6a=512C+α060120180
+0 0.2 0.4 0.6 0.8 1 1.2δ (deg.)a=7a=4 12C+p
+E (MeV)
+Figure 13.12C+p (l= 0) and12C+α(l= 1) phase shifts in the single-pole R-matrix
+approximation computed with the parameters of Table 10 for d ifferent channel radii.
+5.3. Multiresonance elastic scattering
+As mentioned above, the single-pole approximation is often v alid at low energies.
+However, for nuclei with a high level density, several resona nces may appear withTheR-matrix Theory 57
+the same angular momentum and parity. Then the interference s between different
+resonances and/or bound states in the same partial wave may b e important. Typical
+examples are12C+αor14N+p where several partial waves involve more than one state,
+even in a limited energy range.
+In this case, the link between formal and observed parameter s is more complicated,
+and was addressed for many years in a rather indirect way (see , for example, [14, 118]).
+More recently, an iterative procedure was proposed to deter mine the formal R-matrix
+parameters from observed values [23]. This method is valid f or single-channel systems
+only. A generalization to multichannel problems was then de veloped by Brune [119].
+The idea is to propose an alternative parametrization of the Rmatrix, where the input
+parameters are the observed data. It is based on the invarian ce of these values when
+the boundary parameters Bc[see (3.80)] are changed. We briefly summarize Brune’s
+method here, by assuming elastic scattering only. A more gen eral presentation can be
+found in [119].
+Let us assume Nresonances in a given partial wave, with observed energies ERi
+and widths Γ Ri. For each resonance, an observed reduced width is defined, ac cording to
+(5.2), as
+γ2
+obs,i= ΓRi/2Pl(ERi). (5.11)
+In the notations of [119], (5.7) is written as
+˜γ2
+i=γ2
+obs,i
+1−γ2
+obs,iS′
+l(ERi). (5.12)
+From these expressions, the formal pole energies En, used in the N-poleR-matrix
+expansion, are obtained from
+N bn=EnM bn (5.13)
+where matrix elements NijandMijare given by
+Nij=
+
+ERi+ ˜γ2
+iSl(ERi) for i=j,
+˜γi˜γjERiSl(ERj)−ERjSl(ERi)
+ERi−ERjfori/negationslash=j,(5.14)
+and
+Mij=
+
+1 for i=j,
+−˜γi˜γjSl(ERi)−Sl(ERj)
+ERi−ERjfori/negationslash=j.(5.15)
+The generalized eigenvalue problem (5.13) provides the for mal energies En. The formal
+reduced-width amplitudes γnare derived from the eigenvectors bnas
+γn=N/summationdisplay
+j=1bn,j˜γj. (5.16)
+WhenN= 1, it is easy to see that (5.7) and (5.8) are recovered. This m ethod provides
+an efficient way to derive the R-matrix parameters. It represents the starting point of
+an alternative parametrization of the Rmatrix, proposed by Brune [119].TheR-matrix Theory 58
+This formalism is illustrated in Table 11 and Fig. 14 with the14N+p system. In the
+Jπ= 3/2+partial wave ( l= 0), we take three resonances into account. The observed
+valuesERiand Γ Ri[94] are given in Table 11. As for the single-pole approximati on,
+the formal parameters ( En,γn) do depend on the channel radius. Fig. 14 shows that
+the influence of aon the corresponding phase shifts is weak near the narrow res onance
+at 0.987 MeV (this resonance presents a small θ2value and has thus a complicated
+structure). As expected, it is more important in the vicinity of the broader states at 2.2
+and 3.2 MeV.
+Table 11. R-matrix parameters (in MeV) with 3 poles for the Jπ= 3/2+,l= 0
+partial wave in14N+p.
+observed values a= 5 fma= 6 fma= 7 fm
+ER1= 0.987γ2
+obs,10.0082 0.0054 0.0039
+ΓR1= 0.00367θ2
+obs,10.0031 0.0029 0.0028
+γ2
+10.0097 0.0061 0.0042
+E10.981 0.983 0.985
+ER2= 2.187γ2
+obs,20.114 0.086 0.069
+ΓR2= 0.2θ2
+obs,20.043 0.047 0.050
+γ2
+20.119 0.089 0.070
+E22.153 2.165 2.172
+ER3= 3.209γ2
+obs,30.053 0.042 0.034
+ΓR3= 0.14θ2
+obs,30.020 0.023 0.025
+γ2
+30.051 0.041 0.034
+E33.199 3.202 3.204
+0180360540
+0 1 2 3 4 5δ (deg.)14N+p, J=3/2+ a=5
+a=7a=6
+E (MeV)
+Figure 14.14N+pR-matrix phase shifts ( Jπ= 3/2+) computed with the parameters
+of Table 11.TheR-matrix Theory 59
+5.4. Phenomenological parametrization of multichannel co llisions
+To deal with inelastic or transfer cross sections, the R-matrix formalism must be
+extended to several channels. Rigorously this is even true i n elastic scattering involving
+particles with non-zero spins. When the spins are different f rom zero, several ( lI) values
+[see (2.29)] are to be considered, each of them being charact erized by a partial reduced
+width.
+Let us consider the multichannel Rmatrix (3.86). In the calculable variant, the
+number of poles is equal to the product of the number of basis f unctionsNby the
+number of channels. In the phenomenological approach, incl uding all partial waves is
+often not realistic, since too many parameters may be involv ed. At low energies, the
+choice of the relative angular momenta lis guided by the penetration factor, i.e. the
+minimumlvalue for given Jandπis often adopted. Such a selection mechanism does
+not exist for the channel spin I. In general, all possible values should be considered.
+The choice of the relevant partial waves is made according to the quality of the fit.
+For these reasons, the multichannel approach in the phenome nological variant of the R
+matrix is most often restricted to two channels.
+Here we limit the presentation to a single pole at energy E1, the most frequently
+used approximation. In that case, the Rmatrix (3.86) is reduced to
+Rcc′(E) =γcγc′
+E1−E, (5.17)
+whereγcis the reduced-width amplitude of the pole in channel c(for simplicity we drop
+the index 1 of the reduced widths). These parameters are real . In this approximation,
+the property
+R2
+cc′(E) =Rcc(E)Rc′c′(E), (5.18)
+provides the multichannel Zmatrix (3.89) as
+Zcc′= (akc′)−1/2Oc′/bracketleftbigg
+δcc′−Lc′/radicalBig
+RccRc′c′/bracketrightbigg
+. (5.19)
+With the help of (B2), we obtain the inverse as
+(Z−1)cc′= (akc)1/2O−1
+c/bracketleftbigg
+δcc′+Lc′√RccRc′c′
+1−/summationtext
+cRccLc/bracketrightbigg
+, (5.20)
+where the logarithmic derivatives Lc(3.28) and outgoing functions Ocare defined for
+each channel. Using definition (3.88) for a real Rmatrix, one obtains
+Ucc′=ei(φc+φc′)/bracketleftBigg
+δcc′+2i√PcPc′RccRc′c′
+1−/summationtext
+cRccLc/bracketrightBigg
+, (5.21)
+whereφcis the hard-sphere phase shift (3.26) in channel c.
+From the denominator in (5.21), the resonance energy and the reduced width are
+direct generalizations of (5.5) and (5.6),
+ER=E1−/summationtext
+cγ2
+cSc(E1)
+1+/summationtext
+cγ2
+cS′
+c(E1), (5.22)
+γ2
+obs,c=γ2
+c
+1+/summationtext
+cγ2
+cS′
+c(E1), (5.23)TheR-matrix Theory 60
+whereindex linthepenetrationandshiftfactorsisreplacedbythemoreg eneralchannel
+indexc. With (5.17), the collision matrix is parametrized at the ge neralized Breit-
+Wigner approximation [96] as
+UBW
+cc′=ei(φc+φc′)
+δcc′+i/radicalBig
+Γc(E)Γc′(E)
+ER−E−iΓ(E)/2
+, (5.24)
+where Γ cis the observed partial width in channel cand Γ =/summationtext
+cΓc. A simple calculation
+gives
+Γc(E) = 2γ2
+obs,cPc(E) (5.25)
+and Γ cR= Γc(ER). Again, the R-matrix parameters ( E1,γc) can be deduced from the
+observed values as
+γ2
+c=γ2
+obs,c
+1−/summationtext
+cγ2
+obs,cS′
+c(ER), (5.26)
+E1=ER+/summationdisplay
+cγ2
+cSc(ER). (5.27)
+These analytical formulas are direct extensions of (5.7) an d (5.8) obtained for elastic
+scattering in the single-pole approximation.
+5.5. Application to the12C+p elastic scattering
+Data on12C+p exist for many years. In particular, elastic scattering cross sections
+have been measured with a high precision [120]. This system i s well adapted to the
+phenomenological R-matrix approach, since the level density near threshold is quite
+low. This example should be considered as a typical applicat ion of the method. Similar
+fits have been done on the16O+pl= 0 elastic phase shifts [121] and on the14O+p cross
+section [48].
+From [94], three resonances are expected in the energy range covered by the data:
+1/2+at 0.421 MeV, 3 /2−at 1.558 MeV and 5 /2+at 1.603 MeV. The pole corresponding
+to the bound state is neglected. Data sets at three c.m. angle s (θ= 89.1◦,118.7◦,146.9◦)
+are available. The smallest and largest angles are fitted sim ultaneously by using the
+single-pole approximation (5.3) for the resonant partial w aves. For other partial waves,
+the hard-sphere phase shift is used. This is consistent with the absence of resonance
+(Rl= 0), but plays a minor role in the cross sections. Replacing t he hard-sphere phase
+shifts by zero provides essentially the same fits.
+The observed resonance properties are given in Table 12 for d ifferent channel radii.
+Clearly the results are almost independent of a, as expected from physical arguments.
+The fitted values are consistent with the literature [120], a nd the corresponding cross
+sections are shown in Fig. 15 for both scattering angles. The three channel radii provide
+fits which are indistinguishable at the scale of the figure. As e xpected [120], the R-
+matrix parametrization reproduces the data very well, not o nly in the vicinity of the
+resonances, but also between them, where the process is most ly non-resonant. ThisTheR-matrix Theory 61
+technique is very successful in the analysis of recent data i nvolving radioactive beams
+(see, for example, [110, 111, 112]).
+Table 12. R-matrix parameters from a simultaneous fit of12C+p scattering data [120]
+atθ= 89.1◦and 146.9◦. Resonance energies ERare expressed in MeV and widths Γ R
+in keV.
+Jπ= 1/2+Jπ= 3/2−Jπ= 5/2+
+ERΓRERΓRERΓR
+a= 4 fm 0.427 33.8 1.560 51.4 1.603 48.1
+a= 5 fm 0.427 32.9 1.559 51.4 1.604 48.1
+a= 6 fm 0.427 30.9 1.558 51.3 1.606 47.8
+Exp. [120] 0.424 33 1.558 55 1.604 50
+050010001500
+0 0.5 1 1.5 2dσ/dΩ (mb/sr)θ=146.9o05001000150020002500
+0 0.5 1 1.5 2dσ/dΩ (mb/sr)θ=89.1o
+E (MeV)
+Figure 15. R-matrix fits of12C+p experimental excitation functions at two c.m.
+angles [120] with the parameters of Table 12.
+5.6. Application to the18Ne(p,p’)18Ne(2+)inelastic scattering
+We present here an application of the phenomenological R-matrix method to inelastic
+scattering. The18Ne(p,p’)18Ne(2+,1.887 MeV) cross section has been measured in
+parallel with elastic cross sections [113]. These data were obtained at various angles andTheR-matrix Theory 62
+complementedapreviousdataset,obtainedatlowerenergie s,andaimedatinvestigating
+elastic scattering only [122].
+Here our goal is not to repeat the analysis of [113], where seve ral angles were
+simultaneously included, but where previous elastic data w ere not considered. Instead,
+we select a single angle but cover a broader energy range by in cluding data sets of
+[122,113]inaglobalfit. Bothexperimentsmeasuredtheelas ticcrosssectionsindifferent
+energy ranges, but also at slightly different angles. We sele ct the elastic data sets of
+[113] atθlab= 6.2◦and of [122] at θlab= 4.9◦. As the angular dependence is weak, we
+combine these both data sets at a common angle, taken as the av erage (θlab= 5.6◦).
+For the inelastic cross section, the experimental angle [11 3]θlab= 6.2◦is used.
+0510152025
+2.5 2.6 2.7 2.8 2.9 3 3.1 3.218Ne(p,p')18Ne(2+)0100200300400500
+0 1 2 3 418Ne(p,p)18Ne
+E (MeV)dσ/dΩ (mb/sr)
+Figure 16. R-matrix fits of18Ne(p,p)18Ne elastic (upper panel) and
+18Ne(p,p’)18Ne(2+) inelastic (lower panel) cross sections with the parameter s of Table
+13. The data are from [122] (full circles) and [113] (open cir cles). The fits are done
+witha= 4.5 fm (dashed lines), a= 5.0 fm (solid lines), and a= 5.5 fm (dotted lines).
+Theexperimentaldataandthecorresponding R-matrixfitsarepresentedinFig.16.
+As suggested in [113], the fits are performed by including thre e resonances ( Jπ=
+1/2+,5/2+,3/2+), which are characterized by their energy ERand their partial widths
+Γ1and Γ 2corresponding to the p+18Ne(0+) and p+18Ne(2+) channels, respectively.
+The fitted parameters are given in Table 13 for different chann el radii. As expected
+we essentially reproduce the results of [122, 113]. The 1 /2+resonance at 1.06 MeV is
+below the inelastic threshold (Γ 2= 0), and corresponds to a single-particle state, with
+a large reduced width. The higher-lying resonances (5 /2+,3/2+) correspond to sstates
+in the p+18Ne(2+) channel. They present a dominant width in that channel (Γ 2≫Γ1)TheR-matrix Theory 63
+and correspond to a significant fraction of the Wigner limit ( 5.9). These resonances
+are hardly visible in the elastic data, and could not be prope rly analyzed without the
+inelastic cross sections.
+Owing to the use of a radioactive beam, the error bars are rath er large in the
+inelastic cross sections, and the sensitivity of the R-matrix parameters to the channel
+radius is slightly stronger than in12C+p. InR-matrix analyses, this sensitivity should
+be taken into account in the evaluation of the recommended er ror bars.
+Table 13. R-matrixparametersfromasimultaneousfitofelasticandine lastic18Ne+p
+cross sections (see text). Resonance energies are given in M eV and widths in keV. The
+bracketed values represent the dimensionless reduced widt hs.
+Jπa= 4.5 fma= 5 fma= 5.5 fm
+1/2+ER1.064 1.063 1.062
+Γ1101 (0.26) 98 (0.23) 94 (0.20)
+5/2+ER2.773 2.768 2.766
+Γ17 (0.01) 6 (0.01) 7 (0.01)
+Γ285 (0.46) 85 (0.40) 80 (0.34)
+3/2+ER3.108 3.080 3.036
+Γ111 (0.01) 11 (0.01) 9 (0.01)
+Γ2300 (0.47) 379 (0.57) 306 (0.48)
+χ20.43 0.44 0.54
+5.7. Radiative capture reactions
+5.7.1. Extension of R-matrix formalism
+The general formalism of radiative-capture reactions has b een given in section 2.4.
+In theR-matrix theory, the matrix element ˜U(2.58) is split in internal and external
+contributions as [21]
+˜U=˜Uint+˜Uext, (5.28)
+where we have dropped all indices for the sake of clarity. The terms˜Uintand˜Uextof the
+r.h.s. involve the internal and external wave functions, re spectively. We assume here
+single-channel calculations, but the spins may be different from zero.
+By using expansion (3.36) for the initial radial wave functi onuint
+liandCli= 1, the
+internal part of (2.58) becomes
+˜Uint=ei(δJπ
+li−π
+2)
+|1−LliRJπ
+li|/summationdisplay
+nǫn/radicalBig
+Γγ,n(E)Γn(E)
+En−E, (5.29)
+where we have introduced the formal gamma width of pole nas
+Γγ,n=2Jf+1
+2J+18π(λ+1)k2λ+1
+γ,n
+λ(2λ+1)!!2/vextendsingle/vextendsingle/vextendsingle/angbracketleftΨJfπf||Mσλ||ΦJπ
+n/angbracketright/vextendsingle/vextendsingle/vextendsingle2(5.30)TheR-matrix Theory 64
+withkγ,n= (En−Ef)/¯hc. The matrix element is calculated over the internal region
+only. Its definition involves a state ΦJπ
+ncorresponding to the pole En, whose radial part
+r−1φJπ
+n(r) is defined like in the orthogonal basis (3.22) but for partia l waveJπ. Matrix
+element (5.29) depends on the particle width Γ n(E) (3.45) and on the gamma width at
+collision energy E,
+Γγ,n(E) =/parenleftBiggE−Ef
+En−Ef/parenrightBigg2λ+1
+Γγ,n(En) (5.31)
+(for simplicity, Γ γ,n(En) is often denoted as Γ γ,n). In (5.29), ǫn=±1 is the product
+of the signs of the matrix element in (5.30) and of the reduced width amplitude. For
+a single-pole approximation, this sign does not play a role b ut determines interference
+effects in multi-pole calculations.
+The external contribution is determined as in (3.63). A calc ulation similar to the
+one leading to (2.64) provides
+˜Uext=eZ(Eλ)
+effCJfπf
+lf1
+(2J+1)1/2(−1)If−Jf+λ/parenleftBigg2(λ+1)k2λ+1
+γ
+¯hvλ(2λ+1)!!2/parenrightBigg1/2
+×Z(liJlfJf,Ifλ)/integraldisplay∞
+aWlf(2κBr)rλ(Ili(kr)−UJπ
+liOli(kr))dr, (5.32)
+whereCJfπf
+lfis the ANC of the final bound state and UJπ
+lithe collision matrix at energy
+E. This contribution is often referred to as “direct capture” . In fact it is closely related
+to the internal term through the collision matrix. A resonan t behaviour of the collision
+matrix affects the external term (5.32) (see also Table 9). As u sual, in the calculable R-
+matrix theory, the total matrix element (5.28) should not de pend on the channel radius
+a, although each individual term does depend on a. In addition, one easily shows that
+both terms present an identical phase factor. The calculati on can therefore be reduced
+to real expressions.
+In the calculable R-matrix, the gamma width (5.30) is computed from basis
+functions. In the phenomenological variant, the constant Γ γ,n(En) appearing in (5.31)
+becomes a parameter. The treatment of radiative-capture re actions therefore requires
+one additional parameter for each pole, the gamma width Γ γ,n(En) (and the associated
+interference sign), and a global parameter, the ANC CJfπf
+lfof the final bound state. The
+latter parameter is sometimes available independently. As f or elastic widths, the fitted
+values of the gamma widths may depend on the channel radius. T he importance of this
+dependence will vary with the amplitude of the external cont ribution.
+5.7.2. Isolated resonance approximation
+Let us consider the single-pole approximation (5.3). Start ing from expression (5.29)
+of the internal matrix element ˜Uint, a simple calculation using definition (3.34) of the
+phase shift provides near the resonance energy ERthe approximation
+˜Uint(E)≈ei(φli−π
+2)/radicalBig
+Γγ,R(E)ΓR(E)
+ER−E−iΓR(E)/2, (5.33)TheR-matrix Theory 65
+where the observed particle width Γ Ris defined from (5.25) and where the observed
+gamma width Γ γ,Ris given by
+Γγ,R(E) =Γγ,1(E)
+1+γ2
+1S′
+li(E1). (5.34)
+The correction factor is thus identical for the particle and gamma widths [see (5.6)].
+For a resonant process, the main part of the wave function is l ocated at short distances,
+and the external term (5.32) can often be neglected to a good a pproximation. In that
+case, the capture cross section (2.57) takes the usual Breit -Wigner form
+σσλ
+Jfπf,Jπ(E)≈π
+k22J+1
+(2I1+1)(2I2+1)Γγ,R(E)ΓR(E)
+(ER−E)2+(ΓR(E)/2)2.(5.35)
+As the electromagnetic interaction is weak, it is implicitly assumed that Γ γ,R(E)≪
+ΓR(E). This formula is of course an approximation which assumes t hat (i) there is
+no background or other resonances interfering and (ii) the e xternal contribution is
+negligible. Going beyond these two approximations can be do ne by using the more
+general formulas (5.29) and (5.32). Notice that the relative roles of˜Uintand˜Uextdepend
+on energy. When Ebecomes small, the internal part of the initial state become s smaller
+andsmaller, andtheimportanceoftheexternalcontributio nincreases. Thecontribution
+of the external term also depends on the binding energy of the final state. If Efis small,
+theasymptoticdecreaseofthebound-statewavefunctionis slowandtheexternalmatrix
+element (5.32) may be important.
+In the single-channel approximation, the internal and exte rnal components can be
+combined, which yields a slightly different definition for th e gamma width [17]. In [123],
+the term involving the collision matrix in (5.32) is recast w ith the internal contribution.
+Thisprovidesmodifiedelectromagneticmatrixelements. Asl ongastheexternalcapture
+is negligible, all definitions of the gamma width are equival ent.
+5.7.3. Application to12C(p,γ)13N
+As discussed in section 4.12, the12C(p,γ)13NS-factor at low energies is essentially
+determined by the properties of the 1 /2+(l= 0) resonance at ER= 0.42 MeV in
+13N [108, 17]. As usual in nuclear astrophysics the main issue is to extrapolate the
+available data down to stellar energies (around 24 keV at the typical temperature
+1.5×107K). We use the phenomenological R-matrix approach with the single-pole
+approximation. In that case, four parameters are to be consi dered: the energy, proton
+widtf Γ R, gamma widths Γ γ,R(ER) of the resonance, and the ANC Clfof the13N ground
+state (Jf= 1/2−,lf= 1). Since we are dealing with very low scattering energies a nd
+since13N is not strongly bound ( Ef=−1.94 MeV), the ANC should be included. The
+fit is performed using (2.57), (5.32), (5.33) at a channel rad ius ofa= 5 fm, and the
+resulting observed parameters are given in Table 14.
+Figure 17 presents the R-matrixSfactor compared with experimental data. With
+the single-pole approximation the fit is not perfect, in part icular above the resonance,
+where the R-matrix calculation slightly overestimates the data. This was alreadyTheR-matrix Theory 66
+Table 14. Observed R-matrix parameters at the resonance energy ERfor the
+12C(p,γ)13N reaction.
+ER(MeV) Γ R(keV) Γ γ,R(eV)C1(fm−1/2)
+0.415 31 0.4 1.1
+observed in Fig. 12, with the calculable approach. This prob lem has been addressed
+by Barker and Ferdous [17] who showed that an excellent fit of t he data requires at
+least two poles in the R-matrix expansion. In addition to the total Sfactor, we also
+present the internal and external contributions independe ntly. This analysis is done for
+a= 5 fm, but also for a= 6 fm, where the same observed parameters are used. The
+internal part (dashed line) corresponds to the Breit-Wigne r approximation (5.35); it is
+almost insensitive to the choice of the channel radius. On th e contrary, the external part
+(dotted lines) does depend on a. Its influence near the resonance energy is weak but, as
+expected, it increases at low energies. Neglecting the exter nal term when extrapolating
+down to stellar energies would provide a strong underestima tion of the Sfactor.
+0 0.1 0.2 0.3 0.4 0.5 0.6
+E (MeV)S-factor (MeV-b)
+a=6 fma=5 fma=6 fma=5 fm
+10-410-1
+10-2
+10-3100
+Figure 17.12C(p,γ)13NS-factor computed with the parameters of Table 14 at a= 5
+and 6 fm. The dashed line corresponds to the internal contrib ution (a=5 fm and a= 6
+fm are indistinguishable), and the dotted lines to the exter nal term.TheR-matrix Theory 67
+6. Recent applications of the R-matrix method
+6.1. Introduction
+In section 4 we gave simple examples of the calculable R-matrix method, in order to
+illustrate the theoretical framework with applications wh ich can be easily reproduced
+by the reader. However, in most cases, alternative methods, s ometimes simpler, are
+available.
+In this section, we present more ambitious applications of t heR-matrix theory in
+nuclear physics. The first deals with microscopic cluster mo dels [96], where the relative
+motion between the colliding particles is not given by a pote ntial, but by a nucleon-
+nucleon interaction. The R-matrix theory is also very efficient to solve coupled-channe l
+problems [124]. In various models, the Schr¨ odinger equati on is reduced to a system of
+coupled differential equations. This can be solved, for exam ple, with the generalized
+Numerov algorithm, but this method looses stability when the size of the system
+increases [125]. In that case, the R-matrix theory provides an efficient alternative, in
+particular when it is associated with the Lagrange-mesh met hod [93]. Two applications
+concerning the three-body continuum are presented: (i) the Continuum Discretized
+Coupled Channel (CDCC) method [126, 87], and (ii) the three- body hyperspherical
+formalism [127, 128, 83]. Recent applications of the R-matrix theory in atomic physics
+can be found, for example, in [72, 73, 129, 130].
+6.2. Microscopic cluster models
+6.2.1. General presentation
+Innuclearphysics, amicroscopictheoryisbasedonadescri ptionofallnucleonstaking
+full account of antisymmetrization and derived from intera ctions between nucleons. The
+Hamiltonian (2.25) reads
+H=A/summationdisplay
+i=1p2
+i
+2mN+A/summationdisplay
+i>j=1Vij+A/summationdisplay
+i>j>k=1Vijk−Tcm, (6.1)
+whereAis the nucleon number, piis the momentum of nucleon iandVijand
+Vijkare two- and three-nucleon interactions. In (6.1), subtrac ting the center-of-mass
+energyTcmguarantees that the wave function is free of spurious c.m. co mponents
+[131]. The two-body interactions Vijinvolve a nuclear term with spin-orbit, tensor
+and other components, and the Coulomb interaction. The thre e-body interactions Vijk
+are necessary to explain the binding energies of the3H,3He and4He nuclei. Realistic
+interactionsarederived from field theoriesand partlyfitte don propertiesofthenucleon-
+nucleonsystem. Forsmallnucleonnumbers(i.e. A≤4),differenttechniquesexisttofind
+numericallyexactsolutionsoftheSchr¨ odingerequation[ 132]. Few-bodycalculationscan
+be performed by using various realistic interactions and co mpare well with experiment.
+Forheaviersystems, abinitiocalculations[133,134]becomeavailable. Differentvarian ts
+are being developed, but are currently limited to A≈12, and their application to
+continuum states is quite difficult [135, 136].TheR-matrix Theory 68
+The microscopic cluster approach is based on an assumed clus ter structure, i.e. on
+the existence of correlated subsystems in the fully antisym metric wave function of the
+A-nucleon system [96]. The microscopic cluster model provid es a unified framework for
+the description of nuclear spectroscopy and of nuclear reac tions. The cluster assumption
+allows the application of a microscopic theory to heavier sy stems (typically up to
+A∼20−24), but requires the use of effective nucleon-nucleon two-b ody interactions
+(see, for example, [137, 138]) which are adapted to the clust er approximation. We
+use here the Minnesota effective interaction [138] which doe s not include tensor forces,
+but simulates their contribution in the binding energy of th e deuteron by the central
+term. Three-body effects are in general also approximately s imulated in these effective
+interactions.
+6.2.2. The Resonating Group Method
+The Resonating Group Method (RGM) assumes that the wave func tion can be
+expressed in terms of normed cluster wave functions φiinvolvingAiamong the
+Anucleons and depending on translation-invariant internal coordinates, and of an
+unknown wave function gl(r) for their relative motion. Using the isospin formalism,
+theA-nucleon approximate wave function reads
+Ψlm=A!
+A1!A2!Aφ1φ2gl(r)Ym
+l(Ω), (6.2)
+wherer= (r,Ω) is the relative coordinate between the c.m. of the cluster s, andAis
+theA-nucleon antisymmetrization projector
+A=1
+A!A!/summationdisplay
+p=1(−1)pPp, (6.3)
+where the operator Ppperforms the permutation pamongAparticles. For simplicity,
+we assume a two-cluster structure, with zero-spin clusters [see (2.32) for a multichannel
+definition]. More general presentations can be found in [96, 36, 37, 139, 140]. The
+antisymmetric internal wave functions φ1andφ2are normalized to unity and defined in
+the harmonic-oscillator shell model, with a common oscilla tor parameter b[96]. First
+applications were limited to s-shell clusters [35], but progressively, heavier systems w ere
+considered [141, 142].
+Theunknownrelativefunction gl(r)isobtainedbysolvingtheSchr¨ odingerequation
+HΨlm=EΨlm, (6.4)
+with Hamiltonian (6.1) and approximate wave function (6.2). By projecting the
+Schr¨ odinger equation over /angbracketleftφ1φ2Ym
+l(Ω)|, this technique provides an integro-differential
+equation involving local and non-local potentials [139]
+(Tr+VD(r)−E)gl(r)+/integraldisplay∞
+0/parenleftbigg
+KHl(r,r′)−EKNl(r,r′)/parenrightbigg
+gl(r′)dr′= 0,(6.5)
+whereTris the relative kinetic energy operator (2.39), VD(r) is the direct potential
+(2.40), and KHl(r,r′) andKNl(r,r′) are the Hamiltonian and overlap exchange kernels,
+respectively.TheR-matrix Theory 69
+At large distances, the Hamiltonian (6.1) can be written as
+H−→r→∞H1+H2+Tr+VC(r), (6.6)
+whereH1andH2are the internal Hamiltonians of the clusters, similar to (6. 1), and
+VC(r) is the Coulomb potential between charges Z1eandZ2e. The internal energies Ei
+of clustersi= 1,2 are defined by the variational expressions
+Ei=/angbracketleftφi|Hi|φi/angbracketright. (6.7)
+In parallel, the wave function (6.2) tends to
+Ψlm−→r→∞φ1φ2gl(r)Ym
+l(Ω), (6.8)
+since the antisymmetrization operator Aacts at short distances only. For large rvalues,
+the radial wave function rgl(r) is given by the Coulomb equation (2.4). Although it is
+also suitable for bound states, the ansatz (6.2) of the total wave function is therefore
+well adapted to the treatment of scattering states.
+The main problem of the RGM is not to solve (6.5). This can be do ne by standard
+techniques, orbythe R-matrixmethod[42]. Recentcalculations, usingrealistic nucleon-
+nucleon interactions, have been performed on the3He+p and3H+n scattering [143] and
+on theα+nucleon scattering [136]. In these references, the RGM equ ation (6.5) is
+extended to a multichannel generalization. In [136], it is s olved on a Lagrange mesh.
+ThedrawbackoftheRGMisthatthecalculationoftheoverlap andHamiltoniankernels
+is not systematic. The reason is that the relative coordinat erand the cluster internal
+coordinates in (6.2) are modified in different ways by the term s of the antisymmetrizer
+A. This method requires heavy analytical calculations [96, 1 39] (see also [42, 144] for
+examples of kernels). This problem is simplified by using the Generator Coordinate
+Method (GCM), described in the next subsection.
+6.2.3. The Generator Coordinate Method
+In the GCM, the relative wave function gl(r) is expanded over projected Gaussian
+functions [145] as
+gl(r) =/integraldisplay
+fl(R)Γl(r,R)dR, (6.9)
+whereRis the generator coordinate and Γ l(r,R) is defined as
+Γl(r,R) =/parenleftbiggµ0
+πb2/parenrightbigg3/4
+exp/parenleftBigg
+−µ0r2+R2
+2b2/parenrightBigg
+il/parenleftbiggµ0rR
+b2/parenrightbigg
+. (6.10)
+In this equation, µ0is the reduced mass in units of the nucleon mass, and il(x) =/radicalBig
+π/2xIl+1/2(x),Il+1/2(x) being the modified spherical Bessel function of the first kin d
+[49]. The calculation of gl(r) is therefore replaced by the calculation of the generator
+functionfl(R).
+Inserting (6.9) in the RGM definition (6.2) provides
+Ψlm=/integraldisplay
+fl(R)Φlm(R)dR, (6.11)TheR-matrix Theory 70
+where
+Φlm(R) =A!
+A1!A2!Aφ1φ2Γl(r,R)Ym
+l(Ω). (6.12)
+After multiplication by an appropriate factor depending on t he c.m. coordinate of the A
+nucleons, the basis function Φ lm(R) can be expressed as a projected Slater determinant
+provided that the oscillator parameters of the clusters are identical [145]. The Slater
+determinant is defined from A1andA2shell-model orbitals centered at −A2R/Aand
+A1R/Afor the first and second cluster, respectively. This propert y is well adapted to
+systematic and numerical calculations since it involves ma trix elements of single-particle
+orbitals only. Well-known techniques exist to determine ma trix elements between Slater
+determinantsfrom single-particle matrix elements[146, 1 47]. They allow a rather simple
+extensionoftheclustermodeltoheavysystems[148]andtom ultichannelproblems[149].
+The projection over angular momentum lcan be performed numerically [146].
+In practice, the integral in (6.9) is replaced by a finite sum o ver a set of values
+Rnof the generator coordinate. This means that, at large dista ncesr, the radial
+wave function gl(r) presents a Gaussian behaviour, not consistent with the phy sical
+asymptotic behaviour. This problem can be addressed by usin g the microscopic R-
+matrix method [36]. The wave function is approximated in the internal region by a
+discretized version of (6.11) as
+Ψint
+lm=N/summationdisplay
+n=1fl(Rn)Φlm(Rn). (6.13)
+In the external region, it is approximated by the asymptotic expression (6.8) as
+Ψext
+lm=φ1φ2gext
+l(r)Ym
+l(Ω) (6.14)
+where the external radial function rgext
+l(r) is a linear combination of Coulomb functions,
+as in (3.3).
+The application of the R-matrix method to the GCM is straightforward and follows
+the method detailed in section 3.2. The generalized matrix Cdefined in (3.14) involves
+matrix elements of the Hamiltonian in the internal region onl y. This is achieved by
+subtracting the external contributions [36]. By definition of the channel radius a,
+antisymmetrization effects and the nuclear interaction are negligible in the external
+region. The relevant matrix elements are therefore given by
+/angbracketleftΦl(Rn)|Φl(Rn′)/angbracketrightint=/angbracketleftΦl(Rn)|Φl(Rn′)/angbracketright−/integraldisplay∞
+aΓl(r,Rn)Γl(r,Rn′)r2dr,
+/angbracketleftΦl(Rn)|H|Φl(Rn′)/angbracketrightint=/angbracketleftΦl(Rn)|H|Φl(Rn′)/angbracketright
+−/integraldisplay∞
+aΓl(r,Rn)(Tr+VC(r)+E1+E2)Γl(r,Rn′)r2dr, (6.15)
+where the first terms in the r.h.s. are matrix elements over th e whole space, involving
+Slater determinants. The second terms represent the extern al contributions of the basis
+functions (6.10) and can easily be computed numerically. Th en theR-matrix and the
+associated collision matrix are obtained as in section 3.2. Similarly, the collision matrix
+should not depend on the choice of the channel radius a, provided it is large enoughTheR-matrix Theory 71
+to make the nuclear interaction and the antisymmetrization effects negligible in the
+external region. A generalization to multichannel systems can be found in [36, 37].
+6.2.4. Applications: α+αand12C+p
+We present three typical applications of the GCM associated with the microscopic R-
+matrix method. The first deals with the well known α+αphase shifts (see section 4.4).
+Then we compute the12C+p elastic cross section, as well as the12C(p,γ)13NSfactor
+at low energies. In all cases we use the Minnesota (MN) effectiv e interaction [138] as
+central nucleon-nucleon force. The Minnesota potential pr ovides the correct binding
+energy of the deuteron (without tensor force) and reproduce s fairly well some properties
+of nucleon-nucleon scattering. It involves the admixture p arameteruwhose standard
+value isu= 1, but which can be slightly modified to fit important physica l quantities,
+such as the energy of a bound state or of a resonance. For the12C+p system, a zero-
+range spin-orbit force (with amplitude S0) is added [150].
+Theα+αsystem is described by cluster wave functions φidefined from four 0 s
+harmonic-oscillator orbitals with an oscillator paramete rb= 1.36 fm for the two protons
+and the two neutrons. With the MN force, the binding energy of theαparticle is 24.28
+MeV, which is smaller than the experimental value 28.30 MeV. Th is difference does
+not play an important role, since all theoretical energies a re defined with respect to the
+α+αthreshold. The admixture parameter uis taken as u= 0.94687, as recommended
+in [144]. This value provides an excellent description of th eα+αphase shifts in a wide
+energy range. We use N= 10 basis functions in (6.13) with Rnvalues ranging from 0.8
+fm to 8 fm by steps of 0.8 fm. In Table 15, we give the 0+,2+,4+phase shifts at typical
+energies, and for various conditions of the calculation. Th e channel radius ais taken
+asa= 6.4 fm ora= 7.2 fm, andN= 9 or 10 are considered. In all cases, the phase
+shifts are very stable when the conditions are changed. They can be obtained with an
+accuracy better than 0 .1◦. Fig. 18 shows the phase shifts as a function of energy. It is
+known that a cluster model is well adapted to the α+αsystem since the αparticle has
+a large binding energy and since the first open threshold (7Li+p) is near 17 MeV. The
+GCM phase shifts are therefore in very good agreement with ex periment. The same
+results can be obtained from the RGM equation (6.5) [144].
+The second application deals with the12C+p elastic scattering. The12C internal
+wavefunction( b= 1.65fm)isdescribedinthe pshelllimitedtozeroangularmomentum
+and isospin. This provides four independent shell-model st ates defined from a linear
+combination of Slater determinants [152]. The calculation is therefore performed with
+four channels, obtained from the diagonalization of the12C basis. The first eigenstate
+corresponds to the ground state, and the three additional ei genvalues are considered as
+virtual excitations, which simulate the distortion of12C in the12C+p system.
+The spin-orbit amplitude is fixed as S0= 36.3 MeV.fm5, which allows to reproduce
+the1/2−and3/2−bindingenergiesof13Nwithu= 0.77. Thisvalueisusedfornegative-
+parity partial waves. In positive parity, u= 0.998 reproduces the experimental energy
+of the 1/2+resonance. The GCM widths of the 1 /2+and 3/2−resonances are 43 keVTheR-matrix Theory 72
+Table 15. Microscopic α+αphase shifts (in degrees) for different conditions of
+calculation.
+N= 9 N= 10
+E(MeV)a= 6.4 fma= 7.2 fma= 6.4 fma= 7.2 fm
+l= 0 1 146.00 145.93 146.00 146.00
+5 47.48 47.42 47.49 47.48
+10 −5.67 −5.79 −5.67 −5.67
+15 −38.47−38.52−38.46−38.46
+l= 2 1 0.56 0.53 0.56 0.56
+5 112.00 111.90 112.00 112.00
+10 94.97 94.94 94.98 94.97
+15 77.50 77.38 77.50 77.50
+l= 4 1 0.00 0.00 0.00 0.00
+5 1.00 0.84 1.01 1.00
+10 26.66 26.64 26.66 26.65
+15 118.40 118.40 118.40 118.40
+-60060120180
+0 2 4 6 8 10 12 14
+E (MeV)δ (deg.)α+α
+=2
+=4=0
+Figure 18. α+αphase shifts with the microscopic cluster model. Experimen tal data
+are taken from [151].
+and 99 keV, somewhat larger than the experimental values (31 .7±0.8 keV and 62 ±4
+keV, respectively [94]).
+The12C+p excitation functions at θ= 89.1◦andθ= 146.9◦are shown in Fig. 19.
+Below 1.5 MeV, the agreement with experiment is quite good. How ever the resonant
+structure near 1.6 MeV is not well described since the experi mental data involve a 5 /2+
+resonance, which cannot be described with a simple cluster s tructure. This problem
+often occurs in microscopic cluster models, where some expe rimental states do not have
+a cluster structure, and are therefore not present in the mod el (see, for example, [153]TheR-matrix Theory 73
+05001000
+0 0.5 1 1.5 2dσ/dΩ (mb/sr)θ=146.9o05001000150020002500
+0 0.5 1 1.5 2dσ/dΩ (mb/sr)θ=89.1o
+E (MeV)
+Figure 19.12C+p excitation functions at two c.m. angles. The data are fro m [120].
+for the16O+p system). Of course the phenomenological approach (see F ig. 15) provides
+a better description, but the parameters are adjusted to the data, and all resonances
+are included from the very beginning.
+Our third application of the GCM is the12C(p,γ)13N cross section at astrophysical
+energies. The matrix elements of the E1 operator are determined as explained in
+section 5.7 (see [37] for detail). The capture cross section requires the13N ground
+state wave function, as well as12C+p scattering states. Only swaves, corresponding to
+the 1/2+resonant partial wave are included in the scattering state. Calculations with d
+waves show that this component is of the order of 15% at 1 MeV, bu t becomes negligible
+as soon as the energy decreases, because of the higher centri fugal barrier.
+The resulting Sfactor (2.55) is shown in Fig. 20. As expected from the
+overestimation of the proton width, the peak near the 1 /2+resonance at 0.42 MeV is
+slightly too broad. This type of calculation cannot be expec ted to perfectly reproduce
+the data. However the Sfactor is quite satisfactory considering the fact that no
+parameter is fitted to capture data. The model has thus a predi ctive power which
+is very useful for capture reactions that have not been measu red yet or can not be
+measured. The GCM has been used to study many reactions of ast rophysical interest,
+to compute either the cross section or the properties of low- energy resonances (see for
+example [154]).TheR-matrix Theory 74
+0 0.2 0.4 0.6 0.8 1
+E (MeV)S (MeV-b)12C(p,γ)13N100
+10-1
+10-2
+10-3
+10-4
+Figure 20.12C(p,γ)13N GCM S-factor. The experimental data are from [107] (open
+circles) and [108] (closed circles).
+6.3. The Continuum Discretized Coupled Channel (CDCC) meth od
+6.3.1. Formalism
+The purpose of the CDCC method is to determine, as accurately as possible, the
+scattering and dissociation cross sections of a nucleus whi ch can be easily broken up
+in the nuclear or Coulomb field of a target. The final states may thus involve three
+particles: the target and the fragments of the projectile. T he relative motion of these
+fragments is described by approximate continuum wave funct ions at discrete energies.
+The CDCC method was suggested by Rawitscher [126] and first ap plied to deuteron +
+nucleus elastic scattering and breakup reactions. It was th en extensively developed and
+used by several groups [87, 125]. Its interest has been still revived by the availability of
+radioactive beams of weakly bound nuclei dissociating into three fragments, such as6He
+whose first dissociation channel is α+n+n [155, 156]. Although variants of the CDCC
+methodalsoexistinatomicphysics[157],wefocushereonap plicationsinnuclearphysics
+with two-body projectiles. To simplify the presentation, w e assume a spin zero for the
+constituents of the projectile, and for the target t. As usual in CDCC calculations, the
+internal structure of the three particles is neglected.
+Let us consider the coordinate system of Fig. 21: Ris the internal coordinate of the
+projectile and ris the coordinate of the relative motion between projectile and target.
+The three-body Hamiltonian is given by
+H=H0+Tr+Vt1/parenleftbigg
+r+A2
+AR/parenrightbigg
++Vt2/parenleftbigg
+r−A1
+AR/parenrightbigg
+, (6.16)
+whereH0is the two-body Hamiltonian of the projectile
+H0=TR+V12(R). (6.17)
+In general, potential V12(R) associated with the projectile is real, whereas the
+interactions Vtibetween the fragments iand the target tare derived from the optical
+model, and thus complex. In a schematic notation, the wave fu nction associated withTheR-matrix Theory 75
+(6.16) can be expanded as
+Ψ(r,R) =/summationdisplay
+BΦB(R)uB(r)+/integraldisplay
+Φk(R)uk(r)dk, (6.18)
+whereBdenotes the bound states of the projectile, and φk(R) are two-body scattering
+wave functions with wave number k. The first term represents the elastic and inelastic
+channels, and the second term is associated with the breakup contribution.
+In practice, two methods are available to perform the contin uum discretization, i.e.
+discretize the integral over k. In the “pseudo-state” approach, it is replaced by a sum
+over square-integrable positive-energy eigenstates of Ham iltonian (6.17). The projectile
+Hamiltonian H0is diagonalized over a finite basis, yielding the square-int egrable radial
+functions ΦL
+i(R) at energies EL
+i,
+H0ΦL
+i(R)YM
+L(ΩR) =EL
+iΦL
+i(R)YM
+L(ΩR). (6.19)
+These functions are associated with bound states ( i=B,Ei<0), or represent square-
+integrable approximations of continuum wave functions ( Ei>0).
+The alternative is to replace the integral over kby averages of exact scattering
+states over a range of energies (“bin” method) [87]. This app roach also provides square-
+integrable basis functions. As far as the applicability of th eR-matrix is concerned, both
+methods are treated in the same way. We use here the pseudo-st ate method.
+1
+2tRr
+Figure 21. Coordinate system used for CDCC three-body calculations.
+The total wave function (6.18) is then rewritten, for an angu lar momentum Jand
+parityπ= (−1)l+L, as
+ΨJMπ(r,R) =/summationdisplay
+lLiYJM
+lL(Ωr,ΩR)ΦL
+i(R)uJπ
+lLi(r), (6.20)
+whereJresults from the coupling of orbital momenta landL, and
+YJM
+lL(Ωr,ΩR) =il+L[Yl(Ωr)⊗YL(ΩR)]JM. (6.21)
+The relative wave functions uJπ
+lLi(r) are given by a set of coupled equations
+/bracketleftBigg
+−¯h2
+2µ/parenleftBiggd2
+dr2−l(l+1)
+r2/parenrightBigg
++EL
+i−E/bracketrightBigg
+uJπ
+c(r)+/summationdisplay
+c′VJπ
+cc′(r)uJπ
+c′(r) = 0, (6.22)
+where the channel index cstands for ( lLi). Of course, the sum over Lmust be truncated
+at some value Lmax. The sum over the pseudo-states iis limited by the number of basisTheR-matrix Theory 76
+states and can be reduced further by eliminating states abov e a maximum energy Emax.
+The CDCC problem is therefore equivalent to a system of coupl ed equations where the
+potentialsVJπ
+cc′(r) are given by
+VJπ
+cc′(r) =
+/angbracketleftYJM
+lL(Ωr,ΩR)ΦL
+i(R)|Vt1/parenleftbigg
+r+A2
+AR/parenrightbigg
++Vt2/parenleftbigg
+r−A1
+AR/parenrightbigg
+|YJM
+l′L′(Ωr,ΩR)ΦL′
+i′(R)/angbracketright.(6.23)
+This matrix element represents a 5-dimensional integral ov er (Ωr,ΩR,R). In practice
+the potentials are expanded into multipoles as
+Vt1/parenleftbigg
+r+A2
+AR/parenrightbigg
++Vt2/parenleftbigg
+r−A1
+AR/parenrightbigg
+=/summationdisplay
+λVλ(r,R)Pλ(cosθRr), (6.24)
+whereθRris the angle between Randr, andPλ(x) a Legendre polynomial. In practice
+the number of λvalues is limited by angular-momentum couplings. The four a ngular
+integralsin(6.23)areperformedanalytically, whereasth eintegrationover Rmayrequire
+a numerical approach.
+The system (6.22) can be solved by various methods [124], in p articular with the R-
+matrix formalism. We expand the radial wave functions uJπ
+c(r) over Lagrange functions
+(4.7), and the calculation of potential (6.23) is therefore limited to the mesh points as in
+(4.12). Further simplifications are possible by describing the projectile wave functions
+ΦL
+i(R) [see (6.19)] in a Lagrange basis as well. In this well known t wo-body problem
+[93], the wave function is expanded in a basis differing from ( 4.7) by the fact that it
+is constructed from Laguerre polynomials. The matrix eleme nts involving ΦL
+i(R) [as,
+for example, in the potentials (6.23)] are determined in a fa st and accurate way. The
+extension to three-body projectiles [155, 156] can also be c onsidered in this approach.
+The calculations are much more time-consuming since the pro jectile wave functions
+depend on two radial coordinates, and are more difficult to han dle. Consequently,
+even though the formulation is similar, the calculation of t he potential matrix elements
+(6.23) raises important numerical difficulties, which could be addressed by the present
+technique.
+6.3.2. Application to the d+58Ni elastic scattering
+The CDCC theory, associated with the R-matrix method, is applied to the elastic
+scattering of deuterons on58Ni atElab= 80 MeV. This collision has been widely covered
+in the literature [87, 158]. The p+nand nucleon-58Ni interactions are chosen as in these
+references.
+The first step is to determine the deuteron ground state, and t hep+npseudo-
+states, from (6.19). These wave functions are expanded over a Lagrange-Laguerre basis,
+involving a scaling parameter hwhich is adapted to the size of the system [93]. Typical
+values areh∼0.3−0.4 fm with 15 ∼20 basis functions. We include partial waves
+L= 0,2,4. In a second step, the potentials Vcc′(r) [see (6.22)] are computed at the
+mesh points of the d+58Ni relative motion. These mesh points are defined from the
+channel radius aand from the size of the basis N[see (4.8)].TheR-matrix Theory 77
+Table 16. Amplitude ηJπand phase shift δJπfor elastic d+58Ni elastic scattering
+(Lmax= 4,Emax= 40 MeV) for Jπ= 0+and 17−.
+a(fm)Nη0+δ0+η17−δ17−
+15 30 0.117 31.5 0.5958 20.454
+17 30 0.105 29.7 0.5958 20.454
+15 40 0.116 31.2 0.5958 20.454
+17 40 0.116 31.1 0.5959 20.454
+[158] 0.5956 19.9
+In Table 16 and Fig. 22, we present the elastic part of the coll ision matrix
+UJπ
+11=ηJπexp(2iδJπ)
+forJπ= 0+and 17−. Values for Jπ= 17−can be compared with the literature [158].
+The amplitude ηand phase shift δare computed for different channel radii and numbers
+of wave functions. For the pseudo-states, we truncate at Lmax= 4 andEmax= 50
+MeV. The number of pseudo-states depends on the size of the bas is (here, typically
+their number is about 20). We have checked that changing the s ize of the basis does
+not affect the phase shifts. Table 16 shows that, for high part ial waves (Jπ= 17−), the
+stability with the numerical conditions is virtually perfe ct. The amplitude is in excellent
+agreementwith[158],butthephaseshiftisslightlydiffere nt(0.5◦). Forlowpartialwaves
+(Jπ= 0+), the stability is still acceptable but higher Nvalues are necessary. This can
+be understood by the fact that the internal part is more and mo re important as J
+decreases.
+0.580.600.620.640.66
+0 20 40 60
+Emax (MeV)η17-d+58Ni, Elab=80 MeV
+0
+24
+Figure 22. Amplitude η17−for different Lmaxvalues (labels) as a function of the
+truncation energy Emax.
+Fig. 22 displays the amplitude η17−as a function of the p+ntruncation energy
+Emax, and for different Lmaxvalues. The convergence with respect to Emaxis reachedTheR-matrix Theory 78
+nearEmax≈40 MeV, which corresponds to kmax≈1 fm−1. This result agrees with the
+conclusionof[158]. Fromthefigure, itisclearthat L= 0pseudo-statesarenotsufficient
+to provide accurate values. However, a very good convergence is already obtained with
+Lmax= 2.
+The elastic cross section is presented in Fig. 23, and compar ed with experimental
+data (quoted in [159]). A first calculation, referred to as “n o-breakup” approximation,
+is done by including only the ground state of the deuteron (do tted line). This
+approximation clearly overestimates the data above 30◦. WithL= 0 deuteron pseudo-
+states, the agreement in the region 30◦−50◦is significantly improved. The present
+result is very close to the CDCC calculation of [159]. The cur ves withLmax= 2 and
+Lmax= 4 are indistinguishable at the scale of the figure.
+0 20 40 60 80 100 120
+θ (deg.)σ/σR
+Lmax=4Lmax=0
+d+58Ni, Elab=80 MeV101
+100
+10-1
+10-2
+10-3
+Figure 23. d+58Ni elastic cross section relative to the Rutherford cross se ction at
+Elab= 80 MeV. Solid lines correspond to Emax= 40 MeV and to Lmax= 0,4. The
+dottedlinecorrespondstotheno-breakupapproximation. E xperimentaldataaretaken
+from [159].
+From this example, it turns out that the R-matrix method is a very efficient tool
+to solve the CDCC equations. By using Lagrange functions, th e calculation of the
+coupling potentials Vcc′(r) is very fast, since only values at the mesh points are requir ed
+(typicallyN∼30−40). Consequently, the main part of the computer time is devo ted
+to the inversion of the (complex) matrix C[see (3.15)]. Fast techniques are available for
+matrix inversion, and the total computer time is therefore v ery short (typically a few
+seconds for the angular distribution of Fig. 23). This opens encouraging perspectives
+for CDCC calculations involving three-body projectiles wh ich require large computer
+times with standard techniques [155, 156].TheR-matrix Theory 79
+6.4. Three-body continuum states
+6.4.1. Hyperspherical formalism
+The separation energy being very low in exotic nuclei, a prec ise treatment of
+the continuum is necessary for the description of reactions leading to a three-body
+dissociation. The R-matrix method provides an efficient way to treat three-body
+continuum states [128, 83]. We use the hyperspherical forma lism, well adapted to three-
+body systems [127, 88, 160]. This model is well known, and is j ust briefly outlined
+here.
+In a three-body system, the Hamiltonian is defined, after remo val of the c.m.
+motion, as
+H=3/summationdisplay
+i=1Ti−Tcm+3/summationdisplay
+i>j=1Vij(rj−ri), (6.25)
+whereriare the space coordinates of the particles, Titheir kinetic energy and Vijthe
+interaction between particles iandj. The scaled Jacobi coordinates [ x= (x,Ωx) and
+y= (y,Ωy)] are defined from the coordinates ri, and provide the hyperradius ρand
+hyperangle αas
+ρ=/radicalBig
+x2+y2,
+α= arctan(y/x). (6.26)
+In this coordinate system, the 3-body kinetic energy involv es the operator K2which
+generalizes the concept of angular momentum in 2-body syste ms. It commutes with
+l2
+xandl2
+yand their common eigenfunctions YJM
+γK(Ω5) are known analytically (see
+[88] for details). The eigenvalue of K2isK(K+ 4) where the integer Kis the
+hypermomentum quantum number. In these expressions, Ω 5= (Ωx,Ωy,α) andγstands
+forγ= (lx,ly,L,S) where (lx,ly) are the angular momenta associated with ( x,y),Land
+Sare the total orbital momentum and spin, respectively. The t otal angular momentum
+Jresults from the coupling of LandS, and the parity is given by π= (−1)K.
+The wave function of partial wave Jπassociated with Hamiltonian (6.25) is then
+expanded over hyperspherical harmonics as
+ΨJMπ(ρ,Ω5) =ρ−5/2/summationdisplay
+γKχJπ
+γK(ρ)YJM
+γK(Ω5), (6.27)
+where the radial functions χJπ
+γK(ρ) have to be determined. The Schr¨ odinger equation is
+then reduced to a system of coupled differential equations
+/bracketleftBigg
+−¯h2
+2mN/parenleftBiggd2
+dρ2−(K+3/2)(K+5/2)
+ρ2/parenrightBigg
+−E/bracketrightBigg
+χJπ
+γK(ρ)
++/summationdisplay
+K′γ′VJπ
+K′γ′,Kγ(ρ)χJπ
+γ′K′(ρ) = 0, (6.28)
+where the potentials matrix elements are defined as
+VJπ
+K′γ′,Kγ(ρ) =/angbracketleftYJM
+γ′K′(Ω5)|3/summationdisplay
+i>j=1Vij(rj−ri)|YJM
+γK(Ω5)/angbracketright. (6.29)TheR-matrix Theory 80
+The integral over Ω xand Ω yare performed analytically, whereas a numerical quadratur e
+is used for the integral over the hyperangle α. With the Raynal-Revai coefficients
+[161, 162] the evaluation of (6.29) is rather easy. A truncat ion must be done in the
+summation over K; this provides a maximum Kvalue, denoted as Kmax. The number
+of components in (6.27) increases rapidly when Kmaxincreases.
+This formalism has been extensively applied to three-body b ound states [88]. In
+[163], we have addressed this problem by expanding the radia l wave functions over
+Lagrange functions. This approach provides a very fast meth od to evaluate matrix
+elements (6.29).
+One of the main issues associated with nuclear three-body pr oblems is the presence
+of forbidden states in the nucleus-nucleus interaction [96 , 98]. These two-body forbidden
+states introduce spurious states in the three-body problem and should be removed.
+This problem has been discussed in detail by Thompson et al.[128], and is usually
+solved, either by introducing a projector over the forbidde n states [164], or by using
+supersymmetric potentials [165].
+6.4.2. The R-matrix method for three-body states
+The treatment of three-body slates, with exact three-body a symptotic conditions, is
+more recent [128, 83]. In the R-matrix method, the solutions of the system (6.28) are
+split in two regions,
+χJπ
+γK,int(ρ) =N/summationdisplay
+i=1cJπ
+γKiϕi(ρ), (6.30)
+forρ<awhereϕi(ρ) are basis functions and
+χJπ
+γK,ext(ρ) =CJπ
+γK/bracketleftBig
+H−
+γK(kρ)δγγ′δKK′−UJπ
+γK,γ′K′H+
+γK(kρ)/bracketrightBig
+(6.31)
+forρ≥a. In this equation, CJπ
+γKis a normalization coefficient, UJπis the three-body
+collision matrix and the incoming and outgoing functions H±
+γK(x) are defined as
+H±
+γK(x) =±i/parenleftbiggπx
+2/parenrightbigg1/2
+[JK+2(x)±iYK+2(x)],
+whereJn(x) andYn(x) are Bessel functions of first and second kind, respectively . We
+assume here systems without two-body Coulomb interaction.
+As in previous applications, we choose Lagrange functions fo r the basis states ϕi(ρ)
+[83]. TheR-matrix method is then used to solve the coupled system (6.28 ). Formally
+this is equivalent to the CDCC system [see (6.22)], although the external wave function
+involvesBesselfunctions. Anotherdifferencearisesfromth elongrangeofthethree-body
+potential (6.29). As shown in [83], this potential behaves as
+VJπ
+K′γ′,Kγ(ρ)→VJπ
+0,K′γ′,Kγ
+ρ3, (6.32)
+even with short-range two-body interactions. This arises f rom the definition of the
+hyperspherical coordinates. Even for large ρvalues, two particles can still be close
+to each other and interact strongly. Constants VJπ
+0,K′γ′,Kγcan take rather large valuesTheR-matrix Theory 81
+(examples are given in [83]). For this reason, the R-matrix radius should take large
+values (typically a∼200−300 fm) to ensure that potential (6.32) is negligible compar ed
+with the centrifugal term in (6.28). In those conditions, th e propagation techniques
+described in section 3.9 are necessary to avoid huge basis si zes.
+6.4.3. Application to α+n+nthree-body scattering
+The6He nucleus is an ideal test case for three-body continuum stat es and has been
+considered in previous studies [128, 83]. Accurate α+nandn+ninteractions exist in
+the literature [166, 138]. The α+nsubsystem presents one forbidden slate for l= 0,
+which is removed by using a pair of supersymmetric transform ations [165]. Details can
+be found in [83].
+Figure 24 illustrates the need for propagation techniques. Dotted lines correspond
+to channel radii a= 20 fm and a= 30 fm, and are compared with the exact values
+(solid line, a= 250 fm) obtained by propagation. As expected from the long ra nge of
+the three-body potential (6.29), large values for the chann el radius are necessary. As
+soon as this condition is satisfied, the R-matrix phase shifts are very stable against
+variations of a.
+In Fig. 25, we display the 0+eigenphases as a function of Kmax. The eigenphases
+are obtained after diagonalization of the collision matrix [see (2.37)]. In each case
+we select the eigenphase with the dominant resonant structu re. Above 4 MeV, a
+fair convergence is obtained, but high hypermomenta are nec essary near 2 MeV. The
+0+,1−,2+eigenphases are displayed in Fig. 26 with Kmax= 24,19,and 16, respectively.
+The 2+phase shift presents an experimentally well known narrow re sonance at low
+energies. For the 0+and 1−partial waves, the calculation shows a broad structure near
+1.5 MeV. The existence of three-body resonances at low energi es, and in particular for
+Jπ= 1−, is still an open debate (see, for example, the discussion in [167]), from the
+experimental as well as from the theoretical viewpoints.
+In the future, this formalism could be applied to the 3 αsystem. Current
+experimental [168] and theoretical [169, 170, 171] results are rather controversial
+concerning the existence (or non-existence) of broad 0+and 2+resonances above the
+3αthreshold. This issue is crucial in nuclear astrophysics, s ince the Hoyle state (0+
+2)
+in12C is known to be the main resonance in Helium burning. If the pro perties of this
+resonance are well known, the12C level scheme above this resonance is still not clear.
+In this context, the calculation of 3 αphase shifts would help clarifying the situation.
+However, existing α+αlocal potentials do not provide a satisfactory description of the
+12Cspectroscopy[172]. Non-localpotentialsaremorepromisi ng[172,173], butalthough
+bound states can be easily investigated with Lagrange meshe s [83], their application to
+three-body continuum states remains a challenge for the fut ure.TheR-matrix Theory 82
+60120180
+0 2 4 6 8 10δ (deg.)a=30
+a=20K=0, x=y=0
+0510152025
+0 2 4 6 8 10δ (deg.)
+K=8, x=y=0a=30a=20
+-120-60060120180240300
+0 2 4 6 8 10δ (deg.)
+a=20a=30K=8, x=y=4
+E (MeV)
+Figure 24. α+n+n phase shifts ( Jπ= 0+) for channel radii a= 20 fm ( N= 20) and
+a= 30 fm ( N= 30) without propagation (dashed lines). Solid lines are ob tained with
+propagation up to a= 250 fm (from [83]).
+0306090120150
+0 2 4 6 8 10
+E (MeV)δ (deg.)J=0+
+416
+12
+82024
+Figure 25. Energy dependence of α+n+n eigenphases ( Jπ= 0+) for different Kmax
+values (from [83]).TheR-matrix Theory 83
+060120180240300
+0 2 4 6 8 10δ (deg.) 1-
+0+2+6He
+E (MeV)
+Figure 26. Eigenphases of6He for different Jvalues (from [83]).TheR-matrix Theory 84
+7. Conclusion
+TheRmatrix method was born sixty years ago with the important but rather limited
+goal of describing resonances in nuclear reactions. Today i t has evolved into powerful
+toolscoveringseveralsubfieldsofatomic,molecularandnu clearcollisions. Theliterature
+is so enormous that it is not possible to master it and to cover it in a single review.
+We have emphasized a fact that is often unknown to Rmatrix practitioners: two
+variants exist. The phenomenological Rmatrix remains close to the original spirit
+and is very much used in nuclear physics to parametrize low-e nergy cross sections. Its
+main merits are that all parameters are real and that they hav e a physical meaning.
+Although resonances often play a crucial role in these parame trizations, non-resonant
+cross sections are accurately described as well. The calcul ableRmatrix is an efficient
+technique to solve the stationary or time-dependent Schr¨ o dinger equation in various
+situations as well as its relativistic extensions. It under went most of its developments
+in atomic physics but we have shown in section 6 that it can als o be useful in nuclear
+physics.
+Because of the variety and complexity of its applications, t heR-matrix theory has
+been and is still sometimes misunderstood or misjudged for d ifferent reasons. A first
+often-made criticism concerns the role of the channel radiu s. TheR-matrix formulas
+depend on this radius which has no strict physical meaning. T his criticism can not be
+addressed to the calculable Rmatrix where the independence of the physical results
+on this radius, provided that it is large enough, is a useful v alidity test. In the
+phenomenological R-matrix however, the channel radius is indeed a parameter wh ose
+value is disputable. This arises from the truncation of the Rmatrix to a small number
+of poles. In spite of this truncation, this approximation of ten provides excellent fits
+to the data. The standard option is to optimize the channel ra dius together with the
+other parameters. This radius should however be larger than the sum of the radii of
+the colliding nuclei. Its value should always be mentioned b ecause the other parameters
+depend on its choice.
+Another criticism deals with a reputation of poor convergenc e of the calculable R
+matrix. As we have shown, this reputation is undeserved. Its o rigin lies in the choice of
+a common logarithmic derivative for the basis states in the f ounding papers. While this
+choice is acceptable (although with some discontinuity dra wback) for an infinite basis,
+it leads to inaccuracies when the basis is truncated. The int roduction of the Bloch
+operator has opened a way to the use of finite bases providing a sufficient variety of
+behaviours at the boundary for which this problem disappear s, a fact not yet known
+enough. Modern R-matrix codes can employ different types of such bases. They r each
+an excellent convergence and do not require the use of the But tle correction.
+The calculable Rmatrix provides bound-state and scattering-state wave fun ctions
+that can be used in a variety of applications with sometimes t echnical complications due
+to the existence of two regions to define the wave function. In atomic, molecular and
+nuclear physics, the challenge is now to reach the same level of accuracy for processesTheR-matrix Theory 85
+with more than two unbound particles in the final states. Advan ces have been made
+for double ionization by a photon in atomic and molecular phy sics and for the breakup
+of two- and three-body halo nuclei in nuclear physics. Progr esses ofR-matrix theory in
+these directions should still be expected.
+Acknowledgments
+We are grateful to our present and former colleagues of the de partment ”Physique
+Nucl´ eaire Th´ eorique et Physique Math´ ematique” for many c ommon works and help-
+ful discussions. This text presents research results of the Belgian program P6/23 on
+interuniversity attraction poles initiated by the Belgian -state Federal Services for Sci-
+entific, Technical and Cultural Affairs (FSTC).TheR-matrix Theory 86
+A. Appendix: Collision matrix and Kmatrix
+System (2.38) possesses Nlinearly independent solutions that vanish at the origin. A
+matrix solution uis obtained by putting those independent solutions as colum ns of a
+square matrix. By multiplication on the right by any inverti ble matrix, one obtains
+another matrix solution of system (2.38) which is physicall y equivalent. The most
+general asymptotic expression of such a matrix generalizes (2.34) as
+u−→r→∞v−1/2(I−OU)C, (A1)
+whereUis the collision matrix, IandOare complex conjugate diagonal matrices
+involving incoming and outgoing Coulomb functions (2.8) on the diagonal, and vis a
+diagonal matrix of velocities. Complex matrix Cis arbitrary non singular. In (2.34),
+we have chosen Cdiagonal for simplicity.
+WithC=i(1 +U)−1C′where matrix C′is also non singular, an equivalent
+asymptotic form which is often used is obtained with (2.8) as
+u−→r→∞v−1/2(F+GK)C′(A2)
+whereFandGare real diagonal matrices involving regular and irregular Coulomb
+functionsFlandGlon the diagonal. This asymptotic form is real if C′is real. It is
+then the most general asymptotic form of real solutions.
+Matrices UandKare related by
+U= (1−iK)−1(1+iK) (A3)
+or
+K=i(1−U)(1+U)−1. (A4)
+MatrixKis real and symmetric if Uis unitary and symmetric.
+B. Appendix: Proof of relation (3.27)
+LetBbe an invertible N×Nmatrix and uandvbe vectors with Ncomponents. The
+inverse of the square matrix
+A=B+uvT(B1)
+is given by
+A−1=B−1−B−1uvTB−1
+1+vTB−1u, (B2)
+where the denominator is a scalar. A corollary of (B2) reads
+A−1u=B−1u
+1+vTB−1u. (B3)
+Another corollary is the relation
+(vTA−1u)−1= 1+(vTB−1u)−1(B4)
+from which (3.27) follows.TheR-matrix Theory 87
+C. Appendix: Matrix elements for various basis functions
+Here we present the matrix elements used for different basis fu nctions in Sect. 4. Unless
+specified otherwise the matrix elements of the kinetic energ y are given for l= 0 and in
+units of ¯h2/2µ.
+(i)Sine functions
+The overlap matrix elements between basis functions (4.3) a re given by
+/angbracketleftϕi|ϕj/angbracketright=a
+2δij. (C1)
+The matrix elements for the kinetic energy are simple,
+/angbracketleftϕi|T0|ϕj/angbracketright=π2
+2a/parenleftbigg
+i−1
+2/parenrightbigg2
+δij. (C2)
+Because of property (4.4) at the boundary, matrix elements o f the Bloch operator
+L(0) vanish. The matrix elements of 1 /r2are also analytical but involve Sine
+Integral functions. For the potential, a numerical treatme nt is necessary.
+(ii)Gaussian functions
+Let us define
+Ik(ν) =/integraldisplaya
+0rkexp(−νr2)dr
+=γ((k+1)/2,νa2)/2ν(k+1)/2, (C3)
+whereγis the incomplete gamma function and ais implied. The overlap matrix
+elements between basis functions (4.5) are given by
+/angbracketleftϕi|ϕj/angbracketright=I2l+2(νi+νj), (C4)
+withνi= 1/b2
+i. For the kinetic energy, we have
+/angbracketleftϕi|Tl+L(0)|ϕj/angbracketright= 4νiνjI2l+4(νi+νj)
+−2(l+1)(νi+νj)I2l+2(νi+νj)
++(l+1)(2l+1)I2l(νi+νj). (C5)
+For a Gaussian potential and for the Coulomb potential, the m atrix elements read
+/angbracketleftϕi|exp(−(r/r0)2)|ϕj/angbracketright=I2l+2(νi+νj+1/r2
+0), (C6)
+/angbracketleftϕi|1/r|ϕj/angbracketright=I2l+1(νi+νj), (C7)
+and therefore do not require any numerical integration. Of c ourse, other potentials
+can be considered, but the matrix elements must be, in genera l, obtained from a
+numerical integration.
+(iii)Lagrange functions
+Let us start with matrix elements in interval (0 ,a). The regularization coefficient
+nin (4.7) is taken as n= 1. This ensures that the wave function vanishes at
+the origin and allows that the Coulomb potential is treated a ccurately at theTheR-matrix Theory 88
+Gauss approximation. Using wave functions (4.9) and the corr esponding Gauss
+approximation the matrix elements take a very simple form
+/angbracketleftϕi|ϕj/angbracketright=δij (C8)
+/angbracketleftϕi|V|ϕj/angbracketright=V(axi)δij (C9)
+i.e. they only require the evaluation of the potential at the mesh points. For the
+kinetic energy, a simple calculation [46] provides for i=j
+/angbracketleftϕi|T0+L(0)|ϕi/angbracketright=(4N2+4N+3)xi(1−xi)−6xi+1
+3a2x2
+i(1−xi)2 (C10)
+and fori/negationslash=j,
+/angbracketleftϕi|T0+L(0)|ϕj/angbracketright=(−1)i+j
+a2[xixj(1−xi)(1−xj)]1/2
+×/bracketleftBigg
+N2+N+1+xi+xj−2xixj
+(xi−xj)2−1
+1−xi−1
+1−xj/bracketrightBigg
+. (C11)
+Thanks to the Bloch operator, this matrix element is symmetr ic. Forl/negationslash= 0, the
+centrifugal term is included in the potential (C9).
+Next we consider ( a1,a2) intervals which are used in the propagation method. The
+basis functions (4.7) with n= 0 are extended to
+ϕi(r) = (−1)N+i(xi(1−xi)∆a)1/2PN((2r−a1−a2)/∆a)
+r−xi∆a−a1,(C12)
+with ∆a=a2−a1, and the Lagrange condition becomes
+ϕi(a1+xj∆a) = (λi∆a)−1/2δij. (C13)
+The matrix elements of the potential read
+/angbracketleftϕi|V|ϕj/angbracketright=V(a1+xi∆a)δij, (C14)
+and are still given by a simple evaluation of the potential at the mesh points. The
+matrix elements of the kinetic energy are given at the Gauss a pproximation by
+/angbracketleftϕi|T0|ϕi/angbracketright=1
+3∆a2xi(1−xi)/bracketleftBigg
+N2+N+6−2
+xi(1−xi)/bracketrightBigg
+(C15)
+fori=jand
+/angbracketleftϕi|T0|ϕj/angbracketright=(−1)i+j
+∆a2/radicaltp/radicalvertex/radicalvertex/radicalbtxj(1−xj)
+xi(1−xi)2xixj+3xi−xj−4x2
+i
+xi(1−xi)(xj−xi)2(C16)
+fori/negationslash=j. The matrix elements of the Bloch operators read
+/angbracketleftϕi|La2|ϕj/angbracketright=(−1)i+j
+∆a2/radicalBiggxixj
+(1−xi)(1−xj)/bracketleftBigg
+N(N+1)−1
+1−xj/bracketrightBigg
+(C17)
+and
+/angbracketleftϕi|La1|ϕj/angbracketright=(−1)i+j
+∆a2/radicaltp/radicalvertex/radicalvertex/radicalbt(1−xi)(1−xj)
+xixj/bracketleftBigg
+−N(N+1)+1
+xj/bracketrightBigg
+.(C18)
+Although this does not appear clearly, one can verify that the matrix elements of
+operatorT0+La2−La1are symmetric in accord with the fact that this operator is
+Hermitian over the region ( a1,a2).TheR-matrix Theory 89
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\ No newline at end of file
diff --git a/1001.0679.txt b/1001.0679.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d2fbeac286745d6175f1fa66359a4935ad6c13e1
--- /dev/null
+++ b/1001.0679.txt
@@ -0,0 +1,366 @@
+ 1 Highly efficient acceleration and collimation of hi gh-density plasma using  
+laser-induced cavity pressure  
+ 
+J. Badziak 1a) , S. Borodziuk 1, T. Pisarczyk 1, T. Chodukowski 1, E. Krousky 2, K. Masek 2,  
+J. Skala 2, J. Ullschmied 2, Yong-Joo Rhee 3,  
+1 Institute of Plasma Physics and Laser Microfusion, 01-497 Warsaw, Poland  
+2 PALS Research Centre ASCR, 18200 Prague 8, Czech Re public 
+3 KAERI, Daejeon, 305 – 353 Korea  
+A novel efficient scheme of acceleration and collim ation of dense plasma is proposed and 
+examined. In the proposed scheme, a target placed i n a cavity at the entrance of a guiding 
+channel is irradiated by a laser beam introduced in to the cavity through a hole and accelerated 
+along the channel by the pressure created and accum ulated in the cavity by the hot plasma 
+expanding from the target and the cavity walls. Usi ng 1.315- µm, 0.3-ns laser pulse of energy 
+up to 200J and  a thin CH target,  it was shown tha t the forward accelerated dense plasma 
+projectile produced from the target can be effectiv ely guided and collimated in the 2-mm 
+cylindrical guiding channel and the energetic effic iency of acceleration in this scheme is an 
+order of magnitude higher than in the case of conve ntional ablative acceleration.  
+ 
+The ablative pressure of expanding plasma produced at 
+the interaction of laser or X-ray radiation with th e outer 
+layer of a fusion target is commonly used to accele rate 
+and compress DT fuel in all currently considered 
+approaches to inertial confinement fusion (ICF) [1,  2]. 
+The ablative acceleration (AA) has also been propos ed to 
+be used to accelerate a macroparticle to hyperveloc ity to 
+ignite the fuel in the impact fast ignition (IFI) f usion 
+scheme [3, 4] as well as for other, non-fusion 
+applications [5]. The energetic efficiency of accel eration, 
+ηacc , defined as the ratio of kinetic energy of the 
+accelerated target to energy of radiation used for the 
+acceleration (producing the ablating plasma in case  of 
+AA), is limited by two factors: the absorption coef ficient 
+of radiation in the plasma, ηabs , and the hydrodynamic 
+efficiency, ηh. Even for a short-wavelength radiation 
+(e.g. a 3 ω beam of Nd:glass laser), for which these 
+coefficients can be relatively high ( ηabs   
+≤ 70 – 80%, ηh ~ 10 – 20% [1, 2]), the efficiency 
+ηacc  = ηabs  ηh   is rather low:  ηacc ≤ 7 – 16%. Actually,  a 
+total acceleration efficiency t
+acc η– which also takes into 
+account the energy conversion efficiency, ηc, from the 
+primary laser beam (e.g. a 1 ω Nd:glass laser beam) to 
+the short-wavelength radiation (a 3 ω beam, X-rays) – is 
+yet lower, and for the Nd:glass laser t
+acc η= ηacc ηc ≤ 4 – 
+8%, since, usually, ηc ≤ 50%. One of the important 
+consequences of the low energetic efficiency of 
+acceleration is very high laser energy required for  high-
+gain laser fusion [1, 2, 6]. 
+Fortunately, AA is not the only possible laser 
+method of acceleration of macroparticles or fusion 
+plasmas. About 30 years ago, a cannonball-like 
+acceleration mechanism was proposed to compress a 
+spherical fusion target [7] and its high energetic  
+ 
+a) E-mail: badziak@ifpilm.waw.pl   efficiency was revealed [7, 8]. In its original for m, 
+applied in firearms, this mechanism employs the 
+pressure produced in a closed cavity by chemical 
+explosives to accelerate a projectile e.g. in a can non. In 
+the case of cannonball-like acceleration driven by a 
+laser, the pressure accelerating  a projectile (a t arget) is 
+created in the cavity by the plasma ablating from t he 
+laser-irradiated target placed in the cavity [7 – 9 ] . Such 
+a kind of acceleration can be referred to as laser- induced 
+cavity pressure acceleration (LICPA). Though, for m any 
+reasons, the use of LICPA for effective spherical 
+compression of a fusion target is questionable, LIC PA 
+has still a great potential to be successfully used  for other 
+fusion geometries and/or schemes (e.g. for IFI fusi on 
+[3]) as well as for some non-fusion applications .  
+In this paper, a highly efficient scheme of 
+acceleration and collimation of dense plasma using 
+LICPA is proposed and experimental results 
+demonstrating its very high energetic efficiency ar e 
+presented.  
+ 
+ 
+Fig.1. A scheme of laser-driven accelerator using L ICPA (see the text). 
+ 
+In the proposed scheme (Fig. 1), a projectile place d 
+in a cavity at the entrance of the guiding channel is 
+irradiated by a laser beam through a small hole in the 
+cavity wall and accelerated along the channel by th e 
+pressure produced and accumulated in the cavity by the 
+hot plasma expanding from the irradiated part of th e 
+projectile (from the ablator) and from the cavity w alls. 
+An important part of the scheme (not considered in 
+previously proposed LICPA schemes) is the guiding  2 channel, which plays a role similar to that of a ba rrel in a 
+conventional cannon. In particular, it prevents an 
+“escape” of the pressure from the cavity (which all ows 
+for acceleration of the projectile for a long time)  and, 
+moreover, it makes it possible to collimate and com press 
+the accelerated plasma.  
+The motion of a projectile in the cylindrical, 
+one-dimensional LICPA accelerator without losses ca n 
+be described by the equation:  
+p
+dt zdσ22
+= ,     (1) 
+where p is the pressure in the cavity, σ is the areal mass 
+density of the projectile and z is the position of the 
+projectile measured from its initial position. The cavity 
+volume changes during the projectile acceleration a s 
+(Fig.1): V c = S c [L c+(1+ σ/Σ)z], where L c is the initial 
+cavity length, S c is the projectile (cavity) area, and Σ is 
+the areal mass density of the cavity wall. Assuming  that 
+the pressure in the cavity changes adiabatically wi th an 
+increase in the volume, we arrive at the equation:  
+[ ]γσ )c0
+22
+)(z/L / (1 1p
+dt zdσ
+Σ++= ,    (2) 
+where ) 1 )( /( − = γcca
+L LSE0p  is the initial pressure in the 
+cavity, a
+LE is the laser energy absorbed in the cavity and 
+γ is the adiabatic exponent. In practical units, the  initial 
+pressure can be written as ca
+LL F/) 1 (10 − ≈ γ0p  , [bar, 
+J/cm 2, cm], where ca
+La
+L SE F / =  is the laser energy 
+fluence in the cavity. From this equation, the foll owing 
+expressions for the projectile velocity and the 
+hydrodynamic efficiency as a function of the distan ce z 
+of acceleration can be derived:  
+[ ]2 / 11
+max )/ 1 (1γ−+− =czz vv    (3) 
+ 
+[ ]γη η−+− =1 max )/ 1 (1c h h zz    (4) 
+where:  
+1 max 2 / 1
+max )/ 1 ( ,)/ 1 (2−Σ += 
+
+
+
+Σ += σ ησ σha
+LFv,  (5) 
+and z c = L c(1+ σ/Σ)-1. It can be seen from (5) that in the 
+case σ << Σ (the areal mass density of the projectile is 
+much smaller than that of the cavity wall), almost all 
+laser energy absorbed in the cavity can be transfor med in 
+the projectile kinetic energy and the hydrodynamic 
+efficiency  of acceleration can approach 1, thus it  can be 
+significantly higher than in the case of ablative 
+acceleration.  
+The experiment aimed to demonstrate efficient 
+acceleration and collimation of dense plasma in the  
+proposed scheme was performed at the PALS laser 
+facility [10] in Prague. Both the cavity and the ch annel 
+of the accelerator were hollowed-out in the massive  Al 
+cylinder and their length (L c, LCh ) and diameters (d c, d Ch ) 
+were equal to: L c = 0.1mm, L Ch  = 2mm, d c = d Ch  = 0.3mm (Fig. 2). The diameter of the hole in the cav ity 
+wall was d hole  = 0.18 mm and the  thickness  of  this  wall 
+ 
+ 
+ 
+Fig. 2. A simplified scheme of the experimental set -up (out of scale). 
+ 
+was 0.1mm. A 10- µm polystyrene (CH) foil placed at the 
+channel entrance was irradiated by a 1.315 µm, 0.3-ns 
+laser pulse of energy up to 200 J and intensity up to 
+10 15 W/cm 2 injected into the cavity through the hole. The 
+parameters of the forward-accelerated part of the f oil 
+(actually – the dense CH plasma), playing the role of a 
+projectile, were measured either in configuration M  or N 
+(Fig. 2). The volume of the crater produced in the Al 
+massive target at the output of the channel (config uration 
+M) was assumed to be a measure of energy deposited in 
+the target by forward moving dense plasma and three -
+frame interferometry [11] as well as ion diagnostic s (ion 
+charge collectors) [12] (configuration N) were used  to 
+estimate the plasma velocity and temperature. The 
+results of the measurements performed in the LICPA 
+scheme, in particular, the measurements of the crat er 
+volume and depth, were compared to the ones obtaine d 
+for the AA scheme (CH foil accelerated in the chann el 
+without the use of the cavity) as well as for a dir ect laser-
+Al target interaction.  
+The pictures of the craters and the replicas of the  
+craters produced in the Al target in the LICPA sche me, 
+the AA scheme and the L-T scheme (the direct laser- Al 
+target interaction) – obtained at the same laser be am 
+parameters – are shown in Fig. 3. It can be seen th at the 
+craters produced with LICPA are considerably larger  and 
+deeper than in the case of AA or L-T.  
+A quantitative comparison of the volume and depth 
+of craters produced in the LICPA, AA and L-T scheme s 
+is presented in Figs. 4 and 5. The laser intensity values 
+marked in the figures were calculated assuming 
+  3   
+ 
+ 
+Fig. 3. Craters produced in the Al target by a dire ct laser-target interaction (L-T) as well as by hig h-density plasma driven by LICPA or ablative 
+acceleration (AA) and guided in the cylindrical cha nnel. E L ≈ 130 J, τL = 0.3 ns, I L ≈ 8 × 10 14  W/cm 2, L Ch = 2mm,  d Ch  = 0.3mm.  
+ 
+ 
+ 
+Fig.4. The volume of the craters produced in the Al  target by a direct 
+laser-target interaction  (L–T) as well as   by hig h-density plasma 
+accelerated in the cylindrical channel by LICPA or AA as a function of 
+laser energy (intensity). Note that the crater volu me produced by AA or  
+L – T is magnified 10 times in the figure. L Ch = 2mm, d Ch  = 0.3mm.  
+ 
+that all laser energy (focused to h
+Ld ≈ 0.5 d hole ) was 
+injected into the cavity through the hole in the LI CPA 
+scheme. The volume of the craters produced in the A l 
+target by high-density plasma accelerated in the LI CPA 
+scheme is more than 30 times greater than that for the 
+AA scheme and 12 – 15 times greater than in the cas e of 
+direct laser-target interaction. The crater volume for the 
+LICPA scheme increases approximately linearly with an 
+increase in laser energy and the saturation, seen i n the 
+plot for the AA scheme, does not appear in the 
+considered energy range. As the crater volume is a 
+measure of the energy deposited in the target and, 
+indirectly, a measure of the kinetic energy of the 
+forward-accelerated plasma, these results demonstra te 
+that the energetic efficiency of acceleration ( ηacc ) in the 
+LICPA scheme is significantly (an order of magnitud e) 
+higher than in the case of ablative acceleration. A lso, the 
+energy fluence of the plasma accelerated in the LIC PA 
+scheme is remarkably higher than that in the AA sch eme, 
+which results in a few times greater crater depth ( Fig. 5).  
+ 
+ 
+Fig.5. The depth of the craters produced in the Al target by high-
+density plasma accelerated in the cylindrical chann el by LICPA or AA 
+or by a direct laser-target interaction (L – T) as a function o laser 
+energy (intensity). L Ch = 2mm, d Ch  = 0.3mm.  
+ 
+It should be underlined that, as it results from ou r 
+measurements, using a guiding channel in the LICPA 
+scheme is of key importance for production of a fas t and 
+dense plasma bunch since it ensures both efficient 
+acceleration and collimation of the plasma. We have  
+observed that in the case of employing LICPA withou t 
+the channel, a very shallow crater or no crater was  
+produced in the Al target placed at distances from the 
+CH foil comparable to the channel length (1or 2mm).  It 
+indicates that in such a case the energy accumulate d in 
+the cavity is finally dispersed in a large angle li ke in the 
+case of using AA for the foil acceleration in free space 
+[13] .  
+The three-frame interferometry using the 2 ω PALS 
+laser beam reveals that, at laser energy ~ 130 J, t he 
+forward-accelerated plasma covers the distance of 
+2 mm in ~10 – 15 ns in the channel (Fig. 6), which 
+means that the average plasma velocity in the chann el is 
+<v> ~ (1.5 – 2) ×10 7cm/s. This velocity was found to be 
+comparable to the velocity of a plasma jet of relat ively 
+low electron density (~10 18  – 10 20  cm -3) observed at  
+the channel  output  in  the  AA  scheme  [13].  Ho wever,   
+L - T  LICPA  
+AA   4 
+ 
+Fig. 6. Interferograms of plasma accelerated in the  LICPA accelerator 
+recorded at the output of the cylindrical channel. EL ≈ 130 J, 
+τL = 0.3 ns, I L ≈ 8 × 10 14  W/cm 2, L Ch = 2mm,  d Ch  = 0.3mm.  
+ 
+contrary to the AA case, such a plasma jet was not 
+recorded in the LICPA scheme. As it results from ou r 
+numerical simulations using 1D HYDES code, a 
+plausible reason for this difference is the signifi cantly 
+higher density of plasma in the LICPA scheme caused  
+by additional strong plasma compression by the pres sure 
+accumulated in the cavity.  
+The observed high energetic efficiency of 
+acceleration in the LICPA scheme is a consequence o f 
+high hydrodynamic efficiency and high laser radiati on 
+absorption in the cavity which are higher than in t he case 
+of AA or direct L-T interaction. The higher absorpt ion 
+results from the fact that a significant part of la ser 
+radiation scattered and reflected back by the ablat ive 
+plasma can be confined in the cavity and its energy  can 
+be transformed in the ablative pressure of the plas ma 
+expanding from the irradiated cavity walls. When th e 
+area of the hole in the cavity wall is much smaller  than 
+the area of the cavity walls, almost all laser radi ation can 
+be absorbed in the cavity. In addition, soft X-rays  and 
+fast ions emitted from the plasma, which normally a re a 
+source of plasma energy loss, are absorbed in the c avity 
+and contribute to the cavity pressure production. F or the 
+cavity geometry used in our experiment we can expec t 
+that the cavity absorption coefficient is ηabs ~ 0.5 – 0.7. 
+Taking ηabs  = 0.6 and moreover: σCH  ≈ 1.1 × 10 -3g/cm 2, 
+ΣAl  ≈ 2.7 ×10 -2g/cm 2, L c= 0.1 mm, d c= 0.3 mm, E L = 
+130 J, from expressions (5) we obtain for the LICPA  
+scheme: max 
+hη ≈ 0.96,  v max  ≈ 4.4 × 10 7cm/s and, 
+moreover, p o ≈ 73 Mbar. Assuming z = L Ch = 2mm and  
+γ = 5/3, from (3) and (4) we obtain v ≈ 4.2 × 10 7cm/s and 
+ηh ≈ 0.86 for the channel output, which result in the 
+acceleration efficiency ηacc  = ηabs ηh ≈ 0.52 and the 
+average velocity in the channel <v> ≈ 3.6×10 7cm/s. The 
+last value is a factor 2 higher than the value esti mated 
+from the measurement, likely due to simplicity of t he 
+model which, in particular, does not take into acco unt 
+any energy losses in the considered LICPA accelerat or.  
+ 
+ 
+ 
+ 
+ 
+ In conclusion, a novel efficient scheme of high-
+density plasma acceleration and collimation using l aser-
+induced cavity pressure has been proposed and 
+demonstrated. Due to higher hydrodynamic efficiency  
+and higher absorption of laser radiation in the cav ity the 
+energetic efficiency of acceleration in this scheme  can be 
+an order of magnitude greater than in the case of t he 
+conventional ablative acceleration using the “rocke t 
+effect”. The proposed LICPA accelerator has a poten tial 
+to be highly useful for fusion-related applications  (e.g. 
+for IFI fusion) as well as for other fields of rese arch such 
+as high energy-density physics, laboratory astrophy sics 
+or material processing.  
+ 
+This work was supported in part by the HiPER projec t 
+under Grant Agreement No 211737.  The experiment was 
+performed within the Access to Research Infrastruct ure 
+activity in the Seventh Framework Programme of the EU 
+(Contract No 212025, Laserlab Europe-Continuation)  
+ 
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+    Murakami, H. Nagatomo, T. Johzaki, J. Gardner, D. G. Colombant,  
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+     Plasmas 8, 2495 (2001).  
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+     Plasmas 13 , 062704 (2006).  
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+13. J. Badziak, T. Pisarczyk, T. Chodukowski, A. Ka sperczuk, P.  
+      Parys, M. Rosi ński,  J. Wołowski, E. Krousky, J. Krasa, K. Mašek, 
+      M. Pfeifer, J. Skala, J. Ullschmied, A. Velyh an, L. J. Dhareshwar,  
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+ 
\ No newline at end of file
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+arXiv:1001.0680v1  [math.SG]  5 Jan 2010NON-SYMPLECTIC SMOOTH CIRCLE ACTIONS
+ON SYMPLECTIC MANIFOLDS
+BOGUS/suppress LAW HAJDUK, KRZYSZTOF PAWA/suppress LOWSKI,
+AND ALEKSY TRALLE
+Abstract. We present some methods to construct smooth circle actions on sy mplectic
+manifolds with non-symplectic fixed point sets or non-symplectic cyc lic isotropy point
+sets. All such actions are not compatible with any symplectic form. T o cover the case
+of non-symplectic fixed point sets, we use non-symplectic 4-manifo lds which become
+symplectic after taking the product of the manifold and the 2-sphe re. In turn, the case
+of non-symplectic cyclic isotropy point set is obtained by construct ing smooth circle
+actions on spheres with non-symplectic cyclic isotropy point sets, a nd then by taking
+the equivariant connected sum of the sphere and the cartesian pr oduct of copies of
+the 2-sphere. By using mapping tori, we can convert smooth cyclic g roup actions into
+non-symplectic smooth fixed point free circle actions on symplectic m anifolds.
+Keywords : circle action, symplectic form
+AMS classification (2000) : Primary 53D05; Secondary 57S25.
+1.Introduction
+Oneoftheintriguingquestions onsymplectic manifoldsistogive, fora closedmanifold
+X, sufficient and necessary conditions on M=X×S1to admit a symplectic structure.
+IfMadmits a symplectic form ωinvariant with respect to the obvious action of the
+circle, then on Xwe have a non-vanishing and closed 1-form ιVω,whereVis the vector
+field generating the action. This implies that Xfibres over a circle. Conversely, if X
+admits a symplectic fibration over the circle, then X×S1admits a symplectic structure
+(compare the proof of Proposition 4.8). No other examples are kno wn. In dimension 4,
+an answer to the described problem is known.
+Theorem 1.1. [FV]IfXis a closed irreducible 3-manifold with vanishing Thurston
+norm, then X×S1is symplectic if and only if Xfibres over a circle.
+A more general conjecture was stated by Scott Baldridge in [B].
+Conjecture 1.2. Every closed 4-manifold that admits a symplectic form and a smooth
+circle action also admits a symplectic circle action (with r espect to a possibly different
+symplectic form).
+In the same paper [B], Baldgidge proved that the answer is positive fo r circle actions
+with non-empty fixed point sets.
+Theorem 1.3. [B]IfMis a closed symplectic 4-manifold with a circle action such that
+the fixed point set is non-empty, then there exists a symplect ic circle action on M.
+12 BOGUS/suppress LAW HAJDUK, KRZYSZTOF PAWA/suppress LOWSKI, AND ALEKSY TRAL LE
+It seems unlikely that this continue to be true in higher dimensions. On the other side,
+symplecticness of the action does imply some topological restriction s on a manifold. For
+example, one can repeat the argument above to show that if a circle action is free and
+compatible with a symplectic form, then X/S1also fibers over a circle. Thus, one can
+ask the following question: given a closed symplectic manifold Mwith a smooth circle
+action, when does Madmit a symplectic circle action?
+Wewishtogainsomeunderstanding ofthesecompatibilityproblems by lookingfirst at
+constructions of Lie group actions which are non-compatible with an y symplectic form.
+Note that actions which are non-compatible with a given symplectic fo rm obviously
+exist. Simply deform an invariant symplectic form by a non-equivarian t diffeomorphism
+or deform the action by a non-symplectic diffeomorphism. In [A], Allda y constructed
+examples of cohomologically symplectic manifolds with circle actions whic h cannot be
+symplectic for cohomological reasons. However, there is no metho d in sight to see if
+manifolds in his examples are symplectic, since the surgery used in the construction
+is apparently non-symplectic. Hence, it is still interesting to look for constructions of
+symplectic manifolds with exotic, from the point of view of symplectic topology, s mooth
+circle actions.
+Yet there are other reasons to study examples along the borderlin e between symplectic
+and topological properties of group actions. In fact, if a symplect ic manifold does admit
+a symplectic circle action, there are various restrictions on topolog y ofM, e.g. expressed
+in cohomological terms (cf. [A], [Oz]), on characteristic classes (cf. [Fe]), or on Novikov
+cohomology (cf. [Fa]). However, as in [A], some of these results can b e proved by purely
+cohomological means.
+Aspecificexampleofthisphenomenonaremanifoldswhichareasymme tric, i.e. donot
+admit anyactionof acompact Liegroup. It was discovered some 40y ears agothat closed
+asymmetricmanifoldsexist, andthereareplentyofthem, seeVolke rPuppe’ssurvey[Pu].
+Symplectic manifolds with no symplectic action of any compact connec ted Lie group are
+also known. Namely, Polterovich [Po] has shown that for any closed s ymplectic manifold
+(M,ω) withπ2(M) = 0 and centerless π1(M),the identity component Symp0(M,ω) of
+the symplectomorphism group is torsion free. However, it seems th at these manifolds do
+not admit circle actions.
+Inthispaperweconstruct examples oftwo types. First, therear esmoothcircle actions
+on closed symplectic manifolds with non-symplectic sets of fixed point s, resulting from
+an action on a manifold Xwith isolated fixed points and two 4-manifolds MandNsuch
+thatM×Xis diffeomorphic to N×X, while only one of MandNis symplectic.
+The other type of examples is obtained from smooth actions of the c ircleS1on spheres
+withprescribedfixedpointsetsand Zpq-isotropypointsetsfordistinctprimes pandq. By
+forming the equivariant connected sum of the sphere and the cart esian product of copies
+of the 2-sphere with some action of S1, we obtain smooth actions of S1on symplectic
+manifolds with non-symplectic Zpq-isotropy point sets inherited from the spheres.
+Once we have a specific smooth action of a cyclic group Zron a symplectic manifold
+Mwith fixed point set Fsuch that S1×S1×Fis not symplectic, the mapping torus
+construction allows us to convert Minto symplectic manifold with a smooth fixed point
+free action of S1such that the Zr-isotropy point set is diffeomorphic to S1×S1×F, and
+therefore the constructed action of S1is not symplectic (see Proposition 4.8).NON-SYMPLECTIC SMOOTH CIRCLE ACTIONS 3
+2.Fixed point sets of symplectic actions
+The following is well-known, cf. [GuSt], Lemma 27.1.
+Lemma 2.1. LetGbe a compact Lie group. If Gacts symplectically on a symplectic
+manifold (M,ω), then the fixed point set MGis a symplectic submanifold.
+Proof.Letx∈MG.Then, with respect to a chosen invariant Riemannian metric, Gacts
+on a normal slice via a faithful orthogonal representation. Thus, a vector UofTx(M)
+belongs to Tx(MG) if and only if g∗U=Ufor every g∈G. Moreover, vectors of the
+formV−g∗Vspan a subspace of Tx(M) transversal to MG.Hence, for U∈Tx(MG),
+we have ω(U,V) =ω(g∗U,g∗V) =ω(U,g∗V), and therefore ω(U,V−g∗V) = 0 for any
+g∈GandV∈Tx(M). So, ifω(U,W) = 0 for all W∈Tx(MG), then also ω(U,W′) = 0
+for allW′∈Tx(M) and this implies that U= 0. Thus ω|T(MG) is symplectic. /square
+Corollary 2.2. LetGbe a compact Lie group and let Hbe a closed subgroup of G. If
+Gacts symplectically on a symplectic manifold M, then the set of points with isotropy
+equal toHis a symplectic manifold.
+An analogous property for almost complex manifolds and actions is st raightforward.
+Lemma 2.3. If a compact Lie group Gacts smoothly on an almost complex manifold M
+preserving an almost complex structure J, then the fixed point set MGis a J-holomorphic
+submanifold of M.
+Proof.IfJisG-invariant, g∗(JU) =g∗Jg−1
+∗g∗U=JUfor anyU∈TxMGandg∈G./square
+3.Circle actions with non-symplectic fixed point sets
+Forclosedsimplyconnected6-manifolds MwhosesecondStiefel–Whitney class w2(M)
+vanishes, thediffeomorphism typeof Miscompletelydetermined bythecohomologyring
+M∗(M,Z) and the first Pontriagin class p1(M). More precisely, Theorem 3 in the book
+of Wall [W] can be stated as follows.
+Theorem 3.1. The diffeomorphism classes of closed simply connected 6-manifolds M
+with torsion free H∗(M,Z)andw2(M) = 0correspond bijectively to the isomorphism
+classes of an algebraic invariant consisting of:
+(1)two free abelian groups H=H2(M;Z)andG=H3(M;Z),
+(2)a symmetric trilinear map µ:H×H×H→Zgiven by the cup product,
+(3)a homomorphism p1:H→Zdetermined by the first Pontriagin class p1.
+In the sequel, for a space X, the product of ncopies of Xis denoted by/producttextnX, and
+the disjoint union of ncopies of Xis denoted by/coproducttextnX.
+Proposition 3.2. LetMandNbe two closed simply connected 4-dimensional smooth
+manifolds such that the following condition holds.
+(1)MandNare homeomorphic, but MandNare not diffeomorphic.
+(2)Mis not a symplectic manifold and Nadmits a symplectic structure.
+(3)The second Stiefel–Whitney class w2(M)vanishes, w2(M) = 0.
+(4)The cohomology ring H∗(M,Z)is torsion free.4 BOGUS/suppress LAW HAJDUK, KRZYSZTOF PAWA/suppress LOWSKI, AND ALEKSY TRAL LE
+Then for any n≥1, the manifold M×/producttextnS2is symplectic and admits a smooth action
+ofS1not compatible with any symplectic form.
+Proof.It follows from Theorem 3.1 that M×S2isdiffeomorphic toN×S2. Indeed,
+under our assumptions, w2(M×S2) =w2(N×S2) = 0. Recall that p1(M×S2) is
+inherited from M,andp1is a topological invariant for closed 4-manifolds. Therefore
+p1(M×S2) =p1(N×S2), andM×S2is diffeomorphic to N×S2.As the manifold
+N×S2is symplectic, so are M×S2andM×/producttextnS2for anyn≥1.
+Now, consider the diagonal action of S1onM×/producttextnS2, whereS1acts trivially on M
+andS1acts smoothly on/producttextnS2with a finite number kof fixed points. Then the fixed
+point set of the diagonal action of S1onM×/producttextnS2is diffeomorphic to/coproducttextkM. So, by
+Lemma 2.1, the action is not symplectic with respect to any symplectic structure. /square
+Example 3.3. Some examples of manifolds MandNas required above are obtained
+by applying to symplectic 4-manifolds constructions such as logarith mic transformation
+or knot surgery. To detect both non-diffeomorphism and non-sym plecticness one uses
+Taubes’ theorem that for a symplectic manifold N, the Seiberg–Witten invariant SWN
+is equal to ±1 on a class u∈H2(M;Z). There are examples of symplectic manifolds N
+such that it is possible to obtain from Na smooth manifold Mof the same topological
+type but with SWM(u)/\e}a⊔io\slash=±1 for any u(see Sec. 12.4 of [S], or [P]). An explicit example
+is the Barlow surface which is non-symplectic and homeomorphic to CP2blown up in 8
+points. There exists also a non-symplectic manifold homeomorphic to the K3 surface.
+4.Circle actions with non-symplectic cyclic isotropy point s ets
+For any integer k≥0, we shall consider the representation tk:S1→U(1) =S1given
+bytk(z) =zkfor allz∈S1, and we write tk+tℓ:S1→U(2) to denote the direct sum
+oftkandtℓfork,ℓ≥0. Moreover, ntkdenotes the direct sum tk+···+tk,n-times.
+ForG=S1andH=Zpqfor two distinct primes pandq, we wish to give examples
+of smooth actions of Gon symplectic manifolds Msuch that the H-isotropy point set
+MH={x∈M|Gx=H}is not a symplectic manifold, and thus the action of GonM
+is not symplectic with respect to any possible symplectic structure o nM.
+Hereafter, Dndenotes the n-dimensional disk. First, we construct a smooth action of
+S1on the disk D7with prescribed properties.
+Theorem 4.1. LetG=S1andH=Zpqfor two distinct primes pandq. Then there
+exists a smooth action of Gon the disk D7such that the following conclusions hold.
+(1)The family of isotropy subgroups in D7consists of G,H,Zp,Zq, and{1}.
+(2)The manifold D7
+Gis diffeomorphic to D1.
+(3)The manifold D7
+HisG-diffeomorphic to/coproducttextkG/H∼=/coproducttextkS1for anyk≥1.
+(4)For any x∈D7
+G(resp.D7
+H), the representation of G(resp.H)on the normal
+space to D7
+G(resp.D7
+H)inD7atxis isomorphic to V(resp. ResG
+H(V)), where
+V=R6with the action of Ggiven by the representation t1+tp+tq.
+Proof.As in the proof of [Pa, Lemma 1], consider the action of Gon the disk D3by
+identifying D3withD2×D1, whereGacts via the representation t1onD2and trivially
+onD1. Now, form the quotient space Y=D3/∼by the following identifications on theNON-SYMPLECTIC SMOOTH CIRCLE ACTIONS 5
+boundary S2=S2
++∪S2
+−ofD3:
+S2
++/Zp, S2
+−/Zq,and (S2
++∩S2
+−)/H.
+ThenYis a finite contractible G-CW complex built up from two disjoint cells G/G×D1
+andG/H, by attaching three cells G/Zp×D1,G/Zq×D1, andG×D2. In particular,
+YG=D1andYH=G/H∼=S1.
+For an integer k≥1, setX=Y∪···∪Y, the union of k-copies of YalongYG. Then
+Xis a finite contractible G-CW complex such that XH=XG⊔XHwith
+XG=D1andXH=/coproductdisplay
+kG/H∼=/coproductdisplay
+kS1.
+Clearly, the family of isotropy subgroups in X\XHconsists of Zp,Zq, and{1}.
+LetV=R6with the action of Ggiven by the representation t1+tp+tq:G→U(3).
+Then the product G-vector bundle X×(R⊕V) overX, whereGacts trivially on R,
+when restricted over XGandXH, splits as follows:
+XG×(R⊕V)∼=T(XG)⊕(XG×V),
+XH×(R⊕V)∼=T(XH)⊕(XH×V).
+The total space M=XH×D(V) of the disk bundle XH×D(V) overXHis a compact
+smooth 7-manifold upon which Gacts smoothly in such a way that
+MG=XG, MH=XH,andMH=MG⊔MH=XH,
+and the family of isotropy subgroups in M\MHconsists of Zp,Zq, and{1}. Moreover,
+for any point x∈MG(resp.MH), the representation of G(resp.H) on the normal
+space to MG(resp.MH) inMatxis isomorphic to V(resp. ResG
+H(V)).
+To complete the proof, we can apply the equivariant thickening of [P a, Proposition 1].
+This procedure allows us to replace the G-cells inX\XH,
+G/Zp×D1, G/Zq×D1,andG×D2,
+byG-handles to convert Minto a compact contractible smooth 7-manifold Dequipped
+with a smooth actionof Gsuch that D⊃MasaG-invariant submanifold and the family
+of isotropy subgroups in D\Mconsists of Zp,Zq, and{1}. As∂Dis simply connected
+by the construction, Dis diffeomorphic to D7by the h-Cobordism Theorem. /square
+Corollary 4.2. LetG=S1andH=Zpqfor two distinct primes pandq. Then for any
+given integer n≥1, there exists a smooth action of Gon the sphere S2n+6such that the
+following four conclusions hold.
+(1)The family of isotropy subgroups in S2n+6consists of G,H,Zp,Zq, and{1}.
+(2)The manifold (S2n+6)G=S2n+6
+Gis diffeomorphic to S2n.
+(3)The manifold S2n+6
+HisG-diffeomorphic to/coproducttextk(G/H×S2n−1)for anyk≥1.
+(4)For anyx∈S2n+6
+G(resp.S2n+6
+H), the representation of G(resp.H)on the normal
+space toS2n+6
+G(resp.S2n+6
+H)inS2n+6atxis isomorphic to V(resp. ResG
+H(V)),
+whereV=R6with the action of Ggiven by the representation t1+tp+tq.
+Proof.For a given integer n≥1, consider S2n+6=∂D2n+7∼=∂(D2n×D7), where G
+acts trivially on D2n, and on D7as in Theorem 4.1. This yields a smooth action of G
+onS2n+6such that the conclusions (1)–(4) all hold. /square6 BOGUS/suppress LAW HAJDUK, KRZYSZTOF PAWA/suppress LOWSKI, AND ALEKSY TRAL LE
+Remark 4.3. In Corollary 4.2, remove a point from S2n+6fixed under the action of G
+to obtain a smooth action of GonR2n+6. AsS1×S2n−1is not a symplectic manifold
+forn≥2, andR2n+6
+Hconsists of a number of copies of S1×S2n−1, the action of Gon
+the symplectic manifold R2n+6is not symplectic for n≥2.
+Lemma 4.4. LetG=S1and letρ:G→U(n)⊂O(2n)be an orthogonal representation.
+Then there exists a smooth action of GonX2n=S2×···×S2,n-times, such that for
+any point x∈X2n, the representation of GxonTx(M)is isomorphic ResG
+Gx(ρ).
+Proof.To obtain the required conclusion, consider the diagonal action of Gon
+R2n=R2⊕···⊕R2, n-times,
+obtained from the decomposition of ρinto the irreducible complex summands and taking
+their realifications. For every G-summand R2, the one point compactification
+R2∪{∞}∼=S2
+has the obvious action of G, either trivial if Gacts trivially on R2, or with exactly two
+fixed points, otherwise. The diagonal action of GonX2nhas the required property. /square
+We refer to the action of S1on the symplectic manifold X2ndescribed in the proof of
+Lemma 4.4 as to the ρ-associated action ofS1onX2n.
+LetMandNbe two smooth G-manifolds. Assume that for some points x∈MGand
+y∈NG, the representations of GonTx(M) andTy(N) are isomorphic. This allows us to
+choose some closed disks in MandNcentered at xandywith equivalent linear actions
+ofG. By removing their interiors and gluing together their boundaries, w e can form the
+connected sum M#x,yNwhich admits a smooth action of Gsuch that
+(M#x,yN)G∼=MG#x,yNG.
+Wewish to obtainnonsymplectic smoothactions of S1onclosedsymplectic manifolds.
+As in Lemma 4.4, for n≥1, letX2ndenote the product S2×···×S2,n-times.
+Theorem 4.5. LetG=S1andH=Zpqfor two distinct primes pandq. Then for any
+integern≥1, there exists a smooth action of GonX2n+6such that X2n+6
+G=X2nand
+X2n+6
+H∼=G/coproductdisplayk(G/H×S2n−1)∼=/coproductdisplayk(S1×S2n−1)
+for any given integer k≥1. Ifn≥2, the product S1×S2n−1does not admit a symplectic
+structure, and thus the action of Gon the symplectic manifold X2n+6is not symplectic
+with respect to any possible symplectic structure on X2n+6.
+Proof.According to Corollary 4.2, for any integer n≥1, there exists a smooth action of
+GonS2n+6such that the following four conclusions hold.
+(1) The family of isotropy subgroups in S2n+6consists of G,H,Zp,Zq, and{1}.
+(2) The manifold ( S2n+6)G=S2n+6
+Gis diffeomorphic to S2n.
+(3) The manifold S2n+6
+HisG-diffeomorphic to/coproducttextk(G/H×S2n−1) for any k≥1.
+(4) For any x∈S2n+6
+G(resp.S2n+6
+H), the representation of G(resp.H) on the normal
+space to S2n+6
+G(resp.S2n+6
+H) inS2n+6atxis isomorphic to V(resp. ResG
+H(V)),
+whereV=R6with the action of Ggiven by the representation t1+tp+tq.NON-SYMPLECTIC SMOOTH CIRCLE ACTIONS 7
+Choose a point y∈S2n+6
+Gand consider the representation ρofGonTy(S2n+6). Clearly,
+ρis isomorphic to the realification of nt0+t1+tp+tq. Consider the ρ-associated action
+ofGonX2n+6(cf. Lemma 4.4). In particular, for any x∈X2n+6
+G, the representation of
+GonTx(X2n+6) is isomorphic to ρ. Form the G-equivariant connected sum
+Y2n+6:=X2n+6#x,yS2n+6
+and note that
+Y2n+6
+G∼=X2n+6
+G#x,yS2n+6
+G∼=X2n#x,yS2n∼=X2n.
+AsX2n+6
+H=∅, it follows that
+Y2n+6
+H=S2n+6
+H∼=G/coproductdisplayk(G/H×S2n−1)∼=/coproductdisplayk(S1×S2n−1).
+Now, the conclusion that Y2n+6is diffeomorphic to X2n+6yields the required action of
+GonX2n+6. /square
+Corollary 4.6. LetG=S1andH=Zpqfor two distinct primes pandq. Then for any
+integern≥1, there exists a smooth action of Gon the manifold M2n+8=T2×X2n+6
+such that M2n+8
+G=∅and
+M2n+8
+H∼=(T2×X2n)⊔/coproductdisplayk(T3×S2n−1)
+for any given integer k≥1. Ifn≥2, the product T3×S2n−1does not admit a symplectic
+structure, and thus the action of Gon the symplectic manifold M2n+8is not symplectic
+with respect to any possible symplectic structure on M2n+8.
+Proof.In order to obtain the required action of GonM2n+8, consider the action of G
+onX2n+6as in the conclusion of Theorem 4.5, and then take the diagonal actio n ofG
+onG/H×G/H×X2n+6∼=T2×X2n+6, whereGacts onG/Hin the obvious way. /square
+Now, we wish to discuss some procedures which allow us to obtain smoo th fixed point
+free actions of S1on symplectic manifolds from smooth actions of Zron manifolds M
+with non-empty fixed point set F.
+Proposition 4.7. LetMbea smooth manifoldsuch that S1×Mis a symplectic manifold.
+Assume a cyclic group Zracts smoothly on Mwith non-empty fixed point set F, and for
+the generator gofZr, the diffeomorphism
+g:M→M, x/maps⊔o→gx
+is isotopic to the identity on M. Then there exists a smooth fixed point free action of S1
+onS1×Msuch that the Zr-isotropy point set is diffeomorphic to S1×F. In particular,
+ifS1×Fis not a symplectic manifold, the action of GonS1×Mis not symplectic.
+Proof.The projection S1×M→S1yields a fibration S1×ZrM→S1/Zrwith fiber M
+and the gluing map g:M→M,x/maps⊔o→gx, wheregis the generator of Zr. Asgis isotopic
+to the identity on M,S1×ZrMis diffeomorphic to S1/Zr×M∼=S1×M. The obvious
+action of S1on the twisted product S1×ZrMis fixed point free. Moreover,
+(S1×ZrM)Zr∼=S1/Zr×MZr∼=S1×F,
+completing the proof. /square8 BOGUS/suppress LAW HAJDUK, KRZYSZTOF PAWA/suppress LOWSKI, AND ALEKSY TRAL LE
+We wish to present a variation of Proposition 4.7. Let gbe a periodic (with order r)
+diffeomorphism of a smooth manifold M. Consider the mapping torus T(g) =S1×gM,
+i.e., the fiber bundle over S1with fiber Mand the gluing diffeomorphism g.
+Sincegr= id,T(g) =S1×ZrM. Now,S1acting on itself provides an action on T(g).
+One can easily see that this action is fixed point free and the Zr-isotropy point set is
+diffeomorphic to S1/Zr×F∼=S1×F, whereF=MZr.
+Proposition 4.8. LetMbe a closed symplectic manifold upon which Zracts smoothly
+with non-empty fixed point set F, and for the generator gofZr, the diffeomorphism
+g:M→M, x/maps⊔o→gx,
+is isotopic to a symplectomorphism of M. ThenS1×T(g)is a symplectic manifold with
+a smooth fixed point free action of S1such that the Zr-isotropy point set is diffeomorphic
+toS1×S1×F. In particular, if S1×S1×Fis not a symplectic manifold, the action of
+GonS1×T(g)is not symplectic.
+Proof.As noted above, S1has a smooth fixed point free action on the mapping torus
+T(g) such that the Zr-isotropy point set is diffeommorphic to S1×F. Note that the
+diagonal action of S1onS1×T(g), where S1acts trivially on the first factor S1, is fixed
+point free and the Zr-isotropy point set is diffeomorphic to S1×S1×F.
+Iff:M→Mis a symplectomorphism isotopic to g, thenT(f) is diffeomorphic
+toT(g), andT(f) is a symplectic fibration over a circle, i.e., it possesses a well-defined
+symplectic structure on fibers. This enables ustoapplyThurston’s theorem(see [McDS],
+chapter 6) to get a symplectic structure on S1×T(f), a symplectic fibration over S1×S1
+with fibre M. It suffices to check the claim that the cohomology class of the symp lectic
+form onMis in the image of the cohomology homomorphism i∗,wherei:M→T(f) is
+the inclusion. The claim follows from the Mayer–Vietoris exact sequen ce resulting from
+a decomposition of the base space of the fibration T(f) into two intervals (elements in
+cohomology which are invariant under the gluing map all are in the image ofi∗), or from
+the Wang exact sequence. /square
+Propositions 4.7 and 4.8 allows us to obtain smooth fixed point free act ions ofS1on
+symplectic manifolds with non-symplectic Zpq-isotropy point sets.
+For example, consider the action of S1on the symplectic manifold X2n+6constructed
+in Theorem 4.5, and restrict the action of S1toZpq. Then the diffeomorphism
+g:X2n+6→X2n+6, x/maps⊔o→gx
+given bythe generator gofZpqisisotopic totheidentity on X2n+6, asymplectomorphism
+ofX2n+6. Therefore, we may apply Proposition 4.8 to obtain the required act ion ofS1
+on the symplectic manifold S1×T(g)∼=S1×S1×X2n+6.
+Also, we may replace X2n+6byS1×X2n+6with the diagonal action of Zpq, whereZpq
+acts trivially on S1, andZpqacts onX2n+6by restricting of the action of S1onX2n+6.
+Then we may apply Proposition 4.7 for M=S1×X2n+6, to obtain the required action
+ofS1on the symplectic manifold S1×M=S1×S1×X2n+6.
+Propositions 4.7 and 4.8 are expected to be useful also in the case wh ere the manifold
+Min question admits a smooth action of Zpqwhich does not extend to a smooth action
+ofS1onM. Examples of such actions of ZpqonMcan be obtained from smooth actions
+ofZron disks, whose fixed point sets are well-understood by the work of Oliver [O].NON-SYMPLECTIC SMOOTH CIRCLE ACTIONS 9
+Acknowledgements. This work was partially supported by the grant 1P03A 03330 of
+the Ministry of Science and Higher Education, Poland. The third auth or is grateful to
+the Max-Planck-Institute for Mathematics in Bonn for hospitality a nd excellent working
+conditions.
+References
+[A] C. Allday, Examples of circle actions on symplectic space, Banach C enter Publ. 45(1998),
+87–90.
+[B] S. Baldridge, Seiberg-Witten vanishing theorem for S1-manifolds with fixed points , Pacif. J.
+Math. 217(2004), 1–10.
+[Fa] M. Farber, Topology of closed one-forms , Math. Surveys and Monogr.108, Amer. Math. Soc.,
+Providence, RI, 2004.
+[Fe] K. Feldman, Hirzebruch genera of manifolds supporting Hamiltonian cir cle actions , Russian
+Math. Surveys 56(2001), 978–979.
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+[FV] S. Friedl, S. Vidussi, Symplectic S1×N3, subgroup separability, and vanishing Thurston
+norm , J. Amer. Math. Soc. 21(2008), 597–610.
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+[McDS] D. McDuff, S. Salamon, Introduction to symplectic topology , Oxford Univ. Press, 1998.
+[O] B. Oliver, Fixed point sets and tangent bundles of actions on disks and E uclidean spaces ,
+Topology 35(1996), 583–615.
+[Oz] Y. Ozan, On cohomology of invariant submanifolds of Hamiltonian act ions, Michigan Math.
+J. 53(2005), 579–584.
+[P] J. Park, Exotic smooth structures on 4-manifolds , Forum. Math. 14(2002), 915–929.
+[Pa] K. Pawa/suppress lowski, Smooth circle actions on highly symmetric manifolds , Math. Ann. 341(2008),
+845–858.
+[Pu] V. Puppe, Do manifolds have little symmetry? , J. Fixed Point Theory Appl. 2(2007), 85–96.
+[Po] L. Polterovich, Growth of maps, distortion in groups and symplectic geometr y, Invent. Math.
+150(2002), 655–686.
+[S] A. Scorpan, The Wild World of 4-Manifolds. AMS. Providence, RI, 2005.
+[T] C.H. Taubes, The Seiberg–Witten invariants and symplectic forms , Math. Res. Letters 2
+(1994), 9–14.
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+vent. Math. 1(1966), 355-374. Acad. Press, New York, 1970.
+BH:Mathematical Institute, Wroc/suppress law University,
+pl. Grunwaldzki 2/4,
+50-384 Wroc/suppress law, Poland
+BH and AT: Department of Mathematics and Information Technology,
+University of Warmia and Mazury,
+˙Zo/suppress lnierska 14A, 10-561 Olsztyn, Poland
+KP:Faculty of Mathematics and Computer Science,
+Adam Mickiewicz University,
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+tralle@matman.uwm.edu.pl
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+arXiv:1001.0681v2  [cond-mat.stat-mech]  6 Jan 2010Fractional Brownian motion and generalized Langevin equat ion motion in confined
+geometries
+Jae-Hyung Jeon1,∗and Ralf Metzler1,†
+1Department of Physics, Technical University of Munich,
+James-Franck Straße, 85747 Garching, Germany
+Motivated by subdiffusive motion of bio-molecules observed in living cells we study the stochastic
+propertiesofanon-Brownianparticle whose motionis gover nedbyeitherfractional Brownian motion
+or the fractional Langevin equation and restricted to a finit e domain. We investigate by analytic
+calculations and simulations how time-averaged observabl es (e.g., the time averaged mean squared
+displacement and displacement correlation) are affected by spatial confinement and dimensionality.
+In particular we study the degree of weak ergodicity breakin g and scatter between different single
+trajectories for this confined motion in the subdiffusive dom ain. The general trend is that deviations
+from ergodicity are decreased with decreasing size of the mo vement volume, and with increasing
+dimensionality. We define the displacement correlation fun ction and find that this quantity shows
+distinct features for fractional Brownian motion, fractio nal Langevin equation, and continuous time
+subdiffusion, such that it appears an efficient measure to dist inguish these different processes based
+on single particle trajectory data.
+I. INTRODUCTION
+Anomalous diffusion denominates deviations from the
+regular linear growth of the mean squared displacement
+/angbracketleftr2(t)/angbracketright ≃K1tas a function of time t, where the propor-
+tionality factor K1is the diffusion constant of dimension
+cm2/sec. Often these deviations are of power-law form,
+and in this case the mean squared displacement in ddi-
+mensions
+/angbracketleftr2(t)/angbracketright=2dKα
+Γ(1+α)tα(1.1)
+describes subdiffusion when the anomalous diffusion ex-
+ponent is in the range 0 <α<1, and superdiffusion for
+α>1 [1]. The generalized diffusion constant Kαhas di-
+mension cm2/secα. We here concentrate on subdiffusion
+phenomena. Power-law mean squared displacements of
+the form (1.1) with 0 < α <1 have been observed in a
+multitude of systems such as amorphous semiconductors
+[2], subsurface tracer dispersion [3], or in financial mar-
+ket dynamics [4]. Subdiffusion is quite abundant in small
+systems. These include the motion of small probe beads
+in actin networks [5], local dynamics in polymer melts
+[6], or the motion of particles in colloidal glasses [7]. In
+vivo, crowding-induced subdiffusion has been reported
+for RNA motion in E.colicells [8], the diffusion of lipid
+granules embedded in the cytoplasm [9, 10], the propa-
+gation of virus shells in cells [11], the motion of telomeres
+in mammalian cells [12], as well as the diffusion of mem-
+brane proteins and of dextrane probes in HeLa cells [13].
+While themeansquareddisplacementEq.(1.1) iscom-
+monly used to classify a process as subdiffusive, it does
+not provide any information on the physical mechanism
+∗jae-hyung.jeon@ph.tum.de
+†metz@ph.tum.deunderlying this subdiffusion. In fact there are several
+pathwaysalongwhich this subdiffusion mayemerge. The
+most common are:
+(i) The continuous time random walk (CTRW) [2] in
+which the step length has a finite variance /angbracketleftδr2/angbracketrightof jump
+lengths, but the waiting time τelapsing between succes-
+sivejumps is distributed asa powerlaw ψ(τ)≃τα
+0/τ1+α,
+with 0< α <1. The diverging characteristic waiting
+time gives rise to subdiffusion of the form (1.1) [2]. The
+subdiffusive CTRW in the diffusion limit is equivalent
+to the fractional Fokker-Planck equation, that directly
+shows the long-ranged memory intrinsic to the process
+[1]. Waiting times of the form ψ(τ) were, for instance,
+observed in the motion of tracer beads in an actin net-
+work [5]. We note that recently subdiffusion was also
+demonstrated in a coupled CTRW model [14].
+(ii) A random walk on a fractal support meets bot-
+tlenecks and dead ends on all scales and is subdiffusive.
+The resulting subdiffusion is also of the form (1.1), and
+the anomalousdiffusion exponent is related to the fractal
+and spectral dimensions, dfandds, characteristic of the
+fractal, through α=ds/df[15]. A typical example is the
+subdiffusion on a percolation cluster near criticality that
+was actually verified experimentally [16].
+(iii) Fractional Brownian motion (FBM) and the frac-
+tional Langevin equation (FLE) that will be in the fo-
+cus of this work and will be defined in Sec. II. The un-
+derstanding of these types of stochastic motions is up
+to date somewhat fragmentary. Thus the first passage
+behavior of FBM is known analytically only in one di-
+mension on a semi-infinite domain [17]; the escape from
+potential wells in the framework of FLE has been stud-
+ied analytically [18, 19] and numerically [20, 21]; and a
+priori unexpected critical exponents have been identified
+for the FLE [22]. Here we address a fundamental ques-
+tion related to FBM and FLE. Namely, what is their
+behavior under confinement? Two main aspects of this
+question will be addressed. One is the study of the re-
+laxation towards stationarity in a finite box by means of2
+the ensemble averaged mean squared displacement. We
+also investigate a new quantity used to characterize the
+motion, the displacement correlation function. For these
+aspects we also study the dependence on the dimension-
+ality of the motion.
+The second aspect concerns how FBM and FLE mo-
+tions under confinement behave with respect to ergodic-
+ity. Experimentally the recording of single particle tra-
+jectories has become a standard tool, producing time se-
+ries of data that are then analyzed by time rather than
+ensemble averages. For subdiffusion processes both are
+not necessarily identical. In fact for CTRW subdiffusion
+with an ensemble averaged mean squared displacement
+of the form (1.1) the time averaged mean squared dis-
+placement
+δ2(∆,T) =1
+T−∆/integraldisplayT−∆
+0/bracketleftbig
+x(t+∆)−x(t)/bracketrightbig2dt(1.2)
+on average scales like/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+≃∆/T1−α[23, 24].
+That is, the anomalous diffusion is manifested only in
+the dependence on the overall measurement time Tand
+not in the lag time ∆, that defines a window swept along
+the time series. Thus time and ensemble averages are
+indeed different. In contrast, for normal Brownian dif-
+fusion/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+≃∆ is independent of T, and time
+and ensemble averages become identical, i.e., the sys-
+tem is ergodic. Different from CTRW subdiffusion, sys-
+tems governed by FBM or FLE are ergodic. The er-
+godicity breaking parameter measured from time aver-
+agedmeansquareddisplacementsconvergesalgebraically
+to zero [ergodic behavior] as the measurement time in-
+creases, the convergence speed depending on the Hurst
+exponentH=α/2 [25]. For the FLE case, however, it
+wasalsoshownthat the ergodicitymeasuredfrom the ve-
+locity variance can be broken for a class of colored noises
+[26].
+One of the open questions in the context of ergodic-
+ity breaking for FBM and FLE in the above sense is the
+influence of boundary conditions on the time averages.
+It was shown for CTRW subdiffusion that confinement
+changes the short time scaling/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+≃∆/T1−αto
+the long time behavior/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+≃(∆/T)1−α[27, 28].
+Although we expect that FBM and FLE processes be-
+come stationary under confinement and, for instance,
+attain the same long-time mean squared displacement
+dictated by the size of the confinement volume, we in-
+vestigate how fast this relaxation actually is, and how
+it depends on the volume and the dimensionality. To
+this end we study the ergodicity breaking parameter for
+the system. We find that for both FBM and FLE con-
+finement actually decreases the value of the ergodicity
+breaking parameter with respect to unbounded motion,
+i.e., the process becomes more ergodic. Ergodicity is also
+enhancedwith increasingdimensionality. We alsodiscuss
+how FBM and FLE motions can be distinguished from
+time series from single particle trajectories.The paper is organized as follows. In Sec. II, we in-
+troduce FBM and FLE motions, and review briefly their
+basic statistical properties. In Sec. III, we describe the
+numerical scheme for simulating FBM and FLE in con-
+fined space. Simulations results are presented in Sec. IV
+and V, where we discuss the effects of confinement and
+dimensionality on time-averaged mean squared displace-
+menttrajectory,ergodicity,anddisplacementcorrelation.
+We draw our Conclusions in Sec. VI.
+II. THEORETICAL MODEL
+We here define FBM and FLE. These two stochas-
+tic models share many common features, however, their
+physical nature is different. In the following we will see,
+in particular, how the two can be distinguished on the
+basis of experimental or simulations data.
+A. Fractional Brownian motion
+FBM was originallyintroduced by Kolmogorovin 1940
+[29] and further studied by Yaglom [30]. In a different
+context it was introduced by Mandelbrot in 1965 [31]
+and fully described by Mandelbrot and van Ness in 1968
+in terms of a stochastic integral representation [32]. In
+the latter reference the authors wrote that ”We believe
+FBMs do provideuseful models for a host of natural time
+series”. This study was motivated by Hurst’s analysis of
+annual river discharges [33], the observation that in eco-
+nomic time series cycles of all orders of magnitude oc-
+cur [34], and that many experimental studies exhibit the
+now famed 1 /fnoise. FBM by now is widely used across
+fields. Among many others FBM has been identified as
+the underlying stochastic process of the subdiffusion of
+large molecules in biological cells [13, 35, 36]. We note
+that FBM is neither a semimartingale nor a Markov pro-
+cess, which makes it quite intricate to study with the
+tools of stochastic calculus [37, 38].
+FBM,xH(t), is a Gaussian process with stationary in-
+crements which satisfies the following statistical proper-
+ties: the process is symmetric,
+/angbracketleftxH(t)/angbracketright= 0, (2.1)
+withxH(0) = 0; and the second moment scales like
+Eq. (1.1):
+/angbracketleftxH(t)2/angbracketright= 2KHt2H. (2.2)
+Foreasiercomparisonwithotherliteratureweintroduced
+the Hurst exponent Hthat is related to the anomalous
+diffusion exponent via H=α/2. The Hurst exponent
+may vary in the range 0 < H <1, such that FBM de-
+scribes both subdiffusion (0 <H <1/2) and superdiffu-
+sion (1/2<H <1). The limits H= 1/2 andH= 1 cor-
+respond to Brownian and ballistic motion, respectively.3
+Finally, the two point correlation behaves as
+/angbracketleftxH(t1)xH(t2)/angbracketright=KH(t2H
+1+t2H
+2−|t1−t2|2H).(2.3)
+Here/angbracketleft·/angbracketrightrepresents the ensemble average. It is con-
+venient to introduce fractional Gaussian noise (FGN),
+ξH(t), from which the FBM is generated by
+xH(t) =/integraldisplayt
+0dt′ξH(t′). (2.4)
+FGN has the properties of zero mean
+/angbracketleftξH(t)/angbracketright= 0 (2.5)
+and autocorrelation [39, 40]
+/angbracketleftξH(t1)ξH(t2)/angbracketright= 2KHH(2H−1)|t1−t2|2H−2
++ 4KHH|t1−t2|2H−1δ(t1−t2),
+(2.6)
+as can be seen by differentiation of Eq. (2.3) with re-
+spect tot1andt2. Here we see that KHplays the role
+of a noise strength. For subdiffusion (0 <H <1/2) the
+autocorrelation is negative for t1/negationslash=t2, i.e., the process
+is anti-correlated or antipersistent [41]. In contrast, for
+1/2< H <1, the noise is positively correlated (persis-
+tent) and the motion becomes superdiffusive. For nor-
+mal diffusion ( H= 1/2), the noise is uncorrelated, i.e.,
+/angbracketleftξH(t1)ξH(t2)/angbracketright= 2KHδ(t1−t2). For further details com-
+pare the discussions in Ref. [32, 42].
+We define d-dimensional FBM as a superposition of
+independent FBMs for each Cartesian coordinate, such
+that
+xH(t) =d/summationdisplay
+i=1/integraldisplayt
+0dt′ξH
+i(t′)ˆxi, (2.7)
+where ˆxiistheCartesiancoordinateofthe ithcomponent
+andξH
+iis FGN which satisfies
+/angbracketleftξH
+i(t)/angbracketright= 0 (2.8)
+and
+/angbracketleftξH
+i(t1)ξH
+j(t2)/angbracketright= 2KHH(2H−1)|t1−t2|2H−2δij
++ 4KHH|t1−t2|2H−1δ(t1−t2)δij.
+(2.9)
+From this definition, d-dimensional FBM xH(t) has the
+properties of zero mean
+/angbracketleftxH(t)/angbracketright= 0, (2.10)
+variance
+/angbracketleftxH(t)2/angbracketright= 2dKHt2H, (2.11)
+and autocorrelation
+/angbracketleftxH(t1)·xH(t2)/angbracketright=dKH(t2H
+1+t2H
+2−|t1−t2|2H).(2.12)Note that |xH(t)|cannot satisfy these properties, thus it
+is not an FBM.
+A few remarks on this multidimensional extension of
+FBM are in order. We note that, albeit intuitive due
+to the Gaussian nature of FBM, this multidimensional
+extension is not necessarily unique. In mathematical lit-
+erature higher dimensional FBM in the above sense was
+defined in Refs. [43, 44]. In physics literature an analo-
+gous extension to higher dimensions was used in Ref. [13]
+based on the Weierstrass-Mandelbrot method. To ver-
+ify that this d-dimensional extension is meaningful, we
+checked from our simulations of d-dimensional FBM that
+the fractal dimension of FBM, df= 1/HforH >1/d
+[45], is preservedin higherdimensions. Moreoverweused
+an alternative method to create FBM in d-dimensions,
+namely, to use 1D FBM to choose the length of a radius
+and then choose the space angle randomly. The results
+were equivalent to the above definition to use indepen-
+dent FBMs foreveryCartesiancoordinate. We arethere-
+fore confident that our definition ofFBM in ddimensions
+is a proper extension of regular 1D FBM.
+B. Fractional Langevin equation motion
+An alternative approach to Brownian motion is based
+on the Langevin equation [46–48]
+md2y(t)
+dt2=−γdy(t)
+dt+ξ(t), (2.13)
+whereξ(t) corresponds to white Gaussian noise.
+When the randomnoise ξ(t) isnon-white, the resulting
+motionis described bythe generalizedLangevinequation
+(GLE)
+md2y(t)
+dt2=−γ/integraldisplayt
+0K(t−t′)dy
+dt′dt′+ξ(t),(2.14)
+wheremis the test particle mass, Kis the memory
+kernel [49–51] which satisfies the fluctuation-dissipation
+theorem /angbracketleftξ(t)ξ(t′)/angbracketright=kBTγK(t−t′). WhenξistheFGN
+introducedabove, Kdecaysalgebraically,andEq.(2.14)
+becomes the fractional Langevin equation (FLE)
+md2y(t)
+dt2=−γ/integraldisplayt
+0(t−t′)2H−2dy
+dt′dt′+ηξH(t)
+=−γΓ(2H−1)d2−2H
+dt2−2Hy(t)+ηξH(t).(2.15)
+Here,γisageneralizedfrictioncoefficient. Wealsodefine
+the coupling constant
+η=/radicalBigg
+kBTγ
+2KHH(2H−1)(2.16)
+imposed by the fluctuation dissipation theorem, and the
+Caputo fractional derivative [52]
+d2−2H
+dt2−2Hy(t) =1
+Γ(2H−1)/integraldisplayt
+0dt′(t−t′)2H−2dy
+dt′.(2.17)4
+Note that in Eq. (2.15) the memory integral diverges for
+Hsmaller than 1 /2, such that the Hurst exponent in the
+FLE is restricted to the range 1 /2<H <1.
+It can be shown that the relaxationdynamics governed
+by the FLE (2.15) follows the form
+/angbracketlefty(t)/angbracketright=v0tE2H,2/parenleftBig
+−γt2H/parenrightBig
+(2.18)
+for the first moment, where v0is the initial particleveloc-
+ity. Therescaledfrictioncoefficientis γ=γΓ(2H−1)/m.
+The coordinate variance behaves as
+/angbracketlefty2(t)/angbracketright= 2kBT
+mt2E2H,3/parenleftBig
+−γt2H/parenrightBig
+(2.19)
+where/angbracketleftv2
+0/angbracketright=kBT/mis assumed, and we employed the
+generalized Mittag-Leffler function [53]
+Eα,β(z) =∞/summationdisplay
+n=0zn
+Γ(αn+β)(2.20)
+whose asymptotic behavior for large zis
+Eα,β(z) =−∞/summationdisplay
+n=1z−n
+Γ(β−αn). (2.21)
+Thus the mean squared displacement shows a turnover
+from short time ballistic motion to long time anomalous
+diffusion of the form [54, 55]
+/angbracketlefty2(t)/angbracketright ∼/braceleftbiggt2, t→0
+t2−2H, t→ ∞. (2.22)
+Therefore, for persistent noise with 1 /2<H <1 the re-
+sulting motion is in fact subdiffusive, i.e., the persistence
+of the noise has the opposite effect than in FBM.
+InanalogytoourdiscussionofFBMina d-dimensional
+embedding the FLE is generalized to
+md2y(t)
+dt2=−γ/integraldisplayt
+0/vector/vectorK·dy
+dt′dt′+ξ(t),(2.23)
+wherey(t) =/summationtext
+iyi(t)ˆxi,ξ(t) =/summationtext
+iξH
+i(t)ˆxi, and/vector/vectorKis
+the memory tensor which is in diagonal form (i.e., Kij=
+K(t−t′)δij) in the absenceofmotionalcouplingbetween
+different coordinates.
+III. SIMULATIONS SCHEME
+We here briefly review the simulations scheme used to
+produce time series for FBM and FLE motion.
+A. Fractional Brownian motion
+d-dimensional FBM is simulated via Eq. (2.7) by nu-
+merical integration of ξH
+i(t). The underlying FGN wasgenerated by the Hosking method which is known to be
+an exact but time-consuming algorithm [56]. We checked
+that in the one-dimensional case the generated FBM in
+free space successfully reproduces the theoretically ex-
+pected behavior, the mean squared displacement (1.1),
+the fractaldimension df= 2−α/2ofthe resultingtrajec-
+tory, and the first passage time distribution. To simulate
+the confined motion, reflecting walls were considered at
+locations ±Lfor each coordinate. For instance in the 1D
+case, if|xH(t)|>Lat some time t, the particle bounces
+back to the position xH(t)−2|xH(t)−sign(xH)L|. Sim-
+ilar reflecting conditions were taken into account in the
+multi-dimensional case.
+B. Fractional Langevin equation motion
+In simulating FLE motion, we follow the numerical
+method presented by Deng and Barkai [25]. First, inte-
+grating Eq. (2.15) from 0 to t, we obtain the Volterra
+integral equation for velocity field v(t) =dy(t)/dt
+v(t) =−γ
+(2H−1)m/integraldisplayt
+0(t−t′)2H−1v(t′)dt′
++v0+η
+mxH(t), (3.1)
+wherev(t= 0) =v0. This stochastic integral equa-
+tion can be evaluated by the predictor-corrector algo-
+rithm presented in Ref. [57] with the FBM xH(t) inde-
+pendently obtained by the Hosking method. We calcu-
+latedy(t) =y0+/integraltextt
+0v(t′)dt′by the trapezoidal algorithm.
+For discrete time steps, the equation of motion is given
+by
+yn+1=y0+dh
+2(v0+vn+1)+dhn/summationdisplay
+i=1vi,
+=dh
+2(vn+vn+1)+yn, (3.2)
+wheredhis the time increment. When evaluating
+Eq. (3.2), a reflecting boundary condition was considered
+in the sense that yn→yn−2|yn−sign(yn)L|if|yn|>L.
+To show the reliability of our simulation, we compare
+the simulation result with the well-known solution for
+free space motion. In Fig. 1, we simulate the subdiffu-
+sion case with the parameters values, H= 5/8,m= 1,
+v0= 1,y0= 0,γ= 10,kBT= 1, anddh= 0.01. From
+200 simulated trajectories, we obtain the ensemble av-
+eraged and time averaged mean squared displacements,
+/angbracketlefty2(t)/angbracketrightandδ2(∆,T), and compare them with the exact
+solution Eq. (2.19). Note that /angbracketlefty2(t)/angbracketrightshould be identical
+toδ2(∆,T), Eq. (5.2), with tregarded as the lag time ∆
+duetotheergodicityoftheFLEmotioninfreespace[25].
+The deviation from the exact solution is markedly re-
+duced with decreasing time increment dh. With our cho-
+sen valuedh= 0.01 the mean squared displacements ob-
+tained from simulation appear to be in good agreement
+with the theory.5
+/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48
+/s32/s50
+/s40 /s44/s84/s41
+/s32/s60/s121/s50
+/s40/s116/s41/s62
+/s32/s101/s120/s97/s99/s116/s32/s115/s111/s108/s117/s116/s105/s111/s110/s32/s69/s113/s46/s32/s40/s50/s46/s49/s57/s41
+/s32/s50
+/s40 /s44/s84/s41/s32/s119/s105/s116/s104/s32/s100/s104/s61/s48/s46/s49
+/s32/s60/s121/s50
+/s40/s116/s41/s62/s32/s119/s105/s116/s104/s32/s100/s104/s61/s48/s46/s49
+/s32/s32/s101/s110/s115/s101/s109/s98/s108/s101/s32/s97/s110/s100/s32/s116/s105/s109/s101/s32/s97/s118/s101/s114/s97/s103/s101/s100/s32/s77/s83/s68/s115
+/s116/s32/s111/s114/s32
+FIG. 1. (Color online) The mean squared displacements
+(MSD) for FLE motion in free space. The ensemble av-
+eraged and time averaged MSDs, /angbracketlefty2(t)/angbracketrightandδ2(∆,T), ob-
+tained from simulation are compared with the exact solution
+2t2E5/4,3[−10Γ(5/4−1)t5/4] given by Eq. (2.19). The ensem-
+ble averaged value was obtained from 200 simulated trajecto -
+ries. Inthe simulation, we chose theHurstexponent H= 5/8,
+time increment dh= 0.01, particle mass m= 1, initial veloc-
+ityv0= 1, initial position y0= 0, friction coefficient γ= 10,
+andkBT= 1.
+IV. FRACTIONAL BROWNIAN MOTION IN
+CONFINED SPACE
+We now turn to the investigation of the behavior of
+FBM under confinement, analyzing the mean squared
+displacement and potential ergodicity breaking. We then
+define the displacement correlation function, and finally
+study the influence of dimensionality.
+A. Mean squared displacement
+For FBM in free space /angbracketleftxH(t)2/angbracketrightcan be estimated by
+the time averaged mean squared displacement Eq. (1.2)
+via the exact relation [25]
+/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+= 2KH∆2H. (4.1)
+Here/angbracketleft·/angbracketrightdenotes the ensemble average. In contrast to
+CTRW subdiffusion, in FBM this quantity is ergodic.
+However, as mentioned above, the approach to ergod-
+icity is algebraically slow, and we want to explore here
+whether boundary conditions have an impact on the er-
+godic behavior. Let us now analyze the behaviorin a box
+of size 2L.
+In Fig. 2 we show typical curves for the time averaged
+mean squared displacement for Hurst exponent H= 1/4/s49/s48 /s49/s48/s48 /s49/s48/s48/s48 /s49/s48/s48/s48/s48 /s49/s48/s48/s48/s48/s48/s48/s46/s49/s49/s49/s48
+/s51
+/s52/s47/s51
+/s49/s47/s51/s76/s61/s51/s76/s61/s105/s110/s102/s105/s110/s105/s116/s121
+/s76/s61/s50
+/s76/s61/s49
+/s32/s32/s84/s105/s109/s101/s45/s97/s118/s101/s114/s97/s103/s101/s100/s32/s77/s83/s68
+/s108/s97/s103/s32/s116/s105/s109/s101 /s32
+FIG. 2. (Color online) Time-averaged mean squared displace -
+ment (MSD) versus the lag time ∆ for given values of L. The
+drawn line has slope 1 /2, corresponding to the expected short
+lag time behavior for the used value H= 1/4 of the Hurst
+exponent. For L= 1, 2, and 3, five different trajectories each
+are drawn to be able to see whether the trajectories scatter.
+The simulation time is T= 217≈1.3·105.
+and three different interval lengths L. Regardless of the
+size ofL, the confined environment does not affect the
+power law with exponent 2 Hfor short lag times. More-
+overat long lag times we observesaturation of the curves
+to a value that depends on L. This behavior is distinct
+from that of the CTRW case where/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+shows a
+power law with slope 1 −α[27, 28].
+One can estimate the saturated value as a function of
+L. For long ∆ and measurement time T, the probability
+p(x) to find the particle located at xis independent of x
+due to the equilibration between the reflecting walls, and
+thus/integraltextL
+−Lx2p(x)dx=L2/3. The dotted lines in Fig. 2
+represent these values.
+We observe that the scatter between different single
+trajectories becomes more pronounced when the interval
+length is increased. In fact the scatter is negligible for
+L= 1 while it is quite appreciablefor L= 3, eventhough
+the slope of all curves at finite Lconverges to a horizon-
+tal slope, with an amplitude close to the predicted value
+L2/3. We also note that the scatter depends on the total
+measurement time T. For given Lit tends to be reduced
+as we increase T. This effect will be discussed quantita-
+tively in detail using the ergodicity breaking parameter.
+B. Ergodicity breaking parameter
+In contrast to CTRW subdiffusion, FBM in free space
+is known to be ergodic [25]. The time averaged mean
+squared displacement traces displayed in Fig. 2 exhibit6
+no extreme scatter as known from the CTRW case. This
+implies that ergodicity is indeed preserved for confined
+FBM.Wequantifythisstatementmorepreciselyinterms
+of the ergodicity breaking parameter [23]
+EB(∆,T) =/angbracketleftbigg/parenleftBig
+δ2(∆,T)/parenrightBig2/angbracketrightbigg
+−/angbracketleftBig
+δ2(∆,T)/angbracketrightBig2
+/angbracketleftBig
+δ2(∆,T)/angbracketrightBig2,(4.2)
+where lim T→∞EB(T) = 0 is expected for ergodic sys-
+tems. For the case of free FBM, Deng and Barkai ana-
+lytically derived that EBdecays to zero as
+EB(∆,T)∼
+
+∆
+Tfor 0<H <3
+4,
+∆
+TlogTforH=3
+4,
+/parenleftbigg∆
+T/parenrightbigg4−4H
+for3
+4<H <1,(4.3)
+for long measurement time T[25].
+We numericallyinvestigatethe boundaryeffects on the
+ergodicity breaking parameter. First, in Fig. 3 we eval-
+uateEBas function of the lag time ∆ from 200 FBM
+simulations for each given L. The dotted line represents
+the expected free space behavior EB∼∆, which is nicely
+fulfilled bythe dataat shortertimes and sufficiently large
+L. At longer times or small Lthe results show that
+EBbehaves very differently when confinement effects are
+present. The plateau in EBis related to the saturation
+of the curves for the mean squared displacement, Fig. 2.
+As the motion is restricted by the walls roughly above a
+crossover lag time ∆ cr= (L2/2KH)1/2H, the ergodicity
+breaking parameter EBlevels off at ∆ /greaterorsimilar∆cr. The sharp
+increase at the end of the curve is due to the singularity
+when the lag time reaches the size of the overall mea-
+surement time T, which would disappear in the infinite
+measurement time.
+In Fig. 4 we show EBfor given ∆ as function of
+the measurement time Tfor the same choice of interval
+lengths,L= 1,3,5, and 10. For short lag times ∆, all
+EBcurves coincide and decay as T−1, in complete anal-
+ogy to the free space motion (dotted line). In the case of
+longer ∆ the general trend is that EBdecays like T−1,
+unaltered with respect to the free case. However, there is
+a sudden decrease in EBfor the smallest interval size, for
+L= 1. One can understand this behavior by observing
+theEBcurve forL= 1 in Fig. 3; as the fluctuations of
+the mean squared displacement are strongly suppressed
+due to the tight confinement in this case, EBhas almost
+no dependence on ∆ for ∆ cr/lessorsimilar∆/lessorsimilarTand the saturated
+value is quite small compared to those for other cases.
+Therefore,the curvesfor L= 1appeardisconnectedfrom
+the other curves. Corresponding to the approximate in-
+dependence of the L= 1 curve for ∆ /greaterorsimilar10 in Fig. 3,
+we observe in Fig. 4 that at longer times TtheL= 1
+curves approach each other. Only at T≈∆ these curves/s49 /s49/s48 /s49/s48/s48 /s49/s48/s48/s48 /s49/s48/s48/s48/s48/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49
+/s40/s50/s53/s47/s50/s75
+/s72/s41/s49/s47/s50/s72
+/s40/s57/s47/s50/s75
+/s72/s41/s49/s47/s50/s72/s32/s76/s61/s49/s48
+/s32/s76/s61/s53
+/s32/s76/s61/s51
+/s32/s76/s61/s49
+/s32/s32/s69
+/s66
+/s108/s97/s103/s32/s116/s105/s109/s101/s32/s126
+/s40/s49/s47/s50/s75
+/s72/s41/s49/s47/s50/s72
+FIG. 3. (Color online) Ergodicity breakingparameter EBver-
+sus lag time ∆ for L= 1, 3, 5, and 10 (from bottom to top)
+with Hurstexponent H= 1/4. The overall measurement time
+isT= 214≈1.6·104. The dotted line with slope 1 represents
+the theoretical expectation EB≃∆ in free space. For each L
+the curve was obtained from 200 single trajectories. The ver -
+tical lines show the crossover lag time ∆ cr= (L2/2KH)1/2H
+forL= 1,3, and 5.
+separate, as then δ2
+i(T,T) = [xH
+i(t+T)−xH
+i(t)]2, and
+EBis evaluated with the same small number of squared
+displacement data. Note that the splitting of the EB
+curve can be also observed for larger Lat ∆s larger than
+∆crunder longer total measurement time Tas otherEB
+curves also have corresponding constant saturation val-
+ues for ∆ /greaterorsimilar∆cr(= (L2/2KH)1/2H) which increases with
+the sizeL.
+C. Displacement correlation function
+As explained for the stochastic properties of FBM in
+Sec. II, the position autocorrelation /angbracketleftxH(t1)xH(t2)/angbracketrightex-
+plicitly depends on t1andt2as well as their difference,
+|t1−t2|. It is therefore not an efficient quantity to es-
+timate directly from experimental or simulations data.
+However, the correlation function of the displacements
+δxH(t,∆) =xH(t+∆)−xH(t) (4.4)
+depends only on the time interval ∆ of the displacement,
+/angbracketleftδxH(t,∆)δxH(t−∆,∆)/angbracketright=KH(22H−2)∆2H(4.5)
+for free FBM. This relation is derived in App. A. This
+quantity is anticorrelated for 0 < H < 1/2 (subdiffu-
+sive motion), uncorrelated for H= 1/2 (normal Brown-
+ian motion), and positively correlated for 1 /2< H <1
+(superdiffusion). As Eq. (4.5) does not depend on the
+measurement time Tthe value of the ensemble averaged
+value is identical to the corresponding time average.7
+/s49 /s49/s48 /s49/s48/s48 /s49/s48/s48/s48 /s49/s48/s48/s48/s48 /s49/s48/s48/s48/s48/s48/s49/s48/s45/s54/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49
+/s126/s84/s45/s49/s61/s49/s48/s48/s48/s61/s49/s48/s48
+/s32/s76/s61/s49
+/s32/s76/s61/s51
+/s32/s76/s61/s53
+/s32/s76/s61/s49/s48
+/s32/s32/s69
+/s66
+/s84/s61/s49
+FIG. 4. (Color online) Ergodicity breaking parameter EB
+versus overall measurement time Tat given lag time ∆ = 1,
+100, and 1000. The dotted line depicts a power law with slope
+-1, representingthe analytic behavior EB≃T−1in free space.
+For each ∆, the EBcurves are drawn for different interval
+lengthsL= 1, 3, 5, and 10. Each curve was obtained from
+200 single trajectories, and the Hurst exponent was H= 1/4.
+/s49 /s49/s48 /s49/s48/s48 /s49/s48/s48/s48 /s49/s48/s48/s48/s48/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49
+/s32/s32/s124/s68/s67/s124
+/s108/s97/s103/s32/s116/s105/s109/s101/s32
+FIG. 5. (Color online) Absolute value of the displacement
+correlation (DC) versus lag time ∆ for L= 1, 2, and 3 (from
+bottomtotop)withaslope proportional to∆2H(dottedline).
+Total measurement time is T= 217≈1.3·105, and the Hurst
+exponent is H= 1/4. Each curve was obtained via time
+averaging from a single particle trajectory.
+We present the time averageddisplacement correlation
+for confined subdiffusive motion ( H= 1/4) in Fig. 5. Be-
+cause of the negativity of expression (4.5) the absolute
+value of the displacement correlation is drawn in the log-
+log representation. At short lag times the slope of the
+correlation functions is proportional to 2 Has expected/s49/s48 /s49/s48/s48 /s49/s48/s48/s48 /s49/s48/s48/s48/s48/s48/s46/s49/s49/s49/s48/s84/s105/s109/s101/s45/s97/s118/s101/s114/s97/s103/s101/s100/s32/s77/s83/s68
+/s108/s97/s103/s32/s116/s105/s109/s101/s32
+/s32/s32
+FIG. 6. (Color online) Time-averaged mean squared displace -
+ment (MSD) curves versus lag time ∆ for L= 1 in 1D, 2D,
+and 3D space (from bottom to top), with a slope ∆2H(dot-
+ted line). For each dimension, 5 trajectories were drawn wit h
+total measurement time T= 217andH= 1/4.
+from Eq. (4.5). However, at long lag times, we inter-
+estingly observe fluctuations of the correlations around a
+constant value, reflecting the confinement of the motion.
+D. Dimensionality
+To mimic the anomalous diffusion of particles in-
+side biological cells, we also simulate two- and three-
+dimensional FBM based on Eq. (2.7) in the presence of
+reflecting walls. In free space, the ensemble average of
+the time averaged mean squared displacement is simply
+given by
+/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+=1
+T−∆/integraldisplayT−∆
+0/angbracketleftbig
+[xH(t+∆)−xH(t)]2/angbracketrightbig
+dt,
+= 2dKH∆2H, (4.6)
+i.e., it is additive as for the ensemble average. This be-
+havior is indeed observed in Fig. 6 where five different
+mean squared displacement curves are drawn for L= 1
+in 1D, 2D, and 3D, respectively. Only the height of the
+curves are affected by the dimensionality. There is no
+noticeable difference in the scatter of the curves.
+We further investigate the effects of dimensionality on
+the scatter of the mean squared displacement curves, as
+possibly the strong scatter observed in experiments [8–
+10] may also occur for FBM in higher dimensions. To see
+the effect of dimensionality on the ergodicity behavior
+we measure EBversus lag time for one-, two-, and three-
+dimensional embedding dimension for the same values of
+LandH. Interestingly,theresultshowsthat EBtendsto
+decrease with increasing dimensionality d, meaning that8
+/s49/s48/s48 /s49/s48/s48/s48 /s49/s48/s48/s48/s48/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48
+/s32/s51/s68
+/s32/s50/s68
+/s32/s49/s68
+/s108/s97/s103/s32/s116/s105/s109/s101/s32
+/s32/s32/s100/s32/s69
+/s66/s40/s100/s41
+/s32
+FIG. 7. (Color online) Rescaled ergodicity breaking param-
+eterdEBversus lag time ∆ for interval size L= 5 for 1D,
+2D, and 3D space (from top to bottom), with Hurst exponent
+H= 1/4 and measurement time T= 214.EBwas evaluated
+from 200 trajectories with different initial positions.
+for FBM big scatter is not caused by higher dimensions
+in presence of reflecting walls. In fact, from Eqs. (4.2)
+and (4.6), we can analytically derive the relation
+EB(d) =EB(d= 1)
+d, (4.7)
+which still holds in the case of confined motion (see ap-
+pendix B for the derivation). This relation is numerically
+confirmed in Fig. 7 where three EBcurves collapse upon
+rescaling by dEB(d). According to this relation, we ex-
+pect that ergodic behavior obtained in one-dimensional
+confined motion (Figs. 3 and 4) will also be present in
+multiple dimensions with a factor of 1 /d.
+V. FRACTIONAL LANGEVIN EQUATION
+MOTION IN CONFINED SPACE
+In this section we analyze FLE motion under confine-
+ment. Due to the different physical basis compared to
+FBM, in particular, the occurrence of inertia, we observe
+interestingvariationsonthepropertiesstudiedinthepre-
+vious section.
+A. Mean squared displacement
+Using the correlation function [55]
+/angbracketlefty(t1)y(t2)/angbracketright=kBT
+m[t2
+1E2H,3(−γt2H
+1)+t2
+2E2H,3(−γt2H
+2)
+−(t2−t1)2E2H,3(−γ|t2−t1|2H)], (5.1)/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49
+/s76/s61/s49
+/s49/s47/s51
+/s76/s61/s49/s48/s48/s76/s61/s51
+/s51
+/s49/s47/s49/s50
+/s76/s61/s49/s47/s50
+/s32/s32/s84/s105/s109/s101/s45/s97/s118/s101/s114/s97/s103/s101/s100/s32/s77/s83/s68
+/s108/s97/s103/s32/s116/s105/s109/s101 /s32
+FIG. 8. (Color online) Time-averaged mean squared displace -
+ment (MSD) versus lag time ∆. The two dashed lines repre-
+sent the two asymptotic scaling behaviors/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+≃∆2
+and/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+≃∆2−2H. For each given L= 1/2, 1, and 3,
+five different trajectories are drawn to visualize the scatte r.
+As a reference curve for motion in free space, the curve for
+L= 100 (line) is drawn. Inthe simulation, we chose the Hurst
+exponent H= 5/8 [NB: for the FLE this means subdiffusion],
+time increment dh= 0.01, particle mass m= 1, initial veloc-
+ityv0= 1, initial position y0= 0, friction coefficient γ= 10,
+andkBT= 1.
+onecan showanalyticallythat, similarlyto the FBM, the
+ensemble averaged second moment /angbracketlefty2(t)/angbracketrightis identical to
+its time averaged analog/angbracketleftBig
+δ2(∆)/angbracketrightBig
+, for all ∆ in free space,
+namely
+/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+= 2kBT
+m∆2E2H,3(−γ∆2H).(5.2)
+Thus, the time averaged mean squared displacement
+turns over from a ballistic motion
+/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+≃∆2(5.3)
+at short lag time to the subdiffusive behavior
+/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+≃∆2−2H(5.4)
+at long lag times, in free space.
+We numerically study how this scaling behavior is af-
+fected by the confinement. Figure 8 shows typical curves
+for the time averaged mean squared displacement, for in-
+terval sizes L= 1/2, 1, 3, and 100 (regardedas free space
+motion) with identical initial conditions and Hurst expo-
+nentH= 5/8. The results are summarized as follows:
+(1) We observe both scaling behaviors,/angbracketleftBig
+δ2(∆,T)/angbracketrightBig
+≃∆29
+turning over to ≃∆2−2H, for confined FLE motions.
+(2) For narrow intervals, the curves eventually reach the
+saturation plateau within the chosen total measurement
+timeT. The saturation values are approximately L2/3.
+For interval size L= 1/2, the saturated value is notice-
+ably larger than L2/3, which appears to be caused by
+multiple reflection events on the walls. The same behav-
+ior is observed in the FBM case when considering a large
+value ofH≥1/2, or very narrow intervals for the given
+H= 1/4. (3) As in the case of FBM, the scatter becomes
+pronounced as the interval length increases.
+B. Ergodicity breaking parameter
+From a simple argument and simulations it was shown
+in Ref. [25] that the FLE and FBM mean squared
+displacements are asymptotically equal,/angbracketleftBig
+δ2(y)/angbracketrightBig
+∼/angbracketleftBig
+δ2(xH)/angbracketrightBig
+, similarly for the ergodicity breaking param-
+eter,EB(y)∼EB(xH). Here the asymptotic equivalence
+is valid at long measurement times T, and the derivation
+holds for motion in free space. From 200 trajectories of
+the mean squared displacement we measure the ergodic-
+ity breaking parameter EBas function of lag time ∆ for
+interval lengths L= 1/2, 1, 3, and 100 in Fig. 9. The
+behavior is similar to the corresponding curves for FBM,
+displayed in Fig. 3: EBsignificantly deviates from the
+reference curve for free space motion (i.e., the longer ∆
+behavior for L= 100 and the drawn power law ≃∆),
+due to the confinement effect. EBtends to decrease with
+smallerLfor the same value of ∆. However, the plateau
+at short lag times that is still observed for L= 100 (re-
+garded as free space motion), is due to the initial ballistic
+motion of FLE. In that regime the initially directed mo-
+tion renders the random noise effect negligible.
+C. Displacement correlation function
+Using the correlation function /angbracketlefty(t1)y(t2)/angbracketright, we analyti-
+cally obtain the displacement correlation function in free
+space in the form (refer to App. A for the derivation)
+/angbracketleftδy(t,∆)δy(t−∆,∆)/angbracketright= 4kBT
+m∆2E2H,3[−γ(2∆)2H]
+−2kBT
+m∆2E2H,3[−γ∆2H],(5.5)
+so that we observe the following asymptotic behavior
+/angbracketleftδy(t,∆)δy(t−∆,∆)/angbracketright
+∼
+
+2kBT
+mΓ(3)∆2for ∆→0,
+(22−2H−2)kBT
+mγΓ(3−2H)∆2−2Hfor ∆→ ∞,(5.6)
+whereδy(t,∆) =y(t+ ∆)−y(t). Above expression
+shows that the displacement correlation has two distinct/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48 /s76/s61/s49/s48/s48
+/s76/s61/s49/s76/s61/s51
+/s76/s61/s49/s47/s50/s69
+/s66
+/s108/s97/s103/s32/s116/s105/s109/s101/s32
+/s32/s32
+FIG. 9. (Color online) Ergodicity breaking parameter EB
+versuslagtime∆forintervalsizes L= 1/2, 1, 3, and100. The
+straight line corresponds to the free space behavior EB≃∆.
+Each curve was obtained from 200 single trajectories, with
+the same parameter values used in Fig. 8.
+scaling behaviors. At short lag times, it grows like ∆2
+and is positive, due to the ballistic motion. At long lag
+times, it is negative in the domain 1 /2<H <1, ex-
+hibiting the same subdiffusive behavior as observed for
+FBM [cf. Eq. (4.5)] when we replace H→2−2H. Note
+that to bridge these two scaling behaviors the displace-
+ment correlation passes the zero axis at ∆ = ∆ cthat
+satisfies 2E2H,3/bracketleftBig
+−γ(2∆c)2H/bracketrightBig
+=E2H,3/bracketleftBig
+−γ∆2H
+c/bracketrightBig
+in free
+space. For small γ, we find approximately
+∆c≈/parenleftbiggΓ(2H+3)
+2Γ(2H−1)(22H+1−1)m
+γ/parenrightbigg1/2H
+,(5.7)
+such that it becomes exactly the momentum relaxation
+timem/γfornormalBrownianmotion( H= 1/2). Inthe
+limitH→1, ∆cgoes to infinity to satisfy the equality
+2E2H,3/bracketleftBig
+−γ(2∆c)2H/bracketrightBig
+=E2H,3/bracketleftBig
+−γ∆2H
+c/bracketrightBig
+. Thus, ∆ ccan
+be interpreted as the typical timescale for the persistence
+of the ballistic motion.
+Figure 10 shows (a) the displacement correlation ver-
+sus lag time ∆, and (b) the absolute value of the dis-
+placement correlation as function of ∆, for L= 1/2, 1,
+3, and 100. The scaling properties derived in Eq. (5.6)
+are indeed observed. At short lag times all curves are
+positive and scale like ∼∆2, before decreasing to zero.
+In the long lag time regime the displacement correla-
+tion becomes negative and the predicted scaling behavior
+≃∆2−2His observed. For small intervals ( L= 1/2 and
+1), it is saturated due to the confinement effect as seen
+in the case of FBM.10
+/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48
+/s76/s61/s49/s48/s48/s76/s61/s51
+/s76/s61/s49
+/s76/s61/s49/s47/s50
+/s32/s32/s60/s124/s68/s67/s124/s62
+/s108/s97/s103/s32/s116/s105/s109/s101/s32/s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s45/s48/s46/s48/s49/s48/s45/s48/s46/s48/s48/s53/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53
+/s32/s76/s61/s49/s47/s50
+/s32/s76/s61/s49
+/s32/s76/s61/s51
+/s32/s76/s61/s49/s48/s48/s40/s97/s41
+/s40/s98/s41/s60/s68/s67/s62
+/s32/s32/s32
+FIG. 10. (Color online) (a) Displacement correlation (DC)
+versus lag time ∆ for L= 1/2, 1, 3, and 100 (from bottom
+to top). (b) Absolute value of the displacement correlation
+as a function of ∆ for the given values of L. The two slopes
+correspond to the limiting behavior ∆2and ∆2−2H. In (a)
+and (b), the ensemble averaged curves were obtained from
+200 different time averaged displacement correlation curve s.
+Same parameter values as in Fig. 8.
+D. Dimensionality
+In the case when the memory tensor is diagonalized,
+each coordinate motion is independent and FLE mo-
+tion exhibits qualitatively the same behavior as shown
+in the case of FBM with increasing dimensionality. From
+the mean squared displacement curves, the same scal-
+ing behavior is expected with more elevated amplitude
+for higher dimensionality. In fact, when each coordinate
+motion is decoupled, a d-dimensional motion effectively
+increases the number of single trajectories dtimes com-
+pared to the one-dimensionalcase. Therefore, the scatter
+in the mean square displacement curves decreases with
+increasingdimensionalityandthe ergodicitybreakingpa-
+rameterEBis expected to follow the relation Eq. (4.7).
+VI. CONCLUSION
+Motivated by recent single particle tracking experi-
+ments in biological cells, in which confinement due to
+the rather small cell size becomes relevant, we studiedFBM and FLE motions in confined space. In particular
+we analyzed the effects of confinement and dimensional-
+ity on the stochastic and ergodic properties of the two
+processes. Interestingly for both stochastic models, the
+confinement tends to decrease the value of the ergodicity
+breaking parameter EBcompared to that in free space.
+The same trend is observedfor increasing dimensionality.
+Correspondingly the scatter of time averaged quantities
+between individual trajectories is quite small, apart from
+regimes when the lag time ∆ becomes close to the over-
+all measurement time Tand the sampling statistics for
+the corresponding time average become poor. The relax-
+ation of the ergodicity breaking parameter as function of
+measurement time is quite similar to previous results in
+free space. We conclude that neither confinement nor di-
+mensionality effects lead to the appearance of significant
+ergodicitybreakingorscatterbetween singletrajectories.
+The displacement correlation function introduced here
+is a useful quantity that can be easily obtained from sin-
+gle particle trajectories. It can be used as a tool to dis-
+criminate one stochastic model from another. For subd-
+iffusive motion governed by FBM and FLE motion, the
+displacement correlationshould be negative and saturate
+in the long measurement time limit due to the confine-
+ment. Notably, the negative decrease ( ∼ −∆α) with lag
+time ∆ and anomalous diffusion exponent αis an intrin-
+sic property of FBM and FLE displacement correlations
+which is clearlydistinguished fromthat ofCTRW subdif-
+fusion. In the latter case, the subdiffusive motion occurs
+due to the long waiting time distribution between suc-
+cessive jumps and there is no spatial correlation between
+them, so that displacement correlation only fluctuates
+around zero with time. FLE motion can be distinguished
+from FBM motion since the displacement correlation has
+a positive value at short times due to the ballistic motion
+in the FLE model.
+ACKNOWLEDGMENTS
+We thank Stas Burov and Eli Barkai for helpful and
+enjoyable discussions. Financial support from the DFG
+is acknowledged.
+Appendix A: Derivation of the displacement
+correlation function
+In this appendix we derive analytical expressions for
+the displacement correlations, Eqs. (4.5) and (5.5). For
+a stochastic variable x, we define
+δx(t,∆) =x(t+∆)−x(t). (A1)
+The displacement correlation is then given by
+/angbracketleftδx(t,∆)δx(t−∆,∆)/angbracketright=
+/angbracketleftx(t+2∆)x(t+∆)/angbracketright−/angbracketleftx(t+2∆)x(t)/angbracketright
+−/angbracketleftx(t+∆)2/angbracketright+/angbracketleftx(t+∆)x(t)/angbracketright. (A2)11
+We now calculate this expression for FBM and FLE mo-
+tions.
+1. FBM
+For FBM (x(t) =xH(t)), we use the expression
+/angbracketleftx(t1)x(t2)/angbracketright=KH(t2H
+1+t2H
+2−|t2−t1|2H) (A3)
+for the autocorrelation. With this we readily obtain the
+result
+/angbracketleftδx(t,∆)δx(t−∆,∆)/angbracketright=KH(22H−2)∆2H.(A4)
+2. FLE
+For FLE (x(t) =y(t)), we use the correlation function
+[55]
+/angbracketleftx(t1)x(t2)/angbracketright=kBT
+m[t2
+1E2H,3(−γt2H
+1)+t2
+2E2H,3(−γt2H
+2)
+−(t2−t1)2E2H,3(−γ|t2−t1|2H)].(A5)
+The displacement correlation is then obtained as
+/angbracketleftδx(t,∆)δx(t−∆,∆)/angbracketright
+=4kBT
+m∆2E2H,3/bracketleftBig
+−γ(2∆)2H/bracketrightBig
+−2kBT
+m∆2E2H,3[−γ∆2H]. (A6)Expanding the generalized Mittag-Leffler function
+E2H,3(x)≈1/Γ(3) +x/Γ(2H+ 3) +···forx≪1 the
+displacement correlation is approximated as
+/angbracketleftδx(t,∆)δx(t−∆,∆)/angbracketright ∼2
+Γ(3)kBT
+m∆2(A7)
+at short lag times. With the expansion E2H,3(−x)≈
+1/xΓ(3−2H) forx≫1 the long lag time behavior of
+the displacement correlation is obtained as
+/angbracketleftδx(t,∆)δx(t−∆,∆)/angbracketright ∼22−2H−2
+γΓ(3−2H)kBT
+m∆2−2H.(A8)
+Note that the prefactor (22−2H−2)/Γ(3−2H) is zero
+forH= 1/2 and then becomes increasingly negative,
+saturating at the value 1 for H= 1.
+Appendix B: Derivation of Equation (4.7)
+From the definition of the time averagedmean squared
+displacement, Eq. (4.6), we expand/angbracketleftbigg/parenleftBig
+δ2(∆,T)/parenrightBig2/angbracketrightbigg
+in
+the form
+/angbracketleftbigg/parenleftBig
+δ2(∆,T)/parenrightBig2/angbracketrightbigg
+=1
+(T−∆)2/integraldisplayT−∆
+0/integraldisplayT−∆
+0/braceleftBigd/summationdisplay
+i=1/angbracketleft[xH
+i(t1+∆)−xH
+i(t1)]2[xH
+i(t2+∆)−xH
+i(t2)]2/angbracketright
++/summationdisplay
+i/negationslash=j/angbracketleft[xH
+i(t1+∆)−xH
+i(t1)]2/angbracketright/angbracketleft[xH
+j(t2+∆)−xH
+j(t2)]2/angbracketright/bracerightBig
+dt1dt2.(B1)
+Using the Isserlis theorem for Gaussian process with zero mean [48]:
+/angbracketleftx(t1)x(t2)x(t3)x(t4)/angbracketright=/angbracketleftx(t1)x(t2)/angbracketright/angbracketleftx(t3)x(t4)/angbracketright+/angbracketleftx(t1)x(t3)/angbracketright/angbracketleftx(t2)x(t4)/angbracketright+/angbracketleftx(t1)x(t4)/angbracketright/angbracketleftx(t2)x(t3)/angbracketright,(B2)
+the first term in the braces in Eq. (B1) can be rewritten as
+d/summationdisplay
+i=1/angbracketleft[xH
+i(t1+∆)−xH
+i(t1)]2[xH
+i(t2+∆)−xH
+i(t2)]2/angbracketright=d/summationdisplay
+i=1/angbracketleft[xH
+i(t1+∆)−xH
+i(t1)]2/angbracketright/angbracketleft[xH
+i(t2+∆)−xH
+i(t2)]2/angbracketright
++2d/summationdisplay
+i=1/angbracketleft[xH
+i(t1+∆)−xH
+i(t1)][xH
+i(t2+∆)−xH
+i(t2)]/angbracketright2. (B3)
+In this expression, we note that the sum of the second term in Eq. ( B1) and the first term in Eq. (B3) yields/angbracketleftBig
+δ2(∆,T)/angbracketrightBig2
+:
+/angbracketleftBig
+δ2(∆,T)/angbracketrightBig2
+=d2
+(T−∆)2/integraldisplayT−∆
+0/integraldisplayT−∆
+0/angbracketleft[xH(t1+∆)−xH(t1)]2/angbracketright/angbracketleft[xH(t2+∆)−xH(t2)]2/angbracketrightdt1dt2
+=d2/angbracketleftδ2/angbracketright2
+1D, (B4)12
+where we used the property
+/summationdisplay
+i,j/angbracketleft[xH
+i(t1+∆)−xH
+i(t1)]2/angbracketright/angbracketleft[xH
+j(t2+∆)−xH
+j(t2)]2/angbracketright=d2/angbracketleft[xH(t1+∆)−xH(t1)]2/angbracketright/angbracketleft[xH(t2+∆)−xH(t2)]2/angbracketright(B5)
+duetotheindependence ofthemotionineachcoordinatedirection. We alsonotethat theexpression/angbracketleftbigg/parenleftBig
+δ2(∆,T)/parenrightBig2/angbracketrightbigg
+−
+/angbracketleftBig
+δ2(∆,T)/angbracketrightBig2
+simplifies to
+/angbracketleftBig
+(δ2(∆,T)2/angbracketrightBig
+−/angbracketleftδ2/angbracketright2=2
+(T−∆)2d/summationdisplay
+i=1/integraldisplayT−∆
+0/integraldisplayT−∆
+0/angbracketleft[xH
+i(t1+∆)−xH
+i(t1)]×[xH
+i(t2+∆)−xH
+i(t2)]/angbracketright2dt1dt2
+=d[/angbracketleft(δ2)2/angbracketright1D−/angbracketleftδ2/angbracketright2
+1D]. (B6)
+From Eqs. (B4) and (B6), the ergodicity breaking parameter follow s the general relation
+EB(d) =/angbracketleftbigg/parenleftBig
+δ2(∆,T)/parenrightBig2/angbracketrightbigg
+−/angbracketleftBig
+δ2(∆,T)/angbracketrightBig2
+/angbracketleftBig
+δ2(∆,T)/angbracketrightBig2=EB(d= 1)
+d. (B7)
+[1] R. Metzler and J. Klafter, Phys. Rep. 3391 (2000); J.
+Phys. A 37, R161 (2004).
+[2] H. Scher and E. W. Montroll, Phys. Rev. B 12, 2455
+(1975).
+[3] J. W. Kirchner, X. Feng, and C. Neal, Nature 403, 524
+(2000); H. Scher, G. Margolin, R. Metzler, J. Klafter,
+and B. Berkowitz, Geophys. Res. Lett. 29, 1061 (2002).
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\ No newline at end of file
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+arXiv:1001.0682v1  [astro-ph.EP]  5 Jan 2010Astronomy& Astrophysics manuscriptno.article c∝circleco√†rtESO 2018
+October24,2018
+The SOPHIE searchfor northern extrasolarplanets⋆
+II.Amulti-planetsystem around HD9446
+G. H´ ebrard1,X.Bonfils2, D.S´ egransan3,C.Moutou4,X.Delfosse2,F.Bouchy1,5, I. Boisse1, L.Arnold5,M.Desort2,
+R.F.D´ ıaz1,A.Eggenberger2, D.Ehrenreich2,T. Forveille2,A.-M.Lagrange2,C.Lovis3, F.Pepe3,C.Perrier2,
+F.Pont6,D.Queloz3, N.C.Santos3,7, S.Udry3, A.Vidal-Madjar1
+1Institutd’Astrophysique deParis,UMR7095 CNRS,Universi t´ e Pierre&Marie Curie,98bis boulevard Arago, 75014 Paris ,France
+2Universit´ e J. Fourier(Grenoble 1) /CNRS,Laboratoire d’Astrophysique de Grenoble (LAOG,UMR5 571), France
+3Observatoire de Gen` eve, Universit´ e de Gen` eve, 51 Chemin des Maillettes, 1290 Sauverny, Switzerland
+4Laboratoire d’Astrophysique de Marseille, Universit´ e de Provence, CNRS(UMR 6110), BP8, 13376 Marseille Cedex12, Fr ance
+5Observatoire de Haute-Provence, CNRS /OAMP,04870 Saint-Michel-l’Observatoire, France
+6School of Physics,Universityof Exeter,Exeter,EX44QL,UK
+7Centrode Astrof´ ısica,Universidade doPorto, Rua das Estr elas,4150-762 Porto,Portugal
+Received TBC;accepted TBC
+ABSTRACT
+We report the discovery of a planetary system around HD9446, performed from radial velocity measurements secured with t he
+spectrograph SOPHIE at the 193-cm telescope of the Haute-Provence Observatory d uring more than two years. At least two planets
+orbitthisG5V,activestar:HD9446b hasaminimum massof0.7 MJupandaslightlyeccentricorbitwithaperiodof30days, where as
+HD9446c has a minimum mass of 1.8M Jupand a circular orbit with a period of 193 days. As for most of th e known multi-planet
+systems, the HD9446-system presents a hierarchical dispos ition, witha massive outer planet and a lighter inner planet .
+Key words. planetary systems –techniques: radial velocities –stars: individual: HD9446
+1. Introduction
+Among the more than 400 exoplanets known so far, most of
+themhavebeendiscoveredfromthereflexmotiontheycause to
+theirhost-star,whichcanbedetectedfromstellarradialv elocity
+wobble. Thus, accurate radial velocity measurements remai n a
+particularlyefficientandpowerfultechnicforresearchandchar-
+acterization of exoplanetarysystems. They allow the stati stic of
+systemstobeextendedbycompletingtheminimummass-perio d
+diagram of exoplanets, in particular towards lower masses a nd
+longerperiods,asthemeasurementaccuracyimproves.
+Together with the advent of the new SOPHIE spectrograph
+at the 1.93-m telescope of Haute-ProvenceObservatory(OHP ),
+France, the SOPHIE Consortium (Bouchy et al. 2009) started
+in late-2006 a large observationalprogramof exoplanetsse arch
+and characterization, using the radial velocity technique . In the
+present paper we announce the discovery of two exoplanets
+aroundHD9446,from radial velocity measurementssecured a s
+partofthesecondsub-programofthe SOPHIEConsortium.This
+sub-program is a giant-planet survey on a volume-limited sa m-
+ple around 2000 FGK stars, requiring moderate accuracy, typ i-
+callyintherange5 −10ms−1(Bouchyetal.2009).Itsgoalisto
+improvethestatisticsontheexoplanetparametersandthei rhost-
+ing stars by increasing the number of known Jupiter-mass pla n-
+ets, as well as offer a chance to find new transitinggiant planets
+in front of bright stars. SOPHIE sub-program-2 data were al-
+ready used to report the detection of several planets (Da Sil va
+⋆Based on observations collected with the SOPHIE spectrograph
+on the 1.93-m telescope at Observatoire de Haute-Provence ( CNRS),
+France, bythe SOPHIEConsortium (program 07A.PNP.CONS).et al. 2008, Santos et al. 2008, Bouchyet al. 2009) and to stud y
+stellaractivity(Boisseetal.2009a).Thissub-programal soaims
+atfollowinguptransitinggiantexoplanets;thisalloweds pectro-
+scopic transits to be observed (Loeillet et al. 2008), inclu ding
+the detection of the two first cases of spin-orbit misalignme nt,
+namely XO-3b (H´ ebrard et al. 2008) then HD80606b (Moutou
+et al. 2009, Pont et al. 2009), simultaneouslywith the disco very
+ofthetransitingnatureoftheplanetinthislast case.
+TheSOPHIE observationsof HD9446 that allow the detec-
+tionoftwonewplanetsarepresentedinsection2.Wederivea nd
+discussthestellarandplanetarypropertiesinsections3a nd4re-
+spectively,andconcludeinsection5.
+2. Observations
+We observed HD9446 with the OHP 1.93-m telescope and
+SOPHIE,whichisa cross-dispersed,environmentallystabilized
+echelle spectrograph dedicated to high-precision radial v eloc-
+ity measurements (Perruchot et al. 2008, Bouchy et al. 2009) .
+Observations were secured in high-resolution mode, allowing
+the resolution power λ/∆λ=75000 to be reached. The spec-
+tra were obtained in three seasons, from November 2006 to
+March2009.Dependingofvariableatmosphericconditions, the
+exposuretimes rangedbetween3 and 18 minutesand signal-to -
+noiseratiosperpixelat 550nmbetween32and94,withtypica l
+values of 5.5 minutes and 55, respectively. Exposure time an d
+signal-to-noiseratio wereslightlygreaterduringthe firs t season
+of observation. Three exposures performed through too clou dy
+conditionswereexcludedfromthefinaldataset,thatinclud es79
+spectra.Thetotal exposuretimeis about7hours.2 H´ ebrardet al.:Amulti-planet system around HD9446
+Thespectrographisfedbytwoopticalfibers,thefirstonebe-
+ingusedforstarlight.Duringthefirstseasonthesecond SOPHIE
+entrancefiberwasfedbyathoriumlampforsimultaneouswave -
+lengthcalibration.Thereafterwe estimated that waveleng thcal-
+ibration performed with a ∼2-hour frequency each night (al-
+lowinginterpolationforthetimeoftheexposure)wassu fficient,
+andthattheinstrumentwasstableenoughtoavoidsimultane ous
+calibration for this moderate-accuracyprogram. So for the sec-
+ondandthirdseasons,nosimultaneousthoriumcalibration were
+performed, avoiding pollution of the first-entrance spectr um by
+the calibration light. The second entrance fiber was instead put
+on the sky; this allowed us to check that none of the spectra
+were significantly a ffected by sky background pollution, espe-
+ciallyduetomoonlight.
+Weusedthe SOPHIEpipeline(Bouchyetal.2009)toextract
+the spectra from the detector images, cross-correlate them with
+a G2-type numerical mask, then fit the cross-correlation fun c-
+tions(CCFs)byGaussianstogettheradialvelocities(Bara nneet
+al.1996,Pepeetal.2002).EachspectrumproducesaclearCC F,
+witha8.47±0.05kms−1fullwidthathalfmaximumandacon-
+trastrepresenting38 .0±0.8%ofthecontinuum.Onlythe33first
+spectral orders of the 39 available ones were used for the cro ss
+correlation; this allows the full dataset to be reduced with the
+same procedure, as the last red orders of the HD9446 spectra
+obtained during the first season were polluted by argonspect ral
+linesofthe simultaneouswavelength-calibration.
+The derived radial velocities are reported in Table 1. The
+accuracies are between 5.4 and 9.7ms−1, typically around
+6.5ms−1. This includes photon noise (typically ∼3.5 ms−1),
+wavelength calibration ( ∼3 ms−1), and guiding errors ( ∼
+4ms−1)thatproducemotionsoftheinputimagewithinthefiber
+(Boisse et al. 2009b). These computed uncertainties do not i n-
+cludeany“jitter”duetostellar activity(seebelow).
+Table 1.Radial velocities of HD9446 measured with SOPHIE
+(fulltableavailableelectronically).
+BJD RV ±1σ
+-2400000 (kms−1) (kms−1)
+54043.4849 21.7152 0.0063
+54045.4850 21.7451 0.0055
+54047.4598 21.7747 0.0056
+54048.5033 21.7969 0.0054
+... ... ...
+... ... ...
+54889.2599 21.7201 0.0073
+54890.2628 21.7409 0.0064
+54893.2986 21.7222 0.0090
+54894.2961 21.7074 0.0097
+3. StellarpropertiesofHD9446
+We used the 50 SOPHIE spectra secured without simultaneous
+thoriumexposureto obtain an averagedspectrum, and we man-
+aged a spectral analysis from it. Table 2 summarizes the stel -
+lar parameters. According to the SIMBAD database, HD9446
+(HIP7245,BD+28253)isa V=8.35,highproper-motionG5V
+star.ItsHipparcosparallax( π=19.92±1.06mas)impliesadis-
+tanceof53±3pc.TheHipparcoscoloris B−V=0.680±0.015
+(Perrymanet al.1997).
+Fromspectralanalysisofthe SOPHIEdatausingthemethod
+presented in Santos et al. (2004), we derived the temperatur eTable 2.AdoptedstellarparametersforHD9446.
+Parameters Values
+mv 8.35
+Spectral type G5V
+B−V 0.680±0.015
+Parallax[mas] 19 .92±1.06
+Distance [pc] 53 ±3
+vsini⋆[kms−1] 4±1
+logR′
+HK−4.5±0.1
+[Fe/H] 0.09±0.05
+Teff[K] 5793 ±22
+logg[cgi] 4.53±0.16
+Mass [M⊙] 1.0±0.1
+Radius [R⊙] 1.0
+Luminosity[L ⊙] 1.1
+Teff=5793±22 K, the gravity log g=4.53±0.16, [Fe/H]=
++0.09±0.05, andM∗=1.0±0.1M⊙. The 10% uncertaintyon
+thestellarmassisanestimation,systematice ffectsbeingdifficult
+to quantify (Fernandes & Santos 2004). We derive a projected
+rotational velocity vsini⋆=4±1 kms−1from the parameters
+oftheCCFusingthecalibrationofBoisseetal.(inpreparat ion),
+which is similar to that presented by Santos et al. (2002). We
+alsoobtained[Fe/H]=+0.12±0.10fromtheCCF,whichagrees
+with, but is less accurate than the metallicity obtained fro m our
+spectralanalysis.
+Fig.1.H and K Caiilines of HD9446 on the averaged
+SOPHIE spectra. Chromospheric emissions are detected, yield-
+ingalogR′
+HK=−4.5±0.1.
+The cores of the large Ca iiabsorption lines of HD9446
+show small emissions (Fig. 1), which are the signature of an
+active chromosphere. Such stellar activity would imply a si g-
+nificant “jitter” on the stellar radial velocity measuremen t. The
+level of the Caiiemission corresponds to log R′
+HK=−4.5 with
+a±0.1 dispersion according to the SOPHIE calibration (Boisse
+et al. in preparation). For a G-type star with this level of ac tiv-
+ity,Santoset al.(2000)predictadispersionoftheorderof 10to
+20 ms−1for the stellar jitter. According to Noyes et al. (1984)
+and Mamajek & Hillenbrand (2008), this level of activity im-H´ ebrardet al.:Amulti-planet system around HD9446 3
+plies a stellar rotation period Prot≃10 days. This agrees with
+ourvsini⋆measurement, which translates into PRot<17 days
+(Bouchy et al. 2005), depending on the unknown inclination i⋆
+ofthestellar rotationaxes.
+4. AplanetarysystemaroundHD9446
+TheSOPHIE radial velocities of HD9446 are plotted in Fig. 2.
+Spanningmorethantwo years,theyshowclearvariationsoft he
+order of 200 ms−1, implying a dispersion σRV=58ms−1; this
+is well over the expected stellar jitter due to chromospheri c ac-
+tivity (10 to 20 ms−1, see above). In addition, the bisectors of
+the CCF are stable (Fig. 3, upper panel), showing dispersion of
+the order ofσBIS=20 ms−1, well below that of the radial ve-
+locities. An anticorrelation between the bisector and the r adial
+velocity is usually the signature of radial velocity variat ions in-
+duced by stellar activity (see, e.g., Queloz et al. 2001, Boi sse
+et al. 2009a). The bisectors are flat by comparison with the ra -
+dial velocities,whichsuggeststhatthe radialvelocityva riations
+are mainly due to Doppler shifts of the stellar lines rather t han
+stellar profile variations. This leads to conclude that refle x mo-
+tionduetocompanion(s)arethelikelycauseofthestellarr adial
+velocityvariations.
+Fig.2.Top:Radialvelocity SOPHIE measurementsofHD9446
+asafunctionoftime,andKeplerianfitwithtwoplanets.Theo r-
+bitalparameterscorrespondingtothisfitarereportedinTa ble3.
+Bottom:Residualsofthefit with1- σerrorbars.
+These facts were known in late 2007, after two seasons of
+SOPHIE observations of HD9446. A search of Keplerian fits
+thenproduceda solutionwith two Jupiter-likeplanets,ono rbits
+of30and190daysofperiod,with loweccentricities.Thisso lu-
+tionwasthereafterconfirmedbythethirdseasonofobservat ion;
+together with the “flat” bisectors, this providesa strong su pport
+tothetwo-planetinterpretationofthe radialvelocityvar iations.
+Figs. 2and4 show thefinal fit ofthe 851-dayspan SOPHIE
+radialvelocitiesofHD9446.ThisKeplerianmodelincludes two
+planetswithout mutual interactions,which are negligible in this
+case (see Sect. 5). All the parametersare free to varyduring theFig.3.Bisectorspanasafunctionoftheradialvelocity(top)and
+the radial velocity residuals after the 2-planet fit (bottom ). For
+clarity,errorbarsarenotplottedinthebottompanel.Ther anges
+havethesameextentsinthe x- andy-axesonbothpanels.
+fit. The derived orbital parameters are reported in Table 3, t o-
+getherwitherrorbars,whichwerecomputedfrom χ2variations
+andMonteCarloexperiments.
+The inner planet, HD9446b, produces radial velocity varia-
+tionswithasemi-amplitude K=46.6±3.0ms−1,corresponding
+to a planet with a minimum mass Mpsini=0.70±0.06 MJup
+(assuming M⋆=1.0±0.1M⊙forthehoststar).Itsorbithasape-
+riodof30.052±0.027days,andissignificantlynon-circular( e=
+0.20±0.06).Thisperiodislongerthanthestellarrotationperiod,
+as determined above from the log R′
+HKand thevsini⋆. AProt-
+valueof30dayswouldcorrespondsto vsini⋆<2kms−1,which
+is incompatible with our data. The outer planet, HD9446c,
+yields a semi-amplitude K=63.9±4.3 ms−1, corresponding
+to a planet with a projected mass Mpsini=1.82±0.17 MJup.
+Theorbitalperiodis192 .9±0.9days.ThisisabouthalfaEarth-4 H´ ebrardet al.:Amulti-planet system around HD9446
+Fig.4.Phase-folded radial velocity curves for HD9446b ( P=
+30d, top) and HD9446c ( P=193d, bottom) after removing
+the effect of the other planet. The SOPHIE radial velocity mea-
+surements are presented with 1- σerror bars, and the Keplerian
+fits are the solid lines. Orbital parameters corresponding t o the
+fitsarereportedinTable3.Thecolorsindicatethemeasurem ent
+dates.
+year, which made di fficult a good phase coveragefor the obser-
+vations. As seen in the lower panel of Fig. 4, the rise of the ra -
+dialvelocityduetoHD9446clacksofmeasurements,fororbi tal
+phases between 0.0 and 0.3. This implies significant uncerta in-
+ties on the shape of the orbit. Circularity can not be exclude d
+(e=0.06±0.06); furthermore,if the orbit actually is eccentric,
+there are nearly no constraints with the present dataset on t he
+orientationoftheellipsewithrespecttothelineofsight. There-
+sultingerrorbarsonthelongitude ωoftheperiastronandonthe
+timeT0atperiastronarethuslarge;therearehowevercorrelated,
+and the timing of a possible transit for this planet is better con-
+strained than T0in Table 3. Our estimations of vsini⋆andProt
+allow theconstraint i⋆>30◦to beput. So,if we assume a spin-
+orbit alignment for the HD9446-system, i⋆=iand sini>0.5;Table 3.Fitted orbitsandplanetaryparametersforthe HD9446
+system,with1-σerrorbars.
+Parameters HD9446b HD9446c
+P[days] 30 .052±0.027 192.9±0.9
+e 0.20±0.06 0.06±0.06
+ω[◦] −145±30−260±130
+K[ms−1] 46 .6±3.0 63.9±4.3
+T0(periastron) [BJD] 2454854 .4±2.0 2454510±70
+Mpsini[MJup] 0 .70±0.06†1.82±0.17†
+a[AU] 0 .189±0.006†0.654±0.022†
+Vr[kms−1] 21 .715±0.005
+N 79
+reducedχ22.6
+σO−C[ms−1] 15.1
+Typical RV accuracy [ms−1] 6.5
+span [days] 851
+†: usingM⋆=1.0±0.1M⊙
+this implies projected masses that translate into actual ma sses
+clearlyintheplanetaryrange.
+The reducedχ2of the Keplerian fit is 2.6, and the standard
+deviation of the residuals is σO−C=15.1 ms−1. This is bet-
+ter than the 58-ms−1dispersion of the original radial velocities,
+but this remains higher than the 6.5-ms−1typical error bars on
+the individual measurements, suggesting an additional noi se of
+∼13.5ms−1. Suchdispersionispreciselyin the rangeof the10
+to20ms−1expectedjitterforaG-typestarwiththislevelofac-
+tivity(Sect.3).Stellaractivityisthuslikelytobethema incause
+oftheremainingdispersion,aswellasthe ∼20-ms−1dispersion
+ofthe bisectors.Theresidualsof thefits donotshowsignific ant
+anticorrelationwiththebisectors(Fig.3,lowerpanel),a sitcould
+be expected in such cases (see, e.g., Melo et al. 2007, Boisse et
+al. 2009a); this is however at the limit of detection accordi ngto
+the error bars. A few bisectors values are larger than the oth er
+ones. They could be due to a particularly active phase of the
+star, as they are localized in a short time interval (between late
+JanuaryandearlyFebruary2007).Excludingtheseoutliers from
+the analysisdoesnotsignificantlychangetheresults. Fina lly,as
+it can be seen on lower panel of Fig. 2, the residualsare signi fi-
+cantlyless scattered duringthe first season than duringthe third
+one. This can be mainly explainedby the highersignal-to-no ise
+ratioreachedwith longerexposuretimesduringthe firstsea son,
+as well as the simultaneous thorium calibration secured for the
+first measurements.
+Figure5showsLomb-Scargleperiodogramsoftheradialve-
+locitymeasurementsofHD9446infourdi fferentcases:without
+any planet removed, with one or the other planet removed, and
+with both planets removed. A similar study was performed in
+thecaseofBD−08◦2823,anotherstarwithtwodetectedplanets
+(H´ ebrard et al. 2009). In the upper panel of Fig. 5 that prese nts
+the periodogram of the raw radial velocity measurements of
+HD9446, periodic signals at ∼30 days and∼195 days are
+clearly detected with peaks at those periods, correspondin g to
+the two planets reported above, with the same amplitudes. Th e
+peak at∼1 day corresponds to the aliases of all the detected
+signals,asthesamplingisbiasedtowards“onepointpernig ht”.
+A fourth, weaker peak is detected at ∼13.3 days. A Keplerian
+fit of this signal would providea semi-amplitude K≃11 ms−1,
+corresponding to a projected mass of 40 Earth masses. We do
+not conclude, however, that we detect a third, low-mass plan et
+withinthecurrentdata.
+Indeed,firstlythis13.3-dayperiodisnearthestellarrota tion
+period (∼10 days, Sect. 3), so it could be at least partially dueH´ ebrardet al.:Amulti-planet system around HD9446 5
+to stellar rotation. However,no significant peaks are detec ted at
+this period (nor at 30 days or 193 days) on the bisectors peri-
+odograms. We interpret this 13.3-d signal as being more like ly
+due to aliases. In order to validate this, we constructed a fa ke
+radialvelocitydatasetwiththesametimesamplingasourac tual
+data,andthatincludesonlytheKeplerianmodelofthetwopl an-
+ets found above.The periodogramof this fake dataset is almo st
+identical as the one plottedin the upperpanel Fig. 5: it incl udes
+of course the two peakscorrespondingto the periodsof the tw o
+planets,butalsothe peakat 13.3days.
+In additionto the one-daypeak, the window functionof our
+data shows a peak at ∼14.3 days, indicating that this inter-
+val is favored in our time sampling. The 13.3-d signal could
+thusbe mainlydue to the 14.3-dayalias of the 192.9-daysign al
+(1/13.3≃1/14.3+1/192.9). On the second panel of Fig. 5 is
+plottedtheperiodogramoftheresidualsaftersubtraction ofafit
+including the 30-day-period planet only. The peak at 193 day s
+is visible, as well as these aliases at 1, 13.3, and 15.4 days
+(1/15.4≃1/14.3−1/192.9).Inthesamemanner,thethirdpanel
+of Fig. 5 shows the periodogram of the residuals after a fit in-
+cludingthe 193-day-periodplanet only.The peakat 193days is
+no longer visible, and neither are the three aliases seen on t he
+upper panel. This time the peak at 30 days is visible, togethe r
+withthesetwoaliasesduetothe1-dayfavoredsampling,at0 .97
+and1.03day.ThebottompanelofFig.5showstheperiodogram
+of the residuals after subtraction of the Keplerian fit inclu ding
+HD9446b and HD9446c. There are no remaining strong peaks
+on this periodogram;even the 1-day alias disappeared,show ing
+that most of the periodic signals have been removed from the
+data.Theremainingpeaksarebelow10m /samplitude,showing
+that the main part of the detected periodical signals in our d ata
+aredueto thetwo planets.Theremainingsignalin theresidu als
+are at the limit ofdetectionaccordingto ouraccuracy.Asin ad-
+dition the stellar rotation periodis close to an alias of the signal
+of HD9446c, this makes tough any characterization of radial -
+velocity signal due to stellar activity, which is expected m ainly
+at the stellar rotation period. As the stellar jitter on the r adial
+velocities is of the order of 10 ms−1, this effect on the derived
+parametersofthetwodetectedplanetsisnegligible.
+Theresidualsofthemeasurementssecuredduringthesecond
+observational season are preferentially negative (Fig. 2, lower
+panel).This may suggest a possible additionalcomponent,w ith
+an orbital period of the same order or larger than the time spa n
+ofourdataset(2.3years).Suchanadditionalplanetcouldn otbe
+establishedwith theavailable data.Fora ∼2-yrperiod,the pro-
+jected mass of such an hypothetic planet should be lower than
+one Jupiter mass. On the other hand on short periods, a hot-
+Jupiter is excluded in this system, the accuracy of our datas et
+being good enough to detect it if there were any. The data al-
+lowplanetswith masseslargerthan0.3M Jupandorbitalperiods
+shorterthan10daystobeexcludedin theHD9446system.
+5. Discussion
+The data we presented allow us to conclude there is a planetar y
+systemaroundHD9446,withatleasttwoJupiter-likeplanet s,on
+30 and 193-day orbits. HD9446b has a projected mass slightly
+lower than Jupiter; it is on a 0.2-eccentricityorbit, showi ng that
+tidal effects were not strong enough to circularize it. HD9446c
+isatleast1.8timesmoremassivethanJupiter,andisonanea rly-
+circularorbit.Thehost-starofthissystemisslightlymor emetal-
+lic than the Sun, in agreement with the tendencyfoundfor sta rs
+harboringJupiter-massplanets(see,e.g.,Santoset al.20 05).Fig.5.Lomb-Scargleperiodogramsofthe SOPHIEradialveloc-
+ities. The upper panel shows the periodogram computed on the
+initialradialvelocities,withoutanyfitremoved.Theseco ndand
+third panels show the periodograms computed on the residual s
+of the fits including HD9446b or HD9446c only, respectively.
+The bottom panel shows the periodogram after the subtractio n
+ofthe2-planetfit.Thetwoverticaldottedlinesshowtheper iods
+ofthetwo planets.
+ThemutualgravitationalinteractionsbetweenHD9446band
+HD9446careweak.Theinnerplanetisstabilizedonitsorbit by
+the strong gravity of the star. Following Correia et al. (200 5), a
+simulation of the two orbits from the current solution was ru n
+for 106years, in order to estimate their evolution from mutual
+interactions.Thisshowsnosignificantchangesintheeccen trici-
+ties,whichremainintheranges[0 .18−0.23]and[0.03−0.075],
+forHD9446bandHD9446crespectively.Thereforethissyste m
+is stable for 106years, and it seems to be stable for longer time
+scales also. We estimated the order of magnitude of the poten -
+tial transit timing variations due to those weak mutual inte rac-
+tions, if any of the planets of the system does transit. For th at
+purpose we performed another 3-body simulation of the sys-
+tem, assuming the masses of the planets are equal to the min-
+imum masses and that the orbitsare coplanar.We employedthe
+Burlisch-StoeralgorithmimplementedintheMercury6pack age
+(Chambers 1999) and integrated the system for 2000 days -i.e .
+around10orbitsoftheexteriorplanet.Wefoundthattheint erac-6 H´ ebrardet al.:Amulti-planet system around HD9446
+tion between the planets producesvariations in the central time
+oftransitswithsmallamplitude,thatdoesnotexceed0.4se cond
+foranyofthetwobodies.
+No photometric search for transits have been managed for
+HD9446; depending on the unknown inclination iof the orbit,
+the transit probability for HD9446b and HD9446c are about
+2% and 1%, respectively. There are more than 200 exoplan-
+ets detected from radial velocity surveys with orbital peri ods
+longer than 50 days, so with transit probabilities of the lev el of
+the percent. Only one is known to transit, namely HD80606b
+(Moutou et al. 2009). It is likely that at least one or two more
+of these knownlong-periodexoplanetsare actuallytransit ingas
+seen fromthe Earth.Their searchis challenging,as the time sof
+the possible transits are not known accurately, especially a few
+yearsafterthesecurementoftheradialvelocitydata.
+Among the more than 400 exoplanets discovered so far, al-
+most 25% are located in the ∼40 known multiple-planet sys-
+tems.Mostofthemhavebeendetectedfromradialvelocityme a-
+surements. Additional planetary companions around HD9446
+can not be detected with the available data besides the two
+planets reported, but they are of course possible, as multip le-
+planet systems are common.For example HD155358(Cochran
+et al. 2007) has a Jupiter-mass planet on an orbit similar to
+HD9446c, and another planet with a 530-day orbital period, o r
+HD69830(Lovisetal.2006)havetwoNeptune-massplanetson
+orbits similar to those of the two detected planets of HD9446 ,
+and a third one on a 8.7-day orbit. More data are needed, and
+the monitoring of HD9446 should thus be maintained. As low-
+mass planets are preferentially found in multiple planetar y sys-
+tems (see, e.g., McArthur et al. 2004, Pepe et al. 2007, Mayor
+et al. 2009), HD9446 should be considered for high-precisio n
+radial-velocityprograms,despiteitsactivitylevel.
+The HD9446 system presents a hierarchical disposition,
+with the inner planet being the less massive one and the outer
+planet being the more massive. Fig. 6 displays the mass –
+semi-major axis relation for known multi-planetary system s.
+The data are taken from the compilation of the Extrasolar
+Planets Encyclopedia1. Most of the known multi-planetary sys-
+tems show such hierarchical disposition, as roughly the Sol ar
+System. The fitted relation between those two parameters pro -
+vides a planet mass proportional to a0.3(∝a1.5for the Solar
+System). The plot suggests the slope could be deeper for sys-
+tems including low-massplanets. A positive slope could be d ue
+to a higher migration e fficiency for low-mass planets, and /or to
+thefact thatgiantplanetsarepreferentiallyformedat lar gerdis-
+tances of their host stars than low-mass planets. However, o b-
+servational biases are important here, as low-mass planets are
+easier to detected at short orbital periods from radial velo city
+variations.Thesemi-amplitudeofthereflexmotionofastar due
+to a planetary companion is proportional to√a×Mpsini, so
+one could expect a a2-dependance in Fig. 6. As the averaged
+slope is lower, this could suggest there is actually no stron g de-
+pendance on average between those two parameters for multi-
+planetsystems.
+Fig. 6 also shows that only a few multi-planetary sys-
+tems include close-in giant planets. This agrees with Wrigh t et
+al.(2009),whoreportedthatsingle-planetsystemsshowap ileup
+at 3-dayperiodanda jumpat a≃1 AU, while multi-planetsys-
+temsshowamoreuniformdistribution.Still,theclose-inp lanets
+in knownmulti-planetsystem aremainlylow-massplanets.H ot
+Jupiters appear to be sparse in multi-planet systems, showi ng
+here again a distribution which is di fferent from single-planet
+1http://exoplanet.euFig.6.Semi-majoraxesasafunctionoftheprojectedmassesfor
+planetsinmulti-planetsystems.38knownextrasolarsyste msare
+plotted, with the planets of a given system that are linked by a
+solid line. The HD9446-system is shown with filled diamonds.
+The fitted relation is plotted in dotted line (massproportio nalto
+a0.3); the dashed line show the a2relation. The 8 planets of the
+Solar System are also plotted for comparison (circles). Sys tems
+with two, three and more planets are in black, blue and red, re -
+spectively.
+systems.Onlyfivehot-Jupitersareknowntobeinplanetarys ys-
+tems,namelyHIP14810b,upsAndb,HAT-P-13b,HD187123b,
+and HD217107b. Single- and multi-planet systems thus appea r
+to have significant di fferences in some of their properties. Such
+differences may provide clues allowing a better understanding
+of the formation and evolution of those systems. Improvingt he
+statistic of extra-solarplanets should be continued,in pa rticular
+inmulti-planetsystemsandwith radialvelocitysurveys.
+Acknowledgements. We thank A. C. M. Correia for helps and discussions, as
+wellasallthestaffofHaute-ProvenceObservatoryfortheirsupportatthe1.93 -m
+telescope andon SOPHIE.Wethank the“ProgrammeNational dePlan´ etologie”
+(PNP) of CNRS/INSU, the Swiss National Science Foundation, and the French
+National Research Agency (ANR-08-JCJC-0102-01 and ANR-NT 05-4-44463)
+for their support to our planet-search programs. NCS would l ike to thank the
+supportbytheEuropean Research Council /European Community under theFP7
+through a Starting Grant, as well from Fundac ¸˜ ao para a Ciˆ e ncia e a Tecnologia
+(FCT), Portugal, through a Ciˆ encia2007 contract funded by FCT/MCTES
+(Portugal)andPOPH /FSE(EC),andintheformofgrantsreferencePTDC /CTE-
+AST/098528/2008 and PTDC/CTE-AST/098604/2008 from FCT/MCTES.
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\ No newline at end of file
diff --git a/1001.0683.txt b/1001.0683.txt
new file mode 100644
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+Introducing Automated Regression Testing in Open Source Projects
+Christopher Oezbek
+Freie Universität Berlin
+Institut für Informatik
+Takustr. 9, 14195 Berlin, Germany
+christopher.oezbek@fu-berlin.de
+Abstract
+To learn how to introduce automated regression testing
+to existing medium scale Open Source projects, a long-
+term field experiment was performed with the Open Source
+project FreeCol. Results indicate that (1) introducing test-
+ing is both beneficial for the project and feasible for an out-
+side innovator, (2) testing can enhance communication be-
+tween developers, (3) signaling is important for engaging
+the project participants to fill a newly vacant position left by
+a withdrawal of the innovator. Five prescriptive strategies
+are extracted for the innovator and two conjectures offered
+about the ability of an Open Source project to learn about
+innovations.
+Contents
+1 Introduction 1
+2 Automated testing 2
+3 Methodology 2
+3.1 A model of external innovation introduction 3
+3.2 Choosing a project . . . . . . . . . . . . . 3
+3.2.1 FreeCol . . . . . . . . . . . . . . . 4
+3.3 Conducting the innovation introduction . . 4
+3.4 Data analysis methodology . . . . . . . . . 5
+4 Results 5
+4.1 Insights into automated testing . . . . . . . 8
+4.2 Insights into innovation introduction . . . . 12
+5 Limitations and conclusion 14
+5.1 Acknowledgements . . . . . . . . . . . . . 17
+References 17
+A Innovator Diary 201 Introduction
+The Open Source development paradigm based on copy-
+left licenses, global distributed development and volunteer
+participation has become an alternative development model
+for software, competing on par with proprietary solutions
+in many areas. Open Source software especially has es-
+tablished a good track record related to quality measures
+such as number of post-release defects or time to resolution
+for bug reports [50, 57]. These achievements are typically
+seen as hailing from open access to source code and from
+openness to participation, which leads to increased peer re-
+view and involvement of bug-fixing experts as summarized
+in Raymond’s quote “Given enough eyeballs, all bugs be-
+come shallow” [60].
+To further improve the quality assurance methods em-
+ployed in Open Source projects and learn about the mech-
+anisms by which Open Source projects adopt new innova-
+tions in general, the introduction of automated regression
+testing into an Open Source project is studied. Regression
+testing was chosen because it represents a well-known and
+established quality assurance practice from industry. The
+perspective assumed in this study is from an individual on
+the outskirts of the project who has used the project but not
+actively participated and who aims to strengthen the quality
+attributes of the project. This could be a developer in a com-
+pany who has decided to use an Open Source software as an
+off-the-shelf component for building a software application
+and is now interested in long-term quality improvements in
+the project.
+For this study, introduction is defined as the set of activ-
+ities meant to achieve adoption of a technology, practice or
+tool with the boundaries of a social system such as a soft-
+ware development project. In the case of this study, the in-
+novation is thus the practice of using automated regression
+testing. Adoption for this matter has been achieved, when
+the uses of an innovation have become “embodied recur-
+rent actions taken without conscious thought” [13], i.e. the
+writing and executing of test cases in the case of regressionarXiv:1001.0683v1  [cs.SE]  5 Jan 2010testing have become a habit or routine.
+To develop an understanding of how to introduce au-
+tomated regression testing to Open Source projects and
+whether usage is as expected, an exploratory field experi-
+ment was conducted (the methodology is explained in full
+detail in Section 3). The study was performed with the
+Open Source project FreeCol described in more detail in
+Section 3.2.1 and had the following research questions in
+mind:
+1. Is the introduction of a quality assurance process inno-
+vation such as automated regression testing a feasible
+possibility for an external innovator?
+2. Which aspects of the introduction and adoption of an
+innovation need particular assistance?
+3. How should the innovator behave and which strategic
+advice can be deduced?
+4. What can be learned about automated regression test-
+ing within the scope of Open Source software develop-
+ment?
+The remainder of the article presents a more detailed
+overview of the practice of automated regression testing, the
+project chosen to introduce testing with and the methodol-
+ogy used in the study. The last two sections then describe
+the actual introduction effort and the results achieved.
+2 Automated testing
+Automated testing is the practice of writing executable
+specification against which an application can automatically
+be tested. “Automated testing” is used as a term to distin-
+guish from manual testing in which software response is
+verified by a human against manually entered input.1We
+typically distinguish unit testing of individual classes and
+modules, integration testing of the interaction of several
+modules, and system testing of the application in the con-
+text of later use [76, p.77]. Since the boundaries between
+these different testing scopes are often hard to draw pre-
+cisely, the term unit testing, which was originally reserved
+for testing the smallest possible unit in isolation and allow-
+ing also manual execution, has become near synonymous
+with automated testing independent of the granularity of the
+tests contained in a test suite.
+While automated testing is primarily a quality assurance
+technique to achieve compliance with specification, it also
+has become popular to view it (a) as a specification in its
+own right, for instance when translating an ambiguous bug
+1From now on, we will use the unqualified term of testing as referring
+toautomated testing exclusively and will qualify all occurrences of testing
+as pertaining to manual testing respectively.report into a specific test case, and (b) as a means to track
+development progress by the individual developer (“you
+will be done developing when the test runs” [5]).
+It is not necessarily defined at which time the test cases
+are created with relation to the code they are testing. In
+a waterfall or V model software development process it is
+likely that unit and integration tests will be created bottom-
+up in a “sequential fashion” after implementation and in-
+tegration respectively [38] and in a dedicated testing de-
+partment. Yet, with the advent of more agile develop-
+ment processes, developers have become increasingly in-
+volved in writing tests during or even before implementa-
+tion. Such test-driven development (TDD, also “test-first”)
+has received a lot of attention in academia and industry [36]
+and is characterized by repeated cycles of writing tests, im-
+plementing functionality to pass the tests and subsequent
+refactoring to improve the quality of the code.
+The last step in the TDD cycle is aided by the regression
+detecting capabilities of a test suite. Since tests ensure com-
+pliance to the behavior coded in them, deviations from this
+behavior will cause affected tests to fail. This capability of
+tests is cited to increase developers’ confidence to refactor-
+ing code [49, p.129].
+Lastly, for the relationship between testing and finding
+bugs in software, the black-box properties of the test exe-
+cution must be regarded. This is famously summarized by
+Dijktra’s remark that “Program testing can be used to show
+the presence of bugs, but never to show their absence” [14],
+which reminds us that tests compare expected and actual
+output of the program for a defined set of scenarios and do
+not prove correctness of the code. In practical terms this
+means that even though the tester derives confidence about
+the quality of the code by successfully testing a program, it
+still might be that the tests (1) do not cover all the scenarios
+in which the code might be used, (2) contain bugs them-
+selves, and (3) do not detect interrelating defects that result
+in seemingly correct behavior.
+At this point we do not further qualify what kinds of
+means and ends are targeted by the habit of automated test-
+ing (for instance whether developers translate bug reports to
+test cases or use it for test-driven development), but rather
+leave the exact kind of usage up to the Open Source project
+and its specific requirements.
+3 Methodology
+This study is a long-term field experiment [31] with post-
+hoc data analysis being performed both quantitatively and
+qualitatively. The study proceeded in four steps: First some
+time was spent on building a theoretical model of how an
+innovation introduction with regard to automated testing
+should proceed. Second, a project was selected to conduct
+an innovation introduction following this model with, the
+2result of which was the project FreeCol. Third, testing was
+introduced to this project, which took place from April to
+September 2007. Last, the outcome of the introduction was
+analyzed by means of data mining the source code reposi-
+tory of the project and analyzing the mailing-list communi-
+cation on testing up until August 2009.
+Even though Open Source projects are robust against
+negative external influence, it was attempted to minimize
+risk toward the project and to protect the autonomy of the
+subjects [8, 53] by creating an atmosphere of collabora-
+tion, involvement and participation between project and re-
+searcher, and protecting privacy and confidentiality [4, 22].
+3.1 A model of external innovation intro-
+duction
+To give the innovator a plan for introducing automated
+testing according to which to operate, first a prescriptive
+stage model of introduction behavior was designed. The
+goal was to guide the innovator and make the process repro-
+ducible in other projects and context. The resulting model
+is presented in the following paragraphs and summarized in
+Figure 1.
+The goal of the first phase is to get to know the Open
+Source project and establish the technical requirements of
+participating in the project. This entails subscribing to and
+reading the mailing-list of the project ( lurking [52, 58]) with
+a focus on community and power structures of the project,
+downloading the source code from the repository, setting up
+the development environment, reading mission statements
+and task trackers, and building the project.
+In the next phase the innovator is to become active by
+writing test cases and contributing them unsolicitedly to the
+project using the mailing-list (see [63] for a discussion on
+contact strategies). This strategy is to be used to create
+something valuable for the project [16, 72] and as a side-
+effect to become known and possibly gain write access to
+the project repository. Areas to test will be selected to max-
+imize the learning experience for the innovator regarding
+the code base, while still providing benefit to the project.
+Previous work had shown that to be beneficial, close inte-
+gration with existing code structure and coding conventions
+is important [59]. To make these test contributions usable
+for the project, documentation on testing will be written and
+the build-structure of the project improved to better accom-
+modate testing. An upper limit of 10-15 hours per week
+in time spent on communicating and writing test code will
+be set, so that the researcher is not overtaxing the capacity
+of the project and is behaving in line with the average time
+commitment of OSS project participants [26].
+In the third phase the focus will be shifted away from
+solitary activity, which is communicated primarily via the
+project leadership, to collaboration with other project mem-bers, based on three different collaboration models for au-
+tomated testing: (1) The innovation will codify recent bug
+tracker entries into failing test cases and contact the devel-
+opers who had previously worked in that area. (2) The in-
+novator will add test cases for code recently committed by
+other members ex post. (3) The innovator will write fail-
+ing test-first code for features that members are discussing
+to implement in the very near future. Each of these areas
+should generate a possibility for interaction and collabora-
+tion between one or several developers and the innovator.
+The goal here is to build a social network between the in-
+troducer and the developers, to spread knowledge about au-
+tomated testing and demonstrate benefit of the innovation to
+the developers.
+In the fourth phase the activity of the developer will be
+phased-out slowly with an emphasis on support and main-
+tenance. The introducer will actively monitor test contri-
+bution by other developers and guide them to develop high
+quality tests with large benefit for the project.
+3.2 Choosing a project
+Using the project hosting site SourceForge.net, I first
+randomly chose among projects fitting the following set of
+criteria: (1) the project uses an Open Source license and de-
+velopment model (in particular this means distributed devel-
+opment and openness to outside contributions [7]), (2) the
+project has between 5 and 15 active developers who have
+committed code to the CVS repository within the last three
+months to ensure both interesting interactions and suffi-
+cient possibilities for the researcher to act upon, (3) the
+project uses C/C++, Java, C#, Ruby, Perl or Python, (4) the
+code-base contains less than 15 test cases prior to being ap-
+proached, (5) the project is not in process of adopting any
+time-consuming innovations, and (6) the project consists
+not primarily of GUI code (which is harder to test [48]).
+These criteria were used to ensure that the case is (a) typ-
+ical enough to generalize to other projects, (b) suitable for
+automated testing, (c) has potential for interesting interac-
+tion regarding the introduction, and (d) automated testing is
+relatively new with regard to the project [54].
+Ten projects were then randomly chosen from a list of
+projects with at least 5 developers as given by Source-
+Forge.net and analyzed for the remaining criteria. It turned
+out that none of the these projects fulfilled all criteria and it
+was decided not to continue by randomly selecting projects
+(three projects were inactive in the last 3 months, one
+project already had more than 15 test cases, three projects
+were moving to another source code management system,
+which was seen as major innovation introduction underway,
+two projects did not have any public communication, and
+one project did have more than 5 members but consisted of
+several completely independent sub-projects none of which
+3 2 wks 2 wks 2 wks 2 wks 
+1 wk 1 wk 1 wk 1 wk 1 wk 1 wk 1 wk 1 wk          
+Period:  Lurking Activity Collaboration Phase-Out 
+Activities:  • Subscribe to mailing-list 
+• Check-out project 
+• Build project 
+• Analyse power structure and 
+mission statement • Write test cases 
+• Contribute tests on mailing-list 
+ Contribute tests for... 
+• demonstrating current bug tracker entries 
+• recently checked-in code 
+• test-first development • Improve test cases 
+• Maintain infrastructure 
+Goals:  • Get to know the project 
+• Establish infrastructure for testing • Demonstrate value of testing 
+• Understand code base 
+• Gain commit access • Introduce innovation to individual members 
+• Build social network • Sustain usage of technology Figure 1. Phases in the introduction process.
+passed the five-member-hurdle). At this time, the Source-
+Forge.net newsletter arrived which features a “Project of the
+Month”2chosen by the SourceForge.net staff. As the first
+project listed in this newsletter already complied with all
+set criteria, choosing from the list of Projects of the Month
+offered a suitable alternative to a random sample.
+3.2.1 FreeCol
+FreeCol is a project striving to recreate the turn-based strat-
+egy game Sid Meier’s Colonization – a sequel to Sid Meier’s
+successful empire building game Civilization [23]. The
+project was started in March 2002 and had its first re-
+lease on January 2, 20033. FreeCol is structured as client-
+server application to facilitate multi-user play and is writ-
+ten entirely in Java. The project is regularly ranked in
+the top 50 of Open Source projects hosted at the project
+hoster SourceForge.net and 16,500 copies have been down-
+loaded per month on average over the last six months. The
+project is lead by the two maintainers Stian Grenborgen
+and Michael Burschik and has 60 members enlisted on the
+SourceForge.net project page4of which 46 are designated
+as developers and 13 of which are deemed active5. The
+project already had one (1) test case using JUnit which was
+integrated into the Ant build-scripts.
+3.3 Conducting the innovation introduc-
+tion
+The introduction of automated testing into the project
+FreeCol according to the conceptual model described in the
+Section 3.1 was begun at the beginning of April 2007 and
+lasted for six weeks until May 15, 2007. The lurking in par-
+ticular was shortened to just 2 days, because I had become
+sufficiently familiar with the code to find a defect, provide a
+2http://sourceforge.net/community/potm/
+3http://www.freecol.org/history.html
+4http://sourceforge.net/project/memberlist.php?
+group_id=43225
+5http://www.freecol.org/team-and-credits.htmltest case for it and fix the associated issue. I then thus pro-
+ceeded directly to actively contribute on the mailing-list and
+was granted commit privileges within the first week of do-
+ing so. This phase of writing and contributing test code nat-
+urally evolved into the more collaborative phase 3 of my en-
+gagement. First, one of the test cases I had written caught a
+regression caused by a previous check-in which allowed me
+to start communicating with the developer who had caused
+the failure. Second, when new developers asked on the list
+about open tasks, I noted that writing test code could be
+a good way to get started, which convinced one new de-
+veloper to provide two test cases. Third, I created several
+tests to show which aspects of a reported bug had not yet
+been fixed. Rather than fixing the bug myself to make the
+tests pass, I advertised it as opportunity for working on a
+well specified problem. After 4 weeks of engagement the
+number of test cases in FreeCol had then increased from
+1 to 57 (see Figure 8) and this was deemed sufficient for
+phase 4 to start. As part of phasing out over the next two
+weeks, I added more documentation regarding testing, cre-
+ated a short video for setting-up Eclipse to run the test suite
+for FreeCol and fixed the Ant build-scripts to execute the
+test cases for those not using Eclipse. May 15th marks the
+last day of the phase-out.
+Returning at the beginning of June and July each, I had to
+notice that the introduction so far was unsuccessful with no
+new tests having been created and the number of tests being
+stuck at 65 tests. Two months later, in August 2007, one of
+the project maintainers informed me that the test suite was
+now “spectacularly broken” after a large feature addition
+and refactoring had changed substantial parts of the soft-
+ware (see [71] for an overview how refactoring and testing
+relate to each other). During a discussion about the useful-
+ness of testing, I then agreed to repair the test suite, which
+was finished mid September 2007, before ending my en-
+gagement as a committer in the project.
+Over the next two years, my engagement on the mailing-
+list was restricted to answering several questions with re-
+gard to using Eclipse for developing FreeCol and running
+the tests using Ant. When returning to the project in Au-
+gust 2009 and analyzing both the repository and mailing-
+4list, it turned out that the number of test cases had increased
+markedly, which will be discussed in Section 4 on Results
+after the data analysis methodology has been presented be-
+low.
+3.4 Data analysis methodology
+The ex-post data analysis of the introduction of testing
+was performed using two primary data sources. First the
+source code repository of FreeCol was mined6to discover
+the number of test cases in FreeCol, failure rates and state-
+ment coverage (see [80] for an overview regarding measur-
+ing test case coverage). Second, e-mail lists were down-
+loaded from SourceForge.net as permitted by one of the
+project maintainers and were analyzed briefly and qualita-
+tively for social interaction regarding testing.
+The detailed steps of the repository mining were:7
+1. To perform data analysis efficiently, the FreeCol
+source code repository (managed via Subversion [9])
+was first mirrored locally using svnsync8.
+2. An XML version history log was next extracted for the
+whole repository and converted to a CSV file including
+revision, author, commit messages, date and affected
+paths via xml2csv9. Using a regular expression search,
+the commits which affected test classes were identi-
+fied, the result of which was appended as a last column
+to the table.
+3. This data was then used to produce Figures 2, 3, 4 us-
+ing the R project for statistical computing [61].
+4. Using a Ruby [19] script as a driver, FreeCol was
+then checked out with the appropriate version at each
+month since April 2007. For each check-out, a cus-
+tomized and overlaying Ant build-script10was run,
+which would compile and then execute the test cases,
+thereby generating JUnit test results. Test coverage
+was calculated using Cobertura11. Using the Nokogiri
+XML API12, tests results were converted to CSV .
+5. The concluding analysis in R resulted in Figures 5, 6,
+7, 8.
+6See [39] for a survey of mining source code repositories, [79, 81, 24,
+47, 46] for selected articles, and [1] for the dangers associated with repos-
+itory mining
+7All scripts used for producing the results in this study as well as in-
+termediate data to reproduce the statistical analysis is available at http:
+//www.inf.fu-berlin.de/inst/ag-se/pubs/test-intro2009data.zip
+8http://svn.collab.net/repos/svn/trunk/notes/
+svnsync.txt
+9http://www.a7soft.com/xml2csv.html
+10http://ant.apache.org
+11http://cobertura.sourceforge.net
+12http://nokogiri.rubyforge.org/4 Results
+If we first look back quantitatively at the introduction
+from April 2007 to August 2009, we find it a clear success:
+First, the number of test cases has increased almost con-
+stantly from 73 at the departure of the innovator to 277 as
+of August 2009. Figure 8 shows this linear growth of on
+average 9.9 testcases being added per month (95% confi-
+dence interval: 6.3–13.4) with no noticeable plateau. Sup-
+porting this linear growth is the percentage of monthly com-
+mits affecting test cases which is between 10.0% to 15.1%
+with 95% confidence, depicted in Figure 3) and implies that
+writing novel tests was a constantly exercised practice in
+the project. Second, existing test cases were maintained
+to execute successfully. This indicates that besides writing
+new test cases, the developers assumed the responsibility of
+keeping the test suite in working condition. Only in rare
+cases did the monthly snapshot copies reveal broken test
+cases, such as in September 2007 when all tests were bro-
+ken, or in March 2008 when 15 out of 234 test cases failed
+(see Figure 8). Third, writing test code is not an activity
+performed only by one or two developers. Out of 32 devel-
+opers having ever committed to the FreeCol repository for
+a total of 5,030 commits and thus disregarding patches sub-
+mitted by other developers and committed for instance by
+the maintainer, 16 made at least one of the 449 test-affecting
+commits in their career at FreeCol (11 at least 5, 6 at least
+10, 4 at least 20 commits). An overview of testing activities
+by developer is shown in Figure 4. Fourth, while the project
+grew from 31,800 non-comment source lines of code (NC-
+SLOC) to 48,20013over the course of the last two years, the
+associated code being covered by tests was expanded from
+200 to 11,000 lines (see Figure 5), which represents an in-
+crease from 0.5% coverage to 23% (see Figure 6).
+Moving away from the quantitative assessment of the
+source code repository to an analysis of the communication
+on the mailing-list, we also note multiple incidents in which
+developers voiced their positive attitude towards testing.
+For example, after a successful bug resolution episode, one
+core developer asked whether additional test cases should
+be added and one of the maintainers answered enthusiasti-
+cally:
+That goes without saying! We should have unit
+tests for absolutely everything. Everyone gets
+extra bonus points for writing unit tests. [free-
+col:2518]14
+13Source code lines for this study only include executable lines, i.e. this
+excludes documentation, whitespace and tests, but also method signature
+and class definitions.
+14Citations such as [freecol:2518] are hyperlinks to the respective e-
+mails from the Freecol Developer Mailing-list and are numbered in the
+order the e-mails appeared in the mbox archive file from SourceForge.Net.
+5MonthCommits per month
+50100150200250
+AMJ
+2004JASONDJFMAMJ
+2005JASONDJFMAJ
+2006JASONDJFMAMJ
+2007JASONDJFMAMJ
+2008JASONDJFMAMJ
+2009JATest−affecting commits
+Non test−affecting commitsFigure 2. A time series display of the number of commits per month since the beginning of the project being managed in
+a source code repository. Commits which affect test cases are shown as a subset in green. It is well visible that testing
+activity in the project is on-going since April 2007.
+MonthPercentage of commits being tests10203040
+JFMAJ
+2006JASONDJFMAMJ
+2007JASONDJFMAMJ
+2008JASONDJFMAMJ
+2009JA
+Figure 3. A time series display of the percentage of commits affecting test cases per month since the beginning of the
+project. It can be seen that testing activity per month since April 2007 was on average above 10% (mean 12.5%, first
+quartile at 8.9%, third quartile at 15.5%)
+6MonthCommits per month
+51015
+JFMAJ
+2006JASONDJFMAMJ
+2007JASONDJFMAMJ
+2008JASONDJFMAMJ
+2009JAOthers51015Core Developers51015Test Masters51015MaintainersFigure 4. A time series display of absolute number of commits affecting test cases per month separated for each de-
+veloper with commit access in the project. The developers have been split into the four groups of “maintainers”, “test
+masters”, “core developers” and “others”. A test master for this matter was defined as somebody with more than 30 test
+commits and a core developer as a developer with more than 80 total commits. All test masters were also core develop-
+ers. The engagement of the innovator is well visible as two initial bursts of testing activity in April/May of 2007 and then
+in September 2007. Also the complementary patterns of activity between one of the maintainers with the test masters
+can be seen, when his testing activity increased during winter 2007/2008 and late summer 2008 to offset the notable
+absence of testing commits from the test masters. The display was cropped for the activity of one of the maintainers
+who made 22 test affecting commits in February 2008.
+7MonthLOC
+1000020000300004000050000
+AMJ
+2007JASONDJFMAMJ
+2008JASONDJFMAMJ
+2009JACovered Lines of Code
+Total Lines of CodeFigure 5. A time series display of the test coverage achieved over the course of the innovation introduction. Coverage is
+shown as the number of lines of the project’s source code which were executed as part of running all tests which could
+be compiled for the first commit of each month displayed. We can see that while the total lines of (non-documented, non-
+test, non-whitespace) source code for the software increased from 31,800 lines to 48,200, the number of lines exercised
+by testing code increased from 200 to 11,000.
+In another example, one core developer was able to catch
+several bugs in the existing code during the implementation
+of a new feature, which made him praise the test cases:
+[I] Also found a couple of existing bugs (while
+implementing the tests, god bless them) [free-
+col:3351]
+While this assessment gives us reasonable certainty
+to declare the introduction of automated testing into the
+project FreeCol a success, we should now look for more
+general insights to be gathered from the case study. These
+insights can be split into (1) results regarding automated
+testing and (2) results regarding innovation introduction,
+which will be discussed in turn.
+4.1 Insights into automated testing
+The most interesting insight regarding the use of testing
+in Open Source projects is that test cases have been used
+repeatedly to enhance communication in the project in two
+major ways: (1) If facing a defect in the software without
+the necessary knowledge to repair it, or even when unable
+to understand the general problem, we have seen developers
+write failing test cases which reproduce or narrow down the
+failure caused by the defect and use the test case as a moreconcise alternative for communicating the circumstances of
+the failure (for instance [freecol:2606] [freecol:2610] [free-
+col:2640] [freecol:2696] [freecol:3983]). (2) When facing
+ambiguity about how FreeCol should behave, we have seen
+developers codify their opinion as test cases [freecol:3276]
+[freecol:3056] or existing tests being the starting-point
+for discussions about how FreeCol should behave [free-
+col:1935].
+Thus, test cases become explicit articulations of intent
+during requirements arbitrage inside the project. If we ab-
+stract, we can say that in both cases the developers construct
+a.) an expectation against the behavior of the software but
+also b.) an expectation towards the behavior of the fellow
+project members as to fix the failing test cases. This latter
+aspect of communicating expectations seems to be a second
+major advantage beside the regression detecting abilities of
+having a test suite in an Open Source project (see for in-
+stance [freecol:3961] or [freecol:4431]).
+It is outside the scope of this research to assess the rel-
+ative advantage of communicating an informal and vague
+bug description or aspect of a specification using an ex-
+ecutable test case beyond the observed episodes on the
+mailing-list. In particular, it would be hard to extract how
+information has flown inside the project and deduce the role
+that the test cases actually played from commit messages
+8MonthTest coverage in percent of total source code5101520
+AMJ
+2007JASONDJFMAMJ
+2008JASONDJFMAMJ
+2009JAFigure 6. A time series display of the test coverage achieved over the course of the innovation introduction. Coverage
+is shown as the percentage of lines of code exercised by running the test suite over the total number of lines of code.
+We can take note that there are two very pronounced increases in coverage. One was induced by the innovator when
+introducing automated testing into the project in April to September of 2007, which resulted in the 10% margin being
+attained, and the second was in April to June 2008 when testing was hugely expanded to reach 20% coverage. After
+these jumps in coverage, there is a slow increase in coverage, reaching 23% in August 2009.
+9MonthPercentage of LOC covered
+by tests per package
+1020304050
+AMJ
+2007JASONDJFMAMJ
+2008JASONDJFMAMJ
+2009JABusiness Model
+Server
+Artificial Intelligence
+Other
+User InterfaceFigure 7. A time series display of the test coverage by major application module as a percentage of lines of code being
+exercised by running the test suite over the total number of lines of code. Noticeably, the application module most tested
+is the business model with around 50% which accounts for 7,400 lines of code being tested, with server code coming
+second at 40%. and the artificial intelligence routines at 22%. Also striking is the absence of any user interface testing
+(only 58 of 20,937 lines are tested in the latest check-out). The drop in coverage in March 2009 is induced by fifteen
+failing test cases (see Figure 8) and the missing results for September 2007 where the test suite was so broken that it
+did not compile.
+10MonthTests
+50100150200250
+AMJ
+2007JASONDJFMAMJ
+2008JASONDJFMAMJ
+2009JAFailing test cases
+Passing test casesFigure 8. A time series display of the number of test cases in the FreeCol test suite distinguishing passing and failing
+tests on the first commit for each month. It can be seen that a.) the number of test cases is expanding on average by
+9.9 test cases per month (median 9.0, standard deviation 8.9) and that b.) the source code is passing the test suite with
+very low numbers of failing tests. One notable exception to the latter is September 2004 when the whole test suite did
+no longer compile.
+and e-mails on the mailing-list. For instance, while we have
+seen defects encoded as test cases often being fixed within
+24 hours [freecol:3057] [freecol:2610] [freecol:2606], we
+also have at least one case in which the defect highlighted
+by a test was fixed while this test was just being writ-
+ten [freecol:2640]. Working out the mechanisms and actual
+pathways of causation would require quite some detective
+work.
+Instead, we want to (1) assess the magnitude of code-
+centric “hard” contributions vs. communication-based
+“soft” contributions and (2) abstract from the innovation of
+automated testing to innovations in general.
+If we turn to the first aspect, we can note that if one is to
+distinguish between the project activities of communicating
+on the mailing-list (“soft” contributions) and committing to
+the project repository (“hard” contributions), then a slightly
+higher ratio between the number of discussion threads and
+the total commits (848 threads containing 3,163 e-mails
+over 3,676 commits, i.e. a ratio of 4.3 commits to 1 thread)
+and the number of threads relating to testing and the test
+affecting commits (71 threads containing 423 e-mails over
+441 commits, i.e. 6.2 commits to 1 thread) can be found.
+These ratios are in line with results from literature [66] and
+help to put the communication aspect of testing into per-
+spective: Open Source projects are focused on hard contri-butions (removing user requests, off-topic discussions and
+SPAM should bring the general ratio even more closely to
+the testing one as well) and the majority of commits are
+never discussed on the mailing-list, a phenomenon which
+has been called the bias for action of Open Source software
+development [78, pp.334f]. Thus one possible explanation
+for the relative success of testing in an Open Source project
+could derive from bridging hard and soft contributions via
+test cases which can be committed to the project repository
+while at the same time offering the possibility for commu-
+nication.
+As a second aspect and abstracting from testing as a par-
+ticular innovation, we can deduce that reinvention — the us-
+age of an innovation in an unexpected way [62] — can pro-
+vide valuable benefits to the project not obvious to the in-
+novator. Testing in this case had been targeted by the inno-
+vator at improving (1) the quality of the software produced
+in the Open Source project, and (2) its resistance against re-
+gression. Yet, it was used instinctively and in addition as a
+means of communicating complex expectations against the
+code or the behavior of others inside the project.
+Strategy 1. The innovator should keep an open, supportive
+mind about innovations being reinvented to gain the most
+tangible benefits for a project.
+11As a second insight we found that testing varies largely
+by module, based on its technical complexity regarding test-
+ing. While the FreeCol business logic including the game
+objects attained more than 50% coverage, other areas such
+as the server module at 40% and the artificial intelligence
+module at 22% are less tested and UI testing is absent com-
+pletely from FreeCol (see Figure 7). How to expand the
+coverage of underrepresented modules substantially is an
+open question for automated and unit testing in particular.
+Finally, a failing of the whole test suite due to insuffi-
+cient memory being allocated when running it — an event
+which occurred no less than five times over the two years
+— highlights the relationship between testing and specify-
+ing a software. Little memory was assigned primarily to
+the tests to prevent a memory leak from slipping into the
+software, but each failure due to insufficient memory also
+caused the question whether the failure was due to FreeCol
+having actually outgrown the memory limitations assigned
+to it [freecol:4276]. Developers needed to assess each time
+whether a failure is an acceptable consequence of FreeCol
+being expanded or a regression necessary to be rectified.
+This ambiguity regarding the specification of FreeCol then
+turned out to be difficult to resolve for the developers, one
+of whom remarked: “I don’t remember why we decided to
+restrict memory during tests. I’m not sure whether it is a
+good idea, or not.” [freecol:4272]. One conclusion for the
+innovator should be that the role of testing with regard to
+specifying the behavior of the software is explained in more
+detail to the developers so as to reduce the ambiguity if the
+specification can only be vague such as with limited mem-
+ory.
+4.2 Insights into innovation introduction
+On introducing innovations we have found two main re-
+sults in this study for projects comparable to FreeCol (see
+Section 5 on generalizability):
+1. Open Source projects excel at incrementally expand-
+ing innovation usage over long time and maintaining
+an existing code base, yet does require assistance by an
+innovator or particularly skilled individual to achieve
+radical expansion with regard to an innovation.
+2. When detaching from an Open Source project, the in-
+novator should signal this to release ownership of re-
+sponsibilities and code.
+The first insight was deduced by analyzing the increase
+in coverage in the project (see Figure 6), which shows two
+notable expansions over the last two years. The first was
+the expansion of coverage from 0.5% to 10% by this au-
+thor (as the innovator) when introducing automated testing
+to the project in 2007, and the second in April and Mayof 2008 when one developer expanded coverage from 13%
+to 20% by starting testing the artificial intelligence module
+(see Figure 7). Yet, beside these notable increases, which
+occurred over a total of only four months, coverage re-
+mained relatively stable over the two years. This is unlike
+the number of test cases which constantly increased with
+a remarkable rate of passing tests (see Figure 8). On the
+mailing-list a hint can be found that this is due to the dif-
+ficulty of constructing scaffolding for new testing scenar-
+ios (see for instance the discussion following [freecol:4147]
+as to the problems of expanding client/server testing) and
+thus indirectly with unfamiliarity with testing in the project.
+This thus poses a question to our understanding of Open
+Source projects: If – as studies consistently show – learn-
+ing ranks highly among Open Source developers’ priorities
+for participation [25, 32, 34], then why is it that coverage
+expansion was conducted by just two project participants?
+Even worse, it seems that the author as an innovator and
+the one developer both brought existing knowledge about
+testing into the project and that project participants’ affinity
+for testing and their knowledge about it expanded only very
+slowly.
+Conjecture 1. Knowledge about and affinity for innova-
+tions by individual developers is primarily gathered outside
+of Open Source projects.
+This conjecture can be strengthened by results from Hah-
+sler who studied adoption and use of design pattern by Open
+Source developers. He found that for most projects only one
+developer — independently of project maturity and size up
+until very large projects — used patterns [27, p.121], which
+should strike us as strange if sharing of best practices and
+knowledge did occur substantially.15This argument then
+on the one hand can thus emphasize the importance of an
+Schumpeterian entrepreneur or innovator who pushes for
+radical changes:
+Conjecture 2. To expand the use and usefulness of an in-
+novation radically, an innovator or highly skilled individual
+needs to be involved.
+On the other hand, if we put the innovator into a more
+strategic role as to acquire such highly skilled and knowl-
+edgeable project participants with regard to an innovation,
+then we can deduce two different options:
+1.) Taking an optimistic view of the technical and in-
+tellectual skill of the developers participating in an OSS
+project, we might conclude for the innovator that additional
+training and support is necessary to counteract the lack of
+progress with regard to learning about a new innovation.
+15Certainly to really confirm such a conjecture an in-depth study or in-
+terviews with the developers would be necessary [28].
+12Strategy 2. The innovator needs to actively promote learn-
+ing about an innovation to achieve levels of comprehension
+necessary for radical improvements using this innovation.
+This strategy has been used only very little by the innova-
+tor by providing a tutorial video about setting up Eclipse for
+testing and by writing a small tutorial document explaining
+how to write test cases and explaining some rationales for
+it. Yet beyond these actions by the innovator, no other at-
+tempts were made to systematically learn about the innova-
+tion. This poses an open question for Open Source research
+to which we can only suggest some initial ideas:
+Open Question 1. How can knowledge acquisition and
+sharing in Open Source projects be maximized?
+This open research question needs to be seen in addition
+to the results of existing research about information gener-
+ated inside the project (see [55] for an innovation targeting
+increased information management capabilities in a project)
+such as information about tasks, social relationships and
+contextual factors [10]. It was found that Open Source
+projects are well positioned for such information because
+of their nature as Communities of Practice [41, 73, 70]
+which foster both re-experience and participatory learn-
+ing [33, 32] and by use of information management in-
+frastructures such as bug trackers, source code management
+systems or wikis [43]. Yet, we find that for innovation in-
+troductions and radical changes, the knowledge acquisition
+from the outside must be considered separately. This study
+did not aim to answer this question, but some general ideas
+come to mind if assuming the role of an at least somewhat
+proficient innovator and pondering how to increase knowl-
+edge about an innovation: First, the most obvious way and
+certainly the status quo in an Open Source project for the
+innovator to enhance knowledge is by explaining and tutor-
+ing the other project members via e-mail and references to
+external literature. Yet, presenting information in a static,
+non-interactive way via e-mail or the web has several draw-
+backs, such as for instance being more work for the author
+than verbal communication, equipped with less cues about
+which aspects are particular important, and with greater
+ambiguity about the success of transferring information to
+the recipients’ side [10, p.220f]. To alleviate these prob-
+lems, the innovator might want to strive for a more direct
+communication channel of which two particular kinds ex-
+ist: (1) Real world meetings and conferences such as the
+Free Software Developer European Meeting (FOSDEM) or
+the KDE project’s yearly aKademy16can provide a yearly
+get-together in which information about a new innovation
+can be distributed to all central project participants. (2)
+With the increasing maturity of collaboration awareness and
+16For some analysis on participation in community events such as these,
+see [11, 6].screen sharing solutions, it becomes possible to conduct dis-
+tributed pair programming (DPP) sessions for demonstra-
+tion and teaching purposes (see for instance [3, 67, 30] for
+empirical results, [35, 51, 65, 15] for several tool solutions
+supporting DPP, and [77] for a comparison of tools) which
+can provide an efficient high-bandwidth synchronous com-
+munication channel between a knowledgeable project par-
+ticipant and those to be taught.
+2.) Taking a more pessimistic view on the other hand,
+we might discard such a teaching approach when noting
+high turn around of participants in Open Source projects,
+which would foil any long term strategies to spread knowl-
+edge in OSS projects. Rather the innovator should focus
+on the acquisition of new developers already skilled with
+desired innovations from outside the project and focus on
+their inclusion into the project.
+Strategy 3. An innovator can strengthen innovation intro-
+ductions by lowering entry barriers and acquiring highly
+skilled individuals.
+The second insight regarding innovation introduction
+arose from the circumstance and events surrounding the
+phasing-out of the innovator’s involvement both in May and
+September 2007. As previously discussed, the first attempt
+of detaching from the project failed and the test suite as a
+consequence was unmaintained during a large-scale refac-
+toring and thus soon “spectacularly broken”, as one of the
+maintainers stated. Comparing this phase-out with the sec-
+ond much more successful one in September 2007, which
+resulted in the role of maintaining the tests being picked up
+by one of the maintainers, we find that the primary differ-
+ence in behavior is one of signaling and ownership. When I
+— as the innovator — first detached from the role of man-
+aging test cases, I did neither consider ownership of the
+test code just being created nor communicating my with-
+drawal to the project an important priority. In fact, I did not
+consider myself the code owner of the testing code. Yet,
+just as Mockus et al. found in their case study of Apache
+and Mozilla, code ownership is achieved implicitly for code
+the developer is “known to have created or to have main-
+tained consistently" [50, p.318]. While such code owner-
+ship “doesn’t give them [the owner] any special rights over
+change control", it stipulates a barrier for other developers
+to engage with the code (see for instance [40] for a discus-
+sion on code ownership as an important part of the mental
+model of developers). Having not signaled the phase-out
+then prevented the other project participants from under-
+standing that the test code should be perceived as under
+shared ownership. Shared ownership should be assumed
+because, unlike separate modules of the code being main-
+tained, test code is deeply dependent upon the rest of the
+software as evident to it being broken completely due to the
+refactoring.
+13Only when the test suite decayed to be broken com-
+pletely after the refactoring must it have become apparent
+that the suite was unmaintained. Abstracting from the case
+of introducing automated testing, this might have been a
+major factor in the success of the whole introduction: Run-
+ning the test cases automatically provided a way to know
+whether the test suite was still being maintained by being
+signaled with a red bar of failing tests.
+Thus, when phasing-out the innovator’s engagement
+again after having fixed the test-suite in September 2007, a
+discussion (see [freecol:2182]) was sufficient to create this
+shared obligation for testing. When the innovator then dis-
+engaged, one of the maintainers picked up the role of main-
+taining the test cases successfully (see Figure 4), keeping
+the percentage of test affecting commits at around 10% of
+the total commits (see Figure 3) until another developer as-
+sumed a more active role in testing. We can conclude:
+Strategy 4. An innovator should explicitly signal his en-
+gagement and disengagement in the project to facilitate
+shared ownership of the innovation introduction. In partic-
+ular this will be necessary if the innovation itself has no sig-
+naling mechanisms to show individual or general engage-
+ment or disengagement.
+When analyzing the contributions of developers to the
+testing effort, we find that besides myself as the innovator
+and the aforementioned maintainer there were two notable
+individuals who contributed extensively to testing. Interest-
+ingly, as their contribution increased and waned according
+to their time contribution to the project, the maintainer who
+had already picked up the testing effort from me seemed
+to adjust his own contribution to testing accordingly, which
+was best visible from January to May 2008, where the in-
+creasing contribution of one developer caused the main-
+tainer to invest less into testing until the developer was in-
+active in June to August 200817which caused the main-
+tainer to resume testing activities (see Figure 4). As con-
+tributions of the other core and peripheral developers never
+exceeded five testing commits per month and never contin-
+ued for an extended period of time, we can interpret that the
+project adopted a more flexible code ownership strategy. In
+this approach, the role of a “test master” exists who con-
+tributes heavily to testing and is pivotal to the expansion of
+test coverage and development of knowledge regarding test-
+ing. This role is not formally but rather implicitly assigned
+and acknowledged explicitly in the project only for instance
+when a core developer — stumped by a difficulty regarding
+testing — asked: “Any suggestions, particularly from the
+resident test expert [name of developer]?” [freecol:4446].
+Strategy 5. To maintain constant activity with regard to in-
+troducing an innovation in the face of fluctuating developer
+17Due to vacation [freecol:2919].contributions, the project leadership and/or the innovator
+needs to be flexible to resume and cease their own contribu-
+tions.
+5 Limitations and conclusion
+In the last section, the limitations of this research regard-
+ing internal and external validity shall be discussed before
+summing up the results.
+We first want to turn to external validity, i.e. the ques-
+tion whether the results of this study can be generalized
+to other settings, in particular whether the introduction of
+testing can be achieved in another project than FreeCol and
+with a different innovator. For other Open Source projects
+many aspects might be different to FreeCol, in particular
+(a) attitude toward testing and similar software process im-
+provements, (b) a programming language other than Java
+might be used for which there is no well established auto-
+mated testing framework such as JUnit, (c) the project might
+be differently sized, causing many other aspects of interac-
+tion to occur and have developers showing less interest in
+testing, and (d) the software produced by the project might
+have different characteristics regarding testing. For other
+innovators similarly (a) their level of knowledge about in-
+novation introduction and (b) testing might be substantially
+different, as well as their (c) available time and (d) moti-
+vation to achieve the introduction successfully. We discuss
+these in turn.
+Different attitudes towards testing specifically and in-
+novations in general might influence adoption and rep-
+resents the most important threat to generalizability.
+If nobody would have found testing to be a worth-
+while innovation the introduction would have failed
+with tests breaking as the source code continues to
+evolve. Looking at the case of FreeCol gives us one
+defenses against this threat: Contributing to testing
+seems to be an individual decision. This at least can
+be concluded by the large variability in testing com-
+mitment where some developers contribute while oth-
+ers do not. I then think it is reasonable to argue that (a)
+finding a developer interested in testing is correlated to
+project size and (b) that we did not see any indication
+in the case of FreeCol that the fraction should be lower
+in another project.
+If a different programming language than Java is used
+in a project in which testing is to be established, this
+may hurt external validity on several counts. For in-
+stance, one could argue that Java provides many fea-
+tures for modularizing a software into well-testable
+units, such as clearly separated interface definitions,
+package boundaries and visibility modifiers which
+might lack in other languages. Furthermore, one could
+14argue that JUnit as a well-established automated test-
+ing framework created by even more well-known prac-
+titioners must make a Java project more likely to adopt
+testing. Arguing against this validity restriction, one
+could note that untyped programming languages such
+as Python, Ruby and Perl in particular have a tradi-
+tion to replace the safety of a compiled language with
+strong test suites [17] and thus projects’ using them
+should be even more inclined to use automated regres-
+sion testing, and that all major programming language
+have comparable automated testing libraries such as
+CppUnit for C++, PerlUnit for Perl, PyUnit for Python,
+NUnit for C# or Test::Unit for Ruby [45, p.14].
+The size of the project as well as the characteristics of
+its participants might influence the results of the study
+to a degree which makes a transfer to other projects
+difficult. We think there are essentially three sub-types
+in this threat which need to be addressed: (1) The tar-
+get project might be substantially smaller than Free-
+Col, (2) the project might be substantially larger than
+FreeCol, and (3) the participants in the project might
+have a negative attitude towards automated testing to
+start with. We do not believe that the first and sec-
+ond threat are fundamentally challenging the results
+of this study, because for projects sufficiently smaller
+than FreeCol (i.e. a maintainer and at most a handful
+of developers) the contribution of a single additional
+person such as an innovator will always have a much
+more marked influence on the project as in a smaller
+project. It is thus not likely that the contribution of the
+innovator is rejected and rather possible that the inno-
+vator will alone be able to achieve high levels of cover-
+age within a short amount of time. Testing should then
+show its benefits in uncovering defects in the software
+and achieve adoption easier than with a medium-sized
+software such as FreeCol, where high levels of cov-
+erage are harder to attain. While achieving an initial
+introduction thus should be easier to attain in a smaller
+project, it should be more difficult to withdraw from
+the project because the number of developers avail-
+able to assume the role of testing will be smaller. In
+a substantially larger project (i.e. around 20 core de-
+velopers) we think it is reasonable to assume a higher
+level of software engineering knowledge and thus with
+a high probability either already an existing test suite
+or at least some experience with it. Both should in-
+crease the likelihood that an innovator willing to con-
+tribute test cases or expand on an existing test suite
+will be welcomed to do so. Conversely to a small
+project, we would argue that initial introduction might
+be harder but withdrawing would be easier. For the
+third threat, that of negatively minded developers, we
+must conclude that introduction will be more of a chal-lenge and would primarily refer to existing research
+about how to deal with resistance to change in OSS
+projects [69]. Secondly, we should note that the in-
+troduction with FreeCol contained at least two aspects
+which mark it an introduction with some resistance:
+For one, the first attempt to withdraw failed and the in-
+novator needed to expedite additional effort to achieve
+adoption, and second, only one of the maintainers sup-
+ported the adoption actively while the other one had
+only a minor number of testing commits (see Figure 4).
+While an introduction might still be made much more
+difficult if no maintainer is championing the introduc-
+tion, we think it reasonable that results should general-
+ize.
+As a third threat to external validity one might note
+that a different kind of software being produced by the
+OSS project should lead to different adoption behav-
+ior. We agree it should, but we think that FreeCol as
+a desktop application including client/server network
+communication, artificial intelligence and complicated
+user interface should be at the one end regarding dif-
+ficulty in testing. Other types of software such as for
+instance algorithmic and utility libraries, web frame-
+works [69] or programming languages should be far
+easier to test. Only in operating systems and systems
+libraries with their dependence on different configura-
+tions of hardware do we see a higher testing complex-
+ity than in desktop applications.
+Regarding validity concerns about the person of the in-
+novator we must conclude that this author had proba-
+bly more knowledge about how to introduce innova-
+tions than most innovators as described in Section 1.
+Yet, since he acted using the phase model as described
+in Section 3.1, it does not seem too difficult to repli-
+cate his actions in a different project. Rather, the in-
+sights presented in this study should already increase
+the likelihood for innovator success, for instance by
+appropriately signaling withdrawal from the project as
+described in Section 4.2. As for the other mentioned
+concerns of comparable motivation, testing knowledge
+and available time, we think them reasonable to as-
+sume for any innovator committed to establish testing
+in an OSS project. Time spent on introduction was de-
+liberately kept at around 10-15 hours per week, and
+testing knowledge can be acquired before an introduc-
+tion easily from existing technical literature. As for
+motivation, we can note that while this innovator was
+driven to achieve research results, the motivation by in-
+creased quality and probably a desire to participate in
+a project which one is dependent upon as assumed in
+the introduction should establish sufficient motivation.
+Before moving on to the discussion about internal va-
+15lidity, external validity of the results should be examined
+when abstracting from the concrete case of introducing the
+innovation of automated testing to the general set of inno-
+vations which could be introduced into an OSS project. To
+this end, testing can be categorized as (1) a knowledge in-
+tensive innovation with (2) medium up-front costs to estab-
+lish benefit for a project, (3) high levels of trialability be-
+cause of the orthogonality of testing technology with other
+infrastructure and pre-existing innovations, (4) high levels
+of independence regarding individual innovation decisions
+to adopt testing, (5) as requiring ongoing maintenance ef-
+fort particular in the face of code refactorings, and (6) as
+transparently regarding the current state of the implementa-
+tion of the innovation in the project as visible by the fail-
+ure state of the test suite. Other interesting innovations,
+such as for instance a decentralized source code manage-
+ment system such as Git [29] or a quality assurance process
+of pre-commit peer reviews such as practiced by the Apache
+project [18], will likely differ in at least one of these aspects.
+For instance, for the introduction of a new source code man-
+agement system, trialability is markedly reduced, as migrat-
+ing the data from the existing repository to a novel one will
+either require duplicate effort to maintain both repositories
+or will lead to wasted efforts. For a process improvement
+such as peer review on the other hand, a different type of in-
+novation decision is likely, because peer review in particular
+becomes useful if applied uniformly to all commits and thus
+should need a project-wide consensus18. Nevertheless it is
+hard to see why a result such as the importance of signal-
+ing departure or the strategic dichotomy between teaching
+existing project participants vs. recruiting new ones should
+not transfer to other innovation introductions.
+Two major threats need to be discussed when turning
+now to internal validity: (1) This study relied exclusively
+on information which was publicly available in mailing-lists
+and the project source code management repository, possi-
+bly leading to conclusions deviating from the events as they
+actually occurred. (2) The long-term analysis performed on
+the repository and the mailing-list is coarsely grained by
+only performing monthly check-outs of the repository and
+by using keyword searches on the mailing-list.
+The first threat arises from the methodology used, which
+is markedly in contrast to fieldwork and ethnographic stud-
+ies conducted with companies (see for instance [42]). In
+this study we only regarded intermediates and process re-
+sults, such as the mailing-list messages, source code com-
+mits and — to some limited extent — bug reports. One
+can argue against this being a real threat as the whole com-
+munication in the Open Source world is based on these
+18While the argument here is that peer review is likely to be adopted after
+a project-wide organizational innovation decision, Fogel notes an interest-
+ing introduction episode in the Subversion project in which one particular
+individual leading by example has achieved the introduction of peer review
+almost single-handedly [20, p.39ff].forms of encoded information and no face-to-face com-
+munication exists. Thus, while we might not learn what
+one project participant actually did, we can nevertheless
+assume that most other project participants will have re-
+ceived a similarly restricted view on the participant’s ac-
+tivities. If one would want to strengthen the research with
+regard to this threat, two major remedies can be offered:
+(a) Using a more collaboration-oriented research method-
+ology such as action research based on a researcher client
+agreement [44, 68, 2, 12] should allow for a more de-
+tailed and comprehensive communication between the open
+source project participants and the researcher. Alternatively,
+research by West and O’Mahonycan be named as partic-
+ular successful examples of using interviews with project
+members to derive interesting results about Open Source
+projects and their governance [56] and firm–community re-
+lations [75, 74]. (b) The researcher could use appropriate
+instrumentation in agreement with the project participants
+to capture more detailed information about their daily work
+processes. For instance using a tool such as Hackystat [37]
+or the ElectroCodeoGram [64], it should be possible to cap-
+ture in detail each time a project participant executed test
+cases or made changes to them.A further threat to inter-
+nal validity arises from the coarsely-grained data analysis.
+First, we only took monthly snapshots from the repository
+to graph the evolution of coverage and failure rate of the test
+suite over time19. A more fine-grained analysis could re-
+veal notable stretches in-between those check-outs in which
+the test suite was broken, and should show in more detail
+how coverage was expanded by the project. Yet, we find it
+hard to see how this threat should attack the primary results
+drawn from the coverage and failure rate analysis, namely
+that coverage is steadily expanding and that the failure rate
+is actively controlled by the project members. Secondly,
+when searching the mailing-list for e-mails regarding au-
+tomated testing, we employed a set of keywords20which
+might have missed important events on the mailing-list re-
+garding testing which did not contain any of the keywords.
+While it is easy to guard against this threat by a more com-
+plete analysis of the messages sent to the mailing-list, we
+found that perusing over 3,100 e-mails a markedly less ef-
+fective way than keyword searching.
+When looking to future work it seems best to prioritize
+conducting a second case with another project that is dif-
+ferent from FreeCol and then secondly use additional data
+sources such as in particular interview with project mem-
+bers to strengthen internal validity.
+To conclude, this study has shown that the introduction
+of code-centric process innovation such as automated test-
+19To be more precise: out of 3,776 commits during the observation pe-
+riod, only those 29 representing the first commit of each month were ana-
+lyzed for coverage and test failures, which is less than 1% of commits.
+20Keywords did include: junit, test*, regression, failure, automated
+16ing into an OSS project is feasible and can be successfully
+achieved. In particular, this study revealed that an innovator
+(1) from the outside of the project and (2) with a relatively
+short period of activity can trigger a beneficial innovation
+adoption curve based on a simple four-stage model of inno-
+vator activities.
+Regarding automated testing, this study has found a sur-
+prising number of episodes in which test cases were used
+for communicating bug reports and opinions about spec-
+ification more precisely and as technical “hard” contribu-
+tions. Secondly, the state of the practice regarding auto-
+mated testing has been found to be lacking in modules such
+as the user interface, the artificial intelligence reasoner and
+client/server communication, which prevents further expan-
+sion in coverage and thus quality improvements.
+For an innovator and innovation introduction in general,
+results agree with [55] in that reinvention can be a major
+source of benefit of an innovation, and that the innovator
+should thus seek or at least not prevent such from occurring.
+As to the radical expansion of innovation use in a project,
+two strategies are deducible from the events observed in
+the project FreeCol. These events highlight the importance
+of external participants for radical expansions, as both of
+the pushes in coverage were achieved by participants with
+knowledge about automated testing which precedes their in-
+volvement in FreeCol. This leads to the first strategy of re-
+ducing entry barriers for external developers as to leverage
+such pre-existing knowledge to the fullest. Yet, since inside
+the project the amount of coaching and teaching of auto-
+mated testing specific knowledge did only occur minimally,
+we thus propose that an innovator or project leader could
+also attempt to employ such a teaching strategy to increase
+knowledge about testing and have coverage expanded as a
+result of the increased skill of existing project members.
+As a third and last insight, signaling the departure of the
+innovator is important even for an innovation such as au-
+tomated testing which has explicit signaling mechanisms
+such as test cases failing. Thus, when reducing effort or
+leaving the project, the innovator should inform the project
+and potentially even look for a successor for sustaining an
+innovation.
+5.1 Acknowledgements
+Dan Delorey provided the author with a list of all java
+projects on Sourceforge.net that had more than 5 active de-
+velopers over the course of 2006. Many thanks also to Ge-
+sine Milde, Florian Thiel, Lutz Prechelt and the FreeCol
+maintainers and test masters who read a draft version of this
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+2005.
+A Innovator Diary
+2007-04-02 Checked-out project, subscribed to mail-
+ing-list and explored the application.
+2007-04-03 Wrote a first test case to find a bug in the
+MapGenerator.
+2007-04-04 Wrote a first e-mail to the mailing-list con-
+cerning test case for MapGenerator and accompanied
+it with a fix. This ended phase #1 (lurking) which only
+lasted 2 days thus and started phase #2 of actively con-
+tributing.
+2007-04-05 The MapGenerator patch was included in
+FreeCol and I got a direct reply from Core Developer I
+of FreeCol. Began to structure a testing framework for
+FreeCol.
+Got a reply from the Core Developer I indicating that
+the project did not have a lot of exposure to automated
+testing so far. I volunteered to write a little introduc-
+tory document regarding testing.
+2007-04-06 Contacted the other maintainer (Main-
+tainer I) for commit privileges after prompted by Core
+Developer I to do so.
+2007-04-09 Received answer from Maintainer I. Com-
+mit privileges were granted to me, and I was intro-
+duced on the mailing-list.
+2007-04-10 Wrote tests for pioneer work.
+2007-04-11 Finished writing tests for pioneer work
+and committed them.
+Writing to the mailing-list about the deviations from
+the original game specification represented by these
+tests did elicit only a response from Core Developer
+I. While on technical level the discussion was success-
+ful in resolving the question of whether this deviation
+from the original FreeCol specification was intentional
+or not, from social perspective it would have been ben-
+eficial to achieve communication with the project more
+broadly.2007-04-22 Running the test cases enabled me to catch
+a regression. In an e-mail to the mailing-list I men-
+tioned the failure as a good example of how to write
+a test case to New Developer I. The next two weeks
+make up phase #3 of the innovation introduction and
+consisted of communicating and collaborating with
+others developers about testing.
+2007-04-23 New Developer I used my suggestion and
+wrote two small test cases based on the code I had sent
+to him the previous day. The test case showed the unfa-
+miliarity with testing in general for most of the people
+in the project. It helped to understand that there are
+several hurdles for the adoption of an innovation. In
+particular, his test cases returned the correct result and
+passed, but not for the reason he anticipated but rather
+because of incorrectly setting up the testing environ-
+ment.
+2007-04-23 Another New Developer II wrote a mes-
+sage to the list of being interested in helping out with
+the website, to which he got an reply one day later to
+write to the current webmaster and offer him his help.
+2007-04-27 New Developer II made another join-
+attempt, this time based on an infrastructure suggestion
+(switching from the hosting platform SourceForge.net
+to Launchpad21) to which he got a short reply that his
+suggestion sounded nice but would be a lot of work,
+at which point the discussion stalled. This is indicative
+of a wrong join-strategy by newbies, who try to ask the
+members of the project for too much change.
+2007-05-03 Used a bug report22to write twelve test
+cases of which 3 succeeded and 9 failed and posted it
+on the mailing-list. Directly got a reply by a lurker
+(New Developer III) who saw it as an opportunity to
+get involved. He said he had questions and I gave him
+my contact details (e-mail, IM).
+2007-05-03 By now, a total of 57 tests have been writ-
+ten and it was deemed sufficient for self-sustaining
+growth. The next two weeks were decided to consisti-
+tute phase #4, phasing out the innovator’s involvement.
+2007-05-07 I produced a screencast of how to set
+up FreeCol for developing and regression testing in
+Eclipse.
+2007-05-15 The build-scripts of the project were fixed
+to let those who do not use Eclipse for developing and
+21Launchpad is a project hosting platform started by Canonical Inc. as
+part of their development on the Linux distribution Ubuntu. Homepage:
+https://launchpad.net/
+22Bug #1616384 https://sourceforge.net/tracker/
+?func=detail&atid=435578&aid=1616384&group_id=
+43225
+20testing run the tests as well. This was intended to mark
+the end of my phase-out from FreeCol.
+2007-05-16 The issues for which test cases were con-
+tributed on 2007-05-03 was fixed.
+2007-05-22 Core Developer I was granted maintainer
+status and is now referred to as Maintainer II.
+2007-05-30 Innovation introduction so far a failure.
+The number of test cases is stalled at 65 with no ac-
+tivity in the two weeks in which I did not participate.
+Without the engagement of the innovator no new test
+cases got written.
+2007-08-28 I sent a literature suggestion to the list23
+which I thought had some relevance for a current de-
+sign discussion and also declared my status as phased
+out. Maintainer II then informed me that this was a
+pity since the test cases were “spectacularly broken”
+since the refactoring of using a softcoded configura-
+tion. Figure 8 shows that since June the number of test
+cases had not increased anymore and that there was
+some decay in July already causing several failing tests
+at the beginning of August. The refactoring in August
+then caused both the highest number of commits per
+month in FreeCol ever (See Figure 2) and the whole
+test suite to break (as indicated by zero test cases for
+September 2007 in Figure 8).
+2007-09-12 I asked the project how much they valued
+testing before agreeing to go about fixing the test cases
+broken by the restructuring.
+2007-09-24 I repaired all test cases, but never indicated
+that I was intending to resume activities in the project.
+This thus can be seen as second attempt to phase out
+involvement.
+2007-09-26 Helped Maintainer II to resolve an out-of-
+memory error, which occurred when running the test
+cases and was due to a memory leak in FreeCol.
+2007-10-15 Wrote a test to be used for Test-Driven De-
+velopment.
+2008-01-09 Number of test cases surpasses 100, all of
+which were written by Maintainer II.
+2008-01-12 A core developer who would become Test
+Master II later on (if the innovator is seen as Test Mas-
+ter I) contributed his first patch against the unit tests.
+2008-05-18 A new developer who would later become
+Test Master III contributed his first commit for the unit
+tests.
+23Martin Fowler’s “Dealing with Roles” [21]2008-06-01 Number of test cases surpasses 150.
+2008-10-14 New Developer III contacted me to ask
+about information about how to develop test cases.
+2008-12-01 Number of test cases surpasses 200.
+2009-05-01 Number of test cases surpasses 250.
+2009-08-23 I started a full dump both of the Free-
+Col Subversion repository and requested a snapshot in
+mbox format of the mailing-list from Maintainer II.
+2009-09-04 I finished the analysis of the Subversion
+logs and snapshot check-outs.
+21
\ No newline at end of file
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@@ -0,0 +1,270 @@
+arXiv:1001.0684v2  [math.NT]  23 Apr 2010Non-singular points on Hypersurfaces over Fq
+Jahan Zahid
+1 Introduction
+In this survey we investigate hypersurfaces defined over finite fie ldsFq. More
+specifically we wish to determine for which hypersurfaces one can en sure the
+existence of a non-singular point taking the cardinality qof our ambient field
+large if need be. Additionally for such hypersurfaces we will find a lowe r
+bound on qfor which a non-singular point is guaranteed.
+We shall assume that our hypersurface is projective and is defined by the
+homogeneous polynomial
+F(x1,...,x n)∈Fq[x1,...,x n],
+inn≥3 variables. Recall that a non-singular point x∈Fn
+qis one in which
+∇F(x) :=/parenleftbigg∂F
+∂x1(x),...,∂F
+∂xn(x)/parenrightbigg
+/ne}ationslash=0.
+1.1 Motivation
+It is a classical theorem of Chevalley & Warning that every form Fof degree
+dinn > dvariables has a non-trivial zero in Fq. However there seems to be
+much less in the literature giving conditions which ensure that Fhas a non-
+singular point. This question is worth addressing for a number of rea sons.
+For example suppose we have a form Fdefined over the p-adic numbers Qp,
+we may wish to determine whether or not Fhas a non-trivial zero. This is
+of course a requisite for the corresponding form to have a non-tr ivial zero in
+the rational numbers Q. A standard and well known method for determining
+the existence of a p-adic point is by an application of Hensel’s lemma viz.
+assume without loss of generality Fis defined over Zp, then if there is some
+xoverZpsuch that
+F(x)≡0 (mod p) and ∇F(x)/ne}ationslash≡0(modp)
+1thenFhas a non-trivial p-adic point.
+A conjecture of Artin [1, Preface] states that every p-adic form of degree
+dinn > d2variables has a non-trivial zero. There exist forms in n=d2
+variables with only the trivial zero in every p-adic field, so this is the best
+we can hope for. The conjecture is false in general (see for examp le [10]).
+However by a rather remarkable theorem of Ax & Kochen [2] it is know n to
+be true for any fixed degree d, provided the characteristic of the residue class
+fieldislargeenough. Areasonablelower boundforthecharacterist ic required
+is only known in a handful of cases. For example, recently Wooley [11] has
+shown that for d= 7 and 11 we require the size of the residue class field
+to respectively exceed 883 and 8053. These estimates depend on t he ability
+to ascertain the existence of a non-singular point on the Fq-varieties under
+consideration. It is often in this regard that we are motivated to se ek non-
+singular points on varieties.
+Theexistenceofa p-adicpointisalsocrucialforapplicationsoftheHardy-
+Littlewood circle method where one requires a certain “singular serie s” to be
+positive. The singular series of some function is positive precisely whe n that
+function has a non-singular p-adic point for every p.
+1.2 Varieties with a non-singular point
+We necessarily require some kind of classification of our variety F(x) = 0,
+since there are examples such as F(x) =G(x)2with only singular zeros.
+After factorizing Fover the algebraic closure ¯Fqwe may write
+F(x) =F1(x)a1···Fr(x)ar(1)
+whereai∈Nand each Fiis absolutely irreducible and pairwise coprime. If
+ai≥2 for each i, then it is clear that Fhas only singular points. We may
+therefore assume henceforth that a1= 1. Consequently we can write
+F(x) =G(x)H(x)
+whereGis absolutely irreducible and Gdoes not divide H. Since we are
+interested in finding non-singular points over the base field Fq, for simplicity
+we shall assume that both GandHare defined over Fq.
+It is clear that we can find a non-singular zero of F, provided that we can
+find a non-singular zero of Gwhich is not a zero of H. Since for any such
+pointx∈Fn
+q, clearlyF(x) = 0 and
+∇F(x) =H(x)∇G(x)/ne}ationslash=0.
+This exact question has been studied by Lewis & Schuur [8, Theorem 1 ] who
+prove the following result in their paper.
+2Theorem 1 (Lewis & Schuur, 1973) .LetGbe an absolutely irreducible form
+of degree doverFqandHa form of degree eoverFqnot divisible by G. Then
+there exists a function A(d,e)such that if Fqis a finite field of cardinality
+q > A(d,e), then there exists an Fq-point which is a non-singular zero of G
+and which is not a zero of H.
+We shall apply a recent result of Cafure & Matera [3] in order to find a
+permissible value for A(d,e) in the Lewis & Schuur theorem. More precisely
+we will prove
+Theorem 2. LetGbe an absolutely irreducible form of degree doverFqand
+Ha form of degree eoverFqnot divisible by G. Then provided that
+q >1
+4/parenleftbig
+α+/radicalbig
+α2+4β)2
+where
+α= (d−1)(d−2)andβ= 5d13/3+d(d+e−1)
+there is a non-singular zero of Gwhich is not a zero of H.
+When the form Fis absolutely irreducible we can do slightly better,
+by applying an effective version of the first Bertini theorem, along w ith the
+sharper estimates which areavailableforpointcounting oncurves. Forusthe
+first Bertini theorem says that given any absolutely irreducible hyp ersurface,
+there exists a linear variety of dimension 2 whose intersection with th eformer
+is absolutely irreducible. The existence of this linear variety has been made
+quantitative by Kaltofen [5] which allows us to prove
+Theorem 3. LetFbe an absolutely irreducible form of degree doverFqthen
+provided that,
+q >1
+2(3d4−4d3+5d2)
+there is a non-singular zero of F.
+We may also apply a flexible version of the Bertini theorem (as given
+in [3, Corollary 3.4]) in the general case i.e. whenever F(x) =G(x)H(x).
+However there is more delicacy involved dealing with the degrees of ea ch
+irreducible factor Fiin (1). In this respect it seems difficult to asymptotically
+improve on the permissible lower bound for q, given in Theorem 2. One may
+wish to read [11] for further details.
+32 Some preliminaries
+In this section we shall review some of the elements which are available for
+point counting on varieties.
+2.1 Lang–Weil estimates
+Arguably the genesis for much subsequent work on estimating the n umber of
+points on Fq-varieties, is due to a theorem of Lang & Weil [6].
+Theorem 4 (Lang–Weil, 1954) .Given any absolutely irreducible variety V
+of degree dand dimension rinn-dimensional projective space over Fq, there
+exists a constant C=C(n,r,d)such that the number of Fq-pointsNofV
+satisfies
+|N−qr| ≤(d−1)(d−2)qr−1/2+Cqr−1.
+Inthecaseofwheneverthevariety Visdefinedbyahypersurfaceofdegree
+dinnvariables over Fn
+q, there has been much work on finding a permissible
+value for the constant C=C(n,r,d) in the Lang & Weil theorem. For
+example Schmidt [9, Theorem 5A, p.210] proved that we can take
+C= 6d2k2k, (2)
+wherek=1
+2d(d+1). More recently Cafure& Matera [3, Theorem 5.2] proved
+the following
+Theorem 5 (Cafure & Matera, 2006) .The number of points N, on an
+absolutely irreducible Fqhypersurface in nvariables of degree dsatisfies
+|N−qn−1| ≤(d−1)(d−2)qn−3/2+5d13/3qn−2.
+Thissubstantial improvement of(2)owesmuchtotheworkofKalto fen[5]
+forprovidinganeffectiveversionofthefirstBertinitheorem. Wes halldiscuss
+this further in the next section.
+2.2 Upper bounds on the number of Fq-points
+We will also require bounding from above, the number of points on an a ffine
+variety of prescribed degree. Thus note the following well known re sult (for
+example see [3, Lemma 2.1]).
+Lemma 1. AnyFq-varietyV, of dimension rand degree dinFn
+qsatisfies
+#(V∩Fn
+q)≤dqr.
+4In order to keep some control on the number of singular points on o ur
+variety we need to estimate the number of points on the intersectio n of two
+hypersurfaces, with no common component. To prove the next lem ma it will
+be necessary to utilize the B´ezout inequality (for example see [4, p.148]): if
+VandWare affine varieties in Fn
+qthen
+deg(V∩W)≤degVdegW. (3)
+Lemma 2. LetF1,F2∈Fq[x]be non-zero polynomials of degrees d1and
+d2without a common factor in ¯Fq[x]then the variety Vdefined by F1,F2
+satisfies
+#(V∩Fn
+q)≤d1d2qn−2.
+Proof.SinceF1andF2have no common factors in ¯Fq[x], thenVis anFq
+variety of dimension n−2. From the B´ezout inequality (3), degV≤d1d2.
+The result then follows by Lemma 1.
+2.3 Non-singular points on curves
+Finally we will also need to estimate the number of non-singular points o n
+an absolutely irreducible curve. The following lemma is due to the work o f
+Leep & Yeomans [7].
+Lemma 3. LetP∈Fq[x,y]be an absolutely irreducible polynomial of degree
+d. Then the number N′of non-singular zeros of Psatisfies
+N′≥q+1−1
+2(d−1)(d−2)[2√q],
+where[γ]denotes the least integer not exceeding γ.
+Proof.WriteSto denote the number of Fqsingular zeros of P. Then if the
+curve defined by P(x,y) = 0 has genus g, it follows from [7, Corollary 1] that
+|N′+S−(q+1)| ≤g([2√q]−1)+1
+2(d−1)(d−2).
+Next we use the above estimate together with the following bound on the
+genus
+g≤1
+2(d−1)(d−2)−S,
+which comes from [7, Lemma 1], to obtain the required bound
+N′≥q+1−1
+2(d−1)(d−2)[2√q].
+53 Proof of Theorems 2 and 3
+Our strategy will be relatively straight forward: to prove Theorem 2 we shall
+bound from below the total number of points on the hypersurface G(x) = 0,
+whilst simultaneously bounding from above, the number of singular po ints
+ofG(x) = 0 plus the number of points on the intersection G(x) =H(x) = 0;
+for Theorem 3 we shall use an effective version of the first Berini th eorem
+and apply Lemma 3 to obtain the desired conclusion.
+Proof of Theorem 2. If we let
+S1= #{x∈Fn
+q:G(x) = 0and∇G(x) =0}
+and
+S2= #{x∈Fn
+q:G(x) = 0and H(x) = 0},
+then it follows that we have a non-singular point of Gwhich not a zero of H
+provided that N−S1−S2>0, where Ndenotes the number of points of G.
+If∇Gisidenticallyzerothenitfollowsthat G(x1,...,x n) =M(xp
+1,...,xp
+n),
+wherepdenotesthecharacteristic of Fq, forsomeform M. Consequently over
+¯Fq[x],G(x) factorsinto M′(x1,...,x n)pfor some form M′, contradicting that
+Gis absolutely irreducible. Therefore some component of ∇Gis non-zero,
+∂G
+∂x1say.
+It is clear that Gand∂G
+∂x1have no common factor in ¯Fq[x], therefore by
+Lemma 2 it follows that S1≤d(d−1)qn−2. Moreover since GandHdo
+not have a common component it also follows that S2≤deqn−2. Hence by
+Theorem 5
+N−S1−S2≥qn−1−(d−1)(d−2)qn−3/2−(5d13/3+d(d−1)+de)qn−2.
+It is clear that N−S1−S2>0 provided that
+q−(d−1)(d−2)√q−5d13/3−d(d+e−1)>0,
+from which the conclusion follows.
+Before proving Theorem 3, we need to introduce some notation. Le tL
+be a field and consider a polynomial f∈L[x0,x1,...,x n]. When ξ∈L3n+1,
+we write f|ξ=f|ξ(X,Y) to denote the sliced polynomial
+f(ξ0+X,ξ1+ξn+1X+ξ2n+1Y,...,ξ n+ξ2nX+ξ3nY).
+Next notethefollowing result ofKaltofen[5,p.285]which ismade explic it
+by Cafure & Matera [3, Corollary 3.2].
+6Lemma 4. Letf∈Fq[x0,...,x n]be an absolutely irreducible polynomial of
+degreed≥2. Then the number of slices ξ∈F3n+1
+q, for which the polynomial
+f|ξis not absolutely irreducible is at most1
+2(3d4−4d3+5d2)q3n.
+Proof of Theorem 3. We know that Fis absolutely irreducible, so by Lemma
+4 provided that
+q >1
+2(3d4−4d3+5d2),
+there exists a slice ξ∈F3n+1
+qsuch that F|ξ(x) = 0 is an absolutely irreducible
+curve. HencebyLemma 3thenumber N′ofnon-singularpointsonitsatisfies
+N′≥q−1
+2(d−1)(d−2)[2√q]+1.
+Note that N′>0 provided that
+q−(d−1)(d−2)√q+1>0.
+The above inequality clearly holds for d= 1,2. Consequently if we assume
+d/ne}ationslash= 1,2 and take
+q >1
+4(α+√
+α2−4)2,
+whereα= (d−1)(d−2) then the curve F|ξhas a non-singular point. Finally
+it is not difficult to see that for d/ne}ationslash= 1,2
+6d4−8d3+10d2>(α+√
+α2−4)2.
+HenceF(x) = 0 will always have a non-singular point whenever
+q >1
+2(3d4−4d3+5d2),
+as required.
+Acknowledgments: It is a pleasure to thank the Hausdorff Research
+Institute for Mathematics, for their kind hospitality where part of this work
+took place. I would also like to thank Prof. Roger Heath-Brown for t he
+valuable and interesting discussions I have had with him on this topic.
+7References
+[1] E. Artin. The collected papers of Emil Artin . Addison-Wesley, London,
+1965.
+[2] J. Ax and S. Kochen. Diophantine problems over local fields. I. Amer.
+J. Math., 87:605–630, 1965.
+[3] A. Cafure and G. Matera. Improved explicit estimates on the num ber of
+solutions of equations over a finite field. Finite Fields Appl. , 12(2):155–
+185, 2006.
+[4] W. Fulton. Intersection theory , volume 2 of Ergebnisse der Mathematik
+und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Math-
+ematics. Springer-Verlag, Berlin, second edition, 1998.
+[5] E. Kaltofen. Effective Noether irreducibility forms and application s.
+J. Comput. System Sci. , 50(2):274–295, 1995. 23rd Symposium on the
+Theory of Computing (New Orleans, LA, 1991).
+[6] S. Lang and A. Weil. Number of points of varieties in finite fields. Amer.
+J. Math., 76:819–827, 1954.
+[7] D. B. Leep and C. C. Yeomans. The number of points on a singular
+curve over a finite field. Arch. Math. (Basel) , 63(5):420–426, 1994.
+[8] D. J. Lewis and S. E. Schuur. Varieties of small degree over finite fields.
+J. Reine Angew. Math. , 262/263:293–306, 1973. Collection of articles
+dedicated to Helmut Hasse on his seventy-fifth birthday.
+[9] W. M. Schmidt. Equations over finite fields. An elementary approach .
+Lecture Notes in Mathematics, Vol. 536. Springer-Verlag, Berlin, 1 976.
+[10] G. Terjanian. Un contre-exemple ` a une conjecture d’Artin. C. R. Acad.
+Sci. Paris S´ er. A-B , 262:A612, 1966.
+[11] T. D. Wooley. Artin’s conjecture for septic and unidecic forms. Acta
+Arith., 133(1):25–35, 2008.
+8
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+arXiv:1001.0685v1  [q-bio.QM]  5 Jan 2010Noise and nonlinearities in high-throughput data.
+Viet-Anh Nguyen(1), Zdena Koukol´ ıkov´ a-Nicola(2), Franco
+Bagnoli(3), Pietro Li´ o(1)
+1Computer Laboratory, University of Cambridge, Cambridge CB3 0F D, UK
+2Fachhochschule Nordwestschweiz, Hochschule fr Technik, Steina ckerstrasse 5,
+CH-5210 Windisch, Switzerland
+3Department of Energy, University of Florence, S. Marta, 3 50139 Firenze.
+Also CSDC and INFN, sez. Firenze.
+E-mail:franco.bagnoli@unifi.it
+Abstract. High throughput data analysis are becoming common in biology,
+communications, economics and sociology. This vast amount of data are usually
+representedinformofmatricesandcanbeconsideredasknowledg enetworks. Spectral-
+basedapproacheshavebeen proveduseful in extractinghidden in formationwithin such
+networks and to estimate missing data, but these methods are bas ed essentially on
+linear assumptions. The physical models of matching, when available, often suggest
+nonlinear mechanisms, that may sometimes be identified as noise. The use of nonlinear
+models in data analysis, however, may require the introduction of ma ny parameters,
+that lowers the statistical weight of the model. According with the q uality of data, a
+simpler linear analysis may be more convenient than more complex appr oaches.
+In this paper, we show how a simple nonparametric Bayesian model ma y be used
+to explore the role of nonlinearities and noise in synthetic and experim ental datasets.
+Keywords : Special issue, Data mining (Theory)
+1. Introduction
+Thereiscurrentlyatremendousgrowthintheamountofhighthrou ghputdataextracted
+from life sciences, electronic communications, economics and social sciences. Examples
+of high throughput life science data are the large number of complet ely sequenced
+genomes, 3D protein structures, DNA chips, and mass spectrosc opy data. Large
+amounts of data are distributed across many sources over the we b, with high degree
+of semantic heterogeneity and different levels of quality. These dat a must be combined
+with other data and processed by statistical tools for patterns, similarities, and unusual
+occurrences to be observed.
+The results of many experiments can be summarized in a large matrix, where
+rows represent repetition of the experiment in different contexts , and the columns are
+the output of a single measurement. Let us consider the following ca ses: microarray
+sampling, protein-substrate affinity, socio-psychological survey s. Let us illustrate the
+similarities of these examples.Noise and nonlinearities in high-throughput data. 2
+1.1. Examples of datasets
+1.1.1. Microarray data A DNA microarray (gene chip) can be seen as an ordered
+collectionofspots, oneachofwhichthereisadifferent probeforme dbyknownsequences
+of cDNA. A sample of mRNA, supposed to represent the gene expre ssed in a given tissue
+under investigation, is let hybridize with the probes. Fluorescent te chniques allows to
+detect the hybridized spots. The idea is that of using probes specifi c for a unique region
+ofagene, detecting thegenes expressed inatissue. Theexperime nt isrepeatedformany
+tissue, from different parts of the body, from different patients, or from a different phase
+of the cellular cycle. The data can therefore be arranged using the probe numbering as
+column index, and tissue numbering as row index. The goal is that of id entifying the
+difference in gene expressions in the different cases.
+There are many problems in extracting information from these data . Some data
+may be missing, control spots are sometimes more hybridized that n ormal ones, low-
+intensity data cannot be easily distinguished from noise.
+1.1.2. Protein-substrate affinity A similar problem is that of investigating the shape
+of a protein or of a peptide. The interaction of proteins with the out er world (in
+particular concerning the immune response) depends on the shape . At present, it is not
+possible to reconstruct the tri-dimensional shape of a protein fro m its primary sequence
+(easily obtained by mRNA sequence). Moreover, proteins very oft en glycosylated, and
+these sugar chains attached to the outer surface may be the mos t important factor for
+inflammation. On the other hand, direct visualization of protein surf ace, using NMR,
+electronic microscopy, etc. is a very slow and costly process.
+A method for obtaining information about this shape is that of using p rotein or
+antibodyarrays, similartoDNAmicroarrays. Again, inthiscase, the patternofmatches
+can be represented as a matrix, with columns corresponding to sub strates (probing
+proteins or antibodies) and rows to different proteins under invest igation.
+1.1.3. Questionnaires and other socio-psychological data The high-level investigation
+ofthehumanmindtakeoftentheformofthestudyofresponses t ostimuli. Thestimulus
+maybeplannedandtargeted, like inthecaseofquestionnaires, oro ccasional/unplanned
+like for instance those that lead to choosing some good. In this case , it is economically
+advantageous to study the patterns that emerge, for instance in renting DVDs [1],
+opinions onbooks [2], supermarket tickets. Also Googlepagerank[3] may beconsidered
+in this class; in this case the “opinions” are the links that bring to the p age under
+investigation.
+Allthese datamaybe(ideally) represented inmatrixform, withrows corresponding
+to customers, and columns corresponding to items or goods.
+In this case, in addition to the usual problems of consistency and no ise, there is a
+special meaning in missing data: an accurate method for “anticipatin g” them from the
+knowledge stored inthe matrixwould constitute avaluable toolfor p ersonal advertising.Noise and nonlinearities in high-throughput data. 3
+However, in the case of humans, one should consider also that tast es change and
+evolve in time.
+1.2. Knowledge networks
+The extraction of information about the properties of the gene, p roteins and humans
+is performed using statistical tools, mainly based on variations of sin gular value
+decomposition [4]. The goal is that of extracting the most robust ch aracteristics of
+patterns, clustering the data in similarity classes, reconstruct mis sing data, detect
+outliers, reduce noise. it is rather unusual to take into considerat ion an explicit model
+for the generation of data, i.e., for the matching mechanism.
+The problem may be reformulated in geometrical terms. We shall den ote with the
+word “probe” the substrates or the questionnaires, and with the word “subject” the
+mRNAs, the proteins and the individuals of the three examples. Actu ally, the role of
+probes and subjects are symmetric, the only difference is that i sg eneral one has more
+“a priori” knowledge of probes that of subjects.
+A subject can be visualized as an array of M“tastes”, and the probes as a
+complementary array of “characteristics”. In the case of mRNA, this space is just the
+sequence space of basis, in case of proteins it is a way of coding the s urface (motifs), in
+caseofpsychological data, thesearemental modules, oftencalle dfactorial“dimensions”.
+The match between tastes and characteristics is denoted “opinion ”.
+The match between tastes and characteristics may be linear, like a s calar product,
+or highly nonlinear, as in the case of protein-antibody or microarray interactions. In
+case of one-or-none interaction, there is no noise and no inferenc e can be performed on
+missing data.
+In the linear case, the results are much more blurred, there is a non -zero overlap
+between different samples, and it has been shown [5] that, if one kno ws a sufficient
+number of overlaps between subjects, there is a rigid percolation t hreshold that in
+principle allows the reconstruction of any “taste” once that one is k nown. However,
+tastes are in general hidden or difficult to be obtained. If one has at his disposition a
+sufficient amount of data, it can be shown [6] that the correlation b etween expressed
+opinions approximates the real overlap among tastes. This would in p rinciple allow the
+perfect reconstruction of missing opinions and detection of outlier s.
+1.3. Paper outline
+In this paper we investigate the role of nonlinearities and noise in the m atching phase.
+In particular, it is shown that nonlinearities appear as noise when linea r investigation
+tools are used. We investigate the influence of nonlinearities in the rig id percolation
+transition. Our recent proceeding paper [7] could be used as a comp lementary source,
+from which we suggest the use of correlation distribution among sub jects to predict the
+difference between random noise and nonlinearity.Noise and nonlinearities in high-throughput data. 4
+The goal of this paper is to explore the limits of linear prediction, coup led with
+Bayesiantuningofparameters, inpredictingthefeaturesoflinear andnonlinearsystems,
+in the presence of an eventual noise. As expected, the reconstr uction of missing values
+works quite well with the linear model, since it actually consists ina linear interpolation.
+However, it works also for nonlinear models. Indeed, sometimes we h ave information
+about the real physical process of matching, like in the case of micr oarrays or proteins,
+while in other cases (opinions) this information is unknown. So, let us c oncentrate
+on the more “favorable” case. Even in this case, we are not sure to have taken into
+consideration all possible sources of nonlinearities. Since it is rather difficult to develop
+a complete matching model, one has to be content with a ”phenomeno logical” one, with
+a certain number of phenomenological parameters to be tuned. As usual, the preference
+ofa morecomplex model, withmoreparameters, withrespect to asim pler oneshould be
+justified onthebasis ofthequality ofavailable data. Therefore, on ehasto startwith the
+simplest model, usually a linear one, and tune the parameters (done h ere by Bayesian
+estimators). Given the amount of “explained” signal by this model, o ne can decide if it
+is worth to use a more complex model. In this paper we show that even for synthetic
+data, where the matching model is perfectly controlled, most of th e information in the
+signal may be captured by linear methods also in the presence of non linearities. In a
+future work we shall explore the properties of nonlinear analysis.
+We apply a recursive technique [5, 6] to synthetic data, obtained fr om linear
+matching models, and investigate the limits of the “black box” recons truction model.
+We also investigate the role of nonlinearities and noise in the matching p hase. In
+particular, it is shown that nonlinearities appear as noise when linear in vestigation
+tools are used. We propose the use of Bayesian statistics to autom atically determine
+the most appropriate value at each iteration of the learning proces s. As the result, the
+value ofMis changed accordingly to the amount of information available at each step,
+and approaches a fixed value when the predictions start to conver ge.
+2. The model
+Let us denote by Nthe number of subjets, and by Dthenumber of probesor substrates.
+We assume that subject iis represented in a (hidden) M-dimensional space as a vector
+xi= (x(1)
+i,x(2)
+i,...,x(M)
+i). The quantities x(k)
+ican in principle be arbitrary, but we can
+assume that they are normalized in the interval [ −1,1]. A substrate jis represented
+similarly by a vector wj={w(k)
+j}in a dual space, with j= 1,...,Dandk= 1,...,M.
+The match yijbetween a subject and a substrate is supposed to be a function of a
+the concordance between the characteristics
+yij=f/parenleftBigg/summationdisplay
+ka(k)w(k)
+jx(k)
+i/parenrightBigg
+,
+wherea(k)denotes the weight assigned to component k(this weight could be eliminated
+by using non-normalized characteristics). For simplicity in the followin g we assume
+a(k)= 1/M, so that the argument of the function fis also normalized.Noise and nonlinearities in high-throughput data. 5
+The match yijis the output of the experiment, and is the subject of data analysis
+in order to extract hidden information. For instance, one would like t o know if data are
+consistent, how to reconstruct missing entries, how to use data f or clustering subjects
+and substrates in terms of similarities, how to choose the optimal se t of subjects for a
+given classification task.
+The problems in data analysis arises from the role of noise and nonlinea rities. We
+study three topical cases:
+(i) linearcase: f(x) =x, withnonoiseadded. Thisisareferencecase, andcorresponds
+to the maximum of information that can be extracted.
+(ii) noisy case: f(x) =x+ǫ, whereǫis some gaussian noise. This is the model that is
+tacitly assumed in most of data analysis.
+(iii) nonlinear case: f(x) is some nonlinear function, for instance f(x) = tanh( βx),
+whereβcontrols the influence of nonlinearities. This particular choice of the
+matching function corresponds to a case of thresholded match, but other choices
+could be more suited, see for instance [8].
+(iv) nonlinear case with noise: a combination of case 2 and 3, which is pr esumably the
+nearer to reality.
+For convenience, we rewrite the standard assumption (case 2) in m atrical form
+yi=Wxi+ǫ. (1)
+Oneof thefirst target problemis theconsistency check, or recon struction of missing
+items. Suppose that some data yi∗j∗is missing. Can it be reconstructed from the rest of
+data? This should be possible if the number of subjects and substra tes is large enough,
+and if their characteristics cover evenly enough the available space . On the contrary, if
+themissing datacorrespondtoaparticularmatch, notpresent ina llotherdata, recovery
+is impossible. For instance, let us assume that only subject i∗has characteristic k∗
+different from zero, and has average values for all other charact eristics. Therefore, this
+subject is the only sampling the “dimension” k∗, and its corresponding entry contains
+information not present in the rest of the database. On the contr ary, if individual i∗is
+exactly the same of another individual, this similarity should emerge (e xcept for noise),
+and allow the perfect reconstruction.
+The problem of reconstruction can therefore serve as a consiste ncy check, and also
+as a tool for pointing out the data that deserve further investiga tions, either because
+they are spurious, or because they contain “original” information.
+3. Inference on missing values
+3.1. Estimating the length of (hidden) feature space
+Looking at the model in (1) from the regression view, we consider yas the observed
+variables from the experimental space, xas the hidden variables from the feature space,Noise and nonlinearities in high-throughput data. 6
+Was the model parameter that relates these two sets of variables, and error ǫto follow
+the Gaussian distribution N(0,σ2I).
+We assume xito follow the standard Gaussian distribution N(0,I). The diagonal
+unit variance implies that all vector components are independent, w hich is reasonable
+enough. By integrating out x, we get the likelihood distribution of the data given all
+parameters:
+p(Y|W,σ2) = (2π)−ND/2|V|−N/2exp/parenleftbigg
+−1
+2tr/parenleftbig
+(V)−1P/parenrightbig/parenrightbigg
+, (2)
+whereV=WWT+σ2IandP=YYT.
+Tipping and Bishop [9] has used maximum likelihood to estimate Was:
+˜W=UM(ΛM−σ2I)1/2(3)
+where the N×MmatrixUMis constructed by Mprincipal eigenvectors of YYT, the
+M×MmatrixΛMcontains Mlargest eigenvalues of YYT. The arbitrary rotation
+matrix as in [9] was effectively selected as Ifor simplicity. The square root operation is
+safe with the corresponding estimation of σ2.
+However, Mis normally not a known and fixed property in real systems. Even in
+the case in which we have information about the matching mechanism, it is more correct
+to treatMas an unknown quantity, especially in the case in which we exploit linear
+correlationstoobtaininformationaboutanonlinearmatchingmodel. MaslovandZhang
+[5] has proposed a conjecture to effectively estimate Musing the knowledge of portion
+of missing values of symmetric data matrix. Although the conjectur e sometimes prove
+useful [10], its usability limits to the case of symmetric input data. Her e, we propose
+the use of Bayesian approach to estimate Mfrom its posterior distribution. In other
+words, we want to calculate:
+p(M|Y) =p(Y|M)p(M)/integraltext
+p(Y|M)p(M)dM(4)
+where the likelihood of the data given Mis computed by integrating over all unknown
+parameters:
+p(Y|M) =/integraldisplay
+X,W,σ2p(Y|W,σ2)p(W|σ2,M)p(σ2|M)dWdσ2(5)
+There is no closed-form solution to the model. A few papers [11, 12, 1 3] proposing
+different ways to estimate the sufficient number of principal compon ents to keep
+when performing PCA, which is very close in nature to our problem. To keep the
+lightweight characteristic of the spectral algorithm, we base our c alculation on the
+Laplace approximation of p(Y|M) proposed by [13], which leads to the following
+estimation:
+p(Y|M)≈N−M
+2(2π)(D−M+1)M
+2/parenleftBiggM/productdisplay
+i=1λi/parenrightBigg−N
+2/parenleftBigg/summationtextD
+i=M+1λi
+D−M/parenrightBigg−N(D−M)
+2
+|A|−1
+2(6)
+where|A|=/producttextM
+i=1/producttextD
+j=i+1N(λ−1
+j−λ−1
+i)(λi−λj) andλi,i= 1..Dare the square root of
+the eigenvalues of YYT.Noise and nonlinearities in high-throughput data. 7
+Maslov and Zhang [5] provided an estimation of Meff- the sufficient number of
+eigenvalues to keep during matrix reconstruction, taking into acco unt the proportion
+of missing values of the data matrix. Here we adapt this conjecture to the case of
+asymmetric data, and use it as the suggestion for the upper bound ary ofMgivenm
+missing elements. We define the prior distribution p(M) as:
+p(M) =
+
+k
+(k−1)Meff+DforM <=Meff
+1
+(k−1)Meff+DforMeff< M <=D
+0 otherwise(7)
+whereMeff=ND−m
+Nand the empirical value k= 3.
+3.2. The algorithm
+Our proposed approximation algorithm to infer on missing values of da ta matrix Yis
+as follows:
+(1) Construct the initial estimation ˜YofYby assigning 0 to all unknown positions.
+(2) Estimate the sufficient ˜Mfor˜Yusing equations (4), (6), and (7), the denominator
+in (4) is ignored since it is a constant to M.
+(3) Perform SVD on ˜Y, and construct the matrix ˜Y′by keeping the ˜Mlargest singular
+values and corresponding eigenvectors.
+(4) Reconstruct ˜Yfrom˜Y′by filling known positions with their original values.
+(5) Go to step (2).
+We repeat this process until either there is no significant change on estimated values or
+a maximum number of iterations has been reached.
+The goal of the algorithm is to find a rank-M matrix that best approx imates the
+data matrix, or mathematically we want to find a solution for the follow ing problem:
+min
+UM,ΛM,VM∝bardblY−UMΛMVT
+M∝bardbl2
+When there is no missing values, the solution is simply the SVD of the com plete data
+matrix. For our iterative algorithm of reconstructing missing positio ns, the solution to
+this problem will not change once it is found. More specifically, if we are given the
+solution and use it to fill in the missing positions, the SVD of the resulte d matrix will
+be exactly the given solution. Hence, it is reasonable to believe that t he algorithm
+will converge to the solution. Our numerical experiments actually ac hieved very good
+convergence rate under various proportions of missing data.
+4. Experiments and Discussion
+4.1. Synthetic Data
+UsingN,D,Mas free parameters, we randomly generated two matrices XandW
+with each components to be either 1 or -1. The match matrix Ywas then computed
+as in section 2. We then investigated the capabilities of our algorithm in reconstructingNoise and nonlinearities in high-throughput data. 8
+(a)0 1 2 3 4 500.10.20.30.40.50.60.70.80.91
+# iterationsNRMSE
+(b)0 1 2 3 4 500.10.20.30.40.50.60.70.80.91
+# iterationsNRMSE
+(c)0 1 2 3 4 50.40.50.60.70.80.91
+# iterationsNRMSE
+(d)0 1 2 3 4 50.40.50.60.70.80.91
+# iterationsNRMSE
+Figure 1. Prediction error evolvement of Bayesian spectral method for N= 100,
+D= 30,M= 10. (a) linear matching, (b) thresholded ( f(x) = tanh( x)) match
+function ( β= 1)(c) noisy linear matching, , (d) noisy thresholded matching.
+the full matrix for 4 different cases: nearly perfect linear matching (little noise with
+σ= 0.01 added), noisy linear matching ( σ= 0.1), nearly perfect thresholded match
+function ( f(x) = tanh( x)), and noisy thresholded matching.
+FixingN= 100 and D= 30, we applied the algorithm to two different data
+sets ofM= 5 and M= 10. For each case, the unknown positions of the data
+matrix were randomly picked with missing percentages from 10% to 90 %. For each
+missing percentage, an average result from 1000 runs were then o btained. The accuracy
+of predictions was measured by the popular used normalised root me an square error
+(NRMSE) after each iteration.
+Figure 1 shows the evolvement of the algorithm prediction error thr ough the first 5
+iterations. The error at iteration 0 was calculated by replacing each missing position by
+themeanvalueofitscorrespondingsubject. The9linescorrespon dto9different missing
+percentages, ranging from 10% to 90%. It can be seen that the alg orithm converge
+quickly to reasonable accuracy after 5 iterations for up to 40% miss ing percentages
+in all cases. With too much missing data (80% upward for perfect linea r data), the
+matrix could not be reconstructed. This phase transition corresp onds to a rigidity
+percolation threshold. The noise moves this threshold down to 60%, but under perfectNoise and nonlinearities in high-throughput data. 9
+(a)10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.70.80.9
+missing percentageNRMSE
+  
+linear0.01
+tanh0.01
+linear0.1
+tanh0.1
+(b)10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.70.80.91
+missing percentageNRMSE
+  
+linear0.01
+tanh0.01
+linear0.1
+tanh0.1
+Figure 2. Prediction errors of Bayesian spectral method for the 4 referen ce cases.
+(a)M= 5, (b)M= 10
+data collection condition, the nonlinear match function ( f(x) = tanh( x)) does not make
+any considerable effect. It comes to action however in the case of n oisy data, moving
+the percolation threshold to 50%. In the case of linear matching dat a with very little
+noise, the rigid percolation threshold actually agrees with the theor etical result in [5],
+which equals to p= 1−2M/N≃80%
+Figure2presentstheeffectofnoiseandnonlinearityonpredictiona ccuracy. Itcould
+beeasily seen the dramatic increase of error at the corresponding percolationthresholds.
+Noise shows critical effect on prediction accuracy compared to the thresholded match
+function, increasing by the complexity of data (large M).
+4.2. Real data
+To get a reasonable view of the phase transition, we applied the Baye sian spectral
+methodontwo different datasets. The firstoneisthemeasuremen ts ofthetranscription
+levels under different experimental conditions of 215 mutants of es sential yeast genes
+[18]. The dataset was extracted by the authors from microarray m easurements of about
+5000 genes of the budding yeast S.cerevisiae. The second dataset is the small-scale
+measurements of binding energies between the bacteria E.coli ligand s and enzymes of
+the bacteria E.coli [20]. We removed all columns with missing values from the data,
+leaving two complete data matrices of size 215 ×15 and 119 ×15.
+Following similar procedure to synthetic data tests, we randomly gen erated a list
+of missing positions in the complete data matrix, and cleared the know n values out of
+those cells. We then applied the Bayesian spectral method to recon struct the missing
+values on each data set, and compared to the original ones to calcu late the prediction
+error. The procedure was applied for various missing percentage f rom 5% to 90%, for
+each case the average result from 1000 runs was obtained.
+The prediction results over increasing missing percentages of the t wo datasets are
+shown in figure 3a. The algorithm performs considerably better on t he enzyme-ligandNoise and nonlinearities in high-throughput data. 10
+(a)510152025304050607080900.10.20.30.40.50.60.70.80.91
+missing percentageNRMSE
+  
+microarray
+enzyme ligand
+(b)0 1 2 3 4 5 6 7 8 90.650.70.750.80.850.90.951
+# iterationsNRMSE
+  
+5% missing
+10% missing
+15% missing
+20% missing
+25% missing
+30% missing
+35% missing
+Figure 3. Missing value reconstruction of microarrays by the Bayesian spect ral
+method. (a)prediction error for increasing missing percentage, ( b)error evolvement
+of microarray data
+data than microarray data. Setting a threshold at a value of the NR MSE of 0 .7, where
+there is a big jump for the microarray data, we see that the algorith m is able to
+reconstruct enzyme-ligand values up to 70% missing, while the trans ition occurs at
+20% missing for the case of microarray. Since enzyme-ligand data se t was acquired by
+focused biochemistry studies of each enzyme, its reliability is a lot high er compared
+to the high-thoughput technique used to obtain gene expressions [21]. Apart from the
+common inaccuracy due to background noises and diversity of samp les, the limitation
+of the measuring technology largely reduces microarray data reliab ility.
+Figure 3b shows the prediction evolvement though early iterations f or microarray
+data. Agreeing with the final prediction results, the algorithm kept improving its
+prediction by iterations with up to 20% missing data, while not much info rmation
+can be reconstructed for data with 25% upward missing. Interest ingly, in this case
+the error diminishes during the first 2 iterations, but then increase s again. A similar
+effect was observed [22] in reconstructing missing values in synthet ic data by means
+of linear regression and multi-entry correlations. Using the simplest algorithm [6, 8],
+missing values ˜ yijare reconstructed by using two-entry correlations: ˜ yij∝/summationtext
+kCjkyki,
+whereCijis the Pearson correlation among entries of subjects iandj. For a large
+percentage of missing values, the correlation is affected by a low sta tistics, so one may
+try to exploit three- and higher-entry correlations. According wit h the level of noise,
+the number of hidden components and the nonlinearity of the match ing function, this
+procedure improves the results only up to a certain point, after wh ich the multi-entry
+contributions essentially furnish more noise than information, in a wa y similar to what
+is observed in Figure 3b. By looking at Figure 1, one can realize that in t he case of
+noisy match (subfigures c and d), the evolvement curves of the er ror above the phase
+boundary (not converging phase) slightly rise after the first two it erations, while in the
+absence of noise (subfigures a and b) it keeps decreasing. This beh avior suggests that
+in case of microarray, the level of noise is much greater that the lev el of nonlinearity.Noise and nonlinearities in high-throughput data. 11
+5. Conclusions
+We have applied a Bayesian spectral algorithm in order to investigate the nature of
+noise in synthetic and real datasets. The investigation on synthet ic data shows that
+the approach is very robust in handling large percentage of missing d ata both in terms
+of accurate prediction and quick convergence rate. Although a so lid conclusion on the
+ability of our method in predicting systems noise nature has not been made, the result
+on synthetic data and on some experimental datasets are very pr omising. A systematic
+investigation with specific consideration on different kinds of noises w ould prove useful.
+References
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+of annotation errors in a database of protein sequences. Bioinformatics 18(12), 1641–9.
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+in Proteobacteria genomes. BMC Evolutionary Biology 7(1):S7.Noise and nonlinearities in high-throughput data. 12
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\ No newline at end of file
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+arXiv:1001.0686v1  [cond-mat.other]  5 Jan 2010physica statussolidi, 22 June 2018
+Angular distributions of electrons
+emitted from free and depositedNa 8
+clusters
+Matthias B ¨ar1, Phuong Mai Dinh2, Lyudmila V. Moskaleva3, Paul-Gerhard Reinhard*,1,2, Notker R ¨osch3, Eric
+Suraud2
+1Institutf¨ ur Theoretische Physik,Universit¨ at Erlangen , Staudtstrasse 7, D-91058 Erlangen, Germany
+2Laboratoire de Physique Th´ eorique, IRSAMC, UPS and CNRS, U niversit´ e de Toulouse, 118 Rte de Narbonne, F-31062 Toulou se
+cedex, France
+3Department Chemie andCatalysis ResearchCenter, Theoreti sche Chemie, Technische Universit¨ atM¨ unchen, D-85748 Ga rching, Ger-
+many
+ReceivedXXXX, revisedXXXX,accepted XXXX
+Publishedonline XXXX
+PACS31.15.ee, 33.60.+q, 36.40.Vz, 36.90.+f, 68.47.Jn, 79.60. -i
+∗Corresponding author: e-mail reinhard@theorie2.physik.uni-erlangen.de , Phone: +49-9131-85-28462, Fax:+49-9131-85-28907
+Weexplorefromatheoreticalperspectiveangulardistribu tionsofelectronsemittedfromaNa 8clusterafterexcitation
+by a short laser pulse. The tool of the study is time-dependen t density-functionaltheory (TDDFT) at the level of the
+local-density approximation (LDA) augmented by a self-int eraction correction (SIC) to put emission properties in
+order. We consider free Na 8and Na 8deposited on the surfaces MgO(001) or Ar(001). For the case o f free Na 8, we
+distinguishbetweenahypotheticalsituationofknownclus terorientationandamorerealisticensembleoforientatio ns.
+We alsoconsiderthe angulardistributionsforemissionfro mseparatesingle-electronlevels.
+Copyrightlinewillbe provided by the publisher
+1 Introduction Optical methods have provided the
+key analyzing tools in cluster physics over decades. In
+the first stage, optical absorption measurements allowed
+one to collect rich information on structure and dynamics
+of these small, nano-sized particles; for an overview see,
+e.g., [1–4]. More information can be gathered when one
+additionally measures the features of reaction products in
+more detail. A most important channel in this context is
+electron emission and thus there meanwhile exist many
+investigations that analyze the properties of the electron s
+emitted after irradiation by a short laser pulse. The first
+step in that direction is photoelectron spectroscopy (PES)
+where the distribution of the kinetic energy of the emitted
+electrons is recorded. This method has been applied in
+cluster physics since long, see e.g. [5,6]. Stepping furthe r
+in refinement, one can determine the angular distribution
+of the outgoing electrons which, in fact, is mostly done
+simultaneously together with PES [7–12]. Further chal-
+lenging aspects come into play when considering clusters
+incontactwithsubstrates.Thatcombinationisoftenmoti-
+vated(ifnotdictated)byeasierexperimentalhandling.It isofgreatimportanceforpossiblepracticalapplications,a nd
+the effectsat interfacesare also an interestingproblemfo r
+basic research, for up to date collections see e.g. [13,14].
+It is obvious that the deposition on a surface modifies the
+emission properties, in particular angular distributions ,
+thus calling for dedicated theoretical studies. This paper
+is devoted to a theoretical exploration of angular distribu -
+tions of laser excited metal clusters, free and deposited on
+insulatingsurfaces,MgO(001)andAr(001).
+There is a broad choice of approachesfor the descrip-
+tion of clusters on surfaces, from fully detailed quantum
+mechanical treatments of all constituents [15] to a robust
+jellium treatment of cluster and interface [16]. We are go-
+inghereforanintermediatestrategy,detailedatthesideo f
+thehighlyreactiveclusterandlesssofortheinertsubstra te.
+Asabasisofthedescription,weemploydensity-functional
+theory[17],whichprovidesamostefficienttheoreticalde-
+scriptionfortheelectronicstructureanddynamicsofclus -
+ters [18]. We simulate the detailed dynamicsof laser exci-
+tationandsubsequentelectronemissionbytime-dependent
+density-functionaltheory(TDDFT)solvedonagridinco-
+Copyrightlinewillbe provided by the publisher2 Matthias B ¨ar et al.:Angular distributions of electrons from Na 8
+ordinate space. The substrates are inert and will be de-
+scribed classically in terms of polarizable atoms [19,20].
+The computation of angular distributions requires rather
+large numerical boxes. Therefore, previous explorations
+dealt with approximations, a semi-classical approach [21]
+whichisconfinedtoveryintenselaserpulseoraquantum-
+mechanical, but two-dimensional axial, treatment [22,23]
+for lower intensities and short pulses. Here we are going
+forafullythreedimensionalTDDFTanalysisbecauseboth
+simplifications are not applicable anymore for the studies
+intended here and because the progress of computational
+resourcesmeanwhileallowsaTDDFTanalysisinfullthree
+spatial dimensions. We will take Na 8as a test case and
+compare free Na 8with Na 8on MgO(001) as well as Na 8
+onAr(001).
+The study concentrates on direct electron emission,
+i.e. those electrons which are leaving the cluster with or
+directly after laser impact. In practice, there is a com-
+petition between direct and thermal emission. The pre-
+ferredexitchanneldependsonthelengthofthelaserpulse.
+For pulses longer than the collisional relaxation time in-
+duced by two-electron processes, thermal emission domi-
+nates, while shorter pulses change the weight towards di-
+rect emission [24–27]. We will use here pulses with full
+width at half maximum of 20 fs, which surely are at the
+sideofdominantdirectemission.
+There is also a subtle point about the orientation of
+the cluster relative to the laser polarization. Experiment s
+with clusters in the gas phase deal, in fact, with an equi-
+distributedensembleoforientations,while depositedclu s-
+ters have a well defined orientation due to the known sur-
+face structure. We will briefly address this question, con-
+sidering both free clusters with known orientation and av-
+eragingoverclusterorientations.Wewillthenreturntoor i-
+entedclustersandinvestigatethechangescausedbydepo-
+sitionon a surface.We will discusshow thespecific angu-
+lar distributions for emission of each single-electron sta te
+separately combine to the total distributions and how this
+helps to understand the pattern observedin emission from
+deposited clusters. Furthermore, we will investigate and
+compare two different substrates, MgO(001) and Ar(001)
+surfaces. Both are insulators with a rather large band gap.
+The MgO(001) surface shows more corrugation and has
+a stronger interface attraction, while Ar(001) is softer an d
+dominatedbycorerepulsion.
+The paper is outlined as follows. In section 2, we
+brieflysummarizetheformalframework,numericalstrate-
+gies, and the basic properties of the test cases. In section
+3, we present and discuss results on free Na 8and also ex-
+plore the double differential distributions. Section 4 com -
+pares results for free Na 8to those for Na 8deposited on
+MgO(001) and Ar(001). Conclusions are given in section
+5.
+2 Briefsummary ofthe model2.1 The degrees of freedom The hierarchical
+quantum-mechanical-molecular-mechanical (QM/MM)
+model has been describedin detail elsewhere (see [20,28]
+and[29]forarecentdetailedreview).Herewegiveabrief
+outline. The constituents of the system and their degrees
+of freedom are: ϕn(r,t), n= 1,...,Nelfor valence elec-
+trons of the Na cluster, Ri(Na),i(Na)= 1,...,Nifor the
+positions of the Na+ions,Ri(c), i(c)= 1,...,Mfor the
+positions of the O cores, Ri(v), i(v)= 1,...,Mfor the
+centersoftheOvalenceclouds, Ri(k),i(k)= 1,...,Mfor
+thepositionsoftheMg2+ions.TheNaclusteristreatedin
+standard fashion: The valence electrons quantum-mecha-
+nicallyandtheionsclassically[18,30].TheMgOsubstrate
+is composed of two species : Mg2+cations and O2−an-
+ions. The cations are electrically inert and can be treated
+as charged point particles; they are labeled by i(k). The
+anions are easily polarizable; this fact is accounted for by
+associating them with two constituents : A valence elec-
+tron distribution (labeled by i(v)) and the complementing
+core (labeled by i(c)). All ions of the MgO substrate are
+described as classical degrees of freedom in terms of po-
+sitionsRi(type). The difference R(c)−R(v)describes the
+electrical dipole moment of an O2−anion. The Mg and
+O ions are arranged in crystalline order corresponding to
+bulk MgO.The dynamicaldegreesoffreedomforMg and
+O aretakenintoaccountinanactivecell oftheMgO(001)
+surface region underneath the Na cluster. The active cell
+is embedded in an “outer region” of MgO material where
+only ions are kept fixed, while oxygen dipoles remain
+fully dynamical. Beyond that region, only the Madelung
+potential is considered. The effect of the outer region on
+the active part is given by a time-independentshell-model
+potentialtakenoverfrom[31].Actually,thesubstratecon -
+sists of six layerseach containing784Mg2+ions and784
+O2−ions. The ions in the lowest layer are fixed to pre-
+vent them from relaxing and forming an artificial second
+surface. The active cell consists of three layers, each con-
+tainingsquarearrangementsof242Mg2+cationsand242
+O2−anions.
+In the case of Ar, the modeling follows a similar, al-
+thoughsimplified, track [19,32]. The Ar substrate is com-
+posed of only once species, neutral Ar atoms, each of
+which being characterized by its position and dipole mo-
+ment(exactlyasO2−anions).Thesubstratecomprises384
+atoms;thismodelhasbeenshowntoprovideafairapprox-
+imationtobulk[33,34].
+2.2 Energy and fields The total energy is decom-
+posedasE=ENa+ESurf+EcouplwhereENadescribes
+an isolated Na cluster, ESurfthe MgO(001) or Ar (001)
+substrate,and Ecouplthecouplingbetweenthetwosubsys-
+tems. ForENa,we take thestandardfunctionalof TDDFT
+atthelevelofthelocaldensityapproximation(LDA),cou-
+pledwithmoleculardynamics(MD),asinpreviousstudies
+of free clusters [18,30] including an average density self-
+interactioncorrection[35].Theenergyofthesubstratean d
+the coupling to the Na cluster consists of the long-range
+Copyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 3
+Coulomb energy and some short-range repulsion which is
+modeledthrougheffectivelocalcore-potentials[31,36,3 7].
+To avoid the Coulomb singularity and to simulate the fi-
+nite extension of Ar, Mg2+and O2−ions, we associate
+a smooth charge distribution ρ(r)∝exp(−r2/σ2)with
+eachoftheseioniccenters.ThisyieldsasoftCoulombpo-
+tentialtobeusedforallactiveparticles.Themodelparam-
+eters were calibrated to represent results of calculations
+where a fully quantum-mechanical description of the ac-
+tive MgO cell was employed in the case of MgO, for de-
+tails see [20]. The model parameters for the Ar substrate
+were chosen to reproduce model calculations of the NaAr
+dimer[38],inturnfittedto ab-initiodata.
+The field equations obtained by variation of the above
+energyareaugmentedbytheexternallaserfield
+Ulas=−er·npolE0sin2/parenleftbiggt
+Tpulseπ/parenrightbigg
+sin(ωlast)(1)
+which is activated only in the time interval 0≤t≤
+Tpulse. The field strength E0is related to the intensity I
+asI/(Wcm−2) = 27.8/bracketleftbig
+E0/(Vcm−1)/bracketrightbig2. The total pulse
+lengthTpulsecorrespondsto a full width at half (intensity)
+maximum as FWHM ≈Tpulse/3. The laser polarization
+npolcan take any direction. We will consider npol=nz,
+the direction perpendicular to the surface, i.e. along the
+symmetry axis of Na 8, and one direction orthogonal to it
+withnpol=nx(forthegeometry,seesection2.5).Alaser
+with polarization nzis, of course, an idealization because
+itwouldcorrespondtoaplanewaverunningparalleltothe
+surface, but it serves to model rather flat impact as com-
+paredtotheperpendicularimpactof x-polarization.
+From the energy functional, once established, one de-
+rives the static and dynamical equations variationally in a
+standardmanner[18].
+2.3 Solution scheme The numerical solution of the
+coupled quantum-classical system proceeds as described
+in [19, 20, 39]. The electronic wave functions and the
+spatial fields are represented on a Cartesian grid in three-
+dimensional coordinate space. The numerical box em-
+ployed here has a size of (64a0)3. The spatial derivatives
+are evaluated via fast Fourier transformation. The ground
+state configurations were found by accelerated gradient
+iterationsfor the electronic wave functions[40] and simu-
+latedannealingfortheionsintheclusterandthesubstrate .
+Propagation is done by the time-splitting method for the
+electronic wave functions [41] and by the velocity Verlet
+algorithm for the classical coordinates of Na+ions and
+MgOorArconstituents.
+2.4 Gathering angular distributions Electrons
+emittedfromtheclusterwilleventuallyarriveatthebound -
+ariesofthe box.In orderto suppressre-feedof these elec-
+trons back into the box, we employ absorbing boundary
+conditions[30,42]. This is achieved by the mask functionM(r)definedas:
+M(r) =
+
+1 for|r|<Rcut/bracketleftbigg
+sin/parenleftbiggRbox−|r|
+Rbox−Rcutπ
+2/parenrightbigg/bracketrightbigg1/4
+forRcut≤ |r| ≤Rbox
+0 for|r|>Rbox
+(2a)
+whereRcutis the cut-off radius outside which absorption
+starts andRboxis the minimal radial distance from the
+origin to the closest point on the boundaries. The Kohn-
+Sham time step actually performed with the time-splitting
+method[41]isthusaugmentedbyanabsorbingstepas
+˜ϕ=ˆUTVϕ(t)→ϕ(t+δt) =M(r)˜ϕ(2b)
+whereˆUTVis the unitary propagationoperator. Applying
+themaskfunction Mtotheorbitalsgentlyremovesdensity
+approaching the box boundary and prevents it from being
+reflected.
+To compute the angular distribution of emitted elec-
+trons, the absorbed density is accumulated for each state
+andeach(absorbing)gridpointas
+Γi(r) =/integraldisplay∞
+0dt/vextendsingle/vextendsingle/vextendsingle[1−M(r)]ˆUTVϕi(t)/vextendsingle/vextendsingle/vextendsingle2
+.(3a)
+By definition of M, the fieldΓ(r)is non-vanishing only
+inthesphericalabsorbingzone.Theangulardistributiono f
+emittedelectronsisfinallygatheredbydividingtheabsorb -
+ing zone into radial segments Aν, and integrating Γi(r)
+overthose segments.The PhotoelectronAngularDistribu-
+tion(PAD) thenbecomes
+dNi
+esc(θ,φ)
+dΩ∝1
+||Aν(θ,φ)||/integraldisplay
+Aνd3rΓi(r),(3b)
+dNesc(θ,φ)
+dΩ=Nel/summationdisplay
+i=1dNi
+esc(θ,φ)
+dΩ(3c)
+where||Aν(θ,φ)||denotes the area of the segment Aν
+on the surface of a unit sphere. Eq. (3c) provides the total
+cross-section,whileEq.(3b)thecross-sectionforemissi on
+froma specificquantumstate ϕi.
+We will sometimespresentsimpler polardistributions,
+averagingthefulldistributionsd Nesc(θ,φ)/dΩoverφand
+dividingby the polar volume element sinθ(see Figure 1).
+This is convenient when the variation in φis weak. How-
+ever, even if the azimuthal distribution may carry interest -
+ing details, particularly for deposited clusters, we will s ee
+that thereducedviewcanbringsomeusefulinformation.
+It is to be noted that we consider here PAD which are
+integrated over all outgoing electron momenta. The state-
+specific PAD carry automaticallysome informationon the
+electron spectra because the dominant emission strength
+goes into the first (multi-)photon peak above continuum
+threshold.Moredetailedinformationcanbe gatheredwith
+techniques as proposed in [22, 43]. They also allow to
+Copyrightlinewillbe provided by the publisher4 Matthias B ¨ar et al.:Angular distributions of electrons from Na 8
+produce a double differential cross-section providing the
+photo-electron spectra in each angular bin separately. An
+explanationof this techniqueand the much more complex
+analysisoftheemergingdataispostponedtoa laterpubli-
+cation.
+2.5 Structure of the test cases As test cases, we
+will consider the free Na 8cluster and Na 8deposited on
+MgO(001)orAr(001)surfaces.Startingpointofthe laser-
+induced dynamics is a well relaxed structure of each sys-
+tem. These structures had been discussed extensively in
+[20].Figure1illustratesthestructureofNa 8onMgO(001).
+xyθ
+φz
+Figure 1 ThestructureofNa 8onMgO(001).Na ionsare
+indicated by black spheres, O ions by white ones, and Mg
+ions by gray ones. The structure of Na 8on Ar(001) is es-
+sentially similar. The bond distance in the two four-fold
+rings of Na 8is 6.2a0and the distance between the two
+ringsis5.8a0.Theequilibriumdistancebetweenthelower
+clusterplaneandthefirst surfacelayeris 5 a0.
+The symmetry axis of Na 8which coincides for the de-
+posited Na 8with the axis perpendicular to the surface is
+taken asz-axis. The MgO surface is a cut from the cubic
+crystal structure. On the surface, the O and Mg ions are
+arranged in squares where next neighbors are always the
+otherspecies.
+The Na 8cluster has a highly symmetric configuration
+out of two rings of four ions each, twisted relative to each
+otherby 45◦to minimize the Coulombenergy.The Na 8is
+ratherrigidandonlyverylittleaffectedbythesurface.Fr ee
+Na8is very similar to the deposited cluster shown here,
+with bond lengths differing by less than 3%. The same
+holdsforNa 8onAr(001).TheelectroniccloudoffreeNa 8
+exhibitsclosetosphericalshapebecause Nel= 8electrons
+correspondstoastrongelectronicshellclosureforNaclus -
+ters [44]. That changes for the deposited cluster. The sur-
+facedestroysthereflectionsymmetrywhich,inturn,mixes
+single-electron states of different z-parities. The presence
+ofthe surfacedoesalso shift the single-electronlevelsan d
+theionizationpotentials(IP).Therelationsaresketched in
+figure 2. The IP is 4.3eV for free Na 8,3.8eV for Na 8oninto Ar into MgOtransmission single−particle
+energy [eV]
+1pz
+1pxy
+1s
+free deposited
+on MgO on Ar0
+−2
+−4
+−62
+vacuum threshold
+4.33.85.6
+4.25.4
+Figure2 Thesingle-electronlevelsoffreeNa 8andofNa 8
+depositedonMgO(001)orAr(001).Thedegeneracyofthe
+1pxand1pylevels in free Na 8is indicated by a heavier
+line. The vacuum threshold is at zero energy. The IPs are
+indicated by vertical lines with open arrowsand the trans-
+mission barriers by vertical lines with filles arrows. The
+numbersbeneatharetheIPsorbarriersin unitsofeV.
+MgO(001) and 4.2eV on Ar(001). The transmission bar-
+rierforemissionfromdepositedNa 8intothesubstratelies
+at+5.6eV for MgO(001) and +5.4eV for Ar(001): both
+are substantially larger than the IP. The 1pxand1pylev-
+elsoffreeNa 8are perfectlydegenerate,slightlysplit from
+1pzdue to a very small quadrupole deformation of Na 8.
+The ionic structure of Na 8deposited on MgO(001)hardly
+changes because the magic electron configuration renders
+that cluster very rigid. There is an overall up-shift due to
+core repulsion from MgO. The symmetry breaking due to
+the surface orientation slightly splits the 1px,ylevels, by
+0.204eV.MuchlessshiftsareseenforNa 8onAr(001)and
+the splitting between 1pxand1py, 0.05 eV, is also much
+smaller.Thatdifferencecorrelatestothemuchlowerinter -
+face energyof the Na-Ar system. Indeed,the bottomlayer
+of Na8is within 5.1 a0from the MgO(001)surface, much
+closerto thesubstratethanin thecase ofAr(001)(6.4 a0).
+The optical absorption spectrum of our test cases is
+showninfigure3.OnerecognizesthepronouncedMiesur-
+faceplasmonresonance[1,3,18].ForfreeNa 8,thereisone
+cleanpeakat 2.45eValmost thesameforeachmode.The
+surfacesleadtosomespectralfragmentation,about1.4eV
+forMgO(001)and0.2eVforAr(001).Thehighlypolariz-
+ableMgO(001)alsoinducesasmalldown-shiftofthereso-
+nancecenterto2.31eVwhilecorerepulsionoutweighspo-
+larizationforAr(001)leadingtoasmallup-shiftto2.54eV .
+The MgO(001) surface leads to a strong fragmentation of
+thezmodeessentiallyduetosymmetrybreaking[20].
+In the following studies, we will vary laser frequency
+ωlasand polarization. We work in all cases with the
+same pulse length Tpulse= 60fs, which corresponds to
+FWHM= 20fs. The angular distributions in the high-
+intensity regime are always concentrated on forward
+and backward emission along the laser polarization axis,
+Copyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 5
+ 2.1  2.2  2.3  2.4  2.5  2.6  2.7  2.8  2.9
+frequency [eV]Na8Ar(001)optical absorption strength  [arb.units]Na8MgO(001)Na8 freex mode
+y mode
+z mode
+Figure 3 The optical absorption spectrum (dipole re-
+sponse) of free Na 8and of Na 8deposited on MgO(001)
+or Ar(001). The spectra from the modes in the three ba-
+sicdirectionsareshownseparatelyasindicated.The zaxis
+stands for the direction perpendicular to the surface and
+alongthesymmetryaxisofNa 8.
+whereas a high sensitivity to all details of the excitation
+is found in the regime of weak perturbations [23]. The
+present study thus aims at staying close to the perturba-
+tiveregimeto exploretherichvarietyofdistributions.Th e
+intensity should be low enough for a perturbative regime
+beingvalid[22,26,45],butalsosufficientlyhightoprovid e
+signals safely above numerical noise. We found an inten-
+sity ofI= 109W/cm2to be a good compromise and we
+usedthat value in most of the test cases. Use ofa different
+valuewill beclearlyindicated.
+3 Brief survey of free Na 8Detailed studies of the
+PAD of free clusters will be discussed in a forthcoming
+publication.We brieflysummarizeheretheresults.
+Variation of the laser intensity shows almost constant
+pattern of the PAD throughoutthe regime of moderate in-
+tensities. For further increasing intensities, the struct ures
+changesteadily towardssimple forward-backwardscatter-
+ing. This is related to the transition from the frequency-
+dominatedregimeofmoderateintensitiestothefielddom-
+inatedhighlynon-linearregime[27,30,46,47].
+Variationoffrequencyleadstodramaticchangesinthe
+angulardistributions.Theyvaryamongstthreetypicalpat -
+terns. We have chosen three frequencies to have an ex-
+ample for each type. Results are shown in figure 4. The
+most common case is forward scattering (along the axis
+of polarization, θ= 0,π), seen here in the lowest panel 0 0.02 0.04 0.06 0.08 0.1
+φθω=2.5 eV
+360o270o180o90o 0180o
+135o
+90o
+45o
+ 0 0 0.005 0.01 0.015 0.02 0.025 0.03θω=3.0 eV
+180o
+135o
+90o
+45o
+ 0 0 0.005 0.01 0.015 0.02 0.025 0.03θω=5.4 eV
+180o
+135o
+90o
+45o
+ 0
+Figure 4 Gray scale plot of angular distributions for free
+Na8excited with three different laser frequencies,as indi-
+cated.Thegrayscale isused inarbitraryunits.High emis-
+sion is signified by white and no emission correlates to
+deep black. The frequencies had been selected to display
+thethreedifferentpatternswhichcouldbefound.Thelaser
+ispolarizedalongthe z-axis,thesymmetryaxisofNa 8.
+forωlas= 2.5eV. Sometimes one encounters “diagonal
+scattering” where electron emission is concentrated on a
+doubleconearoundangle π/4(withrespectofthe z-axis),
+see upper panel for ωlas= 5.4eV. Sideward scattering as
+exemplified in the middle panel, is observed occasionally,
+here forωlas= 3.0eV. Note that the spectral relations of
+these three cases are very different. For 5.4 eV, we have
+a one-photon process for the 1pstates but a (much sup-
+pressed) two-photon process from the 1sstate. For 3 eV,
+we have a two-photon process for both, the 1pand the1s
+shell.Andfor2.5eV,wehaveatwo-photonemissionfrom
+the1pwhilethe 1sshell,again,requiresonemorephoton.
+The strong frequency dependence of the PADs can be
+relatedtothefactthatthestrongestemittingsingle-elec tron
+state dependssensitively on the relationshipbetweenlase r
+frequency and IP, and that the emission from each state
+looks very differently. This is demonstrated by analyzing
+the PADs of individual single-electron states. Experimen-
+Copyrightlinewillbe provided by the publisher6 Matthias B ¨ar et al.:Angular distributions of electrons from Na 8
+tally, this is achieved by measuring the PAD simultane-
+ously with the photoelectron spectrum (PES) [8,10,48].
+The energy selection by PES allows one to associate the
+PAD to the single-electron states from which these were
+produced. Theoretical calculations have the separate in-
+formation trivially at hand, as seen in Eq. (3c). Figure 5
+ 0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05
+φθ
+1s
+360o270o180 90o 00o
+45o
+90o
+135o
+180o 0 3e-06 6e-06 9e-06 1.2e-05θ
+1px0o
+45o
+90o
+135o
+180o 0 3e-06 6e-06 9e-06 1.2e-05θ
+1py0o
+45o
+90o
+135o
+180o 0 2e-05 4e-05 6e-05 8e-05θ
+1pz0o
+45o
+90o
+135o
+180o
+Figure 5 State selective angular distribution
+dNesc,i(θ,φ)/dΩof electrons which were emitted
+from single-electron state ϕiof free Na 8as indicated,
+given in arbitrary units. The laser was polarized along the
+zaxisandhada frequencyof ωlas= 2.5eV.
+showsthestate-specificPADoffreeNa 8forthe(hypothet-
+ical) case that the clusters symmetry axis is aligned with
+the laser polarization.The separate distributions are muc h
+more structured and show their maxima at different an-
+gles. Particularly noteworthy are the pattern from the 1px
+and1pystates. Both have their emission maxima at po-
+lar angleθ= 45◦and135◦and both show pronounced
+structures in azimuthal angle φ; the azimuthal pattern is
+shifted by 90◦between 1pxand1py. That feature will
+play a role again for deposited Na 8, see section 4. The
+maxima atθ= 45◦and135◦can be qualitatively under-
+stoodinapictureusingwavefunctionsofasphericalmean
+field. The 1pxyhave then the angular wave function interms of the spherical harmonics Y±1
+1. The dipole excita-
+tioncomeswithangulardistribution Y0
+1.For1px,theemit-
+ted wave is driven by the product |(Y+1
+1+Y−1
+1)Y0
+1|2∝
+sin2(2θ)sin2φ,havingmaximaat φ= 0◦and90◦,andfor
+1pyby|(Y+1
+1−Y−1
+1)Y0
+1|2∝sin2(2θ)cos2φwith max-
+imaatφ= 45◦and135◦,whilebothwavesaremaximized
+atθ= 45◦and135◦.Thispictureagreeswiththeobtained
+patternsinfigure5.Notethatthetotalcross-sectionoffre e
+Na8doesnot exhibitanystructurein φasthe1pxand1py
+distributions add up to a constant value in φ(besides a
+faint perturbationby the ions). A similar finding holds for
+the energy-resolveddistributionsasthe states 1pxand1py
+have exactly the same single-electron energy and thus are
+added up with equal weight. We also separately explored
+thefrequencydependenceofthedistributionsforemission
+fromeachsingle-electronstate. Theyturnedouttodepend
+muchless on the frequencythanthe total PAD. We refrain
+frompresentingtheseresultsindetailasthiswouldrequir e
+quitea bitspace.
+ 0 0.2 0.4 0.6 0.8
+ 0  30  60  90  120  150  180emission probability [arb.u.]
+polar angle θ1s
+1pxy1pz
+Figure 6 Angular distributions, averaged over orienta-
+tions, along polar angle θfor emission from the different
+single-electron states of free Na 8. The laser frequency is
+ωlas= 5.44eV.
+Considering free clusters with known orientation al-
+lows one to investigate characteristic structural feature s
+and to provide basic information for deposited clusters to
+be discussed later on. An alignment of clusters in the gas
+phaseremainsanopenexperimentalproblem(“orientation
+burning”maybeanoption[49]).Present-daysexperiments
+inthegasphasemeasureanensembleofclusterswhereall
+orientations are equally distributed. A simulation of that
+situation requires one to calculate an ensemble of orien-
+tations and to average the emerging angular distributions.
+The procedure is straightforward. We computed angular
+distributions on a grid of orientationswith grid spacing of
+22.5◦in Euler angles θandφ, ignoringψdue to the near
+axial symmetry of Na 8, and added them with appropriate
+geometrical weights. The orientation averaging wipes out
+thesub-structuresalong φdirection.Figure6showsthere-
+sulting (orientationaveraged)distributionsalongpolar an-
+gleθ. Note that we have also averaged over 1pxand1py
+Copyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 7
+ 0 0.0002 0.0004 0.0006
+φθ
+ω=2.6 eV
+360o270o180 90o 0180o135o90o45o 0 0 0.0005 0.001θ
+ω=3.7 eV0o
+45o
+90o
+135o
+180o 0 0.001 0.002 0.003 0.004θ
+ω=4.2 eV0o
+45o
+90o
+135o
+180o
+Figure 7 Angular distributions for emission from Na 8
+on MgO(001) for three different laser frequencies as indi-
+cated.Thelaserispolarizedalongthe z-axis,perpendicular
+tothesurface.
+distributionssincethesetwostatesareenergeticallydeg en-
+erate and could not be discriminatedby PES. The orienta-
+tionaveragingyieldsmuchdifferentandsimplerstructure s
+than those seen in the case of known orientation, see fig-
+ure 5. The dominance of purely forward-backward struc-
+ture holds only for the small system Na 8. Larger clusters
+showricherstructures[10,50,51].
+4 Emissionpatterns for depositedNa 8
+4.1 Deposition onMgO(001) Aspointedoutin sec-
+tion 3, the detailed analysis of free PAD leads to remark-
+able insights into the structures of clusters. However, the
+orientation problem inherent to clusters in the gas phase
+blurssomeofthedetailedinformation.Itisthusinteresti ng
+to consider the complementing case of deposited clusters
+where the orientation is well defined. On the other hand,
+the presenceof the substrate complicatesthe PAD. Analy-
+sisofthiseffectisthusakeyissue.
+Figure 7 displays angular distributions of Na 8on
+MgO(001) for three different frequencies. Again, the fre-
+quencies were selected to show three different and typical
+emission patterns.The sequenceis comparableto the case
+of free Na 8discussed above, but having somewhat lower
+values to accommodatefor the lower IP (see figure 2) and
+stay below the transmission threshold. The substrate ob-viously has a very strong influence such that the patterns
+are quite different from those of free Na 8. The dominant
+forward emission ( θ= 0◦) is still observed; backward
+emission (θ= 180◦) is, of course, totally suppressed by
+the presenceoftheinsulatingsubstrate.Moresurprisingi s
+the appearance of a strong azimuthal dependence. Recall
+the pronounced azimuthal structures for emission from
+the1px,ylevels of free (aligned) clusters, see figure 5. In
+the total orenergy-resolvedcross-sectionsof free cluste rs,
+thesestructuresarewipedoutbythedegeneracy,aseachof
+thetwolevelscontributesequallytotheemissionstrength ,
+see the discussion of figure 5. This 1px–1pydegeneracy
+is now split by the surface leading to an energy difference
+between1pxand1pystate of 0.2 eV. Because these states
+areclosetotheemissionthreshold,thesmallenergydiffer -
+ence has a large effect on the relative emission strengths.
+One of the two states dominates emission and its pattern
+affectsthetotaldistribution.Comparisontofigure5makes
+it clear that the dominant state is 1pxforωlas= 2.6eV
+(figure 7, lower panel) and 1pyforωlas= 3.7eV (figure
+7, middle panel). There is also some effect on the po-
+lar distribution. The emission cones are neither perfectly
+alignedalong θ= 90◦,asinasidewardsemittingcase,nor
+alongθ= 45◦, as was the case for 1px,yorbitals of free
+Na8. The compositionof emission strengthsfrom the four
+single electron levels is different for the deposited case
+as compared to free Na 8. Furthermore, the polarization
+attraction bends the outwards cones towards the substrate.
+Both effects together widen the emission cone such that
+the angle of inclination of the emitted electronsrelative t o
+thesurfacedecreases.
+This example demonstrates that the detailed structure
+oftheangulardistribution,inthepolaraswellasintheaz-
+imuthalangles,sensitivelydependsonthelaserfrequency .
+There is much more structure compared to the emission
+spectraoffreeclusters,thusmoreinformation,worthtobe
+unraveled.Thatinformation,however,ismaskedbythein-
+volvedinterplayofcluster and surfacewhich inhibitssim-
+pleone-to-onecorrespondencesandrequiresacarefulcor-
+relationanalysis.
+4.2 Comparison with argon substrate We also in-
+vestigated the case for another combination, namely Na 8
+on Ar(001).Frozen Ar is an insulator as MgO is. We thus
+expected, as in the case of Na 8on MgO(001), heavily
+modifiedangulardistributionscomparedwiththefreeNa 8
+case.Aris,however,aVan-der-Waalsboundsystemwitha
+smallersurfacecorrugationandsmallerinitialpolarizat ion
+fields,incontrasttotheioniccrystalMgO.Figure8shows
+angular distributions for Na 8on Ar(001) at two frequen-
+cies of the laser field. The effects are very similar to the
+case of MgO. Backward scattering is totally suppressed,
+forward scattering is accordingly enhanced, there appear
+pronouncedazimuthal structures, and the distributionsde -
+pend sensitively on the frequency.With 0.14eV, the split-
+tingofthe1px,ylevelsduetoAr(001)surfaceissimilarto
+thatatMgO,whichisprobablyresponsibleforthesimilar-
+Copyrightlinewillbe provided by the publisher8 Matthias B ¨ar et al.:Angular distributions of electrons from Na 8
+ 0 0.001 0.002 0.003 0.004 0.005 0.006
+φθ
+ω=2.6 eV
+360o270o180 90o 0180o135o90o45o 0 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012θ
+ω=4.3 eV0o
+45o
+90o
+135o
+180o
+Figure 8 Angulardistributionsforemission fromNa 8on
+Ar(001)fortwolaserfrequenciesasindicated.Thelaseris
+polarizedalongthe z-axis,perpendiculartothe surface.
+ities of the emission patternsbetween Ar and MgO. There
+is, however, one difference to the case of MgO. The side-
+ward cone (lower panel of figure 8) is not so close to the
+surface and points towards a polar angle of θ= 45◦as in
+the case of free Na 8. That is probably due to the smaller
+surfacepolarizationforAr.
+4.3 Frequencyandintensitydependences Inthis
+section we explore more systematically the influence of
+laser frequency and intensity on the angular distributions
+for deposited Na 8. To simplify the graphical represen-
+tation, we consider φ-averaged distribution along polar
+angleθ.Figure9showsazimuthallyaverageddistributions
+for emission from Na 8on MgO(001). The upper panel
+presents the variation with ωlasfor moderate intensity.
+The prevailing effect is the strong suppression of back-
+wardemissionthroughthesubstrate.Inspite ofreflection-
+enhancedsurfacedominance,weseeagainthevariationof
+pattern with frequency qualitatively similar to the case of
+free Na 8(see figure 4). The pattern is, however, more in-
+volvedthanwhatwouldemergefromasimplerescattering
+ofthereflectedelectronstoangle θ−→180◦−θ.Thesur-
+face attraction shifts the former diagonal maximum from
+45◦towards the substrate and produces sizeable emission
+under rather flat angles θ≈90◦. A first glimpse of trans-
+mission into the substrate can be seen for ωlas= 5.44eV.
+Thatis veryclose tothetransmissionthresholdforNa 8on
+Mg(001)atabout5.58eV, see figure2.
+The lower panel of figure 9 shows the changes with
+increasing intensity for fixed frequency. We see again the
+expected typical increasing focus towards forward emis-
+sion. A somewhat surprisingand most interesting effect is
+that backward emission gathers rather large strength with
+further increasing intensity. The high field strengths make 0 0.05 0.1 0.15 0.2 0.25 0.3
+ 0  45  90  135  180dNesc/dθ/Nesc  [arb. u.]
+polar angle θ  [degrees]108 Wcm-2
+1010 Wcm-2
+1012 Wcm-2
+2.5 x 1013 Wcm-2
+6.2 x 1013 Wcm-2 0 0.1 0.2 0.3 0.4 0.5dNesc/dθ/Nesc  [arb. u.]ω=2.18 eV
+ω=2.45 eV
+ω=2.58 eV
+ω=3.00 eV
+ω=4.08 eV
+ω=5.44 eV
+Figure 9 Top : Azimuthally averaged angular distribu-
+tion (arbitrary units) of electrons emitted from Na 8on
+MgO(001)irradiatedbylaserpulseswithintensity I= 109
+W/cm2andvaryingfrequency.Thelaserispolarizedalong
+thez-axis,perpendiculartothesurface.Bottom:Distribu-
+tion for fixedfrequency ωlas= 4.76eVwith varyinglaser
+intensity.
+two-andmore-photonprocessescompetitivewhich,inturn
+overrules the frequency counting. However, one has to be
+cautious with interpreting these processes where substan-
+tialamountsofelectronchargearepenetratingintothesub -
+strate. The present QM/MM model is not set up for that
+situation and likely is no longer accurate at a quantitative
+level.
+Figure 10 shows some azimuthally averaged distribu-
+tionsforNa 8onAr(001).Thegeneraltrendsareverysim-
+ilartothecaseofMgO,seeforexampletheupperpanelof
+figure9.Weseeagainthestrongforwarddominancedueto
+reflection from the substrate and the strong dependenceof
+the distribution on laser frequencyas was observedbefore
+forallcases.Thereisaminordifferenceinthatthechances
+fortransmissionintothesubstrateareevenslightlylower .
+5 Conclusion We have explored from a theoretical
+perspectiveangulardistributionsofelectronsemittedfr om
+Na8clusters, free and deposited on Ar(001) or MgO(001)
+surfaces.Therebyweconcentratedondirectelectronemis-
+sion which takes place within few fs after laser excitation
+andwhichisthedominantprocessforshortlaserpulses(up
+toabout50fs).Severallaserfrequencieswereused,higher
+ones above the threshold for one-photon emission, lower
+frequencies in the two-photon regime close to the Mie
+Copyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 9
+ 0 0.1 0.2 0.3 0.4 0.5 0.6
+ 0  45  90  135  180dNesc/dθ/Nesc
+polar angle θ  [degree]ω=2.6 eV
+ω=3.1 eV
+ω=4.2 eV
+Figure10 Azimuthallyaveragedangulardistributions(ar-
+bitrary units) for Na 8on Ar(001) for intensity I= 109
+W/cm2and three different laser frequencies, as indicated.
+Thelaserispolarizedalongthe z-axis,perpendiculartothe
+surface.
+plasmon resonance, and cases safely off-resonance. The
+laser intensity was in most cases I= 109W/cm2, large
+enoughto overcomenumericalnoise, but safelybelow the
+regime of violent and highly non-linear processes. The
+numerical simulations employed a hierarchical quantum-
+mechanical-molecular-mechanical (QM/MM) model ap-
+proach. The cluster electrons were described by time-de-
+pendentdensity-functionaltheory(TDDFT)at the levelof
+the local-density approximation(LDA). A self-interactio n
+correctionhas been added to put the single-electronlevels
+totheappropriateenergyrelativetotheparticlecontinuu m.
+The cluster ions were propagated by classical molecular
+dynamics. Similarly, the Ar(001) or MgO(001) substrates
+were treated as classical systems with atomic positions
+anddipolepolarizabilityasdynamicaldegreesoffreedom.
+TheTDDFTequationsweresolvedonathree-dimensional
+coordinate-space grid without any symmetry restriction.
+Absorbingboundarieswere appliedand the angulardistri-
+butionswereobtainedbyrecordingtheelectronabsorption
+ateachgridpointinthe absorbingzone.
+We first briefly explored free Na 8. The state-specific
+distributions for Na 8aligned with laser polarization show
+pronounced patterns in both angles, θandφ, which ex-
+hibit clear footprints of the emitting state. Simulating th e
+experimental situation which deals with orientation aver-
+aged ensembles renders the distributions independent of
+the azimuthal angle φand reduces structures in the polar
+anglesθ. A study of averaging effects, size-, frequency-,
+andintensity-dependenceforfreeclusterswillfollowina n
+upcomingpublication.
+The present study focused on angular distributions
+from Na 8deposited on MgO(001) and Ar(001). The at-
+tachment to the surface provides a well defined cluster
+orientation and it adds substantial perturbations from the
+surface interaction. The large band gaps of both materials
+manifested in high transmission barriers, almost totally
+suppress backward emission, focusing electrons into aforward cone. On the other hand, the long-rangepolariza-
+tion attraction can bend sideward flowing electrons down
+towards the surface, thus widening the opening angle of
+the emission cone. Polarization is strong for MgO(001)
+which induces a sizeable trend to emission parallel to the
+surface.RepulsiondominatesforAr(001)whichstabilizes
+the forward cone. The surfaces do also break the four-
+fold symmetry ( C4) of Na 8which, in turn, removes the
+degeneracy of the two single-electron levels with nodes
+orthogonal to the symmetry axis, and thus inhibits the
+compensation of φdependence which was observed for
+(oriented) free Na 8. Pronounced azimuthal structures are
+foundforthedistributionsofthedepositedcases.
+Altogether, the computational study revealed interest-
+ing structures which are, however, somewhat complicated
+bysurface(particularlytheelectronicandgeometricstru c-
+ture of the clusters) containedin the calculated photoelec -
+tronangulardistributionsandtheirenergydependence.
+Acknowledgements This work was supported by the
+Deutsche Forschungsgemeinschaft (RE 322/10, RO 293/27),
+Fonds der Chemischen Industrie (Germany), a Bessel-Humbol dt
+prize, and aGay-Lussac prize.
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+arXiv:1001.0687v1  [astro-ph.HE]  5 Jan 2010To appear in ApJ Letters
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+RADIATIVE SIGNATURES OF RELATIVISTIC SHOCKS
+John G. Kirk and Brian Reville
+Max-Planck-Institut f¨ ur Kernphysik, Postfach 10 39 80, 69 029 Heidelberg, Germany
+To appear in ApJ Letters
+ABSTRACT
+Particle-in-cell simulations of relativistic, weakly magnetized collisionle ss shocks show that particles
+can gain energy by repeatedly crossing the shock front. This requ ires scattering off self-generated
+small length-scale magnetic fluctuations. The radiative signature o f this first-order Fermi acceleration
+mechanism is important for models of both the prompt and afterglow emission in gamma-ray bursts
+and depends on the strength parameter a=λe|δB|/mc2of the fluctuations ( λis the length-scale
+and|δB|the magnitude of the fluctuations.) For electrons (and positrons) , acceleration saturates
+when the radiative losses produced by the scattering cannot be co mpensated by the energy gained
+on crossing the shock. We show that this sets an upper limit on both t he electron Lorentz factor:
+γ <106/parenleftbig
+n/1cm−3/parenrightbig−1/6¯γ1/6and on the energy of the photons radiated during the scattering p rocess:
+/planckover2pi1ωmax<40Max(a,1)/parenleftbig
+n/1cm−3/parenrightbig1/6¯γ−1/6eV, where nis the number density of the plasma and ¯ γ
+the thermal Lorentz factor of the downstream plasma, provided a < a crit∼106. This rules out
+‘jitter’ radiation on self-excited fluctuations with a <1 as a source of gamma-rays, although high-
+energy photons might still be produced when the jitter photons ar e upscattered in an analog of the
+synchrotron self-Compton process. In fluctuations with a >1, radiation is generated by the standard
+synchrotron mechanism, and the maximum photon energy rises linea rly with a, until saturating at
+70MeV, when a=acrit.
+Subject headings: radiation mechanisms: non-thermal — acceleration of particles — ga mma rays:
+bursts
+1.INTRODUCTION
+In astrophysics, the most widely discussed mechanism
+of particle acceleration is the first-order Fermi process
+operating at collisionless shocks. It is based on the
+idea that particles undergo stochastic elastic scatterings
+both upstream and downstream of the shock front. This
+causes particles to wander across the shock repeatedly.
+On each crossing, they receive an energy boost as a re-
+sult of the relative motion of the upstream and down-
+stream plasmas. At non-relativistic shocks, scattering
+causes particles to diffuse in space, and the mechanism,
+termed ‘diffusive shock acceleration’, is widely thought
+to be responsible for the acceleration of cosmic rays in
+supernova remnants. At relativistic shocks, the trans-
+port process is not spatial diffusion, but the first-order
+Fermi mechanism operates nevertheless (for reviews see
+Kirk & Duffy 1999; Hillas 2005). In fact, the first ab ini-
+tiodemonstrations of this process using particle-in-cell
+(PIC) simulations have recently been presented for the
+relativistic case (Spitkovsky 2008b; Martins et al. 2009;
+Sironi & Spitkovsky 2009).
+Several factors, such as the lifetime of the shock front,
+or its spatial extent, can limit the energy to which par-
+ticles can be accelerated in this process. However, even
+in the absence of these, acceleration will ultimately cease
+when the radiative energy losses that are inevitably asso-
+ciated with the scattering process overwhelm the energy
+gains obtained upon crossing the shock. Exactly when
+this happensdepends on the detailsofthe scatteringpro-
+cess.
+In the non-relativistic case, the diffusion coefficient
+john.kirk@mpi-hd.mpg.deis frequently parameterized in terms of its lower limit,
+knownasBohmdiffusion, atwhichtheparticlemeanfree
+path equals the gyro radius. In this case, particles that
+achieve the highest possible energy radiate synchrotron
+photons up to an energy ∼150η(vs/c)2MeV, where vs
+is the speed of the shock front and η(≤1) is the inverse
+ratio of the diffusion coefficient to its value in the Bohm
+limit. Interestingly, this photon energy is independent of
+the magnetic field strength.
+In PIC simulations of weakly magnetized shocks, how-
+ever, particles appear to undergo small-angle scatterings
+on fluctuations in the electromagnetic fields that are
+driven by the Weibel instability. These have a length
+scale roughly equal to the inertial length c/ωp, where
+ωp=/radicalbig
+4πne2/¯γmis the relativistic electron plasma fre-
+quency,and ¯ γistheLorentzfactoroftheaveragethermal
+motion. As we show below (see Eq. (6)) the resulting
+mean free path ofa particle of Lorentzfactor γis propor-
+tional to γ2, rather than the linear dependence expected
+in Bohm diffusion. This modifies the maximum energy
+of the synchrotron photons (Derishev 2007), but the ra-
+diation emitted when a particle is deflected through very
+small angles differs significantly from synchrotron radi-
+ation (Landau & Lifshitz 1975), and can, in principle,
+produce more energetic photons. This has led to the
+suggestion that “jitter” radiation from shock-accelerated
+particles is responsible for both the prompt emission
+(Medvedev 2006) and the afterglow (Medvedev et al.
+2007) from gamma-ray bursts. In this Letterwe show
+that the inherent weakness of the scattering produced
+by Weibel-driven turbulence implies that radiationlosses
+quench first-order Fermi acceleration relatively quickly.2
+In this case, the maximumphotonenergyisgivenby(16)
+and particles are unable to produce X-ray or gamma-ray
+photons in the downstream rest frame.
+2.RELATIVISTIC SHOCK FRONTS
+Recent particle-in-cellsimulationsofrelativisticshocks
+have revealed fundamental differences between the mag-
+netized and unmagnetized cases. The most detailed re-
+sults are available for electron-positron pair plasmas,
+which are likely to be found in gamma-ray bursts and
+pulsar winds. The upstream and downstream plasmas
+can be characterized by magnetic field strengths Bu,d
+andnumberdensities nu,dmeasuredintherespectiverest
+frames. The upstream plasma is cold, and streams into
+the shock with a Lorentz factor ¯ γ, roughly equal to the
+thermal Lorentz factor of the downstream plasma. The
+magnetizationparametersineachregioncanbe written:
+σu=B2
+u/(4πnumc2)
+σd=B2
+d/(4π¯γndmc2)(1)
+At a parallel shock, Bu=Bdandσd≪σu. If the mag-
+netic field is compressed at an oblique shock front, one
+expectsσu∼σd, sinceBd/Bu∼nd/nu∼3¯γ. However,
+if magnetic field is generated, then σd≫σu. It is found
+(Sironi & Spitkovsky 2009) that ‘unmagnetized’ shocks,
+withσu<10−3, are mediated by the Weibel instability.
+This is true also for shocks with an upstream magnetic
+field parallel to the shock normal, provided σu<0.1.
+On the other hand, more strongly magnetized or oblique
+shocks are mediated by the synchrotron maser instabil-
+ity that operates when particles streaming into the shock
+start to gyrate about the compressed downstream mag-
+neticfield. Ineachcase, it ispossibletoestimatealength
+scale on which turbulence is initially generated in the
+electromagnetic field. For Weibel-mediated (unmagne-
+tized) shocks, this is
+λw=ℓwc/ωp (2)
+withℓw∼10, according to Sironi & Spitkovsky (2009).
+Here,ωpis the local plasma frequency. Because nd/nu∼
+¯γ, the plasma frequencies in the upstream and down-
+stream media are roughly equal, and so the wavelength
+of the turbulence generated is approximately the same.
+For the purpose of estimating radiative signatures of ac-
+celerated particles, it is more convenient to characterize
+the fluctuations in terms of their ‘strength’, or ‘wiggler’
+parameter a, defined as the ratio of their length scale
+to the length defined by the turbulent field strength:
+a=λe|δB|/mc2. In the upstream and downstream me-
+dia (subscripts “u” and “d”) one has
+aw,u≈ℓwσ1/2
+u
+aw,d≈ℓw¯γσ1/2
+d(3)
+This suggests that for Weibel-mediated shocks in pair
+plasmas the strength parameters are likely to be small
+(a/lessorsimilar1), although moderate values ( a/greaterorsimilar1) are possible
+if the self-generated field grows locally to levels close to
+equipartition. Carrying out the same analysis for un-
+magnetized electron-ion shocks results in an additional
+factor of ( mi/me) in the value of the strength parame-
+ter, since the dominant length-scale in this case is the
+ion skin-depth (Spitkovsky 2008a). Provided σu,d≪1,
+moderate strength parameters can still be expected.For magnetized shocks mediated by the synchrotron
+maser instability (Lyubarsky 2006), the characteristic
+length in the downstream plasma λs,dis dictated by the
+requirement that the incoming particles be significantly
+deflected:
+λs,d=ℓs¯γmc2/eBd (4)
+withℓs∼1. This generates electromagnetic waves of
+strength parameter
+as,d=ℓs¯γ (5)
+These waves propagate into the upstream medium with-
+out change of their (Lorentz invariant) strength param-
+eter:as,u=as,d.
+To summarize, the strength parameters associated
+with shock-generated turbulence are likely to be small
+(a/lessorsimilar1) in the case of Weibel-mediated pair shocks, but
+moderate a∼¯γin the case of electron-ion shocks and
+magnetized shocks mediated by the synchrotron insta-
+bility.
+3.PARTICLE ACCELERATION
+Fermi acceleration at relativistic shocks differs from
+that at nonrelativistic shocks in that it is sensitive
+to the physics of the particle scattering process (e.g.,
+Kirk & Duffy 1999). Typically, it is assumed that
+the particles undergo stochastic small-angle deflections
+through interactions with magnetic irregularities. Ac-
+cording to the strength parameter of these irregularities,
+the transport of the particles can then be divided into
+two distinct regimes, which we call ‘ballistic’ and ‘heli-
+cal’. In ballistic transport, the scattering mean free path
+is shorter than the gyroradius in the local field, so that
+the curvature radius of a trajectory is larger than the
+length scale on which this quantity fluctuates. This kind
+of transport is produced by fluctuations with a < γ. On
+the other hand, in the helical transport regime, gyro mo-
+tion is only slightly perturbed. Particles have sufficient
+time to gyrate about the field while their pitch angles
+and guiding-center positions diffuse. This requires fluc-
+tuations with a > γ. This has a strong influence on
+the acceleration process and its interplay with radiation
+losses.
+3.1.Ballistic transport regime
+At a relativistic shock, a particle remains in the up-
+stream medium until it has been deflected, on average,
+through an angle of 1 /¯γin the upstream rest frame. A
+turbulent fluctuation of strength parameter a, deflects a
+particle of Lorentz factor γthrough an angle a/γ. Pro-
+vided this angle is small, the diffusion coefficient is sim-
+plyDθ=a2νsc/γ2, whereνscis the mean scattering fre-
+quency. The average number of scatterings in the up-
+stream medium between shock encounters is therefore
+Nscatt,u≈(γ/au¯γ)2(6)
+At each scattering, the power radiated in photons by
+the energetic particle can be estimated from Larmor’s
+formula:
+∆γ
+γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+loss,u≈2a2
+ue2γ
+3mc2λu(7)
+assuming inverse Compton losses can be neglected. For
+kinematic reasons, the average energy gain per cycle isRadiative signatures of relativistic shocks 3
+roughly a factor of two (Achterberg et al. 2001), so that
+the acceleration process will saturate when the energy
+lost in the upstream medium is roughly γmc2. This im-
+pliesNscatt,u∆γ/γ|loss,u<1, or
+γ</parenleftbigg3mc2λu¯γ2
+2e2/parenrightbigg1/3
+(8)
+Applying the same argument, energy loss by scatter-
+ing in the downstream medium, where particles must be
+scattered through an angle of roughly π/2, imposes the
+condition
+γ</parenleftbigg3mc2λd
+2e2/parenrightbigg1/3
+(9)
+which, since λd∼λu, is more restrictive.
+The ballistic regime requires au,d< γ, which, accord-
+ing to (3), is fulfilled for unmagnetized pair shocks for
+all particles with γ >¯γ, provided ℓw/lessorsimilar10, andσu,d<1.
+For electron-ion shocks, this condition is unlikely to be
+satisfied for electrons with γ∼¯γ, but is easily satisfied
+for electrons that achieve energies comparable to those
+of thermal ions or higher, provided σu,d≪1.
+3.2.Helical transport regime
+In the helical transport regime, energy losses are im-
+portant at all points along a trajectory, and, if Bohm
+diffusion operates, the time taken to return to the shock
+front is approximately the gyro period. Under these con-
+ditions, the maximum Lorentzfactor has been calculated
+by Achterberg et al. (2001):
+γ<3m2c3
+2e3B(10)
+At magnetized, relativistic shocks, particle accelera-
+tion by the first-order Fermi mechanism is less plau-
+sible, since, at least for superluminal shocks, it relies
+on strong cross-field diffusion (e.g., Baring & Summerlin
+2009). However,ifthe processdoesoperate, particlescan
+move out of the helical regime into the ballistic regime,
+as their Lorentz factor increases. Because of this, it is
+convenient to define a critical strength parameter acrit
+such that when a=acritthe maximum Lorentz factor
+γmaxpermitted by radiation losses is achieved just at
+the point at which the transport changes character from
+helical to ballistic. If a > a crit, all particles remain in
+the the helical regime. On the other hand, if a < acrit,
+particles of the maximum Lorentz factor undergo ballis-
+tic transport, but lower energy particles may be in the
+helical regime. Since the transition from helical to bal-
+listic occurs at γ=a, the critical strength parameter in
+the downstream region follows from (9):
+acrit=/parenleftbigg3mc2λd
+2e2/parenrightbigg1/3
+(11)
+Expressed in these terms, the maximum Lorentz factor
+in the helical transport regime (10) is γmax=/radicalbig
+a3
+crit/a.
+Typically, acrit≫1 in both unmagnetized and magne-
+tized shocks: inserting, for example, the length scale ap-
+propriate for Weibel-mediated shocks (2) one finds
+acrit=1.4×106ℓ1/3
+w¯γ1/6/parenleftbig
+n/1cm3/parenrightbig−1/6(12)In the helical regime, the first-order Fermi process has
+been observed, so far, only in PIC simulations of sublu-
+minal pair shocks (Sironi & Spitkovsky 2009). The fail-
+ure of the mechanism at superluminal shocks may be
+because the helical character of orbits prevents particles
+from recrossing the shock front by sweeping them away
+into the downstream medium (Begelman & Kirk 1990;
+Bednarz & Ostrowski 1996; Lemoine et al. 2006). How-
+ever, first-order Fermi is not the only possible acceler-
+ation mechanism at superluminal shocks. Strong elec-
+tromagnetic wave fields are generated at these shocks,
+and, at least in plasmas that contain some protons, these
+are thought to be responsible for particle acceleration
+(Amato & Arons 2006; Lyubarsky 2006). This can only
+happen if the nonthermal particles are confined within
+the source, i.e., if they are deflected through an angle
+∼1 before losing their energy to radiation. In this case,
+demanding that a particle be able to complete a gyra-
+tion before radiating its energy leads to the the same
+constraint as obtained for first order Fermi acceleration
+(10).
+Combiningtheconstraintsfromthe ballisticregime(9)
+and the helical regime (10) gives:
+γmax=
+
+acrit fora < acrit
+acrit/radicalbig
+acrit/afora > acrit(13)
+4.RADIATIVE SIGNATURES
+The spectrum of photons that are emitted when a par-
+ticle is scattered by turbulent fluctuations is, in the gen-
+eral case, quite complex (Toptygin & Fleishman 1987).
+However, to estimate the radiative signature of a particle
+accelerated at a relativistic shock front several simplifi-
+cations can be made.
+Consider the radiation emitted when a particle is scat-
+tered by a single fluctuation of length λand strength
+parameter a, and is unperturbed before and after in-
+teraction. The character of the emission depends cru-
+cially on the “formation” or “coherence” length of the
+radiated photons Akhiezer & Shul’ga (1987). At low fre-
+quencies, this length becomes large, but we confine our
+estimates to the highest frequency photons produced in
+the interaction. If a >1, the formation length is roughly
+λcoh≈mc2/eB. This is much smaller than the wave-
+length of the turbulence, so that the individual photons
+arecreatedinregionsinwhichthefieldisalmostconstant
+and homogeneous. The result is synchrotron radiation,
+with the emissivity defined by the local value of the field.
+The emission extends up to the roll-overfrequency of the
+highest energy electrons:
+ωmax≈0.5γ2
+maxeB/mc
+=0.5aγ2c/λfora >1 (14)
+These photons are radiated into a forward directed cone
+of opening angle 1 /γ. Well below this frequency, the
+emissivity is proportional to ω1/3, provided the coher-
+ence length remains shorter than the wavelength of the
+turbulence.
+If, on the other hand, a <1, the particle is deflected
+through an angle that is small compared to 1 /γ. In this
+case, the coherence length is no longer limited by de-
+flection, but is given by the distance moved by the par-4
+a= 1 a=acritlog (a)→log (ˆωmax,γmax)→
+Ballistic
+HelicalJitterSynchrotron
+γmax=acrit γmax∝a−1/2
+ˆωmax= 1/αfacritˆωmax= 1/αf
+1Fig. 1.— The maximum Lorentz factor γmaxof electrons and
+the maximum photon energy ˆ ωmax=/planckover2pi1ωmax/mc2they radiate
+when scattered by magnetic fluctuations of strength a(see Eqs. (3)
+and (5)) at a relativistic shock. The critical strength para meter
+acritis defined in (12), αfis the fine-structure constant. The jit-
+ter/synchrotron regimes are separated by the vertical a= 1 line;
+the ballistic/helical transport regimes by the a=acritline. See the
+electronic edition of the Journal for a color version of this figure.
+ticle in the lab. frame during the time it takes for the
+photon to move one wavelength ahead of the particle:
+λcoh≈γ2c/ω. The spectrum of radiated photons rolls
+over when λcoh≈λ, and is flat ( ∝ω0) towards lower fre-
+quency, as in the case of bremsstrahlung. The maximum
+frequency is roughly
+ωmax≈0.5γ2
+maxc/λfora <1 (15)
+In each case, most of the power radiated by an indi-
+vidual electron emerges within a decade of the roll-over
+frequency that corresponds to its Lorentz factor. There-
+fore, to estimate the spectrum radiated by a power-law
+distribution of electrons, with differential number den-
+sity dn/dγ∝γ−p, one can simply make a one-to-one
+correspondence between radiated frequency and Lorentz
+factor:ω= Max(a,1)γ2c/λ. Larmor’s formula, which
+holds for all values of a, states that the radiated power
+is proportional to the Lorentz factor squared, so that
+the power radiated by d nelectrons of Lorentz factor γ
+is dL∝γ2dn. A simple change of variables then shows
+that in both the synchrotron and jitter cases, the spec-
+trum at frequencies below the roll-over frequency of the
+highest energy electrons is d L/dω∝ω−(p−1)/2.
+Combining the limit on the Lorentz factor (9) with the
+expressions for the roll-over frequency (14) and (15), one
+finds for the maximum frequency that can be radiated
+by particles accelerated at a relativistic shock front:
+/planckover2pi1ωmax
+mc2=
+
+(αfacrit)−1a <1
+a(αfacrit)−11< a < a crit
+α−1
+fa > acrit(16)
+whereαf=e2//planckover2pi1cis the fine structure constant. There
+are two transport regimes: that in which the unper-
+turbed motion of the highest energy particles is ballistic
+(a < γmax) and that in which it is helical ( a > γmax) andtwo radiation regimes: those of jitter ( a <1) and syn-
+chrotron ( a >1) radiation. This is illustrated in Fig. 1.
+5.DISCUSSION
+Radiation emitted by relativistic electrons scattering
+in the small-scale turbulent magnetic fields generated at
+Weibel-mediated relativisticshockshasbeen proposedas
+the mechanism responsible for both the prompt and af-
+terglow emission of gamma-ray bursts (Medvedev 2006;
+Medvedev et al. 2007). The evidence in favour of this
+suggestion is based on modelling the observed spec-
+tra assuming an electron distribution of power-law type
+with arbitrary high and low energy cut-offs. Power-
+law distributions are expected on theoretical grounds
+(Achterberg et al.2001; Kirk et al.2000), andareindeed
+observed in simulations of weakly magnetized, relativis-
+tic shocks (Spitkovsky 2008b; Sironi & Spitkovsky 2009;
+Martins et al. 2009). However, the constraint on the
+maximum photon energy imposed by the above analy-
+sis (16) suggests that this picture is not self-consistent,
+because the scatterings are too weak to accelerate elec-
+trons to the required Lorentz factor. In order to radiate
+photonsofenergy ∼mc2inthe plasmarestframe, strong
+fluctuations of large length-scale with a∼αfacrit∼104
+are required.
+The above conclusion rests on the assumption that
+the same fluctuations are responsible for both the parti-
+cle transport and radiation. In terms of the strength
+parameter aand length scale λthat we use to char-
+acterize the fluctuations, the deflection angle scales as
+∆θ∝aand the radiation losses as ∆ γ∝a2/λ. If,
+therefore,thescatteringresponsibleforisotropizationoc-
+curs on fluctuations of comparable strength, but much
+larger length scale than those responsible for the radi-
+ation losses, the limit on the maximum photon energy
+(16) is relaxed. In principle, the fluctuations induced
+by the Weibel instability could be responsible for pho-
+ton production, provided longer wavelength fluctuations
+are present to provide the necessary isotropization and
+transport. The accelerated particles themselves appear
+to generate longer wavelength fluctuations downstream
+of the shock (Keshet et al. 2009), but this is a relatively
+small effect compared to that needed to significantly in-
+fluence the maximum photon energy.
+On the other hand, if, as simulations suggest, the
+Weibel-inducedfluctuationsareresponsibleforthetrans-
+port, the bulk of the radiation must be produced by in-
+teraction with fluctuations of much shorter wavelength.
+An obvious candidate is the soft photon field pro-
+duced by the interaction of thermal electrons with the
+Weibel-induced fluctuations — the ‘jitter’ analog of the
+synchrotron photons produced by relativistic thermal
+electrons (Baring & Braby 2004; Giannios & Spitkovsky
+2009). With these photons as targets, the radia-
+tion mechanism is analogous to the synchrotron self-
+Compton mechanism, which has been discussed in con-
+nection with the problem of rapidly decaying mag-
+netic fluctuations (Rossi & Rees 2003). While this
+model offers a simple explanation for the produc-
+tion of high energy photons, it does not easily ac-
+commodate those low-frequency BATSE burst spectra
+(Baring & Braby 2004), that violate the so-called syn-
+chrotron ‘line of death’ (Preece et al. 1998). These rare
+cases might be accounted for if the upscattering occursRadiative signatures of relativistic shocks 5
+deep in the Klein-Nishina regime (Derishev et al. 2001;
+Boˇ snjak et al. 2009), an explanation which is facilitated
+ifthetargetphotonsarisenotfromsynchrotronemission,
+but from relatively hard jitter radiation.This research was supported in part by the National
+Science Foundation under Grant No. PHY05-51164. BR
+gratefully acknowledges support from the Alexander von
+Humboldt foundation.
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\ No newline at end of file
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+arXiv:1001.0688v4  [math-ph]  22 Nov 2010Absence of Embedded Mass Shells: Cerenkov
+Radiation and Quantum Friction
+Wojciech De Roeck
+Institut f¨ ur Theoretische Physik Universit¨ at Heidelber g,
+Philosophenweg 19 D69120 Heidelberg, Germany
+(w.deroeck@thphys.uni-heidelberg.de)
+J¨ urg Fr¨ ohlich
+Institute of Theoretical Physics; ETH Z¨ urich;
+CH-8093 Z¨ urich, Switzerland
+(juerg@itp.phys.ethz.ch)
+Alessandro Pizzo
+Department of Mathematics, University of California Davis ;
+One Shields Avenue, Davis, California 95616, USA
+(pizzo@math.ucdavis.edu)
+February-8-2010
+Abstract
+We show that, in a model where a non-relativistic particle is cou-
+pled to a quantized relativistic scalar Bose field, the embed ded mass
+shell of the particle dissolves in the continuum when the int eraction is
+turned on, provided the coupling constant is sufficiently sma ll. More
+precisely, undertheassumptionthatthefibereigenvectors correspond-
+ingtotheputativemassshellaredifferentiableasfunctions ofthetotal
+momentum of the system, we show that a mass shell could exist o nly
+at a strictly positive distance from the unperturbed embedd ed mass
+shell near the boundary of the energy-momentum spectrum.DFP-Abs. Mass. Shell., February-8-2010 1
+I Introduction
+The model studied in this paper describes a system consisting of a no n-
+relativistic quantum particle coupled to a quantized relativistic field of scalar
+massless bosons through an interaction term linear in creation- and annihila-
+tion operators. The system is invariant under space translations. Therefore
+its total momentum is conserved. In states where the initial partic le momen-
+tum is larger than mc, wheremis the mass of the non-relativistic particle
+andcthe propagation speed of the bosonic modes, we expect that the p arti-
+cle will emit Cerenkov radiation, because its group velocity is larger th an the
+speed of thebosons. We arethus interested inthespectral regio n(E,/vectorP) with
+|/vectorP|>1; using units such that m=c= 1. HereE,/vectorPare the spectral vari-
+ables of the Hamiltonian and of the total momentum operator, resp ectively.
+In this region, we expect that a mass shell of the non-relativistic pa rticle
+does not exist. Put differently, we expect that the mass shell, which in the
+unperturbed system is described by the equation E=/vectorP2/2, disappears, as
+soon as the interaction is switched on. This would show that one-par ticle
+states of the non-relativistic particle are unstable for values of |/vectorP|larger than
+1.
+Our main result is as follows. We assume that, for |/vectorP|>1, a mass
+shell exists with the property that the corresponding fiber eigenv ectors are
+differentiable as functions of the total momentum of the system. T hen we
+show that, for sufficiently small values of the coupling constant, su ch a mass
+shell may exist only at a strictly positive distance ( >O(1)) from the un-
+perturbed mass shell in the energy-momentum spectrum. More pr ecisely,
+one-particle states might only exist in a region around the three-dim ensional
+surfaceE=|/vectorP| −1
+2, whose width tends to zero, as the coupling constant
+approaches 0. Our results are proven for models with a fixed ultrav iolet
+cutoff that turns off interactions with high-energy bosons, and un der the as-
+sumption of appropriate infrared regularity of the form factor th at models
+the interaction.
+In the literature, many results are concerned with the existence o f a mass
+shell for |/vectorP|<1, depending on the behavior of the coupling between the
+non-relativistic particle and the relativistic boson field in the infrared region.
+These results clarify and extend the notion of stable particle by pro viding
+a scattering picture for infraparticles , for which a mass shell does not exist
+(i.e., the single-particle states are not normalizable in the Hilbert spac e of
+pure states of the system); see [10], [11], [18], [19], [4], [6], [7], [3], [13], [16 ].
+Toourknowledge, forthespectralregionstudiedinthispaper,no rigorous
+results have yet appeared in the literature concerning the existen ce or non-
+existence of an embedded mass shell. However, in [8], for the model studiedDFP-Abs. Mass. Shell., February-8-2010 2
+in this paper, it is proven that the electron motion in the kinetic limit is
+described byaBoltzmannequationthatexhibits theslowdown ofthe particle
+by emitting Cerenkov radiation, as long as its velocity is greater than 1. This
+supports the thesis that there is no mass shell for |/vectorP|>1.
+We also stress that the conclusions of our paper leave open an inter esting
+question: Our analysis does not exclude the existence of single-par ticle states
+near the boundary of the energy-momentum spectrum (which, fo r|/vectorP|>1, is
+approximately linear in |/vectorP|). In this respect, we recall that the existence of
+the groundstate eigenvalue for the fiber Hamiltonians, in the region |/vectorP|>1,
+has been studied in [20] and [17] (see also [1], [2] for some related spec tral
+problems) but under some assumptions on the boson dispersion rela tion that
+change the physical phenomenon we are interested in. In fact, in t hese pa-
+pers, the bosons are massive and their energy dispersion relation is strictly
+subadditive (see [17]). In particular, in [17], it is proven that, for spa tial
+dimensiond= 3, the fiber Hamiltonian has no groundstate whenever the in-
+fimum of its spectrum equals the infimum of its essential spectrum. H owever,
+because of the assumptions above, this result does not apply to th e model
+studied in this paper.
+In the following, the spin of the electron is neglected, and the boson s are
+scalar.
+Acknowledgement We thank an anonymous referee who pointed out the
+Remark on page 41. At the time when this work was finished, W.D.R. was
+supported by the European Research Council and the Academy of Finland.
+A.P. is supported by NSF grant DMS-0905988.
+II Description of the model and result
+II.1 Hilbert space
+The Hilbert space of pure states of the system is given by
+H=L2(R3)⊗F, (II.1)
+whereFis the Fock space of scalar bosons,
+F:=∞/circleplusdisplay
+N=0F(N),F(0)=CΩ, (II.2)
+with Ω the vacuum vector, i.e., the state without any bosons, and th e state
+space,F(N), ofNbosons is given by
+F(N):=SNh⊗N, N≥1. (II.3)DFP-Abs. Mass. Shell., February-8-2010 3
+Here the Hilbert space, h, of state vectors of a single boson is given by
+h:=L2[R3], (II.4)
+andSNdenotes symmetrization. We introduce the usual creation- and an ni-
+hilation operators, a∗
+/vectorkanda/vectork, obeying the canonical commutation relations
+[a∗
+/vectork, a∗
+/vectork′] = [a/vectork, a/vectork′] = 0, (II.5)
+[a/vectork, a∗
+/vectork′] =δ(/vectork−/vectork′), (II.6)
+a/vectorkΩ = 0, (II.7)
+for all/vectork,/vectork′∈R3.
+II.2 Fiber decomposition
+We may write Has a direct integral
+H=/integraldisplay⊕
+H/vectorPd3P. (II.8)
+Given any /vectorP∈R3, there is an isomorphism, I/vectorP,
+I/vectorP:H/vectorP−→ Fb, (II.9)
+fromthefiberspace H/vectorPtotheFockspace Fb, acteduponbytheannihilation-
+and creation operators b/vectork,b∗
+/vectork, whereb/vectorkcorresponds to ei/vectork·/vector xa/vectork, andb∗
+/vectorkto
+e−i/vectork·/vector xa∗
+/vectork, and with vacuum Ω f:=I/vectorP(ei/vectorP·/vector x). To define I/vectorPmore precisely, we
+consider a vector ψ(f(n);/vectorP)∈ H/vectorPwith a definite total momentum describing
+an electron and nbosons. Its wave function in the variables ( /vector x;/vectork1,...,/vectorkn) is
+given by
+ei(/vectorP−/vectork1−···−/vectorkn)·/vector xf(n)(/vectork1,...,/vectorkn), (II.10)
+wheref(n)is totally symmetric in its narguments. The isomorphism I/vectorPacts
+by way of
+I/vectorP/parenleftbig
+ei(/vectorP−/vectork1−···−/vectorkn)·/vector xf(n)(/vectork1,...,/vectorkn)/parenrightbig
+(II.11)
+=1√
+n!/integraldisplay
+d3k1...d3knf(n)(/vectork1,...,/vectorkn)b∗
+/vectork1···b∗
+/vectorknΩf.DFP-Abs. Mass. Shell., February-8-2010 4
+II.3 Hamiltonians
+We consider a non-relativistic particle moving in a medium of relativistic
+bosons. The Hamiltonian of the system is given by
+H:=1
+2/vector p2+gφ(ρ/vector x) +Hf, (II.12)
+where:
+•The operators /vector x, /vector pdescribe the electron position and momentum, re-
+spectively;
+•Hf:=dΓ(ω(|/vectork|)) (see Section II.5), where ω(|/vectork|) :=|/vectork|, is the free
+field Hamiltonian. In physicist’s notation
+Hf=/integraldisplay
+d3k|/vectork|a∗
+/vectorka/vectork.
+•The real number g,|g|>0, is a coupling constant.
+•The interaction Hamiltonian is
+φ(ρ/vector x) :=/integraldisplay
+d3kρ(/vectork)(a∗
+/vectorke−i/vectork·/vector x+a/vectorkei/vectork·/vector x),(II.13)
+where the form factor ρ(/vectork)∈Rsatisfies the following conditions
+1. There is an ultraviolet cutoff Λ, i.e. ρ(/vectork) = 0 whenever |/vectork|>Λ.
+2. The function ρis rotationally invariant, i.e., ρ(/vectork) =ρ(|/vectork|), conti-
+nously differentiable, ρ∈C1. For expository convenience, when
+we will describe the decay mechanism in Theorem V.1, we will
+also assume that ρ(/vectork)/\e}atio\slash= 0 for 0<|/vectork|<Λ. Actually, this as-
+sumption is not necessary to state the main result of the theorem,
+but simplifies the construction of the trial state in Eq. (V.2) of
+Theorem V.1.
+3. The following infrared regularity condition holds:
+|ρ(/vectork)| ≤ O(|/vectork|β),and|/vector∇/vectorkρ(/vectork)| ≤ O(|/vectork|β−1),as/vectork→0
+(II.14)
+for an exponent β >11/2. We believe that the critical value,
+β= 11/2, is not optimal. From physical considerations, the result
+concerning the instability of the mass shell should hold for any
+exponentβ≥ −1/2. Forβ=−1/2, the Hamiltonian describes
+the interaction of the electron with the quantized relativistic field
+with no infrared regularization.DFP-Abs. Mass. Shell., February-8-2010 5
+The operator His self-adjoint, because φ(ρ/vector x) is an infinitesimal perturbation
+ofH0:=Hf+/vector p2
+2, and Dom( H) = Dom(H0), i.e., the domains of self-
+adjointness coincide. Since the Hamiltonian Hcommutes with the total
+momentum, it preserves the fiber spaces H/vectorP, for all/vectorP∈R3. Thus, we can
+write
+H=/integraldisplay⊕
+H/vectorPd3P, (II.15)
+where
+H/vectorP:H/vectorP−→ H /vectorP. (II.16)
+In terms of the operators b/vectork,b∗
+/vectork, and of the variable /vectorP, the fiber Hamiltonian
+H/vectorPis given by
+H/vectorP:=H0
+/vectorP+gφb(ρ), (II.17)
+with
+H0
+/vectorP:=/parenleftbig/vectorP−/vectorPf/parenrightbig2
+2+Hf, (II.18)
+where, as operators on the fiber space H/vectorP,
+/vectorPf=/integraldisplay
+d3k/vectorkb∗
+/vectorkb/vectork, (II.19)
+Hf=dΓb(ω(|/vectork|)) =/integraldisplay
+d3kω(|/vectork|)b∗
+/vectorkb/vectork, (II.20)
+and
+φb(ρ) :=/integraldisplay
+d3kρ(/vectork)(b∗
+/vectork+b/vectork). (II.21)
+II.4 Result
+The absence of a mass-shell for |/vectorP|>1 is expressed by the following state-
+ment: The equation
+H/vectorPΨ/vectorP=E/vectorPΨ/vectorP(II.22)
+hasnonormalizablesolutionforanyvalueof E/vectorPandforalmostevery /vectorP∈R3,
+|/vectorP|>1. What we actually prove in this paper is the absence of regularmass
+shells as formulated in the theorem below (see also Figure 1).
+More concretely, we address the question whether, for a given re gion
+I×∆Iin the momentum-energy space (see (ii) below), there is an open
+intervalIg,Ig⊂I, of size at least O(|g|γ),γ >0, where the mass shell exists,
+withE/vectorP∈∆Iand with the regularity property specified in the theorem.DFP-Abs. Mass. Shell., February-8-2010 6
+Recall that βdetermines the infrared behaviour of the form factor ρ, see
+(II.14).
+Theorem II.1. Assume that the form factor ρsatisfies (II.14), withβ >
+11/2, and fix an interval Iof the form I:= (1 +δ, σ),δ >0,σ <∞
+and a bounded interval ∆I. Fix constants 0< CI,cI<∞and exponents
+0< γ <1/4and0< ǫ < γ/ 4. Then, there is a g∗>0such that, for all g
+satisfying 0<|g|<g∗,the following is ruled out :
+There exist normalizable solutions to equation (II.22), fo r all|/vectorP| ∈Ig, such
+that:
+(i)Igis an interval of length larger than |g|γ/2(|Ig| ≥ |g|γ/2).
+(ii)Ig⊂IandE/vectorP⊂∆I, for all|/vectorP| ∈Ig.
+(iii) For all |/vectorP| ∈Ig,/vextenddouble/vextenddouble/vextenddouble/vector∇/vectorPΨ/vectorP,E/vectorP/vextenddouble/vextenddouble/vextenddouble< CI.
+(iv) For all |/vectorP| ∈Ig,
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleE/vectorP−(|/vectorP|−1
+2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>cI|g|γ/4−ǫ.
+We note that it is an interesting open problem to understand whethe r
+single-particle states could emerge at the boundary of the energy -momentum
+spectrum, i.e. near E/vector p=|/vectorP|−1
+2. Our results only rule out the existence of
+single particle states whose energies are embedded in the energy-momentum
+spectrum and with suitable regularity properties as far as their dep endence
+on/vectorPis concerned.
+Remark In the following Theorems, Lemmas, and Corollaries, we always
+assume that the Main Hypothesis in Section III.1.1 holds. Furthermore, |g|
+“sufficiently small” means 0 <|g|<g∗, whereg∗depends only on I, on ∆ I,
+and onγ, but with the form factor ρand the ultraviolet cutoff Λ kept fixed.
+II.4.1 Main ingredients of the proof
+(a) If Ψ /vectorP,E/vectorPexisted with properties (i)-(iv) above, and ||/vector∇E/vectorP| −1|>
+3
+2|g|γ/3, then
+/ba∇dblΨ0
+/vectorP−Ψ/vectorP,E/vectorP/ba∇dbl ≤ O(|g|(1−2γ)/6), (II.23)DFP-Abs. Mass. Shell., February-8-2010 7
+PSfrag replacements
+1 1 1δ
+|/vectorP| |/vectorP| |/vectorP|E E E
+1
+2|/vectorP|2 1
+2|/vectorP|2
+|/vectorP|−1
+2|/vectorP|−1
+2∼ |g|γ/4−ǫ
+Figure 1: The joint energy-momentum spectrum. By the rotation symmetry , it suffices
+to plot the ( E,|/vectorP|)- plane. In the leftmost figure, we have drawn the spectrum of th e
+uncoupled system. The parabola1
+2|/vectorP|2(in boldface) is the mass shell and the spectrum
+lies above the three-dimensional surface consisting of1
+2|/vectorP|2, for|/vectorP|<1, and|/vectorP|−1/2, for
+|/vectorP|>1. Hence, for |/vectorP|>1, the mass shell is embedded in the continuum. In the middle
+figure, we represent the situation when the coupling is switched on, according to formal
+perturbation theory . The mass shell has dissappeared (drawn as a dashed line) for |/vectorP|>1.
+For|/vectorP|<1, the mass shell persists but gets deformed (mass renormalizatio n). In the
+rightmost figure, we represent what is know rigorously: a regular m ass shell is excluded
+in the coloured area (result of the present paper) and there is a re normalized mass shell
+for small |P|(earlier works, see Section I ).
+where Ψ0
+/vectorPis the bare one-particle state, (i.e., Ψ0
+/vectorP= Ωf), and
+/vextendsingle/vextendsingle/vectorP2
+2−E/vectorP/vextendsingle/vextendsingle≤ O(|g|(1−2γ)/6). (II.24)
+(b) If Ψ /vectorP,E/vectorP, as in (a), existed then it could decay into a state consisting
+of an unperturbed single particle state and a boson with momentum /vectork
+in a region of momentum space away from the ray {λ/vectorP|0<λ≤ ∞}.
+(c) If||/vector∇E/vectorP|−1| ≤3
+2|g|γ/3then|E/vectorP−(|/vectorP|−1
+2)|<const|g|γ/4. In other
+words, a mass shell with group velocity close to one, necessarily lies
+near the boundary of the energy momentum spectrum.
+II.5 Notation
+Here is a list of notations used in subsequent sections.
+1. Given any vector /vector u∈R3, ˆu:=/vector u
+|/vector u|.DFP-Abs. Mass. Shell., February-8-2010 8
+2.Ffinis the dense subspace of Fobtained as the span of vectors con-
+taining finitely many bosons.
+3.1(a,b)(/vectork) is the characteristic function of the set
+{/vectork∈R3:|/vectork| ∈(a, b)}.
+4. For any function w∈h,/ba∇dblw/ba∇dbl2is the corresponding L2-norm.
+5.dΓ(A) is the second quantization of an operator Aacting on h;dΓ(A)
+is an operator on F. Analogously, dΓb(A) is defined on Fb.
+6. We define the (boson) number operators by N:=dΓ(1(/vectork)) andNb:=
+dΓb(1(/vectork)), where 1(/vectork) is the identity operator on L2(R3;d3k).
+7. We use the notation
+a∗(f/vector x) :=/integraldisplay
+d3kf/vector x(/vectork)a∗
+/vectork, a(f/vector x) :=/integraldisplay
+d3kf/vector x(/vectork)a/vectork
+for smeared creation/annihilation operators, depending also on th e
+(electron) position /vector x.
+8. Expressions like ( /vectorP,E/vectorP)∈Ig×∆Iare interpreted as follows: /vectorP∈R3
+with|/vectorP| ∈Ig, andE/vectorP∈∆I.
+II.6 Structure of the paper
+In Section III below, we state a Main Hypothesis (Section III.1). The up-
+shot of our analysis is Theorem V.4 in Section V. This theorem describe s
+the possible location of a mass shell, under the assumption that the Main
+Hypothesis holds true. In other words, the implication
+Main Hypothesis =⇒
+Assumptions
+in Section II.3Theorem V.4 (II.25)
+is our main result, and this implication gives rise to Theorem II.1.
+In the remainder of Section III.1, we state some immediate consequ ences
+of theMain Hypothesis , and in Section III.2, we put the technical tools in
+place. Section III.3 contains a rather detailed description of the st rategy of
+our proofs. The proofs themselves are presented in Sections IV a nd V. An
+appendix contains the proofs of some preliminary results used in Sec tion IV.DFP-Abs. Mass. Shell., February-8-2010 9
+III Strategy of the proof
+III.1 Main Hypothesis and key properties
+The proof of our result, Theorem II.1, is by contradiction. We will as sume
+that a regular mass shell exists, and subsequently, we derive that it cannot
+be located anywhere else than near the boundary of the energy-m omentum
+spectrum. Our assumption is stated in Section III.1.1 below and it will b e
+referred to as the Main Hypothesis . Throughout the rest of the paper, we
+assume that the Main Hypothesis holds. In Section III.1.2, we derive some
+consequences of the Main Hypothesis , namely Properties P1, P2 and P3.
+III.1.1 Main Hypothesis
+LetRbearotationmatrixin R3andU(R)theunitaryoperatorimplementing
+the transformation
+b/vectork→bR−1/vectork=:U∗(R)b/vectorkU(R). (III.1)
+The identity
+U∗(R)HR/vectorPU(R) =H/vectorP, (III.2)
+implies that if Ψ /vectorP,E/vectorPis a normalized eigenvector of H/vectorPwith eigenvalue E/vectorP
+thenU(R)Ψ/vectorP,E/vectorPis an eigenvector of HR/vectorPwith the same eigenvalue, i.e.,
+HR/vectorPU(R)Ψ/vectorP,E/vectorP=E/vectorPU(R)Ψ/vectorP,E/vectorP. (III.3)
+In particular, the existence of an eigenvector, Ψ /vectorP,E/vectorP, ofH/vectorPfor all/vectorPin a
+given direction, ˆP, yields a mass shell with energy function E/vectorP≡E|/vectorP|.
+Main hypothesis :We temporarilyassume thatsingle-particlestates, Ψ/vectorP,E/vectorP,
+exist, i.e.,
+H/vectorPΨ/vectorP,E/vectorP=E/vectorPΨ/vectorP,E/vectorP,/ba∇dblΨ/vectorP,E/vectorP/ba∇dbl= 1, (III.4)
+such that the vector Ψ/vectorP,E/vectorPis differentiable in /vectorPwith
+/vextenddouble/vextenddouble/vextenddouble/vector∇/vectorPΨ/vectorP,E/vectorP/vextenddouble/vextenddouble/vextenddouble< CI, (III.5)
+where the constant CI<∞, for all/vectorPsuch that |/vectorP| ∈Ig⊂Iand for
+E/vectorP∈∆I, where∆Iis a bounded interval. Here, Igis an open interval,
+|Ig|>|g|γ/2, andI:= (1+δ,σ), σ−1>δ>0.DFP-Abs. Mass. Shell., February-8-2010 10
+From the assumption in Eq. (III.5), the following properties follow fo r
+|/vectorP| ∈Ig, E/vectorP∈∆I.
+III.1.2 Properties (P1), (P2) and (P3)
+(P1)E/vectorP=E|/vectorP|is differentiable and the Feynman-Hellman formula holds
+/vector∇E/vectorP= (Ψ/vectorP,E/vectorP,(/vectorP−/vectorPf)Ψ/vectorP,E/vectorP). (III.6)
+The expression on the R.H.S. (right-hand side) of (III.6) is continuo us
+in/vectorP. Thus/vector∇E/vectorPis a continuous function of /vectorP. Moreover, |/vector∇E/vectorP|<C′
+I
+for someC′
+I<∞, and, because of rotation invariance, /vector∇E/vectorPand/vectorPare
+colinear.
+(P2) For some 0 <C′′
+I<∞,
+/vextendsingle/vextendsingle/vextendsingle∂2
+|/vectorP|E/vectorP
+∂|/vectorP|2/vextendsingle/vextendsingle/vextendsingle≤C′′
+I. (III.7)
+Starting from the derivative of the R.H.S. of (III.6), this bound can be
+easily obtained using (III.5) and that H0
+/vectorPisH/vectorP-bounded.
+(P3) From Hf=dΓb(|/vectork|) and Eq. (III.5), it follows that
+|/vector∇/vectorP(Ψ/vectorP,E/vectorP, dΓb(1(1
+n+1,1
+n)(/vectork))Ψ/vectorP,E/vectorP)| ≤ O(nCI[(sup
+/vectorP∈I|E/vectorP|)+1];
+(III.8)
+here we use the inequality
+/ba∇dbldΓb(1(1
+n+1,1
+n)(/vectork))ψ/ba∇dbl ≤(n+1)/ba∇dblHfψ/ba∇dbl,∀ψ∈Dom(Hf),
+and thatHfisH/vectorP-bounded.
+III.2 Technical Tools
+We will use two different virial arguments to expand Ψ /vectorP,E/vectorPin the coupling
+constantg,|g| ≪1. For this purpose, we must introduce single-particle
+“wave packets”, Ψ fg
+/vectorQ, defined below.
+(I)Single-particle “wave packets”, Ψfg
+/vectorQ, and the interval I′
+g.DFP-Abs. Mass. Shell., February-8-2010 11
+For|g|small enough, we define the open interval I′
+gsuch that
+(|Ig|>)|I′
+g|>4|g|γ, (III.9)
+with the property that
+|/vectorQ|+|/vector z| ∈Ig, (III.10)
+for all|/vectorQ| ∈I′
+gand for all/vector zsuch that |/vector z|<|g|γ.
+We consider single-particle “wave packets”, Ψ fg
+/vectorQ, with wave function,
+fg
+/vectorQ, centered around vectors /vectorQ,|/vectorQ| ∈I′
+g. The vector Ψfg
+/vectorQis defined by
+Ψfg
+/vectorQ:=/integraldisplay
+fg
+/vectorQ(/vectorP)Ψ/vectorP,E/vectorPd3P (III.11)
+wherefg
+/vectorQ(/vectorP) :=R(|/vectorP|−|/vectorQ|
+|g|γ)A(θˆQP
+|g|γ),θˆQPis the angle between /vectorQand/vectorP,
+andR(z),A(θ) are defined as follows.
+1)R(z),z∈R, is non-negative, smooth and compactly supported in
+the interval ( −1,1),R(z) = 1 forz∈(−1
+2,1
+2),
+2)A(θ),θ∈R, is non-negative, smooth and compactly supported in
+the interval ( −1,1),A(θ) = 1 forθ∈(−1
+2,1
+2). Therefore, the angular
+restriction
+ˆP·ˆP′≥cos(|g|γ)
+holds for any /vectorP,/vectorP′∈suppfg
+/vectorQ.
+3) Since|/vectorQ|>1, it follows from the definitions of R(z) andA(θ) that:
+|/vector∇/vectorPfg
+/vectorQ(/vectorP)| ≤ O(|g|−γ),
+for any/vectorP∈suppfg
+/vectorQ.
+(II)Multi-scale virial argument on the Hilbert space Hfor the Hamiltonian
+H.
+We define dilatation operators on the one-particle space h, constrained
+to a suitable range of frequencies and to a suitable angular sector
+around a direction ˆ u. We introduce the conjugate operator
+Dˆu,ˆQ
+n,⊥:=dΓ(dˆu,ˆQ
+n,⊥), (III.12)
+with
+dˆu,ˆQ
+n,⊥:=χn(|/vectork|)ξg
+ˆu(ˆk)1
+2(/vectork⊥·i/vector∇/vectork⊥+i/vector∇/vectork⊥·/vectork⊥)ξg
+ˆu(ˆk)χn(|/vectork|),(III.13)
+where:DFP-Abs. Mass. Shell., February-8-2010 12
+(a)/vectork⊥is the component of the vector /vectorkorthogonal to /vectorQ, i.e.,/vectork⊥:=
+/vectork−/vectork·/vectorQ
+|/vectorQ|2/vectorQ;
+χn(|/vectork|),n∈N, are non-negative, C∞(R+) functions with the
+properties:
+(i)χn(|/vectork|) = 0 for |/vectork| ≤1
+2(n+1)and for|/vectork| ≥3
+2n;
+(ii)χn(|/vectork|) = 1 for1
+n+1≤ |/vectork| ≤1
+n;
+(iii)|χ′
+n(|/vectork|)| ≤Cχn, for alln∈N, where the constant Cχis
+independent of n.
+(b)ξg
+ˆu(ˆk) (see Figure 2), 0 ≤ξg
+ˆu(ˆk)≤1, is a smooth function with
+support in the g-dependent cone
+Cˆu:={ˆk:ˆk·ˆu≥cos(|g|γ)}, (III.14)
+such that:
+i)
+ξg
+ˆu(ˆk) = 1 for {ˆk:ˆk·ˆu≥cos(1
+2|g|γ)}; (III.15)
+ii)
+ξg
+ˆu(ˆk) = 0 for {ˆk:ˆk·ˆu<cos(|g|γ)}; (III.16)
+iii)
+|∂θˆkuξg
+ˆu(ˆk)| ≤Cξ|g|−γ, (III.17)
+whereθˆkuis the angle between ˆkand ˆu, and the constant Cξ
+is independent of g.
+We also define
+dˆu
+n:=1
+2χn(|/vectork|)ξg
+ˆu(ˆk)[/vectork·i/vector∇/vectork+i/vector∇/vectork·/vectork]χn(|/vectork|)ξg
+ˆu(ˆk),(III.18)
+and we introduce the second quantized operator
+Dˆu
+n:=dΓ(dˆu
+n). (III.19)
+Later on in the paper, when we implement the virial argument, we will
+make use of the creation/annihilation operators
+a∗(idˆu
+nρ/vector x) =/integraldisplay
+d3ki(dˆu
+nρ/vector x)(/vectork)a∗
+/vectork
+a(idˆu
+nρ/vector x) =/integraldisplay
+d3ki(dˆunρ/vector x)(/vectork)a/vectork,
+and, analogously, a(idˆu,ˆQ
+n,⊥ρ/vector x),a∗(idˆu,ˆQ
+n,⊥ρ/vector x). In Lemma IV.1, we show
+that the vector Ψfg
+/vectorQbelongs to the form domain of these operators.DFP-Abs. Mass. Shell., February-8-2010 13
+Figure 2: The cone Cˆucorresponding to the support of the smooth charac-
+teristic function ξg
+ˆu(ˆk).
+(III)Virial argument in each fiber space H/vectorP.
+Here we consider
+Db
+1
+κ,κ:=dΓb(d1
+κ,κ)
+as the conjugate operator, where
+d1
+κ,κ:=χ[1
+κ,κ](|/vectork|)1
+2(/vectork·i/vector∇/vectork+i/vector∇/vectork·/vectork)χ[1
+κ,κ](|/vectork|).(III.20)
+χ[1
+κ,κ](|/vectork|),∞> κ >max{Λ,1}, are non-negative, C∞(R+) functions
+with the properties:
+(i)χ[1
+κ,κ](|/vectork|) = 0 for |/vectork| ≥2κ,|/vectork| ≤1
+2κ;
+(ii)χ[1
+κ,κ](|/vectork|) = 1 for1
+κ≤ |/vectork| ≤κ;
+(iii) for some C >0,|χ′
+[1
+κ,κ](|/vectork|)|<Cκ;
+Analogously to a∗(idˆu
+nρ/vector x),a(idˆu
+nρ/vector x), we will use
+b∗(id1
+κ,κρ) =/integraldisplay
+d3k(id1
+κ,κρ)(/vectork)b∗
+/vectorkDFP-Abs. Mass. Shell., February-8-2010 14
+b(id1
+κ,κρ) =/integraldisplay
+d3k(id1
+κ,κρ)(/vectork)b/vectork.
+We will also consider the g-dependent cones (see Figure 3),
+Ca
+ˆP:={/vectork:|ˆk·ˆP| ≤cos(a|g|γ/8)}, (III.21)
+and use the smooth functions ξg
+Ca
+ˆP(ˆk),a=1
+2,2, defined below.
+The functions ξg
+Ca
+ˆP(ˆk), 0≤ξg
+Ca
+ˆP(ˆk)≤1, are chosen such that
+i)
+ξg
+Ca
+ˆP(ˆk) = 1 for {ˆk:|ˆk·ˆP| ≤cos(2a|g|γ/8)}; (III.22)
+ii)
+ξg
+Ca
+ˆP(ˆk) = 0 for {ˆk:|ˆk·ˆP|>cos(a|g|γ/8)}; (III.23)
+iii)
+|∂θˆkPξg
+Ca
+ˆP(ˆk)| ≤Cξ|g|−γ/8, (III.24)
+for a constant Cξindependent of g, whereθˆkPis the angle between ˆk
+andˆP.DFP-Abs. Mass. Shell., February-8-2010 15
+Figure 3: The double cone Ca
+ˆPis the complement in R3of the inner double
+cone around ˆPof angular width a|g|γ/8.
+III.3 Description of strategy
+To exclude the existence of eigenvalues E/vectorP,|/vectorP|>1, we elaborate on an
+argument introduced in [12]. The idea ofthe proofis asfollows. Oneas sumes
+that an eigenvector Ψ /vectorP,E/vectorP∈ H/vectorPofH/vectorPexists, for some energy E/vectorPin a
+compact set. Then, using a multiscale virial argument, one intends t o prove
+that
+(Ψ/vectorP,E/vectorP,NbΨ/vectorP,E/vectorP)≤ O(g2), (III.25)
+whereNbis the boson number operator in the fiber spaces. The multiscale
+virial argument involves the dilatation operators
+dn:=1
+2χn(|/vectork|)[/vectork·i/vector∇/vectork+i/vector∇/vectork·/vectork]χn(|/vectork|), (III.26)
+on the one-particle space h, whereχnis a suitable smooth approximation to
+the characteristic function of the interval [1
+n+1,1
+n] contained in the positive
+frequency half axis, n= 1,2,.... After introducing the second quantized
+dilatation operators Db
+n:=dΓb(dn), one starts from the formalvirial identity
+0 = (Ψ /vectorP,E/vectorP, i[H/vectorP, Db
+n]Ψ/vectorP,E/vectorP) (III.27)DFP-Abs. Mass. Shell., February-8-2010 16
+to establish the scale-by-scale inequality below, in a rigorous way:
+(Ψ/vectorP,E/vectorP, Nb
+nΨ/vectorP,E/vectorP)≤ O(g2n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2), (III.28)
+whereNb
+n:=/integraltext
+d3ka∗(/vectork)χ2
+n(|/vectork|)a(/vectork),n= 1,2,3,.... If (III.28) holds true,
+for sufficiently large values of the exponent βin the form factor ρ, one can
+sum overnand conclude that (Ψ /vectorP,E/vectorP, NbΨ/vectorP,E/vectorP)≤ O(g2). Next, the eigen-
+value equation (II.22) and the inequality in Eq. (III.25) can be combin ed
+to conclude that the vector Ψ /vectorP,E/vectorPand the eigenvalue E/vectorPmust fulfill the
+following estimates:
+/ba∇dblΨ/vectorP,E/vectorP−Ψ0
+/vectorP/ba∇dbl2≤ O(g2), (III.29)
+where Ψ0
+/vectorP:= Ωfis the unperturbed eigenstate, and
+|E/vectorP−/vectorP2
+2| ≤ O(g2). (III.30)
+This result would imply that putative eigenvalues of H/vectorPlie in an O(g2)-
+neighborhood of the eigenvalue of the Hamiltonian Hg=0
+/vectorP.
+Then the argument proceeds with the construction of suitable tria l states of
+the type
+η/vectorP:=/integraldisplay
+d3k1
+ǫ1
+2h/parenleftbig(/vectorP−/vectork)2/2+|/vectork|−E/vectorP
+ǫ/parenrightbig
+b∗
+/vectorkΨ0
+/vectorP, (III.31)
+whereǫ>0 andh(z)∈C∞
+0(R),h(z)≥0. One then exploits the identity
+(η/vectorP,(H/vectorP−E/vectorP)Ψ/vectorP,E/vectorP) = 0 (III.32)
+that must hold true if Ψ /vectorP,E/vectorPis an eigenvector of H/vectorP. Starting from Eqs.
+(III.29)-(III.30), and using that the equation
+(/vectorP−/vectork)2/2+|/vectork|−E/vectorP= 0 (III.33)
+has solutions for |/vectorP|>1, provided |g|is small enough, one arrives at a
+contradiction, for ǫand|g|small enough.
+However, the procedure just outlined (mimicking the treatment of atomic
+resonances in [12]) will not work without some important modification s. We
+will therefore implement analogous, but more elaborate strategy.
+The first problem ecountered is that we cannot control the expec tation value
+(Ψ/vectorP,E/vectorP,NbΨ/vectorP,E/vectorP)DFP-Abs. Mass. Shell., February-8-2010 17
+by a multiscale virial argument in the fiber space H/vectorP, because of the term
+(/vectorPf)2inH/vectorP. The commutator of ( /vectorPf)2withdΓb(dn), formally given by
+/vectorPf·dΓb(χ2
+n(/vectork)/vectork)+dΓb(χ2
+n(/vectork)/vectork)·/vectorPf, (III.34)
+cannot be controlled in terms of the commutator of HfwithdΓb(dn) . Con-
+sequently, the estimate in Eq. (III.28) cannot be justified startin g from the
+virial identity in Eq. (III.27).
+At the price of limiting our analysis to regularmass shells (see Main Theo-
+remin Section II.4), this problem can be circumvented by implementing a
+multiscale virial argument in the full Hilbert space, by using single-par ticle
+“wave packets” rather than fiber eigenvectors, i.e., vectors in Hof the type
+Ψf:=/integraldisplay
+f(/vectorP)Ψ/vectorP,E/vectorPd3P, (III.35)
+wheref(/vectorP) is a smooth function with support in Ig(the region of momenta
+for which an eigenstate was assumed to exist). In practice, we cho osef=fg
+/vectorQ
+to be sharply peaked around a given momentum /vectorQ, see definition below
+(III.11). In the full Hilbert space, we can essentially mimick the trea tement
+of atomic resonances to derive the following result (see Section IV) .
+Theorem (IV.3).For|g|sufficiently small,
+(Ψfg
+/vectorQ, Nn,C1/2
+ˆQΨfg
+/vectorQ)
+(Ψfg
+/vectorQ,Ψfg
+/vectorQ)≤ O(g2(1−2γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2),(III.36)
+whereNn,C1/2
+ˆQ:=dΓ(χ2
+n(|/vectork|)ξg2
+C1/2
+ˆQ(ˆk))and|/vectorQ| ∈I′
+g.
+Furthermore, if for all /vectorP∈suppfg
+/vectorQthe inequality
+||/vector∇E/vectorP|−1|>|g|γ/3
+holds true, then
+(Ψfg
+/vectorQ, NnΨfg
+/vectorQ)
+(Ψfg
+/vectorQ,Ψfg
+/vectorQ)≤ O(g2(1−2γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2),(III.37)
+whereNn:=dΓ(χ2
+n(|/vectork|).
+The constants in (III.36), (III.37) can be chosen uniformly in/vectorQ,|/vectorQ| ∈I′
+g⊂I
+(I′
+gis defined in Section III.2, Eqs. (III.9),(III.10)). They on ly depend on I
+and on∆I.DFP-Abs. Mass. Shell., February-8-2010 18
+By exploiting the g-dependence of the wavefunctions fg
+/vectorQand the assump-
+tion on the regularity in /vectorPof Ψ/vectorP,E/vectorP, one can convert a bound for the number
+operatorNon single-particle wave packets to a bound that holds pointwise
+in/vectorPon the number operator Nbacting on the fiber eigenvectors Ψ /vectorP,E/vectorP. In
+essence, this follows from the fundamental theorem of calculus. T hese argu-
+ments are implemented in Section IV and give the following results.
+Theorem (IV.5).For|g|sufficiently small and (/vectorP,E/vectorP)∈I′
+g×∆I,
+(Ψ/vectorP,E/vectorP, Nb
+n,C2
+ˆPΨ/vectorP,E/vectorP)≤ O(|g|(1−2γ)
+3|g|−γ/8n4
+3/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl1
+3
+2),
+(III.38)
+where
+Nb
+n,C2
+ˆP:=dΓb(χ2
+n(|/vectork|)ξg2
+C2
+ˆP(ˆk)). (III.39)
+Furthermore, if in addition ||/vector∇E/vectorP|−1|>3
+2|g|γ/3then
+(Ψ/vectorP,E/vectorP, Nb
+nΨ/vectorP,E/vectorP)≤ O(|g|(1−2γ)
+3n4
+3/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl1
+3
+2)) (III.40)
+where
+Nb
+n:=dΓb(χ2
+n(|/vectork|)). (III.41)
+The constants in (III.38), (III.40) can be chosen uniformly in/vectorP,|/vectorP| ∈I′
+g⊂I
+(I′
+gis defined in Section III.2, Eqs. (III.9), (III.10)). They on ly depend on
+Iand on∆I.
+We now comment on the contents of Theorem IV.5. Inequality (III.3 8)
+means that we can bound the boson number operator if we exclude a double
+cone (see Figure 3) and the definition of C2
+ˆPin Eqs. (III.22)-(III.24), Section
+III.2) around the direction of the particle velocity, provided the fo rm factor
+ρ(/vectork) scales like |/vectork|βwithβ >11/2, i.e. (II.14).
+The second result (see (III.40)) says that, for putative mass sh ells (/vectorP,E/vectorP)
+such that ||/vector∇E/vectorP| −1|is not too small (i.e., ||/vector∇E/vectorP|−1|>3
+2|g|γ/3), we can
+bound the boson number without any angular restrictions, again us ing that
+ρ(/vectork) scales like |/vectork|βwithβ >11/2. The constraint means that the forward
+emission of bosons by the (massive) particle cannot be controlled if it s speed
+is too close to the boson propagation speed.
+The estimates on the number operator obtained in Section IV are us ed in
+Section V, where we will establish the following two results regarding t he
+regionIg×∆I, whereIgis any open interval contained in Isuch that |Ig|>
+|g|γ/2.DFP-Abs. Mass. Shell., February-8-2010 19
+(i) The first result is that we can exclude all the regular mass shells ex cept
+those with slope close to 1, i.e., all the regularmass shells such that
+(/vectorP,E/vectorP)∈Ig×∆Iand||/vector∇E/vectorP|−1|>3
+2|g|γ/3.(III.42)
+(ii) The second result shows that a regularmass shell might exist only for
+(/vectorP,E/vectorP) such that
+E/vectorP=|/vectorP|−1
+2+O(|g|γ/4). (III.43)
+More precisely, we use that:
+(1) The expectation value in Ψ /vectorP,E/vectorP,|/vectorP| ∈I′
+g, of the operator Nbrestricted
+to the angular sector C2
+ˆPvanishes as g→0 (see Theorem IV.5).
+(2) For(/vectorP,E/vectorP)∈I′
+g×∆Isuchthat ||/vector∇EP|−1|>O(|g|γ/3),theexpectation
+value of the number operators Nbon Ψ/vectorP,E/vectorPvanishes as g→0; see
+TheoremIV.5. Analogously, if suppfg
+/vectorQ⊂ {(/vectorP,E/vectorP)∈I′
+g×∆I| ||/vector∇EP|−
+1|>O(|g|γ/3)}then the expectation value of the number operator N
+on Ψg
+f/vectorQvanishes as g→0; see Theorem IV.3.
+(3) The results in (2) imply that the putative fiber eigenvectors Ψ /vectorP,E/vectorP,
+|/vectorP| ∈I′
+g, and the corresponding energies E/vectorPare pertubative in g(see
+Corollary IV.6), provided that β >11/2.
+We derive (i) in Theorem V.1 by mimicking the argument with the trial
+states employed for the treatment of the atomic resonances [12], which was
+anticipated in Section III.3, Eqs (III.31)–(III.33). To this end, we make us
+of (2) and (3).
+The result in (ii) follows thanks to a stronger version of (1) (for det ails, see
+Lemma V.2, Lemma V.3) where only the forward cone around the direc tion
+of the particle velocity is excluded in the definition of the restricted n umber
+operator, and by combining the eigenvalue equation with a standard (i.e.,
+not a multi-scale analysis) virial argument in the fiber space H/vectorP, where the
+conjugate operator is Db
+1
+κ,κ:=dΓb(d1
+κ,κ); see (III.20) and Theorem V.4.
+Thevirial identity exploited inTheorem V.4isactuallyenoughtoexclude
+that, fiber by fiber, the eigenvalue lies at a distance larger than O(g) above
+the unperturbed eigenvalue. This observation is explained in the Rem arkDFP-Abs. Mass. Shell., February-8-2010 20
+after Theorem V.4 in Section V. However, the instability of the unper turbed
+mass shell proven in this paper requires a detailed analysis of the con figu-
+ration of bosons in the putative eigenvector whose momenta are co ntained
+in different cones of momentum space. The decay mechanism exploite d in
+Theorem V.1 combined with the assumed continuity of the mass shell is
+responsible for the absence of single-particle states except for t he region
+E/vectorP=|/vectorP| −1
+2+O(|g|γ/4). This is because if the particle propagated at
+the critical velocity, i.e., |/vector∇EP|= 1, then there would be no kinematical con-
+straint preventing the emission of an arbitrarily large number of sof t bosons
+in the forward direction (the direction of /vectorP).
+IV Boson number estimates
+The main results in this section are Theorem IV.3, Theorem IV.5, and C orol-
+lary IV.6. Two preparatory results, contained in Section IV.1, are n eeded.
+In particular, in Lemma IV.2, we provide a rigorous justification of a v irial
+identity employed in Lemma IV.4 and in Theorem IV.3.
+Since the proof of Theorem IV.3 is lengthy, we present it in two differe nt
+smaller sections: (a) In Section IV.2.1, we outline the proof of the th eorem
+and, in Lemma IV.4, we introduce an important ingredient used later o n. (b)
+In Section IV.2.2, we complete the steps of the proof by assuming th e result
+obtained in Lemma IV.4.
+In Section IV.3, by using the regularity properties that follow from t he
+Main Hypothesis , wederivesomeestimatesforthenumberoperator Nbevalu-
+ated on the fiber eigenvectors Ψ /vectorP,E/vectorPanalogous to those obtained in Theorem
+IV.3 for the number operator Nevaluated on the single-particle states Ψfg
+/vectorQ.
+In Corollary IV.6, we then finally show that E/vectorPand Ψ /vectorP,E/vectorPare perturbative
+ing, provided |/vectorP| ∈I′
+g,E/vectorP∈∆I,β >11/2, and||/vector∇E/vectorP|−1|>3
+2|g|γ/3.
+IV.1 Preparatory results on virial identities
+The following two lemmas are repeated and proven in Sections VI.1 and VI.2
+of the appendix, respectively.
+Lemma IV.1. The vector Ψfg
+/vectorQbelongs to the domain of the position operator
+/vector xand
+/ba∇dblxjΨfg
+/vectorQ/ba∇dbl ≤ O(|g|−γ/ba∇dblΨfg
+/vectorQ/ba∇dbl), j= 1,2,3. (IV.1)
+Proof.See the appendixDFP-Abs. Mass. Shell., February-8-2010 21
+Lemma IV.2statesavirialtheoremforourmodel. Weobserve thatb yformal
+steps one can derive the identity
+i[H−E/vectorP, Dˆu
+n] =dΓ(i[|/vectork|, dˆu
+n])−/vector∇E/vectorP·dΓ(i[/vectork, dˆu
+n]) (IV.2)
+−g[a∗(idˆu
+nρ/vector x)+a(idˆu
+nρ/vector x)],
+whereE/vectorP, and/vector∇E/vectorPare operator-valued functions of the total momentum
+operator/vectorP. Another formal step would imply that
+0 = (Ψfg
+/vectorQ, i[H−E/vectorP, Dˆu
+n]Ψfg
+/vectorQ), (IV.3)
+and, hence,
+0 = (Ψfg
+/vectorQ, dΓ(i[|/vectork|, dˆu
+n])Ψfg
+/vectorQ)−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(i[/vectork, dˆu
+n])Ψfg
+/vectorQ)
+−g(Ψfg
+/vectorQ,[a∗(idˆu
+nρ/vector x)+a(idˆu
+nρ/vector x)]Ψfg
+/vectorQ). (IV.4)
+The next Lemma shows that all terms on the RHS of Eq. (IV.4) can be given
+a well-defined meaning such that the equality is true.
+Lemma IV.2. The identity
+0 = (Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork|)Ψfg
+/vectorQ)−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/vectork)Ψfg
+/vectorQ)
+−g(Ψfg
+/vectorQ,[a∗(idˆu
+nρ/vector x)+a(idˆu
+nρ/vector x)]Ψfg
+/vectorQ) (IV.5)
+holds true. As the one-particle state Ψfg
+/vectorQbelongs to the form domain of all
+operators on the RHS of (IV.5), this RHS is well-defined.
+Proof.See the appendix
+Note that the formal equality of the RHS of (IV.4) and (IV.5) is stra ight-
+forward. The virial identity of the Lemma above is first used in item (i) of
+IV.2.1. A similar virial identity in item (ii) is proven analogously.
+IV.2 Number operator estimates in putative single-
+particle states
+We now proceed to proving the following theorem, where the expect ation of
+the boson number operator in the state Ψfg
+/vectorQis bounded scale by scale.DFP-Abs. Mass. Shell., February-8-2010 22
+Theorem IV.3. For|g|sufficiently small,
+(Ψfg
+/vectorQ, Nn,C1/2
+ˆQΨfg
+/vectorQ)
+(Ψfg
+/vectorQ,Ψfg
+/vectorQ)≤ O(|g|2(1−2γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2),(IV.6)
+whereNn,C1/2
+ˆQ:=dΓ(χ2
+n(|/vectork|)ξg2
+C1/2
+ˆQ(ˆk))and/vectorQ∈I′
+g.
+Furthermore, if for all /vectorP∈suppfg
+/vectorQthe inequality
+||/vector∇E/vectorP|−1|>|g|γ/3
+holds true then
+(Ψfg
+/vectorQ, NnΨfg
+/vectorQ)
+(Ψfg
+/vectorQ,Ψfg
+/vectorQ)≤ O(|g|2(1−2γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2),(IV.7)
+whereNn:=dΓ(χ2
+n(|/vectork|).
+The (implicit) constants in (IV.6)-(IV.7) can be chosen to b e uniform in /vectorQ,
+|/vectorQ| ∈I′
+g⊂I; for the definition of I′
+gsee Eqs. (III.9), (III.10)). They only
+depend onIand on∆I.
+IV.2.1 Outline of the proof of Theorem IV.3
+To prove inequalities (IV.6), (IV.7) we exploit two different virial argu ments
+and properties (P1), (P2), and (P3) of Sect. III.1.2. More precis ely, we
+employ both conjugate operators Dˆu
+n:=dΓ(dˆu
+n) andDˆu,ˆQ
+n,⊥:=dΓ(dˆu,ˆQ
+n,⊥), with
+dˆu
+nanddˆu,ˆQ
+n,⊥defined in Eqs. (III.18) and (III.13), respectively. The virial
+identities (see Lemma IV.2 for a rigorous treatment of the identities below)
+corresponding to Dˆu
+nandDˆu,ˆQ
+n,⊥are:
+i)
+0 = (Ψfg
+/vectorQ, dΓ(i[|/vectork|, dˆu
+n])Ψfg
+/vectorQ) (IV.8)
+−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(i[/vectork, dˆu
+n])Ψfg
+/vectorQ) (IV.9)
+−g(Ψfg
+/vectorQ,[a∗(idˆu
+nρ/vector x)+a(idˆu
+nρ/vector x)]Ψfg
+/vectorQ) (IV.10)
+= (Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork|)Ψfg
+/vectorQ) (IV.11)
+−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/vectork)Ψfg
+/vectorQ) (IV.12)
+−g(Ψfg
+/vectorQ,[a∗(idˆu
+nρ/vector x)+a(idˆu
+nρ/vector x)]Ψfg
+/vectorQ); (IV.13)DFP-Abs. Mass. Shell., February-8-2010 23
+ii)
+0 = (Ψ fg
+/vectorQ, dΓ(i[|/vectork|, dˆu,ˆQ
+n,⊥])Ψfg
+/vectorQ) (IV.14)
+−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(i[/vectork, dˆu,ˆQ
+n,⊥])Ψfg
+/vectorQ) (IV.15)
+−g(Ψfg
+/vectorQ,[a∗(idˆu,ˆQ
+n,⊥ρ/vector x)+a(idˆu,ˆQ
+n,⊥ρ/vector x)]Ψfg
+/vectorQ) (IV.16)
+= (Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork⊥|2
+|/vectork|)Ψfg
+/vectorQ) (IV.17)
+−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/vectork⊥)Ψfg
+/vectorQ) (IV.18)
+−g(Ψfg
+/vectorQ,[a∗(idˆu,ˆQ
+n,⊥ρ/vector x)+a(idˆu,ˆQ
+n,⊥ρ/vector x)]Ψfg
+/vectorQ).(IV.19)
+(For the definition of the functions χn(|/vectork|),ξg
+ˆu(ˆk) see (a) and (b), in Section
+III.2)
+Next, we explain in detail the key role of the virial identities. In order to
+arrive at inequalities (IV.6), (IV.7), we study (see Lemma IV.4) the n umber
+operator restricted to the sector associated with the unit vecto r ˆu, and derive
+the estimate
+(Ψfg
+/vectorQ, Nn,ˆuΨfg
+/vectorQ)≤(Ψfg
+/vectorQ,Ψfg
+/vectorQ)O(|g|2(1−γ−˜γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2),
+(IV.20)
+whereNn,ˆu:=dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)), for some ˜ γ, 0<˜γ < γ; we will eventually
+choose ˜γ=γ/2. In doing this, we start from the bound
+|(Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork|)Ψfg
+/vectorQ)−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/vectork)Ψfg
+/vectorQ)|
+≥ O(|g|˜γ)(Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork|)Ψfg
+/vectorQ) (IV.21)
+that holds if, for all /vectorP∈suppfg
+/vectorQand for all ˆkin the sector,
+|1−ˆk·/vector∇E/vectorP|>|g|˜γ>0. (IV.22)
+Given(IV.21),itisstraightfowardtocontroltheterm(see(IV.13 ))associated
+with the interaction part of the Hamiltonian, and to derive the inequa lity in
+(IV.20). Therefore, the bound in (IV.22) is crucial, and we must iden tify the
+sectors where it is violated. We recall that /vector∇E/vectorPis collinear to /vectorP, and we
+may assume that they are parallel; the other case can be treated in the same
+way.DFP-Abs. Mass. Shell., February-8-2010 24
+First, note that the angle between /vectorP∈suppfg
+/vectorQand/vectorQ, as well as the
+angle between ˆ uand a vector /vectorkthat belongs to the sector associated with
+ˆu, areO(|g|γ). This follows from the definitions of the function fg
+/vectorQand the
+conesCˆu, given in Section III.2. It implies that, roughly speaking, we can
+identify ˆP=ˆQandˆk= ˆu, since, for |g|small enough, |g|γis much smaller
+than|g|˜γin (IV.22).
+The vectors /vectorkfor which (IV.22) fails, satisfy
+/vextendsingle/vextendsingle/vextendsingleˆk·ˆP−|/vector∇E/vectorP|−1/vextendsingle/vextendsingle/vextendsingle≤ O(|g|˜γ(=γ/2)),|/vector∇E/vectorP|>0. (IV.23)
+Hence, if |/vector∇E/vectorP|is bounded away from 1, either - for |/vector∇E/vectorP|<1; see also
+(B)in Section IV.2.2 - the condition (IV.22) is always satisfied, or - for
+|/vector∇E/vectorP|>1; see also (C)in Section IV.2.2 - such /vectorkhave a nonvanishing
+component, /vectork⊥, (of order/radicalBig
+1−|/vector∇E/vectorP|−2) in the orthogonal complement of
+/vectorQ(=/vectorP). In particular, they satisfy
+/vextendsingle/vextendsingle|/vectork⊥|2
+|/vectork|2−|/vectork⊥|
+|/vectork|ˆk⊥·/vector∇E/vectorP/vextendsingle/vextendsingle>O(|g|γ/3)>0, (IV.24)
+Note that the second term on the LHS of (IV.24) actually vanishes if our
+approximation ˆP=ˆQwere to hold exactly. In Section IV.2.2, we establish
+(IV.24) rigorously. The bound (IV.24) immediately implies that, for th e
+sectors ˆufor which (IV.22) fails, the following bound holds true
+|(Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork⊥|2
+|/vectork|)Ψfg
+/vectorQ)
+−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/vectork⊥)Ψfg
+/vectorQ)|
+≥ O(|g|γ/3)(Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork|)Ψfg
+/vectorQ),(IV.25)
+Starting from this bound, we can use the second virial identity (IV.1 7, IV.18,
+IV.19)toderivetheinequality(IV.20)forthesectors ˆ uforwhich(IV.22)fails.
+The conclusion is that, under the condition that |/vector∇E/vectorP|,/vectorP∈suppfg
+/vectorQ, dif-
+fers from 1 by a quantity >O(|g|γ/3), we can cover all the sectors by the two
+virial identities above. Without the restriction on |/vector∇E/vectorP|, these arguments
+only show that Eq. (IV.20) holds for all ˆ u-dependent sectors contained in
+the cone C1/2
+ˆQ.
+In implementing this strategy, we make use of the following lemma.DFP-Abs. Mass. Shell., February-8-2010 25
+Lemma IV.4. Fix a unit vector ˆuand assume that, for all /vectorP∈suppfg
+/vectorQand
+for allˆk∈suppξg
+ˆu,
+|1−ˆk·/vector∇E/vectorP|>|g|˜γ>0,0<˜γ <γ, (IV.26)
+where˜γisg- and/vectorQ-independent. Then, for |g|small enough, the following
+bound holds true
+(Ψfg
+/vectorQ, Nn,ˆuΨfg
+/vectorQ)≤(Ψfg
+/vectorQ,Ψfg
+/vectorQ)O(|g|2(1−γ−˜γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2),
+(IV.27)
+whereNn,ˆu:=dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)).
+Proof
+We assume that (IV.26) holds with
+1−ˆk·/vector∇E/vectorP<0;
+the other case, 1 −ˆk·/vector∇E/vectorP>0, can be treated similarly. We get
+0 = (Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork|)Ψfg
+/vectorQ) (IV.28)
+−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/vectork)Ψfg
+/vectorQ)
+−g(Ψfg
+/vectorQ,[a∗(idˆu
+nρ/vector x)+a(idˆu
+nρ/vector x)]Ψfg
+/vectorQ)
+≤ −|g|˜γ(Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork|)Ψfg
+/vectorQ) (IV.29)
++c|g|1−γ/ba∇dblΨfg
+/vectorQ/ba∇dbl/ba∇dbldΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk))1/2Ψfg
+/vectorQ/ba∇dbl× (IV.30)
+×(/integraldisplay
+|/vectork|2β1(1
+2(n+1),3
+2n)(/vectork)d3k)1/2
+for some constant c,c >0, uniform in /vectorQ,|/vectorQ| ∈I′
+g. To do the step from
+(IV.28) to (IV.29), we split
+i(dˆu
+nρ/vector x)(/vectork) =−χn(|/vectork|)ξg
+ˆu(ˆk)/parenleftbig/vectork·/vector∇/vectork/parenleftbig
+χn(|/vectork|)ξg
+ˆu(ˆk)/parenrightbig/parenrightbig
+ρ(/vectork)e−i/vectork·/vector x
+−χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/parenleftbig/vectork·/vector∇/vectorkρ(/vectork)/parenrightbig
+e−i/vectork·/vector x
+−1
+2χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/parenleftbig/vector∇/vectork·/vectork/parenrightbig
+ρ(/vectork)e−i/vectork·/vector x
+−χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)ρ(/vectork)/parenleftbig/vectork·/vector∇/vectorke−i/vectork·/vector x/parenrightbig
+(IV.31)
+and we may justify this step for each of the four terms separately , using the
+Schwarz inequality andDFP-Abs. Mass. Shell., February-8-2010 26
+i) The assumption/vextendsingle/vextendsingle/vextendsingle1−ˆk·/vector∇E/vectorP/vextendsingle/vextendsingle/vextendsingle>|g|˜γfor/vectorP∈suppfg
+/vectorQ;
+ii) The infrared behavior of ρ(/vectork) as assumed in (II.14), i.e., |ρ(/vectork)| ≤
+O(|/vectork|β) and|/vector∇/vectorkρ(/vectork)| ≤ O(|/vectork|β−1);
+iii) Lemma IV.1.
+As an example, for the term proportional to
+−χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)ρ(/vectork)/parenleftbig/vectork·/vector∇/vectorke−i/vectork·/vector x/parenrightbig
+we proceed as follows:
+/vextendsingle/vextendsingle/vextendsingle(Ψfg
+/vectorQ, a∗/parenleftbig
+−χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)ρ(/vectork)/parenleftbig/vectork·/vector∇/vectorke−i/vectork·/vector x/parenrightbig/parenrightbig
+Ψfg
+/vectorQ)/vextendsingle/vextendsingle/vextendsingle
+=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)ρ(/vectork)/vectork·(a/vectorkΨfg
+/vectorQ,/vector∇/vectorke−i/vectork·/vector xΨfg
+/vectorQ)d3k/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤/integraldisplay
+χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|ρ(/vectork)||/vectork||(a/vectorkΨfg
+/vectorQ,/vector∇/vectorke−i/vectork·/vector xΨfg
+/vectorQ)|d3k
+≤/integraldisplay
+χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|ρ(/vectork)||/vectork|/ba∇dbla/vectorkΨfg
+/vectorQ/ba∇dbl/ba∇dbl/vector∇/vectorke−i/vectork·/vector xΨfg
+/vectorQ/ba∇dbld3k
+≤/parenleftbig/integraldisplay
+χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/ba∇dbla/vectorkΨfg
+/vectorQ/ba∇dbl2d3k/parenrightbig1/2×
+×/parenleftbig/integraldisplay
+/ba∇dbl/vector∇/vectorke−i/vectork·/vector xΨfg
+/vectorQ/ba∇dbl2|/vectork|2β1(1
+2(n+1),3
+2n)(/vectork)|/vectork|2d3k/parenrightbig1/2.(IV.32)
+We notice that
+/parenleftbig/integraldisplay
+χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/ba∇dbla/vectorkΨfg
+/vectorQ/ba∇dbl2d3k/parenrightbig1/2=/ba∇dbldΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk))1/2Ψfg
+/vectorQ/ba∇dbl(IV.33)
+and, since /ba∇dbl/vector∇/vectorke−i/vectork·/vector xΨfg
+/vectorQ/ba∇dbl ≤ O(|g|−γ/ba∇dblΨfg
+/vectorQ/ba∇dbl) (Lemma IV.1),
+/parenleftbig/integraldisplay
+/ba∇dbl/vector∇/vectorke−i/vectork·/vector xΨfg
+/vectorQ/ba∇dbl2|/vectork|2β1(1
+2(n+1),3
+2n)(/vectork)|/vectork|2d3k/parenrightbig1/2(IV.34)
+≤C|g|−γ(/integraldisplay
+|/vectork|2β1(1
+2(n+1),3
+2n)(/vectork)d3k)1/2/ba∇dblΨfg
+/vectorQ/ba∇dbl.(IV.35)
+Then, starting from (IV.29), the bound from above takes the for m
+(Ψfg
+/vectorQ, Nn,ˆuΨfg
+/vectorQ)≤(Ψfg
+/vectorQ,Ψfg
+/vectorQ)O(|g|2(1−γ−˜γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2),
+(IV.36)DFP-Abs. Mass. Shell., February-8-2010 27
+whereNn,ˆu:=dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)), because
+χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork| ≥1
+2(n+1)χ2
+n(|/vectork|)ξg2
+ˆu(ˆk). (IV.37)
+IV.2.2 Proof of Theorem IV.3
+Notice that, starting from Lemma IV.4, we can fill the region
+{ˆk:|1−ˆk·/vector∇E/vectorP|>|g˜γ| ∀/vectorP∈suppfg
+/vectorQ} (IV.38)
+with sectors corresponding to functions ξg
+ˆujwhere 1≤j≤¯j≤ O(|g|−γ), so
+that we obtain
+¯j/summationdisplay
+j=1(Ψfg
+/vectorQ, Nn,ˆujΨfg
+/vectorQ)≤(Ψfg
+/vectorQ,Ψfg
+/vectorQ)O(|g|2(1−3γ
+2−˜γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2).
+(IV.39)
+We observe that if, for some /vectorP∈suppfg
+/vectorQ,
+||/vector∇E/vectorP|−1| ≤ |g|γ/3
+then, for |g|small enough,
+||/vector∇E/vectorP′|−1| ≤2|g|γ/3
+for all/vectorP′∈suppfg
+/vectorQ. This holds because
+•of the constraints on the support of fg
+/vectorQ(see Section III.2);
+• |∂2E/vectorP
+∂|/vectorP|2| ≤C′′
+I; (see Property (P2) in Section III.1).
+After the result in Eq. (IV.39), which holds for sectors such that ( IV.26)
+(Lemma(IV.4))isfulfilled, wemaydistinguishthreepossiblesituations ,(A),
+(B), and(C), depending on the length of the vector /vector∇E/vectorP,/vectorP∈suppfg
+/vectorQ.
+(A)For some/vectorP∈suppfg
+/vectorQ,||/vector∇E/vectorP|−1| ≤ |g|γ/3.
+In this case, ∀ˆk∈ C1/2
+ˆQ, the inequality in (IV.26) holds true for ˜ γ=γ/2
+and|g|small enough, becauseDFP-Abs. Mass. Shell., February-8-2010 28
+i)||/vector∇E/vectorP′|−1| ≤2|g|γ/3,∀/vectorP′∈suppfg
+/vectorQ.
+ii) by definition
+C1/2
+ˆQ:={ˆk:|ˆk·ˆQ| ≤cos(1
+2|g|γ/8)}.
+Thus, we can use the estimate in Eq. (IV.39) with ˜ γ=γ/2,
+(Ψfg
+/vectorQ, Nn,C1/2
+ˆQΨfg
+/vectorQ)
+(Ψfg
+/vectorQ,Ψfg
+/vectorQ)≤ O(|g|2(1−2γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2),(IV.40)
+whereNn,C1/2
+ˆQ:=dΓ(χ2
+n(|/vectork|)ξg2
+C1/2
+ˆQ(ˆk)) (χn(|/vectork|) andξg
+C1/2
+ˆQ(ˆk) are defined in
+(a) and (b) of Section (III.2)).
+(B)For some/vectorP∈suppfg
+/vectorQ,|/vector∇E/vectorP|<1−|g|γ/3.
+The constraint (IV.26) with ˜ γ=γ/2 is fulfilled for all angular sectors.
+(C)For all/vectorP∈suppfg
+/vectorQ,|/vector∇E/vectorP|>1+|g|γ/3.
+First we notice that we can restrict our analysis to an angular secto r
+labeled by a direction ˆ usuch that, for some /vectorP∈suppfg
+/vectorQ, the inequality
+|1−ˆk·/vector∇E/vectorP| ≤ |g|˜γ(=γ/2)(IV.41)
+holds true for some ˆkbelonging to the sector under consideration. This
+is because, if
+|1−ˆk·/vector∇E/vectorP|>|g|˜γ(=γ/2), (IV.42)
+for allˆkbelonging to the given sector, then the result in (IV.27) holds,
+as we have proven in Lemma IV.4.
+We now show that the combination of |/vector∇E/vectorP|>1+|g|γ/3and (IV.41)
+yields the useful inequality (IV.46) below.
+We notice that, assuming the bound in Eq. (IV.41) for some ˆkbelong-
+ing to the sector, for |g|small enough,
+|1−ˆk·/vector∇E/vectorP| ≤2|g|˜γ(=γ/2), (IV.43)
+for allˆkin the sector labeled by ˆ u. Furthermore, ˆP·ˆQ≥cos(|g|γ),
+for all/vectorP∈suppfg
+/vectorQ, by construction, and we may assume that /vector∇E/vectorPis
+parallel to /vectorP; the other case, /vectorP·/vector∇E/vectorP=−|/vectorP||/vector∇E/vectorP|, can be treated inDFP-Abs. Mass. Shell., February-8-2010 29
+an analogous way. Let ηbe the angle between ˆkandˆQ. Then, (IV.43)
+means that
+−2|g|γ/2≤1−cos(η+ǫ)|/vector∇E/vectorP| ≤2|g|γ/2, (IV.44)
+whereǫ=O(|g|γ) and, for |g|small enough,
+1−c′|g|γ/3≥[1+2|g|γ/2]
+|/vector∇E/vectorP|≥cos(η+ǫ)≥[1−2|g|γ/2]
+|/vector∇E/vectorP|(IV.45)
+for some constant c′>0. Hence we have η≥c′′|g|γ/6>0 wherec′′>0,
+and we find that
+/vextendsingle/vextendsingle|/vectork⊥|2
+|/vectork|2−|/vectork⊥|
+|/vectork|ˆk⊥·/vector∇E/vectorP/vextendsingle/vextendsingle>O(sin2(η))≥ O(|g|γ/3)>0,(IV.46)
+for all/vectorkin the sector, where /vectork⊥:=/vectork−/vectork·/vectorQ
+|/vectorQ|2/vectorQ, because
+i) by assumption, ˆP·ˆQ>cos(|g|γ);
+ii)/vector∇E/vectorPis parallel (or antiparallel) to /vectorPand and |/vector∇E/vectorP|<C′
+I;
+iii)|/vectork⊥|
+|/vectork|= sin(η);
+iv)|ˆk⊥·/vector∇E/vectorP| ≤ |/vector∇E/vectorP|×O(|g|γ) using that ˆP·ˆQ>cos(|g|γ).
+Assuming, for example, that (IV.46) holds, because
+|/vectork⊥|2
+|/vectork|2−|/vectork⊥|
+|/vectork|ˆk⊥·/vector∇E/vectorP<−c|g|γ/3, (IV.47)
+wherec >0, we use the second virial identity (see Eq. (IV.14)) to
+obtain
+0 = (Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork⊥|2
+|/vectork|)Ψfg
+/vectorQ) (IV.48)
+−(Ψfg
+/vectorQ,/vector∇/vectorPE/vectorP·dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/vectork⊥)Ψfg
+/vectorQ)
+−g(Ψfg
+/vectorQ,[a∗(idˆu,ˆP′
+n,⊥ρ/vector x)+a(idˆu,ˆP′
+n,⊥ρ/vector x)]Ψfg
+/vectorQ),
+≤ −c|g|γ/3(Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork|)Ψfg
+/vectorQ) (IV.49)
++c′|g|1−γ/ba∇dblΨfg
+/vectorQ/ba∇dbl/ba∇dbldΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk))1/2Ψfg
+/vectorQ/ba∇dbl×(IV.50)
+×(/integraldisplay
+|/vectork|2β1(1
+2(n+1),3
+2n)(/vectork)d3k)1/2
+forsomec′>0. Nowweestimate(IV.49)similarlyto(IV.29)inLemma
+IV.4.DFP-Abs. Mass. Shell., February-8-2010 30
+Conclusions
+For|g|small enough, we have proven that:
+i) By combining cases (A),(B), and(C),
+(Ψfg
+/vectorQ, Nn,C1/2
+ˆQΨfg
+/vectorQ)
+(Ψfg
+/vectorQ,Ψfg
+/vectorQ)≤ O(|g|2(1−2γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2).(IV.51)
+Note that, in cases (B)and(C), the angular restriction was not used.
+ii) Under the assumption that
+||/vector∇E/vectorP|−1|>|g|γ/3,for all/vectorP∈fg
+/vectorQ,
+we have that
+(Ψfg
+/vectorQNnΨfg
+/vectorQ)
+(Ψfg
+/vectorQ,Ψfg
+/vectorQ)≤ O(|g|2(1−2γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2),(IV.52)
+whereNn:=dΓ(χ2
+n(|/vectork|). This follows from cases (B)and(C).
+IV.3 Number operator estimates in putative fiber eigen-
+vectors
+Using the results in Theorem IV.3 and Property (P3), we are now in a p o-
+sition to state some bounds on the expectation value of the boson n umber
+operator restricted to the fiber spaces. These bounds hold point wise in/vectorP, for
+|/vectorP|in the open interval I′
+g⊂Igintroduced in Section III.2; see Eqs. (III.9,
+III.10).
+Theorem IV.5. For|g|sufficiently small, and (/vectorP,E/vectorP)∈I′
+g×∆I:
+(Ψ/vectorP,E/vectorP, Nb
+n,C2
+ˆPΨ/vectorP,E/vectorP)≤ O(|g|(1−2γ)
+3|g|−γ/8n4
+3/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl1
+3
+2),
+(IV.53)
+where
+Nb
+n,C2
+ˆP:=dΓb(χ2
+n(|/vectork|)ξg2
+C2
+ˆP(ˆk)). (IV.54)DFP-Abs. Mass. Shell., February-8-2010 31
+Furthermore, if in addition ||/vector∇E/vectorP|−1|>3
+2|g|γ/3then
+(Ψ/vectorP,E/vectorP, Nb
+nΨ/vectorP,E/vectorP)≤ O(|g|(1−2γ)
+3n4
+3/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl1
+3
+2) (IV.55)
+where
+Nb
+n:=dΓb(χ2
+n(|/vectork|)). (IV.56)
+The constants in (IV.53), (IV.55) can be chosen uniformly in /vectorP,|/vectorP| ∈I′
+g⊂I
+(I′
+gis defined in Section III.2, Eqs. (III.9), (III.10)). They on ly depend on
+Iand on∆I.
+Proof
+First of all, we observe that, for /vectorPsuch thatfg
+/vectorQ(/vectorP) = 1, the inequality
+(Ψ/vectorP,E/vectorP, Nb
+n,C1/2
+ˆQΨ/vectorP,E/vectorP)≤ O(|g|(1−2γ)n/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2) (IV.57)
+can fail to hold true only for /vectorPin a setI∗
+fg
+/vectorQof measure bounded above by
+(Ψfg
+/vectorQ,Ψfg
+/vectorQ)O(g(1−2γ)n/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2), (IV.58)
+i.e.,
+/integraldisplay
+I∗
+fg
+/vectorQd3P≤(Ψfg
+/vectorQ,Ψfg
+/vectorQ)O(|g|(1−2γ)n/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2).(IV.59)
+This follows from inequality (IV.6), which we can write as
+/integraldisplay
+d3P¯fg
+/vectorQ(/vectorP)fg
+/vectorQ(/vectorP)(Ψ/vectorP,E/vectorP, Nb
+n,C1/2
+ˆQΨ/vectorP,E/vectorP) (IV.60)
+≤(Ψfg
+/vectorQ,Ψfg
+/vectorQ)O(|g|2(1−2γ)n2/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2
+2).(IV.61)
+Next, we make use of the following inequality, which holds in the sense o f
+quadratic forms,
+Nb
+n,C2
+ˆP≤Nb
+n,C1/2
+ˆQ(IV.62)
+for/vectorPin the support of fg
+/vectorQ. This inequality can be easily derived from the
+definitions of the smooth functions ξg
+C1/2
+ˆQ,ξg
+C2
+ˆP(see Section III.2) with support
+in the sets
+C1/2
+ˆQ:={ˆk:|ˆk·ˆQ| ≤cos(1
+2|g|γ/8)}, (IV.63)DFP-Abs. Mass. Shell., February-8-2010 32
+C2
+ˆP:={ˆk:|ˆk·ˆP| ≤cos(2|g|γ/8)}, (IV.64)
+respectively, and from the constraint ˆP·ˆQ≥cos(|g|γ).
+Hence, for /vectorP∈suppfg
+/vectorQ\I∗
+fg
+/vectorQsuch thatfg
+/vectorQ(/vectorP) = 1, we have that
+(Ψ/vectorP,E/vectorP, Nb
+n,C2
+ˆPΨ/vectorP,E/vectorP)≤(Ψ/vectorP,E/vectorP, Nb
+n,C1/2
+ˆQΨ/vectorP,E/vectorP) (IV.65)
+≤ O(|g|(1−2γ)n/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2),(IV.66)
+by definition of I∗
+f/vectorQ.
+Because of Eq. (IV.59), any point /vectorPbelonging to the set I∗
+fg
+/vectorQ, and such that
+fg
+/vectorQ(/vectorP) = 1, is at a distance at most
+(Ψfg
+/vectorQ,Ψfg
+/vectorQ)1/3O(|g|(1−2γ)
+3n1
+3/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl1
+3
+2) (IV.67)
+≤ O(|g|(1−2γ)
+3n1
+3/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl1
+3
+2) (IV.68)
+from an arbitrary point in suppfg
+/vectorQ\I∗
+fg
+/vectorQ. Thus we consider a slightly modified
+version of property (P3) for the operator Nb
+n,C2
+ˆP, namely
+|/vector∇/vectorP(Ψ/vectorP,E/vectorP, Nb
+n,C2
+ˆPΨ/vectorP,E/vectorP)| ≤ O(nCI[(sup
+/vectorP∈I|E/vectorP|)+1]|g|−γ/8) (IV.69)
+where, following the derivation of property (P3), the term |g|−γ/8comes from
+the derivative of the smooth function ξg
+C2
+ˆP. Using the fundamental theorem
+of calculus, we can finally state that
+(Ψ/vectorP,E/vectorP, Nb
+n,C2
+ˆPΨ/vectorP,E/vectorP)≤ O(|g|(1−2γ)
+3|g|−γ/8n4
+3/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl1
+3
+2|),/vectorP∈I∗
+fg
+/vectorQ.
+(IV.70)
+We remark that the bounds in Eqs. (IV.65), (IV.70) hold uniformly in /vectorQ,
+|/vectorQ| ∈I′
+g. The bounds in Eqs. (IV.65), (IV.70) hold for /vectorP≡/vectorQ, because
+f/vectorQ(/vectorQ) = 1 by definition. Thus, we arrive at the estimate in Eq. (IV.53) for
+any/vectorP∈I′
+g.
+Now, assume that for ( /vectorP∗,E/vectorP∗)∈I′
+g×∆I, we have ||/vector∇E/vectorP∗|−1|>3
+2|g|γ/3.
+Then we can consider a wave function fg
+/vectorQwith/vectorQ≡/vectorP∗. Thanks to Property
+P2, and for |g|small enough, i.e., less than some value |¯g|uniform in /vectorP∗,
+|/vectorP∗| ∈I, we have that ||/vector∇E/vectorP| −1|>|g|γ/3,for all/vectorP∈fg
+/vectorP∗. Thus, we
+can apply Theorem IV.3. Finally, following the same steps used before , oneDFP-Abs. Mass. Shell., February-8-2010 33
+arrives at the inequality in Eq. (IV.55) for /vectorP≡/vectorP∗. Notice that, in this case,
+since there is no angular restriction, no term proportional to |g|−γ/8appears
+on the RHS of Eq. (IV.55).
+The bound in Eq. (IV.55) trivially implies the corollary below.
+Corollary IV.6. Forβ >11/2, and for (/vectorP,E/vectorP)∈I′
+g×∆Iwith||/vector∇E/vectorP|−1|>
+3
+2|g|γ/3, the putative eigenvector Ψ/vectorP,E/vectorP(up to a suitable phase) is asymptotic
+to the vacuum vector Ψ0
+/vectorPinH/vectorP, asgtends to0. Likewise, the energy E/vectorPis
+asymptotic to /vectorP2/2. More precisely,
+/ba∇dblΨ0
+/vectorP−Ψ/vectorP,E/vectorP/ba∇dbl ≤ O(|g|(1−2γ)/6) (IV.71)
+and
+/vextendsingle/vextendsingle/vectorP2
+2−E/vectorP/vextendsingle/vextendsingle≤ O(|g|(1−2γ)/6). (IV.72)
+Proof
+The norm estimate in (IV.71) follows from Theorem IV.5. Without loss o f
+generality, we can start from the identity below, for some realandpositive
+coefficientc(g),
+Ψ/vectorP,E/vectorP=c(g)Ψ0
+/vectorP+Ψ(≥1)
+/vectorP,E/vectorP(IV.73)
+where Ψ /vectorP,E/vectorPand Ψ0
+/vectorPare normalized, and Ψ(≥1)
+/vectorP,E/vectorPcontains at least one boson.
+Then we can write:
+/ba∇dblΨ/vectorP,E/vectorP−Ψ0
+/vectorP/ba∇dbl2= (c(g)−1)2+/ba∇dblΨ(≥1)
+/vectorP,E/vectorP/ba∇dbl2(IV.74)
+=c(g)2+1−2c(g)+/ba∇dblΨ(≥1)
+/vectorP,E/vectorP/ba∇dbl2.(IV.75)
+Using the normalization condition,
+/ba∇dblΨ/vectorP,E/vectorP/ba∇dbl2= 1 =c(g)2+/ba∇dblΨ(≥1)
+/vectorP,E/vectorP/ba∇dbl2, (IV.76)
+we have
+c(g) =|1−/ba∇dblΨ(≥1)
+/vectorP,E/vectorP/ba∇dbl2|1/2(IV.77)
+and
+/ba∇dblΨ/vectorP,E/vectorP−Ψ0
+/vectorP/ba∇dbl2= 2−2c(g). (IV.78)
+From Theorem IV.5, it follows that
+/ba∇dblΨ(≥1)
+/vectorP,E/vectorP/ba∇dbl2≤ /ba∇dbl(Nb)1/2Ψ(≥1)
+/vectorP,E/vectorP/ba∇dbl2≤ O(|g|(1−2γ)/3),(IV.79)DFP-Abs. Mass. Shell., February-8-2010 34
+since the sum over nin (IV.55) can be estimated as
+/summationdisplay
+n≥1n4
+3/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl1
+3
+2≤/summationdisplay
+n≥1n4
+3n−2β+3
+6≤const., ifβ >11/2
+(IV.80)
+where we used that /ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl2=O(n−2β+3
+2), as follows by the size
+O(1/n) of the support of the function χn, and the spatial dimension d= 3.
+We remind the reader that the expectation value in Ψ /vectorP,E/vectorPof the number
+operator associated with boson momenta above |/vectork|= 1 can be bounded
+above by using the form inequality Hf<aH/vectorP+b, for somea,b>0.
+Consequently, the estimate in Eq. (IV.71) is easily obtained.
+For the inequality in Eq. (IV.72) consider
+E/vectorP−/vectorP2
+2= (Ψ /vectorP,E/vectorP,H/vectorP(Ψ/vectorP,E/vectorP−Ψ0
+/vectorP)) (IV.81)
++(Ψ/vectorP,E/vectorP,(H/vectorP−H0
+/vectorP)Ψ0
+/vectorP) (IV.82)
++(Ψ/vectorP,E/vectorP−Ψ0
+/vectorP,H0
+/vectorPΨ0
+/vectorP). (IV.83)
+Then use (IV.72) and the fact that H/vectorP−H0
+/vectorP=gφb(ρ) isH0
+/vectorP-bounded.
+V Absence of regular mass shells
+In this section, we first make use of the results obtained in Section I V to
+arrive at an argument that shows a contradiction to the existence of a mass
+shell (/vectorP,E/vectorP)∈Ig×∆Iassuming that ||/vector∇E/vectorP|−1|>3
+2|g|γ/3for/vectorP∈I′
+g. In
+implementing theargument weemploy suitabletrialstates; see Theo remV.1.
+Then we proceed to showing that, if we remove the assumption ||/vector∇E/vectorP|−1|>
+3
+2|g|γ/3, a mass shell might exist for ( /vectorP,E/vectorP)∈Ig×∆Isuch that
+E/vectorP=|/vectorP|−1
+2+O(|g|γ/4). (V.1)
+This result is completed in Theorem V.4.
+We recall that so far we have assumed the existence of a mass shell for/vectorPin
+the open interval Ig, and we have defined another open interval I′
+g⊂Igwith
+the properties specified in Section III.2. The results of Corollary IV .6, which
+will be used in the following theorem, hold for /vectorP∈I′
+g.DFP-Abs. Mass. Shell., February-8-2010 35
+Theorem V.1. Forβ >11/2, and for |g|small enough, no regular (i.e.,
+fulfilling the MainHypothesis in Section III.1.1) mass shell (/vectorP,E/vectorP)can exist
+with the properties:
+i)|/vectorP| ∈Ig,|Ig|>|g|γ/2;
+ii)|E/vectorP| ∈∆I;
+iii)||/vector∇E/vectorP|−1|>3
+2|g|γ/3for/vectorP∈I′
+g.
+ProofThe proof is by contradiction. For |g|sufficiently small (depending
+on the exponent γ), we pick an open interval I′′
+g⊂I′
+gfulfilling the following
+properties:
+(a)|I′′
+g|>|g|γ;
+(b) If|/vectorQ| ∈I′′
+gthen|/vectorP| ∈I′
+gfor any/vectorP∈suppfg
+/vectorQ.
+Notice that the definition of I′′
+gis meaningful for |g|small enough. For
+|/vectorQ| ∈I′′
+g, we introduce the trial vector
+η/vectorQ:=/integraldisplay
+d3P/integraldisplay
+d3kfg
+/vectorQ(/vectorP)1
+ǫ1
+2h/parenleftbig(/vectorP−/vectork)2/2+|/vectork|−E/vectorP
+ǫ/parenrightbig
+b∗
+/vectorkΨ0
+/vectorP,(V.2)
+where:
+•ǫ>0;
+•h(z)∈C∞
+0(R),h(z)≥0.
+Since Ψ fg
+/vectorQis a single-particle state, we have that
+(η/vectorQ,(H0−E/vectorP)Ψfg
+/vectorQ) =−(η/vectorQ, gφ(ρ/vector x)Ψfg
+/vectorQ), (V.3)
+whereH0:=/vector p2
+2+HfandE/vectorPis a (operator-valued) function of the total
+momentum operator /vectorP. This equation implies that
+g(η/vectorQ, φ(ρ/vector x)PΩΨfg
+/vectorQ) (V.4)
+=−(η/vectorQ,(H0−E/vectorP)P⊥
+ΩΨfg
+/vectorQ) (V.5)
+−g(η/vectorQ, φ(ρ/vector x)P⊥
+ΩΨfg
+/vectorQ), (V.6)DFP-Abs. Mass. Shell., February-8-2010 36
+where, as usual, the expressions PΩ,P⊥
+Ωacting on Hstand for 1Hel⊗PΩ,
+1Hel⊗P⊥
+Ω, respectively. We observe that
+(η/vectorQ, φ(ρ/vector x)PΩΨfg
+/vectorQ) (V.7)
+=c(g)/integraldisplay
+d3P/integraldisplay
+d3k|fg
+/vectorQ(/vectorP)|21
+ǫ1
+2h/parenleftbig(/vectorP−/vectork)2/2+|/vectork|−E/vectorP
+ǫ/parenrightbig
+ρ(|/vectork|),
+wherec(g)→1, asg→0, because of Corollary IV.6. Notice that, for
+|/vectorP|>1+δ, whereδ>0 isg-independent, the equation
+(/vectorP−/vectork)2/2+|/vectork|−/vectorP2/2 = 0,|/vectork|>0 (V.8)
+has the one-parameter family of solutions
+|/vectork|= 2(|/vectorP|cosθ−1)>0
+for cos(θ)−1
+|/vectorP|>0, where cos θ=/vectorP·/vectork
+|/vectorP||/vectork|.
+Notice that, for /vectorP∈I,ρ(2(|/vectorP|cosθ−1))/\e}atio\slash= 0 for some 0 < θ < π, see the
+conditions on ρin Section II.3. Hence, using (IV.72), for ǫand|g|small
+enough, we arrive at the following bound
+|(η/vectorQ, φ(ρ/vector x)PΩΨfg
+/vectorQ)|>D1ǫ1
+2/ba∇dblfg
+/vectorQ/ba∇dbl2
+2, (V.9)
+whereD1is anǫ- andg- independent (positive) constant; (hint: for each θ
+in Eq. (V.8), implement the change of variable |/vectork| →zθwithzθ:= [(/vectorP−
+/vectork)2/2+|/vectork|−E/vectorP]/ǫ).
+Using the Schwarz inequality, we find that
+|(η/vectorQ,N1
+2(H0−E/vectorP)P⊥
+ΩΨfg
+/vectorQ)| (V.10)
+≤ /ba∇dbl(H0−E/vectorP)η/vectorQ/ba∇dbl/ba∇dblN1
+2P⊥
+ΩΨfg
+/vectorQ/ba∇dbl. (V.11)
+We then observe that
+/ba∇dbl(H0−E/vectorP)η/vectorQ/ba∇dbl ≤ O(/ba∇dblfg
+/vectorQ/ba∇dbl2ǫ). (V.12)
+Using Eq. (IV.7), one may easily derive the inequalities
+/ba∇dblN1
+2P⊥
+ΩΨfg
+/vectorQ/ba∇dbl ≤ O(|g|2(1−2γ)
+2/ba∇dblfg
+/vectorQ/ba∇dbl2) (V.13)DFP-Abs. Mass. Shell., February-8-2010 37
+and
+|(η/vectorQ, φ(ρ/vector x)P⊥
+ΩΨfg
+/vectorQ)| (V.14)
+=|(η/vectorQ, Nφ(ρ/vector x)P⊥
+ΩΨfg
+/vectorQ)| (V.15)
+≤ O(|g|2(1−2γ)
+2/ba∇dblfg
+/vectorQ/ba∇dbl2). (V.16)
+For the step from (V.15) to (V.16), one may use that
+(η/vectorQ, Nφ(ρ/vector x)P⊥
+ΩΨfg
+/vectorQ) = (η/vectorQ, Nφ(−)(ρ/vector x)P⊥
+ΩΨfg
+/vectorQ) (V.17)
+whereφ(−)(ρ/vector x),φ(+)(ρ/vector x) stand for the part proportional to the annihilation-
+and to the creation operator, respectively; i.e., φ(ρ/vector x) =φ(−)(ρ/vector x)+φ(+)(ρ/vector x).
+Then, we observe that
+(η/vectorQ, Nφ(−)(ρ/vector x)P⊥
+ΩΨfg
+/vectorQ) = (V.18)
+= (η/vectorQ, φ(−)(ρ/vector x)NP⊥
+ΩΨfg
+/vectorQ) (V.19)
+−(η/vectorQ,[φ(−)(ρ/vector x), N]P⊥
+ΩΨfg
+/vectorQ), (V.20)
+and we finally use Theorem IV.3 together with the estimates
+/ba∇dblN1
+2φ(+)(ρ/vector x)η/vectorQ/ba∇dbl ≤ O(/ba∇dblfg
+/vectorQ/ba∇dbl2), (V.21)
+/ba∇dbl[φ(−)(ρ/vector x), N]P⊥
+ΩΨfg
+/vectorQ/ba∇dbl ≤ O(/ba∇dblN1
+2P⊥
+ΩΨfg
+/vectorQ/ba∇dbl). (V.22)
+Finally, we arrive at
+D1|g|ǫ1
+2/ba∇dblfg
+/vectorQ/ba∇dbl2
+2≤ O(ǫ|g|(1−2γ)/ba∇dblfg
+/vectorQ/ba∇dbl2
+2)+O(|g|2−2γ/ba∇dblfg
+/vectorQ/ba∇dbl2
+2).(V.23)
+This inequality is violated whenever
+c1|g|1−2γ<ǫ1
+2<c2|g|2γ(V.24)
+for somec1,c2>0. We note that the inequality in Eq. (V.24) can be fulfilled
+if 0<γ <1/4 and|g|is small enough.
+From the argument above, we conclude that, for |g|small enough, a mass
+shell cannot exist in Ig×∆Iwith the assumed regularity properties, because
+I′′
+g⊂I′
+g⊂Ig.
+We need two preparatory lemmas to state our final result, Theore m V.4,
+concerning the absence of a mass shell anywhere but near the bou ndary of
+the energy-momentum spectrum.DFP-Abs. Mass. Shell., February-8-2010 38
+From property (P1), we know that the vector /vector∇E/vectorPis collinear to /vectorP. In the
+first of the two lemmas below, Lemma V.2, assuming that ||/vector∇E/vectorP| −1| ≤
+3
+2|g|γ/3andβ >11/2, we show that /vector∇E/vectorPand/vectorPare in fact parallel.
+The second lemma, Lemma V.3, states that the boson number opera tor, re-
+stricted to the cone {ˆk:−ˆk·ˆP <cos(2|g|γ/8)}andevaluated onthe putative
+fibereigenvectorΨ /vectorP,E/vectorP,|/vectorP| ∈I′
+g, isalsoboundedaboveby O(|g|(1−2γ)
+3|g|−1/8),
+forβ >11/2.
+Lemma V.2. Forβ >11/2, and forgin an interval {g: 0<|g| ≤g∗}
+withg∗>0small enough, if (/vectorP,E/vectorP)∈Ig×∆Ifulfills the constraint
+||/vector∇E/vectorP|−1| ≤3
+2|g|γ/3(V.25)
+then the bound∂E/vectorP
+∂|/vectorP|≥1−3
+2|g|γ/3holds true.
+Proof
+The proof is indirect. We assume that there exists g∗>0 such that, for some
+|g|<g∗and for some /vectorP∗∈Ig,
+∂E/vectorP
+∂|/vectorP||/vectorP=/vectorP∗<−1+3
+2|g|γ/3<0. (V.26)
+We also assume that g∗is small enough to apply Lemma IV.4 and Theorem
+IV.5 later on. We shall show that the assumption in Eq. (V.26) yields a
+contradiction. Consider the function fg
+/vectorQ≡/vectorP∗. By Property P2
+∂E/vectorP
+∂|/vectorP|<c|g|γ,withc>0, (V.27)
+for all/vectorP∈suppfg
+/vectorQ≡/vectorP∗. Now, for all ˆ u-dependent sectors such that
+ˆu·ˆP∗>0,
+we consider thefirst virial identity of Section IV.2.1 (see Eqs. (IV.8) -(IV.13))
+and observe that
+−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/vectork)Ψfg
+/vectorQ) (V.28)
+≥ −c|g|γ(Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork|)Ψfg
+/vectorQ), (V.29)
+for all/vectorP∈suppfg
+/vectorQ≡/vectorP∗. Then, for |g|< g∗andg∗small enough, one can
+proceed as in Lemma IV.4, and finally apply the argument used in Theor em
+IV.5 to obtain that
+(Ψ/vectorP∗,E/vectorP∗, Nb
+n,ˆuΨ/vectorP∗,E/vectorP∗) (V.30)DFP-Abs. Mass. Shell., February-8-2010 39
+can be summed over n, yielding a quantity bounded by O(|g|γ/2). This result
+readily implies that
+(Ψ/vectorP∗,E/vectorP∗,/vectorPfΨ/vectorP∗,E/vectorP∗)·ˆP∗≤C|g|γ/2(V.31)
+for some positive constant C, hence
+−(Ψ/vectorP∗,E/vectorP∗,/vectorPfΨ/vectorP∗,E/vectorP∗)·ˆP∗≥ −C|g|γ/2(V.32)
+Using the Feynman-Hellman formula
+/vector∇E/vectorP=∂E/vectorP
+∂|/vectorP|ˆP=/vectorP−(Ψ/vectorP,E/vectorP,/vectorPfΨ/vectorP,E/vectorP), (V.33)
+we deduce that
+∂E/vectorP
+∂|/vectorP||/vectorP=/vectorP∗≥ |/vectorP∗|−C|g|γ/2>1−C|g|γ/2. (V.34)
+This yields a a contradiction for g∗small enough, therefore we conclude that
+the bound∂E/vectorP
+∂|/vectorP||/vectorP=/vectorP∗≥1−3
+2|g|γ/3(V.35)
+holdsfor {g|0<|g| ≤g∗}, forsomeg∗>0, becauseof(V.25).
+We are now in a position to extend the result in Eq. (IV.53).
+Lemma V.3. For(/vectorP,E/vectorP)∈I′
+g×∆I, with||/vector∇E/vectorP| −1| ≤3
+2|g|γ/3, and for
+β >11/2and|g|small enough,
+(Ψ/vectorP,E/vectorP, Nb
+n,C2
+ˆP∪C2,−
+ˆPΨ/vectorP,E/vectorP)≤ O(|g|(1−2γ)
+3|g|−1/8n4
+3/ba∇dbl|/vectork|β1(1
+2(n+1),3
+2n)(/vectork)/ba∇dbl1
+3
+2),
+(V.36)
+where
+Nb
+n,C2
+ˆP∪C2,−
+ˆP:=dΓb(χ2
+n(|/vectork|)ξg2
+Cg2
+ˆP∪C2,−
+ˆP(ˆk)) (V.37)
+andξg
+C2
+ˆP∪C2,−
+ˆP(ˆk),0≤ξg
+C2
+ˆP∪C2,−
+ˆP(ˆk)≤1, is a smooth function with support in
+C2
+ˆP∪ C2,−
+ˆP, (V.38)
+whereC2,−
+ˆP:={ˆk:−ˆk·ˆP≥cos(2|g|γ/8)}.ξg
+C2
+ˆP∪C2,−
+ˆP(ˆk)is defined as follows
+i)
+ξg
+C2
+ˆP∪C2,−
+ˆP(ˆk) = 1for{ˆk:ˆk·ˆP≤cos(4|g|γ/8)}; (V.39)DFP-Abs. Mass. Shell., February-8-2010 40
+ii)
+ξg
+C2
+ˆP∪C2,−
+ˆP(ˆk) = 0for{ˆk:ˆk·ˆP >cos(2|g|γ/8)}; (V.40)
+iii)
+|∂θˆkPξg
+C2
+ˆP∪C2,−
+ˆP(ˆk)| ≤Cξ|g|−γ/8, (V.41)
+whereθˆkPis the angle between ˆkandˆP, and the constant Cξis inde-
+pendent of g.
+Proof
+Because of Lemma V.2, for ˆkin the sector C2,−
+ˆQ≡/vectorPand/vectorP′∈suppfg
+/vectorQ≡/vectorP, with
+/vectorQ≡/vectorP∈I′
+g, theconditionin(IV.26)ofLemmaIV.4isfulfilled. Thenonecan
+repeat the arguments of Theorem IV.5 for the number operator r estricted to
+the sector C2,−
+ˆQ≡ˆP, and derive the inequality in Eq. (V.36) for all /vectorQ≡/vectorP∈I′
+g.
+Theorem V.4. Forβ >11/2and|g|small enough, if a regular mass shell
+(i.e. fulfilling the Main Hypothesis in Section III.1.1) exists in an interval
+Ig, and if for some (/vectorP,E/vectorP)∈I′
+g×∆I
+||/vector∇E/vectorP|−1| ≤3
+2|g|γ/3, (V.42)
+then, for all /vectorP∈Ig,
+E/vectorP=|/vectorP|−1
+2+O(|g|γ/4). (V.43)
+Proof
+We consider ( /vectorP,E/vectorP)∈I′
+g×∆Isuch that
+||/vector∇E/vectorP|−1| ≤3
+2|g|γ/3. (V.44)
+From the Feynman-Hellman formula (see Eq.(III.6))
+/vectorP·/vector∇E/vectorP=|/vectorP|2−/vectorP·(Ψ/vectorP,E/vectorP,/vectorPfΨ/vectorP,E/vectorP). (V.45)
+From the result in Lemma V.2, we can derive the following identity
+/vectorP·/vector∇E/vectorP=|/vectorP|(1+O(|g|γ/3)). (V.46)DFP-Abs. Mass. Shell., February-8-2010 41
+From Lemma V.3, for the expectation values in the equation below, we can
+restrict/vectorPfandHfto the sector C2,+
+ˆP:={ˆk:ˆk·ˆP≥cos(2|g|γ/8)}up to an
+o((|g|γ/4) remainder, and we deduce that
+ˆP·(Ψ/vectorP,E/vectorP,/vectorPfΨ/vectorP,E/vectorP) = (Ψ /vectorP,E/vectorP, HfΨ/vectorP,E/vectorP)+O(|g|γ/4).(V.47)
+Hence, by combining (V.45)-(V.47), one arrives at
+(Ψ/vectorP,E/vectorP, HfΨ/vectorP,E/vectorP)−/vectorP·(Ψ/vectorP,E/vectorP,/vectorPfΨ/vectorP,E/vectorP)
+=|/vectorP|−1+|/vectorP|−|/vectorP|2+O(|g|γ/4). (V.48)
+Next, starting from the formalvirial identity
+(Ψ/vectorP,E/vectorP, i[H/vectorP, Db
+1
+κ,κ]Ψ/vectorP,E/vectorP) = 0, (V.49)
+whereDb
+1
+κ,κ=dΓb(d1
+κ,κ) is defined in Section III.2 ( III), we derive
+0 = (Ψ /vectorP,E/vectorP, dΓb(i[|/vectork|, d1
+κ,κ])Ψ/vectorP,E/vectorP) (V.50)
++(Ψ/vectorP,E/vectorP, dΓb(i[/vectork, d1
+κ,κ])·dΓb(/vectork)Ψ/vectorP,E/vectorP)
+−/vectorP·(Ψ/vectorP,E/vectorP, dΓb(i[/vectork, d1
+κ,κ])Ψ/vectorP,E/vectorP)
+−g(Ψ/vectorP,E/vectorP,[b∗(id1
+κ,κρ)+b(id1
+κ,κρ)]Ψ/vectorP,E/vectorP).
+The virial identity in Eq. (V.50) needs to be justified and this is done in
+Section VI.3 in the Appendix.
+By taking the limit κ↑+∞on the RHS of (V.50), it follows that
+0 = (Ψ /vectorP,E/vectorP, HfΨ/vectorP,E/vectorP) (V.51)
++(Ψ/vectorP,E/vectorP,/vectorPf·/vectorPfΨ/vectorP,E/vectorP)
+−/vectorP·(Ψ/vectorP,E/vectorP,/vectorPfΨ/vectorP,E/vectorP)
+−g(Ψ/vectorP,E/vectorP,[b∗(id∞ρ)+b(id∞ρ)]Ψ/vectorP,E/vectorP),
+where
+d∞:=1
+2(/vectork·i/vector∇/vectork+i/vector∇/vectork·/vectork). (V.52)
+Eq. (V.51) follows from (V.50) thanks to
+1. the infrared behavior of the form factor ρ(/vectork), namely for any β >−1.
+2. the ultraviolet cut-off Λ; see Eq. (II.14).DFP-Abs. Mass. Shell., February-8-2010 42
+3. the fact that dΓb(i[|/vectork|, d1
+κ,κ]) anddΓb(i[/vectork, d1
+κ,κ]) are bounded by Hf
+and Ψ /vectorP,E/vectorPbelongs to the domain of Hf.
+Therefore, we can express the expectation value of ( /vectorPf)2in the state Ψ /vectorP,E/vectorP
+as a function of |/vectorP|up tog-dependent corrections
+(Ψ/vectorP,E/vectorP,(/vectorPf)2Ψ/vectorP,E/vectorP) = (|/vectorP|−1)2+O(|g|γ/4). (V.53)
+Using the eigenvalue equation (II.22), we obtain
+E/vectorP=1
+2/bracketleftbig
+(|/vectorP|−1)2+2|/vectorP|−|/vectorP|2+O(|g|γ)/bracketrightbig
++|/vectorP|−1+O(|g|γ/4)
+=|/vectorP|−1
+2+O(|g|γ/4). (V.54)
+Finally, because of the constraint on /vector∇E/vectorP(see Property P1, Section III.1),
+if Eq. (V.54) holds for |/vectorP| ∈I′
+g, either it is also true for |/vectorP| ∈Igor the mass
+shell cannot be defined on Igwith the assumed regularity properties. This
+can be explained considering the following two cases:
+a) if|Ig|<2|g|γ/4, use that |/vector∇E/vectorP|< C′
+Iand conclude that Eq. (V.54)
+holds onIg;
+b) if|Ig| ≥2|g|γ/4, writeIgasIg=∪jIj
+g, with{Ij
+g}disjoints, and 2 |g|γ/4>
+|Ij
+g|>|g|γ/2. ForeachIj
+g, either onecanrepeat theargument developed
+in Eqs. (V.44)-(V.54), and proceed as in a), or one concludes that
+the mass shell does not exists for /vectorP∈Ij
+g. In the latter case, since
+Ij
+g⊂Ig, the mass shell does not exist in Igwith the assumed regularity
+properties.
+Remark
+It is easy to see that
+E/vectorP≤/vectorP2
+2+O(|g|). (V.55)
+The proof follows from Eq. (V.51) by adding and subtracting /vectorP2on the
+right-hand side. In fact, one gets
+0 = (Ψ /vectorP,E/vectorP, H/vectorPΨ/vectorP,E/vectorP)−/vectorP2
+2(V.56)
++1
+2(Ψ/vectorP,E/vectorP,/vectorPf·/vectorPfΨ/vectorP,E/vectorP)
+−g(Ψ/vectorP,E/vectorP,[b∗(id∞ρ)+b(id∞ρ)]Ψ/vectorP,E/vectorP).DFP-Abs. Mass. Shell., February-8-2010 43
+Furthermore, assuming the validity of the Feynman-Helman formula , we
+see that
+/vectorP·/vector∇E/vectorP=/vectorP·(Ψ/vectorP,E/vectorP,(/vectorP−/vectorPf)Ψ/vectorP,E/vectorP) (V.57)
+≥/vectorP2−|/vectorP|/ba∇dbl/vectorPfΨ/vectorP,E/vectorP/ba∇dbl (V.58)
+From Eq. (V.56)
+/ba∇dbl/vectorPfΨ/vectorP,E/vectorP/ba∇dbl2≤/vectorP2−2E/vectorP+C|g|, C >0, (V.59)
+and then
+/vectorP·/vector∇E/vectorP≥/vectorP2−|/vectorP|/radicalBig
+/vectorP2−2E/vectorP+C|g| (V.60)
+For|/vectorP| ≥1 +δ, because of the constraint E/vectorP≥ |/vectorP| −1
+2+O(|g|), we can
+conclude that
+ˆP·/vector∇E/vectorP≥1−C′|g| (V.61)
+for some positive constant C′. This yields an alternative proof of Lemma
+V.2.
+VI Appendix
+In Sections VI.1 and VI.2, we provide the proofs of Lemmas IV.1 and I V.2
+in Section IV. For the convenience of the reader, these lemmas are repeated
+below. In Section VI.3, we prove the equality (V.50) in Section V.
+Lemma IV.2 and the equality (V.50) are virial identities whose justifica -
+tion is, in general, a hard task. We refer the reader to [9, 14] and [1 5] for
+more background.
+VI.1 Proof of Lemma IV.1
+Lemma (IV.1).The vector Ψfg
+/vectorQbelongs to the domain of the position oper-
+ator/vector xand
+/ba∇dblxiΨfg
+/vectorQ/ba∇dbl ≤ O(|g|−γ/ba∇dblΨfg
+/vectorQ/ba∇dbl), i= 1,2,3 (VI.1)DFP-Abs. Mass. Shell., February-8-2010 44
+Proof.It suffices to estimate, in the limit ∆ i→0,
+e−i∆ixiΨfg
+/vectorQ−Ψfg
+/vectorQ
+∆i(VI.2)
+=1
+∆i/bracketleftbig
+e−i∆ixi/integraldisplay
+fg
+/vectorQ(/vectorP)Ψ/vectorP,E/vectorPd3P−/integraldisplay
+fg
+/vectorQ(/vectorP)Ψ/vectorP,E/vectorPd3P/bracketrightbig
+=1
+∆i/bracketleftbig/integraldisplay
+fg
+/vectorQ(/vectorP)e−i∆ixiΨ/vectorP,E/vectorPd3P−/integraldisplay
+fg
+/vectorQ(/vectorP)Ψ/vectorP−∆iˆi,E/vectorP−∆iˆid3P/bracketrightbig
+(VI.3)
++1
+∆i/bracketleftbig/integraldisplay
+(fg
+/vectorQ(/vectorP)−fg
+/vectorQ(/vectorP−∆iˆi))Ψ/vectorP−∆iˆi,E/vectorP−∆iˆid3P/bracketrightbig
+(VI.4)
++1
+∆i/bracketleftbig/integraldisplay
+fg
+/vectorQ(/vectorP−∆iˆi)Ψ/vectorP−∆iˆi,E/vectorP−∆iˆid3P−/integraldisplay
+fg
+/vectorQ(/vectorP)Ψ/vectorP,E/vectorPd3P/bracketrightbig
+(VI.5)
+We notice that e−i∆ixiΨ/vectorP,E/vectorP∈ H/vectorP−∆iˆi(in (VI.3)), and
+I/vectorP−∆iˆi(e−i∆ixiΨ/vectorP,E/vectorP) =I/vectorP(Ψ/vectorP,E/vectorP) (VI.6)
+as vectors in Fb. The term in (VI.5) is identically zero, by a change of vari-
+ables. We now derive bounds for (VI.3), (VI.4), as ∆ i→0.
+Byitem(iii)intheMain Hypothesis (which, strictlyspeaking, meansthat
+/ba∇dbl/vector∇/vectorPI/vectorP(Ψ/vectorP,E/vectorP)/ba∇dbl ≤CI)andtheCauchy-Schwartz inequality, weconcludethat
+(VI.3) is bounded by CI/ba∇dblfg
+/vectorQ(/vectorP)/ba∇dbl2.
+For (VI.4), we use again Cauchy-Schwartz and the bound (for som e con-
+stantC)
+/ba∇dbl/vector∇/vectorPfg
+/vectorQ(/vectorP)/ba∇dbl2≤C|sup/vector∇/vectorPfg
+/vectorQ(/vectorP)|/ba∇dblfg
+/vectorQ(/vectorP)/ba∇dbl2=O(|g|−γ/ba∇dblfg
+/vectorQ(/vectorP)/ba∇dbl2),(VI.7)
+which can be checked from the construction of the functions fg
+/vectorQ(see below
+Eq. (III.11)).
+Collecting the bounds on (VI.3, VI.4, VI.5), we have proven the lemma .
+.
+VI.2 Proof of Lemma IV.2
+We now proceed with the proof of Lemma IV.2 in Section IV.DFP-Abs. Mass. Shell., February-8-2010 45
+Lemma (IV.2).The identity
+0 = (Ψfg
+/vectorQ, dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork|)Ψfg
+/vectorQ) (VI.8)
+−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/vectork)Ψfg
+/vectorQ) (VI.9)
+−g(Ψfg
+/vectorQ,[a∗(idˆu
+nρ/vector x)+a(idˆu
+nρ/vector x)]Ψfg
+/vectorQ) (VI.10)
+holds true. As the one-particle state Ψfg
+/vectorQbelongs to the form domain of all
+operators in (VI.8, VI.9, VI.10), this RHS is well-defined.
+Since the dilation operator is unbounded, we must check that a regu -
+larized expression for the commutator i[H−E/vectorP, Dˆu
+n] in Eq. (IV.2) is well
+defined and that, upon the removal of the regularization, the exp ectation
+value of that commutator in the state Ψfg
+/vectorQcorresponds to the right-hand
+side above, i.e. (VI.8, VI.9, VI.10). We show that, provided βis sufficiently
+large, the same strategy as implemented in [12] justifies this identity . Most
+of the arguments below, with the exception of the one in Section VI.2 .5, are
+standard in the literature.
+However, compared to the literature, our virial theorem has a little twist.
+Thisisduetothefactthatwedonotattempt toruleoutanyeigenve ctor, but
+merely an eigenvector with a certain regularity property. This is exp loited
+in Lemma IV.1 and it is a crucial ingredient of the justification of the vir ial
+identity in Lemma IV.2.
+In Section VI.2.1, we prove that the expressions in (VI.8, VI.9, VI.10 ) are
+well-defined. In Section VI.2.2, we start the proof of the equality in L emma
+IV.2.
+VI.2.1 Well-definedness of the terms (VI.8, VI.9, VI.10)
+The operators
+dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)|/vectork|) and/vector∇/vectorPE/vectorP·dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/vectork) (VI.11)
+are bounded by a (multiple of) Hf. In fact, the operator /vector∇E/vectorPis surely
+bounded if we restricted the total Hilbert space to the fibers /vectorP∈I. This
+restriction can be done since the function fg
+/vectorQhas support in I. Since
+Ψfg
+/vectorQ∈Dom(H)⇒Ψfg
+/vectorQ∈Dom(Hf) (VI.12)
+the expressions (VI.8) and (VI.9) are well-defined. Next, from the expression
+in (IV.31) and the fact that ρ∈C1, we have
+/integraldisplay
+d3k1
+|/vectork|sup
+/vector x/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
+|/vector x|+1(dˆu
+nρ/vector x)(/vectork)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+<∞. (VI.13)DFP-Abs. Mass. Shell., February-8-2010 46
+and hence, by a standard argument for bounding creation/annihila tion op-
+erators,
+/ba∇dbl1
+|/vector x|+1a(idˆu
+nρ/vector x)1
+(Hf+1)/ba∇dbl<∞. (VI.14)
+Since Ψfg
+/vectorQ∈Dom(/vector x)∩Dom(Hf) by Lemma IV.1 and (VI.12), it follows that
+also the expression (VI.10) makes sense.
+VI.2.2 Virial Identity with a regularized dilation operato r
+We introduce the regularized gradient
+/vector∇ǫ
+/vectork:=/vector∇/vectork
+1−ǫ∆/vectork, (VI.15)
+where the parameter ǫ >0 will be eventually removed. Consequently, we
+also define Dˆu,ǫ
+n:=dΓ(dˆu,ǫ
+n) wheredˆu,ǫ
+ncorresponds to dˆu
+nwith/vector∇/vectorkreplaced
+by/vector∇ǫ
+/vectork. Since, thanks to the regularization, Dˆu,ǫ
+nis bounded w.r.t. to Hf, we
+deduce that Ψfg
+/vectorQ∈Dom(Dˆu,ǫ
+n) (cfr. (VI.12)).
+We claim that
+i((H−E/vectorP)Ψfg
+/vectorQ, Dˆu,ǫ
+nΨfg
+/vectorQ)−i(Dˆu,ǫ
+nΨfg
+/vectorQ,(H−E/vectorP)Ψfg
+/vectorQ) (VI.16)
+= (Ψfg
+/vectorQ, dΓ(i[|/vectork|, dˆu,ǫ
+n])Ψfg
+/vectorQ) (VI.17)
+−i(Ψfg
+/vectorQ[E/vectorP,Dˆu,ǫ
+n]Ψfg
+/vectorQ) (VI.18)
+−g(Ψfg
+/vectorQ,[a∗(idˆu,ǫ
+nρ/vector x)+a(idˆu,ǫ
+nρ/vector x)]Ψfg
+/vectorQ) (VI.19)
+where the LHS makes sense since Ψfg
+/vectorQ∈Dom(Dˆu,ǫ
+n) and the RHS is obtained
+byformalevaluation of the commutator [ H−E/vectorP,Dˆu,ǫ
+n]. All terms on the
+RHS are well-defined by similar (but easier) arguments as those in Sec tion
+VI.2.1(forexample, notethat[ |/vectork|, dˆu,ǫ
+n]isaboundedoperator). Nevertheless,
+the equality above requires a justification. In the case at hand, a p edestrian
+way to provide such a justification is to introduce cutoffs in /vector x,/vectorkandN(the
+number operator) such that all operators involved are bounded, calculate the
+commutator and finally remove the cutoffs.
+Since (H−E/vectorP)Ψfg
+/vectorQ= 0 by assumption, the expression (VI.16) vanishes.
+Thus, it is enough to prove that the expressions (VI.17, VI.18, VI.1 9) con-
+verge to (VI.8, VI.9, VI.10), respectively, as ǫtends to 0. These three con-
+vergence statements will be established Sections VI.2.4, VI.2.5 and V I.2.6,
+respectively.DFP-Abs. Mass. Shell., February-8-2010 47
+VI.2.3 Some properties of the regularized dilation operato r
+In this preparatory section, we state some estimates on
+ei/vector z·/vectorPDˆu,ǫ
+ne−i/vector z·/vectorP−Dˆu,ǫ
+n (VI.20)
+that will be useful in taking the limit ǫ→0. First, we remark that
+ei/vector z·/vectorPDˆu,ǫ
+ne−i/vector z·/vectorP=dΓ(dˆu,ǫ
+n,/vector z), dˆu,ǫ
+n,/vector z:=ei/vector z·/vectorkdˆu,ǫ
+ne−i/vector z·/vectork(VI.21)
+on the appropriate domain. Explicitly,
+dˆu,ǫ
+n,/vector z=χn(|/vectork|)ξg
+ˆu(ˆk)1
+2(/vectork·/vectorFǫ(i/vector∇/vectork+/vector z)+/vectorFǫ(i/vector∇/vectork+/vector z)·/vectork)ξg
+ˆu(ˆk)χn(|/vectork|) (VI.22)
+and/vectorFǫis the family of R3/ma√sto→R3functions given by (cfr. (VI.15))
+/vectorFǫ(/vector y) =/vector y
+1+ǫ|/vector y|2. (VI.23)
+We define the vector operator /vectordˆu,ǫ
+n,/vector zsuch that it satisfies
+/vector z·/vectordˆu,ǫ
+n,/vector z=dˆu,ǫ
+n,/vector z−dˆu,ǫ
+n. (VI.24)
+Namely,
+(/vectordˆu,ǫ
+n,/vector z)j:=χn(|/vectork|)ξg
+ˆu(ˆk)1
+2/summationdisplay
+l/parenleftbig
+kl/integraldisplay1
+0dt(/vector∇Fǫ,l)j(i/vector∇/vectork+t/vector z)/parenrightbig
+ξg
+ˆu(ˆk)χn(|/vectork|)+h.c. .
+(VI.25)
+where the subscripts landjlabel vector components. To check that (VI.24)
+holds, we substitute the line integral
+Fǫ,l(/vector y+/vector z)−Fǫ,l(/vector y) =/vector z·/integraldisplay1
+0dt/vector∇Fǫ,l(/vector y+t/vector z), l= 1,2,3,(VI.26)
+into the explicit expression for (VI.22), using the functional calculu s.
+We derive immediately the following properties
+1. The operator norms
+/ba∇dbl/vectordˆu,ǫ
+n,/vector z/ba∇dbl,/ba∇dbl/vectorkχ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/ba∇dbl, (VI.27)
+and hence also
+/ba∇dbldΓ(/vectordˆu,ǫ
+n,/vector z)1
+(Hf+1)/ba∇dbl /ba∇dbldΓ(/vectorkχ2
+n(|/vectork|)ξg2
+ˆu(/vectork))1
+(Hf+1)/ba∇dbl,(VI.28)
+are bounded uniformly in ǫand in/vector z∈R3. For the operators on the
+left (involving /vectordˆu,ǫ
+n,/vector z), this follows from the fact that sup/vector y,ǫ/ba∇dbl/vector∇Fǫ,j(/vector y)/ba∇dblis
+bounded. For the operators on the right, this is a trivial conseque nce
+of the momentum cutoff functions.DFP-Abs. Mass. Shell., February-8-2010 48
+2. For each /vector z,
+/vectordˆu,ǫ
+n,/vector zstrongly−→
+ǫ→0/vectorkχ2
+n(|/vectork|)ξg2
+ˆu(ˆk). (VI.29)
+dΓ(/vectordˆu,ǫ
+n,/vector z)1
+(Hf+1)strongly−→
+ǫ→0dΓ(/vectorkχ2
+n(|/vectork|)ξg2
+ˆu(/vectork))1
+(Hf+1)(VI.30)
+This convergence on Dom( /vector∇/vectork) and Dom( dΓ(/vector∇/vectork))∩ Ffinfollows by
+/vector∇Fǫ,j(/vector y)→ˆyj, asǫ→0, pointwise in /vector y. Convergence on all vectors
+then follows by using the uniform boundedness (VI.27, VI.28) above .
+VI.2.4 The term [Hf,Dˆu
+n]
+In this section, we show that (VI.17) converges to (VI.8), as ǫ→0.
+We derive
+dΓ(i[|/vectork|, dˆu,ǫ
+n])1
+Hf+1strongly−→
+ǫ→0dΓ/parenleftBig
+|/vectork|χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/parenrightBig1
+Hf+1
+(VI.31)
+in exactly the same way as we did to arrive at (VI.30). That is, we first
+establish (using properties of Fǫ) that
+sup
+ǫ/ba∇dbli[|/vectork|, dˆu,ǫ
+n]/ba∇dbl<∞,
+and that, on the dense domain Dom( /vector∇/vectork), the operator i[|/vectork|, dˆu,ǫ
+n] converges
+to|/vectork|χ2
+n(|/vectork|)ξg2
+ˆu(ˆk). Since Ψfg
+/vectorQ∈Dom(Hf), we conclude that
+dΓ/parenleftBig
+i[|/vectork|, dˆu,ǫ
+n]/parenrightBig
+Ψfg
+/vectorQ−→
+ǫ→0dΓ/parenleftBig
+|/vectork|χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/parenrightBig
+Ψfg
+/vectorQ(VI.32)
+We have proven that the difference between (VI.17) and (VI.8) van ishes as
+ǫ→0.
+VI.2.5 The term [E/vectorP,Dˆu
+n]
+In this section, we show that (VI.18) converges to (VI.9), as ǫ→0.
+We consider an extension of the function E/vectorP, that is twice differentiable
+(see Section III.1.2) and of compact support K(i.e.,{/vectorP||/vectorP| ∈I} ⊂ K). We
+use the same symbol, E/vectorP, for the function extended to K, and we write
+E/vectorP=/integraldisplay
+d3zˆE(/vector z)ei/vector z·/vectorP, (VI.33)DFP-Abs. Mass. Shell., February-8-2010 49
+whereˆE(/vector z) is the Fourier transform of E/vectorP(up to the prefactor (2 π)−3/2).
+SinceE/vectorPis twice differentiable and of compact support, |/vector z|2ˆE(/vector z) belongs
+toL2(R3;d3z) and, by Cauchy-Schwarz, ˆE(/vector z) is inL1(R3;d3z). Therefore,
+using the functional calculus, we can write,
+(Ψfg
+/vectorQ,[E/vectorP, Dˆu,ǫ
+n]Ψfg
+/vectorQ) =/integraldisplay
+d3zˆE(/vector z)(Ψfg
+/vectorQ,[ei/vector z·/vectorP, Dˆu,ǫ
+n]Ψfg
+/vectorQ).(VI.34)
+Then we observe that, on e.g. the domain Dom( Hf),
+ei/vector z·/vectorPDˆu,ǫ
+n−Dˆu,ǫ
+nei/vector z·/vectorP= (ei/vector z·/vectorPDˆu,ǫ
+ne−i/vector z·/vectorP−Dˆu,ǫ
+n)ei/vector z·/vectorP=/vector z·dΓ(/vectordˆu,ǫ
+n,/vector z)ei/vector z·/vectorP(VI.35)
+with the bounded operator /vectordˆu,ǫ
+n,/vector zas defined in Section VI.2.3. We are now
+ready to compare (VI.18) with (VI.9):
+i(Ψfg
+/vectorQ,[E/vectorP, Dˆu,ǫ
+n]Ψfg
+/vectorQ)−(Ψfg
+/vectorQ,/vector∇E/vectorP·dΓ(χ2
+n(|/vectork|)ξg2
+ˆu(ˆk)/vectork)Ψfg
+/vectorQ) (VI.36)
+=i/integraldisplay
+d3zˆE(/vector z)(Ψfg
+/vectorQ,/parenleftBig
+dΓ(/vectordˆu,ǫ
+n,/vector z)−dΓ(/vectorkχ2
+n(|/vectork|)ξg2
+ˆu(/vectork)/parenrightBig
+·/vector zei/vector z·/vectorPΨfg
+/vectorQ) (VI.37)
+=−i/integraldisplay
+d3zˆE(/vector z)(/vector xΨfg
+/vectorQ,[dΓ(/vectordˆu,ǫ
+n,/vector z)−dΓ(/vectorkχ2
+n(|/vectork|)ξg2
+ˆu(/vectork))]ei/vector z·/vectorPΨfg
+/vectorQ) (VI.38)
++i/integraldisplay
+d3zˆE(/vector z)(Ψfg
+/vectorQ,[dΓ(/vectordˆu,ǫ
+n,/vector z)−dΓ(/vectorkχ2
+n(|/vectork|)ξg2
+ˆu(/vectork))]ei/vector z·/vectorP/vector xΨfg
+/vectorQ) (VI.39)
+The first equality follows by (VI.34, VI.35, VI.24) and the fact that t he
+Fourier transform sends differentiation into multiplication. To obtain the
+second equality, we used the canonical commutation relation
+/vector zei/vector z·/vectorP= [ei/vector z·/vectorP,/vector x] (VI.40)
+which holds e.g. on Dom( /vector x)∩Dom(Hf).
+SinceˆE(/vector z)∈L1(R3;d3z), we can estimate (VI.38)
+|(VI.38)| ≤/integraldisplay
+d3z|ˆE(/vector z)|/vextendsingle/vextendsingle/vextendsingle(/vector xΨfg
+/vectorQ,[dΓ(/vectordˆu,ǫ
+n,/vector z)−dΓ(/vectorkχ2
+n(|/vectork|)ξg2
+ˆu(/vectork))]ei/vector z·/vectorPΨfg
+/vectorQ)/vextendsingle/vextendsingle/vextendsingle
+For each/vector z, the second factor vanishes as ǫ→0 by (VI.30) and the fact that
+Ψfg
+/vectorQ∈Dom(/vector x)∩Dom(Hf). Hence we conclude that (VI.38 ) tends to zero as
+ǫtends tozero, by dominated convergence. Obviously, (VI.39)can betreated
+in exactly the same way and hence we have proven that (VI.37) vanis hes as
+ǫ→0.
+Hence, we have shown that the difference between (VI.18) and (VI .9)
+vanishes, as ǫ→0.DFP-Abs. Mass. Shell., February-8-2010 50
+VI.2.6 The term [gφ(ρ/vector x),Dˆu
+n]
+In this section, we prove that (VI.19) converges to (VI.10) as ǫ↓0.
+First, we note that
+sup
+ǫ/integraldisplay
+d3k1
+|/vectork|sup
+/vector x/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
+|/vector x|+1(dˆu,ǫ
+nρ/vector x)(/vectork)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+<∞, (VI.41)
+This follows in the same way as (VI.13) , established in Section VI.2.1. To -
+gether, (VI.41) and (VI.13) imply that the operator norms of
+Rˆu
+n,ǫ:=1
+Hf+1a∗(i(dˆu,ǫ
+n−dˆu
+n)ρ/vector x)1
+|/vector x|+1(VI.42)
+(Rˆu
+n,ǫ)∗:=1
+|/vector x|+1a(i(dˆu,ǫ
+n−dˆu
+n)ρ/vector x)1
+Hf+1(VI.43)
+are uniformly bounded in ǫ. We can now take advantage of the fact that
+Ψfg
+/vectorQ∈Dom(/vector x)∩Dom(Hf) to write
+(Ψfg
+/vectorQ,[a∗(i(dˆu,ǫ
+n−dˆu
+n)ρ/vector x)+a(i(dˆu,ǫ
+n−dˆu
+n)ρ/vector x)]Ψfg
+/vectorQ) (VI.44)
+= ((Hf+1)Ψfg
+/vectorQ, Rˆu
+n,ǫχKδ(|/vector x|+1)Ψfg
+/vectorQ) (VI.45)
++((Hf+1)Ψ fg
+/vectorQ,Rˆu
+n,ǫ(|/vector x|+1)(1−χKδ)Ψfg
+/vectorQ) (VI.46)
++((|/vector x|+1)Ψfg
+/vectorQ, χKδ(Rˆu
+n,ǫ)∗(Hf+1)Ψfg
+/vectorQ) (VI.47)
++((1−χKδ)(|/vector x|+1)Ψ fg
+/vectorQ,(Rˆu
+n,ǫ)∗(Hf+1)Ψ fg
+/vectorQ) (VI.48)
+whereχKδ=χKδ(/vector x) is the characteristic function of a compact set Kδ⊂R3,
+chosen such that the |(VI.46)|,|(VI.48)|are smaller than δ. This can be done
+by the uniform bound on /ba∇dblRˆu
+n,ǫ/ba∇dbl, and the fact that /ba∇dbl(χKδ−1)(|/vector x|+1)Ψfg
+/vectorQ/ba∇dbl
+can be made arbitrarily small by choosing Kδbig enough. Moreover, for any
+compactK,
+lim
+ǫ→0/integraldisplay
+d3k1
+|/vectork|[sup
+/vector x∈K|((idˆu,ǫ
+n−idˆu
+n)ρ/vector x)(/vectork)|]2= 0,(VI.49)
+This implies that /ba∇dblχKRˆu
+n,ǫ/ba∇dbl,/ba∇dblχK(Rˆu
+n,ǫ)∗/ba∇dbland hence (VI.45), (VI.47) vanish,
+asǫ→0. Together, the bounds on (VI.45), (VI.47) and on (VI.46), (VI.4 8)
+prove that (VI.44) vanishes in the limit ǫ→0. Hence, the difference of
+(VI.19) and (VI.10) vanishes as ǫ↓0.DFP-Abs. Mass. Shell., February-8-2010 51
+VI.3 Proof of the fiber virial identity in (V.50)
+The justification of the virial identity in (V.50) is largely analogous to t hat
+of the virial identity in Lemma IV.2. To avoid repetitive arguments, we just
+sketch the main strategy of the proof.
+First one introduces a regularized dilation operator dǫ
+1
+κ,κand the corre-
+sponding second quantized operator Db,ǫ
+1
+κ,κ:=dΓb(dǫ
+1
+κ,κ). The operator dǫ
+1
+κ,κis
+obtained from d1
+κ,κ(see Eq. (III.20)) by replacing the gradient, /vector∇/vectork, with
+/vector∇ǫ
+/vectork:=/vector∇/vectork
+1−ǫ∆/vectork, ǫ>0. (VI.50)
+Then one exploits the following properties:
+i) On the dense subspace Dom( /vector∇/vectork)∈h,
+i[|/vectork|, dǫ
+1
+κ,κ]→ |/vectork|χ2
+[1
+κ,κ](|/vectork|), (VI.51)
+i[/vectork, dǫ
+1
+κ,κ]→/vectorkχ2
+[1
+κ,κ](|/vectork|) (VI.52)
+asǫ→0. (Strong convergence on the whole of hfollows than from ii)
+below).
+ii) The operator norms
+/ba∇dbl[|/vectork|, dǫ
+1
+κ,κ]/ba∇dbl,/ba∇dbl[/vectork, dǫ
+1
+κ,κ]/ba∇dbl (VI.53)
+/ba∇dbldΓb(i[|/vectork|, dǫ
+1
+κ,κ])1
+(1+Hf)/ba∇dbl,/ba∇dbldΓb(i[/vectork, dǫ
+1
+κ,κ])1
+(1+Hf)/ba∇dbl(VI.54)
+are bounded uniformly in ǫ.
+iii)
+lim
+ǫ→0/integraldisplay
+d3k1
+|/vectork||(idǫ
+1
+κ,κ−id1
+κ,κ)ρ(/vectork)|2= 0.(VI.55)
+iv) the operator norm
+/ba∇dblb(idǫ
+1
+κ,κρ)1
+(Hf+1)1/2/ba∇dbl (VI.56)
+is uniformly bounded in ǫ.
+v)
+/vextenddouble/vextenddouble/vextenddouble/vextenddoubleb(idǫ
+(1
+κ,κ−id1
+κ,κ)ρ(/vectork))1
+(1+Hf)1/2/vextenddouble/vextenddouble/vextenddouble/vextenddouble2
+≤/integraldisplay
+d3k1
+|/vectork||(idǫ
+1
+κ,κ−id1
+κ,κ)ρ(/vectork))|2
+(VI.57)DFP-Abs. Mass. Shell., February-8-2010 52
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+[18] A. Pizzo. One Particle (improper) States in Nelson’s Massless Mod el.
+Ann. H. Poincar´ e ,4(3), 439–486 (2003).
+[19] A. Pizzo. Scattering of an Infraparticle: The One Particle Sect or in
+Nelson’s Massless Model. Ann. H. Poincar´ e ,6, 553–606 (2005).
+[20] H. Spohn. The polaron at large total momentum. J. Phys. A ,21,
+1199–1212 (1988).
\ No newline at end of file
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+arXiv:1001.0689v1  [nlin.CD]  5 Jan 2010INTERACTION AND CHAOTIC DYNAMICS OF THE
+CLASSICAL HYDROGEN ATOM IN AN ELECTROMAGNETIC
+FIELD
+M. Alaburda, V. Gontis and B. Kaulakys
+Institute of Theoretical Physics and Astronomy, A. Goštaut o 12, 2600 Vilnius, Lithuania
+Expressions for energy and angular momentum changes of the h ydrogen atom
+due to interaction with the electromagnetic field during the period of the
+electron motion in the Coulomb field are derived. It is shown t hat only
+the energy change for the motion between two subsequent pass ings of the
+pericenter is related to the quasiclassical dipole matrix e lement for transitions
+between excited states.
+1 Introduction
+Classical hydrogen atom in a monochromatic electromagneti c field is one of the simplest
+real nonlinear system which dynamics may be regular or chaot ic [1, 2], depending on the
+relative field strength and frequency. Even one-dimensiona l classical model of highly excited
+atom yields results sufficiently close to the experimental fin dings. For theoretical analysis
+approximate mapping equations of motion, rather than differ ential equations, are most con-
+venient [2–7]. Here a two-dimensional map (for the scaled en ergy and for relative phase of
+the field) is generalized for two-dimensional hydrogen atom , i.e. we calculate energy and
+angular momentum changes of the atom interacting with the el ectromagnetic field.
+2 One-dimensional atom in monochromatic field
+The Hamiltonian of the hydrogen atom in a linearly polarized monochromatic electromag-
+netic field (in atomic units) is [4, 7]
+H=1
+2/parenleftbigg
+P+1
+cA/parenrightbigg2
+−1
+r. (1)
+HerePis the generalized momentum, cis the light velocity,
+A=−cF
+ωsin(ωt+ϑ) (2)
+is the vector potential of the field, F,ωandϑare the field strength amplitude, field fre-
+quency and phase, respectively. The change of the electron e nergy can be obtained from the
+Hamiltonian equations of motion [8]
+˙E=−r·Fcos(ωt+ϑ). (3)
+One can introduce the scaled energy Es=E/ω2/3and the scaled field strength Fs=
+F/ω4/3. However, it is convenient [2–7] to introduce the positive s caled energy ε=−2Es
+and the relative field strength F0=Fs/ε2
+0, withε0being the initial scaled energy.
+1Integration of eq. (3) over the period of time between two sub sequent passages of the
+electron at the apocenter results in the change of the electr on energy [3, 4]
+εj+1=εj−πF0ε2
+0h(εj+1)sinϑj, (4)
+where
+h(εj+1) =4
+εj+1J′
+sj+1(sj+1). (5)
+Heres=ε−3/2=ω/(−2E)3/2is the relative frequency of the field, i.e. the ratio of the fie ld
+frequency to the electron Kepler orbital frequency and J′
+s(s)is the derivative of the Anger
+function. Introducing a generating function G(εj+1,ϑj)[9, 10] one can calculate the phase
+ϑchange over the period
+ϑj+1=ϑj+2πε−3/2
+j+1−πF0ε2
+0η(εj+1)cosϑj, (6)
+where
+η(εj+1) =dh(εj+1)
+dεj+1. (7)
+Figure 1. Trajectories (ε,ϑ)for the map (4) and (6) with the parameter πF0ε2
+0= 0.0035and initial condi-
+tionsϑ0=π,ε0= 0.3−0.003i(i= 0,1,2,...).
+Equations (4) and (6) describe the changes of the energy and p hase in time. This map
+greatly facilitates numerical investigation of dynamics a nd ionization process. In figures 1
+and figure 2 results of the numerical analysis of map (4) - (6) a re presented.
+3 Two-dimensional atom in monochromatic field
+For calculation of the energy change of the arbitrary orient ated two-dimensional atom in the
+electromagnetic field according to equation (3) one should p erform the transformation of the
+coordinates (figure 3). The change of the angular momentum of the atom it follows from the
+Hamiltonian equations of motion
+˙M=−∂H
+∂α=−rFsin(α+ϕ)cos(ωt+ϑ). (8)
+20,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,60,000,020,040,060,080,100,120,14
+ 1
+ 2
+F0c
+s0
+1 2 3 4 50,0150,0200,0250,0300,0350,0400,0450,050
+ 1
+ 2
+F0c
+s0
+Figure 2. Ionization threshold field dependence on the relative frequ encys0. Continuous curve represents
+results calculated with variation of the initial phase ϑ0, while dotted curve is for ϑ0= 0.
+By analogy with the scaled energy one can introduce the scale d angular momentum
+µ= 2Ms= 2Mω1/3. Moreover, it is convenient to introduce the parametric equ ations of
+motion for the Coulomb potential [6–8].
+Integration of equations (3) and (8) for the half of the perio d of the electron motion, i.e.
+for transition from apocenter to pericenter yield the scale d energy and angular momentum
+changes in the linearly polarized electromagnetic field
+εj+1=εj+2πF0ε2
+0
+εj+1{−[J′
+s(z)sinϑj+E′
+s(z)cosϑj]cosϕ
++√
+1−e2
+e/bracketleftbigg/parenleftbigg
+Js(z)−sin(sπ)
+sπ/parenrightbigg
+cosϑj−/parenleftbigg
+Es(z)−1−cos(sπ)
+sπ/parenrightbigg
+sinϑj/bracketrightbigg
+sinϕ/bracerightbigg
+(9)
+3Figure 3. Two-dimensional atom in the electromagnetic field.
+and
+µj+1=µj+2πF0ε2
+0
+εj+1/braceleftbigg√
+1−e2
+e/bracketleftbigg/parenleftbigg
+Js(z)−sin(sπ)
+sπ/parenrightbigg
+sinϑj
++/parenleftbigg
+Es(z)−1−cos(sπ)
+sπ/parenrightbigg
+cosϑj/bracketrightbigg
+cosϕ+/bracketleftbigg/parenleftbigg
+−J′
+s(z)+(1+e)sin(sπ)
+π/parenrightbigg
+cosϑj
++/parenleftbigg
+E′
+s(z)+(1+e)cos(sπ)
+π+1−e
+π/parenrightbigg
+sinϑj/bracketrightbigg
+sinϕ/bracerightbigg
+.(10)
+HereEs(z)is the Weber function, z=esand e is the eccentricity of the ellipsis. By
+analogy or from equations (9) and (10) choosing appropriate initial phases of the field one can
+calculate the energy and angular momentum changes for the el ectron motion from pericenter
+to apocenter as well as for the complete period with different initial conditions.
+So, for motion between two subsequent passages at the apocen ter we have generalization
+of eq. (4) for the energy change
+εj+1=εj+4πF0ε2
+0
+εj+1/braceleftbigg
+−J′
+s(z)sinϑjcosϕ+√
+1−e2
+e/bracketleftbigg
+Js(z)−sin(sπ)
+sπ/bracketrightbigg
+cosϑjsinϕ/bracerightbigg
+(11)
+and the angular momentum change
+µj+1=µj+4πF0ε2
+0
+εj+1/braceleftbigg√
+1−e2
+e/bracketleftbigg
+Js(z)−sin(sπ)
+sπ/bracketrightbigg
+sinϑjcosϕ
++/bracketleftbigg
+−J′
+s(z)+(1+e)sin(sπ)
+π/bracketrightbigg
+cosϑjsinϕ/bracerightbigg
+.(12)
+One can calculate the energy and angular momentum changes fo r the electron motion
+between two subsequent passages at the pericenter in the sim ilar way.
+43.1 Approximation for relatively high frequency s
+For relatively high frequency of the field, s≫1, the asymptotic form of the Anger function
+Js(se)and its derivative J′
+s(se)may be used [7]. Then eq. (11) may be written in the form
+εj+1=εj+πF0ε2
+0/braceleftBigg
+(e−2)/bracketleftBigg
+4b−4a
+5εj+1−sin(ε−3/2
+j+1π)
+πε2
+j+1/bracketrightBigg
+sinϑjcosϕ
++√
+1−e2
+e/bracketleftBigg
+4aε−1/2
+j+1−2sin(ε−3/2
+j+1π)
+πε1/2
+j+1−2b
+35ε3/2
+j+1
++(e−1)/parenleftBigg
+4bε−3/2
+j+1−4a
+5ε−1/2
+j+1−sin(ε−3/2
+j+1π)
+πε1/2
+j+1/parenrightBigg/bracketrightBigg
+cosϑjsinϕ/bracerightBigg
+.(13)
+Introducing the generating function we can obtain the itera tive equation for the phase ϑjas
+a generalization of eq. (6)
+ϑj+1=ϑj+2πε−3/2
+j+1+πF0ε2
+0/braceleftBigg
+(2−e)/bracketleftBigg
+4a
+5−3cos(ε−3/2
+j+1π)
+2ε−1/2
+j+1
++2sin(ε−3/2
+j+1π)
+πεj+1/bracketrightBigg
+cosϑjcosϕ+√
+1−e2
+e/bracketleftBig
+2aε−3/2
+j+1−3cos(ε−3/2
+j+1π)ε−2
+j+1
++sin(ε−3/2
+j+1π)
+πε−1/2
+j+1+3b
+35ε1/2
+j+1+(e−1)/parenleftBig
+6bε−5/2
+j+1
+−2aε−3/2
+j+1−3cos(ε−3/2
+j+1π)
+2ε−2
+j+1+sin(ε−3/2
+j+1π)
+2πε−1/2
+j+1/parenrightBigg/bracketrightBigg
+sinϑjsinϕ/bracerightBigg
+.(14)
+Using mapping equations (13) and (14) we can represent the en ergy and phase dynamics
+(figure 4) and calculate numerically the relative threshold field strength for the ionization of
+the two-dimensional hydrogen atom (figure 5).
+3.2 Limiting cases for energy and angular momentum changes
+3.2.1 Approximations for very extended orbits
+For very extended orbits, e→1, the expansions for the energy (11) and angular momentum
+(12) in powers of β=√
+1−e2≪1are
+εj+1=εj+4πF0ε2
+0
+εj+1/braceleftbig
+−/parenleftbig
+1+β2/parenrightbig
+J′
+s(s)sinϑjcosϕ
++β/bracketleftbigg
+Js(s)−sin(sπ)
+sπ/bracketrightbigg
+cosϑjsinϕ/bracerightbigg
+,(15)
+5Figure 4. Phase plane (ε,ϑ)of two-dimensional map (13) and (14) with parameter πF0ε2
+0= 0.0035, initial
+conditions ϑ0=π,ε0= 0.3−0.003i(i= 0,1,2,...) and orientation angle (a) ϕ= 0and (b)
+ϕ=π/6.
+µj+1=µj+4πF0ε2
+0
+εj+1/braceleftbigg
+β/bracketleftbigg
+Js(s)−sin(sπ)
+sπ/bracketrightbigg
+sinϑjcosϕ
++/bracketleftbigg
+−/parenleftbig
+1+β2/parenrightbig
+J′
+s(s)+/parenleftbig
+2−β2/parenrightbigsin(sπ)
+π/bracketrightbigg
+cosϑjsinϕ/bracerightbigg
+.(16)
+Fore= 1eq. (15) coincides with eq. (4) for the one-dimensional atom , while eq. (16)
+represents change of the angular momentum, resulting to tra nsition to the elliptic states with
+nonzero angular momentum.
+3.2.2 Approximations for almost circular orbits
+For almost circular orbits the eccentricity is small, e→0. Expansion of expressions (11)
+and (12) in powers of emay be obtained from equations (3) and (11). The results are
+62 4 6 8 100,0120,0140,0160,0180,020
+ 1
+ 2
+F0c
+s0
+Figure 5. Ionization threshold field dependence on the relative frequ encys0for two-dimensional atom with
+the eccentricity e= 0.9and orientation angle ϕ= 0(curve 1) and ϕ=π/6(curve 2).
+εj+1=εj+4F0ε2
+0sin(sπ)
+εj+1/braceleftbigg/bracketleftbigge
+2+3
+4e2s2−1
+1−s2+1
+2es2
+4−s2−3
+4e2s2
+9−s2/bracketrightbigg
+sinϑjcosϕ
++s/bracketleftbigg1−1
+4e2(s2−4)
+1−s2−e
+4−s2+1
+4e2s2
+9−s2/bracketrightbigg
+cosϑjsinϕ/bracerightbigg
+,(17)
+µj+1=µj+4F0ε2
+0sin(sπ)
+εj+1
+×/braceleftbigg
+s/bracketleftbigg
+−e
+2+1−e2(1+1
+4s2)
+1−s2+e(1−1
+2s2)
+4−s2+1
+4e2s2
+9−s2/bracketrightbigg
+sinϑjcosϕ
++/bracketleftbigg3e
+2+s2(1
+2e2(1+1
+2s2)−1)
+1−s2−1
+2es2
+4−s2+1
+2e2s2(3−1
+2s2)
+9−s2/bracketrightbigg
+cosϑjsinϕ/bracerightbigg
+.(18)
+These expansions are not valid for s= 1,2,3,....
+4 Atom in the circularly polarized field
+The equation for the energy change of the atom in the circular ly polarized field is
+˙Ek=−eF(vxcos(ωt+ϑ)±vysin(ωt+ϑ)). (19)
+Here sign ”+”or”−”corresponds to the same and opposite directions of the elect ron and
+field rotations, respectively.
+7For the electron, moving between two subsequent pericenter the energy and angular
+momentum changes are
+εj+1=εj−4πF0ε2
+0
+εj+1/braceleftbigg
+J′
+−s(z)±√
+1−e2
+e/bracketleftbigg
+J−s(z)−sin(sπ)
+sπ/bracketrightbigg/bracerightbigg
+sin(sπ+ϑj∓ϕ),(20)
+µj+1=µj+4πF0ε2
+0
+εj+1/braceleftbigg/bracketleftbigg√
+1−e2
+e/parenleftbigg
+J−s(z)−sin(sπ)
+sπ/parenrightbigg
+∓/parenleftbigg
+−J′
+−s(z)+(e−1)sin(sπ)
+π/parenrightbigg/bracketrightbigg
+sin(sπ+ϑj∓ϕ)
+±(1+1/e)cos(sπ)+1−1/e
+πcos(sπ+ϑj∓ϕ)/bracerightbigg
+.(21)
+It should be noted, that expression in the curly brackets in e q. (20) coincides with the
+expression in the quasiclassical radial dipole matrix elem ent in the velocity representation
+[6]
+D±
+p=1
+s/braceleftbigg
+J′
+−s(z)±√
+1−e2
+e/bracketleftbigg
+J−s(z)−sin(sπ)
+sπ/bracketrightbigg/bracerightbigg
+. (22)
+This correspondence, however, takes place only for interac tion of the hydrogen atom
+with the circularly polarized microwave field and for integr ation of the equations of motion
+between two subsequent pericenters. In general, the energy [3, 7] and angular momentum
+changes depend on the integration interval. So, for motion o f the electron between two
+subsequent apocenters, i.e., the most distant from the nucl eus points, where the electron’s
+energy change is minimal, the energy change is described by t he expression similar to eq.
+(20) but instead of J−s(es)andJ′
+−s(es)we have the Anger function and its derivative of the
+positive order, Js(es)andJ′
+s(es). This interval has been used in [3, 4, 5] for derivation of
+the Kepler map for the one-dimensional hydrogen atom.
+5 Conclusion
+Analytical expressions for the energy and angular momentum changes of the two-dimensional
+hydrogen atom in linearly and circularly polarized electro magnetic fields are derived. It
+should be noted that in general the expressions are rather co mplicated. The approximate
+expressions for limiting cases of the parameters are more co nvenient for analytical and nu-
+merical analysis of the dynamics.
+The derived expressions are suitable for the three-dimensi on hydrogen atom as well, and
+may be generalized for analysis of the chaotic motion (due to the Jupiter perturbations) of
+comets and asteroids in the Sun system.
+References
+[1]N. B. Delone, V. P. Krainov and D. L. Shepelyansky , Usp. Fiz. Nauk 140, 355
+(1983) [Sov. Phys.-Usp. 26, 551 (1983)].
+8[2]P. M. Koch and K. A. H. van Leeuwen , Phys. Rep., V. 255 , 289 (1995).
+[3]V. Gontis and B. Kaulakys , J. Phys. B: At. Mol. Opt. Phys., V. 20, 5051 (1987).
+[4]B. Kaulakys and G. Vilutis , Phys. Scripta, V. 59, 251 (1999) .
+[5]B. Kaulakys, D. Grauzhinis and G. Vilutis , Europhys. Lett., V. 43, 123 (1998).
+[6]B. Kaulakys , J. Phys. B: At. Mol. Opt. Phys., V. 28, 4963 (1995).
+[7]B. Kaulakys , J. Phys. B: At. Mol. Opt. Phys., V. 24, 571 (1991).
+[8]L. D. Landau and E. M. Lifshitz ,Classical Field Theory (Pergamon, New York,
+1975).
+[9]A. J. Lichtenberg and M. A. Lieberman ,Regular and Stochastic Motion (Springer-
+Verlag, New York, 1983 and 1992).
+[10]G. M. Zaslavskii ,Stochastic Behavior of Dynamical Systems (Nauka, Moscow, 1984;
+Harwood, New York, 1985).
+9
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+arXiv:1001.0690v1  [physics.plasm-ph]  5 Jan 2010Contributions toPlasmaPhysics, tobe published invol.50, issue 1(Jan. 2010)
+Thermodynamic functions of dense plasmas: analytic approx i-
+mationsfor astrophysicalapplications
+AlexanderY. Potekhin∗1,2andGillesChabrier2
+1Ioffe Physical-Technical Institute,194021 St.Petersbur g, Russia
+2Ecole Normale Sup´ erieure de Lyon, CRAL(UMR CNRSNo. 5574), 69364 LyonCedex 07, France
+Received 18September 2009, revised 18November 2009, accep ted 18November 2009
+Key words Thermodynamics of plasmas, strongly-coupled plasmas, deg enerate stars.
+PACS52.25.Kn, 05.70.Ce, 52.27.Gr, 97.20.Rp
+We briefly review analytic approximations of thermodynamic functions of fully ionized nonideal electron-
+ion plasmas, applicable in a wide range of plasma parameters , including the domains of nondegenerate and
+degenerate, nonrelativistic and relativistic electrons, weakly and strongly coupled Coulomb liquids, classical
+and quantum Coulomb crystals. We present improvements to pr eviously published approximations. Our code
+for calculation ofthermodynamic functions based onthe rev iewed approximations ismade publiclyavailable.
+Copyrightline willbeprovidedbythepublisher
+1 Introduction
+In a previous work [1,2], we performed hypernetted chain (HN C) calculations and proposed analytic formulae
+fortheequationofstate (EOS)ofelectron-ionplasmas(EIP ).AnalternativeanalyticapproximationfortheEOS
+ofEIPwasproposedin[3,4]. Acomparison(e.g.,[1,4])show sthattheformulaein[2]haveahigheraccuracy,in
+particularforthethermodynamiccontributionsofion-ion andion-electroncorrelationsintheregimeofmoderate
+Coulomb coupling. Recently [5,6], we studied classical ion mixtures and proposed a correction to the linear
+mixingrule. Inthispaperwereviewtheanalyticexpression sforallcontributionstothermodynamicfunctionsand
+introducesome practical modifications to the previouslypu blished formulae. The reviewed analytic description
+ofthethermodynamicfunctionsofCoulombplasmasisrealiz edin apubliclyavailablecomputercode.
+Letneandnibetheelectronandionnumberdensities, AandZtheionmassandchargenumbers,respectively.
+The electric neutrality implies ne=Zni. In this paper we neglect positrons (they can be described us ing the
+same formulae as the electrons; see, e.g., Ref. [7]) and free neutrons (see, e.g., Ref. [8]), and consider mainly
+plasmascontaininga singletypeofions(fortheextensiont omulticomponentmixtures,see [5,6]).
+The state of a free electron gas is determined by the electron number density neand temperature T. Instead
+ofneit is convenientto introducethe dimensionlessdensity par ameterrs=ae/a0, wherea0is the Bohr radius
+andae= (4
+3πne)−1/3. The parameter rscan be easily evaluated from the relations rs= 1.1723n−1/3
+24,where
+n24≡ne/1024cm−3, orrs= (ρ0/ρ)1/3,whereρ0= 2.6752(A/Z)gcm−3. Theanalogousdensityparameter
+fortheionone-componentplasma(OCP)is RS=aimi(Ze)2//planckover2pi12= 1822.89rsAZ7/3,wheremiistheionmass
+andai≡(4
+3πni)−1/3=aeZ1/3istheionsphereradius.
+At stellar densities it is convenient to use, instead of rs, the relativity parameter [9] xr=pF/mec=
+1.00884 (ρ6Z/A)1/3= 0.014005r−1
+s,wherepF=/planckover2pi1(3π2ne)1/3is the electron Fermi momentum and ρ6≡
+ρ/106g cm−3. The Fermi kinetic energy is ǫF=c/radicalbig
+(mec)2+p2
+F−mec2,and the Fermi temperature equals
+TF≡ǫF/kB=Tr(γr−1),whereTr≡mec2/kB= 5.93×109K,γr≡/radicalbig
+1+x2r, andkBis the Boltzmann
+constant. If xr≪1, thenTF≈1.163×106r−2
+sK. The effects of special relativity are controlled by xrin
+degenerateplasmas( T≪TF) andbyτ≡T/Trin nondegenerateplasmas( T≫TF).
+The ions are nonrelativistic in most applications. The stre ngth of the Coulomb interaction of ions is char-
+acterized by the Coulomb coupling parameter Γ = (Ze)2/aikBT= ΓeZ5/3,whereΓe≡e2/aekBT≈
+22.747T−1
+6(ρ6Z/A)1/3andT6≡T/106K.
+∗Corresponding author: e-mail: palex@astro.ioffe.ru
+Copyrightlinewillbe providedby thepublisher2 A.Y. Potekhinand G.Chabrier: Analytic equation of state o f fullyionized plasmas
+ThermaldeBrogliewavelengthsoftheionsandelectronsare usuallydefinedas λi=/parenleftbig
+2π/planckover2pi12/mikBT/parenrightbig1/2and
+λe=/parenleftbig
+2π/planckover2pi12/mekBT/parenrightbig1/2.The quantum effects on ion motion are important either at λi/greaterorsimilaraior atT≪Tp,
+whereTp≡(/planckover2pi1ωp/kB)≈7.832×106(Z/A)√ρ6K is theplasma temperature determined by the ion plasma
+frequency ωp=/parenleftbig
+4πe2niZ2/mi/parenrightbig1/2.Thecorrespondingdimensionlessparameteris η≡Tp/T.
+Assuming commutativity of the kinetic and potential energy operators and separation of the traces of the
+electronic and ionic parts of the Hamiltonian, the total Hel mholtz free energy Fcan be convenientlywritten as
+F=F(i)
+id+F(e)
+id+Fee+Fii+Fie,whereF(i)
+idandF(e)
+iddenote the ideal free energy of ions and electrons,
+and the last three terms represent an excess free energy aris ing from the electron-electron, ion-ion, and ion-
+electroninteractions,respectively. Thisdecomposition inducesanalogousdecompositionsofpressure P,internal
+energyU, entropy S, the heat capacity CV, and the pressure derivatives χT= (∂lnP/∂lnT)Vandχρ=
+−(∂lnP/∂lnV)T.Othersecond-orderfunctionscanbe expressedthroughthes e onesbyMaxwellrelations.
+2 Idealpart ofthe free energy
+Thefree energyof a gasof Ni=niVnonrelativisticclassical ionsis F(i)
+id=NikBT/bracketleftbig
+ln(niλ3
+i/M)−1/bracketrightbig
+,where
+Mis the spin multiplicity. In the OCP, it can be written in term s of the dimensionless plasma parameters as
+F(i)
+id=NikBT[3lnη−1.5lnΓ−0.5ln(6/π)−lnM−1].
+The free energy of the electron gas is given by F(e)
+id=µeNe−P(e)
+idV,whereµeis the electron chemical
+potential. Thepressureandthenumberdensityarefunction sofµeandT:
+P(e)
+id=8
+3√πkBT
+λ3e/bracketleftBig
+I3/2(χe,τ)+τ
+2I5/2(χe,τ)/bracketrightBig
+, ne=4√πλ3e/bracketleftbig
+I1/2(χe,τ)+τI3/2(χe,τ)/bracketrightbig
+,(1)
+whereχe=µe/kBT(here,we donotincludethe restenergy mec2inµe)and
+Iν(χe,τ)≡/integraldisplay∞
+0xν(1+τx/2)1/2
+exp(x−χe)+1dx (2)
+isa Fermi-Diracintegral. Ananalyticapproximationfor µe(ne)hasbeenderivedin[1].
+In Ref.[1] we calculated Iν(χe,τ)usinganalyticalapproximations[7]. These approximation sarepiecewise:
+below, within, and above the interval 0.6≤χe<14. Their typical fractional accuracy is a few ×10−4, the
+maximum error and discontinuities at the boundaries reach ∼0.2%. For the first and second derivatives, the
+errorsanddiscontinuitieslie within1.5%. At χe≥14weuse theSommerfeldexpansion(e.g.,[10])
+Iν(χe,τ)≈ I(0)
+ν(˜µ)+π2
+6τ2I(2)
+ν(˜µ)+7π4
+360τ4I(4)
+ν(˜µ)+..., (3)
+wherewe havedefined ˜µ=χeτ=µe/mec2,
+I(0)
+ν(ǫ) =/integraldisplayǫ
+0I(1)
+ν(ǫ′)dǫ′=/integraldisplayx0
+0/parenleftbig/radicalbig
+1+x2−1/parenrightbigν−1/2x2dx√
+1+x2,I(n+1)
+ν(˜µ) =dI(n)
+ν(˜µ)
+d˜µ,(4)
+I(1)
+ν(ǫ) =ǫν√2+ǫandx0≡/radicalbig
+˜µ(2+ ˜µ). Inparticular,
+I(0)
+1/2(˜µ) = [x0γ0−ln(x0+γ0)]/2,I(0)
+3/2(˜µ) =x3
+0/3−I(0)
+1/2(˜µ),I(0)
+5/2(˜µ) =x3
+0γ0/4−2x3
+0/3+1.25I(0)
+1/2(˜µ),
+whereγ0≡/radicalbig
+1+x2
+0= 1+ ˜µ(notethat,if ˜µ= ˜ǫ,where˜ǫ≡ǫF/mec2, thenx0=xrandγ0=γr).
+Atsmall˜µ,accuracycanbelostbecauseofnumericalcancellationsof closetermsofoppositesignsinEq.(3)
+andin therespectivepartialderivatives. Inthiscasewe us ethe expansion
+Iν(χ,τ) =Inr
+ν(χ)+∞/summationdisplay
+m=0(−1)m(2m−1)!!τm+1
+4m+1m!Inr
+ν+m+1(χ), Inr
+ν(χ) =/integraldisplay∞
+0xνdx
+ex−χ+1,(5)
+Copyrightlinewillbe providedby thepublishercppheader willbe provided bythe publisher 3
+where(2m−1)!!≡/producttextm
+k=1(2k−1)should be replaced by 1 for m= 0. At large χ, the nonrelativistic Fermi
+integrals Inr
+ν(χ)are calculated with the use of the Sommerfeld expansion. Suf ficiently smooth and accurate
+overall approximations for functions Iν(χe,τ)withν=1
+2,3
+2, and5
+2are provided by switching to Eq. (5) at
+χe≥14and˜µ <0.1,whileretainingthetermsupto m= 3.
+Thechemicalpotentialatfixed neisobtainedwithfractionalaccuracy ∼T2/T2
+FbyusingEqs.(1),(3),setting
+I(n)
+ν(˜µ)≈ I(n)
+ν(˜ǫ) +I(n+1)
+ν(˜ǫ)(˜µ−˜ǫ),and dropping the terms that contain the product (˜µ−˜ǫ)τ2. Then at
+T≪TF
+∆˜ǫ≡˜ǫ−˜µ≈π2τ2
+6˜ǫ1+2x2
+r
+γr(1+γr),∆F≡F(e)
+id−F(e)
+0≈ −1
+2TS(e)
+id≈ −PrVxrγrτ2
+6,(6)
+whereF(e)
+0= (PrV/8π2)[xr(1+2x2
+r)γr−ln(xr+γr)]−Nemec2isthezero-temperaturelimit[11](withoutthe
+restenergy Nemec2),andPr≡mec2(mec//planckover2pi1)3= 1.4218×1025dyncm−2is therelativisticunitofpressure.
+Accordingly, P(e)
+id≈P(e)
+0+ ∆P,whereP(e)
+0= (Pr/8π2)/bracketleftbig
+xr/parenleftbig2
+3x2
+r−1/parenrightbig
+γr+ln(xr+γr)/bracketrightbig
+,and∆P=
+(Pr/18)τ2xr(2+x2
+r)/γr. In this case, χ(e)
+ρ≈Prx5
+r/9π2γrP(e)
+id, χ(e)
+T≈2∆P/P(e)
+id, C(e)
+V≈π2kBNeγrτ/x2
+r.
+Inordertominimizenumericaljumpsatthetransitionbetwe enthefit at χe<14andtheSommerfeldexpansion
+atχe>14,wemultiplytheexpressionsfor ∆Fbyempiricalcorrectionfactor (1+∆˜ǫ/˜ǫ)−1,andthosefor χ(e)
+T
+andC(e)
+Vby[1+(4−2xr/γr)∆˜ǫ/˜ǫ]−1.
+Atxr<10−5we replace F(e)
+0andP(e)
+0by their nonrelativistic limits, F(e)
+0/V=Prx5
+r/10π2andP(e)
+0=
+Prx5
+r/15π2∝n5/3
+e. Intheoppositecaseof xr≫1,onehas P(e)
+0=Prx4
+r/12π2∝n4/3
+e.
+3 Electron exchange andcorrelation
+Electron exchange-correlationeffects were studied by man y authors. For the reasons explainedin [1], we adopt
+thefit toFeepresentedinRef.[12]. It isvalidat anydensitiesandtempe ratures,providedthat xr≪1.
+InRef.[2]weimplementedaninterpolationbetweenthenonr elativisticfit[12]andapproximation[3,4]valid
+for strongly degenerate relativistic electrons. However, later on we found that such an interpolation may cause
+anunphysicalbehavioroftheheatcapacityinacertaindens ity-temperaturedomain. Thereforewehavereverted
+totheoriginalformula[12],takingintoaccountthatinapp lications,aslongastheelectronsarerelativistic,their
+exchange and correlation contributionsare orders of magni tude smaller than the other contributions to the EOS
+ofEIP(see,e.g.,[8,13]).
+4 One-component plasma
+4.1 Coulombliquid
+For the reducedfree energyof the ion-ioninteraction fii≡Fii/NikBTin the liquid OCP at anyvalues of Γwe
+use the analytic approximation derived in Ref. [2]. In order to extend its applicability range from T≫Tpto
+T∼Tp, we add the lowest-order quantum corrections to the Helmhol tz free energy [14]: f(2)
+q=η2/24.The
+next-order correction ∝/planckover2pi14has been obtained in [15]. These corrections have limited ap plicability, because as
+soon asηbecomes large, the Wigner expansion diverges and the plasma forms a quantum liquid, whose free
+energyisnotknowninananalyticform.
+A classical OCP freezes at temperature Tmcorrespondingto Γ = 175, but in real plasmas Tmis affected by
+electron polarization[2,16] and quantumeffects (e.g., [1 7–19]). Hence the liquid does not freeze, regardlessof
+T, at densities largerthanthe critical one. The critical den sityvaluesin the OCP correspondto RS≈140–160
+forbosonsand RS≈90–110forfermions. Inastrophysicalapplications,the appe aranceofquantumliquidcan
+beimportantforhydrogenandheliumplasmas(see,e.g.,Ref .[18]fordiscussion).
+4.2 Coulombcrystal
+ThereducedfreeenergyoftheCoulombcrystalis flat≡Flat/NikBT=C0Γ+1.5u1η+fth+fah.Here,thefirst
+term is the Madelung energy ( C0≈ −0.9), the second represents zero-point ion vibrations energy ( u1≈0.5),
+Copyrightlinewillbe providedby thepublisher4 A.Y. Potekhinand G.Chabrier: Analytic equation of state o f fullyionized plasmas
+Fig.1Anharmonic contribution tothe reduced free energy
+of an ion lattice as a function of the quantum parameter
+η=Tp/Tfor two values of the Coulomb coupling param-
+eter,Γ = 175 (upper curves) and 1000 (lower curves). A
+comparison of different approximations (see text): Wigner
+expansion (long-dashed lines); an approximation from [25]
+(IOI, dotted lines); the same approximation with the coeffi-
+cientsadjustedtoRef.[21](short-dashed lines);andpres ent
+interpolation (8) (solid lines). The points with errorbars
+show simulation resultsfrom [25]for Γ = 1000 .Fig. 2Harmonic and anharmonic lattice contributions to
+the reduced heat capacity at Γ = 175 and 1000. The har-
+moniclatticecontributionaccordingtoRef.[20](dot-das hed
+lines) is compared to the model [18] (long-short-dash lines )
+andtotheanharmoniccorrectionindifferentapproximatio ns
+(see text): a derivative of the IOI fit[25] (dotted lines: ori g-
+inal; short-dashed lines: improved; see text) and a corre-
+sponding derivative of Eq. (8)(solidlines).
+fthis the thermal correction in the harmonic approximation, an dfahis the anharmonic correction. We use the
+mostaccuratevaluesof C0,u1,andanalyticapproximationsto fthforbccandfccCoulombOCPlattices,which
+havebeenobtainedin[20].
+For theclassical anharmonic corrections , 11 fitting expressions were given in [21]. We have chosen one of
+them:f(0)
+ah(Γ) =a1/Γ +a2/2Γ2+a3/3Γ3,witha1= 10.9,a2= 247, anda3= 1.765×105, because this
+choiceismost consistentwith theperturbationtheory[17, 22].
+In applications one needs a continuous extension for the fre e energy to η∝negationslash= 0. With the leading quantum
+anharmoniccorrectionat small ηonehas[15]
+fah≈f(0)
+ah(Γ)−(0.0018/Γ+0.085/Γ2)η4. (7)
+AtT→0, the quantum anharmonic corrections were studied in [17,23 ,24], where an expansion in powers of
+R−1/2
+Swasobtained,assuming RS≫1. Theleadingtermofthisexpansioncanbewrittenintheform fah,T→0=
+−b1η2/Γ.In the literatureone findsdifferentestimates for b1; we useb1= 0.12as an approximationconsistent
+with [23,24]. Free and internal energies of finite-temperature quantum crystals were calculated using quantum
+Monte Carlo methods in [19,25,26]. Iyetomi et al. in Ref. [25 ] (hereafter IOI) proposed an analytic expression
+for the quantum anharmonic corrections. However, differen ces between the numerical results in [19,25,26]
+are comparable to the differences between the numerical res ults and the harmonic approximation. Therefore,
+finite-temperatureanharmoniccorrectionscannotbeaccur atelydeterminedfromthelisted results.
+In order to reproducethe zero-temperatureand classical li mits, we multiply f(0)
+ahby exponential suppression
+factore−c1η2andaddfah,T→0. Thenwe have
+fah=f(0)
+ah(Γ)e−c1η2−b1η2/Γ. (8)
+Copyrightlinewillbe providedby thepublishercppheader willbe provided bythe publisher 5
+According to Eq. (7), the Taylor expansion coefficient at η2/Γequals zero. In order to reproducethis property,
+we setc1=b1/a1≈0.0112. In Fig. 1, the resulting approximationfor the free energya s a function of η(solid
+lines) is compared to the semiclassical expression [Eq. (7) ; long-dashed lines] and to the elaborated IOI fitting
+formula: the dotted lines correspondto the original coeffic ients of this fit, chosen by IOI so as to reproducethe
+resultsof Ref. [22] at η→0, while the short-dashedcurvesshow the same fit, but with the coefficientsadjusted
+to more recent results [21] at η→0. The curvesof the two latter types nearly coincide at Γ = 1000 , but differ
+nearthe meltingpoint Γ = 175.
+Interpolation(8)(unlike,forexample,Pad´ eapproximati ons[27])doesnotproduceanunphysicalbehaviorof
+thermodynamicfunctions. In particular, anharmoniccorre ctions to the heat capacity and entropy do not exceed
+thereference(harmonic-lattice)valuesofthesequantiti esatanyη. InFig.2,weshowtheioncontributionstothe
+reducedheat capacityofthe ionlattice cV,i≡CV,i/NikB,calculatedthroughderivativesofEq.(8) (solidlines),
+and compare it to the contributions calculated through deri vatives of the fitting formula in [25] (short-dashed
+and dotted lines). In the same figure we plot the harmonic-cry stalcontributionto the reducedheat capacity: the
+short-dash–long-dashlinecorrespondstothemodelofRef. [18](whichweadoptedin[2]),whilethedot-dashed
+linecorrespondstothe mostaccurateformula[20].
+Interpolation Eq. (8) should be replaced by a more accurate f ormula in the future when accurate finite-
+temperatureanharmonicquantumcorrectionsbecomeavaila ble.
+5 Electron polarization
+5.1 Coulombliquid
+Electron polarization in Coulomb liquid was studied by pert urbation[13,28] and HNC [1,2,29,30] techniques.
+The results for Fieare fitted by analytic expressions in [2]. This fit is accurate within several percents, and it
+exactlyrecoverstheDebye-H¨ uckellimitforEIPat Γ→0andtheThomas-Fermiresult[9]at Γ≫1andZ≫1.
+5.2 Coulombcrystal
+Calculation of thermodynamicfunctionsfor a Coulomb cryst al with allowance for the electron polarization is a
+complexproblem. Forclassicalions,thesimplestscreenin gmodelconsistsinreplacingtheCoulombpotentialby
+theYukawapotential. Forinstance,thereweremoleculardy namicssimulationsofclassicalYukawasystems[16]
+andpath-integralMonteCarlosimulationsofaquantumYuka wacrystalat RS= 200[31]. However,theYukawa
+interactionreflectsonlythesmall-wavenumberasymptoteo ftheelectrondielectricfunction. Arigoroustreatment
+wouldconsistincalculatingthedynamicalmatrixandsolvi ngacorrespondingdispersionrelationforthephonon
+spectrum. Thefirst-orderperturbationapproximationfort hedynamicalmatrixofaclassicalCoulombsolidwith
+thepolarizationcorrectionswasdevelopedinRef.[32]. Th ephononspectruminsuchaquantumcrystalhasbeen
+calculated only in the harmonic approximation[33], which h as a restricted applicability: for example, it cannot
+reproducethe knownclassical ( T≫Tp)limit oftheanharmonic iecontributionto theheatcapacity.
+A semiclassical perturbationapproachwasusedin[2]. Ther esultswerefittedbytheanalyticexpression
+fie=−f∞(xr)Γ[1+A(xr)(Q(η)/Γ)s]. (9)
+In the classical Coulomb crystal, Q= 1,f∞(x) =b1/radicalbig
+1+b2/x2,andA(x) = (b3+a3x2)/(1 +b4x2).
+Parameters sandb1–b4depend on Zand are chosen so as to fit the perturbational results; a3is a constant. All
+theparametersweaklydependonthelattice type;theyareex plicitin[2] forthebccandfcclattices.
+Thefactor Q(η)inEq.(9)isdesignedtoreproducethesuppressionofthedep endenceof FieonTatT≪Tp.
+For the classical solid, Q(0) = 1. In the quantum limit ( η→ ∞)Q(η)≃qη, so that the ratio Q/Γin Eq. (9)
+becomes independent of T. The proportionality coefficient qwas found numerically in [2]; it equals 0.205 for
+thebcclattice.
+The form of Q(η)suggested in [2] assumed a too slow decrease of the heat capac ity (CV,ie∝η−1∝T) at
+η→ ∞, which signals the violation of the employed semiclassical perturbation theory in the strong quantum
+limit,relatedtotheapproximateformoftheionstructuref actorusedinthecalculations,asdiscussedin[2,8]. In
+ordertofixthisproblem,we havechangedtheformof Q(η)to
+Q(η) =/bracketleftBig
+ln/parenleftBig
+1+e(qη)2/parenrightBig/bracketrightBig1/2/bracketleftBig
+ln/parenleftBig
+e−(e−2)e−(qη)2/parenrightBig/bracketrightBig−1/2
+. (10)
+Copyrightlinewillbe providedby thepublisher6 A.Y. Potekhinand G.Chabrier: Analytic equation of state o f fullyionized plasmas
+This formof Q(η)has the correct limits at η→0andη→ ∞and is compatiblewith the numericalresults [2].
+In addition, it eliminates the problematic iecontributions to the heat capacity and entropy at η≫1. It can
+be improved in the future when the polarization corrections for the quantum Coulomb crystal are accurately
+evaluated.
+6 Conclusions
+We have reviewed analytic approximations for the EOS of full y ionized electron-ion plasmas and described
+recentimprovementstothepreviouslypublishedapproxima tions,takingintoaccountnonidealityduetoion-ion,
+electron-electron,andelectron-ioninteractions. Thepr esentedformulaeareapplicableinawiderangeofplasma
+parameters, including the domains of nondegenerate and deg enerate, nonrelativistic and relativistic electrons,
+weaklyandstronglycoupledCoulombliquids,classical and quantumCoulombcrystals.
+For brevity we have considered plasmas composed of electron s and one type of ions. Extension to the case
+whereseveraltypesofionsarepresentisprovidedby[5,6].
+We have made the Fortran code that realizes the analytical ap proximationsfor the free energyand its deriva-
+tives,describedinthispaper,freelyavailablein theInte rnet1.
+Acknowledgements The work of A.Y.P. was partially supported by the Rosnauka Gr ant NSh-2600.2008.2 and the RFBR
+Grant 08-02-00837, and byCompStar,a Research Networking P rogramme of the European Science Foundation.
+References
+[1] G.Chabrier and A.Y. Potekhin, Phys.Rev. E 58, 4941 (1998).
+[2] A.Y. PotekhinandG. Chabrier, Phys.Rev. E 62, 8554 (2000).
+[3] W.Stolzmannand T.Bl¨ ocker, Phys. Lett.A 221, 99 (1996).
+[4] W.Stolzmannand T.Bl¨ ocker, Astron. Astrophys. 361, 1152 (2000).
+[5] A.Y. Potekhin, G.Chabrier, andF.J.Rogers, Phys. Rev. E 79, 016411 (2009).
+[6] A.Y. Potekhin, G.Chabrier, A.I.Chugunov, H.E.DeWitt, and F.J.Rogers, Phys.Rev. E 80, 047401 (2009).
+[7] S.I. Blinnikov, N.V. Dunina-Barkovskaya, and D.K. Nady ozhin, Astrophys. J. Suppl. Ser. 106171 (1996); erratum:
+ibid.118, 603 (1998).
+[8] P.Haensel, A.Y. Potekhin, and D.G. Yakovlev, Neutron Stars 1: Equation of State and Structure (Springer, New York,
+2007).
+[9] E.E.Salpeter,Astrophys. J. 134, 669 (1961).
+[10] S.Chandrasekhar, AnIntroductiontotheStudyofStellarStructure (UniversityofChicagoPress,Chicago,1939; Dover,
+New York,1957), Chap. X.
+[11] J.Frenkel, Z.f.Physik 50, 234 (1928).
+[12] S.Ichimaru, H.Iyetomi, andS.Tanaka, Phys.Rep. 149, 91 (1987).
+[13] D.G.Yakovlev and D.A.Shalybkov, Sov. Astron. Lett. 13, 308(1987); Astrophys. Space Phys. Rev. 7,311 (1989).
+[14] J.P.Hansen, Phys.Rev. A 8, 3096 (1973).
+[15] J.P.Hansenand P.Vieillefosse,Phys. Lett.A 53, 187 (1975).
+[16] S.Hamaguchi, R.T. Farouki, andD.H.E.Dubin, Phys.Rev . E56, 4671 (1997).
+[17] H.Nagara, Y.Nagata, and T.Nakamura, Phys.Rev. A 36, 1859 (1987).
+[18] G.Chabrier, Astrophys. J. 414695 (1993).
+[19] M.D. Jones and D.M. Ceperley, Phys.Rev. Lett. 76, 4572 (1996).
+[20] D.A.Baiko, A.Y. Potekhin, and D.G.Yakovlev, Phys.Rev . E64, 057402 (2001).
+[21] R.T.Farouki and S.Hamaguchi, Phys.Rev. E 47, 4330 (1993).
+[22] D.H.E.Dubin, Phys.Rev. A 42, 4972 (1990).
+[23] W.J.Carr,Jr.,R.A.Coldwell-Horsfall, andA.E.Fein, Phys.Rev. 124, 747 (1961).
+[24] R.C.Albers andJ.E.Gubernatis, Phys.Rev. B 33, 5180 (1986); erratum: ibid.42, 11373 (1990).
+[25] H.Iyetomi, S.Ogata,and S.Ichimaru, Phys.Rev. B 47, 11703 (1993).
+[26] G.Chabrier, F.Douchin, andA.Y. Potekhin, J.Phys.: Co ndens. Matter 14, 9133 (2002).
+[27] A.I.Chugunov and D.A.Baiko, Physica A 352, 397 (2005)
+[28] S.Galam andJ.P.Hansen, Phys. Rev. A 14, 816 (1976).
+[29] G.Chabrier, J.Phys. (France) 51, 1607 (1990).
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+[31] B.Militzerand R.L.Graham, J.Phys. Chem. Sol. 67, 2136 (2006).
+[32] E.L.PollockandJ.P.Hansen, Phys. Rev. A 8, 3110 (1973).
+[33] D.A.Baiko, Phys. Rev. E 66, 056405 (2002).
+1http://www.ioffe.ru/astro/EIP/
+Copyrightlinewillbe providedby thepublisher
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+arXiv:1001.0691v1  [cond-mat.soft]  5 Jan 2010An extended analysis of the viscosity kernel for monatomic a nd diatomic fluids
+R. M. Puscasu∗and B. D. Todd†
+Centre for Molecular Simulation, Swinburne University of T echnology,
+PO Box 218, Hawthorn, Victoria 3122, Australia
+P. J. Daivis‡
+Applied Physics, School of Applied Sciences, RMIT Universi ty,
+G.P.O. Box 2476, Melbourne, Victoria 3001, Australia
+J. S. Hansen§
+Centre for Molecular Simulation, Swinburne University of T echnology,
+PO Box 218, Hawthorn, Victoria 3122, Australia
+(Dated: November 2, 2018)
+We present an extended analysis of the wave-vector dependen t shear viscosity of monatomic and
+diatomic (liquid chlorine) fluids over a wide range of wave-v ectors and for a variety of state points.
+The analysis is based on equilibrium molecular dynamics sim ulations, which involves the evaluation
+of transverse momentum density and shear stress autocorrel ation functions. For liquid chlorine we
+present the results in both atomic and molecular formalisms . We find that the viscosity kernel
+of chlorine is statistically indistinguishable with respe ct to atomic and molecular formalisms. The
+results furthersuggest that thereal space viscosity kerne lsof monatomic anddiatomic fluidsdepends
+sensitivelyonthedensity, thepotential energyfunctiona ndthechoice offittingfunctioninreciprocal
+space. It is also shown that the reciprocal space shear visco sity data can be fitted to two different
+simple functional forms over the entire density, temperatu re and wave-vector range: a function
+composed of n-Gaussian terms and a Lorentzian type function. Overall, th e real space viscosity
+kernel has a width of 3 to 6 atomic diameters which means that t he generalized hydrodynamic
+constitutive relation is required for fluids with strain rat es that vary nonlinearly over distances of
+the order of atomic dimensions.
+I. INTRODUCTION
+Fluid dynamics at atomic and molecular scales still
+present a challenge for theoreticians as well as for experi-
+mentalists. Moleculardynamics(MD) isacomputational
+tool that has contributed significantly to the fundamen-
+tal understanding of these systems by providing informa-
+tion about processes not directly approachable by exper-
+imental studies. A central problem in the study of fluids
+at such small length and time scales is the computation
+of meaningful transport properties. Many equilibrium
+molecular dynamics (EMD) as well as nonequilibrium
+molecular dynamics (NEMD) simulations of nanofluids
+have been performed since the early 1980s [1–5]. In most
+of these simulations the stress was treated as being de-
+pendent of the local strain rate rather than the entire
+strain rate distribution in the system. Todd et al.have
+recentlyshownthatin allbut thesimplestflows(e.g. pla-
+nar Couette and Poiseuille flows) and for velocity fields
+with high gradients in the strain rate over the width of
+the real space viscosity kernel, non-locality can play a
+significant role [6, 7]. In the case of a homogeneous fluid,
+∗Electronic address: rpuscasu@swin.edu.au
+†Electronic address: btodd@swin.edu.au
+‡Electronic address: peter.daivis@rmit.edu.au
+§Electronic address: jehansen@swin.edu.aua local viscosity defined by Newton’s viscosity law as
+Pxy(r,t) =−η0t/integraldisplay
+0∞/integraldisplay
+−∞δ(r−r′,t−t′)˙γ(r′,t′)dr′dt′(1)
+even exhibits singularities at points where the strain rate
+is zero [8–11]. In Eq. (1) Pxy(r,t) represents the ( x,y)
+off-diagonal component of the pressure tensor, ˙ γ(r,t) is
+theshearstrainrateatposition randtime t, andη0isthe
+local shear viscosity. In general, a nonlocal constitutive
+equationthatallowsforspatialandtemporalnon-locality
+can be expressed as [12, 13]
+Pxy(r,t) =−t/integraldisplay
+0∞/integraldisplay
+−∞η(r−r′,t−t′)˙γ(r′,t′)dr′dt′,(2)
+forahomogeneousfluid. Inthesituationwherethestrain
+rate is constant in time and only varies with respect to
+the spatial coordinate y, Eq. (2) can be written as
+Pxy(y) =−∞/integraldisplay
+−∞η(y−y′)˙γ(y′)dy′. (3)
+In reciprocal space Eq. (3), can be expressed as
+˜Pxy(ky) =−˜η(ky)˜˙γ(ky), (4)2
+wherekyis theycomponent of the wave-vector as de-
+fined later in Section III. Such a constitutive equation is
+expected to be necessary for the description of flows in
+highly confined systems, due to the large change in the
+strain rate with position in the vicinity of the wall [8].
+The best available theoretical predictions of the wave-
+vector dependent viscosity are based on mode-coupling
+theory and generalized Enskog theory [14–16]. However,
+these theories do not quantitatively agree with data ob-
+tained via computer simulations [12]. The theoretical
+predictions focus on the transverse momentum density
+autocorrelation function, which is found by an iterative
+numerical solution of a system of nonlinear equations.
+Consequently, the theories do not result in analytical ex-
+pressions for the correlation functions or the wave-vector
+dependent transport coefficients, which are the focus of
+the present study. More recently, a modified collective
+modeapproachhasbeensuccessfullyappliedbyOmelyan
+et al.[17] to the TIP4 model of water. In contrast
+to other semi-phenomenological approaches used for in-
+stanceinTIP4PandSPC/Emodels ofwaterbyBertolini
+et al.[18] and Palmer [19], Omelyan et al.reproduced
+the reciprocal space kernel using a relatively small num-
+ber of modes.
+In this paper, we extend the work done by Hansen et
+al.[20] and focus on computing the spatially non-local
+viscosity kernel for monatomic and diatomic fluids over
+a wider range of wave-vectors, state points and potential
+energy functions. We are specifically interested in identi-
+fying functional forms that fit the reciprocal space kernel
+data. On the basis of these results, we are be able to
+assess the length scale (i.e. the width of the real space
+kernel) over which the governing generalized constitutive
+relation Eq. (3) must be used. We expect that non-local
+transport phenomena would be relevant in shock waves
+[12, 21–24], shearbanding [25], flows ofmicellar solutions
+[26], suspensions of rigid fibers [27] and jammed or glassy
+systems [28].
+This paper is structured as follows: In Section IIA
+we give an overview of the general formulation and the
+expressions for the complex wave-vector and frequency
+dependent viscosity are given. In Section III we describe
+the simulation methodology and conditions. In Section
+IV we present our molecular dynamics simulation data
+and compare the results of our monatomic and diatomic
+viscosity kernels, particularly the shape of the kernels.
+Finally, we summarize and conclude our analysis in Sec-
+tion 5.
+II. METHODOLOGY
+We shall briefly introduce the main conceptual back-
+ground used in this work, namely, the wave-vector de-
+pendent momentum density, stress and viscosity in the
+atomic and molecular formulations.A. Wave-vector dependent momentum density for
+atomic and molecular fluids
+For a single component atomic fluid the real space mi-
+croscopic momentum density is given by [13]
+J(r,t) =ρa(r,t)v(r,t) =Na/summationdisplay
+i=1mivi(t)δ(r−ri) (5)
+whereρa(r,t) =/summationtextNa
+i=1miδ(r−ri) is the mass density,
+mi,riandviare the mass, position and velocity of atom
+i. The summation runs over the number of atoms Na
+in the system. The Fourier transform of the momentum
+density is
+˜J(k,t) =Na/summationdisplay
+i=1mivi(t)eik·ri(6)
+while the Fourier transform of the mass density is
+˜ρa(k,t) =/summationtextNa
+i=1mieik·ri. We define the Fourier
+transform of a function f(r) asF[f(r)] =˜f(k) =/integraltext∞
+−∞eikrf(r)dr.
+The atomic representation of the momentum density
+for a molecular fluid can be written in real space as [29]:
+JA(r,t) =ρa(r,t)v(r,t) =Nm/summationdisplay
+i=1Ns/summationdisplay
+α=1miαviα(t)δ(r−riα)
+(7)
+where the mass density is defined as ρa(r,t) =/summationtextNm
+i=1/summationtextNs
+α=1miαδ(r−riα). The inner summationextends
+overtheNsmass points in a molecule and the outer sum-
+mation extends over the number of molecules Nmin the
+system. In general, Nsdepends on the molecule index i
+for a multicomponent system, but in our systems Nsis
+the same for all molecules and the interaction sites are
+assumed to have identical mass, namely miα.
+The Fourier transform of the momentum density is
+˜JA(k,t) =Nm/summationdisplay
+i=1Ns/summationdisplay
+α=1miαviα(t)eik·riα(8)
+where the transformed mass density is ˜ ρa(k,t) =/summationtextNm
+i=1/summationtextNs
+α=1miαeik·riα. For molecules composed of Ns
+atoms we can define the mass of molecule i,Mi=/summationtextNs
+α=1miα, position of the molecular center of mass as
+ri=/summationtextNs
+α=1miαriα/Mi, position of site αof molecule i
+relative to the center of mass of molecule iasRiα=
+riα−ri, and center of mass momentum of the molecule
+aspi=/summationtextNs
+α=1piα. This means that the atomic
+mass density can be written in k-space as ˜ ρa(k,t) =/summationtextNm
+i=1/summationtextNs
+α=1miαeik·(ri+Riα). If we expand this rela-
+tion further we can express the atomic mass density
+in terms of molecular mass density in which we de-
+fine the mass density in the molecular representation
+as ˜ρm(k,t) =/summationtextNm
+i=1Mieik·riin reciprocal space and3
+asρm(r,t) =/summationtextNm
+i=1Miδ(r−ri) in real space, respec-
+tively. In a similar way we can expand the atomic mo-
+mentum density about the molecular center of mass:
+˜JA(k,t) =/summationtextNm
+i=1/summationtextNs
+α=1miαviα(1+ik·Riα+...)eik·riα.
+The Fouriertransformofthe momentum density in the
+molecular representation can then be defined as
+˜JM(k,t) =Nm/summationdisplay
+i=1Mivi(t)eik·ri(9)
+A complete procedure for expressing the mass and mo-
+mentum densities in physical and reciprocal space for
+atomic and molecular fluids has been discussed in more
+detail by Todd and Daivis [29].
+B. Wave-vector dependent pressure tensor
+For a monatomic system the wave-vector dependent
+pressure tensor is defined as [29]
+˜P(k,t) =N/summationdisplay
+i=1pipi
+mieik·ri−1
+2N/summationdisplay
+i=1N/summationdisplay
+j/negationslash=irijFijg(k)eik·ri
+(10)
+whereFiis the force on atom idue to atom jandg(k) =
+(eik·r−1)/ik·r=/summationtext∞
+n=0(ik·r)n/(n+1)! is the Fourier
+transform of the Irving-Kirkwood Oijoperator [30].
+For a molecular system the molecular pressure tensor
+is the pressure calculated using the intermolecular forces
+and the molecular center of mass momenta. The atomic
+pressure on the other hand includes all atomic momenta
+andall interatomicforcesandconstraintforces. Thusthe
+wave-vectordependentpressuretensorforconstraineddi-
+atomic fluid can be written in atomic representation as
+˜PA(k,t) =Nm/summationdisplay
+i=12/summationdisplay
+α=1piαpiα
+miαeik·riα
+−1
+2Nm/summationdisplay
+i=12/summationdisplay
+α=1Nm/summationdisplay
+j/negationslash=i2/summationdisplay
+β=1riαjβFiαjβg(k)eik·riαjβ
++Nm/summationdisplay
+i=12/summationdisplay
+α=1riαFC
+iαg(k)eik·riα(11)
+whereFiαjβis the LJ force acting on site αof molecule i
+duetosite βofmolecule j, andFC
+iαisthe bondconstraint
+force on site αof molecule i.riαjβ=rjβ−riαis the
+minimum image separation of site αof molecule ifrom
+siteβof molecule j.
+The pressure tensor for a diatomic molecule in the
+molecular representation is defined as
+˜PM(k,t) =Nm/summationdisplay
+i=1pipi
+Mieik·ri−1
+2Nm/summationdisplay
+i=1Nm/summationdisplay
+j/negationslash=irijFN
+ijg(k)eik·rij(12)
+where,FN
+ijrepresents the total intermolecular force on
+molecule idue to molecule j.rij=rj−riis the mini-
+mum image separation of the center of mass of moleculeifrom the center of mass of molecule j. In cases where
+two sites on two different periodic images of the same
+molecule may interact, the correct value of rij=rj−ri
+corresponding to the particular images of molecule iand
+jinFiαjβmust be used. Though this is unlikely to hap-
+pen in diatomic fluids it is particularly important in sim-
+ulation of long molecules. The momenta appearing in
+these equations, piα,pi, are the momenta appearing in
+the respective atomic and molecular equations of motion
+Eqs. (21) and (24).
+C. Wave-vector and frequency dependent viscosity
+Thecomplexwave-vectorandfrequencydependentvis-
+cosity can be evaluated by using two different expres-
+sions in terms of the Fourier-Laplace transform of the
+transverse momentum density autocorrelation function
+(ACF),C⊥(k,t), and the Fourier-Laplace transform of
+the stress tensor autocorrelation function, N(k,t) [13].
+We define the complex Laplace transform (one-sided
+Fourier transform) as L[f(t)] =˜f(w) =/integraltext∞
+0f(t)e−iωtdt.
+We also note that for the sake of simplicity of notation
+and consistency with the notation used in previous pub-
+lications, in what follows, we drop the tilde sign over the
+Fourier transformed correlation functions and keep the
+tilde notation over the Fourier-Laplace transformed cor-
+relation functions only. If we set Γ k= (0,ky,0) andJx
+is the component of the momentum density in the xdi-
+rection, the expression for the wave-vectorand frequency
+dependent viscosity in terms of ˜C⊥(ky,ω) takes the form
+[13]:
+˜η(ky,ω) =ρ
+k2yC⊥(ky,t= 0)−iω˜C⊥(ky,ω)
+˜C⊥(ky,ω)(13)
+whereρis the number density of the fluid and ˜C⊥(ky,ω)
+is the Laplace transform of the ensemble averaged
+transverse momentum density autocorrelation function
+C⊥(ky,t), which is defined as
+C⊥(ky,t) =1
+V/angbracketleftBig
+Jx(ky,t)Jx(ky,t= 0)/angbracketrightBig
+.(14)
+The zero time value of C⊥(ky,t= 0) in the thermody-
+namic limit is
+C⊥(ky,t= 0) =ρkBT. (15)
+kBis Boltzmann’s constant. Due to finite time aver-
+aging and finite system size, the value of C(ky,t= 0)
+obtained from simulation differs insubstantially from the
+exact value given by
+C⊥(ky,t= 0) =ρkBT3N−4
+3N(16)
+because the total peculiar kinetic energy and three com-
+ponents of the momenta are constants of the motion in4
+our simulations. We also note that the number of degrees
+of freedom in the simulated system has not been taken
+into account in Eq. (15), therefore we use the simulated
+value in our calculations to ensure numerical consistency
+of the computed properties.
+The expression for the wave-vector and frequency de-
+pendent viscosityintermsoftheautocorrelationfunction
+of the shear stress ˜N(ky,ω) takes the form:
+˜η(ky,ω) =˜N(ky,ω)
+C(ky,t= 0)/ρkBT−k2˜N(ky,ω)/iωρ(17)
+where
+˜N(ky,ω) =1
+VkBTL/bracketleftBig/angbracketleftBig
+Pxy(ky,t)Pxy(ky,0)/angbracketrightBig/bracketrightBig
+.(18)
+In the zero wave-vector limit, a generalization of the
+Green-Kubo expression for the shear viscosity allows the
+transversemomentum flux to be in an arbitrarydirection
+rather than along a coordinate axis and can be written
+in terms of the stress tensor as [31, 32]:
+η=V
+10kBT/integraldisplay∞
+0/angbracketleftBig
+Pos(t) :Pos(0)/angbracketrightBig
+dt(19)
+where the ossuperscript denotes the traceless symmet-
+ric part of the stress tensor Pos(t) =1
+2[P(t) +PT(t)]−
+1
+3tr[P(t)]1andVis the simulation volume. In an
+isotropic fluid, because the tensor, Pos, appearing in
+Eq. (19) is traceless and symmetric, the shear viscosity
+is also traceless and symmetric. Consequently, the shear
+viscosity can be expressed in terms of the invariant Iof
+the shear viscosity tensor as η=I/10.
+In the case ω→0 andky→0, relation (17) reduces
+to the Green-Kubo formula [31]. All the non-zero wave-
+vector integrals, Eq. (18), converge to zero [13] there-
+fore the relation in Eq. (13) must be used when non-
+zero wave-vector viscosities are calculated. By comput-
+ing the integrals one can computationally verify the zero
+values of the zero-frequency limit of the function ˜N(k,ω)
+and thus demonstrate why neither substitution of ω= 0
+into Eq.(17) nor evaluation of Eq. (18) at non-zero wave-
+vector yields the zero-frequency wave-vector dependent
+viscosity.
+III. SIMULATION DETAILS
+We use the Edberg, Evans, and Morriss algorithm
+[33–35] with an improved cell neighbour list construc-
+tion algorithm [36] to perform equilibrium simulations
+at constant N,V,TM.Nis either the number of atoms
+or molecules and TMis the molecular temperature as
+defined by Eq. (26). For the atomic fluids studied in
+this work, the atoms interact via a Lennard-Jones (LJ)
+or Weeks-Chandler-Andersen (WCA) potential energy
+function [37]. The LJ atoms have an interaction poten-
+tial truncated at rc= 2.5σand WCA atoms have aninteraction potential truncated at rc= 21/6σ[37]. In
+general:
+Φij(rij) =
+
+4ǫ/bracketleftBigg/parenleftbigg
+σrij/parenrightbigg12
+−/parenleftbigg
+σrij/parenrightbigg6/bracketrightBigg
+−Φc, rij< rc
+0, r ij≥rc
+(20)
+whererijis the interatomic separation, ǫis the poten-
+tial well depth, and σis the value of rijat which the
+unshifted potential is zero. The shift Φ cis the value of
+the unshifted potential at the cutoff rij=rc, and is in-
+troduced to eliminate the discontinuity in the potential
+energy. At distances greater than the cutoff distance rc,
+the potential is zero.
+Our diatomic model of liquid chlorine is similar the
+the one used by Edberg et al.[38], Hounkonnou et al.
+[39], Travis et al.[40, 41] and more recently by Matin
+et al.[42, 43] to allow a direct comparison of our results
+with previous work. This model represents chlorine as a
+diatomic LJ molecule with rc= 2.5σand a fixed bond
+length of 0 .63σ. For an adequate representation of the
+properties of chlorine the LJ parameters are: σ= 3.332˚A
+andǫ/kB= 178.3K. Liquid chlorine systems of 108 and
+864 molecules are studied at a reduced site number den-
+sity ofρa= 1.088 and a reduced molecular temperature
+TM= 0.970. We summarize the most important simu-
+lation parameters in Table I. All our simulations were
+TABLE I: Simulation details
+WCA LJ Chlorine
+Site number density, ρa0.375,0.480,0.840 0 .840 1 .088
+Temperature, T 0.765,1.0 0 .765,1.0 0 .97
+Number of atoms, Na108,2048,6912 108 ,2048,6912 216 ,1728
+Number of sites, Ns 1 1 2
+Bond length, l - - 0 .63
+LJ cutoff, rc 21/62.5 2 .5
+carried out in a cubic box with periodic boundary condi-
+tions. The fifth-order Gear predictor corrector algorithm
+[44, 45] with time step δt= 0.001 was employed to solve
+the equations of motion. The equations of motion can be
+written for a monatomic fluid in the isokinetic ensemble
+(at equilibrium) as [13]:
+˙ ri=pi
+mi,˙ pi=Fi−ζApi (21)
+whereidenotes atom i.riis the position, piis the
+momentum and miis the mass of the designated atom.
+Fiis the force on atom idue to other atoms and ζAis the
+atomic thermostat multiplier. The thermostat multiplier
+is chosen so as to fix the kinetic temperature. We use the
+value of ζAthat results from the application of Gauss’
+principle of least constraint to the imposition of constant
+kinetic temperature:
+ζA=/summationtextN
+i=1Fi·pi/summationtextN
+i=1p2
+i. (22)5
+The atomic temperature TAfor a system of Naatoms
+with no internal degrees of freedom is defined as
+TA=1
+(dNa−Nc)kB/angbracketleftbiggNa/summationdisplay
+i=1p2
+i
+mi/angbracketrightbigg
+(23)
+Hereangledbracketsdenoteanensembleaverage, disthe
+dimensionality of the atomic system, Ncis the number
+of constraints on the system (including constraints for
+conserved quantities).
+The equations of motion (EOM) for a molecular fluid
+can be formulated in either atomic or molecular ver-
+sions. In fact the molecular versions of the homogeneous
+isothermal EOM with a molecular thermostat at equilib-
+rium are similar to atomic EOM with a molecular ther-
+mostat, provided that all of the relevant forces are in-
+cluded [29]. The thermostatted EOM for molecular sys-
+tems are given by [29]
+˙ riα=piα
+miα,˙ piα=Fiα+FC
+iα−ζMmiα
+Mipi(24)
+whereiαdenotes site αon molecule i.riαis the posi-
+tion,piαis the momentum and miαis the mass of the
+site. The force on a site is separated into two terms:
+Fiαis the contribution due to the Lennard-Jones type
+interactions on site αof molecule iandFC
+iαis either the
+constraintforceorthebondingforce. ζMisthe molecular
+thermostat multiplier, given by
+ζM=/summationtextNm
+i=1Fi·pi/Mi/summationtextNM
+i=1p2
+i/Mi(25)
+whereFiis the total force acting on molecule iandMiis
+the mass of molecule i.ζMis derived from Gauss’ prin-
+ciple of least constraint and acts to keep the molecular
+center of mass kinetic temperature TMconstant. Sev-
+eral algorithms are available for this purpose [47]. The
+molecular temperature TMis defined by
+TM=1
+(dNm−Nc)kB/angbracketleftbiggN/summationdisplay
+i=1p2
+i
+mi/angbracketrightbigg
+(26)
+whereNcis the number of translational center-of-mass
+degrees of freedom and depends on the total number
+of sites and the number of constraints on the system.
+The details of the constraint algorithm used to calcu-
+lateFC
+iαhave been discussed previously [33, 38, 46]. All
+the systems were equilibrated for at least 106time steps.
+The results from production runs were ensemble aver-
+aged over 14 runs, each of length 106steps (i.e. a total
+of 1.4×107time steps). The transverse momentum den-
+sity ACFs were computed over at least 20 reduced time
+units and the stress ACFs were computed over at least
+40 reduced time units. Both the transverse momentum
+density and stress ACFs were computed at wave-vectors
+kyn= 2πn/Lywhere mode number nis from 0 to 40
+with increment 2 and Ly= [Na/ρ]1/3. For the remainderof this paper we drop the nindex in kynfor simplic-
+ity. The ACFs were Laplace transformed with respect
+to time using Filon’s rule [47]. We do not report the
+frequency dependent viscosities in this work. The wave-
+vector and frequency dependent viscosities were calcu-
+lated using Eqs. (13) and (17). Eq. (13) was used to
+obtain the non-zero wave-vector and frequency depen-
+dent viscosities and Eq. (17) was used to obtain the zero
+wave-vector viscosity.
+In this work, all quantities are expressed in reduced
+Lennard-Jones units. Thus, our reference units are
+the: reduced length r∗=r/σ, reduced number density
+ρ∗=ρ/σ3, reduced temperature T∗=kBT/ǫ, reduced
+timet∗=t/(σ(m/ǫ)1/2, reduced pressure P∗=P(σ3/ǫ),
+reduced energy E∗=E/ǫand reduced viscosity η∗=
+ησ2//radicalbig
+(mǫ). For the remainder of this paper we apply
+these units and omit writing the asterisk. We will not
+distinguish between TMandTA, but simply use Tto
+indicate the temperature.
+IV. RESULTS AND DISCUSSION
+The autocorrelation functions were evaluated for both
+108 and 864 molecule system in order to determine
+whether the results were system size dependent. There
+were no observed differences within the statistical errors
+in both monatomic and diatomic systems. We also note
+that in order to limit the number of figures, we do not
+display the results for the transverse momentum density
+and stress autocorrelation functions. However, we must
+mention that for monatomic systems both quantities (i.e.
+the transversemomentum density and stress ACFs) were
+in good agreement with those previously observed for
+Lennard-Jones monatomic liquids and their running in-
+tegrals have fully converged which suggests that the cor-
+relation functions have decayed to zero.
+A. Viscosity kernels in reciprocal space
+The reciprocal space kernels for monatomic and di-
+atomic fluids are plotted in figures (1-3). The error bars
+are smaller than the symbol sizes and therefore omit-
+ted in figures 2 and 3. Generally the statistical relia-
+bility of reciprocal space kernel data increases as kyin-
+creases. Our zero wave-vector, zero frequency viscosities
+for monatomic fluids agree well with those available in
+the literature. For the WCA system at the state point
+(ρa= 0.375,T= 0.765) we found η0= 0.27±0.01 which
+agrees with the results of Hansen et al.0.273 [12, 20],
+while at the state point ( ρa= 0.840,T= 1.0) we found
+η0= 2.29±0.07, inagreementwith the resultsofMatin et
+al.(2.1±0.2)[42]. Forchlorinewefound η0= 6.89±0.32,
+which agrees with the limiting values (6.7 ±0.4) of the
+shear or elongational viscosities at zero strain rate [42].
+The wave-vector dependent viscosity for diatomic sys-
+tems, figure 1, depicts a similar behaviour within the6
+TABLE II: Zero frequency, zero wave-vector shear viscosity and fitted parameter values for monatomic and diatomic syste ms
+WCA WCA WCA LJ LJ Chlorine
+State Point ρa0.375 0 .480 0.840 0.840 0.840 1.088
+T 0.765 0 .765 1.000 0.765 1.0 0.97
+System size Na 2048 1728
+η00.265(0.273[20]) 0 .392 2.290 2.810 2.650 6.889
+2-term Gaussian ,A2= 1−A1Eq. (27) A0.189(0.440[20]) 0 .309 0.155 0.093 0.107 0.407
+σ112.48(4.750[20]) 6 .916 8.122 10.04 9.088 5.377
+σ22.116(1.376[20]) 1 .835 2.592 2.778 2.759 1.236
+sr0.007(0.005[20]) 0 .013 0.044 0.021 0.027 0.082
+2-term Gaussian Eq. (27) A10.792 0 .687 0.874 0.907 0.892 0.592
+A20.174 0 .254 0.155 0.094 0.106 0.407
+σ12.245 2 .113 2.592 2.776 2.765 1.237
+σ213.36 7 .745 8.124 10.02 9.127 5.377
+sr0.007 0 .011 0.035 0.022 0.031 0.081
+4-term Gaussian Eq. (27) A10.432 0 .566 0.778 0.689 0.868 0.398
+A20.394 0 .248 0.118 0.190 0.047 0.538
+A30.120 0 .138 0.088 0.114 0.089 0.055
+A40.056 0 .047 0.017 0.017 0.020 0.008
+σ13.228 2 .826 2.950 2.709 2.814 4.355
+σ21.261 0 .821 0.651 2.709 0.145 1.155
+σ38.165 6 .973 8.496 7.037 7.628 10 .46
+σ415.19 14 .74 23.66 24.94 19.99 37.56
+sr0.008 0 .002 0.012 0.014 0.024 0.018
+Lorentzian-type Eq. (28) α0.198(0.180[20]) 0 .170 0.062 0.041 0.043 0.239
+β1.562(1.662[20]) 1 .715 2.326 2.602 2.572 1.667
+sr0.002(0.005[20]) 0 .005 0.042 0.018 0.042 0.016
+ 0 1 2 3 4 5 6 7 8
+ 0 5 10 15 20 25 30 35 40 η~(ky)
+kyChlorine
+Atomic formalism
+Molecular formalism
+FIG. 1: ˜ η(ky) versus kyfor chlorine calculated using atomic
+and molecular formalisms ( ρa= 1.088,T= 0.97,Na= 1728).
+statistical uncertainty in both atomic and molecular for-
+malisms.
+It has been shown previously that numerous one pa-
+rameter functions failed to capture the behaviour of the
+reciprocal space kernel data [20]. We therefore present
+the best fits with two or more fitting parameters. We
+haveidentifiedtwofunctional formsthatfit thedatawell:anNGterm Gaussian function
+˜ηG(ky) =η0NG/summationdisplay
+jAjexp(−k2
+y/2σ2
+j)Aj,σj∈R+(27)
+and a Lorentzian type function
+˜ηL(ky) =η0
+1+α|ky|βα,β∈R+,(28)
+We present the best fits of the data to (i) a two-term
+Gaussian function with freely estimated amplitudes (i.e.
+unconstrained fitting) termed as ˜ ηG2, (ii) to a two-term
+Gaussian function with interdependent amplitudes (i.e.
+constrained fitting/summationtextNG
+jAj= 1) given by Hansen et
+al.[20] and termed as ˜ ηG2H, (iii) to a four-term Gaus-
+sianfunction withfreelyestimatedamplitudes, termedas
+˜ηG4and (iv) to the Lorentzian type function, Eq. (28).
+In order to measure the magnitude of the residuals we
+use the residual standard deviation defined as sr=/radicalbig/summationtextns
+n=1r2/(ns−np) where nsis the number of data
+points,npis the number of fitting parameters, and ris
+the residual [48]. After an iterative curve fitting proce-
+duretheaccurateestimationof η0waskeptfixedallowing
+all other parameters in Eqs. (27) and (28) to be used as
+fitting parameters. In Table II we have listed the fitting
+parameters for monatomic and diatomic molecular fluids
+and compared to the previous results where possible.7
+00.20.40.60.81
+05101520253035 η~ (ky) / η~ (0)
+kyChlorine
+data
+ηG2HηG4ηL
+(a)
+00.20.40.60.81
+05101520253035η~
+ (ky) / η~
+ (0)
+kyLJ
+data
+ηG2HηG4ηL
+(b)
+-0.06-0.04-0.0200.020.04
+0510152025303540Δ η~(ky) / η~ (0)
+kyChlorine
+ηG2H - data
+ ηG4 - data
+ηL - data(c)
+-0.06-0.04-0.0200.020.040.06
+0510152025303540Δ η~
+(ky) / η~
+ (0)
+kyChlorine
+ηG2H - ηG4ηL - ηG4ηL - ηG2H(d)
+FIG. 2: Normalized kernel data, best fit of Eq. (27) with NG= 2 and 4, and Eq. (28) and difference between the fits: (a) Best
+fits to normalized kernel for chlorine fluid ( ρa= 1.088,T= 0.97,Na= 1728); (b) Best fits to normalized kernel for LJ fluid
+(ρa= 0.840,T= 1.0,Na= 2048); (c) Differences between the kernels and simulation d ata (a); (d) Differences between the
+fitted kernels (b).
+TABLE III: Total amplitude for Gaussian functional form
+WCA LJ Chlorine
+State Point ρa0.375 0.480 0.840 0.840 0.840 1.088
+T 0.765 0.765 1.000 0.765 1.0 0.97
+2-term Gaussian/summationtext2
+j=1A0.966 0.941 1.029 1.001 0.998 0.999
+4-term Gaussian/summationtext4
+j=1A1.002 0.999 1.001 1.010 1.024 0.999
+Auseful checkofthe fitting canbe performed bycalcu-
+lating the total Gaussian amplitudes which should con-
+verge to the value of 1, Table III.
+The reciprocal space results presented in figure 2 for
+LJ and chlorine systems demonstrate that the four-term
+Gaussian function fits the data much better than the
+other two forms with a difference between the data and
+the fit of less than 0.5%, see figure 2(c). The two-
+term Gaussian ˜ ηG2Hfits the kernel data better than the
+Lorentzian-typefunctioninthelow- kyregion,figure2(a),
+which suggests a more Gaussian-like behaviour in the
+low-kyregion, a fact previously observed by Hansen et.
+al[20] for atomic fluids modeled with WCA potentials.
+Nevertheless the difference between the two-term Gaus-
+sian fit and data is less than 2% which still makes the
+˜ηG2Ha good analytical three parameter approximation
+of the reciprocal space viscosity kernel. The maximum
+difference between the Lorentzian-type fit and Gaussian
+fitsarearound4%whilethemaximumdifferencebetween
+the Gaussian fits is about 2%, see figure 2(d). Essen-tially, this suggests that, when computing the real space
+kernels, the four-term Gaussian functional form is to be
+trusted. It is obvious that eight parameters in the four-
+term Gaussian make its use less convenient, but on the
+other hand the Gaussian function can analytically be in-
+verseFouriertransformedwhilethe inverseFouriertrans-
+form of the Lorentzian-type function can only be evalu-
+ated numerically for general values of β.
+Figure 3(a) shows the kernel data for a WCA fluid at
+two different densities along with two sets of data pub-
+lished previously by Hansen et al.[20]. EMD is the
+set obtained from an equilibrium MD simulation at the
+same state point ( ρa= 0.375,T= 0.765) and NEMD is
+the set obtained from a nonequilibrium MD simulation
+based on the sinusoidal transverse force (STF) method.
+An excellent agreement between both sets of data was
+found. Figure 3(b) shows the normalized fit to Eq. (28).
+The normalized kernels, figure 3(b), show a similar be-
+havior for kn≤4. Though the higher density kernel is
+slightly lower for kn≥4 they show a similar limiting be-
+havior. This effect was not seen by Hansen et al.due
+to lack of data for high wave vectors. Figure 3(d) indi-
+cates that despite the difference between the interaction
+potentials, the results for LJ and WCA fluids are very
+close. This confirms that transport is dominated by re-
+pulsive interactions, rather than attractive. The sharper
+kernel for the diatomic system, figure 3(d), suggests a
+more Lorentzian-type behavior in the low wave-vector
+region. It is also important to mention that even though8
+00.050.10.150.20.250.30.350.4
+05101520253035 η~ (ky)
+kykernel data
+WCA (ρ=0.375, T=0.765)
+WCA (ρ=0.480, T=0.765)
+EMD
+NEMD
+(a)
+0.10.20.30.40.50.60.70.80.91
+05101520253035 η~
+L (ky) / η~
+L (0)
+kyLorentzian-type fit
+WCA (ρ=0.375, T=0.765)
+WCA (ρ=0.480, T=0.765)
+(b)
+0123456
+05101520253035 η~ (ky)
+kykernel data
+WCA (ρ=0.840, T=1.0)
+LJ (ρ=0.840, T=1.0)
+Cl2 (ρ=1.088, T=0.97)
+(c)
+00.20.40.60.81
+05101520253035 η~
+L (ky) / η~
+L (0)
+kyLorentzian-type fit
+WCA (ρ=0.840, T=1.0)
+LJ (ρ=0.840, T=1.0)
+Cl2 (ρ=1.088, T=0.97)
+(d)
+FIG. 3: Reciprocal-space kernels of monatomic and diatomic fluids: (a) Kernel data of a WCA fluid at two different densities
+(T= 0.765,Na= 2048); (b) Best fit of normalized kernel data (a) to Lorentzi an-type function Eq. (28); (c) Kernel data of
+a WCA fluid and LJ fluid at the same state point ( ρa= 0.840,T= 1.0,Na= 2048), Chlorine at ( ρa= 1.088,T= 0.97,
+Na= 1728); (d) Best fit of the normalized kernel data (c) to the Lo rentzian-type function Eq. (28).
+the fitting parameters are significantly affected by tem-
+perature, the resultingkernelsvaryweaklyoverthe range
+of temperatures chosen here. This was also observed by
+Hansenet al.for WCA monatomic fluids.
+B. Viscosity kernels in physical space
+The viscositykernelin reciprocalspaceisan evenfunc-
+tion since it is symmetric about the origin; thus the the
+real space kernel is symmetric because the Fourier trans-
+form keeps the even properties of the function. This
+means that the viscosity kernel in physical space can be
+foundviaaninverseFouriercosinetransform, F−1
+c[...](a
+special case of the continuous Fourier transform arising
+naturally when attempting to transform an even func-
+tion), of the viscosity kernel in reciprocal space. Since
+the integral is being computed over an interval symmet-
+ric about the origin (i.e. - ∞to +∞), the second integral
+must vanish to zero, and the first may be simplified to
+give:
+F−1
+c[˜η(ky)] =η(y) =/radicalbigg
+2
+π∞/integraldisplay
+0˜η(ky)cos(kyy)dky.(29)
+The inverse Fourier cosine transform of the Gaussian
+function, Eq. (27), exists [49] and it is even possible to
+obtain an analytical expression. For an NGterm Gaus-sian function the inverse Fourier cosine transform is
+ηG(y) =η0√
+2πNG/summationdisplay
+jAjσjexp[−(σjy)2/2]Aj,σj∈R+.
+(30)
+Though the Lorentzian-type function given in Eq. (28)
+fulfillsthe criteriaforhavinganinverseFouriertransform
+ηL(y) (i.e. the function is absolutely integrable, square
+integrable and the function and its derivative are piece-
+wise continuous), the integral in Eq. (29) is not readily
+obtained analytically in the general case. However, the
+integral can be evaluated numerically. In this work, a
+Simpson method has been employed for this purpose.
+The real space kernels for atomic and diatomic flu-
+ids at zero frequency are presented in figure 4, 5 and 6.
+Figure 4(a) shows the resulting kernels for chlorine and
+figure 4(b) shows the resulting kernels for a LJ fluid ex-
+tracted from two-term and four-term Gaussian functions
+Eq. (30), and inverse Fourier transform of Eq. (28). We
+find very little difference between the kernels obtained
+via two- and four-term Gaussians for these systems. Fig-
+ures 4(c) and 4(d) show the differences between all three
+fits. It can be seen that there exists a significant differ-
+ence (almost 25%) between the kernels extracted from
+Gaussian and Lorentzian type functions for small y. The
+discrepancy decreasesrapidly as yincreasesand becomes
+approximately zero for y≥1.5. The width of the kernel
+for chlorine is roughly 4-6 atomic diameters, figure 4(a),
+and 3-5 atomic diameters for monatomic LJ and WCA9
+0246810
+00.511.522.53η(y)
+yChlorine
+ηG2HηG4ηL
+(a)
+00.511.522.533.54
+00.511.522.53η(y)
+yLJ
+ηG2HηG4ηL
+(b)
+-2-1.5-1-0.50
+012345Δη(y)
+yChlorine
+ηG2H - ηG4ηL - ηG4ηL - ηG2H(c)
+-1.2-1-0.8-0.6-0.4-0.200.2
+012345Δη(y)
+yLJ
+ηG2H - ηG4ηL - ηG4ηL - ηG2H(d)
+FIG. 4: Real space kernel of monatomic and diatomic fluids as p redicted by Gaussian and Lorentzian fits of reciprocal space
+kernel data: (a) chlorine ( ρa= 1.088,T= 0.97); (b) LJ ( ρa= 0.840,T= 1.0); (c) Differences between the kernels shown in
+(a); (d) Differences between the kernels shown in (b).
+fluids, figure 4(b).
+For monatomic systems at relatively low densities
+(ρa= 0.375−0.480), the real space kernels are affected
+considerably by the functional form chosen to fit the re-
+ciprocal space kernel, figure 5. For instance, the equally
+weighted two-term Gaussian function, figure 5(a), dis-
+torts the real space kernels and predicts a noticeably
+higherη(y= 0)value. Asweincreasethedensity, thedis-
+crepancy between Gaussian functions, as well as between
+Gaussian and Lorentzian-type functions, only partially
+reduces, figure 5(b,d). The width of the kernel for WCA
+fluids at low density is roughly 2-4 atomic diameters.
+In figures 6(a) and 6(b) we compare the unnormalized
+kernel data in yspace extracted from four-term Gaussian
+and Lorentzian-type functional forms for all the simu-
+lated systems. Despite the fact that the difference be-
+tween the reciprocal kernels is less than 4%, figure 2(d)
+(e.g. chlorine - dashed line), the kernels for the corre-
+sponding systems in real space look noticeably different
+(figure6(a)andfigure6(b)), forallthesystems(e.g. chlo-
+rine - dashed dotted line). However, the zero wave-vector
+viscosities obtained from both functional forms are very
+close, with less than 2% error.
+We can determine the zerowave-vectorviscosities η0=
+η(k= 0,ω= 0) by integrating the real space kernel over
+y, and thus test our numerical analysis. The zero wave-
+vectorviscosity η0obtainedbyaGaussianfunction ηG(y)TABLE IV: Effective viscosities evaluated from ηG4(y),
+ηG2H(y) and numerically from ˜ ηL(k). The values can be
+compared to the zero frequency, zero wave-vector viscositi es
+shown in Table II.
+WCA LJ Chlorine
+State Point ρa0.375 0.480 0.840 0.840 0.840 1.088
+T0.765 0.765 1.000 0.765 1.000 0.97
+2-term Gaussian η00.265 0.392 2.290 2.723 2.614 6.881
+4-term Gaussian η00.265 0.390 2.288 2.807 2.653 6.897
+Lorentzian η00.269 0.428 2.320 2.913 2.711 7.049
+is
+η0=∞/integraldisplay
+−∞ηG(y)dy=η0√
+2π∞/integraldisplay
+−∞/braceleftBig/summationdisplay
+jAjσjexp[−(σjy)2/2]/bracerightBig
+dy.
+(31)
+Since the general analytical expression for ηL(y) does
+not exist [20] we evaluate the integral numerically and
+present the results from all functional forms in Table IV.
+A comparison of the viscosities in Table IV with the sim-
+ulated zero frequency zero wave-vector shear viscosities
+given in Table II shows an integration error of less than
+3%. This confirmsthe accuracyofour numerical analysis
+techniques.
+It is of interest to discuss the real space viscosity ker-
+nels for monatomic and diatomic systems from a struc-
+tural point of view. For this purpose we define a struc-10
+00.050.10.150.20.250.30.350.40.45
+00.5 11.5 2η(y)
+yGaussian fit
+ηG4ηG2HWCA ρ=0.375
+(a)
+00.10.20.30.40.50.6
+0 0.5 1 1.5 2η(y)
+yGaussian fit
+ηG4ηG2HWCA ρ=0.480
+(b)
+-0.2-0.15-0.1-0.0500.050.10.150.2
+00.511.522.53Δ η(y)
+yGaussian fit
+ηG4 - ηG2HηG4 - ηL
+ηL - ηG2HWCA ρ=0.375
+(c)
+-0.2-0.15-0.1-0.0500.050.10.150.2
+00.511.522.53Δ η(y)
+yGaussian fit
+ηG4 - ηG2HηG4 - ηL
+ηL - ηG2HWCA ρ=0.480
+(d)
+FIG. 5: Real space kernel of monatomic WCA fluids as predicted by Gaussian fits of the reciprocal space kernel data, Eq. (30) :
+(a) Kernels obtained from two- and four-term Gaussian fits fo r a WCA fluid at ρa= 0.375 and T= 0.765; (b) Kernels obtained
+from two- and four-term Gaussian fits for a WCA fluid at ρa= 0.480 and T= 0.765; (c) Differences between the kernels for a
+WCA fluid at ρa= 0.375 and T= 0.765; (d) Differences between the kernels for a WCA fluid at ρa= 0.480 and T= 0.765.
+0123456789
+00.511.522.53ηG4(y)
+y4-term Gaussian fit
+WCA ρ=0.375
+WCA ρ=0.480
+WCA ρ=0.840
+LJ ρ=0.840
+Cl2 ρ=1.088
+(a)
+0123456789
+00.511.522.53ηL(y)
+yLorentzian-type fit
+WCA ρ=0.375
+WCA ρ=0.480
+WCA ρ=0.840
+LJ ρ=0.840
+Cl2 ρ=1.088
+(b)
+FIG. 6: Real space viscosity kernels of monatomic and diatom ic fluids, WCA ( ρa= 0.375,T= 0.765), WCA ( ρa= 0.480,
+T= 0.765), WCA ( ρa= 0.840,T= 1.0), LJ (ρa= 0.840,T= 1.0), chlorine ( ρa= 1.088,T= 0.97): (a) Kernels obtained from
+the four-term Gaussian functional form Eq. (30); (b) Kernel s obtained numerically from the Lorentzian-type functiona l form
+Eq. (28).
+tural normalization factor
+ξg=∞/integraltext
+0r[g(r)−1]2dr
+∞/integraltext
+0[g(r)−1]2dr(32)
+whereg(r) is the radial distribution function (RDF).
+Eq. (32) is a measure of the range over which the cor-
+relation function decays to zero and therefore could be
+regarded as a correlation length of the radial distribu-
+tion function. The RDF (or structure factor in reciprocalspace) can be defined either in terms of the vector norm
+rijbetween the atoms iandjor between the centres of
+mass of molecules iandj:g(r) =/angbracketleftBig/summationtextN
+i=1/summationtextN
+j>1δ(|r−rij|)
+4πr2Nρ/angbracketrightBig
+,
+whereNis the total number of atoms or molecules, and
+ρis the atomic or molecular number density.
+The radial distribution functions and normalization
+factors are presented in figure 7. We can see that the
+RDF are typical monatomic and diatomic Lennard-Jones
+paircorrelationfunctions. ξggenerallyincreasesaswein-
+crease the density and temperature from 0 .605 at state11
+ 0 2 4 6 8 10 12 14
+ 0 1 2 3 4 5g(r)
+r(a) WCA ρ = 0.375, ξg=0.605(b) WCA ρ = 0.480, ξg=0.628(c) WCA ρ = 0.840, ξg=0.730(d) LJ ρ = 0.840, ξg=0.739(e) Chlorine ρ = 1.088, ξg=0.585
+FIG.7: Pair-distance correlation function g(r)andnormaliza-
+tion factors ξg, Eq. (32): WCA[(a) ρa= 0.375, (b)ρa= 0.480
+both at T= 0.765], (c) WCA ( ρa= 0.840,T= 1.0); LJ (d)
+(ρa= 0.840,T= 1.0); chlorine (e) ( ρa= 1.088,T= 0.97).
+For clarity, the RDFs are shifted upwards by 3 units.
+pointρa= 0.375,T= 0.756 to 0 .730 atρa= 0.480,
+T= 1.0 and only slightly increases as we increase the
+cutoff distance, i.e. switch from WCA system to a LJ
+system at the same state point. ξgfor chlorine at state
+pointρa= 1.088,T= 0.97 was found to be 0 .585.
+The normalized kernels with respect to η(y= 0) and
+normalization factor ξgare shown in figures 8(a) and
+8(b). While generally the width of unnormalized ker-
+nels increases as we increase the density (figures 6(a) and
+6(b)), the width of the normalized kernels of WCA flu-
+ids decreases marginally as we increase the density from
+0.376 (continuous line) to 0 .840 (short-dashed line). The
+LJ system shows a slightly narrower kernel (dotted line)
+compared to the WCA system at the same state point
+(short-dashed line). Though the kernels obtained from
+both functional forms are quite close to each other (al-
+most identical for values of yof about half of the atomic
+or molecular diameters, i.e. y= 0.5σ), we can see in
+figure 8(a) and 8(b) that the structural normalization
+did not completely remove the discrepancy between the
+normalized kernelsof the WCA system at different densi-
+ties, and the normalizedkernelsofWCA, LJ andchlorine
+systems, for values higher than y=σ. If we recall that
+figure 8(b) is based on a Lorentzian-type fit, a further
+question as to whether the kernel differences are due to
+numerical analysis, i.e. the choice of the fitting functionor due to improper structural factor arises. A four-term
+Gaussian only shows a slightly narrower kernels which
+suggests a need for a more comprehensive structural nor-
+malization.
+V. CONCLUSION
+The wave-vector dependent viscosity of monatomic
+Lennard-Jones, monatomic Weeks-Chandler-Andersen
+and diatomic (liquid chlorine) fluids over a large wave-
+vector range and for a variety of state points has been
+computed. The equilibrium molecular dynamics calcula-
+tion involved the evaluation of the transverse momentum
+densityandshearstressautocorrelationfunctionsin both
+atomic and molecular hydrodynamic representations for
+molecular fluids. The main results can be summarized as
+follows:
+(i) For monatomic fluids the shape of the normalized
+viscositykernelin reciprocalspacein the low wave-vector
+region is the same for a whole range of densities consid-
+ered here. Though the normalized reciprocal kernels in-
+significantly decreases with the density they show a sim-
+ilar limiting behaviour at high kyvalues. For the LJ
+potential compared to a WCA potential we find higher
+viscosities in the low wave-vector region but the normal-
+ized shape of the kernels are almost identical.
+(ii) For liquid chlorine, the wave-vector dependent vis-
+cosity shows a similar behaviour in both atomic and
+molecular formalisms within statistical uncertainty.
+(iii) While a relatively simple Lorentzian-type function
+fits the atomic and diatomic data well over the entire
+range of kyat all the state points it is not possible to
+analytically inverse Fourier transform it to the real space
+domain. Therefore one may consider an expansion up
+to a four-term Gaussian which gives better accuracy in
+reciprocal space compared to the Lorentzian-type func-
+tion. Our analysis of the high kyregime reveals that the
+two-term equally weighted Gaussian functional form is
+inaccurate in predicting the real space kernels whilst the
+unequally weighted Gaussian only slightly improves the
+fit.
+(iv) Theoverallconclusionis thatthe realspaceviscos-
+ity kernel for chlorine has a width of roughly 4-6 atomic
+diameters while for monatomic systems at high densities
+the width is about 3-5 atomic diameters and 2-4 atomic
+diameters at low densities. This means that generalized
+hydrodynamics must be used in predicting the flow prop-
+erties of molecular fluids on length scales where the gra-
+dientin the strainratevariessignificantlyonthesescales.
+Consequentlyanonlocalconstitutiveequationsshouldbe
+invoked for a complete description of flows at atomic and
+molecular scales under such conditions.
+Finally, ourresultsformolecularfluids shouldalsopro-
+vide a good test for more complex molecular systems and
+the methodology can easily be used for instance in chain-
+like molecules.12
+00.20.40.60.81
+0123456ηG4(y) / ηG4(0)
+y/ξg4-term Gaussian fit
+WCA ρ=0.375
+WCA ρ=0.480
+WCA ρ=0.840
+LJ ρ=0.840
+Cl2 ρ=1.088
+(a)
+00.20.40.60.81
+0123456ηL(y) / ηL (0)
+y/ξgLorentzian-type fit
+WCA ρ=0.375
+WCA ρ=0.480
+WCA ρ=0.840
+LJ ρ=0.840
+Cl2 ρ=1.088
+(b)
+FIG. 8: Normalized real space viscosity kernels of monatomi c and diatomic fluids, WCA ( ρa= 0.375,T= 0.765), WCA
+(ρa= 0.480,T= 0.765), WCA ( ρa= 0.840,T= 1.0), LJ ( ρa= 0.840,T= 1.0), chlorine ( ρa= 1.088,T= 0.97):
+(a) Normalized kernels, shown in Fig. 6(a), obtained from th e four-term Gaussian functional form Eq. (30); (b) Normaliz ed
+kernels, shown in Fig. 6(b), obtained numerically from the L orentzian-type functional form Eq. (28).
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\ No newline at end of file
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+arXiv:1001.0692v3  [hep-ph]  20 May 2010On the critical exponentof η/sand a newexponent-less
+measureof fluidity
+RaktimAbir,MunshiG. Mustafa
+TheoryDivision, Saha Institute of Nuclear Physics
+1/AFBidhan Nagar, Kolkata 700064, India
+Abstract
+Wediscussonthecriticalexponentof η/sforafluid,andproposeanewexponent-less
+measureoffluiditybasedonamode-modecouplingtheory. Thi sexhibitsaremarkable
+universality for fluids obeying a liquid-gas phase transiti on both in hydrodynamic as
+well as in nonhydrodynamic region. We show that this result i s independent of the
+choice of the fluid dynamics, viz., relativistic or nonrelativistic. Quark-GluonPlasma,
+being a hot relativistic and a nearly perfect fluid produced i n relativistic heavy-ion
+collisions, is expected to obey the same universality const rained by both the viscous
+and the thermal flow modes in it. We also show that if the ellipt ic flow data in RHIC
+putsaconstrainton η/sthenthenewfluiditymeasureforQuark-GluonPlasmainturn
+alsorestrictstheothertransportcoefficient, viz., thethermalconductivity.
+Keywords: Relativistic Fluid,Quark-GluonPlasma, Heavy-IonCollis ion,Viscosity,
+ThermalConductivity
+Recentresults[1,2]fromtheRelativisticHeavyIonCollid er(RHIC)atBNLreveal
+surprising and intriguing dynamical properties of the Quar k-Gluon Plasma (QGP). In
+non-centralcollisions,theanisotropywithrespecttothe reactionplane, i.e.,theelliptic
+flow coefficient v2[2], can well be described up to transverse momenta of order 1 .5
+GeV/cby the nearly ideal hydrodynamicswith small shear viscosit y,η. This is much
+smaller than that obtained in the perturbative quark-gluon plasma [3]. This suggests
+that the matter produced in the early phase of the RHIC collis ions is strongly inter-
+acting with nearly perfect fluidity represented by a very sma ll ratio of shear viscosity
+to entropy density, η/s. The supporting arguments for it have come from various di-
+rections: viz, viscous hydrodynamics [4, 5] sets an upper bound, η/s≤(5/4π); the
+gauge-gravitydual theory [6] conjectures at universal low er bound, η/s≥1/4π; and
+the Heisenberg’s uncertainty principle implies a lower bou nd [6, 7], η/s∼1 (we use
+units with ¯h=kB=c=1). The fluidity of QCD fluid has also been computed from
+numericalsimulationinlattice [8],non-perturbativemod els[9, 10]andhasbeencom-
+pared [12, 11, 13] with commonly known fluids. Moreover, for fl uidsηis either a
+Email addresses: raktim.abir@saha.ac.in (Raktim Abir), munshigolam.mustafa@saha.ac.in
+(MunshiG. Mustafa)
+Preprintsubmitted to Elsevier November 5,2018nondivergingor a very weakly diverging quantity [14, 7] wit h an exponent ∼−0.04.
+Thus, the fluidity of a hot viscous QCD matter remains an open a s well as interesting
+problemthatrequiresamuchdesiredexplorationofthefluid mechanics.
+Onecannowaskwhatisthephysicalquantitythatwouldreflec tagoodmeasureof
+fluidityofa hotviscousfluid orin thiscontexthowmuchrelev antisthe η/s, whichis
+formedby two quantities, completely differentin nature as ηis a dynamicaltransport
+coefficient whereas sis a thermodynamicone. A possible explanation lies within t he
+general structure of any diffusion constant may it be therma l diffusion, Dκrelated to
+thediffusivedecayoflongitudinalcomponentsofmomentum fluctuationortheviscous
+diffusion, Dηrelated to the diffusive decay of transverse components of m omentum
+fluctuationof fluids. In this letter we try to addressthis imp ortantaspect of a hot fluid
+basedonfluiddynamics.
+Dynamical propertiesof a manyparticle system can be invest igatedby employing
+an externalprobe, which disturbsthe system only slightly i n its equilibriumstate, and
+bymeasuringthe responseof the system to thisexternalpert urbation. A largenumber
+ofexperimentsbelongtothis category,such asstudiesof va riousline shapes, acoustic
+attenuation, and transport behaviour. In all these experim ents, one probes the dynam-
+ical behaviour of the spontaneousfluctuationsin the equili briumstate. In general, the
+fluctuationsare related to the correlationfunction, which provideimportant inputsfor
+quantitativecalculationsofcomplicatedmany-bodysyste m.
+Thedensity-densitycorrelationfunctionisdefinedas
+σnn(r,t)=/angbracketleftδn(r,t)δn(0,0)/angbracketright, (1)
+where the angular bracket denotes an equilibrium ensemble a verage, and δn(r,t) =
+n(r,t)−/angbracketleftn/angbracketright,is the local deviation of the number density n(r,t)from the equilibrium
+valueofthenumberdensity. Thespectralfunctioncanbeobt ainedbyFouriertransfor-
+mationof(1) as
+ρnn(q,ω)=/integraldisplay
+d3r/integraldisplay∞
+−∞dtσnn(r,t)e−i(q.r−ωt), (2)
+andthe static correlationfunctioncanbe definedas
+τnn(q)=/integraldisplay+∞
+−∞dω
+2πρnn(q,ω). (3)
+Now, within hydrodynamical description the correlation fu nction can be calculated
+only when wave number of the dynamical density fluctuation, qis much smaller that
+the inversecorrelationlength, ξ(qξ<<1), or, equivalentlywavelength, λof density
+fluctuation is appreciably larger than the correlation leng th (λ>>ξ). In the case of
+nonrelativistic fluids, the dynamical density fluctuation b ased on Navier-Stokes equa-
+tion[15]inthehydrodynamiclimitaswellinthenonhydrody namiclimit ( viz.,around
+critical point) has been studied [16] for Newtonian fluids in details. Recently, a rel-
+ativistic generalisation of density fluctuation is obtaine d [17] using dissipative rela-
+tivistic fluid dynamics coupled with conserved quantities a nd the dynamical structure
+2functionis
+ρnn(q,ω)/τnn(q) =/parenleftbigg
+1−1
+γ/parenrightbigg2Dκq2
+ω2+(Dκq2)2+1
+γ/braceleftBigg
+Dsq2/2
+(ω−csq)2+(Dsq2
+2)2+Dsq2/2
+(ω+csq)2+(Dsq2
+2)2/bracerightBigg
+, (4)
+whereγis the ratio of specific heats, ˜CP=T(∂S/∂T)Pat constant pressure to ˜Cn=
+T(∂S/∂T)nat constant density with Sis the equilibrium entropy per particle, csis
+the speed of sound propagation with the damping constant DsandDκis the thermal
+diffusivity. The higher-order terms involving the quantit iesDκq/csandDsq/csare
+neglected1. Thethermaldiffusivityisdefinedas,
+Dκ=κ
+n0˜CP, (5)
+whereκis the thermal conductivityand n0is the equilibriumnumberdensity of fluid.
+Note that the nonrelativistic equivalence of n0˜CPin (5) isρ0Cp, whereρ0is the equi-
+libriummassdensity.
+Thesoundwavedampingconstantisgivenas
+Ds=Dκ(γ−1)+4
+3η+ζ
+w0+c2
+sT2/parenleftbiggκ
+w0−2DκαP/parenrightbigg
+, (6)
+whereηandζare, respectively, shear and bulk viscosities, w0is the equilibrium en-
+thalpydensityand αPisthethermalexpansivityatconstantpressure. Thethirdt ermin
+(6) is purely relativistic correctionwhereas the second te rm is the minimal relativistic
+oneduetothe appearanceof w0insteadof ρ0.
+The dynamical structure function of density fluctuation for relativistic dissipative
+fluidsin(4)isasumofthreeLorentzianlineshapeswithgene ralformf(ω)=2Γ/[Γ2+
+(ω−ω′)2], centered at the frequency ω′with a half-width at half maximum given by
+Γ. TheRayleighcomponent,iscenteredabout ω=0,witha halfwidth
+ΓR=Dκq2. (7)
+TheBrillouindoublets,arelocatedsymmetricallyabout ω=0atthefrequencies ω±
+B=
+±csq,eachonewitha half-widthgivenby
+ΓB=1
+2Dsq2. (8)
+1Arelativistic firstorderequation, suchasthatbyLandauor byEckart,isparabolicandformallyviolates
+thecausality,andhenceacausal. Thecausalityproblemcan becircumventedwiththeinclusionofthesecond-
+order in derivative expansion [4, 17, 18]. However, this doe s not affect the spectral function of dynamical
+density fluctuation.
+3Figure 1: (Colour online) The structure function ρnn/τnnas a function of ωin the minimal relativistic fluid
+(red curve), nonrelativistic fluid (black curve) and their d ifference (blue curve) for parameters [17] q=
+0.1 fm−1,chemical potential µ=200 MeV, T=200 MeV, η/(n0S)=ζ/(n0S)=0.3 andκT/(n0S)=0.6.
+The minimal relativistic correction shows up in the Brillou in doublets but not in the Rayleigh component
+centered at ω=0 (seetext).
+We note that in (4) the transversecomponentof the momentumfl uctuationdecou-
+ples from the density fluctuation but admits a diffusive solu tion that relaxes exponen-
+tiallywith theviscousdiffusion, Dηas
+Dη=η
+w0, (9)
+which reduces to η/ρ0in the nonrelativistic limit. The knowledge of the line widt hs
+in (7) and (8) along with the three diffusion equations(5), ( 6) and (9) are sufficient to
+determinethe threetransportcoefficients, κ,ηandζ.
+NowwenotethatthepeakandthewidthoftheBrillouindouble tsobviouslychange
+duetotherelativisticcorrectionsofthesoundmode, Dsin(6). IncontrasttheRayleigh
+peakaswellasthewidth ΓRremainsameforbothnonrelativisticandrelativisticfluid s
+because the thermal diffusion Dκdoes not change due to the equivalence between
+n0˜CPandρ0CP. We demonstratethisaspectinFig.1withonlytheminimalre lativistic
+correctionsin(6). Further,thelinewidth ΓRiscontrolledbythedominantbehaviourof
+eitherκor˜CPinDκ. The behaviourof Dswill control that of ΓBthroughthe relative
+behaviours of κ/˜Cn,ηandζ. Also the strength of the various peaks in (4) depends
+onγ. Thishydrodynamicspredictionshavebeenusedinthe limit T→TCto studythe
+behaviour of relativistic fluid [17] around critical point. The fundamental assumption
+of hydrodynamic, qξ<<1, close to TCis no longer valid but it was argued that for a
+givenqtherewillbealwaysatemperaturerangeoverwhichhydrodyn amicspredictions
+shouldbereliable. Aroundthe critical point γdivergesfaster thanthe viscosities ( viz.,
+4ζ), one can easily find from (4) that the sound mode gets attenua ted and the only
+survivingsoft modeisthe diffusivethermalmode. Thishasi mportantconsequenceas
+wewill see below.
+Generally,the longwavelengthpart of the density fluctuati onaroundcritical point
+isveryintensethatcouldinduceavelocitygradientintheb oundaryofthefluid,which
+woulddissipateenergy. Thisdissipationcanbeinterprete dintermsofthemode-mode
+coupling[19,20]. Forinstance,asdiscussedabove,thehea tflowmode(thermaldiffu-
+sion) decays into shear viscousmode plus a thermal mode. One can obtain this decay
+modeintermsofthe relevanttransportcoefficients.
+Information on critical behaviour of fluids are generally ex tracted from scatter-
+ing experiments performed on the fluids [21]. To analyse thes e data Swinney and
+Henry[22],basedonmode-mode-couplingtheory,obtaineda particulardimensionless
+combinationof measured quantities for wide range of fluids o beyingliquid-gasphase
+transitionas
+Γ∗=6πηΓR
+Tq3. (10)
+Usingthe measuredvaluesof ΓR,ηand thecorrelationlength, ξfordifferenttemper-
+aturesandscatteringangles( q=4πsin(θ/2)/λ,θistheangleofscattering),obtained
+for various sample of multiple component fluids [21], when pl ottedΓ∗as a function
+ofqξall fluids obeying liquid-gas phase transition fall on a sing le universal curve as
+shown in Fig. 2. The single curve describes the critical beha viour not only along the
+critical isochore and the coexistence curve, but also along any thermodynamicpath in
+thecriticalregion. ItisclearfromFig. 2that,
+Γ∗=/braceleftbigg1
+qξforqξ<1 (hydrodynamicdomain),
+1 forqξ≥1 (nearcriticalpoint),(11)
+provided6πηΓRξ
+Tq2=1,andusing(7)it becomes
+Fηκ=η
+(Dκξ)−1T=1
+6π, (12)
+whichis independentofthe natureofthe fluiddespite the var iousmicroscopicdetails.
+This is also consistent with Stokes formula for the mobility of a sphere of radius ξ
+movingthrougha viscousliquidwitha nondivergingshearvi scosity,η.
+Now, the equality, Γ∗=1 in (11) basically holds at and near the critical point,
+indicatingthelengthscalematchingaroundthecriticalpo int. Thisimpliesthatinstead
+of diverging without any upper bound, the correlation lengt h of fluid becomes of the
+order of the wavelength of the density fluctuation ( ξ∼λ). On the other hand this
+also suggests that Dκξin together is a nondiverging quantity around critical poin t
+since both of them are expected to have same critical exponen t [15] but opposite in
+sign. The dimensionless quantity, Fηκassociated with the viscous and the thermal
+modes together, could then be regarded as a good fluidity meas ure of a hot viscous
+fluid obeying liquid-gas phase transition, which remains at a universal value 1 /(6π)
+forbothnonhydrodynamicalaswell asinhydrodynamicalreg ion.
+5Figure 2: Γ∗as a function of qξfor several several substances that cover a wide range of mol ar mass,
+chemical structure and complexity and itisadopted from[22 ]. Notethat themeasured data arefromvarious
+experiments and thedetailed references of thosecan also be found in [21, 22,23].
+Wenownotethattheimportantquantitythatentersin(11)an d(12)istheRayleigh
+width,ΓR. Theanalysis,basedontheNavier-Stokestheorypresented inthebeginning
+of this letter, clearly exhibits that the relativistic corr ection does not show up in the
+Rayleighwidth ΓRin (7). It indicatesthat the fluidity, Fηκis, obviously,independent
+of the choice of the ( viz., relativistic or nonrelativistic) fluid dynamics. This possibly
+indicates that a good fluid, either relativistic or nonrelat ivistic, is always a good fluid
+irrespectiveof theotherdetailsofthe microscopicdegree soffreedom.
+In a heavy-ioncollisions, the producedmatter has a strong fl ow initially along the
+beam axis. If the matter is viscous the viscosity then counte racts this by reducing the
+effectivelongitudinalpressureandthusenhancingitinth etransverseplane. Therefore,
+the viscosity leads to a larger radial flow than that of the ide al fluid and also to an
+angular modulated radial flow ( v2) due to the velocity gradient. This is indeed the
+situationintheexperimentsatRHIC[1,2]thatprobablypro bedaQGPatatemperature
+little above TCwith a small value of η/s[4, 5, 12, 11] implying that the QCD matter
+produced in such experiments is a nearly perfect fluid and lik ely to obey a liquid-gas
+phase transition with a critical end point [24]. The spectra l analysis for relativistic
+fluid in (4) or in Fig. 1 represents that a small density pertur bation propagates at the
+speed of sound and is eventually damped near the critical poi nt. The only surviving
+modeisthediffusiveheatflowmode. Theobservedfluiditycou ldbeinterpretedonthe
+basis of mode-modecoupling theory as the decay of a longitud inalheat flow mode in
+QGP to a viscous and a thermal mode in which the transverse vis cous mode reduces
+the thermal flow in the QGP. The QGP fluid produced in RHIC belon gs to the same
+universality class defined by the fluidity measure in (12) tha t interrelates the relevant
+6transportcoefficients, viz.,the shearviscosity η, andthethermalconductivity κ.
+We nowrecallthatforfluids ηiseitherbenondivergingorweaklydiverging(with
+anexponent ∼−0.04)[7, 14]. Onecan expectthat η/sshouldalso be a nondiverging
+orweaklydivergingquantityasentropygenerallydoesnots howanycriticalexponent.
+Surprisingly, for simple fluids as well as for QGP η/s(as obtained from lattice QCD
+simulation [8] and also fromthe viscoushydrodynamiccalcu lations[4, 5, 12] in view
+of recent RHIC data [2]) shows an exponent (see Fig.3 of Ref. [ 12]) of the order of
+’one’. Thisobservedfeatureof η/scanclearlybe seenfrom(12)as
+η
+s=Fηκµ
+S(κξ)−1, (13)
+where inverse of κξin together has an exponent ∼1.2 for liquid-gas phase transi-
+tion[15].
+For various fluids as discussed in Refs. [12, 23] one can param etriseη/swith the
+reducedtemperature, ε=|T/TC−1|,inthe domain TC≤T≤2TCas
+η
+s∼(aεδ+b), (14)
+whereaandbare different for different fluids. For QGP [8, 12] the ellipt ic flow data
+in RHIC [2] restrict those as δ∼1,a∼0.64∼2/3 andb∼0.1∼1/(4π), which
+describe the viscous mode despite the various microscopic f eatures in it. Obviously,
+the new fluidity measure Fηκin (12) then puts a constraint on the heat flow mode in
+QGP,whichinturnrestrictstheassociatedtransportcoeffi cientκas
+κ=Fηκ(aεδ+b)−1µ
+Sξ, (15)
+wherethecorrelationlength, ξisrelatedtovariousmomentsofthenetbaryonnumber
+fluctuations[25].
+In view of nearly perfect fluidity nature of QGP observed in RH IC we suggest
+througha newexponent-freefluiditythat the thermalconduc tivity,whichisa measure
+of transport of energy by the particles, also becomes an impo rtant quantity to look at.
+Thishasnotbeen discussedbeforein the literaturein thisc ontext. Interestingly,it has
+beenproposedto measurethethermalconductivityinRHICwi thits upgradation[26],
+whichcouldthentesttheaboveestimationofthisquantityb othinhydrodynamicaland
+nonhydrodynamicalregions, and in turn our proposal of new fl uidity measure. It will
+alsobeveryinteresting,ifonecanfindapossiblewaytochec kthisuniversalbehaviour
+offluiditythroughlattice QCD calculationsandinnonpertu rbativemodels. TheLarge
+Hadron Collider (LHC) at CERN will produce a QGP at much highe r temperature(at
+least twice ofRHIC) and µclose to zero, whichwouldthen coolandpass throughthe
+criticalregionanditwouldbeinterestingtoseewhetherth erecouldstillbeaRayleigh
+peakin thislimit.
+We notethatouranalysisisbasedonthemode-modecouplingt heoryandtherela-
+tivistic Navier-Stokestheory. A basic assumptionof thism ode-mode-couplingformu-
+lationis the nondivergenceofshear viscosity whereasa pro perRG calculationreveals
+that shear viscosity has a very weak divergencegovernedby t he exponent −0.04. On
+7the otherhanda Navier-Stokestheoryis a kindofmeanfield eq uationandmaynot be
+very appropriate to discuss the dynamics around critical po int. The dynamics of the
+fluid in and around the critical point should, however, be des cribed by the evolution
+equation derived from a microscopic theory. Nonetheless, t he present understanding
+of the critical dynamics of the produced QCD matter in RHIC ex perimentsis meagre
+andoneneedstodependonakindofmeanfielddescriptiontoqu alitativelyunderstand
+thebehaviourofthesystem inandaroundthecritical pointb ecausethehydrodynamic
+variablesbecomecomparabletothetimescaleofanorderpar ameteraroundthecritical
+pointduetocriticalslowingdown.
+Acknowledgments:. Authors are thankful to S. Chattopadhyaya for very fruitful dis-
+cussion. In particlar, RA is also thankful to A. Basu and A. Mi dya for very useful
+discussionandhelpduringthe courseofthiswork.
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+[24] M.Stephanov,K.RajagopalandE. Shuryak,Phys.Rev.Le tt.81,4816(1998).
+[25] M.A. Stephanov,Phys.Rev.Lett. 102,032301(2009).
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+10
\ No newline at end of file
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+arXiv:1001.0693v1  [cond-mat.stat-mech]  5 Jan 2010Kibble-Zurek mechanism and infinitely slow annealing throu gh critical points
+Giulio Biroli,1Leticia F. Cugliandolo,2and Alberto Sicilia3
+1Institut de Physique Th´ eorique, CEA, IPhT, F-91191 Gif-su r-Yvette, France and CNRS, URA 2306
+2Universit´ e Pierre et Marie Curie – Paris VI, LPTHE UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
+3Department of Chemistry, University of Cambridge, Lensfiel d Road, CB2 1EW, Cambridge, UK
+(Dated: November 17, 2018)
+We revisit the Kibble-Zurek mechanism by analyzing the dyna mics of phase ordering systems
+during an infinitely slow annealing across a second order pha se transition. We elucidate the time
+and cooling rate dependence of the typical growing length an d we use it to predict the number of
+topological defects left over in the symmetry broken phase a s a function of time, both close and far
+from the critical region. Our results extend the Kibble-Zur ek mechanism and reveal its limitations.
+PACS numbers:
+The out of equilibrium dynamics induced by a quench
+are the focus of intense research [1, 2]. Interesting re-
+alizations are quenches through a second order phase
+transition, which take the system from the symmetric
+phase into the symmetry broken one. Below the tran-
+sition, times scaling with the system size are needed to
+reach equilibrium and to realize the spontaneous symme-
+try breaking process. Before this–typically unreachable–
+asymptoticlimitthesymmetryisbrokenonlylocally: the
+system is formed by ordered regions of size growing with
+time [3]. Only when this size reaches the order of the vol-
+ume of the sample the symmetry is broken globally and
+the spatial average of the order parameter deviates from
+zero. The majority of theoretical studies focused on the
+dynamics after infinitely rapid quenches although experi-
+mentally quenches are performed at finite speed. Indeed,
+since the typical time-scale on which the system evolves
+is its age, i.e. the time elapsed since crossing the critical
+point, finite quench time-scales ( τQ) eventually become
+short compared to the relaxation time. Thus, they al-
+ter the out of equilibrium dynamics at short times only.
+The opposite limit of an extremely slow annealing, cor-
+responding to very long τQ, needs a separate treatment.
+Surprisingly,thishasnotbeenstudiedindetailinthesta-
+tistical physics literature with, however, some exceptions
+for disordered systems [4–6]. It has, instead, attracted a
+lot of attention within the cosmology and, more recently,
+the condensed matter communities. An explanation of
+the slow dynamics induced by this protocol was given by
+the so-called Kibble-Zurek (KZ) mechanism [7–10]. This
+isanequilibrium scaling argument that yieldsanestimate
+for the density of topological defects left over in the or-
+dered phase as a function of the quenching rate close to
+the critical point. The argument has been recently gen-
+eralized to study very slow ‘quantum annealing’ across a
+quantum phase transitions in isolated systems [11–13].
+The aim of this work is to obtain a more complete pic-
+ture of the slow dynamics induced by an extremely slow
+annealing. With numerical and analytical arguments we
+unveil the limitations of the KZ approach and we obtain
+a full scaling description of the slow dynamics. Our mainresult is that the dynamic evolution is characterized by
+a first adiabatic regime in agreement with KZ, followed
+by critical coarseningand, finally, standard coarseningat
+very long times. We find a newuniversal scaling function
+that characterizes the growth of the correlation length
+out of equilibrium under slow cooling procedures and we
+relate it to the number of topological defects in cases in
+which these exist.
+We start our discussion by recalling the KZ mecha-
+nism [8–10]. We take a system in equilibrium at equi-
+librium at a value g0> gcof the control parameter in
+the symmetric phase and subsequently anneal it at finite
+rate (in typical situations gcorrespondsto temperature).
+As KZ, we focus on the protocol g(t) =gc(1−t/τQ)
+starting from, say, g0=g(−τQ) = 2gc. Henceforth we
+use the standard notation of dynamical critical phenom-
+ena [14] and we set the microsocopic time and length
+scales to one. Far from the critical point the equilib-
+rium relaxation time, τeq, is barely larger than the mi-
+croscopic time. Thus, for very small annealing rate, i.e.
+very long τQ, the system evolves adiabatically and re-
+mains in equilibrium at the running g(t). However, this
+regime must inevitably break down since τeqdiverges at
+the critical point as |∆g|−νzeqwith ∆g≡g−gc. KZ
+argued that the end of the adiabatic regime occurs when
+theremainingtime, ˆt, neededtoreach gcbecomessmaller
+thanτeq. This is certainly a lower bound and yields
+ˆt∝τeq(ˆg)∝τνzeq/(1+νzeq)
+Q with ˆg=g(−ˆt). The dis-
+tance from the critical point at −ˆtis ∆ˆg∝τ−1/(1+νzeq)
+Q.
+KZ assumed that after −ˆtthe topological defect con-
+figuration remains frozen, in the sense that the order
+parameter ceases to evolve. In this so called ‘impulse’
+regime the effect of lowering gis to reduce fluctuations,
+which are in general of thermal origin since very often
+gis related to the temperature. The main prediction
+of KZ is the number of topological defects, N, at the
+symmetric instant ˆtwhere the coupling constant equals
+gc−∆ˆg. Within their approach Nis inherited from the
+configuration at −ˆt, it is therefore equal to the number
+of defects in equilibrium at ˆ g, and it is estimated to be2
+N(ˆt)≃[f2ξeq(ˆg)]−d≃f−2d|∆ˆg|dνwithfof the order
+of one. Knowing the τQ-dependence of ∆ˆ gallows one to
+derive the τQ-dependence of N:
+N(ˆt)∝τ−dν/(1+νzeq)
+Qatˆt∝τνzeq/(1+νzeq)
+Q.(1)
+We stress that for each τQthis expression should be mea-
+sured at the special instant ˆt(τQ) after the phase tran-
+sition. The system’s behaviour at t >ˆtis not fully ad-
+dressed by KZ. In some publications it is assumed that
+any further evolution [15, 16] can be neglected, whereas
+in others it is reckoned that after ˆtthe system resumes
+its out of equilibrium evolution with a mechanism that
+depends onthe problemat hand(domaingrowth, vortex-
+anti-vortex diffusion and annihilation, etc.) [9] and the
+density of defects may therefore continue to decrease al-
+though no detailed study was performed. Numerous nu-
+merical [15–19] and experimental [20–24] papers tested
+the quantitative consequences of the Kibble-Zurek mech-
+anism with variable results. While the numerical studies
+claimed that they successfully verified the predictions,
+the conclusions are less clear in the experimental works
+that studied vortex formation in superfluid4He and3He
+with null results in the former [20] and agreement with
+the KZ prediction in the latter [21]. See [22–24] for dis-
+cussions of some recent experimental results disagreeing
+with the KZ prediction.
+In the following we revisit the KZ scaling analysis.
+We focus on the dynamics of classical systems coupled
+to an environment, a setting in which the dynamics are
+stochastic and the energy, directly linked to the num-
+ber of topological defects, is not conserved. We use the
+temperature of the thermal bath as the control parame-
+ter driving the second order phase transition and a linear
+cooling rate T(t) =Tc(1−t/τQ). These choices are made
+to keep the discussion simple; extensions to more compli-
+cated protocols are straightforward and will be partially
+addressed later. Moreover, we restrict to systems with
+a unique equilibrium correlation length and a single dy-
+namic counterpart, the typical growinglength. We solely
+dealwith problemswith power-lawscalinglaws, that link
+e.g.thecorrelationlengthtothedistancefromcriticality,
+and length-scales to time-scales [25]. These restrictions
+exclude from the analysis complex systems with several
+competing lengths and problems with quenched disorder.
+Henceforth we focus on the growth law of the size of the
+correlated regions, R(t). This will allow us to discuss
+systems characterized by topological defects as well as
+those that are not from the same point of view. We shall
+explain below how the density of topological defects can
+be obtained from R(t).
+Let us start our analysis with some simple remarks.
+First, although the initial adiabatic regime and the de-
+parture from it are expected, the existence of the im-
+puse regime is questionable. During this regime, which
+would take place between −ˆtandˆt, the system is sup-
+posed not to evolve [7–10]. However, it is well-knownthat after a rapid quench into the critical region any
+system undergoes critical coarsening described by the
+growth of a typical linear length-scale for correlated re-
+gions,R(∆t)∼∆t1/zeqwhere ∆ tis the time spent in
+the critical region and zeqis the exponent that links the
+equilibrium relaxation time and correlation length close
+to criticality [26, 27]. Above criticality, the major differ-
+ence between slow and rapid quenches is in the extension
+ofthe adiabaticregime. The slowerthe quench orthe an-
+nealing, the closer the system gets to the critical point in
+equilibrium. However, also for very slow annealing, the
+system eventually departs from the adiabatic evolution
+and has to undergo critical coarsening.
+Our second remark is that once getting across the crit-
+ical point, when the running temperature T(t) is far
+enough from Tc, the dynamics crosses over to standard
+coarsening. In order to get a better insight into this pro-
+cess, let us recallthat aninfinitely rapidquenchto atem-
+perature T < T cleads to a growth law R≃λ(T)∆t1/zd
+[1]. Now zdis thedynamic exponent that, quite gener-
+ally, is different from zeqand depends on the dynamic
+rules. The prefactor vanishes at Tcand is characterized
+by a singular power law λ(T)≃ |T−Tc|ν(−1+zeq/zd)=
+ξ1−zeq/zdeq [28]. If the annealing rate is finite, one natu-
+rally expects the growth law at long times and far from
+the critical point to be R≃λ(T(t))∆t1/zd. The reason
+is that the dynamical process renormalizing the value of
+λ(T(t)) should be finite and, hence, evolve on a much
+faster timescale than the coarsening one which instead is
+of the order of the age of the system and diverges with t.
+We now endeavor to connect the dots and propose a
+general scenario for infinitely slow annealing. Our main
+conjecture, that is motivated by the previous discussion
+and the fact that the system stays for a very long time
+in the vicinity of the critical point, involves the growth
+of the length-scale R(t):
+R(t)≃ξeq(T(t))f/bracketleftbiggt
+τeq(T(t))/bracketrightbigg
+. (2)
+Thisasymptoticformencompassesequilibrium abovethe
+critical point, x≡t/τeq(T(t))≪ −1, critical coarsening,
+x∝O(1), and the cross-overto standard coarsening x≫
+1. The limits of f(x) are obtained by requiring to find
+the expected adiabatic behavior and standard coarsening
+on the two extremes,
+R(t)≃/braceleftBigg
+ξeq(T(t)) t≪ −τeq(T(t)),
+[ξeq(T(t))]1−zeq
+zdt1
+zdt≫τeq(T(t)).(3)
+This imposes that f(x) be a constant for x≪ −1 and
+proportional to x1/zdforx≫1. We expect the scaling
+function f(x) to be universal since it describes evolution
+on diverging time and length scales close to the critical
+point. Asketchof ξeqandRis shownin Fig. 1. Ourscal-
+ing assumption applies to coarsening with and without3
+(a)∆g(t)
+τQ2 τQ1 0 −ˆt1−ˆt2 −ˆt3
+RτQ4RτQ3RτQ2RτQ1ξeq
+(b)ξeq,R
+ˆg4 ˆg1gc
+FIG. 1: (Colour online.) (a) The control parameter, ∆ g(t) =
+[g(t)−gc]/gc, for different cooling rates. The crossover be-
+tween adiabatic and out of equilibrium dynamics are signale d
+as−ˆtiforτQi< τQi+1. (b) Sketch of the control parameter
+dependence of the equilibrium correlation length (thick re d
+line) and the dynamic growing length Rfor four linear cool-
+ing rate proceedures with τQi< τQi+1. Values of the control
+parameter at which the dependence changes from adiabatic to
+critical areshown as ˆ gi(for simplicity weplot themas singular
+points in the evolution of R. In reality they just correspond
+to cross-overs). For comparison the assumption of constanc y
+during the impulse and subcritical regimes are shown with
+thin horizontal lines.
+topological defects. In the former case, the decaying typ-
+ical number of topological defects ,N(t)≃R(t)−d, reads
+fort/greaterorsimilarτQ
+N(t)≃τdν
+zd(zeq−zd)
+Qt−d
+zd[1+ν(zeq−zd)].(4)
+Note that a qualitatively similar dependence on tandτQ
+was found numerically in [29] in a system with vortex-
+antivortex pairs. Evaluating the above expression at
+t=ˆt=τνzeq/(1+νzeq)
+Qwe recover KZ’s result, eq (1).
+Ours, however, is more general since it applies to any
+timetand it allows one to describe all the slow annealing
+evolution. For example, N(t)∝τ−d/zd
+Qon times of the
+order of the inverse annealing rate, t≃τQ, thus show-
+ing that a substantial decrease takes place after ˆt. In
+comparison with the KZ mechanism, our arguments al-
+low one to understand why R(ˆt)≃fR(−ˆt) with a factorfthat can be as large as 10 in some cases [15]. This was
+somewhat mysterious in the KZ scenario where defects
+are frozen out in the impulse regime. We understand
+the reduction in the number of topological defects as due
+to critical coarsening. Taking this phenomenon into ac-
+count is crucial for more general annealing protocols, e.g.
+T(t) =θ(−t)Tc(1−t/τ(1)
+Q)+θ(t)Tc(1−t/τ(2)
+Q). Foralarge
+ratioτ(2)
+Q/τ(1)
+Qthe system spends a long time in the criti-
+cal region and R(t) evolves during the critical coarsening
+from [τ(1)
+Q]ν/(1+νzeq)to [τ(2)
+Q]ν/(1+νzeq).
+In the following we provide numerical evidence for
+the conclusions outlined above by presenting results of
+a Monte Carlo simulation (using the heat bath algorithm
+with random sequential updates) of the 2 dIsing model
+(IM) on square and triangular lattices. In particular,
+we check eqs. (2)-(4) and the universality of the scaling
+function f(x). We equilibrate the system at T0= 2Tc
+and we use a linear cooling rate that takes the temper-
+ature of the bath from T0att=−τQtoT= 0 at
+t=τQ. We find that the system undergoes an adia-
+batic evolution until it falls out of equilibrium close to
+Tc. The typical correlation length, R(t), is extracted
+from the analysis of the space-time correlation C(r,t)≡
+∝angbracketlefts(/vector x)s(/vector x+/vector r)∝angbracketright ≃g(r/R(t)) with the average taken over
+100 initial conditions and noise realizations. We use vari-
+ouswaysto determine Rand verifythat they yield equiv-
+alent results. Two of them are C(R(t),t) = 1/2 and
+R(t) =/integraltext
+d2r rζC(r,t)//integraltext
+d2r rζ−1C(r,t) withζa pa-
+rameter that is chosen for convenience, namely to weigth
+differently shorter or longer distances. In the top panel
+of Fig. 2 we test the scaling hypothesis, eq. (2), and the
+limits of the scaling function f(x), eq. (3). For both the
+square and triangular lattices we find very good agree-
+ment between numerical data and theoretical expecta-
+tion. The square root growth at positive times demon-
+strates that standard coarsening cannot be ignored. The
+scaling collapse improves, as expected, restricting the
+range of |x|. A zoom on the small |x|region is shown
+in the inset. Moreover, we find that the scaling functions
+for square and triangular lattices coincide within numer-
+ical accuracy, confirming the universality of f(x). The
+bottom panel of Fig 2 displays N(τQ) and confirms the
+τ−d/zd
+Qdecay - the power is shown as a guide-to-the-eye
+next to the data.
+A further test is provided by the analytic solution to
+theevolutionofthe O(N) model inthelarge Nlimit fora
+very slow annealing. This is a λφ4field theory in which
+the order parameter is upgraded to an N-dimensional
+vector and the fourth order term in the double-well po-
+tential is conveniently normalizedto allowfor an N→ ∞
+limit in which the model becomes solvable but still non-
+trivial [1]. Note that although there are no topological
+defects, since the large Nlimit is taken at fixed dimen-
+sion,N≫d, the dynamics are still characterized by a
+growing correlation length R(t). The analysis of a finite4
+21028tr 2621028sq 26x1/2
+t|∆g|νzeqR|∆g|ν
+10 5 0 -5 -106
+5
+4
+3
+2
+1
+0t|∆g|νzeqR|∆g|ν
+0.5 0 -0.52
+1
+0
+τQ−1Simulations
+τQN(t=τQ)
+1000 1001000
+100
+10
+FIG. 2: (Colour online.) Top panel: Test of the dynamic
+scaling hypothesis (2) and the limits (3) in the 2 dIM on a
+triangular and a square lattice, annealed at different cooli ng
+rates given in the key. A zoom on the critical region |x|<∼1
+is shown in the inset. The exponents are ν= 1,zeq= 2.17
+andzd= 2. Bottom panel: Number of deffects at t=tQfor
+different cooling rates. Points represent the numerical dat a
+while the line corresponds to the prediction N(τQ) =τQ−d/zd
+rate quench is a simple generalization of the treatment
+of infinite rate ones (see, e.g.[1]). We find that the scal-
+ing (3) holds with ν= 1/2 andzd=zeq= 2 in all d >2.
+Due to the coincidence of the zexponents the prefactor
+in the bottom expression of eq. (3) equals one and the
+dependence on τQdisappears.
+As a summary we analyzed the dynamic evolution in-
+duced by annealing with rate 1 /τQ(τQ→ ∞) in pure
+systems characterized by conventional dynamic scaling
+and standard low temperature coarsening. We obtained
+a complete picture of the dynamics which is character-
+ized by three regimes: adiabatic, critical coarsening and
+standard coarsening. Using scaling arguments we found
+the growth law of the correlation length during the an-
+nealing and its τQscaling dependence. The cross-over
+between adiabatic and coarsening regimes is governed by
+a universal scaling function. We tested our findings with
+numerical simulations of the 2 dIsing model and a large
+Nanalysis of the O(N) model in d >2. Our results gen-eralizetheKZmechanismand, atthesametime, showits
+limitation. In particular we find that the defect dynam-
+ics are not frozen in the so-called impulse regime, as it
+can be found by using more general annealing protocols
+than a linear ramp in temperature.
+Physical situations in which understanding the evolu-
+tion during a slow annealing is important and which we
+plan to study in the future are disordered and quantum
+systems. Several studies have dealt with the former, see
+e.g. [4–6]. The latter have only recently received atten-
+tion in connection with quantum quenches and annealing
+in cold atoms. In these cases the conditions are differ-
+ent from the ones analyzed in this paper since isolated
+systems in which a coupling is slowly changed through a
+quantum critical point are usually considered. The ab-
+sence of the thermal bath may change drastically the
+physics. The KZ mechanism has been argued to ap-
+plymutatis mutandis to the isolated quantum case as
+well [11, 12]. This has been verified in some integrable
+cases [13].
+We close with a note on an exact study of the cool-
+ing rate effects in the relaxation of the classical Ising
+chainwith GlauberdynamicsbyP.Krapivskythatshows
+qualitative but not quantitative agreement with the KZ
+mechanism [30].
+This work was financially supported by ANR-BLAN-
+0346 (FAMOUS). We thank C. Kollath, C. Godreche, A.
+Gambassi, D. A. Huse, A. Jelic, P. Krapivsky, J. Kur-
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\ No newline at end of file
diff --git a/1001.0694.txt b/1001.0694.txt
new file mode 100644
index 0000000000000000000000000000000000000000..fa8aeff0b30f8216ecc796e251d6e2db28118c76
--- /dev/null
+++ b/1001.0694.txt
@@ -0,0 +1,1192 @@
+arXiv:1001.0694v2  [quant-ph]  7 Jan 2010JOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 1
+Photon Counting OTDR :
+Advantages and Limitations
+Patrick Eraerds, Matthieu Legr´ e, Jun Zhang, Hugo Zbinden, Nicolas Gisin
+c∝circleco√yrt20xx IEEE. Personal use of this material is permitted.
+However, permission to reprint/republish this material fo r
+advertising or promotional purposes or for creating new
+collective works for resale or redistribution to servers or lists,
+or to reuse any copyrighted component of this work in other
+works must be obtained from the IEEE.
+Abstract—We give detailed insightinto photoncountingOTDR
+(ν−OTDR) operation, ranging from Geiger mode operation of
+avalanche photodiodes (APD), analysis of different APD bia s
+schemes, to the discussion of OTDR perspectives. Our result s
+demonstrate that an InGaAs/InP APD based ν−OTDR has the
+potential of outperforming the dynamic range of a conventio nal
+state-of-the-art OTDR by 10 dB as well as the 2-point resolut ion
+by a factor of 20. Considering the trace acquisition speed of
+ν−OTDRs, we find that a combination of rapid gating for high
+photon flux and free running mode for low photon flux is the
+mostefficientsolution.Concerningdeadzones,ourresults areless
+promising. Without additional measures, e.g. an optical sh utter,
+the photon counting approach is not competitive.
+Index Terms —Distributed detection, fiber metrology, optical
+time-domain reflectometry, photon counting
+I. INTRODUCTION
+OPTICAL Time Domain Reflectometry [1] is a well
+known technique for fiber link characterization. Most
+of today’s commercially available optical time domain refle c-
+tometers (OTDRs) are based on linear photon detectors, such
+as p-i-n or avalanche photodiodes (APDs). Although single
+photondetectionfeaturesunmatchedsensitivity,OTDRs ba sed
+onthistechnique( ν−OTDR)[2] havereachedthe marketonly
+in niches [3].
+Several single photon detection techniques are possible [4 ]-
+[9], but only few of them are suitable for in-field measure-
+ments. Geiger-mode operated InGaAs/InP APDs (for telecom
+wavelengths) [5] [10] [11] are the most promising candidate s,
+due to their robustness and manageable cooling.
+Inthispaperwediscusstheadvantagesandlimitationsofth ese
+devices, when used in an ν−OTDR. We concentrate in par-
+ticular on the dynamic range, 2-point resolution, measurem ent
+time and dead zone. All ν−OTDR measurements are supple-
+mented by measurements using a conventional state-of-the-
+art OTDR (Exfo, FTB7600). This makes it easier to evaluate
+P. Eraerds, J. Zhang, H. Zbinden, N. Gisin are with the Group o f Ap-
+plied Physics, University of Geneva, 1211 Geneva 4 Switzerl and, e-mail:
+Patrick.Eraerds@unige.ch
+M. Legr´ e is with idQuantique SA, 1227 Carouge/Geneva, Swit zerland
+Manuscript received April x, 200y; revised January x, 200y. Financial
+supports from the Swiss Federal Department for Education an d Science
+(OFES), in the framework of the European COST299 project, fr om the Swiss
+NCCR ”Quantum photonics” are acknowledged.theν−OTDR performance. Our discussion also contains the
+possible yield of newly emerged gating techniques like rapid
+gating[12] [13] [14].
+We notethatsomeyearsago ν-OTDRsbasedonsiliconAPDs,
+suitable for C-band operation, were demonstrated [15] [16] .
+Although silicon APDs show superior behavior, concerning
+afterpulsing and timing jitter, the upconversion of teleco m
+photons to the visible regime demands more expensive optics
+and more sophisticated alignment. Therefore we believe tha t
+they are less suitable when robustness is required.
+Paper organization : In Sect.II we provide information abou t
+Geiger-mode operation of InGaAs/InP APDs and discuss its
+major impairment, the afterpulsing effect. Sect.III focus ses on
+ν−OTDR operation and performance (dynamic range, 2-point
+resolution,measurementtime,deadzone)andcomparesitwi th
+the performance of a conventional state-of-the-art long ha ul
+OTDR (Exfo FTB7600). Sect.IV considers time efficient bias
+schemes( rapidgating,free running )andfinallywesummarize
+our results in Sect.V.
+II. GEIGER-MODEAPD
+A. Basic operation
+In Geiger-mode, the APD is biased beyond its breakdown
+voltage,typicallybya few percent.Thisprovidesa sufficie ntly
+large gain (order of 106) to detect a single incident photon
+(with detection efficiency η). In contrast to a linear mode
+APD, the output signal is no longer proportional to the
+number of primary charges. Whenever an avalanche occurs
+and the current reaches a certain discrimination level, a
+detection is counted, independent of how many primary
+charges caused or were created during the avalanche.
+To reset the APD for the next detection, the avalanche needs
+to be quenched. This is typically done by lowering the bias
+voltage, either actively or passively [17].
+An APD based on InGaAs/InP is particularly well suited for
+use with the principal telecom wavelength bands. Although
+the dark count1rate is higher than in silicon based APDs,
+high sensitivity can be regained by cooling, typically arou nd
+−50◦C (see also Sect.II-B).
+Thereare differentwaysof applyingthe overbias( Vbias> Vbd
+(breakdown voltage)) to the diode. The most common ones
+are thegated mode and thefree running mode [18]. Ingated
+modethe overbias is applied only during a short time ∆tgate
+(called gate), in a repetitive manner with frequency fgate
+(respecting fgate<1
+∆tgate). Typically ∆tgate∈[2ns,20µs]
+1A detection which was not initiated by a signal photon but the rmal
+excitation or tunneling.JOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 2
+andfgate∈[100Hz,10MHz]. Infree running mode , the
+overbias is applied until a photon or noise initiates an
+avalanche.
+While the gated mode achieves high signal to noise ratios
+when a synchronizedsignal is being detected, the free running
+modeis most suited when the photon arrival time is not
+known (e.g. in OTDR).
+More recent developments, summarized by the name rapid
+gating[12] [13] [14], apply very short gates ( ≈200 ps)
+in order to severely limit avalanche evolution and reduce
+afterpulsing (see Sect.II-C). The technical challenge con sists
+in discriminating the rather small avalanche signal from
+the capacitive response to overbias of the diode itself. In
+”classical gating”, described in the previous paragraph, o ne
+usually waits until the avalanche signal is easy to discrimi nate.
+Typical gating frequencies in rapid gating are of the order of
+1 GHz.
+In Sect.IV we will discuss pros and cons of these different
+approaches, in particular concerning their applicability for
+ν−OTDRs.
+B. Detection sensitivity
+A figure of merit for the sensitivity of a detector is its
+noise equivalent power ( NEP). For example, the bandwidth
+normalized NEPnormof a linear photo detector is given by
+[21] [22]
+NEPnorm=∆Inoise
+S·G[W√
+Hz] (1)
+where∆Inoise[A/√
+Hz] is the standard deviation of the total
+noise current (thermal-, dark-, signal shot- and in case of g ain
+also gain noise), normalized with respect to the bandwidth o f
+the detector, S[A/W] is the detector photosensitivity and
+Gis the gain of the diode (p-i-n diode : G= 1, linear APD
+(typically) : G= 10−100).
+APDs are superior to p-i-n diodes in the circuit noise limite d
+regime2[23], but lose their advantage when the gain noise
+becomes important, i.e. at stronger signal powers. The
+minimal detectable power NEPnorm,0[W/√
+Hz] is obtained
+by setting the signal power and thus the signal shot-noise
+equal to zero. NEPnorm,0can usually be found in the data
+sheet of the diode, typically 10−15−10−13[W/√
+Hz] for
+InGaAs APDs at 25◦C.
+A similar expression can be derived for Geiger-mode APDs
+(see App.B, Eq.20):
+NEPnorm=hν
+η·/radicalbig
+2·ˆpnoise [W√
+Hz](2)
+whereηis the detection efficiency and ˆpnoiseis the noise
+detection probability per gate (includingsignal and dark c ount
+shot noise), normalized with respect to the gate width ∆tgate
+2Circuit noise results for example from thermal motion of cha rges in
+resistors or charge fluctuation in transistors in the receiv er amplifier.-50 -40 -30 -20 -10 0 10 201x10-1610-15Hz]/c214
+AAAA
+NEPnorm,0[W/
+T [°C]
+Fig. 1. Bandwidth normalized noise equivalent power ( NEPnorm,0, see
+Eq.3)asfunction ofGeiger-mode APDtemperature. Thedetec tion efficiency η
+is kept constant at 10%. At ambient temperatures the noise eq uivalent power
+is increased by almost a factor of 10 with respect to the usual operating
+temperature of −50◦C.
+in seconds. Again, setting the input optical power equal to
+zero, we infer the minimal detectable power (App.B, Eq.21)
+NEPnorm,0=hν
+η·/radicalbig
+2·ˆpdc[W√
+Hz](3)
+whereˆpdcdenotes the dark count probability per gate, nor-
+malized with respect to the gate width ∆tgatein seconds.
+Inserting the parameters of the Geiger mode APD used in
+our experiments ( ˆpdc= 2000s−1,η= 10%,T=−50◦C,
+Sect.III-A), we estimate NEPnorm,0≈10−16[W/√
+Hz].
+In Fig.1 we see the evolution of NEPnorm,0as function
+of temperature. We observe that when approaching ambient
+temperatures, we almost reach the regime of the best linear
+mode diodes. Conversely, one might be tempted to cool linear
+diodes to −50◦C to reach the NEPof Geiger mode APDs.
+Even if this might be in general achievable, one should not
+forget, that the output signal still needs to be amplified. Ev en
+atambienttemperaturesthesmallpulseamplifiernoiseusua lly
+constitutesthedominatingnoisesourceleadingtomuchhig her
+effective NEPs.
+By analyzing the performance of a conventional OTDR in
+Sect.III, we will gain more insight into the sensitivity lim its
+of linear mode APD detection systems.
+C. Afterpulsing
+One of the major impairments of InGaAs/InP APDs is
+the afterpulsing effect. Imperfections and impurities in t he
+semiconductormaterialareresponsibleforintermediatee nergy
+levels (also called trap levels), located between the valen ce
+band and the conduction band. During an avalanche, these
+levels get overpopulated with respect to the thermal equi-
+librium population. If the APD gets reactivated right after
+the quenching of an avalanche, the probability of thermal
+excitation or tunneling of one of these charges into the
+conduction band and the subsequent initiation of an afterpu lseJOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 3
+0 10 20 30 40 501E-41E-30.010.11
+ ∆tgate=10 ns
+τpAP,gate 
+dead time     [ µs]
+Fig. 2. Afterpulse probability as function of dead time τ. The detection
+efficiency ηis equal to 10% and the temperature T= −50◦C. An active
+quenching application specific integrated circuit (ASIC) [ 19] was used.
+avalanche, is high. Although fundamentally the improvemen t
+of semiconductor purity and thus the reduction of the number
+of trap levels is preferable, different mitigation measure s can
+be carried out :
+a) dead time : A purely passive measure is the introduction
+of a dead time. The trap population decreases exponentially
+with time, due to thermal diffusion. Finally the thermal equ i-
+librium configuration is restored. The impact of afterpulsi ng
+can therefore be mitigated by maintaining the bias voltage
+below breakdown, i.e. the application of a dead time τ, after a
+detection takes place. Dead times severely limit the maximu m
+achievable detection rate.
+b) heating : An increasedtemperatureacceleratesthe diffusion
+of trapped charges. However, at the same time charge excita-
+tion fromthe valenceinto the conductionbandincreases, le ad-
+ing to globally increased noise, which eventually reduces t he
+detector sensitivity. Thus one cannot achieve low afterpul sing
+and high sensitivity at the same time. It is necessary to find a
+trade-off depending on the particular application.
+c) quenching technique : As soon as the avalanche has gained
+enough strength such that the current pulse can be detected, it
+needs to be quenched.The quenchingspeed is crucial to limit -
+ing the number of secondary charges which can populate trap
+levels. Here, fully integrated active quenching circuits y ield
+much better results than non-integrated ones [19]. Another
+approach is rapid gating (Sect.II-A). Avalanche evolution is
+terminated by short gate durations (200 ps) and the number
+of secondary charges is kept low.
+In Fig.2 we plot an example of afterpulse probability as
+function of dead time τ, using a fully integrated ASIC based
+active quenching circuit [19]. Whenever a detection takes
+place, we activate a second gate of width ∆tgate= 10ns
+with a temporal delay of τ. In this second gate, no incident
+photons are present. If there is a detection it is either a dar k
+count or an afterpulse. Since for large τonly the actual
+dark counts remain, we can subtract it from the total countrate and obtain the pure afterpulsing probability ( →Fig.2).
+During larger gates, the afterpulse probability sums up and
+afterpulsing increases. One can easily calculate the after pulse
+probability of a gate of width ∆tgate(≤10µs) by
+pAP,∆tgate(τ) = 1−(1−pAP,10ns(τ))m(4)
+wherepAP,10ns(τ)is the afterpulse probability in a gate of 10
+ns width and m=∆tgate
+10ns.
+We note that if no afterpulse occurs in the first activated
+gate after a detection, it can also happen in any succeeding
+gate. However, the probability decreases due to trap charge
+diffusion. To get the total afterpulse probability, or rath er
+the signal to afterpulse ratio, one needs to account for this
+summingeffectaswell.Thelowerthesignaldetectionrate, the
+more summing-up takes place. Thus a higher signal detection
+rate improves the signal to afterpulse ratio.
+In Fig.3 we illustrate the impact afterpulsing can have in a
+ν−OTDRmeasurement.Firstly and most importantlywe must
+consider dead zones (see Sect.III-D for definition, not to be
+confused with dead time). Whenever an important loss (at 25
+km) or a reflection (at 36 km) occurs, it is followed by a
+tail which prevents the detection of the Rayleigh backscatt er
+directly behind it. Secondly, the backscatter trace is shif ted to
+higher values, since more detections than in the pure signal
+case occur (pile-up effect). Thirdly, the slope of the trace is
+flatter than it should be.
+05101520253035404550−30−25−20−15−10−50
+distance [km]dB
+  
+ afterpulsing
+ no afterpulsingreflection
+Fig. 3. Illustration of the afterpulsing effect on ν−OTDR trace. Most severe
+are the dead zones after large loss events (at 25 km) and reflec tions (at 35
+km). More subtle is the change of the slope of the trace which i s usually
+smaller than what is measured when afterpulsing can be negle cted.
+How much afterpulsing can be tolerated, generally depends
+on the particular measurement. For instance in a coarse
+measurement on a long span of fiber, where only peak
+positions or large loss events are of interest, one can toler ate
+a fairly high afterpulsing contribution. On the other hand,
+in the case of short links, where high precision for fiber
+attenuation measurement and dead zone minimization is
+desired, afterpulsing must be kept to a few percent or even
+lower, depending on required precision of the measurement.
+Numerical afterpulse correction methods were also analyze d
+[20], but it was found that for high precision measurements
+the algorithm lacks robustness due to possible variations i nJOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 4
+the afterpulse probability. It should therefore be used onl y
+when the requirements on precision are not stringent.
+III. PHOTON COUNTING VS .CONVENTIONAL OTDR
+Althoughthissectionismainlyconcernedwiththe ν-OTDR
+technique, we also perform measurements using a state-of-
+the-art conventional3OTDR (FTB-7600, EXFO), a product
+especially designed for long-haul applications (up to 50 dB
+dynamic range). This makes it easier for us to highlight the
+advantages and drawbacks of the photon counting approach.
+The experimental setup and a detailed explanation is given i n
+Fig.4.
+Fig. 4. Basic ν-OTDR setup. The laser (we use the laser of the FTB-7600,
+Ppeak= 400mW) emits pulses with a frequency fpulseadapted to the
+length of the fiber under test Lfiber(→fpulse=c
+2·Lfiber). The signal
+is split at a 99/1-coupler. The 99% part is launched into the fi ber under test
+(FUT) via a circulator. Backscattered light from the fiber ex its the lower port
+of the circulator and illuminates the InGaAs/InP-APD. The 1 % part is used
+to measure the time of departure t0of the laser pulse (for synchronization
+reasons) using a conventional photodiode (Newport, 1GHz). The output signal
+is sent to a delay generator. A delayed signal at t0+tdelayis sent to the APD
+to apply a detection gate of length ∆tgateand the backscattered intensity
+corresponding to the applied delay is measured. TheAPD reve rse bias is equal
+48.7 V, yielding a detection efficiency η= 10% at−57◦C (minimal value).
+We measure a dark count probability per gate equal to ˆpdc= 2000 s−1
+(normalized with respect to the gate width, Eq.3). This corr esponds to a dark
+count probability of 2·10−5for a gate width of 10 ns and 2·10−2for a
+10µs gate.
+For a fixed delay, a number of laser pulses Npulse(repeti-
+tion frequency fpulse) are sent and Ngate(=Npulse) gates
+are activated. A counter records the number of detections.
+The incident signal power can be inferred from the ratio of
+detections to activated gates (for details, see App.A).
+To get information on the backscatter of the entire fiber, the
+delay needs to be scanned, repeating the procedure explaine d
+beforefor each single delay position.The samplingresolut ion,
+i.e. the delay step ∆tdelay(tdelay=i·∆tdelay,i= 1,2,3...),
+needs to be adapted to the requirements of the particular
+measurement (e.g. zoomingor coarse full trace measurement ).
+The detection bandwidth is given by B=1
+2∆tgate.
+We note that this is only the most basic version of a photon
+counting OTDR. One of the advantages of this system is
+that due to the low gating frequency, we can totally exclude
+3Based on linear-mode APD detection.afterpulse effects (dead time 2 ms). Therefore we can deter-
+mine the unadulterated dynamic range and 2-point resolutio n.
+Nevertheless, data acquisition is very time consuming. For
+examplethemeasurementofthe entire200kmfiber,discussed
+in the next section (Fig.5), took about 6 hours. In Sect.IV we
+will see, how it can be performed more efficiently.
+A. Dynamic range
+To measure the dynamic range of both devices for different
+laser pulse widths, we take a 200 km fiber, composed of a
+50 km spool and an installed fiber link of 150 km (Swisscom,
+Geneva-Neuchatel),which itself consists of several fibers . The
+length of the fiber allows a maximal laser pulse repetition ra te
+offlaser=c
+2·Lfiber= 500Hz, where cis the speed of light
+in standard optical fiber.
+We start measuring the trace with the FTB-7600 for 3 min-
+utes4with a laser pulse width of 1 µs. The device acquires
+180s·500Hz= 9·104differenttraces.Thefinaloutputtraceis
+the numerical average of these single traces (Fig.5, light g rey
+curve). For a fair comparison the detection bandwidth shoul d
+be equal for the two devices. For the conventional OTDR it
+is automatically chosen by the device and not available to us .
+We infer its value by looking at the noise period at the end of
+the measurement range. For a pulse width of 1µswe obtain 4
+MHz. Under these conditions the dynamic range is found to
+be 34.5 dB.
+We then perform the ν-OTDR measurement, ensuring that
+we do not saturate the detector with the backscatter from
+the beginning of the fiber. Therefore we insert an additional
+attenuatorin frontof the APD to reach the unsaturatedregim e.
+We adjust the attenuation to yield a detection rate of about
+90% of the gate rate for the first delay position. At each delay
+position we count the number of detections within 3 minutes,
+which yields the same statistics per sampling point as in the
+previous case ( 180s·500Hz= 9·104samplings). We choose
+∆tdelayequal to 3 µs (=300 m sampling point separation
+in fiber) and ∆tgate= ∆tpulse. With increasing delay the
+backscatter power and thus the detection rate decreases. Wh en
+we start to approach the noise level of the detector, we
+pause the measurement and remove a part of the attenuation
+(to regain 90% detection rate), reduce the delay for a few
+kilometers (to get an overlap with the previous part) and
+resume the measurement. In this way we obtain several single
+traces of adjacent parts of the fiber. In the following we
+will refer to this as partial trace measurement. By means
+of the overlaps, the entire trace can be reconstructed. Each
+partial trace measurement contributes approximately 20 dB to
+the overall ν-OTDR dynamic range. For example, to cover
+50 dB of fiber loss, we need to perform three partial trace
+measurements5.
+The result of the ν-OTDR measurements is also shown in
+Fig.5 (blue curve). It is important to note that we adapt the
+4We choose 3 min because this is the time specified in the definit ion of
+OTDR dynamic range for conventional OTDRs [24].
+5The first measurement covers 0-20 dB, the second 15-35 dB and t he third
+30-50 dB respecting the necessary overlap between differen t partial trace
+measurements.JOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 5
+0 20 40 60 80 100 120 140 160 180 200-50-40-30-20-10010
+-46 dB-34.5 dBdB
+distance [km] conventional OTDR
+ photon counting OTDR
+Fig. 5. OTDR traces of 200 km fiber link using a laser pulse widt h of
+1µs. The light gray curve represents the result of the conventio nal OTDR
+(Exfo FTB-7600) after 3 minutes of measurement in standard c onfiguration.
+In this configuration it uses a detection band width of 4 MHz at the end of the
+measurement range. The ν−OTDR result is represented by the blue curve.
+The measurement bandwidth is 500 kHz using the same number of samplings
+for each point as the conventional device. The bandwidth ada pted results for
+different pulse widths can be found in Fig.6.
+gate width to the laser pulse width to obtain the minimal
+NEP0(App.B, Eq.19) withoutaffectingthe 2-pointresolution
+(limited by the laser pulse width). This means that in the cas e
+of Fig.5, the detection bandwidth of the ν−OTDR is B=
+1
+2·∆tgate= 500kHz6. To be able to compare the measured
+resultsina representativemanner,weaveragethe conventi onal
+OTDR trace in order to obtain the same bandwidth as was
+used in the ν−OTDR measurement. We gain 2.5 dB, yielding
+a corrected dynamic range of 37 dB. The ν−OTDR advantage
+is found to be roughly 9 dB in this case.
+We repeat the measurement for different pulse widths keepin g
+all other parameters unchanged. The final results are shown
+in Fig.6. The detection bandwidths were adapted as before. I t
+holds that B=1
+2∆tpulsefor both devices.
+For pulse widths between 30 ns and 1 µs the dynamic range
+difference is about 9-10 dB. This is a direct consequence of
+the smaller NEPnorm,0(see Sect.II-B) of the Geiger-mode
+APD, since bandwidthand integrationtime per samplingpoin t
+were equally chosen. This means that the NEPnorm,0of
+theν−OTDR is roughly a factor 63-100 smaller7than the
+NEPnorm,0of the conventional OTDR (noise dominated by
+the small pulse amplifier). However, going to larger laser
+pulses, we observe increased noise for the ν-OTDR and the
+advantage gets smaller. We suppose that this happens due to
+the increased backscatter power from the beginning of the
+fiber. Although the diode is not active, the charge persisten ce
+effect (also sometimes called charge subsistence) can have a
+non negligible impact on the noise counts in a subsequently
+activated gate (for more details see also Sect.III-D).
+In summary, by adapting sampling statistics and detection
+6The conventional device uses a higher bandwidth since highe r sampling
+resolution is useful when the position of an event needs to be determined with
+higher precision.
+7Respecting the functional dependence between NEP and dynam ic range
+given in Eq.25, using NEP0∝NEPnorm,010n 100n 1µ 10µ2025303540455055photon counting OTDR
+conventional OTDRdynamic Range [dB]
+/c68tpulse[s]
+Fig. 6. Dynamic ranges of FTB-7600 and ν−OTDR for different laser pulse
+widths. The length of the used fiber was 200 km. The detection b andwidths
+Bwere adapted in each case, it holds that B=1
+2∆tpulse.
+bandwidth of both devices we find an advantage of about 9-
+10 dB in dynamic range for the ν-OTDR. By increasing the
+laser pulse width we observe increased detector noise and th e
+effective advantage gets smaller. We uncouple the question of
+measurement time since it is highly dependent on the gating
+technique used in the ν−OTDR. This discussion is postponed
+to section IV.
+B. 2-point resolution
+When considering the 2-point resolution8, one can divide
+OTDR operation into two regimes a) the receiver limited and
+b) the laser peak power limited regime. In case a) the ultimat e
+timing resolution is either given by the amplifier bandwidth or
+the detector jitter (using fine laser pulses), whereas in cas e b)
+the limited laser peak power makes it necessary to use larger
+pulse widths (larger than the limit given in case a)) in order
+to reach high dynamic ranges. In the receiver limited regime ,
+the advantages of photon counting were already discussed in
+[20], yielding a maximal 2-point resolution of 10 cm for the
+ν-OTDR and 1 m for the conventional device. In long haul
+OTDRapplications,weoperatein thelaserpeakpowerlimite d
+regime.
+It is easy to see that in this regime the sensitivity advantag e
+(NEPnorm,0) of photon counting translates directly into an
+advantage in 2-point resolution.
+Example : we consider a reflective event at the end of the
+dynamic range (with a certain pulse width and measurement
+time) of the FTB-7600, see Fig.7.
+Theν−OTDR can achieve the same dynamic range with a
+much smaller pulse width, see Fig.6. According to App.D,
+an advantage of 10 dB in dynamic range9corresponds to an
+advantage in 2-point resolution by a factor of 20. In Fig.8 we
+see the results of three ν−OTDR measurements focusing on
+8By 2-point resolution we mean the minimal distance, necessa ry between
+two reflective events, in order to be able to recognize them as distinct peaks
+on the OTDR trace output (e.g. dip between peaks at least 1 dB l ower than
+the peaks themselves)
+9Bandwidth normalized.JOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 6
+0 20 40 60 80 100 120 140-40-30-20-10010
+102.70 102.75 102.80 102.85 102.90-30-28-26-24dB
+distance [km]dB
+distance [km]
+Fig. 7. OTDR trace measured using the FTB-7600 with a laser pu lse width
+of 300 ns, acquisition time = 3min. The achieved dynamic rang e is about
+30 dB. At the transition to the noise level (at about 102.8 km) , we can see
+a reflection peak (maybe two, see inset). The FTB-7600 cannot achieve a
+better resolution of the peak without decreasing the laser p ulse width. As the
+dynamic range would drop (keeping the acquisition time unch anged) and the
+peak would not be seen anymore.
+102.55 102.60 102.65 102.70 102.75 102.80-10-9-8-7-6-5-4-3-2
+∆102.70 102.75 102.80 102.85 102.90-9-8-7-6-5-4-3∆tpulse=100 nsdB
+distance [km]102.76 102.78 102.80 102.82 102.84-11-10-9-8-7-6-5∆tpulse=30 nsdB
+distance [km]
+102.800 102.805 102.810 102.815 102.820-15-14-13-12-11∆tpulse=10 nsdB
+distance [km]
+Fig.8. Step bystep reduction oflaser pulse width in ν−OTDRmeasurement,
+when zooming onthereflective event seenbytheFTB-7600 (Fig .7).Thelower
+graph is again a zoom on the two reflective events which were re vealed by
+the second graph (peak 2 and 3). Due to its larger sensitivity theν−OTDR
+can afford smaller laser pulses at distances where the FTB-7 600 reaches its
+limits.
+the reflective event at about 102.8 km. With each reduction
+in pulse width more and more structure is revealed and we
+actually find 3 reflections. The ratio of the peak widths agree s
+well with the calculated one.
+In summary, when the OTDRs are operated in the laser
+peak power limited regime, the sensitivity advantage of the
+ν−OTDR translates directly into an advantage in 2-point
+resolution. Its amount is described by Eq.27 in App.D.
+C. Measurement time
+We start discussing the measurement time by taking a look
+at the time necessary to obtain a sufficient signal to noise ra tio(SNR) for a specific delay position in the fiber, from which
+we receive a backscatter power Popt(see App.E):
+t=1
+fpulse·/parenleftBigg
+SNR·NEPnorm·√
+B
+Popt/parenrightBigg2
+(5)
+whereNEPnorm(see Eq.20) is the noise equivalent power
+normalized with respect to detector bandwidth in [W/√
+Hz],
+Bthe detector bandwidth in [Hz] and fpulsethe laser pulse
+repetition rate in [Hz]. This formulaapplies to both the Gei ger
+and linear mode operation as long as linearity between input
+and output signal is guaranteed10.
+While the Geiger mode exhibits linearity only in a relativel y
+restricted domain of Popt, the linear mode is able to cover
+several orders of magnitude of input power. Therefore Eq.5
+is applicable for a much wider range of optical powers and
+showsthe significantadvantageofthe linearmodewhenlarge r
+powers need to be measured ( ∝P−2
+opt).
+More interesting from the ν−OTDR perspective is the case
+whenEq.5appliesaswell forGeiger-mode,i.e.forsufficien tly
+small powers (order -100 dBm). We can then easily calculate
+the ratio of measurement times (assuming fpulse,BandPopt
+to be equal):
+t(conv)
+t(pc)=/parenleftBigg
+NEP(conv)
+norm
+NEP(pc)
+norm/parenrightBigg2
+(6)
+where the superscript pcsignifies photon counting (i.e. Geiger
+mode), and convrepresents the conventional detection mode
+(i.e. linear mode).
+NEPnormcanbesplitintoasignalinitiatednoisecontribution
+NEPnorm,sig(e.g.duetosignalshotnoise)anda contribution
+from signal independent sources (dark current, dark counts )
+represented here by NEPnorm,0:
+NEP2
+norm=NEP2
+norm,sig+NEP2
+norm,0(7)
+For sufficiently small input power, NEPnormbecomes
+NEPnorm,0. In Sect.III-A we estimated the ratio of the
+NEPnorm,0of conventional and ν−OTDR to lie within 63
+and 100. Thus the ratio given in Eq.6 approaches a value
+between 4000 and 10000. This means that we continuously
+pass from a huge linear mode advantage ( Poptlarge) to a
+huge Geiger mode advantage ( Poptsmall).
+We stress at this point that this result applies for
+the measurement of a specific delay position. OTDR
+measurements consist of scanning a range of different
+powers, i.e. the exponentially decreasing backscatter pow er.
+In case of the conventional OTDR the achievement of a
+sufficient SNR for a certain delay position, e.g. far down the
+fiber, means that all delays closer to the beginning of the
+fiber have been scanned at least with the same SNR. Thus we
+already obtain the full trace up to that point. How large the
+actually scanned interval is in the case of photon counting
+depends on the gating technique. For example, using the basi c
+approach, explained earlier (Fig.4), one would scan exactl y
+10Linearity in Geiger mode applies if the signal detection pro bability per
+gatepsig,gatedepends linearly on the input power Popt, see App.A Eq.9 &
+10. For sufficiently small Poptit holds that : psig,gate=η·Popt·∆tgate
+hνJOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 7
+one position. In Sect.IV we discuss how the NEP advantage
+of photon counting can be used more efficiently.
+D. Dead zone
+Dead zones are parts of a fiber link where the OTDR trace
+does not display the actual Rayleigh backscatter, but a sign al
+induced by another source. One example that we have already
+encountered is the afterpulsing effect. If not accounted fo r it
+leads to tails after large loss events or reflections (see Fig .3).
+Unfortunately this is not the only origin of dead zones. If
+we mitigate the afterpulsing effect by using an appropriate
+dead time, another effect, very similar to afterpulsing get s
+dominant : charge persistence. Even though the APD is not
+biased beyondbreakdownbetween adjacent gates, there is st ill
+a bias which can weakly multiply primary charges created
+by photons incident during that time. These weak avalanches
+might howeverlead to increased trap populationand increas ed
+noise avalanche probability in the next gate ( Vbias> Vbdown).
+This effect, although less severe than afterpulsing, becom es
+visible under the same circumstances, namely after large lo ss
+events and reflections.
+To estimate the impact of this effect on the OTDR output,
+we perform a measurement on a fiber link containing a large
+loss event (17 dB), with a reflection (-45 dB) just before it.
+This link simulates a typical situation encountered in pass ive
+opticalnetworks,whereasplitterofhighmultiplicityind ucesa
+significant loss. Weak reflections right in front can be induc ed
+by bad connectors.
+We perform a measurement of this particular fiber link situa-
+tion, using our ν−OTDR in basic mode, which ensures that
+no afterpulsing effect is present and the charge persistenc e
+effectbecomesvisible.Ourresults, includingthemeasure ment
+using the conventionalOTDR, are shown in Fig.9. We observe
+2 4 6 8 10 12-16-12-8-4048
+RBS level 2RBS level 1dB
+distance [km] conventional OTDR
+ photon counting OTDR-45dB Refl .
+17 dB loss
+Fig. 9. Behavior of conventional and photon counting OTDR wh en subjected
+to significant change in backscatter power level (here : 17 dB ). We introduced
+a reflection of -45dB in front of the loss, simulating for exam ple a bad
+connector. The black line represents a fit of the backscatter behind 8 km
+and can be used as a reference to assess the magnitude of the de ad zone.the emergence of a tail approximately 10 dB below the loss
+edge, which decays by approximately 3.5 dB/km. The lower
+Rayleigh backscatter level gets visible again after about 2 km
+(20µs).
+The result obtained with the conventional OTDR is much
+better. The emergenceof the tail starts on a significantly lo wer
+level. Full sensitivity is regained after 1 km. The results f ound
+for theν−OTDR, confirm the observations made in [20] and
+[25].Onepossibilitytomitigatedeadzones,inducedbycha rge
+persistence, is the use of an optical shutter as performed in
+[25]. If the initial backscatter is blocked by the shutter an d
+the gate gets activated, as well as the shutter deactivated r ight
+after the loss event, much better results can be obtained.
+In summary, concerning dead zones, the conventional OTDR
+shows superior behavior, when no additional measures are
+takenin thecase of the ν−OTDR, e.g.usingan opticalshutter.
+IV. TIME EFFICIENT BIAS SCHEMES
+The way we implemented the ν-OTDR in Sect.III is one
+of the simplest and trace acquisition is time consuming. It
+is well suited to study general characteristics, but not for
+other applications. Its apparent drawback is the wasting of
+backscattered signal, due to the fact that fgate=fpulse, i.e.
+only one gate per laser pulse is activated.
+A more efficient approach is the train of gates scheme [20].
+Unlike to the basic mode, the gating frequency is higher than
+the laser pulse repetition rate and more than one gate gets
+activated per laser pulse, see Fig.10. In the ideal case we
+could choose a gating frequency fgatein the way that the
+designated sampling resolution is obtained. Unfortunatel y we
+have to account for the afterpulsing effect and thus need to
+apply a dead time τ(whose length depends on the tolerable
+afterpulsing)11. In App.F we discuss the impact of dead time
+on the detection statistics in detail. To follow the general
+discussion here it can be skipped though.
+A measureforthe acquisitionspeedis theachievabledetect ion
+ratefdet. We state a linearized formula for fdet, which
+illustrates the most important relations very well :
+fdet=1
+1
+η·µ·Γ+τ(8)
+whereηis the detection efficiency ,µis the incident photon
+flux [photons/sec], Γ =fgate·∆tgateis the detection duty
+cycle and τis the dead time.
+a) high flux : If the photon flux is large, the detection rate
+is limited by the dead time, i.e. fdet,max=1
+τ. In order
+to increase the detection rate, the dead time needs to be
+decreased. In Sect.II-C we have already started to discuss t he
+possibilitiesofafterpulsemitigation.Thequenchingtec hnique,
+that yields to date the lowest afterpulsing, is rapid gating [12]
+[13] [14]. Dead times of the order of 10 ns can be considered
+realistic. This is approximately a factor 1000 better than t he
+best actively quenched circuits can deliver. To maintain a
+11Like depicted in Fig.10, we can number each gate in the train o f gates,
+1...n. In the ideal case (no dead time) each of these gates gets acti vated for
+each laser pulse sent into the fiber. In reality, when a dead ti me needs to be
+applied, it is not certain that gate i gets activated. It is po ssible that it falls
+into the dead time application range (gate 4-7 in Fig.10).JOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 8
+reasonable duty cycle, gating frequencies of the order of 1
+GHz are used ( ∆tpulse≈200ps).
+b) low flux : If the flux is small, the dead time is insignificant
+Fig. 10. Different bias schemes for backscatter measuremen t : basic, train
+of gates and free running mode. τrepresents the dead time.
+andfdet=η·µ·Γ. Here the most important parameter is
+the duty cycle Γwhich should be preferably high. Increasing
+the duty cycle will finally lead to a situation that is called
+free running mode [18], Fig.10. The overbias is applied until
+a signal photon or a noise effect initiates an avalanche. The
+photon flux µbelow which the free running mode yields
+higher detection rates is roughly1
+η·τ, see also App.F. We
+see that in this case improved afterpulsing and thus smaller
+dead times, would extend the application range to higher
+photon fluxes. To date the best low afterpulsing solution for
+thefree running mode is the earlier discussed integrated active
+quenching approach [19]. Fig.11 illustrates our discussio n.
+At this point we want to demonstrate, that using the free
+running mode in its application regime (low flux), the acqui-
+sition speed of the conventional OTDR can be considerably
+outperformed.
+Example : We want to scan a 10 km interval of a 200 km
+fiber (⇒fpulse= 500Hz). We assume τ= 10µs,η= 0.1
+⇒the maximal flux is µ= 106photons/s, which corresponds
+to a power of Pstart=−99dBm12. At the end of the 10
+km interval the power drops to about Pstop=−103dBm
+(assuming regular fiber behavior, attenuation =0.2 dB/km).
+FromPstopone can infer ˆpsig(Eq.22) and using ˆpdc= 2000
+s−1we obtain NEP(pc)
+norm= 3.6·10−16W/√
+Hz (see Eq.20),
+which enters into Eq.5 for measurement time calculation. Th e
+bandwidth Bof the measurement is managed by appropriate
+averaging of adjacent points after the full data is acquired .
+12It depends on the laser power, pulse width and the fiber link qu ality, to
+which distance this corresponds. Choosing Ppeak= 400mW and assuming
+that this is also the effective power that reaches the fiber (n o internal loss),
+∆tpulse= 100ns and a regular fiber behavior (loss = 0.2 dB/km), then -99
+dBm correspond to backscatter power coming from a distance o f 158 km.Fig. 11. According to Eq.8 we can determine the flux regimes wh ere each
+of the techniques at disposal deliver their best performanc e. In the low flux
+regimeµ <1
+η·τthe detection duty cycle is most important, therefore the
+free running mode is the optimal solution. As soon as the dead time becomes
+the limiting factor for the detection rate, rapid gating can take advantage of
+its significantly lower afterpulsing, resulting in much sma ller required dead
+times.
+Here we suppose a bandwidth Bof 10 MHz (averaging on
+50 ns intervals) and a signal to noise ratio (SNR) of 4. All
+together we compute a measurement time of approximately
+20 s.
+To obtain the time of the conventional OTDR we calculate
+theNEP(conv)
+normand the ratio of Eq.6. In the beginning of
+Sec.III-C, we stated that the NEP(conv)
+norm,0was about 63-100
+times larger than the NEP(pc)
+norm,0= 10−16W/√
+Hz. The
+signal power is low enough to neglect the signal induced
+noise contribution and we obtain simply NEP(conv)
+norm≈63·
+NEP(pc)
+norm,0= 6.3·10−15W/√
+Hz
+⇒t(conv)
+t(pc)=/parenleftbigg6.3·10−15
+3.6·10−16/parenrightbigg2
+≈300
+yielding a total measurement time for the conventional devi ce
+of about 1.5 h, assuming the same detection bandwidth B.
+We stress thatduringthistime, theconventionaldeviceobt ains
+information not only on our 10 km measurement range, but
+also about the entire range before. The ν−OTDR only scans
+the designated 10 km, but finishes doing this in around 20 s.
+V. CONCLUSION
+The huge advantage of Geiger-mode APDs with respect to
+its linear mode counterparts is the small core noise equival ent
+powerNEPnorm,0. Our comparison of a state-of-the-art con-
+ventional OTDR (based on linear mode APD) and a photon
+countingOTDR(basedon Geiger-modeAPD) revealsa differ-
+ence of roughlytwo ordersof magnitude.We demonstratethat
+thisresourcehasthepotentialtoimprovethedynamicrange by
+10dBas well as the 2-pointresolutionby a factorof20 (inthe
+laser peak power limited regime). The important question is ,
+how efficient can it be used in OTDR applications(concerning
+the measurement time)? To sample the backscatter of a fiber
+we have different possibilities at hand, i.e. train of gates
+(with classical gating), free running mode orrapid gating .
+For sufficiently low backscatter power (order of -100 dBm;
+long fibers) we show that the free running mode is capable of
+efficiently using the NEP advantage. For example, measuring
+a 10 km interval far down the fiber, yielded a measurementJOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 9
+time of around 20 s, while the conventional device needs to
+integrate for about 1.5 h to average out the noise sufficientl y.
+At higher backscatter power, i.e. closer to the beginning of
+the fiber, we show that rapid gating can largely profit from its
+reduced afterpulsing, which makes dead times of the order of
+10 ns realistic.
+We see a combination of rapid gating for the beginning of the
+fiber and free running mode for its end part as the currently
+bestν−OTDR solution. Alternatively one can also imagine
+a hybrid of conventional OTDR for high flux and photon
+counting for low backscatter power, including high resolut ion
+scans with fine laser pulses on short distances.
+Concerning dead zones, the conventional OTDR is clearly
+superior. It is more robust to sudden changes in backscatter
+power,while the ν−OTDR suffersfromthe chargepersistence
+effect. This effect can for example be mitigated by using an
+additional optical shutter.
+APPENDIX A
+POWER MEASUREMENT WITH GATED APD
+We consider coherent light, with a mean photon number
+per second µ, incident on the diode (detection efficiency η).
+We apply gates of duration ∆tgate, which means that on
+averageµ·∆tgatephotons hit the diode during a gate. Due
+to the limited detection efficiency not every photon leads to a
+detection. If our detector would be photon number resolving ,
+the average number of signal detections per gate would be
+givenbyη·µ·∆tgate. SinceourAPDdoesnothavethisability,
+all cases where more than one signal detection would occur,
+results in only one detection output. According to Poissoni an
+statistics, the probabilityof having no signal detection i s given
+bye−η·µ·∆tgate,hencetheprobabilityofhavinganAPD signal
+detection output is given by :
+psig,gate= 1−e−η·µ·∆tgate. (9)
+µcan be expressed by the incident optical power Poptas
+µ=Popt
+hν(10)
+Solving for Poptyields
+Popt=−hν
+η·∆tgateln(1−psig,gate) (11)
+Measuring the ratio of the number of detections Ndet(in-
+cluding signal detections Nsigand dark counts Ndc) and the
+number of activated gates ( Ngate), yields the signal detection
+probability per gate
+psig,gate=Ndet−Ndc
+Ngate
+and finally the incident optical power Popt.
+If the signal is weak, then it is apparent that first of all
+the signal needs to be separated from noise by applying a
+sufficientlylargenumberofgates Ngate, see also App.B. Once
+this is achieved, one has to consider the precision or statis tical
+error of the result, which also is a function of Ngate. Two
+examples : a) Ngates= 10,∆tgate= 100ns, pdc,gate=
+2·10−4, psig,gate= 0.2, the probability of a dark count toappear is negligible and we obtain 2±√
+2signal counts=
+total counts, from which we can calculate an optical input
+power lying in [0.7 pW, 5.3 pW]. b) Ngates= 10000 and
+the other parameters like before, we obtain a number of
+total counts of 2002±√
+2002from which 2±√
+2are dark
+counts and 2000±√
+2000are signal counts. From the signal
+counts we infer an optical input power lying within [2.8 pW,
+2.9 pW]. The statistical error of the second measurement is
+much smaller.
+APPENDIX B
+NOISE EQUIVALENT POWER OF GEIGER-MODEAPD
+In the treatment of the APD noise we mainly consider two
+contributions, i.e. the shot noise of a) the signal counts an d b)
+the dark counts (assuming that afterpulse contributions ca n be
+neglected, for example by choosing a sufficiently large dead
+time).
+LetPoptbe the incident optical power on the diode. With the
+energy per photon hν, the detection efficiency ηand the gate
+width∆tgate,we inferthesignaldetectionprobabilitypergate
+(linearized version of Eq.9, for sufficiently small Popt) :
+psig,gate=η·Popt·∆tgate
+hν(12)
+Afterapplying Ngategatesofthesamewidththemeannumber
+of signal detections Nsigis
+Nsig=psig,gate·Ngate (13)
+Assuming Poissonian statistics we calculate the fluctuatio n
+∆Nsig=/radicalbig
+psig,gate·Ngate (14)
+The same derivation holds for the dark counts : we introduce
+a dark count probability per gate pdc,gate(which will be
+measured directly) leading to
+∆Ndc=/radicalbig
+pdc,gate·Ngate (15)
+thus the total noise fluctuation :
+∆Ntot=/radicalBig
+∆N2
+sig+∆N2
+dc(16)
+The noise equivalent power ( NEP) is inferred by calculating
+the optical power necessary to produce ∆Ntotcounts when
+applying Ngategates. In order to achieve this we replace Popt
+byNEPin Eq.12 and multiply by Ngate:
+∆Ntot=η·NEP·∆tgate
+hν·Ngate (17)
+Using Eq.14-16 and solving for NEPyields
+NEP=hν
+η·/radicalBigg
+psig,gate+pdc,gate
+Ngate∆t2
+gate[W](18)
+The minimal detectable power NEP0is obtained by setting
+the signal shot noise contribution equal to zero :
+NEP0=hν
+η·/radicalbiggpdc,gate
+Ngate∆t2
+gate[W](19)
+The existence of Ngatein these equations represents the
+iteration of a measurement and is a function of timeJOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 10
+(Ngate=fpulse·t)). The elemental measurement time
+is represented by ∆tgate, the duration of a single gate,
+which can be interpreted as the detection bandwidth via
+B:=1
+2·∆tgate. These formulas are practical when a NEP
+for a particular measurement needs to be calculated (see als o
+App.C). In order to obtain a formula which makes it easy
+to compare different detectors, we normalize with respect t o
+Ngate(which represents nothing else than the measurement
+time) and bandwidth B:
+NEPnorm=hν
+η·/radicalBig
+2·(ˆpsig+ ˆpdc) [W√
+Hz](20)
+and
+NEPnorm,0=hν
+η·/radicalbig
+2·ˆpdc[W√
+Hz](21)
+where
+ˆpdc:=pdc,gate
+∆tgate,ˆpsig:=psig,gate
+∆tgate(22)
+are the signal and dark count probability per gate, normaliz ed
+with respect to the gate width in seconds13.
+APPENDIX C
+DYNAMIC RANGE OF OPTICALTIMEDOMAIN
+REFLECTOMETER
+The strongest backscatter signal is observed right after th e
+emission at time t0=∆lp
+c, coming from the fiber locations
+within the interval [0;∆lp
+2], wherecis the speed of light in the
+fiberand∆lpisthewidthofthelaserpulse.Thecorresponding
+backscatter power is given by [24]
+PBS,0=S·P0,eff·e−2αsL(1−e−αs∆lp)(23)
+whereSis the fibers caption ratio, P0,effis the effective laser
+peak power corrected for internal component (e.g. circulat or)
+and connector loss and αsthe scattering coefficient. If we
+assumethat αs∆lp<<1,whichistrueinstandardfiber( αs≈
+0.04km−1) and∆lp<2km (∆tp<10µs) we can expand
+the exponential and get
+PBS,0≈S·P0,eff·αs·∆lp (24)
+The dynamic range is then given by the ratio of PBS,0and
+the minimal detectable power NEP0[W](Eq.19):
+dynR= 5log(PBS,0
+NEP0) (25)
+where the factor 5 accounts for the roundtrip in the fiber.
+Finally we obtain :
+dynR≈5log(S·P0,eff·αs·∆lp
+NEP0) (26)
+We note that NEP0like used here, includes the measurement
+time and decreases ∝√
+t(see also Eq.19 in the case of the
+13Therelation between signal count/dark count probability p er gate and gate
+width is almost linear over a large range of practical gate wi dths (typically
+ranging from nanoseconds to microseconds).Knowing ˆpsigandˆpdcmakes it
+easy to calculate the signal count and dark count probabilit y of a particular
+gate of widths ∆tgate, just by multiplying it by ∆tgate.ν−OTDR, where the measurement time tis represented by
+the number of applied gates, Ngate=fpulse·t).
+The operational definition of the dynamic range of an OTDR,
+given for example in [24], contains a measurement time of
+3 minutes. It is apparent that an extended measurement time
+enhancesthe NEP0and thusthe dynamicrange.In general,if
+the measurement time is increased by a factor d, the standard
+deviation of the noise is lowered by a factor√
+dand thus the
+NEP0by the same amount.
+APPENDIX D
+2-POINT RESOLUTION ADVANTAGE OF ν-OTDR
+We assume an xdBν-OTDR advantage in dynamic range
+(with respect to conventional OTDR, using the same laser
+pulse width ∆lp). Now we look for a factor αsuch that α·∆lp
+yields a reduction of the ν-OTDR dynamic range by xdB.
+Using Eq.26,19 and ∆tgate=∆lp
+c(adapting laser pulse width
+and gate width) we have to fulfill
+5log((α·∆lp)3
+2) = 5log((∆lp)3
+2)−x
+yielding
+α= 10−2
+15x(27)
+For example : x= 10dB→α= 0.046. Thus the ν-OTDR
+can achieve the same dynamic range with a 20 times smaller
+pulse width.
+APPENDIX E
+SIGNAL TO NOISE RATIO AS FUNCTION OF MEASUREMENT
+TIME
+We derive a formula for the SNR ratio as a function of
+time from the photon counting perspective. However, the fina l
+result, after linearization will not contain any photon cou nting
+specific quantity and is therefore generally applicable.
+We define the signal to noise ratio (SNR)as the ratio of
+signal counts Nsigto the total fluctuation of the counts ∆Ntot
+including fluctuation of signal and dark counts (like defined
+in App.B).
+SNR=Nsig
+∆Ntot=psig,gate·Ngate/radicalbig
+(psig,gate+pdc,gate)·Ngate
+using Eq.13 and 16. Now we introduce the bandwidth normal-
+ized NEP (Eq.20) and use Ngate=fpulse·twheretis the
+measurement time
+⇒SNR=√
+2·hν
+η·psig,gate·/radicalbig
+fpulse·t
+NEPnorm·/radicalbig
+∆tgate
+then replacing psig,gateusing Eq.9 and 10:
+SNR=√
+2·hν
+η·(1−e−η
+hν·Popt·∆tgate)·/radicalbig
+fpulse·t
+NEPnorm·/radicalbig
+∆tgate(28)
+If the optical input power is sufficiently small, the signal
+detection probability increases linearly with the optical power
+and we can expand the exponential to obtain
+SNR=Popt·/radicalbig
+fpulse·t·/radicalbig
+2·∆tgate
+NEPnormJOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 11
+=Popt·/radicalbig
+fpulse·t
+NEPnorm·√
+B(29)
+where we used B=1
+2·∆tgate, the detection bandwidth.
+This final formula is independent of any photon counting
+quantities and does also apply for the general case, includi ng
+linearAPDdetection.Inthelinearregimethereisevennosu ch
+severe restrictions as in photon counting mode since much
+higherPoptcan be processed.
+On the other hand,if measurementtime needsto be calculated
+as a function of SNR, we obtain straight forward
+t=1
+fpulse·/parenleftBigg
+SNR·NEPnorm·√
+B
+Popt/parenrightBigg2
+(30)
+APPENDIX F
+TRAIN OF GATES DISCUSSION
+Theapplicationofa deadtimecanhaveconsiderableimpact
+on the gate activation statistics. Fig.12 shows what happen s if
+the product fgate·τis chosen too large.
+If we assume, that the probabilityof detecting a signal phot on
+02468101214161820222400.10.20.30.40.50.60.70.80.91
+gate number [1/12] or distance [km]gate activation probability
+  
+activation minimumfgate= 1.2 MHz, τ= 10µ s, psig,gate1=0.2
+Fig. 12. Dead time application (here : τ= 10µscorresponding to 1 km
+in terms of sampling distance) has considerable impact on th e gate activation
+statistics. If the gating frequency fgateis chosen too high, then the activation
+minimum approaches zero, no signal can be acquired there.
+in the first gate of the train of gates is psig,gate 1, then gate 2
+gets activated with probability (1−psig,gate 1)(otherwise it
+falls into the dead time of gate 1). The probability that gate i
+gets activated ( i≤fgate·τ) is then given by
+pact,gate i = (1−psig,gate 1)(i−1)(31)
+where we assume, that the signal detection probability is
+almost constant at the beginning of the fiber. This expressio n
+approaches 0 when i is large. In the worst case, ”activation
+holes” appear in a repetitive manner (see Fig.12, in the case
+where the minima of the periodic structure at the beginning
+touch zero probability) and therefore detections around th ese
+locations are impossible or only possible with very low stat is-
+tics.
+To avoid this, we can define a criteria, which ensures that
+each gate has a sufficient number of activations. The first00.10.20.30.40.50.60.70.8100101102
+  
+psig,gate1fgate,max ⋅ τactivation min > 0.2
+activation min > 0.4
+activation min > 0.6
+activation min > 0.8
+Fig. 13. The maximally allowed gating frequency fgate,max , in order to
+avoid activation holes, is a function of the photon detectio n probability in the
+first gate pph,gate 1and the designated dead time, which itself is set by the
+designated afterpulsing probability. Depending on the cho ice of the activation
+minimum value one can infer the product fgate,max ·τ, which also signifies
+the number of non activated gates after a detection. An examp le is given in
+the text.
+minimum plays the role of a bottleneck. When it is above
+some threshold (to be defined), all the other minima are as
+well14. The minimum depends on psig,gate 1andfgate·τ.
+Using Eq.31 we can calculate, the maximally suitable gating
+frequency fgate,max, dependingon a designated threshold, see
+Fig.13.
+Example : assuming τ= 1µs, psig,gate 1= 0.25and an
+activation minimum of 0.4. Then we infer fgate,max·τ= 4
+(see arrows in Fig.13) and therefore fgate,max=4
+τ= 4MHz.
+If the maximally suitable gating frequency fgate,max is not
+large enough to obtain the designated sampling resolution,
+it is necessary to shift the start delay of the train of gates.
+For instance, if we want a sampling resolution of 5 m, but
+fgate,max = 4MHz, yielding only 25 m, we need to delay
+the train (with respect to the laser pulse departure) four ti mes
+by 50 ns. This of course increases the total measurement time
+by a factor 5.
+We note that fgate,max, is in principle bounded by 1/∆tgate.
+If the suggested fgate,max, according to Fig.13, is larger than
+1/∆tgate,we arenaturallyledtothe free runningmode ,where
+the diode stays active until a detection is obtained [18], se e
+Fig.10.
+This happens if
+µ <b
+η·τ(32)
+or equivalently
+Popt< hν·b
+η·τ(33)
+whereµis the photon flux (number of photons per second,
+cw) and Poptthe corresponding power, ηthe detector
+efficiency, τthe detector dead time and ba constant
+depending on the activation minimum criteria, explicitly :
+14The reason for this is the fiber loss, which decreases the back scattered
+signal and therefore yields less detections and dead time ap plications. The
+dead time effect gets less severe. When the backscatter powe r is very low
+there is almost no difference to the ideal caseJOURNAL OF LIGHTWAVE TECHNOLOGY , VOL. X, NO. Y, JANUARY 200Z 12
+for an activation minimum of 0.2(0.4,0.6,0.8), one obtains
+b= 1.61(0.92,0.51,0.23). Due to its superior duty cycle, the
+free running mode is the ideal low power solution.
+ACKNOWLEDGMENT
+The authors would like to thank EXFO for providing the
+FTB-7600 OTDR module and support, in particular S. Perron,
+J. Gagnon and G. Schinn. We also like to thank J.D. Gautier
+for providing technical support and Bruno Sanguinetti for
+thorough cross-reading. We thank Swisscom for providing
+access to the fiber link between Geneva and Neuchatel.
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+arXiv:1001.0695v2  [cs.CG]  6 Jan 2012The Geodesic Diameter of Polygonal Domains∗
+Sang Won Bae†Matias Korman‡Yoshio Okamoto§
+31stOctober, 2018 00:39
+Abstract
+This paper studies the geodesic diameter of polygonal domains havin ghholes and n
+corners. For simple polygons (i.e., h= 0), the geodesic diameter is determined by a pair of
+corners of a given polygon and can be computed in linear time, as know n by Hershberger
+and Suri. For general polygonal domains with h≥1, however, no algorithm for computing
+the geodesic diameter was known prior to this paper. In this paper, we present the first
+algorithms that compute the geodesic diameter of a given polygonal domain in worst-case
+timeO(n7.73) orO(n7(logn+h)). The main difficulty unlike the simple polygon case relies
+on the following observation revealed in this paper: two interior point s can determine the
+geodesic diameter and in that case there exist at least five distinct s hortest paths between
+the two.
+1 Introduction
+Apolygonal domain Pwithhholes and ncorners is a connected and closed subset of R2having
+hholes whose boundary consists of h+1 simple closed polygonal chains of ntotal line segments.
+Given a polygonal domain P, the geodesic distance d( p,q) between two points pandqofPis
+defined as the length of a shortest path that connects pandqand stays within P.
+This paper addresses the geodesic diameter problem in polyg onal domains having one or
+more holes. The geodesic diameter diam( P) of domain Pis defined as the largest possible
+geodesic distance between any two points of P, that is, diam( P) = max s,t∈Pd(s,t).
+For simple polygons (i.e., domains with no hole), the geodes ic diameter has been extensively
+studied. Chazelle [7] provided the first O(n2)-time algorithm computing the geodesic diameter
+of a simple polygon. Afterwards, Suri [20] presented an O(nlogn)-time algorithm that solves
+the all-geodesic-farthest neighbors problem, computing t he farthest neighbor of every corner
+and thus finding the geodesic diameter. At last, Hershberger and Suri [12] showed that the
+diameter can be computed in linear time using fast matrix sea rch techniques.
+On the other hand, the geodesic diameter of a domain having on e or more holes is less
+understood. Mitchell [16] has posed an open problem asking a n algorithm for computing the
+geodesic diameter of a polygonal domain. However, even for t he corner-to-corner diameter
+∗A preliminary version of this paper was presented at the 18th Annual European Symposium on Algorithms
+(ESA 2010). Work by S.W. Bae was supported by National Resear ch Foundation of Korea (NRF) grant funded
+by the Korea government (MEST) (No. 2010-0005974). Work by Y . Okamoto was supported by Global COE
+Program “Computationism as a Foundation for the Sciences” a nd Grant-in-Aid for Scientific Research from
+Ministry of Education, Science and Culture, Japan, and Japa n Society for the Promotion of Science.
+†Department of Computer Science, Kyonggi University, Suwon , Korea. Email: swbae@kgu.ac.kr
+‡Computer Science Department, Universit´ e Libre de Bruxell es (ULB), Belgium. Email: mkormanc@ulb.ac.be
+§Center for Graduate Education Initiative, Japan AdvancedI nstituteof Science and Technology, Nomi, Japan.
+Email:okamotoy@jaist.ac.jp
+1t∗s∗
+(a) (b) (c)u1u2
+u3 v3v2
+v1
+s∗t∗s∗
+t∗
+Figure 1: Three polygonal domains where the geodesic diameter is determined by a pair ( s∗,t∗) of
+non-corner points; Gray-shaded regions depict the interior of th e holes and dark gray segments depict
+the boundary ∂P. Recall that P, as a set, contains its boundary ∂P. (a) Both s∗andt∗lie on∂P.
+There are three shortest paths between s∗andt∗. In this domain, there are two (symmetric) diametral
+pairs (only one is depicted for clarity). (b) s∗∈∂Pandt∗∈intP. Three triangular holes are placed
+in a symmetric way, obtaining four shortest paths between s∗andt∗. (c) Both s∗andt∗lie in the
+interior int P. Here, the five holes are packed like jigsaw puzzle pieces, forming na rrow corridors (dark
+gray paths) and two empty, regular triangles. Observe that d( u1,v1) = d(u1,v2) = d(u2,v2) = d(u2,v3)
+= d(u3,v3) = d(u3,v1). The points s∗andt∗lie at the centers of the triangles formed by the uiand the
+vi, respectively. There are six shortest paths between s∗andt∗.
+maxu,v∈Vd(u,v), where Vdenotes the set of corners of P, we know nothing better than a brute-
+force algorithm that takes O(n2logn) time, checking all the geodesic distances between every
+pair of corners.1Prior to our results, there was no known algorithm for comput ing the geodesic
+diameter in domains with holes. We should also mention that K oivisto and Polishchuk [14] had
+claimed an improved algorithm after a preliminary report of our work [5], but it was shown to
+be a failed trial through conversations with the authors.2
+This fairly wide gap between simple polygons and polygonal d omains with holes is seemingly
+due to the uniqueness of the shortest path between any two poi nts. When a domain Phas no
+hole, it is well known that there is a unique shortest path bet ween any two points [10]. Using
+this uniqueness, one can show that the diameter diam( P) is realized by a pair of corners [12,20].
+For general polygonal domains, however, this is not the case . In this paper, we exhibit several
+examples where the diameters are realized by non-corner poi nts on∂Por even by interior
+points of P. See Figure 1. Such examples were constructed based on the mu ltiplicity of shortest
+paths and, to our best knowledge, never known prior to this wo rk. This observation also
+shows an immediate difficulty in devising any exhaustive algo rithm since one sees no intuitive
+discretization of the search space.
+Thestatus of thegeodesiccenter problemisalso similar. A p oint inPis definedas a geodesic
+centerif it minimizes the maximum geodesic distance from it to any o ther point of P. Asano
+and Toussaint [3] introduced the first O(n4logn)-time algorithm for computing the geodesic
+center of a simple polygon (i.e., when h= 0), and Pollack, Sharir and Rote [19] improved it
+toO(nlogn) time. As with the diameter problem, there is no known algori thm for domains
+with holes. See O’Rourke and Suri [18] and Mitchell [16] for m ore references on the geodesic
+diameter/center problem.
+Since the geodesic diameter/center of a simple polygon is de termined by its corners, one
+can exploit the geodesic farthest-site Voronoi diagram of the set Vof corners to compute the
+diameter/center, whichcanbebuiltin O(nlogn)time[2]. Recently, BaeandChwa[4] presented
+1Personal communication with Joseph S. B. Mitchell.
+2Personal communication with Valentin Polishchuk.
+2anO(nklog3(n+k))-time algorithm for computing the geodesic farthest-sit e Voronoi diagram
+ofksites in polygonal domains with holes. This result can be use d to compute the geodesic
+diameter max p,q∈Sd(p,q) of afinitesetSof points in P. However, this approach cannot be
+directly used for computing diam( P) without any characterization of the diameter. Moreover,
+whenS=V, this approach is no better than the brute-force O(n2logn)-time algorithm for
+computing the corner-to-corner diameter max u,v∈Vd(u,v).
+In this paper, we present the first algorithms that compute th e geodesic diameter of a given
+polygonal domain in O(n7.73) orO(n7(logn+h)) time in the worst case. Our new geometric
+results underlying the algorithms show that the existence o f any diametral pair consisting of
+non-corner points implies multiple shortest paths between the pair; among other results, we
+show that if(s,t)is a diametral pair and both sandtlie in the interior of P, then there are at
+least five shortest paths between sandt.
+Someanalogies betweenpolygonaldomainsandconvexpolyto pesinR3canbeseen. O’Rourke
+and Schevon [17] proved that if the geodesic diameter on a con vex 3-polytope is realized by two
+non-corner points, then at least five shortest paths exist be tween the two; see also Zalgaller [21]
+forsimplerarguments. Basedonthisobservation, theypres entedan O(n14logn)-timealgorithm
+for computing the geodesic diameter on a convex 3-polytope. Afterwards, the time bound was
+improved to O(n8logn) by Agarwal et al. [1] and recently to O(n7logn) by Cook IV and
+Wenk [9].
+The rest of the paper is organized as follows: After introduc ing preliminary definitions and
+concepts in Section 2, we investigate local maxima of the low er envelope of convex functions in
+Section 3, resulting in Theorem 1. Section 4 extensively exp loits the intermediate result to show
+lower bounds on the number of shortest paths between a diamet ral pair for every possible case,
+and then Section 5 describes our algorithms for the geodesic diameter. We finally concludes
+the paper with summary, some remarks, and open issues in Sect ion 6. Also, we exhibit several
+interesting examples that cover all possible combinatoria l cases in Appendix A. We hope that
+the readers will enjoy them.
+2 Preliminaries
+Throughout the paper, we frequently use several topologica l concepts such as open and closed
+subsets, neighborhoods, and the boundary ∂Aand the interior int Aof a setA; unless stated
+otherwise, all of them are supposed to be derived with respec t to the standard topology on Rd
+with the Euclidean norm /ba∇dbl·/ba∇dblfor fixed d≥1. We also denote the straight line segment joining
+two points a,bbyab.
+Apolygonal domain Pwithhholes and ncorners3is a connected and closed subset of R2
+withhholes whose boundary ∂Pconsists of h+1 simple closed polygonal chains of ntotal line
+segments. The boundary ∂Pof a polygonal domain Pis regarded as a series of obstacles so
+that any feasible path in Pis not allowed to cross ∂P. Thegeodesic distance d(p,q) between
+any two points p,qin a polygonal domain Pis defined as the length of a shortest feasible path
+between pandq, where the lengthof a path is the sum of the Euclidean lengths of its segments.
+It is well known from earlier work [15] that there always exis ts a shortest feasible path between
+any two points p,q∈ P, and thus the geodesic distance function d( ·,·) is well defined. The
+geodesic diameter diam( P) of a polygonal domain Pis defined as the largest geodesic distance
+between any two points of P, that is,
+diam(P) = max
+s,t∈Pd(s,t).
+3We reserve the term “vertex” for 0-dimensional faces of subd ivisions of a certain space.
+3A pair (s,t) of points in Pthat realizes the geodesic diameter diam( P) is called a diametral
+pair.
+Shortest path map. LetVbethesetofallcornersof Pandπ(s,t)beashortestpathbetween
+s∈ Pandt∈ P. Such a path π(s,t) is represented as a sequence π(s,t) = (s,v1,...,vk,t) for
+somev1,...,vk∈V; that is, a polygonal chain through a sequence of corners [15 ]. Note that we
+can have k= 0 when d( s,t) =/ba∇dbls−t/ba∇dbl. If two paths (with possibly different endpoints) induce
+the same sequence of corners ( v1,...,vk), then they are said to have the same combinatorial
+structure .
+Theshortest path map SPM(s) for a fixed s∈ Pis a decomposition of Pinto cells such that
+every point in a common cell can be reached from sby shortest paths of the same combinatorial
+structure. Each cell σs(v) ofSPM(s) is associated with a corner v∈Vwhich is the last
+corner of π(s,t) for any tin the cell σs(v). We also define the cell σs(s) as the set of points
+t∈ Psuch that π(s,t) passes through no corner of P, soπ(s,t) =st. Each edge of SPM(s)
+is an arc on the boundary of two incident cells σs(v1) andσs(v2) determined by two corners
+v1,v2∈V∪{s}. Similarly, each vertex of SPM(s) is determined by at least three distinct corners
+v1,v2,v3∈V∪{s}.
+Note that, for fixed s∈ P, a point farthest apart from slies at either (1) a vertex of
+SPM(s), (2) an intersection between the boundary ∂Pand an edge of SPM(s), or (3) a corner
+inV. The shortest path map SPM(s) hasO(n) total number of cells, edges, and vertices and
+can be computed in O(nlogn) time using O(nlogn) working space [13]. For more details on
+shortest path maps, see [13,15,16].
+Path-length function. Ifπ(s,t)/\e}atio\slash=st, then there are two corners u,v∈Vsuch that uand
+vare the first and last corners along π(s,t) fromstot, respectively. Here, the path π(s,t) is
+formed as the union of su,vtand a shortest path π(u,v) fromutov. Note that uandvare
+not necessarily distinct. In order to realize such a path, we assert that sis visible from uand
+tis visible from v. That is, s∈VR(u) andt∈VR(v), where VR(p) for any p∈ Pis defined to
+be the set of all points q∈ Psuch that pq⊂ P, also called the visibility region ofp∈ P.
+We now define the path-length function lenu,v:VR(u)×VR(v)→Rfor any fixed pair of
+cornersu,v∈Vto be
+lenu,v(s,t) :=/ba∇dbls−u/ba∇dbl+d(u,v)+/ba∇dblv−t/ba∇dbl.
+That is, len u,v(s,t) represents the length of paths from stotthat have a common combinatorial
+structure; going straight from stou, following a shortest path from utov, and going straight
+tot. Also, unless d( s,t) =/ba∇dbls−t/ba∇dbl(equivalently, s∈VR(t)), the geodesic distance d( s,t) can be
+expressed as the pointwise minimum of some path-length func tions:
+d(s,t) = min
+u∈VR(s), v∈VR(t)lenu,v(s,t).
+Consequently, wehavetwopossibilitiesforadiametralpai r(s∗,t∗); eitherwehaved( s∗,t∗) =
+/ba∇dbls∗−t∗/ba∇dblor the pair ( s∗,t∗) is a local maximum of the lower envelope of several path-len gth
+functions. In the following, we will mainly study the latter case, since the former can be easily
+handled.
+3 Local Maxima of the Lower Envelope of Convex Functions
+In this section, we give an interesting property of the lower envelope of a family of convex
+functions which will afterwards be used in our geodesic diam eter environment. We start with a
+4basic observation on the intersection of hemispheres on a un it hypersphere in the d-dimensional
+spaceRd. For any fixed positive integer d, letSd−1:={x∈Rd| /ba∇dblx/ba∇dbl= 1}be the unit
+hypersphere in Rdcentered at the origin. A closed(oropen)hemisphere onSd−1is defined
+to be the intersection of Sd−1and a closed (open, respectively) half-space of Rdbounded by a
+hyperplane that contains the origin.
+We call a k-dimensional affine subspace of Rdak-flat. Note that a hyperplane in Rdis a
+(d−1)-flat and a line in Rdis a 1-flat. Also, the intersection of Sd−1and ak-flat through the
+origin in Rdis called a great(k−1)-sphereonSd−1. Note that a great 1-sphere is called a great
+circle and a great 0-sphere consists of two antipodal points .
+Lemma 1 For any two positive integers dandm≤d, a set of any mclosed hemispheres on
+Sd−1has a nonempty common intersection. Moreover, if the inters ection has an empty interior
+relative to Sd−1, then it includes a great (d−m)-sphere on Sd−1.
+Proof.We only give a proof for the second statement, which implies t he first. The case of
+d= 1 is trivial, so we assume d >1. LetH1,...,H mbe anymclosed hemispheres on Sd−1, and
+hibe the hyperplane through the origin in Rdsuch that Hilies in a closed half-space supported
+byhi. In this proof, we denote byÙHithe open hemisphere, defined to beÙHi=Hi\hi. Also,
+letHj:=/intersectiontext
+1≤i≤jHiandÙHj:=/intersectiontext
+1≤i≤jÙHi.
+Suppose thatÙHm=∅. Letkbe the smallest integer such thatÙHk=∅. By definition,
+k≥2 andÙHk−1/\e}atio\slash=∅. Note that the intersection of any k−1 non-parallel hyperplanes of
+Rdincludes a ( d−k+ 1)-flat and each hicontains the origin. Hence,/intersectiontext
+1≤i≤k−1hiincludes a
+(d−k+1)-flat through the origin and thus Hk−1includes a great ( d−k)-sphere GonSd−1.
+Sincex∈Gimplies−x∈Gfor anyx∈Sd−1, we must have G⊆hk, in order to have an empty
+intersectionÙHk. This implies that/intersectiontext
+1≤i≤khialso includes a ( d−k+1)-flat through the origin,
+and further that/intersectiontext
+1≤i≤mhiincludes a ( d−m+1)-flat through the origin. We hence conclude
+thatHm=/intersectiontext
+1≤i≤mHiincludes a great ( d−m)-sphere on Sd−1.
+Using Lemma 1 we prove the following theorem.
+Theorem 1 For any fixed positive integer d, letFbe a finite family of real-valued convex
+functions defined on a convex subset C⊆Rdandg(x) := min f∈Ff(x)be their pointwise
+minimum. Suppose that gattains a local maximum at x∗∈Cand there are exactly m≤d
+functions f1,...,fm∈ Fsuch that fi(x∗) =g(x∗)for each i= 1,...,m. Then, there exists a
+(d+1−m)-flatϕ⊂Rdthrough x∗such that gis constant on ϕ∩Ufor some neighborhood
+U⊂Rdofx∗withU⊂C.
+Proof.First, we give a sketchy idea of our proof for the theorem. All functions f∈ Fother
+thanf1,...,fmmust satisfy f(x)> g(x) in a small neighborhood of x∗. In particular, the
+function gis the lower envelope of the mconvex functions fiin a small neighborhood of x∗.
+By convexity, we will show that for each i, there is a hemisphere Hiof directions in Sd−1in
+whichfidoes not decrease. (Note that the sphere Sd−1represents the space of all directions in
+Rd.) This result combined with Lemma 1 gives that the intersect ion of hemispheres will be a
+(d+1−m)-flat in which neither of the mfunctions (nor g) can decrease. Since x∗is a local
+maximum of g, the only possibility is that gremains constant near x∗along the flat.
+A more detailed proof is given as follows. Let x∗∈Candf1,...,fm∈ Fbe as in the
+statement. For each i, consider the sublevel set Li:={x∈C|fi(x)≤fi(x∗)}. Here, we
+consider two cases: (i) x∗lies in the interior of Lior (ii) on its boundary ∂Li. Note that
+Li⊆Cis convex since fiis a convex function. For the latter case (ii), there exists a supporting
+hyperplane hitoLiatx∗sinceLiis convex and x∗∈∂Li. Denote by h⊕
+ithe closed half-space
+that is bounded by hiand does not contain Li. For the former case (i), we choose hito be any
+hyperplane of Rdthrough x∗andh⊕
+ito be any closed half-space supported by hi. Then, we
+have that fi(x∗)≤fi(x) for any x∈h⊕
+i∩C, regardless of the cases; in particular for Case (i),
+5observe that fi(x) =fi(x∗) for any x∈Liby convexity so that we can choose any hyperplane
+ashi.
+Now, we let
+Hi:={x−x∗|x∈h⊕
+i,/ba∇dblx−x∗/ba∇dbl= 1}
+be a closed hemisphere of the unit sphere Sd−1centered at the origin. Note that fidoes
+not decrease if we move from x∗locally in any direction in Hi. Sinceg(x∗) =fi(x∗) for any
+i∈ {1,...,m}andx∗is a local maximum of g, the intersection/intersectiontextm
+i=1Hihas an empty interior
+relative to Sd−1; otherwise, there exists y∈int/intersectiontextm
+i=1Hisuch that fi(x∗+λy)≥fi(x∗) for any
+i∈ {1,...,m}and any λ >0 withx∗+λy∈C.
+Hence, by Lemma 1,/intersectiontextm
+i=1Hihas a nonempty intersection including a great ( d−m)-sphere
+GonSd−1. Letϕbe the corresponding ( d−m+1)-flat in Rdthrough x∗defined as
+ϕ:={x∗+λy∈Rd|y∈Gandλ∈R}.
+Consider the restriction fi|ϕ∩Coffionϕ∩C. Sincefiis convex and ϕis an affine sub-
+space (thus, convex), fi|ϕ∩Cis also convex and their pointwise minimum g|ϕ∩Cattains a local
+maximum at x∗∈ϕ∩C. On the other hand, each fi|ϕ∩Cattains a local minimum at x∗; since
+ϕ⊆h⊕
+i, we have fi(x∗)≤fi(x) for any point x∈ϕ∩C. Hence, g|ϕ∩Calso attains a local
+minimum at x∗sinceg(x∗) =fi(x∗) for any i∈ {1,...,m}. Consequently, gis locally constant
+atx∗onϕ; more precisely, there is a sufficiently small neighborhood U⊂Rdofx∗withU⊂C
+such that gis constant on U∩ϕ, completing the proof.
+4 Properties of Geodesic-Maximal Pairs
+We call a pair ( s∗,t∗)∈ P ×P maximal if (s∗,t∗) is a local maximum of the geodesic distance
+function d. That is, ( s∗,t∗) is maximal if and only if there are two neighborhoods Us,Ut⊂R2
+ofs∗and oft∗, respectively, such that for any s∈Us∩ Pand any t∈Ut∩ Pwe have
+d(s∗,t∗)≥d(s,t). Clearly, any diametral pair is maximal.
+Consider any maximal pair ( s∗,t∗) inP. Let Π(s∗,t∗) be the set of all shortest paths from
+s∗tot∗. Then, each path π∈Π(s∗,t∗) is associated with a pair of corners ( u,v) that are its
+first and last corners as discussed in Section 2. Note that suc h a pair ( u,v) of corners always
+exists for any π∈Π(s∗,t∗); even if d( s∗,t∗) =/ba∇dbls∗−t∗/ba∇dbl, then both endpoints s∗andt∗must
+be corners in Vby its maximality. We now focus on the set of such pairs of the fi rst and last
+corners, defined to be
+V(s∗,t∗) :={(u,v)| ∃π∈Π(s∗,t∗) s.t.u,v∈Vare the first and last corners along π, resp.}.
+Giving an arbitrary ordering, we set V(s∗,t∗) ={(u1,v1),...,(um,vm)}, wheremis the cardi-
+nality of V(s∗,t∗). Also, we let
+Vs∗:={u1,...,u m}, Vt∗:={v1,...,vm}.
+Some immediate bounds are |Π(s∗,t∗)| ≥m,|Vs∗| ≤m, and|Vt∗| ≤m. Observe that it is not
+true that we always have the equality |Π(s∗,t∗)|=m; in some cases, there can be multiple
+shortest paths between a pair of corners. In the following, w e show the tight bound on the
+cardinality mof the set V(s∗,t∗), provided that ( s∗,t∗) is maximal.
+LetEbe the set of all sides of Pwithout their endpoints and Bbe their union. Note that
+B=∂P \Vis the boundary of Pexcept the corners V. The goal of this section is to prove the
+following theorem, which is the main combinatorial result o f this paper.
+6Theorem 2 Suppose that (s∗,t∗)is a maximal pair in P, and that V(s∗,t∗),Vs∗, andVt∗are
+defined as above. Then, we have the following implications.
+(V-V) s∗∈V,t∗∈Vimplies |V(s∗,t∗)| ≥1,|Vs∗| ≥1,|Vt∗| ≥1;
+(V-B) s∗∈V,t∗∈ B implies |V(s∗,t∗)| ≥2,|Vs∗| ≥1,|Vt∗| ≥2;
+(V-I) s∗∈V,t∗∈intPimplies |V(s∗,t∗)| ≥3,|Vs∗| ≥1,|Vt∗| ≥3;
+(B-B) s∗∈ B,t∗∈ B implies |V(s∗,t∗)| ≥3,|Vs∗| ≥2,|Vt∗| ≥2;
+(B-I) s∗∈ B,t∗∈intPimplies |V(s∗,t∗)| ≥4,|Vs∗| ≥2,|Vt∗| ≥3;
+(I-I) s∗∈intP,t∗∈intPimplies |V(s∗,t∗)| ≥5,|Vs∗| ≥3,|Vt∗| ≥3.
+Moreover, each of the above bounds is tight.
+Together with the bound |Π(s∗,t∗)| ≥ |V(s∗,t∗)|, Theorem 2 immediately implies tight lower
+bounds on the number of shortest paths between any maximal pa ir.
+Corollary 1 For anyp∈ P, letδ(p) := 0ifp∈V;δ(p) := 1ifp∈ B;δ(p) := 2ifp∈intP. If
+(s∗,t∗)is a maximal pair in P, then we have
+|Π(s∗,t∗)| ≥δ(s∗)+δ(t∗)+1.
+Moreover, the above bound is tight.
+To see the tightness of the bounds, we present examples with r emarks in Figure 1 and
+Appendix A. In particular, one can easily see the tightness o f the bounds on |Vs∗|and|Vt∗|
+from shortest path maps SPM(s∗) andSPM(t∗), whenV∪{s∗,t∗}is in general position.
+We first give an overview of the proof. The general reasoning i s roughly the same for all the
+different scenarios, and we thus focus on the case in which ( s∗,t∗) is a maximal pair and both s∗
+andt∗are interior points (Case (I-I)). Regard the geodesic distance function d as a four-variate
+function in a small convex neighborhood of ( s∗,t∗). As mentioned in Section 2, the geodesic
+distance is the pointwise minimum of a finite number of path-l ength functions. Since the pair
+(s∗,t∗) is maximal, we will apply Theorem 1 and obtain that the geode sic distance is constant
+in a flat of dimension d+1−m= 5−m, wherem=|V(s∗,t∗)|. On the other hand, we will
+also show that the geodesic distance function can only remai n constant in a zero-dimensional
+flat (i.e., at a point), hence m≥5. In the other cases (boundary-interior, boundary-bounda ry,
+etc.) the boundary of Pintroduces additional constraints that reduce the degrees of freedom
+of the geodesic distance function. Hence, fewer paths are en ough topinthe solution.
+The main technical difficulty of the proof is the fact that the p ath-length functions len u,v
+are not globally defined. Thus, we must properly extend them i n a way that all conditions of
+Theorem 1 are satisfied.
+4.1 Proof of Theorem 2
+We start with several basic observations. The proof of Theor em 2 will be done separately for
+each case.
+The following lemma proves the bounds on |Vs∗|and|Vt∗|of Theorem 2.
+Lemma 2 Let(s∗,t∗)be a maximal pair.
+1. Ift∗∈ B, then|Vt∗| ≥2. Moreover, if t∗∈e∈E, then there exists v∈Vt∗such that vis
+off the line supporting e.
+2. Ift∗∈intP, then|Vt∗| ≥3andt∗lies in the interior of the convex hull of Vt∗.
+7s∗uiu′
+iπi
+s∗ui=u′
+iπi
+s∗uiu′
+iπi
+(a)uiu′
+iπi
+(b)s∗s∗uiu′
+iπi
+Figure 2: (a) How to determine u′
+i. (left to right) ui=u′
+i;s∗,ui, and the second corner are collinear;
+s∗and the first three corners are collinear (b) For points in a small disk Bcentered at s∗withB⊂
+VR(u′
+i)∪VR(ui), the function αimeasures the length of the shortest path from u′
+ito each.
+Proof.Since (s∗,t∗) is a maximal pair, the function ds∗(t) := d(s∗,t) is maximized at t=t∗on
+a sufficiently small subset U⊂ Pwitht∗∈U. As discussed in Section 2, if t∗/∈V, thent∗must
+be either a vertex of SPM(s∗) or an intersection point between an edge of SPM(s∗) and∂P.
+Ift∗∈intP, thent∗should fall into the former case and hence we have at least thr ee corners
+v1,v2,v3∈Vdetermining the vertex t∗ofSPM(s∗). Ift∗∈ B, thent∗may also occur at the
+latter case. In that case, t∗lies on an edge of SPM(s∗) and thus we have at least two corners
+v1,v2∈Vdetermining an edge of SPM(s∗).
+The other claims of the lemma can be shown as follows. If t∗∈intPbutt∗lies out of the
+interior of theconvex hull of Vt∗, then we can findanother point t∈ Parbitrarily close to t∗such
+that/ba∇dblt−vi/ba∇dbl>/ba∇dblt∗−vi/ba∇dblfor every vi∈Vt∗. This implies that d( s∗,t)>d(s∗,t∗), contradicting
+the maximality of ( s∗,t∗). Ift∗∈e∈Ebut every vi∈Vt∗lies on the supportingline ℓofe, then
+we obtain a strictly larger distance than d( s∗,t∗), as moving t∗in a perpendicular direction to
+ℓ. (Notice that a similar argument can be also found in [17, Lem ma 2.2].)
+Lemma 2 immediately implies the lower bound on |V(s∗,t∗)|whens∗∈Vort∗∈Vsince
+|V(s∗,t∗)| ≥max{|Vs∗|,|Vt∗|}. This completes Cases (V-*). Note that the bounds for Case
+(V-V)are trivial.
+From now on, we assume that neither s∗nort∗is a corner of P. This assumption, together
+with Lemma 2, implies multiple shortest paths between s∗andt∗, and thus d( s∗,t∗)>/ba∇dbls∗−t∗/ba∇dbl.
+Hence, as discussed in Section 2, any maximal pair falling in to one of Cases (B-B),(B-I), and
+(I-I)appears as a local maximum of the lower envelope of some path- length functions.
+Case (I-I): When both s∗andt∗lie inintP.We will apply Theorem 1 to prove Theorem 2
+for Case (I-I). Recall the definition of V(s∗,t∗) ={(u1,v1),...,(um,vm)}andm=|V(s∗,t∗)|.
+For each ( ui,vi)∈ V(s∗,t∗), we have the corresponding shortest path πibetween s∗andt∗and
+lenui,vi(s∗,t∗) = d(s∗,t∗). Thus, we have at least mfunctions famong{lenu,v|u,v∈V}such
+thatf(s∗,t∗) = d(s∗,t∗). If the number of such path-length functions are exactly m, we can
+apply Theorem 1 directly.
+Unfortunately, this is not always the case. A single shortes t pathπi∈Π(s∗,t∗) may give
+additional pairs ( u,v) of corners with u,v∈πisuch that ( u,v)/\e}atio\slash= (ui,vi) and len u,v(s∗,t∗) =
+d(s∗,t∗). This situation can occur even when the corners of Pare in general position. Observe
+that this happens only when u,ui,s∗orv,vi,t∗are collinear. In order to resolve this problem,
+we define the mergedpath-length functions that satisfy all the requirements of Theorem 1 even
+in degenerate cases.
+Recall that the combinatorial structure of each shortest pa thπican be represented by a
+sequence ( ui=ui,1,...,u i,k=vi) of corners in V. We define u′
+ito be one of the ui,jas follows.
+8Ifs∗does not lie on the line ℓthrough uiandui,2, thenu′
+i:=ui; otherwise, if s∗∈ℓ, then
+u′
+i:=ui,j, wherejis the largest index such that for any open neighborhood U⊂R2ofs∗
+there exists a point s∈(U∩VR(ui,j))\ℓ. Note that such u′
+ialways exists, and if no three of
+Vare collinear, then we always have either u′
+i=uioru′
+i=ui,2. Figure 2(a) illustrates how
+to determine u′
+i. Also, we define v′
+iin an analogous way. Let αi:VR(u′
+i)∪VR(ui)→Rand
+ωi:VR(v′
+i)∪VR(vi)→Rbe two functions defined as
+αi(s) :=/braceleftBigg
+/ba∇dbls−u′
+i/ba∇dbl ifs∈VR(u′
+i),
+/ba∇dbls−ui/ba∇dbl+/ba∇dblui−u′
+i/ba∇dblifs∈VR(ui)\VR(u′
+i);
+ωi(t) :=/braceleftBigg
+/ba∇dblt−v′
+i/ba∇dbl ift∈VR(v′
+i),
+/ba∇dblt−vi/ba∇dbl+/ba∇dblvi−v′
+i/ba∇dblift∈VR(vi)\VR(v′
+i).
+This allows us to define the merged path-length function fi:Di→Ras
+fi(s,t) :=αi(s)+d(u′
+i,v′
+i)+ωi(t),
+whereDi:= (VR(u′
+i)∪VR(ui))×(VR(v′
+i)∪VR(vi))⊆ P×P; see Figure 2(b). We consider P×P
+as a subset of R4and each pair ( s,t)∈ P ×P as a point in R4. Also, we denote by ( sx,sy) the
+coordinates of a point s∈ Pand we write s= (sx,sy) or (s,t) = (sx,sy,tx,ty) by an abuse of
+notation. Observe that
+fi(s,t) = min{lenui,vi(s,t),lenu′
+i,vi(s,t),lenui,v′
+i(s,t),lenu′
+i,v′
+i(s,t)}
+for any ( s,t)∈Diif we define len u,v(s,t) =∞whens/\e}atio\slash∈VR(u) ort/\e}atio\slash∈VR(v).
+Lemma 3 The following properties hold for the functions fi.
+(i)fi(s∗,t∗) = d(s∗,t∗)for anyi∈ {1,...,m}.
+(ii) There exists a convex neighborhood C⊂R4of(s∗,t∗)withC⊆/intersectiontextm
+i=1Disuch that
+d(s,t) = min i∈{1,...,m}fi(s,t)for any(s,t)∈C.
+(iii) Each of the functions fifori∈ {1,...,m}is convex on C.
+(iv) For any i∈ {1,...,m}, there exists a unique line ℓi⊂R4through (s∗,t∗)∈R4such that
+fiis constant on ℓi∩C. Moreover, there exists at most one index j/\e}atio\slash=isuch that ℓi=ℓj.
+(v) For any i,j∈ {1,...,m}, any(s,t)∈C, and any neighborhood U⊆Cof(s,t), there
+exists(s′,t′)∈Usuch that fi(s,t)< fi(s′,t′)andfj(s,t)< fj(s′,t′).
+Proof.(i)This immediately follows from the fact that fi(s∗,t∗) = len ui,vi(s∗,t∗).
+(ii)In this proof, we extend len u,vto any (s,t)∈ P ×P where len u,v(s,t) =∞ifs /∈VR(u) or
+t /∈VR(v). By thedefinitionof fi, thereexists asmall neighborhood Ui⊂Diof (s∗,t∗) such that
+fi(s,t) = min{lenui,vi(s,t),lenu′
+i,vi(s,t),lenui,v′
+i(s,t),lenu′
+i,v′
+i(s,t)}= min u,v∈πi∩Vlenu,v(s,t) for
+all (s,t)∈Ui. We claim that there exists an open convex neighborhood C⊂/intersectiontext
+iUisuch that
+for any ( s,t)∈C
+d(s,t) = min
+1≤i≤mfi(s,t).
+To prove our claim, assumeto thecontrary that for every open convex neighborhood C⊂R4
+of (s∗,t∗)∈R4there exist a pair ( u,v) of corners and ( s,t)∈Csuch that d( s,t) = len u,v(s,t)<
+minifi(s,t). Note that none of the shortest paths πi∈Π(s∗,t∗) between s∗andt∗pass through
+both of such uandvsince, otherwise, we must have ( u,v)∈ V(s∗,t∗) and thus ( u,v) = (uj,vj)
+for some 1 ≤j≤m. This implies that d( s,t) = len uj,vj(s,t)<minifi(s,t) = d(s,t), a
+contradiction.
+9ss′ t′tuiπi
+viδδ
+s′′t′′
+Figure 3: Illustration to Lemma 3(v); for any ( s,t)∈Diand any sufficiently small δif we pick ( s′,t′)
+such that s′isδcloser to uithansandt′isδfarther from vithant, then we have fi(s′,t′) =fi(s,t).
+Symmetrically, fi(s′′,t′′) =fi(s,t) with/ba∇dbls′′−ui/ba∇dbl=/ba∇dbls−ui/ba∇dbl+δand/ba∇dblt′′−vi/ba∇dbl=/ba∇dblt−vi/ba∇dbl−δ.
+Consider a sequence C1,C2,...of neighborhoods of ( s∗,t∗)∈R4that converges to the
+singleton {(s∗,t∗)}. Since there are only n2pairs of corners, there exist a fixed pair ( u0,v0) of
+corners and a subsequence Ck1,Ck2,...converging to the singleton {(s∗,t∗)}such that none of
+theπipass through both u0andv0, and for any integer j >0 there exists ( sj,tj)∈Ckjwith
+d(sj,tj) = len u0,v0(sj,tj)<min
+1≤i≤mfi(sj,tj).
+Sincelim j→∞(sj,tj) = (s∗,t∗), it holdsthatlim j→∞d(sj,tj) = lim j→∞minifi(sj,tj) = d(s∗,t∗)
+by Property (i). By the sandwich theorem, we have
+lim
+j→∞lenu0,v0(sj,tj) = len u0,v0(s∗,t∗) = d(s∗,t∗).
+This implies the existence of the ( m+1)-st shortest path between s∗andt∗since none of the
+πi∈Π(s∗,t∗) contains both u0andv0, a contradiction.
+(iii)Since the sum of convex functions is a convex function, it suffi ces to show that αiandωi
+are convex. More precisely, for any ( s1,t1),(s2,t2)∈Cand 0≤λ≤1, we have
+fi(λ(s1,t1)+(1−λ)(s2,t2)) =αi(λs1+(1−λ)s2)+d(u′
+i,v′
+i)+ωi(λt1+(1−λ)t2)
+≤λαi(s1)+(1−λ)αi(s2)+d(u′
+i,v′
+i)+λωi(t1)+(1−λ)ωi(t2)
+=λfi(s1,t1)+(1−λ)fi(s2,t2)
+ifαiandωiare convex.
+We now show the convexity of αion any convex subset C⊂VR(u′
+i)∪VR(ui). Note that the
+convexity of ωican be shown in the same way. There are two cases: u′
+i=uioru′
+i/\e}atio\slash=ui. For the
+former case, αiis convex on Csince it measures the Euclidean distance between uiand a given
+point in C. For the latter case, let ℓ0be the line through ui,u′
+i, and also s∗. Then,Cmay be
+partitioned by ℓ0into two regions A1andA2, whereA1=C∩VR(u′
+i) andA2=C\A1. Note
+thatαiis convex on A1and onA2. Thus, we are done by checking every point on ℓ0∩C.
+Pick any s∈ℓ0∩Cand any line ℓ⊂R2through s. Letθbe the angle between ℓ0andℓ. If
+we restrict the domain of αionℓ∩C, then one can check with elementary calculus that both
+the derivatives of /ba∇dbls−ui/ba∇dbl+/ba∇dblui−u′
+i/ba∇dbland of/ba∇dbls−u′
+i/ba∇dblare equal to ccosθatsfor some constant
+c. Hence, αiis smooth and convex along ℓ. Since we have taken any line ℓthrough any point
+onℓ0∩C, this suffices to prove the convexity of αionC.
+(iv)Fix any i∈ {1,...,m}. Any ray γ⊂R4with endpoint ( s∗,t∗)∈R4can be determined
+by three parameters ( θs∗,θt∗,λ) with 0 ≤θs∗,θt∗≤πandλ≥0 as follows: Let γs∗andγt∗be
+the projections of γonto the ( sx,sy)-plane and the ( tx,ty)-plane, respectively. Note that γs∗is
+a ray in the ( sx,sy)-plane with endpoint s∗andγt∗is a ray in the ( tx,ty)-plane with endpoint
+10t∗. Letθs∗be the smaller angle at s∗made by γs∗and another ray starting from s∗in direction
+away from ui. Define θt∗analogously with γt∗,t∗, andvi. The derivative of fiat (s∗,t∗) along
+γis represented as c(cosθs∗+λcosθt∗) for some constants λ≥0 andc >0 depending only on
+i. Note that the second derivative of fiat (s∗,t∗) alongγis derived as c/parenleftBigsin2θs∗
+/bardbls∗−ui/bardbl+λsin2θt∗
+/bardblt∗−vi/bardbl/parenrightBig
+.
+Suppose that fiis constant along γlocally around ( s∗,t∗). Then, its first and second
+derivatives along γshould be zero in a small neighborhood U⊂R4of (s∗,t∗) withU⊂Di.
+First, we observe that λshould be positive; if λ= 0, then t=t∗is fixed while smoves from
+s∗alongγs∗, and hence fidoes not stay constant. Since every term of the second deriva tive is
+nonnegative and λ >0, we only obtain two solutions ( θs∗,θt∗) = (0,π) or (π,0). Consequently,
+we have two such rays γ= (0,π,1) or (π,0,1) thatfiremains constant along γ. These two rays
+form a unique line ℓi⊂R4through ( s,t) such that fiis constant on ℓi∩U. See Figure 3 for
+more intuitive and geometric description of ℓi.
+The projections of ℓionto the ( sx,sy)-plane and the ( tx,ty)-plane appear the lines through
+s∗anduiand through t∗andvi, respectively. Hence, one can easily check that firemains
+constant on ℓi∩Di, which completes the proof of the first part of the claim.
+We now show the second part of the claim. As observed above, we have that the projection
+ofℓionto the ( sx,sy)-plane is the line through s∗andui. Also, the projection of ℓionto the
+(tx,ty)-plane is the line through t∗andvi. Hence, ℓi=ℓjimplies that ui,uj,s∗are collinear and
+vi,vj,t∗are collinear. First, since the pairs ( ui,vi) are all distinct, we have ui/\e}atio\slash=ujorvi/\e}atio\slash=vj.
+Ifui=ujandvi/\e}atio\slash=vj, then one can easily check that ℓi/\e}atio\slash=ℓjfrom geometric interpretation of
+ℓias shown in Figure 3. We hence have ui/\e}atio\slash=ujandvi/\e}atio\slash=vj. Moreover, s∗must lie in between
+uiandujandt∗must lie in between viandvjby definition; if ujlies in between uiands∗,
+then the first corner of πifroms∗becomes ujsince the three are collinear. Therefore, for each
+i∈ {1,...,m}, there is at most one index j∈ {1,...,m}such that i/\e}atio\slash=jandℓi=ℓj.
+(v)Pickany i,j∈ {1,...,m}andconsiderthesublevelsets Li={(˜s,˜t)∈R4|fi(˜s,˜t)≤fi(s,t)}
+andLj={(˜s,˜t)∈R4|fj(˜s,˜t)≤fj(s,t)}. Since fiandfjare convex and non-constant
+functions, LiandLjare closed convex sets that have ( s,t) on their boundaries. Therefore,
+there exist hyperplanes hiandhjtangent to LiandLj, respectively, at ( s,t). Leth⊕
+ibe a
+closed half-space bounded by hithat avoids LiandH′
+i:={(s′,t′)∈h⊕
+i| /ba∇dbl(s′,t′)−(s,t)/ba∇dbl= 1}
+be a closed hemisphere on the unit sphere centered at ( s,t). Define H′
+janalogously for hj.
+SinceH′
+iandH′
+jareclosedhemisphereswithacommoncenter, H′
+i∩H′
+j/\e}atio\slash=∅. Byconstruction,
+we have fi(s,t)≤fi(s′,t′) for any ( s′,t′)∈H′
+i, andfj(s,t)≤fj(s′,t′) for any ( s′,t′)∈H′
+j.
+On the other hand, by Property (iv) of the lemma, the equality holds only when ( s′,t′) lies
+on lineℓiorℓj, respectively. Therefore, for any ( s′,t′)∈(H′
+i∩H′
+j)\(ℓi∪ℓj), the claimed
+inequalities fi(s,t)< fi(s′,t′) andfj(s,t)< fj(s′,t′) hold strictly. The last task is to check that
+(H′
+i∩H′
+j)\(ℓi∪ℓj)/\e}atio\slash=∅, which follows clear by Lemma 1.
+Back to the proof of Theorem 2, we take a convex neighborhood Cof (s∗,t∗) satisfying
+Property (ii) of Lemma 3 and apply Theorem 1. Note that Proper ties (i)–(iii) of Lemma 3
+ensure that the preconditions of Theorem 1 are satisfied.
+Suppose that m <5. Then, by Theorem 1, there exists at least one line ℓ∈R4through
+(s∗,t∗) such that d is constant on ℓ∩C. Since (s∗,t∗) is a local maximum, there exists a small
+neighborhood U⊂Cof (s∗,t∗) such that d( s,t)≤d(s∗,t∗) for all ( s,t)∈U. By Property (iv)
+of Lemma 3, at most two functions fiare constant on ℓ∩U. Without loss of generality, we
+can assume that functions f3,...,fmare not constant. Since the geodesic distance function d
+is constant on ℓ∩Uand d(s,t) = min i∈{1,...,m}fi(s,t), any of f3,...,fmmust strictly increase
+in both directions along ℓ. That is, for any ( s′,t′)∈ℓ∩Uwith (s′,t′)/\e}atio\slash= (s∗,t∗) and for all
+i≥3, we have min {f1(s′,t′),f2(s′,t′)}< fi(s′,t′). Thus, there exists a small neighborhood
+U′⊆Uof (s′,t′) such that d( s,t) = min {f1(s,t),f2(s,t)}for all (s,t)∈U′. However, by
+11(s∗, t∗) (s′, t′)ℓU
+U2
+(s′′, t′′)
+Figure 4: Proof of Theorem 2 for the (I-I)case; If d( s,t) is constant on ℓ, we can find pairs of points
+(s′′,t′′) arbitrarily close to ( s∗,t∗) whose geodesic distance is larger than d( s∗,t∗).
+Property (v) of Lemma 3, there exists a pair ( s′′,t′′)∈U′such that f1(s′,t′)< f1(s′′,t′′) and
+f2(s′,t′)< f2(s′′,t′′), contradicting the maximality of ( s∗,t∗). See Figure 4. Hence, we achieve
+a bound m=|V(s∗,t∗)| ≥5, as claimed in Case (I-I)of Theorem 2.
+Case (B-B): When both s∗andt∗lie onB.In this case, we assume that s∗∈es∈Eand
+t∗∈et∈E. The outline of proof is analogous to the above discussion fo r Case(I-I); the only
+difference is that the search space has a lower dimension.
+Letpbe an endpoint of esandlsbe the length of es. We denote by s(ζs) the unique point
+onessuch that /ba∇dbls(ζs)−p/ba∇dbl=ζsfor any 0 < ζs< ls. Thus,s: (0,ls)→esestablishes a bijection
+between the open interval (0 ,ls)⊂Rand the segment es⊂R2except its endpoints. We also
+definet(ζt), analogously. Then, we let ¯fi:Di→Rbe a function defined as the composition of
+fiand the two bijections:
+¯fi(ζs,ζt) :=αi(s(ζs))+d(u′
+i,v′
+i)+ωi(t(ζt)),
+where the domain of ¯fiisDi:=s−1((VR(u′
+i)∪VR(ui))∩es)×t−1((VR(v′
+i)∪VR(vi))∩et). We
+consider Dias a subset of R2and each pair ( ζs,ζt)∈Dias a point in R2. Letζ∗
+sandζ∗
+tbe real
+numbers such that s∗=s(ζ∗
+s) andt∗=t(ζ∗
+t). We obtain the analogue of Lemma 3.
+Lemma 4 The following properties hold for the functions ¯fi.
+(i)¯fi(ζ∗
+s,ζ∗
+t) = d(s(ζ∗
+s),t(ζ∗
+t))for anyi∈ {1,...,m}.
+(ii) There exists a convex neighborhood C⊂R2of(ζ∗
+s,ζ∗
+t)withC⊆/intersectiontextm
+i=1Disuch that
+d(s(ζs),t(ζt)) = min i∈{1,...,m}¯fi(ζs,ζt)for any(ζs,ζt)∈C.
+(iii) Each of the functions ¯fifori∈ {1,...,m}is convex on C.
+(iv) If there exists a line ℓi⊂R2such that ¯fiis constant on ℓi∩C, thenuilies on the line
+supporting esandvilies on the line supporting et.
+(v) For any i∈ {1,...,m}, any(ζs,ζt)∈C, and any neighborhood U⊆Cof(ζs,ζt), there
+exists(ζ′
+s,ζ′
+t)∈Usuch that ¯fi(ζs,ζt)<¯fi(ζ′
+s,ζ′
+t).
+Note that the above claims are almost identical to those of Le mma 3. The results have been
+adapted taking into account that ¯fiis the composition of fiand both ζsandζt. Proofs follow
+verbatim, thus we omit them. Property (v) is the only excepti on: since the degrees of freedom
+have decreased, we cannot certify the existence of points ar bitrarily close that increase two
+functions ¯fi. Instead, we will use the second property of Lemma 2 to lead to a contradiction.
+Recall that by the first claim of Lemma 2 we have m≥2. Thus, we are done by showing
+that the case m= 2 is not possible. Suppose that m= 2. Then, by Theorem 1, there exists
+12a lineℓ⊂R2through ( ζ∗
+s,ζ∗
+t)∈R2such that d is constant on ℓ∩C. By the second claim of
+Lemma 2, there exists a vertex v∈Vs∗off the line supporting es. Without loss of generality,
+we assume that v=v2. By Property (iv) of Lemma 4, function ¯f2cannot remain constant in
+any line.
+Now, we proceed as in Case (I-I). Consider any small neighborhood U⊆Cof (ζ∗
+s,ζ∗
+t).
+Any point ( ζ′
+s,ζ′
+t)∈ℓ∩Cwith (ζ′
+s,ζ′
+t)/\e}atio\slash= (ζ∗
+s,ζ∗
+t) satisfies the strict inequality d( s(ζ′
+s),t(ζ′
+t)) =
+¯f1(ζ′
+s,ζ′
+t)<¯f2(ζ′
+s,ζ′
+t), since¯f2cannot remain constant and d is a local maximum. Thus, there
+exists a sufficiently small neighborhood U′⊆Uof (ζ′
+s,ζ′
+t) such that d( s(ζs),t(ζt)) =¯f1(ζs,ζt)
+for all (ζs,ζt)∈U′.
+Now, we apply Property (v) of Lemma 4 to obtain a point ( ζ′′
+s,ζ′′
+t) arbitrarily close to ( ζ∗
+s,ζ∗
+t)
+with strict inequality d( s(ζ′′
+s),t(ζ′′
+t))>d(s(ζ∗
+s),t(ζ∗
+t)) = d(s∗,t∗), contradicting the maximality
+of (s∗,t∗). We hence conclude that m=|V(s∗,t∗)| ≥3 for Case (B-B)when both s∗andt∗lie
+onB.
+Case (B-I): When s∗∈ Bandt∗∈intP.This case is a mixture of the two previous cases.
+Without loss of generality, we can also assume that s∗∈es∈Eandt∗∈intP. We define
+s(ζs) as in Case (B-B)withs(ζ∗
+s) =s∗. We now define function ˆfi:Di→Rasˆfi(ζs,tx,ty) :=
+αi(s(ζs))+d(u′
+i,v′
+i)+ωi(tx,ty), where Di:=s−1((VR(u′
+i)∪VR(ui))∩es)×(VR(v′
+i)∪VR(vi)) is
+a subset of R3.
+We obtain another analogy of Lemmas 3 and 4.
+Lemma 5 The following properties hold for the functions ˆfi.
+(i)ˆfi(ζ∗
+s,t∗) = d(s(ζ∗
+s),t)for anyi∈ {1,...,m}.
+(ii) There exists a convex neighborhood C⊂R3of(ζ∗
+s,t∗)withC⊆/intersectiontextm
+i=1Disuch that
+d(s(ζs),t) = min i∈{1,...,m}ˆfi(ζs,t)for any(ζs,t)∈C.
+(iii) Each of the functions ˆfifori∈ {1,...,m}is convex on C.
+(iv) For any i∈ {1,...,m}, there exists a unique line ℓi⊂R3through (ζ∗
+s,t∗)∈R3such that
+ˆfiis constant on ℓi∩C. Moreover, there is at most one index j/\e}atio\slash=isuch that ℓi=ℓj.
+(v) For any i,j∈ {1,...,m}, any(ζs,t)∈C, and any neighborhood U⊆Cof(ζs,t), there
+exists(ζ′
+s,t′)∈Usuch that ˆfi(ζs,t)<ˆfi(ζ′
+s,t′)andˆfj(ζs,t)<ˆfj(ζ′
+s,t′).
+We proceed as in Case (B-B). Suppose m≤3 and apply Theorem 1. Then, we obtain a line
+ℓsuch that the geodesic distance (composed with ζs) is constant on ℓ∩C. However, since at
+most two functions fican remain constant on ℓby Property (iv) of Lemma 5, there must exist a
+point arbitrarily close to ( ζ∗
+s,t∗) with strictly larger function value. Details are almost id entical
+to the previous cases, and we get the claimed bound m=|V(s∗,t∗)| ≥4 for Case (B-I).
+The claimed bounds on |Vs∗|and|Vt∗|are shown by Lemma 2, which completes the proof
+of Theorem 2.
+5 Computing the Geodesic Diameter
+Since a diametral pair is in fact maximal, it falls into one of the cases shown in Theorem 2. In
+order to find a diametral pair we examine all possible scenari os accordingly.
+Cases(V-*), where at least one point is a corner in V, can be handled in O(n2logn) time
+by computing SPM(v) for every v∈Vand traversing it to find the farthest point from v, as
+discussed in Section 2. We thus focus on Cases (B-B),(B-I), and(I-I), where a diametral
+pair consists of two non-corner points.
+From the computational point of view, the most difficult case c orresponds to Case (I-I)
+of Theorem 2. In particular, if |Vs∗|=|Vt∗|= 5, ten corners of Vare involved and thus
+13any exhaustive method would check O(n10) possibilities to find maximal pairs of this case.
+Observe that such a case can happen even under a general posit ion assumption as shown in
+Appendix A.3. By Theorem 2, in Case (I-I), it is guaranteed that there are at least five
+distinct pairs ( u1,v1),...,(u5,v5) of corners in Vsuch that len ui,vi(s∗,t∗) = d(s∗,t∗) for any
+i∈ {1,...,5}and the system of equations len u1,v1(s,t) =···= lenu5,v5(s,t) determines a 0-
+dimensional zero set, corresponding to a constant number of candidate pairs in int P×intP. On
+the other hand, each path-length function len u,vis an algebraic function of degree at most 4.
+Thus, given fivedistinctpairs( ui,vi) ofcorners, wecancomputeall candidatepairs( s,t) inO(1)
+time by solving the system.4For each candidate pair we compute the geodesic distance bet ween
+the pair to check its validity. Since the geodesic distance b etween any two points s,t∈ Pcan be
+computed in O(nlogn) time [13], we obtain a brute-force O(n11logn)-time algorithm, checking
+O(n10) candidate pairs obtained from all possible combinations o f 10 corners in V.
+As a different approach, one can exploit the SPM-equivalence decomposition ofP, which
+subdivides Pinto regions such that the shortest path map of any two points in a common
+region are topologically equivalent [8]. It is not difficult to see that if ( s,t) is a pair of points
+that equalizes any five path-length functions, then both sandtappear as vertices of the de-
+composition. However, the current best upper bound on the co mplexity of the SPM-equivalence
+decomposition is O(n10) [8], and thus this approach hardly leads to a remarkable imp rovement.
+Instead, we do the following for Case (I-I)with|Vs∗|= 5. We choose any five corners
+u1,...,u 5∈V(as a candidate for the set Vs∗) and overlay their shortest path maps SPM(ui).
+Since each SPM(ui) hasO(n) complexity, the overlay consists of O(n2) cells. Any cell of the
+overlay is the intersection of five cells associated with v1,...,v5∈VinSPM(u1),...,SPM(u5),
+respectively. Choosing a cell of the overlay, we get five (pos sibly, not distinct) corners v1,...,v5
+andaconstantnumberofcandidatepairsbysolvingthesyste mlenu1,v1(s,t) =···= lenu5,v5(s,t).
+We iterate this process for all possibletuples of five corner su1,...,u 5, to obtain a total of O(n7)
+candidate pairs, roughly spending O(n7logn) time. Note that the other subcases with |Vs∗| ≤4
+can be handled similarly, resulting in O(n6) candidate pairs.
+The validity of each candidate pair ( s,t) is examined by checking if the paths from sthrough
+uiandvitotare indeed shortest. For the purpose, we evaluate its geodes ic distance d( s,t)
+using a two-point query structure of Chiang and Mitchell [8] . For a fixed parameter 0 < δ≤1
+and any fixed ǫ >0, one can construct, in O(n5+10δ+ǫ) time, a data structure that supports
+O(n1−δlogn)-time two-point shortest path queries. The total running t ime isO(n7logn) +
+O(n5+10δ+ǫ)+O(n7)×O(n1−δlogn). We set δ=3
+11to optimize therunningtime to O(n7+8
+11+ǫ).
+Also, wecanuseanalternativetwo-point querydatastructu rewhoseperformanceissensitive
+to the number hof holes [8]: after O(n5) preprocessing time using O(n5) storage, two-point
+queries can beansweredin O(logn+h)time.5Usingthis alternative structure, thetotal running
+time of our algorithm amounts to O(n7(logn+h)). Note that this method gives a better bound
+than the previous one when h=O(n8
+11).
+The other cases can be handled analogously with strictly bet ter time bound. For Case (B-
+I), by Theorem 2, we have |V(s∗,t∗)| ≥4 and thus there are at least four distinct pairs ( ui,vi)
+of corners with len ui,vi(s∗,t∗) = d(s∗,t∗). Here, we handle only the case of |Vt∗|= 3 or 4. For
+the subcase with |Vt∗|= 4, we choose any four corners from Vasv1,...,v4as a candidate for
+Vt∗and overlay their shortest path maps SPM(vi). The overlay, together with V, decomposes
+4Here, we assume that fundamental operations on a constant nu mber of polynomials of constant degree with
+a constant number of variables can be performed in constant t ime.
+5Ifhis relatively small, one could use the structure of Guo, Mahe shwari and Sack [11] which answers a two-
+point query in O(hlogn) time after O(n2logn) preprocessing time using O(n2) storage, or another structure by
+Chiang and Mitchell [8] that supports a two-point query in O(hlogn) time, spending O(n+h5) preprocessing
+time and storage.
+14∂PintoO(n) intervals. Each such interval determines u1,...,u 4as above, and the side es∈E
+on which s∗should lie. Now, we have a system of four equations on four var iables: three from
+the corresponding path-length functions len ui,viwith 1≤i≤4 which should be equalized at
+(s∗,t∗), and the fourth from the supporting line of es. Solving the system, we get a constant
+number of candidate maximal pairs, again by Theorem 2. In tot al, we obtain O(n5) candidate
+pairs. The other subcase with |Vt∗|= 3 can be handled similarly, resulting in O(n4) candidate
+pairs. As above, we can exploit two different structures for tw o-point queries. Consequently,
+we can handle Case (B-I)inO(n5+10
+11+ǫ) orO(n5(logn+h)) time.
+In Case (B-B)whens∗,t∗∈ B, we have |Vs∗|= 2 or 3. For the subcase with |Vs∗|= 3,
+we choose three corners as a candidate of Vs∗and take the overlay of their shortest path maps
+SPM(ui). It decomposes ∂PintoO(n) intervals. Each such interval determines three corners
+v1,v2,v3formingVt∗and a side et∈Eon which t∗should lie. Note that we have only three
+equations sofar; two fromthethreepath-length functions a ndthethirdfromthelinesupporting
+toet. Sinces∗also should lie on a side es∈Ewithes/\e}atio\slash=et, we need to fix such a side esthat/intersectiontext
+1≤i≤3VR(ui) intersects es. In the worst case, the number of such sides esis Θ(n). Thus, we
+haveO(n5) candidate pairs for Case (B-B); again, the other subcase with |Vs∗|= 2 contributes
+to a smaller number O(n4) of candidate pairs. Testing each candidate pair can be done as
+above, resulting in O(n5+10
+11+ǫ) orO(n5(logn+h)) total running time.
+Alternatively, one can exploit a two-point query structure only for boundary points on ∂P
+for Case (B-B). The two-point query structure by Bae and Okamato [6] builds an explicit
+representation of the graph of the lower envelope of the path -length functions len u,vrestricted
+on∂P×∂PinO(n5lognlog∗n) time.6Since|V(s∗,t∗)| ≥3 in Case (B-B), such a pair appears
+as a vertex on the lower envelope. Hence, we are done by traver sing all the vertices of the lower
+envelope.
+The following table summarizes the discussion so far.
+Case Independent of hDependent on h
+(V-*) O(n2logn)
+(B-B) O(n5lognlog∗n)O(n5(logn+h))
+(B-I) O(n5+10
+11+ǫ)O(n5(logn+h))
+(I-I) O(n7+8
+11+ǫ)O(n7(logn+h))
+As Case (I-I)is the bottleneck, we conclude the following.
+Theorem 3 Given a polygonal domain having ncorners and hholes, the geodesic diameter
+and a diametral pair can be computed in O(n7+8
+11+ǫ)orO(n7(logn+h))time in the worst case,
+whereǫis any fixed positive number.
+6 Concluding Remarks
+We have presented the first algorithms that compute the geode sic diameter of a given polygonal
+domain. As mentioned in the introduction, a similar result f or convex 3-polytopes was shown
+in [17]. We note that, although the main result of this paper i s similar, the techniques used
+in the proof are quite different. Indeed, the key requirement f or our proof is the fact that
+shortest paths in our environment are polygonal chains whos e vertices are in V, a claim that
+does not hold in higher dimensions (even in 2.5-D surfaces). It would be interesting to find
+other environments in which similar result holds.
+Another interesting question would be finding out how many ma ximal pairs a polygonal
+domain can have. The analysis of Section 5 gives an O(n7) upper bound. On the other hand,
+6More precisely, in O(n4λ65(n)logn) time, where λm(n) stands for the maximum length of a Davenport-
+Schinzel sequence of order monnsymbols.
+15one can easily construct a simple polygon in which the number of maximal pairs is Ω( n2). Any
+improvement on the O(n7) upper bound would lead to an improvement in the running time of
+our algorithm.
+Though in this paper we have focused on exactgeodesic diameters only, an efficient algo-
+rithm for finding an approximate geodesic diameter would be also interesting. Notice that an y
+points∈ Pand its farthest point t∈ Pyield a 2-approximate diameter; that is, diam( P)≤
+2maxt∈Pd(s,t) for any s∈ P. Also, based on a standard technique using a rectangular gri d
+with a specified parameter 0 < ǫ <1, one can obtain a (1 + ǫ)-approximate diameter in
+O((n
+ǫ2+n2
+ǫ)logn) time as follows. Scale Pso thatPcan fit into a unit square, and partition P
+with a grid of size ǫ−1×ǫ−1. We define the set Das the point set that has the center of grid
+squares (that have a nonempty intersection with P) and intersection points between boundary
+edges and grid segments. We now can discretize the diameter p roblem by considering only
+geodesic distances between pairs of points of D. It turns out that the distance between any
+two points sandtinPis within a (1 + ǫ) factor of the distance between two points of D.7
+Breaking the quadratic bound in nfor the (1+ ǫ)-approximate diameter seems a challenge at
+this stage. We conclude by posing the following problem: for any or some 0 < ǫ <1, is there
+any algorithm that finds a (1 + ǫ)-approximate diametral pair in O(n2−δ·poly(1/ǫ)) time for
+some positive δ >0?
+Acknowledgements We thank Hee-Kap Ahn, Jiongxin Jin, Christian Knauer, and Jo seph
+Mitchell for fruitful discussion. We also thank Joseph O’Ro urke for pointing out the reference
+[21].
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+17APPENDIX
+A More Examples and Remarks
+In this section, we show more constructions of polygonal dom ains and their diametral pairs with
+remarks. In the figures, we keep the following rules: the boun dary∂Pis depicted by dark gray
+segments and the interior of holes by light gray region. A dia metral pair is given as ( s∗,t∗) and
+shortest paths between s∗andt∗are described as black dashed polygonal chains.
+A.1 Examples where at least one point of a diametral pair lies on∂P
+s∗
+t∗s∗t∗
+v3v2
+v1t∗ s∗
+(c) (a) (b)
+v3 v2t∗s∗
+v1
+v3 v2t∗s∗
+v1
+(e) (f) (d)v1u1
+t∗s∗
+u2 u3
+v3 v2
+(g)t∗s∗
+v1
+v3 v2
+Figure 5: (a–c) Polygonal domains whose geodesic diameter is determined by a corners∗and (d–g)
+variations of the construction (c). (a) When both s∗andt∗are corners; (b) When t∗is a point on ∂P;
+(c) When t∗∈intP. This polygonal domain consists of two holes, forming a narrow corr idor and three
+shortest paths between s∗andt∗. Here, we have d( s∗,v1) = d(s∗,v2) = d(s∗,v3) andt∗is indeed the
+vertex of SPM(s∗) defined by v1,v2,v3; (d) Variation of (c) with all convex holes; (e) Three shortest
+paths are not enough to determine a boundary-interior diametral pair; (f) If we add one more hole, then
+the diameter is determined by s∗∈ Bandt∗∈intPwith four shortest paths; (g) A polygonal domain
+made by attaching two copies of (e) and modifying it to have d( u1,v1) = d(u2,v2) = d(u3,v3). Observe
+that, in this polygonal domain, the diameter is determined by two bou ndary points with three shortest
+paths.
+Note that, as expected, every example in Figure 5 obeys Theor em 2. An interesting con-
+struction is Figure 5(g), where neither of the two centers of △u1u2u3and of△v1v2v3appears
+in any diametral pair. Also note that Figure 5(d) consists of convexholes only. We think that
+any complicated construction can be “convexified” in a simil ar fashion. This would suggest that
+computing the diameter in polygonal domains with convex hol es only might be as difficult as
+the general case.
+18A.2 A proof for Figure 1(c): Case (I-I) with 6 shortest paths
+u1u2
+u3 v3v2
+v1u1 u2 u3
+v3 v2 v1s∗t∗
+Figure 6: A schematic diagram corresponding to the polygonal domain shown in Figure 1(c).
+Claim 1 In the polygonal domain described in Figure 1(c), (s∗,t∗)is the unique diametral
+pair.
+Proof of Claim. Recall that by construction of the problem instance, the tri angles△u1u2u3
+and△v1v2v3are regular and d( u1,v1) = d(u1,v2) = d(u2,v2) = d(u2,v3) = d(u3,v3) =
+d(u3,v1) =L, for some arbitrarily large value L >0. Also, s∗andt∗are the centers of
+△u1u2u3and△v1v2v3, respectively.
+We assume that both triangles △u1u2u3and△v1v2v3are inscribed in a unit circle (and
+thus d(s∗,t∗) = 2+L). For any point son any shortest path between uiandvj, it is easy to see
+that d(s,t)≤√
+3+L <d(s∗,t∗) for every point t∈ P. In particular, no point on those paths
+cannot contribute to the diameter.
+(1) First, observe that max t∈△v1v2v3d(s∗,t) = max s∈△u1u2u3d(s,t∗) = d(s∗,t∗).
+(2) For any s∈ △u1u2u3, its farthest point t∈ △v1v2v3is on the angle bisector of some vi.
+Considerany s∈ △u1u2u3. Withoutlossofgenerality weassumethat /ba∇dbls−u1/ba∇dbl ≤mini{/ba∇dbls−ui/ba∇dbl}.
+Both shortest paths to v1and tov2fromspass through u1. We have d( s,v1) = d(s,v2) by
+constructionanditsfarthestpoint t∈ △v1v2v3mustbeintheanglebisectorof v3. Bysymmetry,
+the same property holds when the closest corner from s∗is either u2oru3.
+Conversely, for any t, its farthest point s∈ △u1u2u3must be on a bisector of some ui. In
+any diametral pair ( s,t), we have that tis the farthest point of s(and vice versa), so both must
+be on one of the angle bisectors.
+(3)If(s,t)isadiametralpair, then s∈uis∗andt∈vjt∗, forsome iandj. Supposethat slies
+on the bisector of u1but not in between u1ands∗. We then have /ba∇dbls−u2/ba∇dbl=/ba∇dbls−u3/ba∇dbl</ba∇dbls−u1/ba∇dbl
+and d(s,v1) = d(s,v2) = d(s,v3) =/ba∇dbls−u2/ba∇dbl+Lby construction. This implies that t∗is the
+farthest point of such s. Since/ba∇dbls−u2/ba∇dbl<1 and d(s,t∗)<2+L, (s,t∗) is not a diametral pair.
+(4) Now, pick any point s∈u1s∗withs/\e}atio\slash=s∗. Suppose that t∈ △v1v2v3is the farthest
+point from s. We know that t∈v3t∗by above discussions. In this case, we have four shortest
+paths between sandtthrough ( u1,v1), (u1,v2), (u2,v3), and (u3,v3); the other two are strictly
+longer unless s=s∗. By Theorem 2, such s∈u1s∗withs/\e}atio\slash=s∗and its farthest point tcannot
+form a maximal pair. By symmetry, the other cases where s∈uis∗can be handled.
+Hence, (s∗,t∗) is a unique diametral pair and the geodesic diameter is 2+ L.
+19A.3 Diametral pair of Case (I-I) with exactly 5 shortest path s
+Here, we present a polygonal domain in which the diameter is d etermined by two interior
+points and exactly five shortest paths between them. This pro ves the tightness of Case (I-I)in
+Theorem 2.
+u1=u2=u3
+v3=v4v2
+v1=v5
+s∗t∗π1π2π3
+π4π5
+u4 u5cucv
+Figure 7: A schematic diagram of a polygonal domain in which |Vs∗|=|Vt∗|= 3 and |Π(s∗,t∗)|= 5.
+Figure 7 shows a schematic description of a polygonal domain P. We assume that only
+the position of the vertices uiand theviare geometrically precise. We construct the problem
+instance such that we have u1=u2=u3,v1=v5, andv3=v4, and the convex hulls of
+theuiand of the viform isosceles triangles △uand△v. Each of △uand△vis inscribed in
+a unit circle centered at cuandcv. Moreover, the bases of both triangles are horizontal and
+the angles opposite to the bases are 18◦and 112◦, respectively. Note that the side lengths of
+the triangles △uand△vare as follows: /ba∇dblu1−u4/ba∇dbl= 1.97537···and/ba∇dblu4−u5/ba∇dbl= 0.61803···;
+/ba∇dblv2−v1/ba∇dbl= 1.11833···and/ba∇dblv1−v3/ba∇dbl= 1.85436···.
+In this configuration, we set the constants as follows: letti ngL:= d(u1,v1) = d(u3,v3) be
+some sufficiently large number, we set d( u2,v2) =L+0.5 and d( u4,v4) = d(u5,v5) =L+0.2.
+Note that this configuration can be realized with four obstac les in a similar way as Figure 1(c).
+Since we have fixed all necessary parameters, we have a fully e xplicit description of the
+lenui,vi. Due to the difficulty of finding an exact analytical solution, we used numerical methods
+to solve the system of equations len u1,v1(s,t) =···= lenu5,v5(s,t). We have foundthat there is a
+uniquesolution ( s∗,t∗) suchthat s∗∈ △uandt∗∈ △v; weobtained s∗=cu+(0,−0.102795···),
+t∗=cv+(0,0.555361···) and d(s∗,t∗) = 2.047433734 ···+L. (See Figure 7.)
+We first checked that ( s∗,t∗) is a maximal pair based on the following lemma, which can be
+shown using elementary linear algebra together with the con vexity of the path-length functions.
+Lemma 6 Suppose that (s,t)is a solution to the system lenu1,v1(s,t) =···= lenu5,v5(s,t).
+If any four of the five gradients ∇lenui,viat(s,t)are linearly independent (as vectors in a 4-
+dimensional space) and one of them is represented as a linear combination of the other four
+with all “negative” coefficients, then (s,t)is a local maximum of the pointwise minimum of the
+five functions lenui,vi.
+Next, to see that ( s∗,t∗) is a diametral pair, we have run our algorithm for each of Cas es
+(B-B),(B-I), and(I-I); as a result, there are 44 candidate pairs, including ( s∗,t∗), falling
+into those cases among which at most 11 are maximal and only ( s∗,t∗) is diametral. Note that
+the pair ( s∗,t∗) is the only candidate pair of Case (I-I). Also, observe that any point on the
+shortest path between uiandvicannot belong to a diametral pair. This implies that none of
+theuiand the vibelongs to a diametral pair. In particular, we have that none of the Cases
+(V-*)can happen. In addition, we also sampled about 350,000 point s uniformly from each of
+△uand△v, and evaluated the geodesic distances of the 350,0002pairs.
+20Note that one can modify the construction to have |Vs∗|=|Vt∗|=|Π(s∗,t∗)|= 5. For
+the purpose, we can split u1,u2,u3into three close corners (analogously for corners, v1,v5and
+v3,v4). The splitting process should preserve the differences betw een the distances d( ui,vi) for
+alli= 1,...,5 (and increase other distances). We have tested such an exam ple in the same
+way as above and concluded that a solution equalizing the five path-length functions is indeed
+a diametral pair.
+21
\ No newline at end of file
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+arXiv:1001.0696v1  [astro-ph.SR]  5 Jan 2010Astronomy& Astrophysics manuscriptno.12780 c/circlecop†rtESO 2018
+October15,2018
+Aphotometric andspectroscopic investigationofstar form ation in
+the veryyoungopen cluster NGC6383⋆,⋆⋆
+G.Rauw1,⋆⋆⋆,J. Manfroid1,†, and M.DeBecker1,‡
+GAPHE,Institut d’Astrophysique et de G´ eophysique, Unive rsit´ e de Li` ege, All´ eedu 6Aoˆ ut 17, Bˆ at B5c,4000 Li` ege, Belgium
+Received date/Accepted date
+ABSTRACT
+Context. The very young open cluster NGC6383 centered on the O-star bi nary HD159176 is an interesting place for studying the
+impact of early-type stars withstrongradiationfields and p owerful winds on the formationprocesses of low-mass stars.
+Aims.Toinvestigate this process, itis necessary todetermine th e characteristics (age, presence, or absence of circumstel lar material)
+of the population of low-mass pre-main-sequence (PMS)star s inthe cluster.
+Methods. Weobtaineddeep U BV(RI)cHαphotometric dataoftheentireclusteraswellasmedium-res olutionopticalspectroscopy
+of a subsample of X-rayselectedobjects.
+Results.OurspectroscopicdatarevealonlyveryweakH αemissionlinesinafewX-rayselectedPMScandidates.Wepho tometrically
+identifya number of H αemissioncandidates but their cluster membership is uncert ain. We findthat the fainter objects inthe fieldof
+view have a wide range ofextinction (upto AV=20), one X-rayselected OBstar having AV≃8.
+Conclusions. Our investigation uncovers a population of PMS stars in NGC6 383 that are probably coeval with HD159176. In
+addition, we detect a population of reddened objects that ar e probably located at di fferent depths within the natal molecular cloud
+of the cluster. Finally, we identify a rather complex spatia l distribution of H αemitters, which is probably indicative of a severe
+contamination by foreground and background stars.
+Key words. Stars:pre-main sequence –X-rays: stars –open clusters and associations: individual: NGC6383
+1. Introduction
+Substantial efforts have been dedicated to the investigation of
+thelow-massstarformationprocessinveryyoungopenclust ers
+harbouringa populationofearly-typestars. Manyof theses tud-
+ies were inspired by X-ray observations of these open cluste rs
+(with either Chandra orXMM-Newton ) that found a wealth of
+moderately bright X-ray sources associated with low-mass p re-
+main-sequence (PMS) stars (e.g., Prisinzano et al. 2005, Sa na
+et al. 2007). A strong and often variable X-ray emission is fr e-
+quentlyobservedforPMSstarsintheclassicalandtheweak- line
+TTauri(hereaftercTTsandwTTsrespectively)stage.Howev er,
+enhanced X-ray emission is only one characteristic of PMS
+stars; in addition to a Li overabundance, position in a colou r-
+magnitude diagram, photometric variability, H αemission, and
+near-infrared excesses that can help us identify the popula tion
+of PMS stars in a given cluster. Unfortunately,all these cri teria
+are biased to some extent: X-ray and H αemission are strongly
+variableandcouldbeweakeroversomepartofthePMSlifetim e
+or over some stellar mass ranges; the Li lines can only be mea-
+sured for rather bright objects observableusing medium res olu-
+Send offprint requests to : G.Rauw
+⋆Based on observations collected at the European Southern
+Observatory (ESO,La Silla,Chile).
+⋆⋆Table 2 is only available in electronic form at the CDS
+via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) o r via
+http://cdsweb.u-strasbg.fr /cgi-bin/gcat?J/A+A/
+⋆⋆⋆Research Associate FRS-FNRS(Belgium)
+†Research Director FRS-FNRS(Belgium)
+‡Postdoctoral Researcher FRS-FNRS(Belgium)
+Correspondence to :rauw@astro.ulg.ac.betion spectroscopy; photometric surveys are heavily a ffected by
+foregroundor backgroundobjects that are unrelatedto the c lus-
+ter, and IR-excesses are only detected for a subset of the PMS
+objects.Therefore,itisimportanttoconsiderasmanycrit eriaas
+possiblewhenidentifyingthestarsinthePMS stage.
+In this paper, we consider the case of the very young clus-
+terNGC6383.Theclusterdistancewasconsistentlydetermi ned
+to be 1.3±0.1kpc, while the reddening towards the brighter
+cluster members corresponds to E(B−V)=0.32±0.02 (for
+a review of the cluster properties, see Rauw & DeBecker 2008
+and references therein). NGC6383 is centered on the massive
+O7((f))V+O7((f))VbinaryHD159176(Linderetal.2007)and
+anXMM-Newton observation detected 76 X-ray sources (Rauw
+et al. 2003) in addition to HD159176 (De Becker et al. 2004).
+Most of these secondary sources were considered to be PMS
+stars or at least PMS candidates (Rauw et al. 2003, hereafter
+Paper I). To ascertain the status of these objects, we obtain ed
+optical spectroscopy of a subsample of the X-ray selected ob -
+jectsaswellasmulti-bandphotometryoftheclusterasawho le.
+Thepresentpaperreportstheanalysisofthese data.
+2. Observations
+Low-resolution spectra of a set of X-ray selected stars in
+NGC6383, taken from Paper I, were obtained on the night
+of June 22-23, 2004 with the EMMI instrument mounted on
+ESO’s 3.5m New Technology Telescope (NTT) at La Silla.
+The EMMI instrument was used in the Red Imaging and
+Low Dispersion Spectroscopy (RILD) mode with grism # 5
+(600 groovesmm−1), providing a wavelength coverage from
+about 3800 to 7020Å. This instrumental configuration yields2 G.Rauw et al.:Thevery young open cluster NGC6383
+Table 1.The filters used in the photometriccampaignswith the
+WFI.
+Filter ESOnumber
+UBB#U/50ESO877
+BBB#B/123ESO878
+VBB#V/89ESO843
+RcBB#Rc/162ESO844
+IcBB#I/203ESO879
+HαNB#Halpha/7ESO856
+a spectral resolution of 5.0Å ( FWHMof the He-Ar lines in the
+wavelength calibration exposures). The targets selected f or
+spectroscopy were the brightest X-ray sources found to have a
+single optical counterpart within the 8arcsec correlation radius
+used in Paper I. Their Vmagnitudes range from about 11.3 to
+17.9.Theexposuretimeswerebetween5and40mindepending
+on the magnitude of the targets. The data were reduced in the
+standard way using the longcontext of themidaspackage.
+The observing conditions were photometric over most parts o f
+the night and we thus performed a relative flux calibration of
+our spectra against an observation of the spectrophotometr ic
+standardstar LTT9239(Hamuyetal. 1992).
+Photometric observations of the cluster were obtained with
+theWideFieldImager(WFI)instrumentattheESO /MPG2.2m
+telescope at La Silla during two observing runs in September
+2003(servicemode)andJune2004(visitormode).TheWFIin-
+strumenthasafieldofviewofabout34′×33′coveredbyamo-
+saic of 4×2 CCD chips with a pixel size of 0.238arcsec /pixel.
+The frames were taken through a set of U BV(RI)cHαfilters
+(see Table1) with various exposure times (from 1 second to
+about 10 minutes, depending on the filter) to allow measure-
+ments of both faint and moderately bright objects. Our photo -
+metricdataarecompletetomagnitudesofabout20.5,22.5,2 2.0,
+21.5, 20.5, and 20.5 for the U,B,V,Rc,Ic, and Hαfilters, re-
+spectively.
+The data were bias subtracted and flat fielded using the iraf
+package. The raw images consist of dithered observations pe r-
+formedto fill thegapsbetweentheWFI CCDs. Thephotometry
+in the natural system was obtained with the daophotsoftware
+in two steps: a first pass on stacked, registered images in all
+filters allowed us to produce a deep catalogue of sources. In a
+second pass, the photometry of these sources was obtained fo r
+the individual CCDs with apertures of 4, 5, 8, and 11 pixel di-
+ameter (i.e., 0.95, 1.2, 1.9, and 2.6arcsec). The larger ape rture
+data were used to perform all-sky photometry. After a number
+of iterations, satisfactory zeropoints and extinction coe fficients
+were obtained. To determine the latter, seven Stetson ( BVRI)
+and Landolt ( BVRI) standard fields were observed during the
+samenights.Inaddition, U BVRI datafor8starsinNGC6383
+were obtained from FitzGerald et al. (1978), Th´ e et al. (198 5)
+andZwintzet al.(2005).Thedataoftheindividualframeswe re
+then homogenizedusing brightisolated stars as secondarys tan-
+dards with values determined from the all-sky photometry. T he
+astrometry was established by matching the instrumental co or-
+dinates with the 2MASS point source catalogue. The 1.2arcse c
+aperture is well suited to our analysis in terms of both accu-
+racyandsensitivitytocrowdinginthefield.Inthefollowin g,we
+use the photometrycorrespondingto this aperture,except w hen
+statedotherwise.
+ThepassbandsoftheWFIfilters,andinparticularthatofthe
+Bfilter,arerathernon-standard.Unfortunately,thelimite dnum-
+berofobservedstandardstarsdidnotcoverasu fficientrangeofFig.1.Comparison between the B−Vcolour index in the WFI
+systemandthestandardsystemfor156starsselectedastheb est
+observed among those in common with Paunzen et al. (2007).
+Thesolidlineyieldsourbest-fitcolourtransformation.
+spectral types to establish precise colour transformation s from
+these standards. Therefore, we compared our WFI photometri c
+datain thenaturalsystemwith publishedJohnson-Cousinsp ho-
+tometry of stars in common in NGC6383. In this context, we
+established a colour transformation (Fig.1) based on the co m-
+parison between our VWFImagnitudes and ( B−V)WFIcolours
+and the standard BVphotometryfor 156 stars in common with
+the (online)Table1 of Paunzenet al. (2007). The resultingr ela-
+tionsare1
+(B−V)std=−0.227+1.418(B−V)WFI,
+Vstd=VWFI+0.053−0.101(B−V)WFI.
+Thecoefficientsofthecolourtermsoftheserelationsareinrea-
+sonable agreement with those proposed by ESO on the WFI in-
+strument website2. Unfortunately, there are much fewer avail-
+abledataforthecalibrationofthe U,Rc,andIcfilters.Wethere-
+fore adopted the coe fficients of the colour terms following the
+transformations given by ESO and only determined the zero-
+pointsby comparingthe WFI withthe existingstandardcolou rs
+ofsevenstars. Theresultingrelationsare
+(U−V)std=1.08(U−V)WFI+0.002+0.02(B−V)std,
+(V−Rc)std=0.98(V−R)WFI+0.017−0.09(B−V)std,
+(V−Ic)std=0.94(V−I)WFI+0.023−0.08(B−V)std.
+1Thezeropoints intheserelationsonlyapplytoourreductio nofthis
+specific data set and cannot be compared to other reductions o f other
+WFIdata.
+2http://www.eso.org/sci/facilities/lasilla//instruments/wfi/inst/zero-
+points/ColorEquationsG.Rauw et al.:Thevery young open cluster NGC6383 3
+Fig.2.Left and middle panel: flux-calibrated spectra of the F-K typ e stars in our spectroscopic sample of NGC6383. The fluxes
+were arbitrarily normalized to unity at the wavelength of 68 20Å and the spectra were shifted vertically by 0.4 units for c larity.
+The spectral types become progressivelylater from left to r ight and from top to bottom. The rightmost panel shows a zoomo n the
+spectralregionaroundtheH αlineforthosestarsthatareemittersoremissioncandidate s.
+In the following, we applied these transformations to all of our
+data.Unlessstatedotherwise,allcoloursgiveninthispap errefer
+tothetransformedcolours.3
+Because ofits specialpassbanddefinitionandalack ofsuit-
+able standard fields, the H αfilter in use at the WFI cannot be
+directlytied to anyexisting standardphotometricsystem. In the
+naturalsystem,the Rc−Hαzeropointwasatfirstfixedarbitrarily
+to zero for the standard stars. In a second step, we re-calibr ated
+the zeropoint by comparing the observed Rc−Hαand dered-
+denedV−Icindicesof 9 stars whose spectra do not exhibitH α
+emissionwiththe Rc−HαversusV−Icrelationofemission-free
+main-sequence stars in NGC2264 as determined by Sung et al.
+(1997).Toexpressthe Rc−Hαindicesinthesystem ofSunget
+al. (1997), we need to subtract (4 .71±0.06) from the observed
+indices. All values of the Rc−Hαindex quoted in this paper
+and all plots involving this index refer to the Sung et al. (19 97)
+system.
+3. Spectroscopy
+To establish the MK spectral types of the targets in NGC6383,
+weusedthedigitalspectralclassificationatlascompiledb yR.O.
+Gray available on the web4. The details of the classification are
+given for each star in AppendixA. The results are summarized
+inTable3.
+The majority of the spectroscopically studied objects are
+late-type (F-K) main-sequence or slightly more luminous st ars
+(seeFig.2andTable3).TheexceptionsarethemidB-typesta r5
+#56andtheearly-typeB star#76(see Fig.3).
+3Table2withthestandardphotometricdataofthosesourcest hatare
+detected inall filtersisavailable at the CDS.
+4http://nedwww.ipac.caltech.edu /level/Gray/frames.html
+5Throughout this paper, we shall use the numbering conventio n in-
+troduced inPaper Itodesignate theX-raysources inthe clus ter,except
+whenstatedotherwise.3.1. Late-type stars
+For most stars of our spectroscopic sample, H αis detected in
+absorption. The exceptions are stars #20, 25, 50, and possib ly
+#57 and 39. We checked the existence of these emission lines
+onthe originaltwo-dimensionallong-slitspectra toavoid incor-
+rectly identifying a cosmic ray hit or poorly corrected nebu lar
+Hαemission. Although nebular emission exists around all tar-
+getsofoursample,it isremovedbysky subractionandthe neb -
+ular line is too weak to account for the strength of the observ ed
+stellar emission features.We emphasizethat the strongest emis-
+sion is seen in the spectrum of the counterpart of X-ray sourc e
+#50. This object experienced an X-ray flare during the XMM-
+Newtonobservation (see Paper I). The situations for stars #57
+and #39 are not entirely clear. In the former, we observe two
+weakemissionpeaksbluewardsofthe rest-framewavelength of
+theHαlinewithheliocentricvelocitiesof −564and−71kms−1.
+Although our inspection of the 2D-spectra confirmed that the se
+featuresare not artefacts, the detection of this blueshift ed emis-
+sion structureneedsto be confirmedbyadditionalobservati ons.
+Finally, in the spectrum of star #39, a broad absorption feat ure
+is observed blueward of the H αrest wavelength, while the line
+itself seemsto befilledin byemission.
+All spectroscopically observed objects with H αemission in
+their spectrumare of spectral type K (see Fig.2), while all s tars
+ofearlierspectraltypedisplaytheH αlineinabsorption.Wealso
+notethattheequivalentwidths(EWs)oftheseemissionline sare
+rathermodest.Forinstance,theyaremuchweakerthanthe10 Å
+equivalentwidthconventionallyadoptedasthelimitbetwe enthe
+objects of the wTTS and cTTs categories (e.g., Preibisch et a l.
+2001).Therefore,theseobjectsarecandidatewTTs.
+Although the spectral resolution of our data is insu fficient
+to resolve the possible Li iλ6708 line from the neighbouring
+Feilines, we note that the spectra of the majority of the targets
+later than spectral type F (especially stars #1, 16, 25, 39, 5 5,
+57, 65, 67, and 70) display a rather strong absorption featur e at
+the wavelength of the Li iline. The unresolved blend with the
+Feilinesprobablyaccountsforthe asymmetricshapeofthe ob-
+servedprofiles.ThepossibledetectionofLiinthespectrao fthe4 G.Rauw et al.:Thevery young open cluster NGC6383
+late-type stars of our sample is compatible with a PMS status .
+We note that all spectra (except #56 and #76) display a signifi -
+cant absorptionfeatureat about6104Å. Thisfeatureis unli kely
+to be associated with the Li line observedin PopulationII st ars,
+whoseintensityshouldbeveryweak(seee.g.,Fordetal.200 2).
+ThisabsorptioncouldinsteadbecausedbyablendofCa iλ6103
+withFei.
+Finally, we emphasize that star #55 ( =FJL23) was previ-
+ously suggested to be a foreground object (Th´ e et al. 1985;
+Rauw et al. 2003), but this conclusion is not confirmed by our
+present analysis. The photometric and spectroscopic prope rties
+of this star are indeed consistent with it being a member of
+NGC6383.
+3.2. Early-type stars
+The OB star #76 appears heavily absorbed and must therefore
+belong to a population of stars located behind NGC6383. Our
+photometricmeasurementsindeedyield E(B−V)≃2.5forthis
+star.TheNGC6383clusterisunderstoodtobelocatedinfron tof
+a dust cloud(LloydEvans1978) and star #76is thereforelike ly
+a backgroundobjectlocatedeitherinsideorbehindthisclo ud.
+Fig.3.Flux-calibrated spectra of the two early-type stars in our
+spectroscopic sample of NGC6383. Star #76 probably has an
+earlier spectral type than #56, but is much more absorbed. Th e
+fluxesarepresentedinthesame wayasinFig.2.
+Thecaseofstar#56( =FJL24,FitzGeraldetal.1978)isalso
+quite interesting. Mid to late-type B stars are usually not s trong
+X-rayemitters.UsingthebolometriccorrectionforaB5-7V star
+and the X-ray flux from Paper I, we derive a log LX/Lbolvalue
+of−5.13±0.11 for this star. To evaluate the X-ray luminosity,
+we converted the EPIC-pn count rate given in Paper I into an
+unabsorbed flux of (8 .2±1.9)×10−14ergcm−2s−1in the 0.5 –
+10keV band,assuminga thermalplasmaat a temperatureofkT
+=0.5keV and an absorptionby an interstellar neutral hydroge n
+columnof1.91×1021cm−2.WenotethatourestimateoftheX-
+rayfluxwouldchangebylessthan20%regardlessofthetemper -ature of the emittingplasma in the range0.3 – 2keV. Assuming
+a distance of 1.3kpc, the resulting X-ray luminosityamount sto
+(1.7±0.4)×1031ergs−1,whichplacesthisobjectclearlyamong
+theX-raybrightlateB-typestars(seee.g.,Bergh¨ ofereta l.1997,
+Sanaetal.2006b,andNaz´ e2009).Wenotethatthisvalueoft he
+X-ray luminosity would also be rather high if the emission wa s
+originatinginan unresolvedPMS companion.
+4. Photometry
+Photometric data for a total of about 80000 sources were ex-
+tracted. We have cross-correlated these data with the 2MASS
+near-IRcatalogue(Skrutskieetal.2006)andfoundabout18 000
+matches within 0.5arcsec, 7000 of which have good quality
+near-IR photometry. Among the latter, 2639 objects have rel i-
+ableWFI photometryinall filtersincludingtheH αfilter.
+Cluster members fainter than V∼11 are expected to be
+pre-main-sequence objects. Unfortunately, these objects popu-
+latearegionofthecolour-magnitudediagramsthatispoten tially
+highly contaminated by field stars unrelated to NGC6383 (see
+Fig.4). Since the cluster is probably located in front of a du st
+cloud, we expect the background objects to appear redder tha n
+thetrueclustermembers.
+4.1. Cross-correlation with theX-raycatalogue
+We cross-correlated the catalogue of X-ray sources from Pap er
+I with the list of sources detected in our photometric data. T o
+derive the optimal correlation radius, we applied the techn ique
+of Jeffries et al. (1997). In this approach, the distribution of the
+cumulative number of catalogued sources as a function of the
+cross-correlationradius rXisgivenby
+Φ(d≤rX)=A1−exp−r2
+X
+2σ2
++(N−A)/bracketleftig
+1−exp(−πBr2
+X)/bracketrightig
+,
+whereN,A,σ, andBindicate, respectively, the total number
+of cross-correlated X-ray sources ( N=76), the number of true
+correlations, the uncertainty in the X-ray source position , and
+thesurfacedensityofthecatalogueofphotometricsources .The
+best-fit modelparametersare A=57.8,σ=1.8arcsec,and B=
+1.22×10−2arcsec−2.Thelattervalue(independentlydetermined
+fromthebestfittothecross-correlationprocess)isinreas onable
+agreementwiththetruemeansurfacedensityofourphotomet ric
+catalogue(1.8×10−2arcsec−2).
+The optimal correlation radius that includes the majority o f
+the true correlations whilst simultaneously limiting cont amina-
+tionbyspuriouscoincidencestoabout15%,isfoundtobeabo ut
+5arcsec. The following X-ray sources have no optical counte r-
+part in our photometric data: #4, 5, 18, 19, 21, 26, 28, 40, 71,
+and #72. Some of the sources (e.g. #14 and #36) which had no
+counterpart in the USNO catalogue (see Paper I), now have a
+faint optical counterpart in our data. We note that HD159176
+(X-ray source #44) is too bright to be reliably measured in ou r
+photometric data and was thus not included in the correlatio n
+process.
+Thecolour-magnitudediagramsshowninFig.4indeedcon-
+firm that the bulk of the X-ray sources with an optical counter -
+partpopulatealocuswherePMSstarsareexpected.Wenoteth at
+thesediagramsalsoincludethe44PMScandidatesidentified by
+Paunzenetal. (2007)onthegroundsofphotometriccriteria .G.Rauw et al.:Thevery young open cluster NGC6383 5
+Table 3.Summaryof the propertiesof the targetsof our spectroscopi csample.The last columnyieldsthe cross identificationwit h
+objectspreviouslystudiedbyFitzGeraldet al.(1978) aswe llassomeinformationaboutobjectswith H αemission.
+Star Spectral Type H α V B−V V−IcRc−Hα Notes
+#1 K0V abs. 15.04 1.16 1.45 −4.70
+#3 F5V abs. 13.51 0.77 1.07 −4.77
+#16 G0–5V–IV abs. 14.96 1.10 1.37 −4.72
+#20 K0V em. 15.84 1.25 1.67 −4.72 EW=−1.0Å,v=85kms−1
+#25 K2V em. 15.09 1.25 1.62 −4.54 EW=−0.5Å,v=5kms−1
+#39 K2–4V P-Cyg? 15.15 1.25 1.54 −4.70 =FJL9
+#50 K7V em. 17.88 1.61 2.09 −4.65 EW=−3.0Å,v=−14kms−1
+#54 G7V abs. 12.77 0.85 1.00 −4.71
+#55 G0–1IV abs. 13.81 0.97 1.20 −4.72 =FJL23
+#56 B5–7V abs. 11.35 0.26 0.40 −4.85 =FJL24
+#57 K0–5III double peak em.? 15.49 1.25 1.56 −4.72EW=−0.3Å,v=−564,−71kms−1
+#65 G5IV–III abs. 14.02 1.05 1.30 −4.72
+#67 G5–K0V abs. 14.21 0.91 1.12 −4.65
+#70 G5–K0IV–III abs. 15.19 1.08 1.33 −4.75
+#76 O9–B5 abs. 17.81 2.21 2.89 E(B−V)≃2.5
+Fig.4.Colour-magnitudediagramsoftheWFIphotometricdatawith errorsoflessthan0.15on Vandlessthan0.21intherelevant
+colour.SingleandmultiplecounterpartsoftheX-raysourc esfromPaperIareshownasfilledandopensquares,respectiv ely,while
+the PMS candidatesfromPaunzenet al. (2007) areshownas ope ncircles. Themain-sequencerelationwasshifted to accoun tfor a
+distance modulus of 10.57 and a reddening of E(B−V)=0.32. The reddening vectors are shown for each diagram and the s hort
+andlong-dashedlinesyieldthe1.5and10Myrisochrones,re spectively,fromSiess etal. (2000).
+4.2. Nearinfraredphotometry
+Based on the cross-correlation with the 2MASS data, we can
+compile a near-IR colour-colour diagram. For this purpose, we
+applied the colour transformationsof Carpenter (2001, upd ated
+in 20036) to convert the 2MASS colours into the homogenized
+JHKphotometricsystemofBessell&Brett(1988).Theresults
+are illustrated in Fig.5. In the left panel of this figure, we s how
+theJHKcolour-colourdiagram.Theheavysolidlinesyieldthe
+intrinsic coloursof main-sequencestars and giantsaccord ingto
+Bessell & Brett (1988), while the dashed straight line yield s the
+locus of dereddenedcolours of classical TTauri stars accor ding
+toMeyeret al. (1997).
+6http://www.ipac.caltech.edu /2mass/index.htmlWithl=355.69◦andb=+0.04◦, the NGC6383 field lies
+just outside the area investigated by Nishiyama et al. (2009 )
+in their study of the extinction law towards the Galactic cen -
+tre.Nevertheless,we assumeherethattheirresultsalsoap plyto
+our targets. In doing so, we find that the apparent reddening o f
+the objects in the field of NGC6383 spans a very wide range:
+about 2.4 in AKs, which corresponds roughly to 20 magnitudes
+inAV. It is most interesting that there appears to be a continu-
+ousdistributionofreddeningvalues.Wealreadypointedou tthat
+the cluster is likelylocated at the frontside of a largemole cular
+cloud.TheresultsinFig.5implythatweobserveapopulatio nof
+sources from different depths inside this molecular cloud. This
+situation compromises any attempt to define a straightforwa rd
+clustermembershipcriterionbasedonthereddening.6 G.Rauw et al.:Thevery young open cluster NGC6383
+Fig.5.Left panel: colour-colourdiagramof the 2MASS counterpart sof the WFI objects. Only those sources with reliable near-I R
+photometry are shown. The solid lines yield the locus of the m ain-sequence and of the giant branch according to Bessell & B rett
+(1988),whilethedashedstraightlineindicatesthelocuso ftheunreddenedcTTsfollowingMeyeretal.(1997).Theredd eningvector
+is shown for AKs=0.5. Middle panel: Locationof the X-raysources fromPaper I (fi lled and open squares),PMS candidatesfrom
+Paunzen et al. (2007, open circles) and H αemitters (×symbols) and candidates ( +symbols) in the JHKcolour-colour diagram.
+The other symbolshave the same meaningas in Fig.4. Right pan el:colour-magnitudediagram.The main-sequenceis shown f or a
+distancemodulusof10.57andforthreedi fferentvaluesofthereddening(fromlefttoright, AKs=0.0,0.5,and1.0).
+In the middle panel of Fig.5, we have also plotted the near-
+IR counterparts of those stars that are either selected as X- ray
+sources,PMScandidatesfromPaunzenetal.(2007),orH αemit-
+ters(see nextsubsection).Thisfigureshowsthat thevast ma jor-
+ityofthe X-raysourcesaresubjecttoa rathermoderateredd en-
+ing. At the same time, we find no obvious correlation between
+Hαemissionandtheexistenceofanear-IRemission:themajor-
+ityofthepossibleH αemitters(seeSect.4.3)haverathernormal
+near-IRcolours.
+Finally, the rightmost panel of Fig.5 displays the colour-
+magnitudediagram,where the main-sequencerelation has be en
+shifted for a distance modulusof 10.57. Again, the existenc e of
+a wide, continuousrange of interstellar reddeningis quite obvi-
+ous. We note that the reddening of the bright cluster members
+(E(B−V)=0.32)correspondsroughlyto AKs=0.11.
+4.3. Hαemissioncandidatesidentifiedfrom photometry
+AccordingtoourcurrentunderstandingofclassicalTTauri stars,
+these objectsaresurroundedbya circumstellardiskthat is trun-
+catedat itsinneredgebythestellar magnetosphere.Most of the
+time, cTTs accrete at a rate well below 10−6M⊙yr−1and their
+Hαemission originates in accretion streams that are controll ed
+bythemagneticfield(see e.g.,Schulz2005).
+Weusedthe Rc−Hαindex,calibratedagainsttheSungetal.
+(1997)Rc−HαversusV−Icmain-sequencerelation(seeSect.2),
+toidentifypotentialH αemitters.TherelevanceoftheSungetal.
+(1997)relationforourdatasetwascheckedusingsynthetic pho-
+tometry, obtained by numerical integration of the spectrop ho-
+tometric data of main-sequence stars from Jacoby et al. (198 4)
+through the WFI VandHαfilter passbands. The operation was
+performed for stars of spectral types from B4V to M5V. Our
+synthetic photometryis in excellentagreementwith the Sun get
+al. main-sequence relation. We then repeated the integrati on by
+addinganHαemissionlineofincreasingequivalentwidthtothe
+Jacoby et al. (1984) spectra. In this way, we found that an H α
+equivalentwidthof10Å,whichisusuallyadoptedasthethre sh-oldforcTTs,correspondstoan Rc−Hαindex0.24±0.04above
+themain-sequencerelation.
+We therefore consider a star as an H αemission candidate if
+itsRc−Hαcolour index is between 0.12 and 0.24 above the
+main-sequence relation (corresponding roughly to an EW be-
+tween 6 and 10Å) and as an H αemitter if the corresponding
+colour index is more than 0.24 above the main-sequence rela-
+tion. In this way, we arrive at a list of 924 H αcandidates and
+592 emitters. If we restrict ourselves to stars with photome tric
+errorsof less than 0.15mag in Vand less than 0.21in Rc−Hα,
+these numbers become 475 and 189 for the H αcandidates and
+emitters,respectively.The0.06maguncertaintyinthezer opoint
+oftheRc−Hαindextranslatesintoanuncertaintyofafactor1.8
+inthe numbersofemittersandcandidates.
+Since the objects in our data appear to have a range of ex-
+tinction values, we checked that our selection criterion fo r Hα
+emissionisnotbiasedbyfalsedetectionswhichcouldbecau sed
+by a heavyreddeningthat producesan artificially high Rc−Hα
+index.Todoso,weevaluatedthe(Rc+Ic)
+2−Hαcolourindex,which
+hasadifferentsensitivitytoreddeningthanour Rc−Hαcriterion.
+Thevastmajorityofobjectsselectedwiththe Sunget al.(19 97)
+criterion are also identified as candidates or emitters acco rding
+tothe(Rc+Ic)
+2−Hαcolour.
+From Fig.6, we see that neither the Paunzen et al. (2007)
+PMS candidatesnor the X-ray sources display strong H αemis-
+sion:thereareonlyhalfadozenX-raysourcesintheH αemitter
+region.Wenotethatthoseobjectsforwhichourspectroscop yre-
+vealed (rather weak) H αemission are not identified as emitters
+in our photometric data. This suggests that wTTs will be di ffi-
+cultto identifywith narrow-bandphotometry(seealso Dahm &
+Simon 2005 for a comparison of the e fficiency of narrow-band
+photometryandslitless grismspectroscopytoidentifywea kHα
+emitters).
+Whilst many emission candidates and emitters are rather
+faintobjects,weseethatH αemissionisapparentlynotrestricted
+to only the faintest (and hence least massive) stars. A few ob -
+jects asbrightas V∼10 – 11are identifiedas H αemitters.The
+brightest is the Be star HDE317861 (B3 - 5Vne, Lloyd EvansG.Rauw et al.:Thevery young open cluster NGC6383 7
+Fig.6.Left:theRc−Hαindexisshownasafunctionofthe V−Iccolour.ThesymbolshavethesamemeaningasinFig.4.Thesol id
+line indicatesthe relation for main-sequencestars as take n fromSung et al. (1997) and reddenedby E(V−Ic)=0.512.The short-
+andlong-dashedlinesyieldthethresholdsforH αcandidatesandemitters,respectively.Right: VversusRc−Hαcolour-magnitude
+diagram.The+and×symbolsrepresentH αemittercandidatesandemitters,respectivley.
+1978)with V=9.76andR−Hα=−4.32.WhilsttheHerbigAe
+starFJL4(=V486Sco,A5IIIp,Th´ eetal.1985)isclassifiedas
+acandidateHαemitterwithourcriteria,indicationsofemission
+are found in neither our photometric nor spectroscopic data of
+FJL24 (=X-ray source #56, B5-7). We note that the latter ob-
+ject was classified as B8Vne by FitzGerald et al. (1978), sinc e
+one observation out of six analysed by these authors display ed
+Hβemission. Our spectrum of this star exhibits strong Balmer
+absorptionlines(seeFig.3).
+The presence of H αemission does not automatically imply
+that a star is a member of the cluster. For instance, our sampl e
+ofHαemittersandcandidatesmaycontainforeground,magnet-
+ically active, dMe stars, unrelated to the cluster. The inci dence
+of dMe stars (with an H αEW>1Å) increases towards lower
+masses and hence later spectral types (see Hawley et al. 1996
+and referencestherein):the observedfrequencyis about10 % at
+spectral typesearlier thanM2, but increasestowardsabout 60%
+aroundspectral typeM5. Furthermore,the dMestars of the ea r-
+lierspectraltypesappearonaverage0.5magbrighterin MVand
+MKthan their dM counterparts (Hawley et al. 1996). An esti-
+mate ofthe numberofforegrounddMestars canbe obtainedby
+following the same approach as Dahm & Simon (2005), based
+on the stellar luminosity function of Jahreiß & Wielen (1997 )
+and the dMe incidence of Hawley et al. (1996). In this way, we
+arrive at an estimated numberof 151 foregrounddMe stars. We
+note that this number is an upper limit, since the reddeningd ue
+tointerstellarmaterialalongthelineofsighttowardsNGC 6383
+wasnottakenintoaccountin thisevaluation.
+A strong contaminationby foregrounddMe objectsis a pri-
+oriexpected to produce a rather uniform spatial distribution o f
+Hαselected stars across the field of view. However, the spatial
+distribution of the H αemitters and candidates is far from uni-
+form (see Fig.7). Indeed, the density of emitters is higher t o-
+wards the north of the field of view and especially in the north -westernpart,whilethereisapaucityofH αemittersinthesouth-
+west and to the east of HD159176. This can be quantified for
+instance by comparingthe north-westand south-east quadra nts:
+theformercontainsa totalof1951objectswithgoodphotome t-
+ricdata,18%ofwhichareeitheremittersorcandidates,whe reas
+the latter contains 3056 good quality objects only 3% of whic h
+are selected as Hαemitters or candidates.The lack of H αemit-
+ters and candidates in the south-east is particularly surpr ising
+because this region has the highest density of stars. Theref ore,
+the paucity of emitters cannot be attributed to the e ffect of in-
+terstellar absorption. Although projection e ffects could play a
+role, we note that unlike our findings for X-ray emitters (see
+Paper I), there is no clear concentration of H αselected stars in
+the cluster core7. Outside the core, the mean spatial density of
+Hαemitters and candidates are 0.08 and 0.26arcmin−2, respec-
+tively. Taken together, these numbers are higher than expec ted
+from the predicted maximum number of foreground dMe stars
+(0.13arcmin−2), indicating that the distribution of H αemitters
+andcandidatesmustberelatedtothepropertiesoftheclust eror
+itsbackground.
+What is the reason for the lack of accreting PMS stars in
+the south-westernandeasternpartsofthe haloofNGC6383,a s
+inferred from the position of H αemitters and candidates? One
+possibilityisphotoevaporationoftheprotoplanetarydis ksbyei-
+thertheextreme-UV(EUV)andfar-UV(FUV)irradiationfrom
+the O-type binary HD159176 or the EUV, FUV and X-rays of
+the central PMS star itself. The formere ffect was predictedthe-
+oretically, and evaporating disks with cometary tails were ob-
+served with Spitzerin three cases at distances between 0.1 and
+0.35pc from an O-star (Balog et al. 2006, 2008 and references
+therein). However, in NGC6383, several H αemitters are rela-
+7Kharchenko etal.(2005)estimateacore radius of4.8arcmin anda
+corona radius of 15arcminfor NGC6383, while Piskunovet al. (2008)
+evaluate a tidalradius ofat least 30arcmin.8 G.Rauw et al.:Thevery young open cluster NGC6383
+Fig.7.Left: three-colourimage of the NGC6383 cluster compiled fr omour WFI photometry.Right: spatial distribution of the H α
+emitters (×signs) and candidates ( +signs). The large dot in the centre of the field indicates the p osition of HD159176, while the
+opencirclesrepresenttheX-raysourcestakenfromPaperI. Northisat thetopandeast isontheleft.
+tivelyclosetotheO-starbinary.Wefound8H αemittersand19
+candidates within 5armin of HD159176, which corresponds to
+alinearradiusof1.9pcatthedistanceof1.3kpc.Therefore ,we
+would have to assume that the photoevaporation occurs prefe r-
+entiallyalongcertaindirectionsanda ffectstheclusterhalomore
+efficientlythanthecore,whichisratherunlikely.Concerning the
+secondeffect,Gorti&Hollenbach(2009)predictphotoevapora-
+tion of the accretion disk by irradiation of the PMS star with in
+a rather shorttimescale ofabout 1Myr. If the emission measu re
+of the X-ray emitting plasma correlates with the EUV emissio n
+measure of the PMS star, this second scenario could, at least
+qualitatively,explainwhythemostprominentX-rayemissi onis
+observed from wTTs. This photoevaporationcaused by irradi a-
+tion by the central star itself should however a ffect PMS stars
+irrespectiveoftheirpositioninthecluster.Thisscenari oaloneis
+therefore insufficient to explain the complex distribution of H α
+emitters.
+An alternative could be that our sample is heavily contam-
+inated by background objects. For instance, we could be ob-
+serving the edge of another star-forming region to the north of
+HD159176.We testedthispossibilitybyconsideringthespa tial
+distributionofobjectswithhigh J−Kindices(J−K>2.5)that
+hence are potentially highly reddened background stars. Th is
+distributionrevealsalackofobjectsinthesouth-west,bu tisoth-
+erwiseratheruniform.Therefore,whileit islikelythatou rsam-
+ple is highlycontaminatedby a populationof backgroundsta rs,
+we see no peculiar pattern in the spatial distribution of hig hly
+reddenedstars.
+5. TheHertzsprung-Russelldiagram
+From the various colour-magnitude diagrams presented in th e
+previoussection,wefindthattheX-raysourcesandthePaunz en
+etal.(2007)PMScandidates(hereafterthePNZobjects)are the
+best candidatecluster members,whereasthe H αemission crite-
+riaselectapopulationthatcouldcontainasignificantfrac tionof
+backgroundobjectsaswellassomefield dMestars.Assuming a distance modulus of 10.57 and a reddening
+E(B−V)=0.32fortheclustermembers(seeRauw&DeBecker
+2008 and Sect.1), we built the Hertzsprung-Russell diagram of
+the X-ray sources, the PNZ objects, and the H αselected stars.
+To do so, we evaluated the e ffective temperatures and bolomet-
+ric correctionsby meansof a linear interpolationof the Ken yon
+&Hartmann(1995)tablesasafunctionof( V−Ic)0.Theresults
+areshowninFig.8.WhilstsomeoftheH αselectedobjectsform
+acontinuationofthelocusoftheX-rayandPNZobjects,them a-
+jorityoftheHαemittersarelocatedbelowthislocus.Thisresult
+mustbe causedbythe contaminationof ourH αsamplebyfore-
+grounddMestarsorbackgroundobjectslocatedinsideorbeh ind
+the molecular cloud. Since our attempts to establish an e fficient
+and self-consistent membership criterion failed, we inste ad ap-
+plied a rough selection based upon the J−Kcolour and the
+position with respect to the cluster core. By selecting only stars
+withJ−K<1.5, which is roughly equivalent to AK<0.54,
+and within the 5arcmin radius of the cluster core, we obtain a
+cleanerHertzsprung-RusselldiagramwheretheH αemittersare
+now mainly in a lower mass continuation of the locus of PNZ
+andX-rayselectedPMS stars.
+There are a number of available theoretical PMS tracks that
+we can compare with our data. The various evolutionary codes
+differ in many respects and the most important di fferences con-
+cerntheopacities,theequationofstate,thetreatmentoft hestel-
+lar atmosphere, and most importantly the treatment of conve c-
+tion (see the discussions in Siess et al. 2000). Here, we chos e
+to compare our data with the results of four di fferent models:
+the 1997 version of the tracks of D’Antona& Mazzitelli (1997 ,
+hereafter DM97), the Siess et al. (2000, hereafter SDF00) ca l-
+culations, and both the standard and rotating, models of Lan din
+et al. (2009, hereafterLMV09).Whilst the DM97computation s
+use thefull spectrumof turbulence(FTS)approachto treat c on-
+vection, the other calculations considered here rely inste ad on
+the mixing length theory (MLT). These di fferent treatments ob-G.Rauw et al.:Thevery young open cluster NGC6383 9
+Fig.8.Top: Hertzsprung-Russell diagram of the X-ray sources,
+PNZ objects, Hαemitters, and candidates. The PMS evolution-
+ary tracks are taken from Siess et al. (2000) for masses of 0.2 ,
+0.3,0.4,0.5,0.7,1.0,1.5,2.0,2.5,3.0,4.0,5.0,6.0,and 7.0M⊙.
+The thick solid line illustrates the zero-age main-sequenc e and
+the dashed lines yield isochrones for ages of 0.5, 1.5, 4.0, 1 0.0,
+and 20.0Myr. The symbolshave the same meaning as in Fig.6.
+Bottom: same, but with the selection criteria J−K<1.5 and
+positionwithin the5arcminradiusofthecluster coreappli edto
+theHαemittersandcandidates.
+viously lead to quantitative di fferences in the PMS tracks, and
+oneexpectsconflictingresultswhenempiricallydetermini ngthe
+agesofobservedPMSstars.Forinstance,D’Antona&Mazitel li
+(1997)arguethattheFTSapproachgenerallyyieldsalowera ge
+spread than the MLT. With these limitations in mind, we com-piledthe histogramsof theagesofthe PMS candidatesbycom-
+paringwiththeisochronesofthevariousmodels(seetheres ults
+in Fig.9 and Table4). We emphasizethat the agedistribution of
+theHαselectedstarswouldactuallybemeaninglessbecause,as
+stated above, this selection criterion probably fails to pr operly
+identifyclustermembers.
+Table4.Meanagesandstandardagedeviations(inMyr)forthe
+varioussamplesofPMS candidatesin NGC6383.
+X-rayselected PNZobjects
+mean stand. dev. mean stand. dev.
+DM97 1.9 2.5 2.6 4.0
+SDF00 3.8 3.0 2.8 2.9
+LMV09std. 2.1 2.6 1.9 2.9
+LMV09rot. 1.7 2.3 1.8 3.0
+Figure9 shows that the DM97 isochrones indeed yield a
+strongly peaked age distribution (especially for the PNZ ob -
+jects). We note that this information is not obvious from the
+standard deviations listed in Table4, since the few outlier s at
+ages of 10Myr and older have a strong impact on these param-
+eters. Although the quantitative picture depends upon the e vo-
+lutionary models in use, we note that the X-ray sources and th e
+PNZ objects have essentially identical age distributions w hich
+are strongly peaked below 5Myr and the likely PMS stars have
+a mean age between 2 and 3Myr. From the observational point
+of view, the combined uncertainty in cluster distance and re d-
+dening (as inferred for the brighter cluster members) amoun ts
+to 0.18mag, which introduces an uncertainty of about 0.4Myr
+in the cluster age. The latter number is well within the dispe r-
+sion of the ages implied from the comparisonwith the di fferent
+evolutionarymodels.
+Our results are in reasonable agreement with the upper age
+limit of 4Myr inferred by Paunzen et al. (2007). We note that
+FitzGerald et al.(1978) derived a younger cluster age of 1 .7±
+0.4Myr. The difference may originate from the use of di fferent
+PMS isochrones or the work of FitzGerald et al.(1978) being
+restrictedtothebrighterclustermembersonly.
+Finally,weestimatedtheageofthecentralearly-typebina ry
+HD159176 from the e ffective temperature and bolometric
+luminosity determined by the spectral analysis of Linder et
+al. (2007). The bolometric luminosities were computed by
+assuming that AV=0.96 andDM=10.57, while the effec-
+tive temperatures were adapted from Martins et al. (2005)
+following the spectral types derived by Linder et al. (2007) .
+Since HD159176 is probably a detached binary system, binary
+evolution effects are unlikely to have played a major role in
+the evolution of the system to date. Therefore, we compared
+the position of the stars in the Hertzsprung-Russell diagra m
+with the single-star evolutionary tracks of Meynet & Maeder
+(2003). Depending on whether we use the evolutionary tracks
+with an initial equatorial rotational velocity of 0 or 300km s−1,
+we estimate the age of the system to be in the range 2.3 –
+2.7Myr, in good agreement with the 2 .8±0.5Myr estimated
+by FitzGerald et al. (1978). As a result, we find that the age
+of the massive binary and the mean age of the PMS stars are
+reasonablyconsistent with a single star-formationeventa bout2
+–3Myrago.
+We emphasize that many objects meet more than one PMS
+criterion simultaneously. This is the case for the X-ray sou rces
+#16, 20, 25, 39, 42, 47, 51, 57, and 63 that appear in the list of10 G.Rauw et al.:Thevery young open cluster NGC6383
+Fig.9.Distributionofagesasinferredfromthecomparisonwith
+PMS isochrones. The evolutionary models are from D’Antona
+& Mazzitelli (1997, top left), Siess et al. (2000, top right) , and
+Landin et al. (2009) without rotation (lower left) and with r ota-
+tion(lowerright).
+PMS candidates of Paunzen et al. (2007). Five of these object s
+wereobservedspectroscopicallyandwerefoundtoexhibita sig-
+nificant absorption probably caused by Li iλ6708. For three of
+them, we also found weak H αemission (#20, 25 and 57) in the
+spectra. Finally, we note that our X-ray source #51 is identi cal
+with WEBDA object 382, which is understood to be a rapidly
+rotatingPMS star (Paunzenet al. 2007,Zwintzet al. 2005,th eir
+object71).
+6. Conclusions
+Our photometric and spectroscopic investigation of NGC638 3
+has detected a significant population of reddened objects wi th
+extinctionsspanningacontinuousrangeupto AV≃20.Wesug-
+gest that these objects are located at di fferent depths within the
+natal molecular cloud of the cluster. NGC6383 could therefo re
+be the visible part of a significantly larger star formation e vent,
+probablyaffectinglargepartsofthismolecularcloud.
+We have found that the best PMS candidates display little
+or no Hαemission in their spectrum.The mean age of the most
+likelylow-massPMScandidatesinNGC6383isfoundbycom-
+paringwithvariousPMSevolutionarytracks,tobe2–3Myr,i n
+reasonable agreement with the estimated age of HD159176 at
+the centre of the cluster. It therefore appears that the low- mass
+starsandtheO-typebinarywereborninthesamestar-format ion
+event.
+Acknowledgements. We acknowledge support from the Fonds de Recherche
+Scientifique (FRS/FNRS),through theXMM /INTEGRALPRODEXcontract as
+well as by the Communaut´ e Franc ¸aise de Belgique - Action de recherche con-
+cert´ ee - Acad´ emie Wallonie - Europe. This research made us e of data products
+from the Two Micron All Sky Survey, which is a joint project of the Universityof Massachusetts and the Infrared Processing and Analysis C enter/California
+Institute of Technology, funded by NASAand NSF.
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+AppendixA: Noteson individualstars
+Star #1: the strength of the G-band and the ratio Fe iλ4325/Hγsuggest a K0
+spectral type. The strength of Y iiλ4376 relative to Fe iλ4383 and the general
+appearance of the violet-system CN bands and Ca iiH and K lines indicate a
+main-sequence or slightly brighter luminosity class. Weth usadopt aK0Vspec-
+tral type with an uncertainty ofabout two spectral subtypes .
+Star #3: the strength of the H8 line compared to theCa iiH and Klines, the
+strength ofthe(weak) G-band,and therelative importance o f metallic linespec-
+trumindicate anF5spectral-type. Theweakness oftheSr iiλ4077lineindicatesG.Rauw et al.:Thevery young open cluster NGC6383 11
+a main-sequence or subgiant luminosity class. We thus class ify this star as F5V
+with an uncertainty of two spectral subtypes.
+Star #16: the strength of the G-band and the Ca iλ4227 line as well as the
+ratioFeiλ4325/Hγsuggestanearly G(G0-5)spectraltype.Thestrength ofSr ii
+λ4077 suggests either a subgiant or giant luminosity class. T he strength of Yii
+λ4376relative toFe iλ4383isinstead indicative ofamain-sequence luminosity
+class. Wethus adopt aG0-5V-IV spectral type.
+Star #20: the strength of the G-band and the ratio Fe iλ4325/Hγyield a
+spectral type of about K0. The strength of Sr iiλ4077 compared to the nearby
+Feilinessuggestsamain-sequence orslightly brighter lumino sityclass.Wethus
+classify this star as K0V with two subclasses of uncertainty . The star exhibits
+a rather strong Hαemission above the continuum with an equivalent width of
+about−1.0Å.
+Star #25: the same criteria as for star #20 yield an early to mi d K spectral
+type. The strength of Sr iiλ4077 compared to the nearby Fe ilines suggests a
+main-sequence luminosity class. This is also in agreement w ith the strength of
+Yiiλ4376 relative to Fe iλ4383. We thus adopt a K2V spectral type with an
+uncertainty ofabouttwospectralsubtypes.TheH αlineisseeninemissionabove
+the continuum with an EWof −0.5Å.
+Star #39: the strength of the G-band indicates that this star must be earlier
+than K5,whilstthe weakness of theBalmer lines and theratio of Feiλ4325/Hγ
+suggestaspectral typelater than K0.Considering alsotheg eneralappearance of
+the spectrum, we favour a spectral type K2-4. The same criter ia as for star #25
+indicateamain-sequenceluminosityclass.WethusadoptaK 2-4Vspectraltype.
+While no Hαemission is seen above the continuum, there is also no absorp tion
+at the wavelength of H α. There is an absorption towards the blue, suggesting a
+kind of P-Cygni profile.
+Star #50:thestrength oftheCa iλ4227 lineand theMgHfeature at4780Å
+suggest a late-K main-sequence type, whilst the lack of visi ble TiO bands indi-
+cates that the star must be earlier than M0. We adopt a K7V spec tral type with
+an uncertainty of atleast two subtypes. A strong (EW =−3.0Å)Hαemission is
+detected.
+Star #54: the strength of the G-band and the Ca iλ4277 line as well as the
+ratio Feiλ4325/HγsuggestalateGspectral type.ThestrengthofSr iiλ4077as
+wellasthestrengthofY iiλ4376comparedtoFe iλ4383yieldamain-sequence
+orgiantluminosity class.WethusadoptaG7Vspectral typew ithanuncertainty
+of two subtypes in spectral type.
+Star #55: the same criteria as for star #54 yield an early-G ty pe, G0-5, but
+closer to G0. The strength of the Ti ii, Feiiλλ4172-78 blend, as well as of
+the Sriiλ4077 line suggest a subgiant luminosity class. We classify t his star
+accordingly as G0-1IV.
+Star #56: as inferred by the presence of He ilines, this object is a B-star.
+The intensity ratio He iλ4471 versus Mgiiλ4481 as well as the strength of
+the CaiiK line indicate a spectral type B5-7. The rather broad Balmer lines
+suggest a main-sequence luminosity class. We thus classify this star as B5-7V.
+Thespectrum also exhibits the di ffuse interstellar band (DIB) at 4430Å.
+Star #57: the same criteria as for star #39 suggest a spectral type K0-5.
+The strength of Sr iiλ4077 compared to the nearby Fe ilines suggests a giant
+luminosity class,whilstthestrength ofY iiλ4376comparedtoFe iλ4383could
+be consistent with either a main-sequence or giant classific ation. Considering
+also the general appearance of the spectrum, we adopt a K0-5I II classification.
+Our spectrum displays a blue-shifted H αemission above the continuum with an
+equivalent width of about −0.3Å.
+Star #65: the same criteria as for star #16 indicate a G5 spect ral type. The
+strength of the Sriiλ4077 line suggests that the star is a subgiant or giant. We
+thus classify this star as G5IV-III.
+Star#67:thesamecriteria asforstars#16and#25yieldaG5- K0Vspectral
+type.
+Star#70:thesamecriteriaasforstar#16indicatealateGsp ectraltype.The
+strength of Sriiλ4077 compared to the nearby Fe ilines and the strength of Y ii
+λ4376 relative to Fe iλ4383 suggest a subgiant or giant luminosity class. We
+thus classify the star as G5-K0IV-III.
+Star #76: the spectrum of this source is very red with almost n o flux be-
+low∼4300Å. However, this must be a rather hot object, since in add ition to
+someBalmerabsorptionlines,wealsoseeseveralHe iabsorptionlines(λλ4471,
+4922, 5876, 6678). The Mg iiλ4481 line is weaker than He iλ4471. The spec-
+trum also exhibits a number of strong di ffuse interstellar bands. This star must
+therefore be aheavily extinguished object of spectral type O9-B5.
\ No newline at end of file
diff --git a/1001.0697.txt b/1001.0697.txt
new file mode 100644
index 0000000000000000000000000000000000000000..604fbe6138df52eab8d514761f5f55fa2ba52418
--- /dev/null
+++ b/1001.0697.txt
@@ -0,0 +1,1109 @@
+arXiv:1001.0697v1  [cond-mat.other]  5 Jan 2010Deposition of Na Clusters on MgO(001)
+M. B¨ ar1, P. M. Dinh2,3, L. V. Moskaleva4, P.-G. Reinhard1, N. R¨ osch4, and E. Suraud2,3
+1) Institut f¨ ur Theoretische Physik, Universit¨ at Erlang en, Staudtstrasse 7, 91058 Erlangen, Germany
+2) Universit´ e de Toulouse; UPS; Laboratoire de Physique Th ´ eorique (IRSAMC); F-31062 Toulouse, France
+3) CNRS; LPT (IRSAMC); F-31062 Toulouse, France
+4) Technische Universit¨ at M¨ unchen, Department Chemie an d Catalysis Research Center, 85747 Garching, Germany
+(Dated: November 7, 2018)
+We investigate the dynamics of deposition of small Na cluste rs on MgO(001) surface. A hier-
+archical modeling is used combining Quantum Mechanical wit h Molecular Mechanical (QM/MM)
+description. Full time-dependent density-functional the ory is used for the cluster electrons while
+the substrate atoms are treated at a classical level. We cons ider Na 6and Na 8at various impact
+energies. We analyze the dependence on cluster geometry, tr ends with impact energy, and energy
+balance. We compare the results with deposit on the much soft er Ar(001) surface.
+PACS numbers: 34.50.Lf, 36.40.Sx, 68.49.Fg
+I. INTRODUCTION
+A major branch of present-days cluster research com-
+prises clusters in contact with solid surfaces, for an
+overview see, e.g., [1–4]. The interaction of these
+two entities gives rise to a rich scenery of effects such
+as, e.g., chemical reactions at surfaces [5–7], particu-
+larly, catalytic applications [8–10], or modified optical
+response[11–13]. One crucial aspect here is the process
+of cluster deposition which is relevant for synthesis and
+for analysis of clusters in contact with surfaces. More-
+over, the deposition dynamics as such is an interesting
+and demanding process due to the subtle interplay of
+the impact of interface energy, electronic band struc-
+ture of the substrate, and surface corrugation. Accord-
+ingly, thereisawealthofinvestigationsonclusterdeposi-
+tion, experimentally oriented [14–21], theoretically with
+moleculardynamics (MD) techniques [22–26] or more de-
+tailed quantum mechanical methods [27–33], for reviews
+see [2, 8, 34]. Although addressing the same physical
+processes, these various theoretical approaches, relying
+on different approximations, often provide useful com-
+plementary information. Recently, we have investi-
+gated deposition dynamics of Na clusters on Ar(001) sur-
+face [33, 35, 36]. The aim of this paper is to continue
+these theoretical studies now considering deposition dy-
+namics of Na clusters on a much ”harder” surface than
+Ar, namely MgO(001) insulator surfaces. The struc-
+tural properties and optical response of Na non MgO
+have already been studied in great detail in using the
+present computational approach [37]. Both Ar and MgO
+are similar in that they are both insulators with a large
+band gap, but they differ significantly in other important
+properties. There are, however, large differences in other
+properties. Ar is a Van-der-Waals bound material, thus
+very soft with little surface corrugation. On the other
+hand, MgO is an ionic crystal, well bound and with large
+surface corrugation. It is thus most interesting to see
+how deposition dynamics proceeds in that case, as such,
+and at variance with the Ar case.
+The theoretical description of clusters on surfaces isvery involved due to the huge number of degrees-of-
+freedom of these systems. This holds the more so for
+dynamics. The vast majority of theoretical studies thus
+resorts to MD simulations using effective force fields be-
+tween the atoms, as mentioned above [22–26, 34]. These
+are comparatively inexpensive and can provide a perti-
+nent picture of the leading atomic transport processes.
+Metal clusters are more than an ensemble of atoms be-
+cause they held together by delocalized bonds (due to a
+common electron cloud); in consequence they show pro-
+nounced shell effects [38–40]. This makes a quantum me-
+chanical description of cluster dynamics advisable. Most
+of the fully quantum mechanical pictures make compro-
+mises in concluding on dynamical features from a series
+of static calculations. A true Born-Oppenheimer MD for
+deposition of Pd clusters on MgO substrate can be found
+in [29]. The enormous expense of such high level cal-
+culations limits the size of the systems, particularly the
+size of the representative for the substrate. On the other
+hand, there are many situations in which the substrate
+is much more inert than the cluster. Our test case of Na
+clusters on MgO(001) belongs to that class. This sug-
+gests to use a hierarchical description where the cluster
+electronsaretreatedquantum-mechanicallybyfullTime-
+Dependent Density-Functional theory (TDDFT) while
+the substrate atoms are handled at a lowerlevel of refine-
+ment by classical motion. This modeling belongs to the
+family of coupled Quantum-Mechanical with Molecular-
+Mechanical methods (QM/MM) which are often used
+in other fields as, e.g., bio-chemistry [41–43] or surface
+physics [44, 45]. In earlier studies, we developed and ap-
+plied a QM/MM model for Na clusters in contact with
+Ar [35, 36, 46–48]. We have shown that it was most cru-
+cial to include properly the dynamical polarizability of
+the substrate when exploring truly dynamical processes
+as we aim at. Recently, we extended the modeling to Na
+clusters on MgO surfaces, again including dynamical po-
+larizability [37]. Here we take up that model and apply
+it to a study of deposition dynamics. We will consider
+Na6and Na 8as test cases. These two clusters have very
+different geometries and binding properties which allows2
+ϕn(/vector r), n= 1,...Nelvalence electrons of the Na cluster
+/vectorRi(Na), i(Na)= 1...Nipositions of the Na+ions
+/vectorRi(c), i(c)= 1...Mpositions of the O cores
+/vectorRi(v), i(v)= 1...Mcenter of the O valence cloud
+/vectorRi(k), i(k)= 1...Mpositions of the Mg2+cations
+TABLE I: The dynamical degrees-of-freedom of the model.
+Upper block : Na cluster. Lower block : Active cell of the
+MgO substrate. See text for details.
+to explore qualitatively the impact of cluster properties
+on the deposition process.
+The paper is organized as follows: In section II, we
+summarize the QM/MM model for Na clusters on MgO.
+Section III presents results tracking the detailed dynam-
+ics in terms of trajectories and analyzing the processes
+with respect to energy transfer and energy balance. Con-
+clusions are summarized in section IV.
+II. BRIEF SUMMARY OF THE MODEL
+A. The degrees-of-freedom
+The hierarchical QM/MM model has been detailed in
+[37]. We review that here briefly. The various con-
+stituents and their degrees-of-freedom are summarized
+in table I. The Na cluster is treated in standard fash-
+ion [49, 50]. Valence electrons are described in terms of
+single-particle wavefunctions ϕn(/vector r) and the complement-
+ing Na+ions are handled as charged classical point par-
+ticles characterized by their positions Ri(Na), see upper
+block of table I. The electrons are described by TDDFT
+atthelevelofthelocaldensityapproximation(LDA)The
+substrate is composed of two species: Mg2+cations and
+O2−anions. The cations are electrically inert and can
+be treated as charged point particles; they are labeled by
+i(k). The anions are easily polarizable, an aspect which is
+describedbyallowingfortwoconstituents: avalenceelec-
+trondistribution(labeledby i(v))andthe complementing
+core (labeled by i(c)). Each of these three types of con-
+stituents is described as a classical degree-of-freedom in
+terms of positions /vectorRi(type), see the lower block of table I.
+The difference R(c)−R(v)represents the electrical dipole
+moment of the O2−anion and is thus allowed, by con-
+struction, to explicitly evolvein time, as a function of the
+local electric field due to all constituents of the system
+at a given instant.
+The combined system is sorted in four stages of de-
+creasing activity, as sketched in figure 1. The Na cluster
+is treated at the highest level oftheorywith full TDLDA-
+MD. The Mg and O ions of the substrate are arranged
+in fcc crystalline order corresponding to bulk MgO, with
+a lattice parameter of 7.94 a0. All dynamical degrees-
+of-freedom for Mg and O, as listed in table I, are taken
+into account in an active cell of the MgO(001) surface re-
+gion underneath the Na cluster, denoted “zone I” in thesketch. The active cell is continued by an outer region of
+MgO material (“zone IIa”) where the ionic centers of Mg
+andOarekeptfixed, whileoxygendipolesstillremainac-
+tive degrees-of-freedom. Thus zone I together with zone
+IIa constitute the “active cell”. Anything farther out
+(“zone IIb”) is totally frozen at crystalline configuration
+and only its Madelung potential is considered. The ef-
+fect of the outer region on the active part is given by
+a time-independent shell-model potential [51]; the actual
+parametersofthisforcefieldwereadoptedfrom[45]. The
+surfaceNa cluster
+shell model
+zone I
+zone IIa
+zone IIb(frozen cores, free shells)
+Gaussian charge densities
+(free cores & shells)
+point charges
+Madelung potential onlyQM active
+FIG. 1: Schematic view of the hierarchical model for Na Non
+MgO(001) surface.
+active cell consists of three layers, each containing square
+arrangements of 242 Mg2+cations and 242 O2−anions.
+Theionsinthelowestlayerarefixedatthebulkstructure
+to prevent them from relaxing and forming an artificial
+second surface. The volume where ions and electrons
+are mobile (zone I) has a diameter of 24 a0, all layers
+together (zone I+IIa) extend to 42 a0from the surface.
+Bulk structure farther out is modeled by the Madelung
+potential. Checks with models of a larger number of lay-
+ers showed that three layersprovidean adequate descrip-
+tion of the very inert MgO material. (The soft Ar(001)
+substrate, used for comparison later on, is more critical
+and requires at least four active layers plus two frozen
+ones.)
+B. The energy
+The total energy is composed as E=ENa+EMgO+
+EcouplwhereENadescribes an isolated Na cluster, EMgO
+the MgO(001) substrate, and Ecouplthe coupling be-
+tween the two subsystems. For ENa, we take the stan-
+dard TDLDA-MD functional as in previous studies of
+free clusters [49, 50] including an average self-interaction
+correction [52]. The energy of the substrate and the cou-
+pling to the Na cluster consist of long-range Coulomb
+energy and some short-range repulsion which is modeled
+through effective local core-potentials [45]. To avoid the
+Coulomb singularity and to simulate the finite extension
+of Mg2+and O2−ions, we associate a smooth charge
+distribution ρ(/vector r)∝exp(−/vector r2/σ2) with each of these ionic3
+centers. We associate a similar smooth charge distribu-
+tion to the O2−valence cloud as well. This altogether
+yields a soft Coulomb potential to be used for all active
+particles.
+C. Calibration of the QM/MM model
+Thecalibrationofthewholemodelhastoaddressthree
+issues: The cluster as such, the environment as such,
+and the coupling between both. The modeling for the
+cluster is taken over from work on free clusters [49, 50].
+The model parameters for the pure environment are the
+same as in previous studies of MgO(001) [45, 53]. The
+parameters for the coupling between environment and
+Na cluster were calibrated from scratch. The tuning
+for Na@MgO(001) was performed using fully quantum-
+mechanically computed Born-Oppenheimer surfaces for
+Na atoms and Na+ions on MgO(001) from [53]. These
+surfaces were computed at four different substrate sites
+(O2−, Mg2+, hollow, bridge) down to close distances
+where the full substrate repulsion was felt. For further
+details and the actual model parameters, see [37].
+Two quantitative points are worth to be mentioned.
+The modeling achieves a barrier for penetration of clus-
+ter electrons into the substrate which reproduces nicely
+the largeband gap of6.9 eV for MgO. The fully quantum
+mechanical calculations show that electron transfer from
+the substrates O2−anions to a Na atom remains below
+0.1 charge units down to the closest distances consid-
+ered (where the repulsive energy comes about the band
+gap). Transfer from the Na atom to the Mg2+cation is
+totally ignorable. This nice decoupling of ad-atoms and
+substrate is probably a feature of simple metals. Noble
+metals, e.g., can develop a more involved surface chem-
+istry due to the closeness of the dshell [29].
+D. Solution scheme
+From the energy functional, once established, one de-
+rives the static and dynamical equations variationally
+in a standard manner. The numerical solution of the
+coupled quantum-classical system proceeds as described
+in [37, 54, 55]. The electronic wavefunctions and spa-
+tial fields are represented on a Cartesian grid in three-
+dimensional coordinate space. The numerical box em-
+ployed here has a size of (64 a0)3. The spatial derivatives
+are evaluated via fast Fourier transform. The ground
+state configurations were found by interlaced accelerated
+gradient iterations for the electronic wavefunctions [56]
+and simulated annealing for the ions in the cluster and
+the substrate. Propagation is done by the time-splitting
+method for the electronic wavefunctions [57] and by the
+velocity Verlet algorithm for the classical coordinates of
+Na+ions and MgO constituents.
+All the collisional processes studied in the current pa-
+per proceed on an ionic time scale, i.e. slow as com-pared to electronic motion. True electronic excitations
+are thus extremely small. For example, ionization stays
+safely below a fraction of 0.001 electrons. It would then
+be well justified to use Born-Oppenheimer-MD rather
+than full TDLDA-MD, as long as one carefully main-
+tains the crucial dipole polarizability of the substrate.
+But the TDLDA-MD scheme is so efficient that it is still
+preferable for reasons of computing time. Remind that
+the dipole response of the substrate needs to be propa-
+gated at electronic time scale and dipole stepping is more
+economic than fully relaxing the dipoles in each Born-
+Oppenheimer step.
+E. Preparation of the system
+First, the ground state structures of the pure MgO
+surface and of the free Na cluster are determined for the
+given model by simulated annealing. The Na cluster is
+then placed at a certain distance from the surface of the
+substrate. A distance of about 15 a0has turned out to be
+sufficient. After that aGalileantransformationisapplied
+to the cluster. This means that each of the cluster ions
+is given a momentum /vectorP0in the direction towards the
+surface and the electronic wave functions are boosted by
+an equivalent momentum as
+ϕi(/vector r)→exp(ı/vector p0·/vector r)ϕi(/vector r) (1)
+where/vector p0=me
+M/vectorP0,meandMare the electron and Na
+ion mass, respectively. This provides the initial state
+fromwhichonthesystempropagatesinastraightforward
+manner according to the TDLDA-MD equations.
+F. Structure of the test cases
+FIG. 2: The structures of free Na 6(left) and Na 8(right). The
+bond distance along the five-fold ring of Na 6is 6.5a0and the
+topion resides 3.1 a0abovethe ring. The bonddistance inthe
+two four-fold rings of Na 8is 6.2a0and the distance between
+the two rings is 5.8 a0.
+The starting point of deposition dynamics are well re-
+laxed structures for the clusters and pure MgO(001) sur-
+face. These had been discussed extensively in [37]. The
+MgO surface is a cut through cubic crystal structure.
+From the top, one sees a chess-board structure with al-
+ternating Mg and O ions. Forthe Na clusters, we will use
+here Na 6and Na 8as examples. The initial state starts4
+from free clusters. Their structures are shown in figure 2.
+Note that the vertical axis in the figure will represent the
+direction perpendicular to the surface in the forthcom-
+ing deposition processes ( zaxis). Na 6is strongly oblate
+consisting out of a ring of five ions topped by one sin-
+gle ion. Na 8has a highly symmetric configuration out of
+two rings of each four ions tilted relative to each other by
+45◦to minimize Coulomb energy. The electronic cloud of
+Na8is close to spherical shape because N= 8 electrons
+correspond to a strong shell closure for Na clusters [40].
+It is important to note that the bond distances for Na 8
+(6.2a0) are not far from the diagonal distance between
+oxygen sites in the MgO(001) surface (5.7 a0) while the
+dimensions of the fivefold ring in Na 6do not fit well to
+the surface. That will play a role in the dynamical evo-
+lution studied later on. The equilibrium distance of the
+lower cluster plane (facing towards the surface) and the
+first surface layer is 5 a0.
+III. RESULTS AND DISCUSSION
+A. Na monomer on MgO – the influence of sites
+A two component system like MgO has more possible
+adsorption sites than a homogeneous material like an ar-
+gon substrate. The properties of an oxygen site are much
+different from those of the magnesium site because of the
+much larger polarizability of oxygen. We will thus con-
+sider four positions with respect to the surface : O site,
+Mg site, hollow and bridge. The structure calculations
+of [37] have shown that O, due to its large polarizability,
+is the most attractive site while Mg acts like a repulsive
+site on the cluster. This is in agreement with quantum
+chemical ground state calculations of transition metals
+on MgO [58].
+In order to check the adsorption properties of the var-
+ious sites, we first study the deposition dynamics of a
+Na monomer. We briefly remind the static properties
+of Na@MgO(001). The O site is most attractive, bind-
+ing Na 5 a0above the surface with energy 0.25 eV. The
+Mg site is dominantly repulsive. The hollow and bridge
+sites lie in between these extremes. For deposition dy-
+namics, the atom was initialized 15 a0away from the
+substrate above an O site, Mg site or hollow site respec-
+tively, each with an initial momentum along zdirection,
+pointing perpendicular towards the surface with a mag-
+nitude corresponding to a kinetic energy E0
+kin= 0.136
+eV. Figure 3 shows the results of the simulation. The
+z-coordinates are chosen such that the (average) MgO
+surface layer resides at z= 0. The simplest case is the
+impact on the O site. The atom approaches the surface
+up to a distance of 4 .5a0which is reached at about 500fs
+and transferspart ofits momentum to the substrate ions.
+The transfer proceeds at a very short time scale. The
+surface itself is excited mainly by the first collision which
+initially only affects the ions in the immediate vicinity
+of the atom at closest impact. The perturbation quicklyspreads over the surface, but the associated sound wave
+does not penetrate very deep into the surface. The os-
+cillations in the third layer are already almost negligible.
+After the instant of closest contact, the atom bounces
+back, but it has already lost so much energy that it can-
+not escape from the surface anymore. Thus it performs
+damped oscillations, with each bounce transferring some
+momentum to the surface and being practically adsorbed
+within the first 2ps. The final distance approachesnicely
+the equilibrium distance of 5 a0. The right upper column
+of figure 3 shows the corresponding kinetic energy con-
+tributions. In the first 400fs, the attraction from the
+MgO substrate leads to a rapid increase of the kinetic
+energy of the atom up to 0 .45 eV. At the point of clos-
+est contact, the repulsive part of the interface potential
+stops the atom abruptly. That first collision transfers by
+far the largest amount of energy to MgO, whereas at all
+subsequent collisions the energy decreases more slowly.
+In order to check that the oscillations proceed only per-
+pendicular to the surface, the kinetic energy of the Na
+atom has been split into contributions from perpendicu-
+lar (or vertical) and parallel (or transverse) motion. The
+latter is too small to be visible in figure 3 and practi-
+cally negligible. Thus the motion of Na proceeds strictly
+perpendicular to the surface. The kinetic energy trans-
+ferred to the MgO can also be read off from figure 3
+(see right upper panel). The contributions from oxygen
+and magnesium are given separately. Oxygen ions are
+the lighter species and therefore react first being quickly
+accelerated. About 100 fs later, the energy has already
+been distributed almost equally over both ion types.
+The dynamics behaves totally different if the atom im-
+pinges on the (repulsive) Mg site, see middle panels of
+figure 3. At first glance, the z-component of the Na tra-
+jectory looks quite similar to the case before. But one
+notes that the motion is not damped after the first re-
+flection. The kinetic energies (middle right panel) give a
+clue on the process. There is much less energy transfer
+at first impact which is related to the fact that the Mg2+
+ion is more inert. And there is a significant amount of
+lateral kinetic energy for the Na atom creeping up after
+impact time at 500 fs. In fact, most of the kinetic energy
+is now in lateral motion. The atom is deflected by the
+Mg2+ion. It is to be noted that the annealing of the
+substrate configuration leaves a small amount of symme-
+try breaking with fluctuations of the atomic positions of
+about 0.05 a 0. This small symmetry breaking allows the
+atom to acquire sidewards momentum and so it bounces
+away in sideward direction, hops over the surface sev-
+eral times changing direction whenever it comes close to
+another surface ion. The motion is almost undamped be-
+cause little energy is transferred to the surface after the
+first collision. The atom has thus still too much energy
+to be caught by a certain site of the surface. But as the
+atom cannot escape the surface as a whole, it will con-
+tinue to lose slowly energy and finally be attached to an
+oxygen site, long after the simulation time of 3 ps.
+The bottom panels of figure 3 show the case of impact5
+-5 0 5 10 ionic z coord. (a0)O site
+Na
+O
+Mg
+-5 0 5 10 ionic z coord. (a0)Mg site
+-5 0 5 10
+ 0.5  1  1.5  2  2.5ionic z coord. (a0)
+time (ps)hollow site
+ 0  0.5  1  1.5  2  2.5 0 0.1 0.2 0.3 0.4
+kinetic energy (ev)
+time (ps) 0 0.1 0.2 0.3 0.4
+kinetic energy (ev) 0 0.1 0.2 0.3 0.4
+kinetic energy (ev)Na
+Na trans.
+Na vert.
+O
+Mg
+FIG. 3: Time evolution of the ionic coordinates and kinetic e nergiesEkinfor the deposition of a Na monomer on MgO(001)
+with initial kinetic energy E0
+kin= 0.136 eV, impinging on various sites : O site (top), Mg site (cen ter) and hollow site (bottom).
+Left :zcoordinates of Na (thick line), Mg (thin curve), and O (gray l ine) cores. Right : Total Ekinof Na (thick gray or red
+curve), lateral Ekinof Na (magenta dots), vertical Ekinof Na (black dashes), and total Ekinof Mg (thin dark or blue curve)
+and O (thin light or green curve) cores.
+at an hollow site. We see again the immediate reflection
+at impact time associated with fast energy transfer. Less
+energy is transferred than on the other sites (see upper
+and middle panels) and thus the bounce-backhas a much
+larger amplitude than in both other cases. The Na mo-
+tionremainsstrictlyperpendiculartothe surfaceasprac-
+tically no lateral kinetic energy can be seen. The vertical
+kinetic energy is almost approaching zero because the
+departing Na atom has to work against the polarization
+potential. The case is at the limits of our box size and
+energyresolutionsuchthat we cannotdecidewhether the
+atom will finally escape with extremely small kinetic en-
+ergy, or will bounce back and relax to an adsorption siteon a very long time scale. Nevertheless, we find it worth-
+notingthat the hollowsiteseemssufficiently attractiveto
+hinder deflection towardsthe still moreattractive oxygen
+site.
+B. Cluster deposition
+1. The case of symmetric Na 8
+The analysisof section IIIA has shown the importance
+of surface site nature in the deposition process. Deposit-
+ing an extended object such as a cluster will lead to a6
+-5 0 5 10 15 20
+ 0  0.5  1  1.5  2  2.5ionic z-coord. (a0)
+time (ps)Na8 leaves boxE0
+kin=10.9 eV
+NaOMg-5 0 5 10 15 20ionic z-coord. (a0)E0
+kin=1.09 eV-5 0 5 10 15 20ionic z-coord. (a0)E0
+kin=0.109 eV
+-10-5 0 5 10
+-10 -5  0  5  10ionic y-coord. (a0)
+ionic x-coord. (a0)-10-5 0 5 10ionic y-coord. (a0)-10-5 0 5 10ionic y-coord. (a0)
+FIG. 4: Left : Time evolution of the ionic z-coordinates of
+Na8approaching MgO(001). Right : Projection ofthe Naand
+MgO trajectories into the x-yplane; the figures are vertically
+sorted by increasing initial kinetic energy : Soft depositi on
+(top), robust deposition (center), reflection (bottom).
+mixed situation because the ions of the cluster will nec-
+essarily be placed above different sites. In the following,
+we will discuss deposition of Na 8and Na 6which have
+very different structures and so promise to show different
+deposition scenarios. As a first step in the analysis, we
+shall consider detailed ionic trajectories both perpendic-
+ular and parallel to the surface. The case ofNa 8is shown
+in figure 4. The cluster was injected with its symmetry
+axis pointing through a hollow site and with the lower
+ring facing closer to bridge sites. The top panels show a
+softdepositionwheretheinitialkineticenergyis0 .109eV
+(0.0136 eV per Na ion). The left upper panel shows that
+the cluster is slightly accelerated in the initial phase, due
+to the attraction from the surface. But that acceleration
+differs for the different ions on the lower ring because
+they approach different sites on the surface. At the same
+time, the cluster rotates in the x-yplane to bring the
+four ions of the lower ring closer to the attractive oxy-
+gen sites. One may spot that from the top view in the
+right upper panel. At the point of closest impact around
+900 fs, the cluster transfers some momentum to the sur-
+face. The substrate ions are slightly displaced from their
+equilibrium positions and oscillate around their new po-
+sitions. The disturbance quickly decreases from layer to
+layer. Theperturbationisnegligiblealreadyinthefourth
+layer. The complicated detailed dynamics of the Na ions
+indicates that a major part of the translational kinetic
+energy is converted into heat, i.e.kinetic energy of the
+motionrelativetothe centerofmass, aswill beconfirmedin section IIID. Nevertheless, the cluster basically keeps
+its original structure during the whole simulation period
+of 9 ps. In particular the two rings, each made of four
+ions, always stay clearly separated from each other. The
+top-down projection of the trajectories (see right-hand
+side of figure 4) shows that the cluster as a whole (or
+its center of mass) remains oscillating around the point
+of impact. The remaining kinetic energy of the cluster
+does apparently not suffice to overcome the surface cor-
+rugation barriers. This is related to the fact that the
+Na8structure fits approximately well to the structure
+and binding distance of MgO, see section IIF.
+The middle panels of figure 4 show a more robust
+deposition dynamics with initial kinetic energy E0
+kin=
+1.09 eV. The initial velocity is higher and the impact
+time comes earlier, now at 450 fs. The pattern remains,
+in principle, similar to the softer deposition. There is lit-
+tle momentum transfer to the substrate, strong internal
+excitation of the cluster, and the cluster is not departing
+too far from the impact point. However, perturbations
+are much larger, yielding larger amplitudes in vertical
+and lateral motion. As a consequence, the two rings of
+Na8are now not always clearly separated. Nevertheless
+the typical structure of Na 8reappears from time to time
+as we will see later. The top-down projection of the tra-
+jectories (middle right panel of figure 4) indicates a new
+effect, a sideward drift from one adsorption site to the
+next equivalent site. This sideward drift is again induced
+by the interplay between attractive O and repulsive Mg
+sites. The chaotically moving Na ions explore a strongly
+corrugated surface which leads to occasional side kicks
+from the repulsive Mg sites.
+The bottom panels of figure 4 show a hard collision
+with initial kinetic energy E0
+kin= 10.9 eV. Internal clus-
+terandexcitationandsurfaceperturbationare,ofcourse,
+again larger. The new feature is that the cluster is re-
+flected from the surface and leaves the numerical box at
+about 1 ps, however with huge internal excitation. It is
+not clear whether the departing cluster will stay asymp-
+totically stable. That is beyond our simulation capacity.
+Figure 5 complements the view within showing a se-
+quence of snapshots of the detailed structure for each of
+the three cases discussed above. The uppermost panel
+for soft deposition nicely shows the initial rotation of
+the cluster to match the attractive oxygen sites. The
+further snapshots indicate the sizeable internal excita-
+tion, however remaining small enough to see at all times
+clearly the two-ring structure of Na 8. The middle panel
+for more robust deposition also presents the much larger
+cluster oscillations where the original cluster structure is
+often blurred, but reappearsshortly at other times. That
+demonstrates the surprisingly good binding of Na clus-
+ters, particularly the Na 8cluster with its magic electron
+configuration. The lowest panel shows the case of re-
+flection. Obviously, some ions would like to stick to the
+surface, but are finally caught back by the cluster which
+departs in a highly excited state.7
+FIG. 5: Snapshots of the time evolution for three cases of col -
+lisions of Na 8on MgO(001) with different initial kinetic ener-
+giesEkin. Upperpanel: Softdepositionwith E0
+kin= 0.109eV.
+Middle panel : Robust deposition with E0
+kin= 1.09 eV. Lower
+panel : Reflection with E0
+kin= 10.9 eV.
+2. The case of strongly oblate Na 6
+Results for the deposition of Na 6are shown in figure
+6. The impact energy is varied and all three cases start
+from the same initial configuration where the top ion of
+Na6(see section IIF) is facing away from the substrate
+and the fivefold ring is parallel to the surface. The gen-
+eral features are similar to the case of Na 8. One observes
+a large internal excitation of the cluster while compara-
+tively little perturbation goes to the substrate and there
+is againthe cleardistinction between deposition for lower
+impact energies and reflection for higher ones. But there
+are several interesting differences in detail. Most of all,
+there is a strong lateral drift in all cases. Indeed the
+pentagonal ring of Na 6does not match the rectangular-5 0 5 10 15 20
+ 0  0.5  1  1.5  2  2.5ionic z-coord. (a0)
+time (ps)Na6 leaves boxE0
+kin=8.16 eV
+NaOMg-5 0 5 10 15 20ionic z-coord. (a0)E0
+kin=0.816 eV
+Na6leaves
+box-5 0 5 10 15 20ionic z-coord. (a0)E0
+kin=0.0816 eV
+-10-5 0 5 10
+-10 -5  0  5  10ionic y-coord. (a0)
+ionic x-coord. (a0)-10-5 0 5 10ionic y-coord. (a0)-10-5 0 5 10ionic y-coord. (a0)
+FIG. 6: Left : Time evolution of the ionic z-coordinates of
+Na6approaching MgO(001). Right : Projection of the Naand
+MgO trajectories into the x-y-plane; the figures are vertica lly
+sorted by increasing initial kinetic energy : Soft depositi on
+(top), robust deposition (center), reflection (top).
+structureofMgO,which hindersit from fully accomodat-
+ing the attractive oxygen sites. Thus one or two corners
+of the pentagon are bent up during the deposition pro-
+cess, and this, in turn, induces a sizeable lateral momen-
+tum (see right panels of figure 6), and a strong perturba-
+tion of the pentagon, as can be deduced from the motion
+ofz-coordinates shown in the left panels. In the case
+of the robust deposition, the cluster even rolls over the
+surface. The stronger lateral excitation leaves also some-
+what more perturbation to the substrate than in the case
+of Na8, as may be spotted when comparing figures 6 and
+4. That will become more obvious when checking ener-
+gies in section IIID. Finally, it is interesting to note that
+the cluster orientation is also reverted in the case of re-
+flection. The zcoordinates (upper left panel in figure 6)
+suggest a process where the ring and the former top ion
+are reflected ”independently” such that the topping ion
+is departing ”behind/after” the ring, and thus reverting
+the cluster orientation.
+As noted above, when deposited, the Na 6experiences
+a sizable drift due to the mismatch of its structure with
+the crystalline structure of MgO. One thus expects that
+direction and strength of the lateral motion depend sen-
+sitively on the initial position and orientation of Na 6rel-
+ative to the surface. Indeed figure 7 shows the trajec-
+tory of the center-of-mass of the Na 6cluster projected
+onto the x-y-plane for three different initial orientations.
+There are obvisouly dramatic differences. The cluster is
+kicked to a strong lateral motion for initial impact at8
+-15-10-5 0 5 10 15
+-15-10-5 0 5 10 15y [a0]
+x [a0]top-down hollow
+top-up O
+top-up Mg
+FIG. 7: The trajectories of the Na 6center of mass, projected
+on a plane parallel to the surface. The lateral drift sensiti vely
+depends on the site above which the cluster impinges (hollow ,
+O or Mg site) and on its orientation (top ion above – up –
+and top ion below – down – the ring).
+repulsive sites (Mg, hollow) while only moderate lateral
+drift appears for impact on the attractive O site.
+One can learn more about the electronic charge distri-
+bution in the cluster during a collision by a direct multi-
+pole analysis of this distribution. We discuss here briefly
+the lowest non-trivial moments, the dipoles. Figure 8
+-2 0 2 4 6 8
+ 0  1  2  3ionic c.m. in z (a0)
+time  (ps)-0.2 0 0.2 0.4 0.6 0.8 1electronic dipole (a0)Ekin,0=0.16 eV
+z
+y
+x
+ 0  0.5-2 0 2 4 6 8
+time  (ps)-0.2 0 0.2 0.4 0.6 0.8 1Ekin,0=8.16 eV
+FIG. 8: Dipole polarization of the cluster electrons during
+collision of Na 6on MgO(001). Upperpanels: Dipole moments
+of the electron cloud of the Na 6cluster in x-y- (horizontal)
+andz-direction as function of time. Lower panels: Time-
+evolution of the zcomponente of the center-of-mass of the
+Na6cluster; the line z= 0 corresponds to the center of mass
+of a fully relaxed configuration of Na 6on MgO(001). Left
+panels: case of soft deposition at rather low initial kineti c
+energy (as indicated). Right panels: case of more energetic
+collision which leads to immediate reflection.
+shows the time evolution of dipole polarization for two
+different scenarios, deposition vs. reflection. The figure
+is augmented by the time-evolution of the center-of-mass
+as globalindicator ofthe dynamical situation (lowerpan-
+els). The two scenarios differ by the initial kinetic ener-
+gies, the lower value related to a more or less soft depo-
+sition, while the higher initial velocity leads to immedi-
+ate reflection of the cluster, as discussed above. In theslow deposition process (left panels), there is a strong in-
+crease of the z-polarization at the time of closest impact.
+This polarization remains during the further evolution
+at about 10% of the Wigner-Seitz radius, hence repre-
+sents a considerable internal polarization. Also, some
+x-ypolarization builds up during the ongoing deposition
+oscillations. In contrast, in the case of a reflection (right
+panels), oneseesalargeinstantaneouspolarizationatthe
+time of closest approach, but only a very small remain-
+ing effect when the cluster has departed from the surface.
+The very short interaction time limits the internal exci-
+tation.
+C. MgO versus Ar Substrate
+In previous works [33, 47], the deposition of Na 6on
+a cold, condensed argon substrate Ar(001) was investi-
+gated. Like MgO, solid Ar has a large band gap. But
+apart from this insulating nature, the two materials have
+much different properties. The attractive interaction be-
+tween MgO and Na is much stronger than between Ar
+and Na, due to the larger polarizability of the oxygen
+ion. On the other hand, frozen Ar material is a very
+soft solid due to the weak Ar-Ar binding. The melting
+point of Ar is 83 .78K [59], much lower than that of MgO
+around 3073K [60]. The softness of the Ar material thus
+changes the energy balance to the extent that the Ar
+substrate takes up most of the impinging energy, leaving
+ratherlittle internalexcitationforthe clusteritself. Thus
+Ar substrates are very efficient soft stopper materials. In
+the former analysis [33, 47], we had run a similar series of
+impact energies as above and we also found soft deposi-
+tion for energies up to at least E0
+kin/Nion= 0.272 eV. An
+attempt to reacha reflectionregime by further increasing
+the impact energy then led to a significant destruction of
+the substrate. Figure 9 illustrates that violent collisionof
+-30-20-10 0 10 20 30 40 50
+ 0 0.5 1 1.5 2 2.5 3ionic z-coordinate [a0]
+time [ps]Na6 at Ar, E0
+kin=8.16 eV
+FIG. 9: Time evolution of zcoordinates during collision of
+Na6with initial total kinetic energy E0
+kin= 8.16 eV (1.36 eV
+per Na atom) on an Ar(001) surface. Data taken from [47].
+Na6on an Ar(001) surface. The cluster is indeed finally
+reflected, but the process evolves much different from the
+case of the collision with MgO shown in figure 6. The9
+cluster is not reflected instantaneously, as for MgO, but
+with a delay of about 500 fs. It requires an additional
+boost from momentum reflected by the first Ar layer to
+finally release the cluster. Moreover, so much energy has
+been deposited in the weakly bound Ar material that
+the substrate is seriously damaged by that forced “reflec-
+tion”. These significant differences between Ar and MgO
+substrate will also be seen in the energy analysislater on.
+D. Energy Transfer
+1. Time evolution of energy components
+ 0 0.2 0.4 0.6 0.8
+ 0 0.5 1 1.5 2 2.5 3Ekin (eV)
+time (ps)E0
+kin=0.109 eV 0 0.4 0.8 1.2 1.6Ekin (eV)E0
+kin=1.09 eV 0 2 4 6 8 10 12 Ekin (eV)
+Na8 leaves boxE0
+kin=10.9 eV
+Na
+Na, transl.
+Na, relat.
+MgO
+FIG. 10: Time evolution of various contributions to kinetic
+energy for the collision of Na 8with MgO(001) at three differ-
+ent initial kinetic energies as indicated. Contributions a re :
+Total kineticenergyofthecluster ions(Na, thickcurves), con-
+tributions due to center of mass motion (Na transl., dashes)
+and relative motion or heat (Na relat., dots), and total kine tic
+energy of the substrate (MgO, thin curves).
+A complementing view of the deposition dynamics is
+given by the kinetic energies. Figure 10 shows the time
+evolution of the kinetic energies for Na and MgO. The
+kinetic energy for the Na cluster is furthermore splittedinto center-of-mass energy and intrinsic kinetic energy
+(from the motion relative to the center-of-mass). Let us
+first consider the case of reflection (upper panel). In the
+approaching phase, the cluster is accelerated by about
+0.68 eV which is small compared to the initial energy
+10.9 eV. Dramatic and fast changes emerge at impact
+time at 200 fs. The cluster kinetic energy exhibits a deep
+minimum. In that stage, almost all energy is stored in
+deformation. A large part of that deformation energy
+is quickly released showing up now as intrinsic kinetic
+energy of the cluster plus a smaller bit in translational
+energy. Another small fraction of energy is transferred
+to the substrate. The translational kinetic energy de-
+creases further on, because the departing cluster has to
+workagainsttheattractivepolarizationinteraction. Still,
+there remains sufficient translational energy to allow the
+cluster to finally escape, as in a very inelastic collision.
+Similar results are found for the soft (lower panel in
+figure 10) and robust deposition (middle panel). Again,
+only a small fraction of the energy is transferred to
+the substrate, another small fraction goes to the cluster
+center-of-mass oscillations, and the major part to intrin-
+sic energy of the cluster. Particularly interesting is the
+caseofrobust deposition(middle panel) where it requires
+a second bounce to stir up intrinsic cluster motion. Be-
+fore that, there is still enough energy in translation to
+allow a lateral hopping from one attractive MgO site to
+the next (see also figure 4). It is worthnoting that the
+average trend of the kinetic energy of MgO in figure 10
+has a small, but nonvanishing, slope. The cluster con-
+tinues to exchange energy with the substrate on a very
+slowpace. That indicates a thermalization processwhich
+eventually leads to equidistribution of kinetic energies af-
+ter long time, however much beyond our simulation ca-
+pabilities.
+ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
+ 0  100  200  300  400  500  600dipole energy [eV]
+time  [fs]Na6 with MgO(001)
+Ekin=8.16 eV
+FIG. 11: Time evolution of the dipole polarization energy
+in the MgO(001) substrate for the collision for Na 6with
+MgO(001) at an initial kinetic energy of 8.16 eV (reflection
+case).
+The present modeling includes an independent dynam-
+ics of the dipole moments of the oxygen anions in the
+substrate. It has been shown recently that a signifi-
+cant amount of energy can be stored in these degrees-10
+of-freedom when a metal cluster is deposited on an Ar
+surface[35, 36]. One can define a dipole energy which
+scales as the square of the dipole amplitudes. Figure 11
+shows a typical result for the time evolution of the en-
+ergy contained in the oscillating dipoles in the case of
+reflection of Na 6deposited on MgO. There is a small
+initial value which corresponds well to the finite initial
+distance of the Na cluster to the substrate. There is a
+large contribution at the time of closest impact. This
+part is dominated by (instantaneous) static polarization
+which would also be contained in a Born-Oppenheimer
+MD. The dipole energy falls back to lower values when
+the cluster departs from the substrate (see figure 6). But
+there remains some offset which corresponds to the en-
+ergy finally transferred to the dipole degrees of freedom.
+It amounts to about 2% of the impact energy, which is
+small compared to the other energetic observables, see
+figure 14 and corresponding discussion in section IIID3.
+In contrast to the case of Ar substrate, energy transfer
+to MgO dipoles has actually only a small effect for the
+overall ionic dynamics. But more subtle properties as
+optical response and trajectories of free charges (to be
+discussed in a subsequent publication) will be sensitive
+to such details.
+2. Energy transfers ”at” impact
+Notwithstanding asymptotic thermalization, the fast
+energy transfer to the substrate in the early stages is
+an interesting observable characterizingthe collision pro-
+cess. Figure 12 shows the kinetic energy of the substrate
+ 0 0.5 1 1.5 2 2.5 3 3.5
+ 0 2 4 6 8 10Ekin,substrate (eV)
+E0
+kin (eV)Na6@MgO
+Na8@MgO
+Na6@Ar
+Na8@Ar
+FIG. 12: Heating of the MgO or Ar substrate after impact
+of Na6and Na 8. The energy transfer to Ar is independent of
+the cluster structure, whereas the energy transfer to MgO is
+not. Na 6transfers about twice as muchenergyas thecompact
+Na8.
+soon after the collision, i.e. averaged over the first 2 ps
+after impact, as a function of the initial kinetic energy of
+the cluster E0
+kin. Apparently the energy absorbed by the
+substrate is proportional to E0
+kin. But the slope depends
+very much on cluster and surface types. The soft Ar sub-
+strate absorbs much more energy than MgO, typically a
+bit more than 50% of the initial kinetic energy. The
+softness and the rather small surface corrugation of Armake the process insensitive to the actual cluster which
+is approaching. That is different for MgO. There is al-
+ways less energy absorption by the substrate and there
+is a strong dependence on the cluster configuration. Na 6
+transfers more than twice as much energy as Na 8. The
+strong surface corrugation of MgO induces that sensitiv-
+ityto cluster geometry. Remind that Na 6does notmatch
+very well to the MgO surface while Na 8does (see section
+IIF).
+ 0 1 2 3 4 5 6
+ 0 2 4 6 8 10inst. energy loss (eV)
+E0
+kin  (eV)Na6
+Na8
+0.55*E0
+kin 0.1 1 10init. gain:  (Ekinmax-E0kin)/E0kinNa6
+Na8
+FIG. 13: Upper panel : The gain in kinetic energy of the clus-
+ter when approaching the MgO surface; lower panel : Instan-
+taneous energy loss, defined as the difference between max-
+imum kinetic energy before the impact and next maximum
+after impact.
+More information about what is happening directly
+around impact time can be obtained by reading off ob-
+servables at shorter time scales (shorter than the 2 ps
+used above). Figure 13 shows two such observables as a
+function of initial kinetic energy. The upper panel shows
+the energy gain in the approaching phase due to the ac-
+celeration by the polarization potential. It is defined as
+the difference of the first maximum of the cluster kinetic
+energy and the initial energy. Na 6acquires slighlty more
+energy than Na 8because it has a non-vanishing dipole
+moment which, in turn, enhances the polarizationattrac-
+tion. There is a large gain in the low energy range (the
+regime of soft deposit), while the trend becomes very
+flat for fast collisions. This is probably due to a move
+from adiabatic to non-adiabatic relaxation processes in
+the surface. For very low impact velocities, the sur-
+face ions have time to follow the forces from the cluster,
+whereas for very high velocities the surfaces ions do not
+have enough time to respond before the cluster collides.
+Thelowerpaneloffigure13showsanattempttoquantify
+an “instantaneous energy loss”. To that end, we take dif-
+ference between the maximum kinetic energy before the
+impact and the next maximum after the impact. Obvi-
+ously the instantaneous energy transfer is practically the11
+same for Na 6and Na 8independent of their differences in
+structure; and the energy loss is approximately propor-
+tional to E0
+kin. About half (more precisely around 55%)
+of the impact energy is withdrawn from the cluster in
+that first round.
+3. Redistribution of initial kinetic energy
+ 0 0.2 0.4 0.6 0.8 1
+ 0  2  4  6  8  10E / Ekinmax
+E0
+kin (eV)Na6 0 0.2 0.4 0.6 0.8 1E / Ekinmax
+Na8Na, kin. transl.
+Na, kin. relat.
+MgO, kin.
+MgO, pot.
+dip, pot.
+FIG. 14: Distribution of impact energy amongst various com-
+ponents : Kinetic energy of the cluster (split into intrinsi c,
+open squares, and translational, close squares, motion), k i-
+netic energy of the MgO substrate (open circles), potential
+energy in the oxygen dipoles (triangles), and other potenti al
+energy in the substrate (close circles). Upper panel : Na 8
+with MgO, lower panel : Na 6with MgO.
+It is, furthermore, interesting to see how the ini-
+tially available energyis distributed overthe various con-
+stituents. Such energy balance is shown in figure 14 as
+a function of E0
+kin. All energies have been averaged over
+2 ps after impact time. They are drawn relative to the
+maximum kinetic energy of the cluster before impact.
+The energy terms do not necessarily sum up to one be-
+cause the interaction between substrate and cluster is
+omitted, as well as the intrinsic potential energy of the
+cluster. Low energies ( <2.72 eV for Na 8and<2.18 eV
+for Na 6) represent the regime of deposition. The largest
+amount of energy is used up here for intrinsic cluster
+motion combined with potential energy of the substrate.
+The latter is responsible for the attachment of the clus-
+ter to the surface. The higher energies represent the
+regime of reflection. The share of energies depends here
+on the cluster geometry. For Na 6, the translational mo-
+tion takes the lead, whereas the intrinsic motion shrinks
+to almost zero. That complies with the trajectories in
+figure 6 which show that the cluster is repelled from the
+surface with opposite orientation but only weakly per-turbed structure. The reflection of the top ion seems to
+proceed independently from the five-fold ring. The ring
+hits first and departs first while the top ion is reflected
+later thus departing behind the ring.
+For both clusters, kinetic and potential energies of the
+MgO behave similar. For high E0
+kinin the reflection
+regime, potential and kinetic energy are equal. The clus-
+ter quickly transfers some momentum to the substrate
+at impact and then disappears. This leaves the sub-
+strate ions oscillating around their equilibrium positions
+in harmonic motion, associated to equipartition between
+kinetic and potential energy. For high E0
+kinin the soft
+landing domain, the situation is different. The poten-
+tial energy becomes the dominant contribution because
+the cluster is adsorbed and remains in contact with the
+surface. This distorts the surface and leads to a large po-
+tential energy. Polarization energy is dominating in the
+potential energy. But the isolated contribution from the
+oxygen dipoles, also shown in figure 14, is comparatively
+small. The polarization within the oxygen ions in fact
+remains small as compared to the polarization caused by
+the displacement of O versus Mg, each one carrying a net
+charge of ±2e.
+IV. CONCLUSIONS
+We have analyzed the dynamics of deposition of small
+Na clusters on an MgO surface taking up a well tested hi-
+erachical QM/MM modeling where the cluster electrons
+are treated quantum mechanically by time-dependent
+local-density approximation and the cluster ions as well
+as the substrate atoms by classical molecular mechanics.
+The dynamical polarizability of the substrate atoms is
+taken into account to describe correctly the strong po-
+larization effects in the cluster–material interaction. The
+results are compared to deposition on Ar surface which
+is much softer than MgO.
+Test cases were Na 6, which is a strongly oblate cluster
+with a finite dipole momentum, and the well bound and
+highly symmetrical Na 8. The general pattern are simi-
+lar : The clusters are very quickly stopped by the sub-
+strate, they transfer a rather small amount of energy to
+the substrate while acquiring strong internal excitation.
+For largerimpact energies, the clusters are reflected from
+the surface. This reflection is, of course, inelastic and
+leaves the clusters departing in highly excited intrinsic
+motion. There are, on the other hand, significant differ-
+ences between the two cases, since the cluster geometry
+has a large influence on the dynamics. The main effects
+come from the strong surface corrugation of MgO(001).
+Na6does not match the surface structure and thus ac-
+quires significant lateral motion in contrast to Na 8which
+keeps better on a vertical track. Moreover, Na 6transfers
+more energy to the substrate than Na 8.
+In comparison to Ar(001) surface, we find a simi-
+lar energy range for deposition and reflection. The de-
+tails, however,differdramatically. DepositiononAr(001)12
+transfers most of the energy to the substrate leaving a
+rather mildly excited cluster on the surface while there is
+very little energy transfer to MgO(001) and large intrin-
+sic excitation of the cluster. Reflection from Ar(001) is
+achieved at the price of severe surface destruction while
+MgOremainsintact atthe dangerthat the highlyexcited
+departing cluster may fragment later on.
+The detailed energy balance differs in the deposition
+and reflection regime. In case of deposit, most energy
+is going to intrinsic cluster excitation and substrate po-
+larization. In case of reflection, there is, of course, more
+translational energy left for the cluster and the substratedevelops equipartition of kinetic and potential energy re-
+lated to the remaining small, nearly harmonic, oscilla-
+tions.
+Acknowledgements
+This work was supported by the Deutsche Forschungs-
+gemeinschaft (RE 322/10-1, RO 293/27-2), Fonds der
+Chemischen Industrie (Germany), a Bessel-Humboldt
+prize, and a Gay-Lussac prize.
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\ No newline at end of file
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+arXiv:1001.0698v1  [astro-ph.CO]  5 Jan 2010Mon. Not. R. Astron. Soc. 000, 1–9 (2009) Printed 24 October 2018 (MN L ATEX style file v2.2)
+Ionized gas in the starburst core and halo of NGC 1140⋆
+M. S. Westmoquette1†, J. S. Gallagher III2, L. de Poitiers1
+1Department of Physics and Astronomy, University College Lo ndon, Gower Street, London, WC1E 6BT
+2Department of Astronomy, University of Wisconsin-Madison , 5534 Sterling, 475 North Charter St., Madison WI 53706, USA
+Accepted 2009 December 17. Received 2009 December 3; in orig inal form 2009 October 15
+ABSTRACT
+We present deep WIYN H αSparsePak and DensePak spatially-resolved optical spec-
+troscopy of the dwarf irregular starburst galaxy NGC 1140. The d ifferent spatial reso-
+lutions and coverage of the two sets of observations have allowed u s to investigate the
+properties and kinematics of the warm ionized gas within both the cen tral regions of
+the galaxy and the inner halo. We find that the position angle of the H αrotation axis
+for the main body of the galaxy is consistent with the H irotation axis at PA = 39◦,
+but that the ionized gas in the central 20 ×20 arcsecs ( ∼2×2 kpc) is kinematically
+decoupled from the rest of the system, and rotates at a PA appro ximately perpendic-
+ular to that of the main body of the galaxy at +40◦. We find no evidence of coherent
+large-scale galactic outflows. Instead multiple narrow emission line co mponents seen
+within a radius of ∼1–1.5 kpc, and high [S ii]/Hαratios found beyond ∼2 kpc im-
+plying a strong contribution from shocks, suggest that the intens e star formation is
+driving material outwards from the main star forming zone in the for m of a series of
+interacting superbubbles/shells.
+A broad component (100 /lessorsimilarFWHM/lessorsimilar230 kms−1) to the H αline is identified
+throughout galaxy disk out to >2 kpc. Based on recent work looking at the origins of
+this component, we conclude that it is produced in turbulent mixing lay ers on the sur-
+faces of cool gas knots embedded within the ISM, set up by the impa ct of the ionizing
+radiation and fast-flowing winds from young massive star clusters. Our data suggest
+a physical limit to the radius where the broad emission line component is significant,
+and we propose that this limit marks a significant transition point in the development
+of the galactic outflow, where turbulent motion becomes less domina nt. This mirrors
+what has recently been found in another similar irregular starburst galaxy NGC 1569.
+Key words: galaxies: individual (NGC 1140) – galaxies: starburst – galaxies: ISM –
+ISM: kinematics and dynamics – ISM: jets and outflows.
+1 INTRODUCTION
+Low-metallicity, starbursting, dwarf galaxies are a parti c-
+ularly interesting class of galaxy for a number of reasons:
+(1) their low metallicities make them excellent analogues t o
+intensely star forming galaxies at high-redshift, which in hi-
+erarchical galaxy formation models are thought to be the
+building blocks of present-day massive galaxies, and (2) th e
+starbursteventcanhavemuchmoreofadestructiveeffecton
+the galaxy than for more massive systems; their low gravita-
+tional potentials may also mean that for those with flattened
+morphologies, outflowing material is much more likely to
+escape altogether (Larson 1974; Mac Low & Ferrara 1999)
+⋆Based on observations made with the WIYN SparsePak and
+DensePak instruments.
+†E-mail:msw@star.ucl.ac.uk(note the reverse may be true for those with more irreg-
+ular morphologies; Silich & Tenorio-Tagle 2001). Although
+the ejection of the ISM (including the freshly chemically en -
+riched material) has clear consequences for the future evol u-
+tion of the system in terms of its dynamics and star for-
+mation rate (Dekel & Silk 1986), and for the enrichment
+of the intergalactic medium (Aguirre et al. 2001; Cen et al.
+2005), some studies suggest that ejection of hot gas through
+bubble blow-out may not be as efficient as first thought
+(De Young & Heckman 1994; Martin 1998). It is therefore
+important to study such systems to understand how gas is
+removed and what effects this has in the evolution of the
+galaxy.
+NGC 1140 is a nearby (20 Mpc; Moll et al. 2007),
+low metallicity (LMC-like, 0.4 Z ⊙; Izotov & Thuan 2004;
+Nagao, Maiolino & Marconi 2006; Moll et al. 2007) blue
+compact dwarf (BCD) galaxy with a total star formation
+c/circlecop†rt2009 RAS2M.S. Westmoquette et al.
+rate of 0 .7±0.3 M⊙yr−1(Hunter, van Woerden & Gal-
+lagher 1994, hereafter H94b; Moll et al. 2007). The cur-
+rent star formation is dominated by an extremely H α-
+luminous central concentration that is known to host a num-
+ber of young massive clusters (YMCs) with ages <5 Myr
+andmasses >105M⊙(Hunter, O’Connell & Gallagher 1994;
+Moll et al. 2007); this extreme youth is supported by the
+detection of strong WR star signatures (Moll et al. 2007).
+It is thought that these clusters are the most recent
+products of a galaxy-wide starburst induced by a merger
+or interaction with a low luminosity, gas-rich companion
+(H94b): the optical colours (H94b), broad-band photometry
+(Cid Fernandes et al. 2003), and photometric star cluster
+age-dating (de Grijs et al. 2004) all suggest extensive star
+formation throughout the galaxy in the last 1 Gyr, and par-
+ticularly in the last 20 Myr.
+The misalignment of the galaxy’s structural compo-
+nents are consistent with the merger scenario. On large
+scales (/lessorsimilar40 kpc), the H igas is elongated along a position
+angle (PA) of −51◦(although strongly warped on the south-
+eastern side). In the inner regions (central ∼10 kpc), the H i
+gas is concentrated along an approximately perpendicularl y
+oriented ridge (PA ≈+22◦).To first order this neutral
+gas all rotates in almost solid-body rotation, with
+the line of nodes at the PA of −51◦(i.e. Hirotation
+axis PA = + 39◦), and exhibits a high velocity dispersion
+indicating that the system is not relaxed (H94b). In the
+V-band however, the morphology of this inner region is sig-
+nificantly different, being roughly rectangular in shape wit h
+a PA of 0◦(H94b). At the south-west end, at a distance
+of∼2.5 kpc, the rectangle has a hook that extends towards
+the west, traced by a chain of H iiregions (see Fig. 1). This
+chain is aligned with the PA ≈+22◦Hiridge, and it is
+these Hiiregions that are coincident with the peak of the
+Hiemission (H94b), not the central H αconcentration as
+might be expected. Further out at fainter V-band surface
+brightness levels, the galaxy appears elongated towards th e
+north-west and south-east along a PA ≈ −15◦. Further out
+still, a number of low surface brightness shells are observe d
+that may be linked with either past star-formation events or
+the merger/interaction (H94b).
+HSTimaging(Fig.1)showsthatthecentralH αconcen-
+tration extends into a network of complex shells, filaments
+and bubbles that fill the ISM of the galaxy out to a distance
+of∼2 kpc. High [S ii]/Hαline ratios (H94b; Burgh et al.
+2006) and strong [Fe ii] emission (de Grijs et al. 2004) are
+both indicators that shocks play an important role in the
+halo. Thus the presence of multiple YMCs, the high H αlu-
+minosity,strongevidenceofshocks,andthis“froth”ofstr uc-
+ture in the ionized gas all suggest strong starburst-driven
+feedback is taking place, although to date no kinematic ev-
+idence has been found for a coherent, galaxy-wide outflow.
+With all this in mind, we have obtained deep spatially-
+resolved optical spectra of NGC 1140 from two instruments
+with differentspatial resolutions andcoverage, toinvesti gate
+theproperties andkinematicsofthewarm ionizedgas within
+both the central regions of the galaxy and the inner halo.
+In this work, we adopt a systemic velocity for NGC
+1140 of 1475 kms−1giving a distance of 20 Mpc (Moll et al.
+2007), meaning 1′′∼97 pc.2 OBSERVATIONS AND DATA REDUCTION
+We observed NGC 1140 on two separate occasions with the
+WIYN1SparsePak and DensePak instruments. These two
+data sets, providing observations at complimentary spatia l
+resolutions, have allowed us to examine the kinematics and
+nebular properties of the galaxy’s disk, halo and nuclear
+regions in some detail.
+2.1 SparsePak
+Observations of NGC 1140 were obtained with the
+SparsePak instrument (Bershady et al. 2004) on the WIYN
+3.5-m telescope. SparsePak is a “formatted field unit” sim-
+ilar in design to a traditional IFU, except that its 82
+fibers are arranged in a sparsely packed grid, with a small,
+nearly-integral core (Bershady et al. 2004). SparsePak was
+designed to maximise throughput and spectral resolution at
+the expense of spatial coverage/resolution, and as such has a
+total light throughput of ∼90 per cent longwards of 500 nm.
+Each fibre has a diameter of 500 µm, corresponding to 4 .′′69
+on the sky; the formatted field has approximate dimensions
+of 72×71.3 arcsecs, including seven sky fibres located on the
+north and west side of the main array separated by ∼25′′.
+The mapping order of fibers between telescope and spectro-
+graph focal planes was purposefully designed in a fairly ran -
+domised fashion inorder todistribute skyfibresevenlyalon g
+the slit and minimise the effects of spectrograph vignetting
+on the summed sky spectrum. SparsePak is connected to
+the Hydra bench-mounted echelle spectrograph which uses
+a T2KC 2048 ×2048 CCD detector.
+We obtained observations of NGC 1140 centred on the
+Hαpeak atα=2h54m33.s43,δ=−10◦01′39.′′3 (J2000) during
+the period 13–16th December 2004 with a total exposure
+time of 3 ×1800 secs. The position of the SparsePak foot-
+print is shown in Fig. 1, overlaid on an HST/ACS-WFC
+continuum subtracted F658N image (Prop ID: 9892, PI:
+Jansen). Using an order 8 grating with an angle of 63.25◦
+(giving a spectral coverage of 6450–6865 ˚A and dispersion
+of 0.20˚Apix−1), we were able to cover the nebular emis-
+sion lines of H α, [Nii]λλ6548,6583, and [S ii]λλ6716,6731.
+A number of bias frames, flat-fields and arc calibration ex-
+posures were also taken together with the science frames.
+Basic reduction was achieved using the ccdproc task
+within the noao iraf package. Instrument-specific reduc-
+tionwasthenachievedusingthe hydratasksalsowithinthe
+noaopackage. The first step was to run apfindon the flat-
+field exposure to automatically detect the individual spect ra
+on the CCD frame. The output of this task is an aperture
+identification table which can be used for each science frame
+to extract out the individual spectra. The task dohydra
+was then used to perform the bias subtraction, flat-fielding
+and wavelength calibration. The datafile at this stage con-
+tained 82 reduced and wavelength calibrated spectra. A rep-
+resentative sky spectrum was created by averaging the seven
+sky fibre spectra, and sky-subtraction was achieved by re-
+running dohydra with the sky-subtraction option switched
+on. Cosmic-rays were cleaned from the data using lacosmic
+1The WIYN Observatory is a joint facility of the University of
+Wisconsin-Madison, Indiana University, Yale University, and the
+National Optical Astronomy Observatories.
+c/circlecop†rt2009 RAS, MNRAS 000, 1–9Ionized gas in NGC 1140 3
+(van Dokkum 2001), before final combination of the individ-
+ual frames was achieved using imcombine . The final datafile
+now contained 82 reduced, wavelength calibrated and sky-
+subtracted spectra. An example reduced and labelled spec-
+trum is shown in the top panel of Fig. 3.
+In order to determine an accurate measurement of the
+instrumental contribution to the spectrum broadening, we
+selectedhighS/Nspectrallinesfromawavelengthcalibrat ed
+arc exposure that were close to the H αline in wavelength,
+and sufficiently isolated to avoid blends. After fitting these
+lines with Gaussians in all 82 apertures, we find the average
+instrumental width is 0 .7±0.02˚A = 31.4±0.5 kms−1.
+2.1.1 Emission line fitting
+The S/N and spectral resolution of the data have allowed
+us to accurately quantify the line profile shapes of the emis-
+sion lines. In the central regions where the S/N is highest,
+we find the emission lines to be composed of two bright,
+narrow components (FWHM ∼20–100 kms−1;corrected
+for instrumental broadening ) overlaid on afainter broad
+component (FWHM ∼100–230 kms−1;also corrected for
+instrumental effects ). In some spaxels further out, we see
+only the two narrow components. Two split narrow compo-
+nents are indicative of expanding gas motions.
+To quantify the emission line profile shapes, we fitted
+model Gaussian profiles to each line detected in each spaxel
+of the SparsePak field using a customised IDL-based χ2fit-
+ting routine called pan(Peak ANalysis; Dimeo 2005). A de-
+tailed description of the program and the customisations we
+have made to it are given in Westmoquette et al. (2007a).
+As mentioned above, in many cases we found that addi-
+tional Gaussian components were needed to satisfactorily
+fit the integrated line profile; the number of components
+fitted to each line was determined using a combination of
+visual inspection and the χ2fit statistic output by pan. For
+the multi-component cases, we specified a number of rules
+to identify the different line components. This consistent
+approach helped both in the minimisation of the fit and in
+the subsequent analysis. For double-components, the secon d
+component (hereafter C2) was sometimes required to fit a
+broad underlyingprofile, and in others tofit the fainter com-
+ponent of a split line. Where three components were iden-
+tified, the broadest component was assigned to component
+2 (C2), and after that, the brightest to component 1 (C1)
+and the tertiary component to C3. We note here that in the
+double-component cases where a broad component was not
+detected, the secondary narrow component can be thought
+of as equivalent to C3 in the triple-component cases (where
+C2represents thebroad component). In all cases the indi-
+vidual Gaussian models were constrained to have a
+width greater than the aforementioned instrumental
+width.
+AnumberofexampleH αlineprofilesfromvariousspax-
+els are shown in Fig. 4, together with the individual Gaus-
+sian fits required to model the integrated line-shapes. Thes e
+were compiled to show the variation in profile shapes found.
+2.2 DensePak
+DensePak (Barden et al. 1998) was a small fibre-fed integral
+fieldarraythatwas usuallyattachedattheNasmythfocus ofthe WIYNtelescope2. Duringour run DensePak was instead
+attached to the f13.7 modified Cassegrain port, thus chang-
+ing the detector plate-scale from the documented value. The
+array has 91 fibres, each with a diameter of 300 µm (1.3′′
+on the sky at the Cassegrain focus). The fibre-to-fibre spac-
+ing is 400 µm making the overall dimensions of the array
+12.4×19.8 arcsecs. Four additional fibres are offset by ∼30′′
+from the array centre and serve as dedicated sky fibres. The
+arrangement of the DensePak fibre array on the sky, includ-
+ing the sky fibres, is shown in Sawyer (1997, note the di-
+mensions given in this reference apply to DensePak at the
+Nasmyth focus). At the time of observation, there were 8
+damaged and unusable fibres in the main array making a
+usable total of 83. DensePak’s fibre bundle was reformatted
+into a ‘pseudo-slit’ to feed the Hydra bench-mounted echell e
+spectrograph, just as for SparsePak.
+On 30th September 2003, we observed the nuclear re-
+gions of NGC 1140 with DensePak by centring the array
+on the coordinates 02h54m33.s43,−10◦01′40.′′6 with a PA
+of +90◦. The total exposure time was 1800 secs. Fig. 2
+shows the footprint of the array on the HST/ACS F658N
+image. The 600 line mm−1grating at an angle of 26.82◦
+gave a wavelength range of 5260–8110 ˚A with a dispersion
+of 1.40˚Apix−1, allowing us access to a number of optical
+nebular lines. Contemporaneous bias frames, flat-fields and
+arc calibration exposures were also observed.
+2.2.1 Reduction
+Reduction was achieved using the same methodology as out-
+lined for SparsePak, using the ccdproc andhydratasks
+withinthe noaoiraf package. Traces oftheindividualspec-
+tra were identified from the master flat-field frame, and the
+resulting aperture identification table was used to extract
+theobject spectrafrom thescience frames. After flat-fieldi ng
+and wavelength calibrating the dataset, we formed a repre-
+sentative sky spectrum by averaging the data from all four
+sky fibres from all three exposures. This sky spectrum was
+then subtracted from each of the object spectra. Final com-
+bination of the individual frames was done using imcom-
+bineafter cosmic-rays were removed with lacosmic . The
+final datafiles contained 83 reduced and sky-subtracted ob-
+ject spectra.
+In order to determine an accurate measurement of the
+instrumental broadening, we fitted a single Gaussian to a
+number of high S/N, isolated arc lines for all 83 aper-
+tures and took the average: the resulting measurement was
+165±10 kms−1. Note that this is considerably higher than
+that of the SparsePak data, and that, for the most part, the
+emission line profiles are unresolved.
+Fig. 2 shows the position of the DensePak footprint on a
+cut-out of the HSTF658N image with the SparsePak fibres
+from Fig. 1. An example reduced spectrum is shown in the
+bottom panel of Fig. 3. A number of residuals remain (e.g.
+around [O i]λ5577 and Na iλλ5890,5896) as a result of an
+imperfect sky subtraction.
+Given the spectral resolution of these data, we fitted
+the emission lines of H α, [Nii]λ6583 and [S ii]λλ6717,6731
+in each spectrum with only a single Gaussian component
+2DensePak was decommissioned in 2008.
+c/circlecop†rt2009 RAS, MNRAS 000, 1–94M.S. Westmoquette et al.
+(constrained to have a width greater than the afore-
+mentioned instrumental width ). Twofibres coveringthe
+brightest, central-most regions show evidence of multiple
+components in the H αemission line; in both cases they
+are resolved into two components separated by ∼100 kms−1
+(these additional components are not shown in the emission
+line maps presented below).
+3 EMISSION LINE MAPS
+In this section we present and describe the SparsePak and
+DensePak emission line maps created from the line profile
+fits described above. We begin by looking at the SparsePak
+results.
+3.1 SparsePak
+3.1.1 Kinematics
+The spatial distribution FWHM and radial velocity of each
+Gaussian component identified are shown in Figs. 5 and 6,
+respectively.
+Narrow/lessorsimilar100 kms−1Hαcomponents are detected out
+to>40′′(∼3.8 kpc) from the nucleus (Fig. 5 left panel),
+corresponding well to the limit of emission seen in the deep
+Hαimage presentedbyH94b.InC1,theimprintofthelarge-
+scale rotation of the galaxy can clearly be seen (Fig. 6 left
+panel), and a polynomial surface fit to these data shows that
+thepeakrotation amplitude( ∼20kms−1kpc−1)isalongPA
+=−51◦(i.e. rotation axis PA = + 39◦), in agreement
+with the H idata of (H94b). In C1, the inner ∼20×20′′
+of the galaxy shows hints of rotating at a different position
+angle. This is discussed further in light of the DensePak dat a
+presented in Section 3.2.1. We would also like to note that
+the movement of H α-emitting gas associated with the chain
+of Hiiregions (Fig 1) is consistent with the rest of the halo
+(and with the findings of H94b).
+An underlying broad component (C2; 100–230 kms−1)
+can be identified throughout the disk and inner halo, ex-
+tending out to a radius of /greaterorsimilar2 kpc (Fig. 5 central panel).
+In general, broader line widths are seen on the western side
+of the galaxy. This is mirrored in both C1 and C2, where
+C2 widths increase up to FWHM = 230 kms−1(Fig. 5;
+see also Fig. 4 top-right panel). No broad H αemission is
+associated with the chain of H iiregions towards the south-
+west. Here only two narrow components are present (note
+that the second, fainter component has been assigned to C2
+– our assignment convention is described in Section 2.1.1).
+In general, the radial velocities of the broad C2 component
+are blueshifted by ∼10–50 kms−1compared to C1, suggest-
+ing that it is associated with gas expanding out of the star
+forming zones. These speeds are more typical for expanding
+supershells as compared to the much larger velocities seen
+in galactic winds.
+At this point we would like to note that in NGC 1569
+we compared the range of emission line radial velocities
+measured in individual spaxels over one 5 ×3.5 arcsecs
+GMOS IFU field-of-view (Westmoquette et al. 2007b) with
+the width of lines from SparsePak fibres that covered ap-
+proximately the same area (Westmoquette et al. 2008), and
+found that the SparsePak FWHM measurements were verylikely artificially elevated at the level of a few tens of kms−1
+by multiple (unresolved) emission components along the
+line-of-sight. An effect similar to this is also very likely t o be
+at work here, particularly in the central regions. Here, the
+presence of complex filamentary emission line structures im -
+plies multiple overlapping shells or dynamical components
+on scales less than the DensePak or SparsePak fibre sizes.
+The presence of a second narrow component (assigned
+to C3 where a broad component is identified and C2 oth-
+erwise) is indicative of expanding gas motions. These split
+lines are seen within the nuclear regions (central part of
+array) and in the chain of H iiregions, where the velocity
+differences between the two components are 10–60 kms−1.
+These kind of velocity differences are consistent with those
+expected from regions of intense star formation, and sugges t
+the presence of outflowing gas.
+3.1.2 Nebular diagnostics
+The [Sii]λλ6717,6731 lines were used to derive the electron
+densities of the ionized gas. For the majority of the galaxy
+where the the [S ii] lines were detected (the central ∼2 kpc
+and along the chain of H iiregions), the derived values fell
+below the low-density limit at ∼100 cm−3. However, in an
+area encompassing the central few spaxels and those extend-
+ing∼20′′to the south, we find C1 densities rise to a few
+100 cm−3. In the spaxels located in the central part of the
+array (inner ∼1 kpc) a second [S ii] component could also be
+identified. The densities measured in this component were
+again mostly at or just above the low-density limit.
+These results are consistent with those of previous stud-
+ies: H94b find densities of a few 100 cm−3in the central
+20–40′′of the galaxy, and Moll et al. (2007) use oxygen line
+diagnostics to find a density of 60 ±50 cm−3in the nebular
+region directly surrounding starburst knot A (the northern
+star cluster complex). Burgh et al. (2006) find electron den-
+sities of just over 100 cm−3from the [S ii] ratio measured in
+the nuclear regions.
+The forbidden/recomination line flux ratio of
+[Sii](λ6717+λ6731)/Hαcan be used as an indicator
+of the number of ionizations per unit volume, and thus
+the ionization parameter, U(Veilleux & Osterbrock 1987;
+Dopita et al. 2000, 2006). [S ii]/Hαis also particularly
+sensitive to shock ionization because relatively high-
+density, partially ionized regions form behind shock
+fronts which emit strongly in [S ii] thus producing an
+enhancement in [S ii]/Hα(Dopita 1997; Oey et al. 2000;
+Osterbrock & Ferland 2006). Maps of this ratio in the three
+line components are shown in Fig. 7. In the scale bar, we
+have indicated a fiducial value of log([S ii]/Hα) =−0.4
+above which non-photoionized emission is thought to play
+a dominant role (Dopita & Sutherland 1995; Kewley et al.
+2001; Calzetti et al. 2004). In environments such as these,
+the most likely excitation mechanism after photoionizatio n
+is that of shocks. Outside of the central 2 kpc region,
+the ratios in all three line components rise significantly
+above this non-photoionization threshold. This suggests
+that shocks dominate the gas excitation in the halo (as
+previously noted by Calzetti et al. 2004 in the outer regions
+of other moderate luminosity dwarf starburst galaxies) and
+thus may play a central role in driving the gas outward.
+The ratio of [N ii]λ6583/Hαcan be used in a sim-
+c/circlecop†rt2009 RAS, MNRAS 000, 1–9Ionized gas in NGC 1140 5
+ilar fashion to [S ii]/Hαas a diagnostic of the gas ex-
+citation level and to search for the presence of shocks
+(Veilleux & Osterbrock 1987; Dopita & Sutherland 1995).
+Fig. 8 shows maps of this ratio for all three components;
+here the marker in the scale bar represents a fiducial non-
+photoionized threshold of log([N ii]/Hα) =−0.1. Again we
+see that the line ratios increase from the central region out -
+wards, but in this case never rise above our adopted non-
+photoionization threshold. Here we should note that metal-
+licity plays a strong part in setting the H iiregion/shock
+boundary for these diagnostic ratios, in the sense that a
+lower N abundance (as would be expected at the low metal-
+licity of NGC 1140) would decrease the non-photoionization
+threshold (Dopita et al. 2006).
+Again our results agree very well with previous stud-
+ies: H94b measured log([S ii]/Hα)∼ −0.7 to−0.3 and
+log([Nii]/Hα)∼ −1 to−0.7 in the central regions of NGC
+1140, and Burgh et al. (2006) found [S ii]/Hαto vary with
+radius from a minimum of log([S ii]/Hα) =−0.9 at the peak
+of the Hαemission, to log([S ii]/Hα) =−0.3 at the edges of
+the galaxy.
+3.2 DensePak results
+As described in Section 2.2.1, the emission lines in the
+DensePak spectra were almost all unresolved meaning that
+a single Gaussian component fit was all that was required.
+The spatial distribution of H αflux and radial velocity, and
+the [Sii]/Hαand [Nii]/Hαline ratios, resulting from these
+fits are shown in Fig. 9. The location of the DensePak array
+on theHSTHαimage is shown in Fig. 2.
+3.2.1 Kinematics
+Fig. 9b shows the H αradial velocity map. It is clear that
+the velocity gradient observed does not follow what is seen
+on large scales in, for example, our SparsePak maps (Fig. 6).
+It is instead offset by approximately 90◦to a PA of ∼+40◦.
+This corroborates what was hinted at in the SparsePak data
+(Section3.1.1;andinthelong-slit dataofBurgh et al. 2006 ),
+and suggests the presence of a counter-rotating core within
+the central ∼20×20′′with a rotation axis aligned roughly
+perpendicular to the main body rotation axis. This was also
+found in the long-slit data of Burgh et al. (2006).
+3.2.2 Nebular diagnostics
+Most DensePak spaxels exhibit [S ii]-derived electron den-
+sities below the low-density limit (100 cm−3), although a
+small number show densities of a few hundred cm−3. These
+results are consistent with our SparsePak measurements and
+with previous studies as described above.
+In both [S ii]/Hαand [Nii]/Hα(Fig. 9c and d) the low-
+est ratios are coincident with the location of the H αpeak
+(Fig. 9a), then increase radially outward (the similarity o f
+the two maps is striking). No evidence for shocked line ra-
+tios (in either [S ii]/Hαor [Nii]/Hα) are found in these cen-
+tral regions. Our measurements are again very consistent
+with those from the SparsePak data (once the finer spatial
+sampling has been taken into account), and with previous
+studies (H94b; Burgh et al. 2006).4 DISCUSSION AND CONCLUSIONS
+In this paper we have presented deep spatially-resolved
+optical spectra of NGC 1140 obtained with the WIYN
+SparsePak and DensePak instruments. The different spa-
+tial resolutions and coverage of the two sets of observation s
+have allowed us to investigate the properties and kinematic s
+of the warm ionized gas within both the central regions of
+the galaxy and the inner halo, and have revealed a num-
+ber of interesting results. We will now summarise these and
+discuss their implications in the context of the galaxy as a
+whole.
+4.1 Kinematical structure of the disk and halo
+We have measured the position angle of the H αrotation axis
+for main body of the galaxy, and findit tobe consistent with
+the Hirotation axis at PA = 39◦(H94b).
+The peak of the optical emission lies roughly at the
+centre of the integrated H imorphology (H94b). The H i
+ridge feature is oriented roughly perpendicularly to the ma -
+jor axis, and contains the peak of the H iemission, which
+lies to the south-west of the optical peak, coincident with
+the chain of H iiregions (H94b). Despite their appearance,
+the chain of H iiregions located towards the south-west are
+not part of a tidal tail (which might be expected to be kine-
+matically distinct), but appear to rotate together with the
+rest of the galaxy. This makes sense since we know they
+are coincident with the H iridge identified by H94b, and
+the peak of the H iemission. What do the chain of H iire-
+gions represent? Split H αlines with velocity separations of
+a few tens of kms−1are seen at the location of the chain
+of Hiiregions indicating that they are young, star forming,
+and dynamically active. It is possible that we are witness-
+ing a sequential progression of star formation through the
+disk (perhaps the H iiregions represent the stub of a tidally
+induced spiral arm), and that in a few Myr the galaxy will
+look elongated in H αalong the +22◦ridge direction as star
+formation takes hold.
+Ourdata clearly show thatthe ionized gas in the central
+20×20 arcsecs (2 ×2 kpc) region of the galaxy is kinemat-
+ically decoupled from the rest of the system, and rotates
+at a PA approximately perpendicular to that of the main
+body of the galaxy at +40◦. This finding of yet another mis-
+aligned structural component is not unsurprising, and only
+adds weight to the interaction/merger scenario proposed by
+H94b.
+4.2 The starburst environment and the properties
+of the outflowing gas
+We see no evidence of coherent large-scale outflows within
+NGC 1140. Instead, the H αmorphology (Fig 1), the fact
+that we find multiple narrow line components with velocity
+separations of 10–60 kms−1within a radius of ∼1–1.5 kpc,
+and the fact that beyond ∼2 kpc the [S ii]/Hαratios rise sig-
+nificantly implying a strong contribution to the excitation
+levels by shocks, all suggest that locally intense star form a-
+tion is driving material outwards in the form of a series of
+interactingsuperbubbles/shells(i.e.the“froth”). Thel ackof
+any coherent wind may be due to the disturbed structure of
+the galaxy disk, and/or the modest thermal pressures in the
+c/circlecop†rt2009 RAS, MNRAS 000, 1–96M.S. Westmoquette et al.
+starburst core (implied by the low electron densities) and/ or
+the low global average star formation rate ( <1 M⊙yr−1;
+H94b; Moll et al. 2007).
+Abroadcomponent(100 /lessorsimilarFWHM/lessorsimilar230kms−1)tothe
+Hαline is seen throughout the galaxy disk out to >2 kpc.
+Broad emission line components have been observed before
+in the Hβline profiles extracted from the nebularegions sur-
+rounding the SSC knots A and B by Moll et al. (2007), and
+in many other nearby starburst galaxies (e.g. Izotov et al.
+1996; Homeier & Gallagher 1999; Marlowe et al. 1995;
+Mendez & Esteban 1997; Sidoli, Smith & Crowther 2006).
+For many years the nature of the energy source for these
+broad lines has been contested, but recent detailed IFU
+studies of the ionized ISM in the starburst galaxies NGC
+1569 and M82 (Westmoquette et al. 2007a,b, 2009a,b) have
+allowed us to address this problem. Accurately decompos-
+ing the emission line profiles and mapping out their prop-
+erties led us to the conclusion that the broad underlying
+component is produced in turbulent mixing layers (TMLs;
+Slavin et al. 1993; Binette et al. 2009)onthesurfaces ofcoo l
+gas knots, set up by the impact of the ionizing radiation and
+fast-flowing winds from the YMCs (Pittard et al. 2005). In
+both these galaxies, we found evidence for a highly frag-
+mented ISM that provides copious cloud surfaces on which
+TMLs can form thus explaining the pervasiveness of this
+broad component. The existence of multiple YMCs in the
+central region, the complex, “frothy” H αmorphology of the
+disk and halo (Fig. 1), and the similarity in width and mor-
+phologyofthebroadcomponentobservedhere,strongly sug-
+gest a similar origin for this component. The same conclu-
+sion was reached by Moll et al. (2007).
+Can we determine if the radius at which we stop see-
+ing the broad component is physical or simply an effect of
+the data? Examination of a number of H αline profiles with
+similar S/N levels from spaxels near this boundary reveals
+slight evidence for the former: spaxels 82 (coordinates +20 ,
+0)and37( −25,−8)bothexhibitabroadH αcomponentand
+have integrated H αS/N ratios of ∼55 and∼37, respectively,
+whereas spaxels 28 ( −20, 0) and 63 (15,8), with integrated
+HαS/N ratios of ∼85 and∼45, respectively, do not. Finding
+a physical limit to the broad component region echoes what
+we found in NGC 1569 (Westmoquette et al. 2008). Here
+we found that the radius at which the broad line component
+ceased to exist roughly corresponded to the point that the
+Hαprofile started showing a secondary narrow component.
+Clearly the situation is somewhat different in NGC 1140
+since the broad component region is much larger ( ∼2 kpc
+vs. 500 pc), and we see split narrow components within the
+central regions, but this limit to the broad component re-
+gion may still mark a significant transition point in the de-
+velopment of the galactic outflow, where turbulent motion
+becomes less dominant. Further investigation at higher spa -
+tial resolution and S/N and with comparison to deeper H α
+imaging would be needed to better quantify this potential
+transition region.
+We can also try to assess the relationship between
+the state of the outflow in NGC 1140 and that of NGC
+1569 and the well-known starburst M82. On galaxy size-
+scales, the NGC 1569 outflow exhibits an irregular su-
+pershell morphology with very little collimation or pre-
+ferred outflow direction (Hunter et al. 1993; Martin 1998;
+Westmoquette et al. 2008). M82’s outflow, on the otherhand, consists of two large bi-polar plumes of outflow-
+ing gas that are relatively well structured and colli-
+mated (Shopbell & Bland-Hawthorn 1998; Ohyama et al.
+2002; Westmoquette et al. 2009a). The halo of NGC 1140,
+with its complex, frothy morphology, seems mid-way be-
+tween these two examples.
+Firstly, NGC 1140 is almost three orders of magnitude
+more massive than NGC 1569 and has a rotational velocity
+vrot∼100 kms−1(H94b) compared to ∼35 kms−1in NGC
+1569 (Stil & Israel 2002). Since the escape speed, vesc, scales
+asv2R,vescin NGC 1140 is >3 times greater than that
+in NGC 1569. The two galaxies have similar star formation
+rates per unit area so it may not be surprising that the burst
+in NGC 1140 is producing less in the way of an outflow.
+NGC 1140 and M82 have similar vrotand sizes, so
+should have roughly similar values of vesc. However, both
+the thermal and turbulent pressures at the base of the M82
+wind are >10 times that in NGC 1140 meaning that M82
+must have a significantly higher energy density in its ISM
+and is therefore able to drive a large-scale superwind in a
+way NGC 1140 cannot. The presence of a broad emission
+line component in both cases suggests that despite this dif-
+ference, the underlying structure of cloud boundaries and
+turbulent mixing layers are similar, and that these feature s
+are not directly linked to the magnitude of the winds.
+ACKNOWLEDGMENTS
+We thank the anonymous referee for comments that led to
+an improvement of the clarity of this paper. MSW would
+like to thank Daniel Harbeck and the staff at the WIYN
+Observatory for their support during the observing runs,
+and Phillip Cigan for reducing the DensePak observations.
+MSW also thanks the University of Wisconsin–Madison for
+the warm hospitality received in support of this project.
+JSG’s research was partially funded by the National Science
+Foundation through grant AST-0708967 to the University
+of Wisconsin-Madison.
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+Thispaperhas beentypesetfrom aT EX/LATEXfileprepared
+by the author.
+c/circlecop†rt2009 RAS, MNRAS 000, 1–98M.S. Westmoquette et al.
+Figure 1. HST/ACS-WFC continuum subtracted H αimage of NGC 1140 (inverse log scaled) with SparsePak footpr int overlaid.
+c/circlecop†rt2009 RAS, MNRAS 000, 1–9Ionized gas in NGC 1140 9
+Figure 2. HST/ACS continuum subtracted H αimage of the
+central regions of NGC 1140 (inverse log scaled) with DenseP ak
+footprint overlaid (bold) on the SparsePak footprint from F ig. 1.
+Figure 3. ExampleSparsePak andDensePak spectra showingthe
+full spectral range of each observation (in arbitrary flux un its).
+The identified nebular lines are labelled. A number of residu als
+remain in the DensePak spectrum as a result of an imperfect sk y
+subtraction.
+c/circlecop†rt2009 RAS, MNRAS 000, 1–910M.S. Westmoquette et al.
+Figure 4. Example H αline profiles chosen to represent the main types of profile sha pes observed over the field. Observed data is shown
+by a solid black line, individual Gaussian fits by dashed red l ines (including the straight-line continuum level fit), and the summed model
+profile in solid red. Flux units are arbitrary but on the same s cale. Below each spectrum is the residual plot.
+c/circlecop†rt2009 RAS, MNRAS 000, 1–9Ionized gas in NGC 1140 11
+Figure 5. SparsePak H αFWHM maps in the three H αline components. Left:C1,centre:C2 andright:C3. A scale bar is given in units
+of kms−1, corrected for instrumental broadening. The box in the left panel represents the DensePak footprint outline; and the da shed
+circle in the central panel indicates a radius of 2 kpc.
+Figure 6. SparsePak H αradial velocity maps in the three line components. Left:C1 (the dashed line represents the large-scale H i
+rotation axis, PA=39◦; H94b), centre:C2 and right:C3. A scale bar is given in units of kms−1, relative to vsys, and the vertical bar
+indicates zero. The box in the left panel represents the Dens ePak footprint outline.
+c/circlecop†rt2009 RAS, MNRAS 000, 1–912M.S. Westmoquette et al.
+Figure 7. SparsePak [S ii]λλ6717,6731/Hαratio maps in the three line components. Left:C1,centre:C2 and right:C3. The vertical
+line in the scale bar represents a fiducial ratio (log([S ii]/Hα) =−0.4) above which it is expected that a significant proportion of the
+ionization is achieved through non-photoionizing process es.
+Figure 8. SparsePak [N ii]λ6583/Hαratio maps in the three line components. Left:C1,centre:C2 and right:C3. The vertical line in
+the scale bar represents a fiducial ratio (log([N ii]/Hα) =−0.2) above which it is expected that a significant proportion of the ionization
+is achieved through non-photoionizing processes.
+c/circlecop†rt2009 RAS, MNRAS 000, 1–9Ionized gas in NGC 1140 13
+Figure 9. DensePak maps: (a) H αfluxes; (b) H αradial velocities (the dashed line indicates the axis of rot ation of the counter-rotating
+core,PA≈ −50◦); (c) [S ii]/Hαratio; (d) [N ii]/Hαratio. Scale bars are shown for each map individually. Unlik e with SparsePak, only
+one line component can be detected with DensePak due to the mu ch lower spectral resolution of these observations.
+c/circlecop†rt2009 RAS, MNRAS 000, 1–9
\ No newline at end of file
diff --git a/1001.0699.txt b/1001.0699.txt
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+arXiv:1001.0699v1  [math.DG]  5 Jan 2010Lamarle Formula in 3-Dimensional Lorentz Space
+Soley ERSOY, Murat TOSUN
+Department of Mathematics, Faculty of Arts and Sciences
+Sakarya University, 54187 Sakarya/TURKEY
+October 30, 2018
+Abstract
+The Lamarle Formula, given by Kruppa in [8], is known as a rela tion-
+ship between the Gaussian curvature and the distribution pa rameter of a
+ruled surface in the surface theory. The ruled surfaces were investigated
+in 3 different classes with respect to the character of base cu rves and rul-
+ings, [14],[15]. In this paper on account of these studies, t he relationships
+between the Gaussian curvatures and distribution paramete rs of space-
+like ruled surface, timelike ruled surface with spacelike r uling and time-
+like ruled surface with timelike ruling are obtained, respe ctively. These
+relationships are called as Lorentzian Lamarle formulas. F inally some ex-
+amples concerning with these relations are given.
+Subject Classification : 53B30, 53C50, 14J26
+Key Words : Ruled surface, distribution parameter, Gaussian curvatu re,
+Lamarle formula.
+1 Introduction
+The study of ruled surface in R3is classical subject in differential geometry. It
+has again been studied in some areas (i.e. Projective geometry, [13], Computer-
+aided design, [11], etc.) Also, it is well known that the geometry of rule d surface
+is very important of kinematics or spatial mechanisms in R3, [4], [7]. A ruled
+surface is one which can be generated by sweeping a line through spa ce. Devel-
+opable surfaces are special cases of ruled surfaces, [10]. Cylindric al surfaces are
+examples of developable surfaces. On a developable surface at leas t one of the
+two principal curvatures is zero at all points.Consequently the Gau ssian curva-
+ture is zero everywhere too. So it is meaningful for us to study non -cylindrical
+ruled surfaces.
+Lorentz metrics in 3 −dimensional Lorentz space R3
+1is indefinite. In the theory
+of relativity, geometry of indefinite metric is very crucial. Hence, th e theory of
+ruled surface in Lorentz space R3
+1, which has the metric ds2=dx2
+1+dx2
+2−dx2
+3,
+attracted much attention. The situation is much more complicated t han the
+Euclidean case, since the ruled surfaces may have a definite metric ( spacelike
+surfaces), Lorentz metric (timelike surfaces) or mixed metric. So me character-
+izations for ruled surfaces are obtained by [9]. Timelike and spacelike r uled
+surfaces are defined and the characterizations of timelike and spa celike ruled
+surfaces are found in [2], [5], [6], [14] and [15].
+12 Preliminaries
+LetR3
+1denote the 3 −dimensional Lorentz space, i.e. the Euclidean space E3
+with standard flat metric given by
+g=dx2
+1+dx2
+2−dx2
+3 (2.1)
+where (x1,x2,x3) is rectangular coordinate system of R3
+1. Sincegis indefinite
+metric, recall that a vector /vector vinR3
+1can have one of three casual characters:
+it can be space-like if g(/vector v,/vector v)>0 or/vector v= 0, time-like if g(/vector v,/vector v)<0 and null
+g(/vector v,/vector v) = 0 and /vector v/ne}ationslash= 0. Similarly, an arbitrary curve /vector α=/vector α(s)⊂R3
+1can locally
+be space-like, time-like or null (light-like), if all of its velocity vectors /vector α′(s) are
+respectively space-like, time-like or null (light-like). The norm of a ve ctor/vector v
+is given by /bardbl/vector v/bardbl=/radicalbig
+|g(/vector v,/vector v)|. Therefore, /vector vis a unit vector if g(/vector v,/vector v) =∓1.
+Furthermore, vectors /vector vand/vector ware said to be orthogonal if g(/vector v, /vector w) = 0, [10].
+Let the set of all timelike vectors in R3
+1be Γ. For /vector u∈Γ, we call
+C(/vector u) ={/vector v∈Γ| /an}bracketle{t/vector v,/vector u/an}bracketri}ht<0}
+as time-conic of Lorentz space R3
+1including vector /vector u, [10].
+Let/vector vand/vector wbe two time-like vectors in Lorentz space R3
+1. In this case there
+exists the following inequality
+|g(/vector v, /vector w)| ≥ /bardbl/vector v/bardbl./bardbl/vector w/bardbl.
+In this inequality if one wishes the equality condition, then it is necessa ry for/vector v
+and/vector wbe linear dependent.
+If time-like vectors /vector vand/vector wstay inside the same time-conic then there is a
+unique non-negative real number of θ≥0 such that
+g(/vector v, /vector w) =−/bardbl/vector v/bardbl./bardbl/vector w/bardbl.coshθ (2.2)
+where the number θis called an angle between the timelike vectors, [10].
+Let/vector vand/vector wbe spacelike vectors in R3
+1that span a spacelike subspace. We have
+that
+|g(/vector v, /vector w)| ≤ /bardbl/vector v/bardbl./bardbl/vector w/bardbl
+with equality if and only if /vector vand/vector ware linearly dependent. Hence, there is a
+unique angle 0 ≤θ≤πsuch that
+g(/vector v, /vector w) =/bardbl/vector v/bardbl./bardbl/vector w/bardbl.cosθ (2.3)
+where the number θis called the Lorentzian spacelike angle between spacelike
+vectors/vector vand/vector w, [12].
+Let/vector vand/vector wbe spacelike vectors in R3
+1that span a timelike subspace. We have
+that
+|g(/vector v, /vector w)|>/bardbl/vector v/bardbl./bardbl/vector w/bardbl.
+Hence, there is a unique real number θ >0 such that
+g(/vector v, /vector w) =/bardbl/vector v/bardbl./bardbl/vector w/bardbl.coshθ. (2.4)
+The Lorentzian timelike angle between spacelike vectors /vector vand/vector wis defined to
+beθ, [12].
+2Let/vector vbe a spacelike vector and /vector wbe a timelike vector in R3
+1. Then there is a
+unique real number θ≥0 such that
+g(/vector v, /vector w) =/bardbl/vector v/bardbl./bardbl/vector w/bardbl.sinhθ. (2.5)
+The Lorentzian timelike angle between /vector vand/vector wis defined to be θ, [12].
+For any vectors /vector v= (v1,v2,v3),/vector w= (w1,w2,w3)∈R3
+1, the Lorentzian product
+/vector v∧/vector wof/vector vand/vector wis defined as [1]
+/vector v∧/vector w= (v3w2−v2w3,v1w3−v3w1,v1w2−v2w1). (2.6)
+3 Ruled Surface in R3
+1
+A ruled surface M∈R3
+1is a regular surface that has a parametrization ϕ:
+(I×R)→R3
+1of the form
+ϕ(u,v) =/vector α(u)+v/vector γ(u) (3.1)
+where/vector αand/vector γare curves in R3
+1with/vector α′never vanishes. The curve αis called the
+base curve. The rulings of ruled surface are the straight lines v→/vector α(u)+v/vector γ(u).
+Ifconsecutiverulingsofaruled surfacein R3
+1intersect, then the surfaceis saidto
+bedevelopable. All otherruledsurfacesarecalledskewsurfaces. Ifthereexistsa
+common perpendicular to two constructive rulings in the skew surfa ce, then the
+foot of the common perpendicular on the main ruling is called a striction point.
+The set of striction points on a ruled surface defines the striction c urve,[15].
+The striction curve, β(u) can be written in terms of the base curve α(u) as
+/vectorβ(u) =/vector α(u)−g(/vector α′(u),/vector γ′(u))
+/an}bracketle{t/vector γ′,/vector γ′/an}bracketri}ht/vector γ(u). (3.2)
+A ruled surface given by (3.1) is called non-cylindrical if /vector γ∧/vector γ′is nowhere
+zero. Thus, the rulings are always changing directions on a non-cylin drical
+ruled surface. A non-cylindrical ruled surface always has a parame terization of
+the form
+˜ϕ(u,v) =/vectorβ(u)+v /vector e(u) (3.3)
+where/bardbl/vector e(u)/bardbl=/vector γ(u)
+/bardbl/vector γ(u)/bardbl= 1,/angbracketleftBig
+/vectorβ′(u),/vector e′(u)/angbracketrightBig
+= 0 and /vectorβ(u) is striction curve of
+˜ϕ, [3].
+The distribution parameter (or drall) of a non-cylindrical ruled surf ace given by
+equation (3.3), is a function Pdefined by
+P=det/parenleftBig
+/vectorβ′,/vector e,/vector e′/parenrightBig
+/an}bracketle{t/vector e′,/vector e′/an}bracketri}ht. (3.4)
+where/vectorβis the striction curve and /vector eis the director curve. Moreover, Gaussian
+curvature of non-cylindrical ruled surface ˜ ϕ(u,v) is
+K=LM−N2
+EG−F2(3.5)
+3whereE,FandGare the coefficients of the first fundamental form, whereas L,
+NandMare the coefficients of the second fundamental form, of non-cylin drical
+ruled surface, [3].
+The unit normal vector of non-cylindrical ruled surface ˜ ϕis given by
+η(u,v) =˜ϕu∧˜ϕv
+/bardbl˜ϕu∧˜ϕv/bardbl. (3.6)
+A surface in the 3 −dimensional Minkowski space-time R3
+1is called a time-like
+surface if induced metric on the surface is a Lorentzian metric i.e. th e normal
+on the surface is a space-like vector, [15].
+InR3
+1, according to the character of the non-null base curve and the n on-null
+ruling, ruled surfaces are classified into three different groups. As a spacelike
+ruling moves along a spacelike curve it generates a spacelike ruled sur face, that
+will be denoted by M1. Furthermore, the movement of a timelike ruling along
+a spacelike curve and the movement of a spacelike ruling along a timelike curve
+generate timelike ruled surfaces. Let us denote these timelike ruled surfaces
+byM2andM3, respectively. Now, we will establish Lamarle formula for these
+ruled surfaces M1,M2,M3separately.
+4 Lamarle Formula for the Spacelike Ruled Sur-
+face
+LetM1be a spacelike ruled surface parametrized by
+ϕ1: I×R→R3
+1
+(u,v)→ϕ1(u,v) =/vector α1(u)+v /vector e1(u).
+If we choose /bardbl/vector e1/bardbl= 1 ,/vector n1=/vector e′
+1
+/bardbl/vector e′
+1/bardbland/vectorξ1=/vector e1∧/vector e′
+1
+/bardbl/vector e1∧/vector e′
+1/bardbl, we obtain the orthonormal
+frame field/braceleftBig
+/vector e1,/vector n1,/vectorξ1/bracerightBig
+. Suppose that these orthonormal frame field forms right
+handed system and is {space,time,space}type. In this case we may write
+/an}bracketle{t/vector e1,/vector e1/an}bracketri}ht= 1,/an}bracketle{t/vector n1,/vector n1/an}bracketri}ht=−1,/angbracketleftBig
+/vectorξ1,/vectorξ1/angbracketrightBig
+= 1,
+/an}bracketle{t/vector e1,/vector n1/an}bracketri}ht=/angbracketleftBig
+/vector n1,/vectorξ1/angbracketrightBig
+=/angbracketleftBig
+/vectorξ1,/vector e1/angbracketrightBig
+= 0(4.1)
+and
+/vector e1∧/vector n1=−/vectorξ1, /vector n1∧/vectorξ1=−/vector e1,/vectorξ1∧/vector e1=/vector n1. (4.2)
+The Frenet formulae of this orthonormal frame along e1become
+/vector e′
+1=κ1/vector n1, /vector n′
+1=κ1/vector e1+τ1/vectorξ1,/vectorξ′
+1=τ1/vector n1. (4.3)
+Let/vectorβ1(u) be a striction curve of spacelike ruled surface M1given by equation
+(3.2) inR3
+1. In this case the tangent vector /vectorβ′
+1of this curve stays in spacelike
+plane. Taking the angle σ1to be the angle between /vectorβ′
+1and/vector e1since the tangent
+vector of striction curve of M1is
+/vectorβ′
+1=/vector e1cosσ1+/vectorξ1sinσ1
+4we find the striction curve of M1to be
+/vectorβ1=/integraldisplay/parenleftBig
+cosσ1/vector e1+ sinσ1/vectorξ1/parenrightBig
+du.
+The spacelike non-cylindrical ruled surface M1is parametrized by
+˜ϕ1(u,v) =/integraldisplay/parenleftBig
+cosσ1/vector e1+sinσ1/vectorξ1/parenrightBig
+du+v/vector e1.
+From the equation (3.4) the distribution parameter of M1is found to be
+P=det/parenleftBig
+cosσ1/vector e1+sinσ1/vectorξ1,/vector e1,κ1/vector n1/parenrightBig
+/an}bracketle{tκ1/vector n1,κ1/vector n1/an}bracketri}ht=sinσ1
+κ1.
+Adopting κ1=1
+ρ1we get the distribution parameter as follows
+P=ρ1sinσ1. (4.4)
+Consideringequation(4.2) fromequation (3.6) wewrite the unit norm altangent
+vector of spacelike non-cylindrical ruled surface M1
+/vector η1=sinσ1/vector n1+vκ1/vectorξ1/radicalBig/vextendsingle/vextendsingle−sin2σ1+v2κ2
+1/vextendsingle/vextendsingle. (4.5)
+Taking into consideration that κ1=1
+ρ1and equation (4.4) we obtain
+/vector η1=P/vector n1+v/vectorξ1/radicalbig
+|−P2+v2|. (4.6)
+Furthermore, since the unit normal tangent vector η1of a spacelike surface M1
+is timelike we find that −P2+v2<0, that is |v|<|P|.
+The partial differentiation of M1with respect to uandvfrom equation (4.3)
+are as follows
+˜ϕ1u= cosσ1/vector e1+sinσ1/vectorξ1+vκ1/vector n1,
+˜ϕ1v=/vector e1.(4.7)
+Therefore, we find the first fundamental form’s coefficients of M1to be
+E=/an}bracketle{t˜ϕ1u,˜ϕ1u/an}bracketri}ht= cos2σ1+sin2σ1−v2κ2
+1= 1−v2κ2
+1,
+F=/an}bracketle{t˜ϕ1u,˜ϕ1v/an}bracketri}ht= cosσ1,
+G=/an}bracketle{t˜ϕ1v,˜ϕ1v/an}bracketri}ht= 1.(4.8)
+In addition to these, the second order partial differentials of M1are found to be
+˜ϕ1uu=/parenleftbig
+−σ′
+1sinσ1+vκ2
+1/parenrightbig
+/vector e1+(κ1cosσ1+τ1sinσ1+vκ′
+1)/vector n1+(σ′
+1cosσ1+vκ1τ1)/vectorξ1,
+˜ϕ1uv=κ1/vector n1,
+˜ϕ1vv= 0.
+From equation (4.5) and the last equations we get the coefficients of second
+fundamental of M1as
+5L=/an}bracketle{t˜ϕ1uu,/vector η/an}bracketri}ht=−κ1cosσ1sinσ1−τ1sin2σ1−vκ′
+1sinσ1+σ′
+1cosσ1vκ1+v2κ2
+1τ1/radicalBig
+|−sin2σ1+v2κ2
+1|,
+N=/an}bracketle{t˜ϕ1uv,/vector η/an}bracketri}ht=−κ1sinσ1 /radicalBig
+|−sin2σ1+v2κ2
+1|,
+M=/an}bracketle{t˜ϕ1vv,/vector η/an}bracketri}ht= 0.(4.9)
+Considering equation (4.8) and (4.9) together, we give the following t heorem for
+the Gaussian curvature of spacelike ruled surface M1.
+Theorem 4.1 LetM1be spacelike non-cylindrical ruled surface in R3
+1. The
+Gaussian curvature of spacelike non-cylindrical ruled sur faceM1is given in
+terms of its distribution parameter Pby
+K=−P2
+(P2−v2)2(4.10)
+where|v|<|P|.
+Proof.Substituting equations (4.8) and (4.9) into equation (3.5) and making
+appropriate simplifications we find the Gaussian curvature of M1to be
+K=−κ2
+1sin2σ1/parenleftbig
+sin2σ1−v2κ2
+1/parenrightbig2.
+Considering κ1=1
+ρ1and equation (4.4) completes the proof.
+The relation between Gaussian curvature and the distribution para meter of
+M1given by equation (4.10) is called Lorentzian Lamarle formula for the
+spacelike non-cylindrical ruled surface M1.
+The Lorentzian Lamarle formula for the spacelike ruled surface in R3
+1is non-
+positive. Therefore we give the following corollary.
+Corollary 4.1 LetM1be a spacelike non-cylindrical ruled surface with distri-
+bution parameter Pand Gaussian curvature KinR3
+1.
+1. Along a ruling the Gaussian curvature K(u,v)→0asv→ ∓∞.
+2.K(u,v) = 0if and only if P= 0.
+3. If the distribution parameter is Pnever vanishes then K(u,v)is contin-
+uous and when v= 0i.e. at the central point on each ruling, K(u,v)
+assumes its maximum value.
+Example 4.1 In3−dimensional Lorentzspace R3
+1let us define a non-cylindrical
+ruled surface as
+ϕ(u,v) = (−vcoshu, u,−vsinhu)
+that is a 2ndtype helicoid and a spacelike surface where −1< v <1, see Figure
+4.1.
+6The Gaussian curvature of this 2ndtype helicoid is K=−1
+(1−v2)2,|v|<1,
+see Figure 4.2.
+5 Lamarle Formula for the Timelike Ruled Sur-
+face with Spacelike Base Curve and Timelike
+Ruling
+LetM2be timelike ruled surface with spacelike base curve and timelike ruling
+in 3−dimensional Lorentz space, R3
+1. Thus, this ruled surface is parametrized
+as follows
+ϕ2: I×R→R3
+1
+(u,v)→ϕ2(u,v) =/vector α2(u)+v /vector e2(u).
+Here, taking /bardbl/vector e2/bardbl= 1 ,/vector n2=/vector e′
+2
+/bardbl/vector e′
+2/bardbland/vectorξ2=/vector e2∧/vector e′
+2
+/bardbl/vector e2∧/vector e′
+2/bardbl, we reach the or-
+thonormal frame field/braceleftBig
+/vector e2,/vector n2,/vectorξ2/bracerightBig
+.This forms a right handed system and in
+{time,space,space}type. Therefore,
+−/an}bracketle{t/vector e2,/vector e2/an}bracketri}ht=/an}bracketle{t/vector n2,/vector n2/an}bracketri}ht=/angbracketleftBig
+/vectorξ2,/vectorξ2/angbracketrightBig
+= 1
+/an}bracketle{t/vector e2,/vector n2/an}bracketri}ht=/angbracketleftBig
+/vector n2,/vectorξ2/angbracketrightBig
+=/angbracketleftBig
+/vectorξ2,/vector e2/angbracketrightBig
+= 0(5.1)
+and
+/vector e2∧/vector n2=−/vectorξ2, /vector n2∧/vectorξ2=/vector e2,/vectorξ2∧/vector e2=−/vector n2. (5.2)
+The differential formulae of this orthonormal system are
+/vector e′
+2=κ2/vector n2, /vector n′
+2=κ2/vector e2−τ2/vectorξ2,/vectorξ′
+2=τ2/vector n2. (5.3)
+Now, let the striction curve given by equation (3.2) of timelike ruled su rfaceM2
+be/vectorβ2(u)./vectorβ2(u) is a spacelike curve and the tangent vector of this curve /vectorβ′
+2
+stays in the timelike plane/parenleftBig
+/vector e2,/vectorξ2/parenrightBig
+. Adopting the hyperbolic angle σ2between
+/vectorβ′
+2and/vector e2we write
+/vectorβ′
+2= sinhσ2/vector e2+coshσ2/vectorξ2.
+7From the last equation we write for the striction curve of M2
+/vectorβ2=/integraldisplay
+(sinhσ2/vector e2+coshσ2ξ2)du.
+LetM2be timelike non-cylindrical ruled surface with spacelike base curve an d
+timelike ruling in R3
+1. In this case we reparametrize M2such as
+˜ϕ2(u,v) =/integraldisplay/parenleftBig
+sinhσ2/vector e2+ coshσ2/vectorξ2/parenrightBig
+du+v/vector e2.
+Considering equation (3.4) we find the distribution parameter of time like non-
+cylindrical ruled surface M2to be
+P=det/parenleftBig
+sinhσ2/vector e2+coshσ2/vectorξ2,/vector e2,κ2/vector n2/parenrightBig
+/an}bracketle{tκ2/vector n2,κ2/vector n2/an}bracketri}ht=coshσ2
+κ2.
+Takingκ2=1
+ρ2we rewrite the distribution parameter of M2as
+P=ρ2coshσ2. (5.4)
+If we consider equation (5.2), from equation (3.6) we see that the t imelike non-
+cylindrical ruled surface’s unit normal vector becomes
+/vector η2=−coshσ2/vector n2+vκ2/vectorξ2/radicalBig/vextendsingle/vextendsinglecosh2σ2+v2κ2
+2/vextendsingle/vextendsingle. (5.5)
+Sinceκ2=1
+ρ2, from equation (5.4) we find
+/vector η2=−P/vector n2+v/vectorξ2√
+P2+v2. (5.6)
+From equation (5.3), partial differential of M2with respect to uandvare
+˜ϕ2u= sinhσ2/vector e2+ coshσ2/vectorξ2+vκ2/vector n2,
+˜ϕ2v=/vector e2.(5.7)
+Considering the last equations with equation (5.1) we find the coefficie nts of
+first fundamental form of M2to be
+E=/an}bracketle{t˜ϕ2u,˜ϕ2u/an}bracketri}ht=−sinh2σ2+cosh2σ2+v2κ2
+2= 1+v2κ2
+2,
+F=/an}bracketle{t˜ϕ2u,˜ϕ2v/an}bracketri}ht=−sinhσ2,
+G=/an}bracketle{t˜ϕ2v,˜ϕ2v/an}bracketri}ht=−1.(5.8)
+Furthermore, if we consider equation, (5.7) we reach that the sec ond order
+partial differentials of M2
+˜ϕ2uu=/parenleftbig
+σ′
+2coshσ2+vκ2
+2/parenrightbig
+/vector e2+(κ2sinhσ2+τ2coshσ2+vκ′
+2)/vector n2+(σ′
+2sinhσ2−vκ2τ2)/vectorξ2,
+˜ϕ2uv=κ2/vector n2,
+ϕ2vv= 0.
+8From equation (5.5) and the last equations we find the second funda mentals
+form’s coefficients as follows
+L=/an}bracketle{t˜ϕ2uu,/vector η2/an}bracketri}ht=−κ2sinhσ2coshσ2−τ2cosh2σ2−vκ′
+2coshσ2−vκ2σ′
+2sinhσ2+v2κ2
+2τ2√
+cosh2σ2+v2κ2
+2,
+N=/an}bracketle{t˜ϕ2uv,/vector η2/an}bracketri}ht=κ2coshσ2 √
+cosh2σ2+vκ2
+2,
+M=/an}bracketle{t˜ϕ2vv,/vector η2/an}bracketri}ht= 0.
+(5.9)
+Therefore, for the Gaussian curvature of timelike ruled surface M2, we give the
+following theorem.
+Theorem 5.1 LetM2be a timelike non-cylindrical ruled surface with spacelike
+base curve and timelike ruling in R3
+1. Taking Pto be the distribution parameter
+ofM2we see that the Gaussian curvature of M2is
+K=P2
+(P2+v2)2. (5.10)
+Proof.Substituting equations (5.8) and (5.9) into equation (3.5) we find the
+Gaussian curvature of M2to be
+K=κ2
+2cosh2σ2/parenleftbig
+cosh2σ2+κ2
+2v2/parenrightbig2.
+Here considering κ2=1
+ρ2and equation (5.4) completes the proof.
+The relation between Gaussian curvature and the distribution para meter of
+M2given by equation (5.10) is called Lorentzian Lamarle formula for the
+timelike non-cylindrical ruled surface with spacelike base curve and t imelike
+ruling.
+The Lamarle formula for the timelike ruled surface in R3
+1is non–negative. So,
+we give the following corollary.
+Corollary 5.1 LetPbe a distribution parameter and Kbe a Gaussian curva-
+ture of a timelike non-cylindrical ruled surface M2with spacelike base curve and
+timelike ruling in R3
+1. In this case
+1. Along ruling as v→ ∓∞,K(u,v)→0.
+2.K(u,v) = 0if and only if P= 0.
+3. If the distribution parameter of M2never vanishes, then K(u,v)is con-
+tinuous and as v= 0i.e. at the central point on each ruling, K(u,v)
+takes its minimum value.
+Example 5.1 In3−dimensional Lorentz space R3
+1.
+ϕ(u,v) = (−vsinhu, u,−vcoshu)
+is a3rdtype helicoid and a timelike non-cylindrical ruled surface with spacelike
+base curve and timelike ruling, see: Figure 5.1.
+9The Gaussian curvature of this 3rdtype helicoid is K=−1
+(1+v2)2, see Figure
+5.2.
+6 Lamarle Formula for Timelike Ruled Surface
+with Timelike Base Curve and Spacelike Rul-
+ing
+Suppose that the timelike ruled surface M3with timelike base curve and space-
+like ruling in three dimensional Lorentz space R3
+1is parametrized as follows
+ϕ3: I×R→R3
+1
+(u,v)→ϕ3(u,v) =/vector α3(u)+v /vector e3(u).
+Considering that /bardbl/vector e3/bardbl= 1 ,/vector n3=/vector e′
+3
+/bardbl/vector e′
+3/bardbland/vectorξ3=/vector e3∧/vector e′
+3
+/bardbl/vector e3∧/vector e′
+3/bardbl, we reach the orthonor-
+mal frame field/braceleftBig
+/vector e3,/vector n3,/vectorξ3/bracerightBig
+.This forms a right handed system which is in type
+{space,space,time}. Thus we write
+/an}bracketle{t/vector e3,/vector e3/an}bracketri}ht=/an}bracketle{t/vector n3,/vector n3/an}bracketri}ht=−/angbracketleftBig
+/vectorξ3,/vectorξ3/angbracketrightBig
+= 1
+/an}bracketle{t/vector e3,/vector n3/an}bracketri}ht=/angbracketleftBig
+/vector n3,/vectorξ3/angbracketrightBig
+=/angbracketleftBig
+/vectorξ3,/vector e3/angbracketrightBig
+= 0(6.1)
+and cross product is defined to be
+/vector e3∧/vector n3=/vectorξ3, /vector n3∧/vectorξ3=−/vector e3,/vectorξ3∧/vector e3=−/vector n3. (6.2)
+Differential formulae for this orthonormal system is expressed by
+/vector e′
+3=κ3/vector n3, /vector n′
+3=−κ3/vector e3+τ3/vectorξ3,/vectorξ′
+3=τ3/vector n3. (6.3)
+Let thestrictioncurveoftimelike ruledsurfacegivenbyequation(3 .2) be/vectorβ3(u).
+This curve is a timelike curve and the tangent vector of this curve st ays within
+the timelike plane/parenleftBig
+/vector e3,/vectorξ3/parenrightBig
+. Adopting the hyperbolic angle σ3to be the angle
+between /vectorβ′
+3and/vector e3we may write
+/vectorβ′
+3= sinhσ3/vector e3+ coshσ3/vectorξ3
+10yielding the striction curve of M3to be
+/vectorβ3=/integraldisplay/parenleftBig
+sinhσ3/vector e3+ coshσ3/vectorξ3/parenrightBig
+du.
+The timelike non-cylindrical ruled surface M3with timelike base curve and
+spacelike ruling is reparametrized by
+˜ϕ3(u,v) =/integraldisplay/parenleftBig
+sinhσ3/vector e3+ coshσ3/vectorξ3/parenrightBig
+du+v/vector e3.
+The distribution parameter of this ruled surface is found to be
+P=det/parenleftBig
+sinhσ3/vector e3+ coshσ3/vectorξ3,/vector e3,κ3/vector n3/parenrightBig
+/an}bracketle{tκ3/vector n3,κ3/vector n3/an}bracketri}ht=coshσ3
+κ3.
+The distribution parameter of M3becomes
+P=ρ3coshσ3 (6.4)
+whereκ3=1
+ρ3. Taking equation (6.2) into consideration, we find from equation
+(3.6) that the unit normal vector of timelike non-cylindrical ruled su rfaceM3is
+/vector η3=−coshσ3/vector n3−vκ3/vectorξ3/radicalBig/vextendsingle/vextendsinglecosh2σ3−v2κ2
+3/vextendsingle/vextendsingle. (6.5)
+Sinceκ3=1
+ρ3, from equation (6.4) we find
+/vector η3=−P/vector n3+v/vectorξ3/radicalbig
+|P2−v2|. (6.6)
+In addition to these, since the unit normal vector /vector η3of timelike ruled surface
+M3is spacelike, that is, P2−v2>0 i.e.|P|>|v|.
+The partial differentials of M3with respect to uandv(from equation (6.3))
+becomes
+˜ϕ3u= sinhσ3/vector e3+ coshσ3/vectorξ3+vκ3/vector n3,
+˜ϕ3v=/vector e3.(6.7)
+Considering the last equations together with equation (6.1) the coe fficients of
+first fundamental form of M3are
+E=/an}bracketle{t˜ϕ3u,˜ϕ3u/an}bracketri}ht= sinh2σ3−cosh2σ3+v2κ2
+3=−1+v2κ2
+3,
+F=/an}bracketle{t˜ϕ3u,˜ϕ3v/an}bracketri}ht= sinhσ3,
+G=/an}bracketle{t˜ϕ3v,˜ϕ3v/an}bracketri}ht= 1.(6.8)
+Furthermore, considering equation (6.7) we find for the second or der partial
+differentials of M3as
+˜ϕ3uu=e3/parenleftbig
+σ′
+3coshσ3−vκ2
+3/parenrightbig
++n3(κ3sinhσ3+τ3coshσ3+vκ′
+3)+ξ3(σ′
+3sinhσ3+vκ3τ3),
+˜ϕ3uv=n3κ3,
+ϕ3vv= 0.
+11From equation (6.5) and the last equations, the coefficients of the s econd order
+principal form read to be
+L=/an}bracketle{t˜ϕ3uu,/vector η3/an}bracketri}ht=−κ3sinhσ3coshσ3−τ3cosh2σ3−vκ′
+3coshσ3+vκ3σ′
+3sinhσ3+v2κ2
+3τ3/radicalBig
+|cosh2σ3−v2κ2
+3|,
+N=/an}bracketle{t˜ϕ3uv,/vector η3/an}bracketri}ht=κ3coshσ3 /radicalBig
+|cosh2σ3−vκ2
+3|,
+M=/an}bracketle{t˜ϕ3vv,/vector η3/an}bracketri}ht= 0.
+(6.9)
+Takingequations(6.8) and(6.9) togetherinto account, we cangive the following
+theorem for the Gaussian curvature of timelike ruled surface M3.
+Theorem 6.1 LetM3be a timelike non-cylindrical ruled surface with timelike
+base curve and spacelike ruling in R3
+1. Adopting that the distribution parameter
+ofM3isP, the Gaussian curvature of M3becomes
+K=P2
+(P2−v2)2(6.10)
+whereP2−v2>0.
+Proof.Substituting equations (6.8) and (6.9) into equation (3.5) we find the
+Gaussian curvature of M3to be
+K=κ2
+3cosh2σ3/parenleftbig
+cosh2σ3+κ2
+3v2/parenrightbig2.
+Considering equation (6.4) together with κ3=1
+ρ3completes the proof.
+The relation between the Gaussian curvature and the distribution p arameter
+oftimelike ruledsurfacegivenbyequation(6.10) iscalled Lorentzian Lamarle
+formula for the timelike non-cylindrical ruled surface with timelike base curve
+and spacelike ruling.
+The Lamarle formula for the timelike ruled surface in R3
+1is non–negative. So,
+we give the following corollary.
+Corollary 6.1 LetM3be a timelike non-cylindrical ruled surface with timelike
+base curve and spacelike ruling in R3
+1. Considering that Pis the distribution
+parameter and Kis the Gaussian curvature we see that
+1. Along ruling v→ ∓∞the Gaussian curvature K(u,v)→0.
+2.K(u,v) = 0if and only if P= 0.
+3. If the distribution parameter of M3never vanishes, then K(u,v)is con-
+tinuous and as v= 0i.e. at the central point on each ruling, K(u,v)
+takes its minimum value.
+Example 6.1 Let us parametrize a 1sttype helicoid as
+ϕ(u,v) = (−vcosu ,−vsinu, u)
+12which is timelike non-cylindrical ruled surface with timel ike base curve and
+spacelike ruling in 3−dimensional Lorentz space R3
+1and here −1< v <1,
+see: Figure 6.1.
+The Gaussian curvature of this 1sttype helicoid is K=1
+(1−v2)2,|v|<1,
+see Figure 6.2.
+Acknowledgement 1 The authors are very grateful to Prof.Dr. Ibrahim Okur
+for improving presentation of the paper.
+References
+[1] Akutagawa, K., Nishikawa, S., The Gauss map and space-like surfaces with
+prescribed mean curvature in Minkowski 3-space , Thoku Math. J. 42, 67-82,
+1990.
+[2] Aydemir, S., Kasap, E., Timelike ruled surfaces with spacelike rulings in
+R3
+1, Kuwait Journal of Science and Engineering, Vol:32, Issue:2, Pages :1324,
+2005.
+[3] Gray, A., Modern Differential Geometry of Curves and Surfaces with Mat h-
+ematica, Boca Raton, FL: CRC Press, 1993.
+[4] Gursoy, O., On the integral invariants of a closed ruled surface , Journal of
+Geometry, 39, (1990), 80-91.
+[5] Kasap, E., Ruled surfaces with timelike rulings,(vol 147, pg 241, 2004 ), Ap-
+plied Mathematics and Computation, Volume:179, Issue:1, Pages:402 -405,
+2006.
+[6] Kasap, E., Aydemir, I., Kuruo˜ glu, N., Ruled surfaces with time like rul-
+ings (vol 147, pg 241, 2004) , Applied Mathematics and Computation, Vol-
+ume:174, Issue:2, Pages:16601667, 2006.
+[7] Kose O., Contribution to the theory of integral invariants of a close d ruled
+surface, Mechanism and Machine,Theory 32 (1997), 261-77.
+[8] Kruppa, E. Analytische und Konstruktive differentialgeometrie , Wien
+Springer-Verlag, 1957.
+13[9] Izumiya S., Takeuchi N., Special curves and ruled surfaces , Beitrage Algebra
+Geom. 44, (2003) 203-212.
+[10] O’Neill, B., Semi-Riemannian Geometry , Academic Press, New York 1983.
+[11] Pottman H., Wallner J., Computational Line Geometry , Springer-Verlag,
+Berlin, 2001.
+[12] Ratcliffe, J. G., Foundations of Hyperbolic Manifolds , Department ofMath-
+ematics, Vanderbilt University, 1994.
+[13] Sasaki T., Projective Differential Geometry and Linear Homogeneous Di f-
+ferential Equations , Rokko Lectures in Mathematics, Kobe University, vol.
+5, 199.
+[14] Turgut, A., Hacisalihoglu H. H., Spacelike ruled surfaces in the Minkowski
+3-space, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 46 (1997), no.
+1-2, 83–91 (1998).
+[15] Turgut, A., Hacisalihoglu, Time-like ruled surfaces in the Minkowski 3-
+space, Far East J. Math. Sci. 5 (1) (1997) 83-90.
+14
\ No newline at end of file
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+ 
+1 Vandalism Detection in Wikipedia : a Bag -of-Words C lassifi er A pproach  
+Amit Belani  
+ab422@cornell.edu  
+November 11, 2009  
+Abstract  
+A bag -of-words based probabilistic classifier is trained using regularized logistic regression to 
+detect vandalism in the English Wikipedia. Isotonic regression is used to calibrate the class 
+membership  probabilities . Learning curve, r eliability, ROC , and cost analysis are performed.  
+1. Motivation  
+Wikipedia is a collaborative encyclopedia that anyone can edit  and improve . Although it has been 
+argued that this openness is one of the reasons for its success, it does allow some unscrupulous 
+editors to in troduce damaging edits. At the present time, most of these edits are identified and 
+reverted either by human editors or by automated anti -vandal bots . 
+Prominent anti -vandal bots such as ClueBot and VoABot II use lists of regular expressions and user 
+blacklists to detect vandalism. These regular expression rules are created manually and are difficult 
+to maintain. They detect only 30% of the committed vandalism. Their combined R ecall was 
+observed to be 33%, with a P recision of 100%. [1] 
+Since 2008, m achine learning algorithms have been attempted for this  task ,[1-5] but with limited 
+success, leaving room for improvement.  Moreover, t he methods attempted have either utilized only 
+a small sample of the available training data, or have generally not considered a  cost -sensitive 
+approach to model development.  Such an  approach aids  in improv ing the true positives detection 
+rate while maintaining low false positives. I t is believed that  a false positive would have a hi gher 
+cost than a false negative in  any practical application of the learning algorithm for this classification 
+problem.  
+2. Task  
+An e dit is a sequential revision to an article. A revert  is an edit that returns the article to a previous 
+version. That is,  for versions  , where   in chronological order, if , then k is a revert, 
+and j is most likely a damaging edit. These events often occur when an article is vandalized, or when 
+an edit does not follow  the conventions of the article.[2] Not all damaging  edits are immediately 
+reverted, i.e. versions  may not  immediately follow each other, although they usually do.  
+Two categories of editors exist  – anony mous and registered.  Anonymous users commit most of the 
+vandalism, although their overall legitimate contributions are rather small. [1] 
+For each rev ision, the author, editorial comment and full text of the version of the page a re 
+available.[6] This data is used  to create a classifier with l ow false positives that can predict whether  
+2 an edit by an anonymous editor is vandalism.  This model  can be utilized by an anti -vandal bot  or 
+other quality review  software  such as Huggle .[7] 
+3. I/O behavior  
+3.1. Input  
+The quality of an edit is predicted using features of the edit, its author, and the article being 
+edited.[2] The input to the classification system will be : 
+• Change s to the article by the respective edit, expressed as a bag -of-words feature vector . 
+• Metadata associated with the edit , including:  
+o The edit sum mary statement, also expressed as  a bag -of-words feature vector . 
+o The IP address  of the anonymous edit or. 
+All of the above inputs are merged into a single feature vector . 
+3.2. Output 
+The probabilistic classifier ’s output is the predicted probability of the edit being vandalism.  Based 
+on a predetermined threshold, this probability is  transformed into a class , predicting whether the 
+respective edit is vandalism or not . 
+4. Dataset  
+4.1. Raw dataset  transformation  
+The most recent version of the required pages -meta -history.xml .bz2  dataset [8] for the task is from 
+March 2008. Due to technical hurdles, creation of newer versions of this dataset by Wikimedia has 
+been halted , although newer data can be queried incrementally using a web API . The size of the 
+compressed version of the dataset is 147 GB. A sufficiently large  sample  of this dataset was used. 
+Using the entire dataset was not feasible, as the implementation of the learning algorithm requires 
+the training data to be held in memory . 
+The XML dataset was  parsed using a serial -access parser . Two parsers were evalua ted: SAX (Simple 
+API for XML), and ElementTree. [9] The latter function ed faster and require s less code . 
+Interlacing r evisions were scanned for reverts  by comparing their article text s for equality , in order 
+to detect vandalism.  As indicated earlier, because anonymous e ditors  commit most of the 
+vandalism, only revisions by these e ditors  were considered for further usage . Multiple successive 
+revisions by the same e ditor  were merged into a single revision . 
+The article and comment text in  each revision were  tokenized into lowercased words.  A custom 
+regular expression based tokenizer was developed for this purpose.  HTML tags and Wikipedia ’s 
+various syntactical elements were detected as words, as these are believed to be predictive. Long 
+words containing repetitions of smaller words, such as “hihihi…” or “ funfunfun… ” were detected  
+and split into the ir respectively contained smaller words.   
+3 For each article, the number of instances of each unique word in each revision was determined . 
+Using a ll word counts of  all revisions  for model development, however, would have been 
+computationally prohibitive . Besides, only  those select few words whose counts are change d from 
+one revision to the next are believed to contain the majority of the predictive value.  For these 
+reasons, only this subset of words was  considered for further use.  
+The differences in word counts from one revision to the next were calculated. These have values in 
+the set of all integers . The ratio of the word counts of the two revisions was calculated as well, given  
+that the se ratios may add to the predictive value of the model . Using this  ratio vector in addition to 
+the difference vector resulted in slightly better performance, although this double d the dataset size. 
+These difference and ratio vectors are believed to have more predictive value than the two 
+individual word count vectors themselves.  In order to maintain and benefit from sparsity, i.e. non -
+representation of zero values, t he ratios were transformed to be center ed on zero, and be in the set 
+of all real numbers.  
+While this representation results in a loss of information of  the order of words,  it allow s text to be 
+represented numerically , as this is necessary for many learning algorithms . Term -weighting 
+technique s such as tf -idf (term frequency, inverse document frequency) which are commonly used 
+were  not applied,  as it would result in a n unacceptable  loss of sparsity when computing the 
+difference and ratio vectors.  Word stemming was not deemed necessary , as different derivations of 
+a stem may be different ly predictive , given that  the dataset is large.  
+4.1.1.  Example  
+The following tables show  an example demonstrating  the aforementioned  transformation  of text 
+into numerical vector s: 
+Version  Article text  
+ Multiverses have been hypothesized in many fields of science, including cosmology, 
+physics, and astronomy . 
+ Multiverses have been hypothesized in cosmology, physics, astronomy, philosophy, and 
+fiction , particularly in science  fiction and fantasy.  
+  
+4 Word  Sparse word vectors  
+   
+(Difference)   
+(Transformed ratio ) 
+, 3 5 2 1.6667  
+. 1 1   
+and 1 2 1 2.0 
+astronomy  1 1   
+been  1 1   
+cosmology  1 1   
+fantasy   1 1 1.0 
+fiction   2 2 2.0 
+fields  1  –1 –1.0 
+have  1 1   
+hypothesized  1 1   
+in 1 2 1 2.0 
+including  1  –1 –1.0 
+many  1  –1 –1.0 
+multiverses  1 1   
+of 1  –1  
+particularly   1 1 1.0 
+philosophy   1 1 1.0 
+physics  1 1   
+science  1 1   
+ 
+Positive and negative numbers in the above word vectors refer to addition and removal of words 
+respectively.  Zero values are not listed.  
+4.1.2.  Ancillary features  
+Several potentially predictive metadata features for each revision were also computed . These are:  
+• a Boolean  (0 or 1) indicating whether the revision has any text or is empty, as revisions with 
+no text are likely to be vandalism  
+• a Boolean indicating whether the editor entered a summary comment, as one is usually  not 
+supplied for a revision that is vandalism  
+• a Boolean indicating whether the editor marked the revision as a minor edit, as such 
+revisions are unlikely to be vandalism  
+• a Boolean indicating whether the revision  was a change to the “External l inks” section of the 
+article, as articles are often vandalized by adding a spam link  to this section  
+• the first of the four octets of the IP v4 address of the anonymous editor , as different values 
+for this correspond to different geographical regions and may be differently predictive  
+• the number of revisions in the collapsed sequence of successive revisio ns by the editor , 
+minus one , as vandalism is likely to be just a single revision and  not part of a sequence  
+• the number of characters and words in the previous and current revision texts and 
+comments, and their respective differences  
+All of t he transformed data w as stored in a relational database.   
+5 4.2. Further transformation and scaling  
+Addition s and subtractions  of a word were processed as two separate unsigned features, instead of 
+as one signed feature. This resulted in slightly better performance with out affecting the dataset 
+size.  
+Learning algorithms generally require that the values of features in the training data be in the range 
+of 0 to 1  in order to achieve a reduced training time and optimal performance.  Three scaling 
+functions  were experimented with : 
+Scaling function name  Formula  for , with   Notes  
+atan    
+binary    
+log-lin   derived from training data.  
+If any test  , then  . 
+ 
+As required, f or each function , , and  . For the binary  function, because the 
+difference  and ratio  vector s are equal after being  scaled , only the difference vector needs to be 
+used , and so the dataset size is halved.  
+4.3. Dataset summary  
+2,000,000 cases from the years 2001 -08 were extracted  from articles starting with the letters A to 
+M. 1,585,397 unique wo rds and an average of 104 words per case were extracted. Because 
+additions and subtractions of words were processed as separate features, the number of features in 
+the dataset is at least twice the number of unique words , or 3,170,794 . Additionally, becaus e atan  
+and log-lin scaled datasets also include word ratio  feature s, the number of features in them is at 
+least four times the number of unique words , or 6,341,588 . 
+43% of the cases  belong to the positive class, i.e. vandalism, and the remaining 57% to the negative 
+class, i.e. not vandalism. The baseline accuracy and RMSE on the dataset are  therefore 57% and 
+66% respectively . 
+The data was split into training, validation, and test set s. 50% of the cases were randomly assigned 
+to the training set for use by the learning algorithm. 25% were randomly assigned to the validation set, to be used for model parameter  optimization. The remaining 25% were assigned to the test set, 
+to be used onl y for evaluating the performance o f the chosen  model.  Cross -validation was not used 
+for parameter optimization due to the large size of the datasets and the prohibitive cost of training 
+multiple models.  
+5. Model training and optimization  
+5.1. Logistic regression training  
+Given that the number of features in the training dataset  is much larger than the number of cases, 
+simple and highly regularized approaches are the method s of choice. [10] The Liblinear  package was 
+used to train L2 -regularized logistic regression models  using a trust region Newton method. Linear  
+6 methods such as logistic regression tend to work well in very high -dimensional applications 
+including document classification .[11]  Regularization aids in generalization.[12]  Liblinear allows 
+for large -scale linear classification, and is efficient on large datasets.[13]  
+5.1.1.  Formulation  
+Given data x , weights  , and class label y , if training instances are  and labels are 
+, then  are estimated by  
+ 
+where C > 0 is a penalty parameter .[12]  
+5.1.2.  Optimization  
+As recommended by the  author s of Liblinear , models were trained on  the training set , with C varied 
+on a log scale [14]  from 2–5 to 211. A bias value   was used. The p erformance of each model was 
+measured on both the training and the validation sets.  
+Since the exact misclassification costs are not known as they depend upon the application in which 
+the model is used, the RMSE (root mean squared error) metric was optimized  instead.  This is 
+because MS E tends to be correlated with profit, and so it can be used when exact costs are 
+unavailable. [15]  This allows the desired misclassification costs to be selected at a later time  by 
+varying the threshold for classification .[16]  
+5.2. Calibration  
+MSE can be decomposed into two components, one measuring calibration and the other measuring 
+refinemen t.[17]  The calibration process improves the estimate of the probability that each test 
+example is a member of the class of interest. [18]  Isotonic regression using the PAV ( pair -adjacent  
+violators ) algorithm was used to calibrate each model. It is a non -parametric form of regression  for 
+reducing calibration error  in the class membership probabilities returned  by the logistic model. 
+Given predictions  from a model and true targets  , the basic assumption in  isotonic regression is 
+that  where m is an isotonic (monotonically increasing) function . Given training set 
+, 
+ 
+PAV finds the stepwise -constant isotonic  function that best fits the data according to a mean -
+squared  error criterion.  It can be viewed as a binning algorithm where  the position of the 
+boundaries and the size s of the bins are chosen  according to how well the classifier ranks the 
+examples.[15]  
+Even though logistic regression models tend to be reasonably calibrated by default , they are known 
+to still benefit from calibration in most cases. [11]  If the learning algorithm does not overfit the 
+training data, the same data can be used to learn a model for calibration, without risking that th is  
+7 calibration model will overfit the training data .[15]  Because it was observed that logistic regres sion 
+did not overfit the training data greatly , specifically by an RMSE of 1.2% on average, the same 
+training data was used to learn the calibration model.  
+5.3. Results  
+Experiments were performed for each of the three scaling functions. The following results we re 
+obtained:  
+Scaling 
+function  Best C 
+for 
+RMSE  1 – RMSE  Accuracy  (calibrated)  
+Training  Training 
+(calibrated)  Validation  Validation (calibrated)  Validation  Best Threshold  Rank  
+atan  16 61.77% 62.1 3% 60.27% 60.52 % 78.03 % 0.5041  1 
+binary  0.125  61.72 % 62.1 % 60.24 % 60.49%  77.98 % 0.5087  1 
+log-lin 128  61.23 % 61.58 % 60.07 % 60.34%  77.72 % 0.5151  2 
+5.3.1.  1 – RMSE 
+For the 1 – RMSE  performance metric , higher values are better. The calibrated scores for 1 – RMSE 
+over the validation set for the three scaling functions are comparable, with a maximum difference of 
+less than a quarter of a percent. Calibration always improved the RMSE, although only slightly . The 
+baseline performance for 1 – RMSE is 34.12%.  
+Changing C had only a negligible impact on  the performance on the validation set.  
+5.3.2.  Accuracy  
+The probability threshold for accuracy was optimized over the training set. The optimal thresholds 
+on the calibration predictions are close to the defaults of 0.5. The learnt thresholds were then used 
+to compute the accuracy on the validation set. A ccuracy is not a useful metric by itself, since it 
+assumes equal misclassification costs , and this assumption is rarely true . The observed accuracies 
+are seen to be correlated with R MSE. The baseline for accuracy is 56.6%.  
+McNemar’s Test was performed over the validation set to compare the three classifiers with each 
+other  and with the baseline . McNemar’s Test is a significance test for comparing two classifiers. It is 
+used to accept or reject  the hypothesis that they have the same error rate at a given significance 
+level. [19]  Based on the test, the classifiers trained over the atan  and binary  scaled data have the 
+same error rate. The classifier trained over the log -lin scaled data has a n error rate that is different 
+from the others.  
+The table below lists the computed  chi-square statistics for all six pairs of classifiers, as were  
+required for the tests. Values less than 3.84 imply the hypothesis that  the two classifiers have the 
+same err or rate at significance 0.05.  
+Classifier  Baseline  atan  binary  
+atan  57,380   
+binary  57,424 1.01   
+log-lin 56,173  202  106   
+8 5.3.3.  Model selection  
+For 1 – RMSE, t he binary  scaling function perform ed almost as good  as the atan  scaling function.  
+Because t he binary  scaling function is simpler and needs half as many  features  as atan , it results in  a 
+faster prediction time . For these reasons , binary  was chosen  over atan . 
+ 
+6. Analysis  
+6.1. Learning curve  
+Learning curves are used in machine learning to see the gen eralization performance as a function of 
+the number of training examples.  The curve is steeper for smaller training  set sizes . In other words,  
+the in crease in performance  is lesser  for larger training  set sizes. [20]  
+The training set size was changed on a log scale from 101 to its full size of 106. Performance was 
+measured on the entire validation set. Ten iterations  were performed  for each training set size 
+ 
+9 except the largest , in which case the full dataset was used . Training d ata was selected  using simple 
+random sampling without replacement.  
+For the learning curve, calibration using  isotonic regression was not applied,  as this requires at 
+least 1,000 training cases,[15]  and these were not available for all instance s of the training data . 
+The value of C that previously yielded  the best RMSE  on the validation set without calibration  (with 
+binary  scaling) was used . 
+ 
+Error bars in the plot represent one standard deviation.  
+It is observed that the learning method does indeed learn a better model when given increasing 
+amounts of training data . As expected, the in crease in performance  is lesser  for larger training  set 
+sizes . 
+6.2. Reliability diagram  
+The calibration of a classifier c an be visualized through a reliability  diagram .[17]  To construct a  
+reliability diagram, the  prediction space is discretized into ten bins. Cases with  a predicted value 
+ 
+10 between 0 and 0.1 fall in the first bin, between  0.1 and 0.2 in the second bin, etc. For each bin, the  
+mean predicted value is plotted against the true fr action of  positive cases. If the model is well 
+calibrated the points  fall near the diagonal line.[18]  
+ 
+A reliability diagram was constructed over the validation set using the previously chosen model . 
+The pre and post calibration points were plotted. As seen in the diagram, the poin ts belonging to 
+the calibrated classifier are closer to the diagonal.  
+7. Thresholding  
+Different approaches exist for selecting a threshold between 0 and 1 for classification. Predictions 
+over the operating threshold are classified as belonging to the positive class, and the rest to the 
+negative class.  
+ 
+11 7.1. ROC and PR curves  
+Receiver Operator Characteristic  (ROC ) and Precision -Recall  (PR)  curves show different  tradeoff s 
+achieved by varying the threshold for classification.  
+Terms  Definition  Interpretation  
+TPR, sensitivity, recall  TP/(TP+FN)  Fraction of  positives that are correctly classified  
+FPR, fallout  FP/(TN+FP)  Fraction of negatives misclassified as positive  
+Precision, PPV  TP/(TP+FP)  Fraction classified  as positive that are truly positive  
+ 
+The ROC curve shows false positive rate vs. true positive rate.  PR curves show Recall vs. Precision , 
+and are used in Information Retrieval.  The goal of a learning  algorithm is to be in the upper -left and 
+upper -right corners of the ROC and PR curves respectively.  The area under the curve (AUC) can be 
+used as a metri c for comparison of algorithms, with higher values being better. A n algorithm that 
+optimizes the  area under the ROC curve (AUC -ROC) is not guaranteed to optimize  the area under 
+the PR curve  (AUC -PR).[21]  
+ 
+While the model  was not directly optimized for AUC -ROC or AUC -PR, their values were measured 
+for the selected model over the validation set. These were observed to be 84.73% and 80.46% 
+respectively.  
+7.2. Cost anal ysis  
+The essence of cost -sensitive decision -making is that it can be optimal to act  as if one class is true 
+even when another  class is more  probable. For example, it can be rational not to approve a large 
+credit card transaction even if the transaction is most  likely legitimate.[22]  As stated earlier, it is 
+believed  that  a false positive would have a hi gher cost than a false negative in any practical 
+application of the learning algorithm for this classification problem.  The following cost matrix is 
+proposed:  
+ 
+12  actual negative  actual positive  
+predicted negative  c00 = 0 (true -negative cost)  c01 = 1 (false -negative cost)  
+predicted positive  c10 >1 (false -positive cost)  c11 = 0 (true -positive cost)  
+ 
+This cost matrix meets the reasonableness condition  for cost matrices , namely , that the cost of 
+labeling an example incorrectly should always be greater than the cost of labeling it correctly .[22]  
+Given the p roposed cost matrix, e ffectively, only the ratio of the two misclassification costs in it is 
+uncertain. With  , multiple values for  can be considered.  
+The optimal prediction is the positive class if and  only if the expected cost of this prediction is less 
+than or equal to the expected cost of predicting the negative class.[22]  Given , the 
+theoretical thr eshold for making an  optimal decision on classifying instances as positive  is 
+. In practice however, an empirically derived threshold yield s a lower cost than the 
+theoretical threshold.  The validation set can be used to search for th is empirical threshold. [16]  
+ 
+ 
+13 Given the suggested cost matrix with an uncertain false -positive cost, values from 1 to 50 for the 
+false -positive cost were plotted on a log scale against the respective theoretical and empirical 
+thresholds.  It is observed that the empirical threshold is consistently higher than the theoretical. 
+For false -positive  cost s greater than 30, the empirical threshold is 1 , and therefore the model would 
+not be discriminative  for th ose costs.  
+8. Evaluation  on test set  
+The selected model, i.e. with binary  scaling , C = 0.125 , and calibration, was evaluated on the test set.  
+Accuracy was computed using the previously selected threshold of 0.5087. The following results 
+were obtained:  
+1 – RMSE  Accuracy  AUC -ROC  AUC -PR 
+60.47%  77.88%  ± 0.15% (99% CI)  84.72%  80.32%  
+8.1. Confusion matrix  
+Given the previously suggested cost matrix , along  with a supposed false -positive cost of 4, the 
+empirical threshold for this specific cost matrix was previously observed to be 0.808.  
+A confusion  matri x is a contingency table showing the  differences between the true and predicted 
+classes for  a set of labeled examples .[23]  The following is the confusion matrix  for the 
+aforementioned threshold  on the test set, with values expressed as percents : 
+ actual negative  actual po sitive  sum  
+predicted negative  53.84% 25.44%  79.28%  
+predicted positive  2.82%  17.91% 20.72%  
+sum  56.65%  43.35%   
+9. Conclusions  
+The L2-regularized logistic regression model with binary  scaling , C = 0.125 and calibration using 
+isotonic regression was the preferred model on the English Wikipedia vandalism prediction 
+dataset.  
+New words are continually introduced in to the Wikipedia corpus , and a failure to capture their 
+predictive potential may lead to suboptimal predictions.  The obvious  solution to  this concern is of 
+course to retrain  the model frequently  with up -to-date data . More generally, t he extent to which 
+concept drift , i.e. a change over time in the statistical properties of the target variable, may exist  in 
+the data is unknown, and is a top ic for future study.  
+Because there is very limited overfitting of the model to the training set, the entire dataset of 
+2,000,000 cases can be used to train a model using the previously estimated optimal parameters . 
+Given sufficient memory, a  further margin al improvement in performance should be possible by 
+parsing and utilizing the entire corpus  of several million cases .  
+14 Other learning methods such as linear  SVM or Random Forests with calibration may  perform 
+better. [11]  Using an ensemble method such as Random Forests will result in a slightly longer 
+prediction time, however, despite opportunities for parallelization. Nonlinear SVM kernels are not 
+believed to be feasible or useful for a dataset of this size . 
+References  
+1. Smets, K., B. Goethals, and B. Verdonk, Automatic Vandalism Detection in Wikipedia: Towards 
+a Machine Learning Approach.  2008.  
+2. Druck, G., G. Miklau, and A. McCallum, Learning to Predict the Quality of Contributions to 
+Wikipedia.  2008.  
+3. Potthast, M., B. Stein, and R. Gerling, Automatic Vandalism Detection in Wikipedia , in 
+Advances in Information Retrieval . 2008. p. 663 -668.  
+4. Chin, S., et al., Using Language Models to Detect Wik ipedia Vandalism . 2009.  
+5. Itakura, K.Y. and C.L.A. Clarke, Using dynamic markov compression to detect vandalism in the wikipedia , in Proceedings of the 32nd international ACM SIGIR conference on Research and 
+development in information retrieval . 2009, ACM : Boston, MA, USA.  
+6. Buriol, L.S., et al., Temporal Analysis of the Wikigraph.  Web Intelligence, 2006: p. 45 -51. 
+7. Krieger, M., E.M. Stark, and S.R. Klemmer, Coordinating tasks on the commons: designing for 
+personal goals, expertise and serendipity , in Proceedings of the 27th international conference 
+on Human factors in computing systems . 2009, ACM: Boston, MA, USA.  
+8. Data dumps .   [11/6/2009]; Available from: 
+http://meta.wikimedia.org/wiki/Data_dumps . 
+9. Garabík, R., Processing XML Text with Python and ElementTree –a Practical Experience.  2006: 
+p. 160 -165.  
+10. Hastie, T., R. Tibshirani, and J. Friedman, The Elements of Statistical Learning; 2nd ed.  2009: 
+Springe r. 
+11. Caruana, R., N. Karampatziakis, and A. Yessenalina, An empirical evaluation of supervised learning in high dimensions , in Proceedings of the 25th international conference on Machine 
+learning . 2008, ACM: Helsinki, Finland.  
+12. Lin, C., R. Weng, and S . Keerthi, Trust region newton method for large -scale logistic 
+regression.  The Journal of Machine Learning Research, 2008. 9 : p. 627 -650.  
+13. Fan, R., et al., LIBLINEAR: A library for large linear classification.  The Journal of Machine 
+Learning Research, 2008. 9: p. 1871 -1874.  
+14. Hsu, C., C. Chang, and C. Lin, A practical guide to support vector classification . 2009, National 
+Taiwan University: Taipei, Taiwan.  
+15. Zadrozny, B. and C. Elkan. Transforming classifie r scores into accurate multiclass probability 
+estimates . in SIGKDD ’02 . 2002. Edmonton, Alberta, Canada: ACM New York, NY, USA.  
+16. Sheng, V.S. and C.X. Ling, Thresholding for Making Classifiers Cost -sensitive , in AAAI . 2006, 
+AAAI Press.  
+17. DeGroot, M. an d S. Fienberg, The comparison and evaluation of forecasters.  The Statistician, 
+1983: p. 12 -22. 
+18. Niculescu -Mizil, A. and R. Caruana. Predicting good probabilities with supervised learning . in 
+Proceedings of the 22nd International Conference on Machine Le arning . 2005. Bonn, 
+Germany: ACM.  
+19. Alpaydin, E., Introduction to Machine Learning . 2004: The MIT Press.  
+20. Perlich, C., Learning Curves in Machine Learning . 2009, IBM Research Division: Yorktown 
+Heights, NY.   
+15 21. Davis, J. and M. Goadrich, The relations hip between Precision -Recall and ROC curves , in 
+Proceedings of the 23rd international conference on Machine learning . 2006, ACM: 
+Pittsburgh, Pennsylvania.  
+22. Elkan, C. The Foundations of Cost -Sensitive Learning . in Proceedings of the Seventeenth 
+Internati onal Joint Conference on Artificial Intelligence, IJCAI 2001, Seattle, Washington, USA, 
+August 4 -10, 2001 . 2001: Morgan Kaufmann.  
+23. Bradley, A., The use of the area under the ROC curve in the evaluation of machine learning 
+algorithms.  Pattern Recognition , 1997. 30(7): p. 1145 -1159.  
+  
\ No newline at end of file
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+arXiv:1001.0701v2  [math.CA]  9 Mar 2010Existence results for non-smooth second order differential
+inclusions, convergence result for a numerical scheme and
+application to the modelling of inelastic collisions
+Fr´ ed´ eric Bernicot
+CNRS - Universit´ e Lille 1
+Laboratoire Paul Painlev´ e
+59655 Villeneuve d’Ascq Cedex, France
+frederic.bernicot@math.univ-lille1.frAline Lefebvre-Lepot
+CNRS - Ecole Polytechnique
+CMAP
+91128 Palaiseau Cedex, France
+aline.lefebvre@polytechnique.edu
+March 9, 2010
+Abstract
+We are interested in existence results for second order differentia l inclusions, involving
+finite number of unilateral constraints in an abstract framework. These constraints are de-
+scribed by a set-valued operator, more precisely a proximal norma l cone to a time-dependent
+set. In order to prove these existence results, we study an exte nsion of the numerical scheme
+introduced in [8] and prove a convergence result for this scheme.
+Key-words: Second order differential inclusions ; Proximal normal cone ; Inelastic collisions ;
+Numerical scheme.
+MSC:34A60 ; 34A12 ; 65L20.
+1 Introduction
+We consider second order differential inclusions, involving proximal normal cones. These ones
+were firstly treated by M. Schatzman [24] in the framework of e lastic impacts and later by J.J.
+Moreau [13, 14] to model inelastic impacts for a mechanical s ystem in order to describe contact
+dynamics. The impact law describing the dynamics leads to a n on-increasing kinetic energy at
+impacts. These second order problems appear in several mode ls of mechanical systems with a
+finite number of degrees of freedom and dealing with friction less and inelastic contacts.
+Let us specify this class of problems. Let Ibe a bounded time-interval, f:I×Rd→Rdbe
+a map and C:I⇒Rdbe a multi-valued map. The main question concerns the existe nce for
+solutions to the following second order differential inclusi on:
+
+
+∀t∈I, q(t)∈C(t)
+d2q
+dt2+N(C(·),q(·))∋f(·,q(·))
+∀t∈I,˙q(t+) = PCt,q(t)˙q(t−)
+q(0) =q0∈int[C(0)]
+˙q(0) =u0.(1)
+1We denote by int[ C(0)] the interior of the set C(0), by N the proximal normal cone and for
+q∈C(t), byCt,qthe set of feasible velocities:
+Ct,q:={u, q+ǫu∈C(t+ǫ) for small enough ǫ >0}. (2)
+We refer the reader to [3] and [4] for details concerning differ ent normal cones (“limiting cone”,
+“Clarke cone”, ...). Here we will deal with “uniform prox-re gular sets” C(t) so, according to
+[23], all these cones coincide.
+Remark 1.1 We are looking for solutions qsuch that ˙qhas a bounded variation, in order that
+the second order differential equation in (1) should be thoug ht in the distributional sense. More
+precisely, we will solve it for time-measure ˙q∈BV(I)and it should be written with time-measures
+d˙q+N(C(·),q(·))dt∋f(·,q(·))dt.
+In all this work, the second order differential inclusion will be written in the distributional sense
+for easiness. However, we emphasize that we consider time-m easures.
+This differential inclusion can be thought as follows: the poi ntq(t), submitted to the external
+forcef(t,q(t)), has to live in the set C(t) and so to follow its time-evolution. The unilateral
+constraint “ q(t)∈C(t)” may lead to some discontinuities for the velocity ˙ q. For example, fric-
+tionless impacts can be modelled by a second order differentia l inclusion involving the proximal
+normal cone (see [13, 14]). This differential inclusion does n ot uniquely define the evolution of
+the velocity during an impact. To complete the description, we impose the impact law
+˙q(t+) = PCt,q(t)˙q(t−),
+introduced by J.J. Moreau in [13] and justified by L. Paoli and M. Schatzman in [17, 19] (using
+a penalty method) for inelastic impacts.
+The setC(t) corresponds to a set of “admissible configurations” for q. In physical problems, it
+is generally described by several constaints ( gi)ias follows :
+C(t) :=p/intersectiondisplay
+i=1{q, gi(t,q)≥0}. (3)
+The existence of a solution for such second-order problems i s still open in a general framework.
+The first positive results were obtained by M.P.D. Monteiro M arques [10] and L. Paoli and
+M. Schatzman [18] in the case of a smooth time-independent ad missible set (which locally
+corresponds to the single constraint case p= 1 in (3)). The proof relies on a numerical method
+involving a time-discretization of (1) in order to compute a pproximate solutions and is based on
+the study of its convergence . The multi-constraint case wit h analytical data was then treated by
+P. Ballard with a different method in [1], where a positive resu lt of uniqueness for such problems
+was obtained. Thenin [20, 21, 22], an existence resultis pro ved inthecase of anon-smoothtime-
+independent convex admissible set (given by multiple const raints). There, the active constraints
+are supposed to be linearly independent in the following sen se: for each configuration q∈∂C,
+the gradients ( ∇gi(q))i∈Iassociated to active constraints I:={i, gi(q) = 0}are supposed to be
+linearly independent.
+Inthecase of non-convex admissiblesets, someresults were obtained for asingleconstraint p= 1
+(for example in [6] or in [11] and [26] for the first result conc erning time-dependent constraints).
+Recently in [8], B. Maury has proposed a numerical scheme for time-independent multiple and
+2convex constraints gi. The admissible set Cis not supposed to be convex, however at each
+time step, the numerical scheme uses a local convex approxim ation ofC. This improvement is
+interesting as it permits to define an implementable scheme, since the projection onto a convex
+set can be performed with efficient algorithms.
+A first result of convergence for this scheme was proved in [8] for a single constraint and appli-
+cations to the numerical simulation of sytems of particles s ubmitted to inelastic collisions are
+studied in [7] by the second author.
+We emphasize that in the previously mentioned works, the diffe rent numerical schemes (per-
+mitting to discretize (1)) are written (or can be written) in a multi-constraint case. The main
+difficulties consist in proving in a one hand the existence of s olutions for (1) and in the other
+hand a convergence result for the associated numerical sche mes for such multi-constrained prob-
+lems. Concerning the uniqueness, we know from [24] and [1] th at even with smooth data the
+uniquenessdoesnot hold. Theonly positive results areprov ed in [25] for one-dimensional impact
+problems and in [1] in the context of analytic data. This crit ical question of uniqueness is not
+studied here.
+The framework
+In this work, we are interested in extending the previous wor k [8], in order to prove the existence
+of solutions and to get a convergence result of the scheme in t he case of multiple time-dependent
+constraints. Moreover we give applications in modelling in elastic collisions.
+First of all, let us precise some notations. We write W1,∞(I,Rd) (resp. W1,1(I,Rd)) for the
+Sobolev space of functions in L∞(I,Rd) (resp.L1(I,Rd)) whose derivative is also in L∞(I,Rd)
+(resp.L1(I,Rd)).BV(I,Rd) is the space of functions in L∞(I,Rd) with bounded variations on
+I. We define the dual space M(I) := (Cc(I))′whereCc(I) is the space of continuous functions
+with compact support (corresponding to the set of Radon meas ure due to Riesz Theorem). We
+setM+(I) for the subset of positive measures.
+We consider second-order differential inclusions involving a set-valued map Q: [0,T]⇒Rd
+satisfying that for every t∈[0,T],Q(t) is the intersection of complements of smooth convex
+sets. Let us first specify the set-valued map Q. – This general framework has already been
+described by J. Venel in [28] for first order differential inclu sions (fitting into the so-called
+sweeping process theory) and in [2] for a stochastic perturb ation of such problems. –
+Fori∈ {1,..,p}, letgi: [0,T]×Rd→Rbea convex function with respect to the second variable.
+For every t∈[0,T], we introduce the sets Qi(t) defined by:
+Qi(t) :=/braceleftBig
+q∈Rd, gi(t,q)≥0/bracerightBig
+, (4)
+and the feasible set Q(t) (supposed to be nonempty)
+Q(t) :=p/intersectiondisplay
+i=1Qi(t). (5)
+We denote by I= [0,T] the time interval. The considered problem is the following one: we are
+looking for a solution q∈W1,∞(I,Rd),˙q∈BV(I,Rd) such that
+
+
+∀t∈I, q(t)∈Q(t)
+d2q
+dt2+N(Q(·),q(·))∋f(·,q(·))
+∀t∈I,˙q(t+) = PCt,q(t)˙q(t−)
+q(0) =q0∈int[Q(0)]
+˙q(0) =u0,(6)
+3where N( Q(t),q(t)) is the proximal normal cone of Q(t) atq(t) andCt,qis the set of admissible
+velocities :
+Ct,q:={u, ∂tgi(t,q)+/a\}b∇acke⊔le{⊔∇qgi(t,q),u/a\}b∇acke⊔∇i}h⊔ ≥0 ifgi(t,q) = 0}, (7)
+which corresponds to (2) in our framework.
+Wehavetomakeassumptionsontheconstraints gi. Firstwerequiresomeregularity: wesuppose
+that there exist c >0 and open sets Ui(t)⊃Qi(t) for alltin [0,T] verifying
+dH(Qi(t),Rd\Ui(t))> c, (A0)
+wheredHdenotes the Hausdorff distance. Moreover we assume that ther e exist constants
+α,β,M > 0 such that for all tin [0,T],gi(t,·) belongs to C2(Ui(t)) and satisfies
+∀q∈Ui(t), α≤ |∇qgi(t,q)| ≤β, (A1)
+∀q∈Ui(t),|∂tgi(t,q)| ≤β, (A2)
+∀q∈Ui(t),|∂t∇qgi(t,q)| ≤M, (A3)
+∀q∈Ui(t),|D2
+qgi(t,q)| ≤M, (A4)
+and
+∀q∈Ui(t),|∂2
+tgi(t,q)| ≤M. (A5)
+In comparison with [28] and [2] where first order differential i nclusion are studied, we require
+the new and natural assumption (A5), due to the fact that we co nsider second order differential
+inclusions.
+Note that these assumptions can slightly be weakened. Indee d, the lower bound in (A1) is only
+required in a neighborhood of q∈∂Q(t). Moreover, we have assumed C2-smoothness in (A4)
+and (A5) for the sake of simplicity, but we only need C1+ǫregularity.
+Furthermore, we require a kind of independence for the activ e gradients. For all t∈[0,T] and
+q∈Q(t), we denote by I(t,q) the active set at q
+I(t,q) :={i∈ {1,..,p}, gi(t,q) = 0}, (8)
+corresponding to the active constraints. For every ρ >0, we define the following set:
+Iρ(t,q) :={i∈ {1,..,p}, gi(t,q)≤ρ}. (9)
+We assume there exist γ >0 andρ >0 such that for all t∈[0,T],
+∀q∈Q(t),∀λi≥0,/summationdisplay
+i∈Iρ(t,q)λi|∇qgi(t,q)| ≤γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+i∈Iρ(t,q)λi∇qgi(t,q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (A6)
+We will use the following weaker assumption too:
+∀q∈Q(t),∀λi≥0,/summationdisplay
+i∈I(t,q)λi|∇qgi(t,q)| ≤γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+i∈I(t,q)λi∇qgi(t,q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (A6’)
+Note that assumptions (A6) and (A6’) describe a kind of “posi tive linear independence” of the
+almost active gradients. In the time-independent case, (A6 ’) is lightly weaker than the linear
+independence assumption made in [20, 21, 22]. In fact, such a ssumptions imply a “uniform
+prox-regularity” of the admissible set Q(t), which is a weaker property than the convexity.
+Under these assumptions, we have a characterization of the p roximal normal cone N( Q(t),·).
+4Proposition 1.2 (Prop 2.8 of [28]) In this framework, we know that for every t∈I, and
+everyq∈∂Q(t),
+N(Q(t),q) :=
+
+−/summationdisplay
+i∈I(t,q)λi∇qgi(t,q), λi≥0
+
+.
+So our Problem (6) can be written as follows: we are looking fo r solutions q∈W1,∞(I,Rd),˙q∈
+BV(I,Rd) and time-measures λi∈ M+(I) such that
+
+
+∀t∈I, q(t)∈Q(t)
+d2q
+dt2=f(·,q(·))+p/summationdisplay
+i=1λi∇qgi(·,q(·))
+supp(λi)⊂ {t, gi(t,q(t)) = 0}for alli
+∀t∈I,˙q(t+) = PCt,q(t)˙q(t−)
+q(0) =q0∈int[Q(0)]
+˙q(0) =u0.(10)
+We denote by λ= (λ1,···,λp)∈Rpthe vector of the Lagrange multipliers associated to these
+pconstraints.
+As usual, we obtain existence results for (10) by proving the convergence of a sequence of dis-
+cretized solutions.
+To do so, we extend the algorithm proposed by B. Maury in [8] fo r modelling inelastic collisions,
+to the case of abstract and time-dependent constraints. In [ 8], the convergence (up to a subse-
+quence) is proved in the case of a single constraint. Here, we show that this convergence still
+holds in the multi-constraint case.
+Let us describe the numerical scheme. Let h=T/Nbe the time step. We denote by qn
+h∈Rd
+andun
+h∈Rdthe approximated solution and velocity at time tn
+hhforn∈ {0,..,N}.
+The discretization of the continuous constraints Ctn
+h,q(tn
+h)proposed in [8] corresponds to a first
+order approximation of the constraints in the space variabl e: fort∈Iandq∈U(t), we set
+Kh(t,q) :={u, gi(t,q)+h/a\}b∇acke⊔le{⊔∇qgi(t,q),u/a\}b∇acke⊔∇i}h⊔ ≥0}. (11)
+The reason why we do not expand the time variable in this discr ete admissible set is that we
+will use a semi-implicit numerical scheme, with directly im pliciting in time the set Kh(t,q) (See
+(13)).
+The approximated solutions are built using the following sc heme:
+1. Initialization :
+(q0
+h,u0
+h) := (q0,u0) (12)
+2. Time iterations: qn
+handun
+hare given. We define fn
+h:=1
+h/integraldisplaytn+1
+h
+tn
+hf(s,qn
+h)ds,
+un+1
+h:= PKh(tn+1
+h,qn
+h)[un
+h+hfn
+h] (13)
+and
+qn+1
+h:=qn
+h+hun+1
+h, (14)
+5where P Cis the Euclidean projection onto the set C. This algorithm is a “prediction-correction
+algorithm”: the predicted velocity un
+h+hfn
+h, that may not be admissible, is projected onto the
+approximate set of admissible velocities.
+Since the projection PKh(tn+1
+h,qn
+h)consists in a constrained minimization problem, with a finit e
+number of affine constraints, it involves Lagrange multiplie rs (λn+1
+h)∈Rpcorresponding to the
+pconstraints. It can be checked that we have a discrete counte rpart of the momentum balance
+appearing in (10):
+un+1
+h−un
+h
+h=fn
+h+/summationdisplay
+λn+1
+h,i∇qgi(tn+1
+h,qn
+h) (15)
+withλn+1
+h,i≥0 andλn+1
+h,i= 0 when gi(tn+1
+h,qn
+h)+h/a\}b∇acke⊔le{⊔∇qgi(tn+1
+h,qn
+h),un+1
+h/a\}b∇acke⊔∇i}h⊔>0.
+In [8], the scheme is shown to be stable, robust and to present a good behaviour for large time-
+steps. That is why, we are interested in continuing its numer ical analysis in the multi-constraint
+case, with proposing some extensions like the time-depende nce of the constraints.
+Results
+We recall that I= [0,T] is the time interval and his the constant time step ( tn
+hhforn= 0...N).
+We denote by qhthe piecewise affine function with qh(tn
+h) =qn
+h. We denote by uhthe derivative
+ofqh, piecewise constant equal to un+1
+hon ]tn
+h,tn+1
+h[. Finally, we define λh, piecewise constant
+equal to λn+1
+hon ]tn
+h,tn+1
+h[.
+The convergence theorem is the following one.
+Theorem 1.3 Let (qh,uh,λh)be the sequence of solutions constructed from the scheme (12 -15)
+and suppose that f:I×Rd→Rdis a measurable map satisfying:
+∃KL>0,∀t∈I,∀q,˜q∈U(t),|f(t,q)−f(t,˜q)| ≤KL|q−˜q| (16)
+∃F∈L1(I),∀t∈I,∀q∈U(t),|f(t,q)| ≤F(t). (17)
+Then, when hgoes to zero, there exist subsequences, still denoted by (qh)h,(uh)h,(λh)h, and
+(q,u,λ)∈W1,∞(I,Rd)×BV(I,Rd)×M+(I)p
+such that
+uh−→uinL1(I,Rd),
+qh−→qinW1,1(I,Rd)andL∞(I,Rd)with˙q=u,
+λh⋆−⇀ λinM+(I)p
+where(q,u,λ)is solution to (10) and so (q,u)is a solution to (6).
+We emphasize that, up to our knowledge, this result is the firs t one concerning such multi-
+constrained second order differential inclusions with on the one handtime-dependent constraints
+and on the other hand a non-convex and non smooth admissible s et.
+Remark 1.4 For time-independent constraints, Assumption (A6’) is req uired but Assumption
+(A6) is not necessary.
+The proof is quite long and technical. We refer the reader to [ 8] for a first proof dealing with
+the case of one ( p= 1) time-independent constraint g. We will follow the same reasoning with
+some new arguments (appearing in [28]) in order to solve the d ifficulties raised by the multiple
+constraints and the time-dependence. Section 2 is devoted t o the outline and the main ideas
+of the proof. For the sake of readibility, the demonstration s of some technical Propositions are
+postponed to Section 3. In Section 4, we describe an applicat ion to the modelling of inelastic
+collisions.
+62 Convergence result
+This section is devoted to the proof of Theorem 1.3. It is divi ded in 7 steps and for readibility
+reasons we have postponed some technical proofs in the next s ection.
+•Step 1: The scheme is well-defined and produces feasible configurations
+Proposition 2.1 For a small enough parameter h, the scheme is well-defined. Moreover the
+computed configurations are feasible :
+∀h >0,∀n∈ {0,..,N}, qh(tn
+h)∈Q(tn
+h).
+Proof : Lethbe smaller than c/c0wherecandc0are given in Lemma 2.2 (below stated)
+and Assumption (A0) respectively. By assuming that qh(tn
+h)∈Q(tn
+h), we also deduce that
+qh(tn
+h)∈Q(tn+1
+h)+c0hB(0,1)⊂U(tn+1
+h). Then the gradient ∇qgi(tn+1
+h,qn
+h) is well-defined and
+so is the set Kh(tn+1
+h,qn
+h). The step 2 of the scheme can be performed and due to the conve xity
+of function gi(tn+1
+h,·),
+un+1
+h∈Kh(tn+1
+h,qn
+h) =⇒qn+1
+h∈Q(tn+1
+h).
+Then we conclude by iteration. ⊓⊔
+Lemma 2.2 The set-valued map Qis Lipschitz continuous with a constant c0, for the Hausdorff
+distance.
+We refer the reader to Proposition 2.11 of [28] for a detailed proof of this result.
+For the intermediate times t∈]tn
+h,tn+1
+h[, the point qh(t) may not belong to Q(t). However from
+Proposition 2.1 and Lemma 2.2, we have the following estimat e :
+∀h >0,∀t∈I, d(qh(t),Q(t))≤max{d(qn
+h,Q(t)),d(qn+1
+h,Q(t))} ≤c0h. (18)
+•Step 2: uhis bounded in BV(I,Rd)
+First, we check that the velocities are uniformly bounded (p roved later in Subsection 3.1).
+Proposition 2.3 The sequence of computed velocities (uh)his bounded in L∞(I,Rd). We set
+K:= sup
+h/ba∇dbluh/ba∇dblL∞(I)<∞.
+Proposition 2.4 The sequence of computed velocities (uh)his bounded in BV(I,Rd).
+Proof : Sinceu0
+h=u0and using Proposition 2.3, it suffices to show that ( uh)hhas bounded
+variations on I. This has been proved for a single constraint in [8]. Unfortu nately, this proof
+cannot be extended to the multi-constraint case. To obtain a n estimate on the total variation,
+we use a similar technique to the one proposed in [5] and [6]. T hese ideas rest on the following
+property: all the cones Kh(tn+1
+h,qn
+h) contain a ball of fixed radius with a bounded center, which
+describes the fact that the solid angles of the cones N( Q(t),qh(t)) are not too small. This
+property is proved by using a “good direction” (see Lemma 3.1 ) which permits to increase all
+the almost active constraints. The details of the proof are p ostponed to Subsection 3.1. ⊓⊔
+•Step 3: Extraction of convergent subsequences
+Proposition 2.4 directly implies the following convergenc e result:
+7Proposition 2.5 There exist qinW1,∞(I,Rd)anduinBV(I,Rd)such that, up to a subse-
+quence,
+uh− −− →
+h→0uinL1(I,Rd),
+qh− −− →
+h→0qinW1,1(I,Rd)andL∞(I,Rd)with˙q=u.
+Furthermore (18) yields
+∀t∈I, q(t)∈Q(t).
+In addition, we show that the sequence of Lagrange multiplie rs converges:
+Proposition 2.6 There exists λinM+(I)psuch that, up to a subsequence,
+λh⋆−⇀ λinM+(I)p.
+Proof : From (15), we have
+/summationdisplay
+ihλn+1
+h,i∇qgi(tn+1
+h,qn
+h) =un+1
+h−un
+h−hfn
+h
+withλn+1
+h,i≥0 andλn+1
+h,i= 0 when
+gi(tn+1
+h,qn
+h)+h/a\}b∇acke⊔le{⊔∇qgi(tn+1
+h,qn
+h),un+1
+h/a\}b∇acke⊔∇i}h⊔>0. (19)
+Remark that for a small enough parameter h,gi(tn+1
+h,qn
+h) +h/a\}b∇acke⊔le{⊔∇qgi(tn+1
+h,qn
+h),un
+h+hfn
+h/a\}b∇acke⊔∇i}h⊔ ≤0
+implies that i∈Iρ(tn+1
+h,qn
+h). Consequently, the reverse triangle inequality (Assumpt ion (A6’))
+with (A1) and the Lipschitz regularity (Assumption (A3)) im ply for a small enough parameter
+h
+h/summationdisplay
+iλn+1
+h,i/lessorsimilar|un+1
+h−un
+h−hfn
+h|.
+Therefore, we obtain for this small enough parameter h
+/ba∇dblλh/ba∇dblL1(I)/lessorsimilarVarI(uh)+/ba∇dblF/ba∇dblL1(I).
+Hence from Proposition 2.4 together with hypothesis (17), ( λh)his bounded in L1(I), which
+concludes the proof. ⊓⊔
+•Step 4: Momentum balance
+As in Step 6 of Theorem 1 in [8], using Proposition 2.5 togethe r with Proposition 2.6, we pass
+to the limit in the discrete momentum balance (15) to obtain
+Proposition 2.7 The momentum balance is verified by the limits uandλ: in the sense of
+time-measure
+˙u=f(·,q(·))+/summationdisplay
+iλi∇qgi(·,q(·)).
+Note that this equation has to be thought in term of time-meas ure (see Remark 1.1):
+du=f(·)dt+/summationdisplay
+i∇qgi(·,q(·))λi,
+where we denote by duthe differential measure of the BV-function u.
+•Step 5: Support of the measures λi
+From the uniform convergence of qhand the Lipschitz regularity of gi(Assumptions (A1) and
+(A2)), it can be checked, as in Step 7 of Theorem 1 in [8], that
+8Proposition 2.8
+∀i,supp(λi)⊂ {t, gi(t,q(t)) = 0}.
+Indeed (19) describes a similar property for the discretize d multipliers. The uniformconvergence
+allows us to go to the limit in (19) and to prove the previous pr oposition.
+This property describes the fact that the measure λihas a contribution only when the associated
+constraint giis saturated.
+•Step 6: Initial condition
+As in Step 8 of Theorem 1 in [8], using again the uniform conver gence of qhit can be shown
+that
+q(0) =q0andu(0) =u0.
+We emphasize that to prove this point, we use the property q0∈Int[Q(0)]. From this, it can
+be checked that for tn
+h< swithsa small enough parameter the desired velocity un
+h+hfn
+hstill
+remains admissible and so we don’t need to project. That allo ws us to deal with any initial
+velocityu0∈Rd. Ifq0∈∂Q(0), this property still holds if we assume that the initial v elocity is
+admissible: u0∈ C0,q0. Else, we would get
+u+(0) = P C0,q0(u0)
+according to the next proposition.
+•Step 7: Collision law
+Finally, Theorem 1.3 will follow, provided that we check the collision law for the limits uandq,
+which is given by the following proposition.
+Proposition 2.9
+∀t0∈I, u+(t0) = PCt0,q(t0)(u−(t0)).
+Proof : The idea is to let hgo to zero in the discrete collision law,
+un+1
+h= PKh(tn+1
+h,qn
+h)[un
+h+hfn
+h].
+The main difficulty comes from the fact that the mapping q→Kh(t,q) is not Lipschitzean. The
+details of the proof are postponed to Subsection 3.2. ⊓⊔
+3 Auxiliary results
+Before proving the technical Propositions 2.3, 2.4 and 2.9, we recall the following main Lemma.
+This technical result is very important and all our proofs re st on this idea. It says that, for each
+timetn
+h, one can find a “good direction” increasing all the constrain ts which are almost active,
+the corresponding increase being independent of n.
+Lemma 3.1 There exist constants δ,κ,θandτ >0such that for all t∈I, for all q∈∂Q(t)
+there exists a unit vector v:=v(t,q)satisfying :
+•for alls∈[t−τ,t+τ],y∈Q(s)∩B(q,θ)andi∈Iκρ(s,y),
+/a\}b∇acke⊔le{⊔∇qgi(s,y),v/a\}b∇acke⊔∇i}h⊔ ≥δ,
+whereρis the constant defined in (A6).
+This lemma is a consequence of the reverse triangle inequali ty (Assumption (A6)). We do not
+give the proof here and refer the reader to Lemma 2.10 in [28] f or a detailed proof with τ= 0 and
+to Proposition 4.2 in [2] for a complete proof with some τ >0. Indeed there is in the previously
+mentioned papers, a detailed construction of such “good dir ections”.
+93.1 BV estimate for uh(Propositions 2.3 and 2.4)
+We first prove a uniform bound of the computed velocities uhinL∞(I).
+Proof of Proposition 2.3
+1−) For any t∈Iandq∈Q(t), construction of a specific point w∈ Ct,q.
+From Lemma 3.1 (stated at Section 3), it exists a “good direct ion”: a unit vector vsatisfying,
+∀i∈Il(t,q),/a\}b∇acke⊔le{⊔∇qgi(t,q),v/a\}b∇acke⊔∇i}h⊔ ≥δ, (20)
+for some numerical constants δ,l >0 non-depending on tandq. Fork >0 a large enough real,
+we claim that kvbelongs to Ct,q. Indeed for i∈Il(t,q) (⊃I(t,q)), withk≥(β+δ)/δ, it comes
+∂tgi(t,q)+/a\}b∇acke⊔le{⊔∇qgi(t,q),kv/a\}b∇acke⊔∇i}h⊔ ≥∂tgi(t,q)+kδ≥δ >0 (21)
+thanks to Assumption (A2).
+Choosing k= (β+δ)/δ, we have built a point w=kvbelonging to Ct,qwith|w|= (β+δ)/δ.
+2−) This vector wbelongs to Kh(s+h,˜q) for (s,˜q) close to ( t,q) andhsmall enough.
+Let fix a point ( t,q) (for example the initial condition ( t,q) = (0,q0)). From the previous point,
+we know that there exists a bounded admissible velocity w∈ Ct,q, which satisfies the stronger
+property (21).
+More precisely if s∈Iand ˜q∈Q(s) satisfy
+|s−t|+|˜q−q| ≤min{l/(2β),ǫ}:=ν (22)
+withǫa small parameter verifying 2 ǫM(1+(β+δ)/δ)≤δthenw∈Kh(s+h,˜q). Indeed, from
+(22) we have
+Il/2(s,˜q)⊂Il(t,q).
+This, together with (21) and (22) gives, for all i∈Il/2(s,˜q)
+∂tgi(s,˜q)+/a\}b∇acke⊔le{⊔∇qgi(s,˜q),w/a\}b∇acke⊔∇i}h⊔ ≥δ−ǫM(1+|w|)≥δ/2.
+Moreover, for the other indices i /∈Il/2(s,˜q) we have
+gi(s,˜q)+h[∂tgi(s,˜q)+/a\}b∇acke⊔le{⊔∇qgi(s,˜q),w/a\}b∇acke⊔∇i}h⊔]≥l
+2−hβ(1+|w|).
+Consequently, for hsmall enough ( h≤h0:=l/(2β(1+(β+δ)/δ) +δ/2)), we obtain
+∀i, g i(s,˜q)+h[∂tgi(s,˜q)+/a\}b∇acke⊔le{⊔∇qgi(s,˜q),w/a\}b∇acke⊔∇i}h⊔]≥hδ
+2,
+which, by a first order expansion in time gives:
+∀i, g i(s+h,˜q)+h/a\}b∇acke⊔le{⊔∇qgi(s+h,˜q),w/a\}b∇acke⊔∇i}h⊔ ≥hδ
+2+Oh→0(h2).
+We deduce that there exists h1≤h0such that, for h≤h1,
+∀i, g i(s+h,˜q)+h/a\}b∇acke⊔le{⊔∇qgi(s+h,˜q),w/a\}b∇acke⊔∇i}h⊔ ≥0,
+and consequently w∈Kh(s+h,˜q).
+3−) Estimate on the velocities for small time intervals.
+Let us fix h≤h1(given in the previous point) and a small time interval [ t−,t+]⊂Iof length
+|t+−t−| ≤ν
+2/parenleftBig
+|un0
+h|+2β+δ
+δ+/integraltextT
+0F(t)dt/parenrightBig
+10wheren0is the smallest integer nsuch that tn
+h≥t−. We suppose that tn0
+h∈[t−,t+]. We are
+looking for a bound on the velocity on this time interval. Fro m the first two points, setting
+(t,q) = (tn0
+h,qn0
+h), we have an admissible velocity w∈ Ct,qsuch that for all s∈Iand ˜q∈Q(s)
+we have
+w∈Kh(s+h,˜q)
+as soon as |s−t|+|˜q−q| ≤ν. Since for s=tn
+h∈[t−,t+] ,|s−t| ≤ν/2, we deduce that for all
+integernsuch that tn
+h∈[t−,t+], if
+|qn
+h−qn0
+h| ≤ν/2 (23)
+then
+w∈Kh(tn+1
+h,qn
+h).
+Considering such an integer nsatisfying (23), since un+1
+his the Euclidean projection of un
+h+hfn
+h
+on the convex set Kh(tn+1
+h,qn
+h) (containing the point w), we deduce that
+|un+1
+h−w| ≤ |un
+h+hfn
+h−w|,
+which implies
+|un+1
+h−w| ≤ |un
+h−w|+/integraldisplaytn+1
+h
+tn
+hF(t)dt.
+We setmthe smallest integer (bigger than n0) such that m+1 does not satisfy (23) or tm+1
+h/∈
+[t−,t+]. By summing these inequalities from n0ton=p−1 withn0≤p≤m, we get
+∀p∈[n0,m],|up
+h−w| ≤ |un0
+h−w|+/integraldisplayT
+0F(t)dt.
+Finally, it comes
+sup
+n0≤p≤m|up
+h| ≤ |un0
+h|+2β+δ
+δ+/integraldisplayT
+0F(t)dt. (24)
+By integrating in time, we deduce
+|qm+1
+h−qn0
+h| ≤/parenleftbigg
+|un0
+h|+2β+δ
+δ+/integraldisplayT
+0F(t)dt/parenrightbigg
+|t+−t−| ≤ν/2
+by the assumption on the length of the time interval. As a cons equence, we get that n=m+1
+satisfies (23) which by definition of m, yieldstm
+h≤t+< tm+1
+h. Hence, from (24), we have
+sup
+t−≤tn
+h≤t+|un
+h| ≤ |un0
+h|+2β+δ
+δ+/integraldisplayT
+0F(t)dt.
+4−) End of the proof.
+The parameter h < h1being fixed, we are now looking for a bound on uhon the whole time
+intervalI= [0,T]. Let us start with t−=t(0) := 0. From the previous point we know that with
+t+=t(1) := min
+
+ν
+2/parenleftBig
+|u0|+2β+δ
+δ+/integraltextT
+0F(t)dt/parenrightBig,T
+
+
+we have
+sup
+0≤tn
+h≤t(1)|uh(tn
+h)| ≤ |u0|+2β+δ
+δ+/integraldisplayT
+0F(t)dt.
+11Then, let us suppose that there exists n1such that t(0)< tn1
+h≤t(1)< tn1+1
+h. We have
+0≤δ1:=t(1)−tn1
+h< h. In that case, we set t−=tn1
+hand
+t+=t(2) := min
+
+tn1+ν
+2/parenleftBig
+|u0|+4β+δ
+δ+2/integraltextT
+0F(t)dt/parenrightBig,T
+
+
+= min
+
+t(1)−δ1+ν
+2/parenleftBig
+|u0|+4β+δ
+δ+2/integraltextT
+0F(t)dt/parenrightBig,T
+
+.
+From the previous point, we deduce that
+sup
+t(1)≤tn
+h≤t(2)|uh(tn
+h)| ≤sup
+tn1≤tn
+h≤t(2)|uh(tn
+h)| ≤ |u0|+4β+δ
+δ+2/integraldisplayT
+0F(t)dt
+and so
+sup
+0≤tn
+h≤t(2)|uh(tn
+h)| ≤ |u0|+4β+δ
+δ+2/integraldisplayT
+0F(t)dt.
+By iterating this reasoning, for any integer k≥1 we set
+t(k) := min
+
+t(k−1)−δk−1+ν
+2/parenleftBig
+|u0|+2kβ+δ
+δ+2k/integraltextT
+0F(t)dt/parenrightBig,T
+
+
+= min
+
+−k−1/summationdisplay
+i=1δi+k/summationdisplay
+i=1ν
+2/parenleftBig
+|u0|+2iβ+δ
+δ+2i/integraltextT
+0F(t)dt/parenrightBig,T
+
+.
+whereδk< hfor allk. This construction of t(k) can be made while there exists nksuch that
+t(k−2)< tnk−1
+h≤t(k−1)< tnk−1+1
+h. That is, while t(k−1)−t(k−2)> h. This condition will
+be verified as long as
+−δk−2+ν
+2/parenleftBig
+|u0|+2(k−1)β+δ
+δ+2(k−1)/integraltextT
+0F(t)dt/parenrightBig> h.
+Therefore, using the fact that 0 ≤δk−2< h, we see that we can construct t(k) fork < N
+verifyingν
+2/parenleftBig
+|u0|+2(k−1)β+δ
+δ+2(k−1)/integraltextT
+0F(t)dt/parenrightBig>2h,
+which is equivalent to
+k < k0(h) := 1+/parenleftbiggν
+8h−|u0|
+2/parenrightbigg/parenleftbiggβ+δ
+δ+/integraldisplayT
+0F(t)dt/parenrightbigg−1
+. (25)
+Consequently, we know that the velocities can be bounded on [ 0,t(k0(h))] where
+t(k0(h)) = min
+
+−k0(h)−1/summationdisplay
+i=1δi+k0(h)/summationdisplay
+i=1ν
+2/parenleftBig
+|u0|+2iβ+1
+δ+2i/integraltextT
+0F(t)dt/parenrightBig,T
+
+.
+Now, using the fact that k0(h) goes to infinity when hgoes to zero, that the harmonic serie
+diverges and that (25) yields/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay
+i=1δi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤hk0(h)≤C,
+12we see that t(k0(h)) is equal to Tforhsmall enough. Therefore, there exists h2< h1such that,
+forh < h2,T=t(k0(h2)) =t(k0(h)). Finally, we see that, for h < h2,t(k) can be constructed
+untilk=k0(h2) anduhcan be bounded as follows
+sup
+h≤h2sup
+0≤tn
+h≤T|uh(tn
+h)| ≤ |u0|+2k0(h2)β+δ
+δ+2k0(h2)/integraldisplayT
+0F(t)dt (26)
+which concludes the proof of the existence of a uniform bound inL∞for the velocities uh.⊓⊔
+To prove Proposition 2.4, it suffices now to show that the seque nce (uh)hhas bounded variation.
+Theorem 3.2 The sequence (uh)hhas a bounded variation on I.
+Proof : In order to study the variation of uhonI, we split Iinto smallest intervals. We define
+(sj)jforjfrom 0 to Psuch that:
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingles0= 0, sP=T,
+|sj+1−sj|=1
+2min/braceleftbigg
+τ,θ
+K/bracerightbigg
+,forj= 0...P−2,
+|sP−sP−1| ≤1
+2min/braceleftbigg
+τ,θ
+K/bracerightbigg
+,
+whereτandθare given by Lemma 3.1 and Kis the bound on /ba∇dbluh/ba∇dblL∞(I)(see Proposition 2.3).
+All these constants do not depend on hand such a construction gives
+P=/bracketleftBigg
+2T
+min/braceleftbig
+τ,θ
+K/bracerightbig/bracketrightBigg
++1, (27)
+which is independent of h. Then, for all h, we define nj
+hforjfrom 0 to P−1 as the first time
+step strictly greater than sj:
+tnj
+h−1
+h≤sj< tnj
+h
+h,
+andnP
+his set equal to N(tN
+h=tnP
+h
+h=T).
+In the following, we suppose h <min{|sj+1−sj|}/2. Doing so, we obtain a strictly increasing
+sequence of ( tnj
+h
+h)jwith
+|tnj
+h
+h−tnj−1
+h
+h| ≤min/braceleftbigg
+τ,θ
+K/bracerightbigg
+. (28)
+The variation of uhonIcan be written as follows
+VarI(uh) =N−1/summationdisplay
+n=0|un+1
+h−un
+h|=P−1/summationdisplay
+j=0Varjuh
+where
+Varj(uh) :=nj+1
+h−1/summationdisplay
+nj
+h|un+1
+h−un
+h|
+corresponds to the variation on [ tnj
+h
+h,tnj+1
+h
+h[. To study these terms, we recall that
+un+1
+h= PKh(tn+1
+h,qn
+h)[un
+h+hfn] (29)
+by construction and state the following lemma:
+13Lemma 3.3 There exist η >0and uniformly bounded vectors ynj
+hsuch that, for all small
+enoughh, for allj= 0...Pandn∈[nj
+h,nj+1
+h[,we have
+x1= PKh(tn+1
+h,qn
+h)[x0] =⇒ |x1−x0| ≤1
+2η/parenleftBig
+|x0−ynj
+h|2−|x1−ynj
+h|2/parenrightBig
+Proof : The outline of the proof is the following: first, we prove that there exist unit vectors
+vnj
+hsuch that
+n∈[nj
+h,nj+1
+h[ =⇒B(2βK
+δvnj
+h,η)⊂Kh(tn+1
+h,qn
+h) withη:=K
+2, (30)
+whereKis a bound on /ba∇dbluh/ba∇dblL∞(I)(see Proposition 2.3). Then, we conclude using similar
+arguments to the ones exposed in [5, 6].
+Step 1: From Lemma 3.1 with t=tnj
+h
+handq=qnj
+h
+h, we have a unit “good direction” written
+vnj
+h. Letnbelong to [ nj
+h,nj+1
+h[. From Proposition 2.1, we know that qn+1
+hbelongs to Q(tn+1
+h).
+Moreover, (28) gives |tn+1
+h−tnj
+h
+h| ≤τand|qn+1
+h−qnj
+h
+h| ≤θ. Consequently, Lemma 3.1 gives
+∀i∈Iκρ(tn+1
+h,qn+1
+h),/angbracketleftBig
+∇qgi(tn+1
+h,qn+1
+h),vnj
+h/angbracketrightBig
+≥δ. (31)
+We deduce that for all index i∈ {1,..,p}and a small enough parameter h:
+gi(tn+1
+h,qn
+h)+2βK
+δh/a\}b∇acke⊔le{⊔∇qgi(tn+1
+h,qn+1
+h),vnj
+h/a\}b∇acke⊔∇i}h⊔ ≥ −hβK+2hβK=hβK. (32)
+Indeed, we write
+gi(tn+1
+h,qn
+h)+2βK
+δh/a\}b∇acke⊔le{⊔∇qgi(tn+1
+h,qn+1
+h),vnj
+h/a\}b∇acke⊔∇i}h⊔=
+/braceleftbig
+gi(tn+1
+h,qn
+h)−gi(tn+1
+h,qn+1
+h)/bracerightbig
++/braceleftbigg
+gi(tn+1
+h,qn+1
+h)+2βK
+δh/a\}b∇acke⊔le{⊔∇qgi(tn+1
+h,qn+1
+h),vnj
+h/a\}b∇acke⊔∇i}h⊔/bracerightbigg
+.
+The first term can be estimated using (A1) and the bound Kon/ba∇dbluh/ba∇dblL∞(I). In order to estimate
+the second term, if i∈Iκρ(tn+1
+h,qn+1
+h), we use (31) together with the fact that gi(tn+1
+h,qn+1
+h)≥0,
+which gives the required bound. In the case i /∈Iκρ(tn+1
+h,qn+1
+h), we use gi(tn+1
+h,qn+1
+h)≥κρ
+and (A1), which also gives the required bound for hsmall enough.
+Finally, (32) together with assumption (A1) implies that fo r allv∈B(2βK
+δvnj
+h,K/2)
+gi(tn+1
+h,qn
+h)+h/a\}b∇acke⊔le{⊔∇qgi(tn+1
+h,qn
+h),v/a\}b∇acke⊔∇i}h⊔ ≥hβK−hβK
+2=hβK
+2≥0,
+which proves (30).
+Step 2: Letnbelong to [ nj
+h,nj+1
+h[. We define
+znj
+h:=ynj
+h+ηx0−x1
+|x0−x1|whereynj
+h:=2βK
+δvnj
+h.
+(Here we suppose x0/\e}a⊔io\slash=x1, else the desired result is obvious.) From the previous step we have
+znj
+h∈B(2βK
+δvnj
+h,η)⊂Kh(tn+1
+h,qn
+h).
+The point x1being the projection of x0onto the closed convex set Kh(tn+1
+h,qn
+h), we get
+/a\}b∇acke⊔le{⊔x0−x1,znj
+h−x1/a\}b∇acke⊔∇i}h⊔ ≤0.
+14From this we have
+|x0−ynj
+h|2=|x1−ynj
+h|2+|x0−x1|2+2/a\}b∇acke⊔le{⊔znj
+h−ynj
+h,x0−x1/a\}b∇acke⊔∇i}h⊔+2/a\}b∇acke⊔le{⊔x1−znj
+h,x0−x1/a\}b∇acke⊔∇i}h⊔
+≥ |x1−ynj
+h|2+2/a\}b∇acke⊔le{⊔znj
+h−ynj
+h,x0−x1/a\}b∇acke⊔∇i}h⊔
+≥ |x1−ynj
+h|2+2η|x0−x1|.
+This, together with the fact that the vectors ynj
+hare uniformly bounded by2βK
+δ, ends the proof
+of Lemma 3.3. ⊓⊔
+We now come back to the proof of Theorem 3.2. For nin [nj
+h,nj+1
+h[, using (29) and the previous
+lemma (with x0=un
+h+hfn
+handx1=un+1
+h), it comes
+|un+1
+h−un
+h−hfn
+h| ≤1
+2η/parenleftBig
+|x0−ynj
+h|2−|x1−ynj
+h|2/parenrightBig
+≤1
+2η/parenleftBig
+|un
+h+hfn
+h−ynj
+h|2−|un+1
+h−ynj
+h|2/parenrightBig
+≤1
+2η/parenleftBig
+|un
+h−ynj
+h|2−|un+1
+h−ynj
+h|2/parenrightBig
++1
+2η|hfn
+h|2+1
+η|hfn
+h||un
+h−ynj
+h|
+≤1
+2η/parenleftBig
+|un
+h−ynj
+h|2−|un+1
+h−ynj
+h|2/parenrightBig
++1
+2η|hfn
+h|2+1
+η|hfn
+h|(K+L)
+whereL:= 2βK/δ(see Proposition 2.3 for the definition of K). By summing up these terms
+fornfromnj
+htonj+1
+h−1 we get
+Varj(uh) =nj+1
+h−1/summationdisplay
+nj
+h|un+1
+h−un
+h| ≤1
+2η/parenleftbigg
+|unj
+h
+h−ynj
+h|2−|unj+1
+h
+h−ynj
+h|2/parenrightbigg
++nj+1
+h−1/summationdisplay
+nj
+h1
+2η|hfn
+h|2
++1
+η(K+L+η)nj+1
+h−1/summationdisplay
+nj
+h|hfn
+h|
+and finally
+Var(uh) =P−1/summationdisplay
+j=0Varj(uh)≤1
+2ηP−1/summationdisplay
+j=0/parenleftbigg
+|unj
+h
+h−ynj
+h|2−|unj+1
+h
+h−ynj
+h|2/parenrightbigg
++1
+2η/ba∇dblF/ba∇dbl2
+L1(I)
++1
+η(K+L+η)/ba∇dblF/ba∇dblL1(I)
+≤1
+η(K+L)2P+1
+2η/ba∇dblF/ba∇dbl2
+L1(I)+1
+η(K+L+η)/ba∇dblF/ba∇dblL1(I).
+This completes the proof of Theorem 3.2, since Pdoes not depend on hfrom (27). ⊓⊔
+3.2 Collision law for the limits uandq(Proposition 2.9)
+This subsection is devoted to the proof of Proposition 2.9, r ecalled in the following Theorem :
+Theorem 3.4 Lett0∈Ibe fixed. The limit function uverifies:
+u+(t0) = PCt0,q(t0)(u−(t0)).
+15Note that, from Proposition 2.5, u∈BV(I), so that the left-sided u−(t0) and the right-sided
+u+(t0) limits are well-defined.
+The proof is quite technical so for an easy reference, we reme mber the definitions of the sets Ct,q
+(given in (7)):
+Ct,q:={u, ∂tgi(t,q)+/a\}b∇acke⊔le{⊔∇qgi(t,q),u/a\}b∇acke⊔∇i}h⊔ ≥0,ifgi(t,q) = 0}
+andKh(t,q) (given in (11)):
+Kh(t,q) :={u, gi(t,q)+h/a\}b∇acke⊔le{⊔∇qgi(t,q),u/a\}b∇acke⊔∇i}h⊔ ≥0}.
+Moreover, we recall that
+K:= sup
+h/ba∇dbluh/ba∇dblL∞(I)<∞.
+The desired property
+u+(t0) = PCt0,q(t0)(u−(t0)) (33)
+can be seen as the limit (for hgoing to 0) of the “discretized property”
+un+1
+h= PKh(tn+1
+h,qn
+h)[un
+h+hfn]. (34)
+Proof : First we claim that
+u+(t0)∈Ct0,q(t0). (35)
+To verify this property, let us consider an index isuch that gi(t0,q(t0)) = 0. Then a first order
+expansion gives :
+gi(t0+ǫ,q(t0+ǫ)) =ǫ/bracketleftbig
+∂tgi(t0,q(t0))+/a\}b∇acke⊔le{⊔u+(t0),∇qgi(t0,q(t0))/a\}b∇acke⊔∇i}h⊔/bracketrightbig
++oǫ→0(ǫ).
+The feasibility of q(t0+ǫ) (see Proposition 2.5) yields
+∂tgi(t0,q(t0))+/a\}b∇acke⊔le{⊔u+(t0),∇qgi(t0,q(t0))/a\}b∇acke⊔∇i}h⊔ ≥0
+which corresponds to (35).
+Let us now come back to the proof of (33). As we just proved u+(t0)∈ Ct0,q(t0)and since Ct0,q(t0)
+is a convex set, (33) is equivalent to
+∀w∈ Ct0,q(t0),/a\}b∇acke⊔le{⊔u−(t0)−u+(t0),w−u+(t0)/a\}b∇acke⊔∇i}h⊔ ≤0. (36)
+So, in the following, let us choose w∈ Ct0,q(t0). To prove (36), we construct a family of points
+wνforν >0 such that wνtends to wwhenνgoes to zero and satisfies wν∈Kh(t+h,q) forh
+sufficiently small and ( t,q) close to ( t0,q(t0)). Then, for each ν, we go to the limit on h,tandq
+to show that /a\}b∇acke⊔le{⊔u−(t0)−u+(t0),wν−u+(t0)/a\}b∇acke⊔∇i}h⊔ ≤0 and finally, we make νgo to zero to conclude.
+Step 1: From Lemma 3.1, there exists a neighborhood U⊂I×Rdaround (t0,q(t0)) andv∈Rd
+such that for all t∈Iandq∈Q(t)
+(t,q)∈U=⇒ ∀i∈Iκρ(t,q),/a\}b∇acke⊔le{⊔∇qgi(t,q),v/a\}b∇acke⊔∇i}h⊔ ≥δ, (37)
+with a numerical constant δ >0. Forν >0, we consider the point wν:=w+νvwithν >0.
+For alli∈Iκρ(t,q)∩I(t0,q(t0)), (37) together with w∈ Ct0,q(t0)gives
+∂tgi(t0,q(t0))+/a\}b∇acke⊔le{⊔∇qgi(t,q),wν/a\}b∇acke⊔∇i}h⊔=∂tgi(t0,q(t0))+/a\}b∇acke⊔le{⊔∇qgi(t,q),w/a\}b∇acke⊔∇i}h⊔+ν/a\}b∇acke⊔le{⊔∇qgi(t,q),v/a\}b∇acke⊔∇i}h⊔
+≥ /a\}b∇acke⊔le{⊔∇qgi(t,q)−∇qgi(t0,q(t0)),w/a\}b∇acke⊔∇i}h⊔+νδ
+≥νδ−M|w|[|t−t0|+|q−q(t0)|]
+16and consequently from Assumptions (A3) and (A5)
+∂tgi(t,q)+/a\}b∇acke⊔le{⊔∇qgi(t,q),wν/a\}b∇acke⊔∇i}h⊔ ≥νδ−(M|w|+M)[|t−t0|+|q−q(t0)|].
+So for every ν >0, if (t,q) is closed enough to ( t0,q(t0)), we deduce that for all i∈Iκρ(t,q)∩
+I(t0,q(t0))
+∂tgi(t,q)+/a\}b∇acke⊔le{⊔∇qgi(t,q),wν/a\}b∇acke⊔∇i}h⊔ ≥νδ
+2.
+For the indices i /∈Iκρ(t,q), we have
+gi(t,q)+h[∂tgi(t,q)+/a\}b∇acke⊔le{⊔∇qgi(t,q),wν/a\}b∇acke⊔∇i}h⊔]≥κρ−hβ(1+|w|+ν).
+Finally for i /∈I(t0,q(t0)),
+gi(t,q)+h[∂tgi(t,q)+/a\}b∇acke⊔le{⊔∇qgi(t,q),wν/a\}b∇acke⊔∇i}h⊔]≥σ−hβ(1+|w|+ν)−β[|t−t0|+|q−q(t0)|],
+with
+σ:= min
+i/∈I(t0,q(t0))gi(t0,q(t0))>0.
+We conclude that for each fixed ν >0, there are ǫνandhνsuch that for every h < hν, (t,q)∈U
+with|t−t0|+|q−q(t0)| ≤ǫνandq∈Q(t) we have
+∀i, g i(t,q)+h[∂tgi(t,q)+/a\}b∇acke⊔le{⊔∇qgi(t,q),wν/a\}b∇acke⊔∇i}h⊔]≥hνδ
+2,
+which by a first order expansion in time gives:
+∀i, g i(t+h,q)+h/a\}b∇acke⊔le{⊔∇qgi(t+h,q),wν/a\}b∇acke⊔∇i}h⊔ ≥hνδ
+2+Oh→0(h).
+At the cost of decreasing hν, it comes for h < hν,
+∀i, g i(t+h,q)+h/a\}b∇acke⊔le{⊔∇qgi(t+h,q),wν/a\}b∇acke⊔∇i}h⊔ ≥0,
+and consequently, wν∈Kh(t+h,q) for every h < hν, (t,q)∈Uwith|t−t0|+|q−q(t0)| ≤ǫν
+andq∈Q(t).
+Step 2: Let us now fix the parameter ν.
+ThankstotheuniformLipschitzregularity ofthemaps qhandtheiruniformconvergence towards
+q, there exists ˜hν≤hνsuch that for ǫ≤ǫν/(2+2K) andh≤˜hν,
+tk
+h,tk+1
+h∈[t0−ǫ,t0+ǫ] =⇒ |tk+1
+h−t0|+|qk
+h−q(t0)| ≤ǫν.
+From this, as qk
+h∈Q(tk
+h) from Proposition 2.1, the previous step (with t=tk
+h) giveswν∈
+Kh(tk+1
+h,qk
+h). Therefore, Kh(tk+1
+h,qk
+h) being convex, we have
+/a\}b∇acke⊔le{⊔uk
+h+hfk
+h−uk+1
+h,wν−uk+1
+h/a\}b∇acke⊔∇i}h⊔ ≤0. (38)
+We sum up these inequalities for kfromntop, integers chosen such that tn
+his the first time
+step in [t0−ǫ,t0−ǫ+h] andtp+1
+hthe last one in [ t0+ǫ−h,t0+ǫ]. First, we know that
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep/summationdisplay
+kh/a\}b∇acke⊔le{⊔fk,wν−uk+1
+h/a\}b∇acke⊔∇i}h⊔/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤(|wν|+K)/integraldisplayt0+ǫ
+t0−ǫF(t)dt, (39)
+17withK:= suph/ba∇dbluh/ba∇dbl∞. We also have
+p/summationdisplay
+k/a\}b∇acke⊔le{⊔uk
+h−uk+1
+h,wν/a\}b∇acke⊔∇i}h⊔=/a\}b∇acke⊔le{⊔uh(tn
+h)−uh(tp+1
+h),wν/a\}b∇acke⊔∇i}h⊔. (40)
+We deal with the remainder as follows: we write
+p/summationdisplay
+k/a\}b∇acke⊔le{⊔uk
+h−uk+1
+h,−uk+1
+h/a\}b∇acke⊔∇i}h⊔=p/summationdisplay
+k/a\}b∇acke⊔le{⊔uk
+h−uk+1
+h,uk
+h/a\}b∇acke⊔∇i}h⊔−|un
+h|2+|up+1
+h|2,
+which gives
+p/summationdisplay
+k/a\}b∇acke⊔le{⊔uk
+h−uk+1
+h,−uk+1
+h/a\}b∇acke⊔∇i}h⊔=1
+2p/summationdisplay
+k|uk
+h−uk+1
+h|2+1
+2/bracketleftBig
+−|uh(tn
+h)|2+|uh(tp+1
+h)|2/bracketrightBig
+=1
+2Var2(uh)2
+[tn
+h,tp
+h]+1
+2/bracketleftbig
+−|uh(tn
+h)|2+|uh(tp
+h)|2/bracketrightbig
+, (41)
+where we wrote Var 2for theL2-variation of a function. Using (38), (39), (40) and (41), we
+finally get :
+1
+2Var2(uh)2
+[tn
+h,tp
+h]+1
+2/bracketleftBig
+−|uh(tn
+h)|2+|uh(tp+1
+h)|2/bracketrightBig
++/a\}b∇acke⊔le{⊔uh(tn
+h)−uh(tp
+h),wν/a\}b∇acke⊔∇i}h⊔/lessorsimilar/integraldisplayt0+ǫ
+t0−ǫF(t)dt.
+Let us now choose a sequence of ǫmgoing to zero, such that uhpointwisely converges to uat the
+instants t0−ǫmandt0+ǫm(which is possible as uhconverges almost everywhere towards u).
+For each ǫmandhsmall enough, we have shown that the last inequality holds. T hen, passing
+to the limit for h→0 we get
+1
+2Var2(u)2
+[t0−ǫm,t0+ǫm]+1
+2/bracketleftbig
+−|u(t0−ǫm)|2+|u(t0+ǫm)|2/bracketrightbig
++/a\}b∇acke⊔le{⊔u(t0−ǫm)−u(t0+ǫm),wν/a\}b∇acke⊔∇i}h⊔/lessorsimilar/integraldisplayt0+ǫm
+t0−ǫmF(t)dt,
+which gives for ǫm→0
+1
+2Var2(u)2
+[t−
+0,t+
+0]+1
+2/bracketleftbig
+−|u−(t0)|2+|u+(t0)|2/bracketrightbig
++/a\}b∇acke⊔le{⊔u−(t0)−u+(t0),wν/a\}b∇acke⊔∇i}h⊔ ≤0.
+Finally we obtain
+1
+2/vextendsingle/vextendsingleu+(t0)−u−(t0)/vextendsingle/vextendsingle2+1
+2/bracketleftbig
+−|u−(t0)|2+|u+(t0)|2/bracketrightbig
++/a\}b∇acke⊔le{⊔u−(t0)−u+(t0),wν/a\}b∇acke⊔∇i}h⊔ ≤0.
+By expanding the square quantities, this can be written as fo llows
+/a\}b∇acke⊔le{⊔u−(t0)−u+(t0),wν−u+(t0)/a\}b∇acke⊔∇i}h⊔ ≤0. (42)
+To conclude the proof, it now suffices to remember that wν=w+νvand, since for each ν >0,
+the previous reasoning holds, we obtain (36) by letting νgo to 0 in (42). ⊓⊔
+18PSfrag replacements qiqjeij(q)
+Dij(q)
+Figure 1: Particles iandj: notations.
+4 Application to the modelling of inelastic collisions
+The continuous model
+We consider a mechanical system of Nspherical rigid particles in three-dimensions. We denote
+byqi∈R3the position of the center of particle i, byriits radius, by miits mass and by fi∈R3
+the external force exerted on it. Let q∈R3Nbe defined by q:= (...,qi,...) andf∈R3Nby
+f:= (...,fi,...). We denote by Dij(q) the signed distance between particles iandj:
+Dij(q) :=|qi−qj|−(ri+rj),
+and we set eij(q) = (qj−qi)/|qj−qi|(see Fig. 1).
+The problem we are interested in is to describe the path of the configuration qsubmitted to the
+force-field fand undergoing inelastic collisions. This inelastic colli sion law can be modelled by
+imposingnon-overlappingcontraints ontheparticles(see theworkofJ.JMoreau[15]introducing
+this concept). Therefore, we write that the positions of the particles have to belong to a set of
+admissible configurations Q0avoiding overlappings:
+q∈Q0:=/intersectiondisplay
+i,j{q, Dij(q)≥0}.
+We define Mas themassmatrixof dimension3 N×3N,M=diag(...,mi,mi,mi,...). Then, we
+denote by Gij∈R3Nthe gradient of distance Dijwith respect to the positions of the particles:
+Gij(q) = (...,0,−eij,0,...,0, eij,0,...,0)t.
+i j
+The setCqis the set of admissible velocities:
+Cq:={u,/a\}b∇acke⊔le{⊔Gij(q),u/a\}b∇acke⊔∇i}h⊔ ≥0,ifDij(q) = 0}. (43)
+To finish with notations, we denote by λ= (...,λij,...)∈RN(N−1)/2the vector made of the
+Lagrange multipliers associated to the N(N−1)/2 constraints “ Dij(q)≥0”.
+LetI=]0,T[ bethe time interval. Themulti-particle model we are inter ested in may beformally
+phrased as follows:
+
+
+q∈W1,∞(I,R3N),˙q∈BV(I,R3N), λ∈(M+(I))N(N−1)/2,
+∀t∈I,˙q(t+) = PCq(t)˙q(t−)
+M¨q=f+/summationdisplay
+i<jλijGij(q)
+supp(λij)⊂ {t, Dij(q(t)) = 0}for alli,j
+Dij(q(t))≥0 for all i,j
+q(0) =q0such that Dij(q0)>0 for all i,j,˙q(0) =u0(44)
+19The main equation
+M¨q−f=/summationdisplay
+i<jλijGij(q)∈ −N(Q0,q) (45)
+expresses the fact that overlapping is prevented by a repuls ive force (the impulsion) acting on
+each sphere along the normal vector at the contact point. Whe n there is no contact, N( Q0,q) is
+reduced to {0}, so that (45) reads as M¨q=f, which is the Fundamental Principle of Dynamics
+applied on each sphere. Equation ˙ q(t+) = PCq(t)˙q(t−) provides the inelastic collision model. It
+can be extended to an elastic collision model with a restitut ion coefficient eby writing
+˙q(t+) = PCq(t)˙q(t−)−ePN(Q0,q(t))˙q(t−).
+We assume for simplicity that each mass miis equal to 1. Then Problem (44) fits into the
+previously studied framework.
+Remark 4.1 The case of different masses can be taken into account by using t he adapted scalar
+product(u,v)M=/a\}b∇acke⊔le{⊔Mu,v/a\}b∇acke⊔∇i}h⊔, as done in [8]. It turns back to replace the projection step in the
+numerical algorithm by
+un+1= PKh(tn+1
+h,qn
+h)/parenleftbig
+un+hM−1fn/parenrightbig
+,
+wherePhere denotes the projection relatively to this new norm.
+Mbeing a diagonal matrix with non-negative diagonal coefficie nts, it is easy to show that the
+following results still hold true in that case.
+We emphasize that Assumption (A0) is satisfied as soon as
+min
+iri>0,
+and then Assumptions (A1) and (A4) hold true.
+In order to apply our previous results, it remains to check As sumption (A6). As explained in
+[28], that corresponds to estimate the Kuhn-Tucker multipl iers. Such an estimate is given in the
+following lemma.
+Lemma 4.2 There exists a >0(depending on Nand on the radii ri) such that for all q∈R3N,
+F∈R3Nand Lagrange multipliers (µij)∈RN(N−1)/2satisfying
+/summationdisplay
+µijGij(q) =Fwithµij≥0andµij= 0whenDij(q)>0,
+then
+µij≤a|F|.
+Concerningtheproofof this lemma, wereferthereader toPro position 4.7 of [28] (forageometric
+proof) and to Proposition 2.18 of [9] (for a more “physical” p roof). These proofs are written in
+a 2-dimensional framework but they can be easily extended in our 3-dimensional case. Actually,
+Lemma 4.2 is equivalent to Assumption (A6) with ρ= 0. However, it can be extended and
+still holds for ρsmall enough (for example ρ <infiri), see Remark 4.11 of [28]. Consequently,
+Assumption (A6) is satisfied for some small enough ρ >0.
+According to our main Theorem (Theorem 1.3), it follows that Problem (44) has solutions and
+that the associated numerical scheme converges (up to a subs equence). We can allow the radii
+to depend on time as soon as riis uniformly twice-differentiable in time and
+inf
+t∈[0,T]inf
+iri(t)>0.
+Thesetheoretical results permitto legitimate the impleme ntation of this numerical scheme. This
+was performed by the second author by creating SCoPI Softwar e [27]. We refer the reader to [8]
+for some good properties of stability and robustness for the algorithm and efficiency for large
+time steps.
+20Remark 4.3 We refer the reader to [7], where the second author extends th is model in order to
+consider gluey particles. In this case, she add an extra param eter (depending on q) for describing
+the corresponding admissible set. This new operation does no t keep the necessary regularity of
+the admissible set. She has already obtained a result of conv ergence for the associated numerical
+scheme in the single-constraint case. We plan in a forthcomi ng work to extend this proof with
+the ideas presented here in order to deal with the multi-cons traint case.
+References
+[1] P. Ballard, The dynamics of discrete mechanical systems with perfect unilateral con-
+straints. Arch. Rational Mech. Anal. 154(2000), 199-274.
+[2] F. Bernicot, J. Venel, Stochastic perturbations of swee ping process. Submitted and
+available at http://fr.arxiv.org/abs/1001.3128 (2010).
+[3] F.H. Clarke, R.J. Stern and P.R. Wolenski, Proximal smoo thness and the lower- C2
+property. J. Convex Anal. 2(1995), 117-144.
+[4] F.H. Clarke, Y.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and
+Control Theory, Springer-Verlag , 1998, New York, Inc.
+[5] G. Colombo and M.D.P. Monteiro Marques, Sweeping by a con tinuous prox-regular set,
+J. Diff. Equa. 187(2003) no.1 , 46–62.
+[6] R. Dzonou, M.D.P. Monteiro Marques, L. Paoli, A converge nce result for a vibro-impact
+problem with a general inertia operator, Nonlinear Dyn. 58 (2009), 361–384.
+[7] A. Lefebvre, Numerical simulation of gluey particles, Model. Math. Anal. Numer. 43
+(2009), 53–80.
+[8] B. Maury, A time-stepping scheme for inelastic collisio ns,Numer. Math. 102(2006),
+649–679.
+[9] B. Maury, J. Venel, A discrete contact model for crowd mot ion,Model. Math. Anal.
+Numer.to appear 2010. Available at http://arxiv.org/abs/0901.0984
+[10] M.D.P. Monteiro-Marques, Differential inclusions in no n-smooth mechanical problems:
+shocks and dry friction. PNLDE,9, Birkhauser, Basel, 1993.
+[11] M.D.P. Monteiro-Marques and L. Paoli, An existence res ult in non-smooth dynamics.
+Nonsmooth mechanics and analysis , Chap 23, 279–288, Adv. Mech. Math. 12, Springer,
+New York, 2006.
+[12] J.J. Moreau, D´ ecomposition orthogonale d’un espace H ilbertien selon deux cˆ ones
+mutuellement polaires. C. R. Acad. Sci, Ser. I ,255(1962), 238–240.
+[13] J.J. Moreau, Liaisons unilat´ erales sans frottements et chocs in´ elastiques. C. R. Acad.
+Sci, Ser. II ,296(1983), 1473–1476.
+[14] J.J. Moreau, Standard inelastic shocks and the dynamic s of unilateral constraints. CISM
+Courses and Lectures ,288Springer, Berlin, 1985, 173–221.
+[15] J.J. Moreau, Some numerical methods in multibody dynam ics: Application to granular
+materials. Eur. J. Mech. A/Solids ,13(1994), 93–114.
+21[16] L. Paoli and M. Schatzman, Mouvement ` a un nombre fini de d egr´ es de libert´ e avec
+contraintes unilat´ erales; cas avec perte d’´ energie. Model. Math. Anal. Numer. ,27-6
+(1993), 673–717.
+[17] L. Paoli and M. Schatzman, Penalty approximation for no n smooth constraints in vibro-
+impact.J. Diff. Equa. ,177(2001), 375–418.
+[18] L. Paoli and M. Schatzman, A numerical scheme for impact problems I and II. SIAM J.
+Numer. Anal. ,40-2(2002), 702–768.
+[19] L. Paoli and M. Schatzman, Penalty approximation for dy namical systems submitted to
+multiple non smooth constraints. Multibody System Dynamics ,8no.3 (2002), 347–366.
+[20] L. Paoli, An existence result for non-smooth vibro-imp act problems. J. Diff Equa. ,211
+(2005), 247–281.
+[21] L. Paoli, Time-stepping approximation of rigid-body d ynamics with perfect unilateral
+constraints. I-The inelastic impact case. A.R.M.A. , to appear.
+[22] L. Paoli, Time-stepping approximation of rigid-body d ynamics with perfect unilateral
+constraints. II-The partially inelastic impact case. A.R.M.A. , to appear.
+[23] R.A. Poliquin, R.T. Rockafellar and L.Thibault, Local differentiability of distance func-
+tions.Trans. Amer. Math. Soc. 352(2000), 5231–5249.
+[24] M. Schatzman, A class of nonlinear differential equation s of second order in time. Non-
+linear Anal. 2(1978), no. 3, 355–373.
+[25] M.Schatzman, Uniqueness and continuous dependence on data for one-dimensional im-
+pact problems. Math. Comput. Modelling 28(1998), 1–18.
+[26] M.Schatzman, Penalty method for impact in generalized coordinates. Phil. Trans. Roy.
+Soc. London A 359(2001), 2429–2446.
+[27] SCoPI Software, Presentation available at
+http://www.projet-plume.org/relier/scopi , and numerical simulations available at
+http://www.cmap.polytechnique.fr/$\sim$lefebvre/SCo PI.htm
+[28] J. Venel, A numerical scheme for a whole class of sweepin g process. submitted , 2009.
+Available at http://arxiv.org/abs/0904.2694
+22
\ No newline at end of file
diff --git a/1001.0702.txt b/1001.0702.txt
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+arXiv:1001.0702v3  [hep-ph]  31 Mar 2010TECHNION-PH-2009-34
+arXiv:1001.0702
+January 2010
+CP ASYMMETRIES IN B→Kπ,K∗π,ρKDECAYS
+Michael Gronau
+Physics Department, Technion – Israel Institute of Technol ogy
+32000 Haifa, Israel
+Dan Pirjol
+Department of Particle Physics, National Institute for Phy sics and Engineering
+077125 Bucharest, Romania
+Jure Zupan
+Faculty of Mathematics and Physics, University of Ljubljan a
+Jadranska 19, 1000 Ljubljana, Slovenia
+and
+Josef Stefan Institute, Jamova 39, 1000 Ljubljana, Sloveni a
+We show that ratios of tree and penguin amplitudes in B→K∗πandB→ρK
+are two to three times larger than in B→Kπ. This allows for considerably
+larger CP asymmetries in the former processes than the 10% asymm etry mea-
+sured inB0→K+π−. We study isospin sum rules for rate asymmetries in
+B→Kπ,K∗π,ρK, estimating small violation from interference of tree and
+electroweak penguin amplitudes. The breaking of the Kπasymmetry sum rule
+is estimated to be one to two percent and negative. Violation of K∗πandρK
+sum rules can be estimated from B→ρπamplitudes using flavor SU(3), while
+breaking of a sum rule combining K∗πandρKasymmetries can be measured
+directly in a Dalitz analysis of B0→K+π−π0. The three sum rules can be
+tested using complete sets of data taken at e+e−Bfactories and in experiments
+at the LHCb and at a future Super Flavor Factory, providing precis ion searches
+for new ∆S= ∆I= 1 operators in the low energy effective Hamiltonian.
+I. INTRODUCTION
+TheCabibbo-Kobayashi-Maskawa (CKM) frameworkforflavor phy sics andCPviolation
+has been confirmed successfully in numerous experiments involving a variety ofBmeson
+decays [1, 2, 3, 4]. The measured oscillating CP asymmetry in B0→J/ψK S,Lhas a
+theoreticallypreciseinterpretationintermsoftheCKMphase β, atalevel of10−3, including
+a smallb→u¯usamplitude whose calculation involves a perturbative term [5] and long
+distance rescattering corrections [6]. This asymmetry provided an accurate value for β
+with an experimental precision of 1◦. Using isospin symmetry [7, 8], time-dependent CP
+symmetries measured in B0→π+π−,ρ+ρ−,ρ±π∓led to a determination of the phase α
+with an error of 4◦. The values obtained for the two CP-violating phases are in good
+agreement with CP-conserving CKM constraints. Other asymmetr ies probing the phase
+βwere measured in loop-dominated processes from b→q¯qs(q=u,d,s) including B0→
+1π0KS,η′KS,φKS,ρ0KS. These processes are sensitive to corrections from flavor chang ing
+∆S= 1 operators appearing in extensions of the low energy theory [9, 1 0]. Uncertainties in
+calculating hadronic amplitudes by applying QCD dynamics [11, 12] proh ibit distinguishing
+between New Physics and hadronic effects on these asymmetries.
+Direct CP asymmetries have been observed in several charmless ∆ S= 1Bdecays, with
+a high confidence level above 8 σinB0→K+π−and at a lower confidence level of 3-4 σin
+B+→ηK+,ρ0K+andB0→ηK∗0[4]. Theseobservationsareinteresting bythemselves but
+do not provide clean tests for the CKM framework. An interpretat ion of direct asymmetries
+in terms of fundamental CP phases requires calculating nonleptonic decay amplitudes and
+strong phases. Remarkable progress has been achieved in the pas t ten years in the theory
+of hadronic Bdecays, starting with work advocating QCD factorization [13] and p ertur-
+bative QCD [14]. While great progress has been made recently in calcula ting higher order
+terms inαs[15, 16], 1/mbcorrections remain a serious difficulty. Thus, nonfactorizable
+hadronic matrix elements of charming penguin operators [17, 18] an d of color-suppressed
+tree amplitudes lead to difficulties in calculating the decay rate for B0→π0π0[13, 16, 19].
+This example and an intrinsic failure to account for individual strong p hases (as discussed
+briefly in Sec. II for one example), seem to originate in incalculable long distance final state
+rescattering effects.
+Whereas individual direct CP asymmetries in hadronic Bdecays cannot be calculated
+reliably, there are certain classes of decays in which asymmetries ca n be related to each
+other within the CKM framework in a model independent way using sym metry arguments.
+Isospin symmetry, which is expected to hold within a couple of percen t, has been shown to
+imply an approximate sum rule for asymmetries in the four B→Kπdecay processes [20],
+ACP(K+π−)+ACP(K0π+)≈ACP(K+π0)+ACP(K0π0). (1)
+Aviolationof thissum rulewould beevidence foranew ∆ S= ∆I= 1 terminthelowenery
+effective Hamiltonian. A similar sum rule may hold approximately in the CKM framework
+forB→K∗πandB→ρKdecays, where individual CP asymmetries may be larger than
+inB→Kπ.
+In this paper we will study asymmetries in penguin-dominated B→Kπ,K∗πand
+B→ρKandwill comparetheir threeisospinsum rules. These sumrules involve corrections
+from interference of subleading tree and electroweak penguin (EW P) amplitudes which are
+related to each other in a flavor SU(3) limit. Our aim will be to estimate t hese second order
+corrections in a model-independent way, or to propose methods in w hich the corrections can
+be measured elsewhere. Our main results are summarized in Eqs. (65 ), (72), (81)-(83). In
+order to present a self-contained analysis for the three sum rules inB→Kπ,B→K∗πand
+B→ρKwe will introduce and study in some detail amplitudes and ratios of amp litudes
+occurring in the three classes of decay processes.
+Section II gives B→Kπdecay amplitudes in terms of their diagramatic contributions,
+discussing the role of these contributions in B→Kπasymmetries. Sec. III generalizes this
+description to B→K∗πandB→ρKdecays. In Sec. IV we use broken flavor SU(3) to
+calculate ratios of tree-to-penguin amplitudes in B→Kπ,K∗π,ρK, providing estimates
+for maximal possible CP asymmetries in these three classes of proce sses. Sec. V studies
+experimental tests for the broken SU(3) symmetry assumption. SU(3) relations between
+subleading tree and EWP amplitudes are discussed in Sec. VI, and are used in Sec. VII
+2for estimating deviations from exact sum rules among asymmetries f orB→Kπ,K∗πand
+B→ρKdecays. Sec. VIII concludes.
+II. AMPLITUDES AND ASYMMETRIES IN B→Kπ
+Decay amplitudes for B→Kπmay be described generally in a model-independent way
+using topological graphical contributions [21],
+−A(K+π−) =λ(s)
+t(Ptc+2
+3PC
+EW)+λ(s)
+u(Puc+T), (2)
+A(K0π+) =λ(s)
+t(Ptc−1
+3PC
+EW)+λ(s)
+u(Puc+A), (3)
+−√
+2A(K+π0) =λ(s)
+t(Ptc+PEW+2
+3PC
+EW)+λ(s)
+u(Puc+T+C+A),(4)
+√
+2A(K0π0) =λ(s)
+t(Ptc−PEW−1
+3PC
+EW)+λ(s)
+u(Puc−C), (5)
+whereλ(q′)
+q≡V∗
+qbVqq′(q=u,t;q′=d,s) are Cabibbo-Kobayashi-Maskawa (CKM) factors
+with a small ratio |λ(s)
+u|/|λ(s)
+t| ∼0.02. Each of the seven amplitudes, Ptc,PEW,PC
+EW,T,C,
+Puc,Ainvolves an unknown strong phase.
+The common penguin amplitude P′≡λ(s)
+tPtc[or the linear combination λ(s)
+t(Ptc−
+PC
+EW/3)] with weak phase arg( λ(s)
+t) =π, dominates all four B→Kπamplitudes. This
+dominance is tested by simple approximate ratios,
+B(K+π−) :B(K0π+)/rτ:B(K+π0)/rτ:B(K0π0)≃1 : 1 :1
+2:1
+2, (6)
+The four branching ratios given in Table I, and the B+toB0lifetime ratio, rτ≡τ+/τ0=
+1.071±0.009 [4], imply experimental ratios:
+(0.899±0.048) : 1 : (0 .558±0.035) : (0.454±0.034). (7)
+Thus Eq. (6) holds reasonably well with several percent errors lea ving little space for non-
+penguin amplitudes.
+Let us now discuss the subdominant terms occurring in Eqs. (2)-(5 ), their relative mag-
+nitudes with respect to PtcorP′, and their effects on CP asymmetries in B→Kπdecays.
+•The largest subdominant term with weak phase arg( λ(s)
+u)≡γis expected to be the
+color-favoredtreeamplitude T′≡λ(s)
+uT. Aninterferencebetween P′andT′dominates
+the asymmetry in B0→K+π−for which a negative value has been measured at a
+level of 10%. We will show in the next section that T′is about an order of magnitude
+times smaller that P′.
+•The difference between the asymmetry in B0→K+π−and the preferably positive
+asymmetry measured in B+→K+π0, obtaining contributions from interference of P′
+with bothT′and and with the formally color-suppressed C′≡λ(s)
+uC, has shown [22,
+23] that the ratio |C|/|T|is not much smaller than one as naively anticipated, and
+that the relative strong phase Arg( CT∗) is sizable and negative. (We use a convention
+in which strong phase differences lie between −πandπ.) We note that Arg( CT∗)≃0
+3Table I: Branching fractions and CP asymmetries for B→Kπ,K∗π,ρK[4].
+Mode B(10−6)ACP
+B0→K+π−19.4±0.6−0.098+0.012
+−0.011
+B+→K0π+23.1±1.0 0.009±0.025
+B+→K+π012.9±0.6 0.050±0.025
+B0→K0π09.8±0.6−0.01±0.10
+B0→K∗+π−8.6+0.9
+−1.0−0.18±0.08
+B+→K∗0π+9.9+0.8
+−0.9−0.038±0.042
+B+→K∗+π06.9±2.3 0.04±0.29
+B0→K∗0π02.4±0.7−0.15±0.12
+B0→ρ−K+8.6+0.9
+−1.10.15±0.06
+B+→ρ+K08.0+1.5
+−1.4−0.12±0.17
+B+→ρ0K+3.81+0.48
+−0.460.37±0.11
+B0→ρ0K04.7±0.7 0.06±0.20
+is predicted at leading order in 1 /mbby QCD factorization [24] and in a Soft Collinear
+Effective Theory approach [18]. The large strong phase may be inter preted as either
+large 1/mbcorrections or sizable nonfractorizable contributions to Cfrom rescattering
+through color-favored intermediate states.
+•EWP terms PEW,PC
+EW, which are higher order in the electroweak coupling, are an
+order of magnitude times smaller than Ptc. A quantitative discussion of these con-
+tributions relating them to penguin and tree amplitudes will be presen ted in Section
+VI. Effects of EWP terms on CP asymmetries through their interfer ence with tree
+amplitudes are of second order because these two kinds of amplitud es are an order of
+magnitude smaller than Ptc. Interference of EWP amplitudes with Ptcleads to no CP
+asymmetry as these two contributions carry the same weak phase .
+•The very small asymmetry measured in B+→K0π+, consistent with zero, indicates
+that bothPucand the annihilation amplitude Aare much smaller than T. A large
+enhancement by rescatterring of these amplitudes, which are exp ected to be intrinsi-
+cally very small [25], would have created a sizable strong phase in thes e amplitudes
+relative toPtc, thereby leading to a non-nlegligible CP asymmetry in B+→K0π+.
+III. AMPLITUDES IN B→K∗πandB→ρK
+Amplitudes for the eight processes B→K∗πandB→ρKwith final charges as in B→
+Kπmay be decomposed into expressions similar to Eqs. (2)–(5), where P,T,C,A are now
+replaced by PP,TP,CV,APinB→K∗πand byPV,TV,CP,AVinB→ρK[26]. The suffix
+M=P,Vdenotes whether the spectator quark is included in a pseudoscalar (P) or vector
+meson (V). Electroweak contributions, P′
+EW,M≡λ(s)
+tPEW,MandP′C
+EW,M≡λ(s)
+tPC
+EW,M, are
+introduced as in B→Kπthrough the substitution [21]
+T′
+M→T′
+M+P′C
+EW,M, C′
+M→C′
+M+P′
+EW,M, P′
+M→P′
+M−1
+3P′C
+EW,M.(8)
+4Thus,B→K∗πamplitudes analogous to Eqs. (2)-(5) are given by:
+−A(K∗+π−) =λ(s)
+t(Ptc,P+2
+3PC
+EW,P)+λ(s)
+u(Puc,P+TP), (9)
+A(K∗0π+) =λ(s)
+t(Ptc,P−1
+3PC
+EW,P)+λ(s)
+u(Puc,P+AP), (10)
+−√
+2A(K∗+π0) =λ(s)
+t(Ptc,P+PEW,V+2
+3PC
+EW,P)+λ(s)
+u(Puc,P+TP+CV+AP),(11)
+√
+2A(K∗0π0) =λ(s)
+t(Ptc,P−PEW,V−1
+3PC
+EW,P)+λ(s)
+u(Puc,P−CV). (12)
+Corresponding amplitudes for B→Kρare obtained by interchanging subscripts P↔V.
+Penguin dominance in B→K∗πandB→ρKdecays leads to approximate ratios
+of branching ratios in these processes which are similar to the ratios 1 : 1 : 1/2 : 1/2 in
+B→Kπ. Using branching ratios in Table I, involving experimental errors con siderably
+larger than in B→Kπ, one finds,
+B(K∗+π−) :B(K∗0π+)/rτ:B(K∗+π0)/rτ:B(K∗0π0)
+= (0.93±0.13) : 1 : (0.70±0.24) : (0.26±0.08), (13)
+B(ρ−K+) :B(ρ+K0)/rτ:B(ρ0K+)/rτ:B(ρ0K0)
+= (1.15±0.25) : 1 : (0.48±0.10) : (0.63±0.15). (14)
+The value of B(K∗0π0) in (13) seems somewhat small [27] because it involves small values
+withlargeerrorsinoldCLEOandBellemeasurements [28,29]. Usingthe considerablymore
+precise Babar measurement [30], B(K∗0π0) = (3.6±0.7±0.4)×10−6, the last numerical
+factor in (13) becomes 0 .39±0.09 consistent within error with 1 /2. We note that large
+experimental errors in branching ratio measurements leave ample s pace for subdominant
+non-penguin amplitudes in these processes.
+Magnitudes of dominant penguin amplitudes and of subdominant amplit udes, and cer-
+tain relative strong phases between them, have been studied nume rically in Ref. [31], apply-
+ing a global flavor SU(3) fit to data of these processes and corres ponding ∆S= 0B→VP
+decays. Recently χ2fits were performed for B→K∗πandB→ρKdata, concluding
+that current experimental errors are too large for showing an inc onsistency with the CKM
+framework [32]. The purpose of the next section is limited to estimatin g ratios of tree and
+penguin amplitudes determining the potentially largest CP asymmetrie s inB→K∗πand
+B→ρK.
+IV. RATIOS OF TREE-TO-PENGUIN AMPLITUDE IN B→Kπ,K∗π,ρK
+The ratios |T′/P′|,|T′
+P/P′
+P|and|T′
+V/P′
+V|determine the maximal potential asymme-
+tries inB0→K+π−,B0→K∗+π−andB0→ρ−K+, given roughly by 2 |T′/P′|sinγ,
+2|T′
+P/P′
+P|sinγand 2|T′
+V/P′
+V|sinγ. Values of these three ratios may be estimated by re-
+lating within broken flavor SU(3) the amplitudes for these processe s to those for B0→
+π+π−,B0→ρ+π−andB0→ρ−π+. One way of using flavor SU(3) is by applying group
+theory for decomposing ∆ S= 1 and ∆ S= 0 decay amplitudes into five SU(3) reduced
+matrix elements in B→PPand ten reduced matrix elements in B→VP[33]. A more
+5powerful SU(3) method is based on a diagramatic approach [21, 26] in which a certain hi-
+erarchy between amplitudes can be shown to exist [34] and SU(3) br eaking factors in tree
+amplitudes may be motivated by a factorization assumption. In the d iagramatic approach,
+where certain combinations of terms correspond to SU(3) reduce d matrix elements, tree
+amplitudes defines as [26],
+T′
+P=TP+Puc,P, T′
+V=TV+Puc,V, (15)
+are well-defined, and are renormalization-group scale and scheme- independent.
+We keep SU(3) invariant penguin amplitudes as a default because fac torization is not
+expectedtoholdforthesecontributions[17,18]. Testsofthesea ssumptionswillbepresented
+in the next section. The effect of possible SU(3) breaking in penguin a mplitudes, at an
+expected level of 20%, can be easily included in our estimates for the ratios of tree and
+penguin amplitudes.
+Table II: Amplitudes, branching fractions and asymmetries for cer tainB→PP,VP[4].
+Mode Amplitude B(10−6)ACP
+B+→K0π+P′Table I Table I
+B0→K+π−−(P′+T′) Table I Table I
+B0→π+π−˜λP′−˜λ−1T′/parenleftBigfπ
+fK/parenrightBig
+5.16±0.22 0.38±0.06
+B+→K∗0π+P′
+P Table I Table I
+B0→K∗+π−−(P′
+P+T′
+P) Table I Table I
+B0→ρ+π−˜λP′
+P−˜λ−1T′
+P/parenleftBigfρ
+fK∗/parenrightBig
+15.7±1.8a0.11±0.06
+B+→ρ+K0P′
+V Table I Table I
+B0→ρ−K+−(P′
+V+T′
+V) Table I Table I
+B0→ρ−π+˜λP′
+V−˜λ−1T′
+V/parenleftBigfπ
+fK/parenrightBig
+7.3±1.2c−0.18±0.12
+a,cWe useB(ρ±π∓) =1
+2B(1±Aρπ
+CPC±∆C),B ≡ B(ρ+π−)+B(ρ−π+).
+Table II lists dominant terms in amplitudes (in the c-convention [35]), branching ratios
+and asymmetries for ∆ S= 1 penguin-dominated amplitudes, and for ∆ S= 0 processes
+involving the three pairs ( P′,T′),(P′
+P,T′
+P),(P′
+V,T′
+V). We define ˜λ≡λ/(1−λ2/2) = 0.232,
+whereλis the Wolfenstein parameter [36], and use the following ratios of meso n decay
+constants to represent SU(3) breaking factors in color-favore d tree amplitudes, fπ/fK=
+0.84,fρ/fK∗= 0.96 [1]. Other SU(3) breaking factors in tree amplitudes involving ratio s of
+kernels and ratios of their overlaps with meson wave functions (give n roughly by ratios of a
+given form factor at slightly different values of q2) will be neglected.
+In order to give simple estimates for |T′
+(V,P)|/|P′
+(V,P)|we first assume that
+˜λ2|P′
+(V,P)| ≪ |T′
+(V,P)|. (16)
+Thus we estimate the three ratios |T′
+(V,P)|/|P′
+(V,P)|using central values for branching ratios
+6given in Tables I and II,
+|T′|
+|P′|≃˜λ/parenleftBiggfK
+fπ/parenrightBigg/radicaltp/radicalvertex/radicalvertex/radicalbtrτB(π+π−)
+B(K0π+)= 0.13, (17)
+|T′
+P|
+|P′
+P|≃˜λ/parenleftBiggfK∗
+fρ/parenrightBigg/radicaltp/radicalvertex/radicalvertex/radicalbtrτB(ρ+π−)
+B(K∗0π+)= 0.31, (18)
+|T′
+V|
+|P′
+V|≃˜λ/parenleftBiggfK
+fπ/parenrightBigg/radicaltp/radicalvertex/radicalvertex/radicalbtrτB(ρ−π+)
+B(K0ρ+)= 0.27. (19)
+We note that our assumption (16) is a good approximation for the las t two cases where
+indeed|T′
+P,V|/|P′
+P,V| ≫˜λ2= 0.05, but may provide only a crude estimate for |T′|/|P′|.
+One may relax the assumption (16) for |T′|/|P′|by replacing (17) with a quadratic
+relation following from the amplitudes in Table II,
+R≡rτB(π+π−)
+B(K0π+)=˜λ2+/parenleftBigg|T′|
+|P′|/parenrightBigg2/parenleftBigg
+˜λfK
+fπ/parenrightBigg−2
++2|T′|
+|P′|fπ
+fKcosδcosγ , (20)
+whereδis the unknown strong phase difference between P′andT′. Denoting z= cosδcosγ
+one solves for |T′|/|P′|,
+|T′|
+|P′|=˜λ2fK
+fπ/parenleftbigg/radicalBig
+z2+(R−˜λ2)/˜λ2−z/parenrightbigg
+, (21)
+which is a monotonically decreasing function of z. Taking conservative bounds −0.6≤z≤
+0.6, based on a lower bound on γ[2] and on no restrictions on δ, one obtains
+0.09≤|T′|
+|P′|≤0.16, (22)
+which describes a rather narrow range around the value in (17). Sim ilarly, including
+quadratic corrections in estimates of |T′
+P|/|P′
+P|and|T′
+V|/|P′
+V|, one obtains the following
+bounds,
+0.28≤|T′
+P|
+|P′
+P|≤0.35,0.23≤|T′
+V|
+|P′
+V|≤0.31. (23)
+We do not include errors in input branching ratios. The resulting unce rtainties in the above
+two ratios from errors in branching ratios are somewhat smaller tha n the ones shown, which
+are caused by our conservative assumption of completely arbitrar y strong phases δ.
+Our conclusion is that the ratios |T′
+P|/|P′
+P|and|T′
+V|/|P′
+V|of tree-to-penguin amplitudes
+inB→K∗πandB→ρK, respectively, are between two to three times larger than the
+corresponding ratio |T′|/|P′|inB→Kπ. The processes B0→K∗+π−andB+→K∗+π0
+orB0→K+ρ−andB+→K+ρ0involve interference of P′
+PandT′
+Por interference of P′
+V
+andT′
+V. Thus, these processes may potentially involve asymmetries betwe en two to three
+times larger than the 10% asymmetry measured in B0→K+π−.
+Maximal CP asymmetries Amax
+CP≃2|T′
+(P,V)|/|P′
+(P,V)|sinγinB0→K+π−,B0→K∗+π−
+andB0→ρ−K+, are obtained for δ= 90◦orz= 0 corresponding to the central values of
+7|T′
+(P,V)|/|P′
+(P,V)|given in Eqs. (17), (18) and (19). The actual values of these asym metries
+depend, of course, on δ. The measured asymmetry in B0→K+π−indicates |T′/P′| ∼0.10
+within the range (22) and δ(Kπ)∼30◦. The strong phases in B0→K∗+π−andB0→
+ρ−K+may be different which would affect the asymmetries in these process es.
+The asymmetry in B+→K∗+π0(B+→K+ρ0) depends also on the interference of P′
+P
+andC′
+V(P′
+VandC′
+P). UnlikeB+→K+π0, where the effect of C′on the CP asymmetry is
+destructive relative to the effect of T′, these interference effects do not have to act destruc-
+tively with respect to the interference between P′
+PandT′
+P(P′
+VandT′
+V) . Thus they may
+increase the asymmetry in B+→K∗+π0(B+→K+ρ0) relative to that in B0→K∗+π−
+(B0→K+ρ−).
+Wenote inpassing thatone mayuse the rangesofvalue inEqs. (22) a nd(23)to estimate
+ratios of penguin and tree amplitudes in B0→π+π−,B0→ρ+π−andB0→ρ−π+[37, 38].
+Taking into account CKM and SU(3) breaking factors, one finds for these three ratios
+0.40≤˜λ2fK
+fπ|P′|
+|T′|≤0.71,0.16≤˜λ2fK∗
+fρ|P′
+P|
+|T′
+P|≤0.20,0.21≤˜λ2fK
+fπ|P′
+V|
+|T′
+V|≤0.28.(24)
+Thus, whereas the penguin contribution in B0→π+π−is only somewhat smaller than
+that of the tree amplitude, penguin terms contribute only about 20 % to the amplitudes for
+B0→ρ+π−andB0→ρ−π+which are largely dominated by tree amplitudes. We note
+that while calculations in Ref. [39] using QCD factorization obtain value s for the ratios (24)
+which are about a factor two smaller than the above, a more recent update agrees with the
+above ranges [40].
+V. TESTS OF FLAVOR SU(3)
+In the previous section we assumed that penguin amplitudes are SU( 3)-invariant, while
+SU(3) breaking factors involving ratios of meson decay constants were assumed for tree
+amplitudes. In this section we will present experimental tests for t hese assumptions based
+onmeasurements of decay rates andCP asymmetries in pairs of ∆ S= 1 and ∆S= 0 decays
+which are related to each other by flavor SU(3).
+Consider the three pairs of B0decay processes listed in Table II. The structure of these
+amplitudes, involving for a given pair the same penguin and tree amplitu des with different
+CKM factors, leads to simple relations between CP rate differences d efined by
+∆(B→f)≡Γ(¯B→¯f)−Γ(B→f)≡2¯Γ(B→f)ACP(B→f), (25)
+where¯Γ is a CP-averaged decay rate. Denoting by BACPthe product of a charge-averaged
+branching ratio and a CP asymmetry for a given process, one finds [4 1, 42]
+∆(K+π−) =−fK
+fπ∆(π+π−) or [ BACP](K+π−) =−fK
+fπ[BACP](π+π−),(26)
+and
+∆(K∗+π−) =−fK∗
+fρ∆(ρ+π−) or [ BACP](K∗+π−) =−fK∗
+fρ[BACP](ρ+π−),(27)
+∆(ρ−K+) =−fK
+fπ∆(ρ−π+) or [ BACP](ρ−K+) =−fK
+fπ[BACP](ρ−π+).(28)
+8Using branching ratios and asymmetries in Tables I and II, these thr ee equalities read
+respectively in units of 10−6,
+−1.90±0.23 =−2.33±0.38, (29)
+−1.54±0.71 =−2.01±1.07, (30)
+1.29±0.54 = 1.27±1.03. (31)
+The first test involves reasonably small errors and works well within 1σ. It would
+have worked somewhat less well if penguin amplitudes were assumed t o factorize like tree
+amplitudes and to involve an SU(3) breaking factor fK/fπ. The current experimental errors
+inB→VPdecays are still very large and do not provide useful SU(3) tests.
+An independent test for SU(3) invariance of penguin amplitudes is pr ovided by the ratio
+of rates of penguin dominated ∆ S= 0 and ∆S= 1B→PPandB→VPdecays,
+/radicaltp/radicalvertex/radicalvertex/radicalbtB(¯K0K+)
+B(K0π+)≃/radicaltp/radicalvertex/radicalvertex/radicalbtB(¯K∗0K+)
+B(K∗0π+)≃˜λ . (32)
+UsingB(¯K0K+) = (1.36+0.29
+−0.27)×10−6,B(¯K∗0K+) = (0.68±0.19)×10−6[4], andtaking∆ S=
+1branchingratiosinTableI,theaboveratiosofamplitudesarefoun dtobe0.243±0.026and
+0.262±0.038, consistent with ˜λ= 0.232 within reasonable experimental errors. These errors
+must be reduced somewhat in order to distinguish between the assu med SU(3) invariant
+penguin amplitudes and SU(3) breaking in these amplitudes given by a f actorfK/fπ.
+VI. SU(3) RELATIONS BETWEEN TREE AND EWP AMPLITUDES
+It has been noted that in the limit of flavor SU(3) symmetry approxim ate relations hold
+between the subdominant tree and electroweak penguin (EWP) amp litudes inB→Kπ
+transformingas∆ I= 1. Neglecting EWPoperators O7andO8withtinyWilsoncoefficients
+in the effective weak Hamiltonian, and using a quite precise relation for Wilson coefficients
+(true within a few percent) [43],
+K ≡c9+c10
+c1+c2≈c9−c10
+c1−c2≈ −0.0087, (33)
+one obtains [44, 45, 46],
+EWP(B+→K0π+)+√
+2EWP(B+→K+π0) =−(PEW+PC
+EW) =3K
+2(T+C),(34)
+and [45]
+EWP(B0→K+π−)+EWP(B+→K0π+) =−PC
+EW=3K
+2(C−E).(35)
+These relations follow from corresponding properties of operator s in the ∆S= 1,∆I= 1
+effective Hamiltonian behaving as 15and6under SU(3) transformation [45, 47],
+1
+λ(s)
+tH(s)
+EWP(15) =−3K
+21
+λ(s)
+uH(s)
+Tree(15), (36)
+1
+λ(s)
+tH(s)
+EWP(6) =3K
+21
+λ(s)
+uH(s)
+Tree(6), (37)
+9and imply relations similar to (34) and (35) in B→K∗πandB→ρKdecays.
+It is convenient to study tree and EWP amplitudes for definite isospin states. We
+consider ∆I= 1 amplitudes for final states f=Kπ,K∗π,ρKwith isospin I= 1/2,3/2,
+denoting corresponding tree and EWP contributions by Tf
+I, andEf
+I, respectively,
+Af
+I=λ(s)
+uTf
+I+λ(s)
+tEf
+I. (38)
+Thus one has for instance,
+6AK∗π
+1/2=√
+2A(K∗+π0)+4A(K∗0π+)−3√
+2A(K∗0π0)
+≡6[λ(s)
+uTK∗π
+1/2+λ(s)
+tEK∗π
+1/2], (39)
+3AK∗π
+3/2=A(K∗+π−)+√
+2A(K∗0π0) =A(K∗0π+)+√
+2A(K∗+π0)
+≡3[λ(s)
+uTK∗π
+3/2+λ(s)
+tEK∗π
+3/2]. (40)
+We will apply flavor SU(3) to tree and EWP amplitudes in theses three c lasses of
+penguin-dominated decays. In the SU(3) limit both tree and EWP amp litudes may be
+expressed intermsofthesamegraphicalcontributionsdefinedinS ections II[45]andIII[47],
+6TKπ
+1/2=−T+2C+3A , 3TKπ
+3/2=−T−C , (41)
+6EKπ
+1/2=K
+2(−6T+3C−9E),3EKπ
+3/2=3K
+2(T+C), (42)
+6TK∗π
+1/2=−TP+2CV+3AP,3TK∗π
+3/2=−TP−CV, (43)
+6EK∗π
+1/2=K
+2(−6TV+3CP−9EP),3EK∗π
+3/2=3K
+2(TV+CP), (44)
+6TρK
+1/2=−TV+2CP+3AV, 3TρK
+3/2=−TV−CP, (45)
+6EρK
+1/2=K
+2(−6TP+3CV−9EV),3EρK
+3/2=3K
+2(TP+CV). (46)
+Wewillmakeuseoftheseexpressions inthenextsection, assumingt hattheannihilation-like
+amplitudes A(P,V)andE(P,V)are negligible relative to the other amplitudes [25, 34].
+One mayuse the above relationsto estimate the magnitudes of EWP a mplitudes relative
+to those of dominant penguin amplitudes. For instance, noting that the amplitude T+C
+dominatesB+→π+π0[21] with B(π+π0) = (5.59+0.41
+−0.40)×10−6[4] , whilePtc(or actually
+Ptc−PC
+EW/3) governsB+→K0π+in (3), Eq. (34) implies in the limit of flavor SU(3) [48]:
+|PEW+PC
+EW|
+|Ptc|≃3|K|√
+2|λ(s)
+t|
+|λ(d)
+u|/radicaltp/radicalvertex/radicalvertex/radicalbtB(π+π0)
+B(K0π+)= 0.10, (47)
+We use values of CKM factors λ(q′)
+qquoted in Refs. [1, 2]. We have not introduced a flavor
+SU(3) breaking factor fK/fπinT+Cbecause the color-suppressed tree amplitude C, which
+is comparable in magnitude to Twith a large relative strong phase, cannot be assumed to
+factorize.
+10A ratio including CKM factors of an EWP amplitude and a corresponding tree ampli-
+tudes inB+→K+π0, which involve a common strong phase in the flavor SU(3) limit, is
+obtained directly from (34) [44, 45]:
+λ(s)
+t(PEW+PC
+EW)
+λ(s)
+u(T+C)=−3K
+2λ(s)
+t
+λ(s)
+u=−0.61e−iγ. (48)
+Deviations from this pure SU(3) limit have been estimated and were fo und to be at most
+at a level of ten percent in the magnitude of the right-hand-side an d a few degrees in its
+strong phase [40, 49].
+VII. CP ASYMMETRY SUM RULES IN B→Kπ,K∗π,ρK
+A rather precise sum rule among the four CP rate differences in B→Kπdecays has
+been proven in Ref. [20],
+∆(K+π−)+∆(K0π+)−2∆(K+π0)−2∆(K0π0)≈0. (49)
+Using the approximate relation (6) this implies a corresponding sum ru le for CP asymme-
+tries given inEq. (1). This sum rule, which is based primarily onisospin sy mmetry, provides
+a prediction for ACP(K0π0) in terms of the other more precisely measured B→Kπasym-
+metries. In this section we will study this sum rule and similar ones for B→K∗πand
+B→ρKdecays, comparing the precision of the three sum rules to one anot her.
+We follow notations introduced in Section VI to denote by Tf
+IandEf
+I∆I= 1 tree and
+EWP amplitudes for definite isospin states I, wheref=Kπ,K∗π,Kρ. In addition, ∆ I= 0
+amplitudes multiplying CKM factors λ(s)
+t(Ptcand EWP) and λ(s)
+u(Pucand tree) will be
+denoted by Pf
+1/2andtf
+1/2, respectively. Subscripts on amplitudes will refer the charges of
+the two mesons in the final state. Using isospin permits describing all twelve amplitudes
+forB→Kπ,K∗π,Kρ, in the generic form,
+−Af
++−=λ(s)
+t/parenleftBig
+Pf
+1/2−Ef
+1/2−Ef
+3/2/parenrightBig
++λ(s)
+u/parenleftBig
+tf
+1/2−Tf
+1/2−Tf
+3/2/parenrightBig
+, (50)
+Af
+0+=λ(s)
+t/parenleftBig
+Pf
+1/2+Ef
+1/2+Ef
+3/2/parenrightBig
++λ(s)
+u/parenleftBig
+tf
+1/2+Tf
+1/2+Tf
+3/2/parenrightBig
+, (51)
+−√
+2Af
++0=λ(s)
+t/parenleftBig
+Pf
+1/2+Ef
+1/2−2Ef
+3/2/parenrightBig
++λ(s)
+u/parenleftBig
+tf
+1/2+Tf
+1/2−2Tf
+3/2/parenrightBig
+,(52)
+Af
+00=λ(s)
+t/parenleftBig
+Pf
+1/2−Ef
+1/2+2Ef
+3/2/parenrightBig
++λ(s)
+u/parenleftBig
+tf
+1/2−Tf
+1/2+2Tf
+3/2/parenrightBig
+,(53)
+wheref=Kπ,K∗π,Kρ.
+Defining CP rate differences for each of these twelve processes (c ommon phase space
+factors for a given fare omitted) ,
+∆f
+ij≡ |¯Af
+ij|2−|Af
+ij|2, (54)
+we consider the sums
+∆(f)≡∆f
++−+∆f
+0+−2∆f
++0−2∆f
+00. (55)
+A generic amplitude of the form
+A=λ(s)
+tP+λ(s)
+uT , (56)
+11implies a CP rate difference (we omit a phase space factor)
+∆≡ |¯A|2−|A|2= 4Im(λ(s)
+tλ(s)∗
+u)Im(PT∗). (57)
+Thus, using Eqs. (50)-(53) one finds
+∆(f) = 24Im[λ(s)
+tλ(s)∗
+u]Im/parenleftBig
+Ef
+1/2Tf∗
+3/2+Ef
+3/2Tf∗
+1/2−Ef
+3/2Tf∗
+3/2/parenrightBig
+. (58)
+The dominant contributions to ∆( f), involving interference of the penguin amplitude
+(contained in Pf
+1/2) with tree amplitudes, have cancelled. This is the essence of the thr ee
+asymmetry sum rules, shown in Ref. [20] to follow from the isosinglet n ature of the domi-
+nant penguin amplitude and an isospin quadrangle relation among the f ourB→fampli-
+tudes [50, 51],
+−Af
++−+Af
+0++√
+2Af
++0−√
+2Af
+00= 0. (59)
+A new ∆S= ∆I= 0 operator in the effective Hamiltonian would not affect this argumen t
+as such an operator could be absorbed in the penguin operator.
+The remaining terms in the sum rules (58) involve second order interf erence terms of
+tree and EWP amplitudes. We will now estimate these remaining terms f or each of the
+three cases, f=Kπ,K∗π,Kρ, or suggest methods for measuring these terms elsewhere.
+As we noted in Sec. IV, the potential asymmetries in B→K∗+π−andB+→K∗+π0, or
+inB0→K+ρ−andB+→K+ρ0, may be significantly larger than the 10% asymmetry
+measured in B0→K+π−. Thus naively one would expect the remaining terms in the
+B→K∗πandB→Kρsum rules to be correspondingly larger than in the B→Kπsum
+rule.
+a.B→Kπ
+Inserting (41) and (42) into the sum rule (58) for f=Kπand neglecting terms involving
+AandEone obtains
+∆(Kπ) =−12KIm[λ(s)
+tλ(s)∗
+u]Im(CT∗). (60)
+The sign of the remaining term in ∆( Kπ) is predicted to be negative because Im[ λ(s)
+tλ(s)∗
+u] =
+|Vts||Vcb||Vus|sinγis positive while Kand Im(CT∗) are negative. We have argued in Sec.
+II that the difference between ACP(K+π0) andACP(K+π−) implies Arg( CT∗)<0 or
+Im(CT∗)<0 [22].
+The remaining term in ∆( Kπ) should be compared with the negative CP rate difference
+inB0→K+π−which is dominated by an interference of PandT,
+∆(K+π−)≃4Im(λ(s)
+tλ(s)∗
+u)Im(PT∗). (61)
+Using the numerical value of Kin (33) one has
+∆(Kπ)
+∆(K+π−)≃ −3KIm(CT∗)
+Im(PT∗)= 0.026|C|
+|P|sin[Arg(CT∗)]
+sin[Arg(PT∗)]. (62)
+The ratio |C|/|P|, where the numerator and denominator do not include CKM factors , may
+be evaluated by assuming that |C| ≤ |T|and taking our estimate |T′|/|P′| ∼0.1 in Sec.
+IV. Using numerical values of CKM factors [1, 2], this implies
+|C|
+|P|≤|VtbVts|
+|VubVus||T′|
+|P′|∼4.6. (63)
+12One may further assume that the sine of the relative strong phase betweenCandTis
+not substantially larger than that of the relative strong phase bet weenPandT(which we
+argued in Sec. IV to be around 30◦corresponding to sin[Arg( PT∗)] = 0.5). This implies
+0≤∆(Kπ)
+∆(K+π−)≤0.12. (64)
+Although this positive upper bound is not rigid we consider it rather sa fe to conclude that
+the sum rule (1) for B→Kπasymmetries holds within two or perhaps even one percent,
+−0.02 (−0.01)<ACP(K+π−)+ACP(K0π+)−ACP(K+π0)−ACP(K0π0)≤0.(65)
+AnimportantconclusionfollowingfromArg( CT∗)<0isthattheverysmallremainingterm
+in the sum rule must be negative. Using three of the B→Kπasymmetries in Table I leads
+to a prediction for ACP(K0π0) which includes second order corrections in the asymmetry
+sum rule,
+ACP(K0π0) =−0.149±0.037±0.01. (66)
+The first error is purely experimental while the second one is due to a possible small vio-
+lation of the sum rule. We note that the current experimental erro rs inACP(K0π+) and
+ACP(K+π0) dominate the uncertainty in this prediction.
+b.B→K∗πandB→Kρ
+Substituting (43) and (44) in the sum rule (58) for f=K∗πand neglecting small AP
+andEPterms one obtains,
+∆(K∗π) = 6KIm[λ(s)
+tλ(s)∗
+u]Im(TVT∗
+P+2TVC∗
+V+CPC∗
+V). (67)
+As mentioned, expressions for B→Kρamplitudes may be obtained from those for B→
+K∗πby interchanging subscripts V↔P. This applies also to ∆( Kρ) which can beobtained
+from ∆(K∗π) through the same transformation:
+∆(Kρ) = 6KIm[λ(s)
+tλ(s)∗
+u]Im(TPT∗
+V+2TPC∗
+P+CVC∗
+P). (68)
+An interesting relation holds between the difference ∆( Kρ)−∆(K∗π) and a CP rate
+difference for the amplitude 3 AK∗π
+3/2≡A(K∗+π−)+√
+2A(K∗0π0) already defined in (40),
+∆/parenleftBig
+(K∗π)3/2/parenrightBig
+≡ |3¯AK∗π
+3/2|2−|3AK∗π
+3/2|2. (69)
+Using Eqs.(40), (43) and (44),
+3AK∗π
+3/2=−λ(s)
+u(TP+CV)+λ(s)
+t3K
+2(TV+CP), (70)
+and applying (57) one has
+∆/parenleftBig
+(K∗π)3/2/parenrightBig
+= 6KIm[λ(s)
+tλ(s)∗
+u]Im[(TP+CV)(T∗
+V+C∗
+P)], (71)
+which implyes
+∆(Kρ)−∆(K∗π) = 2∆/parenleftBig
+(K∗π)3/2/parenrightBig
+. (72)
+13Note that, while certain individual CP rate asymmetries in B→K∗πandB→Kρ
+(involving interference of penguin andtree amplitudes) are expect ed to be largeas discussed
+in Sec IV, the asymmetry ∆(( K∗π)3/2) involves interference of tree and EWP amplitudes
+and is considerably smaller.
+The CP rate difference ∆(( K∗π)3/2) can be measured in a Dalitz analysis of B0→
+K+π−π0and its charge-conjugate [30, 52]. This analysis yields values for the magnitudes
+ofA(K∗+π−),A(K∗0π0), their relative phase and their charge-conjugates which togeth er fix
+|3AK∗π
+3/2|and its charge conjugate thereby determining ∆(( K∗π)3/2). This quantity is related
+to a ratio of amplitudes, R3/2≡¯AK∗π
+3/2/AK∗π
+3/2, which is measurable in B0→K+π−π0and its
+charge conjugate:
+∆/parenleftBig
+(K∗π)3/2/parenrightBig
+=|3AK∗π
+3/2|2/parenleftBig
+|R3/2|2−1/parenrightBig
+. (73)
+The phase of R3/2has been proposed to give a new constraint on CKM parameters [53, 54].
+Its magnitude |R3/2|and the magnitude |3AK∗π
+3/2|determine |∆/parenleftBig
+(K∗π)3/2/parenrightBig
+|which provides
+a good estimate for the combined violation of the K∗πandKρasymmetry sum rules.
+Experimental errors using current data [52] are large and do not p ermit a useful con-
+straint on ∆(( K∗π)3/2). The current error on ∆(( K∗π)3/2) is comparable to the asymme-
+tries measured in B→K∗πdecays. (This error decreases with increased statistics scaling
+roughly as 1 /√Luminosity.) A four-foldambiguity in the solution for B→K∗πamplitudes
+observed in Ref. [52] may be resolved by requiring destructive inter ference between the two
+penguin-dominated amplitudes A(K∗+π−) and√
+2A(K∗0π0) forming together the ampli-
+tude 3AK∗π
+3/2which involves smaller tree and EWP contributions. This requirement a pplies
+also to charge conjugate amplitudes, thus choosing solution I amon g the four solutions in
+Table V of Ref. [52].
+A somewhat less precise way for obtaining separate estimates for ∆ (K∗π) and ∆(Kρ)
+requires using flavor SU(3) which relates B→K∗πandB→KρtoB→ρπ. In the SU(3)
+symmetry limit, the right-hand-side of (67) may be expressed in ter ms of amplitudes for
+B→ρπdecays which are dominated by TV,PandCV,P(compare the first two equations
+with expressions in Table II) [47]:
+−A(ρ+π−)≃λ(d)
+uTP+λ(d)
+tPP, (74)
+−A(ρ−π+)≃λ(d)
+uTV+λ(d)
+tPV, (75)
+−2A(ρ0π0)≃λ(d)
+u(CV+CP)−λ(d)
+t(PV+PP), (76)
+−√
+2A(ρ+π0)≃λ(d)
+u(TP+CV)−λ(d)
+t(PV−PP), (77)
+−√
+2A(ρ0π+)≃λ(d)
+u(TV+CP)+λ(d)
+t(PV−PP). (78)
+We neglect tiny EWP contributions and very small terms EV,P+PAV,P, just as we have
+neglectedE+PAinA(B0→π+π−) in Table II.
+Working in the SU(3) symmetry limit, which is expected to introduce an uncertainty of
+20−30% in amplitudes, we will neglect contributions of penguin amplitudes in B→ρπ
+which will now be estimated to be at the same level. These contribution s are measured by
+amplitudes for B→K∗¯K,¯K∗K. Using the tree dominated branching ratio for B0→ρ+π−
+in Table II and the penguin dominated branching ratio for B+→¯K∗0K+quoted a line
+14below Eq. (32), one has
+|λ(d)
+tPP|
+|λ(d)
+uTP|≃/radicaltp/radicalvertex/radicalvertex/radicalbtB(¯K∗0K+)
+rτB(ρ+π−)= 0.20±0.03. (79)
+This valueis inagreement with the second range of values in Eq. (24), obtained for the same
+ratio using somewhat different considerations including an SU(3) bre aking factor fK∗/fρ.
+In this approximation one finds
+TVT∗
+P+2TVC∗
+V+CPC∗
+V≃ |λ(d)
+u|−2[2A(ρ0π+)A∗(ρ+π0)
++√
+2A(ρ−π+)A∗(ρ+π0)−√
+2A(ρ0π+)A∗(ρ+π−)],(80)
+implying
+∆(K∗π)≃6KIm[λ(s)
+tλ(s)∗
+u]
+|λ(d)
+u|2Im[2A(ρ0π+)A∗(ρ+π0)
++√
+2A(ρ−π+)A∗(ρ+π0)−√
+2A(ρ0π+)A∗(ρ+π−)]. (81)
+Charge conjugate modes may also be used on the right-hand-side t o increase statistics.
+The same quantity for B→Kρmay be obtained by interchanging subscripts V↔P
+corresponding to interchanging the charges of ρandπinB→ρπamplitudes,
+∆(Kρ)≃6KIm[λ(s)
+tλ(s)∗
+u]
+|λ(d)
+u|2Im[−2A(ρ0π+)A∗(ρ+π0)
++√
+2A(ρ−π+)A∗(ρ+π0)−√
+2A(ρ0π+)A∗(ρ+π−)]. (82)
+Comparing this expression with that of ∆( K∗π) in (81), we note that the only difference
+betweenthetwoexpressions isthesignofthefirstterm. Thusone maycombineasymmetries
+as in (72) to obtain:
+∆(Kρ)−∆(K∗π)≃24KIm[λ(s)
+tλ(s)∗
+u]
+|λ(d)
+u|2Im[A(ρ+π0)A∗(ρ0π+)]. (83)
+Wenowdiscuss awaybywhichtheimaginarypartsofproductsof B→ρπamplitudesin
+Eqs. (81), (82) and (83) can be determined experimentally. Infor mation about magnitudes
+and relative phases of the three B0decay amplitudes, A(ρ+π−),A(ρ−π+) andA(ρ0π0), is
+obtainedinatime-dependent Dalitzanalysisof B0→π±π∓π0[55,56],|A(ρ0π+)|isobtained
+ina Dalitzanalysis of B+→π+π+π−[57], while |A(ρ+π0)|is measured directly in this quasi
+two-body mode [58, 59]. The five amplitudes obey an isospin pentagon relation [50, 51],
+A(ρ+π−)+A(ρ−π+)+2A(ρ0π0) =√
+2A(ρ+π0)+√
+2A(ρ0π+) =D , (84)
+whereDis a diagonal of the pentagon describing an I= 2 amplitude involving tree contri-
+butions but no penguin amplitude. A similar relation holds for ¯Bdecay amplitudes.
+The magnitude |D|can be determined by studying the three B0amplitudes in a Dalitz
+analysis of B0→π+π−π0. The three magnitudes |A(ρ0π+)|,|A(ρ+π0)|and|D|fix the
+15Table III: Branching fractions and CP asymmetries for B→ρπ[4].
+Mode B(10−6)ACP
+B+→ρ+π010.9+1.4
+−1.50.02±0.11
+B+→ρ0π+8.3+1.2
+−1.30.18+0.09
+−0.17
+B0→ρ0π02.0±0.5−0.30±0.38a
+aWe take an average [4] of Belle [55] and Babar [56] results.
+triangle for these amplitudes up to a two-fold ambiguity correspond ing to flipping the
+triangle around D. This gives a value for Im[ A(ρ+π0)A∗(ρ0π+)]. Information gained in
+B0→π+π−π0on magnitudes of A(ρ±π∓) and their phases relative to Dpermits a deter-
+mination of Im[ A(ρ−π+)A∗(ρ+π0)] and Im[A(ρ0π+)A∗(ρ+π−)].
+In the approximation of neglecting penguin amplitudes in Eqs. (74)-( 78), (in which
+Eqs. (81)-(83) were written) or in the limit of vanishing strong phas es between tree and
+penguin amplitudes, all B→ρπCP asymmetries vanish. Current experimental values of
+the five asymmetries quoted in Table II and III are consistent with z ero within experimental
+errors. In this approximation the two pentagons for Band¯Bdecays coincide, and Eq. (84)
+turns into a single pentagon relation for square roots of decay rat es. A flat pentagon would
+correspond to
+/radicalBig
+B(ρ+π−)+/radicalBig
+B(ρ−π+)+2/radicalBig
+B(ρ0π0) =√
+2/radicalBig
+B(ρ+π0)/rτ+√
+2/radicalBig
+B(ρ0π+)/rτ.(85)
+Checking this possibility using branching ratios in Tables II and III, we find the following
+values (in units of 10−3) for the left and right-hand sides,
+9.49±0.48 = 8.45±0.42, (86)
+which holds within 1 .6σ. In the limit of a flat pentagon the imaginary parts of all terms
+in (81) and (82) vanish implying ∆( K∗π) = ∆(Kρ) = 0. Current agreement with a flat
+pentagon may indicate that violation of the K∗πandKρasymmetry sum rules are strongly
+suppressed.
+Current information on B→ρπamplitudes, in particular that obtained from Dalitz
+analysesofB0→π+π−π0isnotsufficiently preciseforausefulquantitativestudyof∆( K∗π)
+and ∆(Kρ). Errors are large in relative phases between amplitudes and in B→(ρπ)0
+asymmetries. Belle measured a large central value for ACP(ρ+π−) and a smaller value
+forACP(ρ−π+) [55],ACP(ρ+π−) = 0.21±0.08±0.04,ACP(ρ−π+) = 0.08±0.16±0.11,
+whereas the situation in the Babar measurements was the opposite [56],ACP(ρ+π−) =
+0.03±0.07±0.04,ACP(ρ−π+) =−0.37+0.16
+−0.09±0.10. Although the Belle and Babar results are
+statistically consistent with each other, this situation indicates larg e uncertainties. Using
+the Babar data and performing a numerical study as described abo ve, we obtain broad
+ranges for ∆( K∗π) and ∆(Kρ) which are consistent both with zero and with ∆( K∗+π−)
+and ∆(K+ρ0), respectively.
+A detailed study based on more precise data must also consider theo retical uncertainties
+in ∆(K∗π) and ∆(Kρ) caused by SU(3) breaking and by neglecting penguin amplitudes
+16inB→ρπ, which are given by decay amplitudes for B→K∗¯KandB→¯K∗K. One
+may include these terms and smaller terms of the forms EV,P+PAV,Pby substituting into
+Eq. (58) the following expressions for the isospin amplitudes TK∗π
+IandEK∗π
+Iin terms of
+∆S= 0 amplitudes [47]:
+6λ(d)
+uTK∗π
+1/2= 3A(ρ+π−)−2√
+2A(ρ+π0)−3A(K∗+K−)+A(¯K∗0K+)+2A(K∗+¯K0),(87)
+3λ(d)
+uTK∗π
+3/2=√
+2A(ρ+π0)+A(¯K∗0K+)−A(K∗+¯K0), (88)
+λ(d)
+uEK∗π
+1/2=K
+4[3A(ρ−π+)−√
+2A(ρ0π+)+3A(K∗0¯K0)+A(¯K∗0K+)−A(K∗+¯K0)],(89)
+λ(d)
+uEK∗π
+3/2=K
+2[−√
+2A(ρ0π+)−A(K∗+¯K0)+A(¯K∗0K+)]. (90)
+Corresponding expressions for the case f=Kρmay be obtained from the above by inter-
+changing the charges of ρandπand ofK∗andK. Both magnitudes and relative phases of
+B→ρπamplitudes may be measured as discussed above. This also applies to B→K∗¯K
+andB→¯K∗Kamplitudes involving a common final ¯KKπstate. However relative phases
+between the latter amplitudes and amplitudes for B→ρπare unmeasurable.
+VIII. CONCLUSION
+A CP asymmetry of 10% has been measured in B0→K+π−. We have argued that
+asymmetries in B0→K∗+π−,K+ρ−andB+→K∗+π0,K+ρ0may be two or three times
+larger because of their larger ratios of tree and penguin amplitudes . A previously proven
+approximate isospin sum rule for CP asymmetries in B→Kπdecays was generalized to
+B→K∗πandB→Kρ. The residues of the three sum rules consist of contributions from
+interference of tree and electroweak penguin amplitudes. We have shown that the residue
+of theKπsum rule is negative at a level of one or at most two percent. Combinin g the
+K∗πandKρasymmetries into a single sum rule, we have shown that its residue is giv en
+by a CP asymmetry in B→(K∗π)I=3/2, which may be measured in a Dalitz analysis of
+B0→K+π−π0. Using flavor SU(3), separate residues for the K∗πandKρasymmetry sum
+rules may be obtained by studying B→ρπamplitudes in Bdecays into three pions.
+Some of the existing experimental results on B→K∗πandB→ρKasymmetries and
+onB→ρπamplitudes do not use the full data samples accumulated for these m odes at
+the twoe+e−Bfactories, and do not study K∗decays in all possible decay modes. We
+are awaiting more complete analyses. The latest results for B0→K+π−π0published by
+Belle [29] used a data sample from an integrated luminosity of only 78 fb−1[29]. By now
+Belle has accumulated at least ten times more data for B0→K+π−π0which they should
+be encouraged to analyze. Measuring asymmetries at Bfactories, at the LHCb detector
+and at a future Super Flavor Factory, in violation of these residues forB→Kπ,B→K∗π
+orB→Kρsum rules would provide evidence for a new ∆ S= ∆I= 1 operator in the
+low energy effective Hamiltonian. Such operators occur in a variety o f extensions of the
+Standard Model [60].
+We thank Andrzej Buras and Andrew Wagner for questions which mo tivated this work
+and we are grateful to Tim Gershon for a few useful comments. Th e work of JZ is supported
+in part by the European Commission RTN network, Contract No. MRT N-CT-2006-035482
+(FLAVIAnet) and by the Slovenian Research Agency.
+17References
+[1] C. Amsler et al.[Particle Data Group], Phys. Lett. B 667, 1 (2008).
+[2] J. Charles et al.[CKMfitter Group], Eur. Phys. J. C 41(2005) 1 [arXiv:hep-
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+arXiv:1001.0704v1  [astro-ph.CO]  5 Jan 2010Propertiesandshort-timeevolutionofnearby
+galaxies
+A. Chuprikov∗
+Astro SpaceCenterof P.N.LebedevPhysicalInstituteofRus sianAcademyof
+Sciences,Profsoyuznaya84/ 32,117997,Moscow,Russia
+E-mail:achupr@asc.rssi.ru
+I. Guirin
+Astro SpaceCenterof P.N.LebedevPhysicalInstituteofRus sianAcademyof
+Sciences,Profsoyuznaya84/ 32,117997,Moscow,Russia
+E-mail:igirin@asc.rssi.ru
+We analyzethe results of processingof data of observations which had beencarried out with the
+VLBA during 10 last years. All the data have been retrieved fr om archive of the National Radio
+Astronomy Observatory(USA NRAO, Socorro, New Mexico). Par ticularly, we examine data of
+VLBA observational sessions titled BK068,BL111,BL137,BL149, andBD086. Objects of our
+interest are near galaxies with z < 0.02. The radio maps of com pact structure around the active
+galacticnuclei (AGN)reconstructedfortwo such galaxies (NGC315and3C274) in U frequency
+band (15 GHz) are presented. Some parametersof these source s are shown in Table 1. We have
+to perform the careful amplitude and phase calibration for a ll the data. Particularly, a correction
+of the delay caused by the Earth atmosphere has been made beca use it is necessary in this fre-
+quency range. Bright quasars close to the target sources (fo r instance, J0136+4751 and 3C279
+correspondingly) are used as atmosphere calibrators. Seco ndly, the Multi Frequency Synthesis
+(MFS) method (Likhachev, 2004) is used for final maps reconst ruction as well as for estimation
+of spectral index values of core and jet of AGN of both objects . As a result, we can make some
+conclusionsaboutthemilliarcsecondstructureofcentral regionsofsourcesthankstotheusingof
+these two methodsof the VLBA data processing. Moreover,we c an revealsome changesof this
+structurein1999-2009,andestimate roughlythevelocitie sandacceleration/decelerationvalues
+for brightest componentsof the nuclei of NGC315 and 3C274. A ny polarizationphenomenaare
+not taken into account. We present results of processing of d ata of LL polarization for all the
+observationalsessions
+PanoramicRadioAstronomy: Wide-field1-2GHzresearchonga laxyevolution-PRA2009
+June02-052009
+Groningen,theNetherlands
+∗Speaker.
+c/circlecopyrtCopyrightownedbytheauthor(s)underthetermsoftheCreat ive CommonsAttribution-NonCommercial-ShareAlikeLicen ce. http://pos.sissa.it/NearGalaxies A.Chuprikov
+1. Results ofdata processing
+The software project Astro Space Locator (ASL for Windows) ( Chuprikov, 2002) has been
+used toprocess all the data. The data processing consists of the following stages :
+•Amplitude calibration of all the data using GCand TYtables
+•Single band fringe fitting (the primary phase calibration) o f all the data. Estimation of the
+optimum value of solution interval
+•Averaging of all the data over each frequency band
+•Multi band fringe fitting of the atmosphere calibrator data. Estimation of gain values for
+every frequency and every time interval
+•Application of gains, compensation of atmosphere delay for all the data
+•Self-calibration
+•Imaging
+Figure 1 demonstrates the compact structure of nuclei of 3C2 74 and NGC315 reconstructed
+according to the scheme above. Evolution of this structure i s obvious for both sources. Table 2
+and Table 3 show the changes of coordinates of components. Th ese changes have the following
+properties :
+1. For 3C274
+•Image consists of 3components (A,B,and C)
+•Thereis not anyvisible proper motion of the brightest compo nent (component A)
+•Velocity value of the B component is minimal between Februar y 6, 2003 and February 5,
+2007. This value is equal approximately to 1 .5·108cm/s , and it increases after February 5,
+2007 up to1 .1·109cm/s
+•Velocity value of the C component also has a minimum between F ebruary 6, 2003 and
+February 5, 2007. Thisminimum value isequal to approximate ly 6.1·108cm/s
+Again, this value increases after February 5, 2007 upto 3 .6·109cm/s
+2. For NGC315
+•Image consists of 2components (A,and B)
+•Thereis not anyvisible proper motion of the brightest compo nent (component A)
+•Velocity of the B component decreases between July 19, 1999 a nd September 12, 2008
+from 6.5·109cm/s downto 1 .7·108cm/s
+2. Conclusions
+Weassumed that using theprocedure of atmosphere delay cali bration canessentially improve
+the image of source in the 2 centimetre wavelength range. The monitoring of near galaxies radio
+structure evolution for the period of approximately 10 year s is possible thanks to such application
+of such procedure to VLBA data. Analysis of evolution of stru cture of two galaxies (3C274, and
+NGC315) demonstrates absence of superluminal motion of any component in both sources. The
+2NearGalaxies A.Chuprikov
+maximum value of component velocity isequal approximately to6.5·109cm/s(or 0 .22·c). Thus,
+we could to propose that strong synchrotron self-absorptio n is significant in both sources. More-
+over, spectral index maps show a highly inverted core (spect ral index is positive in core of both
+sources). This could be another indicator of self-absorpti on.
+Alltheresultspresented inthispaperarepreliminary. Iti snecessary toperformsimilarmonitoring
+of near galaxies in other frequency ranges to make final concl usions. Procedures and techologies
+used during the VLBA data processing also could be very usefu l for processing of data of future
+Space VLBImission titled RADIOASTRON.
+Figure1: Evolutionoftheradiostructurein1999- 2009for3C274(lef t),andNGC0315(right)
+References
+[1] A.Chuprikov2002,Proc.ofthe 6thEuropeanVLBI Network Symposium,June25th-28th2002,
+Bonn,Germany,27
+[2] E.B. Fomalont2000,APJSS, 131,95
+[3] S. Likhachev2004,Proc.ofthe7thEuropeanVLBINetwork SymposiumonNewDevelopmentsin
+VLBIScienceandtechnology,October12-152004,Toledo,Sp ain,325
+3NearGalaxies A.Chuprikov
+Table 1:Redshift(Fomalont,2000)anddistancefor3C274andNGC315
+IAU Common z Distance 1mas corresponds to
+Name Name [Mpc] linear size [cm]
+J1230+1223 3C274 0.004 14.7 2 .20·1017
+J0057+3021 NGC315 0.017 62.5 9 .35·1017
+Table 2:RelativecoordinatesofA, B, andC componentsofthecentral regionof3C274in1999- 2009
+Date JD VLBA RAof DECof RAof DECof RAof DECof
+2450000 + Session A (mas) A (mas) B (mas) B (mas) C (mas) C (mas)
+22/05/1999 1320.51 BK068 0 0 -0.870 0.61 -2.32 0.64
+06/02/2003 2676.80 BL111 0 0 -1.552 0.59 -2.52 1.01
+05/02/2007 4136.85 BL137 0 0 -1.637 0.57 -2.40 0.68
+07/01/2009 4838.85 BL149 0 0 -1.900 0.40 -3.35 0.98
+Table 3:RelativecoordinatesofA andB componentsofthe centralreg ionofNGC315in 1999-2008
+Date JD VLBA RAof DECof RAof DECof
+2450000 + Session A (mas) A (mas) B (mas) B (mas)
+19/07/1999 1378.97 BK068 0 0 -1.618 1.196
+28/12/2000 1907.38 BK068 0 0 -1.437 0.930
+03/10/2003 2915.65 BD086 0 0 -1.525 1.486
+12/09/2008 4721.82 BL149 0 0 -1.500 1.500
+4
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+arXiv:1001.0705v2  [quant-ph]  7 Jan 2010Structural change in multipartite entanglement sharing: a random matrix approach
+G. Gennaro,1∗S. Campbell,2∗M. Paternostro,2and G. M. Palma,3
+1Dipartimento di Scienze Fisiche ed Astronomiche,
+Universita’ degli Studi di Palermo, via Archirafi 36, I-9012 3 Palermo, Italy
+2School of Mathematics and Physics. Queen’s University Belf ast, BT7 1NN, United Kingdom
+3NEST-CNR-INFM & Dipartimento di Scienze Fisiche ed Astrono miche,
+Universita’ degli Studi di Palermo, via Archirafi 36, I-9012 3 Palermo, Italy
+(Dated: November 16, 2018)
+We study the typical entanglement properties of a system com prising two independent qubit envi-
+ronments interacting via a shuttling ancilla. The initial p reparation of the environments is modelled
+using random-matrix techniques. The entanglement measure used in our study is then averaged
+over many histories of randomly prepared environmental sta tes. Under a Heisenberg interaction
+model, the average entanglement between the ancilla and one of the environments remains con-
+stant, regardless of the preparation of the latter and the de tails of the interaction. We also show
+that, upon suitable kinematic and dynamical changes in the a ncilla-environment subsystems, the
+entanglement-sharingstructure undergoes abruptmodifica tions associated with a change in the mul-
+tipartite entanglement class of the overall system’s state . These results are invariant with respect
+to the randomized initial state of the environments.
+PACS numbers: 03.65.Yz, 03.67.Mn,03.67.-a
+Open-system dynamics involving environments com-
+prisingonly afinite numberofelements havebeen proven
+a valuable arena for the study of interesting physical
+phenomena such as quantum chaos [1], quantum ther-
+modynamics [2], entanglement and relaxation [3, 4]. In
+most of these studies, the use of random-matrix theory
+has proven technically very advantageous in modelling
+random collisions between parts of the overall system
+at hand. Random matrices have been helpful in deal-
+ing with many tasks of quantum information processing,
+including quantum data hiding, quantum state distinc-
+tion, superdense coding and noise estimation [5]. Very
+recently, random matrices have found extensive applica-
+tioninthecharacterizationofMarkoviandecoherence[6].
+In this paper we unveil a further interesting situation
+where the theory of random matrices finds fertile appli-
+cations: We study the typical amount of entanglement
+that can be set in a multipartite system comprising two
+environments of arbitrary (finite) size and a shuttling
+two-level ancilla that bridges their cross-talking (Fig. 1).
+Differently from Refs. [3, 4, 6, 7], random matrices are
+used in order to model the initial preparation of the en-
+vironments. These interact with the shuttling ancilla via
+a Hamiltonian model having pre-determined interaction
+strength. This allows us to investigate on the typical
+ancilla-environment as well as all-environment degree of
+entanglement simply by averagingthe values correspond-
+ing to many hystories of random initial preparations.
+We point out the sensitivity of the entanglement-sharing
+structure on the kinematic and dynamical aspects of
+the interactions and its independence from the random
+preparation of the environments. In particular, we show
+that upon tuning of the coupling Hamiltonian regulating
+the ancilla-environment interactions, an abrupt transi-
+tion between two inequivalent classes of multipartite en-
+tanglement is achieved. The dimensions of the environ-
+ments and the number of interactions enter preponder-antly into the determination of the entanglement-sharing
+structure, as we quantitatively reveal.
+It is important to remark that our model does not al-
+low for phenomena of thermalization or homogeneization
+typical of effectively Markovian reservoirs such as those
+considered in Refs. [8]. In fact, the evolution here at
+handisprofoundlydifferentfromaforgetfulopendynam-
+ics comprising environments which are weakly perturbed
+by the interactions with the ancilla. Memory effects are
+clearly seen in our analysis, which reveals how the state
+of each environment is deeply affected by its coupling to
+the shuttling ancilla. As we do not impose restrictions to
+the strength of each interaction, also the Born approxi-
+mation does not hold in our analysis.
+The remainder of this paper is organized as follows.
+In Sec. I we address the case of a single environment
+interacting with the ancilla A. Through numerical sim-
+ulations supported by a clear analytical study we show
+that when the ancilla interacts repeatedly with the same
+environmental qubit, the degree of entanglement is in-
+variant to both the random initial preparation and di-
+mension of the environment. Sec. II extends our study
+to the case of two multi-qubit environments mutually
+connected by the bouncing ancilla. Arepeatedly inter-
+acts (collides, as we often say in this paper) with two
+specific qubits, each beloning to the respective environ-
+ment. In this case, the amount of entanglement that
+can be set is a delicate trade-off between both the di-
+mensions of the two environments and the order of the
+ancilla-environment interaction. This scenario allows us
+to address the main points of our study. First we show
+that, by tuning the form of the coupling Hamiltonian,
+the system undergoes an entanglement-sharing transi-
+tion between two non-equivalent classes of multipartite
+entanglement. As no time is explicitly involved, we re-
+fer to this effect as a “kinematic” transition. We then
+study how repeated sequential collisions are able to af-2
+fect the bipartite and tripartite entanglement in a way
+so as to mark a second, more dynamical entanglement-
+sharing transition. Sec. III summarizes our findings and
+opens up perspectives for future work. Finally, an ap-
+pendix is devoted to the technical description of the way
+random unitaries are built in this work.
+I. SINGLE-ENVIRONMENT CASE:
+PROPAEDEUTIC RESULTS
+In this Section we study the single environment-ancilla
+system, which serves a useful case for the discussion of a
+few preparatory results that will then be reprised when
+the two-enviroment situation is studied. Let us consider
+a pure, initially separable state of an n-qubit environ-
+mentE={e1,e2,..,en}. As a reference-state for our
+analysis, we assume that all the qubits are initially in
+their respective ground state so that we have
+|ψ/an}bracketri}htE=n/circlemultiplydisplay
+k=1|0/an}bracketri}htk. (1)
+We then introduce an ancillary two-level system, A, with
+whichEinteracts. The ancilla is assumed to be prepared
+in its ground state |0/an}bracketri}htAand it is meant to collidewith a
+specific element of E, which we label e. The assumption
+on the preparation of A is simply matter of convenience
+as far as pure states are taken. Moreover, we will show
+that for dim( E) = 1 any state can be considered for the
+ancilla, including mixed states. The Hamiltonian model-
+ing such an interaction is taken to be of the anisotropic
+Heisenberg form
+ˆHAe=3/summationdisplay
+i=1Jiˆσi,A⊗ˆσi,e (2)
+FIG. 1: (Color online). Sketch of the situation considered.
+An ancillary shuttle A, modeled as a two-level system, in-
+teracts with one of the elements of two spin environments,
+EL,R, which are remotely located. The initial preparation of
+each environment state is random. Upon control over the de-
+tails of the interactions and the order of the collisions wit h
+A, effective manipulation of the entanglement-sharing stru c-
+ture of the multipartite system comprising environments an d
+shuttle is achieved. Environment ERis shadowed in order
+to indicate that we consider both the single-environment an d
+two-environment cases.whereJiis an interaction strength and ˆ σi,A(e)’s are the
+Pauli spin operators of the A(e) qubit. We have used
+the labelling ˆ σ1= ˆσx, ˆσ2= ˆσyand ˆσ3= ˆσz. Our task
+is to investigate typical entanglement properties of the
+e−Asystem and we thus aim at removing any depen-
+dence of our analysis from the state of the environment
+E. In order to achieve this we proceed as follows: we
+prepareEin the state resulting from the application of a
+randomn-partite quantum gate [9] constructed using the
+random unitary matrix ˆUrthat we uniformly draw from
+the normalized Haar measure of the unitary group [10–
+12]. In the Appendix we provide the technical details
+for the parameterization of such random unitaries, which
+are the building block of our numerical calulations. We
+then let the e−Asystem evolve as ̺=ˆUhˆUrρˆU†
+rˆU†
+h,
+whereˆUh=e−iˆHAetandρ=|ψ/an}bracketri}htE/an}bracketle{tψ|⊗|0/an}bracketri}htA/an}bracketle{t0|and de-
+termine the associated amount of entanglement. This is
+then statistically averaged over a large ensemble of ran-
+dom initial preparations so as to remove the dependence
+on the initial environmental state.
+We now anticipate a quantitative result achieved via
+our numerical calculations, which will then be justified
+in a fully analytical way: for an isotropic Heisenberg in-
+teraction defined by taking Jx,y,z=Jin Eq. (2), the
+entanglement set between Aandeonly depends on the
+dimensionless interaction time Jtand [as far as the an-
+cilla is prepared in a pure state] is insensitive to the di-
+mension of E, dim(E), and the preparation of A. One
+may think that this result is only due to the fact that A
+collides with a single environmental qubit, thus reducing
+our problem to a two-qubit interaction. However, this
+is definitely not the case. In fact, in general, the ran-
+dom evolution within the environment, encompassed by
+the random unitary matrices Ur’s, creates a multipar-
+tite entangled state. Although the ancilla Aphysically
+collides with eonly, the state of the latter is crucially af-
+fected by the entanglement-sharing structure within E,
+which is highly non trivial. In Ref. [13], for instance, it
+was shown that the behavior of bipartite entanglement
+shared by any two elements of a multipartite register E
+is a decreasing function of dim( E). Intuition would then
+lead one to believe that the entanglement set between
+the interacting qubits would also depend on dim( E) [14].
+Here we show that such a dependence is not in order. In
+order to understand this counter-intutive behavior, we
+take a two-step approach. First, by studying the case
+of dim(E) = 1, we provide an intuition of the reasons
+behind the dependence of the typical amount of A−e
+entanglement on the sole interaction strength J. In this
+simple and yet useful case, a random unitary operation
+built according to the parameterization given in the Ap-
+pendix is represented by the 2 ×2 matrix
+Ur=eiα/parenleftbigg
+cosφeiψsinφeiχ
+−sinφe−iχcosφe−iψ/parenrightbigg
+,(3)
+where the angles α,χ,ψ(φ) are picked up from the range
+[0,2π]([0,π/2]) uniformly with respect to the normalized3
+Haar measure dUr= (8π3)−1sin(2φ)dφdψdχdα [11, 15,
+16]. In order to evaluate the entanglement set within the
+system, we use Wootters’ concurrence [17]. For a general
+bipartite pure or mixed state described by the density
+matrix̺, concurrence is given by
+C= max[0,/radicalbig
+λ1−4/summationdisplay
+k=2/radicalbig
+λk] (4)whereλ1≥λj(j= 2,3,4) are the eigenvalues of
+ρ(σ2⊗σ2)ρ∗(σ2⊗σ2). When Ais prepared in its
+ground state and upon evolution of the e−Asystem
+as̺=ˆUhˆUrρˆU†
+rˆU†
+h, we get the density matrix
+̺=
+C2(φ) −i
+2e−2iJt+i(ψ+χ)S(2Jt)S(2φ)−1
+2e−2iJt+i(ψ+χ)C(2Jt)S(2φ) 0
+i
+2e2iJt−i(ψ+χ)S(2Jt)S(2φ) S2(φ)S2(2Jt) −i
+2S(4Jt)S2(φ) 0
+−1
+2e2iJt−i(ψ+χ)C(2Jt)C(2φ)i
+2S(4Jt)S2(φ) C2(2Jt)S2(φ) 0
+0 0 0 0
+,(5)
+withC(x) = cos(x) andS(x) = sin(x). It is then
+straightforward to see that the concurrence shared by
+eandAafter the application of ˆUrandˆUhresults in the
+elegant expression
+CeA= sin2(φ)|sin(4Jt)|. (6)
+This shows that, given a specific preparation of the en-
+vironment, the only parameters governing the entangle-
+ment areφandJt. The typical value of CeAis obtained
+by averaging the above expression over any possible uni-
+tary matrix Uruniformly drawn according to the proper
+Haar measure. Explicitly, we have to calculate
+CeA=1
+8π3/integraldisplayπ/2
+0sin(2φ)dφ/integraldisplay2π
+0dψ/integraldisplay2π
+0dχ/integraldisplay2π
+0dαCeA
+=1
+2|sin(4Jτ)|,
+(7)
+which may seem a special result arising from the pure-
+state preparation of the ancilla. However, this is defi-
+nitely not the case, as we now demonstrate. By starting
+with the mixed ancilla state
+ρA=/parenleftbigg
+ρ00
+0 1−ρ0/parenrightbigg
+, ρ0∈[0,1] (8)
+andcalculatingtheevolutionofthe e−Asystem, onegets
+a density matrix that is an easy generalization of Eq. (5).
+By inspecting the eigenvalues of ρ(σ2⊗σ2)ρ∗(σ2⊗σ2),
+whose explicit form is too lenghty to be reported here,
+one finds no dependence on χ, ψorα, in full analogy
+with the case of Eq. (7). The explicit calculation of con-
+currence leads us to the expression
+CeA=1
+2|[−1+(2ρ0−1)cos(2φ)]sin(4Jt)|.(9)
+As/integraltextπ/2
+0cos(2φ)sin(2φ)dφ= 0, the average concurrence
+(calculated using the appropriate Haar measure, as done
+before) turns out to be identical to Eq. (6). The studycan be straightforwardly generalised to the case of qubit
+Abeingpreparedinanycoherent-superpositionstate,the
+only difficulty being a slightly more complicated expres-
+sion for the concurrence corresponding to any set prepa-
+ration ofE. The message, however, is rather clear: re-
+gardless of the state into which the ancilla is prepared,
+the typical e−Aentanglement is simply set by the
+rescaled interaction time Jt. This is strictly valid only in
+the statistical sense: if specific instances of preparation
+of bothAandeare taken, such an independence does
+not hold anymore. However, contrary to a naive expec-
+tation, the typical entanglement does not vanish. As we
+see later, this result is the key to understand what occurs
+in the two-environment case.
+We now approach the second step of our proof by
+studying the invariance of the e−Aentanglement with
+respecttodim( E)whenAispreparedinapurestate. For
+this task, we consider a simple extension of the previous
+case to a two-qubit environment E={e1,e2}, initially
+prepared in a state described by
+ρE=1
+4(11e1⊗11e2+3/summationdisplay
+k=1βk,e211e1⊗ˆσk,e2
++3/summationdisplay
+k=1βk,e1ˆσk,e1⊗11e2+3/summationdisplay
+k,l=1χklˆσk,e1⊗ˆσl,e2)(10)
+withχthe elements of the tensor accounting for the cor-
+relations between e1ande2andβej(j= 1,2) the Bloch
+vector of qubit ej(j= 1,2) [9, 18]. This form holds
+for both entangled and separable two-qubit states and
+is thus a formal description of an arbitrary preparation
+ofE. By taking e≡e2(an arbitrary choice that does
+not affect the generality of our discussion) we follow the
+recipe for evolution described above. This time, before
+calculating the e−Aconcurrence, we have to trace over
+e1’s degrees of freedom. Through a tedious but otherwise
+straightforward calculation, we see that although the tri-
+partiteE−Adensity matrix depends on χandβe1, the
+reduced density matrix of the e−Asystem only depends4
+onβe2. By properly averagingoverany possible prepara-
+tion ofthe environmentalqubit e2, we are led to a typical
+e−Aentanglement that is identical to Eq. (6). These
+considerations can be extended to dim( E) =n, although
+the complexity ofan analyticalproofscalesexponentially
+withn. However, our numerical calculations are in per-
+fect agreement with the analytic conclusions, supporting
+the general validity of the arguments used here and thus
+demonstrating the claimed invariance.
+The relevance of this result and its significance become
+evident when one considers that random-matrix theory
+is widely believed to provide a good effective descrip-
+tion of the state of a system in thermal equilibrium at
+a high temperature [19]. In such unfavorable conditions,
+the achievement of entanglement regardless of the initial
+state of the environment ( i.e.its effective temperature)
+anditsrobustnessagainstthecomplexityof E’sstructure
+and the initial (pure) state of Ais a remarkable feature.
+Our study reveals that through the use of a suitable in-
+teraction we can reliably create entanglement between a
+cleanancilla qubit and a mixed-state environment qubit.
+Furthermore, the invariance of the amount of generated
+entanglement with respect to the size of the environment
+stronglycontrastswiththeintuitivebeliefthat CeAwould
+have followed a monogamy constraints similar to those
+responsible for the bipartite entanglement in an n-qubit
+system [13, 20]. Besides its intrinsic interest, such an in-
+variance will be proven crucial for the understanding of
+the study conducted in Sec. II
+II. TWO-ENVIRONMENT CASE:
+ENTANGLEMENT-STRUCTURE TRANSITION
+We now turn our attention to the main scenario of our
+investigation, which consists oftwo independent and spa-
+tially separated environments, ERandEL. This time,
+the ancilla Ashuttles between them, colliding always
+with the same qubit of each environment. Again, we
+are interested in typical values of entanglement and, in
+order to cancel any dependence on the initial prepara-
+tion of the environments, we rely on a random-matrix
+approach. We first show that, in line with the results un-
+veiled in Sec. I, one of the environmentsis entangled with
+Awith a constant typical degree, dependent solely on
+Jt. This is accompanied by a few other results related to
+the details of the environment-ancilla interactions. Most
+importantly, however, we demonstrate that the genuine
+tripartite entanglement established among Aand the en-
+vironmental qubits involved in the collisions is subjected
+to a drastic change in its structure, depending on the
+shape of the interaction Hamiltonian and the number of
+collisions. The fact that these results refer to typical ( i.e.
+statistically averaged) quantities, make them even more
+intriguing. Not only multipartite entanglement can be
+generated: through suitable interaction engineering we
+can actually efficiently dictate its form and nature.A. Quantitative analysis of entanglement
+Let us first consider the case where dim( EL) = 1 (we
+label its only element as {eL}), whileERconsists of
+qubits{e1,e2,..,en}. IfAcollides only once, first with
+ERand thenEL, we find that the degree ofentanglement
+between the single-qubit environment ELand the ancilla
+remains constant against the number of qubits belonging
+toER, as show in Fig. 2 (a). We can easily understand
+this in light of the results discussed in the previous Sec-
+tion, where we have shown that the typical entanglement
+betweenAand a single-qubit environment does not de-
+pend on the input state of the ancilla. As the collision
+withER, followed by the trace over its degrees of free-
+dom, simply prepares Ain a mixed state, the invariance
+ofCeLAagainst dim( ER) is clear: quantitatively, CeLA
+is identical to Eq. (6). On the other hand, by calling
+eRtheERqubit with which Acollides, we have that
+CeRA<CeLA, regardless of dim( ER), a behavior that
+can be rigorously explained as follows. For the sake of
+simplicity, we consider the simplest case where both the
+environments consist of a single qubit. By means of the
+decomposition in Eq. (3), we prepare a general (sepa-
+rable) initial state of eRandeLby specifying two sets
+of parameters {φj,ψj,χj,αj}(j=L,R). The ancilla
+is then let interact with eRandeL, in this order. The
+degrees of freedom of eLare thus traced out so as to as-
+certain the concurrence between AandeR. This turns
+out to be
+CeRA=|sin2(φR)cos(2Jt)sin(4Jt)|.(11)
+There is no dependence on the state of eLin this ex-
+pression, although the second collision has occurred and
+no average has yet been taken. However, this is quite
+clear in virtue of the fact that we are considering only a
+single interaction with each environment. After the col-
+lisionwitheL, there is no way for Ato convey to eR
+information on the state of the left qubit. In fact, if we
+instead trace out eR, we find precisely Eq. (7). Look-
+ing for typical values, i.e.calculating averages over the
+tensor product of any possible unitary matrix by using
+the appropriate Haar measure introduced before, we get
+CeRA=|cos(2Jt)sin(4Jt)|/2<CeLA∀Jt. Quantita-
+tively, for Jt= 1 the typical eR−Aconcurrence is
+≃0.157, which perfectly agrees with the independent
+numerical estimate provided in Fig. 2 (a)(dashed line).
+The ordering relation involving CeRAandCeLAis pre-
+served as the number of qubits in ERgrows, as shown
+in Fig. 2 (a), where the difference CeLA−CeRAappear
+to increase with dim( ER). This is a clear effect of the
+profound differences, stemming from the possibility to
+achieve multipartite entanglement, between the present
+situation and the single-environment case addressed in
+Sec. I. For dim( ER)>1, one cannot exclude that
+the random unitaries taken in order to prepare ERset
+some genuine multipartite entanglement among its ele-
+ments. Consequently, the entanglement set by the dy-
+namics studied here and shared by eRandAwould be5
+(a) (b)
+FIG. 2: (Color online). Typical concurrence between the an-
+cillaAand the environment qubits eL,Rfor dim(EL) = 1 and
+dim(ER) = 1,..,4. In panel (a),Ainteracts first with eR
+and then with eL, for a rescaled interaction time Jt= 1 (see
+inset). The blue dots (red squares) show the average concur-
+rence between AandeR(eL). In agreement with the analytic
+formula in Eq. (6), CeLA=|sin4|/2≃0.378.(b)Same as in
+panel(a), but with a reversed order of interactions.
+bound to follow many-body entanglement monogamy re-
+lations [20], which lower its amount as ERgrows in di-
+mension. This is also the reason behind the behavior
+shown in Fig. 2 (b), where by inverting the order of the
+ancilla-environmentinteractionswe lose the invarianceof
+theeL−Aconcurrence, an effect occurring for precisely
+the same reasonsexplained above. On the other hand, as
+before, the second interaction always sets the lowest de-
+gree of entanglement. It is worth remarkingthat Fig. 2 is
+the result of an independent numerical calculation that,
+although follows the formal recipe highlighted here, does
+not rely on the analytic result in Eq. (11).
+B. Entanglement-Structure Transitions
+The explicit introduction of multipartite entanglement
+in the process described until now directs us towards the
+main point of this study. Through the proper tuning of
+the dynamical and kinematic properties of the system,
+we shall demonstrate the ability to manipulate the shar-
+ing structure of the typical multipartite entanglement set
+amongA,eLandeR. Fortheremainingpartofourstudy,
+we find it computationally more convenient to abandon
+concurrence for negativity [21, 22]. This is an entangle-
+ment measuredevisedfromthe Peres-Horodeckicriterion
+on positivity of partial transposition (PPT). It is a nec-
+essary and sufficient condition for 2 ×2, 2×3 and∞×∞
+systems, and is a sufficient condition for an arbitrarysys-
+tem. For a bipartite case we define the negativity as [22]
+Nbi= max[0,−2λneg] (12)
+whereλnegis the single negative eigenvalue arising after
+the partial transposition of the density matrix. The con-
+venience in using negativity stems for its straightforwardgeneralization to the multipartite scenario: we simply
+haveto consider all the possible bipartite splits in a given
+system and the geometric average of the negativity asso-
+ciated with any of them. This construction ensures that
+our measure remains an entanglement monotone [23]. As
+we are mostly interested in at most three qubits, we con-
+centrate on tripartite negativity , which in our case reads
+Ntri= [NA|eLeRNeL|AeR3NeR|AeL]1
+3(13)
+whereA|eLeR, for instance, indicates the bipartition of
+qubitAagainst the group of qubits eLandeR.
+In light of the analysis performed in Sec. IIA one can
+clearly conclude that, if tripartite entanglement is set
+within theeL−A−eRsystem, this has tobe W-class[24].
+In fact, we find non-zero entanglement in any biparti-
+tion obtained by tracing out one of the components of
+the overall system. In Fig. 2, we have shown the en-
+tanglement between the ancilla and each environmental
+qubit involved in the collisions. Moreover, we have also
+checked that the eL−eRbipartition turns out to be in-
+separable in each of the situations addressed there. This
+is a property not shared by the GHZ-class [25], where
+by tracing one qubit out of a tripartite register, one gets
+separable reduced states. In order to check that W-class
+entanglementis indeed in orderhere, wehaveutilized the
+witness [26, 27]
+ˆW=3
+411−|GHZ/an}bracketri}ht/an}bracketle{tGHZ| (14)
+with|GHZ/an}bracketri}ht= (|000/an}bracketri}ht+|111/an}bracketri}ht)/√
+2 a three-qubit GHZ
+state and 3 /4 the squared maximum overlap between
+a GHZ state and a W-class state. Eq. (14) is a valid
+GHZ witness: a negative expectation value of ˆWsuc-
+cessfully reveals GHZ-class entanglement. On the other
+hand, this construction guarantees that a positive expec-
+tation value is achievedfor W-class states [26]. By gener-
+ating a sample of 1000 random environmental states and
+taking dim( ER) up to 4 (so as to match the situation
+depicted in Fig. 2), we have numerically verified that, as
+far as an isotropic Heisenberg model is used to model the
+collisions, /an}bracketle{tˆW/an}bracketri}ht>0 for every single preparation of the en-
+vironments and regardless of dim( ER). The calculation
+has been performed by maximizing the overlap between
+the sample states and |GHZ/an}bracketri}htover the tensor product of
+three local rotations, each acting on A,eLandeRre-
+spectively. Moreover, the tripartite negativity defined in
+Eq. (13) turns out to be non-zero for any initial prepara-
+tion of our statistical sample, and so is the typical value
+obtained by averaging over it, which guarantees genuine
+tripartite entangled nature of the eL−A−eRstates with
+strong evidence of their W-class nature.
+The task of this Section is to show that the typical
+entanglement-sharing structure of such tripartite state
+can be abruptly modified by biasing the coupling Hamil-
+tonian towards a specific spin non-preserving model. We6
+FIG. 3: (Color online). Bipartite and tripartite negativit y re-
+sulting from a bilocal interaction model ruled by Eq. (15), f or
+single-qubit environments and |000/angbracketrighteLAeRinitial state. The
+full disappearance of entanglement at λ= 1 occurs in virtue
+of this specific choice of initial state. At λ= 0we have a point
+of entanglement-sharing change. Other three such points ar e
+visibile within the shown range of λ’s.
+thus recast Eq. (2) into ( j=L,R)
+ˆHb,j=J[σ1,A⊗σ1,ej+λ3/summationdisplay
+i=2σi,A⊗σi,ej],(15)
+where we have clearly taken J2,3=λJ(λ∈R) andJ1=
+J. The reasons behind this choice are best explained by
+means ofthe followinganalysis. Forthe sakeofargument
+we focus on single-qubit environments and take qubits
+A,eLandeRaspreparedintheir respectivegroundstate.
+By considering an eL−Acollision followed by an eR−A
+one, both ruled by Eq. (15), we get
+|ψ/an}bracketri}hteLAeR=e−2iλJtcos(Jt−λJt)|0/an}bracketri}hteL[cos(Jt−λJt)|00/an}bracketri}ht
+−isin(Jt−λJt)|11/an}bracketri}ht]AeR−sin(Jt−λJt)|1/an}bracketri}hteL⊗
+[sin(λJt+Jt)|01/an}bracketri}ht−icos(λJt+Jt)|10/an}bracketri}ht]AeR.
+(16)
+Despite its innocence, the entanglement-sharing prop-
+erties of Eq. (16) are quite interesting. In Fig. 3 we
+show the bipartite and tripartite negativity against the
+anisotropy parameter λ, forJt= 1 (arbitrary choice).
+The behavior of the bipartite entanglement is strongly
+dependent on the choice of λ. In particular, at λ= 0,
+which leaves us with ˆHb,j=Jσ1,A⊗σ1,ej, only two out
+of three possible bipartitions are inseparable: the eL−eR
+subsystem remains separable regardless of the choice for
+Jt. This is easily checked by studying the partial trans-
+posed of the state resulting from |ψ/an}bracketri}hteLAeRupon trace
+overA. And yet, the tripartite negativity Ntriis rather
+large atλ= 0 (cfr. Fig. 3, dashed line), demonstrating
+that we have a tripartite entangled state that, in evident
+contrast with the results of the previous Section, is out
+of the W-class. [28]. This qualitative feature is typical
+as it survives to a statistical average over random ini-
+tial preparations of the environments, their dimensions
+and the number of A−ejinteractions ( j=L,R). Thislatter feature is shown in Figs. 4, where we plot the aver-
+age bipartite and tripartite negativity against λ∈[−5,5]
+and the number of collisions. Fig. 4 (c)shows that at
+λ= 0 there is no quantum correlationsshared by the two
+environments, even in this statistically typical scenario.
+How can we understand this result and relate it to
+the claimed change in multipartite entanglement struc-
+ture? We refer to the studies conducted by Plesch and
+Buˇ zek [29] on the classification of multiparticle quan-
+tum correlations via entangled graphs. Following their
+lines, we represent a multi-qubit system with an inherent
+structure of shared entanglement as a connected graph.
+Each vertex embodies a qubit while a bond connecting
+two vertices represents bipartite entanglement shared by
+the components of the corresponding reduced state. For
+three qubits, four possible classes areidentified, as shown
+in Fig. 5. In particular, Eq. (16), as well as any of the
+states resulting from the application of random unitaries
+overthechosenreferencestate |000/an}bracketri}hteLAeR, haveasharing
+structure corresponding to the graph in Fig. 5 (c), panel
+(a)being representative of W-class. As it is argued in
+Ref. [29], by following this graph-based classification of
+multipartiteentanglement, astateshowingtripartitecor-
+relations with the two-way residual entanglement struc-
+ture of panel (c)is GHZ-class. Moreover, a remark is
+due: such a two-way entanglement-sharing structure is
+possible, for pure states, only if there exist classical cor-
+relation between the separable qubits ( eLandeRin our
+case). This property is readily confirmed in Eq. (16) and
+any random element of our statistical sample. In fact, we
+find that
+̺eReL/ne}ationslash=̺eL⊗̺eR (17)
+where̺eL,Rdenotes the reduced density matrix of qubit
+eL,Rafter (random) preparation, unitary evolution and
+appropriate partial traces [30]. We can thus rightfully
+claimfortheanticipatedentanglementtransitioninduced
+by the tuning of the effective order parameter embod-
+ied byλ. The naturally generated W-class state arising
+from the use of an unbiased Heisenberg coupling is non-
+trivially changed into a GHZ-type entanglement charac-
+terized by a two-way sharing structure. The transition
+is typical in the sense explained so far: given anypure
+initial environmental state, we are able to create either
+type of tripartite entanglement structures simply by tun-
+ing the interaction properties.
+As anticipated, we also allow the ancilla to interact
+multiple times, in sequence, with each environment (each
+time with the very same eL,Rqubit). Multiple interac-
+tions, however, do not appear to have any qualitative
+relevance on the generated entanglement structure, al-
+though there is an evident quantitative effect on the
+amount of entanglement shared both in bipartite and tri-
+partite sense. In Fig. 4 (a)and(b)we plot the typical
+tripartiteandbipartiteentanglementofthe A−eLsystem
+anditiseasilyverifiedthatthesamebehaviorisfoundfor
+theA−eRsystem. For the wholeconsidered rangeofval-
+ues forλthe bipartite negativity is always non zero, sig-7
+(a) (b) (c)
+FIG. 4: (Color online). (a)Tripartite negativity of the EL−A−ERsystem against λ∈[−5,5] and the number of collisions,
+for a dimensionless interaction strength Jt= 1 in an anisotropic Heisenberg coupling involving single- qubit environments. (b)
+Bipartite negativity for the A−Ejqubit-pair, regardless of j=L,R.(c)Bipartite negativity for the EL−ERsystem. For
+λ= 0 there is no entanglement in this bipartition, while the an cilla-environment ones are inseparable.
+naling that this interaction model guarantees the genera-
+tion of entanglement between the ancilla and an environ-
+mental qubit even under the “minimum-control” condi-
+tions where one is unable to stop the ancilla-environment
+interactions after a desired number of collisions. More-
+over, as claimed above, at λ= 0 theeL−eRsystem is
+indeed separable, regardless of the number of considered
+collisions (see the NeLeR= 0 line in Fig. 4 (c)).
+FIG. 5: (Color online). Entanglement graphs for the tripar-
+tite setting. Each sphere is a qubit, each bond represents a
+non-separable state. (a)W-class states having no separable
+bipartition. (b)Fully separable and GHZ-class states, with
+no residual bipartite entanglement. (c)Entangled-class with
+bipartition-dependentresidualentanglement. Ifthestat e con-
+tains genuine tripartite entanglement, it can be enumerate d
+in the GHZ class. Systems eLandeRshare classical correla-
+tions.(d)Biseparable states encompassing a Bell pair [9].It is very informative to study the effects that increas-
+ing the size of the environments has on the degree and
+type of shared entanglement for λ= 0. We first take
+dim(EL)≥1 and leave ERwith a single qubit. The
+scheme of interaction in the usual one, with eLbeing
+struck first. The features associated with the resulting
+entanglement sharing are shown in Figs. 6. In panel (a)
+weplotthetripartitenegativitysetwithinthestateofthe
+interacting qubits after tracing out the non-interacting
+elements from the ELenvironment. We find that Ntri
+decreases with the dimension of EL, which is a physi-
+cally meaningful result in light of monogamy relations
+that the system should adhere to [20]. A similar behav-
+ior is found for the bipartite eL−Aentanglement, as
+shown in Fig 6 (b), which can be easily interpreted in
+light of the discussion in Sec. IIA. Finally, panel (c)
+shows theeR−Aentanglement, which exhibits a con-
+stant trend that is consistent with our previous results
+on the isotropic Heisenberg Hamiltonian. We omit show-
+ing theeR−eLentanglement as this is exactly null, in
+these conditions. The entanglement structure remains of
+the two-way GHZ class [29] although for dim( EL)/greaterorsimilar8,
+NeLA≃0. Thisimpliesthedisappearanceoftheleftmost
+bond in Fig. 5 (c)but no further change, as genuine tri-
+partite entanglement is still present in the eL−A−eR
+system (see Fig. 6 (a)).
+The above analysis covers only partially the issue of
+increasing-sized environments, as it addresses an asym-
+metricsituation. Anotherinterestingeffect arisesif ERis
+let grow in size. In fact, the rate at which bipartite quan-
+tum correlations decrease in the case considered above
+suggests that, by increasing dim( ER), all residual en-
+tanglement would die off, eventually, leaving us with a
+sharing structure that reminds more of a typical GHZ
+state. In other words, we are looking for a configura-
+tion allowing an additional entanglement transition from8
+(a) (b) (c)
+FIG. 6: (Color online). Entanglement obtained for an increa singly-sized ELenvironment (which comprises of up to eight
+qubits) and a single-qubit ER. We set λ= 0 in Eq. (16) and the rescaled interaction time Jt= 1.(a)Tripartite negativity.
+(b)Bipartite eL−Aentanglement against the dimension of ELand the number of collisions. (c)Same as panel (b)but for
+the single-qubit environment ER.
+a graph of class (c)to one of class (b)in Fig. 5. This is
+indeed possible. We consider dim( EL) = dim(ER) and
+investigate the behavior of both tripartite and bipartite
+negativity against the number of environmental qubits
+and the number of interactions with the ancilla. A typi-
+cal result, achieved by considering 18 pairs of collisions,
+is givenin Fig.7 (a)whereit is evident that, by consider-
+inglargerenvironments,weareabletokeepthe tripartite
+entanglement non-zero using repeated collisions between
+Aand the environments. Any two-qubit quantum cor-
+relation, on the other hand, disappears. The curves are
+the best-fits to the discrete numerical values obtained
+via our simulations. We have found that the functional
+formAe−B(dim(Ej)−1)(withA,B ∈R) gives an excellent
+agreement with the numerical results. Fig. (7) (b)shows
+(a) (b)
+FIG. 7: (Color online). (a)Negativity against the environ-
+ments’ dimension dim( Ej) (j=L,R) forλ= 0, 18 pairs
+of collisions and Jt= 1. The dashed line shows the tripar-
+tite negativity, while the solid lines are for the entanglem ent
+within each bipartite subsystem involving only one environ -
+ment. The eR−eLentanglement is obviously null. Each curve
+is rescaled with respect to its maximum value. (b): We plot
+the number of qubits required in the remote environments in
+order to make the tripartite and bipartite negativity small er
+than10−6whenafixednumberofcollisions is taken. The top-
+most curve is for Ntriwhile the other two curves, which are
+almost superimposed, show the cases associated with NejA.the number of qubits necessary in Ej(j=R,L) in order
+to get negativities smaller than 10−6(arbitrarily chosen)
+at a fixed number of collisions. It is clearly seen that
+the environmental dimension required to get entangle-
+ment lower than the chosen threshold is almost always
+much larger for the tripartite negativity than for the bi-
+partiteones, thus strengtheningourconclusions: wehave
+been able to dynamically drive the system state towards
+typical GHZ-class entanglement, therefore realizing yet
+another sharing-structure transition.
+III. CONCLUSIONS
+Using a random-matrix approach, we have shown that
+the typical entanglement properties of two remote qubit
+environments, indirectly communicating via the media-
+tion operated by a shuttling ancilla, can be effectively
+manipulated both quantitatively and qualitatively. Bi-
+partite as well as genuinely multipartite entanglement
+can be faithfully generated and shown to have intrigu-
+ing statistically averaged properties of resilience against
+arbitrary pure preparations of the environments, their
+dimension and the number of interactions that each has
+with the shuttle. Interestingly, by means of a simple
+tuning of the details of the interaction, we can achieve
+transitions in the entanglement-sharing structure from
+W to two-way GHZ-class [29]. We have studied the ef-
+fects that an increasing dimension of the environments
+have in such transitions. The use of random matrices
+has emerged, here, as a valuable tool for the agile han-
+dling of the numerical simulations required for the pur-
+poses of our study, enlarging the range of applicability
+of the technique to the context of multipartite entangle-
+ment generation under unfavorable conditions. This is
+a central problem in current quantum information and
+computation problems, where we hope that our results9
+will trigger further interest. We are currently investi-
+gating strategies, developed well along the lines of this
+paper, to study the interplay between quantum correla-
+tions and thermodynamical properties of a multipartite
+system.
+Acknowledgments
+MP thanks Dr. M. Guta for interesting discussions
+on the topic of entanglement in random-matrix set-
+tings. This work has been supported by PRIN 2006
+“Quantum noise in mesoscopic systems”, EUROTECH,
+DEL, the British Council/MIUR British-Italian Part-
+nership Programme 2007-2008 and the UK EPSRC
+(EP/G004579/1). SC gratefully acknowledges the hospi-
+tality of the DSFSA, Universita’ degli Studi di Palermo,
+where part of this work has been performed.
+APPENDIX
+Construction of Random Unitary Matrices
+Here we briefly outline the recipe to generatearandom
+unitary matrix. The algorithm has been extensively used
+in recent works [11, 12, 31–33]. The parameterisation
+is based on the original work presented by Hurwitz in
+1897 [10]. Any unitary matrix, Ur, of dimension n, can
+be decomposed as
+Ur=eiαE1E2...En−1 (A-1)whereEiis ann×nmatrix. Matrices Ei’s are readily
+constructed using products of proper rotation matrices
+R(i,j)(φij,ψij,χij), each depending on the respectiveset
+of Euler’s angles {φij,ψij,χij}as follows
+E1=R(1,2)(φ12,ψ12,χ12),
+E2=R(2,3)(φ23,ψ23,0)R(1,3)(φ13,ψ13,χ13),
+E3=R(3,4)(φ34,ψ34,0)R(2,4)(φ24,ψ24,0),
+×R(1,4)(φ14,ψ14,χ14),
+...
+En−1=R(n−1,n)(φn−1n,ψn−1n,0)×...
+×R(1,n)(φ1n,ψ1n,χ1n).(A-2)
+The matrix elements are taken as
+R(i,j)
+k,k= 1 (fork/ne}ationslash=i,j),
+R(i,j)
+i,i=eiψcosφ, R(i,j)
+i,j=eiχsinφ,
+R(i,j)
+j,i=−e−iχsinφ R(i,j)
+i,j=e−iψcosφ,(A-3)
+and zero otherwise. The angles are drawn from the
+ranges 0 ≤φij≤π/2,0≤ψij≤2π,0≤χij≤2π
+and 0≤α≤2π,uniformly with respect to the corre-
+sponding (and properly normalized) Haar measure [11].
+* These authors contributed equally to this paper.
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+[14] It is sufficient to consider the following simple example
+to see that the dimension of a multipartite environment
+prepared in an entangled state can affect the bipartite
+quantum correlations between an ancilla Aand the qubit
+eof the environment it interacts with. For the sake of
+argument, let us assume the e−Ainteraction is embod-
+ied, in this example, by a controlled-NOT gate [9]. While
+Ais prepared in a superposition state |+/angbracketrightAsuch that10
+ˆσx|+/angbracketrightA=|+/angbracketrightA, the environment is off-line prepared in
+anN-qubit W state (1 /√
+N)/summationtextN
+j=1|0102··1j··0N/angbracketright. For
+definiteness, let us take e= 1. It is straightforward to
+see that upon e−Ainteraction and trace over the N−1
+environmental qubits not involved in such a dynamics,
+the bipartite entanglement shared by eand the ancilla
+does indeed depend on N. Quantitatively, CeA= 1/3 for
+N= 3 while it is 1 /2 forN= 4. Clearly, this is sim-
+ply an example, which is nevertheless significant enough
+to illustrate the potential dependence of CeAon dim(E)
+under the assumptions of our work.
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+for the general Haar measure given in [11] should beused.
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+[28] The fact that, at λ= 1, the state is separable both in the
+bipartite and tripartite sense should not surprise in light
+of the choice of initial state |000/angbracketrighteLAeRand the spin-
+preserving nature of the isotropic Heisenberg Hamilto-
+nian.
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\ No newline at end of file
diff --git a/1001.0706.txt b/1001.0706.txt
new file mode 100644
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@@ -0,0 +1,707 @@
+arXiv:1001.0706v1  [hep-th]  5 Jan 2010Jet-quenching of the rotating heavy meson in
+aN=4 SYM plasma in presence of a constant
+electric field
+J. Sadeghia,∗and B. Pourhassana†
+aSciences Faculty, Department of Physics, Mazandaran Unive rsity,
+P .O .Box 47416-95447, Babolsar, Iran
+February 27, 2018
+Abstract
+In this paper, we consider a rotating heavy quark-antiquark (q¯q) pair in a N=4
+SYM thermal plasma. We assume that q¯qcenter of mass moves at the speed vand
+furthermore they rotate around the center of mass. We use the AdS/CFT correspon-
+dence and consider the effect of external electromagnetic fiel d on the motion of the
+rotating meson. Then we calculate the jet-quenching parame ter corresponding to the
+rotating meson in the constant electric field.
+Keywords: AdS/CFT correspondence; Super Yang Mills theory; Black hole; S tring
+theory.
+1 Introduction
+As we know the Maldacena conjecture [1-3] provides useful mathe matical tools for studying
+complicated problems of QCD at strong coupling. In this way one of th e important subjects
+is the motion of charged particles through the strongly coupled the rmal medium. Already
+the subject of a quark in the thermal plasma at weak coupling has be en well studied in
+literature [4-10]. But, QCD at the strong coupling will be a hard proble m, however Malda-
+cena conjecture make it easy. According to the Maldacena conjec ture there is the relation
+between type IIB string theory in AdS5×S5space and N=4 super Yang-Mills (SYM) gauge
+theory on the 4-dimensional boundary of AdS5space. So, this duality will be a candidate
+for the studying strongly coupled plasma. In that case, instead of a heavy quark in the
+∗Email: pouriya@ipm.ir
+†Email: b.pourhassan@umz.ac.ir
+1gauge theory, one may consider dual picture of the heavy quark w hich is an open string in
+AdS space. Also the dual picture of temperature in the gauge theo ry is a black hole (black
+brane) in AdS5space. One of the important fields in the QCD, which is also interesting to
+experiment in the LHC and RHIC [11], is consideration of the moving hea vy quark through
+theN=4 Super Yang-Mills thermal plasma [12-18]. Recently, the same calcu lations are done
+forN=2 supergravity thermal plasma [19]. This subject is important beca use the solutions
+of supergravity theory with N=4 andN=8 supersymmetry may be reduced to the solutions
+of theN=2 supergravity. In Refs. [19] we found that the problem of drag f orce in the N=2
+supergravity thermal plasma at zero non - extremality parameter and finite chemical poten-
+tial [20] is corresponding to the N=4 SYM plasma for heavy quark. Another interesting
+problem is to consider a q¯qpair which may be interpreted as a meson. As we know the
+problem of celebrated Regge behavior of the hadron spectra has b een discussed in literature
+[21]. Also the meson spectrum obtained so far and reasonably descr ibing experiment can be
+seen in the Ref. [22], where it is found that the angular momentum play s an important role
+to obtain the meson spectrum. So, this give us good motivation to co nsider the rotating
+meson.
+Already, the energy of the moving q¯qpair through the N=4 SYM plasma is studied in both
+rest frames of thermal plasma and q¯qpair, which relate to each other by a Lorentz transfor-
+mation [23, 24, 25]. Authors in [25] found that the heavy meson in the plasma feels no drag
+force. Considered system in that paper was an ideal case. Actually , theq¯qpair may have
+more degrees of freedom such as the rotational motion around th e center of mass and the
+oscillation along the connection axis. In the Refs. [26, 27] the descr iption of quark-antiquark
+system instead single quark well explained. Also the problem of the sp inning open string
+(meson) in description of the non-critical string/gauge duality [28] considered. In that paper
+the relationship between the energy and angular momentum of spinn ing open string for the
+Regge trajectory of mesons in a QCD-like theory is studied [29, 30]. I t is important to note
+that, inN=4 SYM there is no dynamical quark hence no dynamical mesons, the refore the
+rotating mesons are non-dynamical in N=4 SYM.
+Already a rotational motion for the q¯qpair considered, then the momentum densities for
+such a system calculated [31]. In that case their method was differen t with Refs. [28] and
+[32]. In the Ref. [32] authors considered a rotating quark and calcu lated drag force on
+a test quark moving through the plasma. In order to obtain the dra g force one needs to
+calculate the components of the energy-momentum density Π1
+Xand Π1
+Y. We assumed that
+q¯qpair moves at a constant speed valongXdirection, and also rotates around the center of
+mass. In [31] authors obtained the effect of the rotational motion in the energy-momentum
+components of the heavy q¯qpair with the non-relativistic velocity. Therefore, they are de-
+termined the motion of the heavy q¯qpair more exactly. As we know, in the case of the single
+quark, the energy of quark goes from D-brane in to the black hole. In the case of quark and
+antiquark the currents of energy and momentum coming from two p oints on D-brane. These
+components cancel each other on the top of the string and there fore effectively the heavy q¯q
+pair feels no drag force, so the string can save its shape in the quar k-gluon plasma. In this
+paper we would like to consider the effect of constant electromagne tic field on the motion of
+rotating heavy meson through the N=4 SYM plasma. Also we are going to use results of
+2the [31] to calculate jet-quenching parameter, which is one of the in teresting properties of
+the strongly coupled plasma [33-38]. The jet-quenching parameter supplies a measurement
+of the dispersion of the plasma. This quantity commonly obtained by u sing perturbation
+theory, but in here by using the AdS/CFT correspondence we find j et-quenching parameter
+in non-perturbative quantum field theory.
+This paper organized as the following. In the section 2 we review some important results of
+the Ref. [31] for completeness. In the section 3 we add an externa l constant electromagnetic
+field and obtain the motion of the rotating string. In the section 4 we calculate the jet-
+quenching parameter in this system and finally in the section 5 we summ arized our results
+and give some suggestions for future works.
+2 Rotating String Equation of Motion
+The dual picture of a meson in the CFT is a string in the AdS space which both endpoints
+of it live on the D-brane. This configuration illustrates the strong int eraction between two
+quarks due to a tube of gluon field and describes the confinement me chanism in QCD. In the
+classical level, these states can be regarded as the rotation of th e system. In another word
+we will dealing with the spinning open string. The spinning string is intere sting because it is
+dual picture of the rotating meson. In the Fig.1 we show configurat ion of the rotating string
+in the AdS space.
+Figure 1: A rotating ∩- shape string dual to a q¯qpair which can be interpreted as a meson.
+The points AandBrepresent quark and antiquark with separating length l. The radial
+coordinate rruns from rh( black hole horizon radius) to r=rm(r=∞)on theD-brane.
+Thercis the critical radius, obtained for single quark solution. The rminis turning point of
+3string and one can obtain rmin≥rc. The parameter θis assumed to be the angle with Y
+axis and the string center of mass moves along Xaxis at velocity v.
+From Maldacena dictionary, we know that, adding temperature to t he system is equal to
+the existence of a black hole in the center of AdSspace. For the dual picture of N=4 SYM
+plasma there is the AdS5black hole solution which is given by [25],
+ds2=1√
+H(−hdt2+d/vector x2)+√
+H
+hdr2,
+h= 1−r4
+h
+r4,
+H=R4
+r4, (1)
+whereRandrhare curvature radius of AdS space and radius of black hole horizon r espec-
+tively, also the motion direction described by /vector x: (X,Y,Z). So we choose motion axis as X
+andY(X-Y plan, Z= 0).
+We know the dynamics of the open string is described by the Nambu-G oto action,
+S=T0/integraldisplay
+dtdrL, (2)
+where we used static gauge ( σ=randτ=t). Therefore the lagrangian density of system
+is given by,
+L=−√−g
+=−/bracketleftbigg
+1+h
+H(X′2+Y′2)−1
+h(˙X2+˙Y2)−1
+H(˙X2Y′2+˙Y2X′2−2˙XX′˙YY′)/bracketrightbigg1
+2
+,(3)
+where prime and dot denote derivative with respect to randtrespectively, also g≡detgab,
+wheregabis the metric on the world sheet of the string. It is most general lagr angian for
+the string which its endpoints lie on D-brane. In that case one can obtain the following
+expressions for the energy and momentum density components,
+
+Π0
+XΠ1
+X
+Π0
+YΠ1
+Y
+Π0
+rΠ1
+r
+Π0
+tΠ1
+t
+=−T0
+H√−g
+˙YY′X′−˙XY′2−H
+h˙X˙X˙YY′−X′˙Y2+hX′
+˙XX′Y′−˙YX′2−H
+h˙Y˙X˙YX′−Y′˙X2+hY′
+H
+h(˙XX′+˙YY′)−H
+h(˙X2+˙Y2−h)
+h(X′2+Y′2+H
+h)−h(˙XX′+˙YY′)
+.(4)
+4In order to obtain the total energy, total momentum and angular momentum of the string
+we use the following relations respectively,
+E=−/integraldisplayr0
+rmindrΠ0
+t,
+PX=/integraldisplayr0
+rmindrΠ0
+X,
+PY=/integraldisplayr0
+rmindrΠ0
+Y,
+J=/integraldisplayr0
+rmindr(XΠ0
+Y−YΠ0
+X). (5)
+ItispossibletostudyReggetrajectorybycalculationofE2
+J. Inthatcaseweshoulddetermine
+the motion of string, it means that we should find explicit expressions forx(r) andy(r).
+Now, we are in the place of applying rotational motion to the string, s o we consider the
+following anstaz as solution of the equation of motion,
+X(r,t) =vt+xsinωt,
+Y(r,t) =ycosωt, (6)
+with constants linear and rotational motion which are denoted by vandωrespectively. we
+chooseθ(t) =ωtas an angle with Yaxis (see Fig. 1).
+Here, we consider the special case of small velocities to find moment um densities. So, we
+consideramovingheavy q¯qwithnon-relativisticspeed v,whichrotatebyangle θ=ωtaround
+the center of mass, therefore we have ω≪1. It is corresponding to motion of the heavy
+meson with large spin. Indeed in the very large angular momentum limit a semiclassical
+approximation is reliable. In this case, as mentioned above, the angu lar velocity of the
+string is very small and the separation value of quark and antiquark is very large. In this
+limits one can obtain,
+√−g≈/bracketleftbigg
+1−v2
+h+h
+Hx′2sin2ωt+h−v2
+Hy′2cos2ωt/bracketrightbigg1
+2
+, (7)
+and from equation (4) one finds the following expression of momentu m currents,
+/parenleftbigg
+Π1
+X
+Π1
+Y/parenrightbigg
+=−T0
+H√−g/parenleftbigg
+hx′sinωt
+(h−v2)y′cosωt/parenrightbigg
+, (8)
+where√−gis given by the equation (7). By using equations (7) and (8) one can o btain the
+following expressions,
+x′sinωt=H(h−v2)
+hΠ1
+X/radicaligg
+1
+(h−v2)(hT2
+0−HΠ1
+X2)−hHΠ1
+Y2,
+y′cosωt=HΠ1
+Y/radicaligg
+1
+(h−v2)(hT2
+0−HΠ1
+X2)−hHΠ1
+Y2. (9)
+5Then one can fix momentum densities by using the conditiony′
+x′= cotωtatr=rmin[31].
+The reality condition for single quark solution yields to the velocity-de pendent critical radius
+[12, 13, 16, 19],
+rc=rh
+(1−v2)1
+4. (10)
+By using the reality condition in equation (9) for the quark-antiquar k system one can find
+special radius, rmin, where functions xandyare not imaginary. Then, by using square root
+quantity in (9) one can obtain,
+rmin=
+r4
+h+b
+2T2
+0(1−v2)
+1−/radicaligg
+1−4T2
+0(1−v2)v2r4
+hR4Π1
+X2
+b2
+
+1
+4
+,(11)
+where we define,
+b≡R4(1−v2)Π1
+X2+R4Π1
+Y2+T2
+0v2r4
+h. (12)
+Indeed, the rminis the turning point of the string. It is easy to check that rmin≥rc. If
+we consider Π1
+X= 0 (l= 0) the special case of rmin=rcwill be satisfied. We note here
+rmin=rccorrespond to the single quark solution [25]. Also one can see that if v= 0 then
+rmin=rhwhich is expected. The turning point rminis an important parameter to determine
+the jet quenching parameter, so we will use relation (11) in the sect ion 4.
+In the next section we addan electromagnetic field to the backgrou nd and will obtain motion
+of the rotating meson.
+3 Effect of the constant electromagnetic field
+Intheprevioussectionweconsideredtherotatingmesonthrough thethermalplasmawithout
+any external field. Now, we introduce a constant electromagnetic field in the background.
+The constant electromagnetic field assumed along the XandYdirections. The constant
+electric field and the constant magnetic field denoted by B01andB12respectively. This
+procedure for the single quark in the N=4 SYM theory originally studied in [39]. In this
+configuration the lagrangian density (3) changes to,
+−g= 1+h
+H(X′2+Y′2)−1
+h(˙X2+˙Y2)−1
+H(˙X2Y′2+˙Y2X′2−2˙XX′˙YY′)
+−/parenleftig
+B01X′+B12(˙XY′−˙YX′)/parenrightig2
+. (13)
+Therefore the equations of motion read as,
+∂
+∂r/bracketleftigg
+1√−g/parenleftigg
+X′˙Y2−˙X˙YY′−hX′
+H+(B01−B12˙Y)/parenleftig
+B01X′+B12(˙XY′−˙YX′)/parenrightig/parenrightigg/bracketrightigg
++1√−g∂
+∂t/bracketleftigg˙X
+h+˙XY′2−X′Y′˙Y
+H+B12Y′/parenleftig
+B01X′+B12(˙XY′−˙YX′)/parenrightig/bracketrightigg
+= 0,(14)
+6and
+∂
+∂r/bracketleftigg
+1√−g/parenleftigg
+Y′˙X2−˙X˙YX′−hY′
+H+B12˙X/parenleftig
+B01X′+B12(˙XY′−˙YX′)/parenrightig/parenrightigg/bracketrightigg
++1√−g∂
+∂t/bracketleftigg˙Y
+h+˙YX′2−X′Y′˙X
+H−B12X′/parenleftig
+B01X′+B12(˙XY′−˙YX′)/parenrightig/bracketrightigg
+= 0,(15)
+forXandYrespectively. In this case the string motion described by the followin g anstaz,
+X(r,t) =vxt+xsinωt,
+Y(r,t) =vyt+ycosωt, (16)
+Therefore the equation (7) extends to the following relation,
+−g≈1−v2
+x+v2
+y
+h+h
+H(x′2sin2ωt+y′2cos2ωt)+1
+H(x′vysinωt−y′vxcosωt)2
+−[B01x′sinωt+B12(y′vxcosωt−x′vysinωt)]2(17)
+Also the energy and momentum currents in the small velocities appro ximation obtained as
+the following,
+Π0
+X=vx
+h+1
+H(vxy′2cos2ωt−vyx′y′sinωtcosωt)
++B12y′cosωt(B01x′sinωt+B12(vxy′cosωt−vyx′sinωt))
+Π0
+Y=vy
+h+1
+H(vyx′2sin2ωt−vxx′y′sinωtcosωt)
++B12x′sinωt(B01x′sinωt+B12(vxy′cosωt−vyx′sinωt))
+Π1
+X=−h
+Hx′sinωt+1
+H(x′v2
+ysinωt−vxvyy′cosωt)
++ (B01−B12vy)(B01x′sinωt+B12(vxy′cosωt−vyx′sinωt))
+Π1
+Y=−h
+Hy′cosωt+1
+H(y′v2
+xcosωt−vxvyx′sinωt)
++B12vx(B01x′sinωt+B12(vxy′cosωt−vyx′sinωt)), (18)
+up to the factorT0√−g. Now one may consider two special cases as the following. In the
+simplest case we can choose vx=vandvy= 0 which is corresponding to the existence of
+only electric field, ie. B12= 0. Then one may choose the case of only magnetic field which
+impliesB01= 0.
+In this paper we consider the case of B12= 0, then for small velocities one can obtain,
+x′sinωt=(h−v2)/radicalbig
+h(h−HB2
+01)HΠ1
+X/radicalig
+(h−v2)[(h−HB2
+01)T2
+0−HΠ1
+X2]−(h−HB2
+01)HΠ1
+Y2,
+y′cosωt=/radicalbigg
+h−HB2
+01
+hHΠ1
+Y/radicalig
+(h−v2)[(h−HB2
+01)T2
+0−HΠ1
+X2]−(h−HB2
+01)HΠ1
+Y2,(19)
+7which are extension of the equation (9) to the case of the existing a constant electric field in
+the background. In that case, similar to the Ref. [31], one can find t he following expression
+for the momentum densities,
+Π1
+X=(h(rmin)−H(rmin)B2
+01)tan2ωt/radicalig
+H(rmin)(h(rmin)−v2)+h(rmin)−H(rmin)B2
+01
+H(rmin)tan4ωt,
+Π1
+Y=/radicaltp/radicalvertex/radicalvertex/radicalbth(rmin)−v2
+H(rmin)+h(rmin)−H(rmin)B2
+01
+H(rmin)(h(rmin)−v2)tan4ωt. (20)
+Now by using reality condition one can obtain,
+rmin=/bracketleftigg
+r4
+h+d
+2T2
+0(1−v2)/parenleftigg
+1−/radicalbigg
+1−4T2
+0(1−v2)R4c
+d2/parenrightigg/bracketrightigg1
+4
+, (21)
+where,
+d≡R4(1−v2)(Π1
+X2+T2
+0B2
+01)+R4Π1
+Y2+T2
+0v2r4
+h,
+c≡v2r4
+h(Π1
+X2+T2
+0B2
+01)+R4B2
+01Π1
+Y2. (22)
+We can see that the effect of the constant electric field is increasing of the radius rmin. Again
+in the case of the B01= 0 we recover the relation (11) and in the case of B01= 0 andv= 0
+we obtain rmin=rhwhich is expected. In the next section we will use relations (11) and
+(21) to obtain the jet-quenching parameter.
+4 Calculation of the Jet-Quenching Parameter
+In ultra-relativistic heavy-ion collisions at LHC or RHIC, interactions between the high-
+momentum Parton and the quark-gluon plasma are expected to lead to jet energy loss,
+which is called jet quenching. It is known that the non-perturbative definition of the jet-
+quenching parameter may be obtained in terms of light-like Wilson loop [3 3]. In that case
+we use results of [33-38] to obtain the jet-quenching parameter in our system. At the first we
+calculate the jet-quenching parameter without any external field . Then we add a constant
+electric field to the system and obtain the jet-quenching paramete r. In order to find the
+jet-quenching parameter, first by introducing the new coordinat esx±=1√
+2(t±x1) , we
+rewrite the line element (1) in the light-cone coordinates,
+ds2=1−h
+2√
+H[(dx+)2+(dx−)2]−1+h√
+Hdx+dx−+1√
+H[(dx2)2+(dx3)2]+√
+H
+hdr2.(23)
+We choose word-sheet coordinates as r(x−,/vector x) and use static gauge x−=τwith length L−
+and/vector x=σwith length L. Because of condition L−≫Lwe can assume that the coordinates
+8ris independent of τand one can consider r(σ) as world-sheet coordinates, so there is
+boundary condition as r(±l
+2) =∞. Other coordinates assuming to be constant. Now,
+definition of the jet-quenching parameter is given by [33],
+ˆq≡8√
+2S−S0
+L−L2, (24)
+whereSobtained from action (2) by using above definitions, so one can obta in,
+S=√
+2T0L−/integraldisplay∞
+rmindr
+r′/radicalbigg
+(1−h)(1
+H+r′2
+h), (25)
+whererminis given by the equation (11). In order to remove r′in the above expression we
+use energy conservation law H=L−∂L
+∂r′r′=Const.≡E, which yield us to the following
+relation,
+r′2=h
+H(r4
+h
+2E2R4−1), (26)
+which implies that E2≤r4
+h
+2R4. We are interested in low energy limit where E≪1. This limit
+is corresponding to L→0 limit which is agree with L≪L−. The action S0in the equation
+(24) interpreted as self-energy of the quark and antiquark which is given by [38],
+S0= 2T0L−/integraldisplay∞
+rmindr√−g−−grr, (27)
+whererminis given by the relation (11). Inserting (26) in to the action (25) and expanding
+for infinitesimal Eyields to the following relation,
+S−S0=√
+2T0L−E2R4
+r2
+h/integraldisplay∞
+rmindr/radicalbig
+r4−r4
+h. (28)
+On the other hand one can integrate (26) and obtain the following re lation between Land
+constant E,
+L
+2=√
+2ER4
+r2
+h/integraldisplay∞
+rmindr/radicalbig
+r4−r4
+h= (√
+2ER4
+rminr2
+h)2F1[1
+4,1
+2;5
+4;r4
+h
+r4
+min]. (29)
+Inserting equations (28) and (29) in the equation (24) leads us to t he expression for the
+jet-quenching parameter,
+ˆq=2T0
+R4rminr2
+h
+2F1[1
+4,1
+2;5
+4;r4
+h
+r4
+min]. (30)
+It is easy to compare our result with the previous work. In the case ofv=ω= 0 one
+can obtain rmin=rhand therefore hypergeometric function reduced to the gamma fu nction
+as2F1[1
+4,1
+2;5
+4;r4
+h
+r4
+min]rmin=rh=√πΓ(5
+4)/Γ(3
+4), then by using relations rh=πR2T,R2=α′√
+λ
+andT0=1
+2πα′we recover the famous relation of the jet-quenching parameter in N=4 SYM
+9theory in the large Ncand large- λlimits, ˆq=π2
+a√
+λT3, wherea≈1.311. However in the
+case of rotational motion the value of the jet-quenching paramet er increases.
+Now, we consider a two form B01dt∧dx1. In that case one can obtain,
+S−S0=√
+2T0L−E2R4I, (31)
+where
+I=/integraldisplay∞
+rmindr/radicalbig
+(r4−r4
+h)(r4
+h−B01R2r2), (32)
+and the radius rminis given by the equation (21). Also one can obtain,
+r′2=h
+H(r4
+h−B01R2r2
+2E2R4−1). (33)
+Therefore the jet-quenching parameter for the rotating heavy meson through the N=4 SYM
+thermal plasma in a constant electric field obtained as the following re lation,
+ˆq=2T0
+R4I−1. (34)
+In order to obtain the explicit expression of the jet-quenching par ameter, including the
+electricfield, weassume thattheelectric fieldisinfinitesimal paramet er. Fortheinfinitesimal
+electric field one can obtain,
+ˆq=π
+α′T2rmin
+2F1[1
+4,1
+2;5
+4;r4
+h
+r4
+min]
+1−rmin
+2π2α′√
+λT2B01
+2F1[1
+4,1
+2;5
+4;r4
+h
+r4
+min]/integraldisplay∞
+rminr2dr/radicalbig
+r4−r4
+h
+.(35)
+It means that the effect of the constant electric field is decreasing of the jet-quenching
+parameter. This is agree with the fact that electric field decreases the drag force [19].
+Therefore we calculated the jet-quenching parameter in rotating meson system under effect
+of the constant electric field. It is easy to check that the above re sults are coincide with the
+previous work, without electric field, if we cancel the electric field, ie .B01= 0.
+5 Conclusion
+In this paper we generalized the problem of the rotation meson in the N=4 SYM thermal
+plasma to the case of the existing constant electromagnetic field. W e considered the con-
+stant electromagnetic field in the background and obtained the lagr angian density for the
+small velocities. Then we calculated the effect of the constant elect ric field on the motion
+of the rotating meson and obtained the momentum densities. We hav e shown that in the
+presence of a constant electric field the distance of the turning po int from the D-brane is
+smaller than the case of without the electric field. Then, without pre sence of any external
+field we obtained the jet-quenching parameter for the rotating me sons in terms of the hy-
+pergeometric function and have shown that our results for the ω= 0 are coincide with the
+10case of non-rotating meson. We found that the rotating mesons h ave larger jet-quenching
+parameter than non-rotating mesons. Finally we calculated the jet -quenching parameter
+under effect of a constant electric field. We found that the electric field decreases the value
+of the jet-quenching parameter.
+Here, there are some interesting problem for future works. For e xample one can obtain jet
+quenching parameter [33-38] for rotating q¯qpair or shear viscosity [40] in other backgrounds
+such asN=2 supergravity [19]. Also it is interesting to consider the effect of hig her deriva-
+tive terms as in the previous cases [41-49]. As a recent work [50] one may consider more
+quarks, such as four quarks in the baryon through N=4 SYM thermal plasma. At the end,
+it may be interesting to consider fluctuations of the quark-antiqua rk pair and obtain the
+exact solution of such a system.
+Acknowledgment: We would like to thanks Dr. K. Bitaghsir Fadafan for useful discussio n
+about the jet-quenching parameter.
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+0812(2008)019, (hep-th/0809.5143).
+14
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+arXiv:1001.0707v3  [hep-ph]  27 Jul 2011November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+International Journal of Modern Physics A
+c/circlecopyrtWorld Scientific Publishing Company
+On the thermal phase structure of QCD at vanishing chemical
+potentials
+Sonia Kabana
+SUBATECH, Ecole des Mines, 4 rue Alfred Kastler, 44307 Nante s, France.
+Peter Minkowski
+Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of
+Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland, and CERN , Theoretical Physics, CH-1211
+Geneva 23, Switzerland.
+Received Day Month Year
+Revised Day Month Year
+The hypothesis is investigated, that the thermal structure of QCD phases at and near
+zero chemical potentials is determined by long range cohere nce, inducing the gauge boson
+pair condensate, and its thermal extension, representing a fundamental order parameter.
+A consistent model for thermal behaviour including interac tions is derived in which the
+condensate does not produce any latent heat as it vanishes at the critical temperature
+inducing a second order phase transition with respect to ene rgy density neglecting even-
+tual numerically small critical exponents. Localization a nd delocalization of color fields
+are thus separated by a unique critical temperature.
+Keywords : QCD; quark-gluon plasma; quantum phase transitions.
+PACS numbers: 12.38.-t,12.38.Mh,05.30.Rt
+1. Introduction
+The existence and nature of the QCD phase transition is theoretica lly accessible
+to a universal description underlying a thermal system with all its ge neral and
+specific restrictions. At the moment key theoretical questions co ncerning the phase
+structure of thermal systems in QCD remain. Lattice QCD calculatio ns lead to
+very clear predictions, establishing the lack of any phase transition at zero chemical
+potentials1,2. The validity of this method and its results is not uncontroversial, fo r
+example see Refs.3,4.
+There is a longstanding experimental effort to measure character istic signatures
+of the phase transition in collisions of heavy ions at several facilities a t AGS at
+BNL, SPS at CERN, RHIC at BNL, LHC at CERN and in the future FAIR a t
+GSI. The present state of these studies yields significant results, well interpretable
+as a strongly interacting partonic matter formed in the earlier stag es of the heavy
+ion collisions5,6. These results suggest the existence of a phase transition in the
+1November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+2Sonia Kabana, Peter Minkowski
+course of the collision. Assuming the correctness of this conclusion , the detailed
+phase structure and order(s) of assocated transition(s) could not be determined
+conclusively so far .
+The experimentally measured reactions cover a large range of collisio n energies and
+chemical potentials in thermal analyses. In particular, with increas ing energies the
+baryochemical potential in hadron-hadron or heavy ion collisions de creases. For a
+comparison reducing these systems to a common baryochemical po tential we refer
+to7,8.
+In this paper we propose to give an ”outline in principle” of the therma l phases of
+QCD at vanishing chemical potentials and their relation to the gauge b oson pair
+condensate of QCD. We proceed to show that the two phases, sep arated by the
+critical temperature Tcratµ= 0 can indeed be understood as a dynamical conse-
+quence ofuniquely the bosonicpair condensateat nonzeroquark m asses,in contrast
+to supersymmetric models with no spontaneous breaking of supers ymmetry.
+The two phases are represented approximatively by a collection of noninteracting
+hadrons for 0 < T < T crand a collection of 8 gauge bosons and 4 flavours of quarks
+and antiquarks (u,d,s,c & anti) for T > T cr, in which interactions are modeled.
+We use here initially two collections of hadron resonances , denoted N type=65
+and Ntype=26 in the following. The Ntype=65 collection, on which most of the
+subsequent calculations of thermodynamic state functions rely, is an extension of
+the smallerNtype=26 one. In the former ( Ntype=65 ) the 4 SU3flmeson nonets
+withJPCn= 0++,1++,2++; 1+−inthelowestmassp-wawe qq′configuration
+areadded,aswellasthelowestmassopencharmandanticharms-w avepseudoscalar
+and vector mesons with composition qcandcqand baryons and antibaryons
+containing one charmed quark and their antiparticles together with overall s-wave
+composition qq′candqq′cwithq,q′=u,d,sare added, together with 3
+representative color gauge boson binaries (glueballs) with JPC= 0++,0−+,2++
+and two hybrid nonets of composition q gq′withJPCn= 1−+.
+The adopted approximation allows to illustrate the thermal phase pr operties in
+principle.
+2. Energy momentum density tensor and gauge boson condensat e
+We begin with the local, symmetric and conserved energy momentum d ensity ten-
+sorϑµ νexisting and renormalized as a basic consequence of QCD . The above
+properties imply exact Poincar´ e invariance.
+{ϑµ ν=ϑν µ}(x) ;∂νϑµ ν= 0 (1)
+The central defining quantities, of the existence and order of the phase transition,
+are pressure, energy density and entropy density for zero chem ical potentials. TheNovember 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+On the thermal phase structure of QCD at vanishing chemical p otentials 3
+localoperator form of the trace anomaly in QCD reads:
+ϑµ
+µ=/summationtext
+fmfS˙ff(x)+δ0(x)
+δ0=/parenleftbig
+−2β(g)/ g3/parenrightbig/bracketleftbig
+−1
+4(:)FA
+µν(x)Fµν A(x)(:)/bracketrightbig
+s.d.
+=−2b0/bracketleftbig1
+4(:)FA
+µν(x)Fµν A(x)(:)/bracketrightbig(2)
+In Eq. 2 the subscript s.d.stands for ”renormalization scale dependent”, while b0=
+9/(16π2). All quantities are defined to be renormalized and renormalization g roup
+invariant, except those with the subscript s.d. β(g) denotes the (Callan-Symanzik-)
+rescaling function, where gis the coupling constant.1
+4(:)FA
+µ νFµ ν A(:) is the
+localscalar density – to be called localgauge boson bilinear – composed bilinearly of
+field strength tensors FA
+µ ν– the latter including multiplicatively the gauge coupling
+constantin theirdefinition, relativeto the perturbativefield stren gth normalization.
+Adenotes the color component within the adjoint representation of SU(3)c. (:)(:)
+stands for a suitablenormal ordering. mfdenotes the mass of quark flavour f : S˙ff
+denotes the localscalar density of antiquark ˙fand quark fflavors :
+S˙ff(x) = (:)q˙f˙cs(x)qs
+fc(x) (:), where s denotes the spin.
+The focus is to consider vacuum expected values of the localoperators in Eq. 2,
+which by translation invariance are independent of the position x, suppressed in
+the following
+ηµν/an}bracketle{tΩ|ϑµν(x)|Ω/an}bracketri}ht=−2b0/an}bracketle{tΩ|1
+4(:)FA
+µνFµνA(x)(:)|Ω/an}bracketri}ht
++mf/an}bracketle{tΩ|S˙ff(x)|Ω/an}bracketri}ht(3)
+/an}bracketle{tΩ|(:)1
+4FA
+µνFµν A(:)|Ω/an}bracketri}ht=B2shall be called the gauge boson pair condensate ,
+abbreviated by B2.
+/an}bracketle{tΩ|S˙f f|Ω/an}bracketri}ht /ne}ationslash= 0 induces spontaneous chiral symmetry breaking and is gen-
+erally called quark condensate.
+In connection with normal ordering ambiguities it is important to admit in the
+precise form of the energy momentum tensor a nontrivial vacuum e xpected value ,
+which as a consequence of exactPoincar´ e invariance must be of the form
+/an}bracketle{tΩ|ϑµν(x)|Ω/an}bracketri}ht=−ηµνpvac/braceleftbig
+ηµν=diag(1,−1,−1,−1) ;pvac=−ρvac/bracerightbig
+independent of x→
+∆ϑµν(x) =ϑµν(x)−/an}bracketle{tΩ|ϑµν(x)|Ω/an}bracketri}ht |Ω/an}bracketri}ht /an}bracketle{tΩ|
+with∂ν∆ϑµ ν(x) = 0 ; /an}bracketle{tΩ|∆ϑµν(x)|Ω/an}bracketri}ht= 0(4)
+In Eq. 4 PΩ=|Ω/an}bracketri}ht /an}bracketle{tΩ|denotes the projector on the ground state .
+From the two local, conserved tensors in Eq. 4 only ∆ ϑµν(x) with vanishing vacuum
+expected value is acceptable as representing the conserved 4 mom entumoperators
+and their densities yielding the integral form
+/hatwidePµ=/integraldisplay
+td3x∆ϑµ0(t,/vector x) (5)November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+4Sonia Kabana, Peter Minkowski
+We use here throughout strictly thermal , ’extension in phase spac e’ associated
+potentials, depending in subtle ways on vacuum condensates. To th ese potentials
+no vacuum associated spontaneous parameters like pvac=−ρvac, defined in Eq.
+4, contribute in a direct way, dominating in the limit T→0 ;µα≡0 .
+From Eqs. 3 and 4 we obtain the relation and estimates ’pour fixer les id ´ ees’
+pvac=9
+32π2B2+1
+4Λ =/braceleftbigg0.00302 GeV4
+0.00658 GeV4
+B2=/braceleftBigg
+0.125 GeV49
+0.250 GeV410
+Λ =−/summationtext
+fmf/an}bracketle{tΩ|S˙ff|Ω/an}bracketri}ht
+∼f2
+π(1
+2m2
+π+m2
+K) = 0.00217 GeV4(6)
+3. Construction of a thermal model including interactions
+We follow the strategy layed out in Refs.7,8taking into account the modifications
+described above, distinguishing two eventual phases
+1) the hadronic (h)-phase , with color localized within stable hadrons and selected
+hadron resonances corresponding to the two collections Ntype=6 5⊂26.
+Thermal potentials of the (h)-phase are approximated by those o f free
+hadrons, neglecting decay widths, as described in Refs.7,8.
+2) the quark-antiquark-gauge boson (qg)-phase, wherein ther mal potentials are
+related but not equal to those of free quarks and antiquarks, re stricted to
+the flavors u,d,s and c, and 8 gauge bosons pertaining to the gauge g roup
+SU3c. Next we describe the modeling of interactions in the (qg)-phase,
+which deviates from noninteracting constituents assumed in Refs.7,8.
+We introduce for later use for the (qg)-phase , the Gibbs density g(0)
+qgand energy
+density ̺(0)
+e qgpertaining to noninteracting tricolored quarks , antiquarks with
+flavors
+u , d , s , c and eightfold colored gauge bosons
+g(0)
+qg(T) =/summationtext
+αqgwαqg/parenleftbig
+1/(2π2)/parenrightbig/integraldisplay∞
+mαqgl E p d E
+wαqg=/parenleftbig
+2spinαqg+1/parenrightbig/braceleftbigg3 forq,q
+8 forg=/braceleftbigg6 forq,q
+16 forg
+β≡1/T;l=∓log[1∓exp (−βE)]
+̺(0)
+e qg(T) =−(d/dβ)g(0)
+qg(T) =T2(d/dT)g(0)
+qg(T)(7)
+In Eq. 7 the index αqgruns over the different constituents of the (qg) phase, while
+wαqgdenotes the multiplicity beyond momentum phase space associated w ith the
+constituent αqg. The sign ( ∓) in the expression for lis - for bosons and + for
+fermions .November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+On the thermal phase structure of QCD at vanishing chemical p otentials 5
+We choose the following masses for quark flavors u, d, s, c
+[mu= 0.00525,md= 0.00875
+ms= 0.175,mc= 1.27 ] GeV(8)
+The absolute masses of the u,d,s light flavors as well as their ratios mu:md:
+ms= 3 : 5 : 100 ,11,12,13, used here play no decisive role in the present
+derivations, within generous ranges of ±20 % .
+The inclusion of the charmed quark serves the purpose to check wh ether it has any
+significant influence on the thermal parameters in the region of Tcr∼0.2 GeV ,
+wich turns out to be in the few percent range .
+We proceed to modify the free quark antiquark gauge boson (qg-) parametrization
+of the Gibbs potential and the energy density, which for µα= 0 must obey the
+exact relation
+gqg=gqg(T≡β−1) ;−(d/dβ)gqg(T) =̺e qg(T)
+andgqg↔gh, ̺e qg↔̺e h(9)
+The Gibbs- and energy-densities in the hadron phase are construc ted from the
+expressions analogous to the ones given in Eq. 7 , where the index αqg→αhruns
+over a suitable choice of hadrons and hadron resonances as define d in Refs.7,8with
+real masses and neglecting interactions among these states .
+The ensuing parametrizationof interactions is understood as repr esentingthe phase
+structure in principle and not in numerical detail. The modeling of inter actions in
+the qg-phase is performed setting two parameters k ,∆g, independent of tem-
+perature, as approximately parametrizing the interaction in the de confined phase
+in a limited region of T≥Tcr
+̺e qg(T;k) =k ̺(0)
+e qg(T)
+gqg(T;k,∆g) =k g(0)
+qg(T)−∆g(10)
+The parameter 0 < k < 1 is taking into account the reduction of Gibbs
+density or pressure relative to the noninteracting (Stefan-Boltz mann) limit , noted
+in perturbative QCD calculations of thermal parameters for T≃Tcrof interest
+here14, while the second parameter ∆ garises as integration constant from
+the differential equation ( Eq. 9 ) , which is clearly satisfied for arbitr ary values of
+(k ,∆g) .
+We proceed in two steps to map out the structure of the phase tra nsition
+I : the condition determining Tcr↔k
+The equality of the energy densities – in the hadron phase ̺e hadforT≤
+Tcras outlined in Refs.7,8and in the qg-phase as defined in Eq. 10
+̺e qgforT≥Tcrdetermine the critical temperature
+̺e had(T) =̺e qg(T;k)↔T=Tcr(k) (11)
+The matching ( Eq. 11 ) is further restricted to yield the value
+Tcr∼0.2 GeV↔k∼0.365 (12)
+in accordance with the estimate of one of us16.November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+6Sonia Kabana, Peter Minkowski
+II : the condition avoiding singular behaviour of pressure gradient
+This condition implies
+gh(Tcr) =gqg(Tcr;k,∆g)↔∆g= ∆g(Tcr(k)) (13)
+and determines ∆ g
+∆g∼0.0062 GeV3= 0.775 fm−3 (14)
+4. Results and discussion
+We begin with a disussion with a comparison of the two colletions of hadr on reso-
+nances Ntype=65 and Ntype=26 whence thermal state functions are extrapolates
+beyond the temperature range 130 MeV ≤T≤170 MeV obtained in studies
+of chemical equilibrium of hadron yield ratios.
+It is expected that approaching a phase transition will require inter actions on the
+hadronicsideaswellasonthequark-gluonside.Themereextension ofthecollection
+of hadron resonances also beyond Ntype=65 here , without introd uction of interac-
+tions among the hadron resonances is unrealistic. If there is no pha se transition at
+all, this does not change the mentioned expectation.
+This is illustrated in Figs. 1 and 2 , in which the two noninteracting collect ions of
+hadron resonances, Ntype=65 and 26, are compared with respec t to the quantities
+dscale = ( ̺e−3p)/ T4and pressure / T4respectively .
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
+ 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
+( rho_{e} - 3 p ) / T^{4}  HRG Ntype 26 , 65
+T [GeV]Ntype=26Ntype=65
+Fig.1. The thermal quantity representing the trace anomaly – dscale ( T) = (̺e−3p)/ T4
+is shown for 0 .1 GeV ≤T≤0.7 GeV . The comparison of HRG collections Ntype=65 and
+26 shows the sensitive temperature regions beyond temperat ures where chemical freeze out takes
+place – T∼150−170 MeV for Pb - Pb collisions at SPS and Au - Au collisions at RH IC.November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+On the thermal phase structure of QCD at vanishing chemical p otentials 7
+ 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5 5.75 6 6.25 6.5 6.75 7 7.25 7.5
+ 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.3 0.325 0.35 0.375 0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5
+p / T^{4} HRG Ntype 26 , 65
+T [GeV]Ntype=26 Ntype=653 fl q qbar and g limit
+S.Borsanyi et al.
+Fig. 2. Pressure / T4versus T in the range 0 .1 GeV≤T≤0.4 GeV , for the two collections
+Ntype=65 and 26 is compared with the results of lattice QCD by S. Borsanyi et al.28.
+As a consequence of the fair agreement of the pressure / T4as obtained from the
+noninteracting hadron resonance collection Ntype=65 with the latt ice simulation
+results of S. Borsanyi et al.1,28, as shown in Fig. 2 , we choose this collection in all
+subsequent calculations representing the hadron phase for T≤Tcr≃200 MeV .
+We mention the fact, that collectionsofhadronresonancesused in recentanalysesof
+especially ratios od hadron yields as observed in Au-Au collisions at RHI C include
+all resonances reported by the PDG15up to a mass of 2 GeV or 2.5 GeV , which
+are larger than the one defined Ntype=65 here. In Ref.1the agreement up to
+T∼170 MeV of all thermal state functions with the HRG quantities of th ese
+large collections is very satisfactory .
+The restriction to noninteracting hadron resonances neverthele ss has a limited va-
+lidity particularly at temperatures above chemical freeze out as T∼200 MeV .November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+8Sonia Kabana, Peter Minkowski
+The result of steps I and II above are illustrated in Figs. 3 - 7 below . F or a
+comparison with thermodynamic analyses in lattice QCD we refer to Re fs.17,
+18and1. The existence of two crossing points, one each for energy densit y and
+pressure, coincident in temperature at acceptable values of all pa rameters near
+Tcr≃0.2 GeV is nontrivial , shown in Fig. 3 below
+-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025
+ 0.1  0.12  0.14  0.16  0.18  0.2  0.22rho_{e} [ GeV^{4} ] ; g [ GeV^{3} ]
+T [ GeV ] 
+ kqg-rho_{e}=k=0.452
+Deltag = 0.00753T_{cr} = 0.190 GeVNtype = 65  g_{ had} ( T )
+  g_{ qg } ( T )
+  rho_{ e had} ( T )
+  rho_{ e qg } ( T )
+Fig. 3. The two thermal quantities energy density ̺eand Gibbs density g=p / Tin units of
+GeV4and GeV4respectively are decomposed into the hadronic (had) ( valid forT≤Tcr)
+and quark-gluon (qg) parts ( valid for T≥Tcr) each : ̺e had: blue curve ; ̺e qg: purple
+curve and ghad: red curve ; gqg: green curve . The two small circles mark the two equalities
+̺e had=̺e qgandghad=gqgcoincident in temperature T=Tc(≡Tcr) .
+In Figs. 4 - 6 the modifications of ̺e qg/T4,pqg/T4and dscale =
+(̺e qg−3pqg)/ T4introduce a subleading critical exponent ν;ν= 0.975
+used here throughout, in the vicinity of T=Tcrallowing the free quark-gluon limits
+to be reached for T→ ∞. The modified quantities are defined as
+/braceleftbigg̺ν
+e qg
+pν
+qg/bracerightbigg
+=/braceleftbiggfmod(ν , T / T c)̺e qg+fmod1 (ν , T / T c)pqg
+fmod(ν , T / T c)pqg/bracerightbigg
+fmod(ν , T / T cr) = 1+ |1−Tcr/ T|2ν( 1/ k−1 )
+fmod1 (ν , T / T c) =T(d / d T)fmod(ν , T / T c)(15)
+Assuming that all thermodynamic potentials are analytic functions o fTin a suffi-
+ciently large neighbourhood of Tcr, with the transition arising from appropriate
+conditional choices below and above Tcr, as is the case for the parametrizations
+adopted here, the orders of the so deduced phase transition ( fo rµα= 0 )
+becomeNovember 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+On the thermal phase structure of QCD at vanishing chemical p otentials 9
+order with respect to quantity
+2. ↔ energy density
+3. ↔ pressure
+1. ↔ sound velocity (16)
+We note the behaviour of the square velocity of sound. Steps I and II imply that
+the sound velocity (square) exhibits a 1. order transition at T=Tcr.
+Let the squares of the sound velocity in the hadron phase ( qg-pha se ), always for
+vanishing chemical potentials, be denoted respectively
+v2
+s had(qg)(T) =˙phad(qg)
+˙̺e had(qg); ˙ =d / d T (17)
+It follows that the sound velocity (square) jumps from a lower value in the hadron
+phase to a higher value in the qg-phase .
+For the qg-phase we have the relations
+pqg=T/parenleftBig
+k g(0)
+qg−∆g/parenrightBig
+, T g(0)
+qg=p(0)
+gq(T)
+˙p(0)
+qg=̺(0)
+s(T) :entropy density
+for noninteracting
+qg-phase
+−→v2
+qg=k˙p(0)
+qg−∆g
+k˙̺(0)
+e qg=/parenleftbig
+v2
+qg/parenrightbig(0)−∆g
+k˙̺(0)
+e qg(18)
+All quantities with superscript(0)refer to the noninteracting qg-phase . The
+introduction of interactions leads to a reduction of the qg-sound v elocity square
+proportional to ∆ g. This is however a small correction for allT > T cras
+shown in Fig. 4, also illustrating the 1. order transition of v2
+s. In the qg-phase
+the sound velocity square approaches the limiting value1
+3and is just after the
+transition already quite near this value. The behaviour of v2
+sis shown in Fig. 7 .November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+10Sonia Kabana, Peter Minkowski
+Fig. 4. The ( dominantly ) second order nature is displayed of the transition – unmodi-
+fied and modified on the quark-gluon-side by subleading criti cal exponents – of the quantity
+̺e(T)/ T4, piecewise described by ̺e hg(−65)/ T4forT≤Tcrfrom the HRG
+side and ρe qg no mod (mod)/ T4forT≥Tcrfrom the quark-gluon side as defined in Eq.
+15. The three curves are represented with dashed lines outsi de their range of validity.
+Fig. 5. The ( dominantly ) third order nature is displayed of t he transition under the same
+conditions as underlying Fig. 4 for the quantity p / T4, piecewise described by pe hg(−65)/ T4
+forT≤Tcrfrom the HRG side and pqg no mod (mod)/ T4forT≥Tcrfrom the quark-
+gluon side as defined in Eq. 15 . The three curves are represent ed with dashed lines outside their
+range of validity.November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+On the thermal phase structure of QCD at vanishing chemical p otentials 11
+Fig. 6. The ( dominantly ) second order nature of the transiti on is displayed with respect to the
+thermal quantity – dscale – forming the shape of an ’indian te nt’ . The same quantity as obtained
+in Ref.28from lattice simulation of QCD under the same thermal condit ions is also plotted for
+comparison . The quantity dscaleν
+qg modis obtained from Eq. 15 , here with ν= 0.975 , the
+same as in Figs. 4 - 5 . The three curves forming the ’indian ten t’ are represented with dashed
+lines outside their range of validity.November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+12Sonia Kabana, Peter Minkowski
+Fig. 7. The ( dominantly ) first order nature is displayed of th e transition with respect to the
+square of the velocity of sound , under the same conditions as for Figs. 4 - 6 , piecewise described
+by the quantities v2
+hg; (hg≡h) forT≤Tcrfrom the HRG side and v2
+qg no modfor
+T≥Tcrfrom the quark-gluon side . Only the unmodified setting is use d forv2
+gq. The two
+curves for v2
+hg, v2
+gqare represented with dashed lines outside their range of val idity.November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+On the thermal phase structure of QCD at vanishing chemical p otentials 13
+The derivations concerning the phase structure of QCD at vanishin g chemical po-
+tentials , laid out in the present discussion , follow within the hypothesis , that the
+gauge boson pair condensate is at the origin of stable embeddings of quark flavors
+– light and heavy – into the ( long range ) dynamics controled primarily by gauge
+boson self-interactions . By the same hypothesis – this embedding depends in a
+minimal way on quark masses , including their chiral as well as anti-chir al limits :
+mχ1,···mχM→0 ;mh1,···mhN→ ∞.
+Extensivelatticesimulationsofthe thermodynamicstate in particula rat zerochem-
+ical potential have been performed in recent years. With realistic q uark masses for
+the three light flavours u,d,s no phase transition but a cross over re gion is found.
+For a review we refer to Ref. [13].
+Forpuregaugetheorieswith gaugegroup SUNc,Nc>2a 1storderphasetransition
+and forNc= 2 a second order transition is found [14,15].
+The latter phase transition agrees with the results pertaining to th e 3-dimensional
+Ising model.
+The argument for – at least – a second order phase transition w ith
+respect to energy density
+In this subsection a short outline is given of how spontaneous param eters atT= 0
+andT >0 are obtained in general renormalizable local field theories.
+We define first in physical space-time, the effective action Γ for the composite clas-
+sical field ϕcl↔ϕassociated with the local gauge boson bilinear, introduced in Eq.
+2
+expiW=/an}bracketle{tΩ|exp/parenleftbigg
+i/integraldisplay
+d4x(L+Jϕ)(x)/parenrightbigg
+|Ω/an}bracketri}htc
+ϕ(x) =1
+4(:)FA
+µν(x)Fµν A(x)(:) ;J(x) : source(19)
+in the presence of a classical local source J.cin Eq. 19 denotes the connected part
+of the generating functionsal W. The effective action Γ( ϕc) is obtained from the
+generating functional Wthrough functional Legendre transform
+ϕcl(x) =δ/δJ(x)W(J)→
+Γ(ϕcl) =/integraldisplay
+d4x(ϕcl(x)J(x))−W;ϕcl=ϕcl(J)
+δ/δϕcl(x) Γ =J(x) ;ϕcl(x)|J→0=B2,(Eqs. 3,6)
+Γ(ϕcl) =/integraldisplay
+d4x v(ϕcl)(20)
+The (nonlocal) density functional vdefined in Eq. 20 determines the spontaneous
+ground state expected value of ϕclin the limit of vanishing sources as absolute
+minimum .
+It is essential to safeguard appropriate boundary and/or integr ability conditions for
+the sources J(x) which translate into corresponding conditions for the LegendreNovember 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+14Sonia Kabana, Peter Minkowski
+transforms ϕcl(x), e.g.
+limx→∞J(x) = 0↔/integraldisplay
+d4y|J|2(y)<∞ (21)
+as described in Ref.20. The functionals W ,Γ in Eqs. 19, 20 can be extended to
+Euclidean space. The absolute thermodynamic limit at zero temperat ure is reached
+from the boundary conditions in Eq. 21 for spontaneous vacuum pa rameters al-
+lowing sources and classical fields to approach alsoconstant values21, whereas
+thermal environment at finite temperature Tcorresponds to a finite Euclidean time
+extension
+∆tE=β= 1/T (22)
+In the finite temperature thermal limit new boundary conditions at t he bounding
+timestE= 0, βhave to be set, periodic for gauge invariant bosonic sources and
+classical fields, as considered above relative to the local operator ϕ(x) defined in Eq.
+19. Hence weface the twothermodynamic limits, treated herefor justthe associated
+quantities
+1
+4(:)FA
+µν(x)Fµν A(x)(:) =ϕ(x)→(ϕcl(x),J(x) )
+J(x) = ( 1/ g2)−( 1/ g2(x) ) =−∆1
+g2(x)(23)
+The external source J(x) represents a space time dependent coupling constant,
+before thermodynamic limits are taken.
+Absolute thermodynamic ( T≡0 ) - and thermal finite temperature ( Tfinite )
+limits yield the relations
+J(x)
+v(ϕcl(x))−→
+TJ∈[−J∗,J∗]→0
+v(ϕcl,T)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT:/braceleftBigg
+≡0
+finite
+∂ϕclv(ϕcl,T) =J→0→ϕcl=ϕcl(T)(24)
+Performing the sequence of limits and letting the (constant) values of the source J
+vary in a suitable interval [ −J∗,J∗] before relaxing it to zero, as shown in Eq. 24,
+defines the hysteresis line of the spontaneous parameter ϕcl(J→0,T). The latter
+emerges as thermal average for Tfinite and as vacuum expected value for T≡0
+(Eqs. 19-21).
+In theTfinite case we have the choice to include chemical potentials , one eac h for
+conserved quark flavor neutral currents, which we set to zero h ere.
+The phase transition underlying the present discussion is associate d – by hypothesis
+– with the vanishinging in a singular way at finite critical temperature, T=Tcr>0
+of the quantity ϕcl(T) defined in Eq. 24
+ϕcl(T)→0 forT→Tcr (25)November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+On the thermal phase structure of QCD at vanishing chemical p otentials 15
+Classical configurations leading to this behaviour inherit a ’Watt-less ’ nature from
+beeing particular to the ground state of QCD22,23and thus should not generate a
+step-like behaviour of thermal energy density . This is borne out in the subtraction
+of the ground state projection of the associated energy moment um density tensor
+ϑµνas shown in Eqs. 4,5.
+This however does not imply that the first derivative of energy dens ity changes by a
+finite amount through the transition, i.e. its corresponding genuine ly second order
+nature. The singularity could well be characterized by critical expo nents without
+relation to a definite step in a given derivative – of thermal energy de nsity.
+The mechanism of gauge boson pair condensation discussed here is a new analysis
+taking its roots in material presented in Refs.22,16and7,8in chronological order.
+It is centered on the thermal average of the field strength bilinear local density
+operator defined in Eq. 2
+1
+4(:)FA
+µν(x)Fµν A(x)(:) (26)
+The nonzero vacuum expected value ( Eq. 3 ) signals a connection be tween the
+localization of color and the structure of Bogoliubov transformatio ns as appropriate
+for gauge fields at long range.
+For QCD, actual calculations of effective potentials at large and inte rmediary range
+are not accessible to perturbation theory. Hence analytic contro l is lacking at
+present.
+5. Conclusions
+Our modeling of strong interactions in the qg-phase, in the vicinity bu t above the
+critical temperature Tcr∼0.2 Mev, as described between Eqs. 10 and 18, is sup-
+ported by observations of strong coupling in central Au-Au collision s at 200 GeV
+at RHIC5,27. The reduction of pressure and energy density relative to noninte r-
+acting (anti-) quarks and gauge bosons then allows for the phase t ransition to be
+(essentially) of second order with respect to energy density, as d emonstrated in the
+related thermal quantities shown in Figs. 2 and 3. The square of the velocity of
+sound reveals most directly the order of the transition through its discontinuity as
+shown in Fig. 4.
+The upcoming lower c.m. energy scan at RHIC, new results expected from LHC
+concerning hadronic physics and more extended studies in lattice QC D promise to
+clarify its phase structure, eventually accompanied by theoretica l insights.
+Acknowledgements
+It is a pleasure to thank the TH-division of CERN for its hospitality, an d Anne
+Perrin as well as Markus Moser, representing the group of comput ing coordinators
+at the ITP in Bern, for their logistics support. Topical discussions w ith Martin
+L¨ uscher and Uwe-Jens Wiese are gratefully acknowledged.November 2, 2018 9:27 WSPC/INSTRUCTION FILE thermal˙april2011
+16Sonia Kabana, Peter Minkowski
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\ No newline at end of file
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+arXiv:1001.0708v1  [math.HO]  5 Jan 2010On the so called Boy or Girl Paradox
+G. D’Agostini
+Universit` a “La Sapienza” and INFN, Roma, Italia
+(giulio.dagostini@roma1.infn.it, http://www.roma1.infn.it/ ~dagos)
+Abstract
+A quite old problem has been recently revitalized by Leonard Mlodinow’s book
+The Drunkard’s Walk , where it is presented in a way that has definitely confused
+several people, that wonder why the prevalence of the name of on e daughter among
+the population should change the probability that the other child is a g irl too. I
+try here to discuss the problem from scratch, showing that the ra rity of the name
+plays no role, unless the strange assumption of two identical names in the same
+family is taken into account. But also the name itself does not matter . What is
+really important is ‘identification’, meant in an acceptation broader th an usual, in
+the sense that a child is characterized by a set of attributes that m ake him/her
+uniquely identifiable (‘that one’) inside a family. The important point of h ow
+the information is acquired is also commented, suggesting an explana tion of why
+several people tend to consider the informations “at least one boy ” and “a well
+defined boy” (elder/youngest or of a given name) equivalent.
+1 Introduction
+A classical series of problems in elementary probability th eory is about the gender
+combinations ( m-m,m-f,f-mandf-f) in a family of two children. Being this an
+academic exercise (in the bad sense of the term), usually one does not attempt to
+assess how much one believes that these combinations happen in a real family. This
+means that the well known male over female birth asymmetry is neglected, as are
+neglected gender correlations within a family, like those i nduced by genetic factors, or
+by the possibility of monovular twins.
+Once the conditions are properly defined, the usual question s, besides the trivial
+one of male/female, are
+1Eldest Youngest
+m f m∪f
+m1/4 1/41/2
+f1/4 1/41/2
+m∪f1/2 1/21
+Table 1: Table of equiprobable cases of the four possible sequences o f child’s gender.
+The symbol ‘ ∪’ stands for ‘OR’.
+Q1) What is the probability of two boys?
+Q2) What is the probability of two boys, if the eldest child is a b oy?
+Q3) What is the probability of two boys, if at least one child is a boy?
+These questions can be promptly answered looking at conting ency table 1 that lists
+the space of the four equiprobable elementary cases.
+A1) The probability of two boys is 1/4, or 25%, since it is just th e probability of each
+elementary event, that all together have to sum up to unity, o r 100%.
+A2) If the eldest child is a boy, the space of possibilities is sq ueezed to the first row of
+the table. We remain with two equiprobable cases, each of whi ch gets probability
+1/2. In formulae:
+P(Em∩Ym|Em,I0) =P(Em∩Ym|I0)
+P(Em|I0)(1)
+=1/4
+1/2=1
+2. (2)
+[The symbol ‘ ∩’ stands for a logical ‘AND’; ‘ |’ stands for ‘given’, or ‘conditioned
+by’; ‘Em’ and ‘Ym’ are short forms for “the eldest is male” and “the youngest
+is male”; the condition I0is thebackground status of information under which
+the probabilities are evaluated, that includes the simplif ying hypotheses stated
+above; when there is a further condition, like EmandI0in the l.h.s. of Eq. (1),
+they are both indicated after the conditional symbol ‘ |’, separated by a comma.]
+A3) Finally, the information that there is at least one boy in th e family reduces the
+space of possibilities to three equiprobable cases, of whic h only one is that of our
+interest, thus getting 1/3 (the symbol ‘ ∪’ in the following formulae indicated a
+logical ‘OR’). Formally
+P(Em∩Ym|Em∪Ym,I0) =P[(Em∩Ym)∩(Em∪Ym)|I0]
+P(Em∪Ym|I0)(3)
+2=P(Em∩Ym|I0
+P(Em∪Ym|I0)(4)
+=1/4
+3/4=1
+3(5)
+Obviously, the problem can been turned into probability of g irl-girl by symmetry.
+The ‘complication’ (mainly induced confusion) comes when t he information about
+the name of one child is provided (‘Florida’ in The Drunkard’s Walk [1]):
+Q4What is the probability of two boys, if one of the children is c alled Mark?1
+2 How the child name changes the probabilities
+At this point the question is how table 1 is changed by the info rmation that one child
+is known by gender and name (the latter usually implying the f ormer).
+The way the problem is often solved is to assume that the same n ame can given
+to two different children in the same family. Frankly, I never h eard of this possibility
+before. And, anyhow, if such a strange behavior occurs in som e very rare cases, it seems
+to me of an importance much lower than all other questions tha t have been neglected
+(male/female asymmetry, genetic biases, etc.). What I foun d annoying is that this
+peculiar solution is not reported as a mathematical curiosi ty (see e.g. the Drunkard’s
+Walk or the Wikipedia page site dedicated to the so called ‘pa radox’ [2] – a puzzle one
+is unable to solve is not necessarily a paradox), but as it wou ld be ‘the solution’.
+Nevertheless, let us first see what happens when this possibi lity is allowed. (By the
+way, onemight thinkof families withchildren comingfrompr eviousmarriages, in which
+case identical names might occur, but this possibility is ex cluded in the often implicit
+assumptions of this kind of puzzles, often formulated as “a l ady has two children, ...”.)
+2.1 Allowing identical names for two children of a family
+Just to stay close to the formulation of the problem in the rec ent disputes that have
+triggered this paper, let us focus on girl probabilities, as suming we know that one of
+the child is a girl of a given name. Splitting the female categ ory into ‘female of that
+given name’ ( fN) and ‘female with any other name’ ( fN), there are now three possible
+cases for each child, {m,fN,f N}, no longer equiprobable.
+Callingrthe fraction of girls owning that name in the population, we g et the
+following probabilities, under the new background conditi onI1:P(m|I1) = 1/2,
+P(fN|I1) =r×1/2 =r/2 andP(fN|I1) = (1−r)/2. The nine possibilities and
+their probabilities, calculated using the product rule (ju stified by elder/youngest name
+independence), are reported in table 2. From the table we can calculate the probability
+1This question can be turned into “what is the probability tha t Mark has a brother or a sister?”
+and any normal and sane person might wonder about the sense of thismadness , as said a friend of
+mine with zero math skill, when he saw a first draft of this pape r on my desk, because – he explained
+– “it is absolutely equally likely that he has either a brothe r or a sister”.
+3Eldest Youngest
+m f m∪f
+fN f N
+m 1/4 r/4(1−r)/41/2
+ffN r/4 r2/4 r(1−r)/4r/21/2fN(1−r)/4r(1−r)/4(1−r)2/4(1−r)/2
+m∪f 1/2 r/2 (1 −r)/21
+1/2
+Table 2: Table of probabilities of the possible cases assuming that e ldest and youngest
+children can have the same name (see text).
+of both females, if we know one girl by name:
+P[(Ef∩Yf)|(EfN∪YfN),I1] =P[(Ef∩Yf)∩(EfN∪YfN)|I1]
+P[(EfN∪YfN)|I1].(6)
+The denominator is given by the five elements emphasized in bo ldface in table 2, whose
+probability sum up to r−r2/4. The numerator is given by the three elements that
+havemneither in the rows nor in the columns, whose probabilities a rer2/4,r(1−r)/4
+andr(1−r)/4, adding up to (2 r−r2)/4. We get then
+P[(Ef∩Yf)|(EfN∪YfN),I1] =(2r−r2)/4
+r−r2/4(7)
+=1
+2/bracketleftbigg1−r/2
+1−r/4/bracketrightbigg
+(8)
+≈1
+2−r
+8(forr≪1) (9)
+As we can see, the probability does depend on r, but it tends rapidly to 1/2 for small
+values of r, as also shown in table 3 for some numerical values of this par ameter.2
+2The value r= 0.02 = 1/50 is that used in Ref. [2], for which the probabilities of tab le 2 acquire
+the following values
+0.2500 0.0050 0.2450 0.5000
+0.0050 0.0001 0.0049 0.0100
+0.2450 0.0049 0.2401 0.4900
+0.5000 0.1000 0.4900 1.0000
+from which we get the following table of expected values in 10000 families
+2500 50 2450 5000
+50 1 49 100
+2450 49 2401 4900
+5000 100 4900 10000
+[By the way, I would like to point out that quoting expected va lues is a way to state ...what we
+4rP(two girls |fN,I1)
+0.3 0.45946
+0.2 0.47368
+0.1 0.48718
+0.02 0.49749
+0.01 0.49875
+0.001 0.49988
+0.0001 0.49999
+Table 3: Probability of two girls in family, if we know by name a daught er, calculated
+as a function of the prevalence of that name within the girls o f that population. [Note,
+just for mathematical curiosity, that if r= 1(all girls have the same name), Eq. (8)
+gives a probability of 1/3, thus recovering Q3. In fact, in this case telling the name
+adds no more information to “at least one is female”.]
+2.2 Unique names of children within a family
+Let us now see what happens if we require that, as it normally h appens, children names
+are unique. The central element of table 2 goes to zero, but th e sums along rows and
+columns have to be preserved3[for example the probability of EfN∩YfNbecomes
+r(1−r)/4+r2/4, that is the same as r/2−r/4, i.e.r/4]. The result is shown in table
+4 (we label the central value of the table by ‘-’ to remark that this case is impossible
+by assumption). [Note that the impossibility of identical c hildren names constrains
+rto be smaller than 1 /2, well above any reasonable value. Remember also that the
+probabilities of table 4 reflect the several simplifying ass umptions of the problem.]
+Eldest Youngest
+m f m∪f
+fN f N
+m 1/4r/4(1−r)/41/2
+ffN r/4-r/4 r/21/2fN(1−r)/4r/4(1−2r)/4(1−r)/2
+m∪f 1/2r/2 (1−r)/21
+1/2
+Table 4: Same as table 2, but not allowing the identical names of the ch ildren.
+expect in a probabilistic sense – and probability theory tea ches how to calculate standard expectation
+uncertainties (the σ’s) - and has little to do with ‘frequentistic approach’, sin ce the probabilities have
+not been evaluated by past frequencies (statistical data). ]
+3If no further information is provided, the probability that any child is a female with the special
+nameNhas to be r/2, no matter if the child in question is the eldest or the young est. Similarly, the
+probability of girl with a name different from Nhas to be (1 −r)/2.
+5Contrary to table 2, the four cases that involve fNare nowequiprobable (each with
+probability r/4). It follows that the probability that the other child is a b oy or a girl
+is 50%,independently of the rarity of the name. In formulae (note the new backgroun d
+condition I2):
+P[(Ef∩Yf)|(EfN∪YfN),I2] =P[(Ef∩Yf)∩(EfN∪YfN)|I2]
+P(EfN∪YfN|I2)(10)
+=2×r/4
+4×r/4(11)
+=1
+2. (12)
+3 Does the name really matter?
+At this point it is easy to understand that we could replace fNin the table by fID,
+where ‘ID’ stands now for ‘uniquely identified within the family’, thu s getting table 5.
+Think, for example to the following statements
+•“the secretary of the department X of hospital Y in Rome is dau ghter of my aunt
+B who has also another child”;
+•“the parents of the actress starring in the last movie I have s een have two chil-
+dren”;
+•“the mother of that lady has got two children”;
+•and so on...
+In all these cases the probability that the female in questio n has a sister is 50%, as
+everybody that is not fooled by probability theory will prom ptly tell us (see footnote
+1). It is not just a question of knowing her name, or knowing th at she is the eldest or
+the youngest (that’s the reason we recover the answer to Q2!).What matters is that
+this person is somehow uniquely ‘identified’ in the family , where ‘identified’ is within
+Eldest Youngest
+m f m∪f
+fID f ID
+m 1/4r/4(1−r)/41/2
+ffID r/4-r/4 r/21/2fID(1−r)/4r/4(1−2r)/4(1−r)/2
+m∪f 1/2r/2 (1−r)/21
+1/2
+Table 5: Same as table 4, but based on ‘identification’ of a girl.
+6Eldest Youngest
+m f m∪f
+fID fID
+m 1/4 r/4(1−r)/41/2
+ffID r/4 r2/4 r(1−r)/4r/21/2fID(1−r)/4r(1−r)/4(1−r)2/4(1−r)/2
+m∪f 1/2 r/2 (1 −r)/21
+1/2
+Table 6: Same as table 2, but with reference to non unique identification (note the
+symbol ‘ID’ instead of ‘ ID’ of table 5) rather then name.
+quote marks because it is not requested we know her passport n umber, but just that
+we are able to point to her as that one.
+If this is not the case, and two children could correspond to t he same description,
+then table 2 holds, assuming no correlation between the desc riptions (if one is blond,
+there is high change that the other is blond too, and so on). Th erefore we recover it
+as table 6, but in terms of non unique identification (‘ID’) rather than of name. Now
+it makes sense. In fact, since we are referring here to ‘ident ification’ in a loose sense,
+it might really occur that two daughters correspond to the sa me description (‘goes to
+college’, or ‘play tennis’, and so on). Finally, the name can be considered a generic
+identification, in order to include the possibility of ident ical names in a family (for
+example in the cases of second marriages).
+4 Some Bayesian flavor
+Someone asks me about theBayesian solution of the problem (because I am supposed
+to bea Bayesian ). But, besides the clarification that “I am not a Bayesian” [3 ], such
+a kind of ‘alternative’ solution of the problem does not exis t. The solution is already
+that provided by Eq. (10), because ‘Bayesians’ just make use of probability theory to
+state the relative beliefs of several hypotheses given some well stated assumptions. In
+particular, the so called Bayes’ rule for this problem is ess entially Eq. (10), that can be
+possibly written in other convenient forms using the rules o f probability.
+4.1 Reconditioning the probability of an hypothesis on the l ight of a
+new status of information
+To make the point clearer, and calling A=Ef∩Yf(“both children are female”) and
+B=EfN∪YfN(“one child is a female of a given particular name”) to simpli fy the
+7notation, we can rewrite Eq. (10) as
+P(A|B,I2) =P(A∩B|I2)
+P(B|I2)(13)
+=P(B|A,I2)P(A|I2)
+P(B|I2). (14)
+The latter expression shows explicitly how the probability ofAis updated, by the extra
+condition B, via the factor P(B|A,I2)/P(B|I2), i.e.
+P(A|B,I2) =P(B|A,I2)
+P(B|I2)×P(A|I2). (15)
+The three ingredients we need to evaluate P(A|B,I2) can be easily read from table 4,
+P(A|I2) =1
+4(16)
+P(B|I2) =r (17)
+P(B|A,I2) = 2r, (18)
+from which we get
+P(A|B,I2) =2r
+r×1
+4= 2×1
+4=1
+2, (19)
+recovering the result of section 2.2 (note that it must beso b ecause we are strictly using
+the probabilities of table 4).
+4.2 Updating the odds
+We can do it in a different way, comparing the probability of “tw o girls” ( A) with that
+of “only one girl” [let us indicate the latter hypothesis as C= (Ef∩Ym)∪(Em∩Yf)].
+The probability of Cconditioned by B, i.e.P(C|B,I2), could be obtained in analogy
+to Eq. (13), reading P(C∩B|I2) from table 4. But it can be more instructive to get it
+the Bayesian way , using the formula that shows how relative probabilities ar e updated
+by theBayes factor to take into account the new piece of information (this secon d
+approach has also the advantage of getting rid of rsince the very beginning):
+P(A|B,I2)
+P(C|B,I2)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+‘updated odds’=P(B|A,I2)
+P(B|C,I2)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+‘updating factor’
+(Bayes factor )×P(A|I2)
+P(C|I2)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+‘initial odds’. (20)
+The initial probability of two girls is one half that of a sing le girl, i.e.
+P(A|I2)
+P(C|I2)=1
+2, (21)
+8while the probability that there is a girl with a precise name is proportional to the
+number of girls in the family (remember that the condition ‘ I2’ does not allow the same
+name), namely
+P(B|A,I2)
+P(B|C,I2)= 2. (22)
+It follows
+P(A|B,I2)
+P(C|B,I2)= 2×1
+2= 1 : (23)
+the girl of which we know the name has equal probability to hav e a sister ( A) or a
+brother ( C), that is
+P(A|B,I2) =P(C|B,I2) =1
+2. (24)
+5 Conclusions
+The probability that, knowing the name of one child in a famil y of two, the other one
+child is of the same gender has nothing to do with the rarity of the name, unless the
+crazy possibility of identical names in a family is assumed ( and if somebody insists
+that this can happen, he/she is invited to calculate more rea listic probabilities that
+take into account male/female asymmetry and genetic correl ations; also the possibility
+of identical names of children coming from previous marriag es are implicitly excluded
+in this kind of puzzles, that usually talk of “a lady having tw o children...”).
+Moreover, what matters is not the knowledge of the name, but r ather something
+that allows us to point to him/her as ‘that one’. For this reas onQ2andQ4have the
+same solution.
+I would like to end with some comments on the last of the three t ext book questions
+reminded in the introduction. It seems to me that the reason t here is quite a broad
+tendency to confuse Q3withQ4(or similar questions involving the child identifica-
+tion, including Q2), is that in normal life the information about boy/girl is acquired
+simultaneously with other attributes that make the identifi cation unique (“my daughter
+Claudia”). People do not express themselves as in math textb ooks, stating that “I have
+two children, and at least one of them is a boy”, or “my childre n are not both boys”.
+We usually gain this information in an indirect way. For this reason several people have
+some initial difficulty to grasp that “that lady has Claudia an d another child” is not
+the same as “that lady has two children, at least one being a gi rl”.
+Moreover, even if a mother says “if I had two boys”, we may unde rstand from
+the context (already knowing she has two children) that she h as two girls, because we
+perceived that she emphasized ‘boys’ instead of ‘two’ (in th e latter case we could think
+she has already a boy). Instead, if she said “if my children we re both boys”, we usually
+understand that she is expressing this way because she has a b oy and a girl. Therefore,
+9besides stereotyped recreational puzzles, the evaluation of probabilities, in the sense
+of how much we have to rationally believe the several hypothe ses, can be not trivial.
+We need to take properly into account all contextual informa tion “when the bare facts
+wont’do” [4]. Indeed, in probability evaluations not only t he ‘facts’ play a role, but
+also the words, their sound and the expression of the person w ho says them, and (too
+often ignored) the question to which they reply [4].
+It is a pleasure to thank Dino Esposito, Enrico Franco, Paolo Agnoli, Serena Ce-
+natiempo and Stefano Testa for the several interactions on t he issues discussed here
+and related ones. The paper has benefitted from comments by Di no and Paolo.
+References
+[1] L.Mlodinow, The Drunkard’s Walk. How Randomness Rules Our Lives ,Vintage
+Books, New York, 2008.
+[2]http://en.wikipedia.org/wiki/Boy_or_Girl_paradox .
+[3] G. D’Agostini, Bayesian reasoning versus conventional statistics in high en-
+ergy physics , Proc. XVIII International Workshop on Maximum Entropy and
+Bayesian Methods, Garching (Germany), July 1998, V. Dose et al. eds., Kluwer
+Academic Publishers, Dordrecht, 1999.
+http://xxx.lanl.gov/abs/physics/9811046 .
+[4] J. Pearl, Probabilistic reasoning in intelligent systems: Networks of Plausible
+Inference , MorganKaufmannPublishers,SanMateo, 1988(inparticula rSection
+2.3.2).
+Appendix — On the direct calculation of the elements of
+table 4
+The elements of table 4 have been evaluated from the condition that the ‘central’ one vanishes
+and that the marginal probabilities have to be preserved (this mean s that, for example, the
+probability that the younger is female with the special name Nisr/2 if no other information is
+provided, because r/2 is the assumed probability that an individual of that population carr ies
+that name). Nevertheless, one might be interested to calculate th e eight non vanishing terms
+in a direct way. But this calculation might reserve surprises, as we sh all see.
+10First row and first column of the table (at least one boy)
+Although the elements that contain at least one boy are the easiest ones to be evaluated, let
+us get them with some pedantic detail, for didascalic purposes and in p reparation of the less
+obvious cases. In particular, we shall rewrite Em∩YfNasEm∩Yf∩YNto remember that
+we require the eldest child to be a boy, the youngest to be a girl and t he name of the girl to be
+the particular one in which we are interested ( YN). Applying the ‘chain rule’ we get
+P(Em∩Ym|I2) =P(Em|I2)×P(Ym|Em,I2) =1
+2×1
+2=1
+4
+P(Em∩YfN|I2) =P(Em∩Yf∩YN|I2)
+=P(Em|I2)×P(Yf|Em,I2)×P(YN|Yf,Em,I 2)
+=1
+2×1
+2×r=r
+4
+P(Em∩YfN|I2) =P(Em∩Yf∩YN|I2)
+=P(Em|I2)×P(Yf|Em,I2)×P(YN|Yf,Em,I 2)
+=1
+2×1
+2×(1−r) =1−r
+4,
+recovering the first row of table 4. (Note that Nstands for ‘a feminine name different from N’
+and not ‘any name but N’!)
+Similarly, the second and third elements of the first column are
+P(EfN∩Ym|I2) =P(Ef∩EN∩Ym|I2)
+=P(Ef|I2)×P(EN|Ef,I2)×P(Ym|Ef,EN,I 2)
+=1
+2×r×1
+2=r
+4
+P(EfN∩Ym|I2) =P(Ef∩EN∩Ym|I2)
+=P(Ef|I2)×P(EN|Ef,I2)×P(Ym|Ef,EN,I2)
+=1
+2×(1−r)×1
+2=1−r
+4.
+The first column of table 4 is also recovered.
+Probabilities of both females
+The probability that two girls have the same name is zero, that is beca use
+P(EfN∩YfN|I2) =P(Ef∩EN∩Yf∩YN|I2)
+=P(Ef|I2)×P(EN|Ef,I2)×P(Yf|Ef,EN,I 2)
+×P(YN|Yf,Ef,EN,I 2)
+=1
+2×r×1
+2×0 = 0.
+The third element of the second row is given by
+P(EfN∩YfN|I2) =P(Ef∩EN∩Yf∩YN|I2)
+=P(Ef|I2)×P(EN|Ef,I2)×P(Yf|Ef,EN,I 2)
+×P(YN|Yf,Ef,EN,I 2)
+=1
+2×r×1
+2×1 =r
+4.
+11(The fourth factor of the r.h.s. of the last equation is 1 because, o nce we know the eldest
+child has the name N, the youngest cannot have that name.) Also the second row of tab le 4 is
+recovered.
+The missing elements of the third row might present some pitfalls. Let us start from
+EfN∩YfN, that is Ef∩EN∩Yf∩YN. It can be calculated as
+P(EfN∩YfN|I2) =P(Ef∩EN∩Yf∩YN|I2)
+=P(Yf∩YN∩Ef∩EN|I2)
+=P(Yf|I2)×P(YN|Yf,I2)×P(Ef|Yf,YN,I 2)
+×P(EN|Ef,Yf,YN,I 2)
+=1
+2×r×1
+2×1 =r
+4.
+thus obtainingthesamevalue ofthe third elementofthe secondrow , that isP(EfN∩YfN|I2),
+as we expect by symmetry and as it was in table 4.
+A pitfall
+But one would like to calculate P(EfN∩YfN|I2) ‘the other way around‘, i.e. applying the
+chain rule starting from P(EfN|I2). If one tries to proceed this way, there is high chance to
+arrive to the following result
+P(EfN∩YfN|I2) =P(Ef∩EN∩Yf∩YN|I2)
+=P(Ef|I2)×P(EN|Ef,I2)×P(Yf|Ef,EN,I2)
+×P(YN|Ef,EN,Yf,I 2)
+=1
+2×(1−r)×1
+2×r
+=r(1−r)
+4=r
+4−r2
+4,
+that differs from the value r/4got previously .
+In a similar way, one could be tempted to evaluate the probability that there are two girls,
+none of them carrying the name N, as
+P(EfN∩YfN|I2) =P(Ef∩EN∩Yf∩YN|I2)
+=P(Ef|I2)×P(EN|Ef,I2)×P(Yf|Ef,EN,I2)
+×P(YN|Ef,EN,Yf,I 2)
+=1
+2×(1−r)×1
+2×(1−r)
+=(1−r)2
+4=(1−2r)
+4+r2
+4,
+thus obtaining table 7, that differs from table 4. Namely, in the case o f two females, it is now
+less probable that the youngest girl has the particular name N. The probabilities differ by r2/4,
+thus being negligible for small r.
+One might think it is right so, because it reflects the order of naming t he children (“since
+the name Ncannot be given twice, the the eldest girl has a kind of first choice”) . But on
+the other hand, we are dealing here with knowledge (or ignorance), and therefore P(EfN|I2)
+12Eldest Youngest
+m f m∪f
+fN f N
+m 1/4 r/4 (1−r)/4 1/2
+ffN r/4 - r/4 r/21/2fN(1−r)/4r(1−r)/4(1−r)2/4(1−r)/2
+m∪f 1/2r/2−r2/4 (1−r)/2+r2/41
+1/2
+Table 7: Same as table 4, but obtained by a wrongreasoning that implicitly assumes
+that the condition EfNdoes not change the probabilities of YfNand ofYfN.
+andP(YfN|I2) must be absolutely equal. It is just a question of symmetry in reaso ning
+in conditions of uncertainty. It doesn’t matter if we start thinking f rom the eldest or from
+the youngest child. Stated in different words, from a probabilistic po int of view ‘eldest’ and
+‘youngest’ are mere labels . The probability ‘matrix’ must besymmetric.
+Moreover, it is curious to realize that table 7 produces a probability o f two females that
+depends on r, as we saw in section 2.1. We get in fact
+P[(Ef∩Yf)|(EfN∪YfN),I2] =r/4+r(1−r)/4
+3×r/4+r(1−r)/4
+=1
+2/bracketleftbigg1−r/2
+1−r/4/bracketrightbigg
+,
+exactly the same result of section 2.1 [see Eq. (8)]. Therefore those who maintain that the
+probability of two girls, provided we know that one child is girl known by name, does depend
+on the rarity of the name either assume that identical names are po ssible inside the same family
+(a bizarre assumption), or have been caught by this pitfall (a mista ke in reasoning).
+Conditional probabilities of female names
+The weak points of the previous evaluations, that lead to table 7, wit h all its consequences, are
+the conditional probabilities P(YN|Ef,EN,Yf,I 2) andP(YN|Ef,EN,Yf,I 2), for which we
+assumed intuitively the values rand (1−r), as if the information that the eldest child is a girl
+with a name different from Ndid not change the probability of the name of the other girl. This
+intuition, roughly but not exactly correct , is due to the fact that we tend to consider rsmall (any
+modern population has a large amount of possible feminine names) suc h that the assumption
+that a girl has any name but the particular one ( N) does not change sizably the probability of
+the name of the other girl. This is the reason why the correct result s are recovered for r→0,
+that was the hidden initial assumption!
+But, strictly speaking, the EfNandYfNare not independent (in probability, or ‘stochas-
+ticly’), as are not independent EfNandYfN: the information that the eldest girl has a name
+different from Nhas toincrease the probability that the youngest girl is called N(and has to
+decrease the probability that also the youngest girl has a name diffe rent from N). Comparing
+tables 4 and 7 we see that the effect goes this direction and has a size that decreases rapidly
+withr, going as r2.
+13From this reasoning we get the following qualitative results:
+P(YN|Ef,EN,Yf,I 2)> P(YN|Yf,I2)
+P(YN|Ef,EN,Yf,I 2)< P(YN|Yf,I2).
+The onlyproblem isthat it is not easyto evaluatethese probabilities. B ut, fortunately, they can
+be calculated from the general rules of probability, remembering th at, as discussed above, the
+probabilities of EfN∩YfNand ofEfN∩YfNcan be obtained in different ways. More
+precisely, the probability of EfN∩YfNcan be calculated either directly, from the easy
+P(YN|Yf,Ef,EN,I 2), as we have done just above in the appendix, or indirectly, requir-
+ingP(YfN|I2) =r/2, as it was done building up table 4 in section 2.2. Instead, the last
+element of the table, P(EfN∩YfN|I2), can by only calculated indirectly, either requiring
+P(YfN|I2) = (1−r)/2, or from the normalization rule, i.e. from the constraint that all
+elements of the table have to sum up to one.
+Therefore, the conditional probabilities of interest can be finally ev aluated from the joint
+probabilities as
+P(YN|Ef,EN,Yf,I 2) =P(YN∩Ef∩EN∩Yf|I2)
+P(Ef∩EN∩Yf|I2)
+=P(YN∩Ef∩EN∩Yf|I2)
+P(Ef∩EN|I2)·P(Yf|Ef∩EN,I2)
+=r/4
+(1−r)/2×1/2=r
+1−r
+≈r (forr≪1)
+P(YN|Ef,EN,Yf,I 2) =P(YN∩Ef∩EN∩Yf|I2)
+P(Ef∩EN∩Yf|I2)
+=P(YN∩Ef∩EN∩Yf|I2)
+P(Ef∩EN|I2)·P(Yf|Ef∩EN,I2)
+=(1−2r)/4
+(1−r)/2×1/2=1−2r
+1−r
+≈1−r (forr≪1).
+[Obviously, the latter probabilitycould havebeen calculated easieras P(YN|Ef,EN,Yf,I 2) =
+1−P(YN|Ef,EN,Yf,I 2) = 1−r/(1−r), getting the same result.]
+Notethatforverysmallvaluesof rwerecover P(YN|Yf,I2) =randP(YN|Yf,I2) = 1−r,
+respectively. That is, in this limit EfNandYfN, as well as EfNandYfN, are approximately
+independent, in agreement with our initial intuition.
+Finally, we remind that the probabilities of the eldest girl name, condit ioned by YfN, can
+be obtained by symmetry, i.e. P(EN|Yf,YN,Ef,I 2) =P(YN|Ef,EN,Yf,I 2) =r/(1−r)
+andP(EN|Yf,YN,Ef,I 2) =P(YN|Ef,EN,Yf,I 2) = (1−2r)/(1−r), and that all these
+expressionsdepend on thesimplifying assumptionsembedded in this k indofrecreationalpuzzle.
+14
\ No newline at end of file
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--- /dev/null
+++ b/1001.0709.txt
@@ -0,0 +1,3959 @@
+  
+   
+ ESA/SRE(2009)6
+December 2009
+Assessment Study ReportSPICA
+Revealing the origins of
+planets and galaxies
+European Space Agency
+i
+SPICA – SPace Infrared telescope for Cosmology and Astrophysics
+Science objectives•Formation and evolution of planetary systems: Gas and dust in proto-planetary discs, includ-
+ing water, and their link to planetary formation; mineralogy of debris discs; gas exoplanetsatmospheres; composition of Kuiper Belt objects.
+•Life cycle of dust: Physics and chemistry of gas and dust in the Milky Way and in nearbygalaxies; dust mineralogy; dust processing in supernova remnants and the origin of interstel-
+lar dust in the early Universe.
+•Formation and evolution of galaxies: AGN/starburst connection over cosmic time and as afunction of the environment; co-evolution of star formation and super-massive black holes;
+star-formation and mass assembly history of galaxies in relation with large scale structures;the nature of the Cosmic Infrared Background.
+Wavelength range Medium to far infrared (5–210 μm)
+Telescope Ritchey-Chr ´etien, 3.5 m diameter cryogenic infrared telescope, with diffraction limited perfor-
+mance at 5μm (5 arcmin radius FOV).
+Instruments•MIRACLE (mid infrared camera and spectrophotometer)
+•SAFARI (far infrared imaging spectrometer)
+•MIRMES (mid infrared medium resolution spectrometer)
+•MIRHES (mid infrared high resolution spectrometer)
+•SCI (mid infrared coronagraph)
+•BLISS (far infrared and sub-mm spectrometer; optional)
+•FPC (dedicated Focal Plane Cameras for guidance)
+On Board Science Data: 24 h*4M b/s=350 Gbit/day
+Launcher and orbit•H-IIB (JAXA) – direct launch in L 2transfer trajectory.
+•Launcher dispersion correction on first day, main trajectory manoeuvre at day L +2 and
+touch up at L+15, L 2final insertion at L +120; monthly orbit correction manoeuvres.
+•Operational orbit is a large amplitude libration orbit around SE-L 2(Halo orbit), with a semi-
+major axis amplitude of 750 000 km.
+•Radiation environment: TID (2 mm Al shield) of 19.5 krad (5 yr), 26.8 krad (10 yr).
+Mission phasesand operations•Launch by end of 2018
+•LEOP duration∼3 days.
+•Commissioning phase ∼2 months. Cool-down ∼168 days
+•Guaranteed mission duration 3 yr (5 yr goal)
+•MOC at JAXA (nominal ground station Usuda)
+•Main SOC at JAXA, with additional SOC at ESA and SAFARI ICC in Europe.
+Spacecraft•Cryogenic PLM supported by octagonal SVM via a dedicated, low thermal conductivity, truss
+structure.
+•Dimensions:∼4.5 m width, 7.5 m high. Total wet mass ∼4000 kg
+•Total maximum power of 2.4 kW (EOL)
+•Total delta-V budget: 110 m s−1
+•Pointing requirements: Standard mode: APE <0.135 arcsec, RPE <0.075 arcsec/200 s;
+Coronagraph mode: APE <0.03 arcsec, RPE<0.03 arcsec/20 min.
+•Attitude control system: 3-axis stabilised spacecraft, RW’s. AOCS sensors complemented bydedicated focal plane cameras for fine guidance.
+•Thermal Control System: Passive cooling via Sun-shield, 3 thermal shields, telescope shelland external telescope baffle. Active cooling via a redundant set of mechanical coolers based
+on 2-stage Stirling units and JT expansion units (4.5 and 1.7 K stage).
+•Propulsion system: Monopropellant (blow down), 4x 23N and 8x 3N thrusters. Propellant:220 kg.
+•On Board Data Handling: Central DPU (common to PLM and SVM), SpaceWire, on boardmass memory>48 Gbyte.
+•Telecommand, Telemetry & Communication: X-Band (HGA, MGA, and 3x LGA for down-link), S-band (2 LGA for up and down-link).ii
+Page intentionally left blankForeword
+In October 2007, the Space Science Advisory Committee selected SPICA (SPace Infrared telescope for Cos-
+mology and Astrophysics) for an assessment study as a candidate M-class mission with the character of “mis-
+sion of opportunity”. SPICA is a JAXA/ISAS led observatory that will operate in the mid and far infrared
+wavelength range with unprecedented sensitivity, thanks to the 3 m-class (3.5 m current baseline) cold tele-
+scope and advanced instruments. SPICA will carry mid infrared camera, spectrometers and coronagraph (built
+by JAXA institutes), a far infrared imager spectrometer (provided by an European/Canadian consortium), and
+a far infrared/sub-millimetre spectrometer (proposed by NASA). The collaboration between the far infrared
+instrument (SAFARI) European/Canadian consortium and JAXA, initiated more than a decade ago when the
+SPICA concept was outlined, led to the Cosmic-Vision proposal in which ESA is invited to become a mission
+partner. In particular, it is proposed that ESA procures the telescope assembly, a ground station, contributes
+to the science operations and manages the JAXA/SAFARI interfaces. From the scientific point of view, the
+participation in SPICA would enable European astronomers to greatly benefit from a first-class observatory
+with unique capabilities that will improve drastically our understanding of planetary systems formation and
+evolution, exoplanets and galaxy evolution.
+As a first step of the assessment phase, the SPICA Telescope Science Study Team was organised to identify
+the science requirements associated with the telescope. Once the science and engineering requirements wereestablished, an internal study on the telescope assembly was carried out between March and April 2008 at
+ESTEC, in close cooperation with JAXA. Following the release of the Invitation to Tender, two parallel com-
+petitive contracts of one year duration were awarded to EADS Astrium (F) and to Thales Alenia Space (F) for
+the industrial assessment study of the SPICA telescope. The two studies showed the overall feasibility of the
+telescope development given the heritage and expertise available in Europe.
+The European instrument SAFARI was the subject of an internal ESA CDF study in May 2008, with a strong
+participation of the SAFARI specialists. The SAFARI Consortium has carried out the instrument assessment
+study until September 2009, including a Technology Readiness Review that was organised in collaboration with
+ESA. The study has led to a consolidation of the instrument design and a clear definition of the detector options
+with a plan for their selection.
+The SPICA science case described in the document is the result of a combined effort from scientists from
+the SAFARI consortium, supported by astronomers from different disciplines, and the Japanese SPICA Task
+Force, which is in charge of the definition of the mission science objectives and requirements. A joint Euro-
+pean/Japanese workshop took place in July 2009 in Oxford, where the science objectives were presented indetail and discussed in the light of their uniqueness and relevance.
+At JAXA, SPICA has been in the Pre-project phase, which is approximately equivalent to ESA’s phase-
+A, since July 2008. It consists of two phases, the Concept Design Phase and System Definition Phase. Theaim of the Concept Design Phase is to define the system requirements and obtain their approval in the System
+Requirements Review that will be held in the period December 2009-March 2010. After the System Definition
+Review in Autumn 2010, SPICA will go through the Project Approval Review by mid 2011, a management
+review by JAXA HeadQuarters, where the decision whether SPICA can proceed though phase-B to the end of
+mission will be made.
+In this document we present the scientific objectives, science requirements and the description of the mis-
+sion which is the result of the ESA and SAFARI assessment studies and of JAXA phase-A activities. Although
+we have put special emphasis on the elements of the European contribution, the mission as a whole is covered
+in the document. The reasons are two-fold. First, to provide the context for the European contribution and
+second, more importantly, to present the full SPICA observatory and its unprecedented capabilities as they will
+be available to the astronomical community.
+iiiiv
+Page intentionally left blankAuthors
+The SPICA Assessment Study Report has been prepared with inputs from the ESA SPICA Study Team, the
+SPICA Telescope Science Study Team, the JAXA/ISAS SPICA Team and the SAFARI Consortium. The
+authors, in alphabetical order, are:
+Marc Ferlet Rutherford Appelton Laboratory (UK); SPICA Telescope Science Study Team
+Norbert Geis Max Planck Institute f ¨ur extraterr. Physik (Germany); SPICA Telescope Science Study Team
+Javier Goicoechea Centro de Astrobiolog ´ıa, CSIC-INTA (Spain);
+SAFARI Consortium, Galactic Science Coordinator
+Douglas K. Griffin Rutherford Appelton Laboratory (UK); SAFARI System Engineer
+Ana M. Heras European Space Agency; SPICA Study Scientist, Document Editor
+Kate Isaak Cardiff University (UK); SAFARI Consortium, Extragalactic Science Coordinator
+Takao Nakagawa JAXA /ISAS; JAXA SPICA Project Leader
+Takashi Onaka University of Tokyo, Japan; SPICA Telescope Science Study Team
+Nicola Rando European Space Agency; SPICA Study Manager
+Bruce Swinyard Rutherford Appleton Laboratory (UK);
+SAFARI PI, SPICA Telescope Science Study Team
+Nobuhiro Takahashi JAXA; SPICA System Engineer
+Sebastien Vives Astronomy Observatory of Marseilles Provence (France);
+SPICA Telescope Science Study Team
+Acknowledgements
+The authors would like to acknowledge the essential contributions to the definition of the science objectives
+from: J.-C. Augereau (UJF, Fr), M. Barlow (UCL, UK), J.-P. Beaulieu (IAP, Fr), A. Belu (U. Nice, Fr), O. Berne
+(CAB/CSIC-INTA, E), B. B ´ezard (Obs. Paris, Fr), A. Boselli (LAM, Fr), V . Buat (LAM, Fr), D. Burgarella
+(LAM, Fr), J. Cernicharo (CAB/CSIC-INTA, E), D. Clements (Imperial College, UK), G. de Zotti (Oss. Astr.
+Padova, It), M. Delbo (UNS, Fr), Y . Doi (U. Tokyo, JP), S. Eales (U. Cardiff, UK), E. Egami (U. Arizona,
+USA), D. Elbaz (CEA, Fr), K. Enya (ISAS/JAXA, JP), A. Franceschini (U. Padova, It), H. Fraser (STRATH,
+UK), M. Fukagawa (Osaka U., JP), H. G ´omez (Cardiff U., UK), J. G ´omez-Elvira (CAB/CSIC-INTA, E),
+O. Groussin (LAM, Fr), C. Gruppioni (INAF, It), S. Hailey-Dunsheath (MPE, Ger), S. Hasegawa (ISAS /JAXA,
+JP), F. Helmich (SRON, NL), M. Honda (KAN-U, JP), Y . Itoh (Kobe U., JP), R. Ivison (U. Edinburgh, UK),
+H. Izumiura (NAO, JP), C. Joblin (CESR, Fr), D. Johnstone (NRC, CA), G. Joncas (ULA V AL, CA), A. Jones
+(IAS, Fr), I. Kamp (U. Groningen, NL), H. Kaneda (Nagoya U., JP), H. Kataza (ISAS/JAXA, JP), C. Kemper(Manchester U., UK), F. Kerschbaum (UNIVIE, AT), Y . Kitamura (ISAS/JAXA, JP), P. Lacerda (Belfast U.,
+UK), F. Levrier (LRA-LERMA, Fr), D. Lutz (MPE, Ger), S. Madden (CEA, Fr), M. Magliocchetti (IFSI, It),
+J. Mart ´ın-Pintado (CAB/CSIC-INTA, E), H. Matsuhara (ISAS/JAXA, JP), R. Meijerinks (Sterrewacht Leiden,
+NL), S. Molinari (IFSI, It), R. Moreno (Obs. Paris, Fr), A. Moro-Mart ´ın (CAB/CSIC-INTA, E), P. Najarro
+(CAB/CSIC-INTA, E), N. Narita (NAO, JP), Y . Okamoto (Ibaraki U., JP), S. Oliver (U. Sussex, UK), T. Onaka
+(U. of Tokyo, JP), T. Ootsubo (ISAS/JAXA, JP), M. Page (MSSL, UK), E. Pantin (CEA, Fr), E. Pascale (U.
+Cardiff, UK), I. P ´erez-Fournon (IAC, E), S. Pezzuto (IFSI, It), A. Poglitsch (MPE, Ger), E. Pointecouteau (U.
+Toulouse, Fr), C. Popescu (U. Lancaster, UK), F. Pozzi (U. Bologna,It), W. Raab (MPE, Ger), G. Raymond(Cardiff U., UK), D. Rigopoulou (Oxford U., UK), I. Roseboom (U. Sussex, UK), H. Rottgering (Leiden U.,
+NL), F. Selsis (LAB, Fr), S. Serjeant (Open U., UK), A. Smith (U. Sussex, UK), M. Spaans (Kapteyn Astr.Inst., Neth), L. Spinoglio (IFSI, It), E. Sturm (MPE, Ger), M. Takami (Academia Sinica,Taiwan), T. Takeuchi
+(Nagoya U., JP), S. Takita (ISAS/JAXA, JP), M. Tamura (NAO, JP), G. Tinetti (UCL, UK), S. Tommasin (IFSI,
+It), J. Torres (CAB/CSIC-INTA, E), R. Tuffs (MPIK, Ger), M. Vaccari (Oss. Astr. Padova, It), P. van der Werf
+(Leiden Obs., NL), S. Viti (UCL, UK), C. Waelkens (KUL, Be), R. Waters (UV A, NL), M. Wyatt (CAM, UK),T. Yamashita (NAO, JP).
+This report has been supported by the ESA Science Planning and Community Coordination Office: Marcello
+Coradini, Philippe Escoubet, Fabio Favata and Timo Prusti.
+vvi
+Page intentionally left blankContents
+Executive Summary 1
+1 Scientific Objectives 5
+1.1 Introduction ............................................ 5
+1.1.1 SPICA in the Cosmic Vision . . . . . .......................... 6
+1.1.2 A critical waveband ................................... 7
+1.2 Planetary systems formation and evolution . . . ......................... 9
+1.2.1 Protoplanetary and debris discs . . . .......................... 1 0
+1.2.2 The inner and outer Solar System “our own debris disc” . . . . . ........... 1 8
+1.3 Exoplanet characterisation in the Infrared . . . ......................... 2 0
+1.3.1 New frontiers in exoplanet research . . . ........................ 2 1
+1.3.2 Mid-infrared coronagraphy: “Direct” imaging and spectroscopy of EPs ........ 2 2
+1.3.3 Mid-IR “transit” photometry and spectroscopy of exoplanets . . . . .......... 2 4
+1.4 The life cycle of gas and dust ................................... 2 4
+1.5 Galaxy evolution . . . . ..................................... 2 6
+1.5.1 Major recent discoveries and open problems . . ..................... 2 8
+1.5.2 MIR/FIR imaging spectroscopy is the key observational technique ........... 3 0
+1.6 The role of SPICA-SAFARI ................................... 3 2
+1.6.1 Spectroscopic cosmological surveys with SAFARI ................... 3 2
+1.6.2 Deep cosmological surveys with SAFARI ........................ 3 7
+1.7 SPICA in the context of ALMA and the JWST . . . ...................... 4 0
+1.8 Discovery Science with SPICA .................................. 4 1
+1.9 Concluding remark . . . . .................................... 4 4
+2 Scientific Requirements 45
+2.1 Telescope scientific requirements ................................. 4 5
+2.1.1 Wavelength coverage ................................... 4 5
+2.1.2 Telescope aperture .................................... 4 5
+2.1.3 Image quality and field of view . . . . ......................... 4 7
+2.1.4 Telescope temperature .................................. 4 8
+2.1.5 Straylight . . ....................................... 4 9
+2.1.6 Surface cleanliness .................................... 5 0
+2.2 Mission definition scientific requirements . . . . . ....................... 5 0
+2.2.1 Wavelength coverage ................................... 5 0
+2.2.2 Observational and environmental requirements ..................... 5 1
+2.2.3 Pointing requirements .................................. 5 1
+2.3 Far infrared instrument scientific requirements . . . . . . . . . ................. 5 2
+2.3.1 Assessment of basic instrument concept . . . ...................... 5 4
+2.3.2 Scientific requirements .................................. 5 5
+2.4 Mid-infrared instruments scientific requirements . . . ...................... 5 6
+2.4.1 Imaging and spectrophotometric capabilities ...................... 5 6
+2.4.2 Spectroscopic capabilities . . . . . . . . . ....................... 5 6
+2.5 Mid-infrared coronagraph scientific requirements ........................ 5 7
+viiviii Contents
+3 Payload 59
+3.1 The SPICA telescope ....................................... 5 9
+3.1.1 STA requirements .................................... 6 0
+3.1.2 Description of interfaces and STA accommodation. . .................. 6 0
+3.1.3 Optical design ...................................... 6 2
+3.1.4 Mechanical design .................................... 6 4
+3.1.5 Thermal design ...................................... 6 6
+3.1.6 Resource budgets . . . .................................. 6 6
+3.1.7 Technology readiness ................................... 6 7
+3.1.8 Assembly Integration & Verification plan . . . ..................... 6 8
+3.1.9 Development plan .................................... 7 1
+3.2 Focal Plane Instruments . . . ................................... 7 1
+3.2.1 Overview . . . . ..................................... 7 1
+3.2.2 Far Infrared Instrument: SAFARI . . . . . ....................... 7 2
+3.2.3 Mid Infrared Camera: MIRACLE . . .......................... 7 7
+3.2.4 Mid-IR Spectrometer: MIRMES & MIRHES . ..................... 7 7
+3.2.5 SPICA Coronagraph Instrument: SCI . . . . . . . . . ................. 7 8
+3.2.6 Focal Plane Finding Camera . . . . . .......................... 7 9
+3.2.7 BLISS ........................................... 8 0
+4 Mission Design 81
+4.1 Mission profile . . . . . . .................................... 8 1
+4.2 Spacecraft design . . ....................................... 8 3
+4.2.1 Bus module and spacecraft sub-systems . . ....................... 8 3
+4.2.2 Payload module – cryogenic system . . ......................... 8 4
+5 Mission Operations 93
+5.1 Science management ....................................... 9 3
+5.1.1 Science teams ....................................... 9 3
+5.1.2 Observing time allocation . . . . . . . . . ....................... 9 4
+5.1.3 Observing programmes .................................. 9 5
+5.1.4 SPICA Observing Time Allocation Committee ..................... 9 5
+5.2 SPICA operational phases .................................... 9 5
+5.3 SPICA operations centres . . . .................................. 9 6
+5.3.1 Mission Operations Centre and ground stations . .................... 9 6
+5.3.2 Science Operations and Instrument Control Centres ................... 9 8
+6 Management 101
+6.1 The SPICA Steering Group .................................... 1 0 1
+6.2 ESA contribution to SPICA .................................... 1 0 1
+6.3 SPICA project activities at JAXA ................................. 1 0 2
+6.4 ESA/JAXA interfaces ....................................... 1 0 3
+6.5 SAFARI procurement and management . . . . ......................... 1 0 3
+6.6 SPICA schedule and coordination with JAXA . . . . . . . . . ................. 1 0 4
+6.7 Industrial organisation of ESA contribution . . . ........................ 1 0 4
+References/Acronyms 105Executive Summary
+Understanding of the origin and evolution of galaxies, stars, planets, our Earth and of life itself are fundamental
+objectives of Science in general and Astronomy in particular. Although impressive advances have been made
+in the last twenty years, our knowledge of how the Universe has come to look as it does today is far from
+complete. A full insight of the processes involved is only possible with observations in the long wavelength
+infrared waveband of the electromagnetic spectrum. It is in this range that astronomical objects emit most of
+their radiation as they form and evolve in regions where obscuration by dust prevents observations in the visible
+and near infrared.
+Over the past quarter of a century successive space infrared observatories (IRAS, ISO, Spitzer and AKARI)
+have revolutionised our understanding of the evolution of stars and galaxies. Mid to far infrared observationshave led to stunning discoveries such as the Ultra Luminous Infrared Galaxies (ULIRGS), the basic processes
+of star formation from “class 0” pre-stellar cores through to the clearing of the gaseous proto-planetary discs
+and the presence of dust excesses around main sequence stars. The Herschel Space Observatory launched this
+year will continue this work in the far infrared and sub-mm and JWST, due for launch in 2014, will provide a
+major boost in observing capability in the 2 – 28 μm range.
+Previous infrared missions have been hampered by the requirement to cool the telescope and instruments
+to<5 K using liquid cryogens. This has limited the size of the apertures to <1 m and our view of the infrared
+Universe has been one of poor spatial resolution and limited sensitivity. The Herschel mission addresses the
+first of these by employing a 3.5 m mirror to dramatically increase the available spatial resolution but, because
+it is only cooled to 80 K, only offers a modest increase in sensitivity in the 55 – 210 μm range compared to
+previous facilities. JWST will provide a major increase in both spatial resolution and sensitivity but only up to28μm.
+It is in this context that the JAXA led mission SPICA (SPace Infrared telescope for Cosmology and As-
+trophysics) is proposed. SPICA is an observatory that will provide imaging and spectroscopic capabilities inthe 5 to 210μm wavelength range with a 3.5 m telescope like Herschel, but now cooled to a temperature less
+than 6 K. In combination with a new generation of highly sensitive detectors, the low telescope temperaturewill allow us to achieve sky-limited sensitivity over the full 5 to 210 μm band for the first time. This unique
+capability means that SPICA will be between one and two orders of magnitude more sensitive than Herschelin the far infrared band. SPICA will cover the full 5 to 210 μm wavelength range, including the missing 28 μm
+to 55μm octave which is out of the Herschel and JWST domains, with unprecedented sensitivity and spatial
+resolution. Furthermore, SPICA will be the only observatory of its era to bridge the wavelength gap betweenJWST and ALMA, and carry out unique science not achievable at sub-mm wavelengths with ALMA. In the
+mid infrared SPICA will be able to carry out medium and high-resolution spectroscopy (R ∼30 000), one order
+of magnitude higher than in JWST, and will include spatial high multiplexing imaging and medium spectralresolution capabilities. In addition, the characteristics of the SPICA monolithic telescope will provide unique
+and optimal conditions for mid infrared coronagraphy in imaging and, uniquely, spectroscopic mode.
+The science objectives of SPICA are closely linked to three of the ESA Cosmic Vision Themes:
+Theme 1 - What are the conditions for planet formation and the emergence of life?;
+Theme2-H o wdoes the Solar System work?;
+Theme4-H o wd i dt h eU n i v erse originate and what is it made of?
+With its powerful scientific capabilities, SPICA will provide unique and ground-breaking answers to these key
+questions, and it is with this goal that the SPICA science objectives have been defined:
+(i) The formation and evolution of planetary systems: By accessing key spectral diagnostic lines, SPICA
+will provide a robust and multidisciplinary approach to determine the conditions for planetary systems forma-
+tion. This will include the detection for the first time of the most relevant species and mineral components in
+the gas and dust of hundreds of transitional protoplanetary discs at the time when planets form. SPICA will also
+be able to trace the warm gas in the inner ( <30 AU) disc regions and by resolving the gas Keplerian rotation,
+will allow us to observe the evolution of disc structure due to planet formation.
+SPICA will study debris discs and make the first unbiased survey of the presence of zodiacal clouds in
+12 Executive Summary
+thousands of exoplanetary systems around all stellar types. It will allow us to detect both the dust continuum
+emission and the brightest grain/ice bands as well as the brightest lines from any gas residual present in the
+disc. SPICA will have the unique capability to observe water ice in all environments, and thus fully explore its
+impact on planetary formation and evolution and in the emergence of habitable planets. In the closest debris
+discs, SPICA will spatially resolve the distribution of water ice and determine the position of the “snow line”,
+which separates the inner disc region of terrestrial planet formation from that of the outer planets.
+SPICA will drastically enhance our knowledge of the Solar System by making the first detailed charac-
+terisation of hundreds of Kuiper Belt Objects, and of different families of inner, hotter centaurs, comets and
+asteroids. SPICA will provide the means to quantify their composition and determine unambiguously their size
+distribution: critical observational evidence for the models of Solar System formation. No other planned orpresent facility will be able to carry out these observations.
+SPICA will provide direct imaging and low resolution mid infrared spectroscopy of outer young giant
+exoplanets (e.g., at ∼9 AU of a star at 10 pc), which will allow us for the first time to study the physics and
+composition of their atmospheres in a wavelength range particularly rich in spectral signatures (e.g., H
+2O, CH 4,
+O3, silicate clouds, NH 3,C O 2) and to compare it with the planets of our Solar System. In addition, mid infrared
+transit photometry and medium or high resolution spectroscopy of “hot Jupiters” will be routine with SPICA.
+SPICA will also provide an unprecedentedly sensitive window into key aspects of the dust life-cycle both
+in the Milky Way and in nearby galaxies, from its formation in evolved stars, its evolution in the ISM, itsprocessing in supernova-generated shock waves and massive stars, to its final incorporation into star forming
+cores and protoplanetary discs.
+(ii) The formation and evolution of galaxies: SPICA observations will provide a unique insight into the
+basic questions about how galaxies form and evolve such as: What drives the evolution of the massive, dusty
+distant galaxy population, and what feedback /interplay exists between the physical processes of mass accretion
+and star formation? How and when do the normal, quiescent galaxies such as our own form, and how do theyrelate to (Ultra) Luminous Infrared Galaxies (ULIRGs)? How do galaxy evolution, star formation rate and
+AGN activity vary with environment and cosmological epoch?
+Substantial progress in this area can only be made by making the transition from large-area photometric to
+large-area spectroscopic surveys in the mid to far infrared, which will be possible with SPICA. This is because
+the mid and far infrared region plays host to a unique suite of diagnostic lines to trace the accretion and star
+formation, and to probe the physical and chemical conditions in different regimes from AGN to star-forming
+regions. While the Herschel-PACS spectrometer will detect the brightest far infrared objects at z∼1, SPICA
+will be able to carry out blind spectroscopic surveys out to z∼3. This will lead to the first statistically unbiased
+determination of the co-evolution of star formation and mass accretion with cosmic time. Spectroscopic surveys
+will provide direct and unbiased information on the evolution of the large scale structure in the Universe from
+z∼3 and the unprecedented possibility to investigate the impact of environment on galaxy formation and
+evolution as a function of redshift.
+The high sensitivity of SPICA will enable photometric surveys beyond z∼4 that will resolve more than
+90% of the Cosmic Infrared Background (in comparison with 50% that Herschel will achieve). SPICA will alsoobserve Milky Way type galaxies in the far infrared out to z∼1, where the cosmic star formation rate peaks.
+For the first time, we will be able to piece together the story of the evolution of our own galaxy and answer thequestion of whether we are in a “special place” in the cosmos.
+The science objectives of SPICA have led to the identification of the mission science requirements and
+these, in turn, have informed the assessment study for the mission and the payload. This study has been carriedout by JAXA, ESA and the consortium of the European far infrared instrument, SAFARI, for their respective
+contributed elements. In particular, ESA’s contribution to the SPICA mission consists of the cyrogenic tele-
+scope assembly, a ground station, a collaboration in the science operations and the management of the SAFARI
+instrument interfaces to JAXA.
+The 3.5 m SPICA telescope is the main ESA contribution to the mission. It needs to operate at a tem-
+perature of<6 K in order for SPICA to achieve the unprecedented sky-limited sensitivity. The image quality
+requires diffraction limited performance at 5 μm with a wavefront error <350 nm rms, and an unvignetted fieldExecutive Summary 3
+of view larger than 12 arcmin. More stringent requirements on image quality, telescope pupil obscuration and
+telescope transmission are imposed by the SPICA coronagraph, and have been considered as a delta in the study
+phase. Two parallel competitive contracts were awarded to EADS Astrium (F) and Thales-Alenia (F) for the
+industrial assessment study. The results have confirmed the Ritchey-Chr ´etien configuration as the most suitable
+to meet the performance requirements. Both contractors have identified a design entirely based on ceramicmaterial (SiC100 for Astrium and HB-Cesic for Thales) for the optical surfaces (M1 and M2), as well as forthe overall telescope structure. Both assessment studies have shown that the mass, heat load, and wavefront
+error budgets are compliant with the JAXA allocations and the scientific requirements, including those from
+the coronagraph. The reference development schedule of the SPICA Telescope assumes an early freezing of
+all design requirements during the ESA Definition Phase (A/B1). In this scenario, the delivery of the telescope
+to JAXA could take place by Q4/2016, leading to a launch in 2018. Inclusion of the coronagraph requirementswould extend polishing of an additional 6 months.
+The scientific instruments on board SPICA are: a mid infrared camera and spectrophotometer, MIRA-
+CLE (5 – 38μm, 6
+/prime×6/primeFOV), a mid infrared medium resolution spectrometer, MIRMES (10.32 – 36.04 μm,
+R∼900−1500), a mid infrared high resolution spectrometer, MIRHES (4 – 18 μm,R∼20 000−30 000), a
+mid infrared coronagraph, SCI (5(3.5) – 27 μm,R∼20−200, contrast 10−6, inner working angle 3.3λ/ Dtel),
+an optional far infrared spectrometer, BLISS (38 – 430 μm,R∼700), and a far infrared imaging spectrometer,
+SAFARI. SAFARI, which is to be built by a consortium of European institutes (with Canadian and Japanese
+participation), is an imaging Fourier Transform Spectrometer designed to provide continuous coverage in pho-tometry and spectroscopy from 34 to 210 μm, with a field of view of 2
+/prime×2/primeand spectral resolution modes
+R=2000 (at 100μm), R∼few hundred and 20 <R<50. The spectral sensitivity is required to be
+∼3×10−19Wm−2at 48μm( 5σ, 1 hour). Four detector technologies are undergoing development to meet
+these requirements: TES bolometers, Ge:Ga photoconductors, Si-bolometers and KID detectors. Several of the
+detector technologies have already demonstrated that they should be capable of delivering the required sensi-
+tivity for SAFARI and all are undergoing a development programme aimed at proving their compatibility with
+the SPICA system - the final detector selection will be made by mid 2010.
+SPICA will be launched with JAXA’s H-IIB from the Tanegashima Space Centre and is planned as a nom-
+inal three year mission (goal 5 years) orbiting at L 2. A large amplitude halo orbit is baselined, with a period of
+around 180 days and a semi-major axis amplitude of about 750 000 km. The thermal environment required by
+the telescope and the instruments is maintained by a combination of passive cooling (via dedicated solar andthermal shields combined with radiators) and active cooling, using a number of mechanical coolers to provide
+base temperatures of 4.5 K and 1.7 K. The total mass of the spacecraft is about 4 tonnes and it has a width of
+4.5 m and a height of 7.5 m.
+SPICA is an observatory open to the world astronomical community. The “Guaranteed” time for the
+instrument builders will be 25% of the total operational time; and 60% will be opened to the scientific com-munity (“Open time”), of which 22% will be reserved to astronomers from the ESA member states, 45% to
+scientists from Japan and other contributing countries and 33% to astronomers of any nationality. Proposals
+for “Legacy” and “normal” observing programmes will be selected by a single Time Allocation Committee
+through Announcements of Opportunity. A Science Advisory Committee will be established by JAXA, which
+will be chaired by the Japanese SPICA Project Scientist and in which the ESA Project Scientist will participate.In parallel, the ESA SPICA Science Team will be organised. JAXA will carry out the satellite and science
+operations in centres in Japan. A European SPICA Data Centre will be established, which will consist of the
+SAFARI Instrument Control Centre and the ESA SPICA Science Centre. The latter, located at ESAC, will be
+responsible for community support to the European astronomers for all observatory capabilities, and will be the
+interface between SAFARI and JAXA for operational and ground segment software development matters.
+JAXA will be responsible for the overall SPICA project. ESA will be the prime project partner, and will be
+responsible for the delivery of the Telescope Assembly and for the formal delivery of the SAFARI instrument.The SAFARI is a nationally funded, Principal Investigator provided instrument, led by SRON (The Nether-
+lands). A SPICA Steering Group will be established early in the programme. It will be chaired by a JAXA
+representative and will consist of members from JAXA, ESA and potentially other funding agencies. In 2010
+the SPICA Steering Group will draft the content of a formal agreement between both agencies.4 Executive Summary
+Page intentionally left blankChapter 1
+Scientific Objectives
+1.1 Introduction
+A full understanding of the formation and evolution of galaxies, stars and planets can only be found through
+the investigation of the cold and obscured parts of the Universe where the basic processes of formation and
+evolution occur. The deep exploration of the cold Universe through high spatial resolution observations in the
+Far Infrared (FIR) and sub-mm started in earnest in 2009 with the launch of the Herschel Space Observatory.
+Herschel will open a new window on this almost unexplored Universe through photometric surveys of star for-
+mation in our own Galaxy and of distant galaxies, however its relatively warm telescope (between 82 and 90 K)
+greatly limits its sensitivity. A new mission reaching better sensitivity must therefore follow to continue the
+work started by Herschel and gain a deeper understanding of the physics of the objects discovered by Herschel:
+i.e., to spectroscopically measure star formation rates in the obscured extra-galactic sources at di fferent cosmic
+epochs and explore the physics and chemistry of planet formation. To obtain this increase in sensitivity we needa cold (< 6 K) telescope of at least the same diameter as Herschel. The Japanese led SPace Infrared telescope
+for Cosmology and Astrophysics (SPICA) mission promises this in 2018 and Europe can play a vital role in itssuccess.
+Progress in mid- and far infrared astronomy has been slow because instruments and telescopes must be
+cooled to cryogenic temperatures to achieve high sensitivity and, for most of the frequency range, the observa-tions can only be made from the space. Only four small space observatories have operated in the past quarter
+of a century (IRAS, ISO, Spitzer and AKARI) offering limited spatial resolution and sensitivity. Herschel, with
+a 3.5-m 80-K telescope, the largest telescope ever put in space, is providing greatly increased spatial resolution
+with modest increases in sensitivity over the 55 – 210 μm band, and JWST, with a ∼6-m 45-K telescope, will
+provide a leap in both spatial resolution and sensitivity in the mid infrared (MIR) up to 28 μm. However, cov-
+erage of the full MIR/FIR band with high sensitivity and spatial resolution will still not have been achieved and
+it is here that SPICA will be a breakthrough mission. SPICA will have a similar size telescope to Herschel but
+cooled to<6 K, thus removing its self-emission and allowing observations limited only by the astronomical
+background.
+SPICA offers a sensitivity up to two orders of magnitude better than Herschel covering the mid-to-far-IR
+(the full 5 – 210μm range, mostly unreachable from the ground) with imaging, spectroscopic and coronagraphic
+instruments (see Figure 1.1). Thanks to this tremendous increase in sensitivity, SPICA will make photometric
+images in a few seconds that would take hours for Herschel and will take a full 5 – 210 μm infrared spectrum
+of an object in one hour that would take several thousand hours for Herschel. We illustrate this major increase
+in sensitivity in Figure 1.2 (left panel) which shows the area covered by a full spatial and spectral survey in
+which we can detect spectroscopically all galaxies down to a luminosity of ∼10
+11L⊙atz=1 and∼1012L⊙
+atz=2 in 900 hours. In theory, in about twice this time, the Herschel-PACS spectrometer would just be able
+to detect a single object over its full waveband to the same sensitivity. We can immediately see that this major
+increase in sensitivity, combined with a wide field of view and coverage of the full 5 – 210 μm waveband,
+will revolutionise our ability to spectroscopically explore the nature of the thousands of objects that Herschel,
+56 Chapter 1. Scientific Objectives
+Figure 1.1: Left panel: Photometric performance expected for SPICA (blue), compared to Herschel, ALMA and
+JWST (black), for a point source (in μJy for 5σ in 1 hour) using the goal sensitivity detectors on SPICA (NEP =2×
+10−19WH z−1/2). Note the∼2orders of magnitude increase in FIR photometric sensitivity compared to Herschel-PACS.
+The effects of confusion, and their mitigation, are discussed later. For illustrative purposes the SED of the starburst galaxy
+M82 as redshifted to the values indicated is shown in the background. Right panel: Spectroscopic performance expected
+for SPICA (blue) compared to predecessor and complementary facilities (black) for an unresolved line for a point source
+in W m−2for 5σ in 1 hour. For ALMA 100 km s−1resolution is assumed. The SPICA MIR sensitivities are scaled by
+telescope area from the JWST and Spitzer-IRS values respectively. Note that SPICA becomes more sensitive than JWST
+beyond 20μm.
+JWST, and SPICA will discover in photometric surveys.
+In the rest of this section we first place SPICA in context within the ESA Cosmic Vision science goals
+before discussing the critical importance of observing in the mid to far infrared and the diagnostic tools that are
+available. We then describe in detail how SPICA will shed light on the processes of planetary formation in thelocal Universe and the formation and evolution of galaxies in the more distant Universe.
+1.1.1 SPICA in the Cosmic Vision
+The ambition set out in the ESA Cosmic Vision (ESA BR-247 2005) is to seek “... the answers to profoundquestions about our existence, and our survival in a tumultuous cosmos”. To do this European scientists have
+identified four grand themes in space science that will bring us closer to understanding how the Universe has
+come to look as it does and the place of our Earth within the Cosmos. Three of these themes directly require
+observations in the mid to far infrared and a space based far infrared observatory is identified as a key facility
+within the aegis of Cosmic Vision. We recapitulate those themes here and briefly state how SPICA will directly
+contribute to furthering our knowledge of the Universe:
+Theme 1: “What are the conditions for planet formation and the emergence of life?” The Cosmic
+Vision calls for a mission that will “place the Solar System into the overall context of planetary formation,
+aiming at comparative planetology” and “Search for planets around stars other than the Sun”. SPICA will have
+a state of the art MIR coronagraph that will allow imaging and spectroscopy of young massive planets for the
+first time.
+There is also the call to, “Investigate star-formation areas, proto-stars and proto-planetary discs and find out
+what kinds of host stars, in which locations in the Galaxy, are the most favourable to the formation of planets”
+and “Investigate the conditions for star formation and evolution”. SPICA will have an FIR spectrometer more
+than an order of magnitude more sensitive than any previous facility to probe further and into a wider range of1.1. Introduction 7
+Figure 1.2: Left panel: Spectroscopic mapping speeds of the SPICA far infrared instrument (SAFARI) and Herschel-
+PACS superposed on a realisation of the Millenium simulation at z ∼1.4(Springel et al. 2006). In the centre are the
+footprints of the instantaneous spectroscopic FOV of PACS (blue) and SAFARI (yellow). The large green box shows the
+area covered by SAFARI in a 900 hour spectral full wavelength spectral survey (∼ 1 degree). In this survey SAFARI will
+see down to 5×10−19Wm−2over the full 34 – 210 μm band. PACS would require approximately twice this time just to
+cover a single pointing to this depth over its full waveband. Right panel: A selection of the fine-structure atomic and
+ionic lines accessible with SPICA, plotted as a function of critical density and ionisation potential. Using ratios between
+lines with different ionisation or critical density, we can trace out a wide range of different physical/excitation conditions.
+objects and regions than ever before.
+Theme 2: “How does the Solar System work?” This theme calls for the study of asteroids and comets
+as they are “the most primitive small bodies that can give clues to the chemical mixture and initial conditions
+from which the planets formed in the early solar nebula”. SPICA’s high sensitivity spectroscopy will allow thechemistry of these objects to be studied by remote sensing to an unprecedented level of detail, for the first time
+allowing us to link our own “debris” to that seen in the formation of planetary systems around other stars.
+Theme 4: “How did the Universe originate and what is it made of?” Here SPICA is ideally suited
+to “. . . trace the subsequent co-evolution of galaxies and super-massive black holes” and can “Resolve the farinfrared background into discrete sources, and the star-formation activity hidden by dust absorption”. The
+combination of MIR and FIR spectroscopy on SPICA is also essential to “Trace the formation and evolution of
+the super-massive black holes at galactic centres – in relation to galaxy and star formation – and trace the life
+cycles of chemical elements through cosmic history”.
+In the following sections we illustrate in detail how observations with the full suite of instruments on SPICA
+are essential to answering the questions posed in these themes and show how a European involvement in SPICA
+is an essential part of Europe’s Cosmic Vision for the 21
+stcentury. We first set the context of why observations
+in the mid to far infrared are critical to providing the observational evidence for galaxy, star and planetary
+formation and evolution.
+1.1.2 A critical waveband
+The composite spectrum shown in Figure 1.3 shows the importance of observations in the mid to far infrared.Here we see the spectral energy distribution (SED) from the X-rays to the radio of a typical galaxy under-
+going a modest amount of star formation, showing the different physical components at work. The bulk of
+the radiation is emitted either in the optical (i.e., starlight) or in the mid to far infrared where the dust in the
+interstellar medium absorbs and re-radiates the starlight as grey body radiation. Superimposed on the contin-8 Chapter 1. Scientific Objectives
+Figure 1.3: A synthetic spectrum of a typical galaxy undergoing modest rates of star formation – similar to the Milky Way
+– showing the parts of the spectrum that will be covered by SPICA and contemporaneous facilities (cf. F . Galliano). The
+major importance and spectral richness of the mid to far infrared region is demonstrated by the detailed spectra from ISO
+shown to the right of the diagram (Rosenthal et al. 2000; Goicoechea et al. 2004; Polehampton et al. 2007).
+uum throughout the5–2 1 0μm range are fine structure lines from ions, atoms and molecular lines which,
+due to their low excitation potentials and because they are not readily absorbed, control the thermal energy
+balance in a large variety of physical conditions. The gas reservoirs cool through an extensive network of
+atomic, ionic and molecular lines, which include the fine-structure lines of carbon and oxygen, ionic lines of
+neon and sulphur as well as many rotational transitions of molecules such as hydrogen (H 2), water, hydroxyl
+and carbon monoxide. The lines can be strong, emitting up to a percent of the total far infrared luminosity.Through observations of combinations of lines it is possible to disentangle the complex dependencies and so
+to start to characterise the physical properties that parameterise star-formation such as the nature and strength
+of the interstellar radiation fields, chemical abundances, local physical temperatures and gas densities. As well
+as the low excitation/ionisation tracers measuring the star formation component in galaxies, the MIR contains
+important diagnostic lines that can reveal the presence of Active Galactic Nuclei (AGN), such as the [Ne v]
+and [Oiv] emission lines. These lines are not severely affected by extinction, while optical and UV lines of
+forbidden and permitted transitions are heavily reddened and even X-rays are blocked by the large columns of
+hydrogen absorption in Compton thick AGN type 2 objects. Therefore observations in the mid-to-far infrared
+gives access to a large set of diagnostic line ratios to measure the physical conditions of astrophysical sources.Figure 1.2 (right panel) shows the mid- and far-IR lines from many different species, with a range of different
+ionisation potentials and excitation conditions. Used together, these transitions place constraints over a wide
+space of physical conditions and phases of the ISM, from the neutral atomic ISM, through the ionised ISM as
+seen in photo-dissociation regions and HII regions to the highly ionised AGN and “coronal” regions.
+These powerful diagnostics can be used to study both star and planetary formation very locally, and star
+formation and the influence of AGN in the very distant Universe. For example, the wavelength region between5 and 40μm contains transitions of many ionised species that can be used to differentiate between the role
+of Active Galactic Nuclei (AGN) and starburst regions in the evolution of galaxies (see section 1.6.1) and, as
+these and other MIR lines are redshifted into the FIR, they can trace the star formation history of the Universe.
+Similarly, the most abundant “metals”, C, N and O, have atomic and ionic fine structure emission lines in the
+FIR that allow direct, and unambiguous, determination of their relative abundance in distant galaxies. In the
+more local Universe, the MIR and FIR line ratios of these species can be used as direct probes of the temper-1.2. Planetary systems formation and evolution 9
+ature, density and UV fields present in star formation regions, Planetary Nebulae, Supernova Remnants and
+proto-planetary discs. In addition to the atomic and ionic species, important molecular reservoirs of H, C, N
+and O, such as H 2, OH, H 2O, CO, HC xN and carbon clusters (C x), have emission features throughout the mid
+to far infrared, which are uniquely diagnostic of the physical conditions of the sources where they arise and, inthe case of water, are critically significant in any discussion about how and where the conditions for life have
+emerged. The H
+2, OH, H 2O and high-J lines of CO can only be observed from space and require high sen-
+sitivity coupled with sufficient spatial and spectral resolution to distinguish the different environments within
+complex emission regions. Importantly for cosmogonological studies, the deuterated species of hydrogen bear-
+ing molecules (HD, HDO, OD, etc.) also have transitions through the mid and far infrared that can only be
+adequately accessed from space. Observation of these molecules in a variety of objects and comparison with
+the terrestrial deuterium ratio gives a direct insight into how the Earth formed and the origins of the oceans.
+Dust emission and absorption dominates the continuum spectrum in the MIR/FIR range with the most
+prominent features of MIR continua spectra being the UV excited emission bands of PolyAromatic Hydrocar-bon molecules (PAHs) at 6.2, 7.7, 8.6 and 11.3 μm. These PAH molecules appear to be omnipresent in all
+phases of dust evolution both in our own galaxy and the most distant galaxies yet seen using MIR spectroscopy.One of the most important features of PAHs is that they are excited by absorption of single UV photons; con-
+sequently, and in contrast to emission from larger grains, their output does not depend on their distribution
+around the heating source. Thus they are relatively insensitive to radiation from older stellar populations, mak-
+ing them good tracers of the UV field and so of star formation as a function of redshift. Also present in the
+5 –100μm range are diagnostic spectral features arising from the minerals that make up the larger interstellar
+dust grains, as well as solid state features from the ices that condense onto them under the right physical condi-tions. Particularly prominent in the MIR range are the “silicate” feature at ∼9.7μm – often seen in absorption
+in extragalactic spectra – and another at 20 – 24 μm. Some solid state features, for instance that at 69 μm from
+forsterite, can also be used as a direct measurement of the dust temperature. All these features are broad, faint
+and difficult to detect with the current or near future generation of facilities, which are either unsuitable for the
+task or do not have the full uninterrupted wavelength coverage required. Only the exquisite, and unrivalled,sensitivity of SPICA in the full mid to far infrared range will allow astronomers to trace the full story of dust
+evolution in our own and distant galaxies.
+This overview of the power of the mid and far infrared “toolbox” is of necessity brief and incomplete; in
+the remainder of this chapter we will give specific examples of how these tools can be used to answer some
+of the most vital questions about how and when planets, stars and galaxies came to form and their subsequentevolution into the Universe we see today.
+1.2 Planetary systems formation and evolution
+Modern astrophysics is just beginning to provide answers to one of the most basic questions about our place
+in the Universe: Are Solar Systems like our own common among the millions of stars in the Milky Way and,
+if so, what implications does this have for the occurrence of exoplanets that might give rise to life? The most
+straightforward method we have to start answering these questions is to compare our current knowledge of our
+own Solar system evolution with observations of planets and planetary disc systems around stars other than theSun. Observations of the most primitive bodies in the Solar System are also critical to infer the physical and
+chemical conditions in the early solar nebula as well as to provide clues about the water abundance, the most
+obvious solvent for life, its distribution, and its transfer from the outer Kuiper Belt regions to the inner terrestrial
+planet regions. The basic building blocks of an extra-solar system, the gas and dust, emit predominantly at mid
+infrared and far infrared wavelengths, and thus this is the critical domain in which to unveil the processes that
+transform the interstellar gas and dust into stars and into planetary systems.
+We discuss below how only the very sensitive observations over the entire mid infrared to far infrared
+(MIR and FIR) wavelength range provided by SPICA will give access to key spectral diagnostics and providea robust and multidisciplinary approach to determine the conditions for planetary systems formation. Such a
+comprehensive study will include the first detailed characterisation of hundreds of pristine bodies in our own10 Chapter 1. Scientific Objectives
+Solar System, the detection for the first time of the most relevant chemical species and mineral components of
+hundreds of transitional protoplanetary discs at the time when planets form, the spatial resolution of the water
+snow line in the closest and more evolved planetesimals debris discs, the first unbiased survey of the presence of
+zodiacal clouds in thousands of exoplanetary systems around all stellar-types and the first direct determinationof the chemical composition of outer exoplanet atmospheres. These and other challenging science goals require
+more than an order of magnitude sensitivity improvement in the FIR compared to Herschel and higher spectralresolution than provided by JWST in the mid infrared. Additionally, direct spectroscopic coronagraphy of
+exoplanets and planetary discs in the critical mid infrared domain is not planned in any space telescope other
+than SPICA.
+1.2.1 Protoplanetary and debris discs
+All planets are thought to form in the accretion discs that develop during the collapse and infall of massivedusty and molecular cocoons (/greaterorsimilar 10 000 AU) where stars are born. However, we still have a very incomplete
+understanding of the physical and chemical conditions in such discs, how they evolve when dusty bodies grow
+and collide, their mineral content, how they clear as a function of time and, ultimately, how planets as diverse
+as the Earth or hot-Jupiters form around different types of stars and at different places of the galaxy.
+Circumstellar discs are classified into 3 classes according to their evolutionary stage. Primordial pro-
+toplanetary discs, which are rich in atomic and molecular gas, and are composed of relatively unprocessed
+interstellar material left over from the star formation process. Such young discs are very optically thick in dust,
+with high radial midplane optical depths in the visual (τ
+V>>1). These protoplanetary discs start to become
+optically thin (τ V/similarequal1) in a few million years after formation (Haisch et al. 2001) and evolve into transitional
+discs as their inner regions begin to clear at /greaterorsimilar10 Myr. This is the critical intermediate stage when planetary
+formation is believed to take place, with dust particles colliding and growing to form larger bodies reducing thedisc opacity. Spitzer has shown that their outer regions (beyond ∼10−20 AU) can remain intact for longer,
+and thus residual gas can exist in the disc and play an important role in its evolution. In spite of its importance,the gas content in transitional discs at the first stages of planet formation is very poorly constrained, and many
+clues on these early stages of planetary formation can be provided by spectral line observations. Discs with
+ages above /greaterorsimilar10 Myr are thought to be practically devoid of gas (Duvert et al. 2000) and the dust in these older
+debris discs is generally not primordial but continuously generated “debris” from planetesimal and rocky bodycollisions. The smallest dust grains have, at this stage, either been dispersed or have coagulated into largergrains and the disc becomes very optically thin (τ
+V<<1).Debris discs are thus younger and more massive
+analogs of our own asteroid (hot inner disc, Td/similarequal200 K) and Kuiper belts (cool outer disc, Td/similarequal60 K) so their
+study is vital to place the Solar System in a broader context.
+The gas content in transitional, planet forming discs
+The physical and chemical conditions in young protoplanetary discs ( primordial andtransitional ) set the bound-
+ary conditions for planet formation, and an understanding of the formation and evolution of such discs will
+finally link star formation and planetary science. Planets are believed to form in transitional protoplanetary
+discs, which emit most of their energy in the MIR/FIR region (Boss 2008). Although the dust is relatively
+easily detected by photometric observations in the far infrared range, very little is known about the gas phase.
+It seems clear that an abrupt transition from massive optically thick discs to tenuous debris discs occurs at
+/greaterorsimilar10 Myr (Meyer et al. 2008), when the majority of at least the giant, gaseous planets (e.g., Jupiters and “hot
+Jupiters”) must have formed. The very fact that these planets are largely gaseous means they must be formed
+before the gaseous disc dissipated, making the study of the gas in discs essential to understand how and where
+they formed. For instance, the recently detected massive “hot Jupiters” orbiting close to the parent star are very
+unlikely to have formed in situ, but rather they must have migrated inwards from outer disc regions beyond the
+snow line. Likewise it seems clear that Neptune and Uranus must have formed in a region closer to the Sun and
+migrated outwards. The mechanism for either of these scenarios is not certain.
+According to our current theories, gas giant planets are thought to form in discs either via accretion of1.2. Planetary systems formation and evolution 11
+2) warm molecular layer
+3) midplane1) photodissociation layer external radiation field
+3) water ice: 44+62um bands2) H2O+OH+CO(high−J)
+migration?1)  [CII]157+[OI]63+[SI]56+[SiII]35
+X−ray, UV, opticalrocky planets formation zone?
+freeze−out of molecules
+("snow lines")gas giants formation zone?mineralogy: silicates, calcite, Fe/Mg ...
+mixing, gas dispersal...Sun−like star"Transitional" protoplanetary disks (~10Myr): spectral diagnostics for SPICA/SAFARI
+Figure 1.4: Diagram showing where radiation arises from a protoplanetary disc (at the time when planets assemble) and
+why the FIR (i.e., SAFARI) and the millimetre (e.g., ALMA) are essential to understanding the full picture of planetary
+formation and the primordial chemistry that leads to the emergence of life.
+gas onto rocky/icy cores of a few earth masses (Lissauer 1993; Pollack et al. 1996; Kornet et al. 2002) or by
+gravitational instability in the disc that triggers the formation of overdense clumps that afterwards compress
+to form giant planets (Boss 2003). In the latter scenario, gas giants form quickly and the gas may dissipate
+early (/lessorsimilar 10 Myr), whereas longer gas disc lifetimes ( /greaterorsimilar10 Myr) may leave enough time for building a rock/ice
+core and ease the subsequent accretion of large amounts of gas. Therefore, observations of the amount of gas in
+transitional discs around a large sample of stars can discriminate between the two most accepted planet forming
+theories. The residual gas content in the innermost regions (< 2 AU for a Sun-like star but much larger for more
+massive stars) at the time terrestrial rocky protoplanets assemble will determine their final mass, chemicalcontent and orbit eccentricity, and therefore its possible habitability (Agnor & Ward 2002).
+Protoplanetary disc models predict a “flared” disc structure (see Figure 1.4) which allows the disc to cap-
+ture a significant portion of the stellar UV and X-ray radiation even at large radii (Qi et al. 2006), boosting theMIR/FIR dust thermal emission and the MIR/FIR lines emission from gas phase ions, atoms and molecules.
+Theoretical models of protoplanetary discs recognise the importance of these X-UV irradiated surface layers,
+which support active photochemistry and are responsible for most of the MIR/FIR line emission. They are, in
+many ways, similar to the well studied interstellar photodissociation regions (PDRs). State-of-the-art 2D and
+3D disc/PDR models are nowadays able to simulate the disc dynamics, thermodynamics, chemistry and radia-
+tive transfer (Gorti & Hollenbach 2004; Aikawa et al. 2002; Gorti & Hollenbach 2008; Woitke et al. 2009a,b;
+Cernicharo et al. 2009) and they guide us in the interpretation of the observed disc emission (generally spa-tially unresolved). In particular, the FIR fine structure lines of the most abundant elements and metals (O, C,
+S, Si...) together with the FIR rotational line emission of light hydrides (H
+2O, OH...) are predicted to be the
+strongest gas coolants (i.e., the brightest lines) of the warm disc, especially close to the star. In particular theFIR [Siii]3 4μm, [O i]6 3μm, [Si]5 6μm and [C ii] 158μm fine structure lines are likely the most intense lines
+emitted in the disc, and thus the best diagnostic of the gas content in discs. By detecting these FIR lines we
+can directly probe a wide range of physical and chemical conditions (e.g., those associated with the UV-
+illuminated disc regions) that are very difficult, if not impossible, to trace at other wavelengths (e.g., by
+ALMA). Herschel can only search for the strongest FIR lines (above a few 10
+−18Wm−2) toward the brightest,
+closest and most massive young protoplanetary discs. In fact, only a few discs will be fully surveyed in the
+FIR domain, i.e., those with strong dust continuum which unfortunately hampers line detection at low spectral
+resolution. In particular, Herschel does not reach the sensitivity to either detect the gas emission in the much
+more tenuous transitional discs or detect the predicted line emission of less massive, and thus more numerous,
+discs around cooler stars.
+SPICA studies of the gas dispersal and of the chemical complexity in protoplanetary discs:
+In order to shed some light on the gas dispersal time scales, and thus on the formation of gaseous Jovian-type
+planets, high sensitivity infrared to sub-mm spectroscopic observations over large statistical stellar samples
+tracking all relevant disc evolutionary stages and stellar types are clearly needed. To date, studies of the gas
+content are biased to young and massive protoplanetary discs (probably not the most representative) through
+observations with ground-based (sub)mm interferometers and optical/near-IR telescopes. Sub-mm observations12 Chapter 1. Scientific Objectives
+allow one to trace the outer cooler disc extending over a few hundred AU where most molecular species start
+to freeze-out onto dust grains (Dutrey et al. 2007). Recently the very inner regions of protoplanetary discs have
+been probed by means of optical/IR observations (Najita et al. 2007). Spitzer and ground-based telescopes have
+just started to show the potential diagnostic power of MIR/FIR spectroscopy in a few “template” discs. This has
+allowed the exploration of the gas content and composition at intermediate radii (1 −30 AU), i.e., the crucial
+region for the formation of planets. Recent MIR detections toward young discs include atomic lines such as
+[Neii], [Feii], etc. (Lahuis et al. 2007), molecules like H 2,H2O, OH, HCN, C 2H2and CO 2(Carr & Najita
+2008; Salyk et al. 2008), and complex organics like PAHs (Geers et al. 2006; Habart et al. 2006).
+Molecular hydrogen (H 2) is the most abundant gas species in a primordial protoplanetary disc (∼ 90% of the
+initial mass). Due to the relatively poor line sensitivity that can be achieved with even the largest ground-basedmid infrared telescopes (a few 10
+−17Wm−2), H 2has been detected only toward a few protoplanetary discs
+so far (Bitner et al. 2008). JWST will improve this situation enormously and many more discs will have beenobserved by the time SPICA is active. To complete and complement the JWST capability, SPICA’s mid infrared
+high resolution echelle spectrometer (MIRHES) will be able to detect the brightest H
+2line (the v=0-0 S(1) line
+at∼17μm) with∼10 km s−1resolution with much higher sensitivities than those achieved from the ground
+and with an order of magnitude higher spectral resolution than JWST. Such a high spectral resolution will be
+enough to start resolving the gas Keplerian rotation, trace the warm gas in the inner ( <30 AU) disc regions and
+be sensitive to H 2masses of the order of ∼0.1 M earth. However, given the peculiar excitation conditions of H 2,
+its emission does not sample the whole circumstellar disc. Instead, the deuterated isotopologue of molecular
+hydrogen, HD (with the lowest energy lines at 112, 56 and 37 μm), although a few thousand times less abundant,
+can serve as a proxy of H 2column density. The D/H abundance ratio of different species can also be used as a
+diagnostic of the interstellar origin of the young disc material.
+In terms of chemistry, the protoplanetary disc is the major reservoir of key species with prebiotical rel-
+evance, such as oxygen, ammonia (NH 3), methane (CH 4) or water (H 2O) to be found later in (exo)planets,
+asteroids and comets. But how the presence and distribution of these species relates to the formation of planets,
+and most particularly, rocky planets with substantial amounts of water present within the so called habitable
+zone, remains open to speculation without a substantial increase in observational evidence. Water is an obvious
+ingredient for life, and thus it is very important that we understand how water transfers from protostellar clouds
+and primordial protoplanetary discs to more evolved asteroids, comets and planets like our own. Ultimately
+one has to understand how the water we see today in our oceans was delivered to the Earth.
+At a distance of∼50 pc from the Sun, a given gas line will produce a line flux of ∼10−19(Lline/10−8L⊙)Wm−2
+where Llineis the radiated power in the line expressed in solar luminosities. SAFARI will be able to detect gas
+lines with luminosities /greaterorsimilar10−8L⊙in nearby, isolated discs, or /greaterorsimilar10−7L⊙in more distant discs in the closest star
+forming regions at ∼150 pc (see Figure 1.5). State-of-the-art models of transitional discs around Sun-like stars
+and with gas masses as low as ∼0.001 M Jup(∼0.3 M earth) predict luminosities around 10−8L⊙for [Oi]145μm,
+[Si]56μm and H 2O, OH or CO high- Jemission lines (Gorti & Hollenbach 2004). The [O i]63μm line is ex-
+pected to be the brightest line (10−6−10−7L⊙), even for modest gas masses, and therefore it is one of the
+most robust gas tracers in transitional discs. The exciting prospect of detecting this small amount of gas in a
+statistically significant sample of discs, e.g., toward the 6 closest (/lessorsimilar 150 pc) young stellar clusters with ages of
+/similarequal1−30 Myr, Taurus, Upper Sco, TW Hya, Tuc Hor, Beta Pic and Eta Cha, will provide a crucial test on the
+lifetime of gas and its dissipation timescales in planet forming discs and therefore on the exoplanet formation
+theory itself.
+Nevertheless, the angular resolution of SPICA (∼ 4/prime/primeat the [O i]63μm line) can not compete with the
+resolution of telescopes with much larger collecting area (e.g., ALMA) and thus most protoplanetary discs willbe spatially unresolved (a young disc like TW Hya, at a distance of ∼50 pc, has a size of ∼10
+/prime/primeon the sky).
+However, SAFARI has access to the critical FIR domain that can not be observed with ALMA or JWST, thusenabling the detection of unique diagnostics of the disk (dust, ice, atomic and molecular features). SAFARI
+will also permit us to: (1) Carry out deep searches of gas towards large samples of discs (including more
+evolved debris discs) by e.g., searching for the bright [O i]63μm line and (2) take advantage of the broadband
+coverage of SAFARI-FTS to obtain unbiased FIR spectral line surveys. Even if the line emission can not
+be fully spatially resolved, SPICA observations will allow us to detect the presence of water vapour and the1.2. Planetary systems formation and evolution 13
+WATER VAPOR ABUNDANCE
+"snow line "a)b )c )Expected gas line emission in transitional protoplanetary disks with SPICA/SAFARI
+Figure 1.5: Predicted fluxes of key FIR cooling lines in transitional protoplanetary discs (∼ 10 Myr) around (a)a cool
+K star and (b)a G solar-type as a function of distance. The horizontal lines indicate 5σ-1hr detection limits of SAFARI.
+Dashed lines correspond to sensitivities using narrow band filters on a restricted set of line wavelengths. The total gas
+mass in the disc is low ∼10−2MJup. SPICA will have the sensitivity to observe the gas content of transitional discs in the
+closest star forming regions (∼ 150 pc) at the time planets are thought to assemble (adapted from Gorti &Hollenbach
+2004). (c)Water abundance model for a young Herbig/Ae protoplanetary disc (Woitke et al. 2009a,b). Three different
+regions with high water vapour abundance can be distinguished and be traced by different MIR/FIR H 2O lines with
+MIRHES and SAFARI: 1) the inner water reservoir where rocky planets form, 2) the distant water belt, and 3) the hotwater layer. The “snow line” refers to the radii where water is expected in ice phase (and can be directly traced by the
+∼44μm and∼62μm water ice bands with SAFARI). Gas giant exoplanets are thought to form beyond there, and migrate
+inwards.
+basic building blocks of the UV illuminated-disc chemistry (C+,C H+, CH, O, OH, Si+, S, HD, ...) for the first
+time in hundreds of transitional discs and, together with disc/PDR models, relate their presence with the main
+parameters of each planetary system (stellar mass, temperature, age ...).
+In summary, SPICA spectrometers will provide continuous coverage from ∼5t o2 1 0μm with medium
+to high spectral resolution and high sensitivity to detect the gas line emission (H 2,[ Oi], H 2O, HD...) from
+hundreds of tenuous planet forming discs and thus to infer their gas masses, chemical composition, etc. Ad-
+ditionally, the MIR coronagraph (SCI) will cover the ∼5−27μm range continuously, providing high stellar
+suppression within a small inner working angle (IWA /similarequal1/prime/prime). Instantaneous images of entire planetary systems
+with SCI will reveal their morphology in great detail and also will allow us to carry out MIR spectroscopy.
+Planetesimals, dust and mineralogy in debris discs
+Dust is a key building block of rocky planets and the cores of giant gas planets. Dust appears to be present
+at all stages of planetary system formation with the mass ratio of gas to dust, the amount of dust present,
+its temperature and its distribution evolving rapidly through the process of planetary formation (Tanaka et al.
+2005; Su et al. 2005). The processing and evolution of dust from protoplanetary discs to evolved solar systems
+like ours is key to understanding the formation and mineralogy of rocky, Earth-like, planets. For instance, it
+appears that in discs around intermediate mass, pre-main sequence stars, the dust in the inner regions of the
+disc (≤ 2 AU) can be more evolved. That is, it shows signatures of grain growth and crystallisation whereas
+the dust in the outer region is often seen to be pristine and similar to the interstellar dust (Natta et al. 2006).
+This implies a strong radial dependence of the dust processing whereby the inner regions are dominated by
+the stellar radiation field, leading to grain heating and crystallisation, and by higher densities, leading to co-
+agulation, whilst the outer regions remain largely unprocessed and carry the signature of the pre-stellar nebula
+from which the star formed. And yet in our own Solar System we see crystalline silicates present in comets
+that clearly originate from regions far from the zones where this processing must have occurred. Only with de-
+tailed mapping of the MIR/FIR mineral and ice band emission/absorption of our own and distant circumstellar
+material, will we be able to understand the evolutionary track that leads to this situation. Observations using14 Chapter 1. Scientific Objectives
+SPICA spectrometers will allow us to carry out mineral and ice spectral studies (started with ISO and followed
+by Spitzer and AKARI) with unprecedented sensitivity and angular resolution.
+The final stage of planetary system formation, the formation of small planetesimals that sweep up much
+of the disc’s material, together with the formation of larger planetary bodies via collisions of planetesimals,often appears to result in an almost gas-free, “second generation” dusty disc, the debris disc. The dust in
+these discs is produced by mutual collisions of planetesimals in the final stages of planetary formation and
+the “heavy bombardment” phase (> 300 Myr) evidenced in the impact craters seen in all rocky Solar System
+planets. Debris discs can survive over billions of years. This points towards the presence of large reservoirsof colliding asteroids and evaporating comet-like bodies. Moreover, since colliding planetesimals need to be
+present to replenish the dust grains in the disc, detecting a debris disc is a strong signature of an emerging
+planetary system and is indicative of the presence of analogous asteroid and Kuiper belts (Wyatt 2008), or of
+regions where the formation of Earth-like or Pluto-like planets is ongoing (Kenyon & Bromley 2008). Indeed,
+the very recent direct detection of exoplanets towards several stars that host bright debris discs (Kalas et al.
+2008; Marois et al. 2008; Lagrange et al. 2009) have quantitatively confirmed that studies of debris discs are
+critical to advance in our understanding of the formation and diversity of extra-solar planetary systems.
+The dust emission in circumstellar discs produces an “excess” of FIR continuum emission that becomes
+apparent and reaches its maximum in the SAFARI wavelength range, which is not seen in the shorter wavelengthmid infrared range covered by JWST (see Figure 1.6a). The exact wavelength location of the disc dust emission
+peak depends on the grains temperature, size distribution and composition. Around ∼300 debris discs have been
+photometrically discovered so far with ISO and Spitzer. Although there are very few debris disc detectionsaround stars later than K2, this is likely an observational bias because these discs would have been too cold and
+faint to be detected. This large discovery space of discs around cool stars is yet to be explored. The most recent
+FIR photometric census suggests that about 10% of lower-mass, Sun-like stars, are surrounded by debris discs,
+at least down the limiting fluxes observables with Spitzer (Meyer et al. 2008).
+The study of debris discs is an emerging field that is currently limited by the available FIR sensitivity
+which makes it impossible to (1)perform large statistical surveys over meaningful star volumes; (2)probe
+more distant discs and/or less massive dusty discs down around all types of stars including young M dwarfs
+and cooler substellar objects. In particular, the sensitivity required to detect the dust disc over stellar luminosityratio ( L
+dust/Lstar) in a planetary system analogous to the Kuiper Belt disc ( Ldust/Lstar/similarequal10−7) or Asteroid Belt
+(Ldust/Lstar/similarequal10−8) in our Solar System remains far below the far infrared capabilities provided by IRAS,
+ISO, AKARI, Spitzer and Herschel (see Figure 1.6b). Note that the flux of a Kuiper Belt disc with a mass of
+∼0.1 M earthat 30 pc corresponds to ∼4 mJy at 70μm (Meyer et al. 2008). In other words, due to sensitivity
+constraints, previous and current FIR telescopes are probably only showing us the “tip of the iceberg” as far asthe detection and characterisation of extra-solar planetary discs is concerned.
+SAFARI offers two powerful methods to detect and characterise the dust emission from debris discs (1)fast
+and ultra sensitive photometric searches in the 48 μm, 85μm and 160μm bands simultaneously reaching the
+/lessorsimilar100μJy level sensitivity for 5σ in 1 min and (2)very sensitive spectral characterisation at modest spectral
+resolution (R∼100) over the full 34 – 210 μm waveband with ∼1 mJy level sensitivity for 5σ in 1 hour.
+Both modes take advantage of a large instantaneous FOV for efficient wide area surveys of multiple objects.
+Figure 1.6c shows the minimum detectable fractional luminosities ( L
+dust/Lstar) for debris discs at different
+distances as observed with Spitzer-MIPS at 70 μm, Herschel-PACS at 70 μm and 100μm, and SPICA/SAFARI
+at 48μm and 85μm. For solar-type stars, we see that Herschel will not detect dust at the Kuiper or Asteroid
+belt levels even for the closest discs, and debris disc detections at 70 μm have generally been limited to ∼100
+times the luminosity of the dust in the Kuiper belt. Hundreds of nearby main sequence stars, ∼1000 stars
+atd<25 pc, without known far infrared excesses will be targets for SAFARI to search for the faintest dust
+excesses and the presence of dusty belts indicating the presence of hitherto undetected planetary systems.
+The ability of SAFARI to survey and detect large numbers of FIR excesses will represent a revolution in
+debris discs studies because of the large number stars that harbour debris discs that may be detected. Modelsand observations indicate that debris discs might be very common and there are ∼10
+5F0 – K2 stars within
+∼150 pc, a number to be compared to the ∼190 debris discs around these type of stars known to date. SAFARI
+will increase the number of debris discs detections around solar-type stars by 3 orders of magnitude, allowing1.2. Planetary systems formation and evolution 15
+SPICA/SAFARI@48,85umDetecting Kuiper and Asteroid Belt analogs in distant extrasolar systems  
+3sigma−1min 3sigma−1hrSPICA/SAFARI@48,85umSpitzer/MIPS@70um
+Herschel/PACS@70um
+Tdust (K) Tdust (K)Ldust / LstarAU Mic
+KBABc)
+KB ABa) Sun−like star (G2 V ) Cool star (M0 V)
+distance =10 pc
+distance = 30 pc
+distance = 140 pcb)
+SAFARI domain
+Figure 1.6: (a)SED of HD105 star. The lack of excess emission at MIR wavelengths is due to the depletion of dust
+at distances<35 AU from the central star. The FIR wavelengths covered by SAFARI are thus critical (adapted from
+Meyer et al. 2008). (b)Distribution of fractional luminosities, L dust/Lstar, for the number of debris discs known to date.
+The characteristic values for the Kuiper and Asteroid belts (KB and AB) are shown at the left (Moro-Mart´ ın 2009). (c)
+Minimum detectable fractional luminosities for discs at different distances around stars of different types: cool M0 V
+(left) and solar G2 V (right) – with characteristic values for the KB and AB shown as black squares. The different colours
+correspond to different far infrared instruments (3σ-1hr): Spitzer-MIPS at 70 μm (blue); Herschel-PACS at 70 and
+100μm (red); and SPICA/SAFARI at 48 and 85 μm (black). The thick/thin lines for SAFARI refer to 1hr/1min integrations
+(Moro-Mart´ ın 2009).
+realistic statistics of the disc frequencies and properties as a function of stellar type, age and environment. In
+particular, SAFARI will be the first instrument to detect Kuiper Belt analogues out to ∼150pc(see Fig-
+ure 1.6c). As discussed before, this is the distance to the closest star forming regions, meaning that SAFARI
+will be particularly adept at studying discs during the brief epoch at /greaterorsimilar10 Myr when the transition from
+protoplanetary to debris disc occurs, a transition which is as yet poorly understood but which is of primeimportance due to its curtailment and direct link with planet formation.
+SAFARI will be so sensitive that it will be perfectly adapted to detect the faintest and maybe the most
+profuse planet forming systems in the Galaxy, those around cool stars and brown dwarfs (BDs) (Rebolo et al.1995; Teixeira et al. 2009). Note that ∼77% of stars in the local neighbourhood are cool M-stars. In fact
+there is observational evidence that at least 3% harbour planets and that “super-earths” are more common than
+giants. Figure 1.6 shows that debris discs orders of magnitude fainter than that around AU-Mic (at 9.9 pc)
+will be detectable, and there are ∼160 M-type stars within ∼10 pc. These studies will open new frontiers in
+planetary studies: What kind of planets could eventually form around such cool stars? Will they be habitable?The detection of the faint far infrared excesses around cool stars and brown dwarfs is a great challenge for
+current instrumentation, with disc fluxes around mJy or less. However, SAFARI will routinely observe them
+and massively increase our knowledge of how these enigmatic objects are formed.
+Very sensitive spectrophotometric characterisation of circumstellar discs:
+Despite the poorer angular resolution of SPICA compared to the future facilities (e.g., ALMA or SKA), much
+will be learnt by going beyond photometric detections and carrying out extensive MIR/FIR spectroscopy stud-
+ies of the strongest dust/ice features in a large sample of debris discs. SAFARI will provide the continuous FIR16 Chapter 1. Scientific Objectives
+SAFARI  FOV~2’x2’
+Vega disk at 70umSAFARI PSF @ 70umSPICA/SAFARI
+Water ice FIR featuresProtoplanetary disk
+ISO (R~150)around HD142527
+Herschel/PACS JWST/MIRIa) b)
+Spitzer PSF
+50 0 −50−50050c)Resolving the "snow line" in nearby Vega−like discs + detailed mineralogy of hundred’s of debris discs
+SAFARI FoV Vega disc @ 70um (Spitzer)
+Figure 1.7: (a)ISO spectrum of the disk around HD142527 (Malfait et al. 1999). Note that amorphous water ice only
+shows bands at∼44μm in the FIR (Moore &Hudson 1992). Such features can not be accessed with Herschel or JWST.
+SPICA will take the equivalent spectra of objects at flux levels less than ∼10 mJy per minute. (b)Image of Vega debris
+disk at 70μm with Spitzer (Su et al. 2005). SAFARI’s large FOV and smaller pixel size ( ∼1.8/prime/primeat∼44μm, red squares)
+will provide very detailed spectroscopic images of nearby disks. (c)Simulated image with SAFARI at 70 μm assuming an
+inner hole in the disk with a radius of ∼11/prime/prime.
+spectral energy distribution (SED) and be able to detect both the dust continuum emission and the brightest
+grain/ice bands as well as the brightest lines from any gas residual present in the disc. The ability of FIR spec-
+troscopy to determine the mineral makeup of dusty discs around young stars is illustrated by the ISO spectrum
+of HD142527 shown in Figure 1.7a. SPICA will be hundreds of times more sensitive than ISO and so will see
+not only the young and massive opaque circumstellar discs like this, but will be able to trace the mineralogy
+of dust within discs at all stages of planetary system formation. SPICA will not only determine the detailed
+mineralogy of discs, but will also trace the variation grain size distribution and temperature, which are both
+expected to evolve with disc age leading to a variation in the disc SED. Models suggest that there is little pre-
+dicted evolution of the mid infrared SED with disc age and a possible ambiguity in the mid infrared intensity
+between age and disc structure which is removed with observations in the far infrared (Tanaka et al. 2005). A
+key step forward needed to cope with all model complexities (Dullemond et al. 2001) and to fully characterise
+the nature of dusty discs is to obtain full SED wavelength sampling (at least in the critical far infrared domain)
+in order to accurately determine the emission peak, slope, and dust spectral features, and so be able to constraingrain sizes and dust opacities, critical ingredients to determine the disc mass. Either in photometric or in spec-
+troscopy modes, SAFARI will be the true “hunter of debris discs” of the next decade.
+Water Ice in discs:
+Below gas temperatures of ∼150 K water vapour freezes-out onto dust grains and the main form of water in
+the cold circumstellar disc midplane and at large disc radii will be ice. The physical location of the point at
+which water freezes out determines the position of the so called “snow line”, i.e., the water ice sublimation
+front, which separates the inner disc region of terrestrial, rocky, planet formation from that of the outer giant
+planets (Nagasawa et al. 2007; Woitke et al. 2009a,b) (see model predictions in Figure 1.5c). Grains covered
+by water icy mantles can play a significant role in planetary formation, enabling the formation of planetesimals
+and the core of gas giants protoplanets beyond the snow line. Observations of the Solar System’s asteroid belt
+suggest that our snow line occurred near a disc radius of ∼2.7 AU (Lecar et al. 2006). In the outer reaches of
+our own Solar System, i.e., beyond the snow line, most of the satellites and small bodies contain a significantfraction of water ice; in the case of comets this fraction is as high as 80% and the presence of water in the
+upper atmospheres of the four gas giants is thought to be highly influenced by cometary impacts such as that
+of Shoemaker-Levy 9 on Jupiter. It is possible that it is during the later phases of planetary formation that the
+atmospheres, and indeed the oceans, of the rocky planets were formed from water ice contained in the comets
+and asteroids that bombarded the inner Solar System. For more distant exoplanetary systems, very little is
+known as (1) we first need to detect the presence of water-ice and infer its abundance in a large sample of
+protoplanetary systems (spanning a broad range of host star types and ages) and (2) even for the closest discs,1.2. Planetary systems formation and evolution 17
+the exact location of the snow line is very hard to resolve spatially with current instrumentation.
+In the far infrared there is a powerful tool for the detection of water ice and determination of the amor-
+phous/crystalline nature: Namely the transverse optical mode at ∼44μm both from crystalline and amorphous
+water ice and the longitudinal acoustic mode at ∼62μm arising only from crystalline water ice (Warren 1984;
+Moore & Hudson 1992; Maldoni et al. 1999). In contrast to the mid infrared ∼6.1μm water ice feature (that has
+a stronger band strength and that SPICA MIR spectrometers will observe with sub arcsec angular resolution),
+the difference between the amorphous and crystalline phase is very well defined in the FIR (see Figure 1.7a)
+and, again unlike the MIR features, the FIR bands are not confused with other solid state features of less abun-dant species. In optically thin discs it is extremely difficult to use MIR absorption to trace water ice and the
+material is too cold to emit in the NIR/MIR bands. Hence, these strong FIR features are robust probes of (1)
+the presence/absence of water ice, even in cold or heavily obscured or cold regions without a MIR background,
+and(2)the amorphous/crystalline state which provides clues on the formation history of water ice. Note that
+JWST cannot access these FIR ice solid-state bands (Moore & Hudson 1994).
+First observed in emission towards the Frosty Leo Nebula (Omont et al. 1990), water ice has been detected
+in young protoplanetary discs in a few bright sources either in the FIR using the ISO-LWS (Dartois et al. 1998;
+Malfait et al. 1999) or MIR (Pontoppidan et al. 2005; Terada et al. 2007). The FIR features were also observed
+using ISO-LWS in comets within our own Solar System (Lellouch et al. 1998). Since the bands change shape
+i.e., they narrow for crystalline ice and the peak shifts in wavelength with the temperature, relatively high
+spectral resolution is needed to extract all the available information from the ice band profiles. In its highest
+resolution mode SAFARI provides R∼4500 at∼44μm which is appropriate for very detailed ice spec-
+troscopy studies. Indeed SPICA is the only planned mission that will allow water ice to be observed in all
+environments and fully explore its impact on planetary formation and evolution and the emergence of habit-
+able planets. In many cases, the absorption from other features (e.g., water ice at ∼6.1μm, but also methanol,
+carbon dioxide, formaldehyde and methane ice) will be studied also with complementary MIR spectra takenwith SCI and MIRMES. In combination with SAFARI observations, such complete MIR/FIR ice spectra will
+accurately constrain disc models, determine for how long ice mantles survive under the irradiation of differentstellar UV fields (Grigorieva et al. 2007) and determine the thermal history of the ice crystallinity.
+Spatially resolved debris discs (resolving the snow line in discs?):
+Images of the few spatially resolved debris discs, either seen in reflection or directly in the FIR or sub-mm,
+can show gaps and ring-like structures indicating the presence of planets which “shepherd” the dust. This has
+been most readily observed so far using ground based near-IR scattered emission and thermal MIR emission
+(e.g., Subaru, Gemini; Fukagawa et al. 2006; Fujiwara et al. 2006) and sub-mm and mm telescopes (JCMT,
+CSO and IRAM). However, these facilities cannot detect specific dust grain spectral features, nor the FIR water
+ice bands. ALMA will undoubtedly add a great deal to the subject by detecting the optically thin sub-mm con-
+tinuum emission of the cold dust, but will not be very efficient in mapping the very extended emission of the
+closest discs which can extend to a few arcmin in size. Spitzer has revealed that the size of some disc systems,such the Vega debris disc, is surprisingly larger in the FIR (e.g., at the emission peak) than in the sub-mm.
+Obviously this is telling us something about the disc nature and it is clear that only through multi-wavelength
+observations that we can fully understand the complete picture of the formation and evolution of these discs.
+SPICA will greatly increase the number of debris discs that can be spatially resolved (only ∼20 currently),
+allowing us to break the degeneracy in the spectral energy distribution between dust location and mineral
+composition. For instance, at the 48 μm photometric band the expected angular resolution is 3.5
+/prime/primeallowing a
+100 AU disc to be just resolvable at ∼28 pc. The number of stars within that distance is approximately ∼210
+(A),∼1100 (F0-K2) and ∼3600 (K2-M). In the closest ones, we will be able to indirectly infer from observa-
+tions of clumpy/warped/asymmetric dust structures that planets are gravitationally perturbing the planetesimals
+and dust (Wyatt et al. 1999; Wyatt 2003; Kalas et al. 2005). In nearby debris discs ( d<10 pc) such as those
+around Vega or Formalhaut ( ∼100 A-type stars), the spatial resolution of SPICA at the 44 μm feature (∼25 AU)
+will be sufficient to spatially resolve the distribution of water ice – the snow line (Figure 1.7). In all these
+“Vega-like” systems the expected snow line is pushed away to ∼22 to 44 AU, or even larger radii (Ida & Lin
+2005; Grigorieva et al. 2007; Kennedy & Kenyon 2008).18 Chapter 1. Scientific Objectives
+Herschel will only be able to observe the ∼62μm water ice band, i.e., only regions where crystalline water
+is present in large quantities. Even then, observing a broad indistinct spectroscopic feature with the narrow
+band Herschel-PACS grating spectrometer will be difficult and the sensitivity will be very much worse than
+SAFARI with inferior spatial resolution. SAFARI will be so sensitive that it will not only be able to producefully sampled spectroscopic images of the ∼44μm and∼62μm water ice bands, but also of any “secondary”
+or “residual” gas content (e.g., by detecting FIR OH and [O i] lines) produced by the photoevaporation of ice
+grain mantles, outgassing of comets or collisional evaporation. Only with SPICA will we be able to confirmthese predictions and provide the critical tests on which to base future models of the formation of planetary
+systems. All in all, the study of discs with SPICA spectrometers will shed light on the diversity of planetary
+systems, the link between circumstellar discs and planets and the link between extra-solar planetary systems
+and our own.
+1.2.2 The inner and outer Solar System “our own debris disc”
+The observation of circumstellar discs around distant stars at different evolutionary stages provide clues on how
+our planetary system was formed, from which materials, and how they were processed. However, in order to
+understand the observed diversity of extra-solar planetary systems, we also need to explain how our own Solar
+System emerged. Clearly both approaches complement each other and represent intimately and increasingly
+interconnected research fields. The study of the Solar System extends from the traditional investigation of the
+planets and their rings and moons, to the most recent characterisation of the different populations of primitive
+leftovers of their construction (comets, asteroids, Kuiper belt bodies, etc.). Finally, investigations of how life
+came to exist on Earth are fundamental and becoming of great general interest.
+Our current view of the Solar System’s early evolution is based on the “Nice Model” (Gomes et al. 2005).
+This model argues that, after a relatively slow evolution, the orbits of Saturn and Jupiter crossed their mutual
+2:1 mean motion resonance about 700 Myr after formation, which caused a violent destabilisation of the orbits
+of planetesimals throughout the disc. This event populated the Kuiper belt (> 40 AU) with different families
+of leftover bodies and delivered pristine, icy planetesimals to the inner Solar System. This model agrees with
+the geochemical evidence (e.g., in the Moon) of a very peaked cratering rate at that time, and is generally
+known as the late heavy bombardment period (or LHB). Because there is also evidence of planet migration
+and planetesimal belts in extra-solar systems, a natural question arises: Are LHB-type events common in other
+planetary systems? From a broader astrobiological point of view, those primitive bodies coming from the outer
+Solar System (with little or no chemical processing) could have delivered significant amounts of volatiles and
+chemical species to the inner rocky planets (e.g., water and organic matter) that are relevant for the habitability
+of such planets. The study of the enigmatic nature of the outer Solar System is very challenging because (1)at
+such large distances from the Sun, rocky and icy bodies are cold, below /similarequal60 K, therefore their thermal emission
+peak occurs at FIR wavelengths that cannot be observed from the ground and (2)the FIR fluxes are very weak
+(a few mJy and below). As we shall describe hereafter, the very sensitive instruments on board SPICA, together
+with their broadband spectroscopic capabilities, will provide a new perspective of the Solar System’s outermost
+belts, the regions that hide a record of the earliest phases of the solar nebula.
+Studying the Kuiper Belt object by object
+Kuiper Belt Objects (KBOs) refer to the physico-chemically unaltered population of bodies beyond Neptune’s
+orbit. Since the discovery of the first KBO (Jewitt & Luu 1993), more than 1200 objects have been detectedso far in the outer Solar System (> 30−40 AU), including planetary-sized objects such as Eris (Bertoldi et al.
+2006). Several thousands of KBOs are expected to exist, especially at high ecliptic latitudes. Unlike asteroidsin the inner Solar System, KBOs must have formed relatively slowly and they are thought to be composed of
+pristine, almost unprocessed chemical material. The detailed study of their physical parameters (temperature,
+size distribution and albedo) and chemical characteristics (mineral/ice content) is thus fundamental to consis-
+tently link the history of our system with that of distant extra-solar systems. Unfortunately, most of the KBOattributes and chemical composition are almost unknown. Together with the inner asteroid belt, these remnant1.2. Planetary systems formation and evolution 19
+a) b)Detection and Characterization of outer Solar System Objects with SPICA/SAFARI
+Herschel/PACS Herschel/PACS
+SPICA/SAFARI SPICA/SAFARI
+Figure 1.8: (a)Diameters (in km) of known outer Solar System objects (SSOs), that emit at FIR wavelengths, as a function
+of their heliocentric distance. The coloured curves display detectability estimations for the different FIR photometric
+bands of Herschel-PACS (dashed) and SPICA/SAFARI (continuous). Predictions take into account that SSOs are movingobjects so that they can be detected below the FIR confusion limit by pair-subtracting within a reasonable time interval.
+(b)Same but for spectroscopy, which will allow us to detect the mineral/ice content of the most primitive SSOs for the first
+time (updated from Hasegawa 2000).
+planetesimal belts are analogues of the cold debris discs observed as FIR photometric excess towards more
+distant stars. In this sense, the outer Solar System provides the closest “template” to study the composition,
+processing and transport of minerals, ices and organic matter by studying debris disc bodies “one by one” .
+Spitzer-MIPS has detected a few KBOs photometrically at 24 and 70 μm by observing from minutes to
+hours per target. Measurements of the far infrared thermal emission of KBOs (where the bulk of the KBOs’
+energy is radiated) reveal the thermal properties of the near-surface layers (Stansberry et al. 2006). Comple-
+mentary observations in the visible are needed to establish the position accurately, determine the objects albedo,
+and, in combination with the FIR observations, determine the object size and mass. About ∼30 KBOs have
+known albedos; low albedo values (such as those in comets) are presumed to be darkened by the presence oforganic matter while high albedos are thought to be associated with ice mantles. Surprisingly, all inner and
+cold KBOs targets detected in the Spitzer sample (∼20) have much higher albedos (Brucker et al. 2009) than
+previously assumed (Jewitt & Luu 1993). The underestimation of the KBO albedos leads to a significant over-estimation of their mass. This is an unexpected result, and it is clear that future FIR studies on the size and
+mass distribution of the outer Solar System will need to provide a much more robust confirmation by observing
+a large sample of thousands of KBOs.
+SAFARI photometry – FIR detection of all known KBOs: While Herschel will require ∼300 hr to detect pho-
+tometrically∼10% of the currently known KBOs (those with diameters larger than >250 km at rate of∼1 per
+hr), SAFARI will detect almost all known KBOs (those with diameters >100 km) in only∼50 hr at a rate of 1
+object per minute. SAFARI’s ultra-deep FIR photometric surveys will provide the crucial observational input to
+determine their sizes and albedos. SAFARI will also detect bodies as small as ∼10 km diameter, thus including
+the new KBOs to be discovered from now to the launch of SPICA, the number which is expected to increase bya factor∼2, to∼2500 by∼2018. To be more precise, Figure 1.8 shows the diameters (in km) of a population
+of known KBOs as a function of their heliocentric distance. We plot on the same figures the detection limitpredictions for the different photometric bands of Herschel-PACS and SAFARI showing the large number of
+KBOs that could be detected photometrically in the FIR in short time exposures by SAFARI. The ability to
+constrain the size and albedo of such a huge number of KBOs is a unique science driver for SAFARI.20 Chapter 1. Scientific Objectives
+SAFARI spectroscopy: What are the most primitive objects in the Solar System made of? Neither Spitzer nor
+Herschel have the required sensitivity to detect spectral features from KBOs in the FIR as the expected flux at
+70μm is below a few mJy and below a few μJy at 24μm (Brucker et al. 2009). SAFARI will be the first FIR
+spectrometer to cover simultaneously the ∼34 – 210μm spectrum of KBOs with the required sensitivity, thus
+opening the outer Solar System studies to FIR spectroscopy. SAFARI low-resolution spectra (R ∼100) will
+be a new powerful tool to constrain KBOs thermal models, because it does not just add a few photometric
+measurements, but fully samples the peak and shape of the thermal emission. Even more importantly, with
+spectroscopy we will be able investigate the presence of the main minerals and ices in a statistically significant
+sample of targets.
+The broad wavelength range and high sensitivity of SAFARI will allow us to detect both the water ice fea-
+tures at∼44 and∼62μm and the broadband FIR emission from Mg-rich crystalline silicates (e.g., forsterite,
+clino-enstatite and diopside; Bowey et al. 2002) and other Fe-rich minerals. As a result, we will be able to
+determine the mineral composition of KBOs and link it to the mineralogy of extra-solar planetary systems. SA-
+FARI promises even more ambitious discoveries such as the detection of the possible tenuous gas atmospheresof some KBOs, plausibly formed through cryo-volcanism as in Enceladus, by looking for the lowest-energy-
+level transitions of water vapour and the FIR lines of elements such as neutral sulphur or oxygen. For the most
+distant/cooler/faintest KBOs, the FIR features will be the best and only way to characterise the outer Solar
+System composition.
+The inner Solar System, its size distribution and its chemical composition:
+SPICA will not only characterise the composition of the cold and outer KBOs, but also of the different families
+of inner, hotter centaurs, comets and asteroids, thus probing the primary/secondary dust processing in the Solar
+System. A full determination of their surface composition requires direct spectroscopic studies over the entire
+critical MIR/FIR range, where specific minerals and ices can be identified unambiguously. The similarity
+between interstellar and cometary ices (Ehrenfreund et al. 1997; Crovisier et al. 1997; Bockel ´ee-Morvan et al.
+2000) may not directly prove that comets accrete unprocessed material, but rather suggest that similar chemical
+processes were at work in the early Solar Nebula and in interstellar clouds (Cernicharo & Crovisier 2005).
+However, abundant water, ammonia and methanol ice mantles in Solar System objects could well be the seeds
+of more complex biogenic molecules, such as the aminoacids, that do form when interstellar ice analogues are
+irradiated with UV radiation (Mu ˜noz Caro et al. 2002). Some of the amino acids identified in the laboratory are
+also found in meteorites which suggests that prebiotic molecules could have been delivered to the early Earthby cometary dust, meteorites or interplanetary dust particles. Spectroscopic studies with SPICA in the critical
+MIR/FIR domain will clearly advance our understanding of the chemical complexity in planetary systems by
+direct searches of the “ingredients for life”.
+In summary, SPICA will provide for the first time the means to quantify the composition of hundreds of
+Solar System objects. Sensitive FIR photometric and spectroscopic observations will give the first unambigu-
+ous determination of their size distribution (and thus mass) and composition providing critical observational
+evidence for models of Solar System formation.
+1.3 Exoplanet characterisation in the Infrared
+The first detection of an exoplanet around a solar-type star (Mayor & Queloz 1995) has been one of the most
+relevant discoveries of the last century. Indeed, the existence of extra-solar planetary systems has global impli-
+cations that go beyond astrophysics, and captures the interest of the general public. Extra-solar systems may
+include a variety of planets even richer than the selection we have in our own Solar System: gas-giants, icy bod-
+ies as well as less massive rocky planets with their oceans, moons and rings (Gillon et al. 2007; Fortney et al.
+2008). Almost 15 years after this outstanding discovery, we know that extra-solar planetary systems clearly
+exist wide in range of masses and orbital parameters, and new exoplanets (EPs) are found at surprisingly in-1.3. Exoplanet characterisation in the Infrared 21
+creasing rates. At the time of writing, 373 EPs have been detected by different methods (radial velocity, transits,
+microlensing, imaging and timing). In terms of individual EP detections, most of them (∼ 294) have been found
+through radial velocity searches, and refer to massive (out to a few Jupiter masses), hot (T eff∼1000−1500 K)
+gas planets, commonly named as “hot Jupiters”, which orbit very close (<< 1 AU) to Sun-like stars (G-type)
+and thus have short periods ( P<10 days). A few lower-mass planets ( M<10 M earth), so called “super-Earths”,
+have also been discovered, which suggests that rocky planets relatively similar to the Earth may not be uncom-mon. Planets with larger orbital axes, e.g., comparable to Jupiter’s orbit, will be detected in the future through
+ground-based direct detection capabilities (e.g., Subaru-HiCIAO, VLT-SPHERE, Gemini-GPI) or astrometry
+methods (e.g., with Gaia). In fact, a few outer EPs (∼ 20 to 120 AU) were directly imaged in 2008 for the first
+time: a planet around Formalhaut at ∼7.7 pc (Kalas et al. 2008) and multiple planets around HR8799 at ∼40 pc
+(Marois et al. 2008).
+A further step forward in our understanding of these intriguing objects requires not just the detection of
+their presence (with many specialized ground- and space-based telescopes such as MEarth, COROT, Kepler,etc., designed to carry out extensive photometric searches of the sky), but also a detailed characterisation of
+their main physical properties and chemical conditions (temperature, densities, climate, atmospheric compo-
+sition and biosignatures). Surprisingly, it has been possible to infer from Hubble and Spitzer observations
+some atmospheric components in a few EPs: sodium (Charbonneau et al. 2002; Redfield et al. 2008), methane
+(Swain et al. 2008b), CO and CO
+2(Swain et al. 2009) and even water vapour (Barman 2007; Tinetti et al.
+2007). All these recent works show that, with more sensitive space instrumentation, a firm characterisation
+of a representative sample of EPs would be possible. Two different observational approaches can be used to
+characterise both outer andinner EPs: direct detection and transit techniques respectively. Direct detection,
+which refers to observations where the star and the planet can be spatially separated on the sky with corona-graphs. Due to the limited size of current telescopes, this technique is only able to image EPs at large orbital
+distances (∼ 50 – 100 AU). Besides, the huge contrast between the host star and the planet flux (e.g., the Sun
+is 10
+9times brighter than the Earth in the visible) makes the direct detection of an Earth-like planet at ∼1A U
+a distant goal for future instrumentation. Fortunately, EPs orbiting very close to the host star (/lessorsimilar 0.05 AU)
+and with a favourable inclination (almost edge-on systems) can be indirectly studied via the transit technique
+by which the dimming of the starlight as the EP transits the star is followed (Seager & Sasselov 2000). Be-
+cause of the precise timing of a transit event, this technique allows high-contrast observations if very sensitive,
+low background and stable observing conditions are available (i.e., from space). Although not specifically
+designed for EP research, transit observations with Hubble and Spitzer have revolutionised and redefined the
+field of EP characterisation over the past 5 years (Barman 2007; Beaulieu et al. 2008; Charbonneau et al. 2002;
+Deming et al. 2005, 2006, 2007; Marley et al. 2007; Demory et al. 2007; Gillon et al. 2007; Grillmair et al.2007; Harrington et al. 2006, 2007; Knutson et al. 2007; Richardson et al. 2006, 2007; Machalek et al. 2008;
+Swain et al. 2008b,a; Tinetti et al. 2007). Collectively, this work has conclusively established that the detailed
+characterisation of EP atmospheres is feasible and today we can discuss the observational signatures including
+weather and atmospheric chemistry (e.g., Selsis et al. 2002; Tinetti & Beaulieu 2009).
+1.3.1 New frontiers in exoplanet research
+The mid infrared region (from ∼3t o∼30μm) is especially important in the study of planetary atmospheres
+as it spans both the peak of thermal emission from the majority of EPs so far discovered, and is particularly
+rich in molecular features that can uniquely identify the composition of planetary atmospheres and trace the
+fingerprints of primitive biological activity. In the coming decades many space and ground based facilities
+are planned that are designed to search for EPs on all scales from massive, young “hot Jupiters”, through
+large rocky “super-Earths”. Few of the planned facilities, however, will have the ability to characterise the
+atmospheres which they discover through the application of infrared spectroscopy.
+The observation of EPs at IR wavelengths offers several advantages compared to traditional studies in the
+visible domain. First, the star-to-planet flux contrast is much lower than in the visible (∼ 103for a typical
+“hot Jupiter” around a Sun-like star) and second, transiting planets around stars much cooler than the Sunhave to be observed in the IR where their emission peaks. Spitzer has measured EP photometric transits22 Chapter 1. Scientific Objectives
+Exoplanet and host star fluxes in the mid− and far−infrared 
+Figure 1.9: Left panel: Fit to HD 209458b “hot Jupiter” (T eff/similarequal1000 K) MIR fluxes inferred from a secondary transit with
+Spitzer (Swain et al. 2008a) around a G0 star (d ∼47 pc, in black) and interpolation to d =10 pc (red). For comparison,
+the emission of a cooler Jupiter-like planet at 5 pc is shown in blue (reflected emission neglected). The thick lines are
+the 5σ-1hr photometric sensitivities of SPICA MIR instruments (cyan) and SAFARI (blue, magenta and red). Dashedlines show sensitivities in spectrophotometric mode (R /similarequal25). SPICA will observe similar inner “hot Jupiter” transits
+routinely and will potentially extract their IR spectrum (rich in H
+2O, O 3,C H 4,N H 3and HD features as in Solar System
+planets). Middle panel: Flux from the host star at different distances. Right panel: Increasing planet-to-star contrast at
+long wavelengths (Goicoechea et al. 2008).
+out to 24μm demonstrating that the light-curve is simpler (“box-like”) than in the visible domain due to the
+negligible role of stellar limb-darkening effects. This allows a robust and precise determination of the EP radius
+as a function of wavelength and provides further strong constraints to the atmospheric properties. SPICA will
+achieve a precision similar or better than that reachable with Spitzer instruments, enabling us to detect the
+atmospheric features of hot-Jupiters and Neptunes. Transit observations with the Spitzer-IRS spectrometer have
+been used to extract the absolute intrinsic spectrum of HD209458b hot Jupiter around a Sun-like star – with
+the resultant spectrum in physical units (e.g., in μJy) as opposed to the typical relative contrast measurements
+(Swain et al. 2008a) (see Figure 1.9). Infrared observations have allowed to characterise temperature-pressureprofiles, chemistry and circulation patterns of a select subset of massive, close-in hot Jupiters along with severalless massive EPs: the hot Saturn HD14926b and the cooler Neptune-mass planet GJ 436b.
+By operating at long wavelengths, SPICA will be well adapted to carry out transit observations of rocky
+planets closely orbiting around the numerous cool M stars (∼ 75 stars at d<5 pc and∼6500 within a d<25 pc
+volume!). Owing to their low mass (∼ 0.09−0.68 M
+⊙) and low effective temperature (2500 −3800 K), M stars
+are better studied at infrared wavelengths. The location of the habitable zone (i.e., the distance from the hoststar to a planet where liquid water could exist) scales with ∼L
+1/2
+star, and thus M stars have habitable zones much
+closer (∼ 0.02−0.5 AU) than Sun-like stars. In addition, the light curve of an earth-like EP around a M star
+is about as deep as for a Jupiter-like planet around a G star. The characterisation of rocky “super-Earths” with
+volatile-rich, maybe prebiotic atmospheres (e.g., with traces of O 3,O2,C O 2,C H 4or H 2O) orbiting within
+the habitable zone of M stars, in spite of being challenging (several transit measurements needed) will have atremendous impact as these cool EPs are much more interesting from an astrobiological point of view. Searches
+for bright M star transit candidates with modest ground-based telescopes are currently underway (e.g., MEarthProject; Nutzman & Charbonneau 2008), and more targets will be searched by new radial velocity surveys. We
+anticipate that several of these will be good targets for detailed characterisation with SPICA.
+1.3.2 Mid-infrared coronagraphy: “Direct” imaging and spectroscopy of EPs
+Although the transiting technique is a powerful method to characterise “hot-Jupiters” spectroscopically, it is
+limited to EPs orbiting very close to the host star and to planetary systems with a favourable viewing geome-
+try. High–contrast “direct” observations with coronographs are needed to study outer and cooler planets. The
+projected coronagraph on SPICA (SCI) is a unique instrument that provides several advantages over JWST
+coronagraphs. First, the monolithic mirror of SPICA will be better optimised for coronagraphy than the seg-1.3. Exoplanet characterisation in the Infrared 23
+mented mirror of JWST due to its much simpler and clean PSF. The segmented geometry of JWST also requires
+complex Lyot stops for the suppression of the light diffracted by each mirror segment, reducing the throughput
+and requiring a high degree of alignment stability. Secondly, the SPICA telescope itself will be actively cooleddown to∼6 K (compared to passive cooling down to 45 K for JWST); it is, therefore, further optimised for
+MIR/FIR astronomy. From the scientific point of view, the main difference from JWST is the possibility
+of undertaking “direct” spectroscopy with SPICA/SCI in the critical MIR domain (using a grism/prism
+providing R∼20−200) in addition to imaging. This spectral capability in a continuous wavelength domain
+rich in chemical signatures (∼ 3.5−27μm) represents a unique science possibility of SPICA (Abe et al. 2007)
+compared to JWST, which will just have quadrant phase and Lyot photometric coronagraphs in the MIR and
+will provide lower contrast than SCI. As we have seen earlier, exo-giant-planets (EGPs) are thought to form
+beyond the “snow line”. Indeed, several of them are known from Doppler shift measurements or direct imaging
+in the visible (Wetherill & Stewart 1989; Kalas et al. 2008; Marois et al. 2008) and more will be detected in
+the near future (with large optical telescopes equipped with coronagraphs, JWST and SPICA itself). This key
+population of outer and young EGPs can not be studied through transit experiments but will be targets for SCI
+direct characterisation. In fact, in the next decade SCI will be the only instrument available to characterise
+outer and cool EPs through direct spectroscopy in the MIR. Therefore, SPICA will add greatly to EP research(and much earlier than the future Terrestrial Planet Finder type missions) by imaging young EPs directly and
+by recording their MIR spectra, thus constraining their temperature and atmospheric composition. The SCI
+simpler binary mask type approach will achieve a contrast of 10
+−6at the equivalent to ∼9A U(∼Saturn’s
+orbit) at∼5μm for a star at 10 pc (IWA∼1/prime/prime). At this wavelength we probe the younger end of the planet
+age range (∼100 Myr to 1 Gyr, see Figure 1.10a). SCI is expected to observe ∼200 targets and produce an
+“spectral atlas” of several outer EPs, therefore completing the discovery and characterisation space of other
+telescopes and methods. According to their expected flux at long wavelengths (λ> 5μm), the main targets for
+SCI spectroscopy will be ∼2−3M Jupplanets around solar-type stars and lower mass planets, ∼1M Jup, around
+M dwarfs. We anticipate ∼200 targets for direct spectroscopy, for which ∼1 hr integration per EP will be
+required. In photometric mode, SCI will perform imaging surveys integrating a few minutes per target (bothyoung M stars:∼300 Myr,<50 pc; and FGK stars: ∼1 Gyr,<25 pc).
+The most significant atmospheric features expected in the EP spectra can be summarised as follows: (1) The
+∼4−5μm “emission bump” due to an opacity window in EGPs and cooler objects with temperatures between
+100 and 1000 K; (2) Molecular vibration bands of H
+2O(∼6–8μm), CH 4(∼7.7μm) , O 3(∼9.6μm), silicate
+clouds (∼ 10μm), NH 3(∼10.7μm), CO 2(∼15μm) and many other trace species such as hydrocarbons and
+1e-071e-061e-051e-041e-031e-021e-011e+001e+01
+ 5  10  15  20  25Flux Density (mJy)
+Wavelength (micron)1 MJ, 1 Gyr
+2 MJ, 1 Gyr
+5 MJ, 1 Gyr
+10 MJ, 1 GyrH2OC H 4 NH3
+Spectral models by Burrows, 
+Sudarsky & Lunine (2003)10   F(G2)-6
+10   F(M0)-6
+SPICA sensitivity
+ (1hr, 5 sig)d = 10 pc
+Wavelength (um)Model, water+methaneObservationsNICMOS
+IRACMIPSAbsorption (%)2.50
+2.452.402.352.30
+51 0 2 0a) b)MIR transitsMIR exoplanet characterization: coronagraphy and transit spectroscopy 
+SPICA at high spectral resolution
+constrain atmospheric models in detail"direct" spectroscopy
+Coronagraph (SCI)
+Figure 1.10: (a)Simulated spectra for a range of exoplanet masses at an age of 1 Gyr and a distance of 10 pc (from
+Burrows et al. (2003) models). SCI is the only planned coronagraph that will carry out MIR “direct” spectroscopy of
+outer (>10 AU) and young EPs. (b)Hubble and Spitzer primary transit of HD 189733b hot Jupiter (photometry) and
+H2O and CH 4models (by G. Tinetti). Transit observations with SPICA spectrometers will characterise the atmospheric
+composition of many EPs at the required higher spectral resolution.24 Chapter 1. Scientific Objectives
+more exotic nitrogen/sulphur-bearing molecules. If detected, the relative abundance of all these species could
+be compared among different EPs and with Solar System planets and bodies. Note that giant planets like Saturn
+in the Solar System show strong NH 3,P H 3and H 2O features around∼5μm (de Graauw et al. 1997). SPICA
+will also have access to the PAHs band features which are excellent tracers of UV-radiation fields; (3) He-H 2
+and H 2-H2collision induced absorption band features as tracer of the He/H relative abundance; (4) Features
+from deuterated molecular species to distinguish cool brown dwarfs from “real” EP (e.g., CH 3Da t∼8.6μm)
+and non-equilibrium species (e.g., PH 3at∼8.9 and∼10.1μm). Were the spectral coverage to be extended
+down to 3.5μm then H+
+3(∼4μm) would also be accessible, and with it the key to understanding the cooling
+processes in the upper atmosphere (Koskinen et al. 2007).
+1.3.3 Mid-IR “transit” photometry and spectroscopy of exoplanets
+The SPICA MIRACLE, MIRMES and MIRHES instruments will cover the mid infrared range with low,
+medium, and high spectral resolution (out to R∼30 000). Following the unexpected but successful obser-
+vations of “hot-Jupiters” EPs made by Spitzer, these instruments will be used to study primary and secondarytransits of EPs orbiting close to the star. SPICA’s spectrometers will be used to perform (1) multi-wavelength
+transit photometry of hot Jupiters routinely and (2) carry out “transmission” and “occultation” spectroscopy
+of an appropriate sample of bright EPs. Hot-Jupiters in particular, are bright and the possibility to carry out
+detailed atmospheric spectroscopy will allow us to study exo-atmospheres in great detail. This will help to de-
+velop the techniques and models that will be needed to interpret the spectra of habitable exo-Earths in the future.
+We estimate that the ∼24μm thermal emission of gas giant planets similar to HD 209458b could be extracted
+from secondary transit observations around stars as far as ∼150 pc (a few hundred star targets) as the contrast
+requirement is relatively modest (∼ 0.1%). We anticipate that several giant EPs, with a great diversity of mass,
+semi-major axis, eccentricities, etc. will be available to SPICA for detailed mid infrared characterisation by
+∼2018. With even higher photometric precision (∼ 0.01%), SPICA will have access to more challenging goals
+such as inferring potential Saturn-like rings around EPs (Barnes & Fortney 2004).
+In the case of EPs around cooler stars, a super-Earth (2 −3R
+earth;Tp∼300 K) orbiting around the habitable
+zone of a cool M8 star (T ∼2500 K) will produce an intrinsic flux of ∼25μJy at MIR wavelengths, roughly a
+few times higher than the projected 1σ-1min photometric sensitivity of SPICA’s MIR instruments (but very hardto achieve with ground-based MIR instruments on “Extremely Large Telescope”-type observatories). These
+photometric flux levels are accessible for SPICA secondary transit studies with a contrast of ∼0.1%. Mid in-
+frared transmission spectroscopy will probe the low-pressure layers of EP atmospheres where non-equilibriumchemical conditions occur. The first optical transmission spectrum taken from space used a medium resolution
+spectrometer (R∼5500; Charbonneau et al. 2002). More recently, high resolution (R ∼60 000) ground-based
+optical transmission spectra have been used to resolve EP spectral features (Redfield et al. 2008). Either atmodest or high spectral resolution, SPICA mid infrared spectra of transiting EPs will provide a unique op-
+portunity to determine the properties of their atmospheres through detailed analysis of band profiles, e.g., to
+extract molecular abundances and isotopic ratios accurately. In summary, SPICA will be an important inter-
+mediate milestone in exoplanet research, both in science achievements (direct spectroscopy of outer EPs for
+the first time in the MIR, atmospheric characterisation of transiting EPs at long wavelengths...) and in the re-quired technological developments for future longer-term missions (e.g., coronagraphic techniques, cryogenic
+systems...).
+1.4 The life cycle of gas and dust
+SPICA will provide an unprecedented window into key aspects of the dust life-cycle both in the Milky Wayand in nearby galaxies: from the dust formation in evolved stars, its evolution in the ISM, its processing in
+supernova-generated shock waves and massive stars (winds, HII regions, etc.), to its final incorporation into
+star forming cores and protoplanetary discs. Dust grains are a key player in determining the energy budget
+of the interstellar and circumstellar media because of their fundamental role in reprocessing stellar UV/visible1.4. The life cycle of gas and dust 25
+photons into IR to sub-mm radiation, and in heating the gas via photo-electron emission from small grains
+and PAHs. SPICA will be the first space telescope since ISO (launched in 1995) covering in spectroscopy the
+uninterrupted MIR/FIR domain where astro-mineralogy studies can be carried out. ISO caused a revolution in
+the field of astromineralogy (Henning 2003) and showed that much can be learnt by following the evolutionof the grains composition. The critical keys to unlock our understanding of the dust evolutionary cycle are
+MIR/FIR spectroscopy and the FIR dust SEDs. The former gives direct access to the poorly known ice andgrain composition and its evolution along the dust life-cycle, and the latter enables us to determine the dust
+temperature uniquely.
+In addition to dust, other very important gas-phase species that are hard, often impossible, to detect from the
+ground have their spectral signatures in the MIR/FIR: ions, atoms, light molecules and heavier organic species.
+Their associated MIR/FIR spectral features provide means to derive physical conditions in environments that
+are difficult to probe at other wavelengths (e.g., in UV/X-ray illuminated PDR/XDRs, shocked regions, galacticnuclei, etc.). They provide clues on the elemental abundances (C, O, S, D, Si, Fe ...) and also provide deep
+insights into the gas/dust chemical interplay: ice formation/evaporation, deuterium enrichment, grain growth
+and metal depletion (Okada et al. 2008). In this context, SAFARI will provide far infrared spectroscopic im-
+ages of the regions that are too obscured for JWST to examine in the mid infrared or too warm/extended to
+be efficiently traced by ALMA and SKA interferometers. Thanks to its superb sensitivity, SPICA will extend
+our knowledge of the physics and chemistry of the gas and dust in our galaxy (e.g., acquired with Herschel)to similar detailed studies in nearby galaxies. In the following we list several fields where we anticipate that
+SPICA will play a unique role. This is not an exhaustive list, and probably does not do justice to the great
+discovery space available to SPICA (see section 1.8):
+Chemical complexity beyond our galaxy: Spectral observations from UV to cm wavelengths reveal a high
+degree of chemical complexity in our Galaxy that was until recently not expected. Such complexity is demon-strated by the diversity of detected species, that range from simple light hydrides, molecules carrying heavymetals, alcohols (methanol, ethanol,...), and a collection of many organic families (acetylenic chains, methane
+or benzene; Cernicharo et al. 2001) that lack permanent electric dipole (i.e., do not have radio spectrum to be
+observed with ALMA) but show MIR/FIR features. Either in the gas phase or as ice mantles, those species can
+be precursors of more complex molecules (sugars, amino acids, etc. Mu ˜noz Caro et al. 2002). SPICA spec-
+trometers will open new windows to probe the chemical complexity in the universe.
+Evolved stars (dust factories): The principal objects injecting dust into the ambient ISM in our Galaxy are
+the evolved stars, mainly red giants and AGB stars that predominantly emit at MIR/FIR wavelengths. SAFARI
+will be specially adapted to study the inner parts of their circumstellar envelopes (CSE), and to study the mass-
+loss history and mineralogy of a large sample of targets in very different environments including the closest
+galaxies. SAFARI observations will include (1) detailed spectroscopic studies of unresolved CSEs, (2) point
+source photometry of unresolved AGB populations in nearby galaxies, and (3) spectroscopic images of spatially
+resolved CSEs and their faint interaction with the ISM. For nearby objects, SAFARI will provide spectroscopic
+images of faint but very extended “detached” shells around evolved stars revealing their physical and chemical
+structure in great detail (for a prototype shell like TT Cyg, ∼5
+/prime/primecorresponds to only ∼100 years of expansion).
+Photometric measurements will constrain the mass-loss rates in AGB stars out to the Large Magellanic Cloud(LMC; d∼50 kpc) in modest integration times. For example, the expected flux at 100 μm of a star like R Cas at
+the distance of the LMC is ∼200μJy, while detached shells like TT Cyg will have fluxes of ∼500μJy, requiring
+an integration time of a few minutes with SAFARI (see Figure 1.1).
+Supernova remnants (dust processing): Stars much more massive than our Sun end their lives with a vi-
+olent explosion where massive flows are blown outward and shock waves propagate through the surrounding
+medium. When supernova (SNe) remnants encounter molecular clouds, they drive slower shock waves that
+compress and heat the molecular material and result in strong MIR/FIR line emission (Neufeld et al. 2007). At
+the same time, dust grains are processed (eroded, fragmented, amorphised ...), their physical state changes, andlarge quantities of refractory elements are sputtered back to the gas phase (Si, Fe, Mg, etc.). Dust grains may26 Chapter 1. Scientific Objectives
+condense after the shock passage, and therefore SNe may contribute to the formation of dust. In fact, high-z
+galaxies seems to contain dust that can only have formed in SNe ejecta because the galaxies are too young for
+AGB stars to have formed dust. Thus, dust condensation must be efficient, requiring about 1 M ⊙of dust per
+SNe. However, the efficiency of SNe dust formation in our own Galaxy is still debated and typically seems to
+be inefficient, for example, SNe 1987a only formed about 10−4M⊙of dust. SPICA spectroscopic images will
+allow us to infer the physical conditions prevailing in the shocked gas while directly tracing the dust emission
+to the faint levels required to measure the dust production, and even its composition.
+ISM (The impact of low metallicity): The ISM is the repository of the metals ejected by dying stars as well
+as the site for the birth of the next generation. Winds of low and high mass stars and SNe explosions continually
+raise the metallicity of the ISM where recycling is ongoing. In local metal-poor ISM, representative of the early
+epochs of star formation, the spectral fingerprints of both dust and gas di ffer markedly from those of gas-rich
+starburst galaxies. The conditions for star formation as well as the effects of star formation on the ISM inmetal-poor ISM can give us insight on the process and the effects of cosmic enrichment. Thus the study of
+the lifecycle of dust and gas in galaxies is incomplete without the dimension of metallicity. Large numbers
+of low-metallicity, actively starforming dwarf galaxies are known in the local Universe, and these provide
+ideal laboratories in which to study the interplay between star formation and the ISM, in environments that are
+reminiscent of conditions likely to be found in very young galaxies in the distant and youthful Universe. Recent
+surveys with ISO and Spitzer have observed the dust and MIR properties of less than 75 dwarf galaxies: only
+a handful of the most metal-poor (1/20 of solar) were detected (Madden et al. 2006; Engelbracht et al. 2006),however. Spectroscopic surveys of the major FIR cooling lines are planned with Herschel: Just detecting the
+[Cii]158μm line in the lowest metallicity galaxy will be a major breakthrough. To determine the properties
+of the dense and diffuse components of the ISM in these unexplored systems we need both the molecular and
+atomic as well as FIR fine structure diagnostics. The factor of 100 gain in sensitivity of SAFARI will bring
+FIR spectroscopy of this enigmatic population well within our reach, and provide an important benchmark
+against which to compare studies of distant objects. While the more massive galaxies have been the targets
+of the majority of high redshift surveys, recent spectroscopic surveys of the low-mass population at z∼1
+(Davies et al. 2009) are beginning to put dwarf galaxies into a clearer perspective in galaxy evolution, favouringthe downsizing scenario of galaxy evolution (Cowie et al. 1996). A SAFARI survey of these targets would be
+the next step in understanding the evolution of metals in the gas and dust, and how these properties a ffect the
+star formation density as a function of redshift. The importance of low metallicity at high redshift is highlightedin recent studies of gamma ray burst hosts (Chen et al. 2009), Lyman-Break galaxies (Mannucci et al. 2009)
+and damped Ly-α systems (Pettini et al. 2003, 2008), which all bear the tell-tale hallmarks of low metallicity
+dwarf galaxies.
+1.5 Galaxy evolution: the coevolution of stars and supermassive black
+holes
+The study of galaxies in their cosmological context has undergone important changes during the past years,with the accumulation of evidence that the standard picture based on a naive interpretation of the hierarchical
+scenario of galaxy formation is not consistent with observations. This may be summarised using simple key-
+words such as the “downsizing” or “bimodality” of galaxies, however it certainly reflects a major limitation of
+our theoretical framework that only observations which can be directly compared to theory will overcome.
+When trying to define the observations required, astronomers have been confronted with two major limi-
+tations – firstly, the increasing importance of obscuration acting both on newly formed stars and supermassive
+black holes (SMBH) in their growing phase, and secondly the so-called confusion limit. The confusion limit
+is a technical limitation that cannot be avoided when observing at long wavelengths. Due to large beam sizes
+and high projected source densities, the very same galaxies we wish to study make the Universe opaque. These
+limitations call for a new generation of infrared instrumentation such as SPICA, as other existing or scheduled
+observatories will all fail to address them. To overcome these two sources of blindness – i.e., intrinsic dust1.5. Galaxy evolution 27
+Figure 1.11: (a) Left panel: A plot of the confusion limit in the FIR/sub-mm, with the wavelength coverage and sensitivity
+of selected instruments overlaid. The inset plot gives the % of the background at a given wavelength that is resolved by
+3.5 metre-class telescope, as a function of wavelength – from this one can see that below 80 μm one can resolve almost all
+of the background; (b) Right Panel: A composite plot of the evolution of the star formation rate (SFR) density and black
+hole accretion rate, adapted from Merloni et al. (2006). The points represent values derived from observations (listed inMerloni et al. 2004) and the dark-blue shaded region the best fit model to the data. The light blue locus is a model of the
+black-hole accretion rate (BHAR, Merloni et al. 2006). Both the SFR and BHAR peak between z ∼1−2.
+absorption and projected overlapping of galaxies – there is an optimal definition of a telescope diameter for a
+given wavelength range. Herschel is limited by confusion in five of its six broadband filters, i.e., above 100 μm,
+and as a result will not resolve more than half of the infrared background light resulting from the cumulativedust obscured star formation and accretion in galaxies over the Hubble time. SPICA, like Herschel, will have
+a 3.5 m telescope due to the limitations imposed by the launch vehicle lift capability and payload shroud. In
+(Figure 1.11a) we show the detection limit versus wavelength at which a 3.5 m aperture becomes limited byconfusion between sources and the amount of the observable Universe that will be resolved at each wavelength.
+We can see two things from these plots: (1) the optimal wavelength at which to observe the extragalactic back-
+ground with a 3.5 m telescope is around 70 μm and (2) that Herschel-PACS will never reach the confusion limit
+at this wavelength but SPICA SAFARI will reach it easily. The extremely high spatial resolution of ALMA, onthe other hand, is not suited to the study of galaxies in their cosmological context, since it will only follow-up
+targeted galaxies or produce pencil beam surveys. These are, by definition, not suited to developing an under-
+standing of the anti-hierarchical behaviour of galaxies, since they will su ffer from cosmic variance and limited
+statistics. Submillimetre observations from ground-based facilities are also limited by confusion to a few mJy,hence to the very brightest tip of the galaxy population (i.e., confined to galaxies more luminous than a hundred
+times L
+∗galaxies, the average galaxy luminosity).
+SAFARI onboard SPICA will survey the Universe in the most efficient waveband with which to study
+galaxy formation with a 3.5 m class telescope in space, and will therefore:
+•Unveil the population of heavily obscured active nuclei responsible for the missing half of the cosmic
+X-ray background (CXB) at its peak around 30 keV (Comastri et al. 1995; Gilli et al. 2001, 2007).
+•Provide, for the first time, a complete census on the star formation history of galaxies over the last 90% of
+the age of the Universe, by measuring dust obscured star formation down to the limits at which galaxies
+become optically thin, and redshifted ultra-violet light becomes a robust tracer of star formation.
+•By combining the points above we will be able to follow the relative pace at which stars and black
+holes formed, to pinpoint the major events of both processes, and therefore to understand the connection
+between the two major sources of light in the Universe after the Big Bang, nucleosynthesis and accretion.28 Chapter 1. Scientific Objectives
+JWST ALMA SAFARIFar-Infrared Spectra of IR-Bright Galaxies
+50 100 150 200
+Rest Wavelength (μm)        υ Fυ
+Cen AArp 220
+Arp 299M 82NGC 253NGC 4945
+IC 342
+M 83
+NGC 1068
+NGC 2146[O III] EP=35 eV
+[N III] EP=30 eV
+[O I] EP=0 eV
+[O III] EP=35 eV
+[N II] EP=15 eV
+[O I] EP=0 eV
+[C II] EP=11 eVH2OOH
+CH
+Figure 1.12: (a) Left Panel: Recent constraints placed on the cosmic infrared background (CIRB) by Spitzer show that the
+integrated extragalactic background flux density at MIR/FIR wavelengths is comparable to that found in the cosmic optical
+background (COB: UV-visible-near-IR). The CIRB peak is a measure of the re-radiated emission from light processed bydust over cosmic time, whilst the COB measures the light emitted by stars and AGN. It is clear that a significant piece of
+the galaxy evolution puzzle is missed if we neglect the IR. The solid line traces out the CIRB spectrum that is predicted by
+the galaxy evolution model of Franceschini et al. (2009). References for the observational data points can also be found in
+this paper. (b) Right panel: The full ISO-LWS spectra of ten local galaxies, illustrating the range of emission line strengths
+observed. These differences will be explored in detail by Herschel, and will provide a zero-redshift benchmark against
+which to compare distant galaxies. The spectra have been offset vertically and plotted as a sequence (bottom-to-top) of
+increasing IR luminosity (Fischer et al. 1999).
+•Resolve the individual sources that make up the cosmic infrared background (CIRB).
+1.5.1 Major recent discoveries and open problems
+Two important discoveries in recent years have changed our perspective of galaxy evolution: the first is the
+strong observed correlation in the local Universe between the mass of the black hole at the centres of mas-
+sive galaxies and the key properties of the host galaxy, and the second the existence of a dusty population of
+galaxies that emits a significant fraction of their bolometric luminosity in the FIR /submillimetre (sub-mm).
+The strong correlation observed in the local Universe between the masses of central black holes and the lu-
+minosities, dynamical masses, and velocity dispersions of their host spheroids (e.g., Magorrian et al. 1998;
+Ferrarese & Merritt 2000; Ferrarese & Ford 2005; Shankar et al. 2009) implies that the processes of blackhole growth (through gravitational accretion) and bulge formation (through star formation (SF)) are intimately
+linked. Optical studies of the local massive galaxy population show that most, if not all, galaxy spheroids host
+massive relic black-holes (Richstone et al. 1998) which, in turn, suggests that a mass-accreting AGN (active
+galactic nucleus) phase is one through which all massive galaxies pass. The origin of the correlation cannot
+be explained by observations of the local Universe, where the two processes appear to be almost independent
+of one another, and so must be seeded at much earlier epochs. The evolution of the two processes with time
+can be seen in Figure 1.11b. Global accretion power, measured using X-rays (Hasinger et al. 2005) and the star1.5. Galaxy evolution 29
+formation power, measured by Hα and rest-frame UV observations (Shim et al. 2009), were both ∼20 times
+higher at z=1−1.5 than today. That both peak at around z∼1−2 strongly suggests the co-evolution of
+SMBHs and the star formation rate (SFR) (e.g., Marconi et al. 2004; Merloni et al. 2004). To understand and
+unravel the relationship between the blackhole growth and bulge formation we need to track back the evolution
+of black hole growth and star formation over cosmic time, and determine whether there is a causal link between
+the growth of the central black hole and its host spheroid. If there is such a link, how and when is it established,and which, if either, process drives or regulates the other?
+We know from galactic and extragalactic studies that the processes of star formation and early stellar evo-
+lution are typically deeply enshrouded in dust. This is also the case for active galactic nuclei (AGN), which areoften optically obscured both locally (e.g., Goulding & Alexander 2009) and at high-z (Hern ´an-Caballero et al.
+2009). Galaxies therefore pass through the most active periods of their lives deeply obscured by dust. Thesecond important discovery – of a population of distant, dusty IR-bright galaxies that were missed by optical
+surveys – is a direct manifestation of this. Luminous and ultra-luminous infrared galaxies (ULIRGs) are rare
+in the local Universe, but become much more common (factor of 1000) at high-redshift. Evidence for this
+first came from the identification of the so-called sub-mm galaxies (SMGs) (Smail et al. 1997; Hughes et al.
+1998) which emit a significant fraction of their rest-frame bolometric luminosity in the FIR/sub-mm, and was
+subsequently confirmed with Spitzer. The integrated luminosity of the (U)LIRG population accounts for themajority of the CIRB (Dole et al. 2006), the infrared component of the extragalactic background light, yet thispopulation is insignificant in the optical. Shown in Figure 1.12a is a plot of the extragalactic background light
+from optical to millimetre wavelengths. The two peaks in the energy density – the peak in the CIRB which
+includes reprocessed light emitted by dust, and the peak in the optical (the cosmic optical background, COB),
+which is made up of contributions from stellar processes – are of approximately equal height, and underline the
+importance of the IR. Without a detailed understanding of the IR-bright, dust-obscured galaxy population we
+miss a substantial piece in the galaxy evolution puzzle.
+The discoveries discussed above shed new light onto one of the most topical problems of cosmology: the
+origin and shaping of the galaxy mass and luminosity functions. On the one hand, the very luminous phases
+of galaxy evolution revealed by FIR /sub-mm observations may explain the formation of spheroidal galaxies
+and galaxy bulges, with their massive stellar populations. On the other, the nuclear activity from gravitationalaccretion, very often found concomitant with the ultra-luminous IR phase, is believed to play a key role in
+shaping the galaxy stellar mass and luminosity functions. The highly energetic feedback from the centralAGN could be responsible for regulating the cooling and collapse of primordial gas into the forming galaxy,
+essentially stopping star formation above a given threshold for the SMBH and host galaxy mass. This feedback
+might be primarily responsible for the well-known exponential shape of the universal mass and luminosity
+functions of galaxies, and its fundamental divergence from the power-law shape of the parent dark-matter halo
+mass functions. Unfortunately, this will remain a mere conjecture without suitable instrumentation to penetrate
+the dust opaque media wherein this activity occurs.
+What are the key questions to be answered? Our current picture of galaxy evolution raises many unanswered
+questions as we attempt to reconcile observational data with models:
+•What drives the evolution of the massive, dusty distant galaxy population, and what feedback /interplay
+exists between the AGN and star-forming phase?
+•What physical processes are responsible for shaping the mass and luminosity functions of galaxies, the
+universal form of which both differ considerably from that of dark-matter halos?
+•How do galaxy evolution, star formation rate and AGN activity vary with environment and cosmological
+epoch?
+•How and when do the normal, quiescent galaxies such as our own form, and how do they relate to
+(U)LIRGs?
+Two key requirements need to be met to address these questions: first, we need access to the MIR/FIR to
+overcome the obscuring effects of dust and to fingerprint and track both star formation and AGN activity through30 Chapter 1. Scientific Objectives
+cosmic time. Second, we need an imaging capability combined with a spectroscopic capability to undertake
+the large-scale surveys required to locate and study these dusty sources during their evolution. These two
+requirements translate into high sensitivity, large instantaneous spectral coverage and spectroscopic imaging
+capabilities. SPICA provides all of these.
+In the remainder of this section we demonstrate how these instrumental features will address the science
+questions posed above: in subsection 1.5.2 we make the case for an MIR/FIR spectroscopic imaging capability,and highlight how this waveband can be used to identify and investigate the physical processes driving energy
+production in individual galaxies; in subsections 1.6.1 and 1.6.2 we illustrate how the deep FIR spectroscopic
+and photometric imaging capabilities of SPICA will revolutionise our picture and understanding of galaxy
+evolution.
+1.5.2 MIR/FIR imaging spectroscopy is the key observational technique
+Hot and young stars and black hole accretion discs show strong differences in the shape of their ionising con-
+tinuum. However the far UV continuum, dominating the total bolometric output luminosity in both processes,
+is in general not observable directly, due to absorption by HI. The best tracers and indeed discriminators of ac-
+cretion and star formation are therefore emission lines from the photo-ionised gas. In both stabursts and AGN afairly constant fraction (10 – 20%) of the ionising continuum gets absorbed by gas and then re-radiated as line
+emission, making spectroscopy a powerful diagnostic tool. Detecting the exact fraction of ionising radiation
+absorbed by gas surrounding the power source is however not crucial if one uses emission line ratios, which are
+therefore the most efficient observable probes of energy production mechanisms. The principle spectroscopic
+tracers of star formation are in the rest-frame optical, however, these can be rendered useless by extinction in
+dusty objects. Emission lines in the near-, mid- and far-IR are much less affected by dust extinction, and thus
+provide powerful diagnostics of obscured regions dominated by star formation and AGN activity. MIR/FIR
+spectroscopy is therefore essential to tracing the energetics and evolution of these regions because it is only
+with MIR/FIR spectroscopy that we can push past the effects of dust obscuration to determine the physical
+processes powering the individual galaxies. A wide field imaging capability with broad instantaneous spectral
+coverage provides the key to surveying large areas of the sky, so tracing galaxy evolution in large numbers of
+objects in an unbiased way.
+By the launch of SPICA, deep cosmological surveys undertaken by ISO, AKARI and Spitzer along with
+those planned for Herschel and SCUBA-2 will have produced catalogues containing the fluxes of many tens ofthousands of faint MIR/FIR/sub-mm sources. Photometric surveys can be used to determine source counts and
+establish the contribution of the observed populations at the given wavelength to the extragalactic background.However, to derive source luminosities, a reliable estimate of the source redshift is required implying a great
+amount of multiwavelength photometric data and the use of galaxy templates derived from local observations.
+Even more importantly, photometric data alone do not allow us to unambiguously differentiate between AGN
+and star formation activity.
+Substantial progress in studying galaxy evolution therefore can only be made by making the transition from
+photometric to spectroscopic surveys as it is only with direct MIR/FIR spectroscopy that we can unambiguouslycharacterise the detected sources. This will not only allow for a direct census of galaxies to be made, but
+will also measure the AGN and starburst contributions to the bolometric luminosity of all detected galaxies
+over a wide range of cosmological epochs. And this will all happen in a single observation. The enormous
+advantage between the photometric surveys that Herschel will perform and those that SPICA-SAFARI willmake is precisely this: the capability of obtaining unbiased spectroscopic data in a rest-frame spectral
+range that have been shown in the local Universe to be mostly unaffected by extinction and to contain
+strong, and unambiguous signatures of both AGN and star formation emission.
+The diagnostic power of the rest-frame MIR and FIR
+Pioneering spectroscopy of a handful of nearby star forming galaxies and AGN with ISO-LWS (Fischer et al.
+1999; Braine & Hughes 1999) has demonstrated the importance and diagnostic power of the rest-frame FIR.1.5. Galaxy evolution 31
+0.0010.010.1110
+0.1 1 10++++++ + 
+AGNAGN
+0%100%50% AGN
+23128
+M8 217208NGC 4151NGC 1068
+Mrk 231Mrk 273
+NGC 5253Arp 220star
+formation
+dominatesAGN
+dominates
+relative strength of 7.7 μm PAH feature[O IV]/Ne II] or [O IV]/(1.7 x [S III])
+Figure 1.13: (a) Left panel: An example of how combinations of MIR line emission/feature emission can be used to deter-
+mine the relative importance of AGN- and starburst-activity in IR-luminous galaxies in the local Universe (Genzel et al.
+1998). This and other line ratio/feature combinations that will be formed from SPICA spectra will provide the key to
+identifying and separating AGN and stabursts in distant galaxies; (b) Right panel: An illustration of the range of MIR
+diagnostic lines and features that can be seen in Seyfert type-1/type-2 AGN (Tommasin et al. 2009), a starburst galaxy
+(Brandl et al. 2006) and a heavily obscured ULIRG (Sturm, priv. comm.). Many of the strongest of these lines will be
+detectable with SAFARI in a single 1-hr observation. Superposed onto the plot is the fraction of the waveband that willbe accessible to SAFARI at z ∼1,z∼2and z∼3. The spectra have been normalised at ∼27μm.
+Shown in Figure 1.12b are ten different spectra that have been observed: from strong atomic/ionic fine-structure
+line emission originated in photon dominated regions (PDR) and HII regions, to spectra showing strong molec-
+ular and even atomic absorption lines. ISO observations show, for example, that FIR fine structure lines can
+be used to separate starburst and AGN (Spinoglio et al. 2003). Star formation produces [O iii]88μm that origi-
+nates from HII regions as well as strong [C ii]158μm emission from PDRs, whilst AGN activity produces strong
+emission of [O iii] in the Narrow Line Regions (NLR), but also transforms a PDR into an X-ray dominated re-
+gion (XDR), significantly enhancing the [O i]63μm emission (Meijerink et al. 2007). The line ratio diagram of
+[Cii]/[Oi] vs. [O iii]/[Oi] can therefore be used to separate the two sources of energy production. Molecular
+emission lines can be used to probe the spatial extent and the geometry of the FIR continuum in the centres of
+galaxies (Gonz ´alez-Alfonso et al. 2008). The high excitation lines of OH and H 2O provide constraints on dust
+opacity models and OH and possibly the high-J lines of CO can be used to detect the molecular tori aroundAGN and study their physical conditions (Gonz ´alez-Alfonso et al. 2008).
+Herschel-PACS will observe complete 55 – 210 μm spectra of a relatively small number of local galaxies
+and confirm statistically the diagnostic power of the FIR molecular and atomic lines. In this way it will calibratelocally the tools with which to study distant galaxies which can only be seen with SPICA.
+The rest-frame MIR waveband contains several emission lines and spectral features which measure the
+contributions from AGN and star formation to the overall energy budget. These features do not suffer the heavy
+extinction that affects the UV , optical and even the near-IR lines, and therefore provide an almost unique di-
+agnostic in highly obscured regions (Spinoglio & Malkan 1992). ISO demonstrated that the ratios of emission
+lines tracing the hard UV field found in the narrow line region of AGN (e.g., [Ne v], [Oiv], [Nevi]) to those
+tracing stellar HII regions (e.g., [S iii], [Neii]) versus the strength of the PAH emission features - indicators
+of star formation - define a diagram which separates well the star formation and the AGN component in ob-
+scured galaxies (e.g., Genzel et al. 1998, Figure 1.13a). Spitzer observations have exploited these results and32 Chapter 1. Scientific Objectives
+have demonstrated that MIR tracers can be used to quantify the properties of AGN (Tommasin et al. 2008,
+Tommasin et al. 2009, submitted), weigh the black holes driving the AGN activity (Dasyra et al. 2008), reveal
+previously unknown populations of highly obscured AGN (Goulding & Alexander 2009), pin down the phys-
+ical properties of the local ULIRG population (Armus et al. 2007), trace directly the luminosity of the AGN
+(Mel ´endez et al. 2008; Rigby et al. 2009), and start to trace the lifecyle of local (U)LIRGs (Veilleux et al. 2009).
+The strong [Ne v] lines are unambiguous tracers of the presence of an AGN, as was shown by the detection of
+[Nev]14.3μm in NGC6240 (Armus et al. 2006), previously classified as a starburst galaxy before the detection
+of a buried AGN in the X-rays by Chandra (Komossa et al. 2003). Spitzer spectra have also demonstrated that
+the MIR hosts several new candidates for indicators of star formation in extragalactic objects: [Ne ii]12.8μm,
+[Siii]34μm, and [Si ii]35μm lines, as well as the PAH features (e.g., 7.7 μm, 8.6μm, 11.25μm) and the H 2
+pure rotational emission lines (e.g., 17.04 μm).
+1.6 The role of SPICA-SAFARI
+The SAFARI instrument on SPICA will exploit the power of MIR/FIR imaging spectroscopy and photometry tostudy the dust-enshrouded processes that drive galaxy evolution. With its broad instantaneous spectral coverage
+and high sensitivity, SAFARI will be able to cover the full 34 – 210 μm band in a few thousandths of the time
+that Herschel-PACS will take to obtain a deep spectrum of comparable resolution. SAFARI will be able to
+detect the key MIR and FIR lines from distant galaxies and use the MIR/FIR diagnostics that have, or will
+have, been revealed and calibrated by Spitzer and Herschel, respectively, in the local Universe. At a redshift of
+z∼1 the rest-frame 11 – 35 μm, which is very rich in ionic fine structure as well as H
+2rotational lines (see
+Figure 1.13b), moves to 22 – 70 μm and then out to 33 – 105 μmb yz∼2 and will still be in the SAFARI
+spectral range out to a z∼6. We will see in the following sections that SAFARI will detect the brightest MIR
+lines to z∼2i nintermediate luminosity objects in a 1 hour integration (see also Figure 1.14b). This should be
+compared with Herschel-PACS, which only has the sensitivity to detect the very brightest MIR (and FIR) lines
+in several hours of integration per line on the most luminous objects at z∼1.
+Photometrically, SAFARI will have the sensitivity to undertake very deep large-area, confusion-limited
+surveys at e.g., 70 μm (see Figure 1.16b), detecting all galaxies with LIR≥1011L⊙at a redshift of z=2, all
+those with LIR≥5×1011L⊙at a redshift of z=3 and the most luminous IR galaxies ( LIR≥1012L⊙) out to
+z=4, as well as the Milky Way-type populations ( LIR∼1010L⊙) out to z∼1.
+The wide SAFARI FOV of 2/prime×2/primegives an important spatial multiplexing advantage that will make it pos-
+sible for the first time to collect blind spectroscopic surveys (Figure 1.14a) that are both wide and deep enough
+to measure the underlying physical processes driving galaxy evolution out to z∼3 and, in the most lumi-
+nous/lensed objects, to even higher redshift. By comparing blind surveys with those targeted around known,
+high-z objects, we will be able to determine the role of environment on galaxy evolution.
+1.6.1 Spectroscopic cosmological surveys with SAFARI
+The first question that we need to answer, before moving to more detailed modelling, is whether the localgalaxies, for which we have detailed knowledge of their IR spectra, can be observed at high redshifts. In order
+to do so, we predicted the line intensities as a function of redshift (in the range z=0.1−5) for three local
+template objects: NGC1068 (Alexander et al. 2000; Spinoglio et al. 2005) ( L
+IR=20×1010L⊙), a prototypical
+Seyfert 2 galaxy containing both an AGN and a starburst; NGC6240 (Lutz et al. 2003) ( LIR=50×1010L⊙),
+a bright starburst with an obscured AGN; and M82 (F ¨orster Schreiber et al. 2001; Colbert et al. 1999) ( LIR=
+4×1010L⊙), the prototypical starburst galaxy.
+Figure 1.14b shows the predicted intensities of a selection of fine-structure emission lines that trace AGN,
+stellar ionisation and PDR regimes. These have been plotted as a function of redshift for the three templates. Itis clear from the figure that the Herschel-PACS spectrometer will be able to observe only the brightest object
+(NGC6240) up to z∼2 in the brightest line ([O i]63μm). In contrast, the SAFARI goal sensitivity (5σ-1hr of
+2×10
+−19Wm−2atR∼2000) will enable us to detect, simultaneously - in a single observation - unambiguous1.6. The role of SPICA-SAFARI 33
+Herschel
+Herschel
+HerschelSPICA
+SPICA
+SPICA
+Figure 1.14: (a) Left panel:A 5/prime×5/primeregion taken from an off-source part of a 250 μm ESA SPIRE publicity image, with
+the2/prime×2/primeFOV of SAFARI outlined (credit: ESA and the SPIRE consortium). With its spectral imaging capability, SAFARI
+will be able to obtain spectral information covering the full 34−210μm range, in multiple sources, simultaneously – in a
+single pointing; (b) Right panel: SAFARI has the sensitivity to detect redshifted MIR/FIR emission out to high-z. Shown is
+a plot of line intensity vs. redshift of a selection of the key MIR/FIR emission lines visible with SPICA in three archetypical
+objects: M82 (L FIR∼4×1010L⊙), NGC1068 (L FIR∼2×1011L⊙) and NGC6240 (L FIR∼5×1011L⊙) - in each panel
+the upper/lower dashed lines denote the 5σ-1 hr sensitivity of Herschel-PACS and SAFARI, respectively.
+signatures of both AGN and star formation and to determine redshifts in these relatively low luminosity objects
+out to z∼1−2 for most lines, and even higher redshifts for the brightest lines.
+Line diagnostic diagrams, such as the one shown in Figure 1.15a (using the [O iv]25.9μm, [Feii]26.0μm
+and [Siii]33.5μm lines) can readily separate AGN-dominated and starburst-dominated galaxies. Critically,
+these redshifted MIR lines all fall into the SAFARI waveband at redshifts of z≥0.4.
+The second task that we must fulfil to prove the feasibility of cosmological spectroscopic surveys, is to
+estimate of the number of galaxies that we expect to detect. We have used models constrained by observational
+data to predict the number of sources that we will be able to detect per SAFARI FOV at different redshifts
+in a blank-sky survey (see Spinoglio et al. 2009, for the details). Throughout, we assume the SAFARI goal
+sensitivity given above.
+Our first simulation uses the galaxy evolution model developed by Gruppioni et al. (2009) and Gruppioni
+et al. (in prep.) that makes use of available IR data to determine luminosity functions from z=0t oz∼4,
+with explicit contributions in the model from starbursts and AGN. To account for the population of spheroids
+(proto-ellipticals) which locally do not appear as substantial far-IR emitting sources, but are expected to be
+relevant at high redshifts (e.g., SMG galaxies), we integrated the model by Granato et al. (2004). This model
+uses an ab-initio approach for galaxy formation and evolution (see also Lapi et al. 2006) capable of predicting
+multi-wavelength luminosity functions and number counts. By following this approach, we calculate that we
+will detect around 7 sources over the redshift range 0 <z<3 in the [Si ii]34.8μm line. The lines of [O iv]26μm
+(a tracer of AGN) and [S iii]33μm (a tracer of star formation) are, in the local galaxies, a factor 1.5 – 2 fainter
+than the [Si ii] line, but comparable on average in LIRGs and ULIRGs (Veilleux et al. 2009). Given this, we
+expect to detect all three lines in these sources.
+In a second approach we adopted the model by Franceschini et al. (2009). This is a backward evolution
+model which fits all available data from Spitzer, ISO, COBE, SCUBA, etc. It includes direct determinations of
+multi-wavelength redshift-dependent luminosity functions from Spitzer and accounts in great detail not only for34 Chapter 1. Scientific Objectives
+star forming galaxies, but also for type-1 and type-2 AGN. Using this model we predict that it will be possible
+to detect about 4 starburst galaxies, 2 type-2 AGN and 1 type-1 AG N - a total of 7 galaxies - in the redshift
+range of 0<z<3 per SAFARI FOV assuming the standard 5σ 1hr sensitivity, and using as examples the
+[Siii]34.8μm line to detect starburst galaxies and the [O iv]26μm line to detect AGNs.
+Using these two independent methods we predict that we will detect of order 5 to 10 objects per SAFARI
+FOV per hour within the redshift range z≤3. These estimates confirm that in a deep SAFARI survey of
+about 500 hours/2000 square arcmin we will detect a few thousand sources in the key MIR/FIR tracers that
+will identify the energy sources powering the detected galaxies. Deep spectroscopic surveys with SAFARIwill therefore provide the first statistical and unbiased determination of the co-evolution of star formation
+and mass accretion with cosmic time. We have assumed a redshift-independent relationship between lineluminosity and continuum/IR bolometric luminosity throughout: Were these to evolve with redshift, then the
+number of sources that will be detected by SAFARI in blind surveys would increase significantly, particularlyat high-redshift. It is important to note that the increased abundance of ULIRGs at higher redshifts is accompa-
+nied by an evolution in their mode of star formation. Mid-IR spectroscopy with Spitzer-IRS (Farrah et al. 2008;
+Men ´endez-Delmestre et al. 2009) and ground-based submillimetre spectroscopy (Hailey-Dunsheath et al. 2008)
+have demonstrated that the emission from ULIRGs at z∼1−2 is markedly different from that observed in local
+ULIRGs, instead resembling that seen in lower luminosity starburst galaxies. This result can be understood if
+the high-redshift ULIRGs represent vigorous star formation extended over a few kiloparsecs, rather than the
+sub-kpc bursts that power local ULIRGs, and may be a consequence of the higher gas mass fractions present at
+these earlier epochs.
+Beside the rest-frame fine-structure MIR lines, other potential lines that can be measured by SAFARI in-
+clude the oxygen lines of [O i]63μmand [Oiii]52μmand the nitrogen [N iii]57μm line, that can all be seen
+in the SAFARI spectral range out to redshifts of z∼2. The enhanced [C ii] emission at high-z (possibly due
+to low-metalliticy effects, Nagao & Maiolino 2009), indicates that these FIR lines at high-z might indeed be
+stronger than in the local Universe. Moreover, Spitzer observations of distant radio galaxies (Egami et al. 2006;
+Ogle et al. 2007), have unveiled a class of objects termed as “H
+2luminous galaxies” whose spectra are dom-
+inated by H 2rotational emission lines. The lack of detection of PAH emission and hydrogen recombination
+lines (Egami 2009, priv. comm.), ruling out UV fluoresence as a possible excitation mechanism, indicates that
+H2is produced by shocks which deposit huge amounts of kinetic energy into the ISM. These H 2luminous
+galaxies may represent a population at high redshifts where interactions and mergers are more common. The-oretical work (Mizusawa et al. 2005) predicts that in these galaxies the pure rotational lines S(1) (17 μm), S(2)
+(12.3μm) and S(3) (9.6 μm) should have luminosities in excess of 10
+35erg s−1and thus within the reach of
+deep SAFARI surveys. Based on detections of H 2in nearby galaxies, we find that the power emitted in the two
+H2lines S(0) (28.2μm) and S(1) is typically around 20–25% of that found in the MIR lines of [Si ii]o r[ Siii].
+For the disc galaxies (Roussel et al. 2007), this corresponds to about 3 −3.5×10−4of the total infrared power,
+a limit that should be attainable in deep SAFARI surveys. Additional observational evidence for the existence
+of large molecular H 2reservoirs in high-redshift galaxies is provided by the tight correlation found between
+H2and PAH emission in local star-forming and disc galaxies (Rigopoulou et al. 2002; Roussel et al. 2007), as
+well as in galaxies out to z∼1 (Dasyra et al. 2009). PAHs are ubiquitous in high-z galaxies (Valiante et al.
+2007; Pope et al. 2008; Huang et al. 2009), and so if the PAH-H 2correlation holds at higher redshifts, then the
+LH2/LFIRcalculated above should serve as a lower limit. Simple calculations suggest that we might see up to 7
+sources per SAFARI field of view in the S(1) line, out to z<3.
+Beating spatial confusion
+The third, spectral dimension in deep spectral line surveys provides an important way by which to overcome
+spatial confusion. Narrow-band line emission from a source at a given redshift appears only at very specific
+and discrete wavelengths, and so the high density of sources in an individual beam that causes confusion is
+drastically reduced relative to all sources emitting in the continuum (Figure 1.15b). This enables individual
+objects, whose continuum may be substantially below the continuum confusion limit, to be detected. Through
+simulations (Clements et al. 2007; Raymond et al. 2009) we have found that sources more than 10 times below1.6. The role of SPICA-SAFARI 35
+10−210−110010110−210−1100101
+●
+●●
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+●●●
+●
+●●
+●●●
+●●
+●●●
+●●●
+[O IV] (25.9 μ μm) / [S III] (33.5 μ μm)[Fe II] (26.0 μ μm) / [O IV] (25.9 μ μm) ●
+●
+●Starbursts
+LINERs
+AGNs
+Supernovae
+Figure 1.15: (a) Left panel: MIR line ratios provide a powerful means by which to determine the dominant mecha-
+nism powering the prodigious IR output in the IR-luminous galaxy population. Plotted is the ratio of [Fe ii]/[Oiv] vs.
+[Oiv]/[Siii] for a selection of different types of galaxies. The overplotted grid represents a simple mixing by a linear
+superposition of starburst, supernovae (shocks) and AGN line spectra. The loci of the grid lines indicate where the con-
+tribution to the [S iii] is 0%, 10%, 20% etc. for supernovae shocks and AGN (Sturm et al. 2000). All lines are accessible
+in the SAFARI band from a redshift of z >0.4, providing a discriminator between star formation and mass accretion; (b)
+Right panel: Deep, spectroscopic surveys with SAFARI will enable us to break through the traditional confusion limit andwill single out and characterise individual galaxies that would otherwise be indistinguishable from the background sky.
+This power of increased spectral resolution is illustrated in the three panels above: Shown in the panel (i) is an image
+covering a single SAFARI FOV , simulating photometric mode at 120 μm, at a spatial resolution of 9
+/prime/prime. In this particular
+realisation of the sky, no discrete sources can be distinguished against the background; shown in the panels (ii) and (iii)
+are the same FOVs made at the same angular resolution, but this time centred at 63.2 and 58.3 μm respectively, and with
+a spectral resolution of R ∼1000. Several sources can be seen in each narrow band.
+the traditional (photometric) confusion limit at, for example, 120 μm can be isolated and detected. In this way,
+the majority of sources that contribute significantly to the CIRB can be detected and classified spectroscopically,
+as described above.
+Clustering and first large scale structures
+The source detection estimates presented in previous subsections for spectroscopic surveys assume random
+Poissonian distributions of galaxies all through the various cosmic epochs. However, extragalactic sources
+are known to be clustered. Significant clustering has been measured in high-redshift (z ∼1−3) Spitzer
+galaxies (Farrah et al. 2006; Magliocchetti & Br ¨uggen 2007; Magliocchetti et al. 2008; Farrah et al. 2006), with
+a strength which increases, as might be expected, with decreasing look-back time. Comparisons with theoretical
+models have provided a direct estimate of the dark matter mass of such sources, and the derived values ( M∼
+1013M⊙) indicate that luminous-IR galaxies at z∼2 are most likely the progenitors of the giant ellipticals which
+reside locally in rich clusters. This implies that studies of the infrared population at redshifts z>1−1.5 can
+provide a unique tool with which to investigate the formation and evolution of super-structures such as proto-
+clusters, clusters and of the galaxies that belong to them from even before the peak of cosmic star formation
+activity. While the results from Spitzer necessarily suffered from some limitations due primarily to the lack of
+measured redshifts, SAFARI in its spectroscopic photometric mode will, for the first time, be able to overcome
+such problems and provide definite answers to a number of crucial issues:
+(i) Direct and unbiased investigation of the evolution of the Large Scale Structure in the Universe
+from z∼3. Thanks to the SAFARI capabilities, no previous selection on the observed population is needed as
+redshifts will be taken for all the objects brighter than some limiting flux which fall in the SAFARI FOV . With36 Chapter 1. Scientific Objectives
+our goal 5σ sensitivity, such a task could be easily reached in about 500 hours by “blindly” surveying an area
+of approximately 0.5 square degree (see page 34). These results will not only be of invaluable importance per
+se, but will also complement those that will possible with NIR surveys which are limited for such studies to
+z∼1.5.
+(ii) Studies of galaxy formation and evolution as a function of environment. Since high-redshift sources
+exhibit strong clustering, galaxies observed by SAFARI in a blind spectroscopic survey will either reside inoverdense or underdense regions. This will give us the unprecedented possibility of investigating the impact
+of environment on galaxy formation and evolution as a function of redshift up - and possibly beyond - the
+epoch which marks the bulk of AGN/stellar activity. By doing this, it will be possible to provide answers to a
+number of questions such as whether there is any large-scale influence between surrounding environment andgalactic AGN/stellar activity and, if so, if seeds for such a phenomenon were already in place by z∼2−3. The
+spectroscopic diagnostics will be essential to measure directly the star formation and AGN luminosities in each
+cluster component.
+(iii) Targeted studies of Proto-clusters. As discussed above, most of high-redshift active galaxies reside in
+overdense regions or what we call proto-clusters. This was not only seen by Spitzer, as a number of other works
+report the discovery of z∼2−3 proto-cluster candidates in the NIR and for instance via the concentration of
+Lyman Break galaxies around powerful radio sources. With the imaging capabilities of SAFARI it will bepossible for the first time to collect a large sample of high redshift proto-clusters in a relatively short time.
+When targeting regions centred on “known” high-redshift sources, we expect the number of sources detectable
+per SAFARI field to be boosted by as much as a factor 10 above the predictions presented earlier in this section.
+Tracing the evolution of Milky Way-type galaxies
+Tracing the cosmic evolution of (U)LIRGs is one of the key goals of SAFARI; however, whilst this population
+makes a very significant contribution to the CIRB, such objects only make up a small fraction of the total
+number of galaxies. With SAFARI we will have, for the first time, the sensitivity to characterise the ISM of the
+complete z∼1 population, and so to follow the evolution as a function of redshift of the much less IR-luminous,
+less massive but more ubiquitous galaxies. Our own Milky Way will be easily detectable in the continuum outtoz∼1 as will be seen in section 1.6.2, whilst the MIR/FIR diagnostic lines will be visible out to z∼0.5 where
+the universal SFR has already increased by close to an order of magnitude as compared to the local Universe.
+The diagnostic power of the PAH and silicate features
+The ability of the UV excited PAH bands to trace any star forming component has been demonstrated for
+local galaxies (Soifer et al. 2002; Peeters et al. 2004). The ratio of the PAH bands has been related to star
+formation activity for a wide variety of galaxies (Galliano et al. 2005, 2008), while the ratio of PAH emission
+line strength to underlying continuum has been established as a proxy for disentangling star-forming and AGN
+contributions (Brandl et al. 2006). In the deepest spectra obtained with Spitzer IRS (Men ´endez-Delmestre et al.
+2009; Siana et al. 2009) these features can be seen out to z>2 in sources that are both optically and MIR-faint
+(S(24μm)∼100μJy). The observed similarities between MIR spectra of LIRGS at the present epoch out to
+z∼2.5 suggest similar dust properties, and thus that significant enrichment and evolution has already taken
+place by z∼2.5 (e.g., Swinbank et al. 2004). When then, did the bulk of dust production and evolution take
+place? Delayed injection of heavy elements from AGB stars would, for example, weaken PAH emission,and
+increase the relative importance of other sources of dust such as supernovae or AGN winds (Maiolino
+2007). Detecting changes in the PAH emission/silicate absorption feature profiles and strengths with epoch
+require modest-resolution spectrometers in both the MIR and FIR (R ∼25) in order to follow the PAHs as they
+move from one waveband to another with redshift (Figure 1.16a). With SPICA instruments we will have the
+sensitivity to exploit the diagnostic power of the PAH/silicate features: By combining the low-resolution modes
+of MIRACLE and the spectrophotometric mode of SAFARI we will be able to make large-scale spectroscopic
+surveys of very faint galaxies, down to ∼20μJy and∼a few 100μJy, respectively with the two instruments.
+A wide-field, spectrophotometric survey with SPICA over an area of 1/4 the size of the COSMOS field1.6. The role of SPICA-SAFARI 37
+(0.5 square degrees) would yield spectra of ∼5000 high-z galaxies in 500 hours, a factor of 10 more than the
+total number of high-z galaxies observed with Spitzer IRS, and a significant increase on those that would have
+been observed by JWST by the launch date of SPICA. The bulk of these flux-limited catalogued objects will
+be found at moderate redshifts (z <2). With deeper integrations (e.g., a 500 hr survey covering 200 sq. arcmin
+with resolution R∼20), we anticipate being able to detect and characterise the spectrum (emission features and
+continuum) of several hundreds of objects at z>2. As mentioned above, these numbers are likely to be boosted
+significantly when targeting regions around “known” high-redshift sources. By measuring the PAH and silicate
+features on top of the dust continuum, we will not only be able to disentangle directly the relative importance
+of starburst and AGN emissions and to estimate the rates of star formation and gravitational accretion in the
+distant sources. With these spectral surveys we will also pin down redshifts in a completely unbiased way,
+with which to evaluate the co-moving space density of these dusty populations and to constrain the bolometric
+luminosity functions of luminous infrared galaxies.
+Spectroscopy of very distant galaxy populations through gravitational lensing
+The strong gravitational lensing of the distant Universe by intervening galaxy clusters has been exploited at
+many wavelengths in order to push the luminosity limits and redshifts out to the farthest limits. In the IR/sub-
+mm, lensing clusters act as transparent lenses, and cluster surveys at these wavelengths have been particularly
+powerful at pushing below the confusion limits of sub-mm surveys to constrain the faint end of the source
+counts (Smail et al. 1997; Blain et al. 1999) and opening up the study of not only some of the most distant
+objects, but also of those that are less extreme at slightly more modest redshifts (Knudsen et al. 2006). DeepHerschel lensing surveys will do this in the FIR, but only photometrically. With SAFARI we will be able
+to exploit the strong lensing power of massive clusters to push the same deep spectroscopic imaging studiesdescribed above from 0 <z<3 out to z∼3 and beyond and will be able to probe the properties of LIRGs
+in the epoch of cosmic reionisation for the first time in the MIR. In a more quantitative way, if we assume that(i) the IR galaxy luminosity function at z∼2 (Caputi et al. 2007) is representative of that at higher redshifts,
+(ii) the fractional IR luminosity in the 7.7 μm PAH feature is at least that observed in z∼2 submillimetre
+galaxies (Pope et al. 2008), (iii) that a typical cluster has a lensing potential similar to that of Abell 773, then in
+a survey of 50 massive clusters with SAFARI in spectrophometry mode we would expect to detect ∼23 LIRGs
+at 7<z<10 using the 7.7μm PAH feature. A blank-field survey with the same areal coverage will produce
+∼8 detections in the same redshift range, so the lensing survey will offer a factor of ∼3 gain in number. The
+gain will be even larger if there is an abundance of low-mass /low-luminosity galaxies at such high redshift as
+for instance predicted by hierarchical structure formation theories. In fact, by accessing an epoch at which the
+space density of IR-luminous galaxies is virtually unknown, SAFARI will provide valuable observational datain the epoch of reionisation, constraining the cosmic histories of star formation, metallicity evolution (e.g., the
+7.7μm PAH strength is sensitive to metallicity), and structure formation. This is especially timely since in the
+coming decade, various new observatories such as ALMA and JWST will push the frontier of extragalactic
+IR/sub-mm astronomy beyond a redshift of 4 and possibly up to 10.
+1.6.2 Deep cosmological surveys with SAFARI
+The existence of a large number of distant sources radiating most of their energy in the IR implies that the
+critical phases of the star-formation and nuclear accretion history took place in heavily obscured systems, em-
+bedded within large amount of gas and dust. The substantial reddening affecting these dust-obscured objects
+makes their characterisation in the optical/UV severely biased and sometimes even impossible. Observations
+in the FIR/sub-mm to very faint flux densities (few 10s μJy) are often the only means to detect star-formation
+and/or AGN activity in most of these heavily obscured sources at large cosmological distances. Deep pho-
+tometric surveys with SAFARI will be sensitive to the important galaxy populations that will be missed both
+by spectroscopic surveys and by the current generation of FIR/sub-mm surveys. Both ground-based sub-mm
+surveys with existing facilities (SCUBA, MAMBO, LABOCA, SCUBA2) and FIR surveys from space (ISO,
+Spitzer) are limited by the confusion that results from modest angular resolution, and are sensitive only to38 Chapter 1. Scientific Objectives
+Figure 1.16: (a) Left panel: Shown in green is a local ULIRG template that has been fitted to the observed spectrum SMM
+J09429 (red) (Valiante et al. 2007), with the SPICA spectrophotometric sensitivities overlaid (for SAFARI this is R ∼50).
+Both spectra have been redshifted to z ∼3for illustrative purposes. SAFARI will have the sensitivity in low-resolution
+mode (R∼50) to detect PAH/silicate features in dusty distant galaxies out to z ≈3in 1 hour, and out to z ∼4in 10
+hours. These features will provide important pointers to the physical processes powering distant dusty galaxies; (b) Right
+panel: Deep photometric surveys will identify many thousands of distant FIR luminous galaxies. Shown in blue is a plot
+of the luminosity-redshift distribution of FIR galaxies that will be detected by Herschel in its deep cosmological survey
+programmes. In red we show the galaxies that will be detected in a SAFARI 70 μm confusion-limited survey of 900 square
+arcmin. For a given IR luminosity, SAFARI will detect galaxies that are much more distant, whilst at a given redshiftSAFARI will detect a much wider range of source luminosities thus identifying the bulk of the galaxy population. The boldblack line traces the redshift evolution of an L
+∗galaxy and represents the knee luminosity of the IR luminosity function
+derived using Spitzer-MIPS 24 and 70 μm (Magnelli et al. 2009).
+the most luminous and distant galaxies, or to those that are very nearby. Such galaxies contribute less than
+half of the background at these wavelengths, and make up an even smaller fraction of the integrated CIRB.
+Herschel-SPIRE and PACS will open new windows in the FIR above 100 μm - a taster of this has been given
+by the BLAST balloon experiment (Devlin et al. 2009; Pascale et al. 2009). However our current knowledge ofsource densities at these wavelengths suggests that we will resolve only 50% of the CIRB. Our ability to identify
+sources against the CIRB is governed by confusion which dominates the performance of Herschel long-ward
+of 100μm. At shorter wavelengths the sensitivity of Herschel-PACS is insufficient to reach the confusion limit
+and precludes our ability to resolve the CIRB. In contrast, SAFARI will be able to resolve the bulk (>90%) ofthe cosmological background over a wide wavelength interval from ∼30 to∼100μm and, as we show below,
+will be unrivalled by either ALMA or JWST.
+Deep surveys at 70 μm – resolving the CIRB
+According to current models, the 70 μm confusion limit for SPICA is around ∼50μJy. To reach such depths
+with Herschel-PACS would take an enormous and unrealistic amount of time (10 000 hours 5σ!), while with
+SAFARI this can be achieved in minutes. Using current assumptions for the intensity of the CIRB at this
+wavelength, a 70μm confusion-limited survey (Figure 1.16b) would resolve more than 90% of the CIRB over
+80% of the Hubble time (z ∼2), detecting galaxies down to a star formation rate regime at which rest-frame
+UV observations meet the IR, i.e., 10 M
+⊙/yr (Lagache et al. 2004) (more luminous galaxies emit the majority of
+their light in the IR). By observing at 70 μm we can avoid the effects of contamination by the strong MIR PAH
+features, and thus make very reliable determinations of the IR flux for sources with z<2. Such a survey would
+increase significantly the range in IR luminosity and redshift over which galaxies will have been detected: at
+the more luminous end, LIRGs out to z∼4 and ULIRGs to beyond, and appreciable numbers of L∗galaxies out
+toz∼3. With SAFARI it will be also be possible to detect galaxies as quiescent as our own ( LIR<1010L⊙)
+out to z∼1 (Figure 1.16b), where the cosmic SFR peaks (see Figure 1.11b). Whilst this less IR-luminous
+population contributes only minimally to the CIRB, these galaxies are extremely abundant, and dominate theextragalactic background light at optical wavelengths. Confusion-limited surveys at 70 μm are unique territory1.6. The role of SPICA-SAFARI 39
+Figure 1.17: (a) Left panel: The broadband SED of NGC1068 (in black) redshifted to z ∼3; overlaid are the different
+constituent emission components that are produced by a torus (red: model by Fritz et al. 2006), a starburst component
+based on the starburst galaxy NGC1482 and a stellar component (orange) based on the stellar evolution models. The
+vertical bars indicate the three bands of SAFARI, and highlight how the combination of 40 μm and 70μm emission can be
+used to differentiate between AGN- and starburst-dominated galaxies; (b) Right panel: A plot to illustrate the accuracy
+with which a multi-filter photometric survey with SAFARI and MIRACLE will be able to recover redshifts. Note the outliersare typically elliptical galaxies and type-1 AGN, both of which have very little PAH emission.
+for SAFARI.
+Deep surveys at 40 μm – Broadband footprints of distant, obscured AGN
+The surveys with SAFARI at 40 μm will not be confusion limited (estimated confusion limit <10μJy). A
+survey at 40μm down to a limit of a few 10s μJy would be sensitive to moderate luminosity ( LIR∼1011L⊙)
+type-2/obscured AGN out to z∼5, and particularly in the 3 ≤z≤5 range, where the co-evolution of star-
+formation and accretion activity is expected to already be in place. Heavily dust-obscured, mostly Compton-
+thick, AGN are believed to be responsible for the unresolved peak of the X-ray background at 30 keV . These
+AGN are missed by the deepest current-generation X-ray surveys (the prototype Compton-thick AGN NGC
+1068 would be missed above z=0.5 even in 2 Ms with Chandra), yet reveal themselves in the MIR when
+stacking techniques are applied at X-ray wavelengths to strongly amplify their extremely dim fluxes. ObscuredAGN can be identified through their infrared signature, which is reprocessed UV emission from the AGN itself.
+By selecting MIR sources with faint NIR and optical emission, both Daddi et al. (2007) and Fiore et al. (2008)
+suggest that the majority of the so called “IR excess” sources in the Chandra Deep Field-South are in fact highly
+obscured AGN at 1 <z<3. Spitzer MIR surveys have been key to complementing X-ray surveys, in identifying
+sizable samples of obscured AGN for z<3 through the 9.7μm silicate absorption (e.g., Weedman et al. 2006).
+The combination of the silicate absorption feature, the PAH features and the broadband contribution from thewarm, dusty tori can be used to disentangle the AGN from starburst populations at redshifts z>3 through
+photometry in the 40 μm( 3 4–6 0μm) band. This applies both at high and low infrared luminosities.The
+combination of both the 40 μm and 70μm bands will provide a unique means in the SPICA era by which to
+disentangle co-evolving AGN and star-formation activity in the very distant universe (Figure 1.17a), and wouldcomplete the census of accreting supermassive black-holes (SMBHs) at z>3.
+SAFARI and IXO will provide a complementary perspective on distant AGN. IXO will characterise SMBHs
+out to z∼7 by measuring the X-ray spectral slopes and luminosities, which provide constraints on accretion
+rates, and so assess directly the growth of supermassive black holes at high redshift. SAFARI+IXO will be ableto measure the bolometric output of AGN out to z∼5 at least, by combining contributions due to star formation
+and accretion, and together will provide the first direct measurement and study of the dust-to-gas ratio and itsevolution from z∼0t oz∼5.40 Chapter 1. Scientific Objectives
+Spectral Energy Distributions and photometric redshifts in the MIR/FIR
+The direct characterisation of the properties of dusty star-forming galaxies and AGN at MIR /FIR wavelengths
+is crucial for estimating the fraction of their bolometric energy output absorbed by dust and re-radiated at longer
+wavelengths. It also provides unique constraints on the nature of the different physical processes driving theirongoing activity (e.g., Smith et al. 2007; Spoon et al. 2007). Dust plays a dual role in extragalactic studies, si-
+multaneously helping (emission by heated dust) and hindering (obscuration) our understanding of the physical
+processes driving galaxy evolution. The deep 40 and 70 μm surveys described above will provide unique insight
+into galaxy evolution at very high redshifts, and as a very important by-product will allow us to track changesin dust attenuation of stellar light in galaxies. GALEX has shown that dust attenuation in individual galaxies
+out to redshifts of z∼1 remains constant, with a hint of a decrease (Zheng et al. 2007; Buat et al. 2007a). This
+is in stark contrast to the mean dust attenuation in the Universe which has been shown by Takeuchi et al. (2005)
+to increase over the same epoch. This apparent contradiction can be attributed to the general brightening of the
+galaxy population with redshift, and the long-established correlation between dust obscuration and total lumi-
+nosity of the galaxies (e.g., Hopkins et al. 2001; Buat et al. 2007b). At some point in cosmic history, obscured
+environments must become less common since it takes time for dust to not only be produced, but also to spread
+through the ISM of a young galaxy. The epoch at which this takes place can be pinpointed by comparing UV
+and rest-frame IR emission of galaxies which are responsible for the bulk of star formation at di fferent epochs.
+The deepest Spitzer observations at 24 μm suggest that at z>2 galaxies with bolometric luminosities L
+bolof
+1011<Lbol<1011.5L⊙contribute substantially (Caputi et al. 2007) or even dominate (Reddy et al. 2008) the
+star formation activity in the Universe. By detecting these galaxies directly in the FIR we can measure the emis-
+sion from dust. SAFARI will detect galaxies ten times fainter than Herschel, corresponding to Lbol=1011.5L⊙
+atz=2.5 (Figure 1.16b). Whilst ALMA and JWST can both detect some fractions of this population, neither
+can make the crucial step of characterising the dust SED, since such observations either sample the redshiftedstellar emission (JWST) or are restricted to the Rayleigh-Jeans dust emission (ALMA), and so cannot be used
+to establish dust temperates and source luminosities.
+The same MIR PAH and silicate absorption features that contaminate and complicate the interpretation of
+deep surveys, on the other hand provide an excellent means by which to determine redshift using SPICA. To-
+date, photometric redshifts of SCUBA and Spitzer sources have been determined using optical/NIR data taken
+with many different surveys. With SPICA we will be able to determine redshifts using the combination of the
+NIR and MIR. Simulations suggest that with as few as 13 (R ≈8) filters, evenly spaced over the 15 - 70 μm
+wavelength, it is possible to determine the redshift and type of a very wide range of galaxies, to an accuracy of afew % (Figure 1.17b). By including SAFARI bands we not only extend the redshift range of the technique, but
+also increase accuracy: with SAFARI bands we sample well the PAH and silicate absorption features which, in
+turn, break most of the degeneracy in the MIRACLE-only estimations. Sources for which the match between
+recovered and true redshift is poor are those lacking distinct PAH/silicate features: in the case of the AGN,
+these are intrinsically optically bright, and thus good candidates for spectroscopy.
+1.7 SPICA in the context of ALMA and the JWST
+SPICA will be the bridge between two other very large astronomical facilities that are due to become operational
+within the next five years - namely ALMA and JWST. The JWST will revolutionise near-IR and MIR astronomy
+through its impressive sensitivity, and ALMA will make a huge impact with its unprecedented sensitivity and
+high angular resolution in the sub-mm. SAFARI probes the crucial wavelength regime between the JWST and
+ALMA regions.
+SPICA will cover critical spectral diagnostics (at their rest frame wavelengths) that are not accessible to
+ALMA. While the study of the oxygen chemistry with ALMA will be limited to observations of CO, HCO+
+and other trace species, SPICA will observe the emission of the major oxygen reservoirs like atomic oxygen,
+water ice, CO 2or the thermal line emission of water vapour. ALMA will detect the millimetre and sub-mm dust
+continuum emission, but only SPICA will be able to observe the specific grain and ice spectral features that areneeded to determine the dust composition and its formation history along the life-cycle of interstellar matter.1.8. Discovery Science with SPICA 41
+ALMA and SPICA will be complementary to address the chemical complexity in the Universe. In particular,
+SPICA will allow us to observe molecules without permanent electric dipole or the mid-IR emission of PAHs,
+neither detectable with ALMA, which are major players in organic chemistry in space.
+Since the discovery of the cosmic infrared background and the important populations of luminous dusty
+galaxies at high redshift, it has become clear that obscured star formation accounts for a large fraction of thestellar mass assembly budget. Enormous progress in infrared/(sub)mm/radio source count surveys will be made
+by Herschel, ALMA, EVLA and ground-based bolometer arrays such as SCUBA2. While ALMA will probe
+the molecular ISM in these objects and JWST the stellar component, the highly diagnostic mid-IR region is
+exclusive SAFARI territory. For instance, for an object at z∼3, JWST will be able to observe the rest-frame
+spectrum only out to 7 μm. The most luminous PAH features, hot dust continuum and key fine-structure
+lines will all be accessible only with SAFARI. This opens up the intermediate redshift Universe (at the peak
+of cosmic star formation activity) for scrutiny with the techniques that have been successfully used on low-z
+galaxies since ISO and Spitzer. The line sequence [Ne ii]/[Neiii]/[Nev] will characterise star formation as well
+as the importance of AGN activity. Many other similar probes are available for SAFARI, but neither ALMA nor
+JWST provides equivalent diagnostics. Thus the unique role of SAFARI is the characterisation of the sources
+of the cosmic infrared background. A further complementarity between ALMA and SAFARI is provided by
+SAFARI’s ability to trace the energy budget of the star forming ISM out to z∼2 using the [O i]63μm line and
+[Cii]158μm at lower redshift, while ALMA will access the [C ii] line only at z>2.
+The relation between SMBH and spheroid mass is one of the most astonishing results of modern astro-
+physics, given the vastly di fferent scales involved. The enormous difference between the Schwarzschild radius
+of the SMBH and the characteristic radius of the stellar population suggest that this relation is related to the co-
+eval formation of both. While theoretical proposals to account for this relation exist, observational constraints
+are far and few between. To improve this situation, observations of galaxies at the peak of the cosmic star
+formation and black hole accretion rate are a key requirement. A fundamental diagnostic is provided by the
+[Nev] line at 14.3μm. This line is a very powerful probe of AGN activity in dusty galaxies, as shown by
+recent Spitzer work which has uncovered a large number of obscured AGN in the nearby Universe, includingCompton-thick sources. Furthermore, as shown by Dasyra et al. (2008), the width of [Ne v] can be used to
+measure black hole mass even at the low spectral resolution of Spitzer. SAFARI will thus play a key role inestablishing the inventory of AGN in obscured galaxies at intermediate redshifts, and trace the mass-buildup of
+the nuclear SMBHs. Again, equivalent diagnostics are not available with either ALMA or JWST.
+Both JWST and ALMA will probe galaxies at the epoch of reionisation (z >6). At these redshifts the most
+luminous CO lines shift out of the observing band, and the [C ii] line becomes the key probe (Walter & Carilli
+2008). In pristine (zero-metallicity) gas, however, this line is not available. The molecular medium in zero
+metallicity galaxies cools principally through H
+2rotational and vibrational lines. As shown by Mizusawa et al.
+(2004), a number of these lines may be detectable with SAFARI, over a wide range of conditions. Targets willbe provided by JWST, which is expected to discover significant numbers of z>6 galaxies, with the prime
+targets for SAFARI being those that are not detected in CO or [C ii] by ALMA: the detection of the signatures
+of a metal-free molecular interstellar medium is again unique SAFARI territory.
+1.8 Discovery Science with SPICA
+The science cases we have presented illustrate the concrete and definitive progress in our understanding of theUniverse that SPICA will allow. However, as with all strides in increased sensitivity or the opening of new
+wavelength bands, new discoveries are certain once SPICA starts observing. To give a flavour of what might
+be expected in the new discovery space opened by SPICA, we give here some examples of speculative research
+projects from both the nearby and distant Universe.
+The “end-point” of extra-Solar planetary systems: During the AGB phase, the radius of a star like our Sun
+will have expanded to a few AU (close to Jupiter’s orbit) and it will be a few hundred times more luminous
+than during the main sequence stage. The associated increase of temperature in the outer Solar System may be42 Chapter 1. Scientific Objectives
+sufficient to cause vapourisation of any icy body orbiting within a few hundred AU, i.e., the KBOs presented
+in section 1.2.2 (Stern et al. 1990). Indeed, the surprising detection of water vapour lines in the C-rich star
+IRC+10216 (Melnick et al. 2001) has been interpreted as the vapourisation of a collection of orbiting ice bod-
+ies caused by the luminosity increase in this evolved star (with an original mass similar to the Sun). SAFARI
+will simultaneously search for atomic oxygen, water vapour and ice features (i.e., the ∼44μm feature) in many
+C-rich evolved stars where the prevailing carbon chemistry can not produce substantial amounts of water. If
+the previous scenario is right, SPICA/SAFARI not only will detect hundreds of planetary disk systems, and
+characterise tens of icy KBOs in our Solar System, but it will also trace the “end-point” of extra-solar systems
+around evolved stars; i.e., the sublimation of entire Kuiper belts at the end of the host star life.
+First identification of PAHs molecules in Space (the “GrandPAHs”): The ubiquitous MIR emission bands
+are attributed to the emission of a family of carbonaceous macromolecules: the PAHs. However, because these
+bands are due to the nearest neighbour vibrations of the C-C or C-H bonds, they are not specific to individ-
+ual PAH species. Therefore, in spite of their relevance for astrophysics (as tracers of UV radiation fields or
+star forming regions in a broader extragalactic context), the identification and characterisation of a given PAHmolecule in space has not been possible yet. The hope for such an identification lies in their FIR lower-energy
+bands (PAH skeleton modes) which are specific fingerprints of individual molecules. The energy that is emitted
+in these bands is expected to be very weak, typically a few tenths of the energy absorbed in the UV (Joblin et al.
+2002; Mulas et al. 2006a,b). For example, it can be shown that if all the MIR emission observed in the ISM
+was due to a unique PAH such as ovalene, its FIR emission should be detectable even with a low (∼ 10) signal-
+to-noise ratio (Mulas et al. 2006a; Bern ´e et al. 2009). Recent progress in the field suggests that there are in fact
+only a few (or at least limited number) of large and compact PAHs in space, that can resist the harsh interstellar
+conditions. These large PAHs are the so called “Grand-PAHs”. The instantaneous broad band coverage of
+SAFARI will be especially adapted for deep searches of the Grand-PAH FIR bands, which is crucial since the
+positions of these bands are not known. Therefore, SAFARI can represent our first chance to identify specific
+PAHs in space.
+A new window in exoplanet research–far infrared observations with SAFARI: In the field of exoplanets
+(EPs), SAFARI will provide capabilities to complement SPICA studies in the MIR (either coronagraphic or
+transit studies). With very stable detectors, and efficient, high cadence observations, SAFARI will be used to
+perform transit photometry and, on some favourable candidates, spectrophotometry for the first time in the FIR
+domain. Therefore, SPICA could be the only planned mission able to study exoplanets in a completely new
+wavelength domain (neither covered by JWST nor by Herschel). This situation is often associated with unex-
+pected discoveries. In the shortest wavelength photometric band (∼ 48μm), primary and secondary eclipses of
+Jupiter-size EPs around F to M stars will be detected in a single transit. FIR measurements will be speciallysuited for EPs around cool M stars. Since cooler EPs show much higher contrast in the FIR than in the NIR/MIR
+(e.g., Jupiter’s effective temperature is ∼110 K), if such EPs are found in the next 10 years, their transit stud-
+ies with SAFARI will help to constrain their main properties, which are much more difficult to infer at shorter
+wavelengths. The SPICA longer wavelength range hosts a variety of interesting atmospheric molecular features
+(e.g., H
+2Oa t3 9μm, HD at 37μm, NH 3at 40 and 42μm, etc.). Strong emission/absorption of these features
+was first detected by ISO in the atmospheres of Jupiter, Saturn, Titan, Uranus and Neptune (Feuchtgruber et al.1999). Suitable EP candidates around M dwarfs for SPICA spectroscopic observations may be scarce, and a
+considerable amount of observing time will be needed to extract their main atmospheric composition. How-
+ever, in a few optimal systems, SPICA observations may allow us to expand the variety of “characterisable”
+extrasolar planets. These are crucial steps to enter in a new era of quantitative characterisation of EPs.
+Searching for the first generation of star-formation and black hole growth: In regions of low metallicity,
+≤10
+−2solar, the pure rotational lines of H 2(at 28, 17, etc.μm) are the primary coolant of UV and X-ray irradi-
+ated gas. Typical line intensities can be of the order of 0 .1−1.0e r gs−1cm−2sr−1, exceeding those of the solar
+metallicity fine-structure and CO lines. Thus, molecular hydrogen emission provides a unique means by which
+to pinpoint the first episodes of star formation (population III stars) and early signatures of black hole growth at1.8. Discovery Science with SPICA 43
+high redshift (z≥6). By definition, the nature of the search for such objects relies on the imaging capability of
+SAFARI, for which simple calculations suggest that between 1-10 of such halos might be detectable in a single
+SAFARI FOV . Whilst alternative cooling mechanisms will exist in more metal-rich systems, the H 2rotational
+lines will still be valuable probes of the warm molecular gas reservoir.
+The diagnostic potential of the very high-J CO lines: The high-J ( J≥13) CO lines provide a yet-to-
+be-demonstrated method of establishing the presence and physical properties of highly-obscured AGN
+(Krolik & Begelman 1988; Meijerink & Spaans 2005). The predicted differences between the expected CO
+spectral line energy distributions in X-ray (XDR) and photon-dominated regions are most significant (3 orders
+of magnitude) in the SAFARI band. By summing the CO, [C ii]158μm and [O i]63μm emission one can
+evaluate the total cooling rate for gas within the locale of the AGN and, assuming thermal balance, derivethe product of black hole mass and accretion efficiency. Thus, one can infer the properties of the black holes
+residing in obscured environments even in the absence of X-ray measurements. Herschel-PACS will make
+the first studies of these sources; the sensitivity of SAFARI will take us to the next stage and give access to
+a representative sample covering a wide range of mass and accretion luminosities, thus tracing the role and
+destiny of torus-covered black holes both locally and out to z∼1.
+Feeding hungry mouths–mass accretion from the IGM: The means by which primordial gas is accreted from
+the IGM to fuel on-going star formation in galaxies remains an unanswered question in galaxy evolution.Any mechanism for this direct fuelling requires a means by which to cool what will be hot gas. The cooling
+process is widely believed to be mediated by the injection of metals into the IGM in galactic winds, which
+themselves cool by inelastic collisions with dust grains at the IGM/wind interaction zone (Dwek & Werner
+1981; Montier & Giard 2004). The recent discovery by Spitzer of a diffuse IR counterpart to the X-ray emitting
+intracluster medium of Stefan’s Quintet supports this hypothesis, its luminosity exceeding that of the X-ray
+emission by two orders of magnitude. Herschel will search for the FIR signatures from hot plasmas in dense
+cluster and group environments; however, exploring this phenomenon in the bulk of the galaxy population will
+fall to SAFARI. SAFARI will be able to trace the FIR emission both photometrically and spectrophometrically,thus enabling not only determinations of the cooling properties of the IGM to be made, but also characterisation
+of the dust grains.
+0 5 10 15
+Time in hours24.624.825.025.225.425.625.826.0Flux in mJy @ 48 micron
+Time in hoursFlux in mJy @ 48umb)
+SAFARI @ 48umNew frontiers in exoplanet research: Far infrared observations
+a)
+Figure 1.18: (a)Signal-to-noise ratio for a primary transit of a Jupiter-mass EP at a distance of 1 AU from the host star
+with SAFARI (photometry at 48 μm). The star-mass vs. planet period parameter space is shown. A single spectral feature
+of 1μm width is assumed in the bandpass. R p: EP radius,μ: mean atmospheric molecular weight, Δλ: spectral feature
+width (by A. Belu&F . Selsis). (b)Specific synthetic light-curve at 48 μm of a Jupiter-like EP around a cool M0 star at
+10 pc (by S. Pezutto).44 Chapter 1. Scientific Objectives
+1.9 Concluding remark
+We have seen in the preceding sections that pushing forward our knowledge and understanding of the forma-
+tion of galaxies and planets requires a leap in sensitivity in mid to far infrared spectroscopic and far infrared
+photometric capabilities. Only by placing a cold (< 6 K) 3.5 m telescope in space with instruments sensitive
+enough to take advantage of the low photon background, can we achieve the detection limits required to fulfil
+the ambitions of the ESA Cosmic Vision and gain a true insight into the conditions for planet formation and
+the emergence of life, how planetary systems, including our own Solar System work and how galaxies orig-
+inated and what the Universe is made of . In other words, SPICA will play a central and vital part in ESA’s
+Science Programme in the next decade.Chapter 2
+Scientific Requirements
+This chapter describes the scientific requirements that should be fulfilled to achieve the scientific objectives de-
+scribed in chapter 1. The requirements are related to the following elements: i) Mission definition; ii) Payload:
+Telescope, mid infrared instruments, far infrared instrument, and mid infrared coronagraph instrument.
+Requirements on the subsystems that are part of the European contribution (Telescope and SAFARI) have
+been explained in more detail and are given first. Requirements on the other satellite elements responsibility of
+JAXA are included here for completeness.
+2.1 Telescope scientific requirements
+The achievement of the scientific objectives is strongly dependent on the specification of the SPICA telescope.
+The telescope operational temperature, wavelength coverage and image quality are defined such that SPICA
+provides unique scientific capabilities with respect to other space missions like AKARI, Herschel, and JWST.
+The “direct detection of exoplanets” scientific objective drives the scientific requirements on the mid in-
+frared coronagraph instrument, which in turn impose the most stringent requirements on telescope image qual-ity. However, since the mid infrared coronagraph is considered by JAXA as optional element of the payload,
+we have specified the requirements that it adds on the telescope separately. In this way it is possible to assess
+properly the delta on the telescope development related to the mid infrared coronagraph needs.
+2.1.1 Wavelength coverage
+The “design and operation” wavelength range of the telescope is defined as the interval 5 – 210 μm, which
+corresponds to the nominal baseline spectral coverage of SPICA core baseline focal plane instrumentation
+spectral range in the mid infrared and far infrared. A possible extension of the mid infrared instrument coverage
+(including the baseline operation range of the mid infrared coronagraph) down to 3.5 μm are considered in the
+definition of the goal “design and operation” wavelength range of 3.5 – 210 μm.
+A “characterisation” wavelength range of 1 – 1000 μm has been defined over which knowledge of the
+telescope characteristics is required. This characterisation can be obtained by analysis, meaning that specific
+testing activities are not necessary. This knowledge is essential because the detector arrays of the core mid
+infrared and far infrared focal plane instruments will still have residual detectivity down to ∼1μm and up to
+∼1 mm respectively.
+2.1.2 Telescope aperture
+For diffraction limited performance, the angular resolution as provided by the telescope optics is equal to
+1.22λ/D. Many of the science cases described in chapter 1 require a 3-m class telescope to achieve the nec-
+essary collecting area and the spatial resolution. An aperture similar to Herschel’s (3.5 m) is required for the
+4546 Chapter 2. Scientific Requirements
+science to go beyond Herschel’s results. In particular, the current launcher selected for SPICA, the JAXA
+H-IIB, will allow a monolithic telescope of 3.5 m circular diameter.
+The monolithic design of the SPICA telescope is particularly relevant for the mid infrared coronagraph. A
+clean and simple telescope Point Spread Function is essential to meet the required scientific performance of thecoronagraph and for the quality of the planet image and spectra. In addition, the telescope size is determinant
+for the value of the inner working angle of the coronagraph (= 3.3λ/ D), or minimum angular separation from
+a star at which an exoplanet can be detected.
+Obscuration and transmission
+In order to maximise the efficiency of the telescope collecting area over the “design and operation” wavelength
+range, and minimise its possibility of self-emission, in particular in the far infrared, the total obscuration of the
+telescope shall be less than 12.5% (goal <10%) in terms of telescope pupil relative surface area, over the full
+telescope field of view (as defined in section 2.1.3).
+The intensity transmission T of the unobstructed region(s) of the telescope pupil aperture shall be as high
+as possible to take advantage of the large collecting area, and more specifically higher than the piecewise
+linear transmission curve. Table 2.1 lists the values of the intensity transmission required over the “design andoperation” wavelength range of the telescope.
+Table 2.1: Required and goal intensity transmission
+SPICA core wavelength (μm) 3.5 5 15 30 110 210
+Ttelminimum requirement 85% 90% 95% 97.5% 99.0% 99.0%
+Ttelgoal 90% 92.5% 96% 98% 99.2% 99.4%
+Mid infrared coronagraph specific requirements
+The mid infrared coronagraph requires that, within the unobscured part of the telescope pupil, the telescope
+transmission is spatially uniform within to 1.5% rms or better over the spatial frequency range corresponding
+to the typical coronagraph high contrast region (i.e., 3.3 to 50 cycles/D), and in the mid infrared spectral
+range 3.5 – 27μm. The non-uniformity of the transmitted flux across the pupil can reduce the coronagraph
+contrast. Although it can be partially compensated internally by the coronagraph, it is preferred to allocatethe compensation budget for a wavefront error correction, assuming that the nominal high transmission of the
+telescope will nominally force possible variations of the transmission to be low.
+The preferred shapes for the telescope pupil obscuration (if any) are, in the order of preference (i.e. best
+adapted to match maximal coronagraph performances):
+1. 2 spider struts (e.g. along a complete pupil diameter)
+2. 4 spider struts (e.g. in a centred orthogonal cross shape)
+3. 3 spider struts
+The exact shape and spatial distribution of the obscuration at the telescope pupil can affect the telescope PSF
+main lobe and (near and far-) sidelobes. A mid infrared coronagraph based on pupil apodisation by re-mapping
+or shaped mask is particularly sensitive to entrance pupil obscuration shape and geometry. No telescope pupil
+obscuration would be overall preferred, but may set overall unrealistic constraints (e.g., specific telescope
+design becoming an observatory-level design driver). For more than 4 spider struts, it is expected that the
+coronagraph rejection performance in its dark region will be too badly affected and therefore these cases are
+not considered.2.1. Telescope scientific requirements 47
+2.1.3 Image quality and field of view
+General
+In order to achieve the maximum angular resolution for the given telescope diameter, the SPICA telescope shall
+have diffraction-limited imaging capability at λ=5μm over a field of view of 5 arcmin radius (with a goal of 6
+arcmin). In addition, the SPICA telescope shall keep diffraction-limited performances at 30 μm over a field of
+view of 10 arcmin radius (goal is 12 arcmin). The mapped telescope field of view shall be unvignetted over a
+radius of 12 arcmin (goal is 15 arcmin).
+The mid infrared instruments will be located in the 5 arcmin radius area of the telescope field of view,
+where the image quality is expected to be higher. The far infrared instrument may be located in the peripherybut still within the unvignetted field of view, not to damage the imaging efficiency nor to generate spurious
+spatial response from diffraction at a structure other than the one defined by obscuration in the telescope pupil.
+Details of telescope aperture phase error distribution
+Takingλ
+0=5μm as the baseline shortest “design and operation” wavelength, and following the above, the main
+requirement is that the overall maximum SPICA telescope wavefront error, over a field o view of 5 arcmin radius
+(goal 6 arcmin), shall be 350 nm rms in order to be compatible with nominal diffraction-limited performancesatλ
+0.
+Beyond this top-level requirement, imaging in certain modes and bands can be affected by phase or wave-
+front error at different pupil spatial scale and this leads to the following break-down of the overall telescope
+pupil phase error:
+•Low spatial frequencies (i.e., below 3.3 cycles/D with D is the telescope pupil diameter): Overall figure
+errors to be limited for the SPICA telescope to <350 nm rms. As an example, this can be spread equally
+between focus and higher order aberrations: <250 nm rms for focus translates into a maximum defocus
+of±0.185 mm at the telescope focal plane. In addition, <250 nm rms for higher but still low order
+aberrations (total telescope) translates into a maximum allocation of about one wave peak-to-valley atthe standard visible test wavelength (e.g., 633 nm) or a surface figure error of <λ
+0/70 per mirror in a
+2-mirror telescope case.
+•Medium spatial frequencies (i.e., from 3.3 to 1000 cycles/D): No specific requirements are defined for
+this interval. The residual mirror surface figure errors in these spatial frequencies range shall be limitedonly by the overall compatibility with the general 350 nm rms requirement. These errors arise fromresidual higher frequency surface error, mount induced stress, print-through effects or general mirror
+substrate medium-scale structure.
+•High spatial frequencies (i.e., >to>>1000 cycles/D):<λ
+0/285 rms per mirror. These mostly arise
+from mirror surface micro-roughness, cracks and small local defects in coatings all of which induce largescale/wide angle scatter as well as generating small off-axis angle diffuse halo by near-specular scatter.
+The specification translates into a <17.5 nm rms specification for each telescope mirror surface leading
+to a TIS<0.2% at<λ
+0.
+Mid infrared coronagraph specific requirements
+The maximum telescope wavefront peak-to-valley (PTV) error, within the spatial frequency range from 0 to
+50 cycles/D and over the entire unobscured part of the telescope pupil, shall be no higher than 2 μm.
+It should be noted that although baselined to operate down to 3.5 μm, the coronagraph does not require
+diffraction-limited performance at this wavelength as it is expected to perform its own wavefront compensa-tion internally to recover the diffraction-limited performance at 3.5 μm. Nevertheless, the limitations of this
+wavefront correction require this extra PTV specification.48 Chapter 2. Scientific Requirements
+The mid infrared coronagraph imposes more stringent requirements on the telescope pupil phase and wave-
+front errors at the mid spatial frequencies (i.e., from 3.3 cycles/D to 1000 cycles/D):
+1. From 3.3 cycles/D to 12 cycles/D: <λ 0/45 (= 111 nm) rms (with a goal <λ 0/133 (= 37 nm) rms) for
+the SPICA telescope in this spatial frequency range;
+2. From 12 cycles/D to 50 cycles/D: <λ 0/71 (= 70 nm) rms (with goal <λ 0/133 (= 37 nm) rms) for the
+SPICA telescope in this spatial frequency range;
+3. from 50 cycles/D to 1000 cycles/D: there is no specific requirement, residual error only limited by overall
+compatibility with the general wavefront requirement.
+The limits of the mid spatial frequency range are defined to match the expected Inner Working Angle ( ∼3.3λ0/D)
+and Outer Working Angle (OWA, potentially extending up to ∼33λ0//D), including a 50% margin beyond the
+OWA for the aliasing effect from the surface error at spatial frequencies around the one equivalent to OWA. In
+this region, the goal telescope pupil phase (or wavefront) error is specified based on an equivalent maximum
+degradation on raw contrast of a factor 10 with respect to an assumed raw contrast of 10−7(with quasi-perfect
+optics and/or active wavefront compensation), leading to the main aim of 10−6raw contrast. In the nominal case
+of the baseline specification, it is assumed that only 10−6is nominally achieved (i.e., 10 time less effective or
+absent active correction); with the telescope specification only 10−5will be then obtained in final raw contrast.
+Beyond 50 cycles/D, the coronagraph dark region is not expected to be affected and therefore the require-
+ment is unconstrained (and known to be not easily accessible to verification by measurements) although theoverall main requirement of diffraction-limited performances at λ
+0still applies.
+2.1.4 Telescope temperature
+The science objectives defined in Chapter 1 can only be achieved by increasing dramatically the sensitivity with
+respect to ISO, AKARI and Herschel. In particular, sections 2.3 and 2.4 give the sensitivity requirements on
+the mid infrared and far infrared instruments. To meet those, the development of more sensitive detectors needs
+to be complemented with the reduction to the maximum extent of the background photon noise generated by
+the telescope. In particular, it is a scientific requirement for SPICA that the observatory sensitivity is limited by
+the sky background.
+Figure 2.1: Thermal emissions from telescopes at different temperatures and the natural sky background.2.1. Telescope scientific requirements 49
+Figure 2.1 shows the thermal emission for several telescope temperatures and the natural background de-
+fined by the zodiacal light, the cirrus and the CMB. The strong dependency of background on the telescope
+temperature is due to the locations of the mid infrared and far infrared wavebands in the Wien region of the
+telescope thermal emission. As can be seen in Figure 2.1, a telescope with a temperature /lessorsimilar4.5 K emits clearly
+below the sky background for wavelengths less than 200 μm. The figure also illustrates that the need for
+such low temperature is mainly related to the performance of the longer wavelength bands of the far infraredinstrument.
+From all the above, it is required that the temperature of the SPICA telescope is less than 6 K, with a goal of
+<5 K. The limit of 6 K has been derived from calculations made with the sensitivity model of SAFARI, which
+show that, taking goal sensitivity detectors, this is the maximum telescope temperature that does not lead to a
+reduction of the instrument sensitivity.
+Note that, in particular, the Herschel telescope emits several orders of magnitude higher than the required
+level for SPICA. The orders of magnitude lower photon noise, combined with the higher sensitivity of the
+SAFARI detectors, will lead to a breakthrough in sensitivity in the far infrared wavelength range. Regarding
+the mid infrared instruments, the low SPICA telescope temperature enables to achieve sensitivities similar to
+JWST in spite of the smaller telescope aperture. Actually, as can be seen in Figure 2.1, sky limited sensitivity
+in the mid infrared is already reached with a temperature of ∼25±5 K, depending on spectral bandwidth.
+Spatial and temporal telescope temperature stability
+As explained above, the sensitivity to telescope elements temperature variations is very high for the far infrared
+instrument. For example, for wavelengths longer than 100 μm it becomes typically ∼300%/K. Telescope tem-
+perature gradients and spatial non-uniformities increase the instrument background associated noise, generate afield-dependent signal leading to a non-uniform background illumination across the far instrument array. While
+allowing the presence of “hotter spots” (typically up to 10 K) on the telescope aperture, their influence on the
+background needs to be minimised by being “weighted” through a smaller relative area and therefore through
+the view factor from the far infrared instrument pixel. Based on calculations of the contribution of higher tem-
+perature zones to the total emission, it is required that for every degree Kelvin above the maximum baseline
+telescope temperature of 5 K, the relative surface area (i.e., compared to the total telescope aperture) of this
+associated “higher” temperature zone shall be further reduced by at least a factor 5.
+The strong dependence of the background noise on the telescope temperature imposes also a requirement
+on the temporal stability of the telescope operating temperature, which shall be stable to within 0.25 K over a
+typical observation duration, which is taken as one hour.
+2.1.5 Straylight
+The high sensitivity requirement of SPICA makes the avoidance of straylight essential. This needs to be tackledboth in the telescope and instrument design. It is required that the system composed of the telescope and
+focal plane instruments shall provide adequate internal straylight control to enable SPICA observations with
+sky-limited sensitivity. In particular, the straylight rejection level shall be such that total background from
+out-of-field stray sources (artificial and natural) shall not increase the in-field optical background signal (from
+zodiacal light and telescope thermal self-emission) by more than 20% (goal 10%). The out-of-field sources,
+potentially seen by reflection, diffraction or scattering mechanisms, typically include the sky emission itself
+(zodiacal light) as well as particular “local” celestial objects such as Sun, Earth, Moon and planets, particular
+zones of the sky (e.g., galactic centre) as well as the SPICA observatory sub-systems self-emission.
+In particular:
+•Any emission from observatory structure at surface temperatures >10 K (respectively >50 K), that en-
+ters the far infrared instrument field of view (respectively the mid infrared instrument field of view) at
+telescope focal plane, shall be attenuated by the telescope system and associated baffles in order to reduce50 Chapter 2. Scientific Requirements
+the view factor to these surfaces (including their in-band emissivity and from the focal plane instrument
+field of view) to less than 1 part in 104.
+•Scattered light from any natural source inside the telescope field of view (but outside the respective fieldof view of the focal plane instruments) shall contribute, into the respective aperture of any focal plane
+instrument, less than 10
+−3of its peak irradiance when imaged by the telescope.
+•Scattered light from any natural source outside the telescope field of view shall be attenuated to the
+maximum relative levels given in Table 2.2 defining the telescope PST (Point Source Transmission) in
+any azimuthal direction.
+2.1.6 Surface cleanliness
+Telescope cleanliness shall be optimised so that it does not affect the telescope collecting efficiency, in particular
+in the science bands where mid infrared and far infrared emission/absorption of particular ice compounds are
+intended to be studied with SPICA. For smooth infrared optical surfaces, surface scattering is dominated by
+contaminating dust particles which can raise the level of far-side lobes (i.e., wide field angle from boresight) of
+ad iffraction-limited telescope PSF. The goal is that the telescope surface cleanliness level shall be kept within
+∼2000 mm2/m2or 2000 ppm for particulate and 3 mg/0.1 m2for molecular based on ECSS-Q-70-01A, or
+equivalently, (400±50)C in terms of MIL-STD-1246C classification.
+2.2 Mission definition scientific requirements
+2.2.1 Wavelength coverage
+SPICA will cover the mid and far infrared wavelength region, which is one of the richest ranges of the electro-
+magnetic spectrum emitted by astrophysical objects. It is in this region where most energy is emitted during
+galaxy evolution. It plays host to a variety of atomic /ionic forbidden transitions, molecular lines and solid
+state features that provide unique diagnostics to understand physical and chemical processes in a multitude ofastrophysical environments, as described in the Science Objectives (section 1.1.2).
+The wavelength region that will be covered by SPICA is 5 – 210 μm. The goal is that the focal plane
+instruments provide continuous coverage over the full range, both for imaging and spectroscopy. A full spec-
+tral coverage has important advantages to investigate and model the physical conditions of the gas and dust in
+Ultraluminous Infrared Galaxies (ULIRGs), circumstellar discs, exoplanets and Solar System bodies. In par-
+ticular, molecules like H
+2,H2O, CO and OH, and ions of Ne, S, Si, O, and C have essential diagnostic lines
+in the whole SPICA range. Broad spectral features associated with PAHs, ices, silicate and carbon grains canbe much better interpreted with observations over both the mid and far infrared. Accurate measurements of
+Table 2.2: Telescope Point Source Transmission, defined as the ratio of the signal level from the stray source reaching the
+telescope focal to its incoming level
+Off-axis angular region
+from telescope boresightFrom 0.25◦to 3◦From 3◦to 30◦From 30◦to 45◦Beyond 45◦
+Max PST level(base requirement) <5×10−4<10−510−6Assumed drop-ping to 10
+−8at
+90◦
+Max PST level (goal) <5×10−510−610−7Assumed drop-ping to 10
+−9at
+90◦2.2. Mission definition scientific requirements 51
+their broad spectral shapes and of their spectral peaks along the mid infrared and far infrared region, enable
+to further constrain the models for an unambiguous identification of the makeup minerals or composites and
+for the determination the physical and chemical conditions. Furthermore, a continuous wavelength coverage
+from the mid to the far infrared allows the study of astronomical objects as a function of redshift, by analysing
+powerful diagnostic lines and spectral features as they shift from the mid infrared to longer wavelengths and
+into the far infrared.
+However, constraints in the satellite resources for instrument allocation may prevent meeting the goal of full
+wavelength coverage both with imaging and spectroscopy. A review and selection of the wavelength regions
+and scientific capabilities to be covered will be carried out by JAXA in 2010, after an assessment of the mission
+requirements priorities and the scientific capability uniqueness with respect to other observatories.
+2.2.2 Observational and environmental requirements
+SPICA must be in an orbit that enables observation of all areas of the sky over the duration of the mission, and
+provides an instantaneous sky visibility of >30% (TBC). SPICA’s orbit shall provide:
+•A benign and stable thermal environment, to facilitate the optimal performance of the cryogenic system
+and to minimise disturbances in the background photon noise.
+•A relatively low-radiation environment, to avoid to the maximum extent radiation effects on the electron-
+ics and glitches on the detector signal that would compromise the required sensitivities.
+2.2.3 Pointing requirements
+The spatial resolution of SPICA, which corresponds to diffraction limited performance, combined with theneed to reduce pointing instabilities that introduce additional noise in the signal, lead to the following pointing
+requirements:
+Standard mode: Absolute Pointing Error <0.135 arcsec (3σ), Relative Pointing Error <0.075 arcsec in 200 s;
+Coronagraph mode: Absolute Pointing Error <0.03 arcsec (3σ), Relative Pointing Error <0.03 arcsec in
+20 min.
+SPICA must be able to track fast moving targets in the Solar System, (e.g., comets), with a speed of
+10 arcsec per minute during a period of 20 minutes. The pointing requirements specified above also apply
+during tracking.
+SPICA shall provide a pointing mode to enable mapping of large areas of the sky. In this mode a station-
+ary target star is measured by continuously shifting the spacecraft attitude with constant speed and direction
+according to the following specification:
+Slow-scan speed range: 10 to 72 arcsec/sec for the far infrared instrument and from 0.054 to 2.28 arcsec/sec
+for the mid infrared camera and spectrometers.
+Slow-scan speed accuracy: Less than 1% for the far infrared instrument and less than 10% for the mid infrared
+camera and spectrometers.
+Slow-scan duration: Less than 600 sec for the far infrared instrument and from 5 to 50 sec for the mid infrared
+camera and spectrometers.
+The absolute pointing at any point in time during scan execution shall be recoverable a posteriori (e.g., from
+housekeeping data) with the accuracy given for the standard pointing mode.52 Chapter 2. Scientific Requirements
+2.3 Far infrared instrument scientific requirements
+The science case presented for the SPICA mission in chapter 1 includes the requirement for the provision of
+a flexible capability for imaging and spectroscopy working in the wavelength range from ∼34μm, where the
+[Siii]34.8μm line appears and the Si:Sb detectors stop operating, out to ∼210μm, where the background
+from the telescope and baffles starts to dominate and the [N ii]205.2μm line is observed. This capability will
+be provided by the core far infrared instrument SAFARI. In Table 2.3 we present an analysis of the instrument
+requirements necessary to address the key science goals of the SPICA mission.
+Table 2.3: Requirements of the SAFARI instrument to match the scientific objectives of the SPICA mission
+Science topic/section Observing Requirement SAFARI Requirement
+Planetary Formation
+Gas in protoplanetary discs Spectroscopy of early stage gas
+discs in the main MIR/FIR gascooling lines, light hydrides and
+organics to study physical con-
+ditions and oxygen/carbon chem-
+istry as planetary systems form.High sensitivity medium resolu-tion (R∼few thousand) spec-
+troscopy mode. Integral field
+spectroscopy over as large a field
+as possible is required for ef-
+ficient surveys and to facilitate
+background subtraction. Unin-
+terrupted wavelength coverage re-
+quired from 34 to 210 μm.
+Water in proto-planetarydiscs Spectral survey of discs of a vari-ety of ages to look for to the tran-sition between gaseous water and
+water ice. In nearby systems spec-
+tral imaging of discs to resolve the
+“snow line”.High sensitivity medium res-
+olution (R∼few thousand)
+mode for gas features and lower
+(R∼hundreds) resolution for
+solid state features. Spectralimaging mode required with field
+of view large enough to efficiently
+map nearby discs. Uninterruptedwavelength coverage required
+from 34 to 200μm.
+Occurrence and mass ofdebris discs Unbiased survey of all stellartypes out to a few hundred pc
+to determine frequency of dusty
+discs.High sensitivity multiband (mini-mum 3) photometric mode with as
+large a field of view as practicable.
+Dust mineralogy in debrisdiscs Spectral survey of dusty debrisdiscs to look for differences in
+mineralogy indicating the chemi-
+cal and thermal history of dust. In
+nearby systems spectral imaging
+of discs.High sensitivity modest resolution(R∼few hundred) spectral imag-
+ing mode with a field of viewlarge enough to efficiently map
+nearby discs. Uninterrupted wave-
+length coverage required from 34
+to 100μm.2.3. Far infrared instrument scientific requirements 53
+Table 2.3: (continued)
+Science topic/section Observing Requirement SAFARI Requirement
+The composition of the
+Kuiper BeltUnbiased survey of Kuiper beltobjects. Mineralogy and physical
+properties of Kuiper belt objects
+and associated dust.High sensitivity photometricmode with as large a field of view
+as practicable. High sensitiv-
+ity modest resolution (R ∼few
+hundred) spectroscopy mode forsolid state features. Uninterruptedwavelength coverage required
+from 34 to 100μm.
+Galaxy formation and Evolution
+The life cycle of gas and dust Imaging low resolution spec-
+troscopy and photometry of faint
+diffuse regions in our own and
+nearby galaxies to determine
+the mineralogical content of thedust in a variety of contexts and
+sources. Imaging spectroscopy of
+FIR cooling lines in faint diffuse
+regions in a variety of contexts
+and sources.High sensitivity modest to moder-
+ate resolution (R ∼few hundred
+to a few thousand) imaging spec-troscopy mode required with as
+large a field of view as practicablegiving uninterrupted wavelength
+coverage from 34 to 210 μm and
+Nyquist sampled imaging.
+AGN/starburst connection
+over cosmic timeSpectroscopy of unresolved diag-
+nostic features of ionic and atomic
+lines over wide wavelength range
+in individual sources with simul-
+taneous characterisation of the
+background. Low resolution spec-troscopy also required for detec-
+tion of solid state PAH and silicate
+features.
+Possible use of higher resolution
+spectroscopy to use line widths as
+a diagnostic of Black hole masses.For line diagnostics we require
+high sensitivity medium resolu-
+tion (R∼few thousand) spec-
+troscopy mode. Integral field
+spectroscopy over as large a field
+as possible is required for efficientsurveys and to facilitate back-
+ground subtraction.
+For PAHs and solid state dust fea-
+tures a spectro-photometric mode
+with R∼25−50 with uninter-
+rupted wavelength coverage from34 to 210μm is required.54 Chapter 2. Scientific Requirements
+Table 2.3: (continued)
+Science topic/section Observing Requirement SAFARI Requirement
+The nature of the CIRB Very large area photometric sur-
+veys to below the confusion limit
+at all bands (minimum 3). Very
+high sensitivity blind spectral sur-
+veys of as large an area as possi-
+ble.High sensitivity multiband (min-imum 3) photometric mode with
+as large a field of view as prac-
+ticable and with Nyquist sampled
+imaging capability. High sensi-
+tivity modest resolution (R ∼few
+hundred) imaging spectroscopy
+mode with as large a field of
+view as practicable giving unin-
+terrupted wavelength coverage re-
+quired from 34 to 210 μm and
+Nyquist sampled imaging.
+2.3.1 Assessment of basic instrument concept
+Inspection of Table 2.3 shows that the core far infrared instrument, i.e. SAFARI, needs to have a flexible
+imaging and spectroscopy capability with as high a sensitivity as possible given the constraints imposed by
+the presence of the photon background from the natural background and the telescope baffles. In principle
+the capabilities required could be provided by a mixture of a photometric camera and an integral field spec-
+trometer such as employed in the PACS instrument of Herschel (Poglitsch et al. 2008) or the MIRI instrument
+on JWST (Wright et al. 2004). However, a study carried out in the pre-assessment study phase of instrument
+design showed that, given that megapixel detector arrays are not available in the FIR (unlike in the MIR), the
+spectroscopy mode of such an instrument would be prohibitively slow to cover the wavelength range required,
+a separate camera channel would be needed, the field of view of the integral field unit would be small and therequirements on the detector sensitivity to match the very low photon background would be extremely difficult
+to meet within the context of a space mission launching in 2017/2018.
+We have therefore pursued an instrument concept based on an imaging Fourier Transform Spectrometer
+(iFTS) design such as used for the spectroscopy channel on the Herschel SPIRE instrument. This concept hasthe advantage of naturally providing both a photometric and spectroscopic imaging mode within the same in-
+strument without the need for extra mechanisms such as switching mirrors etc. This instrument concept was
+also investigated in the ESA sponsored Concurrent Design Facility study carried out at the start of the present
+assessment phase (CDF Study Report, SPICA-SAFARI 2008). The naturally broad wavelength band of the
+instrument means that it will be limited in sensitivity by the natural photon background if the detector “dark”
+noise equivalent power can be kept to <few×10
+−19WH z−1/2. Figure 2.2 illustrates the line sensitivity that
+can be achieved in an iFTS under a number of assumptions about the instrument throughput and photon back-
+ground contributions as a function of the dark NEP of the detectors. We can see that the ultimate performance
+for a detector array that provides instantaneous Nyquist sampled images is reached when the detector NEP fallsbelow∼2−3×10
+−19WH z−1/2. We describe the detector technology necessary to achieve this performance in
+section 3.2.2 of the present report. For the purposes of setting the requirements on the instrument performancewe do not invoke a given detector technology but rather give the requirement in terms of meeting the perfor-
+mance that could be achieved for the goal detector technology. In section 3.2.2 we discuss how the instrument
+design and capabilities would need to change in order to maintain the ultimate sensitivity if the detectors do
+not match the goal performance. Under any circumstances we aim to meet the sensitivity requirement for an
+unresolved spectral line on a point source.2.3. Far infrared instrument scientific requirements 55
+Figure 2.2: Achieved unresolved line sensitivity on SPICA as a function of detector dark NEP for a broad band (R ∼3)
+Fourier Transform Spectrometer operating with three wavelength limited detector bands under the assumption of a photon
+background comprised of the Zodiacal light and the Cosmic IR background. The bands used were centred on 48, 85 and
+160μm and we assume the throughput of the instrument is that appropriate to instantaneous Nyquist sampling of the point
+spread function – i.e. 0.5 f/# λpixels.
+2.3.2 Scientific requirements
+The final scientific requirements for the far infrared instrument, SAFARI, are as follows:
+1. Wavelength coverage over at least 34 to 210 μm with a design driver to achieve 28 – 210 μm (unless
+covered by other instrument). This wavelength range is chosen to match the Si:Sb (or Si:As for 28 μm
+coverage of the mid infrared instruments.
+2. A photometric camera mode with R∼2t o5 .
+3. Range of spectral resolution modes with a resolution of at least R=2000 at 100μm,R∼few hundred
+and a spectrophometric mode of 20 <R<50.
+4. As large an instantaneous wavelength coverage as possible for spectroscopy but not at the expense of
+spectral resolution.
+5. 3-D mapping in photometry and spectroscopy to be as fast as possible.
+6. An instantaneous field of view of at least 2 ×2 arcmin2.
+7. Unresolved line sensitivity to be limited only by the natural background over as wide a wavelength range
+as possible. This equates to ∼3×10−19Wm−2(5σ, 1 hour) at 48 μm and<2×10−19Wm−2for a full
+R∼2 band.
+8. For certain wavelengths higher sensitivity is desirable, which may be achieved only over a narrow spectral
+range through the use, for instance, of narrow band filters inserted into the optical path.
+9. Continuum sensitivity of <50μJy.
+10. Unresolved line sensitivity to be maintained at the expense of spatial sampling in the event of less than
+goal performance from the detector.56 Chapter 2. Scientific Requirements
+2.4 Mid-infrared instruments scientific requirements
+The scientific requirements for the mid infrared instruments have been defined by JAXA/ISAS in the
+SPICA Mission Requirements Document (2009). The achievement of the scientific objectives demands imag-
+ing and spectroscopic capabilities over the SPICA mid infrared range, which is defined as the interval from 5to∼34μm.
+2.4.1 Imaging and spectrophotometric capabilities
+Spectral resolution
+The spectral resolution for the imaging capabilities must be R∼10 and for the spectrophotometric capabilities,
+R∼100.
+SensitivityThe requirement for photometric sensitivity in the mid infrared is 5 μJy, with a goal of 1 μJy (5σ, 1 hour
+observation). A main driver for this specification is to perform imaging of a wide-area survey of galaxy clusters
+and of large scale structures in the early Universe where the star formation activities was at a peak, and whose
+redshifted emitting energy shifts into the mid infrared.
+This sensitivity will also allow us to carry out imaging of debris discs whose amount of dust is comparable
+to our Solar System, and reveal the distribution and physical state of solid materials, particularly ice, in proto-planetary discs and dust discs in the main-sequence stars.
+Field of view
+The mid infrared camera must have a field of view greater than 5 ×5 arcmin
+2, with a goal>6×6 arcmin2. This
+large field of view will facilitate the execution of an efficient survey of galaxy clusters in the early Universe over
+an area corresponding to ∼300 Mpc. The survey will allow us to trace the large scale structures and to reveal
+the star formation history in the early Universe (up to 9 Gyr ago) as well as the mass assembly history and
+its environmental effect on the galaxy evolution. The large field of view will be essential for the mid infrared
+imaging spectroscopy of 50 nearby galaxies of the AKARI sample, such that the ISM emission can be spatiallyresolved, to track galactic-scale material circulation from sources to sinks in the ISM in galaxies. Another
+important driver is the imaging and broad band spectroscopy of the galactic plane, to obtain a point sourcecatalogue of 10
+9sources, a low-resolution spectral catalogue of 107sources and diffuse line emission spectral
+maps.
+Spatial resolution
+The spatial resolution should correspond to diffraction limited performance as given by diameter of the SPICA
+telescope at 5μm, which approximately corresponds to an angular resolution of 0.36/prime/prime/pixel. This resolution
+is necessary to spatially resolve the ISM emission in nearby galaxies; to reveal the distribution and of solidmaterials in proto-planetary discs and discs around main sequence stars; to map in detail faint molecular and
+dust shells of evolved stars in the Milky Way and observe individual objects in the Magellanic cloud for the
+determination of the mass-loss histories and dust formation processes.
+2.4.2 Spectroscopic capabilities
+Sensitivity
+The spectral sensitivity of the mid infrared spectrometers must be at least 5 ×10−20Wm−2for a 5σ, 1 hour
+observation. This sensitivity is required to study the interstellar environment and dust emission characteristics2.5. Mid-infrared coronagraph scientific requirements 57
+of high-redshift galaxies out to z∼3, through PAH emission as well as atomic and molecular emission lines.
+These observations are essential to further understand the AGN/starburst connection at high redshift. It will
+also enable SPICA to make infrared imaging and spectroscopic observations of ∼1,000 candidates in search
+for the forming super-massive black holes (SMBHs), that can not be observed easily in other methods due to
+the obscuration of dust, from the present to the early Universe. Supplementing these results with the results of
+observations for the galaxy formation history, we will understand the role of SMBHs in the galaxy evolution.
+A second main driver for the sensitivity specification is to make surveys for emission lines which could be
+associated with warm gas (100 – 1000 K) in protoplanetary discs, and to correlate the amount and characteristics
+of gas with stellar masses and ages.
+Spectral resolution
+The spectral resolution (R =λ/Δλ) of the mid infrared spectrometers must be greater than 1000, which would
+satisfy most of the scientific objectives.
+There is also the requirement for a high-resolution spectrometer with R>10 000, goal of 30 000. This
+higher resolution is crucial to elucidate the geometric, physical and chemical structure of proto-planetary discs
+by measuring the motion of gas, and therefore to witness the evolution of the disc structure due to planet
+formation. High-resolution spectroscopy will be particularly valuable when studying the atmosphere of gas
+giant planets through the transmission spectroscopy method.
+Higher resolution spectroscopy is also required to examine the properties of molecules in the thick molec-
+ular layer surrounding the photosphere of red giants (discovered by ISO observations) and the properties ofmolecules in cold dense molecular clouds with embedded young stellar objects in the Milky Way. In generalterms, the availability of a high resolution spectrometer on board SPICA will provide a unique scientific capa-
+bility in the mid infrared that was neither available in Spitzer nor will be in JWST, whose spectral resolution is
+one order of magnitude lower.
+Spatial resolution
+The spatial resolution should correspond to diffraction limited performance at 10 μm. The drivers are the same
+as for mid infrared imaging.
+2.5 Mid-infrared coronagraph scientific requirements
+The main scientific driver for the mid infrared coronagraph is the capability to directly image and perform mid
+infrared spectroscopy of exoplanets, in particular young gas giant exoplanets that are within 1 Gyr of their
+formation. To accomplish this objective, the instrument needs to meet the following specification:
+•The instrument wavelength range must be 5 – 27 μm to cover the atmospheric mid infrared spectral
+features from molecules expected in giant exoplanets atmospheres, like water, ammonia and methane, as
+described in section 1.3. As a goal, the minimum wavelength should become 3.5 μm to to reduce the
+inner working angle and to allow us to observe the 4 μm predicted spectral feature due to an opacity
+window in EGPs and cooler objects with temperatures between 100 and 1000 K.
+•The planet/star contrast ratio must be 10−6at all instrument wavelengths.
+•An inner working angle (IWA) of ∼3.3λ/ D(Dis the telescope diameter), which corresponds to the
+capability to detect a planet at a distance of the orbit of ∼9 AU (Saturn’s orbit) at 5 μm around a star at
+10 pc. The IWA is the point where the required contrast ratio is achieved. The goal is to have an IWAof∼1.5λ/ D, to detect at 5μm a planet at∼5 AU from a star at 10 pc. The reduction of the minimum
+wavelength range to 3.5 μm would lower the minimum distance at which a planet could be detected.
+•The spectral resolution must be R=20−200.58 Chapter 2. Scientific Requirements
+•The outer working angle (OWA) should be large enough to allow the observation of planets at 5 μma ta
+distance of∼50 AU from a star at 10 pc, that is, ∼16λ/ D.
+•The field of view must be 1 ×1 arcmin2to ensure the spatial coverage of close planetary systems and
+compatibility with the OWA.
+The coronagraph instrument sets stringent requirements on the SPICA telescope as have been described in
+section 2.1.Chapter 3
+Payload
+3.1 The SPICA telescope
+In order to achieve unprecedented sensitivity levels in the mid and far infrared bands, the SPICA mission relies
+on a 3.5 m diameter cryogenic telescope operating at about 5 K. Such a cryogenic telescope (indicated hereafteras STA, SPICA Telescope Assembly) represents the main ESA contribution to the mission and it is described
+in detail in the paragraphs below.
+After the selection of the SPICA proposal for the assessment phase, initial contacts between ESA and
+JAXA were established in November 2008 and preliminary interface specifications agreed on. Based on
+the work of the SPICA Telescope Science Study Team (STSST), science requirements applicable to the
+assessment activities were defined in SPICA Telescope Science Requirements Document (2009). Finally, a
+SPICA Telescope Requirements Document (2009), containing all engineering requirements applicable to the
+assessment study was compiled and maintained.
+In order to obtain a preview on the main critical issues and properly define the scope of the planned indus-
+trial activities, an internal study was performed by ESA at ESTEC between March and April 2008, in closecooperation with JAXA (see SPICA Internal Assessment Study Report 2008).
+Following the release of the Invitation to Tender, two parallel competitive contracts were awarded to EADS
+Astrium (F) and to Thales Alenia Space (F) for the industrial assessment study of the SPICA telescope. Bothstudies, with duration of 1 year, were kicked-off at the end of July 2008 and completed at the end of July 2009.
+The results of the study activities are summarised in this document.
+The main objectives of the assessment study were: a) consolidation of all requirements applicable to the
+STA; b) further definition of the interfaces between the ESA provided telescope assembly and the JAXA pro-vided mission elements; c) definition of a baseline design and demonstration of its compliance with the re-
+quirements and overall technical feasibility; d) definition of a complete development plan, comprehensive of
+any required technology validation; e) programmatic estimates (schedule and cost).
+Based on the results of the industrial studies, very detailed for Phase 0/A level, we can conclude that all
+objectives have been met and that we have acquired a rather well defined picture of the challenges associatedwith the development of the STA. Based on the existing heritage and expertise, we claim that Europe is very
+well placed to deliver the SPICA telescope to JAXA, meeting the agreed requirements.
+A Ritchey-Chr ´etien configuration is confirmed as the most suitable to meet the performance requirements.
+Both contractors have identified a design entirely based on ceramic material (SiC100 for Astrium and HB-
+Cesic for Thales), for the optical surfaces (M1 and M2) as well as the overall telescope structure. A quadripode
+configuration for the M2 support structure is selected as most appropriate from both a mechanical and optical
+point of view (PSF requirements induced by the Coronagraph instrument). Such a support structure would
+be connected to the Telescope Optical Bench (TOB) and not directly to M1, in order to minimise distortions
+of the mirror surface figure and meet the wavefront error requirements. The need for a separate Instrument
+Optical bench (IOB – to be provided by JAXA) is confirmed to host the Focal Plane Instruments (FPI’s and
+properly manage their interfaces with the rest of the STA. A focus and tip/tilt mechanism acting on M2 will
+5960 Chapter 3. Payload
+Figure 3.1: Overview of the STA design from EADS Astrium (left) and Thales Alenia Space (right).
+allow correcting for any residual deviation from the nominal alignment, while a set of heaters would allow
+decontamination during flight if required.
+3.1.1 STA requirements
+The requirements applicable to the STA assessment study are described in the SPICA Telescope Requirements
+Document (2009) and are divided in different categories: a) general requirements (including a prescription for a
+R-C optical design); b) functional requirements; c) performance requirements; d) environmental requirements;
+e) interface requirements; f) qualification and AIV requirements. Given the collaborative nature of the SPICA
+project, specific attention has been paid to the interface requirements deriving from the JAXA provided elements
+of the spacecraft. Such requirements are based on the SPICA Telescope Interface Control Specifications (2009)
+delivered by JAXA.
+Specific performance requirements (related to obscuration, transmission and image quality) are dictated by
+the inclusion of a coronagraph instrument in the SPICA payload complement. Such requirements have beenapplied to the telescope design activities, but, given their potential impact on the telescope development, indus-try has been also requested to identify the simplifications in case of a de-scoping (i.e. in case the coronagraph
+requirements were not considered).
+A summary of the main telescope requirements is provided in Table 3.1. The additional requirements
+derived from the coronagraph instrument are listed in Table 3.2.
+3.1.2 Description of interfaces and STA accommodation.
+The SPICA Telescope Assembly (STA) includes the following parts under ESA responsibility:
+•SPICA Telescope, including optical elements (primary and secondary mirror), M2 support structure, M2refocusing mechanism with drive electronics, internal telescope baffles, adjustment shims, integration
+alignment devices.
+•Telescope Optical Bench, as required to provide adequate mounting and mechanical support to the tele-scope elements and interface to the S/C structure and the Instrument Optical Bench (IOB).3.1. The SPICA telescope 61
+Table 3.1: Summary of main STA requirements.
+Optical design: Ritchey-Chr ´etien, axi-symmetric, EFL=20 m, M1 dia =3.5 m
+Operating temperature: Nominal<6 K, operating range 4.5 – 10 K
+Wavelength range: 5t o2 1 0μm (goal 3.5 to 210 μm)
+Collecting aperture: 3.5 m diameter (maximum allowed by HII-B fairing)
+Total obscuration: <12.5 % on axis (goal <10%)
+Total transmission: >90% at 5μm,>95% at 15μm,>99% at 110μm
+Image quality: Diffraction limited at 5 μm over a FOV of 5 arcmin radius
+Field Of View: >12 arcmin (unvignetted)
+Launch environment: JAXA launcher H-IIB, warm launch.
+STA mass allocation: <700 kg (including margins, excluding mass of IOB)
+Stray-light rejection: Total background from out-of-field stray sources (artificial and
+natural)<20% of in-field background (zodiacal light and self
+emission from telescope).
+Lifetime: >5 yr in orbit (and>5 yr on ground)
+Functional requirements:•M2 refocusing at nominal operating temperature
+•In-flight decontamination capability
+Table 3.2: Summary of MIR coronagraph specific requirements.
+Telescope transmission: Spatially uniform<1.5% rms over the range 3.5 to 27 μm.
+Telescope pupil obscuration: Preferred shape: 4 M2 spider legs
+Image quality (λ 0=5μm):•0 to 50 cycle/D range: WFE <2μm (Peak-to-peak)
+•3.3 – 12 cycle/D range: WFE <λ0/45 rms (goal:
+<λ0/133 rms)
+•12 – 50 cycle/D range: WFE <λ0/71 rms (goal:
+<λ0/133 rms)62 Chapter 3. Payload
+•Thermal control hardware (MLI, electrical heaters, harness, temperature sensors) as required to monitor
+and control the telescope temperature, including decontamination procedures.
+•Thermal interfaces to the S/C cooling chain (anchoring for thermal straps).
+•Required transport, handling and ground support equipment.
+The Instrument Optical Bench (IOB) and its interface mountings as well as the external baffle are not ESA
+provided. Model designs of both these elements have been used by industry to define in detail all STA elementsand critical interfaces. In particular, the IOB (kept at 4.5 K via dedicated thermal strapping and supporting the
+cold unit of all instruments) has been modelled to have the following characteristics:
+•Total mass of IOB and FPI’s of 200 kg. The centre of mass of the FPI is assumed to be located on the
+optical axis, 250 mm below the STA interface plane.
+•IOB of circular shape, 2m diameter, interfaced to TOB via 3 isostatic mounts.
+•Global (IOB+FPI) stiffness requirements: 1
+staxial (Z axis) natural frequency >100 Hz, 1stlateral (X,Y
+axis) natural frequency >60 Hz.
+The SPICA Telescope Assembly (STA) is a key element of the Payload Module (PLM) and is accommodated
+inside the telescope baffle (maintained at ∼11 K and also supported by the PLM truss) and shielded by the
+telescope tube, three thermal shields and the main Sun shield (see Figure 3.2). The STA is mounted ontothe (PLM) support truss via 8 interface points on a diameter of 2700 mm (see Figure 3.2). A thermal strap
+connects the TOB to the 5 K mechanical cooler. Electrical heaters, thermistors and the M2 focusing mechanism
+are connected to their electronic units via dedicated harness. The reference frame of the SPICA Telescope
+Assembly is defined as follows:
+•the origin (O
+STA) is located on the telescope to S/C interface plane
+•the Z-axis (Z STA) coincides with the telescope bore sight
+•the Y-axis (Y STA) points in the direction of the Sun
+•the X-axis (X STA) completes the right-handed coordinate system
+3.1.3 Optical design
+The optical design of the SPICA telescope did not vary significantly during the study activities, which confirmed
+the axi-symmetric Ritchey-Chr ´etien approach, with the aperture stop located at the secondary mirror. During
+Phase 1 sensitivity analysis were performed on the main optical design parameters, coming to the following
+conclusions:
+•Effective Focal Length optimised at 20 m to reduce size/mass of M2.
+•Aperture stop confirmed on M2 to optimise the stray-light rejection.
+•Back Focal Length defined at 828 mm (although adjustments still possible).
+•M2 support structure based on four legs (driven by PSF characteristics).
+During the design activities, effort concentrated on the definition of the polishing and coating specifications,
+on a preliminary stray-light analysis (with associated baffle design) and on the demonstration of the WFE
+feasibility (WFE budget). Additional sensitivity analyses have been performed, in particular on M2 de-center,
+tilt and de-focus. The main results are summarised in Table 3.3.3.1. The SPICA telescope 63
+Figure 3.2: Right panel: STA accommodation in the SPICA spacecraft. Left panel: STA-PLM interfaces.
+Table 3.3: Summary of optical design specifications.
+Configuration: Ritchey-Chr ´etien, aperture stop on M2
+M1 diameter Mechanical=3500 mm, useful optical ∼3470 mm
+M2 diameter: Mechanical∼660 mm, useful optical ∼640 mm
+M1 – M2 distance: - 2986 mm
+Focal Length, f-number: 20000 mm, f=6.015
+Back Focal Length: 828 mm
+Nominal design WFE: ∼350 nm rms at 5μm, FOV radius of 5 arcmin
+Total transmission: >0.90 at 3.5μm,>0.92 at 5μm,>0.95 between 15
+and 210μm64 Chapter 3. Payload
+Figure 3.3: Telescope internal baffling (Left: Thales Alenia. Right: EADS Astrium)
+The stray-light analysis has been conducted by both contractors at a level commensurate with phase-A, with
+the main objective of proving overall compatibility with the requirements. The approach has been based on a
+preliminary model of the external baffle provided by ESA (on JAXA behalf) and on the design of the inner M1and M2 baffles by each contractor.
+The preliminary stray-light activities have demonstrated the capability to meet the corresponding require-
+ments by adopting the following measures: a) further optimisation of the external baffle thermo-optical proper-
+ties (paint selection); b) leg profile and M2 support design minimising reflections; c) adoption of internal and
+external vanes in the M1 internal baffle; d) selection of paint for the internal baffles; e) adoption of cold stop
+for the instruments operating at λ>70μm (i.e., SAFARI). In the range between 70 and 210 μm (dominated by
+thermal self-emission from the external bafflea t∼ 11 K), compliance with the stray-light requirements depends
+on particulate contamination levels after launch, to be defined in the next project phases.
+3.1.4 Mechanical design
+The mechanical design of the STA design is based on a dedicated Telescope Optical Bench supporting M1as well as the M2 support structure. This approach avoids transferring directly to M1 any loads due to M2,
+minimising distortions. All structural and optical elements are to be built in ceramic material (either HB-Cesic
+or SiC100), which ensures small and highly uniform Coefficients of Thermal Expansion (CTE), combined withhigh strength and low mass. Titanium alloy is used for the mirror fixation devices (bipods).
+Given the different characteristics of SiC100 and HB-Cesic, a different manufacturing approach is base-
+lined respectively for Astrium (M1 built by brazing of 12 segments and TOB in 8 segments) and Thales (M1monolithic, TOB in two halves). In both cases M2 (of smaller size) is built in a single piece. Junctions between
+separate ceramic elements (e.g. M2 struts and TOB) are handled via Invar fittings. The JAXA provided IOB
+would be connected to the TOB via dedicated Ti bipods.
+The mechanical design is driven by the quasi-static loads (particularly demanding at the M2 location) and
+the stiffness requirements provided by JAXA (see Tables 3.4 and 3.5). The quasi-static loads are subject to
+confirmation in the following phases, based on dedicated coupled load analysis (including spacecraft elementsand STA behaviour). Detailed Finite Element Models have been constructed by both companies.
+The mechanical analyses covered four main areas: a) modal analysis to demonstrate compliance with the
+stiffness requirements; b) thermo-elastic analysis to demonstrate proper behaviour during the cool-down fromambient to 5 K without affecting the telescope performance specifications; c) analysis of gravity effects; d)3.1. The SPICA telescope 65
+Table 3.4: Global STA (+ IOB) stiffness requirements.
+1stnatural freq. along X and Y axis: >30 Hz
+1stnatural freq. along Z axis: >60 Hz
+1stnatural freq. around Z axis (torsion): >30 Hz
+Table 3.5: Design limits and quasi static loads.
+M1 and M1 supporting structure (X, Y , Z non simultaneously): 12.5 g
+M2 and M2 supporting structure (X, Y , Z non simultaneously): >20 g
+1stnatural freq. around Z axis (torsion): >30 Hz
+analysis of quilting effects (induced by polishing).
+The modal analysis has been conducted in different configurations: 1) Isolated STA and IOB system on
+rigid interfaces (representing the nominal test configuration in Europe); 2) STA and IOB on PLM support truss
+(closer to the flight configuration). Such analysis have assumed that the IOB (global behaviour including the
+FPI’s) will have a first longitudinal mode (Z axis) at a frequency >100 Hz and a first lateral mode (X, Y axis)
+at a frequency>60 Hz. Both EADS and TAS design show compliance with the requirements of configuration
+1) and 2). In case 2), the analysis of the truss behaviour has been based on a JAXA provided Finite ElementModel.
+The thermo-elastic analysis has been conducted combined the detailed telescope Finite Element Model with
+a corresponding temperature map provided by the thermal mathematical model and with materials data (CTE
+as a function of temperature). The main objective of this analysis was to demonstrate the capability of the
+truss supporting structure to absorb the differential thermal contractions during the cool-down (originated by
+the presence of different materials). Such a demonstration is of specific importance given the nature of the STA
+to PLM interface (based on 8 points). The main results of this analysis are:
+•The supporting truss modelled by JAXA provides adequate radial “breathing” to absorb the differentialcontractions between STA and PLM elements, with a behaviour equivalent to isostatic mounting condi-
+tions.
+•Optimising the TOB to IOB interfaces, it is possible to absorb any residual differential contraction without
+affecting the optical performance (even in case of large CTE differences as in the case of an Al based
+IOB).
+•The telescope wave-front error, with proper shimming and corrective action from the M2 mechanism,
+remains within the allocated budget.
+Industry performed the analysis of the gravity effects on both M1 and M2 WFE, thus being able to predicttheir behaviour during the polishing (with optical axis in vertical position) and telescope testing (with optical
+axis in horizontal position). The mechanical design of the mirrors and of the supporting structure has then
+been optimised to maintain the WFE within acceptable levels in 1g conditions (and different orientations), in
+agreement with the overall WFE budget allocations.
+Quilting effects induced during polishing have been examined in order to establish the optimal back stiff-
+ening (primary and secondary) structure and front skin thickness (in particular for M1). Both contractors have
+identified designs ensuring a reduced impact on WFE (typical ∼50 nm rms) as a result of the applied polishing
+pressure.
+Finally the mechanical analysis has demonstrated compliance (in worst loading conditions) with the max-
+imum stress levels allowed by each STA material, including required safety margins. This analysis included
+thermo-elastic effects induced by large temperature gradients (during cool down). The mechanical design66 Chapter 3. Payload
+showed compliance with the yield and ultimate load levels, including the required factors of safety for each
+different material used in the STA. Specific attention has been paid to the case of ceramic parts and to the
+definition of the related design margins, including latest lessons learnt from similar programs.
+3.1.5 Thermal design
+A first thermal analysis of the STA has been performed at ESA on the basis of a preliminary Thermal Mathemat-
+ical Model and JAXA input. This model focused on determining the temperature distribution on the telescope
+given the boundary conditions imposed by the SPICA spacecraft (including thermal shields and mechanical
+coolers). During the industrial activities this model has been further refined and adapted to the specific STA de-
+signs by each contractor, including numerous sensitivity analyses. The analysis has assumed direct connection
+of the TOB to the 4.5 K cooler via a dedicated strap and thermal decoupling between TOB and IOB. The main
+results obtained with the Thermal Mathematical Model (also used to derive the T maps for the thermo-elastic
+analysis) are the following:
+•In equilibrium conditions (after cool down and with nominal cooling power at 4.5 K and realistic in-terfaces) all telescope elements have a temperature below 7 K. In particular the optical surfaces have a
+temperature<6K .
+•Temperature uniformity on the optical surfaces is very high (typically <0.2 K).
+•Adequate cooling of M2 can be achieved without complex thermal strapping, which could be limited to
+a few interface areas.
+•The thermal behaviour of the STA is dominated by the external boundary conditions (SPICA spacecraft),in particular by the thermal shields, the cooling power of the 4.5 K cooler and heat conduction through
+the supporting truss.
+•At equilibrium the heat load on the 4.5 K cooler is below 20 mW, compatible with the estimated perfor-mance of the JAXA cooler.
+•The transient analysis showed that cool down of the PLM after launch takes approximately 160 days, andthat the STA does not lag behind the rest of the system, while equilibrium conditions after operating the
+M2 mechanism are recovered in about 1-2 days.
+•If required, decontamination of the telescope (up to 315 K) can be performed with a realistic heater powerbudget, compatible with the on-board resources.
+•The heat load on the 4.5 K stage is sensitive to variations in the boundary conditions (e.g., temperatureof heat shields and telescope shell).
+Decontamination could be performed at beginning of life (out-gassing phase) and /or during the flight. Heaters
+would be implemented on the TOB. A summary of the thermal analysis results is provided in Table 3.6
+3.1.6 Resource budgets
+One of the main objectives of the assessment study was to consolidate the resource and performance budgetsassociated to the telescope design. The following budgets have been analysed in detail: a) mass budget; b) heat
+load budget; c) Wave Front Error. The results (all including conservative assumptions and maturity margins)
+are compliant with JAXA allocations and scientific requirements and summarised in Tables 3.7, 3.8 and 3.9.3.1. The SPICA telescope 67
+Table 3.6: Thermal analysis results (equilibrium conditions).
+Industry design EADS Astrium Thales Alenia Space
+Nominal cooler temperature: 5±0.5 K
+Telescope shell (boundary condition): 24 K
+External baffle temperature: 10.5 K 11.3 K
+M1 mirror temperature: 5.3 K 5.4 K
+M2 mirror temperature: 5.5 K 6.0 K
+M2 support structure temperature: 5.4 K 5.9 K
+Telescope Optical Bench tempera-
+ture:5.3 K 5.7 K
+Internal M1 baffle temperature: 6.7 K 6.9 K
+Internal M2 baffle temperature: 5.8 K 6.0 K
+Table 3.7: STA mass budget (including 20% maturity margins on all units)
+Industry design EADS-Astrium Thales Alenia
+[kg] [kg]
+M1 mirror and bipods: 306 313
+Telescope Optical Bench: 190 210
+M2 support structure: 109 100
+M2 mirror and bipods: 11.4 25.6
+M2 mechanism: 12.0 7.0
+Refocusing mechanism electronics: 3.0 4.8
+Thermal hardware: 20.8 17.2
+Internal baffles (M1+M2): 20.0 11.0
+Miscellaneous (screws, bolts, etc.): 8.0 9.6
+TOTAL (including maturity margins): 680 698
+3.1.7 Technology readiness
+The technology readiness of all telescope elements has been reviewed during the assessment study; any remain-ing development risk areas have been identified and corresponding risk mitigation actions have been baselined.
+Given the recent Herschel experience and other scientific and remote sensing space projects, Europe is very
+well placed to develop the SPICA telescope. The progress achieved in the manufacturing of large size ceramic
+mirrors is a main pre-requisite for the STA development, together with the experience acquired in the AIT of
+cryogenic telescopes and in the polishing of large mirrors for astronomical telescopes. No major technology
+developments are required for the SPICA telescope; the remaining critical areas and the corresponding risk
+mitigation actions are summarised in Table 3.10. The planned technology consolidation activities are expected
+to achieve TRL>5 before the final M class selection (end 2011), thus fully meeting the Cosmic Vision 2015-25
+needs.68 Chapter 3. Payload
+Table 3.8: STA heat load budget on 5 K cooler (telescope shell at 24 K).
+Industry design EADS-Astrium Thales Alenia
+[mW] [mW]
+Overall radiative load (from baffle): 5.4 1.5
+STA interface (from truss): 12.8 18
+TOTAL heat load (5 K stage): 18.2 19.5
+Table 3.9: STA WFE RMS budget (including integration, polishing and cool down)
+Industry designMain
+contributorRequirement EADS Astrium Thales Alenia
+[nm] [nm] [nm]
+WFE rms at
+λ0=5μm over
+5/primeradiusM1 cool downDesign aberration
+M1 polishing350 342 258
+WFE rms atλ
+0=30μm over
+10/primeradiusDesign aberration
+M1 cool down
+M1 polishing2100 670 800
+WFE rms atλ
+0=5μm
+3.3 to 12 cycle/DM1 polishing 111 51 89
+WFE rms atλ
+0=5μm
+12 to 50 cycle/DM1 polishing 70 69 42
+WFE PtV at
+λ0=5μm
+0 to 50 cycle/DM1 cool down 2000 1768 1350
+3.1.8 Assembly Integration & Verification plan
+As part of the assessment study activities industry has produced a complete telescope AIV plan. The telescopedevelopment plan is based on the following elements:
+•Proto-flight approach, with a Structural Thermal Model and spare units.
+•Mechanism and drive electronics qualified at unit level prior integration.
+•Risk mitigation via 2 main technology consolidation activities: 1) Light-weight ceramic mirror bread-
+board; 2) M2 cryogenic focusing mechanism.
+•Additional risk mitigations included in the Definition Phase contract (e.g., materials characterisationmeasurements, metrology related aspects, etc.).
+•Cryogenic testing limited to about 50 K (TBC), without major changes to test facilities existing in Europe.Final testing at 10 K in Japan.
+The proposed model philosophy is the optimal solution to balance risk mitigation and cost /schedule con-
+straints. The Structural Thermal Model would allow refining AIV issues, to address mechanical qualification3.1. The SPICA telescope 69
+Table 3.10: STA critical areas and identified risk mitigation actions.
+Critical area: Technical driver: Risk mitigation:
+M2 focusing
+mechanismQualification of linear actuator operating
+at cryogenic temperature (5 K).Mechanism BB
+developed during
+Definition Phase.
+Light-weight, large
+size ceramic mirrors.Fabrication of 3.5 m diameter. Optimisa-
+tion of polishing process. Gravity com-
+pensation.Mirror BB and polishingtests during Definition
+Phase
+Material properties at
+cryo temperature.Detailed knowledge of CTE, Young
+Modulus, thermal conductivity,
+emissivity and BRDF properties of
+selected materials.Perform measurementson representative
+samples during the
+Definition Phase
+aspects and to perform mechanical and thermal system level testing at JAXA. Static proof testing of all struc-
+tural parts is also considered.
+The polishing of the primary mirror is, both from performance and schedule aspects, the single most critical
+activity of the programme. M1 will be designed to achieve a well controlled level of gravity induced distortion
+and its polishing will be performed in a dedicated facility, equipped with gravity compensation devices (vertical
+optical axis) and metrology equipment. The 0-g figure of M1 will be measured by moving the M1 optical axis
+in horizontal configuration and measuring the WFE at different orientations of the mirror around the optical
+axis.
+Matching testing between M1 and M2 in (horizontal axis) is also foreseen, via dedicated Mechanical
+Ground Support Equipment and stitching of interferometric measurements performed on the full aperture of
+M1, via a rotating auto-collimating mirror. The same technique would also be used during the final telescope
+integration and performance verification, with M2 on its supporting structure and M1 on the TOB, including
+gravity compensation devices.
+The test facilities existing in Europe are well suited for the characterisation of the telescope from ambient
+down to T∼50 K. Modest changes (adaptations to the STA size, inclusion of adequate metrology equipment and
+thermal shrouds) are required to perform the integrated STA testing (both vertical and horizontal configurations
+are possible). Preliminary analyses indicated that the extension of testing in Europe to lower temperatures (10 to
+20 K) may be possible, subject to a more detailed analysis in the next project phases. Thermal vacuum testing
+in Europe will include a full characterisation of the optical performance of the telescope close to operating
+conditions before final shipment to Japan. Additional STA testing at 10 K is presently planned in Japan, before
+final integration on the SPICA spacecraft.
+Figure 3.4: Left panel: STA cryo testing in vertical configuration. Right panel: STA cryo testing in horizontal configura-
+tion.70 Chapter 3. Payload
+Figure 3.5: SPICA Telescope Assembly – reference schedule (including M class down-selection milestones).3.2. Focal Plane Instruments 71
+3.1.9 Development plan
+The SPICA development plan takes into consideration the boundary conditions given by the project approval
+process (at JAXA and at ESA) as well as all relevant technical issues. Key events to be accounted for (in
+case of down-selection) are: a) earliest start date of the Definition Phase at ESA (Q2/2010); b) JAXA final
+project approval in Q1/2011; c) earliest start date of the Implementation Phase at ESA (Q2/2012). Based on the
+assessment study, the tasks driving the telescope schedule are:
+•Detailed design of all telescope elements (following freezing of requirements).
+•Manufacturing of M1 blank (including related procurement actions).
+•Polishing of primary mirror (estimated to take no less than 24 months; an additional 6 months would be
+required when including the coronagraph needs).
+•Coating of primary mirror and final WFE characterisation.
+•Telescope integration and test activities (at ambient and cryo temperature).
+The reference development schedule of the SPICA Telescope is reported in Figure 3.5. The project plan takes
+into account all boundary conditions and assumes an early freezing of all design requirements during the ESA
+Definition Phase (A/B1), immediately after JAXA SDR (Q3/10), thus preparing key manufacturing drawings
+before the start of Phase B2-C/D. All technology consolidation and risk mitigation activities are also to be com-
+pleted before starting the Implementation Phase. In this scenario, the delivery of the telescope to JAXA could
+take place by Q4/2016 to Q1/2017, assuming a minimum contingency of 6 months. Delays in the requirements
+definition would require and additional 8-12 months, while inclusion of the Coronagraph requirements wouldextend polishing an additional 6 months.
+The SPICA reference schedule takes into account both the JAXA and ESA mission approval process as
+well as technical schedule drivers. The single most critical activity is the manufacturing (∼ 1 yr) and polishing
+of the primary mirror (∼ 2 yr). An early definition of the interfaces and optical requirements (to be agreed with
+JAXA) would allow anticipating the detailed design of M1, saving about 1 yr. An additional 6 months would
+be required for polishing in case the coronagraph requirements were enforced.
+3.2 Focal Plane Instruments
+3.2.1 Overview
+The following instruments are candidates for the SPICA focal plane: MIRACLE (Mid-InfRAred Camera
+w/o Lens), MIRMES (Mid-IR Medium-resolution Echelle Spectrometer), MIRHES (Mid-IR High-resolution
+Echelle Spectrometer), SCI (SPICA Coronagraph Instrument), SAFARI (SPICA Far infrared Instrument), FPC(Focal Plane finding Camera; FPC-G is a guider camera for attitude control), BLISS (Background-Limited
+Infrared-Sub-millimetre Spectrograph).
+Only one instrument at a time shall be in operation for astronomical observations. This excludes the FPC-G,
+which will be in operation for most of the observations. The Focal Plane Instruments (FPIs) onboard SPICAwill be attached to the STA via the Instrument Optical Bench, which is thermally lifted at ∼4.5 K by a J-T
+cooler. A total mass of 150 kg is allocated for the FPIs. For SAFARI the allocation of 50 kg (cold units,
+including 20% margin) has been agreed. Cooling power constraints are tight: 15 mW total at the 4.5 K stage,
+and 5 mW at 1.7 K, lifted by the J-T coolers. At the 4.5 K stage, the routinely operated FPC-G plus the
+parasitic loads amount to approximately 2 mW. Since the resource allocations are rather stringent to allow the
+accommodation of all instrument candidates, an instrument review/selection process will be followed before
+the SPICA System Design Review (Q4/2010). In this process, BLISS is considered as an “optional” instrument.
+Figure 3.6 shows a planned field-of-view (FOV) configuration (projected on the sky). Within the unvi-
+gnetted field of STA (30 arcmin in diameter), all the FOVs, including two redundant FOVs of the FPC, shall be
+placed.72 Chapter 3. Payload
+Figure 3.6: Planned configuration of the field-of-views of the FPIs projected on the sky. Those of MIRMES, MIRHES, and
+BLISS are not shown.
+3.2.2 Far Infrared Instrument: SAFARI
+The SpicA FAR infrared Instrument (SAFARI) is an imaging Fourier Transform Spectrometer (iFTS) de-
+signed to give continuous wavelength coverage in both photometric and spectroscopic modes from around
+34 to 210μm. The scientific performance requirements of SAFARI are outlined in section 2.3 “Far infrared
+instrument scientific requirements”, together with the rationale for choosing an iFTS concept for the overall in-strument configuration. Figure 3.7 shows the current conceptual design of the SAFARI Focal Plane Instrument.
+It reflects much of the design optimisation carried out by the consortium during the Cosmic Vision Assessment
+study. For more information about the status of the instrument design, refer to the SAFARI Phase-A1 Study Report
+(2009).
+Figure 3.7: SAFARI focal plane conceptual design at the conclusion of the Phase-A1 study.3.2. Focal Plane Instruments 73
+Key design drivers
+Detector performance: The scientific performance of SAFARI is critically dependant on the sensitivity of the
+detector system. Given the importance of detector sensitivity to the ultimate impact of SAFARI science, the
+consortium has prioritised detector technology development activities so that the competing requirements oftechnological maturity /development risk vs. ultimate scientific performance can be optimised. There are cur-
+rently four competing candidate detector technologies which are undergoing targeted development to meet theSAFARI requirements; TES, KID, Silicon bolometers and Ga:Ge Photoconductors. The four technologies are
+all credible candidates for SAFARI although they differ considerably in detail with respect to (1) likely in-flight
+sensitivity, (2) system impact on the overall instrument design and the attendant demands on the spacecraft re-
+sources and (3) technology readiness and development risk. JAXA have established a mission level requirement
+that the FPI detector technology is baselined prior to start of the SDR (Q4/2010) and therefore the consortium
+has planned to converge on a technology baseline in Q2/3 2010. The selection of the detector technology base-
+line will be carried out on the basis of the parameters of scientific performance, spacecraft/instrument interface
+compatibility, development risk and national programmatic support.
+In the period leading up to the eventual selection of the detector technology baseline, the consortium has
+decided to adopt the TES detectors as the reference case. This is due partly to the fact that a prototype detectorwith a sensitivity meeting the SAFARI minimum requirement has successfully been optically characterised andbehaves largely as predicted. It is also due to the fact that it is likely that the TES detectors will place the most
+demanding requirements on the cryogenic spacecraft resources, and can therefore act as a “design stressing”
+reference case.
+Figure 3.8: Left panel: SAFARI flight hardware block diagram. Right panel: Optical elements of SAFARI.
+Mass budget : Due to the non-linear system level impact of added mass on the SPICA IOB to the thermal
+and structural design of the overall STA, the mass budget allocation of 200 kg for the IOB and FPI has verylittle flexibility. The JAXA agreed allocation of 50 kg (which includes 20% margin) for the SAFARI focal
+plane instrument is a significant instrument design driver. The instrument design produced during the ESTEC
+CDF study (CDF Study Report, SPICA-SAFARI 2008) was non-compliant on this basis. Recent evolution of
+the instrument design has reduced the current best estimate mass to 38 kg by (1) removing a mechanism used
+to stabilise the astronomical image on the instrument focal plane, (2) making the optical layout more compact
+and mass efficient and (3) by incorporating a Darwin Optical Delay Line heritage scan mechanism design into
+the instrument, which is lighter than the initial baseline. Although the current best estimate mass is compliant,strict mass growth and requirements creep discipline will have to be enforced during the definition phase to
+ensure that the instrument does not exceed the allocated mass budget. The main path to further optimisation of
+instrument mass is to reduce the volume of the Focal Plane Array, which would allow a more compact optical
+layout and a lighter instrument structure.
+Cryogenic thermal budget: The scientific promise of the SPICA mission is made possible thanks to the74 Chapter 3. Payload
+availability of the JAXA J-T and Stirling cryocoolers to cool the STA and the SPICA instrument suite. Despite
+the availability of these coolers, the available cryogenic heat lift places severe constraints on the design of
+the SAFARI focal plane instrument. Of particular concern is the heat loads on the spacecraft J-T coolers
+during instrument cooler recycle. The cryogenic design of the instrument is currently therefore one of the most
+important design drivers. Moreover, due to the highly coupled behaviour of the spacecraft /instrument thermal
+system, proper attention must be paid to the design optimisation of the overall integrated system.
+Electromagnetic design and analysis: The very demanding SAFARI detector sensitivity requirements
+imply that EMI coupling to the detector system must be stringently controlled. The susceptibility of bolometric
+detectors to EMI currents coupling into the detector system and dissipating power within the bolometers is a
+particular susceptibility of this class of detector technology. The detectors are also susceptible to DC and low
+frequency magnetic fields from the instrument cooler Adiabatic Demagnetisation Refrigerator (ADR) solenoids
+as well as the scan mechanism and other spacecraft systems. Provision for RF and DC/low frequency magnetic
+shielding has been incorporated into the instrument design, but further detailed design and analysis is required
+in order to confirm appropriate margins.
+Stray-light control: The SAFARI detector sensitivity requirements dictate that the level and modulation of
+the stray-light on the detector system is well controlled. This has been made apparent during SAFARI detector
+testing where standard stray-light control methods in test cryostats were found to provide inadequate shielding
+in the dark detector test cavity.
+Verification: The planning and design of the verification activities, particularly at spacecraft level, for most
+of these design activities (e.g., Electro- Magnetic Compatibility, telescope stray-light, cryogenic performance,detector sensitivity) is a challenge due to the difficulties of reproducing a flight-like environment in a cleanroom environment.
+SAFARI system design
+The main flight hardware elements of the SAFARI instrument are shown in the block diagram (see Figure 3.8,
+left panel). The Focal Plane Instrument is accommodated on the SPICA Instrument Optical Bench while the
+Detector Control Unit and the Instrument Control Unit are mounted on the spacecraft bus. The primary sources
+of the high level requirements driving the instrument design are the SAFARI science requirements as containedin the SPICA Mission Requirements Document (2009) and the interface requirement with the spacecraft which
+are being controlled and documented in the SAFARI Interface Control Specification (2009).
+Optical design
+The main optical elements with overall dimensions are shown in Figure 3.8 (right panel). The Pick Off
+Mirror is centred 7
+/primefrom the telescope field of view slightly ahead of the focal plane. At this location, SAFARI
+has diffraction limited imaging at 40 μmo v e ra2/prime×2/primesquare field of view taking into account instrument
+and spacecraft alignment tolerances. The input optics reimages the telescope pupil for stray-light control (Lyot
+stop) and expands and partially collimates the beam entering the spectrometer. The beam splitter creates two
+symmetric optical paths through the interferometer. The retro-reflectors in the scan mechanism are cat’s eye
+mirrors which is a configuration found to be well suited to both the simple design of the mechanism as well as
+achieving a compact and well controlled optical design. There is sufficient clearance between the elements of
+the interferometer to achieve an optical path difference of between -4 mm and +31.5 mm which (together with
+the 33 mm pupil diameter) is compatible with the spectral resolution requirement of the instrument of R=2000
+at 100μm. The beam combiner is the final element of the interferometer and produces the interference beams
+in the two output arms. The camera optics focuses the image onto the three Focal Plane Arrays with f/20
+beams. The overall fringe contrast and optical throughput of the instrument are compatible with the instrument
+sensitivity and spectral resolution requirements. The Focal Plane Arrays have an unvignetted field of view of
+2/prime×2/primeand Nyquist sample the image (F λ/2) at the central wavelength of each band. The three science bands
+are defined by a series of edge filters and one dichroic filter. The nominal bandwidth can be tailored with
+two, six-position filter wheels accommodated within the overall camera optics. The approach for stray-light
+control is to enclose the instrument in a light tight enclosure, use a series of edge filters to attenuate out of3.2. Focal Plane Instruments 75
+band radiation and prevent it reaching the detectors, block off-axis light with pupil stops, locally apply black
+paint to critical locations within the instrument, enclose the Focal Plane Arrays in a light-tight 1.7 K box and
+make an appropriate design of the detector harness to prevent radiation “tunnelling” through the dielectric
+insulator. Considerable quantitative analysis and design will be required to ensure that best use is made of the
+low-background primary telescope.
+Thermal design
+A hybrid3He sorption cooler coupled to a low temperature ADR will be used to cool the TES detectors1
+to around 50 mK and provide heat lift at 300-mK to intercept parasitic heat loads and cool detector front end
+electronics. The cooler will be operated in recycle mode with a duty cycle efficiency of 75% (10 hour recycle /
+30 hours operation). The hybrid sorption /ADR cooler architecture was adopted due the fact that it is far lighter
+than a multi-stage ADR cooler and is therefore compatible with the overall SAFARI mass allocation. High
+purity, annealed Copper straps are used to provide a high conductivity link between the cooler and the detector
+arrays. The detectors will be mounted on light tight enclosures on the main chassis of the instrument which
+will be cooled to 1.7 K. The main chassis of the instrument will be isothermal with the IOB.
+There will be three primary thermal interfaces with the spacecraft:
+1. a 1.7-K interface to the3He J-T cooler: used in the operation of the cooler, cooling the detector boxes
+and intercepting the electrical dissipation from the detector readout electronics,
+2. a 4.5-K interface to the4He J-T cooler: used in the operation of the cooler, intercepting cryoharness
+parasitics and the electrical dissipation in various subsystems mounted on the 4.5-K structure, and,
+3. a 12-K interface to the Stirling cooler (2ST): used to recycle the cooler and possibly intercept dissipation
+from detector pre-amplifier (not part of the current baseline for TES detectors).
+The main point of tension in the thermal interface with the spacecraft is the magnitude and duration of the
+peak loads from the instrument cooler during recycle; in particular the load on the 1.7-K interface from theenthalpy of the “hot” Helium de-absorbed from the sorption cooler pump and the enthalpy of condensation
+in the evaporator. The results of the thermal modelling (reported in detail in SAFARI Phase-A1 Study Report
+2009) shows margins of over 30% for most thermal loads during instrument operations except the 4.5-K stage
+(−3.5%). Also, during cooler recycle, the peak load on the 4.5-K stage shows a margin of only 9%. These
+issues will be addressed during the design and development programme of the instrument and system level
+cooling systems by providing a breadboard cooler system to JAXA to test against the engineering models of the
+J-T coolers. In principle the instantaneous power from the cooler can always be reduced by taking more time
+to recycle the cooler, at the expense of some operational efficiency.
+Mechanical design
+The mechanical interface requirements of the FPI are contained in the ICS. For the purpose of the instrument
+assessment study, the two key requirements used to demonstrate overall compliance were (1) the mass (50 kg
+with 20% margin) and (2) Eigen frequency requirements (150 Hz axial /100 Hz lateral). The conceptual
+structural design of the focal plane instrument is shown in Figure 3.7. The instrument chassis is fabricated
+from Aluminium alloy and mounted on three isostatic bipod mounts on the IOB to allow for the differential
+contraction across the interface (several mm). The optical components are either mounted on the main base
+optical bench or the vertical optical bench. Two covers are used to enclose the instrument and complete the
+stray-light/EMI tight enclosure as well as act as structural elements. As all elements (with the exception of theEMI filters) are mounted on the main chassis, subsystems can largely be integrated/de-integrated independently
+from each other. This approach also allows optical alignment to be carried out with the covers removed giving
+access to the individual components.
+1Both the TES and the Silicon Bolometer detector options need to be cooled to around 50 mK. The KID detectors operate at a base
+temperature of around 150 mK whereas the photoconductors operate at 1.7 K and above and do not need a cooler.76 Chapter 3. Payload
+The detailed breakdown of the various contributions to the mass budget are given in the
+SAFARI Phase-A1 Study Report (2009). In summary the best estimate for the mass of the cold FPI is 38.8 kg,
+giving a 22% margin over the allocation, and 23.6 kg for the warm electronics showing over 21% margin.
+Table 3.11: Status of critical subsystems for SAFARI
+Subsystem Key Functional, Interface and
+Performance RequirementsStatus
+Detector System 5940 detector channels in three arrays,
+Sensitivity: 2×10−19WH z−1/2
+Electrical and sensor 3 dB knee: 20 Hz50 mK base temperatureDemonstrated<2×10−18WH z−1/2optical
+NEP in a single detector.
+Electrical and optical testing of a 5 ×5 sub-
+array with NEP∼2×10−19WH z−1/2on-
+going.
+Baseband feedback readout electronics
+demonstrated in laboratory breadboard test-
+ing.Conceptual design of Focal Plane Arrays
+completed.
+Component level testing and prototyping
+ongoing (RF Notch Filters, SQUIDs, Focal
+Plane Optics, electrical interconnects).
+InstrumentCooler 1μW heat lift at 50 mK
+20μW heat lift at 300 mK
+Thermal interfaces at 1.7 K, 4.5 K and12 KHybrid3He sorption cooler /ADR architec-
+ture confirmed.
+Breadboard cooler similar to the require-
+ments of SAFARI tested at CEA under ESA
+Technology Research Programme study.
+Mechanical CAD model of SAFARI coolerproduced.
+Critical issue is control of heat loads during
+recycle.
+FTS ScanMechanism Total scan length∼35 mm
+Scan velocity noise: <3μm/s 0-20 Hz
+Position measurement precision: 15 nmCryogenic zer0-g dissipation: <1m WBaseline architecture based on TNO Dar-win/Optical Delay Line mechanism.
+Detailed feasibility study completed and no
+stopping issues identified.
+Mirrors >99% reflectivity in SAFARI band
+λ/6 WFE per mirrorGeneral design based on SPIRE technologyand considered low-technical risk.
+Filters, Dichroics
+and Beam split-
+tersBand definition of three arrays, out of
+band rejection and 50:50 amplitude split-
+ting in interferometer.Design based on well established design
+heritage as used on Herschel, Spitzer and
+many other programmes.
+Filter Wheel Two six position,∼35 mm filter element,
+low tolerance mechanismFilter wheel mounted on cryogenic steppermotor.
+Structure First mode>150 Hz and instrument mass
+<50 kg with marginSolid model and Finite element model ofinstrument structure completed and is com-
+pliant with mass and Eigen mode require-
+ments.3.2. Focal Plane Instruments 77
+Table 3.11: (continued)
+Subsystem Key Functional, Interface and Perfor-
+mance RequirementsStatus
+Warm
+ElectronicsMass less than 30 kg and power less than120 W. Bias and readout the detectors and
+focal plane subsystems and provide the
+electrical interface with the spacecraft.Overall architecture defined. Some ele-ments have been bread boarded. Digital
+ASIC development will be required for de-
+tector readout.
+Critical subsystem development
+Table 3.11 gives an overview of the critical subsystems required for the SAFARI instrument and an assess-
+ment of their current development status. Three of these, the detectors, the sub-100-mK cooler and the scan
+mechanism were identified as critical development items during the initial assessment of the instrument design
+by ESTEC in June 2008. Active development programmes are ongoing to raise the technical readiness of these
+items and, as can be seen from the table, significant progress has already been made. The remaining technical
+challenges for the critical subsystems are well understood and Technical Development Plans have been submit-
+ted describing how these subsystems will demonstrate the required Technology Readiness Level ahead of the
+mission downselect in 2011 and entry into Phase B.
+3.2.3 Mid Infrared Camera: MIRACLE
+MIRACLE (Mid InfRAred Camera w/wo LEns) is the focal plane instrument for wide field imaging and low-
+resolution spectroscopic observations (R =λ/Δλ∼5 – 100) over a wide spectral range in the mid infrared (5 –
+40μm). MIRACLE consists of two channels (MIR-S and -L) with almost the same optical design. Each of
+them has fore-optics re-imaging the telescope focal plane. A field mask wheel is installed at the focal plane
+in order to provide optimal slits in the spectroscopic mode. The subsequent camera optics has a pupil position
+where filter wheels with band-pass filters and grisms are installed.
+Table 3.12: Specifications of MIRACLE
+MIR-S MIR-L
+Wavelength coverage 5−26μm 20−38μm
+Detector Si:As 1024×1024 Si:Sb 1024×1024
+Pixel scale 0.36/prime/prime/pixel 0.36/prime/prime/pixel
+FOV 6×6 arcmin26×6 arcmin2
+Spectral resolution (R =λ/Δλ)∼10 (imaging)/∼100 (spec.)∼10 (imaging)/∼100 (spec.)
+Band pass filters/grism Band pass filters/grism
+Size (mm) 125(R) x 50(H) x 30 degree 125(R) x 50(H) x 30 degree
+3.2.4 Mid-IR Spectrometer: MIRMES & MIRHES
+MIRMES (Mid-IR Medium-Resolution Echelle Spectrometer)
+MIRMES performs medium-resolution (R =λ/Δλ∼1500) spectroscopic observations over a wide spectral
+range in the mid infrared (10 – 40 μm) range. The instrument consists of two arms, ARM-S and ARM-L.
+They share the same field of view (FOV) area on the focal plane by means of the dichroic beam splitter, which78 Chapter 3. Payload
+Figure 3.9: Left panel: A successful optical design of MIRACLE. The folding mirror for beam pick-off at the STA focus
+is omitted for simplicity. The final F-number is set to fast value (3.15) in order to match the PSF size to small pixel size
+(18μm) of the detector. Right panel: Optical layout of MIRHES S-mode.
+separates transmission/reflectance at around 18.7 μm. Each arm has the field mask to limit the FOV and each
+arm has an image slicer as an integral field unit.
+Table 3.13: Specifications of the Mid-IR Medium-Resolution Echelle Spectrometer.
+ARM-S ARM-L
+Wavelength coverage 10.32−19.35μm 19.22−36.04μm
+Spectral resolution (R =λ/Δλ)∼1500 ∼900
+Pixel scale 0.37/prime/prime0.72/prime/prime
+Slit width 1.11/prime/prime(3 pixel) 3.6/prime/prime(5 pixel)
+FOV 12.95/prime/prime×5.55/prime/prime25.2/prime/prime×18.0/prime/prime
+(35 pixels×3 pixels×5 rows) (35 pixels×5 pixels×5 rows)
+Size (mm) 125(R)×50(H)×30 degree 125(R)×50(H)×30 degree
+Mass (kg) 17.5 17.5
+MIRHES (Mid-IR High-Resolution Echelle Spectrometer)
+MIRHES performs high-resolution (R =λ/Δλ∼20 000 – 30 000) spectroscopic observations in two wavelength
+ranges, 4 – 8μm (S-mode) and 12 – 18 μm (L-mode). Immersion gratings enable high spectral resolution with
+small mass and volume resources. Two independent spectrographs for S-mode and L-mode are considered
+(Figure 3.9, right panel).
+3.2.5 SPICA Coronagraph Instrument: SCI
+SCI (SPICA Coronagraph Instrument) is a high dynamic-range imager and spectrometer with coronagraph
+optics. The primary target of SCI is the direct observation (imaging and spectroscopy) of Jovian exoplanets
+in the infrared, whilst circum-stellar discs, other type of exoplanets, Active Galactic Nuclei, and any other
+compact systems with high contrast can be potential targets.
+The specifications of SCI are summarised in Table 3.15. Though diffraction limited imaging at 5 μm
+wavelength is a specification for the SPICA telescope, SCI has as goal to observe at shorter wavelengths with3.2. Focal Plane Instruments 79
+Table 3.14: Specifications of the Mid-IR High-Resolution Echelle Spectrometer.
+Short(S)-mode Long(L)-mode
+Wavelength coverage 4–8μm 1 2–1 8μm
+Spectral resolution (R =λ/Δλ) 30 000 20 000 – 30 000
+Slit width 0.72/prime/prime1.20/prime/prime
+Slit length 3.5/prime/prime6.0/prime/prime
+Dispersion element ZeSe immersion grating KRS5 immersion grating
+Cross disperser reflective reflective
+Size (mm) 200(L)×200(W)×100(H) 350(L)×350(W)×200(H)
+Mass (kg) 7.4 7.4
+Table 3.15: Specifications of SCI.
+Wavelength coverage 5–2 7μm (goal: 3.5 – 27 μm)
+(In 1−3.5μm, SCI has sensitivity, though contrast is not guaranteed)
+Observation mode Coronagraphic/Non-coronagrahic,
+Imaging/Spectroscopy
+Coronagraph Binary-shaped pupil mask
+Contrast 10−6
+Inner working angle (IWA) 3.3λ/ D
+Outer working angle (OWA) 16λ/ D
+Detector Si:As 1k×1k (long wavelength channel)
+InSb 1k×1k (short wavelength channel)
+FOV 1×1 arcmin2
+Spectral resolution R=20−200
+wave front correction by a deformable mirror and a tip-tilt mirror. For the coronagraph method, binary-shaped
+pupil mask is the baseline solution because of the robustness against telescope pointing error, achromatic e ffects
+(except image size effects scaling with wavelength), and simplicity. A more challenging method, a hybrid
+solution of Phase Induced Amplitude Apodization (PIAA) and binary-shaped pupil mask, is considered asan option to obtain better Inner Working Angle (IWA) and throughput. With the InSb detector option, the
+observable range can be extended to shorter wavelengths, and simultaneous imaging will be possible with two
+detectors. Band-pass filters will be used for imaging, and transmissive dispersers (e.g., grism) will be used for
+spectroscopy.
+Thanks to a simple pupil shape and active optics, SCI provides the potential to perform coronagraphic
+observations with contrasts significantly higher than the coronagraph of JWST. This can be a unique capabilityto characterise atmospheric features of exoplanets in the 2010s decade. It should be noted that SCI can be
+useful for monitoring and observation of exoplanet transits.
+3.2.6 Focal Plane Finding Camera
+FPC (Focal Plane finding Camera) consists of two components. One is FPC-G that is used for the attitudecontrol system to an accuracy of 0.05
+/prime/prime. FPC-G is a system instrument and will be operated continuously80 Chapter 3. Payload
+Figure 3.10: Design of SCI. A tip-tilt mirror and a deformable mirror is used. All device before focal plane mask is made
+of mirror optics (i.e., no transmissive device). Mechanical changer is used to realise the coronagraphic mode and fine
+camera/spectroscopy mode without mask.
+during the whole SPICA mission. FPC-S is a near infrared camera for astronomical observations. FPC-S has
+a filter wheel with several filters and dispersive materials. FPC-G and FPC-S are integrated into one package
+with beam splitter. FPC specifications and optical design are shown in Table 3.16. The FPC mass is estimated
+at<10 kg.
+Table 3.16: FPC specifications
+FPC-G FPC-S
+Wavelength coverage 1.6μm (H band) 2−5μm
+Detector InSb, 512×412 HgCdTe, 2K×2K
+Pixel scale 0.5/prime/prime0.18/prime/prime
+Slit width 1.11/prime/prime(3 pixel) 2.16/prime/prime(3 pixel)
+FOV 4.3×3.5 arcmin26×6 arcmin2
+Heat generation <1m W<2m W
+3.2.7 BLISS
+BLISS (Background-Limited Infrared-Sub-mm Spectrograph) is an extremely sensitive broad-band far infrared& sub-mm spectrometer proposed mainly by US astronomers. BLISS covers the entire far infrared and sub-mmwavelengths (38 – 430 μm) with a spectral resolution R=700, using 4200 superconductive bolometer arrays
+cooled down to 50 mK by a dedicated adiabatic demagnetisation refrigerator.Chapter 4
+Mission Design
+The SPICA mission design is tailored to achieve the scientific objectives described in chapter 1. To this goal, the
+spacecraft design is based on a 3-m class telescope (the nominal diameter of the primary mirror is 3.5 m in the
+current baseline) operated at a temperature lower than 10 K (the nominal temperature of the telescope assembly
+is<6 K in the current baseline). The telescope is designed to have diffraction limited performance at 5 μm
+(with a WFE<350 nm rms). The main SPICA wavelength range is 5 – 210 μm, covered with five scientific
+instruments: MIRACLE (mid infrared camera and spectrophotometer), SAFARI (far infrared imaging spec-
+trometer), MIRMES (mid infrared medium resolution spectrometer), MIRHES (mid infrared high resolution
+spectrometer), and SCI (mid infrared coronagraph) A sixth instrument, BLISS (FIR and sub-mm spectrometer)
+is considered as optional. The SPICA Telescope Assembly (STA) and the Focal Plane Instruments (FPI) are lo-
+cated in the cryo-module of the spacecraft, supported via a truss structure which is fixed to the Service Module(SVM).
+The thermal environment required to operate the telescope and the instruments is obtained by a combination
+of passive cooling (via dedicated Sun and thermal shields combined to radiators) and active cooling (includinga number of mechanical coolers, ensuring a base temperature of 4.5 K and 1.7 K). The total mass of the S/Ci s
+about 4 ton, in line with the capabilities of the baseline launcher vehicle, JAXA’s H-IIB.
+The Sun – Earth L
+2point is the optimum environment to obtain excellent sky visibility and a stable thermal
+environment to be able to cool the telescope. The onboard data generation rate is of order 4 Mbps, therefore
+requiring a high-speed downlink (compatible with X band). The guaranteed lifetime is 3 yr, with the goal of
+extended operations to 5 yr. The absence of cryogens onboard allows to extend the nominal lifetime beyond
+the nominal duration (finally limited by the AOCS propellant and any onboard failures). The main SPICA
+subsystems are illustrated in Figure 4.1.
+4.1 Mission profile
+The presently envisaged launcher vehicle for SPICA is JAXA’s H-IIB, capable of delivering in excess of 4 ton
+to a SE-L 2transfer orbit. The H-IIB is equipped with a 5S-H fairing with a useful diameter of 4.6 m, compatible
+with the SPICA needs. The launch site is JAXA’s Tanegashima Space Center, while the launch is planned in
+2018.
+The orbit baselined for SPICA is a large amplitude halo orbit, with a period of about 180 days and semi-
+major axis amplitude of about 750 000 km (see Figure 4.3).
+A first launcher dispersion manoeuvre is performed on the first day after launch, with the S/C already on a
+trajectory to SE-L 2. The spacecraft will be in proximity of the libration point after about 30 days from launch.
+The cool-down will start immediately after the out-gassing phase, during the transfer. The final orbit injection
+in the Halo orbit will take place approximately 120 days after launch. The guaranteed mission lifetime is 3 yr
+(nominal Operation Phase, driven by the guaranteed lifetime of the mechanical cooler), with the goal of 5 yr
+(including an Extension Phase of 2 yr).
+8182 Chapter 4. Mission Design
+Figure 4.1: Artist view of the SPICA spacecraft.
+Figure 4.2: H-IIB launcher vehicle and Tanegashima launch center.4.2. Spacecraft design 83
+Figure 4.3: L 2transfer trajectory and Halo Orbit Injection.
+4.2 Spacecraft design
+The SPICA spacecraft consists of the Cryo Payload Module (PLM) and the Bus (or Service) Module (SVM) as
+illustrated in Figure 4.4. The PLM includes the Cryogenic Assembly and the Scientific Instruments Assembly.
+The Cryogenic Assembly includes mechanical coolers and passive thermal shields, required to cool down and
+maintain at about 5 K the Scientific Instruments Assembly. The latter includes the STA and the Instrument
+Optical Bench (IOB). Additional coolers allow reaching the required focal plane detector temperature (down
+to∼100 mK). The overall PLM architecture is dominated by the need to have a large Sun shield, three coaxial
+thermal shields, the telescope shell and the additional telescope baffle. The PLM is connected to the Bus via alow thermal conductivity truss structure, which is also supporting the STA.
+The Bus Module, of a conventional design, hosts most of the spacecraft subsystems (power, propulsion,
+attitude control, data handling, thermal control and telecommunications), as required to operate the observatory.
+A list of the SPICA subsystems can be found in Figure 4.5.
+The total wet mass of the spacecraft (including system level margins) is estimated to be about 4000 kg (see
+Table 4.2), while the total electrical (peak) power required during operations is estimated to be about 2.3 kW
+(see Table 4.3). The spacecraft would have a maximum height of about 7.4 m, of which about 1.3 m for the
+Bus and 6.1 m for the PLM and support structure.
+4.2.1 Bus module and spacecraft sub-systems
+The SPICA bus module has an octagonal cross-section and a structure based on a combination of truss and panel
+elements. The octagonal geometry is reflected in the 8 interface points adopted between the launcher adaptor
+and bus module, between bus module and truss structure as well as between truss structure and telescope
+assembly, thus allowing a balanced transfer of the launch loads. The estimated dry mass of the bus module is
+830 kg: this value takes into account all sub-systems and units, including the solar arrays.
+The lateral panels allow easy access to the equipment and additional area to be used as radiating surface.
+The bus module is almost completely shielded from the direct Sun radiation thanks to the lower part of the PLM
+Sun shield, the Solar Arrays and the High Gain Antenna. The temperature of the upper panel of the bus and of
+the truss interface points determines the conductive loads to the cryo-module and needs to be minimised.
+The Electrical Power Subsystem is based on triple-junction GaAs solar cells sized for a total maximum
+power of 2.4 kW (EOL) at an operating temperature of about 100 °C. The solar arrays, located outside the Sun84 Chapter 4. Mission Design
+shield, are arranged in two wings, each consisting of 3 elements, a deployment hinge, a holding mechanism
+and a yoke. The solar arrays are folded around the bus module and deployed immediately after separation from
+the launch vehicle. The Electrical Power Subsystem, designed for a mission duration of 5 yr, is based on an
+unregulated 50 V bus and a set of Li-ion batteries.
+The Telemetry, Telecommand and Communication system includes a high speed X-band telemetry down-
+link (∼ 10 Mbit/s), a low speed S-band TT&C data link and a low speed X-band TT&C data link. The high
+speed down-link includes steerable HGA and MGA. The low speed S band link is equipped with 2 Low Gain
+Antennas to ensure full coverage at different spacecraft attitudes.
+The Data Handling System is based on an architecture already developed by JAXA for other science mis-
+sions with the purpose of enabling re-use of existing equipment. The baseline configuration is illustrated in
+Figure 4.6. The onboard mass memory is sized for a minimum capacity of 48 Gbytes, corresponding to thedata volume produced in slightly more than one day.
+The architecture of the SPICA Attitude and Orbit Control Subsystem is represented in Figure 4.7. In order
+to achieve the required pointing stability performance, the information provided by the Inertial Reference Unit
+and by the Star Trackers is complemented by dedicated Focal Plane Cameras (FPC-G and C-FPC), located on
+the Instrument Optical Bench and used as Fine Guidance Sensors. FPC-G is used for observations not involving
+the Coronagraph instrument, while C-FPC is used in conjunction with the Coronagraph operations. Reaction
+wheels are baselined as actuators.
+The AOCS performance during operations with the Coronagraph instrument and with the other FPI is
+provided in Table 4.1. The use of these two additional fine guidance sensors allows achieving rather goodAbsolute Pointing Error (APE) and Relative Pointing Error (RPE) performance.
+The SPICA propulsion system is based on a blow-down, mono-propellant (N
+2H4hydrazine) approach.
+The System includes four propellant tanks, four 23N thrusters, eight 3N thrusters and auxiliary equipment
+(such as piping, filters, valves). This simple configuration is well matched to the modest delta-V requirement
+(∼1 0 0ms−1) and the compensation of external torques (mainly related to solar radiation pressure).
+The four tanks carry the propellant and the pressurant gas. The 23N thrusters are used for orbit control
+purposes and provide the thrust required to achieve the required delta-V (Z axis direction). The 3N thrustersare used mainly for attitude control purposes (Down-loading of the reaction wheels). The total mass of the
+propulsion system is estimated to be about 110 kg, with an additional 200 kg of propellant.
+Table 4.1: SPICA AOCS performance (APE and RPE)
+Observation mode Absolute Pointing Accuracy Pointing stability
+Except for coronagraph 0.135 arcsec (3σ) 0.075 arcsec (0-P) /200 sec
+Coronagraph 0.03 arcsec (3σ) 0.03 arcsec (0-P)/20 min
+The mass and power spacecraft resource budgets are reported in Table 4.2 and 4.3. The unit or sub-system
+level mass values listed in the table are comprehensive of individual maturity level margins (depending on their
+design maturity). A system level margin of 20% is applied on top of the maturity margins in order to reflect the
+present level of design definition (commensurate to Phase 0/A).
+Similarly the unit or sub-system level power values listed in the power budget table are comprehensive of
+individual margins and are representative of peak power demands.
+4.2.2 Payload module – cryogenic system
+The SPICA adopts a new concept of cryogenic system that uses no cryogen. The elements maintained at 4.5 K,
+including the STA, the IOB and some focal plane instruments (FPI), are refrigerated by the combined action
+of mechanical cooling and efficient radiative cooling in the stable thermal environment at the Sun-Earth L 2.I n
+consideration of the cooling capacity of a 4 K-class mechanical cooler (4 K-MC) at the end of life (EOL) and4.2. Spacecraft design 85
+Table 4.2: Overall SPICA spacecraft mass budget.
+Subsystem Main units Mass (kg)
+Payload Module (PLM)Cryogenic Assembly 1200
+STA 700
+FPI (including IOB) 200
+STA+FPI warm electronics 100
+Cryogenic electronics 90
+Total PLM (with maturity margins) 2290
+Bus Module (BM)Mechanical structure 330
+Thermal Control System 66
+Power control system 32
+Solar Array assembly 94
+TT&C 115
+Data Handling System 22
+AOCS 106
+Propulsion 110
+Harness 39
+Total BM (with maturity margins) 914
+Total Dry Mass PLM+BM 3204
+System Level Margin 20% on dry mass 640
+Propellant Hydrazine 220
+TOTAL with maturity & system level margins 4064
+on the assumption of Joule heating of the FPI being approximately 15 mW, the baseline design of the Thermal
+Insulation and Radiative Cooling System (TIRCS) was determined by the thermal and the structural analyses,so that the total parasitic heat flow from the higher temperature elements to the 4.5 K stage can remain below
+25 mW.
+The 4.5 K units, such as STA, IOB and FPI, are surrounded by the TIRCS, consisting of the external tele-
+scope baffle, the telescope shell, three shields and a sun shield to reject the heat flow from the outer environment,as depicted in Figure 4.8. Multi-Layer Insulation (MLI) is attached to the sun shield and to the thermal shield
+#3 to block thermal radiation, while a section of each thermal shield acts as a radiator to reject most of the
+absorbed heat from the sun and from the spacecraft service module (SVM) to deep space. The layout of these
+shields and of the solar array paddles is determined to optimise radiative cooling.
+The STA, the telescope shell and the three shields are structurally supported by the SVM with main trusses
+between them. The trusses are made of the Carbon Fiber Reinforced Plastics (CFRP) and Alumina FibreReinforced Plastics (ALFRP) with low thermal conductivity, while the external baffle and the main part of the
+telescope shell are made of high thermal conductivity CFRP to dissipate the heat flow. The sun shield and threeco-axial thermal shields are made of aluminium plates, and their structural frames are connected by means of
+dedicated trusses of ALFRP as well. The shield #3 and the external baffle are supported axially by the SVM
+and the telescope shell, respectively. Figure 4.9 shows a more detailed view of the truss structure separating the
+PLM from the SVM and supporting directly STA, FPI and thermal shields.
+Wire harness between the FPI and their electronics equipments in the SVM is assumed to be made of
+Manganin (up to 1000 lines). The Heat Rejection System transports the heat exhausted from the mechanical86 Chapter 4. Mission Design
+Figure 4.4: Overview of the main elements of the SPICA spacecraft.
+Table 4.3: Total peak power with margins
+Subsystem Main units Power (W)
+Payload Module (PLM)Cryogenic Assembly 900
+STA 0(1)
+FPI (including IOB) 235
+Service Module (SVM)Mechanical structure 0
+Thermal Control System 160(2)
+Power control system 35
+TT&C 185
+Data Handling System 100
+AOCS 130
+Propulsion 185
+Total Power PLM+SVM 1930
+Margin 20% on total power 385
+TOTAL Total power with margins 2315
+(1)During normal operations the STA is passive. The STA is equipped with a refocusing mechanism used only during the observatory
+commissioning. The STA is equipped with heaters to allow for decontamination (with a power is of order several hundreds of W,
+available given the fact that the mechanical coolers would be stopped during decontamination).
+(2)Not including heater power for the telescope decontamination.4.2. Spacecraft design 87
+Figure 4.5: List of the subsystems of the SPICA spacecraft.
+coolers on the cooler base plate to the cooler radiators. A loop heat pipe, which has some advantages such as
+flexibility in components arrangement and adaptability for long-distance transportation of large heat amount,
+is assumed to be used as heat transport device. The radiators dedicated to this purpose are located immediatelyabove the service module, looking in the anti-sun direction.
+A dedicated thermal analysis was carried out on the SPICA TIRCS configuration, showing that the heat
+load to the 4.5 K stage is dominated by conduction through structural supports (truss) and harness. The heatflow analysis for the steady state case shows that the total amount of parasitic heat and heat dissipation (fromall elements of the PLM) is less than 40 mW at the 4.5 K stage, while the temperature of the external baffle
+and of the telescope shell are less than 14 K and 30 K, respectively. On the other hand, the bread board
+model of the upgraded 4 K Mechanical Cooler successfully demonstrated a cooling power larger than 50 mW,
+a cooling power compliant with the results of the TIRCS thermal analysis. Specific attention has been paid by
+the minimisation of vibrations transmission from the mechanical coolers to the PLM (and in particular to the
+FPI).
+Concerning the transient case, a preliminary time-dependent analysis gives a cooling profile at the initial
+phase as shown in Figure 4.10, showing that a cooling time of about 170 days is required to reach less than 5 K
+after launch. The possibility to reduce the cool down time is presently being studied by JAXA.88 Chapter 4. Mission Design
+Figure 4.6: Overall SPICA Data Handling System architecture.
+Figure 4.7: Overall SPICA AOCS architecture (0-P =Zero to Peak).4.2. Spacecraft design 89
+Figure 4.8: SPICA Thermal Insulation and Radiative Cooling System (TIRCS)
+Figure 4.9: Truss structure connecting the SPICA PLM to the SVM.90 Chapter 4. Mission Design
+Figure 4.10: SPICA cool-down profile.
+The SPICA Mechanical Cooling System is based on a set of advanced Stirling and Joule-Thompson coolers.
+The 4K Mechanical Cooler for the 4.5 K stage is a 4K-class Joule-Thomson cooler (4K-JT) connected with a
+pre-cooler of a 20 K-class two-stage Stirling cooler (2ST), which does not use any consumable cryogen. Far
+infrared instruments such as SAFARI on the IOB require further cooling to 1.7 K, reached thanks to a 1K-
+Mechanical Cooler consisting of a dedicated 1K-JT with3He working gas and an additional 2ST pre-cooler.
+The overall architecture of the Mechanical Cooling System is illustrated in Figure 4.11. Specifications ofmechanical coolers for the SPICA are listed in Table 4.4, also showing the good heritage existing, thanks to
+previous missions, such as AKARI and Astro-H.
+Figure 4.11: Block-diagram of the SPICA Mechanical Cooling System.
+Although based on the heritage of the 2ST flown on board the successful AKARI, the 2ST for SPICA is
+required to have higher reliability, for continuous operation during more than 3 years to complete the guaranteedand extended mission phases. At the same time, the cooling capacity at 20 K has to be increased, as this largely
+contributes to increasing the cooling capacity of the 4K-JT and of the 1K-JT. The engineering model of the
+upgraded SPICA 2ST (with lower out-gassing materials) is shown in Figure 4.12. This engineering model was
+proved to provide higher cooling capacity, up to 325 mW at 20 K.
+Based on heritage of the 4K-Mechanical Cooler developed for ISS/JEM/SMILES, the upgraded SPICA
+4K-Mechanical Cooler was designed and fabricated by combining new JT heat exchangers (based on coaxial
+double tubes) with low pressure loss, coupled with the modified 2ST. Test results show that the maximum4.2. Spacecraft design 91
+Table 4.4: Specifications of the SPICA mechanical coolers.
+Figure 4.12: Bread board model of the upgraded SPICA two-stage Stirling cooler.92 Chapter 4. Mission Design
+cooling power of 50.1 mW was efficiently obtained with an electric input power of 55.9 W (AC) for the JT
+compressors and of 89.2 W (AC) for the 2ST stage. The remarkable improvement of the cooling power at the
+4.5 K stage is attributed to the increase of the mass flow rate; this was made possible by high-power pre-cooling
+at 18 K, thanks to the modified 2ST, with the extended 8 mm diameter second displacer.
+Test results for the 1K-MC indicate that the cooling power was successfully improved by 16.0 mW with
+an efficient input power of AC 76.6 W for JT compressors and AC 89.0 W for the powerful 2ST. Higher mass
+flow rate, obtained by cooling the3He gas in the JT circuit at 12 K by the modified 2ST, drastically increases
+the heat lift capacity at 1.7 K.
+It is notable that all the coolers satisfy the cooling requirements at the beginning of life (BOL), whereas
+long life tests for all the coolers are under preparation for the verification of reliability at EOL. In the SPICA
+Mechanical Cooling System baseline, two sets of 4K-Mechanical Coolers and 1K-Mechanical Coolers are
+employed for redundancy, respectively. During the development of the mechanical coolers, significant effort
+has been invested in minimising the generation of vibrations.Chapter 5
+Mission Operations
+SPICA is a JAXA/ESA mission open to the general astronomical community as a multi-user observatory. The
+focal plane instruments are provided by consortia of scientific institutes. In particular, SAFARI is provided by
+a consortium of European, Canadian and Japanese institutes; MIRACLE, MIRHES and MIRMES and SCI are
+provided by Japanese consortia; FPC is provided by a consortium of Korean and Japanese institutes and BLISS
+has been proposed to NASA. The science management and the operations of the mission described here reflect
+the character of open observatory together with the distribution of responsibilities to the parties involved in the
+development of the spacecraft and the instruments.
+5.1 Science management
+The science management of the observatory accounts for the fact that SPICA is a JAXA led mission in which
+ESA is a partner. The scientific organisation and the allocation of observing time are defined to enable European
+astronomers to take maximum benefit of the observatory capabilities as a whole, and in correspondence with
+the relative European contribution.
+5.1.1 Science teams
+JAXA will establish a Science Advisory Committee that will consist of scientists representing all relevant fieldsin astronomy. It will be responsible to review the scientific objectives of the mission, monitor the development
+of the satellite in the context of the scientific performance, and give advice to the JAXA SPICA management
+team on all scientific aspects. The Science Advisory Committee will be chaired by the SPICA Project Scientist
+in Japan and will count with the participation of the ESA SPICA Project Scientist.
+The ESA SPICA Project Scientist will be responsible for representing the interests of the astronomical
+European community in the Science Advisory Committee and in the interactions with the Time AllocationCommittee (TAC). Together with the SPICA Project scientist, he/she will have the responsibility to maximise
+the scientific return of the mission, and to monitor that the development of the European elements (telescope,SAFARI, European SPICA Data Centre) is adequate to achieve the required scientific capabilities. The ESASPICA Project Scientist will interface with JAXA, the ESA SPICA Project Team and the SAFARI PI. He/she
+will participate in the organisation of conferences, workshops or other activities to promote the mission and
+will be the interface with ESA corporate entities in support of Public Relations and outreach activities in col-
+laboration with JAXA.
+To execute these tasks, the ESA SPICA Project Scientist will be supported and advised by the European
+SPICA Science Team. The European SPICA Science Team will identify and advocate the interests of the Euro-
+pean astronomers, provide input to the Science Advisory Committee, and ensure the maximum scientific return
+in relation with the elements of the European contribution, that is, the telescope, SAFARI and the European
+SPICA Data Centre. The membership will include European scientists representing the main fields associated
+with the SPICA science objectives, the SAFARI PI, telescope-optics experts, and a JAXA representative.
+9394 Chapter 5. Mission Operations
+Figure 5.1: SPICA operational time allocation.
+For community support activities, the ESA SPICA Project Scientist will be supported by the ESA SPICA
+Science Centre (described in section 5.3.2).
+5.1.2 Observing time allocation
+The SPICA observing time will be distributed following the principle of “guaranteed” and “open” time. Guar-
+anteed time will be allocated to the scientific institutes directly involved in the building of the focal plane
+instruments. The instrument consortia will decide internally the distribution of their guaranteed time, underthe final responsibility of the instrument PI. Open time will be allocated to proposals submitted by general
+astronomers, through a competitive process in which the TAC will provide the final recommendations.
+The distribution of satellite time aims at keeping an adequate balance among the assignment of guaranteed
+time to the instruments scientific consortia, the availability of open time to astronomers from SPICA associated
+countries and the open time to the world community. In the following, the percentages of SPICA time for each
+category are specified (with respect to total satellite time in the Nominal Observation phase; see Figure 5.1):
+•Engineering/Calibration Time: 10%
+•Director’s time, including ToO observations: 5%
+•Guaranteed time for SPICA team: 25%. It will be shared taking into account the relative contribution of
+each team to the project, and includes the guaranteed time for instrument teams, science teams members,
+and science operations centres.
+•Open Time: 60%, which will be further subdivided as:
+–For astronomers from countries directly involved in the SPICA project: 40%
+–For the general community, without restrictions of nationality: 20%
+The time allocation to European astronomers will be proportional to the European economic contribution to
+SPICA, which we assume will amount to one third of the total. Consequently, the fraction of European open
+plus guaranteed time should also be one third of the “allocated” open and guaranteed time. Within this scenario,
+time reserved to astronomers of the ESA member states is distributed in the following way:
+•European guaranteed Time: 8.3% of total satellite time, of which 95% is reserved for the SAFARI
+consortium, 3.5% for the ESA SPICA Science Centre, and 1.5% for the European mission scientists.
+•European open time: 13% of the total satellite time (or 22% of the total open time).5.2. SPICA operational phases 95
+5.1.3 Observing programmes
+Announcements of Opportunity will be made before and during the mission for SPICA Legacy Programmes
+and normal programmes. Legacy programmes are coherent and comprehensive observation proposals that aim
+to address key questions in Astronomy, and which require a large number of hours for their execution. The
+detailed rules for the SPICA Legacy programmes are under definition. Both Legacy programmes and normalprogrammes can be proposed through guaranteed and open time.
+The observations in Legacy and normal programmes will have associated one year of proprietary time,
+counting from the moment that the observer receives the processed data. For Legacy programmes whose results
+are required for follow-up observations, the proprietary time may be reduced and proposers may be requested
+to make their results public as early as ∼six months after receiving their observational data. Engineering and
+Director’s time is not proprietary, that is, observations will immediately be made available.
+The SPICA Science Operations Centre will distribute to the observers the raw observational data, ancillary
+data required for the processing (e.g., pointing) and calibrated products generated automatically by the instru-
+ment pipelines, in which the instrument effects have been removed. It is also the intention to provide a SPICA
+interactive analysis package for data reduction. Legacy programme teams will be responsible (together withthe SOC) to make a coherent processing of their observational data and to provide highly processed products
+to the general community for observation follow-up and general scientific exploitation.
+5.1.4 SPICA Observing Time Allocation Committee
+A single international scientific Observing Time Allocation Committee will be established by JAXA, ESA andother potential mission partners sufficiently early before launch. The composition of the TAC will be arranged
+such that conflicts of interests with proposers will be avoided. It will be based on scientific excellence.
+The function of the TAC will be to assess observing proposals made by the science community at large as
+part of the open time. In particular, the TAC will establish criteria for open time observing proposal selection,
+will review and categorise proposals on scientific merit, technical feasibility, and priority in light of the SPICA
+scientific objectives, and will recommend to JAXA and ESA the assignment of observing time. The TAC may
+decide to approve/reject full or partial parts of a proposal. The TAC will be supported by the SOC and by the
+ESA SPICA Science Centre, which will assess the proposals technical feasibility prior to the TAC meetings andprovide statistics from the submitted/approved proposals on, e.g., distribution of requested time per instrument
+and per science area.
+5.2 SPICA operational phases
+The nominal lifetime of the SPICA mission is 3 years (guaranteed by the current design), with a goal of 5 years(expected from future developments in the cryogenic system). There are five operation phases from the time
+that the satellite is installed in the launch facility and the end of the mission:
+•Launch Operations phase, which extends from the arrival of the satellite at the Tanegashima Space Centreuntil the end of launch operations.
+•Initial Mission Operations phase, which starts at the end of launch operations. The satellite will beinjected into the S-E L
+2Halo orbit. After 18 hours from the launch, the first orbit manoeuvre will
+be carried out. The following manoeuvres will take place on days 20 to 40. It will take the satellite
+approximately 120 days to be injected in the observational orbit. At this moment, we assume that it takes
+168 days to cool the telescope by using 2 redundant 4K Mechanical Cooler at the same time. During the
+Initial Mission Operations phase, function checkout, performance verification and test observations are
+performed.96 Chapter 5. Mission Operations
+•Nominal Mission Operations phase, which is the “planned operation” period. During this phase, the
+routine scientific observations will be carried out. Nominal duration is 2-2.5 years, with the goal to
+achieve 4-4.5 years.
+•Extended Mission Operations phase: Covers from the end of the Nominal Observation Phase to the time
+of termination of the satellite. This phase will start when one or more of the cryocoolers malfunction
+prevent nominal operations to continue. In this phase extended operation programmes will be executed
+with the instruments for which science operations in warmer conditions are possible.
+5.3 SPICA operations centres
+SPICA is a Japanese-led mission implemented on the basis of international cooperation. Following this princi-
+ple, the mission operations will also be implemented as a collaborative effort from JAXA, ESA and the scientific
+institutes responsible for the provision of the SPICA instruments. The SPICA Ground Segment will consist of
+the following elements:
+•Mission Operations Centre (MOC), located in Japan, and the associated ground stations, one of themsupported from ESOC, Germany
+•Science Operations Centre (SOC), located in Japan
+•Japanese Instrument Control Centres
+•European SPICA Data Centre, with two organisational units:
+–ESA SPICA Science Centre (ESSC), located at ESAC, Spain
+–SAFARI Instrument Control Centre (ICC). The SAFARI ICC will be geographically distributed inseveral European countries. The contact to the distributed ICC will be via the PI institute at SRON
+Groningen.
+5.3.1 Mission Operations Centre and ground stations
+The SPICA Mission Operations Centre (MOC) will be set up in JAXA and will be responsible for all spacecraft
+operations, including routine observations and contingency plans. The main functions of the MOC will be to
+generate and upload commands and receive telemetry data. The operations centre will control the conditions
+of the SPICA observatory by monitoring the housekeeping data and will forward to the Science OperationsCentres the science data. The overall role of the SPICA MOC, its interfaces to the Science Operations Centres
+(located in Japan, Europe and possibly other countries depending on the consolidated cooperation scenario) is
+illustrated in Figure 5.2.
+The prime ground station for the SPICA mission is the 64 m Usuda station. In the present spacecraft
+design, the telemetry downlink will be performed using X-band. A low speed data link in S band will also be
+available. The required downlink capacity is in the order of magnitude of 350 Gbit/day. This is more than the
+316-237 Gbit/pass achievable with Usuda at 11 Mb/s and 8-6 hr dump duration (X-band). The JAXA downlink
+baseline is QPSK modulation and Reed Salomon coding with a maximum data rate of 11 Mbit/s. This wouldin fact lead to a daily downlink volume of about 316 Gbit/day in case ofa8h rwindow.
+The downlink, subject to final agreements, could be supported by an ESA ground station at a different
+longitude than Usuda, i.e., Cebreros and/or the new DS3 ground station in South America. A downlink rateof order of 9 Mb/s seems feasible for Cebreros (the same data rate can also be used for DS3). It is assumed
+that a respective variable coding is provided by the SPICA spacecraft. Given the envisaged loading of the
+ESA Deep Space Network at the time of the SPICA operations, it is realistic to assume a maximum dump
+duration of 4 hr/day. In this case the additional downlink volume given by the ESA ground station is about
+130 Gbit/day. Thus the combined JAXA-ESA daily downlink volume would be about 446 to 367 Gbit/day,5.3. SPICA operations centres 97
+Figure 5.2: Overall MOC and SOC interfaces (*: under discussion).
+compatible with the mission needs. The actual need for an additional ground station needs to be confirmed,
+pending a final estimate of the actual payload demands and a more detailed evaluation of the SPICA telemetry
+data rate/volume needs.
+It should be noted that the SPICA ranging standard is based on digital ranging different from the ECSS or
+CCSDS standards. Thus ranging could not be provided for SPICA by the ESA ground stations. The envisaged
+ESA involvement in the SPICA mission would not include any spacecraft or payload operations tasks (per-
+formed by JAXA). The ESOC involvement in the SPICA mission includes ground segment support in the form
+of ground station data relay services and data routing services to JAXA, as well as the necessary respective
+preparation and operations activities. The following high level requirements are presently considered for the
+ESA contribution to the SPICA ground segment:
+•No ESA coverage during LEOP and transfer phase (ESA coverage starting during the commissioning
+phase).
+•Science data reception at the Cebreros/DS3 ground station and off line data delivery in Japan to JAXA.
+•4h daily data dump duration at the Cebreros/DS3 ground.
+•Firm ground segment availability and data return requirements: 95% (measured over any half year pe-
+riod).
+•Timeliness: 24 hr for science data and real time (latency: TC 5 s TBC, TM 15 s TBC) for TC and HKTM.
+•Planning to be compatible with long term planning (∼ half a year in advance), minor changes can be
+processed up to 1 month ahead of the actual passes.
+•Contingency services for the SPICA mission shall be made available on the basis of the general ESA –
+JAXA mutual support agreements.
+•On line interface to be activated in case of acquisition problems.
+•All facilities established for SPICA shall be based on the existing ground-segment infrastructure (only
+limited tailoring, such as software updates in relation to the final coding choice).98 Chapter 5. Mission Operations
+The ground segment preparations on the ESA side would start 3 years prior to launch. The readiness of
+the ground segment would be validated by a series of test campaigns, e.g., Radio Frequency Compatibility
+Test, Data Flow Test, System Validation Test. Besides software updates related to final coding choice, we are
+presently assuming that no additional infrastructure update is required.
+As the spacecraft is built by and the mission is operated by JAXA, it must be assumed that the JAXA packet
+structure is applied, which is different from ESA’s Packet Utilisation Standard (PUS). This is not a problemfor the ESOC services, because they can be handled on CCSDS packet level, but instrument operations will be
+affected. Changes to existing instruments EGSE (e.g., if Herschel Ground Support Equipment was to be reused)
+may be required. The impact on command generation tools for the instruments needs to be investigated.
+5.3.2 Science Operations and Instrument Control Centres
+Science operations concept
+SPICA is an observatory type mission open to the general astronomical community. Dedicated astronomical
+observations will be carried out following the specifications from the observers. Astronomers will be invitedto submit proposals following Announcements of Opportunity. Observations will be specified following Astro-
+nomical Observation Templates (AOTs) based on pre-defined observing modes that will be optimised for each
+type of astronomical measurement to be performed with SPICA. Submitted proposals will be reviewed by the
+SPICA Time Allocation Committee. Approved programmes will be stored in the mission database with the
+detailed specification of the corresponding observations. It is assumed that the submission of proposals will be
+done electronically at the SOC, where the mission database will be administrated.
+Scientific mission planning will be carried out at the SOC to generate the daily observation schedules.
+Because of the limited duration of the SPICA mission, it is essential that the planning is optimised and thatmaximum observation efficiency is obtained. Scientific schedules will include the sequence of scientific obser-
+vations to be performed and the required associated instrument commands. Scientific schedules will be send tothe MOC, where they will be completed and uploaded to the satellite.
+The SOC will systematically process raw observational data through the pipelines to standard products and
+will add quality information to each observation. Standard products will contain calibrated data in which theinstrument artefacts have been removed. Products should be processed to a level that will allow astronomers to
+perform scientific analysis and extract astronomical information, and such that they can be made available in
+multi-observatory environments like the Virtual Observatory.
+In addition to standard processing, a dedicated interactive analysis package will be provided to the as-
+tronomical community for the interactive data reduction of the SPICA data. Observation standard products
+generated at the SOC will be made available to the observers and the astronomical community through the
+SPICA science archive, following the user rights and proprietary time policy specified in the SPICA Science
+Management Plan. The SPICA science archive will be located in Japan, with a mirror at ESAC. In addition,
+a European SPICA Data Centre Archive will be implemented to store SAFARI and related ancillary data pro-
+duced from the beginning of the instrument level tests until the end of operations.
+Science Operations Centre
+The SOC is the main Science Operations Centre, and it is the interface to the MOC for scheduling, commanding
+and telemetry reception. In particular, responsibilities of the SOC include the of issue AOs with the required
+documentation, the administration of the observations database and the generation of observation schedules to
+be delivered to the MOC. The SOC will be the main responsible for the implementation of the data reduction
+pipelines and the systematic data processing (in collaboration with the ICCs and the ESA SPICA Science
+Centre) to generate the SPICA science products that will be distributed to the observers. The SOC will support
+non-European astronomers on the preparation of proposals and on data reduction.5.3. SPICA operations centres 99
+JAXA 
+European Data Centre SPICA 
+MOC 
+SOC ESA Science 
+Centre 
+Japanese 
+ICCs SAFARI ICC European 
+Astronomers 
+Figure 5.3: The European SPICA Data Centre in the context of the SPICA ground segment
+European SPICA Data Centre
+The European SPICA Data Centre will consist of two elements: The ESA SPICA Science Centre and the
+SAFARI ICC. Figure 5.3 shows the European SPICA Data Centre in the context of SPICA ground segment and
+the general approach for interactions between centres.
+The purpose of the ESA SPICA Science Centre (ESSC) is to act as interface for the European community
+to the SPICA observatory, and to represent the European users in the definition and testing of the scienceoperations systems. In particular, the ESSC will support European observatory users in proposal submission,
+observation preparation and data reduction and analysis for all SPICA instruments.
+The ESA SPICA Science Centre will also be the interface between the SAFARI ICC and the JAXA SOC
+and MOC for all SAFARI operational matters. This will involve the identification and definition of the groundsegment interfaces, both operational and for exchange of data, and the definition with the SAFARI ICC of the
+instrument telecommands and instrument telemetry contents following the JAXA packet structure (which, as
+explained above, is different from ESA’s Packet Utilisation Standard). The ESSC will support the ICC in the
+preparation of the SAFARI operational procedures following JAXA’s operational requirements. With regardto software development, the ESSC will define the S/W interfaces to the JAXA system, carry out integration
+and testing and be responsible of the Quality Assurance for all S/W SAFARI items that will be delivered
+to the MOC/SOC (observing modes, pipeline, documentation). SAFARI documents intended for the general
+observers (Observer’s Manual, Data User’s Manual) will be prepared by ESSC personnel in collaboration withthe SAFARI ICC. As an additional contribution to the observatory, the ESSC will implement the proposalsubmission system by re-using and adapting S/W implemented for the Hershel Space Observatory. Other
+possible contributions in terms of software recycling are under discussion.
+The SAFARI Instrument Control Centre (ICC) will be set under the responsibility of the SAFARI PI,
+and will start its activities early on for the definition and execution of the instrument level tests. It will be
+geographically distributed and several national agencies have expressed an interest in participating in the ICC
+activities. The precise distribution of tasks and responsibilities has not yet been defined but in any event the
+PI institute will take overall management responsibility for the ICC and act as the point of contact between the
+consortium and the ESSC. The SAFARI ICC will be responsible for the calibration and characterisation of the100 Chapter 5. Mission Operations
+instrument before and after launch, for which it will develop the required software, like e.g., the quick-look
+analysis and calibration analysis packages. The SAFARI ICC will provide the instrument command generation
+facilities, and define and validate the observing modes and the astronomical observation templates for the
+execution of the observations. It will also be responsible for the development of the software for the standard
+generation of observational products and for interactive data analysis, in collaboration with the ESSC. During
+operations, the SAFARI ICC will be in charge of the instrument on-board software maintenance, and will bethe final responsible in relation with instrument health and operational procedures.Chapter 6
+Management
+6.1 The SPICA Steering Group
+SPICA is an international cooperation project led by JAXA, which is responsible for the overall project. If
+SPICA is selected as one of the Cosmic Vision programmes, ESA will be the prime project partner, responsiblefor development of the SPICA Telescope Assembly and for the delivery of the SAFARI instrument. Figure 6.1
+shows the presently envisaged international cooperation scheme and the role of each country.
+It is planned to establish a SPICA Steering Group early in the program prior to the formal agreement by
+both space agencies. This Steering Group, which will be chaired by a JAXA representative, will consist of
+members from ESA, JAXA and potentially other national funding agencies. During the System Definition
+Phase, the content of the formal agreement between the two agencies would be drafted by the Steering Group
+(i.e., agreement on the details of the ESA deliverables and overall joint SPICA program level agreements).
+The second role of the Steering Group, which will remain in place until the end of the SPICA operations, is
+to monitor the overall technical, financial and programmatic status of the mission. It will have the authority
+to make high level trade-off decisions when required (impacting on cost, schedule, risk, science performance
+etc.) and take executive actions to deal with problems arising throughout the programme. Additionally, a JointSystems Engineering Team (JSET) will be set up with the aim to provide a forum for discussion and mutualdebriefing on the mission development, thus avoiding any ”hidden gap” in the context of a collaborative project.
+JSET is expected to be co-chaired by JAXA and ESA and to consist of members from the project teams of both
+parties. In parallel a Science Advisory Committee will be established. This Advisory Committee, which will
+be chaired by the SPICA Project Scientist, will consist of a number of scientists representing important fields
+of astronomy and will count with the participation of the ESA Project Scientist. The main role of the Science
+Advisory Committee will be to review the project status and to advice the programme on science related matters.
+6.2 ESA contribution to SPICA
+The ESA contribution to the SPICA project includes: a) the SPICA Telescope Assembly; b) provision ofground segment support; c) interface management and responsibility for the delivery to JAXA of the nationally
+developed SAFARI instrument; d) provision of the ESA SPICA Science Centre. ESA will procure the Tele-
+scope Assembly from the European industry, following a parallel competitive definition phase (A/B1) and the
+selection of a single contractor at the beginning of the implementation phase (B2/C/D). The SPICA Telescope
+Assembly includes the following elements:
+•SPICA Telescope, including optical elements (primary and secondary mirror), M2 support structure, M2refocusing mechanism with drive electronics, internal telescope baffles, adjustment shims, integration
+alignment devices.
+•Telescope Optical Bench, as required to provide adequate mounting and mechanical support to the tele-scope elements and interface to the S/C structure and the Instrument Optical Bench (IOB).
+101102 Chapter 6. Management
+Figure 6.1: SPICA international cooperation scheme.
+•Thermal control hardware (MLI, electrical heaters, harness, temperature sensors) as required to monitor
+and control the telescope temperature, including decontamination procedures.
+•Thermal interfaces to the S/C cooling chain (anchoring for thermal straps).
+•Required transport, handling and Ground Support Equipment.
+The provision of ground segment support (in the form of data relay services from ESOC) is subordinated to
+the availability of a suitable ground station during the SPICA operations. Presently it is envisaged to baseline a
+4 hr time window from Cebreros (or from the planned DS3 as an option).
+ESA will take responsibility for the delivery to JAXA of SAFARI and will play the role of official interface
+between JAXA and the national consortium developing the instrument. In order to properly conduct this role,
+ESA will actively monitor the instrument development activities and the fulfilment of all interface requirements
+agreed with JAXA. In particular ESA will be responsible for conducting the review cycle, Flight Acceptance
+of the instrument, and co-ordination of system level activities with JAXA (integration and test at spacecraft
+level, launch campaign, LEOP and in orbit commissioning). It is presently planned that the European SPICA
+Science Centre will be based at ESAC and will facilitate the interface between the European astronomers and
+the SPICA project, providing adequate support to access the observatory capabilities.
+6.3 SPICA project activities at JAXA
+JAXA approved the SPICA project entering the so called pre-project phase (phase A) on 8 July 2008, with thenomination of a study team. The approved pre-project activities include a 2 year study phase, leading to the
+System Requirements Review (SRR - Q4/2009) and to the System Definition Review (SDR - Q4/2010). The
+next JAXA approval milestone is the formal Project Approval Review, planned by mid 2011 and enabling the
+project to enter the implementation phase (B/C/D). The JAXA SPICA pre-project team is presently working
+with three companies: Sumitomo Heavy Industry (SHI), NEC and MELCO. SHI has advanced space cryogenic
+technology and is studying the Payload Module. NEC and MELCO are satellite system experts, each of whom
+is working with JAXA individually. JAXA has contracts with these two companies and will define the system
+requirement before the SRR, also using elements provided by industry. After SRR, JAXA will submit the
+Request for Proposal (RFP) to industry and select a prime contractor for the SPICA project.6.4. ESA/JAXA interfaces 103
+Figure 6.2: SPICA Management Structure
+6.4 ESA/JAXA interfaces
+A successful SPICA cooperation depends on the effective definition and management of the interfaces between
+JAXA’s led activities and the elements provided by Europe. During the assessment phase, a continuous com-
+munication flow has been maintained between the JAXA and ESA study teams, leading to the definition of
+interface requirements and to the common understanding of the performance requirements applicable to the
+telescope assembly, to the ground segment and to the SAFARI instrument. This work, in addition to the con-
+solidation of the telescope requirements document, has in fact resulted in the compilation of Interface Control
+Specifications applicable to both the STA and the SAFARI instrument. Under the assumption of SPICA en-
+tering the Definition Phase (A/B1), further progress will be made possible by the JAXA System Requirements
+Review (Q4/2009) and by the following System Definition Review (Q4/10). The assessment study has con-firmed the critical role played by an early definition of the interface requirements applicable to the telescope
+in order to deliver the flight model in a timely manner: achieving such an early definition will be one of the
+challenges ahead of the ESA and JAXA teams.
+For the definition phase an “exchange of letters” between ESA and JAXA is envisaged, to define the in-
+tended activities and to prepare the way to a formal Memorandum of Understanding (MoU), which can only
+be signed once the mission is approved for implementation. A single MoU is envisioned to cover all European
+contributions to the JAXA led mission (i.e., including the provision of the SAFARI instrument).
+The issue of space standards applicable to the SPICA project has been only marginally addressed during
+the Assessment Study and will need further discussions in the next project phases. It is presently envisaged tofollow an approach similar to the one already applied by ESA to the James Webb Space Telescope contributions,
+with each Agency applying its own standards to the respective contributions and a mutual recognition of the
+rules applied (without excluding the possibility of detailed analysis/discussions on specific and critical issues).
+6.5 SAFARI procurement and management
+In case of SPICA selection, the procurement of SAFARI will be the subject of an ESA Announcement ofOpportunity to be issued, in early 2010. The SPICA far infrared instrument, SAFARI, is a nationally funded,Principal Investigator (PI) provided instrument. The development of the instrument will be led by a consortium
+of national institutes, under the leadership of an European PI and of the Instrument Project Manager. The PI will
+be responsible for guaranteeing the scientific performance of the instrument and its delivery within the overall
+schedule and financial constraints. The SAFARI consortium is led by SRON (Space Research Organisation of
+the Netherlands). The SAFARI consortium has been studying the scientific potential and technical requirements
+for over five years and has established that the technical capability exists within the consortium to deliver the
+instrument and its constituent subsystems.104 Chapter 6. Management
+Significant contributions to the instrument development are expected to come from Austria, Belgium,
+Canada, Denmark, France, Germany, Ireland, Italy, the Netherlands, Spain, Switzerland, the UK and, pos-
+sibly, Japan. In all cases, these funding agencies are already aware of SAFARI and have been actively engaged
+in discussion during the assessment study. Each institution collaborating on SAFARI will have a nominated
+Co-I or Co-PI depending on the level of funding from a particular nation/institute to manage the work packages
+for that institution. An ESA sponsored Multi-Lateral Agreement (MLA) will be established to be signed by
+the entire consortium at funding agency level. Individual institutes will be asked to produce development plans
+and schedules for their deliverable items, in consultation and agreement between the PI, institute Co-I, SAFARI
+project manager and the institute project manager. Issues associated with resources within the consortium will
+be governed by the MLA and referred to a project steering committee attended by the PI, project manager and
+representatives of the nation funding agencies, ESA and JAXA.
+The SAFARI consortium has performed a preliminary assessment study on the instrument (see chapters 2
+and 3), provided a reference design that envelopes the required instrument resources in a “design stressing”(worst) case, and identified a road map, which shall lead to the selection of the focal plane detector technology
+and the instrument baseline design. As described in section 6.2, it is envisaged that ESA will be responsible for
+the delivery of SAFARI to JAXA and will act as official interface between the instrument consortium and the
+Japanese Agency.
+6.6 SPICA schedule and coordination with JAXA
+In case of SPICA selection, the Definition Phase (A/B1) study on the telescope is expected to start in July 2010
+for a period of 16 months, with the objective to enable the project final adoption in early 2012. It will include
+two major reviews: the Preliminary Requirements Review (PRR), to be held by the mid-term of the study, and
+the System Requirements Review (SRR), which will close the Definition Phase. The Technology DevelopmentActivities (TDA’s) will be initiated as soon as possible after the mission down-selection in February 2010.
+These activities will run in parallel with the Definition Phase and their intermediate results will be fed into the
+telescope study as necessary. Results from the TDA’s which are critical to ensuring the project feasibility or its
+development schedule are expected to be available before the decision for the mission final adoption. At the
+PRR, the mission baseline should be well established and documented. It will be critically reviewed, with the
+aim of confirming the technical and programmatic feasibility of the space segment, and more generally of the
+overall mission concept. The System Requirements Review will close the Definition Phase by consolidating
+the overall mission concept for enabling an efficient start of the Implementation Phase, should the mission be
+finally adopted.
+The coordination of JAXA’s and ESA’s schedules is subordinated to the respective project down-selection
+and approval processes. Despite of the different approval schemes, the coordination of the SPICA schedule is
+quite good, with a final go ahead from JAXA not before mid-2011 and a possible final approval from ESA byFeb 2012 (SPC). Given these two milestones and the parallel competitive approach used at ESA, the start of
+the telescope phase B2/C/D activities could not take place before Q2/2012, leading to a delivery of the flight
+model not before Q4/2016 (depending on specific options). This delivery date would allow a launch by the end
+of 2018 as from JAXA’s schedule. Further improvements to the schedule synchronisation would be enabled by
+the delivery to JAXA of intermediate telescope models, such a structural model, allowing system level tests at
+an earlier stage.
+6.7 Industrial organisation of ESA contribution
+As for all other Cosmic Vision missions, the industrial Phase A/B1 will be opened for competition in early
+2010, with the possibility of running two parallel industrial contracts. The final industrial organisation will be
+completed only in Phase B2, mostly through a process of competitive selection and according to the ESA Best
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+ICS Interface Control Specification IGM InterGalactic Medium
+IOB Instrument Optical bench ISAS Institute of Space and Astronautical Science
+ISO Infrared Space Observatory ISS International Space Station
+IWA Inner Working Angle JAXA Japan Aerospace Exploration Agency
+JEM Japanese Experiment Module JSET Joint Systems Engineering Team
+JWST James Webb Space Telescope KBO Kuiper Belt Object
+KID Kinetic Inductance Detector LEOP Launch and Early Orbit Phase
+LGA Low Gain Antenna LHB Late Heavy Bombardment
+M1, M2 . . . Mirror 1 (primary mirror), Mirror 2 (secondary
+mirror), . . .MGA Medium Gain Antenna
+MIR Mid-InfraRed MIRACLE Mid-InfRAred Camera w/o LEns110 Acronyms
+MIRHES Mid-InfraRed High resolution Echelle
+SpectrometerMIRMES Mid-InfraRed Medium resolution Echelle
+Spectrometer
+MLA Multi Lateral Agreement MLI Multi Layer Insulation
+MOC Mission Operations Centre NEP Noise Equivalent Power
+NIR Near InfraRed NLR Narrow Line Region
+OWA Outer Working Angle QM Qualification Model
+QPSK Quadrature Phase Shift Keying PACS Photodetector Array Camera and Spectrometer
+PAH Polycyclic Aromatic Hydrocarbons PDR Photon Dominated Region
+PI Principal Investigator PIAA Phase Induced Amplitude Apodization
+PLM PayLoad Module PSF Point Spread Function
+PST Point Source Transmittance PTV Peak-To-Valley
+PUS Packet utilisation Standard RD Reference Document
+RF Radio Frequency rms root mean square
+RPE Random Pointing Error Rs Symbol Rate
+RT Room Temperature RW Reaction Wheel
+S/C Spacecraft SAFARI SpicA FAR-infrared Instrument
+SCI SPICA Coronagraph Instrument SCUBA Submillimetre Common-User Bolometer Array
+SDR System Design Review SE Sun-Earth
+SED Spectral Energy Distribution SF Star Formation
+SFR Star Formation Rate SKA Square Kilometre Array
+SMBH Super Massive Black Hole SMG Sub-Millimetre Galaxy
+SMILES Super conducting sub-mm Limb Emission Sounder SNe Super Novae
+SOC Science Operations Centre SPC Science Programme Committee
+SPICA SPace Infrared telescope for Cosmology and
+AstrophysicsSPIRE Spectral and Photometric Imaging REceiver
+SRON Space Research Organisation of the Netherlands SRR System Requirements Review
+SSO Solar System Object STA SPICA Telescope Assembly
+STSST SPICA Telescope Science Study Team SVM Service Module
+TAC Time Allocation Committee TAS Thales Alenia Space
+TBC To Be Confirmed TBD To Be Defined
+TC Telecommand TDA Technology Development Activities
+TES Transition Edge Sensors TID Total Ionising Dose
+TIRCS Thermal Insulation and Radiative Cooling System TIS Total Integrated Scatter
+TM Telemetry TOB Telescope Optical Bench
+ToO Target of Opportunity TT&C Tracking Telemetry and Command
+ULIRG UltraLuminous InfraRed Galaxy UV UltraViolet
+WFE Wave Front Error XDR X-ray Dominated Region
\ No newline at end of file
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+arXiv:1001.0710v1  [cond-mat.str-el]  5 Jan 2010Magnetic frustration in an iron based Cairo pentagonal latt ice
+E. Ressouche,1,∗V. Simonet,2B. Canals,2M. Gospodinov,3and V. Skumryev4
+1CEA/Grenoble, INAC/SPSMS-MDN, 17 rue des Martyrs, 38054 Gr enoble Cedex 9, France
+2Institut N´ eel, CNRS & Universit´ e Joseph Fourier, BP166, 3 8042 Grenoble Cedex 9, France
+3Institute of Solid State Physics, Bulgarian Academy of Scie nces, 1184 Sofia, Bulgaria
+4Instituci´ o Catalana de Recerca i Estudis Avanats (ICREA) a nd Departament de F´ ısica,
+Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Spa in
+(Dated: October 30, 2018)
+The Fe3+lattice intheBi 2Fe4O9compoundis foundtomaterialize the firstanalogue ofamagne tic
+pentagonal lattice. Due to its odd number of bonds per elemen tal brick, this lattice, subject to first
+neighbor antiferromagnetic interactions, is prone to geom etric frustration. The Bi 2Fe4O9magnetic
+properties have been investigated by macroscopic magnetic measurements and neutron diffraction.
+The observed non-collinear magnetic arrangement is relate d to the one stabilized on a perfect tiling
+as obtained from a mean field analysis with direct space magne tic configurations calculations. The
+peculiarity of this structure arises from the complex conne ctivity of the pentagonal lattice, a novel
+feature compared to the well-known case of triangle-based l attices.
+The pentagon, a 5-edges polygon, is an old issue in
+mathematical recreation. It forms the faces of the do-
+decahedron, one of the platonic solids whose shape is re-
+produced in biological viruses and in some metallic clus-
+ters [1]. The main peculiarity of this polygon is that,
+contrary to triangles, squares or hexagons, it is impossi-
+ble to tile a plane with congruent regular pentagons, the
+tilings must involve additional shapes to fill the gaps [2],
+like in the Penrose lattice (Fig. 1). It exists however
+several possibilities of tessellation of a plane with non-
+regular pentagons, a famous one being the Cairo tessel-
+lation whose name was given because it appears in the
+streets of Cairo and in many Islamic decorations(Fig. 1).
+FIG. 1: (color online) Pentagonal lattices. (a) Cairo pen-
+tagonal tiling, (b) Penrose pentagonal tiling, (c) pentago nal
+lattice studied in refs. 6–8, (d) dodecahedron.
+Such a lattice could attract interest in the field of ge-
+ometric frustrated magnetism. Indeed, frustration usu-
+ally arises when all pairs of magnetic interactions are
+not simultaneously satisfied in a system due to the lat-tice topology, resulting in unusual physical properties [3].
+Today, most of the studied systems are mainly based on
+triangles (or tetrahedra in three dimensions), and can
+be made perfect from both structural (regular polygons)
+andmagnetic(equalfirstneighbormagneticinteractions)
+points of view. This is not the case with pentagonal lat-
+tices that must involve only non-regular pentagons to fill
+the space. The experimental realization of such a model
+system is thus a unique opportunity that could bring
+new interesting features in this field of magnetic frus-
+tration. In this letter we show that the Fe3+lattice in
+Bi2Fe4O9materializes the first analogue of a magnetic
+pentagonal lattice, and we interpret its peculiar non-
+collinear magnetic arrangement with respect to magnetic
+frustration.
+Bi2Fe4O9is a common by-product in the synthesis of
+themultiferroiccompoundBiFeO 3, andhasbeenclaimed
+recently to display itself multiferroic properties [4]. This
+was the initial motivation of our study: this finding re-
+quires a precise determination of the symmetries in the
+magnetic ordered phase, since ferroelectricity can only
+arise in polar space groups. Such a precise determination
+hasnever been carriedout on single crystalsofBi 2Fe4O9,
+and only powder neutron diffraction measurements have
+been reported [5] with no definitive conclusions due to
+the complexity of this orthorhombic structure. Actu-
+ally, there are two different sites of four iron atoms in
+Bi2Fe4O9: Fe1occupies a tetrahedral position and Fe 2
+an octahedral one. In the structure, columns of edge-
+sharing Fe 2octahedra form chains along the caxis, and
+these chains are linked together by corner-sharing Fe 1
+tetrahedra and Bi atoms (Fig. 2). As a result, the lattice
+formed by the different Fe3+magnetic atoms is quite re-
+markable. From a simple examination of the structure,
+five main magnetic superexchange interactions, J1toJ5,
+can be identified (Fig. 2). The structure can be simply
+viewed as layers perpendicular to the caxis: the mag-
+netic coupling of these layers along the cdirection in-2
+FIG. 2: (color online) The Bi 2Fe4O9compound (space group Pbam; a = 7.965(4), b = 8.448(5), c = 6.007(3) ˚A at 300 K).
+(left): View of the unit cell with the environment of the two F e ions. Bi atoms are purple, Fe yellow and O red. Projection of
+the structure on the ( a,c) plane (middle) and on the ( a,b) plane (right). The atoms labels are in blue. Black lines mat erialize
+the magnetic exchange interactions.
+volves only Fe 2atoms in the octahedron columns and is
+achieved through two interactions, J1between the atoms
+within the unit cell and J2between the atoms ofadjacent
+cells. Within a layer, each Fe 1interacts with its nearest
+neighbour Fe 1viaJ4and with two nearest neighbour
+Fe2pair (located above and below the mean plane) via
+J3andJ5. The projection of this layer along cforms
+a pentagonal lattice with three slightly different bonds
+per pentagon (Fig. 2). This geometry is equivalent to a
+distorted Cairo lattice, the perfect one involving convex
+equilateral pentagons with equal-length sides, but with
+different associated angles (Fig. 1).
+What should be expected from such a geometry and
+what is actually observed ? Up to now, no experimental
+studies and veryfew theoreticalcalculationsare available
+in the literature [6–8]. They only concern a very particu-
+lar pentagonal tiling, derived from the hexagonal lattice
+(see lattice cof Fig.1). Calculations within the antiferro-
+magnetic Ising model on this lattice yields a disordered
+ground state with a finite entropy per spin [6, 8], whereas
+in the Heisenberg model the classical ground state of the
+perfect pentagonal lattice has not yet been solved [7].
+Single crystals of Bi 2Fe4O9were grown by the high
+temperature solution growth method using a flux of
+Bi2O3[9]. The temperature and magnetic field depen-
+dences ofthe magnetization alongthe principal crystallo-
+graphic axes were measured using a SQUID magnetome-
+ter (Quantum Design) in the temperature range 2-380 K
+and in fields up to 5.5 T. The reported data, measured
+on the very same crystal used for the neutron diffraction
+study (2.5 x 2 x 1.5 mm3), were corrected for the demag-
+netizingfieldeffect. TheDCsusceptibilitybetween300K
+and 800 K was measured on a home made Faraday-type
+magneticbalanceinamagneticfieldof0.7T,usinganas-
+sembly of several randomly oriented single crystals. The
+reported susceptibility was corrected from diamagnetism
+using the Pascal’s constants χdia= -198 10−6emu/mol.
+Above room temperature, the magnetic susceptibility ofBi2Fe4O9is found to obey a Curie-Weiss law with an ex-
+trapolated paramagnetic temperature θp≈- 1670 K and
+an effective magnetic moment µeff= 6.3(3) µBper iron
+atom (see Fig. 3). This later value is compatible with
+the 5.9µBexpected for Fe3+ions (J=S=5/2). When
+decreasing the temperature, an evident drop in the ( a,b)
+planesusceptibility and a wellpronouncedincreaseofthe
+caxis susceptibility are observed at T N= 238(2) K, in-
+dicating a transition towarda magnetic long rangeorder.
+This N´ eel temperature is in agreement with the results
+on single crystal reported in ref. 10 but some 20 K lower
+that the one reported in refs. 4, 5, 11 for sintered sam-
+ples. The observed ratio between the N´ eel temperature
+and the paramagnetic one, θp/TN≈7, suggests, as ex-
+pected from the geometry, a large degree of frustration.
+Further insights concerning the magnetic arrangement
+below T Nwere obtained using neutron diffraction. The
+neutron diffraction experiments were performed on the
+twoCEA-CRG diffractometers D23 and D15 at the Insti-
+tut Laue-Langevin (Grenoble, France) using wavelengths
+of respectively λ= 1.280˚A and 1.173˚A . Complete data
+collections have been made at two different temperatures
+T= 15 K and T= 300 K. The experimental data were
+corrected from absorption using the CCSL library [12].
+The structural and magnetic arrangements were refined
+using the least-square programs MXD [13] and FULL-
+PROF [14], including extinction corrections [15].
+When decreasing the temperature, new reflections,
+characteristic of an antiferromagnetic order, appear
+around T N= 244(2) K. They can be indexed with a
+propagationvector k= (1/2,1/2,1/2),that is adoubling
+of the magnetic cell compared to the nuclear one in the
+three directions. The temperature variation of a mag-
+netic Bragg peak is presented in Fig. 3. The magnetic
+structure was solved from the 654 magnetic Bragg peaks
+collected at T= 15 K taking into account the presence
+of two magnetic domains [9]. There are four Fe 1atoms
+and four Fe 2in the unit cell, whose coordinates are re-3
+FIG. 3: (color online) Linear magnetic susceptibility of
+Bi2Fe4O9as a function of temperature measured in magnetic
+fields of 0.1 T and 0.7 T below and above 300 K respectively.
+Inset: Temperature variation ofthe (1 /2,3/2,−1/2) magnetic
+Bragg peak in zero field from neutron diffraction.
+(Bottom) Graphical representation of the magnetic structu re
+refinement at T = 15 K in Bi 2Fe4O9: observed versus calcu-
+lated intensities.
+ported in Table I. The best refinement led to a residual
+factorRw= 6.4%, with twonearlyequallypopulated do-
+mains (42/58%). The magnetic moment parameters for
+the first domain arereported in Table I. The correspond-
+ing pictures of the magnetic arrangements are presented
+in Fig. 4. The moments on all the atoms are found re-
+stricted in the ( a,b) plane. On each site, the four atoms
+are gathered into two pairs, antiferromagnetically cou-
+pled in the case of Fe 1and ferromagnetically coupled for
+Fe2. The magnetic moments of each pair are oriented
+at 90◦, that is perpendicular to the moments of the
+other pair on the same site. The phase between the two
+sublattices Fe 1and Fe 2is 155◦. The moment amplitudes
+on the two sites Fe 1and Fe 2are slightly different, and
+both much smaller than the expected value for a Fe3+
+ion (5µB). A spin transfer process from the iron to the
+neighbouring oxygen ions [16] could explain this reduc-
+tion.x y z µ θ φ
+Fe110.3540 0.3361 0.5 3.52(1) 100.1(1) 90
+Fe120.6460 0.6639 0.5 3.52(1) θ1
+1+ 180 90
+Fe130.1460 0.8361 0.5 3.52(1) θ1
+1- 90 90
+Fe140.8540 0.1639 0.5 3.52(1) θ1
+1+ 90 90
+Fe210 0.5 0.2582 3.73(1) -105.4(1) 90
+Fe220 0.5 0.7418 3.73(1) θ1
+290
+Fe230.5 0 0.7418 3.73(1) θ1
+2+90 90
+Fe240.5 0 0.2582 3.73(1) θ1
+2+ 90 90
+TABLE I: Coordinates of the Fe3+ions in Bi 2Fe4O9and
+spherical components of their magnetic moments in the first
+magnetic domain. The magnetic moments are defined in
+spherical coordinates by ( µ,θ,φ). In cartesian coordinates
+(X,Y,Z)whereXis along a, Yalong bandZalong c, themag-
+netic moment components are : MX=µ cosθ sinφ,M Y=
+µ sinθ sinφ,M Z=µ cosφ.
+As it can be seen on Fig. 4, this non-collinear mag-
+netic structureformsimbricatedrectanglesofspins. This
+is quite unusual in such an orthorhombic system, where
+all the directions are non-equivalent and where there is
+no fourfold axis. This peculiarity certainly has an ori-
+gin in the frustration of the magnetic interactions. One
+should also notice that for this structure the magnetic
+space group is centrosymmetric, that is non polar. This
+result isa priorinot compatible with ferroelectricity, and
+this shed some doubts on the claimed ferroelectricity ob-
+served in polycrystalline Bi 2Fe4O9[4].
+To check the relevance of the Cairo pentagon lattice
+model and the role of geometric frustration on the ob-
+served magnetic structure of Bi 2Fe4O9, we have under-
+taken a two step theoretical approach based on a lo-
+calized spin model. Calculations have been made both
+for an ideal Cairo lattice (including the perfect case
+J3=J4=J5), and in the real geometry. The angles
+between magnetic moments in a system is imposed by
+the set of magnetic interactions. In a first step, an exact
+Fourier space analysis at zero temperature is performed,
+assumingasingle-klike ordering,which allowsto find the
+periodicity of the magnetic structure [17]. Once the pe-
+riodicity is known, an energy minimization in real space
+gives the respective orientation of the magnetic moments
+within the magnetic unit cell. For these calculations, the
+five exchange interaction parameters J1toJ5already de-
+scribed have been considered, using a Heisenberg Hamil-
+tonianH=−1
+2/summationtext
+i,jJk/vectorSi·/vectorSj. All the moments were
+assumed to be of equal amplitudes.
+In the Cairo pentagonal lattice, a large set of Jivalues
+yield the observed (1/2, 1/2, 1/2) propagation vector.
+The doubling of the magnetic cell along calone requires
+however J2to be negative and J1positive or slightly neg-
+ative. Whatever the k= (1/2,1/2,1/2) solution found,4
+FIG. 4: (color online) Magnetic structure. First and second
+magnetic domains. (Yellow): Fe 1sites. (Orange): Fe 2sites.
+The second domain is deduced from the first one by apply-
+ing the symmetry operation ( x+1/2,¯y+ 1/2,¯z) lost by the
+magnetic structure, taking into account the translational part
+that shifts atoms from one cell to a neighbouring one.
+the magnetic configuration is always made of antiparal-
+lel Fe1pairs of moments and parallel Fe 2ones. On a
+same site, the pairs of moments are perpendicular to
+each other. This is exactly what is observed experimen-
+tally. The only parameter that could vary as a function
+of the particular values of the Ji’s is the phase angle be-
+tween the two sites Fe 1and Fe 2. The∼155◦observed
+value requires J3,J4,J5to be all negative, with a ratio
+J3/J5=2.15for |J4|≥ |J3|. Theobservedmagneticstruc-
+ture of Bi 2Fe4O9and in particular the 90◦phase angle
+between pairs of atoms at the same site, is thus recovered
+by this model with some realistic set of the five super-
+exchange interactions Ji. In the Bi 2Fe4O9real geometry,
+the Fe3+lattice differs from the Cairo pentagonal one by
+the presence of two stacked Fe 2atoms, instead of a sin-
+gle atom, at two vertices in each pentagon. Performing
+the calculations in the real geometry yield very similar
+results than those of the ideal case. The perpendicular
+ferromagnetic and antiferromagnetic Fe 1and Fe 2pairs is
+a robust recovered feature. The only difference between
+both lattices is the domain of existence of this phase that
+extends to lower |J3|and|J5|with respect to |J4|in thereal case. The Bi 2Fe4O9real lattice is therefore topolog-
+ically equivalent to the Cairo one after renormalization
+of theJ3andJ5interactions with respect to the J4one.
+To summarize, we have found that the compound
+Bi2Fe4O9displays a very original non-collinear magnetic
+structure, made of four Fe 1moments and four Fe 2ones
+forming interpenetrating patterns of four-fold spin rota-
+tions. This magnetic arrangement is very peculiar for
+a pentagonal lattice with nearest neighbour interactions
+only and is not a mere propagation of the magnetic ar-
+rangement minimizing the energy in each individual pen-
+tagon, contrary to the case of triangles based lattices. It
+actually results from two ingredients, a high geometrical
+frustration and an additional complex connectivity (two
+kinds of sites with different coordinations). This first
+materialisation of a pentagonal Cairo lattice opens new
+perspectives in the field of magnetic frustration.
+Work at Sofia was supported by the Bulgarian Science
+Foundation under Grants No. TK-X-1712/2007. The
+authors are grateful to Michael Mikhov for the high tem-
+peraturesusceptibility resultsandtoLucValentin forthe
+help with the tilings.
+∗ressouche@ill.fr
+[1] X.J. Kong, Y.P.Ren, L.S.Long, Z.Zheng, R.B.Huang,
+and L. S. Zheng, J. Am. Chem. Soc. 129, 7016 (2007).
+[2] J. Kepler, Harmonices Mundi (J. Planck for G. Tampach,
+Linz, Austria, 1619).
+[3] R. Moessner, A. P. Ramirez, Physics Today, 59 (2), 24
+(2006).
+[4] A. K. Singh, S. D. Kaushik, B. Kumar, P. K. Mishra, A.
+Venimadhav, V. Siruguri, S. Patnaik, Appl. Phys. Lett.,
+92, 132910 (2008).
+[5] N. Shamir, E. Gurewitz, Acta Crystallogr. Sect. A 34,
+662 (1978).
+[6] M. H. Waldor, W. F. Wolff, J. Zittartz, Phys. Lett.
+106A, 261 (1984); Z. Phys. B: Condens. Matter 59, 43
+(1985).
+[7] U. Bhaumik, I. Bose, Phys. Rev. B 58, 73 (1998).
+[8] R. Moessner, S. L. Sondhi, Phys. Rev. B 63, 224401
+(2001).
+[9] E. Ressouche et al., in preparation.
+[10] D. M. Gianchinta, G. C. Papaefthymiou, W. M. Davis,
+H. C. Zur Loye, J. of Solid State Chem. 99, 120 (1992).
+[11] A. G. Tutov, I. E. Mylnikova, N. N. Parfenova, V. A.
+Bokov, S. A. Kizhaev, Sov. Phys. Solid State 6, 963
+(1964).
+[12] J. Brown, J. Matthewman, Cambridge Crystallography
+Subroutine Library , Report No. RAL93-009 (1993).
+[13] P. Wolfers, J. Appl. Crystallogr. 23, 554 (1990).
+[14] J. Rodriguez-Carvajal, Physica B 192, 55 (1993).
+[15] P.J. Becker, P. Coppens, Acta Crystallogr. Sect. A 30,
+129 (1974).
+[16] V. C. Rackhecha, N. S. Sayta Murthy, J. Phys. C.:Solid
+State Phys. 11, 4389 (1978).
+[17] E. F. Bertaut, J. Phys. Chem. Solids 21, 256 (1961).
\ No newline at end of file
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+arXiv:1001.0712v1  [cond-mat.mtrl-sci]  5 Jan 2010Indication of antiferromagnetic interaction between para magnetic
+Co ions in the diluted magnetic semiconductor Zn 1−xCoxO
+M. Kobayashi,1,∗Y. Ishida,1,†J. I. Hwang,1Y. Osafune,1A. Fujimori,1Y. Takeda,2
+T. Okane,2Y. Saitoh,2K. Kobayashi,3H. Saeki,4T. Kawai,4and H. Tabata5,6
+1Department of Physics, University of Tokyo,
+7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
+2Synchrotron Radiation Research Unit,
+Japan Atomic Energy Agency, Sayo-gun, Hyogo 679-5148, Japa n
+3National Institute for Materials Science (NIMS), SPring-8 ,
+1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5148, Japan
+4Institute of Scientific and Industrial Research,
+Osaka University, Ibaraki, Osaka 567-0047, Japan
+5Department of Electronic Engineering,
+School of Engineering, University of Tokyo,
+7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
+6Department of Bioengineering, School of Engineering,
+University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-865 6, Japan
+(Dated: November 16, 2018)
+1Abstract
+The magnetic properties of Zn 1−xCoxO (x= 0.07 and 0.10) thin films, which were homo-
+epitaxially grown on a ZnO(0001) substrates with varying re latively high oxygen pressure, have
+been investigated using x-ray magnetic circular dichroism (XMCD) at Co 2 pcore-level absorption
+edge. The line shapes of the absorption spectra are the same i n all the films and indicate that
+the Co2+ions substitute for the Zn sites. The magnetic-field and temp erature dependences of
+the XMCD intensity are consistent with the magnetization me asurements, indicating that except
+for Co there are no additional sources for the magnetic momen t, and demonstrate the coexistence
+of paramagnetic and ferromagnetic components in the homo-e pitaxial Zn 1−xCoxO thin films, in
+contrast to the ferromagnetism in the hetero-epitaxial Zn 1−xCoxO films studied previously. The
+analysis of the XMCD intensities using the Curie-Weiss law r eveals the presence of antiferromag-
+netic interaction between the paramagnetic Co ions. Missin g XMCD intensities and magnetization
+signals indicate that most of Co ions are non-magnetic proba bly because they are strongly coupled
+antiferromagnetically with each other. Annealing in a high vacuum reduces both the paramagnetic
+and ferromagnetic signals. We attribute the reductions to t hermal diffusion and aggregation of Co
+ions with antiferromagnetic nanoclusters in Zn 1−xCoxO.
+PACS numbers: 75.50.Pp, 75.30.Hx, 78.70.Dm
+2INTRODUCTION
+Ferromagnetic diluted magnetic semiconductors (DMS’s) are key ma terials for future
+“spin electronics” or “spintronics”, in which one manipulates both th e spin and charge de-
+grees offreedom of electrons, because interaction between the spins of host sp-bandelectrons
+and the magnetic moments of doped magnetic ions enables us to cont rol the spin degrees
+of freedom of electrons [1, 2]. In fact, the ferromagnetism of the III-V DMS’s Ga 1−xMnxAs
+and In 1−xMnxAs arises from carrier-mediated interaction between Mn spins [3], an d new
+functional devices utilizing both magnetism and electrical conductiv ity have been fabricated
+based on these materials, e.g., spin-dependent circular light-emittin g diodes [4] and field-
+effect transistors controlling ferromagnetism [5]. New control met hods of magnetization
+have also been demonstrated, such as electrical manipulation of ma gnetization reversal [6]
+and current-induced domain-wall switching [7]. For practical applica tions of such devices,
+ferromagnetic DMS’s having Curie temperatures ( TC’s) above room temperature are nec-
+essary. Recently, oxide-based DMS’s have attracted much atten tion first motivated by the
+theoretical predictions of high TC’s in wide-gap semiconductors [8, 9] followed by various
+reports of ferromagnetism at room temperature [10].
+Ever since the discovery of ferromagnetism in Zn 1−xCoxO thin films [11], ZnO-based
+DMS’s have been investigated intensively because the host material ZnO has potential for
+optical applications due to the wide band gap of 3 .4 eV and the large exciton energy of 60
+meV. In particular, Zn 1−xCoxO is one of the most investigated DMS’s, and there have been
+many reports about the magnetic properties of Zn 1−xCoxO. Schwartz and Gamelin [12] have
+reported reversible switching of room-temperature ferromagne tism in Zn 1−xCoxO by intro-
+ducing and removing interstitial Zn defects. Reproducibly, exposu re of Zn 1−xCoxO thin film
+to Zn vapor creates interstitial Zn defects and induces room-tem perature ferromagnetism,
+and annealing the thin film in air quenches the ferromagnetism due to e limination of the in-
+terstitial Zn by oxidation. Neal et al. [13] have observed magnetic circular dichroism (MCD)
+signals with open hysteresis loop at the ZnO band edge of Co-doped Z nO thin films at room
+temperature, implying a splitting of the conduction band of ZnO thro ugh interaction with
+the magnetic moments of the Co ions. These observations rule out C o-cluster segregations
+or secondary phase formation as the origin of the ferromagnetism . In addition, there are
+some reports about the observation of anomalous Hall effects in Zn 1−xCoxO [14, 15], provid-
+3ing experimental evidence for carrier-mediated ferromagnetism in Zn1−xCoxO. Because the
+preparation of Zn 1−xCoxO thin films has not been highly reproducible yet and secondary
+phases, defects, and carriers likely affect the magnetic behavior o f Zn1−xCoxO, systematic
+measurements on well-defined Zn 1−xCoxO thin films are necessary to understand the origin
+of the ferromagnetism in Zn 1−xCoxO.
+Recently, Matsui and Tabata [16] have succeeded in homo-epitaxia l growth of Co-doped
+ZnO on a ZnO substrate. The homo-epitaxial growth has enabled us to obtain more re-
+producible high-quality Zn 1−xCoxO thin films, providing correlation between the growth
+parameters and the physical properties. The magnetization meas urements of the homo-
+epitaxial Zn 1−xCoxO thin films indicated that the spontaneous magnetization ( Ms) depends
+on electron-carrier concentration ( n) controlled by oxygen pressure during deposition. In
+the Zn 1−xCoxO thin films, Msincreases with increasing n, consistent with the result of
+anomalous Hall effect measurements [14, 15].
+Since the ferromagnetism in Zn 1−xCoxO has been debated between researchers using dif-
+ferent techniques, element-specific magnetization measurement s provide useful information
+in order to address the magnetic properties of Zn 1−xCoxO. X-ray magnetic circular dichro-
+ism (XMCD) is one of the most powerful tools to investigate the magn etic properties of
+materials. XMCD is defined as the difference between absorption spe ctra for different circu-
+lar polarized x rays. In the previous XMCD measurements on a heter o-epitaxial Zn 1−xCoxO
+thin film grown on Al 2O3(0001) substrates [17], we have concluded that the ferromagnet ism
+comes from Co ions substituting the Zn site in ZnO, and proposed tha t non-ferromagnetic
+Co ions are strongly coupled antiferromagnetically with each other f romthe linear magnetic-
+field (H) dependence and temperature ( T) independence of XMCD. Sati et al. [18] have
+reported antiferromagnetic interaction in single crystalline Zn 1−xCoxO thin films with low
+Co concentration ( x= 0.003−0.005) from magnetic and EPR measurements. On the other
+hand, there are some reports that the Co 3 delectrons are not the origins of the ferromag-
+netism [19] or that metallic Co atoms contribute to the ferromagnet ism [20]. Since the
+magnetic behavior of Zn 1−xCoxO may be affected by extrinsic factors such as secondary
+phases and defects, systematic XMCD measurements on well-defin ed Zn1−xCoxO thin films
+will provide us with useful insight into the magnetism.
+Recently, annealing effects on the ferromagnetism in Zn 1−xCoxO have been discussed
+in relation to oxygen vacancies and/or Zn interstitials [21, 22]. Annea ling of Zn 1−xCoxO
+4in reducing atmosphere such as in high vacuum, which may produce ox ygen vacancies,
+have been reported to enhance the ferromagnetism, while annealin g in oxidizing atmosphere
+such as air decreases the magnetic moments of Zn 1−xCoxO [23–26]. In addition, we note
+that ferromagnetism has been reported in some oxide thin films witho ut magnetic ions
+[27–29], probably due to defects such as oxygen vacancies. It is th erefore important to
+investigate annealing effects and/or the role of oxygen vacancies f or the understanding of
+the ferromagnetism in Zn 1−xCoxO.
+In the present work, we report on the results of H- andT-dependent XMCD measure-
+ments on homo-epitaxial Zn 1−xCoxO thin films which show coexisting paramagnetic and
+ferromagnetic properties. Analysis using the Curie-Weiss law sugge sts that antiferromag-
+netic correlation between the Co ions suppress the paramagnetic b ehavior. We have also
+found that the XMCD intensity decreases by high-vacuum annealing . Based on the exper-
+imental findings, a possible origin of the ferromagnetism in the Zn 1−xCoxO thin films shall
+be discussed.
+EXPERIMENTAL
+Zn1−xCoxO thin films were homo-epitaxially grown on Zn-terminated ZnO(0001) sub-
+strates by the pulsed laser deposition technique with varying oxyge n pressure ( PO2) from
+1×10−4to 10−2Pa. Four thin films were prepared: a x=0.07 film deposited at PO2=10−4
+Pa, and x=0.10 films deposited at PO2=10−2, 10−3, and 10−4Pa. During the deposition,
+the substrate temperature was kept at 400◦C. The thickness of each film was 50 −150 nm.
+X-ray diffraction confirmed that the thin films had the wurtzite stru cture and no secondary
+phase was observed. The magnetization of those thin films was meas ured using a supercon-
+ducting quantum interference device (SQUID) magnetometer (Qu antum Design, Co. Ltd).
+The present homo-epitaxial films are expected to have higher crys tal quality [16] than the
+hetero-epitaxial one studied in the previous XMCD measurements [1 7]. However, the homo-
+epitaxial thin films have lower Msthan the hetero-epitaxial ones because of the high PO2
+values [16].
+X-ray absorption spectroscopy (XAS) and XMCD measurements w ere performed at the
+helicalundulatorbeamlineBL23SUofSPring-8[30,31]. Themonochro matorresolutionwas
+E/∆E>10,000. Absorption spectra of right-handed ( µ+) and left-handed ( µ−) circularly
+5polarized x rays were obtained by reversing photon helicity at each p hoton energy in the
+total-electron-yield mode. The µ+andµ−spectra were averaged between spectra for the
+positive and negative applied magnetic fields so as to eliminate spurious signals. The degree
+of circular polarization of x-rays was over 90 %. External magnetic fields were applied
+perpendicular to the sample surface. Circularly-polarized x-ray ab sorption spectra under
+each experimental conditions have been normalized to the maximum h eight of the Co 2 p
+XAS [(µ++µ−)/2] spectrum. Backgrounds of the XAS spectra at the Co 2 pedge were
+assumed to be hyperbolic tangent functions. Annealing of the PO2=10−2Pa sample was
+performed at 700◦C and∼10−7Pa for 5 min. For surface cleaning, Ar+ion sputtering and
+subsequent 250◦C annealing (well below the deposition temperatures) were perform ed in a
+high vacuum of 10−7Pa.
+RESULTS AND DISCUSSION
+Figure 1 shows the magnetic-field ( H) and temperature ( T) dependences of the mag-
+netization ( M) measured using a SQUID magnetometer. Although the raw magnet ization
+data decreases with Hdue to the diamagnetic contribution from the substrate as shown in
+the inset of Fig. 1(a), there are small but finite ferromagnetic sign als. By subtracting the
+linear component at high magnetic fields from the raw data, the ferr omagnetic hysteresis
+loops have been extracted at 10 and 300 K, as shown in the main pane l of Fig. 1(a). The
+hysteresis loop at 300 K implies that TCof the ferromagneitc component is above room
+temperature. Here, the coercive fields are found to be 100-200 O e. TheMsof the films was
+of the order of 10−2µB/Co, indicating that less than 1 % of the Co ions participate in the
+ferromagnetism if the Co ions is in the high spin Co2+state. The Msslightly decreases with
+increasing temperature. The Ms’s of the present homo-epitaxial films are as small as 5 −10
+% of that of the previous hetero-epitaxial one because of the high PO2, although the present
+Zn1−xCoxO thin film has higher crystal quality than the previous one. Assuming that the
+diamagneticandferromagneticresponseshardlychangewithtemp erature, theparamagnetic
+susceptibilities have been obtained by subtracting the slopes of the M-Hcurves taken at
+different T’s, as shown in Fig. 1(b). One can see that the paramagnetic signals in crease with
+decreasing Tin the low Tregion. The observations thus suggest the presence of both the
+paramagnetic and ferromagnetic components in the present Zn 1−xCoxO thin films.
+6Figure 2 shows the Co 2 pXAS and XMCD spectra of the samples measured at H= 7.0
+T andT= 20 K. The XAS and XMCD spectra have been normalized to the positiv e and
+negative peak heights, respectively. We have observed clear XMCD signals for all the thin
+films. As shown in Fig. 2(a), the line shapes of the XAS and XMCD spect ra are identical
+between these films, suggesting that the doped Co ions have almost the same electronic
+structure. The XAS and XMCD spectra are compared with these of the previous report on
+the hetero-epitaxial Zn 1−xCoxO thin film and the cluster-model calculation for the Co2+ion
+under tetrahedral crystal field [17], as shown in Fig. 2(b). The line s hapes of the XAS and
+XMCD spectra well agree with these of the previous ones and are sim ilar to those of the
+calculation for the Co2+ion tetrahedrally coordinated by oxygen atoms, indicating that the
+Co ions substitute for the Zn sites.
+In order to understand the magnetic properties of the present Z n1−xCoxO thin films,
+we have studied the HandTdependences of the XMCD spectra. Figure 3 shows the H
+dependence of the XMCD spectra of the Zn 0.9Co0.1O thin film deposited at PO2=10−3Pa.
+The line shape of the XMCD spectrum is unchanged with the strength of applied magnetic
+field, as shown in the inset of Fig. 3(a). The total magnetization Mtot=Mspin+Morb
+estimated from the XMCD sum rules of these thin films are plotted as a function of Hin
+Fig. 3(b). Here, MspinandMorbare spin and orbital magnetic moments, respectively. In all
+the films, Mtotlinearly increases with H, andMsobtained by extrapolating MtottoH=
+0 (MXMCD
+s) is very small, quaritatively consistent with the magnetization measu rements.
+The observations suggest that in the present Zn 1−xCoxO films the paramagnetic signals are
+predominant comparedwiththeferromagneticonesunlike theprev ioussample[17]. Figure4
+shows the Tdependence of the XMCD spectra of the Zn 0.9Co0.1O thin film deposited at
+PO2=10−3Pa. The line shapes of the XMCD spectra were independent of tempe rature
+although the XMCD intensity decreased with increasing T. TheMXMCD
+s’s of all the samples
+as a function of Tare summarized in Fig. 4(b).
+Althoughthe XMCD intensities were different between these films, all the XMCD spectra
+ofthethinfilmsshowedsimilar HandTdependences. Accordingtotherelationshipbetween
+theMsandnof the homo-epitaxial Zn 1−xCoxO thin films [16], the effective n’s of all the
+samples studied here are estimated to be less than 1018cm−3. The observations that the
+XMCD intensity increases and decreases with increasing HandT, respectively, indicate a
+Curie-Weiss paramagnetic behavior, in spite of almost the same elect ronic structure of the
+7Co ions as that of the ferromagnetic hetero-epitaxial Zn 1−xCoxO thin film [see Fig. 2(b)],
+whichdidnotshowaCurie-Weissparamagneticcomponent. Theresu ltsimplythatalthough
+the Co ions substitute for the Zn sites, the Co ions become paramag netic or ferromagnetic
+in Zn1−xCoxO depending on nand the crystallinity. Considering that Co 2 pXMCD reflects
+only the local electronic states of the Co ion, it is likely that the differe nt magnetic responses
+of the Zn 1−xCoxO films are originated from different nearest and next-nearest neig hbor
+interaction with neighboring cations, defects, and/or carrier con centration. We shall discuss
+about the differences in the magnetic properties below.
+LetusquantitativelydiscusstheparamagneticbehavioroftheCoio nsbyapplyingXMCD
+sum rules to the spectra [32, 33]. The HandTdependences of Mtothave been fitted to the
+formula
+/angbracketleftMtot/angbracketright=CH
+T−θ+Ms, (1)
+whereCis the Curie constant and θis the Weiss temperature. The second term Msin
+Eq. (1) is contributions from the ferromagnetic component. Resu lts of the fit are shown in
+Figs.3(b)and4(b), indicating thatEq.(1)well reproduces botht heHandTdependences of
+Mtot. The fitted parameters are listed in Table I. It should be noted that the obtained Ms’s
+well agree with those deduced from the magnetization measuremen ts as shown in Table I,
+providing experimental evidence that we have probed the intrinsic b ulk magnetic properties
+of the Zn 1−xCoxO thin films. In addition, the agreement suggests that magnetic mom ent of
+the substituted Co ions is predominant source of the ferromagnet ic responses of the samples
+expected to be in the low nregion. The Cvalues estimated from the Curie-Weiss fit are
+close to that of the free Co2+ion with quenched orbital moment of ∼3.36µBK/T, except
+for thePO2=10−2Pa thin film. The Weiss temperature θof all the samples are negative,
+consistent with the report of antiferromagnetic interaction in par amagnetic Zn 1−xCoxO thin
+films [18].
+In order to see the effects of vacuum annealing on the electronic an d magnetic properties
+of the Co ion, we have measured the Co 2 pXMCD spectra of the PO2=10−2Pa film after
+vacuum annealing, too. Figure 5(a) shows the Co 2 p3/2XAS and XMCD spectra of the
+PO2=10−2Pa film before and after the annealing. The line shapes of the XAS and XMCD
+spectra are almost identical before and after the annealing, indica ting that the local elec-
+tronic structure of the Co ion remained unchanged by the annealing . However, the XMCD
+intensity of the annealed sample becomes about half of the value of t he as-grown one. The
+8Msof the annealed film has been estimated by fitting of the Hdependence of the Mtotto a
+linear function
+/angbracketleftMtot/angbracketright=χH+Ms, (2)
+as shown in Fig. 5(b), and the results of the fitting are listed in Table I I. The slopes of the
+MtotversusHcurve, namely, the susceptibility χbecame half of the as-grown sample by
+the annealing. The value of Msof 1.0±0.8×10−2µB/Co became also somewhat smaller
+than that of the as-grown one. The observations indicate that bo th the paramagnetic and
+ferromagnetic components were suppressed by the annealing, in c ontrast to the reports on
+high-vacuum annealing on Zn 1−xCoxO thin films that the ferromagnetism of Zn 1−xCoxO
+thin films increased after annealing in a high vacuum [23, 24, 26]. The ag reement of Ms
+between the XMCD and SQUID measurements implies that the substit uted Co ions are also
+predominantly responsible for the ferromagnetism even after the annealing process. Figure 6
+shows the PO2dependences of Msandnin the homo-epitaxial Zn 1−xCoxO thin films [16].
+TheMsincreases with increasing nin both the O- and Zn-polar Zn 1−xCoxO. Taking into
+account the correlation between Msandnthat the increase in nincreases Msas show in
+Fig. 6, we conclude that the annealing hardly created enough oxyge n vacancies in the thin
+film andnof the film remained less than 10−19cm−3[14–16] and that the suppressions of the
+paramagnetic and ferromagnetic magnetizations by the annealing a re not originated from
+defect formation nor carrier doping.
+Let us then discuss the effects of annealing on the magnetic proper ties of Zn 1−xCoxO
+by considering the antiferromagnetic interaction between the Co io ns. Recently, Dietl et
+al.[34] have observed that the blocking temperature of Zn 1−xCoxO thin films, which show
+both ferromagnetism and paramagnetism, reaches room tempera ture, and have proposed
+that the ferromagnetism of Zn 1−xCoxO results from antiferromagnetic nanoclusters of Co-
+rich wurtzite (Zn,Co)O created by spinodal decomposition, similar to the ferromagnetism in
+NiO nanoparticles [35, 36]. In this model, the finite moments areorigina ted from uncompen-
+sated spins on the surfaces of the nanoclusters and the average moments shall decrease with
+increasing nanocrystal size. Similarly, for bulk Zn 1−xCoxO, it has been reported that the
+dominant interaction between the Co ions is antiferromagnetic and t he magnetic propertiy
+can be explained by asuuming the existence of antiferromagnetic Co clusters [37]. Based on
+the above picture, the annealing effects may be understood as the rmal diffusion and aggre-
+gation of isolated paramagnetic Co ions into antiferromagnetic nano clusters, as follows. In
+9the as-grown sample, there exist isolated Co ions and a small number of nanoclusters, which
+contribute to the Curie-Weiss paramagnetism and the weakly ferro magnetic antiferromag-
+netism, respectively. If the Co ions tend to aggregate in ZnO, ther mal diffusion of Co ions
+by the annealing may lead the decrease of the number of isolated Co io ns and the growth of
+the size of nanoclusters, suppressing the paramagnetic and ferr omagnetic responses, respec-
+tively. Figure 7 shows schematic images of the Co-ion distribution in Zn 1−xCoxO. According
+to this picture, in the ferromagnetic hetero-epitaxial Zn 1−xCoxO thin film previously mea-
+sured by XMCD [17], there were possibly few paramagnetic isolated Co ions while there
+were smaller nanoclusters than the present homo-epitaxial films [se e Fig. 7(c)]. Because the
+present homo-epitaxial films are expected to have higher crystal quality than the previous
+hetero-epitaxial ones, the nanocluster formation (or the aggre gation of the Co ions) may be
+prevented compared to the hetero-epitaxial Zn 1−xCoxO film. Since the annealing suppresses
+the magnetization and the nis probably unchanged by the annealing, the suppression is
+attributed not to be originated from the mechanisms related to oxy gen vacancy such as
+bound magnetic polaron [38] and donor impurity band exchange [39].
+It should be noted that the model based on the spinodal decompos ition cannot explain
+thecarrier-related ferromagneticproperties of Zn 1−xCoxO such asthe anomalousHall effects
+[14, 15]. It is possible that there are two origins of ferromagnetism in Zn1−xCoxO, that is,
+the formation of antiferromagnetic Co-rich wurtzite (Zn,Co)O nan oclusters and the electron
+carrier-mediated ferromagnetic interaction. In typical ferroma gnetic DMS Ga 1−xMnxAs,
+the origin of ferromagnetism is considered to be due to p-dexchange interaction for high
+carrier concentration and formation of bound magnetic polarons f or low carrier concentra-
+tion. Considering that the value of Msof Zn1−xCoxO (<2µB/Co) is smaller than the full
+moment of Co2+(3µB/Co), the antiferromagnetic interaction between neighboring Co io ns
+and the carrier-mediated ferromagnetic interaction between dist ant Co ions may simultane-
+ously affect the magnetic properties of Zn 1−xCoxO. Based on the correlation between Ms
+andnshown in Fig. 6, it is probable that in Zn 1−xCoxO, the spinodal decomposition is
+predominant in low nregion (n/lessorsimilar1019cm−3), while carrier-induced ferromagnetic interac-
+tion is predominant rather than the antiferromagnetic interaction in highnregion (n/greaterorsimilar1019
+cm−3). It follows that electron carrier-induced ferromagnetism in Zn 1−xCoxO possibly needs
+a highly crystallized Zn 1−xCoxO thin film with high electron carrier concentration n/greaterorsimilar1019
+cm−3.
+10CONCLUSION
+Wehave performed XMCD measurements onhomo-epitaxial Zn 1−xCoxOthin films which
+were prepared with various relatively high oxygen pressures and sh owed both ferromagnetic
+and paramagnetic behaviors. The line shapes of the Co 2 pXAS and XMCD spectra are
+the same in all the samples, and indicate that the Co ions substitute f or the Zn sites. The
+HandTdependences of Co 2 pXMCD suggest that paramagnetic behavior is dominant in
+the samples. Analyses using the Curie-Weiss law have revealed that t he Curie constant of
+the samples are close to the value of Co2+and the Weiss temperature are negative, indicat-
+ing that the paramagnetic Co2+ions interact antiferromagnetically with each other. The
+saturation magnetizations estimated using the XMCD sum rules were almost the same as
+the magnetization measurements, suggesting that the ferromag netic responses of the thin
+films are predominantly caused by magnetic moment of the doped Co io ns and confirm-
+ing that we have probed the bulk magnetic properties. The paramag netic susceptibility
+and saturation magnetization decreased by high-vacuum annealing . The suppression of the
+paramagnetism by annealing can be explained by thermal diffusion and aggregation of Co
+ions leading to spinodal decomposition and the antiferromagnetic int eraction between the
+Co ions. Therefore, we propose that the formation of Co-rich nan oclusters coexists and/or
+competes with carrier-mediated ferromagnetic interaction in Zn 1−xCoxO depending on elec-
+tron carrier concentration. Further systematic XMCD measurem ents on Zn 1−xCoxO with
+controlled electron-carrier and/or oxygen-vacancy concentra tions will provide us with un-
+derstandings of the carrier-induced ferromagnetism in Zn 1−xCoxO.
+Acknowledgments
+This work was supported by a Grant-in-Aid for Scientific Research in Priority Area
+“Creation and Control of Spin Current” (19048012) from MEXT, J apan. MK acknowledges
+support from the Japan Society for the Promotion of Science for Y oung Scientists.
+∗Present address: Department of Applied Chemistry, School o f Engineering, University of
+Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
+11†Present address: RIKEN, SPring-8 Center, Sayo-cho, Hyogo 6 79-5148, Japan
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+14TABLE I: Results of the Curie-Weiss analysis for the XMCD spe ctra of the Zn 1−xCoxO thin films.
+Here,Cis the Curie constant, θis the Weiss temperature, MXMCD
+sandMSQUID
+sare the saturation
+magnetizations estimated from the XMCD sum rules and measur ed by the SQUID, respectively.
+Samples C(µBK/T) θ(K)MXMCD
+s(10−2µB/Co)MSQUID
+s(10−2µB/Co)
+x=0.07PO2=10−4Pa3.8±0.5−37.17±5 0 .27±2.0 -
+x=0.10PO2=10−2Pa1.9±0.5−12.3±5 3 .4±0.8 2 .5±0.3
+PO2=10−3Pa3.0±0.5−37.0±5 1 .6±0.8 1 .4±0.3
+PO2=10−4Pa3.8±0.5−8.7±5 1 .7±0.8 -
+TABLE II: Results of the Curie-Weiss analysis for the XMCD sp ectra of the as-grown and annealed
+Zn0.9Co0.1O thin films deposited at PO2= 10−2Pa. Here, χis the paramagnetic susceptibility.
+χ(µB/Co T)MXMCD
+s(10−2µB/Co)MSQUID
+s(10−2µB/Co)
+As-grown 0.059 3 .4±0.8 2 .5±0.3
+Annealed 0.031 1 .0±0.8 1 .7±0.3
+150.15
+0.10
+0.05
+0Paramagnetic Susceptibility ( /s109B/Co T)
+300 200 100 0
+Temperature (K)-0.02-0.0100.010.02Magnetization ( /s109B/Co)
+-2 -1 0 1 2
+Magnetic Field (T)-404
+-2-1012PO2=10−3 Pa 
+  20 K
+  300 K
+PO2=10−2 Pa
+  10 K
+  300 KZn0.9Co0.1O (a) (b)
+ PO2=10−3 Pa
+ PO2=10−2 PaZn0.9Co0.1O
+FIG. 1: (Color online) Magnetic properties of Zn 0.9Co0.1O thin films measured by a SQUID mag-
+netometer. (a) Magnetic-field ( H) dependence of the magnetization. The H-linear components
+deduced from high Hdata have been subtracted. The inset shows raw data containi ng the diamag-
+netic signals from the substrates. (b) Temperature( T) dependenceof the paramagnetic component
+of the magnetization represented in the form of magnetic sus ceptibility, where the T-independent
+component deduced from high- Tdata have been subtracted.
+16-1.0-0.50Normalized Intensity (arb. units)-10
+782780778776
+800795790785780775
+Photon Energy (eV)100
+50
+0100
+50
+0
+-1.00
+800795790785780775
+Photon Energy (eV)100
+50
+0782780778776
+-10
+782780778776T=20 K,
+H=7.0 T
+Zn1-xCoxO x=0.07, PO2=10-4 Pa
+ x=0.10, PO2=10-2 Pa
+ x=0.10, PO2=10-3 Pa
+ x=0.10, PO2=10-4 Pa(a)
+ PO2=10-3 Pa
+ Ferro. ZnCoO
+ CI calculation(b)Zn1-xCoxO
+x=0.10Co 2p
+XAS
+Co 2p3/2Co 2p3/2
+FIG. 2: (Color online) Co 2 pcore-level absorption spectra of Zn 1−xCoxO thin films measured at
+H= 7.0 T andT= 20 K. (a) Comparison of the XAS (upper) and XMCD (lower) spec tra between
+the samples measured at H= 7.0 T and T= 20 K. (b) Comparison of the line shapes of the
+present spectra with that of the ferromagnetic sample and th e calculated spectrum reported in the
+previous study [17]. All the XAS and XMCD spectra have been no rmalized to the positive and
+negative peak heights, respectively. The insets show expan ded plots at the Co 2 p3/2edge.
+17100
+50
+0
+-15-10-50Intensity (arb. units)
+800795790785780775
+Photon Energy (eV)-10 Normalized Intensity
+7827807787761.0
+0.8
+0.6
+0.4
+0.2
+0Magnetization ( /s109B/Co)
+76543210
+Magnetic Field (T)Zn0.9Co0.1O
+PO2=10-3 Pa
+ 4.5 T
+ 7.0 TT = 20 KCo 2p XAS
+XMCDCo 2p3/2 x=0.07, PO2=10-4 Pa,
+ x=0.10, PO2=10-2 Pa,
+ x=0.10, PO2=10-3 Pa,
+ x=0.10, PO2=10-4 Pa(a) (b)
+Fit function:
+CH/(T–/s113) + MsT = 20 K Fitted
+FIG. 3: (Color online) Magnetic-field dependence of the XMCD spectra of Zn 1−xCoxO thin films.
+(a) XMCD spectra of the x= 0.10,PO2= 10−3sample at T= 20 K and at various magnetic fields.
+The inset shows the XMCD spectra normalized to the negative p eak height at hν= 778.5 eV. (b)
+Hdependence of the total magnetization ( Mtot) estimated using the XMCD sum rules at T= 20
+K. Solid curves are the results of fitting to the Curie-Weiss l aw,CH/(T−θ)+Ms.
+18100
+50
+0
+-10-50Intensity (arb. units)
+781780779778777
+Photon Energy (eV)0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1Magnetization ( /s109B/Co)
+10080604020
+Temperature (K)Zn0.9Co0.1OCo 2p3/2
+XAS
+XMCD
+PO2=10–3 Pa 20 K
+ 40 K
+ 100 KH = 4.5 T x=0.07, PO2=10-4 Pa,
+ x=0.10, PO2=10-2 Pa,
+ x=0.10, PO2=10-3 Pa,
+ x=0.10, PO2=10-4 Pa(a) (b)
+Fit function:
+C'H/(T–/s113) + Ms'
+H = 4.5 T Fitted
+FIG. 4: (Color online) Temperature dependence of the XMCD sp ectra of Zn 1−xCoxO thin films.
+(a) XMCDspectraofthe x= 0.10,PO2=10−3Pasampleat H= 4.5Tandat varioustemperatures.
+(b)Tdependence of the total magnetic moment Mtotestimated using the XMCD sum rules at
+H= 4.5 T. Solid curves are the results of fitting to the Curie-Weiss law,C′H/(T−θ)+M′
+s.
+19100
+50
+0
+-20-15-10-50Intensity (arb. units)
+781780779778777
+Photon Energy (eV)0.5
+0.4
+0.3
+0.2
+0.1
+0Magnetization ( /s109B/Co)
+6420
+Magnetic Field (T)Co 2p3/2
+XAS
+XMCD
+PO2=10-2 PaAs grown
+  4.5 T
+  7.0 T
+Annealed
+  4.5 T
+  7.0 TT=20 K
+Zn0.9Co0.1O As-grown
+ Annealed
+—  Fitted
+T = 20 KZn0.9Co0.1O(a) (b)
+FIG. 5: (Color online) Annealing effects on the XMCD spectra of the Zn 0.9Co0.1O thin film de-
+posited at PO2= 10−2Pa. (a) Comparison of the Hdependent XMCD spectra between the
+as-grown and annealed samples. (b) Hdependence of the magnetization of the same samples de-
+duced from the XMCD spectra. The Hdependences have been fitted to linear functions χH+Ms.
+201016101710181019Carrier  concentration (cm-3)
+10-510-410-310-2
+Oxygen Pressure (Pa)610-324610-224610-124Saturation Magnetization ( /s109B/Co)
+ MsXMCD
+x = 0.07
+x = 0.10
+Annealed
+Zn1-xCoxOZn polar 
+      Ms  n
+O polar 
+      Ms  n
+  MsSQUID
+x = 0.10
+Annealed
+ 
+FIG. 6: (Color online) PO2dependences of saturation magnetization Msand carrier concentration
+nin homo-epitaxial Zn 1−xCoxO thin films. The values of O- and Zn-polar Zn 1−xCoxO are taken
+from Ref. [16]. Here, all the experimental values of the pres ent study are plotted based on Msand
+PO2.
+Free Up Down(a) As-grown (b) Annealed (c) Hetero
+FIG. 7: (Color online) Schematic drawings of Co-ion distrib ution in Zn 1−xCoxO. (a), (b) Co
+distribution of the as-grown and annealed homo-epitaxial Z n1−xCoxO thin films, respectively. (c)
+Hetero-epitaxial Zn 1−xCoxO thin film.
+21
\ No newline at end of file
diff --git a/1001.0713.txt b/1001.0713.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6012b5fc86ed3d6437b01df22f292eac1050a98d
--- /dev/null
+++ b/1001.0713.txt
@@ -0,0 +1,1338 @@
+arXiv:1001.0713v1  [math-ph]  5 Jan 2010HYPERFINE SPLITTING OF THE DRESSED HYDROGEN ATOM
+GROUND STATE IN NON-RELATIVISTIC QED
+L. AMOUR AND J. FAUPIN
+Abstract. We consider a spin-1
+2electron and a spin-1
+2nucleus interacting with the quan-
+tized electromagnetic field in the standard model of non-rel ativistic QED. For a fixed total
+momentum sufficiently small, we study the multiplicity of the ground state of the reduced
+Hamiltonian. We prove that the coupling between the spins of the charged particles and the
+electromagnetic field splits the degeneracy of the ground st ate.
+1.Introduction
+This paper is concerned with the spectral analysis of the qua ntum Hamiltonian associated
+witha freehydrogenatom, inthecontext of non-relativisti c QED. Beforedescribingourresult
+more precisely, let us neglect for a while the corrections du e to quantum electrodynamics.
+In the following we recall a few well-known properties of the spectrum of Hydrogen. For
+more details, we refer the reader to classical textbooks on Q uantum Mechanics (see, e.g.,
+[Me, CTDL]). See also [BS].
+Consider a neutral hydrogenoid system composed of one elect ron and one nucleus. The sys-
+tem is supposed to be free, in the sense that no external poten tial acts on it. The two charged
+particles (the electron and the nucleus) are supposed to be n on-relativistic. In particular,
+relativistic corrections will not be taken into account in t he discussion below.
+Under sufficiently strong approximations, the spectrum of th e hydrogen atom Hamiltonian
+is explicitly known. Assume, indeed, that the nucleus is infi nitely heavy and that the two
+charged particles are spinless. Then the Schr¨ odinger oper ator in L2(R3) associated with the
+system reads
+p2
+el
+2mel−α
+|xel|. (1.1)
+Here the units are chosen such that /planckover2pi1=c= 1, where /planckover2pi1=h/2π,his the Planck constant,
+andcis the velocity of light. The mass of the electron is denoted b ymel,α=e2is the
+fine-structure constant (with ethe charge of the electron), and xel, respectively pel=−i∇xel,
+represents the position of the electron, respectively its m omentum.
+The spectrum of (1.1) consists of an infinite increasing sequ ence of negative isolated eigen-
+values (ej)j≥0, and the branch of absolutely continuous spectrum [0 ,∞). The ground state is
+non-degenerate, that is the eigenvalue e0is simple, while the excited eigenvalues ( ej)j≥1are
+degenerate (the degeneracy of the ( j+1)theigenvalue, ej, being equal to ( j+1)2).
+Introducing the degrees of freedom associated with the spin of the electron, the hydrogen
+atom can be described by the following Hamiltonian acting on L2(R3;C2):
+p2
+el
+2mel−α
+|xel|+1
+4m2
+elα
+|xel|3σel·(xel∧pel), (1.2)
+Date: October 13, 2018.
+12 L. AMOUR AND J. FAUPIN
+whereσel= (σel
+1,σel
+2,σel
+3) are the Pauli matrices for the electron spin. The last term i n
+the preceding expression is the spin-orbit interaction . It can be derived from the Dirac
+equation in the non-relativistic regime. Together with oth er relativistic corrections that have
+been neglected in (1.2), the spin-orbit interaction is resp onsible for the fine structure of the
+spectrum of Hydrogen. In particular, the ground state of (1. 2) is twice-degenerate, while
+the spin-orbit coupling splits the degeneracy of the excite d eigenvalues ( ej)j≥1. This can be
+justified by means of standard perturbation theory, the unpe rturbedHamiltonian being given
+by the expression (1.1) seen as an operator on L2(R3;C2).
+Assume now that the nucleus is a fixed particle with a spin equa l to1
+2and a finite mass.
+In order to take the effect of the spin nucleus into account, one can study the following Pauli
+Hamiltonian in L2(R3;C4):
+1
+2mel(pel−α1
+2An(xel))2−α
+|xel|−α1
+2
+2melσel·Bn(xel)+1
+4m2
+elα
+|xel|3σel·(xel∧pel).(1.3)
+HereAn(xel) is the vector potential of the electromagnetic field genera ted by the nucleus,
+andBn(xel) = ipel∧An(xel). Notice that An(xel) can be expressed in terms of the Pauli
+matrices σn= (σn
+1,σn
+2,σn
+3) associated with the spin of the nucleus as An(xel) = cα1/2(σn∧
+xel)/(mn|xel|3), where mnis the mass of the nucleus and c is a positive constant. The
+expression (1.3) includes the spin-orbit interaction.
+TheHamiltonian (1.3)allows onetojustifytheso-called hyperfine structure ofthespectrum
+of Hydrogen. Again, the argument follows from perturbation theory. The unperturbed part
+is still given by (1.1), now seen as an operator on L2(R3;C4). Hence the degeneracy of the
+unperturbedgroundstateeigenvalue, e0, isequal to4. Undertheinfluenceof theperturbation
+terms appearing in (1.3) (more precisely, under the influenc e of the term σel·Bn(xel)),e0
+splits into two parts: a simple eigenvalue associated with a unique ground state, and a 3-fold
+degenerate eigenvalue. Let us mention that this splitting o f the ground state explains the
+famous 21 -cm Hydrogen line . Besides, a similar hyperfine splitting of the excited eigen values
+(ej)j≥1occurs.
+Of course, in order to get a refined picture of the spectrum of H ydrogen, one should also
+treat the nucleus as a moving quantum particle. The correspo nding physical system is then
+translation invariant, and one is led to study the relative H amiltonian in the center of mass
+frame. In other words, one can consider the Hamiltonian obta ined by putting the total
+momentum equal to 0. For instance, in the simplest case where both the electron and the
+nucleus are spinless, the relative Hamiltonian is given by t he expression (1.1), except that xel
+andpelare replaced by the relative position and the relative momen tum respectively, and mel
+is replaced by the reduced mass µ=melmn/(mel+mn) (withmnthe mass of the nucleus).
+In this paper, we investigate the hyperfine structure of the h ydrogen atom Hamiltonian in
+non-relativistic QED. Let us mention that non-relativisti c QEDprovides asuitable framework
+to rigorously justify radiative decay and Bohr’s frequency condition. In particular, save for
+the ground state, all stationary states are expected to turn into metastable states with a
+finite lifetime. Since the unperturbed eigenvalues are embe dded into the essential spectrum,
+usual perturbation theory does not apply, and proving the la tter statement involves highly
+non-trivial analysis (see [BFS1, BFS2, AFFS, Sig] for the ca se of atomic systems with an
+infinitely heavy nucleus). The present work focuses on the qu estion of the nature of the
+ground state, for the model of a freely moving hydrogen atom a t a fixed total momentum.
+The difference between the unperturbed and the perturbed grou nd state energies is referredHYPERFINE SPLITTING IN NON-RELATIVISTIC QED 3
+to asthe Lamb shift . Our main concern is to study the degeneracy of the ground state. More
+precisely, we aim at establishing that a hyperfine splitting of the ground state does occur in
+the framework of non-relativstic QED.
+Let us now describe our main result in more details.
+We consider a non-relativistic hydrogen atom interacting w ith the quantized electromag-
+netic field in the standard model of non-relativistic QED. Bo th the electron and the nucleus
+are treated as moving particles, so that the total Hamiltoni an,Hg, is translation invariant.
+Heregdenotes a coupling parameter depending on the fine-structur e constant, α, related
+to the “strength” of the interaction between the charged par ticles and the electromagnetic
+field (precise definitions will be given in Subsection 2.1 bel ow). The translation invariance
+implies that Hgadmits a direct integral decomposition, Hg∼/integraltext
+R3Hg(P)dP, with respect to
+the total momentum Pof the system. This paper is devoted to the study of the degene r-
+acy of the ground state energy of the dressed hydrogen atom at a fixed total momentum,
+Eg(P) := infσ(Hg(P)).
+In [AGG2], it is established that, for gandPsufficiently small, Eg(P) is an eigenvalue of
+Hg(P), that is Hg(P) has a ground state. We also mention [LMS1] where the existen ce of a
+ground state for Hg(P) is obtained for any value of g, under the assumption that Eg(0)≤
+Eg(P). Using a method due to [Hi2], it is proven in [AGG2] that the m ultiplicity of Eg(P)
+cannot exceed the multiplicity of E0(P) := infσ(H0(P)), where H0(P) :=Hg=0(P) denotes
+the non-interacting Hamiltonian. In other words,
+(0<)dim Ker( Hg(P)−Eg(P))≤dim Ker( H0(P)−E0(P)). (1.4)
+Our purpose is to determine whether the inequality in (1.4) i s strict, or, on the contrary, is
+an equality.
+Of course, in the same way as in the case where the coupling to t he quantized electro-
+magnetic field is neglected (see the discussion at the beginn ing of this introduction), the
+multiplicity of Eg(P) depends on the value of the spins of the charged particles. I f the spin
+of the electron is neglected and the spin of the nucleus is equ al to 0, then E0(P) is simple,
+and hence, according to (1.4), Eg(P) is also a simple eigenvalue. In particular, (1.4) is an
+equality.
+Ifthespinoftheelectronistaken intoaccount, andthespin ofthenucleusisequalto0, then
+E0(P) is twice-degenerate. Using Kramer’s degeneracy theorem ( see [LMS2]), one can prove
+that the multiplicity of Eg(P) is even. Therefore, by (1.4), Eg(P) is also twice-degenerate,
+and hence (1.4) is again an equality. We refer the reader to [H S, Sp, Sa, Hi1, LMS2] for results
+on the twice-degeneracy of the ground state of various QED mo dels.
+Consider now a hydrogen atom composed of a spin-1
+2electron and a spin-1
+2nucleus (e.g. a
+proton). In this case, the multiplicity of E0(P) is equal to 4. Our main result states that
+dim Ker( Hg(P)−Eg(P))<dim Ker( H0(P)−E0(P)) = 4, (1.5)
+forg/ne}ationslash= 0smallenough. Equation(1.5)canbeinterpretedasahyper finesplittingoftheground
+state ofHg(P). In other words, the Hamiltonian of a freely moving hydroge n atom at a fixed
+total momentum in non-relativistic QED contains hyperfine i nteraction terms which split the
+degeneracy of the ground state, in the same way as for the Paul i Hamiltonian of Quantum
+Mechanics mentioned above. Pursuing the analogy with the Pa uli Hamiltonian (1.3), one
+can conjecture that Eg(P) is simple. Generally speaking, a way to establish the uniqu eness
+of the ground state of a given Hamiltonian Hconsists in showing that e−tHis positivity
+improving for all t >0. In several cases, this can be done by constructing a functi onal4 L. AMOUR AND J. FAUPIN
+integral representation forthesemi-group e−tH(seee.g. [Sim] forSchr¨ odingeroperatorsunder
+various general assumptions). To our knowledge, however, s uch a functional representation
+for the model of two spin-1
+2particles minimally coupled to the quantized radiation fiel d does
+not presently exist (we refer to the recent work [HL] for the c ase of one spin-1
+2particle
+minimally coupled to the radiation field). Here, we shall fol low a different approach; Proving
+the simplicity of Eg(P) is beyond the scope of this paper.
+In addition, in relation with the 21-cm hydrogen line mentio ned above, one can expect
+that a resonance appears near the ground state energy Eg(P), with a very small imaginary
+part. Showing this would presumably require the use of compl ex dilatations together with
+renormalization techniques as in [BFS1].
+The case of a nucleus of spin ≥1 is not considered here (for instance, the nucleus of
+deuterium, composed of one proton and one neutron, can be tre ated as a spin-1 particle), but
+we expect that a similar hyperfine splitting of the ground sta te occurs in this case also.
+As for positively charged hydrogenoid ions, the question of the existence of a ground state
+is more subtle than for the hydrogen atom. Indeed, it is prove n in [HH] that the Hamiltonian
+of a positive ion at a fixed total momentum in non-relativisti c QED does not have a ground
+state in Fock space. This result should be compared with the c orresponding one for the model
+of a freely moving, dressed non-relativistic electron in no n-relativistic QED, which has been
+studied recently by several authors (see, among other paper s, [Ch, CF, BCFS2, HH, CFP,
+LMS2, FP]). For the latter model, it is established that a gro und state exists in a non-Fock
+representation (see [CF]).
+The same results (the absence of a ground state in Fock space, and the existence of a
+ground state in a non-Fock representation) are proven in [AF GG1, AFGG2], for a dressed
+non-relativistic electron which interacts with a classica l magnetic field, following an approach
+different from [CF]. We expect that the method of [AFGG2] can be adapted to prove the
+existence of a ground state for a renormalized Hamiltonian ( in a non-Fock representation)
+associated with the dressed helium ion He+, at a fixed total momentum. Since the nucleus of
+He+, composed of two protons, can be treated as a spin-0 particle , one can conjecture that
+the ground state is twice-degenerate according to Kramer’s theorem (see [LMS2]).
+Regarding the model of a hydrogenoid ion whose nucleus has a n on-zero spin, we do not
+know whether it is possible to adapt [AFGG2]. Indeed, an impo rtant ingredient in [AFGG2]
+lies in the regularity of the ground state energy with respec t to the total momentum. For
+the model considered in [AFGG2], the latter property can be e stablished using the method
+developed in [Pi, CFP, FP]. However, if a hyperfine splitting of the ground state occurs, it is
+not known, to our knowledge, how to study the regularity of Eg(P) with respect to P.
+Let us finally mention that the ground state degeneracy of the non-relativistic hydrogen
+atom confined by its center of mass (see [AF, Fa]) could also be analyzed by the techniques
+developed here, provided that both the electron and the nucl eus have a spin equal to1
+2.
+2.Definition of the model and statement of the main result
+Before stating our main theorem, let us precisely define the m odel under consideration.
+2.1.Definition of the model. We consider a mobile, non-relativistic hydrogen atom, inte r-
+acting with the quantized electromagnetic field in the Coulo mb gauge. In the standard model
+of non-relativistic QED, the Hamiltonian associated with t his system acts on the Hilbert
+space
+H:=Hat⊗Hph, (2.1)HYPERFINE SPLITTING IN NON-RELATIVISTIC QED 5
+where
+Hat:= L2(R3;C2)⊗L2(R3;C2)∼L2(R6;C4) (2.2)
+is the Hilbert space for the charged particles (the electron and the nucleus), and
+Hph:=C⊕∞/circleplusdisplay
+n=1Sn/bracketleftbig
+L2(R3×{1,2})⊗n/bracketrightbig
+(2.3)
+is the symmetric Fock space for the photons. Here Sndenotes the symmetrization operator.
+The units are chosen such that the Planck constant /planckover2pi1=h/2πand the velocity of light c
+are equal to 1. The Hamiltonian of the system, HSM, is formally given by the expression
+HSM:=1
+2mel/parenleftBig
+pel−α1
+2A(xel)/parenrightBig2
++1
+2mn/parenleftBig
+pn+α1
+2A(xn)/parenrightBig2
++V(xel,xn)+Hph
+−α1
+2
+2melσel·B(xel)+α1
+2
+2mnσn·B(xn), (2.4)
+wherexel(respectively xn) denotes the position of the electron (respectively the pos ition of
+the nucleus), and pel:=−i∇xel(respectively pn:=−i∇xn) is the momentum operator of the
+electron (respectively of the nucleus). The parameter αdenotes the fine-structure constant.
+Forx∈R3, the vectors A(x) andB(x) are defined by
+A(x) :=1
+2π/summationdisplay
+λ=1,2/integraldisplay
+R3χΛ(k)
+|k|1
+2ελ(k)/bracketleftBig
+e−ik·xa∗
+λ(k)+eik·xaλ(k)/bracketrightBig
+dk, (2.5)
+B(x) :=−i
+2π/summationdisplay
+λ=1,2/integraldisplay
+R3|k|1
+2χΛ(k)/parenleftbigk
+|k|∧ελ(k)/parenrightbig/bracketleftBig
+e−ik·xa∗
+λ(k)−eik·xaλ(k)/bracketrightBig
+dk,(2.6)
+where the polarization vectors are chosen in the following w ay:
+ε1(k) :=(k2,−k1,0)/radicalbig
+k2
+1+k2
+2, ε2(k) :=k
+|k|∧ε1(k) =(−k1k3,−k2k3,k2
+1+k2
+2)/radicalbig
+k2
+1+k2
+2/radicalbig
+k2
+1+k2
+2+k2
+3.(2.7)
+In (2.5) and (2.6), χΛ(k) denotes an ultraviolet cutoff function which, for the sake o f con-
+creteness, we choose as
+χΛ(k) :=1|k|≤Λα2(k). (2.8)
+Here, Λ is supposed to be a given arbitrary (large and) positi ve parameter. As explained in
+[BFS2, Sig], the model is physically relevant if we assume th at 1≪Λ≪α−2. The reason for
+introducing α2into the definition (2.8) will appear below (see (2.21)).
+For anyh∈L2(R3×{1,2}), we set
+a∗(h) :=/summationdisplay
+λ=1,2/integraldisplay
+R3h(k,λ)a∗
+λ(k)dk, a(h) :=/summationdisplay
+λ=1,2/integraldisplay
+R3¯h(k,λ)aλ(k)dk, (2.9)
+and
+Φ(h) :=a∗(h)+a(h), (2.10)
+where the usual creation and annihilation operators, a∗
+λ(k) andaλ(k), obey the canonical
+commutation relations
+[aλ(k),aλ′(k′)] = [a∗
+λ(k),a∗
+λ′(k′)] = 0,[aλ(k),a∗
+λ′(k′)] =δλλ′δ(k−k′). (2.11)
+Hence, in particular, for j∈ {1,2,3}, we have
+Aj(x) = Φ(hA
+j(x)),andBj(x) = Φ(hB
+j(x)), (2.12)6 L. AMOUR AND J. FAUPIN
+with
+hA
+j(x,k,λ) :=1
+2πχΛ(k)
+|k|1
+2ελ
+j(k)e−ik·x, (2.13)
+hB
+j(x,k,λ) :=−i
+2π|k|1
+2χΛ(k)/parenleftbiggk
+|k|∧ελ(k)/parenrightbigg
+je−ik·x. (2.14)
+The Coulomb potential V(xel,xn) is given by
+V(xel,xn)≡V(xel−xn) :=−α
+|xel−xn|, (2.15)
+andHphis the Hamiltonian of the free photon field, defined by
+Hph:=/summationdisplay
+λ=1,2/integraldisplay
+R3|k|a∗
+λ(k)aλ(k)dk. (2.16)
+The 3-uples σel= (σel
+1,σel
+2,σel
+3) andσn= (σn
+1,σn
+2,σn
+3) are the Pauli matrices associated with
+the spins of the electron and the nucleus respectively. They can be written as 4 ×4 matrices
+in the following way:
+σel
+1=
+0 0 1 0
+0 0 0 1
+1 0 0 0
+0 1 0 0
+, σel
+2=
+0 0−i 0
+0 0 0 −i
+i 0 0 0
+0 i 0 0
+, σel
+3=
+1 0 0 0
+0 1 0 0
+0 0−1 0
+0 0 0 −1
+,(2.17)
+σn
+1=
+0 1 0 0
+1 0 0 0
+0 0 0 1
+0 0 1 0
+, σn
+2=
+0−i 0 0
+i 0 0 0
+0 0 0 −i
+0 0 i 0
+, σn
+3=
+1 0 0 0
+0−1 0 0
+0 0 1 0
+0 0 0 −1
+. (2.18)
+In order to exhibit the perturbative behavior of the interac tion between the charged par-
+ticles and the photon field, we proceed to a change of units. Mo re precisely, let U:H → H
+be the unitary operator associated with the scaling
+(xel,xn,k1,λ1,...,kn,λn)/ma√sto→(xel/α,xn/α,α2k1,λ1,...,α2kn,λn).(2.19)
+We have
+1
+α2UHSMU∗=1
+2mel/parenleftBig
+pel−α3
+2˜A(αxel)/parenrightBig2
++1
+2mn/parenleftBig
+pn+α3
+2˜A(αxn)/parenrightBig2
+−1
+|xel−xn|+Hph−α3
+2
+2melσel·˜B(αxel)+α3
+2
+2mnσn·˜B(αxn), (2.20)
+where˜Aand˜Bare defined in the same way as AandB, except that the ultraviolet cutoff
+function χΛ(k) is replaced by
+˜χΛ(k) :=χΛ(α2k) =1|k|≤Λ(k). (2.21)
+To simplify the notations, we redefine ˜ χΛ=χΛ,A=˜AandB=˜B. Setting g:=α3
+2, we are
+thus led to study the Hamiltonian
+HSM
+g:=1
+2mel/parenleftBig
+pel−gA(g2
+3xel)/parenrightBig2
++1
+2mn/parenleftBig
+pn+gA(g2
+3xn)/parenrightBig2
+−1
+|xel−xn|+Hph−g
+2melσel·B(g2
+3xel)+g
+2mnσn·B(g2
+3xn). (2.22)HYPERFINE SPLITTING IN NON-RELATIVISTIC QED 7
+Let the total mass, M, and the reduced mass, µ, be defined respectively by
+M:=mel+mn,1
+µ:=1
+mel+1
+mn. (2.23)
+Let
+r:=xel−xn, R:=mel
+Mxel+mn
+Mxn, (2.24)
+pr
+µ:=pel
+mel−pn
+mn, PR:=pel+pn. (2.25)
+Forg= 0, the Hamiltonian HSM
+0:=HSM
+g=0is given by
+HSM
+0=p2
+el
+2mel+p2
+n
+2mn−1
+|xel−xn|+Hph=HR+Hr+Hph, (2.26)
+where the Schr¨ odinger operators HRandHron L2(R3) are defined by
+HR:=P2
+R
+2M, Hr:=p2
+r
+2µ−1
+|r|. (2.27)
+Let us note that the spectrum of HRconsists of the branch of essential spectrum [0 ,∞),
+whereas the spectrum of Hris composed of an increasing sequence of isolated eigenvalu es
+(e0,e1,e2,...) accumulating at 0, and the essential spectrum [0 ,∞). The first eigenvalue of
+Hris
+e0=−µ
+2, (2.28)
+and a normalized eigenstate associated with e0is given by
+φ0(r) := (π−1µ3)1
+2e−µ|r|. (2.29)
+To conclude this subsection, we recall the definition of the p hoton number operator, Nph,
+which will be used in the sequel:
+Nph:=/summationdisplay
+λ=1,2/integraldisplay
+R3a∗
+λ(k)aλ(k)dk. (2.30)
+2.2.Fiber decomposition. The Hamiltonian HSM
+gis translation invariant in the sense that
+HSM
+gformally commutes with the total momentum operator Ptot:=PR+Pph, wherePph
+denotes the momentum operator of the photon field, given by th e expression
+Pph:=/summationdisplay
+λ=1,2/integraldisplay
+R3ka∗
+λ(k)aλ(k)dk. (2.31)
+In the same way as in [AGG2], it follows that HSM
+gcan be decomposed into a direct integral,
+which is expressed in the following proposition.
+Proposition 2.1 ([AGG2]) .There exists gc>0such that for all |g| ≤gc, the following
+holds: the Hamiltonian HSM
+ggiven by the formal expression (2.22)identifies with a self-
+adjoint operator which is unitarily equivalent to the direc t integral/integraltext⊕
+R3Hg(P)dP,
+HSM
+g∼/integraldisplay⊕
+R3Hg(P)dP. (2.32)8 L. AMOUR AND J. FAUPIN
+For allP∈R3,Hg(P)is a self-adjoint operator acting on the Hilbert space
+H(P) := L2(R3;C4)⊗Hph∼C4⊗L2(R3,dr)⊗Hph, (2.33)
+with domain D(Hg(P)) =D(H0(P)), andHg(P)is given by the expression:
+Hg(P) =1
+2mel/parenleftBigmel
+M(P−Pph)+pr−gA(mel
+Mg2
+3r)/parenrightBig2
++1
+2mn/parenleftBigmn
+M(P−Pph)−pr+gA(−mn
+Mg2
+3r)/parenrightBig2
+−1
+|r|+Hph−g
+2melσel·B(mel
+Mg2
+3r)+g
+2mnσn·B(−mn
+Mg2
+3r). (2.34)
+Let us mention that the direct integral decomposition (2.32 ) remains true for an arbitrary
+value of the coupling constant g(see [LMS1]). However, in this paper, we shall only be
+interested in the small coupling regime.
+Forg= 0, the fiber Hamiltonian H0(P) :=Hg=0(P) reduces to the diagonal operator
+H0(P) =Hr+1
+2M(P−Pph)2+Hph, (2.35)
+whereHris the Schr¨ odinger operator defined in (2.27). Let Ω denote t he photon vacuum in
+Hph. One can verify that
+E0(P) := infσ(H0(P)) =e0+P2
+2M, (2.36)
+and that e0+P2/2Mis an eigenvalue of multiplicity 4 of H0(P). Moreover, the associated
+normalized eigenstates can be written under the form y⊗φ0⊗Ω, where yis an arbitrary
+normalized element in C4.
+The operator H0(P) is treated as an unperturbedHamiltonian, the perturbatio nWg(P) :=
+Hg(P)−H0(P) being given by
+Wg(P) =−g
+mel/parenleftBig/parenleftbigmel
+M(P−Pph)+pr/parenrightbig
+·A(mel
+Mg2
+3r)/parenrightBig
++g
+mn/parenleftBig/parenleftbigmn
+M(P−Pph)−pr/parenrightbig
+·A(−mn
+Mg2
+3r)/parenrightBig
++g2
+2melA(mel
+Mg2
+3r)2+g2
+2mnA(−mn
+Mg2
+3r)2
+−g
+2melσel·B(mel
+Mg2
+3r)+g
+2mnσn·B(−mn
+Mg2
+3r). (2.37)
+Note that, due to the choice of the Coulomb gauge, the operato rsA(melg2/3r/M) and
+A(−mng2/3r/M) commute both with prandPph.
+As mentioned in the introduction, this paper is concerned wi th the nature of the bottom
+of the spectrum of the perturbed Hamiltonian, Eg(P) := infσ(Hg(P)). In other words, we
+would like to determine the behavior of the unperturbed eige nvalueE0(P) under the pertur-
+bationWg(P). We emphasize that, since E0(P) is embedded into the continuous spectrum of
+H0(P), usual perturbation theory of isolated eigenvalues of fini te multiplicity does not apply.
+In [AGG2], it is proven that, for gandPsufficiently small, Hg(P) has a ground state of
+multiplicity ≤4, that is Eg(P) is an (embedded) eigenvalue of multiplicity ≤4 ofHg(P).
+The aim of our work is to study more precisely the multiplicit y ofEg(P), still assuming that
+gandPare sufficiently small.HYPERFINE SPLITTING IN NON-RELATIVISTIC QED 9
+2.3.Main result. Our main result is stated in the following theorem.
+Theorem 2.2. There exists gc>0andpc>0such that, for any 0<|g| ≤gcand0≤ |P| ≤
+pc,
+dimKer( Hg(P)−Eg(P))<4. (2.38)
+This theorem shows that the interaction between the charged particles with spins and the
+photon field splits the degeneracy of the ground state. As exp lained in the introduction,
+this splitting of the ground state can be seen as a manifestat ion of the hyperfine interaction
+between the electron and the nucleus. It may be expected that a renormalization group
+analysis adapted from [BFS1, BCFS2], together with a carefu l study of the second order term
+in the expansion of Eg(P) w.r.t. g, would lead to a more precise result. A feature of the
+method developed in the present work is its brevity.
+Ourproofof Theorem2.2 is basedon acontradiction argument andtheuseof the Feshbach-
+Schur identity . Let ussketch the argument moreprecisely. For technical co nvenience, we shall
+work with the Hamiltonian obtained from Hg(P) by Wick ordering (see Section 3). Let P0
+denote the projection onto the eigenspace associated with t he eigenvalue E0(P) ofH0(P),
+and¯P0:=1−P0. Note that P0=1⊗Pφ0⊗PΩin the tensor product C4⊗L2(R3)⊗Hph,
+wherePφ0denotes the projection onto the eigenspace associated with the ground state φ0of
+Hr, andPΩdenotes the projection onto the Fock vacuum. We also set Pρ:=1⊗Pφ0⊗1Hf≤ρ,
+and¯Pρ:=1−Pρ, whereρis a suitably chosen positive parameter (depending on gin such
+a way that g2≪ρ≪1, see Sections 3 and 4). By the Feshbach-Schur identity, we w ill see
+that, for all ε >0,
+Pρ/bracketleftbig
+Hg(P)−Eg(P)+ε/bracketrightbig−1Pρ=Fρ(ε)−1, (2.39)
+whereFρ(ε) denotes the Feshbach-Schur operator :
+Fρ(ε) =/parenleftbig
+H0(P)−Eg(P)+ε/parenrightbig
+Pρ+PρWg(P)Pρ
+−PρWg(P)/bracketleftbig¯PρHg(P)¯Pρ−Eg(P)+ε/bracketrightbig−1¯PρWg(P)Pρ. (2.40)
+Bythefunctionalcalculus, theprojectionontotheeigensp aceassociatedwiththeeigenvalue
+Eg(P) ofHg(P) can be written as 1{Eg(P)}(Hg(P)). The limit Fρ(0) := lim ε→0Fρ(ε) being
+well-defined (in the norm topology), we will deduce from (2.3 9) that
+Fρ(0)Pρ1{Eg(P)}(Hg(P))Pρ= 0. (2.41)
+Choosing ρ=|g|2−2τwithτ >0 small enough and using the property that
+/ba∇dbl¯P01{Eg(P)}(Hg(P))/ba∇dbl ≤Constg2, (2.42)
+(see Appendix A), we will obtain that
+P0Fρ(0)P01{Eg(P)}(Hg(P))P0=O(|g|4−2τ). (2.43)
+Assuming by contradiction that
+dimKer( Hg(P)−Eg(P)) = 4, (2.44)
+we will then show that (2.43) implies
+P0Fρ(0)P0=O(|g|4−2τ). (2.45)10 L. AMOUR AND J. FAUPIN
+Next, for ρ=|g|2−2τ≫g2, using standard estimates involving creation and annihila tion
+operators, it can be verified, in a way similar to [BFS1, BFS2] , that the reduced resolvent
+[¯PρHg(P)¯Pρ−Eg(P)]−1¯Pρdecomposes into the Neumann series
+/bracketleftbig¯PρHg(P)¯Pρ−Eg(P)/bracketrightbig−1¯Pρ=/bracketleftbig
+H0(P)−Eg(P)/bracketrightbig−1
+/summationdisplay
+n≥0/parenleftbig
+−¯PρWg(P)¯Pρ/bracketleftbig
+H0(P)−Eg(P)/bracketrightbig−1/parenrightbign¯Pρ.(2.46)
+Introducing (2.46) into (2.40) and (2.45), we will deduce th e following identity:
+/summationdisplay
+n≥0P0Wg(P)/bracketleftbig
+H0(P)−Eg(P)/bracketrightbig−1/parenleftbig
+−¯PρWg(P)¯Pρ/bracketleftbig
+H0(P)−Eg(P)/bracketrightbig−1/parenrightbign¯P0Wg(P)P0
+= (E0(P)−Eg(P))P0+O(|g|2+τ). (2.47)
+Hence, identifying the left-hand-side of (2.47) with a 4 ×4 matrix, (2.47) implies in particular
+that all the terms of order g2must be located on the diagonal. Extracting the latter from
+the sum over nwill lead to a contradiction.
+2.4.Organization of the paper. We decompose the proof of Theorem 2.2 into two main
+steps. In Section 3, we introduce and study some properties o f the Feshbach-Schur operator
+Fρ(ε) mentioned in the previous subsection. Next, in Section 4, w e assume that the multi-
+plicity of Eg(P) is equal to 4, and we conclude the proof of Theorem 2.2 by a con tradiction
+argument. In Appendix A, we collect some technical estimate s used in Sections 3 and 4.
+Throughout the paper, C ,C′,C′′will denote positive constants that may differ from one
+line to another.
+3.The Feshbach-Schur operator
+As mentioned above, it is convenient to work with the Hamilto nian˜Hg(P) obtained from
+Hg(P) by Wick ordering, that is ˜Hg(P) = :Hg(P) : , with the usual notations. It is not
+difficult to check that ˜Hg(P) =Hg(P)−g2CΛ, where C Λis a positive constant depending
+on the ultraviolet cutoff parameter Λ. Hence it suffices to prov e Theorem 2.2 with ˜Hg(P)
+replacing Hg(P) and˜Eg(P) := infσ(˜Hg(P)) replacing Eg(P). To simplify the notations, we
+redefine Hg(P) :=˜Hg(P) andEg(P) :=˜Eg(P). Moreover, in what follows, we drop the
+dependence on Peverywhere unless a confusion may arise. In particular, we s et
+Hg=Hg(P), H0=H0(P), Wg=Wg(P),
+Eg=Eg(P), E0=E0(P) =e0+P2
+2M. (3.1)
+Let us begin with the proof of the convergence of the Neumann s eries (2.46).
+Lemma 3.1. There exist gc>0andpc>0such that, for all 0≤ |g| ≤gc,0≤ |P| ≤pc,
+ε≥0andg2≪ρ≪1, the operator ¯PρHg¯Pρ−Eg+ε:D(H0)∩Ran(¯Pρ)→Ran(¯Pρ)is
+invertible and satisfies
+/bracketleftbig¯PρHg¯Pρ−Eg+ε/bracketrightbig−1¯Pρ
+=/bracketleftbig
+H0−Eg+ε/bracketrightbig−1¯Pρ/summationdisplay
+n≥0/parenleftBig
+−Wg¯Pρ/bracketleftbig
+H0−Eg+ε/bracketrightbig−1¯Pρ/parenrightBign
+. (3.2)HYPERFINE SPLITTING IN NON-RELATIVISTIC QED 11
+Proof.Since
+¯Pρ/parenleftbig
+Hg−Eg+ε/parenrightbig¯Pρ=/parenleftbig
+H0−Eg+ε)¯Pρ+¯PρWg¯Pρ, (3.3)
+it suffices to prove that the Neumann series in the right-hand- side of (3.2) is convergent. It
+follows from Lemmata A.2 and A.8 in Appendix A that, for all n∈N,ε≥0 andρ >0,
+/vextenddouble/vextenddouble/vextenddouble/bracketleftbig
+H0−Eg+ε/bracketrightbig−1¯Pρ/parenleftBig
+−Wg¯Pρ/bracketleftbig
+H0−Eg+ε/bracketrightbig−1¯Pρ/parenrightBign/vextenddouble/vextenddouble/vextenddouble
+≤Cρ−1/parenleftbig
+C′|g|ρ−1
+2/parenrightbign, (3.4)
+Therefore, for 1 ≫ρ≫g2, (3.4) implies (3.2). /square
+Using Lemma 3.1, we now verify that the Feshbach-Schur opera torFρ(ε) written in (2.40)
+is well-defined for any ε≥0.
+Lemma 3.2. There exist gc>0andpc>0such that, for all 0≤ |g| ≤gc,0≤ |P| ≤pc,
+ε≥0andg2≪ρ≪1, the Feshbach-Schur operator
+Fρ(ε) = (H0−Eg+ε)Pρ+PρWgPρ−PρWg/bracketleftbig¯PρHg¯Pρ−Eg+ε/bracketrightbig−1¯PρWgPρ.(3.5)
+is a well-defined (bounded) operator on Ran(Pρ). Moreover, Fρ(ε)satisfies
+Fρ(0) = lim
+ε→0+Fρ(ε), (3.6)
+in the norm topology, and
+/ba∇dblFρ(0)/ba∇dbl ≤Cρ. (3.7)
+Proof.By Lemma 3.1 and the fact that Ran( Pρ)⊂D(H0)⊂D(Wg),Fρ(ε) is obviously
+well-defined on Ran( Pρ), for any ε≥0. The boundedness of Fρ(ε) and Equation (3.6) are
+straightforward verifications.
+In order to prove (3.7), we proceed as follows: First, it foll ows from Lemma A.6 that
+/vextenddouble/vextenddouble(H0−Eg)Pρ/vextenddouble/vextenddouble=/vextenddouble/vextenddouble(E0−Eg)Pρ+(−1
+MP·Pph+1
+2MP2
+ph+Hph)Pρ/ba∇dbl
+≤Cg2+C′ρ≤C′′ρ, (3.8)
+since, by assumption, ρ≫g2. Next, by Lemma A.8, we have that
+/ba∇dblPρWgPρ/ba∇dbl ≤C|g|ρ1
+2≤C′ρ. (3.9)
+Lemma 3.1 gives
+PρWg/bracketleftbig¯PρHg¯Pρ−Eg/bracketrightbig−1¯PρWgPρ
+=PρWg/bracketleftbig
+H0−Eg/bracketrightbig−1¯Pρ/summationdisplay
+n≥0/parenleftBig
+−Wg¯Pρ/bracketleftbig
+H0−Eg/bracketrightbig−1¯Pρ/parenrightBign
+WgPρ. (3.10)
+Using again Lemma A.8, we get, for all n≥0,
+/vextenddouble/vextenddoublePρWg/bracketleftbig
+H0−Eg/bracketrightbig−1¯Pρ/parenleftBig
+−Wg¯Pρ/bracketleftbig
+H0−Eg/bracketrightbig−1¯Pρ/parenrightBign
+WgPρ/vextenddouble/vextenddouble
+≤Cg2(C′|g|ρ−1
+2)n, (3.11)
+which implies/vextenddouble/vextenddoublePρWg/bracketleftbig¯PρHg¯Pρ−Eg/bracketrightbig−1¯PρWgPρ/vextenddouble/vextenddouble≤Cg2≤C′ρ. (3.12)
+Equations (3.8), (3.9) and (3.12) give (3.7). /square12 L. AMOUR AND J. FAUPIN
+We now turn to the proof of the Feshbach-Schur identity, Equa tion (2.39). We refer to
+[BFS1, BCFS1, GH] for definitions and properties of the “(smo oth) Feshbach-Schur map”,
+and its use in the context of non-relativistic QED. In our cas e, the operator Hg−Eg+ε
+is obviously invertible (for ε >0), so that the following theorem simply follows from usual
+second order perturbation theory. For the convenience of th e reader, we recall the proof.
+Lemma 3.3. There exist gc>0andpc>0such that, for all 0≤ |g| ≤gc,0≤ |P| ≤pc,
+ε >0andg2≪ρ≪1, the operators Hg−Eg+ε:D(H0)→C4⊗L2(R3)⊗ Hphand
+Fρ(ε) : Ran(Pρ)→Ran(Pρ)are invertible and satisfy
+Pρ[Hg−Eg+ε]−1Pρ=Fρ(ε)−1. (3.13)
+Proof.SinceHg−Eg≥0, for any ε >0, the operator Hg−Eg+εfromD(H0) toC4⊗
+L2(R3,dr)⊗Hphis obviously invertible. Next, since H0commutes with Pρ, we have that
+/parenleftbig
+H0−Eg+ε/parenrightbig
+Pρ+PρWgPρ=Pρ/parenleftbig
+Hg−Eg+ε/parenrightbig
+Pρ. (3.14)
+Combining (3.14) with the facts that ¯PρWgPρ=¯Pρ(Hg−Eg+ε)PρandPρ+¯Pρ=1, we get
+Fρ(ε)Pρ/bracketleftbig
+Hg−Eg+ε/bracketrightbig−1Pρ
+=Pρ/parenleftbig
+Hg−Eg+ε/parenrightbig
+Pρ/bracketleftbig
+Hg−Eg+ε/bracketrightbig−1Pρ
+−PρWg[¯PρHg¯Pρ−Eg+ε]−1¯Pρ/parenleftbig
+Hg−Eg+ε/parenrightbig
+Pρ/bracketleftbig
+Hg−Eg+ε/bracketrightbig−1Pρ
+=Pρ−Pρ/parenleftbig
+Hg−Eg+ε/parenrightbig¯Pρ/bracketleftbig
+Hg−Eg+ε/bracketrightbig−1Pρ
++PρWg[¯PρHg¯Pρ−Eg+ε]−1¯Pρ/parenleftbig
+Hg−Eg+ε/parenrightbig¯Pρ/bracketleftbig
+Hg−Eg+ε/bracketrightbig−1Pρ
+=Pρ−PρWg¯Pρ/bracketleftbig
+Hg−Eg+ε/bracketrightbig−1Pρ+PρWg¯Pρ/bracketleftbig
+Hg−Eg+ε/bracketrightbig−1Pρ
+=Pρ. (3.15)
+The identity Pρ[Hg−Eg+ε]−1PρFρ(ε) =Pρfollows similarly, which proves (3.13). /square
+As a consequence of Lemmata 3.2 and 3.3, we obtain the followi ng lemma.
+Lemma 3.4. There exist gc>0andpc>0such that, for all 0≤ |g| ≤gc,0≤ |P| ≤pc, and
+g2≪ρ≪1,
+Fρ(0)Pρ1{Eg}(Hg)Pρ= 0. (3.16)
+Proof.We obtain from (3.13) that
+Fρ(ε)Pρ[Hg−Eg+ε]−1Pρ=Pρ. (3.17)
+for allε >0. It follows from the functional calculus that
+s−lim
+ε→0+ε[Hg−Eg+ε]−1=1{Eg}(Hg), (3.18)
+where s−lim stands for strong limit. Hence, using (3.6), we obtain (3 .16) by multiplying
+(3.17) by εand letting εgo to 0. /square
+Introducing (3.2) into (3.5), we obtain the following ident ity which holds for any ε≥0 and
+g2≪ρ≪1:
+Fρ(ε) =/parenleftbig
+H0−Eg+ε/parenrightbig
+Pρ+PρWgPρ
+−/summationdisplay
+n≥0PρWg/bracketleftbig
+H0−Eg+ε/bracketrightbig−1/parenleftbig
+−¯PρWg¯Pρ/bracketleftbig
+H0−Eg+ε/bracketrightbig−1/parenrightbign¯PρWgPρ.(3.19)HYPERFINE SPLITTING IN NON-RELATIVISTIC QED 13
+The next lemma will be used in the proof of Theorem 2.2.
+Lemma 3.5. There exist gc>0andpc>0such that, for all 0≤ |g| ≤gcand0≤ |P| ≤pc,
+P0Fρ(0)P0=/parenleftbig
+E0−Eg/parenrightbig
+P0
+−/summationdisplay
+λ=1,2/integraldisplay
+R3P0˜w(r,k,λ)/bracketleftbig
+Hr+1
+2M(P−k)2+|k|−Eg/bracketrightbig−1w(r,k,λ)P0dk
++O(|g|2+τ), (3.20)
+whereρ=|g|2−2τ,τ >0is fixed sufficiently small, and
+w(r,k,λ) :=−g
+mel/parenleftBig/parenleftbigmel
+M(P−Pph)+pr/parenrightbig
+·hA(mel
+Mg2
+3r,k,λ)/parenrightBig
++g
+mn/parenleftBig/parenleftbigmn
+M(P−Pph)−pr/parenrightbig
+·hA(−mn
+Mg2
+3r,k,λ)/parenrightBig
+−g
+2melσel·hB(mel
+Mg2
+3r,k,λ)+g
+2mnσn·hB(−mn
+Mg2
+3r,k,λ),(3.21)
+respectively
+˜w(r,k,λ) :=−g
+mel/parenleftBig/parenleftbigmel
+M(P−Pph)+pr/parenrightbig
+·¯hA(mel
+Mg2
+3r,k,λ)/parenrightBig
++g
+mn/parenleftBig/parenleftbigmn
+M(P−Pph)−pr/parenrightbig
+·¯hA(−mn
+Mg2
+3r,k,λ)/parenrightBig
+−g
+2melσel·¯hB(mel
+Mg2
+3r,k,λ)+g
+2mnσn·¯hB(−mn
+Mg2
+3r,k,λ).(3.22)
+Proof.SinceP0Pρ=PρP0=P0andH0P0=E0P0, we have that
+P0Fρ(0)P0=/parenleftbig
+E0−Eg/parenrightbig
+P0+P0WgP0−P0Wg/bracketleftbig
+H0−Eg/bracketrightbig−1¯PρWgP0
+−/summationdisplay
+n≥1P0Wg/bracketleftbig
+H0−Eg/bracketrightbig−1/parenleftbig
+−¯PρWg¯Pρ/bracketleftbig
+H0−Eg/bracketrightbig−1/parenrightbign¯PρWgP0. (3.23)
+By (A.13), P0WgP0= 0. Moreover, using (3.11) for n≥1, we obtain that
+P0Fρ(0)P0=/parenleftbig
+E0−Eg/parenrightbig
+P0−P0Wg/bracketleftbig
+H0−Eg/bracketrightbig−1¯PρWgP0+O(|g|3ρ−1
+2). (3.24)
+We conclude the proof by applying Lemma A.9 of Appendix A. /square
+4.Proof of Theorem 2.2
+From now on we assume that dimKer( Hg−Eg) = 4, which will lead to a contradiction at
+the end of this section.
+Lemma 4.1. There exist gc>0andpc>0such that, for all 0≤ |g| ≤gcand0≤ |P| ≤pc,
+the following holds: If dimKer( Hg−Eg) = 4, thenP01{Eg}(Hg)P0is invertible on Ran(P0)
+and satisfies
+/vextenddouble/vextenddouble[P01{Eg}(Hg)P0]−1/vextenddouble/vextenddouble≤1
+1−Cg2. (4.1)
+Proof.In order to prove that P01{Eg}(Hg)P0is invertible on Ran( P0), it suffices to show that
+/vextenddouble/vextenddoubleP0−P01{Eg}(Hg)P0/vextenddouble/vextenddouble<1. (4.2)14 L. AMOUR AND J. FAUPIN
+Observe that P0−P01{Eg}(Hg)P0is a finite rank and positive operator. We have that
+/vextenddouble/vextenddoubleP0−P01{Eg}(Hg)P0/vextenddouble/vextenddouble≤tr(P0−P01{Eg}(Hg)P0)
+= tr(P0)−tr(P01{Eg}(Hg))
+= tr(P0)−tr(1{Eg}(Hg))+tr(¯P01{Eg}(Hg))
+= 4−4+tr(¯P01{Eg}(Hg)) = tr(¯P01{Eg}(Hg)).(4.3)
+The projection ¯P0can be decomposed as
+¯P0=1⊗¯Pφ0⊗PΩ+1⊗1⊗¯PΩ. (4.4)
+It follows from Lemma A.7 that
+tr((1⊗¯Pφ0⊗PΩ)1{Eg}(Hg))≤Cg2, (4.5)
+and from Lemma A.5 that
+tr((1⊗1⊗¯PΩ)Pg)≤tr(NphPg)≤Cg2. (4.6)
+Therefore, /ba∇dblP0−P01{Eg}(Hg)P0/ba∇dbl ≤Cg2. The invertibility of P01{Eg}(Hg)P0and Equation
+(4.1) directly follow from the latter estimate. /square
+As a consequence of Lemma 4.1, we obtain the following lemma.
+Lemma 4.2. LetΓdenote the operator on Ran(P0)defined by
+Γ :=/summationdisplay
+λ=1,2/integraldisplay
+R3P0˜w(r,k,λ)/bracketleftbig
+Hr+1
+2M(P−k)2+|k|−Eg/bracketrightbig−1w(r,k,λ)P0dk, (4.7)
+withw(r,k,λ)and˜w(r,k,λ)as in(3.21)–(3.22). There exist gc>0andpc>0such that, for
+all0≤ |g| ≤gcand0≤ |P| ≤pc, the following holds: If dimKer( Hg−Eg) = 4, then
+Γ =/parenleftbig
+E0−Eg/parenrightbig
+P0+O(|g|2+τ), (4.8)
+whereτ >0is fixed sufficiently small.
+Proof.Fixρ=|g|2−2τfor some sufficiently small τ >0. Multiplying both sides of Equation
+(3.16) by P0, we get
+P0Fρ(0)Pρ1{Eg}(Hg)P0= 0. (4.9)
+Introducing the decomposition 1=P0+¯P0into (4.9) and using Lemma 4.1, this yields
+P0Fρ(0)P0=−P0Fρ(0)Pρ¯P01{Eg}(Hg)P0[P01{Eg}(Hg)P0]−1. (4.10)
+By Equations (4.4), (4.5) and (4.6), we learn that
+/vextenddouble/vextenddouble¯P01{Eg}(Hg)/vextenddouble/vextenddouble≤tr(¯P01{Eg}(Hg))≤Cg2, (4.11)
+which, combined with (3.7) and (4.1), implies that
+/vextenddouble/vextenddoubleP0Fρ(0)Pρ¯P01{Eg}(Hg)P0[P01{Eg}(Hg)P0]−1/vextenddouble/vextenddouble≤Cg2ρ= C|g|4−2τ.(4.12)
+We conclude the proof thanks to Lemma 3.5. /square
+Let us consider the canonical orthonormal basis of C4in which the Pauli matrices σel
+j,
+σn
+j,j∈ {1,2,3}, are given by (2.17)–(2.18). Obviously, Γ identifies with a 4 ×4 matrix in
+this basis. In the next theorem, we determine a non-diagonal coefficient of Γ of the form
+−C0g2+o(g2) with C 0>0.HYPERFINE SPLITTING IN NON-RELATIVISTIC QED 15
+Theorem 4.3. LetΓbe given as in (4.7). There exist gc>0andpc>0such that, for all
+0≤ |g| ≤gcand0≤ |P| ≤pc, the coefficient of Γlocated on the third line and second column,
+Γ32, satisfies
+Γ32=−C0g2+O(|g|8
+3), (4.13)
+whereC0is a strictly positive constant independent of g.
+Proof.We view w(r,k,λ) as a linear combination (some coefficients being given by ope rators)
+ofthefunctions hA
+j(···) andhB
+j(···),j∈ {1,2,3}. We introducethecorrespondingexpression
+into (4.7) and consider each term separately.
+Since the coefficients located on the third line and second col umn of the Pauli matrices
+expressed in (2.17)–(2.18) vanish, the terms containing at least one factor hA
+j(···) do not
+contribute to Γ 32. The same holds for the terms containing at least one factor hB
+3(···), since
+the third Pauli matrices, σel
+3andσn
+3, are diagonal.
+Therefore, Γ 32is equal to the coefficient located on the third line and second column of the
+matrix Γ′given by
+Γ′=/summationdisplay
+λ=1,2/integraldisplay
+R3P0/summationdisplay
+j=1,2/parenleftBig
+−g
+2melσel
+j¯hB
+j(mel
+Mg2
+3r,k,λ)+g
+2mnσn
+j¯hB
+j(−mn
+Mg2
+3r,k,λ)/parenrightBig
+/bracketleftBig
+Hr+1
+2M(P−k)2+|k|−Eg/bracketrightBig−1
+/summationdisplay
+j′=1,2/parenleftBig
+−g
+2melσel
+j′hB
+j′(mel
+Mg2
+3r,k,λ)+g
+2mnσn
+j′hB
+j′(−mn
+Mg2
+3r,k,λ)/parenrightBig
+P0dk.(4.14)
+It follows from the definition (2.14) of hB
+jthat
+/vextendsingle/vextendsinglehB
+j(r,k,λ)−hB
+j(0,k,λ)/vextendsingle/vextendsingle≤C|k|3
+2χΛ(k)|r|, (4.15)
+for anyj∈ {1,2,3},λ∈ {1,2},r∈R3andk∈R3. Moreover, the expression (2.29) of φ0
+implies that/vextenddouble/vextenddouble|r|φ0(r)/vextenddouble/vextenddouble≤C. (4.16)
+Hence, using in addition that, for |P|sufficiently small,
+/vextenddouble/vextenddouble/vextenddouble/bracketleftBig
+Hr+(P−k)2
+2M+|k|−Eg/bracketrightBig−1/vextenddouble/vextenddouble/vextenddouble≤C
+|k|, (4.17)
+we obtain from (4.14) and (4.15)–(4.17) that
+Γ′=/summationdisplay
+λ=1,2/integraldisplay
+R3P0/summationdisplay
+j=1,2/parenleftBig
+−g
+2melσel
+j¯hB
+j(0,k,λ)+g
+2mnσn
+j¯hB
+j(0,k,λ)/parenrightBig
+/bracketleftBig
+e0+1
+2M(P−k)2+|k|−Eg/bracketrightBig−1
+/summationdisplay
+j′=1,2/parenleftBig
+−g
+2melσel
+j′hB
+j′(0,k,λ) +g
+2mnσn
+j′hB
+j′(0,k,λ)/parenrightBig
+P0dk+O(|g|8
+3).(4.18)
+Notice now that, for j,j′∈ {1,2}, the coefficient on the third line and second column of
+the products σel
+jσel
+j′andσn
+jσn
+j′vanishes. We thus obtain from (4.18) that
+Γ32= Γ′
+32=γ1+γ2+O(|g|8
+3), (4.19)16 L. AMOUR AND J. FAUPIN
+where
+γ1:=−g2
+4melmn/summationdisplay
+λ=1,2/integraldisplay
+R3(φ0,/bracketleftbig¯hB
+1(0,k,λ)+i¯hB
+2(0,k,λ)/bracketrightbig
+/bracketleftBig
+e0+1
+2M(P−k)2+|k|−Eg/bracketrightBig−1/bracketleftbig
+hB
+1(0,k,λ)−ihB
+2(0,k,λ)/bracketrightbig
+φ0)dk, (4.20)
+and
+γ2:=−g2
+4melmn/summationdisplay
+λ=1,2/integraldisplay
+R3(φ0,/bracketleftbig¯hB
+1(0,k,λ)−i¯hB
+2(0,k,λ)/bracketrightbig
+/bracketleftBig
+e0+1
+2M(P−k)2+|k|−Eg/bracketrightBig−1/bracketleftbig
+hB
+1(0,k,λ) +ihB
+2(0,k,λ)/bracketrightBig
+φ0)dk. (4.21)
+We remark that the cross terms involving hB
+1(0,k,λ) andhB
+2(0,k,λ) vanish. Thus, we obtain
+Γ32=−g2
+2melmn/summationdisplay
+j=1,2/summationdisplay
+λ=1,2/integraldisplay
+R3¯hB
+j(0,k,λ)
+/bracketleftBig
+e0+(P−k)2
+2M+|k|−Eg/bracketrightBig−1
+hB
+j(0,k,λ)dk+O(|g|8
+3).(4.22)
+The integral in the right-hand-side of (4.22) still depends ongthrough the ground state
+energyEg. Nevertheless, one can readily check that
+/vextendsingle/vextendsingle/vextendsingle/bracketleftBig
+e0+(P−k)2
+2M+|k|−Eg/bracketrightBig−1
+−/bracketleftBig
+e0+(P−k)2
+2M+|k|−E0/bracketrightBig−1/vextendsingle/vextendsingle/vextendsingle
+≤ |E0−Eg|C
+|k|2≤C′g2
+|k|2, (4.23)
+where, in the last inequality, we used Lemma A.6. Therefore, since, for any j∈ {1,2}and
+λ∈ {1,2}, the functions hB
+j(0,k,λ) satisfy |hB
+j(0,k,λ)| ≤C|k|1/2χΛ(k), we get
+Γ32=−g2
+2melmn/summationdisplay
+j=1,2/summationdisplay
+λ=1,2/integraldisplay
+R3¯hB
+j(0,k,λ)
+/bracketleftBig
+e0+(P−k)2
+2M+|k|−E0/bracketrightBig−1
+hB
+j(0,k,λ)dk+O(|g|8
+3).(4.24)
+Now, the integrals in the right-hand-side of (4.24) can be ex plicitly computed, which leads to
+Γ32=−g2
+8π2melmn/integraldisplay
+R3|k|χΛ(k)2
+k2/2M−k·P/M+|k|/parenleftbigk2
+3
+|k|2+1/parenrightbig
+dk+O(|g|8
+3). (4.25)
+The integrand in (4.25) is strictly positive (for Psufficiently small), and hence the integral
+does not vanish. This concludes the proof of the theorem. /square
+We are now able to prove Theorem 2.2:
+Proof of Theorem 2.2 . By [AGG2], we know that dimKer( Hg−Eg)≤4. Assume by con-
+tradiction that dimKer( Hg−Eg) = 4. By Lemma 4.2, the matrix Γ defined in (4.7) satisfies
+(4.8). In particular, in any basis of C4, the non-vanishingterms of order g2of Γ are necessarily
+located on the diagonal. However, according to Theorem 4.3, in the canonical orthonormal
+basis ofC4in which the Pauli matrices are given by (2.17)–(2.18), the n on-diagonal coefficientHYPERFINE SPLITTING IN NON-RELATIVISTIC QED 17
+Γ32contains a non-vanishing term of order g2. Hence we get a contradiction and the theorem
+is proven. /square
+Appendix A.
+In this appendix, we collect some estimates which were used i n Sections 3 and 4. Some of
+them are standard (see for instance [BFS1, BFS2]). We begin w ith two lemmata concerning
+the non-interacting Hamiltonian H0defined in (2.35).
+Lemma A.1. There exists pc>0such that for all 0≤ |P| ≤pc,
+Hph≤2(H0−E0). (A.1)
+Proof.Forj∈ {1,2,3}, one can easily verify that |(Pph)j| ≤Hph. Hence, since E0=
+e0+P2/2M, we have that
+H0=Hr+P2
+2M−1
+MP·Pph+1
+2MP2
+ph+Hph≥E0+1
+2Hph, (A.2)
+forPsufficiently small, which proves the lemma. /square
+Lemma A.2. There exists pc>0such that, for all 0≤ |P| ≤pcandρ≥0,
+¯PρH0¯Pρ≥/parenleftbigP2
+2M+min(e0+ρ
+2,e1)/parenrightbig¯Pρ. (A.3)
+Proof.SincePρ=1⊗Pφ0⊗1Hph≤ρin the tensor product C4⊗L2(R3)⊗Hph, we can write
+¯Pρ=1−Pρ=1⊗¯Pφ0⊗1Hph≤ρ+1⊗1⊗1Hph≥ρ, (A.4)
+where¯Pφ0=1−Pφ0. SinceHr¯Pφ0≥e1¯Pφ0, we get that
+H0(1⊗¯Pφ0⊗1Hph≤ρ)≥/parenleftbig
+e1+P2
+2M/parenrightbig
+(1⊗¯Pφ0⊗1Hph≤ρ), (A.5)
+forPsmall enough. Moreover, by Lemma A.1,
+H0(1⊗1⊗1Hph≥ρ)≥/parenleftbig
+e0+P2
+2M+ρ
+2/parenrightbig
+(1⊗1⊗1Hph≥ρ). (A.6)
+Hence (A.3) is proven. /square
+The proof of the next two lemmata being standard, we omit them .
+Lemma A.3. For anyf∈L2(R3×{1,2}), the operators a(f)[Nph¯PΩ]−1/2and[Nph¯PΩ]−1/2a(f)
+extend to bounded operators on Hphsatisfying
+/vextenddouble/vextenddoublea(f)[Nph¯PΩ]−1
+2/vextenddouble/vextenddouble≤ /ba∇dblf/ba∇dbl, (A.7)
+/vextenddouble/vextenddouble[Nph¯PΩ]−1
+2a(f)/vextenddouble/vextenddouble≤√
+2/ba∇dblf/ba∇dbl. (A.8)
+Lemma A.4. Letf∈L2(R3×{1,2})be such that (k,λ)/ma√sto→ |k|−1/2f(k,λ)∈L2(R3×{1,2}).
+Then, for any ρ >0, the operators a(f)[Hph+ρ]−1/2and[Hph+ρ]−1/2a(f)extend to bounded
+operators on Hphsatisfying
+/vextenddouble/vextenddoublea(f)[Hph+ρ]−1
+2/vextenddouble/vextenddouble≤ /ba∇dbl|k|−1
+2f/ba∇dbl, (A.9)
+/vextenddouble/vextenddouble[Hph+ρ]−1
+2a(f)/vextenddouble/vextenddouble≤ /ba∇dbl|k|−1
+2f/ba∇dbl+ρ−1
+2/ba∇dblf/ba∇dbl. (A.10)18 L. AMOUR AND J. FAUPIN
+The following lemma is taken from [AGG2]. Its proof is based o n a “pull-through” formula
+(see [AGG2]).
+Lemma A.5. There exist gc>0andpc>0such that, for all 0≤ |g| ≤gcand0≤ |P| ≤pc,
+the following holds:
+∀Φg∈Ker(Hg−Eg),/ba∇dblΦg/ba∇dbl= 1,we have (Φg,NphΦg)≤Cg2, (A.11)
+whereCis a positive constant independent of g.
+In the next lemma, we estimate the difference between the groun d state energies Eg=
+infσ(Hg) andE0= infσ(H0).
+Lemma A.6. There exist gc>0andpc>0such that, for all 0≤ |g| ≤gcand0≤ |P| ≤pc,
+Eg≤E0≤Eg+Cg2, (A.12)
+whereCis a positive constant independent of g.
+Proof.Note that, since the perturbation Wgis Wick-ordered, we have that
+(1⊗1⊗PΩ)Wg(1⊗1⊗PΩ) = 0, (A.13)
+where, recall, PΩdenotes the orthogonal projection onto the vector space spa nned by the
+Fock vacuum Ω. Hence, by the Rayleigh-Ritz principle,
+Eg≤/parenleftbig
+(y⊗φ0⊗Ω),Hg(y⊗φ0⊗Ω)/parenrightbig
+=/parenleftbig
+(y⊗φ0⊗Ω),H0(y⊗φ0⊗Ω)/parenrightbig
+=E0,(A.14)
+where, as above, ydenotes an arbitrary normalized element in C4.
+In order to prove the second inequality in (A.12), we use Lemm ata A.3 and A.5. More
+precisely, let Φ g∈Ker(Hg−Eg),/ba∇dblΦg/ba∇dbl= 1 (Φ gexists by [AGG2]). We have
+E0−Eg≤(Φg,(H0−Hg)Φg) =−(Φg,WgΦg). (A.15)
+Recall that Wgis given by the Wick-ordered expression obtained from (2.37 ). We express the
+latter in terms of operators of creation and annihilation, a nd estimate each term separately.
+Consider for instance the term
+g
+mel/parenleftBig/parenleftbigmel
+M(P−Pph)+pr/parenrightbig
+·a(hA(mel
+Mg2
+3r))/parenrightBig
+. (A.16)
+It is not difficult to check that
+(P−Pph)2≤aH0+bandp2
+r≤aH0+b, (A.17)
+for some positive constants aandbdepending on µandM. One easily deduces from (A.17)
+that/vextenddouble/vextenddouble/parenleftbigmel
+M(P−Pph)+pr/parenrightbig
+Φg/vextenddouble/vextenddouble≤C. (A.18)
+Moreover, by Lemmata A.3 and A.5, we have that
+/vextenddouble/vextenddoublea(hA(mel
+Mg2
+3r))Φg/vextenddouble/vextenddouble≤C/vextenddouble/vextenddoubleN1
+2
+phΦg/vextenddouble/vextenddouble≤C′|g|. (A.19)
+Equations (A.18) and (A.19) imply that
+/vextendsingle/vextendsingle(Φg,(A.16)Φ g)/vextendsingle/vextendsingle≤Cg2, (A.20)
+and since the other terms in Wgare estimated similarly, this concludes the proof. /squareHYPERFINE SPLITTING IN NON-RELATIVISTIC QED 19
+Lemma A.5 gives an estimation of the overlap of the ground sta te ΦgofHgwith the Fock
+vacuum. We also need to estimate the overlap of Φ gwith the ground state φ0of the electronic
+Hamiltonian Hrin the sense stated in the following lemma.
+Lemma A.7. There exist gc>0andpc>0such that, for all 0≤ |g| ≤gcand0≤ |P| ≤pc,
+the following holds:
+∀Φg∈Ker(Hg−Eg),/ba∇dblΦg/ba∇dbl= 1,we have |(Φg,(1⊗¯Pφ0⊗PΩ)Φg)| ≤Cg2,(A.21)
+whereCis a positive constant independent of g.
+Proof.Let Φgbe a normalized ground state of Hg, that is ( Hg−Eg)Φg= 0,/ba∇dblΦg/ba∇dbl= 1. Since
+E0−Eg=e0+P2/2M−Eg≥0 by Lemma A.6, we have that
+0 =/parenleftbig
+Φg,(1⊗¯Pφ0⊗PΩ)(Hg−Eg)Φg/parenrightbig
+=/parenleftbig
+Φg,(1⊗¯Pφ0⊗PΩ)/parenleftbig
+Hr+P2
+2M−Eg+Wg/parenrightbig
+Φg)
+≥/parenleftbig
+Φg,(1⊗¯Pφ0⊗PΩ)(e1−e0+Wg)Φg/parenrightbig
+, (A.22)
+and hence
+(Φg,(1⊗¯Pφ0⊗PΩ)Φg)≤ −1
+e1−e0(Φg,(1⊗¯Pφ0⊗PΩ)WgΦg). (A.23)
+We conclude the proof thanks to Lemmata A.3 and A.5, by arguin g in the same way as in
+the proof of Lemma A.6. /square
+We now give estimates relating the perturbation WgtoH0.
+Lemma A.8. There exist gc>0andpc>0such that, for all 0≤ |g| ≤gc,0≤ |P| ≤pc,
+0< ρ≪1andε≥0, the following estimates hold:
+/vextenddouble/vextenddouble[H0−Eg+ε]−1
+2¯PρWg¯Pρ[H0−Eg+ε]−1
+2/vextenddouble/vextenddouble≤C|g|ρ−1
+2, (A.24)
+/vextenddouble/vextenddoublePρWg¯Pρ[H0−Eg+ε]−1
+2/vextenddouble/vextenddouble≤C|g|, (A.25)
+/vextenddouble/vextenddouble[H0−Eg+ε]−1
+2¯PρWgPρ/vextenddouble/vextenddouble≤C|g|, (A.26)
+/vextenddouble/vextenddoublePρWgPρ/vextenddouble/vextenddouble≤C|g|ρ1
+2. (A.27)
+Proof.Let us begin with proving (A.24). As in the proof of Lemma A.6, we express Wgin
+terms of creation and annihilation operators from the Wick- ordered expression obtained from
+(2.37), and we estimate each term separately. Let us conside r again the term (A.16) as an
+example. Using (A.17), Lemma A.2, and the fact that E0≥Eg, we obtain
+/vextenddouble/vextenddouble/vextenddouble[H0−Eg+ε]−1
+2¯Pρ/parenleftbigmel
+M(P−Pph)+pr/parenrightbig
+j/vextenddouble/vextenddouble/vextenddouble≤Cρ−1
+2. (A.28)
+forj∈ {1,2,3}. Next, for j∈ {1,2,3}, Lemma A.4 gives
+/vextenddouble/vextenddoublea(hA
+j(mel
+Mg2
+3r))[Hph+ρ]−1/2/vextenddouble/vextenddouble≤C, (A.29)
+and it follows from Lemmata A.1 and A.2 that
+/vextenddouble/vextenddouble[Hph+ρ]1
+2¯Pρ[H0−Eg+ε]−1
+2/vextenddouble/vextenddouble≤C. (A.30)20 L. AMOUR AND J. FAUPIN
+Using (A.28), (A.29) and (A.30), we obtain
+/vextenddouble/vextenddouble[H0−Eg+ε]−1
+2¯Pρ(A.16)¯Pρ[H0−Eg+ε]−1
+2/vextenddouble/vextenddouble≤C|g|ρ−1
+2. (A.31)
+Theothertermsin Wgareestimated similarly, usinginparticularEstimate(A.1 0)(inaddition
+to (A.9)) for the terms quadratic in the annihilation and cre ation operators. Hence (A.24) is
+proven. In order to prove (A.25), (A.26) and (A.27), we proce ed similarly, using the further
+following estimates:
+/vextenddouble/vextenddouble/vextenddouble/parenleftbigmel
+M(P−Pph)+pr/parenrightbig
+jPρ/vextenddouble/vextenddouble/vextenddouble≤C, (A.32)
+/vextenddouble/vextenddouble[Hph+ρ]1
+2Pρ/vextenddouble/vextenddouble≤Cρ1
+2. (A.33)
+Estimate (A.32) follows from (A.17), and (A.33) is an obviou s consequence of the Spectral
+Theorem. /square
+Lemma A.9. There exist gc>0andpc>0such that, for all 0≤ |g| ≤gc,0≤ |P| ≤pc,
+0< ρ≪1, andε≥0, we have
+P0Wg/bracketleftbig
+H0−Eg/bracketrightbig−1¯PρWgP0
+=/summationdisplay
+λ=1,2/integraldisplay
+R3P0˜w(r,k,λ)/bracketleftbig
+Hr+1
+2M(P−k)2+|k|−Eg/bracketrightbig−1w(r,k,λ)P0dk
++O(|g|3)+O(g2ρ), (A.34)
+wherew(r,k,λ)and˜w(r,k,λ)are defined in (3.21)–(3.22).
+Proof.The perturbation Wgappears twice in P0Wg[H0−Eg]−1¯PρWgP0. We introduce the
+expression (2.37) of Wginto the latter operator, and consider each term separately .
+First, the terms containing a creation operator in the “first ”Wgvanish since P0projects
+onto the Fock vaccum. The same holds for the terms containing an annihilation operator in
+the “second” Wg.
+Next, the terms involving the parts of Wgquadratic in the creation and annihilation oper-
+ators are (at least) of order O(|g|3), as follows again from Lemmata A.4 and A.8.
+Therefore, one can compute
+P0Wg/bracketleftbig
+H0−Eg/bracketrightbig−1¯PρWgP0
+=/summationdisplay
+λ=1,2/integraldisplay
+R3P0˜w(r,k,λ)/bracketleftbig
+Hr+1
+2M(P−k)2+|k|−Eg/bracketrightbig−1w(r,k,λ)P0dk
+−/summationdisplay
+λ=1,2/integraldisplay
+|k|≤ρP0˜w(r,k,λ)/bracketleftbig
+e0+1
+2M(P−k)2+|k|−Eg/bracketrightbig−1
+×(1⊗Pφ0⊗1)w(r,k,λ)P0dk+O(|g|3).(A.35)HYPERFINE SPLITTING IN NON-RELATIVISTIC QED 21
+The second term in the right-hand-side of (A.35) is estimate d as follows:
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+λ=1,2/integraldisplay
+|k|≤ρP0˜w(r,k,λ)/bracketleftbig
+e0+1
+2M(P−k)2+|k|−Eg/bracketrightbig−1
+×(1⊗Pφ0⊗1)w(r,k,λ)P0dk/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤/summationdisplay
+λ=1,2/integraldisplay
+|k|≤ρC
+|k|2dk≤C′ρ. (A.36)
+Hence (A.34) is proven. /square
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+Invent. Math., 145, (2001), 557–595.22 L. AMOUR AND J. FAUPIN
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+(L. Amour) Laboratoire de Math ´ematiques EDPPM, FRE-CNRS 3111, Universit ´e de Reims,
+Moulin de la Housse - BP 1039, 51687 REIMS Cedex 2, France
+E-mail address :laurent.amour@univ-reims.fr
+(J. Faupin) Institut de Math ´ematiques de Bordeaux, UMR-CNRS 5251, Universit ´e de Bordeaux
+1, 351 cours de la lib ´eration, 33405 Talence Cedex, France
+E-mail address :jeremy.faupin@math.u-bordeaux1.fr
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+arXiv:1001.0714v1  [math.FA]  5 Jan 2010A convex body whose centroid and Santal´ o
+point are far apart∗
+Mathieu Meyer, Carsten Sch¨ utt and Elisabeth M. Werner†
+Abstract
+We give an example of a convex body whose centroid and Santal´ o poin t
+are “far apart”.
+∗Keywords: Santal´ o point, . 2000 Mathematics Subject Class ification: 52A20, 53A15
+†Partially supported by an NSF grant, a FRG-NSF grant and a BSF grant
+11 Introduction. The main theorem.
+In his survey paper [7], Gr¨ unbaum introduced measures of asymme try for convex bodies
+inRn, i.e. compact, convex sets in Rnwith nonempty interior. Measures of asymmetry
+determine the degree of non-symmetricity of a non symmetric conv ex body. Recall that
+a convex set KinRnis centrally symmetric with respect to the origin if x∈Kimplies
+−x∈K. More generally, Kis centrally symmetric with respect to the point zifx∈K
+implies 2 z−x∈K.
+Gr¨ unbaum argues that a measure of asymmetry should be a funct ion that is 0 for
+centrallysymmetricconvexbodiesand maximalforthe simplex andon lyforthesimplex.
+However, the simplex has enough symmetries to ensure that all affin e invariant points
+are identical. This leads to the question: How far apart can specific a ffine invariant
+points be? This distance can be used as a measure for the asymmetr y of the convex
+body.
+Of particular interest are the centroid g(K) and the Santal´ o point s(K) of a convex
+bodyKinRn. These points are of major importance and have been widely studied . For
+the (less well known) Santal´ o point see e.g. [4, 12, 14, 13, 20]. Rece nt developments in
+this direction include e.g. the works [1, 2, 3, 5, 6, 10, 15]. We show her e that the points
+can be very far apart, namely
+Theorem 1 There is an absolute constant c >0andn0∈Nsuch that for all n≥n0
+there is a convex body C=CninRnwith
+c≤/bardblg(C)−s(C)/bardbl2
+vol1(ℓ∩C)=/bardblg(C)−s(C)/bardbl2
+wC(u). (1)
+/bardbl·/bardbl2is the Euclidean norm, lis the line through g=g(C)ands=s(C)andwC(u) =
+hC(u)+hC(−u)is the width of Cin directon of the unit vector u=g−s
+/bardblg−s/bardbl2.
+The proof actually shows that we can asymptotically determine the c onstantcof the
+theorem.
+/bardblg(C)−s(C)/bardbl2
+vol1(ℓ∩C)
+is asymptotically with respect to the dimension, greater than or equ al to
+/parenleftbigg
+1−1
+e/parenrightbigg√eπ−2√eπ+2
+e−1= 0.142673...
+and is asymptotically with respect to the dimension, less than or equa l to
+/parenleftbigg
+1−1
+e/parenrightbigg√e−1√e+1
+e−1= 0.18383...
+Throughout the paper we use the following notations. Bn
+2(a,r) is the Euclidean ball in
+Rncentered at awith radius r. We denote Bn
+2=Bn
+2(0,1).Sn−1is the boundary ∂Bn
+2
+2of the Euclidean unit ball. For x∈Rn, let/bardblx/bardblp= (/summationtextn
+i=1|xi|p)1
+p, if 1≤p <∞and
+/bardblx/bardbl∞= max 1≤i≤n|xi|, ifp=∞and letBn
+p={x∈Rn:/bardblx/bardblp≤1}be the unit ball of
+the space ln
+p.
+For a convex body KinRnwe denote the volume by vol n(K) (if we want to emphasize
+the dimension) or by |K|. co[A,B] ={λa+(1−λ)b:a∈A,b∈B,0≤λ≤1}is the
+convex hull of AandB.
+hK(u) = max x∈K/an}bracketle{tu,x/an}bracketri}htis the supportfunction of Kin direction u.
+Forx∈K,Kx= (K−x)◦={y∈Rn:/an}bracketle{ty,z−x/an}bracketri}ht ≤1 for all z ∈K}is the polar
+body ofKwith respect to x.
+Letξ∈Rnwith/bardblξ/bardbl2= 1 and s∈R. Then the ( n−1)-dimensional section of K
+orthogonal to ξthrough sξis
+K(s,ξ) ={x∈K|/an}bracketle{tξ,x/an}bracketri}ht=s}.
+We just write K(s) if it is clear which direction ξis meant.
+The centroid g(K) of a convex body KinRnis the point
+g(K) =1
+voln(K)/integraldisplay
+Kxdx.
+The Santal´ o point s(K) of a convex body Kis the unique point s(K) for which
+voln(K)voln(Kx)
+attains its minimum. It is more difficult to compute the Santal´ o point as it is defined
+implicitly.
+Before we go into the construction of the convex bodies that fulfill the properties
+of Theorem 1, we discuss an example that shows that a more elabora te construction is
+necessary to get the statement of Theorem 1.
+For instance, it is not enough to take one half of a Euclidean ball
+B={x∈Rn|/bardblx/bardbl2≤1,x1≥0}.
+We compute the centroid of B. The coordinates of the centroid are g(B)(2) =···=
+g(B)(n) = 0 and
+g(B)(1) =2
+|Bn
+2|/integraldisplay1
+0tvoln−1/parenleftig/radicalbig
+1−t2Bn−1
+2/parenrightig
+dt=2voln−1/parenleftbig
+Bn−1
+2/parenrightbig
+(n+1)|Bn
+2|.
+NowwecomputetheSantal´ opointof B. ThepolarbodyoftheEuclideanball Bn
+2(λe1,1)
+with center λe1, 0≤λ <1, and radius 1 is
+/braceleftigg
+x∈Rn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1−λ2)2/parenleftbigg
+x1+λ
+1−λ2/parenrightbigg2
++(1−λ2)x2
+2+···+(1−λ2)x2
+n≤1/bracerightigg
+.(2)
+Therefore, the polar body of the half ball Bwith respect to λe1is the convex hull of
+the vector −e1
+λand the ellipsoid (2). We estimate the volume of ( B−λe1)◦. Since
+(B−λe1)◦contains the ellipsoid (2)
+(1−λ2)−n+1
+2voln(Bn
+2)≤voln((B−λe1)◦).
+3On the other hand, as ( B−λe1)◦is contained in the union of the ellipsoid (2) and a
+cone with height1
+λand with a base that is a Euclidean ball with radius (1+ λ2)−1,
+voln((B−λe1)◦)≤voln(Bn
+2)
+(1−λ2)n+1
+2+voln−1(Bn−1
+2)
+nλ(1+λ2)n−1
+Forλ=1√nwe get
+voln((B−λe1)◦)≤/parenleftbigg
+1−1
+n/parenrightbigg−n+1
+2
+voln(Bn
+2)+voln−1(Bn−1
+2)√n(1+1
+n)n−1.
+Since
+voln−1(Bn−1
+2)
+voln(Bn
+2)∼/radicalbiggπn
+2
+we get
+voln((B−λe1)◦)/lessorsimilar/parenleftbigg√e+1
+e/radicalbiggπ
+2/parenrightbigg
+voln(Bn
+2).
+On the other hand, for λ=γ√n
+voln((B−λe1)◦)≥/parenleftbigg
+1−γ2
+n/parenrightbigg−n+1
+2
+voln(Bn
+2).
+With 1−t≤e−t
+voln((B−λe1)◦)≥eγ2n+1
+2nvoln(Bn
+2).
+Therefore, there is a γsuch that for all n∈Nwe haves(B)(1)≤γ√n. Thus/bardblg(B)−s(B)/bardbl2
+vol1(ℓ∩B)
+is of order1√n.
+The next lemma is well known (see [20]).
+Lemma 2 For any convex body K, an interior point xofKis the Santal´ o point if and
+only if0is the centroid of (K−x)◦.
+This lemma can be rephrased as follows:
+LetKbe a convex body. Then 0is the Santal´ o point of Kg(K).
+Indeed,Kg(K)= (K−g(K))◦and (K−g(K))◦◦=K−g(K). Since 0 is the centroid of
+(K−g(K))◦◦, it follows by Lemma 2 that 0 is the Santal´ o point of ( K−g(K))◦=Kg(K).
+For convex bodies KandLinRnand natural numbers k, 0≤k≤n, the coefficients
+Vn−k,k(K,L) =V(K,...,K/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+n−k,L,...,L/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+k)
+4in the expansion
+voln(λ1K+λ2L) =n/summationdisplay
+k=0/parenleftbiggn
+k/parenrightbigg
+λn−k
+1λk
+2Vn−k,k(K,L) (3)
+are the mixed volumes of KandL(see [20]).
+Now we introduce the convex bodies which will serve as candidates fo r Theorem 1.
+For convex bodies KandLinRnand real numbers a >0 andb >0, we construct a
+convex body MninRn+1
+Mn= co[(K,−a),(L,b)] ={t(x,−a)+(1−t)(y,b)|x∈K,y∈L,0≤t≤1}.(4)
+The bodies we are using in Theorem 1 will be the polar bodies to Mn. The polar of Mn
+can be described as follows: For −1
+a≤s≤1
+b, the sections of M◦
+northogonal to en+1
+and containing sen+1are
+M◦
+n(s) =M◦
+n(s,en+1) = (1+ s a)K◦∩(1−s b)L◦. (5)
+We show this.
+M◦
+n={(z,s)∈Rn×R|∀x∈K:< z,x > −sa≤1 and∀y∈L:< z,y > +sb≤1}
+={(z,s)∈Rn×R|z∈(1+sa)K◦∩(1−bs)L◦}
+The body Mnis the convexhulloftwo n-dimensionalfaces KandL. In the following
+proposition we choose specific bodies for those faces. We choose t hem in such a way
+that their volume product differs greatly, which has as effect that t he centroid and the
+Santal´ o point of Mg(Mn)
+nare “far apart”. One face is the Euclidean ball and the other
+the unit ball of ℓn
+∞. We normalize both balls so that their volume is 1.
+Proposition 3 Let
+K=Bn
+2
+|Bn
+2|1
+nL=1
+2Bn
+∞.
+anda= 1andb=1
+e−1. LetMnbe the convex body in Rn+1defined in (4). Then
+(i)limn→∞g(Mn) = 0.
+(ii) Let
+s0=−√e−1√e+1
+e−1=−0.290815... s 1=2−√eπ√eπ+2
+e−1=−0.225705...
+Then for every ε >0there isn0such that for all n≥n0then+1-st coordinate of the
+centroid g(M◦
+n)ofM◦
+nsatisfies
+s0−ε≤g(M◦
+n)(n+1)≤s1+ε.
+5The Santal´ o point of Mg(Mn)
+n,s∗
+n=s(Mg(Mn)
+n) = 0and therefore the centroid g∗
+nand
+the Santal´ o point s∗
+nofMg(Mn)
+nsatisfy
+√e−1√e+1
+e−1≥limsup
+n→∞|g∗
+n−s∗
+n| ≥liminf
+n→∞|g∗
+n−s∗
+n|= liminf
+n→∞|g∗
+n| ≥2−√eπ√eπ+2
+e−1.
+As for the proof of Proposition 3, it follows from Lemma 2 that s(Mg(Mn)
+n) = 0. For the
+centroid g∗
+nwe actually estimate the coordinates of g(M◦
+n) and not g(Mg(Mn)
+n). Since
+limg(Mn) = 0 it suffices to use a perturbation argument. The remaining part o f the
+proof of the proposition is in the next section.
+Proof of Theorem 1. The proof follows immediately from Proposition 3: For every
+n, letCn=M◦
+n.
+2 Proof of Proposition 3
+This section is devoted to the proof of Proposition 3. We will need sev eral lemmas and
+then give the proof of the proposition at the end of the section.
+Lemma 4 LetKandLbe convex bodies in Rnsuch that their centroid are at 0. Let
+c >0and letMnbe the convex body in Rn+1
+Mn=co[(K,0),(L,c)].
+Then the (n+1)-coordinate of the centroid g(Mn)(n+1)ofMnsatisfies
+g(Mn)(n+1) =/an}bracketle{tg(Mn),en+1/an}bracketri}ht=c
+n+2/summationtextn
+k=0(k+1)Vn−k,k(K,L)/summationtextn
+k=0Vn−k,k(K,L).
+Proof.By definition
+/an}bracketle{tg(Mn),en+1/an}bracketri}ht=/integraltextc
+0wvoln(Mn(w))dw/integraltextc
+0voln(Mn(w))dw.
+Note that
+co[(K,0),(L,c)] =/braceleftig/parenleftig
+(1−w
+c)K+w
+cL,w/parenrightig/vextendsingle/vextendsingle/vextendsingle0≤w≤c/bracerightig
+.
+Therefore
+/an}bracketle{tg(Mn),en+1/an}bracketri}ht=/integraltextc
+0wvoln((1−w
+c)K+w
+cL)dw/integraltextc
+0voln((1−w
+c)K+w
+cL)dw
+=c/integraltext1
+0tvoln((1−t)K+tL)dt
+/integraltext1
+0voln((1−t)K+tL)dt.
+6Now we use the mixed volume formula (3). Thus
+/an}bracketle{tg(Mn),en+1/an}bracketri}ht=c/integraltext1
+0/summationtextn
+k=0/parenleftbigg/parenleftbign
+k/parenrightbig
+tk+1(1−t)n−kVn−k,k(K,L)/parenrightbigg
+dt
+/integraltext1
+0/parenleftbigg/summationtextn
+k=0/parenleftbign
+k/parenrightbig
+tk(1−t)n−kVn−k,k(K,L)/parenrightbigg
+dt
+=c
+n+2/summationtextn
+k=0(k+1)Vn−k,k(K,L)/summationtextn
+k=0Vn−k,k(K,L),
+where we have also used the Betafunction
+B(k,l) =/integraldisplay1
+0tk−1(1−t)l−1dt=Γ(k)Γ(l)
+Γ(k+l), k,l > 0.
+/square
+The next lemma is well known ([8], p. 216, formula 54).
+Lemma 5 For alln∈Nandt≥0
+voln(Bn
+2+tBn
+∞) =n/summationdisplay
+k=0/parenleftbiggn
+k/parenrightbigg
+2kvoln−k/parenleftbig
+Bn−k
+2/parenrightbig
+tk,
+with the convention that vol0(B0
+2) = 1. Therefore, for 0≤k≤n,
+Vn−k,k(Bn
+2,Bn
+∞) = 2kvoln−k/parenleftbig
+Bn−k
+2/parenrightbig
+.
+Lemma 6 The following formula holds
+lim
+n→∞1
+n+2/summationtextn
+k=0k+1
+Γ(1+n
+2)k
+nΓ(1+n−k
+2)/summationtextn
+k=01
+Γ(1+n
+2)k
+nΓ(1+n−k
+2)= 1−1
+e.
+Proof.One has
+1
+n+2/summationtextn
+k=0k+1
+Γ(1+n
+2)k
+nΓ(1+n−k
+2)/summationtextn
+k=01
+Γ(1+n
+2)k
+nΓ(1+n−k
+2)=1
+n+2/summationtextn
+k=0n−k+1
+Γ(1+n
+2)n−k
+nΓ(1+k
+2)/summationtextn
+k=01
+Γ(1+n
+2)n−k
+nΓ(1+k
+2)
+=1
+n+2/summationtextn
+k=0(n−k+1)Γ(1+n
+2)k
+n
+Γ(1+k
+2)
+/summationtextn
+k=0Γ(1+n
+2)k
+n
+Γ(1+k
+2)=n+1
+n+2−1
+n+2/summationtextn
+k=0kΓ(1+n
+2)k
+n
+Γ(1+k
+2)
+/summationtextn
+k=0Γ(1+n
+2)k
+n
+Γ(1+k
+2).
+It is thus needed to prove that
+lim
+n→∞1
+n/summationtextn
+k=0kΓ(1+n
+2)k
+n
+Γ(1+k
+2)
+/summationtextn
+k=0Γ(1+n
+2)k
+n
+Γ(1+k
+2)=1
+e. (6)
+7For every x≥0,
+n/summationdisplay
+k=0kxk
+Γ(1+k
+2)=n/summationdisplay
+k=1kxk
+k
+2Γ(k
+2)= 2xn/summationdisplay
+k=1xk−1
+Γ(k
+2)
+= 2x/parenleftig1
+Γ(1
+2)+n/summationdisplay
+k=2xk−1
+Γ(k
+2)/parenrightig
+= 2x/parenleftig1
+Γ(1
+2)+xn−2/summationdisplay
+k=0xk
+Γ(1+k
+2)/parenrightig
+.
+Thus for x= Γ(1+n
+2)1
+n
+n/summationdisplay
+k=0kΓ(1+n
+2)k
+n
+Γ(1+k
+2)= 2Γ/parenleftig
+1+n
+2/parenrightig1
+n/parenleftigg
+1
+Γ(1
+2)+Γ/parenleftig
+1+n
+2/parenrightig1
+nn−2/summationdisplay
+k=0Γ(1+n
+2)k
+n
+Γ(1+k
+2)/parenrightigg
+.(7)
+Let
+An=n/summationdisplay
+k=0kΓ(1+n
+2)k
+n
+Γ(1+k
+2)andBn=n/summationdisplay
+k=0Γ(1+n
+2)k
+n
+Γ(1+k
+2).
+(6) is equivalent to
+lim
+n→∞An
+nBn=1
+e.
+By Stirling’s formula
+/parenleftign
+2/parenrightign
+2e−n
+2√πn≤Γ/parenleftig
+1+n
+2/parenrightig
+≤/parenleftign
+2/parenrightign
+2e−n
+2√πne1
+6n−12
+and thus /radicalbiggn
+2e(πn)1
+2n≤Γ/parenleftig
+1+n
+2/parenrightig1
+n≤/radicalbiggn
+2e(πn)1
+2ne1
+n(6n−12)
+Moreover,
+Bn≥Γ(1+n
+2)2
+n
+Γ(2)≥n
+2e.
+(In fact, it can be proved that Bn∼2ec2
+n∼2en
+2e, where we write a(n)∼b(n), to mean
+that there are absolute constants c1,c2>0 such that c1a(n)≤b(n)≤c2a(n). But for
+our purposes the above estimate is enough.) Also, for n≥3
+Γ(1+n
+2)n−1
+n
+Γ(1+n−1
+2)=Γ(1+n
+2)
+Γ(1+n
+2)1
+nΓ(1+n−1
+2)
+≤(n
+2)n
+2e−n
+2√πn e1
+6n−12
+(πn)1
+2n/radicalbign
+2e(n−1
+2)n−1
+2e−n−1
+2/radicalbig
+π(n−1)
+≤(n)n
+2√n
+√n(n−1)n−1
+2/radicalbig
+(n−1)/parenleftige
+πn/parenrightig1
+2n
+=/parenleftige
+πn/parenrightig1
+2n/parenleftbiggn
+n−1/parenrightbiggn
+2
+≤e1
+2n/parenleftbigg
+1+1
+n−1/parenrightbiggn
+2
+≤exp/parenleftbiggn+1
+2(n−1)/parenrightbigg
+≤e.
+8Therefore, by (7) and with cn=/parenleftbig
+Γ(1+n
+2)/parenrightbig1
+n,
+lim
+n→∞An
+nBn= lim
+n→∞2cn
+n
+1
+Γ(1
+2)Bn+cn
+1−Γ(1+n
+2)n−1
+n
+Γ(1+n−1
+2)+Γ(1+n
+2)n
+n
+Γ(1+n
+2)
+Bn
+
+
+= lim
+n→∞2c2
+n
+n= lim
+n→∞2
+n/parenleftbigg/radicalbiggn
+2e/parenrightbigg2
+=1
+e.
+/square
+Lemma 7 Leta,b∈R,a >0,b >0and
+K=Bn
+2
+voln(Bn
+2)1
+nL=Bn
+∞
+2.
+LetMn= co[(K,−a),(L,b)]be defined as in (4). Then the center of gravity g(Mn)
+satisfies
+lim
+n→∞g(Mn) =1
+e(−a)+/parenleftbigg
+1−1
+e/parenrightbigg
+b.
+Proof.By symmetry, the centroid of Mnis an element of the ( n+1)-axis. Therefore
+we only need to compute its ( n+1)-th coordinate.
+Instead of Mn= co[(K,−a),(L,b)], we consider ˜Mn= co[(K,0),(L,a+b)] with
+centroid ˜ g= ˜gn. The centroid gnofMnis then given by gn= ˜gn−a.
+We use Lemma 4 with c=a+bto get
+/an}bracketle{t˜g(˜Mn),en+1/an}bracketri}ht=a+b
+n+2/summationtextn
+k=0(k+1)Vn−k,k(K,L)/summationtextn
+k=0Vn−k,k(K,L).
+By Lemma 5 and the linearity of the mixed volumes in each component, w e get
+/an}bracketle{t˜g(˜Mn),en+1/an}bracketri}ht=a+b
+n+2/summationtextn
+k=0(k+1) vol n(Bn
+2)k
+nvoln−k/parenleftbig
+Bn−k
+2/parenrightbig
+/summationtextn
+k=0voln(Bn
+2)k
+nvoln−k/parenleftbig
+Bn−k
+2/parenrightbig
+=a+b
+n+2/summationtextn
+k=0k+1
+(Γ(1+n/2)k
+nΓ(1+(n−k)/2)/summationtextn
+k=01
+(Γ(1+n/2)k
+nΓ(1+(n−k)/2). (8)
+Now we apply Lemma 6. /square
+Eventually we will have to investigate expressions of the form
+voln/parenleftigg
+Bn
+2
+voln(Bn
+2)1
+n∩tBn
+1
+voln(Bn
+1)1
+n/parenrightigg
+, t≥0.
+9Schechtman, Schmuckenschl¨ ager and Zinn established asymptot ic formulas for large n
+for the volumes of Bn
+p∩tBn
+q[17, 18, 19]. To do so, they considered independent random
+variables hp
+1,...,hp
+nwith densityp
+2Γ(1
+p)e−|t|p(9)
+Here, we need uniform estimates instead of asymptotic ones.
+Lemma 8 Let0< p,q < ∞and lethpbe a random variable with density (9). Then
+E|hp|q=p
+2Γ(1
+p)/integraldisplay∞
+−∞|t|qe−|t|pdt=p
+Γ(1
+p)/integraldisplay∞
+0tqe−tpdt=1
+Γ(1
+p)Γ/parenleftbiggq+1
+p/parenrightbigg
+(10)
+In particular, for p= 1 and q= 1 respectively q= 2 we get
+E|h1|= 1 respectively E|h1|2= 2. (11)
+Forp= 2 and q= 2 we get
+E|h2|2=1
+2.
+The next lemma treats the Gaußian case of a more general stateme nt which we will
+prove below.
+Lemma 9 Let(Ω,P)be a probability space. Let gi: Ω→R,1≤i≤n, be independent
+N(0,1)-random variables. Then, for all γ >0there isn0such that for all n≥n0
+P
+
+ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
+n/summationtextn
+i=1|gi(ω)|
+/parenleftbig1
+n/summationtextn
+i=1|gi(ω)|2/parenrightbig1
+2−/radicalbigg
+2
+π/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤γ
+
+≥1
+2(12)
+Proof.By Chebyshev’s inequality. /square
+Lemma 10 Let(Ω,P)be a probability space. Let h1
+i: Ω→Rn,1≤i≤n, be indepen-
+dent random variables with density e−|t|. Then for all γ >0there isn0such that for all
+n≥n0
+P/braceleftigg
+ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√
+2−γ≤(1
+n/summationtextn
+i=1|h1
+i(ω)|2)1
+2
+1
+n/summationtextn
+i=1|h1
+i(ω)|≤√
+2+γ/bracerightigg
+≥1
+2(13)
+Proof.Again, by Chebyshev’s inequality. /square
+Lemma 11 Let(Ω,P)be a probability space. Let gi: Ω→R,1≤i≤n, be independent
+N(0,1)-random variables. Then
+voln(Bn
+2∩t Bn
+1) =nvoln(Bn
+2)/integraldisplay1
+0rn−1P/parenleftigg/summationtextn
+i=1|gi|
+(/summationtextn
+i=1|gi|2)1
+2≤t
+r/parenrightigg
+dr(14)
+10Again, we state this lemma and its proof in the Gaußian case to illustrat e the ideas.
+Below, we will prove a more general statement. A more general situ ation has also been
+explored in [18].
+Clearly, formula (14) is equivalent to
+voln(Bn
+2∩t Bn
+1) =nvoln(Bn
+2)/integraldisplay1
+0rn−1P/parenleftbigg1
+n/summationtextn
+i=1|gi|
+(1
+n/summationtextn
+i=1|gi|2)1
+2≤t
+r√n/parenrightbigg
+dr(15)
+and to
+voln/parenleftigg
+Bn
+2
+voln(Bn
+2)1/n∩sBn
+1
+voln(Bn
+1)1/n/parenrightigg
+=n/integraldisplay1
+0rn−1P/parenleftbigg1
+n/summationtextn
+i=1|gi|
+(1
+n/summationtextn
+i=1|gi|2)1
+2≤s
+r√n/parenleftbiggvoln(Bn
+2)
+voln(Bn
+1)/parenrightbigg1
+n/parenrightbigg
+dr. (16)
+(16) follows from (15) with the substitution s=t/parenleftbigg
+voln(Bn
+1)
+voln(Bn
+2)/parenrightbigg1/n
+.
+Proof of Lemma 11. Also in the case of a convex body that is not centrally symmetric
+we use
+/bardblξ/bardblK= inf/braceleftbigg1
+ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingleρξ∈K/bracerightbigg
+.
+Using polar coordinates we get for any convex body K⊆Bn
+2
+voln(K) =1
+n/integraldisplay
+∂Bn
+2/bardblξ/bardbl−n
+Kdσn−1(ξ)
+=/integraldisplay
+∂Bn
+2/integraldisplay/bardblξ/bardbl−1
+K
+0rn−1drdσn−1(ξ)
+=/integraldisplay1
+0rn−1voln−1/parenleftbigg
+∂Bn
+2∩1
+rK/parenrightbigg
+dr.
+Hence for K=Bn
+2∩tBn
+1
+voln(Bn
+2∩tBn
+1) =/integraldisplay1
+0rn−1voln−1/parenleftbigg
+∂Bn
+2∩1
+r(Bn
+2∩tBn
+1)/parenrightbigg
+dr
+=/integraldisplay1
+0rn−1voln−1/parenleftbigg
+∂Bn
+2∩t
+rBn
+1/parenrightbigg
+dr
+Letγnbe the normalized Gauss measure on Rn. Then
+voln−1(∂Bn
+2)·γn/parenleftigg/summationtextn
+i=1|xi|
+(/summationtextn
+i=1|xi|2)1
+2≤t
+r/parenrightigg
+= voln−1/parenleftbigg
+∂Bn
+2∩t
+rBn
+1/parenrightbigg
+11Indeed,
+/braceleftigg
+x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationtextn
+i=1|xi|
+(/summationtextn
+i=1|xi|2)1
+2≤t
+r/bracerightigg
+=/braceleftbigg
+sx/vextendsingle/vextendsingle/vextendsingle/vextendsingles >0,/bardblx/bardbl2= 1,/bardblx/bardbl1≤t
+r/bracerightbigg
+=/braceleftbigg
+sx/vextendsingle/vextendsingle/vextendsingle/vextendsingles >0,x∈∂Bn
+2∩t
+rBn
+1/bracerightbigg
+Therefore, using polar coordinates
+γn/parenleftigg/braceleftigg
+x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationtextn
+i=1|xi|
+(/summationtextn
+i=1|xi|2)1
+2≤t
+r/bracerightigg/parenrightigg
+=γn/parenleftbigg/braceleftbigg
+sx/vextendsingle/vextendsingle/vextendsingle/vextendsingles >0,x∈∂Bn
+2∩t
+rBn
+1/bracerightbigg/parenrightbigg
+=1
+(2π)n
+2/integraldisplay
+s>0/integraldisplay
+∂Bn
+2∩t
+rBn
+1sn−1e−s2
+2dx ds
+=Γ/parenleftbign
+2/parenrightbig
+2πn
+2voln−1/parenleftbigg
+∂Bn
+2∩t
+rBn
+1/parenrightbigg
+.
+Thus
+voln(Bn
+2∩tBn
+1) = vol n−1(∂Bn
+2)/integraldisplay1
+0rn−1γn/parenleftigg/summationtextn
+i=1|xi|
+(/summationtextn
+i=1|xi|2)1
+2≤t
+r/parenrightigg
+dr.
+Since the coordinate functionals are independent N(0,1)-random variables with respect
+to the measure γnwe have established (14). /square
+Lemma 12 LetPpbe the probability on Rnwith density fp:Rn→Rdefined by
+fp(x) =fp(x1,...,x n) =1/parenleftbig
+2Γ(1+1
+p)/parenrightbigne−/summationtextn
+i=1|xi|p.
+Then for every starshaped body K, one has
+voln/parenleftbig
+Bn
+p∩K/parenrightbig
+=nvoln/parenleftbig
+Bn
+p/parenrightbig/integraldisplay1
+0rn−1Pp/parenleftbigg/braceleftbigg
+x∈Rn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bardblx/bardblK
+/bardblx/bardblp≤1
+r/bracerightbigg/parenrightbigg
+dr.
+Proof.As vol n/parenleftbig
+Bn
+p/parenrightbig
+=/parenleftbig
+2Γ(1+1
+p)/parenrightbign/parenleftbig
+Γ(1+n
+p)/parenrightbig−1, it follows that
+nvoln/parenleftbig
+Bn
+p/parenrightbig/integraldisplay1
+0rn−1Pp/parenleftbig/braceleftbig
+x∈Rn;/bardblx/bardblK
+/bardblx/bardblp≤1
+r/bracerightbig/parenrightbig
+dr
+=n
+Γ(1+n
+p)/integraldisplay1
+0/parenleftig/integraldisplay
+{x:r/bardblx/bardblK≤/bardblx/bardblp}e−/bardblx/bardblp
+pdx/parenrightig
+rn−1dr
+=n
+Γ(1+n
+p)/integraldisplay
+Rn/parenleftig/integraldisplaymin(1,/bardblx/bardblp
+/bardblx/bardblK)
+0rn−1dr/parenrightig
+e−/bardblx/bardblp
+pdx
+=1
+Γ(1+n
+p)/integraldisplay
+Rn1/parenleftbig
+max(1,/bardblx/bardblK
+/bardblx/bardblp)/parenrightbigne−/bardblx/bardblp
+pdx
+12We pass to polar coordinates. σn−1denotes the (non-normalized) surface measure on
+Sn−1.
+nvoln/parenleftbig
+Bn
+p/parenrightbig/integraldisplay1
+0rn−1Pp/parenleftbig/braceleftbig
+x∈Rn;/bardblx/bardblK
+/bardblx/bardblp≤1
+r/bracerightbig/parenrightbig
+dr
+=1
+Γ(1+n
+p)/integraldisplay
+θ∈Sn−1/parenleftig/integraldisplay+∞
+0e−rp/bardblθ/bardblp
+prn−1dr/parenrightig1/parenleftbig
+max(1,/bardblθ/bardblK
+/bardblθ/bardblp)/parenrightbigndσn−1(θ)
+=1
+Γ(1+n
+p)1
+p/integraldisplay+∞
+0e−ssn
+p−1ds/integraldisplay
+θ∈Sn−11
+/bardblθ/bardblnp1/parenleftbig
+max(1,/bardblθ/bardblK
+/bardblθ/bardblp)/parenrightbigndσn−1(θ)
+=1
+n/integraldisplay
+θ∈Sn−11/parenleftbig
+max(/bardblθ/bardblp,/bardblθ/bardblK)/parenrightbigndσn−1(θ)
+= voln/parenleftbig
+K∩Bn
+p/parenrightbig
+/square
+Corollary 13 Let1≤p,q,<∞and lets≥0. Lethp
+i: Ω→Rn,1≤i≤n, be
+independent random variables with densityp
+Γ(1/p)e−|t|pandh= (hp
+1,...,hp
+n). Then
+voln/parenleftbig
+Bn
+p∩sBn
+q/parenrightbig
+=nvoln/parenleftbig
+Bn
+p/parenrightbig/integraldisplay1
+0rn−1Pp
+
+
+(x1,...,x n)∈Rn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbig1
+n/summationtextn
+i=1|xi|q/parenrightbig1
+q
+/parenleftbig1
+n/summationtextn
+i=1|xi|p/parenrightbig1
+p≤n1
+p−1
+qs
+r
+
+
+dr
+=n/integraldisplay1
+0rn−1P/braceleftbigg
+ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bardblhp(ω)/bardblq
+/bardblhp(ω)/bardblp≤s
+r/bracerightbigg
+dr
+Lemma 14 For allγ >0there isn0such that for all n≥n0and alltwith
+t≥/parenleftig/radicalig
+2
+π+γ/parenrightig√nvoln(Bn
+1)1
+n
+voln(Bn
+2)1
+nwe have
+1
+2≤voln/parenleftigg
+Bn
+2
+voln(Bn
+2)1/n∩tBn
+1
+voln(Bn
+1)1/n/parenrightigg
+≤1.
+Proof.By (15)
+voln(Bn
+2∩t Bn
+1) =nvoln(Bn
+2)/integraldisplay1
+0rn−1P/parenleftbigg1
+n/summationtextn
+i=1|gi|
+(1
+n/summationtextn
+i=1|gi|2)1
+2≤t
+r√n/parenrightbigg
+dr .
+This gives
+voln/parenleftigg
+Bn
+2
+voln(Bn
+2)1
+n∩tBn
+1
+voln(Bn
+2)1
+n/parenrightigg
+=n/integraldisplay1
+0rn−1P/parenleftbigg1
+n/summationtextn
+i=1|gi|
+(1
+n/summationtextn
+i=1|gi|2)1
+2≤t
+r√n/parenrightbigg
+dr.(17)
+13We substitute
+t=svoln(Bn
+2)1
+n
+voln(Bn
+1)1
+n
+and obtain
+voln/parenleftigg
+Bn
+2
+voln(Bn
+2)1
+n∩sBn
+1
+voln(Bn
+1)1
+n/parenrightigg
+=n/integraldisplay1
+0rn−1P/parenleftigg
+1
+n/summationtextn
+i=1|gi|
+(1
+n/summationtextn
+i=1|gi|2)1
+2≤s
+r√nvoln(Bn
+2)1
+n
+voln(Bn
+1)1
+n/parenrightigg
+dr.
+By Lemma 9, for every γ >0 there is n0such that for all n≥n0
+P
+
+ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
+n/summationtextn
+i=1|gi(ω)|
+/parenleftbig1
+n/summationtextn
+i=1|gi(ω)|2/parenrightbig1
+2−/radicalbigg
+2
+π/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤γ
+
+≥1
+2
+Therefore, for every γ >0 there is n0such that for all n≥n0and allswith
+/radicalbigg
+2
+π+γ≤s√nvoln(Bn
+2)1
+n
+voln(Bn
+1)1
+n
+or, equivalently,/parenleftigg/radicalbigg
+2
+π+γ/parenrightigg
+√nvoln(Bn
+1)1
+n
+voln(Bn
+2)1
+n≤s
+we have
+voln/parenleftigg
+Bn
+2
+voln(Bn
+2)1
+n∩sBn
+1
+voln(Bn
+1)1
+n/parenrightigg
+≥1
+2.
+The other inequality is obvious from (17). /square
+Lemma 15 For allγ >0there isn0such that for all n≥n0and allswith
+s≥√
+2+γ√nvoln(Bn
+2)1
+n
+voln(Bn
+1)1
+n∼/radicalbiggπ
+e
+we have
+1
+2≤voln/parenleftigg
+Bn
+1
+voln(Bn
+1)1
+n∩sBn
+2
+voln(Bn
+2)1
+n/parenrightigg
+≤1.
+Proof.By Corollary 13, for q= 2 and p= 1
+voln(Bn
+1∩tBn
+2)
+voln(Bn
+1)=n/integraldisplay1
+0rn−1P/braceleftbigg
+ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bardblh1(ω)/bardbl2
+/bardblh1(ω)/bardbl1≤t
+r/bracerightbigg
+dr .
+14We put
+t=svoln(Bn
+1)1
+n
+voln(Bn
+2)1
+n
+and obtain
+voln/parenleftigg
+Bn
+1
+voln(Bn
+1)1
+n∩sBn
+2
+voln(Bn
+2)1
+n/parenrightigg
+=n/integraldisplay1
+0rn−1P/parenleftigg
+/bardblh1(ω)/bardbl2
+/bardblh1(ω)/bardbl1≤s
+rvoln(Bn
+1)1
+n
+voln(Bn
+2)1
+n/parenrightigg
+dr.(18)
+By Lemma 10, for every γ >0 there is n0such that for all n≥n0
+P/braceleftigg
+ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1
+n/summationtextn
+i=1|h1
+i(ω)|2)1
+2
+1
+n/summationtextn
+i=1|h1
+i(ω)|≤√
+2+γ/bracerightigg
+≥1
+2
+Therefore, for all rwith 0< r≤1 and all swith
+svoln(Bn
+1)1
+n
+voln(Bn
+2)1
+n≥√
+2+γ√n
+or, equivalently,
+s≥√
+2+γ√nvoln(Bn
+2)1
+n
+voln(Bn
+1)1
+n
+we have
+P/braceleftigg
+ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1
+n/summationtextn
+i=1|h1
+i(ω)|2)1
+2
+1
+n/summationtextn
+i=1|h1
+i(ω)|≤svoln(Bn
+1)1
+n
+voln(Bn
+2)1
+n/bracerightigg
+≥1
+2.
+By (18)
+1
+2≤voln/parenleftigg
+Bn
+1
+voln(Bn
+1)1
+n∩sBn
+2
+voln(Bn
+2)1
+n/parenrightigg
+provided that
+s≥√
+2+γ√nvoln(Bn
+2)1
+n
+voln(Bn
+1)1
+n.
+Again, the other inequality follows by (18). /square
+Leta >0,b >0 be real numbers. By (5), for K=Bn
+2
+voln(Bn
+2)1
+n,L=Bn
+∞
+2, for allswith
+−1
+a≤s≤1
+b
+M◦
+n(s) = (1+ s a)voln(Bn
+2)1
+nBn
+2∩(1−s b)2Bn
+1 (19)
+which is the same as
+M◦
+n(s) =
+(1+s a)voln(Bn
+2)2
+n/parenleftbigg
+Bn
+2
+voln(Bn
+2)1
+n∩2(1−s b)
+(1+s a)voln(Bn
+1)1
+n
+voln(Bn
+2)2
+nBn
+1
+voln(Bn
+1)1
+n/parenrightbigg
+.(20)
+15Lemma 16 Leta >0,b >0be real numbers. Then, for all γ >0there isn0such that
+for alln≥n0and allswith
+s≥ −√πe−2
+a√πe+2b+γ
+we have
+2n−1(1−s b)nvoln(Bn
+1)≤voln(M◦
+n(s))≤2n(1−s b)nvoln(Bn
+1).
+Proof.We have
+M◦
+n(s) = 2(1−s b)/parenleftigg
+(1+s a)voln(Bn
+2)1
+n
+2(1−s b)Bn
+2∩Bn
+1/parenrightigg
+Therefore
+voln(M◦
+n(s)) = 2n(1−s b)nvoln/parenleftigg
+(1+s a)voln(Bn
+2)1
+n
+2(1−s b)Bn
+2∩Bn
+1/parenrightigg
+= 2n(1−s b)nvoln(Bn
+1)voln/parenleftigg
+(1+s a)
+2(1−s b)voln(Bn
+2)2
+n
+voln(Bn
+1)1
+nBn
+2
+voln(Bn
+2)1
+n∩Bn
+1
+voln(Bn
+1)1
+n/parenrightigg
+.
+We get immediately
+voln(M◦
+n(s))≤2n(1−s b)nvoln(Bn
+1).
+We show now the opposite inequality. By Lemma 15, for all γ >0 there is n0such that
+for alln≥n0
+1
+2≤voln/parenleftigg
+Bn
+1
+voln(Bn
+1)1
+n∩(1+s a)
+2(1−s b)voln(Bn
+2)2
+n
+voln(Bn
+1)1
+nBn
+2
+voln(Bn
+2)1
+n/parenrightigg
+≤1,(21)
+provided that
+(1+s a)
+2(1−s b)voln(Bn
+2)2
+n
+voln(Bn
+1)1
+n≥√
+2+γ√nvoln(Bn
+2)1
+n
+voln(Bn
+1)1
+n
+or
+1+s a
+1−s b≥2(√
+2+γ)
+√nvoln(Bn
+2)1
+n.
+This means
+s≥ −√nvoln(Bn
+2)1
+n−2(√
+2+γ)
+a√nvoln(Bn
+2)1
+n+2b(√
+2+γ).
+Since
+voln(Bn
+2) =πn
+2
+Γ(n
+2+1)
+and
+kke−k√
+2πk≤Γ(k+1)≤kke−k√
+2πke1
+6(k−2)
+16we have √
+2πe
+√n(πn)1
+2ne1
+6n(n−2)≤voln(Bn
+2)1
+n≤√
+2πe√n(πn)1
+2n. (22)
+Thus (21) holds, provided that
+s≥ −√
+2πe
+(πn)1
+2n−2(√
+2+γ)
+a√
+2πe
+(πn)1
+2ne1
+6n(n−2)+2b(√
+2+γ)=−√
+2πe−2(√
+2+γ)(πn)1
+2n
+a√
+2πe
+e1
+6n(n−2)+2b(√
+2+γ)(πn)1
+2n.
+Now we pass to a new γand obtain: For every γ >0 there is a n0such that for all
+n≥n0and allswith
+s≥ −√
+2πe−2√
+2
+a√
+2πe+2b√
+2+γ=−√πe−2
+a√πe+2b+γ.
+we have
+1
+2≤voln/parenleftigg
+Bn
+1
+voln(Bn
+1)1
+n∩(1+s a)
+2 (1−s b)voln(Bn
+2)2
+n
+voln(Bn
+1)1
+nBn
+2
+voln(Bn
+2)1
+n/parenrightigg
+≤1. (23)
+/square
+Lemma 17 Leta >0andb >0real numbers. Then, for all γ >0there isn0such
+that for all n≥n0and allswith
+s≤ −√e−1
+b+a√e−γ
+we have1
+2(1+s a)nvoln(Bn
+2)2≤voln(M◦
+n(s))≤(1+s a)nvoln(Bn
+2)2
+Proof.By (20), for all swith−1
+a≤s≤1
+b
+M◦
+n(s) = (1+ s a)voln(Bn
+2)2
+n/parenleftigg
+Bn
+2
+voln(Bn
+2)1
+n∩2(1−s b)
+(1+s a)voln(Bn
+1)1
+n
+voln(Bn
+2)2
+nBn
+1
+voln(Bn
+1)1
+n/parenrightigg
+.
+The right hand side inequality follows immediately. We show now the left h and side
+inequality. By Lemma 14, for all γ >0 there is n0such that for all n≥n0and alltwith
+t≥/parenleftig/radicalig
+2
+π+γ/parenrightig√nvoln(Bn
+1)1
+n
+voln(Bn
+2)1
+nwe have
+1
+2≤voln/parenleftigg
+Bn
+2
+voln(Bn
+2)1/n∩tBn
+1
+voln(Bn
+1)1/n/parenrightigg
+≤1.
+17Therefore,
+1
+2≤voln/parenleftigg
+Bn
+2
+voln(Bn
+2)1/n∩2/parenleftbigg1−s b
+1+s a/parenrightbiggvoln(Bn
+1)1
+n
+voln(Bn
+2)2
+nBn
+1
+voln(Bn
+1)1/n/parenrightigg
+≤1
+provided that
+/parenleftigg/radicalbigg
+2
+π+γ/parenrightigg
+√nvoln(Bn
+1)1
+n
+voln(Bn
+2)1
+n≤2/parenleftbigg1−s b
+1+s a/parenrightbiggvoln(Bn
+1)1
+n
+voln(Bn
+2)2
+n.
+The latter inequality is equivalent to
+/parenleftigg/radicalbigg
+2
+π+γ/parenrightigg
+√nvoln(Bn
+2)1
+n≤21−s b
+1+s a.
+By (22), the above inequality holds if
+/parenleftigg/radicalbigg
+2
+π+γ/parenrightigg/radicalbiggπe
+2≤1−s b
+1+s a(πn)1
+2n.
+This is equivalent to
+s≤ −/parenleftig/radicalig
+2
+π+γ/parenrightig/radicalbigπe
+2−(πn)1
+2n
+b(πn)1
+2n+a/parenleftig/radicalig
+2
+π+γ/parenrightig/radicalbigπe
+2.
+We pass to a new γ
+s≤ −√e−1
+b+a√e−γ
+/square
+Lemma 18 Let
+s0=1−√e
+b+a√es1=2−√πe
+a√πe+2b
+(i) For every γ >0there isn0such that for all nwithn≥n0
+/integraldisplay1
+b
+−1
+avoln(M◦
+n(s)ds≤(1+γ)/integraldisplays1+γ
+s0−γvoln(M◦
+n(s))ds
+(ii) For every γ >0there isn0such that for all nwithn≥n0
+/integraldisplays0−γ
+−1
+a|s|voln(M◦
+n(s))ds≤γ/integraldisplay1
+b
+−1
+avoln(M◦
+n(s))ds
+and/integraldisplay1
+b
+s1+γ|s|voln(M◦
+n(s))ds≤γ/integraldisplay1
+b
+−1
+avoln(M◦
+n(s))ds.
+18Please note that the expression/integraltexts1+γ
+s0−γsvoln(M◦
+n(s))dsis negative. This lemma
+means that the volume of M◦
+nis concentrated between the hyperplanes through s0en+1
+ands1en+1.
+Although the inequality s0< s1follows from the above computations, it is comfort-
+ing to verify this directly. The inequality s0< s1is equivalent to
+(a+b)√e(2−√π)>0
+which holds for all positive aandb.
+Proof.(i) By Lemma 17, for all γ >0 there is n0such that for all nwithn≥n0and
+allswith
+−1
+a≤s≤1−√e
+b+a√e−γ=s0−γ
+we have1
+2(1+s a)nvoln(Bn
+2)2≤voln(M◦(s))≤(1+s a)nvoln(Bn
+2)2
+Therefore
+/integraldisplays0−γ
+s0−2γvoln(M◦
+n(s))ds
+≥1
+2/integraldisplays0−γ
+s0−2γ(1+s a)nvoln(Bn
+2)2ds
+=1
+2a(n+1)|Bn
+2|2/parenleftig
+(1+a(s0−γ))n+1−(1+a(s0−2γ))n+1/parenrightig
+=1
+2a(n+1)|Bn
+2|2(1+a(s0−γ))n+1/parenleftigg
+1−/parenleftbigg1+a(s0−2γ)
+1+a(s0−γ)/parenrightbiggn+1/parenrightigg
+.
+For sufficiently large n
+/integraldisplays0−γ
+s0−2γvoln(M◦
+n(s))ds≥1
+4a(n+1)|Bn
+2|2(1+a(s0−γ))n+1.
+On the other hand, by Lemma 17
+/integraldisplays0−2γ
+−1
+avoln(M◦
+n(s))ds
+≤/integraldisplays0−2γ
+−1
+a(1+s a)nvoln(Bn
+2)2ds
+=1
+a(n+1)voln(Bn
+2)2(1+a(s0−2γ))n+1
+≤1
+a(n+1)voln(Bn
+2)2(1+a(s0−γ))n+1/parenleftbigg
+1−aγ
+1+a(s0−γ)/parenrightbiggn+1
+.
+Therefore, for sufficiently large n
+/integraldisplays0−2γ
+−1
+avoln(M◦
+n(s))ds≤4/parenleftbigg
+1−aγ
+1+a(s0−γ)/parenrightbiggn+1/integraldisplays0−γ
+s0−2γvoln(M◦
+n(s))ds.(24)
+19Thus/integraldisplays0−2γ
+−1
+avoln(M◦(s))ds≤γ
+2/integraldisplays0−γ
+s0−2γvoln(M◦(s))ds.
+Now we consider the interval [ s1,1
+b]. By Lemma 17
+/integraldisplays1+2γ
+s1+γvoln(M◦
+n(s))ds
+≥/integraldisplays1+2γ
+s1+γ2n−1(1−s b)nvoln(Bn
+1)ds
+= 2n−1voln(Bn
+1)
+b(n+1)/parenleftbig
+(1−b(s1+γ))n+1−(1−b(s1+2γ))n+1/parenrightbig
+= 2n−1voln(Bn
+1)(1−b(s1+γ))n+1
+b(n+1)/parenleftigg
+1−/parenleftbigg
+1−bγ
+1−b(s1+γ)/parenrightbiggn+1/parenrightigg
+Therefore, for sufficiently large n
+/integraldisplays1+2γ
+s1+γvoln(M◦
+n(s))ds≥2n−2voln(Bn
+1)(1−b(s1+γ))n+1
+b(n+1).
+On the other hand, by Lemma 17
+/integraldisplay1
+b
+s1+2γvoln(M◦
+n(s))ds
+≤/integraldisplay1
+b
+s1+2γ2n(1−s b)nvoln(Bn
+1)ds
+= 2n1
+b(n+1)(1−b(s1+2γ))n+1voln(Bn
+1)
+= 2n1
+b(n+1)(1−b(s1+γ))n+1/parenleftbigg
+1−bγ
+1−b(s1+γ)/parenrightbiggn+1
+voln(Bn
+1).
+Thus
+/integraldisplay1
+b
+s1+2γvoln(M◦
+n(s))ds≤4/parenleftbigg
+1−bγ
+1−b(s1+γ)/parenrightbiggn+1/integraldisplays1+2γ
+s1+γvoln(M◦
+n(s))ds .(25)
+Sinces1<0, for sufficiently big n
+/integraldisplay1
+b
+s1+2γvoln(M◦
+n(s))ds≤γ
+2/integraldisplays1+2γ
+s1+γvoln(M◦
+n(s))ds. (26)
+It is left to pass to a new γ.
+(ii) By (24)
+/integraldisplays0−2γ
+−1
+a|s|voln(M◦
+n(s))ds
+≤4max/braceleftbigg1
+a,1
+b/bracerightbigg/parenleftbigg
+1−aγ
+1+a(s0−γ)/parenrightbiggn+1/integraldisplays0−γ
+s0−2γvoln(M◦
+n(s))ds .
+20It is left to choose nsufficiently big. The other estimate is done in the same way using
+(26)./square
+Proof of Proposition 3. (i) is proved in the remark after the statement of Proposition
+3. We show (ii). By Definition
+g(M◦
+n)(n+1) =/integraltext1
+b
+−1
+asvoln(M◦
+n(s))ds
+/integraltext1
+b
+−1
+avoln(M◦n(s))ds.
+Therefore, by Lemma 18
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg(M◦
+n)(n+1)−/integraltexts1+γ
+s0−γsvoln(M◦
+n(s))ds
+/integraltext1
+b
+−1
+avoln(M◦n(s))ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤/integraltexts0−γ
+−1
+a|s|voln(M◦
+n(s))ds
+/integraltext1
+b
+−1
+avoln(M◦n(s))ds+/integraltext1
+b
+s1+γ|s|voln(M◦
+n(s))ds
+/integraltext1
+b
+−1
+avoln(M◦n(s))ds≤2γ.
+Thus
+g(M◦
+n)(n+1)−2γ≤/integraltexts1+γ
+s0−γsvoln(M◦
+n(s))ds
+/integraltext1
+b
+−1
+avoln(M◦n(s))ds≤g(M◦
+n)(n+1)+2γ.
+Sinces1<0, we may assume that s1+γ <0. Therefore
+/integraldisplays1+γ
+s0−γsvoln(M◦
+n(s))ds <0
+and by Lemma 18 (i)
+g(M◦
+n)(n+1)−2γ≤/integraltexts1+γ
+s0−γsvoln(M◦
+n(s))ds
+/integraltext1
+b
+−1
+avoln(M◦n(s))ds≤/integraltexts1+γ
+s0−γsvoln(M◦
+n(s))ds
+(1+γ)/integraltexts1+γ
+s0−γvoln(M◦n(s))ds≤s1+γ
+1+γ.
+On the other hand
+s0−γ≤/integraltexts1+γ
+s0−γsvoln(M◦
+n(s))ds
+/integraltexts1+γ
+s0−γvoln(M◦n(s))ds≤/integraltexts1+γ
+s0−γsvoln(M◦
+n(s))ds
+/integraltext1
+b
+−1
+avoln(M◦n(s))ds≤g(M◦
+n)(n+1)+2γ .
+Therefore,
+s0−3γ≤g(M◦
+n)(n+1)≤s1+γ
+1+γ+2γ.
+We apply now these estimates to the convex body
+Mn= co/bracketleftbigg
+(K,−1),/parenleftbigg
+L,1
+e−1/parenrightbigg/bracketrightbigg
+.
+21The centroid of Mnis (0,δn) with lim n→∞δn= 0. We get
+M(0,δn)
+n=/braceleftbigg
+(z,s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∀x∈K:/an}bracketle{tz,x/an}bracketri}ht−s(1+δn)≤1 and∀y∈L:/an}bracketle{tz,y/an}bracketri}ht+s/parenleftbigg1
+e−1−δn/parenrightbigg
+≤1)/bracerightbigg
+.
+It is left to apply the above estimates to M◦
+nwitha= 1+δnandb=1
+e−1−δn./square
+References
+[1]S. Artstein, B. Klartag and V. Milman ,On the Santal´ o point of a function
+and a functional Santal´ o inequality , Mathematika 54(2004), 33-48.
+[2]K.M. Ball , PhD dissertation, Cambridge.
+[3]K.M. Ball ,Logarithmically concave functions and sections of convex s ets inRn,
+Studia Math. 88(1988), 69-84.
+[4]K.M. Ball ,An elementary introduction to modern convex geometry , Flavors of
+geometry, 1–58, Math Sci. Res. Inst. Publ. 31, Cambridge University Press, 1997
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+Mathieu Meyer
+Universit´ e de Paris Est - Marne-la-Vall´ ee
+Equipe d’Analyse et de Mathematiques Appliqu´ ees
+Cit Descartes - 5, bd Descartes
+Champs-sur-Marne 77454 Marne-la-Vall´ ee, France
+mathieu.meyer@univ-mlv.fr
+Carsten Sch¨ utt
+Christian Albrechts Universit¨ at
+Mathematisches Seminar
+24098 Kiel, Germany
+schuett@math.uni-kiel.de
+Elisabeth Werner
+Department of Mathematics Universit´ e de Lille 1
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+elisabeth.werner@case.edu
+23
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+arXiv:1001.0715v2  [cond-mat.dis-nn]  27 Jul 2010Aging to Equilibrium Dynamics of SiO 2
+K. Vollmayr-Lee∗and J. A. Roman
+Department of Physics and Astronomy, Bucknell University, Lewisburg, Pennsylvania 17837, USA
+J. Horbach
+Institut f¨ ur Materialphysik im Weltraum, Deutsches Zentru m f¨ ur
+Luft- und Raumfahrt (DLR), Linder H¨ ohe, 51147 K¨ oln, Germa ny
+(Dated: May 3, 2010)
+Molecular dynamics computer simulations are used to study t he aging dynamics of SiO 2(mod-
+eled by the BKS model). Starting from fully equilibrated con figurations at high temperatures
+Ti∈ {5000K,3760K}, the system is quenched to lower temperatures Tf∈ {2500K, 2750K, 3000K,
+3250K}and observed after a waiting time tw. Since the simulation runs are long enough to reach
+equilibrium at Tf, we are able to study the transition from out-of-equilibriu m to equilibrium dynam-
+ics. We present results for the partial structure factors, f or the generalized incoherent intermediate
+scattering function Cq(tw,tw+t), and for themean square displacement ∆ r2(tw,tw+t). Weconclude
+that there are three different twregions: (I) At very short waiting times, Cq(tw,tw+t) decays very
+fast without forming a plateau. Similarly ∆ r2(tw,tw+t) increases without forming a plateau. (II)
+With increasing twa plateau develops in Cq(tw,tw+t) and ∆r2(tw,tw+t). For intermediate waiting
+timestheplateau heightisindependentof twandTi. Time superposition applies, i.e. Cq=Cq(t/tCq
+r)
+wheretCq
+r=tCq
+r(tw) is a waiting time dependent decay time. Furthermore Cq=C(q,tw,tw+t)
+scales as Cq=C(q,z(tw,t) where zis a function of twandtonly, i.e. independent of q. (III) At
+largetwthe system reaches equilibrium, i.e. Cq(tw,tw+t) and ∆r2(tw,tw+t) are independent of
+twandTi. ForCq(tw,tw+t) we find that the time superposition of intermediate waiting times (II)
+includes the equilibrium curve (III).
+PACS numbers: 61.20.Lc, 61.20.Ja, 64.70.ph, 02.70.Ns, 61. 43.Fs
+I. INTRODUCTION
+When a glass-forming liquid is quenched from an
+equilibrium state at a high temperature Tito a non-
+equilibrium state at a lower temperature Tf, “aging pro-
+cesses” set in. Provided that crystallization plays no role
+atTf(e.g. due to very low crystal nucleation rates), the
+transitiontoafinal (metastable)equilibrium stateoccurs
+on a time scale that corresponds to the typical equilib-
+rium relaxation time τeqof the (supercooled) liquid at
+Tf. The dynamics of the system depends on the wait-
+ing time twwhich is the time elapsed after the tempera-
+ture quench. If τeqexceeds the waiting time twthen the
+system is observed in a transient non-equilibrium state
+which corresponds to a glass for tw≪τeq. During the
+aging process, i.e. for tw< τeq, thermodynamic proper-
+ties such as volume and energy are changing and time
+translation invariance does not hold: correlation func-
+tions at time tw+tand the time origin at twdo depend
+not only on the time difference tbut also on the waiting
+timetw.
+Recently this aging process has been investigated ex-
+tensively with experiments [1–6], theoretically [7–9] and
+with computer simulations. For a more complete sum-
+mary of previous results we refer the reader to the ref-
+erences [10, 11] and references therein. Computer sim-
+ulation studies most similar to the work presented here
+∗Electronic address: kvollmay@bucknell.eduare on attractive colloidal systems [12–15], on the Kob-
+Andersen Lennard-Jones (KALJ) mixture [16–25], and
+silica (SiO 2) [25–28]. In the case of silica the interpreta-
+tionoftheresultsislessclearthanfortheKALJmixture;
+e.g. different findings [25, 27] have been reported on the
+violation of the fluctuation-dissipation regime during ag-
+ing [7–9]. Thus, it remains open whether silica, as the
+prototype of a glass-forming system forming a tetrahe-
+dral network structure, exhibits a different aging dynam-
+ics than, e.g. the KALJ model, where the structure is
+similar to that of a closed-packed hard-sphere structure.
+Recent simulation studies on amorphous silica [25–39]
+have widely used the BKS potential [40] to model the
+interactions between the atoms. Although it is a simple
+pair potential, it reproduces various static and dynamic
+properties of amorphous silica very well. For BKS sil-
+ica, the self-diffusion constants Dα(α= Si,O) show two
+different temperature regimes: At high temperatures,
+Dαdecays according to a power law, as predicted by
+the mode coupling theory (MCT) of the glass transition
+(note, however, that also other interpretations have been
+assigned to this high temperature regime). At low tem-
+perature, Dαas well as the shear viscosity ηexhibit an
+Arrheniusbehaviorwithanactivationenergyoftheorder
+of 5eV, in agreementwith experiment (see [30] and refer-
+ences therein). The temperature at which the crossover
+between both regimes occurs is at Tc≈3300K, corre-
+sponding to the critical MCT temperature of BKS silica.
+Previousstudies of the aging dynamics of BKS silica [25–
+27] were performed in two steps. First, the system was
+fully equilibrated at a temperature Ti> Tc. Then, the2
+system was quenched to a low temperature Tf< Tc, fol-
+lowed by the production runs. Wahlen and Rieger [26]
+analyze time-dependent correlation functions at different
+waiting times twand Berthier [25] and Scala et al. [27]
+study the generalized fluctuation dissipation relation and
+the energy landscape [27]. All three studies [25–27] in-
+vestigate the early stages of the aging dynamics, i.e. the
+dynamics was explored on time scales that were much
+smaller than the equilibrium relaxation time τeqat the
+temperature Tf.
+In this work, we also consider quenches in BKS sil-
+ica from a high temperature Tito a low temperature Tf.
+Different from previoussimulation studies, weaim at elu-
+cidating the full transient dynamics at Tffrom the initial
+state attw= 0 to equilibrium. To this end, temperatures
+Tfare chosen such that the system can be fully equili-
+brated on the typical time span of the MD simulation.
+Note that the considered temperatures Tf∈ {2500K,
+2750K, 3000K, 3250K }are below the critical MCT tem-
+peratures Tc. Thus, we have access to the full aging dy-
+namics in the experimentally relevant Arrhenius temper-
+ature regime that we have mentioned above.
+The analysis of time-dependent density correlation
+functions Cq(tw,tw+t) (withqthe wavenumber) and the
+mean square displacement ∆ r2(tw,tw+t) reveal three
+different regimes of waiting times tw: In the case of
+Cq(tw,tw+t) (and similarly for ∆ r2(tw,tw+t)) at early
+tw, a rapid decay to zero is seen, without forming a
+plateau at intermediate times. Then, for larger values
+oftwa plateau is formed. The height of this plateau
+grows with waiting time and becomes more pronounced,
+before in the final regime the plateau height is indepen-
+dentoftwandTi. Inthelatterregime,timesuperposition
+holds, i.e. by scaling time with a decay time tCq
+rtheCq
+for the different values of twfall onto a master curve at
+a given wavenumber q. This behavior is very similar to
+that found for the KALJ mixture. However, it is differ-
+ent from the behavior predicted by mean-field spin glass
+models and the activated dynamics scaling, as proposed
+by Wahlen and Rieger. Thus, these results suggest that
+the aging dynamics in silica, the prototype of a glass-
+former with a tetrahedral network structure, is very sim-
+ilar to that of simple glass-formers with a closed-packed
+hard-sphere-like structure. We find a difference between
+the KALJ mixture and SiO 2, however, in the parametric
+plotofC′
+q(Cq). ForSiO 2C′
+q(Cq)showsadatacollapsefor
+different sufficiently large twand thus Cq=C(q,z(tw,t))
+whereas this data collapse does not hold as well in the
+case of the KALJ mixture.
+The rest of the paper is organized as follows: In the
+next Sec. we give the details of the BKS potential and
+the simulation. Then, we present the results in Sec. III,
+before we summarize in Sec. IV. Appendix A describes
+the implementation of the Nos´ e-Hoover thermostat used
+in our simulation.II. MODEL AND DETAILS OF THE
+SIMULATION
+The interactions between the particles are modeled by
+the BKS potential [40] which has been used frequently
+and has proven to be reliable for the study of the dy-
+namics of amorphous silica [25–39] The functional form
+of the BKS potential is given by a sum of a Coulomb
+term, an exponential and a van der Waals term. Thus
+the potential between particles iandj, a distance rij
+apart, is given by
+φ(rij) =qiqje2
+rij+Aije−Bijrij−Cij
+r6
+ij,(1)
+whereeis the charge of an electron and the con-
+stantsAij,BijandCijareASiSi= 0.0eV,ASiO=
+18003.7572eV, AOO= 1388.7730eV, BSiSi= 0.0˚A−1,
+BSiO= 4.87318˚A−1,BOO= 2.76000˚A−1,CSiSi=
+0.0eV˚A−6,CSiO= 133.5381eV˚A−6andCOO=
+175.0000eV˚A−6[40]. The partial charges qiare
+qSi= 2.4 andqO=−1.2 ande2is given by
+1602.19/(4π8.8542)eV˚A.
+The Coulombic part of the interaction was computed
+by using the Ewald method [41, 42] with a constant
+αL= 6.3452, where Lis the size of the cubic box, and
+by using all q-vectors with |q| ≤6·2π/L. [51]. We
+ensure that the Ewald term in real space is also differ-
+entiable at the cutoff by smoothing similarly to Eq. (3)
+in [43] with rc= 8˚A andd= 0.05˚A2. To increase com-
+putation speed the non-Coulombic contribution to the
+potential was truncated, smoothed and shifted at a dis-
+tance of 5.5 ˚A. Note that this truncation is not negligible
+since it affects the pressure of the system. In Ref. [43]
+further slight variations on the potential are described
+in detail [52]. In order to minimize surface effects pe-
+riodic boundary conditions were used. The masses of
+the Si and O atoms were 28.086u and 15.9994u, re-
+spectively. The number of particles was 336, of which
+112 were silica atoms and 224 were oxygen atoms. For
+all simulation runs the size of the cubic box was fixed
+toL= 16.920468˚A which corresponds to a density of
+ρ= 2.323g/cm3, a value that is very close to the one of
+real silica glass, ρ= 2.2g/cm3[44].
+We investigated the aging dynamics for systems which
+were quenched from a high temperature Tito a low tem-
+perature Tf. To increasethe statistics, foreach( Ti,Tf) 20
+independent simulation runs were performed. To obtain
+20 independent configurations we carried out molecular
+dynamics (MD) simulations using the velocity Verlet al-
+gorithm with a time step of 1 .6fs at 6000K. The tem-
+perature was kept constant at 6000 K with a stochastic
+heat bath by replacing the velocities of all particles by
+new velocities drawn from the corresponding Boltzmann
+distributionevery150timesteps. Independentconfigura-
+tions wereat least 3 .27ns apart. Eachof these configura-
+tions undergoesthe followingsequence ofsimulation runs
+(see also Fig. 1). After fully equilibrating the samples at3
+wtT
+NVT NVE
+(Nose Hoover) (Velocity Verlet)6000 K 
+5000 K 
+3760 K 
+3250 K 
+3000 K 
+2750 K 
+2500 K Ti
+Tf
+measurement waiting timetNVT (stochastic)
+NVT (stochastic)20 initial configurations 
+FIG. 1: Schematic sketch of the protocol for the simulation
+runs. 20 independent initial configurations are obtained vi a one
+long simulation run at 6000K. For each independent config-
+uration the system is quenched instantaneously and then ful ly
+equilibrated at temperatures Ti= 3760K and 5000K, followed
+by instantaneous quenches to the temperatures Tf= 3250K,
+3000K,2750K, and2500K. At each Tf, time-dependent cor-
+relation functions are determined for different waiting tim estw.
+Temperature is kept constant at Tfby coupling the system to a
+Nos´ e-Hoover thermostat. The thermostat is switched off aft er
+0.33ns, followed by the continuation of the simulations in t he
+microcanonical ensemble for 33ns.
+the initial temperatures Ti= 5000K (for 16 .35 ns) and
+Ti= 3760K (for 32 .7 ns), the system was quenched in-
+stantaneously to Tf∈ {2500K,2750K,3000K,3250K}.
+To disturb the dynamics minimally, we used a Nos´ e-
+Hoover thermostat [45, 46] instead of a stochastic heat
+bath to keep the temperature at Tfconstant. A velocity
+Verlet algorithm was used to integrate the Nos´ e-Hoover
+equations of motion (see Appendix A) with a time step
+of 1.02fs. After 0 .33 ns the Nos´ e-Hoover thermostat was
+switchedoffandthesimulationwascontinuedintheNVE
+ensemble for 33 ns using a time step of 1 .6fs. Whereas
+previous simulations used instead the NVT ensemble for
+the whole simulation run, we chose to switch to the NVE
+ensemble to minimize any influence on the dynamics due
+to the chosen heat bath algorithm. For the comparison
+with previous simulations and to check for the lack of
+a temperature drift, we show in Fig. 2 for exemplatory
+simulation runs the temperature T=2Ekin
+3Nas a function
+of time where Ekinis the time averaged kinetic energy
+with fluctuations as indicated with error bars. We find
+that evenafter switching off the heat bath [53] there is no
+temperaturedriftfor T= 3250Kand T= 3000Kandfor
+T= 2500Kand T= 2750Kthereisonlyaslighttemper-
+ature drift which is of the same order as the temperature
+fluctuations and the drift occurs only for t/lessorapproxeql0.6 ns. For
+all times t/greaterorapproxeql0.6 ns and for all investigated temperatures
+there is no temperature drift and thus the comparison
+with previous simulations valid.10-210-1100101
+t [ns]2000250030003500 T [K]Tf=3250 K
+Tf=2500 K
+FIG. 2: (Color online) Temperature T=2Ekin
+3Nas a function
+of timetforTi= 5000K ,Tf= 2500K ,2750K,3000K,3250K
+shown in each case for the 11th independent simulation run.
+Ekinincludes a time average and error bars indicate the corre-
+sponding fluctuations.
+III. RESULTS
+In all following, we investigate how the structure and
+dynamics of the system depend on the waiting time tw
+elapsed after the quench from TitoTf. We varied the
+waiting time in the range 0ns ≤tw≤23.98ns.
+A. Partial Structure Factor
+Figure 3 shows for the temperature quench Ti= 5000
+K toTf= 2500 K the partial structure factors [11]
+Sαβ(q,tw) =1
+N/angbracketleftBiggNα/summationdisplay
+i=1Nβ/summationdisplay
+j=1eiq·/parenleftbig
+ri(tw)−rj(tw)/parenrightbig/angbracketrightBigg
+(2)
+whereriandrjare the positions of particles iandjof
+speciesα,β= O,Si. The partial structure factors for all
+other (Ti,Tf) combinations are very similar. Although
+Fig. 3 shows S(q,tw) for the largest investigated temper-
+ature quench, there is only a slight tw-dependence for
+very short waiting times tw≤0.33ns and almost no tw-
+dependence for tw≥0.33ns.
+B. Generalized Incoherent Intermediate Scattering
+Function
+In this section we focus on the time-dependent gener-
+alized intermediate incoherent scattering function [11]
+Cq(tw,tw+t) =1
+Nα/angbracketleftBiggNα/summationdisplay
+j=1eiq·/parenleftbig
+rj(tw+t)−rj(tw)/parenrightbig/angbracketrightBigg
+(3)4
+00.511.5
+tw=0 ns
+tw=0.33 ns
+tw=16.7 ns
+-0.8-0.6-0.4-0.200.20.4Sαβ(q,tw)
+0 1 2 3 4 5678
+q [Å-1]00.10.20.30.40.50.6O-O
+Si-O
+Si-Si
+FIG. 3: (Color online) Partial structure factors Sαβ(q,tw)as
+defined in Eq. (2) for the temperature quench Ti= 5000K to
+Tf= 2500K. Indicated with arrows are the wave vectors qwhich
+have been used to determine Cq(tw,tw+t)as defined in Eq. (3).
+which is a measure for the correlations of the positions
+at time twand at a later time ( tw+t). We investi-
+gated wave vectors of magnitude q= 1.7,2.7,3.4,4.6,5.5
+and 6.6˚A−1, as indicated with arrows in Fig. 3. We
+show in Fig. 4 and Figs. 6 - 8 results for the first sharp
+diffraction peak at q= 1.7˚A−1. Similar results are found
+for all other investigated wave vectors. Figure 4 shows
+Cq(tw,tw+t) for the largest investigated temperature
+quench from Ti= 5000K to Tf= 2500K for waiting
+timestw= 0 - 23.98ns, as listed in the figure caption of
+Fig. 4. We find that Cq(tw,tw+t) is dependent on tw
+for all but the last three investigated waiting times. For
+very short times t/lessorsimilar5·10−5ns and zero waiting time,
+Cq(tw= 0,t) is well approximatedby Cqofthe high tem-
+perature Ti= 5000 K from which the system has been
+quenched (see dashed line in Fig. 4). Thus, Cq(tw= 0,t)
+for very short times is only dependent on Ti,qand the
+particle type, but independent of Tf. For times of the
+order of t= 10−3ns,Cq(tw,tw+t) is oscillatory due to
+the small system size. For times t/greaterorsimilar10−3ns,Cqdecays
+to zero without forming a plateau for small tw. With
+increasing twa plateau is formed, which is independent
+oftwfortw≥0.33ns.
+To characterize the plateau height we define Fas the
+time average of Cq(tw,tw+t) for times 2 .55ps≤t≤
+6.64ps. The inset of Fig. 5 shows that for large waiting
+timesF(tw) becomesindependent of twandofTi. Totest
+this independence of Tifurther, we show Fas a function
+ofqfortw= 16.67ns in Fig. 5. We find that the plateau
+height is dependent on the particle type and decreases
+with decreasing q, butFis independent of Ti.10-610-510-410-310-210-1100101
+t [ns]00.20.40.60.81Cq(tw,tw+t)tw=24.0 ns
+q=1.7 Å-1Otw=0 ns
+FIG. 4: (Color online) Cq(tw,tw+t)as defined in Eq. (3) for
+the quench Ti= 5000K toTf= 2500K for O-atoms and q=
+1.7˚A−1. Waiting times for the solid lines are from bottom to
+toptw= 0ns,1.63·10−4ns,1.63·10−3ns,1.63·10−2ns,
+0.33ns,0.49ns,1.17ns,1.96ns,8.83ns,16.67ns, and23.98ns.
+The order of solid lines are for increasing twblack thin, green
+(gray) thin, black thick, green (gray) thick, black thin, gr een
+(gray) thin, etc. Error bars are as indicated exemplary. The
+thick dashed line corresponds to Cq(tw,tw+t) =Fs(q,t)at
+5000K.
+The plateau in Cqbecomes more horizontal with de-
+creasing final temperature Tf, as can be seen by the com-
+parison of Fig. 4 ( Tf= 2500K) and Fig. 6 ( Tf= 3000K).
+Timest/greaterorsimilar0.1ns correspond to the α-relaxation, where
+Cq(tw,tw+t) decays from the plateau to zero. For the
+quench from 5000K to 2500K (see Fig. 4) this decay
+depends on twfor alltw<8.83ns. However, for tw≥
+8.83ns(thelargestthree tw),Cq(tw,tw+t)isindependent
+oftwnot only for intermediate times t(plateau) but for
+all times (including the α-relaxation). Thus, the system
+reaches equilibrium during the simulation run. For the
+quench from 3760K to 3000K (see Fig. 6), Cq(tw,tw+t)
+becomes independent of twfortw/greaterorsimilar1ns, which means
+that the time at which equilibrium is reached is depen-
+dent on the temperature quench.
+To estimate the time when the system reaches equilib-
+rium for each ( Ti,Tf) combination, we next quantify the
+decay of Cq(tw,tw+t) . Instead of taking a vertical cut
+inCq(tw,tw+t) as we did for F, we now take a hori-
+zontal cut. We define the decay time tCq
+rto be the time
+t=tCq
+rfor which Cq(tw,tw+tCq
+r) =Ccut. We chose
+Ccut= 0.625/0.41/0.295/0.155/0.085/0.04) for the Si
+particles and Ccut= 0.625/0.305/0.195/0.08/0.04/0.014
+for the O particles at the wavenumbers q=
+1.7/2.7/3.4/4.6/5.5/6.6˚A−1, respectively. The resulting
+decaytimes tCq
+rasafunctionofwaitingtime twareshown
+in Fig. 7a for O-atoms and in Fig. 7b for Si-atoms. Color
+(black or green/gray)indicates the initial temperature Ti5
+2 3 4 5 6 7
+q [Å-1]0.00.20.40.60.8F(q)
+Tf=
+2500 K
+2750 K
+3000 K
+3250 K
+2500 K
+2750 K
+3000 K
+3250 K10-410-310-210-1100101
+tw0.20.40.60.8F(tw)
+Ti=3760 K
+Ti=5000 K O  q=1.7 Å-1
+ O  atoms
+FIG. 5:(Color online) Plateau height, F, as defined in the text,
+as a function of wave vector q(tw= 16.67ns) and in the inset as
+a function of tw(q= 1.7˚A−1). Error barsare of the same size as
+symbols as indicated exemplary for Ti= 5000K andTf= 2500
+K.
+10-610-510-410-310-210-1100101
+t [ns]00.20.40.60.81Cq(tw,tw+t)
+tw=24.0 ns
+q=1.7 Å-1O tw=0 ns
+FIG. 6:(Color online) Cq(tw,tw+t)for the quench Ti= 3760K
+toTf= 3000K. Waiting times and corresponding line styles
+are the same as in Fig. 4. Error bars are of the same order
+as indicated in Fig. 4. The thick dashed line corresponds to
+Cq(tw,tw+t) =Fs(q,t)at3760K.
+and symbol shape indicates the final temperature Tf.
+We find that tCq
+r(tw) is characterized by three differ-
+enttw-windows. (I) For waiting times tw/lessorsimilar0.3ns, de-
+cay times are significantly lower for Ti= 5000K (black
+lines and symbols)than for Ti= 3760K (green/graylines
+and symbols). The dependence of tCq
+rontwis strongly
+dependent on all varied parameters, i.e. Ti,Tf, particle
+type, and q. ForTi= 5000K, Tf= 2500K, 2750 K, and10-410-310-210-1100101
+tw [ns]10-410-310-210-1100101trCq(tw) [ns]
+Tf=
+2500 K
+2750 K
+3000 K
+3250 K
+2500 K
+2750 K
+3000 K
+3250 KO-atoms
+Ti=3760 K
+Ti=5000 Kq=1.7 Å-1(a)
+10-410-310-210-1100101
+tw [ns]10-410-310-210-1100101trCq(tw) [ns]
+Tf=
+2500 K
+2750 K
+3000 K
+3250 K
+2500 K
+2750 K
+3000 K
+3250 K(b)
+Ti=3760 K
+Ti=5000 KSi-atoms
+q=1.7 Å-1
+FIG. 7: (Color online) Decay time tCq
+rforq= 1.7˚A−1and (a)
+O-atoms and (b) Si-atoms. Green (gray) is used for Ti= 3760K
+and black for Ti= 5000K. Symbol shape indicates Tfas given
+in the legend. Error bars are given exemplary for ( Ti= 3760K,
+Tf= 2500K), (Ti= 5000K,Tf= 2500K) and ( Ti= 5000K,
+Tf= 3000K).
+q= 1.7˚A−1, 2.7˚A−1tCq
+r(tw) followsroughlya powerlaw
+with an exponent µ≈1.15 with variations of the order
+of 0.07 dependent on Tf, particle type, and q. (II) For
+intermediate waiting times, tCq
+r(tw) also follows roughly
+a power law with a different exponent than in regime (I).
+We find for Ti= 5000K, Tf= 2500K, 2750Kand q= 1.7
+˚A−1, 2.7˚A−1µ≈0.35with variationsoftheorderof0 .08
+depending on Tf, particle type, and q. Best power law
+fits are for Ti= 3760K, Tf= 2500K with µranging
+fromµ= 0.55/0.57 forq= 1.7˚A−1toµ= 0.69/0.63
+forq= 6.6˚A−1and for Si/O atoms. Kob and Barrat
+find for the binary Lennard-Jones system also a power
+law fortCq
+r(tw), however, with µ= 0.882 [16]. Similar to
+Grigera et al. [47] we find that the transition from small6
+waiting times (I) to intermediate waiting times (II) is
+accompanied by a change of the exponent µ. (III) For
+very long waiting times tCq
+r(tw) is independent of twand
+Ti, i.e. equilibrium is reached. The waiting time t23for
+which the transition from regime (II) to regime (III) oc-
+curs is dependent on Tf:t23≈0.3ns forTf= 3250K,
+t23≈1ns forTf= 3000K, t23≈3ns forTf= 2750K,
+andt23≈10ns for Tf= 2500K.[54]
+Mean-field spin glass models predict [8, 9]
+Cq(tw,tw+t) =CST
+q(t)+CAG
+q/parenleftbiggh(tw+t)
+h(tw)/parenrightbigg
+,(4)
+according to which Cq(tw,tw+t) can be separated into
+a short-time term CST
+q(t) that is independent of twand
+an intermediate-time term that scales as h(tw+t)/h(tw)
+wherehis a monotonously increasing function. It has
+been observedfor different systems that the function h(t)
+followsh(t)≈tα. Thus, the so-called“simple aging” (see
+[16]andreferencestherein)appliesand, asaconsequence,
+Cqas a function of ( t/tw) for different twsuperimpose.
+M¨ ussel and Rieger [48] have proposed activated dy-
+namics for Cq(tw,tw+t),
+Cq(tw,tw+t) =CST
+q(t)+CAG
+q/parenleftbiggln[(tw+t)/τfit]
+ln[tw/τfit]/parenrightbigg
+,(5)
+where the characteristic time scale τfitis a fit parame-
+ter. We find that neither Cq/parenleftBig
+tw+t
+tw/parenrightBig
+, norCq/parenleftBig
+t
+tw/parenrightBig
+, nor
+Cq/parenleftBig
+ln[(tw+t)/τfit]
+ln[tw/τfit]/parenrightBig
+(for any choice of τfit) superimpose for
+different tw. Instead we find, similar tothe resultsofKob
+and Barrat [16] for a binary Lennard-Jones system, that
+time superposition holds, defined by
+Cq(tw,tw+t) =CST
+q(t)+CAG
+q/parenleftBigg
+t
+tCq
+r(tw)/parenrightBigg
+.(6)
+In Fig. 8, we show Cq(t/tCq
+r) for the same set of parame-
+tersasin Fig. 4. When all waitingtimes areincluded (see
+inset of Fig. 8) time superposition does not apply due to
+including too short waiting times. Wahlen and Rieger
+[26] have studied Cq(tw,tw+t) for the same BKS-SiO 2
+system, however for waiting times smaller than 50ps.
+Their results are consistent with the inset of Fig. 8. For
+waiting times tw≥0.49ns (see Fig. 8), however, Eq. (6)
+is a good approximation. Please note that Cq(t/tCq
+r) fol-
+lowstimesuperpositionforallwaitingtimes tw≥0.49ns,
+i.e. not only for the time-range (II), but alsofor the time-
+range (III). That means for the α-relaxation that the
+shapeofthe out-ofequilibrium curvesisthe same(within
+error bars) as the shape of the equilibrium curves. We
+find similar results for all other ( Ti,Tf) combinations, Si-
+particles, and all other q.
+Next we test whether Cq=C(q,tw,tw+t) scales as
+Cq=C(q,z(tw,t) wherezis a function of twandtonly,
+i.e. independent of q. Following an approach of Kob and
+Barrat [17] we show in Figure 9 a parametric plot for10-310-210-1100101
+t/tCq
+r00.20.40.60.8Cq(tw,tw+t)tw=0.49 ns
+1.17 ns
+1.96 ns
+8.83 ns
+16.67 ns
+23.98 ns
+10-610-410-210010200.20.40.60.81
+q=1.7 Å-1Otw=0 ns
+tw=24.0 ns
+FIG. 8: (Color online) Cq(t/tCq
+r)for the quench from Ti=
+2500K toTi= 5000K for O-atoms and q= 1.7˚A−1. Waiting
+times and corresponding lines in the inset are the same as in
+Fig. 4.
+0 0.2 0.4 0.6 0.8 1
+Cq=1.7Å-1 (tw,tw+t)00.20.40.60.81Cq’(tw,tw+t)tw=0.49 ns
+1.17 ns
+1.96 ns
+8.83 ns
+16.67 ns
+23.98 ns
+q’=2.7 Å-1
+q’=3.4 Å-1
+q’=4.6 Å-1
+q’=6.6 Å-1
+q’=5.5 Å-1
+FIG.9:(Color online) Cq′(tw,tw+t)asafunctionof Cq(tw,tw+
+t)forq= 1.7˚A−1andq′= 2.7, 3.4, 4.6, 5.5, 6.6 ˚A−1for O-
+atoms and the temperature quench from 5000K to2500K.
+Cq′(tw,tw+t) as a function of Cq(tw,tw+t) forq=
+1.7˚A−1andq′= 2.7, 3.4, 4.6, 5.5, 6.6 ˚A−1for O-atoms
+and the temperature quench from 5000K to 2500K. For
+sufficiently large twwe find, contrary to the results of
+Kob and Barrat for the Lennard-Jones system, that for
+SiO2the parametric curves superimpose and thus that
+C(q,tw,tw+t) =C(q,z(tw,t) fortw≥0.49 ns. This
+includes, within error bars, also the equilibrated curves
+fortw/greaterorsimilar10ns. Wefindsimilarresultsforallother( Ti,Tf)
+combinations, Si-particles, and all other q.7
+10-610-510-410-310-210-1100101
+t [ns]10-410-310-210-1100101∆r2(tw,tw+t) [Å2]tw=0 ns
+tw=24.0 nsO-atoms
+FIG. 10: (Color online) Mean square displacement ∆r2(tw,tw+
+t)as defined in Eq. (7) for the temperature quench from 5000K
+to2500K and for O-atoms. Waiting times and corresponding
+line styles are the same as in Fig. 4.
+C. Mean Square Displacement
+Intheprevioussection, wehavefocusedontheanalysis
+ofCq(tw,tw+t) and identified different time-windows. In
+this section, we consider the mean square displacement
+∆r2(tw,tw+t) =1
+NN/summationdisplay
+i=1/angbracketleftBig
+(ri(tw+t)−ri(tw))2/angbracketrightBig
+.(7)
+Figure 10 shows ∆ r2(tw,tw+t) for the temperature
+quench from 5000K to 2500K and for O-atoms. As in
+Fig. 4, for times t/lessorsimilar5·10−5ns and zero waiting time,
+the mean square displacement ∆ r2(tw= 0,t) is well ap-
+proximated by ∆ r2of the high temperature Ti= 5000K
+from which the system has been quenched (see dashed
+line in Fig. 10) and thus independent of Tf. For times
+t≈10−3ns, ∆r2(tw,tw+t) is oscillatorydue to the small
+system size [31], while for times t/greaterorsimilar10−3ns and waiting
+timestw≥0.33ns, we find that ∆ r2forms a plateau
+which is independent of tw. As for Cq, we find that the
+plateau is the more horizontal the smaller Tfand the
+plateau height depends on the particle type but is inde-
+pendent of Ti.
+For waiting times tw≥0.33ns and times t/greaterorsimilar0.1ns,
+the mean square displacement leaves the plateau and in-
+creases further. To characterize the dependence of this
+α-relaxation we define the time tmsd
+ras the time t=tmsd
+r
+for which ∆ r2(tw,tw+tmsd
+r) = 1.35˚A2(see Fig. 11). We
+can identify again the three time windows (I) of waiting
+timestw/lessorsimilar0.3nswithadependenceon Ti,Tfandparticle
+type, (II) the aging regime of intermediate waiting times
+wheretmsd
+rfollows roughly a power law, and (III) for
+verylongwaitingtimes whenequilibrium isreached. The
+transition from (II) to (III) occurs at approximately the10-410-310-210-1100101
+tw [ns]10-410-310-210-1100101trmsd(tw) [ns]
+Tf=
+2500 K
+2750 K
+3000 K
+3250 K
+2500 K
+2750 K
+3000 K
+3250 K O-atoms
+Ti=3760 K
+Ti=5000 K
+FIG. 11: (Color online) tmsd
+rfor O-atoms. Symbols for the dif-
+ferent(Ti,Tf)combinations are the same as in Fig. 7. Error bars
+are indicated exemplary for ( 3760K,2500K), (5000K,2500K)
+and (5000K,3000K).
+10-210-1100101
+t /trmsd100101∆r2(tw,tw+t) [Å2]tw=0.49 ns
+1.17 ns
+1.96 ns
+8.83 ns
+16.67 ns
+23.98 ns
+tw=0.49 nstw=24.0 ns
+O-atoms
+FIG. 12: (Color online) ∆r2(t/tmsd
+r)for the temperaturequench
+from5000K to2500K and for O-atoms.
+same times t23as forCq, i.e.t23≈0.3ns forTf= 3250K,
+t23≈1ns forTf= 3000K, t23≈3ns forTf= 2750K
+andt23≈10ns for Tf= 2500K.
+Figure 12 shows the equivalent of Fig. 8 to test time
+superposition. We find for ∆ r2(t/tmsd
+r) that time super-
+position is valid for waiting times 0 .34ns≤tw/lessorsimilar8.83ns,
+i.e. for the time window (II) but not for the time window
+(III).8
+IV. SUMMARY
+Usingmoleculardynamicssimulations, we investigated
+for the strong glass former SiO 2the aging dynamics be-
+low the critical MCT temperature Tc, using the BKS po-
+tential to model the interactionsbetween silicon and oxy-
+gen atoms. After an instantaneous quench from Ti> Tc
+to a temperature Tf< Tcthe dynamics towards equilib-
+rium was studied as a function of waiting time tw. Note
+that the temperatures Tfwere chosen such that equilib-
+rium was reached on the time span of the simulations
+(of the order of 30ns). The central quantities considered
+in this work are the incoherent intermediate scattering
+function Cq(tw,tw+t) and the mean square displace-
+ment ∆r2(tw,tw+t). These functions depend on the
+time origin at twas long as twis smaller than the typical
+relaxation time, τeq, that is required to equilibrate the
+system.
+We find that the decay of Cq(tw,tw+t) (and similarly
+the rise of ∆ r2(tw,tw+t)) exhibit qualitative changes
+from short to long waiting times. At short waiting times,
+relaxation processes are dominant that correspond to the
+earlyβrelaxation regime at the target temperature Tf.
+In thistwregime, no well-defined plateau is found in
+Cq(tw,tw+t) (see Fig. 4). Instead, this function first
+decreases rapidly, followed by a strongly stretched expo-
+nential decay to zero. At long waiting times, the βre-
+laxation seems to be very similar to that at equilibrium.
+The Debye-Waller factor (i.e. the height of the plateau
+inCq(tw,tw+t)) has reached its equilibrium value, al-
+thoughthedecayof Cq(tw,tw+t)fromtheplateautozero
+is faster than that at equilibrium. However, the shape of
+curves describing the long-time decay of Cq(tw,tw+t) is
+the same as that at equilibrium. Thus, Cqfollows a sim-
+ple time superposition for long waiting times, different
+e.g. from the “activated dynamics scaling” proposed by
+Wahlen and Rieger for BKS silica.
+Our results show that the aging dynamics of BKS sil-
+ica is very similar to that of the KALJ mixture. For
+both silica and the KALJ mixture three tw-regimes can
+be identified and Cqfollows time superposition for suffi-
+ciently large tw. The only difference between these two
+systems is that Cqscales as C(q,z(tw,t) for SiO 2but
+less well for the KALJ mixture. So slightly below its
+critical temperature Tcof MCT, the strong glass-former
+silicadoes not seem to be verydifferent from typical frag-
+ile systems, although one has already reached the low
+temperature Arrhenius regime (note that activation en-
+ergies for the self-diffusion, viscosity etc. are of the order
+of 5eV, similar to the corresponding activation energies
+close to the glass transition temperature Tg≈1450K, as
+measured in various experiments). However, the dynam-
+icscouldbe verydifferentat verylowtemperatures(close
+toTg) where the long-time aging regime is not accessi-
+ble by computer simulations. Thus, more experimental
+work on the aging dynamics of silica around Tgwould
+be very desirable. We also leave for future work to test
+whether also for other systems three tw-regimes are iden-tified and whether the equilibrium curve is included in
+the time superposition of Cqat intermediate and large
+tw.
+Acknowledgments
+KVL thanks A. Zippelius and the Institute of Theoret-
+ical Physics, University of G¨ ottingen, for hospitality and
+financial support.
+Appendix A: Velocity Verlet Nos´ e Hoover
+The Nos´ e Hoover equations of motion are for particles
+i= 1,...,Nat position riwith momentum pi
+˙ri=pi
+mi(A1)
+˙pi=Fi−ζpi (A2)
+˙ζ=1
+Q/parenleftBiggN/summationdisplay
+i=1p2
+i
+mi−XkT/parenrightBigg
+(A3)
+and thus
+¨ri=Fi
+mi−ζ˙ri (A4)
+d2lns
+dt2=˙ζ=1
+Q/parenleftBiggN/summationdisplay
+i=1mi˙r2
+i−XkT/parenrightBigg
+(A5)
+whereX= 3N. We integrated with the generalized
+velocity Verlet form of Fox and Andersen [49]
+ri(t+∆t) =ri(t)+∆t˙ri(t) (A6)
++(∆t)2
+2/parenleftbiggFi(t)
+mi−ζ(t)˙ri(t)/parenrightbigg
+lns(t+∆t) = lns(t)+∆tζ(t) (A7)
++(∆t)2
+2Q/parenleftBiggN/summationdisplay
+i=1mi˙r2
+i(t)−XkT/parenrightBigg
+ζapprox(t+∆t) =ζ(t) (A8)
++∆t
+Q/parenleftBiggN/summationdisplay
+i=1mi˙r2
+i(t)−XkT/parenrightBigg
+˙ri(t+∆t) =˙ri(t) (A9)
++∆t
+2/parenleftbiggFi(t)+Fi(t+∆t)
+mi
+−/bracketleftbig
+ζ(t)+ζapprox(t+∆t)/bracketrightbig˙ri(t)/parenrightbigg
+/parenleftbigg
+1−∆t
+2ζapprox(t+∆t)/parenrightbigg
+ζ(t+∆t) =ζ(t)+∆t
+2Q/parenleftBigN/summationdisplay
+i=1mi˙r2
+i(t) (A10)
++N/summationdisplay
+i=1mi˙r2
+i(t+∆t)−2XkT/parenrightBig9
+To ensure that Υ =N/summationtext
+i=1p2
+i
+2mi+U({ri})+Q
+2ζ2+XkTlns
+is conserved (see [46]) we chose Q= 50000 ˚A2u.
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+895.11679eV/ ˚A(instead of a2,O= 90.38499 eV/ ˚A).
+[53] Both the duration of the NVT-simulation run, as well as
+the Nos´ e-Hoover parameter Qwere chosen carefully such
+that Υ and Etotwere constant during the NVT and NVE
+runs respectively.
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+results of Scheidler et al. [39] who determined the re-
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+agreement with experimental data.
\ No newline at end of file
diff --git a/1001.0717.txt b/1001.0717.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c7d3289e1e6b8e917ef9c9c31d9d25c6f212ae1a
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+++ b/1001.0717.txt
@@ -0,0 +1,3831 @@
+arXiv:1001.0717v5  [math.DG]  19 Mar 2013SIAM J. Imaging Sci. 5 (2012), pp. 244-310.
+ALMOST LOCAL METRICS ON SHAPE SPACE OF
+HYPERSURFACES IN n-SPACE
+MARTIN BAUER, PHILIPP HARMS, PETER W. MICHOR
+Abstract. This paper extends parts of the results from [P.W.Michor and D.
+Mumford, Appl. Comput. Harmon. Anal., 23 (2007), pp. 74–113] for plane
+curves to the case of hypersurfaces in Rn. LetMbe a compact connected
+oriented n−1 dimensional manifold without boundary like the sphere or t he
+torus. Then shape space is either the manifold of submanifol ds ofRnof type
+M, or the orbifold of immersions from MtoRnmodulo the group of diffeo-
+morphisms of M. We investigate almost local Riemannian metrics on shape
+space. These are induced by metrics of the following form on t he space of
+immersions:
+Gf(h,k) =/integraldisplay
+MΦ(Vol(f),Tr(L))¯g(h,k)vol(f∗¯g),
+where ¯gis the Euclidean metric on Rn,f∗¯gis the induced metric on M,h,k∈
+C∞(M,Rn)are tangent vectors at fto the space of embeddings or immersions,
+where Φ : R2→R>0is a suitable smooth function, Vol( f) =/integraltext
+Mvol(f∗¯g) is
+the total hypersurface volume of f(M), and the trace Tr( L) of the Weingarten
+mapping is the mean curvature. For these metrics we compute t he geodesic
+equations both on the space of immersions and on shape space, the conserved
+momenta arising from the obvious symmetries, and the sectio nal curvature.
+For special choices of Φ we give complete formulas for the sec tional curvature.
+Numerical experiments illustrate the behavior of these met rics.
+Contents
+1 Introduction
+2 Shape space and the Hamiltonian approach
+3 Differential geometry of surfaces and notation
+4 Variational formulas
+5 The geodesic equation on Imm( M,Rn)
+6 The geodesic equation on Bi(M,Rn)
+7 Sectional curvature on Bi(M,Rn)
+8 Geodesic distance on Bi(M,Rn)
+9 The set of concentric spheres
+10 Special cases of almost local metrics
+11 Numerical results
+References
+The AMPL model file
+Date: October 24, 2018.
+2000Mathematics Subject Classification. 58B20, 58D15, 58E12, 65K10.
+All authors were supported by FWF Project 21030 and by NSF-FR G grant DMS-0456253.
+12 Almost local metrics on shape space of hypersurfaces in n-space
+1.INTRODUCTION
+Many procedures in science, engineering, and medicine produce dat a in the form
+of shapes of point clouds in Rn. If one expects such a cloud to follow roughly a
+submanifold of a certain type in Rn, then it is of utmost importance to describe the
+space of all possible submanifolds of this type (we call it a shape spac e hereafter)
+and equip it with a significant metric which is able to distinguish special fe atures
+of the shapes. Almost local metrics are a contribution towards this aim.
+This paper benefited from discussions with David Mumford, Hermann Schichl,
+who taught us about the use of AMPL, Johannes Wallner and Tilak Rat nanather.
+Parts of this paper can be found in the Ph.D. thesis of Martin Bauer [3 ].
+1.1.Reading suggestions. A reader who wants to see results before immersing
+himself in the theoretical background is recommended to pick up the necessary
+definitions in the introduction and to jump directly to the last two sec tions con-
+taining special cases and numerical results. In section 2 we build the fundaments
+for shape analysis in a Riemannian setting. Section 3 presents backg round material
+in differential geometry and can serve as a reference for our nota tion. Throughout
+this work we will use covariant derivatives of vector fields along immer sions. This
+concept might not be known to all readers; thus we have decided to give a careful
+description in section 3.5. The main results of the work are in sections 6–11.
+In the following introduction we give a non-technical presentation o f our ap-
+proach. Parts of it can also be found in the Ph.D. thesis of Philipp Harm s [8].
+1.2.The Riemannian setting. Most of the metrics used today in data analysis
+and computer vision are of an ad-hoc and naive nature. One embeds shape space in
+some Hilbert space or Banach space and uses the distance therein. Shortest paths
+are then line segments, but they leave shape space quickly. For sev eral reasons the
+Riemannian setting for shape analysis is a better solution.
+•It formalizes an intuitive notion of similarity of shapes: Shapes that differ
+only by a small deformation are similar to each other. To compare shapes,
+we measure the length of a deformation. A deformation of a shape is a path
+in shape space. Remember that in a Riemannian manifold, the geodesic
+distance between two points is the infimum over the length of all path s
+connecting them.
+•Riemannianmetricsonshapespacehavebeen usedsuccessfullyin computer
+visionfor a long time, often without any mention of the underlying metric.
+Gradient flows for shape smoothing are an example. An underlying me tric
+is needed for the definition of a gradient. Often, the metric used imp licitly
+is theL2-metric which has, however, turned out to be too weak.
+•The exponential map (if it exists) that is induced by a Riemannian met-
+ric permits us to linearize shape space : When shapes are represented as
+initial velocities of geodesics connecting them to a fixed reference s hape,
+one effectively works in the linear tangent space over the referenc e shape.M. Bauer, P. Harms, P. Michor 3
+Curvature will play an essential role in quantifying the deviation of cu rved
+shape space from its linearized approximation.
+•The linearization of shape space by the exponential map allows us to d o
+statistics on shape space.
+However a disadvantage of the Riemannian approach is that shapes can be com-
+pared with each other only when there is a deformation between the m, i.e., when
+they have the same topology.
+1.3.Shape spaces. In mathematics and computer vision, shapes have been rep-
+resented in many ways. Point clouds, meshes, level-sets, graphs o f a function,
+currents, and measures are but some of the possibilities. The notio n of shapes
+underlying this work is that of immersed or embedded submanifolds of Rnof co-
+dimension one. Any such submanifold can be represented as a fixed im mersion or
+embedding modulo reparametrizations.
+The space of all immersions is illustrated in figure 1. The colors are the re
+to help the reader visualize parametrizations. Immersions differing o nly by a
+reparametrization are drawn along vertical lines. These lines are th e orbits of
+the reparametrization group. Immersions in the same orbit corres pond to the same
+shape in shape space. In other words, shape space is the space of orbits of the
+reparametrization group acting on the space of immersions.
+Figure 1. Illustration of the space of immersions of S1inR2.
+As mentioned in the previous section, only shapes with the same topo logy can be
+compared. Thus we assume that all shapes (i.e., submanifolds) are d iffeomorphic
+to the same compact connected oriented n−1 dimensional manifold M. We will
+deal only with smooth shapes. They form the core of actual shape space, which
+can be viewed as the Cauchy completion with respect to geodesic dist ance for one
+of the Riemannian metrics that we treat in this paper. See section 2 f or a formal
+definition of shape space.
+1.4.Riemannian metrics on shape space. Riemannian metrics measure infin-
+itesimal deformations . Riemannian metrics on shape space come in two flavors:4 Almost local metrics on shape space of hypersurfaces in n-space
+•Outermetricsmeasurehowmuchambientspacehastobedeformed inorder
+to yield the desired deformation of the shape. An infinitesimal defor mation
+of ambient space is a vector field on ambient space and could be pictur ed
+as a small arrow attached to every point in ambient space; see figur e 2.1
+•Inner metrics measure deformations of the shape itself within a fixe d ambi-
+ent space. An infinitesimal deformation of the shape itself is a vecto r field
+along the shape. It could be pictured as a small arrow attached to e very
+point of the shape; see figure 2.
+The metrics treated in this work are inner metrics.
+Figure 2. Infinitesimal transformation of ambient space (left) as
+measured by an outer metric and infinitesimal transformation of
+the shape itself (right) as measured by an inner metric.
+1.5.Where this paper comes from. In [15], Michor and Mumford investigated
+inner metrics on the shape space of planar curves. The simplest suc h metric is the
+L2-metric given by
+G0
+f(h,k) =/integraldisplay
+S1¯g/parenleftbig
+h(θ),k(θ)/parenrightbig
+|f′(θ)|dθ,
+wheref,h,k:S1→R2are smooth functions. fis the curve representing the
+shape, and h,kare deformation vector fields of f. The Euclidean metric on R2is
+denoted by ¯ g. We use integration by arc length ds=|f′(θ)|dθ. This makes the
+metric invariantunder reparametrizationsof f,h,k. This invarianceis needed when
+factoring out reparametrizations; see section 2. Since the metric has to be positive
+definite, it is natural to require that f′(θ)/negationslash= 0 everywhere, i.e., fis an immersion.
+Unfortunately it turned out that the L2metric induces vanishing geodesic dis-
+tance on shape space [15]. This means that any two shapes can be co nnected by
+an arbitrarily short path in shape space, when path length is measur ed with the L2
+metric. Later in [14] it was found that the vanishing geodesic distanc e phenomenon
+for theL2-metric occurs also in the more general shape space where S1is replaced
+by a compact manifold Mand Euclidean R2is replaced by a Riemannian manifold
+1The graphic is an adaptation by the authors of a graphic in [20 ].M. Bauer, P. Harms, P. Michor 5
+N. It also occurs on the full diffeomorphism group Diff( N).2These results together
+imply vanishing geodesic distance on spaces of immersions.3
+The discovery of the degeneracy of the L2metric was the starting point of a
+quest for better Riemannian metrics. A possibility excluding the dege nerate paths
+encountered in [15] is to penalize high length and/or curvature. This led to the
+investigationofaclassofmetricswhichwerecalledalmostlocalmetric s; see[16,15].
+A better name might be weighted L2-metrics. They are of the form
+GΦ
+f(h,k) =/integraldisplay
+S1Φ(ℓ(f),κf(θ))¯g/parenleftbig
+h(θ),k(θ)/parenrightbig
+|f′(θ)|dθ,
+where Φ : R2→R>0is a suitable smooth function, ℓ(f) =/integraltext
+S1|f′(θ)|dθis the
+length off, andκfis the curvature of f. If Φ = Φ( ℓ(f)), then this is just a
+conformal change of the metric; it was proposed and investigated independently in
+[18] and in [23, 24, 25]. For Φ = 1+ Aκ2
+f, whereAis a positive constant, the metric
+was investigated in great detail in [15].
+1.6.Almost local metrics, geodesics, and curvature. In this paper we take
+up the investigation of almost local metrics from [16] and we genera lize it to higher
+dimensions. For surfaces in R3this leads to metrics of the form
+GΦ
+f(h,k) =/integraldisplay
+MΦ(Area(f),µ)¯g(h,k)dArea.
+HereMis a compact connected oriented two-dimensional manifold, ¯ gis the Eu-
+clidean metric on R3,f:M→R3is an immersion, h,k:M→R3are seen as
+deformation vector fields of the immersion, and Φ : R3→R>0is again a suit-
+able positive smooth function depending on the area of the immersed surfacef(M)
+and on the mean curvature µ. In two dimensions, the mean curvature µ= Tr(L)
+and Gauß curvature κ= det(L) are all the invariants of the Weingarten mapping
+L. However, in this paper we do not treat the metric which involves the Gauß
+curvature. This is done in the paper [1].
+We do not restrict ourselves to surfaces in R3. Instead we treat almost local
+metrics on spaces of hyper-surfaces in Rn. More specifically, these are metrics of
+the form
+GΦ
+f(h,k) =/integraldisplay
+MΦ(Vol(f),Tr(L))¯g(h,k)vol(g).
+HereMis a compact connected oriented n−1 dimensional manifold, f:M→Rn
+is an immersion, h,k:M→Rnare deformation vector fields, and Φ : R2→R>0is
+again a suitable positive smooth function. We denote the pullback of t he Euclidean
+metric ¯gtoMvia the immersion fbyg=f∗¯g. It is sometimes called the first fun-
+damental form of the surface and is given in coordinates by gij= ¯g(∂if,∂jf). The
+naturalreplacementof dAis then−1dimensionalvolume densityvol( g) induced by
+g. In coordinates u1,...un−1onMit is given by/radicalbig
+det(gij)|du1∧···∧dun−1|. The
+totaln−1 dimensional volume of f(M) is denoted by Vol( f) =/integraltext
+Mvol(g). Finally,
+2But not on the subgroup Diff( N,vol) of volume preserving diffeomorphisms, where the geo-
+desic equation for the L2-metric is the Euler equation of an incompressible fluid.
+3This has not been stated in [15, 14], but it follows easily. Fi rst choose a short horizontal path
+going from the immersion f0to the Diff( M)-orbit of the immersion f1. Then choose another short
+path in the Diff( M)-orbit of f1connecting the endpoint of the previous path to f1.6 Almost local metrics on shape space of hypersurfaces in n-space
+Tr(L) is the mean curvature, which is the trace of the Weingarten mappin gL. See
+section 3 for a rigorous definition of the objects that are used in th e definition of
+the metric.
+As mentioned above, metrics of this form will be called almost local as in [16].
+A better name might be weighted H0-metrics or weighted L2-metrics. It is natural
+to consider Gauß-curvature weighted metrics as well. This is done in [ 1]. It might
+also be worth considering other curvature invariants.
+In the theoretical part of this work Φ is general. The special choice s of Φ inves-
+tigated in the more practical two last sections are
+Φ = Volk,Φ =eVol,Φ = 1+ATr(L)2k,Φ = Vol1+n
+1−n+ATr(L)2
+Vol,
+whereA>0 andk∈Nare constants. The last choice of Φ induces a scale-invariant
+metric.
+TheGΦ-metrics are weak Riemannian metrics on the manifold of all immersions
+M→Rn, which is an open subset of the Fr´ echet space of all smooth mappin gs.
+But we are interested in the induced Riemannian metric on shape spac e, which
+is the quotient space under the identification of immersions differing o nly by a
+reparametrization; see 2.8. Shape space is difficult to handle, but ge odesics on
+shape space are images of so-called horizontal geodesics on the space of immersions.
+A geodesic on the manifold of immersions is called horizontal if its velocit y vector
+is a horizontal tangent vector at each time. A tangent vector is ca lled horizontal if
+it isGΦ-perpendicular to the reparametrization orbits. The length of a ho rizontal
+(minimizing) geodesic defines the distance between its endpoints, wh ich is what we
+are interested in.
+In general, geodesics are critical points of the energyfunctional
+E(f) =1
+2/integraldisplay1
+0GΦ
+f(t)(∂tf,∂tf)dt,
+wherefis a smooth curve of immersions. A curve f(t) is a critical point of the
+horizontal energy functional
+Ehor(f) =1
+2/integraldisplay1
+0GΦ
+f(t)/parenleftbig
+(∂tf)hor,(∂tf)hor/parenrightbig
+dt
+if and only if a suitable reparametrization of it is a horizontal geodesic . In the
+above formula ( ∂tf)hordenotes the horizontal part of the velocity ∂tf; see section
+2.8.
+Almost local metrics havethe greatadvantagethat a tangent vec torh:M→Rn
+with footpoint an immersion f:M→Rnis horizontal if and only if h(x) is normal
+toTf(x)f(M) inRnfor allx∈M; see section 6.1. Therefore the horizontal energy
+for almost local metrics is given by an easy and computable formula. T his makes
+the numerical approach in this paper possible; see section 11. But a n analytical
+proof of the existence of critical points for the horizontal energ y (which can be
+viewed as an anisotropic plateau problem) is still lacking. The simple for m of the
+horizontal bundle also opens the way to computations of sectional curvature on
+shape space; see section 7. We are interested in sectional curvat ure because it will
+eventuallybe important for doing statistics on shape space and for the computationM. Bauer, P. Harms, P. Michor 7
+of conjugate points. Furthermore, unbounded positive sectiona l curvature might be
+related to vanishing geodesic distance (see [14]).
+1.7.Contributions of this work. Almost local metrics are generalized to higher
+dimensions. Some estimates for geodesic distance on shape space a re proven. They
+show that almost local metrics with suitable functions Φ overcome th e degeneracy
+of theL2metric. In addition, almost local metrics are compared to the Fr´ ec het
+metric.
+The geodesic equation and conserved quantities are calculated on s hape space
+and on the full space of immersions. For this aim the Hamiltonian forma lism
+developed in [16] is updated to the more general situation here. Fur thermore, the
+Riemann curvature tensor is calculated on shape space. It contain s some negative,
+positive, and indefinite terms. Explicit formulas for special choices o f Φ are given.
+For all these calculations, first and second derivativesofthe metr ic, volume form,
+second fundamental form and other curvature terms are develo ped. The derivatives
+are taken with respect to the immersion inducing these objects.
+The last section contains numerical experiments for geodesics. We do only
+boundary value problems and no initial value problems (it is not clear if t he initial
+valueproblemis wellposed). We use Mathematicatoset up the triang ulationofthe
+surfaces, feed this into AMPL (a modeling software developed for o ptimization),
+and use the solver IPOPT. The numerical results are tested on the totally geodesic
+subspace of concentric spheres where we also have analytic solutio ns. Then we
+study translations and deformations of surfaces for various met rics and discuss the
+appearing phenomena.
+Forthesakeofsimplicitywehaverestrictedourselvestotheshape spaceofhyper-
+surfaces in Rn. The more general case of arbitrary co-dimension and Rnreplaced
+by a non-flat Riemannian manifold ( N,¯g) will be treated in another paper.
+2.Shape space and the Hamiltonian approach
+The aim of this chapter is to develop a rigorous notion of shape space , to derive
+the geodesic equation on shape space, and to calculate the conser ved momenta.
+2.1.Manifolds of immersions and embeddings and the diffeomorphi sm
+group.Mathematically, parametrized surfaces will be modeled as immersions or
+embeddings of one manifold into another. We call immersions and embe ddings
+parametrized since a change in their parametrization (i.e., applying a d iffeomor-
+phism on the domain of the function) results in a different object. We will deal
+with the following sets of functions:
+(1) Emb( M,Rn)⊂Imm(M,Rn)⊂C∞(M,Rn).
+C∞(M,Rn) is the set of smooth functions from MtoRn. Imm(M,Rn) is the set
+of allimmersions ofMintoRn, i.e., all functions f∈C∞(M,Rn) such that Txf8 Almost local metrics on shape space of hypersurfaces in n-space
+is injective for all x∈M. Emb(M,Rn) is the set of all embeddings ofMinto
+Rn, i.e., all immersions fthat are a homeomorphism onto their image. In most
+cases, immersions will be used since this is the most general setting. Working with
+embeddings instead of immersions makes a difference in section 8.
+SinceMis compact, by assumption it follows that C∞(M,Rn) is aFr´ echet
+manifold [11, section 42.3]. All inclusions in (1) are inclusions of open subsets.
+Therefore, all function spaces in (1) are Fr´ echet manifolds as we ll.
+The tangent bundle of the manifold of immersions is
+TImm(M,Rn) = Imm(M,Rn)×C∞(M,Rn)⊂C∞(M,Rn×Rn),
+and the cotangent bundle is T∗Imm(M,Rn) = Imm(M,Rn)×D′(M)n, where the
+second factor consists of n-tuples of distributions in D′(M) =C∞(M)′, which is
+the space of distributional sections of the density bundle.
+By Diff(M) we will denote the group of all smooth diffeomorphisms. Diff( M)
+is a Fr´ echet manifold as well, since it is an open subset of C∞(M,M). It is an
+infinite dimensional Lie group in the sense of [11, section 43]. The diffeo morphism
+group Diff( M) acts smoothly on C∞(M,Rn) and its subspaces Imm( M,Rn) and
+Emb(M,Rn) by composition from the right. The action is given by the mapping
+Imm(M,Rn)×Diff(M)→Imm(M,Rn),(f,ϕ)/ma√sto→r(f,ϕ) =rϕ(f) =f◦ϕ.
+The tangent prolongation of this group action is given by the mapping
+T(rϕ) :TImm(M,Rn)×Diff(M)→TImm(M,Rn),
+(f,h,ϕ)/ma√sto→(f◦ϕ,h◦ϕ).
+We will sometimes use the abbreviations Emb ,Imm, and Diff when the domain
+and co-domain of the functions are clear from the context.
+2.2.Riemannian metrics on the manifold of immersions. In this work we
+consider smooth Riemannian metrics on Imm( M,Rn), i.e., smooth mappings
+G: Imm(M,Rn)×C∞(M,Rn)×C∞(M,Rn)→R,
+(f,h,k)/ma√sto→Gf(h,k),bilinear inh,k,
+Gf(h,h)>0 forh/negationslash= 0.
+Each such metric is weakin the sense that Gf, viewed as bounded linear mapping
+Gf:TfImm(M,Rn) =C∞(M,Rn)→T∗
+fImm(M,Rn) =D′(M)n,
+G:TImm(M,Rn)→T∗Imm(M,Rn),
+G(f,h) = (f,Gf(h, .)),
+is injective but can never be surjective. We shall need also its tange nt mapping
+TG:T(TImm(M,Rn))→T(T∗Imm(M,Rn)).
+Wewriteatangentvectorto TImm(M,Rn)as(f,h;k,v),where(f,h)∈TImm(M,Rn)
+is its footpoint, kis its vector component in the Imm( M,Rn)-direction, and where
+vis its component in the C∞(M,Rn)-direction.M. Bauer, P. Harms, P. Michor 9
+ThenTGis given by
+TG(f,h;k,v) = (f,Gf(h, .);k,D(f,k)Gf(h, .)+Gf(v, .)).
+Note that only these smooth functions on Imm( M,Rn) whose derivatives lie in
+the image of Gin the cotangent bundle have G-gradients. This requirement has
+only to be satisfied for the first derivative; for the higher ones it fo llows (see [11]).
+We shall denote by C∞
+G(Imm(M,Rn)) the space of such smooth functions.
+In what follows we shall further assume that that the weak Riemannian metric
+Gitself admits G-gradients with respect to the variable fin the following sense:
+D(f,m)Gf(h,k) =Gf(m,Hf(h,k)) =Gf(Kf(m,h),k),where
+H,K: Imm(M,Rn)×C∞(M,Rn)×C∞(M,Rn)→C∞(M,Rn)
+(f,h,k)/ma√sto→Hf(h,k),Kf(h,k)
+are smooth and bilinear in h,k.
+Note thatHandKcould be expressed in abstract index notation as gij,kgkland
+gij,kgil. We will check and compute these gradients for several concrete metrics
+below.
+2.3.The fundamental symplectic form on TImm(M,Rn)induced by a
+weak Riemannian metric. The basis of Hamiltonian theory is the natural 1-
+form on the cotangent bundle T∗Imm(M,Rn) given by
+Θ :T(T∗Imm(M,Rn)) = Imm(M,Rn)×D′(M)n×C∞(M,Rn)×D′(M)n→R,
+(f,α;h,β)/ma√sto→ /angb∇acketleftα,h/angb∇acket∇ight.
+The pullback via the mapping G:TImm(M,Rn)→T∗Imm(M,Rn) of Θ is
+(G∗Θ)(f,h)(f,h;k,v) =Gf(h,k).
+Thus the symplectic form ω=−dG∗Θ onTImm(M,Rn) can be computed as
+follows, where we use the constant vector fields ( f,h)/ma√sto→(f,h;k,v):
+ω(f,h)((k1,v1),(k2,v2)) =−d(G∗Θ)((k1,v1),(k2,v2))|(f,h)
+=−D(f,k1)Gf(h,k2)−Gf(v1,k2)+D(f,k2)Gf(h,k1)+Gf(v2,k1)
+=Gf/parenleftbig
+k2,Hf(h,k1)−Kf(k1,h)/parenrightbig
++Gf(v2,k1)−Gf(v1,k2). (1)
+2.4.The Hamiltonian vector field mapping. Here we compute the Hamilton-
+ian vector field gradω(F) associated to a smooth function Fon the tangent space
+TImm(M,Rn); that isF∈C∞
+G(Imm(M,Rn)×C∞(M,Rn)) assuming that it has
+smoothG-gradientsin bothfactors. See [11, section48]. Usingtheexplicit fo rmulas
+in section 2.3, we have
+ω(f,h)(gradω(F)(f,h),(k,v))
+=ω(f,h)((gradω
+1(F)(f,h),gradω
+2(F)(f,h)),(k,v))
+=Gf/parenleftbig
+k,Hf/parenleftbig
+h,gradω
+1(F)(f,h)/parenrightbig/parenrightbig
+−Gf(Kf(gradω
+1(F)(f,h),h),k)
++Gf(v,gradω
+1(F)(f,h))−Gf(gradω
+2(F)(f,h),k).10 Almost local metrics on shape space of hypersurfaces in n-space
+On the other hand, by the definition of the ω-gradient we have
+ω(f,h)(gradω(F)(f,h),(k,v)) =dF(f,h)(k,v) =D(f,k)F(f,h)+D(h,v)F(f,h)
+=Gf(gradG
+1(F)(f,h),k)+Gf(gradG
+2(F)(f,h),v),
+and we get the expression of the Hamiltonian vector field:
+gradω
+1(F)(f,h) = gradG
+2(F)(f,h),
+gradω
+2(F)(f,h) =−gradG
+1(F)(f,h)
++Hf/parenleftbig
+h,gradG
+2(F)(f,h)/parenrightbig
+−Kf(gradG
+2(F)(f,h),h).
+Note that for a smooth function FonTImm(M,Rn) theω-gradient exists if and
+only if both G-gradients exist.
+2.5.The geodesic equation on the manifold of immersions. The geodesic
+flow is defined by a vector field on TImm(M,Rn). One way to define this vector
+field is as the Hamiltonian vector field of the energy function
+E(f,h) =1
+2Gf(h,h), E: Imm(M,Rn)×C∞(M,Rn)→R.
+The two partial G-gradients are
+Gf(gradG
+2(E)(f,h),v) =d2E(f,h)(v) =Gf(h,v),
+gradG
+2(E)(f,h) =h,
+Gf(gradG
+1(E)(f,h),k) =d1E(f,h)(k) =1
+2D(f,k)Gf(h,h)
+=1
+2Gf(k,Hf(h,h)),
+gradG
+1(E)(f,h) =1
+2Hf(h,h).
+Thus the geodesic vector field is
+gradω
+1(E)(f,h) =h
+gradω
+2(E)(f,h) =1
+2Hf(h,h)−Kf(h,h)
+and the geodesic equation becomes
+/braceleftBigg
+ft=h,
+ht=1
+2Hf(h,h)−Kf(h,h),orftt=1
+2Hf(ft,ft)−Kf(ft,ft).
+This is nothing but the usual formula for the geodesic flow using the C hristoffel
+symbols expanded out using the first derivatives of the metric tens or.
+2.6.The momentum mapping for a G-isometric group action. We consider
+now a (possibly infinite dimensional regular) Lie group with Lie algebra gwith a
+right action g/ma√sto→rgby isometries on Imm( M,Rn). Denote by X(Imm(M,Rn)) the
+set of vector fields on Imm( M,Rn). Then we can specify this action by the funda-
+mental vector field mapping ζ:g→X(Imm(M,Rn)), which will be a bounded Lie
+algebra homomorphism. The fundamental vector field ζX,X∈g, is the infinitesi-
+mal action in the sense that
+ζX(f) =∂t|0rexp(tX)(f).M. Bauer, P. Harms, P. Michor 11
+We also consider the tangent prolongation of this action on TImm(M,Rn), where
+the fundamental vector field is given by
+ζTImm
+X: (f,h)/ma√sto→(f,h;ζX(f),D(f,h)(ζX)(f) =:ζ′
+X(f,h)).
+The basic assumption is that the action is by isometries,
+Gf(h,k) = ((rg)∗G)f(h,k) =Grg(f)(Tf(rg)h,Tf(rg)k).
+Differentiating this equation at g=ein the direction X∈g, we get
+(1) 0 = D(f,ζX(f))Gf(h,k)+Gf(ζ′
+X(f,h),k)+Gf(h,ζ′
+X(f,k)).
+The key to the Hamiltonian approach is to define the group action by H amiltonian
+flows. We define the momentum map j:g→C∞
+G(TImm(M,Rn),R) by
+jX(f,h) =Gf(ζX(f),h).
+Equivalently, since this map is linear, it is often written as a map
+J:TImm(M,Rn)→g′,/angb∇acketleftJ(f,h),X/angb∇acket∇ight=jX(f,h).
+Themain propertyofthe momentum mapis that it fits into the following commuta-
+tive diagram and is a homomorphism of Lie algebras:
+H0(TImm)i/d47/d47C∞
+G(TImm,R)gradω/d47/d47X(TImm,ω) /d47/d47H1(TImm)
+gj/d101/d101▲▲▲▲▲▲▲▲▲▲▲ζTImm/d57/d57ttttttttttt
+whereX(TImm,ω) is the space of vector fields on TImm whose flow leaves ωfixed.
+Note also that Jis equivariant for the group action. See [16] for more details.
+By Noether’s theorem, along any geodesic t/ma√sto→f(t, .) this momentum mapping
+is constant; thus for any X∈gwe have
+/angb∇acketleftJ(f,ft),X/angb∇acket∇ight=jX(f,ft) =Gf(ζX(f),ft) is constant in t.
+We can apply this construction to the following group actions:
+•The smooth right action of the group Diff( M) on Imm(M,Rn), given by
+composition from the right: f/ma√sto→f◦ϕforϕ∈Diff(M).
+ForX∈X(M) the fundamental vector field is then given by
+ζDiff
+X(f) =ζX(f) =X(f) =∂t|0(f◦FlX
+t) =df.X,
+where FlX
+tdenotes the flow of X. Thereparametrization momentum , for
+any vector field XonMis thus
+jX(f,h) =Gf(df.X,h).
+Assuming the metric is reparametrization invariant, it follows that on any
+geodesicf(x,t), the expression Gf(df.X,ft) is constant for all X.
+•The left action of the Euclidean motion group Rn⋊SO(n) on Imm(M,Rn)
+given byf/ma√sto→Af+Bfor (B,A)∈Rn×SO(n). The fundamental vector
+field mapping is
+ζ(B,X)(f) =Xf+B.12 Almost local metrics on shape space of hypersurfaces in n-space
+Thelinear-momentum is thusGf(B,h),B∈Rn, and if the metric is trans-
+lation invariant, Gf(B,ft) will be constant along geodesics for every B∈
+Rn. Theangular-momentum is similarly Gf(X.f,h),X∈so(n), and if
+the metric is rotation-invariant, then Gf(X.f,ft) will be constant along
+geodesics for each X∈so(n).
+•The action of the scaling group of Rgiven byc/ma√sto→erf, with fundamental
+vector field ζa(f) =a.f.
+If the metric is scale-invariant, then the scaling momentum Gf(f,ft) will
+also be invariant along geodesics.
+2.7.Shape space. Asdiscussedintheintroduction,byashapewemeanasmoothly
+immersed or embedded hypersurface in Rnwhich is diffeomorphic to a fixed com-
+pact, connected, and oriented manifold Mof dimension n−1. The space of these
+shapes will be denoted Bi(M,Rn) orBe(M,Rn) and viewed as the quotient
+Be(M,Rn) = Emb(M,Rn)/Diff(M) orBi(M,Rn) = Imm(M,Rn)/Diff(M).
+In [11, section 44.1] it is shown that Be(M,Rn) is a manifold again. Bi(M,Rn)
+is, however, no longer a manifold but an orbifold with finite isotropy gr oups; see
+[5]. We will sometimes use the abbreviations BiandBewhen it is clear what the
+domain and co-domain of the functions are.
+More generally, a shape will be an element of the Cauchy completion (i.e ., the
+metriccompletion forthe geodesicdistance)of Bi(M,Rn) with respect toa suitably
+chosen Riemannian metric. This will allow for corners.
+2.8.Riemannian submersions and the metric on shape space. We will al-
+ways assume that a Diff( M)-invariant Riemannian metric on Imm( M,Rn) is given.
+Then there is a unique Riemannian metric on the quotient space Bi(M,Rn) such
+that the quotient map π: Imm(M,Rn)→Bi(M,Rn) is aRiemannian submersion .
+This is the construction that we use to induce a metric on shape spac e.
+Let ker(Tπ)⊂TImm(M,Rn) be the vertical bundle . Thehorizontal bundle is
+its orthogonal complement with respect to the metric G. ThenTπ(f)Bi(M,Rn) is
+isometric to the horizontal bundle at f∈Imm(M,Rn). Note that the horizontal
+bundle depends on the definition of the metric. For almost local metr ics, it consists
+of vector fields along fthat are everywhere normal to f; see section 6.1 .
+By the conservation of the reparametrization momentum, geodes ics in the space
+of immersions with horizontal initial velocity stay horizontal for all t ime. Such
+geodesics project down to geodesics in shape space because πis a Riemannian
+submersion. See [13, section 26] for a proof of this fact. We will sho w in section 6.1
+that almost local metrics have the property that any curve in shap e space can be
+lifted to a horizontal curve of immersions. This implies that instead of solving
+the geodesic equation on shape space one can equivalently solve the equation for
+horizontal geodesics in the space of immersions.
+3.Differential geometry of surfaces and notationM. Bauer, P. Harms, P. Michor 13
+In this section we will present and develop the differential geometric tools that
+are needed to deal with immersed surfaces. The most important po int is a rigorous
+treatment of the covariant derivative and related concepts.
+In [2, section 2] one can find some parts of this section in a more gene ral setting.
+We use the notation of [13]. Some of the definitions can also be found in [9].
+3.1.Tensor bundles and tensor fields. We will deal with the tensor bundles
+Tr
+sM
+/d15/d15Tr
+sM⊗f∗TRn
+/d15/d15
+M M
+HereTr
+sMdenotes the bundle of (r
+s)-tensors on M, i.e.,
+Tr
+sM=r/circlemultiplydisplay
+TM⊗s/circlemultiplydisplay
+T∗M,
+andf∗TRnis the pullback of the bundle TRnviaf; see [13, section 17.5]. A tensor
+fieldis a section of a tensor bundle. Generally, when Eis a bundle, the space of its
+sections will be denoted by Γ( E).
+To clarify the notation that will be used later, some examples of tens or bundles
+and tensor fields are given now.
+•End(TM) =L(TM,TM ) =T1
+1Mis the bundle of endomorphisms of TM.
+•SkT∗M=Lk
+sym(TM;R) is the bundle of symmetric (0
+k)-tensors.
+•S2
+>0T∗Mis the bundle of symmetric positive definite (0
+2)-tensors.
+•ΛkT∗M=Lk
+alt(TM;R) is the bundle of alternating (0
+k)-tensors.
+•Ωr(M) = Γ(ΛrT∗M) is the space of differential forms .
+•X(M) = Γ(TM) is the space of vector fields .
+•Γ(f∗TRn)∼=/braceleftbig
+h∈C∞(M,TRn) :πN◦h=f/bracerightbig
+is the space of vector fields
+alongf.
+ForX∈X(M) the insertion ιXwill always insert Xinto the leftmost covariant
+entry of a tensor.
+3.2.Metric on tensor spaces. Let ¯g∈Γ(S2
+>0T∗Rn) denote the Euclidean metric
+onRn. Themetric induced on Mbyf∈Imm(M,Rn) is the pullback metric
+g=f∗¯g∈Γ(S2
+>0T∗M), g(X,Y) = (f∗¯g)(X,Y) = ¯g(Tf.X,Tf.Y ),
+whereX,Yare vector fields on M. The dependence of gon the immersion f
+should be kept in mind. Let ( u,U) be a fixed chart on Mwith∂i=∂
+∂ui. In these
+coordinates the pullback metric is given by
+g|U=n−1/summationdisplay
+i,jgijdui⊗duj=n−1/summationdisplay
+i,j¯g/parenleftbig
+∂if,∂jf/parenrightbig
+dui⊗duj.
+The metric can be seen as a mapping
+g:TM→T∗M, X /ma√sto→g(X) =:X♭14 Almost local metrics on shape space of hypersurfaces in n-space
+with inverse
+g−1:T∗M→TM, α /ma√sto→g−1(α) =:α♯.
+This defines a metric on the cotangent bundle T0
+1M=T∗Mvia
+g0
+1(α,β) =g−1(α,β) =α(β♯) =g(α♯,β♯)
+forα,β∈T∗M. The product metric
+gr
+s=r/circlemultiplydisplay
+g⊗s/circlemultiplydisplay
+g−1
+extendsgto all tensor spaces Tr
+sM, andgr
+s⊗¯gyields a metric on Tr
+sM⊗f∗TRn.
+3.3.Traces. Thetracecontractspairsofvectorsandcovectorsin a tensorproduct:
+Tr :T∗M⊗TM=L(TM,TM )→M×R
+A special case of this is the operator ιXinserting a vector Xinto a covector or into
+a covariant factor of a tensor product. The inverse of the metric gcan be used to
+define a trace
+Trg:T∗M⊗T∗M→M×R
+contracting pairs of covectors. Note that Trgdepends on the metric whereas Tr
+does not. The following lemma will be useful in many calculations (see [2 , section
+2]).
+Lemma.g0
+2(B,C) = Tr(g−1Bg−1C)forB,C∈T0
+2MifBorCis symmetric.
+In the expression under the trace, BandCare seen as maps TM→T∗M.
+3.4.Volume density. Let Vol(M) be the density bundle overM; see [13, sec-
+tion 10.2]. The volume density onMinduced by f∈Imm(M,Rn) is
+vol(g) = vol(f∗¯g)∈Γ/parenleftbig
+Vol(M)/parenrightbig
+.
+In a chart ( u,U) the volume density reads as
+vol(g) =/radicalBig
+det(¯g(∂if,∂jf))|du1∧···∧dun−1|.
+Thevolumeof the immersion is given by
+Vol(f) =/integraldisplay
+Mvol(f∗¯g) =/integraldisplay
+Mvol(g).
+Theintegraliswelldefinedsince Miscompact. Since Misoriented, wemayidentify
+the volume density with a differential form.
+3.5.Covariant derivative. We will use covariant derivatives on vector bundles
+as explained in [13, sections 19.12, 22.9]. Let ∇g,∇¯gbe theLevi–Civita covariant
+derivatives on (M,g) and (Rn,¯g), respectively. For any manifold Qand vector field
+XonQone has
+∇g
+X:C∞(Q,TM)→C∞(Q,TM), h /ma√sto→ ∇g
+Xh,
+∇¯g
+X:C∞(Q,TRn)→C∞(Q,TRn), h /ma√sto→ ∇¯g
+Xh.
+Usually we will simply write ∇for all covariant derivatives. It should be kept in
+mind that ∇gdepends on the metric g=f∗¯gand therefore also on the immersion
+f. TheRncovariant derivative ∇¯g
+Xhequals the ordinary differential dh(X) butM. Bauer, P. Harms, P. Michor 15
+remembers the footpoint fofh, i.e. ,∇¯g
+X(f,h) = (f,dh(X)) if we write ( f,h)
+instead ofh. The following properties hold [13, section 22.9]:
+(1)∇Xrespects base points, i.e., π◦∇Xh=π◦h, whereπis the projection
+of the tangent space onto the base manifold.
+(2)∇XhisC∞-linear inX. So for a tangent vector Xx∈TxQ,∇Xxhmakes
+sense and equals ( ∇Xh)(x).
+(3)∇XhisR-linear inh.
+(4)∇X(a.h) =da(X).h+a.∇Xhfora∈C∞(Q), the derivation property of
+∇X.
+(5) For any manifold /tildewideQand smooth mapping q:/tildewideQ→QandYy∈Ty/tildewideQone
+has∇Tq.Yyh=∇Yy(h◦q). IfY∈X(Q1) andX∈X(Q) areq-related, then
+∇Y(h◦q) = (∇Xh)◦q.
+The two covariant derivatives ∇g
+Xand∇¯g
+Xcan be combined to yield a covariant
+derivative ∇XactingonC∞(Q,Tr
+sM⊗TRn) byadditionallyrequiringthe following
+properties [13, section 22.12]:
+(6)∇Xdoes not change the grade of tensors, i.e., it induces mappings
+∇X:C∞(Q,Tr
+sM⊗TRn)→C∞(Q,Tr
+sM⊗TRn).
+(7)∇X(h⊗k) = (∇Xh)⊗k+h⊗(∇Xk), a derivation with respect to the
+tensor product.
+(8)∇Xcommutes with any kind of contraction (see [13, section 8.18]). A
+special case of this is
+∇X/parenleftbig
+α(Y)/parenrightbig
+= (∇Xα)(Y)+α(∇XY) forα⊗Y:Q→T1
+1M.
+Property 1 is important because it implies that ∇Xrespects spaces of sections of
+bundles. For example, for Q=Mandf∈C∞(M,Rn), one gets
+∇X: Γ(Tr
+sM⊗f∗TRn)→Γ(Tr
+sM⊗f∗TRn).
+3.6.Swapping covariant derivatives. We will make repeated use of some for-
+mulas, allowing us to swap covariant derivatives. Let fbe an immersion, ha vector
+field along f, andX,Yvector fields on M. Since∇is torsion-free, one has [13,
+section 22.10]
+(1) ∇XTf.Y−∇YTf.X−Tf.[X,Y] = 0.
+Furthermore, one has [13, section 24.5]
+(2) ∇X∇Yh−∇Y∇Xh−∇[X,Y]h= 0,
+sinceRnis flat. These formulas also hold when f:R×M→Rnis a path of
+immersions, h:R×M→TRnis a vector field along f, and the vector fields are
+vector fields on R×M. A case of special importance is when one of the vector
+fields is (∂t,0M) and the other (0 R,Y), whereYis a vector field on M. Since the
+Lie bracket of these vector fields vanishes, (1) and (2) yield
+(3) ∇(∂t,0M)Tf.(0R,Y)−∇(0R,Y)Tf.(∂t,0M) = 0
+and
+(4) ∇(∂t,0M)∇(0R,Y)h−∇(0R,Y)∇(∂t,0M)h= 0.16 Almost local metrics on shape space of hypersurfaces in n-space
+If the context is clear, we shall write ∂tinstead of the more detailed notation
+(∂t,0M) andYinstead of (0 R,Y).
+3.7.Higher covariant derivatives and the Laplace operator. When the co-
+variant derivative is seen as a mapping
+∇: Γ(Tr
+sM)→Γ(Tr
+s+1M) or∇: Γ(Tr
+sM⊗f∗TRn)→Γ(Tr
+s+1M⊗f∗TRn),
+then the second covariant derivative is simply ∇∇=∇2. Since the covariant
+derivative commutes with contractions, ∇2can be expressed as
+∇2
+X,Y:=ιYιX∇2=ιY∇X∇=∇X∇Y−∇∇XYforX,Y∈X(M).
+Higher covariant derivates are defined as ∇k,k≥0. We can use the second co-
+variant derivative to define the Laplace–Bochner operator . It can act on all tensor
+fieldsBand is defined as
+∆B=−Trg(∇2B).
+Forh= (h1,...,hn) :M→Rnone has ∆h= (∆h1,...,∆hn).
+3.8.Normal bundle. Thenormal bundle Nor(f)ofanimmersion fisasubbundle
+off∗TRnwhose fibers consist of all vectors that are orthogonal to the ima ge off,
+i.e.,
+Nor(f)x=/braceleftbig
+Y∈Tf(x)Rn:∀X∈TxM: ¯g(Y,Tf.X) = 0/bracerightbig
+.
+Any vector field halongfcan be decomposed uniquely into parts tangential and
+normaltofas
+h=Tf.h⊤+h⊥,
+whereh⊤is a vector field on Mandh⊥is a section of the normal bundle Nor( f).
+Whenfis orientable, then the unit normal field νoffcan be defined. It is a
+section of the normal bundle with constant ¯ g-length one which is chosen such that
+/parenleftbig
+ν(x),Txf.X1,Txf.X2,...,Txf.Xn−1/parenrightbig
+is a positive oriented basis in Tf(x)RnifX1,...,Xn−1is a positive oriented basis
+inTxM. In this notation the decomposition of a vector field halongfreads as
+h=Tf.h⊤+a.ν.
+The two parts are defined by the relations
+a= ¯g(h,ν)∈C∞(M),
+h⊤∈X(M),such thatg(h⊤,X) = ¯g(h,Tf.X) for allX∈X(M).
+3.9.Second fundamental form and Weingarten mapping. LetXandYbe
+vector fields on M. Then the covariant derivative ∇XTf.Ysplits into tangential
+and normal parts as
+∇XTf.Y=Tf.(∇XTf.Y)⊤+(∇XTf.Y)⊥=Tf.∇XY+S(X,Y).
+S=Sfis thesecond fundamental form of f. It is a symmetric bilinear form with
+values in the normal bundle of f. WhenTfis seen as a section of T∗M⊗f∗TRn,
+one hasS=∇Tfsince
+S(X,Y) =∇XTf.Y−Tf.∇XY= (∇Tf)(X,Y).M. Bauer, P. Harms, P. Michor 17
+Taking the trace of Syields the vector valued mean curvature
+Trg(S)∈Γ/parenleftbig
+Nor(f)/parenrightbig
+.
+One can define the scalar second fundamental form s=sfas
+s(X,Y) = ¯g/parenleftbig
+S(X,Y),ν/parenrightbig
+.
+Moreover, there is the Weingarten mapping orshape operator L=Lf=g−1s. It
+is ag-symmetric bundle mapping defined by
+s(X,Y) =g(LX,Y).
+The eigenvalues of Lare called principal curvatures and the eigenvectors principal
+curvature directions . Tr(L) = Trg(s) is thescalar mean curvature and for surfaces
+inR3theGauß curvature is given by det( L). The covariant derivative ∇Xνof the
+normal vector is related to Lby theWeingarten equation
+∇Xν=−Tf.L.X.
+In a chart ( u,U) the second fundamental form is given by
+sij=s(∂i,∂j) = ¯g(∇∂iTf.∂j,ν) = ¯g/parenleftBig∂2f
+∂i∂j,ν/parenrightBig
+,
+and the mean curvature by Tr( L) =/summationtext
+i,jgijsij.
+3.10.Directional derivatives of functions. We will use the following ways to
+denote directional derivatives of functions, in particular in infinite d imensions.
+Given a function F(x,y), for instance, we will write
+D(x,h)Fas shorthand for ∂t|0F(x+th,y).
+Here (x,h) in the subscript denotes the tangent vector with footpoint xand direc-
+tionh. IfFtakes values in some linear space, we will identify this linear space and
+its tangent space.
+4.Variational formulas
+Recall that many operators such as
+g=f∗¯g, S=Sf, L=Lf,vol(g),∇=∇g,∆ = ∆g
+depend on the immersion f. We want to calculate their derivative with respect to
+f, which we call the first variation. We will use these formulas to calculate the
+metric gradients that are needed for the geodesic equation.
+Some of the formulas can also be found in [2, 4, 14, 21, 8].18 Almost local metrics on shape space of hypersurfaces in n-space
+4.1.Paths of immersions. All of the concepts introduced in section 3 can be
+recastforapath ofimmersionsinsteadofa fixedimmersion. Thisallow sustostudy
+variations of immersions. So let f:R→Imm(M,Rn) be a path of immersions. By
+convenient calculus [11], fcan equivalently be seen as f:R×M→Rnsuch that
+f(t,·) is an immersion for each t. We can replace bundles over Mby bundles over
+R×M:
+pr∗
+2Tr
+sM
+/d15/d15pr∗
+2Tr
+sM⊗f∗TRn
+/d15/d15Nor(f)
+/d15/d15
+R×M R×M R×M
+Here pr2denotes the projection pr2:R×M→M. The covariant derivative ∇Zhis
+now defined for vector fields ZonR×Mand sections hof the above bundles. The
+vector fields ( ∂t,0M) and (0 R,X), whereXis a vector field on M, are of special
+importance. Let
+inst:M→R×M, x /ma√sto→(t,x).
+Then by [13, 22.9.6] one has for vector fields X,YonM
+∇XTf(t,·).Y=∇XT(f◦inst)◦Y=∇XTf◦Tinst◦Y
+=∇XTf◦(0R,Y)◦inst=∇Tinst◦XTf◦(0R,Y)
+=/parenleftbig
+∇(0R,X)Tf◦(0R,Y)/parenrightbig
+◦inst.
+This shows that one can recover the static situation at tby using vector fields on
+R×Mwith vanishing R-component and evaluating at t.
+4.2.Setting for first variations. In the remainder of this section, let fbe an
+immersion and ft∈TfImm a tangent vector to f. The reason for calling the
+tangent vector ftis that in calculations it will often be the derivative of a curve of
+immersions through f. Using the same symbol ffor the fixed immersion and for
+the path of immersions through it, one has in fact that
+D(f,ft)F=∂tF(f(t)).
+For the sake of brevity we will write ∂tinstead of (∂t,0M) andXinstead of (0 R,X),
+whereXis a vector field on M.
+Let the smooth mapping F: Imm(M,N)→Γ(Tr
+sM) take values in some space
+of tensor fields over M, or more generally in any natural bundle over M; see [10].
+4.3.Lemma (tangential variation of equivariant tensor fields). IfFis equi-
+variant with respect to pullbacks by diffeomorphisms of M, i.e.,
+F(f) = (ϕ∗F)(f) =ϕ∗/parenleftBig
+F/parenleftbig
+(ϕ−1)∗f/parenrightbig/parenrightBig
+for allϕ∈Diff(M)andf∈Imm(M,N), then the tangential variation of Fis its
+Lie derivative:
+D(f,Tf.f⊤
+t)F=∂t|0F/parenleftBig
+f◦Flf⊤
+t
+t/parenrightBig
+=∂t|0F/parenleftBig
+(Flf⊤
+t
+t)∗f/parenrightBig
+=∂t|0/parenleftBig
+Flf⊤
+t
+t/parenrightBig∗/parenleftbig
+F(f)/parenrightbig
+=Lf⊤
+t/parenleftbig
+F(f)/parenrightbig
+.M. Bauer, P. Harms, P. Michor 19
+Here Flf⊤
+t
+tdenotes the flow of f⊤
+tandLf⊤
+tdenotes the Lie derivative along the
+vector field f⊤
+tonM. This allows us to calculate the tangential variation of the
+pullback metric and the volume density, because these tensor fields are natural with
+respect to pullbacks by diffeomorphisms.
+4.4.Lemma (variation of the metric). The differential of the pullback metric
+/braceleftbigg
+Imm→Γ(S2
+>0T∗M),
+f/ma√sto→g=f∗¯g
+is given by
+D(f,ft)g=−2¯g(ft,ν).s+Lf⊤
+t(g).
+Proof. Letf:R×M→Rnbe a path of immersions. Swapping covariant
+derivatives as in section 3.6, formula (3), one gets
+∂t/parenleftbig
+g(X,Y)/parenrightbig
+=∂t/parenleftbig
+¯g(Tf.X,Tf.Y )/parenrightbig
+= ¯g(∇∂tTf.X,Tf.Y )+ ¯g(Tf.X,∇∂tTf.Y)
+= ¯g(∇Xft,Tf.Y)+ ¯g(Tf.X,∇Yft) =/parenleftbig
+2Sym¯g(∇ft,Tf)/parenrightbig
+(X,Y).
+Splittingftinto its normal and tangential parts yields
+2Sym¯g(∇ft,Tf) = 2Sym¯g(∇f⊥
+t+∇Tf.f⊤
+t,Tf)
+=−2Sym¯g(f⊥
+t,∇Tf)+2Symg(∇f⊤
+t,·)
+=−2¯g(f⊥
+t,S)+2Sym ∇(f⊤
+t)♭.
+Finally, the relation
+D(f,Tf.f⊤
+t)g= 2Sym ∇(f⊤
+t)♭=Lf⊤
+tg
+follows from the equivariance of g(see section 4.3).
+4.5.Lemma (variation of the inverse of the metric). The differential of the
+inverse of the pullback metric
+/braceleftbiggImm→Γ(L(T∗M,TM)),
+f/ma√sto→g−1= (f∗¯g)−1
+is given by
+D(f,ft)g−1=−2¯g(ft,ν).L.g−1+Lf⊤
+t(g−1).
+Proof.
+∂tg−1=−g−1(∂tg)g−1=−g−1/parenleftbig
+−2¯g(f⊥
+t,S)+Lf⊤
+tg/parenrightbig
+g−1
+= 2g−1¯g(f⊥
+t,S)g−1−g−1(Lf⊤
+tg)g−1= 2¯g(f⊥
+t,g−1Sg−1)+Lf⊤
+t(g−1).20 Almost local metrics on shape space of hypersurfaces in n-space
+4.6.Lemma (variation of the volume density). The differential of the volume
+density/braceleftbigg
+Imm→Γ(Vol(M)),
+f/ma√sto→vol(g) = vol(f∗¯g)
+is given by
+D(f,ft)vol(g) =/parenleftBig
+divg(f⊤
+t)−¯g(f⊥
+t,ν).Tr(L)/parenrightBig
+vol(g).
+Proof. Letg(t)∈Γ(S2
+>0T∗M) be any curve of Riemannian metrics. Then
+∂tvol(g) =1
+2Tr(g−1.∂tg)vol(g).
+This follows from the formula for vol( g) in a local oriented chart ( u1,...,un) on
+M:
+∂tvol(g) =∂t/radicalBig
+det((gij)ij)du1∧···∧dun−1
+=1
+2/radicalbig
+det((gij)ij)Tr(adj(g)∂tg)du1∧···∧dun−1
+=1
+2/radicalbig
+det((gij)ij)Tr(det((gij)ij)g−1∂tg)du1∧···∧dun−1
+=1
+2Tr(g−1.∂tg)vol(g).
+Now we can set g=f∗¯gand plug in the formula for ∂tg=∂t(f∗¯g). This yields
+∂tvol(g) =1
+2Tr(g−1(−2¯g(ft,ν).s+Lh⊤g)).vol(g)
+=−¯g(ft,ν)Tr(g−1.s).vol(g)+1
+2Tr(g−1Lh⊤g).vol(g).
+The same calculation as above with ∂treplaced by Lh⊤shows that
+Lh⊤vol(g) =1
+2Tr(g−1Lh⊤g).vol(g).
+Therefore,
+∂tvol(g) =−¯g(ft,ν)Tr(L).vol(g)+Lh⊤(vol(g))
+=−¯g(ft,ν)Tr(L).vol(g)+divg(h⊤)vol(g).
+4.7.Lemma (variation of the volume). The differential of the total volume
+/braceleftbigg
+Imm→R,
+f/ma√sto→Vol(f) =/integraltext
+Mvol(f∗¯g)
+is given by
+D(f,ft)vol(g) =−/integraldisplay
+M¯g(f⊥
+t,ν).Tr(L)vol(g).M. Bauer, P. Harms, P. Michor 21
+4.8.Lemma (variation of the second fundamental form). The differential
+of the second fundamental form
+/braceleftbiggImm→Γ(S2T∗M),
+f/ma√sto→sf
+is given by
+D(f,ft)s= ¯g(∇2ft,ν) =∇2¯g(ft,ν)−¯g(ft,ν).g(L,L)+Lf⊤
+t.s.
+Proof. By definition s(X,Y) = ¯g/parenleftbig
+S(X,Y),ν/parenrightbig
+= ¯g/parenleftbig
+∇X(Tf.Y)−Tf.∇XY,ν/parenrightbig
+.
+Interchanging covariant derivatives as in section 3.6, formulas (3) and (4), yields
+∂ts(X,Y) = ¯g/parenleftbig
+∂tS(X,Y),ν/parenrightbig
++ ¯g/parenleftbig
+S(X,Y),∂tν/parenrightbig
+= ¯g/parenleftbig
+∇X∇YTf.∂t−∇∇XYTf.∂t,ν/parenrightbig
++0 = ¯g/parenleftbig
+∇2
+X,Yft,ν/parenrightbig
+,
+where the term ¯ g/parenleftbig
+S(X,Y),∂tν/parenrightbig
+vanishes since ∂tνis tangential (see section 4.11).
+Forthenormalpartthisyieldsthefollowing: Togetthesecondform ulawecalculate
+(D(f,¯g(ft,ν).ν)s)(X,Y) = ¯g/parenleftBig
+∇2
+X,Y/parenleftbig
+¯g(ft,ν).ν/parenrightbig
+,ν/parenrightBig
+=∇2
+X,Y¯g(ft,ν)+0+ ¯g(ft,ν).¯g/parenleftbig
+∇2
+X,Yν,ν/parenrightbig
+=∇2
+X,Y¯g(ft,ν)−¯g(ft,ν).¯g/parenleftbig
+∇Xν,∇Yν/parenrightbig
++0
+=∇2
+X,Y¯g(ft,ν)−¯g(ft,ν).g/parenleftbig
+LX,LY/parenrightbig
+Bysection4.3, the formulaforthetangentialvariationfollowsfrom the equivariance
+of the second fundamental form:
+sf◦φ(X,Y) = ¯g/parenleftbig
+∇XT(f◦φ)◦Y,νf◦φ/parenrightbig
+= ¯g/parenleftbig
+∇X(Tf◦(φ∗Y)◦φ),νf◦φ)
+= ¯g/parenleftbig
+∇Tφ◦XTf◦(φ∗Y),νf◦φ)◦φ
+= ¯g/parenleftbig
+∇φ∗XTf◦(φ∗Y),νf)◦φ=sf(φ∗X,φ∗Y)◦φ= (φ∗sf)(X,Y).
+4.9.Lemma (variation of the Weingarten map). The differential of the Wein-
+garten map/braceleftbiggImm→Γ(End(TM)),
+f/ma√sto→Lf
+is given by
+D(f,ft)L=g−1.∇2/parenleftbig
+¯g(ft,ν)/parenrightbig
++ ¯g(ft,ν)L2+Lf⊤
+t(L).
+Proof. FromL=g−1.sfollows
+∂tL=g−1∂ts+∂t(g−1)s
+=g−1/parenleftBig
+∇2(¯g(ft,ν))−¯g(ft,ν)gL2+Lf⊤
+t(s)/parenrightBig
++/parenleftBig
+2¯g(ft,ν)Lg−1+Lf⊤
+t(g−1)/parenrightBig
+.s
+=g−1∇2/parenleftbig
+¯g(ft,ν)/parenrightbig
++ ¯g(ft,ν)L2+Lf⊤
+t(L).22 Almost local metrics on shape space of hypersurfaces in n-space
+4.10.Lemma (variation of the mean curvature). The differential of the mean
+curvature /braceleftbiggImm→C∞(M),
+f/ma√sto→Tr(Lf)
+is given by
+D(f,ft)Tr(L) =−∆/parenleftbig
+¯g(ft,ν)/parenrightbig
++ ¯g(ft,ν).Tr/parenleftbig
+L2/parenrightbig
++d/parenleftbig
+Tr(L)/parenrightbig
+(f⊤
+t).
+Proof. This statement follows from the linearity of the trace operator an d from
+the previous equation for D(f,ft)L.
+4.11.Lemma (variation of the normal vector field). Whenfis a curve of
+immersions, the normal vector field to fis a smooth map ν:R×M→Rn.
+Therefore, as explained in section 3.5, we can take its covar iant derivative in the
+direction of vector fields on R×M. Identifying ∂twith the vector field (∂t,0M)on
+R×M, we get
+∇∂tν=−Tf./parenleftBig
+Lf⊤
+t+gradg/parenleftbig
+¯g(ft,ν)/parenrightbig/parenrightBig
+.
+Proof.∇∂tνis tangential because ¯ g(∇∂tν,ν) =1
+2∂t¯g(ν,ν) = 0. Therefore, one
+can write ∇∂tν=Tf.(∇∂tν)⊤. Then for all X∈X(M) we have
+g((∇∂tν)⊤,X) = ¯g/parenleftbig
+∇∂tν,Tf.X/parenrightbig
+= 0−¯g/parenleftbig
+ν,∇∂tTf.X/parenrightbig
+=−¯g(ν,∇XTf.∂t),
+where in the last step we swapped Xand∂tas in section 3.6, formula (3). Splitting
+into normal and tangential parts yields
+g((∇∂tν)⊤,X) =−¯g(ν,∇Xft) =−¯g/parenleftBig
+ν,∇X(Tf.f⊤
+t+ ¯g(ft,ν).ν)/parenrightBig
+=−¯g/parenleftBig
+ν,∇X(Tf.f⊤
+t+ ¯g(ft,ν).ν)/parenrightBig
+=−s(X,f⊤
+t)−∇X/parenleftbig
+¯g(ft,ν)/parenrightbig
+−0
+=−g/parenleftBig
+Lf⊤
+t+gradg¯g(ft,ν),X/parenrightBig
+.
+4.12.Lemma (variation of the covariant derivative). Let∇=∇g=∇f∗¯g
+be the Levi Civita covariant derivative acting on vector fiel ds onM. Since any two
+covariant derivatives on Mdiffer by a tensor field, the first variation of ∇f∗¯gis
+tensorial. It is given by the tensor field D(f,ft)∇f∗¯g∈Γ(T1
+2M), which is determined
+by the following relation holding for vector fields X,Y,ZonM:
+g/parenleftbig
+(D(f,ft)∇)(X,Y),Z/parenrightbig
+=1
+2(∇D(f,ft)g)/parenleftbig
+X⊗Y⊗Z+Y⊗X⊗Z−Z⊗X⊗Y/parenrightbig
+.
+Proof. The defining formula for the covariant derivative is
+g(∇XY,Z) =1
+2/bracketleftBig
+Xg(Y,Z)+Yg(Z,X)−Zg(X,Y)
+−g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y])/bracketrightBig
+.M. Bauer, P. Harms, P. Michor 23
+Taking the derivative D(f,ft)yields
+(D(f,ft)g)(∇XY,Z)+g/parenleftbig
+(D(f,ft)∇)(X,Y),Z/parenrightbig
+=1
+2/bracketleftBig
+X/parenleftbig
+(D(f,ft)g)(Y,Z)/parenrightbig
++Y/parenleftbig
+(D(f,ft)g)(Z,X)/parenrightbig
+−Z/parenleftbig
+(D(f,ft)g)(X,Y)/parenrightbig
+−(D(f,ft)g)(X,[Y,Z])+(D(f,ft)g)(Y,[Z,X])+(D(f,ft)g)(Z,[X,Y])/bracketrightBig
+.
+Then the result follows by replacing all Lie brackets in the above form ula by co-
+variant derivatives using [ X,Y] =∇XY−∇YXand by expanding all terms of the
+formX/parenleftbig
+(D(f,ft)g)(Y,Z)/parenrightbig
+using
+X/parenleftbig
+(D(f,ft)g)(Y,Z)/parenrightbig
+= (∇XD(f,ft)g)(Y,Z)+(D(f,ft)g)(∇XY,Z)+(D(f,ft)g)(Y,∇XZ).
+4.13.Setting for second variations. All formulas for second derivatives will be
+used in section 7.2. There we consider a curve of immersions
+f(t,x) =f0(x)+t.a(x).νf0(x)
+for a fixed immersion f0. This curve of immersions has the property that at t=
+0 its first derivative and the covariant derivative of the first deriva tive are both
+horizontal, i.e.,
+(1) f|t=0=f0, ∂t|0f=a.νf0,and∇∂tTf.∂t|t=0= 0.
+In all calculations of second variations we will assume that the above properties
+hold.
+4.14.Lemma (second variation of the metric). The second derivative of the
+pullback metric/braceleftbiggImm→Γ(S2
+>0T∗M),
+f/ma√sto→g=f∗¯g
+along a curve of immersions fsatisfying property (1)from section 4.13 is given by
+∂2
+t|0f∗¯g= 2(da⊗da)+2a2g0(Lf0,Lf0).
+Proof. Since∇∂tTf.∂t|0= 0, we have
+∂2
+t|0g(X,Y) =∂2
+t|0¯g(Tf.X,Tf.Y )
+=∂t|0¯g(∇∂tTf.X,Tf.Y )+∂t|0¯g(Tf.X,∇∂tTf.Y)
+= 2¯g(∇∂tTf.X|0,∇∂tTf.Y|0)+0+0 = 2¯ g(∇XTf.∂t,∇YTf.∂t).
+UsingTf.∂t=a.νf0, we get
+∂2
+t|0(g(X,Y)) = 2da(X).da(Y)+2a2¯g(∇Xνf0,∇Yνf0)
+= 2(da⊗da)(X,Y)+2a2.g0(Lf0X,Lf0Y).24 Almost local metrics on shape space of hypersurfaces in n-space
+4.15.Lemma (second variation of the inverse metric). The second derivative
+of the inverse of the pullback metric
+/braceleftbigg
+Imm→Γ(L(T∗M,TM)),
+f/ma√sto→g−1= (f∗¯g)−1
+along a curve of immersions fsatisfying property (1)from section 4.13 is given by
+∂2
+t|0(f∗¯g)−1= 6a2(Lf0)2.g−1
+0−2g−1
+0(da⊗da)g−1
+0.
+Proof. We look at g=f∗¯gas a bundle map from TMtoT∗M. Then
+∂2
+t|0(g−1) =∂t|0/parenleftbig
+−g−1.∂tg.g−1/parenrightbig
+= 2g−1
+0.∂t|0g.g−1
+0.∂t|0g.g−1
+0−g−1
+0.∂2
+t|0g.g−1
+0
+= 2(−2aLf0)2.g−1
+0−g−1
+0./parenleftbig
+2(da⊗da)+2a2g0◦(Lf0⊗Lf0)/parenrightbig
+.g−1
+0
+= 8a2(Lf0)2.g−1
+0−2g−1
+0(da⊗da)g−1
+0−2a2(Lf0)2.g−1
+0.
+4.16.Lemma (second variation of the volume form). The second derivative
+of the volume form/braceleftbigg
+Imm→Γ(Vol(M)),
+f/ma√sto→vol(g) = vol(f∗¯g)
+along a curve of immersions fsatisfying property (1)from section 4.13 is given by
+∂2
+t|0vol(g) =/bracketleftBig
+a2Tr(Lf0)2−a2Tr/parenleftbig
+(Lf0)2/parenrightbig
++/ba∇dblda/ba∇dbl2
+g−1
+0/bracketrightBig
+vol(g0).
+Proof. In section 4.6 we showed that for any curve g(t)∈Γ(S2
+>0T∗M) of Rie-
+mannian metrics, we have
+∂tvol(g) =1
+2Tr/parenleftbig
+g−1.∂tg/parenrightbig
+vol(g).
+Therefore,
+∂2
+tvol(g) =∂t1
+2Tr/parenleftbig
+g−1.∂tg/parenrightbig
+vol(g) =1
+2Tr/parenleftbig
+∂t(g−1).∂tg/parenrightbig
+vol(g)
++1
+2Tr/parenleftbig
+g−1.∂2
+tg/parenrightbig
+vol(g)+1
+2Tr/parenleftbig
+g−1.∂tg/parenrightbig
+∂tvol(g).
+Evaluating at t= 0 and setting g(t) =f∗¯g, we get
+∂2
+t|0vol(g) =1
+2Tr/parenleftbig
+(2aLf0g−1
+0).(−2a.sf0)/parenrightbig
+vol(g0)
++1
+2Tr/parenleftbig
+g−1
+0.2(da⊗da)/parenrightbig
+vol(g0)
++1
+2Tr/parenleftbig
+g−1
+0.2a2g0.(Lf0)2/parenrightbig
+vol(g0)
++1
+2Tr/parenleftbig
+g−1
+0.(−2a.sf0)/parenrightbig
+(−Tr(Lf0).a)vol(g0)
+=/bracketleftBig
+a2Tr(Lf0)2−a2Tr/parenleftbig
+(Lf0)2/parenrightbig
++/ba∇dblda/ba∇dbl2
+g−1/bracketrightBig
+vol(g0).M. Bauer, P. Harms, P. Michor 25
+4.17.Lemma (Second variation of the second fundamental form). The
+second derivative of the second fundamental form/braceleftbiggImm→Γ(S2T∗M),
+f/ma√sto→sf
+along a curve of immersions fsatisfying property 1 from section 4.13 is given by
+∂2
+t|0s= 2(da⊗da)/parenleftbig
+Id⊗Lf0+Lf0⊗Id/parenrightbig
+−/ba∇dblda/ba∇dbl2
+g−1
+0.sf0
++2.a(∇gradg0(a)sf0).
+Proof. From section 4.8 we have
+∂ts(X,Y) = ¯g(∇2
+X,YTf.∂t,ν) = ¯g(∇2
+X,Yft,ν).
+Using∇∂tft= 0, we get
+∂2
+ts(X,Y) = ¯g(∇2
+X,Yft,∇∂tν)+¯g(∇∂t∇X∇Yft−∇∂t∇∇XYft,ν)
+= ¯g(∇2
+X,Yft,∇∂tν)+0−¯g(∇∇XY∇∂tft+∇[∂t,∇XY]ft,ν)
+= ¯g(∇2
+X,Yft,∇∂tν)+0−¯g(∇[∂t,∇XY]ft,ν)
+= ¯g(∇2
+X,Yft,∇∂tν)−¯g(∇(D(f,ft)∇)(X,Y)ft,ν).
+In the last step we used
+[∂t,∇XY] = [(∂t,0M),(0R,∇XY)] =/parenleftbig
+0R,(D(f,ft)∇)(X,Y)/parenrightbig
+= (D(f,ft)∇)(X,Y).
+Evaluating at t= 0 yields
+∂2
+t|0s(X,Y)
+= ¯g(∇2
+X,Y(a.νf0),−Tf0.gradg0a)−¯g(∇(D(f0,a.νf0)∇)(X,Y)(a.νf0),νf0)
+= 0+¯g(da(X).∇Yνf0+da(Y).∇Xνf0,−Tf0.gradg0a)
++ ¯g(a.∇2
+X,Y(νf0),−Tf0.gradg0a)−da/parenleftbig
+(D(f0,a.νf0)∇)(X,Y)/parenrightbig
++0.
+We will treat the three terms separately. Using ∇Zν=−Tf.L.Z, one gets for the
+first term
+¯g(da(X).∇Yνf0+da(Y).∇Xνf0,−Tf0.gradg0a)
+=g0(da(X)Lf0Y+da(Y)Lf0X,gradg0a)
+=da(X).da(Lf0Y)+da(Y).da(Lf0X).
+For the second term one gets
+¯g(a.∇2
+X,Y(νf0),−Tf0.gradg0a) =−a¯g(∇X∇Yνf0−∇∇XYνf0,Tf0.gradg0a)
+=−a¯g(−∇X(Tf0Lf0Y)+Tf0Lf0∇XY,Tf0.gradg0a)
+=−a¯g(−(∇Tf0)(X,Lf0Y)−Tf0∇X(Lf0Y)+Tf0Lf0∇XY,Tf0.gradg0a)
+= 0+a¯g(Tf0(∇XLf0)(Y),Tf0.gradg0a) =ag0/parenleftbig
+(∇XLf0)(Y),gradg0a/parenrightbig
+=a∇X/parenleftbig
+g0(Lf0Y,gradg0a)/parenrightbig
+−ag0(Lf0∇XY,gradg0a)
+−ag0(Lf0Y,∇Xgradg0a)
+=a∇X/parenleftbig
+sf0(Y,gradg0a)/parenrightbig
+−asf0(∇XY,gradg0a)−asf0(Y,∇Xgradg0a)
+=a(∇Xs)(Y,gradg0a).26 Almost local metrics on shape space of hypersurfaces in n-space
+∇2
+X,Yνis symmetric in X,Ybecause the ambient space Rnis flat. Therefore, the
+last formula and the symmetry of simply that
+a(∇Xs)(Y,gradg0a) =a(∇Ys)(X,gradg0a) =a(∇gradg0as)(X,Y).
+The third term yields, using the formula in section 4.12
+−g0/parenleftbig
+(D(f,a.νf0)∇)(X,Y),gradg0(a)/parenrightbig
+=−1
+2(∇(−2a.sf0))(X,Y,gradg0(a))−1
+2(∇(−2a.sf0))(Y,X,gradg0(a))
++1
+2(∇(−2a.sf0))(gradg0(a),X,Y)
+=da(X).sf0(Y,gradg0(a))+a.(∇Xsf0)(Y,gradg0(a))
++da(Y).sf0(X,gradg0(a))+a.(∇Ysf0)(X,gradg0(a))
+−da(gradg0(a)).sf0(X,Y)−a.(∇gradg0(a)sf0)(X,Y)
+=da(X).da(Lf0Y)+da(Y).da(Lf0X)
+−/ba∇dblda/ba∇dbl2
+g−1.sf0(X,Y)+a.(∇gradg0(a)sf0)(X,Y).
+4.18.Lemma (second variation of the mean curvature). The second deriv-
+ative of the mean curvature
+/braceleftbiggImm→C∞(M),
+f/ma√sto→Tr(Lf)
+along a curve of immersions fsatisfying property 1 from section 4.13 is given by
+∂2
+t|0Tr(L) = 2a2Tr/parenleftbig
+(Lf0)3/parenrightbig
++4aTr/parenleftbig
+Lf0g−1
+0.∇2a/parenrightbig
++2Tr(g−1(da⊗da)Lf0)
+−/ba∇dblda/ba∇dbl2
+g−1
+0Tr(Lf0)+2aTrg0/parenleftbig
+∇gradg0asf0/parenrightbig
+.
+Proof. From Tr(L) = Tr(g−1.s) we get
+∂2
+tTr(L) = Tr/parenleftbig
+∂2
+t(g−1).s/parenrightbig
++2Tr/parenleftbig
+∂t(g−1).∂ts/parenrightbig
++Tr/parenleftbig
+g−1.∂2
+ts/parenrightbig
+.
+Evaluating at t= 0, we get
+∂2
+t|0Tr(L) = Tr/parenleftbig
+6a2(Lf0)2.g−1
+0.sf0/parenrightbig
++Tr/parenleftbig
+−2g−1
+0.(da⊗da).g−1
+0.sf0/parenrightbig
++2Tr/parenleftbig
+2aLf0g−1
+0.∇2a/parenrightbig
++2Tr/parenleftbig
+2aLf0g−1
+0.(−ag0(Lf0)2)/parenrightbig
++2.Tr/parenleftbig
+g−1
+0.((da⊗da◦Lf0)+(da◦Lf0⊗da))/parenrightbig
+−/ba∇dblda/ba∇dbl2
+g−1
+0Tr(Lf0)+2aTrg0/parenleftbig
+∇gradg0asf0/parenrightbig
+= 2a2Tr/parenleftbig
+(Lf0)3/parenrightbig
+−2Tr/parenleftbig
+g−1
+0.(da⊗da).Lf0/parenrightbig
++4aTr/parenleftbig
+Lf0g−1
+0.∇2a/parenrightbig
++4Tr/parenleftbig
+g−1
+0.(da⊗da).(Lf0)/parenrightbig
+−/ba∇dblda/ba∇dbl2
+g−1
+0Tr(Lf0)+2aTrg0/parenleftbig
+∇gradg0asf0/parenrightbig
+.M. Bauer, P. Harms, P. Michor 27
+5.The geodesic equation on Imm(M,Rn)
+We recall the definition of the GΦ-metric from section 1.5,
+GΦ
+f(h,k) =/integraldisplay
+MΦ/parenleftbig
+Vol,Tr(L)(x)/parenrightbig
+¯g/parenleftbig
+h(x),k(x)/parenrightbig
+vol(g)(x).
+WewillwriteΦ( v,µ)forthefunctionΦanditsarguments( v,µ)∈R2corresponding
+to the volume and mean curvature.
+5.1.The geodesic equation on Imm(M,Rn).We use the method of section 2.5
+to calculate the geodesic equation. So we need to compute the metr ic gradients.
+The calculation at the same time shows the existence of the gradient s. Letm∈
+TfImm(M,Rn) with
+m=a.νf+Tf.m⊤.
+To shortenthe notation, we will not alwaysnote the dependence on fin expressions
+asνf,Lf,....
+D(f,m)GΦ
+f(h,k) =/integraldisplay
+M(∂vΦ)(D(f,m)Vol)¯g(h,k)vol(g)
++/integraldisplay
+M(∂µΦ)(D(f,m)Tr(L))¯g(h,k)vol(g)
++/integraldisplay
+MΦ.¯g(h,k)(D(f,m)vol(g)).
+To read off the K-gradient of the metric, we write this expression as
+/integraldisplay
+MΦ.¯g/parenleftbigg/bracketleftBig∂vΦ
+Φ(D(f,m)Vol)+∂µΦ
+Φ(D(f,m)Tr(L))+D(f,m)vol(g)
+vol(g)/bracketrightBig
+h,k/parenrightbigg
+vol(g).
+Therefore, using the formulas from section 4, we can calculate the K–gradient:
+Kf(m,h) =/bracketleftBig∂vΦ
+Φ(D(f,m)Vol)+∂µΦ
+Φ(D(f,m)Tr(L))+D(f,m)vol(g)
+vol(g)/bracketrightBig
+h
+=/bracketleftBig∂vΦ
+Φ/parenleftBig/integraldisplay
+M−Tr(L).avol(g)/parenrightBig
++∂µΦ
+Φ/parenleftbig
+−∆a+aTr(L2)+dTr(L)(m⊤)/parenrightbig
++divg(m⊤)−Tr(L).a/bracketrightBig
+h.
+To calculate the H-gradient, we treat the three summands of D(f,m)GΦ
+f(h,k) sep-
+arately. The first summand is
+/integraldisplay
+M(∂vΦ)(D(f,m)Vol(x))¯g(h(x),k(x))vol(g)(x)
+=/integraldisplay
+x∈M(∂vΦ)/parenleftBig/integraldisplay
+y∈M−Tr(L(y)).a(y)vol(g)(y)/parenrightBig
+¯g(h(x),k(x))vol(g)(x)
+=/integraldisplay
+y∈M¯g/parenleftbigg
+a(y).ν(y),−Tr(L(y))/integraldisplay
+x∈M(∂vΦ)¯g(h(x),k(x))vol(g)(x).ν(y)/parenrightbigg
+vol(g)(y)28 Almost local metrics on shape space of hypersurfaces in n-space
+=GΦ
+f/parenleftBig
+m,−1
+ΦTr(L)/integraldisplay
+M(∂vΦ)¯g(h,k)vol(g).ν/parenrightBig
+.
+By the symmetry of the Laplacian
+/integraldisplay
+M∆(a).bvol(g) =/integraldisplay
+Ma.∆(b)vol(g) fora,b∈C∞(M)
+one gets for the second summand
+/integraldisplay
+M(∂µΦ)(D(f,m)Tr(L)) ¯g(h,k)vol(g)
+=/integraldisplay
+M(∂µΦ)/parenleftbig
+−∆a+aTr(L2)+dTr(L)(m⊤)/parenrightbig
+¯g(h,k)vol(g)
+=/integraldisplay
+M−a.∆/parenleftbig
+(∂µΦ)¯g(h,k)/parenrightbig
+vol(g)+/integraldisplay
+Ma.(∂µΦ)Tr(L2)¯g(h,k)vol(g)
++/integraldisplay
+M(∂µΦ)g/parenleftbig
+gradg(Tr(L)),m⊤/parenrightbig
+¯g(h,k)vol(g)
+=/integraldisplay
+M−a.∆/parenleftbig
+(∂µΦ)¯g(h,k)/parenrightbig
+vol(g)+/integraldisplay
+Ma.(∂µΦ)Tr(L2)¯g(h,k)vol(g)
++/integraldisplay
+M(∂µΦ)¯g(Tf.gradg(Tr(L)),Tf.m⊤)¯g(h,k)vol(g)
+=GΦ
+f/parenleftBig
+m,−1
+Φ∆/parenleftbig
+(∂µΦ)¯g(h,k)/parenrightbig
+.ν/parenrightBig
++GΦ
+f/parenleftBig
+m,1
+Φ(∂µΦ)Tr(L2)¯g(h,k).ν/parenrightBig
++GΦ
+f/parenleftBig
+m,1
+Φ(∂µΦ)¯g(h,k)Tf.gradg(Tr(L))/parenrightBig
+.
+In the calculation of the third term of the Hf(m,h)–gradient, we will make use of
+the following formula, which is valid for φ∈C∞(M) andX∈X(M):
+0 =/integraldisplay
+Mdiv(φ.X).vol(g) =/integraldisplay
+MLφ.Xvol(g)
+=/integraldisplay
+M(d◦iφ.X+iφ.X◦d)vol(g) =/integraldisplay
+Md(φ.iXvol(g))
+=/integraldisplay
+Mdφ∧iXvol(g)+/integraldisplay
+Mφ∧d(iXvol(g))
+=/integraldisplay
+M/parenleftbig
+−iX(dφ∧vol(g))+iX◦dφ∧vol(g)/parenrightbig
++/integraldisplay
+Mφ.LXvol(g)
+=/integraldisplay
+Mdφ(X)vol(g)+/integraldisplay
+Mφ.div(X)vol(g).
+Therefore, we can calculate the third summand, which is given by
+/integraldisplay
+MΦ¯g(h,k)(D(f,m)vol(g)) =/integraldisplay
+MΦ¯g(h,k)/parenleftbig
+divg(m⊤)−Tr(L).a/parenrightbig
+vol(g)
+=−/integraldisplay
+Md(Φ¯g(h,k))(m⊤)vol(g)+GΦ
+f(m,−¯g(h,k)Tr(L).ν)
+=−/integraldisplay
+M¯g/parenleftBig
+Tf.gradg(Φ¯g(h,k)),m/parenrightBig
+vol(g)+GΦ
+f(m,−¯g(h,k)Tr(L).ν)
+=GΦ
+f/parenleftBig
+m,−1
+ΦTf.gradg(Φ¯g(h,k))−¯g(h,k)Tr(L).ν/parenrightBig
+.M. Bauer, P. Harms, P. Michor 29
+Summing up all the terms the H-gradient is given by
+Hf(h,k) =/bracketleftBig
+−1
+ΦTr(L)/integraldisplay
+M(∂vΦ)¯g(h,k)vol(g)−1
+Φ∆/parenleftbig
+(∂µΦ)¯g(h,k)/parenrightbig
++1
+Φ(∂µΦ)Tr(L2)¯g(h,k)−¯g(h,k)Tr(L)/bracketrightBig
+νf
++1
+ΦTf./bracketleftBig
+(∂µΦ)¯g(h,k)gradg(Tr(L))−gradg(Φ¯g(h,k))/bracketrightBig
+.
+5.2.Theorem. The geodesic equation for the almost local metrics GΦonImm(M,Rn)
+is then given by
+ft=h=a.νf+Tf.h⊤,
+ht=1
+2/bracketleftBig
+−1
+ΦTr(L)/integraldisplay
+M∂vΦ/ba∇dblh/ba∇dbl2vol(g)−1
+Φ∆/parenleftbig
+(∂µΦ)/ba∇dblh/ba∇dbl2/parenrightbig
++1
+Φ(∂µΦ)Tr(L2)/ba∇dblh/ba∇dbl2−/ba∇dblh/ba∇dbl2Tr(L)/bracketrightBig
+νf
++1
+2ΦTf./bracketleftBig
+(∂µΦ)/ba∇dblh/ba∇dbl2gradg(Tr(L))−gradg(Φ/ba∇dblh/ba∇dbl2)/bracketrightBig
+−/bracketleftbig∂vΦ
+Φ/integraldisplay
+M−Tr(L).avol(g)
++∂µΦ
+Φ/parenleftbig
+−∆a+aTr(L2)+dTr(L)(h⊤)/parenrightbig
++divg(h⊤)−Tr(L).a/bracketrightbig
+h.
+5.3.Momentum mappings. The metric GΦis invariant under the action of the
+reparametrization group Diff( M) and under the Euclidean motion group Rn⋊
+SO(n). According to section 2.6, the momentum mappings for these grou p actions
+are constant along any geodesic in Imm( M,Rn):
+∀X∈X(M) :/integraldisplay
+MΦ(Vol(f),Tr(Lf))¯g(Tf.X,ft)vol(g) reparam. momenta
+or Φ(Vol(f) ,Tr(Lf))g(f⊤
+t)vol(g)∈Γ(T∗M⊗MVol(M)) reparam. momentum
+/integraldisplay
+MΦ(Vol(f),Tr(Lf))ftvol(g) linear momentum
+∀X∈so(n) :/integraldisplay
+MΦ(Vol(f),Tr(Lf))¯g(X.f,ft)vol(g) angular momenta
+or/integraldisplay
+MΦ(Vol(f),Tr(Lf))(f∧ft)vol(g)∈/logicalandtext2Rnangular momentum.
+Here Γ(T∗M⊗MVol(M))⊂Γ(TM)′is the space of cotangent bundle valued densi-
+ties contained in the dual of the Lie algebra Γ( TM). The name angular momentum
+is justified by the natural identification/logicalandtext2Rn∼=so(n)∼=so(n)∗.30 Almost local metrics on shape space of hypersurfaces in n-space
+6.The geodesic equation on Bi(M,Rn)
+6.1.The horizontal bundle and the metric on the quotient space. Since
+vol(f∗¯g) and Tr(Lf) react equivariantly to the action of the group Diff( M), every
+GΦ-metric is Diff( M)-invariant. As described in Section 2.8, it induces a Riemann-
+ian metric on Bi(off the singularities) such that the projection π: Imm→Biis a
+Riemannian submersion.
+By definition, a tangent vector htof∈Imm(M,Rn) is horizontal if and only
+if it isGΦ-perpendicular to the Diff( M)-orbits. This is the case if and only if
+¯g(h(x),Txf.Xx) = 0 at every point x∈M. Therefore, the horizontal bundle at the
+pointfequals the set of sections of the normal bundle (see Section 3.8) alo ngf.
+Thus the metric on the horizontal bundle is given by
+GΦ
+f(hhor,khor) =GΦ
+f(a·ν,b·ν) =/integraldisplay
+MΦ(Vol,Tr(L))a.bvol(g).
+The following lemma shows that every path in Bicorresponds to exactly one
+horizontal path in Imm, and therefore the calculation of the geode sic equation can
+be done on the horizontal bundle instead of on Bi.
+Lemma. For any smooth path finImmthere exists a smooth path ϕinDiff(M)
+withϕ(t,) = IdMdepending smoothly on fsuch that the path f(t,ϕ(t,x))is
+horizontal, i.e., ∂tf(t,ϕ(t,x))lies in the horizontal bundle.
+The basic idea is to write the path ϕas the integral curve of a time dependent
+vector field. This method is called the Moser–Trick, (see [14, Sectio n 2.5]).
+6.2.The geodesicequation on Bi(M,Rn).Asdescribedinsection2.8, geodesics
+inBicorrespond to horizontal geodesics in Imm. A horizontal geodesic fin Imm
+hasft=a.νfwitha∈C∞(R×M). The geodesic equation is given by
+ftt=at.ν/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+normal+a.νt/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+tang.=1
+2H(a.ν,a.ν)−K(a.ν,a.ν);
+see section 2.5. This equation splits into a normal part and a tangent ial part.
+From the conservation of the reparametrization momentum (see s ection 2.6 and
+the previous section) it follows that the tangential part of the geo desic equation is
+satisfied automatically. We will nevertheless check this by hand. Fro m section 5.1,
+where we calculated the metric gradients on Imm, we get
+Kf(a.ν,aν) =/bracketleftbig
+−∂vΦ
+Φ/integraldisplay
+MTr(L).avol(g)
++∂µΦ
+Φ/parenleftbig
+−∆a+aTr(L2)/parenrightbig
+−Tr(L).a/bracketrightbig
+a.ν
+Hf(a.ν,aν) =1
+ΦTf./bracketleftBig
+(∂µΦ)a2gradg(Tr(L))−gradg(Φa2)/bracketrightBig
++/bracketleftBig
+−1
+ΦTr(L)/integraldisplay
+M∂vΦa2vol(g)−1
+Φ∆/parenleftbig
+(∂µΦ)a2/parenrightbigM. Bauer, P. Harms, P. Michor 31
++1
+Φ(∂µΦ)Tr(L2)a2−a2Tr(L)/bracketrightBig
+ν.
+From this we can easily read the tangential part of the geodesic equ ation
+a.νt=1
+2ΦTf./bracketleftBig
+(∂µΦ)a2gradg(Tr(L))−gradg(Φa2)/bracketrightBig
+.
+We expand the right–hand side using a Leibnitz rule for the gradient,
+gradg(a.b) =agradgb+bgradgafora,b∈C∞(M).
+This yields
+a.νt=1
+2ΦTf./bracketleftBig
+(∂µΦ)a2gradg(Tr(L))−gradg(Φa2)/bracketrightBig
+=1
+2ΦTf./bracketleftBig
+(∂µΦ)a2gradg(Tr(L))−Φ.gradg(a2)−a2.gradg(Φ)/bracketrightBig
+=1
+2ΦTf./bracketleftBig
+(∂µΦ)a2gradg(Tr(L))−Φ.gradg(a2)−(∂µΦ)a2gradg(Tr(L))/bracketrightBig
+=−1
+2ΦΦTf.gradg(a2) =−Tf.a.gradg(a).
+By the variational formula for νin section 4.11 this equation is satisfied automati-
+cally. The normal part is given by
+at= ¯g/parenleftbig1
+2H(a.ν,a.ν)−K(a.ν,a.ν),ν/parenrightbig
+=1
+Φ/bracketleftbig1
+2Φa2Tr(Lf)−1
+2Tr(Lf)/integraldisplay
+M(∂vΦ)a2vol(f∗¯g)−1
+2∆((∂µΦ).a2)
++(∂vΦ)a/integraldisplay
+MTr(Lf).avol(f∗¯g)−1
+2(∂µΦ)Tr((Lf)2)a2+(∂µΦ)a∆a/bracketrightbig
+.
+We rewrite this equation by expanding Laplacians of products,
+∆(a.b) = (∆a)b−2Trg(da⊗db)+a(∆b) fora,b∈C∞(M).
+6.3.Theorem. The geodesic equation of the almost local metric GΦonBireads
+as
+ft=a.νf,
+at=1
+Φ/bracketleftbigg1
+2Φa2Tr(Lf)−1
+2Tr(Lf)/integraldisplay
+M(∂vΦ)a2vol(f∗¯g)−1
+2a2∆(∂µΦ)
++2aTrg(d(∂µΦ)⊗da)+(∂µΦ)Trg(da⊗da)
++(∂vΦ)a/integraldisplay
+MTr(Lf).avol(f∗¯g)−1
+2(∂µΦ)Tr((Lf)2)a2/bracketrightbigg
+.
+For the case of curves immersed in R2, this formula specializes to the formula
+given in [16, section 3.4]. (When verifying this, remember that ∆ = −D2
+sin the
+notation of [16].)32 Almost local metrics on shape space of hypersurfaces in n-space
+7.Sectional curvature on shape space
+To compute the sectional curvature we will use the following formula , which is
+valid in a chart at the center 0 of the chart:
+R0(a1,a2,a1,a2) =GΦ
+0(R0(a1,a2)a1,a2)
+=1
+2d2GΦ
+0(a1,a1)(a2,a2)−d2GΦ
+0(a1,a2)(a1,a2)+1
+2d2GΦ
+0(a2,a2)(a1,a1)
++GΦ
+0(Γ0(a1,a1),Γ0(a2,a2))−GΦ
+0(Γ0(a1,a2),Γ0(a1,a2)).
+Sectional curvature is given by
+R0(a1,a2,a2,a1) =−R0(a1,a2,a1,a2).
+Therefore, wehaveto calculate the metric in a chart, calculate its s econdderivative,
+and the value Γ 0at the center 0 of the Christoffel symbols Γ.
+7.1.The almost local metric GΦin a chart. In the following section we will
+follow the method of [14]. First we will construct a local chart for Bi. Letf0:
+M→Rnbe a fixed immersion, which will be the center of our chart. Consider t he
+mapping
+ψ=ψf0:C∞(M,(−ǫ,ǫ))→Imm(M,Rn),
+ψ(a)(x) = exp¯g
+f0(x)(a(x).νf0(x)) =f0(x)+a(x).νf0(x),
+where exp¯gis the exponential mapping on ( Rn,¯g) and where ǫis so small that ψ(a)
+is an immersion for each a.
+Denote byπthe projection from Imm( M,Rn) toBi(M,Rn). The inverse on its
+image ofπ◦ψ:C∞(M,(−ǫ,ǫ))→Bi(M,Rn) is then a smooth chart on Bi(M,Rn).
+We want to calculate the induced metric in this chart, i.e.,
+((π◦ψ)∗GΦ)a(b1,b2)
+for anya∈C∞(M,(−ǫ,ǫ)) andb1,b2∈C∞(M). We shall fix the function aand
+work with the ray of points t.ain this chart. Everything will revolve around the
+map:
+f(t,x) =ψ(t.a)(x) =f0(x)+t.a(x).νf0(x).
+We shall use a fixed chart ( u,U) onMwith∂i=∂
+∂ui. To calculate the metric GΦ
+in this chart we have to understand how
+Tt.aψ.b1=b1(x).νf0(x)
+splits into tangential and horizontal parts with respect to the imme rsionf(t,).
+The tangential part locally has the form
+Tf.(T(t.a)ψ.(b1))⊤=n−1/summationdisplay
+i=1ci∂if(t,x),
+where the coefficients ciare given by
+ci=n−1/summationdisplay
+j=1gij¯g/parenleftbig
+b1(x)νf0(x),∂jf(t,x)/parenrightbig
+.M. Bauer, P. Harms, P. Michor 33
+Thus the horizontal part is
+(Tt.aψ.b1).ν= (Tt.aψ.b1)−Tf.(T(t.a)ψ.(b1))⊤=b1(x)νf0(x)−n−1/summationdisplay
+i=1ci∂if(t,x).
+Lemma. Using the local expression of section 3 the metric GΦin the chart (π◦ψ)−1
+reads as (by an abuse of notation)
+/parenleftbig
+(π◦ψf0)∗GΦ/parenrightbig
+(t.a)(b1,b2) =GΦ
+π(ψ(t.a))/parenleftbig
+Tt.a(π◦ψ).b1,Tt.a(π◦ψ)b2/parenrightbig
+=GΦ
+ψ(t.a)/parenleftbig
+(Tt.aψ.b1)⊥.ν,(Tt.aψ.b2)⊥.ν/parenrightbig
+=/integraldisplay
+MΦ¯g/parenleftbig
+(Tt.aψ.b1)⊥.ν,(Tt.aψ.b2)/parenrightbig
+vol(g)
+=/integraldisplay
+MΦ/parenleftBigg
+b1.b2−n−1/summationdisplay
+i=1ci¯g(∂if(t,x),b2(x).νf0(x))/parenrightBigg
+vol(g).
+7.2.Second derivative of the GΦ-metric in the chart. We will calculate
+∂2
+t|0((π◦ψf0)∗GΦ)(t.a)(b1,b2).
+We will use the following arguments repeatedly:
+∂t|0∂jf=∂j∂t|0f=∂j(a.νf0) = (∂ja)νf0+a(∂jνf0)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+tang.,
+¯g/parenleftbig
+b1(x)νf0(x),∂jf(t,x)/parenrightbig
+|t=0= 0,
+and consequently ci|t=0= 0.
+∂t|0ci=n−1/summationdisplay
+j=1∂t|0(gij).0+n−1/summationdisplay
+j=1gij∂t|0¯g/parenleftbig
+b1νf0,∂jf/parenrightbig
+=n−1/summationdisplay
+j=1gij¯g/parenleftbig
+b1νf0,∂t|0∂jf/parenrightbig
+=n−1/summationdisplay
+j=1gij¯g/parenleftbig
+b1νf0,∂j(a.νf0)/parenrightbig
+=n−1/summationdisplay
+j=1gijb1∂ja.
+Therefore
+/parenleftBig
+b1.b2−n−1/summationdisplay
+i=1ci¯g(∂if,b2.νf0)/parenrightBig/vextendsingle/vextendsingle
+t=0=b1.b2
+∂t|0/parenleftBig
+b1.b2−n−1/summationdisplay
+i=1ci¯g(∂if,b2.νf0)/parenrightBig
+=−n−1/summationdisplay
+i=1(∂t|0ci).0−n−1/summationdisplay
+i=10.∂t|0¯g(∂if,b2.νf0) = 0,
+∂2
+t|0/parenleftBig
+b1.b2−n−1/summationdisplay
+i=1ci¯g(∂if,b2.νf0)/parenrightBig
+=
+=−n−1/summationdisplay
+i=1(∂2
+t|0ci).0−2n−1/summationdisplay
+i=1(∂t|0ci)∂t|0¯g(∂if,b2.νf0)−n−1/summationdisplay
+i=10.∂2
+t|0¯g(∂if,b2.νf0)34 Almost local metrics on shape space of hypersurfaces in n-space
+=−2n−1/summationdisplay
+i=1(∂t|0ci)¯g(∂i(a.νf0),b2.νf0) =−2n−1/summationdisplay
+i=1(∂t|0ci)(∂ia)b2
+=−2b1b2n−1/summationdisplay
+i=1n−1/summationdisplay
+j=1gij∂ja.∂ia=−2b1b2/ba∇dblda/ba∇dbl2
+g−1.
+The derivatives of Φ are
+∂t|0/parenleftbig
+Φ◦(Vol,Tr(L))/parenrightbig
+= (∂vΦ).(∂t|0Vol)+(∂µΦ).(∂t|0Tr(L))
+∂2
+t|0/parenleftbig
+Φ◦(Vol,Tr(L))/parenrightbig
+= (∂2
+vΦ).(∂t|0Vol)2+(∂2
+µΦ).(∂t|0Tr(L))2
++2(∂v∂µΦ).(∂t|0Vol).(∂t|0Tr(L))+(∂vΦ)(∂2
+t|0Vol)+(∂µΦ)(∂2
+t|0Tr(L)).
+Lemma. The second derivative of the GΦ-metric in the chart (π◦ψ)−1is given by
+(1)∂2
+t|0/parenleftbig
+(π◦ψf0)∗GΦ/parenrightbig
+(t.a)(b1,b2) =/parenleftBig
+d2/parenleftbig
+(π◦ψf0)∗GΦ/parenrightbig
+(0)(a,a)/parenrightBig
+(b1,b2)
+=/integraldisplay
+M... b1.b2vol(g)
+over the following expression:
+...= Φ/parenleftBig∂2
+t|0vol
+vol−2/ba∇dblda/ba∇dbl2
+g−1/parenrightBig
++(∂vΦ)./parenleftBig
+(∂2
+t|0Vol)+2(∂t|0Vol)∂t|0vol
+vol/parenrightBig
++(∂µΦ)./parenleftBig
+(∂2
+t|0Tr(L))+2(∂t|0Tr(L))∂t|0vol
+vol/parenrightBig
++(∂2
+vΦ).(∂t|0Vol)2
++2(∂v∂µΦ).(∂t|0Vol)(∂t|0Tr(L))+(∂2
+µΦ).(∂t|0Tr(L))2.
+7.3.Sectional curvature on shape space. To understand the structure of the
+formulas for the sectional curvature tensor, we will use some fac ts from linear
+algebra.
+Sublemma 1. Let V=C∞(M), and letPandQbe bilinear and symmetric maps
+V×V→V. Then
+⊞(P,Q)(a1∧a2,b1∧b2) :=1
+2/parenleftbig
+P(a1,b1)Q(a2,b2)−P(a1,b2)Q(a2,b1)
++P(a2,b2)Q(a1,b1)−P(a2,b1)Q(a1,b2)/parenrightbig
+defines a symmetric, bilinear map (V∧V)⊗(V∧V)→V.
+Also⊞(P,Q) =⊞(Q,P). The symbol ⊞stands for the Young tableau encoding
+the symmetries; see [7]. We have
+⊞(P,Q)(a1∧a2,a1∧a2)
+=1
+2P(a1,a1)Q(a2,a2)−P(a1,a2)Q(a2,a1)+1
+2P(a2,a2)Q(a1,a1).
+Pis called positive semidefinite if for all x∈Manda∈C∞(M),P(a,a)(x)≥0.
+Pis called negative semidefinite if −Pis positive semidefinite. We will write P≥
+0,P≤0,P<
+>0 ifPis positive semidefinite, negative semidefinite, or indefinite.M. Bauer, P. Harms, P. Michor 35
+Sublemma 2. If PandQare positive semidefinite bilinear and symmetric maps
+V×V→V, then⊞(P,Q)also is a positive semidefinite symmetric, bilinear map.
+Proof. To shorten notation, we will write, for instance, P12instead ofP(a1,a2).
+The Cauchy inequality applied to PandQgives us
+P12Q12≤/radicalbig
+P11P22Q11Q22,
+and therefore we have
+⊞(P,Q)(a1∧a2,a1∧a2) =1
+2P11Q11−P12Q12+1
+2P22Q22
+≥1
+2P11Q22−/radicalbig
+P11P22Q11Q22+1
+2P22Q11
+=1
+2/parenleftBig/radicalbig
+P11Q22−/radicalbig
+P22Q11/parenrightBig2
+≥0.
+Letλ,µ:V→V. Then the map λ⊗µ:V⊗V→Vis given by
+(λ⊗µ)(a⊗b) =λ(a).µ(b),
+where the multiplication is in V=C∞(M). Denote by λ∨µthe symmetrization
+of the tensor product given by
+λ∨µ=1
+2(λ⊗µ+µ⊗λ).
+We will make use of the following simplifications.
+Sublemma 3. Let λ,β,µ,ν, :V→V. Then the bilinear symmetric map
+⊞(λ∨β,µ∨ν)
+satisfies the following properties:
+⊞(λ∨µ,λ∨ν)(a1∧a2,a1∧a2) =−1
+4(λ⊗µ)(a1∧a2).(λ⊗ν)(a1∧a2), (S1)
+⊞(λ∨µ,λ⊗λ) = 0, (S2)
+⊞(λ⊗λ,µ∨ν)(a1∧a2,a1∧a2) =1
+2(λ⊗µ)(a1∧a2).(λ⊗ν)(a1∧a2). (S3)
+Proof. For the proof of simplification (S1) we calculate
+⊞(λ∨µ,λ∨ν)(a1∧a2,a1∧a2)
+=1
+2(λ⊗µ⊗λ⊗ν)/bracketleftBig
+a1⊗a1⊗a2⊗a2+a2⊗a2⊗a1⊗a1
+−1
+2a1⊗a2⊗a1⊗a2−1
+2a1⊗a2⊗a2⊗a1
+−1
+2a2⊗a1⊗a1⊗a2−1
+2a2⊗a1⊗a2⊗a1/bracketrightBig
+.
+Using the symmetries of the quasilinear mapping λ⊗µ⊗λ⊗µ, we can swap the
+first and third positions in the tensor product of the two summands in the first line.
+Then the expression inside the square brackets equals −1
+2(a1∧a2)⊗(a1∧a2).
+Sinceλ⊗λvanishes when applied to elements of V∧V, simplification (S2) is a
+direct consequence of (S1).36 Almost local metrics on shape space of hypersurfaces in n-space
+For the proof of simplification (S3) we calculate
+⊞(λ⊗λ,µ∨ν)(a1∧a2,a1∧a2)
+=1
+2(λ⊗λ⊗µ⊗ν)/bracketleftBig
+a1⊗a1⊗a2⊗a2+a2⊗a2⊗a1⊗a1
+−a1⊗a2⊗a1⊗a2−a1⊗a2⊗a2⊗a1/bracketrightBig
+.
+Using symmetries as above, we can replace the third summand a1⊗a2⊗a1⊗a2
+bya2⊗a1⊗a2⊗a1, because the first two tensor components of λ⊗λ⊗µ⊗νare
+equal. Then, swapping the second and third positions in all tensor pr oducts, we
+get
+⊞(λ⊗λ,µ⊗ν)(a1∧a2,a1∧a2)
+=1
+2(λ⊗µ⊗λ⊗ν)/bracketleftBig
+a1⊗a2⊗a1⊗a2+a2⊗a1⊗a2⊗a1
+−a2⊗a1⊗a1⊗a2−a1⊗a2⊗a2⊗a1/bracketrightBig
+.
+The expression inside the square brackets equals ( a1∧a2)⊗(a1∧a2).
+For orthonormal a1,a2the sectional curvature is the negative of the curvature
+tensorR0(a1,a2,a1,a2). We will use the following formula for the curvature tensor :
+(1)R0(a1,a2,a1,a2) =GΦ
+0(R0(a1,a2)a1,a2)
+=1
+2d2GΦ
+0(a1,a1)(a2,a2)−d2GΦ
+0(a1,a2)(a1,a2)+1
+2d2GΦ
+0(a2,a2)(a1,a1)
++GΦ
+0(Γ0(a1,a1),Γ0(a2,a2))−GΦ
+0(Γ0(a1,a2),Γ0(a1,a2)).
+Looking at formula (1) from section 7.2, we can express the second derivative of
+the metricGΦin the chart as/parenleftBig
+d2(π◦ψf0)∗GΦ(0)(a1,a2)/parenrightBig
+(b1,b2)
+=/integraldisplay
+M/parenleftBig
+Φ.P1(a1,a2)+(∂vΦ)P2(a1,a2)+(∂µΦ)P3(a1,a2)+(∂2
+vΦ)P4(a1,a2)
++(∂v∂µΦ)P5(a1,a2)+(∂2
+µΦ)P6(a1,a2)/parenrightBig
+P(b1,b2)vol(g),
+whereP(b1,b2) =b1.b2, soP= id⊗id, and where the Piare obtained by sym-
+metrizing the terms in formula (1) from section 7.2.
+For the rest of this section, we do not note the pullback via the char t anymore,
+writingGΦ
+0instead of/parenleftbig
+(π◦ψf0)∗GΦ/parenrightbig
+(0), for example. To further shorten our
+notation, we write Linstead ofLf0andginstead ofg0. The following terms are
+calculated using the variational formulas from section 4:
+P1(a,a) =∂2
+t|0vol
+vol−2/ba∇dblda/ba∇dbl2
+g−1=a2/parenleftbig
+Tr(L)2−Tr(L2)/parenrightbig
+−/ba∇dblda/ba∇dbl2
+g−1
+P2(a,a) = (∂2
+t|0Vol)+2(∂t|0Vol)∂t|0vol
+vol
+=/integraldisplay
+Ma2/parenleftbig
+Tr(L)2−Tr(L2)/parenrightbig
++/integraldisplay
+M/ba∇dblda/ba∇dbl2
+g−1vol(g)M. Bauer, P. Harms, P. Michor 37
++2Tr(L).a/integraldisplay
+MTr(L).avol(g),
+P3(a,a) = (∂2
+t|0Tr(L))+2(∂t|0Tr(L))∂t|0vol
+vol
+= 2a2Tr(L3)+4aTr/parenleftbig
+L.g−1.∇2a/parenrightbig
++2Tr(g−1(da⊗da)L)
+−/ba∇dblda/ba∇dbl2
+g−1Tr(L)+2aTrg/parenleftbig
+∇gradas/parenrightbig
++2/parenleftbig
+−∆a+aTr(L2)/parenrightbig
+(−Tr(L).a)
+= 2a2Tr(L3)+4aTr/parenleftbig
+L.g−1.∇2a/parenrightbig
++2Tr(g−1(da⊗da)L)
+−/ba∇dblda/ba∇dbl2
+g−1Tr(L)+2aTrg/parenleftbig
+∇gradas/parenrightbig
++2Tr(L)a∆a−2Tr(L)Tr(L2).a2,
+P4(a,a) = (∂t|0Vol)2=/parenleftBig/integraldisplay
+MTr(L).avol(g)/parenrightBig2
+,
+P5(a,a) = 2(∂t|0Vol)(∂t|0Tr(L)) = 2/integraldisplay
+M−Tr(L).avol(g)/parenleftbig
+−∆a+aTr(L2)/parenrightbig
+= 2∆a/integraldisplay
+MTr(L).avol(g)−2Tr(L2)a/integraldisplay
+MTr(L).avol(g),
+P6(a,a) = (∂t|0Tr(L))2=/parenleftbig
+−∆a+aTr(L2)/parenrightbig2
+= (∆a)2−2a∆aTr(L2)+a2Tr(L2)2.
+Then the first part of the curvature tensor is given by
+1
+2d2GΦ
+0(a1,a1)(a2,a2)−d2GΦ
+0(a1,a2)(a1,a2)+1
+2d2GΦ
+0(a2,a2)(a1,a1)
+=/integraldisplay
+M/parenleftbig
+Φ.⊞(P1,P)+(∂vΦ)⊞(P2,P)+(∂µΦ)⊞(P3,P)
++(∂2
+vΦ)⊞(P4,P)+(∂v∂µΦ)⊞(P5,P)+(∂2
+µΦ)⊞(P6,P)/parenrightbig
+vol(g)
+·(a1∧a2,a1∧a2).
+Note thatPis positive definite, so that ⊞(Pi,P) is positive semidefinite if Piis
+positive semidefinite. We can always assume that Φ is positive because otherwise
+GΦwould not be a Riemannian metric.
+P1=P1
+1+P2
+1, P1P
+with
+P1
+1= (Tr(L)2−Tr(L2))id⊗id,
+P2
+1=−Trg(d⊗d).
+Applying simplification (S3) to ⊞(P1
+1,P) and⊞(P2
+1,P), we get
+⊞(P1
+1,P) =1
+2(Tr(L)2−Tr(L2))(id⊗id)2= 038 Almost local metrics on shape space of hypersurfaces in n-space
+on (V∧V)⊗(V∧V) and
+⊞(P2
+1,P) =−1
+2Trg/parenleftbig
+(id⊗d)2/parenrightbig
+,
+⊞(P2
+1,P)(a1∧a2,a1∧a2) =−1
+2/ba∇dbla1da2−a2da1/ba∇dbl2
+g−1≤0.
+Therefore, we have
+/integraldisplay
+MΦ.⊞(P1,P)(a1∧a2,a1∧a2)vol(g)≤0.
+P2=P1
+2+P2
+2+P3
+2 P2P
+with
+P1
+2=/integraldisplay
+M(id⊗id)(Tr(L)2−Tr(L2))vol(g)
+P2
+2= 2Tr(L)(id∨/integraldisplay
+MTr(L)idvol(g))
+P3
+2=/integraldisplay
+MTrg(d⊗d)vol(g)
+P1
+2is indefinite. Applying simplification (S2) we get ⊞(P2
+2,P) = 0.P3
+2, and there-
+fore also ⊞(P3
+2,P) is positive semidefinite. Therefore
+/integraldisplay
+M(∂vΦ)⊞(P1
+2,P)(a1∧a2,a1∧a2)vol(g)<
+>0,
+/integraldisplay
+M(∂vΦ)⊞(P3
+2,P)(a1∧a2,a1∧a2)vol(g)≥0.
+P3=P1
+3+P2
+3+P3
+3, P3P
+with
+P1
+3= 2id∨/parenleftBig
+Tr(L3)id+2Tr(Lg−1∇2(id))+Trg(∇gradids)
++Tr(L)∆(id)−Tr(L)Tr(L)2id/parenrightBig
+,
+P2
+3= 2Tr/parenleftbig
+g−1(d⊗d)L/parenrightbig
+,
+P3
+3=−Trg(d⊗d)Tr(L).
+Applying simplification (S2), we get that ⊞(P1
+3,P) vanishes. Furthermore,
+⊞(P2
+3,P)(a1∧a2,a1∧a2) =a2
+1Tr(g−1(da2⊗da2).L)
+−2a1a2Tr(g−1(da1⊗da2).L)
++a2
+2Tr(g−1(da1⊗da1).L)
+=g0
+2/parenleftbig
+(a1da2−a2da1)⊗(a1da2−a2da1),s/parenrightbig<
+>0,
+⊞(P3
+3,P)(a1∧a2,a1∧a2) =−1
+2/ba∇dbla1da2−a2da1/ba∇dbl2
+g−1Tr(L)<
+>0.
+P4=/integraldisplay
+MTr(L)idvol(g)⊗/integraldisplay
+MTr(L)idvol(g) P4PM. Bauer, P. Harms, P. Michor 39
+Applying simplification (S3), we get
+⊞(P4,P) =1
+2/parenleftBig
+id⊗/integraldisplay
+MTr(L)idvol(g)/parenrightBig2
+.
+Therefore, if ∂2
+vΦ≥0, then
+/integraldisplay
+M(∂2
+vΦ)⊞(P4,P)(a1∧a2,a1∧a2)vol(g)≥0,
+P5=P1
+5+P2
+5 P5P
+with
+P1
+5= 2/parenleftBig
+∆∨/integraldisplay
+MTr(L)idvol(g)/parenrightBig
+,
+P2
+5=−2Tr(L2)/parenleftBig
+id∨/integraldisplay
+MTr(L)a2vol(g)/parenrightBig
+.
+Applying simplification (S3), we get that ⊞(P1
+5,P) is the indefinite form given by
+⊞(P1
+5,P) =/parenleftbig
+id⊗∆/parenrightbig
+⊗/parenleftbig
+id⊗/integraldisplay
+MTr(L)idvol(g)/parenrightbig
+.
+Simplification (S2) gives ⊞(P2
+5,P) = 0. Therefore,
+/integraldisplay
+M(∂v∂µΦ)⊞(P1
+5,P)(a1∧a2,a1∧a2)vol(g)<
+>0.
+P6=P1
+6+P2
+6 P6P
+with
+P1
+6= ∆⊗∆,
+P2
+6= Tr(L2)2id⊗id,
+P3
+6=−2Tr(L2)id∨∆.
+Applying simplification (S2), we get that ⊞(P2
+6,P) and⊞(P3
+6,P) vanish. Simplifi-
+cation (S3) gives
+⊞(P1
+6,P) =1
+2(id⊗∆)2
+If∂2
+µΦ≥0, then we get
+/integraldisplay
+M(∂2
+µΦ)⊞(P6,P)(a1∧a2,a1∧a2)vol(g)≥0.
+Now we come to the second part of the curvature tensor R0(a1,a2,a1,a2), which
+is given by
+G0(Γ0(a1,a1),Γ0(a2,a2))−G0(Γ0(a1,a2),Γ0(a1,a2)).
+From the geodesic equation calculated in section 6, which is given by
+at= Γ0(a,a) =1
+Φ/bracketleftbig1
+2Φa2Tr(L)−1
+2Tr(L)/integraldisplay
+M(∂vΦ)a2vol(g)−1
+2a2∆(∂µΦ)40 Almost local metrics on shape space of hypersurfaces in n-space
++2aTrg(d(∂µΦ)⊗da)+(∂µΦ)Trg(da⊗da)
++(∂vΦ)a/integraldisplay
+MTr(L).avol(g)−1
+2(∂µΦ)Tr(L2)a2/bracketrightbig
+,
+we can extract the Christoffel symbol by symmetrization and get
+Γ0(a1,a2) =1
+Φ5/summationdisplay
+i=1Qi(a1,a2),
+whereQ1,...,Q 5are the symmetrizations of the summands in the geodesic equa-
+tion.Qiare given by
+Q1=1
+2/parenleftBig
+ΦTr(L)−∆(∂µΦ)−(∂µΦ)Tr(L2)/parenrightBig
+id⊗id,
+Q2=−1
+2Tr(L)/integraldisplay
+M(∂vΦ)id⊗idvol(g),
+Q3= 2id∨Trg(d(∂µΦ)⊗d),
+Q4= (∂vΦ)id∨/integraldisplay
+MTr(L)idvol(g),
+Q5= (∂µΦ)Trg(d⊗d).
+Then
+G0(Γ0(a1,a1),Γ0(a2,a2))−G0(Γ0(a1,a2),Γ0(a1,a2))
+=/integraldisplay
+M1
+Φ/summationdisplay
+i⊞(Qi,Qi)(a1∧a2,a1∧a2)vol(g)
++/integraldisplay
+M2
+Φ/summationdisplay
+i<j⊞(Qi,Qj)(a1∧a2,a1∧a2)vol(g).
+The contribution of the following terms to R0(a1,a2,a1,a2) is/integraltext
+M1
+Φ...vol(g)
+over the terms listed.
+⊞(Q1,Q1) = 0 Q1Q1
+according to simplification (S2).
+⊞(Q2,Q2)(a1∧a2,a1∧a2) =Tr(L)2
+4/bracketleftBig
+Q2Q2
+/integraldisplay
+M(∂vΦ)a2
+1vol(g)./integraldisplay
+M(∂vΦ)a2
+2vol(g)−/parenleftbig/integraldisplay
+M(∂vΦ)a1a2vol(g)/parenrightbig2/bracketrightBig
+which is positive by the Cauchy–Schwarz inequality, assuming that ∂vΦ≥0.
+⊞(Q3,Q3)(a1∧a2,a1∧a2) Q3Q3
+=−/parenleftBig/parenleftbig
+id⊗Trg(d(∂µΦ)⊗d)/parenrightbig
+(a1∧a2)/parenrightBig2
+=−g−1/parenleftbig
+d(∂µΦ),a1da2−a2da1/parenrightbig2≤0
+according to simplification (S1).
+⊞(Q4,Q4) =−1
+4(∂vΦ)2(id⊗/integraldisplay
+MTr(L)idvol(g))2≤0 Q4Q4M. Bauer, P. Harms, P. Michor 41
+according to simplification (S1).
+⊞(Q5,Q5) = (∂µΦ)2/parenleftbig
+/ba∇dblda1/ba∇dbl2
+g−1/ba∇dblda2/ba∇dbl2
+g−1−g−1(da1,da2)2/parenrightbig
+Q5Q5
+= (∂µΦ)2/ba∇dblda1∧da2/ba∇dbl2
+g0
+2≥0
+by the Cauchy–Schwarz inequality.
+The contribution of the following terms to R0(a1,a2,a1,a2) is/integraltext
+M2
+Φ...vol(g) over
+the terms listed:
+⊞(Q1,Q2) =−1
+4/parenleftbig
+ΦTr(L)2−Tr(L)∆(∂µΦ)−Tr(L)Tr(L2)(∂µΦ)/parenrightbig
+. Q1Q2
+.⊞/parenleftBig
+id⊗id,/integraldisplay
+M(∂vΦ)id⊗idvol(g)/parenrightBig
+,
+where the second factor is ≥0 assuming that ∂vΦ≥0.
+⊞(Q1,Q3) = 0 Q1Q3
+according to simplification (S2).
+⊞(Q1,Q4) = 0 Q1Q4
+according to simplification (S2).
+⊞(Q1,Q5) =1
+4/parenleftbig
+ΦTr(L)(∂µΦ)−(∂µΦ)∆(∂µΦ)−Tr(L2)(∂µΦ)2/parenrightbig
+. Q1Q5
+./ba∇dbla1da2−a2da1/ba∇dbl2
+g−1
+⊞(Q2,Q3)<
+>0 Q2Q3
+⊞(Q2,Q4) =−1
+2(∂vΦ)Tr(L)· Q2Q4
+·⊞/parenleftbig/integraldisplay
+M(∂vΦ)id⊗idvol(g),id∨/integraldisplay
+MTr(L)idvol(g)/parenrightbig
+This form is indefinite, but we have
+/integraldisplay
+M2
+Φ⊞(Q2,Q4)vol(g) =−⊞(˜Q2,˜Q4),
+with the positive semidefinite form
+˜Q2=/integraldisplay
+M(∂vΦ)id⊗idvol(g)
+and the form
+˜Q4=/integraldisplay
+MTr(L)1
+Φ(∂vΦ)idvol(g)∨/integraldisplay
+MTr(L)idvol(g),
+which is positive semidefinite if∂vΦ
+Φis a non negative constant.
+⊞(Q2,Q5) =−1
+2(∂µΦ)Tr(L)· Q2Q542 Almost local metrics on shape space of hypersurfaces in n-space
+·⊞/parenleftbig/integraldisplay
+M(∂vΦ)id⊗idvol(g),Trg(d⊗d)/parenrightbig<
+>0,
+because of the factor ( ∂µΦ)Tr(L). But the factor
+⊞/parenleftbig/integraldisplay
+M(∂vΦ)id⊗idvol(g),Trg(d⊗d)/parenrightbig
+is positive definite.
+⊞(Q3,Q4)<
+>0 Q3Q4
+⊞(Q3,Q5)(a1∧a2,a1∧a2) Q3Q5
+= (∂µΦ)/parenleftBig
+a1g−1(d(∂µΦ),da1)/ba∇dblda2/ba∇dbl2
+g−1−/parenleftbig
+a1g−1(d(∂µΦ),da2)
++a2g−1(d(∂µΦ),da1)/parenrightbig
+g−1(da1,da2)+a2g−1(d(∂µΦ),da2)/ba∇dblda1/ba∇dbl2
+g−1/parenrightBig
+= (∂µΦ)g0
+2/parenleftbig
+d(∂µΦ)⊗(a1da2−a2da1),da1∧da2/parenrightbig<
+>0,
+⊞(Q4,Q5)<
+>0. Q4Q5
+We are now able to compile a list of all negative, positive, and indefinite t erms of
+the curvature R0(a1,a2,a1,a2). Remember that negative terms of R0(a1,a2,a1,a2)
+make a positive contribution to sectional curvature. Positive sect ional curvature is
+connected to the vanishing ofgeodesic distance because the spac e wrapsup on itself
+in tighter and tighter ways.
+P4PP6PQ2Q2Q5Q5are positive, assuming ∂vΦ,∂2
+vΦ,∂2
+µΦ≥0.
+P1PQ3Q3Q4Q4Q1Q2are negative, assuming ∂vΦ≥0.
+Q2Q4is negative, assuming that∂vΦ
+Φis a non negative constant, and indefinite
+otherwise.
+Q2Q5is negative, assuming that Tr( L)(∂µΦ) is positive, and indefinite otherwise.
+P2PP3PP5PQ1Q5Q2Q3Q3Q4Q3Q5Q4Q5are indefinite.
+8.Geodesic distance on Bi(M,Rn)
+We will state some conditions on Φ ensuring that the almost local metr icGΦ
+induces non vanishing geodesic distance on Bi. The proofs are based on a compar-
+ison between the GΦ-length of a path and its area swept out. In the last part we
+will use the vector space structure of Rnto define a Fr´ echet metric on shape space
+Bi(M,Rn). In section 8.8 it is shown how this metric is related to an L∞Finsler
+metric, and in section 8.9 the Fr´ echet metric is compared to almost lo cal metrics.
+The main results are in sections 8.6 and 8.9.M. Bauer, P. Harms, P. Michor 43
+8.1.Geodesic distance on Bi.Geodesic distance on Biis given by
+distGΦ
+Bi(f0,f1) := inf
+fLenGΦ
+Bi(f),
+where the infimum is taken over all f: [0,1]→Biwithf(0) =f0andf(1) =f1.
+LenGΦ
+Biis the length of paths in Bigiven by
+LenGΦ
+Bi(f) =/integraldisplay1
+0/radicalBig
+GΦ
+f(ft,ft)dtforf: [0,1]→Bi.
+Lettingπ: Imm→Bidenote the projection, we have
+LenGΦ
+Bi(π◦f) = LenGΦ
+Imm(f) =/integraldisplay1
+0/radicalBig
+GΦ
+f(ft,ft)dtfor horizontal f: [0,1]→Imm.
+By non vanishing geodesic distance on Biwe mean that distGΦ
+Biseparates points.
+8.2.Area swept out. Forf: [0,1]→Imm we have
+(area swept out by f) =/integraldisplay
+[0,1]×Mvol(f(·,·)∗¯g).
+Iffis horizontal, then this integral can be rewritten as
+(area swept out by f) =/integraldisplay1
+0/integraldisplay
+M/ba∇dblft/ba∇dblvol(f(t,·)∗¯g)dt=:/integraldisplay1
+0/integraldisplay
+M/ba∇dblft/ba∇dblvol(g)dt.
+8.3.Lemma (first area swept out bound). For an almost local metric GΦ
+satisfying
+Φ(v,µ)≥C1forC1>0
+and a horizontal path f: [0,1]→Imm, we have the area swept out bound
+/radicalbig
+C1(area swept out by f)≤max
+t/radicalBig
+Vol/parenleftbig
+f(t)/parenrightbig
+.LenGΦ
+Imm(f).
+The proof is an adaptation of that given in [14, section 3.4] for the GA-metric.
+Proof.
+LenGΦ
+Imm(f) =/integraldisplay1
+0/radicalBig
+GΦ
+f(ft,ft)dt
+=/integraldisplay1
+0/parenleftBig/integraldisplay
+MΦ/ba∇dblft/ba∇dbl2vol(g)/parenrightBig1
+2dt≥/radicalbig
+C1/integraldisplay1
+0/parenleftBig/integraldisplay
+M/ba∇dblft/ba∇dbl2vol(g)/parenrightBig1
+2dt
+≥/radicalbig
+C1/integraldisplay1
+0/parenleftBig/integraldisplay
+Mvol(g)/parenrightBig−1
+2/integraldisplay
+M1./ba∇dblft/ba∇dblvol(g)dt
+≥/radicalbig
+C1min
+t/parenleftBig/integraldisplay
+Mvol(g)/parenrightBig−1
+2/integraldisplay1
+0/integraldisplay
+M/ba∇dblft/ba∇dblvol(g)dt
+=/radicalbig
+C1/parenleftBig
+max
+t/integraldisplay
+Mvol(g)/parenrightBig−1
+2/integraldisplay1
+0/integraldisplay
+M/ba∇dblft/ba∇dblvol(g)dt.44 Almost local metrics on shape space of hypersurfaces in n-space
+8.4.Lemma (Lipschitz continuity of√
+Vol).For an almost local metric GΦ,
+the condition
+Φ(v,µ)≥C2µ2
+implies the Lipschitz continuity of the map
+√
+Vol : (Bi,distGΦ
+Bi)→R≥0
+by the inequality holding for f1andf2inBi:
+/radicalbig
+Vol(f1)−/radicalbig
+Vol(f2)≤1
+2√C2distGΦ
+Bi(f1,f2).
+The proof is an adaptation of that given in [14, section 3.3] for the GA-metric.
+Proof. Letf: [0,1]→Imm be a horizontal path, and let ft=a.νfdenote its
+derivative. Using the formula from section 4.7 for the variation of th e volume, we
+get
+∂tVol(f) =−/integraldisplay
+MTr(L)avol(g)≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+MTr(L)avol(g)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤/parenleftBig/integraldisplay
+M12vol(g)/parenrightBig1
+2/parenleftBig/integraldisplay
+MTr(L)2a2vol(g)/parenrightBig1
+2
+≤/radicalbig
+Vol(f)/parenleftBig/integraldisplay
+MΦ
+C2a2vol(g)/parenrightBig1
+2≤1√C2/radicalbig
+Vol(f)/radicalBig
+GΦ
+f(ft,ft).
+Thus
+∂t/radicalbig
+Vol(f) =∂tVol(f)
+2/radicalbig
+Vol(f)≤1
+2√C2/radicalBig
+GΦ
+f(ft,ft).
+By integration we get
+/radicalbig
+Vol(f1)−/radicalbig
+Vol(f0) =/integraldisplay1
+0∂t/radicalbig
+Vol(f)dt
+≤/integraldisplay1
+01
+2√C2/radicalBig
+GΦ
+f(ft,ft) =1
+2√C2LenGΦ
+Imm(f).
+Now take the infimum over all horizontal paths fconnecting f1andf2.
+8.5.Lemma (second area swept out bound). For an almost local metric GΦ
+satisfying
+Φ(v,µ)≥CvwithC >0
+and a horizontal path f: [0,1]→Imm, we get the area swept out bound
+√
+C(area swept out by f)≤LenGΦ
+Imm(f).
+The proof is adapted from proofs for the case of planar curves th at can be found
+in [16, section 3.7], [18, Lemma 3.2], [25, Proposition 1] and [24, Theore m 7.5].
+Proof.
+LenGΦ
+Imm(f) =/integraldisplay1
+0/radicalBig
+GΦ
+f(ft,ft)dt=/integraldisplay1
+0/parenleftBig/integraldisplay
+MΦ/ba∇dblft/ba∇dbl2vol(g)/parenrightBig1
+2dt
+≥√
+C/integraldisplay1
+0/radicalbig
+Vol(f)/parenleftBig/integraldisplay
+M/ba∇dblft/ba∇dbl2vol(g)/parenrightBig1
+2dt≥√
+C/integraldisplay1
+0/integraldisplay
+M1./ba∇dblft/ba∇dblvol(g)dtM. Bauer, P. Harms, P. Michor 45
+=√
+C/integraldisplay
+[0,1]×Mvol(f(·,·)∗¯g)dt=√
+C(area swept out by f).
+8.6.Non vanishing geodesic distance. Using the estimates proven above, we
+get the following result.
+8.7.Theorem. At least on Be(M,Rn) = Emb(M,Rn)/Diff(M), the almost lo-
+cal metricGΦinduces non vanishing geodesic distance if at least one of th e two
+following conditions holds:
+Φ(v,µ)≥C1+C2µ2forC1,C2>0, (1)
+Φ(v,µ)≥Cv forC >0. (2)
+8.8.Fr´ echet distance and Finsler metric. The Fr´ echet distance on the shape
+spaceBi(M,Rn) is defined as
+distL∞
+Bi(F0,F1) = inf
+f0,f1/ba∇dblf0−f1/ba∇dblL∞,
+where the infimum is taken over all f0,f1withπ(f0) =F0,π(f1) =F1. As before,
+πdenotes the projection π: Imm→Bi. Fixingf0andf1, one has
+distL∞
+Bi/parenleftbig
+π(f0),π(f1)/parenrightbig
+= inf
+ϕ/ba∇dblf0◦ϕ−f1/ba∇dblL∞,
+where the infimum is taken over all ϕ∈Diff(M). The Fr´ echet distance is related
+to the Finsler metric
+G∞:TImm(M,Rn)→R, h/ma√sto→/vextenddouble/vextenddoubleh⊥/vextenddouble/vextenddouble
+L∞.
+Lemma. The path length distance induced by the Finsler metric G∞provides an
+upper bound for the Fr´ echet distance:
+distL∞
+Bi(F0,F1)≤distG∞
+Bi(F0,F1) = inf
+f/integraldisplay1
+0/ba∇dblft/ba∇dblG∞dt,
+where the infimum is taken over all paths
+f: [0,1]→Imm(M,Rn)withπ(f(0)) =F0, π(f(1)) =F1.
+Proof. Since any path fcan be reparametrized such that ftis normal to f, one
+has
+inf
+f/integraldisplay1
+0/vextenddouble/vextenddoublef⊥
+t/vextenddouble/vextenddouble
+L∞dt= inf
+f/integraldisplay1
+0/ba∇dblft/ba∇dblL∞dt,
+where the infimum is taken over the same class of paths fas described above.
+Therefore,
+distL∞
+Bi(F0,F1) = inf
+f/ba∇dblf(1)−f(0)/ba∇dblL∞= inf
+f/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay1
+0ftdt/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+L∞≤inf
+f/integraldisplay1
+0/ba∇dblft/ba∇dblL∞dt
+= inf
+f/integraldisplay1
+0/vextenddouble/vextenddoublef⊥
+t/vextenddouble/vextenddouble
+L∞dt= distG∞
+Bi(F0,F1).46 Almost local metrics on shape space of hypersurfaces in n-space
+It is claimed in [12, Theorem 13] that distL∞
+Bi= distG∞
+Bi. Unfortunately, the proof
+is not correct because convex combinations of immersions are used , even though
+the space of immersions is not convex.
+8.9.Theorem (almost local versus Fr´ echet distance on shape spa ce).On
+Bi(M,Rn)theGΦdistance cannot be bounded from below by the Fr´ echet distan ce
+if any one of the following conditions holds:
+Φ(v,µ)≤C1+C2µ2kforC1,C2>0andk<(dim(M)+2)/2, (1)
+Φ(v,µ)≤CvkforC >0, (2)
+Φ(v,µ)≤CevforC >0. (3)
+Indeed, then the identity map
+Id :/parenleftbig
+Bi(M,Rn),distGΦ
+Bi/parenrightbig
+→/parenleftbig
+Bi(M,Rn),distG∞
+Bi/parenrightbig
+is not continuous.
+Proof. Letf0be a fixed immersion of MintoRn, and letf1be a translation
+off0by a vector hof lengthℓ. We will show that the GΦ-distance between π(f0)
+andπ(f1) is bounded by a constant that does not depend on ℓ. It follows that
+theGΦ-distance cannot be bounded from below by the Fr´ echet distance , and this
+proves the claim.
+For smallr0, we calculate the GΦ-length of the following path of immersions:
+First scale f0down to a factor r0, then translate it by a vector hof lengthℓ,
+and then scale it up again around the new origin huntil it has reached f1. Let
+m= dim(M).
+For the scaling down part, let rbe a decreasing function such that r(0) = 1 and
+r(1) =r0. Then the length of the path f(t) :=r(t).f0is
+LenGΦ
+Imm(f)
+=/integraldisplay1
+0/radicalBigg/integraldisplay
+MΦ/parenleftbig
+Vol(r.f0),Tr(Lr.f0)/parenrightbig
+¯g/parenleftbig
+rt.f0,rt.f0/parenrightbig
+vol/parenleftbig
+(r.f0)∗¯g/parenrightbig
+dt
+=/integraldisplay1
+0/radicalBigg/integraldisplay
+Mr2
+t.Φ/parenleftbig
+rmVol(f0),1
+rTr(Lf0)/parenrightbig
+¯g/parenleftbig
+f0,f0/parenrightbig
+rmvol/parenleftbig
+f∗
+0¯g/parenrightbig
+dt
+=/integraldisplay1
+r0/radicalBigg/integraldisplay
+MΦ/parenleftbig
+rmVol(f0),1
+rTr(Lf0)/parenrightbig
+¯g/parenleftbig
+f0,f0/parenrightbig
+rmvol/parenleftbig
+f∗
+0¯g/parenrightbig
+dr.
+The last integral converges for r0→0 under any of the above assumptions. So we
+see that the length of the shrinking part is bounded by a constant t hat does not
+depend onℓ.
+The pathf(t) :=r0.f0+t.his a pure translation of the scaled immersion r0.f0
+by the vector hof lengthℓ. The length of this path is
+LenGΦ
+Imm(f) =/integraldisplay1
+0/integraldisplay
+MΦ.¯g(ft,ft)vol(g)dtM. Bauer, P. Harms, P. Michor 47
+=/integraldisplay1
+0/integraldisplay
+MΦ.ℓ2vol(g)dt=ℓ2/integraldisplay
+MΦvol(g)
+=
+
+O(r(m−2k)
+0) if Φ satisfies (1) ,
+O(rm(k+1)
+0) if Φ satisfies (2) ,
+O(er0.m.rm
+0) if Φ satisfies (3) .
+Under the above assumptions, this tends to zero as r0tends to zero.
+To scale the immersion r0.f0+hback up to its original size, we use the path
+f(t) :=r(1−t).f0+hwithr(t) as in the shrinking part. It follows as before that
+the length of this path is bounded by a constant that does not depe nd onℓ.
+Finally, we use
+distGΦ
+Bi/parenleftbig
+π(f0),π(f1)/parenrightbig
+≤distGΦ
+Imm(f0,f1).
+9.The set of concentric spheres
+For an almost local metric, the set of spheres with common center x∈Rnis a
+totally geodesic subspace of Bi. The reason is that it is the fixed point set of a
+group of isometries acting on Bi, namely, the group of rotations of Rnaroundx.
+(We also have to assume uniqueness of solutions to the geodesic equ ation.) For the
+GAmetric where Φ = 1+ ATr(L)2and plane curves, the set of concentric spheres
+has been studied in [15], and for Sobolev type metrics it has been stud ied in [2, 8].
+Some work for the G0-metric has also been done by [17].
+We denote the ( n−1)-dimensional volume of the n−1-dimensional unit sphere
+inRnby
+ωn−1:=nπn
+2
+Γ(1+n
+2).
+9.1.Theorem. Within a set of concentric spheres, any sphere is uniquely de scribed
+by its radius r. Thus the geodesic equation within a set of concentric spher es reduces
+to an ordinary differential equation for the radius. It is giv en by
+rtt=−r2
+tn−1
+Φ/parenleftBig1
+2rΦ+∂vΦ
+2rn−2ωn−1+1
+2r2(∂µΦ)/parenrightBig
+.
+The space of concentric spheres is geodesically complete wi th respect to a GΦmetric
+if and only if
+/integraldisplayr1
+0rn−1
+2/radicalBig
+Φ/parenleftbig
+ωn−1rn−1,−(n−1)/r/parenrightbig
+dr=∞, r 1>0,
+and
+/integraldisplay∞
+r0rn−1
+2/radicalBig
+Φ/parenleftbig
+ωn−1rn−1,−(n−1)/r/parenrightbig
+dr=∞, r 0>0.48 Almost local metrics on shape space of hypersurfaces in n-space
+For the metrics studied in this work, this yields
+Φ(v,µ) =vk: incomplete,
+Φ(v,µ) =ev: incomplete,
+Φ(v,µ) = 1+Aµ2k: complete iff k≥n+1
+2,
+Φ(v,µ) =v1+n
+1−n+Aµ2
+v: complete.
+Proof. The differential equation for the radius can be read from the geod esic
+equation in section 6.2 when it is taken into account that all functions are constant
+on each sphere and that
+Vol =ωn−1rn−1, L=−1
+rIdTM,Tr(Lk) = (−1)kn−1
+rk.
+To determine whether the space of concentric spheres is complete , we calculate the
+length of a path fconnecting a sphere with radius r0to a sphere with radius r1:
+LenGΦ
+Bi(f) =/integraldisplay1
+0/radicalBig
+GΦ
+f(f⊥
+t,f⊥
+t)dt
+=/integraldisplay1
+0/radicalBigg/integraldisplay
+MΦ/parenleftbig
+Vol,Tr(L)/parenrightbig
+r2
+tvol(g)dt
+=/integraldisplay1
+0|rt|/radicalBig
+Φ/parenleftbig
+ωn−1rn−1,−(n−1)/r/parenrightbig
+ωn−1rn−1dt
+=√ωn−1/integraldisplayr1
+r0rn−1
+2/radicalBig
+Φ/parenleftbig
+ωn−1rn−1,−(n−1)/r/parenrightbig
+dr.
+10.Special cases of almost local metrics
+10.1.TheG0-metric. TheG0-metric is the special case of a GΦ-metric with Φ ≡
+1. Thus its geodesic equation can be read from section 5.1. It reads as
+ft=a.ν+Tf.f⊤
+t,
+ftt=−1
+2/parenleftbig
+/ba∇dblft/ba∇dbl2Tr(L).ν+Tf.gradg(/ba∇dblft/ba∇dbl2)/parenrightbig
++/parenleftbig
+Tr(L).a−divg(hf⊤
+t)/parenrightbig
+.ft.
+We have three conserved quantities, namely,
+g(f⊤
+t)vol(g)∈Γ(T∗M⊗Mvol(M)) reparametrization momentum,
+/integraldisplay
+Mftvol(g) linear momentum,
+/integraldisplay
+M(f∧ft)vol(g)∈/logicalandtext2Rn∼=so(n)∗angular momentum.M. Bauer, P. Harms, P. Michor 49
+The geodesic equation on Bi(M,Rn) is well studied. We can read it from section 6.
+ft=a.ν, a t=Tr(L).a2
+2.
+Sectional curvature is given by
+R0(a1,a2,a2,a1) =1
+2/integraldisplay
+M/ba∇dbla1da2−a2da1/ba∇dbl2
+g−1vol(g)≥0.
+This formula is in accordance with [14, section 4.5] since we have codime nsion one
+and a flat ambient space, so that only term(6) remains, and for the case of plain
+curves, it is in accordance with [16, section 3.5].
+TheG0-metric induces vanishing geodesic distance; see section 8.
+10.2.TheGA-metric. For a constant A>0, theGA-metric is defined as
+GA
+f(h,k) =/integraldisplay
+M/parenleftbig
+1+ATr(L)2/parenrightbig
+¯g(h,k)vol(g).
+This metric has been introduced by [15, 14, 16]. It corresponds to a n almost local
+metricGΦwith Φ(v,µ) = (1+Aµ2); thus its geodesic equation on Imm( M,Rn) is
+given by (see section 5.1)
+ft=a.ν+Tf.f⊤
+t,
+ftt=1
+2/bracketleftBig
+−∆/parenleftbig
+(2ATr(L))/ba∇dblft/ba∇dbl2/parenrightbig
+1+ATr(L)2+/ba∇dblft/ba∇dbl2.Tr(L)/parenleftbig2ATr(L2)
+1+ATr(L)2−1/parenrightbig/bracketrightBig
+ν
++Tf./bracketleftBig
+(2ATr(L))/ba∇dblft/ba∇dbl2gradg(Tr(L))−gradg((1+ATr(L)2)/ba∇dblft/ba∇dbl2)/bracketrightBig
+2(1+ATr(L)2)
+−/bracketleftBig(2ATr(L))
+1+ATr(L)2/parenleftbig
+−∆a+aTr(L2)+dTr(L)(f⊤
+t)/parenrightbig
++divg(f⊤
+t)−Tr(L).a/bracketrightBig
+ft.
+The conserved quantities have the form
+(1+ATr(L)2)g(f⊤
+t)vol(g)∈Γ(T∗M⊗Mvol(M)) reparam. momentum,
+/integraldisplay
+M(1+ATr(L)2)ftvol(g) linear momentum,
+/integraldisplay
+M(1+ATr(L)2)(f∧ft)vol(g)∈/logicalandtext2Rn∼=so(n)∗angular momentum.
+The horizontal geodesic equation for the GA–metric reduces to
+ft=a.ν,
+at=1
+2a2Tr(L)+−a2A∆(Tr(L))+4Aag−1(dTr(L),da)
+(1+ATr(L)2)
++2ATr(L)/ba∇dblda/ba∇dbl2
+g−1−ATr(L)Tr(L2)a2
+(1+ATr(L)2).50 Almost local metrics on shape space of hypersurfaces in n-space
+For the case of curves immersed in R2, this formula specializes to the formula given
+in [15, section 4.2]. (When verifying this, remember that ∆ = −D2
+sin the notation
+of [15].)
+The curvature tensor R0(a1,a2,a1,a2) is the sum of
+P1PQ3Q3negative terms,
+P6PQ5Q5positive terms, and
+P3PQ1Q5Q3Q5indefinite terms.
+R0(a1,a2,a1,a2) =/integraldisplay
+MA(a1∆a2−a2∆a1)2vol(g)
++/integraldisplay
+M2ATr(L)g0
+2/parenleftbig
+(a1da2−a2da1)⊗(a1da2−a2da1),s/parenrightbig
+vol(g)
++/integraldisplay
+M1
+1+ATr(L)2/bracketleftbigg
+−4A2g−1/parenleftbig
+dTr(L),a1da2−a2da1/parenrightbig2
+−/parenleftBig1
+2/parenleftbig
+1+ATr(L)2/parenrightbig2+2A2Tr(L)∆(Tr(L))+2A2Tr(L2)Tr(L)2/parenrightBig
+·
+·/ba∇dbla1da2−a2da1/ba∇dbl2
+g−1+(2A2Tr(L)2)/ba∇dblda1∧da2/ba∇dbl2
+g2
+0
++(8A2Tr(L))g0
+2/parenleftbig
+dTr(L)⊗(a1da2−a2da1),da1∧da2/parenrightbig/bracketrightbigg
+vol(g).
+We want to express the curvature in terms of the basic skew symme tric forms.
+Therefore, mimicking the notation of [15, 16], we define
+W2=a1da2−a2da1, W22=a1∆a2−a2∆a1, W12=da1∧da2.
+Then the above equation reads as
+R0(a1,a2,a1,a2) =/integraldisplay
+MAW2
+22vol(g)+/integraldisplay
+M2ATr(L)g0
+2/parenleftbig
+W2⊗W2,s/parenrightbig
+vol(g)
++/integraldisplay
+M1
+1+ATr(L)2/bracketleftbigg
+−4A2g−1/parenleftbig
+dTr(L),W2/parenrightbig2
+−/parenleftBig1
+2/parenleftbig
+1+ATr(L)2/parenrightbig2+2A2Tr(L)∆(Tr(L))+2A2Tr(L2)Tr(L)2/parenrightBig
+/ba∇dblW2/ba∇dbl2
+g−1
++(2A2Tr(L)2)/ba∇dblW12/ba∇dbl2
+g2
+0+(8A2Tr(L))g0
+2/parenleftbig
+dTr(L)⊗W2,W12/parenrightbig/bracketrightbigg
+vol(g).
+For the case of plain curves, this formula specializes to the formula g iven in [16,
+section 3.6].
+TheGA-metric satisfies condition (1) from section 8.6; thus it induces non-
+vanishing geodesic distance.M. Bauer, P. Harms, P. Michor 51
+10.3.Conformal metrics. Theconformalmetricscorrespondtoalmostlocalmet-
+ricsGΦwhere Φ depends only on the volume and not on the mean curvature. For
+the case of planar curves these metrics have been treated in [23, 2 4, 25, 18]. Then
+[18] provides very interesting estimates on geodesic distance induc ed by metrics
+with Φ(v) =vand Φ(v) =ev. The geodesic equation on Imm( M,Rn) is given by
+ft=h=a.ν+Tf.h⊤,
+ht=−1
+2/bracketleftBigΦ′
+Φ/parenleftbigg/integraldisplay
+M/ba∇dblh/ba∇dbl2vol(g)/parenrightbigg
+Tr(L).ν
++/ba∇dblh/ba∇dbl2Tr(L).ν+Tf.gradg(/ba∇dblh/ba∇dbl2)/bracketrightBig
++/bracketleftbiggΦ′
+Φ/parenleftbigg/integraldisplay
+MTr(L).avol(g)/parenrightbigg
++Tr(L).a−divg(h⊤)/bracketrightbigg
+.h.
+The conserved quantities are given by
+Φ(Vol)g(f⊤
+t)vol(g)∈Γ(T∗M⊗Mvol(M)) reparam. momentum,
+Φ(Vol)/integraldisplay
+Mftvol(g) linear momentum,
+Φ(Vol)/integraldisplay
+M(f∧ft)vol(g)∈/logicalandtext2Rn∼=so(n)∗angular momentum.
+The horizontal part of the geodesic equation is given by
+at= ¯g/parenleftBig1
+2H(a.ν,a.ν)−K(a.ν,a.ν),ν/parenrightBig
+=−Φ′
+2Φ/parenleftbigg/integraldisplay
+Ma2vol(g)/parenrightbigg
+Tr(L)+1
+2a2Tr(L)+Φ′
+Φ/parenleftbigg/integraldisplay
+Ma.Tr(L)vol(g)/parenrightbigg
+a.
+To simplify this equation let b(t) = Φ(Vol).a(t). We get
+bt= Φ′.(D(f,a.ν)Vol).a+Φ.at
+=−Φ′.a./integraldisplay
+MTr(L).a.vol(g)+Φ1
+2a2.Tr(L)
+−1
+2Φ′/parenleftbigg/integraldisplay
+Ma2vol(g)/parenrightbigg
+.Tr(L)+Φ′.a./integraldisplay
+MTr(L).avol(g)
+=−1
+2Φ′/integraldisplay
+Ma2vol(g).Tr(L)+1
+2Φa2.Tr(L).
+Thus the geodesic equation of the conformal metric GΦonBiis
+ft=b(t)
+Φ(Vol)ν,
+bt=Tr(L)
+2Φ(Vol)/parenleftbigg
+b2−Φ′(Vol)
+Φ(Vol)/integraldisplay
+Mb2vol(g)/parenrightbigg
+.
+For the case of curves immersed in R2, this formula specializes to the formula given
+in [16, section 3.7].
+Assuming that Φ′and Φ′′are non negative, the curvature tensor consists of the
+following summands.52 Almost local metrics on shape space of hypersurfaces in n-space
+P4PQ2Q2are the positive summands.
+P1PQ4Q4Q1Q2are the negative summands.
+Q2Q4is indefinite, but assuming thatΦ′
+Φis a non negative constant, it is negative.
+Solving the ODEΦ′
+Φ=C >0 leads to Φ(Vol) = eC.Vol. In the case of curves,
+conformal metrics of this type have been studied in [12] and [18].
+P2Pis indefinite.
+Since the formula for sectional curvature with general Φ = Φ(Vol) is still too
+long, we will print only the formula for Φ(Vol) = Vol. To shorten notat ion we will
+writeafor the integral over a∈C∞(M), i.e.,
+a=/integraldisplay
+Mavol(g).
+Then the sectional curvature reads as
+R0(a1,a2,a1,a2) =−1
+2Vol/integraldisplay
+M/ba∇dbla1da2−a2da1/ba∇dbl2
+g−1vol(g)
++1
+4VolTr(L)2/parenleftBig
+a2
+1.a2
+2−a1.a22/parenrightBig
++1
+4/parenleftBig
+a2
+1.Tr(L)2a2
+2−2a1.a2.Tr(L)2a1.a2+a2
+2.Tr(L)2a2
+1/parenrightBig
+−3
+4Vol/parenleftBig
+a2
+1.Tr(L)a22−2a1.a2.Tr(L)a1.Tr(L)a2+a2
+2.Tr(L)a12/parenrightBig
++1
+2/parenleftBig
+a2
+1.Trg((da2)2)−2a1.a2.Trg(da1.da2)+a2
+2Trg((da1)2)/parenrightBig
+−1
+2/parenleftBig
+a2
+1.a2
+2.Tr(L2)−2.a1.a2.a1.a2.Tr(L2)+a2
+2.a2
+1.Tr(L2)/parenrightBig
+.
+For the case of curves immersed in R2, this formula is in accordance with the
+formula given in [16, section 3.7].
+From condition (2) in section 8.6 we read that the conformal metrics induce non
+vanishing geodesic distance if Φ(Vol) ≥C.Vol for some constant C >0.
+10.4.A scale–invariant metric. For a constant A>0 we define the metric
+GSI
+f(h,k) =/integraldisplay
+M/parenleftBig
+Vol1+n
+1−n+ATr(L)2
+Vol/parenrightBig
+¯g(h,k)vol(g).
+This is an almost local metric with Φ( v,µ) =v1+n
+1−n+Aµ2
+v. Scale–invariance means
+that this metric does not change when f,h,kare replaced by λf,λh,λk forλ>0.
+To see that GSIis scale–invariant, we calculate as in [16] how the scaling factor
+λchanges the metric, volume form, and volume and mean curvature. We fix an
+oriented chart ( u1,...,un−1) onM. Then
+(λf)∗¯g(∂i,∂j) = ¯g(T(λf).∂i,T(λf).∂j) =λ2.f∗¯g(∂i,∂j),
+vol((λ.f)∗¯g) =/radicalbig
+det(λ2(f∗¯g)|U)du1∧...∧du−1=λn−1vol(f∗¯g),M. Bauer, P. Harms, P. Michor 53
+Tr(L((λf)∗¯g)) = ((λf)∗¯g)ij¯g(∂2(λf)
+∂i∂j,νλ.f)
+=λ
+λ2(f∗¯g)ij¯g(∂2f
+∂i∂j,νf) =1
+λTr(L(f)).
+The scale-invariance of the metric GSIfollows. Thus along geodesics we have an
+additional conserved quantity (see section 5.3), namely,
+/integraldisplay
+M/parenleftBig
+Vol1+n
+1−n+ATr(L)2
+Vol/parenrightBig
+¯g(f,ft)vol(g) scaling momentum.
+From 6 we can read the geodesic equation for GSIonBi:
+ft=a.ν,
+at=1
+2a2Tr(L)+1
+Vol1+n
+1−n+ATr(L)2
+Vol/bracketleftBig
+−1
+2Tr(L)/integraldisplay
+M/parenleftBig
+1+n
+1−nVol2n
+1−n−ATr(L)2
+Vol2/parenrightBig
+a2vol(g)−A∆(Tr(L)).a2
+Vol
++4A.a
+Volg−1(dTr(L),da)+2ATr(L)
+Vol/ba∇dblda/ba∇dbl2
+g−1
++/parenleftBig
+1+n
+1−nVol2n
+1−n−ATr(L)2
+Vol2/parenrightBig
+a/integraldisplay
+MTr(L).avol(g)−ATr(L2)Tr(L)
+Vola2/bracketrightBig
+.
+For the case of curves immersed in R2, this formula specializes to the formula given
+in [16, section 3.8]. (When verifying this, remember that ∆ = −D2
+sin the notation
+of [16].)
+The metric GSIinduces non vanishing geodesic distance. This follows from the
+fact that log(Vol) is Lipschitz; see [16, section 3.8].
+11.Numerical results
+11.1.Discretizing the horizontal path energy. Wewanttosolvetheboundary
+value problem for geodesics in shape space of surfaces in R3with respect to several
+almost local metrics – more specifically, with respect to GΦ-metrics with
+Φ = Volk,Φ =eVol,Φ = 1+ATr(L)2k
+and the scale-invariant metric
+Φ = Vol1+3
+1−3+ATr(L)2
+Vol.
+In order to solve this infinite-dimensional problem numerically, we will r educe it to
+a finite-dimensional problem by approximatingan immersed surface b y a triangular
+mesh.
+Oneapproachtosolvingtheboundaryvalueproblemisbythemethod ofgeodesic
+shooting. This method is based on iteratively solving the initial value pr oblem for
+geodesics while suitably adapting the initial conditions.54 Almost local metrics on shape space of hypersurfaces in n-space
+Another approach, and the approach we will follow, is to minimize horiz ontal
+path energy
+Ehor(f) =/integraldisplay1
+0/integraldisplay
+MΦ(Vol,Tr(L))¯g(ft,ν)2vol(g)
+overthe set of paths fof immersions with fixed endpoints. Note that, by definition,
+the horizontal path energy does not depend on reparametrizatio ns of the surface.
+Nevertheless we want the triangular mesh to stay regular. This can be achieved by
+adding a penalty functional to the horizontal path energy.
+11.2.Discrete path energy. To discretize the horizontal path energy
+Ehor(f) =/integraldisplay1
+0/integraldisplay
+MΦ(Vol,Tr(L))¯g(ft,ν)2vol(g),
+one has to find a discrete version of all the involved terms, notably t he mean
+curvature. We will follow [19] to do this. Let V,E,Fdenote the vertices, edges,
+and faces of the triangular mesh, and let star( p) be the set of faces surrounding
+vertexp. Then the discrete mean curvature at vertex pcan be defined as
+Tr(L)(p) =/ba∇dblvector mean curvature /ba∇dbl
+/ba∇dblvector area /ba∇dbl=/ba∇dbl∇p(surface area) /ba∇dbl
+/ba∇dbl∇p(enclosed volume) /ba∇dbl.
+Here∇pstands for a discrete gradient, and
+(vector mean curvature) p=∇p(surface area) =/summationdisplay
+(p,pi)∈E(cotαi+cotβi)(p−pi)
+is the vector mean curvature defined by the cotangent formula. I n this formula, αi
+andβiare the angles opposite the edge ( p,pi) in the two adjacent triangles. For the
+numerical simulation it is advantageous to express this formula in ter ms of scalar
+and cross products instead of the cotangents. Furthermore,
+(vector area) p=∇p(enclosed volume) =/summationdisplay
+f∈Star(p)ν(f).(surface area of f)
+is the vector area at vertex p.
+We discretize the time by
+0 =t1<...<tN+1= 1.
+Then the (N−1)(#V) free variables representing the path of immersions fare
+f(ti,p) with 2 ≤i≤N, p∈V.
+f(0,p) andf(1,p) are not free variables, since they define the fixed boundary
+shapes.ftcan be approximated by either forward increments
+ffw
+t(ti,p) =f(ti+1,p)−f(ti,p)
+ti+1−ti
+or backward increments
+fbw
+t(ti,p) =f(ti,p)−f(ti−1,p)
+ti−ti−1.
+We use a combination of both to make path energy symmetric. (Inst ead of this
+we could have used the central difference quotient. However, minim izing an energy
+functional depending on central differences favors oscillations, s ince they are notM. Bauer, P. Harms, P. Michor 55
+felt by the central differences.) Using the discrete definitions of th e normal vector
+and increments we can calculate f⊥
+tat every vertex pand are now able to write
+the discrete horizontal path energy:
+GΦ
+f(h⊥,k⊥) =/summationdisplay
+p∈V/summationdisplay
+F∋pΦ/parenleftBig
+Vol,Tr(L)(p)/parenrightBig
+.¯g/parenleftbig
+h(p),ν(F)/parenrightbig
+.¯g/parenleftbig
+k(p),ν(F)/parenrightbigarea(starf(p))
+3
+Ehor(f) =N/summationdisplay
+i=1ti+1−ti
+2
+./parenleftBig
+GΦ
+f(ti)/parenleftbig
+ffw
+t(ti),ffw
+t(ti)/parenrightbig
++GΦ
+f(ti+1)/parenleftbig
+fbw
+t(ti+1),fbw
+t(ti+1)/parenrightbig/parenrightBig
+.
+This is not the only way to discretize the energy functional. There ar e several
+ways to distribute the discrete energy on faces, vertices, and ed ges. Depending
+on how this was done, the minimizer converged faster, slower, or ev en not at all.
+However, if the minimizer converged to a smooth solution, the result s were quali-
+tatively the same. This increased our belief in the discretization. How ever, we do
+not guarantee the accuracy of the simulations in this section.
+This energy functional does not depend on the parametrization of the surface
+at each instant of time. So we are free to choose a suitable paramet rization. We
+do this by adding to the energy functional a term penalizing irregular meshes. So
+instead of minimizing horizontal path energy, we minimize the sum of ho rizontal
+path energy and a penalty term. The penalty term measuresthe de viation of angles
+from the “perfect angle” 2 πdivided by the number of surrounding triangles, i.e.,
+N/summationdisplay
+t=2/summationdisplay
+p∈V/summationdisplay
+(p,q,r)∈∆/vextendsingle/vextendsingle/vextendsingle∢(pq,pr)−(perfect angle)/vextendsingle/vextendsingle/vextendsinglek
+, k∈N.
+11.3.Numerical implementation. Discrete path energy depends on a very high
+number of real variables, namely, three times the number of vertic es times one less
+than the number of time steps. In the numerical experiments that we have done,
+there were between 5.000 and 50.000 variables. To solve this problem we used
+the nonlinear solver IPOPT (Interior Point OPTimizer [22]). IPOPT use s a filter
+based line search method to compute the minimum. In this process it n eeds the
+gradientand the Hessian ofthe energy. IPOPTwasinvokedby AMPL (A Modeling
+Language for Mathematical Programming [6]). The advantage of u sing AMPL is
+that it is able to automatically and symbolically calculate the gradient an d Hessian
+needed for the optimizer. All the user has to do is to write a model an d data file for
+AMPL in a quite readable notation. The data file containing the definitio n of the
+combinatorics of the triangle mesh was automatically generated by t he computer
+algebra system Mathematica. As an example, some discretizations o f the sphere
+that we used can be seen in figure 3.56 Almost local metrics on shape space of hypersurfaces in n-space
+Figure 3. Triangulations of a sphere with 320, 500 and 720 tri-
+angles, respectively.
+11.4.Scaling a sphere. In section 9 we studied the set of concentric spheres in n
+dimensions. In dimension three the geodesic equation for the radius simplifies to
+rtt=−r2
+t1
+Φ/bracketleftbig1
+rΦ+∂vΦ4r2π+1
+r2(∂µΦ)+1
+r2(∂3Φ)/bracketrightbig
+.
+This equation is in accordance with the numerical results ob tained by minimizing
+the discrete path energy defined in section 11.2. As will be se en, the numerics
+show that the shortest path connecting two concentric spher es in fact consists of
+spheres with the same center, and that the above differential equation is (at least
+qualitatively) satisfied. Furthermore, in our experiments the optimal paths obtained
+were independent of the initial path used as a starting value for the optimization. In
+all numerical experiments of this section we used 50 timeste ps and a triangulation
+with 320 triangles.
+For conformal metrics of type Φ = Volkand Φ =eVol, the ODE for the radius is
+Φ = Volk: rtt=−r2
+tk+1
+r,
+Φ =eVol: rtt=−r2
+t/parenleftBig1
+r+4rπ/parenrightBig
+.
+Note that the equation for Φ = Vol−1isrtt= 0. These equations have explicit
+solutions:
+Φ = Volk: r=C1/parenleftbig
+(k+2)t−C2/parenrightbig1
+k+2,
+Φ =eVol: r=1
+2π/radicalBig
+log/parenleftbig
+C1t+C2/parenrightbig
+.
+A comparison of the numerical results with the exact analytic solutio ns can be seen
+in Figures 4 and 5. The solid lines are the exact solutions. Note that fo r big radii
+as in Figure 4, the solution for Φ = eVolhas a very steep ascent, is more curved,
+and lies above the solutions for Φ = Vol ,Vol2,Vol3. For small radii, it lies below
+these solutions, as can be seen in figure 5. Note also that when the a scent gets too
+steep, the discrete solution is somewhat inexact as in Figure 4.
+For mean curvature weighted metrics, the differential equation fo r the radius is
+Φ = 1+ATr(L)2k:rtt=−r2
+t/parenleftBig1
+r−2kA22k−1
+r2k+1+A22kr/parenrightBig
+.M. Bauer, P. Harms, P. Michor 57
+/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle
+/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare
+/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond
+/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle
+/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star
+/SolidCircle /SolidSquare /MedSolidDiamond /SolidUpTriangle /SolidUpTrianglePSfrag replacements
+Φ =1
+Vol Φ =eVolΦ = Vol2Φ = Vol3Φ = Volr
+t0
+11250
+1000
+1500
+1750
+2
+250
+0.24
+0.4500
+6
+0.6750
+8
+0.80
+1
+1.5
+0.1
+2
+0.2
+0.3
+0.4
+0.50.60.70.8
+−1
+−125
+−100
+−1.5
+−150
+−2
+−25
+−50
+−0.5
+−75
+Figure 4. Geodesics between concentric spheres of radius 0 .4 to
+0.8 for several conformal metrics. Solid lines are the exact solu-
+tions.
+/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle
+/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare
+/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond
+/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle
+/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star/Star
+/SolidCircle /SolidSquare /MedSolidDiamond /SolidUpTriangle /SolidUpTrianglePSfrag replacements
+Φ =1
+Vol Φ =eVolΦ = Vol2Φ = Vol3Φ = Volr
+t0
+11250
+1000
+1500
+1750
+2
+250
+0.24
+0.4500
+6
+0.6750
+8
+0.80
+1
+0.120.141.5
+0.160.180.1
+2
+0.2
+0.3
+0.4
+0.5
+0.6
+−1
+−125
+−100
+−1.5
+−150
+−2
+−25
+−50
+−0.5
+−75
+Figure 5. Geodesics between concentric spheres of radius 0 .1 to
+0.2 for several conformal metrics. Solid lines are the exact solu-
+tions.
+The numerics for these metrics are shown in figure 6 and figure 7. No te that we
+got convergence to a path consisting of concentric spheres even for theG0-metric
+(A= 0), even though we know from the theory that this is not the shor test path.
+In fact, there are no shortest paths for the G0–metric since it has vanishing geodesic
+distance [14].58 Almost local metrics on shape space of hypersurfaces in n-space
+/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle
+/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare
+/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond
+/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle
+/SolidCircle /SolidSquare /MedSolidDiamond /SolidUpTrianglePSfrag replacements
+k= 2 k= 4 k= 6 k= 8r
+t0
+11200
+1400
+1000
+2
+200
+0.24
+400
+0.46
+600
+0.68
+800
+0.80
+1
+1.21.41.5
+1.61.80.1
+20.2
+0.3
+0.4
+0.5
+0.6
+−1
+−125
+−100
+−1.5
+−150
+−2
+−25
+−50
+−0.5
+−75
+Figure 6. Geodesics between concentric spheres for Φ = 1 +
+0.1Tr(L)kand varying k. Solid lines are the exact solutions.
+/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle
+/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare
+/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond
+/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle
+/SolidCircle /SolidSquare /MedSolidDiamond /SolidUpTrianglePSfrag replacements
+A= 0.01 A= 0.1 A= 0 A= 1r
+t0
+11200
+1400
+1000
+2
+200
+0.24
+400
+0.46
+600
+0.68
+800
+0.80
+11.5
+0.1
+2
+0.20.3
+0.40.5
+0.60.8
+−1
+−125
+−100
+−1.5
+−150
+−2
+−25
+−50
+−0.5
+−75
+Figure 7. Geodesics between concentric spheres for Φ = 1 +
+ATr(L)2and varying A. Solid lines are the exact solutions.
+For the scale-invariant metric, the differential equation is given by
+Φ = Vol−2+ATr(L)2
+Vol: rtt=r2
+t
+r.
+This equation has an explicit analytical solution
+Φ = Vol−2+ATr(L)2
+Vol: r=C1eC2t.M. Bauer, P. Harms, P. Michor 59
+/SolidCirclePSfrag replacements
+Φ = Vol−2+ATr(L)2Vol−1r
+t0
+110
+100
+15
+150
+20
+200
+25
+250
+0.2300
+350
+0.45
+50
+0.6 0.80
+1
+0.120.141.5
+0.160.180.1
+20.2
+0.3
+0.4
+0.5
+0.6
+−1
+−10
+−1.5
+−2
+−20
+−30
+−40
+−50
+−0.5
+Figure 8. Geodesics between concentric spheres for the scale-
+invariant metric.
+Note that this equation, and therefore its solution, is independent ofA. Again, this
+is confirmed by the numerics: see Figure 8.
+11.5.Translation of a sphere. In this section we will study geodesics between a
+sphereandatranslatedsphereforvariousalmostlocalmetricsof the typeΦ = Volk,
+Φ =eVol, and Φ = 1+ ATr(L)2k.
+Depending on the distance (relative to the radius) of the two trans lated spheres,
+different behaviors can be observed.
+High distance:
+•Shrink and grow: For some metrics it is possible to shrink a sphere in finite
+time to zero. For these metrics long translation goes via a shrinking p art
+and growing part. Metrics with this behavior are Φ = Volk, Φ =eVol
+and Φ = 1 + ATr(L)2. This phenomenon is studied in more detail in
+section 11.6; see also Figure 10.
+•Moving an optimal middle shape: For some of the metrics translation of
+a sphere with a certain optimal radius is a geodesic. For these metric s
+geodesics for long translations scale the sphere to the optimal rad ius and
+translate the sphere with the optimal radius. Metrics with this beha vior
+are Φ = 1+ ATr(L)2kfork>1. This behavior is studied in section 11.7.
+Low distance:
+•Geodesics of pure translation (Φ = 1+ ATr(L)2kfork>1; c.f. Figure 13).
+•Geodesics that pass through an ellipsoid, where the longer principal axis is
+in the direction of the translation (conformal metrics, c.f. Figure 9 ).60 Almost local metrics on shape space of hypersurfaces in n-space
+•Geodesics that pass through an ellipsoid, where the principal axis in t he
+direction of the translation is shorter (Φ = 1 + ATr(L)2kfork >1, c.f.
+Figure 13).
+•Geodesics that pass through a cigar–shaped figure (Φ = 1+ ATr(L)2,c.f.
+Figure 12).
+Figure 9. Geodesic between two unit spheres translated by dis-
+tance 1.5 for Φ = Vol. 20 timesteps and a triangulation with 500
+triangles were used. Time progresses from left to right. Boundary
+shapest= 0 andt= 1 are not included.
+11.6.Shrink and grow. In section 9 we showed that it is possible to shrink a
+sphere to zero in finite time for some of the metrics, namely, confor mal metrics
+with Φ = Volkor Φ =eVoland for the GA–metric. For these metrics geodesics of
+long translation will go via a shrinking part and growing part, and almos t all of
+the translation will be done with the shrunken version of the shape. An example
+of such a geodesic can be seen in figure 10.
+Figure 10. Geodesic between two unit spheres translated by dis-
+tance 2 for Φ = eVol. 20 timesteps and a triangulation with 500
+triangles were used. Time progresses from left to right. Boundary
+shapest= 0 andt= 1 are not included.
+We could not determine numerically whether a collapse of the sphere t o a point
+occurs or not. But the more time steps were used, the smaller the e llipsoid in the
+middle turned out. Also, the energy of the geodesic path comes ver y close to the
+energy needed to shrink the sphere to a point and blow it up again. It is remarkable
+thatalmostallofthetranslationisconcentratedatasingletimeste p, independently
+of the number of timesteps that were used. The reason for this be havior is that
+high volumes are penalized so much: In the case of figure 10, eVolis more than 1000
+times smaller in the middle than at the boundary shapes.
+We now want find out under what conditions on the distance and radiu s of the
+boundary spheres of the geodesic this behavior can occur. To do t his, we compare
+the energy needed for a pure translation with the energy needed t o first shrink the
+sphere to almost zero, then move it, and then blow it up again.M. Bauer, P. Harms, P. Michor 61
+The energy needed for a pure translation of a sphere with radius rby distance
+ℓin the direction of a unit vector e1is given by
+E=/integraldisplay1
+0/integraldisplay
+S2Φ(Vol,Tr(L))¯g(ℓ.e1,ν)2vol(g)dt
+= Φ(4r2π,−2
+r)/integraldisplayπ
+0/integraldisplay2π
+0¯g
+ℓ.e1,
+cosϕsinθ
+sinϕsinθ
+cosθ
+
+2
+r2sinθ dϕdθ
+= Φ(4r2π,−2
+r)/integraldisplayπ
+0/integraldisplay2π
+0ℓ2.(cosϕsinθ)2r2sinθdϕdθ= Φ(4r2π,−2
+r).4π
+3ℓ2.r2.
+Any other unit vector can be chosen instead of e1, yielding the same result.
+/SolidSquare /SolidSquare /SolidSquare /SolidSquare /SolidSquarePSfrag replacements
+l
+G1Φ =eVolΦ = Vol2Φ = Vol3Φ = Volrr
+t
+0.0 0.0 0.5 0.50
+1.0 1.01
+1250
+10
+1000
+1500
+1.51750
+2
+250
+0.2
+4
+0.4
+500
+6
+0.6
+750
+8
+0.8
+0.0
+0.5
+0.50
+1.01
+1.5
+0.1
+2
+0.2
+0.3
+0.4
+0.5
+0.6
+0.8
+−1
+−120
+−100
+−1.5
+−2
+−20
+−40
+−0.5
+−60
+−80
+Figure 11. Left: Shrinking a sphere to zero along a geodesic
+path and blowing it up again. Right: Pairs of ℓandrsuch that
+translating a sphere of radius rby distance ℓneeds as much energy
+as shrinking it to zero and blowing it up again. G1stands for the
+GAmetric with A= 1.
+We will now calculate the energy needed for shrinking the sphere, mo ving it, and
+blowing it up again. The energy needed for translating a sphere of ra dius almost
+zero can be neglected. Shrinking and blowing up are done using the so lutions to the
+geodesic equation for the radius from the last section, where one h as to adapt the
+constants to the boundary conditions. For the shrinking part we h aver(0) =rand
+r(1
+2) = 0, and for the growing part we have r(1
+2) = 0,r(1) =r; see Figure 11 (left).
+The energy of the path is
+Φ = Volk:E=/integraldisplay1
+0Volk/integraldisplay
+S2r2
+tvol(g)dt=4k+2πk+1
+(k+2)2r2k+4,
+Φ =eVol:E=/integraldisplay1
+0eVol/integraldisplay
+S2r2
+tvol(g)dt=1
+π(e2πr2−1)2.
+The energy of the two different paths are the same when
+Φ = Volk: ℓ=2√
+3r
+k+2,62 Almost local metrics on shape space of hypersurfaces in n-space
+Φ =eVol: ℓ=√
+3(1−e−2πr2)
+2rπ.
+These curves are shown in Figure 11 (right). We did not derive an ana lytic solution
+for theGA–metric, but for A= 1 one can see the solution curves in figure 11.
+11.7.Moving an optimal shape. In the following we want to determine whether
+pure translation of a sphere is a geodesic. Therefore, let ft=f0+b(t)·e1, where
+f0is a sphere of radius rand whereb(t) is constant on M. Plugging this into the
+geodesic equation from section 5.1 yields an ODE for b(t) and a part which has to
+vanish identically. The latter is given by
+(1) ( ∂vΦ)2
+r4r2π+(∂µΦ)2
+r2+Φ2
+r= 0
+For conformal metrics this equation is satisfied only if Φ = Vol−1. Since this metric
+induces vanishing geodesic distance (see section 8) we are not inter ested in this
+case. For curvature weighted metrics the above equation reads a s
+Φ = 1+ATr(L)2k:4kA(k−1)
+r4k= 1
+Solutions to these equations are given by
+Φ = 1+ATr(L)2k:r= 22k/radicalbig
+A(k−1), k≥1.
+For the most prominent example, the GA–metric, this yields r= 0, and therefore
+translation can never be a geodesic for this type of metric. The num erics have
+shown that the GA–metric yields geodesics that resemble the geodesics of the GA–
+metric for planar curves from [15, section 5.2]. Namely, when the two spheres are
+sufficiently far apart, the geodesic passes through a cigar-like midd le shape, see
+figure 12. As predicted by the theory (see section 8.9), geodesics for very high
+distances tend to have behavior similar to that of Volkmetrics; i.e., the geodesic
+first shrinks the sphere, then moves it, and then blows it up again (c f. section 11.6).
+Figure 12. Middle figure of a geodesic between two unit spheres
+translated by distance 3 for Φ = 1+ ATr(L)2. From left to right:
+A= 0.2,A= 0.4,A= 0.6,A= 0.8. In each of the simulations 20
+timesteps and a triangulation with 720 triangles were used.
+For metrics weighted by higher factors of mean curvature, the ab ove equation
+for the radius has a positive solution. For these metrics, geodesics for translations
+tend to scale the sphere until it has reached the optimal radius and then translate
+it. If the radius is already optimal, the resulting geodesic is a pure tra nslation (see
+figure 13).M. Bauer, P. Harms, P. Michor 63
+Figure 13. Geodesic between two unit spheres translated by dis-
+tance3forΦ = 1+1
+16Tr(L)4(firstrow)andΦ = 1+Tr( L)6(second
+row). In each of the experiments 20 timesteps and a triangulation
+with 720 triangles were used. Time progresses from left to right.
+Boundary shapes t= 0 andt= 1 are not included.
+Figure 14. Geodesic between a sphere and a sphere with a small
+bump for Φ = Vol. 20 timesteps and a triangulation with 500
+triangles were used. Time progresses from left to right.
+If the distance is not high enough a scaling towards the optimal size s till occurs,
+but the middle figure is not a perfect sphere anymore. Instead it is a n ellipsoid as
+in figure 13.
+11.8.Deformation of a shape. We will calculate numerically the geodesic be-
+tween a shape and a deformation of the shape for various almost loc al metrics.
+Small deformations are handled well by all metrics, and they all yield s imilar re-
+sults. An example of a geodesic resulting in a small deformation can be seen in
+figure 14, where a small bump is grown out of a sphere. The energy n eeded for this
+deformation is reasonable compared to the energy needed for a pu re translation.
+Taking the metric with Φ = Vol as an example, growing a bump of size 0.4 a s in
+figure 14 costs about a third of a translation of the sphere by 0.4.
+BiggerdeformationsworkwellwithVolk-metricsandcurvatureweightedmetrics,
+but not with the eVol-metric, which tends to shrink the object and to concentrate
+almost all of the deformation at a single time step. In figure 15, a larg e deformation
+can be seen for the case of Φ = Vol and Φ = eVol. Clearly one can see that the
+eVol-metric concentrates almost all of the deformation in a single time st ep. We
+have met this misbehavior of the eVol-metric already with translations. Again, the
+reason is that eVolis so sensitive to changes in volume. In figure 16 one sees that
+geodesics are smoothed further by higher curvature weights.64 Almost local metrics on shape space of hypersurfaces in n-space
+Figure 15. Large deformation of a shape for Φ = Vol and Φ =
+eVol. 20timestepsandatriangulationwith 500triangleswereused.
+Time progresses from left to right.
+Figure 16. Large deformation of a shape for Φ = 1+0 .1Tr(L)2
+(top), Φ = 1+10Tr( L)2(middle), andΦ = 1+Tr( L)6(bottom). 20
+timesteps and a triangulation with 720 triangles were used. Time
+progresses from left to right.
+References
+[1] M. Bauer, P. Harms, and P. W. Michor. Curvature weighted m etrics on shape space
+of hypersurfaces in n-space. To appear in: Differential Geom etry and its Applications.
+arXiv:1102.0678 .
+[2] M. Bauer, P. Harms, and P. W. Michor. Sobolev metrics on sh ape space of surfaces in n-space.
+To appear in: Journal of Geometric Mechanics. arXiv:1009.3616 .
+[3] Martin Bauer. Almost local metrics on shape space of surfaces . PhD thesis, University of
+Vienna, 2010.
+[4] Arthur L. Besse. Einstein manifolds . Classics in Mathematics. Springer-Verlag, Berlin, 2008.
+[5] V. Cervera, F. Mascar´ o, and P. W. Michor. The action of th e diffeomorphism group on the
+space of immersions. Differential Geom. Appl. , 1(4):391–401, 1991.
+[6] R. Fourer and B. W. Kernighan. AMPL: A Modeling Language for Mathematical Program-
+ming. Duxbury Press, 2002.
+[7] William Fulton. Young tableaux , volume 35 of London Mathematical Society Student Texts .
+Cambridge University Press, 1997.M. Bauer, P. Harms, P. Michor 65
+[8] Philipp Harms. Sobolev metrics on shape space of surfaces . PhD thesis, University of Vienna,
+2010.
+[9] Shoshichi Kobayashi and Katsumi Nomizu. Foundations of differential geometry. Vol. I .
+Wiley Classics Library. John Wiley & Sons Inc., New York, 199 6.
+[10] I. Kol´ avr, P. W. Michor, and J. Slov´ ak. Natural operations in differential geometry . Springer-
+Verlag, Berlin, 1993.
+[11] Andreas Kriegl and Peter W. Michor. The convenient setting of global analysis , volume 53
+ofMathematical Surveys and Monographs . American Mathematical Society, Providence, RI,
+1997.
+[12] A. Mennucci, A. Yezzi, and G. Sundaramoorthi. Properti es of Sobolev-type metrics in the
+space of curves. Interfaces Free Bound. , 10(4):423–445, 2008.
+[13] Peter W. Michor. Topics in differential geometry , volume 93 of Graduate Studies in Mathe-
+matics. American Mathematical Society, Providence, RI, 2008.
+[14] Peter W. Michor and David Mumford. Vanishing geodesic d istance on spaces of submanifolds
+and diffeomorphisms. Doc. Math. , 10:217–245 (electronic), 2005.
+[15] Peter W. Michor and David Mumford. Riemannian geometri es on spaces of plane curves. J.
+Eur. Math. Soc. (JEMS) 8 (2006), 1-48 , 2006.
+[16] Peter W. Michor and David Mumford. An overview of the Rie mannian metrics on spaces of
+curves using the Hamiltonian approach. Appl. Comput. Harmon. Anal. , 23(1):74–113, 2007.
+[17] Marcos Salvai. Geodesic paths of circles in the plane. Revista Matem´ atica Complutense , 2009.
+[18] Jayant Shah. H0-type Riemannian metrics on the space of planar curves. Quart. Appl. Math. ,
+66(1):123–137, 2008.
+[19] John M. Sullivan. Curvatures of smooth and discrete sur faces. In Discrete differential geom-
+etry, volume 38 of Oberwolfach Semin. , pages 175–188. Birkh¨ auser, Basel, 2008.
+[20] D’Arcy Thompson. On Growth and Form . Cambridge University Press, 1942.
+[21] Steven Verpoort. The geometry of the second fundamental form: Curvature prop erties and
+variational aspects . PhD thesis, Katholieke Universiteit Leuven, 2008.
+[22] A. W¨ achter. An Interior Point Algorithm for Large-Scale Nonlinear Opti mization with Ap-
+plications in Process Engineering . PhD thesis, Carnegie Mellon University, 2002.
+[23] A. Yezzi and A. Mennucci. Conformal riemannian metrics in space of curves. EUSIPCO,
+2004.
+[24] A. Yezzi and A. Mennucci. Metrics in the space of curves. arXiv:math/0412454 , December
+2004.
+[25] Anthony Yezzi and Andrea Mennucci. Conformal metrics a nd true ”gradient flows” forcurves.
+InProceedings of the Tenth IEEE International Conference on C omputer Vision , volume 1,
+pages 913–919, Washington, 2005. IEEE Computer Society.66 Almost local metrics on shape space of hypersurfaces in n-spaceThe AMPL model file
+Listing 1. AMPL model file
+1paramA default 1;
+paramk default 1;
+3paramB default 1;
+paraml default 1;
+5paramTimestepsN >1 integer;
+paramVerticesN integer;
+7paramPenaltyFactor default 1;
+paramPenaltyExponent default 2;
+9setVerticesI := 1..VerticesN;
+setVerticesOfEdgesI within {VerticesI,VerticesI };
+11setVerticesOfFacesI within {VerticesI,VerticesI,VerticesI };
+setFacesOfVerticesI {v in VerticesI }within VerticesOfFacesI;
+13setLinkOfVerticesI {VerticesI }within{VerticesOfFacesI,VerticesOfEdgesI, {−1,1}};
+setAdjacentEdgesOfVerticesI {VerticesI }within{VerticesOfEdgesI, {1,−1},VerticesOfEdgesI, {1,−1}};
+15setEdgesOfFacesI {VerticesOfFacesI }within VerticesOfEdgesI;
+setEdgesOfVerticesI {v in VerticesI }:= setof {(f1,f2,f3,e1,e2,o) in LinkOfVerticesI[v] }(e1,e2);
+17
+paramPi default 3.141592653589793;
+19paramPerfectAngle {v in VerticesI }default cos(2 ∗Pi/card(FacesOfVerticesI[v]));
+paramInitialVertices {VerticesI ,1..3 };
+21paramFinalVertices {VerticesI,1..3 };
+23varMiddleVertices {2..TimestepsN,VerticesI,1..3 };
+25varVertices {t in 1..TimestepsN+1,v in VerticesI,i in 1..3 }=
+(ift=1thenInitialVertices[v,i]
+27else ift=TimestepsN+1 thenFinalVertices[v,i]M. Bauer, P. Harms, P. Michor 67elseMiddleVertices[t,v,i ]);
+29
+varVectorOfEdges {t in 1..TimestepsN+1, (v1,v2) in VerticesOfEdgesI,i in 1.. 3}=
+31 Vertices[t,v2,i] −Vertices[t,v1,i ];
+33varLengthOfEdges {t in 1..TimestepsN+1, (v1,v2) in VerticesOfEdgesI }=
+sqrt(VectorOfEdges[t,v1,v2,1]ˆ2+VectorOfEdges[t,v1,v2,2 ]ˆ2+VectorOfEdges[t,v1,v2,3]ˆ2);
+35
+varCrossOfFaces {t in 1..TimestepsN+1,(v1,v2,v3) in VerticesOfFacesI,i in 1..3}=
+37ifi=1then(Vertices[t,v2,2] −Vertices[t,v1,2]) ∗(Vertices[t,v3,3] −Vertices[t,v1,3]) −
+(Vertices[t,v2,3] −Vertices[t,v1,3]) ∗(Vertices[t,v3,2] −Vertices[t,v1,2])
+39else if i=2then−(Vertices[t,v2,1] −Vertices[t,v1,1]) ∗(Vertices[t,v3,3] −Vertices[t,v1,3]) +
+(Vertices[t,v2,3] −Vertices[t,v1,3]) ∗(Vertices[t,v3,1] −Vertices[t,v1,1])
+41else(Vertices[t,v2,1] −Vertices[t,v1,1]) ∗(Vertices[t,v3,2] −Vertices[t,v1,2]) −
+(Vertices[t,v2,2] −Vertices[t,v1,2]) ∗(Vertices[t,v3,1] −Vertices[t,v1,1]) ;
+43
+varNormCrossOfFaces {t in 1..TimestepsN+1,(v1,v2,v3) in VerticesOfFacesI }=
+45sqrt(CrossOfFaces[t,v1,v2,v3,1]ˆ2 + CrossOfFaces[t,v1,v2, v3,2]ˆ2 + CrossOfFaces[t,v1,v2,v3,3]ˆ2);
+47varNuOfFaces {t in 1..TimestepsN+1,(v1,v2,v3) in VerticesOfFacesI,i in 1..3}=
+CrossOfFaces[t,v1,v2,v3,i]/NormCrossOfFaces[t,v1,v2 ,v3];
+49
+varAreaOfFaces {t in 1..TimestepsN+1,(v1,v2,v3) in VerticesOfFacesI }=
+51NormCrossOfFaces[t,v1,v2,v3]/2;
+53varAreaOfVertices {t in 1..TimestepsN+1, v in VerticesI }=
+(sum{(f1,f2,f3) in FacesOfVerticesI[v] }AreaOfFaces[t,f1,f2,f3])/3;
+55
+varVectorAreaOfVertices {t in 1..TimestepsN+1, v in VerticesI, i in 1..3 }=
+57(sum{(v1,v2,v3) in FacesOfVerticesI[v] }CrossOfFaces[t,v1,v2,v3,i ])/6;68 Almost local metrics on shape space of hypersurfaces in n-space59varSquareOfNormOfVectorAreaOfVertices {t in 1..TimestepsN+1, v in VerticesI }=
+VectorAreaOfVertices[t,v,1]ˆ2+VectorAreaOfVertices[ t,v,2]ˆ2+VectorAreaOfVertices[t,v,3]ˆ2;
+61
+varNormOfVectorAreaOfVertices {t in 1..TimestepsN+1, v in VerticesI }=
+63sqrt( SquareOfNormOfVectorAreaOfVertices[t,v]);
+65varVolume{t in 1..TimestepsN+1 }=
+sum{(v1,v2,v3) in VerticesOfFacesI }AreaOfFaces[t,v1,v2,v3];
+67
+varVectorMeanCurvatureOfVertices {t in 1..TimestepsN+1, v in VerticesI, i in 1..3 }=
+69ifi=1then
+sum{(f1,f2,f3,e1,e2,o) in LinkOfVerticesI[v] }o∗
+71 ( VectorOfEdges[t,e1,e2,2] ∗NuOfFaces[t,f1,f2,f3,3] −
+VectorOfEdges[t,e1,e2,3] ∗NuOfFaces[t,f1,f2,f3,2] )
+73else if i=2then
+sum{(f1,f2,f3,e1,e2,o) in LinkOfVerticesI[v] }o∗
+75 (−VectorOfEdges[t,e1,e2,1] ∗NuOfFaces[t,f1,f2,f3,3] +
+VectorOfEdges[t,e1,e2,3] ∗NuOfFaces[t,f1,f2,f3,1] )
+77else
+sum{(f1,f2,f3,e1,e2,o) in LinkOfVerticesI[v] }o∗
+79 ( VectorOfEdges[t,e1,e2,1] ∗NuOfFaces[t,f1,f2,f3,2] −
+VectorOfEdges[t,e1,e2,2] ∗NuOfFaces[t,f1,f2,f3,1] ) ;
+81
+varSquareOfScalarMeanCurvatureOfVertices {t in 1..TimestepsN+1, v in VerticesI }=
+83(VectorMeanCurvatureOfVertices[t,v,1]ˆ2+VectorMeanC urvatureOfVertices[t,v,2]ˆ2
++VectorMeanCurvatureOfVertices[t,v,3]ˆ2)/SquareOfNo rmOfVectorAreaOfVertices[t,v];
+85
+varPhiOfVertices {t in 1..TimestepsN+1,v in VerticesI }=
+871+ A∗(SquareOfScalarMeanCurvatureOfVertices[t,v])ˆk +B ∗(Volume[t])ˆl;M. Bauer, P. Harms, P. Michor 6989varIncrementsOfVertices {t in 1..TimestepsN,v in VerticesI,i in 1..3 }=
+TimestepsN ∗(Vertices[t+1,v,i] −Vertices[t,v,i ]);
+91
+varEnergy = 1/ 12 / TimestepsN ∗(
+93sum{t in 1..TimestepsN,v in VerticesI }
+PhiOfVertices[t,v] ∗sum{(w1,w2,w3) in FacesOfVerticesI[v] }
+95 ( IncrementsOfVertices[t,v,1] ∗CrossOfFaces[t,w1,w2,w3,1] +
+IncrementsOfVertices[t,v,2] ∗CrossOfFaces[t,w1,w2,w3,2] +
+97 IncrementsOfVertices[t,v,3] ∗CrossOfFaces[t,w1,w2,w3,3] )ˆ2 /
+NormCrossOfFaces[t,w1,w2,w3] +
+99sum{t in 1..TimestepsN,v in VerticesI }
+PhiOfVertices[t+1,v] ∗sum{(w1,w2,w3) in FacesOfVerticesI[v] }
+101 ( IncrementsOfVertices[t,v,1] ∗CrossOfFaces[t+1,w1,w2,w3,1] +
+IncrementsOfVertices[t,v,2] ∗CrossOfFaces[t+1,w1,w2,w3,2] +
+103 IncrementsOfVertices[t,v,3] ∗CrossOfFaces[t+1,w1,w2,w3,3] )ˆ2 /
+NormCrossOfFaces[t+1,w1,w2,w3] );
+105
+varPenalty =
+107sum{t in 1..TimestepsN+1, v in VerticesI,(v1,w1,o1,v2,w2,o2) in AdjacentEdgesOfVerticesI[v] }
+abs(
+109 ( VectorOfEdges[t,v1,w1,1] ∗VectorOfEdges[t,v2,w2,1] +
+VectorOfEdges[t,v1,w1,2] ∗VectorOfEdges[t,v2,w2,2] +
+111 VectorOfEdges[t,v1,w1,3] ∗VectorOfEdges[t,v2,w2,3] ) ∗o1∗o2
+/ LengthOfEdges[t,v1,w1] / LengthOfEdges[t,v2,w2]
+113 −PerfectAngle[v]
+)ˆPenaltyExponent;
+115
+minimize f:
+117Energy+Penalty ∗PenaltyFactor;70 Almost local metrics on shape space of hypersurfaces in n-space
+Martin Bauer, Peter W. Michor: Fakult ¨at f¨ur Mathematik, Universit ¨at Wien, Nord-
+bergstrasse 15, A-1090 Wien, Austria.
+E-mail address :Bauer.Martin@univie.ac.at
+E-mail address :Peter.Michor@univie.ac.at
+Philipp Harms: EdLabs, Harvard University, 44 Brattle Street, Cambridge, MA
+02138, USA
+E-mail address :pharms@edlabs.harvard.edu
\ No newline at end of file
diff --git a/1001.0718.txt b/1001.0718.txt
new file mode 100644
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@@ -0,0 +1,779 @@
+arXiv:1001.0718v5  [math.FA]  30 Nov 2010INVARIANT EXPECTATIONS AND VANISHING OF BOUNDED
+COHOMOLOGY FOR EXACT GROUPS
+RONALD G.DOUGLASANDPIOTRW.NOWAK
+Abstract. We study exactness of groups and establish a characterizat ion of ex-
+act groups in terms of the existence of a continuous linear op erator, called an
+invariant expectation, whose properties make it a weak coun terpart of an invari-
+ant mean on a group. We apply this operator to show that exactn ess of a finitely
+generated group Gimplies the vanishing of the bounded cohomology of Gwith
+coefficients inanew class ofmodules, whichare definedusingthe Ho pf algebra
+structure ofℓ1(G).
+1. Introduction
+Exactness is a weak amenability-type property of finitely ge nerated groups. It
+was defined in [12] in terms of properties of the minimal tenso r product of the
+reduced group C∗-algebra. Similar to amenability, exactness has a few equiv alent
+definitions which are each of separate interest in di fferent areas of mathematics.
+In particular, exactness is equivalent to the existence of a topologically amenable
+action of the group on a compact space [10] and to Yu’s propert y A [8, 21]. For
+this reason exactness has many interesting applications in analysis, geometry and
+topology. Most notably, Yu [30] proved that groups with prop erty A satisfy the
+Novikov conjecture.
+Amenablegroupsarepreciselytheoneswhichcarryaninvari ant mean;thatis,a
+functional onℓ∞(G) whichispositive, preserves theidentity and isinvariant under
+thenaturalactionof Gonℓ∞(G). Invariant meansallowforaveragingonamenable
+groups, whichisprecisely whatmakes such groups soconveni ent toworkwith. In
+the case of exact groups there was no parallel characterizat ion and in this article
+weinvestigate theexistence of such acounterpart and someo f itsapplications.
+Coarse-geometric versions of classical notions or results in group theory can
+sometimes be obtained by considering the problem “with coe fficients inℓ∞(G)”,
+where the precise meaning of this phrase varies with the cont ext. For instance,
+the coarse Baum-Connes conjecture for a group Gis the Baum-Connes conjecture
+with coefficients in the algebra ℓ∞(G,K), whereKdenotes the compact operators
+on an infinite-dimensional, separable Hilbert space [29], s ee also [27]. Similarly,
+in coarse geometry the reduced crossed product ℓ∞(G)⋊rGis an analogue of the
+Key words and phrases. exact group; bounded cohomology; Hochschild cohomology; c onvolu-
+tionalgebra; invariant expectation.
+During this research the first author was partially supporte d by NSF grant DMS-0600865. The
+second author was partiallysupported byNSFgrant DMS-0900 874.
+12 RONALDG. DOUGLASANDPIOTR W.NOWAK
+reduced group C∗-algebra of G. It is this point of view that suggests the study of
+the space of operators from a given space into the algebra ℓ∞(G), which, loosely
+speaking, plays the role of a“dual withcoe fficients inℓ∞(G)”.
+LetGbe a finitely generated group. Given a left Banach G-moduleXwe con-
+sider the spaceL(X,ℓ∞(G)) of continuous linear operators from Xtoℓ∞(G). This
+space is naturally a bounded Banach G-module by pre- and post-composing with
+the actions of GonXandℓ∞(G). Additionally, we can identify L(X,ℓ∞(G)) with
+thespaceℓ∞(G,X∗),whichisadualspaceto ℓ1(G,X). Usingthisidentification we
+can naturally equip L(X,ℓ∞(G)) with aweak-* topology.
+An invariant expectation on the group Gis then an operator Mfrom the G-
+moduleL(ℓu(G),ℓ∞(G)) intoℓ∞(G), whereℓu(G) is a certain Banach algebra as-
+sociated to the group, obtained as a completion of the algebr aic crossed product
+ℓ∞(G)⋊algG(theprecise definition isgivenin2.1). Theproperties ofth einvariant
+expectation are similar to properties of invariant means on groups; namely, it is
+positive, unital in an appropriate sense and a limit of eleme nts of a special form.
+The most important property of a mean, invariance, is replac ed here by equivari-
+ance withrespect to the G-actions. Werefer toDefinition 3.7for details.
+Theorem 1.1. A finitely generated group G is exact if and only if there exist s an
+invariant expectation onG.
+The existence of an invariant expectation and its relation t o exactness of groups
+relies on the properties of a certain subspace WofL(L(ℓu(G),ℓ∞(G)),ℓ∞(G)). It
+is the choice of this subspace that allows us to carry out appr oximation arguments
+inasetting suitable for exactness.
+Weapplytheabovecharacterization tocomputetheboundedc ohomologyofex-
+act groups with coe fficients in anew class of Banach modules. The motivation for
+studying bounded cohomology in the context of exactness com es from a question
+of N.Higson, whoasked if there isacohomological character ization of exactness.
+The first such characterization was proved in [3], but in term s of a cohomology
+theoryintroduced inthatarticle. Onenatural direction to investigate iswhether the
+characterization ofamenabilityintermsofboundedcohomo logy, duetoB.E.John-
+son[11],couldbegeneralized tooursetting. Hereweshow,u singinvariant expec-
+tations, that bounded cohomology with coe fficients in a class of modules defined
+below, vanishes for exact groups. Characterizations of exa ctness via vanishing of
+bounded cohomology wereprovedlaterin[1]and[16],indepe ndently, usingsome
+of the ideas presented here.
+The spaceL(X,ℓ∞(G)), in addition to being a G-module, also carries a natural
+structure of anℓ∞(G)-module, given bymultiplying the image of an operator Tby
+an element ofℓ∞(G). It is the existence of this second structure that is essent ial
+in our considerations. A G-submoduleE⊆L(X,ℓ∞(G)), which additionally is an
+ℓ∞(G)-module in this sense, will be called a Hopf G-module, as its two module
+structures aredefinedbytheactionsofaHopf-vonNeumannal gebraℓ∞(G)andits
+predual Banach algebra ℓ1(G). ByH1
+b(G,E) we denote the bounded cohomology
+group ofGin degree 1 withcoe fficients in aBanach G-moduleE.INVARIANT EXPECTATIONS AND BOUNDED COHOMOLOGY 3
+Theorem1.2. LetG beafinitely generated group. IfG isexact then H1
+b(G,E)=0
+forany weak-* closed HopfG-module.
+Bounded cohomology allows the use of dimension shifting tec hniques: given a
+G-moduleMthereisa“shifting module” ΣMsuchthat Hn+1
+b(G,M)=Hn
+b(G,ΣM).
+Anargument duetoN.Monodshowsthatshifting modulesofwea k-*closed Hopf
+G-modules are again weak-* closed Hopf G-modules, which allows to conclude
+that all the higher bounded cohomology groups with coe fficients in dual Hopf G-
+modules also vanish for exact groups.
+The above results, after being circulated in preprint form, were followed by a
+rapid development of the relation between bounded cohomolo gy and exactness.
+The papers [1] and [16] mentioned earlier, as well as [2] and [ 6], contain related
+results.
+Wewouldliketothank NicolasMonodformanyvaluable commen ts, aswellas
+suggesting the name Hopf modules and for allowing us to inclu de Proposition 5.1
+inthis paper. Wealso thank thereferee for helpful comments .
+Contents
+1. Introduction 1
+2. Algebras, duality and topologies 3
+2.1. Theuniform convolution algebra ℓu(G) 4
+2.2.G-duality forℓu(G) andℓ∞(G,C) 5
+2.3. Weaktopologies 5
+3. Exactness and invariant expectations 6
+3.1. Exact groups 6
+3.2. Invariant expectations in L(L(ℓu(G),C),C) 7
+4. Bounded cohomology of exact groups 14
+4.1. Hopf G-modules 14
+5. Concluding remarks 16
+5.1. Dimension reduction and higher cohomology groups. 16
+5.2. Relation between MandW. 17
+References 17
+2. Algebras,dualityandtopologies
+LetGbeadiscretegroupgeneratedbyafiniteset S,whichissymmetric; thatis,
+S=S−1. AllBanachspaceswediscussareover R. Sinceinmostofourarguments
+wewillviewthealgebra ℓ∞(G) asanalgebra of coe fficients, throughout thearticle
+wewill shorten the notation to C=ℓ∞(G). Wedenote by 1 Gthe identity inℓ∞(G)
+and by/bardbl·/bardblCthe usual supremum norm. The algebra Cis equipped with a natural
+G-action,
+(1) ( g∗f)(h)=f(g−1h)4 RONALDG. DOUGLASANDPIOTR W.NOWAK
+forg,h∈Gandf∈C. For afunction f:G×G→Rwewill denote
+fg=f(g,·),
+sothatfgisa real function on G. Theset of those g∈Gfor which fg/nequal0 iscalled
+thesupport of fand denoted supp f.
+2.1.Theuniformconvolution algebra ℓu(G).Let
+Cc(G,C)=/braceleftbigf:G→C: #suppf<∞/bracerightbig,
+wheresupp fdenotes thesupport of f. Bythe above, Cc(G,C)can beviewedasa
+linearsubspaceofthespaceofboundedfunctionson G×G,eachofwhichvanishes
+outside of K×G, for some finite K⊆G.Cc(G,C) is a linear space and we equip
+it withthe norm
+/bardblf/bardblu=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+g∈G|fg|/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleC.
+Thespace Cc(G,C) is,inanatural way,asubspace of/parenleftig/circleplustext
+g∈Gℓ1(G)/parenrightig
+∞,theinfinite
+direct sum of copies of ℓ1(G) with the norm/bardblη/bardbl=supg∈G/bardblη(g)/bardbl1, whereη:G→
+ℓ1(G), and these norms agree on Cc(G,C). We define multiplication on Cc(G,C)
+bythe formula
+(f⋆f′)g=/summationdisplay
+h∈Gfh(h∗f′
+h−1g)
+and an involution f∗
+g=g∗fg−1, which turns Cc(G,C) into the algebraic crossed
+product C⋊algG. It can be easily seen that /bardblf⋆f′/bardblu≤/bardblf/bardblu/bardblf′/bardbluforf,f′∈
+Cc(G,C).
+Definition2.1. Theuniform convolution algebra, denoted ℓu(G),isthecompletion
+ofCc(G,C)withrespect to the norm /bardbl·/bardblu.
+Note that there is a natural isometric inclusion of ℓ1(G) inℓu(G). Indeed, con-
+siderf∈ℓ1(G) and defineξg=f(g)1G.
+For each element g∈Gconsiderδg∈ℓu(G) given by
+(δg)h=/braceleftigg1Gifh=g,
+0 otherwise.
+We haveδgh=δg⋆δhand there is a natural action of Gonℓu(G) by isometries,
+also denoted by⋆, such that
+(2) g⋆ξ=δg⋆ξ,
+forξ∈ℓu(G). Observe also that δe, wheree∈Gis the identity element, is the unit
+inℓu(G).INVARIANT EXPECTATIONS AND BOUNDED COHOMOLOGY 5
+2.2.G-duality forℓu(G)andℓ∞(G,C).Consider the Banach space
+ℓ∞(G,C)=f:G→C: sup
+g∈G/bardblfg/bardblC<∞.
+Wedenote by 1Gthe identity inℓ∞(G,C): (1G)h=1Gfor every h∈G.
+This space is naturally isometrically isomorphic to ℓ∞(G×G); however, the
+above notation has advantages in our setting. The space ℓ∞(G,C) is a left G-
+module withan action given by
+(3) ( g⊙f)h=g∗fg−1h,
+whereg,h∈G. There is a natural inclusion ℓu(G)⊆ℓ∞(G,C) and the two actions
+⋆and⊙agree onℓu(G).
+There exists a natural C-valued pairing between the elements of ℓu(G) and
+ℓ∞(G,C),/an}bracketle{t·,·/an}bracketri}htC:ℓu(G)×ℓ∞(G,C)→Cgiven by
+(4) /an}bracketle{tξ,f/an}bracketri}htC=/summationdisplay
+g∈Gξgfg.
+It iswell-defined since
+|/an}bracketle{tξ,f/an}bracketri}htC|≤/summationdisplay
+g∈G|ξg||fg|≤/bardblf/bardblℓ∞(G,C)/bardblξ/bardblu1G.
+Lemma2.2. Forξ∈ℓu(G)and f∈ℓ∞(G,C)wehave
+/an}bracketle{tg⋆ξ,f/an}bracketri}htC=g∗/an}bracketle{tξ,g−1⊙f/an}bracketri}htC.
+Proof.Wehave
+/an}bracketle{tg⋆ξ,f/an}bracketri}htC(h)=/summationdisplay
+k∈G(g⋆ξk(h))fk(h)
+=/summationdisplay
+k∈Gξg−1k(g−1h)fk(h).
+Onthe other hand,
+/parenleftig
+g∗/an}bracketle{tξ,g−1⊙f/an}bracketri}htC/parenrightig
+(h)=/an}bracketle{tξ,g−1⊙f/an}bracketri}htC(g−1h)
+=/summationdisplay
+k∈Gξk(g−1h)/parenleftig
+(g−1⊙f)k(g−1h)/parenrightig
+=/summationdisplay
+k∈Gξk(g−1h)/parenleftig
+fgk(gg−1h)/parenrightig
+Substituting gk=k′wesee that the twoexpressions areequal. /square
+2.3.Weak topologies. LetXbe a Banach space. One of the main objects of our
+study will be the module L(X,C) of bounded linear maps from XtoC, with its
+natural operator norm, which wedenote by /bardbl·/bardblL.
+If not stated otherwise weconsider Casadual ofℓ1(G) withits natural weak-*
+topology. We will denote weak-* limits in Cbyw∗−lim. The action∗defined in
+(2) ofGonCis weak-*-continuous.6 RONALDG. DOUGLASANDPIOTR W.NOWAK
+Theweak-* topology on L(X,C).The spaceL(X,C) is isometrically isomorphic
+to the spaceℓ∞(G,X∗), where the isomorphism I:L(X,C)→ℓ∞(G,X∗) is given
+by
+(I(T)g)(x)=(T(x))g
+forT∈L(X,C),x∈Xandg∈G. Thus our spaceL(X,C) can be identified as
+the Banach space dual of ℓ1(G,X) and as such it can be naturally equipped with a
+weak-* topology.
+The following description of the weak-* topology on L(X,C) is convenient in
+our setting.
+Proposition 2.3. LetXbe a Banach space and let {Tβ}be a net inL(X,C). The
+following conditions are equivalent:
+(a)C−limβTβ=T,
+(b)w∗−limβTβ(x)=T(x)inCfor every x∈X.
+The weak topology on X.Every elementξ∈Xdefines a map ˆξ:L(X,C)→C
+bythe formula
+ˆξ(T)=T(ξ)
+for every T∈L(X,C). This defines anatural embedding
+i:X→L(L(X,C),C).
+Wedenote thenatural norm on L(L(X,C),C)by/bardbl·/bardblLL. Sincethedual space X∗
+ofXis naturally embedded in L(X,C) by defining T(x)=ϕ(x)1G, whereϕ∈X∗,
+weeasily see that the embedding isisometric.
+Definition2.4. LetXbeaBanachspace. Theweaktopologyon Xistherestriction
+toXof the weak-* topology on L(L(X,C),C).
+This gives thefollowing descrition
+Proposition2.5. Anet{xβ}ofelementsofXconverges weaklyto x ∈Xifandonly
+if for every T∈L(X,C)wehave T (x)=w∗−limβT(xβ).
+The weak topology of Definition 2.4 is the same as the weak topo logy onXin
+the classical sense. For our purposes it su ffices to see that the weak topology in
+thesenseof Definition 2.4isformally stronger thantheweak topology onXinthe
+classical sense.
+3. Exactnessandinvariantexpectations
+3.1.Exact groups. The term exact group originates in the theory of C∗-algebras.
+However, in the last decade many new characterizations were discovered and our
+use of this term is not restricted strictly tothe C∗-algebraic definition.
+Originally exact groups were defined by Kirchberg and Wasser mann, see [12,
+13] in the study of group C∗-algebras. A C∗-algebraAis exact if given any exact
+sequence
+0−→I−→B−→B /I−→0INVARIANT EXPECTATIONS AND BOUNDED COHOMOLOGY 7
+thesequence
+0−→I⊗ minA−→B⊗ minA−→B/I⊗minA−→0
+remains exact. Note that the maximal tensor product always p reserves short ex-
+act sequences in the above sense. Exactness of a C∗-algebra is weaker than its
+nuclearity. Indeed, Aisnuclear ifB⊗minA=B⊗maxAfor anyC∗-algebraB.
+A groupGis called exact if its reduced C∗-algebraC∗
+r(G) is exact in the above
+sense. Werefer to [4,12,13, 28] for details.
+Exactness of a group turned out to be equivalent to property A of Yu due to
+work of Guentner and Kaminker [8] and, subsequently, Ozawa [ 21]. Property A
+wasintroduced in[30]asacondition su fficient toenableonetoembedagroup(or,
+more generally, a metric space) coarsely into a Hilbert spac e. At present there are
+no known examples of groups which embed coarsely into the Hil bert space but do
+not have property A (see, however, [18]). Property A in [30] w as defined in terms
+ofaFølner-typecondition whichhighlights thefactthatit canbeviewedasaweak
+amenability-type property. Werefer to [19, 25, 28] for an in troduction to property
+A.
+In [10] Higson and Roe characterized property A and exactnes s of a group G
+in terms of topologically amenable actions on the Stone- ˇCech compactification of
+G. We will use a version of the characterization from [10] as ou r definition of
+exactness.
+Definition 3.1. Afinitely generated groupG isexact if for every ε>0there exists
+anelementξ∈ℓu(G)such that
+(a)ξis finitely supported; that is, ξg=0for all but finitely many g ∈G,
+(b)ξis anC-valued probability measure; that is, ξg≥0for all g∈G and/summationtext
+g∈Gξg=1G,and
+(c)ξisε-invariant; that is, /bardblξ−s⋆ξ/bardblu≤εfor every generator s ∈S.
+Exactness has numerous consequences in the theory of C∗-algebras, index the-
+ory and geometric group theory. In particular, Yu proved tha t ifGis has property
+A or, equivalently, is exact, then the coarse Baum-Connes co njecture holds for G
+[30]. Thisontheother handimpliestheNovikovconjecture f orG,thezero-in-the-
+spectrum conjecture and has applications to the positive sc alar curvature problem.
+More recently exactness was related to isoperimetric inequ alities on finitely gen-
+erated groups and quantitative invariants like decay of the heat kernel and type of
+asymptotic dimension [17, 20].
+Exactgroupsconstitute averylargeclassofgroups. Mostno tablyitincludesall
+amenable groups, hyperbolic groups (both in [30]), linear g roups [7]. We refer to
+[28] for a more complete list. The task of constructing a grou p which is not exact
+turns out to be a di fficult one. At present only one family of examples is known,
+Gromov’s random groups [9]. The question how to find new examp les of groups
+which would not beexact is still open.
+3.2.Invariant expectations in L(L(ℓu(G),C),C).A Banach spaceXis said to
+beabounded leftBanach G-moduleifthereisahomomorphism Φ:G→L(X,X)8 RONALDG. DOUGLASANDPIOTR W.NOWAK
+such that
+sup
+g∈G/bardblΦ(g)/bardblL(X,X)<∞.
+IfXisaleftG-module, then wewill denote the action of g∈Gbygxforx∈X.
+IfXis a leftG-module thenL(X,C) is a bounded left G-module with the left
+action of Ggiven by pre- and post-composing withthe actions of G:
+(g·T)(x)=g∗/parenleftig
+T(g−1x)/parenrightig
+, (5)
+forT∈L(X,C),x∈Xandg∈G.
+Lemma3.2. Theabove action isweak-* continuous.
+Proof.IfT=C−limβTβ, then
+w∗−lim
+βg·Tβ(x)=w∗−lim
+βg∗(Tβ(g−1x))
+=g∗(w∗−lim
+βTβ(g−1x))
+=g·T(x),
+wherethe second equality follows from weak-* continuity of the action on C./square
+Observethatgiven f∈ℓ∞(G,C)thepairing/an}bracketle{tξ,f/an}bracketri}htCforξ∈ℓu(G)givesnaturally
+anoperator inL(ℓu(G),C). Moreover,
+Lemma3.3./bardblf/bardblℓ∞(G,C)=/bardblf/bardblL. In particular, the space ℓ∞(G,C)is isometrically
+embedded inL(ℓu(G),C).
+Proof.Theestimate/bardblf/bardblL≤/bardblf/bardblℓ∞(G,C)followseasily. Toseetheconverse observe
+that for everyε >0 there is g∈Gsuch that/bardblf/bardblℓ∞(G,C)≤ /bardblfg/bardblC+ε. Then
+/an}bracketle{tδg,f/an}bracketri}htC=fgand the required inequality follows by taking εconverging to 0. /square
+Thefollowinglemmashowsthat ℓ∞(G,C)isalsoa G-submoduleofL(ℓu(G),C).
+Lemma3.4. Theactions·and⊙agree onℓ∞(G,C)⊆L(ℓu(G),C).
+Proof.ByLemma2.2, for any f∈ℓ∞(G,C) andξ∈ℓu(G) wehave
+(g·f)(ξ)=g∗(/an}bracketle{tg−1⋆ξ,f/an}bracketri}htC)=(g⊙f)(ξ).
+Takingξ=δhfor anyhgives the equality. /square
+The action of Gon theG-moduleL(L(ℓu(G),C),C) will now be denoted by •
+todistinguish it from theaction ·onL(ℓu(G),C):
+(g•Ξ)(T)=g∗/parenleftig
+Ξ(g−1·T)/parenrightig
+, (6)
+forΞ∈L(L(ℓu(G),C),C) andT∈L(ℓu(G),C).
+Lemma3.5. Theactions•and⋆agree onℓu(G)⊆L(L(ℓu(G),C),C).INVARIANT EXPECTATIONS AND BOUNDED COHOMOLOGY 9
+Proof.Wehave
+(/hatwideg⋆ξ)(T)=T(g⋆ξ)
+=g∗/parenleftig
+g−1∗(T(g⋆ξ))/parenrightig
+=g∗/parenleftig
+g−1·T(ξ)/parenrightig
+=g∗/parenleftigˆξ(g−1·T)/parenrightig
+=(g•ˆξ)(T),
+for every T∈L(ℓu(G),C),ξ∈ℓu(G). /square
+Consider the space
+W00=/braceleftbigξ∈ℓu(G) : #suppξ<∞and/an}bracketle{tξ,1G/an}bracketri}htC=c1Gfor some c∈R/bracerightbig
+and letW0bethe closure ofW00in the norm topology in ℓu(G).
+Definition 3.6. Define the subspace W ⊆L(L(ℓu(G),C),C)to be the weak-*
+closure ofW00.
+Clearly,Wis a Banach subspace of L(L(ℓu(G),C),C). Moreover, it has a
+natural structure of a G-module.
+The above setup allows us to prove now the main theorem charac terizing ex-
+actness. Amenable groups are known to be characterized by a F ølner and Reiter
+conditions, whichcorrespond toourDefinition3.1ofexactn ess (see[22,23]). An-
+other standard definition of amenability is through the exis tence of an invariant
+mean on the group. The next definition provides a weak version of the invariant
+mean.
+Definition3.7. LetG beafinitely generated group. Aninvariant expectation onG
+isabounded linear operator M :L(ℓu(G),C)→Cwhichsatisfies
+(a)M∈W,
+(b)M(1G)=1G, and
+(c)M isG-invariant; that is, g •M=M for every g∈G.
+The importance of the notion of an invariant expectation is i n its relation to
+exactness ofgroups described inTheorem 1.1,whosestateme nt werecall fromthe
+introduction.
+Theorem 1.1. A finitely generated group G is exact if and only if there exist s an
+invariant expectation onG.
+Proof.Consider a sequence {ξn}whereξnisobtained from the definition of exact-
+ness withε=1
+n. Eachξnis an element ofℓu(G) and as such induces a continuous
+linear map ˆξn∈L(L(ℓu(G),C),C). We consider this last space with the weak-*
+topology described in the previous section. Since, by the Ba nach-Alaoglu theo-
+rem, the unit ball of the space L(L(ℓu(G),C),C) is compact with this topology,
+thesequence{ˆξn}has aconvergent subnet {ˆξβ}and wedefine
+M=C−lim
+βˆξβ,10 RONALDG. DOUGLASANDPIOTR W.NOWAK
+which isequivalent to
+M(T)=w∗−lim
+βˆξβ(T)
+for every T∈L(ℓu(G),C), by Corollary 2.3. We will show that Mis an invariant
+expectation on G.
+Clearly,M∈Wand, in particular, since /an}bracketle{tξβ,1G/an}bracketri}htC=1Gfor everyβ, it follows
+thatM(1G)=1G.
+Bylemmas 3.2and 3.5wehave for T∈L(ℓu(G),C) and any generator s∈S,
+(s•M)(T)=s∗/parenleftig
+M(s−1·T)/parenrightig
+=s∗/parenleftigg
+w∗−lim
+β/parenleftigˆξβ(s−1·T)/parenrightig/parenrightigg
+=w∗−lim
+βs∗/parenleftigˆξβ(s−1·T)/parenrightig
+=w∗−lim
+β(s•ˆξβ)(T)
+=w∗−lim
+β(/hatwides⋆ξ)(T).
+Thus
+(7) (M−s•M)(T)=w∗−lim
+β(ˆξβ−/hatwides⋆ξβ)(T)
+and wehave
+/bardbl(ˆξβ−/hatwides⋆ξβ)(T)/bardblC≤ /bardblˆξβ−/hatwides⋆ξβ/bardblLL/bardblT/bardblL
+≤ /bardblˆξβ−/hatwides⋆ξβ/bardblu/bardblT/bardblL
+≤εβ/bardblT/bardblL.
+forevery n. Sinceεβtendsto0thisimpliesthat theweak-*limitin(7)alsois0fo r
+everyT. Thisproves G-invariance of M.
+Conversely, let Mbe an invariant expectation on G. Since Mis inWwe
+can approximate it in the weak-* topology on the module L(L(ℓu(G),C),C))by
+finitely supported elements of W00. More precisely, there exists a net {ξβ}such
+thatξβ∈W00and
+w∗−lim
+β(ˆξβ−s•ˆξβ)(T)=0 inC
+for every T∈L(ℓu(G),C) ands∈S. Since the actions •and⋆agree onℓu(G),
+this isthe sameas
+(8) w∗−lim
+βT(ξβ−s⋆ξβ)=0
+for every T∈L(ℓu(G),C) ands∈S. Moreover,ξβsatisfy
+/an}bracketle{tξβ,1G/an}bracketri}htC=cβ1G,INVARIANT EXPECTATIONS AND BOUNDED COHOMOLOGY 11
+where the net of real numbers {cβ}converges to 1. By passing to a cofinal subnet,
+which wewill also denote by ξβ,wecan assume that
+/an}bracketle{tξβ,1G/an}bracketri}htC≥1
+21G.
+We will now ensure condition (c) of Definition 3.1 and constru ct a sequenceξ′
+n
+of finitely supported elements in W00with similar properties to those of ξβand
+such that, additionally, /bardblξn−s⋆ξn/bardblutends to 0 uniformly for all s∈S. Consider
+thespace/circleplustext
+s∈Sℓu(G) withthe norm
+/bardblσ/bardbl⊕u=sup
+s∈S/bardblσs/bardblu
+whereσ∈/circleplustext
+s∈Sℓu(G),σ=⊕s∈Sσs.
+For eachβconsider the direct sum
+σβ=⊕s∈S/parenleftig
+ξβ−s⋆ξβ/parenrightig
+.
+From equation (8) we deduce that for each s∈S, the net{(σβ)s}converges in the
+weaktopology onℓu(G). Namely, for each generator s∈S,wehave
+ϕ(ξβ−s⋆ξβ)−→0
+for every linear functional ϕ∈ℓu(G)∗. Since the dual spaces satisfy the equality/parenleftig/circleplustext
+s∈Sℓu(G)/parenrightig∗=/circleplustext
+s∈Sℓu(G)∗, the netσβconverges in the weak topology on/circleplustext
+s∈Sℓu(G). Now Mazur’s lemma applied to the closed convex hull ∆of the{σβ}
+gives that the weak and strong closures of ∆are the same and, in particular, 0 ∈
+∆. Thus we can approximate 0 by finite convex combinations of σβin the norm
+topologyon/circleplustext
+s∈Sℓu(G). Thismeansthatthereexistsasequence {σ′
+n}suchthatfor
+eachn∈Nthe elementσ′
+nis a finite convex combination of the {σβ}and with the
+property thatσ′
+nconverges strongly to 0 in/circleplustext
+s∈Sℓu(G). There is a corresponding
+sequenceξ′
+n∈ℓu(G) such thatσ′
+n=⊕s∈S/parenleftbigξ′
+n−s⋆ξ′
+n/parenrightbigwhich satisfies
+sup
+s∈S/bardblξβ−s⋆ξβ/bardblu−→0.
+Sinceeachξ′
+nis afinite convex combination of the {ξβ}, wehave
+/an}bracketle{tξ′
+n,1G/an}bracketri}htC=/angbracketleftiggk/summationdisplay
+i=1ciξβi,1G/angbracketrightigg
+C
+=k/summationdisplay
+i=1ci/angbracketleftig
+ξβi,1G/angbracketrightig
+C
+≥k/summationdisplay
+i=1ci/parenleftigg1
+21G/parenrightigg
+≥1
+21G,
+whereci≥0 and/summationtextci=1. The elementsξ′
+nare also finitely supported and belong
+toW00. Thusthe sequence ξ′
+nsatisfies conditions (a) and (c) of Definition 3.1.12 RONALDG. DOUGLASANDPIOTR W.NOWAK
+We need to ensure condition (b) from Definition 3.1. To this en d consider the
+sequence{ζn}defined as
+(ζn)g=|(ξ′
+n)g|/summationtext
+h∈G|(ξ′n)h|.
+Thenζn∈W00and wehave
+/an}bracketle{tζn,1G/an}bracketri}htC=1G.
+Since
+/summationdisplay
+g∈G|(ξ′
+n)g|≥/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+g∈G(ξ′
+n)g/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥1
+21G
+weconclude that
+(9)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1
+h∗/parenleftig/summationtext
+g∈G|(ξ′n)g|/parenrightig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleC≤2
+for anyh∈G.
+Itremainstoshowthat ζnsatisfytheconditionsofDefinition3.1Clearly,( ζn)g≥
+0and/summationtext
+g∈G(ζn)g=1Gforeveryn∈N. Itisalsoobviousthat ζnisfinitelysupported
+for every n∈N. Weonly need toverify the approximate invariance.
+Lemma3.8./bardbls⋆ζn−ζn/bardblu≤4/vextenddouble/vextenddouble/vextenddoubles⋆ξ′
+n−ξ′
+n/vextenddouble/vextenddouble/vextenddoubleufor every generator s ∈S.
+Proof.Wehave
+(10)/bardbls⋆ζn−ζn/bardblu=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubles⋆|ξ′
+n|
+s∗/summationtext
+g∈G|(ξ′n)g|−|ξ′
+n|/summationtext
+g∈G|(ξ′n)g|/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleu
+Adding aconnecting term, applying the triangle inequality and (9) weobtain
+≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubles⋆|ξ′
+n|
+s∗/summationtext
+g∈G|(ξ′n)g|−|ξ′
+n|
+s∗/summationtext
+g∈G|(ξ′n)g|+|ξ′
+n|
+s∗/summationtext
+g∈G|(ξ′n)g|−|ξ′
+n|/summationtext
+g∈G|(ξ′n)g|/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleu(11)
+≤2/bardbls⋆|ξ′
+n|−|ξ′
+n|/bardblu+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble|ξ′
+n|
+s∗/summationtext
+g∈G|(ξ′n)g|−|ξ′
+n|/summationtext
+g∈G|(ξ′n)g|/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleu. (12)
+Thesecond summand, after cancellation, can be estimated as follows. Weobserve
+that
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble|ξ′
+n|
+s∗/summationtext
+g∈G|(ξ′n)g|−|ξ′
+n|/summationtext
+g∈G|(ξ′n)g|/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleuINVARIANT EXPECTATIONS AND BOUNDED COHOMOLOGY 13
+=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+g∈G|(ξ′
+n)g|s∗/summationtext
+h∈G|(ξ′
+n)h|−/summationtext
+h∈G|(ξ′
+n)h|
+s∗/parenleftig/summationtext
+g∈G|(ξ′n)g|/parenrightig/parenleftig/summationtext
+g∈G|(ξ′n)g|/parenrightig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleC
+=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationtext
+h∈G|(s⋆ξ′
+n)h|−/summationtext
+h∈G|(ξ′
+n)h|
+s∗/parenleftig/summationtext
+g∈G|(ξ′n)g|/parenrightig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleC
+≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+h∈G|(s⋆ξ′
+n)h|−/summationdisplay
+h∈G|(ξ′
+n)h|/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleC/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1
+s∗/parenleftig/summationtext
+g∈G|(ξ′n)g|/parenrightig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleC
+≤2/vextenddouble/vextenddouble/vextenddoubles⋆|ξ′
+n|−|ξ′
+n|/vextenddouble/vextenddouble/vextenddoubleu,
+wherethe last step follow from the triangle inequality and ( 9). Altogether weget
+/bardbls⋆ζn−ζn/bardblu≤4/vextenddouble/vextenddouble/vextenddoubles⋆|ξ′
+n|−|ξ′
+n|/vextenddouble/vextenddouble/vextenddoubleu≤4/vextenddouble/vextenddouble/vextenddoubles⋆ξ′
+n−ξ′
+n/vextenddouble/vextenddouble/vextenddoubleu,
+wherethe last inequality follows again from the triangle in equality. /square
+ThusthesequenceζnsatisfiesallthreeconditionsofDefinition3.1andthegroup
+Gisexact. /square
+We will denote by Mthe subset ofL(L(ℓu(G),C),C) of expectations on G,
+meaning elements Msatisfying only conditions (a) and (b) of Definition 3.7, and
+byMG⊆Mthe subset of invariant expectations. Webelieve that, in ge neral,MG
+is an infinite set. A natural question in this context is under what conditions the
+invariant expectation on Gisunique?
+Remark 3.9 (Automatic positivity) .The above proof establishes one additional
+property of the invariant expectation Mconstructed in the “only if” part of the
+proof. Namely, the restriction of Mtoℓ∞(G,C) is a positive map. Indeed, if
+f∈ℓ∞(G,C),f≥0then/an}bracketle{tξβ,f/an}bracketri}htC≥0for everyβ. Since
+M(f)=w∗−lim
+β/an}bracketle{tξn,f/an}bracketri}htC
+and weak-* limits in Cpreserve positivity, we have M(f)≥0. However, if we
+start with an Mthat is not positive in this sense, by passing through the “if ” and
+then the “only if” part of the proof we obtain a new invariant e xpectation with the
+positivity property.
+We will require one more lemma about invariant expectations for the proof of
+our mainresults.
+Lemma3.10. LetG beanexact, finitelygenerated groupand M ∈Mbeaweak-*
+limit of a net{ξβ}of elements satisfying conditions of Definition 3.1. Let f′∈C
+and define f∈ℓ∞(G,C)by fg=f′forevery g∈G. Then M (f)=f′.
+Proof.For eachh∈Gand everyβwehave
+/an}bracketle{tξβ,f/an}bracketri}htC=/summationdisplay
+g∈G(ξβ)gfg=f′/summationdisplay
+g∈G(ξβ)g=f′,14 RONALDG. DOUGLASANDPIOTR W.NOWAK
+since/summationtext(ξβ)g=1Gand this property ispreserved bythe weak-* limit in C./square
+4. Boundedcohomologyofexactgroups
+Wewillnowusethefactsestablished intheprevious section s toproveavanish-
+ing result for bounded cohomology of exact groups. Recall th at given a group G
+and a bounded Banach G-moduleE, a bounded cocycle is a map b:G→Esuch
+that supg∈G/bardblb(g)/bardbl<∞and
+b(gh)=gb(h)+b(g)
+for allg,h∈G. Such a cocycle bis called a boundary if there exists an element
+φ∈Esuch that
+b(g)=gφ−φ
+for every g∈G. Then the bounded cohomology in degree 1 of Gwith coefficients
+inEis the defined as
+H1
+b(G,E)=Z1(G,E)/B1(G,E),
+whereZ1(G,E) is the space of all bounded cocycles b:G→EandB1(G,E)⊆
+Z1(G,E) is the subspace of all boundaries b:G→E. ThusH1
+b(G,E)=0 if
+and only if every bounded bounded cocycle b:G→Eis a boundary. See [5,
+26] for details in the context of Banach algebras and [14, 15] in the context of
+locally compact groups. The bounded cohomology groups of Gare canonically
+isomorphic to the Hochschild cohomology groups of the Banac h algebraℓ1(G),
+withthe samecoefficients.
+4.1.HopfG-modules. SinceCisaBanachalgebra, forany XthespaceL(X,C)
+carries anatural structure of a C-module. For a∈CandT∈L(X,C) define
+(aT)(x)=aT(x),
+wherethe multiplication on the right isin C.
+Definition 4.1. A Banach G-module Eis called a Hopf G-module if for some left
+bounded Banach G-module Xit is a subspaceE⊆L(X,C)which is both a G-
+module with respect to the action of G and a C-module with respect to the above
+action of C.
+The intuition behind the notion of Hopf G-modules is that they are “large” sub-
+modulesofL(X,C),asthetwostructures are,looselyspeaking, transverse t oeach
+other. Indeed, consider a G-moduleX. The dual space X∗has an induced G-
+module structure and we consider the two natural inclusions ofX∗intoL(X,C).
+The first one is obtained by mapping a functional φtoφ(x)1G. The second one
+is obtained by mapping φtoφ(x)1e, where 1 eis the Dirac delta function at the
+identity element. Note that neither of these inclusions giv es rise to a structure of
+a Hopf-G-module onX∗. Indeed,X∗is not a C-submodule under the first one X∗
+and it isnot a G-submodule under thesecond one.
+Theorem1.2. LetG beafinitely generated group. IfG isexact then H1
+b(G,E)=0
+forany weak-* closed HopfG-module.INVARIANT EXPECTATIONS AND BOUNDED COHOMOLOGY 15
+Proof.LetXbealeftG-module,E⊆L(X,C)beasinTheorem1.2, b:G→Ebe
+a bounded cocycle and let C=supg∈G/bardblb(g)/bardbl. We will show that bis a boundary.
+Defineanoperator Λ:X→ℓ∞(G,C) by setting
+Λ(x)g=/bracketleftbigb(g)/bracketrightbig(x),
+forx∈X. Wehave
+/bardblb(g)(x)/bardblC≤/bardblb(g)/bardblL/bardblx/bardblX≤C/bardblx/bardblX,
+so the map is well-defined and continuous. The group Gis exact so, by Theorem
+1.1, there exists an invariant expectation MonG. Moreover, Mcan be chosen to
+be a weak-* limit of a net {ξβ}, whereξβ∈W00, as in the proof of Theorem 1.1.
+Sinceℓ∞(G,C)⊆L(ℓu(G),C) wedefineφ:X→Cby
+φ(x)=M(Λ(x)g).
+Theprocess isillustrated bythe following diagram
+Xb(g)✲
+φ✲C
+L(ℓu(G),C)M✲
+Λ
+✲
+Obviously,φ∈L(X,C) and we need to show that φ∈E. By the definition of M,
+for every x∈Xwehave
+φ(x)=w∗−lim
+β/an}bracketle{tξβ,Λ(x)g/an}bracketri}ht
+=w∗−lim
+β/summationdisplay
+g∈G(ξβ)gb(g)(x).
+Inother words,
+φ=C−lim
+βbβ,
+wherebβ=/summationtext
+g∈G(ξβ)gb(g). SinceEis a Hopf G-module, each (ξβ)g∈Cand only
+finitely many of them are non-zero, and b(g)∈E, we deduce that bβbelongs toE.
+SinceEisweak-* closed in L(X,C),φis also anelement of E.
+For anyg∈Gandx∈Xwehave
+(g·φ−φ)(x)=g∗φ(g−1x)−φ(x)
+=g∗/parenleftig
+M(Λ(g−1x)h)/parenrightig
+−M(Λ(x)h).
+Notethatfor f∈ℓ∞(G,C),theinvarianceof Mcanalsobewritteninthefollowing
+way
+g∗(M(fh))=M((g⊙f)h)=M/parenleftig
+g∗fg−1h/parenrightig
+.16 RONALDG. DOUGLASANDPIOTR W.NOWAK
+Thus
+(g·φ−φ)(x)=M/parenleftig
+g∗Λ(g−1x)g−1h/parenrightig
+−M(Λ(x)h)
+and
+g∗Λ(g−1x)g−1h=g∗/parenleftig/parenleftig
+b(g−1h)/parenrightig
+(g−1x)/parenrightig
+=g·b(g−1h)(x)
+=g·/parenleftig
+g−1·b(h)+b(g−1)·h/parenrightig
+(x).
+Forcocycles wehave g·b(g−1)=−b(g). Thusfor every h∈Gwehave
+g∗Λ(g−1x)g−1h=(b(h)−b(g))(x)
+= Λ(x)h−b(g)(x).
+In the above expression, b(g)(x) is independent of h. Hence after applying M, by
+Lemma3.10, wehave M(b(g)(x))=b(g)(x) and
+(g·φ−φ)(x)=M(Λ(x)h)−b(g)(x)−M(Λ(x)h)
+=−b(g)(x).
+Finally, settingΞ=−φwegetb(g)=g·Ξ−Ξ,sothatbisaboundary asrequired.
+/square
+5. Concludingremarks
+5.1.Dimension reduction and higher cohomology groups. In the case when G
+isamenable the fact that H1
+b(G,E∗)=0for every Banach G-moduleEimplies that
+Hn
+b(G,E∗)=0 for alln≥1. The method used to prove this is the dimension re-
+duction formula in bounded cohomology, see for example [11] ,[14, Section 10.3],
+[26, Theorem 2.4.6]. Nicolas Monod communicated to us the fo llowing argument
+showing that asimilar fact istrue in thecase of exactness.
+Proposition5.1 (N.Monod) .IfGisexactthenforeveryn ≥1wehaveHn
+b(G,E)=
+0foreveryweak-*closed HopfG-moduleof L(X,C),whereXisaleftG-module.
+Sketch of proof. Forevery moduleEwehave
+Hn+1
+b(G,E)=Hn
+b(G,ΣE),
+whereΣE=ℓ∞(G,E)/E. Note thatΣEis a dual space, namely it is the dual of
+K, the kernel of the summation map ℓ1(G,E∗)→E∗, whereE∗denotes a given
+predual ofE.
+There is anatural inclusion i:ℓ∞(G,E)→ℓ∞(G,ℓ∞(/tildewideG,X∗)), where /tildewideG=Gand
+the notation allows one to keep track of the di fferent copies of G. Wehave natural
+isometric isomorphisms
+ℓ∞(G,ℓ∞(/tildewideG,X∗))=ℓ∞(G×/tildewideG,X∗)=ℓ∞(/tildewideG,ℓ∞(G,X∗)).
+Thisdescends toacanonical inclusion
+ΣE⊆ℓ∞(/tildewideG,ΣX∗)
+andonecanverifythat, under theresulting identification, themoduleΣEisaHopf
+G-module, withrespect to ℓ∞(/tildewideG), ofℓ∞(/tildewideG,ΣX∗)≃L(K,ℓ∞(/tildewideG))./squareINVARIANT EXPECTATIONS AND BOUNDED COHOMOLOGY 17
+5.2.Relation between MandW.It isalso interesting toinvestigate therelation
+between the set of positive expectations M+(in the sense of remark 3.9) and the
+moduleW. One possibility is that Wis generated byM+in the sense that for
+everyΞ∈Wthere exist M,M′∈M+and constants c,c′∈[0,+∞) such that
+Ξ=cM−c′M′.
+Question 5.2. IsWgenerated byM+in the above sense?
+References
+[1] J. Brodzki, G.A. Niblo, P.W. Nowak, N. Wright,Amenable actions, invariant means and
+bounded cohomology , preprint, arXiv:1004.0295v1
+[2] J. Brodzki, G.A. Niblo, P.W.Nowak, N. Wright,A homological characterization of topo-
+logical amenability , preprint arXiv:1008.4154v1 .
+[3] J.Brodzki, G.A.Niblo, N. Wright,Ona cohomological characterization of Yu’sproperty
+A,preprint, arXiv:1002.5040v2
+[4] N. Brown, N. Ozawa,C∗-algebras and finite-dimensional approximations. Graduate Stud-
+iesinMathematics, 88. AmericanMathematical Society,Pro vidence, RI,2008.
+[5] H.G.Dales,P.Aiena,J.Eschmeier,K.Laursen,G.A.Willis,IntroductiontoBanachalge-
+bras, operators, and harmonic analysis. London Mathematical Society Student Texts, 57.
+Cambridge UniversityPress,Cambridge, 2003.
+[6] R.G.Douglas,P.W.Nowak,Everyfinitely generated group isweakly exact ,preprint.
+[7] E. Guentner, N. Higson, S. Weinberger ,The Novikov conjecture for linear groups. Publ.
+Math. Inst.Hautes ´Etudes Sci.No. 101 (2005), 243–268.
+[8] E.Guentner,J.Kaminker,ExactnessandtheNovikov conjecture , Topology41(2002),no.
+2, 411–418.
+[9] M.Gromov,Randomwalkinrandomgroups. Geom.Funct.Anal.13(2003),no.1,73–146.
+[10] N. Higson, J. Roe,Amenable group actions and the Novikov conjecture. J. Reine Angew.
+Math. 519 (2000), 143–153.
+[11] B.E. Johnson,Cohomology in Banach algebras. Memoirs of the American Mathematical
+Society,No. 127. AmericanMathematical Society, Providen ce, R.I.,1972.
+[12] E. Kirchberg, S. Wassermann ,Operations on continuous bundles of C∗-algebras. Math.
+Ann. 303 (1995), no. 4, 677–697.
+[13] E.Kirchberg,S.Wassermann ,ExactgroupsandcontinuousbundlesofC∗-algebras. Math.
+Ann. 315 (1999), no. 2, 169–203.
+[14] N. Monod,Continuous bounded cohomology of locally compact groups. Lecture Notes in
+Mathematics, 1758. Springer-Verlag,Berlin,2001.
+[15] N. Monod,An invitation to bounded cohomology. International Congress of Mathemati-
+cians. Vol.II, 1183–1211, Eur.Math. Soc., Z¨ urich,2006.
+[16] N.Monod,Anote on topological amenability , preprint arXiv:1004.0199v2
+[17] P.W. Nowak,On exactness and isoperimetric profiles of discrete groups. J. Funct. Anal.
+243 (2007), no. 1, 323–344.
+[18] P.W.Nowak,Coarsely embeddable metric spaces without Property A. J. Funct. Anal. 252
+(2007), no. 1, 126–136.
+[19] P.W. Nowak, G.Yu,What is...Property A? Notices Amer. Math. Soc. 55 (2008), no. 4,
+474-475.
+[20] P.W.Nowak,Isoperimetry of group actions. Adv. Math. 219 (2008), no. 1,1–26.
+[21] N.Ozawa,Amenable actions and exactness for discrete groups , C.R. Acad. Sci.ParisSer.
+IMath., 330 (2000), 691–695.
+[22] A.L.T. Paterson,Amenability. Mathematical Surveys and Monographs, 29. American
+Mathematical Society, Providence, RI,1988.18 RONALDG. DOUGLASANDPIOTR W.NOWAK
+[23] J.-P. Pier,Amenable locally compact groups. Pure and Applied Mathematics (New York).
+AWiley-Interscience Publication. John Wiley& Sons, Inc., New York, 1984.
+[24] J.-P.Pier,AmenableBanachalgebras. PitmanResearchNotesinMathematicsSeries,172.
+Longman Scientific& Technical,Harlow; John Wiley& Sons, In c.,New York, 1988.
+[25] J. Roe,Lectures on coarse geometry , University Lecture Series, 31. American Mathemati-
+cal Society,Providence, RI,2003.
+[26] V. Runde,Lectures on amenability . Lecture Notes in Mathematics, 1774. Springer-Verlag,
+Berlin,2002.
+[27] J.ˇSpakula,Uniform K-homology theory. J.Funct. Anal. 257(2009), no. 1, 88–121.
+[28] R. Willett,Some notes on Property A. Limits of graphs in group theory and computer
+science, 191–281, EPFLPress,Lausanne, 2009.
+[29] G.Yu,Baum-Connesconjecture andcoarsegeometry. K -Theory9(1995),no.3,223–231.
+[30] G. Yu,The coarse Baum-Connes conjecture for spaces which admit a u niform embedding
+intoHilbertspace , Invent. Math. (1) 139 (2000), 201-240.
+Departmentof Mathematics , TexasA&M University, CollegeStation, TX77843
+E-mailaddress :rdouglas@math.tamu.edu, pnowak@math.tamu.edu
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+arXiv:1001.0719v2  [astro-ph.IM]  8 Jan 2010Mon. Not.R. Astron. Soc. 000, 1–5 (2008) Printed 10 April 2018 (MN L ATEXstyle file v2.2)
+Commenton “Bayesianevidence:canwe beatMultiNestusing
+traditionalMCMC methods”,byRutgervanHaasteren
+(arXiv:0911.2150)
+F. Feroz1,⋆M.P. Hobson1andR. Trotta2
+1Astrophysics Group, Cavendish Laboratory, JJ Thomson Ave nue, Cambridge CB3 0HE,UK
+2Astrophysics Group, Imperial College, Blackett Laborato ry, PrinceConsort Road,London SW7 2AZ,UK
+Accepted —. Received —; in original form 10 April 2018
+ABSTRACT
+In arXiv:0911.2150, Rutger van Haasteren seeks to criticiz e the nested sampling algorithm
+for Bayesian data analysis in general and its M ULTINESTimplementation in particular. He
+introduces a new method for evidence evaluation based on the idea of Voronoi tessellation
+and requiring samples from the posterior distribution obta ined through MCMC based meth-
+ods. He compares its accuracy and efficiency with M ULTINEST, concluding that it outper-
+formsM ULTINESTinseveralcases.Thiscomparisoniscompletelyunfairsinc etheproposed
+method can not performthe complete Bayesian data analysis i ncluding posteriorexploration
+and evidence evaluation on its own while M ULTINESTallows one to perform Bayesian data
+analysis end to end. Furthermore, their criticism of nested sampling (and in turn M ULTI-
+NEST)is basedon a few conceptualmisunderstandingsofthe algor ithm.Here we seek to set
+therecordstraight.
+Keywords: methods:dataanalysis–methods:statistical
+1 INTRODUCTION
+In a recent paper (vanHaasteren 2009), Rutger van Haasteren has
+criticized the M ULTINESTalgorithm for Bayesian analysis and
+suggested another method to calculate the Bayesian evidenc e with
+the claim that it significantly outperforms M ULTINESTboth in
+termsofaccuracyandcomputationalaccuracy.Ouraiminthi sshort
+note isto highlight the conceptual misunderstandings and t he false
+premise on whichthe comparison is based.
+The outline of the paper is as follows. In Sec. 2 we give an
+introduction to Bayesian inference and describe the nested sam-
+plingalgorithm and itsimplementation in M ULTINESTpackage in
+Sec. 3. In Sec. 4 we give an account of the conceptual misunder -
+standings in van Haasteren 2009 which are the basis of the att ack
+mounted by the author on M ULTINEST. Finally, our conclusions
+are presented inSec.5.
+2 BAYESIANINFERENCE
+Bayesian inference methods provide a consistent approach b oth to
+the estimation of a set of parameters Θin a model (or hypothe-
+sis)Hfor the data Dand to the evaluation of the relative merits
+of different models for the data (see Trotta (2008) for a revi ew of
+⋆E-mail: f.feroz@mrao.cam.ac.ukBayesianmethodsincosmologyandastrophysics).Bayes’th eorem
+statesthat
+Pr(Θ|D,H) =Pr(D|Θ,H)Pr(Θ|H)
+Pr(D|H), (1)
+wherePr(Θ|D,H)≡P(Θ)is the posterior probability distribu-
+tion of the parameters, Pr(D|Θ,H)≡ L(Θ)is the likelihood,
+Pr(Θ|H)≡π(Θ)is the prior distribution, and Pr(D|H)≡ Zis
+the Bayesianevidence.
+Bayesian evidence is the factor required tonormalise the po s-
+teriorover Θ:
+Z=/integraldisplay
+L(Θ)π(Θ)dDΘ, (2)
+whereDis the dimensionality of the parameter space. Since the
+Bayesian evidence is independent of the parameter values Θ, it is
+usuallyignoredinparameterestimationproblemsandthepo sterior
+inferences are obtained by exploring the un–normalized pos terior
+usingstandard MCMC sampling methods.
+Bayesian parameter estimation has been used quite exten-
+sively in a variety of astronomical applications, includin g gravita-
+tionalwaveastronomy,althoughstandardMCMCmethods,suc has
+the basic Metropolis–Hastings algorithm or the Hamiltonia n sam-
+pling technique (see e.g. Mackay 2003), can experience prob lems
+insamplingefficientlyfromamulti–modalposteriordistri butionor
+one with large (curving) degeneracies between parameters. More-
+over, MCMC methods often require careful tuning of the propo sal
+c/circlecopyrt2008 RAS2F.Feroz,M.P.HobsonandR. Trotta
+(a) (b)
+Figure 1. Cartoon illustrating (a) the posterior of a two dimensional prob-
+lem; and (b) the transformed L(X)function where the prior volumes Xi
+are associated with each likelihood Li.
+distribution to sample efficiently, and testing for converg ence can
+be problematic.
+Inordertoselectbetweentwomodels H0andH1oneneedsto
+compare their respective posterior probabilities given th e observed
+data setD,as follows:
+Pr(H1|D)
+Pr(H0|D)=Pr(D|H1)Pr(H1)
+Pr(D|H0)Pr(H0)=Z1
+Z0Pr(H1)
+Pr(H0),(3)
+wherePr(H1)/Pr(H0)is the prior probability ratio for the two
+models, which can often be set to unity but occasionally requ ires
+furtherconsideration(see,forexample,Feroz et al.2009e ,2008for
+thecaseswherethepriorprobabilityratioshouldnotbeset tounity)
+andtheratiooftheevidencesZ1
+Z0iscalledtheBayesfactorbetween
+the two models. It can be seen from Eq. (3) that the Bayesian ev i-
+denceplaysacentralroleinBayesianmodelselection.Asth eaver-
+age of likelihood over the prior, the evidence automaticall y imple-
+ments Occam’s razor: a simpler theory which agrees well enou gh
+with the empirical evidence is preferred. A more complicate d the-
+orywillonlyhaveahigherevidenceifitissignificantlybet teratex-
+plaining the data than a simpler theory (e.g., Liddle (2004) ; Trotta
+(2007a)).
+The evaluation of the Bayesian evidence involves the multi-
+dimensional integral (Eq. 2) and thus presents a challengin g nu-
+merical task. Standard techniques like thermodynamic inte gration
+´O Ruanaidh &Fitzgerald (1996) are usually extremely comput a-
+tionally expensive, which makes evidence evaluation typic ally at
+least an order of magnitude more costly than parameter estim a-
+tion.Recently,aestimationoftheevidenceusingpopulati onMonte
+Carlo techniques has been successfully implemented in the c os-
+mological context Kilbinger etal. (2009). Some fast approx imate
+methods have been used for evidence evaluation, such as trea ting
+theposterior asamultivariateGaussiancentredatitspeak (see,for
+example, Hobson etal. 2002), but this approximation is clea rly a
+poor one for highly non-Gaussian and multi–modal posterior s. A
+computationally cheap and accurate method is the Savage-Di ckey
+densityratioTrotta(2007a,b);Vardanyan et al.(2009),wh ichhow-
+ever can only be used to compute the Bayes factor between nest ed
+models.Variousalternativeinformationcriteriaformode lselection
+are discussed in Liddle (2004); Liddle (2007), but the evide nce re-
+mains the preferred method.
+3 NESTEDSAMPLINGAND THE MULTINEST
+ALGORITHM
+In this section we briefly review the Nested sampling algorit hm
+and the M ULTINESTimplementation. Further details can be found
+inSkilling(2004); Feroz &Hobson (2008);Feroz et al.(2009 c).3.1 NestedSampling
+Nested sampling Skilling (2004) is a Monte Carlo method targ et-
+ted at the efficient calculation of the evidence, but also pro duces
+posterior inferences as a by-product. It calculates the evi dence by
+transforming the multi–dimensional evidence integral int o a one–
+dimensionalintegralthatiseasytoevaluatenumerically. Thisisac-
+complished by defining the prior volume XasdX=π(Θ)dDΘ,
+sothat
+X(λ) =/integraldisplay
+L(Θ)>λπ(Θ)dDΘ, (4)
+where the integral extends over the region(s) of parameter s pace
+contained within the iso-likelihood contour L(Θ) =λ. The evi-
+dence integral,Eq. (2),canthen be writtenas
+Z=/integraldisplay1
+0L(X)dX, (5)
+whereL(X), the inverse of Eq. (4), is a monotonically decreas-
+ing function of X. Thus, if one can evaluate the likelihoods Li=
+L(Xi),whereXiis a sequence of decreasing values,
+0< XM<···< X2< X1< X0= 1, (6)
+asshownschematicallyinFig.1,theevidencecanbeapproxi mated
+numericallyusing standard quadrature methods as a weighte dsum
+Z=M/summationdisplay
+i=1Liwi, (7)
+where the weights wifor the simple trapezium rule are given by
+wi=1
+2(Xi−1−Xi+1). An example of a posterior in two dimen-
+sions and itsassociated function L(X)isshown inFig.1.
+The summation in Eq. (7) is performed as follows. The iter-
+ation counter is first set to i= 0andN‘active’ (or ‘live’) sam-
+plesaredrawnfromthefullprior π(Θ),sotheinitialprior volume
+isX0= 1. The samples are then sorted in order of their likeli-
+hoodandthesmallest(withlikelihood L0)isremovedfromtheac-
+tive set (hence becoming ‘inactive’) and replaced by a point drawn
+from the prior subject to the constraint that the point has a l ikeli-
+hoodL>L0. The corresponding prior volume contained within
+the iso-likelihood contour associated with the new live poi nt will
+be a random variable given by X1=t1X0, wheret1follows the
+distribution Pr(t) =NtN−1(i.e., the probability distribution for
+the largestof Nsamples drawn uniformlyfrom the interval [0,1]).
+Ateachsubsequent iteration i,theremovalofthelowestlikelihood
+pointLiintheactiveset,thedrawingofareplacementwith L>Li
+and the reduction of the corresponding prior volume Xi=tiXi−1
+are repeated, until the entire prior volume has been travers ed. The
+algorithm thus travels through nested shells of likelihood as the
+priorvolume is reduced. The meanandstandard deviationof logt,
+whichdominates the geometrical exploration, are:
+E[logt] =−1/N, σ[logt] = 1/N. (8)
+Since each value of logtis independent, after iiterations the prior
+volume will shrinkdown such that logXi≈ −(i±√
+i)/N. Thus,
+one takes Xi= exp(−i/N).
+The nested sampling algorithm is terminated when the evi-
+dencehasbeencomputedtoapre-specifiedprecision.Theevi dence
+that could be contributed by the remaining live points is est imated
+as∆Zi=LmaxXi,whereLmaxisthe maximum-likelihood value
+oftheremainingliveppoints,and Xiistheremainingpriorvolume.
+The algorithm terminates when ∆Ziis less than a user-defined
+value (we use 0.5inlog-evidence).
+c/circlecopyrt2008 RAS,MNRAS 000, 1–53
+Oncetheevidence Zisfound,posteriorinferencescanbeeas-
+ily generated using the final live points and the full sequenc e of
+discarded points from the nested sampling process, i.e., th e points
+withthelowestlikelihoodvalueateachiteration iofthealgorithm.
+Eachsuch point issimply assigned the probabilityweight
+pi=Liwi
+Z. (9)
+Thesesamplescanthenbeusedtocalculateinferencesofpos terior
+parameters such as means, standard deviations, covariance s and so
+on, or toconstruct marginalised posterior distributions.
+3.2 The MULTINESTAlgorithm
+The most challenging task in implementing nested sampling i s to
+draw samples from the prior within the hard constraint L>Li
+at each iteration i. The M ULTINESTalgorithm Feroz & Hobson
+(2008); Ferozet al. (2009c) tackles this problem through an ellip-
+soidal rejection sampling scheme. The live point set is encl osed
+within a set of (possibly overlapping) ellipsoids and a new p oint is
+thendrawnuniformlyfromtheregionenclosedbytheseellip soids.
+Theellipsoidaldecompositionofthelivepointsetischose ntomin-
+imize the sum of volumes of the ellipsoids. Theellipsoidal d ecom-
+position is well suited to dealing with posteriors that have curving
+degeneracies,andallowsmodeidentificationinmulti-moda lposte-
+riors.Iftherearesubsetsoftheellipsoidsetthatdonotov erlapwith
+theremainingellipsoids,these areidentifiedasadistinct mode and
+subsequently evolved independently.
+The M ULTINESTalgorithm has proven very useful for
+tackling inference problems in cosmology and particle phys ics
+(see e.g. Sekiguchi et al. 2009; Bridges et al. 2009; Vegetti et al.
+2009; Sollomet al. 2009; AbdusSalam etal. 2009b,a; Feroz et al.
+2009e, 2008, 2009d; Feroz etal. 2008a; Trottaet al. 2008;
+L´ opez-Fogliani et al. 2009; Raklev &White 2009; Trottaet a l.
+2009; Roszkowski et al. 2009) typically showing two orders o f
+magnitude improvement in efficiency over conventional tech -
+niques.Morerecently,ithasbeenshowntoperformwellasas earch
+tool for gravitational wave data analysis (Feroz etal. 2009 b,a).
+4 DETAILEDCRITIQUE OFVAN HAASTEREN(2009)
+In vanHaasteren (2009), the author proposes a method based o n
+Vornoi tessellation to calculate the Bayesian evidence (Eq . 2) as
+follows:
+Z=/summationdisplay
+iLiOi, (10)
+where the index iiterates over the samples (may be obtained
+through an MCMC algorithm) in the parameter space and Oiis
+the area of the Vornoi cell occupied by the ithsample. Although
+this approximation converges to the true evidence value, it can not
+beusedinpracticebecauseofthehighcomputationalcostin volved
+in the calculation of Vornoi tesselation. In order to overco me this
+problem, the author suggests to concentrate on a small subse t of
+samplesFt, around the peakof the distribution, occupying volume
+Vtandexploits the following relationship:
+Vt=/summationdisplay
+Θi∈Ftα
+Li, (11)
+whereαis a proportionality constant to be determined. In order to
+calculate α, the author then suggests to make a Gaussian approx-
+imation around the peak so that the volume Vtcan be calculatedas,
+Vt=rDπD/2
+Γ(1+D
+2)/radicalbig
+|C|, (12)
+whereDis the dimensionality of the problem, Cthe covariance
+matrixand ristheMahalanobis distancefromthepeakofthepoint
+with the lowest likelihood value inside the region Ft. Onceαhas
+been calculated using Eqs.(11) and (12), The evidence can be cal-
+culatedas:
+Z=Nα, (13)
+whereNis the total number samples inside the region Ft.The au-
+thorthenappliesthismethodtoafewtoyproblemsandcompar esit
+with MULTINEST,attempting toshow that it outperforms M ULTI-
+NESTinterms of accuracy as wellas computational efficiency.
+Webelievethatthecomparisonof M ULTINESTtothemethod
+proposed invan Haasteren(2009)is completely unfair, as it makes
+a very strong assumption about the availability of suitable poste-
+riorsamples. Moreover, the criticismofnestedsampling al gorithm
+is due to author’s misunderstanding about the method. We dis cuss
+these indetailnow.
+4.1 NestedSamplingsolves thefullinferenceproblem
+The method proposed in vanHaasteren (2009) and briefly sum-
+marized in the previous section requires the region Ftto be suf-
+ficiently and adequately populated by MCMC samples but does
+not propose an MCMC algorithm to satisfy this condition. The re-
+fore, the proposed method does not provide a complete soluti on
+for Bayesian inference problem, but it actually makes the st rong
+assumption that such a solution has already been implemente d us-
+inganother technique.Formanyproblemsofinterest(e.g.p roblem
+exhibitingstrong, curving degeneracies and/or multi-mod ality) ad-
+equatelysamplingtheparameter space isadifficultproblem which
+often proves to be a stumbling block even for the parameter in -
+ference step. The algorithm proposed by van Haasteren (2009 ) as-
+sumes that this part of the problem has already been solved, w hich
+strongly limits the usefulness of the suggested method if no effi-
+cient wayof gathering posterior samples is put forward.
+The MULTINESTalgorithm, on the other hand, provides a
+means to carry out both parameter exploration and the eviden ce
+evaluation in an efficient and robust way. This has been widel y
+demonstrated by applying it very successfully to various re al in-
+ference problems in astrophysics, cosmology and particle p hysics
+phenomenology (see e.g. Sekiguchi et al. 2009; Bridges et al .
+2009; Vegetti etal. 2009; Sollomet al. 2009; AbdusSalam et a l.
+2009b,a; Ferozet al. 2009e, 2008, 2009d; Feroz et al. 2008a;
+Feroz etal.2009b,a;Trottaet al.2008;L´ opez-Fogliani et al.2009;
+Raklev& White2009).
+The eggbox problem (discussed in Sec. 5.2 of van Haasteren
+2009) is a particular example of what we believe is an unfair c om-
+parison. The author assumes that a suitable trick can be foun d en-
+abling the MCMC algorithm to explore all the modes adequatel y
+so that his proposed method can be applied to this highly mult i–
+modal problem, while M ULTINESTnot only finds all the modes
+without making any assumptions but also calculates the evid ence
+accurately.
+A fair comparison in our opinion would require the specifica-
+tion of another algorithm capable of providing not only the m eans
+to calculate the evidence but also to explore the parameter s pace
+without making too many assumptions. In our view, the compar i-
+c/circlecopyrt2008 RAS,MNRAS 000, 1–54F.Feroz,M.P.HobsonandR. Trotta
+sonbetweenthemethodofvanHaasteren(2009)and M ULTINEST
+isfundamentallyunfair,asthelatterhasamuchwiderappli cability
+and solves the fullinference problem from scratch.
+One of the most attractive features of M ULTINESTis its gen-
+eral applicability for moderately dimensional but highly c omplex
+problems. It is completely application–independent in the sense
+thatthespecificproblembeingtackledentersonlythrought helike-
+lihood computation, and does not change how the live point se t
+is updated. It makes no assumptions about the number and natu re
+of modes. Even if a perfect MCMC sampler were available, the
+method proposed in van Haasteren (2009) although can be quit e
+useful for uni–modal Gaussian problems, would nonetheless fail
+for highly non–Gaussian problems. The Gaussian–shell prob lem
+discussed in Sec. 6.2 of Feroz et al. (2009c) would perhaps be the
+mostobviousexample.Comparing M ULTINESTwiththeproposed
+method on problems ideally suited to the proposed method is a n-
+other reason whywe thinkthe comparison is completelyunfai r.
+4.2 MisconceptionsaboutNested Sampling
+Intheintroduction,theauthorarguesthat M ULTINEST(andconse-
+quently nested sampling) algorithm by design, samples from the
+prior distribution and not the posterior distribution, and conse-
+quently suffers more with the ‘curse of dimensionality’ tha n the
+traditional MCMC based algorithms. This statement ignores the
+fact that nested sampling does not sample from the prior dist ribu-
+tion blindly, it samples from the prior within the hard const raint
+L>Liat each iteration i. As discussed in Sec. 3.1, this hard–
+edge sampling scheme results in exponential decrease in the prior
+volumeXioccupied by the live points at the ithiteration with the
+expected value of Xigiven as,
+< Xi>= exp(−i/N), (14)
+Nbeing the total number of live points. This allows the algori thm
+toreach highly localized regions of the parameter space wit hlarge
+likelihood values in a reasonable number of iterations. We c an not
+see why such a sampling scheme would suffer from the ‘curse of
+dimensionality’ more thanthe traditional MCMC schemes.
+Inour opinion, the information content H,given as follows,
+H=/integraldisplay
+log/parenleftbiggdP
+dX/parenrightbigg
+dX, (15)
+wherePdenotes the posterior, is more important than the di-
+mensionality of the problem. Most of the contribution to the ev-
+idence value usually comes from the iterations around the ma xi-
+mum likelihood point, which occurs in the region with prior v ol-
+umeX≈e−Handtherefore because of theexponential shrinkage
+of the prior volume, one could argue that nested sampling has an
+inherent advantage over MCMC schemes. It should be noted tha t
+nested sampling framework itself leaves it to the user to des ign a
+scheme to sample from the iso–likelihood contour and it is ce r-
+tainlypossible tocome upwithhighlyinefficient schemes, t osam-
+ple a new point with L>Li, suffering severely with the ‘curse of
+dimensionality’.
+In Sec. 4.2, the author claims that nested sampling generate s
+samples from the whole of the parameter space, rather than fr om
+the posterior distribution, and therefore it can never reac h the ef-
+ficiency achieved by the traditional MCMC based methods. Thi s
+statement ignores two key points. First, as discussed earli er in this
+section, nested sampling does not sample from the whole of th e
+priordistribution,butinsteaditsamples fromthehard–ed ge region
+inside the prior whose volume is reduced exponentially with eachiteration.Secondly, thesamplesgeneratedbyanestedsamp lingal-
+gorithm are not all equallyweighted as the ones generated th rough
+an MCMC method are. As discussed in Sec. 3.1, one needs to as-
+sign a probability weight given by Eq. (9) to the point with lo west
+likelihoodvalue ateachiterationof thenestedsamplingal gorithm.
+InFig.3ofvanHaasteren(2009),theauthorshowsascatterp lotof
+40,000samplesobtainedthroughanestedsamplingalgorith m.The
+author does not mention whether he has plotted simply the poi nts
+with the lowest likelihood values at each iteration or the pr obabil-
+ity weights of these points have also been taken into account and
+therefore we are unable tocomment on the accuracy of the figur e.
+4.3 Curseof Dimensionality
+Theauthorrepeatedlyattacksnestedsamplingalgorithmin general
+and MULTINESTin particular, saying that it suffers severely from
+the‘curseofdimensionality’.Wehavealreadydiscussedin thepre-
+vious section that as far as the general nested sampling fram ework
+isconcerned, thisis not true.
+MULTINESTimplements the nested sampling algorithm
+through an ellipsoidal rejection sampling scheme to sample uni-
+formly from the iso–likelihood contour as discussed in Sec. 3.2. It
+is well known that all rejection sampling schemes are highly in-
+efficient for high dimensional problems (see e.g. Mackay 200 3).
+MULTINESTwasdesignedtoworkwithproblemswithmoderately
+high number of dimensions and it has proven highly successfu l to
+dealwithhighlymulti–modalandcomplexproblemsincosmol ogy
+andparticlephysicsphenomenology. Itshouldbenotedthat nested
+samplingnotonlyprovidesawaytoevaluatetheBayesianevi dence
+accuratelybutwithacleveralgorithmtosamplefromthehar dcon-
+straint, it also provides a solution to deal with highly dege nerate
+and multi–modal problems. A very good example is the use of
+MULTINESTingravitational wave astronomy (see e.g. Feroz et al.
+2009b,a)wheretheproblemsareinherentlymulti–modaland sofar
+MULTINESThas mainlybeen usedas a parameter exploration tool
+withgreat deal of success.
+For parameter exploration, M ULTINESTworks with reason-
+able efficiency up to ∼100D beyond which the efficiency drops
+appreciablyasexpected. Accurateevidence evaluationreq uiresad-
+equatesamplingfromthewholeofparameterspaceandevensl ight
+inaccuracies inestimatingtheiso–likelihood regionthro ughtheel-
+lipsoidal decomposition can result in large inaccuracies a nd there-
+fore MULTINESTcan calculate the evidence value accurately with
+reasonableefficiencyforproblemsupto ∼50D.Inordertodemon-
+strate this, we ran M ULTINESTon ann–dimensional ellipsoidal
+Gaussianwithlikelihood1,
+L=n/productdisplay
+i=11√2πσiexp/parenleftbigg
+−θi−0.5
+2σi/parenrightbigg
+, (16)
+with
+σi= 0.001i. (17)
+We set uniform prior U(0,1)for all the parameters so that the an-
+alyticallog-evidence value is 0.0 regardless of the dimensionality
+of the problem. We used 1,000 live points with target efficien cye
+set to 1.0 for the 2D problem and reducing it to 0.01 for the 32D
+1We would have liked to test M ULTINESTon the same toy problem de-
+scribed in Sec. 5.1 of van Haasteren (2009) but we were unable to do so
+as the author did not describe the prior distribution he used and hence we
+chose asimilar butnotexactly the sameproblem.
+c/circlecopyrt2008 RAS,MNRAS 000, 1–55
+n N like log(Z)H
+2 14 ,184 0 .04±0.10 10.00
+4 26 ,033−0.07±0.14 19.60
+8 107 ,876−0.24±0.18 32.40
+16 651 ,345 0 .11±0.24 57.60
+32 14,134,227 0 .40±0.31 96.10
+Table 1. The log–evidence ( log(Z)) and information content ( H) values
+obtained by the M ULTINESTalgorithm when applied to the problem de-
+scribed inEq.(16).Theanalytical log(Z)is0.0regardless ofdimensional-
+ityn.
+problem. A value of e= 0.3was suggested in Ferozet al. (2009c)
+forthe standardCMBdata analysis,but itwas accompanied by the
+caveat that the user should check that the evidence value is c on-
+sistent when eis lowered. We list the number of likelihood eval-
+uations, recovered log(Z)and information content Hin Table 1.
+These results clearly show that M ULTINESTis able to correctly
+evaluate the evidence values even for the 32Dproblem with the
+posterior occupying only e−96.10of thepriorvolume, althoughthe
+efficiency does drop appreciably with the increase in dimens ion-
+ality of the problem. This drop in efficiency is mainly due to t he
+exponential increase inthe information content.
+Weshould alsomentionthatfor aGaussianproblem, wehave
+beenabletocalculatetheevidencevalueaccuratelyupto ∼1000D
+using nested sampling with a Hamiltonian sampling scheme to
+sample from the hard–constraint. This method is still under devel-
+opment andwewouldpresent theresultsinaforthcoming publ ica-
+tion.
+5 CONCLUSIONS
+MULTINESThas proven to be a very useful and powerful tool to
+carryout Bayesianinference forawidevarietyofproblems i ncos-
+mology and particle physics phenomenology as well as for gen -
+eralmoderatelydimensionalinferenceproblems.Whilethe method
+proposedinvanHaasteren2009tocalculatetheBayesianevi dence
+is interesting for sufficiently simple problems, it does not have
+general applicability and relies on the availability of sam ples dis-
+tributed according to the posterior distribution. The comp arison of
+thismethodwithM ULTINESTiscompletelyunfairasM ULTINEST
+providesthemeanstoperformfullBayesiananalysiswithou tmak-
+ing any assumptions about the nature of the problem nor does i t
+rely on the availability of posterior samples, in fact it can be used
+toprovide the posterior samples.
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+arXiv:1001.0720v1  [cond-mat.mes-hall]  5 Jan 2010Edge-induced spinpolarizationintwo-dimensional electr on gas
+P. Bokes1,2,∗and F. Horv´ ath1
+1Department of Physics, Faculty of Electrical Engineering a nd Information Technology,
+Slovak University of Technology, Ilkoviˇ cova 3, 812 19 Brat islava, Slovak Republic
+2European Theoretical Spectroscopical Facility (ETSF, www.etsf.eu )
+(Dated: November 9, 2018)
+We characterize the role of the spin-orbit coupling between electrons and the confining potential of the edge
+innonequilibrium2Dhomogeneous electronicgas. Wederive asimpleanalyticalresultforthemagnitudeofthe
+current induced spin polarization at the edge and prove that it is independent of the details of the confinement
+edge potential and the electronic density within realistic values of the parameters of the considered models.
+While the amplitude of the spin accumulation is comparable t o the experimental values of extrinsic spin Hall
+effect in similar samples, the spatial extent of edge induce d effect is restricted to the distances of the order of
+Fermiwavelength ( ∼10nm).
+PACS numbers: 72.25.-b; 85.75.-d;73.63.Hs
+I. INTRODUCTION
+One of the exciting new discoveries in solid state physics
+in the last few years has been the experimental observation
+of the extrinsic spin Hall effect1–4in GaAs heterostructures.
+The mechanism of this effect relies on the spin-orbit cou-
+pling between the spin of electrons with the perturbing po-
+tential of impurities. Another example of similar coupling
+is the Rashba-Bytchkov interaction in asymmetrically dope d
+GaAsquantumwellscontainingtwo-dimensionalelectronga s
+(2DEG) where the coupled potential comes from the internal
+electrostaticelectricfieldinducedbythestructuralasym metry
+of the well. In modeling both of these situations, the atomic
+potential of the ideal bulk semiconductorenters only throu gh
+the renormalization of the effective mass and the spin-orbi t
+interactionstrength.
+Interestingly, the Rashba-Bytchkov interaction has been
+also suggested as a source of spin Hall-like phenomenology5
+for a clean 2DEG. While the effect has been shown to disap-
+pearinextended2Dsystemswitharbitrarilyweakdisorder6,7,
+it is now known that finite size of the sample in combina-
+tion with Rashba-Bytchkovcoupling results in the transver se
+spincurrentandaccumulationofoppositespindensitiesat the
+edges of the sample8,9. Furthermore, it has been found that
+boundaries might directly influence the spin polarization d ue
+to the Rashba-Bytchovcoupling either in very wide 2D elec-
+tronicsystemswithanedge10–14ornarrowRashbastrips15,16.
+However, the physical origin of the latter is quite differen t
+from the former: whereas the bulk Rashba-Bytchkov effect
+arises from accelerations of electrons in the external elec tric
+field,the latterresult fromthereflectionofelectronsfrom the
+sample’sboundary.
+Motivated by this development,it is natural to explore fur-
+ther alterations of the perfectly periodic bulk potential a nd
+its consequences on the spin polarization in the presence of
+the current. Several authors have considered such a situati on
+in quantum wires with a parabolic confining potential17–20,
+wider strips with parabolic21or abrupt22confinement edge.
+Thesealterationsinthepotentiallandscapecanbealsovie wed
+asthe“impurity”whichleadstononzerospinpolarizationi n-duced by the flow of the electric current. Recently, exper-
+imental evidence of this effect of the in-plane field on spin
+polarizationhasbeenreported23. Furtherenhancementofthis
+kindofspin-polarizationhasbeensuggestedusingatransp ort
+through chaotic quantum dot24or resonant tunable scattering
+center25.
+In our previous work22we have compared the magnitude
+ofthe spin polarizationinducedby the latter mechanismwit h
+the one induced by the Rashba-Bytchkov coupling and scat-
+teringof the edge10. It turnsoutthat fora typical 2DEG with
+an edge, it is about three orders of magnitude larger than the
+onecausedbytheedgeandtheRashba-Bytchkovinteraction.
+Thissubstantialdifferenceismostlyduetothefactthatth epa-
+rameter characterizing spin-orbit coupling appears in sec ond
+order in the case of the Rashba-Bytchov-basedmechanism10
+whereasfortheedge-inducedspin-orbitcouplingit comesa l-
+readyinthefirst order.
+Inthispaperweconsidertheedgeofthe2DEGinSidoped
+GaAs quantum well, a system for which the spin Hall effect
+hasbeen experimentallyobserved. Thisallowsus to estimat e
+the valuesof all the parametersenteringand we can compare
+the importanceof thisedge-effectto the contributionof ot her
+mechanismsleadingtocurrent-inducedspinpolarization. The
+models considered here are reasonably realistic yet simple
+enough to be solved almost analytically. We derive a gen-
+erallyvalidsimpleanalyticalformulaforthe inducednumb er
+of electrons with un-compensated spin per unit length of the
+edge,
+mz=−me
+¯hqeαEj, (1)
+wherejis the electrical current density in the 2DEG, αEis
+theparametergivingthestrengthofthespin-orbitcouplin g26,
+andmeandqeare the effective mass and the charge of elec-
+tronrespectively. Noticeably,this result is independent of the
+density of the 2DEG and the details of the edge potential as
+longastheelectronsareconfinedwithinthe sample.2
+II. THE MODELOFTHE EDGEAND SPIN-ORBIT
+COUPLING
+Let us consider a sample with 2DEG created within a
+GaAs-basedquantumwell. Theelectricalcurrentintheplan e
+of the 2DEG will be driven along the ydirection. We will
+be interested in the physics close to one of the two edges of
+the sample along which the current flows. The coordinate in
+the directionperpendicularto the edgewill be x,x>0 corre-
+spondingtotheregionwherethe2DEGispresent(seeFig.1).
+We will describe the electronic states for electrons in
+the 2DEG within the effective mass approximation, treat-
+ing the electrons as noninteracting quasiparticles. Quali ta-
+tive changesin our results introducedwithin a self-consis tent
+mean-field treatment will be studied afterwards. The length
+scale characterizing the electrons is given by their Fermi
+wavelength. It typically attains values33λF=/radicalbig
+2π/n2D∼
+2.5a∗
+B, wheren2Dis the two-dimensional electronic density
+anda∗
+B=9.79nmis the effectiveBohr radiusin GaAs. Since
+this is about two orders of magnitude larger than the inter-
+atomic distances, we can employ the effective mass theory
+and approximate the form of the confining potential with a
+simplefunctionalform.
+In our work will assume that the confining edge potential,
+V(x), is independent of the coordinate y, directed along the
+edge. Within our analytical derivations we model V(x)as an
+abrupt step of height ΔV,V(x) =Vθ(x) =ΔVθ(−x), where
+θ(x)is the unit step function. The actual value of the step
+is much larger than the Fermi energy of the 2DEG. This is
+frequently used to set the potential step to infinity. Howeve r,
+hereitisessentialthatthestepisfiniteasthewholespin-o rbit
+couplingisnonzeroonlyintheregionofthenonzerogradien t
+of the confining potential. Since this value should be of the
+order of the work function of the electrons in the 2DEG, we
+fix this value to ΔV=2.7eV∼230Ha∗. One of the results of
+our work is the demonstration that the current induced spin
+polarization is rather independent of this value so that we d o
+not need to be concerned with its precise numerical value, as
+longasit islarge( >>EF) butfinitevalue.
+Once we establish the analytical result for this model of
+abrupt potential step, we will also consider more general
+forms: step with a linear slope on a distance d,Vd(x)(used
+in Fig. 1), and a partially self-consistent, density-depen dent
+model with a small triangular-shaped dip, Vn(x), close to the
+edge. In the absence of this dip, the decrease in the density
+tozeroattheedgeinthe2DEGresultsina slightlocaldeple-
+tionof electronsthere. A small negativevalueof thedeptho f
+thedip,−V0,thatisfoundself-consistentlyresultsinacharge
+neutraledgewhichis thephysicallyexpectedsituation.
+The essential part of our model is the spin-orbit (SO) cou-
+pling term in the electrons’ Hamiltonian. In general, sever al
+different types of SO interactions can be similar in magni-
+tude: the Rashba-Bytchkov, the Dresselhaus or the impurity -
+induced SO coupling. However, since they are all small per-
+turbationstotheeffectiveHamiltonian,wecanconsiderth em
+separably and superpose their outcomes in the sense of first
+orderperturbationtheoryin termsofSOparameterappearin g
+in the SO interaction. In our paper we will considera specialcaseoftheimpurity-inducedSOcouplingforthespecialcas e
+whenthe“impurity”istheedgepotentialonly. Ingeneral,t he
+SO contributiontotheHamiltoniantakestheform27
+ˆVSO=αE/vectorσ·/parenleftBig
+/vectork×∇V(x)/parenrightBig
+, (2)
+whereαEcharacterizes the strength of the SO coupling, /vectorσ
+are Pauli matrices, /vectorkis the operator of electron’s momen-
+tum andV(x)is the effective one-electron potential energy.
+Within our study the latter corresponds to various models
+of the edge potential: Vθ(x),Vd(x)orVn(x). The strength
+of the SO coupling in GaAs is known to be approximately
+αE≈5.3˚A2=5.53×10−4a∗
+Bwhich is 106times larger than
+thestrengthoftheSO couplinginvacuum26.
+Withintheframeworkofthismodelitisveryeasytounder-
+stand the origin of the current-inducedspin polarization. The
+gradientofthepotentialenergyisdirectedonlyinthe xdirec-
+tion, the momentum is confined only within the xyplane so
+that the only nonzero component of the spin operator comes
+fromitszcomponentduetothe mixedproductformin Eq.2.
+Specifically,theSOinteractionissimplyaspin-dependent po-
+tential
+ˆVSO=−αEσzqV′(x) (3)
+with a well defined quantumnumberfor the projectionof the
+spin in the zdirection, σz, attaining values ±1, andqis they
+component of the electron’s momentum (Fig. 1). Hence, the
+spin-orbittermisapotentialenergywhich,forstateswith pos-
+itive momentum in the +ydirection (the current), is smaller
+(larger)inthespatial regionofnegativederivativeofthe edge
+potential, V′(x)<0, for electrons with their spin down (up).
+Sincecloseto x=0wehave V′(x)=−ΔVδ(x)<0,weexpect
+thatclosetotheedgewewillfindmajorityofspin-downelec-
+trons. Of course, in equilibriumfor zero total current dens ity
+there will be equal number of electron with q>0 andq<0
+so that in this special case zero total spin polarization wil l be
+observed.
+III. SPINPOLARIZATIONANDTHE SPIN-DEPENDENT
+PHASE-SHIFT
+In our previous work22, we have given estimates for the
+totalinducedspinperunitlengthoftheedgeusingargument s
+based on wavepacket propagation. We considered a single-
+electron wavepacket state, ψq,σ
+k,Δk(t), for a specific momentum
+in theydirection q, spinσ, with momentum kof uncertainty
+Δk,
+ψq,σ
+k,Δk(t) =/integraldisplayk+Δk/2
+k−Δk/2dk√
+2πΔk/parenleftBig
+e−ikx+rq,σ(k)eikxeiqy/parenrightBig
+×e−i(k2/2+q2/2)t, (4)
+whererq,σ(k)isthereflectionamplitudeoftheeigenstatewith
+energyE=(1/2)(k2+q2). For large negativetimes only the
+incoming part (moving from the right in the Fig. 1) of this3
+Δ
+y2l
+Va
+xVso,
+Vso,
+FIG.1: (Coloronline)Theedge’sconfinement potentialofde pthΔV
+leads, via the SO coupling (Eq. 3) to attractive potential ( VSO,↓, in
+green) for electrons with spin down in the vicinity of the edg e. The
+initially compensated spins of electrons approaching the e dge be-
+comes locallyun-compensated due tothedifference inthesc attering
+shiftla.
+wavepacket will give rise to nonzero contributions (the re-
+flectedpartwillbeextremelysmallduetothequicklyoscill at-
+ing factors) and the wavepacket will be approachingthe edge
+(Fig.1, movingtothe left). On theotherhand,forlargeposi -
+tive times onlythe reflected wave will givenonzerocontribu -
+tion. The position and form of the reflected wavepacket will
+depend on the spin of the incoming wavepacket since the re-
+flection amplitude, rq,σ(k)will be differentforthe two possi-
+blespinorientations. Specifically,itisshownintheAppen dix
+A that the reflection amplitude has the form rq,σ(k) =eiθq,σ
+k
+wherethephaseshiftupto3rdorderin αEis
+θq,↑/↓
+k=π+2atank
+κ∓2kαEq+2kκα2
+Eq2
+∓4α3
+Eq3k(ΔV−2
+3k2), (5)
+whereκ=√
+2ΔV−k2. Assuming the width Δkof the
+wavepacket ψq,σ
+k,Δk(t)small, onecaneasily findthat the proba-
+bilitydensityofthereflectedwavepacketswiththephase-s hift
+givenbyEq.5 forlargetimeswill takea form
+|ψq,σ
+k,Δk(t)|2≈1
+πsin2(x(t)Δk/2)
+x(t)2Δk/2(6)
+wherex(t)=x+dθq,σ/dk−kt,i.e. themaximumoftheprob-
+ability is at x=kt−dθq,σ/dk. From this it is evident that
+the electron with spin down will be lagging behind that with
+spin up by a distance 2 la=dθq,↑/dk−dθq,↓/dk(indicated
+for the reflected right-goingwavepackets in Fig. 1). This be -
+havior of the scattering of single electron in the wavepacke t
+canbedirectlyextendedtomany-electronsinviewofthestr o-
+boscopic wavepacket basis22. Namely, the above wavepack-
+ets for times tm=t+2πm/(kΔk),m=0,±1,±2,..., wheret
+is arbitrary moment of physical time, form orthogonal set of
+states which are all occupiedwith electrons. Hence the prod -
+uct of the length 2 latimes the number of electrons per area
+must give an estimate of the spin-polarizationper unit leng thoftheedge,asgiveninourpreviouswork. Herewewillshow
+thatthisphysicallymotivatedestimateisinfactarigorou sre-
+sultthatweformallyproveinthefollowing,directlyusing the
+eigenstatesoftheHamiltonianofthestudiedsystem.
+Thespinpolarization perunitlengthoftheedge isgivenin
+termsofthespin-resolveddensityas
+mz=/integraldisplay+∞
+−∞dx/integraldisplaydq
+2π/integraldisplaydk
+2π(nk,q
+↑(x)−nk,q
+↓(x)),(7)
+where the integration in qandkgoes over all the occupied
+eigenstatesand nk,q
+σ(x)isthecontributiontothedensityatpo-
+sitionx,yfromanoccupiedeigenstate
+nk,q
+σ(x)=/vextendsingle/vextendsingle/vextendsingleeiqy/parenleftBig
+e−ikx+rq,σ(k)eikx/parenrightBig/vextendsingle/vextendsingle/vextendsingle2
+=2+2ℜ/braceleftBig
+ei(2kx+klq,σ)/bracerightBig
+.
+(8)
+Sincethespatialshift lq,σisverysmall(givenbythestrength
+oftheSOinteraction),wecanapproximatetheexpressionfo r
+thespindensitytothefirst order
+nk,q
+σ(x) =nk,q
+σ(x)/vextendsingle/vextendsingle/vextendsingle
+l=0+d
+dlnk,q
+σ(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+l=0lq,σ,(9)
+=nk,q
+σ(x)/vextendsingle/vextendsingle/vextendsingle
+l=0+1
+2d
+dxnk,q
+σ(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+l=0lq,σ(10)
+wherewe haveinterchangedthe differentiationin viewof th e
+functional dependence of the density, Eq. 8 on xandlq,σ.
+Hence,usingEq.7andEq.10weobtainforthespinpolariza-
+tion
+mz=/integraldisplaydq
+2π/integraldisplaydk
+2πnk,q
+↑(x)/vextendsingle/vextendsingle/vextendsinglex=+∞
+x=−∞lq
+a,(11)
+where we have introduced the antisymmetric part of the spa-
+tial shift
+lq
+a=θq,↑−θq,↓
+2k=−2αEq−4α3
+Eq3(ΔV−2
+3k2).(12)
+The density at x=−∞is exponentially small, where as the
+contribution in the bulk of the 2DEG is according to Eq. 8
+equalto 2.
+Including higher order terms in Eq. 10 would result in ap-
+pearance of contributions with higher spatial derivatives of
+the density at x=±∞. There, however, is the density con-
+stant(eitherzerofor x=−∞orthehomogeneous2DEGvalue
+forx= +∞) so that the expression for the spin polarization,
+Eq.11iscorrectalsoforlargevaluesof lq,σin principle.
+To complete the derivation we need to integrate over the
+occupied states. The simplest way is to assume that the non-
+equilibrium distribution is well described by a Fermi spher e
+with the radius (Fermi momentum) kFand the correspond-
+ing Fermi energy EF=k2
+F/2 in 2D, shifted by a drift mo-
+mentumqdin the+ydirection. Such a model corresponds
+to the distribution function maximizing the information en -
+tropyfor a fixed averageenergy,densityand current28,29, and
+is also supported by Boltzmann-equation based analysis30.
+In the Appendix B we discuss the results obtained using the
+non-equilibriummodelwithtwoFermienergies,convention al4
+withinthe meso-andnano-scopicquantumtransport. It give s
+identical result for the first order expansion in terms of αE;
+higher order terms in αEdiffer by a numerical prefactor but
+arenegligiblysmall foranyrealistic situations.
+Using the expression for the asymmetric spatial shift,
+Eq. 12, we find for the inducedspin polarizationlinear in the
+driftmomentum(andhencethe currentdensity)
+mz=/integraldisplaykF
+−kFdq
+2π/integraldisplay√
+2EF−q2
+0dk
+π/bracketleftbig
+lq+qda/bracketrightbig
+=/parenleftbigg
+−αE−3α3
+EΔVEF+2
+3α3
+EE2
+F/parenrightbigg
+qdn2D(13)
+wherekF=√2EFis the Fermi momentum, n2D=EF/πis
+thedensityofthe2DEG.(SeeFig.5fortheexplanationofthe
+integration domain for k,q.) On the other hand, the current
+densityis j=qdn2Dsothat wefindthefinal result
+mz=−αEj−3α3
+EΔVEFj+2
+3α3
+EE2
+Fj+O(α5
+E,j3)(14)
+Inatypicalsituation,allexceptthefirst termofthisexpan -
+sionarenegligiblysmallsothattheresultingspinpolariz ation
+isindependentoftheheightoftheconfiningpotential ΔVand,
+according to the numerical results presented in the followi ng
+section, practically independentof other gentle modificat ions
+ofthemodelsothatthefirsttermshouldbeusefulasageneral
+estimateofthemagnitudeofthecurrentinducedspinpolari za-
+tion in heterostructures based on 2DEG. Expressing the first
+orderresultintheS.I.unitswe obtainthecentralresultof our
+work,stated intheintroductioninEq.1.
+IV. EFFECTSOFTHE FINITESLOPEOF THEEDGE
+ANDTHE PARTIALSELF-CONSISTENCY
+Theaimofthissectionistoexploretherigidityoftheresul t
+inEq.1 withrespectto changesin themodelofthe confining
+potential at the edge. We will first address the dependenceof
+our result on finite slope of the edge potentialon the distanc e
+dusingthepotentialenergy
+Vd(x)=
+
+ΔV x <0
+ΔV−(ΔV/d)x0<x<d
+0 x>d(15)
+Qualitatively we can expect this dependence to be similarly
+weak as the dependence on ΔVobtained in the previous sec-
+tion (Eq.14). The reasonbehindthis is a mutualcancellatio n
+oftwo effects: (1)increasing ΔVincreasesthe strengthofthe
+SO coupling in the Hamiltonian and hence the spin polariza-
+tion, (2) increasing ΔVdecreases the amplitude of the wave-
+functionsintheregionofthenonzerogradientoftheconfini ng
+edgepotentialandhenceitssensitivitytothe SO coupling.
+To characterize this dependence quantitatively we have
+evaluated the expression Eq. 7 numerically, where the spin-
+dependent contribution to the density, nk,q
+σ(x), was obtained
+fromthe eigenfunctionsofthe effective1DHamiltoniancon -
+taining the potential energy Eq. 15 and, according to Eq. 3,fitnumerical
+l[a∗
+B]mz×105
+9 8 7 6 5-5.48
+-5.52
+-5.56
+-5.6n↑(x) +n↓(x)n↑(x)−n↓(x)
+x[a∗
+B]
+(n↑+n↓) [a.u.∗](n↑−n↓)×104[a.u.∗]1.2
+1
+0.8
+0.6
+0.4
+0.2
+0
+10 8 6 4 2 00.4
+0.2
+0
+-0.2
+-0.4
+-0.6
+-0.8
+-1
+FIG. 2: (Color online) The difference in spin densities (red -full)
+and the density (green-dashed) of electrons close to the edg e of the
+sample (located at x=0). The former, similarly to the density, ex-
+hibits Friedel-like oscillations. Its first dominant local minimum (at
+x≈0.2) dominates the contribution tothe nonzero spin-polariza tion
+obtained by integrating the difference in spin densities ov er whole
+xaxis. The inset demonstrates the extrapolation of this inte gration
+with respect to its upper limit, l, using a fit to a functional form
+∼Asin(2kFl+φ)/l.
+from it derived spin-orbit potential energy. The eigenfunc -
+tions were obtained by matching the plane-waves in regions
+x<0 andx>dwith the Airy functions in the region of the
+potentialenergywith linear slope,0 <x<d. The integration
+overkin Eq. 7 could be done analytically, whereas the inte-
+gral over qwas done numerically on the regular mesh with
+Nkpoints. We used model parameters that are typical for
+2DEG created within GaAs quantum wells2, electronic den-
+sityn=0.985correspondingto the Fermi energy EF=3.01,
+the current density j=0.1 and the valueof the spin-orbitpa-
+rameterαE=5.53×10−4.
+Ford=0 we confirm our analytical results for abrupt step
+potential, Eq. 14. The difference between the spin-up and
+spin-down densities, shown in Fig. 2, exhibits Friedel-lik e
+oscillations with the first period having a pronounced nega-
+tive amplitude giving the major contribution to the non-zer o
+spin polarization. The proportion of this amplitude to the
+electron is ∼10−4for the current density j=0.1. Since
+the current density in the experimental situation is typica lly
+j=0.1−0.01 (the amplitude scales linearly with the current
+density for these values of j), this value is comparable to the
+amplitude of the spin polarization in the extrinsic spin Hal l
+effect4. Unfortunately,incontracttotheextrinsicspinHallef-
+fect,thenonzerospin-densityislocatedonlywithinfewFe rmi
+wavelengthsawayfromtheedge.
+Integrating the difference in the spin densities along the
+wholexaxisweobtainthespin-polarizationperunitlengthof
+the sample. The numerical result of this integration natura lly
+dependsontheupperlimitoftheintegration;areliableres ult,
+is easily obtainedextrapolatingthe upper limit to infinity (in-
+set of Fig. 2). Using this methodologywe find the numerical5
+n2D[a.u.∗]V0[Ha∗]
+5 4.54 3.53 2.52 1.51 0.510
+8
+6
+4
+2
+0
+FIG. 3: (Color online) The dependence of the self-consisten t value
+ofthepotentialdip V0ontheelectronicdensityshowsmonotonicbe-
+havior. Whilethe value ofthe dipis comparable tothe Fermie nergy
+(EF=3.01), the considered values do not lead to appearance of the
+bound state at the edge.
+result for the spin polarization mz= (5.53±0.003)×10−5,
+which is in perfect agreement with our analytical expressio n,
+Eq.14.
+Using this extrapolation scheme we can address the
+changesinthespin-polarizationwithparametersofthemod el
+discussed in the following. First, we confirm the observatio n
+that the dependence on the magnitude of the confining edge
+potential comesin the 3rd orderof αE: the value of the slope
+of the polarizationvs. the heightof the confinementedge po-
+tential,ΔVis∂mz/∂(ΔV)≈1.6×10−10whichisclosetothe
+analytical value 1 .5×10−10. The difference comesprimarily
+from the extrapolation of the value of mzwith respect to the
+upperlimitofintegrationin x.
+Considering nonzero values of the smearing length of the
+confining potential, d∈(0,λF/2), we find that the resulting
+spinpolarizationischangingonlyverylittle( ∂mz/∂d∼3.3×
+10−8). Thereasonforthisnegligibledependenceis, similarly
+to the dependence on ΔV, the mutual cancellation of the two
+effectsdiscussedearlier.
+The second modification to the confining potential that we
+consider is partial self-consistency of the edge potential that
+guarantees charge neutrality of the edge of the sample. Pres -
+enceofthe modelconfiningedge(Eq.15) leadsto redistribu-
+tion of the charge density characterized with excess positi ve
+charge(perunitlengthofthesample)
+ΔN=/integraldisplay+∞
+−∞dxΔn(x)=/integraldisplay+∞
+−∞dx(EF/π−n(x)),(16)
+wheren(x)is the electronic density and EF/πis the density
+of the positive background charge. The Fermi energy of the
+electronsguaranteesthat for large xwe haven(x)=EF/πso
+that the system is charge neutral in the bulk even through the
+total charge at the edge is not necessarily zero. The behav-
+ior of the density at the edge of a 2D sample is in contrast
+with the typicalsituationfor theedgeof 3D metalswherethe
+work function and the Fermi energy, mutually comparable in
+value, lead to leakage of the electronic density into vacuum
+andtherebytoanaccessofnegativecharge31. Howeveratthe
+edgeofa2DEG,theFermienergy,typicallyfewmeVismuch
+smallerthantheworkfunctionwhichisoforderofseveraleV
+andthesituationisreversed.n2D[a.u.∗]mz×105[a.u.∗]
+4.5 3.5 2.5 1.5 0.5-5.54
+-5.56
+-5.58partial SCFno SCF
+x[a∗
+B](n↑−n↓)×104[a.u.∗]
+10 8 6 4 2 01
+0.5
+0
+-0.5
+-1
+-1.5
+-2
+FIG. 4: (Color online) The difference in the spin densities f or non-
+interacting (red-full) and the partially self-consistent (green-dashed)
+electrons close to the edge of the sample (located at x=0). The
+spatially resolved spin density difference exhibits prono unced dif-
+ferences due to the selfconsistency, the most important eff ect being
+the shift of the curve closer to the edge. However, this resul ts only
+in weak enhancement of the total spin polarization, mz, shown in
+the inset for an interval of electronic densities, due tocou nter-acting
+spin-orbit coupling induced by the self-consistency corre ction in the
+regionx∈(0,λF/2).
+Theoverallchargeoftheedgecanbeneutralizedbya sim-
+plemodelpotential
+Vn(x)=
+
+ΔV x <0
+−V0+(2V0/λF)x0<x<λF/2
+0 x>λF/2,(17)
+whichonadistance λF/2exhibitssmallpotentialdiptoattract
+more electrons. The magnitude of the dip is obtained self-
+consistently to achieve zero total charge at the edge, i.e. t he
+condition ΔN=0. Whileinprinciplethisformofthepotential
+may support new bound states (edge states) for sufficiently
+largeV0, we have checked that for the here, self-consistently
+foundvaluesof V0nosuchstatesexist. Thedependenceofthe
+self-consistentvalueof V0onthe typicalvaluesofthedensity
+ofthe2DEGisshownin Fig.3.
+Includingthe SO interaction, Eq. 3, on top of the potential
+energyVn(x), we obtain aditional repulsive SO-induced con-
+tributionforthespin-downelectronsduetothelinearlyri sing
+potential of the dip for 0 <x<λF/2. This is in competition
+with the character of the edge-confinement potential prefer -
+ringthespin-downelectrons. However,thedipinthepotent ial
+energyalsotendstomovetheelectronsclosertotheconfinin g
+edge and, by increasing the density in this region, effectiv ely
+enhancesthe effect of the SO couplingthere. While the form
+of the spin-density does change significantly due to all thes e
+mechanisms(seeFig.4),theoverallspinpolarizationperu nit
+lengthremainsessentiallyunalteredoverawiderangeofel ec-
+tronicdensities, shownin the inset of Fig. 4. Thisresult on ce
+againconfirmstherigidityoftheresultgivenbyEq.1.6
+V. CONCLUSIONS
+The confining potential close to the edge of a 2D electron
+gastogether with a nonzerocurrentalong this edge inducesa
+nonzerospinpolarizationthat islocalizedwithina few Fer mi
+wavelengths from the edge. To characterize this effect quan -
+titatively we have derived a simple analytical formula for t he
+spinpolarizationperunitlengthoftheedge. Interestingl y,the
+spinpolarizationisindependentoftheheightoftheconfini ng
+potential as well as the electronic density of the 2DEG. Fur-
+thermore, using numerical calculations we have also showed
+that this result is independenton otherpossible modificati ons
+oftheshapeoftheconfiningpotential: thespatialextentof the
+confiningpotential and the partial selfconsistency of the c on-
+finingpotentialwithrespecttochargeneutralityoftheedg e.
+Acknowledgments
+The authors wish to acknowledgefruitful discussions with
+J.T´ obikandL.Kiˇ c´ ınov´ a. P.B.wouldliketothankRexGod by
+for many stimulating discussions. This research has been
+supported by the Slovak grant agency VEGA (project No.
+1/0452/09) and the NANOQUANTA EU Network of Excel-
+lence(NMP4-CT-2004-500198).
+VI. APPENDIXA
+The calculation of the spin-dependent phase-shift follows
+the usual textbook treatment of the wave-function matching
+methodin 1D problems. Forthe separated x−dependentfac-
+tor of the eigenstate we need to solve the 1D Schr¨ odinger
+equation
+/parenleftbigg
+−1
+2d2
+dx2+ΔVθ(−x)±αEqΔVδ(x)/parenrightbigg
+φq,σ
+e(x)=eφq,σ
+e(x),
+(18)
+wherethe‘ +’signisfor σ=↑spinandthe’ −’signfor σ=↓
+spin. We are only interested in the solution well below the
+vacuum,i.e. e<ΔVforwhichwedemandasymptoticforms
+φq,σ
+e(x)=tq,σ(κ)eκx,x<0,κ=/radicalbig
+2(ΔV−e)(19)
+φq,σ
+e(x)=e−ikx+rq,σ(k)eikx,x>0,k=√
+2e(20)
+The reflection amplitude ( rq,σ(k)) and the coefficient tq,σ(κ)
+arethenfoundfromthecontinuityof φq,σ
+e(x)anditsderivative
+atx=0with theresult
+rq,σ(k) =−±2αEqΔV+κ+ik
+±2αEqΔV+κ−ik(21)
+tq,σ(κ) =−2ik
+±2αEqΔV+κ+ik. (22)
+The sought phase shift is obtained from the reflection ampli-
+tuderq,σ
+e=|rq,σ(k)|eiθq,σ
+k,
+θq,σ
+k=π+2atank
+±2αEqΔV+κ. (23)/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
+/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 kFR
+kx kxyk yk
+/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
+/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
+kFkFL
+qd
+FIG. 5: The shifted Fermi sphere (left) and the two-Fermi rad ii
+(right)occupations in2DEG.Theoccupiedstatesareshowna sfilled
+areas, have both identical current density and electronic d ensity but
+differ inhigher moments of k.
+Expandingthelastresultinpowersof αEto3rdoderweobtain
+theexpressionEq.5.
+VII. APPENDIXB
+We have stated that the result for the spin-polarization,
+Eq. 14, is independentof the considered non-equilibriumoc -
+cupationsof electronicstates to the first orderin the the sp in-
+orbit coupling αE. This is true as long as the current density
+in the system is the same for all of these considered occupa-
+tions. Let us consider two simple models of occupations: (1)
+the“two-Fermiradiimodel”(2FR),frequentlyusedwithint he
+coherent transport with the occupancies dictated by the lef t
+andrightmacroscopicelectrodeswithFermimomenta kR
+Fand
+kL
+Frespectively32attached to the sample at its ends (Fig. 5,
+right), and (2) the shifted Fermi distribution function (Sh F),
+usedwithinthemaintext(Fig.5,left)Fromthisoneimmedi-
+ately sees that the contributionto the spin polarization th at is
+linearinαEinEq.11,andthereforealsolinearinthemomen-
+tumky(=q),willbeindependentoftheparticularformofthe
+occupations as long as the zero-th and the first moments are
+identical.
+Apartfromthis,the2FRandShFoccupationsrepresenttwo
+differentsituations: theformerissuitedforveryshortba llistic
+systems and it directly facilitates interpretation of the r esults
+intermsofthedifferenceinelectro-chemicalpotentialso fthe
+electrodes, Δµ= (1/2)/parenleftbig
+(kL
+F)2−(kR
+F)2/parenrightbig
+. The latter is related
+tothecurrentdensityinsidethesamplethrough
+j=/integraldisplaydkxdky
+4π2n(kx,ky)ky=1
+3π2/parenleftbig
+(kR
+F)3−(kL
+F)3/parenrightbig
+(24)
+≈1
+π2/radicalbig
+2EFΔµ, (25)
+where the occupation factor, n(kx,ky) =2 (spin degeneracy
+in the unperturbed system) for the occupied states shown in
+Fig. 5 and n(kx,ky)=0 otherwise. Eq. 25 givesthe linearex-
+pansionintheappliedbias Δµ. Nosuchsimpleconnectionto
+the applied bias voltage can be made for the ShF occupation.
+On the other hand, ShF is suitable for longer samples, where
+electrons’ scattering leads to partial equilibration of th e dis-
+tribution function30. This is then characterized with the drift7
+momentum, qd, whichgivesthecurrentdensity
+j=n2Dqd=EF
+πqd. (26)
+Since the samples we consider are typically longer than the
+electron’s coherence length, we have used the ShF occupa-
+tionswithin the main text. For completenesswe givealso the
+resultsforthe 2FRcase. Thespinpolarizationperunitleng th
+ofthesampletothirdorderin αEis
+mz=−αE
+π2/radicalbig
+2EFΔµ−2α3
+EΔV
+3π2E3/2
+FΔµ−4α3
+E
+45π2E5/2
+FΔµ
+(27)whichafterexpressingtheappliedbiasintermsofthecurre nt
+densityresultsin
+mz=−αEj−2α3
+EΔV
+3EFj+4α3
+E
+45E2
+Fj.(28)
+ComparingthelastequationwithEq.14weseethatwhilethe
+first order term is identical for both occupations, the highe r
+ordersdifferbyanumericalprefactor. Bothofthesearelar ger
+inthecaseofpartiallyequilibratedshiftedFermi-likeoc cupa-
+tions.
+∗Electronic address: peter.bokes@stuba.sk
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+(2005).
+33We will use effective atomic units; the distances are measur ed
+in multiplies of the effective Bohr radius, a∗
+B=εr
+mef=9.79nm,
+energy in the effective Hartrees, Ha∗=mef
+ε2rHa=11.9meV, both
+numerical values are given for GaAs where m∗e=0.067meand
+εr=12.4 are the effective mass of the electron and the relative
+permittivity.
\ No newline at end of file
diff --git a/1001.0722.txt b/1001.0722.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d646d5f29bbbb7718f5690f7d35d1221b0ede281
--- /dev/null
+++ b/1001.0722.txt
@@ -0,0 +1,911 @@
+arXiv:1001.0722v1  [math-ph]  5 Jan 2010Symmetry Classes
+January 5, 2010
+Martin R. Zirnbauer
+Institut f¨ ur Theoretische Physik, Universit¨ at zu K¨ oln,
+Z¨ ulpicher Straße 77, 50937 K¨ oln, Germany
+Abstract
+Physical systems exhibiting stochastic or chaotic behavio r are often amenable
+to treatment by random matrix models. In deciding on a good ch oice of model,
+random matrix physics is constrained and guided by symmetry considerations.
+The notion of ‘symmetry class’ (not to be confused with ‘univ ersality class’)
+expresses the relevance of symmetries as an organizational principle. Dyson,
+in his 1962 paper referred to as The Threefold Way, gave the pr ime classifica-
+tion of random matrix ensembles based on a quantum mechanica l setting with
+symmetries. In this article we review Dyson’s Threefold Way from a modern
+perspective. We then describe a minimal extension of Dyson’ s setting to in-
+corporate the physics of chiral Dirac fermions and disorder ed superconductors.
+In this minimally extended setting, where Hilbert space is r eplaced by Fock
+space equipped with the anti-unitary operation of particle -hole conjugation,
+symmetry classes are in one-to-one correspondence with the large families of
+Riemannian symmetric spaces.
+1 Introduction
+In Chapter 2 of this handbook1, the historical narrative by Bohigas and Wei-
+denm¨ ullerdescribeshowrandommatrixmodelsemergedfrom quantumphysics,
+more precisely from a statistical approach to the strongly i nteracting many-
+body system of the atomic nucleus. Although random matrix th eory is nowa-
+days understood to be of relevance to numerous areas of physi cs, mathematics,
+and beyond, quantum mechanics is still where many of its appl ications lie.
+1The present article is to be Chapter 3 of the Oxford Handbook of R andom Matrix Theory.
+1Quantum mechanics also provides a natural framework in whic h to classify
+random matrix ensembles.
+In this thrust of development, a symmetry classification of r andom matrix
+ensembles was put forth by Dyson in his 1962 paper The Threefold Way: alge-
+braic structure of symmetry groups and ensembles in quantum mechanics , where
+he proved (quote from the abstract of [Dys62]) “that the most general matrix
+ensemble, defined with a symmetry group which may be complete ly arbitrary,
+reduces to a direct product of independent irreducible ense mbles each of which
+belongs to one of the three known types”. The three types know n to Dyson
+wereensemblesofmatriceswhichareeithercomplexHermiti an, orrealsymmet-
+ric, or quaternion self-dual. It is widely acknowledged tha t Dyson’s Threefold
+Way has become fundamental to various areas of theoretical p hysics, includ-
+ing the statistical theory of complex many-body systems, me soscopic physics,
+disordered electron systems, and the field of quantum chaos.
+Overthelastdecade, anumberofrandommatrixensemblesbey ondDyson’s
+classification have come to the fore in physics and mathemati cs. On the physics
+side these emerged from work [Ver94] on the low-energy Dirac spectrum of
+quantum chromodynamics, and also from the mesoscopic physi cs of low-energy
+quasi-particles in disordered superconductors [AZ97]. In the mathematical re-
+search area of number theory, the study of statistical corre lations of the values
+of Riemann zeta and related L-functions has prompted some of the same gen-
+eralizations [KS99]. It was observed early on [AZ97] that th ese post-Dyson
+ensembles, or rather the underlying symmetry classes, are i n one-to-one corre-
+spondence with the large families of symmetric spaces.
+The prime emphasis of the present handbook article will be on describing
+Dyson’sfundamentalresultfromamodernperspective. Asec ondgoalwillbeto
+introduce the post-Dyson ensembles. While there seems to ex ist no unanimous
+view on how these fit into a systematic picture, here we will fo llow [HHZ05] to
+demonstrate that they emerge from Dyson’s setting upon repl acing the plain
+structure of Hilbert space by the more refined structure of Fo ck space.2The
+reader is advised that some aspects of this story are treated in a more leisurely
+manner in the author’s encyclopedia article [Zir04].
+To preclude any misunderstanding, let us issue a clarificati on of language
+right here: ‘symmetry class’ must not be confused with ‘univ ersality class’!
+Indeed, inside a symmetry class as understood in this articl e various types of
+physical behavior are possible. (For example, random matri x models for weakly
+disordered time-reversal invariant metals belong to the so -called Wigner-Dyson
+symmetry class of real symmetric matrices, and so do Anderso n tight-binding
+models with real hopping and strong disorder. The former are believed to
+2We mention in passing that a classification of Dirac Hamilton ians in two dimensions has
+been proposed in [BL02]. Unlike ours, this is not a symmetry cl assification in Dyson’s sense.
+2exhibit the universal energy level statistics given by the G aussian Orthogo-
+nal Ensemble, whereas the latter have localized eigenfunct ions and hence level
+statistics which is expected to approach the Poisson limit w hen the system size
+goes to infinity.) For this reason the present article must re frain from writing
+down explicit formulas for joint eigenvalue distributions , which are available
+only in certain universal limits.
+2 Dyson’s Threefold Way
+Dyson’s classification is formulated in a general and simple mathematical set-
+ting which we now describe. First of all, the framework of qua ntum theory
+calls for the basic structure of a complex vector space Vcarrying a Hermitian
+scalar product /an}bracketle{t·,·/an}bracketri}ht:V×V→C. (Dyson actually argues [Dys62] in favor of
+working over the real numbers, but we will not follow suit in t his respect.) For
+technical simplicity, we do join Dyson in taking Vto be finite-dimensional. In
+applications, V≃Cnwill usually be the truncated Hilbert space of a family of
+disordered or quantum chaotic Hamiltonian systems.
+The Hermitian structure of the vector space Vdetermines a group U( V)
+of unitary transformations of V. Let us recall that the elements g∈U(V) are
+C-linear operators satisfying the condition /an}bracketle{tgv,gv′/an}bracketri}ht=/an}bracketle{tv,v′/an}bracketri}htfor allv,v′∈V.
+Building on the Hermitian vector space V, Dyson’s setting stipulates that
+Vbe equipped with a unitary group action
+G0×V→V,(g,v)/ma√sto→ρV(g)v, ρ V(g)∈U(V). (2.1)
+In other words, there is some group G0whose elements gare represented on V
+by unitary operators ρV(g). This group G0is meant to be the group of joint
+(unitary) symmetries of a family of quantum mechanical Hami ltonian systems
+with Hilbert space V. We will write ρV(g)≡gfor short.
+Now, not every symmetry of a quantum system is of the canonica l unitary
+kind. The prime counterexample is the operation, T, of inverting the time
+direction, called time reversal for short. It is represente d on Hilbert space Vby
+ananti-unitary operator T≡ρV(T), which is to say that Tis complex anti-
+linear and preserves the Hermitian scalar product up to comp lex conjugation:
+T(zv) =zTv,/angbracketleftbig
+Tv,Tv′/angbracketrightbig
+=/an}bracketle{tv,v′/an}bracketri}ht(z∈C;v,v′∈V).(2.2)
+Another operation of this kind is charge conjugation in rela tivistic theories such
+as the Dirac equation for the electron and its anti-particle , the positron.
+Thus in Dyson’s general setting one has a so-called symmetry groupG=
+G0∪G1where the subgroup G0is represented on Vby unitaries, while G1(not
+a group) is represented by anti-unitaries. By the definition of what is meant
+3by a ‘symmetry’, the generator of time evolution, the Hamilt onianH, of the
+quantum system is fixed by conjugation gHg−1=Hwith anyg∈G.
+The setG1may be empty. When it is not, the composition of any two
+elements of G1is unitary, so every g∈G1can be obtained from a fixed element
+ofG1, sayT, by right multiplication with some U∈G0:g=TU. The
+same goes for left multiplication, i.e., for every g∈G1there also exists U′∈
+G0so thatg=U′T. In other words, when G1is non-empty, G0⊂Gis a
+proper normal subgroup and the factor group G/G0≃Z2consists of exactly
+two elements, G0andTG0=G1. For future use we record that conjugation
+U/ma√sto→TUT−1=:a(U) by time reversal is an automorphism of G0.
+Following Dyson [Dys62] we assume that the special element Trepresents
+aninversion symmetry such as time reversal or charge conjugation. Tmust
+then be a (projective) involution, i.e., T2=z×IdVwith 0/ne}ationslash=z∈C, so that
+conjugation by T2is the identity operation. Since Tis anti-unitary, zmust
+have modulus |z|= 1, and by the C-antilinearity of Tthe associative law
+zT=T2·T=T·T2=Tz=zT (2.3)
+forceszto be real, which leaves only two possibilities: T2=±IdV.
+Let us record here a concrete example of some historical impo rtance: the
+Hilbert space Vmight be the space of totally anti-symmetric wave functions
+ofnparticles distributed over the shell-model space of an atom or an atomic
+nucleus, and the symmetry group Gmight beG= O3∪TO3, the full rotation
+group O 3(including parity) together with its translate by time reve rsalT.
+In summary, Dyson’s setting assumes two pieces of data:
+•a finite-dimensional complex vector space Vwith Hermitian structure,
+•a groupG=G0∪TG0acting onVby unitary and anti-unitary operators.
+It should be stressed that, in principle, the primary object is the Hamiltonian,
+andthesymmetries Garesecondaryobjectsderivedfromit. However, adopting
+Dyson’s standpoint we now turn tables to view the symmetries as fundamental
+and given and the Hamiltonians as derived objects. Thus, fixi ng any pair ( V,G)
+our goal is to elucidate the structure of the space of all compatible Hamiltoni-
+ans, i.e., the self-adjoint operators HonVwhich commute with the G-action.
+Such a space is reducible in general: the G-compatible Hamiltonian matrices
+decompose as a direct sum of blocks. The goal of classificatio n is to enumerate
+the irreducible blocks that occur in this setting.
+Whilethemainobjectstoclassifyarethespacesofcompatib leHamiltonians
+H, we find it technically convenient to perform some of the disc ussion at the
+integrated level of time evolutions Ut= e−itH//planckover2pi1instead. This change of focus
+results in no loss, as the Hamiltonians can always be retriev ed by linearization
+4intatt= 0. The compatibility conditions for U≡Utread
+U=g0Ug−1
+0=g1U−1g−1
+1(for allgσ∈Gσ). (2.4)
+The strategy will be to make a reduction to the case of the triv ial group
+G0={Id}. The situation with trivial G0can then be handled by enumeration
+of a finite number of possibilities.
+2.1 Reduction to the case of G0={Id}
+To motivate the technical reduction procedure below, we beg in by elaborating
+the example of the rotation group O 3acting on a Hilbert space of shell-model
+states. Any Hamiltonian which commutes with G0= O3conserves total an-
+gular momentum, L, and parity, π, which means that all Hamiltonian matrix
+elements connecting states in sectors of different quantum n umbers (L,π) van-
+ish identically. Thus, the matrix of the Hamiltonian with re spect to a basis of
+states with definite values of ( L,π) has diagonal block structure. O 3-symmetry
+further implies that the Hamiltonian matrix is diagonal wit h respect to the
+orthogonal projection, M, of total angular momentum on some axis in position
+space. Moreover, for a suitable choice of basis the matrix wi ll be the same for
+eachM-value of a given sector ( L,π).
+To put these words into formulas, we employ the mathematical notions of
+orthogonal sum and tensor product to decompose the shell-mo del space as
+V≃/circleplusdisplay
+L≥0;π=±1V(L,π), V (L,π)=Cm(L,π)⊗C2L+1, (2.5)
+wherem(L,π) is the multiplicity in Vof the O 3-representation with quantum
+numbers (L,π). The statement above is that all symmetry operators and com -
+patible Hamiltonians are diagonal with respect to this dire ct sum, and within
+a fixed block V(L,π)the Hamiltonians act on the first factor Cm(L,π)and are
+trivial on the second factor C2L+1of the tensor product, while the symmetry
+operators act on the second factor and are trivial on the first factor. Thus
+we may picture each sector V(L,π)as a rectangular array of states where the
+Hamiltonians act, say, horizontally and are the same in each row of the array,
+while the symmetries act vertically and are the same in each c olumn.
+This concludes our example, and we now move on to the general c ase of
+any groupG0acting reductively on V. To handle it, we need some language
+and notation as follows. A G0-representation Xis aC-vector space carrying a
+G0-actionG0×X→Xby (g,x)/ma√sto→ρX(g)x. IfXandYareG0-representations,
+then by the space Hom G0(X,Y) ofG0-equivariant homomorphisms from Xto
+Yone means the complex vector space of C-linear maps ψ:X→Ywith
+the intertwining property ρY(g)ψ=ψρX(g) for allg∈G0. IfXis an ir-
+reducibleG0-representation, then Schur’s lemma says that Hom G0(X,X) is
+5one-dimensional, being spanned by the identity, Id X. For two irreducible G0-
+representations XandY, the dimension of Hom G0(X,Y) is either zero or one,
+by an easy corollary of Schur’s lemma. In the latter case XandYare said to
+belong to the same isomorphism class.
+Using the symbol λto denote the isomorphism classes of irreducible G0-
+representations, wefixforeach λastandardrepresentationspace Rλ. Notethat
+dimHom G0(Rλ,V)countsthemultiplicityin Voftheirreduciblerepresentation
+of isomorphism class λ. In our shell-model example with G0= O3we have
+λ= (L,π),Rλ=C2L+1, and dimHom G0(Rλ,V) =m(L,π).
+The following statement can be interpreted as saying that th e example ad-
+equately reflects the general situation.
+Lemma 2.1 LetG0act reductively on V. Then
+/circleplusdisplay
+λHomG0(Rλ,V)⊗Rλ→V,/circleplusdisplay
+λ(ψλ⊗rλ)/ma√sto→/summationdisplay
+λψλ(rλ)
+is aG0-equivariant isomorphism.
+Remark. The decomposition offered by this lemma perfectly separates the
+unitary symmetry multiplets from the dynamical degrees of f reedom and thus
+gives an immediate view of the structure of the space of G0-compatible Hamil-
+tonians. Indeed, the direct sum over isomorphism classes (o r ‘sectors’) λis
+preserved by the symmetries G0as well as the compatible Hamiltonians H;
+andG0is trivial on Hom G0(Rλ,V) while the Hamiltonians are trivial on Rλ.
+Next, we remove the time-evolution trivial factors Rλfrom the picture. To
+do so, we need to go through the step of transferring all given structure to the
+spacesEλ:= Hom G0(Rλ,V).
+2.1.1 Transfer of structure.
+We first transfer the Hermitian structure of V. In the present setting of a
+unitaryG0-action, the Hermitian scalar product of Vreduces to a Hermitian
+scalar product on each sector of the direct-sum decompositi on of Lemma 2.1,
+by orthogonality of the sum. Hence, we may focus attention on a definite sector
+E⊗R≡Eλ⊗Rλ. Fixing a G0-invariant Hermitian scalar product /an}bracketle{t·,·/an}bracketri}htRon
+R=Rλwe define such a product /an}bracketle{t·,·/an}bracketri}htE:E×E→Cby
+/an}bracketle{tψ,ψ′/an}bracketri}htE:=/an}bracketle{tψ(r),ψ′(r)/an}bracketri}htV//an}bracketle{tr,r/an}bracketri}htR, (2.6)
+which is easily checked to be independent of the choice of r∈R,r/ne}ationslash= 0.
+Before carrying on, we note that for any Hermitian vector spa ceVthere
+exists a canonically defined C-antilinear bijection CV:V→V∗to the dual
+vector space V∗byCV(v) :=/an}bracketle{tv,·/an}bracketri}htV. (In Dirac’s language this is the conversion
+from ‘ket’ vector to ‘bra’ vector.) By naturalness of the tran sfer of Hermitian
+structure we have the relation CE⊗R=CE⊗CR.
+6Turning to the more involved step of transferring time rever salT, we begin
+withapreparation. If L:V→Wisalinearmappingbetweenvectorspaces, we
+denote byLt:W∗→V∗the canonical transpose defined by ( Ltf)(v) =f(Lv).
+Let nowVbe our Hilbert space with ket-bra bijection C≡CV. Then for any
+g∈U(V) we have the relation CgC−1= (g−1)tbecause
+C(gv) =/an}bracketle{tgv,·/an}bracketri}ht=/an}bracketle{tv,g−1·/an}bracketri}ht= (g−1)tC(v) (v∈V).(2.7)
+Moreover, recallingthe automorphism G0∋g/ma√sto→a(g) =TgT−1ofG0we obtain
+CTg=a(g−1)tCT(g∈G0). (2.8)
+Thus, since CandTare bijective, the C-linear mapping CT:V→V∗is a
+G0-equivariant isomorphism interchanging the given G0-representation on V
+with the representation on V∗byg/ma√sto→a(g−1)t. In particular, it follows that T
+stabilizes the decomposition V=⊕λVλ≃ ⊕λ(Eλ⊗Rλ) of Lemma 2.1.
+IfTexchanges different sectors Vλ, the situation is very easy to handle (see
+below). The more challenging case is TVλ=Vλ, which we now assume.
+Lemma 2.2 LetTVλ=Vλ. Under the isomorphism Vλ≃Eλ⊗Rλthe time-
+reversal operator transfers to a pure tensor
+T=α⊗β , α:Eλ→Eλ, β:Rλ→Rλ,
+with anti-unitary αandβ.
+Proof. WritingEλ≡EandRλ≡Rfor short, we consider the transferred
+mappingCT:E⊗R→E∗⊗R∗, which expands as CT=/summationtextφi⊗ψiwithC-
+linearmappings φi,ψi. SinceCTisknowntobea G0-equivariantisomorphism,
+so is every map ψi:R→R∗. By the irreducibility of Rand Schur’s lemma,
+there exists only one such map (up to scalar multiples). Henc eCTis a pure
+tensor:CT=φ⊗ψ. UsingC=CE⊗R=CE⊗CRwe obtainT=α⊗β
+withC-antilinear α=C−1
+Eφandβ=C−1
+Rψ. Since the tensor product lets
+you move scalars between factors, the maps αandβare not uniquely defined.
+We may use this freedom to make βanti-unitary. Because Tis anti-unitary, it
+then follows from the definition (2.6) of the Hermitian struc ture ofEthatαis
+anti-unitary.
+Remark. By an elementary argument, which was spelled out for the anti -
+unitary operator Tin Eq. (2.3), it follows that α2=ǫαIdEandβ2=ǫβIdR
+withǫα,ǫβ∈ {±1}. WritingT2=ǫTIdVwe have the relation ǫαǫβ=ǫT. Thus
+whenǫβ=−1 the parity ǫα=−ǫTof the transferred time-reversal operator α
+is opposite to that of the original time reversal T.
+This change of parity occurs, e.g., in the case of G0= SU2. Indeed, let
+R≡Rnbe the irreducible SU 2-representation of dimension n+ 1. It is a
+7standard fact of representation theory that Rnis SU2-equivariantly isomorphic
+toR∗
+nby a symmetric isomorphism ψ=ψtfor evennand skew-symmetric
+isomorphism ψ=−ψtfor oddn. From (−1)nψt=ψ=CRβand
+ψ(v)(v′) =/an}bracketle{tβv,v′/an}bracketri}htR=/an}bracketle{tβ2v,βv′/an}bracketri}htR=/an}bracketle{tβv′,β2v/an}bracketri}htR=ψ(v′)(β2v),(2.9)
+we conclude that β2= (−1)nIdRn.
+2.2 Classification
+By the decomposition of Lemma2.1 the space ZU(V)(G0) ofG0-compatible time
+evolutions in U( V) is a direct product of unitary groups,
+ZU(V)(G0)≃/productdisplay
+λU(Eλ). (2.10)
+We now fix a sector Vλ≃Eλ⊗Rλand run through the different situations
+(of which there exist three, essentially) due to the absence or presence of a
+transferred time-reversal symmetry α:Eλ→Eλ.
+2.2.1 Class A
+The first type of situation occurs when the set G1of anti-unitary symmetries is
+eitheremptyorelsemaps Vλ≃Eλ⊗Rλtoadifferentsector Vλ′,λ/ne}ationslash=λ′. Inboth
+cases, theG-compatible time-evolution operators restricted to Vλconstitute a
+unitary group U( Eλ)≃UNwithN= dimEλbeing the multiplicity of the
+irreducible representation RλinV. The unitary groups U Nor to be precise,
+their simple parts SU N, are symmetric spaces (cf. Section 3.4) of the Afamily
+orAseries in Cartan’s notation – hence the name Class A. In random matrix
+theory, the Lie group U Nequipped with Haar measure is commonly referred to
+as the Circular Unitary Ensemble, CUE N[Dys62a].
+TheHamiltonians HinClassAarerepresentedbycomplexHermitian N×N
+matrices. By putting a U N-invariant Gaussian probability measure
+dµ(H) =c0e−TrH2/2σ2dH , dH =N/productdisplay
+i=1dHii/productdisplay
+j<kdHjkdHkj,(2.11)
+on that space, one gets what is called the GUE – the Gaussian Un itary Ensem-
+ble – defining the Wigner-Dyson universality class of unitary symmetry. An
+important physical realization of that class is by electron s in a disordered metal
+with time-reversal symmetry broken by the presence of a magn etic field.
+2.2.2 Classes AI andAII
+We now turn to the cases where Tis present and TVλ=Vλ≃Eλ⊗Rλ. We
+abbreviate Eλ≡E. From Lemma 2.1 we know that T=α⊗βis a pure tensor
+with anti-unitary α, and we have α2=ǫαIdEwith parity ǫα=ǫTǫβ.
+8Using conjugation by αto define an automorphism
+τ: U(E)→U(E), u/ma√sto→αuα−1, (2.12)
+we transfer the conditions (2.4) to Vλand describe the set ZU(E)(G) ofG-
+compatible time evolutions in U( E) as
+ZU(E)(G) ={x∈U(E)|τ(x) =x−1}. (2.13)
+Now letU≡U(E) for short and denote by K⊂Uthe subgroup of τ-fixed
+elementsk=τ(k)∈U. The setZU(G) is analytically diffeomorphic to the
+coset space U/Kby the mapping
+U/K→ZU(G)⊂U , uK /ma√sto→uτ(u−1), (2.14)
+which is called the Cartan embedding of U/KintoU. The remaining task is
+to determine K. This is done as follows.
+Recalling the definition CEα=φand usingCEk= (k−1)tCEwe express
+the fixed-point condition k=τ(k) =αkα−1as (k−1)t=φkφ−1or, equivalently,
+φ=ktφk, which means that the bilinear form associated with φ:E→E∗,
+Qφ:E×E→C,(e,e′)/ma√sto→φ(e)(e′), (2.15)
+is preserved by k∈K. By running the argument around Eq. (2.9) in reverse
+order (with the obvious substitutions ψ→φandβ→α), we see that the
+non-degenerate form Qφis symmetric if ǫα= +1 and skew if ǫα=−1. In the
+former case it follows that K= O(E)≃ONis an orthogonal group, while in
+the latter case, which occurs only if N∈2N,K= USp(E)≃USpNis unitary
+symplectic. In both cases the coset space U/Kis a symmetric space (cf. Section
+3.4) – a fact first noticed by Dyson in [Dys70].
+Thus in the present setting of TVλ=Vλwe have the following dichotomy
+for the sets of G-compatible time evolutions ZU(E)(G)≃U/K:
+ClassAI :U/K≃UN/ON(ǫα= +1),
+ClassAII :U/K≃UN/USpN(ǫα=−1, N∈2N).
+Again we are referring to symmetric spaces by the names they – or rather their
+simple parts SU N/SONand SU N/USpN– have in the Cartan classification.
+In random matrix theory, the symmetric space U N/ON(or its Cartan embed-
+ding into U Nas the symmetric unitary matrices) equipped with U N-invariant
+probability measure is called the Circular Orthogonal Ense mble, COE N, while
+the Cartan embedding of U N/USpNequipped with U N-invariant probability
+measure is known as the Circular Symplectic Ensemble, CSE N[Dys62a]. (Note
+the confusing fact that the naming goes by the subgroup which is divided out.)
+9Examples for Class AI are provided by time-reversal invariant systems with
+symmetryG0= (SU 2)spin. Indeed,bythefundamentallawsofquantumphysics
+time reversal Tsquares to ( −1)2Stimes the identity on states with spin S. Such
+states transform according to the irreducible SU 2-representation of dimension
+2S+ 1, and from β2= (−1)2S(see the Remark after Lemma 2.1) it follows
+thatǫα=ǫTǫβ= (−1)2S(−1)2S= +1 in all cases. A historically important
+realization of Class AI is furnished by the highly excited states of atomic nuclei
+as observed by neutron scattering just above the neutron thr eshold.
+By breaking SU 2-symmetry (i.e., by taking G0={Id}) while maintaining
+T-symmetry for states with half-integer spin (say single ele ctrons, which carry
+spinS= 1/2), onegets ǫα=ǫT= (−1)2S=−1, thereby realizingClass AII.An
+experimental observation of this class and its characteris tic wave interference
+phenomena was first reported in the early 1980’s [Ber84] for d isordered metallic
+magnesium films with strong spin-orbit scattering caused by gold impurities.
+The Hamiltonians H, obtained by passing to the tangent space of U/Kat
+unity, are represented by Hermitian matrices with entries t hat are real num-
+bers (Class AI) or real quaternions (Class AII). The simplest random matrix
+models result from putting K-invariant Gaussian probability measures on these
+spaces; they are called the Gaussian Orthogonal Ensemble an d Gaussian Sym-
+plectic Ensemble, respectively. Their properties delinea te the Wigner-Dyson
+universality classes of orthogonal and symplectic symmetr y.
+3 Symmetry Classes of Disordered Fermions
+WhileDyson’sThreefoldWayisfundamentalandcompleteini tsgeneralHilbert
+space setting, the early 1990’s witnessed the discovery of v arious new types of
+strong universality, which were begging for an extended sch eme:
+•The introduction of QCD-motivated chiral random matrix ens embles (re-
+viewedbyVerbaarschotinChapter32ofthishandbook)mimic kedDyson’s
+scheme but also transcended it.
+•Number theorists had introduced and studied ensembles of L-functions
+akin to the Riemann zeta function (see the review by Keating a nd Snaith
+in Chapter 24 of this handbook). These display random matrix p henom-
+ena which are absent in the classes A,AI, orAII.
+•The proximity effect due to Andreev reflection, a particle-ho le conversion
+process in mesoscopic hybrid systems involving metallic as well as su-
+perconducting components, was found [AZ97] to give rise to p ost-Dyson
+mechanisms of quantum interference (cf. Chapter 35 by Beena kker).
+By the middle of the 1990’s, it had become clear that there exi sts a unifying
+mathematicalprinciplegoverningthesepost-Dysonrandom matrixphenomena.
+10This principle will be explained in the present section. We m ention in passing
+that a fascinating recent development [Kit08, SRF09] uses t he same principle
+for a homotopy classification of topological insulators and superconductors.
+3.1 Fock space setting
+We now describe an extended setting, which replaces the Herm itian vector
+spaceVby its exterior algebra ∧(V) but otherwise retains Dyson’s setting to
+the fullest extent possible. In physics language we say that we pass from the
+(single-particle) Hilbert space Vto the fermionic Fock space ∧(V) generated
+byV. TheZ-grading ∧(V) =⊕n∧n(V) by the degree nhas the physical
+meaning of particle number. Thus ∧0(V)≡Cis the vacuum, ∧1(V)≡Vis the
+one-particle space, ∧2(V) is the two-particle space, and so on. We adhere to
+the assumption of finite-dimensional V. Particle number nthen is in the range
+0≤n≤N:= dimV. Note that dim ∧n(V) =/parenleftbiggN
+n/parenrightbigg
+.
+Then-particlesubspace ∧n(V)oftheFockspaceofaHermitianvectorspace
+Vcarries an induced Hermitian scalar product defined by
+/angbracketleftbig
+u1∧···∧un,v1∧···∧vn/angbracketrightbig
+∧n(V)= Det
+/an}bracketle{tu1,v1/an}bracketri}htV.../an}bracketle{tu1,vn/an}bracketri}htV
+.........
+/an}bracketle{tun,v1/an}bracketri}htV.../an}bracketle{tun,vn/an}bracketri}htV
+.(3.1)
+Relevant operations on ∧(V) are exterior multiplication (or particle creation)
+ε(v) :∧n(V)→ ∧n+1(V) byv∈Vand contraction (or particle annihilation)
+ι(f) :∧n(V)→ ∧n−1(V) byf∈V∗. The standard physics convention is to
+fix some orthonormal basis {ek}k=1,...,NofVand writea†
+k:=ε(ek) for the
+particle creation operators and ak:=ι(/an}bracketle{tek,·/an}bracketri}htV) for the particle annihilation
+operators. This notation reflects the fact that Hermitian co njugation †in Fock
+space relates ε(v) andι(f) byε(v)†=ι(f) wheref=/an}bracketle{tv,·/an}bracketri}htV. The operators a†
+k
+andaksatisfy the so-called canonical anti-commutation relatio ns
+akal+alak= 0 =a†
+ka†
+l+a†
+la†
+k, a†
+kal+ala†
+k=δkl.(3.2)
+TheserepresentthedefiningrelationsoftheCliffordalgebr aCl(W)ofthevector
+spaceW=V⊕V∗with quadratic form ( v⊕f,v′⊕f′)/ma√sto→f(v′)+f′(v).
+Having introduced the basic Fock space structure, we now tur n to what is
+going to be our definition of a symmetry group Gacting on Fock space ∧(V).
+As before, we assume that we are given a normal subgroup G0⊂G. The action
+of the elements g∈G0is defined by unitary operators on Vwhich are extended
+to∧(V) by
+g(v1∧···∧vn) := (gv1)∧···∧(gvn). (3.3)
+Similarly, the anti-unitary operator of time reversal Tis defined on Vand is
+extended to ∧(V) byT(v1∧···∧vn) := (Tv1)∧···∧(Tvn).
+11Now the Z-grading of Fock space offers the natural option of introduci ng
+another kind of anti-unitary operator, which is a close cous in of the Hodge star
+operator for the deRham complex: particle-hole conjugatio n,C, transforms an
+n-particle state into an ( N−n)-particle state or a state with nholes. (Note
+the change of meaning of the symbol Cas compared to Section 3.3.)
+Definition 3.1 Fix a generator Ω∈ ∧N(V),N= dimV, with normalization
+/an}bracketle{tΩ,Ω/an}bracketri}ht∧N(V)= 1. Then particle-hole conjugation C:∧n(V)→ ∧N−n(V)is the
+anti-unitary operator defined by
+(Cψ)∧ψ′=/an}bracketle{tψ,ψ′/an}bracketri}ht∧n(V)Ω.
+Thus the definition of the operator Cuses the Hermitian scalar product of Fock
+space and a choice of fully occupied state Ω. An elementary ca lculation shows
+thatC2|∧n(V)= (−1)n(N−n).
+What are the commutation relations of CwithTandg∈G0? To answer
+this question, we observe that by dim ∧N(V) = 1 we may always choose Ω to
+beT-invariant (i.e., TΩ = Ω). Then from the following computation,
+(TCψ)∧Tψ′=T/parenleftbig
+(Cψ)∧ψ′/parenrightbig
+=T/parenleftbig
+/an}bracketle{tψ,ψ′/an}bracketri}htΩ/parenrightbig
+=/an}bracketle{tψ,ψ′/an}bracketri}htTΩ =/an}bracketle{tTψ,Tψ′/an}bracketri}htΩ = (CTψ)∧Tψ′,
+we haveCT=TC. Also, making the natural assumption that both the vacuum
+space∧0(V) and the fully occupied space ∧N(V) transform as the trivial G0-
+representation (i.e., gΩ = Ω forg∈G0), a similar calculation gives Cg=gC.
+In order to enlarge the set of possible symmetries and hence t he scope of
+the theory, we now introduce a ‘twisted’ variant of particle -hole conjugation.
+LetS∈U(V) be an involution ( S2= Id) and extend Sto∧(V) by Eq. (3.3).
+To obtain an extension of the group G0, we require that Scommutes with T,
+satisfiesSΩ = Ω, and normalizes G0, i.e.,SG0S−1=G0. (Here we identify G0
+with its action on Fock space.) By twisted particle-hole con jugation we then
+mean the operator ˜C=CS=SC. Note that ˜CG0˜C−1=G0and˜CT=T˜C.
+Definition 3.2 On the Fock space ∧(V)over a Hermitian vector space V, let
+there be the action of a group G=G0∪TG0∪˜CG0∪˜CTG0withG0a normal
+subgroup and ˜CT=T˜C. We call this a minimal extension of Dyson’s setting
+ifG0acts by unitary operators defined on V, time reversal Tacts as an anti-
+unitary operator also defined on V, and twisted particle-hole conjugation ˜Cis
+an anti-unitary bijection ∧n(V)→ ∧N−n(V).
+3.2 Classification goal
+The simplest question to ask now is this: what is the structur e of the set of
+Hamiltonians that operate on Fock space ∧(V) and commute with the given
+12G-action on ∧(V)? Since this question ignores the grading of Fock space by
+particle number, it takes us back to Dyson’s setting and the a nswer is, in fact,
+provided by Dyson’s Threefold Way. (Note that in the absence of restrictions,
+the most general Hamiltonian in Fock space involves n-body interactions of
+arbitrary rank n= 1,2,3,...,N.) So there is nothing new to discover here.
+We shall, however, be guided to a new and interesting answer b y asking a
+somewhat different question: what is the structure of the set ofone-body time
+evolutions of ∧(V) which commute with the given G-action? Here by a one-
+body time evolution we mean any unitary operator obtained by exponentiating
+a self-adjoint Hamiltonian Hwhich is quadratic in the particle creation and
+annihilation operators:
+H=/summationdisplay
+klWkla†
+kal+1
+2/summationdisplay
+kl/parenleftbig
+Zkla†
+ka†
+l+Zklalak/parenrightbig
+. (3.4)
+These operators U= e−itH//planckover2pi1∈U(∧V) form what is called the spin group,
+Spin(WR), of the 2N-dimensional Euclidean R-vector space WRspanned by the
+Majorana operators ak+a†
+k, iak−ia†
+k(k= 1,...,N). Spin(WR)≃Spin2Nis a
+double covering of the real orthogonal group SO( WR)≃SO2N. The spin group
+of most prominence in physics is Spin3≡SU2, a double covering of SO 3. (This
+double covering is known to physicists by the statement that a rotation by 2 π,
+which acts as the neutral element of SO 3, changes the sign of a spinor.)
+Thus, our interest is now in the set
+ZSpin(G) :=ZU(∧V)(G)∩Spin(WR) (3.5)
+ofG-compatible time evolutions in Spin( WR)⊂U(∧V). By adaptation of the
+earlier definition (2.4), the G-compatibility conditions are
+U=g0Ug−1
+0=g1U−1g−1
+1 (3.6)
+for allg0∈G0∪˜CTG0andg1∈TG0∪˜CG0.
+3.3 Reduction to Nambu space
+Toinvestigatetheset ZSpin(G)weusethefollowingfact. Anyinvertibleelement
+A∈Cl(W) determines an automorphism γ/ma√sto→AγA−1of the Clifford algebra
+Cl(W) by conjugation. This conjugation action restricts to a rep resentation
+τ(g)w:=gwg−1(3.7)
+of Spin(WR)⊂Cl(W) onWR⊂Cl(W). Phrased in physics language, the
+set of Majorana field operators w=/summationtext
+k(zka†
+k+zkak)∈WRis closed under
+conjugation w/ma√sto→gwg−1by one-body time evolution operators g∈Spin(WR).
+In fact, by elementary considerations one finds that τ(g) :w/ma√sto→gwg−1for
+13g∈Spin(WR) is an orthogonal transformation τ(g)∈SO(WR) of the Euclidean
+vector space WR. The mapping τ: Spin(WR)→SO(WR),g/ma√sto→τ(g) =τ(−g)
+is two-to-one. It is a covering map, which amounts to saying t hat any path in
+SO(WR)liftsuniquelytoapathinSpin( WR). Notealsothatthelinearmapping
+τ(g) :WR→WRextends to a linear mapping τ(g) :W→WbyC-linearity.
+Thus, instead of studying Spin( WR) as a group of operators on the full
+Fock space ∧(V), we may simplify our work by studying its representation
+τ: Spin→SO on the smaller space W=V⊕V∗, here referred to as Nambu
+space. Of course the object of interest is not Spin( WR) but its intersection with
+theG-compatibility conditions. To keep track of the latter cond itions, we now
+transfer the G-action from ∧(V) toW=V⊕V∗.
+It is immediately clear how to transfer the actions of G0andT, as these are
+defined onV. In the case of the twisted particle-hole conjugation opera tor˜C,
+we do the following computation. Let ψ∈ ∧n(V) andψ′∈ ∧n+1(V). Then
+(˜Ca†
+kψ)∧ψ′=/an}bracketle{tSa†
+kψ,ψ′/an}bracketri}htΩ =/an}bracketle{tSψ,Sa kS−1ψ′/an}bracketri}htΩ
+= (˜Cψ)∧(SakS−1)ψ′= (−1)N−n+1(SakS−1˜Cψ)∧ψ′.
+Thus the twisted particle-hole conjugate of a†
+k∈V⊂Cl(W) is˜Ca†
+k˜C−1=
+±SakS−1∈V∗⊂Cl(W) where the sign alternates with particle number. Note
+that the operation a†
+k/ma√sto→ ±SakS−1is anti-unitary. Note also that the untwisted
+particle-hole conjugation a†
+k/ma√sto→akis none other than the C-antilinear bijection
+CV:V→V∗,v/ma√sto→ /an}bracketle{tv,·/an}bracketri}htV.
+To sum up, we have the following induced structures on Nambu s pace:
+•One-bodytimeevolutions g∈Spin(WR)actonW=V⊕V∗byorthogonal
+transformations τ(g)∈SO(WR).
+•G0is defined on Vand acts on W=V⊕V∗byg(v⊕f) =gv⊕(g−1)tf.
+The same goes for time reversal T.
+•The operator ˜Cof twisted particle-hole conjugation induces on W=
+V⊕V∗an anti-unitary involution V↔V∗.
+The goal of symmetry classification now is to characterize th e setZSO(G) of
+elements in SO( WR) which are compatible with the induced G-action onW.
+This problem was posed and solved in [HHZ05], by using an elab oration of the
+algebraic tools of Section 2 to make a reduction to the case of the trivial group
+G0={Id}. (The involution V↔V∗given by twisted particle-hole conjugation
+is called a mixingsymmetry in [HHZ05].) The outcome is as follows.
+Theorem 3.1 The spaceZSO(G)is a direct product of factors each of which
+is a classical irreducible compact symmetric space. Conver sely, every classical
+irreducible compact symmetric space occurs in this setting .
+14There is no space to reproduce the proof here, but in order to t urn the theorem
+into an intelligible statement we now record a few basic fact s from the theory
+of symmetric spaces [Hel78, CM04].
+3.4 Symmetric spaces
+LetMbe a connected m-dimensional Riemannian manifold and pa point of
+M. In some open subset Npof a neighborhood of pthere exists a map sp:
+Np→Np, the geodesic inversion with respect to p, which sends a point x∈Np
+with normal coordinates ( x1,...,x m) to the point with normal coordinates
+(−x1,...,−xm). The Riemannian manifold Mis called locally symmetric if
+the geodesic inversion is an isometry (i.e., is distance-pr eserving), and is called
+globally symmetric if spextends to an isometry sp:M→M, for allp∈M. A
+globally symmetric Riemannian manifold is called a symmetr ic space for short.
+The Riemann curvature tensor of a symmetric space is covaria ntly constant,
+which leads to three distinct cases: the scalar curvature ca n be positive, zero,
+or negative, and the symmetric space is said to be of compact t ype, Euclidean
+type, or non-compact type, respectively. In random matrix t heory each type
+plays a role: the first one provides us with the scattering mat rices and time
+evolutions, the second one with the Hamiltonians, and the th ird one with the
+transfer matrices. Our focus here will be on compact type, as it is this type
+that houses the unitary time evolution operators of quantum mechanics.
+Symmetric spaces of compact type arise in the following way. LetUbe
+a connected compact Lie group equipped with a Cartan involut ion, i.e., an
+automorphism τ:U→Uwith the involutive property τ2= Id. LetK⊂U
+be the subgroup of τ-fixed points u=τ(u). Then the coset space U/Kis a
+compact symmetric space in a geometry defined as follows. Wri tingu:= Lie(U)
+andk:= Lie(K) for the Lie algebras of the Lie groups involved, let u=k⊕p
+be the decomposition into positive and negative eigenspace s of the involution
+dτ:u→uinduced by linearization of τat unity. Fix on pa Euclidean scalar
+product /an}bracketle{t·,·/an}bracketri}htpwhich is invariant under the adjoint K-action Ad( k) :p→p
+byX/ma√sto→kXk−1. Then the Riemannian metric guKevaluated on vectors v,v′
+tangenttothecoset uKisguK(v,v′) :=/an}bracketle{tdL−1
+u(v),dL−1
+u(v′)/an}bracketri}htpwheredLudenotes
+the differential of the operation of left translation on U/Kbyu∈U.
+It is important for us that the coset space U/Kcan be realized as a subset
+M:={x∈U|τ(x) =x−1} (3.8)
+by the Cartan embedding U/K→M⊂U,uK/ma√sto→uτ(u−1). The metric tensor
+in this realization is given (in a self-explanatory notatio n) byg= Trdxdx−1. It
+is invariant under the K-actionM→Mby twisted conjugation x/ma√sto→uxτ(u−1).
+The geodesic inversion with respect to y∈Missy:M→M,x/ma√sto→yx−1y.
+15family compact type Euclidean type
+AUN Hcomplex Hermitian
+AI U N/ON Hreal symmetric
+AII U 2N/USp2N Hquaternion self-dual
+CUSp2N Zcomplex symmetric
+CI USp2N/UN Zcomplex sym., W= 0
+B,DSON Zcomplex skew
+DIII SO 2N/UN Zcomplex skew, W= 0
+AIII U p+q/(Up×Uq) Zcomplexp×q,W= 0
+BDI O p+q/(Op×Oq) Zrealp×q,W= 0
+CII USp2p+2q/(USp2p×USp2q)Zquaternion 2 p×2q,W= 0
+Table 1: The Cartan table of classical symmetric spaces
+We note that special examples of compact symmetric spaces ar e afforded
+by compact Lie groups K. For these examples, one takes U=K×Kwith flip
+involutionτ(k,k′) = (k′,k) leading to U/K= (K×K)/K≃K. Cartan’s list of
+classical (or large families of) compact symmetric spaces i s presented in Table 1.
+The form of the generator Hof time evolutions u= e−itH//planckover2pi1is indicated in the
+third column, where the notation W,Zrefers to the Fock space expression (3.4)
+which translates to H=/parenleftbiggW Z
+Z†−Wt/parenrightbigg
+by the covering map τ: Spin→SO.
+3.5 Post-Dyson classes
+We now run through the symmetry classes beyond those of Wigne r-Dyson. As
+was mentioned before, these appear in various areas of physi cs and in the ran-
+dom matrix theory of L-functions. For brevity we concentrate on their physical
+realization by quasi-particles in disordered metals and su perconductors.
+3.5.1 Class D
+Considerasuperconductorwithnosymmetriesinitsquasi-p articledynamics,so
+G={Id}. (Someconcretephysicalexamplesfollowbelow.) Thetimee volutions
+u= e−itH//planckover2pi1in this case are constrained only by the condition u∈Spin(WR)
+in Fock space and τ(u)∈SO(WR) in Nambu space. The orthogonal group
+16SO(WR)≃SO2Nis a symmetric space of the Dfamily – hence the name class
+D. In a basis of Majorana fermions ak+a†
+k, iak−ia†
+k, the matrix of i H∈so2N
+is real skew, and that of His imaginary skew.
+Concrete realizations are found in superconductors where t he order param-
+eter transforms under rotations as a spin triplet in spin spa ce and as a p-wave
+in real space. A recent candidate for a quasi-2d (or layered) spin-triplet p-
+wave superconductor is the compound Sr 2RuO4. (A non-charged analog is
+theA-phase of superfluid3He.) Time-reversal symmetry in such a system may
+be broken spontaneously, or else can be broken by an external magnetic field
+creating vortices in the superconductor.
+The simplest random matrix model for class D, the SO-invariant Gaussian
+ensemble of imaginary skew matrices, is analyzed in Mehta’s book [Meh04].
+From the expressions given there it is seen that the level cor relation functions
+at high energy coincide with those of the Wigner-Dyson unive rsality class of
+unitary symmetry (Class A). The level correlations at low energy, however,
+show different behavior defining a separate universality cla ss. This universal
+behavior at low energies has immediate physical relevance, as it is precisely the
+low-energy quasi-particles that determine the thermal tra nsport properties of
+the superconductor at low temperatures.
+3.5.2 Class DIII
+Now, let time reversal Tbe a symmetry: G={Id,T}. Physically speaking this
+implies the absence of magnetic fields, magnetic impurities and other agents
+which distinguish between the forward and backward directi ons of time. Our
+physical degrees of freedom are spin-1/2 particles, so T2=−IdV.
+According to (3.6) we are looking for the intersection ZSO(G) of the condi-
+tionu−1=TuT−1with Spin(WR), or after transfer to Nambu space, SO( WR).
+By introducing the involution τ(u) :=TuT−1we express the wanted set as
+ZSO(G) ={u∈SO(WR)|u−1=τ(u)}. (3.9)
+Following the discussion around Eq. (3.8) we have ZSO(G)≃U/KwhereU=
+SO(WR) andK⊂Uis the subgroup of τ-fixed points in U.
+In order to identify Kwe note that time reversal T:W→Wpreserves the
+real subspace WRof Majorana operators ak+a†
+k, iak−ia†
+k. BecauseT2=−Id,
+theR-linear operator T:WR→WRis a complex structure of the real vector
+spaceWR≃R2N≃CN. In other words, there exists a basis {e1,j,e2,j}j=1,...,N
+ofWRsuch thatTe1,j=e2,jandTe2,j=−e1,j. Now the τ-fixed point
+conditionk=τ(k)saysthatk∈Kcommuteswiththecomplexlinearextension
+J:W→WofT:WR→WRbyJe±,j=±ie±,jwheree±,j=e1,j±ie2,j.
+The general element kwith this property is a U N-transformation which acts
+17on spanC{e+,1,...,e+,N}askand on spanC{e−,1,...,e−,N}ask= (k−1)t.
+HenceK= UNand
+ZSO(G)≃U/K≃SO2N/UN, (3.10)
+which is a symmetric space in the DIII family.
+Known realizations of this symmetry class exist in gapless s uperconductors,
+say with spin-singlet pairing, but with a sufficient concentr ation of spin-orbit
+impurities to break spin-rotation symmetry. In order for qu asi-particle excita-
+tions to exist at low energy, the spatial symmetry of the orde r parameter should
+be different from s-wave. A non-charged realization occurs in the B-phase of
+3He, where the order parameter is spin-triplet without break ing time-reversal
+symmetry. Another candidate are heavy-fermion supercondu ctors, where spin-
+orbit scattering often happens to be strong owing to the pres ence of elements
+with large atomic weights such as uranium and cerium.
+3.5.3 Class C
+Next let the spin of the quasi-particles be conserved, but le t time-reversal sym-
+metry be broken instead. Thus magnetic fields (or some equiva lentT-breaking
+agent) are now present, while the effect of spin-orbit scatte ring is absent. The
+symmetry group of the physical system then is G=G0= Spin3= SU2. Such a
+situation is realized in spin-singlet superconductors in t he vortex phase. Promi-
+nent examples are the cuprate superconductors, which are la yered and exhibit
+an order parameter with d-wave symmetry in their copper-oxide planes.
+The symmetry-compatible time evolutions are identified by g oing through
+the process of eliminating the unitary symmetries G0=G. For that, we de-
+compose the Hilbert space as V=E⊗R,E= Hom G(R,V), whereR:=C2
+is the fundamental representation of G= SU2. Now there exists a skew-
+symmetric SU 2-equivariant isomorphism (known in physics by the name of
+spin-singlet pairing) between the vector space Rand its dual R∗. Therefore
+we haveW=V⊕V∗≃(E⊕E∗)⊗R, and elimination of the conserved factor
+Rtransfers the canonical symmetric form of W=V⊕V∗to the canonical
+alternating form ( e⊕f,e′⊕f′)/ma√sto→f(e′)−f′(e) ofE⊕E∗. On transferring
+also the Hermitian scalar product from V⊕V∗toE⊕E∗, one sees that the
+G-compatible time evolutions form a unitary symplectic grou p,
+ZSO(G) = SO(WR)G≃USp(E⊕E∗), (3.11)
+which is a compact symmetric space of the Cfamily.
+3.5.4 Class CI
+The next class is obtained by taking spin rotations as well as the time reversal
+Tto be symmetries of the quasi-particle system. Thus the symm etry group
+18now isG=G0∪TG0withG0= Spin3= SU2. Like in the previous symmetry
+class, physical realizations are provided by the low-energ y quasi-particles of
+unconventional spin-singlet superconductors. The superc onductor must now be
+in the Meissner phase where magnetic fields are expelled by sc reening currents.
+To identify the relevant symmetric space, we again transfer fromV⊕V∗=
+(E⊕E∗)⊗Rto the reduced space E⊕E∗. By this reduction, the canonical
+form undergoes a change of type from symmetric to alternatin g as before. We
+must also transfer time reversal; because the fundamental r epresentation R=
+C2of SU2is self-dual by a skew-symmetric isomorphism, the parity of the
+time-reversal operator changes from T2=−IdV⊕V∗toT2= +Id E⊕E∗by the
+mechanism explained at the end of Section 2.2.
+We haveZSO(G)≃U/KwhereU= USp(E⊕E∗) andKis the subgroup of
+elements fixed by conjugation with T. Because the reduced Tsquares to +1, we
+may realize it on matrices as the complex conjugation operat or by working in a
+basis ofT-fixed vectors of E⊕E∗. The Lie algebra elements X∈usp(E⊕E∗)
+have the form X=/parenleftbiggA B
+−BA/parenrightbigg
+with anti-Hermitian Aand complex symmetric
+B. They commute with the operation of complex conjugation if Ais real skew
+andBreal symmetric. Matrices Xwith suchAandBspan the Lie algebra
+uN,N= dim(E). At the Lie group level it follows that K= UN. Hence
+ZSO(G)≃USp2N/UN– a symmetric space in the CI family.
+3.5.5 Class AIII
+So far, we have made no use of twisted particle-hole conjugat ion˜Cas a sym-
+metry, but now let the symmetry group be G=G0∪˜CG0whereG0= U1acts
+onW=V⊕V∗byv⊕f/ma√sto→zv⊕z−1f(forz∈C,|z|= 1).
+In order for the elements u∈SO(WR) to commute with the G0-action,
+they must be of the block-diagonal form u=k⊕(k−1)t,k∈U(V). Therefore
+ZSO(G0)≃U(V). The wanted set then is ZSO(G)≃U/KwithU≡U(V) and
+Kthe subgroup of elements which are fixed by conjugation with ˜C.
+Recall from Section 3.1 that ˜C|V=CSwhereS∈U,S2= Id, and un-
+twisted particle-hole conjugation Ccoincides (up to an irrelevant sign) with
+the canonical bijection CV:V→V∗,CV(v) =/an}bracketle{tv,·/an}bracketri}htV. The condition for
+u=k⊕(k−1)tto belong to Kreads
+(k−1)t=˜Ck˜C−1. (3.12)
+Sincek−1=k†andC−1ktC=k†, this condition is equivalent to k=SkS.
+Now letV=V+⊕V−whereV±are orthogonal subspaces with projection
+operators Π ±. Then ifS= Π+−Π−we haveK= U(V+)×U(V−) and hence
+ZSO(G)≃U(V)/(U(V+)×U(V−)) (3.13)
+19orZSO(G)≃UN/(Up×UN−p) withp= dimV+.
+The space (3.13) is a symmetric space of the AIII family. Its symmetry class
+is commonly associated with random-matrix models for the lo w-energy Dirac
+spectrum of quantum chromodynamics with massless quarks [V er94]. An al-
+ternative realization exists [ASZ02] in T-invariant spin-singlet superconductors
+withd-wave pairing and soft impurity scattering.
+3.5.6 Classes BDI andCII
+Finally, let the symmetry group Ghave the full form of Definition 3.2, with
+G0= U1and˜Cas before (Class AIII) and a time-reversal symmetry T,T2=
+±Id. We recall that the elements of ZSO(G0) areu=k⊕(k−1)t,k∈U(V).
+The requirement of commutation with the product φ:=˜CT:V→V∗of
+anti-unitary symmetries is equivalent to the condition φ=ktφk.
+LetU:=ZSO(G0∪˜CTG0). To identify U, we use that φ(v)(v′) =/an}bracketle{tSTv,v′/an}bracketri}ht
+andST=TS. By the computation of (2.9), it follows that the parity of T
+equals the parity of the isomorphism φ:V→V∗. In other words, if T2=ǫId
+thenφt=ǫφ. Thus the condition φ=ktφksingles out an orthogonal group
+U= O(V)≃ONin the symmetric case ( ǫ= +1) and a unitary symplectic
+groupU= USp(V)≃USpNin the alternating case ( ǫ=−1).
+Inbothcases,thewantedsetis ZSO(G) =U/KwithKthesubgroupoffixed
+pointsk=˜C−1(k−1)t˜C=SkS. In the former case we have K≃Op×ON−p,
+and in the latter case K≃USpp×USpN−p(with even N,p). Thus we arrive
+at the final two entries of Cartan’s list:
+ClassBDI :U/K≃ON/(Op×ON−p) ( T2= +1),
+ClassCII :U/K≃USpN/(USpp×USpN−p) (T2=−1).
+These occur as symmetry classes in the context of the massles s Dirac operator
+[Ver94]. Class BDI is realized by taking the gauge group to be SU 2or USp2n,
+ClassCII by taking fermions in the adjoint representation or gauge group SO n.
+4 Discussion
+Given the classification scheme for disordered fermions, it is natural to ask
+whetherananalogousschemecanbedevelopedforthecaseofb osons. Although
+there exists no published account of it (see, however, [LSZ0 6]), we now briefly
+outline the answer to this question.
+The mathematical model for the bosonic Fock space is a symmet ric algebra
+S(V). It is still equipped with a canonical Hermitian structure induced by
+that ofV. The real form WRof Nambu space W=V⊕V∗for bosons has an
+interpretation as a classical phase space spanned by positi onsqj= (aj+a†
+j)/√
+2
+and momenta pj= (aj−a†
+j)/√
+2i. At the level of one-body unitary time
+20evolutions in Fock space, the role of the spin group Spin( WR) for fermions is
+handed over to the metaplectic group Mp( WR) for bosons.
+By the quantum-classical correspondence, a one-parameter group of time
+evolutionsut= e−itH//planckover2pi1∈Mp(WR) in Fock space gets assigned to a linear
+symplectic flow τ(ut)∈Sp(WR) in classical phase space. This correspondence
+τ: Mp(WR)→Sp(WR) is still two-to-one (reflecting, e.g., the well-known
+fact that the sign of the harmonic oscillator wave function i s reversed by time
+evolution over one period). An important difference as compa red to fermions
+is that the classical flow τ(ut)∈Sp(WR) is not unitary in any natural sense.
+In nuclear physics, the differential equation of the flow τ(ut) is called the
+RPAequation. Forexample, inthecasewithoutsymmetriesth isequationreads
+d
+dta†
+k=/summationdisplay
+j/parenleftbig
+a†
+jAjk+ajBjk/parenrightbig
+,d
+dtak=/summationdisplay
+j/parenleftbig
+a†
+jCjk+ajDjk/parenrightbig
+,(4.1)
+where one requires B=Bt,C=Ct, andD=−Atin order for the canonical
+commutation relations of the boson operators a†,ato be conserved. Unitarity
+of the flow (as a time evolution in Fock space) requires A=−A†andC=B†.
+This should be compared with the fermion problem in Class C, where one has
+exactly the same set of equations but for a single sign change: C=−B†. Thus
+the corresponding generator of time evolution is X=/parenleftbiggA B
+±BA/parenrightbigg
+where the
+plus sign applies to bosons and the minus sign to fermions. In either case X
+belongs to the samecomplex Lie algebra, sp(W). The difference is that the
+generator for fermions lies in a compact real form usp(W)⊂sp(W), whereas
+the generator for bosons lies in a non-compact real form sp(WR)⊂sp(W).
+This remains true in the general case with symmetries. Thus i f the word
+‘symmetry class’ is understood in the complex sense, then th e bosonic setting
+does not lead to any new symmetry classes; it just leads to diff erent real forms
+of the known symmetry classes viewed as complex spaces. The s ame statement
+applies to the non-Hermitian situation. Indeed, all of the s paces of [Mag08] are
+complex or non-compact real forms of the symmetric spaces of Cartan’s table.
+Here we must reiterate that the notion of symmetry class is an algebraic one
+whose prime purpose is to inject an organizational principl e into the multitude
+of possibilities. It must not be misunderstood as a cheap veh icle to produce
+immediate predictions of eigenvalue distributions and uni versal behavior!
+Let us end with a few historical remarks. The disordered harm onic chain,
+a model in the post-Dyson Class BDI, was first studied by Dyson [Dys53].
+The systematic field-theoretic study of models with sublatt ice symmetry (later
+recognized as members of the chiral classes AIII,BDI) was initiated by Op-
+permann, Wegner and Gade [OW79, Gad93, GW91]. Gapless super conductors
+were the subject of numerous papers by Oppermann; e.g., [Opp 90] computes
+the one-loop beta function of the non-linear sigma model for ClassCI.
+21The 10-way classification of Section 3 was originally discov ered by a very
+different reasoning: the mapping of random matrix problems t o effective field-
+theorymodels[Zir96]combinedwiththefactthatclosureof therenormalization
+group flow takes place for non-linear sigma models where the t arget is a sym-
+metric space. A less technical early confirmation of the 10-w ay classification
+came from Wegner’s flow equations [Weg94]. These take the for m of a double-
+commutator flow for Hamiltonians Hbelonging to a matrix space p; if the
+double commutator [ p,[p,p]] closes in p, so does Wegner’s flow. The closure
+condition is satisfied precisely if pis the odd part of a Lie algebra u=k⊕p
+with involution, i.e., the infinitesimal model of a symmetri c space.
+Lastbutnotleast, letusmentiontheviewpointofVolovik(s ee, e.g., [Vol03])
+who advocates classifying single-particle Green’s functi ons rather than Hamil-
+tonians. That viewpoint in fact has the advantage that it is n ot tied to non-
+interacting systems but offers a natural framework in which t o include (weak)
+interactions.
+References
+[ASZ02] A. Altland, B.D. Simons, and M.R. Zirnbauer, “Theor ies of low-
+energy quasi-particle states in disordered d-wave superconductors”
+Phys. Rep. 359(2002) 283-354
+[AZ97] A. Altland and M.R. Zirnbauer, “Non-standard symmet ry classes in
+mesoscopic normal-/superconducting hybrid systems”, Phy s. Rev. B
+55(1997) 1142-1161
+[Ber84] G. Bergmann, “Weak localization in thin films, a time -of-flight ex-
+periment with conduction electrons”, Phys. Rep. 107(1984) 1-58
+[BL02] D. Bernard and A. LeClair, “A classification of random Dirac
+fermions”, J. Phys. A 35(2002) 2555-2567
+[CM04] M. Caselle and U. Magnea, “Random matrix theory and sy mmetric
+spaces”, Phys. Rep. 394(2004) 41-156
+[Dys53] F.J. Dyson, “The dynamics of a disordered linear cha in”, Phys. Rev.
+92(1953) 1331-1338
+[Dys62] F.J. Dyson, “The threefold way: algebraic structur e of symmetry
+groups and ensembles in quantum mechanics”, J. Math. Phys. 3
+(1962) 1199-1215
+[Dys62a] F.J. Dyson, “Statistical theory of energy levels o f complex systems”,
+J. Math. Phys. 3(1962) 140-156
+22[Dys70] F.J. Dyson, “Correlations between eigenvalues of a random matrix”,
+Commun. Math. Phys. 19(1970) 235-250
+[Gad93] R. Gade, “Anderson localization for sublattic mode ls”, Nucl. Phys.
+B398(1993) 499-515
+[GW91] R. Gade and F. Wegner, “The n= 0 replica limit of U( n) and
+U(n)/SO(n) models”, Nucl. Phys. B 360(1991) 213-218
+[HHZ05] P. Heinzner, A.H. Huckleberry, and M.R. Zirnbauer, “Symmetry
+classes of disordered fermions”, Commun. Math. Phys. 257(2005)
+725-771
+[Hel78] S. Helgason, Differential geometry, Lie groups and symmetric spaces ,
+Academic Press, New York 1978
+[Kit08] A. Kitaev, “Periodic table for topological insulat ors and supercon-
+ductors”, AIP Conf. Proc. 1134 (2009) 22-30 [arXiv:0901.26 86]
+[KS99] N. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and
+monodromy , American Mathematical Society, Providence, R.I., 1999
+[LSZ06] T. Lueck, H.-J. Sommers, and M.R. Zirnbauer, “Energ y correlations
+for a random matrix model of disordered bosons”, J. Math. Phy s.47
+(2006) 103304
+[Mag08] U. Magnea, “Random matrices beyond the Cartan class ification”, J.
+Phys. A 41(2008) 045203
+[Meh04] M.L. Mehta, Random Matrices , 3rdEdition, Academic Press, London
+2004
+[Opp90] R. Oppermann, “Anderson localization problems in g apless supercon-
+ducting phases”, Physica A 167(1990) 301-312
+[OW79] R. Oppermann and F. Wegner, “Disordered system with norbitals
+per site – 1 /nexpansion”, Z. Phys. B 34(1979) 327-348
+[SRF09] A.P. Schnyder, S. Ryu, A. Furusaki, A.W.W. Ludwig, “ Classification
+of topological insulators and superconductors”, AIP Conf. Proc. 1134
+(2009) 10-21 [arXiv:0905.2029]
+[Ver94] J. Verbaarschot, “Spectrum of the QCD Dirac operato r and chiral
+random-matrix theory”, Phys. Rev. Lett. 72(1994) 2531-2533
+[Vol03] G.E. Volovik, The universe in a helium droplet , Clarendon Press,
+Oxford 2003
+23[Weg94] F. Wegner, “Flow equations for Hamiltonians”, Anna len d. Physik 3
+(1994) 77-91
+[Zir96] M.R.Zirnbauer, “Riemanniansymmetricsuperspace sandtheirorigin
+in random matrix theory”, J. Math. Phys. 37(1996) 4986-5018
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+24
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+arXiv:1001.0724v1  [nlin.SI]  5 Jan 2010Infinitely many symmetries and conservation laws
+for quad-graph equations via the Gardner method
+Alexander G. Rasin
+Department of Mathematics,
+Bar-Ilan University, Ramat Gan, 52900, Israel
+rasin@math.biu.ac.il
+November 6, 2018
+Abstract
+Theapplication of theGardnermethod forgeneration of cons ervation laws to
+all the ABS equations is considered. It is shown that all the n ecessary informa-
+tion for the application of the Gardner method, namely B¨ ack lund transforma-
+tions and initial conservation laws, follow from the multid imensional consistency
+of ABS equations. We also apply the Gardner method to an asymm etric equa-
+tion which is not included in the ABS classification. An analo g of the Gardner
+method for generation of symmetries is developed and applie d to discrete KdV.
+It can also be applied to all the other ABS equations.
+1 Introduction
+The first integrable example of a partial difference equation (P∆E) g oes back to [4],
+whowrotedownanequationonaquad-graphrelatedbyasimple chan geofvariablesto
+the discrete KdV equation. However, the subject has only been int ensively developed
+in the last few years. A particular milestone was the so-called ABS clas sification of
+integrable scalar quad-graph equations based on the principle of co nsistency on the
+cube [1]. All the equations in this classification have the discrete analo gue of a matrix
+Lax pair and also a natural auto-B¨ acklund transformation [2]. Oth er integrability
+properties (discrete analoguesofscalar Laxpairs, bihamiltonian st ructures and infinite
+numbers of conservation laws) have been studied less.
+The topic of conservation laws for the P∆E has become very importa nt recently.
+This started in the work of Orphanidis [12] in which he presents cons ervation laws for
+the fully discrete sine-Gordon equation. The next step in the resea rch of conservation
+laws was the introduction of a systematic computation method. The first systematic
+method developed was the direct method [6, 15, 16]. This method allow s use of
+computer algebra, which makes it more constructive. But it has som e disadvantages:
+it requires massive computations and cannot produce infinite numbe rs of conservation
+laws. The next method for computing conservation laws was propos ed in [17]. It
+produces infinite numbers of conservation laws by acting with a mast ersymmmetry on
+basic conservation laws.
+1A new approach to the computation of conservation laws for P∆E ha s appeared
+very recently [14]. It is called the Gardner method and is an analog of t he Gardner
+method for partial differential equations. The method uses a B¨ ac klund transformation
+(BT) togenerateconservation laws. Itseems thatGardner meth odisthemost efficient
+among the methods described above.
+Symmetries for P∆E were also researched recently. They first app eared as sim-
+ilarity constraints for integrable lattices [10]. Tongas et al.pointed out that the
+similarity constraints for quad-graph equations obtained previous ly are equivalent to
+characteristics of symmetries [20]. Symmetries of several quad-g raph equations have
+been found in [7, 8, 13, 19]. Hydon developed a direct method for find ing symmetries
+based on solving functional equations by creating an associated sy stem of differential
+equations that can be solved [5, 6]. In [18] this method was applied t o P∆E. There
+the authors compute five-point symmetries for all the ABS equatio ns (equations from
+[1]). Symmetries for P∆E were also discussed in other articles [9, 21].
+Our intention is to show that the Gardner method for generation of conservation
+laws for P∆E can be applied systematically. We apply it to all the ABS equ ations
+and to an asymmetric quad-graph equation. We also show that the G ardner method
+can be applied for symmetries. Namely, we use it to generate an infinit e number of
+symmetries from known one.
+2 The Gardner method
+Before presenting the Gardner method for generating conserva tion laws for P∆E, we
+review themethodforthecontinuum Korteweg-deVriesequation ( KdV).The Gardner
+method for KdV starts with the BT. The BT states that if usolves KdV then so does
+u+vxwherevis a solution of the system
+vx=θ−2u−v2
+vt=−1
+2uxx+(θ+u)/parenleftbig
+θ−2u−v2/parenrightbig
++uxv
+Hereθisaparameter. Itisstraightforwardtocheck that visactuallythe Gcomponent
+of a conservation law. Specifically, we have
+∂tv+∂x/parenleftbigg1
+2ux−(u+θ)v/parenrightbigg
+= 0.
+Butvdepends onthe parameter θ. Expanding ina suitable series in θyields aninfinite
+number of conservation laws. For KdV the appropriate series is
+v=θ1/2−u
+θ1/2+ux
+2θ−uxx+2u2
+4θ3/2+uxxx+8uux
+8θ2−uxxxx+8u3+10u2
+x+12uuxx
+16θ5/2+O/parenleftbig
+θ−3/parenrightbig
+.
+In[14]theimplementation oftheGardnermethodtothedKdVequat ionwasgiven.
+Here we apply the Gardner method to the following asymmetric quad- graph equa-
+tion [11]
+αu0,0u1,0−β(u0,0u0,1+u1,0u1,1) = 0. (1)
+Herek,l∈Z2are independent variables and u0,0=u(k,l) is a dependent variable that
+is defined on the domain Z2. We denote the values of this variable on other points by
+2ui,j=u(k+i,l+j) =Si
+kSj
+lu0,0, whereSk, Slare the unit forward shift operators in k
+andlrespectively. Equation (1) is not included in the ABS list, since it is asym metric.
+A BT for (1) is
+α(u0,0u1,0+ ˜u0,0˜u1,0)−θ(u0,0˜u0,0+u1,0˜u1,0) = 0, (2)
+θu0,0˜u0,0−β(u0,0u0,1+ ˜u0,0˜u0,1) = 0. (3)
+Hereθis a parameter. This BT follows from the multidimensional consistency of
+equations (1,2,3) embedded in three dimensions. Let us introduce th e third variable
+mso that
+uk,l,m=u0,0,0=u0,0, u0,0,1= ˜u0,0, u1,0,1= ˜u1,0, etc. (4)
+Then, equations (1,2,3) can be written as
+P1:αu0,0,0u1,0,0−β(u0,0,0u0,1,0+u1,0,0u1,1,0) = 0,
+P2:α(u0,0,0u1,0,0+u0,0,1u1,0,1)−θ(u0,0,0u0,0,1+u1,0,0u1,0,1) = 0,(5)
+P3:θu0,0,0u0,0,1−β(u0,0,0u0,1,0+u0,0,1u0,1,1) = 0.
+Equations P2andP3form a BT for P1. In the sequel we use the notation (4), writing
+u0,0,1andu0,0,−1for the BT andinverse BT of u0,0,0, where possible, however, we revert
+to 2 dimensional notation.
+In general we cannot solve the equations of the BT to write u0,0,1in terms of u0,0,0.
+However there is a special case θ=αfor which u0,0,1=u1,0,0oru−1,0,0. Consider
+θ=α+ǫwhereǫis small, and look for a solution of BT in the form
+u0,0,1=u1,0+∞/summationdisplay
+i=1v(i)
+0,0ǫi. (6)
+We just look at the first equation of the BT. This reads
+∞/summationdisplay
+i=1v(i)
+0,0ǫi/parenleftBigg
+αu2,0+α∞/summationdisplay
+i=1v(i)
+1,0ǫi−αu0,0−ǫu0,0/parenrightBigg
+=ǫu1,0/parenleftBigg
+u0,0+u2,0+∞/summationdisplay
+i=1v(i)
+1,0ǫi/parenrightBigg
+.
+The leading order approximation gives
+v(1)
+0,0=u1,0(u0,0+u2,0)
+α(u2,0−u0,0).
+Higher order terms give
+v(i)
+0,0=u0,0v(i−1)
+0,0+u1,0v(i−1)
+1,0−α/summationtexti−1
+j=1v(i)
+0,0v(j−i)
+0,0
+α(u2,0−u0,0), i= 2,3.... (7)
+AsinthecaseofthedKdVequationalltheseformulasareonthehor izontalline, i.e. all
+thev(i)
+0,0only depend on values of ui,jwithj= 0. An infinite sequence of conservation
+laws can be obtained starting from the ǫexpansion of a single conservation law. It is
+straightforward to check that if we define
+F= ln/parenleftbiggu0,0,1
+u0,0,0/parenrightbigg
+, G=−ln/parenleftbiggαu0,0,1−θu1,0,0
+u0,0,0/parenrightbigg
+, (8)
+3then
+(Sk−I)F+(Sl−I)G= 0,
+onsolutionsof(5). Byplugging(6)and θ=α+ǫinto(8)andexpanding F=/summationtext∞
+i=0Fiǫi
+andG=−ln(2ǫ)+/summationtext∞
+i=0Giǫiwe obtain
+F0= ln/parenleftbiggu1,0
+u0,0/parenrightbigg
+, F 1=u2,0+u0,0
+u2,0−u0,0,
+G0=−ln/parenleftbiggu1,0
+u2,0−u0,0/parenrightbigg
+, G 1=u1,0(3u2,0+u0,0)+u3,0(u2,0−u0,0)
+2u0,0(u3,0−u0,0)(u2,0−u0,0),
+F2=4u2,0u0,0(u3,0+u1,0)+(u2
+2,0−u2
+0,0)(u3,0−u1,0)
+2(u2,0−u0,0)2(u1,0−u3,0),
+G2=(u3,0u2,0+3u2,0u1,0+u1,0u0,0−u3,0u0,0)2
+8(u2,0−u0,0)2(u3,0−u1,0)2, etc.
+Thus we see how the expansion of the BT around the point θ=αyields an infinite
+sequence of conservation laws on the horizontal line. Expansion ar oundθ=βdoes
+not seem to yield an infinite sequence of conservation laws on the ver tical line, since
+(1) is asymmetric. In the case of the dKdV equation expansion arou ndθ=βalso
+gives an infinite number of conservation laws.
+3 The Gardner method for conservation laws of in-
+tegrable equations on the quad-graph
+In this section we present all the necessary information for the ap plication of the
+Gardner method to the quad-graphs that are listed in [1]. The gener al form of ABS
+equations is
+P1:P(u0,0,0,u1,0,0,u0,1,0,u1,1,0,α,β) = 0. (9)
+For the application of the Gardner method we need a special BT and t he initial
+conservation law. For all the ABS equations the necessary BTs are known and these
+are so called natural auto-BTs [2]. The general form of the natura l auto-BTs for ABS
+equations can be obtained from the form of the equation (9). For ( 9) the form of the
+natural auto-BT is
+P2:P(u0,0,0,u1,0,0,u0,0,1,u1,0,1,α,θ) = 0, (10)
+P3:P(u0,0,0,u0,0,1,u0,1,0,u0,1,1,θ,β) = 0. (11)
+This is the BT which we need for the application of the Gardner method . Note that
+equations (9,10,11) are one equation embedded in three dimensions.
+The initial conservation law(ICL) also follows from multidimensional co nsistency.
+Expressions for conservation laws for P1, P2, P3are respectively
+(Sk−I)F1+(Sl−I)G1= 0,
+(Sk−I)F2+(Sm−I)H2= 0, (12)
+(Sl−I)G3+(Sm−I)H3= 0.
+4Five-point conservation laws for ABS equations were found in [17], wh ere the authors
+show that each ABS equation has three five-point conservation law s. Two five-point
+conservation laws for P1can be always presented as
+F=f(u0,−1,0,u0,0,0,u0,1,0,β), G =g(u0,−1,0,u0,0,0,u1,0,0,α,β),(13)
+F′=g(u−1,0,0,u0,0,0,u0,1,0,β,α), G′=f(u−1,0,0,u0,0,0,u1,0,0,α).(14)
+As we said before, equations P1, P2, P3are one equation which is embedded in three
+dimensions. Therefore conservation laws for P1, P2can be obtained from conservation
+laws forP1by rewriting them on the appropriate plane. The conservation law fo rP2
+which corresponds to F, Gis
+F2=f(u0,0,−1,u0,0,0,u0,0,1,θ), H2=g(u0,0,−1,u0,0,0,u1,0,0,α,θ).(15)
+The conservation law for P3which corresponds to F, Gis
+G3=f(u0,0,−1,u0,0,0,u0,0,1,θ), H3=g(u0,0,−1,u0,0,0,u0,1,0,β,θ).(16)
+F2is equal to G3, so with the help of linear combination of expressions from (12) we
+obtain that
+(Sl−I)(Sk−I)F2+(Sl−I)(Sm−I)H2−(Sk−I)(Sl−I)G3
+−(Sk−I)(Sm−I)H3= (Sm−I)((Sl−I)H2−(Sk−I)H3) = 0
+is true on solutions of P2andP3. This expression can be integrated with respect to m
+to give
+(Sl−I)H2−(Sk−I)H3=C(k,l) (17)
+whereC(k,l) is a function which has to be determined. Equations P2andP3are
+symmetric with respect to the reflection of the coordinate m, that is
+P(u0,0,0,u1,0,0,u0,0,1,u1,0,1,α,θ) =±P(u0,0,1,u1,0,1,u0,0,0,u1,0,0,α,θ)
+=±SmP(u0,0,0,u1,0,0,u0,0,−1,u1,0,−1,α,θ).
+Therefore (17) is also symmetric with respect to the reflection of c oordinate m, so
+(Sl−I)g(u0,0,1,u0,0,0,u1,0,0,α,θ)−(Sk−I)g(u0,0,1,u0,0,0,u0,1,0,β,θ) =C(k,l).(18)
+By substitution of P2andP3into (18) we obtain that C(k,l) = 0 for all the ABS
+equations. Therefore
+(Sl−I)H2−(Sk−I)H3= 0,
+for all the ABS equations. Thus, the components of ICLs for all th e ABS equations
+are
+FICL=−g(u0,0,1,u0,0,0,u0,1,0,α,θ), GICL=g(u0,0,1,u0,0,0,u1,0,0,β,θ).
+We present a table with ICLs for all the ABS equations bellow. The equ ations from
+the ABS classification are as follows; for convenience, we have used the form of Q4
+5that was discovered by Hietarinta [3].
+Q1: α(u0,0−u0,1)(u1,0−u1,1)−β(u0,0−u1,0)(u0,1−u1,1)+δ2αβ(α−β) = 0,
+Q2: α(u0,0−u0,1)(u1,0−u1,1)−β(u0,0−u1,0)(u0,1−u1,1)
++αβ(α−β)(u0,0+u1,0+u0,1+u1,1)−αβ(α−β)(α2−αβ+β2) = 0,
+Q3: ( β2−α2)(u0,0u1,1+u1,0u0,1)+β(α2−1)(u0,0u1,0+u0,1u1,1)
+−α(β2−1)(u0,0u0,1+u1,0u1,1)−δ2(α2−β2)(α2−1)(β2−1)/(4αβ) = 0,
+Q4: sn(α)(u0,0u1,0+u0,1u1,1)−sn(β)(u0,0u0,1+u1,0u1,1)−sn(α−β)(u0,0u1,1+u1,0u0,1)
++sn(α−β)sn(α)sn(β)(1+K2u0,0u1,0u0,1u1,1) = 0, (19)
+H1: ( u0,0−u1,1)(u1,0−u0,1)+β−α= 0,
+H2: ( u0,0−u1,1)(u1,0−u0,1)+(β−α)(u0,0+u1,0+u0,1+u1,1)+β2−α2= 0,
+H3: α(u0,0u1,0+u0,1u1,1)−β(u0,0u0,1+u1,0u1,1)+δ2(α2−β2) = 0,
+A1: α(u0,0+u0,1)(u1,0+u1,1)−β(u0,0+u1,0)(u0,1+u1,1)−δ2αβ(α−β) = 0,
+A2: ( β2−α2)(u0,0u1,0u0,1u1,1+1)+β(α2−1)(u0,0u0,1+u1,0u1,1)
+−α(β2−1)(u0,0u1,0+u0,1u1,1) = 0.
+Here sn(α) = sn(α;K) is a Jacobi elliptic function with modulus K. Without loss
+of generality, the parameter δis restricted to the values 0 and 1. Obtained ICLs are
+summarized in table 1, in which we list the components FandGfor each of the ABS
+equations.
+Table 1: Initial conservation laws for the equations from the ABS cla ssifica-
+tion
+Eq. Components
+Q1F= ln(−˜u0,0+u0,0−δθ)−ln(δ(β−θ)+u0,1−˜u0,0),
+G=−ln(u0,0−˜u0,0−δθ)+ln(δ(α−θ)+u1,0−˜u0,0),
+Q2F= ln((θβ2+β(˜u0,0−u0,0−θ2)+θ(u0,0−u0,1))2)
+−ln(β4−2β2(u0,1+u0,0)+(u0,0−u0,1)2),
+G=−ln((θα2+α(˜u0,0−u0,0−θ2)+θ(u0,0−u1,0))2)
++ln(α4−2α2(u1,0+u0,0)+(u0,0−u1,0)2),
+Q3F= ln((θ2(u0,0−βu0,1))2−θ(1−α2)˜u0,0−α2u0,0+α2u0,1)
+−ln(4β(βu0,0−u0,1)(u0,0−βu0,1)+δ(1−β2)2),
+G=−ln((θ2(u0,0−αu1,0)−θ˜u0,0(1−α2)−α2u0,0+αu1,0)2)
++ln(4α(αu0,0−u1,0)(u0,0−αu1,0)+δ(1−α2)2),
+Q4F= ln(sn(θ)2(1+K2u0,0˜u0,0)+2cn(θ)dn(θ)u0,0˜u0,0−u2
+0,0−˜u2
+0,0)
+−ln(sn(θ−β)2(u0,1˜u0,0−sn(θ)sn(β))(sn(θ)sn(β)K2˜u0,0u0,1−1)
+−(sn(θ)u0,1−sn(β)˜u0,0)(sn(β)u0,1−sn(θ)˜u0,0)),
+G=−ln(sn(θ)2(1+K2u0,0˜u0,0)+2cn(θ)dn(θ)u0,0˜u0,0−u2
+0,0−˜u2
+0,0)
++ln(sn(α−θ)2(u1,0˜u0,0−sn(θ)sn(α))(sn(θ)sn(α)K2˜u0,0u1,0−1)
+−(sn(θ)u1,0−sn(α)˜u0,0)(sn(α)u1,0−sn(θ)˜u0,0)),
+Continued on next page
+6Eq. Components
+H1F=−ln(˜u0,0−u0,1),
+G= ln(˜u0,0−u1,0),
+H2F= ln((β−θ+u0,1−˜u0,0)2)−ln(β+u0,0+u0,1),
+G=−ln((α−θ+u1,0−˜u0,0)2)+ln(α+u0,0+u1,0),
+H3F= ln((β˜u0,0−θu0,1)2)−ln(δβ+u0,0u0,1),
+G=−ln((α˜u0,0−θu1,0)2)+ln(δα+u0,0u1,0),
+A1F= ln((u0,0+ ˜u0,0)2−δθ2)−ln((u0,1−˜u0,0)2−δ(θ−β)2),
+G=−ln((u0,0+ ˜u0,0)2−δθ2)+ln((u1,0−˜u0,0)2−δ(α−θ)2),
+A2F= ln(˜u0,0u0,0−θ)(θ˜u0,0u0,0−1)−ln((θu0,1−β˜u0,0)(βu0,1−θ˜u0,0)),
+G=−ln(˜u0,0u0,0−θ)(θ˜u0,0u0,0−1)+ln((θu1,0−α˜u0,0)(αu1,0−θ˜u0,0)),
+Another problem which can appear during the application of the Gard ner method
+is thatv(i)
+0,0has to be found for i= 1,2,...and for all the ABS equations. We prove
+that it is always possible to explicitly find v(i)
+0,0fori= 1,2,...for all the ABS equations
+in the application of the Gardner method in the kdirection. For the ldirection the
+proof is similar. Consider θ=α+ǫwhereǫis small, and look for a solution of BT in
+the form
+u0,0,1=u1,0+∞/summationdisplay
+i=1v(i)
+0,0ǫi. (20)
+We look only at the first equation of BT. This reads
+P/parenleftBigg
+u0,0,u1,0,u1,0+∞/summationdisplay
+i=1v(i)
+0,0ǫi,u2,0+∞/summationdisplay
+i=1v(i)
+1,0ǫi,α,α+ǫ/parenrightBigg
+= 0. (21)
+LetCnbe the coefficient next to the ǫnin the expansion of (21) in the Taylor series
+aroundǫ= 0. At first sight Cndepends upon v(i)
+0,0,v(i)
+1,0, i= 1,2,...nand to find v(i)
+0,0one
+has to solve non-trivial difference equations. We show that that Cndepends just upon
+v(i)
+0,0,v(j)
+1,0, i= 1,2,...n, j= 1,2,...n−1, soCncan be solved explicitly with respect to
+v(n)
+0,0. First of all let us show
+C0=P(u0,0,u1,0,u1,0,u2,0,α,α) = 0
+SincePis an ABS equation it has the symmetry property [1]
+P(x,u,v,y,α,β )=−P(x,v,u,y,α,β ).
+According to this symmetry property we obtain
+P(u0,0,u1,0,u1,0,u2,0,α,α) =−P(u0,0,u1,0,u1,0,u2,0,α,α) = 0.
+The derivative of (21) with respect ǫis
+d
+dǫP=∞/summationdisplay
+i=1iv(i)
+0,0ǫi−1P3+∞/summationdisplay
+i=1iv(i)
+1,0ǫi−1P4+P6, (22)
+7wherePiis the derivative of Pwith respect to its ith argument. For example
+P3=∂
+∂aP/parenleftBigg
+u0,0,u1,0,a,u2,0+∞/summationdisplay
+i=1v(i)
+1,0ǫi,α,α+ǫ/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+a=u1,0+/summationtext∞
+i=1v(i)
+0,0ǫi.
+Forǫ= 0 we obtain C1
+C1=d
+dǫP/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ǫ=v(1)
+0,0P3(u0,0,u1,0,u1,0,u2,0,α,α)+v(1)
+1,0P4(u0,0,u1,0,u1,0,u2,0,α,α)
++P6(u0,0,u1,0,u1,0,u2,0,α,α).(23)
+We checked that for all the ABS equations
+P3(u0,0,u1,0,u1,0,u2,0,α,α)/negationslash= 0, P4(u0,0,u1,0,u1,0,u2,0,α,α) = 0.
+So,C1depends upon v(1)
+0,0and does not depend upon v(1)
+1,0. From (22) it is seen that
+generally the form of Cnis similar to the form of C1, namely
+Cn=n!v(n)
+0,0P3(u0,0,u1,0,u1,0,u2,0,α,α)+n!v(n)
+1,0P4(u0,0,u1,0,u1,0,u2,0,α,α)+R
+HereRdoesnotdependupon v(n)
+0,0, v(n)
+1,0. Aswesaidbefore P4(u0,0,u1,0,u1,0,u2,0,α,α) =
+0 therefore Cncan be solved with respect to v(n)
+0,0. By solving Ciwith respect to v(i)
+0,0
+iteratively for i= 1,2,...we can find as many v(i)
+0,0as necessary. Once v(i)
+0,0, i= 1...n
+are known, we plug (20) into the initial conservation law and expand it aroundǫ= 0
+up to order n. In this way nconservation laws can be found.
+4 The Gardner method for the symmetries of in-
+tegrable equations on the quad-graph
+In [18] five-point symmetries were found for ABS equations. The au thors show there
+that each ABS equation P1has four five-point symmetries. Two of these symmetries
+can always be presented as
+Xh=η(u−1,0,0,u0,0,0,u1,0,0,α)∂u0,0,0, (24)
+Xv=η(u0,−1,0,u0,0,0,u0,1,0,β)∂u0,0,0. (25)
+Therefore equations P2andP3both have a symmetry
+X=η(u0,0,−1,u0,0,0,u0,0,1,θ)∂u0,0,0. (26)
+SinceP1, P2andP3are consistent in three dimensions we obtain that Xis also a
+symmetry for P1. Let us call Xtheinitial symmetry forP1. Let us apply the Gardner
+method to Xor, in other words, expand Xin a series. Xinvolves the BT and inverse
+BTu0,0,1andu0,0,−1, therefore we are looking for the solution of those for θ=α±ǫ
+whereǫis small in the forms
+u0,0,1=u1,0,0+∞/summationdisplay
+i=1v(i)
+0,0ǫi, (27)
+u0,0,−1=u−1,0,0+∞/summationdisplay
+i=1w(i)
+0,0ǫi. (28)
+8By plugging this in the characteristic ηof the symmetry Xwe obtain
+η=η(u−1,0,0+∞/summationdisplay
+i=1w(i)
+0,0ǫi,u0,0,0,u1,0,0+∞/summationdisplay
+i=1v(i)
+0,0ǫi,α+ǫ). (29)
+After expansion of ηin Taylor series around ǫ= 0 we obtain
+η=∞/summationdisplay
+i=1ηiǫi. (30)
+Eachηi, i= 1,2,...is the characteristic of a symmetry for P1. For example let us
+consider the discrete KdV equation
+(u0,0,0−u1,1,0)(u1,0,0−u0,1,0)+β−α= 0. (31)
+Two of the symmetries for this equation are
+Xh=1
+u1,0,0−u−1,0,0∂u0,0,0, (32)
+Xv=1
+u0,1,0−u0,−1,0∂u0,0,0. (33)
+Therefore
+X=1
+u0,0,1−u0,0,−1∂u0,0,0. (34)
+is also a symmetry. Expression (29) for Xis
+η=1
+u1,0,0−u−1,0,0+/summationtext∞
+i=1(v(i)
+0,0−w(i)
+0,0)ǫi.
+We just look at the first equation of BT which is P2. This reads
+ǫ=/parenleftBigg∞/summationdisplay
+i=1v(i)
+0,0ǫi/parenrightBigg/parenleftBigg
+u0,0,0−u2,0,0−∞/summationdisplay
+i=1v(i)
+1,0ǫi/parenrightBigg
+, (35)
+ǫ=/parenleftBigg∞/summationdisplay
+i=1w(i)
+1,0ǫi/parenrightBigg/parenleftBigg
+u−1,0,0−u1,0,0+∞/summationdisplay
+i=1w(i)
+0,0ǫi/parenrightBigg
+. (36)
+The leading order approximation gives
+v(1)
+0,0=1
+u0,0,0−u2,0,0, (37)
+w(1)
+1,0=1
+u−1,0,0−u1,0,0. (38)
+Higher order terms give
+v(i)
+0,0=1
+u0,0,0−u2,0,0i−1/summationdisplay
+j=1v(j)
+0,0v(i−j)
+1,0, i= 2,3,... , (39)
+w(i)
+1,0=−1
+u−1,0,0−u1,0,0i−1/summationdisplay
+j=1w(j)
+0,0w(i−j)
+1,0, i= 2,3,.... (40)
+9The first three coefficients in the expansion of ηaroundǫ= 0 are
+η1=1
+u1,0−u−1,0,
+η2=u−2,0−u2,0
+(u−1,0−u1,0)2(u0,0−u2,0)(u0,0−u−2,0),
+η3=u−1,0−u3,0
+(u0,0−u2,0)2(u1,0−u3,0)(u−1,0−u1,0)3+2
+(u0,0−u2,0)(u−2,0−u2,0)(u−1,0−u1,0)3
++u−3,0−u1,0
+(u0,0−u−2,0)2(u−3,0−u−1,0)(u−1,0−u1,0)3−2
+(u0,0−u−2,0)(u−2,0−u2,0)(u−1,0−u1,0)3.
+These are characteristics of symmetries for (31). η1andη2were already presented in
+[20].
+5 Concluding remarks
+In this article we showed how to apply the Gardner method for gener ation of conser-
+vation laws to all the ABS equation. It is shown that all the necessar y information for
+the application of the Gardner method follows from the multidimension al consistency
+of ABS equations. Namely the natural auto-BT for ABS equation is t he equation itself
+embedded in three dimensions. This construction is possible because of the multidi-
+mensional consistency of the equation. The initial conservation law for ABS equations
+follows from the linear combinations of conservation laws for the cor responding em-
+bedded equations.
+Another important result of this article is the introduction of the Ga rdner method
+for symmetry generation for ABS equations. In the Gardner meth od for the symmetry
+generation one has to substitute expressions
+u0,0,1=u1,0,0+∞/summationdisplay
+i=1v(i)
+0,0ǫi, u0,0,−1=u−1,0,0+∞/summationdisplay
+i=1w(i)
+0,0ǫi.
+into the initial symmetry. The expansion of this symmetry in a Taylor s eries gives an
+infinite number of symmetries.
+Application of the Gardner method for generation of conservation laws for the
+asymmetric equation
+αu0,0u1,0−β(u0,0u0,1+u1,0u1,1) = 0. (41)
+is also considered. In this case we can again say that all necessary in formation for
+application of the method follows from multidimensional consistency. Equation (41)
+is consistent with the symmetric equation
+α(u0,0u1,0+u0,1u1,1)−β(u0,0u0,1+u1,0u1,1) = 0. (42)
+We obtain a three dimensional consistent system by imposing (42) on 2 opposite sides
+of a cube and (41) on the remaining 4 sides. From this consistency fo llows the Lax
+pair for (41)
+L=
+1αu1,0
+αλ2
+u0,0−u1,0
+u0,0
+, M=
+1βu0,1
+βλ2
+u0,00
+.
+10Equation (41) satisfies all the criteria of integrability [22], namely the re exist infinite
+number of conservation laws and symmetries, and a Lax pair. As is se en from this
+article all these properties are connected with multidimensional con sistency. So, mul-
+tidimensional consistency is a universal criterion of integrability. It is not necessary
+for an equation to be consistent with itself, as we showed for (41) it can be consistent
+with other equations.
+Here are some interesting topics for future research:
+•The conditions for the application of the Gardner method to the con sistent equa-
+tion (equation and its BT).
+•The classification of quad-graph equations according to these con ditions.
+References
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+arXiv:1001.0725v3  [gr-qc]  8 Nov 2010A useful guide for gravitational wave observers to test modi fied gravity models
+E. O. Kahya1,∗
+1Theoretisch-Physikalisches Institut, Friedrich-Schill er-Universit¨ at Jena, Max-Wien-Platz 1, D-07743 Jena, Ger many
+(Dated: October 30, 2018)
+We present an extension of a previously suggested test of all modified theories of gravity that
+would reproduce MOND at low accelerations. In a class of mode ls, called “dark matter emulators,”
+gravitational waves and other particles couple to different metrics. This leads to a detectable time
+lag between their detection at Earth from the same source. We calculate this time lag numerically
+for any event that occurs in our galaxy up to 400 kpc, and prese nt a graph of this possible time lag.
+This suggests that, gravitational wave observers might hav e to consider the possibility of extending
+their analysis to non-coincident gravitational and electr omagnetic signals, and the graph that we
+present might be a useful guideline for this effort.
+PACS numbers: 98.80.Cq, 98.80.Hw, 04.62.+v
+I. INTRODUCTION
+The last forty years have been a very active and an
+exciting period for cosmologists. The breakthrough was
+the discovery of cosmic microwave background radiation
+(CMBR), an open window to the very early stages of
+the universe [1]. We realized that the universe is homo-
+geneous and isotropic on large scales [2, 3]. And very
+recently we learned that the universe is accelerating con-
+trary to the common belief [4, 5].
+Another very exciting observational discovery was the
+peculiar shape of the so called “galaxy rotation curve”.
+Instead of the velocityof starsorbiting spiral galaxiesbe-
+ing inversely proportional to the distance away from the
+center of the galaxy, it was observed to stay almost con-
+stant [6–8] after certain point. Then even more suprising
+observational fact was the excessive amount of distortion
+inthebackgroundgalaxiesnearaforegroundmass[9–14].
+Assuming that there is only baryonic matter in the uni-
+verse, it is not possible to explain these two observations
+with Einstein’s equations,
+Gµν=8πG
+c4Tµν. (1)
+Two of the following can be done to avoid this discrep-
+ancy. We can modify the right hand side by assuming
+additional dark matter, that would interact only gravi-
+tationally. Or we can modify the left hand side of the
+equation, namely the Einstein’s tensor. The latter ap-
+proach suggests to modify general relativity on cosmo-
+logical scales. And it was partially achieved in 1983 by
+Milgrom [15] with the so called “MOND” for Modified
+Newtonian Dynamics. Partially, because it is not a mod-
+ification of generalrelativity but Newtonian dynamics on
+cosmological scales. The modification of GR would then
+be the underlying fundamental theory that would give
+MOND in the non-relativistic limit.
+∗emre-onur.kahya@uni-jena.deIn order to understand Eq.(1), one needs to be able
+to handle both sides of the equation. That means iden-
+tifying dark matter directly, by a non-gravitational ex-
+periment. Therefore direct detection is crucial for dark
+matter. For more than 30 years, experimentalists have
+been trying to detect dark matter without any success.
+Very recentlysome excitement aroseabout the detection,
+but turned out to be almost no signal with excluding
+new parameter space[16]. On the other hand, the early
+relativistic [17], and the Scalar-Tensor [18] formulations
+of MOND led to, so called TeVeS for “Tensor-Vector-
+Scalar”, which was recently proposed by Bekenstein [19].
+It reproduces the MOND in the non-relativistic limit and
+does a good job on cosmological evolution, strong grav-
+itational lensing as well as post-Newtonian parameters
+[20, 21]. But weak lensing is problematic in the case of
+colliding clusters, such as the Bullet Cluster [22–25]. The
+observedthirdpeakofnew5-yearWMAPdata[26]isalso
+problematic in the context of MOND-like models. But
+still one can not claim that these models are ruled out
+[27, 28]. Thefore it is very important to find a way of
+testing these two explanations by means of observation.
+The possibility for comparing dark matter with
+MOND-like models by means of a model-independent
+test was suggested before[30–32]. It was proposed that
+one would see a time lag between arrival times of gravita-
+tional waves and other massless particles. And the time
+lag was calculated for three different sources; a particu-
+lar gamma-rayburst(GRB) GRB070201, the low mass x-
+raybinary(LMXB) SCo-X1, and the supernovaSN1987a.
+Otherproposed tests ofTeVes with gravitywavesare dis-
+cussed in [29] and references therein. In Sec. 2 we sum-
+marize the formulation of this suggested test. In Sec.
+3 we extend this calculation numerically to any source
+in Milky Way galaxy. In Sec. 4 we discuss the conse-
+quences of this effect to the externally triggered searches
+for GWs. In Sec. 5 we investigate the ambiguities in this
+calculation, and see how sensitive the time lag is to the
+choice of using different data sets. Our conclusions are
+summarized in Sec. 6.2
+II. FORMULATION
+TeVeSisanimpressiveachievementbutanaturalques-
+tion arises when one thinks about multiple metric for-
+malisms. Why do we need to have multiple metrics? It
+was realized that[33] if one wants to generalize MOND,
+toget Tully-Fisherrelation[34] andsufficient lensing, to a
+stable, relativistic, covariant pure metric theory without
+dark matter, it would not be possible. This no-go theo-
+rem restricts the modified gravity models, that are con-
+sistent with Tully-Fisher relationship and the observed
+amount of lensing. One can violate one of the assump-
+tions of the no-go theorem by introducing another metric
+which would carry the MOND force. These class of mod-
+ified gravity models with multiple metrics, which would
+give MOND in the non-relativistic limit, were called as
+“dark matter emulators”. TeVeS and the scalar-vector-
+tensorgravitytheorythat wasproposedbyMoffat[35,36]
+are examples of dark matter emulators. These models
+all have the following property: All the ordinary matter
+should move on a geometry as if there is dark matter,
+whereas the gravity waves move on a geometry without
+any. The total time needed for a gravitational wave to
+reach the Earth from a given source is the sum of dis-
+tance divided by speed of light and additional delay due
+togravitatiionalpotentialofinterveningmatteralongthe
+line of sight (also known as Shapiro delay) [37, 38]. This
+can be restated as follows: Gravity waves couple to the
+usual metric of GR without dark matter and the rest of
+the matter couples to a different metric that would be
+produced with GR plus dark matter. So the answer to
+the question at the beginning of this section is, we might
+have to introduce multiple metrics for a modified grav-
+ity model in order to explain Tully-Fisher relationship
+and the observed amount of lensing consistently, without
+dark matter.
+Heuristically the calculation can be understood with a
+simple picture. Let us compare the arrival times of two
+massless particles that are produced by a distant source.
+The gravity wave is going to move on a geometry where
+there is only ordinary matter, and the other ordinary
+massless particle, which can be a photon or a neutrino,
+will move in a geometry where there is dark matter as
+well as the ordinary matter. Because of this additional
+mass the ordinary massless particle would have to travel
+a geometryasif the distance is longer, which would mean
+that gravity waves arrive earlier than all the rest of the
+massless particles. In other words,, because of this addi-
+tional mass, the ordinary massless particles feel an extra
+Shapiro delay as compared to gravity waves. Therefore
+the procedure is the following: Calculate the arrival time
+of a massless particle by solving the geodesic equation
+with a massdensity being just that ofdarkmatter, which
+would be equal to the difference of arrival times of these
+two massless particles.
+We will calculate the time lag in the context of bi-
+metric theories mimicking cold dark matter which has a
+spherically symmetric distribution. One can express thetime lag of an event in terms of its initial ( /vector x1) and final
+(/vector x2) positions, by solving corresponding geodesic equa-
+tions perturbatively. The result [32] for the time lag is,
+c∆t=∆/vector x·/vector x1
+2∆x∆B(r1)−∆/vector x·/vector x2
+2∆x∆B(r2)
++/integraldisplayr1
+r2dr2GM(r)
+c2r/radicalbigg
+1−r2
+1∆x2−(/vector x1·∆/vector x)2
+r2,(2)
+where
+∆B(r) =−2G
+c2M(r)
+r−2G
+c2/integraldisplay∞
+rdr′M(r′)
+r′.(3)
+M(r) is the mass function,
+M(r)≡4π/integraldisplayr
+0dr′ρ(r′). (4)
+Therefore, in order to mimick dark matter one has to
+choose a dark matter density profile ρ(r) and that will
+determine the amount of time lag.
+III. CALCULATION
+In this Section, we revisit a calculation that was done
+in a previous work. The time lag of photons from
+GRB070201[32], a short duration Gamma Ray Burst
+(GRB) which is believed to have happened at the An-
+dromeda Galaxy, was calculated. To evaluate the total
+time lag, one needs to include contributions from both
+the Milky Way and Andromeda galaxies. In the men-
+tioned work, the data for the density profiles of these two
+galaxies were taken from two different groups. Therefore
+we decided to do the calculation again, based on the data
+of a group where they did a numerical work on the den-
+sity profile of both of the galaxies [39].
+As stated above, to calculate the time lag we need to
+know the form of the dark matter density profile. The
+NFW profile can be used for this purpose, which assumes
+the following density profile of dark matter halos in nu-
+merical simulations [40],
+ρ(r) =ρ0
+r
+r0[1+r
+r0]2. (5)
+The parameters, that we need to calculate the time
+lag, areρ0the characteristic density, and r0the radius of
+the halo. The summary of the Shapiro delays for GRB
+070201 is given in Table I. One can notice that the time
+lag becomes larger with increasing virial mass of the An-
+dromeda galaxy.
+The next step is considering a source which is in the
+Milky Way and estimating the time lag by solving Eq.(2)
+for ∆t. At this point it is useful to analyze an ap-
+proximate expression for ∆ tusing the previous work of
+Woodard and Soussa [43]. Forcing the dark matter em-
+ulators to mimick dark matter isothermal halo distribu-
+tion, results into the following approximate values for3
+Data Set ρ(GeV/cm3)Mvir(M⊙)∆t(days)
+MW−Klypin [39] 0.185 1.0×1012426
+MW−Ascasibar [41] 0.347 1.0×1012421
+M31−Klypin [39] 0.188 1.6×1012634
+M31−Tempel [42] 0.661 1.0×1012383
+TABLE I: Shapiro delays for GRB 070201 from the NFW
+profiles of the Milky Way and Andromeda using two different
+data sets.
+∆A(r) and ∆B(r):
+A(r)≈1 +ǫ+ǫ∗, B(r)≈1−ǫ+ǫ∗ln(r
+rs).(6)
+Hereǫis the usual Schwarzschild term ǫ≡(2GM/c2r)
+and theǫ∗is the extra small parameter due to this effect,
+ǫ∗≡2v2
+∗/c2where given the asymptotic rotation speed
+v∗for Milky Way ǫ∗is at the order of 6 ×10−7. Since
+rs≈8kpcthe logarithm will be of order one. Looking
+at Eq.(2) and observing the direct proportionality of ∆ t
+with ∆A(r) and ∆B(r), one can see that the time delay
+would approximately be equal to 10−6times the total
+time of flight in flat space.
+But our aim is to calculate the time lag numerically,
+duetothedifficultyofsolvingtheEq.(2)analytically. Us-
+ing the relevant parameters for the NFW density profile
+of the Milky Way [39], the numerical solution is depicted
+in Figure 1.
+100 200 300 400distance/LParen1kpc/RParen150100150200250300daysTimeLag
+FIG. 1: Shapiro delays for sources located in Milky Way.
+If we look at Figure 1 we can see that the time lag for
+an event that is located at 100 kpc is approximately 100
+days, which is 10−6times 3×105light years (the total
+time of flight of a photon in flat space).
+The time lag would surely depend on the angular po-
+sition of the source, since we are not the center of the
+Milky Way. Therefore one would like to see the effect of
+the direction of the position of a source in the whole sky.
+This was done for a source that is located up to 400 kpcaway from the Earth, and it is demonstrated in Figure
+2. It can be seen that the angular position of the source
+might have a maximum of 5% effect on the Shapiro delay
+of the considered source.
+TimeLag
+0.0 0.5 1.0 1.52.0RA/Minus0.50.00.5 Dec
+295300305
+days
+FIG. 2: The angular dependence of Shapiro delays for sources
+located in Milky Way. The units of RA(Right ascension) and
+Dec(Declination) is coverted to radians.
+IV. EXTERNALLY TRIGGERED SEARCHES
+FOR GWS
+Externally triggered search of gravitational waves is a
+strategyto look for possible signalsaround sourceswhich
+might produce both GWs and other particles(external
+triggers). The information provided by an external trig-
+geris extremelyuseful to impose extrarequirementsfor a
+possible GW signals as well as to increase the confidence
+of the detection. Knowing the source direction would al-
+low us to look for only relevant portion of the sky. One
+can even infer the frequency of the possible GW signal
+based on the information provided by an external trigger
+and make a data analysis at a specific band [44].
+Gamma ray bursts (GRBs) are one of the possible
+external triggers. A neutron star(NS) black hole or a
+NS-NS merger might be a good source for a GRB. A
+highly magnetized NS might also produce short and in-
+tense gamma rays, which are called as soft gamma ray
+Repeaters (SGRs). One sixth of short GRBs are thought
+tobebecauseofSGRs, whereduringS5ofLIGO,electro-
+magnetically several hundred of them are were observed
+[45]. Disruption of neutron star’s crust would produce an
+increase in the rotation energy of a neutron star which
+results in increase of freqency, so called a “pulsar glitch”.
+This might also lead to emission of GWs. Last but not
+least, supernova explosions would surely produce both
+GWs and neutrinos. This triggered detection on the
+other hand, would provide us more information about
+neutrino mass as well as the mechanism of supernova ex-
+plosions.
+But still, there has not been a single direct detection
+of GWs. The search is based on the assumption that,
+the GW signal and the external trigger are coincident in4
+M31 Mtot(1012M⊙)Milky Way Mtot(1012M⊙)
+Corteau [46] 1.33+0.18
+−0.18 Xue [51] 1.0+0.3
+−0.2
+Evans [47] 1.23+1.8
+−0.6Smith [52] 1.42+1.14
+−0.54
+Fardal [48] 0.74+0.12
+−0.12Wilkinson [53] 1.9+3.6
+−1.7
+Seigar [49] 0.73+0.02
+−0.02Sakamoto [54] 1.8+0.4
+−0.7
+Ibata [50] 0.75+0.25
+−0.13Battaglia [55] 1.5+0.2
+−0.2
+TABLE II: Different mass estimates of the Milky Way and
+the Andromeda with the corresponding error bars.
+time within a small time window. Therefore it is possible
+that, if“darkmatter emulators”describegravitythen we
+might be lookingat the wrongplace. This possibilitywas
+suggested before and the possible time lag was calculated
+for certain events [32]. But it is crucial for GW observers
+(especially the people who are involved with externally
+triggered search) to know how big this effect is for a par-
+ticulat object and the amount of error. In this work we
+provide an estimate of this possible time lag for any ob-
+ject located at Milky Way and current uncertainties are
+discussed in the next section.
+V. UNCERTAINTIES IN THE CALCULATION
+There are three kinds of uncertainties in the time lag
+calculation. First one is the position of the assumed
+source. The uncertainity in the angular position of the
+GRB’s and other sources are determined by a relatively
+good accuracy. In a previous work For GRB-070201 the
+errorwasshownto be ofthe orderof2% [32]. The second
+kind of uncertainty is the choice of the dark matter den-
+sity profile and this choice had an effect which wass less
+than 1% [32]. The third kind of uncertainty is the data
+input(parameters) coming from different groups who are
+working on numerical simulations of dark matter density
+profiles. And this choice has the biggest effect on the
+calculation.
+In order to calculate the time lag in Eq.(2) we need
+to use these parameters coming from numerical simula-
+tions. They enter into the calculation through the mass
+function. In this work, to get the mass function, we used
+NFW dark matter density profile for modified gravity
+models to mimic dark matter. One can see from Table
+I. that using a data set of a different group changed the
+time lag from 383 days to 634 days for the Andromeda
+galaxy. And this difference is expected; since the virial
+mass estimation of M31 is almost twice of one another.
+Therefore the biggest uncertainty comes from the de-
+termination ofthe mass ofthe galaxies. Lookingat Table
+I. the results for our galaxy are not very different, unlike
+M31. But still one naturally ould start wondering about
+the case of mass estimate for our galaxy as well. Someof the recent mass estimates for both M31 and Milky-
+Way galaxy are shown in Table II. The results of virial
+massestimatesarequite different fromone another. This
+might be misleading since the virial radius which is con-
+sidered in these works are sometimes different from one
+another. But even taking that into consideration, the er-
+ror bars of these estimates are too big, varying from 13%
+to 140% ,that we can not give a precise estimate of the
+time lag for objects that are far from us.
+To summarize, our prediction of this time lag has cer-
+tain uncertainties, where the biggest one is the input
+coming from the numerical simulations of dark matter
+density profiles of the galaxies. For an event which will
+occur at a very distant location, more than 100kpc, we
+canonlyconcludethatthe time lagwill bebig. Therefore
+the predictivity of this test is much better for an event
+which is closer to us ( ∼10kpc). The errors of the most
+recent mass estimates are at the order of 20% and nu-
+merical simulation community is hopeful that these mass
+estimates will get much better in recent future for our
+galaxy [56].
+VI. CONCLUSIONS
+A peculiar property of dark matter emulators is that
+gravitational waves couple to a different metric than the
+ordinary matter. This results in a time lag between the
+arrival times of these particles coming from the same
+source, which would be a powerful check of these models
+with a single detection of a gravitational wave. A si-
+multaneous detection of gravitational wave and a photon
+would rule out entire class of modified gravity models.
+But if a time lag is observed which is at the order magni-
+tude of the one mentioned in this work, that would mean
+general relativity is wrong and there is no dark matter.
+By a numerical analysis we made a rough estimation
+of the time lag for any source in our galaxy as far as 400
+kpc, and demonstrated it in Figure 1. The time lag de-
+pends almost linearly on the distance of the source from
+us, and is approximately 10−6times the total time of
+flight. Estimation of the time lag depends on the data
+input(mass function of Milky-Way) which is provided by
+numerical simulations. Therefore the accuracy of this,
+highly depends on the error bars coming from these sim-
+ulations. The current state of affairs of the numerical
+simulations of Milky-Way galaxy puts some constraints
+on our prediction. Therefore, at this stage we can con-
+clude that the best objects, for this proposed test, are
+the ones that are close to us( ∼10 kpc).
+We also have to point that Bekenstein takes the ap-
+proach as velocity of GWs is smaller than photons [19].
+But this would lead to an immediate invalidation of
+TeVeS from the boundary of Moore and Nelson[57] via
+gravitational cherenkov-radiation. Therefore we do not
+agreewith Bekenstein’s point of view and force the “dark
+matter emulators” to agree with the observed rotation
+curve and weak lensing results.5
+Besides its own significance, gravitational wave obser-
+vation is very important since it will open new windows
+in our understanding of the universe. One can test whole
+class of alternate theories of gravity which would results
+into peculiar imprints such as additional polarization
+states. Higher order theories of gravity, Brans-Dicke the-
+ory, massive graviton theories and Chern-Simons modi-
+fiedgravityarejustsomeofthealternatetheoriesofgrav-
+ity that we can test (see [58, 59] and references therein).
+This work should serve as a guide for gravita-
+tional wave observers who would like to search for im-
+prints of modified gravity models without dark matter.
+LIGO/VIRGO are currently doing joint science opera-
+tions at design sensitivities and searches for gravitationalwaves with electromagnetic counterparts is a key project
+[60]. And this test is just another motivation to look for
+it.
+Acknowledgements
+I am grateful to Richard Woodard and Shantanu Desai
+for stimulating discussions. I would like to thank Szabi
+Marka for suggesting this problem to me. I also would
+like to thank Hongsheng Zhao and Anatoly Klypin for
+illuminating comments and discussions. I am also grate-
+ful for the hospitality of the Physics Department of Ko¸ c
+University. This work was partially supported by DFG-
+Research Training Group ”Quantum and Gravitational
+Fields” GRK 1523/1,by Marie Curie Grant IRG-247803.
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\ No newline at end of file
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+ 
+Multiverse Scenarios in Cosmology:  
+Classification, Cause, Challenge, Controversy, and Criticism  
+ 
+ 
+Rüdiger Vaas  
+ 
+Center for Philosophy and Foundations of Science,  
+University of Giessen, Germany  
+ 
+Ruediger.Vaas@t -online.de  
+ 
+ 
+ 
+ 
+ 
+Abstract  
+ 
+Multiverse scenarios in cosmology assume that other universes exist "beyond" our own universe. 
+They are an exciting challenge both for empirical and theoretical research as well a s for philosophy 
+of science. They could be necessary to understand why the big bang occurred, why (some of) the 
+laws of nature and the values of certain physical constants are the way they are, and why there is 
+an arrow of time. This essay clarifies compet ing notions of "universe" and "multiverse"; it proposes 
+a classification of different multiverse types according to various aspects how the universes are or 
+are not separated from each other; it reviews the main reasons for assuming the existence of other 
+universes: empirical evidence, theoretical explanation, and philosophical arguments; and, finally, it 
+argues that some attempts to criticize multiverse scenarios as "unscientific", insisting on a narrow 
+understanding of falsification, is neither appropriat e nor convincing from a philosophy of science 
+point of view.  
+ 
+ 
+Keywords:  big bang, universe, multiverse, cosmic inflation, time, quantum gravity, string theory, 
+laws of nature, physical constants, fine -tuning, anthropic principle, philosophy of science, 
+metaphysics, falsificationism  
+ 
+ 
+  
+ 
+1. Introduction  
+ 
+Why is there something rather than nothing? And why is that which is, the way it actually is? T he 
+multiverse hypothesis (henceforth called M) is an at least partial attempt to answer these two 
+questions, which may be the most fundamental (meta)physical mysteries altogether. No approach 
+can possibly provide an exhaustive or ultimate answer to these q uestions (Vaas 2006), and neither 
+does M seek to do that. But if empirically confirmed in some way, or rigorously derived 
+theoretically, M would be one of the most radical and far -reaching insights ever. Its explanatory 
+power would be extraordinary, if M t urns out to be necessary for an understanding why the big 
+bang occurred, why (some of) the laws of nature and the values of certain physical constants are 
+as they are (even fine -tuned for life?), and why there is an arrow of time. Furthermore, M could be 
+an implication of an otherwise confirmed, well -established theory, i.e. not only a postulated but also 
+a derived (part of an) explanans .  
+ 
+Despite these exciting prospects M is controversial and under attack both from sceptical scientists 
+and critical philo sophers of science (Vaas 2008a & 2010). This is not surprising and, indeed, it is to 
+be appreciated because extraordinary claims require extraordinary evidence. And surely such a 
+kind of evidence remains to be discovered.  So what is the status of M?  2 
+ 
+ 
+This essay provides a classification of different multiverse types, categorized according to the way 
+in which individual universes are assumed to be separated or not separated from another. Then 
+the main reasons for assuming the existence of other universes (e mpirical, theoretical, and 
+philosophical) are reviewed, and it is argued that some attempts to criticize multiverse scenarios as 
+"unscientific", insisting on a narrow understanding of falsification, are neither appropriate nor 
+convincing from a philosophy of science point of view.  
+ 
+ 
+2. Different notions of "universe" and "multiverse"  
+ 
+"The universe is a big place, perhaps the biggest", Kurt Vonnegut once wrote, well -known as a 
+science fiction author among other achievements. But science is often stranger th an fiction, and 
+the universe could be, even though infinite, only a tiny part of what exists. According to M, that is 
+indeed the case. However, this opens up both conceptual and empirical problems.   
+ 
+The term "universe" (or "world"), as it is used today, is ambiguous. There are many different 
+meanings of "universe", especially (Vaas 2004a):  
+ 
+(1) Everything (physically) in existence, ever, anywhere. With this meaning there are no other 
+universes – by definition.  
+ 
+(2) The observable region we inhabit (the  Hubble volume, almost 100 billion light years in 
+diameter), plus everything that has interacted (for example due to a common origin) or will ever or 
+at least in the next few billion years interact with this region.  
+ 
+(3) Any gigantic system of causally int eracting things that is wholly (or to a very large extent, or for 
+a long time) isolated from others; sometimes such a locally causally connected collection is called 
+a multi -domain universe , consisting of the ensemble of all sub -regions of a larger connect ed 
+spacetime, the "universe as a whole", and this is opposed to the multiverse in a stronger sense, i.e. 
+the set of genuinely disconnected universes, which are not causally related at all.  
+ 
+(4) Any system that might  well have become gigantic, etc., even i f it does in fact recollapse while it 
+is still very small.  
+ 
+(5) Other branches of the wavefunction (if it never collapses, cf. Wheeler & Zurek 1983, Barrett 
+1999, Vaas 2001) in unitary quantum physics, i.e. different histories of the universe (e.g. Gell -Mann 
+& Hartle 1990 & 1993) or different classical worlds which are in superposition (e.g. DeWitt & 
+Graham 1973).  
+ 
+(6) Completely disconnected systems consisting of universes in one of the former meanings, which 
+do or do not share the same boundary conditions , constants, parameters, vacuum states, effective 
+low-energy laws, or even fundamental laws, e.g. different physically realized mathematical 
+structures (cf. Tegmark 2004 & 2010).  
+ 
+Nowadays, "multiverse" (or "world" as a whole) is often used to refer to ev erything in existence (at 
+least from a physical point of view), while the term "universe" permits to talk of several universes 
+(worlds) within the multiverse.  
+ 
+In principle, these universes – mostly conceived in the meaning of (2), (3), or (4) – might or might 
+not be spatially, temporally, dimensionally, causally, nomologically and/or mathematically 
+separated from each other (see Table 1). Thus, there are not necessarily sharp boundaries 
+between them.  
+ 
+One might call the whole set of different universes t he multiverse. But it could be true that there are 
+even different sets of totally spatiotemporally and strictly causally separated multiverses, e.g. 
+different bunches of chaotically inflating multiverses. In that case it remains useful to have a term 3 
+ 
+with a still broader extension, namely omniverse  or cosmos . So it shall be taken as the all -
+embracing term for everything in existence which might or might not be the set of different 
+multiverses, while a (or the) multiverse consists of different universes whic h are not separated in 
+every respect.  
+ 
+ 
+3. Towards a multiversal taxonomy  
+ 
+Multiverse classifications should be abstract enough to include all kinds of cosmological scenarios. 
+It was suggested to categorize them with regard to separation/distinction of the  different universes 
+(Vaas 2004a; Table 1 is an extension of this). Other classifications suggested four different levels 
+(Tegmark 2004) or three different types ( Mersini -Houghton 2010) (cf. also Ellis, Kirchner & Stoeger 
+2004). Of course, definitions and taxonomies are just conceptual issues. They are needed for 
+clarification and to avoid misunderstandings, but they explain and prove nothing. Nevertheless it 
+has to be discussed and, in the long run, shown, whether they are useful, sufficiently complete, 
+and not too arbitrary.  
+ 
+ 
+Table 1:  A multiversal taxonomy: Multiverse scenarios differ with respec t to the kind and 
+degree of separation between the individual universes that they assume  and can be 
+categorized accordingly. Note that these aspects of sep aration do not necessarily exclude 
+each other. Some multiverse types fit in several categories. For e xample eternal inflation 
+scenarios describe universes which are spatially exclusive, but could be embedded in a 
+common spacetime; they are not dimensionally separated, only by different vacuum states; 
+they are strictly causally separated with respect to th e future (though not in every case 
+because there are models predicting bubble collisions), but not with respect to the past 
+since they share a common mechanism which generates their existence.  
+ 
+separation  aspects   examples and comments  
+spatiotemporal  spatial 
+• exclusive  
+• inclusive       see also causal separation  
+eternal inflation, stringscape, different quantum tunnel universes  
+embedding:  universes in atoms, black holes, computer 
+simulations...  
+temporal  
+ 
+• linear  
+• cyclic  
+• branching  oscillating universe, cyclic universe, recycling universe,  
+universes (or parts) with different arrows of time  
+     in a causal or acausal series  
+     within circular time or due to exact, global recurrences  
+many quantum worlds/histories  
+dimensional  
+• strict 
+• inclusive  
+ 
+ 
+• abstract  mostly spatial, but there are also two -time-dimensional scenarios  
+tachyon universe?  
+lower -dimensional world as part or boundary of a higher -
+dimensional world:  flatland, brane -worlds, large extradimensions, 
+holographic universe  
+"leafs in superspace"  
+causal  strict  
+• without a common  
+   generator  
+• genealogical  "parallel universes", many worlds in quantum superposition  
+different universes/multiverses in instanton, big bounce, soft 
+bang etc. scenarios; different  "bundles" of (eternal) inflation  
+eternal inflation, cosmic darwinism, cosmic natural or artificial 
+selection, many quantum worlds/histories without interactions  
+continuous  
+• always  
+• past  
+• future       due to an increasing horizon  
+infinite spa ce, eternal inflation, infinite branes  
+     because of inflation  
+     because of accelerated expansion due to dark energy  
+modal  
+ potential (possible)       separated in imagination or conceptual representation 
+(otherwise not real!)  
+actual (rea l) modal realism: physically (nomologically), metaphysically or 
+logically separated  
+nomological  structural/regularities  different laws or different constants of nature  
+mathematical  structural/axiomatic  Platonism, mathematical democracy, ultimate ensem ble 4 
+ 
+ 
+However, as Ernest Rutherford used to provoke, "science is either physics or collecting stamps". 
+So what are the arguments for presuming the existence of other universes?  
+ 
+ 
+4. Is our universe fine -tune d? 
+ 
+Life as we know it depends crucially on the laws and constants of nature as well as the boundary 
+conditions (e.g. Leslie 1989, Vaas 2004b, Carr 2007). Nevertheless it is difficult to judge how "fine -
+tuned" it really is, both because it is unclear how m odifications of many values together might 
+compensate each other and whether laws, constants and initial conditions really could have been 
+otherwise to begin with. It is also unclear how specific and improbable those values need to be in 
+order for informat ion-processing structures – and, hence, intelligent observers – to develop. If we 
+accept, for the sake of argument, that at least some values are fine -tuned, we must ask how this 
+can be explained.  
+ 
+In principle, there are many options for answering this qu estion ( Table 2). Fine -tuning might (1) just 
+be an illusion  if life could adapt to very different conditions or if modifications of many values of the 
+constants would compensate each other; or (2) it might be a result of (incomprehensible, 
+irreducible) chance, thus inexplicable (Vaas 1993); or (3) it might be nonexistent  because nature 
+could not have been otherwise, and with a fundamental theory  we would be able to prove this; or 
+(4) it might be a product of selection : either observational selection  within a vast multiverse of 
+(infinitely?) many different realizations of those values ( weak anthropic principle ), or a kind of 
+cosmological natural selection  making the measured values (compared to possible other ones) 
+quite likely within a multiverse of many dif ferent values, or even a teleological or intentional 
+selection . Even worse, these alternatives are not mutually exclusive – for example it is logically 
+possible that there is a multiverse, created according to a fundamental theory by a cosmic designer 
+who is not self -sustaining, but ultimately contingent, i.e. an instance of chance.  
+ 
+ 
+Table 2:  Digging deeper: Laws, constants and boundary conditions are the basic 
+constituents of cosmology and physics from a formal point of view (besides spacetime, 
+energy, matter, fields and forces or more fundamental entities like strings or sp in-networks 
+and their properties with regards to content). An ambitious goal and historically at least a 
+successful heuristic attitude is reduction, derivation and unification to achieve more 
+fundamental, far -reaching and simple descriptions and explanatio ns. While uniqueness is 
+much more economical and predictive, multiple realizations – presumably within a 
+multiverse – have recently been proposed as an opposing (but not mutually exclusive) 
+alternative. This table provides a summary of different approach es, possibilities and 
+problems; it is neither complete nor the only conceivable system. (From Vaas 2009b.)  5 
+ 
+ 
+ 
+ fundamental laws (L)  
+of nature  fundamental constants 
+(C) boundary conditions (BC)  
+uniqueness?  
+(and just one 
+universe?)  (1) irreducible and 
+disconnected?  
+(2) or derived from or unified 
+within or reducible to one 
+fundamental theory ("Theory 
+of Everything", TOE)?  
+• "logically isolated"?  
+• or even logically sufficient 
+(self-consistent)? (Bootstrap 
+principle)  
+• ultimately ( logically?) 
+deducible? (th us without 
+empirical content if 
+analytically true? natural 
+science as pure 
+mathematics?)  
+(3) or nonexistent because 
+emerging via self -organization 
+from an underlying chaos 
+("law without law" approach)  (1) irreducibly many?  
+(2) or only a few or just one 
+(e.g. string length)?  
+• with a unique value? (just 
+random or determined by 
+L?) 
+• with (infinitely?) many 
+possible different values? 
+(according to a probability 
+distribution determined by 
+L?) 
+(3) or ultimately none?  
+• because C just as 
+conversion factors (e .g. in a 
+TOE)  
+• and/or better 
+understandable as initial 
+conditions (with many 
+different realizations in a 
+multiverse?)  (1) as initial conditions?  
+• irreducible?  
+• not accessible? (due to 
+inflation, mixmaster 
+universe, BKL chaos, 
+"cosmic forgetfulness", 
+decoherence etc.)  
+• nonexistent (in eternal 
+universe models)?  
+• explanatory irrelevant? 
+(because convergence due 
+to an attractor or replaced 
+by present BC?)  
+(2) or as present 
+conditions?  
+• sufficient?  
+• or the only useful ones? 
+(e.g. as loop quantum 
+cosmol ogy constraints or 
+according to the top down 
+approach in Euclidean 
+quantum gravity)  
+• or as (additional?) final 
+conditions? (e.g. constraints 
+in some quantum 
+cosmology models)  
+(3) or nonexistent because 
+determined by L? (e.g. the 
+no-boundary proposal)  
+multiverse?  (1) many realizations  
+• as BC if not reducible  
+• separated in principle or with 
+a common origin (cause)?  
+• restricted by or derived from 
+a TOE, or truly 
+random/irreducible?  
+(2) or with every (logically? 
+metaphysically?) possible 
+realization? (m athematical 
+democracy, ultimate principle 
+of plentitude)  
+(3) TOE as a "multiverse 
+generator"?  (1) many realizations  
+• truly random  
+• restricted by L or BC?  
+(2) or every possible 
+combination of values? 
+(equally or randomly 
+distributed?)  
+(3) or with some absolute 
+frequency according to a 
+probability distribution 
+(determined by L, e.g. string 
+statistics and/or within a 
+framework like cosmological 
+natural selection?)  
+(4) or just observational 
+bias/ selection (weak 
+anthropic principle) from a 
+random set?  (1) many realizations  
+• restricted variance due to 
+an attractor (determined by 
+L)?  
+• or truly random  
+(2) or every possible 
+combination? (equally or 
+randomly distributed?)  
+(3) or with some absolute 
+frequency according to a 
+probability distribution 
+(determine d by L and/or 
+within a framework like 
+cosmological natural 
+selection)?  
+(4) or just observational 
+bias/ selection (weak 
+anthropic principle) from a 
+random set?  
+design?  • transcendent realization? (nonphysical causation)  
+• random creation or cosmological  artificial selection by cosmic engineers?  
+• universal simulation/emulation? (or just a subjective illusion?)  
+however: design of one universe or of a multiverse? and what have caused the designer? 
+are there even transcendent (Platonic) laws? why is there  someone rather than no one?  
+randomness?  ultimately existing (even if 
+there is only one self -
+consistent TOE)  
+• Gödel -Turing -Chaitin 
+theorems  not necessarily (if 
+determined by L or reduced 
+to BC)  not necessarily (if determined 
+by L)  6 
+ 
+ 
+From both a scientific and philosophical perspective the fundamental theory approach and the 
+multiverse scenario are most plausible and heurist ically promising (Vaas 2004a & 2004b).  
+ 
+ 
+5. Unique -universe versus multiverse accounts  
+ 
+Unique -universe accounts  take our universe as the only one (or at least the only one ever relevant 
+for cosmological explanations and theories). It might have had a pred ecessor or even infinitely 
+many (before the big bang) and/or a successor or infinitely many (after a big crunch), but then the 
+whole series can be taken as one single universe with spatiotemporal phase transitions. From the 
+perspective of simplicity, parsi mony and testability it is favourable  to try to explain as much as 
+possible with a unique -universe account. A both straightforward and very ambitious approach is 
+the searching for a fundamental theory with just one self -consistent solution that represents ( or 
+predicts) our universe. (Of course one could always argue that there are other, strictly causally 
+separated universes too, which do not even share a common generator or a meta -law; but then 
+they do not have any explanatory power at all and the claims fo r their existence cannot be 
+motivated in a scientifically useful way, only perhaps by philosophical arguments.)  
+ 
+Future theories of physics might reveal the relations between fundamental constants in a similar 
+way as James Clerk Maxwell did by unifying el ectric and magnetic forces: he showed that three 
+until then independent constants – the velocity of light c, the electric constant ε o (vacuum 
+permittivity), and the magnetic constant µ o (vacuum permeability) – are connected with each other: 
+c = (µ o . εo)-0,5. Indeed some candidates for a grand unified theory of the strong, weak and 
+electromagnetic interaction suggest that most of the parameters in the standard model of particle 
+physics are mathematically fixed, except for three: a coupling constant (the e lectromagnetic fine -
+structure constant) and two particle masses (namely that of down and up quarks) (Hogan 2000). A 
+promise of string theory is even to get rid of any free parameter – if so, all constants could be 
+calculated from first principles (Kane et  al. 2002). However, this is still mainly wishful thinking at the 
+moment. But it is a direction very worth following and, from a theoretical and historical point of 
+view, perhaps the most promising.  
+ 
+So even without an ultimate explanation fine -tuning mig ht be explained away within a (more) 
+fundamental theory. Most of the values of the physical constants should be derived from it, for 
+example. This would turn the amazement about the anthropic coincidences into insight – like the 
+surprise of a student abou t the relationship ei
+ = -1 between the numbers e, i and 
+  in mathematics 
+is replaced by understanding once he comprehends the proof. Perhaps the fact that the mass of 
+the proton is 1836 times the mass of the electron could be similarly explained. If so, t his number 
+would be part of the rigid formal structure of a physical law which cannot be modified without 
+destroying the theory behind it. An example for such a number is the ratio of any circle's 
+circumference to its diameter. It is the same for all circl es in Euclidean space: the circular constant 
+.  
+ 
+But even if all dimensionless constants of nature could be reduced to only one, a pure number in a 
+theory of everything, its value would still be arbitrary, i.e. unexplained. No doubt, such a universal 
+reduction would be an enormous success. However, the basic questions would remain: Why this 
+constant, why this value? If infinitely many values were possible, then even the multitude of 
+possibilities would stay unrestricted. So, again, why should such a univer sal constant have the 
+value of, say, 42 and not any other?  
+ 
+If there were just one constant (or even many of them) whose value can be derived  from first 
+principles, i.e. from the ultimate theory or a law within this theory, then it would be completely 
+explained or reduced at last. Then there would be no mystery of fine -tuning anymore, because 
+there never was a fine -tuning of the constants in the first place. And then an appreciable amount of 
+contingency would be expelled.  
+ 7 
+ 
+But what would such a spectacular  success really mean? First, it could simply shift the problem, 
+i.e. transfer the unexplained contingency either to the laws themselves or to the boundary 
+conditions or both. This would not be a Pyrrhic victory, but not a big deal either. Second, one might  
+interpret it as an analytic solution. Then the values of the constants would represent no empirical 
+information; they would not be property of the physical world, but simply a mathematical result, a 
+property of the structure of the theory. This, however, still could and should have empirical content, 
+although not encoded in the constants. Otherwise fundamental physics as an empirical science 
+would come to an end. But an exclusively mathematical universe, or at least an entirely complete 
+formal description of everything there is, derivable from and contained within an all -embracing 
+logical system without any free parameter or contingent part, might seem either incredible (and 
+runs into severe logical problems due to Kurt Gödel's incompleteness theorems) or t he ultimate 
+promise of the widest and deepest conceivable explanation. Empirical research, then, would only 
+be a temporary expedient like Ludwig Wittgenstein's (1922, 6.54) famous ladder: The physicist, 
+after he has used empirical data as elucidatory steps , would proceed beyond them. "He must so to 
+speak throw away the ladder, after he has climbed up on it."  
+ 
+That there is no contingency at all seems very unlikely. So why are some features realized but not 
+others? Or, on the contrary, is every feature reali zed? Both questions are strong motivations for M.  
+ 
+Multiverse accounts  assume the existence of other universes as defined above and characterized 
+with respect to their  separation in Table 1. This is no longer viewed as mad metaphysical 
+speculation beyond empirical rationality. There are many different multiverse accounts (see, e.g., 
+Smolin 1997, Deutsch 1997, Rees 2001, Tegmark 2004, Davies 2004, Vaas 2004b, 2005, 2008b & 
+2010, Vilenkin 2006, Carr 2007, Linde 2008, Mersini -Houghton 2010) and even some atte mpts to 
+classify them quantitatively (s ee, e.g., Deutsch 2001 for many  worlds in quantum physics, and 
+Ellis, Kirchner & Stoeger 2004 for physical cosmology). They flourish especially in the context of 
+cosmic inflation (Linde 2005 & 2008, Vaas 2008b, Aguirr e 2010), string theory (Chalmers 2007, 
+Douglas 2010) and a combination of both as well as in different quantum gravity scenarios that 
+seek to resolve the big bang singularity and, thus, explain the origin of our universe (for recent 
+reviews see Vaas 2010).   
+ 
+If there is a kind of selection , new possibilities emerge within M. One possibility is just 
+observational selection (Barrow & Tipler 1986, Leslie 1989, Vaas 2004b). Another possibility is a 
+kind of cosmological natural selection making the measured valu es (compared to possible other 
+ones) quite likely within a multiverse of many different values (García -Bellido 1995, Smolin 1992, 
+1997 & 2010, for a critical discussion e.g. Vaas 1998 & 2003). Finally there is the possibility of 
+teleological or intentional  selection (e.g. Leslie 1989, Harrison 1995, Ansoldi & Guendelman 2006, 
+Vidal 2008 & 2010, for a critical discussion e.g. Vaas 2004b & 2009ab).  
+ 
+The strongest version of M is related to the principle of plentitude or principle  of fecundity  
+(advocated e.g. by Richard Feynman, Dennis Sciama). According to this principle everything is 
+real, if it is not explicitly forbidden by laws of nature, e.g. symmetry principles. As Terence H. White 
+wrote in his novel The Once and Future King  (1958): "everything not forbi dden is compulsory". But 
+the question remains: What is forbidden, i.e. physically or nomologically not possible and thus not 
+"allowed" by natural laws? (This is a slippery slope, as Paul Davies 2007 argued: could there even 
+be universes containing magic, a  theistic God, or simulations of every weird fantasy? Otherwise 
+some restrictions are necessary!) And is it really true that everything which is physically or 
+nomologically possible is also re alized somewhere? Answers to th ese questions are not known. 
+And i t is even controversial whether they belong to physical cosmology or to metaphysics. (It is 
+also unclear whether "metaphysically possible" makes truly sense in contrast to both "logically 
+possible" and "physically possible", cf. Meixner 2008).   
+ 
+ 
+6. Why (s hould) we believe in other universes (?)  
+ 
+There are – or could be – at least three main reasons for assuming the existence of other 
+universes: empirical evidence, theoretical explanation, and philosophical arguments (Table 3). 8 
+ 
+These three kinds of reasoni ng are independent from each other, but ideally entangled. As far as 
+there is at least some connection with or embedding into a theoretical framework of physics or 
+cosmology, M is part of the scientific endeavour, not only of philosophy. This is also the c ase in the 
+absence of empirical evidence, if a theoretical embedding exists. And philosophical arguments 
+might at least motivate scientific speculation (Table 4).  
+ 
+ 
+Table 3:  Argument s for the existence of other universes (cf. Vaas 2010).  
+ 
+Arguments for the multiverse hypothesis  comments  
+Empirical evidence? Signs of other universes?  
+ they are also theory -dependent  
+• Wormholes as gates to other universes?  detectable via gravitation al lensing?  
+• Bubble collisions?  signs in the cosmic microwave 
+background (CMB)?  
+• A Void in front of the WMAP Cold Spot  
+  (CMB and radio observations)   the gravitational pull of other universe?  
+• Signs of Other Universes: Gates?  
+  (a different speculation a bout the WMAP Cold Spot)  
+ A type of textures known as brane -
+skyrmions can be understood as holes in 
+the brane which make possible to pass 
+through them along the extra -
+dimensional space.  
+• Dark Flow: more than 10 00 galaxy clusters (CMB imprints of  
+  hot X-ray emitting  gas), up to 5 billion light -years away, are  
+  moving with up to 1000 km/s towards Centaurus/Vela  is it real? due to a mass concentration 
+beyond horizon? a sign of a fractal 
+universe? a sign of a domain wall due to 
+bubble collision? or the gravitational pull 
+of other  universe?  
+• supports for cosmic inflation   
+• gravitational imprints of other branes?  
+  (gravitons travelling through the higher dimensional bulk)   
+• dark matter as a shadow effect from matter in  
+  another dimension?   
+• relics from a precursor  universe? (CMB signatures)  different quantum cosmology predictions  
+• value of certain constants of nature, e.g. cosmological  
+  constant, due to a relaxation process in a cyclic universe?  prediction of the cyclic universe scenario  
+• value of certain constants pred ictable by  
+  principle of mediocrity?     
+• explaining features of the standard model of  
+  elementary particles? (via M theory)   
+• quantum computers for the exploration  of other branches of the  
+  quantum many -worlds?     
+Theory forces us to do so?  if established a nd with this implication  
+• quantum theory  many worlds, many histories...  
+• inflationary cosmology  bubble/pocket universes  
+• string theory  landscape (different string vacua)  
+• big bang explanations  precursor universe or fluctuation models  
+• arrow of time  explanations  precursor universe or fluctuation models  
+Philosophical  arguments? (... and prejudice)  just philosophy or also science?  
+• implication of a slippery -slope argument          
+  e.g. galaxies beyond our telescopes, beyond the horizon...   
+• explanatory power and depth – not-just-so-stories,  
+  anthropic reasoning ,  
+  selection  principles (e. g. cosmic darwinism ) ... explaining the big bang, the "fine -tuning" 
+of nature’s constants, quantum 
+measurement problem, the arrow(s) of 
+time, no time -paradoxes...  
+• anti -copernicanism, principle of mediocrity ...  Copernican principle completed  
+• principle of fecundity/plentitude   
+ 9 
+ 
+ 
+ 
+Table 4:  Controverses about universes (cf. Vaas 2010).  
+ 
+ 
+Are other universes …  
+• physically extravagant?  
+– yes they are, but this is often the case in science (think of extravagancies e.g. in relativi ty, 
+quantum theory...)  
+• transcend ing speculative reasoning?  
+– perhaps, but they could also be a straightforward extrapolation  
+• an explanation of anything, and therefore nothing?  
+– perhaps, but more probably they can explain only something, not every thing  
+• against simplicity and parsimony (Occam’s razor)?  
+– not necessarily, because M admittedly assumes the existence of many objects (i.e. universes), 
+but there is (or might be) a parsimony in relation to principles, restrictions, algorithms, kinds of 
+entities, and M still is in accordance with philosophical naturalism/physicalism and indeed favours it  
+• leading to actual infinities  
+– perhaps, but is this a problem? ( this is controversial ) 
+• an implication of the principle of plentitude /fecundity: e verything is real, if it is not explicitly "forbidden"  
+  by laws of nature  
+ 
+ 
+However, many cosmologist s and philosophers alike argue that M does not belong to science 
+because it cannot be falsified. In the following it shall be argued briefly why this criticism is not 
+valid.  
+ 
+ 
+7. Falsificationism is both too much and not enough  
+ 
+"Scientific statements, re fering to reality, must be falsifiable", Karl R. Popper (1930 -1933 & 1935) 
+already argued in 1932. Falsifiability has been one of the most important and successful criteria of 
+science ever since. Popper even understood it as a demarcation criterium of scie nce (in contrast to 
+metaphysics, logic, pseudoscience...). But what is usually meant here is falsifiability of theoretical 
+systems, or of parts of such a system, not of single statements.  
+ 
+So there is an important distinction: Scientific laws  on the one h and must be falsifiable. Take for 
+example Newton’s law – it can be tested, was tested rigorously indeed, and is still under 
+investigation (at very large and very small scales). Hypothetical universal existential statements  on 
+the other hand need not and ca nnot be falsified, but must be verified.  
+ 
+Take for instance the discovery of the last two naturally occurring stable elements: Hafnium, atomic 
+number 72, was detected by Dirk Coster and George de Hevesy in 1922 through X -ray 
+spectroscopy analysis, after N iels Bohr had predicted it, or its properties. (Actually Dimitri 
+Mendeleev ha d already predicted it implicit ly in 1869, in his report on The Periodic Law of the 
+Chemical Elements ). Similarly, Rhenium, atomic number 75, was found by Walter Noddack, Ida 
+Tacke, and Otto Berg in 1925, after Henry Moseley had predicted it in 1914. So in science 
+verification is obviously extremely important. But it is not sufficient in controversial and hypothetical 
+situations even if there are concise predictions. Otherwise fic tive ghosts or unicorns would also be 
+part of science, for we cannot know a priori whether they exist or not, but we can imagine ways to 
+detect them, that is to verify their existence in principle. So what is missing here? It is theoretical 
+embedding . To b e reasonably part of science, hypothetical universal existence statements must 
+not only be verifiable, they must also be part of a sufficiently confirmed or established scientific 
+theory or theoretical framework (this is somewhat fuzzy of course). That was  the case in the 
+Hafnium and Rhenium examples, where Mosley and Bohr had their theoretical models about the 
+atomic numbers and the properties of the atoms, explaining regularities in the periodic table of the 
+chemical elements.   
+ 10 
+ 
+Statements about the exis tence of other universes are not like statements about scientific laws. 
+The latter must be falsifiable, while the former  should be taken as universal existential statements 
+which cannot be falsified, but must be verified. So ultimately a multiverse scenari o might only be 
+accepted, strictly speaking, if there is empirical evidence for it, i.e. observational data of another 
+universe or its effects. A weaker argument for M would be if a – falsifiable, rigorously tested – 
+theory predicts the existence of other universes, and this theory is well established according to the 
+usual scientific criteria. Still weaker are philosophical reasons; whether they could suffice if they 
+are stronger than alternative statements is a controversial issue and lies at – or beyond – the 
+boundaries of physics and cosmology.  
+ 
+ 
+8. Research programs and systematicity  
+ 
+From a philosophy of science perspective an exaggerated falsificationism is not useful anyway. As 
+Imre Lakatos (1976 & 1977) pointed out, the scientific endeavour usually takes place in the form of 
+research programs (which include not only the main assumptions and theoretical ideas but also 
+methodological issues) rather than in respect of single hypotheses or theories. And refutations are 
+not the straightforward sign of emp irical progress, because research programs grow in a 
+permanent ocean of anomalies.  
+ 
+First, data are theory -dependent, not just empirical; they might simply be false or interpreted 
+wrongly. Thus, theories should not been thrown away too quickly. For exampl e the measured 
+anomalies in Uranus' orbit were not a falsification of Newton’s law of gravity but an indication of 
+Neptune's existence and the starting point for its discovery.  
+ 
+Second, theories are often ahead of data. Thus, theorists should make testabl e predictions of 
+course, and observers and experimenters should obtain data. Supersymmetry models, for 
+instance, were theories in search of applications initially, and physicists (and even mathematicians) 
+learned a lot from them although there has still no t been a single supersymmetric particle 
+discovered (but might be soon). So testability, a falsifiable prediction, is important, but not 
+necessarily from the very beginning. Thus, theories ought to be given a chance: a chance to 
+develop, to be purged of err ors, to become more complete.  
+ 
+What Lakatos advocated was, therefore, a sophisticated falsificationism, referring to a  (quasi -
+darwinistic) struggle between theories and data interpretation. If theoretical or empirical 
+contradictions, anomalies or data dev iances occur, it is reasonable to keep the theory for a while, 
+especially its "hard core" (at least if there is no alternative – an experimentum crucis  is very rarely 
+accomplished). Thus the "protective belt" of auxiliary hypotheses should be modified firs t. Often 
+they are taken as responsible for failed predictions too . Of course this effort to save the core could 
+lead to immunization strategies which would in the end expel the theory from science. 
+Furthermore, falsification is often difficult to achieve b ecause core commitments of scientific 
+theories are rarely directly testable and predictive without further assumptions.  
+ 
+But with time some research programs are more successful than others; ad hoc hypotheses to 
+save a hard core could led to program degen erations – especially if they are problem shifts not 
+pointing to other fruitful areas. So there is some reason behind the evolution and replacement of 
+research programs, it is not just chance or a postmodern temporary fashion. A progressive 
+research progra m is characterized by its growth, along with stunning novel facts being discovered, 
+more precise predictions being made, new experimental techniques being developed, while a 
+degenerating research program is marked by lack of growth, or by an increase of th e protective 
+belt that does not lead to novel facts.  
+ 
+Cosmology provides a lot of examples for these complex interplays between competing theories, 
+data acquisition and interpretation, immunization ("epicycles" are proverbial now), and even 
+paradigm change s. The most prominent were geocentrism versus heliocentrism, metagalaxy 
+versus island universe, and steady state versus big bang – universe versus multiverse seems to be 
+the next challenge in this direction.  11 
+ 
+Though the demarcation criteria for science ha ve sometimes been criticized (Laudan 1988) and 
+are somewhat fuzzy indeed, it is usually rationality that rules. And t here are many well -established 
+criteria for successful research programs, especially the following: many applications; novel 
+predictions; n ew technologies; answering unsolved questions; consistency; elegance; simplicity; 
+explanatory power/depth; unification of distinct phenomena; and truth (but how can we know the 
+latter – only via the criteria mentioned above?).  
+ 
+These criteria are also use ful in analysing modern theoretical cosmology beyond the falsifiability 
+doctrine. Are statements about the "cause" of the big bang and about other universes unscientific if 
+they cannot provide falsifiable predictions (yet)? No. At least not if they are see n as part of a 
+research program which is otherwise well -established and at least partly confirmed by observations 
+or experiments. And indeed (not all, but at least) some of the just -mentioned criteria are fulfilled by 
+the ambitious big bang and multiverse scenarios: many applications? well, no; novel predictions? 
+perhaps yes, partly; new technologies? definitely not (yet); answering unsolved questions? yes; 
+consistency? hopefully; elegance? depends on taste; simplicity? often yes; explanatory 
+power/depth? c ertainly yes; unification of distinct phenomena? yes indeed; truth? nobody knows, 
+but this is of course the main source of the controversies.  
+ 
+Last but not least there are other approaches to science. For example science can, negatively, 
+characterized by comparing it with pseudoscience ( Table 5, see Vaas 2008).  
+ 
+ 
+Table 5:  Features of science in  contrast to pseudoscience ( Vaas 2008). Most or all of the 
+following criteria apply to scientific research aiming to discover new phenomena of nature. 
+Multiverse scenarios fulfil many  of these criteria and, hence, should not be viewed as 
+pseudoscience.  
+ 
+ 
+Indicators of serious science:  
+– no vague, exaggerated, obscure claims  
+– no undefin ed, vague or ambiguous vocabulary  
+– embedding in establishe d scientific theories  
+– embedding in established scientific research program  
+– logical (internal) coherence and consistence  
+– open source: no secret data, methods, knowledge...  
+– Occam’s razor  
+– methodological reflexivity  
+– no questionable "factoids" (No rman Mailer)  
+– no reversal of the burd en of proof  
+– testability (verification, falsification)  
+– rigorously derived predictions  
+– no over emphasis of verifications, anec dotes, rumours, ignorance...  
+– not belief, faith, hope, obedience, but o bservations, measurements,  
+   arguments, mathematical reasoning, inference to the best explanation...  
+– replication, reproduction of measurements, calculations...  
+– statistical significance, double -blind studies ...  
+– distinction between correlation and ca usality  
+– progress, self -corrections, revisions, error analysis  
+– publications in scientific journals etc. (peer reviewed)  
+– quotations of scientific literature, no dubious references  
+– no selective quotes of obsolete or questionable experiments  
+– demarca tion of popular science  
+– demarcation of pseudoscepticism  
+– systematicity  
+ 
+ 
+And science can, positively, defined as a specific sort of systematicity  according to the following 
+nine criteria: descriptions, explanations, predictions, the def ence of knowledge claims, epistemic 
+connectedness, an ideal of completeness, knowledge generation, the representation of knowledge, 12 
+ 
+critical discourse (Hoyningen -Huene 2008). According to these criteria, M should also be taken as 
+a serious part of science (Vaas 2008a).  
+ 
+ 
+9. Outlook  
+ 
+Perhaps in a thousand years or already in ten years cosmologists will wonder about us, asking 
+either how we could have been so blind as not to see or accept the signs of other universes, or 
+how we could have been so crazy in our beliefs in (a science of) other universes. Right now 
+however it is an open issue, and therefore not unreasonable to defend and advance a scientific 
+analysis of M. At least in principle there are some possibilities both for a verification of other 
+universes  understood as hypothetical universal existential claims and for theoretical embeddings of 
+those claims. This should remind us of what Steven Weinberg (1977)  wrote long ago:  "our mistake 
+is not that we take our theories too seriously, but that we do not ta ke them seriously enough."  
+ 
+ 
+Acknowledgements:  It is a pleasure to thank  Angela Lahee and especially André Spiegel for 
+many comments and suggestions. I am also grateful to the people attending and discussing my 
+talks about multiversal issues, especially a t the Debate in Cosmology: The Multiverse conference 
+(Perimeter Institute, Waterloo/Canada, September 2008) and the Challenges in Theoretical 
+Cosmology  conference (Tufts Institute, Talloires/France, September 2009).  
+ 
+ 
+ 
+ 
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+Solitary wave solution to the generalized nonlinear Schrodinger equation for 
+dispersive permittivity and permeability  
+Amarendra K.Sarma  
+Department of Physics, Indian Institute of Technology Guwahati,  
+Guwahati -781039, India.  
+ 
+We present a solitary wave solution of the generalized nonlinear Schrodinger  equation  for 
+dispersive permittivity and permeability using a scaling transformation and coupled amplitude -
+phase  formulation. We have considered the third -order dispersion effect  (TOD)  into our model  
+and sh ow that soliton shift may be suppressed in a negative index material  by a judicious choice 
+of the TOD and self -steepening parameter .  
+ 
+PACS number(s): 42.65.Tg, 42.25.Bs, 05.45.Yv  
+ 
+Veselago ’s landmark paper [1] on left-handed material  predicted the possibility of a 
+negative index material (NIM ). He showed theoretically that if the dielectric permittivity 
+  and 
+the magnetic permeability 
+ are simultaneously negative, then one must take the index of 
+refraction t o be
+00/n , where 
+0 and 
+0 are respectively the free space electric 
+permittivity and magnetic permeability. Though naturally existing NIM s are  yet to be 
+discovered, astonishing progress have been made in artificially or  man-made manufactured 
+NIMs [ 2-9].The first experimental demonstration of a Veselago medium done by Smith et al. [2] . 
+Soon afterwards, the work by Pendry  on perfect lens [3] represents the initial attempt to fill the 
+gap between novel metamaterials and excitin g applications . Since  then, left-handed materials or 
+NIMs have been studied in many different contexts ranging from second harmonic generation 
+[10-11], propagation  of ele ctromagnetic waves [ 12-16] to modulation instability [ 17-20] and so 
+on.  
+One very important difference between a NIM and an  ordinary medium is that the latter 
+has a constant permeability while the former has a frequency dependent or dispersive 
+permeability . In order to find new features of ultra short  pulse propagation resulted from the 
+dispersive permeability, a re -examination of the pulse propagation equation is done by many 
+researchers [1 7-23].Recently, Scalora et al. [2 3] and then more rigorously by Wen  et al. [1 7], have derived a generalized nonlinear Schrodinger equation (GNLSE)  for dispersive permeability 
+and permittivity  suitable for few -cycle pulse propagation in NIMs.  This newly derived equation 
+has been studied so far mainly in the context of modu lation instability only;  however an att empt 
+is made by Zhang and Yi [24 ] to find the exact solution of a generalized NLSE of Scalora type. 
+Zhang and Yi have obtained the exact chirped soliton solution of the GNLSE using the so called 
+variable parametric me thod.  In this work we are finding the solitary wave solution to the GNLSE 
+using a scaling transformation and coupled - amplitude -phase formulation [25]. We have included 
+the effect of third -order dispersion (TOD) also in our studies. It is worth mentioning that a 
+generalized NLSE including the effect of TOD and self -steepening , in the context of 
+femtosecond solitary waves in  ordinary optical fibers,  was first studied by Christodoulides and 
+Joseph [26] .     
+Considering the diffraction effects to be negligible , and including the third -order 
+dispersion, the genera lized NLSE due to Wen et al. [17 ] could be put in the following form:  
+AAtAAitA
+tAizA 2 2
+33
+3 22
+2
+61
+2 
+                                                                   (1) 
+where the group velocity dispersion(GVD),
+2 ,the third order dispersion(TOD),
+3 ,the self -phase 
+modulation (SPM) coefficient ,
+ , and the self -steepening parameter ,
+ , is defined ,respectively,  
+as 
+  
+
+
+
+
+
+
+
+ 
+
+
+
+ 2
+4
+02 2
+02 4
+02 2
+0 02 11 311
+
+ 
+pm pe pm pe
+n nc
+                                                           (2) 
+6
+02 2
+0312
+
+c npm pe
+                                                                                                                   (3) 
+
+ 
+
+
+
+ 2
+02
+003
+12 
+pm
+c n
+                                                                                                                 (4) 
+ 
+
+
+
+2
+022 2
+0
+4
+0 024
+02 2
+01
+
+
+pmpm pm pe
+n
+                                                                                     (5) 
+All the parameters are evaluated at t he frequency
+0 .
+3  is the so called third -order 
+susceptibility. As in Ref. [1 7], in this work also we have used the so -called lossless Drude model 
+to describe the frequency dispersion of 
+  and 
+  , i.e. 
+
+
+
+
+22
+01pe, 
+
+
+
+
+
+22
+01pm                                                                                     (6) 
+where 
+ is the angular frequency,
+pe and 
+pm are the respective electric and magnetic plasma 
+frequencies. In  Fig.1 (a) we plot the variation of
+n ,
+ and 
+ with the normalized frequency,
+pe/0
+, for 
+8.0 /pe pm , while Fig.1 (b) depicts the corresponding variation for 
+2  and
+3 .  
+ 
+FIG.1 (a) (Color online) Variation of 
+,n and 
+ with normalized frequency for
+8.0 /pm pe . 
+ is 
+calculated in the units of 
+cpe/3 and 
+ is calculated in the units of 
+pe/1 . 
+ 
+FIG.1 (b) (Color online ) Variation  of 
+2 and 
+3 with normalized frequency for
+8.0 /pm pe , in the 
+units of 
+pec/1 and 
+2/1pec  respectively.  
+It can be easily seen that these plots are consistent with that of Ref. [1 7].The TOD  parameter is 
+positive in the entire given frequency regime. In  order to find the solitary wave solution of Eq . 
+(1), we start with scaling Eq . (1), following Ref. [25] , by taking:  
+3 2 1 , , btbzubA  , where 
+2 1,bb
+and 
+3b are so chosen that the coefficients corresponding to GVD, TOD and SPM become 
+unity. Eq. (1) can now be rewritten in the anomalous dispersion regime as  
+uuuuuiuiu 2
+33
+2
+22
+  
+                                                                                        (7) 
+with  
+32 3
+                                                                                                                           (8) 
+We assume a solution to the Eq. (7) in the form  
+   i U u ex p ,
+                                                                                              (9) 
+where
+U , with
+ , is a real  quantity.
+ is a real parameter to be determined later.  
+Substituting (9) in (7) and then separating the real and imaginary parts, we obtain  
+3 3 21 31 U U U  
+                                                                            (10) 
+    UU U U2 23 2 3 
+                                                                                        (11) 
+where 
+U is the first derivative of 
+U with respect to 
+  and so on.  
+Eqs. (10) and (11) are consistent only if the following conditions are satisfied  
+ 312 33 2
+2
+                                                                                                  (12) 
+41
+                                                                                                                                    (13) 
+From Eqs. (12) and (13) we obtain  
+ 3 2 2312 3 
+                                                                                       (14) 
+23
+1241
+
+
+                                                                                                                        (15) 
+One can easily obtain the following ordinary differential equation from Eq. (11)  
+03 U U U
+                                                                                                                 (16) 
+with 
+.2 32  Eq. (16) can be solved very easily by using standard methods to obtain the following shape 
+preserving solution:  
+      i h u ex p sec2,
+                                                                     (17) 
+It is important to note from Eq. (17) that localized bright soliton solution s exist only if 
+0 and
+0
+. In view of these constraints our analysis shows that bright soliton solution does not exist 
+for 
+625.00pe  with
+8.0 /pm pe . From Eq. (17 ) the soliton peak power,
+0P  and the 
+soliton width  
+0T can easily be estimated, after some simple algebra, as follows:  
+3 2
+33
+2
+09
+
+P
+                                                                                                                          (18) 
+
+2 31
+32
+23
+0T
+                                                                                           (19) 
+For a given
+0T ,
+2 and 
+3  one can easily work out the parameter
+  from Eq. (19).  
+ 
+FIG.2 Spatio -temporal evolution of a soliton pulse in a NIM based waveguide with 
+mpe pe 1 , 6.00  
+ and 
+fs T 1000  
+ 
+In Fig.2 we depict the spatio -temporal evolution of a soliton  inside the waveguide  over a distance 
+of nearly 6.5 
+m  for the following typical parameters:  
+. 100 ,1 , 6.00 0 fs Tmpe pe   
+  
+It can be  observed that the soliton is shifted towards the positive time axis during its evolution 
+owing to a slight positive self -steepening (SS)  effect (~0.46 in normalized unit at
+pe 6.00 ).It 
+would be interesting to check two extreme cases carefully.  In Fig.3  and 4  we plot the input and 
+output amplitude profiles of soliton pulses for 
+pe 62.00  and 
+pe 01.00  respectively. The 
+distance of propagation in the first case is 6.5
+m while in the later it is 20 nm.  
+ 
+FIG.3 ( Color online) Input and output soliton pulse in a NIM based waveguide with 
+mpe pe 1 , 62.00  
+ and 
+fs T 1000  
+ 
+FIG.4 ( Color online) Input and output soliton pulse in a NIM based waveguide with 
+mpe pe 1 , 01.00  
+ and 
+fs T 1000  
+ 
+It has been predicted in Ref . [19] with regard to the negative index material  and other literatures 
+[27] in the context of ordinary material,  that as the propagation distance increases , the positive 
+SS shifts the center of the generated p ulse towards the trailing edge. T he higher the value of SS 
+the larger would be the shift. We observe that even though SS~ 0.19 in Fig.3 as against 0.46 in 
+Fig.2, the shift of the soliton pulse is larger for the same propagation distance of the waveguide.  
+It is important to note that the respective  values of the TOD coefficient are 140 and 131 (in 
+normalized units) for  
+pe 6.00  and
+pe  62.00 . Again in Fig.4 we find that the soliton is 
+shifted towards the leading edge in spite of having a large positive SS (~100) , but even larger 
+TOD coefficient (~
+910 ) for 
+pe 01.00 .From  these results i t may  not be unreasonable to 
+conclude  that positive SS has the tendency to shift the center of the pulse towards the trailing  
+edge while positive TOD does the opposite. In an ordinary waveguide  like optical fiber, positive 
+TOD slows down the soliton during its propagation and thereby shifts it towards the trailing 
+edge.  
+In conclusion, we have found the solitary wave solution of the generalized Nonlinear 
+Schrodinger equation including the third -order dispersion term, using a scaling transformation 
+and coupled amplitude -phase formulation. Our study shows that the third order dispersion shifts 
+the center of the soliton towards the leading edge of the pulse unlike in an ordinary waveguide. 
+Hence a ju dicious choice of the TOD  and SS parameter might be effective in  control ling the 
+soliton shift in a NIM based waveguide.  This work may motivate the researchers to have a closer 
+look at the role of the higher order dispersive effects in  nonlinear pulse prop agation and other 
+related issues in NIMs.  
+ 
+ 
+[1] V. G. Veselago, Sov. Phys. Usp. 10, 509(1968).  
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+84, 4184 (2000).  
+[3]J.B.Pendry, Phys. Rev.Lett. 85, 3966 (2000 ). 
+[4]R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77(2001).  
+[5] V.M.Shalaev,Nature Photonics 1,41(2007)  
+[6] J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B.Pendry, and C. M. Soukoulis, 
+Phys. Rev. Lett. 95, 223902(2005).  [7] A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev.Lett. 91, 037401(2003).  
+[8]S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry,Phys. Rev. B 69, 241101(R) 
+(2004).  
+[9] M. Lapine, M. Gorkunov, and K. H. Ringhofer, Phys. Rev. E 67, 065601( R) (2003).  
+[10]V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A.Zakhidov, Phys. Rev. B 69, 
+165112(2004).  
+[11]M.W.Klein,M.Wegner,F.Feth and S.Linden,Science 313,502(2006)  
+[12]R. W. Ziolkowski, Phys. Rev. E 63, 046604(2001)  
+[13]G. D’Aguanno, N. Mattiuc ci, M. Scalora, and M. J. Bloemer,Phys. Rev. Lett. 93, 213902 
+(2004).  
+[14] W. T. Lu, J. B. Sokoloff, and S. Sridhar, Phys. Rev. E 69,026604(2004).  
+[15]J. R. Thomas and A. Ishimaru, IEEE Trans. Antennas Propag. 53, 1591 (2005).  
+[16] G. D’Aguanno, N. Akozbek,  N. Mattiucci, M. Scalora, M. J.Bloemer, and A. M. Zheltikov, 
+Opt. Lett. 30, 1998(2005).  
+[17]S.Wen, Y.Wang, W.Su, Y.Xiang, X.Fu and D.Fan, Phys.Rev.E 73, 036617(2006).  
+[18]X.Dai, Y.Xiang, S.Wen and D.Fan, J. Opt. Soc. Am. B 26, 564(2009).  
+[19]M.Marklund, P .K.Shukla and L.Stenflo, Phys.Rev.E. 73, 037601(2006).  
+[20] S.Wen, Y.Xiang, W.Su, Y.Hu, X.Fu and D.Fan, Opt. Express  14, 1568(2006).  
+[21]N. Lazarides and G. P. Tsironis, Phys. Rev. E 71, 036614(2005).  
+[22] I. Kourakis and P. K. Shukla, Phys. Rev. E 72, 016626(2005).  
+[23]M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G.D’Aguanno, N. Mattiucci, M. J. 
+Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902(2005).  
+[24]S.Zhang and L.Yi, Phys.Rev.E. 78, 026602(2008)  
+[25]M.Gedalin, T.C.Scott and Y.B.B and, Phys.Rev.Lett. 78,448(1997)  
+[26] D.N.Christodoulides and R.I.Joseph, Electon.Lett.20, 659(1984)  
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+ 
+ 
\ No newline at end of file
diff --git a/1001.0728.txt b/1001.0728.txt
new file mode 100644
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+++ b/1001.0728.txt
@@ -0,0 +1,283 @@
+arXiv:1001.0728v1  [hep-ph]  5 Jan 2010NewextendedCrewther-typerelation
+A.L.Kataev∗†
+InstituteforNucleaerResearchof theAcademyofScienceso fRussia,117312,Moscow,Russia
+E-mail:kataev@ms2.inr.ac.ru
+S.V.Mikhailov
+BogoliubovLaboratoryofTheoreticalPhysics,JINR,14198 0,Dubna,Russia
+E-mail:mikhs@theor.jinr.ru
+We propose a conjecture about the detailed structure of the c onformal symmetry breaking term
+in the generalized Crewther relation. We conclude that this conjecture leads to new relations
+between the QCD expansioncoefficientsof the Adler D-functi onand the polarizedBjorkensum
+ruleBjp.
+RADCOR 2009- 9thInternationalSymposiumonRadiativeCorr ections(ApplicationsofQuantumField
+Theoryto Phenomenology),
+October25-302009
+Ascona,Switzerland
+∗Speaker.
+†SecondpartofthetalkpresentedatatRADCOR2009-slightly improvedversionofpreprintCERN-PH-TH/2009-
+203, by A.L. Kataev and S.V. Mikhailov, issued October 23, 20 09, 6 pages The first part, based on the work done in
+collaboratrion withV.T.Kim(PNPI,Gatchina) willbe submi ttedelswhere
+c/circlecopyrtCopyrightownedbytheauthor(s)underthetermsoftheCreat ive CommonsAttribution-NonCommercial-ShareAlikeLicen ce. http://pos.sissa.it/New extendedCrewther-typerelation A.L.Kataev
+1. Introduction
+The concept of conformal symmetry is lying below definite mod ern theoretical and phe-
+nomenological investigations in various massless quantum field models (for the most recent re-
+views see [1, 2]). Among them are quantum electrodynamics (Q ED)and quantum chromodynam-
+ics (QCD).Moreover, even before the formulation of QCDthe s tudy of axial-vector-vector (AVV)
+triangle amplitude intheconformal symmetrylimitreveale ddefinitefundamental consequences of
+this symmetry [3], [4]. In particular the simple relation be tween basic characteristics of two main
+inclusiveprocesses, namelybetweenflavour non-singlet pa rtsoftheAdlerD-function DNS
+A(as)and
+thenon-singlet(NS)coefficientfunction K(as),whichenterintodefinitionsoftheBjorkensumrule
+SBjpof thepolarized lepton-hadron scattering andinto theNSco ntribution oftheGross-Llewellyn
+Smith sum rule SNS
+GLSfor the neutrino-nucleon scattering was found. It can be wri tten down in the
+following form
+D(as)·K(as)=1 (1.1)
+whereas=αs/(4π)andαsistheQCDcoupling. Theentries intheLHSof thisbasic equat ion are
+defined as
+DNS
+A=/parenleftBigg
+Nc∑
+fQ2
+f/parenrightBigg
+·D(as);SBjp=/parenleftbigg1
+6gA
+gV/parenrightbigg
+·K(as)SNS
+GLS=3·K(as). (1.2)
+However,itisknownthatintherenormalized quantumfieldmo delsconformal symmetryisbroken
+by the procedure of renormalizations of the coupling consta nt and the trace of energy-momentum
+tensor. Inthefirstcasetheeffectofconformalsymmetrybre akingisdescribedbytherenormalization-
+groupβ-function [5],whileinthesecond caseitleadstotheappear ance of theconformal anomaly
+which is proportional to the factor β(as)/as[6, 7, 8, 9, 10]. Careful studies of the structure of
+analytical results of NNLO perturbative QCD calculations o f the D-function [11, 12] and of the
+Bjorken polarized sumrule[14],written downthrough Casim ircolor factors CF,CAoftheSUc(N)
+gaugegroupandtheflavourfactor TFNF,allowedtorevealthatatthe a3
+s-levelthe“Crewtherunity"
+is receiving additional conformal-symmetry breaking (CSB )contribution, [15],
+D·K=1+β(as)
+as[power series in as] (1.3)
+which is proportional to the 2-loop approximation of the β-function. The application of the
+operator-product expansion method to the AVV diagram in the momentum space [16] gave first
+indications, that the factor β(as)/asmay be really factorized in the RHS of Eq.(1.3) in all orders
+of perturbation theory. This property was proved later on in the coordinate space [17], [18]. The
+subsequent theoretical andphenomenological studiesofth econsequences ofthediscoveredinRef.
+[15] modification of the original Crewther relation weremad e inRefs. [19] and Ref.[20].
+Inthisworkweareproposingnewextensionofthegeneralize d CrewtherrelationofRefs.[15],
+[17],specifying thestructure ofthepower seriesin asintheRHSof Eq.(1.3). Usingthemethodof
+the“detailed β-expansion”[21]wearepredictingsomerelationsbetween α4
+scontributionstothe D
+andK-functions, additionaltothose,whichfollowfromtheorig inalCrewtherrelationinQED[22].
+The letter ones were already checked by the analytical calcu lations of the INR-Karlsruhe-SINP
+collaboration [23]. We also emphasize the importance of kno wledge of the complete analytical
+2New extendedCrewther-typerelation A.L.Kataev
+a4
+sresults of DandK-functions for fixing some still unknown additional coeffici ents in our new
+relations,andatthefirstlevelinthenewextendedCrewther -typerelation. Moredetailedtheoretical
+explanations are beyond the scope of this paper.
+2. New extended relation
+Inthissection wepropose thefollowing new“conjectured" e xtended form of theCSBtermin
+the RHSof Eq.(1.3):
+D(as)·K(as)=1+∑
+n/parenleftbiggβ(as)
+as/parenrightbiggn
+Pn(as), (2.1)
+where
+Pn(x):/braceleftBigg
+Pn(x)=x PnCF+O(x2)
+polynomial Pn(x)does not include the coefficients of β-function
+The coefficient functions D(as)andK(as)are taken in the MS-scheme at Q2=µ2, whereµis the
+renormalization scale of the MS-scheme and
+µ2d
+dµ2as=β(as)=−/parenleftbig
+b0a2
+s+b1a3
+s+b2a4
+s+.../parenrightbig
+(2.2)
+Atthelevelof order O(a3
+s)-corrections Eq.(2.1)wasobtained byrequiring theindepe ndence ofthe
+secondtermofthepowerseriesinthebracketsoftheRHSofEq .(1.3)fromthefirstcoefficient b0of
+the QCD β-function, which both contain TFNF-terms given in Ref.[15]. The concrete expressions
+for the polynomials Pn(as)have thefollowing form:
+P1(as) =−as3CF/braceleftbigg/parenleftbigg7
+2−4ζ3/parenrightbigg
++as/bracketleftbigg/parenleftbigg47
+9−16
+3ζ3/parenrightbigg
+CA−/parenleftbigg397
+18+136
+3ζ3−80ζ5/parenrightbigg
+CF/bracketrightbigg
++O(a2
+s)/bracerightbigg
+(2.3)
+P2(as) =as3CF/parenleftbigg163
+6−76
+3ζ3/parenrightbigg
++O(a2
+s) (2.4)
+P3(as) =O(as) (2.5)
+The results were confirmed with the help “detailed β-function expansion” formalism of Ref.[21].
+Thecalculations oftheorder O(a4
+s)-corrections to DandK-functions should giveusthepossibility
+to fix the order O(a3
+s),O(a2
+s)andO(as)- coefficients Eqs.(2.3-2.5) respectively. We expect that
+they will have the following structure
+P(3)
+1(as) =a3
+sCF/bracketleftbigg
+C2
+Fas1+CFCAas2+C2
+Aas3/bracketrightbigg
+(2.6)
+P(2)
+2(as) =a2
+sCF/bracketleftbigg
+CFas4+CAas5/bracketrightbigg
+(2.7)
+P(1)
+3(as) =asCFas6 (2.8)
+where as ishould contain rational numbers and the terms, proportiona l toζ3,ζ5andζ7-functions,
+observed inthe analytical calculations of order O(a4
+s)-correction tothe Adler D-function [24].
+3New extendedCrewther-typerelation A.L.Kataev
+3. Relations between D andK coefficients at a4
+s
+Let us specify our expectations from the analytical a4
+scalculations of DandKfunctions in a
+bit different form. Wewill define their perturbative as
+D=1+∑
+n(as)ndn,K=1+∑
+l(as)lkl. (3.1)
+Following the analysis of [21], we consider an expansion of t he perturbative coefficients dnin a
+power series in b0,b1,b2,..., as
+d2=b0d2[1]+d2[0], (3.2)
+d3=b2
+0d3[2,0]+b1d3[0,1]+b0d3[1,0]+d3[0,0], (3.3)
+as opposed tothe standard expansion inapower series in NF,
+d3=N2
+Fd3(2)+N1
+Fd3(1)+N0
+Fd3(0).
+Herethefirstargument n0ofthecoefficients dn[n0,n1,...]corresponds tothepower of b0,whereas
+the second one n1corresponds to the power of b1, etc. The coefficient dn[0,0,...,0]represents
+“genuine” corrections with powers ni=0 for all the coefficients bi. If all the arguments of the
+coefficient dn[...,m,0,...,0]to the right of the index mare equal to zero, then, for the sake of
+a simplified notation, we will omit these arguments and write insteaddn[...,m]. Thus one can
+write-down the analogous representation for thenext coeffi cientd4in thefollowing form:
+d4=b3
+0d4[3]+b1b0d4[1,1]+b2d4[0,0,1]+b2
+0d4[2]+b1d4[0,1]+b0d4[1]+d4[0].(3.4)
+The same ordering in the β-function elements applies for all higher coefficients dnand forklas
+well. Thestandard naive non-abelianization (NNA)approxi mation estimates dnfrom thefirst term
+intheequationsabove,namely, dn≃bn−1
+0dn[n−1](seeRefs.[25,26]). Notethattheconsiderations
+of Ref.[25] are taking into account the terms in the coefficie ntsdn, which are proportional to bi
+0
+with 1≤i≤(n−1), but the concrete expansions in higher order coefficients of β-function with
+bi(i≥1)arenot included inthis renormalon-inspired approach.
+The conformal limit in (2.1) provides the relation between u nknown yet elements of 5-loop
+termsd4,k4, and already known parts of the 4-loop results. Indeed, Eq.( 1.1) is satisfied at β=0,
+whenallthecoefficientshavegenuinecontentonly, dn(kn)=dn[0](kn[0]),thatprovidestheevident
+relation between these elements at any loops
+kn[0]+dn[0]+n−1
+∑
+l=1dl[0]kn−l[0]=0. (3.5)
+In particular, this gives a relation of the results of 5-loop calculations k4[0],d4[0]in the LHS of
+Eq.(3.5)
+k4[0]+d4[0]=2d1d3[0]−3d2
+1d2[0]+(d2[0])2+d4
+1, (3.6)
+and the expressions for di[0]up to4-loop level in itsRHS.Theexplicit form of CFandCA- depen-
+dence of the coefficients d2[0],d3[0], which enter into Eq.(3.6), is derived in Ref. [21]. Note, th at
+4New extendedCrewther-typerelation A.L.Kataev
+the terms d4[0]andk4[0]should contain the dependence on two Casimir operators CFandCA. In
+viewofthisthedirectcheckofEq.(3.6)isstrongerthanalr eadyperformedinRef.[23]confirmation
+of the study of the validity of its the projection onto the max imum power of CF, i.e.C4
+F, suggested
+in [22] as the check of the containing ζ3-term results for the QED part of d4-term, available from
+publication in Ref.[27]
+Using the conjecture of Eq.(2.1) one may predict others rela tions between the of d4(dn)and
+k4(kn)elements. For example, in virtue of the βfactorization, one can put for the coefficient in
+front ofa2
+sb0,a3
+sb1,a4
+sb2,...,(for the term (β(as)/as)1in (2.1)) the chain of equations
+−P1=k2[1]+d2[1] =k3[0,1]+d3[0,1]=k4[0,0,1]+d4[0,0,1] =...
+=kn[0,0,...,1]+dn[0,0,...,1]=3CF/parenleftbigg7
+2−4ζ3/parenrightbigg
+,(3.7)
+that fixestheterm P1in theP1. Theterm at a2
+sinEq.(2.3) generates another relation at a4
+slevel
+k4[0,1]+d4[0,1]=k3[1]+d3[1]+d1(k2[1]−k3[0,1]+d3[0,1]−d2[1]), (3.8)
+wheretheRHScanbefixedafterapplyingthemethodof[21]tot hecoefficientsofthe SBjpaswell.
+However, to use this equation in practise weshould know the a nalytical 5-loop results for the DNS
+A
+andSBjp(for later presentation see [28]).
+Acknowledgments
+This work was started at Dubna in August of 2009, when both of u s were participating at Bo-
+golyubov Conference on Quantum Field Theory and Elementary Particle Physics. We are grateful
+to K.G. Chetyrkin for discussions during this Conference. W e also wish to thank V.M.Braun and
+D.J. Broadhurst for useful communications prior and after t his event. One of us (ALK) is grateful
+to his colleagues for creating rather stimulating atmosphe re during his stay at Th Unit of CERN
+up to 24 October, 2009 and to the Members of Organizing Commit tee of RADCOR-2009 for the
+invitation, hospitality andfinancial support duringhisst ayatthisimportant Symposium. Thework
+of both of us was supported in part by the RFBR grant No. 08-01- 00686, while the work of ALK
+in part by thegrant of President of RFNS-1616.2008.2 aswell .
+4. Noteadded
+AfterourworkandtheoneofRef.[28]weresubsequentlypres entedandthevalidityofthegen-
+eralized Crewther relation, written down in the form of Eq.( 1.3) was demonstrated at the α4
+s-level
+[28], we checked that proposed by us new representation for t he conformal symmetry breaking
+term, written down in the form of Eq.(2.1), is valid at the α4
+s-level. The explicit analytical expres-
+sions of the coefficients in Eqs.(2.6-2.8) were fixed directl y from the analytical α4
+s-expression for
+Eq.(1.3),whichweretakenfromtheresultsofRef.[28]only . Itshouldbenotedthatafterthetalkof
+Ref.[28]wasreportedwelearnedthattheseresultswerepre sentedbeforeRADCOR2009aswellin
+thetalkofRef.[29]. Unfortunately, wewereunabletocheck ournewguesses prior RADCOR2009
+Workshop in view of the delay of the kind information by Prof. K.G.Chetyrkin about the previous
+report. Thus, considering this fact as the “closed-box” che ck, we are grateful to him for the real
+interest inour work.
+5New extendedCrewther-typerelation A.L.Kataev
+References
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+[2] V. M.Braun,G. P.KorchemskyandD.Mueller,Prog.Part.N ucl.Phys. 51(2003)311
+[3] R. J. Crewther,Phys.Rev.Lett. 28(1972)1421.
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+[5] N.N.BogoliubovandD.V. Shirkov,Introductiontothe Th eoryofQuantizedFields,IV edition
+(1984),v10inN. N.Bogoliubov,CollectionofScientificWor ksin12volumes,Moscow,Nauka,
+2008,ed.A.D.Sukhanov.
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+[7] N.K. Nielsen,Nucl.Phys.B 120(1977)212.
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+[9] J. C. Collins,A. DuncanandS. D.Joglekar,Phys.Rev.D 16(1977)438.
+[10] P.Minkowski,BernePrint-76-0813(1976)
+[11] S. G.Gorishny,A.L. KataevandS. A.Larin,Phys.Lett.B 259(1991)144.
+[12] L.R. SurguladzeandM.A.Samuel,Phys.Rev.Lett. 66(1991)560[Erratum-ibid. 66(1991)2416].
+[13] K.G. Chetyrkin,Phys.Lett.B 391(1997)402
+[14] S. A.LarinandJ. A. M.Vermaseren,Phys.Lett.B 259(1991)345.
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+[16] G.T. GabadadzeandA.L.Kataev,JETPLett. 61(1995)448[PismaZh.Eksp.Teor.Fiz. 61(1995)
+439]
+[17] R. J. Crewther,Phys.Lett. B 397(1997)137
+[18] D.Muller(1996)-privatecommunicationto A.L.K.(unp ublished)
+[19] S. J.Brodsky,G.T.Gabadadze,A.L.KataevandH.J. Lu,P hys.Lett. B 372(1996)133
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+[20] J. Rathsman,Phys.Rev.D 54(1996)3420
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+[22] A.L. Kataev,Phys.Lett. B 668(2008)350
+[23] P.A. Baikov,K.G.ChetyrkinandJ.H. Kuhn,
+TalkpresentedbyK.G.Chetrkinat BogolyubovConference,D ubna,August,2009
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+[25] C. N.Lovett-TurnerandC. J. Maxwell,Nucl.Phys.B 452(1995)188
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+[27] P.A. Baikov,K.G. ChetyrkinandJ. H.Kuhn,PoS RADCOR2007 (2007)023
+[28] P.A.Baikov,K.G.ChetyrkinandJ.H.Kuhn,talkpresent edat 9thInternationalSymposiumon
+RadiativeCorrections(RADCOR2009),30October2009,Asco na,Switzerland.
+[29] P.A.Baikov,K.G.ChetyrkinandJ.H.Kuhn,talkpresent edat theConference“QCD: TheModern
+View ofthe StrongInteractions”5-9October2009,Berlin,G ermany.
+6
\ No newline at end of file
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+arXiv:1001.0730v1  [math.OA]  5 Jan 2010OPERATOR ALGEBRAS ASSOCIATED TO INTEGRAL
+DOMAINS
+BENTON L. DUNCAN
+Abstract. We study operator algebras associated to integral do-
+mains. In particular, with respect to a set of natural identities
+we look at the possible nonselfadjoint operator algebras which en-
+code the ring structure of an integral domain. We show that these
+algebras give a new class of examples of semicrossed products by
+discrete semigroups. We investigate the structure of these algeb ras
+together with a particular class of representations.
+Recently, in [4] and [8] the notion of a regular C∗-algebra associated
+to an integral domain was introduced as a generalization of a con-
+struction due to Cuntz, [3]. In these papers the authors associat e a
+C∗-algebra by representing ℓ2(R) and viewing the operators given by
+the regular representation Racting on ℓ2(R). In addition they show
+that theC∗-algebras so constructed are universal with respect to a col-
+lection of identities that encode information about the integral dom ain.
+Now one can view an integral domain as an additive group together
+withanactionontheadditivegroupgivenbymultiplicationbynonzero
+elements of the ring. This suggests that an important viewpoint for
+studying operator algebras associated to integral domains is thro ugh
+the use of crossed products. More importantly, since crossed pr oducts
+are well understood many of the significant results can be made brie f
+through the technology of crossed products.
+For this paper we wish to investigate the operator algebras with
+slightly less restrictive identities imposed by only natural ring-theor etic
+constraints. This gives rise to operator algebras with a more natur al
+crossed product structure. However since the multiplication in a rin g
+need not act as automorphisms on the additive group, crossed pro ducts
+are not entirely appropriate. To avoid this we use the nonselfadjoin t
+operator algebras where possible. This goes back to a constructio n of
+ArvesonandJosephson[1]whichwasgeneralizedbyPetersin[10]. T his
+semicrossed product is a nonselfadjoint operator algebra which en codes
+2000Mathematics Subject Classification. 47L74, 47L40.
+Key words and phrases. integral domains, semicrossed products.
+12 BENTON L. DUNCAN
+the same dynamics as the crossed product but does not require th at
+the action on a topological space be via homeomorphisms.
+While one may worry that we lose too much information when we
+lose the∗-structure of the C∗-algebra, in recent work [5] it was shown
+that the semicrossed products of Peters in fact encode the actio n of a
+continuous self map on a topological space in a manner which is unique
+up to conjugacy of the map. This is even true when the map is not a
+homeomorphism and hence unlike with C∗-algebras the nonselfadjoint
+operator algebras can be used as a topological invariant.
+It is these motivating examples which have led us to study the semi-
+crossed product algebras in the context of integral domains. In t his
+paper we have defined the universal operator algebra associated to an
+integral domain (note the different conditions we require from thos e of
+Cuntz and Li). We then study the situation in the case that our inte-
+gral domain is a field. Here the semicrossed product and the crosse d
+product coincide and we can use standard results for crossed pro ducts
+to prove facts about the algebra. After viewing the case of the int egral
+domain being a field we focus on the situation where this may not be
+true. Here the semicrossed product technology is necessary, ho wever
+similar results carry through. After defining the requisite notion of
+semicrossed product and proving some first results in the context of in-
+tegraldomainsweprovesomeresultswhichshowthatthealgebras thus
+defined are distinct from the algebras studied by Peters. In the las t
+section we analyze what we call unitary representations of an integ ral
+domainR. We show that every such representation factors through a
+regular unitary representation.
+We describe some standard notation we intend to use. If Ris an
+integral domain we write Q(R) for the field of quotients. We write
+(R,+) for the additive group on R. This group is a locally compact
+discrete group. We denote the Pontryagin dual of this group by ( /hatwidestR,+)
+and denote by /hatwideathe element in the Pontryagin dual corresponding to
+an element ain (R,+).
+The author would like to thank Jim Coykendall and Sean Sather-
+Wagstaff for helpful discussions about and examples of integral do -
+mains.
+1.Universal algebras of integral domains
+LetRbe an integral domain. Given a Hilbert space Hwe define an
+isometric representation of Rto be{Sr∈B(H) :r∈R×}a collection
+of isometries together with unitaries {Un∈B(H) :n∈R}that satisfy
+the relations:OPERATOR ALGEBRAS ASSOCIATED TO INTEGRAL DOMAINS 3
+(1)SrSt=Srtfor allr,t∈R×,
+(2)UnUm=Um+nfor allm,n∈R, and
+(3)UnSr=SrUrnfor allr∈R×,n∈R.
+If theSrare unitaries for all r∈R×then we say that the represen-
+tation is a unitary representation .
+We present first two examples:
+Example. (The regular representation of an integral domain)
+LetHbe equal to ℓ2(R), witheqdenoting the characteristic function
+of{q} ⊆R. Define operators UnandSras follows:
+Un/parenleftBigg/summationdisplay
+q∈Rζqeq/parenrightBigg
+=/summationdisplay
+q∈Rζqeq+n
+Sr/parenleftBigg/summationdisplay
+q∈Rζqeq/parenrightBigg
+=/summationdisplay
+q∈Rζqerq.
+It is not difficult to see that {Un}and{Sr}give rise to an isometric
+representation of R.
+Example. (The regular unitary representation of an integral domain)
+LetKbe equal to ℓ2(Q(R)), witheqdenoting the characteristic func-
+tion of{q} ⊆Q(R). We use the same formulas to define /tildewiderUnand/tildewideSr.
+Notice this time however that for all r,/tildewideSris onto and hence a unitary.
+(To see this notice that /tildewideSr(eq
+r) =eqfor every q∈Q(R)).
+An important point to notice is that H ⊆ K and further His
+an invariant subspace for the collections {/tildewideSr}and{/tildewiderUn}. Further
+Sr=PH/tildewideSr|Hfor allrandUn=PH/tildewiderUn|H. For this reason we call
+this representation the regular unitary representation of R.
+We letA(R) be the norm closed operator algebra generated by uni-
+taries{un:n∈R}and isometries {sr:r∈R×}which is universal for
+isometric representations of R. We will denote the elements of A(R)
+with lower case letters to distinguish from a representation of Rfor
+which we will use upper case letters.
+We notice some initial facts about the algebra A(R).
+Lemma 1. A(R)is unital with u0=s1.
+Proof.Let{Ur,Sr}be an isometric representation of R. Then notice
+thatU0is an idempotent unitary, hence 1 = U∗
+0U0. ThenU0= 1·U0=
+U∗
+0U2
+0=U∗
+0U0= 1. A similar argument yields the same result for S1.
+Since this is true for an arbitrary isometric representation of R, the
+result follows for A(R). /squaresolid4 BENTON L. DUNCAN
+Lemma 2. Ifris invertible then sris a unitary with s∗
+r=sr−1
+Proof.This follows from the previous lemma since SrSr−1=S1=
+Sr−1Sr, for any isometric representation of R. /squaresolid
+It follows that if Ris a field then A(R) is aC∗-algebra. In addition,
+in this case, the regular representation is a unitary representatio n. In
+fact we have the following:
+Proposition 1. Ris a field if and only if every isometric representa-
+tion is a unitary representation.
+Proof.This comes from the fact that for the regular representation
+(Sr)∗is in the algebra if and only if ris invertible. /squaresolid
+Finally we can see that A(R) is functorial for ring monomorphisms
+since any isometric representation of a ring R2will give rise to an
+isometric representation of a ring R1if there is a ring monomorphism
+fromR1intoR2..
+Proposition 2. A(R)is functorial in the sense that if π:R1→R2
+is a unital ring monomorphism then there is an induced comple tely
+contractive representation ˜π:A(R1)→A(R2)
+2.The universal C∗-algebra for a field
+We now analyze the case where Ris a field. Here any isometric
+representation is, in fact, a unitary representation. We let ( R,+) de-
+note the additive group in R. Notice that R×acts onC∗(R,+) as
+∗-automorphisms via the mapping αλ(Un) =UλnwhereUnis the uni-
+taryinC∗(R,+)corresponding to n∈(R,+). This allowsustorewrite
+C∗(R) as a crossed product.
+Proposition 3. LetRbe a field, then A(R)∼=C∗(R,+)⋊R×.
+Proof.We begin by noting (see [2, II.10.3.10]) that since C∗(R,+) is
+unital and R×is discrete we have C∗(R,+)⊂C∗(R,+)⋊R×via a
+representation π0and there is a natural map ρ0:R×→C∗(R,×)⋊R×.
+Together ( π0,ρ0) give rise to a covariant representation of the triple
+(C∗(R,+),R×,α).
+Now analyzing the covariance conditions that define C∗(R,+)⋊R×
+we see that ρ0(r)π0(n)ρ0(r)∗=π0(rn). Now for each n,π0(n) is
+a unitary and for each r,ρ0(r) is a unitary and hence the natural
+covariant representation π0⋊ρ0gives rise to a unitary representa-
+tion ofR, and hence there is a completely contractive representation
+ι:A(R)→C∗(R,+)⋊R×.OPERATOR ALGEBRAS ASSOCIATED TO INTEGRAL DOMAINS 5
+Next notice that any unitary representation of A(R) gives rise to a
+covariant representation of ( C∗(R,+),R×,α) and hence /bardblx/bardbl ≤ /bardblι(x)/bardbl
+so thatιis faithful. /squaresolid
+SinceC∗(R,+) andR×are both abelian we have that the universal
+norm on the crossed product C∗(R,+)⋊R×coincides with the reduced
+norm [11, Theorem 7.13]. In addition, we have that the algebra A(R)
+is nuclear [11, Corollary 7.18], when Ris a field. We next analyze the
+regular representation of A(R) whereRis a field.
+Proposition 4. LetRbe a field, then the regular representation of
+A(R)is faithful.
+Proof.This follows from analyzing the regular representation of the
+algebraC∗(R,+)⋊αR×, which is faithful since R×is amenable. In
+effect, we take the left regular representation of C∗(R,+) acting on
+L2(R,+) and add to this the action of R×via∗-automorphisms. This
+is exactly the construction of the regular representation of Rand hence
+the two coincide. /squaresolid
+Other factsabout A(R)canalsobeexplained using thecrossed prod-
+uct machinery.
+Proposition 5. For a field Rthe algebra A(R)is not simple.
+Proof.Notice that C∗(R,+) =C(/hatwidestR,+) where (/hatwidestR,+) is the Pontryagin
+dual of the locally compact abelian group ( R,+). Notice that the ∗-
+automorphisms αλinduce a homeomorphism /hatwiderαλon (/hatwidestR,+) which has
+a fixed point for each λ; in particular, /hatwiderαλ(/hatwide0) =/hatwide0) for all λ. It follows
+that there is a nontrivial invariant ideal in C∗(R,+) for the action by
+R×and hence, see [11, Section 3.5] there is a nontrivial induced ideal
+inC∗(R,+)⋊R×. /squaresolid
+We can, however completely describe the ideal structure of A(R) in
+the case of a field.
+Theorem 1. A(R)is∗-isomorphic to C⊕AwhereAis a simple C∗-
+algebra. In fact, Ais∗-isomorphic to C0((/hatwidestR,+)\{/hatwide0})⋊R×.
+Proof.We use the nontrivial invariant ideal from the previous propo-
+sition. In particular, since C∗(R,+) =C(/hatwidestR,+) letπbe the multi-
+plicative linear function given on C(/hatwidestR,+) by evaluation at /hatwide0. Fur-
+ther, if we let /hatwiderαλbe the induced homeomorphism on ( /hatwidestR,+) given
+by the∗-automorphism αλfor allλ∈R×. The induced represen-
+tation,πis a multiplicative linear functional and hence has range
+C. Hence, A(R)∼=C⊕kerπ. We now wish to describe π. So let6 BENTON L. DUNCAN
+σ:C0(/hatwidestR,+)→C0((/hatwidestR,+)\{0}) be the restriction mapping. Further,
+ifλ∈R×then/hatwiderαλ, the homeomorphism on /hatwidestR,+ induced by the au-
+tomorphism αλ, then the range of σis invariant under /hatwiderαλand hence
+there is a map τ:C0((/hatwidestR,+)\{0})⋊R×→kerπ. But since R×acts
+transitively on ( /hatwidestR,+) the crossed product C0((/hatwidestR,+)\ {0})⋊R×is
+simple and hence the map τmust be an isomorphism. /squaresolid
+3.Semicrossed products for discrete semigroups
+The preceding construction suggests that for non-field integral do-
+mains the crossed product may be replaced by a semicrossed produ ct.
+We quickly outline the relevant construction referring to [10] for mo ti-
+vation and to [7] for more information about this semicrossed prod uct.
+Given a compact Hausdorff space Xwe say that a semigroup Sacts
+onXvia continuous maps if for each s∈Sthere is a continuous map
+τs:X→Xwithτs◦τt=τst. IfSis unital with identity 0 we
+will assume that τ0is the identity map. Say that a pair ( π,St) is an
+isometric covariant representation of ( X,S,τ s) ifπis a representation
+ofC(X) on a Hilbert space Hand for each t∈S,Stis an isometry in
+B(H) such that Stπ(f(x)) =π(f(τt(x)))Stfor allx∈X.
+It is not hard to see that given the triple ( X,S,τ s), there is a non-
+trivial isometric covariant representation. The construction follo ws
+in the same manner as in [10], we only outline the idea here. Let
+H=ℓ2(X,S) where this latter Hilbert space is sequences indexed over
+elements of Swith entries from x, with canonical basis {es}. Define
+π:C(X)→B(H) byπ(f(x)) = (f(τs(x)))s∈S. Then set St(es) =est
+and extend by linearity. Then ( π,St) is an isometric covariant repre-
+sentation of ( X,S,τ s).
+We say that theuniversal operator algebra generated by all isome tric
+covariant representations of ( X,S,τ s) is the semicrossed product of X
+bySviaτ. We denote this algebra as C(X)⋊τS.
+Asexamplesnoticethatif αisasingleendomorphismofa C∗-algebra
+then we are in the situation described in Peter’s original work [10],
+where the semigroup is Z+. For an example on the opposite end of the
+spectrum we can view the semicrossed product of [6] as a semicross ed
+product where the monoid is the free monoid on ngenerators.
+Returning toanintegral domain R, we letαrbethe∗-endomorphism
+ofC∗(R,+) induced by the groupendomorphism given by left multipli-
+cationby r. This gives a map from R×into the set of ∗-endomorphisms
+ofC∗(R,+). Notice that since we are in an integral domain each of
+these endomorphisms is injective. However they are only surjectiv e
+whenris a unit.OPERATOR ALGEBRAS ASSOCIATED TO INTEGRAL DOMAINS 7
+Proposition 6. The algebra A(R)is completely isometrically isomor-
+phic toC∗(R,+)⋊αR×.
+Proof.We will show that any isometric representation of Rgives rise
+to an isometric covariant representation of the pair ( C∗(R,+)⋊αR×)
+and vice-versa, hence the two algebras will be completely isometrica lly
+isomorphic.
+So let{Un:n∈R}and{Sr∈R×}be an isometric representation
+ofR. Then the map n/mapsto→Ungives rise to a representation of C∗(R,+),
+call itπ. Further the isometries Srwill satisfy SrUnS∗
+r=Urnand
+hence the pair ( π,{Sr}) will be an isometric covariant representation
+of (C∗(R,+),R×,α).
+Let (π,{Sr}) be an isometric covariant representation of the alge-
+bra (C∗(R,+),R×,α). Then {π(un)}is a collection of unitaries and
+{Sr}is a collection of isometries that trivially satisfy the first two con-
+ditions of an isometric representation of R. Further SrUnS∗
+r=Urn
+so thatSrUn=UrnSrfor allr∈R×so that we have an isometric
+representation of R. /squaresolid
+Notice that in the case of the algebra A(R) for an integral domain
+Rthe semigroup R×will always be commutative with no torsion. In
+addition the semigroup R×can be viewed as a spanning cone for the
+groupQ(R)×(see [9, Page 60] for the definition of a spanning cone for
+a group).
+We can actually improve our characterization of A(R) as a semi-
+crossed product by looking at a more tractable semigroup. Let U(R)
+denote the group of units in R. NowR×is a commutative monoid
+which contains U(R) as a normal submonoid. We let M(R) denote
+the monoid R×/U(R). Notice that R×⊆Q(R)×and further that
+M(R)⊆Q(R)×/U(R), this latter group we call G(R).
+Foru∈U(R) letαu:C∗(R,+)→C∗(R,+) be the ∗-automorphism
+induced by the automorphism of ( R,+) that corresponds to left multi-
+plication by u. Next for r∈M(R) define a covariant representation of
+the triple ( C∗(R,+),U(R),α) byβr(Un) =Unr,ρ(Su) =Su. This
+covariant representation induces a ∗-endomorphism of C∗(R,+)⋊α
+U(R). Hence, βgives rise to a map from M(R) into the set of ∗-
+endomorphisms of C∗(R,+)⋊αU(R).
+Theorem 2. The algebra A(R)is completely isometrically isomorphic
+to(C∗(R,+)⋊αU(R))⋊βM(R), and the diagonal algebra A(R)∩
+A(R)∗= (C∗(R,+)⋊αU(R)).8 BENTON L. DUNCAN
+Proof.We will show that any isometric representation of Rgives rise
+to an isometric covariant representation of the pair
+(C∗(R,+)⋊αU(R),M(R))
+and vice-versa, hence the two algebras will be completely isometrica lly
+isomorphic.
+Solet{Un:n∈R}and{Sr:r∈R×}beanisometricrepresentation
+ofR. Then define a covariant representation of ( C∗(R,+),U(R),α) by
+n/mapsto→Unandr/mapsto→Srfor allr∈U(R). This yields a representation πof
+C∗(R,+)⋊αU(R). Next notice that Sλπ(Un) =π(Uλn)Sλfor allλ∈
+M(R), andSλSr=SrSλfor allr∈U(R),λ∈M(R). Hence we have
+an isometric covariant representation of ( C∗(R,+)⋊αU(R),M(R),β).
+Finally we take an isometric covariant representation ( π,S) of the
+triple (C∗(R,+)⋊αU(R),M(R),β). Define Un=π(n) andSr=π(Sr)
+ifr∈U(R), elseSr=Sr. This gives rise to an isometric representation
+ofR.
+The last result follows from Corollary 2 of [7]. /squaresolid
+Notice that if Ris not a field U(R) does not act transitively on the
+nonunital subalgebra of C∗(R,+), as in the case of a field. In fact we
+have the following fact.
+Corollary 1. The diagonal is isomorphic to C⊕Awhere
+A∼=C0((/hatwidestG,+)\/hatwide0)×U(R).
+Further, Ais simple if and only if Ris a field.
+Proof.ThatAis not simple follows from the fact that U(R) does not
+act transitively on C0(/hatwiderC∗(G,+)\0) unless R×=U(R). /squaresolid
+We now prove some other facts about the relationship between the
+integral domain Rand the structure of the algebra A(R).
+Proposition 7. A(R)∼=C(X)⋊S, whereSis a monoid with no
+nontrivial invertible elements if and only if the identity o fRis the only
+unit. Further if U(R) ={1}thenM(R)will not be finitely generated.
+Proof.If the identity of Ris the only unit, then we have
+C∗(R,+)×U(R)∼=C∗(R,+)
+andM(R) has no nontrivial invertible elements, else M(R)∩U(R)/\e}atio\slash=
+{1}.
+Notice that if A(R)∼=C(X)×SwhereSis a monoid with no
+nontrivialinvertibleelementsthenthediagonalalgebra A(R)∩A(R)∗=
+C(X). However, if x∈U(R) withx/\e}atio\slash= 1, then x/\e}atio\slash∈M(R) and henceOPERATOR ALGEBRAS ASSOCIATED TO INTEGRAL DOMAINS 9
+Sx∈C∗(R,+)×U(R) =A(R)∩A(R)∗. But notice that SxUn/\e}atio\slash=UnSx
+unlessx= 1 and hence C∗(R,+)×U(R) is not commutative.
+Assume now that U(R) ={1}andM(R) is finitely generated by the
+set{x1,x2,···,xn}, then 1+ x1x2···xn/\e}atio\slash∈U(S) so
+1+x1x2···xn=xk1
+1xk2
+2···xkn
+n
+with at least one kj/\e}atio\slash= 0. We will assume without loss of generality
+thatk1/\e}atio\slash= 0. Then 1 = x1(x2x3···xn−xk1−1
+1xk2
+2···xknn) which implies
+thatx1is a unit yielding a contradiction. /squaresolid
+As a corollary we have the following.
+Proposition 8. IfRis a unique factorization domain and A(R)∼=
+A×Z+whereAis aC∗-algebra, then Ais not commutative and U(R)
+is not trivial.
+Proof.Wewillassumethat Aiscommutativeandhence U(R)istrivial.
+In particular A=C∗(R,+) =C0(/hatwidestR,+). Now let x1andx2be two
+irreducible elements of M(R), then define two two-dimensional nest
+representations of A(R) by
+πi(f) =/bracketleftbiggf(/hatwidexi) 0
+0f(/hatwide1)/bracketrightbigg
+,forf∈C0(/hatwidestR,+)
+πi(Sxi) =/bracketleftbigg
+0 1
+0 0/bracketrightbigg
+πi(Sr) = 0 for r/\e}atio\slash=xi.
+It follows from the description of the two-dimensional nest repres en-
+tations of A×Z+, see [5], that x1=x2which contradicts the fact that
+M(R) must be infinitely generated. /squaresolid
+It follows that if Ris a unique factorization domain A(R) is never
+a semicrossed product in the sense of [10] and hence this collection o f
+algebras presents a unique type of semicrossed product.
+4.Unitary representations of R
+Finally we wish to analyze the unitary representations of A(R).
+Lemma 3. There is a canonical completely contractive representatio n
+i:A(R)→C∗(Q(R)).
+Proof.AsR⊆Q(R) the inclusion map provides an isometric represen-
+tation of RinsideC∗(Q(R)) and hence the induced map on A(R) is
+completely contractive. /squaresolid10 BENTON L. DUNCAN
+Proposition 9. Letπ:A(R)→B(H)be a unitary representation.
+There exists a ∗-representation τπ:C∗(Q(R))→C∗(π(A(R))which is
+onto and satisfies τπ◦i(x) =π(x)for allx∈A(R).
+Proof.Sinceπis a unitary representation we know that π(sr) =Tris
+a unitary for all r∈R×. For all/bracketleftBig
+p
+q/bracketrightBig
+∈Q(R)×we define ˜ π/parenleftBig
+s[p
+q]/parenrightBig
+=
+TpT∗
+q. We also define ˜ π/parenleftBig
+u[p
+q]/parenrightBig
+=TqVpT∗
+q. We need only show that the
+unitaries TpT∗
+qandTqVpT∗
+qsatisfy the relations for Q(R) and hence
+the induced representation τπwill be the required ∗-representation.
+Notice first that TpTq=TqTpandT∗
+pT∗
+q=T∗
+qT∗
+psinceπis a unitary
+representation of A(R). It then follows that
+TpT∗
+q=T∗
+qTqTpT∗
+q
+=T∗
+qTpTqT∗
+q
+=T∗
+qTp
+for allq,p∈R×. It follows that ˜ π/parenleftBig
+s[p1
+q1]/parenrightBig
+˜π/parenleftBig
+s[p2
+q2]/parenrightBig
+= ˜π/parenleftBig
+s[p1p2
+q1q2]/parenrightBig
+for all
+[p1
+q1] and [p2
+q2] inQ(R)×.
+Next notice that VpTq=TqVpqandT∗
+qVp=VpqT∗
+qand hence
+Tq1Vp1T∗
+q1Tq2Vp2T∗
+q2=Tq1Tq2Vp1q2Vq1p2T∗
+q1T∗
+q2
+=Tq1q2Vp1q2+q1p2Tq1q2.
+In other words ˜ π/parenleftBig
+u[p1
+q1]/parenrightBig
+˜π/parenleftBig
+u[p2
+q2]/parenrightBig
+= ˜π/parenleftBig
+u[p1
+q1]+[p2
+q2]/parenrightBig
+.
+Next we have that
+˜π/parenleftBig
+u[p1
+q1]/parenrightBig
+˜π/parenleftBig
+s[p2
+q2]/parenrightBig
+=Tq1Vp1T∗
+q1Tp2T∗
+q2
+=Tp2T∗
+q2Tq1Tq2Vp1p2T∗
+q2T∗
+q1
+=Tp2T∗
+q2Tq1q2Vp1p2T∗
+q1q2
+= ˜π/parenleftBig
+s[p2
+q2]/parenrightBig
+˜π/parenleftBig
+u[p1
+q1][p2
+q2]/parenrightBig
+.
+Hence the C∗-algebra generated by the TpandVnsatisfies the relations
+forQ(R). We call the induced representation τπ.
+Finally we note that τπ◦i(sp) =τπ/parenleftBig
+s[p
+1]/parenrightBig
+=TpT∗
+1=Tpfor all
+p∈Q×andτπ◦i(un) =τπ/parenleftbig
+u[n
+1]/parenrightbig
+=T1VnT∗
+1=Vnfor alln∈Rand
+henceτπ◦i(x) =π(x) for allx∈A(R). /squaresolid
+Itwouldfollowthatifeveryisometricrepresentation of Rdilatedtoa
+unitaryrepresentation (asforexampletheregularrepresentat ion does),
+then we could identify the C∗-envelope of A(R) as a crossed product,
+since the canonical representation iwould be completely isometric. WeOPERATOR ALGEBRAS ASSOCIATED TO INTEGRAL DOMAINS 11
+do not think this is likely since this does not even work in the case of
+C(X)⋊αZ+whereαis a non-surjective continuous mapping, see [6].
+References
+[1] W. Arveson and K. Josephson. Operator algebras and measure preserving au-
+tomorphisms. II J. Functional Analysis 4(1969) 100–134.
+[2] B. Blackadar, Operator algebras. Theory of C∗-algebras and von Neumann
+algebras.EncyclopaediaofMathematicalSciences, 122.Operato rAlgebrasand
+Non-commutative Geometry, III. Springer-Verlag, Berlin , 2006.
+[3] J. Cuntz, C∗-algebras associated with the ax+b-semigroup over NK-theory
+and noncommutative geometry , 201–215, EMS Ser. Congr. Rep. Eur. Math.
+Soc. Zrich , 2008.
+[4] J. Cuntz and X. Li, The regular C∗-algebra of an integral domain, preprint,
+2008.
+[5] K. Davidson and E. Katsoulis, Isomorphisms between topological conjugacy
+algebras, J. reine angew. Math. 621, (2008), 29–51.
+[6] K. Davidson and E. Katsoulis, Operator algebras for multivariable dynamics,
+to appear in Mem. Amer. Math. Soc. , 2007.
+[7] B. Duncan and J. Peters, Dynamics and semicrossed products b y discrete
+semigroups, preprint, 2010.
+[8] X. Li, Ring C∗-algebras, preprint, 2009.
+[9] V. Paulsen, Completely bounded maps and operator algebras Cambridge Un i-
+versity Press, Cambridge , 2002.
+[10] J. Peters, Semicrossed products of C∗-algebras, J. Funct. Anal. 59(1984)
+498–534.
+[11] D. Williams, Crossed products of C∗-algebras American Math. Soc. Provi-
+dence, 2007.
+Department ofMathematics, North Dakota State University, Fargo,
+North Dakota, USA
+E-mail address :benton.duncan@ndsu.edu
\ No newline at end of file
diff --git a/1001.0731.txt b/1001.0731.txt
new file mode 100644
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@@ -0,0 +1,1398 @@
+arXiv:1001.0731v1  [astro-ph.CO]  5 Jan 2010Extended radio emission in MOJAVE Blazars: Challenges to
+Unification
+P. Kharb1 1,2∗
+Dept of Physics, Rochester Institute of Technology, Roches ter, NY 14623
+kharb@cis.rit.edu
+M. L. Lister2
+Dept of Physics, Purdue University, 525 Northwestern Ave, W est Lafayette, IN 47907
+and
+N. J. Cooper2
+ABSTRACT
+We present the results of a study on the kiloparsec-scale radio emis sion in the
+completefluxdensitylimitedMOJAVEsample, comprising135radio-loud AGNs.
+New 1.4 GHz VLA radio images of six quasars, and previously unpublishe d im-
+ages of 21 blazars, are presented, along with an analysis of the high resolution
+(VLA A-array) 1.4 GHz emission for the entire sample. While extended emission
+is detected in the majority of the sources, about 7% of the source s exhibit only
+radio core emission. We expect more sensitive radio observations, h owever, to
+detect faint emission in these sources, as we have detected in the e rstwhile “core-
+only” source, 1548+056. The kiloparsec-scale radio morphology va ries widely
+across the sample. Many BL Lacs exhibit extended radio power and k iloparsec-
+scale morphology typical of powerful FRII jets, while a substantia l number of
+quasars possess radio powers intermediate between FRIs and FRI Is. This poses
+challenges to the simple radio-loud unified scheme, which links BL Lacs t o FRIs
+and quasars to FRIIs. We find a significant correlation between ext ended radio
+emission and parsec-scale jet speeds: the more radio powerful so urces possess
+faster jets. This indicates that the 1.4 GHz (or low frequency) rad io emission
+is indeed related to jet kinetic power. Various properties such as ex tended ra-
+dio power and apparent parsec-scale jet speeds vary smoothly be tween different
+blazar subclasses, suggesting that, at least in terms of radio jet p roperties, the
+distinction between quasars and BL Lac objects, at an emission-line equivalent
+width of 5 ˚Ais essentially an arbitrary one. While the two blazar subclasses– 2 –
+display a smooth continuation in properties, they often reveal diffe rences in the
+correlation test results, when considered separately. This can be understood if,
+unlike quasars, BL Lacs do not constitute a homogeneous populatio n, but rather
+include both FRI and FRII radio galaxies for their parent population. It could
+also just be due to small number statistics. We find that the ratio of the radio
+core luminosity to the k-corrected optical luminosity ( Rv) appears to be a better
+indicator of orientation for this blazar sample, than the traditionally used radio
+core prominence parameter ( Rc). Based on the assumption that the extended ra-
+dio luminosity is affected by the kiloparsec-scale environment, we defi ne the ratio
+of extended radio power to absolute optical magnitude ( Lext/Mabs) as a proxy for
+environmental effects. Trends with this parameter suggest that the parsec-scale
+jet speeds and the parsec-to-kiloparsec jet misalignments are no t affected by the
+large-scale environment, but are more likely to depend upon factor s intrinsic to
+the AGN, or its local parsec-scale environment. The jet speeds co uld, for in-
+stance, be related to the black hole spins, while jet misalignments cou ld arise due
+to the presence of binary black holes, or kicks imparted to black hole s via black
+hole mergers, consistent both with radio morphologies resembling pr ecessing jet
+models observed in some MOJAVE blazars, and the signature of a 90◦bump
+in the jet misalignment distribution, attributed to low-pitch helical pa rsec-scale
+jets in the literature. We suggest that some of the extremely misalig ned MO-
+JAVE blazar jets could be “hybrid” morphology sources, with an FRI jet on
+one side and an FRII jet on the other. Finally, it is tempting to specula te that
+environmental radio boosting (as proposed for Cygnus A) could be responsible
+for blurring the Fanaroff-Riley dividing line in the MOJAVE blazars, prov ided a
+substantial fraction of them reside in dense (cluster) environmen ts.
+Subject headings: galaxies: active — quasars: general — BL Lacertae objects:
+general — radio continuum: galaxies
+1. INTRODUCTION
+Radio-loud active galactic nuclei (AGNs) that are characterised by extreme variability
+in their radio cores, high and variable polarization, superluminal jet s peeds and compact
+1∗Major part of this work completed at Department of Physics, Purd ue University, 525 Northwestern
+Ave, West Lafayette, IN 47907– 3 –
+radio emission, are collectively referred to as ‘blazars’ (Angel & Sto ckman 1980). Blazars
+comprise flat-spectrumradio-loudquasars andBLLacobjects. T heir extreme characteristics
+have commonly been understood to be a consequence of relativistic beaming effects due to
+a fast jet aligned close to our line of sight (Blandford & Konigl 1979).
+The radio-loud unified scheme postulates that quasars and BL Lacs are the beamed
+end-on counterparts of the Fanaroff-Riley (FR, Fanaroff & Riley 19 74) type II and type I
+radio galaxies, respectively (Urry & Padovani 1995). The relatively lower radio luminosity
+FRI radio galaxies are referred to as “edge-darkened” since their brightest lobe emission
+lies closer to the radio cores and fades further out, while the higher radio luminosity FRII
+radio galaxies are “edge-brightened” because of the presence of bright and compact hot
+spots where the kpc-scale jets terminate. This definition turns ou t to be useful for FRI-FRII
+classification when hot spots are not clearly delineated ( cf.§3.2).
+The Fanaroff-Riley dichotomy has been proposed to arise due to diffe rences in one
+or more of the following properties: host galaxy environment (Pres tage & Peacock 1988)
+and jet-medium interaction (Bicknell 1995), jet composition (Reyn olds et al. 1996), black
+hole mass (Ghisellini & Celotti 2001), black hole spin (Meier 1999), acc retion rate and/or
+mode (Baum et al. 1995; Ghisellini & Celotti 2001). Many recent findin gs are, however,
+posing challenges for the standard unified scheme. These include th e discovery of FRI
+quasars (Heywood et al. 2007), FRII BL Lacs (Landt et al. 2006), and “hybrid” radio mor-
+phology sources with an FRI jet on one side of the core and an FRII j et on the other
+(Gopal-Krishna & Wiita 2000). One of the primary cited differences be tween quasars and
+BL Lacs has been the presence of strong, broad emission lines in the quasar spectra and
+their apparent absence in BL Lacs. Stickel et al. (1991) and Stock e et al. (1991) have de-
+fined the distinction between BL Lacs and quasars at an emission line e quivalent width
+of 5˚A. However, this distinction has been questioned (e.g., Scarpa & Falom o 1997; Urry
+1999; Landt et al. 2004), and it is known that some BL Lacs have a br oad-line region (e.g.,
+Miller et al. 1978; Stickel et al. 1991; Vermeulen et al. 1995; Corbett et al. 1996).
+In this paper, we examine the unified scheme and FR dichotomy in the M OJAVE2
+sample of blazars. MOJAVE is a long-term program to monitor radio br ightness and po-
+larization variations in the jets associated with active galaxies visible in the northern sky
+on parsec-scales with Very Long Baseline Interferometry (VLBI) (Lister et al. 2009a). The
+MOJAVE sample consists of 135 sources satisfying the following crite ria: (1) J2000.0 dec-
+lination>−20◦; (2) galactic latitude |b|>2.5◦; (3) VLBA 2 cm correlated flux density
+2Monitoring Of Jets in Active galactic nuclei with VLBA Experiments.
+http://www.physics.purdue.edu/MOJAVE/– 4 –
+exceeding 1.5 Jy (2 Jy for declination south of 0◦) at any epoch between 1994.0 and 2004.0.
+Of the 135 AGNs, 101 are classified as quasars, 22 as BL Lac object s, and eight as radio
+galaxies of mostly the FRII-type. Four radio sources are yet to be identified on the basis
+of their emission line spectra and have no redshift information. Over all, eight sources (four
+BL Lacs and four unidentified) have unknown or unreliable redshifts . Based on the syn-
+chrotron peak in the spectral energy distributions (SEDs), BL La cs have been divided into
+low, high, and intermediate energy peaked classes (LBLs, HBLs, IB Ls, Padovani & Giommi
+1995; Laurent-Muehleisen et al. 1999). All but three MOJAVE BL La cs are classified as
+LBLs (see Nieppola et al. 2006, NED). The three BL Lacs, viz.,0422+004, 1538+149 and
+1807+698, are classified as IBLs. There are no HBLs in the MOJAVE s ample.
+The compact flux density selection criteria of the MOJAVE sample bias es it heavily to-
+ward highly beamed blazars with high Lorentz factors and small viewin g angles. While
+relativistic boosting effects are likely to dominate the source charac teristics, the exten-
+sive multi-epoch and multi-wavelength data available for the MOJAVE s ample on both
+parsec- and kiloparsec-scales, provides us with unique constraint s to test the unified scheme.
+Throughout the paper, we adopt the cosmology in which H0=71 km s−1Mpc−1, Ωm=0.27
+and Ω Λ=0.73. Spectral index, α, is defined such that, flux density Sνat frequency ν, is
+Sν∝ν−α.
+2. DATA REDUCTION AND ANALYSIS
+We observed seven MOJAVE quasars at 1.46 GHz with the Very Large Array (VLA,
+Napier et al. 1983) in the A-array configuration (typical synthesiz ed beam ∼1.5′′) on June
+30, 2007 (Program ID:AC874). Sixty MOJAVE sources were previou sly observed by us with
+the VLA A-arrayand presented in Cooper et al. (2007). Data for t he remaining sources were
+reduced directly using archival VLA A-array data, or obtained thr ough published papers.
+The data reduction was carried out following standard calibration an d reduction procedures
+in the Astronomical Image Processing System (AIPS).
+Our new observations included some new expanded VLA (EVLA) ante nnas and the
+observationsweremadeinaspectralline(4-channelpseudo-con tinuum)mode, whichresulted
+in a total effective bandwidth of (2 ×21.875 =) 43.75 MHz. Four ≈9 minute scans of each
+source were interspersed with 1 minute scans of a suitable phase ca librator. 0713+438 was
+used as the bandpass calibrator for the experiment. After the init ial amplitude and phase
+calibration, AIPS tasks CALIB and IMAGR were used iteratively to se lf-calibrate (Schwab
+1980) and image the sources. The resultant rmsnoise in the maps was typically of the order
+of 0.06 mJy beam−1. The quasar 1124 −186 was observed with an incorrect source position,– 5 –DECLINATION (J2000)
+RIGHT ASCENSION (J2000)07 33 54.0 53.5 53.0 52.5 52.0 51.550 22 20
+15
+10
+05
+00
+DECLINATION (J2000)
+RIGHT ASCENSION (J2000)08 08 16.5 16.0 15.5 15.0 14.5-07 50 55
+51 00
+05
+10
+15
+20
+25
+Fig. 1.— 1.4 GHz VLA images of the quasars 0730+504 (Left) and 0805 −077 (Right). The
+contours are in percentage of the peak surface brightness an d increase in steps of 2. The lowest
+contour levels and peak surface brightness are (Left) ±0.042, 672 mJy beam−1and (Right) ±0.010,
+1.43 Jy beam−1.
+which resulted in significant beam smearing. We instead obtained arch ival VLA data for
+this source. The new radio images of the six sources are presented in Figures 1, 2, and 3.
+Since most of the MOJAVE sources have highly compact radio struct ures, most of the
+archival data comprised of snapshot observations where the sam ple sources were observed
+as phase calibrators. By choosing archival datasets with exposur e times/greaterorsimilar10 minutes, we
+were able to obtain maps with typical rmsnoise levels of ∼0.15 mJy beam−1. We present
+previously unpublished images of 21 blazars in Appendix A.
+Acompilationofthe basicparameters foreach sourceis given inTable 1. The integrated
+flux densities for the cores were obtained in AIPS using the Gaussian -fitting task JMFIT,
+while the total flux densities were obtained by putting a box around t he source, using the
+AIPS verbs TVWINDOW and IMSTAT. The extended radio flux density was obtained by
+subtracting the core flux density from the total radio flux density .
+All the maps were created with uniform weighting using a ROBUST para meter of 0 in
+theAIPStaskIMAGR.Wealsoobtainedcoreandlobefluxdensitiesfr ommapswithdifferent
+weighting schemes (by using ROBUST parameters −5 and +5, respectively) for a fraction
+of the sample. We found that the integrated core flux density estim ates differed typically by– 6 –DECLINATION (J2000)
+RIGHT ASCENSION (J2000)10 38 47.5 47.0 46.5 46.005 12 45
+40
+35
+30
+25
+20
+15
+DECLINATION (J2000)
+RIGHT ASCENSION (J2000)10 48 07.5 07.0 06.5 06.0 05.5-19 09 20
+25
+30
+35
+40
+45
+50
+Fig. 2.— 1.4 GHz VLA images of 1036+054 (Left) and 1045 −188 (Right). The contours are in
+percentage of the peak surface brightness and increase in st eps of 2. The lowest contour levels and
+peak surface brightness are (Left) ±0.021, 892 mJy beam−1and (Right) ±0.042, 724 mJy beam−1.
+less than 1%, between different weighting schemes. The extended fl ux density values differed
+typically by <2%, with a large majority of sources showing less than a 5% difference . Only
+in a handful of sources, where there seemed to be an unresolved c ompact component close to
+the core (e.g., 1038+064, 1417+385), was the difference between different weighting schemes
+significant (10% −20%). However, following our radio galaxy study (Kharb et al. 2008b ), we
+havefoundthatnearly15% −20%oftheextendedfluxdensitycouldbegettinglostinA-array
+observations, comparedtocombined-arrayobservations. Ther efore, weconcludethatthelack
+of short spacings in the A-array observations is far more detrimen tal to the determination
+of accurate extended flux values than (not) adopting different we ighting schemes on the
+A-array data to obtain core and extended flux densities. The weigh ting scheme approach
+does suggest that the errors in the extended flux density values a re typically of the order
+of 2%−5%, but could be of the order of 10% −20% for sources where a compact component
+close to the core is not clearly resolved.– 7 –DECLINATION (J2000)
+RIGHT ASCENSION (J2000)12 15 47.6 47.447.247.046.846.646.446.246.0-17 31 35
+40
+45
+50
+55
+DECLINATION (J2000)
+RIGHT ASCENSION (J2000)12 22 23.4 23.223.022.822.622.422.222.021.804 13 25
+20
+15
+10
+05
+Fig. 3.— 1.4 GHz VLA images of 1213 −172 (Left) and 1219+044 (Right). The contours are in
+percentage of the peak surface brightness and increase in st eps of 2. The lowest contour levels and
+peak surface brightness are (Left) ±0.021, 1.66 Jy beam−1and (Right) ±0.042, 549 mJy beam−1.
+3. RESULTS
+3.1. Extended Radio Power
+The 1.4 GHz extended radio luminosities for the MOJAVE sources are p lotted against
+redshift in Figure 4. Sources with no discernible extended emission ar e represented as upper
+limits. The solid lines indicate the FRI −FRII divide (extrapolated from 178 MHz to 1.4
+GHz assuming a spectral index, α= 0.8), following Ledlow & Owen (1996) and Landt et al.
+(2006). The right hand panel of Figure 4 demonstrates the close r elation between the radio
+core and extended luminosity (Table 2). Using the partial correlatio n regression analysis
+routine in IDL (P CORRELATE), we found that the linear correlation between log Lcore
+and logLextis strong even with the effects of luminosity distance (log DL) removed (partial
+correlation coefficient, rXY.Z, = 0.303, tstatistic = 3.41, two-tailed probability that the
+variables are not correlated3,p= 0.0009). We note that the 18 BL Lacs (ones with redshift
+information) alone fail to show a correlation between log Lcoreand logLext, when the effects
+of luminosity distance are removed. The implication of this finding is disc ussed ahead in
+3Calculated using the VassarStats statistical computation website ,
+http://faculty.vassar.edu/lowry/VassarStats.html– 8 –
+§3.3.
+0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
+ Redshift 212223242526272829 log L ext [W Hz -1 ] 
+FRIFRI/IIFRII
+22 23 24 25 26 27 28 29 30
+ log L core [W Hz -1 ] 21222324252627282930 log L ext [W Hz -1 ] Cyg A
+Fig. 4.— (Left) 1.4GHzextendedluminosityversusredshift. (Right ) 1.4GHzcoreversusextended
+luminosity. Open and filled circles denote quasars and BL Lac s, respectively, while open squares
+denote radio galaxies. Core-only sources are represented a s upper limits.
+The salient features of the extended radio emission in the MOJAVE bla zars are:
+(i) Six of the 22 BL Lacs ( ∼27%) have FRII radio powers (log L1.4
+ext>26). Four of these
+(viz.,0235+164, 0716+714, 0808+019, 2131 −021) also have hot spots like quasars.
+(ii) Five BL Lacs ( ∼23%) fall in the FRI/II power range (24 .5<logL1.4
+ext<26). Two of
+these (viz.,0823+033, 1803+784) also appear to have hot spots.
+(iii) Seven BL Lacs have extended luminosity log L1.4
+ext<24.5 and could be regarded as
+the true beamed counterparts of FRI radio galaxies. However, th ree of these ( viz.,
+0754+100, 0851+202, 1807+698) also appear to exhibit hot spot- like features. Hot
+spots may also be present in all four BL Lacs with no redshift informa tion. Overall,
+nearly 60% of the MOJAVE BL Lacs appear to have hot spots. The ho t spots in some
+BL Lacs are however not as bright or compact as those observed in quasars.
+(iv) Excluding the upper limits, 22 quasars ( ∼22%) fall in the FRI/II power range.
+(v) 10 sources ( ∼7%, 9 quasars, 1 BL Lac) do not show any extended emission −these
+are discussed in §3.4.– 9 –
+Based on the extended emission at 1.4 GHz, we can conclude that a su bstantial frac-
+tion of the MOJAVE BL Lacs have both radio powers and radio morpho logies like FRIIs
+or quasars. A substantial fraction of the MOJAVE quasars lie in the intermediate (FRI/II)
+luminosity range. These results are consistent with a number of pre vious radio studies.
+Using a large sample of radio core-dominated AGNs, Murphy et al. (19 93) observed that
+many high redshift ( z >0.5) BL Lacs lie above the FRI/FRII luminosity division. Using
+a sample of 17 BL Lacs, mostly belonging to the 1-Jy sample, Kollgaard et al. (1992) sug-
+gested that about 12% −30% of BL Lacs in a radio flux-limited sample may be bona fide
+FRIIs. Based on extended radio emission and strong emission lines at one or more epochs,
+Rector & Stocke (2001) concluded that many radio-selected BL La cs belonging to the 1 Jy
+sample (Stickel et al. 1991) cannot bebeamed FRIs, but aremorelik ely to be beamed FRIIs.
+Cara & Lister (2008) found similar intrinsic parent luminosity function s for the MOJAVE
+sample (consistent with FRIIs) irrespective of whether the BL Lac s were included or not.
+3.2. Radio Morphology
+We observe from Figures 1–3 that all but two sources show two-sid ed radio structures.
+The variety of radio structures observed here are reasonably re presentative of the entire
+sample (e.g., see Cooper et al. 2007), as well as those previously obs erved in other quasar
+surveys (e.g., Gower & Hutchings 1984; Antonucci & Ulvestad 1985 ; Pearson & Readhead
+1988; Murphy et al. 1993). The quasars often show two-sided rad io structures with hot
+spots on one or both sides of the core. However, one-sided morph ologies with no discernable
+compact or diffuse emission on the other side of the core, are also co mmon.
+While determining if the blazars had an FRII type radio morphology, we relied initially
+on the presence of one or more compact hot spots at the leading ed ge of the radio lobes.
+When no clear hot spots were visible on either side of the cores, we re sorted to the traditional
+“edge-brightened” definition for FRII sources, i.e.,when the brightest radio emission was
+furthest from the core, we classified the source as an FRII. The r emaining sources with no
+clear hot spots, and the brightest extended emission closest to th e cores, were classified as
+FRI types. Even then, it was sometimes difficult to consign the sourc es to FRI or FRII
+classes, and we have listed more than one morphology for some sour ces in Table 1.
+Many of the MOJAVE sources show distinctly curved kiloparsec-sca le jets. Straight jets
+are a rarity in the sample. Many MOJAVE quasars exhibit large parsec -to-kiloparsec jet
+misalignments. While any intrinsic curvature in the jets is likely to be high ly exaggerated
+by projection effects in these low-viewing angle blazars, many sourc es seem intrinsically
+distorted. We discuss in §4.1 how the MOJAVE selection criteria might be preferentially– 10 –
+picking up bent jets. Apart from highly curved jets, many sources show distinct hot spot-like
+features both closer to the core and at the jet termination points , similar to the wide angle
+tail (WAT) radio galaxies (e.g., O’Donoghue et al. 1990).
+Four MOJAVE blazars have not yet been identified as quasars or BL L acs (Table 1).
+A new VLA image of one of the unidentified sources ( viz.,1213−172) is presented in Fig-
+ure3. Anotherunidentified source(0648 −165)hasasimilarcore-dominantradiomorphology
+(Cooper et al. 2007). A third source (2021+317) has a distinct cor e-halo morphology resem-
+bling a BL Lac object (see Appendix A), while the fourth (0446+112) has a straight jet with
+a possible hot spot at the end (Cooper et al. 2007).
+3.3. Parsec-scale Jet Speeds and Extended Luminosity
+21 22 23 24 25 26 27 28 29 30
+ log L core [W Hz -1 ] 0102030405060 Apparent Speed 
+21 22 23 24 25 26 27 28 29 30
+ log L ext [W Hz -1 ] 0102030405060 Apparent Speed 0805-077
+Fig. 5.— (Left) 1.4 GHz core luminosity versus the apparent jet speed . The aspect curve assumes
+γ= 52,Lint= 5×1024andp≈2. (Right) 1.4 GHz extended luminosity versus the apparent j et
+speed. Open and filled circles denote quasars and BL Lacs, res pectively, while open squares denote
+radio galaxies. Core-only sources are represented as upper limits.
+The MOJAVE and 2 cm survey programs have provided apparent par sec-scale jet
+speeds (βapp) for nearly the entire MOJAVE sample, by monitoring structural ch anges over
+timescalesofapproximatelyadecade. Thesehavebeenpreviouslyt abulatedbyKellermann et al.
+(2004), and most recently by Lister et al. (2009b). For our analys is below, we use a βapp
+value that represents the fastest moving feature in that source .
+The parsec-scale apparent jet speeds and kiloparsec-scale radio core luminosity (Fig. 5)– 11 –
+are related by standard beaming relations. The aspect curve in Figu re 5 assumes: L=
+Lint×δp,βapp=βsinθ/(1−βcosθ), where the Doppler factor, δ= 1/(γ(1−βcosθ)) and
+β=v/c. The best-fit values for the curve are γ= 52,Lint= 5×1024andp≈2. The relation
+between parsec-scale apparent jet speeds and parsec-scale co re luminosity for the MOJAVE
+sample is discussed in greater detail by Cohen et al. (2007), and List er et al. (2009b).
+We find that the parsec-scale apparent jet speeds are correlate d significantly with the
+extended radio luminosity (Fig. 5, Table 2). Partial regression analy sis shows that the linear
+correlation between log Lextandβappwith the effects of luminosity distance removed, is still
+highly significant ( rXY.Z= 0.273,tstatistic = 3.05, two-tailed probability, p= 0.0028).
+This implies that more radio powerful sources have faster radio jet s. Since extended radio
+luminosity is correlated with the core luminosity (Fig. 4, Table 2), and t he core luminosity
+is related to the apparent jet speed, we tested the correlation wit h a partial correlation
+test between extended radio luminosity and apparent jet speeds, after removing the effects
+of radio core luminosity. This too yielded a statistically significant corr elation between
+extended radio luminosity and parsec-scale apparent jet speeds ( rXY.Z= 0.246,tstatistic =
+2.72,p= 0.0075).
+The significant implication of this result is that faster jets are launch ed in AGNs with
+larger kiloparsec-scale lobe luminosities. It would therefore appear that the fate of the AGN
+is decided at its conception. This result undermines the role of the kilo parsec environment
+on the radio power of a given source. It also indicates that the 1.4 GH z extended emission
+is indeed related to jet kinetic power, contrary to some suggestion s in the literature (e.g.,
+Bˆ ırzan et al. 2004). Given that there is a large overlap in radio power s between quasars and
+BL Lac objects, it can be concluded that most quasars have faste r jets than most BL Lac
+objects.
+The second noteworthy inference that can be drawn from Figure 5 is that there is
+a continuous distribution of parsec-scale jet speeds going from BL Lacs to quasars. This
+supports the idea that the distinction between the BL Lac and quas ar population, at an
+equivalent width of 5 ˚A, is essentially an arbitrary one at least in terms of radio jet propert ies
+(see also Scarpa & Falomo 1997).
+However, we note that inspite of the two blazar classes displaying a s mooth continuation
+in properties (as is evident in Fig. 5 and other plots), they sometimes reveal differences
+in the correlation test results when considered separately. In Tab le 2 we have listed the
+correlation test results separately for quasars and BL Lacs, if th ey differed from results for
+the combined blazar population. BL Lacs considered alone sometimes failed to exhibit the
+correlations observed in quasars (also see §3.1). Apart from the possible effects of small
+number statistics, a simple interpretation of this finding could be tha t unlike the quasars,– 12 –
+the BL Lacs do not constitute a homogeneous population. As has be en previously suggested
+in the literature (e.g., Owen et al. 1996; Cara & Lister 2008; Landt & B ignall 2008), the BL
+Lacs might constitute the beamed population of both FRI and FRII r adio galaxies. We note
+that Giroletti et al. (2004) have demonstrated that HBLs (which a re absent in MOJAVE)
+conform to the standard unification scheme with respect to FRI ra dio galaxies. This again
+supports the proposed inhomogenity within the BL Lac class.
+PreliminaryMonteCarlosimulationsofapopulationofAGNsimilartothat ofMOJAVE
+also show a correlation between apparent jet speed and extended luminosity (Cooper et al.,
+in preparation). The simulations are based on the luminosity function of Padovani & Urry
+(1992), and assume that the extended luminosity is unbeamed and p roportional to the in-
+trinsic unbeamed parsec-scale luminosity. We are currently incorpo rating the luminosity
+function derived from the MOJAVE sample by Cara & Lister (2008) int o the simulations.
+3.4. Parsec-to-kiloparsec Jet Misalignment
+Apparent misalignment in the jet direction from parsec to kiloparsec scales is commonly
+observed in blazars (e.g., Pearson & Readhead 1988). This misalignme nt could either be due
+to actual large bends in the jets, or due to small bends that are am plified by projection.
+We present the parsec- to kiloparsec-scale jet misalignment for th e MOJAVE sources in
+Figure6. The procedure for estimating the parsec-scale jet posit ion angles(PAs) is described
+in Lister et al. (2009a). Kiloparsec-scale jet position angles were de termined for the FRII
+sources using the brightest hot spots, especially when no jet was c learly visible, under the
+assumptionthatthebrightest hotspotindicates theapproaching jetdirectionduetoDoppler
+boosting. TheAIPSprocedureTVMAXFIT wasusedtoobtainaccur atepeakpixel positions
+of the core and hot spot, which were then used in the AIPS verb IMD IST to obtain the jet
+position angle. For the FRI-type sources we used TVMAXFIT for th e core pixel position,
+and the AIPS verb CURVALUE to get the pixel position roughly towar ds the center and
+end of the broad jet/lobe. Finally IMDIST was used to obtain the jet position angle. Note
+that unlike regular radio galaxies, FRI blazars are typically one-sided , which makes the
+approaching jet direction easy to identify. Due to the wide jet/lobe s in FRIs, the kiloparsec-
+scale jet position angle could be uncertain by 10◦to 15◦. When a jet feature appeared only
+partially resolved ( viz.,1038+064, 1417+385), a two Gaussian component model was used
+in JMFIT to obtain pixel positions of the core and jet feature.
+We note that our approach ( viz.,of using the brightest hot spot to determine the
+approaching jet direction, when no jet was clearly visible) differs fro m that adopted by some
+other authors (e.g., Xu et al. 1994) who assume that, when no jet is visible, the hot spot on– 13 –
+the side of the parsec-scale jet, represents the kiloparsec-sca le jet direction. Our approach
+has led us to identify many more sources ( ∼30%) with misalignment angles greater than
+90◦.
+020406080100120140160180
+ Jet Misalignment [Degrees] 0510152025303540 Number of Sources 
+Fig. 6.— Misalignment anglebetweentheparsec-scaleandthekilopa rsec-scale jetfortheMOJAVE
+sources. There are signatures of a weak 90◦bump.
+For an otherwise straight jet having a single bend, the apparent be nding angle ( η) is
+related to the intrinsic bending angle ( ψ) through the relation, cot η=cotψsinθ−cosθcosφ
+sinφ,
+whereθis the angle to line of sight, and φis the azimuth of the bend (Appl et al. 1996). The
+fact that we see a majority of sources having close to zero appare nt misalignment implies
+one of the following scenarios: (i) the intrinsic misalignment angle is sma ll; (ii) both the
+intrinsic misalignment and jet inclination angles are large; (iii) the azimut h of the bend is
+close to 180◦,i.e.,the plane of the bend is perpendicular to the sky plane. Since scenar io (ii)
+is unlikely for these blazars, possibilities (i) and (iii) are more favorable . In§4, we explain
+how the MOJAVE selection criteria could make it biased towards bent j ets.
+Pearson & Readhead(1988)andConway & Murphy (1993)reporte danunexpected sec-
+ondary peak at ∆PA = 90◦in the misalignment angle distribution. The signatures of a weak
+90◦bump appear to be present in Figure 6. Conway & Murphy (1993) con cluded that while
+the largely aligned population could be explained by straight parsec-s cale jets and small
+intrinsic bends between parsec- and kiloparsec-scales, the secon dary peak sources must have
+gently curving (low-pitch) helical parsec-scale jets. This helical dis tortion could arise due
+to Kelvin-Hemlholtz instabilities or precession of the jet ejection axe s. We reach a similar
+conclusion on the jet misalignments, following a different approach in §4.3.– 14 –
+4. DISCUSSION
+4.1. Selection Effects in MOJAVE
+The MOJAVE survey appears to select many sources with intermedia te radio powers
+and radio morphology (e.g., high radio power BL Lacs, some with hot sp ots, and low radio
+power quasars). This could be a result of the MOJAVE selection crite ria, which are based
+on the relativistically-boosted, high radio frequency (15 GHz) pars ec-scale flux densities.
+This would make it more likely to encompass a much larger range in intrins ic radio powers,
+compared to other samples selected on the basis of total kiloparse c-scale flux-densities (e.g.,
+the 3CR sample). The intrinsic radio luminosity function derived for th e MOJAVE sample
+by Cara & Lister (2008) also supports this view.
+Furthermore, as the MOJAVE survey picks up bright radio cores, m ost of the sources
+could either be fast jets pointed towards us, or curved/bent jet s that have at least some por-
+tion of the jet aligned directly into our line of sight, at at least one epo ch (e.g., Alberdi et al.
+1993). The MOJAVE survey could therefore be biased towards ben t jets. Note that in the
+wide angle tail quasars, the hot spots closer to the cores could be p roduced at the base
+of the plumes (e.g., Hardcastle & Sakelliou 2004), or indicate a sharp b end in the jet (e.g.,
+Alberdi et al. 1993; Jetha et al. 2006). We discuss the possibility of b eamed WAT quasars
+in the MOJAVE sample further in §4.3.
+4.2. Orientation Effects in the Sample
+Since the MOJAVE sample consists almost entirely of blazars, the sou rces are expected
+to have jets aligned close to our line of sight. Our examination of orien tation effects in the
+sample, however, using two different statistical orientation indicat ors,RcandRv, reveals
+that a range of orientations must in fact be present in order to acc ount for several observed
+trends.RcandRvalso reveal, to some extent, dissimilar trends. We finally reach the
+conclusion that Rvis a better indicator of orientation for this blazar sample.
+The ratio of the beamed radio core flux density ( Score) to the unbeamed extended radio
+flux density ( Sext),viz.,the radio core prominence parameter ( Rc), has routinely been used
+as a statistical indicator of Doppler beaming and thereby orientatio n (Orr & Browne 1982;
+Kapahi & Saikia 1982; Kharb & Shastri 2004). The k-correctedRc(=Score
+Sext(1+z)αcore−αext,
+withαcore= 0,αext= 0.8) is plotted against the radio core luminosity in Figure 7. As
+expected,Rcis correlated with the radio core luminosity (Table 2). Rchowever, shows a
+significant anti-correlation with respect to the extended radio lumin osity (Fig. 8). Rcdoes– 15 –
+-4 -3 -2 -1 0 1 2 3 4 5
+ log R C 2224262830 log L core [W Hz -1 ] 
+0 1 2 3 4 5 6 7 8
+ log R V 24252627282930 log L core [W Hz -1 ] 
+Fig. 7.— The orientation indicators, Rc(Left) and Rv(Right) versus the 1.4 GHz core luminosity.
+Open and filled circles denote quasars and BL Lacs, respectiv ely, while open squares denote radio
+galaxies. Core-only sources are represented as upper limit s.
+not show a correlation with jet misalignment (Table 2). In fact the qu asars considered alone
+showed a weak anti-correlation between Rcand jet misalignment (implying that the more
+core-dominant sources have smaller jet misalignments). These res ults are unexpected, be-
+cause in the former case, the extended radio luminosity is expected to be largely unbeamed,
+while in the latter, the jet misalignment is affected by projection and t herefore orienta-
+tion (in the sense that the more core-dominant sources show large r jet misalignments, e.g.,
+Kapahi & Saikia 1982). As an aside, Brotherton (1996) have noted that the anti-correlation
+between log Rc−logLextmay be suggesting that core-dominated quasars are intrinsically
+fainter than their lobe-dominated counterparts. This would be con sistent with the idea that
+the MOJAVE sources span a large range in intrinsic radio powers ( cf.§4.1).
+We also examined the relationship between Rcand apparent jet speed. While both
+Rcandβappdepend on the Lorentz factor and orientation, many of the viewing angles
+may be inside the critical angle for maximum superluminal speed, which would spoil any
+correlation. However, the quasars considered alone do show a sign ificant anti-correlation,
+contrary to expectations. But as we see below, the alternate orie ntation indicator, Rv, does
+show the expected behavior.
+Wills & Brotherton (1995) defined Rvas the ratio of the radio core luminosity to the k-
+correctedabsoluteV-bandmagnitude( Mabs):logRv=logLcore
+Lopt= (logLcore+Mabs/2.5)−13.7,
+whereMabs=MV−k, and thek-correction is, k=−2.5log(1+z)1−αoptwith the optical– 16 –
+-4 -3 -2 -1 0 1 2 3 4 5
+ log R C 2224262830 log L ext [W Hz -1 ] 
+0 1 2 3 4 5 6 7 8
+ log R V 2224262830 log L ext [W Hz -1 ] 
+Fig. 8.— The orientation indicators, Rc(Left) and Rv(Right) versus the 1.4 GHz extended
+luminosity. Open and filled circles denote quasars and BL Lac s, respectively, while open squares
+denote radio galaxies. Core-only sources are represented a s upper limits.
+spectral index, αopt= 0.5.Rvis suggested to be a better orientation indicator then Rc
+since the optical luminosity is likely to be a better measure of intrinsic j et power (e.g.,
+Maraschi et al. 2008; Ghisellini et al. 2009) than extended radio lumin osity. This is due to
+the fact that the optical continuum luminosity is correlated with the emission-line luminosity
+over four orders of magnitude (Yee & Oke 1978), and the emission- line luminosity is tightly
+correlated with the total jet kinetic power (Rawlings & Saunders 19 91). The extended radio
+luminosity, ontheotherhand, issuggestedtobeaffectedbyintera ctionwiththeenvironment
+on kiloparsec-scales.
+Figure 7 suggests that Rvis indeed a better indicator of orientation as the correlation
+with radio core luminosity gains in prominence (see Table 2). Rvshows a weak positive
+correlation with the extended radio luminosity (Fig. 8). Partial regr ession analysis however,
+shows that the linear correlation between log Rvand logLextis no longer significant when the
+effects of log Lcoreare removed ( rXY.Z= 0.078,tstatistic = 0.84, p= 0.4013). This implies
+that the extended radio emission is largely unbeamed. Rvis correlated with the parsec-to-
+kiloparsec jet misalignment, suggesting that orientation does play a role in the observed jet
+misalignments, at least for the quasars. The lack of such a correlat ion for the BL Lacs could
+again be interpreted to be consequence of their comprising of not o nly beamed FRIs (which
+could have intrinsically distorted jets) but beamed FRIIsas well. We d iscuss this point again
+in§4.5. Finally, there appears to be no correlation between Rvandβapp, as is expected if
+many blazar viewing angles are inside the critical angle for maximum sup erluminal speeds,– 17 –
+which would then result in smaller apparent speeds. This result is again consistent with Rv
+being a better indicator of orientation than Rc.
+Here we would like to note that the correlation between apparent je t speed and absolute
+optical magnitude does not seem to be as significant as the one obse rved between apparent
+jet speed and extended radio luminosity (Table 2). This could in princip le undermine the
+suggestion that the optical luminosity is a better indicator of jet kin etic power than extended
+radio luminosity. However, since the optical luminosity is morelikely to b eaffected by strong
+variability, thantheextendedradioluminosity, thelackofastrongc orrelationcanperhapsbe
+accounted for, in these non-contemporaneous radio and optical observations. Nevertheless,
+this is an important result to bear in mind, that requires further tes ting.
+4.3. Environmental Effects
+-16 -18 -20 -22 -24 -26 -28 -30
+ M abs  21222324252627282930 log L ext [W Hz -1 ] 
+-1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8
+ [log L ext ]/M abs  0.00.51.01.52.02.53.03.5 Redshift 0742+103
+2037+5110642+449
+Fig. 9.— (Left) 1.4 GHz extended radio luminosity versus the absolut e optical magnitude. (Right)
+The environment indicator versus redshift. Open and filled c ircles denote quasars and BL Lacs,
+respectively, while open squares denote radio galaxies. Co re-only sources are represented as upper
+limits.
+It has been suggested that apart from projection and relativistic beaming effects, the
+complex radio structures in quasars could arise due to interactions with the surrounding
+medium (Pearson & Readhead 1988; Barthel & Miley 1988). As is obse rved in Figure 9,
+the extended radio luminosity appears to be correlated with absolut e optical luminosity.
+However, a partial regression analysis shows that the linear corre lation becomes weak or non-– 18 –
+existent, after the effects of luminosity distance are removed (fo r quasars,rXY.Z=−0.169,
+tstatistic = −1.62,p= 0.1088; for BL Lacs, rXY.Z= 0.079,tstatistic = 0.31, p= 0.7608).
+The lack of a strong correlation may again be attributable to strong optical variability
+in these blazars. If the extended radio luminosity is indeed affected b y interaction with the
+kiloparsec-scale environment, andthe optical luminosity is closely re lated to theAGN power,
+the ratio [log Lext]/Mabs, can serve as a probe for environmental effects on kiloparsec-sc ales
+(as suggested by Wills & Brotherton 1995). Henceforth we will refe r to this ratio as the
+“environment indicator” (we will drop the “log” term in the ratio for c onvenience, in the
+following text).
+A higher value of the ratio could indicate greater jet-medium interac tion, (as say, from
+an asymetrically dense source environment). Dense environments can decrease expansion
+losses in the source, but increase radiative losses, making the sour ces brighter at low radio
+frequencies (e.g., inCygnus A,seeBarthel & Arnaud1996). Notet hattheopticalmagnitude
+in blazars is expected to be related to the AGN (accretion disk or jet ), rather than to
+starlight. This might not be true for the nearby radio galaxies, wher e starlight might indeed
+be a significant contributor to the optical magnitude. The Lext/Mabsratio might therefore
+not be a good environment proxy for the radio galaxies. We have the refore excluded radio
+galaxies from the correlation tests related to the environment pro xy indicator.
+We plot the environment proxy ratio with respect to redshift in the r ight hand panel
+of Figure 9 and find a strong positive correlation. The “alignment effe ct” between the radio
+source axis andthe emission line region in highredshift radio sourcesdemonstrates that local
+galacticasymmetries (onscalesofseveral kiloparsecs) increasew ithredshift(McCarthy 1993;
+Best et al.1999). Thisresult therefore, isconsistent withthesug gestionoftheenvironmental
+dependence of the extended luminosity. We note that the BL Lacs c onsidered alone fail to
+show a strong correlation (Table 2), which could be a consequence o f an inhomogeneous
+parent population, as discussed in §3.3.
+We find that while the parsec-scale jet speeds appear to be tightly c orrelated with
+the extended radio luminosity (Fig. 5), and weakly correlated with th e optical luminosity,
+they seem to not show any correlation with the kpc-scale environme nt indicator (Fig. 10).
+This could imply that the parsec-scale jet speeds are not driven by t he kiloparsec-scale
+environment, but rather by factors intrinsic to the AGN, or the AG N’s local parsec-scale
+environment. For instance, the jet speeds could be related to the black hole spins (e.g., Meier
+1999). However, like other results pertaining to optical luminosity, this lack of a correlation
+needs further testing with contemporaneous optical and radio da ta.
+We find that the parsec-to-kiloparsec jet misalignment is not corre lated with redshift
+(Fig. 11). This goes against the idea that the bending of the jets is in fluenced by interaction– 19 –
+-1.4 -1.3 -1.2 -1.1 -1.0 -0.9
+ [log L ext ]/M abs  0102030405060 Apparent Speed 0805-077
+Fig. 10.— Apparent parsec-scale jet speed versus the environment ind icator. Open and filled
+circles denote quasars and BL Lacs, respectively, while ope n squares denote radio galaxies.
+with the medium; an effect which should get more prominent with increa sing redshift. This
+result is consistent with the suggestions of Hintzen et al. (1983) an d Appl et al. (1996). We
+note that Hintzen et al. (1983) used the kiloparsec-scale jet-to- counterjet misalignment in
+their study. Kharb et al.(2008b)founda similar lackofcorrelationb etween jet-to-counterjet
+misalignment andredshift foranFRIIradiogalaxysample. Thus, the jetmisalignment could
+also be related to factorsclosely associated with the AGNitself. For instance, the presence of
+binary black holes or kicks imparted to the black hole via a black hole mer ger, could switch
+theejectiondirectiononashorttimescale(e.g.,Begelman et al.1980 ;Merritt & Ekers2002).
+Black hole re-alignment due to a warped accretion disk could also give r ise to jet precession
+(Pringle1997). Wenotethata fewsources have morphologiesrese mbling fast, precessing jets
+(“banana” shaped) as modelled by Gower et al. (1982). Alternative ly, the jet misalignment
+could be influenced by the AGN’s local parsec-scale environment.
+The jet misalignment is inversely correlated with the kpc-scale enviro nment indicator
+(see right hand panel of Fig. 11). The inverse relation suggests th at jets with smaller mis-
+alignments( i.e.,straighterjets)showsignaturesofgreaterenvironmental effe ctsontheirlobe
+emission. This could either be suggestive of a uniformly dense confinin g medium on different
+spatial scales aroundthe source, or, of projection effects in the sample, which could boost the
+optical emission and decrease the Lext/Mabsratio in sources with larger jet misalignments.
+If the latter is true, then this is an important caveat −the environment indicator could
+also be affected by source orientation in blazars. However, the orie ntation-dependence alone– 20 –
+0.0 0.5 1.0 1.5 2.0 2.5 3.0
+ Redshift 04080120160 Jet Misalignment [Degrees] 
+-1.4 -1.3 -1.2 -1.1 -1.0 -0.9
+ [log L ext ]/M abs  04080120160 Jet Misalignment [Degrees] 
+Fig. 11.— (Left) Jet misalignment versus the environment indicator. (Right) Misalignment angle
+versusredshift. Openandfilledcircles denotequasarsandB LLacs, respectively, whileopensquares
+denote radio galaxies.
+cannot satisfactorily explain the correlation between the environm ent indicator and redshift
+(Fig. 9).
+Many MOJAVE blazars exhibit radio structures resembling those of wide angle tail
+radio galaxies. Observations have indicated that WAT quasars could be members of rich
+galaxy clusters (e.g., Hintzen et al. 1983; Harris et al. 1983; Blanton et al. 2000). Using the
+NASA/IPAC extragalatic database (NED) we found that nearly 7% o f the MOJAVE sources
+have a galaxy cluster ∼5′away, while 14% have a galaxy cluster ∼10′−15′away. While
+this fraction appears to be small, we must keep in mind that most of th e blazars are at high
+redshifts, while the cluster information is highly incomplete even at re dshiftsz >0.1 (e.g.,
+Ebeling et al. 1998; B¨ ohringer et al. 2004). Thus the possibility rema ins that a large number
+of MOJAVE blazars inhabit galaxy clusters. This suggestion has inter esting ramifications,
+as discussed below. Furthermore, this could help us understand so me of the results discussed
+in this section.
+4.4. Relation Between Radio Power and Emission-line Lumino sity
+Rawlings & Saunders (1991) demonstrated a tight correlation betw een emission-line lu-
+minosity and total jet kinetic power in a sample of radio-loud AGNs. Th is correlation
+was tighter than the one observed between emission-line luminosity a nd radio luminosity.– 21 –
+Landt et al. (2006) discovered a large number of “blue quasars” wit h strong emission lines,
+but relatively low radio powers in the Deep X-ray Radio Blazar Survey ( DXRBS) and the
+RGB samples, and suggested that this went against the close relatio nship between emission-
+line and jet power. These and other observations (e.g., Xu et al. 199 9) could be suggesting
+thatthemechanismfortheproductionoftheoptical-UVphotonst hationisetheemission-line
+clouds is decoupled from the mechanism that produces powerful ra dio jets. Many MOJAVE
+blazars also seem to not follow the relation between emission-line and r adio luminosity; that
+is, some quasars andBL Lacshave relatively low andhigh radio powers , respectively, and the
+suggestion of decoupled mechanisms might hold true. However, an a lternate hypothesis by
+which their emission-line luminosity could still be correlated with their int rinsic jet kinetic
+power, could come about if the outlying BL Lac objects were presen t in dense confining en-
+vironments, while the outlying quasars were old, field sources. Confi nement would increase
+the synchrotron radiative losses due to the amplification of the mag netic field strength, mak-
+ing the sources brighter (see Barthel & Arnaud 1996), while expan sion losses and gradual
+radiative losses could presumably reduce the low frequency radio em ission in the old, field
+quasars.
+It has been suggested that a large fraction ( ∼50%) of the high redshift radio sources
+could be GPS sources (e.g., O’Dea et al. 1991), which are likely to be pre sent in confined
+environments. One of the highest redshift MOJAVE sources, 0742 +103 (z= 2.624), is a
+GPS quasar. There are several other MOJAVE sources that have peaked spectra, but do not
+show the low-variability or the double-lobed parsec-scale structur es that are characteristic
+of the GPS class. These could be “masquerading GPS” sources, who se overall spectrum
+happens to be dominated by an unusually bright feature in the jet, lo cated downstream
+from the core.
+4.5. Blazar Division Based on Radio Power
+In order to gain more insight into the primary factors that drive the FR dichotomy,
+in this section we disregard the emission-line division (i.e., quasars, BL L acs), and divide
+sources into FRI, FRI/II and FRII classes based solely on their 1.4 G Hz extended powers
+(following Fig. 4). Figure 12 plots the environment indicator, Lext/Mabs, with respect to
+redshift, using the new classification. A quick comparison with Fig. 9 r eveals that many
+of the low redshift quasars fall into the intermediate FRI/II class. The low redshifts of
+these sources discounts the possibility of a (1 + z)4surface brightness dimming effect in
+them, which could potentially reduce their extended luminosity, makin g them fall in the
+intermediate luminosity class. This finding is consistent with the discov ery of “blue quasars”– 22 –
+by Landt et al. (2006), which have radio powers and spectral ener gy distributions (SEDs)
+similar to the high energy peaked BL Lacs.
+-1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8
+ [log L ext ]/M abs  0.00.51.01.52.02.53.03.5 Redshift 22
+222
+22
+22
+222
+2
+22
+22
+2222
+22
+2
+22
+22
+22
+2222
+22
+2
+2222
+222
+22
+222
+22
+2 22222
+22222
+22 22
+22
+2
+22
+222
+2
+2
+22
+111
+11111111
+11
+1
+Fig. 12.— Theenvironment indicator versus redshift, withsources di vided on thebasis of extended
+radio luminosity into FRI (=1), FRII (=2), and FRI/II (=fille d circles).
+Most beamed FRIs lie at redshifts below 0.5 in Fig. 12. This could be a limita tion
+of the flux limited sample. Interestingly, the FRI/II sources lie at all redshift ranges, and
+throughout the Lext/Mabsdistribution. This suggests that these “intermediate” luminosity
+sources are not produced as a result of different environmental c onditions compared to, say,
+other FRIIs. Similarly, sources divided on the basis of the presence or absence of hot spots,
+show no discernable difference in the Lext/Mabsvs.zplane. It is important to note that
+environmental radio boosting, as mentioned in §4.4 (also Barthel & Arnaud 1996) could be
+instrumental in blurring the Fanaroff-Riley dividing line in the MOJAVE bla zars.
+Lastly, the orientation indicator, Rv, seems to be correlated with misalignment only
+for FRIIs, but not for FRIs or FRI/IIs (Table 2). This implies that p rojection is playing a
+significant role in the jet misalignment of FRII sources, but the jets are intrinsically more
+distorted in FRIs and FRI/IIs. A corollary to this is that the differen ces between FRIs
+and FRIIs could be related to factors affecting the black hole spin dir ection, since we have
+previously concluded that jet mislignments appear to be largely unre lated to kiloparsec-scale
+environmental effects.– 23 –
+4.6. Extremely Misaligned Sources
+Six MOJAVE sources ( viz.,0224+671, 0814+425, 1219+044, 1510 −089, 1828+487,
+2145+067) have jet misalignments greater than 160◦. Such extremely misaligned sources
+could be bent jets viewed nearly head-on, as suggested for the ga mma-ray blazar PKS
+1510−089(Homan et al. 2002). Analternatescenario could bethatsome o fthese misaligned
+sources are “hybrid” morphology sources (e.g., Gopal-Krishna & Wiit a 2000). Three of six
+sources with largemisalignments ( viz.,0224+671, 1219+044,1510 −089)indeed seem topos-
+sess radio morphologies that could be classified as “hybrid”. The larg e misalignment then
+makes sense: the VLBI jet (used to determine the parsec-scale j et position angle) could be
+on the FRI side, while the sole hot spot (used to determine the kilopar sec-scale jet position
+angle) could be on the FRII side.
+One interesting source that shows extreme misalignment ( ∼135◦) is M87 (1228+126).
+M87 appears to exhibit a weak hot spot on the counterjet side (e.g., Owen et al. 1990;
+Sparks et al. 1992). Since the brightest emission on the counterje t side is at the extremity
+of the radio lobe (like in FRIIs), M87 could be classified as a hybrid radio morphology
+source. Its radio luminosity ( L178= 1×1025W m−2) also places it close to the FRI/FRII
+division (Biretta et al. 1999). Kovalev et al. (2007) have observed a very short, weak radio
+counterjet onparsec-scales inM87, butperhapsthatisjustaslo wer outersheaththatquickly
+terminates.
+Ifindeedsomeoftheextremelymisalignedsourcesare“hybrid”sou rces, thelackofacor-
+relationbetween jetmisalignment andenvironment appearstocont radict thesuggestionthat
+asymmetries in galactic environments, which increase with redshift ( e.g., Best et al. 1999),
+give rise to hybrid morphology sources (e.g., Gopal-Krishna & Wiita 200 0; Miller & Brandt
+2009). Moreover, the fraction of such sources would be higher in t he MOJAVE (6% −8%),
+than that reported for the FIRST survey (1%, Gawro´ nski et al. 2006), suggesting that hy-
+brid sources may not be as rare as previously supposed. It is also int eresting to note that
+not many MOJAVE “hybrid” morphology sources fall in the FRI/II lum inosity range, but
+rather have FRII luminosities. In other words, the MOJAVE “hybrid ” morphology sources
+are not necessarily “hybrid” in terms of radio power, but mostly fall into the FRII luminos-
+ity class. Chandra X-ray observations of these sources could res olve some of these apparent
+contradictions.– 24 –
+4.7. Core-only Sources
+About 7% of the MOJAVE sources do not show any extended radio em ission in our
+1.4 GHz VLA A-array images. However the ratio of their 5 GHz radio flu x density to their
+B-band optical flux density is of the order of a few hundred, placing them firmly in the
+radio-loud AGN class (e.g., Kellermann et al. 1989). The dynamic range achieved in the
+radio images of these sources varied from about 6000:1 ( e.g.,0955+476) to 30,000:1 ( e.g.,
+0202+149),beingtypicallyoftheorderof10,000:1. Whilesomesourc es, like1749+096,failed
+to reveal any extended emission in a number of different datasets ( see also Rector & Stocke
+2001), we did detect faint halo-like emission around 1548+056, which was listed as a core-
+only source by Murphy et al. (1993). Note that observations with t he VLA A-array-only
+are likely to miss very diffuse radio structure, due to the lack of shor t baselines. We believe
+therefore that these core-only sources probably have associat ed faint emission which require
+more sensitive observations to detect.
+The observed fraction of core-only sources in MOJAVE is smaller tha n that reported
+for other quasar surveys (20%, Gower & Hutchings 1984; Padrielli et al. 1988). While the
+core-only sources could have been affected by the redshift surfa ce brightness dimming effect,
+our discussion in the previous section argues against this conjectu re. Alternatively, these
+sources could just be normal quasars at very small angles to line of sight, so that the VLA
+resolution is insufficient to delineate the various components. This ap pears to be the case in
+1038+064, and 1417+385, where a two Gaussian component model fits the “core” emission
+better than a single one. The hypothesis that at least some of thes e quasars could be beamed
+FRIs also cannot be ruled out (e.g., Heywood et al. 2007).
+4.8. Optical Polarization Subclasses
+Low optical polarization radio quasars (LPRQs) consistently revea l core optical frac-
+tional polarization, mopt<3%, while the high optical polarization quasars (HPQs) routinely
+reveal highly polarized optical cores with mopt≥3% (Angel & Stockman 1980). The classi-
+fication of the MOJAVE sources into these categories is listed in Table 1, and were obtained
+from Impey et al. (1991) and Lister & Smith (2000). In all, there are 38 HPQs and 16
+LPRQs in the MOJAVE sample. We find no statistically significant differen ces in the core
+or extended radio luminosity between LPRQs and HPQs. In Kharb et a l. (2008a) we had
+noted a difference in the integrated (single-dish) 1.4 GHz radio luminos ity between LPRQs
+and HPQs for redshifts less than one. We do not find a difference in th e total radio lumi-
+nosity using the present higher resolution VLA observations. Furt hermore, we do not find
+a statistically significant difference in jet misalignment between LPRQs and HPQs. The– 25 –
+parsec-scale apparent jet speeds of HPQs, however, are syste matically higher than those
+in LPRQs (Kolmogorov-Smirnov test probability that the speeds in HP Qs and LPRQs are
+similar, isp=0.03). Additional optical polarimetry information on the sample is ne eded to
+further investigate these trends.
+5. SUMMARY AND CONCLUSIONS
+1. We describe the results from a study of the extended emission at 1.4 GHz in the
+MOJAVE sample of 135 blazars. New VLA A-array images of six MOJAVE quasars,
+and previously unpublished VLA A-array archival images of 21 blazar s, are presented.
+The kiloparsec-scale structures are varied, with many sources dis playing peculiar, dis-
+torted morphologies. Many blazar jets show parsec-to-kiloparse c-scale jet misalign-
+ments greater than 90◦. These characteristics have been reported in other quasar sur-
+veys as well. The 90◦bump in the jet misalignment distribution (of which we observe
+a weak signature) has been suggested to be a result of low-pitch he lical parsec-scale
+jets in the literature.
+2. AsubstantialnumberofMOJAVEquasars( ∼22%)andBLLacs( ∼23%)displayradio
+powers that are intermediate between FRIs and FRIIs. Many BL La cs have extended
+luminosities ( ∼27%) and hot spots ( ∼60%) like quasars. The hypothesis that at least
+some of the core-only quasars are in fact FRI quasars, cannot be discarded. In terms
+of radio properties alone, it is difficult to draw a sharp dividing line betwe en BL Lacs
+and quasars, in the MOJAVE sample. This could be a result of the MOJA VE selection
+effects which naturally pick sources with a large range in intrinsic radio power (low
+powerquasarsandhighpowerBLLacs), andpreferentiallybentje ts. Whilethequasars
+and BL Lacs display a smooth continuation in all their properties, the correlation test
+results differ sometimes when they are considered separately. This can be understood
+if the BL Lacs did not constitute a homogeneous population like the qu asars, but had
+both FRI and FRII radio galaxies as their parent population. These fi ndings challenge
+the simple radio-loud unified scheme, which links FRI sources to BL Lac objects and
+FRIIs to quasars.
+3. We find that there is a significant correlation between extended r adio luminosity and
+apparent parsec-scale jet speeds. This implies that most quasars have faster jets than
+most BL Lac objects. The large overlap between many properties o f quasars and
+BL Lacs (e.g., 1.4 GHz radio power, parsec-scale jet speeds) howev er suggests that, at
+least in terms of radio jet properties, the distinction between thes e two AGN classes,
+at an emission-line equivalent width of 5 ˚A, is essentially an arbitrary one.– 26 –
+4. These observations suggest that the mechanism for the produ ction of optical-UV pho-
+tonsthationisetheemission-linecloudsisdecoupledfromthemechan ismthatproduces
+powerful radio jets (e.g., Xu et al. 1999; Landt et al. 2006). An alte rnate hypothesis
+could be that, BL Lacs with high radio powers are present in dense co nfining envi-
+ronments, which would increase the radiative losses in them and make them brighter,
+while the radio quasars with low radio powers are old, field sources. En vironmental
+radio boosting (Barthel & Arnaud 1996) could also explain why the Fa naroff-Riley di-
+viding line is blurred in the MOJAVE sample. X-ray cluster studies could b e used to
+test these ideas.
+5. The ratio of the radio core luminosity to the k-corrected optical luminosity ( Rv) ap-
+pears to be a better indicator of jet orientation than the tradition ally used radio core
+prominence parameter ( Rc). This seems to be a consequence of the environmental
+contribution to the extended radio luminosity, used in Rc. Trends with Rvreveal that
+even though the sample comprises largely of blazars, there is a reas onable range of
+orientations present in the sample.
+6. Trends with the “environment” proxy indicator, Lext/Mabs, reveal that jet speeds seem
+tonotdependontheenvironmentonkiloparsecscales, whilethejet misalignmentsseem
+to be inversely correlated. The jet misalignments are also not corre lated with redshift.
+It appears that the parsec-to-kiloparsec jet misalignment, the p arsec-scale jet speeds
+and to an extent the extended emission (which is related to jet spee d) are controlled
+by factors intrinsic to the AGN. Black hole spins could dictate the jet speeds, while the
+presence of binary black holes, or kicks imparted to black holes via bla ck hole mergers
+could influence the jet direction. This is consistent with radio morpho logies similar to
+precessing jet models of Gower et al. (1982) present in the MOJAVE sample, and the
+signature of the 90◦bump in the jet misalignment distribution, which is attributed to
+low-pitch helical parsec-scale jets.
+7. Ifsomeofthehighlymisalignedsources(jetmisalignment ∼180◦)are“hybrid”FRI+II
+morphology sources, then the fraction of such sources is higher in the MOJAVE survey
+(6%−8%), than that reported for the FIRST survey (1%, Gawro´ nski et al. 2006).
+Furthermore, the lack of a correlation between jet misalignment an d the environmental
+indicator would appear to contradict the suggestion that hybrid mo rphology is a result
+of the jet interacting with an asymmetric environment. X-ray obse rvations of hybrid
+sources are needed to resolve these issues.
+8. About 7% of the MOJAVE quasars show no extended radio emission . We expect
+more sensitive radio observations to detect faint emission in these s ources, as we have
+detected in the case of 1548+056, previously classified as a core-o nly quasar. The– 27 –
+hypothesis that at least some of these quasars could be beamed FR Is cannot be ruled
+out.
+9. While the extended radio power or jet misalignments do not show st atistically signif-
+icant differences between the two quasar optical polarization subc lasses,viz.,LPRQs
+and HPQs, the parsec-scale apparent jet speeds of HPQs are sys tematically higher
+than those in LPRQs.– 28 –
+Table 1. The MOJAVE sample
+Source z V Type Opt Ipeak Irms Freq βapp Score Sext Radio PcjetPA KpcjetPA Ref
+name mag Pol Jy beam−1Jy beam−1GHz Jy mJy Morph Degree Degree
+(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
+0003−066 0.347 18.50 B Y 2.605 1.6 ×10−41.4000 2.89 2.66 43.9 2 282 15 AL634 ∗
+0007+106 0.0893 15.40 G ... 0.075 4.0 ×10−51.4000 0.97 0.08 17.6 2hs 284 239 AL634 ∗
+0016+731 1.781 19.00 Q N 0.398 4.2 ×10−41.4000 6.74 0.40 7.8 1hs 132 2 AL634 ∗
+0048−097 .... 17.44 B Y 0.562 1.8 ×10−41.4250 .... 0.57 139.7 1hs+2, 2hs 337 190 AB1141
+0059+581 0.644 17.30 Q ... 1.554 1.1 ×10−41.4000 11.08 1.57 19.4 ch+1 236 .... AL634 ∗
+0106+013 2.099 18.39 Q Y 2.760 1.4 ×10−41.4000 26.50 2.81 530.6 1hs, 2 238 184 AL634 ∗
+0109+224 0.265 15.66 B Y 0.361 7.8 ×10−51.4000 .... 0.36 3.9 1 82 93 AL634 ∗
+0119+115 0.57 19.70 Q ... 1.188 9.8 ×10−51.4000 17.09 1.24 113.7 1hs, ch 6 33 AL634 ∗
+0133+476 0.859 19.50 Q Y 1.872 9.6 ×10−51.4000 12.98 1.88 8.7 1hs 331 340 AL634 ∗
+0202+149c0.405 21.00 Q Y 3.808 1.4 ×10−41.4000 6.41 3.86 1.1 c 319 .... AL634 ∗
+0202+319 1.466 17.40 Q N 0.648 8.4 ×10−51.4000 8.30 0.65 11.7 1hs 357 2 AL634 ∗
+0212+735 2.367 19.00 Q Y 2.458 1.0 ×10−41.4000 7.63 2.47 1.7 1hs, 2hs 122 253 AL634 ∗
+0215+015 1.715 16.09 B Y 0.416 6.9 ×10−51.4000 34.16 0.45 71.1 2hs 104 92 AL634 ∗
+0224+671 0.523 19.50 Q ... 1.476 1.0 ×10−41.4000 11.63 1.48 149.2 2hs 355 184 AL634 ∗
+0234+285 1.213 19.30 Q Y 2.274 1.0 ×10−41.4000 12.31 2.33 99.9 1hs 349 331 AL634 ∗
+0235+164 0.94 15.50 B Y 1.501 5.8 ×10−51.4000 .... 1.51 25.5 1hs+2 325 343 AL634 ∗
+0238−084 0.005 12.31 G ... 1.091 6.0 ×10−51.4000 0.34 1.11 111.3 2 66 275 AL634 ∗
+0300+470 .... 16.95 B Y 1.166 5.7 ×10−51.4000 .... 1.18 60.1 1, ch 148 225 AL634 ∗
+0316+413 0.0176 12.48 G N 20.76 1.0 ×10−31.3121 0.31 21.40 1103.0 2hs 178 158 BT024
+0333+321 1.263 17.50 Q N 2.994 1.0 ×10−41.4000 12.78 3.02 71.8 1hs 126 149 AL634 ∗
+0336−019 0.852 18.41 Q Y 2.908 1.5 ×10−41.4000 22.36 2.92 70.3 1hs 67 344 AL634 ∗
+0403−132 0.571 17.09 Q Y 4.043 3.3 ×10−41.4000 19.69 4.33 9.1 1hs+2 155 202 AL634 ∗
+0415+379 0.0491 18.05 G ... 0.542 5.7 ×10−41.4899 5.86 0.57 2700.0 2hs 71 62 AR102
+0420−014 0.914 17.00 Q Y 2.887 1.0 ×10−41.4000 7.34 2.91 70.2 2hs, 1hs+2 183 189 AL634 ∗
+0422+004 .... 16.98 B Y 1.083 1.3 ×10−41.4000 .... 1.09 6.1 2hs 357 200 AL634 ∗
+0430+052 0.033 15.05 G ... 2.845 1.7 ×10−41.4649 5.38 2.95 159.5 1, 2 252 293 AB379
+0446+112 .... 20.00 U ... 1.527 1.2 ×10−41.4000 .... 1.56 15.4 1hs 103 207 AL634 ∗
+0458−020 2.286 18.06 Q Y 1.629 3.8 ×10−41.4899 16.51 1.66 148.1 1hs, 2hs 321 230 AN047
+0528+134 2.06 20.00 Q ... 2.204 1.2 ×10−41.4000 19.14 2.24 60.1 2hs, 1hs+2 19 261 AL634 ∗
+0529+075 1.254 19.00 Q ... 1.518 5.7 ×10−51.4000 12.66 1.54 126.8 2hs, 1hs+2 353 221 AL634 ∗
+0529+483 1.162 19.90 Q ... 0.641 6.7 ×10−51.4000 19.78 0.65 21.5 1hs+2 32 75 AL634 ∗
+0552+398c2.363 18.30 Q ... 1.539 1.0 ×10−41.4000 0.36 1.55 1.2 c 290 .... AL634 ∗
+0605−085 0.872 17.60 Q ... 2.344 2.8 ×10−41.4250 19.79 1.20 123.9 1hs 109 99 AD298
+0607−157c0.324 18.00 Q ... 3.011 1.4 ×10−41.4000 3.94 3.02 1.1 c 52 .... AL634 ∗
+0642+449c3.396 18.49 Q ... 0.644 7.5 ×10−51.4000 0.75 0.65 1.4 c 90 .... AL634 ∗
+0648−165 .... .... U ... 2.085 1.4 ×10−41.4000 .... 2.12 11.2 c, 1 273 272 AL634 ∗
+0716+714 0.31 15.50 B Y 0.645 6.8 ×10−51.4000 10.06 0.69 376.4 2hs, 1hs+2 40 300 AQ006
+0727−115 1.591 20.30 Q ... 3.182 9.4 ×10−51.4250 .... 3.22 2.4 c, 1hs 292 .... AC874 †
+0730+504 0.72 19.30 Q ... 0.676 6.0 ×10−51.4250 14.06 0.69 82.5 2hs 210 82 AC874 †
+0735+178 .... 16.22 B Y 1.92 0.7 ×10−31.4650 .... 1.91 20.6 1 77 165 Murphy
+0736+017 0.191 16.47 Q Y 2.327 3.2 ×10−41.4899 14.44 2.34 40.9 2hs 273 297 AA025
+0738+313 0.631 16.92 Q ... 2.16 0.4 ×10−31.4650 10.75 2.16 65.0 2hs 156 166 Murphy
+0742+103 2.624 24.00 Q ... 3.600 2.3 ×10−41.4899 .... 3.62 5.8 1, 2 354 81 AY012
+0748+126 0.889 18.70 Q ... 1.43 0.5 ×10−31.4650 18.36 1.43 27.0 2hs 110 125 Murphy
+0754+100 0.266 15.00 B Y 2.07 0.4 ×10−31.4650 14.40 2.07 6.7 2hs 10 293 Murphy
+0804+499 1.436 19.20 Q Y 0.64 0.3 ×10−31.4650 1.82 0.64 5.3 1hs 134 .... Murphy
+0805−077 1.837 19.80 Q ... 1.439 5.8 ×10−51.4250 50.60 1.58 59.8 1hs 332 278 AC874 †
+0808+019 1.148 17.20 B Y 0.449 9.1 ×10−51.4899 12.99 0.46 18.2 2hs 182 200 AA025
+0814+425 0.245 18.18 B Y 1.015 3.1 ×10−41.5159 1.70 1.03 76.8 2hs 136 322 AK460
+0823+033 0.506 16.80 B Y 1.33 0.4 ×10−31.4650 17.80 1.32 4.1 1hs 15 .... Murphy
+0827+243 0.94 17.26 Q ... 0.743 1.4 ×10−41.5100 21.97 0.76 62.7 1hs+2 135 199 AV150
+0829+046 0.174 16.40 B Y 0.788 1.9 ×10−41.4000 10.10 0.80 150.8 2, 1hs+2 60 151 AG618
+0836+710 2.218 17.30 Q N 3.168 1.7 ×10−41.4000 25.38 3.34 73.6 1 211 205 AL634 ∗
+0838+133 0.681 18.15 Q ... 0.287 8.0 ×10−51.4250 12.93 0.35 2216.6 2hs 76 94 AH480
+0851+202 0.306 15.43 B Y 1.563 1.4 ×10−41.5315 15.17 1.57 10.7 1hs, 1 260 251 AS764
+0906+015 1.024 17.31 Q Y 1.00 0.4 ×10−31.4650 20.66 1.00 38.0 1hs 46 54 Murphy
+0917+624 1.446 19.50 Q ... 1.11 0.4 ×10−31.4650 15.57 1.11 6.4 1hs+2 341 234 Murphy
+0923+392 0.695 17.03 Q N 2.399 3.1 ×10−41.5524 4.29 2.83 361.8 1hs+2 98 83 AB310
+0945+408 1.249 18.05 Q N 1.23 0.9 ×10−31.4650 18.60 1.23 95.0 1hs 114 32 Murphy
+0955+476c1.882 18.65 Q ... 0.606 1.0 ×10−41.4899 2.48 0.62 1.1 c 124 270 AP188
+1036+054 0.473 .... Q ... 0.916 4.9 ×10−51.4250 6.14 0.94 57.0 1hs+2 350 226 AC874 †
+1038+064 1.265 16.70 Q ... 1.465 1.5 ×10−41.6649 11.86 1.49 11.0 1 146 166 GX007A
+1045−188 0.595 18.20 Q ... 0.726 6.6 ×10−51.4250 8.57 0.76 509.4 1hs+2 146 125 AC874 †
+1055+018 0.89 18.28 Q Y 2.675 1.5 ×10−41.4250 10.99 2.70 230.8 2hs 300 186 AB631
+1124−186 1.048 18.65 Q ... 0.652 1.9 ×10−41.4250 .... 0.66 12.3 1hs 170 134 AD337– 29 –
+Table 1—Continued
+Source z V Type Opt Ipeak Irms Freq βapp Score Sext Radio PcjetPA KpcjetPA Ref
+name mag Pol Jy beam−1Jy beam−1GHz Jy mJy Morph Degree Degree
+(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
+1127−145 1.184 16.90 Q ... 4.460 2.3 ×10−41.4649 14.17 4.58 59.3 1hs 66 44 AB379
+1150+812 1.25 19.40 Q ... 1.874 6.5 ×10−41.4000 7.08 1.89 89.2 1hs+2, 2 163 257 AL634 ∗
+1156+295 0.729 14.41 Q Y 1.433 2.4 ×10−41.4899 24.85 1.55 196.1 2, 1hs+2 14 344 AA025
+1213−172 .... 21.40 U ... 1.676 1.0 ×10−41.4250 .... 1.85 119.3 1 117 266 AC874 †
+1219+044 0.965 17.98 Q ... 0.548 5.9 ×10−51.4250 2.34 0.60 155.5 1hs+2 172 0 AC874 †
+1222+216 0.432 17.50 Q ... 0.930 8.0 ×10−51.4000 21.02 1.10 956.4 2hs 351 76 AL634 ∗
+1226+023 0.158 12.85 Q N 34.26 1.5 ×10−31.3659 13.44 34.89 17671 1hs 238 222 AE104
+1228+126 0.0044 12.86 G ... 3.558 2.3 ×10−41.6649 0.032 3.88 115012 2 291 67 BC079
+1253−055 0.536 17.75 Q Y 10.17 6.4 ×10−41.6649 20.58 10.56 2095.0 2hs 245 201 W088D5
+1308+326 0.996 15.24 Q Y 1.321 1.5 ×10−41.4000 27.14 1.33 69.1 2hs, 1hs+2 284 359 AL634 ∗
+1324+224 1.4 18.90 Q ... 1.128 9.6 ×10−51.4000 .... 1.14 20.4 1hs 343 237 AL634 ∗
+1334−127 0.539 19.00 Q Y 1.992 9.9 ×10−51.4899 10.26 2.07 151.0 1hs, 1hs+2 147 106 AD176
+1413+135 0.247 20.50 B ... 1.056 1.1 ×10−41.4899 1.79 1.08 5.8 2 55 256 AC301
+1417+385 1.831 19.69 Q ... 0.506 6.4 ×10−51.4000 15.44 0.52 2.5 1 164 123 AL634 ∗
+1458+718 0.905 16.78 Q N 5.760 8.7 ×10−41.4000 7.05 7.57 68.9 c, 1 164 11 AL634 ∗
+1502+106 1.839 18.56 Q Y 1.795 7.9 ×10−51.4000 14.77 1.82 38.3 1hs 116 159 AL634 ∗
+1504−166 0.876 18.50 Q Y 2.354 4.0 ×10−41.4899 4.31 2.39 11.4 1hs, 2hs 165 164 AY012
+1510−089 0.36 16.54 Q Y 1.380 1.0 ×10−41.4899 20.15 1.45 180.2 1hs, 2hs 328 163 AY012
+1538+149 0.605 17.30 B Y 1.479 6.0 ×10−51.4000 8.73 1.67 71.4 ch, 1 323 320 AL634 ∗
+1546+027 0.414 17.45 Q Y 1.15 0.4 ×10−31.4650 12.07 1.15 18.8 1hs+2 170 162 Murphy
+1548+056 1.422 19.50 Q Y 2.106 1.6 ×10−41.5524 11.56 2.21 42.9 ch 8 .... AB310
+1606+106 1.226 18.70 Q ... 1.35 1.3 ×10−31.4650 18.90 1.35 26.5 1hs 332 271 Murphy
+1611+343 1.397 18.11 Q N 2.83 0.5 ×10−31.4650 14.09 2.83 20.6 2hs 166 194 Murphy
+1633+382 1.814 18.00 Q Y 2.17 0.4 ×10−31.4650 29.46 2.17 32.0 1hs+2 284 176 Murphy
+1637+574 0.751 16.90 Q N 0.996 1.1 ×10−41.5524 10.61 1.01 71.2 1hs 198 282 AB310
+1638+398 1.666 19.37 Q ... 1.147 2.0 ×10−41.4899 12.27 1.17 27.6 2hs 300 150 AR250
+1641+399 0.593 16.62 Q Y 7.153 1.3 ×10−31.5149 19.27 7.95 1476.9 1hs+2, 2hs 288 327 AS396
+1655+077 0.621 20.00 Q Y 1.199 1.4 ×10−41.5524 14.44 1.27 199.1 1hs+2 314 269 AB310
+1726+455 0.717 18.10 Q ... 0.993 1.3 ×10−41.4899 1.81 1.00 55.3 1hs+2 271 328 AK139
+1730−130 0.902 19.50 Q ... 6.101 3.4 ×10−41.4250 35.69 6.13 517.8 2hs, 1hs+2 2 273 AL269
+1739+522 1.375 18.70 Q Y 1.571 1.5 ×10−41.4899 .... 1.61 27.6 1hs+2 15 262 AC301
+1741−038c1.054 20.40 Q Y 1.677 1.5 ×10−41.4250 .... 1.70 3.5 c 201 134 AL269
+1749+096c0.322 16.78 B Y 1.039 1.6 ×10−41.4899 6.84 1.05 4.9 c 38 .... AC301
+1751+288 1.118 19.60 Q ... 0.265 1.7 ×10−41.4350 3.07 0.27 8.1 2, 1 349 40 AS659
+1758+388 2.092 17.98 Q ... 0.327 1.7 ×10−41.5149 2.38 0.33 3.2 1hs+2 266 270 AS396
+1800+440 0.663 17.90 Q ... 0.465 2.1 ×10−41.5149 15.41 0.50 246.6 1hs+2 203 239 AS396
+1803+784 0.68 15.90 B Y 1.969 1.0 ×10−41.4000 8.97 1.98 20.8 2hs 266 192 AL634 ∗
+1807+698 0.051 14.22 B Y 1.133 1.1 ×10−41.5149 0.10 1.20 368.7 1hs 261 242 AB700
+1823+568 0.664 19.30 B Y 0.873 6.1 ×10−41.4250 20.85 0.95 137.4 2 201 94 AM672
+1828+487 0.692 16.81 Q N 5.140 6.3 ×10−41.5524 13.65 8.21 5431.2 2hs 310 119 AB310
+1849+670 0.657 16.90 Q ... 0.468 1.1 ×10−41.4000 30.63 0.47 101.0 1hs+2 308 209 AL634 ∗
+1928+738 0.302 16.06 Q N 3.186 1.6 ×10−41.4000 8.43 3.22 356.5 1hs+2, 2hs 164 165 AL634 ∗
+1936−155 1.657 20.30 Q Y 1.064 1.8 ×10−41.4899 2.59 1.08 10.5 1hs+2, 2 180 214 AD167
+1957+405 0.0561 15.10 G ... 33.56‡6.5×10−31.5245 0.21 0.65 1.4 ×1062hs 282 291 AC166
+1958−179c0.65 18.60 Q Y 1.789 1.5 ×10−41.4899 1.89 1.82 9.4 c 207 .... AA072
+2005+403 1.736 19.00 Q ... 2.420 5.1 ×10−41.6657 12.21 2.47 10.6 1hs+2 127 137 AK4010
+2008−159 1.18 18.30 Q ... 0.546 1.0 ×10−41.4899 7.98 0.55 7.5 1hs 7 5 AK240
+2021+317 .... .... U ... 2.968 6.8 ×10−41.6657 .... 3.06 132.9 ch, 1 201 .... AK4010
+2021+614 0.227 19.00 G N 2.647 2.9 ×10−41.4899 0.41 2.67 1.3 2 33 .... AM178
+2037+511 1.686 21.00 Q ... 4.604 6.4 ×10−41.6657 3.29 4.97 657.6 1hs 214 327 AK4010
+2121+053 1.941 20.40 Q Y 1.08 0.6 ×10−31.4650 13.28 1.08 4.8 1, 1hs 271 .... Murphy
+2128−123 0.501 16.11 Q N 1.393 1.3 ×10−41.4899 6.94 1.42 39.8 1hs+2 209 190 AY012
+2131−021 1.285 19.00 B Y 1.320 1.3 ×10−41.5149 20.03 1.37 151.9 2hs 108 137 AB700
+2134+004 1.932 17.11 Q N 4.82 0.4 ×10−31.4650 5.93 4.82 6.6 1hs 266 298 Murphy
+2136+141c2.427 18.90 Q ... 1.131 1.4 ×10−41.5524 5.43 1.14 0.8 c 205 .... AB310
+2145+067 0.99 16.47 Q N 2.840 1.7 ×10−41.4000 2.49 2.87 27.7 1hs, 1 119 313 AL634 ∗
+2155−152 0.672 18.30 Q Y 2.627 1.5 ×10−41.4000 18.11 2.70 304.7 1hs+2 214 195 AL634 ∗
+2200+420 0.0686 14.72 B Y 1.970 7.0 ×10−51.4000 10.57 1.99 14.2 2, 1hs+2 182 143 AL634 ∗
+2201+171 1.075 19.50 Q ... 0.820 5.8 ×10−51.4000 2.54 0.87 74.6 1hs, 2hs 45 267 AL634 ∗
+2201+315 0.295 15.58 Q N 1.517 1.3 ×10−41.4000 7.87 1.54 378.4 2hs 219 251 AL634 ∗
+2209+236c1.125 20.66 Q ... 0.425 4.9 ×10−51.4000 3.43 0.43 0.9 c 54 .... AL634 ∗
+2216−038 0.901 16.38 Q ... 1.722 1.1 ×10−41.4000 5.62 1.76 312.7 2hs 190 143 AL634 ∗
+2223−052 1.404 18.39 Q Y 6.769 3.4 ×10−41.4000 17.33 7.13 91.6 1 99 335 AL634 ∗
+2227−088 1.56 17.43 Q Y 0.920 4.2 ×10−51.4000 8.14 0.93 8.4 1hs 348 314 AL634 ∗
+2230+114 1.037 17.33 Q Y 6.633 3.7 ×10−41.4000 15.41 6.99 148.0 1 147 144 AL634 ∗– 30 –
+Wewouldliketothanktheanonymousrefereeforacarefulassess ment ofthemanuscript,
+which has led to significant improvement. PK would like to thank John Pe terson for useful
+discussions on galaxy clusters, and Chris O’Dea for insightful sugge stions. The MOJAVE
+project is supported under National Science Foundation grant 08 07860-AST and NASA-
+Fermi grant NNX08AV67G. This research has made use of the NASA /IPAC Extragalactic
+Database (NED) which is operated by the Jet Propulsion Laborator y, California Institute
+of Technology, under contract with the National Aeronautics and Space Administration.
+The National Radio Astronomy Observatory is a facility of the Nation al Science Foundation
+operated under cooperative agreement by Associated Universitie s, Inc.
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+A. PREVIOUSLY UNPUBLISHED RADIO IMAGES FROM THE
+ARCHIVE
+This preprint was prepared with the AAS L ATEX macros v5.2.– 35 –
+Table 1—Continued
+Source z V Type Opt Ipeak Irms Freq βapp Score Sext Radio PcjetPA KpcjetPA Ref
+name mag Pol Jy beam−1Jy beam−1GHz Jy mJy Morph Degree Degree
+(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
+2243−123 0.632 16.45 Q Y 2.247 8.4 ×10−51.4000 5.49 2.27 27.7 1hs 29 44 AL634 ∗
+2251+158 0.859 16.10 Q Y 13.88 4.9 ×10−41.4000 14.19 14.09 822.0 1hs 297 311 AL634 ∗
+2331+073 0.401 16.04 Q ... 0.602 4.3 ×10−51.4000 4.47 0.61 38.4 1hs+2 225 251 AL634 ∗
+2345−167 0.576 18.41 Q Y 1.921 8.1 ×10−51.4000 13.45 1.99 142.7 1hs 141 227 AL634 ∗
+2351+456 1.986 20.60 Q ... 2.260 6.3 ×10−51.4000 27.09 2.35 6.9 c, 1hs 278 260 AL634 ∗
+Note. — Col. 1 & 2: IAU B1950 names of the MOJAVE sources and their reds hifts.c= sources do not show any discernable extended
+emission. Col. 3: Apparent V-band magnitude obtained from t he AGN catalog of V´ eron-Cetty & V´ eron (2006) or NED. Col. 4: Blazar
+classification −Q = quasar, B = BL Lac object, G = radio galaxy, U = Unidentified. Col. 5: Highly optically polarized −Y = Yes, N =
+No, .... = no data available. Col. 6 & 7: Peak and rmsintensity in radio map, respectively. Irmsfor the sources in Murphy et al. (1993) are
+the CLEV values in their Table 2. ‡for Cygnus A, the peak intensity corresponds to the south-ea stern hot spot. Col. 8: Central observing
+frequency in GHz. Col. 9: Apparent parsec-scale speed of the fastest jet feature. Col. 10: Integrated core flux density at ∼1.4 GHz obtained
+with JMFIT. Col. 11: Extended flux density obtained by subtra cting the core from the total radio flux density. Col. 12: Radi o morphology −
+c = core only, ch = core-halo, 1hs = 1 hot spot, 2hs = 2 hot spots, 1 = 1 sided, 2 = 2 sided, 1hs+2 = 2 sided structure with hot spot o n one
+side. Alternate morphologies are listed separated by comma s. Col. 13: Parsec-scale jet position angle. Col. 14: Kilopa rsec-scale jet position
+angle. Col. 15: References and VLA archive Project IDs for th e radio data − †=New data presented in this paper. ∗=Data presented in
+Cooper et al. (2007). Murphy = Murphy et al. (1993).– 36 –
+Table 2. Correlation Results
+Param 1 Param 2 Class Spearman Spearman Kendall τKendallτCorrel ?
+Coeff Prob Coeff Prob
+(1) (2) (3) (4) (5) (6) (7) (8)
+Lcore Lext ALL 0.61 1.6 ×10−130.45 <1×10−5YES
+Mabs Lext ALL −0.39 9.8 ×10−6−0.27 1.1 ×10−6YES
+Lext βapp ALL 0.44 5.2 ×10−70.31 5.3 ×10−7YES
+Q 0.36 0.0004 0.24 0.0005 YES
+B 0.32 0.193 0.24 0.148 NO
+Mabs βapp ALL −0.27 0.0025 −0.18 0.0031 YES
+Rc Lcore ALL 0.26 0.003 0.19 0.002 YES
+Q 0.39 8.5 ×10−50.27 8.8 ×10−5YES
+B −0.02 0.912 −0.01 0.909 NO
+Rc Lext ALL −0.49 1.5 ×10−8−0.35 <1×10−5YES
+Rc βapp ALL −0.18 0.046 −0.12 0.044 YES?
+Q −0.28 0.006 −0.19 0.006 YES
+B 0.02 0.922 0.01 0.939 NO
+Rc ∆PA ALL −0.13 0.146 −0.09 0.142 NO
+Q −0.18 0.087 −0.12 0.082 YES?
+B −0.15 0.579 −0.10 0.589 NO
+Rv Lcore ALL 0.62 6.5 ×10−150.45 <1×10−5YES
+Rv βapp ALL 0.11 0.200 0.07 0.226 NO
+Rv ∆PA ALL 0.23 0.011 0.15 0.017 YES?
+Q 0.26 0.011 0.17 0.015 YES?
+B 0.38 0.143 0.23 0.207 NO
+FRII 0.24 0.029 0.16 0.032 YES?
+FRI/II 0.18 0.277 0.11 0.313 NO
+FRI 0.12 0.658 0.07 0.701 NO
+∆PA z ALL 0.06 0.514 0.04 0.498 NO
+Lext/Mabsz Q+B 0.21 0.023 0.14 0.021 YES?
+Q 0.27 0.007 0.19 0.007 YES
+B 0.31 0.209 0.16 0.343 NO
+Lext/Mabsβapp Q+B −0.02 0.810 −0.01 0.768 NO
+Lext/Mabs∆PA Q+B −0.34 0.0003 −0.23 0.0005 YES
+Note. — Cols. 1 & 2: Parameters being examined for correlatio ns. Results from a partial regression
+analysis are cited in the main text. The ten core-only source s have been excluded from correlations
+with extended luminosity. Including the upper limits in ext ended luminosity as detections, does not
+alter any of the observed trends. Col. 3: The correlation tes t results have been presented separately
+if they were different for quasars and BL Lacs considered alon e. ALL = quasars + BL Lacs + radio
+galaxies, Q = only quasars, B = only BL Lacs, Q+B = quasars + BL L acs. FRII, FRI/II, and FRI are
+as defined in §4.5. Cols. 4 & 5: Spearman rank correlation coefficient and cha nce probability. Cols. 6
+& 7: Kendall Tau correlation coefficient and chance probabili ty. Col. 8: Indicates if the parameters are
+significantly correlated. ‘YES?’ indicates a marginal corr elation. We note that at the 95% confidence
+level, 1.7 spurious correlations could arise from the ≈35 correlations that we tested.– 37 –
+(a)
+DECLINATION (J2000)
+RIGHT ASCENSION (J2000)07 30 19.8 19.619.419.219.018.818.618.4-11 41 02
+04
+06
+08
+10
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+(b)
+DECLINATION (B1950)
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+05
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+43 55
+50
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+(c)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)07 42 49.0 48.8 48.6 48.4 48.2 48.010 18 42
+40
+38
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+(d)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)08 38 02.6 02.402.202.001.801.601.401.201.013 23 15
+10
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+(e)
+DECLINATION (J2000)
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+05
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+(f)
+DECLINATION (B1950)
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+04
+06
+08
+10
+12
+14
+16
+18
+Fig. 13.— New radio images from the VLA archive of (a) 0727 −115, (b) 0736+017, (c) 0742+103,
+(d) 0838+133, (e) 1124 −186, and (f) 1334 −127. The contours are in percentage of the peak surface
+brightness and increase in steps of 2. The lowest contour lev els and peak surface brightness are,
+(a)±0.01, 3.15 Jy beam−1; (b)±0.042, 2.33 Jy beam−1; (c)±0.021, 3.61 Jy beam−1; (d)±0.17,
+289.3 mJy beam−1; (e)±0.085, 651.2 mJy beam−1; (f)±0.042, 1.94 Jy beam−1.– 38 –
+(a)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)15 04 16.8 16.6 16.4 16.2 16.0-16 40 54
+56
+58
+41 00
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+(b)
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+20
+15
+10
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+00
+(c)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)17 26 02.0 01.801.601.401.201.000.800.600.445 33 14
+12
+10
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+06
+04
+02
+00
+32 58
+56
+(d)
+DECLINATION (J2000)
+RIGHT ASCENSION (J2000)17 33 04.0 03.5 03.0 02.5 02.0 01.5-13 04 35
+40
+45
+50
+55
+05 00
+(e)
+DECLINATION (J2000)
+RIGHT ASCENSION (J2000)17 43 59.6 59.459.259.058.858.658.4-03 49 56
+58
+50 00
+02
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+10
+12
+14
+(f)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)17 51 46.0 45.8 45.6 45.4 45.2 45.0 44.828 48 46
+44
+42
+40
+38
+36
+34
+32
+30
+Fig. 14.— New radio images from the VLA archive of (a) 1504 −166, (b) 1548+056, (c) 1726+455,
+(d) 1733−130, (e) 1741 −038, and (f) 1751+288. The contours are in percentage of the p eak surface
+brightness and increase in steps of 2. The lowest contour lev els and peak surface brightness are, (a)
+±0.042, 2.36 Jy beam−1; (b)±0.021, 2.09 Jy beam−1; (c)±0.042, 998.9 mJy beam−1; (d)±0.021,
+6.11 Jy beam−1; (e)±0.042, 1.68 Jy beam−1; (f)±0.17, 263.5 mJy beam−1.– 39 –
+(a)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)18 07 20 1918171615141369 49 10
+05
+00
+48 55
+50
+45
+40
+35
+(b)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)19 36 36.8 36.636.436.236.035.835.635.4-15 32 28
+30
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+42
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+(c)
+DECLINATION (B1950)
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+10
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+(d)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)20 05 59.6 59.4 59.2 59.0 58.840 21 06
+04
+02
+00
+20 58
+56
+54
+(e)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)20 08 27.0 26.5 26.0 25.5 25.0-15 55 20
+25
+30
+35
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+(f)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)20 21 19.2 19.018.818.618.418.218.031 43 28
+26
+24
+22
+20
+18
+16
+14
+12
+10
+Fig. 15.— New radio images from the VLA archive of (a) 1807+698, (b) 193 6−155, (c) 1958 −179,
+(d) 2005+403, (e) 2008 −159, and (f) 2021+317. The contours are in percentage of the p eak surface
+brightness and increase in steps of 2. The lowest contour lev els and peak surface brightness are,
+(a)±0.042, 1.14 Jy beam−1; (b)±0.085, 1.05 Jy beam−1; (c)±0.085, 1.77 Jy beam−1; (d)±0.085,
+2.43 Jy beam−1; (e)±0.085, 544.5 mJy beam−1; (f)±0.085, 2.97 Jy beam−1.– 40 –
+(a)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)20 21 14.5 14.0 13.5 13.0 12.561 27 28
+26
+24
+22
+20
+18
+16
+14
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+10
+(b)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)20 37 07.8 07.607.407.207.006.806.606.406.251 08 44
+42
+40
+38
+36
+34
+32
+30
+28
+(c)
+DECLINATION (B1950)
+RIGHT ASCENSION (B1950)21 28 55 54 53 52 51 50-12 19 45
+20 00
+15
+30
+45
+21 00
+Fig. 16.— New radio images from the VLA archive of (a) 2021+614, (b) 203 7+511, and (c)
+2128−123. The contours are in percentage of the peak surface brigh tness and increase in steps of 2.
+The lowest contour levels and peak surface brightness are, ( a)±0.042, 2.64 Jy beam−1; (b)±0.042,
+4.61 Jy beam−1; (c)±0.042, 1.37 Jy beam−1.
\ No newline at end of file
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+arXiv:1001.0732v1  [cond-mat.stat-mech]  5 Jan 2010Diffusion of Tagged Particle in an Exclusion Process
+E. Barkai
+Department of Physics, Bar Ilan University, Ramat-Gan 5290 0, Israel
+R. Silbey
+Department of Chemistry, Massachusetts Institute of Techn ology, Cambridge, Massachusetts 02139, USA
+We study the diffusion of tagged hard core interacting partic les under the influence of an external
+force field. Using the Jepsen line we map this many particle pr oblem onto a single particle one.
+We obtain general equations for the distribution and the mea n square displacement ∝angbracketleft(xT)2∝angbracketrightof the
+tagged center particle valid for rather general external fo rce fields and initial conditions. A wide
+range of physical behaviors emerge which are very different t han the classical single file sub-diffusion
+∝angbracketleft(xT)2∝angbracketright ∼t1/2found for uniformly distributed particles in an infinite spa ce and in the absence of
+force fields. For symmetric initial conditions andpotentia l fieldswe find ∝angbracketleft(xT)2∝angbracketright=R(1−R)
+2Nr2where 2N
+is the (large) number of particles in the system, Ris a single particle reflection coefficient obtained
+from the single particle Green function and initial conditi ons, and rits derivative. We show that
+this equation is related to the mathematical theory of order statistics and it can be used to find
+∝angbracketleft(xT)2∝angbracketrighteven when the motion between collision events is not Brownia n (e.g. it might be ballistic,
+or anomalous diffusion). As an example we derive the Percus re lation for non Gaussian diffusion.
+PACS numbers: 05.60.k, 02.50.Ey, 05.40.Jc, 05.70.Ln
+I. INTRODUCTION
+Systems of particles governed by stochastic dynamics
+and exclusion interactions have been studied for a long
+time [1–3]. One aspect of this problem is the motion of a
+tagged particle sometimes called the tracerparticle [4–7].
+The diffusion of a tagged particle, in a one dimensional
+system of Brownian particles, interacting via hard core
+interaction is a model for the motion of a single molecule
+in a crowded one dimensional environment such as a bi-
+ological pore or channel [8–11], and in experimentally
+studied physical systems such as zeolites [12], confined
+colloid particles [13–15] and charged spheres in circular
+channels [16]. Since particles do not pass each other such
+diffusion processes are called single file diffusion.
+Confinement of a tagged Brownian particle, due to
+its interaction with other Brownian particles, leads to
+a slow-down of the diffusion of the tagged particle
+∝angbracketleft(xT)2∝angbracketright ∝t1/2instead of normal diffusion ∝angbracketleft(xT)2∝angbracketright ∝t
+when the particles in the system are uniformly dis-
+tributed [4, 6]. Such many body problems can be treated
+using methods which exploit the relation between the dy-
+namics of the interacting tagged particle and the motion
+of a particle free of interactions [5, 6, 17–20]. In recent
+years at-least six new directions of research emerged. (i)
+The effect of an external field acting on all the particles
+[21] or on the tagged particle only [22, 23] has attracted
+attention since pores induce entropic barriers [10] and
+are generally inhomogeneous. In this category the ex-
+amples of single file motion in a periodic potential [24]
+or in a box [25] were considered in detail. (ii) Initial
+conditions may have a profound impact on diffusion of
+the tagged particle. For example if particles start as a
+narrow Gaussian packet the diffusion of the tagged cen-
+ter particle increases linearly in time ∝angbracketleft(xT)2∝angbracketright ∝t[26]
+(see details below). Power law initial conditions induce∝angbracketleft(xT)2∝angbracketright ∝tξandξis neither 1 nor 1 /2 [27] (see further
+details in text). When an external field is acting on the
+system, it is important to consider non uniform initial
+conditions where the density of particles is determined
+naturally from Boltzmann’s distribution. (iii) In some
+investigations the underlying motion is not normal diffu-
+sion, instead the particlesmay be non-Brownian, and fol-
+lowinganomalouskinetics[27,28]. (iv)Usuallyhardcore
+interactions are considered, although the hard problem
+of more general interactions has been recently treated in
+[23, 29, 30]. Screened hydrodynamic interactions which
+seem important at short times were investigated in [31]
+both theoretically and experimentally. Granular single
+file diffusion with inelastic collisions shows the typical
+t1/2sub-diffusion [32, 33]. (v) Interacting particles in
+systems with quenched disorder is yet another challenge.
+The motion of a tagged particle was recently treated in
+the context of single file diffusion in the Sinai model
+[34]. (vi) Finally, if the motion of the tagged particle
+is not normal Brownian motion, what stochastic theory
+replaces the usual Brownian-Langevin framework? In
+this direction interesting connections to fractional cal-
+culus emerged [23, 35, 36]. Roughly speaking and under
+certainconditionshalfordertimederivatives( d1/2/dt1/2)
+enter in the Langevin equation [23] and fractional Brow-
+nian noise replaces white noise.
+Here we provide a very general theory of single file dif-
+fusion of the center tagged particle, valid in the presence
+or the absence of an external potential field V(x), as well
+as for thermal and non-thermal initial conditions. Our
+main results reproduce previously obtained formulas and
+many new ones, by mapping the many particle problem
+onto a single particle model. Our method, explained in
+Sec. II, exploits the theoretical concept of the Jepsen
+line [5], is limited to hard core point particles, but, as
+we show in Sec. V, is not limited to Brownian particles.2
+After providing general results in Sec. III we limit our
+attention to symmetric initial conditions and potentials
+where the tagged particle has no average drift. A general
+relation between the mean square displacement of the
+taggedparticleandreflectionprobabilityofthenoninter-
+acting particle is given in Eq. (37). Detailed calculations
+of the mean square displacement of the tagged particle
+then follow, for special choices of force fields and initial
+conditions, in Sec. IV. As wediscuss insub-sectionIIID,
+in certain limits our problem is related to order statistics
+[37], a fact worth mentioning since it allows us to solve
+our problem and related ones using known methods. In
+Sec. V, we discuss non-Brownian kinds of motion and
+the Percus relation. A brief report of part of our results
+was recently published [21].
+II. MODEL AND METHODS
+Inourmodel, 2 N+1identicalpointparticleswithhard
+core particle-particle interactions are undergoing Brow-
+nian motion in one dimension, so particles cannot pass
+each other. The diffusion constant of particles free of
+interaction is D. As mentioned, towards the end of the
+paperwe discussthe moregeneralcasewhere the dynam-
+ics between collision events is not necessarily Brownian.
+An external potential V(x) acts on the particles. The
+system stretches from −LtoL; however, unless stated
+otherwise we will let L→ ∞and obtain a thermody-
+namic limit where N/Lis fixed. We tag the center parti-
+cle, which clearly has Nparticles to its left and Nto
+it right. Initially the tagged particle is at the origin
+x= 0. The motion of a single particle in the absence of
+interactions with other particles is described by a single
+particle Green function g(x,x0,t), with the initial condi-
+tionsg(x,x0,0) =δ(x−x0). In the case of over damped
+Brownian motion the Green function is the solution of
+the Fokker-Planck equation [38]
+∂g(x,x0,t)
+∂t=D/bracketleftbigg∂2
+∂x2−1
+kbT∂
+∂xF(x)/bracketrightbigg
+g(x,x0,t),(1)
+whereF(x) =−V′(x) is the force field, Tis the tem-
+perature and kbis Boltzmann’s constant. The initial
+conditions of Nparticles residing initially to the right
+(left) of the test particle are drawn from the probability
+density function (PDF) fR(x0) (fL(x0)) respectively. We
+consideran ensemble oftrajectoriesand averageovertra-
+jectories and initial conditions (see details below). Our
+goalisto obtain the PDF ofthe position xTofthe tagged
+center particle P(xT) in the large Nlimit.
+A. The Jepsen Line
+A schematic diagram of the problem is presented in
+Fig. 1 for particles in a box. Initial positions of particles
+are given by xj
+0wherej=−N,···,0,···Nwherejis/MT45/MT50 /MT45/MT49 /MT48 /MT50 /MT49/MT116
+FIG. 1: Schematic motion of Brownian particles in a box,
+where particles cannot penetrate through each other. The
+center tagged particle label is 0, its trajectory is restric ted by
+collisions with neighboring particles. The straight line i s the
+Jepsenline, itfollows vtas explainedinthetext. Asexplained
+in the text, in an equivalent non-interacting picture, we al low
+particles to pass through each other, and at time twe search
+for the position of the particle which has Nparticles to its
+right and Nto its left (i.e. the center particle).
+the label of the interacting particles (see Fig. 1). The
+tagged particle whose coordinate is denoted with xT(t),
+is the center particle j= 0 (bold line in Fig. 1). Initially
+the tagged particle is at the origin xT(0) = 0. Since
+particles do not pass each other, their order is clearly
+maintained, and the number of particles to the left and
+right of the tagged particle Nis fixed.
+In Fig. 1 the straight line which starts at x= 0 is
+called the Jepsen line and follows x(t) =vt, wherevis
+a test velocity [5, 6]. We label the interacting particles
+according to their initial position, increasing to the right
+(see Fig. 1). The tagged particle starts just to the right
+of the Jepsen line so at t= 0 the label of the particle
+to the right of the Jepsen line is zero. In this system
+we have 2 N+ 1 particles. Hence initially we have N
+particles to the left of the Jepsen line and including the
+tagged particle N+1 particles to the right.
+Let ˜α(t) be the label number of the first particle sit-
+uated to the right of the Jepsen line. According to our
+rules, at t= 0 we have ˜ α= 0, and then the random
+variable ˜ αwill increase or decrease in steps of +1 or −1
+according to:
+i) if a particle crosses the Jepsen line from left to right
+˜α→˜α−1
+ii) if a particle crosses the Jepsen line from right to left
+˜α→˜α+ 1. Thus the counter ˜ αis performing a ran-3
+/MT61/MT106
+/MT106/MT83/MT119/MT105/MT116/MT99/MT104
+/MT108/MT97/MT98/MT101/MT108/MT115/MT106
+/MT106/MT106/MT43/MT49 /MT106
+/MT106
+/MT99/MT111/MT108/MT108/MT105/MT115/MT105/MT111/MT110/MT125
+/MT110/MT111/MT110/MT32/MT105/MT110/MT116/MT101/MT114/MT97/MT99/MT116/MT105/MT110/MT103/MT125/MT106/MT43/MT49
+/MT106/MT43/MT49 /MT106/MT43/MT49/MT106/MT43/MT49
+/MT106/MT43/MT49
+FIG. 2: In one dimension, the paths of a pair of particles
+in a hard core collision event can be represented by two non
+interacting particles which pass through each other, and th en
+their label numbers are switched.
+dom walk decreasing or increasing its value +1 or −1 at
+random times.
+A collision between two hard core particles is repre-
+sented schematically in Fig. 2. In one dimension a hard
+core collision event is equivalent to two particles that
+pass through each other, i.e. non interacting particles,
+and then after the particles cross each other, the labels
+of the pair of particles are switched (see Fig. 2). In-
+stead of relabeling particles after each collision, we let
+particles pass through each other, and then at time twe
+label our particles (or if we are interested only in tagged
+particle, locate the central particle). Operationally this
+means that in the time interval (0 ,t) we view the parti-
+cles as non interacting, and then find the particle with N
+particles to its right and Nto its left, which is equivalent
+to the tagged particle in the interacting system. Hence
+the problem is related to the mathematical topic of order
+statistics [37] as we will discuss briefly later.
+Following Jepsen [5] and Levitt [6] we introduce the
+stochastic process for the non interacting particles α(t)
+where:
+i) if a particle crosses the Jepsen line from left to right
+α→α−1
+ii) if a particle crosses the Jepsen line from right to left
+α→α+1. The process α(t) is the same as the process
+˜α(t), in statistical sense.
+The event where α(t) = 0 implies that the number
+of particles crossing the Jepsen line from right to left is
+the same as those crossing from left to right. Switching
+back to the interacting system, the event α(t) = 0 means
+that the particle to the right of the Jepsen line at time
+t= 0 is also there at time t. Hence the probability that
+α(t) = 0 is the same as the probability of finding the
+tagged interacting particle to be the first particle to the
+right of the Jepsen line. Therefore we will calculate theprobability for α(t) = 0 and with it obtain in the large
+Nlimit, the probability P(xT)dxTof finding the tagged
+particle in a small interval xT,xT+dxT.
+The random variable αis a sum of many random vari-
+ables
+α=N/summationdisplay
+j=−N′
+δαj (2)
+whereδαjis the number of times particle jcrossed the
+Jepsenlinefromrighttoleftminusthenumberoftimesit
+crossedthe line from left to right. Clearly δαjmayattain
+the values −1 or 0 if the particle started on the left, or
+1 or 0 if it started on the right. Since we are interested
+in the large Nlimit we may neglect the contribution of
+δα0, which is indicated by the prime in the sum [39].
+Wecalculatetheprobability PN(α)oftherandomvari-
+ableα. For that we designate PLL(x−j
+0), the probability
+that particle −j, starting to the left of the Jepsen line
+att= 0 atx−j
+0<0, is found at time tto the left of
+the Jepsen line. Clearly for the corresponding trajectory
+δα−j= 0, since the particle crossed the Jepsen line an
+even number oftimes, or did not crossit at all. Similarly,
+PLR(x−j
+0) is the probability to start to the left of the line
+and end to its right, and PRR(xj
+0),PRL(xj
+0) are defined
+similarly for particles starting on xj
+0>0 to the right of
+the line.
+In our non interacting picture, the motion of particles
+is independent, hence we can use random walk theory
+[40] and Fourier series to find PN(α). It is convenient to
+rewrite Eq. (2)
+α=N/summationdisplay
+j=1∆αj (3)
+where ∆αj=δαj+δα−jmay attain the values 1 ,0,−1.
+Eachsummand∆ αjtakesintoaccountoneparticlestart-
+ing to the left of the Jepsen line and one to the right.
+First consider N= 1, that is one particle that starts
+on the left of the Jepsen line and one to its right. Then
+α= ∆α1=δα1+δα−1and as mentioned α= 1 or 0 or
+−1. Examples for possible trajectories are shown in Fig.
+3. It is easy to see that
+PN=1(α) =
+
+α= 1PRL(x1
+0)PLL(x−1
+0)
+α= 0PLL(x−1
+0)PRR(x1
+0)+PLR(x−1
+0)PRL(x1
+0)
+α=−1PRR(x1
+0)PLR(x−1
+0).
+(4)
+We define the structure function
+λ(φ,x−j
+0,xj
+0) =eiφPLL(x−j
+0)PRL(xj
+0)+/bracketleftBig
+PLL(x−j
+0)PRR(xj
+0)+PLR(x−j
+0)PRL(xj
+0)/bracketrightBig
++e−iφPLR(x−j
+0)PRR(xj
+0).(5)
+The structure function describes a single step in the ran- dom walk E q. (3) in the following usual way: the co-4
+efficient of exp( iφ) is the probability that ∆ αjis equal
+one (i.e. PLLPRL), the coefficient of exp( iφ) = 1 (i.e.
+φ= 0) yields the probability ∆ αj= 0, and similarly for
+exp(−iφ). Since the summands in Eq. (3) are indepen-
+dent, Fourier analysis gives:
+PN(α) =1
+2π/integraldisplayπ
+−πdφΠN
+j=1λ/parenleftBig
+φ,x−j
+0,xj
+0/parenrightBig
+e−iαφ.(6)
+We average Eq. (6) with respect to the initial conditions
+x0whichareassumedindependent identicallydistributed
+random variable and we find
+∝angbracketleftPN(α)∝angbracketright=1
+2π/integraldisplayπ
+−πdφ∝angbracketleftλ(φ)∝angbracketrightNe−iαφ(7)
+where from Eq. (5)
+∝angbracketleftλ(φ)∝angbracketright=
+eiφ∝angbracketleftPLL∝angbracketright∝angbracketleftPRL∝angbracketright+∝angbracketleftPLL∝angbracketright∝angbracketleftPRR∝angbracketright+∝angbracketleftPLR∝angbracketright∝angbracketleftPRL∝angbracketright+e−iφ∝angbracketleftPLR∝angbracketright∝angbracketleftPRR∝angbracketright.
+(8)
+Here∝angbracketleftPij∝angbracketrightis the probability of starting in i=L,R(rel-
+ative to the the Jepsen line) and ending in j=L,Rand
+∝angbracketleft···∝angbracketrightdenotes an averageover initial condition. The prob-
+ability∝angbracketleftPLR∝angbracketrightis given in terms of the Green function of
+the non interactingparticle g(x,x0,t) and the initial den-
+sity of particles situated initially to the left of the Jepsen
+linefL(x0)
+∝angbracketleftPLR(vt)∝angbracketright=/integraldisplay0
+−LfL(x0)/integraldisplayL
+vtg(x,x0,t)dxdx0.(9)
+We see that for ∝angbracketleftPLR∝angbracketrightwe average over initial conditions
+in the domain ( −L,0) weighted by fL(x0) (since the
+starting point is to the left of the Jepsen line) and also
+integrate over xin the domain ( vt,L) with the weight
+g(x,x0,t) (since the end point is to the right of the line).
+Similarly
+∝angbracketleftPRR(vt)∝angbracketright=/integraldisplayL
+0fR(x0)/integraldisplayL
+vtg(x,x0,t)dxdx0.(10)
+As mentioned fL(x0) [fR(x0)] is the PDF of initial po-
+sitions of particles that initially are at x0<0 (x0>
+0) respectively, hence fL(x0) = 0 when x0>0 and/integraltext0
+−LfL(x0)dx0= 1, while fR(x0) is normalized and non
+zero only in (0 ,L).∝angbracketleftPLL∝angbracketrightand∝angbracketleftPRR∝angbracketrightare defined simi-
+larly, and they satisfy ∝angbracketleftPLL∝angbracketright= 1−∝angbracketleftPLR∝angbracketright, and∝angbracketleftPRL∝angbracketright=
+1−∝angbracketleftPRR∝angbracketright.
+III. DYNAMICS OF THE TAGGED PARTICLE
+We now apply the central limit theorem to analyze the
+sum Eq. (3) using Eq. (7) when N→ ∞. The small φ
+expansion of the structure function Eq. (8) is
+∝angbracketleftλ(φ)∝angbracketright= 1+i∝angbracketleft∆α∝angbracketrightφ−1
+2∝angbracketleft(∆α)2∝angbracketrightφ2+O(φ3) (11)/MT116
+FIG. 3: Trajectories crossing the Jepsen line. In the left
+panel one particle started on Land ended on Lthe other
+onRand ended on R. These two paths are assigned the
+probability PLL(x−1
+0)PRR(x1
+0) and yield in Eq. (4) α= 0.
+Similarly the trajectories on the right panel correspond to
+PLL(x−1
+0)PRL(x1
+0) and they contribute α= 1.
+where
+∝angbracketleft∆α∝angbracketright=∝angbracketleftPRL∝angbracketright−∝angbracketleftPLR∝angbracketright (12)
+as expected, and the variance σ2
+∆α=∝angbracketleft(∆α)2∝angbracketright−∝angbracketleft∆α∝angbracketright2is
+σ2
+∆α=∝angbracketleftPRL∝angbracketright∝angbracketleftPRR∝angbracketright+∝angbracketleftPLR∝angbracketright∝angbracketleftPLL∝angbracketright.(13)
+Using the central limit theorem we have the probability
+of zero total crossing of the Jepsen line, namely α= 0 in
+the limit N→ ∞
+PN(α= 0)∼1√
+2πNσ∆αexp/parenleftbigg
+−N∝angbracketleft∆α∝angbracketright2
+2σ2
+∆α/parenrightbigg
+.(14)
+This simple result, valid for a large class of Green func-
+tions and initial conditions, is suited for the investigation
+of a large number of single file problems.
+The probability PN(α= 0) is according to our earlier
+discussion equal to the probability that the tagged par-
+ticle is in the interval vt < x T< vt+δx, whereδxis the
+distance between the Jepsen line and the position of the
+second particle to the right of the Jepsen line. We as-
+sume that δxis small relative to the typical length scale
+the tagged particle traveled. For example if the den-
+sity of interacting particles ρis a constant, we assume
+1/ρ <</radicalbig
+∝angbracketleft(xT)2∝angbracketright. Hence we may replace vt→xTto
+transform PN(α= 0) to the PDF of the tagged particle
+P(xT) [41].
+Making the replacement vt→xT, using Eq. (14) we
+have in the limit of large N
+P(xT)∼
+Cexp/braceleftBigg
+−N[∝angbracketleftPLR(xT)∝angbracketright−∝angbracketleftPRL(xT)∝angbracketright]2
+2[∝angbracketleftPLL(xT)∝angbracketright∝angbracketleftPLR(xT)∝angbracketright+∝angbracketleftPRR(xT)∝angbracketright∝angbracketleftPRL(xT)∝angbracketright]/bracerightBigg
+(15)
+whereCis a normalization constant. In Eq. (15) and
+what follows ∝angbracketleftPij(xT)∝angbracketrightis given by Eqs. (9,10) with
+vt→xT. Eq. (15) yields the PDF of the center par-
+ticleP(xT) in terms of ∝angbracketleftPij(xT)∝angbracketrightwhich according to Eq.
+(10) depends on the free particle green function and the
+initial conditions. Thus the information contained in the
+non-interacting Green function is sufficient for the deter-
+mination of the single file diffusion of the tagged particle.5
+A. Thermal Equilibrium
+In the long time limit, and in the presence of a binding
+potential field, e.g. harmonic field or particles in a box,
+an equilibrium is reached. Then initial conditions do not
+play a role. For example ∝angbracketleftPLR(xT)∝angbracketright=Peq
+R(xT) with
+Peq
+R(xT) =1
+Z/integraldisplayL
+xTexp/parenleftbigg
+−V(x)
+kbT/parenrightbigg
+dx(16)
+whereZis the normalizing partition function
+Z=/integraldisplayL
+−Lexp/bracketleftbigg
+−V(x)
+kbT/bracketrightbigg
+dx. (17)
+Similarly
+Peq
+L(xT) =1
+Z/integraldisplayxT
+−Lexp/parenleftbigg
+−V(x)
+kbT/parenrightbigg
+dx.(18)
+In Eqs. (16,18) we used the steady state solution,
+limt→∞g(x,x0,t) = exp[−V(x)/kbT]/Z, which is Boltz-
+mann’s distribution suited for a system in thermal equi-
+librium. Using Eq. (15) the position PDF of the tagged
+particle is lim t→∞P(xT) =Peq(xT)
+Peq(xT)∼Cexp/braceleftBigg
+−N[∝angbracketleftPeq
+R(xT)∝angbracketright−∝angbracketleftPeq
+L(xT)∝angbracketright]2
+4∝angbracketleftPeq
+L(xT)∝angbracketright∝angbracketleftPeq
+R(xT)∝angbracketright/bracerightBigg
+.
+(19)
+If the potential is symmetric V(x) =V(−x), e.g. parti-
+cles in a box or harmonic field, we have for not too large
+xT,Peq
+L(xT)≃1/2,Peq
+R(xT)≃1/2 hence from Eq. (15)
+Peq(xT)∼Cexp/braceleftBig
+−N[∝angbracketleftPeq
+R(xT)∝angbracketright−∝angbracketleftPL(xT)∝angbracketright]2/bracerightBig
+.(20)
+Expanding the expression in the exponent in xT(since
+Nis large) we find using Eqs. (16,18,19)
+Peq(xT)∼2√
+N√πZexp/bracketleftbigg
+−4N
+Z2(xT)2/bracketrightbigg
+(21)
+where with out loss of generality we assigned V(x= 0) =
+0. Hence the standard deviation is
+∝angbracketleft(xT)2∝angbracketright ∼Z2
+8N. (22)
+The same expression is found in Appendix A using the
+many body Boltzmann distribution, and integrating over
+all the particles except the tagged particle.
+B. Simple Illustration
+We now consider the situation of particles free of a
+forceF(x) = 0 with open boundary conditions L→ ∞
+where initially all the particles are on the vicinity of the
+origin. More precisely, the tagged particle is initially sit-
+uated at xT= 0,Nparticles to its right on ǫ→0+andNparticles on −ǫ. This problem was solved already by
+Aslangul [26], and here we recover the known result us-
+ing our formulas. The Green function g(x,x0,t) of a free
+particle is
+g(x,x0,t) =exp/bracketleftBig
+−(x−x0)2
+4Dt/bracketrightBig
+√
+4πDt(23)
+where as mentioned Dis the diffusion coefficient of the
+freeparticle. Withthespecifiedinitialconditionswehave
+∝angbracketleftPRL(xT)∝angbracketright= lim
+ǫ→0/integraldisplay∞
+0δ(x0−ǫ)/integraldisplayxT
+−∞exp/bracketleftBig
+−(x−x0)2
+4Dt/bracketrightBig
+√
+4πDtdxdx0=
+1
+2+/integraldisplayxT
+0e−x2
+4Dt√
+4πDtdx. (24)
+Similarly
+∝angbracketleftPLR(xt)∝angbracketright=1
+2−/integraldisplayxT
+0e−x2
+4Dt√
+4πDtdx. (25)
+WhenxT≪√
+2Dtwe Taylor expand in xTto find
+(∝angbracketleftPLR∝angbracketright − ∝angbracketleftPRL∝angbracketright)2∼(xT)2/πDt, using Eq. (15) we re-
+cover the result in [26]
+P(xT)∼1
+π/radicalbigg
+N
+Dtexp/bracketleftbigg
+−N(xT)2
+πDt/bracketrightbigg
+,(26)
+hence
+∝angbracketleft(xT)2∝angbracketright ∼πDt
+2N. (27)
+Thediffusionisnormal,inthesensethatthemeansquare
+displacement increases linearly in time. However the dif-
+fusion of the tagged particle is slowed down compared
+with a free particle, by a factor of 1 /Nwhich is due to
+the collisionswith all otherBrownianparticlesin the sys-
+tem. Clearly the approximation breaks down if one is in-
+terested in the tails of P(xT), since we used xT≪√
+2Dt.
+Though clearly when Nis large, the probability of find-
+ing such a particle is extremely small (i.e. use Eq. (26)
+P(xT=√
+2Dt)∼exp(−N2/π)).
+C. Formula for ∝angbracketleft(xT)2∝angbracketright
+We now consider symmetric potential fields V(x) =
+V(−x), and symmetric initial conditions. The latter
+means that the density of particles at time t= 0 to
+the left of the tagged particle, i.e. those residing in
+x0<0, is the same as for those residing to the right,
+fR(x0) =fL(−x0) (e.g. uniform initial conditions). In
+this case the subscript RandLis redundant and we
+usef(x0) =fR(x0) =fL(−x0), where f(x0) = 0 if6
+x0<0,f(x0)≥0 and/integraltext∞
+0f(x0)dx0= 1. From symme-
+try it is clear that the taggedparticle is unbiased, namely
+∝angbracketleftxT∝angbracketright= 0. Further, since Nis large we may expand the
+expressions in the exponent in Eq. (15) in xTto obtain
+the leading term
+∝angbracketleftPRL∝angbracketright−∝angbracketleftPLR∝angbracketright=
+∂
+∂xT[∝angbracketleftPRL(xT)∝angbracketright−∝angbracketleftPLR(xT)∝angbracketright]|xT=0xT+O[(xT)2].
+(28)
+Similarly for small xT
+∝angbracketleftPLL∝angbracketright∝angbracketleftPLR∝angbracketright+∝angbracketleftPRR∝angbracketright∝angbracketleftPRL∝angbracketright=
+2∝angbracketleftPRR(xT)∝angbracketright[1−∝angbracketleftPRR(xT)∝angbracketright]|xT=0+O(xT)(29)
+To derive Eq. (28) we used the symmetry of the prob-
+lem, which means that the probability of crossing the
+pointxT= 0 from left to right is the same as the proba-
+bility to cross from right to left ∝angbracketleftPLR∝angbracketrightxT=0=∝angbracketleftPRL∝angbracketrightxT=0
+and similarly ∝angbracketleftPLL∝angbracketrightxT=0=∝angbracketleftPRR∝angbracketrightxT=0by symmetry. We
+designate the reflection coefficient,
+R=∝angbracketleftPRR∝angbracketrightxT=0 (30)
+[orR=∝angbracketleftPLL∝angbracketrightxT=0] since it is the probability that a
+particle starting at x0>0, is found in x >0 at time t
+when an average over initial conditions is made
+R=/integraldisplayL
+0f(x0)/integraldisplayL
+0g(x,x0,t)dxdx0.(31)
+A transmission coefficient is defined through T=
+∝angbracketleftPRL∝angbracketrightxT=0=∝angbracketleftPLR∝angbracketrightxT=0which is related to the reflec-
+tion coefficient in the usual way T= 1−R.
+Turning our attention to Eq. (28), from left-right sym-
+metry we have
+∂
+∂xT∝angbracketleftPRL(xT)∝angbracketright|xT=0=−∂
+∂xT∝angbracketleftPLR(xT)∝angbracketright|xT=0.(32)
+Hence we define
+r=∂
+∂xT∝angbracketleftPRL(xT)∝angbracketright|xT=0 (33)
+where from Eq. (10)
+r=/integraldisplayL
+0f(x0)g(0,x0,t)dx0. (34)
+Soris the density of non interacting particles at x= 0
+for an initial density f(x0). Note that since ∝angbracketleftPRL(xT)∝angbracketright+
+∝angbracketleftPRR(xT)∝angbracketright= 1 we have
+r=−∂
+∂xT∝angbracketleftPRR(xT)∝angbracketright|xT=0. (35)
+Inserting Eq. (33) in Eq. (28) using Eq. (32) we
+have∝angbracketleftPRL∝angbracketright−∝angbracketleftPLR∝angbracketright= 2rxT+···. Inserting Eq. (30) inEq. (29) we find our main result: the probability density
+function of the position of the tagged central particle
+P(xT)∼1/radicalbig
+2π∝angbracketleft(xT)2∝angbracketrightexp/bracketleftBigg
+−(xT)2
+2∝angbracketleft(xT)2∝angbracketright/bracketrightBigg
+,(36)
+where
+∝angbracketleft(xT)2∝angbracketright=R(1−R)
+2Nr2(37)
+is the mean square displacement of the tagged particle.
+The single particle probability ∝angbracketleftPRR(xT)∝angbracketrightgivesREq.
+(30) and rEq. (35) which in turn yield the mean square
+displacement of the tagged particle Eq. (37). We will
+soon use this equation to demonstrate a variety of phys-
+ical behaviors. However first we establish a connection
+between our work and the theory of order statistics.
+D. Order Statistics
+Generate nindependent identically distributed ran-
+dom variables drawn from a PDF ˆ r(x) and arrange them
+in increasing order. Order statistics deals with the m-th
+observable m= 1,···,namongnobservations taken in
+the increasing order, which is denoted xm. The PDF of
+xm,φ(xm) depends on the PDF ˆ r(x), the sample size
+nand the order m. LetˆR(x) be the cumulative distri-
+bution of x, e.g. if the domain of xis−∞< x <∞,
+ˆR(x) =/integraltextx
+−∞ˆr(x)dxas usual. Following well known re-
+sult [37], define ˆ xwith
+ˆR(ˆx) =m
+n+1(38)
+then when nis large and m/nis of the order 1 /2
+φ(xm) = constexp
+
+−n(x−ˆx)2ˆr2(ˆx)
+2ˆR(ˆx)/bracketleftBig
+1−ˆR(ˆx)/bracketrightBig
+
+.(39)
+Hence the variance of xm
+(σm)2=ˆR(ˆx)/bracketleftBig
+1−ˆR(ˆx)/bracketrightBig
+nˆr2(ˆx)(40)
+which has some resemblance to Eq. (37).
+Theproblemofthemotionofataggedparticleismath-
+ematically identical to the problem of order statistics in
+two cases: (i) in the presence of a binding potential and
+in the long time limit and (ii) when all the particles start
+on the same point. In both cases the single particle PDF
+g(x,x0,t) ofall the particles areidentical since it is either
+independent of x0(case (i)) or we have a unique initial
+condition (case treated in sub section IIIB). For exam-
+ple in the presence of a binding field lim t→∞g(x,x0,t) =
+exp[−V(x)/kbT]/Zwhich is independent of the initial
+position of the particle. Hence in equilibrium, to find7
+the center particle we may draw (2 N+1) random vari-
+ables from Boltzmann’s distribution, and search for the
+center particle (which will give the position of the inter-
+acting tagged particle). Or, using the language of order
+statistics we have ˆ r(x) = exp[ −V(x)/kbT]/Z. Using a
+symmetric potential V(x) =V(−x) and Eq. (38) we in-
+sertn= 2N+1, and m=N+1; hence, when Nis large
+we have
+ˆR(ˆx) =/integraldisplayˆx
+−∞exp/bracketleftBig
+−V(ˆx)
+kbT/bracketrightBig
+Zdx=1
+2.(41)
+Thus, ˆx= 0. Then using Eq. (40) we find
+∝angbracketleft(xT)2∝angbracketright=Z2
+8N(42)
+which is the same as Eq. (22) (recall V(0) = 0).
+While Eq. (40) has superficially a structure similar to
+Eq. (37) they are different. Our REq. (30) is gener-
+ally not equal half, neither must it be close to that value
+[so Eq. (38) is not generally related to our problem].
+In fact when t→0 we must have ∝angbracketleft(xT)2∝angbracketright →0 which
+is found when lim t→0R= 1 (see examples below). In
+the problem of motion of a tagged particle, the number
+of particles which can interact with the center particle
+is usually increasing with time (hence nis not fixed).
+For example consider the classical case of uniformly dis-
+tributed particles in infinite space and in the absence of
+forces. Roughly speaking particles at distances of the
+order of leff=√
+Dtor shorter can influence the tagged
+particle motion via collisions. So in this case roughly
+Neff=ρ√
+Dtparticles participate in the process. In con-
+trast, if all particles are initially at the vicinity of the
+originNparticles participate i.e. influence the motion
+of the tagged particle. Interestingly this gives an argu-
+ment for the well known behavior ofthe mean square dis-
+placement of the tagged particle, with uniform density of
+particles, namely use Eq. (27) ∝angbracketleft(xT)2∝angbracketright ∼πDt/2Nand
+replaceNwithNeff=ρ√
+Dtto find∝angbracketleft(xT)2∝angbracketright ≃√
+Dt/ρ
+which of course misses the correct numerical prefactor
+(see Eq. 50 below).
+IV. PHYSICAL ILLUSTRATIONS
+A. Particles in a Box
+Considerparticles in a finite box extending from −Lto
+Lwhich was recently treated with the Bethe ansatz and
+numerical simulations by Lizana and Ambj¨ ornson [25].
+The tagged particle initially at xT= 0 has Nparticles
+to its right and Nto its left. These particles are assumed
+uniformly distributed, hence
+f(x0) =
+
+1
+L0< x0<L
+0 otherwise .(43)In the limit L→ ∞andN→ ∞in such a way that the
+densityρ=N/Lis fixed, we obtain single file diffusion
+in an infinite system, a case well studied long ago [4, 6].
+The single particle Green function of a parti-
+cle in a box, with reflecting boundary conditions
+∂g(x,x0,t)/∂x|x=±L= 0 is solved using an eigenfunc-
+tion expansion [38]
+g(x,x0,t) =1
+2L+
+1
+L∞/summationdisplay
+n=1cos/bracketleftbiggnπ
+2L/parenleftbig
+x+L/parenrightbig/bracketrightbigg
+cos/bracketleftbiggnπ
+2L/parenleftbig
+x0+L/parenrightbig/bracketrightbigg
+exp/parenleftbigg
+−Dn2π2
+4L2t/parenrightbigg
+.
+(44)
+With Eqs. (31,43,44) we find
+R(t) =1
+2+4
+π2∞/summationdisplay
+n=1,Oddexp/parenleftBig
+−Dn2π2
+4L2t/parenrightBig
+n2(45)
+where the summation is over odd n. Att= 0,R= 1
+since all particles initially in (0 ,L) did not have time to
+move to the other side of the box, and lim t→∞R= 1/2
+since in the long time limit there is equal probability for
+a non interacting particle to occupy each half of the box.
+Using Eqs. (34,43,44) we find
+r=1
+2L(46)
+hence Eq. (37) gives the mean square displacement of
+the tagged particle
+∝angbracketleft(xT)2∝angbracketright ∼2R(t)[1−R(t)]L2
+N. (47)
+The eigenvaluesof the non-interacting particle determine
+the multi exponential type of decay of R(t) with time,
+which in turn determines the dynamics of the interacting
+tagged particle [43].
+Letδ2=Dπ2t/4L2hence the limit δ≪1 gives the
+short time dynamics before the particles interact with
+the walls, or equivalently the limit of an infinite system.
+The reflection coefficient is rewritten
+R=1
+2+4
+π2
+δ∞/summationdisplay
+n=1,Odde−δ2n2−1
+δ2n2δ+∞/summationdisplay
+n=1,Odd1
+n2
+.
+(48)
+Whenδ≪1 we may replace the first summation with
+integration
+∞/summationdisplay
+n=1,Odde−δ2n2−1
+δ2n2δ≃1
+2/integraldisplay∞
+0e−y2−1
+y2dy=−√π
+2
+(49)
+where the factor 1 /2 on the RHS comes from the
+summation over only odd non the LHS. Using8
+FIG. 4: For particles in a box we show ∝angbracketleft(xT)2∝angbracketrightN/L2of the
+tagged particle, in equilibrium versus N. Exact expression
+obtained in [25] valid for all N(dots) converges in the limit
+ofN→ ∞to the behavior predicted by our theory (dashed
+line) Eq. (52). For N= 70 clear deviations between our
+asymptotic theory and the exact result in [25] are found.
+/summationtext∞
+n=1,Odd1/n2=π2/8 we have R= 1−√
+Dt
+L√π+···,
+and hence
+∝angbracketleft(xT)2∝angbracketright ∼2√π√
+Dt
+ρ. (50)
+This result was obtained in [4, 6], and it describes the
+dynamics of the tagged particle in a box, before particles
+have time to interact with the walls, namely when δis
+small.
+In the opposite limit of long times and finite systems
+we attain equilibrium, then lim t→∞R= 1/2 and
+P(xT)∼√
+N√πLe−N(xT)2/L2
+, (51)
+hence
+lim
+t→∞∝angbracketleft(xT)2∝angbracketright=L2/2N. (52)
+Eq. (51) is a special case of the more general Eq. (21)
+since the single particle normalizing partition function
+for our example is Z= 2L.
+As mentioned, in [25] the Bethe ansatz was used to
+solve the problem of tagged particle motion in a box.
+Among otherthings an exact expressionfor the longtime
+limit of∝angbracketleft(xT)2∝angbracketright, valid for all Nwas found [25]
+lim
+t→∞∝angbracketleft(xT)2∝angbracketright=L2/parenleftbigg1
+4/parenrightbiggN+1Γ/parenleftbig1
+2/parenrightbig
+Γ[2(N+1)]
+Γ(N+1)Γ/parenleftbig
+N+5
+2/parenrightbig.(53)Since our formalism is valid only in the large Nlimit,
+comparison of our solution to an exact result like Eq.
+(53) provides insight to the question of convergence.
+In Fig. 4 we plot lim t→∞∝angbracketleft(xT)2∝angbracketrightN/L2versusNus-
+ing Eq. (53) comparing it to our Eq. (52) which gives
+limt→∞∝angbracketleft(xT)2∝angbracketrightN/L2= 1/2. Not surprisinglywe see that
+the two results yield asymptotically the same result. The
+figure illustrates that even for N= 100 deviations be-
+tween exact results and the present theory are observ-
+able.
+In [25] numerical simulation of systems with N=
+1,10,70 (maximum of 141 particles) were performed,
+and favorably compared with the Bethe ansatz solution.
+Thesesimulationsweremade forfinite size hardcorepar-
+ticles whose diameter is ∆. To makecomparisonbetween
+our theory and simulation we scale the size of the box
+according to 2 L→2L−(2N)∆. In Fig. 5 the scaled
+mean square displacement versus t/τeqis shown, where
+τeq= 4L2/D. Some general features of our theory can
+now be discussed. First, at short times simulations show
+normal behavior ∝angbracketleft(xT)2∝angbracketright= 2Dt, which is a trivial ef-
+fect: particles did not have time to collide and hence
+the tagged particle diffuses normally as if it is free. After
+particlesstarttocollide, deviationsfromnormaldiffusion
+are observed(for N= 70but as expected not for N= 1).
+These are mainly due to collisions with other particles.
+Finally saturation due to the finite size of the system is
+found. While our theory shows the general trend of sim-
+ulations (say for N= 70 and for not too short times) it
+is clearly not in perfect agreement. We argue that this
+is due to the small number of particles N= 70, since as
+we showed in Fig. 4, at least for large times and large
+Nour theory gives the exact result. It would be nice
+to have simulations with point particles, with larger N
+and since there are three phases of the motion: normal
+diffusion, single file diffusion without the influence of the
+walls and finally saturation to equilibrium. Separation
+of time scales is needed to demonstrate these behaviors
+clearly.
+B. Gaussian Packet
+Consider particles without external forces V(x) = 0,
+in an infinite system. Initially particles are spread with
+a Gaussian packet with width ξ:
+f(x0) =√
+2√πξexp/bracketleftbigg
+−(x0)2
+2ξ2/bracketrightbigg
+(54)
+forx0>0. As before we consider the tagged particle
+motion, which is initially at xT= 0, with Nparticles to
+its left and Nto its right. With the free particle Green
+function g(x,x0,t) Eq. (23) we proceed to find ∝angbracketleft(xT)2∝angbracketright.
+The reflection probability Eq. (31) is
+R=/integraldisplay∞
+0dx0/radicalbigg2
+ξ2πe−(x0)2
+2ξ2/integraldisplay∞
+0dx√
+4πDte−(x−x0)2
+4Dt.(55)9
+FIG. 5: Motion of tagged particle in a box is shown for
+N= 1,10,70. Simulation results (squares) are taken from
+[25] (see text). At short time simulations exhibit normal di f-
+fusion∝angbracketleft(xT)2∝angbracketright= 2Dtas illustrated by the straight dashed
+line which is a guide to the eye. Our theory is expected to
+work well in the large Nlimit and when many collision events
+between tagged particle and surrounding Brownian particle s
+took place. Hence agreement between theory (solid line) and
+simulation is reasonable at most only for N= 70 and not for
+too short times.
+FIG. 6: Scaled mean square displacement of the tagged par-
+ticle with Gaussian initial conditions of the packet of part icles
+exhibits a transition between short time ∝angbracketleft(xT)2∝angbracketright ∝t1/2law to
+∝angbracketleft(xT)2∝angbracketright ∝tbehavior. Dashed lines are short and long time
+asymptotic behavior Eq. (60), circles represent Eq. (59).Changing variables according to y2/2 = (x−x0)2/4Dt
+and using dimensionless parameter ˜ξ=ξ/√
+2Dt, we find
+R=1
+2+1
+˜ξ/radicalbigg
+2
+π/integraldisplay∞
+0d˜x0e−(˜x0)2
+2˜ξ2Erf/parenleftbig
+˜x0/√
+2/parenrightbig
+2(56)
+where Erf(˜ x0/√
+2)/2 =/integraltext˜x0
+0e−y2/2dy/√
+2πis the error
+function [42]. Mathematica solves the integral in Eq.
+(56) and we find
+R=1
+2+1
+πarccot/parenleftBigg√
+2Dt
+ξ/parenrightBigg
+. (57)
+For short times√
+Dt≪ξwe haveR ∼1−√
+2Dt
+πξ, namely
+most particlesdid not havetime to crossthe origin, while
+in the opposite limit lim t→∞R= 1/2 due to the sym-
+metry of initial conditions. The calculation of rEq. (34)
+using Eqs. (23,54) is straightforward
+r=1
+2√
+πDt/radicalbig
+1+ξ2/2Dt, (58)
+Inserting Eqs. (57,58) in Eq. (37) we find
+∝angbracketleft(xT)2∝angbracketright ∼ξ2π
+N/parenleftbigg
+1+2Dt
+ξ2/parenrightbigg/bracketleftBigg
+1
+4−1
+π2arccot2/parenleftBigg/radicalBigg
+2Dt
+ξ2/parenrightBigg/bracketrightBigg
+,
+(59)
+This solution is shown in Fig. 6 with its two limiting
+behaviors
+∝angbracketleft(xT)2∝angbracketright ∼/braceleftbigg
+ξ√
+2Dt
+Nshort times 2 Dt≪ξ2
+πD
+2Ntlong times 2 Dt≫ξ2.(60)
+Forshorttimesthe particlesdonothavetime todisperse;
+hence, the motion of the tagged particle is slower than
+normal, increasing as t1/2which is similar to the single
+file diffusion with a uniform density Eq. (50). Roughly
+speaking, for short times the tagged particle sees a uni-
+form density of particles with ρ=N/ξ. For long times
+we recover the behavior in Eq. (27) since the scale of
+diffusion is much larger than ξ. Hence if we start with
+a Gaussian or delta function packet we get in the long
+time limit similar behavior, as we showed.
+C. Particles in Harmonic Oscillator
+Consider particles in a harmonic potential V(x) =
+mω2x2/2 where ω >0 is the harmonic frequency. The
+single particle undergoes an Ornstein Uhlenbeck process
+[38] and the corresponding single particle Green function
+is
+g(x,x0,t) =
+1/radicalBig
+2πDτ/parenleftbig
+1−e−2t/τ/parenrightbigexp/bracketleftBigg
+−/parenleftbig
+x−x0e−t/τ/parenrightbig2
+2Dτ/parenleftbig
+1−e−2t/τ/parenrightbig/bracketrightBigg
+,(61)10
+FIG. 7: Scaled mean square displacement of tagged particle
+in harmonic field Eq. (66) exhibits a transition between a
+short time ∝angbracketleft(xT)2∝angbracketright ∝t1/2law to saturation due to the binding
+field. The horizontal dashed line is the long time ∝angbracketleft(xT)2∝angbracketright=
+πξ2
+th/4Nbehavior, the dashed line is the short time limit Eq.
+(67), and the dotted dashed line is Eq. (66).
+where (τ)−1=Dmω2/kbTis the inverse relaxation time.
+We assume thermal initial conditions
+f(x0) =2√
+mω2
+√2πkbTexp/parenleftbigg
+−mω2x2
+2kbT/parenrightbigg
+, x0>0.(62)
+Using Eq. (34) it is easy to show that
+r=1√
+2πξth, (63)
+where the thermal length is ξth=√
+Dτ=/radicalbig
+mω2/kbT.
+Note that one can write r= 1/Zwhich as we will soon
+prove is a general result valid for all potential fields satis-
+fyingV(x) =V(−x), provided that the initial condition
+f(x0) is the thermal equilibrium . The reflection coeffi-
+cient [Eq. (31)] is
+R=1
+2+/radicalbigg
+1
+2πη/integraldisplay∞
+0e−η2y2
+2Erf/parenleftbiggy√
+2/parenrightbigg
+dy,(64)
+whereη=et/τ√
+1−e−2t/τ. Using Mathematica
+R=1
+2+1
+πarccot/parenleftBig/radicalbig
+e2t/τ−1/parenrightBig
+.(65)
+For short times t/τ <<1,R ∼1−√
+2t/(π√τ), and for
+long time R ∼1/2+e−t/τ/π.
+Using Eqs.(37,63,65) the mean square displacement of
+the tagged particle is
+∝angbracketleft(xT)2∝angbracketright=π
+Nξ2
+th/bracketleftbigg1
+4−1
+π2arccot2/parenleftBig/radicalbig
+e2t/τ−1/parenrightBig/bracketrightbigg
+.(66)For short times t << τ
+∝angbracketleft(xT)2∝angbracketright ∼ξth
+N√
+2Dt (67)
+Suchabehaviorisexpected, aswellknownforshorttimes
+the diffusion process of the Ornstein Uhlenbeck process
+x∼t1/2is faster than the drift x∼tand dominates
+the process, i.e. take t << τ in Eq. (61) and get the
+Green function of the force free particle Eq. (23). Hence
+for short times we can neglect the external field, but use
+of course the Gaussian initial condition Eq. (62). Then
+the problem reduces to the short time behavior of the
+Gaussian packet, free of external forces (compare short
+time behavior in Eq. (60) with Eq. (67) when ξ→ξth).
+The long time limit of Eq. (66) gives ∝angbracketleft(xT)2∝angbracketright ∼πξ2
+th/4N
+whichisthethermalequilibriumbehaviorpredictedmore
+generallyinEq. (21). Behaviorofthescaledmeansquare
+displacementanditsasymptoticbehaviorsispresentedin
+Fig. 7.
+D. Thermal Initial Conditions
+If we assume that initially the particles are in thermal
+equilibrium, our simple formulas simplify even more. If
+f(x0) = 2exp[ −V(x0)/kbT]/Zthenr= 1/Z. To see this
+notice that for symmetric potentials V(x) =V(−x) we
+haveg(0,−x0,t) =g(0,x0,t) and
+r=2
+Z/integraldisplay∞
+0exp/bracketleftbigg
+−V(x0)
+kbT/bracketrightbigg
+g(0,x0,t)dx0(68)
+yields
+r=/integraldisplay∞
+−∞exp/bracketleftBig
+−V(x0)
+kbT/bracketrightBig
+Zg(0,x0,t)dx0,(69)
+where we assumed that the potential is binding so a sta-
+tionarysolutionoftheFokker-Planckequationisreached;
+i.e. the free particle is excluded. Therefore r[Eq. (69)]
+is the probability of finding a non interacting particle
+at the origin, with thermal initial conditions. Since the
+thermal equilibrium density is the stationary solution of
+the Fokker Planck operator, ris time independent and
+equal to r= exp[−V(0)/kbT]/Z. We can always take
+V(0) = 0 and then r= 1/Z. Examples for this behavior
+are the already analyzed cases of particles in a box with
+uniform initial density [Eq. (46)] (since Z= 2Lfor that
+case) and similarly for particles in a harmonic potential
+with initial thermal density [Eqs. (62, 63)]. Hence using
+Eq. (37)
+∝angbracketleft(xT)2∝angbracketright=R(1−R)Z2
+2N. (70)
+For the symmetric and binding potentials under consid-
+eration, we have lim t→∞R= 1/2 and hence Eq. (70) in
+the long time limit yields the equilibrium behavior Eq.
+(22).11
+E. Power Law Type of Initial Conditions
+Flomenbom and Taloni [27] considered particles free of
+external forces, hence the free particle Green function is
+g(x,x0,t) =1√
+4πDtexp/bracketleftBigg
+−(x−x0)2
+4Dt/bracketrightBigg
+,(71)
+with initial conditions of power law type
+f(x0) =B|x0|−β0< x0< L (72)
+for 0< β <1. Here the system size L→ ∞and the
+reader should not confuse LwithL. The normalization
+of the PDF of Eq. (72) yields B= (1−β)Lβ−1. In the
+limitβ→0 andL→ ∞in such a way that ρ=N/L
+remains fixed, we anticipate the classical case of single
+file diffusion in the presence of a uniform density of par-
+ticles:∝angbracketleftx2∝angbracketright ∝t1/2[Eq. (50)]. In the opposite limit of
+β→1 the particles are initially concentrated at the ori-
+gin, and we expect the behavior ∝angbracketleftx2∝angbracketright ∝t[Eq. (27)].
+Thus 0< β <1 bridges between these two known be-
+haviors, indeed one finds ∝angbracketleft(xT)2∝angbracketright ∝t(β+1)/2as shown in
+[27]. The latter is valid for times√
+4Dt≪Las discussed
+below. This indicates that specially chosen initial condi-
+tion may control the qualitative behavior of the diffusion
+of the tagged particle. Here we analyze this case using
+ourformalism finding analyticalexpressionsfor the mean
+square displacement.
+To obtain ∝angbracketleft(xT)2∝angbracketrightall we need to do is to find randR.
+Using Eqs. (34,71,72)
+r=(1−β)√πLβ−1
+/parenleftBig√
+4Dt/parenrightBigβ/integraldisplayL√
+4Dt
+0y−βe−y2dy,(73)
+solving the integral
+r=(1−β)√πLβ−1
+/parenleftBig√
+4Dt/parenrightBigβ1
+2/bracketleftbigg
+Γ/parenleftbigg1−β
+2/parenrightbigg
+−Γ/parenleftbigg1−β
+2,L2
+4Dt/parenrightbigg/bracketrightbigg
+,
+(74)
+where Γ( a,z) is the incomplete Gamma function [42].
+The following behaviors are found for short and long
+times
+r=
+
+crLβ−1
+(√
+4Dt)β√
+4Dt≪L
+1√
+4πDt√
+4Dt≫L.(75)
+In the short time limit we took the upper bound in inte-
+gration in Eq. (73) to ∞so
+cr=(1−β)√π/integraldisplay∞
+0y−βe−y2dy=1−β√πΓ(1−β/2)
+2.
+(76)
+InsertingEqs. (71,72)inEq. (31)wefindthereflection
+coefficient
+R=1
+2+1
+2(1−β)/parenleftbiggL√
+4Dt/parenrightbiggβ−1/integraldisplayL/√
+4Dt
+0dyy−βErf(y).
+(77)In the limit of long times we get the expected behavior
+limt→∞R= 1/2sincethen halfofthe particlesareto the
+leftoftheoriginandhalftotheright(instatisticalsense).
+Mathematica solves the integral in the last equation in
+terms of tabulated functions
+/integraltextx
+0y−βErf(y)dy=
+x−β
+(1−β)√π/braceleftBig
+x/bracketleftbig√πErf(x)+xEβ/2/parenleftbig
+x2/parenrightbig/bracketrightbig
+−xβΓ/parenleftBig
+1−β
+2/parenrightBig/bracerightBig
+,
+(78)
+where E n(z) =/integraltext∞
+1dtexp(−zt)/tnis the exponential in-
+tegral function.
+To analyze the short time behavior we use integration
+by parts in Eq. (77) and find
+R=1
+2+1
+2/parenleftBig
+L√
+4Dt/parenrightBigβ−1
+×/bracketleftbigg/parenleftBig
+L√
+4Dt/parenrightBig1−β
+Erf/parenleftBig
+L√
+4Dt/parenrightBig
+−2√π/integraltextL/√
+4Dt
+0y1−βe−y2dy/bracketrightbigg
+.
+(79)
+Using asymptotic properties of the Erf function [42], ne-
+glecting terms of the order of exp( −L2/4Dt), Eq. (79)
+for√
+4Dt≪Lgives
+R ∼1−cR/parenleftBigg√
+4Dt
+L/parenrightBigg1−β
+, (80)
+with
+cR=1√π/integraldisplay∞
+0y1−βe−y2dy=Γ/parenleftBig
+1−β
+2/parenrightBig
+2√π.(81)
+Eqs. (74,77,78) yield the mean square displacement
+Eq. (37). The short time behavior√
+4Dt≪L
+∝angbracketleft(xT)2∝angbracketright ∼1
+2N2√π
+(1−β)2Γ(1−β/2)L1−β/parenleftBig√
+4Dt/parenrightBig1+β
+,
+(82)
+while for long times√
+4Dt≫L
+∝angbracketleft(xT)2∝angbracketright ∼πDt
+2N. (83)
+Thusafteralongtimethediffusionisnormal ∝angbracketleft(xT)2∝angbracketright ∝t,
+exactly like the case where all particles started initially
+on the origin Eq. (27). While for short times ∝angbracketleft(xT)2∝angbracketright ∝
+t(1+β)/2.
+Taking the limit β→0 in Eq. (82), we get the well
+known result of single file motion in uniform density of
+particles [4, 6] Eq. (50), with the density ρ=N/L. In
+the opposite limit β→1 we have ∝angbracketleftx2∝angbracketright=πDt
+2Nfor all
+times which is the same as in Eq. (27). Note that the
+limitβ→1 must be treated with care, the short time
+limit and the β→1 limit do not commute due to the
+the 1/(1−β)2divergence in Eq. (82). When β→1 all
+particlesarecenteredon the originhencewe get the same
+behavior as in Eq. (27).12
+V. PERCUS RELATION: BEYOND BROWNIAN
+PARTICLES
+Sofarweconsideredthecasewhereparticlesarediffus-
+ing according to the laws of Brownian motion in between
+collision events. What happens for other types of mo-
+tion? For example what happens when the underlying
+motion itself is anomalous [27, 28].
+Percus [18] (see also [27, 44]) investigated a general
+relation between the diffusion of the tagged particle
+and motion in the absence of interactions (free motion)
+∝angbracketleft(xT)2∝angbracketright ∼ ∝angbracketleft|x|∝angbracketrightfree/ρ. Such a relation was suggested
+for normal diffusion where we have ∝angbracketleft|x|∝angbracketrightfree∝t1/2, so
+∝angbracketleft(xT)2∝angbracketright ∝t1/2and for a particle moving ballistically be-
+tween collision events hence ∝angbracketleft|x|∝angbracketrightfree∝tand therefore
+when we turn on the interactions ∝angbracketleft(xT)∝angbracketright2∝t. This sim-
+ple relation between free particle motion and the mean
+square displacement of the tagged interacting particle, is
+expected to work for an infinite system, with a uniform
+densityρof particles, and when external forces are zero
+[F(x) = 0]. Here we will derive the Percus relation from
+our formalism, for more general dynamics.
+Assume that the Green function of the non interacting
+particle can be written in the scaling form
+g(x,x0,t) =1√Kαtγ/2G/parenleftBigg
+x−x0/radicalbig
+Kγtγ/2/parenrightBigg
+.(84)
+We assume G(y) =G(−y) and from normalization/integraltext∞
+−∞G(y)dy= 1. Herethefreeparticlemotionisanoma-
+lous, so that ∝angbracketleft|x|∝angbracketrightfree∝tγ/2for 0< γand not equal
+unity. For example the underlying motion might be sub-
+diffusive continuous time random walk (CTRW) [45–47]
+or fractional Brownian motion, where in the latter case
+G(y) is Gaussian and in the former G(y) is expressed
+in terms of L´ evy distributions (see some details below).
+We assume that moments of the process are finite. The
+constant Kγhas units of m2/secγ.
+To obtain the mean square displacement of the tagged
+interacting particle we calculate the reflection coefficient
+Eq. (31)
+R=1
+L/integraldisplayL
+0dx0/integraldisplayL
+0G/parenleftBigg
+x−x0/radicalbigKγtγ/2/parenrightBigg
+dx/radicalbigKγtγ/2.(85)
+Here we used uniform density of particle f(x0) = 1/L,
+and we will consider the limit L→ ∞with the den-
+sityρbeing kept fixed. Change of variables y= (x−
+x0)//radicalbig
+Kγtγ/2and using L//radicalbig
+Kγtγ/2→ ∞we have
+R ∼1
+L/integraldisplayL
+0dx0/integraldisplay∞
+−x0√
+Kγtγ/2G(y)dy. (86)
+Integrating by parts, changing variables y=
+x0//radicalbig
+Kγtγ/2, and using the normalization condition, we
+have
+R ∼1−∝angbracketleft|x|∝angbracketrightfree
+2L(87)where the mean of the absolute value of the free particle
+motion is by its definition
+∝angbracketleft|x|∝angbracketrightfree=/integraldisplay∞
+−∞|x|G/parenleftBigg
+x/radicalbig
+Kγtγ/2/parenrightBigg
+dx/radicalbig
+Kγtγ/2.(88)
+Here we used the assumed symmetry G(y) =G(−y)
+which means that the underlying random walk is not bi-
+ased, as a result ∝angbracketleft|x|∝angbracketright=/radicalbig
+Kγtγ/22/integraltext∞
+0yG(y)dy. It is
+easy to see that r= 1/2L. In the limit where N→ ∞
+andL→ ∞we find using Eqs. (37,87) the Percus for-
+mula for a general class of stochastic dynamics
+∝angbracketleft(xT)2∝angbracketright=∝angbracketleft|x|∝angbracketrightfree
+ρ. (89)
+Clearly this equation gives a useful relationship between
+motion of a free particle and the same particle moving
+in single file when it is surrounded by identical particles
+whose density is ρ,
+As a simple example consider frac-
+tional Brownian motion where g(x,x0,t) =
+(/radicalbig
+4πKγtγ)−1exp/bracketleftbig
+−(x−x0)2/4Kγtγ/bracketrightbig
+, where
+0< γ <2. Then Eq. (89) gives
+∝angbracketleft(xT)2∝angbracketright=2/radicalbigKγtγ
+ρ√π. (90)
+Hence if the underlying motion is ballistic γ→2 the
+motion of the tagged particle undergoing single file dy-
+namics is normal with respect to time ∝angbracketleft(xT)2∝angbracketright ∝t. For
+a CTRW particle in the continuum approximation the
+non-interacting single particle green function is governed
+by the fractional diffusion equation [47–49]
+∂γ
+∂tγg(x,x0,t) =Kγ∂2
+∂x2g(x,x0,t) (91)
+with 0< γ <1. From this equation it is easy to find [50]
+∝angbracketleft|x|∝angbracketrightfree=/radicalBig
+Kγtγ/Γ(1+γ/2). Hence using Eq. (89)
+∝angbracketleft(xT)2∝angbracketright=/radicalBig
+Kγtγ/2
+ρΓ(1+γ/2). (92)
+We see that independent of the mechanism of the under-
+lying anomalous diffusion (i.e. fractional Brownian mo-
+tion, or CTRW) we get for the tagged particle ∝angbracketleft(xT)2∝angbracketright ∼
+tγ/2. Further our general results show that the Green
+function of the tagged particle is Gaussian even though
+the Green function of non-interacting CTRW particles is
+highly non-Gaussian.
+Warning: One should be careful when applying our
+results to the CTRW model, namely Eq. (92) should not
+be abused. One mechanism of anomalous diffusion are
+the power law waiting times of Scher and Montroll [45]
+which yield slow dynamics, as captured by the CTRW
+[46, 47] and the fractional Eq. (91) [50]. Single file diffu-
+sion with such sub-diffusive motion as the starting point13
+was considered recently theoretically and with simula-
+tions in [27, 28]. It should be noted that the meaning
+of collision in such a model should be taken with care.
+For a random walk on a lattice with waiting times on
+each lattice point, one can envisionseveralcollision rules.
+For example one might allow two particles to occupy the
+same site at a given time, or one may consider a mech-
+anism were a particle once hopping into a trap already
+occupied will eject the particle previously residing in the
+trap. Or a particle is allowed to jump only into an empty
+site, as is usually assumed. These type of collision rules
+might yield behaviors different than ours. For example a
+particle stuck with a very large sojourn time, might be
+ejected by anotherparticle, hence one can imagine asitu-
+ation where some form of interaction causes the particles
+to move faster. In our work the collision implies that we
+can let particles go past one another as if they were non
+interacting (so on a lattice two particles may occupy the
+same point at the same time) and eventually we look for
+the center particle. Even more interesting will be to in-
+vestigate interacting particles in systems with quenched
+disorder, since the latter, when strong enough, is known
+to lead to non Gaussian sub-diffusion [45, 46]. There
+the simple formula Eq. (89) is generally not expected to
+hold. Indeed in [34] single file motion of a tagged particle
+in the Sinai model was considered, the results are much
+richer when compared with Eq. (89).
+VI. DISCUSSION
+The one dimensional problem of motion of a tagged
+particle interacting via hard core interactions with other
+particleswas solvedusing the Jepsen line. The formalism
+we developed treats both Brownian and non Brownian
+motion in between collision events, is suited for rather
+general external fields acting on the particles, for open
+and closed system, and handles also different types of
+initial conditions. Following others we have mapped theproblem onto a non interacting problem using the Jepsen
+line. The motion of the tagged particle belongs to the
+general problem of order statistics. The problem reduces
+to considering a list of 2 N+1 random variables and find-
+ing the distribution of the variable which has Nvariables
+smaller than it and Nlargercorrespondingto center par-
+ticle (note that the right most particle we will have an
+extreme value problem). Classical theory of order statis-
+tics deals however with the case where all the random
+variables have identical distributions. In contrast in the
+exclusion process under consideration, particles have non
+identical distribution. Thus except for two cases: i) all
+the particles initially on the same position and ii) equi-
+librium state, the problem deals with non identically dis-
+tributed random variables (since the initial condition are
+non identical). While we treated the problem of sym-
+metric potential and symmetric initial condition for the
+center particle in detail, it is left for future work to con-
+sider non-symmetric potential fields, non symmetric ini-
+tial conditions, and the dynamics of the particle in the
+tails of the packet. We believe that the methods devel-
+oped here with some modifications can treat these cases
+too.
+Acknowledgment EB is supported by the Israel Sci-
+ence Foundation, and RS by the NSF. We thank Ludvig
+Lizana and Tobias Ambj¨ ornson for sharing with us their
+data in Fig. 5 and for providing helpful insight. We also
+thank M. Lomholt and A. Taloni for useful discussions.
+VII. APPENDIX A
+In this Appendix we use Boltzmann’s distribution for
+the interacting system to find the PDF of the tagged
+center particle in equilibrium. The multi dimensional
+PDF for 2 N+ 1 interacting particles, in the presence
+of an external binding field V(x), withV(x) =V(−x),
+acting on all of them is
+P(x−N,···,x−1,x0,x1,···,xN) =1
+Z2N+1exp
+−N/summationdisplay
+j=−NV(xj)
+kbT
+θ(x−N+1−x−N)θ(x−N+2−x−N+1)···θ(xN−xN−1)
+(93)
+whereZ2N+1is a normalizing factor, and θ(x) is a step function: θ(x) = 0 ifx <0,θ(x) = 1 for x≥0. The center
+tagged particle is x0=xT. To find the PDF of xTin equilibrium, which we call Peq(xT), we must integrate Eq. (93)
+over all coordinates besides x0→xT
+Peq(xT) =exp/bracketleftBig
+−V(xT)
+kbT/bracketrightBig
+Z2N+1×
+/integraltextxT
+−∞dx−N/integraltextxT
+x−Ndx−N+1···/integraltextxT
+x−2dx−1exp/bracketleftbigg
+−/summationtext−1
+j=−NV(xj)
+kbT/bracketrightbigg/integraltext∞
+xTdx1/integraltext∞
+x1dx2···/integraltext∞
+xN−1dxNexp/bracketleftbigg
+−/summationtextN
+j=1V(xj)
+kbT/bracketrightbigg
+.(94)14
+We rearrange the integration limits, as explained in Fig. 8
+/integraldisplayxT
+−∞dx−N/integraldisplayxT
+x−Ndx−N+1···=/integraldisplayxT
+−∞dx−N/integraldisplayx−N
+−∞dx−N+1···. (95)
+Hence we can use
+/integraldisplayxT
+−∞dx−N/integraldisplayxT
+x−Ndx−N+1···=1
+2/bracketleftBigg/integraldisplayxT
+−∞dx−N/integraldisplayxT
+x−Ndx−N+1···+/integraldisplayxT
+−∞dx−N/integraldisplayx−N
+−∞dx−N+1···/bracketrightBigg
+=1
+2/integraldisplayxT
+−∞dxN/integraldisplayxT
+−∞dx−N+1···.
+(96)
+Repeating this procedure we rewrite Eq. (94) as
+Peq(xT) = Nor
+
+/integraldisplayxT
+−∞dxexp/bracketleftBig
+−V(x)
+kBT/bracketrightBig
+Z
+
+N
+
+/integraldisplay∞
+xTdxexp/bracketleftBig
+−V(x)
+kBT/bracketrightBig
+Z
+
+N
+exp/bracketleftBig
+−V(xT)
+kBT/bracketrightBig
+Z. (97)
+/MT88/MT45/MT78/MT43/MT49
+/MT88/MT84
+/MT88/MT84/MT88/MT45/MT78/MT88/MT84
+/MT88/MT84/MT88/MT45/MT78/MT88/MT45/MT78/MT43/MT49
+FIG. 8: Left panel integration in the domain −∞< x−N<
+xT,x−N< x−N+1< xTis equivalent to integration −∞<
+x−N< xT,−∞< x−N+1< x−N(right panel).
+where Nor is a normalization constant and Zis defined
+in Eq. (17).
+Thus Eq. (97) describes a problem of order statistics
+which is extensively investigated by mathematicians, as
+mentioned in Sec. IIID. Peq(xT) Eq. (97) is the PDF of
+the random variable which has exactly Nrandom vari-
+able larger than it and Nsmaller. In this sense we have
+transformed the problem to a non-interacting system,
+similar to the non interacting picture in the main text.
+The information contained in the single non-interacting
+particle, i.e. the single particle Boltzmann distribution
+exp[−V(x)/kbT]/Zis all what is needed for the calcula-
+tion of the position of the tagged particle.
+Using the symmetry V(x) =V(−x) we have
+/integraldisplayxT
+−∞e−V(x)
+kBT
+Zdx=1
+2+/integraldisplayxT
+0e−V(x)
+kBT
+Zdx,(98)and
+/integraldisplay∞
+xTe−V(x)
+kBT
+Zdx=1
+2−/integraldisplayxT
+0e−V(x)
+kBT
+Zdx.(99)
+Using Eqs. (98,99), we rewrite Eq. (97)
+Peq(xT) = Nore−V(xT)/kbT
+ZeNln1
+4
+exp/braceleftBigg
+Nln/bracketleftBigg
+1−4/parenleftbigg/integraltextxT
+0e−V(x)/kbTdx
+Zdx/parenrightbigg2/bracketrightBigg/bracerightBigg
+.(100)
+In the limit N→ ∞ onlyxTwith/integraltextxT
+0exp[−V(x)/kbT]dx/Z≪1 will have a measur-
+able contribution to Peq(xT), since if this condition is
+not satisfied, the value of Peq(xT) is exponentially small
+inN. Expanding the ln in Eq. (100) we have
+Peq(xT)∝exp/bracketleftbigg
+−V(xT)
+kbT/bracketrightbigg
+exp
+−4N
+/integraltextxT
+0dxe−V(x)
+kbT
+Z
+2
+.
+(101)
+SinceN >>1 we expand/integraltextxT
+0exp(−V(x)/kbT)dx=xT
+where we used V(0) = 0. We use V(xT)/kbT≪
+N(xT)2/Z2which holds in the center part of the PDF
+of the tagged particle (since N >>1,V(0) = 0, and
+V(x) is analytic) Peq(xT)∼Cexp/bracketleftBig
+−4N(xT)2
+Z2/bracketrightBig
+whereC
+is a normalization constant. This final approximate re-
+sult is the same as Eq. (21), justifying the tricks used to
+derive our main results.
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+(2008).
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+(2008).
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+89188302 (2002).
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+J. Chem. Phys. 120, 35 (2004).[33] D. Villamaina, A. Puglisi, and A. Vulpiani J. of Statisti-
+cal Mechanics: Theory and Experiment L10001 (2008).
+[34] E. Ben-Naim, and P. L. Krapivsky, Phys. Rev. Lett. 102,
+190602 (2009).
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+(2008).
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+and Experiment P08015 (2009).
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+(New York) (2004).
+[38] H. Risken The Fokker-Planck Equation Springer (Berlin)
+(1996).
+[39] For short times particles do not collide and then the dif -
+fusion of the tagged particle is normal. This short time
+behavior is not of interest here. For particles with uni-
+form density ρ, and free of external forces, short times
+meanst≪1/(Dρ2).
+[40] E. W.Montroll andB. J.Westin Fluctuation Phenomena
+Studies inStatistical Mechanics editedbyE. W. Montroll
+and J. L. Lebowitz, Vol. VII(North Holland, Amsterdam
+1979).
+[41] Note that if the potential is unstable and non binding,
+e.g.V(x) =−mω2x2/2 withmω2>0, the center parti-
+cle at long times can be at very large distance from its
+neighbors, and the method does not work.
+[42] M. Abramovitch, and I. A. Stegun Handbook of Mathe-
+matical Functions Dover (New York) 1972.
+[43] As mentioned we treat the case where all the particles ex -
+perience the external force. In [23] the case were only the
+tagged particle is influenced by the force is considered.
+In the latter case the relaxation to equilibrium follows
+Mittag-Leffler behavior, i.e. slow relaxation described by
+fractional Langevin equation. In contrast we find that
+eigen values of the Fokker Planck operator describe the
+exponential approach to equilibrium, and hence frac-
+tional Langevin equation is not valid for our case. Power
+law relaxation describes also the relative coordinate be-
+tween two tagged particles [23].
+[44] K. Hahn, and J. K¨ arger J. Math. Gen 283061 (1995).
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+(1989).
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+3563 (1999).
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+132 (2000).
\ No newline at end of file
diff --git a/1001.0733.txt b/1001.0733.txt
new file mode 100644
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+arXiv:1001.0733v2  [math.QA]  6 Jan 2010HEISENBERG DOUBLE H /D4B
+/A6/D5AS A BRAIDED COMMUTATIVE
+YETTER–DRINFELD MODULE ALGEBRA OVER THE DRINFELD
+DOUBLE
+A.M.SEMIKHATOV
+ABSTRACT . WestudytheYetter–Drinfeld D /D4B /D5-modulealgebrastructureontheHeisen-
+bergdouble H /D4B
+/A6/D5endowedwitha“heterotic”actionoftheDrinfelddouble D /D4B /D5. This
+actioncanbeinterpretedinthespiritofLu’sdescriptiono fH /D4B
+/A6/D5asatwist of D /D4B /D5. In
+termsofthebraidingofYetter–Drinfeldmodules, H /D4B
+/A6/D5isbraidedcommutative. Bythe
+Brzezi´ nski–Militarutheorem, H /D4B
+/A6/D5 /AZD /D4B /D5isthena Hopfalgebroidover H /D4B
+/A6/D5. For
+BaparticularTaftHopfalgebraata2 pthrootofunity,theconstructionisadaptedtoyield
+Yetter–Drinfeld module algebras over the 2 p3-dimensional quantum group Uqsℓ /D42 /D5. In
+particular, it follows that Mat p
+/D4C /D5is a braided commutative Yetter–Drinfeld Uqsℓ /D42 /D5-
+modulealgebraandMat p
+/D4Uqsℓ /D42 /D5/D5isa HopfalgebroidoverMat p
+/D4C /D5.
+1.INTRODUCTION
+For a Hopfalgebra B, the Heisenberg double H /D4B
+/A6/D5is the smash product B
+/A6/AZBwith
+respecttotheleftregularaction b /G9β /AG /DCβ
+/BE,b /DDβ
+/BDofBonB
+/A6;thecompositionin H /D4B
+/A6/D5
+isgivenby
+(1.1) /D4α /AZa /D5/D4β /AZb /D5 /AGα /D4a
+/BD/G9β /D5 /AZa
+/BEb,α,β /C8B
+/A6,a,b /C8B.
+LetD /D4B /D5be the Drinfeld double of B, with its elements written as µ /CQm, whereµ /C8B
+/A6
+andm /C8B; the composition in D /D4B /D5is /D4µ /CQm /D5/D4ν /CQn /D5 /AGµ /D4m
+/BD/G9ν /G8S
+/A11/D4m
+/BF/D5/D5 /CQm
+/BEn
+(andthecoalgebrastructureisthatof B
+/A6cop/CQB). We definea D /D4B /D5actionon H /D4B
+/A6/D5as/D4µ /CQm /D5⊲ /D4β /AZb /D5 /AGµ
+/BF/D4m
+/BD/G9β /D5S
+/A6
+/A11/D4µ
+/BE/D5 /AZ
+/A0/D4m
+/BEbS /D4m
+/BF/D5/D5 /G8S
+/A6
+/A11/D4µ
+/BD/D5
+/A8
+, (1.2)
+µ /CQm /C8D /D4B /D5,β /AZb /C8H /D4B
+/A6/D5,
+whereb /G8µ /AG /DCµ,b
+/BD/DDb
+/BEistherightregularactionof B
+/A6onB(and /DC, /DDistheevaluation).
+1.1. Theorem. For a Hopf algebra B with bijective antipode, H /D4B
+/A6/D5endowed with ac-
+tion(1.2)and thecoaction
+(1.3) δ:H /D4B
+/A6/D5 /FTD /D4B /D5 /CQH /D4B
+/A6/D5
+β /AZb /G6/FT /D4β
+/BE/CQb
+/BD/D5 /CQ /D4β
+/BD/AZb
+/BE/D5
+isa(left–left)Yetter–Drinfeld D /D4B /D5-modulealgebra.
+By aYetter–Drinfeldmodulealgebrawemean amodulecomodul ealgebrathat isalso
+aYetter–Drinfeldmodule,i.e.,acompatibilitycondition betweentheactionandthecoac-
+tionholdsin theform2 SEMIKHATOV
+(1.4) /D4M
+/BD⊲A /D5/D4/A11 /D5M
+/BE/CQ /D4M
+/BD⊲A /D5/D40 /D5
+/AGM
+/BDA/D4/A11 /D5
+/CQ /D4M
+/BE⊲A/D40 /D5
+/D5,
+where, inourcase, M /C8D /D4B /D5andA /C8H /D4B
+/A6/D5.1
+Werecallfrom[1,2,3]thatforaHopfalgebra H,aleftH-moduleandleft H-comodule
+algebraXissaid tobe braidedcommutative (orH-commutative)if
+(1.5) yx /AG /D4y/D4/A11 /D5⊲x /D5y/D40 /D5,x,y /C8X.
+Also, for any two (left–left) Yetter–Drinfeld H-module algebras XandY, theirbraided
+product X /B3Yisdefined as thetensorproductwiththecomposition
+(1.6) /D4x /B3y /D5/D4v /B3u /D5 /AGx /D4y/D4/A11 /D5⊲v /D5 /B3y/D40 /D5u,x,v /C8X,y,u /C8Y.
+(ThisgivesaYetter–Drinfeldmodulealgebra.)
+1.2. Theorem. H /D4B
+/A6/D5is a braided (D /D4B /D5-)commutative algebra. Moreover, H /D4B
+/A6/D5is
+thebraidedproduct
+H /D4B
+/A6/D5 /AGB
+/A6cop/B3B,
+where B
+/A6copand B are (braided commutative )Yetter–Drinfeld D /D4B /D5modulealgebras by
+restriction,i.e.,withthe D /D4B /D5action/D4µ /CQm /D5⊲β /AGµ
+/BE/D4m /G9β /D5S
+/A6
+/A11/D4µ
+/BD/D5, /D4µ /CQm /D5⊲b /AG /D4m
+/BDbS /D4m
+/BE/D5/D5 /G8S
+/A6
+/A11/D4µ /D5
+andcoaction δ:β /G6/FT /D4β
+/BE/CQ1 /D5 /CQβ
+/BD,δ:b /G6/FT /D4ε /CQb
+/BD/D5 /CQb
+/BE(β /C8B
+/A6, b /C8B).
+1.2.1.As a corollary, the Brzezi´ nski–Militaru theorem [2] then “ provides one with a
+rich source of examples of bialgebroids.” In particular, for any Hopf algebra B with
+bijective antipode, the “quadruple” H /D4B
+/A6/D5 /AZD /D4B /D5, where the smash product is defined
+withrespect toaction(1.2), isa Hopfalgebroidover H /D4B
+/A6/D5.
+1.2.2. A“pseudoadjoint”interpretation of (1.2).TheD /D4B /D5-action(1.2)firstappeared
+in [4]. To borrow a popular term from string theory [5] (where it was also a borrowing
+originally),thisaction may betermed “heterotic”because itis constructedby combining
+left and right D /D4B /D5actions, as we describe in 2.2.2(and the heterotic string famously
+combines“left”and“right”). Orbecause(1.2)“cross-bree ds”regularandadjointactions.
+Trying to quantify how “far” (1.2) is from the adjoint action , we arrive at a useful
+interpretation of our “heterotic” action by extending Lu’s description of the product on
+H /D4B
+/A6/D5as a twist of the product on D /D4B /D5[6]. The two algebraic structures, D /D4B /D5and
+H /D4B
+/A6/D5, are defined on the same vector space B
+/A6/CQB, and the product (1.1) in H /D4B
+/A6/D5,
+temporarilydenoted by Æ, can bewrittenas
+(1.7) M ÆN /AGM
+/BDN
+/BDη /D4M
+/BE,N
+/BE/D5,M,N /C8D /D4B /D5
+for a certain 2-cocycle η:D /D4B /D5 /CQD /D4B /D5 /FTk[6]. In the same vein, the D /D4B /D5action on
+1ForaHopfalgebra Handaleft H-comodule X,wewritethecoaction δ:X /FTH /CQXasδ /D4x /D5 /AGx/D4/A11 /D5
+/CQ
+x/D40 /D5;thenthecomoduleaxiomsare /DCε,x/D4/A11 /D5
+/DDx/D40 /D5
+/AGxandx
+/BD/D4/A11 /D5
+/CQx
+/BE/D4/A11 /D5
+/CQx/D40 /D5
+/AGx/D4/A11 /D5
+/CQx/D40 /D5 /D4/A11 /D5
+/CQx/D40 /D5 /D40 /D5.YETTER–DRINFELD MODULE ALGEBRAS 3
+H /D4B
+/A6/D5in(1.2)can berewritteninthe“pseudoadjoint”form
+(1.8) /D4M,A /D5 /G6/FTM
+/BDÆA Æs /D4M
+/BE/D5,M /C8D /D4B /D5,A /C8H /D4B
+/A6/D5,
+wheres /D4M /D5 /AGη /D4M
+/BD,M
+/BE/D5S /D4M
+/BF/D5. Some “antipode-like” properties of sallow indepen-
+dently verifying that the right-hand side here isan action, as we show in 2.4.3, where
+furtherdetailsaregiven.
+1.2.3.The Heisenberg double H /D4B
+/A6/D5 /AGB
+/A6cop/B3Bcan be regarded as the lowest term,
+H /D4B
+/A6/D5 /AGH2, in a series of Heisenberg n-tuples , orchainsHn—the Yetter–Drinfeld
+D /D4B /D5-modules
+H2n
+/AGB
+/A6cop/B3B /B3B
+/A6cop/B3B /B3... /B3B,
+H2n /A01
+/AGB
+/A6cop/B3B /B3B
+/A6cop/B3B /B3... /B3B /B3B
+/A6cop
+(with2nand 2n /A01factors), withtherelations
+b /D62i /D7β /D62j /A01 /D7 /AG /D4b
+/BD/G9β /D5/D62j /A01 /D7b
+/BE/D62i /D7foralliandj, (1.9)
+(whereB
+/A6cop/FTB
+/A6cop/D62j /A01 /D7andB /FTB /D62i /D7are the morphisms onto the respective fac-
+tors,and weomit /B3forsimplicity),and
+α /D62i /A01 /D7β /D62j /A01 /D7 /AG /D4α
+/BFβS
+/A6
+/A11/D4α
+/BE/D5/D5/D62j /A01 /D7α
+/BD/D62i /A01 /D7,i/greaterorequalslantj (1.10)
+a /D62i /D7b /D62j /D7 /AG /D4a
+/BDbS /D4a
+/BE/D5/D5/D62j /D7a
+/BF/D62i /D7,i/greaterorequalslantj, (1.11)
+wherea,b /C8B,α,β /C8B
+/A6cop.
+1.3.As regards the popular subject of Yetter–Drinfeld modules, we note Refs. [8, 9,
+10, 11, 12, 1]. Heisenberg doubles [13, 14, 15, 6], among vari ous smash products, have
+attracted some attention, notably in relation to Hopf algeb roid constructions [16, 17, 2]
+(the basic observation being that H /D4B
+/A6/D5is a Hopf algebroid over B
+/A6[16]) and also from
+various other standpoints and for different purposes [18, 7 , 19, 20]. (A relatively recent
+paper where Yetter–Drinfeld-like structures are studied i n relation to “smash” products
+is[21].)
+1.4.Theaboveresultsareprovedquitestraightforwardly. Thep roofsaregiveninSec.2;
+there,Bdenotes a Hopf algebra with bijectiveantipode. When we pass to an example in
+Sec. 3,BbecomesaparticularTaft Hopfalgebra.
+1.5.The example worked out in Sec. 3 is that of the 2 p3-dimensional quantum group
+Uqsℓ /D42 /D5at the2pthroot ofunity
+q /AGeiπ
+p,
+(p /AG2,3,...). ThisistheHopfalgebrawithgenerators E,K,andFand therelations
+KEK
+/A11/AGq2E,KFK
+/A11/AGq
+/A12F, /D6E,F /D7 /AGK /A1K
+/A11
+q /A1q
+/A11,4 SEMIKHATOV
+Ep/AGFp/AG0,K2p/AG1,
+and the Hopf algebra structure Δ /D4E /D5 /AGE /CQK /A01 /CQE,Δ /D4K /D5 /AGK /CQK,Δ /D4F /D5 /AGF /CQ1 /A0
+K
+/A11/CQF,ε /D4E /D5 /AGε /D4F /D5 /AG0,ε /D4K /D5 /AG1,S /D4E /D5 /AG /A1EK
+/A11,S /D4K /D5 /AGK
+/A11,S /D4F /D5 /AG /A1KF.
+1.5.1.Uqsℓ /D42 /D5is “almost”the Drinfeld doubleof a 4 p2-dimensional Taft Hopf algebra
+B, more precisely, a “truncation” of the double obtained by ta king a quotient and then
+restrictingto a subalgebra. This close kinshipof Uqsℓ /D42 /D5to a Drinfeld doubleextends to
+the “Heisenberg side”: it turns out that the pair /D4D /D4B /D5,H /D4B
+/A6/D5/D5can also be “truncated”
+to a pair /D4Uqsℓ /D42 /D5,Hqsℓ /D42 /D5/D5of 2p3-dimensional algebras, where Hqsℓ /D42 /D5is a braided
+commutativeYetter–Drinfeld Uqsℓ /D42 /D5-modulealgebra.
+1.5.2.Interestingly,the2 p3-dimensionalbraidedcommutativeYetter–Drinfeld Uqsℓ /D42 /D5-
+modulealgebra Hqsℓ /D42 /D5can bedescribed as
+Hqsℓ /D42 /D5 /ALMatp
+/A0
+C2p
+/D6λ /D7
+/A8
+,C2p
+/D6λ /D7 /AHC /D6λ /D7/DF/D4λ2p/A11 /D5,
+whichaddsamatrixflavortoourexample. Inthematrixlangua ge,therelevantstructures
+aredescribed as follows.
+First, the Uqsℓ /D42 /D5actionon matrices X /AG /D4xij
+/D5withλ-dependent entriesis givenby/D4K⊲X /D5ij
+/AGq2 /D4i /A1j /D5
+/A0
+xij
+/A7/A7
+λ /FTq
+/A11λ
+/A8
+, (1.12)
+and
+E⊲X /AG1
+q /A1q
+/A11
+/A0
+XZ /A1Z /D4K⊲X /D5
+/A8
+, (1.13)
+F⊲X /AG1
+q /A1q
+/A11
+/A0
+DX /A1 /D4K
+/A11⊲X /D5D
+/A8
+, (1.14)
+where
+(1.15) Z /AG
+/A4/A6/A6/A6/A6/A6/A6/A50........... 0
+1 0....... 0
+0 1 0 ...0
+............
+0...... 1 0
+/ACÆÆÆÆÆÆ/AD,D /AG /D4q /A1q
+/A11/D5
+/A4/A6/A6/A6/A6/A6/A6/A50 1........... 0
+0 0q
+/A11/D62 /D7...0
+............
+0......... 0q2 /A1p/D6p /A11 /D7
+0.............. 0
+/ACÆÆÆÆÆÆ/AD
+andweusethestandardnotation/D6n /D7 /AGqn/A1q
+/A1n
+q /A1q
+/A11, /D6n /D7! /AG /D61 /D7/D62 /D7... /D6n /D7,
+/AIm
+n
+/AQ/AG
+/D6m /D7!/D6m /A1n /D7! /D6n /D7!.
+Next, to describe the coaction δ:Matp
+/D4C2p
+/D6λ /D7/D5 /FTUqsℓ /D42 /D5 /CQMatp
+/D4C2p
+/D6λ /D7/D5, we first
+note that C2p
+/D6λ /D7is the algebra of coinvariants, δ:λ /G6/FT1 /CQλ. It therefore remains to
+defineδon “constant” matrices Mat p
+/D4C /D5. But the full matrix algebra Mat p
+/D4C /D5is alge-
+braicallygenerated by theabove ZandD, and wehave
+(1.16) δ:Z /G6/FTK
+/A11/CQZ /A1 /D4q /A1q
+/A11/D5EK
+/A11/CQ1,
+D /G6/FTK
+/A11/CQD /A0 /D4q /A1q
+/A11/D5F /CQ1.YETTER–DRINFELD MODULE ALGEBRAS 5
+Tosummarize,
+1.5.3.Theorem. Withtheabove Uqsℓ /D42 /D5actionandcoaction, Matp
+/D4C2p
+/D6λ /D7/D5andMatp
+/D4C /D5
+arebraidedcommutativeleft–leftYetter–Drinfeld Uqsℓ /D42 /D5-modulealgebras.
+1.5.4.By Theorem 4.1 in [2], as already noted in 1.2.1, we then have examples of
+bialgebroids:
+Matp
+/D4C2p
+/D6λ /D7/D5 /AZUqsℓ /D42 /D5and Mat p
+/D4C /D5 /AZUqsℓ /D42 /D5
+are Hopf algebroids over the respective algebras Mat p
+/D4C2p
+/D6λ /D7/D5and Mat p
+/D4C /D5; further
+detailsaregivenin 3.4.
+1.6. Hopf algebras and logarithmic conformal field theory. An additional source of
+interest in Uqsℓ /D42 /D5is its occurrence in a version of the Kazhdan–Lusztig dualit y [22],
+specifically, as the quantum group dual to a class of logarithmic models of conformal
+field theory[23, 24, 25, 26, 27].
+In the “logarithmic” Kazhdan–Lusztig duality, Uqsℓ /D42 /D5appeared in [23, 24]; subse-
+quently,itgraduallytranspired(withthefinalpicturehav ingemergedfrom[28])thatthat
+was just a continuationof a series of previous(re)discover ies of this quantum group [29,
+30, 31] (also see [32]). The ribbon and (somewhat stretching the definition) factorizable
+structuresof Uqsℓ /D42 /D5were workedout in[23].
+ThatUqsℓ /D42 /D5is Kazhdan–Lusztig-dual to logarithmic models of conforma l field the-
+ory—specifically, Uqsℓ /D42 /D5atq /AGeiπ
+pisdualtothe /D4p,1 /D5logarithmicmodel[33]—means
+severalthings,inparticular,(i)the SL /D42,Z /D5representationonthe Uqsℓ /D42 /D5centercoincides
+with theSL /D42,Z /D5representation generated from the characters of the symmet ry algebra
+ofthelogarithmicmodel[23], theso-called triplet W /D4p /D5algebra [34, 33, 35, 36, 37, 38],
+and(ii)the Uqsℓ /D42 /D5andW /D4p /D5representation categoriesareequivalent[24, 26,27].
+The “Heisenberg counterpart” of Uqsℓ /D42 /D5, its braided commutative Yetter–Drinfeld
+modulealgebra Hqsℓ /D42 /D5,isalsolikelytoplayaroleintheKazhdan–Lusztigcontext [39,
+4], but thisis asubjectoffuturework.
+2.H /D4B
+/A6/D5AS AYETTER–DRINFELD D /D4B /D5-MODULE ALGEBRA
+WebeginwithsimplefactsaboutYetter–Drinfeldmodulealg ebras,concentratingin 2.1
+ontheconstructionofabraidedcommutativeYetter–Drinfe ldmodulealgebraasabraided
+productX /B3Yof two such algebras XandY. In2.2, we then specialize to X /AGB
+/A6cop
+andY /AGB, viewed as D /D4B /D5module algebras under the heterotic action. We verify that
+all the necessary conditions are then satisfied, hence our co nclusion in 2.3. In2.4, we
+givea “pseudoadjoint”interpretationoftheheteroticact ion, and in 2.5considermultiple
+“alternating”braided products.6 SEMIKHATOV
+2.1.The category of Yetter–Drinfeld modules over a Hopf algebra with bijective an-
+tipodeiswellknowntobebraided, withthebraiding cX,Y:X /CQY /FTY /CQXgivenby
+cX,Y:x /CQy /G6/FT /D4x/D4/A11 /D5⊲y /D5 /CQx/D40 /D5.
+Theinverseis c
+/A11
+X,Y:y /CQx /G6/FTx/D40 /D5
+/CQS
+/A11/D4x/D4/A11 /D5
+/D5⊲y.
+Wesay thattwo Yetter–Drinfeldmodules XandYarebraidedsymmetric if
+cY,X
+/AGc
+/A11
+X,Y
+(notethat bothsideshereare maps Y /CQX /FTX /CQY), that is,/D4y/D4/A11 /D5⊲x /D5 /CQy/D40 /D5
+/AGx/D40 /D5
+/CQ
+/A0
+S
+/A11/D4x/D4/A11 /D5
+/D5⊲y
+/A8
+.
+2.1.1.Lemma. LetX andY bebraidedsymmetricYetter–Drinfeldmodules,ea chofwhich
+is a braided commutative Yetter–Drinfeld module algebra. T hen their braided product
+X /B3Y is abraidedcommutativeYetter–Drinfeldmodulealgebra.
+2.1.2. Proof. Beyondthestandardfacts,wehavetoshowthebraidedcommut ativity,i.e.,
+(2.1)
+/A0/D4x /B3y /D5/D4/A11 /D5⊲ /D4v /B3u /D5
+/A8/D4x /B3y /D5/D40 /D5
+/AG /D4x /B3y /D5/D4v /B3u /D5
+forallx,v /C8Xandy,u /C8Y. Forthis,wewritethecondition cX,Y
+/AGc
+/A11
+Y,Xas/D4x/D4/A11 /D5⊲y /D5 /CQx/D40 /D5
+/AGy/D40 /D5
+/CQ
+/A0
+S
+/A11/D4y/D4/A11 /D5
+/D5⊲x
+/A8
+andusethistoestablishan auxiliaryidentity,
+(2.2)
+/A0/D4x/D4/A11 /D5⊲y /D5/D4/A11 /D5⊲x/D40 /D5
+/A8/CQ /D4x/D4/A11 /D5⊲y /D5/D40 /D5
+/AG
+/A0
+y/D40 /D5 /D4/A11 /D5⊲
+/A0
+S
+/A11/D4y/D4/A11 /D5
+/D5⊲x
+/A8 /A8/CQy/D40 /D5 /D40 /D5/AG
+/A0
+y
+/BE/D4/A11 /D5S
+/A11/D4y
+/BD/D4/A11 /D5
+/D5⊲x
+/A8/CQy/D40 /D5/AGx /CQy.
+Theleft-hand sideof(2.1)can then becalculatedas/A0/D4x /B3y /D5/D4/A11 /D5⊲ /D4v /B3u /D5
+/A8/D4x /B3y /D5/D40 /D5/AG
+/A0
+x/D4/A11 /D5y/D4/A11 /D5⊲ /D4v /B3u /D5
+/A8/D4x/D40 /D5
+/B3y/D40 /D5
+/D5/AG
+/A0/D4x
+/BD/D4/A11 /D5y
+/BD/D4/A11 /D5⊲v /D5 /B3 /D4x
+/BE/D4/A11 /D5y
+/BE/D4/A11 /D5⊲u /D5
+/A8/D4x/D40 /D5
+/B3y/D40 /D5
+/D5/AG /D4x
+/BD/D4/A11 /D5y
+/BD/D4/A11 /D5⊲v /D5
+/A0/D4x
+/BE/D4/A11 /D5y
+/BE/D4/A11 /D5⊲u /D5/D4/A11 /D5⊲x/D40 /D5
+/A8/B3 /D4x
+/BE/D4/A11 /D5y
+/BE/D4/A11 /D5⊲u /D5/D40 /D5y/D40 /D5/AG /D4x/D4/A11 /D5y
+/BD/D4/A11 /D5⊲v /D5
+/A0/D4x/D40 /D5 /D4/A11 /D5⊲ /D4y
+/BE/D4/A11 /D5⊲u /D5/D5/D4/A11 /D5⊲x/D40 /D5 /D40 /D5
+/A8/B3 /D4x/D40 /D5 /D4/A11 /D5⊲ /D4y
+/BE/D4/A11 /D5⊲u /D5/D5/D40 /D5y/D40 /D5/AG /D4x/D4/A11 /D5y
+/BD/D4/A11 /D5⊲v /D5x/D40 /D5
+/B3 /D4y
+/BE/D4/A11 /D5⊲u /D5y/D40 /D5,
+justbecause of(2.2)in thelast equality. But theright-han dsideof(2.1) is/D4x /B3y /D5/D4v /B3u /D5 /AGx /D4y/D4/A11 /D5⊲v /D5 /B3y/D40 /D5u/AG /D4x/D4/A11 /D5y/D4/A11 /D5⊲v /D5x/D40 /D5
+/B3 /D4y/D40 /D5 /D4/A11 /D5⊲u /D5y/D40 /D5 /D40 /D5
+becauseXandYarebothbraided commutative. Thetwoexpressionscoincide .YETTER–DRINFELD MODULE ALGEBRAS 7
+2.1.3. Remark. Because the braided symmetry condition is symmetric with re spect to
+the two modules, we also have the braided symmetric Yetter–D rinfeld module algebra
+Y /B3X,withtheproduct/D4y /B3x /D5/D4u /B3v /D5 /AGy /D4x/D4/A11 /D5⊲u /D5 /B3x/D40 /D5v.
+Inadditiontothemultiplicationinside Yandinside X,thisformulaexpressestherelations
+xu /AG /D4x/D4/A11 /D5⊲u /D5x/D40 /D5satisfied in Y /B3Xbyx /C8Xandu /C8Y. Because cX,Y
+/AGc
+/A11
+Y,X, these are
+the same relations ux /AG /D4u/D4/A11 /D5⊲x /D5u/D40 /D5that we have in X /B3Y. Somewhat more formally,
+theisomorphism
+φ:X /B3Y /FTY /B3X
+is given by φ:x /B3y /G6/FT /D4x/D4/A11 /D5⊲y /D5 /B3x/D40 /D5. This is a module map by virtue of the Yetter–
+Drinfeld condition, and it is immediate to verify that δ /D4φ /D4x /B3y /D5/D5 /AG /D4id /CQφ /D5/D4δ /D4x /B3y /D5/D5.
+Thatφisan algebramap followsby calculating
+φ /D4x /B3y /D5φ /D4v /B3u /D5 /AG /D4/D4x/D4/A11 /D5⊲y /D5 /B3x/D40 /D5
+/D5/D4/D4v/D4/A11 /D5⊲u /D5 /B3v/D40 /D5
+/D5/AG /D4x/D4/A11 /D5⊲y /D5/D4x/D40 /D5 /D4/A11 /D5v/D4/A11 /D5⊲u /D5 /B3x/D40 /D5 /D40 /D5v/D40 /D5/AG /D4x
+/BD/D4/A11 /D5⊲y /D5/D4x
+/BE/D4/A11 /D5v/D4/A11 /D5⊲u /D5 /B3x/D40 /D5v/D40 /D5/AGx/D4/A11 /D5⊲
+/A0
+y /D4v/D4/A11 /D5⊲u /D5
+/A8/B3x/D40 /D5v/D40 /D5
+and
+φ /D4/D4x /B3y /D5/D4v /B3u /D5/D5 /AGφ
+/A0
+x /D4y/D4/A11 /D5⊲v /D5 /B3y/D40 /D5u
+/A8/AG /D4x/D4/A11 /D5
+/D4y/D4/A11 /D5⊲v /D5/D4/A11 /D5⊲ /D4y/D40 /D5u /D5/D5 /B3x/D40 /D5
+/D4y/D4/A11 /D5⊲v /D5/D40 /D5
+/check/AGx/D4/A11 /D5⊲ /D4y/D40 /D5u /D5/D40 /D5
+/B3x/D40 /D5
+/A0
+S
+/A11/D4y/D40 /D5 /D4/A11 /D5u/D4/A11 /D5
+/D5⊲ /D4y/D4/A11 /D5⊲v /D5
+/A8/AGx/D4/A11 /D5⊲ /D4y/D40 /D5u/D40 /D5
+/D5 /B3x/D40 /D5
+/A0
+S
+/A11/D4y
+/BE/D4/A11 /D5u/D4/A11 /D5
+/D5y
+/BD/D4/A11 /D5⊲v
+/A8/AGx/D4/A11 /D5⊲ /D4yu/D40 /D5
+/D5 /B3x/D40 /D5
+/A0
+S
+/A11/D4u/D4/A11 /D5
+/D5⊲v
+/A8
+/check/AGx/D4/A11 /D5⊲
+/A0
+y /D4v/D4/A11 /D5⊲u /D5
+/A8/B3x/D40 /D5v/D40 /D5,
+wherethebraided symmetryconditionwas usedin each ofthe/check/AGequalities.
+2.2.Weintendto use 2.1.1in thecase where X /AGB
+/A6copandY /AGB. Thisrequires some
+preparations.
+2.2.1.Lemma. Fora HopfalgebraBwith bijectiveantipode,theformulas/D4µ /CQm /D5⊲β /AGµ
+/BE/D4m /G9β /D5S
+/A6
+/A11/D4µ
+/BD/D5, /D4µ /CQm /D5⊲b /AG /D4m
+/BDbS /D4m
+/BE/D5/D5 /G8S
+/A6
+/A11/D4µ /D5
+makeB
+/A6copand Bintoleft D /D4B /D5-modulealgebras.
+2.2.2.This is known, e.g., from [17], where both these actions are d iscussed and refer-
+ences to the previous works are given. The D /D4B /D5action on B
+/A6is obtained by restricting8 SEMIKHATOV
+theleftregularactionof D /D4B /D5onD /D4B /D5
+/A6/ALB /CQB
+/A6[6],/D4µ /CQm /D5 /G9 /D4b /CQβ /D5 /AG /D4µ
+/BE/G9b /D5 /CQµ
+/BF/D4m /G9β /D5S
+/A6
+/A11/D4µ
+/BD/D5,
+to1 /CQB
+/A6. Similarly,the D /D4B /D5actionon Bisobtained[40]byrestrictingthe rightregular
+actionofD /D4B /D5onD /D4B /D5
+/A6/ALB /CQB
+/A6toB /CQεandusingtheantipodetoconvertitintoaleft
+action. Therightregularaction of D /D4B /D5onD /D4B /D5
+/A6is[6, 17]/D4b /CQβ /D5 /G8 /D4µ /CQm /D5 /AGS
+/A11/D4m
+/BF/D5/D4b /G8µ /D5m
+/BD/CQ /D4β /G8m
+/BE/D5,
+whereβ /G8m /AG /DCβ
+/BD,m /DDβ
+/BEis the right regular action of BonB
+/A6. Restricting to Band
+replacing µ /CQmwith /D4S /D4m
+/BF/D5 /G9S
+/A6
+/A11/D4µ /D5 /G8m
+/BD/D5 /CQS /D4m
+/BE/D5then givesthesecond formulain
+thelemma.
+Thefollowingstatementisobvious.
+2.2.3.Lemma. Withtherespectivecoactions
+δ:β /G6/FT /D4β
+/BE/CQ1 /D5 /CQβ
+/BD,δ:b /G6/FT /D4ε /CQb
+/BD/D5 /CQb
+/BE,
+B
+/A6copand Bare D /D4B /D5-comodulealgebras.
+2.2.4. Lemma. With the action and coaction in 2.2.1and2.2.3, both B
+/A6copand B are
+Yetter–Drinfeldmodulealgebras.
+It only remains to verify the Yetter–Drinfeld condition in e ach case. For B
+/A6cop, we
+calculatetheleft-hand sideof(1.4)with M /AGµ /CQmas/D4/D4µ
+/BE/CQm
+/BD/D5⊲β /D5/D4/A11 /D5
+/D4µ
+/BD/CQm
+/BE/D5 /CQ /D4/D4µ
+/BE/CQm
+/BD/D5⊲β /D5/D40 /D5/AG
+/A0/A0
+µ
+/BF/D4m
+/BD/G9β /D5S
+/A6
+/A11/D4µ
+/BE/D5
+/A8/BEµ
+/BD/CQm
+/BE
+/A8/CQ /D4µ
+/BF/D4m
+/BD/G9β /D5S
+/A6
+/A11/D4µ
+/BE/D5/D5
+/BD/AG
+/A0
+µ
+/D45 /D5/D4m
+/BD/G9β
+/BE/D5S
+/A6
+/A11/D4µ
+/D42 /D5/D5µ
+/D41 /D5/CQm
+/BE
+/A8/CQµ
+/D44 /D5β
+/BDS
+/A6
+/A11/D4µ
+/D43 /D5/D5/AG
+/A0
+µ
+/BF/D4m
+/BD/G9β
+/BE/D5 /CQm
+/BE
+/A8/CQµ
+/BEβ
+/BDS
+/A6
+/A11/D4µ
+/BD/D5,
+buttheright-handsideof(1.4)is/D4µ /CQm /D5
+/BDβ/D4/A11 /D5
+/CQ
+/A0/D4µ /CQm /D5
+/BE⊲β/D40 /D5
+/A8/AG /D4µ
+/BF/CQm
+/BD/D5/D4β
+/BE/CQ1 /D5 /CQ
+/A0
+µ
+/BE/D4m
+/BE/G9β
+/BD/D5S
+/A6
+/A11/D4µ
+/BD/D5
+/A8/AG /D4µ
+/BF/CQm
+/BD/D5/D4/D4β
+/BE/G8m
+/BE/D5 /CQ1 /D5 /CQµ
+/BEβ
+/BDS
+/A6
+/A11/D4µ
+/BD/D5
+(because β
+/BE/CQ /D4m /G9β
+/BD/D5 /AG /D4β
+/BE/G8m /D5 /CQβ
+/BD)/AG
+/A0
+µ
+/BF/D4m
+/D41 /D5/G9β
+/BE/G8m
+/D44 /D5S
+/A11/D4m
+/D43 /D5/D5/D5 /CQm
+/D42 /D5
+/A8/CQµ
+/BEβ
+/BDS
+/A6
+/A11/D4µ
+/BD/D5,
+whichisthesame. For B,similarly,theleft-handsideof(1.4)is(assumingthepre cedence
+ab /G8β /AG /D4ab /D5 /G8β, and soon)/D4/D4µ
+/BE/CQm
+/BD/D5⊲b /D5/D4/A11 /D5
+/D4µ
+/BD/CQm
+/BE/D5 /CQ /D4/D4µ
+/BE/CQm
+/BD/D5⊲b /D5/D40 /D5/AG
+/A0
+ε /CQ
+/A0/D4m
+/BDbS /D4m
+/BE/D5/D5 /G8S
+/A6
+/A11/D4µ
+/BE/D5
+/A8/BD
+/A8/D4µ
+/BD/CQm
+/BF/D5 /CQ
+/A0/D4m
+/BDbS /D4m
+/BE/D5/D5 /G8S
+/A6
+/A11/D4µ
+/BE/D5
+/A8/BEYETTER–DRINFELD MODULE ALGEBRAS 9/AG
+/A0
+ε /CQ
+/A0/D4m
+/BDbS /D4m
+/BE/D5/D5
+/BD/G8S
+/A6
+/A11/D4µ
+/BE/D5
+/A8 /A8/D4µ
+/BD/CQm
+/BF/D5 /CQ /D4m
+/BDbS /D4m
+/BE/D5/D5
+/BE
+(becauseΔ /D4a /G8µ /D5 /AG /D4a
+/BD/G8µ /D5 /CQa
+/BE)/AG
+/A0
+µ
+/BE/CQ
+/A0
+S
+/A6
+/A11/D4µ
+/BD/D5 /G9 /D4m
+/BDbS /D4m
+/BE/D5/D5
+/BD
+/A8
+m
+/BF
+/A8/CQ /D4m
+/BDbS /D4m
+/BE/D5/D5
+/BE
+(usingthe D /D4B /D5-identity
+/A0
+ε /CQ /D4b /G8S
+/A6
+/A11/D4µ
+/BE/D5/D5
+/A8/D4µ
+/BD/CQ1 /D5 /AGµ
+/BE/CQ /D4S
+/A6
+/A11/D4µ
+/BD/D5 /G9b /D5)/AG /DCS
+/A6
+/A11/D4µ
+/BD/D5,m
+/D42 /D5b
+/BES /D4m
+/D45 /D5/D5/DD
+/A0
+µ
+/BE/CQ /D4m
+/D41 /D5b
+/BDS /D4m
+/D46 /D5/D5/D5m
+/D47 /D5
+/A8/CQm
+/D43 /D5b
+/BFS /D4m
+/D44 /D5/D5/AG
+/A0
+µ
+/BE/CQm
+/D41 /D5b
+/BD
+/A8/CQ
+/A0
+m
+/D42 /D5b
+/BES /D4m
+/D43 /D5/D5 /G8S
+/A6
+/A11/D4µ
+/BD/D5
+/A8/AG
+/A0/D4µ
+/BE/CQm
+/BD/D5/D4ε /CQb
+/BD/D5
+/A8/CQ
+/A0/D4µ
+/BD/CQm
+/BE/D5⊲b
+/BE
+/A8/AG
+/A0/D4µ /CQm /D5
+/BDb/D4/A11 /D5
+/A8/CQ
+/A0/D4µ /CQm /D5
+/BE⊲b/D40 /D5
+/A8
+,
+whichistheright-handside.
+2.2.5.Lemma. B
+/A6copandB arebraidedcommutative D /D4B /D5-modulealgebras.
+This is entirely obvious once we note that when the D /D4B /D5action on B
+/A6copin2.2.1is
+restrictedtotheactionof B
+/A6cop/CQ1,itbecomestheadjointaction;thesameistrueforthe
+D /D4B /D5actionon Brestrictedtotheactionof ε /CQB;therefore,forexample, /D4a/D4/A11 /D5⊲b /D5a/D40 /D5
+/AG/D4a
+/BD⊲b /D5a
+/BE/AG /D4a
+/BDbS /D4a
+/BE/D5/D5a
+/BF/AGab.
+2.2.6.Lemma. B
+/A6copandB arebraidedsymmetric.
+Wemustshowthat cB
+/A6cop,B
+/AGc
+/A11
+B,B
+/A6cop, i.e.,/D4b/D4/A11 /D5⊲β /D5 /CQb/D40 /D5
+/AGβ/D40 /D5
+/CQ /D4S
+/A11
+D
+/D4β/D4/A11 /D5
+/D5⊲b /D5.
+The antipode here is that of D /D4B /D5, and therefore the right-hand side evaluates as β
+/BD/CQ/D4S
+/A6/D4β
+/BE/D5⊲b /D5 /AGβ
+/BD/CQ /D4b /G8S
+/A6
+/A11/D4S
+/A6/D4β
+/BE/D5/D5/D5 /AGβ
+/BD/CQ /D4b /G8β
+/BE/D5, which is immediately seen to
+coincidewiththeleft-handside.
+2.3.Itnowfollowsfrom 2.1.1thatB
+/A6cop/B3Bisabraided commutativeYetter–Drinfeld
+D /D4B /D5-module algebra. But the product in B
+/A6cop/B3Bactually evaluates as the product
+inH /D4B
+/A6/D5:/D4α /B3a /D5/D4β /B3b /D5 /AGα /D4a/D4/A11 /D5⊲β /D5 /B3a/D40 /D5b /AGα /D4/D4ε /CQa
+/BD/D5⊲β /D5 /B3a
+/BEb /AGα /D4a
+/BD/G9β /D5 /B3a
+/BEb.
+We therefore conclude that with theD /D4B /D5action and coaction in (1.2)and(1.3),H /D4B
+/A6/D5
+isa braidedcommutativeYetter–Drinfeld D /D4B /D5-modulealgebra .
+2.4. A “pseudo-adjoint” interpretation of the D /D4B /D5action on H /D4B
+/A6/D5.The action de-
+fined in(1.2)can bewrittenin the“pseudo-adjoint”form
+(2.3) /D4µ /CQm /D5⊲ /D4α /AZa /D5 /AG /D4µ
+/BE/AZm
+/BD/D5 Æ /D4α /AZa /D5 Æs /D4µ
+/BD/CQm
+/BE/D5,
+where Ætemporarilydenotesthecompositionin H /D4B
+/A6/D5, and
+s /D4µ /CQm /D5 /AG /D4ε /AZS /D4m /D5/D5 Æ /D4S
+/A6
+/A11/D4µ /D5 /AZ1 /D510 SEMIKHATOV/AG /D4S /D4m
+/BE/D5 /G9S
+/A6
+/A11/D4µ /D5/D5 /AZS /D4m
+/BD/D5.
+Theright-handsideof(2.3)is tobecompared withtheadjoin taction of D /D4B /D5onitself,/D4µ /CQm /D5◮ /D4ν /CQn /D5 /AG /D4µ
+/BE/CQm
+/BD/D5 /D4ν /CQn /D5SD /D4B /D5
+/D4µ
+/BD/CQm
+/BE/D5,
+whereSD /D4B /D5
+/D4µ /CQm /D5 /AG /D4ε /CQS /D4m /D5/D5/D4S
+/A6
+/A11/D4µ /D5 /CQ1 /D5 /AG /D4S /D4m
+/BF/D5 /G9S
+/A6
+/A11/D4µ /D5 /G8m
+/BD/D5 /CQS /D4m
+/BE/D5.
+2.4.1.To show(2.3),wecalculateitsright-handsideas/D4µ
+/BE/AZm
+/BD/D5 Æ /D4α /AZa /D5 Æ
+/A0/D4S /D4m
+/BF/D5 /G9S
+/A6
+/A11/D4µ
+/BD/D5/D5 /AZS /D4m
+/BE/D5
+/A8/AG
+/A0
+µ
+/BE/D4m
+/D41 /D5/G9α /D5 /AZm
+/D42 /D5a
+/A8Æ
+/A0/D4S /D4m
+/D44 /D5/D5 /G9S
+/A6
+/A11/D4µ
+/BD/D5/D5 /AZS /D4m
+/D43 /D5/D5
+/A8/AGµ
+/BE/D4m
+/D41 /D5/G9α /D5
+/A0
+m
+/D42 /D5a
+/BDS /D4m
+/D45 /D5/D5 /G9S
+/A6
+/A11/D4µ
+/BD/D5
+/A8/AZm
+/D43 /D5a
+/BES /D4m
+/D44 /D5/D5/AGµ
+/BE/D4m
+/BD/G9α /D5
+/A0/D4m
+/BEaS /D4m
+/BF/D5/D5
+/BD/G9S
+/A6
+/A11/D4µ
+/BD/D5
+/A8/AZ /D4m
+/BEaS /D4m
+/BF/D5/D5
+/BE/AGµ
+/BF/D4m
+/BD/G9α /D5S
+/A6
+/A11/D4µ
+/BE/D5 /AZ /D4m
+/BEaS /D4m
+/BF/D5 /G8S
+/A6
+/A11/D4µ
+/BD/D5/D5
+(because /D4a
+/BD/G9µ /D5 /CQa
+/BE/AGµ
+/BD/CQ /D4a /G8µ
+/BE/D5).
+2.4.2.It maybeinterestingtoseeinmoredetail whythemock-adjointactionin(2.3)is
+aD /D4B /D5action. Werecall from[6]thatEq.(1.7)holdsfortheproduc tonH /D4B
+/A6/D5,withthe
+D /D4B /D5product in the right-hand side and with the 2-cocycle η:D /D4B /D5 /CQD /D4B /D5 /FTkgiven
+by
+η /D4µ /CQm,ν /CQn /D5 /AG /DCµ,1 /DD /DCν,m /DD /DCε,n /DD.
+Ofcourse, /D4M,A /D5 /G6/FTM ÆAisnotaleft action and /D4M,A /D5 /G6/FTA Æs /D4M /D5isnot arightaction
+ofD /D4B /D5; instead, we have the associativity of the Æproduct,M Æ /D4A ÆN /D5 /AG /D4M ÆA /D5 ÆN
+for allM,A,N /C8B
+/A6/CQB. But the identity η /D4M
+/BD,N
+/BD/D5s /D4N
+/BE/D5 Æs /D4M
+/BE/D5 /AGs /D4MN /D5satisfied by
+Lu’s cocycle ηand the “pseudo-antipode” sensures that (2.3) (i.e., (1.8)) is nevertheless
+aD /D4B /D5action.
+Fromthisperspective,furthermore,the D /D4B /D5modulealgebrapropertyof H /D4B
+/A6/D5isen-
+sured by another“antipode-like”property of s,s /D4M
+/BD/D5 ÆM
+/BE/AGε /D4M /D51,M /C8D /D4B /D5. And the
+Yetter–Drinfeldconditioneasilyfollowsforthe“pseudo- adjoint”actionbecause δs /D4M /D5 /AG
+S /D4M
+/BE/D5 /CQs /D4M
+/BD/D5(whereδis thesameas ΔD /D4B /D5and theright-handsideis viewed as an ele-
+ment ofD /D4B /D5 /CQH /D4B
+/A6/D5) and, of course, because H /D4B
+/A6/D5is aD /D4B /D5comodulealgebra [6].
+We somewhat formalize this simple argument as the following theorem (all of whose
+conditionsholdforLu’scocycle).
+2.4.3.Theorem. ForaHopfalgebra /D4H,Δ,S,ε /D5withbijectiveantipode,let ηbeanormal
+right2-cocycle[6], i.e.,a bilinearmapH /CQH /FTk suchthat
+η /D4f
+/BDg
+/BD,h /D5η /D4f
+/BE,g
+/BE/D5 /AGη /D4f,g
+/BDh
+/BD/D5η /D4g
+/BE,h
+/BE/D5,η /D41,h /D5 /AGη /D4h,1 /D5 /AGε /D4h /D5
+forall f,g,h /C8H,and letHÆ
+/AG /D4H, Æ /D5denotetheassociativealgebrawith theproduct
+g Æh /AGg
+/BDh
+/BDη /D4g
+/BE,h
+/BE/D5.YETTER–DRINFELD MODULE ALGEBRAS 11
+Lets:H /FTH begivenby
+(2.4) s /D4h /D5 /AGη /D4h
+/BD,h
+/BE/D5S /D4h
+/BF/D5.
+Iftheconditions
+η /D4s /D4h
+/BD/D5,h
+/BE/D5 /AGε /D4h /D5, (2.5)
+η /D4h
+/BD,s /D4h
+/BE/D5/D5 /AGε /D4h /D5, (2.6)
+η /D4g
+/BD,h
+/BD/D5η /D4s /D4h
+/BE/D5,s /D4g
+/BE/D5/D5 /AGη /D4g
+/BDh
+/BD,g
+/BEh
+/BE/D5 (2.7)
+hold for all g ,h /C8H, then HÆis a left–left Yetter–Drinfeld H-module algebra under the
+leftH-action
+(2.8) g⊲h /AGg
+/BDÆh Æs /D4g
+/BE/D5
+and left coaction δ /AGΔ, viewed as a map HÆ
+/FTH /CQHÆ. Moreover, HÆis braided com-
+mutative.
+Conditions(2.5)–(2.7)can bereformulated as
+s /D4h
+/BD/D5 Æh
+/BE/AGε /D4h /D51, (2.9)
+h
+/BDÆs /D4h
+/BE/D5 /AGε /D4h /D51, (2.10)
+η /D4g
+/BD,h
+/BD/D5s /D4h
+/BE/D5 Æs /D4g
+/BE/D5 /AGs /D4gh /D5. (2.11)
+Also,itfollowsfrom (2.4)that Δ /D4s /D4h /D5/D5 /AGS /D4h
+/BE/D5 /CQs /D4h
+/BD/D5.
+That (2.8) is an Haction immediately follows from (2.11). The module algebra prop-
+erty follows from (2.9). The left coaction δmakesHÆinto a comodule algebra for any
+rightcocycle η[6]. TheYetter–Drinfeldaxiomisthenverifiedasstraightf orwardlyasfor
+thetrueadjointaction:/D4h
+/BD⊲g /D5/D4/A11 /D5h
+/BE/CQ /D4h
+/BD⊲g /D5/D40 /D5
+/AG /D4h
+/BDÆg Æs /D4h
+/BE/D5/D5
+/BDh
+/BF/CQ /D4h
+/BDÆg Æs /D4h
+/BE/D5/D5
+/BE/AGh
+/D41 /D5g
+/BDS /D4h
+/D44 /D5/D5h
+/D45 /D5/CQh
+/D42 /D5Æg
+/BEÆs /D4h
+/D43 /D5/D5 /AGh
+/BDg/D4/A11 /D5
+/CQ /D4h
+/BE⊲g/D40 /D5
+/D5.
+Thebraided commutativityis alsoimmediate:/D4h/D4/A11 /D5⊲g /D5 Æh/D40 /D5
+/AGh
+/BD/D4/A11 /D5
+Æg Æs /D4h
+/BE/D4/A11 /D5
+/D5 Æh/D40 /D5
+/AGh
+/BDÆg Æs /D4h
+/BE/D5 Æh
+/BF/AGh
+/BDÆg Æ1ε /D4h
+/BE/D5 /AGh Æg.
+2.5. Multiple braided products. Further examples of Yetter–Drinfeld modulealgebras
+areproducedbyextendingtheHeisenbergdouble H /D4B
+/A6/D5tomultiple“alternating”braided
+products. We first returnto thesettingof 2.1.
+2.5.1.Multiple braided products X1
+/B3... /B3XNof Yetter–Drinfeld H-module algebras
+Xiarethecorrespondingtensorproductswiththediagonalact ionandcodiagonalcoaction
+ofH, and withtherelations12 SEMIKHATOV
+(2.12) x /D6i /D7 /B3y /D6j /D7 /AG /D4x/D4/A11 /D5⊲y /D5/D6j /D7 /B3x/D40 /D5
+/D6i /D7,i /EHj,
+wherez /D6i /D7 /C8Xi. (The inverse relation is x /D6i /D7 /B3y /D6j /D7 /AGy/D40 /D5
+/D6j /D7 /B3 /D4S
+/A11/D4y/D4/A11 /D5
+/D5⊲x /D5/D6i /D7,i /EGj.)
+Itreadily followsfromtheYetter–Drinfeldmodulealgebra axiomsforeach ofthe Xithat
+X1
+/B3... /B3XNisan associativealgebraand, infact, aYetter–Drinfeld H-modulealgebra.
+Inparticular, itfollowsthat/D4x1
+/D6i1
+/D7 /B3... /B3xm
+/D6im
+/D7/D5 /B3 /D4y1
+/D6j1
+/D7 /B3... /B3yn
+/D6jn
+/D7/D5/AG
+/A0/D4x1/D4/A11 /D5...xm/D4/A11 /D5
+/D5⊲ /D4y1
+/D6j1
+/D7 /B3... /B3yn
+/D6jn
+/D7/D5
+/A8/B3
+/A0
+x1/D40 /D5
+/D6i1
+/D7 /B3... /B3xm/D40 /D5
+/D6im
+/D7
+/A8
+whenever ia
+/EHjbforalla /AG1,...,mandb /AG1,...,n.
+2.5.2. “Alternating”braidedproducts. Next,letXandYbebraidedsymmetricYetter–
+DrinfeldH-modulealgebras, andconsiderthe“alternating”products
+X /B3Y /B3X /B3Y /B3...,
+with an arbitrary number of factors (or a similar product wit h the leftmost Y, or actually
+theirinductivelimitswithrespect totheobviousembeddin gs). We let X /D6i /D7denotethe ith
+copyofX,andsimilarlywith Y /D6j /D7. Forarbitrary x /D6i /D7 /C8X /D6i /D7andy /D6j /D7 /C8Y /D6j /D7,wethenhave
+relations(2.12), i.e.,
+(2.13) x /D62i /A01 /D7 /B3y /D62j /D7 /AG /D4x/D4/A11 /D5⊲y /D5/D62j /D7 /B3x/D40 /D5
+/D62i /A01 /D7,
+for alli/greaterorequalslantj, but by the braided symmetry condition, relations (2.13)—r eplicas of the
+relationsbetweenelementsof Xandelementsof YinX /B3Y—holdforalliand j . Inthe
+multipleproducts,inaddition,wealso havetherelations
+(2.14)x /D62i /A01 /D7 /B3v /D62j /A01 /D7 /AG /D4x/D4/A11 /D5⊲v /D5/D62j /A01 /D7 /B3x/D40 /D5
+/D62i /A01 /D7,x,v /C8X,
+y /D62i /D7 /B3u /D62j /D7 /AG /D4y/D4/A11 /D5⊲u /D5/D62j /D7 /B3y/D40 /D5
+/D62i /D7,y,u /C8Y,i /EHj
+(whichalsohold for i /AGjifXandYare braidedcommutative.)
+2.5.3. Heisenberg n-tuples /DFchains.Generalizing H /D4B
+/A6/D5 /ALB
+/A6cop/B3B /ALB /B3B
+/A6cop, we
+have“Heisenberg n-tuples /DFchains”—thealternatingproducts
+H2n
+/AGB
+/A6cop/B3B /B3B
+/A6cop/B3B /B3... /B3B,
+H2n /A01
+/AGB
+/A6cop/B3B /B3B
+/A6cop/B3B /B3... /B3B /B3B
+/A6cop.
+Aswesaw in 2.5.2, thefollowingrelationsholdhere:
+b /D62i /D7β /D62j /A01 /D7 /AG /D4b
+/BD/G9β /D5/D62j /A01 /D7b
+/BE/D62i /D7,b /C8B,β /C8B
+/A6cop,foralliandj
+(whereB
+/A6cop/FTB
+/A6cop/D62j /A01 /D7andB /FTB /D62i /D7are the morphisms onto the respective fac-
+tors,and weomitthe /B3symbolforbrevity),and
+α /D62i /A01 /D7β /D62j /A01 /D7 /AG /D4α
+/BFβS
+/A6
+/A11/D4α
+/BE/D5/D5/D62j /A01 /D7α
+/BD/D62i /A01 /D7,α,β /C8B
+/A6cop,i/greaterorequalslantj,YETTER–DRINFELD MODULE ALGEBRAS 13
+a /D62i /D7b /D62j /D7 /AG /D4a
+/BDbS /D4a
+/BE/D5/D5/D62j /D7a
+/BF/D62i /D7, a,b /C8B,i/greaterorequalslantj.
+TheD /D4B /D5actionisdiagonaland thecoactionis codiagonal,forexamp le,
+δ /D4α /B3a /B3β /B3b /D5 /AG
+/A0/D4α
+/BE/CQ1 /D5/D4ε /CQa
+/BD/D5/D4β
+/BE/CQ1 /D5/D4ε /CQb
+/BD/D5
+/A8/CQ
+/A0
+α
+/BD/B3a
+/BE/B3β
+/BD/B3b
+/BE
+/A8/AG
+/A0/D4α
+/BE/CQa
+/BD/D5/D4β
+/BE/CQb
+/BD/D5
+/A8/CQ
+/A0
+α
+/BD/B3a
+/BE/B3β
+/BD/B3b
+/BE
+/A8
+.
+The chains with the leftmost Bfactor are defined entirely similarly. The obvious em-
+beddings allow defining (one-sided or two-sided) inductive limits of alternating chains.
+AllthechainsareYetter–Drinfeldmodulealgebras,butnon ewith/greaterorequalslant3factorsarebraided
+commutativeingeneral.
+3.YETTER–DRINFELD MODULE ALGEBRAS AND THE ASSOCIATED HOPF
+ALGEBROID FOR Uqsℓ /D42 /D5
+In this section, we construct Yetter–Drinfeld module algeb ras forUqsℓ /D42 /D5at the 2pth
+root of unity for an integer p/greaterorequalslant2 (see1.5), and also consider the Hopf algebroid associ-
+ated with a braided commutativeYetter–Drinfeld module alg ebra in accordance with the
+constructionin[2].
+Uqsℓ /D42 /D5can be obtained as a subquotient of the Drinfeld double of a Ta ft Hopf alge-
+braB[23, 24] (a trick also used, e.g., in [43] for a closely relate d quantum group). On
+the“Heisenberg side,” H /D4B
+/A6/D5similarlyyields Hqsℓ /D42 /D5,a 2p3-dimensionalbraided com-
+mutativeYetter–Drinfeld Uqsℓ /D42 /D5-modulealgebra. This is worked out in 3.1–3.2below;
+in3.3, dropping the coinvariants in Hqsℓ /D42 /D5, we obtain the algebra of p /A2pmatrices,
+which is also a braided commutative Yetter–Drinfeld Uqsℓ /D42 /D5-module algebra. In 3.4,
+we use the Brzezi´ nski–Militaru theorem to construct the co rresponding Hopf algebroid.
+Multiplealternatingbraidedproductsare consideredin 3.5.
+3.1.D /D4B /D5andH /D4B
+/A6/D5forthe TaftHopfalgebra B.
+3.1.1. TheTaft Hopfalgebra B.Let
+B /AGSpan /D4Emkn/D5,0/lessorequalslantm/lessorequalslantp /A11,0/lessorequalslantn/lessorequalslant4p /A11,
+bethe4p2-dimensionalHopfalgebragenerated by Eandkwiththerelations
+kE /AGqEk,Ep/AG0,k4p/AG1, (3.1)
+andwiththecomultiplication,counit,and antipodegivenb y
+Δ /D4E /D5 /AG1 /CQE /A0E /CQk2,Δ /D4k /D5 /AGk /CQk,ε /D4E /D5 /AG0,ε /D4k /D5 /AG1,
+S /D4E /D5 /AG /A1Ek
+/A12,S /D4k /D5 /AGk
+/A11.(3.2)
+Wedefine F,κ /C8B
+/A6by/DCF,Emkn/DD /AGδm,1q
+/A1n
+q /A1q
+/A11, /DCκ,Emkn/DD /AGδm,0q
+/A1n /DF2.14 SEMIKHATOV
+Then[23]
+B
+/A6/AGSpan /D4Faκb/D5,0/lessorequalslanta/lessorequalslantp /A11,0/lessorequalslantb/lessorequalslant4p /A11.
+3.1.2. The Drinfeld double D /D4B /D5.Direct calculation shows [23] that the Drinfeld dou-
+bleD /D4B /D5istheHopfalgebragenerated by E,F,k, andκwiththerelationsgivenby
+i) relations(3.1)in B,
+ii) therelations κF /AGqFκ,Fp/AG0,andκ4p/AG1 inB
+/A6, and
+iii) thecross-relations
+kκ /AGκk,kFk
+/A11/AGq
+/A11F,κEκ
+/A11/AGq
+/A11E, /D6E,F /D7 /AGk2/A1κ2
+q /A1q
+/A11.
+TheHopf-algebrastructure /D4ΔD,εD,SD
+/D5ofD /D4B /D5isgivenby(3.2)and
+ΔD
+/D4F /D5 /AGκ2/CQF /A0F /CQ1,ΔD
+/D4κ /D5 /AGκ /CQκ,εD
+/D4F /D5 /AG0,εD
+/D4κ /D5 /AG1,
+SD
+/D4F /D5 /AG /A1κ
+/A12F,SD
+/D4κ /D5 /AGκ
+/A11.
+3.1.3. TheHeisenberg double H /D4B
+/A6/D5.Fortheabove B,H /D4B
+/A6/D5is spannedby
+(3.3) Faκb/AZEckd,a,c /AG0,...,p /A11,b,d /C8Z /DF/D44pZ /D5,
+whereκ4p/AG1,k4p/AG1,Fp/AG0,andEp/AG0. Aconvenientbasisin H /D4B
+/A6/D5canbechosen
+as /D4κ,z,λ, /BU/D5,whereκisunderstoodas κ /AZ1and
+z /AG /A1/D4q /A1q
+/A11/D5ε /AZEk
+/A12,
+λ /AGκ /AZk,/BU /AG /D4q /A1q
+/A11/D5F /AZ1.
+Therelationsin H /D4B
+/A6/D5thenbecome κz /AGq
+/A11zκ,κλ /AGq1
+2λκ,κ /BU /AGq /BUκ,κ4p/AG1,and
+(3.4)λ4p/AG1,zp/AG0, /BUp/AG0,
+λz /AGzλ,λ /BU /AG /BUλ,/BUz /AG /D4q /A1q
+/A11/D51 /A0q
+/A12z /BU.
+ThentheD /D4B /D5actionon H /D4B
+/A6/D5in(1.2)becomes κ⊲κn/AGκn,κ⊲ /BUn/AGqn/BUn,κ⊲λn/AG
+qn
+2λn,κ⊲zn/AGq
+/A1nzn,and
+(3.5)E⊲κ /AG0, k⊲κn/AGq
+/A1n
+2κ,F⊲κn/AG /A1qn
+2/D6n
+2
+/D7/BUκn,
+E⊲λn/AGq
+/A1n
+2/D6n
+2
+/D7λnz,k⊲λn/AGq
+/A1n
+2λ,F⊲λn/AG /A1qn
+2/D6n
+2
+/D7λn/BU,
+E⊲zn/AG /A1qn/D6n /D7zn /A01,k⊲zn/AGqnzn,F⊲zn/AG /D6n /D7q1 /A1nzn /A11,
+E⊲ /BUn/AGq1 /A1n/D6n /D7/BUn /A11,k⊲ /BUn/AGq
+/A1n/BUn,F⊲ /BUn/AG /A1qn/D6n /D7/BUn /A01.
+3.2. The /D4Uqsℓ /D42 /D5,Hqsℓ /D42 /D5/D5pair.
+3.2.1. From D /D4B /D5toUqsℓ /D42 /D5.The “truncation” whereby D /D4B /D5yieldsUqsℓ /D42 /D5consistsYETTER–DRINFELD MODULE ALGEBRAS 15
+oftwo steps[23]: first, takingthequotient
+D /D4B /D5 /AGD /D4B /D5/DF/D4κk /A11 /D5 (3.6)
+bytheHopfidealgeneratedbythecentralelement κ /CQk /A1ε /CQ1and,second,identifying
+Uqsℓ /D42 /D5as the subalgebra in D /D4B /D5spanned by FℓEmk2nwithℓ,m /AG0,...,p /A11 and
+n /AG0,...,2p /A11. It then follows that Uqsℓ /D42 /D5is a Hopf algebra—the one described
+in1.5, whereK /AGk2.
+Thecategoryoffinite-dimensional Uqsℓ /D42 /D5representationsisnot braided[28].
+3.2.2. From H /D4B
+/A6/D5toHqsℓ /D42 /D5.InH /D4B
+/A6/D5, dually to the two steps just mentioned, we
+take a subalgebra and then a quotient [4]. In the basis chosen above, the subalgebra
+(which is also a Uqsℓ /D42 /D5submodule) is the one generated by z, /BU, andλ. Its quotient by
+λ2p/AG1 givesthe2 p3-dimensionalalgebra
+Hqsℓ /D42 /D5 /AGC /D6z, /BU,λ /D7
+/C4/D4(3.4)and /D4λ2p/A11 /D5/D5.
+As an associativealgebra,
+Hqsℓ /D42 /D5 /AGCq
+/D6z, /BU/D7 /CQ /D4C /D6λ /D7/DF/D4λ2p/A11 /D5/D5,
+withthep2-dimensionalalgebra
+(3.7) Cq
+/D6z, /BU/D7 /AGC /D6z, /BU/D7/DF/D4zp, /BUp, /BUz /A1 /D4q /A1q
+/A11/D5 /A1q
+/A12z /BU/D5.
+TheUqsℓ /D42 /D5action on Hqsℓ /D42 /D5is given by the last three lines in (3.5), with the central
+column rewritten for K /AGk2. The coaction δ:Hqsℓ /D42 /D5 /FTUqsℓ /D42 /D5 /CQHqsℓ /D42 /D5follows
+from(1.3)as
+λ /G6/FT1 /CQλ,
+zm/G6/FTm/F4
+s /AG0
+/D4/A11 /D5sqs /D41 /A1m /D5/D4q /A1q
+/A11/D5s
+/AIm
+s
+/AQ
+EsK
+/A1m/CQzm /A1s,/BUm/G6/FTm/F4
+s /AG0qs /D4m /A1s /D5/D4q /A1q
+/A11/D5s
+/AIm
+s
+/AQ
+FsKs /A1m/CQ /BUm /A1s.
+3.2.3.WiththeUqsℓ /D42 /D5actionandcoactiongiven above, Hqsℓ /D42 /D5isabraidedcommu-
+tativeYetter–Drinfeld Uqsℓ /D42 /D5-modulealgebra .
+3.3. Matrix braided commutative Yetter–Drinfeld module al gebras.It follows that
+Cq
+/D6z, /BU/D7in (3.7)—the algebra of “quantum differential operators on a line”—is also a
+braidedcommutativeYetter–Drinfeld Uqsℓ /D42 /D5-modulealgebra. Itisinfactthefullmatrix
+algebra[39],
+(3.8) Cq
+/D6z, /BU/D7 /ALMatp
+/D4C /D5.16 SEMIKHATOV
+3.3.1.ThatCq
+/D6z, /BU/D7is (semisimpleand) isomorphicto Mat p
+/D4C /D5already followsfrom a
+more general picture elegantly developed in [44], where “pa ra-Grassmann” algebras of
+the formC /D6z, /BU/D7/DF/D4zp, /BUp/D5withvariousadditional relations on the zi/BUjwere studied. The
+relationsbetweenour zand /BU,/BUmzn/AG
+/F4
+i/greaterorequalslant0q
+/A1/D42m /A1i /D5n /A0im /A1i /D4i /A11 /D5
+2
+/AIm
+i
+/AQ/AIn
+i
+/AQ/D6i /D7!
+/A0
+q /A1q
+/A11
+/A8izn /A1i/BUm /A1i
+(where the range of iis bounded above by min /D4m,n /D5because of the q-binomial coeffi-
+cients), are nondegenerate in terms of the classification in [44], hence the isomorphism
+withthefull matrixalgebra.
+We describe(3.8) as an isomorphism ofUqsℓ /D42 /D5modulecomodulealgebras . Thegen-
+eratorszand /BUhave the respective matrix representations ZandDin (1.15) (where we
+donotreducetheexpressionsusingthat qp/AG /A11and /D6p /A1i /D7 /AG /D6i /D7tohighlightapattern).
+Coaction (1.16) is then just the m /AG1 case of the formulas in 3.2.2, and it is not difficult
+toseethatthelastthreelinesin(3.5)yieldformulas(1.12 )–(1.14)—sofar,withnoeffect
+oftherescaling of λin (1.12).
+3.3.2.OnceCq
+/D6z, /BU/D7is thusidentifiedwithMat p
+/D4C /D5,wecan write
+Hqsℓ /D42 /D5 /AGMatp
+/A0
+C2p
+/D6λ /D7
+/A8
+(where we recall that C2p
+/D6λ /D7 /AGC /D6λ /D7/DF/D4λ2p/A11 /D5), and it is immediate to see from (3.5)
+thatλenteringthematrixentriesrescalesunderthe Uqsℓ /D42 /D5actionasindicatedin(1.12).
+Thisestablishesformulas(1.12)–(1.14).
+3.3.3.Forexample,for p /AG3, choosing xij
+/AGλnijyijwithλ-independent yij, wehave
+F⊲
+/A4/A6/A5λn11y11λn12y12λn13y13
+λn21y21λn22y22λn23y23
+λn31y31λn32y32λn33y33
+/ACÆ/AD/AG
+/A4/A6/A5λn21y21λn22y22
+/A1qn11λn11y11λn23y23
+/A0qn12
+/A12λn12y12
+q
+/A11λn31y31q
+/A11λn32y32
+/A1qn21
+/A12λn21y21q
+/A11λn33y33
+/A0qn22
+/A14λn22y22
+0 /A1qn31
+/A02λn31y31 qn32λn32y32
+/ACÆ/AD.
+3.3.4.Asanexampleofthecoactioninmatrixform, δ:Matp
+/D4C /D5 /FTUqsℓ /D42 /D5 /CQMatp
+/D4C /D5,
+we give the only typographically manageable case, that of p /AG2. Writing elements of
+Uqsℓ /D42 /D5 /CQMat2
+/D4C /D5as matrices with Uqsℓ /D42 /D5-valuedentries,wehave
+δX/AG
+/A3/D41 /A12iEFK3/D5x11
+/A0Fx12
+/A12iEK3x21
+/A02iEFK3x222iEK2x11
+/A0K3x12
+/A12iEK2x22
+FK3x11
+/A0K3x21
+/A1FK3x22
+/D41 /A1K2/A12iEFK3/D5x11
+/A0Fx12
+/A12iEK3x21
+/A0 /D4K2/A02iEFK3/D5x22
+/AB
+.YETTER–DRINFELD MODULE ALGEBRAS 17
+3.3.5.It would be interesting to find a direct matrixderivation of the Yetter–Drinfeld
+axiomforMat p
+/D4C /D5and thebraided commutativityproperty/D4X/D4/A11 /D5⊲Y /D5X/D40 /D5
+/AGXY,X,Y /C8Matp
+/D4C /D5.
+Weillustratethestructureoccurring in theleft-hand side here before thematrixmultipli-
+cation,withtheknownresult,isevaluated(again,necessa rilyrestrictingourselfto p /AG2):/D4X/D4/A11 /D5⊲Y /D5 /CQX/D40 /D5
+/AG
+/A3
+y11
+/A1y12/A1y21y22
+/AB/CQ
+/A3
+0x12
+x210
+/AB/A0
+/A3
+y11y12
+y21y22
+/AB/CQ
+/A3
+x110
+0x22
+/AB/A0
+/A3/A1i
+2y12 0
+i
+2
+/D4y11
+/A1y22
+/D5 /A1i
+2y12
+/AB/CQ
+/A3
+0 2i /D4x11
+/A1x22
+/D5
+0 0
+/AB/A0
+/A3
+i
+2y12 0
+i
+2
+/D4y11
+/A1y22
+/D5i
+2y12
+/AB/CQ
+/A3/A12ix210
+0 /A12ix21
+/AB/A0
+/A3
+y21y11
+/A1y22
+0y21
+/AB/CQ
+/A3
+0 0
+x22
+/A1x110
+/AB/A0
+/A3
+y21y22
+/A1y11
+0y21
+/AB/CQ
+/A3
+x120
+0x12
+/AB/A0
+/A3
+i
+2
+/D4y22
+/A1y11
+/D50
+0i
+2
+/D4y22
+/A1y11
+/D5
+/AB/CQ
+/A3
+2i /D4x11
+/A1x22
+/D50
+0 2 i /D4x11
+/A1x22
+/D5
+/AB
+.
+3.4. Hopf algebroid with the Matp
+/D4C /D5base.Theorem 4.1 in [2] nicely reinterprets
+the structure of a braided commutative Yetter–Drinfeld H-module algebra Aas a bial-
+gebroidstructureon A /AZH. (Wereferthereaderto[2]foracomprehensivediscussiono f/D4Hopf /DGbi /D5algebroids,also in relation toLu’s bialgebroids[16], Xu’ s bialgebroidswithan
+anchor[45], and Takeuchi’s /A2A-bialgebras [46], as well as forreferences tootherrelated
+works.)
+TheexamplesofHopfalgebroids A /AZHwithourbraidedcommutativeYetter–Drinfeld
+modulealgebras A /AGMatp
+/D4C2p
+/D6λ /D7/D5orA /AGMatp
+/D4C /D5may be ofsomeinterest becauseof
+the explicit matrix structure of the base algebra A. Below, we follow [2], adapting the
+formulas there to a leftcomodule algebra by duly inserting the antipodes. To somewh at
+simplify the notation, we discuss the “ λ-independent” example, i.e., the Hopf algebroid
+structure of A /AGMatp
+/D4C /D5 /AZUqsℓ /D42 /D5; reintroducing C2p
+/D6λ /D7on the matrix side is left to
+thereader.
+As a vector space, A /ALMatp
+/D4Uqsℓ /D42 /D5/D5, matrices with Uqsℓ /D42 /D5-valued entries; we can
+therefore write 1 /AZh /AG111h(h /C8Uqsℓ /D42 /D5), where111 is the unit p /A2pmatrix; with a slight
+abuse of notation, similarly, X /AZ1 /AGX, understood as a “constant” p /A2pmatrix. An
+arbitrary element of Acan be written as
+/EWp
+i,j /AG1eijhij, where the eijare the standard
+elementarymatrices and hij
+/C8Uqsℓ /D42 /D5. Thesmash-productcompositionisthen givenby/A1p/F4
+i,j /AG1eijhij
+/A9 /A1p/F4
+m,n /AG1emngmn
+/A9/AGp/F4
+i,j /AG1p/F4
+m,n /AG1eij
+/D4h
+/BD
+ij⊲emn
+/D5h
+/BE
+ijgmn,hij,gmn
+/C8Uqsℓ /D42 /D5,
+with the left action ⊲to be evaluated in accordance with (1.12)–(1.14). We write A /AG
+Matp
+/D4Uqsℓ /D42 /D5/D5/AZ, with the subscript reminding of the smash-product composi tion in this
+algebra(whichis highlynonstandardfrom thematrixstandp oint).18 SEMIKHATOV
+Therelevantstructures
+ε:A /FTMatp
+/D4C /D5,
+s,t:Matp
+/D4C /D5 /FTA,
+Δ:A /FTA /CQMatp
+/D4C /D5A,
+τ:A /FTA
+(thecounit,thesourceand target maps,thecoproduct, andt heantipode)are as follows.
+Thecounit ε:A /FTMatp
+/D4C /D5acts componentwise,
+ε
+/A1p/F4
+i,j /AG1eijhij
+/A9/AGp/F4
+i,j /AG1eijε /D4hij
+/D5.
+The source map s: Matp
+/D4C /D5 /FTAis the identical map onto constant matrices. The
+targetmap t:Matp
+/D4C /D5 /FTA /AGMatp
+/D4C /D5 /AZUqsℓ /D42 /D5isgivenby
+t /D4X /D5 /AGX/D40 /D5
+/AZS
+/A11/D4X/D4/A11 /D5
+/D5,
+whereδ /D4X /D5 /AGX/D4/A11 /D5
+/CQX/D40 /D5
+/C8Uqsℓ /D42 /D5 /CQMatp
+/D4C /D5is the coaction defined in (1.16). It then
+followsthat ZandDin(1.15)mapunder tintothefollowingtwo-diagonalmatriceswith
+Uqsℓ /D42 /D5-valuedentries:
+t /D4Z /D5 /AG
+/A4/A6/A6/A6/A6/A6/A6/A5
+/D4q /A1q
+/A11/D5E0
+K /D4q /A1q
+/A11/D5E0
+............
+0...K /D4q /A1q
+/A11/D5E0
+0............ K /D4q /A1q
+/A11/D5E
+/ACÆÆÆÆÆÆ/AD, (3.9)
+t /D4D /D5 /AG /D4q /A1q
+/A11/D5
+/A4/A6/A6/A6/A6/A6/A6/A5
+/A1FK K
+0 /A1FKq
+/A11/D62 /D7K
+............
+0...0 /A1FKq2 /A1p/D6p /A11 /D7K
+0..... 0 /A1FK
+/ACÆÆÆÆÆÆ/AD. (3.10)
+For any complex matrix Y /AG
+/EW
+m,nymnZmDn, we use (3.9) and (3.10) to calculate t /D4Y /D5 /AG/EW
+m,nymnt /D4D /D5nt /D4Z /D5m/C8A(evidently, with the smash-product multiplication unders tood).
+Furthermore,elementary calculationusingthebraided com mutativityshowsthat
+t /D4Y /D5/D4X /AZh /D5 /AGX /A4t /D4Y /D5 /A4h,X,Y /C8Matp
+/D4C /D5,
+where, abusing the notation, the dot denotes both matrix product and the product in
+Uqsℓ /D42 /D5, with articulately no “smash” effects because multiplicat ionwith a constant ma-
+trixison theleft and witha Uqsℓ /D42 /D5elementontheright.YETTER–DRINFELD MODULE ALGEBRAS 19
+Thecoproduct Δ:A /FTA /CQMatp
+/D4C /D5Ais(co)componentwise,
+Δ
+/A1p/F4
+i,j /AG1eijhij
+/A9/AGp/F4
+i,j /AG1eijh
+/BD
+ij
+/CQMatp
+/D4C /D5111h
+/BE
+ij,hij
+/C8Uqsℓ /D42 /D5.
+The /CQMatp
+/D4C /D5product is here defined with respect to theright action of Mat p
+/D4C /D5onAvia/D4X /AZh /D5.Y /AGt /D4Y /D5/D4X /AZh /D5and theleft actionvia Y. /D4X /AZh /D5 /AGs /D4Y /D5/D4X /AZh /D5,and hence/D4X /A4t /D4A /D5 /A4h /D5 /CQMatp
+/D4C /D5
+/D4Y /AZg /D5 /AG /D4X /AZh /D5 /CQMatp
+/D4C /D5
+/D4AY /AZg /D5
+holds for all X,A,Y /C8Matp
+/D4C /D5andg,h /C8Uqsℓ /D42 /D5(where the first factor in the left-hand
+side,again,involvesmatrixand Uqsℓ /D42 /D5productson thedifferentsidesof t /D4A /D5).
+Theantipode τ:A /FTAis givenbyanothersimpleadaptationofaformulain[2]:
+τ /D4X /AZh /D5 /AG /D41 /AZS /D4h /D5/D5
+/A0/D4S /D4X
+/BE/D4/A11 /D5
+/D5⊲X/D40 /D5
+/D5 /AZS /D4X
+/BD/D4/A11 /D5
+/D5
+/A8
+,X /C8Matp
+/D4C /D5,h /C8Uqsℓ /D42 /D5,
+with theproduct in the right-hand sideto be taken in A.2On 1 /AZUqsℓ /D42 /D5, this is just the
+Uqsℓ /D42 /D5antipode, and on Mat p
+/D4C /D5,τ /D4X /D5 /AG /D41 /AZS /D4X/D4/A11 /D5
+/D5/D5/D4X/D40 /D5
+/AZ1 /D5; asimplecalculation
+thenshowsthat
+τ /D4Z /D5 /AGq2t /D4Z /D5,
+τ /D4D /D5 /AGq
+/A12t /D4D /D5.
+Beingan anti-algebramap,again, thisextendsto allofMat p
+/D4C /D5 /C9
+/EW
+m,nymnZmDn.
+SomeoftheHopfalgebroidproperties(see[2,Defnition2.2 ]foranicelyrefinedlistof
+axioms), e.g., τ /D4t /D4X /D5/D5 /AGs /D4X /D5andt /D4X /D5s /D4Y /D5 /AGs /D4Y /D5t /D4X /D5, are evident for A /AGMatp
+/D4C /D5 /AZ
+Uqsℓ /D42 /D5described in matrix form; with others, it is not entirely obv ious how far one
+can proceed with verifying them in a purely matrixlanguage, i.e., not following [2] in
+resorting to the Yetter–Drinfeld module algebra propertie s; so much more interesting is
+thefact that Mat p
+/D4Uqsℓ /D42 /D5/D5/AZatq /AGeiπ /DFpisaHopfalgebroidoverMat p
+/D4C /D5.
+As already noted, it is entirely straightforward to extend t he above formulas to de-
+scribe Mat p
+/D4C2p
+/D6λ /D7/D5 /AZUqsℓ /D42 /D5 /AGMatp
+/D4Uqsℓ /D42 /D5 /CQC2p
+/D6λ /D7/D5/AZas a Hopf algebroid over
+Matp
+/D4C2p
+/D6λ /D7/D5 /AHMatp
+/D4C /D6λ /D7/DF/D4λ2p/A11 /D5/D5.
+3.5. Heisenberg “chains.” TheHeisenberg n-tuples /DFchainsdefined in 2.5.3can also be
+“truncated”similarlytohowwepassedfrom H /D4B
+/A6/D5toHqsℓ /D42 /D5. Anadditionalpossibility
+2Andthesection γofthenaturalprojection A /CQA /FTA /CQMatp
+/D4C /D5A,requiredinthedefinitionofaHopf
+algebroid [2] to satisfy the condition m /A5 /D4id /CQτ /D5 /A5γ /A5Δ /AGs /A5ε, is given by γ: /D4X /AZh /D5 /CQMatp
+/D4C /D5
+/D4Y /AZg /D5 /G6/FT/D4X /A4t /D4Y /D5 /A4h /D5 /CQ /D4111g /D5.20 SEMIKHATOV
+here is to drop the coinvariant λaltogether, which leaves us with the “ truly Heisenberg ”
+Yetter–Drinfeld Uqsℓ /D42 /D5-modulealgebras
+HHH2
+/AGC
+/A6p
+q
+/D6/BU1
+/D7 /B3Cp
+q
+/D6z2
+/D7 /AGCq
+/D6z2, /BU1
+/D7,
+HHH2n
+/AGC
+/A6p
+q
+/D6/BU1
+/D7 /B3Cp
+q
+/D6z2
+/D7 /B3... /B3C
+/A6p
+q
+/D6/BU2n /A11
+/D7 /B3Cp
+q
+/D6z2n
+/D7,
+HHH2n /A01
+/AGC
+/A6p
+q
+/D6/BU1
+/D7 /B3Cp
+q
+/D6z2
+/D7 /B3... /B3C
+/A6p
+q
+/D6/BU2n /A11
+/D7 /B3Cp
+q
+/D6z2n
+/D7 /B3C
+/A6p
+q
+/D6/BU2n /A01
+/D7
+(ortheirinfiniteversions),where C
+/A6p
+q
+/D6/BU/D7 /AGC /D6/BU/D7/DF/BUpandCp
+q
+/D6z /D7 /AGC /D6z /D7/DFzpwiththebraiding
+inheritedfrom 2.5.3, which amountsto usingtherelations/BUizj
+/AGq /A1q
+/A11/A0q
+/A12zj
+/BUi
+forall(odd)iand (even) j, and
+zizj
+/AGq
+/A12zjzi
+/A0 /D41 /A1q
+/A12/D5z2
+j,/BUi
+/BUj
+/AGq2/BUj
+/BUi
+/A0 /D41 /A1q2/D5/BU2
+j,i/greaterorequalslantj
+(andzp
+i
+/AGand /BUp
+i
+/AG0; our relations may be interestingly compared with those in para-
+Grassmannalgebras studiedin [47]).
+Acknowledgments. I am grateful to J. Fjelstad, J. Fuchs, and A. Isaev for the use ful
+comments. ThisworkwassupportedinpartbytheRFBRgrant08 -01-00737,theRFBR–
+CNRS grant09-01-93105,and thegrant LSS-1615.2008.2.
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+LEBEDEVPHYSICSINSTITUTE
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+arXiv:1001.0734v1  [astro-ph.SR]  5 Jan 2010An Investigation on the Morphological Structure of IC59:
+A New Type of Morphology of BRCs?
+Jingqi Miao1, Koji Sugitani2, Glenn, J. White3,4, Richard P. Nelson5
+ABSTRACT
+With references to recent observational results onthenebula IC 59, we applied
+our previously developed Smoothed Particle Hydrodynamics (SPH) c ode which
+is based on Radiative Driven Implosion (RDI) model, to investigate the possible
+formation mechanism for the observed morphological structures of differently
+shaped BRCs. The simulation results confirmed the existence of the 4th type
+morphology of BRCs – type M BRC. We are able to find the necessary c ondition
+for the appearance of the type M BRC based on the fact that the s imulated
+physical properties of the cloud are consistent with observations on IC59. More
+importantly, the prospect of RDI triggered star formation by RDI model in all
+of the observed type M BRCs is ruthlessly eliminated.
+Subject headings: star: formation – ISM: evolution – ISM: HII regions – ISM:
+kinematics and dynamics – radiative transfer.
+1. Introduction
+Material at the star-facing surface of dense molecular clouds is int ensively ionized by the
+ultraviolet (UV) radiation from nearby OB stars. The emission lines fr om recombination of
+electrons with ions create a bright rim surrounding the star facing s urface of the molecular
+1CentreforAstrophysics&PlanetaryScience,SchoolofPhysical Sciences, UniversityofKent,Canterbury,
+Kent CT2 7NR, UK, J.Miao@kent.ac.uk
+2Institute of Natural Sciences, Nagoya City University, Mizuho-ku , Nagoya 467-8501, Japan
+3Centre for Earth, Planetary, Space & Astronomical Research, T he Open University, Walton Hall, Milton
+Keynes, MK7 6AA
+4Space PhysicsDivision, Space Science & TechnologyDivision, CCLRC Ru therford Appleton Laboratory,
+Chilton, Didcot, Oxfordshire, OX11 0QX, UK
+5School of Mathematical Sciences, Queen Mary College, University o f London, Mile End Road, London
+E1 4NS, UK– 2 –
+cloud. On the other hand, ionization heating drives a strong shock w ave into the molecular
+cloud so that the compressed gas forms a condensed core behind t he bright rim, which is the
+typical structural feature of a Bright Rimmed Cloud (BRC). The Fo rmation of BRCs opens
+a window for our investigation on the physical processes involved in t he interaction of UV
+photons with the gas particles in molecular clouds.
+The variety of the morphology of BRCs has interested both astron omers and theoreti-
+cianssincethelastdecade. MostoftheobservedBRCscouldbecat egorizedbythreedifferent
+morphologies dependent on the curvature of their bright rims: typ e A, B and C with an in-
+creased order of degree of the rim curvature (Sugitani et al 199 1; Sugitani & Ogura 1994).
+The curved bright rims of type A, B and C BRCs are usually convex hav ing the apex of its
+rim aligned with the center of the condensed core and the radial dire ction of the illuminating
+star as shown by the IC63 structure in both panels of Figure 1. Sev eral theoretical models
+of the UV radiation effect on the dynamical evolution of molecular clou ds have successfully
+re-produced the formation process of the above three types of BRCs (Lefloch & Lazareff
+1994; Kessel-Deynet & Burkert 2000; Willams et al. 2001; Miao et al 2 006, 2009), hence we
+have gained a basic understanding on the mechanism of the formatio n of BRCs with type
+A, B and C morphologies.
+However, there is another interesting type cometary clouds, who se axial line is not
+aligned with but offset from the radial direction of the illuminating star by an angle θ
+(Osterbrock 1957), as shown by the right cap structure of the lo wer left object in the left
+panel of Figure1. By a further inspection, one can find that the ob served cometary structure
+is one of the two caps of a bigger cloud structure whose star facing surface is concave to the
+radial direction of the illuminating star as shown by the lower left stru cture in the left panel
+of Figure 1. Therefore the symmetrical line of any one of the two ca ps is neither aligned
+with the radial direction of the illuminating star, nor the condensed m olecular core behind
+the concave rim as shown by the lower left object in the right panel o f Figure 1. This type
+of cometary structure does not belong to any of the three previo usly defined morphological
+types. Quite a few known cometary clouds bear the same structur al feature, e.g., the No. 8
+source in IC 1396 (Osterbrock 1957) and SFO 80 in RCW 108 (Urquha rt et al 2009). The
+Hαimages of IC59 and IC63 in the same region are shown in Figure 1, which also describes
+an offset angle θof the symmetrical line of IC59 from the radial direction of the illumina ting
+star (for the time being we use the conventional name IC59 for the right cap structure only
+and later on we will redefine the structure range of IC59). The phy sical mechanism for the
+appearance of this type of morphology in molecular clouds, and its re lation to the possibility
+of triggered star formation by RDI have not been fully understood .
+Although the offset of the apex of IC59 from the radial direction of the illuminating– 3 –
+star was tentatively explained as the projection effect of a three d imensional space onto a
+two dimensional observation plane (Blouin et al 1997; Karr et al 200 5), the observed CO
+emission core from IC59 by Karr et al (2005), which usually traces t he compressed dense
+molecular materials behind the apex of the bright rim of a BRC, does no t seem to support
+this explanation. The CO emission core presented in the lower part st ructure in the right
+panel ofFigure1reveals adense corebehindtheconcave rim, which isalignedwiththeradial
+direction of the illuminating star. If the offset angle of the apex of IC 59 is a consequence
+of projection effect, the CO emission core should also be subject to the projection effect so
+that it should appear just behind the apex of the bright rim of the IC 59, as the CO emission
+core in IC63 does, as shown in the upper cometary structure in the right panel of Figure
+1. Since the center of the CO core in IC59 is aligned with the radial dire ction ofγCAS,
+it seems more likely that this type of morphological structure, i.e., a c ondensed molecular
+core behind the concave rim which has two apexes at its two sides, is d eveloped during the
+evolutionary process of a molecular cloud by RDI mode, just as the f ormation of type A,
+B and C BRCs, whose morphology formation mechanism has been well u nderstood through
+the theoretical simulations (Miao et al 2009). We would categorize t his type of structure as
+type M morphology for its geometrical analogy.
+In this paper, we will explore the possibility of forming a BRC with a conc ave rim by
+RDI mechanism using our previously developed SPH (Smoothed Partic le Hydrodynamic)
+code (Miao et al 2006, 2009), and discuss the connection between the appearance of type
+M BRC and triggered star formation within it. We will structure the pa per in the following
+way: 1.Gather the structural and physical properties of IC59 an d the illuminating star from
+available literature and our own observation; 2.Present the simulatio n results which reveals
+the formation of type M morphology in the evolutionary process of a molecular cloud by
+RDI mode; 3. Discuss the kinematics behind the formation of type M m orphology in BRCs;
+4. Discuss the possibility of RDI triggered star formation in type M BR Cs; 5. Derive a
+conclusion for result of our investigation.
+2. Physical properties of IC59 and γCAS
+As seen from Figure 1, IC59 is one of the pair nebulae in Sh 2-185 and is at a distance
+about 1.3 pc from the illuminating B0 IV star γCas, which is approximately 190 pc away
+from us. Our understanding on the physical properties of IC59 is g radually improved with
+the availability of the progressively advanced observing facilities. IC 59 was once defined
+as a reflection nebula (Osterbrock 1957) for there was little molecu lar emission detected
+(Jansen et al 1994, 1995) especially when compared with that of its neighbor IC63 in the– 4 –
+same region. Later the radio continuum and HI study by (Blouin et al 1997) revealed the
+existence of atomic hydrogen in IC59, which were produced throug h the dissociation of H 2,
+thus IC59 was taken as an emission nebula. The multiwavelength inves tigation on IC59
+(Karr et al 2005) confirmed the presence of H 2in IC59 and its similarity to IC63 in other
+spectrum features except that an ionization front (a bright rim) in IC59 was not clearly
+detected.
+Recently we obtained H αimages with the Wide Field Grism Spectrograph 2 (Uehara
+et al. 1994) and Tektronix 2048 ×2048 CCD mounted on the University of Hawaii 2.2-m
+telescope on 2009 September 10 UT. It is seen from Figure 2 that ar ound the star facing side
+of the nebula, a bright rim covers the east cap, the concave surfa ce and the west cap. The
+brightest part of the rim is the concave part whose center is aligned with the radial direction
+of the star so that it receives stronger UV radiation than the surf aces around the two side-
+caps. Therefore it seems more reasonable to consider the two ape xes and the concave part
+as one whole piece of the nebula structure and name it as IC59, rath er than the east part
+only. We will use this new definition of IC59 in our following discussions. W ith the position
+of CO emission core in IC59 defined by Karr et al (2005) and the lates t Hαimage of IC59
+we obtained, we believe that we have gathered enough evidence to a scertain that the newly
+defined IC59 structure should be a type of BRC structure with typ e M morphology, although
+which has never been investigated.
+The geometrical and physical parameters we can collect from differ ent literature’s for
+the east cap of IC59 are as follows. The width of the cap w∼0.18 pc; length l∼
+0.12 pc (Osterbrock 1957) where the dimensions are defined by the convention for BRCs
+(Osterbrock 1957; Sugitani et al 1991; Sugitani & Ogura 1994); its column density is 3 .4×
+1017cm−2(Karr et al 2005); the ionization flux at the star facing surface of IC59 isFUV=
+5×109cm−2s−1, under the premise that γCas is a B0 star with a temperature 33,340 K
+(Vacca et al 1996; Karr et al 2005). An average temperature of the east cap of IC59 is
+estimated as T= 590 K (Karr et al 2005).
+The above collected properties of IC59 are used as a guidence for t he selection of an
+appropriate candidate from our simulation experiment in order for u s to address the for-
+mation mechanism of type M BRC from an initially uniform molecular cloud. We run our
+SPH code (named as NEWcloud) for molecular clouds of different initial conditions. From
+the results obtained, we are able to reveal the possible origin and fu ture evolution of IC59,
+and especially address the possibility of RDI triggered star formatio n within IC59 and other
+similar BRCs. It is our intention to focus on the formation mechanism o f type M BRCs in
+this investigation, therefore the properties of IC63 is not included in our investigation, for
+which we will address its physical properties and evolutionary featu res in a separate paper– 5 –
+in preparation.
+3. Results and discussions
+3.1. The formation of type M morphology
+The SPH code we developed (Miao et al 2006, 2009) solves the compr essible fluid dy-
+namic equations, with inclusion of self-gravity of the cloud, the most important heating and
+cooling processes plus a chemical network of 12 basic chemical comp onents in BRCs, which
+has been fully explained in our previous two papers and will not be repe ated here. Interested
+readers could read the above two cited papers to know the details o f the code. The radia-
+tion fields consists of an standard isotropic interstellar backgroun d radiation (with photon’s
+energy between 6.4 and 13.6 eV)(Habing 1968) and a UV radiation field from nearby star
+(with photon’s energy higher than the hydrogen ionizing energy of 1 3.6 eV). The cloud is
+initially situated at the center of a xyzcoordinate and the UV radiation is set along −z
+direction and incident on the surface of the upper hemisphere of th e simulated cloud, with
+a constant energy flux FUVspecified in the last section.
+Thesimulatedcloudisinitiallysituatedinawarmanddefuseinterstellar m edium, which
+is assumed to compose primarily of atomic hydrogen with n(HI) = 10 cm−3andT= 100 K
+(Nelson & Langer 1999; Miao et al 2006). The molecular cloud is of an in itial temperature
+of 60 K, mass of 2 M ⊙and a radius of 1.4 pc (corresponding to an initial number density
+ni(H2) = 3.48 cm−3, which is within the observed range of volume densities of star formin g
+molecular clouds (Heiner et al 2008)) In all of the simulations presen ted in the following,
+20,000 particles are used.
+The hydrogen number density profile in the up-left panel in Figure 3 s hows that 0.47
+My after the radiation fields were switched on, the initially spherical c loud evolved into
+a type A BRC having its front surface (star facing side) squashed f rom a hemispherical
+surface and a maximum density of 30 cm−3was achieved at the head of the upper hemi-
+sphere as a consequence of UV radiation induced shock propogatio n into the cloud, whose
+physical implications have been fully understood (Bertoldi 1989; Le floch & Lazareff 1994;
+Kessel-Deynet & Burkert 2000; Willams et al. 2001; Miao et al 2009). The up-middle and
+-right panels show that the front surface of the cloud gets more s quashed at t= 0.62 My
+and then become very flat at t= 0.82 My, whilst a more and more condensed core appeared
+with a centre density of 136 cm−3at t = 0.83 My. When t= 1.01 My, the front surface of
+the cloud started to become concave to the radial direction of the star which is above the
+cloud at positive zaxis. A further condensed core formed behind the concave surfa ce and– 6 –
+had a centre density of 1127 cm−3. The degree of concavity of the front surface got enhanced
+att= 1.11 My. It is seen from the bottom-middle panel in Figure 3, that the m orphology
+of the cloud structure is very similar to the observed morphology of IC59 as shown in both
+panels in Figure 2, i.e., a concave surface whose center is aligned with zaxis (the radial
+direction of the star), is in between two caps whose geometrical sy mmetrical line is offset an
+angle from the zaxis. Thus a type M BRC has formed and the centre density of the co re
+behind the concave part of the rim is 807 cm−3which is less denser than that at t= 1.01
+My.
+It is obviously seen that the core behind the concave surface expa nded in volume from
+t= 1.01 tot= 1.11 My, therefore the centre density of the core decreased. The bottom right
+panel in Figure 3 shows that the whole structure retained a type M m orphology but further
+expanded at t= 1.24 My and the centre density of the core further decreased to 58 1 cm−3.
+Further simulation sequences after t= 1.24 tells the whole structure contines expansion and
+finally expands away as large piece of very defused cloud at t= 1.7 My.
+Taking any one of two caps in the formed type M BRC at t= 1.11 My as an example,
+we can estimate its average column density and then compare it with t he observational data.
+We take the east cap structure in the middle-bottom panel of Figur e 3 as the corresponding
+structure previously defined as IC59 by astronomers. Simulation r esult revealed that the
+structure of the east cap has a mass of 3 ×10−4M⊙. If we assume the mass distributes
+uniformly in a cap of a sphere of height 0.13 pc and radius 0.10 pc accor ding to the simulated
+dimension, which is very close to the measurement (0.12, 0.09) pc by O sterbrock (1957), a
+minimum value of the column density N= 2.2×1017cm−2is derived which is consistent
+with the observed value of 3 .4×1017cm−2(Karr et al 2005). The result for the west cap is
+similar to the east cap.
+3.2. Kinematics for the formation of type M BRCs
+A qualitative analysis on the kinematics of the evolution of the cloud wo uld reveal a
+physical picture for the formation of type M BRC. The cloud is not st rongly bound by self-
+gravity since it has an initial Jeans number α(the ratio of the gravitational to the thermal
+energies) of 0.025 which is far from the value of 1, the necessary co ndition for a molecular
+cloud to be unstable to gravitational collapse. After the initial stag e of the evolution , the
+head of the front hemisphere of the cloud is modestly compressed a tt= 0.62 My so that the
+gravitational center of the cloud has moved to the head of the upp er hemisphere as shown
+in the up-middle panel in Figure 3. The gas particles at the apex and off -apex points on the
+surface of the upper hemisphere have approximately similar gravita tional acceleration, hence– 7 –
+similar radial velocity VGas illustrated in Figure 4 by the solid arrowed lines. One the other
+hand, the UV radiation induced shock velocity V′
+s(θ) (by the dot arrowed lines in Figure
+4 ) at the point ( R,θ) on the front surface is given by (Bertoldi 1989; Lefloch & Lazare ff
+1994),
+V′
+s(θ) =Vs(0)(cos(θ))1/4(1)
+whereθis the angle of a surface point from zaxis and 0 ≤θ≤π/2 for all of positions at the
+front surface of the upper hemisphere of the cloud; Vs(0) is the shock velocity at the apex
+r=R,θ= 0 and Vs(0)∼[FUV/n]1/2withnbeing the number density of the site (Bertoldi
+1989).
+For the molecular cloud we simulated, because of its very low initial den sity, the shock
+velocityVs(0) is one order of magnitude higher and the radial velocity VGis much lower
+than that in those clouds in which type A, B and C BRCs formed with a hig h probability
+of triggered star formation in their shocked cores (Miao et al 2009 ). Therefore the total
+velocityV′
+T(by the dot-dash arrowed lines in Figure 4) of a gas particle at the fr ont surface
+forms a very small angle δfrom the direction of V′
+sas shown by Figure 4, so that V′
+Tis
+actually dominated by the component V′
+Swhich is dependent on the angular distance θof
+the position from zaxis. Consequently V′
+Tis determined by θand will decrease with θ. Now
+it is clear that the gasparticles at the apex of the front surface ha ve the highest total velocity
+VT=V′
+T(0) whose direction is aligned with −zaxis as shown by Figure 4. After some period
+of time, the gas particles at or close to the apex of the front surfa ce are in advance of the
+particles at two sides of the apex along −zaxis, which results in formation of a concave
+surface on the star facing side of the molecular cloud as shown by th e lower left, middle and
+right panels in Figure 3.
+3.3. Other fingerprints of the simulated cloud
+Figure 5 displays a sequence of formation of a CO core behind the con cave surface.
+Simulation data reveals that the fractional concentration XCOvaried with the centre density
+of the compressed core since CO is a very important ingredient for t racing the dense gas
+in a molecular cloud. From t= 0.47 to 1.01 My, the centre concentration of XCOin the
+compressed core increased from 1 .55×10−11to 7.9×10−7and then decreased to 2.2 ×10−7
+att= 1.11 My, to 3.08 ×10−8att= 1.24 My. The variation of the CO fraction reflects the
+change of the centre density in the cloud structure over the whole evolutionary sequence.
+The position of the CO core at t= 1.11 My is very similar to the observed one as shown in
+IC59 structure in lower part of the right panel of Figure 1 (Karr et al 2005).
+In order to investigate the thermal properties of the caps, the e ast cap was chosen– 8 –
+for keeping a consistency with the column density estimation in the las t section. Figure 6
+presents a temperature distribution of the east cap within a slice of ∆y= 0.056 pc centered
+aty= 0. Because of the low initial density of the cloud (3.48 cm−3), ionizing flux easily
+penetrates into the cloud structure and gas particles in the cap ar e heated dominantly by
+ionizing hydrogen atoms. The average temperature over a region o f 0.43< x <0.63 pc,
+−2.08< z <−1.95 pc is 520 K, which is comparable with the value of 590 K estimated by
+Karr et al (2005), while the average temperature in the core behin d the concave rim is only
+65 K due to its higher density (807 cm−3) than that in the cap structure ( ˜40 cm−3). The
+dimension of the region with such a temperature distribution is also co nsistent with that was
+determined by Osterbrock (1957) for the dimensions of the east c ap: a diameter of 0.18 pc
+and height of 0.12 pc. Beyond the front surface of the east cap, t he temperature reaches 104,
+typical to HII region. Around the front surface of the east cap, recombination of electrons
+with hydrogen ions creates the bright rim as shown in Figure 2, althou gh dimmer than that
+around the concave part of the rim, where the ionisation/recombin ation are most active.
+3.4. The structure formation and the fate of IC59
+Taking the consistent results derived from our simulation for the co lumn density, the
+position of CO core, the dimension and mean temperature of the eas t cap with that from
+observations, we could confidentlly construct a whole evolutionary pictrue for the formation
+of the whole structure of IC59: An initially uniform and spherical mole cular cloud evolved
+into a type M BRC after being exposed to the ionizing radiation from th e nearby star γCAS
+for 1.11 My; the observed cap structure is one of its two caps form ed at the above specified
+time, where there is no condensed cores inside due to its high temper ature state (520 K);
+the whole structure will expand away after t = 1.7 My.
+centre of the cap Therefore no IRAS point sources should be obse rved in both of caps’
+structures. Our simulation results strongly support the comment made by Karr et al (2005)
+on the the nature of the previously reported IRAS point source ins ide the east cap, i.e., ’the
+spuriously detected IRAS point source inside is merely an unresolved dust feature’. From
+the simulation result, we conclude that there is no prospect for trig gered star formation in
+the whole structure of IC59 at all. We will address a general criteria in the next subsection
+for the prospect of RDI triggered star formation in type M BRCs.– 9 –
+3.5. Formation of type M BRC and trigger star formation
+Inordertofindthephysical conditionsfortheformationoftypeM BRCsanditsrelation
+to the possibility of triggered star formation, we firstly define the in itial position of our
+simulated molecular cloud in Lefloch’s two-parameter space, the ioniz ation parameter ∆ and
+recombination parameter Γ which are expressed in the following form s (Lefloch & Lazareff
+1994):
+∆ =ni
+n0=f1(R,FUV,ci,n0) (2)
+Γ =ηαBniR
+ci=f2(R,FUV,ci,n0) (3)
+whereniandn0is the initial densities of ionized and neutral hydrogen atoms respec tively;
+ciis the isothermal sound speed in the ionized gas; η≈0.2 is a parameter to describe
+the effective thickness of the recombination layer around the molec ular cloud; αBis the
+effective recombination coefficient under the the assumption of ’on t he spot’ approximation
+(Dyson & Williams 1997); Ris the initial radius of the cloud; f1andf2are two functions of
+theR,FUV,ciandn0. This two dimensional parameter space was divided into five different
+regions depending on the prospects of the evolution of molecular clo uds under the effect
+of UV radiation. According to lefloch’s RDI modeling without inclusion of the self-gravity
+of the system, the clouds in region I (Γ
+∆≤1.4×10−2) and II (∆ <10−2−10−3) are too
+trivial to discuss because the effect of the ionization is too weak to p roduce any noticeable
+dynamical effects; clouds in region III (defined as ∆ >2) is entirely photo-ionized by an
+R-weak ionization front and there is no possibility for RDI triggered s tar formation. Recent
+investigation based on a SPH code developed by the authors of this p aper has proved that
+the lower boundary of region III should be shifted to ∆ >23 as the result of inclusion of
+the self-gravity of the cloud in the our code (Miao et al 2009). Class ifications of Region IV
+and V are not very relevant to the clouds we are interested in this wo rk, therefore will not
+be described here.
+The cloud we simulated above (termed as cloud A in the following) has Γ = 21 and
+∆ = 57, so that its initial location in Lefloch’s two-parameter space is in region III (∆ >23),
+which means that there should be no prospect for triggered star f ormation in a cloud of an
+initial mass of 2 M ⊙and a radius of 1.4 pc, while F UV= 5×109cm−2s−1. The simulation
+result presented in the last section on the future evolution of the I C59 is consistent with the
+predicted prospect of the BRCs by the modifiled Lefloch’s 2-dimensio nal parameter space
+model.
+In order to show that type M BRC is indeed one of morphologies result ed from UV
+radiation effect and type M BRC formation in cloud A is not by chance, w e made further– 10 –
+explorationto afew moremolecular clouds of different initial physical conditions. We further
+found another two molecular clouds (termed as B, C clouds in the follo wing) which would
+developtypeMBRCsandfinallyexpandandsplitintopiecesascloudAdo es. Welistedtheir
+initial physical properties in Table 1 with the calculated values of ioniza tion/recombination
+parameters. It is clearly seen that their initial physical status mak e them all locate in region
+III. Following the way we did our exploration, we can, in principle, foun d infinite number
+of molecular clouds with different physical conditions and under differ ent strength of UV
+radiations, which would form a type M BRC by RDI mode. Although we ar e not able to
+prove that all of the molecular clouds located in region III in Lefloch’s two-parameter space
+will form type M BRCs, we can at least conclude that an initial location in region III in
+the ∆/Γ parameter space is a necessary condition for a molecular cloud to d evelop a type
+M BRC morphology, i.e, a molecular cloud has to be in a very high initial ioniz ation state,
+as the definition of Region III described. Furthermore it is also true that the triggered star
+formation by RDI mode is not possibly to occur in all of type M BRCs obs erved, which
+may form a good guidance for astronomers who are looking for the s ign of triggered star
+formation in different BRCs.
+4. Conclusion
+Based on our latest observation and numeriacl simulation results to the nebula IC59
+in Sh-158, we conclude that the observed cap structure (so called IC59 in Sh-158) is actu-
+ally one of the two caps of a type M BRC which is a transient morphology from type A
+morphology to a full expansion, which is developed over the evolution of a molecular cloud
+of an initial mass of 2 M ⊙and a initial radius of 1.4 pc, under the effect of UV radiation
+from nearby star γCAS. The molecular cloud is at a very high initial ionization state in the
+ionization flash region III in Lefloch’s two dimensional parameter spa ce. Comparison of the
+simulated physical properties of the type M BRC with the observatio ns on IC59 structure
+tells that the observed IC59 has been exposed to the ionizing radiat ion from γCAS for
+about 1.11 My. Simulation result predicts that the observed IC59 st ructure will continue
+expanding and finally split into pieces after another 0.59 My. The futu re evolution of IC59
+revealed by numerical result is consistent with the analytical conclu sion on the evolutionary
+prospect of a molecular cloud located in region III in the two-parame ter space defined by
+Lefloch & Lazareff (1994), i.e., all of the molecular clouds with their init ial status in region
+III will be finally evaporated with no prospect for triggered star fo rmation at all.
+Further investigation found more molecular clouds of different initial conditions could
+evolved into a type M BRC after a type A morphology was developed th rough RDI mode.– 11 –
+However these molecular clouds share a common character, i.e., no m atter how different their
+initial masses or radii, their initial conditions make them all locate initially in region III in
+Lefloch’s two parameter space. Therefore we are able to define a n ecessary condition for a
+molecular cloud to develop a type MBRC based onthe analytical result s (Lefloch & Lazareff
+1994) and our numerical simulations: the molecular cloud should have the initial physical
+conditions that make it an initial state in region III in Lefloch’s two-pa rameter space.
+Although the formation of type M BRC may not account for all of the obseerved offsets
+of the apexes of BRCs to the radial directions of their illuminating sta rs, our simulation
+results reveal that if there are two caps connected by a concave front surface in an observed
+BRC structure and the concave part is centrally aligned with the rad ial direction of the star,
+then the offset of the apex of the observed caps’ structure to t he radial direction of the star
+maybecaused by theformationof atype MBRC. Inaseperated pap er, we will address some
+other reasons which could possibly cause the offset of the apex of a single structure BRC
+to the radial direction of the illuminating star. More importantly, the presented simulation
+results in this paper reveal that triggered star formation will not o ccur in the observed type
+M BRC, neither in its cap structures nor behind the concave front s urface.
+The above results based on our simulations strongly support the ar gument made by
+Karr et al (2005) on the nature of the structure of IC59: the pr eviously detected potential
+example of triggered star formation is spurious and the observed I RAS source(s) in IC59 is
+merely an unresolved dust feature.
+REFERENCES
+Bertoldi F, 1989, AJ, 735
+Blouin D, McCutcheon W H, Dewdney P E, Roger R S, Purton C R, Keste r D, Bontekoe
+Tj R, 1997, MNRAS, 287, 455
+Dyson J.E., Williams D A., 1997, The physics of the interstellar medium, 2n d Edi.IOP
+Publishing, Bristol and Philadelphia
+Habing D J, 1968, Bulletin Astro. Inst. Netherlands, 19, 421
+Heiner J S, Allen R J, Wong O I and van der Kruit P C, 2008,˚ a, 489, 533
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+Jansen D J, van Dishoeck E F, Black J H, Spaans M, Sosin C,1995,˚ a, 3 02,223– 12 –
+Karr J L, Noriega-Crespo A, Martin P G, 2005, AJ, 129, 954
+Kessel-Deynet O, Burkert A, 2000, MNRAS, 315, 713
+Lefloch B, Lazareff, B, 1994, A&A, 289, 559
+Miao J, White Glenn J, Nelson R, Thompson M, Morgan L, 2006, MNRAS, 369, 143
+Miao J, White Glenn J, Nelson R, Thompson M, 2009, ApJ, 629,382
+Nelson Richard, Langer William D, 1999, ApJ, 524, 923
+Osterbrock Donald E., 1957, ApJ, 125, 622
+Sugitani K, Fukui Y, Ogura K, 1991, ApJS, 77, 59
+Sugitani K, Ogura K, 1994, ApJS, 92, 163
+Uehara M, et al. 2004, SPIE, 5492. 661
+Urquhart J S, Morgan L K, Thompson M A, 2009, A&A, 497, 789
+Vacca William D, Garmany Catharine D, Shull J Michael, 1996, ApJ,460, 9 14
+Williams R J R, Ward-Thompson D, Whitworth A P, 2001, MNRAS, 327, 78 8
+This preprint was prepared with the AAS L ATEX macros v5.2.– 13 –
+Fig. 1.— A geometrical picture of Sh-158 with both IC59 and IC63 sur rounds the illumi-
+nating star γCAS. On the left is theH αimages for IC63 and IC59 and one right is position
+of the detected CO emission cores in both clouds Karr et al (2005).– 14 –
+Fig. 2.— H αimage of IC59 overlaid with CO emissions. The area of the image is ∼11.5′×
+11.5′. South is at the top, West to the left. Arrows indicate the direction s of UV light from
+γCas. The pixel scale is 0′′.34 pixel−1, providing a field of view of 11.5′×11.5′. Three 180
+s exposures dithered by 5′′were taken with H αnarrow band filter. Dome flat fielding was
+applied and a combined image was obtained with these three images. Th e CO contours is
+derived by suming up 5 channels of V(LSR) = -0.839798, -0.0272369, 0.785339, 1.59790 and
+2.41046 km/s. The central LSR velocity of the integrated intensity map is about 0.8 km/s
+and the contour interval is 1 K km/s.– 15 –
+Fig. 3.— The evolutionary sequence of an uniform molecular cloud with a n initial mass of 2
+M⊙and radius of 1.4 pc under the effect of UV radiation field with an ionising photon flux
+of 5×109cm−3. The UV radiation is along the −zdirection from the above of the cloud.
+The images and contours are both for number density of hydrogen atoms within the slice of
+∆y= 0.056pccentered at y= 0. A scale on the numbersity is shown on the gray bars.– 16 –
+Fig. 4.— The diagram of velocities of the gas particles at the apex and o ff-apex point on
+the front surface of the BRC in our simulation.– 17 –
+Fig. 5.— The evolution of the fraction of CO molecules is shown by the co ntours which is
+overlaid the number density images of the gas particles within the slice of ∆y= 0.056pc
+centered at y= 0. A scale on the number density images is shown on the gray bars.– 18 –
+Fig. 6.— The temperature distribution of the east-cap of IC59 (with in the slice of ∆ y=
+0.056pccentered at y= 0) att= 1.11 My.– 19 –
+Table 1. Parameters of clouds which form type M BRC
+Cloud ∆ Γ n0(cm−3) R(pc) Mass(M ⊙) Region
+A 57 21 3.48 1.4 2 III
+B 52 18 4.37 1.03 1 III
+C 35 13 8.8 0.56 0.4 III
\ No newline at end of file
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+arXiv:1001.0735v2  [cs.LO]  3 Feb 2010Symposium on TheoreticalAspects of Computer Science 2010( Nancy, France), pp. 645-656
+www.stacs-conf.org
+NAMED MODELS IN COALGEBRAIC HYBRID LOGIC
+LUTZSCHR ¨ODER1ANDDIRKPATTINSON2
+1DFKIBremenand Department of Computer Science, Universit¨ atBremen
+E-mailaddress :Lutz.Schroeder@dfki.de
+2Department ofComputing, Imperial College London
+E-mailaddress :dirk@doc.ic.ac.uk
+ABSTRACT . Hybrid logic extends modal logic with support for reasonin g about individual states,
+designated by so-called nominals. We study hybrid logic int he broad context of coalgebraic seman-
+tics,whereKripkeframesarereplacedwithcoalgebrasfora givenfunctor,thuscoveringawiderange
+ofreasoningprinciplesincluding, e.g.,probabilistic,g raded, default,orcoalitionaloperators. Specif-
+ically, we establish generic criteria for a given coalgebra ic hybrid logic to admit named canonical
+models, with ensuing completeness proofs for pure extensio ns on the one hand, and for an extended
+hybrid language with local binding on the other. We instanti ate our framework with a number of
+examples. Notably, we prove completeness of graded hybridl ogic withlocal binding.
+Introduction
+Modal logics have traditionally played a central role in Com puter Science, appearing, e.g., in the
+guise of temporal logics, program logics such as PDL, episte mic logics, and later as description
+logics. Thedevelopment ofmodallogicshasseenextensions along(atleast)twoaxes: theenhance-
+ment of the expressive power of basic (relational) modal log ic on the one hand, and the continual
+extension,beyondthepurelyrelational realm,oftheclass ofstructuresdescribedusingmodallogics
+on the other hand. Hybrid logic falls into the first category, extending modal logic with the ability
+to reason about individual states in models. This feature, o riginally suggested by Prior and first
+studied in the context of tense logics and PDL (see [5] for ref erences), is of particular relevance in
+knowledge representation languages and as such has found it s way into modern description logics,
+where it isdenoted by theletter Ointhe standard naming scheme [2].
+Extensions along the second axis – semantics beyond Kripke s tructures and neighbourhood
+models – include various probabilistic modal logics, inter preted over probabilistic transition sys-
+tems,gradedmodallogicovermultigraphs[8],conditional logicsoverselectionfunctionframes[6],
+and coalition logic [17], interpreted over so-called game f rames. As a unifying semantic bracket
+covering all these logics and many further ones, coalgebrai c modal logic has emerged ([7] gives a
+survey). Thescope of coalgebraic modal logic hasrecently b een expanded toencompass nominals;
+we refer to the arising class of logics as coalgebraic hybrid logics . Existing results include a finite
+1998ACMSubjectClassification: F.4.1[MathematicalLogicandFormalLanguages]: Mathemat icalLogic—modal
+logic; I.2.4[ArtificialIntelligence]: Knowledge Represe ntation Formalisms and Methods —modal logic, representati on
+languages.
+Key words and phrases: Logic incomputer science, semantics, deduction, modal log ic, coalgebra.
+c/circlecopyrtL.Schr¨oderandD.Pattinson
+CC/circlecopyrtCreativeCommonsAttribution-NoDerivsLicense646 L.SCHR ¨ODER ANDD.PATTINSON
+model result, an internalized tableaux calculus, and gener icPSPACE upper bounds, but are so
+far limited to logics that exclude frame conditions and loca l binding [14]. What is missing from
+this picture technically is a theory of named canonical models [5]. Named canonical models yield
+not only strong completeness of the basic hybrid logic, but a lso completeness of pure extensions ,
+defined by axioms that do not contain propositional variable s (but may contain nominals; e.g. in
+Kripke semantics, the pure axiom ♦♦i→♦i, withianominal, defines transitive frames). More-
+over, named canonical models establish completeness for an extended hybrid language with alocal
+bindingoperator ↓x.φ(x),readas“thecurrentstate xsatisfiesφ(x)”. Bothpureextensions andthe
+languagewith ↓(notaddressedin[14])are,ingeneral, undecidable[1](it shouldbenoted,however,
+thatfragmentsofthelanguage with ↓overKripkeframesaredecidable andassuchplayarole,e.g. ,
+in conjunctive query answering in description logic [11]). As aconsequence, completeness of pure
+extensions and local binding is the best we can hope for – it es tablishes recursive enumerability of
+the set of valid formulas, and it enables automated reasonin g, if not decision procedures.
+Specifically, we establish two separate criteria for the exi stence of named models. Although
+these criteria are (in all likelihood necessarily) less wid ely applicable than some previous coalge-
+braic results including those of [14], the generic results a llow us to establish new completeness
+results for a wide variety of logics; in particular, we prove strong completeness of graded hybrid
+logic, and ultimately an extension of the description logic SHOQ, with the ↓binder over a wide
+variety of frame classes.
+1. CoalgebraicHybridLogic
+To make our treatment parametric in the syntax, we fix a modal s imilarity type Λconsisting of
+modal operators with associated arities throughout. For gi ven countably infinite and disjoint sets
+Pof propositional variables and Nof nominals, the set (Λ)ofhybridΛ-formulas is given by the
+grammar
+(Λ)∋φ,ψ::=p|i|φ∧ψ| ¬φ| ♥(φ1,...,φ n)|@iφ
+wherep∈P,i∈Nand♥ ∈Λisann-arymodaloperator. (Alternatively, wecouldregard proposi-
+tionalvariables asnullarymodaloperators,thusavoidingtheirexplicitme ntionaltogether. Wekeep
+them explicit here, following standard practice in modal lo gic, as we have to deal with valuations
+anywayduetothepresenceofnominals.) Weusethestandardd efinitionsfortheotherpropositional
+connectives →,↔,∨. The set of nominals occurring in a formula φis denoted by N(φ), similarly
+for sets of formulas. A formula of the form @iφis called an @-formula. Semantically, nominals i
+denote individual states ina model, and @iφstipulates that φholds at state i.
+Toreflectparametricity alsosemantically, weequiphybrid logicswitha coalgebraic semantics
+extending thestandard coalgebraic semantics ofmodal logi cs [16]: wefixthroughout a Λ-structure
+consisting of an endofunctor T:Set→Seton the category of sets, together with an assignment
+of ann-ary predicate lifting /llbracket♥/rrbracketto everyn-ary modal operator ♥ ∈Λ, i.e. a set-indexed family of
+mappings (/llbracket♥/rrbracketX:P(X)n→ P(TX))X∈Setthat satisfies
+/llbracket♥/rrbracketX◦(f−1)n= (Tf)−1◦/llbracket♥/rrbracketY
+for allf:X→Y. In categorical terms, [[♥]]is a natural transformation Qn→ Q ◦Topwhere
+Q:Setop→Setisthe contravariant powerset functor.
+In this setting, T-coalgebras play the roles of frames. AT-coalgebra is a pair(C,γ)where
+Cis a set of statesandγ:C→TCis thetransition function . Whenγis clear from the context,
+we refer to (C,γ)just asC. A(hybrid)T-modelM= (C,γ,V)consists of a T-coalgebra (C,γ)
+together with a hybrid valuation V, i.e. a map P∪N→ P(C)that assigns singleton sets to allNAMED MODELS IN COALGEBRAIC HYBRID LOGIC 647
+nominalsi∈N. We say that Misbasedon the frame (C,γ). The singleton set V(i)is tacitly
+identified with itsunique element.
+The semantics of F(Λ)is a satisfaction relation |=between states c∈Cin hybridT-models
+M= (C,γ,V)and formulas φ∈(Λ), inductively defined as follows. For x∈N∪Pandi∈N,
+M,c|=xiffc∈V(x)andM,c|= @iφiffM,V(i)|=φ.
+Modal operators are interpreted using their associated pre dicate liftings, that is,
+M,c|=♥(φ1,...,φ n)⇐⇒γ(c)∈/llbracket♥/rrbracketC(/llbracketφ1/rrbracketM,...,/llbracketφn/rrbracketM)
+where♥ ∈Λisn-ary and /llbracketφ/rrbracketM={c∈C|M,c|=φ}denotes the truth-set of φrelative toM.
+We writeM|=φifM,c|=φfor allc∈C. For a set Φ⊆(Λ)of formulas, we write M,c|= Φif
+M,c|=φforallφ∈Φ,andM|= ΦifM|=φforallφ∈Φ. Wesaythat Φissatisfiable inamodel
+Mif there exists a state cinMsuch thatM,c|= Φ. IfA ⊆(Λ)isa set of axioms, also referred to
+asframe conditions , a frame (C,γ)is anA-frameif(C,γ,V)|=φfor all hybrid valuations Vand
+allφ∈ A, and a model is an A-modelif it is based on an A-frame. A frame condition is pureif it
+does not contain anypropositional variables (it mayhoweve r contain nominals). Werecall notation
+from earlier work:
+Notation 1. As usual, application of substitutions σ:P→(Λ)to formulas φis denotedφσ.
+For a set Σof formulas and a set Oof operators, we write OΣorO(Σ)for the set of formulas
+arising by prefixing elements of Σwith an operator from O; e.g.Λ(Σ) ={♥(φ1,...,φ n)| ♥ ∈
+Λn-ary,φ1,...,φ n∈Σ}and@Σ :={@i|i∈N}(Σ) ={@iφ|i∈N,φ∈Σ}. Moreover,
+Prop(Z)denotesthesetofpropositionalcombinationsofelementso fsomesetZ. Forφ∈Prop(Z),
+we writeX,τ|=φifφevaluates to ⊤in the boolean algebra P(X)under a valuation τ:Z→
+P(X). Forψ∈Prop(Λ(Z)), the interpretation /llbracketψ/rrbracketTX,τofψintheboolean algebra P(TX)under
+τis the inductive extension of the assignment /llbracket♥(p1,...,pn)/rrbracketTX,τ=/llbracket♥/rrbracketX(τ(p1),...,τ(pn)).
+We writeTX,τ|=ψif/llbracketψ/rrbracketTX,τ=TX, andt|=TX,τψift∈/llbracketψ/rrbracketTX,τ. A set of formulas
+Ξ⊆Prop(Λ(Z))isone-step satisfiable w.r.t.τif/intersectiontext
+φ∈Ξ/llbracketφ/rrbracketTX,τ/ne}ationslash=∅. We occasionally apply this
+notation tosets Z⊆ P(X)withτbeing just inclusion, in which case mention of τissuppressed.
+In the sequel, we will be interested in both localandglobalsemantic consequence, where local
+consequence refers to satisfaction in a single state and glo bal consequence to satisfaction in entire
+models. In fact, we consider local reasoning under global as sumptions: given a set Φ⊆(Λ)of
+global assumptions (a TBoxin description logic terminology) and a class Cof models, we say that
+φis a local consequence of Ψunder global assumptions ΦforC-models, in symbols Φ;Ψ|=Cφ,
+if for allM∈ Csuch thatM|= Φ,M,c|=φwheneverM,c|= Ψ(here, both ΦandΨare
+sets of arbitrary formulas, in particular not subject toany restrictions on the nesting depth of modal
+operators). The standard notions of local and global conseq uence are regained from this general
+definition by taking ΦorΨto beempty, respectively.
+The distinguishing feature of the coalgebraic approach to h ybrid and modal logics is the para-
+metricityinboththelogical language andthenotionoffram e: concreteinstantiations ofthegeneral
+framework, in other words a choice of modal operators Λand aΛ-structureT, capture the syntax
+and semantics of awiderange of modal logics, aswitnessed by the following examples.
+Examples 1.1. 1. The hybrid version of the modal logic K,hybridKfor short, has a single
+unary modal operator /square,interpreted over the structure consisting of the powerset functorP(which
+takes a setXto its powerset P(X)) and the predicate lifting /llbracket/square/rrbracketX(A) ={B∈ P(X)|B⊆A}.
+Itisclearthat P-coalgebras (C,γ:C→ P(C))arein1-1correspondence withKripkeframes,and
+that the coalgebraic definition of satisfaction specialize s tothe usual semantics of the box operator.648 L.SCHR ¨ODER ANDD.PATTINSON
+2.Graded hybrid logic has modal operators ♦k‘in more than ksuccessors, it holds that’. It is
+interpreted overthefunctor Bthattakesaset Xtotheset B(X) =X→N∪{∞}ofmultisetsover
+Xby[[♦k]]X(A) ={B∈ B(X)|/summationtext
+x∈AB(x)>k}. Thiscaptures thesemantics ofgradedmodal-
+ities over multigraphs [8], which are precisely the B-coalgebras. A more general set of operators
+is that of Presburger logic [9], which admits integer linear inequalities/summationtextai·#(φi)≥kamong
+formulas. Unlike in the purely modal case [19], hybrid multi graph semantics visibly differs from
+the more standard Kripke semantics of graded modalities, as the latter validates all formulas ¬♦1i,
+i∈N. However, both semantics agree if we additionally stipulat e¬♦1ias a global (pure) axiom.
+Thus,ourcompletenessresultsformultigraphsemanticsde rivedbelowdotransfertoKripkeseman-
+tics. In particular they apply to many description logics, w hich commonly feature both nominals
+and graded modal operators in theguise of qualified number restrictions .
+3.HybridCK, the hybrid extension of the basic conditional logic CK, has a single binary
+modal operator ⇒, written in infix notation. Hybrid CKis interpreted over the functor Cfthat
+maps a setXto the set P(X)→ P(X), whose coalgebras are selection function models [6], by
+putting[[⇒]]X(A,B) ={f:P(X)→ P(X)|f(A)⊆B}.
+4.Classicalhybridlogic (thehybridversionofthelogic Eofneighbourhood frames,referredto
+as(theminimal)classicalmodallogicin[6])hasasingle,u narymodaloperator /squareandisinterpreted
+overneighbourhood frames , that is, coalgebras for the functor NX=P(P(X))(more precisely,
+the double contravariant powerset functor). The semantics of classical modal logic is defined by
+the lifting /llbracket/square/rrbracketX(A) ={S∈NX|A∈S}.Monotone hybrid logic has the same similarity
+type, but is interpreted over upwards closed neighbourhood frames, or coalgebras for the functor
+MX={S∈NX|Supwards closed }where upwards closure refers tosubset inclusion.
+5. Thesyntaxofcoalition logicoveraset Nofagentsisgivenbythesimilaritytype {[C]|C⊆
+N}, and the operator [C]reads as “coalition Chas ajoint strategy to enforce ...”. Theformulas of
+(hybrid) coalition logic areinterpreted over game frames, i.e., coalgebras for the functor
+G(X) ={(f,(Si)i∈N)|/producttext
+i∈NSi/ne}ationslash=∅,f:/producttext
+i∈NSi→X}
+(aclass-valued functor, technically speaking, whichhowe verdoesnotcauseproblems). Theseman-
+tics arises via theliftings
+/llbracket[C]/rrbracketX(A) ={(f,(Si)i∈N)∈G(X)| ∃(si)i∈C∀(si)i∈N\C(f((si)i∈N)∈A}.
+We proceed to present a Hilbert-style proof system for coalg ebraic hybrid logics, which we prove
+to be sound and strongly complete. This requires that the log ic at hand satisfies certain coherence
+conditions between the axiomatization and the semantics — i n fact the sameconditions as in the
+purely modal case, which are easily verified localproperties that can be verified without reference
+toT-models and are already known tohold for alarge variety of lo gics [16,19].
+Proofsystemsforcoalgebraiclogicsaremostconveniently describedintermsofone-steprules,
+asfollows.
+Definition 1.2. Aone-step rule overΛis a ruleφ/ψwhereφ∈Prop(P)andψ∈Prop(Λ(P))
+(in fact,ψmay be restricted to be a disjunctive clause, which however i s not relevant here). The
+ruleφ/ψisone-step sound ifTX,τ|=ψwheneverX,τ|=φfor a valuation τ:P→ P(X).
+Given a set Rof one-step rules and a valuation τ:P→ P(X), a setΞ⊆Prop(Λ(P))isone-step
+consistent [20] if the set Ξ∪ {ψσ|σ:P→Prop(P);φ/ψ∈ R;X,τ|=φσ}is propositionally
+consistent.
+One-step soundrulesaresound, andwewillassumeone-step s oundness tacitly inthesequel. Com-
+pleteness hinges on variants of the notion of one-step compl eteness [19], which we define furtherNAMED MODELS IN COALGEBRAIC HYBRID LOGIC 649
+below. Asthe notion of one-step rule does not involve hybrid features, suitable rule sets can just be
+inherited from the corresponding modal systems; for graded logics, conditional logics, and many
+others, such rule sets are found, e.g., in [22, 21]. We recall that the one-step complete rule set for
+(hybrid)Kconsists of the rules
+a
+/squareaa∧b→c
+/squarea∧/squareb→/squarec.
+AsetRofone-step rulesnowgivesrisetoaHilbert system LRbyadjoining propositional tautolo-
+giesandthehybridaxioms, andclosing undermodusponens, r uleapplication, and @-necessitation.
+Formally, we write Φ⊢LRφfor a setΦof formulas, the global assumptions (or theTBox), and a
+formulaφifφiscontained in thesmallest set that
+•containsΦand all instances of propositional tautologies
+•contains all instances of @-introduction i∧φ→@iφand make-or-break
+(mob) @ ip→(♥(q1,...,qn)↔ ♥(@ip∧q1,...,@ip∧qn))
+together with all instances of the axioms ¬@i⊥,¬@iφ↔@i¬φ,@i(φ∧ψ)↔(@iφ∧
+@iψ)),@ii,@ij↔@ji,@ik∧@jp→@ip; and
+•is closed under instances of @-generalization p/@ip, instances of rules in R, and modus
+ponens.
+Thesecond group ofaxioms ensures that i∼j:≡@ijdefines anequivalence relation onnominals
+and that@idistributes over propositional connectives. The (mob) axi om captures the fact that the
+truthsetofan @-formulaiseitheremptyorthewholemodel;inthecaseofhyb ridK,itisequivalent
+tothe standard back axiom @iφ→/square@iφ.
+We write Φ;Ψ⊢LRφif there areψ1,...,ψ n∈Ψsuch that Φ⊢LRψ1∧ ··· ∧ψn→φ.
+That is,Φ;Ψ⊢LRφif there is a proof of φfrom global assumptions Φthat additionally assumes
+Ψlocally. As we assume that all one-step rules in Rare one-step sound, soundness for both local
+andglobal consequence isimmediate: wehave Φ;Ψ|=Cφ(forCtheclass of all models) whenever
+Φ;Ψ⊢LRφ. In [14], a criterion has been given for LRto beweakly complete , i.e. complete for
+the case where both the TBox Φand the set Ψof local assumptions are empty. Here, we extend
+this result to combined strong globaland strong localcompleteness, i.e. to cover both an arbitrary
+TBoxand an arbitrary set of local assumptions, even if LRis extended with pure frame conditions
+and local binding.
+2. Strong Completeness ofPure Extensions
+Pure completeness is a celebrated result in hybrid logic [3, Chapter 7.3]. In a nutshell, adding
+pure axioms to an already complete proof system for the hybri d extension of the modal logic K
+(Example1.1),oneretainscompletenesswithrespecttothe classofframesthatsatisfytheadditional
+axioms. In contrast to arbitrary modal axioms, pure axioms d o not contain propositional variables,
+and therefore define – in the classical setting of hybrid K– first-order frame conditions. Here, we
+show that thesametheorem isvalidfor amuchlarger classof l ogics, namelyall coalgebraic hybrid
+logics satisfying one of two suitable sets of conditions. Fo r the sake of readability, we restrict
+the technical development (not the examples) to the case of u nary operators from now on until
+Section 2.2.
+Definition 2.1. IfAis a set of pure formulas and Ris a set of one-step rules, we write
+Φ;Ψ⊢LRA+Nameφif there are ψ1,...,ψ n∈Ψsuch thatψ1∧...ψn→φisLR-derivable650 L.SCHR ¨ODER ANDD.PATTINSON
+from assumptions in Φwhereadditionally all substitution instances of axioms in Aand the rule
+(Name)i→φ
+φ(i /∈N(φ))
+maybe used in deductions. Asbefore, wewrite Φ⊢LRA+NameφifΦ;∅ ⊢LRA+Nameφ.
+Inthe above system, the rule (Name′) @iφ/φ(i /∈N(φ)and the rule
+(NameCong )@j(φ↔ψ)
+♥φ↔ ♥ψ(j /∈N(φ,ψ))
+are derivable. The system is clearly sound for both global an d local consequence over A-models in
+the samesense that LRissound over T-models.
+Definition 2.2. LetA ⊆(Λ)be a set of pure axioms, and let Φ⊆(Λ)be a TBox. A set
+Ψ⊆(Λ)is(LRA+Name)-Φ-inconsistent if there areψ1,...,ψ n∈Ψsuch that Φ⊢LRA+Name
+¬(ψ1∧ ··· ∧ψn). Otherwise, Ψis(LRA+Name)-Φ-consistent . A subset of @F(Λ), i.e. a set
+of@-formulas, is called an ABox(again borrowing terminology from description logic). A maxi-
+mally(LRA+Name)-Φ-consistent ABox is a maximal element Kamong the (LRA+Name)-
+Φ-consistent ABoxes, ordered by inclusion. For such a K, we writeSK={Ki|i∈N}, where
+Ki={φ∈(Λ)|@iφ∈K}, and putVK(i) ={Ki}={Kj∈SK|i∈Kj}.
+Fortheconstructionofanamedmodel,wenowfixamaximally (LRA+Name)-Φ-consistentABox
+K. Later,wewilltake Ktobeamaximallyconsistent extensionofagivenset Φofformulas,where
+wemayassume, thanks to therule (Name′), thatΦ⊆@(Λ). Wenote the following trivial facts:
+Lemma2.3. Wehaveψσ∈Kifor allψ∈ Aand all substitutions σ, and moreover K∪Φ⊆Ki.
+Our goal is the construction of named canonical models inthe following sense:
+Definition 2.4. Anamed canonical K-modelis amodel (SK,γ,VK)such that
+γ(Ki)∈[[♥]]ˆφiff♥φ∈Ki
+for every nominal i, whereˆφ={Kj∈SK|φ∈Kj}.
+It is clear that named canonical models arecountable, as the re are only countably manynominals.
+Lemma2.5 (Truthlemmafornamedcanonicalmodels) .IfM= (SK,γ,VK)isanamedcanonical
+K-model andφisahybrid formula, then for every Ki∈SK,
+M,Ki|=φiffφ∈Ki.
+Hence,M|= Φ,andMis anA-model.
+Thelastclause ofthetruthlemmafollowsfromLemma2.3,the crucial point beingthat satisfaction
+of all substitution instances of Aimplies frame satisfaction of Abecause every state in the model
+is denoted by some nominal. We now establish two criteria for the existence of named canonical
+models. Thefirstcriterionassumesastrongerformofone-st epcompletenessthanthesecond,which
+instead demands that the modalities are bounded.NAMED MODELS IN COALGEBRAIC HYBRID LOGIC 651
+2.1. PureCompleteness forStrongly One-StepCompleteLogi cs
+Theconstruction ofnamedmodelshingesonthefollowingnot ion ofpastedness, whichassures that
+nominals interact correctly across the whole model. For the rest of the section, we fix a one-step
+completeruleset R,asetAofpureaxioms,andaset Φ⊆(Λ)ofglobal assumptions, andwewrite
+‘consistent’ instead of ‘ (LRA+Name)-Φ–consistent’.
+Definition 2.6. An ABoxKis0-pastedif whenever @j(φ↔ψ)∈Kfor all nominals j, then
+@i(♥φ↔ ♥ψ)∈Kfor all nominals i.
+It is clear that Kcan induce a named model only if Kis0-pasted. The construction of pasted
+ABoxes requires a Henkin-like extension of the logical lang uage by adding new nominals. Gener-
+ally, wedenote by (Λ)+anextended language withcountably manynewnominals not ap pearing in
+(Λ).Wenote thefact (slightly glossed over in the literature) th at this extension isconservative:
+Lemma2.7. IfΨ⊆(Λ)is consistent, then Ψremains consistent in (Λ)+.
+Lemma 2.8 (Extended Lindenbaum lemma for 0-Pasted Sets) .IfΨ⊆ F(Λ)is consistent, then
+there exists a 0-pasted maximally consistent ABox K⊆@(Λ)+and a nominal iin(Λ)+such that
+@iΨ⊆K.
+(TheproofoftheaboveversionoftheLindenbaum lemmausesL emma2.7,andexploitsthe Name′
+ruletointroduce thenominal i.) Asweareaiming for strong completeness results, (weak) o ne-step
+completeness as employed in weak completeness proofs using finitemodels [14, 19] is no longer
+adequate. Accordingly, our first criterion assumes astrong er condition:
+Definition 2.9. A rule set Risstrongly one-step complete if for every set X, every one-step con-
+sistent subset of Prop(Λ(P(X)))is one-step satisfiable.
+Lemma 2.10 (Named existence lemma, Version 1) .IfKis0-pasted and Ris strongly one-step
+complete, then there exists anamed canonical K-model.
+Insummary, wehave:
+Theorem 2.11. IfRis strongly one-step complete, then every extension of LRby pure axioms is
+both globally and locally strongly complete over countable hybrid models when equipped with the
+Namerule. That is, if Φ,Ψ⊆(Λ)andφ∈(Λ), thenΦ;Ψ⊢LRA+Nameφwhenever Φ;Ψ|=Cφ,
+whereCis the class of countable A-models.
+Proof.As usual, we show that every (LRA+Name)-Φ-consistent set Ψ⊆(Λ)is satisfiable in
+a countable A-modelMsuch thatM|= Φ(where satisfiability is clearly invariant under passing
+from(Λ)to(Λ)+). TheextendedLindenbaumlemmayieldsa 0-pastedmaximallyconsistent subset
+ABoxK⊆(Λ)+and a nominal iin(Λ)+such that @iΨ⊆K. By the named existence lemma,
+we find a named, hence countable, canonical K-modelM= (SK,γ,VK), and by the truth lemma
+(Lemma2.5), MisanA-model,M|= Φ,andM,Ki|= Ψ.
+Remark 2.12. In the literature (e.g. [3, Theorem 7.29]), the above comple teness theorem is some-
+times phrased as “completeness with respect to named models ”, i.e. models where every state is
+the denotation of some nominal; such models also played acen tral role in theearly development of
+hybrid logic by the Sofia school (see e.g. [15]). In detail, th is means that every state of the model
+is the denotation of a nominal in a language extended with countably many new nominals . This
+extension is necessary, as otherwise the consistent set {¬n|n∈N}would be satisfiable in a
+model where every state is named by a nominal n∈Nof the original language, which is clearly652 L.SCHR ¨ODER ANDD.PATTINSON
+impossible. Completeness with respect to models where ever y state is named by a nominal in an
+extended language, on the other hand, is an immediate conseq uence of completeness with respect
+tocountable models.
+Example 2.13. The previous theorem establishes strong completeness resu lts for pure extensions
+of all hybrid logics with neighbourhood semantics (Example 1.1.4) that are defined by rank-1 ax-
+ioms [20], i.e. modal formulas where the nesting depth of mod alities is uniformly equal to 1(such
+asthemonotonicity axiom /square(a∧b)→/squareb). For themonotonic cases, i.e.extensions of monotonic
+hybrid logic, these results are essentially known [24], whi le they seem to be new for the non-
+monotonic cases, i.e. extensions of classical hybrid logic not containing the monotonicity axiom,
+including, e.g., various deontic logics [12]. Moreover, th e theorem newly proves strong complete-
+ness of the hybridization of coalition logic, as Theorem 3.2 of [17] essentially states that coalition
+logic satisfies strong one-step completeness.
+2.2. PureCompleteness forBoundedLogics
+The condition of strong one-step completeness used in the pr evious section is a comparatively rare
+phenomenon [20];thestrength ofthecondition becomesclea rinthefact that,unlikeintheclassical
+case of Kripke semantics, the above did not require a notion o f1-pastedness [5]. We proceed to
+present an alternative approach for the case where one does h ave an analogue of the ( Paste-1) rule
+— this is the case if the operators are bounded, i.e., their satisfaction hinges, in each case, on only
+finitely and boundedly manystates of a model.
+Definition 2.14. A modal operator ♥isk-bounded fork∈Nwithrespect to a Λ-structureTif for
+every setXand everyA⊆X,
+[[♥]]X(A) =/uniontext
+B⊆A,#B≤k[[♥]]X(B).
+(This implies in particular that ♥is monotonic.) We say that Λis bounded w.r.t. Tif every modal
+operator♥inΛisk♥-bounded for some k♥.
+Theboundedness of an operator can now be internalized in the logical deduction system. In partic-
+ular, fork-bounded operators ♥,one has the paste rule
+(Paste♥(k))@j1φ∧···∧@jkφ∧@i♥(j1∨···∨jk)→ψ
+@i♥φ→ψ
+with the side condition that the jrare pairwise distinct fresh nominals. We write
+Φ⊢LRA+Name+Pasteφifφis derivable from assumptions in Φin the system LR+Namewhere
+additionally the rule (Paste♥(k))may be used in deductions for k-bounded operators ♥. This in-
+ducesthenotionof (LRA+Name+Paste)-Φ-consistency, whichwebrieflyrefertoasconsistency
+as we fix Φ,A, andRthroughout. Again, the system is clearly sound, i.e. Φ;Ψ|=Cφwhenever
+Φ;Ψ⊢LRA+Name+Pasteφ, whereCis theclass of A-models.
+Examples 2.15. 1.HybridK.The modal operator ♦is1-bounded. The arising paste rule
+(Paste♦(1))isprecisely the rule (paste♦)of [4].
+2.Graded hybrid logic. Themodal operator ♦kis(k+1)-bounded. Onethus has apaste rule
+(Paste♦k(k+1))@j1φ∧···∧@jk+1φ∧@i♦k(j1∨···∨jk+1)→ψ
+@i♦kφ→ψ
+withside conditions as before.NAMED MODELS IN COALGEBRAIC HYBRID LOGIC 653
+3.Positive Presburger hybrid logic. A Presburger operator/summationtextai·#(i)≥k(Example 1.1) is
+k-bounded if the aiare positive. E.g., this still allows expressing the statem ent, generally believed
+tobe valid in theGerman national football league, that atea m that has at least 37points will not be
+relegated: 3·#win+1·#draw≥37→ ¬relegated .
+Thegeneralized 1-pastedness condition for bounded operators isas follows.
+Definition 2.16. LetΛbe bounded. An ABox Kis1-pastedif whenever ♥isk-bounded and
+@i♥φ∈K,then{@j1φ,...,@jkφ,@i♥(j1∨···∨jk)} ⊆Kfor some nominals j1,...,jk.
+Again,itisclear thatif Λisbounded, then Kcaninduceanamedmodelonlyof Kis1-pasted. Itis
+easy to see that if Ris one-step complete and Λis bounded (in fact already if Rderives monotony
+for every ♥ ∈Λ),then every 1-pasted set isalso 0-pasted (Definition 2.6).
+Lemma2.17 (Extended Lindenbaum lemmafor 1-pasted sets) .LetΛbebounded. If Ψ⊆ F(Λ)is
+consistent, then there exist a 1-pasted maximally consistent ABox K⊆@(Λ)+and a nominal iin
+(Λ)+such that @iΨ⊆K,where(Λ)+isas inSection 2.1.
+Bounded operators now allow us to use a weaker version of one- step completeness. Instead of
+requiringthatallone-stepconsistent setsareone-stepsa tisfiable, wemayrestrictto finiteextensions
+of propositional variables.
+Definition 2.18. We say that Risstrongly finitary one-step complete if for every set X, every
+one-step consistent subset of Prop(Λ(Pfin(X)))isone-step satisfiable.
+Clearly, any strongly one-step complete rule set is also str ongly finitary one-step complete, but the
+example of graded hybrid logic witnesses that the converse i s not true. We note that the weaker
+criterionstillfailsforprobabilistic logicsduetoinher ent non-compactness [23];probabilistic logics
+also fail tobe bounded, asagiven probability p∈[0,1]can besplit into any number of summands.
+Together with boundedness, the above condition enables a se cond version of the named existence
+lemma.
+Lemma2.19 (Namedexistence lemma,Version 2) .IfΛisbounded, Risstrongly finitary one-step
+complete, and Kis1-pasted, then there exists anamed canonical K-model.
+Summarizing theabove, wehave the following extended compl eteness result.
+Theorem 2.20. LetΛbe bounded, and let Rbe strongly finitary one-step complete. Then every
+extension of LRby pure axioms is globally and locally strongly complete ove r countable hybrid
+models when equipped with the NameandPasterules. In other words, if Φ,Ψ⊆(Λ),φ∈(Λ),
+andCistheclass of all countable A-models, then Φ;Ψ⊢LRA+Name+Pasteφwhenever Φ;Ψ|=Cφ.
+The proof follows the same route via extended Lindenbaum lem ma, existence lemma, and truth
+lemmaas for Theorem 2.11.
+Example 2.21. By Example 2.15 and the fact that the known complete axiomati zations of the
+associated modal logics areinfact strongly finitaryone-st ep complete, theprevious theorem proves
+completeness of pure extensions of hybrid K, graded hybrid logic, and positive Presburger hybrid
+logic. Except for the standard case of hybrid K, these results seem to be new. In particular, we
+obtain completeness of pure extensions of graded (or positi ve Presburger) hybrid logic defining the
+following frame classes inmultigraph semantics:
+•Theclassof Kripkeframes ,seenastheclassofmultigraphswherethetransitionmulti plicity
+between twoindividual states is always at most 1, defined by the pure axiom ¬♦1i.654 L.SCHR ¨ODER ANDD.PATTINSON
+•Theclass of reflexivemultigraphs, defined by the pure axiom i→♦0i.
+•Theclass of transitive multigraphs, defined bythe pure axioms ♦0♦ni→♦ni,n≥0.
+•Theclass of symmetric multigraphs, i.e.,those where thetransition multiplicit y fromxtoy
+always equals the one from ytox, which isdefined by the pure axioms i∧♦kj→@j♦ki.
+Otherframeclassesofinterest,seee.g. [3,Section7.3],c anbecharacterizedsimilarlybytranslating
+the corresponding frameconditions from Kripke tomultigra ph semantics.
+2.3. TheMixed Case
+In some cases, the two methods laid out in the preceding secti ons can be combined for modal
+operators with several arguments that adhere, in each of the ir arguments, to one of the respective
+sets of semantic conditions. For the sake of readability, we formulate this explicitly only for the
+mixed binary case with a single modal operator, i.e. we assum e in this section that Λ ={♥}with
+♥binary; thegeneralization toarbitrary numbersof argumen ts, several modal operators etc. should
+be obvious, and essentially only requires more elaborate te rminology and notation.
+Definition 2.22. We say that Ris(strongly, strongly finitary) one-step complete if every one-step
+consistent subset of Prop(Λ(P(X)×Pfin(X)))isone-step satisfiable. Moreover, wesay that ♥is
+k-bounded in the second argument fork∈Nif for every set Xand allA,B⊆X,[[♥]]X(A,B) =/uniontext
+C⊆A,#C≤k[[♥]]X(A,C).
+Inthe same manner as for Theorems 2.11and 2.20,wederive:
+Theorem 2.23. IfRis (strongly, strongly finitary) one-step complete and ♥isk-bounded in the
+second argument, then every extension of LRby pure axioms is both locally and globally strongly
+completeovercountable hybridmodelswhenequippedwithth eappropriate NameandPasterules.
+Example2.24. HybridCK(Example 1.1) is easily seen tobe (strongly, strongly finita ry) one-step
+complete, and the operator >defined from the conditional operator ⇒bya > b:↔ ¬(a⇒ ¬b)
+is1-bounded in the second argument. By the above, it follows tha t every pure extension of hybrid
+CKis strongly complete over countable hybrid selection funct ion models. E.g. we may define
+the class of conditional frames where all expressible condi tions induce transitive relations by pure
+axioms(φ > φ > i )→(φ > i). Such frames satisfy also the dual axiom (using a propositio nal
+variablea)(φ⇒a)→(φ⇒(φ⇒a)),anaxiomforduplicating conditional assumptions. Simila r
+statementsapplytoacombinationofgradedandconditional logic(obtainablecompositionallyusing
+the methods of [21]), which has operators of the form a⇒kb“ifa, then one normally has more
+thankinstances of b”.
+Thesemantics of conditional logics ingeneral hascomplex r amifications, involving, e.g.,pref-
+erenceorderingsorsystemsofspheres(see,e.g.,[10,18]) ;applicationofourmethodstoconditional
+logicsbeyond CKisthesubjectoffurtherinvestigation. Wenotethatpureco mpletenessofahybrid
+extension of Lewis’ logic of counterfactuals has been estab lished recently [18].
+3. LocalBinding
+We next investigate completeness of a stronger hybrid langu age that includes the ↓binder, which
+binds astate variable tothecurrent state. Concretely, wea llow formulas oftheform ↓i.φ,wherein
+the nominal iis locally bound (for compactness of presentation, we give u p the usual distinction
+between nominals and state variables). Given a modal simila rity type Λ, we write ↓(Λ)for theNAMED MODELS IN COALGEBRAIC HYBRID LOGIC 655
+ensuing extension of (Λ). The reading of the formula ↓i.φis “φholds for the current state i”. The
+satisfaction relation in the extended logic isdefined by an a dditional clause for the ↓binder,
+(C,γ,V)|=↓i.φiff(C,γ,V[c/i])|=φ
+wherecis a state in a coalgebra CandV[c/i]is obtained from Vby modifying the value of itoc.
+Thesemantics of the ↓binder immediately translates into the axiom scheme (see e. g. [4])
+(DA) @ i((↓j.φ)↔φ[i/j]).
+Given a set RofΛ-rules, a set Φ⊆↓(Λ)of formulas and a set A ⊆@↓(Λ)of pure axioms,
+we write Φ⊢LRA+Name+Paste+DAφfor the extension of the associated provability predicate
+⊢LRA+Name+Pastewith(DA). Using (DA), one easily proves an extension of the truth lemma
+fornamedmodels(Lemma2.5)to ↓(Λ),sothatthecompleteness resultsforpureextensions prove d
+before (Theorems 2.11, 2.20, and 2.23) transfer immediatel y toL↓. We make this explicit for the
+bounded case:
+Theorem 3.1. IfΛis bounded and Ris strongly finitary one-step complete, then every pure exte n-
+sion ofL↓is strongly locally and globally complete over countable hy brid models. In other words,
+Φ;Ψ|=CφiffΦ;Ψ⊢LRA+Name+Paste+DAφfor allφ∈↓(Λ)and allΦ,Ψ⊆(Λ), whereCis the
+class of all countable A-models.
+Remark 3.2. As noted in [24], the named model construction more generall y yields completeness
+for anylocally definable extension of the hybrid language, i.e. any extension whose s emantics at
+named states isdefined byaformula similar to (DA).
+Example 3.3. Continuing Example 2.15, Theorem 3.1 reproves not only the k nown completeness
+of pure extensions of hybrid Kwith↓, but also the completeness of pure extensions of graded (or
+positive Presburger) hybrid logic with ↓. This extends easily to the multi-agent case, or, in descrip -
+tionlogicterminology, todescriptionlogicswithmultipl eroles. As,moreover,botharolehierarchy
+and transitivity of roles can be defined using pure axioms, we thus arrive at a complete axiomati-
+zation of an extension of the description logic SHOQwith satisfaction operators and ↓, which has
+been used in connection with conjunctive query answering [1 1], and allows, e.g., talking about the
+number of stepchildren of a stepmother, in continuation of t hestepmother example from [13], .
+4. Conclusions
+We have laid out two criteria for the existence of named canon ical models in coalgebraic hybrid
+logics — one that applies to cases where one has an analogue of the so-called Paste- 1rule of stan-
+dard hybrid logic, and one which applies to cases where one do es not need any such rule. While
+the latter means essentially that the logic is equipped with a neighbourhood semantics, the former
+requires that all modal operators of the logic are bounded, i .e. there isalways only abounded num-
+ber of states relevant for their satisfaction at each point. Our main novel example of this type is
+graded hybrid logic (and an extension of it using certain Pre sburger modalities [9]). The named
+model construction entails completeness of pure extension s and completeness of extended hybrid
+languages with the local binder ↓(of which the I–me construct of [13] is a single-variable res tric-
+tion), which we thus obtain as new results for, e.g., hybrid c oalition logic, hybrid classical modal
+logic, several hybrid deontic logics, hybrid conditional l ogic, graded hybrid logic, and anextension
+of the description logic SHOQ. An open question that remains is the existence of so-called ortho-
+dox axiomatizations [4] in the presence of ↓, as well as to find an analogue of the characterization
+result of [24] stating that a variant of the Paste- 1rule characterizes the Kripke models among the656 L.SCHR ¨ODER ANDD.PATTINSON
+topological models of S4. A further topic of investigation is to find decidable fragme nts of the lan-
+guage with ↓; wenoteslightly speculatively that thefragment usedin[1 3]may,inour terminology,
+be seen as requiring that a suitably defined NNFof aformula co ntains only positive occurrences of
+bound nominals under bounded modal operators.
+References
+[1] C. Areces, P. Blackburn, and M. Marx. A road-map on comple xity for hybrid logics. In Computer Science Logic,
+CSL99, vol. 1683 of LNCS,pp. 307–321, 1999.
+[2] F.Baader, D.Calvanese, D. L.McGuinness, D.Nardi, and P .F.Patel-Schneider,eds. The Description Logic Hand-
+book. Cambridge UniversityPress,2003.
+[3] P.Blackburn, M.de Rijke,and Y.Venema. Modal Logic . Cambridge UniversityPress,2001.
+[4] P.Blackburn andB. tenCate.Pure extensions, proof rule s,and hybridaxiomatics. Stud. Log. ,84:277–322, 2006.
+[5] P.Blackburn andM. Tzakova. Hybrid languages and tempor al logic.Logic J.IGPL ,7:27–54, 1999.
+[6] B.Chellas. Modal Logic . Cambridge UniversityPress,1980.
+[7] C.Cirstea,A.Kurz,D.Pattinson,L.Schr¨ oder,andY.Ve nema.Modallogicsarecoalgebraic. TheComputerJournal ,
+2009. Inprint.
+[8] G.D’AgostinoandA.Visser.Finalityregained: Acoalge braic studyofScott-setsandmultisets. Arch.Math.Logic ,
+41:267–298, 2002.
+[9] S. Demri and D. Lugiez. Presburger modal logic is only PSP ACE-complete. In Automated Reasoning, IJCAR 06 ,
+vol.4130 of LNAI,pp. 541–556. Springer, 2006.
+[10] N.FriedmanandJ.Y.Halpern.Onthecomplexityofcondi tional logics.In Knowledge RepresentationandReason-
+ing, KR94 , pp. 202–213. Morgan Kaufmann, 1994.
+[11] B. Glimm, I. Horrocks, and U. Sattler. Conjunctive quer y answering for description logics with transitive roles. I n
+DescriptionLogics, DL06 , vol. 189 of CEURWorkshop Proceedings . CEUR-WS.org,2006.
+[12] L. Goble. A proposal for dealing with deontic dilemmas. InDeontic Logic in Computer Science, DEON 04 , vol.
+3065 ofLNAI,pp. 74–113. Springer,2004.
+[13] M. Marx. Narcissists,stepmothers andspies. In Description Logics,DL 02 , CEURWorkshop Proceedings, 2002.
+[14] R. Myers, D. Pattinson, and L. Schr¨ oder. Coalgebraic h ybrid logic. In Foundations of Software Science and Com-
+putation Structures, FOSSACS09 , LNCS.Springer, 2009. Toappear.
+[15] S.PassyandT. Tinchev. PDLwithdata constants. Inf. Process.Lett. ,20:35–41, 1985.
+[16] D. Pattinson. Coalgebraic modal logic: Soundness, com pleteness and decidability of local consequence. Theoret.
+Comput. Sci. ,309:177–193, 2003.
+[17] M. Pauly.A modal logic for coalitional power ingames. J.Logic Comput. , 12:149–166, 2002.
+[18] K.Sano. Hybridcounterfactual logics. J. Log. Lang. Inf. ,18:515539, 2009.
+[19] L.Schr¨ oder. Afinite model construction for coalgebra ic modal logic. J. Log. Algebr. Prog. ,73:97–110, 2007.
+[20] L.Schr¨ oder and D.Pattinson. Rank-1modal logics are c oalgebraic. J.Logic Comput. Inprint.
+[21] L. Schr¨ oder and D. Pattinson. Modular algorithms for h eterogeneous modal logics. In Automata, Languages and
+Programming, ICALP07 , vol.4596 of LNCS,pp. 459–471. Springer, 2007.
+[22] L.Schr¨ oder and D.Pattinson. PSPACEbounds for rank-1 modal logics. ACM Trans. Comput. Log. , 10(2:13):1–33,
+2009.
+[23] L.Schr¨ oderandD.Pattinson.Strongcompletenessofc oalgebraicmodallogics.In TheoreticalAspectsofComputer
+Science, STACS 09 , Leibniz International Proceedings in Informatics, pp. 67 3–684. Schloss Dagstuhl – Leibniz-
+Zentrum f¨ ur Informatik, 2009.
+[24] B. ten Cate and T. Litak. Topological perspective on the hybrid proof rules. In Hybrid Logic, HyLo 06 , vol. 174 of
+ENTCS,pp. 79–94, 2007.
+This work is licensed under the Creative Commons Attributio n-NoDerivsLicense. To view a
+copyofthislicense,visit http://creativecommons.org/licenses/by-nd/3.0/ .
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+arXiv:1001.0736v1  [math.ST]  5 Jan 2010A note on the group lasso and a sparse group
+lasso
+Jerome Friedman∗
+Trevor Hastie†
+andRobert Tibshirani‡
+January 5, 2010
+Abstract
+We consider the group lasso penalty for the linear model. We n ote that
+the standard algorithm for solving the problem assumes that the model
+matrices in each group are orthonormal. Here we consider a mo re general
+penalty that blends the lasso ( L1) with the group lasso (“two-norm”). This
+penalty yields solutions that are sparse at both the group an d individual
+feature levels. We derive an efficient algorithm for the resul ting convex
+problem based on coordinate descent. This algorithm can als o be used
+to solve the general form of the group lasso, with non-orthon ormal model
+matrices.
+1 Introduction
+In this note, we consider the problem of prediction using a linear mode l. Our data
+consist of y, a vector of Nobservations, and X, aN×pmatrix of features.
+Suppose that the ppredictors are divided into Lgroups, with pℓthe number
+in group ℓ. For ease of notation, we use a matrix Xℓto represent the predictors
+corresponding to the ℓth group, with corresponding coefficient vector βℓ. Assume
+thatyandXhas been centered, that is, all variables have mean zero.
+∗Dept. of Statistics, Stanford Univ., CA 94305, jhf@stanford.edu
+†Depts. of Statistics, and Health, Research & Policy, Stanford Univ ., CA 94305,
+hastie@stanford.edu
+‡Depts. of Health, Research & Policy, and Statistics, Stanford Univ , tibs@stanford.edu
+1In an elegant paper, Yuan & Lin (2007) proposed the group lasso wh ich solves
+the convex optimization problem
+min
+β∈Rp/parenleftBigg
+||y−1−L/summationdisplay
+ℓ=1Xℓβℓ||2
+2+λL/summationdisplay
+ℓ=1√pℓ||βℓ||2/parenrightBigg
+, (1)
+where the√pℓterms accounts for the varying group sizes, and ||· ||2is the Eu-
+clidean norm (not squared). This procedure acts like the lasso at th e group level:
+depending on λ, an entire group of predictors may drop out of the model. In fact
+if the group sizes are all one, it reduces to the lasso. Meier et al. (20 08) extend
+the group lasso to logistic regression.
+The group lasso does not, however, yield sparsity within a group. Th at is,
+if a group of parameters is non-zero, they will all be non-zero. In t his note
+we propose a more general penalty that yields sparsity at both the group and
+individual feature levels, in order to select groups and predictors w ithin a group.
+We also point out that the algorithm proposed by Yuan & Lin (2007) fo r fitting
+the group lasso assumes that the model matrices in each group are orthonormal.
+The algorithm that we provide for our more general criterion also wo rks for the
+standard group lasso with non-orthonormal model matrices.
+We consider the sparse group lasso criterion
+min
+β∈Rp/parenleftBigg
+||y−L/summationdisplay
+ℓ=1Xℓβℓ||2
+2+λ1L/summationdisplay
+ℓ=1||βℓ||2+λ2||β||1/parenrightBigg
+. (2)
+whereβ= (β1,β2,...βℓ) is the entire parameter vector. For notational simplicity
+we omit the weights√pℓ. Expression (2) is the sum of convex functions and is
+therefore convex. Figure 1 shows the constraint region for the g roup lasso, lasso
+and sparse group lasso. A similar penalty involving both group lasso an d lasso
+terms is discussed in Peng et al. (2009). When λ2= 0, criterion (2) reduces to
+the group lasso, whose computation we discuss next.
+2 Computation for the group lasso
+Here we briefly review the computation for the group lasso of Yuan & Lin (2007).
+In the process we clarify a confusing issue regarding orthonormalit y of predictors
+within a group.
+The subgradient equations (see e.g. Bertsekas (1999)) for the g roup lasso are
+−XT
+ℓ(y−/summationdisplay
+ℓXℓβℓ)+λ·sℓ= 0;ℓ= 1,2,...L, (3)
+2−1.0 −0.5 0.0 0.5 1.0−1.0 −0.5 0.0 0.5 1.0
+β1β2
+Figure 1: Contour lines for the penalty for the group lasso (dotted), la sso (dashed) and
+sparse group lasso penalty (solid), for a single group with t wo predictors.
+wheresℓ=βℓ/||βℓ||ifβℓ/negationslash= 0 andsℓis a vector with ||sℓ||2<1 otherwise. Let the
+solutions be ˆβ1,ˆβ2...ˆβℓ. If
+||XT
+ℓ(y−/summationdisplay
+k/negationslash=ℓXkˆβk)||< λ (4)
+thenˆβℓis zero; otherwise it satisfies
+ˆβℓ= (XT
+ℓXℓ+λ/||ˆβℓ||)−1XT
+ℓrℓ (5)
+where
+rℓ=y−/summationdisplay
+k/negationslash=ℓXkˆβk
+Now if we assume that XT
+ℓXℓ=I, and let sℓ=XT
+ℓrℓ, then (5) simplifies to
+ˆβℓ= (1−λ/||sℓ||)sℓ. This leads to an algorithm that cycles through the groups
+k, and is a blockwise coordinate descent procedure. It is given in Yuan & Lin
+(2007).
+Ifhowever thepredictors arenotorthonormal, oneapproachist oorthonormal-
+ize them before applying the group lasso. However this will not gener ally provide
+a solution to the original problem. In detail, if Xℓ=UDVT, then the columns of
+U=XℓVD−1are orthonormal. Then Xℓβℓ=UVD−1βℓ=U[VD−1βℓ] =Uβℓ∗.
+3But||βℓ∗||=||βℓ||only ifD=I. This will not be true in general, e.g. if Xis a
+set of dummy varables for a factor, this is true only if the number of observations
+in each category is equal.
+Hence an alternative approach is needed. In the non-orthonorma l case, we can
+think of equation (5) as a ridge regression, with the ridge paramete r depending
+on||ˆβℓ||. A complicated scalar equation can be derived for ||ˆβℓ||from (5); then
+substituting into the right-hand side of (5) gives the solution. Howe ver this is not
+a good approach numerically, as it can involve dividing by the norm of a v ector
+that is very close to zero. It is also not guaranteed to converge. I n the next section
+we provide a better solution to this problem, and to the sparse grou p lasso.
+3 Computation for the sparse group lasso
+The criterion (1) is separable so that block coordinate descent can be used for its
+optimization. Therefore we focus on just one group ℓ, and denote the predictors
+byXℓ=Z= (Z1,Z2,...Zk), the coefficients by βℓ=θ= (θ1,θ2,...θk) and the
+residual by r=y−/summationtext
+k/negationslash=ℓXkβk. The subgradient equations are
+−ZT
+j(r−/summationdisplay
+jZjθj)+λ1sj+λ2tj= 0 (6)
+forj= 1,2,...kwheresj=θj/||θ||ifθℓ/negationslash= 0 andsis a vector satisfying ||s||2≤1
+otherwise, and tj∈sign(θj), that is tj=sign(θj) ifθj/negationslash= 0 and tj∈[−1,1] if
+θj= 0. Letting a=Xℓr, then a necessary and sufficient condition for θto be zero
+is that the system of equations
+aj=λ1sj+λ2tj (7)
+have a solution with ||s||2≤1 andtj∈[−1,1]. We can determine this by
+minimizing
+J(t) = (1/λ2
+1)k/summationdisplay
+j=1(aj−λ2tj)2=k/summationdisplay
+j=1s2
+j (8)
+with respect to the tj∈[−1,1] and then checking if J(ˆt)≤1. The minimizer is
+easily seen to be
+ˆtj=/braceleftBiggaj
+λ2if|aj
+λ2| ≤1,
+sign(aj
+λ2) if|aj
+λ2|>1.
+Now ifJ(ˆt)>1, then we must minimize the criterion
+1
+2N/summationdisplay
+i=1/parenleftBig
+ri−k/summationdisplay
+j=1Zijθj/parenrightBig2
++λ1||θ||2+λ2k/summationdisplay
+j=1|θj| (9)
+4Thisisthesumofaconvexdifferentiablefunction(firsttwoterms)a ndaseparable
+penalty, and hence we can use coordinate descent to obtain the glo bal minimum.
+Here are the details of the coordinate descent procedure. For ea chjletrj=
+r−/summationtext
+k/negationslash=jZkˆθk. Thenˆθj= 0 if|ZT
+jrj|< λ2. This follows easily by examining the
+subgradient equation corresponding to (9). Otherwise if |ZT
+jrj| ≥λ2we minimize
+(9) by a one-dimensional search over θj. We use the optimize function in the R
+package, which is a combination of golden section search and succes sive parabolic
+interpolation.
+This leads to the following algorithm:
+Algorithm for the sparse group lasso
+1. Start with ˆβ=β0
+2. Ingroup ℓdefinerℓ=y−/summationtext
+k/negationslash=ℓXkβk,Xℓ= (Z1,Z2,...Zk),βℓ= (θ1,θ2,...θk)
+andrj=y′−/summationtext
+k/negationslash=jZkθk. Check if J(ˆt)≤1 according to (8) and if so set
+ˆβℓ= 0. Otherwise for j= 1,2,...k, if|ZT
+jrj|< λ2thenˆθj= 0; if instead
+|ZT
+jrj| ≥λ2then minimize
+1
+2N/summationdisplay
+i=1(y′
+i−k/summationdisplay
+j=1Zijθj)2+λ1||θ||2+λ2k/summationdisplay
+j=1|θj| (10)
+overθjby a one-dimensional optimization.
+3. Iterate step (2) over groups ℓ= 1,2,...Luntil convergence.
+Ifλ2iszero, weinsteadusecondition(4)forthegroup-level test andw edon’tneed
+to check the condition |ZT
+jrj|< λ2. With these modifications, this algorithm also
+gives a effective method for solving the group lasso with non-orthog onal model
+matrices.
+Note that in the special case where XT
+ℓXℓ=I, withXℓ= (Z1,Z2,...Zk) then
+its is easy to show that
+ˆθj=/parenleftBig
+||S(ZT
+jy,λ2)||2−λ1/parenrightBig
++S(ZT
+jy,λ2)
+||S(ZT
+jy,λ2)||2(11)
+and this reduces to the algorithm of Yuan & Lin (2007).
+4 An example
+We generated n= 200 observations with p= 100 predictors, in ten blocks of ten.
+The second fifty predictors iall have coefficients of zero. The numb er of non-zero
+5coefficients in the first five blocks of 10 are (10, 8, 6, 4, 2, 1) respe ctively, with
+coefficients equal to ±1, the sign chosen at random. The predictors are standard
+Gaussianwithcorrelation0.2withinagroupandzerootherwise. Finally , Gaussian
+noise with standard deviation 4.0 was added to each observation.
+Figure2showsthesignsoftheestimatedcoefficientsfromthelasso , grouplasso
+and sparse group lasso, using a well chosen tuning parameter for e ach method (we
+setλ1=λ2for the sparse group lasso). The corresponding misclassification r ates
+for the groups and individual features are shown in Figure 3. We see that the
+sparse group lasso strikes an effective compromise between the las so and group
+lasso, yielding sparseness at the group and individual predictor leve ls.
+References
+Bertsekas, D. (1999), Nonlinear programming , Athena Scientific.
+Meier, L., van de Geer, S. & B¨ uhlmann, P. (2008), ‘The group lasso f or logistic
+regression’, Journal of the Royal Statistical Society B 70, 53–71.
+Peng, J., Zhu, J., Bergamaschi, A., Han, W., Noh, D.-Y., Pollack, J. R. & Wang,
+P. (2009), ‘Regularized multivariate regression for identifying mast er predic-
+tors with application to integrative genomics study of breast cance r’,Annals
+of Applied Statistics (to appear) .
+Yuan, M. & Lin, Y. (2007), ‘Model selection and estimation in regress ion
+with grouped variables’, Journal of the Royal Statistical Society, Series B
+68(1), 49–67.
+60 20 40 60 80 100−3 −2 −1 0 1 2 3
+PredictorCoefficientGroup lasso
+0 20 40 60 80 100−3 −2 −1 0 1 2 3
+PredictorCoefficientSparse Group lasso
+0 20 40 60 80 100−3 −2 −1 0 1 2 3
+PredictorCoefficientLasso
+Figure 2: Results for the simulated example. True coefficients are indi cated by the open
+triangles while the filled green circles indicate the sign of the estimated coefficients from
+each method.
+72 4 6 8 10 120 1 2 3 4 5
+Regularization parameterNumber of groups misclassified Lasso
+Group lasso
+Sparse Group lasso
+2 4 6 8 10 1210 20 30 40 50 60 70
+Regularization parameterNumber of features misclassified
+Figure 3: Results for the simulated example. The top panel shows the num ber of groups
+that are misclassified as the regularization parameter is va ried. A misclassified group
+is one with at least one nonzero coefficient whose estimated co efficients are all set to
+zero, or vice versa. The bottom panel shows the number of indiv idual coefficients that
+are misclassified, that is, estimated to be zero when the true coefficient is nonzero or
+vice-versa.
+8
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+arXiv:1001.0737v5  [math.CA]  28 Dec 2010Existence and uniqueness of solutions to
+systems of delay dynamic equations on time
+scales
+Ba¸ sak KARPUZ∗
+2000 AMS Subject Classification : Primary: 39A10, Secondary: 34C10.
+Keywords and Phrases : Delay dynamic equations, existence and uniqueness,
+Picard-Lindel¨ of theorem, variation of parameters formul a.
+Abstract
+The purpose of this paper is to establish Picard-Lindel¨ of t heorem
+for local uniqueness and existence results for first-order s ystems of
+nonlinear delay dynamic equations. In the linear case, we ex tend our
+results to global existence and uniqueness of solutions on t he entire
+interval (allowed to be unbounded above), and prove the vari ation
+of parameters formula for the unique solution in terms of the princi-
+pal solution. A simple example concerning the ordinary case is also
+provided.
+1 Introduction
+In [1], the theory of time scale calculus is introduced by S. Hilger in orde r
+to unify the discrete and the continuous analysis. This theory allows the
+researchers to study dynamic equations, which include both differe nce and
+differential equations. In the past decade, many papers appeare d focusing
+∗Address : AfyonKocatepeUniversity,DepartmentofMathematics, Facult yofScience
+and Arts, ANS Campus, 03200 Afyonkarahisar, Turkey.
+Email:bkarpuz@gmail.com
+Web:http://www2.aku.edu.tr/ ~bkarpuz
+1on the investigation of the asymptotic and oscillatory properties of solutions
+of delay dynamic equations of the form
+
+
+x∆(t) =/summationdisplay
+i∈[1,n]Npi(t)x(τi(t))+q(t) fort∈[β,∞)T
+x(t) =ϕ(t) fort∈[α,β]T,(1)
+wheren∈N,Tis a time scale with sup T=∞,α,β∈Twithα≤
+mini∈[1,n]Ninft∈[β,∞)T{τi(t)},pi,q∈Crd([β,∞)T,R),τi∈Crd([β,∞)T,T) sat-
+isfies lim t→∞τi(t) =∞andτi(t)≤tfor allt∈[β,∞)Tand alli∈[1,n]N,
+andϕ∈Crd([α,β]T,R), for instance, see the papers [2, 3, 4, 5, 6]. All the
+mentioned papers assume existence of solutions to (1), however w e have not
+met any result guaranteeing the existence and/or uniqueness of s olutions for
+this type of equations in the literature yet. In this paper, to fill the men-
+tioned gap, we will prove Picard-Lindel¨ of existence and uniqueness theorem
+for the following nonlinear delay dynamic equation
+/braceleftBigg
+x∆(t) =f/parenleftbig
+t,x(τ1(t)),x(τ2(t)),...,x(τn(t))/parenrightbig
+fort∈[β,γ]κ
+T
+x(t) =ϕ(t) fort∈[α,β]T,(2)
+wheren∈N,Tis a time scale, Xis a Banach space, α,β,γ∈Twith
+γ≥βandα≤mini∈[1,n]Ninft∈[β,γ]κ
+T{τi(t)},f∈Crd([β,γ]κ
+T×Xn,X),τi∈
+Crd([β,γ]κ
+T,T) satisfies τi(t)≤tfort∈[β,γ]Tand alli∈[1,n]N, and
+ϕ∈Crd([α,β]T,X). Then, we extend our results to prove existence and
+uniqueness for global solutions of the linear initial value problem (1). It is
+clear that letting τi(t) =tfor allt∈[β,γ]Tand alli∈[1,n]N, (2) reduces to
+the ordinary dynamic equation
+/braceleftBigg
+x∆(t) =f/parenleftbig
+t,x(t)/parenrightbig
+fort∈[β,γ]κ
+T
+x(β) =x0,(3)
+wherex0∈X. Some existence and uniqueness results for (3) can be found in
+[7,§8.2], [8,§2] and in the papers [9, 10]. In the continuous case ( T=R),
+thereadersmayfindPicard-Lindel¨ oftheoremfor(3)in[11, Theor em8.13]. It
+should be also mentioned that our results (see Corollary 3.1 and Rema rk 3.1
+below) particularly salvages the one given in [12, Theorem 1.1.1] for de lay
+differential equations.
+For the completeness in the paper, we find useful to remind some ba sic
+concepts of the time scale theory as follows. A time scale , which inherits the
+2standard topology on R, is a nonempty closed subset of reals. Throughout
+the paper, the notation Twill be used to denote time scales. On a time scale
+T, theforward jump operator , thebackward jump operator and thegraininess
+function are respectively defined by
+σ(t) := inf(t,∞)T, ρ(t) := sup( −∞,t)Tandµ(t) :=σ(t)−t
+fort∈T, where the intervals with the subscript Tdenote the intersection of
+the usual interval with the time scale. A point t∈Tis called right-dense if
+t <supTandσ(t) =t, while is called left-dense ift >infTandρ(t) =t.
+Also, a point t∈Tis called right-scattered ifσ(t)> t, and is called left-
+scattered ifρ(t)< t. TheHilger derivative (in short derivative) of a function
+f:T→Ris defined by
+f∆(t) :=
+
+f/parenleftbig
+σ(t)/parenrightbig
+−f(t)
+µ(t), µ(t)>0
+lim
+s∈T
+s→tf(t)−f(s)
+σ(t)−s, µ(t) = 0
+fort∈Tκ(providedthatlimitexists), and Tκ:=T\{supT}ifsupT= maxT
+and satisfies ρ(maxT)/\e}atio\slash= maxT; otherwise, Tκ:=T. A function fis called
+rd-continuous provided that it is continuous at right-dense points in T, and
+has finite limit at left-dense points, and the set of rd-continuous are denoted
+by Crd(T,R), its subset C1
+rd(T,R) involves functions whose derivative is also
+in Crd(T,R). For a function f∈Crd(T,R), theCauchy integral is defined by
+/integraldisplayt
+sf(η)∆η=F(t)−F(s) fors,t∈T,
+whereF∈C1
+rd(T,R) is an anti-derivative of the function fonT. A function
+f∈Crd(T,R) is called regressive if 1 +µ(t)f(t)/\e}atio\slash= 0 for all t∈T, and
+f∈Crd(T,R) is called positively regressive if 1+µ(t)f(t)>0 for allt∈T.
+Theset of regressive functions and theset of positively regressive functions
+are defined by R(T,R) andR+(T,R), respectively, and the set of negatively
+regressive functions R−(T,R) is defined similarly. Let f∈ R(T,R), then the
+generalized exponential function ef(·,s) on a time scale Tis defined to be the
+unique solution of the initial value problem
+/braceleftBigg
+x∆(t) =f(t)x(t) fort∈Tκ
+x(s) = 1
+3for some fixed s∈T. Forh >0, setCh:={z∈C:z/\e}atio\slash=−1/h},Zh:=
+{z∈C:−π/h <Im(z)≤π/h}, andC0:=Z0:=C. Forh≥0, define the
+cylinder transformation ξh:Ch→Zhby
+ξh(z) :=
+
+z, h = 0
+1
+hLog(1+hz), h >0
+forz∈Chwith 1+hz/\e}atio\slash= 0. Iff∈ R([s,t]T,R), then the exponential function
+can also be written in the form
+ef(t,s) := exp/braceleftbigg/integraldisplayt
+sξµ(η)(f(η))∆η/bracerightbigg
+fors,t∈T.
+Itisknownthattheexponential functione f(·,s)isstrictlypositiveon[ s,∞)T
+provided that f∈ R+([s,∞)T,R), while e f(·,s) alternates in sign at right-
+scattered points of the interval [ s,∞)Tprovided that f∈ R−([s,∞)T,R).
+The readers are referred to the books [7, 8] for further intere sting details in
+the time scale theory.
+The paper is organized as follows: In §2, we state and prove Picard-
+Lindel¨ of local existence and uniqueness of theorem to first-orde r nonlinear
+delay dynamic equation of the form (2); in §3, we focus our attention to
+linear equationsinordertoprove globalexistence anduniqueness o fsolutions
+together with the solution representation formula. At the end of t he paper,
+we provide a simple example.
+2 Picard-Lindel¨ of theorem for nonlinear de-
+lay dynamic equations
+In this section, we prove Picard-Lindel¨ of theorem for the solution s of the
+nonlinear initial valueproblem (2). Just as inthecontinuous andthe d iscrete
+case the solution of the initial value problem in (2) is equivalent to that of
+the corresponding integral equation
+x(t) =
+
+ϕ(β)+/integraldisplayt
+βf/parenleftbig
+η,x(τ1(η)),x(τ2(η)),...,x(τn(η))/parenrightbig
+∆ηfort∈[β,γ]T
+ϕ(t) fort∈[α,β]T.
+(4)
+4Actually, we will prove the existence and uniqueness of solutions to t he delay
+∆-integral equation (4). The key idea of the proof depends on con structing
+a sequence of so-called successive Picard approximations, which un iformly
+converges to the unique solution of (4). For x0∈Xandε∈R+, we introduce
+the notation B(x0,ε) :={x∈X:/bardblx−x0/bardbl ≤ε}and define I(ϕ,ε) :=/uniontext
+t∈[α,β]TB(ϕ(t),ε). The main result of the paper reads as follows.
+Theorem 2.1 (Picard-Lindel¨ oftheorem) .Assume that for some ε∈R+,f∈
+Crd([β,γ]κ
+T×In(ϕ,ε),X), and that for some M∈R+,/bardblf(t,u1,u2,...,u n)/bardbl ≤
+Mfor allt∈[β,γ]κ
+Tand allu1,u2,...,u n∈I(ϕ,ε), and that for some
+L∈R+,fsatisfies the Lipschitz condition
+/vextenddouble/vextenddoublef(t,u1,u2,...,u n)−f(t,v1,v2,...,v n)/vextenddouble/vextenddouble≤L/summationdisplay
+i∈[1,n]N/vextenddouble/vextenddoubleui−vi/vextenddouble/vextenddouble(5)
+for allt∈[β,γ]κ
+Tand allu1,u2,...,u n,v1,v2,...,v n∈I(ϕ,ε). Then, the
+initial value problem (2)has a unique solution xon the interval [α,σ(ζ)]T⊂
+[α,γ]T, whereζ:= max[β,β+δ]Tandδ:= min{γ−β,ε/M}.
+Proof.Ifβ=γ, then trivially the unique solution to (2) solution is ϕ, we
+therefore consider the case γ > βbelow. We shall first prove the existence of
+a unique solution on [ α,ζ]T, and then extend to [ α,σ(ζ)]T. Define the space
+of functions Ω with the functions satisfying the following two proper ties:
+(i)x∈Crd([α,β]T,X) andx∈C1
+rd([β,ζ]T,X),
+(ii)x(t) =ϕ(t) for allt∈[α,β]Tandx(t)∈B(ϕ(β),ε) for allt∈[β,ζ]T.
+In view of (4), define the operator T: Ω→Ω as follows:
+(Tx)(t) :=
+
+ϕ(β)+/integraldisplayt
+βf/parenleftbig
+η,x(τ1(η)),x(τ2(η)),...,x(τn(η))/parenrightbig
+∆ηfort∈[β,ζ]T
+ϕ(t) fort∈[α,β]T.
+(6)
+It is clear that TΩ⊂Ω, and Ω is a closed subset of the Banach space
+Crd([α,β]T,X)∩C1
+rd([β,ζ]T,X) endowed with the supremum norm. Define
+the sequence of functions {xk}k∈Nby
+xk+1(t) := (Txk)(t) fort∈[α,ζ]Tandk∈N, (7)
+5where
+x0(t) :=/braceleftBigg
+ϕ(β) fort∈[β,ζ]T
+ϕ(t) fort∈[α,β]T.
+By applying induction on k∈N, we will prove that
+/vextenddouble/vextenddoublexk(t)−xk−1(t)/vextenddouble/vextenddouble≤MLk−1hk(t,β) for all t∈[α,ζ]Tand allk∈N,(8)
+wherehk:T2→Ris the generalized polynomial defined by
+hk(t,s) :=
+
+1, k = 0/integraldisplayt
+shk−1(η,s)∆η, k∈N
+fors,t∈T(see [7,§1.9]). By the definition of the sequence {xk}k∈N, (8)
+holds on [ α,β]T. Hence, it remains to prove (8) on [ β,ζ]T. For all t∈[β,ζ]T,
+we have
+/vextenddouble/vextenddoublex1(t)−x0(t)/vextenddouble/vextenddouble=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt
+βf/parenleftbig
+η,x0(τ1(η)),x0(τ2(η)),...,x 0(τn(η)/parenrightbig
+∆η/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+≤M(t−β) =Mh1(t,β),
+which proves (8) for k= 1. Suppose now that (8) is true for some k∈N,
+then for all t∈[β,ζ]Twe have
+/vextenddouble/vextenddoublexk+1(t)−xk(t)/vextenddouble/vextenddouble=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt
+β/bracketleftBig
+f/parenleftbig
+η,xk(τ1(η)),xk(τ2(η)),...,x k(τn(η)/parenrightbig
+−f/parenleftbig
+η,xk−1(τ1(η)),xk−1(τ2(η)),...,x k−1(τn(η)/parenrightbig/bracketrightBig
+∆η/vextenddouble/vextenddouble/vextenddouble/vextenddouble
+≤L/integraldisplayt
+β/summationdisplay
+i∈[1,n]N/vextenddouble/vextenddoublexk(τi(η))−xk−1(τi(η))/vextenddouble/vextenddouble∆η
+≤MLk/integraldisplayt
+βhk(η,β)∆η=MLkhk+1(t,β),
+which proves that (8) is true when kis replaced with ( k+1). Hence, we have
+just proved (8). Now, we prove that the sequence
+/braceleftBigg
+x0(t)+/summationdisplay
+ℓ∈[0,k)N0/bracketleftbig
+xℓ+1(t)−xℓ(t)/bracketrightbig/bracerightBigg
+k∈N={xk(t)}k∈Nfort∈[β,ζ]T(9)
+6converges uniformly. For all t∈[β,ζ]Tand allk∈N, we have
+/vextenddouble/vextenddoublexk(t)/vextenddouble/vextenddouble≤/vextenddouble/vextenddoublex0(t)/vextenddouble/vextenddouble+/summationdisplay
+ℓ∈[0,k)N0/vextenddouble/vextenddoublexℓ+1(t)−xℓ(t)/vextenddouble/vextenddouble
+≤/vextenddouble/vextenddoublex0(t)/vextenddouble/vextenddouble+M/summationdisplay
+ℓ∈[0,k)N0Lℓhℓ+1(t,β),
+and for the majorant series, we have
+/vextenddouble/vextenddoublex0(t)/vextenddouble/vextenddouble+M/summationdisplay
+ℓ∈N0Lℓhℓ+1(t,β) =/vextenddouble/vextenddoubleϕ(β)/vextenddouble/vextenddouble+M
+L/parenleftbig
+eL(t,β)−1/parenrightbig
+≤/vextenddouble/vextenddoubleϕ(β)/vextenddouble/vextenddouble+M
+L/parenleftbig
+eL(ζ,β)−1/parenrightbig
+foranyt∈[β,ζ]T(see[7,Theorem1.113,Theorem2.35]and[13,Lemma4.4]).
+Recall that for all s,t∈Twitht≥s, we have e L(t,s)≥1 from [7, Theo-
+rem 1.76, Theorem 2.36, Exercise 2.46, Theorem 2.48]. It follows from the
+Weierstrass M-test that the infinite series
+x0(t)+/summationdisplay
+ℓ∈N0/bracketleftbig
+xℓ+1(t)−xℓ(t)/bracketrightbig
+fort∈[α,ζ]T,
+whose partial sum sequence is (9), converges uniformly. So that t he sequence
+of Picard iterates {xk}k∈Nconverges uniformly on [ α,ζ]T, let
+x(t) := lim
+k→∞xk(t) fort∈[α,ζ]T. (10)
+For allt∈[β,ζ]Tand allk∈N0, we have
+/vextenddouble/vextenddoublef/parenleftbig
+t,xk(τ1(t)),xk(τ2(t)),...,x k(τn(t))/parenrightbig
+−f/parenleftbig
+t,xk(τ1(t)),xk(τ2(t)),...,x k(τn(t))/parenrightbig/vextenddouble/vextenddouble≤L/summationdisplay
+i∈[1,n]N/vextenddouble/vextenddoublexk(τi(t))−x(τi(t))/vextenddouble/vextenddouble.
+Thisprovesthatthesequence offunctions {f(·,xk◦τ1,xk◦τ2,...,x k◦τn)}k∈N0
+converges uniformlytothelimit function f(·,x◦τ1,x◦τ2,...,x◦τn)on[β,ζ]T.
+Lettingk→ ∞in (7) and using [14, Theorem 3.11], we learn that xis the
+fixed point of the operator Tdefined by (6), i.e., x=Txon [α,ζ]T. There-
+fore, we have just proved that xdefined by (10) solves (4) or equivalently
+(2), i.e., solutions to (2) and/or (4) exist.
+7To show uniqueness, assume for the sake of contradiction that xandyare
+two different solutions of (2) on [ α,ζ]T. By the definition of the initial value
+problem x=yon [α,β]T. Setz(t) := supη∈[α,t]T/bardblx(η)−y(η)/bardblfort∈[α,ζ]T,
+thenz∈Crd([α,ζ]T,R) is nonnegative and not identically zero. Using (4)
+and (5), we get
+z(t)≤nL/integraldisplayt
+βz(η)∆ηfor allt∈[β,ζ]T,
+which yields by applying the well-known Gr¨ onwall inequality (see [7, Th eo-
+rem 6.4]) that zis nonpositive on [ β,ζ]T. This is a contradiction and thus
+the solution of (2) is unique on [ α,ζ]T. The proof is completed here if ζis
+right-dense.
+Now let ζbe right-scattered, and ybe the solution of (2) which exists
+uniquely on [ α,ζ]T, and define x: [α,σ(ζ)]T→Xby
+x(t) :=/braceleftBigg
+y(ζ)+µ(ζ)f/parenleftbig
+ζ,y(τ1(ζ)),y(τ2(ζ)),...,y(τn(ζ))/parenrightbig
+fort=σ(ζ)
+y(t) fort∈[α,ζ]T.
+Then,xis the unique solution of (2) on [ α,σ(ζ)]T, and this completes the
+proof.
+Note that for (3), the conclusion of Theorem 2.1 coincides with that of
+[8, Theorem 2.1.1].
+Remark 2.1. Although Theorem 2.1 is a local existence result, we may in-
+terpret the solution as a new initial function, and apply the result repeatedly
+to obtain the unique solution on a larger interval.
+Remark 2.2. If the time scale Tconsists of only isolated points then the
+solution exists and is unique on the entire interval [α,β]T. To see this, one
+may apply Theorem 2.1 for a total of several times (at most num ber of the
+elements in [α,β]T).
+3 More on linear equations
+In this section, we restrict our attention to linear equations and ex tend the
+results in the previous section to prove existence and uniqueness o f solutions
+on the whole interval, where the equation is defined. Later on, we giv e
+the definition of the principal solution and prove the solution repres entation
+formula in terms of the principal solution and the initial function.
+83.1 Global existence and uniqueness of linear delay dy-
+namic equations
+LetXbe a Banach space and B(X) be the Banach algebra on X. InB(X),
+we now consider the following linear delay dynamic equation:
+
+
+x∆(t) =/summationdisplay
+i∈[1,n]Npi(t)x(τi(t))+q(t) fort∈[β,γ]κ
+T
+x(t) =ϕ(t) fort∈[α,β]T,(11)
+where [1,n]Nis a starting bounded segment of N,α,β,γ∈Twithγ > βand
+α≤mini∈[1,n]Ninft∈[β,γ]κ
+T{τi(t)},pi∈Crd([β,γ]κ
+T,B(X)),q∈Crd([β,γ]κ
+T,X),
+τi∈Crd([β,γ]κ
+T,T) satisfies τi(t)≤tfor allt∈[β,γ]Tand alli∈[1,n]Nand
+ϕ∈Crd([α,β]T,X).
+The main result of this subsection is the following.
+Theorem 3.1. The linear initial value problem (11)admits exactly one so-
+lution on the entire interval [α,γ]T.
+Proof.WeshallapplythemethodofstepsmentionedinRemark2.1toobtain
+the unique solution on [ α,γ]T. PickM1,M2∈R+such that
+sup
+η∈[β,γ]T/summationdisplay
+i∈[1,n]N/bardblpi(η)/bardbl ≤M1and sup
+η∈[β,γ]T/bardblq(η)/bardbl ≤M2(12)
+Recall that every rd-continuous (more truly regulated) function defined on a
+compact interval of Tis bounded (see [7, Theorem 1.65]). Define
+f(t,u1,u2,...,u n) :=/summationdisplay
+i∈[1,n]Npi(t)ui+q(t)
+fort∈[β,γ]Tandu1,u2,...,u n∈X. Then, itiseasytoseethattheLipschitz
+conditionholdstrivially on[ β,γ]T×Xn(withthe Lipschitz constant M1>0).
+For some fixed k0∈N, letβ0:=β < β 1<···< βk0:=γbe a partition
+of the interval [ β,γ]Tsatisfying βk=σ(βk−1) ifµ(βk−1)>1/(2M1) and
+|βk−βk−1| ≤1/(2M1) ifµ(βk−1)≤1/(2M1) for all k∈[1,k0]N(see [15,
+Lemma 2.6]). For convenience in the notation, we define y0:=ϕ,β−1:=α
+andν0:= supη∈[β−1,β0]T/bardbly0(η)/bardbl. We may find ε0∈R+such that ε0/(M1(ε0+
+ν0)+M2)≥1/(2M1)(notethatas ε0→ ∞theleft-handsideoftheinequality
+tends to 1 /M1). Clearly, /bardblf(t,u1,u2,...,u n)/bardbl ≤M1(ε0+ν0) +M2for all
+9t∈[β,γ]κ
+Tandallu1,u2,...,u n∈B(0X,ε0+ν0), where0 Xisthezerovectorin
+X. First consider the simple case µ(β0)>1/(2M1), then we have σ(β0) =β1.
+Applying Theorem 2.1 to the initial value problem
+
+
+x∆(t) =/summationdisplay
+i∈[1,n]Npi(t)x(τi(t))+q(t) fort∈[β0,β1]κ
+T
+x(t) =y0(t) fort∈[β−1,β0]T,(13)
+we learn that (13) admits a unique solution y1on [β−1,β1]T. Next consider
+the case µ(β0)≤1/(2M1), then applying Theorem 2.1 to (13), we again see
+that there exists a unique solution y1on [β−1,β1]T. It should be mentioned
+herethat I(y0,ε0)⊂B(0X,ε0+ν0)andmin {β1−β0,ε0/(M1(ε0+ν0)+M2)} ≥
+min{β1−β0,1/(2M1)}, andthisshows thatthewidthoftheexistence interval
+for the unique solution of (14) guaranteed by Theorem 2.1 can not b e less
+than we have proved. In the next step, we may pick ε1∈R+such that
+ε1/(M1(ε1+ν1)+M2)≥1/(2M1), where ν1:= supη∈[β−1,β1]T/bardbly1(η)/bardbl. Then
+we have /bardblf(t,u1,u2,...,u n)/bardbl ≤M1(ε1+ν1)+M2for allt∈[β,γ]κ
+Tand all
+u1,u2,...,u n∈B(0X,ε1+ν1). It follows from Theorem 2.1 (in both of the
+possible cases µ(β1)>1/(2M1) andµ(β1)≤1/(2M1) as shown in the first
+step) that the initial value problem
+
+
+x∆(t) =/summationdisplay
+i∈[1,n]Npi(t)x(τi(t))+q(t) fort∈[β1,β2]κ
+T
+x(t) =y1(t) fort∈[β−1,β1]T(14)
+admits a unique solution y2on [β−1,β2]TsinceI(y1,ε1)⊂B(0X,ε1+ν1) and
+min{β2−β1,ε1/(M1(ε1+ν1) +M2)} ≥min{β2−β1,1/(2M1)}. Repeating
+in this manner, we obtain the unique solution ykon each of the intervals
+[β−1,βk]Tfor allk∈[1,k0]N. Finally, we deduce that the unique solution x
+of (11) on the whole interval [ α,γ]Tisx=yk0, and the proof is therefore
+completed.
+Remark 3.1. The claim of Theorem 3.1 remains true for (2)provided that
+there exist constants p1,p2,...,p n,q∈Crd([β,γ]T,R+)such that
+/bardblf(t,u1,u2,...,u n)/bardbl ≤/summationdisplay
+i∈[1,n]Npi(t)/bardblui/bardbl+q(t)
+for allt∈[β,γ]Tand allu1,u2,...,u n∈X.
+10Remark 3.2. Theorem 3.1 can be used to prove existence and uniqueness
+of global solutions to the initial value problem (1)by the method of steps.
+More precisely, pick an increasing divergent sequence {βk}k∈N⊂[β,∞)Tsuch
+thatβk−1≤mini∈[1,n]Ninft∈[βk,∞)T{τi(t)}for allk∈N(with the convention
+β−1:=αandβ0:=β). Successively for k∈N, apply Theorem 3.1 on
+[βk−2,βk]Tby considering the unique solution obtained in the previous step as
+the initial function on [βk−2,βk−1]T, and denote this solution by yk. Finally,
+we obtain the unique global solution xof(1)by letting x(t) =yk(t)for
+t∈[βk−1,βk]Tandk∈N0.
+3.2 Representation of solutions by means of the prin-
+cipal solution
+In this section, we give the definition of the principal solution and pro ve the
+representationformulafortheuniquesolutionof (11)intermsoft heprincipal
+solution. We need to recall the definition of the characteristic func tionχ.
+LetU⊂R, and define the so-called characteristic function χU:R→ {0,1}
+to be
+χU(t) :=/braceleftBigg
+1, t∈U
+0,otherwise
+fort∈R.
+Now we can give the definition of the principal solution to (11).
+Definition 3.1 (Principal solution) .Letζ∈[β,γ]T. The solution X=
+X(·,ζ) : [α,γ]T→Xof the problem
+
+
+x∆(t) =/summationdisplay
+i∈[1,n]Npi(t)x(τi(t))fort∈[ζ,γ]κ
+T
+x(t) =χ{ζ}(t)1Xfort∈[α,ζ]T,
+where1Xis the unity of the Banach algebra B(X), which satisfies X(·,ζ)∈
+C1
+rd([ζ,γ]T,X), is called the principal solution of(11).
+The following result, which plays the major role (both in the continuou s
+and in the discrete cases) in the qualitative theory by suggesting a r epresen-
+tation formula to solutions of (11) by the means of the principal solu tionX,
+is proven below. This result is also known as the “variation of paramet ers
+formula”.
+11Theorem 3.2 (Solutionrepresentation formula) .Letxbe a solution of (11),
+thenxcan be written in the following form:
+x(t) =X(t,β)ϕ(β)+/integraldisplayt
+βX(t,σ(η))q(η)∆η
++/integraldisplayt
+βX(t,σ(η))/summationdisplay
+i∈[1,n]Npi(η)χ[α,β)T(τi(η))ϕ(τi(η))∆η(15)
+fort∈[β,γ]T.
+Proof.In the light of Theorem 3.1, it suffices to prove that
+y(t) :=
+
+X(t,β)ϕ(β)+/integraldisplayt
+βX(t,σ(η))q(η)∆η
++/integraldisplayt
+βX(t,σ(η))/summationdisplay
+i∈[1,n]Npi(η)χ[α,β)T(τi(η))ϕ(τi(η))∆η, t∈[β,γ]T
+ϕ(t), t∈[α,β]T
+solves (11). For t∈[β,γ]T, setI(t) ={j∈[1,n]N:χ[β,γ]T(τj(t)) = 1}and
+J(t) :={j∈[1,n]N:χ[α,β)T(τj(t)) = 1}. Considering the definition of the
+principal solution X, we have
+y∆(t) =X∆(t,β)ϕ(β)+/integraldisplayt
+βX∆(t,σ(η))q(η)∆η+X(σ(t),σ(t))q(t)
++/integraldisplayt
+βX∆(t,σ(η))/summationdisplay
+i∈[1,n]Npi(η)χ[α,β)T(τi(η))ϕ(τi(η))∆η
++X(σ(t),σ(t))/summationdisplay
+i∈[1,n]Npi(t)χ[α,β)T(τi(t))ϕ(τi(t))
+=/summationdisplay
+j∈I(t)pj(t)/bracketleftBigg
+X(τj(t),β)ϕ(β)+/integraldisplayt
+βX(τj(t),σ(η))q(η)∆η
++/integraldisplayt
+βX(τj(t),σ(η))/summationdisplay
+i∈[1,n]Npi(η)χ[α,β)T(τi(η))ϕ(τi(η))/bracketrightbig
+∆η/bracketrightBigg
++/summationdisplay
+j∈J(t)pj(t)ϕ(τj(t))+q(t)
+12for allt∈[β,γ]κ
+T. Making some arrangements, we get
+y∆(t) =/summationdisplay
+j∈I(t)pj(t)/bracketleftBigg
+X(τj(t),β)ϕ(β)+/integraldisplayτj(t)
+βX(τj(t),σ(η))q(η)∆η
++/integraldisplayτj(t)
+βX(τj(t),σ(η))/summationdisplay
+i∈[1,n]Npi(η)χ[α,β)T(τi(η))ϕ(τi(η))/bracketrightbig
+∆η/bracketrightBigg
++/summationdisplay
+j∈J(t)pj(t)ϕ(τj(t))+q(t)
+=/summationdisplay
+j∈I(t)pj(t)y(τj(t))+/summationdisplay
+j∈J(t)pj(t)y(τj(t))+q(t),
+which proves that ysatisfies (11) for all t∈[β,γ]κ
+TsinceI(t)∩J(t) =∅and
+I(t)∪J(t) = [1,n]Nforeacht∈[β,γ]T. Theproofisthereforecompleted.
+We conclude the paper with the following nice and simple example, which
+considers first-order ordinary dynamic equations.
+Example 3.1. Consider the following scalar first-order dynamic equation :
+/braceleftBigg
+x∆(t) =p(t)x(t)+q(t)fort∈[β,γ]κ
+T
+x(β) =x0.(16)
+The principal solution of (16)isX(t,s) =χ[s,γ]T(t)ep(t,s)fors,t∈[β,γ]T
+provided that p∈ R([β,γ]T,R)(see [7, Theorem 2.71]). Note here that the
+need for the regressivity condition is just to make the expon ential function
+meaningful. Thus, due to Theorem 3.2 the unique solution can be written in
+the form
+x(t) =x0ep(t,β)+/integraldisplayt
+βep(t,σ(η))q(η)∆η
+fort∈[β,γ]T(see [7, Theorem 2.77]).
+References
+[1] S. Hilger. Ein Maßkettenkalk¨ ul mit Anwendung auf Zentrumsmannig-
+faltigkeiten . Ph.D. Thesis. Universit¨ at W¨ urzburg, 1988.
+13[2] H. Agwo. On the oscillation of first order delay dynamic equations w ith
+variable coefficients. Rocky Mountain J. Math. , vol. 38, no. 1, pp. 1–17,
+(2008).
+[3] M. Bohner. Some oscillation criteria for first order delay dynamic e qua-
+tions.Far East J. Appl. Math. , vol. 18, no. 3, pp. 289–304, (2005).
+[4] M. Bohner, B. Karpuz and ¨O.¨Ocalan. Iterated oscillation criteria for
+delay dynamic equations of first order. Adv. Difference Equ. , vol. 2008,
+art. 458687, 12 pp., (2008).
+[5] Y. S ¸ahiner and I. P. Stavroulakis. Oscillations of first order dela y dy-
+namic equations. Dynam. Syst. Appl. , vol. 15, pp. 645–656, (2006).
+[6] B. G. Zhang and X. Deng. Oscillation of delay differential equations on
+time scales. Math. Comput. Model. , vol. 36, no. 11-13, pp. 1307–1318,
+(2002).
+[7] M. Bohner and A. Peterson. Dynamic Equations on Time Scales: An
+Introduction with Applications . Birkh¨ auser, Boston, 2001.
+[8] V. Lakshmikantham, S. Sivasundaram and B. Kaymak¸ calan. Dynamic
+Systems on Measure Chains . Kluwer Academic Publishers, 1996.
+[9] J. Dibl´ ık, M. R˙ uˇ ziˇ ckov´ a and Z. ˇSmarda. Wa˙ zewski’s method for systems
+of dynamic equations on time scales. Nonlinear Anal. , vol. 71, no. 12,
+pp. e1124–e1131, (2009).
+[10] B. Kaymak¸ calan. Existence andcomparison results for dynam ic systems
+on a time scale. J. Math. Anal. Appl. , vol. 172, no. 1, pp. 243–255,
+(1993).
+[11] A. Peterson and W. Kelley. The Theory of Differential Equations: Clas-
+sical and Qualitative . Prentice Hall, New Jersey, 2003.
+[12] I. Gy˝ ori and G. Ladas. Oscillation Theory of Delay Differential Equa-
+tions: With Applications . Oxford Science Publications. The Clarendon
+Press, Oxford University Press, New York, 1991.
+[13] M. Bohner and G. Sh. Guseinov. The convolution on time scales. Abstr.
+Appl. Anal. , Art. ID 58373, 24 pp., (2007).
+14[14] G. Sh. Guseinov. Integration on time scales J. Math. Anal. Appl. ,
+vol. 285, no. 1, pp.107–127, (2003).
+[15] G. Sh. Guseinov and B. Kaymak¸ calan. Basics of Riemann delta an d
+nabla integration on time scales. J. Difference Equ. Appl. , vol. 8, no. 11,
+pp. 1001–1017, (2002).
+15
\ No newline at end of file
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+arXiv:1001.0738v2  [astro-ph.CO]  13 May 2010Accepted by ApJ
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+PHOTOMETRIC ESTIMATES OF REDSHIFTS AND DISTANCE MODULI FOR TYPE IA SUPERNOVAE
+Richard Kessler,1,2David Cinabro,3Bruce Bassett,11,12Benjamin Dilday,4Joshua A. Frieman,1,2,5
+Peter M. Garnavich,6Saurabh Jha,4John Marriner,5Robert C. Nichol,7Masao Sako,9Mathew Smith,11
+Joseph P. Bernstein,8Dmitry Bizyaev,13Ariel Goobar,14,15Stephen Kuhlmann,8Donald P. Schneider,10
+Maximilian Stritzinger16,17
+Accepted by ApJ
+ABSTRACT
+Large planned photometric surveys will discover hundreds of thou sands of supernovae (SNe), out-
+stripping the resources available for spectroscopic follow-up and n ecessitating the development of
+purely photometric methods to exploit these events for cosmologic al study. We present a light curve
+fitting technique for type Ia supernova (SN Ia) photometric reds hift (photo- z) estimation in which the
+redshift is determined simultaneously with the other fit parameters . We implement this “ lcfit+z ”
+technique within the frameworks of the mlcs2k2 andsaltiilight curve fit methods and determine
+the precision on the redshift and distance modulus. This method is ap plied to a spectroscopically
+confirmed sample of 296 SNe Ia from the Sloan Digital Sky Survey-II (SDSS–II) SN Survey and
+37 publicly available SNe Ia from the Supernova Legacy Survey (SNLS ). We have also applied the
+method to a large suite of realistic simulated light curves for existing a nd planned surveys, including
+the SDSS, SNLS, and Large Synoptic Survey Telescope (LSST). Wh en intrinsic SN color fluctuations
+are included, the photo- zprecision for the simulation is consistent with that in the data. Finally, we
+comparethe lcfit+z photo-zprecisionwith previousresultsusingcolor-basedSNphoto- zestimates.
+Subject headings: supernova light curve fitting
+1.INTRODUCTION
+To investigate the expansion history of the universe,
+increasingly large samples of high-quality type Ia su-
+kessler@kicp.uchicago.edu
+1Department of Astronomy and Astrophysics, The Univer-
+sity of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637
+2Kavli Institute for Cosmological Physics, The University o f
+Chicago, 5640 South Ellis Avenue Chicago, IL 60637
+3Department of Physics and Astronomy, Wayne State Uni-
+versity, Detroit, MI 48202
+4Department of Physics and Astronomy, Rutgers University,
+136 Frelinghuysen Road, Piscataway, NJ 08854
+5Center for Particle Astrophysics, Fermi National Accelera -
+tor Laboratory, P.O. Box 500, Batavia, IL 60510
+6University of Notre Dame, 225 Nieuwland Science, Notre
+Dame, IN 46556-5670
+7Institute of Cosmology and Gravitation, Mercantile House,
+Hampshire Terrace, University of Portsmouth, Portsmouth P O1
+2EG, UK
+9Department of Physics and Astronomy, University of Penn-
+sylvania, 203 South 33rd Street, Philadelphia, PA 19104
+8Argonne National Laboratory, 9700 S. Cass Avenue,
+Lemont, IL 60437
+10Department of Astronomy and Astrophysics, The Pennsyl-
+vania State University, 525 Davey Laboratory, University P ark,
+PA 16802.
+11Department of Mathematics and Applied Mathematics,
+University of Cape Town, Rondebosch 7701, South Africa
+12South African Astronomical Observatory, P.O. Box 9, Ob-
+servatory 7935, South Africa.
+13Apache Point Observatory, P.O. Box 59, Sunspot, NM
+88349.
+14Department of Physics, Stockholm University, Albanova
+University Center, SE–106 91 Stockholm, Sweden
+15The Oskar Klein Centre for Cosmoparticle Physics, De-
+partment of Physics, AlbaNova, Stockholm University, SE-1 06
+91 Stockholm, Sweden
+16Las Campanas Observatory, Carnegie Observatories,
+Casilla 601, La Serena, Chile
+17Dark Cosmology Centre, Niels Bohr Institute, University of
+Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen Ø, Den-
+markpernova (SN Ia) light curves are being used to mea-
+sure luminosity distances as a function of redshift (the
+SN Ia Hubble diagram). Expected SN Ia samples
+will be in the thousands for the Dark Energy Sur-
+vey (DES: Bernstein et al. (2009)) and in the hun-
+dreds of thousands for the surveys to be carried out
+by the Panoramic Survey Telescope and Rapid Re-
+sponse System (Pan-STARRS)18and by the Large
+Synoptic Survey Telescope (LSST: Ivezi´ c et al. (2008);
+LSST Science Collaborations(2009)). Forthelattertwo,
+only a small fraction of the SNe will have spectroscopi-
+cally determined redshifts from SN or host-galaxy spec-
+tra for the foreseeable future. To make use of these large
+SNsamples,the redshiftswillhavetobedeterminedpho-
+tometrically using both the SN light curves and the host-
+galaxy photometric observables.
+Methods for estimating galaxy photo- z’s have been
+developed over many years (for a review, see, e.g.,
+Abdalla et al. (2008)). They generally fall into two cat-
+egories: (1) an empirical approach, in which one trans-
+lates observed colors, magnitudes, or other photometric
+observables into a redshift estimate, training the algo-
+rithmonasubsetofgalaxieswithspectroscopicredshifts;
+and (2) template fitting, in which observed colors are
+matched to redshifted template galaxy spectral energy
+distributions until the best match for galaxy redshift and
+type is obtained. The development of photo- zmethods
+using SN data is more recent. Some have followed and
+adapted the empirical approach to galaxy photo- zesti-
+mation, e.g., using observed SN colors near the epoch
+of peak brightness to estimate the redshift (Wang 2007;
+Wang et al. 2007) or early-epoch colors to select SNe in
+particular redshift regions (Dahlen & Goobar 2002).
+In this paper, we present and describe a method of
+18http://pan-starrs.ifa.hawaii.edu/public2
+SN Ia photo- zestimation that is analogous to the tem-
+plate fitting method for galaxy photo- z’s. In models
+used to fit SN Ia light curves, one typically uses a spec-
+troscopically determined redshift, and the fit parame-
+ters are usually taken to be the epoch of peak bright-
+ness, the light curve shape or stretch, a color or dust
+extinction estimate, and the distance modulus. In our
+approach, we extend the usual methods of fitting light
+curves to include the redshift as a fifth fit parame-
+ter. We apply this “ lcfit+z ” method to determine
+photo-z’s for the spectroscopically confirmed SDSS–II
+(Frieman et al. 2008; Kessler et al. 2009a) and SNLS
+(Astier et al. (2006); hereafter A06) SNe, and compare
+the resulting photo- zprecision to that obtained from
+simulated samples. Once we have verified the reliabil-
+ity of the simulations by comparison with the data, we
+apply the lcfit+z method to simulated LSST SN ob-
+servations. In all cases, we use both mlcs2k2 (Jha et al.
+2007; Kessler et al. 2009a) and saltii(Guy et al. 2007)
+light curve fitting models.
+Variantsof the light curve fit approachto SN photo- z’s
+have been used before, by both the SNLS (Sullivan et al.
+2006) and SDSS (Sako et al. 2008) surveys, to select
+SN Ia candidates for spectroscopic follow-up after only
+a few photometric epochs. Kim & Miquel (2007) used
+thesaltiimodel and a Fisher matrix analysis to es-
+timate uncertainties on photometric redshifts and dis-
+tances. Using a technique similar to our lcfit+z
+method, Gong et al. (2010) studied LSST simulations,
+focusing mainly on contamination from non-Ia SNe and
+the resulting precision on cosmological parameters. In
+contrast, we focus here on the precision and bias for the
+photo-zand distance modulus. We also use more real-
+istic simulations based on the LSST cadence for both
+the deep and wide surveys and illustrate some differ-
+ences between these two components of the LSST sur-
+vey. Our estimates of non-Ia contamination and cos-
+mological precision will be presented in a future work.
+Recently, Palanque-Delabrouille et al. (2009) (hereafter
+PD09) employed the light curve fit photo- ztechnique
+within the saltiiframework. For nearly 300 SNe Ia
+from the SNLS, they evaluated both the photo- zpre-
+cision and the fraction of catastrophic redshift outliers.
+The PD09 SN sample is by far the largest to date used
+to study SN photo- zmethods; our SDSS–II sample, at
+lower redshifts, is of comparable size.
+The plan of this paper is as follows. We introduce the
+spectroscopically confirmed SN data samples (SDSS–II
+and SNLS) in §2. In§3, we describe the simulation, and
+we present the lcfit+z method in detail in §4. The
+photo-zprecision and fit-parameter correlations for the
+data samples, along with the corresponding results from
+simulated samples, are presented in §5. We use the sim-
+ulation to make predictions for the LSST survey in §6.
+In§7 we make direct comparisons with the color-based
+photo-zmethod presented in previous works.
+As described in Kessler et al. (2009b), all light curve
+fitting andsimulationsoftwareispublicly availablein the
+SNANApackage.19
+2.THE SDSS–II AND SNLS DATA SAMPLES
+19http://www.sdss.org/supernova/SNANA.htmlTo test the lcfit+z method, we use the full three-
+season sample from the SDSS–II Supernova (SN) Survey
+(Frieman et al. 2008), and the publicly available sample
+from the first season of the SNLS (A06). Below we give
+a brief description of these samples.
+The SDSS–II SN Survey used the SDSS camera
+(Gunn et al. 1998) on the SDSS 2.5 m telescope
+(Gunn et al. 2006) at Apache Point Observatory to
+search for SNe in the Fall seasons (September 1 through
+November 30) of 2005–2007. This survey scanned a re-
+gion(designatedstripe82)centeredonthecelestialequa-
+torin the Southern Galactichemisphere that is 2.5◦wide
+and runs between right ascensions of 20hrand 4hr, cov-
+ering a total area of 300 deg2with a typical cadence of
+every four nights per region. Images were obtained in
+five broad passbands, ugriz(Fukugita et al. 1996), with
+55 s exposures and processed through the PHOTO pho-
+tometric pipeline (Lupton et al. 2001). Within 24 hours
+of collecting the data, the images were searched for SN
+candidates that were selected for spectroscopic follow-up
+observations in a program involving about a dozen tele-
+scopes. The SDSS–II SN Survey discovered and spectro-
+scopically confirmed a total of ∼500 SNe Ia. A larger
+sample of photometrically identified but spectroscopi-
+cally unobserved SNe Ia was also compiled, and host-
+galaxy redshifts for several hundred of these photomet-
+ric candidates have been obtained to date. The SDSS-
+III Survey (Schlegel et al. 2009), as a small part of its
+earlyprogram, is in the process of measuringhost-galaxy
+redshifts for more than 1000 of these photometrically
+identified SNe Ia. The telescope aperture, focal plane,
+and exposure time of the SDSS system (York et al. 2000)
+were ideal for discovering SNe in the previously under-
+explored redshift range 0 .1< z <0.3. Details of the
+SDSS-II SN Survey are given in Frieman et al. (2008);
+Sako et al. (2008), the procedures for spectroscopic iden-
+tification and redshift determinations are described in
+Zheng et al. (2008), and the SN photometry is described
+in Holtzman et al. (2008). A condensed summary of the
+SDSS–II survey, SN typing, redshift determination, pho-
+tometry, and calibration can be found in Kessler et al.
+(2009a).
+The SNLS was a five-year survey covering 4 deg2us-
+ing the MegaCam imager on the 3.6 m Canada–France–
+Hawaii Telescope (CFHT). Images were taken in four
+bandssimilartothoseusedbytheSDSS: gM,rM,iM,zM,
+where the subscript Mdenotes the MegaCam system.
+The SNLS exposures were ∼1 hr in order to discover
+SNe at redshifts up to z∼1. The SNLS images were
+processed in a fashion similar to the SDSS–II so that
+spectroscopic observations could be used to confirm the
+identities and determine the redshifts of the SN candi-
+dates. We use the publicly available sample from their
+first year of operations that ended July 15, 2004. De-
+tailed information about the SNLS can be found in A06
+and references therein.
+Since high-quality light curves are needed for the
+lcfit+z method, we apply the following selection re-
+quirements to the photometric data for inclusion in our
+analysis samples: (1) spectroscopic confirmation of type
+Ia, (2) a measurement with Trest<−3 days, (3) a mea-
+surement with Trest>+10 days, (4) measurements in
+at least three observer-frame filters have signal-to-noise3
+ratio (S/N) greater than 8, and (5) the probability cor-
+responding to the fit- χ2/Ndof(§4) isPχ2>0.02. Here
+Trestis the epoch in the SN rest frame relative to peak
+brightness in the Bband, and we note that it depends
+on the fitted photo- zvalue. For the SDSS–II, we use
+only the gripassbands. The number of SNe Ia satisfying
+these selection requirements is nearly 300 and 40 for the
+SDSS–II and SNLS, respectively; the exact numbers of
+SNe depend on the fitting model ( mlcs2k2 orsaltii)
+and will be given in §5. The selection criteria above are
+not based on optimizations, but are instead motivated
+by the strong correlationbetween redshift and color. Re-
+quiring two colors and at least one measurement in each
+of three passbands with S /N>8 explicitly ensures good
+color measurements. The Trestrequirements ( <−3 and
+>+10 days) ensure a good determination of the time of
+peak brightness ( t0); since SNe become redder after peak
+light, a mismeasurement of t0translates directly into a
+mismeasurement of color, and hence redshift.
+Here we provide some additional motivation for the
+above requirements. Relaxing the S/N requirement (no.
+4 above) to S /N>5 results in about 10% more SNe Ia
+in the SDSS–II sample, and a 20% degradation in the
+photo-zprecision. Making a more restrictive cut of
+S/N>10 results in a 20% loss of events and a negli-
+gible improvement in the precision. We have therefore
+chosen S /N>8 as a reasonable compromise between
+sample statistics and precision. To motivate the sam-
+pling requirements, we have applied the lcfit+z fitting
+method ( §4) to theSDSS–II samplein whichall measure-
+ments prior to peak brightness have been rejected; the
+resulting precision and bias on the fitted t0and photo- z
+are significantly degraded. The sampling requirements
+above (nos. 2 and 3) are therefore designed to ensure a
+good determination of t0.
+3.SIMULATIONS
+We use the SNANAsimulation code to generate realis-
+tic SN Ia light curves that can be analyzed in exactly
+the same manner as the data. The simulation is used to
+compare with the data, to compare with previous photo-
+zstudies based on simulations, and to make predictions
+for LSST. All SN simulations are based on a standard
+ΛCDM cosmology ( w=−1, ΩM= 0.3, ΩΛ= 0.7) and
+they are generated and fit using the same light curve
+model:mlcs2k2 orsaltii. This strategy explicitly as-
+sumes that the light curve model is correct, and will
+therefore yield the most optimistic results. As discussed
+below, the modelsareadjustedto accountforthe anoma-
+lous Hubble scatter. However, these models have not
+been adjusted to account for the discrepancies in the ul-
+traviolet region (Kessler et al. (2009a): hereafter K09)
+Details of the simulation are described in Kessler et al.
+(2009b) and in §6 of K09; here we give a brief overview.
+Formlcs2k2 , rest-frame model magnitudes are gen-
+erated from light curve templates and then dimmed
+by host-galaxy dust extinction. Using SN Ia spec-
+tral templates from Hsiao et al. (2007), K-corrections
+(Nugent et al.2002) areusedto transformthe rest-frame
+(UBVRI) model magnitudes into observer-framemagni-
+tudes. For saltii, the model is based on a time-sequence
+of rest-frame spectra, and observer-frame magnitudes
+are computed by convolution with the appropriate filter-
+response curves.Since light curve models are defined over a specific
+wavelength range in the rest frame, one usually checks
+the rest-frame wavelength ¯λf/(1 +z), where ¯λfis the
+mean wavelength of the observer-frame filter and zis
+the redshift. If the rest-frame wavelength is outside the
+valid range of the model, then the corresponding fil-
+ter is typically ignored. For photo- zapplications, this
+method of ignoring filters clearly cannot be used, since
+the redshift is not known ahead of time. To be realis-
+tic, we should not use this filter-ignoring procedure in
+the simulations either. Therefore, our simulations use
+wavelength-extended models that generate fluxes for fil-
+ters with ¯λf/(1+z) beyond the nominally defined wave-
+length range. For both mlcs2k2 andsaltii, the rest-
+frame wavelengths are extended down to 2500 ˚A. For
+saltii, the 7000 ˚A upper limit hasbeen raisedto 8700 ˚A.
+We note that the simulation and fitter use the same
+wavelength-extended model, and therefore these tests do
+not probe potential problems if the extended part of the
+model is wrong.
+We have implemented two models of intrinsic SN mag-
+nitude variations that produce “anomalous” scatter in
+the Hubble diagram. The source of anomalous scatter
+is unknown, and it is not clear if the scatter can be
+reduced with an improved light curve model, or if this
+scatter is due to some random physical process such as
+brightness variations as a function of viewing angle in an
+asymmetric explosion. For simulations with 104times
+the nominal exposure time and z <0.5, i.e., for which
+photon noise is negligible, we define the anomalous scat-
+ter to be the rms (RMS µ) of the difference between the
+fitted and generated distance modulus ( µfit−µgen) from
+the four-parameter light curve fit using spectroscopically
+determined redshifts. The default model, called “color-
+smearing,” introduces an independent magnitude fluctu-
+ation in each passband, and the fluctuation is the same
+for all epochs within each passband. A random num-
+berrffrom a unit-variance Gaussian distribution is cho-
+sen for each passband f, and a magnitude fluctuation
+δmf=rfσfis added to the generated magnitude at all
+epochs. As described in §5, a scatter of σf= 0.1 mag
+is needed in order for the simulated photo- zprecision
+to match that of the data. The resulting Hubble scat-
+ter is consistent with that seen in analyses of spectro-
+scopically confirmed data samples; RMS µ≃0.16 for
+the SNLS ( griz) simulations, and RMS µ≃0.19 mag
+for the SDSS–II ( gri) simulations. The simulated SNLS
+scatter is slightly smaller because this survey results in
+larger S/N values. The second model of intrinsic vari-
+ations is called “coherent luminosity smearing:” a co-
+herent random magnitude shift, drawn from a Gaussian
+withσcoh∼0.15 mag, is added to the model magnitude
+for all epochs and passbands. In the coherent smearing
+method the intrinsic model colors are not varied, and the
+resulting anomalous scatter is RMS µ=σcoh.
+Although both models of intrinsic magnitude variation
+result in the expected scatter in the Hubble diagram,
+only the color-smearingmodel can generate the observed
+photo-zprecision in the SDSS–II and SNLS data sam-
+ples (§5). We have not investigated simulations in which
+both models of intrinsic variation contribute, nor have
+we investigated variations in the color parameter ( RV
+formlcs2k2 orβforsaltii) that could also introduce4
+anomalous scatter.
+To simulate non-photometric conditions and varying
+time intervals between observations due to bad weather,
+actual observing conditions are used for an existing sur-
+vey, or an estimate of such conditions for a planned (fu-
+ture) survey. For each simulated observation, the noise
+is determined from the measured point spread function
+(PSF),20zero point, CCD gain, and sky background.
+Noise from the host-galaxy background is not included.
+The simulated flux in CCD counts is based on a mag-
+to-flux zero point and a random fluctuation drawn from
+the noise estimate. For the SDSS–II and SNLS surveys,
+a detailed treatment of the search efficiency, including
+spectroscopicselectioneffects, isdescribedin §6.2ofK09.
+The quality of the simulation is illustrated in Fig-
+ures 1 and 2 for the SDSS–II and SNLS surveys, re-
+spectively, using the SN selection requirements described
+in§2. The parameters from each sample have been de-
+termined using the conventional fitting method with a
+fixed spectroscopically determined redshift. Each fig-
+ure shows data-simulation comparisons for the distribu-
+tions of spectroscopic redshift, color parameter ( AVfor
+mlcs2k2 ,cforsaltii), and shape-luminosity parameter
+(∆ formlcs2k2 ,x1forsaltii). The color and shape
+parameters are defined in §4. There is good overall con-
+sistency between the measured and simulated distribu-
+tions.
+0204060
+0 0.1 0.2 0.3 0.40204060
+0 0.1 0.2 0.3 0.4
+02040
+-1 0 1 2020406080
+-0.5 -0.25 0 0.25 0.5
+0204060
+-0.5 0 0.5 1 1.5Zspec   MLCS                                 SALTIIEntries    SDSS data
+SNANA sim
+Zspec
+fitted AVEntries    
+fitted c
+fitted DEntries    
+fitted x102040
+-4 -2 0 2 4
+Fig. 1.— For the SDSS–II SN Survey, comparison of data distri-
+butions (dots) with those from the simulation (histograms) . Fitted
+parameters for mlcs2k2 (z,AV,∆) are on the left, and for saltii
+(z,c,x1) on the right. Fits were done with the redshift fixed to
+the true redshift. The simulated histograms are scaled to ha ve the
+same number of entries as the data.
+4.THElcfit+z METHOD
+The general principle behind the SN Ia photo- zdeter-
+mination is illustrated in Fig. 3 using colors at the epoch
+20The PSF is described by a double-Gaussian function for the
+SDSS–II, and by a single Gaussian for the other surveys.of peak brightness. Increasing the redshift causes uni-
+form reddening at all wavelengths, while intrinsic red-
+dening (or extinction) causes more reddening at bluer
+wavelengths. A time-dependent light curve model (e.g.,
+mlcs2k2 orsaltii) is used to account for the known
+color dependence on the light curve shape (commonly
+known as the stretch) and to use all epochs in order
+to maximize the photostatics in the photo- zmeasure-
+ment. Since SNe become redder with increasing epoch,
+any error in the epoch of peak brightness results in the
+wrong template colors, and hence an increased error in
+the photo- z.
+For real observations, the ideal color-color bands in
+Fig. 3 are smeared by photon statistics and by intrinsic
+SN color variations that are not described by the light
+curve model. With sufficient smearing, the colors of a
+very reddened SN at z≃0.1 are degenerate with a blue
+SN atz≃0.3 (see the region enclosed by dotted oval in
+Fig. 3) and we indeed see this degeneracy in the SDSS–II
+photo-zmeasurements. From data-simulation compar-
+isons, we find that smearing from photon statistics does
+not fully describe the photo- zprecision, and we therefore
+propose that the modeling of intrinsic color variations is
+not adequate. ( §5).
+Whilehost-galaxyphotometricredshiftsdepend onthe
+determination of the 4000 ˚A break, a similar ∼2800˚A
+break in the SN spectrum has little impact on the photo-
+zmeasurement because this feature is either inaccessible
+at low redshifts, or it is poorly measured at higher red-
+shifts. In current SN Ia models, fluxes at these very blue
+wavelengthsare either ignored, orthey areheavily down-
+weighted relative to the optical bands.
+The basic idea of the lcfit+z method is to start
+with a light curve fit model that has four free param-
+eters when the redshift is fixed to an accurately mea-
+sured value, and simply float the redshift as a fifth fitted
+parameter. We refer to these methods as mlcs2k2+z
+andsaltii+z , wheremlcs2k2 andsaltiirefer to the
+conventional models in which the redshift is fixed to a
+02468
+0.2 0.4 0.6 0.8 10510
+0.2 0.4 0.6 0.8 1
+0510
+-1 0 1 2051015
+-0.5 -0.25 0 0.25 0.5
+0510
+-0.5 0 0.5 1 1.5Zspec   MLCS                                 SALTIIEntries    SNLS data
+SNANA sim
+Zspec
+fitted AVEntries    
+fitted c
+fitted DEntries    
+fitted x10510
+-4 -2 0 2 4
+Fig. 2.— Same as Figure 1, but for the SNLS survey, using the
+data sample from A06.5
+z = 0.1z = 0.2z = 0.3
+high stretch (x1 = +0.5)
+low stretch (x1 = -0.5)
+g - r          r - i          
+-0.5-0.4-0.3-0.2-0.100.1
+-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
+Fig. 3.— Observer-frame r−ivs.g−rat peak brightness for
+SNe Ia using the saltiimodel. Each set of color bands corresponds
+to the indicated redshift (0.1, 0.2, 0.3), and the variation within
+each band corresponds to variations in the intrinsic SN colo r (i.e.,
+thesaltiicparameter). The solid and open circles correspond to
+a high-stretch and low-stretch SN, respectively. The dotte d oval
+shows a degenerate region discussed in the text.
+precisely measured value. Compared to the color-based
+photo-zmethod, advantages of the lcfit+z method in-
+clude a natural framework for tracking correlations be-
+tween redshift and distance modulus ( §5.2), and using
+all of the light curve information (instead of just peak
+flux) so that in principle the intrinsic SN color varia-
+tions can be accounted for. The main advantage of the
+color-based method is that it works over a broader red-
+shift range, and there is no need to worry about which
+observer-frame filters map into a valid wavelength range
+in the rest frame.
+Formlcs2k2+z , the five fitted parameters (which
+we denote with the vector /vector x5) are the time of max-
+imum brightness in the (rest frame) B-band (t0), the
+shape-luminosity parameter (∆), the host-galaxy extinc-
+tion in the Vband (AV), the distance modulus ( µ),
+and the redshift ( zphot). We use a flat AVprior and
+RV=AV/E(B−V) = 2.2. Forsaltii+z the five pa-
+rameters are the time of maximum brightness in the rest
+frameB-band (t0), the shape-luminosityparameter ( x1),
+theB−Vcolor (c), the flux normalization ( x0), and the
+redshift ( zphot). The following light curve fit χ2is mini-
+mized using minuit21:
+χ2=/summationdisplay
+i/braceleftBigg/bracketleftbig
+Fdata
+i−Fmodel
+i(/vector x5)/bracketrightbig2
+σ2
+i+2ln(σi/˜σi)/bracerightBigg
+,(1)
+and the corresponding probability ( e−χ2/2) is used to
+marginalize as described in Appendix B. Here, Fdata
+i
+is the SN flux of the ith observation, Fmodel
+i(/vector x5) is the
+predicted flux using the five model parameters ( /vector x5), and
+σ2
+i=σ2
+i,stat+σ2
+i,modelis the quadrature sum of the mea-
+sured and model uncertainties, respectively. The index
+iruns over all epochs and filters. The second term in
+21http://wwwasdoc.web.cern.ch/wwwasdoc/minuit/minmain .htmlTrest (days)                   model error (mag)        
+BUMLCS
+00.10.20.30.40.50.6
+-20 -10 0 10 20 30 40 50
+Trest (days)                   model error (mag)        
+BUSALT2
+00.10.20.30.40.50.6
+-20 -10 0 10 20 30 40 50
+Fig. 4.— Model uncertainty vs. rest-frame epoch for the U
+(solid) and B(dashed) passbands.
+Eq. 1 accounts for model uncertainties that depend on
+therest-framepassbandandepoch, whichinturndepend
+on the photo- zvalue. The model uncertaintiesare shown
+in Fig. 4 for the rest-frame UandBpassbands. The ref-
+erenceuncertainty(˜ σi) fromthefirst iterationofthe fit is
+used in the denominator so that the second term is close
+to zero in the second iteration; although the ˜ σido not
+affect the minimization, these terms reduce the changein
+the calculated fit probability. As explained below, the χ2
+is minimized twice in order to include the appropriatefil-
+ters and epochs. The minimized values and uncertainties
+are then used to estimate the integration ranges needed
+to obtain marginalized results.
+In this study, we use the mlcs2k2 andsaltiilight
+curve fitters that have been implemented in the SNANA
+package. The main reasons for using the SNANAcode are
+(1) exactly the same light curve model is guaranteed to
+be used in both the light curve fits and in the generated
+simulations, (2) there are several improvements to the
+mlcs2k2 light curve fitter as explained in K09, (3) the
+photo-zimplementation is identical for both models, (4)
+theSNANAfitter is significantly faster than the original
+fitting software. To check the saltiiimplementation in
+SNANA, we have repeated the light curve fits and cosmol-
+ogy analysis for the six sample combinations in K09 and
+find that the dark energy equation of state parameter w
+is always within a few hundredths of the value obtained
+with the original code; these discrepancies are well below
+the statistical uncertainties.
+Although it is straightforward to include the redshift
+as a free parameter in the light curve fit, there are sub-
+tle pre-fit issues related to the unknown redshift value:
+(1) SN selection criteria that depend on knowing Trest
+=Tobs/(1 +z),22such as requiring measurements with
+a minimum and maximum Trestvalue; (2) as noted
+above, determining which observer-frame filters (with
+mean wavelength ¯λf) have¯λf/(1+zphot) within the valid
+wavelength range of the fitting model; (3) determining
+the valid rest-frame epoch range for the fitting model;
+(4) as¯λf/(1 +zphot) maps into a different rest-frame
+filter (for mlcs2k2 ) there is a discontinuous change in
+the model error, and therefore the χ2is not a continu-
+ous function of zphot; and (5) determining robust initial
+fit-parameter values. Our treatment of these issues is
+described in Appendix A.
+We end this section with a discussion of the processing
+time. For the SDSS–II light curves, which have nearly
+50 measurements on average, all of the minimization fit-
+22TrestandTobsare the rest-frame and observer-frame times in
+days since peak brightness in the Bband.6
+iterationstake ∼1sperSNusing minuit. Themarginal-
+ization (Appendix B) takes close to half a minute per SN
+usinganintegrationgridof11pointsperfitparameter,or
+a total of 115integration cells. However, the integration
+ranges usually need adjustment after marginalizing, and
+thereforethe marginalizationtypically runs twice, taking
+nearly a minute per SN. The processing time scales lin-
+early with the number of measurements and as the fifth
+power of the number of grid points per fit parameter.
+For the integration grid above, the Monte Carlo Markov
+Chain technique requires about the same amount of pro-
+cessing time.
+Using the SNANAimplementation of mlcs2k2+z and
+saltii+z , we find that minuitgivesadequateminimized
+values, but that theuncertaintiesarenotreliablebecause
+of subtle discontinuities in the χ2derivative with respect
+tothephoto- z. Wemustthereforemarginalizeinorderto
+getusefuluncertaintiesandcovariances. Toillustratethe
+computationalissue more clearly, consider an LSST sam-
+ple of 500,000 SNe Ia. The marginalization for all of the
+SNe in this sample requires a total of 1 CPU-year. A fac-
+tor of 100 is probably needed for code development and
+systematic studies and another factor of several for sim-
+ulation studies. The total computing needs are therefore
+a few CPU centuries with today’s processors, assuming
+that 115integration cells give sufficient accuracy. To use
+the minimization, which reduces the computing needs to
+a few CPU years, the light curve model-magnitudes and
+errors must be continuous functions of redshift, as well
+as their derivatives.
+5.RESULTS FOR SDSS–II AND SNLS
+Here we present results for the three-season SDSS–II
+data and the first-season SNLS data described in §2, and
+we compare with results from simulations of those same
+samples. There are two fit minimizations ( §4) to deter-
+mine the appropriate filters to include. The photo- zand
+distance-modulus results are taken to be the mean of
+their respective probability distribution functions (pdf)
+marginalized over the other fit parameters using a grid
+of 115integration cells (Appendix B). The uncertainty is
+takentobethermsofthepdf. Aprioronthehost-galaxy
+photo-zor SN color could potentially improve the pre-
+cision of the method and reduce the frequency of catas-
+trophicSN photo- zoutliers; we havenot used such priors
+here in order to better illustrate the performance of the
+lcfit+z method on its own.
+Following a commonly used practice in the literature,
+wecharacterizetheprecisionofthe lcfit+z photo-zpre-
+cision with the quantity
+∆z≡(zphot−zspec)/(1+zspec), (2)
+and we use RMS ∆zto denote the root mean square of
+the distribution of∆ z. The SDSS–II results are shown in
+Figures 5 and 6 for the mlcs2k2+z andsaltii+z meth-
+ods, respectively. There are fewer SNe in the sample fit-
+ted with saltiibecause the more restrictive rest-frame
+model wavelength range (2900-7000 ˚A) rejects i-band
+data for redshifts below about 0.1; without i-band, these
+low-redshift SNe fail the requirement of three observer-
+frame filters. The SNLS results are shown in Figures 7
+and 8. Distributions from the data and simulated sam-
+ples are shown side-by-side in these Figures, illustrating
+the reliability of the simulations in predicting the dis-persions and bias. Each plot shows the number of SNe
+satisfying the selection criteria, RMS ∆z(overall and ver-
+suszspec), andthe redshift bias(overalland versus zspec).
+The RMS ∆zvalues are ∼0.04 for both the SDSS–II and
+SNLS samples.
+For themlcs2k2+z method applied to the SDSS–II
+sample(Fig. 5), the overall∆ zbiasin the datais notably
+larger than in the simulation, and there is a redshift-
+dependent bias that is partially predicted by the sim-
+ulation. There are two contributions to the ∆ zbias.
+First, there is a degeneracy between intrinsic redden-
+ing (due to extinction or color) and redshift. For red-
+shiftszspec>0.3, the best-fit photo- zis sometimes near
+zphot∼0.15, and the best-fit color corresponds to a very
+red and intrinsically dim SN Ia. This degeneracy is sen-
+sitive to the S/N; the resulting bias is redshift dependent
+and is well modeled by the simulation.
+To potentially identify fits with a catastrophic photo-
+zerror resulting from a strong color-redshift degener-
+acy, we have looked for a second maximum in the one-
+dimensional marginalizedpdf. For mlcs2k2+z ,∼9% of
+the fits have a second maximum in both the photo- zand
+color pdf, and for saltii+z the corresponding fraction
+is∼4%. For both models, the photo- zprecision is the
+same for the subset with a second maximum in the pdf,
+and therefore this approach cannot be used to identify
+photo-zoutliers in the SDSS–II sample. We also find no
+correlation between the photo- zuncertainty and catas-
+trophic outliers. Clearly a reliable host-galaxy photo- z
+prior will help reduce catastrophic outliers, and this in-
+formation will be used in future analyses that include
+spectroscopically unconfirmed SNe Ia in the Hubble dia-
+gram.
+Thesecondcontributiontothe ∆ zbiasisrelatedtothe
+U-bandanomalydiscussedinK09,andthissourceofbias
+is not modeled in the simulation. The sub-sample with
+z >0.2, where the observer frame gband corresponds to
+the rest-frame UVregion, has a bias larger than the av-
+erage. The sub-sample with z <0.2 has a much smaller
+bias.
+For cosmological applications, it is of interest to study
+how the use of photo- z’s in place of spectroscopic red-
+shifts impacts the determination of SN distances. Dis-
+tance modulus ( µ) dispersions for fits with both spec-
+troscopic and photometric redshifts are shown in Fig. 9,
+where RMS µis the rms scatter of the distribution of
+µfit−µref. The reference distance modulus ( µref) is cal-
+culated from the spectroscopic redshift and the same
+standard cosmology used in the simulation: w=−1,
+ΩM= 0.3, ΩΛ= 0.7. Formlcs2k2 ,µfitis the fitted
+distance modulus. For saltii,µfitis defined to be
+µSALT2
+fit≃30−2.5log10(x0)+α·x1−β·c ,(3)
+whereα= 0.11 andβ= 2.6 are fixed parameters from
+the simulation. Note that the particular choice of αand
+βdoesnot affect the µdispersion. Comparedto the four-
+parameter light curve fits using spectroscopic redshifts,
+thelcfit+z method increases RMS µby 0.1 to 0.2 mag.
+To gauge the appropriate level of intrinsic magnitude
+variations in the simulation, we compare the ∆ zpre-
+cision in the data to that in two different simulated
+samples. The first sample is generated with 0.1 mag
+color-smearing, and the second sample is generated with7
+0.2 mag coherent smearing. The width of a Gaussian fit
+to the ∆ zdistribution ( σ∆z) is used instead ofRMS ∆zto
+reduce sensitivity to outliers. The σ∆zresults are shown
+in Table 1 for the SDSS–II and SNLS data and for the
+simulated samples. Using coherent smearing, the simu-
+lated photo- zprecision is significantly better than that
+of the data, while the color smearing model matches the
+data well. For the small SNLS data sample fitted with
+mlcs2k2+z , the large uncertainty on σ∆zis due to a
+statistical anomaly in the distribution that results in a
+poor fit to a Gaussian.
+This empirical estimate of random intrinsic color dis-
+persionneededinthesimulationdoesnotnecessarilysug-
+gest that there are random color variations in SN light
+curves, but rather that there are additional sources of
+color variation that are not captured by the light curve
+models. Using the nearby SN Ia sample ( z <0.1),
+Nobili & Goobar (2008) also found evidence for intrin-
+sic color dispersion. They fit the SNe with a light curve
+model that includes many more color parameters than
+mlcs2k2 orsaltii, and their estimate of the color dis-
+persion is considerably smaller than our empirical esti-
+mate based on matching the photo- zprecision.
+TABLE 1
+Photo-zPrecision σ∆zafor Data and Simulations
+SDSS–II SDSS–II SNLS SNLS
+sample (mlcs2k2+z ) (saltii+z ) (mlcs2k2+z ) (saltii+z )
+DATA 0.031±0.003 0.027±0.002 0.051±0.020 0.032±0.006
+SIMb0.030±0.001 0.030±0.001 0.027±0.002 0.028±0.002
+SIMc0.020±0.001 0.017±0.001 0.021±0.001 0.014±0.001
+a∆z≡(zphot−zspec)/(1 +zspec)
+bUses color-smearing model with σf= 0.1 mag.
+cUses 0.2 mag coherent mag-smearing.
+5.1.Comparison with Recent SNLS Photo- zResults
+Using data and simulations for the SNLS, we compare
+our photo- zprecision with recent results from PD09.
+They use the saltii+z method in a manner very sim-
+ilar to ours. The differences between our method and
+theirs are: (1) they use all four grizfilters, while we use
+only those filters that correspond to the valid rest-frame
+wavelength range; (2) their initial parameter scan is in
+redshift only (∆ z= 0.1 bins), while our initial scan is
+over a two-dimensional grid of redshift (∆ z= 0.04 bins)
+and color (∆ c= 0.2 bins); (3) they impose priors on the
+color and redshift, while we do not use priors; and (4)
+they impose less restrictive light curve selection require-
+ments, including the addition of photometrically iden-
+tified SNe Ia (i.e., spectroscopically unconfirmed SNe).
+The trade-off between the use of priors versus selection
+criteria mainly affects the rate of catastrophic photo- z
+outliers. Our choice of using flat priors is intended to
+betterillustratetheperformanceofthe saltii+z method
+on higher-quality light curves. While the relaxed cuts in
+PD09 have the advantage of increasing the sample size,
+using a prior on color (or on host-galaxy extinction) re-
+quires a detailed understanding of the underlying color
+distribution and survey selection function. The optimalchoice between priors and selection criteria is not ad-
+dressed here and will need further study.
+The PD09 sample is based on nearly 300 SNe Ia cor-
+responding to the first three seasons of SNLS, while we
+use the publicly available sample from A06. To make
+the selection criteria more similar for this comparison,
+we have relaxed our requirement on the maximum S/N:
+three filters must have at least one measurement each
+with S/N >5 (instead of 8). Our modified selection re-
+sultsin55SNeIa: 13with z <0.45and42with z >0.45.
+To evaluate the photo- zprecision, we use the PD09
+110102
+-0.2 0 0.2110102
+-0.2 0 0.2
+-0.200.2
+0 0.1 0.2 0.3 0.4-0.200.2
+0 0.1 0.2 0.3 0.4
+-0.0500.05
+0 0.1 0.2 0.3 0.4-0.0500.05
+0 0.1 0.2 0.3 0.4(Zphot - Zspec) / (1+Zspec)SDSS - MLCS
+NDATA = 296
+rms = 0.04
+1000 x bias =
+-8.2 ± 2.3Entries    DATA
+(Zphot - Zspec) / (1+Zspec)NSIM = 744
+rms = 0.037
+1000 x bias =
+-3.6 ± 1.4SIM
+Zspec       Dz    
+Zspec       
+rms
+bias
+Zspec       Dz    rms
+bias
+Zspec       
+Fig. 5.— Redshift precision for SDSS–II sample using
+mlcs2k2+z . Left plots are for data; right plots for simulation. Top
+plots show the ∆ zdistribution for the entire sample; the number
+of events, rms, and bias are indicated on the plot. Middle plo ts
+show ∆ zvs.zspec. Bottom plots show the bias and RMS ∆zin
+redshift bins.
+110102
+-0.2 0 0.2110102
+-0.2 0 0.2
+-0.200.2
+0 0.1 0.2 0.3 0.4-0.200.2
+0 0.1 0.2 0.3 0.4
+-0.0500.05
+0 0.1 0.2 0.3 0.4-0.0500.05
+0 0.1 0.2 0.3 0.4(Zphot - Zspec) / (1+Zspec)SDSS - SALT2
+NDATA = 251
+rms = 0.032
+1000 x bias =
+-4.7 ± 2Entries    DATA
+(Zphot - Zspec) / (1+Zspec)NSIM = 629
+rms = 0.039
+1000 x bias =
+-2.1 ± 1.5SIM
+Zspec       Dz    
+Zspec       
+rms
+bias
+Zspec       Dz    rms
+bias
+Zspec       
+Fig. 6.— Same as Fig. 5, but using saltii+z .8
+metricσ∆z/(1+z)≡1.48×median|∆z|, where ∆ zis de-
+fined in Eq. 2. This quantity is much closer to the Gaus-
+sian sigma of ∆ zthan to RMS ∆z. To quantify the rate
+of catastrophic photo- zoutliers, we define ηxas the frac-
+tion of SNe with a photo- zthat satisfies |∆z|> x, and
+we follow PD09 in using x= 0.15.
+Table 2 compares our precision metrics with those in
+PD09. We alsogivethe σ∆z/(1+z)breakdownforthelow-
+redshift ( zspec<0.45) and high-redshift ( zspec>0.45)
+ranges. We caution that these comparisons are based on
+different data samples, different selection criteria, and
+different priors. For the data comparison, the two analy-
+ses are reasonably consistent in both σ∆z/(1+z)andη0.15
+for both redshift ranges. For the simulation compar-
+110
+-0.2 0 0.2110102
+-0.2 0 0.2
+-0.200.2
+0.2 0.4 0.6 0.8 1-0.200.2
+0.2 0.4 0.6 0.8 1
+-0.0500.05
+0.2 0.4 0.6 0.8 1-0.0500.05
+0.2 0.4 0.6 0.8 1(Zphot - Zspec) / (1+Zspec)SNLS - MLCS
+NDATA = 37
+rms = 0.045
+1000 x bias =
+-9.4 ± 7.4Entries    DATA
+(Zphot - Zspec) / (1+Zspec)NSIM = 224
+rms = 0.035
+1000 x bias =
+-0.7 ± 2.3SIM
+Zspec       Dz    
+Zspec       
+rms
+bias
+Zspec       Dz    rms
+bias
+Zspec       
+Fig. 7.— Same as Fig. 5, but using the SNLS sample.
+110
+-0.2 0 0.2110102
+-0.2 0 0.2
+-0.200.2
+0.2 0.4 0.6 0.8 1-0.200.2
+0.2 0.4 0.6 0.8 1
+-0.0500.05
+0.2 0.4 0.6 0.8 1-0.0500.05
+0.2 0.4 0.6 0.8 1(Zphot - Zspec) / (1+Zspec)SNLS - SALT2
+NDATA = 37
+rms = 0.04
+1000 x bias =
+-16.4 ± 6.6Entries    DATA
+(Zphot - Zspec) / (1+Zspec)NSIM = 344
+rms = 0.04
+1000 x bias =
+-0.4 ± 2.2SIM
+Zspec       Dz    
+Zspec       
+rms
+bias
+Zspec       Dz    rms
+bias
+Zspec       
+Fig. 8.— Same as Fig. 5, but using the SNLS sample and
+saltii+z .ison there is a subtle disagreement. The PD09 sim-
+ulation, which uses coherent mag-smearing, underesti-
+matesthescatterinthe low-redshiftrangebutaccurately
+predicts the precision in the high-redshift range. Our
+simulation using coherent mag-smearing underestimates
+σ∆z/(1+z)(in PD09) for both redshift ranges, but our
+simulation based on color-smearing predicts σ∆z/(1+z)
+fairly well, perhaps with a slight overestimate of the
+scatter. Our simulation supports our earlier conclu-
+sion that color-smearing is needed to model the photo- z
+precision; the PD09 simulation supports our conclusion
+in the low-redshift range but not in the high-redshift
+range. Finally, our simulation underestimates the frac-
+tion of catastrophic outliers ( η0.15∼0.002), while the
+PD09 simulation gives good agreement with the data
+(η0.15∼0.01).
+TABLE 2
+photo-zPrecision for SNLS Data and Simulation. Results
+for PD09 and this work are shown. Our results include
+the uncertainty in parentheses.
+σ∆z/(1+z)σ∆z/(1+z)η0.15a
+Reference (z <0.45) (z >0.45) (all z)
+PD09 using
+DATAb(3 seasons) 0.016 0.025 0.014
+SIMb0.006 0.027 0.010
+this work using
+A06 DATA 0.005(5) 0.036(7) 0.02(2)
+SIM(coherent smear) 0.004(2) 0.016(1) 0.002(2)
+SIM(color-smear) 0.019(3) 0.030(3) 0(2)
+aη0.15= fraction of SNe with |zphot−zspec|/(1 +zspec)>0.15.
+bSeezphobcolumns in Table 1 of PD09.
+5.2.Photo-zCorrelations
+Herewebrieflydiscussphoto- zcorrelationsthatshould
+be propagated in a Hubble diagram analysis. Using
+00.050.10.150.20.250.30.350.40.45
+00.050.10.150.20.250.30.350.40.45SDSS-MLCS
+SDSS-SALT2
+SNLS-MLCS
+SNLS-SALT2float z
+z=ZspecDATARMSm     
+SDSS-MLCS
+SDSS-SALT2
+SNLS-MLCS
+SNLS-SALT2float z
+z=ZspecSIM
+Fig. 9.— RMSµfor four-parameter mlcs2k2 andsaltiifits
+(bottom of each arrow) and for five-parameter mlcs2k2+z and
+saltii+z fits (top of each arrow). The left panel shows RMS µfor
+the data; the right panel shows the simulations.9
+thelcfit+z results from the SDSS–II sample, Fig. 10
+shows reduced correlations ( ρ) between the photo- zand
+each of the other four light curve fit parameters. The
+mlcs2k2+z photo-zcorrelation with t0and distance
+modulus (upper plots) are both peaked at large positive
+values, the correlation with extinction ( AV) is negative,
+and there is little correlation with the shape parame-
+ter ∆. For saltii+z (lower plots) the photo- zcorrela-
+tionswith t0, shape/stretchparameter( x1), andcolorare
+qualitatively similar to those based on the mlcs2k2+z
+method. The ρ(x0,z)correlation has a very broad dis-
+tribution with an average near zero. For mlcs2k2+z ,
+the simulated distributions match the data well, while
+forsaltii+z there is a slight discrepancy in the distri-
+butions of ρ(c,z)andρx0,z. This discrepancy occurs only
+forzspec>0.2 and could be an artifact of the subset with
+smaller S/N.
+0102030405060
+-1 0 10102030405060
+-1 0 1010203040506070
+-1 0 1r(t0 ,Z)MLCS2k2+Z  Fit Correlations for SDSSEntries   DATA
+SIM
+r(D ,Z)r(AV ,Z)r(m ,Z)020406080100
+-1 0 1
+01020304050
+-1 0 1051015202530354045
+-1 0 1020406080100
+-1 0 1r(t0 ,Z)SALT2+Z  Fit Correlations for SDSSEntries   DATA
+SIM
+r(x1 ,Z)r(c ,Z)r(x0 ,Z)051015202530
+-1 0 1
+Fig. 10.— Reduced correlations ( ρ) between the photo- z(Z) and
+the other four light curve fit parameters as indicated in the t itle of
+each plot. Top plots are for the SDSS–II sample using mlcs2k2+z ;
+bottom plots are for the saltii+z method. The data are shown by
+dots; the simulation is shown by the histogram.
+6.PREDICTIONS FOR LSST
+Since we have demonstrated that the SNANAsimulation
+can be used to reliably determine the photo- zprecision
+for the SDSS–II and SNLS samples, we now turn our
+attention to forecasts for the LSST survey. LSST will
+discover far more SNe than spectroscopic resources can
+target, andphotometric methodswill be needed to deter-
+mine both the redshift and SN type for the majority of
+events. Here we determine the precision (bias and rms)
+on the photo- zand distance modulus using the lcfit+z
+method on more than 104simulated SNe Ia (after se-
+lection requirements) corresponding to the LSST-DEEP
+and LSST-MAIN surveys. Contamination from non-Ia
+SNe and the resulting precision in cosmological parame-
+ters will be presented in a future work.
+The DEEP survey comprises seven fields, each cover-
+ing nearly 10 deg2, that is densely time-sampled in all of
+theugrizYLSST filters. After rejecting passbands with
+invalid¯λf/(1+zphot), the averagenumberofobservations
+per SN Ia used in the light curve fit is 66. The MAIN
+survey covers more than 20 ,000 deg2but is not opti-
+mized for SN observations, so the light curve samplingoften has large temporal gaps for each filter. The mean
+number of fitted observations per SN for the MAIN sur-
+vey is 20, more than a factor of 3 fewer compared to the
+DEEP fields. More details about the LSST are given in
+Ivezi´ c et al.(2008); LSST Science Collaborations(2009).
+Due to the extensive computing resources needed to
+marginalize these large LSST samples, we have only per-
+formed the minimizations that are adequate for deter-
+mining central values for the fitted parameters.
+To simulate observing conditions, we use the out-
+put of version OPSIM1.29 of the LSST Operations
+Cadence Simulator (Delgado et al. (2006) and §3.1 of
+LSST Science Collaborations (2009)). for the cadence,
+sky noise, and 5 σlimiting magnitude for each measure-
+mentForeachobservation,the SNANAsimulationrequires
+a zero point (Z p.e.) to translate the simulated SN mag-
+nitude (m) into an observed CCD flux measured in pho-
+toelectrons, F= 10−0.4(m−Zp.e.). In terms of the OP-
+SIM1.29 parameters, we calculate this zero point to be
+Zp.e.=2M5σps−Msky+2.5log10(A·(S/N)2)
++2.5log10/bracketleftBig
+1+A−1×100.4(Msky−M5σps)/bracketrightBig
+,(4)
+whereM5σpsis the 5 σlimiting magnitude, Msky
+is the Perry sky brightness (mag/arcsec2),A=
+[2π/integraltext
+[PSF(r)]2rdr]−1= (1.51·FWHM)2is the effective
+aperture (in arcsec2) where FWHM describes the seeing,
+and S/N= 5 is the signal-to-noise ratio corresponding to
+M5σps.
+For this study, we carry out the light curve fits both
+with and without a host-galaxy photo- zprior; no other
+priorsareused. Thehost-galaxypriorisdeterminedfrom
+the Bayesian Photometric Redshift Estimation (BPZ)
+technique (Benitez 2000) applied to a preliminary set
+of simulated galaxies. The galaxy colors and lumi-
+nosities are generated to match observed distributions
+as a function of redshift. For more details, see §3.8
+of LSST Science Collaborations (2009). The signal to
+noise as a function of apparent magnitude for the host-
+galaxy simulation is based on co-added exposures for 10
+years of the MAIN survey, and photo- z’s are determined
+for galaxies with rmagnitudes down to 25. The av-
+erage host-galaxy photo- zprecision from BPZ is 0.02,
+and there are some variations with redshift as shown in
+Fig. 11. The zphot−zgenbias versus redshift has wiggles
+of order 0.005, but the true bias could be larger if the
+SN host galaxies are not a random subset of the galaxies
+used for photo- ztraining. Note that in this LSST dis-
+cussion of the host galaxy and SNe Ia, we characterize
+the photo- zprecision in terms of zphot−zgeninstead of
+∆z.
+These host-galaxy photo- zvalues are stored in a li-
+brary for the SNANAsimulation. For each simulated SN
+with true redshift zSN, the host galaxy with true red-
+shift (zgal) closest to zSNis selected. The corresponding
+host-galaxy photo- zis then scaled by the ratio zSN/zgal
+to correct for the slight redshift mismatch between the
+SN and the host galaxy. The scaled host-galaxy photo- z
+and its uncertainty are used to impose a Gaussian prior
+inlcfit+z .
+To ensure well-sampled light curves for the lcfit+z
+method, we apply the following selection requirements to
+the simulated LSST SN data: (1) at least two filters with10
+a measurement satisfying Trest<−5 days; (2) at least
+two filters with a measurement satisfying Trest>+20
+days; (3) largest rest-frame gap (that overlaps −5 to +20
+days) is<15 days; (4) at least three observer-frame fil-
+tershaveanepochwithS /N>10; and(5)thelightcurve
+fit probability satisfies Pχ2>0.02. The requirement of
+at leasttwo filters(cuts (2) and (3)) removespoorlysam-
+pled light curves predominantly from the MAIN survey.
+Using these requirements, the number of SNe per year
+is 1900 and 5 ×104for the DEEP and MAIN surveys,
+respectively. We note that these selection requirements
+are based on educated guesses rather than an optimiza-
+tion procedure. Example light curves from the DEEP
+and MAIN surveys are shown in Fig. 12.
+As a test of the fitting software, we first simulate ideal
+DEEP-field samples with no intrinsic mag-smearing and
+the exposure time artificially increased by a factor of
+104compared to the nominal exposure. The resulting
+photo-zbias is less than 0.001 at all redshifts for both
+mlcs2k2+z andsaltii+z , and the photo- zdispersion
+(rms) is less than 0.003. The bias on the distance modu-
+lus varies between 0 and 0.01 mag for mlcs2k2+z , and
+islessthan ±0.005magfor saltii+z . Thedistancemod-
+ulus dispersion is ∼0.02 mag for both fitting models.
+6.1.Results for LSST Simulations
+The SN photo- zresiduals as a function of redshift are
+shown in Fig. 13 for a simulation without intrinsic mag-
+nitude fluctuations and for a simulation using the same
+intrinsic color fluctuations needed to match the photo- z
+precision for the SDSS–II and SNLS data samples ( §5).
+Each panel shows the residuals as a function of fitting
+method ( mlcs2k2+z orsaltii+z ), survey field (DEEP
+or MAIN) and redshift prior (flat or host-galaxy photo-
+z). With no intrinsic fluctuations, the most notable
+effects are: (1) with a flat redshift prior, the extreme
+photo-zoutliers extend up to |zphot−zgen| ∼0.2, and
+(2) the host-galaxy photo- zprior significantly reduces
+the number of outliers. With default color fluctuations,
+thereisanotableincreaseinthephoto- zoutliers. Inboth
+cases, the redshift range is higher for saltii+z , because
+itextendstoalowerrest-framewavelength(2900 ˚A)than
+mlcs2k2 (3200˚A).
+-0.0200.020.040.060.08
+0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
+ZgenZphot - Zgen for host galaxy    rms
+bias
+Fig. 11.— For 6×104simulated LSST host galaxies, BPZ bias
+(crosses) and rms (curve) of zphot−zgenvs. the true host-galaxy
+redshift ( zgen) in redshift bins of width ∆ z= 0.03.Wequantifytherateofcatastrophicoutliersusing η0.15
+andη0.10(see§5.1). With ∼104SNe per sample, the
+approximate uncertainty is ση≃√η/100. Without in-
+trinsic mag-smearing (Fig. 13-left), η0.15= 0 in all cases.
+In this case, for the MAIN survey, η0.10∼0.01 with-
+out a host-galaxy photo- zprior, and η0.10is×10 smaller
+when the host-galaxy photo- zprior is used. Using the
+default color-smearing model in the simulation (Fig. 13-
+right), and fitting without a host-galaxy photo- zprior,
+η0.15<0.001 for the DEEP survey, and it is somewhat
+larger for the MAIN survey: 0 .004 and 0 .012 using the
+mlcs2k2+z andsaltii+z methods, respectively. For
+η0.10, the corresponding fractions are ×4 larger. Using
+thesaltii+z method, the numberofoutliersisabout ×3
+larger compared to mlcs2k2+z ; this difference could be
+related to the model, but it could also be an artifact of
+our implementation. We therefore make no claims that
+either method is more precise or has fewer catastrophic
+outliers. When the host-galaxy photo- zprior is used,
+η0.10=η0.15= 0 in all cases.
+To quantify the precision of the lcfit+z method, we
+have evaluated the bias and rms spread as a function of
+redshift for both the photo- zand distance modulus ( µ).
+Figure 14 shows the results based on simulations using
+the coherent mag-smearingmodel, and Fig. 15 shows the
+resultsusingthedefaultcolor-smearingmodel. Themain
+differences between using these two models of intrinsic
+variations are: (1) the photo- zrms goes to nearly zero
+at low redshift for the coherentmag-smearingmodel, but
+has a floor of about 0.01 for the color-smearing model,
+and (2) the rms is slightly larger at high redshift for the
+color-smearing model.
+Here we briefly summarize the precision based on sim-
+ulations using the default color-smearing model and fit-
+ting without a host-galaxy photo- zprior (see “FLATZ”
+panels in Fig. 15). In the DEEP survey, the photo- zrms
+precisionis ∼0.01atlowredshiftsandrisestoabout0.04
+at the highest redshifts. The RMS µprecision is near the
+0.15 mag floor at low redshifts and roughly doubles at
+the highest redshifts. In the MAIN survey, the photo-
+zprecision is about ×2 worse compared to the DEEP
+field survey. The corresponding RMS µprecision is about
+0.2 mag at the lowest redshifts and also roughly doubles
+at the highest redshift. The photo- zbias in the DEEP
+survey has wiggles of amplitude ∼0.01 as a function of
+redshift. The µ-bias wiggles are at most at the 0.01 mag
+level, and are notably less apparent than those seen in
+the photo- zbias. In the MAIN survey, the bias is no-
+ticeably larger and redshift-dependent; the photo- zbias
+reaches 0.02 and the µ-bias reaches 0.1 mag. Both the
+photo-zandµbiases are largest at zgen∼0.6.
+When fitting with a host-galaxy photo- zprior (see
+“HOSTZ”panels in Fig. 15), the precisionis significantly
+improved for both the photo- zandµ. In the MAIN
+survey, however, a redshift-dependent bias remains for
+zgen>0.6.
+Although the LSST photo- zprecision looks promising,
+we urge some caution in the interpretation of these simu-
+lations. If we consider the LSST DEEP-field subset over
+the same redshift range as the SDSS–II sample ( z <0.4),
+the forecast LSST photo- zprecision (rms) is about ×3
+better than that for the SDSS–II data sample described
+in§5, reflecting the higher expected signal to noise and11
+050
+0100
+0100
+0100LSST DEEP   z=0.3669
+g
+r
+i
+z
+Y
+Tobs - 49652.7relative flux0100
+-20 -10 0 10 20 30 40 50 60 70051015
+01020
+01020LSST DEEP   z=0.7623
+r
+i
+z
+Y
+Tobs - 49432.8relative flux02040
+-20 0 20 40 60 80 100050100
+-50050100150
+050100LSST MAIN   z=0.3669
+r
+i
+z
+Y
+Tobs - 49653.2relative flux050100
+-20 -10 0 10 20 30 40 50 6002040
+-200204060
+0204060LSST MAIN   z=0.6359
+r
+i
+z
+Y
+Tobs - 49632.3relative flux
+-250255075
+-20 0 20 40 60 80 100
+Fig. 12.— Left two panels show typical SN Ia light curves simulated for the LSST-DEEP fields; right two panels show typical light
+curves for the LSST-MAIN fields. The redshift is indicated on the top of each plot. Dots are simulated fluxes, the solid curv e is the best-fit
+mlcs2k2+z model, and the dashed curves are the ±1σerror bands for the model.
+broader wavelength coverage. Although we are confident
+in extrapolating the rms precision based on the treat-
+ment of photon statistics and color smearing, we cannot
+rule out unknown systematic effects, primarily from the
+unknown source of intrinsic brightness variations, that
+could limit the photo- zprecision and accuracy. Con-
+cerning the photo- zbias presented here, this should be
+considered a lower limit because the same light curve
+model has been used in both the simulation and in the
+lcfit+z fit. The current mlcs2k2 andsaltiimodels
+have been shown to differ significantly in the ultraviolet
+region (K09), but such modeling errors have not been
+considered here. In future studies, fitting saltiisimu-
+lations with mlcs2k2 (and vice-versa) would likely give
+an upper limit on the photo- zbias, and may lead to ad-
+ditional clues about problems in the light curve models.
+7.COMPARISONS WITH COLOR-BASED REDSHIFT
+ESTIMATES
+Herewe compareour lcfit+z resultswith color-based
+SN redshift estimates for the SNLS sample and for an
+mlcs2k2 -based simulation.
+7.1.Comparison with the SNLS Sample
+Wang (2007) determined color-based photometric red-
+shifts for 40 SNe Ia from the SNLS A06 sample. For the
+20 SNe used in the training, the author finds RMS ∆z=
+0.03; for the remaining 20 SNe, RMS ∆z= 0.05. In
+ourmlcs2k2+z analysis, 37 SNe satisfy the selec-
+tion criteria, and RMS ∆z= 0.045. For the saltii+z
+method, 37 SNe satisfy the selection requirements, and
+RMS∆z= 0.040. By this metric, our lcfit+z method
+works equally well compared to the color-based method.
+Since a list of SNe used by Wang (2007) is not available,
+we cannot make a more detailed comparison with the
+same subset.
+7.2.Comparison with mlcs2k2 -based Simulations
+We compare photo- zresults of the two methods on
+simulations as described in Wang et al. (2007) (hereafterWNW07). FollowingtheprocedureinWNW07,wesimu-
+late observer-frame filters rizusing the mlcs2k2 model
+for redshifts z <0.95 (so that r-band data are always
+well defined within the model), fix the shape-luminosity
+parameter ∆ = 0, generate a flat redshift distribution,
+and ignore intrinsic magnitude variations that introduce
+anomalous Hubble scatter. For each SN and each filter,
+the exposure time is adjusted so that S /N = 25 at the
+epoch of peak brightness. In WNW07, only the peak
+fluxes are used and therefore the light curve sampling
+doesnot matter; to investigatethe comparativeeffective-
+ness of our mlcs2k2+z method, simulated light curves
+are sampled every 7 days in the observer frame.
+Intheidealcaseofnohost-galaxyextinction( AV= 0),
+WNW07 find RMS ∆z= 0.004 with a mean bias of
+5.4×10−4; usingmlcs2k2+z under similar assump-
+tions, we find RMS ∆z= 0.006 and a mean bias of
+(−1.7±2.1)×10−4. Including host-galaxy extinction in
+the simulation with a pdf P(AV) = exp(−AV/0.46) and
+reddening parameter RV= 3.1, WNW07 find RMS ∆z=
+0.044 with a mean bias of 0 .008; using mlcs2k2+z , un-
+der the same conditions we find RMS ∆z= 0.009 and a
+mean bias of (0 .5±2.9)×10−4. The significantly im-
+proved precision with our mlcs2k2+z method in these
+more realistic conditions is in part due to the increase in
+effective signal to noise that comes from using the entire
+lightcurve. Inaddition,shapeandcolorinformationcon-
+tainedin the lightcurveenablesthe mlcs2k2+z method
+topartiallyuntanglecolorvariationsfromextinction ver-
+sus those produced by redshift.
+Since the color-based method in WNV07 cannot un-
+tangle reddening from extinction and redshift, we have
+attempted a more fair comparison to the mlcs2k2+z
+method in which AVis fixed to the mean generated ex-
+tinction of 0.46 mag; in this case RMS ∆znearly doubles
+to 0.015 for the mlcs2k2+z method, yet is nearly a fac-
+tor of 3 smaller than the dispersion in WNV07. This
+difference is either due to the enhanced photon statis-
+tics from using the entire light curve in the mlcs2k2+z12
+-0.4-0.3-0.2-0.100.10.20.30.4
+-0.4-0.3-0.2-0.100.10.20.30.4
+0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1No intrinsic mag-smearingZphot - Zgen     MLCS
+DEEP
+FLATZ
+h(0.10 , 0.15) =
+0 , 0MLCS
+DEEP
+HOSTZ
+h(0.10 , 0.15) =
+0 , 0MLCS
+MAIN
+FLATZ
+h(0.10 , 0.15) =
+0.009 , 0MLCS
+MAIN
+HOSTZ
+h(0.10 , 0.15) =
+0.001 , 0
+Zgen   Zphot - Zgen     SALT2
+DEEP
+FLATZ
+h(0.10 , 0.15) =
+0.001 , 0
+Zgen   SALT2
+DEEP
+HOSTZ
+h(0.10 , 0.15) =
+0 , 0
+Zgen   SALT2
+MAIN
+FLATZ
+h(0.10 , 0.15) =
+0.018 , 0.003
+Zgen   SALT2
+MAIN
+HOSTZ
+h(0.10 , 0.15) =
+0.002 , 0-0.4-0.3-0.2-0.100.10.20.30.4
+-0.4-0.3-0.2-0.100.10.20.30.4
+0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1Default color-smearingZphot - Zgen     MLCS
+DEEP
+FLATZ
+h(0.10 , 0.15) =
+0.004 , 0MLCS
+DEEP
+HOSTZ
+h(0.10 , 0.15) =
+0 , 0MLCS
+MAIN
+FLATZ
+h(0.10 , 0.15) =
+0.018 , 0.004MLCS
+MAIN
+HOSTZ
+h(0.10 , 0.15) =
+0 , 0
+Zgen   Zphot - Zgen     SALT2
+DEEP
+FLATZ
+h(0.10 , 0.15) =
+0.008 , 0.001
+Zgen   SALT2
+DEEP
+HOSTZ
+h(0.10 , 0.15) =
+0 , 0
+Zgen   SALT2
+MAIN
+FLATZ
+h(0.10 , 0.15) =
+0.051 , 0.012
+Zgen   SALT2
+MAIN
+HOSTZ
+h(0.10 , 0.15) =
+0 , 0
+Fig. 13.— zphot−zgenresiduals vs. zgenfor the model ( mlcs2k2 orsaltii), survey (DEEP or MAIN), and photo- zprior (host galaxy
+or flat) indicated in each panel for LSST simulations. Each pa ir of plots compares residuals with no photo- zprior (FLATZ) to residuals
+with host-galaxy photo- zprior (HOSTZ). The fraction of catastrophic outliers ( η0.10,0.15), indicated in each panel, is defined in the text.
+The simulations are generated without intrinsic mag-smear ing (left plots) and with the default color-smearing model ( right plots).
+-0.0200.020.040.060.08
+-0.100.10.20.30.4
+0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8MLCS    Coherent mag-smearing in simZphot - ZgenDEEP-FLATZ
+rms
+biasDEEP-HOSTZ MAIN-FLATZ MAIN-HOSTZ
+Zgen    mfit - mgen  
+DEEP-FLATZ
+Zgen    DEEP-HOSTZ
+Zgen    MAIN-FLATZ
+Zgen    MAIN-HOSTZ-0.0200.020.040.060.08
+-0.100.10.20.30.4
+0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8SALT2    Coherent mag-smearing in simZphot - ZgenDEEP-FLATZ
+rms
+biasDEEP-HOSTZ MAIN-FLATZ MAIN-HOSTZ
+Zgen    mfit - mgen  
+DEEP-FLATZ
+Zgen    DEEP-HOSTZ
+Zgen    MAIN-FLATZ
+Zgen    MAIN-HOSTZ
+Fig. 14.— Bias (crosses) and rms (curve) on photo- zresidual ( zphot−zgen), and on distance modulus residual ( µfit−µgen) vs. the
+true redshift ( zgen) inz-bins of width ∆ Z= 0.03 for the LSST simulations. Each plot indicates DEEP or MAIN fields, and FLATZ (SN
+only) or HOSTZ (host-galaxy photo- zprior). The simulation uses coherent (0.15 mag) brightness fluctuations in all filters and is fitted
+withmlcs2k2+z (left) and saltii+z (right). For z <0.15, thesaltiicurves drop to zero because there are only two valid filters ( gr),
+and the data fail the three-filter requirement. The dashed ho rizontal lines in the µfit−µgenplots show the rms contribution from intrinsic
+variations.
+method or from non-optimal training in the color-based
+method. To increase RMS ∆zto the WNV07 value of
+0.044, the peak S/N must be reduced from 25 to less
+than 10.
+In themlcs2k2+z fits we have used RV= 3.1, the
+same value used in generating the simulation. Using the
+correct value for RVis an apparently unfair advantage
+over the WNW07 treatment, in which no assumptions
+aremadeaboutcolorvariations. However,inpracticethe
+assumption about the value of RVmakes little difference
+to the results: fitting with RV= 2.2 produces the same
+precision for ∆ z, although the resulting distance moduli
+are biased by about 0.1 mag.
+Even though the effects of host-galaxy extinction are
+included the simulation described in WNW07, this sim-
+ulation is not realistic because the shape-luminosity pa-
+rameter ∆ is fixed to zero, and there are no intrinsicmagnitude variations. Simulating our best estimate for
+theseeffects ( §3), the biasand scatterin the fitted photo-
+zand distance modulus are shown as a function of zspec
+in Fig. 16. For redshifts below 0.8 there is no signifi-
+cant bias in either the photo- zor distance modulus. For
+z >0.8 the bias and scatter increase significantly. The
+nature of this high-redshift bias is illustrated in Fig 17,
+which shows the correlation between the fitted versus
+simulated photo- zdifference and the fitted versus sim-
+ulated difference for time of peak ( t0), extinction, and
+shape-luminosity parameter. The photo- zoutliers are
+clearly correlated with t0outliers. Since the SN Ia color
+becomes redder with epoch, a misestimate of t0changes
+the apparent SN color, which is translated into an error
+inthe redshift. The biasin t0isan artifactofthe discrete
+mlcs2k2 passbands used to characterize the rest-frame
+light curves. For SNe simulated at redshifts z >0.8, the13
+-0.0200.020.040.060.08
+-0.100.10.20.30.4
+0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8MLCS    Color mag-smearing in simZphot - ZgenDEEP-FLATZ
+rms
+biasDEEP-HOSTZ MAIN-FLATZ MAIN-HOSTZ
+Zgen    mfit - mgen  
+DEEP-FLATZ
+Zgen    DEEP-HOSTZ
+Zgen    MAIN-FLATZ
+Zgen    MAIN-HOSTZ-0.0200.020.040.060.08
+-0.100.10.20.30.4
+0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8SALT2    Color mag-smearing in simZphot - ZgenDEEP-FLATZ
+rms
+biasDEEP-HOSTZ MAIN-FLATZ MAIN-HOSTZ
+Zgen    mfit - mgen  
+DEEP-FLATZ
+Zgen    DEEP-HOSTZ
+Zgen    MAIN-FLATZ
+Zgen    MAIN-HOSTZ
+Fig. 15.— Same as Fig. 14, except that the default color-smearing mode l is used in the simulation.
+three observer-frame filters map into only two rest-frame
+filters:riz→UBB. In a small fraction of the light curve
+fits, however, the wrong filter-mapping ( riz→UBV) re-
+sults in a smaller χ2. Since the BandV-band templates
+have different shapes, as well as a 2-day shift in the time
+of peak brightness, the fitted t0is biased.
+-0.01-0.00500.0050.010.0150.020.0250.030.0350.04
+0 0.2 0.4 0.6 0.8 1-0.200.20.40.60.8
+0 0.2 0.4 0.6 0.8 1
+ZspecSimulated MLCS with Peak S/N = 25(Zphot - Zspec) / ( 1 + Zspec )     rms
+bias(includes color smearing and D variation)
+Zspecmfit - mgen     rms
+bias
+Fig. 16.— Formlcs2k2 simulations with peak S /N = 25, fitted
+photo-zbias and RMS ∆zvs.zspec(left), and distance modulus
+bias and RMS µvs.zspec(right). The simulations are generated
+according to the prescription of WNW07, but also including n on-
+zero ∆ and intrinsic color variations.
+8.CONCLUSIONS
+We have developed and tested a photometric redshift
+estimation method using SN Ia light curves within the
+framework of the mlcs2k2 andsaltiimodels and find
+that they result in similar photo- zprecision. We used
+an iterative fitting procedure to determine the valid
+observer-frame filters to use in the fits. Applying this
+method to SDSS–II and SNLS data, we obtained an av-
+erage photo- zprecision of RMS ∆z∼0.04. To repro-
+duce a comparable level of precision in simulations, in-
+trinsic color-smearingis needed ( §3) at the level of about
+0.1 mag per passband or 0 .14 mag per color. This em-
+pirical estimate of color smearing is consistent with butdoes not necessarily imply that there are random color
+variationsin SNIa lightcurves. However,this effect does
+indicate that there are additional sources of color varia-
+tion that are not captured by the mlcs2k2 andsaltii
+light curve models.
+We applied the lcfit+z method to simulated LSST
+samples ( §6). For the DEEP fields, the rms scatter of
+zphot−zgenvaries from 0.01 to 0.04 without using a host-
+galaxy photo- zprior. For the MAIN survey the photo- z
+precision is about ×2 worse. Using a host-galaxy photo-
+zprior significantly reduces outliers and improves the
+overall precision. The next critical step is to apply this
+method to simulations that include non-Ia type SNe and
+estimate the resulting contamination of photometric SN
+Ia samples by core-collapse SNe.
+We gratefully acknowledge support from the Kavli
+Institute of Cosmological Physics at the University of
+Chicago, the National Science Foundation at Wayne
+State, and the Department of Energy at Fermilab, the
+-0.2-0.15-0.1-0.0500.050.10.15
+-1 0 1-0.2-0.15-0.1-0.0500.050.10.15
+-3 -2 -1 0 1
+-0.2-0.15-0.1-0.0500.050.10.15
+-1 0 1 2Fit Correlations for MLCS Simulations with Peak S/N=25
+t0fit - t0sim     Zphot - Zsim     
+mfit - msim     Zphot - Zsim     
+AVfit - AVsim     Zphot - Zsim     
+Dfit - Dsim     Zphot - Zsim     
+-0.2-0.15-0.1-0.0500.050.10.15
+-0.4 -0.2 0 0.2
+Fig. 17.— Fit-simulation photo- zresidual ( zphot−zgen) vs.
+fit-simulation residual for time of peak brightness ( t0), distance
+modulus ( µ), host-galaxy extinction ( AV), and shape-luminosity
+parameter (∆).14
+University of Chicago, and Rutgers University. S.J.
+is grateful for the support of DOE grant DE-FG02-
+08ER41562. Funding for the creation and distribu-
+tion of the SDSS and SDSS-II has been provided by
+the Alfred P. Sloan Foundation, the Participating In-
+stitutions, the National Science Foundation, the U.S.
+Department of Energy, the National Aeronautics and
+Space Administration, the Japanese Monbukagakusho,
+the Max Planck Society, and the Higher Education
+Funding Council for England. The SDSS Web site
+ishttp://www.sdss.org/ .
+The SDSS is managed by the Astrophysical Research
+Consortium for the Participating Institutions. The Par-
+ticipating Institutions are the American Museum of Nat-
+ural History, Astrophysical Institute Potsdam, Univer-
+sity of Basel, Cambridge University, Case Western Re-
+serve University, University of Chicago, Drexel Univer-
+sity, Fermilab, the Institute for Advanced Study, the
+Japan Participation Group, Johns Hopkins University,
+the Joint Institute for Nuclear Astrophysics, the Kavli
+Institute for Particle Astrophysics and Cosmology, the
+Korean Scientist Group, the Chinese Academy of Sci-
+ences (LAMOST), Los Alamos National Laboratory, the
+Max-Planck-Institute for Astronomy (MPA), the Max-
+Planck-Institute for Astrophysics (MPiA), New Mexico
+State University, Ohio State University, University of
+Pittsburgh, University of Portsmouth, Princeton Uni-
+versity, the United States Naval Observatory, and the
+University of Washington.This work is based in part on observations made at
+the following telescopes. The Hobby-Eberly Telescope
+(HET) is a joint project of the University of Texas
+at Austin, the Pennsylvania State University, Stanford
+University, Ludwig-Maximillians-Universit¨ at M¨ unchen,
+and Georg-August-Universit¨ at G¨ ottingen. The HET is
+named in honor of its principal benefactors, William
+P. Hobby and Robert E. Eberly. The Marcario Low-
+Resolution Spectrograph is named for Mike Marcario of
+High Lonesome Optics, who fabricated several optical
+elements for the instrument but died before its comple-
+tion; it is a joint project of the Hobby-Eberly Telescope
+partnership and the Instituto de Astronom´ ıa de la Uni-
+versidad Nacional Aut´ onoma de M´ exico. The Apache
+PointObservatory3.5m telescopeisownedandoperated
+by the Astrophysical Research Consortium. We thank
+the observatory director, Suzanne Hawley, and site man-
+ager, Bruce Gillespie, for their support of this project.
+The Subaru Telescope is operated by the National As-
+tronomical Observatory of Japan. The William Herschel
+Telescope is operated by the Isaac Newton Group on
+the island of La Palma in the Spanish Observatorio del
+Roque de los Muchachos of the Instituto de Astrofisica
+de Canarias. The W.M. Keck Observatory is operated
+as a scientific partnership among the California Institute
+of Technology, the University of California, and the Na-
+tional Aeronautics and Space Administration. The Ob-
+servatory was made possible by the generous financial
+support of the W. M. Keck Foundation.
+APPENDIX
+A.FITTING ISSUES WITH UNKNOWN REDSHIFT
+Here we discuss our implementation of the five pre-fit issues mention ed in§4: (1) SN selection criteria that depend
+on knowing Trest=Tobs/(1 +z) such as requiring measurements with a minimum and maximum Trestvalue; (2)
+determining which observer-frame filters (with mean wavelength ¯λf) have¯λf/(1+zphot) within the valid wavelength
+range of the fitting model; (3) determining the valid rest-frame epo ch range for the fitting model; (4) as ¯λf/(1+zphot)
+maps into a different rest-frame filter (for mlcs2k2 ) there is a discontinuous change in the model error, and therefor e
+theχ2is not a continuous function of zphot; and (5) determining robust initial fit parameter values.
+The simplest way to handle SN selection criteria (issue 1) is to postpon e such requirements until the fit has finished
+and then use the fitted photo- zto determine the Trestvalues. This solution is not practical, however, because we often
+wish to remove poorly sampled light curves before fitting, thereby a voiding pathological fits that are of no interest. A
+safe way to apply Trest-dependent requirements before fitting is to relax such cuts by a f actor of 1+ Zmax, whereZmax
+is a safe upper bound on all redshifts. For example, consider the re quirements of a measurement with Trest<−6 days
+and 21< Trest<60 days. Using Zmax= 0.5 for the SDSS–II, the pre-fitted requirements are Trest<−4 days and
+14< Trest<90 days. Any initial redshift value can be used to determine Trestas long as it is less than Zmax. Although
+theseTrest-related requirements are relaxed, they are still useful for reje cting poorly sampled light curves; the nominal
+cut is applied after the fit using the fitted photo- zvalue.
+To determine the valid observer-frame filters (issue 2), the first- iteration photo- zfit uses all filters, with a possible
+exception for those covering the ultraviolet region with a mean wave length below about 4000 ˚A. The ultraviolet filter
+should be left out of the first iteration if it is rarely used, noting that it can be added back for the second fit iteration.
+The fundamental assumption about the fitting model is that extra polating beyond the defined wavelength range gives
+reasonable magnitude estimates so that the fit converges and tha t the photo- zbias is not too large. The (biased)
+photo-zestimate is then used to determine which observer-frame filters to retain and to reject for the second fit
+iteration. Note that if an ultraviolet filter is excluded initially, then a low photo-zvalue will result in the inclusion of
+this filter for the second iteration. If a filter is excluded, the Trest-related requirements are re-tested; the fit stops if
+the light curve no longer has adequate sampling.
+Since the first-iteration photo- z(z1
+phot) is biased and has some uncertainty, a safety margin is used to dete rmine
+which observer-frame filters to keep. For each observer-frame filter (f) with mean wavelength ¯λf, the valid redshift
+range for this filter is defined by
+zmin,f=¯λf/λmodel
+max−1 zmax,f=¯λf/λmodel
+min−1, (A1)
+whereλmodel
+min,maxare the minimum and maximum rest-frame wavelengths defined by the model. For example, the saltii15
+model is defined for rest-frame wavelengths 2900–7000 ˚A; the observer-frame i-band (¯λi= 7500˚A) is therefore valid
+for redshifts above zmin,i= 0.071, and g-band (¯λg= 4720˚A) is valid for redshifts below zmax,g= 0.63. For this
+analysis, a filter is kept in the second iteration if the first-iteration fi tted photo- zsatisfies
+zmin,f+∆zcut
+phot< z1
+phot< zmax,f−∆zcut
+phot. (A2)
+We have set ∆ zcut
+phot= 0.04 and 0 .03 for the SNLS and SDSS surveys, respectively. ∆ zcut
+photis smaller for the SDSS–II
+because the S/N for SNe at zmin,i= 0.07 is larger than the S/N for SNLS SNe at zmax,g= 0.063. The ∆ zcut
+photfilter-
+selection cut has not been optimized; the optimal cut is likely to depen d on the uncertainty of the host-galaxy photo- z
+and may also depend on redshift.
+This filter selection algorithm is illustrated in Fig. 18. The solid histogram s in the top two plots (SDSS data and
+simulation) show the true redshift ( zspec) distributions near zmin,iwhen the i-band is kept in the saltii+z fit. When
+thei-band is discarded (dashed), the entire light curve is discarded bec ause of the requirement of having three filters.
+The gap between zmin,i= 0.071 and the bulk distribution is due to the ∆ zcut
+photcut. The bottom two plots in Fig. 18
+illustrate cases for LSST when a dropped filter does not result in rej ecting the entire light curve. When the LSST
+g-band is included in the mlcs2k2+z fit (solid curve, lower left), the true redshift almost always satisfie s the model-
+validity requirement ( zspec<0.5). The dashed curve shows that the g-band has been correctly excluded when the
+true redshift is above 0.5, but we have also excluded this passband in some cases where it is valid (i.e., zspec<0.5).
+The situation is similar for the LSST Y-band (lower right plot). For both passbands, a small number of ligh t curve
+fits include these bands when they should have been excluded; pote ntial biases from these invalid passbands should
+be accounted for in the assessment of systematic uncertainties o n cosmological parameters.
+The selection of rest-frame epochs (issue 3), which depends on Tobs/(1 +zphot), is made in the same way as the
+selection of filters (issue 2). All measurements are included in the fir st fit iteration, and the initial photo- zestimate is
+then used to select which epochs satisfy the epoch range of the mo del.
+Theχ2continuity (issue 4) is an issue because the SNANAminimization is based on minuit, and this programrelieson
+computing local derivatives with respect to the fitting parameters . This means that discontinuities in the χ2function
+and its derivative can lead to fitting failures or pathological results. The problem can be solved by marginalizing,
+but the necessary five-dimensional integration requires ∼100 times more CPU processing. In addition to the utility
+of using the much faster minimization program, it is useful to identify χ2discontinuities because such functional
+pathologies usually correspond to unphysical behavior in the light cu rve model, and the model should be improved
+accordingly.
+There are two main sources of χ2discontinuity: (1) interpolating lookup tables, and (2) filter-depen dent model
+parameters. For class (1), appropriate numerical methods must be used to ensure continuity in both the function and
+its derivative. For class (2), the problem occurs for rest-frame m odels such as mlcs2k2 , in which an infinitesimal
+change in zphotresults in a different rest-frame filter for the model or a different c olor to use for spectral warping in the
+K-corrections. This step-function change in the model parameter s results in a discontinuity in the model magnitude
+(or its error) as a function of zphot. To prevent such sharp discontinuities, SNANAuses a smooth transition function
+(A+Btan−1(λ/τλ)) to smoothly vary model parameters between neighboring filters .
+Since the redshift and SN color are somewhat degenerate, initial pa rameter estimation (issue 5) is important so
+thatminuitwill find a global, rather than local, minimum. The initial color and photo- zare determined from a
+crudeχ2minimization on a coarse grid: the color is varied in bins of 0 .2 and the photo- zis varied in bins of 0 .04.
+For each photo- zvalue, an initial distance modulus estimate ( µini) is needed to ensure robust convergence of minuit.
+Although µiniis calculated from a standard ΛCDM cosmology using a specific set of c osmological parameters, the
+fitted distance modulus is unconstrained and is therefore not biase d by the µinicalculation. For saltii, the initial x0
+value is estimated by inverting Eq. 3. The shape-luminosity paramete r is initialized to an average value: ∆ = −0.1 for
+mlcs2k2 , andx1= 0.0 forsaltii. After the first fit iteration, the fitted parameters are used as in itial values for the
+second fit iteration. When a filter is dropped and the first fit iteratio n is repeated, the fitted parameters are used as
+initial values, even though an invalid filter was used in the fit. If we do n ot use a coarse grid to initialize the color and
+photo-z, and simply start with average values, the photo- zprecision for the SDSS–II sample is only slightly degraded.
+For the SNLS sample, however, which has a much larger redshift ran ge compared to the SDSS–II, the photo- zprecision
+is degraded by a factor of 2 due to a significant number of catastro phic outliers.
+B.MARGINALIZATION
+Here we describe some of the details related to the marginalization. T he minimized values and uncertainties are used
+to estimate the integration ranges: ±4σaround the minimized value for each parameter. The integration grid consists
+ofNgridpoints per fit-parameter or a total of Ngrid5integration cells. We find that Ngrid= 11 is a good compromise
+between precision and computing time. To improve calculation speed p er integration cell, the χ2calculation stops
+when the probability falls below 10−5.
+The marginalization is repeated for either of the following cases: (1) the probability at any integration boundary
+is greater than 0.03, or (2) more than three one-dimensional bins ( i.e., marginalized over the other four parameters)
+have a probability less than 10−4. In the first case the integration range is extended, while in the sec ond case the
+integration range is reduced.16
+051015202530
+0 0.05 0.1 0.15 0.2010203040506070
+0 0.05 0.1 0.15 0.2
+050100150200250300350400450500
+0.3 0.4 0.5 0.6 0.7020406080100120140
+0.3 0.4 0.5 0.6Zspeczmin,iSDSS-i (data, SALT2)Entries per 0.02    
+Zspeczmin,iSDSS-i (sim, SALT2)
+Zspeczmax,gLSST-g (sim, MLCS)Entries per 0.02    
+Zspeczmin,YLSST-Y (sim, SALT2)
+Fig. 18.— Illustration of filters excluded after the first fit iteration . Each plot shows the spectroscopic (i.e., true) redshift di stribution
+when the indicated filter is kept in the photo- zfit (solid) and when the same filter has been dropped after the fi rst fit iteration (dashed).
+The vertical arrow shows either zmin,forzmax,ffor the filter indicated. zspecis used only in making these plots, and is not used in the
+photo-zfits.
+REFERENCES
+Abdalla, F. et al. 2008, submitted, arXiv:0812.3831
+Astier, P. et al. 2006, A&A, 447, 31
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\ No newline at end of file
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+arXiv:1001.0739v1  [astro-ph.SR]  5 Jan 2010To appearin:
+Proceedingsofthe25thNSO Workshop ChromosphericStructureandDynamics
+Sept 2009, Sunspot NM.
+TheChromosphereDuringSolarFlares
+L. Fletcher
+Department of PhysicsandAstronomy, Universityof Glasgow , Glasgow G128QQ, United
+Kingdom e-mail:lyndsay@astro.gla.ac.uk
+Abstract. The emphasis of observational and theoretical flare studies in the last decade or
+twohasbeenontheflarecorona,andattentionhasshiftedsub stantiallyawayfromtheflare’s
+chromospheric aspects. However, although the pre-flare ene rgy is stored in the corona, the
+radiative flare is primarily a chromospheric phenomenon, an d its chromospheric emission
+presents a wealth of diagnostics for the thermal and non-the rmal components of the flare.
+I will here review the chromospheric signatures of flare ener gy release and the problems
+thrownupbytheapplicationofthesediagnostics inthecont ext ofthestandardflaremodel.
+Iwillpresentsomeideasaboutthetransportofenergytothe chromosphere byothermeans,
+and calculations of the electronacceleration thatone migh t expect inone such model.
+Key words. Sun: flares–Sun: chromosphere –
+1. Introduction
+The first recorded observation of a solar flare
+was made on 1st September 1859 (Carrington
+1859) and showed features which we would
+now recognise as broadly chromospheric in
+origin. The two small ‘patches of intensely
+bright and white light’, now termed a white-
+light flare, mapped the locations from which
+the bulk of the solar flare energywas radiated.
+Although Carrington could not have known it
+at the time, the flare was accompanied also
+by the emission of ultraviolet to X-ray and
+most probably also γ-ray radiation. These ra-
+diations, also primarily chromospheric in ori-
+gin, produced changes in the ionospheric ion-
+isation level, leading to currents disturbances
+and strong geomagnetic activity, recorded as
+an abrupt magnetometer deflection at Kew
+GardensinLondon.
+Send offprint requests to :L.FletcherFor decades, studies of solar flares were
+made primarily via their lower atmospheric,
+and particularly their chromospheric signa-
+tures.Duetoitslowcontrastwiththesurround-
+ing photosphere, white-light flares are hard to
+observe, but the flare dynamics and evolution
+are shown wonderfully well by the bright H α
+emission, usually organised into two or more
+elongated‘ribbons’. Observationsof these H α
+ribbons, as well as the motions and disappear-
+ance of the Hαfilaments and the post-flare
+growth of Hαloops was in great part respon-
+sibleforthedevelopmentoftheflare‘standard
+flaremodel’.
+As solar physics instrumentation moved
+into space, much of the attention of the solar
+flare community shifted to imaging and spec-
+tral observations at higher energies: the ultra-
+violet,extremeUV, X-rayand γ-ray.In partic-
+ular,thespectacularEUVandsoftX-rayloops
+ofthereconfiguringcoronalplasma,withtheir
+close link to the all-important coronal mag-2 Fletcher:Flare Chromospheres
+neticfield,havebeenverycompelling.Buten-
+ergeticallythe EUVandX-rayemissionsare a
+sideshow (Emslieet al. 2004, 2005). The opti-
+cal to ultraviolet continuum and lines are the
+primary means by which the solar atmosphere
+rids itself of energy during a flare. To under-
+stand solar flares, we must link what we have
+learned of coronal processes during the last
+decades of space-based observation with what
+the chromospherehastotell us.
+2. Flarechromosphericsources
+The impulsive phase flare morphology in H α,
+UV and chromospheric EUV is characterised
+by elongated ribbons of emission which start
+out close together, on either side of a mag-
+netic neutral line, and lengthen and spread
+outwards with time. Within these ribbons
+there are compact, bright sources (kernels)
+which tend to be located on the outer edges
+of the ribbons. Overlaying images made in
+hard X-rays (HXRs) with the RHESSI mis-
+sion (Temmeret al. 2007, e.g.) shows clearly
+that kernels are co-spatial with HXR chromo-
+sphericsources,generallyknownasflare‘foot-
+point’ sources. Only a small number of HXR
+footpoint sources are visible in a flare, proba-
+blydueinparttothelimiteddynamicrangeof-
+feredbythe indirectimagingtechniquesinthe
+HXR part of the spectrum. However the very
+fact that there are sources with significantly
+higher photonfluxes points to the existence of
+speciallocationswithintheoverallflaregeom-
+etry.Itisalsonotsurprisingthattheselocations
+arewherethestrongestenhancementsinwhite
+light are to be found.This has been shown us-
+ing TRACE observations (Metcalfetal. 2003;
+Hudsonet al. 2006; Fletcheret al. 2007) taken
+through the ‘white light’ (WL) open filter. In
+the case of this filter, ‘WL’ may be a mis-
+nomerduetothehighfluxofUVradiationthat
+it also admits. However, emission in TRACE
+WL and in UV shows very di fferent spatial
+characteristics,withribbonsinUVthatareex-
+tendedcomparedtothe concentrated,compact
+WLsourceswheremostoftheflareenergyen-
+tersthechromosphere.Figure1showsthespa-
+tialrelationshipbetweenUVflareribbons,WL
+sourcesandHXRsources.The locations of strong flare chromo-
+spheric emission correspond to locations of
+particular significance in the overall magnetic
+structure of the flare. The standard ‘CSHKP’
+model links ribbons, flare loops, and the sup-
+posed coronal magnetic reconnection site, in
+a simple 2-D geometry. This has been trans-
+ported into 3D and put on a firmer the-
+oretical footing, starting with the work of
+Demoulinet al. (1992) who found correspon-
+dencebetween the photosphericprojectionsof
+strongquasi-separatrixlayers,andthechromo-
+spheric Hαsources. Subsequent work, for ex-
+ample by Metcalfet al. (2003) has confirmed
+this, and the most recent thinking in theory
+(DesJardinsetal. 2009, e.g.) associates the
+HXR and by implication the WL sources with
+theloweratmospheremappingofsingularfield
+lines known as spines, which join two mag-
+netic nulls. Reconnection at such structures is
+often proposed to lead to flare particle accel-
+eration, but to provide the accelerated elec-
+trons required to explain the observed HXR
+radiation in the collisional thick target model
+(CTTM)needsavolumeofflaringcoronaout-
+linedroughlybythe extentofthe flareribbons
+and the flare loops to be ‘processed’ by the
+accelerator each second (see Section 4). It is
+unlikely that acceleration can take place in a
+current sheet, null, separator or QSL, because
+the throughputof plasma through these singu-
+lar structures is - even assuming 100% accel-
+eration efficiency and a high Alfv´ en speed -
+just too small for reasonable impulsive phase
+source dimensions. The electron requirements
+point towards a truly volumetric acceleration
+mechanism operating throughout a large por-
+tion of the corona local to the flare. But then
+how can all the electrons accelerated there be
+directed towardsthe footpoints?Electronsfol-
+low the magnetic field, and the field permeat-
+ingtheflaringcoronacertainlyisnotallrooted
+inthe flarefootpoints.
+3. ChromosphericDynamics
+The response of the chromosphere to the en-
+ergy deposited there during a solar flare is
+to radiate, conduct and expand - the latter
+is usually known as chromospheric ‘evapo-Fletcher:Flare Chromospheres 3
+Fig.1.A composite image showing the GOES M8.5 flare on 04-October-2 002. The background image is
+the TRACE 1700 Å channel (log scaled), on which are superpose d contours showing the strongest white
+light patches (yellow /light) and the RHESSI 25-50 keV sources, made using the Pixon s technique. The
+relative pointing between TRACE and RHESSI is uncertain, so the RHESSI sources have been translated
+by (-5,1) arcseconds toobtain agreement withthe strongest white-lightsources.
+ration’ although it has nothing to do with
+the usual meaning of the word. Evaporation
+can happen gradually or explosively, depend-
+ing on whether the heating timescale is larger
+or smaller than the hydrodynamic expansion
+time of the chromosphere. Evaporation has
+been observed using spectroscopy, originally
+withHαand unresolved SXR line profiles
+(Zarro& Lemen1988)andlatterlyintheEUV
+with imaging spectrometers. In the few flare
+impulsive footpoints which have been ob-
+servedusingEUV spectroscopy,the usual pat-
+tern is that higher temperature lines show
+a blueshift from the upflowing plasma, and
+lower temperature lines are redshifted. The
+redshift is interpreted as due to a downwards-
+moving ‘chromospheric condensation’, driven
+byapressurepulsewhichoccurswhentheup-
+per chromosphere is heated rapidly past the
+point where it can radiate e fficiently, and ex-
+pands quickly. There is rough momentum bal-
+ancebetweenthesetwocomponents.
+However, with Hinode/EIS, Milligan
+(2008) and Milligan&Dennis (2009) have
+recently observed redshifted emission compo-nentsupto1.5MK-muchhighertemperatures
+fordownward-movingplasmathanisexpected
+from theoretical calculations. This would
+require heating of the upper chromosphere,
+which could be delivered- in accordance with
+the flare coronal electron beam (Section 4)
+by a soft spectrum of electrons in a beam
+from the corona, such as is suggested by
+the large electron spectral index deduced by
+Milligan& Dennis (2009). But most myste-
+riously, at high energies Milligan&Dennis
+(2009) also find dominant, stationary high
+temperature (>∼12 MK) footpoint sources in
+theimpulsivephase,whicharenotexpectedin
+a beamheatingmodelatall.
+4. Flarechromosphereenergetics
+Thechromosphereishighlystructuredintem-
+perature and density, horizontally as well as
+vertically,and also changesfrom being high- β
+to low-β, from almost neutral to fully ionised,
+and from optically thick to optically thin - at
+all wavelengths - over a matter of a few thou-
+sand kilometers. It is a complicated enough4 Fletcher:Flare Chromospheres
+matter to interpret chromospheric radiation in
+thenominally‘quiet’chromosphere.Inaflare,
+the chromospheric energetics is entirely dom-
+inated by the flare energy release, which ac-
+counts for a power per unit area of around
+1011erg cm−2s−1beingdumpedinthechromo-
+sphere (this number may increase, as we have
+notyetresolvedchromosphericX-rayandopti-
+cal/UV sources).At least duringthe impulsive
+phase,withsuchadominantandrapidheating,
+and interruption to the normal chromospheric
+state of affairs, the normal detailed chromo-
+sphericstructuringmaybecomeirrelevant,and
+we should think instead of a fairly homoge-
+neousplasma,whichrapidlyheats,ionises,ra-
+diatesandexpands-achromospheric‘fireball’
+(terminologydueto H.Hudson).
+The origin,transportandconversionof the
+energy powering the chromospheric flare is
+a primary question in solar physics. It is in-
+disputably the case that the flare energy is
+stored in the stressed magnetic fields of the
+pre-flare corona. In the standard flare model
+electrons (and possibly also ions), acceler-
+ated in the corona and channeled by the mag-
+netic field into the chromosphere are the pri-
+mary agents that transport the energy to the
+chromosphere, where it is radiated. Since the
+1970s, the beam model and the associated
+‘collisional thick target model’ (CTTM) inter-
+pretation of HXR radiation has been widely
+accepted( ˇSvestka 1970; Brown 1971; Hudson
+1972). The bremsstrahlung emission from
+electrons propagating in the dense chromo-
+sphere, though itself energetically insignifi-
+cant, is a major diagnostic of the flare elec-
+tron total energy, spectral and spatial distribu-
+tion, and also more recently their angular dis-
+tribution (Kontar& Brown 2006). Within the
+framework of the CTTM it has been possi-
+ble to deduce the energy and number fluxes
+of electrons arriving from the corona at the
+chromosphere.However,here a possible prob-
+lem emerges, in the deduction that the num-
+ber of electrons required per second to power
+the bremsstrahlungemission inferredfrom the
+CTTM is large compared to the number of
+electronsavailableinareasonablecoronalvol-
+ume and, more seriously, that the return cur-
+rentgeneratedbysuchabeamtravelssofastinthe corona that it is - according to our current
+understanding - unstable (Brown&Melrose
+1977; Fletcher& Warren 2003). The e ffect of
+an instabilityis not clear,butit is boundto ex-
+tractenergyfromthepropagatingelectrondis-
+tribution resulting in heating as it propagates
+(Cromwelletal.1988;Petkakietal.2003)and
+may halt it altogether. If the return current
+drift speed exceeds the sound speed, the re-
+sult is the onset of ion acoustic turbulence,
+as discussed by Hoynget al. (1976), while
+the Buneman instability results if the return
+current drift speed exceeds the electron ther-
+mal speed (Buneman 1959). Under certain
+circumstances, the Buneman instability can,
+on saturating, itself accelerate electrons (e.g
+Drakeet al. 2003) but the initial energy con-
+verted to heating has already sapped the beam
+energy.
+Stabilitydependsonthebeamnumberflux
+rate (number per second per unit area). The
+question of stability against ion acoustic tur-
+bulence was raised with early observations
+(Hoyngetal. 1976) and now with improving
+observational techniques the observed small
+sizesofHXRsourcesmeansthattheBuneman
+instability also starts to be of concern. For
+example, collisional thick-target interpretation
+of the 23 July 2003 flare implies a few (up
+to around 5)×1036electrons s−1at the chro-
+mospheric HXR source location at the peak
+of the flare (Holmanet al. 2003). During flare
+maximum,there are three main HXR chromo-
+spheric sources, each around 5” ×5” giving a
+reasonablevalueforthebeamareainthechro-
+mosphereof4×1017cm−2(thesourcesizemay
+in fact reflect the RHESSI angular resolution,
+andwhite lightfootpointsalso suggestsmaller
+sources than this). Thus an electron flux per
+unit area of around 1019cm−2s−1is required.
+The low energy cuto ffat this time is 20 keV,
+and the power law index around 6, so the bulk
+of the electronshave energiesfrom 20-40keV
+and are traveling at 8 .4−11.9×109cm s−1-
+we takevbeam=1010cm s−1as representative.
+Thus the beam density at the location in the
+chromosphere where the HXRs are generated
+is around nbeam=1010cm−3. Even for the up-
+per chromosphere this is not a trivial density
+perturbation and, ignoring for now the mag-Fletcher:Flare Chromospheres 5
+netic field convergence between corona and
+HXR source location, it implies that the coro-
+nal beam density is a substantial fraction, or
+even in excess of, expected coronal densities.
+Typically ncoris a few×109cm−3early in a
+flare,increasingtoafewtimes1011cm−3,once
+evaporation starts. The return current speed.
+vrc, given by nbeamvbeam=ncorvrcis there-
+fore a substantial fraction of the beam speed
+- vastly in excess of the ion sound speed, and
+greater than the electron thermal speed which
+at 20 MK is 3×109cm s−1. For a given to-
+tal electron rate the density of beamed elec-
+tronsin the coronacan bereducedif onetakes
+into account magnetic field convergence - the
+beam density varies inversely with the mag-
+netic mirror ratio. But then magnetic mirror-
+ing results in a higher overall number of non-
+thermal electrons (i.e. in a non-beam distribu-
+tion), including those trapped in the corona.
+The beam can travel stably if the density of
+the loop in which it moves is large (see e.g.
+vandenOord (1990) for details) but though
+this is possible later in the flare, onceevapora-
+tionhasstarted,thereisnoimagingorspectro-
+scopic evidence before the flare for such loop
+densities (except in coronal thick target loop
+flares (Veronig&Brown 2004), which do not
+show footpoints). Invoking evaporative resup-
+ply also precludes any model involving coro-
+nal acceleration onto loops which are con-
+nected with a reconnection region, since once
+theevaporationstartstheloopshavelongsince
+detached from the accelerator. So, it is clear
+that there are reasons for seeking alternatives
+to thisstandardbeammodel.
+5. Chromosphericaccelerationin
+solarflares
+In view of the difficulties with supplying and
+stabilising a coronal electron beam with the
+flux required to explain collisional thick tar-
+get hard X-rays, new models have recently
+emerged for solar flare energy transport and
+electron acceleration, with the chromosphere
+attheirheart.Fletcher& Hudson(2008)inves-
+tigated a model in which the stored coronal
+magnetic energy is transmitted as an Alfv´ enic
+’pulse’ to the chromosphere, there to be dissi-pated, resulting in chromospheric heating and
+electronacceleration.Brownet al.(2009)have
+suggested that electrons are accelerated, or re-
+accelerated, locally in the chromosphere by
+parallelelectricfieldsinmanysmall-scalecur-
+rent sheets. The two ideas have several as-
+pects in common - the requirement to develop
+small-scale structure in the chromosphere, the
+need to impart magnetic stress to the chromo-
+sphere (i.e. to generate the current sheets in
+the Brown et al. model) and of course the no-
+tion of tapping into the chromospheric elec-
+tron reservoir to supply the electrons that pro-
+duce the X-rays. It should be highlighted here
+that evidence for coronal electron accelera-
+tion most certainly exists, in the form of non-
+thermal coronal HXR sources (Kruckeret al.
+2008),radioTypeIIIbursts(Aschwandenet al.
+1995a, e.g.), and the energy-dependent HXR
+delays (Aschwandenetal. 1995b) interpreted
+asfasterelectronsacceleratedinthecoronaar-
+riving at a chromospherictarget before slower
+ones. However, the demands on the elec-
+tron number implied by all of these signa-
+tures are modest. Although the coronal HXR
+sources may in some cases require that a
+large fraction of coronal electrons in the
+source are accelerated, magnetic trapping and
+a long coronal collision time means that there
+is no resupply problem. Type III bursts are
+non-linear plasma radiation, and are thought
+not to require substantial electron numbers.
+And the HXR bursts from which the energy-
+dependent delays are deduced are typically
+small (rarely more than 20% total HXR flu-
+ence) on top of a more slowly varying back-
+ground (Aschwandenet al. 1996) which has
+an opposite dependence of delay on energy
+(Aschwandenet al. 1997). So, while electron
+acceleration in the corona is almost certainly
+taking place (and indeed fits well within the
+framework of the Fletcher&Hudson (2008)
+model),thefactremainsthatitisdi fficulttosee
+how this can provide also the chromospheric
+electrons.
+Chromospheric electron acceleration, as
+opposed to heating, is feasible only under
+certain circumstances, as the chromosphere
+is dense and highly collisional. Even with-
+out knowing in detail the physics of the pos-6 Fletcher:Flare Chromospheres
+Fig.2.LH Panel : the collisional timescales based on electron-ion scatter ing (solid lines), and the cascade
+timescales (dashed lines) calculated for the VAL-C model at mosphere parameters. RH Panel: The non-
+thermal column available in the accelerating volume as a fun ction of its temperature, at di fferent values of
+the energy at whicha particle isconsidered ’non-thermal’ a nd collisionless.
+sible acceleration mechanisms some general
+statements can be made. A mechanism which
+energises the chromospheric electrons in an
+isotropic manner will first result in heating,
+as the electron-electron collisional timescale
+is short, and the electron component will
+rapidly adopt a Maxwellian distribution. This
+timescaleisgivenby τee=3.4×105T3/2
+e/neΛee
+seconds, where Teis expressed in eV and
+Λeeistheelectron-electronCoulomblogarithm
+(see NRL Plasma Formulary). In a chromo-
+spheric density of 1011cm−3, Coulomb log-
+arithmΛee∼20 with a pre-flare chromo-
+spheric temperature of 104K or 1eV, we have
+τee=1.7×10−7s. As theelectrondistribution
+heats the timescale will increase. At ∼1 MK
+τee=1.7×10−4s.
+Ifthe accelerationmechanismresultsin an
+anisotropicelectrondistribution(e.g.as would
+be the case for field-aligned current sheets or
+from the electric fields generated by kinetic or
+inertial Alfv´ en waves) then the physics will
+be slightly different, as a drifting Maxwellian
+willresult,withallelectronsreceivinganaddi-
+tionallargevelocityboostinthesamedirection
+(and all ions in the opposite direction). Then
+the more relevant timescale for thermalisation
+of the electron population is that on which
+the electrons scatter on ions, which also have
+anoppositelydriftingMaxwelliandistribution.
+Thisscatteringrenderstheelectrondistribution
+oncemoreisotropicandabletorelaxcollision-ally to a hotter Maxwellian. This timescale, in
+the case of the kinetic energy ǫof the acceler-
+ated particles being substantially greater than
+the temperature of the Maxwellian, is τ⊥,ei=
+1.3×105ǫ1/2/niΛeiwhereΛeiis the electron-
+ion Coulomb logarithm. This is also a short
+timescale.
+It is well-known that the chromosphere
+heatssubstantiallyduringtheimpulsivephase,
+for example the footpoint stationary compo-
+nents in highly-ionised lines of iron observed
+byHinode/EIS (Milligan& Dennis 2009) up
+toTe=16 MK (assuming ionisation equilib-
+rium, which is an assumption requiring ex-
+amination in this context) and possibly higher
+(Milligan&Dennis 2009). Also Yohkoh/SXT
+observationsofso-called‘impulsivesoftX-ray
+footpoints’ Mrozek&Tomczak (2004) show
+footpointtemperaturesaveraging8-10MKand
+densities from 4 −10×1010cm−3. As the
+footpoint temperature increases, an increas-
+ing fraction of the electron distribution be-
+comes effectively collisionless. For example,
+at 10 MK, close to 1% of electrons have en-
+ergyǫabove 5 keV (at 15 MK this is 5%).
+If we assume anisotropic acceleration in a
+plasma of density 1011cm−3, the electron-ion,
+andthe electron-electronrelaxationtimescales
+for electrons above this energy is ∼0.02s.
+Thus, an accelerator operating on a shorter
+timescale will accelerate these electrons fur-
+ther.Fletcher:Flare Chromospheres 7
+As an example, consider acceleration by
+a wave-based mechanism, starting with the
+arrival of a macroscopic MHD perturbation
+at the top of the chromosphere, which cas-
+cades to shorter spatial scales where energy
+can be picked up by particles. As in the case
+ofwave-particleaccelerationinthecorona,the
+longest timescale in the system is the turnover
+time of the largest perpendicular wavelength,
+τcasc=(λmax/vA)(B/δB) which we take as
+an upper limit to the acceleration timescale.
+The perpendicular wavelength is important
+since the chromospheric magnetic field is so
+strong that a fully 3D cascade will not hap-
+pen. Figure 1 shows the perpendicular cas-
+cade timescale and collisional timescale (us-
+ing electron-ion collisions) calculated in the
+VAL-C model Vernazzaet al. (1981), plotted
+foraperpendicularwavelengthof10km,vari-
+ousvaluesofthechromosphericmagneticfield
+strength, and a field perturbationof 5%. Note,
+in the VAL-C model the chromospheric elec-
+tron density varies between 2 ×1010cm−3at
+2200kmand6×1010cm−3at1600km.Thefig-
+ure shows that the acceleration timescale cal-
+culatedinthiswayissmallerthanthethermal-
+isation timescale throughout the top few hun-
+dredkmofchromosphere.
+So substantial chromospheric acceleration
+is clearly a possibility. In the ‘local reacceler-
+ation’ scenario of (Brownet al. 2009), the en-
+ergy gain by particles in a chromospheric ac-
+celeration volumeo ffsets the collisional losses
+that they experience there, but the electrons
+could also escape the acceleration volume and
+radiate in the surroundingcollisional thicktar-
+getasnormal,withtheusualattendantrequire-
+ment on replenishment of the accelerator. The
+total requirement for the non-thermal emis-
+sion measure of a flare (Brownetal. 2009)
+isfnhneV∼1046, where fis the fraction
+of all electrons in volume Vthat are accel-
+erated, and nh,neare the hydrogen and elec-
+tron numberdensities Brownetal. (2009). We
+note that the Bremsstrahlung cross-section is
+almostthesameforneutralandforionisedhy-
+drogen, which is why nhis used rather than
+the proton number density. Looking first at
+the case that the acceleration and the radiation
+volume are one and the same, and taking thelower boundary of the acceleration volume as
+thedepthatwhichthethermalisationtimescale
+at 5 keV equals the cascade timescale, results
+in a column-integrated value/integraltext
+nh×nedz∼
+4×1030cm−5.Forf=0.01throughoutmostof
+thisvolume,tomatchtherequirednon-thermal
+emissionmeasureimpliesthata sourceareaof
+2.5×1017cm2isrequired.Varyingthechromo-
+spheric plasma temperatureresults in di fferent
+fractions of electrons in the high-energy tail,
+and different requirements on the flare area.
+The right-hand panel of Figure 5, shows the
+non-thermalcolumn fnhneVdzasafunctionof
+electron temperature, at di fferent values of the
+minimum energy above which the number of
+particles in the Maxwellian tail is evaluated.
+For example, if the electron temperature is
+15MKandassumingamagneticfieldstrength
+of 1 kG, and magnetic perturbation as before,
+a non-thermal column of 2 ×1029cm−3can be
+produced,requiringaflareareaof5 ×1016cm2
+to produce the overall non-thermal emission
+measure.
+Accelerated electrons can also escape the
+acceleration volume and enter denser regions
+lower down, where they undergo collisional
+thick target radiation as usual, without fur-
+ther acceleration. In this case the non-thermal
+emission measure would be NstopfneAwhere
+Nstopis the collisional stopping depth, fne
+the density of fast particles from above and
+Athe flare area. For a 30 keV electron, the
+collisional stopping depth in a fully ionised
+plasma is∼3×1020cm−2, and somewhat
+larger in a partially-ionised plasma because of
+the smaller effective Coulomb logarithm. In
+the VAL-C model, the location of most of the
+electron acceleration is at a density of around
+4×1010cm−3, so that fne=4×108cm−3,
+and the non-thermal emission measure would
+be∼1029Acm−3. So again, an area of
+1017cm2would provide the necessary radia-
+tion. The stability of electron re-supply in the
+formofa returncurrentismorelikely because
+of the low value of the accelerated fraction f.
+However, it should be noted that this scenario
+is not necessarily consistent with the results
+of Kontar& Brown (2006) who find evidence
+for a basically isotropic distribution of radiat-
+ingelectrons.Itremainstobeseenwhetherthe8 Fletcher:Flare Chromospheres
+combinationof electron scattering and mirror-
+ing could generate such a distribution, though
+wenotethatinthelowerchromosphere,where
+the number of free electrons is reduced com-
+pared to the number of ions /hydrogens, that
+electron scattering will happen more rapidly
+than electron energy loss (c.f. Emslie (1978).
+Eq.30),whichisafavourablesituationforpro-
+ducing an isotropic but still energetic distribu-
+tion.
+6. Conclusions
+This brief summary sketches out the observa-
+tional basics of chromospheric flares, though
+ignoring various aspects such as what is know
+from optical and UV spectroscopy (see arti-
+cle by Hudson et al. in this volume), and γ-
+rayradiation.Itoutlineshowthe normalmodel
+of flare energy transport by a beam of elec-
+trons from the corona may run into di fficulty,
+and discusses alternatives to this model which
+invoke chromospheric electron acceleration.
+Such a model appears to be very feasible, as-
+sumingthatthechromosphericplasmacanalso
+be heated as part of the process - wave- or
+current-driven - that transports energy to the
+chromosphere. This article has not touched at
+all on the issue of generating the flare white
+light signature, which is where the chromo-
+spheric flare story started, but since the prob-
+lemwithexistingmodelshasalwaysbeenhow
+to get sufficient energy to a su fficient depth in
+the chromosphere using an electron beam ac-
+celeratedinthecorona,it isclearthatchromo-
+spheric acceleration models o ffer potential so-
+lutionstothisproblem,aswaspointedoutalso
+byBrownet al.(2009).
+Acknowledgements. Theauthorisgratefulfortravel
+support from the NSO, which enabled her to attend
+the verystimulatingworkshop atwhichthis presen-
+tation was made. This work is also supported by
+the EUs SOLAIREResearch and Training Network
+at the University of Glasgow (MTRN-CT-2006-
+035484) and by Rolling Grant ST /F002637/1 from
+theUKsScienceandTechnologyFacilitiesCouncil.References
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+1 
+ Conformational Dynamics of Supramolecul ar Protein Assemblies in the EMDB 
+Do-Nyun Kim, Cong-Tri Nguyen, Mark Bathe* 
+Department of Biological Engineering 
+Massachusetts Institute of Technology 
+ 
+ABSTRACT 
+The Electron Microscopy Data Bank (EMDB) is a rapidly growing repository for the 
+dissemination of structural data from single-part icle reconstructions of supramolecular protein 
+assemblies including motors, chaperones, cytoskel etal assemblies, and viral capsids. While the 
+static structure of these assemb lies provides essential insight into their bi ological function, their 
+conformational dynamics and mechanics provide a dditional important information regarding the 
+mechanism of their biological function. Here, we present an unsupervised computational 
+framework to analyze and store for public access the conformational dynamics of supramolecular 
+protein assemblies deposited in  the EMDB. Conformational dynamics are analyzed using normal 
+mode analysis in the finite element framework,  which is used to compute equilibrium thermal 
+fluctuations, cross-correlations in molecular motions, and strain  energy distributions for 452 of 
+the 681 entries stored in the EMDB at present.  Results for the viral capsid of hepatitis B, 
+ribosome-bound termination factor RF2, and GroEL ar e presented in detail and validated with 
+all-atom based models. The conformational dynami cs of protein assemblies in the EMDB may 
+be useful in the interpretation of their biological function, as well as in the classification and 
+refinement of EM-based structures.  
+ 
+KEY WORDS  
+Electron Microscopy Data Bank, normal mode  analysis, finite element method 
+ 
+*Corresponding author: 
+Mark Bathe 77 Massachusetts Avenue Building NE47, Room 223 Cambridge, MA 02139, U.S.A. Tel. +1–617–324–5685 Fax +1–617–324–7554 mark.bathe@mit.edu
+ 
+ 2 
+ INTRODUCTION  
+Single-particle reconstructi ons of supramolecular protein assemblies deposited in the 
+publically accessible Electron Microscopy Data Bank (EMDB http://www.emdatabank.org/  and 
+http://www.ebi.ac.uk/pdbe/emdb/ ) have been growing rapidly in re cent years, representing a total 
+of approximately 250 distinct structures in 2009 [1, 2]. Recent growth of the EMDB parallels 
+early growth of the Protein Data Bank (PDB), wh ich has developed to include tens of thousands 
+of protein crystal structures since its inception in 1971 [3, 4]. While the static structure of 
+proteins provides invaluable insight into thei r biological function, thei r conformational dynamics 
+often play an additional important role in  understanding their func tion mechanistically. 
+Normal mode analysis (NMA) has proven to be  an effective computational approach to 
+investigate biologically relevant collective motions  about a representative  ground-state structure, 
+or ensemble thereof [5]. The primary advantage of NMA over molecular dynamics is its relative 
+computational efficiency, which is a result of  the harmonic approximation of atomic motions 
+about the ground-state conformation, as well as th e neglect of explicit solvent degrees of 
+freedom. Computational efficien cy is further enhanced in NMA by using coarse-grained 
+modeling approaches that reduce the number of protein degree s of freedom, which has been 
+essential to facilitating the analysis of high molecular weight  supramolecular assemblies. Popular 
+approaches include the Rotational Translationa l Blocks (RTB) procedure [6], which requires 
+atomic coordinates for the underlyi ng protein structure, the Elasti c Network Model [7], and more 
+recently the Finite Element Me thod (FEM) [8]. The FEM provides a natural framework for the 
+computation of conformational dynamics and mech anics of high molecular weight proteins and 
+their assemblies based on EM reconstructions becau se the FE model is defined using its closed 
+molecular surface, which is naturally provided by  single-particle reconstructions. In the FE 3 
+ framework, proteins are modeled as homogeneous isotropic elastic  materials characterized by a 
+mean mass density and elastic stiffness. 
+ While several data banks and servers [9-14] exist to disseminate publically the conformational dynamics of protein structures de posited in the PDB, similar data banks do not 
+exist at present for the EMDB. Such data bank would support both further computational 
+analyses to gain mechanistic insight into the bi ological function of these assemblies, as well as 
+potentially serve as a basis set for classification in single-particle recons truction. Toward this end, 
+here we present the Electron Microscopy Norm al Modes Data Bank (EM-NMDB) to provide 
+conformational dynamics of struct ures present in the EMDB. The FE framework is used to 
+calculate the lowest normal modes of all structures for which a well defined molecular surface can be calculated based either on the EM density  contour level or molecular weight provided in 
+the entry [8]. The EM-NMDB includes normal mode shapes and their root-mean-square 
+amplitudes in thermal equilibrium, their equi librium mechanical strain energy, and the 
+discretized molecular surfaces us ed to perform the analyses. Results for the viral capsid of 
+hepatitis B, ribosome-bound termination factor  RF2, and GroEL are presented in detail, 
+including quantitative comparison of the lowest no rmal modes with those of atomic-level models, 
+equilibrium thermal fluctuations, and correlations  in dynamical motions between distant domains. 
+Effects of EM resolution are additionally examined. Results for the EM-NMDB are available at http://lcbb.mit.edu/~em-nmdb/
+. 
+ 
+METHODS 
+ The EM-NMDB is maintained through an auto mated procedure that consists of several 
+distinct computational steps (Figure 1): (1) retrieval of the EM density map; (2) molecular 4 
+ surface computation and discretiza tion; (3) discretized molecular surface evaluation and repair; 
+(4) finite element mesh generation and normal mode analysis; and (5) results processing. The 
+EM-NMDB server monitors the EMDB regularly to  determine when new structures suitable for 
+conformational dynamics analysis have been deposited. To date, 594 maps out of 681 EMDB 
+entries have been downloaded and analyzed suc cessfully. Of the remaining depositions, 55 maps 
+are on hold by the EMDB, one map file is broke n (unreadable), and 31 maps are tomograms. 
+ For computation of the molecular surface in step 2, the suggested contour level provided 
+by the EMDB is used unless no such  contour level is provided. In this case, the molecular weight 
+is used instead (four entries), where the c ontour level corresponding to the given molecular 
+volume assuming a protein mass density of 1.35 g/cm3 is employed [15]. If neither the contour 
+level nor the molecular weight of the complex is provided, the EMDB entry is classified as “molecular surface indeterminable” and no an alysis is performed (ten entries, Table S1 ). 
+Additionally, in several cases st ructures consisting of disconnect ed multiple bodies are obtained 
+using the suggested contour level, in which cas e the entry is flagged as having “disconnected 
+multiple bodies” and no analysis is performed (87 entries, Table S1). Several examples of such 
+maps are presented in Supplementary Material (F igure S1 and Table S1).  Discretization of the 
+molecular surface is performed using the ma rching cubes algorithm [16] implemented in 
+Chimera [17]. The triangulated surface is subse quently exported in OBJ format, a geometry 
+definition file format originally developed by Wa vefront Technologies, Inc., Santa Barbara, CA.  
+In cases where both the suggested contour le vel and the molecular weight are provided, 
+the molecular volume enclosed by the molecu lar surface discretized by the FE mesh (
+FEV) is 
+compared with the molecular volume pred icted by the molecular weight provided (MWV) 
+assuming a typical protein mass density of 1.35 g/cm3. The relative difference between these 5 
+ values, 100 (%) 1MW
+FEV
+V relV=−Δ× , is computed and denoted using 10 for  0 10%relV<≤Δ , 20 
+for 10% 20%relV<Δ< , 50 for 20% 50%relV<Δ< , 100 for 50% 100%relV<Δ≤  and 100+ for 
+100%relV>Δ .  
+In general, the triangulated molecular surface generated using Chimera is not closed and 
+contains small isolated fragments where an “isola ted fragment” is defined to be a closed surface 
+that consists of fewer than 10% of the number of  triangular faces that forms the largest structure 
+in the map. It additionally often contains in tersecting, overlapping, degenerate, and/or non-
+manifold surface triangles. Because the FEM requir es unique, closed surfaces for the generation 
+of a volumetric mesh, surface mesh repair is requ ired in step 3 prior to performing FE-based 
+NMA (Figure 2). Surface mesh filters available in Meshlab [18] are used for this purpose. Meshlab reads the OBJ file format exported fr om Chimera and exports the filtered molecular 
+surface in STL file format, which is native to the stereolithography CAD software created by 3D 
+Systems, Inc., Rock Hill, SC. The resulting S TL file from Meshlab is imported to the 
+commercially available FEA progr am ADINA (ADINA R&D, Inc., Watertown, MA), which is 
+used to generate the 3D FE volum e mesh consisting of 4-node tetrah edral finite elements [19]. If 
+mesh repair is impossible using Meshlab, then the EMDB entry is classified as “failed in 
+molecular surface repair” and no further analysis is performed (45 entries, 
+Table S1 ). 
+The mesh filtering and repair scheme employs several filters available in Meshlab. The 
+original surface mesh obtained from Chimera is first processed using basic filters with default 
+parameters that remove duplicate faces, unreferen ced vertices, zero-area faces, self-intersecting 
+faces, isolated fragments, and non-manifold faces . Default parameters are additionally used to 
+close holes that are in the original surface mesh  and are created by removing defective faces. In 6 
+ case where closing holes re-int roduce problems with the su rface, the surface mesh is 
+successively refined, smoothened and coarsened wh ere “Midpoint subdivision [20]”, “Laplacian 
+smooth [21]” and “Quadratic edge  collapse decimation [22]” are us ed for refinement, smoothing, 
+and coarsening, respectively. 
+In step 3, FE analysis is used to calcul ate the lowest twenty normal modes based on the 
+three-dimensional volume mesh. While twenty nor mal modes are chosen for the initial database 
+because they generally describe approximately  80% of the total magnitude of equilibrium 
+thermal fluctuations (Figure S3), additional norma l modes may easily be calculated using the FE 
+model as required. Proteins are modeled as homog eneous linear isotropic materials characterized 
+by three independent effective materi al parameters: the Young’s modulus ( E), the mass density 
+(ρ), and Poisson’s ratio ( ν), where proteins are assumed to have mass density 1.35 g/cm3 and 
+Poisson’s ratio 0.3 [23], which is typical of crystalline solids. While the effective Young’s 
+modulus is generally unknown for proteins, it can be obtained by fitting thermal fluctuations of 
+α-carbon atoms in the FE model to those obtai ned using either the all-atom normal mode 
+analysis or the RTB procedure when atomic c oordinates are available, which generally ranges 
+from 2 GPa to 5 GPa [8, 24]. Becau se most structures in the EMDB lack atomic coordinates, 
+normal mode amplitudes and dependent propertie s are computed using a Young’s modulus of 2 
+GPa, representing a lower bound on the protein sti ffness [8]. The precise value of the Young’s 
+modulus affects linearly the ma gnitude of thermal fluctuatio ns, and therefore all results 
+presented may be scaled linearly to calculat e their value correspondi ng to higher or lower 
+Young’s moduli. 
+The subspace iteration procedure [24, 25] is used to solve the eigenvalue problem using 
+2mN starting iteration vectors, where mN denotes the number of eige nmodes to be calculated.  7 
+ The computed number of rigid body modes is comp ared with the number of  isolated molecular 
+volumes calculated in the surface discretizati on step by successively removing the largest 
+components from the molecular surface until no co mponent remains and counting the number of 
+those steps, where each isolated fragment has six rigid body modes. 
+ Each  normal mode amplitude is scaled according to the equipartition theorem [26], 
+which requires that the equilibrium mean elastic energy associated with each normal mode 
+equals 1
+2BkT, where Bk is the Boltzmann constant and T is temperature, taken here to be 300 K. 
+The equilibrium mean elastic ener gy associated with each mode k is given by 
+() ()11 1
+22
+22T
+kk k k kk k B kT αα α λ== = K V xx  where kx denotes the mass normalized 
+eigenvector, kα is its amplitude, K is the stiffness matrix and kλ is the eigenvalue associated 
+with mode k. Scaled normal modes are interpolated to each voxel of the original density map and 
+stored in ASCII format. Eigenv ector magnitudes are stored in the MRC density map format so 
+that both the original density map and the ei genvector magnitude map may be viewed at the 
+same time (for example using Chimera, Figure S2 ). The root mean squared fluctuation (RMSF) 
+amplitude are computed at each FE nodal point using the lowest twenty normal modes (Figure 
+S3), and are interpolated to each voxel to be stored in ASCII format. The mean squared 
+fluctuation for FE node i is given by  22 2 2
+ii ki k k kikmα Δ= Δ=∑∑ rrx  where im denotes the 
+effective mass of node i, which is assumed to be same fo r all FE nodal points. In addition, 
+initial and deformed molecular su rfaces for each mode are stored in STL file format for use in 
+programs such as Maya (Autodesk, Inc., San Rafael, CA). Molecular animations in high 
+(640x320 pixels) and low (320x160 pixe ls) resolutions are provided for four sub-frames in 8 
+ orthogonal views: ISO-3D, XY-plan e, XZ-plane and YZ-plane to illustrate the dynamical 
+motions associated with these modes. 
+ 
+Normal mode validation 
+EM-based normal modes computed using th e FEM are compared with all-atom based 
+results using the molecular surface from the atomic  crystal structure for several validation cases, 
+including the viral capsid of hepatitis B and Gr oEL. Similarity in normal modes is computed 
+using the symmetric mean overlap, ( ),, ,0.5 max maxab ab ba
+ii jin e jin e in e ji eijnOP P
+− ≤≤+ − ≤≤+=+  for mode i where 
+,·ab a b a b
+ij i j i jP=xxx x , ne is the number of neighboring m odes used to account for potential 
+mode-swapping between models with  closely spaced eigenvalues, a and b denote the different 
+models to be compared, and a
+ix represents the eigenvector of the i-th mode  [27]. 
+ 
+Correlations in molecular motions 
+The Linearized Mutual Information (LMI) is us ed to calculate correl ations in molecular 
+motions [28]. The Mutual Information (MI) in atomic displacements is defined as 
+[]12
+1,, ( ) l n()
+(,
+)iN N
+i ip
+ppd
+=ΔΔΔ Δ = Δ Δ
+Δ∫∏" Irrr r r r
+r where iΔr denotes the positional fluctuation 
+vector of atom i, and ( )iipΔr and )(pΔr are their marginal and joint probability distributions, 
+respectively. The LMI can be written in term s of equilibrium conformational properties as 
+() () ( ) ( )1, l nd e t l nd e t l nd e t2lin i j i ij j⎡⎤ΔΔ = + −⎣⎦CCC I rr  where ()( ) () ,,T
+ij ij ij=ΔΔ ΔΔC rr rr  and 
+() iT
+ii=Δ ΔC rr . The generalized corr elation coefficient, 9 
+ (){ }1/2
+,1 e x p 2,j LMI i lin i jrd⎡⎤ ⎡⎤ΔΔ =− − ΔΔ⎣⎦ ⎣⎦I rr rr , is employed, where d is dimensionality ( d 
+=1,2,3 ) [28]. Accordingly, LMI com ponents are computed in the standard way using the normal 
+modes and 
+1T
+ik jk T Bnm
+j
+ki
+k ijkT
+mm λ=ΔΔ = ∑xxrr  where im is the mass of atom i, nm is the number of 
+normal modes and ikx is the displacement vector of atom i due to mode k [29, 30]. The MI 
+metric is employed due to its higher sensitivit y in detecting correlations in molecular motions 
+than the more commonly used Pearson correlati on coefficient, whic h does not account for non-
+colinear correlated motions [28, 30].  To identify molecular domains that are hi ghly correlated in motion, hierarchical 
+clustering is performed using 1
+LMIr−  as a distance metric [31] and defining the distance between 
+clusters as the average distance between all pair s of atoms in any two clusters. Although this 
+process naturally forms clusters of atoms with respect to their magnitude of MI, in most cases 
+clusters with high MI consist of atoms that are spatially near one another because direct 
+bonded/contact interactions introdu ce highly correlated motions, which are trivial and not of 
+interest. Instead, we seek to identify atomic clus ters that are both highl y correlated and spatially 
+distant, which could not be iden tified from molecular structure/geometry alone. To achieve this, 
+N clusters are formed such that  the Kelley–Gardner–Sutcliffe (KGS ) penalty function reaches at 
+its minimum [32], which measures the balance between the average variation within each cluster 
+and across all clusters. Then each cluster is divi ded into its two sub-clusters and checked to 
+determine whether they are composed of “d istant” sub-clusters using the criterion, 
+() ,, ij dist g i g jDC R R×≥+  where ijD is the distance between the m ean positions of (sub-)clusters i 
+and j, ,giR is the radius of gyration of cluster i, and distC is an empirical parameter that is used to 10 
+ define “distant.” Among pairs of distant clusters, the pair with the highest mean correlation is 
+chosen. If no distant sub-clusters exist based on the above criterion, the same procedure is 
+applied to the original clusters to identify di stant clusters that are highly correlated. 1.2distC=  is 
+taken as the default value. This approach is  tested for T4 lysozyme (PDB ID 3LZM) and 
+Adenylate kinase (PDB ID 4AKE, open conformer).  Residue clusters obtained for T4 lysozyme 
+correspond to residues correlated due to hinge-bending [8, 28] a nd those for Adenylate kinase 
+are active residues in the confor mational change from its open (P DB ID 4AKE) to its closed 
+states (PDB ID 1AKE) [24, 33]  (Figure S4 and Figure S5). 
+ 
+EM-NMDB Statistics 
+ The preceding analysis approach is applie d to 497 EMDB entrie s. 452 structures are 
+solved, where 45 structures failed in our mol ecular surface repair pr ocedure (Figure 3 and Table 
+S1). Proteins are classified according to biologic al function by title and sample name key words 
+provided in the EMDB (Figure 4). The EMDB covers  a range of proteins in cluding viruses as the 
+dominant class, and RNA binding proteins and protein kinases as major subclasses of protein 
+complexes. 
+ 
+RESULTS 
+ Conformational dynamics of structures in  the EMDB are publical ly accessible through 
+the EM-NMDB server ( http://lcbb.mit.edu/~em-nmdb ). Results include normal modes and their 
+equilibrium magnitudes, triangul ated molecular surfaces and the deformed surface meshes, the 
+RMSF and the elastic strain energy distributions. Results for the lowest twenty modes for each 
+structure are currently deposited, with the possibility  of increasing the number in the near future. 11 
+ The RMSFs are also computed and stored using the lowest 20 normal modes (Figure 5). These 
+results may be used to provide in sight into biologically relevant motions or to suggest alternate 
+conformations that may be used in multi-referenc e refinement in single- particle reconstruction 
+[34]. The elastic strain energy distributions as  well as displacements for each mode are also 
+presented, potentially identifyi ng domain motions that correspond to high deformation energies 
+in flexible regions linking rigid domains. Result s are presented for hepa titis B virus, ribosome-
+bound termination factor RF2, and GroEL, where the effect of EM re solution on the normal 
+modes and associated conformationa l properties are also evaluated for these structures [35, 36]. 
+ 
+Hepatitis B virus  
+Hepatitis B virus (HBV) is a small virus that  consists of a nucleo capsid core enclosing 
+DNA and an outer lipid envelope, and is physiologically relevant to liver infection in humans 
+[37]. The HBV capsid has icosahedral symmetry consisting of 240 subunits in a T = 4 state, 
+where T denotes the triangulation number [38], with 12 pentamer ic and 30 hexameric capsomers. 
+Cryo-EM-based normal mode results for the HB V nucleocapsid (EMD-1402)  are compared with 
+the reference model obtained from the atomic st ructure, where the molecular surface is given by 
+the global density field that is a superposition of  the densities of each atom constructed using a 
+Gaussian density distribution [39]. The EM-bas ed FE model consists of 18,766 nodes and 72,898 
+tetrahedra while the reference FE model contai ns 36,985 nodes and 173,621 te trahedra (Figure 6). 
+The mean modal overlap (Figure 7)  shows that the lowest normal modes are robust to variations 
+in molecular surface resolution, as found previous ly [40]. RMSFs in molecular motions suggest 
+that hexameric capsomers are more flexible than  their pentameric count erparts (Figure 8). The 
+relative rigidity of pentamers may be attributed to the structural fact that they lack pores. 12 
+  
+Ribosome-bound termination factor RF2  
+Release factors (RFs) are proteins that re cognize a stop codon that  terminates protein 
+synthesis when it arrives at th e ribosomal decoding center [41, 42]. The EM-based FE model for 
+ribosome-bound termination factor RF2 (EMD-1010)  is constructed with 1,509 nodal points and 
+6,155 tetrahedral elements (Figure 9A). A regi on of high RMSFs (Figur e 9B) corresponds to a 
+conserved GGQ amino-acid motif that is thoug ht to be important for peptide release by 
+interacting directly with the peptidyl-transfe rase centre (PTC) [41] . A major conformational 
+shape change is also observed in this region betw een the wild type and a mutant that inhibits 
+peptide release [41]. Seven compact clusters ar e obtained using the KGS criterion (Figure 10). 
+Maximally correlated distant clusters consist of two domains that e xhibit highly correlated 
+collective motions when the clos ed-state crystal structure is fit into the corresponding cryo-EM 
+map in the open state [41].  
+GroEL  
+ GroEL is an extensively studied bacterial ch aperonin that consists of two rings composed 
+of seven identical subunits [43-48]. Three domains  form one subunit: the equatorial domain that 
+connects the two rings and where ATP binds, th e apical domain where ligands bind, and the 
+intermediate domain connecting the equatorial and apical domains. 
+GroEL is analyzed at 6 Ǻ (EMD-1081), 11.5 Ǻ (EMD-1080), and 25 Ǻ (EMD-1095) 
+resolutions and results are comp ared with the reference atomic model based on the crystal 
+structure (PDB ID 1OEL). The molecular surf ace of the reference model is obtained from 
+MSMS using a 1.4 Ǻ radius probe [49]. FE models and their mean mode overlaps (Figure 11, 13 
+ Figure 12, and Table 1) demonstrate a lack of  sensitivity of the lowest normal modes to 
+molecular surface resolution [40]. This high degree of similarity in the lowest normal modes is 
+attributable to the fact that the overall shape de termines the global/collective motions of proteins 
+[8, 40]. However, unlike the low reso lution models, the high resolution, 6 Ǻ EM-based model of 
+GroEL contains bending motions of the two ri ngs mixed with their stretching and shearing 
+motions, which deteriorates the mean overlap sligh tly. This is due to the fact that the suggested 
+contour level provided in the EMDB results in  considerably weaker connection between the 
+rings as compared with the other models, incl uding the reference atomic model (Figure S6). 
+To investigate correlations in molecular motions, the lowest 20 normal modes of the 11.5 
+Ǻ EM-based GroEL model are proj ected onto the alpha carbon at oms of the reference atomic 
+structure and the generalized co rrelation coefficient computed, LMIr. Hierarchical clustering with 
+28 clusters leads to the division of each subunit into tw o clusters, one with the equatorial domain 
+and the other with the apical and intermediate  domains (top-right of Figure 13). Among adjacent 
+clusters (data points in the horiz ontally shaded area in  Figure 13), those in  neighboring subunits 
+within the same ring or between  two rings show the highest co rrelations (Figure 13A-B), while 
+clusters within one subunit are somewhat less co rrelated (Figure 13C). Additionally, next-nearest 
+clusters either in the same ring or in different rings (Figure 13D-E ) are as correlated as clusters 
+within one subunit (data poi nts in the vertically shaded area in Figure 13). These correlated intra- 
+and inter-ring interactions are important to the allosteric mechanism of GroEL, corresponding to 
+positive and negative cooperativity, respectively, in the binding and hydrolysis of ATP [43, 45, 
+48]. 
+ 14 
+ CONCLUSIONS  
+ An unsupervised computational framework based on the FEM is presented to compute 
+the conformational dynamics of supramolecula r assemblies of unknown atomic structures 
+deposited in the EMDB. The FE-based approach  is a natural choice fo r computation of NMs 
+from molecular surface representati ons such as obtained from EM density maps due to the fact 
+that it requires a closed surface to form th e dynamical equations of motion governing the 
+protein’s conformational dynamics. The procedure deve loped here is adequate for the majority of 
+EMDB structures that have a dens ity level that leads to a well de fined molecular surface. Results 
+are publically accessible via a data  bank that is configured to update automatically when new 
+structures are deposited in the EMDB, and ar e expected to be useful in understanding 
+biologically relevant functional mo tions of EM-based structures. 
+ 
+ACKNOWLEDGEMENTS 
+Useful discussions with Gaël McGill, Christoph  Best, Janet Iwasa, Wah Chiu, Joachim Frank, 
+Andres Leschziner, Grant Jensen, Martin Karplu s, and Qiang Cui are gratefully acknowledged. 
+Funding for this work was provided by MIT Faculty Start-up Funds and the Samuel A. Goldblith 
+Career Development Professorship awarded to MB.  15 
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+Explored by Network Models: Ap plication to Chaperonin GroEL.  Plos Computational 
+Biology, 2009. 5(4). 
+45. Ma, J.P. and M. Karplus, The allosteric mechanism of th e chaperonin GroEL: A dynamic 
+analysis.  Proceedings of the National Academy of  Sciences of the United States of 
+America, 1998. 95(15): p. 8502-8507. 
+46. Hyeon, C., G.H. Lorimer, and D. Thirumalai, Dynamics of allosteric transitions in 
+GroEL.  Proceedings of the National Academy of  Sciences of the United States of 
+America, 2006. 103(50): p. 18939-18944. 
+47. Tehver, R., J. Chen, and D. Thirumalai, Allostery Wiring Diagrams in the Transitions 
+that Drive the GroEL Reaction Cycle.  Journal Of Molecular Biology, 2009. 387(2): p. 
+390-406. 
+48. Cui, Q. and M. Karplus, Allostery and coopera tivity revisited.  Protein Science, 2008. 
+17(8): p. 1295-1307. 
+49. Sanner, M.F., A.J. Olson, and J.C. Spehner, Reduced surface: An efficient way to 
+compute molecular surfaces.  Biopolymers, 1996. 38(3): p. 305-320. 
+50. Brooks, B.R., et al., CHARMM—A program for macromol ecular energy, minimization, 
+and dynamics calculations.  Journal of Computational Chemistry, 1983. 4(2): p. 187-217. 
+  18 
+ TABLES 
+ 
+Table 1. GroEL FE models 
+FE models PDB-1OEL EMD-1081 EMD-1080 EMD-1095 
+Number of nodes 50,738 40,182 72,919 72,439 
+Number of elements 216,950 155,039 351,941 356,521 
+Resolution (Å) 2.8 6 11.5 25 
+Volume (Å3) 902,388 937,623 942,211 989,727 19 
+ FIGURES 
+ 
+Figure 1. Automated procedure for computati on of conformational dynamics using the FE 
+framework.  
+Figure 2. Discretized molecular surface of ki nesin dimers bound to a microtubule (EMD-1030) 
+(A) generated by Chimera at the suggested contour level of 67.1 prior to surface mesh repair and 
+(B) after surface mesh repair using Meshlab. (R emoved isolated fragments are highlighted in 
+red.) 
+Figure 3. EM-NMDB statistics for a ll EMDB structures (681 entries).  
+Figure 4. Classification of the EM -NMDB according to biological f unction for solved structures 
+(452 entries)  
+Figure 5. Representative EM-N MDB structures with molecula r surfaces, lowest normal mode 
+with relative magnitude  of the normal mode and correspondi ng relative strain energy, and the 
+root mean squared fluctuations (red denotes high values while bl ue denotes low values). (EMD-
+1231) connector of bacteriophage T7; (EMD-140 2) hepatitis B viral capsid; (EMD-1080) 
+GroEL; (EMD-1554) 70S E. coli  ribosome in the pretranslo cation state; and (EMD-1030) 
+kinesin dimers bound to a microtubule.  
+Figure 6. FE models of the hepatitis B viral ca psid. Molecular surfaces are obtained (A) from the 
+atomic structure assuming a Gaussian electron de nsity distribution for each atom and (B) from 
+the cryo-EM map (EMD-1402). Model (A) is obt ained from M. M. Gi bbons and W. S. Klug 
+(Mechanical and Aerospace Engineering, UCLA).  
+Figure 7. The three lowest normal modes of the hepatitis B viral capsid with relative magnitude of the normal modes displayed (first, second, a nd third columns), red denotes large relative 
+displacement while blue denotes small relative displacement) and mean modal overlap between 
+the two FE models (last row). Six neighboring m odes are used to compute the mean overlap for 
+each mode.
+ 
+Figure 8. Relative total RMSF of the capsid of hepatitis B (EMD-1402) with (A) hexamer-
+centered view and (B) pentamer-centered view . Red denotes large relative RMSF while blue 
+denotes low relative RMSF.  
+Figure 9. Ribosome-bound termination factor RF 2 (EMD-1010). (A) FE mesh and (B) relative 
+RMSF using 20 normal modes (red denotes high  and blue denotes low relative RMSF).  
+Figure 10. (A) Clusters of correlated mo lecular motions found for the ribosome-bound 
+termination factor RF2 (EMD-1010) using the KGS criterion (7 clus ters). (B) Maximally 
+correlated distant clusters from (A) colored in re d with the generalized correlation coefficient of 20 
+ 0.6810. The generalized correlation coefficients fo r other distant clusters are 0.45, 0.58, 0.66, 
+0.62, 0.57, 0.62, 0.61, 0.57, and 0.58 corresponding to  clusters (1, 4), (1, 5) , (1, 7), (2, 4), (2, 7), 
+(3, 4), (3, 7), (4, 6), and (4, 7), respectively.  
+Figure 11. FE models of GroEL obtained from (A) the atomic crystal structure PDB ID 1OEL 
+and three EM resolutions: (B) EMD-1081 at 6 Ǻ resolution; (C) EMD-1080  at 11.5 Ǻ 
+resolution; and (D) EMD-1095 at 25 Ǻ resolution. The 6 Ǻ EM-based model has weak 
+connectivity between the rings of GroEL that is  not present in the reference atomic model 
+(Figure S6).  
+Figure 12. The lowest three normal modes of Gro EL with relative magnitude of the normal mode 
+displayed (red denotes large relative displacement while blue denotes small relative 
+displacement) and mean modal overlap betw een the EMDB-based FE models (EMD-1081, 
+EMD-1080 and EMD-1095) and the PDB-based FE model (PDB-1OEL) of GroEL structures. 
+Six neighboring modes are used to com pute the mean overlap for each mode.  
+Figure 13. Correlations between molecular domains  in GroEL (EMD-1080, 28 clusters in total). 
+Each subunit is composed of two clusters (one wi th the equatorial domain and the other with the 
+apical and the intermediate domains as shown in the top-right figure) . Cluster correlation is 
+defined as the mean correlation between the residues in the clusters and cluster separation is 
+defined as ()ij g,i g, jDR + R  where ijD denotes the distance between the mean position of cluster 
+i and j, and g,iR represents the radius of gyration of cluster i. 
+ 21 
+ 
+ 
+ 
+Figure 1. Automated procedure for computation of conformational dynamics using the FE 
+framework. 22 
+ 
+ 
+Figure 2. Discretized molecular surface of kinesin dimers bound to a microtubule (EMD-
+1030) (A) generated by Chimera at the sugges ted contour level of 67.1 prior to surface 
+mesh repair and (B) after su rface mesh repair using Meshlab. (Removed isolated 
+fragments are highlighted in red.) 23 
+ 
+ 
+Figure 3.  EM-NMDB statistics for all EM DB structures (681 entries). 24 
+ 
+ 
+Figure 4. Classification of the EM-NMDB according to biological function for solved 
+structures (452 entries) 25 
+ 
+ 
+Figure 5. Representative EM-NMDB structures  with molecular surfaces, lowest normal 
+mode with relative magnitude of the normal mode and corresponding relative strain energy, 
+and the root mean squared fluc tuations (red denotes high values while blue denotes low 
+values). (EMD-1231) connector of  bacteriophage T7; (EMD-1402)  hepatitis B viral capsid; 
+(EMD-1080) GroEL; (EMD-1554) 70S E. coli  ribosome in the pretra nslocation state; and 
+(EMD-1030) kinesin dimers  bound to a microtubule. 26 
+ 
+ 
+Figure 6. FE models of the hepatitis B viral capsid. Molecular surfa ces are obtained (A) 
+from the atomic structure assu ming a Gaussian electron dens ity distribution for each atom 
+and (B) from the cryo-EM map (EMD-1402). Mode l (A) is obtained from M. M. Gibbons 
+and W. S. Klug (Mechanical a nd Aerospace Engineering, UCLA). 27 
+ 
+ 
+Figure 7. The three lowest normal modes of the hepatitis B viral capsid with relative 
+magnitude of the normal mode s displayed (first, second, a nd third columns), red denotes 
+large relative displacement while blue deno tes small relative displacement) and mean 
+modal overlap between the two FE models (las t row). Six neighboring modes are used to 
+compute the mean overlap for each mode. 28 
+ 
+ 
+Figure 8. Relative total RMSF of the capsid  of hepatitis B (EMD-1402) with (A) hexamer-
+centered view and (B) pentamer-centered vi ew. Red denotes large relative RMSF while 
+blue denotes low relative RMSF. 29 
+ 
+ 
+Figure 9. Ribosome-bound termination fact or RF2 (EMD-1010). (A ) FE mesh and (B) 
+relative RMSF using 20 normal modes (red de notes high and blue denotes low relative 
+RMSF). 30 
+ 
+ 
+Figure 10. (A) Clusters of correlated mole cular motions found for the ribosome-bound 
+termination factor RF2 (EMD-1010) using the KGS criterion (7 clusters). (B) Maximally 
+correlated distant clusters from (A) colore d in red with the ge neralized correlation 
+coefficient of 0.6810. The genera lized correlation coeffi cients for other distant clusters are 
+0.45, 0.58, 0.66, 0.62, 0.57, 0.62, 0.61, 0.57, a nd 0.58 corresponding to clusters (1, 4), (1, 5), 
+(1, 7), (2, 4), (2, 7), (3, 4), (3, 7), (4, 6), and (4, 7), respectively. 31 
+ 
+ 
+Figure 11. FE models of Gro EL obtained from (A) the atom ic crystal structure PDB ID 
+1OEL and three EM resolutions: (B) EMD-1081 at 6 Ǻ resolution; (C) EMD-1080  at 11.5 
+Ǻ resolution; and (D) EMD-1095 at 25 Ǻ resolution. The 6 Ǻ EM-based model has weak 
+connectivity between the rings of GroEL that is  not present in the reference atomic model 
+(Figure S6). 32 
+ 
+ 
+Figure 12. The lowest three normal modes of Gr oEL with relative magnitude of the normal 
+mode displayed (red denotes large relative di splacement while blue denotes small relative 
+displacement) and mean modal overlap between  the EMDB-based FE models (EMD-1081, 
+EMD-1080 and EMD-1095) and the PDB-base d FE model (PDB-1OEL) of GroEL 
+structures. Six neighboring modes are used to  compute the mean overlap for each mode. 33 
+ 
+ 
+Figure 13. Correlations between molecular do mains in GroEL (EMD-1080, 28 clusters in 
+total). Each subunit is composed of two clust ers (one with the equatorial domain and the 
+other with the apical and the intermediate domains as shown in the top-right figure). 
+Cluster correlation is defined as the mean co rrelation between the residues in the clusters 
+and cluster separation is defined as () ij g,i g, jDR + R  where ijD denotes the distance 
+between the mean position of cluster i and j, and g,iR represents the radius of gyration of 
+cluster i. 
+  34 
+  SUPPLEMENTARY MATERIAL 
+ 
+Table S1. EMDB entries excluded from NMA.  
+ Figure S1. Examples of EMDB entries classifi ed as disconnected multiple bodies and excluded 
+from NMA (colors highlight all or parts of disconnected fragments). (A) EMD-1428 
+(Microtubule) (B) EMD-5104 (ATP ases)  (C) EMD-1111 (Virus)  (D) EMD-1458 (GroEL)  (E) 
+EMD-1073 (Ribosome)  (F)  EMD-1088 (Acros omal actin bundle)  (G) EMD-1061 (Inositol 
+1,4,5-triphosphate receptor)  (H) EMD-1335 (T ail of bacteriophage K1-5)  (I) EMD-1591 
+(Anaphase promoting complex)
+ 
+Figure S2. Sample EM-NMDB structures. Mobile  regions of the lowest normal mode are 
+highlighted in red corresponding to top 20% of  mode magnitude. Figur es are created using 
+Chimera by importing the original density map (t ransparent) and the lowest mode magnitude 
+map (red) together. (A) kinesin dimers bound to a microtubule (EMD-1030); (B) a GroEL 
+(EMD-1080); (C) bacteriophage P22 tail machine (EMD-1119); (D) connect or of bacteriophage 
+T7 (EMD-1231); (E) human RNA polymerase II  (EMD-1283); (F) nitrila se from Rhodococcus 
+rhodochrous J1 (EMD-1313); (G ) a chaperonin, cpn60 (EMD- 1397); (H) parvovirus capsid 
+(EMD-5105)  
+Figure S3. The root mean square fluctuations of  T4 lysozyme (PDB ID 3LZM) computed using 
+20, 50, 100 and 200 modes. The inserted graph (top-right) shows the va riation of the mean 
+relative RMSF as a function of the number of modes used for the RMSF calculations. The RMSF 
+with 200 modes is chosen as the reference. With 20 modes, 77% of the reference RMSF is 
+obtained on average. Normal modes are obtai ned from all-atom nor mal mode analysis 
+implemented in CHARMM [50].  
+Figure S4. Clusters of correlated residues of T4 lysozyme (PDB ID 3LZM).  (A) All clusters resulted from KGS criterion (14 clusters show n in different colors) and (B) the maximally 
+correlated distant clusters, resi dues 33-53 and residues 81-90 colo red in red, corresponding to 
+residues correlated due to hinge-bending.
+ 
+Figure S5. Clusters of correlated residues of Adenyl ate kinase (PDB ID 4AKE).  (A) All clusters 
+resulted from KGS criterion (8 clusters show n in different colors) and (B) the maximally 
+correlated distant clusters, resi dues 30-73 and residues 113-175 co lored in red, corresponding to 
+residues active in the co nformational change from its open (PDB  ID 4AKE) to its closed states 
+(PDB ID 1AKE).  35 
+ Figure S6. Cross-sections of where two rings of GroEL are connected. The FE models are 
+obtained from (A) the atomic crystal structur e PDB ID 1OEL and three EM resolutions: (B) 
+EMD-1081 at 6 Ǻ resolution; (C) EMD-1080  at 11.5 Ǻ resolution; and (D) EMD-1095 at 25 Ǻ 
+resolution. 6 Ǻ EM-based model (B) presents weaker inter-ring connection compared to other 
+models. 36 
+ Table S1. EMDB entries excluded from NMA. 
+ Number 
+of entries EMDB ID 
+Molecular surface 
+indeterminable 1 10 1087, 1151, 1233, 1259, 1533, 1596, 1599, 1601, 1609, 
+5037 
+Disconnected 
+multiple bodies 2 87 1015, 1018, 1021, 1025, 1036, 1042, 1052, 1061, 1073, 
+1079, 1088, 1101, 1106, 1111, 1112, 1118, 1123, 1134, 1137, 1145, 1165, 1176, 1177, 1203, 1207, 1221, 1226, 1229, 1234, 1236, 1237, 1238, 1244, 1254, 1256, 1267, 1268, 1299, 1314, 1320, 1331, 1333, 1334, 1335, 1340, 1341, 1343, 1353, 1374, 1375, 1377, 1379, 1383, 1385, 1387, 1389, 1401, 1415, 1425, 1427, 1428, 1431, 1437, 1442, 1443, 1447, 1458, 1462, 1469, 1471, 1529, 1531, 1532, 1557, 1579, 1580, 1582, 1591, 1617, 5003, 5010, 5012, 5021, 5022, 5023, 5100, 5104 
+Failed in molecular 
+surface repair 3 45 1016, 1026, 1060, 1075, 1083, 1113, 1115, 1130, 1133, 
+1152, 1164, 1179, 1181, 1201, 1206, 1235, 1239, 1264, 
+1265, 1285, 1309, 1316, 1321, 1354, 1371, 1381, 1392, 
+1412, 1420, 1441, 1444, 1461, 1480, 1489, 1490, 1503, 1509, 1511, 1544, 1549, 1552, 1581, 1593, 5001, 5038 
+1 Molecular surface indeterminab le: Neither the contour level nor the molecular weight is 
+provided in the EMDB which is necessary for the molecular surface determination. 
+2 Disconnected multiple bodies: Structures cons ist of disconnected multiple bodies obtained 
+using the suggested cont our level (Figure S1). 
+3 Failed in molecular surface repair : The automated, unsupervised pr ocedure fails in repairing the 
+molecular surfaces. The molecular surfaces may be repaired by applying filters manually or in a 
+supervised way, but it is not tried in this study. 37 
+ 
+ 
+Figure S1. Examples of EMDB entries classifi ed as disconnected multiple bodies and 
+excluded from NMA (colors highlight all or parts of disconnected fragments). (A) EMD-
+1428 (Microtubule) (B) EMD-5104 (ATPases )  (C) EMD-1111 (Virus)  (D) EMD-1458 
+(GroEL)  (E) EMD-1073 (Ribosome)  (F)  EMD-1088 (Acrosomal actin bundle)  (G) EMD-
+1061 (Inositol 1,4,5-triphosphate receptor)  (H ) EMD-1335 (Tail of bacteriophage K1-5)  (I) 
+EMD-1591 (Anaphase promoting complex) 38 
+ 
+ 
+Figure S2. Sample EM-NMDB stru ctures. Mobile regions of th e lowest normal mode are 
+highlighted in red corresponding to top 20% of mode magnitude. Figures are created using 
+Chimera by importing the original densit y map (transparent) and the lowest mode 
+magnitude map (red) together. (A) kinesi n dimers bound to a microtubule (EMD-1030); 
+(B) a GroEL (EMD-1080); (C) ba cteriophage P22 tail machin e (EMD-1119); (D) connector 
+of bacteriophage T7 (EMD -1231); (E) human RNA poly merase II (EMD-1283); (F) 
+nitrilase from Rhodococcus rhodochrous J1 (EMD-1313); (G) a chaperonin, cpn60 (EMD-
+1397); (H) parvovirus capsid (EMD-5105) 39 
+ 
+ 
+Figure S3. The root mean square fluctuations of T4 lysozyme (PDB ID 3LZM) computed 
+using 20, 50, 100 and 200 modes. The inserted graph (top-right) shows the variation of the 
+mean relative RMSF as a func tion of the number of modes us ed for the RMSF calculations. 
+The RMSF with 200 modes is chosen as the ref erence. With 20 modes, 77% of the reference 
+RMSF is obtained on average. Normal modes are obtained from a ll-atom normal mode 
+analysis implemente d in CHARMM [50]. 40 
+ 
+ 
+Figure S4. Clusters of correlated residues of T4 lysozyme (PDB ID 3LZM).  (A) All clusters 
+resulted from KGS criterion (14 clusters show n in different colors) and (B) the maximally 
+correlated distant clusters, res idues 33-53 and residues 81-90 colored in red, corresponding 
+to residues correlated due to hinge-bending. 41 
+ 
+ 
+Figure S5. Clusters of correlated residues of Adenylate kinase (PDB ID 4AKE).  (A) All 
+clusters resulted from KGS criterion (8 clusters shown in different colors) and (B) the maximally correlated distant clusters, resid ues 30-73 and residues 113-175 colored in red, 
+corresponding to residues active in the confor mational change from its open (PDB ID 
+4AKE) to its closed st ates (PDB ID 1AKE). 42 
+ 
+ 
+Figure S6. Cross-sections of where two rings of GroEL are connected. The FE models are 
+obtained from (A) th e atomic crystal structure PDB ID 1OEL and three EM resolutions: 
+(B) EMD-1081 at 6 Ǻ resolution; (C) EMD-1080  at 11.5 Ǻ resolution; and (D) EMD-1095 
+at 25 Ǻ resolution. 6 Ǻ EM-based model (B) presents weaker inter-ring connection 
+compared to other models. 
\ No newline at end of file
diff --git a/1001.0741.txt b/1001.0741.txt
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--- /dev/null
+++ b/1001.0741.txt
@@ -0,0 +1,650 @@
+arXiv:1001.0741v1  [cond-mat.stat-mech]  5 Jan 2010Slow Cooling of an Ising Ferromagnet
+P. L. Krapivsky1
+1Department of Physics, Boston University, Boston, MA 02215 , USA
+Abstract. A ferromagnetic Ising chain which is endowed with a single-s pin-flip Glauber dynamics
+is investigated. For an arbitrary annealing protocol, we de rive an exact integral equation for the
+domain wall density. This integral equation admits an asymp totic solution in the limit of extremely
+slow cooling. For instance, we extract an asymptotic of the d ensity of domain walls at the end of
+the cooling procedure when the temperature vanishes. Slow a nnealing is usually studied using a
+Kibble-Zurek argument; in our setting, this argument leads to approximate predictions which are
+in good agreement with exact asymptotics.
+Keywords: phase transitions, slow annealing, Glauber dynamics
+I. INTRODUCTION
+The quenching procedure that is used in numerous
+studies of coarsening is usually based on the following
+protocol: (i) Start at a high initial temperature Ti> Tc,
+where spins are disordered; (ii) Instantaneously cool the
+system to a final temperature that is lower than the crit-
+ical,Tf< Tc. The most popular choice of the initial
+temperature is Ti=∞. Cooling to the critical tem-
+perature, Tf=Tc, is also often studied, especially for
+one-dimensional systems where Tc= 0 or in quantum
+quenches. After such instantaneous quench, ordered re-
+gions begin to grow and it takes a long time (an infinite
+time in the thermodynamic limit) to reach the equilib-
+rium. Understanding of the dynamics of phase-ordering
+kinetics, particularly the scaling regime that develops
+long after the quench, is the main goal [1].
+Although the very fast cooling is natural in a num-
+ber of settings, there are numerous applications (ranging
+from the formation of glasses to the expansion of the
+Universe) where slow cooling is appropriate. A popu-
+lar gradual cooling scheme posits that the temperature
+vanishes linearly in time,
+T(t) =T0·/parenleftbigg
+1−t
+τ/parenrightbigg
+(1)
+so as time varies in the interval (0 ,τ), the temperature
+falls from Ti=T0toTf= 0. To characterize the cooling
+scheme (1) or other gradual cooling procedure, it is nat-
+ural to focus on the properties of the state at the end of
+the cooling procedure, particularly to find the difference
+between this finalstate [forthe coolingscheme(1), it cor-
+responds to t=τ] and a ground state. If the cooling rate
+1/τis large, the problem is not interesting as during time
+interval (0 ,τ) the system has hardly evolved. Interesting
+behaviors arise if cooling is very slow, τ≫1, so that the
+system has evolved over long time interval (0 ,τ) and its
+state is close to a ground state. Then the problem is to
+describe the deviation from the ground state.
+The limit of slow cooling has been investigated in nu-
+merous papers. Earlier work has been mostly driven by
+applicationsto cosmologyand proposalstostudy analogs
+of the cosmological phase transitions in the laboratory[2, 3]. Recently, similar ideas and methods have been
+applied to the dynamics of slow quantum annealing, see
+e.g. [4–7]. A surprisingly little work has been done in
+the more standard context of thermal phase transitions.
+Among a few exceptions is Ref. [8] that studies the Ising
+chainsupplemented with Glauber’sspin-flip dynamics[9]
+and Ref. [10] which investigates high-dimensional sys-
+tems where Tc>0, so that the critical temperature is
+passed during the quenching procedure.
+Overall, studies of slow quenching tend to rely on un-
+controlledapproximations. Thisis notsosurprisingsince
+little can be done analytically in the high-dimensional
+setting. Some of the uncontrolled approximations, e.g.
+the so-called Kibble-Zurek mechanism [2, 3], rely on
+sound physical ideas, yet we still want to better under-
+stand their possible limitations. Numerical (see e.g. [11–
+13]) and laboratory(see e.g. [14–16]) experiments help in
+this regard, but large size simulations can be challenging
+(particularly for systems exhibiting quantum phase tran-
+sitions), while real system always have features beyond
+those that are captured by theoretical models.
+Unfortunatelyexactresultshaven’tbeenobtainedeven
+for the Ising-Glauber spin chain [8]. Here we show that
+this model is tractable for essentially arbitrary cooling
+procedure. More precisely, we establish an integral equa-
+tion for the density of domain walls. This equation is
+closedand linear. Although it is generally unsolvable, in
+the interesting slow cooling limit it is possible to deduce
+analytically an exact asymptotic for the final density of
+domain walls.
+The following section II we write-down the governing
+equations for the two-spin correlation function, outline
+the problems in solving these equations, and present our
+chief results. We then provide a complete derivation for
+a special cooling scheme (section III). In section IV we
+analyze an arbitrary cooling scheme and derive various
+asymptotic results announced in section II. Section V
+is devoted to the comparison of exact results with ap-
+proximate predictions following from the Kibble-Zurek
+argument. Finally in section VI we give a summary
+and briefly discuss how the cooling protocol can affect
+the probability for high-dimensional Ising ferromagnets
+to (eventually) fall into a ground state.2
+II. COOLING OF A FERROMAGNETIC ISING
+CHAIN: MAIN RESULTS
+In one dimension, Tc= 0 and hence to assure coars-
+ening we must cool the Ising chain to zero temperature:
+Tf= 0. The zero-temperature evolution will continue
+as long as there are domain walls, so if we let the Ising
+chain to evolve during the time interval τ < t < ∞, it
+will eventually reach a ground state. Therefore the den-
+sity of domain walls at the end of the cooling procedure,
+t=τ, gives the quantitative measure of the distance
+between the state at the end of cooling and the ground
+state. The deriving of this final density is ourmajor goal.
+For concreteness, we always assume that the initial
+condition is spatially homogeneous. Then the chain will
+remain (on average) spatially homogeneous. Therefore
+the two-spin correlation function depends only on the
+separation between the spins, Gk(t) =/angbracketleftσi(t)σi+k(t)/angbracketright.
+The two-spin correlation function satisfies [9, 17]
+dGk
+dt=−2Gk+γ(Gk−1+Gk+1) (2)
+fork≥1. Relation
+G0(t) =/angbracketleftσ2
+i/angbracketright ≡1 (3)
+plays the role of a boundary condition. Note also that
+the density ρ(t) of domain walls can be expressed via
+G1(t). Indeed, the bond ( i,i+1) hosts a domain wall if
+1
+2(1−sisi+1) = 1 and therefore
+ρ=/angbracketleftbigg1
+2(1−σiσi+1)/angbracketrightbigg
+=1−G1
+2(4)
+Theparameter γin (2) isgivenby γ= tanh(2 /T). [We
+setthe ferromagneticcouplingstrength Jtounity; other-
+wise we would have had γ= tanh(2 J/T).] Even if we are
+mostly interested by the density of domain walls, there
+is no closed equation for this quantity. Mathematically,
+we need to solve an infinite set of coupled linear ordi-
+nary differential equations (2). Glauber solved Eqs. (2)
+in the situation when γis a constant, see [9, 17]. For the
+gradual cooling scheme, however, we must deal with the
+time-dependent parameter γ(t). Fortunately, a lot can
+be done analytically.
+The cooling scheme (1) is characterized by a simple
+linear time evolution of temperature, yet the governing
+equations (2) depend on time via parameter γwhich has
+a more complicated time dependence:
+γ= tanh/bracketleftBigg
+2
+T0/parenleftbigg
+1−t
+τ/parenrightbigg−1/bracketrightBigg
+(5)
+As a warm-up, in the next section we shall investigate an
+alternative cooling scheme with a linear behavior of the
+parameter γthat appears in the governing equations:
+γ(t) =t/τ (6)For this cooling scheme the temperature decays from in-
+finity (at t= 0) to zero (at t=τ) according to
+T(t) =2
+tanh−1(t/τ)(7)
+The cooling schemes (1) and (7) are morally equivalent.
+The formulas for the latter scheme are a bit simpler and
+for that procedure we can start with Ti=∞when initial
+conditions are particularly simple.
+Thus for the cooling scheme (7) we must solve
+dGk
+dt=−2Gk+t
+τ(Gk−1+Gk+1), k≥1 (8)
+on the time interval 0 ≤t≤τ. The initial conditions are
+Gk(t= 0) =δk,0 (9)
+and the boundary condition is (3). The analysis of
+Eqs. (8) shows that in the case of slow cooling, τ≫1,
+the final density of domain walls is
+ρ(τ)≃Cτ−1/4, C=√π
+Γ/parenleftbig1
+4/parenrightbig (10)
+Remarkably, an Ising chain supplemented with essen-
+tiallyarbitrary cooling scheme is tractable. For concrete-
+ness,weshallassumethatthetemperaturemonotonously
+decreases from T0toT(τ) = 0; then γ=γ(t) is a
+monotonously increasing function that varies from γ0to
+γ(τ) = 1. Actually we do not need to postulate that
+γ(t) is monotonous. We consider the situation when
+γ(t) varies from γ0= tanh(2 /T0) toγ(τ) = 1; since
+γ= tanh(2 /T), it must obey 0 ≤γ≤1, no other as-
+sumptions are required. The monotonicity of γ(t) re-
+flects that in typical cooling schemes the temperature is
+monotonously decreased, and having made such an as-
+sumption we are assured that γlies within the bounds
+γ0≤γ≤1.
+It turns out that in the limit of slow cooling, τ≫1,
+only the behavior of γ(t) near the end of the cooling
+procedure plays a role. For instance, in the case of the
+algebraic behavior,
+1−γ(t)≃A/parenleftbigg
+1−t
+τ/parenrightbiggα
+whent→τ,(11)
+the final density of domain walls depends only on two
+parameters Aandα, namely
+ρ(τ)≃C(2τ)−α/2(1+α)(12)
+with amplitude
+C=/radicalbiggπ
+8/bracketleftbigg
+Γ/parenleftbigg3+2α
+2+2α/parenrightbigg/bracketrightbigg−1/bracketleftbiggA
+1+α/bracketrightbigg1
+2(1+α)
+(13)
+Analgebraicapproach(11)of γ(t)tounityisquitenat-
+ural. However, for the cooling scheme (1) the behavior of3
+γ(t) near the end of the cooling procedure is exponential
+rather than algebraic
+1−γ(t)≃2exp/braceleftbigg
+−4/T0
+1−t/τ/bracerightbigg
+whent→τ
+This suggests to analyze a family of cooling laws with an
+asymptotic behavior of γ(t) of the form
+1−γ(t)≃Bexp/braceleftbigg
+−b
+(1−t/τ)β/bracerightbigg
+whent→τ(14)
+with arbitrary positive parameters B,b,β. In this sit-
+uation, the asymptotic behavior of the final density of
+domain walls is
+ρ(τ)≃1
+4/radicalbiggπ
+τ/parenleftbigglnτ
+b/parenrightbigg1
+2β
+(15)
+This is a remarkably universal asymptotic: The leading
+τ−1/2algebraicfactordoesnotdepend ontheparameters
+B,b,βand the logarithmicprefactordepends only on one
+parameter β.
+We now turn to derivations of the announced results.
+III. SPECIAL COOLING SCHEME
+In this section we consider the special cooling proce-
+dure (6). First, we employ an exact analysis and then
+turn to the asymptotic behavior of the final density of
+domain walls.
+A. Exact Analysis
+To handle an infinite set of ordinary differential equa-
+tions (8) we recast it into a single differential equation
+using a generating function technique. Multiplying (8)
+byzkand summing over all k≥1 we find that the gen-
+erating function
+G(z,t) =/summationdisplay
+k≥1Gk(t)zk(16)
+satisfies
+∂G
+∂t=/bracketleftbigg
+−2+t
+τ/parenleftbig
+z+z−1/parenrightbig/bracketrightbigg
+G+t
+τ[z−G1(t)] (17)
+Solving (17) subject to G(z,t= 0) = 0, we get
+G=/integraldisplayt
+0dt′t′
+τ[z−G1(t′)]exp/bracketleftbigg
+2(t′−t)+ηz+z−1
+2/bracketrightbigg
+whereη= (t2−t′2)/τ. Recalling that the exponential
+factor exp/bracketleftBig
+ηz+z−1
+2/bracketrightBig
+is the generating function for the
+modified Bessel functions
+exp/bracketleftbigg
+ηz+z−1
+2/bracketrightbigg
+=∞/summationdisplay
+n=−∞znIn(η)we conclude that
+Gk(t) =/integraldisplayt
+0dt′t′
+τe2(t′−t)[Ik−1(η)−G1(t′)Ik(η)] (18)
+Thus to determine G1we need to solve an integral
+equation
+G1(t) =/integraldisplayt
+0dt′t′
+τe2(t′−t)[I0(η)−G1(t′)I1(η)] (19)
+Higher correlation functions Gkwithk≥2 are then ex-
+pressed via G1and (18).
+The above results are exact. Now we turn to the most
+interesting case of slow cooling, τ≫1, and compute the
+final density of domain walls.
+B. Final Density of Domain Walls
+Using (4), we re-write the integral equation (19) in
+terms of the density of domain walls:
+1−/integraldisplayt
+0dt′t′
+τe2(t′−t)[I0(η)−I1(η)]
+= 2ρ(t)+2/integraldisplayt
+0dt′t′
+τe2(t′−t)I1(η)ρ(t′) (20)
+We are mostly interested in the final density ρ(τ). Hence
+wespecialize(20)to t=τ. It isalsoconvenienttochange
+the integration variable, t′→η= (τ2−t′2)/τ. We arrive
+at the integral equation
+1−1
+2/integraldisplayτ
+0dηE(η,τ)[I0(η)−I1(η)]
+= 2ρ(τ)+/integraldisplayτ
+0dηE(η,τ)I1(η)ρ/parenleftBig
+τ/radicalbig
+1−η/τ/parenrightBig
+(21)
+where we used the shorthand notation
+E(η,τ)≡exp/bracketleftBig
+2τ/parenleftBig/radicalbig
+1−η/τ−1/parenrightBig/bracketrightBig
+(22)
+Thechiefcontributiontotheintegralontheright-hand
+sideof (21)isgatheredwhen η∼√τ[seeEq.(26)below].
+Thereforewe can replace ρ/parenleftBig
+τ/radicalbig
+1−η/τ/parenrightBig
+byρ(τ) and the
+exponential factor E(η,τ) by
+E(η,τ) =e−η−η2/4τ(23)
+as it follows by expanding the term inside the square
+brackets in Eq. (22). Thus the integral on the right-hand
+side of (21) becomes
+ρ(τ)/integraldisplayτ
+0dηe−η−η2/4τI1(η) (24)
+We can further use the asymptotic formula for the Bessel
+function,
+I1(η)≃eη
+√2πηwhenη≫1, (25)4
+and replace the upper limit in (24) by infinity. Thus we
+obtain
+ρ(τ)/integraldisplay∞
+0dηe−η2/4τ(2πη)−1/2=ρ(τ)Γ/parenleftbig1
+4/parenrightbig
+τ1/4
+√
+4π(26)
+This obviously dominates the second term 2 ρ(τ) on the
+right-hand side of (10).
+On the left-hand side of (10) both terms are compara-
+ble. The integral
+/integraldisplayτ
+0dηE(η,τ)[I0(η)−I1(η)]
+is gathered in the region η=O(1). Thus we can replace
+E(η,τ) bye−ηand integrate up to infinity. Using
+/integraldisplay∞
+0dηe−η[I0(η)−I1(η)] = 1 (27)
+we conclude that the left-hand side of (21) is asymp-
+totically equal to 1 −1/2 = 1/2. Equating this to the
+asymptotic expression (26) for the right-hand side we ar-
+rive at the announced results (10) for the final density of
+domain walls.
+IV. GENERAL COOLING PROCEDURE
+For a cooling scheme characterized by an arbitrary
+function γ(t), the solution is found using the same pro-
+cedure as in Sect. IIIA. Instead of the integral equation
+(19) we get
+G1(t) =/integraldisplayt
+0dt′γ(t′)e2(t′−t)[I0(η)−G1(t′)I1(η)] (28)
+withη= 2/integraltextt
+t′dt′′γ(t′′). Thefinaldensity ofdomainwalls
+satisfies
+1−1
+2/integraldisplayζ
+0dηe2(t′−τ)[I0(η)−I1(η)]
+= 2ρ(τ)+/integraldisplayζ
+0dηe2(t′−τ)I1(η)ρ(t′) (29)
+Note that t′that appears in ρ(t′) ande2(t′−τ)should be
+expressed via ηby inverting
+η(t′) = 2/integraldisplayτ
+t′dt′′γ(t′′) (30)
+Further, the upper limit in integrals in Eq. (29) is given
+byζ=η(t′= 0) = 2/integraltextτ
+0dt′′γ(t′′).
+The dominant contributions to both integrals in (29)
+are gathered in the region where 1 −t′/τ≪1. In this
+region the relation between t′andηsimplifies. We now
+perform the calculations for two families of cooling pro-
+cedures introduced in Sect. II.A. Algebraic Cooling
+For the family of cooling procedures (11) we have
+η= 2/integraldisplayτ
+t′dt′′γ(t′′)
+= 2(τ−t′)−A
+1+ατ/parenleftbigg
+1−t′
+τ/parenrightbigg1+α
++...
+or equivalently
+1−t′
+τ=η
+2τ+A
+1+α/parenleftBigη
+2τ/parenrightBig1+α
++...
+which leads to
+e2(t′−τ)= exp/bracketleftbigg
+−η−A
+1+αη1+α
+(2τ)α/bracketrightbigg
+(31)
+Using (31) and(25) we find that the integralonthe right-
+hand side of Eq. (29) is asymptotically
+ρ(τ)/integraldisplay∞
+0dη√2πηexp/bracketleftbigg
+−A
+1+αη1+α
+(2τ)α/bracketrightbigg
+(32)
+Computing the integral in (32) we find that the asymp-
+totic of the right-hand side of Eq. (29) is given by
+RHS =/radicalbigg
+2
+πΓ/parenleftbigg3+2α
+2+2α/parenrightbigg/parenleftbigg1+α
+A/parenrightbigg1
+2(1+α)
+(2τ)α
+2(1+α)ρ(τ)
+The left-hand side of Eq. (29) is asymptotically
+LHS = 1 −1
+2/integraldisplay∞
+0dηe−η[I0(η)−I1(η)] =1
+2
+where in the last step we have used (27). Equating the
+RHS to the LHS we arrive at (12)–(13).
+B. Exponential Cooling
+For the family of cooling procedures (14) we have
+η= 2(τ−t′)−2B/integraldisplayτ
+t′dtexp/braceleftbigg
+−b
+(1−t/τ)β/bracerightbigg
+Computing the integral and inverting we find (keeping
+two leading terms)
+1−t′
+τ=η
+2τ+B
+bβ/parenleftBigη
+2τ/parenrightBig1+β
+exp/braceleftbigg
+−b/parenleftBigη
+2τ/parenrightBig−β/bracerightbigg
+Hence the exponential factor in the integrands in (29) is
+e2(t′−τ)= exp/bracketleftbigg
+−η−B
+bβ/parenleftBigη
+2τ/parenrightBigβ
+ηexp/braceleftbigg
+−b/parenleftBigη
+2τ/parenrightBig−β/bracerightbigg/bracketrightbigg
+and therefore the RHS of (29) is asymptotically
+ρ(τ)/integraldisplay∞
+0dη√2πηexp/bracketleftbigg
+−B
+bβ/parenleftBigη
+2τ/parenrightBigβ
+ηexp/braceleftbigg
+−b/parenleftBigη
+2τ/parenrightBig−β/bracerightbigg/bracketrightbigg5
+The integral has a simple asymptotic behavior. Indeed,
+changing η→ξviaη= 2τ(ξ/lnτ)1/βwe recast the
+exponential factor in the integrand into
+exp/bracketleftBigg
+−2B
+bβ/parenleftbiggξ
+lnτ/parenrightbigg1+1/β
+τ1−b/ξ/bracketrightBigg
+(33)
+In theτ→ ∞limit, the exponentialfactor(33) is asymp-
+totically equal to 1 when ξ < band zero when ξ > b.
+Therefore the RHS is asymptotically
+ρ(τ)/integraldisplay2τ(b/lnτ)1/β
+0dη√2πη= 2ρ(τ)/radicalBigg
+τ
+π/parenleftbiggb
+lnτ/parenrightbigg1/β
+(34)
+The LHS of (29) is asymptotically 1/2 as before. Equat-
+ing it to (34) we arrive at the announced result (15).
+V. APPROXIMATE ANALYSIS
+A Kibble-Zurek (KZ) argument has been proposed
+[2, 3] to determine the final density of defects (domain
+walls in our case). The KZ argument leads to qualita-
+tively correct results, yet it involves an uncontrolled ap-
+proximation and cannot predict more subtle quantitative
+features, e.g. amplitudes in the scaling laws. In most ex-
+amples, it is impossible to estimate the quantitativeerror
+as we do not possess (asymptotically) exact results. The
+ferromagnetic Ising chain provides a rare exception and
+hence it allows us to probe the quality of an approxima-
+tion provided by the KZ argument.
+The KZ argument suggests to estimate ρ(τ) in the fol-
+lowing way:
+1. During the initial regime the density of domain
+wallsρ(t) is assumed to be given by its equi-
+librium value at the corresponding temperature,
+ρ(t) =ρeq[T(t)]. This assumption is natural since
+the cooling rate is very slow.
+2. During the final regime the density of domain walls
+ρ(t) is assumed to be constant, ρ(t) =ρ∗where
+ρ∗=ρeq(T∗) withT∗=T(t∗). This freezing also
+seems plausible as the density ρ∗is very small and
+during the final regime the density should not de-
+crease much.
+3. The Ising model endowed with a Glauber dynam-
+ics is characterized by the typical time req(T) to
+relax to equilibrium. We now postulate that the
+separation time is determined from the criterion
+req(T∗) =τ−t∗.
+First,weneedtofindtheequilibriumsolution. Inother
+words, we must solve the stationary version of Eqs. (2).
+These equations admit an exponential equilibrium solu-
+tion: By inserting Gk=ηkinto (2) we find that theright-hand side vanishes when 2 = γ(η+η−1). From this
+relation, η=γ−1/bracketleftBig
+1−/radicalbig
+1−γ2/bracketrightBig
+, and therefore
+ρeq=1−η
+2=1
+2/bracketleftBigg
+1−1−/radicalbig
+1−γ2
+γ/bracketrightBigg
+(35)
+The relaxation time is to a degree the matter of agree-
+ment. Equation (2) suggests that
+req=1
+2[1−γ](36)
+as a reasonable definition. Combining (36) and the cri-
+terionreq(T∗) =τ−t∗we arrive at
+1
+2[1−γ∗]=τ−t∗, γ∗=γ(t∗) (37)
+Let us begin with the special cooling procedure (1).
+Thenγ∗=t∗/τand therefore (37) gives 1 −γ∗= 1/√
+2τ.
+Hence
+ρ∗=1
+2/bracketleftBigg
+1−1−/radicalbig
+1−γ2∗
+γ∗/bracketrightBigg
+≃2−3/4τ−1/4(38)
+Comparing(10) and (38) wesee that the exactamplitude
+C=π1/2/Γ(1/4)/equalsdots0.48887 is smaller than the predic-
+tion from the KZ argument CKZ= 2−3/4/equalsdots0.59460.
+A. Algebraic Cooling
+For the family of cooling schemes (11) we recover the
+correct scaling law (12), but with a slightly erroneous
+amplitude
+CKZ= 2−1/2A1/(2+2α)(39)
+Interestingly, the KZ argument correctly predicts even
+the dependence on A: The ratio of the exact and ap-
+proximate amplitudes is the function of αalone
+C
+CKZ=√π
+2/bracketleftbigg
+Γ/parenleftbigg3+2α
+2+2α/parenrightbigg/bracketrightbigg−1/bracketleftbigg1
+1+α/bracketrightbigg1
+2(1+α)
+(40)
+This ratio approaches to 1 as α→0 and to/radicalbig
+π/4/equalsdots
+0.8862269 when α→ ∞. The largest deviation from
+unity is 0 .7913..., so the maximal quantitative mistake
+of the KZ argument is about 20%.
+B. Exponential Cooling
+For the family of cooling procedures (14) we find
+ρ∗≃1√
+4τ/parenleftbigglnτ
+b/parenrightbigg1
+2β
+(41)6
+Note that the ratio of the asymptotically exact density
+given by (15) to the prediction (41) of the KZ argument
+is equal to/radicalbig
+π/4; it is a universal number as it does not
+depend on the parameters B,b,βof the cooling proce-
+dures (14). The same ratio characterizes algebraic cool-
+ing schemes (11) in the α→ ∞limit. This seems to
+be a general feature: If γ(t) approaches to unity faster
+than any power law, then the KZ argument quickly gives
+a prediction and the exact answer is just/radicalbig
+π/4 times
+smaller.
+VI. DISCUSSION
+Thisworkwasinspiredbythedesiretotestthe validity
+of the Kibble-Zurek argument for the spin chain supple-
+mented with non-conservative (Glauber) dynamics. The
+conclusion is that in this setting, the KZ argument is
+very good, namely it gives correct exponents and even
+the amplitudes are approximated reasonably well. The
+chiefoutcomeofthis workisactuallyunrelatedtothe KZ
+argument or any other approximation, namely we have
+shown that the Ising-Glauber chain remains solvable in
+thegeneralcaseofanarbitrary(externallyimposed)tem-
+perature drive T=T(t). The term ‘solvable’ here means
+that for the most interesting characteristic, the density
+of domain walls, we have found a closed linear inhomo-geneous integral equation. It does not seem feasible to
+express a solution in quadratures, but interesting asymp-
+totically exact results have been extracted. We have also
+given expressionsof other two-spin correlationsfunctions
+via the density of domain walls.
+What would happen if we shall continue beyond the
+pointt=τ, that is, we shall follow the zero-temperature
+evolution for t > τ. In one dimension, domain walls
+will continue to diffuse and annihilate upon collisions;
+eventually the system will reach a ground state. In the
+case of instantaneous cooling, higher-dimensional behav-
+iors are different: In two dimensions, the Ising ferromag-
+net may not reach a ground state (it can get trapped in
+a metastable stripe state); in three dimensions, a ground
+state is never reached [18]. It seems plausible that for
+slow cooling the system may not get trapped, or the
+probability of getting trapped decreases as τ→ ∞. This
+appears an interesting challenge and in two dimensions
+at least, one may even hope for theoretical progress [19].
+Acknowledgments
+I am grateful to Leticia Cugliandolo for discussions
+that revitalized my interest to the cooling problem. This
+work has been supported by NSF grant CCF-0829541.
+[1] A. J. Bray, Adv. Phys. 43, 357 (1994).
+[2] T. W. B. Kibble, J. Phys. A 9, 1387 (1976); Phys. Rep.
+67, 183 (1980).
+[3] W. H. Zurek, Nature 317, 505 (9185); Phys. Rep. 276,
+177 (1996).
+[4] W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett.
+95, 105701 (2005).
+[5] A. Polkovnikov, Phys. Rev. 72, 161201(R) (2005).
+[6] J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).
+[7] R.W.CherngandL. S.Levitov, Phys.Rev.A 73, 043614
+(2006).
+[8] S. Suzuki, J. Stat. Mech. P03032 (2009).
+[9] R. J. Glauber, J. Math. Phys. 4, 294 (1963).
+[10] G. Biroli, L. F. Cugliandolo, and A. Sicilia,
+arXiv:1001.0693.
+[11] P. Laguna and W. H. Zurek, Phys. Rev. Lett. 78, 2519
+(1997).[12] A. Yates and W. H. Zurek, Phys. Rev. Lett. 80, 5477
+(1998).
+[13] M. Hindmarsh and A. Rajantie, Phys. Rev. Lett. 85,
+4660 (2000).
+[14] M. I. Bowick et al., Science 263, 943 (1994).
+[15] M. E. Dodd et al., Phys. Rev. Lett. 81, 3703 (1998).
+[16] A. Maniv, E. Polturak, and G. Koren, Phys. Rev. Lett.
+91, 197001 (2003).
+[17] P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic
+View of Statistical Physics (Cambridge University Press,
+Cambridge, 2010). Chapter 8 describes the Glauber dy-
+namics and an exact analysis of the Ising-Glauber chain.
+[18] V. Spirin, P. L. Krapivsky, and S. Redner, Phys. Rev. E
+63, 036118 (2001); Phys. Rev. E 65, 016119 (2002).
+[19] K. Barros, P. L. Krapivsky, and S. Redner, Phys. Rev. E
+80, 040101(R);
\ No newline at end of file
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+arXiv:1001.0742v1  [astro-ph.HE]  5 Jan 2010Highlights of Astronomy, Volume 15
+XXVIth IAU General Assembly, August 2006
+Ian F Corbett, ed.c/circlecopyrt2009 International Astronomical Union
+DOI: 00.0000/X000000000000000X
+QPO-jet relation in X-ray binaries
+Tomaso M. Belloni1
+1INAF–Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy
+email: tomaso.belloni@brera.inaf.it
+Abstract.
+In the past years, a clear picture of the evolution of outburs ts of black-hole X-ray binaries has
+emerged. While the X-ray properties can be classified into ou r distinct states, based on spectral
+and timing properties, the observations in the radio band ha ve shown strong links between
+accretion and ejection properties. Here I briefly outline th e association between X-ray timing
+and jet properties.
+Keywords. accretion, accretion disks, X-rays: binaries, stars: outfl ows
+1. Fast time variability
+The fast time variability observed in the X-ray emission from Black-Ho le Binaries
+(BHB) can be extremely strong and complex. It is clearly connected to the spectral evo-
+lution throughout their outbursts, which can be described throug h the use of Hardness-
+Intensity Diagrams (HID; see Belloni 2009 and references therein ). A total fractional ms
+variability of ∼40% is a major “disturbance” of the accretion flow that can hardly b e
+ignored when trying to understand its properties. Concentrating on the most basic prop-
+erties,wecanidentifytwocategories: loudstates(LHSandHIMSinBelloni (2009)),char-
+acterized by strong flat-top noise components in the power spect ra, with total fractional
+variability 10-40%, and quietstates (SIMS and HSS), with less variability in the form
+of a power law component. Quasi-Periodic Oscillations (QPO) are obse rved in all states,
+with a complex phenomenology. However, the HIMS-SIMS transition is very abrupt and
+involves the interplay between two very different “flavors” of QPO. This transition can
+be marked in a HID with a it QPO line (see Fig. 1). At the same time, the hig h-energy
+part of the X-ray spectrum undergoes abrupt changes through the transition (see Motta
+et al. 2009).
+2. Jet ejection
+The radio properties of BHB display an evident connection with the X- ray states and
+transitions (see e.g. Fender 2006). A relation with the states evolu tion was presented by
+Fender, Belloni & Gallo (2004) on the basis of four well-studied syste ms. At its basis,
+the unified picture of disk-jet coupling presented there identifies t wo regions of the HID:
+the hard region where a steady, compact and mildly relativistic jet is o bserved, and the
+soft region where there is no evidence of nuclear emission from the b inary (see Fig. 1).
+The transition between these two regions marks the ejection of a f ast relativistic jet,
+observed as a bright radio flare or, when imaged, as a superluminal j et. The position of
+this transition was dubbed “jet line”.
+119120 T. M. Belloni
+Figure 1. Left: schematic HID with the two regions identified through t ime variability and the
+‘QPO line’ between them. Right: same HID, with the radio regi ons and the jet line’. The two
+fundamental lines do not coincide.
+3. How do they connect?
+Fender, Belloni & Gallo (2004) identified the jet line with the QPO line. Th is identifi-
+cation would lead to the attractive conclusion that the plasma respo nsible for the noise
+would be the one ejected inform of a jet. However, Fender, Homan & Belloni (2009)
+recently reported, on the basis of a larger sample of sources, tha t the two lines do not
+always coincide, but are close. There seems not to be a direct causa l connection, as some-
+times one precedes the other and vice versa. However, the close a ssociation suggests that
+both ejection and change in timing properties are the outcome of a c omplex physical
+transition that takes place on a longer time scale.
+4. Discussion
+The scheme outlined above, based on the HID and fast timing proper ties can be
+extended to neutron-star systems and even to white-dwarf bina ries (see Belloni 2009;
+K¨ ording et al. 2008; Fender 2009; Tudose et al. 2009). It is clear t hat its properties
+are intimately connected to the spectral state and to the charac teristics of the jet. An
+important key is the study of the correlated variability at optical wa velengths (see e.g.
+Kanbach et al. (2001), Gandhi et al. (2001)), which can shed light o n this connection.
+References
+Belloni, T.M. 2009, in: T.M. Belloni (ed.), The Jet Paradigm: from Microquasars to Quasars ,
+Lecture Notes in Physics (Heidelberg: Springer), in press ( arXiv:0909.2474)
+Fender, R. 2006, in: Lewin, W.H.G. & van der Klis, M. (eds.), Compact stellar X-ray sources ,
+Cambridge Astrophysics Series, No. 39, Cambridge Universi ty Press, p. 381
+Fender, R.P. 2009, in: T.M. Belloni (ed.), The Jet Paradigm: from Microquasars to Quasars ,
+Lecture Notes in Physics (Heidelberg: Springer), in press ( arXiv:0909.2572)
+Fender, R.P., Belloni, T. & Gallo, E. 2004, MNRAS , 355, 1105
+Kanbach, G., et al., 2004, Nature, 324, 23
+Gandhi, P., et al., 2008, MNRAS , 390, L29
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+Motta, S., Belloni, T. & Homan, J. 2009, MNRAS in press (arXiv:0908.2451)
+Tudose, V., et al., 2009, MNRAS in press (arXiv:0909.3604)
\ No newline at end of file
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+arXiv:1001.0743v2  [nucl-th]  27 Jan 2010Microscopic evaluation of the pairing gap
+M Baldo1, U Lombardo2, S S Pankratov3, and E E Saperstein3
+1INFN, Sezione di Catania, 64 Via S.-Sofia, I-95125 Catania, Italy
+E-mail:baldo@ct.infn.it
+2INFN-LNS and University of Catania, 44 Via S.-Sofia, I-95125 Catan ia, Italy
+E-mail:lombardo@lns.infn.it
+3Kurchatov Institute, 123182, Moscow, Russia
+E-mail:pankratov@pretty.mbslab.kiae.ru
+E-mail:saper@mbslab.kiae.ru
+Abstract. We discuss the relevant progress that has been made in the last few years
+on the microscopic theory of the pairing correlation in nuclei and the open problems
+that still must be solved in order to reach a satisfactory descriptio n and understanding
+of the nuclear pairing. The similarities and differences with the nuclear matter
+case are emphasized and described by few illustrative examples. The comparison of
+calculations of different groups on the same set of nuclei show, bes ides agreements,
+also discrepancies that remain to be clarified. The role of the many-b ody correlations,
+like screening, that go beyond the BCS scheme, is still uncertain and requires further
+investigation.
+PACS numbers: 21.60.-n, 21.65.+f, 26.60.+c, 97.60.JdMicroscopic evaluation of the pairing gap 2
+1. Introduction
+Despite the pairing in nuclei was established more than fifty years ag o, the underlying
+interaction processes that are responsible of this remarkable and important correlation
+are not yet understood completely. The main reason of the difficulty is that the
+microscopic theory of nuclear pairing must rely on the effective inter action among
+nucleons, that is not known from first principles. Furthermore it is s trong and therefore
+it requires thesolutionof a complex many-bodyproblem. Even innucle ar matter, where
+the problem is hoped to be simplified to some extent, an accurate micr oscopic theory is
+not available. Numerical Monte-Carlo ”exact” calculations are pres ent in the literature
+onlyforlow-densitypureneutronmatter[1]. Unfortunatelythes eMonte-Carloestimates
+are not in satisfactory agreement among each others [1] and in any case they can hardly
+elucidate the microscopic mechanisms which are at the basis of the on set of pairing.
+Besides the many-body aspects of the problem, at least other two features of the
+nuclear pairing have to be mentioned. One is related to the fact that the pairing
+phenomenon occurs close to the Fermi surface, while the bare nuc leon-nucleon (NN)
+potential necessarily involves also scattering to high energy (or mo mentum) due to its
+strong hard core component, which is one of the main characterist ics of the nuclear
+interaction. It looks therefore natural to develop a procedure w hich removes the high
+energy states and ”renormalize” the interaction into a region close to the Fermi energy.
+This can be done in different ways, among which the most commonly use d seems to
+be the Renormalization Group (RG) Method [2]. A second feature is t he relevance of
+the single particle spectrum, not only because the density of state s at the Fermi surface
+plays of course a major role, but also because the whole single partic le spectrum has
+influence on the effective pairing interaction.
+Despite these uncertainties of the microscopic theory of pairing in n uclear matter,
+there is a commonly accepted point of view that the gap value ∆ is quite small at the
+normal density ρ0, much smaller than typical values of heavy atomic nuclei. This is an
+indication that the nuclear surface must play a major role in establish ing the value of
+the pairing gap in finite nuclei.
+In the last few years relevant progresse have been made in the micr oscopic
+calculations of pairing gap in nuclei [3, 4, 5, 6, 7, 8, 9, 10]. The main est ablished results
+is that the bare NN interaction, renormalized by projecting out the high momenta, is a
+reasonablestartingpointthatisabletoproduceapairinggapwhichs howsadiscrepancy
+with respect to the experimental value not larger than a factor 2. In view of the great
+sensitivity of the gap to the effective interaction this result does no t appear obvious.
+The effective pairing interaction constructed within this renormaliza tion scheme [11]
+explains qualitatively also the surface relevance. Indeed, this inter action at the surface
+can exceed the value inside by one order of magnitude. Direct effect of the surface
+enhancement of the gap was presented in [12, 13] for semi-infinite n uclear matter and
+in [14] for a nuclear slab.
+The role of the surface is also apparent in explaining the puzzling value of theMicroscopic evaluation of the pairing gap 3
+/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s49/s50/s51/s52
+/s32 /s32
+/s32/s32 /s112/s40/s82/s41/s44/s32/s102/s109/s45/s49
+/s82/s44/s32/s102/s109/s32/s68/s70/s51
+Figure 1. Cooper pair distribution for the120Sn nucleus. The self-consistent mean
+field is found with the Energy Density Functional DF3 of [18] .
+coherence length extracted from the pairing gap. In fact, for a p airing gap of the order
+of1MeV, thecoherencelengthisexpectedtobemuchlargerthant hesizeofthenucleus.
+It can be argued that at the surface the pairing gap is larger and th e coherence length
+shorter, i.e. the size of the Cooper pairs should shrink. This was exp licitly shown in a
+recent paper by N. Pillet, N. Sandulescu, and P. Schuck [15] on the b asis of the HFB
+approach and the effective D1S Gogny force. It was shown that Co oper pairs in nuclei
+preferentially are located in the surface region, with a small size (2 −3 fm). In [16],
+it was examined to what extent the effect found in [15] is general and independent of
+the specific choice of NN force. This investigation was carried out fo r a slab of nuclear
+matter, and the pairing characteristics obtained with the Gogny fo rce were compared
+with those with the realistic Paris and Argonne v 18forces. The results obtained with
+the two realistic forces agree with each other within an accuracy of about 10% and agree
+qualitatively with those of the Gogny force. In [17] ananalogous ana lysis, with the Paris
+potential, was made for the nucleus120Sn which is a standard polygon for examining
+nuclear pairing. Again the results turned out to be very close to tho se of [15]. One of
+them is shown in figure 1, where the “probability” distribution of Coop er pairs,
+p(R) = 4πR2/integraldisplay
+κ2(R,r)d3r, (1)
+is displayed. Here R,rare c.m. and relative coordinates. In Fig. 1 this quantity is
+normalized to the total number of Cooper pairs, NCp=/integraltext
+p(R)dR≃10. One can see
+that Cooper pairs, indeed, are strongly concentrated on the sur face.
+Besides the renormalization of the bare interaction of the high mome ntum
+components, other physical effects should be included in a microsco pic approach, like
+the ones related to the effective mass, or more generally, the single particle spectrum
+and the many-body renormalization of the pairing interaction. The r ole of these and
+other features of the microscopic scheme are still to be clarified.
+In this paper we will try to summarize the recent achievements and t he main openMicroscopic evaluation of the pairing gap 4
+problems that hinder the development of an accurate microscopic m any-body theory of
+the nuclear pairing.
+2. Renormalization of the high energy components
+The pairing gap in nuclei is of the order of 1-2 MeV, while the typical en ergy scale of the
+bare NN interaction is of few hundreds MeV. This indicates the difficult y of controlling
+the microscopic construction of the effective pairing interaction Veffat the Fermi surface.
+IntheBCSapproachasimplifiedestimateiscommonlyused, thesocalle dweakcoupling
+limit. In this approximation the relationship between the gap ∆ Fat the Fermi energy
+and the effective interaction has the form
+∆F≈2εFexp(1/νFVeff), (2)
+whereεFis the Fermi energy and νF=m∗kF/π2is the density of state at the Fermi
+energy. The exponential dependence makes the problem particula rly delicate, since a
+small change in the effective interaction can result in substantial ch ange of the pairing
+gap. In phenomenological theories, like the Finite Fermi System (FF S) theory or the
+Hartree-Fock-Bogolyubovmethodwitheffective forces, thevalu eofVeffisconsideredasa
+phenomenological parameter (or a set of parameters) to be used to fit the data. Despite
+the undoubt success of the phenomenological approaches, the c hallenge of ab initio
+evaluation of the pairing gap remains one of the most fundamental a nd not completely
+solved problems in nuclear physics.
+Let us first consider the problem of reducing the interaction to an e ffective one
+close to the Fermi energy. In general language this can be seen as a typical case
+that can be approached by an ”Effective Theory”, where the low en ergy phenomena
+are decoupled from the high energy components. In this procedur e the resulting low
+energy effective interaction is expected to be independent of the p articular form of the
+high energy components. However the procedure is not unique. Th e RG method has
+been particularly developed for the reduction of the general NN int eraction keeping
+the deuteron properties and NN phase shifts up to the energy whe re they are well
+established. In this way a potential, phase equivalent to a known rea listic NN potential,
+can be obtained, which contains only momenta up to a certain cut-off . The extension of
+this procedure to the many-body problem appears in general to re quire the introduction
+ofstrongthree-bodyforces. Ithasbeenappliedalsotothepairin gproblem. Toillustrate
+thedifficultyofdecouplinglowandhighmomentumcomponentsforthe pairingproblem,
+we consider the simple case of nuclear matter in the BCS approximatio n. The BCS
+equation for the gap ∆( k) can be written as
+∆(k) =−/summationdisplay
+k′V(k,k′)
+2/radicalbig
+ε(k′)2+∆(k′)2∆(k′), (3)
+whereVis the free NN potential, ε(k) =e(k)−µ,e(k) is the single particle spectrum
+andµ, the chemical potential. It is possible to project out the momenta la rger than aMicroscopic evaluation of the pairing gap 5
+cutoffkcbyintroducing theinteraction Veff(k,k′), which isrestricted tomomenta k < kc.
+It satisfies the integral equation
+Veff(k,k′) =V(k,k′)−/summationdisplay
+k′′>kcV(k,k′′)Veff(k′′,k′)
+2Ek′′, (4)
+whereE(k) =/radicalbig
+ε(k)2+∆(k)2. The gap equation, restricted to momenta k < k cand
+with the original interaction Vreplaced by Veff, is exactly equivalent to the original gap
+equation. The relevance of this equation is that for a not too small c ut-off the gap ∆( k)
+can be neglected in E(k) to a very good approximation and the effective interaction Veff
+depends only on the normal single particle spectrum above the cuto ffkc. In the RG
+approach the low momenta interaction Vlow−k(k,k′) is constructed in such a way to keep
+the half-on-shell two-body T-matrix in free space and therefore the phase shifts. To the
+extent that the pairing gap is determined only by the phase shifts, t he two approaches
+are therefore equivalent.
+3. The single particle spectrum and the effective mass
+The pairing gap value is strongly affected by the density of state at t he Fermi level, as
+the weak coupling limit (2) indicates. In turn, the density of state is p roportional to
+the effective mass. As it is well known, one has to distinguish between the so-called k-
+mass, which is generated by the momentum dependence of the single particle potential,
+and the e-mass, which is generated by the energy dependence of t he single particle self-
+energy. The total effective mass is just the product of these two effective masses. The
+inclusion of an energy dependent single particle self-energy produc es also the so-called
+Z-factor, i.e. the quasi-particle strength.
+In nuclear matter the four-dimensional gap equation incorporatin g the momentum
+and energy dependent irreducible particle-particle vertex V(k,ε;k′,ε′) and the self-
+energy Σ( k,ε), reads [19, 20, 21]
+∆(k,ε) =−/integraldisplayd3k′
+(2π)3/integraldisplaydε′
+2πiV(k,ε;k′,ε′)∆(k′,ε′)
+D(k′,ε′), (5)
+with
+D(k,ε) = (GGs)−1= [M(k,+ε)−ε−i0][M(k,−ε)+ε−i0]+∆2(k,ε) (6)
+and
+M(k,ε) =k2
+2m+Σ(k,µ+ε)−µ, (7)
+where we define for convenience the energy εrelative to the chemical potential µ.
+GsandGin (6) are, correspondingly, the single particle Green functions with and
+without pairing. For realistic systems, Vand Σ cannot be determined in exact way
+and significant approximations have to be performed. The usual BC S approximation
+amounts to replacing the interaction vertex by the (energy indepe ndent) bare nucleon-
+nucleon potential V, and the nucleon self-energy by some realistic s.p. spectrum. TheMicroscopic evaluation of the pairing gap 6
+/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48
+/s48/s46/s56
+/s48/s46/s57
+/s49/s46/s48
+/s49/s46/s49
+/s49/s46/s50
+/s49/s46/s51
+/s49/s46/s52
+/s48/s49/s50/s51/s52/s53/s54/s55/s42
+/s109/s32/s47/s109
+/s107/s44/s32/s102/s109/s45/s49/s107
+/s70 /s44/s32/s102/s109
+/s45/s49
+Figure 2. Momentum dependence of the effective mass m∗(k) at different Fermi
+momentum values kF.
+latter is characterized by the single particle potential, that can hav e, in nuclear matter,
+a momentum dependence which persists even at high momenta. To illus trate this point
+we have reported in Fig. 2 the effective mass obtained within the BHF a pproximation
+as a function of momentum at different Fermi momenta. The effectiv e mass has usually
+a peak close to the Fermi momentum, but it is substantially different f rom the bare one
+even at high momenta.
+As shown in detail in Ref. [21], energy dependence of the self-energ y can be taken
+into account explicitly provided the interaction vertex remains stat ic,V(k,ε;k′,ε′) =
+˜V(k,k′). In this case, the general gap equation (6) is reduced to the for m
+∆(k) =−/summationdisplay
+k′˜V(k,k′)Z(k′)
+2/radicalbig
+Ms(k′)2+∆(k′)2∆(k′), (8)
+which reminds the BCS gap equation (3), with the “symmetrized” s.p. energy
+Ms(k)≡Re/parenleftbiggM(k,+ek)+M(k,−ek)
+2/parenrightbigg
+(9)
+appearing in the denominator and with the kernel modified by the spe ctral factor
+Z(k)≡/radicalbig
+Ms(k)2+∆(k)22
+π/integraldisplay∞
+0dεIm/parenleftbigg1
+D(k,ε)/parenrightbigg
+. (10)
+In table 1 the neutron effective masses at the Fermi surface m∗(kF) of symmetric
+and asymmetric, ( N−Z)/A= 1/6, nuclear matter found within the BHF approach
+at the equilibrium density ρ0is compared with those of Skyrme forces. In the second
+line, the difference δ(m∗/m)anm= (m∗/m)anm−(m∗/m)snmis given. As one can see,
+the SLy4 effective mass is rather close to the BHF one at the Fermi s urface. OtherMicroscopic evaluation of the pairing gap 7
+/s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48
+/s32/s66/s114/s117/s101/s99/s107/s110/s101/s114
+/s32/s83/s76/s121/s52
+/s32/s83/s75/s77
+/s32/s83/s75/s80/s42
+/s107
+/s70/s44/s32/s102/s109/s45/s49
+/s32/s32
+/s32/s32/s109/s32/s47/s109/s42
+Figure 3. Effectivemass m∗
+Fat the Fermi surfacedepending on the Fermi momentum
+kF.
+Table 1. Comparison of the neutron effective mass in the symmetric and asym metric
+nuclear matter from the BHF calculation with that of different kinds o f the Skyrme
+force.
+SKP [22] SKM* [23] SLy4 [24] BHF
+(m∗/m)snm 0.85 0.71 0.74 0.78
+δ(m∗/m)anm 0.22 0.12 -0.04 −(0.03÷0.05)
+two kinds of Skyrme force give absolutely different isotopic asymmet ry dependence of
+m∗. The density dependence of the effective mass in symmetric nuclear matter for
+BHF and the considered Skyrme forces is reported in figure 3. Arou nd saturation the
+trends are smooth, but at lower density the BHF effective mass has a behavior that
+interpolates between different Skyrme forces, and therefore it c annot be reproduced by
+a given Skyrme force.
+To illustrate the relevance of the high momenta components k > kcon determining
+the effective pairing interaction Veffaround the Fermi momentum, we present in figure
+4 the value of Veffat the Fermi momentum obtained by solving Eq. (4). We take
+kc=√
+2kFand consider three cases : i) the free spectrum, ii) the spectrum o btained
+from the BHF calculation, and iii) the spectrum with a constant effect ive mass, taken
+at the Fermi momentum and from the BHF results. The calculations a re performed in
+the density range relevant for the bulk and surface regions of finit e nuclei.Microscopic evaluation of the pairing gap 8
+/s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s86
+/s101/s102/s102/s40/s107
+/s70/s41/s44/s32/s77/s101/s86
+/s107
+/s70/s44/s32/s102/s109/s45/s49
+/s32/s32
+/s32/s66/s72/s70
+/s32/s109/s42/s61/s99/s111/s110/s115/s116
+/s32/s102/s114/s101/s101
+Figure 4. Effective pairing interaction for the free single particle spectrum (d otted
+line), the BHF spectrum (full line) and the constant effective mass s pectrum (dashed
+line).
+/s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52/s48/s49/s50/s51
+/s32/s70/s40/s107
+/s70/s41/s44/s32/s77/s101/s86
+/s107
+/s70/s44/s32/s102/s109/s45/s49/s32/s66/s72/s70
+/s32/s109/s42/s61/s99/s111/s110/s115/s116
+/s32/s102/s114/s101/s101
+Figure 5. Pairing gap at the Fermi momentum for the three cases of free sing le
+particle spectrum considered in Fig. (4).
+The substantial difference between cases i) and iii) shows the releva nce of the high
+momentum components of the single particle spectrum. On the othe r hand, the strong
+overlap of the curves corresponding to the cases ii) and iii) suggest s that the details of
+the momentum dependence of the effective mass are less relevant. This will be helpful
+for the analysis in finite nuclei, where the approximation of a constan t effective mass
+will be made. The corresponding pairing gaps at the Fermi momenta a re reported in
+figure 5.
+4. The many-body problem
+In atomic nuclei, a new branch of low-laying collective excitations appe ar, the surface
+vibrations (”phonons”). They could be interpreted as a Goldstone mode corresponding
+to spontaneous breaking of translation invariance in nuclei [25]. Th e ghost dipole 1−
+1-Microscopic evaluation of the pairing gap 9
+phonon is the head of this branch. In the self-consistent FFS theo ry or other self-
+consistent approaches, say, the SHF+RPA (or SHFB+qRPA) meth od, the constraint
+ω1−
+1= 0 is fulfilled identically and the next members of the branch of natura l parity
+states, 2+
+1,3−
+1andsoon, have small excitationenergies, less ofatypical distance between
+neighboring shells, ω0≃εF/A1/3. Up to now, only phenomenological approaches were
+used to describe the surface vibrations and their role inthe gapequ ation via the induced
+interaction. The latter was examined by the Milano group in a series of papers using
+various approaches, with the application to the nucleus120Sn. In Ref. [4] the study
+was performed within the Nuclear Field Theory (see [26] and Referen ces therein) in the
+two-phonon approximation. One and two-phonon terms of the indu ced interaction were
+taken into account, as well as the corresponding corrections to t he single-particle states.
+The latter includes both shifts of the single-particle energies and sp read of their strength
+which is analogous to the Z-factor in (8). The ”direct” term of the gap, ∆ dir≃0.7 MeV,
+was found in [4] with the Argonne v 14NN-potential and the mean field generated by the
+SLy4 Skyrme force with the effective mass m∗≃0.7m. It is one half of the experimental
+gap (in [4], which is estimated as ∆ exp≃1.4 MeV, while the simplest 3-point formula
+yields ∆(3p)
+exp≃1.3 MeV). By including all the corrections due to low-laying surface
+vibrations, the complete value ∆ = ∆ dir+indturns out to be in agreement with the
+experiment.
+In [27] the induced interaction itself was analyzed in detail, again conc entrating on
+low-lying ( ωL<5 MeV) surface vibrations. The Vlow−kpotential was used as the free
+NN-interaction, instead of Argonne v 14in [4], and again the experimental value of the
+gap was obtained as a sum of practically equal ”direct” and ”induced ” contributions
+multiplied by the Z-factor representing the quasiparticle strength at the Fermi su rface.
+The latter was estimated on the basis of calculations in [4] as Z= 0.7.
+An essentially different approach was used in [5] where all collective st ates with
+spin values J≤5 and excitation energies ωJ<30 MeV were included into the induced
+interaction term. The surface vibrations discussed above are only a part of them. Both
+the natural and unnatural parity states contribute to the sum. The qRPA method with
+the SLy4 Skyrme force was used for every Jπ-channel. The fragmentation and self-
+energy effects were approximately taken into account with the follo wing relation for the
+total effective pairing interaction:
+<12|Vdir+ind|34>=Z <12|Vdir+Vind|34>, (11)
+whereZ= 0.7 value again was used. A short notation |12>is used in (11) for the two-
+particlebasisstates. Tomakeeasiertheanalysisandcomparisonwit hothercalculations,
+separate calculations of the gap from Vdir(Argonne v 14) andVind, both without the Z-
+factor, were made in [5]. They give ∆ dir= 1.04 MeV and ∆ ind= 1.11 MeV. Using the
+recipe of Eq. (11), yields the average gap ∆ F= 1.47 MeV which is again close to the
+experiment.
+In our opinion, each method used in these papers has some weak poin ts which could
+be criticized. In the first case, the contribution of so-called tadpo le diagrams [25, 28]Microscopic evaluation of the pairing gap 10
+was neglected, see also discussion in [29]. In the last case, the use of Skyrme force in the
+particle-hole spin and spin-isospin channels (unnatural parity stat es) is questionable.
+Indeed, the Skyrme parameters were fitted only to phenomena wh ich are related to
+”scalar” Landau–Migdal amplitudes, F,F′. The combinations which determine the
+spin-dependent amplitudes G,G′were not checked up to now by an attempt to describe
+corresponding phenomena, e.g. magnetic moments, M1-transitions, and so on. This
+problem is discussed also in [5], and some complementary calculation wa s made with
+G=G′= 0. It resulted in a very strong induced interaction yielding very big g ap
+∆F= 2.12 MeV. This could be considered as an estimate of the uncertainty o f such
+calculations with the use of phenomenological forces. Indeed, eve n the amplitudes
+F,F′which are known sufficiently well in vicinity of the Fermi surface could c hange
+significantly when high energy excitations are considered. Another questionable point
+istheuseinthiscalculationscheme ofthe Z-factorfoundonlyfromthelow-lying surface
+vibrations. The high energy response function included into the Vindwill contribute to
+theZ-factor as well. This contribution comes mainly from the spin-isospin c hannel and
+could be estimated as Znm≃0.8 [30].
+We see that the problem of the screening effect is, indeed, very diffic ult, and some
+approximations must be introduced in the calculations. In our opinion , the fact that
+essentially different methods were used in [4] and [5] with different re sults for ∆ ind
+shows by itself that the problem of finding contribution of the induce d interaction into
+the pairing gap in atomic nuclei is far from being solved.
+5. Solution of the “ ab initio” gap equation in finite nuclei
+The explicit form of the gap equation (5) in the coordinate represen tation for a non-
+uniform system is as follows [31]:
+∆(r1,r2,ε) =/integraldisplay
+V(r1,r2,r3,r4;E= 2µ,ε,ε′)×
+×G(r3,r5,ε′)Gs(r4,r6,−ε′)∆(r5,r6,ε′)dε′
+2πidr3dr4dr5dr6. (12)
+As in Sections 2-4, single-particle energies ε,ε′are counted off the chemical potential
+µ. Dealing with nuclear matter, we set µ=µ0≃ −16 MeV (the leading term in the
+Weizsaecker mass formula), whereas we have µ≃ −8 MeV for stable atomic nuclei, as
+the120Sn nucleus considered below. As it was discussed above, in this Sectio n we limit
+ourselves with the simplest Brueckner-like approach in which the irre ducible vertex V
+coincides with the free NN-potential, V=V, which is independent of energy. In
+this case, the gap ∆ is also independent of energy; hence, the prod uct of two Green’s
+functions in (12) can be integrated with respect to ε′:
+As(r1,r2,r3,r4) =/integraldisplaydε′
+2πiG(r1,r2,ε′)Gs(r3,r4,−ε′). (13)
+The gap equation (12) can be written in a compact form as
+∆ =VAs∆. (14)Microscopic evaluation of the pairing gap 11
+Below we deal with the BCS gap equation, not the general one, Eq. ( 12). This explains
+why the term ab initio in the title of the Section is in quotes. In fact, we speak just
+about the first step into the problem, i.e. the solution of the BCS-like gap equation
+with the free NN-potential as the pairing interaction. The mean field potential (or
+more general, the mass operator) used in this solution is taken as a p henomenological
+input. To go beyond the BCS approximation it is necessary, within an ab initio method,
+to calculate, first, the mass operator and, second, corrections to the interaction block V
+in the gap equation (12). The latter includes the induced interaction discussed above
+(see also Ref. [32]) and three-body forces [33]. It turns out that, even at the level of
+this simplest ab initio calculations, serious contradictions are still present.
+The integral equation (14) can be reduced to the form adopted in t he Bogoliubov
+method,
+∆ =−Vκ, (15)
+wheretheabnormaldensitymatrix κ=As∆canbeexpressedintermsof u,v-functions,
+κ(r1,r2) =/summationdisplay
+iui(r1)vi(r2), (16)
+which satisfy the system of Bogoliubov equations. The summation in ( 16) is over the
+complete set of Bogoliubov functions with the eigen energies Ei>0.
+The Milano group was the first who, in a series of papers [3, 4], [5] and Refs.
+therein, solved the gap equation (15) with the realistic Argonne NN-force v 14for the
+nucleus120Sn. The latter was chosen for a definite reason. Indeed, the chain of semi-
+magic tin isotopes is a traditional polygon for examining nuclear pairing [22], [18].
+The nucleus under discussion is in the middle of the chain and the numbe r of neutrons
+participating inthe pairing, those abovethe closed shell N= 50, issufficiently big touse
+the approximationof the”developed” pairing, used, infact, in (15) . This approximation
+implies neglecting the particle number fluctuations [34] typical of the BCS-like theories.
+The set of Bogoliubov equations was solved directly in the basis {λ}of the states with a
+fixed limiting energy Emax. Such direct method is difficult because of a slow convergence
+of sums over intermediate states λin the gap equation. These sums are analogous to
+integrals in the momentum space in the gap equation for infinite nuclea r matter, see
+Section 2. In [3] the value of Emax=600 MeV was used, and in [4, 5], Emax=800 MeV.
+The analysis of [7] showed that the use of such big value of Emaxpermits to find ∆ only
+with accuracy of 10%. In [3] the Shell Model basis was used with the S axon-Woods
+potential and the bare mass, m∗=m, and the value ∆=2 .2 MeV was obtained which is
+by a factor one and half greater than the experimental one ( ≃1.3÷1.4 MeV). Evidently,
+this contradiction forced the authors to use in further works the self-consistent basis
+of the Skyrme-Hartree-Fock (SHF) method with the density depe nding effective mass
+m∗(ρ)/ne}ationslash=m. In particular, the popular Sly4 force was used, which is character ized by
+a small effective mass, equal to m∗≃0.7min nuclear matter at the normal nuclear
+density. Solving the BCS gap equation with such a basis the value of ∆ ≃0.7 MeV was
+obtained in [4] and ≃1.0 MeV in [5]. A close value for the gap in the BCS equationMicroscopic evaluation of the pairing gap 12
+was found in [7] for the nuclear slab with parameters which mimic120Sn nucleus. This
+calculation was done with Argonne v 18potential which differs only slightly from the
+v14one, but the single-particle basis with m∗=mwas used. Remind that in [4, 5]
+corrections to the BCS gap due to induced interaction were conside red bringing the
+results rather close to the experimental data as discussed in Sect ion 4. The crucial
+dependence of the gap on m∗, which is easily seen in weak coupling limit (2) for the
+gap in nuclear matter, is, of course, the main reason of so strong v ariation of the BCS
+gap from [3] to [5]. In finite nuclei, this dependence is weaker than in nu clear matter,
+as the surface region plays here the main role and one has m∗(ρ)→min this region.
+However, the m∗effect remains strong.
+Recently, results from the ab initio BCS equation (15) for a number of semi-magic
+nuclei were published [9, 10]. They are based on the soft realistic low- k force discussed
+above which was calculated starting from the Argonne v 18potential, with the same self-
+consistent Sly4 basis, i.e. the same effective mass, as in [5]. For the nu cleus120Sn under
+consideration the value ∆ ≃1.6 MeV was obtained. This value exceeds the experimental
+one, which raises some questions. Indeed, although there are disc repancies in absolute
+value of the corrections to the BCS gap (see for instance [5] and [35 ]), their sign is
+more or less definite. All calculations of these corrections, to our k nowledge, increase
+∆ considerably. Hence the BCS equation has to lead to the gap value s maller than the
+experimental one. In addition, there is a direct contradiction betw een the results of the
+Milano group and the ones of Duguet with co-authors, despite the B CS problem was
+solved with very similar inputs. Evidently, there is some difference in th e method of
+including the effective mass, which is hidden in [9, 10] under the renor malization of v 18
+intoVlow−k.
+In [16] an attempt was taken to clarify the reasons of this contrad iction. The BCS
+gap equation (15) was solved for the same120Sn nucleus with the separable form of
+Paris potential, which simplifies calculations. Our experience of calcula tions for nuclear
+slab [6, 7] shows that the difference between the Paris potential an d the Argonne v 18
+one for the gap value is of the order of 0.1 MeV which is significantly sma ller than
+the deviation under discussion. For projecting out the high moment a contribution, the
+so-called Local Potential Approximation (LPA) method was used. T his new version of
+the local approximation was introduced by our group for semi-infinit e nuclear matter
+and nuclear slab system, see the reviews [29, 36]. In general, this me thod is analogous
+to the renormalization scheme for solving the gap equation in infinite n uclear matter
+described in Section 2. To solve the gap equation in the form (14) for finite systems,
+we split the complete Hilbert space Sof the pairing problem into the model subspace
+S0, including the single-particle states with energies less than a fixed va lueE0, and the
+subsidiary one, S′. Correspondingly, the two-particle propagator (13) is the sum
+As=As
+0+A′. (17)Microscopic evaluation of the pairing gap 13
+Thenotationbecomesobviousifoneexpands(13)inthebasisofsing le-particlefunctions
+φλ(r),
+As(r1,r2,r3,r4) =/summationdisplay
+λ1λ2λ3λ4As
+λ1λ2λ3λ4φ∗
+λ1(r1)φ∗
+λ2(r2)φλ3(r3)φλ4(r4).(18)
+The model space propagator As
+0includes the terms of the sum (18) with single-particle
+energiesελ< E0,A′being the remaining part.
+The gap equation is reduced to the one in the model space,
+∆ =VeffAs
+0∆, (19)
+with the effective pairing interaction obeying the integral equation in the subsidiary
+space,
+Veff=V+VA′Veff. (20)
+The LPA method concerns the solution of (20). Solving this equation directly in
+coordinate space is rather complicated, not much simpler than the g ap equation (14)
+in the complete Hilbert space. The problem could be simplified with the us e of the
+LPA. It turned out that, with very high accuracy, for each point R, one can use the
+formulas for the infinite system in the potential field U(R) (it explains the term LPA).
+This simplifies equation for Veffsignificantly, in comparison with the initial equation for
+∆. As a consequence, the subspace S′could be chosen as large as necessary. Validity
+of the LPA could be justified by finding that, beginning from some valu e ofE0, the
+result for the gap doesn’t change under additional increase of E0. It turns out that
+for a nuclear slab, the value of E0≃20÷30 MeV is sufficient [6, 7]. For finite nuclei, it
+should be taken a little bigger, E0= 40 MeV [8]. In the slab calculations, we used the
+bare nucleon mass, m∗=m.In this case, the reduction of high momenta components of
+the NN-force in (20) is, in fact, quite similar to the renormalization pr ocedure resulting
+in a low-k interaction [37]. The difference is that in the LPA the renorma lization is
+coordinate dependent.
+The gap equation in the model space S0was solved in the λ-representation with
+the use of different single-particle bases φλ. A discretization method of the continuum
+spectrumwasusedwiththewall radius L=16 fm. Increasing theradiusto L=24 fmdoes
+not practically influence the results. The radial eigen-functions Rnlj(r) were found with
+a steph=0.05 fm. We have used the Shell-Model basis with the Saxon–Woods pot ential
+with a standard set of parameters and also with several self-cons istent basis obtained
+with different methods : the Generalized Energy Density Functional method by Fayans
+et al. [18] with the functional DF3, and the SHF method as well with diff erent kinds of
+Skyrme forces. The bare mass, m∗=m, is used in the first method, just as in the Shell
+Model, whereas in the SHF method the effective mass is not equal to mand is density
+dependent. To clarify the role of the effective mass, we chose two k inds of the Skyrme
+force, SKP and SKM*, for which the difference between m∗andmis quite moderate,
+and the popular SLy4 force, with significant difference of m∗fromm, see Fig. 6. We
+see that the SKP effective mass deviates significantly from that for symmetric nuclearMicroscopic evaluation of the pairing gap 14
+/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50
+/s32/s32
+/s42
+/s109/s32/s40/s82/s41/s47/s109
+/s82/s44/s32/s102/s109/s32/s83/s108/s121/s52/s32
+/s32/s83/s75/s80/s32
+/s32/s83/s75/s77/s32/s42
+Figure 6. Coordinate dependence of the effective mass for different Skyrme forces in
+the120Sn nucleus.
+matter, Fig. 3. This is due to the strong isospin asymmetry effect (( N−Z)/A= 1/6 for
+120Sn nucleus) for this kind of Skyrme force. For SLy4 and SKM* this eff ect is rather
+modest. Remind that the Sly4 basis was used in calculations of ∆ by Milan o group and
+by Duguet with coauthors as well. As to the calculation of the effectiv e interaction in
+the subspace S′, we first put m∗=m. In table 2 the diagonal matrix elements ∆ λλof
+the neutron gap in the120Sn nucleus, for 5 levels nearby the Fermi level, are given for
+each basis under consideration. The quantity ∆ Fis the corresponding Fermi-average
+value: ∆ F=/summationtext
+λ(2j+1)∆ λλ//summationtext
+λ(2j+1). As we see, in all cases except of the last one
+the found value of the gap exceeds the experimental one, 1.4 MeV, significantly. This
+indicates the necessity to take into account the difference betwee n the effective and bare
+masses in the ab initio BCS gap equation.
+In Fig. 7, for each kind of mean field, the anomalous density ν(R) =κ(R,r= 0)
+is displayed. Here the notation R= (r1+r2)/2,r=r2−r1is used. As the gap ∆ is
+proportional to this quantity, ν(R) depends on the effective mass in the same way as
+the matrix elements ∆ λλin table 2. Indeed, the anomalous densities for the effective
+forces DF3, SKP and SKM* are rather close to each other, wherea s for the SLy4 force
+with a small effective mass the anomalous density is suppressed signifi cantly. There
+is a pronounced surface maximum of ν(R) for all the cases. This leads to the surface
+enhancement of the pairing gapfound in [14] for the slab and in [15], f or spherical nuclei.Microscopic evaluation of the pairing gap 15
+Table 2. Diagonal matrix elements ∆ λλ(MeV) with the Paris potential for several
+kinds of the self-consistent basis.
+λSWDF3SKPSKM* SLy4
+3s1/21.521.641.551.551.17
+2d5/21.601.731.681.641.24
+2d3/21.641.801.711.681.26
+1g7/21.852.112.021.911.37
+2h11/21.581.791.691.641.18
+∆F1.651.851.761.711.25
+/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52
+/s32/s68/s70/s51
+/s32/s83/s75/s80
+/s32/s83/s75/s77
+/s32/s83/s76/s121/s52/s32/s42
+/s32/s32
+/s82/s44/s32/s102/s109/s40/s82/s41/s44/s32/s102/s109/s45/s51
+Figure 7. Anomalous density for the120Sn nucleus calculated with different self-
+consistent mean fields.
+Until now, in the case of the Skyrme forces, we took into account t he difference
+between the effective and bare masses only inside the model space, whereas we put
+m∗=mfor calculating Veff. At this point, there is a principal difference between our LPA
+method and calculations of [4, 5] and [9, 10], where the SLy4 effective mass was used for
+all theλ-states. For a closer comparison, we made a modification of the LPA method,
+which permits to take into account the density dependent effective massm∗
+n(ρn,ρp) for
+a part of the space S′, including momenta k <Λ, where Λ is a parameter. Following
+the idea of LPA, with the potentials Un(R),Up(R) , it is natural to determine at each
+pointRthe densities of each type τ=n,pof nucleons with quasi-classical formulas:
+ρτ(R) = (kτ
+F(R))3/3π2,kτ
+F(R) = [2m∗
+τ(ρn(R),ρp(R))(µτ−Uτ(R))]1/2, whereµn,µpare
+the chemical potentials of neutrons and protons in the nucleus und er consideration. ForMicroscopic evaluation of the pairing gap 16
+Table 3. Diagonal matrix elements ∆ λλwith the Paris potential for the Sly4 basis
+depending on the way of account for the effective mass in equation f orVeff.
+λSLy4Sly4-1 Sly4-2 Sly4-3
+3s1/21.171.07 0.88 0.76
+2d5/21.241.13 0.93 0.80
+2d3/21.261.15 0.95 0.83
+1g7/21.371.23 0.99 0.85
+2h11/21.181.08 0.88 0.75
+∆F1.251.14 0.92 0.80
+Table 4. The same as in table 3, but for the Argonne v 18potential.
+λSLy4Sly4-1 Sly4-2 Sly4-3
+3s1/21.231.10 0.83 0.56
+2d5/21.321.18 0.89 0.61
+2d3/21.341.20 0.92 0.63
+1g7/21.481.31 0.96 0.64
+2h11/21.271.13 0.85 0.57
+∆F1.341.19 0.89 0.60
+the functional SLy4 we made several alternative calculations with d ifferent values of Λ.
+They aredenoted asSLy4-1 (Λ=3 fm−1), SLy4-2(Λ=4 fm−1) andSLy4-3 (Λ=6 .2 fm−1).
+The first two versions mimic calculations of [9, 10], the latter, of [4, 5]. The obtained
+gap values are given in table 3.
+Let us now repeat calculations with the SLy4 basis, but for Argonne force v 18.
+Results are given in table 4. Comparison with table 3 shows that the diff erence between
+gap values for the Paris and Argonne force is of the same order ( ≃0.1 MeV) as for
+the slab calculations [7], with the exception of the SLy4-3 version with the cutoff for
+m∗in the equation for Veffequal to Λ 3= 6.2 fm−1. In this case, the difference is
+≃0.2 MeV. It is worth of noticing that the sign of the difference changes depending
+on Λ. Namely the Argonne gap exceeds the Paris one for SLy4 (Λ = 0) and SLy4-1
+(Λ1= 3 fm−1) versions, but becomes smaller for SLy4-2 (Λ 2= 4 fm−1) and SLy4-3
+runs. Such behavior is qualitatively clear.Indeed, the Paris potentia l is much harder of
+the Argonne one, therefore the relative contribution to Veffof the momentum region,
+say, between Λ 2and Λ 3, is less than for the Argonne potential. Therefore, for the gap
+equation itself, the role of the corresponding suppression of Veffdue to putting m∗< m
+is less in the Paris case than in the Argonne one.
+It is rather difficult to make a direct comparison of table 4 with results of [9]. In
+the first column, the effective mass m∗/ne}ationslash=mis introduced only in the model space,
+forελ< E0= 40 MeV. In the free space, outside the nucleus, it corresponds t o the
+momentum cutoff Λ = 1 .4 fm−1; inside the nucleus we have Λ ≃2 fm−1.We see that the
+”average” value of Λ is less a bit of Λ = 2 fm−1in [9]. In the second column (SLy4-1) we
+deal with Λ = 3 fm−1. Thus, we should attribute to the gap value of [9] (∆ ≃1.6 MeV)Microscopic evaluation of the pairing gap 17
+Table 5. The same as in table 4, but for the SKM* Skyrme force.
+λSKM* SKM*-1 SKM*-2 SKM*-3
+3s1/21.58 1.52 1.40 1.29
+2d5/21.71 1.63 1.49 1.38
+2d3/21.75 1.68 1.55 1.43
+1g7/22.01 1.92 1.76 1.62
+2h11/21.72 1.64 1.51 1.40
+∆F1.78 1.71 1.57 1.44
+an average value of these two columns, ∆ ≃1.25 MeV, which is noticeably smaller.
+For comparison with [4, 5] we should be guided by the last column (SLy4 -3), as the
+corresponding value Λ = 6 .2 fm−1was chosen to reproduce Emax= 800 MeV from these
+calculations. Again, this correspondence is not literal as it takes pla ce only outside the
+nucleus where we have m∗=m. Inside, due to m∗/ne}ationslash=m, the same value of Emaxshould
+correspond to smaller value of Λ ≃5.5 fm−1. Again we should take a value between
+those of the 3-rd column and the last one, closer to the latter. In a ny case, it will be
+closer to the result of [4] ( ≃0.7 MeV) than of [5] ( ≃1 MeV).
+The main observation, common to table 3 (Paris) and table 4 (Argonn e), is a
+drastic dependence of the gap on the m∗(k) behavior in the subsidiary space. In fact, a
+set of calculations with different cutoff Λ imitates, very roughly, the k-dependence of the
+effective mass.In the case that the asymptotic limit m∗(k)→moccurs sufficiently far
+away, the pairing gap in nuclei does depend on the effect of m∗/ne}ationslash=mat high momenta.
+It is absolutely lost in calculations of ∆ with low-k force found for small cutoff values.
+Evidently, this is the main reason why gap values found in [9] by solving the BCS gap
+equation are so high, keeping in mind corrections due to the induced in teraction.
+To confirm the leading role of the effective mass in the problem, we mad e a series
+of analogous calculations for the SKM* force for which the effective mass in the nucleus
+under consideration is much closer to m, than for the SLy4 one, see Fig. 6. The results
+are given in table 5 with the notation similar to that of table 4. We see th at even for the
+SKM*-3 version with Λ = 6 .2 fm−1the result does not leave any room for contribution
+of the induced interaction.
+6. Discussion and conclusions
+In this paper we briefly reviewed the status of the microscopic theo ry of nuclear pairing.
+Although to date there is no consistent theory of nuclear matter p airing, progress has
+been made in developing approximated methods yielding comparable re sults on the
+density dependence of the pairing gap. However, the results cann ot be applied to finite
+nuclei directly because the nuclear surface plays a leading role in nuc lear pairing and the
+standard LDA fails at the nuclear surface. As to finite nuclei, some p rogress also exists
+especially in solving the simplest, in fact BCS-type, ” ab initio” gap equation, where
+the free NN potential is used as the effective pairing interaction. Ev en this equationMicroscopic evaluation of the pairing gap 18
+is not completely microscopic as far as the phenomenological mean fie ld is used. The
+inherent technical problems were overcome, first, by the Milan gro up [3, 4, 5] and more
+recently by other teams [9, 10] and [16]. In the first and the last cas es, the nucleus
+120Sn was considered as ”testing sample”, whereas in [9] several isoto pic and isotonic
+chains were considered. The Argonne v 14force was used in [3, 4, 5], the low-k force
+with the cutoff Λ = 2 fm−1in [9], and the Paris potential, in [16]. In this paper
+we performed calculations analogous to [16] for the Argonne v 18potential which differ
+only a bit from the v 14one. All the results differ from the experimental gap in120Sn,
+∆exp≃1.3÷1.4 MeV, not more than by a factor two which shows relevance of the
+ab initio BCS gap equation as a starting point for the microscopic theory of n uclear
+pairing. However, more detailed comparison shows that there are c ontradictions even
+at this ”first level” of the problem. Indeed, the BCS gap is ≃1 MeV in [5] and is
+≃1.6 MeV in [9], whereas inputs look quite similar. Namely, both the calculatio ns use
+the SLy4 Skyrme force with the effective mass m∗≃0.7m; the Argonne v 14force is
+used in [5] and the low-k force in [9], but the latter could be obtained fr om the first
+one with the RGM procedure. The only important difference of the inp uts is the size
+of the momentum space where the effective mass contributes: k < k max≃6 fm−1in
+[5] and only k <Λ = 2 fm−1in [9]. Indeed, the RGM equation is defined for the free
+NN scattering where the equality m∗=mis postulated. In fact, we deal with different
+k-dependence of theeffective mass. The equality m∗≃0.7mtakes place forall momenta
+in [5] and only for k <Λ, in [9]. This reason of the contradictory results of [5] and [9]
+was discussed in [10] and was confirmed with the analysis of [16] for th e Paris force and
+in this paper, for the Argonne v 18force. We use the so-called LPA method developed by
+us previously in which high momenta are excluded via some renormalizat ion procedure
+which recall the RGM one but is coordinate dependent and permits to introduce in
+the high momentum space the effective mass into the equation for th e effective pairing
+interaction. For the same SLy4 basis, we changed ”by hands” the s ize of the space
+where the equality m∗≃0.7mtakes place, putting m∗=moutside. Changing this
+dividing point we evolve from the situation close to that of [9] to the o ne of [5], although
+the correspondence, of course, is not literal. In the first limit, we o btained the value
+of ∆F≃1.25 MeV which is noticeably smaller than that in [9]. In the second one,
+we obtained the result closer to that of [4] than of [5]. But, in our opin ion, fixing
+some numerical contradictions is not of primary importance. This dis agreement can be
+resolved. Much more important is the very high sensitivity of the pair ing gap to the k-
+dependence ofthe effective mass foundin our analysis. This depend ence could behardly
+guessed phenomenologically by a lucky Skyrme-type ansatz. Indee d, any such ansatz
+deals with a number, not a function! Therefore an additional micros copic ingredient,
+namely a theory of the k-dependent effective mass, is necessary even at this first level
+of the problem.
+Thenextopenproblemistheinclusionofthemany-bodycorrections totheBCSgap
+equation. The induced pairing interaction due to exchange by virtua l surface vibrations
+and other particle-hole excitations is the main of them [4, 5]. Up to now , this problemMicroscopic evaluation of the pairing gap 19
+was studied only within phenomenological approaches and is far from being solved. Any
+attempt to attack it from first principles hits the absence of the ge neral microscopic
+nuclear theory.
+7. Acknowledgments
+We thank H.-J. Schulze for useful discussions. Two of the authors (S.S.P. and E.E.S.)
+thank INFN (Sezione di Catania) for hospitality during their stay in C atania. This
+research was partially supported by the Grants of the Russian Minis try for Science
+and Education NSh-3004.2008.2 and 2.1.1/4540, the joint Grant of R FBR and DFG,
+Germany, No. 09-02-91352-NNIO a, 436 RUS 113/994/0-1(R), by the RFBR grants
+07-02-00553-a, 09-02-01284-a and 09-02-12168-ofi m.Microscopic evaluation of the pairing gap 20
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+and Zuo Wei 2002 Phys. Rev. C66, 054304; Muther H and Dickhoff W H 2005 Phys. Rev C72
+054316 ; Bozek P 2003 Phys. Lett. B551, 93
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+[24] Chabanat E, Bonche P, Haensel P, Meyer J, Schaeffer R 1997 Nucl. Phys. A627710
+[25] Khodel V A and Saperstein E E 1982 Phys. Rep. 92 183
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+[29] Baldo M, Lombardo U, Saperstein E E, and Zverev M V 2004 Phys. Rep. 391261
+[30] Saperstein E E and Tolokonnikov S V 1998 JETP Lett. 68553
+[31] Migdal A B Theory of finite Fermi systems and applications to atomic nuc lei(Wiley, New York,
+1967).
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+[33] W. Zuo and U. Lombardo 2009, Phys. Atomic Nuclei 72, 1359
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\ No newline at end of file
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+arXiv:1001.0744v1  [astro-ph.GA]  5 Jan 2010To appear in The Astrophysical Journal Letters
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+NITROGEN ISOTOPIC FRACTIONATION IN INTERSTELLAR AMMONIA
+D. C. Lis1, A. Wootten2, M. Gerin3, and E. Roueff4
+(Received 2009 November 16; Accepted 2009 December 30)
+To appear in The Astrophysical Journal Letters
+ABSTRACT
+Using the Green Bank Telescope (GBT), we have obtained accurate measurements of the14N/15N
+isotopic ratio in ammonia in two nearby cold, dense molecular clouds, Ba rnard 1 and NGC 1333. The
+14N/15N ratio in Barnard 1, 334 ±50 (3σ), is particularly well constrained and falls in between the
+local interstellar medium/proto-solar value of ∼450 and the terrestrial atmospheric value of 272. The
+NGC 1333 measurement is consistent with the Barnard 1 result, but has a larger uncertainty. We do
+not see evidence for the very high15N enhancements seen in cometary CN. Sensitive observations of
+a larger, carefully selected sample of prestellar cores with varying t emperatures and gas densities can
+significantly improve our understanding of the nitrogen fractionat ion in the local interstellar medium
+and its relation to the isotopic ratios measured in various solar syste m reservoirs.
+Subject headings: astrochemistry — ISM: abundances — ISM: individual (Barnard 1, N GC 1333) —
+ISM: molecules — molecular processes — radio lines: ISM
+1.INTRODUCTION
+Molecular isotopic ratios are an important tool in the
+studyoftheoriginofinterstellarandsolarsystemmateri-
+als. Afterhydrogen,nitrogendisplaysthelargestisotopic
+variationsinthesolarsystem,typicallyexplainedbymix-
+ing of various proto-solar or pre-solar reservoirs. Earth,
+Mars interior, Venus and most primitive meteorites have
+nitrogenisotopicratioswithin 5%ofthe terrestrialatmo-
+spheric value,14N/15N = 272 (see Marty et al. 2009).
+However, the proto-solar nebula was poorer in15N, with
+14N/15N≃450, as evidenced by infrared and in-situ
+measurements in the Jupiter atmosphere (530+380
+−170, ISO,
+Fouchet et al. 2000; 435 ±57, Galileo, Owen et al.
+2001; 448 ±62, Cassini, Abbas et al. 2004; 450 ±106,
+Cassini, Fouchet et al. 2004) and recentsolarwind mea-
+surements (442 ±131, 2σ, Genesis, Marty et al. 2009).
+The proto-solar14N/15N ratio is in agreement with
+the local interstellar medium (ISM) value (450 ±22,
+Wilson & Rood 1994; or 414 ±32 at the birth place of
+the Sun, Wielen & Wilson 1997).
+Very low nitrogen isotopic ratios,14N/15N = 148 ±
+6, have been measured in CN in a large sample of
+comets (Manfroid et al. 2009). Similar isotopic ratios
+have been recently derived by Bockel´ ee-Morvan et al.
+(2008) in comet 17P/Holmes in CN and HCN, the pre-
+sumed parent ofthe CNradicalin cometaryatmospheres
+(14N/15N = 165 ±40 and 139 ±26, respectively). Al-
+though higher ratios in HCN were initially reported in
+comet Hale-Bopp, Bockel´ ee-Morvan’s re-analysis of the
+archival data suggests that the14N/15N ratio in HCN in
+1California Institute of Technology, Cahill Center for As-
+tronomy and Astrophysics 301-17, Pasadena, CA 91125;
+dcl@caltech.edu
+2National Radio Astronomy Observatory, 520 Edgemont
+Road, Charlottesville, VA 22903; awootten@nrao.edu
+3LERMA, CNRS UMR8112, Observatoire de Paris and Ecole
+Normale Superieure, 24 Rue Lhomond, 75231 Paris cedex 05,
+France; maryvonne.gerin@lra.ens.fr
+4LUTh, CNRS UMR8102, Observatoire de Paris, Section
+de Meudon, Place J. Janssen, 92195 Meudon, France; eve-
+lyne.roueff@obspm.frthis comet is also consistent with the CN value, given
+the measurement uncertainties. The exact chemical net-
+works explaining the HCN isotopic anomaly in comets
+have yet to be proposed. However, one clear conclu-
+sion is that HCN has never isotopically equilibrated with
+the nebular N 2at the later phases of the evolution of
+the solar system (Bockel´ ee-Morvan et al. 2008). Low
+nitrogen isotopic ratios are also found in some mete-
+orites, interplanetary dust particles (IDPs) and samples
+ofmaterialreturned from comet 81P/Wild2by the Star-
+dust spacecraft; amine groups in particular often show
+low14N/15N ratios (Engel & Macko 1997; Keller et al.
+2004; Floss et al. 2006; McKeegan et al. 2006).
+In the ISM, nitrogen bearing molecules are important
+tracers of cold, dense gas, as they do not freeze-out
+onto grain mantles at densities below a few ×106cm−3,
+contrary to carbon and oxygen species, such as CO
+or CS. In cold, dense, CO depleted ISM regions, high
+15N enhancements have been suggested to be present
+in gas-phase molecules, in particular in ammonia (see
+Rodgers & Charnley 2008a,b, and references therein).
+These isotopic enhancements could be preserved as spa-
+tial heterogeneityin ammonia ice mantles, as monolayers
+deposited at different times would have different isotopic
+compositions and may in turn explain the15N enhance-
+ments seen in primitive solar system refractory organics,
+if they are synthesized from ammonia.
+Measurements of nitrogen isotopic ratios in ammonia
+incoldmolecularcloudsareofgreatinterest, astheypro-
+vide quantitative observational constraints for the chem-
+ical fractionationmodels. Howeverthe data availableare
+scarce. In a recent study, Gerin et al. (2009) detected a
+doubly-fractionated ammonia isotopologue,15NH2D, in
+severalISM regions. They derive14N/15N isotopic ratios
+∼350–850, similar to the proto-solar value of 450. How-
+ever, the15NH2D lines are weak and the corresponding
+measurementuncertaintiesarelarge. Tobetter constrain
+the nitrogen isotopicratios in the solarneighborhood, we
+haveobservedthe14NH3and15NH3inversionlines using
+theNRAO 100m RobertC. ByrdGreen BankTelescope.2 Lis et al.
+Fig. 1.— Spectra of the ammonia inversion lines in Barnard 1-
+b (αJ2000= 03h33m20.s8,δJ2000= +31◦07′34′′). Gray lines
+show hyperfine structure fits—a single velocity component fo r the
+14NH3(1,1), (2,2), and15NH3(1,1) transitions and two compo-
+nents for the14NH3(3,3) transition, as described in the text. The
+15NH3(2,2) spectrum, shown with a Gaussian line fit, is formally a
+3.5σupper limits. The bottom-rightpanels shows the LTE rotatio n
+diagram, as described inthe text. LHS stands forthe left-ha nd-side
+in eq. (2) of Blake et al. (1987).
+We present here results for two nearby clouds, Barnard 1
+and NGC 1333, and discuss prospects for future studies
+of nitrogen isotopic ratios using the GBT.
+2.OBSERVATIONS
+Observationsofammoniainversionlinespresentedhere
+were carried out in 2009 February using the GBT. Con-
+ditions were calm under a lightly overcast sky resulting
+in a stable system temperature of ∼48–50 K. The K-
+band receiver feed/amplifier set covering 22–26.5 GHz
+was employed. The GBT spectrometer was configured
+in its eight 50 MHz intermediate frequency (IF) band-
+width, nine-level mode, which allowssimultaneouscover-
+age of four 50 MHz frequency bands in two polarizations,
+through the use of offset oscillators in the IF. This mode
+gives 3.051 kHz channel separation. The IF No. 0 was
+centered between the (1,1) and (2,2) lines of14NH3, the
+(3,3) line was centered in IF No. 1 and the (2,1) line in
+IF No. 3; IF No. 2 was tuned midway between the (1,1)
+and (2,2) lines of15NH3.
+The antenna temperature calibration was achieved
+by injecting a calibrated, broadband signal from a
+noise diode. The data were placed on the T∗
+Ascale
+(Ulich & Haas 1976) by multiplying by a factor of
+eτA/ηl, whereτisthezenithopacity( ∼0.065at23.5GHz
+and 0.075 at 22.5 GHz, from an atmospheric model), AFig. 2.— Spectra of the ammonia inversion lines in NGC 1333
+(αJ2000= 03h29m11.s6,δJ2000= +31◦13′26′′). The14NH3(2,1)
+spectrum is formally a 3.7 σupper limit. The bottom-right panel
+shows the LTE rotation diagram, as described in the text. Lin es
+as in Fig. 1.
+is the air mass of the observations, and ηlis the rear
+spillover efficiency ( ∼0.99 for the unblocked GBT aper-
+ture). The source 0336+3218 was used for pointing and
+absolute calibration at the beginning of the observation
+set. The absolute calibration accuracy is conservatively
+estimated at 20%. Relative accuracy among lines in
+spectral bandpasses measured at the same time is much
+better, typically dominated by the random noise of the
+spectra. The conventional beam efficiency of the GBT
+at 20 GHz is ηmb≃0.79 and the FWHM beam size is
+∼33′′. Data were taken in the frequency switching mode,
+with a frequency displacement of 6 MHz.
+Automatically updated dynamic pointing and focusing
+corrections were employed based on real-time tempera-
+ture measurements and a thermal model of the GBT;
+zero points were adjusted typically every 2 hours or less,
+using 0336+3218 as a calibrator. The two polarization
+outputs from the spectrometer were averaged in the fi-
+nal data reduction process to improve the signal-to-noise
+ratio. The total integration times for Barnard 1 and
+NGC 1333 are 174 and 48 min, respectively.
+3.RESULTS
+Figure 1 shows spectra of the14NH3and15NH3in-
+version lines toward Barnard 1. The satellite hyper-
+fine lines are detected with high signal-to-noise ratios
+for the (1,1) and (2,2) transitions of14NH3. Results of
+hyperfine structure (HFS) fits, obtained using the IRAM
+CLASSsoftwarepackage,areshowninTable1. The(1,1)
+and (2,2)14NH3lines have the same line width of 0.80Nitrogen Fractionation 3
+TABLE 1
+HFS Fit Results and Ammonia Column Densities.
+Transition T∗
+RdvaVLSR ∆V τ N mol
+(K km s−1) (km s−1) (km s−1) (cm−2)
+Barnard 1-b
+14NH3(1,1) 37 .7±0.006 6.7 0.80 4.0 2 .8×1015
+14NH3(2,2) 1 .90±0.005 6.5 0.80 0.3 2 .8×1015
+14NH3(3,3) 0 .076±0.029 6.5 0.80b— 8.0×1015
+0.26±0.11 7.1 3.5 — —
+14NH3(2,1) 0 .012±0.004 — — — —
+15NH3(1,1)c0.106±0.004 6.6 0.63 — 8 .4×1012
+15NH3(2,2) 0 .013±0.004 — — — —
+NGC 1333
+14NH3(1,1) 16 .4±0.015 7.5 1.2 1.2 8 .9×1014
+14NH3(2,2) 2 .22±0.07 7.3 1.3 1.1 9 .9×1014
+14NH3(3,3) 0 .37±0.06 6.8 1.3b— 2.6×1015
+0.61±0.04 4.7 9.5 — —
+14NH3(2,1) 0 .023±0.007 — — — —
+15NH3(1,1)c0.045±0.007 7.2 0.85 — 2 .6×1012
+15NH3(2,2) 0 .012±0.008 — — — —
+Notes: Upper limit for the14NH3(2,1) and15NH3(2,2) transitions included for completeness. Molecular co lumn densities are computed
+assuming LTE with temperatures of 10.9 and 14.7 K for Barnard 1 and NGC 1333, respectively, as described in the text.aUncertainties are
+from the HFS fit for the14NH3(1,1)–(3,3) lines and computed from the rms in the spectra fo r the remaining weak lines and upper limits.
+bFixed parameter.cThe hyperfine structure of15NH3has been recently revised by Bethlem et al. (2008) who give th e level energies with
+unprecedented accuracy. We have calculated the correspond ing transition strengths using the theory of Kukolich (1967 ) and assuming
+half-integer values of the quantum numbers of the levels inv olved.
+kms−1, while the (3,3) spectrum is much broader, indi-
+cating the presence of a high-velocity component origi-
+nating in warmer gas. Consequently, we fitted the (3,3)
+line with two velocity components, fixing the line width
+of the first component at the value obtained for the (1,1)
+and (2,2) lines. All parametersfor the secondcomponent
+were allowed to vary in the fit. The15NH3(1,1) line is
+slightly narrowerthan the14NH3(1,1) and (2,2) lines, as
+expected for an optically thin line. Figure 2 shows the
+NGC 1333 spectra. The lines are weaker and broader
+than in Barnard 1. The (3,3) transition shows strong
+blueshifted emission from the molecular outflow driven
+by IRAS 4A. The line width derived from the HFS fit to
+the15NH3data is significantly smaller than the values
+obtained for14NH3. However, the signal to noise ratio
+in the15NH3spectrum is relatively low, formally 6.2 σ.
+We use the rotation diagram technique (see, e.g.,
+eq. 2 of Blake et al. 1987) to derive the excitation
+temperatures and molecular column densities. We use
+the spectroscopic constants and partition functions for
+14NH3and15NH3from the JPL Molecular Spectroscopy
+database (http://spec.jpl.nasa.gov/). The resulting ro-
+tation diagrams are shown in Figures 1 and 2, bottom-
+right panels. The least-squares fit to the line intensities
+of the (1,1)–(3,3) data (narrow component) in Barnard 1
+gives a rotation temperature of 12.4 K in (dotted line
+in Fig. 1). However, since the (3,3) spectrum is con-
+taminated by the high-velocity emission, only the (1,1)
+transitionisdetected for15NH3, andgivenourinterestin
+the cold gas component on the line of sight, a more con-
+sistent approach is to use only the (1,1) and (2,2)14NH3
+data for the temperature determination. This approach
+also mitigates possible effects of the non-LTE ortho-to-
+para ratio (e.g., Umemoto et al. 1999) and results in a
+slightly lower rotation temperature of 10 .9±0.2 K (as-suming 2.7% uncertainties for the integrated line inten-
+sities, as discussed in §4; dashed line in Fig. 1, bottom-
+right panel). This value is subsequently used to derive
+the14NH3and15NH3column densities from the (1,1)
+integrated line intensities (Table 1). Given the low rota-
+tion temperature, we include the correction for the 2.7 K
+cosmic background radiation in the column density com-
+putations. This correction has no effect on the tempera-
+ture determination, since all the lines observed have very
+closerest frequencies. The NH 3temperature and column
+density in Barnard 1 we derive here are in good agree-
+ment with the results of Bachiller et al. (1990) based
+on observations with the Effelsberg telescope (12 K and
+2.5×1015cm−2).
+For NGC 1333, we derive a rotational temperature of
+17.6 K using the (1,1)–(3,3) lines and 14 .7±0.5 K using
+only the (1,1) and (2,2) transitions (dotted and dashed
+lines, respectively in Fig. 2, bottom-right panel). We use
+the latter value for the column density determination.
+4.DISCUSSION
+The uncertainty of the derived column densities is dif-
+ficult to estimate quantitatively. As discussed in §2, all
+lines are observed simultaneously and the relative cali-
+bration is largely determined by the signal-to-noise ratio
+in the spectra—24and 6.2 σfor15NH3(1,1) in Barnard1
+and NGC 1333, respectively. The14NH3/15NH3column
+density ratio, when computed from the observed inten-
+sities of the (1,1) transitions of the two isotopologues, is
+insensitive to the exact value of the rotation tempera-
+ture used in the calculations, as long as the temperature
+is the same for the two isotopologues. The remaining
+source of uncertainty is the determination of the14NH3
+optical depth. The formal uncertainties for the opacity
+correctedlineintensitiesgivenbytheHFSfitsarenegligi-
+ble (see Table 1). In order to independently estimate the4 Lis et al.
+uncertainty of the14NH3(1,1) integrated line intensities
+obtained from the HFS fit, we divide the full Barnard 1
+data set into three subsets, 58 min integration time each,
+and fit each subset independently. From the three mea-
+surements we derive a 2.7% uncertainty for the mean
+which we use as an estimate of the uncertainty of our fi-
+nal14NH3˜(1,1) measurement. Adding in quadrature the
+corresponding statistical uncertainty for15NH3of 4.2%,
+we derive a 3 σuncertainty for the14NH3/15NH3ratio of
+15.0%inBarnard1. Assuminga5.2%uncertaintyforthe
+14NH3(1,1) measurement in the shorter NGC 1333 in-
+tegration (the Barnard 1 value scaled by the square root
+of the integration time ratio), and adding in quadrature
+a 16% uncertainty for the15NH3measurement, we de-
+rive a 50% uncertainty (3 σ) of the14NH3/15NH3ratio
+in this source. Our final estimates of the14NH3/15NH3
+abundance ratio are thus 334 ±50 and 344 ±173 (3σ) for
+Barnard 1 and NGC 1333, respectively.
+The14N/15N fractionation ratios in Barnard 1 and
+NGC 1333 derived from deuterated ammonia measure-
+ments by Gerin et al. (2009) are 470+170
+−100and 360+260
+−110,
+respectively. The NH 3and NH 2D results arethus consis-
+tent, giventhe NH 2Dmeasurementuncertainties. Owing
+to the relatively short integration and low SNR in the
+15NH3spectrum, our result for NGC 1333 is not very
+constraining. However, our Barnard 1 data are of excel-
+lent quality. The derived14N/15N ratio falls in-between
+the local ISM/proto-solar value ( ∼450, as discussed in
+§1) and the terrestrial atmospheric ratio of 272. We do
+not see evidence for the very high15N enhancements, as
+suggested by the cometary CN measurements ( ∼150).
+A comparison of our fractionation measurements
+with the nitrogen “superfractionation” models of
+Rodgers & Charnley (2008b) is of interest. In their Fig-
+ure 3, Rodgers & Charnley (2008b) present results of
+time-dependent models of the evolution of the14N/15N
+enhancement ratios in key species, including ammonia.
+For a temperature of 10 K, the gas-phase15N fraction-
+ation in ammonia can reach a maximum enhancement
+of a factor ∼7–10, much higher than the value we de-
+rive for Barnard 1. However, the enhancement factor
+decreases with time to a factor of a few belowthe initial
+isotopic ratio assumed in the model. The time at which
+the maximum fractionation is reached depends strongly
+on the initial fraction of nitrogen in molecular form, butis typically of order 1 Myr. The enhancements of the
+magnitude that we measure can be present very early
+on, or a few tenths of a Myr after the peak enhancement
+time.
+The Rodgers & Charnley (2008b) model results de-
+pend strongly on the gas temperature—at 7 K, sig-
+nificantly higher enhancements are found in the gas-
+phase ammonia fractionation, as compared to 10 K.
+The sources we have studied are somewhat warmer than
+the model clouds considered by Rodgers & Charnley
+(2008b). Itwouldthusbeextremelyinterestingtoextend
+our measurements to colder clouds, such as LDN 1544
+or LDN 134N, with central temperatures of order 7 K.
+Our present observations are exploratory in nature, but
+show that accurate measurements of nitrogen fractiona-
+tion in ammonia in cold ISM sources are clearly feasible
+using the GBT. A focused observing program of a larger,
+carefully selected sample of prestellar cores with varying
+temperatures and gas densities would allow significant
+progress in our understanding of nitrogen fractionation
+in local ISM and its possible implications for the existing
+measurements in various solar system reservoirs.
+The nitrogen fractionation enhancement levels derived
+here are well reproduced by the gas-phase models of
+Roueff et al. (see, e.g., Fig. 2 of Gerin et al. 2009).
+For an H 2density of 105cm−3and a 10 K temperature,
+the gas-phase models predict14NH3/15NH3= 389 and
+14NH2D/15NH2D = 372. The models of Roueff et al.
+are steady-state models that focus on the gas-phase pro-
+cesses and approximate the adsorption of the species on
+grain surfaces by reducing the elemental abundances in
+thegas, ascomparedtothe dynamicalmodelsofRodgers
+& Charnley that explicitely take into account the mantle
+formation and gas-grain interactions.
+We thank T. Minter and the GBT operator, D. Strick-
+lin, for their assistance during the observations and
+J. Braatz for his advice in setting up the observing
+scripts. The National Radio Astronomy Observatory is
+a facility of the National Science Foundation operated
+under cooperative agreement by Associated Universities,
+Inc. D. C. Lisissupportedbythe NationalScienceFoun-
+dation grant AST-0540882 to the Caltech Submillime-
+ter Observatory. M. Gerin and E. Roueff acknowledge
+support from the CNRS-INSU French National Program
+PCMI (Physique et Chimie du Milieu Interstellaire).
+REFERENCES
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\ No newline at end of file
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+Quantum Darwinism as a Darwinian process  
+John Campbell  
+ 
+Abstract  
+The Darwinian nature of Wojciech Zurek’s theory of Quantum Darwinism is evaluated 
+against the criteria of a Darwinian process as understood within Universal Darwinism. 
+The characteristics of a Darwinian process are developed including the consequences of 
+accumulated adaptations resulting in adaptive system s operating in accordance with 
+Friston’s free energy principle and employing environmental simulations. Quantum 
+theory, a s developed in Zurek’s research program and encapsulated by his theory of 
+Quantum Darwinism is discussed from the view that Zurek’s derivation of the 
+measurement axioms implies that the evolution of a quantum system entangled with 
+environmental entities is  determined solely by the nature of the entangled system. There 
+need be no further logical foundation.  Quantum Darwinism is found to conform to the 
+Darwinian paradigm in unexpected detail and is thus may be considered a theory within 
+the framework of Univ ersal Darwinism. With the inclusion of Quantum Darwinism 
+within Universal Darwinism and the explanatory power of Darwinian processes 
+extended beyond biology and the social sciences to include the creation and evolution of 
+scientific subject matter within p article physics, atomic physics and chemistry, it is 
+suggested that Universal Darwinism may be considered a candidate ‘Theory of 
+Everything’ as anticipated by David Deutsch.  
+Introduction  
+Wojciech Zurek’s theory of Quantum Darwinism  (Zurek, 2009) posits a Darwinian 
+process responsible for the emergence of classical reality from its quantum substrate. 
+This theory explains information transfer between the quantum and classical realm 
+during the process of decoherence as involvin g a ‘copy with selective retention ’ 
+mechanism characteristic  of Darwinian processes.  
+ 
+Quantum theory is frequently  applied to m icroscopic  systems near the limits of 
+experimental perception . Such systems are often characterized as fundamental but this 
+migh t be due more to the limitations of scien tific technology than to an actual 
+fundamental  or simple  nature.  It is clear that our leading candidates for the next 
+generation of fundamental physical theories will have to account for phenomena 
+occurring at the Plank scale, that is about 10-35 meters  (Smolin, 2001 ; Green 1999) .  
+When we consider that the diameter of a proton is about 10-14 meters we see that there is 
+a greater relative difference in scale between the Plank scale and that of 'fundamental' 
+particle s than there is between that of fundamental particles and phenomena in our 
+every day experience. There are well developed physical models that indicate quantum 
+theory may apply at scales close to the plank scale  (Markopoulou & Sm olin, 2004) . We 
+might well expect there to be a good deal of emergent complexity and perhaps even 
+layers of emergent phenomena below the scale of 'fundamental’ particles. This 
+consideration may caution us against making assumptions concerning the complex ity of 
+quantum systems.   
+Universal Darwinism, the collection of scientific theories that employ a Darwinian 
+process to explain the creation and evolution of their subject matter, offers such 
+explanation s throughout  a wide range of the biological and social  sciences  (Campbell, 
+2009) .  The question examined here is whether  Quantum Darwinism might be 
+interpreted as a theory extending the scope of Universal Darwinism into the realm of 
+quantum phenomena: particle physics, atomic phy sics and chemistry amongst others.   
+ 
+Answering  this question may be reduced to determining if the Darwinian process of 
+Quantum Darwinism as elucidated by Zurek  and colleagues  is a Darwinian process as 
+understood by Universal Darwinism.  
+ 
+Universal Darwinism consider s a Darwinian process as a substrate -neutral algorithm 
+that may be extracted from Darwin’s theory of natural selection. This algorithm is 
+composed of three steps: replication or copying, variations amongst the copies , and 
+selective surviv al of the copies determined by which  variations  they possess . It has been 
+shown that any entity embodying this algorithm will evolve and will evolve to be better 
+at surviving.  A Darwinian process operating as a system’s evolution ary mechanism  may 
+often res ult in the system becoming an adaptive system . Adaptive systems may be 
+characterized as  having an internal model of their  environment which is maintained 
+through Bayesian updating and subject to the free energy principle  (Friston , 2007) . 
+They also tend to  accumula te adaptations  that function  through the principle of 
+maximum entropy to place constraints on the system ’s evolution to states of higher 
+entropy  (Campbell , 2009) . 
+ 
+Zurek  (2009)  has indicated that  in his view  Quantum Darwinism meets at least some of 
+these criteria : 
+ 
+In the end one might ask: “How Darwinian is Quantum Darwinism?” Clearly, 
+there is survival of the fittest, and fitness is defined as in natural selection .  
+ 
+However he also makes it clear the exact relationship between Quantum Darwinism and 
+natural selection is murky and requires some clarification  (Zurek, 2009) : 
+ 
+Is Quantum Darwinism (a process of multiplication of information about 
+certain favored states that seem to be a “fact of quantum life”) in some way 
+behind the familiar natural selection? I cannot answer this question, but neither 
+can I resist raising it.  
+ 
+This paper is  an attempt to shed some light on th ese question s. Section I will examine 
+the nat ure of Darwinian Process es as they  have come to be developed within Universal 
+Darwinism. Section II will examine the nature of Quantum Darwinism  and Section III 
+will evaluate Quantum Darwinism as a Darwinian process. Section IV discusses some of 
+the many i nterpretational issues for quantum theory raised by Zurek’s  work . 
+ I.  Darwinian Processes  
+ 
+The algorithmic nature of natural selection allows its essential mechanism to be 
+abstracted and hypothesized as a possible mechanism operating in the evolution of 
+subj ect matter often other than biological. Numerous theories of this type abound in the 
+social sciences and even in the hard sciences such as physics  (Campbell, 2009) .    
+ 
+The essential abstraction from natural selection, which we  are calling a Darwinian 
+process, has been developed in the work of Richard Dawkins, Daniel Dennett and Susan 
+Blackmore (amongst others) to consist of a three -step process:  
+1) Replication of system.  
+2) Inheritance of some characteristics that have variatio n amongst the offspring.  
+3) Differential survival of the offspring according to which variable characteristics they 
+possess.  
+ 
+It has been proposed that any system adhering to this three -step algorithm, regardless 
+of its substrate, must evolve and will evol ve in the direction of an increased ability to 
+survive  (Dawkins, 1976; Dennett, 1995; Blackmore, 1995) . 
+ 
+The physical mechanisms instantiated by Darwinian processes that bestow enhanced 
+survivability are refer red to as adaptations. Adaptations are usually discovered through 
+processes with a random 'trial and error' component, such as genetic mutation, but the 
+greater survivability they bestow allows them to become widespread within the 
+population. Systems with long evolutionary histories come to accumulate many 
+adaptations. Indeed any organism may be considered as largely an accumulation of 
+adaptations built up over evolutionary time.  
+ 
+Given the second law of thermodynamics , the survivability of a low entropy s ystem  is 
+highly unlikely  and must entail an intricate balancing of low entropy within the system  
+against an increase of environmental entropy. This complex balancing requirement 
+implies a deep ‘knowledge’ of the environment by the system. Theorists such as Henry 
+Plotkin (1993) have concluded that adaptations must contain detailed information 
+concerning their environment to the extent  that they can  be considered as a form of 
+knowledge.  
+ 
+For this reason a Darwinian process that has been in operation for a sign ificant period of 
+time may become a good deal more sophisticated than the bare bones Darwinian 
+algorithm might indicate. Take for example natural selection, the basic Darwinian 
+process operating in the biological realm. The base ‘copy with selective retent ion’ 
+algorithm is still in effect but has been complemented with an array of embellishments 
+increas ing its effectiveness. For instance it is a consensus amongst those who research 
+the subject that DNA, the substance used in constructing genetic copies, is not the 
+original substance used. At some point the use of DNA evolved and completely replaced 
+the original mechanism. Other embellishments such as sexual reproduction have also 
+added to the complexity with which the basic algorithm is carried out.  
+ Such en hancements to the Darwinian process also bestow greater survivability and are 
+thus also adaptations. Once a Darwinian process has evolved a sufficient depth of 
+adaptations it may be considered an adaptive system. Karl F riston (2007) defines the 
+general cha racteristics of an adaptive system as one that can react to changes in its 
+environment by changing its interactions with the environment to optimize the results 
+for itself. He examines the system in term of three features or variables: the system 
+itself, t he effect of the environment on the system and the effect of the system on the 
+environment. These three features are integrated  via an internal model that resides 
+within the system. Friston has shown  that the measures taken by adaptive systems in 
+order to optimize their interaction with the environment can be understood via the Free 
+Energy Principle: an adaptive system will attempt to minimize the free energy which 
+bounds the ‘surprise’ resulting from  its actual experience in  the environment. In other 
+words the system will act so as to minimiz e the prediction error of its internal model.  
+ 
+The internal model is constructed in a Bayesian process that makes inferences from 
+environmental data and also utilize s knowledge of the action s which the system can take 
+to influence outcomes in the environment. Examples of such internal models residing 
+within adaptive systems may include genetics residing within organisms, mental models 
+residing within brains and business plans residing within corporations.  
+ 
+Let's say we see something out of the corner of our eye that might be important. Because 
+we cannot see the object very well we may have trouble assigning accurate  probabilities 
+to the numerous imaginable possibilities. Being adaptive systems  we might  turn our 
+head and/or eyeballs and bring the thing into focus. We gain a pertinent data set and 
+update, in a Bayesian manner, the probabilities associated with our internal model of 
+the external world  in conformance with this data . We change the w ay we interact with 
+our environment by bringing a potentially important aspect of our environment into 
+focus. We are now better informed and in a better position to optimize our response.  
+ 
+The system’s interaction with its environment must be designed to optimize the 
+outcome for itself.  Thus an adaptive system must envision the optimal but realistic 
+outcome of environmental interactions and execute those actions under its control 
+necessary to achieve the envisioned outcome.  
+ 
+From this view  the modeling per formed by an adaptive system functions as a simulation 
+of its environmental interactions and is thus a form of computation. The theory of 
+computation as outlined by Turing and the Church -Turing -Deutsch principle makes the 
+claim that there exist universal c omputing machines  capable of processing all 
+computable functions. David Deutsch has shown that  the composition of th is underlying 
+class of computable functions is not defined by logic or mathematics but rather by the 
+laws of nature. What is comput able  are those functions where  nature has provided the 
+machine ry capable of performing the algorithm . In Deutsch’s words  (Deutsch, 1985) :  
+ 
+The reason why we find it possible to construct, say, electronic calculators, and  
+indeed why we can perform mental arithmetic, cannot be found in mathematics 
+or logic. The reason is that the laws of physics „happen to‟ permit the existence of physical models for the operations of arithmetic such as addition, subtraction 
+and multiplicati on. 
+ 
+Some  researchers consider  the computational aspect of adaptive systems as central to 
+understanding them. Nobel laureate Sydney Brenner  (1999)  has called for biological 
+science to focus on developing the theoretical capabilities required to compute 
+organisms’ phenotype from their genotype. This approach is judged to have explanatory 
+potential as he characterizes biological systems themselves as essentially information 
+processing machines.  
+ 
+It is interesting that we find computational machinery central t o the functioning of 
+adaptive systems found in nature and that it is such computational machinery found in 
+nature that underlies the theory of computation.  
+ 
+Adaptive systems, by this definition, contain a good deal of internal machinery. They are 
+complex, low entropy structures.  The primary outcome of environmental interactions 
+that any adaptive system must achieve in order for it to be an optimal encounter is to 
+retain its low entropy structure, to survive intact. This is not a trivial accomplishment , as 
+the second law of thermodynamics states that such outcomes are extremely rare  and 
+that their rarity increases exponentially with time and  the amount of matter within the 
+system .  
+ 
+An adaptive system must orchestrate encounters with its environment i n such a  way as 
+to maintain its own low entropy state at the expense of increased entropy in the 
+environment. Selecting and executing these rare outcomes  requires knowledge ; both 
+knowledge of the causes operating in the environment and of how other entities existi ng 
+in the environment can be exploited to maintain the system’s  low entropy state.  
+ 
+For example if we consider life forms as adaptive system s we should expect them to 
+utilize adaptive mechanisms for lowering entropy.  Sunlight is a n abundant  energy 
+source a t the earth’s surface but only extremely specific encounters with this energy will 
+result in work being done in the service of local entropy reduction . Life acquired 
+knowledge to perform this feat billions of years ago when it evolved photosynthesis  and 
+this knowledge has since become ubiquitous amongst the plant kingdom . 
+Photosynthesis causes energy in sunlight to be transferred to a high energy bond in 
+molecules of ATP. The energy in this molecular bond is then used as a resource for 
+performing wor k through  complex chemical processes which  in the end serve to retain 
+the organism’s low entropy state.  
+ 
+The knowledge required to perform photosynthesis and to use the ATP molecule is 
+contained in the genetic code of organisms and this knowledge is passed  between 
+generations through copying  of the organisms ’ genome s. These adaptations were 
+created and have evolved to their present states due to the operation of the Darwinian 
+process called natural selection.  
+ 
+All life forms possess these internal models in  the form of genetic molecules, which are 
+central to their functioning as adaptive systems.   
+The principle of Maximum Entropy requires systems to move to states of highest 
+entropy consistent with the constraints operating on the system. The ability of ada ptive 
+system s to maintain their low entropy s tate must therefore be explainable in terms of 
+the constraints they deploy against increasing entropy. These constraints are precisely 
+the adaptations developed during the evolution of the system. In our example  of the 
+production and consumption of ATP molecules within organisms those constraints 
+often take the form of enzymes which constrain the system ’s chemical pathways to 
+follow a specifi ed course. These enzymes are directly coded for in the organism ’s DNA. 
+Thus the knowledge implicit with their entropy -reducing abilities is contained in the 
+organisms ’ internal model s. 
+ 
+At the heart of Bayesian probability is the notion of updating : our beliefs in a hypothesis 
+should be updated whenever pertinent new data beco mes available. Bayes’ Theorem 
+provides the exact measure by which our confidence in a hypothesis should change with 
+a given piece  of new data.  The essence of the theorem is the common sense notion that 
+our confidence should be adjusted in accordance with the extent  to which the  hypothesis 
+or model enhances our ability to predict the new data . 
+ 
+This theorem has been shown to be the unique mathematical description of how 
+knowledge may be accumulated  (Jaynes, 1985) . It is a signature characteristic of a 
+Darwinian process and is a central mechanism in an adaptive system’s ability to model  
+its situation  (Campbell, 2009) . A consequence of the importance of Bayesian updating is 
+that we see evidence of its operation wherever knowledge accumulates in nature , 
+whether in genetics, brain functioning or science.  
+ 
+In the course of  this analysis of Darwinian processes we have  developed  a detailed a set 
+of criteria against  which  any prospective Darwinian proc ess, such as Quantum 
+Darwinism, may be compared . These criteria include:  
+1. The operation of a three step ‘copy with selective retention’ algorithm.  
+2. Operation as an adaptive system where the system, the effect of the environment 
+on the system and the effect o f the system on the environment  are modeled 
+internally  and the system conforms to F riston’s free energy principle . The 
+simulation of the system’s environmental interactions may be understood as 
+computation.  
+3. Maintenance of the internal model’s accuracy through Bayesian updating . 
+4. Production of adaptations  which utilize the principle of maximum entropy to 
+maintain the system’s low entropy state.  
+ 
+II. Quantum Darwinism  
+The research program into the nature of quan tum systems conducted by Wojciech Zurek 
+of Los Alamos National Laboratory and colleagues has spanned more than 25 years and 
+has identified  and developed  a number of novel processes and constructs. The most 
+important of these might include: decoherence, ein selection, envariance and the theory 
+of Quantum Darwinism. The theoretical advances gained by this program have  enabled Zurek to derive what have commonly been considered two axioms of quantum theory  
+from the remaining  three.  
+ 
+Quantum theory is  often pres ented as a derivation from a set of axioms. A common set 
+of axioms are:  
+1) The s tate of a quantum system is represented by a vec tor in its Hilbert space . 
+2) Evolutions are unitary (i.e., generated by Schrodinger equation).  
+3) Immediate repetition of a measurement y ields the same outcome.  
+4) Measurement outcome is one of the orthonormal  states , the  eigenstates of the 
+measured observab le. 
+5) The p robability pk of finding an outcome   sk   in a measurement of a quantum 
+system that was previously prepared  in the state   ψ   is given by  sk ψ  2. 
+ 
+Decoherence occurs when information is copied from a quantum system to its 
+environment. This process, often referred to as a measurement, has prove n an 
+unresolved interpretational quandary for quantum theory  since the Bohr -Einstein 
+debates  occurri ng during the first part of the last century and has become known as the 
+measurement problem. The problem arises as many consider the axioms to contain 
+seeming contradictions ; axiom 2  requires a quantum system to evolve via the 
+Schrodinger equation in a ma thematically smooth , deterministic  and continuous 
+manner  but axiom 4 and axiom 5 require the quantum system to jump in a 
+discontinuous manner, upon measurement, to a specific state, one often  much different 
+from the state to which it had evolved prior to t he measurement.  
+ 
+Zurek’s resolution is to show that the two problem axioms are implied by the other three  
+(Zurek, 2009) .  The proof of this derivation reveals that much of the information 
+contained in the state vector after a period of unitary evolution cannot survive its being 
+copied into the environment. Only a subset can survive the transfer. Axiom 3 requires 
+that the information that survives in the systems state vector after the transfer is 
+consistent with that cop ied to the environment resulting in a seemingly discontinuous 
+change in the state vector.  
+ 
+Theoretically decoherence has been understood as consisting of two processes roughly 
+equivalent to axioms 4 and 5  (Zurek, 2009) . The first process  is named ‘environment 
+induced superselection ’ or einselection. When a quantum system becomes entangled 
+with an entity in its environment it is properties of the environmental entity that 
+determine the type of information which can be copie d from the quantum system  into 
+that environment. For instance some environment s may be structured to receive 
+information on position and others to receive information on momentum. 
+Mathematically t he subset of information that can survive transfer to an env ironment is 
+restricted to pointer states of the system. These pointer states are the eigenstates of 
+axiom 4 and may be predicted as they are those  which will leave the system in the lowest 
+entropy state available  (Zurek, 200 9). Conversely the ability of the environment to 
+receive information from the quantum system is de pendent on  its ability to increase 
+entropy  (Zwolak, Quan, & Zurek, 2009) . 
+ The second major simplification of quantum theory produced by Zurek’s program was 
+the derivation of axiom 5 using symmetry properties of the entangled quantum system 
+and environmental entities. This process, environment –assisted invariance  or 
+envariance, provides  a probabilistic prediction of the measurement value .  
+ 
+Importantly both of these simplifications provide predictions regardin g measurement; 
+they predict  the type of information that will be measured and the value of the 
+measurement. Given Zurek’s d emonstration that, i n quantum processes , information is 
+physi cal and that  there is no information without  representation  (Zurek, 1998 )  and that 
+axiom 1  posits  that the state of the quantum system is represented by the Hilbert s tate 
+vector , we are compelled to conclude that the quantum system must contain a physical 
+representation of its state vector. Crucially, this physical representation may be 
+considered an internal model residing within the quantum system and this model, 
+through the measurement predictions it contains , simulates  the quantum system’s 
+interactions with its environment.  
+ 
+Recently Zurek’s research has focused on the fate of quantum information that has been 
+copied into its environment . His major finding is that t he information best able to 
+survive the process of decoherence is also the information having the highest 
+reproductive success in the environment , thereby  causing it to be come  the most 
+widespread. This principle, along with decoherence, is encapsulated by the theory of 
+Quantum Darwinism.  
+III. Quantum Darwinism as a Darwinian process  
+Given these brief summaries of Darwinian processes and of Quantum Darwinism we are 
+in a position  to evaluate the extent to which Quantum Darwinism might be considered a 
+Darwinian process. Clearly, in Zurek’s view, there is little question o f the Darwinian 
+nature of  quantum processes or of its central importance; he sees Quantum Darwinism 
+as conformin g to the  Darwinian paradigm and identifies it as the mechanism 
+responsible for the emergence of classical reality from the quantum substrate: (Zurek, 
+2004)  
+ 
+The aim of this paper is to show that the emergence of the classical reality can 
+be viewed as a res ult of the emergence of the preferred states from within the 
+quantum substrate thorough the Darwinian paradigm, once the survival of the 
+fittest quantum states and selective proliferation of the information about them 
+are properly taken into account.  
+ 
+The Darwinian paradigm, as defined within Universal Darwinism, encompasses  details 
+in addition to survival of the fittest and selective proliferation. The following  evaluation 
+will consist of a comparison of four characteristics of Darwinian processes (listed at the 
+end of section 1 ) with the characteristic s of quantum systems as described by  Quantum 
+Darwinism.  
+ 
+1. The operation of a three step ‘copy with selective retention’ algorithm.   
+ Decoherence essentially describes a process by which information is copied or 
+transferred from a quantum system to its environment. Much of the varied 
+information contained in the state vector is copied but most has extremely short 
+period s of survival. The fittest quantum states, or pointer states, are selected  
+(Zurek, 1998 ) . The selected quantum states are those capable of the greatest 
+reproductive success  (Zurek, 2004) . Clearly Quantum Darwinism is formulated 
+in a manner consistent with the three step algorithm of a Darwinian process.  
+ 
+2. Operation as an adaptive system where the system, the effect of the 
+environment on the system and the effect of the system on the 
+environment are modeled internally and the system conforms to 
+Friston’s free energy principle. The simulation of the system’s 
+environmental interactions may be understo od as computation.  
+ 
+Axiom 1 of quantum theory tells us that all information concerning a quantum 
+system is contained within the state vector making it an excellent candidate for 
+an internal model.  As there can be no information without representation this 
+information must physically reside within the quantum system.  Axiom 2 tells us 
+that this state vector, and the information it contains, will evolve according to the 
+Schrödinger equation:  
+ 
+ 
+   
+Where ψ x,t is the state  vector  and H  is the Hamiltonian  operator . 
+ 
+As the Hamiltonian operator encapsulates  internal effects of the system, the 
+effect of the system on the environment and the effects of the environment on the 
+system this model of the quantu m system meets the criteria of an internal model 
+operating within an adaptive system.  
+ 
+Further , we can consider  Friston’s free energy principle to be  satisfied by this 
+adaptive system if we  note that  Zurek has shown the measurement prediction s of 
+quantum t heory are implied by the first three axioms. As the predictions of 
+quantum theory are amongst the most accurate in science we might conclude that 
+the prediction error or free energy between the model and the system’s actual 
+experiences  is minimized.  
+ 
+In Seth Lloyd’s  (2007)  treatment of quantum computation the interactions of 
+quantum systems are interpreted  as quantum computations. The result of the 
+computation is the outcome of the interaction. In this sense the state vector 
+precisely predicts the outcome and the prediction error is zero. We note in this 
+context that the noiseless subsystems by which a quantum computation comes to 
+its answer have been shown to be mathematically isomorphic with Zurek’s 
+process of einselection  (Blum e-Kohout, 2008) . Given the isomorphism between 
+these two processes we are entitled to view quantum computation as a process 
+where the ‘answer’ to a computation emerges via a Darwinian process.  
+ 
+Given Lloyd’s interpretation that  outcomes of  quantum  interaction s are 
+equivalent to simulations carried out by the quantum system  and the fact that all 
+physical interactions are quantum interactions we are led to the Church -Turing -
+Deutsch principle which states that e very finitely realizable physical system  can 
+be perfectly simulated by a quantum computation.  
+ 
+A quantum system , as described by Zurek’s theory,  meets the criteria of an 
+adaptive system. The internal model formed by a representation of the state 
+vector integrates information concerning the syste m, the effect of the 
+environment on the system and the effect of the system on the environment. It 
+also operates in accordance with Friston’s free energy principle for adaptive 
+systems.  The state vector, considered as an  internal model , may be interpreted as 
+performing computational simulations of the system’s environmental 
+interactions.  
+ 
+3. Maintenance of the internal model’s accuracy through Bayesian 
+updating.  
+ 
+ In order for an internal model of an adaptive system to reduce its prediction 
+error it must accum ulate knowledge  through inference  from  environment al data .  
+It has been demonstrated that there is a single method of conducting inference , 
+and that is Bayes' Theorem , derived from the basic sum and product rule of 
+Bayesian probability  (Jaynes, 1985) .  Bayes’ Theorem provides a method of 
+calculating how confidence in a hypothesis or a model should be updated when 
+new data becomes available:  
+𝐏 𝐇 𝐃𝐗 =𝐏(𝐇|𝐗)𝐏(𝐃|𝐇𝐗)
+𝐏(𝐃|𝐗) 
+Where we are considering the Probability of Hypothesis (H) given prior data 
+(X) and new data (D)  
+ 
+In other words we should update our probability for the hypothesis or model 
+being true by  the extent to which the hypothesis enhances our ability to 
+predict the new data.  
+ 
+What was formerly axiom 4 of quantum theory reveals that every possible 
+measurable state of the quantum system is predicted by the system state. 
+The correct weighting that should be given to each possible outcome is given 
+by Born’s rule. With the occurrenc e of a measurement of the system the state 
+vector is updated. The probabilistic weightings for the various outcome 
+hypotheses  are reassigned in a Bayesian manner in accordance with axiom 3 
+and the hypothesis associated with a repeat of the previous measure ment 
+result is assigned probability one and all other hypothesis are assigned 
+probability zero.  In other words our confidence as to the state of a quantum system immediately upon receiving pertinent data is strengthened to 
+certainty.  
+ 
+We can conclude that the internal model of quantum systems undergoes 
+Bayesian updating through the system’s interactions with its environment.  
+ 
+4. Production of adaptations which utilize the principle of maximum 
+entropy to maintain the system’s low entropy state.   
+ 
+Zurek’s progra m has shown that the interaction outcomes involving quantum 
+systems can be predicted by a method named ‘predictability sieve ’ (Zurek, 2003) . 
+This method essentially ranks all potential outcomes of the state vector according 
+to their entropy production. Those outcomes that should be predicted  (in 
+accordance with former axiom 4 ) are those having the lowest entropy.  This 
+ability of a system to find rare low entropy states which are also reproductively 
+successful is a hallmark of Darwinian processes.  
+ 
+As prevalence towards  decoherence is strongly influenced by mass , relatively 
+massive quantum systems such as those considered by atomic physics and 
+chemistry are almost  continuously decohered by their environments and the ir 
+existence assumes what Zurek describes as  a classical trajectory. This trajectory is 
+characterized by stability, predictability an d reproductive success , and should be 
+understood as a near continuous ‘copy with selective retention’ process which 
+unremit tingly prob es potential environmental interactions for low entropy 
+solutions and instantiat es those solutions once found.  The system’s low entropy 
+state, maintained by these interactions is balanced by an increase in 
+environmental entropy  (Zwolak, Quan, & Zurek, 2009) .  
+ 
+The evolution of these intricate quantum systems, which may be composed of 
+many ‘fundamental’ particles, is governed by a Hamiltonian operator  which 
+models  the complex interactions between the system and its en vironment. The 
+principle of maximum entropy requires that t he ability of these low entropy 
+structures to have a well advertised and persistent presence within their classical 
+environments be explained in terms of  the system’s ability to impose constraints 
+on the increas e of entropy. These constraints might be considered adaptations 
+and appear as scientific laws including those governing atomic orbitals and 
+chemical bonds .  
+ 
+Quantum systems evolve by continuously probing their environments to discover 
+and in stantiate rare low entropy outcomes  at the expense of increasing 
+environmental entropy . The variations adopted by the quantum system in its 
+evolution may be considered adaptations and function to provide constraints 
+against increasing entropy. M uch of the subject matter of atomic physics and 
+chemistry can be considered  adaptations discovered and instantiated through a 
+Darwinian process.  
+ In this perhaps radical view quantum systems emerge into classical reality and evolve 
+there through a successi on of information transfers to the environment. Each successive 
+generation of offspring information contain variations. The variations which survive and 
+are accumulated over generations are those that succeed in minimizing system entropy.   
+In a suitable e nvironment a quantum system may accumulate atomic and molecular 
+adaptations and function as a relatively complex composite system. Adaptive systems 
+are characterized by internal models which simulate and orchestrate their 
+environmental interactions. Quantu m theory  tells us that quantum system s must 
+include  a physical  implementation of an information  processing model equivalent to the 
+evolution of a state vector in Hilbert space.  While the nature of this model remains 
+outside of experimental verification we  are left in a similar situation to biology prior to 
+1953 when many characteristics of life’s internal model w ere known  and predictions 
+could be made by invoking laws such as Mendelian inheritance  but the manner o f the 
+model’s  implementation  in molecular DNA was not  known .  
+ 
+This evaluation indicates that quantum systems as described by Quantum Darwinism  
+meet the listed criteria and  may well be considered Darwinian process es as defined 
+within Universal Darwinism.  
+ 
+IV. Interpretational  implications of Quantu m Darwinism  
+Zurek’s research program, encapsulated by the theory of Quantum Darwinism, offers a 
+detailed explanation of the relationship between classical and quantum reality. In th e 
+author’s opinion it provides the greatest theoretical improvement in our understanding 
+of this relationship since Bohr’s development of the correspondence principle in 1920.  
+Einstein engaged Bohr i n a series of debates claiming that quantum theory must be 
+considered incomplete as long as it was unable to explain more clearly th e nature of 
+ontological reality , which for Einstein was classical reality. Philosopher C .P. Snow 
+described these debates: ‘No more profound intellectual debate has ever been 
+conducted’  (Isaacson, 2008) . We might judge the importance of Zurek’s program by 
+considering that the explanation s it offer s would likely have satisf ied both Bohr and 
+Einstein.  
+ 
+The major interpretational issues for quantum theory raised by Quantum Darwinism 
+result from its simplification and clarification of the subject through reducing the 
+number of axioms required for its foundation . This radical reshaping of the foundations 
+of quantum theory leaves most predictions intact  but provides a much more thorough 
+explanation of the processes in volved.  
+ 
+Axiom 4 of quantum theory has now been explained as due to the process of 
+einselection. In the course of developing this explanation a number of further important 
+mechanisms have been identified . One of these is ‘ predictability sieve ’ which clarif ies the 
+ability of quantum systems to probe their environment for low entropy states which 
+have  potential  existence  and to instantiate th ose states once found.  
+ 
+ Axiom 5 of quantum theory has now been derived through the process of envariance, 
+which produc es Born’s rule.  Crucially, taken together, einselection and envariance demonstrate that the measurement predictions of quantum theory are implied by the 
+remaining three axioms of the theory. Zurek’s principle of ‘no information without 
+representation ’ leads us to conclude that sufficient information for the predictions of 
+quantum theory resides in some physical form  within the quantum system.  
+ 
+The emergence of classical reality from the quantum substrate, the topic often most fully 
+informed by two seemingl y arbitrary axiom s, has been explained in detail.  
+ 
+Hopefully this treatment  will finally lay to rest the interpretational confusion around the 
+role of a human observer in quantum measurements that has been prevalent in many 
+treatments and taken to anthropo morphic extremes by some such as Wigner  (1970) . 
+Zurek’s work makes it clear that decoherence takes place whenever there is an 
+information transfer to the environment  (Zurek, 2009) . No human observer need be in 
+attendance.  
+ 
+The word measurement currently contains  an anthropomorphic component but it is also 
+much more nimble in usage than are many of the alternatives such as ‘monitoring by the 
+environment’.  The many variants  or affixes  of the root ‘ measure ’ such as measurement , 
+measurable, measuring etc. make the word attractive for use in explanations.  A possible 
+resolution might be to create an additional scientific definition of measurement based 
+on decoherence  and devoid of references to a human observer . 
+ 
+Conclusion  
+Given the evaluation that Quantum Darwinism is a theory utilizing a Darwinian process 
+and thus a component of Universal Darwinism we find Universal Darwinism extended 
+beyond theories in biology and the social sciences to contain  explanatory theories for 
+particle  physics, atomic physics and chemistry. Universal Darwinism may thus be seen 
+as a paradigm offering a single comprehensive explanation of much of scientific reality.  
+ 
+David Deutsch’s ground breaking work The Fabric of Reality  (Deutsch, 1997)  predicted  
+that science would soon produce ‘a theory of all subjects: a Theory of Everything’. This 
+theory would provide a single paradigm explaining scientific subject matter across all 
+subjects. He ventured to provide details concernin g four explanatory  strands  which the 
+theory might integrate : 
+1) Quantum Physics  
+2) Epistemology  
+3) The theory of computation  
+4) The theory of Darwinian evolution  
+ 
+I suggest that this discussion  has shown the 1st and 4th of these strands to be integrated 
+within the theory of Universal Darwinism.  
+ 
+Epistemology referred to by Deutsch  as the 2nd strand  is the philosophy of science. 
+Evolutionary epistemology, a branch of philosophy within the philosophy of science , 
+contains a number of well-established theories that ascribe  the evolution  of science  to the operation of Darwinian processes  (D. Campbell 1965; J. Campbell, 2009; Popper, 
+1972)  and thus Deutsch’s 2nd strand is integrated within Universal Darwinism . 
+ 
+Deutsch has argued that computability is a property bestowed by the natural world not 
+by logic or mathematics , and that those functions that are computable are those for 
+which nature has designed the appropriate computational machinery. The most basic 
+computational machinery so far found in nature is that allowing quantum computation. 
+This paper makes arguments that quantum computation may be viewed as the 
+simulations of an adaptive system and that the ‘answers’ to a quantum computation 
+emerge through a Darwinian process.  These are arguments that De utsch’s 3rd strand is 
+integrated within Universal Darwinism.  
+ 
+As all physical interactions are quantum interactions we might expect quantum 
+computations to be utilized within other adaptive systems. Some evidence is 
+accumulating  in support of this notion . It has been shown  that photosynthesis may 
+utilize quantum computations to calculate  the most efficient chemical pathways for the 
+conversion of energy contained within a specific photon  (Fleming et al., 2007) . 
+ 
+The theory of Universal Darwinism, as it int egrates Deutsch’s four strands within a 
+single explanatory paradigm, may serve as a candidate for the ‘Theory of Everything’ as 
+anticipated by Deutsch.  
+ 
+References  
+Blackmore, S. (1995). The Meme Machine.  Oxford: Oxford University Press.  
+Blume -Kohout, R. a. (2008). Characterizing the Structure of Preserved Infromation in 
+Quantum Processes. Physical Review Letters, 100 (3), Art. No. 030501  . 
+Brenner, S. (1999). Theoretical biology in the third millennium. Philos Trans R Soc 
+Lond B Biol Sci. December 29; 354(1392)  , 1963 –1965.  
+Campbell, D. (1965). Variation and selective retention in socio -cultural evolution. In R. 
+B. G.I. Herbert, Social change in developing areas: A reinterpretation of evolutionary 
+theory  (pp. 19 -49). Cambridge, Mass.  
+Campbell, J. (2009). Bayesian Methods and Universal Darwinism. Bayesian Methods 
+and Maximum Entropy in Science and Technology (In Press).   
+Dawkins, R. (1976). The Selfish Gene.  Oxford: Oxford University Press.  
+Dennett, D. (1995 ). Darwin's Dangerous Idea.  New York: Touchstone Publishing.  
+Deutsch, D. (1985). Quantum theory, the Church -Turing principle and the universal 
+quantum computer. Proceedings of the Royal Society of London A 400  , 97-117. 
+Deutsch, D. (1997). The Fabric of Re ality.  London: The Penguin Press.  
+Fleming, G. R., Blankenship, R. E., Engel, G. S., Calhoun, T. R., Read, E. L., Ahn, T. -K., 
+et al. (2007). Evidence for wavelike energy transfer through quantum coherence in 
+photosynthetic systems. Nature: Zoology and wildl ife conservation  . 
+Friston, K. (2007). Free Energy and the Brain. Synthese , 159  , 417 -458.  
+Green, B. (1999). The Elegant Universe.  New York: W.W. Norton & Company Inc.  
+Isaacson, W. (2008). Einstein: His Life and Universe.  Simon and Schuster.  Jaynes, E. (1 985). Bayesian Methods: General Background. In E. J.H. Justice, 
+Maximum Entropy and Bayesian Methods in Applied Statistics (Proceedings Volume)  
+(pp. 1 -25). Cambridge University Press.  
+Lloyd, S. (2007). Computing the Universe.  Vintage; Reprint edition.  
+Mark opoulou, F., & Smolin, L. (2004). Quantum Theory from Quantum Gravity . 
+Phys.Rev. D70 (2004) 124029 . 
+Plotkin, H. C. (1993). Darwin Machines.  Cambridge, Massachusetts: Harvard 
+University Press.  
+Popper, K. (1972). Objective Knowledge.  Claredon Press.  
+Smolin, L. (2001). Three Roads to Quantum Gravity.  New Youk: Basic Books.  
+Wigner, E. (1970). Symmetries and Reflections: Scientific Essays.  MIT Press.  
+Zurek, W. H. (1998 ). Decoherence, Einselection and the Existential Interpretation (the 
+Rough Guide). Philosophical Transactions: Mathematical, Physical and Engineering 
+Sciences  , 1793 -1821.  
+Zurek, W. H. (2003). Decoherence, einselection, and the quantum origins of the 
+classical. Review of Modern Physics, 75, 715  . 
+Zurek, W. H. (2009). Quantum Darwinism. Natu re Physics, vol. 5  , 181 -188.  
+Zurek, W. H. (2004). Quantum Darwinism and Envariance. In P. C. J. D. Barrow, 
+Science and Ultimate Reality: From Quantum to Cosmos.  Cambridge University Press.  
+Zwolak, M., Quan, H. T., &  Zurek, W. H. (2009). Quantum Darwinism in non -ideal 
+environments. Preprint; arXiv:0911.4307  . 
+ 
+ 
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+arXiv:1001.0746v3  [cs.CC]  3 Feb 2010Symposium on TheoreticalAspects of Computer Science 2010( Nancy, France), pp. 669-680
+www.stacs-conf.org
+ALTERNATION-TRADING PROOFS, LINEAR PROGRAMMING, AND
+LOWER BOUNDS
+(EXTENDED ABSTRACT)
+RYAN WILLIAMS1
+1IBM Almaden Research Center
+650 Harry Road, San Jose, CA, USA 95120
+E-mail address :ryanwill@us.ibm.com
+URL:http://www.cs.cmu.edu/ ryanw/
+Abstract. A fertile area of recent research has demonstrated concrete polynomial time
+lowerboundsforsolvingnaturalhardproblemsonrestricte dcomputationalmodels. Among
+these problems are Satisfiability, Vertex Cover, Hamilton P ath, MOD 6-SAT, Majority-of-
+Majority-SAT, and Tautologies, to name a few. The proofs of t hese lower bounds follow
+a certain proof-by-contradiction strategy that we call alternation-trading . An important
+open problem is to determine how powerful such proofs can pos sibly be.
+We propose a methodology for studyingthese proofs that make s them amenable toboth
+formal analysis and automated theorem proving. We prove tha t the search for better lower
+bounds can often be turned into a problem of solving a large se ries of linear programming
+instances. Implementing a small-scale theorem prover base d on this result, we extract new
+human-readable time lower bounds for several problems. Thi s framework can also be used
+to prove concrete limitations on the current techniques.
+1. Introduction
+Many known lower bounds for natural problems follow a type of algorithmic argument
+that we call a resource-trading proof . Such a proof assumes that a hard problem canbe
+solved by a “good” algorithm, and tries to derive a contradic tion by combining two essential
+components. One is a speedup lemma , which simulates all good algorithms super-efficiently
+on some “interesting” computational model, trading time fo r some resource. The second
+component is a slowdown lemma , which uses the assumed good algorithm for the hard
+problemtosimulatecomputationsfromthe“interesting” mo delbygoodalgorithms, thereby
+trading the “interesting” resource for more time. Clever co mbinations of speedup and
+slowdown lemmas are used to contradict a known result, in par ticular some complexity
+1998 ACM Subject Classification: F.2.3, I.2.3.
+Key words and phrases: time-space tradeoffs, lower bounds, alternation, linear pr ogramming.
+This material is based on work supported in part by NSF grant C CR-0122581 while the author was a
+student at Carnegie Mellon University, and NSF grant CCF-08 32797 while the author was a member of the
+Institute for Advanced Study.
+c/circlecop†rtR.R. Williams
+CC/circlecopyrtCreativeCommonsAttribution-NoDerivsLicense670 R. R. WILLIAMS
+hierarchy theorem. That is, by assuming a “good” algorithm f or a hard problem, we derive
+something like TIME[n2]⊆TIME[n], a contradiction.
+As an example, one can prove a time-space tradeoff for satisfia bility (SAT) as follows.
+Assume SAT has an algorithm running in nctime and poly(log n) space, for some c >1.
+One speedup lemma is that computations running in natime and poly(log n) space can
+be simulated by an alternating machine that switches from co-nondeterministic mode to
+nondeterministic mode once (i.e., a Π2machine), and runs in na/2+o(1)time. This speedup
+lemmatrades time for alternations . The relevant slowdown lemma is: if SAT has an nc
+time, poly(log n) space algorithm, then (by a strengthening of the Cook-Levi n theorem)
+every language in NTIME[t] hastc+o(1)time, poly(log t) space algorithms. Consequently,
+an alternating machine running in ttime and making k−1 alternations has tck+o(1)time,
+poly(logt) space algorithms. Combining these speedup and slowdown le mmas, we derive
+Σ2TIME[t]⊆DTISP[tc2+o(1),poly(logt)]⊆Π2TIME[tc2/2],
+where the first inclusion holds by slowdown and the second hol ds by speedup. Now observe
+that the alternating time hierarchy is contradicted when c2<2. This proof is the n√
+2−ε
+time lower bound of Lipton and Viglas [LV99].
+Some of the best known separations in complexity theory use r esource-trading proofs.
+Hopcroft, Paul, and Valiant [HPV77] showed that SPACE[n]/notsubseteqlDTIME[o(nlogn)] for mul-
+titape Turing machines, by proving the “speedup lemma” that DTIME[t]⊆SPACE[t/logt]
+and invoking diagonalization. Their result was later exten ded to general models [PR81,
+HLMW86]. Paul, Pippenger, Szemeredi, and Trotter [PPST83] proved that NTIME[n]/ne}ationslash=
+DTIME[n] for multitape Turing machines. The key component in the pro of is the “speedup
+lemma”DTIME[t]⊆Σ4TIME[t/log∗t] for multitape TMs. Despite their age, the above sep-
+arations still constitute the best known progress on PvsPSPACE andPvsNP, respectively.
+In more recent years, resource-trading proofs have establi shed time-space lower bounds
+forNP-complete problems and problems higher in the polynomial hi erarchy [Kan84, For97,
+LV99, FvM00, FLvMV05, Wil06, Wil08]. For instance, thebest knowntimelower boundfor
+solving SATwith no(1)-space algorithms is n2cos(π/7)−o(1)≥n1.801, obtained witharesource-
+trading proof [Wil08]. (Note if one could improve the 1 .801 exponent to arbitrary constants,
+onewouldseparate LOGSPACE fromNP.) Fornondeterministicalgorithmsusing no(1)space,
+the best known time lower bound for solving the coNP-complete Tautology problem
+wasn√
+2−o(1)for several years [FvM00]. Certain time-space lower bounds for probabilistic
+and quantum computations also follow the resource-trading paradigm [AKRRV01, DvM06,
+Vio09, vMW07]. Resource-trading proofs are also abound in t he multidimensional “hybrid”
+Turing machine model, which has read-only random access to i ts input and an no(1)read-
+write store, as well as read-write two-way access to a d-dimensional tape for some d≥1.
+This is the most powerful (and physically realistic) model k nown where we still know non-
+trivial time lower bounds for problems such as SAT. Multidim ensional TMs have a long
+history; e.g., [Lou80, PR81, Kan83, MS87, vMR05, Wil06] pro ved lower bounds for them.
+(For a more complete literature review, please see the full v ersion of the paper.)
+1.1. Main Results
+We introduce a methodology for reasoning about resource-tr ading proofs that is also
+practically implementable for finding short proofs. Inform ally, the “hard work” in theseALTERNATION-TRADING PROOFS, LINEAR PROGRAMMING, AND LOWE R BOUNDS 671
+proofs can be replaced by solving a series of linear programm ing problems. This perspective
+not only aids us practically in the search for new lower bound s, but also allows us to show
+non-trivial limitations on what can be proved.
+This methodology is applied to several lower bound problems . In all cases considered
+here, the resource being “traded” is alternations, so we cal l the proofs alternation-trading .
+Deterministic Time-Space Lower Bounds. Aided by results of acomputer program, weshow
+thatanySATalgorithmrunningin t(n)timeand s(n)spacesatisfies t·s≥Ω(n2cos(π/7)−o(1)).
+Previously, the best known result was t·s≥Ω(n1.573) [FLvMV05]. It has been conjectured
+that the current framework sufficed to prove a n2−o(1)time lower bound for SAT, against
+algorithms using no(1)space. We prove that it is not possible to obtain n2with the frame-
+work, formalizing a conjecture of [FLvMV05].∗A computer search over proofs of short
+length suggests that the best known n2cos(π/7)−o(1)lower bound [Wil08] is already optimal
+for the framework. We also prove lower boundson QBFk(quantified Boolean formulas with
+at mostkquantifier blocks), showing that the problem requires Ω( nk+1−δk) time for no(1)
+space algorithms, where δk<0.2 and lim k→∞δk= 0.†
+Nondeterministic Time-Space Lower Bounds. Adapting our ideas to proving lower bounds
+forTautologies , a computer program found a very short proof improving upon F ortnow
+and Van Melkebeek’s lower bound. Longer proofs suggested an interesting pattern. Joint
+work with Diehl and Van Melkebeek on this observation result ed in an n41/3−o(1)≥n1.587
+time lower bound [DvMW09]. Computer search suggests that th is lower bound is best
+possible for the framework. We prove that it is not possible t o obtain an nφtime lower
+bound, where φ= 1.618...is the golden ratio. This is surprising since we have known fo r
+some time that an nφlower bound isprovable for deterministic algorithms [FvM00].
+Multidimensional Turing Machine Lower Bounds. Here our method uncovers peculiar be-
+havior in the best lower bound proofs, regardless of the dime nsion. Studying computer
+search results, we extract an Ω( nrd) time lower bound for the d-dimensional case, where
+rd≥1 is the root of a particular quintic pd(x) with coefficients depending on d. For ex-
+ample,r1≈1.3009,r2≈1.1887, and r3≈1.1372. Again, our search suggests this is best
+possible, and we can prove it is not possible to improve the bo und ford-dimensional TMs
+ton1+1/(d+1)with the current tools.
+These limitations also hold for other NPandcoNP-hard problems; the only property
+required is that all languages in NTIME[n] (respectively, coNTIME [n]) have sufficiently ef-
+ficient reductions to the problem. Also our linear programmi ng approach is not limited to
+the above, and can be applied to the league of lower bounds dis cussed in Van Melkebeek’s
+surveys [vM04, vM07].
+1.2. Some Remarks on the Reduction to Linear Programming
+The key to our formulation is to separate the discrete choices in an alternation-trading
+proof from the real-valued choices . The discrete choices consist of the sequence of lemmas
+to apply in each step, and what sort of hierarchy theorem to us e in the contradiction. We
+present several simplifications that greatly reduce the num ber of discrete choices, without
+loss of generality. The real-valued choices are the running time exponents that arise from
+∗That is, we formalize the statement: “...some complexity th eorists feel that improving the golden ratio
+exponent beyond 2 would require a breakthrough” in Section 8 of [FLvMV05].
+†Note the QBFkresults appeared in the author’s PhD thesis in 2007 but have b een unpublished to date.672 R. R. WILLIAMS
+the choices of time bounds and rule applications. We prove th at once the discrete choices
+are made, the remaining real-valued problem can be expresse d as an instance of linear
+programming. This makes it possible to search for new proofs via computer, and it also
+gives us a formal handle on the limitations of these proofs.
+One cannot easily search over all possible proofs, as the num ber of discrete choices is still
+about 2n/n3/2for proofs of nlines (proportional to the nth Catalan number). Nevertheless
+it is still feasible to try all 24+ line proofs. These proof se arches reveal patterns, indicating
+that certain strategies will be most successful in proving l ower bounds; in each case we
+study, the resulting strategies differ. Following the strate gies, we establish new lower bound
+proofs. The patterns also suggest how to show limitations on the proof systems.
+Note:Due to space limitations, we can only describe how our method s apply to SAT time-
+space lower bounds. Please see the full version of the paper f or proofs and more details.
+2. Preliminaries
+We assume familiarity with Complexity Theory, especially t he notion of alternationWe
+usebig-Ωnotation intheinfinitelyoftensense, sostatemen ts like“SATisnotin O(nc)time”
+areequivalent to“SAT requiresΩ( nc) time.” All functionsareassumed constructiblewithin
+the appropriate bounds. Our default computational model is the random access machine,
+broadly construed: particular variants do not affect the resu lts.DTISP[t(n),s(n)] is the
+class of languages accepted by a RAM running in t(n) time and s(n) space, simultaneously.
+For convenience, we set DTS[t(n)] :=DTISP[t(n)1+o(1),no(1)] to omit negligible o(1) factors.
+In order to properly formalize alternation-trading proofs , we introduce notation for alter-
+nating complexity classes that include input constraints between alternations. Let us start
+with an example of the notation, then give a general definitio n. Define ( ∃f(n))bDTS[na] to
+be the class of languages recognized by a machine which, on an inputxof length n, writes
+af(n)1+o(1)bit string ynondeterministically, copies at most nb+o(1)bitszfrom the pair
+/an}bracketle{tx,y/an}bracketri}ht(inO(nb+o(1)) time), then feeds zas input to a machine Mrunning in na+o(1)time
+andno(1)space. Note the runtime of Mis measured with respect to the initial input length
+n, not the latter input length nb+o(1)ofz.
+We generalize this definition as follows. Let Cbe a complexity class. For i= 1,...,k, let
+Qi∈ {∃,∀}andai,bi≥0. Define
+(Q1na1)b2(Q2na2)···bk(Qknak)bk+1C
+to be the class of languages recognized by a machine Mthat, on input xof length n, has
+the following general behavior on input x:
+Setz0:=x.
+Fori= 1,...,k,
+IfQi=∃, switch to existential mode .
+IfQi=∀, switch to universal mode .
+Guess an nai+o(1)bit string y(universally or existentially).
+Copy at most nbi+1+o(1)bitszifrom the pair /an}bracketle{tzi−1,y/an}bracketri}ht.
+End for
+Run a machine recognizing a language in class Con the input zk.ALTERNATION-TRADING PROOFS, LINEAR PROGRAMMING, AND LOWE R BOUNDS 673
+When an input constraint biis unspecified, its default value is max {ai,1}. We say that
+the existential and universal modes of an alternating compu tation are quantifier blocks , to
+reflect the complexity class notation. It is crucial to obser ve that the time bound in the ith
+quantifier block is measured with respect to n, the input to the first quantifier block .
+Notice that by simple properties of nondeterminism and cono ndeterminism, we can com-
+bine adjacent quantifier blocks that are of the same type, e.g ., (∃na)a(∃nb)bDTS[nc] =
+(∃nmax{a,b})bDTS[nc]. This useful property is exploited in alternation-tradin g proofs.
+2.1. A Short Introduction to Alternation-Trading Proofs
+Here we give a brief overview of the tools used in alternation -trading proofs. In this ex-
+tended abstract we focus on deterministic time lower bounds for satisfiability for algorithms
+usingno(1)workspace; the other lower bound problems use similar tools .
+It is known that satisfiability of Boolean formulas in conjun ctive normal form (SAT) is
+a complete problem under tight reductions for a small nondet erministic complexity class.
+The class NQL, callednondeterministic quasilinear time , is defined as
+NQL:=/uniondisplay
+c≥0NTIME[n·(logn)c] =NTIME[n·poly(logn)].
+Theorem 2.1 ([Coo88, Sch78, Tou01, FLvMV05]) .SAT isNQL-complete under quasilin-
+ear time O(logn)space reductions, for both multitape and random access mach ine models.
+Moreover, each bit of the reduction can be computed in O(poly(logn))time and O(logn)
+space in both machine models.‡
+LetC[t(n)] represent a time t(n) complexity class under one of the three models:
+•deterministic RAM using time tandto(1)space,
+•co-nondeterministic RAM using time tandto(1)space,
+•d-dimensional Turing machine using time t.
+Theorem 2.1 implies that if NTIME[n]/notsubseteqlC[t], then SAT /∈ C[t/poly(log t)].
+Corollary 2.2. IfNTIME[n]/notsubseteqlC[t(n)], then SAT /∈ C[t(n)/logkt(n)]for some k >0.
+Hence we wish to prove NTIME[n]/notsubseteqlC[nc] for large c >1. To prove time-space lower
+bounds,weworkwith C[nc] =DTS[nc] =DTISP[nc,no(1)]. VanMelkebeekandRaz[vMR05]
+observed that a similar corollary holds for any problem Π suc h that SAT reduces to Π under
+highly efficient reductions, e.g.Vertex Cover ,Hamilton Path ,3-SAT, andMax-2-
+Sat. Therefore similar time lower bounds hold for these problem s as well.
+Speedups, Slowdowns, and Contradictions. Now that ourgoal is toprove NTIME[n]/notsubseteql
+DTS[nc], how can we do this? In an alternation-trading proof, we att empt to establish a
+contradiction from assuming NTIME[n]⊆DTS[nc], by applying two lemmas which comple-
+ment one another. A speedup lemma takes aDTS[t] class and places it in an alternating
+class with runtime o(t). Aslowdown lemma takes an alternating class with runtime tand
+places it in a class with one less alternation and runtime O(tc). The Speedup Lemma dates
+back to Nepomnjascii [Nep70] and Kannan [Kan84].
+‡In the multitape Turing machine model we assume that the tape heads are already oriented on the
+appropriate cells, otherwise it may take linear time to find t he appropriate cells on a tape.674 R. R. WILLIAMS
+Lemma 2.3 (Speedup Lemma) .Leta≥1,e≥0and0≤x≤a. Then
+DTISP[na,ne]⊆(Q1nx+e)max{1,x+e}(Q2logn)max{1,e}DTISP[na−x,ne],
+forQi∈ {∃,∀}whereQ1/ne}ationslash=Q2. In particular,
+DTS[na]⊆(Q1nx)max{1,x}(Q2logn)1DTS[na−x].
+Proof.LetMusenatime and nespace. Let ybe an input of length n. A complete
+description (i.e. configuration ) ofM(y) at any step can be described in O(ne+logn) space.
+To simulate Min (∃nx+e)max{1,x+e}(∀logn)max{1,e}DTISP[na−x,ne], the algorithm N(y)
+existentially guesses a sequence of configurations C1,...,C nxofM(x). ThenN(y) appends
+the initial configuration C0ofM(y) to the beginning of the sequence, and an accepting
+configuration Cnx+1to the end. N(y) universally guesses a i∈ {0,...,nx}, erases all
+configurations except CiandCi+1, then simulates M(y) starting from Ci, accepting if and
+only ifCi+1is reached within na−xsteps. It is easy to see the simulation is correct. The
+input constraints on the quantifier blocks are satisfied sinc e after the universal guess, the
+input is only y,Ci, andCi+1, which is of size n+2ne+o(1)≤nmax{1,e}+o(1).
+Observe in the above alternating simulation, the input to th e finalDTISPcomputation
+is linear in n+ne,regardless of the choice of x. This is a subtle property that is exploited
+heavily in alternation-trading proofs. The Slowdown Lemma is the following simple result:
+Lemma 2.4 (Slowdown Lemma) .Leta≥1,e≥0,a′≥0, andb≥1. IfNTIME[n]⊆
+DTISP[nc,ne], then for both Q∈ {∃,∀},
+(Q na′)bDTIME[na]⊆DTISP[nc·max{a,a′,b},ne·max{a,a′,b}].
+In particular, if NTIME[n]⊆DTS[nc], then
+(Q na′)bDTIME[na]⊆DTS[nc·max{a,a′,b}].
+Proof.LetLbe a problem in ( Q na′)bDTIME[na], and let Abe an algorithm recognizing
+L. On an input xof length n,Aguesses a string yof length na′+o(1)and feeds an nb+o(1)
+bit string ztoA′(z), where A′is a deterministic algorithm that runs in natime. Since
+NTIME[n]⊆DTISP[nc,ne] andDTISPis closed under complement, by padding we have
+NTIME[p(n)]∪coNTIME [p(n)]⊆DTISP[p(n)c,p(n)e] for polynomials p(n)≥n. Therefore
+Acan be simulated with a deterministic algorithm B. Since the runtime of Aisna′+o(1)+
+nb+o(1)+na, the runtime of Bisnc·max{a,a′,b}+o(1)and the space usage is similar.
+Thefinal component of an alternation-trading proof is a time hierarchy theorem, the most
+general of which is the following, provable by a simple diago nalization.
+Theorem 2.5 (Alternating Time Hierarchy) .Fork≥0, for all Qi∈ {∃,∀},1≤a′
+i< ai,
+and1≤b′
+i≤bi,
+(Q1na1)b2···bk(Qknak)bk+1DTS[nak+1]/notsubseteql(R1na′
+1)b′
+2···b′
+k(Rkna′
+k)b′
+k+1DTS[na′
+k+1],
+whereRi∈ {∃,∀}andRi/ne}ationslash=Qifor alli= 2,...,k+1.
+Two Examples. Letusgiveacoupleofexamplesofalternation-tradingproo fs. Tosimplify
+the presentation we do not specify the input constraints to q uantifiers in the below.ALTERNATION-TRADING PROOFS, LINEAR PROGRAMMING, AND LOWE R BOUNDS 675
+(1) In FOCS’99, Lipton and Viglas proved that SAT cannot be so lved by algorithms
+running in n√
+2−εtime and no(1)space, for all ε >0. By Theorem 2.1, if SAT is in n√
+2−ε
+time and no(1)space then NTIME[n]⊆DTS[nc] withc2<2. We have
+(∃n2/c2)(∀n2/c2)DTS[n2/c2]⊆(∃n2/c2)DTS[n2/c] (Slowdown Lemma)
+⊆DTS[n2] (Slowdown Lemma)
+⊆(∀n)(∃logn)DTS[n].(Speedup Lemma, with x= 1)
+But (∃n2/c2)(∀n2/c2)DTS[n2/c2]⊆(∀n)(∃logn)DTS[n] contradicts Theorem 2.5. In
+fact, one can show that if c2= 2, we still have a contradiction with NTIME[n]⊆DTS[nc],
+so theεcan be removed from the previous statement and state that SAT cannot be solved
+inn√
+2time and no(1)exactly.§
+(2) Improving on the previous example, one can show SAT /∈DTS[n1.6004]. IfNTIME[n]⊆
+DTS[nc] and√
+2≤c <2, then applyingtheSpeedupand Slowdown Lemmas onecan deri ve:
+DTS[nc2/2+2]⊆(∃nc2/2)(∀logn)DTS[n2] (Speedup)
+⊆(∃nc2/2)(∀logn)(∀n)(∃logn)DTS[n] (Speedup)
+= (∃nc2/2)(∀n)(∃logn)DTS[n] (Combining ∀Quantifiers)
+⊆(∃nc2/2)(∀n)DTS[nc] (Slowdown)
+⊆(∃nc2/2)DTS[nc2] (Slowdown)
+⊆(∃nc2/2)(∃nc2/2)(∀logn)DTS[nc2/2] (Speedup)
+= (∃nc2/2)(∀logn)DTS[nc2/2] (Combining ∃Quantifiers)
+⊆(∃nc2/2)DTS[nc3/2] (Slowdown)
+⊆DTS[nc4/2] (Slowdown)
+Whenc2/2 + 2> c4/2 (which happens if c <1.6004), we have DTS[na]⊆DTS[na′] for
+somea > a′. One can show by a translation argument (similar to the footn ote) that either
+DTS[na]/notsubseteqlDTS[na′] orNTIME[n]/notsubseteqlDTS[nc], concluding the proof.
+Example (2) was discovered by a computer program. By “discov ered”, we mean that the
+program applied speedups and slowdowns in precisely the sam e way, having only minimum
+knowledge of the lemmas. Furthermore, the program verified t hat above is the best possible
+alternation-trading proof that applies the Speedup and Slo wdown Lemmas at most 7 times.
+A more formal definition of “alternation-trading proof” is g iven in the next section.
+3. Formalizing Alternation-Trading Proofs
+We formalize alternation-trading proofs of lower bounds on DTSclasses as follows:¶
+Definition 3.1. Letc >1. Analternation-trading proof for cis a list of complexity classes
+of the form:
+(Q1na1)b2(Q2na2)···bk(Qknak)bk+1DTS[nak+1], (3.1)
+§Suppose NTIME[n]⊆DTS[nc] andΣ2TIME[n]⊆Π2TIME[n1+o(1)]. The first assumption, along with the
+SpeedupandSlowdown Lemmas, implies thatfor every kthere’sa Ksatisfying Σ2TIME[nk]⊆NTIME[nkc]⊆
+ΣKTIME[n]. But the second assumption implies that ΣKTIME[n] =Σ2TIME[n1+o(1)]. Hence Σ2TIME[nk]⊆
+Σ2TIME[n1+o(1)], which contradicts the time hierarchy for Σ2TIME.
+¶Thisformalization has implicitly appearedinseveralpriorworks, butnottothedegreethatwe investigate
+in this paper.676 R. R. WILLIAMS
+wherek≥0,Qi∈ {∃,∀},Qi/ne}ationslash=Qi+1,ai>0, andbi≥1, for all i. (When k= 0, the class
+is deterministic.) The items of the list are called lines of the proof . Each line is obtained
+from the previous line by applying either a speedup rule or aslowdown rule . More precisely,
+if theith line is
+(Q1na1)b2(Q2na2)···bk(Qknak)bk+1DTS[nak+1],
+then the ( i+1)st line has one of four possible forms:
+Speedup Rule 0: Fork= 0 and any x∈(0,a1), (Q0nx)max{x,1}(Q1n0)1DTS[na1−x]./bardbl
+Speedup Rule 1: Fork >0 and any x∈(0,ak+1),
+(Q1na1)b2(Q2na2)···bk(Qknmax{ak,x})max{x,bk+1}(Qk+1n0)bk+1DTS[nak+1−x].
+Speedup Rule 2: Fork >0 and any x∈(0,ak+1),
+(Q1na1)b2···bk(Qknak)bk+1(Qk+1nx)max{x,bk+1}(Qk+2n0)bk+1DTS[nak+1−x].
+Slowdown Rule: Fork >0,
+(Q1na1)b2(Q2na2)···bk−1(Qk−1nak−1)bkDTS[nc·max{ak+1,ak,bk,bk+1}].
+An alternation-trading proof shows(NTIME[n]⊆DTS[nc] =⇒A1⊆A2) if its first line is
+A1and its last line is A2.
+Theabovedefinitioncomes directlyfromtheSpeedupLemma(L emma2.3)andSlowdown
+Lemma (Lemma 2.4). The rules are easily verified to be syntact ic formulations of the
+corresponding lemmas. For instance, Speedup Rule 1 holds, a s
+(Q1na1)b2(Q2na2)···bk(Qknak)bk+1DTS[nak+1]
+⊆(Q1na1)b2(Q2na2)···bk(Qknak)bk+1(Qknx)max{bk+1,x}(Qk+1n0)bk+1DTS[nak+1]
+⊆(Q1na1)b2(Q2na2)···bk(Qknmax{ak,x})max{bk+1,x}(Qk+1n0)bk+1DTS[nak+1].
+Rule 2 is akin to Rule 1, except that it uses opposite quantifie rs in its invocation of the
+Speedup Lemma. The Slowdown Rule works analogously to Lemma 2.4. It follows that
+alternation-trading proofs are sound.
+NoteSpeedupRules0and2addtwoquantifierblocks, SpeedupR ule1addsonequantifier,
+andall threerulesintroduceaparameter x. Byconsidering“normalform”proofs(definedin
+the following paragraphs), we can prove that Rule 2 can alway s be replaced by applications
+of Rule 1. (A proof is in the full version of the paper.) For thi s reason we just refer to the
+Speedup Rule , depending on which of Rule 0 or Rule 1 applies.
+Defineaclassoftheform(3.1)tobe simple. Defineclasses A1andA2tobecomplementary
+ifA1is the class of complements of languages in A2. Every known (model-independent)
+time-space lower bound for SAT shows “ NTIME[n]⊆DTS[nc] implies A1⊆A2”, for some
+complementary simple classes A1andA2, contradicting a time hierarchy (cf. Theorem 2.5).
+A similar claim holds for nondeterministic time-space lowe r bounds against tautologies
+(which prove “ NTIME[n]⊆coNTS[nc] implies A1⊆A2”), ford-dimensional TM lower
+bounds (which prove “ NTIME[n]⊆DTIME d[nc] implies A1⊆A2”), and other problems.
+Normal Form. It will be very convenient to introduce a normal form for alternation-
+trading proofs. We show that any lower bound provable with co mplementary simple classes
+can also be established with a normal form proof. This greatl y reduces the degrees of
+freedom in a proof, as we no longer need to worry about whichtime hierarchy to contradict.
+/bardblPlease note that the ( k+1)th quantifier is n0in order to account for the O(logn) size of the quantifier.ALTERNATION-TRADING PROOFS, LINEAR PROGRAMMING, AND LOWE R BOUNDS 677
+Definition 3.2. Letc≥1. An alternation-trading proof for cis innormal form if (a) the
+first and last lines are DTS[na] andDTS[na′] respectively, for some a≥a′, and (b) no other
+lines are DTSclasses.
+We show that a normal form proof for cimplies that NTIME[n]/notsubseteqlDTS[nc].
+Lemma 3.3. Letc≥1. If there is an alternation-trading proof for cin normal form having
+at least two lines, then NTIME[n]/notsubseteqlDTS[nc].
+Theorem 3.4. LetA1andA2be complementary. If there is an alternation-trading proof
+Pforcthat shows (NTIME[n]⊆DTS[nc] =⇒A1⊆A2), then there is a normal form proof
+forc, of length at most that of P.
+Proofs of Lemma 3.3 and Theorem 3.4 are in the full version. Th e upshot of these results
+is that we may focus our proof search on normal form proofs. Fo r the remainder of this
+section, we assume all alternation-trading proofs are in no rmal form.
+Proof Annotations. Different lower bound proofs can result in quite different seque nces
+of speedups and slowdowns. A proof annotation represents such a sequence.
+Definition 3.5. Aproof annotation for an alternation-trading proof of ℓlines is the ( ℓ−1)-
+bit vector Awhere for all i= 1,...,ℓ−1,A[i] = 1 (respectively, A[i] = 0) if the ith line
+applies a Speedup Rule (respectively, a Slowdown Rule).
+An (ℓ−1)-bit proof annotation corresponds to a “strategy” for an ℓ-line proof. For a
+normal form proofof ℓlines, it is not hardtoshow that its annotation Amust have A[1] = 1,
+A[ℓ−2] = 0, and A[ℓ−1] = 0.
+Note that an annotation does not determine a proof entirely, as other parameters need
+optimizing. (The problem of optimizing them is tackled in th e next section.) To illustrate
+the annotation concept, we give four examples.
+•Then√
+2lower bound of Lipton and Viglas has the annotation [1 ,0,0].
+•Then1.6004bound from Section 2.1 corresponds to [1 ,1,0,0,1,0,0].
+•Thenφbound of Fortnow and Van Melkebeek [FvM00] is an inductive pr oof, cor-
+responding to an infinite sequence of annotations. In normal form, the sequence is
+[1,0,0],[1,1,0,0,0],[1,1,1,0,0,0,0],...
+•Then2cos(π/7)bound [Wil08] has two inductive stages. Let A= 1,0,1,0,...,1,0,0,
+where the ‘ ...’ contain any number of repetitions. The sequence is
+[A],[1,A,A],[1,1,A,A,A],[1,1,1,A,A,A,A ],...
+That is, the proof performs many speedups, then a sequence of many slowdown-
+speedup alternations, then two consecutive slowdowns, rep eating this until all the
+quantifiers have been removed.
+3.1. Translation To Linear Programming
+Given a (normal form) proof annotation, how can we determine the best proof possible
+with it? We need to optimally set the runtimes of the first and l astDTSclasses in the proof,
+as well as the xiparameters that arise from each application of a Speedup Rul e. It turns
+out that an annotation Aandc >1 can be reduced to a polynomial size linear program
+that is feasible if and only if there is an alternation-tradi ng proof of NTIME[n]/notsubseteqlDTS[nc]678 R. R. WILLIAMS
+with annotation A. More precisely, the problem of optimizing parameters can b e viewed as
+an arithmetic circuit evaluation, where the circuit has max gates, addition gates, and input
+gates that may multiply their input by c. Such circuits can be evaluated using a linear
+program that minimizes the sum of the gate values (cf. [Der72 ]).
+LetAbe an annotation of ℓ−1 bits, and let mbe the maximum number of quantifier
+blocks in any line of A(notemis easily computed in linear time). The target LP has
+variables ai,j,bi,j, andxi, for all i= 0,...,ℓ−1 andj= 1,...,m. The variables ai,j
+represent the runtime exponent of the jth quantifier block in the class on the ith line,bi,jis
+the input exponent to the jth quantifier block of the class on the ith line, and for all lines
+ithat use a Speedup Rule, xiis the choice of xin the Speedup Rule. For example:
+•If thekth line of a proof is DTS[na], the corresponding constraints are
+ak,1=a, bk,1= 1,(∀k >0)ak,i=bk,i= 0.
+•If thekth line of a proof is ( ∃na′)bDTS[na], then the constraints are
+ak,0=a, bk,1=b, ak,1=a′, bk,1= 1,(∀k >1)ak,i=bk,i= 0.
+The objective is to minimize/summationtext
+i,j(ai,j+bi,j)+/summationtext
+ixi. The LP constraints depend on the
+lines of the annotation, as follows.
+Initial Constraints. For the 0th and ( ℓ−1)th lines we have a0,1≥aℓ−1,1, and
+a0,1≥1, b0,1= 1,(∀k >1)a0,k=b0,k= 0,andaℓ,1≥1, bℓ,1= 1,(∀k >1)aℓ,k=bℓ,k= 0,
+representing DTS[na0,1] andDTS[naℓ−1,0], respectively. The1st line of a proof always applies
+Speedup Rule 1, having the form ( Q1nx)max{x,1}(Q2n0)1DTS[na−x]. So the constraints for
+the 1st line are:
+a1,1=a0,1−x1, b1,1= 1, a1,2= 0, b1,2≥x1, b1,2≥1, a1,3=x3, b1,3= 1,
+(∀k: 4≤k≤m)a1,k=b1,k= 0.
+The below constraint sets simulate the Speedup and Slowdown Rules:
+Speedup Rule Constraints. For theith line where i >1 andA[i] = 1, we have
+ai,1≥1, ai,1≥ai−1,1−xi, bi,1=bi−1,1, ai,2= 0, bi,2≥xi, bi,2≥bi−1,1, ai,3≥ai−1,2,
+ai,3≥xi, bi,3≥bi−1,2,(∀k: 4≤k≤m)ai,k=ai−1,k−1,bi,k=bi−1,k−1.
+The constraints express that ···b2(Q2na2)b1DTS[na1] in the ( i−1)th line is replaced by
+···b2(Q2nmax{a2,x})max{x,b1}(Q1n0)b1DTS[nmax{a1−x,1}]
+in theith line, where Q1is opposite to Q2.
+Slowdown Rule Constraints. For theith line where A[i] = 0, the constraints are
+ai,1≥c·ai−1,1, ai,1≥c·ai−1,2, ai,1≥c·bi−1,1, ai,1≥c·bi−1,2, bi,1=bi−1,2
+(∀k: 2≤k≤m−1)ai,k=ai−1,k+1, bi,k=bi−1,k+1, ai,m=bi,m= 0.
+These express the replacement of ···b2(Q1na2)b1DTS[na1] in the ( i−1)th line with
+···b2DTS[nc·max{a1,a2,b1,b2}]
+in theith line.
+This concludes the description of the linear program. To find the largest cthat still yields
+a feasible LP, we can simply binary search for it. The followi ng summarizes this section.
+Theorem 3.6. Given an annotation of nlines, the best possible alternation-trading proof
+following the annotation can be determined up to ndigits of precision, in poly(n)time.ALTERNATION-TRADING PROOFS, LINEAR PROGRAMMING, AND LOWE R BOUNDS 679
+3.2. Results
+Following the above formulation, we wrote proof search rout ines in Maple. Many millions
+of proof annotations were tried, including all those corres ponding to prior work, with no
+success beyond the 2cos( π/7) exponent. The best lower bounds followed a highly regular
+pattern; see the full version for more on this. We are led to:
+Conjecture 3.7. There is no alternation-trading proof that NTIME[n]/notsubseteqlDTS[nc]for all
+c >2cos(π/7).
+Proving the conjecture seems currently out of reach. Howeve r, we can show:
+Theorem 3.8. There is no alternation-trading proof that NTIME[n]/notsubseteqlDTS[n2].
+A proof is in the full version. At a high level, the proof argue s that any minimum length
+proof of a quadratic lower bound could be shortened, giving a contradiction.
+Despite this bad news, the theorem prover did provide enough insight to aid in a new
+lower bound of n2cos(π/7)−o(1)on the time-space product of any SAT algorithm.
+Theorem 3.9. Lett(n)ands(n)be bounded above by polynomials. Any algorithm solving
+SAT in time tand space srequirest·s= Ω(n2cos(π/7)−ε)for allε >0.
+These lower bounds have also been generalized to the QBF prob lem:
+Theorem 3.10. For allk≥1,QBFkrequires Ω(nc)time on no(1)space RAMs, where
+c3/k−c2−2c+k <0.
+4. Discussion
+We introduced a methodology for reasoning about alternatio n-trading proofs of lower
+bounds. It provides a generic means for computers to help us a ttack lower bound problems,
+and lets us establish limitations on known techniques. We no w have a better understanding
+of what these techniques can and cannot do, and a tool for addr essing future problems.
+Previously, the problem of setting parameters to achieve a g ood lower bound was a highly
+technical exercise. Our work should facilitate further res earch: once a new speedup or slow-
+down lemma is found, one only needs to find the relevant linear programming formulation
+to begin understanding its power. We conclude with two open- ended problems.
+(1)Establish tight limitations for alternation-trading proo fs.That is, show that the best
+possible alternation-trading proofs match those we have pr ovided. Our computer
+search results have been met with healthy skepticism. It is c ritical to verify these
+perceived limitations with formal proof. We have managed to prove non-trivial
+limitations; it is possible that the ideas in those can be ext ended.
+(2)Discover new ingredients to add to the framework. One possibility is to find new
+separation results that lead to new contradictions. Anothe r is to find improved
+Speedup and/or Slowdown Lemmas. The Slowdown Lemmas are the “blandest” of
+the ingredients, in that they are the most elementary (and th ey relativize).
+Acknowledgements. I am grateful to my thesis committee for their invaluable feedback o n my
+PhD thesis, which included preliminary results on this work. Thanks to Scott Aaronson for useful
+discussions about irrelativization, and thanks to the STACS refere es for very thoughtful comments.680 R. R. WILLIAMS
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+copyofthislicense,visit http://creativecommons.org/licenses/by-nd/3.0/ .
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+arXiv:1001.0747v2  [astro-ph.SR]  12 Jan 2010Draft version November 12, 2018
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+HYBRID γDORADUS – δSCUTI PULSATORS:
+NEW INSIGHTS INTO THE PHYSICS OF THE OSCILLATIONS FROM KeplerOBSERVATIONS
+A. Grigahc `ene1, V. Antoci2, L. Balona3, G. Catanzaro4, J. Daszy ´nska-Daszkiewicz5, J. A. Guzik6, G. Handler2,
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+K. Uytterhoeven11, W. J. Borucki12, T. M. Brown13, J. Christensen-Dalsgaard14, R. L. Gilliland15,
+J. M. Jenkins16, H. Kjeldsen14, D. Koch12, S. Bernabei17, P. Bradley18, M. Breger2, M. Di Criscienzo19,
+M.-A. Dupret20, R. A. Garc ´ıa11, A. Garc ´ıa Hern ´andez10, J. Jackiewicz21, A. Kaiser2, H. Lehmann22,
+S. Mart ´ın-Ruiz10, P. Mathias23, J. Molenda- ˙Zakowicz5, J. M. Nemec24, J. Nuspl25, M. Papar ´o25, M. Roth25,
+R. Szab ´o25, M.D. Suran27, R. Ventura4
+Draft version November 12, 2018
+ABSTRACT
+Observationsofthepulsationsofstarscanbeusedtoinfertheirin teriorstructureandtesttheoretical
+models. The main sequence γDoradus (Dor) and δScuti (Sct) stars with masses 1.2-2.5 M⊙are
+particularly useful for these studies. The γDor stars pulsate in high-order gmodes with periods of
+order 1 day, driven by convective blocking at the base of their enve lope convection zone. The δSct
+stars pulsate in low-order gandpmodes with periods of order 2 hours, driven by the κmechanism
+operating in the He iiionization zone. Theory predicts an overlap region in the Hertzspru ng-Russell
+diagram between instability regions, where ’hybrid’ stars pulsating in both types of modes should
+exist. The two types of modes with properties governed by differen t portions of the stellar interior
+provide complementary model constraints. Among the known γDor and δSct stars, only four have
+been confirmed as hybrids. Now, analysis of combined Quarter 0 and Quarter 1 Keplerdata for
+hundreds of variable stars shows that the frequency spectra ar e so rich that there are practically
+no pure δSct orγDor pulsators, i.e. essentially all of the stars show frequencies in bo th theδSct
+andγDor frequency range. A new observational classification scheme is proposed that takes into
+account the amplitude as well as the frequency, and is applied to cat egorize 234 stars as δSct,γDor,
+δSct/γDor orγDor/δSct hybrids.
+Subject headings: space vehicle - stars: oscillations - stars: variables: δScuti - stars: variables:
+γDoradus
+1Centro de Astrof´ ısica, Faculdade de Ciˆ encias, Universid ade
+do Porto, Rua das Estrelas, 4150-762 Porto, Portugal
+2Institut fuer Astronomie, Tuerkenschanzstr. 17, A-1180
+Wien, Austria
+3South African Astronomical Observatory, P.O. Box 9,
+Observatory 7935, South Africa
+4INAF - Osservatorio Astrofisico di Catania, Via S. Sofia 78,
+95123 Catania, Italy
+5Instytut Astronomiczny, Uniwersytet Wroc/suppress lawski,
+Kopernika 11, 51-622 Wroc/suppress law, Poland
+6Los Alamos National Laboratory, X-2 MS T-086, Los
+Alamos, NM 87545-2345, USA
+7Jeremiah Horrocks Institute of Astrophysics, University o f
+Central Lancashire, Preston PR1 2HE, UK
+8INAF - Osservatorio Astronomico di Capodimonte, Via
+Moiariello 16, 80131 Naples, Italy
+9Laboratorio de Astrof´ ısica Estelar y Exoplanetas, LAEX-
+CAB (INTA-CSIC), PO BOX 78, 28691 Villanueva de la
+Ca˜ nada, Madrid, Spain
+10Instituto de Astrof´ ısica de Andaluc´ ıa (CSIC), CP3004,
+Granada, Spain
+11Laboratoire AIM, CEA/DSM CNRS - U. Paris Diderot,
+IRFU/SAp, Centre de Saclay, 91191 Gif-sur-Yvette Cedex,
+France
+12NASA Ames Research Center, MS 244-30, Moffett Field,
+CA 94035, USA
+13Las Cumbres Observatory Global Telescope, Goleta, CA
+93117, USA
+14Department of Physics and Astronomy, Aarhus University,
+DK-8000 Aarhus C, Denmark
+15Space Telescope Science Institute, Baltimore, MD 21218,
+USA
+16SETI Institute/NASA Ames Research Center, MS 244-30,
+Moffett Field, CA 94035, USA17INAF - Osservatorio Astronomico di Bologna, Via Ranzani
+1, 40127 Bologna, Italy
+18Los Alamos National Laboratory, X-4 MS T-087, Los
+Alamos, NM 87545-2345, USA
+19INAF - Osservatorio Astronomico di Roma, via Frascati
+33, 00040 Monte Porzio Catone, Rome, Italy
+20Institut d’Astrophysique et de G´ eophysique, Universit´ e de
+Li` ege, All´ ee du 6 Aoˆ ut 17-B 4000 Li` ege, Belgium
+21Department of Astronomy, New Mexico State University,
+Las Cruces, NM 88001, USA
+22Thueringer Landessternwarte, 07778 Tautenburg, Germany
+23Dpt Fizeau, UMR 6525 Observatoire de la C¨ ote
+d’Azur/CNRS, BP 4229, F06304 Nice Cedex 4, France
+24Deptartment of Physics & Astronomy, Camosun College,
+Victoria, British Columbia, Canada
+25Konkoly Observatory of the Hungarian Academy of Sci-
+ences, P.O. Box 67, H-1525 Budapest, Hungary
+26Kiepenheuer Institut f¨ ur Sonnenphysik, Sch¨ oneckstr. 6,
+79104, Freiburg, Germany
+27Astronomical Institute of the Romanian Academy, Str.
+Cutitul de Argint 5, RO 40557, Bucharest, Romenia2 Grigahc` ene et al.
+1.INTRODUCTION
+Many stars oscillate in multiple simultaneous modes
+during some portion of their evolution. The frequency
+range, spacing, amplitude, or other properties can be
+used to infer the interior structure of these stars and test
+the input physics and predictions of theoretical models,
+a process called asteroseismology. The γDoradus (Dor)
+andδScuti (Sct) starsthat pulsatein manysimultaneous
+frequencies are particularly useful for asteroseismic stud-
+ies. These stars are core hydrogen-burning and some-
+what more massive than the Sun, at 1.2-2.5 M⊙. How-
+ever, unlike the Sun, they have convective cores, shal-
+lower convective envelopes, and often rapid rotation.
+While helioseismologists have been searching for solar
+gmodes (e.g., Garc´ ıa et al. 2007), stellar astronomers
+have found them in several groups of pulsating stars in-
+cluding the γDor variables. The γDorgmodes have
+periods of the order of one day, making them difficult to
+observe continuously from the ground. In addition, the
+gmodes reach the stellar surface with small amplitudes
+which make their detection a challenge.
+From a pulsational point of view, there is a clear dis-
+tinction between δSct and γDor stars. The former are
+short-period pulsating stars, with periods between 0.014
+and 0.333 day (d), while the latter are long-period pul-
+sators, with periods between about 0.3 and 3d. The
+distinction is even clearer if one considers the pulsation
+constant Q(Handler & Shobbrook 2002); the δSct stars
+haveQ <0.055d whereas for γDor stars Q >0.24d.
+When the γDor variables were first recognized as a new
+group (Balona et al. 1994; Kaye et al. 1999), their posi-
+tion on the Hertzsprung-Russell diagram (HRD) already
+suggested some relationship with the δSct group. The
+γDor stars lie in a zone close to the cool border of the
+classical instability strip, partially overlapping with the
+δSct pulsators. This latter property led astronomers to
+investigate the possible existence of hybrid stars exhibit-
+ing both types of pulsations (Breger & Beichbuchner
+1996).
+These hybrid objects are of great interest because they
+offer additional constraints on stellar structure. The
+γDor starspulsate in gmodes, giving us the opportunity
+to probe the stellar core, while the δSct stars pulsate in
+pmodes, that help to probe the stellar envelope. More-
+over, since γDor stars pulsate in high-order gmodes,
+the asymptotic approximation can be used, where we
+can benefit from an analytical solution of the equations
+(Smeyers & Moya 2007). Hence, the existence of hybrid
+stars provides a unique opportunity to extend this ad-
+vantage to δSct stars (Handler & Shobbrook 2002).
+Furthermore, in the same region of the HRD where
+theδSct and γDor instability strips overlap, solar-like
+oscillations are also predicted to occur for δSct stars
+(Houdek et al. 1999; Samadi et al. 2002). So far, no de-
+tections of solar-like oscillations in such hot stars have
+been reported (Antoci et al. 2009), which is probably
+due to the detection limits imposed by earthbound mea-
+surements. Stochastic excitation as an excitation mech-
+anism in γDor stars has also been searched for (e.g. see
+Pereira et al. 2007), but so far not confirmed. Kepler
+data are expected to reach the precision needed to ob-
+serve stochastically excited pulsation. A positive or a
+significant null detection would set strong constraints onFig. 1.— Color-Magnitude diagram of δSct (plus signs,
+Rodr´ ıguez et al. 2000), γDor (open circles, Henry et al. 2007)
+and stars that show both types of pulsation (star symbols,
+Henry & Fekel 2005; Rowe et al. 2006; Uytterhoeven et al. 2008 ;
+Handler 2009, Rowe, priv. comm.). The Zero-Age Main Sequenc e
+(dashed-dotted line, Crawford 1979) as well as the observed bor-
+ders of the δSct (solid lines, Rodr´ ıguez & Breger 2001) and γDor
+(dashed lines, Handler & Shobbrook 2002) instability strip s, are
+indicated. Lines on the left indicate blue edges while those on the
+right give the red edges of the instability strips.
+the parametrization of the convective envelopes.
+Poretti et al. (2009), using data from the French-led
+CoRoTmission, show that overmost of the periodogram
+of HD 292790 (V = 9.5), the amplitude of the noise level
+is 0.01 mmag for an observational time span of 57 d.
+InKeplerobservations of the combined Quarter 0 and
+Quarter 1 data of a 9.5 mag star (time span of 44 d) the
+amplitude of the noise level is 0.001 mmag. The noise
+level inKeplerobservations is therefore approximately
+10 times smaller than in CoRoTobservations.
+The large number of stars to be studied by Kepler
+will also allow for control of long-term (low-frequency)
+instrumental changes; understanding these is important
+for confident identification of the low-frequency gmodes.
+Most importantly, the 3.5-y time span of the mission
+will give data with unprecedented frequency resolution.
+This is known to be necessary from ground-based studies
+that show closely spaced frequencies in data sets span-
+ning years.
+2.CURRENT OBSERVATIONAL STATUS
+Among the hundreds of known δSct stars and sev-
+eral dozen confirmed γDor variables, only 4 have
+been previously observationally identified as hybrids
+and interpreted afterward using theoretical models
+(Bouabid et al. 2009).
+One of the major contributions came from the Cana-
+dian Space Mission MOST(Matthews 2007), while
+CoRoT(Auvergne et al. 2009) is expected to add new
+confirmed cases in the near future, especially since
+the preliminary results indicate the possible detection
+of hybrid γDor and δSct pulsators in the exofield
+(Mathias et al. 2009).
+The known hybrid pulsators are depicted, together
+with single-type pulsators, in Fig. 1. The first case
+recorded was HD 209295 (Handler & Shobbrook 2002),
+but it turned out afterward that this star is a close
+binary and the long period modes could be tidallyHybridγDor –δSct pulsators from Kepler 3
+Fig. 2.— Hybrid theoretical instability domain for ℓ= 0−3g
+andpmodes. Models have: Z= 0.02,αMLT = 2.0,αov= 0.2.
+The vertical line corresponds to the effective temperature o f the
+Kepler target: KIC 11445913 ( Teff= 7200±200 K).
+excited (Handler et al. 2002). Following this work,
+Henry & Fekel (2005) discovered both γDor and δSct-
+type pulsations in the single Am star HD 8801 (Handler
+2009).
+MOST found two additional hybrid pulsators:
+HD 114839 (King et al. 2006) and BD+18 4914
+(Rowe et al. 2006). Both of these are also Am stars. In
+preparation for the CoRoTmission, Uytterhoeven et al.
+(2008) identified HD 49434 as a candidate hybrid pul-
+sator.
+It is very interesting to note that three of the four
+known hybrids show Am type peculiarities, even though
+Am stars are less likely to show pulsational variability
+than A/F-type main-sequence stars (Kurtz 1989). In-
+creasing the number of hybrids is expected to shed light
+on the relationship between hybrid pulsation and Am
+chemical peculiarity.
+3.THEORETICAL EXPECTATIONS AND MODELING
+It is well known that the κ-mechanism operating in
+the Heiiionization zone is responsible for the pulsa-
+tional driving of δSct stars (Chevalier 1971). With this
+mechanism,thetheoreticalblueedgeofthe δSctinstabil-
+ity strip is predicted correctly with linear non-adiabatic
+models for radial as well as for non-radial modes. But
+determining the theoretical red edge is more difficult, be-
+causeit requiresatime-dependent treatmentofthe inter-
+action between convection and pulsation. In this stage
+of their evolution, cool δSct stars have relatively deep
+surface convection zones which typically extend beyond
+the location of the second stage of helium ionization,
+the principal driving region of δSct pulsations. Conse-
+quently, mode stability is cruciallyaffected by convection
+dynamics for stars located near the red edge of the clas-
+sical instability strip in the HRD. The location of the
+red edge can be described theoretically only by means
+of a time-dependent treatment of the turbulent fluxes
+(e.g., Deupree 1977; Baker & Gough 1979; Gonzi 1982;
+Stellingwerf 1984). Moreover, the observed location of
+the instabilitystrip providesindependent informationfor
+calibrating convection models.
+Several authors (e.g., Bono et al. 1999; Houdek 2000;
+Dupret et al. 2005; Xiong & Deng 2007) have success-
+fullymodeledtherededgeoftheclassicalinstabilitystripbut reported that different physical mechanisms are re-
+sponsible for stabilizing the pulsations. It is therefore of
+great importance that our calculations for mode stabil-
+ity be reassessed and Kepleris in the position to pro-
+vide the necessary high-quality data for calibrating and
+then testing the models for convection in classical pul-
+sators. Moreover, analysis of the follow-up ground-based
+multi-color photometry can yield valuable constraints on
+efficiency of the convective transport in these pulsators
+(Daszy´ nska-Daszkiewicz et al. 2003).
+Guzik et al. (2000), using frozen-convection models,
+and Dupret et al. (2005, 2006), using time-dependent
+convection(Grigahc` ene et al.2005), haveshownthat the
+positionoftheconvectiveenvelopebaseisthekeytodriv-
+ingγDorgmodes. The convective blocking mechanism,
+operating when the convective timescale becomes com-
+parable to the gmode period, can explain the location
+of the blue edge of the γDor instability strip and its sen-
+sitivity to the mixing length parameter αMLT. On the
+other hand, the red edge of the γDor instability strip is
+caused by radiative damping in the gmode cavity, which
+dominates over the excitation near the convective enve-
+lope base. As an example, we show in Fig. 2 the hybrid
+theoreticalinstabilitydomainfor ℓ= 0−3gandpmodes.
+For the calculated grid ( Z=0.02,αMLT=2.0), the effec-
+tive temperature range of hybrids is 6900 K /lessorsimilarTeff/lessorsimilar
+7500 K. The vertical line in Fig. 2 represents a model of
+a hybrid pulsator for which we have unstable high order
+gmodes, corresponding to γDor range as well as unsta-
+ble low order pandgmodes, corresponding to typical
+δSct pulsations separated by a gap of stable modes.
+We also note that the stability of modes in the gap
+decreases as degree ℓincreases. This is due to the fact
+that the Lamb frequency S2
+ℓ∝ℓ(ℓ+ 1), so the size of
+the propagation cavity, of gmodes increases with ℓfor a
+given frequency.
+Therefore, the direct comparison of the observed fre-
+quency spectrum with models is a test of the excitation
+models and can improve our knowledge about important
+physical characteristics of the stars, including the age,
+chemical composition, etc.
+Another more general problem in stellar pulsation
+modeling is connected with the effects of rotation on
+eigenfrequencies and eigenfunctions. The commonly
+adopted approach is perturbation theory which assumes
+that oscillation frequencies are much larger than the an-
+gular rotation rate Ω and that the star is not signifi-
+cantly deformed by the centrifugal force (Ω ≪Ωcrit≡/radicalbig
+GM/R3) (Reese et al. 2008). This assumption is vi-
+olated for many δSct and γDor stars, due to their rel-
+atively rapid rotation. This implies that rotation will
+significantly affect the pulsation frequency spectrum and
+surface amplitude distribution. This in turn will affect
+howlimb darkeningand lightcancellationaffectsthe pul-
+sation mode visibility and radial velocity variations.
+Hybridscanbeextremelyusefulwheninvestigatingthe
+internal rotation profile of stars. In Su´ arez et al. (2006)
+the authors examine the effect of rotation on the oscilla-
+tion spectrum for intermediate-mass rotating stars. Sig-
+nificant effects for both pand mixed modes (very low
+radial order gandpmodes) are predicted. Analysis of
+hybrid stars will allow us to constrain the modeling of
+their rotation profiles. Better constraints on the internal4 Grigahc` ene et al.
+rotation profile will help us better understand angular
+momentum transport processes in the stellar interior.
+NeitherδSct stars nor hybrid stars are expected to
+pulsate in asymptotic pmode oscillations. However,
+recently Garc´ ıa-Hern´ andez et al. (2009) found regular
+spacings in the oscillation spectrum of the CoRoTδSct
+star HD174936. Analysis of the 422 detected oscilla-
+tion frequencies revealed a spacing periodicity of around
+52µHz. Although the modes considered were not in
+the asymptotic regime, a comparison with stellar models
+confirmed that this signature may stem from a quasi-
+periodic pattern similar to the so-called large separa-
+tion in solar-like stars. Other frequency spacings have
+also been observed in other stars using ground based ob-
+servations (e.g. Handler et al. 2000; Breger et al. 2009).
+From numerical results, this approximateasymptotic be-
+haviour has been found to hold for p modes of low order
+(e.g. Christensen-Dalsgaard 2000). The presence of such
+spacings in hybrid stars may allow us to analyze simul-
+taneously pandgmodes with some characteristics of
+the asymptotic regime and even solar-like oscillations, if
+detected.
+Interestingly, asteroseismology of hybrid stars can also
+benefit from techniques used for analyzing and interpret-
+ing the oscillation spectra of both type of pulsations. For
+instance, if gmodes in the asymptotic regime are de-
+tected, techniques based on the asymptotic properties of
+gmodes can be applied, namely the Frequency Ratio
+Method (FRM – Moya et al. 2005; Su´ arez et al. 2005).
+The FRM is particularly adapted for obtaining astero-
+seismic information on γDor pulsating stars showing at
+least three pulsation frequencies. The method provides
+an identification of the radial order nand degree ℓof
+observed frequencies and an estimate of the integral of
+the buoyancy frequency (Brunt-V¨ ais¨ al¨ a) weighted over
+the stellar radius along the radiative zone. Miglio et al.
+(2008) also considered high-order gmodes to probe the
+properties of convective cores in Slowly Pulsating B and
+γDor stars.
+4.KEPLER ’S FIRST OBSERVATIONS
+The main result from the inspection of this first Ke-
+plerdata is that there are practically no pure δSct or
+γDor pulsators. This finding raises a new problem: why
+is it that there is a clear distinction between the two
+types of pulsator in ground-based observations but not
+inKeplerobservations? Partof the answercould be that
+in ground-based photometric observations the modes of
+highest visibility are modes of low degree whereas at the
+photometric precision of Keplermany more modes of
+high degree are now visible.
+Now we must consider whether we can construct a
+classification scheme that allows us to retain the useful
+δSct/γDor categories. It is important to notice that re-
+lying only on the mode frequencies, Kepler’s data show
+that all stars with variability in these frequency ranges
+are hybrids. The use of amplitude in addition to fre-
+quency can help by potentially filtering out the high-
+degree modes that generally would be expected to have
+a lower photometric amplitude. Considering both am-
+plitude and frequency, we propose a new observational
+classification scheme for δSct andγDor stars:
+•δSct: most of the frequencies are ≥5 d−1, and the 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
+ 3.8  3.82 3.84 3.86 3.88 3.9  3.92 3.94log L/L⊙
+log Teff1R4R1B4B
+Fig. 3.— Theoretical HRD showing classification of Kepler tar-
+get stars. Filled circles are δSct; open circles δSct/γDor; crosses
+γDor/δSct and plus signs γDor. The solid lines show the Zero-
+Age Main Sequence and the radial fundamental red and blue edg es
+(1R, 1B) and the 4-th overtone radial red and blue edges (4R, 4 B)
+(Dupret et al. 2005). The dashed lines are the red and blue edg es
+of theγDor instability strip ( ℓ= 1 and mixing length αMLT =
+2.0) (Dupret et al. 2005).
+TABLE 1
+Number of stars and percentage in each class. For each
+class the mean values of the effective temperature is
+given.
+Class Number Percent <logTeff>
+δSct 67 27 3.885 ±0.003
+δSct/γDor 32 14 3.883 ±0.006
+γDor/δSct 19 9 3.868 ±0.006
+γDor 116 50 3.853 ±0.005
+lower frequencies are of relatively low amplitude;
+•δSct/γDor hybrid: most of the frequencies are ≥
+5 d−1, but there are some lower frequencies which
+are of comparable amplitude;
+•γDor: most ofthe frequenciesare ≤5 d−1, and the
+higher frequencies are of relatively low amplitude.
+•γDor/δSct hybrid: most of the frequencies are ≤
+5 d−1, but there are some higher frequencies which
+are of comparable amplitude;
+We apply this scheme to a sample of 554 stars selected
+from the Keplerworking group targets, where we have
+excluded stars that are constant or eclipsing variables.
+Among them, 234 show γDor orδSct frequencies and
+have an effective temperature near the instability strips
+for these types of pulsation. Table 1 summarizes the
+categorization and mean properties for these stars. The
+same stars are shown in the theoretical HRD in Fig. 3
+using the Teffand radii derived from the KeplerInput
+Catalogue(KIC). Also shownare calculatedred and blue
+edges of the instability strips (Dupret et al. 2005). We
+see that there is some separation between categories, but
+it is not as clean as one might have hoped. We also see
+that the theoretical red and blue edges are not a good
+match for the observations.HybridγDor –δSct pulsators from Kepler 5
+Fig. 4.— Frequency spectra of the long-cadence data obtained
+byKepler for the stars KIC 9775454 and KIC 11445913 that dis-
+play hybrid pulsations in the γDor region (below 5 d−1) and in the
+δSct region (above 12 d−1). The details of the spectra at higher
+frequencies require short-cadence data to be confirmed and a na-
+lyzed.
+The amplitude spectrum of two hybrid stars
+(KIC 9775454 and KIC 11445913) are shown in Fig. 4.
+The vertical line in Fig. 2 indicates the position in effec-
+tive temperature of the star KIC 11445913, confirming
+it as a potential hybrid candidate. Fig. 4 also shows the
+very low level of noise measured in the gap between the
+frequency ranges for the pmodes and gmodes. This
+gap, corresponding to a lack of unstable modes, is ex-
+pected in the theory for hybrid pulsators. However, in
+some stars observed by CoRoT, several frequencies can
+be found in the “gap”. The same behavior is also found
+for some of the Keplertargets. An explanation for these
+frequencies still needs to be found from a more detailed
+study of the observations and by modeling the stars be-
+ing observed. In some cases, rotational splitting seems
+to be the cause of modes in the frequency gap between
+pandgmodes (Bouabid et al. 2009). Preliminary anal-
+ysis ofKeplerdata for candidate hybrid stars gives very
+encouraging results for the number of hybrid stars with
+veryhigh qualityfrequency spectra. Thelargenumberof
+pulsational frequencies makes it possible to study global
+characteristicsof the frequency spectrum such as asymp-
+totic properties, and also individual frequency fittings.
+5.CONCLUSIONS
+Our first look at the Keplerdata provides encouraging
+results on the number of hybrid stars thanks to veryhighquality frequency spectra coupled with unprecedented
+photometric precision. The Keplerdata are essential
+to overcome the aliasing and photometric stability prob-
+lems affecting the γDor frequency range associated with
+ground-based photometric campaigns.
+The additional frequencies found in hybrid stars will
+helptoovercomedifficulties with modeidentificationand
+the detection of many γDorgmodes will enable analy-
+sis by asymptotic techniques. The Keplerdata onγDor
+andδSct stars promise to help us to answer a number of
+outstanding questions: why have Kepler(andCoRoT)
+detected a significant number of hybrids, whereas theory
+predicts the existence of hybrids in only a small over-
+lapping region of the instability strips? What is the
+connection between Am chemical peculiarity and hybrid
+behavior? Why are modes observed between the δSct
+andγDor frequency ranges, where theory predicts a fre-
+quency gap? Will we be able to confirm stochastic ex-
+citation or other pulsation driving mechanisms? Can
+the high-photometric precision of Keplerdetect higher-
+degree modes ( ℓ>2) that could fill in the frequency gap?
+Will these stars help us to distinguish between explana-
+tions for the red edge of the δSct instability strip? Will
+we be able to discern rotational frequency splittings and
+derive internal differential rotation rates for these stars?
+Perhapsjust as important are the many Keplertargets
+that lie within the γDor orδSct instability strips that
+show no periodic variations at micromagnitude photo-
+metric precision levels obtainable by Kepler. Monitoring
+over several years with Keplerwill provide insight on
+amplitude limiting and mode selection mechanisms for
+these stars that is not possible with ground-based obser-
+vations.
+The analysis presented here of the first and second
+Quarter data obtained by Keplerconfirms its capabil-
+ity to provide very high quality data that will allow us
+to study hybrid pulsators of the δSct/γDor type. More-
+over, the very rich frequency spectra of these stars leads
+us to propose a new classification scheme to overcome
+the ambiguityof the older, pure frequency-basedscheme.
+The close frequency spacing of relatively short period
+modes in δSct stars requires short-cadence data. We ex-
+pect to obtain short-cadence data for selected targets of
+mixed character in the near future.
+Funding for this Discovery mission is provided by
+NASA’s Science Mission Directorate. The authors grate-
+fully acknowledge the entire Keplerteam, whose out-
+standing efforts have made these results possible. The
+authors also thank all funding councils and agencies that
+havesupportedtheactivitiesofKASCWorkingGroups4
+and 10.
+Facilities: TheKeplerMission.
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+arXiv:1001.0748v3  [cond-mat.mtrl-sci]  10 Sep 2010Conditions for free magnetic monopoles in nanoscale square arrays of dipolar spin ice
+L. A. S. M´ ol,∗W. A. Moura-Melo,†and A. R. Pereira‡
+Departamento de F´ ısica, Universidade Federal de Vi¸ cosa, Vi¸ cosa, 36570-000, Minas Gerais, Brazil
+We study a modified frustrated dipolar array recently propos ed by M¨ oller and Moessner [Phys.
+Rev. Lett. 96, 237202 (2006)], which is based on an array manufactured lit hographically byWang et
+al.[Nature (London) 439, 303 (2006)] and consists of introducing a height offset hbetween islands
+(dipoles) pointing along the two different lattice directio ns. The ground-states and excitations are
+studied as a function of h. We have found, in qualitative agreement with the results of M¨ oller and
+Moessner, that the ground-state changes for h > h 1, whereh1= 0.444a(ais the lattice parameter or
+distance between islands). In addition, the excitations ab ove the ground-state behave like magnetic
+poles but confined by a string, whose tension decreases as hincreases, in such a way that for h≈h1
+its value is around 20 times smaller than that for h= 0. The system exhibits an anisotropy in the
+sense that the string tension and magnetic charge depends si gnificantly on the directions in which
+the monopoles are separated. In turn, the intensity of the ma gnetic charge abruptly changes when
+the monopoles are separated along the direction of the longe st axis of the islands. Such a gap is
+attributed to the transition from the anti to the ferromagne tic ground-state when h=h1.
+PACS numbers: 75.75.-c, 75.60.Ch, 75.60.Jk
+I. INTRODUCTION
+Geometrical frustration in magnetic materials occurs
+when the spins are constrained by geometry in such a
+way that the pairwise interaction energy cannot be si-
+multaneously minimized for all constituents. A special
+example is an exotic class of crystalline solid known as
+spin ice ( Dy2Ti2O7,Ho2Ti2O7). Recently, Castelnovo
+et al.1have proposed that these materials are the repos-
+itory of some elegant physical phenomena: for instance,
+collective excitations above its frustrated ground-state
+surprisinglybehaveaspoint-like objectsthat arethe con-
+densed matter analogues of magnetic monopoles. Some
+recent experiments2–5have reported the observation and
+even the measurement of the magnetic charge and cur-
+rent of these monopoles in spin ice materials; in addition,
+simulations also support these ideas6,7. Besides, to turn
+the research of monopoles into a proper applied science,
+it will be necessary to ask if the basic ideas of dipole
+fractionalization1,8that give an usual spin-ice material
+its special properties can be realized in other magnetic
+settings. One of the most promising candidates for ac-
+complishing that, is the artificial version of spin ices re-
+cently produced by Wang et al.9. In this system, elon-
+gated magnetic nano-islands are regularly distributed in
+a two-dimensional (2 d) square lattice. The longest axis
+of the islands alternate its orientation pointing in the
+direction of the two principal axis of the lattice9. The
+magnetocrystalline anisotropy of Permalloy (the mag-
+netic material commonly used to fabricate artificial spin
+ice) is effectively zero, so that the shape anisotropy of
+each island forces its magnetic moment to align along
+the largest axis, making the islands effectively Ising-like.
+Actually, the fabrication and study of this kind of lower
+dimensional analogues of spin ice have received a lot of
+attention9–16. Indeed, the ability to manipulate the con-
+stituent degrees of freedom in condensed matter systems
+andtheirinteractionsismuchimportanttowardsadvanc-
+FIG. 1: (Color online) The modified square lattice studied
+in this work. Top: top view of the system. The arrows rep-
+resent the local dipole moments ( /vectorSα(i)or/vectorSβ(i)). Bottom:
+lateral view of the system showing the height offset between
+islands. The original material produced by Wang et al.9is
+two-dimensional with h= 0.
+ing the understanding of a variety of natural phenom-
+ena. Particularly in this context, the possibility of ob-
+serving magnetic monopoles in artificial spin ices14,16is
+a timely problem given that these magnetic compounds
+could provide the opportunity to see them up close and
+also watch them move (for example, with the aid of mag-
+netic force microscopy). Very recently, the direct obser-
+vation of these defects in an artificial kagome lattice was
+reported by Ladak et al.17. However, there is a stim-
+ulating challenging for such an observation (or not) in
+artificial square lattices as pointed out in advance.
+In a previous work14we have pointed out that
+monopoles do not appear as effective low-energy degrees
+of freedom in two-dimensional square spin ices, as they
+do in the three-dimensional materials {Dy,Ho}2Ti2O7.2
+FIG. 2: (Color online) Up: In the artificial spin ice proposed
+in Ref. 10, the spins obeying the ice rule do not point along
+directions passing by the center of a tetrahedron as they do
+in the natural spin ice compounds. Down: Configurations of
+the spins obeying the ice rule in a tetrahedron in the artifi-
+cial (left) and the natural (right) spin ices. This small dis -
+tortion of the spins configuration causes a residual orderin g
+and consequently, an outstanding energetic string connect s
+the monopoles in the modified artificial system.
+Due to the antiferromagnetic order in the ground-state,
+the constituents of a pair monopole-antimonopole be-
+come confined by a string which forbids them to move
+independently. However, we have also argued that above
+a critical temperature, the string configurational entropy
+may lose its tension leaving the monopoles free. The
+quantitative analysis of such a possible phase transition
+isundercurrentinvestigation18. Meanwhile,otherstrate-
+gies to find monopoles in synthetic spin ices have been
+proposed. M¨ oller and Moessner16have suggested a mod-
+ification of the square lattice geometry in which they ar-
+gue that, considering a special condition, the string ten-
+sion vanishes at any temperature. This modification in
+the system produced by Wang et al.9consists of intro-
+ducing a height offset hbetween islands pointing along
+the two different directions10,16(see Fig. 1; such a sys-
+tem is currently under experimental planning19). Their
+idea comprises basically the following: if his chosen so
+that the energies of all vertices obeying ice rule become
+degenerate, then, an ice regime is established leaving the
+monopoles “free” to move (indeed, there is a Coulom-
+bic interaction between the monopoles)16. For point-like
+dipoles they considered that a degenerate state is ob-
+tained when the interactions between nearest-neighbors
+(J1) and next-nearest-neighbors ( J2) are equal, leading
+to the following value for the height offset where “free”
+monopoles occur: hice≈0.419a(whereais the lat-
+tice spacing)10. Taking into account the finite exten-
+sion of the dipoles, the height offset diminishes and as
+ǫ≡1−l/a→0 (lis the length of the island), the
+endpoints of the islands form a tetrahedron, so that at
+h=ǫa/√
+2 the ordering disappears, and the monopolesbecome free to move16.
+Here we numerically calculate the energetics of the
+ground-states and excitations in the modified square lat-
+tice as a function of h. In our calculations we consider
+point-like dipoles forming the lattice. Although the main
+physical aspects of the system must be correct with this
+approximation, someparametervalues (such as magnetic
+charge,stringtension, criticalheightetc)shouldbequan-
+titatively altered for the realistic case in which lhas a
+finite length. On the other hand, since we take into ac-
+count all the long-range dipole-dipole interactions, it is
+expected that our results could better describe the ac-
+tual system. For instance, while in the calculations of
+Refs. 10,16, the ground-state changes its configuration
+ath= 0.419a, our results indicate that it occurs at
+h=h1= 0.444a. Besides, we noted that at least one
+of the several configurations that satisfy the ice rule does
+not have the same energy of the “ground-states” ( GS1
+andGS2) at this very height, indicating that for h=h1
+the system is not in a completely degenerate state. We
+have also shown that the string tension decreases rapidly
+ashincreasesbut it does not vanish at anyvalue ( h≤a):
+rather, at h=h1, its strength reads about 20 times
+smaller than that of the usual case for h= 0. A pos-
+sible cause of the finite strength of the string tension
+even ath1is the fact that, concerning the spin configu-
+rationsin atetrahedron, the artificialspin ice has a slight
+difference with its natural counterpart. For the artificial
+compounds proposed in Ref. 16, the localized magnetic
+moments forming a corner-sharingtetrahedral lattice are
+forced to point along the longest axis of the islands (here,
+x- ory-directions, see Fig.2) while in the original 3 dspin
+ices, they point along a <111>axis (indeed, in this
+case, the magnetic dipoles point along axes that meet
+at the centers of tetrahedra). As a result of this mis-
+match, there is always a single ordered ground state in
+the artificial systems , which is responsible for the resid-
+ual value of the string tension and its anisotropy. An-
+other interesting result obtained here with the point-like
+dipole approximation is that the magnetic charge of the
+monopoles jumps as the system undergoes a transition in
+its ground state. In addition, in general, this strength of
+the interactionbetween amonopoleand itsantimonopole
+is anisotropic, depending on the lattice direction and on
+the type of order. However, as expected from the above
+discussions, we note that the system anisotropy dimin-
+ishes ashgoes toh1. Actually, as hincreases from zero,
+the differences found in the values of the “charges” (as
+distinctdirectionsforthemonopolesseparationaretaken
+into account) decreases, and they tend to disappear as
+h→h1, i.e., in the ice regime (nevertheless, h=h1is
+not really an optimal ice regime, at least for point-like
+dipoles).3
+FIG. 3: (Color online) (a) Ground-stateconfiguration for h <
+h1= 0.444a,GS1. Note that this is exactly the same state
+obtained in Refs.10,14. (b) Configuration of the ground-sta te
+GS2obtained for h >0.444. In GS2, each vertex has a net
+magnetization but globally the magnetization vanishes. No te
+that the ice rule is manifested in every vertex. (c) Another
+configuration that satisfy theice rule buthas anenergy high er
+than the configurations shown in (a) and (b) when h=h1.
+II. THE MODEL AND RESULTS
+We model the system suggested in Refs. 10,16 assum-
+ing: the magnetic moment (“spin”) of the island is re-
+placed by a point dipole at its center. At each site
+(xi,yi,zi) of a “square” lattice two spin variables are de-
+fined:/vectorSα(i)with components Sx=±1,Sy= 0,Sz= 0
+located at /vector rα= (xi+a/2,yi,h), and/vectorSβ(i)with compo-
+nentsSx= 0,Sy=±1,Sz= 0 at/vector rβ= (xi,yi+a/2,0).
+Spins pointing along the y-direction and spins pointing
+along the x-direction are in different planes, separated
+by a height h(see Fig. 1). Hence, in a lattice of volume
+L2=n2a2one gets 2 ×n2spins (we have studied systems
+withn= 20,30,40,50,60,70). Representing the spins of
+the islands by /vectorSi, assuming either /vectorSα(i)or/vectorSβ(i), then the
+modified artificial spin ice is described by the following
+Hamiltonian
+HSI=Da3/summationdisplay
+i/negationslash=j/bracketleftBigg/vectorSi·/vectorSj
+r3
+ij−3(/vectorSi·/vector rij)(/vectorSj·/vector rij)
+r5
+ij/bracketrightBigg
+,(1)
+whereD=µ0µ2/4πa3is the coupling constant of the
+dipolar interaction. The sum is performed over all
+n2(2n2−1) pairs of spins in the lattice for open bound-
+ary conditions (OBC), while for periodic boundary con-
+ditions (PBC) a cut-off radius was introduced when
+rij> n/2a.
+The results presented here consider a lattice with n=
+70, which contains 9800 dipoles (islands) and PBC. We
+observed exactly the same behavior for OBC and PBC
+and the size dependence of the results is not appreciable.0 0,2 0,4 0,6 0,8 1
+h-8-6-4-20
+E
+GS1
+GS2
+Fig. 3 (c)
+FIG. 4: (Color online) The energy per island of the two
+ground-states ( GS1andGS2) and of the configuration shown
+in Fig. 3 (c) (in units of D) as a function of h(in units of
+the lattice spacing a). Black circles represent the GS1energy
+while red squares concern GS2and blue diamonds are for the
+configuration shown in Fig. 3 (c).
+By using a simulating annealing process (see Ref. 14),
+the first thing to notice is that the ground-state configu-
+ration changes for a critical value of h. Indeed, as shown
+in Fig. 3 (a), for all values h < h1= 0.444a, the system
+ground-state (hereafter referred to as GS1) has exactly
+the same form as that of the usual case in which h= 0.
+However, for h > h 1, the ground-state changes to GS2
+(see Fig. 3 (b)). Really, as h→h1, the energies of both
+states are comparable, whereas for h > h1the state GS2
+is less energetic (see Fig. 4). Such a result is in qualita-
+tive agreement with findings of Ref. 10, which presents
+the transition at h=hice= 0.419a. As expected, both
+configurations obey the ice rule (two spins point in and
+twopoint out in everyvertex), but while in GS1the mag-
+netization is zero at each vertex, in GS2it points diago-
+nally, but with net vanishing magnetization. As shownin
+Fig. 4, the energy of GS1increases rapidly as hincreases
+whiletheenergyof GS2isconstant. Actually, forthislat-
+ter configuration, the horizontal and vertical sub-lattices
+are decoupled. We note that these two ground-states are
+metastable in the sense that they are local minima and
+cannot be continuously deformed one into another with-
+out spending a considerable amount of energy; trying to
+align the dipoles from one state to another costs the in-
+version of two spins by vertex (half of the spins have to
+be inverted in the whole system). This changing has an
+h-dependent energy barrier which is roughly of the or-
+der of 160 D(forh= 0.444aandn= 70), making this
+process much improbable to occur spontaneously. Thus,
+considering the system in the GS1state and increasing
+continuously the height from h= 0,GS1may persist
+even for h > h1because of the large energy necessary to
+change to GS2. Besides, in Fig. 4 we also present the en-
+ergy of the configuration shown in Fig. 3 (c), which also
+satisfy the ice rule but has an energy higher than those of
+GS1andGS2even for h=h1. Consequently, the states
+satisfying the ice rule are not completely degenerate.
+Now, we consider the excitations above the ground-
+state. In the two-in/two-out configuration, the effective
+magnetic charge Qi,j
+M(number of spins pointing inward4
+FIG. 5: (Color online) Three of the four basic shortest strin gs
+used in the separation process of the magnetic charges. Pic-
+tures (1) and (2) exhibit strings 1 and 2, respectively used
+forh < h 1= 0.444a. The red circle is the positive charge
+(north pole) while the blue circle is the negative (south pol e).
+Forh > h 1the ground-state is GS2and we used a linear
+string-path (not shown above) and a diagonal path (picture
+(3)).
+0 0,2 0,4 0,6 0,8 1
+h-4-3,8-3,6-3,4
+q
+Path 1
+Path 2
+Linear path
+Path 3GS1GS2
+FIG. 6: (Color online) The monopole “charge” q(see Eq. 2)
+obtained analyzing the energy in the separation process of t he
+charges for the two string shapes shown in Fig.5 for h < h 1.
+Whenh > h 1, the charges variation is shown for a linear
+and diagonal string-paths. Here, qis in units of Dawhile
+hin units of a. Note how the anisotropy of the monopole
+interaction decreases considerably as h→h1from below.minus the number of spins pointing outward on each ver-
+tex (i,j)) is zero everywhere for h < h 1(GS1) and for
+h > h 1(GS2). The most elementary excited state is
+obtained by inverting a single dipole to generate local-
+ized “dipole magnetic charges”. Such an inversion cor-
+responds to two adjacent sites with net magnetic charge
+Qi,j
+M=±1, which is alike a nearest-neighbor monopole-
+antimonopole pair1,14. Following the same method of
+Ref. 14, it is easy to observe that such “monopoles” can
+be separated from each other without violating the lo-
+cal neutrality by flipping a chain of adjacent spins. We
+choose four different ways they may be separated (see
+Fig. 5). Firstly, using the string shape 1 and starting in
+the ground-state GS1(forh < h1= 0.444a)we choosean
+arbitrary site and then the spins marked in dark gray in
+Fig.5areflipped, creatingamonopole-antimonopolesep-
+arated by R= 2a. Next, the spins marked in light gray
+are flipped and the separation distance becomes R= 4a,
+and so on. In this case, the string length ( X) is related
+to the charges separation distance RbyX= 4R/2 (the
+monopole and the antimonopole will be found along the
+same horizontal or vertical line). Secondly, we also con-
+sider a string-path of form 2 (for h < h 1), making the
+separated monopoles to be found in different lines (di-
+agonally positioned; now we have X= 2R/√
+2). More
+two equivalent ways were studied for h > h 1in which
+GS2is the ground-state. In this case, however, dif-
+ferently from the situation in the GS1state, now the
+monopoles can be separated by using a linear string-path
+(so that X=R) without any violation of the ice rule.
+Finally, another monopoles separation studied for GS2
+is the “diagonal-path” (or path 3), in which the charges
+areput in different lines. Our analysisshowsthat besides
+the Coulombic-type term q(h)/R(whereq=µ0
+4πq1q2<0
+is the a coupling constant which gives the strength of
+the interaction), the total energy cost of a monopole-
+antimonopole pair has an extra contribution behaving
+likeb(h)X, brought about by the string-like excitation
+that binds the monopoles, say,
+V(R,h) =q(h)/R+b(h)X(R)+V0(h) (2)
+whereV0(h) is ah-depended constant related to the
+monopole pair creation (for instance, for h= 0V0(0)≈
+23D and V(a,0)≈29D). The results for the “charge”
+q(h) are shown in Figure 6 for the range 0 < h < a .
+Whenh < h1, the excitations are considered above GS1,
+and we observe that there is a small h-dependent differ-
+ence in the q-value for paths 1 and 2, which vanishes as
+h→h1. At higher heights, h > h 1,qis valued with
+respect to GS2, and ish-independent for a linear string-
+path. However, for path 3, it comes back to increase
+ashincreases. Therefore, the interaction of a monopole
+with its partner (antimonopole) is anisotropic in artifi-
+cial spin ices. Perhaps, it would be more appropriate to
+redefine things in such a way that q=µ0
+4πQ1Q2α(h,φ),
+whereq1q2=Q1Q2α(h,φ) and the actual value of the
+chargesQ1=−Q2is independent of the angle φthat
+the line connecting the poles makes with the x-axis. In5
+0 0,2 0,4 0,6 0,8 1
+h0246810
+bPath 1
+Path 2
+Linear path
+Path 3
+GS1GS2
+FIG. 7: (Color online) The string tension for the two string
+shapes shown in Fig.5 for h < h 1and for a linear string-path
+and path 3 (diagonal) for h > h 1. The green dot and the
+dashed lines represents an extrapolation of our data.
+this case, the anisotropy of the interaction (coming from
+the background) is implicitly considered in the function
+α(h,φ) but its complete expression was not evaluated
+here. Since α(h1,φ) tends to be a constant (indepen-
+dent of φ, we set α(h1,φ) = 1 and so, only around
+the ice regime (i.e., h≈h1), the interaction tends to
+be isotropic. Thus we can find the genuine strength of
+the magnetic charge in this artificial compound as being
+Q1=±/radicalbig
+4π|q(h1)|/µ0≈ ±1.95µ/a, where we have
+used|q(h1)|= 3.8Da. Just for effect of comparison, us-
+ingsomeparametersofRef.9suchas a= 320nm,wegeta
+charge value which is about 80 times larger than the typ-
+ical value found for the original 3 dspin ices1(or about
+100 times smaller than the Dirac fundamental charge).
+Besides its anisotropy, another interesting fact about the
+Coulombic interaction in the artificial compounds is that
+it jumps at h=h1. Indeed, at this point, qabruptly
+changes from q<≈ −3.8Datoq>≈ −3.4Dawhen the
+linear path is taken into account. Such a discontinu-
+ity may be attributed to the ground-state transition and
+thatabove GS2theCoulombicinteractionbetweenapair
+somewhatincorporatesthe residualmagnetizationstored
+in each vertex. On the other hand, keeping a diagonal
+separation of the monopoles along the ground state tran-
+sition, the magnetic charge parameter qincreases almost
+continuously.
+How the string tension bdepends upon his shown in
+Fig.7. Note that, while GS1is the ground-state( h < h1),
+bdiminishes as hincreases. At higher heights, and being
+evaluated over GS2, the tension remains a non-vanishing
+small constant for linear path and turns back to increase
+for diagonal separation (path 3). In general, since bis
+also a function of φ(i.e.,b(h,φ)), it is more favorable en-
+ergetically that a pole and its antipole reside at the same
+line in the array. It should be remarked that, near the
+ice regime, b(h1,φ) is almost independent of φ(almost
+isotropic limit) and its value is reduced around 20 times
+whenever this modified system is compared to its coun-
+terpart at h= 0 (at zero temperature). In principle, this
+result indicates that free monopoles do not appear in this
+system. Then, the modified array16faces a small obsta-
+cle by the fact that the islands are placed in such a waythat the spins can not point to the center of a tetrahe-
+dron as they do in the 3 dmaterials. Indeed, as pointed
+out before, the spins in the artificial compound point
+along its edges (see Fig.2); the islands are rigid objects
+that do permit the spins to point only along their longest
+axis. This disparity causes an ordering in the artificial
+material, which diminishes as hincreases; eventually it
+becomes tiny but persists at h=h1. This persistent or-
+dering contributes for the residual string tension at the
+ice regime and also for the different string tension values
+as the monopoles are located at different angular posi-
+tions in the array. Such a difficulty may be overcome
+when one takes the limit l→ain the modified array.
+As pointed out in Ref.10, the mechanism responsible for
+the equivalence between the artificial (2 d) and natural
+3dspin ices is not operational in d= 2, as it requires
+also the dimensionality of the dipolar interaction to coin-
+cide with that of the underlying lattice. Here, we have a
+d= 3 dipolar (1 /r3) and Coulombic (“monopolar”, 1 /R)
+interactions in a two-dimensional array. Independent of
+this, since the state GS1is metastable one could imagine
+if the excitations could be considered to lie in GS1forh
+slightly greater than h1. In this case the extrapolation
+of our results indicate that the string tension may vanish
+ath≈0.502a(see Fig. 7).
+III. SUMMARY
+In summary, we have investigated the energetics of the
+modified artificial spin ice expressing several quantities,
+suchasgroundstatesenergy,magneticchargesandstring
+tension, as a function of the height offset h. Our analysis
+showthat theground-statechangesfromanorderedanti-
+ferromagneticto aferromagneticoneat h=h1≈0.444a,
+which is in good agreement with the value obtained in
+Refs. 10,16, hice≈0.419a. We claim that such a small
+difference comes about from the fact that in these cited
+works,authorsassumedequalnearest-neighborandnext-
+nearest-neighbor interactions, whereas we have taken all
+the dipole interactions into account. For the excitations
+above the ground-state we have found that the magnetic
+charges interact through the Coulomb potential added
+by a linear confining term with tension b(h), which de-
+creases rapidly as hincreases, from 0 to h1, assuming
+a non-vanishing constant value at higher h. Actually,
+the system presents an anisotropy that manifests itself in
+both the Coulombic and linear interactions and it tends
+todiminishas hincreases,almostdisappearingat h=h1.
+Thesourceofthisanisotropyisaresidualordering,which
+still persists even in the ice regime (at h=h1for point-
+like dipoles). Ordering and anisotropy may disappear
+completely in the ideal limit l→a,h→0. Another
+interesting result is that the magnetic charge jumps, de-
+pending on the direction in which the monopoles are sep-
+arated,asthe systemundergoesatransitionin itsground
+state. For a separation of the monopoles, vertex by ver-
+tex, along the same line of vertices, which is possible6
+only for the GS2ground state, the coupling qexhibits
+considerably discontinuity in relation to its limit value
+in theGS1ground state. On the other hand, it tends
+to grow up continuously for the diagonal path along the
+transition. Although the residualorderingleadsto a con-
+fining scenario for monopoles, its very small strength,
+whenever h≈h1, signalizes a significant tendency of
+monopole-pair unbinding at a critical (optimal) height
+offset, even at zero-temperature. Further improvements
+in model (1), for instance, taking the actual finite-size
+of the dipoles into consideration, could shed some extra
+light to this issue. Additionally, temperature effects may
+also facilitate the conditions for free monopoles. Indeed,
+the string configurational entropy is also proportional to
+the string size and therefore, at a critical temperature14,on the order of ba, the monopoles may become free. In
+view of that, for small b, the monopoles should be found
+unbind at very low temperatures. As a final remark we
+would like to stressthat these results show that the back-
+ground configuration of spins has a deep effect in the
+charges interactions, being responsible for the string ten-
+sion, anisotropies and a kind of screening of the charges.
+Acknowledgments
+The authors thank CNPq, FAPEMIG and CAPES
+(Brazilian agencies) for financial support.
+∗Electronic address: lucasmol@ufv.br
+†Electronic address: winder@ufv.br
+‡Electronic address: apereira@ufv.br
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\ No newline at end of file
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+An Ideal Mean-Field Transition in a Modulated Cold Atom System
+Myoung-Sun Heo,1Yonghee Kim,1Kihwan Kim,1Geol Moon,1Junhyun
+Lee,1Heung-Ryoul Noh,2M. I. Dykman,3,∗and Wonho Jhe1,†
+1Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
+2Department of Physics, Chonnam National University, Gwangju 500-757, Korea
+3Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
+(Dated: November 17, 2018)
+We show that an atomic system in a periodically modulated optical trap displays an ideal mean-
+field symmetry-breaking transition. The symmetry is broken with respect to time translation by the
+modulation period. The transition is due to the interplay of the long-range interatomic interaction
+and nonequilibrium fluctuations. The observed critical dynamics, including anomalous fluctuations
+in the symmetry broken phase, are fully described by the proposed microscopic theory.
+PACS numbers: 05.70.Fh, 05.70.Ln, 32.10.Ee, 05.40.-a
+The mean-field approach has been instrumental for de-
+veloping an insight into symmetry breaking transitions in
+thermal equilibrium systems [1]. It has been broadly ap-
+plied also to nonequilibrium systems, the studies of pat-
+tern formation being an example [2]. However, close to
+the phase transition point the mean field approximation
+usually breaks down. This happens even in finite-size
+systems provided they are sufficiently large.
+In this paper we study a nonequilibrium system of
+∼107particles which, as we show, displays an ideal
+mean-field symmetry breaking transition. It is accompa-
+nied by anomalous fluctuations, which are also described
+by a mean-field theory. The system is formed by moder-
+ately cold atoms in a magneto-optical trap (MOT) [3, 4].
+The atoms are periodically modulated in time [5]. Peri-
+odically modulated systems form one of the most impor-
+tant classes of nonequilibrium systems, both conceptually
+and in terms of applications. They have discrete time-
+translation symmetry: they are invariant with respect to
+time translation by modulation period τF. Nevertheless,
+they may have stable vibrational states with periods that
+are multiples of τF, in particular 2 τF. Period doubling is
+well-known from parametric resonance, where a system
+has two identical vibrational states shifted in phase by
+π. It is broadly used in classical and quantum optics and
+has attracted significant attention recently in the context
+of nano- and micro-mechanical resonators [6–9].
+In a many-body system, dynamical period doubling in
+itself does not break the time translation symmetry. This
+is a consequence of fluctuations. Even though each vi-
+brational state has a lower symmetry, fluctuations make
+the states equally populated and the system as a whole
+remains symmetric. However, if as a result of the inter-
+action the state populations become different, the sym-
+metry is broken. It is reminiscent of the Ising transition
+where the interaction leads to preferred occupation of
+one of the two equivalent spin orientations, except that
+∗Electronic address: dykman@pa.msu.edu
+†Electronic address: whjhe@snu.ac.krthe symmetry is broken in time. For atoms in a para-
+metrically modulated MOT spontaneous breaking of the
+discrete time-translation symmetry was observed exper-
+imentally by Kim et al. [10].
+Here we show that the time-translation symmetry
+breaking in a modulated MOT results from the coop-
+eration and competition between the inter-particle inter-
+action and the fluctuations that lead to atom switching
+between the vibrational states. The interaction is weak.
+On its own, it cannot change the state populations. How-
+ever, it may become strong enough to change the rates
+of fluctuation induced switching, which in turn will cause
+the population change. We provide a quantitative theory
+of the phenomenon. We measure the critical exponents
+and the frequency dispersion of the susceptibility. The
+observations are consistent with the mean-field behavior
+and are in good agreement with the theory.
+In the experiment,85Rb atoms in a MOT were cooled
+down to≈0.4 mK. Full three-dimensional confinement
+was achieved with three pairs of counter-propagating
+laser beams. The intensity of the beams along the MOT
+axiszwas 0.19 mW/cm2, the transverse beam intensities
+were 5 to 10 times larger. The magnetic field gradient at
+the trap center was 10 G/cm. The transverse beams were
+detuned from the atomic transition by δ≈−2.3Γp, the
+longitudinal beams were further detuned by 5 MHz (the
+atomic decay rate is Γ p/2π≈5.9 MHz). The atomic
+cloud motion was essentially one-dimensional, along z-
+axis, with frequency ω0/2π≈45 Hz and the damping
+rate Γ≈36 s−1. The total number of trapped atoms Ntot
+was varied by changing the hyperfine-repumping laser in-
+tensity at a decrease rate of 0.5% per second. For the or-
+der parameter and the variance measurements very close
+to criticality, the decrease rate was reduced to 0.03%.
+The beam intensities were modulated at frequency
+ωF= 2ω0by acousto-optic modulators. When the modu-
+lation amplitude exceeded a threshold value, after a tran-
+sient, the atomic cloud split into two clouds [5], which
+were vibrating in counter-phase with frequency ωF/2, see
+Fig. 1a. We took snapshots at the maximum separation
+of the clouds at the frame rate of 1-50 Hz to obtain thearXiv:1001.0749v1  [cond-mat.stat-mech]  5 Jan 20102
+population of each cloud. For a small total number of
+trapped atoms Ntotthe populations were equal and the
+cloud vibrations had equal amplitudes.
+FIG. 1: (Color) (a) The two-dimensional density profiles of
+the vibrating clouds for the maximal displacement along the
+MOT axis (≈4.5 mm). The lower, middle, and upper panels
+refer to the symmetric, close to critical, and broken-symmetry
+states, with Ntot≈1.5×106, 5.2×106, and 6.7×106, respec-
+tively;Nc≈5.6×106. The cloud centers are indicated by
+the ticks at the bottom of the panels and are sketched in the
+right panel. (b) The amplitude changes of the cloud vibra-
+tions as functions of the population of the other cloud. The
+red dashed lines show the expected linear dependence. The
+black dashed line shows N1=N2=Nc/2. The amplitude
+of the cloud with smaller population beyond the transition,
+cloud 1, monotonically increases with Ntot, whereas that of
+the larger cloud, cloud 2, is maximal at criticality. The error
+bars show the standard deviation for 20 measurements.
+OnceNtotexceeded a critical value Nc(Nc∼107in
+our experiment), the average populations of the clouds
+became different, as observed earlier [10]. The vibra-
+tion amplitudes became different, too, see Fig. 1b. This
+is spontaneous breaking of the discrete time-translation
+symmetry, the system becomes invariant with respect to
+time translation by 2 τF, not the modulation period τF.
+As phase transitions in equilibrium systems, the ob-
+served transition is a many-body effect. This is evidenced
+by the required critical number of atoms, which is finite
+but large, typical of phase transitions in cold atom sys-
+tems [11–15]. However, in contrast to systems in ther-
+mal equilibrium, our system is not characterized by the
+free energy and has a discrete time-translation symme-
+try. Most importantly, there are no long-range corre-
+lations of spatial density fluctuations. The intra-cloud
+density fluctuations are irrelevant, they are uncorrelated
+with the inter-cloud fluctuations, which, as we show, are
+responsible for the transition and display critical slowingdown. As a result, the symmetry breaking is quantita-
+tively described by the mean-field theory.
+The irrelevance of intra-cloud density fluctuations is a
+consequence of the time scale separation and the weak-
+ness of the interaction. The decay of correlations of intr-
+acloud density fluctuations is characterized by the damp-
+ing rate Γ of atomic motion, which is determined by the
+Doppler effect in the MOT [4] and does not change at
+the phase transition. On the other hand, switching be-
+tween periodic states in modulated systems requires a
+large fluctuation and involves overcoming an effective ac-
+tivation barrier [8, 16–20]. For low temperature, switch-
+ing is a rare event. We measured the rates Wnmof inter-
+cloudn→mswitching (n,m = 1,2) directly for low Ntot
+from the decay time of the difference of the cloud pop-
+ulations, which gave W12=W21∼1 s−1. Therefore, in
+the experiment there is a strong inequality between the
+characteristic reciprocal times, ωF/greatermuchΓ/greatermuchWnm.
+During switching an atom is far from the clouds. Its
+interaction energy with individual atoms in the clouds
+is much smaller than kBT. However, the total energy of
+interaction with all atoms in the clouds may exceed kBT,
+and then it changes the inter-cloud switching rate. The
+interaction depends on the density distribution within
+the clouds and ultimately on the cloud populations N1,2.
+The switching rates modification by the atom-atom in-
+teraction is the underlying mechanism of the symmetry
+breaking. Since these are the cloud populations N1,2that
+control the switching rates and that are changed as a re-
+sult of switching, we choose their mean relative difference
+η=/angbracketleftN2−N1/angbracketright/Ntotas the order parameter. A natural
+control parameter is the reduced total number of atoms
+θ= (Ntot−Nc)/Nc.
+Our main observations are presented in Figs. 2 and 3.
+They show that all measured quantities display mean-
+field critical behavior. The order parameter scales as
+|η| ∝θβforθ > 0, withβ≈1/2, see Fig. 2a, and
+η= 0 forθ <0 (this scaling was also seen earlier [10],
+but now we have come much closer to the critical point,
+and the precision is much higher). Fluctuations of the
+population difference are characterized by the variance
+σ2=N−2
+tot/angbracketleft(N2−N1)2/angbracketright−η2. It scales as σ∝|θ|−˜γwith
+˜γ≈1 for positive and negative θ, see Fig. 2b. Equi-
+librium systems at criticality show a strongly nonlinear
+response to the symmetry-breaking static field, like a
+magnetic field at the Ising transition. For our system
+an analogue of such field is an additional periodic driv-
+ing at half the strong-field frequency. With this field,
+the overall time-translation period is 2 τF, and the vibra-
+tional states of period 2 τFbecome non-equivalent [21].
+We studied the effect of the dynamic symmetry breaking
+field at criticality, θ= 0, by asymmetrically modulat-
+ing the counter propagating beams. Fig. 2c shows that
+the cloud population difference scales with the additional
+field amplitude has|η|∝h1/δwithδ≈3, in agreement
+with the mean field theory.
+The critical slowing down should lead to a strong re-
+sponse of the population difference to a field detuned by3
+FIG. 2: (Color) (a) The relative difference of the cloud pop-
+ulationsη=/angbracketleftN2−N1/angbracketright/Ntotas a function of the control
+parameterθ= (Ntot−Nc)/Nc; the critical total number of
+atomsNc= 1.6×107. The solid curves show the mean-
+field theory, η= tanh [(θ+ 1)η]. For 0< θ/lessmuch1|η|∝θβ
+withβ= 0.51±0.01 (inset). (b) The variance of the order
+parameter ˜σ2= 103σ2and its scaling (inset). The critical ex-
+ponents in the symmetric and broken-symmetry phases are,
+respectively, ˜ γ= 1.04±0.21 and ˜γ= 1.11±0.13. (c) The
+order parameter at criticality ( θ= 0) as a function of ampli-
+tudehof the additional modulation at frequency ωF/2;his
+scaled by the strong modulation amplitude. The solid line is
+|η|∝h1/δ, withδ= 3.0±0.8. (d) Order parameter |η|as a
+function of Ntot(in units of 107atoms) and the intensity of
+the additional resonant radiation (scaled by the saturation in-
+tensity, 3.78 mW/cm2). The red curve is the phase transition
+line. The error bars in panels (a) and (c) show the standard
+deviations of 10 measurements.
+a small frequency Ω from ωF/2. This is an analog of ap-
+plying a weak slowly varying field to thermal equilibrium
+systems. The amplitude and phase of the forced oscilla-
+tions ofηat frequency Ω as functions of θare shown in
+Fig. 3. They display characteristic asymmetric resonant
+features, again in excellent agreement with the mean-field
+theory.
+In general, one should not expect that our oscillating
+system can be described by the Landau-type theory with
+standard critical exponents, and in fact the theory has
+to be extended. We explain the observations as due to
+kinetic many-body effects. They turn out to be strong as
+a consequence of the exponential sensitivity of the inter-
+cloud switching rates Wnm.
+If the atom-atom coupling can be disregarded, the
+switching rates are equal by symmetry, W(0)
+12=W(0)
+21≡
+W(0)=Cexp/parenleftbig
+−R(0)/kBT/parenrightbig
+withC∼ωcl, whereωcl≈
+3Γ is the frequency of small-amplitude damped atom
+vibrations about the cloud center. Of primary inter-
+est is the activation energy R(0). For weekly non-
+sinusoidal period-2 vibrations it was calculated and mea-
+sured for single-oscillator systems with cubic nonlinear-
+FIG. 3: (Color online) The amplitude (a) and phase (b) of os-
+cillations of the order parameter ηinduced by an extra modu-
+lation at frequency ωF/2+Ω with Ω=0.1 Hz. The solid curves
+show the theory with the Ntot→0 switching rate used as a
+fitting parameter. The error bars show the standard devia-
+tions of 100 measurements.
+ity [8, 16, 17]. If we use the same model, for the
+present experimental parameters the theory [16] gives
+R(0)/kB≈3.4 mK. This is within a factor of 2 of the
+value extracted from the measured W(0)and estimated
+T, which is satisfactory given that in the present case the
+parametric modulation was comparatively strong (the
+squared MOT eigenfrequency ω2
+0was modulated with rel-
+ative amplitude 0.9), and therefore the vibrations were
+noticeably non-sinusoidal.
+The atom-atom interaction changes the switching ac-
+tivation energy. The change becomes significant only
+because of the cumulative effect of the interaction of a
+switching atom with the atoms in the clouds. For not
+too strong interaction the rate of 1 →2 switching be-
+comes
+W12(N1;Ntot) =W(0)exp[α(N2−N1) +βNtot] (1)
+withα,β∝1/T(to obtain W12one should replace
+N1↔N2). The parameters α,β are determined by
+the generalized work of the interatomic force done on
+an atom during its intercloud switching, see Supplemen-
+tary Material. The probability P1to haveN1atoms
+in cloud 1 for given Ntotcan be found from the bal-
+ance equation. For Ntot/greatermuch1, the stationary probabil-
+ityPst
+1(N1) is a function of the reduced population dif-
+ferencex= (N2−N1)/Ntotand the control parameter
+θ=αNtot−1. It has a sharp maximum whose position
+gives the order parameter η=/angbracketleftx/angbracketright.
+For fixedNtot, near criticality (|x|,|θ|/lessmuch1) the station-
+ary distribution has a characteristic Landau mean-field
+formPst
+1∝exp[−Ntot(x4−6θx2)/12] [22]. For θ < 0
+it has a peak at x=η= 0, which corresponds to equal
+mean cloud populations. For θ > 0 and|θ|/greatermuchN−1/2
+tot
+the symmetry is broken and /angbracketleftx/angbracketright=η=±(3θ)1/2. This
+describes the dependence of ηonθin Fig. 2a. The form
+ofPst
+1explains also the scaling σ2∝|θ|−1of the variance
+σ2=/angbracketleftx2/angbracketright−/angbracketleftx/angbracketright2in Fig. 2b.
+The critical total number of atoms (where θ= 0) is
+Nc=α−1. Thus,Nc∝T. This dependence was tested
+by heating up the atoms with additional resonant radi-
+ation. Indeed, Ncwas found to increase almost linearly4
+with the radiation intensity, see Fig. 2d.
+An extra modulation of the beam intensities
+hcos(ωFt/2+φh) leads to factors exp( h12) and exp(−h12)
+in the switching rates W12andW21, withh12∝h/T[21].
+In calculating these factors one can disregard the weak
+atom-atom interaction. As a result, Pst
+1is multiplied by
+exp(Ntoth12x). This gives η∝h1/3at criticality ( θ= 0),
+in agreement with the experiment. If the frequency of
+the extra modulation is slightly detuned from ωF/2, the
+linear response does not diverge at criticality. The result
+of the mean-field calculation in the Supplementary Ma-
+terial is in excellent agreement with the experiment, as
+seen from Fig. 3.
+An independent estimate of the interaction strength
+can be obtained from the dependence of the vibration
+amplitudes of the clouds on the cloud populations. For
+comparatively weak interaction, the interaction-induced
+change of the amplitude of an nth cloud should be pro-
+portional to the number of atoms in the other cloud N3−n
+(n= 1,2). This is indeed seen in Fig. 1c. The main
+contribution comes from the long-range interaction, the
+shadow effect [4, 23, 24]. This interaction also gives the
+main contribution to the parameters α,βand thus deter-
+mines the critical total number of atoms Nc. The value of
+Ncobtained in a simple one-dimensional model [16, 21]
+that assumes sinusoidal vibrations is within a factor of 2
+from the experimental data, as are also the slopes of the
+straight lines in Fig. 1c.
+In the experiment, the total number of trapped atoms
+was slowly fluctuating. The mean-field theory still re-
+mains applicable, but needs to be extended, see Sup-
+plementary Material. An important consequence is that
+in the low-symmetry phase, 1 /greatermuchθ/greatermuchN−1/2
+c, the vari-anceσ2is modified, Ncσ2= (2θ)−1+ 3(4θ+ε)−1, where
+ε=Wout/W(0)exp(βNc),ε/lessmuch1, withWoutbeing the
+probability for an atom to leave the trap per unit time.
+The variance is larger than in the symmetric phase for
+the same|θ|by factor 5/4 for θ/greatermuchεand is smaller by
+factor 2 for smaller θ, i.e., closer to Nc. The scaling
+σ2∝θ−1holds in the both limits. This is the scaling
+seen in Fig. 2b; we could not come close enough to Ncto
+observe the crossover; σ2in the broken-symmetry phase
+remained larger than in the symmetric phase.
+In conclusion, we demonstrate the occurrence of an
+ideal mean-field transition far from equilibrium, the
+spontaneous breaking of the discrete time-translation
+symmetry in a system of periodically modulated trapped
+atoms. The mean-field behavior is evidenced by the crit-
+ical exponents of the order parameter and its variance,
+as well as the nonlinear resonant response at criticality
+and the susceptibility as a function of the distance to the
+critical point. We explain the effect as resulting from the
+interplay of the interaction and nonequilibrium fluctua-
+tions, where the interaction, even though comparatively
+weak, is strong enough to affect the rates of fluctuation-
+induced transitions between coexisting vibrational states.
+The proposed theory is in full agreement with the obser-
+vations.
+We thank M. C. Cross, G. S. Jeon, W. D. Phillips, D.
+L. Stein, and T. G. Walker for helpful comments. This
+work was supported by the Acceleration Research Pro-
+gram of Korean Ministry of Science and Technology. MD
+was supported in part by NSF grant PHY-0555346. WJ
+acknowledges support from Yeonam Foundation.
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+64, 408 (1990).
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+Phys. Rev. A 78, 013408 (2008).Supplementary Material for “An Ideal Mean-Field Transition in a Modulated Cold
+Atom System”
+Myoung-Sun Heo,1Yonghee Kim,1Kihwan Kim,1Geol Moon,1
+Junhyun Lee,1Heung-Ryoul Noh,2M. I. Dykman,3and Wonho Jhe4
+1Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
+2Department of Physics, Chonnam National University, Gwangju 500-757, Korea
+3Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
+4Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
+(Dated: November 17, 2018)
+PACS numbers:
+I. THEORY OF INTER-CLOUD SWITCHING
+A. Single-atom switching
+Here we provide a detailed closed-form description of
+the critical phenomena in a parametrically modulated
+MOT. We start with the single-atom dynamics for reso-
+nant MOT modulation, where the modulation frequency
+ωFis close to twice the eigenfrequency ω0of atomic vi-
+brations in the MOT. The modulation can lead to the
+onset of period-two vibrations at frequency ωF/2. These
+vibrations are close to sinusoidal for moderately strong
+modulation, there is no dynamical chaos in the range of
+phase space of interest. In the single-atom approximation
+the equation of motion can be written as
+ma¨r=G(r,˙r;t) +f(t) (1)
+where ris the position vector of an atom, mais the
+atomic mass, and Gis the well-understood force from the
+laser beams [1–3]; for modulated beams it periodically
+depends on time, G(t+τF) =G(t), withτF= 2π/ωF
+being the modulation period. The force f(t) in Eq. (1)
+is the zero-mean noise from spontaneous light emis-
+sion, which is often modeled by white Gaussian noise,
+/angbracketleftfκ(t)fκ/prime(t/prime)/angbracketright= 4ΓmakBTδκκ/primeδ(t−t/prime), where Γ is the
+effective friction coefficient, Tis the characteristic gas
+temperature, and κ,κ/prime= 1,2,3 enumerate the atomic
+coordinates.
+We are interested in the parameter range where, in
+the absence of noise, Eq. (1) has two stable periodic so-
+lutions r1,2(t), with rn(t+ 2τF) =rn(t) (n= 1,2) and
+r2(t) =r1(t+τF). These solutions give the positions
+of the centers of the vibrating atomic clouds in Fig. 1a
+of the main paper. By linearizing Eq. (1) about rn(t)
+one finds the typical frequencies ωcland relaxation rates
+(∼Γ) of the intracloud motion. Noise leads to fluctua-
+tions about rn(t) with variance∝Tthat determines the
+size of the clouds in Fig. 1.
+Switching between the clouds results from large rare
+fluctuations. It is central for the analysis that the switch-
+ing rates are much smaller than ωFand thanωcl,Γ.
+Therefore of interest are period-averaged rates Wnmofn→mswitching,n,m = 1,2. Quite generally [4, 5]
+Wnm=CWexp(−Rn/kBT), (2)
+where the prefactor CW∼max(ωcl,Γ). The most im-
+portant inWnmis the exponential factor, which has the
+activation form. The activation energy Rnof switching
+from the periodic state nis given by the solution of a
+variational problem Rn= minR, where the functional
+Ris [5],
+R= (8Γma)−1/integraldisplay
+dtf(t)2+/integraldisplay
+dtλ(t) [m¨r−G−f(t)] (3)
+Here, the term quadratic in fdetermines the probability
+density of a realization of the random force that leads to
+switching [6]; λ(t) is the Lagrange multiplier that relates
+the dynamics of the atom and the force to each other.
+For switching from nth periodic state, the trajectory r(t)
+should go from rn(t) fort→−∞ to the unstable periodic
+saddle-type state rb(t) on the boundary of the basin of
+attraction to rn(t) fort→∞ ;λ(t),f(t)→0fort→±∞ .
+Since in the single-atom case the period-two vibra-
+tional states differ only by phase, the activation ener-
+giesR1,2are equal,R1=R2=R(0). The variational
+problem is significantly simplified if the vibrations rn(t)
+are almost sinusoidal, which corresponds to small damp-
+ing and comparatively weak nonlinearity, see below. For
+a one-dimensional weakly nonlinear oscillator R(0)was
+found and measured earlier [7–9].
+B. Switching rate modification by the atom-atom
+interaction
+The interaction between atoms leads to an extra force
+Gint(r) in Eq. (1). This force is weak in the sense that
+it weakly affects the intracloud atomic dynamics. In the
+experiment, the radius of the clouds changed by ∼10%
+where the number of atoms changed by a factor of 2.
+The change of the vibration amplitude was also small,
+see Fig. 1 of the main paper.
+The force Gintdepends on the positions of all atoms.
+In the analysis of the effect of the interaction on the in-
+tracloud dynamics it can be separated into the mean-
+field and fluctuating parts. The mean-field part is duearXiv:1001.0749v1  [cond-mat.stat-mech]  5 Jan 20102
+to the long-range interaction and is described in terms of
+the continuous atomic density [1]. The fluctuating part
+is dominated by short-range elastic collisions. For com-
+paratively low atomic densities and high temperatures
+used in the experiment the random part of Gintis small
+compared to the thermal noise and will be disregarded.
+The inter-cloud interaction is determined by long-range
+forces and can be described in the mean-field approxima-
+tion: the force Gint(r) from another cloud changes only
+slightly when atoms in this cloud shift by an interatomic
+distance.
+From the above arguments, the interatomic force on
+an atom in cloud n,Gint(r;n), can be written as
+Gint(r;n) =/summationdisplay
+m=1,2NmGint(r;n|m), (4)
+whereNmis the number of atoms in cloud m. The force
+Gint(r;n|m) is determined by the density distribution in
+the cloudm, and the terms with n=mandn/negationslash=m
+describe the force from the atoms in the same and in the
+different cloud, respectively.
+Even though Gintis small compared to the single-
+atom force G, it may strongly affect the switching rate,
+leading to the spontaneous symmetry breaking. This
+is because Gintchanges the switching activation energy
+Rn. Extending to the many-atom system the approach
+of Ref. 5, to first order in the interaction we obtain
+Rn=R(0)+R(1)
+nwith
+R(1)
+n=/summationdisplay
+mαnmNm,
+αnm=−/integraldisplay∞
+−∞dtλ(opt)
+n(t)Gint/parenleftbig
+r(opt)
+n(t);n|m/parenrightbig
+.(5)
+Here, r(opt)
+n(t) and λ(opt)
+n(t) describe the trajectory that
+minimizes functional R, Eq. (3), for switching from state
+n. Function r(opt)
+n(t) gives the most probable path fol-
+lowed in switching; for a parametrically excited oscillator
+this path was recently observed in experiment [10]. Since
+the states 1 and 2 differ only in phase, we have α11=α22
+andα12=α21.
+The following arguments were used in deriving Eq. (5).
+First, because the interatomic interaction is weak, atoms
+switch one by one, not in groups; switchings of individual
+atoms are uncorrelated. Second, and most importantly,
+in the optimal fluctuation leading to switching of an atom
+the atomic clouds practically do not change their shape.
+This is also because the interatomic interaction is weak,and the force from a single atom in the cloud very weakly
+affects the motion of the switching atom, |αnm|/lessmuchkBT.
+Only the effect from all atoms in the cloud is apprecia-
+ble, the product αnmNm(N1,2/greatermuch1) can become larger
+thankBT. On the other hand, a significant change of
+the shapes of the clouds, even though it would change
+the switching activation energy, has an exceedingly small
+probability, which can not compensate the decrease of
+R(1)
+1,2. Indeed, if the cloud radius is a0∼(kBT/maω2
+cl)1/2,
+the probability density to have the atoms in nth cloud
+displaced by ∆ rn/greatermucha0is∝exp(−Nn∆r2
+n/2a2
+0). There-
+fore in calculating Gint/parenleftbig
+r(opt)
+n(t);n|m/parenrightbig
+one should assume
+that the atoms in clouds 1 and 2 are at their stable equi-
+librium positions r1(t) and r2(t). In other words, there is
+no fluctuational barrier preparation [11, 12] for intercloud
+switching, even though the switching barrier is strongly
+affected by the many-body interaction. This significantly
+simplifies the explicit calculation.
+Equation (5) shows that the effective switching activa-
+tion energy linearly depends on the numbers of atoms in
+the clouds, for weak interatomic coupling. Even though
+|R(1)
+n|is small compared to the single-atom activation en-
+ergyR(0), it can largely exceed kBT, in which case the
+change of the switching rate will be large. From Eqs. (2)
+and (5) we obtain for the many-atom switching rate
+Wnm≡Wnm(Nn;Ntot) =W(0)e(α+β)Ntot−2αNn,
+Ntot=N1+N2, (6)
+whereW(0)=CWexp(−R(0)/T) is the switching rate in
+the neglect of the interatomic interaction; α= (α11−
+α12)/2kBTandβ=−(α11+α12)/2kBT. Equation (6)
+coincides with Eq. (1) of the paper and provides a general
+form of the coefficients α,β.
+II. THE SYMMETRY BREAKING
+TRANSITION
+A. The master equation
+Because atoms move fast within the clouds compared
+to inter-cloud switching, on time scale ∼W−1
+nmthe state
+of the system is fully characterized by the numbers of
+atoms in each cloud N1,N2. For a given Ntot, the system
+is described by probability P1(N1) to haveN1atoms in
+cloud 1 at time t. This probability can be found from
+the master equation
+˙P1(N1) =−[µ(N1) +ν(N1)]P1(N1) +ν(N1−1)P1(N1−1) +µ(N1+ 1)P1(N1+ 1),
+µ(N1) =N1W12(N1;Ntot), ν(N1) = (Ntot−N1)W12(Ntot−N1;Ntot). (7)
+Here we have taken into account that, as a result of a
+n→mtransition, the number of atoms in cloud nde-creases by 1 and in cloud mincreases by 1, and that the3
+total probability of such a transition per unit time for Nn
+atoms in cloud nisNnWnm.
+Prior to making a transition the atom will move ran-
+domly within the cloud for a long time and will com-
+pletely “forget” its initial state. It is this randomization
+that makes the many-particle system zero-dimensional
+and leads to the mean-field approximation being exact.
+The stationary solution of Eq. (7) is
+Pst
+1(N1) =Z−1/parenleftbiggNtot
+N1/parenrightbigg
+exp [−2αN1(Ntot−N1)],(8)
+whereZ≡Z(Ntot) is the normalization constant given
+by condition/summationtext
+N1P1(N1) = 1.
+For largeNtot, the function Pst
+1(N1) has one or two
+sharp peaks. The location of the peak(s) depends on the
+total number of atoms. We will study the critical region
+whereNtotis close to 1 /α, in which case near the peak(s)
+|N1−N2|/lessmuchNtot. It is convenient then to introduce a
+quasi-continuous variable x= (N2−N1)/Ntot= 1−
+2N1/Ntot. For|x|/lessmuch1 from Eq. (8) we find
+Pst
+1(N1)≈˜Z−1exp/bracketleftbig
+−Ntot(x4−6θx2)/12/bracketrightbig
+,
+θ=αNtot−1, (9)
+where ˜Zis the normalization constant.
+Equation (9) has the standard form of the mean-field
+probability distribution near a symmetry-breaking tran-
+sition [13]. The parameter θis the control parameter,
+which plays the role of the deviation from the critical
+temperature in systems in thermal equilibrium, whereas
+xplays the role of the order parameter. Parameter θlin-
+early depends on the total number of atoms; it also lin-
+early depends on the reciprocal temperature. The critical
+number of atoms Ncis determined by condition θ= 0,
+Nc= 1/α∝T. (10)
+B. Critical exponents
+The mean reduced difference of the cloud populations
+η=/angbracketleftx/angbracketrightis determined by the position x0of the maximum
+of the distribution Pst
+1(N1). Forθ <0 the distribution
+has one maximum at x0=η= 0, whereas at θ > 0 it
+has two symmetric maxima. The system occupies one of
+them, which corresponds to the spontaneous symmetry
+breaking. Close to the critical point
+η=N−1
+tot/angbracketleftN2−N1/angbracketright=±(3θ)1/2for 0<θ/lessmuch1,(11)
+which is the familiar mean-field scaling with the critical
+exponent 1/2.
+Close, but not too close to the critical point, 1 /greatermuch
+|θ| /greatermuchN−1/2
+c, from Eq. (9) we obtain for the mean
+square fluctuations of the populations difference σ2=
+N−2
+tot/angbracketleft(N2−N1)2/angbracketright−η2
+σ2= (Nc|θ|)−1forθ<0;
+σ2= (2Nc|θ|)−1forθ>0 (12)(|θ|/lessmuch 1). This shows the familiar θ−1scaling of the
+variance of the order parameter on the both sides of
+the critical point. At the critical point, θ= 0, we have
+σ2= 2π√
+6N−1/2
+c/Γ(1/4)2≈1.2N−1/2
+c.Note that/angbracketleftx2/angbracketright
+remains finite in a finite-size system at the critical point,
+but its dependence on Ntot≈Ncis given by factor N−1/2
+tot
+instead ofN−1
+totfar from criticality.
+C. Response to the symmetry-breaking field
+The symmetry of the period-two vibrational states can
+be lifted if the system is driven by an extra additive force
+h(t) =hcos(ωFt/2 +φh). Such force is an analog of a
+static force in a thermal equilibrium system, it is “static”
+in the frame oscillating at frequency ωF/2. In the experi-
+ment the force was produced by modulating the counter-
+propagating laser beams at frequency ωF/2 with different
+amplitude; the phase φhis counted off from the phase of
+the strong modulation with period τF. Response to a
+period-2τFforce becomes strongly nonlinear at the criti-
+cal pointθ= 0 even for small amplitude |h|, because the
+force modifies the switching activation energies R1,2[14].
+The latter can be found from Eq. (3) with G→G+h(t).
+Extending the analysis of Refs. 5, 14, we obtain to first
+order in the inter-atomic interaction and the symmetry-
+lifting field
+Rn=R(0)+R(1)
+n+R(h)
+n,
+R(h)
+n=−/integraldisplay
+dtλ(opt)
+n(t)h(t) =¯hcosφn.(13)
+The terms R(h)
+1,2are proportional to the field amplitude,
+¯h∝|h|. The phases φ1,2are the same as for a single
+parametrically modulated oscillator additionally driven
+by a force at frequency ωF/2 [14]. They are linear in φh,
+withφ2=φ1+π. This can be immediately seen from
+Eq. (13), since by symmetry λ(opt)
+2(t) =λ(opt)
+1(t+τF),
+whereas h(t+τF) =−h(t). As a consequence, R(h)
+1=
+−R(h)
+2. The difference in the activation energies of 1 →2
+and 2→1 transitions lifts the degeneracy of the states
+1 and 2, making their populations different even in the
+absence of the inter-atomic coupling.
+Even though|R(h)
+n|/lessmuchR(0)for small field amplitude
+|h|, the ratioR(h)
+n/kBTshould not be small, and there-
+fore the field-induced change of the switching rates may
+be substantial. At the same time, the change of the am-
+plitudes of cloud vibrations remains small for small hand
+will not be discussed. Equation (13) for R(h)
+nresembles
+the change of the free energy of an Ising spin by a mag-
+netic field∝¯htilted by an angle φ1with respect to the
+quantization axis.
+Formally, the terms R(h)
+1,2lead, respectively, to fac-
+tors exp[−(¯h/kBT) cosφ1] and exp[( ¯h/kBT) cosφ1] in the
+rate coefficients µ(N1) andν(N1) in master equation (7).
+As a result the stationary distribution Pst
+1(N1), Eq. (9),4
+acquires an extra factor
+Pst
+1(N1)→Pst
+1(N1) exp/bracketleftbig
+−(Ntot¯hx/kBT) cosφ1/bracketrightbig
+(14)
+[the normalization constant in Pst
+1is changed appropri-
+ately; we remind that x= (N2−N1)/Ntot.].
+The equation for the maximum of the distribution
+x0=ηnear criticality now becomes θx0−1
+3x3
+0−
+(¯h/kBT) cosφ1= 0.For small|¯h/kBT|/lessmuch1 and not too
+close to the phase-transition point, 1 /greatermuch|θ|/greatermuchN−1/2
+c, we
+have
+η≈θ−1(¯h/kBT) cosφ1(θ<0), (15)
+η≈±(3θ)1/2−(2θ)−1(¯h/kBT) cosφ1(θ>0).
+The generalized resonant susceptibility Ξ(0) = −dη/d ¯h
+scales as|θ|−1on the both sides of the phase transition
+point and diverges for θ→0. We note that, even though
+we consider a system far from equilibrium and the sus-
+ceptibility is calculated with respect to a time-dependent
+field, forNtot= const it is simply related to the variance
+of the order parameter σ2as given by Eq. (12).
+At the phase transition point, θ= 0, there is no linear
+static susceptibility, and η=x0∝¯h1/3, as seen in the
+experiment.
+D. Frequency dispersion of resonant response
+The above analysis can be extended also to the case of
+an extra additive force with frequency ( ωF/2) + Ω that
+differs from, but remains close to ωF/2, i.e.,|Ω|/lessmuchΓ/lessmuch
+ωF. To find the distribution P1near the maximum, one
+can transform Eq. (7) into a Fokker-Planck equation by
+expandingP1(N1±1)−P1(N1)≈±∂N1P1+ (1/2)∂2
+N1P1
+and using similar expansions for µ,ν[15]. In the absence
+of extra modulation, for small x= (N2−N1)/Ntotto
+leading order in xthe equation reads
+˜W−1˙P1=ˆLP1, ˜W=W(0)exp(βNtot),
+ˆLP1=−2∂x/bracketleftbig/parenleftbig
+θx−1
+3x3/parenrightbig
+P1/bracketrightbig
++ 2N−1
+tot∂2
+xP1.(16)
+The stationary solution of Eq. (16) Pst
+1has the form (9).
+An extra additive force hcos/bracketleftbig/parenleftbig1
+2ωF+ Ω/parenrightbig
+t/bracketrightbig
+with small
+Ω adiabatically modulates the rates Wnmof the inter-
+cloud transitions, i.e., one can think of the rates Wnmas
+parametrically dependent on time. The major contribu-
+tion to the rate modulation comes from the modulation
+of the activation energies, which is described by Eq. (13)
+with
+R(h)
+1≡R(h)
+1(t) =−R(h)
+2(t) =¯hcos(Ωt) (17)
+(we have disregarded a time-independent term in the ar-
+gument of the cosine; for quantities that do not oscillate
+at frequency ωF/2, like cloud populations, this term can
+be eliminated by changing time origin).
+For weak modulation one can linearize the transition
+rates in ¯hand consider the terms ∝¯has a perturba-
+tion. Then the distribution can be sought in the formP1=Pst
+1+δP1, where the field-induced term δP1∝¯his
+oscillating at frequency Ω. It is given by equation
+˜W−1δ˙P1−ˆLδP 1
+= 2¯h
+kBTcos(Ωt)Ntot/parenleftbigg
+θx−1
+3x3/parenrightbigg
+Pst
+1.(18)
+Equation (18) allows one to find the periodically oscil-
+lating term in the order parameter δη(t) =/angbracketleftx/angbracketright−x0(x0is
+the position of the maximum of the stationary distribu-
+tion) and the generalized susceptibility Ξ(Ω), which can
+be defined by expression
+δη(t) =−1
+2kBT/bracketleftbig
+Ξ(Ω) ¯hexp(−iΩt) + c.c./bracketrightbig
+.
+This can be done by multiplying Eq. (18) by xand inte-
+grating over x. Expanding θx−x3/3 in ˆLaboutx0, for
+1/greatermuch|θ|/greatermuchN−1/2
+tot one finds
+Ξ(Ω) = 2 ˜W/(2|θ|˜W−iΩ), θ< 0,
+Ξ(Ω) = 2 ˜W/(4θ˜W−iΩ), θ> 0. (19)
+[in the critical region, in Eq. (16) for ˜Wone should re-
+placeNtotwithNc]. For Ω/negationslash= 0 the susceptibility Eq. (19)
+remains finite even at the phase transition, θ= 0. Here
+Ξ(Ω) = 2i˜W/Ω. The susceptibility diverges with fre-
+quency, with critical exponent equal to −1.
+III. FLUCTUATIONS OF THE TOTAL TRAP
+POPULATION
+In the above analysis we disregarded inelastic collisions
+between trapped atoms and collisions with atoms out-
+side the trap. These processes lead to fluctuations of
+the total number of atoms in the trap. In the exper-
+imental conditions they were slow, with rates an order
+of magnitude smaller than the rate of inter-cloud transi-
+tionsWnm. However, near the critical point inter-cloud
+population fluctuations are slowed down, and then fluc-
+tuations of the total number of trapped atoms Ntotmay
+become important.
+In the presence of fluctuations of Ntotthe state of the
+system should be characterized by the probability P(N)
+[N≡(N1,N2)] to have N1atoms in cloud 1 and N2
+atoms in cloud 2, with Ntot=N1+N2. For compara-
+tively small Ntotand low temperatures used in the experi-
+ment, evaporation from the trap due to inelastic collisions
+between trapped atoms was rare. The major mechanism
+of the change of Ntotwas collisions with atoms outside
+the trap. In a simple model of such collisions, the rate of
+loss of atoms from an nth cloudWoutNnis proportional
+to the number of atoms in the cloud Nn, whereas the
+rate of capture of outside atoms by a cloud WoutN0/2
+is independent of Nnand depends only on the density
+of outside atoms (which determines N0) and the cross-
+section of their capture. We disregard fluctuations in5
+the density of outside atoms and choose parameter N0in
+such a way that it provides the typical scale of Ntot, aswill be seen below.
+The master equation for P(N) has the form
+˙P(N) = ˜WˆLclP+WoutˆLoutP, ˆLclP=−/summationdisplay
+nµ/prime(Nn;Ntot)P(N)
++µ/prime(N1+ 1;Ntot)P(N1+ 1,N2−1) +µ/prime(N2+ 1;Ntot)P(N1−1,N2+ 1), (20)
+ˆLoutP=−(N1+N2+N0)P(N) +/summationdisplay
+n(Nn+ 1)P(N+δn) + (N0/2)/summationdisplay
+nP(N−δn),
+where
+µ/prime(Nn;Ntot) =Nnexp (αNtot−2αNn)≡˜W−1µ(Nn)
+andδ1= (1,0) and δ2= (0,1) [the inter-cloud switching
+rate ˜Wis defined in Eqs. (6), (16)]. The terms ˆLcland
+ˆLoutdescribe inter-cloud transitions and exchange with
+atoms outside the trap, respectively. We assume that the
+typical rate of this exchange Wout/lessmuch˜W.
+We now consider the critical region where Ntotis close
+toNc. We will be interested in the range of N1,N2where
+P(N1,N2) is close to maximum, |N1−N2|/lessmuchNtot, and
+introduce quasi-continuous variables
+x=N2−N1
+N0, u =N1+N2
+N0−1, (21)
+θ=αN0−1.
+As we will see, variables xandθcoincide with the vari-
+ables used earlier if fluctuations of the total number of
+trapped atoms can be disregarded; variable ugives the
+deviation of NtotfromN0.
+For|x|/lessmuch1,|θ|/lessmuch1, and|u|/lessmuch1 to leading order in
+x,θ,u , andN−1
+0the operators ˆLclandˆLoutbecome
+ˆLclP=−2∂x/braceleftbigg/bracketleftbigg
+x(θ+u)−1
+3x3/bracketrightbigg
+P/bracerightbigg
++2N−1
+0∂2
+xP, (22)
+ˆLoutP=∂x(xP) +∂u(uP) +N−1
+0/bracketleftbig
+(∂2
+x+∂2
+u)P/bracketrightbig
+.
+Equations (20) - (22) allow one to study the stationary
+distribution of the modulated system in the presence of
+fluctuations of the total population. We note first that
+exchange of atoms with the surrounding leads to effec-
+tive exchange of atoms between the clouds. In turn, this
+leads to renormalization of inter-cloud transition rates
+and, effectively, of the control parameter,
+θ→θ−Wout
+2˜W, Nc→α−1/parenleftbigg
+1 +Wout
+2˜W/parenrightbigg
+.(23)
+The change of θis negative, which should be expected,
+since exchange with the surrounding should stabilize the
+symmetric phase, and therefore a larger number of atoms
+is required for the symmetry breaking transition than inthe case where there is no such exchange. Ncis now the
+critical value of the parameter N0, the mean number of
+trapped atoms is /angbracketleftNtot/angbracketright=N0(1+/angbracketleftu/angbracketright). The coefficient of
+diffusion over xis also renormalized, 2 N−1
+0→2N−1
+0(1 +
+Wout/2˜W).
+In the stationary regime fluctuations of the total num-
+ber of atoms in the trap are small, /angbracketleftu2/angbracketright∼1/N0. How-
+ever, the distribution over the scaled difference of the
+cloud populations xdiffers from the standard Landau-
+type distribution (9). This is in spite the fact that the
+system is described by the mean-field theory, no spatial
+correlations are involved. The dynamics of xanduare
+very different. There is no critical slowing down for u,
+and in the critical region |θ|/lessorsimilarN−1/2
+c the width of the
+distribution over xis∼N−1/4
+c/greatermuchN−1/2
+c.
+A. The variance of the order parameter
+The major effect of the coupling between fluctuations
+ofxanduis the change of the variance σ2=/angbracketleftx2/angbracketright−/angbracketleftx/angbracketright2.
+The latter occurs in the broken-symmetry state where
+θ/greatermuchN−1/2
+c. It can be calculated in the stationary regime
+by decoupling the chain of equations for the moments of
+the stationary distribution /angbracketleftxkul/angbracketrightwith integer k,l. For
+N−1/2
+c/lessmuchθ/lessmuch1
+Ncσ2≈1
+2θ+3
+4θ+ (Wout/˜W). (24)
+Fluctuations of uandx(i.e., of the total population
+of the clouds and the population difference) are corre-
+lated in the broken-symmetry phase, with /angbracketleftu(x−x0)/angbracketright≈
+(2x0/Nc)[4θ+ (Wout/˜W)]−1, wherex0=±(3θ)1/2.
+In contrast, in the symmetric phase for 1 /greatermuch−θ/greatermuch
+N−1/2
+c fluctuations of uandxare uncorrelated, with
+Ncσ2≈|θ|−1/bracketleftBig
+1 + (Wout/2˜W)/bracketrightBig
+≈|θ|−1.
+It is seen from Eq. (24) that, for θ/greatermuchWout/4˜W,
+the variance of the order parameter fluctuations in the
+broken-symmetry state scales with the control parame-
+ter in the same way as without fluctuations of the total
+population, Ncσ2≈5/4θ. However, the factor in front
+of 1/θis now larger than in the symmetric phase. This
+corresponds to what was seen in the experiment. As θ6
+decreases below Wout/4˜W, there occurs a crossover to
+σ2≈1/2Ncθ, i.e.,σ2scales with θwith the same scaling
+exponent, but with smaller amplitude than in the sym-
+metric phase.
+We note that there are other fluctuations that can con-
+tribute to the variance σ2. For example, slow fluctuations
+of the trap parameters may lead to small and slow fluc-
+tuations of the total population, which can be thought
+of as fluctuations of the control parameter θ. They give
+an extra contribution 3 /angbracketleft(δθ)2/angbracketright/4/angbracketleftθ/angbracketrighttoσ2in Eq. (12) for
+the symmetry-broken phase, where /angbracketleftδθ2/angbracketrightis the variance
+ofθ. Interestingly, this contribution also scales as θ−1
+away from the critical point. It further increases the dif-
+ference between the variance of the order parameter in
+the broken-symmetry and symmetric phase.
+IV. ONE-DIMENSIONAL MODEL
+A. The rotating wave approximation
+Calculation of the switching rates and response coef-
+ficients is simplified if the atomic motion in MOT is as-
+sumed one-dimensional, along the MOT axis z. This cor-
+responds to averaging over the motion transverse to the
+axis, which is largely small-amplitude fluctuations about
+the vibrating centers of the clouds. In the experiment
+the radiation that confined the atoms transverse to the
+axis had large intensity, so that the correlation time of
+these fluctuations was short. In the single-atom approxi-
+mation the atomic motion can be modeled by the Duffing
+oscillator,
+¨z+ 2Γ ˙z+ω2
+0(1 +/epsilon1FcosωFt)z+γz3=f(t)/ma,(25)
+Here,ω0and Γ are the MOT eigenfrequency and viscous
+friction coefficient, respectively, γis the nonlinearity pa-
+rameter, and /epsilon1Fis determined by the amplitude of the
+laser beam modulation [16]. These parameters are given
+in the main paper. The terms nonlinear in the veloc-
+ity, which are disregarded in Eq. (25), are comparatively
+small for the experimental conditions. The function f(t)
+is white thermal Gaussian noise, cf. Eq. (1).
+For the reduced modulation strength /epsilon1Fω0/4Γ>1,
+resonant modulation ( |ωF−2ω0|/lessmuchω0) can excite period-
+two atomic vibrations [17]. The vibrations are nearly si-
+nusoidal in the absence of noise, if the modulation is not
+too strong. The atomic dynamics can be conveniently de-
+scribed by switching to the rotating frame using a stan-
+dard transformation
+z=CRWA[q2cos(ωFt/2)−q1sin(ωFt/2)],(26)
+˙z=−CRWA(ωF/2) [q2sin(ωFt/2) +q1cos(ωFt/2)]
+withCRWA = (2ω2
+0/epsilon1F/3γ)1/2(the modulation sign is cho-
+sen in such a way that /epsilon1F/γ > 0). In the rotating waveapproximation (RWA), the equations for slow variables
+q≡(q1,q2) in slow time τare
+dq
+dτ=K(q) +f(τ), τ =ω0/epsilon1Ft/4. (27)
+Here, f(τ) is white Gaussian noise with two asymp-
+totically independent components, /angbracketleftfκ(τ)fκ/prime(τ/prime)/angbracketright=
+2DτkBTδκκ/primeδ(τ−τ/prime), withκ,κ/prime= 1,2 andDτ=
+6γΓ/maω5
+0/epsilon12
+F. We use vector notations here formally ,
+qis not a vector in real space, its components are com-
+binations of atomic displacement and momentum along
+the MOT axis.
+Functions K(q) are cubic polynomials and are given
+explicitly in Ref. 14 [they were denoted by K(0)(q)]. The
+zeros of Koccur at the stationary states in the rotating
+frame and, in turn, at the period-two vibrational states in
+the laboratory frame. We will be interested in the param-
+eter range where Eq. (27) has two attractors qA
+1=−qA
+2.
+Functions Kare also equal to zero for q=0, which corre-
+sponds to the unstable state of zero-amplitude period-two
+vibrations in the laboratory frame.
+For low temperatures, the motion described by
+Eq. (27) is primarily small-amplitude fluctuations about
+states qA
+1,2. Switching between the states requires a large
+rare fluctuation. The single-atom switching rate has ac-
+tivation form W(0)=Cexp(−R(0)/kBT), Eq. (2), where
+as in Eq. (3), R(0)= minR(0)with
+R(0)= (4Dτ)−1/integraldisplay
+dτf2(τ)
++/integraldisplay
+dτλ(τ)/bracketleftbiggdq
+dτ−K−f(τ)/bracketrightbigg
+. (28)
+The major distinction from the general formulation (3)
+is that function Kdoes not explicitly depend on time.
+This made it possible to solve the variational problem
+(28) and to find both the optimal trajectories followed in
+switching from nth stable state q(opt)
+n(τ) [7] and functions
+λn(τ), which are equal to the logarithmic susceptibilities
+of Ref. 14 multiplied by −1/Dτ.
+B. Interatomic interaction
+The major inter-atomic interaction that affects the
+switching rate is the long-range interaction, since in
+switching atoms move far away from the clouds. It comes
+from the shadow effect, which is due to “shielding” of
+atoms from the laser light by other atoms [1, 2, 18–20].
+In the one-dimensional picture, the force on ith atom
+with coordinate zifromjth atom with coordinate zjcan
+be approximately modeled as Fi=−fsh/summationtext
+jsgn(zi−zj).
+This force is weak, much smaller than the Doppler force
+that confines atoms to the trap. Multiple scattering of
+light is disregarded in the above expression.
+To estimate fshfor a switching atom we have to take
+into account that the atomic clouds are 3-dimensional.
+We consider a simple model in which a laser beam with7
+intensityIpropagating along z-axis passes on its way
+through an atomic cloud with density distribution ρ(r).
+The resulting intensity change as a function of transverse
+coordinates x,yis ∆I(x,y) =IσL/integraltext
+dzρ(r), where inte-
+gration is done over the length of the cloud and σLis
+the absorption cross-section. This cross-section depends
+in the standard way on the light intensity and the fre-
+quency detuning. Generally, because of the magnetic-
+field induced frequency shift and the Doppler shift σL
+oscillates in time. The light intensity Ialso oscillates in
+time. However, for comparatively small modulation am-
+plitudes, we can disregard these oscillations. Then we get
+for typical experimental conditions σL≈5.6×10−15m2.
+The atomic density distribution ρ(r) can be assumed
+Gaussian with the same width wt≈1 mm in all direc-
+tions, which was achieved in the experiment by tuning
+the transverse beam intensities. According to the op-
+timal path picture, during switching atoms most likely
+move along the MOT axis, and for such atoms the light
+intensity change is determined by ρ(r) on the axis. The
+extra force on the switching atom as it moves between
+the clouds is then directed toward the more populated
+cloud and is equal to fsh|N1−N2|, with
+fsh=/planckover2pi1kΓpσLs/4π(s+ 1)w2
+t. (29)
+Here,kis the photon wave number, Γ pis the recipro-
+cal lifetime of the excited state, and s= (I/Is)[1 +
+(2δ/Γp)2]−1is the resonant absorption strength ( Isis the
+saturation intensity and δis the detuning of the radiation
+frequency from the atomic transition frequency). Equa-
+tion (29) gives fsh≈2.5×10−32N. Note that, since
+the clouds are oscillating, the force ∝fshis oscillating
+as well, its time dependence is determined by the sgn-
+function.
+C. The shadow effect in the rotating frame
+The effect of the shadow force can be conveniently an-
+alyzed in the rotating frame by changing from ( zi,˙zi) to
+slowly varying two-component vectors qi, Eq. (26). In
+the RWA the equation of motion for qibecomes
+dqi
+dτ=Ki(qi) + ˆ/epsilon1∂qiHsh+fi(τ) (30)
+(i= 1,...,N tot). Here, fi(τ) is the random force on
+ith atom (random forces on different atoms are uncorre-
+lated), ˆ/epsilon1is the permutation tensor, and Hshis the inter-
+action Hamiltonian in the slow variables,
+Hsh=1
+2/summationdisplay/prime
+ijVsh(qi−qj),
+Vsh(q) =8fsh
+πmaω2
+0/epsilon1FCRWA|q|, (31)where the prime indicates that i/negationslash=j.
+The general expression for the interaction-induced cor-
+rection to the activation energy Rn, Eq. (5), in the RWA
+has the form
+αnm=/integraldisplay∞
+−∞dτλn(τ)ˆ/epsilon1∂qAmVsh(q(opt)
+n(τ)−qA
+m),(32)
+where λn(τ) and q(opt)
+n(τ) are the solutions of the single-
+atom problem of minimizing the functional (28) for
+switching from the state n.
+Equations (28) and (32) were used to find the coef-
+ficientsαnmand the critical value of the total number
+of atomsNc= 2kBT/(α11−α12) where the symmetry
+breaking transition occurs. The obtained Ncwas within
+a factor of 2 from the value of Ncobserved in the exper-
+iment. This is reasonable given the uncertainty in the
+temperature, the renormalization of Ncby the finite life-
+time of atoms in the MOT, cf. Eq. (23), and the fact
+that the light intensity modulation was not weak and,
+respectively, the atomic vibrations noticeably deviated
+from sinusoidal.
+An important effect that allows one to independently
+compare the many-atomic theory with experiment is the
+change of the vibration amplitudes due to the interac-
+tion. This change is determined by the change of the
+equilibrium positions qA
+nin the rotating frame. For weak
+inter-atomic interaction it can be found by linearizing
+equations of motion (30). The average force on an atom
+from other atoms in the same cloud is equal to zero. The
+shift of the equilibrium position is due only to the force
+from the atoms in the other cloud. Disregarding small
+fluctuations about the equilibrium positions, for the shift
+ofnth attractor in the rotating frame δqA
+nwe obtain
+(δqA
+n∂qAn)K/parenleftbig
+qA
+n/parenrightbig
++N3−nˆ/epsilon1∂qAn¯Vsh(qA
+n−qA
+3−n) = 0.(33)
+Here, ¯Vshis given by Eq. (31) for Vshin whichfshis
+replaced with ¯fsh. The quantity ¯fshcharacterizes the
+force from the shadow effect which is averaged over the
+cross-section of the cloud transverse to the MOT axis.
+Such averaging occurs as atoms are moving within the
+cloud. For Gaussian clouds of the same cross-section,
+¯fsh=fsh/2.
+The shift of the nth attractor ( n= 1,2) given by
+Eq. (33) and the corresponding change of the vibration
+amplitude of the nth cloud is proportional to the num-
+ber of atoms in the other cloud, N3−n. Therefore the
+vibration amplitudes of the clouds become different in
+the broken-symmetry state. This prediction and the lin-
+ear dependence of the amplitude change on the number
+of atoms in the other cloud are in full agreement with
+the experiment; the numerical estimate for the simplified
+model of sinusoidal 1D vibrations is within a factor of 2
+from the measured value.8
+[1] D. W. Sesko, T. G. Walker, and C. E. Wieman, J. Opt.
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+Soc. B 9, 2142 (1992).
+[3] A. S. Arnold and P. J. Manson, J. Opt. Soc. B 17, 497
+(2000).
+[4] M. I. Freidlin and A. D. Wentzell, Random Perturba-
+tions of Dynamical Systems (Springer-Verlag, New York,
+1998), 2nd ed.
+[5] M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyan-
+skiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClin-
+tock, Chaos 11, 587 (2001).
+[6] R. P. Feynman and A. R. Hibbs, Quantum Mechanics
+and Path Integrals (McGraw-Hill, New-York, 1965).
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+Lett. 83, 899 (1999).
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+060601 (2007).
+[10] H. B. Chan, M. I. Dykman, and C. Stambaugh, Phys.Rev. Lett. 100, 130602 (2008).
+[11] Y. Kagan and M. I. Klinger, Zh. Eksper. Teor. Fiz. 70,
+255 (1976).
+[12] Y. Kagan, Journal of Low Temperature Physics 87, 525
+(1992).
+[13] L. Landau and E. M. Lifshitz, Statistical Physics. Part 1
+(Pergamon Press, New York, 1980), 3rd ed.
+[14] D. Ryvkine and M. I. Dykman, Phys. Rev. E 74, 061118
+(2006).
+[15] N. G. Van Kampen, Stochastic Processes in Physics and
+Chemistry (Elsevier, Amsterdam, 2007), 3rd ed.
+[16] K. Kim, H. R. Noh, and W. Jhe, Phys. Rev. A 71, 033413
+(2005).
+[17] L. D. Landau and E. M. Lifshitz, Mechanics (Elsevier,
+Amsterdam, 2004), 3rd ed.
+[18] J. Dalibard, Opt. Comm. 68, 203 (1988).
+[19] T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett.
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\ No newline at end of file
diff --git a/1001.0750.txt b/1001.0750.txt
new file mode 100644
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+arXiv:1001.0750v1  [astro-ph.SR]  5 Jan 2010THEASTROPHYSICAL JOURNAL, 2010, IN PRESS
+Preprint typesetusingL ATEX styleemulateapjv. 03/07/07
+A DEEPCHANDRA X-RAY SPECTRUM OF THEACCRETING YOUNGSTAR TWH YDRAE
+N. S. B RICKHOUSE , S. R. C RANMER, A. K. D UPREE, G. J. M. L UNA,ANDS. WOLK
+Harvard-Smithsonian Center for Astrophysics, 60 Garden St reet, Cambridge, MA02138
+Draftversion September 10,2018
+ABSTRACT
+We present X-ray spectral analysis of the accreting youngst ar TW Hydraefrom a 489 ks observationusing
+theChandra High EnergyTransmission Grating. The spectrum providesa r ich set of diagnosticsfor electron
+temperature Te,electrondensity Ne,hydrogencolumndensity NH,relativeelementalabundancesandvelocities
+andrevealsitssourcein3distinctregionsofthestellarat mosphere: thestellarcorona,theaccretionshock,and
+averylargeextendedvolumeofwarmpostshockplasma. Thepr esenceofMg XII,SiXIII,andSiXIVemission
+linesinthespectrumrequirescoronalstructuresat ∼10MK.Lowertemperaturelines(e.g.,fromO VIII,NeIX,
+and MgXI) formed at 2.5 MK appear more consistent with emission from a n accretion shock. He-like Ne IX
+lineratiodiagnosticsindicatethat Te=2.50±0.25MKand Ne=3.0±0.2×1012cm−3intheshock. Thesevalues
+agreewell with standardmagneticaccretionmodels. Howeve r,theChandra observationssignificantlydiverge
+from current model predictions for the postshock plasma. Th is gas is expected to cool radiatively, producing
+OVIIasitflowsintoanincreasinglydensestellaratmosphere. Su rprisingly,O VIIindicates Ne=5.7+4.4
+−1.2×1011
+cm−3,fivetimeslowerthan Neintheaccretionshockitself,and ∼seventimeslowerthanthemodelprediction.
+WeestimatethatthepostshockregionproducingO VIIhasroughly300timeslargervolume,and30timesmore
+emitting mass than the shock itself. Apparently, the shocke d plasma heats the surroundingstellar atmosphere
+tosoftX-rayemittingtemperaturesandsuppliesthismater ialtonearbylargemagneticstructures–whichmay
+be closed magnetic loops or open magnetic field leading to mas s outflow. Our model explains the soft X-ray
+excessfoundinmanyaccretingsystemsaswellasthefailure toobservehigh Nesignaturesinsomestars. Such
+accretion-fedcoronaemaybeubiquitousin theatmospheres ofaccretingyoungstars.
+Subject headings: accretion — stars: coronae — stars: formation — stars: indiv idual (TW Hydrae) — tech-
+niques: spectroscopic— X-rays: stars
+1.INTRODUCTION
+Low mass stars in star-forming regions produce strong X-
+rayemission fromcoronalmagneticactivity,asevidencedb y
+high temperature ( ∼10 MK) emission from flares and active
+regions (Feigelson & Montmerle 1999; Gagné et al. 2004;
+Preibischet al.2005). Theroleofaccretionin theproducti on
+ofX-raysfromyoungstarsisless wellunderstood. Magnetic
+structures on stars can extend all the way to the inner accre-
+tion disk, as suggestedfrom modelsof decayingstellar flare s
+(Favata et al. 2005). Extended X-ray emission from jets also
+occurs, as observed in DG Tau, a system oriented so that the
+circumstellar disk plane aligns with our line of sight (Güde l
+et al. 2008). Soft X-ray emission may arise from shocks at
+the stellar base of this extended X-ray jet (Güdel et al. 2008 ;
+Schneider&Schmitt2008).
+X-rays can also be produced in accreting systems when
+theaccretingmaterialacceleratestosupersonicvelociti esand
+shocks near the stellar surface, heating the gas to a few MK.
+In the standard model of accreting Classical T Tauri Stars
+(CTTS), the magnetic field of the star connects to the accre-
+tion disk near the corotation radius, channeling the accret ion
+streamfromthedisktoasmallareaonthestar(Calvet&Gull-
+bring1998). X-raysfromthe shockmaybe difficultto detect
+iftheshockisformedtoodeepinthephotosphere(Hartmann
+1998)oriftheyareabsorbedbythestreamofpreshockedneu-
+tral or near-neutralgas. Furthermore, the X-ray shock sign a-
+turemaybedifficulttodistinguishfromsoftcoronalemissi on
+without the high resolution spectroscopy necessary to dete r-
+mine temperature and density and identify the emitting re-
+gions. Shockmodelspredicthighelectrondensity Ne(∼1013
+cm−3) at relatively low electron temperature Te(a few MK),
+featuresthat canbetestedwith X-raylineratiodiagnostic s.TW Hydrae (TW Hya) was the first CTTS found to show
+these characteristics of high density and low temperature
+(Kastneret al. 2002;hereafter,K02),based ona 48 ks Chan-
+draHigh Energy Transmission Grating (HETG) spectrum.
+TW Hya is one of the oldest known stars ( ∼10 Myr) still
+in the CTTS (accreting) phase. It is uniquely poised in an
+interesting state of evolution when it will soon stop accret -
+ing,lose its disk, andperhapsformplanets(e.g.,Calvet et al.
+2002). Atadistanceofonly57pc,itisalsooneofthebright-
+est TTauristars inX-rays. Interstellarabsorptionismini mal,
+andtheobservedabsorptioncanbeconsideredintrinsictot he
+system (K02). TW Hya’scircumstellardisk is nearlyface-on
+with an inclination angleof 7o(Qi et al. 2004). Thusthe star
+appearspole-on.
+Keyto K02’sargumentforaccretionistheelectrondensity
+(Ne∼6×1012cm−3)determinedfromHe-likeNe IXemission
+line ratios, and confirmed by subsequent observations with
+theXMM-Newton Reflection Grating Spectrometer (Stelzer
+& Schmitt2004)andthe Chandra Low EnergyTransmission
+Grating (LETG; Raassen 2009). While Nefor some active
+coolstarcoronaeapproachessuchhighvaluesathigh Te(∼10
+MK), e.g., 44 Boo, UX Ari, and II Peg (Sanz-Forcada et al.
+2003;Testaetal.2004),thehigh NeofTWHyaisproducedat
+thesignificantlylower Te(∼3MK)expectedfortheaccretion
+shock. BP Tau (Schmitt et al. 2005), V4046 Sag (Gunther et
+al. 2006), RU Lup (Robrade & Schmitt 2007), and MP Mus
+(Argiroffi et al. 2007) also show high values of the electron
+density. On the other hand, T Tau and AB Aur have a low
+Ne(Güdel et al. 2007a; Robrade & Schmitt 2007); and, the
+binaryHen3-600appearstolieatanintermediatevalueof Ne
+(Huenemoerderet al. 2007). Despite large differences in Ne,
+all of these stars show an excess ratio of soft X-ray emission2 BRICKHOUSE ETAL.
+tohardX-rayemission,manifestedbytheirlargeO VII/OVIII
+ratiosas comparedwith active mainsequencestars (Güdel&
+Telleschi2007;Robrade&Schmitt2007). AdditionalsoftX-
+raydiagnosticsavailablewithadeep Chandraexposuremight
+beexpectedtoshedlightontheaccretionprocessandtest th e
+shockmodels.
+Sinceaccretionisexpectedtoresultinhigherdensitiesth an
+found in an active stellar corona, it is critical that the den sity
+determinationbesecure. InterpretationoftheNe IXlineratio
+in terms of high density is not unique, since photoexcitatio n
+byultravioletradiationmimicstheeffectsofhighdensity . De-
+termining the density from Mg XIis not subject to the same
+degeneracy, since the relevant ultraviolet radiation, obs erved
+withtheFarUltravioletExplorer(FUSE;Dupreeetal.2005) ,
+istoo weakto photoexciteitsmetastable level. The threeHe -
+like systems (O VII, NeIX, and Mg XI) together can provide
+independent determinations not only of the electron densit y,
+but also of the electron temperature and hydrogen column
+density. Deep spectroscopy can, first, put the accretion in-
+terpretationonfirmgroundandsecond,allowustodetermine
+thestructureoftheaccretionregionitself.
+We were awarded a Chandra Large Observing Project to
+establish definitively whether the X-ray emission in TW Hya
+is associated with accretion, using high signal-to-noise r a-
+tio spectroscopy with the HETG, and, if so, to determine
+the emission measuredistribution and elemental abundance s,
+search for X-raysignaturesof infall, outflow, and turbulen ce,
+and constrain the properties of the accretion hot spot. Dur-
+ing theChandra observations we also conducted a ground-
+based observingcampaign,which will be reportedelsewhere
+(Dupree et al. 2010, in preparation). Here, we describe the
+X-ray observations in Section 2 and report analysis results
+in Section 3, where the densities, temperatures, and column
+densities can be extracted from this rich spectrum. Section 3
+also presents elemental abundances derived from fits to the
+spectrum, as well as the measurement of turbulent velocity.
+Section 4 describes spectral predictions from a model of the
+TW Hya shock. In Section 5 we discuss evidence for a new
+typeofcoronalstructure. Section6givesasummaryofresul ts
+andconclusions.
+2.OBSERVATIONSANDSPECTRALANALYSIS
+Chandra observedTWHyawiththeHETGincombination
+with the ACIS-S detector array for a total of 489.5 ks inter-
+mittentlyspanningtheperiodbetween2007Feb 15and2007
+Mar 3. The observation consists of 4 segments, with expo-
+suretimesasfollows: 153.3ks(Obsid7435),157.0ks(Obsid
+7437),158.4ks(Obsid7436),and20.7ks(Obsid7438). The
+summed spectrum has about ten times the exposure time of
+the spectrumanalyzedbyK02. Figure1 showsthetotal first-
+order light curve, with flux variations of about a factor two
+during the observation. Assuming a distance of 57 pc, the
+observed X-ray luminosity LX, measured using the Medium
+EnergyGratingbetween2.0and27.5Å,is1 .3×1030ergs−1.
+One clear flare, with total X-ray luminosity of 2.1 ×1030erg
+s−1, persists for ∼15 ks, producinga total energy of 4 ×1034
+ergs. The flare is retained in the following analysis since it s
+contributiontolinefluxesisnegligible(seeSection3.4).
+We have extracted spectra from the event files, and pro-
+ducedcalibrationfilesusingthe Chandra InteractiveAnalysis
+ofObservations(CIAOver. 3.4)softwarepackage(Fruscion e
+et al. 2006). With response matrices combined from the four
+observations, Sherpa(Freeman et al. 2001) was used to ex-
+tractindividualemissionlinefluxesfromtheco-addedobse r-0 200 1000 1200 1400
+Time (ks)050100150200CountsTotal (divided by 3)
+AccretionCorona
+FIG. 1.— X-ray light curves (first order counts vs time since star t) for the
+fourobserving segments ofthe Chandra observation. Thehashed arearepre-
+sentsagapof650ks. Theuppercurveshowsthelightcurvefor thetotalfirst
+order counts. This light curve includes both line and contin uum emission.
+Themiddlecurveusesthecombined linecountsfromN VII,OVII,OVIII,
+NeIX,FeXVII,andMgXI(identified asthe“accretion” lines usingModel
+D, discussed in Section 3.4). The lower curve uses the combin ed line counts
+fromMgXII,SiXIII,and SiXIV(identified as thelines from the“corona”
+fromModelD).Notethattheline-based lightcurves alsoinc lude anycontin-
+uumcounts within theline profile. Thedata are binned over 3 k s.
+vations, maintaining separate first order spectra for the fo ur
+gratingarms(plusandminus;MEG andHEG)duringthefit-
+ting procedure. A model for the continuum plus background
+wasfoundfromaglobalfittoline-freeregionsandusedinad-
+dition to the line models. Background rejection is high with
+ACIS order sorting. Thus the background contributes rela-
+tively little to this model except at the long wavelength end
+of the spectrum,where the effectivearea is low. In the O VII
+region, for example, the background contains about half as
+many counts as the continuum. For the relatively weak but
+important O VIIline, background, continuum, and line have
+about1,2,and16counts,respectively.
+LinemodelsareGaussianfunctionswithnegligibleorsmall
+widths relative to the width of the instrumental line respon se
+functions ( ∼0.012 Å and 0.023 Å, for HEG and MEG, re-
+spectively). Table 1 gives the identified lines and their ob-
+served fluxes. For most lines the fit widths are less than half
+theinstrumentalwidthandarenotstatisticallysignifican t. We
+discussthewidthsofthe strongestlinesinSection3.5.
+All line fluxesare within 3 σof the values reportedby K02
+exceptforNe IXλ13.437,whichhasslightlymorethantwice
+as much flux in our observation. Our line fluxes are also
+within3σof the valuesreportedby Stelzer & Schmitt (2004)
+for lines observed with both instruments, except for Ne IX
+λ13.553, for which our flux is about half. For lines in com-
+mon with those reported by Raassen (2009) all fluxes agree
+within 3σ. With the higher signal-to-noise ratio of our data,
+newdiagnosticinformationis available.
+A major goal of this study is the determination of electron
+density from the He-like ions. We first discuss the diagnos-
+tics, and then modelthe global spectrum. Figure 2 showsthe
+high quality in the He-like O VIIand NeIXline regions. In-
+terestingly,the Ne IXregionshowsno apparentFe XIXlines,
+which can contaminate the Ne IXdiagnostics (e.g., Ness et
+al. 2003). Upper limits on the strongest Fe XIXline at 13.52
+Å imply that Fe XIXcontributes 1% or less to the Ne IXline
+fluxes.
+The Mg XItriplet line region is more complex (Fig. 3),
+with lower signal-to-noise ratio and apparent line blendin gACCRETION ANDCORONA IN TWHYA 3
+TABLE1
+SELECTED EMISSION LINES
+Line λrefaFluxbEff AreacOrigind
+(Å) (10−6ph cm−2s−1) (cm2)
+SiXIV 6.182 1.87 ±0.19 163.0 corona
+SiXIII 6.648 2.72 ±0.17 146.6 corona
+SiXIII 6.688 0.74 ±0.14 138.3 corona
+SiXIII 6.740 1.66 ±0.15 151.6 corona
+MgXII 8.421 2.24 ±0.20 162.4 corona
+MgXI 9.169 2.25 ±0.22 135.4 shock
+MgXI 9.231 0.90 ±0.17 133.2 shock
+MgXI 9.314 1.27 ±0.19 130.1 shock
+NeX 9.481 1.64 ±0.20 114.5 corona
+NeX 9.708 2.75 ±0.27 86.2 corona
+NeX 10.239 9.23 ±0.53 81.1 corona
+NeIXe10.765 3.54 ±0.43 67.9 shock
+NeIX 11.001 8.48 ±0.63 67.4 shock
+NeIX 11.544 23.7 ±0.99 55.3 shock
+FeXXII 11.783 3.28 ±0.50 52.3 corona
+FeXXII 11.932 3.01 ±0.50 49.6 corona
+NeX 12.134 76.1 ±1.9 48.5 corona
+FeXVII 12.266 4.3 ±0.6 46.6 mixedf
+NeIX 13.447 177. ±4.2 22.3 shock
+NeIX 13.553 114. ±3.3 22.7 shock
+NeIX 13.699 58.7 ±2.3 25.0 shock
+FeXVIII 14.208 4.59 ±0.76 29.0 corona
+FeXVIII 14.256 1.63 ±0.53 28.5 corona
+OVIII 14.820 5.04 ±0.80 23.7 shock
+FeXVII 15.014 36.5 ±1.8 21.9 shock
+OVIII 15.176 10.4 ±1.2 20.8 shock
+FeXVII 15.261 17.4 ±1.4 20.0 shock
+OVIIIg16.006 30.6 ±2.3 15.9 shock
+FeXVIII 16.071 5.17 ±1.0 16.3 corona
+FeXVII 16.780 25.5 ±3.0 12.1 shock
+FeXVII 17.051 27.9 ±2.3 11.1 shock
+FeXVII 17.096 26.3 ±2.3 11.2 shock
+OVII 17.207 4.68 ±1.4 10.7 shock
+OVII 17.396 4.68 ±1.5 9.8 shock
+OVII 17.768 3.66 ±1.6 7.7 shock
+OVII 18.627 17.9 ±1.4 7.3 shock
+OVIII 18.969 213. ±8.4 6.6 shock
+NVII 20.910 3.8 ±2.5 3.3 shock
+OVII 21.601 117. ±10.0 2.8 postshockh
+OVII 21.804 72.4 ±9.1 2.7 postshockh
+OVII 22.098 15.2 ±4.4 2.1 postshockh
+NVII 24.781 68.5 ±7.6 2.6 shock
+aReference wavelength from ATOMDB. Wavelengths for multipl ets are
+given as intensity-weighted averages.
+bObserved fluxes at Earth, with no correction for absorption. Flux errors
+are 1σ.
+cEffective areas are estimate by combining the four grating a rms and aver-
+aging the effective areas over the line width.
+dTheorigin of theemission lines isbased onModel Dparameter s (Section
+3.4).
+eBlend with Fe XVIIλ10.77.
+fIt is interesting to note that this Fe XVIIline is mixed, while the longer
+wavelength lines are primarily from the shock. The reason fo r this differ-
+ence is that the 12.266 Å line flux is relatively stronger at hi gher tempera-
+ture.
+gBlend with Fe XVIIIλ16.004; however, the predicted Fe XVIIIcontri-
+bution is negligible.
+hThislineisprimarily formedinthepostshock cooling plasm a(Section 5).
+In Model D the EMdistribution has only one component represe nting both
+the shock and postshock regions.
+between 9.2 and 9.4 Å. For this region, the fit is constrained
+such that all known lines are fixed in wavelength relative to
+the MgXIresonance line λ9.169, which is fit separately for
+each grating arm. Lines are Gaussian functions with widths
+set to zero (i.e., only the instrumental line response funct ion
+is used). Line fluxes of the Ne XLyman series lines n= 5
+to 10 are scaled according to their oscillator strengths, si nce
+collision strengthsare not available in the literature. We note21.621.721.821.922.022.1
+Wavelength (Angstroms)01020304050CountsO VIIr
+i
+f
+13.4013.4513.5013.5513.6013.6513.7013.75
+Wavelength (Angstroms)050100150200250300CountsNe IX
+r
+i
+f
+FIG. 2.—Top:Observed MEG spectrum in the O VIIspectral region,
+shownas ahistogram, with best-fitmodel(3 Gaussian lines) o verlaid. Wave-
+length positions are fixed relative to each other and the inst rumental line re-
+sponse function is included in the model. Resonance (r), int ercombination
+(i) and forbidden (f) lines are marked. Bottom:Observed MEG ( upper) and
+HEG (lower) spectra in the Ne IXregion, shown as histograms, with best-
+fit model overlaid. For this fit, the positions, fluxes and line widths are free
+parameters, yielding the constraints on velocities descri bed in the text.
+thatsystematicerrorsfromlineblendsarenotincludedint he
+errorsgiveninTable1.
+3.RESULTS
+One of the advantages of X-ray spectroscopy with high
+signal-to-noise characteristics is the availability of nu mer-
+ous line ratio diagnostics to assess conditions in the radia t-
+ing plasma. In particular, He-like systems provide valuabl e
+diagnostic opportunity, as discussed extensively in the li ter-
+ature (Porquet et al. 2001; Smith et al. 2001; and references
+therein). The G-ratio, defined as the flux ratio of the forbid-
+den(1s2s3S1→1s2 1S0)plusintercombination(1 s2p3P1,2→
+1s2 1S0) lines to the resonance (1 s2p1P1→1s2 1S0) line, is
+sensitive to the electron temperature ( Te), while the R-ratio
+(flux ratio of the forbidden to intercombination lines) is se n-
+sitive toNe. Ratios of emission lines in the resonance line
+series (transitionsfrom 2p, 3p, etc. to ground,also known a s
+Heα1, Heβ, and so on) are sensitive both to Te, through their
+relative Boltzmann factors, and to the absorptionby an inte r-
+vening hydrogen column density ( NH) through their energy
+separation.
+1Heαis the resonance line sometimes labeled rorw.4 BRICKHOUSE ETAL.
+9.15 9.20 9.25 9.30 9.35
+Wavelength (Angstroms)010203040CountsMEG
+HEGMg XIr
+i f
+FIG. 3.— Observed MEG ( upper) and HEG ( lower) spectra in the Mg XI
+spectralregion,shownashistograms,withbest-fitmodelsp ectrum,described
+in text, overlaid. Positions of resonance (r), intercombin ation (i), and forbid-
+den (f) lines are marked.
+This analysis emphasizes the use of line ratio diagnostics,
+including those discussed above. We use the Astrophysi-
+cal Plasma Emission Code (APEC) with the atomic data in
+ATOMDB v1.32(Smith et al. 2001) to calculate line emis-
+sivities, except for He-like Ne IX, for which we use the more
+accuratecalculationsofChenetal.(2006). Themodelforab -
+sorptionbygaswithcosmicabundancesistakenfromMorri-
+son&McCammon(1983).
+In this section we present the physical conditions of the
+emittingplasmaasdeterminedfromthelineratiosofspecifi c
+He-like and other ions, in particular Ne,Te, andNH. We use
+information determined from line ratios to constrain an em-
+pirical model of the emission measure distribution (EMD),
+and then allow additional parametersof the model to be con-
+strainedbyglobalfits tothe spectrum. We givetheelemental
+abundances determined from these fits. Finally we present
+velocityconstraintsfromline centroidsandwidths.
+3.1.Electrontemperature
+We use G-ratios from multiple ions to determine the elec-
+trontemperature. Figure4showstheobservedG-ratiosofHe -
+likeO,Ne andMg,overplottedontheirrespectivetheoretic al
+functionsof Te. Forcomparison,theemissivitiesoftheirreso-
+nancelinespeakat Te=2,4,and6MK,respectively,asshown
+inthefigure. Table2givesthederived Tefromtheseions. The
+NeIXG-ratioindicatesthat Te=2.5±0.25MK.TheNe IXTe
+determination is highly reliable, given that recent theore tical
+calculations,whichincludetheresonancecontributionst othe
+collisional excitation cross sections, agree with experim ental
+measurements to within 7% (Chen et al. 2006; Smith et al.
+2009a,2009b). Ontheotherhand,thetheoreticalcalculati ons
+forOVIIandMgXIhavehighersystematicuncertainties,and
+the observationalerrorsare larger,such that the Tevaluesde-
+rivedfromO VIIandMgXIcarrylessweight(butareconsis-
+tent with Ne IX). Both Ne IXand MgXIare found far below
+theirtemperatureofmaximumemissivity Tmax, supportingan
+accretionshockscenariofortheirformation.
+3.2.Electrondensity
+We use R-ratios from multiple ions to determine the elec-
+tron density. Figure 5 shows the observedR-ratios for O VII,
+NeIX, and Mg XI, overplottedon the theoretical functionsof
+2http://cxc.harvard.edu/atomdb6.2 6.4 6.6 6.8 7.0
+log [Te (K)]0.00.20.40.60.81.01.21.4G-Ratio [(f+i)/r]O VIINe IXMg XI
+Scaled Emissivities
+FIG. 4.— Theoretical G-ratios (i.e., ratios of forbidden plus i ntercombina-
+tion to resonance line fluxes) as functions of Te, for OVII(dashed), NeIX
+(solid), and Mg XI(dash-dotted ). Overplotted are the observed ratios with
+1σerrors. The model for Ne IXis from Chen et al. (2006). The models for
+OVIIand MgXIare from APEC using ATOMDBv1.3 (Smith et al. 2001).
+Scaled emissivities (ph cm3s−1) for the resonance lines of the three ions are
+also shown. For Ne IXand MgXIthe derived temperatures are well below
+the temperatures of their peak emissivities, consistent wi th an intepretation
+of their formation in an accretion shock.
+TABLE2
+HE-LIKEG-RATIOTeDIAGNOSTICS
+Ion Observed RatioaTe
+(K)
+OVII 0.75±0.13 2.4+1.2
+−0.6×106
+NeIX 0.98±0.04 2.5+0.25
+−0.25×106
+MgXI0.97±0.19 2.1+2.1
+−0.5×106
+aErrors are1 σ.
+TABLE3
+HE-LIKER-RATIONeDIAGNOSTICS
+Ion Observed RatioaNe
+(cm−3)
+OVII 0.21±0.07 5.7+4.4
+−1.2×1011
+NeIX 0.51±0.03 3.0+0.2
+−0.2×1012
+MgXI1.41±0.34 5.8+3.8
+−2.4×1012
+aErrors are1 σ.
+electron density Nefrom APEC. Table 3 presents the Neval-
+uesderivedfromtheseR-ratios. Sensitivityto Teisnegligible
+near the observed R-ratio values. Smith et al. (2009) show
+that the APEC R-ratio for Ne IXin the high density range is
+indistinguishable from the R-ratio calculated by Chen et al .
+(2006). ThetheoreticalR-ratiosfromAPECforallthreeion s
+arein excellentagreementwith Porquetet al. (2001)overth e
+densityrangereportedhere.
+From Ne IXwe derive Ne= 3.0±0.2×1012cm−3, while
+OVIIgivesNe= 5.7+4.4
+−1.2×1011cm−3. The 3σrange from
+NeIX,between2.5and3.9 ×1012cm−3,providesatightcon-
+straint.3Thebest-fitvalueforMg XI,usingallthelinesinthe
+region as described in Section 2, is Ne=5.8+3.8
+−2.4×1012cm−3.
+3ThemeasuredvalueoftheNe IXR-ratio is0.515 ±0.033,quitecloseto
+thevalueof0.446 ±0.124reported byK02. Thedifference ofnearly afactor
+of2in thederived Neappears tobeduetothedifference inatomic dataused,
+with their models possibly based on the Raymond & Smith code ( Raymond
+1988).ACCRETION ANDCORONA IN TWHYA 5
+101110121013
+Electron Density (cm-3)0.00.51.01.52.02.53.0R-Ratio (f/i)
+O VIINe IXMg XI
+FIG. 5.—Theoretical R-ratios (i.e.,ratios offorbidden to int ercombination
+line fluxes) as functions of Nefor He-like O, Ne and Mg. Overplotted are
+the observed ratios with 1 σerrors. Models are from APEC using ATOMDB
+v1.3. WhileFigure4showsthattheseionsareformedatsimil artemperature,
+they are found at significantly different densities.
+For comparison, we have also fit only the 3 lines of Mg XI,
+again with line widths and relative positions fixed, obtaini ng
+Ne= 7±2×1012cm−3, in good agreement with the best-fit
+value. This comparison suggests that line blending does not
+undulyaffecttheresult.
+At lowNethe population of the metastable 2 s3S1level
+builds up since the decay rate is extremely low. At high Ne
+electron impact excites the metastable level population up to
+the 2p3Plevels. In principle, low R-ratios may also be pro-
+ducedatlow Nebyphotoexcitation. Stelzer&Schmitt(2004)
+and K02 dismiss photoexcitationbased on the stellar temper -
+ature (∼4000 K), but Drake (2005) suggests that a hot spot
+above the stellar surface, where the accretion stream shock s,
+could produce sufficient photoexciting radiation. We have
+measuredthefluxattheexcitationwavelengths1647Å,1270
+Åand1034ÅforO VII,NeIX,andMg XI,respectively,from
+theHubble Space Telescope STIS (Herczeg et al. 2002) and
+the FUSE spectra (Dupreeet al. 2005). The intensity at 1034
+Å is an order of magnitude lower than at the longer wave-
+lengths, constraining the black body temperature of the hot
+spot to be less than 10,000 K, in good agreement with Inter-
+national Ultraviolet Explorer analysis showing a blackbody
+at 7900 K covering about 5% of the stellar surface (Costa et
+al. 2000). Thus the ultraviolet measurements and the Mg XI
+R-ratioruleoutphotoexcitationandsupporttheinterpret ation
+of highNe. Together with the temperature derived from the
+NeIXG-ratio, the density supports an accretion origin for
+NeIX.
+TheratioofFe XVIIλ17.096to λ17.051becomessensitive
+toNeabove∼1013cm−3(Mauche et al. 2001). The mea-
+suredratioof1.06 ±0.13isatthelow Nelimit,rulingoutthe
+highdensityfromthislineratiosuggestedbyNess&Schmitt
+(2005).
+3.3.Hydrogencolumndensity
+The HETG spectrum contains several series of resonance
+lines that indicate absorption by neutral gas along the line
+of sight. The most useful of these diagnostics are the O VII
+and NeIXHeα/Heβratios because their formation tempera-
+turesare independentlydeterminedfromthe G-ratio. Table 4
+presents the observed line ratios, and their use as diagnos-
+tics forNH. Figure 6 shows the theoretical He α/Heβratios
+as functions of Tefor OVIIand NeIX, for both unabsorbedTABLE4
+LINERATIONHDIAGNOSTICS
+Identification Observed Predicted Ratio for log NH(cm−2):
+Ratioa0.00 20.61 20.89 21.26
+OVIIHeα/Heβb6.54±0.76 8.54 6.52 5.22 2.49
+OVIIILyα/Lyβb6.95±0.58 9.63 8.07 6.82 4.55
+NeIXHeα/Heβb7.38±0.35 11.14 10.07 9.23 7.34
+NeXLyα/Lyβb8.24±0.52 12.93 11.89 10.91 8.93
+NeXLyα/Lyβc8.24±0.52 7.061 6.49 5.96 4.88
+aErrorsare 1 σ.
+bPredictions assume Te=2.5×106K.
+cPredictions assume Te=1.0×107K.
+6.1 6.2 6.3 6.4 6.5
+log [Te (K)]05101520Heα / HeβNe IX
+O VII
+UnabsorbedAbsorbed
+FIG. 6.—Theoretical modelsofHe αto Heβlinefluxratios asfunctions of
+Te, withsolidcurve for Ne IXanddashedcurve for O VII. The filled circle
+is the observed ratio for Ne IXand the open circle is for O VII, with 1σ
+error bars. Unabsorbed and best absorbed models for each ion are labeled.
+The absorbed models shown have NH= 1.8×1021and 4.1×1020cm−2for
+NeIXand OVII,respectively.
+and absorbed cases. The observed ratios are overplotted us-
+ingTedeterminedfromtheirG-ratios(which,asnotedabove,
+is more reliable for Ne IXthan for O VII). For the best-fit
+Te, NeIXgivesNH= 1.8±0.2×1021cm−2and OVIIgives
+NH= 4.1+1.9
+−1.6×1020cm−2. These two values show a mean-
+ingful difference. For Ne IXat 2.5 MK, the observed ratio is
+11σbelow the unabsorbedmodel, making the presence of an
+absorbing column quite definitive. The column density from
+OVIIisconsistentwiththelowerlimitof NH=2×1020cm−2
+determinedfromtheabsenceofNe VIIIandSiXIIlineslong-
+wardof40Åinthe ChandraLETGspectrum(Raassen2009).
+Ratios of the H-like series lines (Ly α, Lyβ, etc.) are also
+sensitive to both absorption and temperature, but unlike th e
+He-like ions, H-like ions do not present an independent tem-
+peraturediagnostic. If the absorptioncolumn dependson th e
+ion,asimpliedbytheinconsistencybetweenNe IXandOVII,
+thenweneedtoassumeamodel Tetoobtainacolumndensity
+from the H-like line ratios. Assuming 2.5 MK, the observed
+ratioOVIIILyα/Lyβindicates NH=7.8+1.1
+−0.9×1020cm−2. This
+value is consistent within the errors with NHfrom OVIIand
+thus a third absorber is not required. For Ne Xat 2.5 MK,
+the column density would be larger than the column density
+for NeIX, as shown in Table 4; however, Table 4 also shows
+that no absorption is required if the Ne Xlines are formed at
+10 MK. We must further consider the effects of temperature
+onNeXline ratios.
+Wecanalsoestimatethetemperatureusingthechargestate
+balance. Table 5 presents the He α/Lyαratios for oxygenand6 BRICKHOUSE ETAL.
+TABLE5
+TeFROMCHARGESTATEBALANCE
+Identification ObservedaCorrected for log NH(cm−2):
+20.61 20.89 21.26
+OVIIHeα/OVIIILyα0.55±0.052 0.70 0.85 1.67
+NeIXHeα/NeXLyα2.32±0.08 2.48 2.60 3.04
+Identification Derived Te(106K)b
+OVIIHeα/OVIIILyα3.03 2.83 2.66 2.23
+NeIXHeα/NeXLyα 4.16 4.08 4.01 3.87
+aErrorsare 1 σ.
+bThe derived Tevalues are based on the ratios listed above in the corre-
+sponding observed and corrected columns.
+neon. First,theobservedratiosarecorrectedfordifferen tcol-
+umn densities as found above, and then Teis derived from
+the respectivecorrectedratios. For oxygen,goodconsiste ncy
+is found for the temperatures derived from the G-ratio and
+the OVIIHeα/OVIIILyαratio, given a column density of
+about 8×1020cm−2. For neon, on the other hand, all the de-
+rived temperatures are larger than Tederived above from the
+NeIXG-ratio. Thus it appears that some of the Ne Xemis-
+sion originates at higher temperature. Using NH= 2×1021
+cm−2andTe=2.5 MK, and the observedNe IXHeαflux, we
+predict the flux of the Ne X Ly αto be only 18% of the ob-
+servedvalue. In this case, the rest of the Ne Xemission must
+be formed at higher temperature. If the trend of increasing
+NHwith charge state continues, then NHfor NeXwould be
+higherthan2 ×1021cm−2,andthefractionofNeXformedat
+2.5MK wouldbesomewhatlarger.
+The emissivity functions for lines of Fe XVIIand Ne X
+bothpeaknear5MK,suggestingthepossibilityofcomparing
+their formation temperatures. The ratio of Fe XVIIλ15.014
+toλ15.261 is sensitive to Te, due to a blend of an inner shell
+excitation line of Fe XVIat 15.261 Å (Brown et al. 2001).
+Thetemperaturesensitivityessentiallycomesfromthecha rge
+state balance of Fe XVIand FeXVII. Using the reported ex-
+perimental dependence of the line ratio on the electron beam
+energyoftheElectronBeamIonTrapatLawrenceLivermore
+NationalLaboratory(Brownetal.2001),theobservedHETG
+value gives Te= 2.34+0.09
+−0.06MK. Despite similar peak temper-
+atures, Fe XVIIand NeXare not formed together. Appar-
+ently, the long tail of the Ne XLyαemissivity function to-
+ward high temperature contributes significantly to its emis -
+sion. The Fe XVIIlines are predominantly formed in the ac-
+cretion shock, while the Ne Xlines are formed both in the
+shockandinthehottercorona.
+One might expect that the ions formed at low temperature
+(OVII, OVIII, and Ne IX) should all experience similar ab-
+sorption. Using NHderived from Ne IX, we can derive char-
+acteristic emitting Tefor OVIIIand OVIIof 1.4 MK and
+<1.0 MK, respectively. These temperatures are far lower
+thanTederived from the O VIIG-ratios. They are both also
+far lower than Te>2 MK derived from the charge state ratio
+of oxygen represented over the range of NHfound here, as
+shown in Table 5. We conclude that an increase in NHwith
+increasing charge state (presumably temperature) up throu gh
+NeIXisconsistentwithall thedata.
+WenotethatArgiroffietal.(2009)claimevidenceforreso-
+nancescatteringofthestrongestlinesintheXMM-RGSspec-
+trum of MP Mus, though their RGS spectrum of TW Hya is
+too weak to show the signatures of absorption we find here.
+We rule out resonancescattering in TW Hya usingthe strongseries of Ne IX. An optical depth τof about 0.4 for Ne IX
+Heαwould be required to model the observed He α/Heβra-
+tio. Forasingleion,theopticaldepth τscalesasgfoscλ, such
+that we can then predict the other Ne IXline ratios. We find
+then that He α/Heγratio is overpredicted by 50%. Since the
+oscillator strengths decrease rapidly with increasing pri nci-
+pal quantum number n, only the He αline might be affected
+by resonance scattering. We find instead, that all the Ne IX
+lines are affected by absorption, and rule out resonance sca t-
+teringastheabsorptionmechanism. Weconcludethattheab-
+sorberis neutralor near neutral, consistent with the presh ock
+accreting gas, as suggested by theoretical studies. For exa m-
+ple, Lamzin (1999) considers the absorption by the preshock
+gasoftheX-rayemittingplasmafordifferentgeometriesan d
+orientations. Gregoryet al. (2007)use realistic coronal m ag-
+neticfieldscoupledwith a radiativetransfercodeto calcul ate
+the obscurationof X-ray emission by accretion columns, and
+suggestthatthiseffectcanexplaintheobservedlowX-rayl u-
+minositiesofaccretingyoungstarsrelativeto non-accret ors.
+3.4.Emissionmeasure(EM)distributionandelemental
+abundances
+We present here four models for the emission measure
+(EM) distribution4and elemental abundances (Table 6).
+These models serve to illustrate how values derived from the
+line ratios affect the global fit to the spectrum and to show
+whether abundance determinations are robust. For all four
+models, the first-order HEG and MEG spectra are fit to a set
+ofvariableabundanceAPECmodels,using Sherpaandapply-
+ing different constraints to the fitted parameters. All mode ls
+have acceptable goodness-of-fit values. Model A has two Te
+componentsandasingleabsorber. ModelAshouldbeappro-
+priate for comparison with other X-ray spectra of cool stars
+obtainedat lowerspectralresolution.
+Model B is also a two- Temodel, but with constraints im-
+posed from the line ratios. The lower Teis fixed at 2.5 MK,
+and the single absorber NHis fixed at 1.0 ×1021cm−2(a
+rough average of the values obtained from line ratios). The
+Ne-sensitive forbidden and intercombination line regions ar e
+excludedfromthefit.
+Table 6 also gives results from multicomponent Models C
+and D, each with the low Teand a single NHfixed, as in
+Model B. Assuming that the hotter component is coronal in
+nature, a broad distribution of Teis reasonable, although the
+shapeisnotwellconstrained. Webroadenthehotcomponent
+using a Gaussian-shaped function centered around 12.5 MK.
+For Model C the entire spectrum is fit simultaneously, with
+onlytheforbiddenandintercombinationlineregionsexclu ded
+asforModelB.
+For Model D the emission measures for the two compo-
+nentsare fit to line-freeregions, identified visually and us ing
+APEC models. The thermal continuum emission fit to line-
+free regions is strongly temperature-dependentand thus co n-
+strainstheemissionmeasuredistribution. Theabundances are
+then fit to narrow regionscentered on the strong lines. These
+separate fitting procedures are iterated until the values st op
+changing. The resulting Model D EM distribution is shown
+4Emission measure EM is defined as/integraltext
+0.8N2
+edV, where the factor 0.8
+accounts for the hydgrogen to electron density ratio and the integral is taken
+over the volume V. The intensity of a spectral feature I=εEM/(4πD2)
+whereεis the emissivity in units of ph cm3s−1andDis the distance to the
+source. The function εdepends on Te, and is given in steps of log [ Te(K)] =
+0.1in ATOMDB(Smith etal. 2001).ACCRETION ANDCORONA IN TWHYA 7
+TABLE6
+MODELPARAMETERSaFROMGLOBALSPECTRAL FITTING
+Parameter Model A Model B Model C Model D
+T1 (MK) 3.58 ±0.04 2.5b2.5b2.5b
+EM1 (1053cm−3) 1.11±0.13 1.49 ±0.09 1.56 ±0.10 4.22 ±0.70
+T2 (MK) 18.8 ±0.5 11.2 ±0.2 12.6c12.6d
+EM2 (1053cm−3) 0.33±0.01 0.47 ±0.01 0.121 ±0.004c0.101±0.005d
+NH(1021cm−2) 0.15 ±0.22 1.0b1.0b1.0b
+N (8.05)e0.66±0.14 0.28 ±0.06 0.53 ±0.10 0.20 ±0.04
+O (8.93)e0.23±0.02 0.14 ±0.01 0.24 ±0.02 0.09 ±0.01
+Ne(8.09)e1.23±0.08 2.04 ±0.11 2.50 ±0.13 1.04 ±0.02
+Mg (7.58)e0.18±0.02 0.16 ±0.02 0.21 ±0.02 0.18 ±0.02
+Si(7.55)e0.33±0.03 0.24 ±0.02 0.29 ±0.03 0.34 ±0.03
+Fe(7.67)e0.13±0.01 0.10 ±0.01 0.20 ±0.01 0.16 ±0.01
+aAllerrors given are statistical errors from thefit.
+bValueis fixed.
+cEM2 given is the fitto the peak of the EM distribution. Shape of the hotcompo-
+nent is thesame as for Model D,as described in the text.
+dEM2given is the fitto the peak of the EMdistribution. Shape of the hotcompo-
+nent EMdistribution is shown in Figure 7 and described in the text.
+eAbundance of element relative to the solar values of Anders & Grevesse (1989).
+The abundances in logarithmic form with H=12.0 are given in p arenthesis next to
+the element label.
+6.46.66.87.07.27.4
+log [Te (K)]51.051.552.052.553.053.5log [Emission Measure (cm-3)]
+FIG. 7.—EMdistribution for Model D as described in text.
+in Figure 7. The larger statistical error for the low Teemis-
+sion measure in Model D compared with Model C (Table 6)
+stems from the smaller number of bins used in the fits; how-
+ever,weexpectthesystematicerrorsforModelDtobelower,
+given that we are only using bins whose information content
+issecure,andthuswechooseModelD toillustratethemodel
+spectrum.5Figure 8 comparesthe ModelD spectra predicted
+by the broad “coronal” component and the soft “accretion”
+componentwith the observedspectra. Table 1 lists the origi n
+ofeachlineasaccretionorcoronabasedonModelD.Anad-
+ditional component with Telower than 2.5 MK can be added
+to both Models C and D but is not required, as it does not
+improvethefitsignificantly. Similarresultsareobtainedw ith
+somewhat different choices of width and peak Tefor the hot
+component.
+Since the line ratios indicate increasing NHwith higher
+charge state, we tried adding a second absorber to the high
+5Thefour models areall statistically acceptable; however, it is interesting
+to note that none of the models do a good job of fitting all the em ission line
+fluxes. For example, Model D does underpredicts Ne XLyαby almost a
+factor of 2. We attribute these difficulties primarily to the complexity of the
+absorption, and secondarily to the few constraints on the sh ape of the EM
+distribution.5 10 15 20
+Wavelength (Angstroms)0200400600800          CountsO VIIO VIIIO VIIFe XVIIFe XVII
+O VIIINe IXNe XNe XFe XXIINe IXNe XMg XIMg XIISi XIIISi XIV
+ObservationShock ModelCoronal Model
+FIG. 8.— Comparison of predicted and observed spectrum from the
+MediumEnergyGrating (MEG).Uppercurveisthepredicted sp ectrum from
+the Model D EM distribution above 5 MK (the “coronal model”), shown in
+Figure7. Middlecurveisthepredicted spectrum fromtheMod elDemission
+measure at 2.5 MK (the “shock model”). The lower curve is the o bserved
+spectrum. Asnoted inthetext, ModelDsignificantly underpr edicts theNe X
+Lyα. Alsonotethattheforbiddenandintercombination linepre dictions using
+theAPEC models in Sherpaare attheir low density values (e.g.,Ne IX).
+TeGaussian component for Models B, C, and D. Robust so-
+lutions are difficult to obtain, as the absorption, abundanc es,
+andnormalizationsofthe emissionmeasuresdonotindepen-
+dentlydeterminetheglobalX-rayspectrum. Wealsoexplore d
+the possibility that the low and high Tecomponentshave dif-
+ferentmetalabundancesbutagain,withoutrobustresults.
+UsingTe=2.5 MK, as derivedfrom the Ne IXline ratio, in-
+steadofTe=3.58MK asfoundinModelA, hasimportantim-
+plicationsfor interpretingthe formationof some of the emi s-
+sion lines. The ratio of the strong Ne IXHeαand NeXLyα
+lineemissiondrivestheModelAfittomorethan3.0MKwith
+very little absorption. Specifically, only ∼15% of the Ne X
+Lyαemission arises from the 2.5 MK component, whereas a
+3.58MK componentcan producemore than half of the Ne X
+emission. The 2.5MK componentproducesessentially all of
+the NVII, OVII, OVIII, and Ne IXemission and more than
+half the emission from Fe XVII. The fluxes from these lines
+formed at 2.5 MK show no increase over their average value8 BRICKHOUSE ETAL.
+duringtheflare,asshowninFigure1,indicatingthattheflar e
+is associated with the hotter corona rather than the accreti on
+shock,andjustifyingits retentionin ouranalysis. Linesf rom
+MgXII, SiXIII, SiXIV(Fig. 1), and Fe XXIIare producedby
+the hotter componentand may participate in the flare. While
+K02 foundthat Mg XIwas overpredictedbytheir modelby a
+factorofabout3,ourModelsB, C, andD havenosuchprob-
+lem. In fact,forModelsCandD, the small fluxof theMg XI
+resonance line forces the emission measure to be negligible
+between∼3and5MK(Fig.7).
+TherelativeelementalabundancesgiveninTable6arerea-
+sonably consistent for the different models. The abundance s
+are also similar to those found by K02, Stelzer & Schmitt
+(2004), and Drake et al. (2005). It is important to note that
+thecontinuumemissionarisesprimarilyfromthehotcompo-
+nent, except at the longest wavelengths (Fig. 8). Thus ab-
+solute abundances in the low Tecomponent for Ne, O, Fe
+and other species cannot be determined reliably. Neverthe-
+less, the extremely large Ne/O abundance ratio appears to be
+a very robust result. Differences in the absolute abundance s
+derivedhere reflect bothdifferencesin Teand the degeneracy
+between emission measure and NH. For example, nitrogen
+and oxygen are formed entirely in the low temperature com-
+ponentandthustheabundancedifferencesamongthemodels
+reflect differencesamongthe low temperatureemission mea-
+sure and NH, as does the difference in the neon abundance
+between Models A and B. The difference in the neon abun-
+dance between Models C and D also reflects the differences
+intheiremissionmeasuresat2.5MK.
+Weak emission is apparent from the S and Ar complexes,
+but we have not included line fluxes from these elements in
+Table 1 because the individual lines (in particular the He-
+like resonance line) cannot not be cleanly determined in the
+four grating arms. Instead we have used the Model C global
+fit to obtain 0.21 ±0.09 and 1.41 ±0.49 times solar (An-
+ders & Grevesse 1989) for the S and Ar abundances, respec-
+tively. A high Ar/O abundancewould addweight to Drake et
+al.’s(2005)suggestionthattheaccretionstreamisdeplet edof
+grain-forming elements; however, the high Ar abundance is
+unfortunatelynotstatistically significant.
+3.5.Velocityconstraints
+Inthissection,wediscussvelocityconstraintsfromtheli ne
+measurements. Themeasurementsoftheemissionlinefluxes
+given in Table 1 use Gaussian lines in addition to the stan-
+dardHETGcalibrationlinespreadfunction. Table7givesth e
+lines with the highest signal-to-noise ratios, for which Ga us-
+sian widths are measured. Lines not listed do not show sta-
+tistically significant widths, but are consistent with the r ange
+of widthsfoundhere. For these fits the continuumlevel is al-
+lowedtovarytoprovideabetterlocalfit. AssumingtheNe IX
+linesforminthesameregion,andusingaflat continuumflux
+level from 13.45 to 13.75 Å, we force the 3 triplet lines to
+have the same Gaussian width to improve the significance of
+the velocity measurement. Figure 2 shows the best fit for the
+NeIXlines. As before, the centroids of the lines measured
+in each grating arm are fit independently. We thus obtain a
+velocity of 183 ±16 km s−1. The thermal Doppler FWHM
+velocity for neon at 2.5 MK is 75 km s−1. The error in the
+calibration is estimated to be about 5%. Thus we determine
+a turbulent velocity of 165 ±18 km s−1, including calibra-
+tion error in quadraturewith the 1 σstatistical error. The line
+profiles are consistent with each other within the errors and
+suggest a range of velocitiesfrom zero up to aboutthe soundspeed of the gas (185 km s−1at 2.5 MK). The turbulent ve-
+locity is well below the preshock gas velocity of ∼500 km
+s−1. Line centroidsare consistent with referencewavelengths
+from ATOMDB to within the calibration uncertainty on the
+absolutewavelengthscale.
+3.6.Physicalconditionsofthe X-rayplasma
+We summarize here the physical conditions of the X-ray
+plasmafroma consistentanalysisof thespectra. Thenew di-
+agnosticmeasurementsstronglysupportK02’sargumenttha t
+thelowtemperatureX-raycomponent( Te∼2.5MK)arisesin
+theaccretionshock. While the densitywe derivefromNe IX,
+Ne= 3×1012cm−3, is within the range found in active stars
+at higher Te(∼6 MK), it is at least an order of magnitude
+largerthanthefewNe IXR-ratiosreported(Huenemoerderet
+al. 2001;Ness et al. 2002;Osten et al. 2003). The Nederived
+from the O VIIR-ratio is also more than an order of magni-
+tude larger than derived from O VIIin other cool stars (see
+Sanz-Forcadaet al.2003;Nesset al. 2004;Testa et al.2004) .
+We also measure the Mg XIR-ratio and rule out photoexcita-
+tion, since the observed Mg XIR-ratio can only be produced
+byhighNe,namely5 .8×1012cm−3.
+The G-ratios from O VII, NeIX, and Mg XIgive similar
+Te∼2.5 MK, significantly lower than the Teof peak emis-
+sivity for Ne IX(4 MK) and Mg XI(6.3 MK). The G-ratio
+from Ne IXis particularly reliable because of good statistics
+and accurate atomic data. In our multi- Temodels (Models C
+and D), this low Tecomponent is isolated and does not con-
+nectcontinuouslywiththe hotter,coronalcomponent. Acti ve
+main-sequencestarstendtohaveemissionmeasuresrisingu p
+to6MKandabove,suchthat,ifbiasedatall, Te-sensitiveline
+ratios should be biased toward higher rather than lower tem-
+perature. Thus the Temeasurements also support the accre-
+tion shock scenario. More accurate G-ratio models for O VII
+andMgXIare requiredto establish any Tedifferencesamong
+theseions.
+We also measure strong absorption using line ratios from
+several ions, and require at least two different absorbing c ol-
+umn densities, with the higher charge states experiencing
+moreabsorption. Itseemslikelythatthegapbetweenthetwo
+componentsoftheEMdistributioniscausedbyneutralHab-
+sorptionofthesoftX-rayspectrumthatisproducedbytheho t
+coronal component. The pattern of absorption is reminiscen t
+of the “two-absorberX-ray sources” reported by Güdel et al.
+(2007b),wherethecoronalcomponenthastentimesmoreab-
+sorption than the soft component; however, unlike TW Hya,
+these sources have high accretion rates and are believed to
+drive jets. We note that the absorption in TW Hya is identi-
+fiedherefromlineratiodiagnostics,whereastheabsorptio nin
+Güdel et al. (2007b) was determined from CCD spectra. We
+were not able to determinemultiple absorbersusing standar d
+global fitting methods, even with the high resolution spec-
+trumpresentedhere,becausethecontinuumspectrumhaslow
+signal-to-noise ratio. On the other hand, it seems likely th at
+withlowspectralresolution,multipleabsorberscouldals obe
+easily missed. Thus the presence of more than one absorber
+may be a more universal characteristic than we can presently
+establishwithoutreliablelineratiodiagnostics.
+Allthevaluesof NHfoundherearehigherthanthosefound
+using the hydrogenLy αprofile, for which a conservativeup-
+perlimitis6 ×1019cm−2(Herczegetal.2004),indicatingthat
+the absorption is not due to the interstellar medium but is in -
+trinsic to the stellar system. Our columndensitiesare cons is-
+tentwiththerangeofvaluespreviouslyreportedfromROSATACCRETION ANDCORONA IN TWHYA 9
+TABLE7
+VELOCITIES FROM STRONGEMISSION LINES
+Line λref FWHM V obsaVthbVturbc
+(Å) (Å) (kms−1) (kms−1) (kms−1)
+NeX10.239 0.0042 ±0.0030 122 ±87 75d96
+NeIX11.544 0.0024 ±0.0024 62 ±62 75 0
+NeX12.134 0.0082 ±0.0009 203 ±23 75d189
+NeIX13.447 0.0103 ±0.0010 229 ±22 75 216
+NeIX13.553 0.0057 ±0.0014 126 ±32 75 101
+NeIX13.699 0.0073 ±0.0019 159 ±41 75 140
+FeXVII15.014 0.0065 ±0.0040 129 ±80 45 121
+OVIII18.969 0.0074 ±0.0026 116 ±41 85 79
+aVobsis the observed FWHMvelocity.
+bVthisthe thermal FWHMvelocity calculated at 2.5 MK.
+cVturbis the turbulent FWHMvelocity assuming Vturb=/radicalBig
+(V2
+obs−V2
+th).
+dNote thethermal velocity for neon at 10 MK is150 kms−1.
+andASCA(Kastneretal.1999). Infactthetwodifferentval-
+uesfromROSATandASCA areconsistentwith ourdifferent
+values and may reflect the complex absorption coupled with
+the different instrument responses rather than time-varia ble
+absorption. As noted also by Güdel et al. (2007b) for the
+highlyabsorbedcoronaeof DG Tau A, GV Tau, DP Tau, our
+X-ray measurements are higher than expected from the op-
+tical extinction assuming standard gas-to-dust ratios. Fo r X-
+rayemissionthatishighlylocalizedcomparedwiththestel lar
+Lyα, and preferentially absorbed by the accretion stream di-
+rectlyaboveit,adifferencein NHderivedbythetwomethods
+seemsentirelyreasonable.
+4.THEACCRETION SHOCKMODEL
+WecantestthehypothesisthatsomeofthemeasuredX-ray
+emission comes from the accretion shock by constructing a
+standard one-dimensional(1D) model of the magnetospheric
+infall, shock heating, and postshock cooling. This model is a
+simplifiedversionofsimilar 1Dmodelsintheliterature(Ca l-
+vet & Gullbring1998; Güntheret al. 2007),andis tailoredto
+specificmeasuredpropertiesofTWHya.
+The magnetospheric accretion is assumed to follow a set
+of dipole magnetic field lines that thread the accretion disk .
+We use Königl’s (1991) expression for the inner truncation
+radius of the disk to determine the distance at which parcels
+ofgasarelaunchedfromrest. Thisexpressiondependsonthe
+mass accretion rate ˙Macc, the surface magnetic field strength
+B∗, and the stellar mass M∗and radius R∗. For TW Hya,
+we takeM∗= 0.7M⊙andR∗= 0.8R⊙(Batalha et al. 2002).
+As in Cranmer (2008), we assume a canonical T Tauri star
+magneticfieldstrengthof B∗≈1000G. A rangeof accretion
+rates for TW Hya has been reported, from 4 ×10−10M⊙yr−1
+(Muzerolle et al. 2000) to 5 ×10−9M⊙yr−1(Batalha et al.
+2002). Thelowendofthisrangeproducesthebestagreement
+with the observations presented here, so the Muzerolle et al .
+accretion rate will be adopted for the cooling-zone models
+below.
+With the abovevalues, the Königl (1991)truncationradius
+of the disk is found to be approximately rt= 4.5R∗, which
+impliesa ballisticfree-fallvelocityatthe stellarsurfa ceof
+vff=/bracketleftbigg2GM∗
+R∗/parenleftbigg
+1−R∗
+rt/parenrightbigg/bracketrightbigg1/2
+≈509kms−1.(1)
+Sincethepreshockgasisoftenassumedtohaveasoundspeed
+only of order ∼10 km s−1(e.g., Muzerolle et al. 2001), thehighly supersonic flow should produce a strong shock at the
+stellarsurface,heatingtheplasmatoapostshocktemperat ure
+ofTpost≈3mv2
+ff/16k,wherekistheBoltzmannconstantand m
+isthemeanatomicmass. Fortheaboveparameters,thisvalue
+isTpost= 3.4 MK. [We neglect any preshock heating due to
+photoionization since the X-ray luminosity is relatively l ow.
+SeeCalvet &Gullbring(1998).]
+The preshockmass density ρprein the accretionstream can
+beestimatedviamassfluxconservationatthesurface,i.e.,
+˙Macc=4πfR2
+∗ρprevff (2)
+where the fis the filling factor of the stellar photosphere
+heated by the accretion stream. The value of fwas esti-
+mated to be 1.1% for the dipole magnetic geometry assumed
+above (see also Cranmer 2008). Assuming complete ioniza-
+tion(with10%Hebynumber),thepreshockelectronnumber
+densityNpre= 5.7×1011cm−3. Because the shock is strong,
+we obtain an immediate postshock density Npost=2.3×1012
+cm−3(i.e.,a factorof fourincrease acrossthe shock,with the
+assumedadiabaticindexof5 /3). Thepostshockvelocity vpost
+isthusapproximately130kms−1.
+As one proceedsdeeperthroughthe coolingzone, the den-
+sityincreasesandthevelocityandtemperaturedecrease,u ntil
+a depth is reached at which there is no longer any influence
+from the shock. The “bottom” density ρbotcan be estimated
+usingram pressurebalance betweenthe accretionstream and
+the unperturbed atmosphere. Using Equation 2 the pressure
+ofthe accretionstreamisgivenby
+Pram=ρprev2
+ff
+2=vff˙Macc
+8πfR2∗(3)
+(e.g., Hartmann et al. 1997; Calvet & Gullbring 1998). As-
+suming the accretion is stopped in the first few scale heights
+above the photosphere (at which radiative equilibrium driv es
+thetemperatureto ∼(4/5)Teff),thepressureoftheatmosphere
+is0.8ρbotkTeff. By equatingthetwo pressures,thedensitycan
+befoundbysolvingfor
+ρbot≈Pram
+0.8kTeff(4)
+withTeff=4070 K for TW Hya (Batalha et al. 2002; see also
+Cranmer 2008). Converting to electron density, this corre-
+sponds to 2 .6×1015cm−3, a factor of 1000 higher than the
+immediatepostshockdensity.10 BRICKHOUSE ET AL.
+0 100 200 300 400
+Depth (km)01230 0.5 1.0 Te (MK)              Intensity (scaled)Ne IX r
+O VIII Ly α
+O VII r
+051015     
+Ne (1012cm-3)                            
+FIG. 9.—Lower: T e(dotted) and Ne(solid) as functions of depth in the
+postshock cooling column model, where 0 corresponds to the p osition of the
+shock. The model is described in Section 4. Upper:Fractional contributions
+to the line intensities as labeled, as functions of depth in t he model.
+Finally, we computed the physical depth and structure of
+the postshockcoolingzone using the analytic modelof Feld-
+meier et al. (1997), which assumes that the radiative coolin g
+rateQrad≈N2
+eΛ(T)hasa power-lawtemperaturedependence
+(Λ∝T−1/2). Thethicknessofthecoolingzoneisproportional
+tov4
+post/ρpost,andforthecaseoftheTWHyaparametersused
+here, it has a value of 385 km. The spatial dependence of
+density,temperature,andvelocityinthecoolingzoneissp ec-
+ified by equations (8)–(11) of Feldmeier et al. (1997), and
+the shapes of these curves resemble those of Günther et al.
+(2007)—which were computed using a more detailed radia-
+tivecoolingrate—reasonablywell (see Fig.9).
+Note that choosing the stronger mass accretion rate of
+Batalha et al. (2002) would have led to a truncation ra-
+dius closer to the star ( rt= 2.2R∗), a lower freefall velocity
+(vff= 430 km s−1), and thus a lower postshock temperature
+(2.4×106K). The densities would all have been higher than
+their counterpartsabovebyabout an orderof magnitude,and
+thethicknessofthecoolingzonewouldhavebeensmaller(32
+km).
+Using the temperature and density in the model, we com-
+pute the line intensities as functions of distance from the a c-
+cretion shock. Figure 9 shows line intensities throughthe r e-
+gion for the strongest lines of Ne IX, OVIII, and OVII. We
+compare the integrated intensities to the observations. As -
+sumingthe abundancesdeterminedfromModelD, andusing
+NH= 1.0×1021cm−2, the predicted line intensities for the
+NeIX, OVIIIand OVIIresonance lines are 0.25, 0.16, and
+0.40 times the observed fluxes, respectively. Higher abun-
+dancesimprovethe agreement,as dolowercolumndensities.
+Nevertheless,agreementwithin a factorof 4 seems quiterea -
+sonable,giventhedifficultyofobtainingabsoluteabundan ces
+andthe complexityoftheabsorption.
+The predictedG-ratios give Te= 2.5 and 2.2 MK for Ne IX
+and OVII, respectively. The Necalculated for Ne IXis
+2.6×1012cm−3,inexcellentagreementwith theobservation;
+however,forO VII,thecalculated Neis4.1×1012cm−3,seven
+times larger than observed. The O VIIdensity is clearly dis-
+crepantwiththestandardmodel.
+5.ANALYSISOFTHEPOSTSHOCKCOOLINGPLASMA
+The physical conditions predicted at the shock front are in
+excellent agreement with the observations; however, the pr e-
+dictions for the postshock cooling plasma are in stark dis-TABLE 8
+TWO-REGIONMODEL FOR ACCRETION SHOCK AND
+POSTSHOCK COOLING
+Model Parametersa
+Ne Te V NH
+(cm−3) (MK) (cm3) (cm−2)
+Region 1 6 .0×10123.00 1 .5×10281.0×1021
+Region 2 2 .0×10111.75 4 .5×10304.0×1020
+Predicted Unabsorbed FluxesbfromRegion 1
+r i f Ly α
+O 260. 166. 3.6 512.
+Ne 447. 260. 73.9 ···
+Mg 3.6 1.3 1.8 ···
+Predicted Unabsorbed FluxesbfromRegion 2
+r i f Ly α
+O 124. 70.8 36.7 25.1
+Ne 16.1 4.8 10.1 ···
+Mg 0.0 0.0 0.0 ···
+Ratio of Predicted Absorbed Fluxesbto Observed Fluxesb
+r i f Ly α
+O 1.00 0.69 0.53 0.69
+Ne 1.31 0.85c0.75c···
+Mg 1.28 1.14 1.13 ···
+aAbundances fromModel Dare 0.09,1.04,and 0.18relative
+to solar (Anders & Grevesse 1989), for oxygen, neon, and
+magnesium, respectively.
+bFluxes atEarth are in units of 10−6ph cm−2s−1.
+cChen et al. (2006) is about 25% larger than APEC.
+agreement with the observations. Most problematic is that
+the electron density derivedfrom O VIIislowerthan that for
+NeIX, when the postshock gas should be compressing as it
+cools and slows and hence the density from O VIIshould be
+larger than from Ne IX. Furthermore, NHdetermined from
+OVIIis four times smallerthan that determined from Ne IX,
+whereasinthestandardpostshockmodelO VIIshouldbeob-
+servedthroughthesameorlargercolumndensityofinterven -
+ing material. In this section, we explore the propertiesof t he
+postshockcoolingplasmabasedonournewresults.
+UsingtheAPECcodetocalculateemissivitiesforarelevant
+gridofTeandNe,weconstructa set oftwo-componentshock
+models for comparison with the diagnostic lines from the
+shock. The two components represent Region 1, the higher
+Teregionnear the shockfront, andRegion 2, the lower Tere-
+gion of postshock plasma. We do not discuss the hot coronal
+plasma here, but consider the diagnostic lines from ions tha t
+are formed below 3 MK, in particular O VII, OVIII, NeIX,
+and Mg XI. Each component in the model has four parame-
+ters:Te,Ne,NH, and volume V. The two regions experience
+different absorption with a factor of 2.5 larger NHat higher
+Te, as requiredby the resonanceserieslines. The abundances
+are taken from the Model D fits. The predicted fluxes from
+thetworegionsaresummedforcomparisonwiththeobserva-
+tions. Table 8 givesthe parametersfor the best-fitting mode l,
+the predicted fluxes for each of the two regions, and the ra-
+tios of predicted to observed line fluxes. For O VIII, NeIX,
+and Mg XIthe predictions agree to better than 30% with the
+observed line fluxes. For O VIIthe predictions are low by a
+factoroflessthan2,butthelineratiosagreewithin30%. Th e
+diagnosticlinefluxesplacetightconstraintsonthemodel.
+In this simple model, all of the emission from Mg XIand
+mostoftheemissionfromNe IXcomefromRegion1,which
+we associate with the shock front. If we assume a cooling
+length of 200 km, based on the accretion shock model de-
+scribed above (see Fig. 9), and use the derived emitting vol-
+ume of 1 .5×1028cm3, we obtain an area filling factor forACCRETIONAND CORONA INTWHYA 11
+Region1of1.5%ofthesurfaceareaofthestar.
+Region 2 has a larger volume than Region 1 by a factor of
+300 as determined primarily from O VII. If we assume that
+the cooling zone is the continuation of Region 1, we again
+obtain a length of about 200 km; however, this short length
+implies a filling factor larger than the stellar surface area and
+is not plausible. Instead, we use the scale height of 0.1 R⋆
+for 1.75 MK plasma to estimate a filling factor of 6.8% for
+Region 2, a value in goodagreementwith estimates from the
+ultraviolet (Costa et al. 2000) and optical continuum veili ng
+(Batalhaetal.2002). Scalingthemassas NeVimpliesthatthe
+massofRegion2is30timesthemassofRegion1.
+Romanova et al. (2004) have used three-dimensionalmag-
+netohydrodynamic (MHD) simulations to show that the hot
+spotsformedonthesurfacebyaccretionare inhomogeneous,
+withthelowerdensityandtemperatureregionsfillingalarg er
+surfaceareathanthedensercentralregion,seeminglysugg es-
+tive of Region 2. However, the larger mass of Region 2 also
+implies a higher accretion rate than for Region 1, and thus a
+larger density, in contradictionto our diagnostics. There fore,
+wedonotinterpretRegion2asasecondaccretionshockwith
+adifferentaccretionrate,butratherasa newaccretion-dr iven
+phenomena. The impact from the accretion presumably pro-
+videstheenergytoheatadditionalmaterialinthestellara tmo-
+sphere to over 1 MK. Very recently, two-dimensional MHD
+simulationshaveshownthatviolentoutflowsofshock-heate d
+material can propagate from the base of the accretion shock
+for cases with a high thermal to magnetic pressure ratio β
+(Orlandoet al.2009).
+One of the key concepts of the standard one-dimensional
+accretionmodelsuggestsasimplepicture(Fig.10). Thefiel d
+line that truncates the disk separates the open magnetic fiel d
+lines of the polar regionfromthe closed magnetic field loops
+intheequatorialregions. Theregionheatedbytheshock(ou r
+Region 2) provides a large supply of ionized material avail-
+able to form the soft X-ray emitting corona presented here, a
+warmwind(Dupreeetal.2005),andsoftX-rayjets(Güdelet
+al. 2008). This corona exists because magnetic loops confine
+it,butitisfedbytheaccretionprocessandisthusunlikest el-
+lar coronae in dwarf stars on the main sequence. Figure 10
+illustratesthe shockanditssurroundings.
+This model provides an explanation for the soft X-ray ex-
+cessdiscoveredinthe XMM-Newton ExtendedSurveyofthe
+TaurusMolecularCloud(XEST;Güdeletal.2007c;Telleschi
+et al. 2007). Grating spectra show that this excess emis-
+sion manifests itself in CTTS as enhanced O VIIrelative to
+OVIIIascomparedwithweak-lineTTauri(WTTS)andmain-
+sequence stars (Güdel & Telleschi 2007; Robrade & Schmitt
+2007). Our O VIImeasurements also demonstrate that the
+postshockcoolinggascan,insomecases,dilutethesignatu re
+of highNeand explain why not all accreting systems show
+highdensities.
+Having carefully distinguished between the accretion and
+corona in Section 3, we reconsider the assumption that these
+two components arise from different processes. Güdel &
+Telleschi (2007) find that the soft X-ray excess dependsboth
+onthe levelof magneticactivityas measuredby thetotal (in -
+cluding coronal) LXand on the presence of accretion. While
+we suggest that the role of accretion is to heat and ionize the
+surrounding stellar atmosphere to create a soft X-ray (post -
+shock cooling) plasma, it seems possible that this plasma
+couldbefurtherheatedto ∼10MK,eitherbytheusualcoro-
+nal heating (MHD) processes, or possibly by the accretion
+energy. Cranmer (2009) finds that in some systems, accre-
+FIG. 10.— Illustration of the shock and its surrounding environ ment. Re-
+gion 1 in the text corresponds to the shock and Region 2 to the p ostshock
+plasma. Figure courtesy of A.Szentgyorgyi.
+tion can provide the energy for coronal heating to such high
+temperatures.
+Studies of star forming regions have found that young
+starsshow strongX-rayactivity,with energeticflaresande x-
+tremely high X-ray emitting temperatures; however, the in-
+crease in activity with rotation rate breaks down for the ac-
+creting systems, leading to the conjecture that the stellar dy-
+namomightoperatedifferentlyin these systems(Preibisch et
+al. 2005). We suggest that the accretion process may play a
+moreprominentrolethanhasgenerallybeenbelieved. While
+X-raysignaturesoftheaccretionshockitselfmaynotusual ly
+dominatethe emission,the energyfromaccretionmayinfact
+contributeto feedingandheatinga newtypeofcorona.
+6.SUMMARYAND CONCLUSIONS
+We have analyzed the deep Chandra HETG spectrum of
+TW Hya to assess the relative contributions of accretion and
+coronal emission. We summarize the most important of our
+findings:
+1. Thespectrumshowslineandcontinuumemissionfrom
+∼10 MK coronal plasma, which may have a broad
+emission measure distribution. Unlike active main se-
+quencestars, the coronalEM distributionabruptlycuts
+off below ∼5 MK, probably due to absorption by the
+accretionstream.
+2. The spectrum also shows emission lines from plasma
+atTe=2.5 MK, as predictedby standard modelsof the
+accretion shock. TeandNediagnostics from He-like
+NeIXareinexcellentagreementwithaccretionmodels
+forTWHya.
+3. We measure the flux of the O VIIforbidden line in
+TW Hya for the first time to better than 3 σ, allowing a
+determinationofelectrondensity. Thisdensityislower
+thanNefromNe IXbyafactoroffourandlowerthan Ne
+from the standard postshock cooling model by a factor
+of seven, and leads to a new model of an accretion-fed
+corona.12 BRICKHOUSE ET AL.
+4. The high Teions suggest a stellar corona. By con-
+trast, line ratio diagnostics from the lower Teions,
+indicate two regions with different densities. In our
+model, the accretion shock itself has Te= 3.0 MK and
+Ne= 6.0×1012cm−3, while the postshock cooling re-
+gion hasTe=1.75 MK and Ne=2.0×1011cm−3. The
+postshock plasma has 30 times more mass than the
+shockitself. Thesurfaceareafillingfactoroftheshock
+is 1.5% while the postshock filling factor is approxi-
+mately6.8%.
+5. Line ratio diagnostics require at least two different ab-
+sorbers, with higher column density ( NH) for higher
+charge state. Absorption by the accretion stream is
+largerbya factorof2.5fortheshockthanforthepost-
+shockregion.
+6. A rangeofmodelsfortheEMdistributionandelemen-
+tal abundances provides acceptable fits to the global
+spectrum. While absorption, emission measure, and
+abundances are not independently determined by such
+methods,thehighNe/Oabundanceseemstobearobust
+result.
+7. From line profile analysis of Ne IXlines we determine
+a subsonicturbulentvelocityofabout165kms−1.
+This high resolution X-ray spectrum of TW Hya presents
+a rich set of diagnostic emission lines for characterizing t he
+physical conditions of the high energy plasma and under-
+standing the dominant physical processes. The observation sstrongly support the model of an accretion shock producing
+a few MK gas at high density, as first suggested by Kastner
+et al. (2002) from an earlier HETG spectrum. The diagnos-
+ticsalsosupporttheroleofaccretioninproducingthesoft X-
+rayexcessat O VIIpreviouslydiscovered(Güdel& Telleschi
+2007; Robrade & Schmitt 2007). This excess emission re-
+quiresthat theaccretionshockheat alargevolumeinthesur -
+rounding stellar atmosphere. In TW Hya the filling factor at
+OVIIislargerthanthefillingfactoroftheshockatNe IX,and
+is consistent with the filling factors of the ultraviolet and op-
+ticalcontinua. Bothopenandclosedmagneticfieldlinesmay
+emergefromthissurroundingareatochannelthisionizedga s
+back into the corona. While it is not yet clear whether the
+energyfrom the impact of accretion at the star can heat a hot
+(10 MK) corona, accelerate a highly ionized stellar wind, or
+drive hot jets, we suggest that accretion-fed coronae may be
+ubiquitousinaccretingyoungstars.
+We acknowledge support from NASA to the Smithso-
+nian AstrophysicalObservatory(SAO) under Chandra GO7-
+8018X for GJML. NSB and SJW were supported by NASA
+contract NAS8-03060 to SAO for the Chandra X-ray Center.
+SRC’scontributiontothisworkwassupportedbytheSprague
+FundoftheSmithonianInstitutionResearchEndowment,and
+by NASA grant NNG04GE77G to SAO. We thank Randall
+Smith for supporting the customized APEC code runs. We
+thank the Chandra Mission Planning team for their effortsto
+accommodatethe ground-basedobservingcampaign.
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+arXiv:1001.0751v2  [hep-th]  2 Jun 2010The CPTgroup of thespin-3/2 field
+B.Carballo P ´erez∗andM. Socolovsky†
+Instituto de Ciencias Nucleares, Universidad Nacional Aut ´ onoma deM´ exico,
+Circuito exterior, Ciudad Universitaria, 04510, M´ exico D .F.,M´ exico
+Abstract
+We find out that both the matrix and the operator CPT groups for the spin-3/2
+field (withor withoutmass)are respectively isomorphicto D4⋊Z2andQ×Z2.
+ThesegroupsareexactlythesamegroupsasfortheDiracfield ,thoughthereisno
+apriorireason whytheyshouldcoincide.
+Keywords: discretesymmetries;spin3/2-field;finitegroups
+∗brendacp@nucleares.unam.mx
+†socolovs@nucleares.unam.mx1 Introduction 2
+1 Introduction
+The CPT group of the Dirac field in Minkowski space-time was ob tained by
+Socolovsky in 2004 [1]. It were found two sets of consistent s olutions for the
+matrices of charge conjugation( C), parity (P), and timereversal ( T), which give
+the transformation of fields ˆψC(x) =Cˆ¯ψT(x),ˆψΠ(xΠ) =Pˆψ(x)andˆψτ(xτ) =
+Tˆψ(x)∗, wherexΠ= (t,−x)andxτ= (−t,x). Thesesetsare givenby:
+a)CD=±γ2γ0,PD=±iγ0,TD=±iγ3γ1,
+b)CD=±iγ2γ0,PD=±iγ0,TD=±γ3γ1.
+Each of these sets generates a non abelian group of sixteen el ements, respec-
+tively,G(1)
+θ∼=D4×Z2andG(2)
+θ∼=16E, whereD4is the group of symmetries
+of the square and 16Eis a non trivial extension of D4byZ2, isomorphic to a
+semidirectproductofthesegroups.
+On the other hand, the quantum operators ˆC,ˆPandˆT, acting on the Hilbert
+space, generateauniquegroup Gˆθ∼=Q×Z2, whereQisthequaterniongroup.
+With this in mind, we decided to find the CPTgroup of the spin-3/2 field
+(Rarita-Schwinger field), for both massive and massless cas es. This field could
+be useful for the description of compound objects (neglecti ng its structure in a
+first approximation), like the baryon decuplet components f or spin-3/2+[2], or
+forelementaryfields such as thegravitino.
+In orderto describe3/2-spinparticles,thesetofequation s
+(iγα∂α−m)ˆψµ(x) = 0, (1)
+γµˆψµ(x) = 0, (2)
+where∂α=∂
+∂xα,isrequired. Theseequationsareknownas theRarita-Schwi nger
+equation [3]. The first of these equation is a Dirac equation f or each vector com-
+ponent of the vector spinor ˆψµand the second one is known as the subsidiary
+condition. Precisely due to the more complexity of these equ ations with respect
+to the Dirac field equation, there is not a priori any apparent reason why the CPT
+groupfortheRarita-Schwingerfieldshouldcoincidewithth osefortheDiracfield.
+2 Parity
+If we want to study the Pinvariance of the Rarita-Schwinger equation, we
+needtorepeattheanalysisdonefortheDiracequationin[1] andalsoconsiderthe
+Pinvarianceofthesubsidiaryequation.2 Parity 3
+Multiplyingthisequationfromtheleftby P,changing x→ −xand inserting
+theunity,oneobtains:
+Pγ0P−1Pˆψ0(t,−x)+Pγ1P−1Pˆψ1(t,−x)+Pγ2P−1Pˆψ2(t,−x)
++Pγ3P−1Pˆψ3(t,−x) = 0,(3)
+but we need to take into account that the vector spinor change s by parity in the
+followingway:
+ˆψµ(t,x) =
+ˆψ0(t,x)
+ˆψ1(t,x)
+ˆψ2(t,x)
+ψ3(t,x)
+−→ˆψµπ(t,x) =
+Pˆψ0(t,−x)
+−Pˆψ1(t,−x)
+−Pˆψ2(t,−x)
+−Pψ3(t,−x)
+;(4)
+whereˆψµπ(t,x)can be written as ˆψµπ(t,x) =PPˆψµ(t,−x), withP ∈O(1,3)
+givenby:
+P=
+1 0 0 0
+0−1 0 0
+0 0−1 0
+0 0 0 −1
+(5)
+andP∈D16, whereD16istheDiracalgebra.
+Substitutingthecomponentsof ˆψµπ(t,x)in (3)oneobtains:
+Pγ0P−1ˆψ0π(t,x)−PγkP−1ˆψkπ(t,x) = 0. (6)
+Thisimpliestheconstraintson P:
+Pγ0P−1=γ0, PγkP−1=−γk, (7)
+or
+Pγ0P−1=−γ0, PγkP−1=γk. (8)
+The relations (7) are the same as those obtained from the Dira c equation,
+whose already known solution is PD=±iγ0, while from the relations (8) is
+obtainedP=P′=zγ3γ2γ1, withz∈C∗=C\{0}.
+Followingthesameanalysisas in[1], thereare twopossibil itiesforeach P:3 Chargeconjugation 4
+a)P2= +1⇒z=±1⇒P′=±γ3γ2γ1;
+b)P2=−1⇒z=±i⇒P′=±iγ3γ2γ1.
+In the first case: P′†=P′=P′−1=−P′T=−P′∗and in the second one:
+P′†=−P′=P′−1=P′T=−P′∗.
+3 Chargeconjugation
+Tostudythe Cinvarianceofthesubsidiaryequation,wemusttakethecomp lex
+conjugateofthisequation,multiplyfromtheleftby Cγ0andinserttheunitmatrix.
+Thisis:
+(Cγ0)γµ∗(Cγ0)−1ˆψµC(x) = 0, (9)
+whereˆψµC(x) =Cγ0ˆψ∗
+µ(x).
+In thiscase, theconstraintson Care:
+(Cγ0)γµ∗(Cγ0)−1=γµ, (10)
+or
+(Cγ0)γµ∗(Cγ0)−1=−γµ. (11)
+Taking into account that γ0γµ∗γ0=γµT, one can find the solutions for the C
+matrices. Theequationwiththenegativesignisthesameasi ntheDiraccaseand
+leadstoC=CD=ηγ2γ0;on theotherhand, fromequation(10)wearriveto:
+CγµTC−1=γµ, (12)
+whichleads tothesolution C=C′=ηγ3γ1.
+ForC′,asecondapplicationofthechargeconjugationtransforma tionisgiven
+by:
+(ˆψC)C=ˆψC2=C′ˆ¯ψT
+C=C′(ˆψ†
+Cγ0)T=C′γ0ˆψ∗
+C=C′γ0C′∗γ0ˆψ=−C′C′∗ˆψ
+=−|η|2γ3γ1γ3∗γ1∗ˆψ=|η|2(γ3)2(γ1)2ˆψ=|η|2ˆψ. (13)
+Since the effect on ˆψcan be, at most, a multiplication by a phase, then η∈
+U(1)andC′isunitary. Thisis:4 Timereversal 5
+C′C′†=ηγ3γ1¯η(γ3γ1)†=|η|2γ3γ1γ1γ3=|η|2γ3(γ1)2γ3=|η|21 = 1.(14)
+Hence, forC=C′wealsofind that ˆψC2=ˆψ.
+As in [1], due to a symmetry consideration between ˆψC=C′ˆ¯ψTandˆ¯ψC=
+−¯η2ˆψTC′, it follows that −¯η2=±1and taking into account that |η|4= 1, one
+obtainsη2=±1, whichimpliesthat η=±1,±i.
+Then, forthespin-3/2field therearetwo possibilitiesfore achC:
+a)C′=±γ3γ1, C D=±γ2γ0;
+b)C′=±iγ3γ1, C D=±iγ2γ0.
+4 Timereversal
+We start again from the subsidiary equation, change t→ −tand take the
+complexconjugate:
+Tγ0∗T−1Tˆψ0(−t,x)∗+Tγ1∗T−1Tˆψ1(−t,x)∗+Tγ2∗T−1Tˆψ2(−t,x)∗
++Tγ3∗T−1Tˆψ3(−t,x)∗= 0,(15)
+but we need to take into account that the vector spinorchange s by timeinversion
+inthefollowingway:
+ˆψµ(t,x) =
+ˆψ0(t,x)
+ˆψ1(t,x)
+ˆψ2(t,x)
+ψ3(t,x)
+−→ˆψµτ(t,x) =
+−Tˆψ0(−t,x)∗
+Tˆψ1(−t,x)∗
+Tˆψ2(−t,x)∗
+Tψ3(−t,x)∗
+;(16)
+whereˆψµτ(t,x)can be written as ˆψµτ(t,x) =TTˆψµ(−t,x)∗, withT ∈O(1,3)
+givenby:
+T=
+−1 0 0 0
+0 1 0 0
+0 0 1 0
+0 0 0 1
+(17)
+andT∈D16.5 Matrixand operatorCPTgroups 6
+Substitutingthecomponentsof ˆψµτ(t,x)in(15)oneobtains:
+−Tγ0∗T−1ˆψ0τ(t,x)+Tγk∗T−1ˆψkτ(t,x) = 0, (18)
+fromwhich wecan deducetheconstraintson T:
+Tγ0T−1=γ0, Tγk∗T−1=−γk, (19)
+or
+Tγ0T−1=−γ0, Tγk∗T−1=γk. (20)
+The relations (19) are the same as those obtained from the Dir ac equation,
+whose already known solution is TD=eiλγ3γ1, while from the relations (20) is
+obtainedT=T′=wγ2γ0,withw∈C∗=C\{0}.
+ForT=T′, applyingτtwice,we arriveto:
+ˆψ(t,x)→ˆψτ(t,x) =T′ˆψ(−t,x)∗→T′(T′ˆψ(t,x)∗)∗=T′T′∗ˆψ(t,x).(21)
+and takingintoaccount that
+T′T′∗=|w|2γ2γ0(γ2γ0)∗=−|w|2γ2γ0γ2γ0=|w|2γ2γ2γ0γ0=−|w|21,(22)
+itfollowsthat ˆψτ2=−ˆψ,by asimilarargumentto thatusedfor C.
+Thus,T′T′∗=−1, which implies T′∗=−T′−1andw∈U(1). Then,T′=
+eiλγ2γ0yT′†=e−iλγ2γ0.
+5 MatrixandoperatorCPTgroups
+In summary,onehasboththetwosets ofmatrices:
+a)CD=±γ2γ0, P D=±iγ0, T D=±iγ3γ1,
+b)CD=±iγ2γ0, P D=±iγ0, T D=±γ3γ1;
+and the set: C′,P′,T′, satisfying the subsidiary equation. But, only the matrice s
+CD,PD,TD,alsosatisfytheDiractypeequation. Thatiswhytheyareth ematrices
+whichconform thematrixCPT groupofthespin-3/2field withm ass.
+Ifwetakethezero masslimitoftheDiractypeequation,
+iγα∂αˆψ(x) = 0, (23)5 Matrixand operatorCPTgroups 7
+andanalyzeitsbehaviorunderparity,chargeconjugationa ndtimereversal,aswe
+didforthesubsidiarycondition,wehave, respectively:
+i(Pγ0P−1∂0−PγiP−1∂i)ˆψΠ(x) = 0, (24)
+(i∂µ+qAµ)(Cγ0)γµ∗(Cγ0)−1ˆψC(x) = 0. (25)
+i(γ0∗∂
+∂t−γk∗∂
+∂xk)ˆψτ(x) = 0. (26)
+From the above equations we can then obtain the correspondin g restrictions
+ontheC,PandTmatrices, respectively:
+Pγ0P−1=±γ0, PγkP−1=∓γk,
+CγµTC−1=±γµ,
+Tγ0T−1=±γ0, Tγk∗T−1=∓γk. (27)
+These relations generate the same sets of matrices CD,PD,TDandC′,P′,T′
+which gave the subsidiary condition. But if we take into acco unt thatˆ¯ψˆψis the
+chargedensityoperator, for C=C′,it followsthat:
+(ˆ¯ψˆψ)C=ˆ¯ψCˆψC=−¯η2ˆψTC′C′ˆ¯ψT=−¯η2C′2(ˆ¯ψˆψ)T
+=−¯η2C′2ˆ¯ψˆψ= (¯ηη)2ˆ¯ψˆψ=|η|4ˆ¯ψˆψ=−ˆ¯ψˆψ; (28)
+from which |η|4=−1, which is a contradiction. Hence, the matrix C=C′must
+bediscarted.
+It was demostratedin[1]that CandPmustfulfilltherelation:
+C(P−1)TC−1=P, (29)
+whileCandTare related by:
+CT∗=TC∗. (30)
+Dueto thefact thateach componentofthevectorspinorsatis fies aDiractype
+equation,theaboverelationsalsoholdinourcase. Thatisw hythesetofmatrices:
+C′,P′,T′arealso discardedforthemasslesscase.6 Acknowledgment 8
+In order to find the operator CPT group for the spin-3/2 field (w ith or with-
+out mass), we follow the same procedure developed in [1], for the case of the
+correspondingCPT group oftheDiracfield.
+TakingˆAandˆBas any of the operators ˆCD,ˆPDandˆTD; andˆψ, as each
+componentofthevectorspinor ˆψµ(x), therelations:
+ˆA·ˆψ=ˆA†ˆψˆA (31)
+and
+(ˆA∗ˆB)·ˆψ= (ˆAˆB)†ˆψ(ˆAˆB), (32)
+can bedefined.
+UsingtheaboveexpressionsandwithsupportinthematrixCP Tgroup,through
+theformulas thatlinkmatriceswithoperators:
+Pˆψ(t,−x) =ˆP†ˆψ(t,x)ˆP,
+Cˆ¯ψT(x) =ˆC†ˆψ(x)ˆC,
+Tˆψ(−t,x)∗=ˆT†ˆψ(t,x)†ˆT; (33)
+therelations:
+ˆPD∗ˆPD=−1,ˆCD∗ˆCD= 1,ˆTD∗ˆTD=−1,
+ˆTD∗ˆPD=−ˆPD∗ˆTD,ˆCD∗ˆPD=ˆPD∗ˆCD,ˆCD∗ˆTD=ˆTD∗ˆCD,(34)
+wereobtainedin[1]; fromwhichitispossibletobuild,also usingtheproperty of
+associativity,themultiplicationtablefortheoperatorC PT group.
+Itwasalsodemostratedin[1]thatonlythesecondofthetwos olutionsforthe
+matrix group ( G(2)
+θ∼=16E∼=D4⋊ Z2), is compatible with the operator group
+(Gˆθ∼=Q×Z2).
+In summary,we showedthat boththematrixand theoperatorCP T groupsfor
+the spin-3/2 field (with or without mass) coincide with the co rresponding groups
+fortheDiracfield.
+6 Acknowledgment
+ThisworkwaspartiallysupportbytheprojectPAPIITIN1186 09-2,DGAPA-
+UNAM, M´ exico. B. Carballo P´ erez also acknowledge financia l support from
+CONACyT, M´ exico. The authors thank Prof. O. S. Zandr´ on fro m IFIR, Rosario,
+Argentina,forsuggestingthiswork.REFERENCES 9
+References
+[1] M.Socolovsky,2004, TheCPTgroupoftheDiracField ,InternationalJour-
+nalofTheoretical Physics, 43, 1941-1967;arXiv: math-ph/0404038.
+[2] F. Riazuddin, 2000, A modern introductionto ParticlePhysics, Second Edi-
+tion(World Scientific), 121-122.
+[3] W. Rarita and J. Schwinger, 1941, On a Theory of Particles with Half-
+IntegralSpin ,PhysicalReview, 60,61.
\ No newline at end of file
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+arXiv:1001.0753v1  [physics.plasm-ph]  5 Jan 2010Power law behavior for the zigzag transition in a Yukawa clus ter
+T. E. Sheridan∗and Andrew L. Magyar†
+Department of Physics and Astronomy,
+Ohio Northern University, Ada, OH 45810
+Abstract
+We provide direct experimental evidence that the one-dimen sional (1D) to two-dimensional (2D)
+zigzag transition in a Yukawa cluster exhibits power law beh avior. Configurations of a six-particle
+dusty (complex) plasma confined in a biharmonic potential we ll are characterized as the well
+anisotropy is reduced. When the anisotropy is large the part icles are in a 1D straight line config-
+uration. As the anisotropy is decreased the cluster undergo es a zigzag transition to a 2D configu-
+ration. The measured dependence of cluster width on anisotr opy is well described by a power law.
+A second transition from the zigzag to an elliptical configur ation is also observed. The results are
+in very good agreement with a model for particles interactin g through a Yukawa potential.
+∗Electronic address: t-sheridan@onu.edu
+†Present address: Department of Physics, Penn State University , University Park, PA 16802
+1Plasma is a quasi-neutral gas that contains electrons, ions and (of ten) neutral atoms and
+which exhibits collective behavior [1]. A dusty (complex) plasma is a syst ems of interacting
+microscopic dust particles immersed in a standard electron-ion plasm a. For typical labora-
+tory conditions, dust particles acquire a net charge q <0 from the electron and ion currents.
+The Coulomb interaction between the charged dust particles is shield ed by the response of
+the free charge in the plasma. In the plane of a two-dimensional clus ter, the inter-particle
+potential is well approximated by a Yukawa (i.e., Debye or shielded Cou lomb) potential [2]
+V(r) =1
+4πǫ0q
+re−r/λD, (1)
+whereλDis the Debye screening length. An unshielded Coulomb interaction is re covered
+asλD→ ∞. Dusty plasma is an exceptional experimental system for examining the static
+and dynamic properties of Yukawa clusters–small clusters of part icles interacting through a
+Yukawa potential.
+In laboratory experiments the dusty plasma is usually confined, typ ically through a
+combination of gravitational and electrical forces. Monodisperse microspheres may then
+form strongly-coupled one-dimensional (1D) [3–5], two-dimensiona l (2D) [6–8] or even three-
+dimensional systems [9]. The general 2Dpotential well isbiharmonic f orsmall displacements
+from the potential energy minimum [10–12]. A dusty plasma confined in a 2D biharmonic
+well may be either in a 1D straight line configuration, or a 2D configura tion, where the
+transition from 1D to 2D states is via the zigzag instability [10, 11, 13– 15].
+A cold 2D dusty plasma can be modeled [10, 12, 16] by assuming nidentical particles
+with mass mand charge q, where the ith particle is at ( xi,yi). The potential energy Uof a
+configuration {xi,yi}is given by the sum of the particles’ potential energy with respect t o
+the biharmonic well plus the sum of the potential energy for each pa ir-wise interaction,
+U=n/summationdisplay
+i=11
+2m/parenleftbig
+ω2
+xx2
+i+ω2
+yy2
+i/parenrightbig
++n/summationdisplay
+i=1n/summationdisplay
+j>i1
+4πǫ0q2
+rije−rij/λD, (2)
+whererij=/radicalBig
+(xi−xj)2+(yi−yj)2is the separation between particles iandj, andωxand
+ωyare the single-particle oscillation frequencies in the xandydirections, respectively. The
+equilibrium configuration minimizes U.
+We nondimensionalize Eq. (2) using the variables ξi=xi/r0,ηi=yi/r0,ρij=rij/r0and
+parameters
+α2=ω2
+y
+ω2x, κ=r0
+λD. (3)
+2Here the characteristic length
+r0=/parenleftbigg2
+mω2
+xq2
+4πǫ0/parenrightbigg1/3
+(4)
+is defined using the longitudinal frequency ωx. The dimensionless potential energy is then
+U
+U0=n/summationdisplay
+i=1/parenleftbig
+ξ2
+i+α2η2
+i/parenrightbig
++n/summationdisplay
+i=1n/summationdisplay
+j>ie−κρij
+ρij, (5)
+whereU0is the characteristic potential energy. Equilibrium configurations {ηi,ξi}of the
+model [Eq. (5)] depend on three parameters: n,κandα2. Hereκis the Debye shielding
+parameterand κ= 0correspondstoanunshielded Coulombinteraction. Thewellaniso tropy
+parameter is α2. Weassume ωy≥ωxorα2≥1. Straight line configurations yi=ηi= 0 then
+lie in the x(longitudinal) direction and are independent of α2. An equilibrium configuration
+{ξi,ηi}canbecomparedtoanexperimentally measured configuration {xi,yi}bymultiplying
+each coordinate by r0.
+Within the context of this model [Eq. (5)], abrupt 1D to 2D transition s [10, 16] are
+predicted to occur for critical values of the three parameters n,κandα2due to a zigzag
+instability. This has been confirmed experimentally as nwas varied by Melzer [11] and
+Sheridan and Wells [16]. Melzer [11] also observed a zigzag transition a s the neutral gas
+pressure was changed, which presumably changed the plasma dens ity and therefore both κ
+andα2. Sheridan and Wells [16] confirmed that the zigzag transition caused by increasing
+nin small clusters behaves as a continuous phase transition between 1D and 2D states,
+where the cluster width near the transition obeys a power law. Using the model of Eq.
+(5), they also predicted [16] that the zigzag transition in small clust ers exhibits power law
+behavior as either κorα2is varied. Piacente, et al. [17] reached a similar conclusion from
+a computational study of unbounded Yukawa systems ( n→ ∞) with a transverse parabolic
+well and longitudinal periodic boundaries. Structural transitions d ue to changes in κorα2
+have not been characterized experimentally.
+In this work we investigate the transitions a Yukawa cluster (i.e., a sm all dusty plasma)
+undergoes as the well anisotropy α2is varied. We use a novel experimental method in which
+the physical geometry of the confining potential well is changed wit hout disrupting the
+plasma discharge. Thedischarge parametersarenearly constant , so that theDebye shielding
+parameter κis also nearly constant. The well anisotropy is accurately determine d using
+measured center-of-mass frequencies for the dusty plasma. As predicted [16], the transverse
+3xyz
+50.8 mm55 mm f/2.8
+telecentric lens2 x adapter2/3 in. CMOS
+video camera
+red laser sheet
+powered electrodeddust particlesAl confining bars
+6.35 mmleadscrew12.7 mm
+Figure 1: Schematic of the experimental geometry (not to sca le). Six dust particles are confined in
+the biharmonic potential well created by a rectangular depr ession placed on the powered electrode
+in an rfdischarge. Theconfiningwell anisotropy is varied by changing thewidth dwhilethe plasma
+is on, and without losing the particles, using a lead screw.
+cluster width yrmsexhibits power law behavior vs α2following the zigzag transition. The
+Debye shielding parameter is found by comparison to the model, and D ebye shielding is
+shown to be important.
+The dusty plasma was studied in the DONUT (Dusty Ohio Northern Univ ersity exper-
+imenT) apparatus (Fig. 1) [7, 8, 12, 16, 18, 19]. Argon gas was leake d into the vacuum
+chamber and the pressure was stabilized at 15.4 mtorr. At this pres sure normal modes of the
+dust clusters are under damped [18]. Inside the chamber a capacitiv ely-coupled rf electrode
+is used both to sustain the discharge and to levitate the dust partic les. The rf frequency
+was 13.56 MHz, the forward rf power was ∼7 W and the dc self-bias on the electrode was
+−90 V.
+The confining well (Fig. 1) is produced by a rectangular aperture pla ced on the 89-mm
+diameter aluminum electrode, creating a biharmonic well in the horizon tal (x-y) plane [3].
+Theapertureismadeoffouraluminumbars, eachwithacrosssectio nof6.35mm ×12.7mm.
+Thedepthofthetrapis6.35mm, thelengthisfixedat50.8mm, andthe widthdisadjustable
+4from≈0 to 38 mm. Here dcan be adjusted while the plasma is on, and without losing the
+dust particles, by mean of a lead screw which is rotated from outside the vacuum. The lead
+screw is made of joined left-handed and right-handed threaded ro ds [unified national coarse
+(UNC) thread size 6-32] that move the two confining bars simultane ously closer together or
+farther apart a distance of 1/16 inch ( ≈1.59 mm) per revolution, thus keeping the minimum
+of the potential well centered with respect to the disk electrode. This arrangement allows us
+to continuously vary the anisotropy parameter α2while maintaining nearly the same plasma
+discharge parameters.
+The dust particles used were melamine formaldehyde resin spheres w ith a nominal diam-
+eter of 9.62±0.09µm. As noted previously [18], we believe that the particle diameter is
+actually 8 .94µm, giving m= 5.65×10−13kg. Confined dust particles are illuminated by a
+red diode laser sheet. A top-view camera mounted about 30 cm abov e the electrode is used
+to acquire dust particle position data. For this experiment, the res olution of the camera
+and associated optics was 16.47 µm/pixel. Sequences of 2048 frames were recorded at 27.9
+frames/s. A side-view camera was used to confirm that there were no out-of-plane particles.
+We acquired 20 data sets with the same n= 6 particles for trap widths d= 12.8 to 36.6
+mm.
+Representative measured configurations are shown in Fig. 2 for inc reasingd. As seen in
+Fig. 2(a), the particles are in a straight line configuration when dis small, or equivalently,
+α2is large. As we increase dthe cluster goes through two structural transitions, straight
+line to zigzag [Figs. 2(b) and (c)], and then zigzag to elliptical [Fig. 2(d) ]. The transition
+to the zigzag state occurs at d≈23 mm and the transition to the elliptical state occurs at
+d≈30 mm. In the elliptical state all six particles lie on the convex hull of th e cluster.
+To analyze the experimental data, we first find the dependence of the anisotropy param-
+eterα2ond. This is done by measuring the frequencies ωxandωyof the c.m. modes from
+the particles’ Brownian motions [8, 12, 16]. The power spectra of th e time histories of the
+c.m. coordinates, xcm= (1/n)/summationtextxiandycm= (1/n)/summationtextyi, were fitted with the expression
+for a driven damped harmonic oscillator to determine the correspon ding normal mode fre-
+quencies ωxandωy. For each d, the dust particle configuration was characterized by the
+root-mean-squared values xrmsandyrmsaveraged over all frames, where
+x2
+rms=1
+nn/summationdisplay
+i=1(xi−xcm)2, y2
+rms=1
+nn/summationdisplay
+i=1(yi−ycm)2. (6)
+5-2 -1 0 1 2
+x (mm)1
+0
+-1
+1
+0
+-1
+1
+0
+-1
+1
+0
+-1(d) d = 31.9 mm, α2 = 5.0(c) d = 25.5 mm, α2 = 8.9(b) d = 23.2 mm, α2 = 11.0(a) d = 20.8 mm, α2 = 13.6y (mm) y (mm) y (mm) y (mm)
+Figure 2: Measured particle configurations vs increasing tr ap width d, or equivalently, decreasing
+well anisotropy α2. (a) Straight line configuration, (b) and (c) zigzag configur ations and (d)
+elliptical configuration. Values of the anisotropy paramet erα2were determined from measured
+c.m. frequencies.
+Herexrmsrepresents the longitudinal cluster size (i.e., its length), while yrmsis the trans-
+verse cluster width. In particular, yrmscan be used as an order parameter for the zigzag
+transition since yrms= 0 in the 1D straight line configuration and yrms>0 for the 2D zigzag
+configuration [16].
+The measured c.m. frequencies ωxandωyare plotted vs din Fig. 3(a). Here ωx(the
+frequency associated with the fixed 50.8 mm separation) shows a we ak linear increase with
+6d, whileωyis roughly constant for d/lessorsimilar20 mm and then decreases as dincreases. That is,
+for these experimental conditions decreasing dbelow≈20 mm does not increase ωybecause
+the sheath edge can no longer conform to the rectangular depres sion, but rather, is pushed
+upward [11]. This explanation is given further weight by measurement s of the cluster height
+habove the top of the confining bars [Fig. 3(a)]. Here hwas determined by the height of the
+laser sheet used to illuminated the particles and has an uncertainty ±0.2 mm. As dincreases
+hdecreases, indicating that the sheath edge is moving into the recta ngular depression.
+Rather than taking ωxandωyfor eachdand directly calculating α2, we fitted a line to ωx
+and a quadratic curve to ωyford >20 mm, which spans the zigzag and elliptical transitions.
+The average value of ωxover this range is 7.47 rad/s. By dividing the two fitted curves we
+determined α2as a function of d[Fig. 3(b)]. We cover a wide range of anisotropies, where
+α2decreases with increasing dfromα2≈14 to 2.5. If we could increase dto 50.8 mm to
+achieve an isotropic well, we would expect α2= 1. The slight increase of ωxwithdimplies
+thatκ∝ω−2/3
+xincreases weakly with α2. If the only change in κis due to ωx, then we
+estimate that κincreases by ≈6% over the range of α2values considered.
+The measured cluster width yrmsand length xrmsare plotted vs α2in Figs. 4(a) and (b),
+respectively. From the yrmsdata we observe a transition from the 1D configuration to a 2D
+zigzag configuration at α2≈12. From α2≈12 to 6.5, the yrmsdata is well fitted by a power
+law with a critical value α2
+c= 12.1 and an exponent of 0.35. The model solution shows the
+same power law behavior [16]. The divergence from the power law fit at α2/lessorsimilar6.5 indicates
+a second transition [16], from the zigzag configuration to an elliptical configuration. The
+elliptical structure [12] occurs when the second (fifth) particle in t he cluster moves between
+the first and third (fourth and sixth) particles, thereby moving on to the convex hull of the
+cluster. Since the equilibrium shell structure for six particles in an iso tropic well is (1, 5), a
+further symmetry breaking transition is needed to reach the α2= 1 configuration.
+For the first value of dafter the zigzag transition ( α2= 11.8), the cluster was seen to flip
+between the two anti-symmetric zigzag configurations. (Since we c an varyα2continuously
+it is possible to tune for this behavior.) A zigzag configuration has two degenerate states
+since the energy is invariant under a reflection of particle coordinat esy→ −y. In a stable
+zigzag configuration these two states are separated by a potent ial barrier corresponding to
+the higher energy of the unstable straight line configuration and th e zigzag state is bistable.
+Just above the zigzag transition the barrier between the two stat es is low, and thermal
+7051015202530
+0246810frequency (rad/s)cluster height h (mm)(a)
+02468101214
+0 5 10 15 20 25 30 35 40
+bar separation d (mm)ωy
+ωxh
+(b)anisotropy parameter α2
+Figure 3: (a) Dependence of measured c.m. frequencies ωxandωyand the height hof the cluster
+above the top of the well on the width d. Hereωxis fitted with a linear function and ωywith a
+quadratic curve. (b) Dependence of the smoothed anisotropy parameter α2=ω2
+y/ω2
+xond.
+fluctuations or small periodic perturbations can cause the system to flip between the two
+zigzagstates, aswasobserved. Forthenextlowervalueof α2flippingwasnotseen, indicating
+that the potential barrier between the two degenerate zigzag st ates was then too high.
+Theoretical configurations which can be compared to the experime nt are found by mini-
+mizing Eq. (5), allowing us to predict xrms=r0ξrmsandyrms=r0ηrmsas a function of α2.
+To calculate a configuration three parameters, α2,κandn, are required. We know n, and
+α2has been measured, so we only need to determine the shielding param eterκ. To do this
+we assume that κis independent of α2and then minimize the sum of the squared differences
+between the measured and computed values of yrms, givingκ= 2.4 andr0= 1.25 mm. The
+Debye length λD=r0/κ= 0.52 mm is comparable to the inter-particle distance and the
+average charge q=−1.2×104e. These values are consistent with those found in previous
+experiments for similar plasma conditions [8, 12, 16, 18, 19].
+800.20.40.60.81
+0 2 4 6 8 10 12 1400.10.20.3(a)
+(b)
+anisotropy parameter α2cluster width yrms (mm) cluster length xrms (mm)
+-10123456
+0 2 4 6 8 10 12 14
+α2% error, δr0
+Figure 4: (a) Measured cluster width yrms(open circles), power law fit to data near the zigzag
+transition (dashed line) and widths computed from the model forκ= 2.4 andr0= 1.25 mm (solid
+line) vs the measured anisotropy parameter α2. (b) Measured cluster length xrms(open circles)
+and lengths computed from the model (solid line) vs α2. (Inset) Comparison of the percentage
+difference between the xrmsdata and the model (open circles) with the percentage change in
+r0∼ω−2/3
+x(broken line) vs α2.
+As shown in Fig. 4(a), the cluster width yrmscalculated using the model for κ= 2.4 and
+r0= 1.25 mm exhibits excellent agreement with the experimental data, inclu ding reproduc-
+ing the transition between the zigzag and elliptical configurations. T he agreement between
+the measured and predicted values of xrmsprovides a cross check on the model. We find
+that the predicted values of xrms[Fig. 4(b)] show good agreement for α2/lessorsimilar4, but system-
+atically underestimate the measured cluster length as α2increases. For the longest clusters,
+which are in a straight line configuration, the predicted cluster lengt hs are about 6% below
+9the measured values. As shown in the Fig. 4(inset), this difference c an be attributed to
+an experimental increase in r0∼ω−2/3
+xdue to the measured decrease of ωxwithα2. This
+implies that the average particle charge qis effectively constant, as previously observed for
+constant neutral pressure [8, 19]. This increase in r0withα2is not as apparent for yrms
+sinceyrms/lessorsimilar0.2xrmsandyrms→0 asα2increases.
+Insummary, we have provided direct evidence that the width ofa Yu kawa cluster exhibits
+power law behavior for the 1D to 2D zigzag transition caused by decr easing the confining
+well anisotropy parameter α2, confirming a previous prediction [16]. Experiments were
+performed using a dusty plasma with n= 6 particles confined in the biharmonic well above
+a rectangular depression. The width dof the rectangular depression was increased while
+the plasma remained on to decrease α2while the Debye shielding parameter κremained
+essentially constant. The dependence of α2ondwas accurately determined by measuring
+the c.m. frequencies of the dusty plasma. A transition from the zigz ag configuration to an
+elliptical configuration was also observed. The cluster width was fou nd to be in excellent
+agreement with the predictions of a model which assumes identical p articles confined in a 2D
+biharmonic well and interacting through a Yukawa potential. From th e fit to the model we
+found the Debye length is comparable to the inter-particle distance , so that Debye shielding
+significantly effects the physics of these clusters.
+Acknowledgments
+Portions of this paper are taken from A. L. M.’s undergraduate phy sics thesis.
+[1] F. F. Chen, Introduction to Plasma Physics (Plenum, New Y ork, 1974), p. 3.
+[2] M. Lampe, G. Joyce, G. Ganguli and V. Gavrishchaka, Phys. Plasmas 7, 3851 (2000).
+[3] A. Homann, A. Melzer, S. Peters and A. Piel, Phys. Rev. E 56, 7138 (1997).
+[4] T. Misawa, N. Ohno, K. Asano, M. Sawai, S. Takamura and P. K . Kaw, Phys. Rev. Lett. 86,
+1219 (2001).
+[5] B. Liu and J. Goree, Phys. Rev. E 71, 046410 (2005).
+[6] W.-T. Juan, Z.-H. Huang, J.-W. Hsu, Y.-J. Lai and L. I, Phy s. Rev. E 58, R6947 (1998).
+[7] T. E. Sheridan, J. Phys. D: Appl. Phys 39, 693 (2006).
+10[8] T. E. Sheridan and W. L. Theisen, Phys. Plasmas 13, 062110 (2006).
+[9] O. Arp, D. Block, A. Piel and A. Melzer, Phys. Rev. Lett. 93, 165004 (2004).
+[10] L. Cˆ andido, J.-P. Rino, N. Studart and F. M. Peeters, J. Phys.: Condens. Matter 10, 11627
+(1998).
+[11] A. Melzer, Phys. Rev. E 73, 056404 (2006).
+[12] T. E. Sheridan, K. D. Wells, M. J. Garee and A. C. Herrick, J. Appl. Phys. 101, 113309
+(2007).
+[13] J. P. Schiffer, Phys. Rev. Lett. 70, 818 (1993).
+[14] S. W. S. Apolinario, B. Partoens and F. M. Peeters, Phys. Rev. E74, 031107 (2006).
+[15] T. E. Sheridan, Phys. Scr. 80, 065502 (2009).
+[16] T. E. Sheridan and K. D. Wells, Phys. Rev. E (in press).
+[17] B. Piacente, I. V. Schweigert, J. J. Betouras and F. M. Pe eters, Phys. Rev. B 69, 045324
+(2004).
+[18] T. E. Sheridan, Phys. Rev. E 72, 026405 (2005).
+[19] T. E. Sheridan, J. Appl. Phys. 106, 033303 (2009).
+11
\ No newline at end of file
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+arXiv:1001.0754v1  [physics.flu-dyn]  5 Jan 2010Manuscript submitted to Website: http://AIMsciences.org
+AIMS’ Journals
+Volume X, Number 0X, XX200X pp.X–XX
+LABORATORY EXPERIMENTS ON
+WAVE TURBULENCE
+Eric Falcon
+Laboratoire Mati` ere et Syst` emes Complexes (MSC)
+Universit´ e Paris Diderot, CNRS (UMR 7057)
+10 rue A. Domon & L. Duquet, 75 013 Paris, France
+Abstract. This review paper is devoted to a presentation of recent prog ress in
+wave turbulence. I first present the context and state of the a rt of this field of
+research both experimentally and theoretically. I then foc us on the case of wave
+turbulence on the surface of a fluid, and I discuss the main res ults obtained by
+our group: caracterization ofthe gravity and capillarywav e turbulence regimes,
+the first observation of intermittency in wave turbulence, t he occurrence of
+strong fluctuations of injected power in the fluid, the observ ation of a pure
+capillary wave turbulence in low gravity environment and th e observation of
+magnetic wave turbulence on the surface of a ferrofluid. Fina lly, open questions
+in wave turbulence are discussed.
+1.Context and state of the art. When strong enough surface waves propagate
+in a medium, their interactions can generate waves of different wave lengths. This
+energy transfer through the different spatial scales can occur o n a wide range of
+wavelengths. This stationary out-of-equilibrium state is called the w ave turbulence:
+the energy of the system cascades through the scales from a sca le where energy
+is injected up to a small scale where energy is dissipated. Wave turbu lence thus
+concerns the study of dynamical and statistical properties of an ensemble of in-
+teracting nonlinear waves. The archetype of wave turbulence is th e study of the
+random state of ocean surface waves generated by wind or curre nt. But this is a
+very common phenomenon that is present in various situations: sur face waves on
+the sea [28,42,50,90], internal waves in the ocean [ 64], Alfv´ en waves in solar winds
+[82], radar waves in ionosphere, spin waves in solids, Rossby waves in geo physics,
+ion waves [ 66] and Langmuir waves [ 49,96] in plasmas, waves in nonlinear optics
+[59], quantum waves in Bose condensates [ 9,24,30,80] ...
+Wave turbulence is thus an interdisciplinary subject that involves diff erent com-
+munity: astrophysics, geophysics, mathematics, fluid mechanics a nd different fields
+of physics. Since the early developments of theoretical tools by ap plied mathe-
+maticians and physicists, the first ones to be interested on wave tu rbulence were
+oceanographers and meteorologists. Their motivations are numer ous such as to de-
+velop climatic models, predict the sea state with more precision, or ex tract some
+energy from ocean waves as a source of alternative energy... At a m ore fundamental
+level, the goal is to understand the energy transfers between no nlinear interacting
+2000Mathematics Subject Classification. Primary: 7605, 76B15; Secondary: 76F99, 76D33.
+Key words and phrases. Wave turbulence, N-wave interaction process, weak turbule nce, surface
+waves, experiments, intermittency, power spectrum, energ y flux, gravity waves, capillary waves.
+12 ERIC FALCON
+waves by means of generic laws whatever the particular medium in whic h these
+waves propagate.
+Contrarily to a widespread belief, the analogy between wave turbule nce and
+hydrodynamicturbulencewithin afluid (3Dturbulence) orwithin afilmo ffluid(2D
+turbulence) is quite limited. Although being also governed by the nonlin ear effects
+and studied in a statistical and non-determinist way, wave turbulen ce is described
+by equations which strongly differs from Navier–Stokes ones of usu al turbulence.
+Wave turbulence theory, also known as weak turbulence (WT), is a s tatistical
+theory describing an ensemble of weakly nonlinear interacting waves . The first ana-
+lytical studies began in the sixties [ 7,8,46]. They were motivated by oceanographic
+measurements of Fourier spectra of the surface wave elevation a s a function of the
+wavenumber. The aim was to understand how nonlinear interactions between the
+waves generate transfer of energy among different scales thus e xplaining the shapes
+of the observed Fourier spectra. In the early seventies, Zakhar ov and co-workers
+understood that, besides equilibrium spectra, nonlinearly interact ing waves also in-
+volve spectra that correspond to the transfer of a finite energy flux from the large
+scales to small ones [ 97,98,99]. These stationary out-of-equilibrium solutions car-
+rying fluxes of conserved quantities are the analogue of the Kolmog orov spectra of
+hydrodynamic turbulence. However, in the case of interacting wav es, they can be
+computed analytically. To wit, let us consider a wave with a wavenumbe r/vectorkand a
+frequency ωk. Its energy E/vectorkcan be read,
+E/vectork=n/vectorkωk (1)
+wheren/vectorkis called the “wave action”. WT then leads to the expression of the
+temporal evolution of the spectral density of the wave action by m eans of a kinetic
+equation such as
+∂n/vectork
+∂t=C/vectork−D/vectork+I/vectork. (2)
+whereC/vectorkis an interaction integral between waves ( e.g.analogue of the collision
+integral of the Boltzmann formalism of the kinetic theory of gases) ,D/vectorka damping
+term representingthe energydissipation and I/vectorka forcingterm describingthe energy
+injection which generates the waves. When the nonlinearity in C/vectorkis weak and the
+number of excited modes is large, a statistical description of wave t urbulence makes
+sense. Indeed, by means of ensemble averages,the combination o fweak nonlinearity
+anddispersionensuresthatthe longtime asymptoticsofthe proba bilitydistribution
+of the wave field are universal (close to a Gaussian) for a broad ban d of initial
+distributions. This means, that over sufficiently long times, the syst em relaxes
+towards Gaussian statistics insensitive to the initial probability distr ibution that is
+unkown a priori. This is called the asymptotic closure. The stationary solutions
+of the kinetic equations of weak turbulence can be then computed a nalytically at
+the equilibrium or in an out-of-equilibrium regime using perturbation me thods of
+a small parameter quantifying the strength of the nonlinear term r elative to the
+linear one. The stationary solutions are found scale invariant for iso tropic and
+homogeneous systems by using a set of conformal transformatio ns [99]. At the
+equilibrium ( D/vectork=I/vectork≡0), the classical solution is the so-called Rayleigh-Jeans
+spectrum corresponding to the equipartition of energy between m odes. An out-of-
+equilibrium solution ( D/vectork∝negationslash= 0,I/vectork∝negationslash= 0) is tractable when the scale of the energy
+injection (large scale) and the scale of the dissipation (small scale) a re assumed
+widely separated to obtain an inertial regime independent of both th e injection andLABORATORY EXPERIMENTS ON WAVE TURBULENCE 3
+the dissipation. Eq. ( 2) then has conserved quantities (energy flux ε≡∂
+∂t/integraltext
+n/vectorkωkd/vectork
+or “particle” flux). For the finite-energy flux solution, the non zer o first order
+expansionofthe interactionintegralallowsto exactlyworksout th e spectral density
+of the wave action n/vectorkor the spectral density of the energy E/vectorksuch as1
+E/vectork∼ε1/(N−1)k−αwithα >1, (3)
+where the exponent of the energy flux depends on the N-wave interaction process
+whichis itselffixed by both the wavedispersionrelationand the domina nt nonlinear
+interaction2. This out-of-equilibrium solution is called the Kolmogorov-Zakharov
+spectrum(byanalogywiththepower-lawspectrumobtaineddimens ionallyforusual
+turbulence). Consequently, a direct “energy cascade” occurs: due to the interac-
+tions between nonlinear waves, the energy flux injected at large sc ale is transfers
+towards smaller scales. Similarly, for the finite-particle flux solution, the wave ac-
+tionN ≡/integraltext
+n/vectorkd/vectorkis conserved, an invariant scale solution is found and an inverse
+cascade occurs (from small scales to large scales)3. These spectra have been exactly
+computed in nearly all fields of physics involving wave dynamics [ 99]. A lot of data
+obtained by remote sensing of the atmosphere or the ocean as well as satellite mea-
+surements in astrophysics, have thus been analyzed using the fra mework of wave
+turbulence.
+The main assumptions for the derivation of the weak turbulence the ory are:
+•A weak nonlinearity is required due to the perturbation methods: th e non-
+linear term has to be much smaller than linear one (the fast linear time s cale
+is separated from the slow nonlinear time scale). In practice, this co ndition
+corresponds to waves with enough amplitudes but not too high.
+•The size of the system is assumed infinite. No boundary effect is take n into
+account theoretically.
+•The conserved quantity (energy flux) in wave turbulence is assume d constant
+with no fluctuation during the cascade through the scales.
+•The hamiltonian formulation of weak turbulence assumes no dissipatio n at
+least in a transparency window in between the scale of injection (usu ally as-
+sumed at /vectork= 0 or at the scale of the system), and the scale of dissipation at
+/vectork→ ∞. In practice, this means that the forcing of waves has not to be at all
+scales.
+•The system is supposed isotropic and homogeneous spatially, and no coherent
+structures4are involved.
+•Waves have random phases. This conditions allows to derive a statist ical
+equation for the ensemble averaged correlations of wave amplitude s.
+•The single assumption on the initial distribution of the wave field is that its
+cumulants have to decay at infinity.
+At the moment, theoretical works try to model some finite size effe cts [51,52,53,
+70,88,100]. The role of coherent structures on the dynamics also begins to ta ke
+into account theoretically [ 23]. During the last decade, theoretical or numerical
+1Note that, for isotropic systems, a frequency power law spec trum corresponds to the spatial
+one of Eq. ( 3) by using the dispersion relation.
+2For instance, the first non-zero resonant term for gravity wa ves is a N= 4 wave interaction
+process, whereas for capillary waves N= 3.
+3This is the analogue, in 2D turbulence, of the enstrophy cons ervation for the direct cascade,
+and the energy conservation for the indirect cascade.
+4For example, in hydrodynamics, vortices, cusps or wave brea kings.4 ERIC FALCON
+works go beyond spectrum prediction. Transient behaviors during the evolution
+towards the stationary regime (or decaying wave turbulence) [ 55,56], condensation
+of classical waves (and its analogy with Bose condensation) [ 9,24,30,58,80] has
+been performed as well as the introduction of an empirical dissipatio n to model
+some strongly nonlinear processes (such as wave breakings) [ 101].
+In spite of the numerous analytical predictions for more than 40 ye ars, wave
+turbulenceisnotsomuchstudiedexperimentally. Thisisthustheopp ositesituation
+of the usual turbulence in which numerous experimental results ex ist, but very few
+analytical results can be carefully derived from master equations.
+Oceanography and atmospheric sciences provide more and more in situdata
+nowadays given by weather balloons, surface buoys, radar and sa tellite measure-
+ments [72,92]. However, these large-scale systems depend on numerous param eters
+(namely for the ocean surface waves: wind, oceanic current, fet ch...) leading to
+empirical determinations of the energy spectral density [ 72]. Wind on the ocean
+surface also generates a wave forcing at various scales, and thus without scale sep-
+aration as imposed by the theory. Laboratory experiments are mu ch more relevant
+to accuratelytune and controlthe system parameters. Surpris ingly, such laboratory
+experiments are rare, but are fundamental in order to compare t hem with in situ
+measurements and theoretical predictions of weak turbulence. O nly few laboratory
+experiments have been performed since 1996, mostly on wave turb ulence on the
+surface of a fluid [ 14,15,47,48,62,93,94]. These works were focused on the
+measurement of the frequency power spectrum of capillary waves in roughly good
+agreement with the theoretical prediction of weak turbulence [ 98]. Note that, some
+recent experiments of elastic wave turbulence on the surface of a thin plate [ 10,68]
+show that the energy spectrum is in strong disagreement with the p redictions of
+weak turbulence [ 29].
+In this paper, I will focus on the hydrodynamic case of wave turbule nce. In§2,
+I will first review the previous results on this issue. Then, the main re sults of our
+experimental works on wave turbulence on the surface of a fluid will be described
+in§3. Finally, §4will be devoted to the conclusion and a discussion on open
+questions in wave turbulence.
+2.Hydrodynamic wave turbulence. Waves on the surface of a fluid are in a
+gravity regime when their wavelength λis large with respect to the capillary length
+λc≡2πlc= 2π/radicalbig
+γ/ρgwheregis the acceleration of gravity, hthe fluid depth, ρ
+the density and γthe surface tension. Otherwise, they are in a capillary regime.
+The crossover between gravity and capillary regimes occurs for a w avelength close
+to the centimeter whatever the working fluid. The dispersion relatio n of linear
+inviscid surface waves reads ω2(k) =/bracketleftbig
+gk+(γ/ρ)k3/bracketrightbig
+tanh(kh). In a deep fluid
+approximation, when λ≪2πh, one has
+ω2=gk+γ
+ρk3(4)
+By balancing both terms of the second hand of Eq. ( 4), one finds the critical
+wavenumber kc≡1/lcfor the transition between both regimes. If waves are weakly
+nonlinear, the weakturbulence theoryleadsto the expressionoft he powerspectrumLABORATORY EXPERIMENTS ON WAVE TURBULENCE 5
+of surface wave amplitude such as
+Sgrav
+η(ω)∝ε1
+3g ω−4for gravity waves [97 ,103] (5)
+Scap
+η(ω)∝ε1
+2/parenleftbiggγ
+ρ/parenrightbigg1
+6
+ω−17
+6 for capillary waves [98] (6)
+These relations can be also computed by dimensional analysis. In the gravity
+regime, parameters involved are the power spectrum of wave amplit udeSη, the
+gravityg, the energy flux εand the frequency ω, of dimensions: [ Sη] =L2T;
+[ε] =L3T−3; [g] =LT−2; [ω] =T−1. One can conclude by dimensional analysis
+only if the scaling law between Sgrav
+ηandεis assumed. As explained above in §1,
+this scaling is known as soon as the number Nof interacting waves is known [ i.e,
+the order of the first non-zero term in the interaction integral C/vectorkof Eq. (2)]. Thus,
+for aN-wave resonant process, the power spectrum scales as ε1/(N−1). Gravity
+waves involving a 4-wave interaction process, one has Sgrav
+η∝ǫ1/3. One thus finds
+dimensionally the power spectrum of gravity wave as Eq. ( 5). Similar dimensional
+analysis can be performed for capillary waves assuming a 3-wave inte raction pro-
+cess. Note that this dimensional analysis can be done whatever the wave dispersion
+relation of the system [ 23].
+Oceansurfacewavesaremainlygeneratedbywindsandcurrentsa tvariousscales.
+In the literature, in situmeasurements of the exponent of the frequency-power law
+of the wave amplitude spectrum varies considerably, even if certain measurements
+roughly leads to a ω−4scaling [28,42,50,90] thus suggesting a possible agree-
+ment with weak turbulence theory [ 97,103] and numerical simulations [ 31,73,78]
+of gravity wave turbulence. However, the wave spectrum depend s on numerous
+uncontrolled parameters such as duration of wind blowing, fetch len gth, stage of
+storm growth and decay, and existence of a swell, leading to fit the s pectrum with
+too many parameters [ 72].
+When gravity wave turbulence is not forced by wind, mechanisms of e nergy
+cascade in the inertial regime change, as well as the scaling law of the spectrum
+[54,60,73]. Indeed, if Sηdoes not depend on the energy flux, Phillips has shown
+dimensionally since 1958 that Sη(ω)∝ε0g2ω−5[75]. This situation corresponds to
+gravity waves of high amplitudes dissipating all the injected power (b y wave break-
+ings for instance). More recently, Kuznetsov has taken into acco unt the presence
+of possible sharp wave-crests (cusps) on the surface that are a ssumed to propagate
+without deformation ( i.e.usingω∝kinstead of ω=√gk). If these sharp-crested
+structuresoccuralongridges(lines) then Sη(ω)∝ω−4[60]. Thisexponentissimilar
+to the one in Eq. ( 5) computed by weak turbulence theory (where no crested waves
+are involved). In the same way, when these wave slope divergences are assumed to
+be isolated peaks (points) propagating as ω=√gk, the Phillips spectrum is found
+again [75]. Recently, theoretical and numerical attempts try to take into a ccount
+the quantization of wavenumber kin finite size systems that can strongly affect
+energy transfers [ 51,52,53,70,88,100]. Notably, for gravity waves in a tank of
+characteristic size L, Nazarenko predicts that Sη(ω)∝ε0g5/2L−1/2ω−6[70].
+Experimentally,thecapillarywaveturbulenceregimehasbeenobser vedbymeans
+of optical methods [ 14,15,47,48,62,93,94] and with a parametric forcing of the
+waves by shaking the whole container holding the fluid. It has been re ported that
+the height of the surface displays a power-law frequency spectru m inω−17/6in
+agreement with weak turbulence theory [ 98] and simulations [ 79]. However, peaks6 ERIC FALCON
+ContainerCapacitive
+wire gauge170180190200210 220-6-4-2024681012
+Time (s) Surface wave
+ amplitude η (mm) 
+CapacimeterVibration
+exciterVibration
+exciter
+CoilF(t)v(t)F(t)*v(t)
+η(t)
+Wave      makerAnalogic
+multiplier
+Force sensor
+Figure 1. Photo and schematic view of the experimental setup.
+Wave makers are on both sides of the capacitive sensor measuring
+the wave amplitude at a given location. Tank size: 20 cm ×20 cm
+filled with water or mercury.
+and its harmonics (related to the parametric forcing) are observe d on the spectrum
+with maximal amplitudes decreasing as a frequency power-law [ 14,15,87,93,94].
+The frequency-spectrum exponent estimated in this way is thus no t very accurate.
+A recent study has even found an exponent in disagreement with we ak turbulence
+underlyingthedifficultytoreachawaveturbulenceregimewithapara metricforcing
+[87]. Finally, note that new experiments of gravity wave turbulence in a la rge tank
+have been recently carried out [ 27,83,84].
+3.Overview of our results. We have developed experiments to study wave tur-
+bulence on the surface of a fluid forced by wave makers [ 11,34,35,36,38]. This
+investigation is of particular relevance since it allows to study on the s ame system
+two different regimes, those of gravity and capillary waves, that ar e characterized
+by different nonlinearities, 4-wave and 3-wave interactions respec tively, leading to
+different poweer-law spectra. The crossover between the two re gimes is also inter-
+esting, since it presents deviations from the weak turbulence theo ry5. The role of
+the forcing parameters on the spectra had never been reported , nor the existence of
+a possible intermittency. The energy flux and its fluctuations had ne ver been mea-
+sured in wave turbulence, although it is the conserved quantity in wa ve turbulence.
+Another motivation is also to understand the statistical propertie s of energy flux
+necessaryto maintain a dissipative system in an out-of-equilibrium st ationarystate.
+This is a general issue that is not restricted to the single case of wav e turbulence.
+3.1.Spectrum and probability distribution of wave amplitudes [34].As
+shown in Fig. 1, surface waves are generated by two wave makers driven with a
+random excitation (both in frequency and amplitude) by means of ele ctromagnetic
+vibration exciters. This random forcing is in a narrow low-frequency range selected
+by a low-pass filter (typically 0.1 – 5 Hz). The square tank, 20 cm side, is filled
+either with water or mercury up to a height of 2 cm6. The wave amplitudes are
+measured at a given location by means of a capacitive method. This en ergy injected
+to the system by large wavelengths is transferred towards small s tructures due to
+5Close to this transition, the nonlinearity is not weak and th e dynamics is dominated by fully
+nonlinear structures [ 23].
+6Waves of wavelengths λ≪2πh≃12 cm are thus in a deep-water regime.LABORATORY EXPERIMENTS ON WAVE TURBULENCE 7
+Figure 2. Typical time recording of the surface waves, η(t), dur-
+ing 50 s. ∝angbracketleftη∝angbracketright= 0.
+10 10010−1010−810−610−410−2
+Frequency (Hz)Power Spectrum η2f (Vrms2 / Hz)
+4 200      f    
+crossover capillary
+cascade  gravity cascade 
+sensor cutoff
+forcing
+c 
+Figure 3. Powerspectrumofwaveamplitude showingthe regimes
+of gravity and capillary wave turbulence. The transition ≃17 Hz
+between both regimes corresponds to λc= 2πlc≃1 cm, where lc
+is the capillary length. Dashed lines have slopes −6.1 and−3.2.
+Random forcing with a narrow bandwidth ≤4 Hz.
+nonlinearities. This process is characterized by measuring the freq uency spectrum
+and the distributions of wave amplitude fluctuations.
+A typical recording of the surface wave amplitude, η(t), is displayed in Fig. 2
+as a function of time. The signal is very erratic and the asymmetry o bserved is a
+signature of the steepness of the waves: high crest waves are mo re probable than
+deep trough waves. By Fourier transform of such a signal, one can obtain the power
+spectrum density of the wave amplitudes Sη(ω)≡/integraltext∝angbracketleftη(t+τ)η(t)∝angbracketrighte−iωτdτ. Waves
+being in nonlinear interactions, a wave turbulence regime is thus expe cted, that is
+the observation of a scale invariant spectrum.
+AsshowninFig. 3, wehaveobserved,thecrossoverbetweentheregimesofgravit y
+(at large scales) and capillary (at small scales) wave turbulence [ 34]. Spectrum
+displays two different power laws corresponding to each regime. In t he capillary
+regime, the frequency power-law exponent is in agreement with wea k turbulence
+theoryofEq.( 6)inω−17/6. Ourestimationofthecapillary-wavespectrumexponent8 ERIC FALCON
+00.2 0.4 0.6 0.8−8−7−6−5−4−3−2
+σV2 (V)Spectrum slopes
+0.4 0.6 0.8 1101520253035
+σV2 (V)Crossover frequency (Hz)
+Figure 4. Slopes of spectra of capillary (open symbols) and grav-
+ity(solidsymbols)waveturbulencefordifferent forcingamplitudes,
+σ2
+V, and bandwidths [( ◦) 0 to 4 Hz, ( ▽) 0 to 5 Hz and ( /square) 0 to 6
+Hz]. Dashed lines are weak turbulence predictions [see Eqs. ( 5) &
+(6)].Inset:Crossoverfrequencybetween both regimesas afunction
+of the forcing parameters.
+−3−2−101234510−310−210−1
+ η / < η2 >1/2Probability Density Functions 
+−5051010−310−210−1100
+ η (mm)PDFs [ η ]
+Figure 5. Inset:Probability density functions of the wave ampli-
+tude,η(t), for different increasing amplitude of the forcing (from
+black to red curves). Main:Same distributions rescaled by its stan-
+dard deviation/radicalbig
+∝angbracketleftη2∝angbracketright. Dashed lines: Gaussian fit with zero mean
+and unit standard deviation.
+is much more accurate than in experiments using a parametric and mo nochromatic
+forcing [14,15,47,48,62,93,94]. The transition between both regimes corresponds
+to a wavenumber of the order of the inverse of the capillary length lc. For usual
+fluids, this corresponds to a critical frequency of fc=/radicalbig
+g/2lc/π≃17 Hz, that is
+a wavelength of the order of 1 cm. In the gravity regime, the spect rum exponent
+is found to be dependent on the forcing parameters (amplitude and bandwidth of
+the random forcing) as shown in Fig. 4, and consequently in disagreement with the
+weak turbulence prediction of Eq. ( 5) inω−4. The crossover frequency between
+both regimes also depends on the forcing parameters (see inset of Fig.4). This
+dependence of the exponent of the gravity wave spectrum has be en recently foundLABORATORY EXPERIMENTS ON WAVE TURBULENCE 9
+again in a much larger tank [ 27]. Its origin is still an open problem. It could
+stem from finite size effects of the tank [ 100]. It has been also shown numerically
+that gravity wave spectrum is very sensitive to the condensation o f long waves
+background (or “condensate” following analogy with Bose-Einstein condensation in
+condensed matter physics) [ 9,24,30,58,80].
+At low forcing amplitude, the probability density function (PDF) of th e wave
+amplitudes is found to be Gaussian (see Fig. 5). At high enough forcing, the PDF
+becomes asymmetric thus showing the non-Gaussian feature of wa ve turbulence
+[34]. This phenomenon is well known by oceanographers which measure c rest-to-
+trough wave distributions which deviates from the usual Rayleigh dis tribution7[43,
+72]. However, only few reports have mentioned this PDF asymmetry in la boratory
+experiments [ 74].
+3.2.Intermittency in wave turbulence [35].Oneofthe most strikingfeature of
+turbulence is the occurrence of bursts of intense motion within mor e quiescent fluid
+flow. This generates an intermittent behavior [ 18,57]. Indeed, a signal is called
+intermittent as soon as it displays more and more intense events whe n it is probed
+on a more and more short duration, leading thus to a strongly non Ga ussian statis-
+tics at small time scales. The origin of non-Gaussian statistics in 3D hy drodynamic
+turbulence has been ascribed to the formation of coherent struc tures (strong vor-
+tices) since the early work of Batchelor and Townsend [ 18]. However, the physical
+mechanism of intermittency is still an open question that motivates a lot of studies
+in 3D turbulence [ 2,19]. Intermittency has also been observed in a lot of prob-
+lems involving transport by a turbulent flow [passive scalar, Burgers turbulence of
+strongly compressible fluid] [ 39], in magnetohydrodynamic turbulence in geophysics
+[26] or in the solar wind [ 1]. The possible observation of intermittency in wave
+turbulence that strongly differs from high Reynolds number hydrod ynamic turbu-
+lence is of primary interest. It could indeed motivate explanations of intermittency
+different than the ones considering the dynamics of the Navier-Sto kes equation.
+We have reported the first observation of intermittency in wave tu rbulence [ 35].
+By measuring the temporal fluctuations of the surface wave amplit ude,η(t), at
+a given location (cf. Fig. 1), one can compute the local slope increments of the
+gravity surface waves at a certain time scale τ: ∆η(τ)≡η(t+τ)−2η(t)+η(t−τ).
+As shown in Fig. 6, starting from a roughly Gaussian statistical distribution of
+these increments at large scale τ, the PDFs undergoes a continuous deformation
+towards strongly non-Gaussian shape at smaller scales. This shape deformation of
+the distributions across the scales is a direct signature of intermitt ency. Figure 7
+shows the structure functions of these increments, defined as Sp(τ)≡ ∝angbracketleft|∆η(τ)|p∝angbracketright,
+which are found to be power laws of the time scale τon more than one decade, such
+asSp(τ)∼τξpwhereξpis an increasing function of the order p. The evolution of
+the exponents ξpwithpis found to be a nonlinear function of pas shown in Fig. 8.
+The nonlinearity of ξpis another direct signature of intermittency. The slope at the
+origin of the curve ξpvs.p(the non-intermittent part) is close to 3 p/2, this value
+being easily derived by dimensional analysis8. However, there exists any model that
+can describe the intermittent nonlinear part of ξpat largep.
+7A Gaussian process with negative and positive values has a Ra yleigh distribution for its crest-
+to-trough amplitude.
+8One has indeed Sp(τ)∼εp/6gp/2τ3p/2.10 ERIC FALCON
+−15−10 −5 05101510−710−510−310−1101103
+[η(t+τ) −2η(t) + η(t−τ)] / στPDF( [η(t+τ) −2η(t) + η(t−τ)] / στ )
+  τ = 6 ms
+τ = 12 ms
+τ = 18 ms
+τ = 27 ms
+τ = 50 ms
+τ = 60 ms
+τ = 80 ms
+τ = 100 ms
+Gaussian
+Figure 6. Probabilitydensity functions ofthe local slopes ofwave
+amplitudes [ η(t+τ)−2η(t) +η(t−τ)]/στfor different time lags
+6≤τ≤100 ms (from top to bottom). Gaussian fit with zero
+mean and unit standard deviation (dashed line). Correlation time:
+τc≃63 ms. Each curve has been shifted for clarity.
+4 10 2030405010−2100102104106
+ τ (ms)Sp(τ)  (mmp)
+3102030
+ τ (ms)S4  /  S22
+4 10 100S6*500
+S5*100
+S4*50
+S3*10
+S2*1
+S1*0.1   τc 
+Figure 7. Structure functions of order pof the local slopes of
+wave amplitudes, Sp(τ), as a function of time lag τ, for 1≤p≤6.
+(−): Power-law fits, Sp(τ)∼τξp, where the slopes ξpdepend on
+the order p(see Fig. 8). Curves have been shifted for clarity. Inset:
+Flatness S4/S2
+2as a function of τfitted by a power law of slope
+−0.88 (−). Correlation time: τc≃63 ms.
+Recently, it has been proposed that theoretical corrections sho uld be taken into
+account in weak turbulence to describe a possible intermittency tha t may be con-
+nected to singularities or coherent structures [ 20,23] such as wave breaking [ 95] or
+whitecaps [ 23] on the fluid surface. However, intermittency in wave turbulence is
+often related to non-Gaussianstatistics of lowwavenumber Fourie r amplitudes [ 20],
+and thus it is not obviously related to small-scale intermittency of usu al hydrody-
+namic turbulence. Here, we have shown the intermittent feature o f the local slope
+of turbulent gravity waves in a same way as intermittency in usual tu rbulence. A
+challenge is to understand the physical origin of intermittency in wav e turbulence.LABORATORY EXPERIMENTS ON WAVE TURBULENCE 11
+Figure 8. Exponents ξpof the structure functions as a function
+of the order p.ξpis computed from the slopes of Fig. 7, and is
+fitted by ( −−)ξp=c1p−c2
+2p2withc1= 1.65 andc2= 0.2. Solid
+line: dimensional analysis ξp= 3p/2.
+3.3.Fluctuations of the energy flux [36,37].The key parameter of wave tur-
+bulence is the energy flux injected in the system. It is assumed to be conserved
+during the cascade across the scales. For the first time in wave tur bulence, we have
+measured the instantaneous injected power in the fluid I(t)≡F×V, whereF(t) is
+the force applied by the wave maker on the fluid, and V(t) the wave maker velocity
+(see Fig. 1). The typical temporal evolution of this energy flux is shown in the
+inset of Fig. 9. It displays fluctuations much stronger than its mean value ∝angbracketleftI∝angbracketright(see
+Fig.9). These fluctuations of energy flux have not been taken into acco unt in the
+theoretical models that have been developed up to date. To wit, th e knowledge of
+their evolution during the whole cascade would be necessary. Here, one studies the
+statistical properties of the energy flux fluctuations measured o n the wave maker
+(integral scale). Numerous events of negative energy flux ( I(t)<0) are observed
+on Fig.9with a fairly high probability. These are events for which the wave field
+gives back energy to the wave maker. Understanding this phenome non is necessary
+to develop marine renewable energy such as ocean wave energy. It is thus of first
+interest to model this empirical statistical distribution of the ener gy flux. To wit,
+the wave maker is described by a Langevin-type model with a pink nois e (Ornstein-
+Uhlenbeck process) that mimics the experimental narrow-bandra ndom forcing [ 36].
+Two linear coupled Langevin-type equation for VandFare then obtained; these
+two variables being Gaussian (due to the forcing) as observed expe rimentally (see
+Fig.10). The computation of the distribution of the product of these two corre-
+lated Gaussian distributions thus gives the analytical expression of the energy flux
+distribution with no adjustable parameter [ 4,36,37]. Figure 9shows a very good
+agreement between our model and the experiment. The asymmetr y of the probabil-
+ity distribution is driven by the mean energy flux ∝angbracketleftI∝angbracketright, that is equal, in a stationary
+state, to the mean dissipated power. This model is not restricted t o wave turbu-
+lence, but is very generic since it also describes the energy flux distr ibution in other
+dissipative out-of-equilibrium systems such as turbulent convectio n, granular gases,
+bouncing ball with a random vibration [ 4], and electronic circuit with a random
+voltage (see below and Ref. [ 37]).12 ERIC FALCON
+Figure 9. Inset:Timerecordingofinjected power, I(t), driventhe
+surface waves. ∝angbracketleftI∝angbracketright= 2 mW. Main:Probability density function
+of injected power showing two exponential tails. Its asymmetry
+is related to its mean value ∝angbracketleftI∝angbracketright(vertical solid line). Note that
+fluctuations are much more larger the mean value. Blue dashed
+line: prediction of our model with any adjustable parameter.
+402 403 404 405−0.15−0.1−0.0500.050.10.15
+Time ( s )Velocity ( m / s )
+402 403 404 405−4−2024
+Time ( s )Force ( N )
+−4−202410−410−2100
+ V  / σ VPDF( V  / σ V )
+−5 0 510−410−2100
+ F  / σ FPDF( F  / σ F )
+Figure 10. Timerecordingsofthevelocity V(t)ofthewavemaker
+(left) and the forceapplied F(t) to the wavemakerby the vibration
+exciter (right). Both probability distributions are quasi Gaussian
+(see dashed lines) with zero mean value. σVandσFare the stan-
+dard deviations.
+We have also emphasized the bias that can result from the system ine rtia when
+onetries a direct measurementof the fluctuations ofinjected pow er. When the wave
+maker inertia is not negligible9, the measurement of the instantaneous force has to
+be corrected by the inertia before performing the product with th e velocity signal
+in order to deduce the power. Otherwise, it may lead to wrong estima tions of the
+fluctuations of injected power, the mean value being not obviously a ffected. There
+exists only a few previous direct measurements of injected power in 3D turbulent
+flows, and this type of inertial bias has never been taken into accou nt [61,76,89].
+9For our experiments, this is the case with water but not with m ercury.LABORATORY EXPERIMENTS ON WAVE TURBULENCE 13
+10 10010−810−610−410−2
+Frequency (Hz)PSD / σV2  (s3)
+0510152005101520
+σV2 (cm/s)2< I > (mW)
+4 
+Figure 11. Spectra of the surface wave amplitude divided by the
+variance σ2
+Vof the wave maker velocity for different forcing ampli-
+tudes:σV= 2.1,2.6,3.5 et 4.1 cm/s. The dashed line has slope
+−5.5, whereas the solid line has slope −17/6.Inset :The mean
+injected power ∝angbracketleftI∝angbracketrightis found proportional to σ2
+V.
+The mean injected power ∝angbracketleftI∝angbracketrightis found to be proportional to the fluid density10
+ρand toσ2
+V, the rms value squared of the wave-maker velocity fluctuations. T he
+rms value of the injected power fluctuations also scales as σI∼ ∝angbracketleftI∝angbracketright ∼ρσ2
+V. The
+energy transfer occurs via 4-wave interactions in the gravity reg ime and via 3-wave
+interactions in the capillary regime. Thus, if these two regimes are ind ependent as
+in Eqs. ( 5) and (6), the dependence of the spectrum amplitude Sηon the mean
+energy flux ∝angbracketleftI∝angbracketrightshould display different scaling laws. Our measurements show that
+a single scaling law, Sη∼ ∝angbracketleftI∝angbracketright, leads to the collapse on a single master curve of our
+experimental spectra for different values of the forcing (see Fig. 11). This strong
+disagreement with the weak turbulence theory [cf. Eqs. ( 5) & (6)] thus shows that
+interactions between these regimes are very important although t hey are not taken
+into account theoretically.
+The experimental distribution of the injected power averaged on a time lag τ,
+P(Iτ), has also been studied [ 36] in the framework of the Fluctuation Theorem
+of the out-of-equilibrium statistical physics (also called the Gallavot ti-Cohen rela-
+tionship) [ 33,44]. This theorem states thatP[+Iτ//angbracketleftI/angbracketright]
+P[−Iτ//angbracketleftI/angbracketright]∼e−β(τ)Iτ//angbracketleftI/angbracketrightwhereP[x]
+is the probability to have the value xandβ(τ) is a constant related to the en-
+ergy of the system. The theorem hypotheses (such as the tempo ral reversibility)
+are generally not fulfilled for a real dissipative system. The apparen t verification
+of this theorem in numerous previous experiments [ 21,22,41,85,86] is due to
+their small range of explored fluctuation amplitudes Iτ/∝angbracketleftI∝angbracketrightat long averaging time
+τ[3,5,6,45,77,91]. Our wave turbulence experiment allows us to reach large
+enough fluctuations of injected power, and we have shown that th ey do not verify
+the theorem [ 36], but are found in good agreement with the analytical calculation
+performed with a Langevin-type equation forced with a Gaussian wh ite noise [ 40].
+We have also studied experimentally the energy flux fluctuations in an electronic
+RC circuit driven with a stochastic voltage [ 37]. This out-of-equilibrium model
+system allows us to easily control the system parameters (notably , its damping rate
+10This has been verified for only two different fluids (water and m ercury) but with a density
+ratio close to 14.14 ERIC FALCON
+−20 −10 0 10 20 3010−410−310−210−1100
+I/<I> PDF(I/<I>)
+Figure 12. Fluctuations of injected power I(t) in an electronic
+RC circuit driven with a stochastic voltage for two different values
+of the damping rate ( RC)−1= 2000 and 200 Hz (resp. blue and
+red curves). Comparison between experiment ( −) and theoretical
+prediction of the Langevin-type model ( −−). As in wave turbu-
+lence, the more dissipative the system is, the more asymmetric the
+PDF ofI(t) is.
+1/(RC)). We show that the large fluctuations reached do not verify the F luctuation
+Theorem even for long averaging time τ. As shown in Fig. 12, the probability
+distribution of the instantaneous energy flux is qualitatively similar to the one of
+wave turbulence of Fig. 9, and is indeed well described by our above model (see
+dashed lines in Fig. 12). This electronic circuit is thus one of the simplest system
+to understand certain properties of the energy flux fluctuations shared by other
+dissipative out-of-equilibrium systems, such as in granular gases [ 41], convection
+[85,86], or wave turbulence [ 36].
+3.4.Wave turbulence in low-gravity environment [38].Experiments of wave
+turbulence in low gravity are relevant to study pure capillary waves o n a large
+range of wavelengths without interfering with the gravity waves. D uring ground
+experiments, the upper limit is the capillary length, and its lower limit is re lated to
+viscous dissipation. Our experiment in weightlessness has been carr ied out during
+CNES or ESA campaign of parabolic flights on board of an specially modifi ed
+Airbus A300. A campaign consists of 3 flights, each flight consists of a series of 30
+parabolic trajectories that result in low-gravity periods, each of 2 2 s. Such a low-
+gravity environment (about ±5×10−2g) allows us to also study the behavior of a
+spherical fluid layer on which waves can be propagates with no bound ary condition
+(see the experimental setup in Fig. 13).
+Insuchlow-gravityconditions, wehaveobservedthe capillarywave turbulenceon
+the surface ofa fluid layerthat coversall the internal surfaceof a sphericalcontainer
+which is submitted to random forcing [ 38]. Figure 14displays the capillary wave
+spectrum over a large range of wavelengths, notably in wavelength s where gravity
+waves were present during ground experiments. This spectrum ov er two decades in
+frequency (corresponding to wavelength from mmto fewcm) is found in roughly
+good agreement with the frequency-exponent prediction of weak turbulence theory
+of the capillary regime in −17/6≃ −2.8 [see Eq. ( 6)]. As shown in Fig. 14, thisLABORATORY EXPERIMENTS ON WAVE TURBULENCE 15
+Figure 13. Experimental setup for the wave turbulence in low
+gravity. In low-gravity periods, the fluid covers all the internal
+surface of the tank which is submitted to vibrations.
+10 100 50010−1010−910−810−710−610−510−410−3
+Frequency (Hz)Power Spectrum Density ( V 2 / Hz )
+10 10020010−1110−910−710−510−3
+Frequency (Hz)PSD ( V 2 / Hz )
+4
+4
+Figure 14. Powerspectraofsurfacewaveamplitudes inlowgravity
+for two different forcings: random forcing from 0 to 6 Hz (lower),
+or sinusoidal forcing at 3 Hz (upper). Dashed lines have slopes of
+-3.1 (lower) et -3.2 (upper). Inset :Samewith gravity . Two power
+laws appear corresponding to the gravity (slope -5) and capillary
+(slope -3) regimes.
+exponent is also found to be independent of the large-scaleforcing parameters (low-
+frequency sinusoidal or random forcing) as expected within the ine rtial range.
+3.5.Wave turbulence on the surface of a ferrofluid [11].A ferrofluid is a
+colloidal suspension of nanometric ferromagnetic particles diluted in a liquid dis-
+playing thus both liquid and strongly magnetic feature. In contrast with usual
+liquids, the dispersion relation of surface waves on a ferrofluid is non monotonic and
+displays a minimum (depending on the amplitude of the applied magnetic fi eldB
+[16,17,32,65,69]) for which an instability occurs. Above a critical field Bc, a static
+hexagonalpattern ofpeaksoccurson the ferrofluidsurface, t he so-calledRosensweig
+instability [ 25]. The amplitude of the magnetic field, B, can thus easily tune the16 ERIC FALCON
+Figure 15. Experimental setup for the study of wave turbulence
+on the surface of a ferrofluid in a normal magnetic field generated
+by two coaxial coils. Surface waves on the ferrofluid are generate d
+by the horizontal motion of a plunging wave maker driven by an vi-
+bration exciter. The amplitude of surface waves at a given location
+is measured by a capacitive wire gauge.
+dispersion relation of the surface wave which reads [ 13,81]
+ω2=gk−f[χ]
+ρµ0B2k2+γ
+ρk3(7)
+whereµ0= 4π×10−7H/m is the magnetic permeability of the vacuum, and f[χ]
+a known function of the magnetic susceptibility of the ferrofluid11. One can thus
+easily tune, with just one single control parameter B, the dispersion relation of
+surface wave from a dispersive one ( kork3term) to a nondispersive one ( k2term).
+Besides, some theoretical questions are open notably about the p ossible existence of
+power-law solution for the energy wave spectrum for two-dimensio nal nondispersive
+systems [ 23,63,71,102]. Wave turbulence on the surface of a magnetic fluid should
+thus display striking properties depending on whether the system is below, close or
+above the onset of the Rosensweig instability.
+We have studied the waveturbulence on the surface of a ferrofluid submitted to a
+normalmagneticfield(seeFig. 15) [11]. We haveshownthatmagneticsurfacewaves
+ariseonlyabovea criticalfield, but belowthe onset Bcofthe Rosensweiginstability.
+As shown in Fig. 16, the power spectrum of the magnetic wave amplitudes displays
+a single power law over the whole range of accessible frequencies inst ead of two
+regimes (gravity and capillary) observed with no applied magnetic field . On can
+compute by dimensional analysis the power spectrum of magnetic wa ve turbulence
+as
+Smag
+η(ω)∼εα/parenleftbiggB2
+ρµ0/parenrightbigg2−3α
+2
+ω−3, (8)
+This predicted frequency exponent ω−3is too close to the one predicted for the
+capillary regime in ω−17/6to be distinguished experimentally with our experimen-
+tal accuracy. This could explain that only one single slope is observed in Fig.16
+which thus corresponds to a “magneto-capillary” wave turbulence regime. Unlike
+the above dispersive systems [see Eq. ( 5) & (6) and§3.1], this−3 exponent does
+11Indeed, χ(H) depends on the applied magnetic field, H, through Langevin’s classical law,
+and thus on the magnetic induction B, sinceB=µ0(1 +χ)H. The magnetic induction, B, will
+be called hereafter the magnetic field for simplicity.LABORATORY EXPERIMENTS ON WAVE TURBULENCE 17
+3      10 100 40010−610−410−2100102
+Frequency (Hz)Power spectrum of η ( mm2 / Hz )
+B = 0.9 Bc3      10 10040010−610−410−2100102
+Frequency (Hz)PSD η ( mm2 / Hz )B = 0
+Figure 16. Power spectrum of wave amplitude, η(t), on the fer-
+rofluid surface for two applied magnetic field. Inset:B= 0 : Grav-
+ity andcapillarywaveturbulenceregimes. Dashedlineshaveslopes
+-4.6 and -2.9. Vertical dashed line: crossover frequency fgcHz.
+Main:B= 0.9Bc: Magnetocapillary wave turbulence. Dashed
+line has slope -3.3. Crossover is decreased towards fgc≃5 Hz.
+not depend on the energy flux exponent α. However, the exponent αis still not
+deduced by dimensional analysis. By measuring the scaling law betwee n the spec-
+trum amplitude Smag
+ηand the magnetic field B, one then shows that α= 1/3,i.e.
+that the magnetic wave turbulence involves a 4-wave interaction pr ocess. One has
+thus obtain the whole expression of the spectrum of magnetic wave turbulence.
+0 0.5 1 1.5 2510152025
+0.65ftfgc
+fgmfmc
+Capillary Wave Turbulence
+Gravity Wave
+TurbulenceMagnetic Wave Turbulence
+Rosensweig
+instability
+ B / BcCrossover frequencies (Hz)
+Figure 17. Crossover frequencies between gravity, magnetic and
+capillary wave turbulence regimes as a function of the applied mag-
+netic field. Different forcing parameters: ( ×) 1 to 4 Hz, ( ⋄) 1 to 5
+Hz, and (+ or ◦) 1 to 6 Hz. Theoretical curves fgc,fgmandfmc
+can be computed analytically from Eq. ( 7) [11]. The triple point
+readsft=1
+2π(g3ρ
+γ)1/4≃10.8 Hz and B2
+t=µ0√ρgγ/f[χ(Ht)], that
+isBt/Bc= 0.65 for our ferrofluid, where Bcis the onset of the
+Rosensweig instability.
+The existence of the regimes of gravity, magnetic and capillary wave turbulence
+is reported in the phase space parameters as well as a triple point of coexistence of18 ERIC FALCON
+these three regimes (see Fig. 17) [11]. The theoretical crossover frequency curves,
+as well as the triple point, can be computed from the dispersion relat ion of Eq. ( 7),
+and are found in good agreement with the experimental data with no adjustable
+parameter (see solid lines in Fig. 17). Finally, the statistical distribution of the
+wave amplitudes and their change with the magnetic field has been also studied:
+our measurements confirm the nonlinear nature of the wave intera ction in a wave
+turbulence regime, and also show the onset of existence of the mag netic waves [ 11].
+4.Conclusions and open problems. I have first presented the context and the
+state of the art in wave turbulence. I have then focused on the wa ve turbulence
+on the surface of a fluid, and I have discussed the main experimenta l results of our
+group on this subject. We have measured, for the first time, the k ey governing
+parameter of wave turbulence: the energy flux that drives the wa ves and cascades
+to small scales through nonlinear interactions. Fluctuations much la rger than the
+mean value have been observed as well as instantaneous negative e nergy flux events
+that occurs with a fairly large probability. Taking into account these energy flux
+fluctuations in theoreticalmodels of cascadesremainsan open pro blem. The scaling
+of the power spectra of capillary wave turbulence is found in agreem ent with weak
+turbulence theory both in laboratory experiments or in low-gravity environment
+(to study pure capillary waves). A disagreement with weak turbulen ce theory has
+been observed for the gravity wave spectrum. It has been also fo und that gravity
+waves are intermittent, i.e.the statistics of velocity increments of the interface
+are strongly non Gaussian at small scales. This first observation of intermittency in
+waveturbulencecouldberelatedtothelargefluctuationsoftheen ergyflux. Finally,
+magnetic wave turbulence has been observed on the surface of a m agnetic fluid
+(ferrofluid) as well as a triple point of coexistence of gravity, capilla ry and magnetic
+waveturbulence regimes. These features areunderstood usingd imensional analysis,
+but no weak turbulence prediction exists for this magnetic case.
+As emphasized in this review, there has been a renewal of interest in wave tur-
+bulence since last years. However, open problems are still numerou s. It is crucial to
+considerin a preciseway the rangeof validity of theoreticalpredict ions and to study
+experimental effects associated to breakdown of existing theorie s. Finite size effects
+in experiments lead to a quantization of wave number that can stron gly affect en-
+ergy transfers. Understanding the observed dependence of th e frequency-exponent
+of the gravity wave spectrum with the forcing parameters is thus o f primary inter-
+est. Besides, most in situor laboratory measurements on wave turbulence are local
+in space whereas theoretical predictions often concern Fourier s pace,e.g.power
+spectra or probability density functions of the fluctuations of the Fourier amplitude
+of a wave component at a given wave number. Thus, an important ch allenge is to
+develop a direct probe in Fourier space. Some recent studies measu re the ampli-
+tudes of gentle interacting waves on a spatial zone [ 67,87,93,94]. However, these
+optical methods cannot measure amplitudes of steep nonlinear wav es as involved
+in wave turbulence. Nevertheless, one need to access to such ste ep wave amplitude
+in the Fourier space in order to get a better understanding of the e lementary dy-
+namical processes involved in the energy cascade. An other challen ge is to observe
+a possible inverse cascade in wave turbulence that has been predict ed theoretically
+[97,99] and numerically [ 58]. Finally, a significant issue is to understand the physi-
+cal mechanisms of intermittency in waveturbulence that has been r ecently observed
+[35].LABORATORY EXPERIMENTS ON WAVE TURBULENCE 19
+Acknowledgements. I thank S. Fauve and C. Laroche who have been strongly
+involved in the experiments of wave turbulence on the surface of a fl uid, as well as
+S. Aumaˆ ıtre, U. Bortolozzo, F. Boyer and C. Falc´ on [ 11,34,35,36,37,38]. This
+work has been supported by ANR Turbonde BLAN07-3-197846, an d by CNES.
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\ No newline at end of file
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+arXiv:1001.0755v1  [math.FA]  5 Jan 2010Sharp estimates for the commutators of the Hilbert, Riesz
+transforms and the Beurling-Ahlfors operator on weighted
+Lebesgue spaces
+Daewon Chung
+October 16, 2018
+Abstract
+We prove that the operator norm on weighted Lebesgue space L2(w)of the commu-
+tators of the Hilbert, Riesz and Beurling transforms with a BMO functionbdepends
+quadratically on the A2-characteristic of the weight, as opposed to the linear depe ndence
+known to hold for the operators themselves. It is known that t he operator norms of these
+commutators can be controlled by the norm of the commutator w ith appropriate Haar
+shift operators, and we prove the estimate for these commuta tors. For the shift operator
+corresponding to the Hilbert transform we use Bellman funct ion methods, however there
+is now a general theorem for a class of Haar shift operators th at can be used instead to
+deduce similar results. We invoke this general theorem to ob tain the corresponding result
+for the Riesz transforms and the Beurling-Ahlfors operator . We can then extrapolate to
+Lp(w), and the results are sharp for 1<p<∞.12
+1 Introduction
+We saywis a weight if it is positive almost everywhere and locally in tegrable. The norm of
+f∈Lp(w)is
+/ba∇dblf/ba∇dblLp(w):=/parenleftbigg/integraldisplay
+R|f(x)|pw(x)dx/parenrightbigg1/p
+.
+Helson and Szegö first found necessary and sufficient conditio ns for the boundedness of the Hilbert
+transform
+Hf(x) =p.v.1
+π/integraldisplayf(y)
+x−ydy
+in weighted Lebesgue spaces in [11]. Hunt, Muckenhoupt, and Wheeden showed in [13] that a new
+necessary and sufficient condition for boundedness of the Hil bert transform in Lp(w)is that the
+weight satisfies the Apcondition, namely:
+[w]Ap:= sup
+I/a\}b∇acketle{tw/a\}b∇acket∇i}htI/a\}b∇acketle{tw−1/(p−1)/a\}b∇acket∇i}htp−1
+I<∞, (1.1)
+where we denote the average over the interval Iby/a\}b∇acketle{t·/a\}b∇acket∇i}htI,and we take the supremum over all intervals
+inR.After one year, Coifman and Fefferman extended in [5] the resu lt to a larger class of convolution
+singular integrals with standard kernels. Recently, many a uthors have been interested in finding the
+1Key words and phrases: Operator-weighted inequalities, Co mmutator, Hilbert transform, Paraproduct
+22000 Mathematics Subject Classification. Primary 42A50; Se condary 47B47.
+1sharp bounds for the operator norms in terms of the Ap-characteristic [w]Apof the weight. That is,
+one looks for a function φ(x), sharp in terms of its growth, such that
+/ba∇dblTf/ba∇dblLp(w)≤Cφ/parenleftbig
+[w]Ap/parenrightbig
+/ba∇dblf/ba∇dblLp(w).
+ForT=M,the Hardy-Littlewood maximal function, S. Buckley [3] showe d thatφ(x) =x1/(p−1)
+is the sharp rate of growth for all 1< p <∞.He also showed in [3] that φ(x) =x2works for
+the Hilbert transform in L2(w).S. Petermichl and S. Pott improved the result to φ(x) =x3/2,
+for the Hilbert transform in L2(w)in [28]. More recently, S. Petermichl proved in [26] the line ar
+dependence, φ(x) =x,for the Hilbert transform in L2(w)
+/ba∇dblHf/ba∇dblL2(w)≤C[w]A2/ba∇dblf/ba∇dblL2(w),
+by estimating the operator norm of the dyadic shift. Linear b ounds inL2(w)were also obtained
+by O. Beznosova for the dyadic paraproduct [2]. Most recently there are new proofs for the linear
+estimates in L2(w)for some operators including the Hilbert transform and the d yadic paraproduct,
+[17] and [7]. The conjecture is that all the Calderón-Zygmun d singular integral operator obey linear
+bounds in weighted L2.So far this is known only for the Beurling-Ahlfors operator [9 ], [30], the
+Hilbert transform [26], Riesz transforms [27], the marting ale transform [32], the square function [12],
+[24] and well localized dyadic operators [15], [17], [7]. It is now also known for Calderón-Zygmund
+convolutions operators that are smooth averages of well loc alized operators [31].
+In this paper, we are interested in obtaining sharp weight in equalities for the commutators of
+the Hilbert, Riesz, and Beurling transforms with multiplica tion by a locally integrable function
+b∈BMO.
+1.1 Commutators: main results
+Commutator operators are widely encountered and studied in many problems in PDEs, and Har-
+monic Analysis. One classical result of Coifman, Rochberg, and Weiss states in [6] that, for the
+Calderón-Zygmund singular integral operator with a smooth kernel,[b,T]f:=bT(f)−T(bf)is a
+bounded operator on Lp
+Rn,1< p <∞,whenbis a BMO function. Weighted estimates for the
+commutator have been studied in [1], [20], [21], and [23]. No te that the commutator [b,T]is more
+singular than the associated singular integral operator T, in particular, it does not satisfy the cor-
+responding weak (1,1)estimate. However one can find a weaker estimate in [21]. In 19 97, C. Pérez
+[21] obtained the following result concerning commutators of singular integrals, for 1<p<∞,
+/ba∇dbl[b,T]f/ba∇dblLp(w)≤C/ba∇dblb/ba∇dblBMO[w]2
+A∞/ba∇dblM2f/ba∇dblLp(w),
+whereM2=M◦Mdenotes the Hardy-Littlewood maximal function iterated tw ice. With this
+result and Buckley’s sharp estimate for the maximal function [3] one can immediately conclude that
+/ba∇dbl[b,T]/ba∇dblLp(w)→Lp(w)≤C[w]2
+A∞[w]2
+p−1
+Ap/ba∇dblb/ba∇dblBMO.
+In this paper we show that for Tthe Hilbert, Riesz, Beurling transform, for 1<p≤2one can drop
+[w]A∞term, in the above estimate, and this is sharp. However for p>2,theLp(w)-norm of [b,T]
+is bounded above by /ba∇dblb/ba∇dblBMOd[w]2
+A2p.ForT=Hthe Hilbert transform we prove, using Bellman
+function techniques similar to those used in [2], [26], the f ollowing Theorem.
+2Theorem 1.1. There exists a constant C >0, such that
+/ba∇dbl[b,H]f/ba∇dblLp(w)→Lp(w)≤C/ba∇dblb/ba∇dblBMO[w]2max{1,1
+p−1}
+Ap/ba∇dblf/ba∇dblLp(w),
+and this is sharp for 1<p<∞.
+Most of the work goes into showing the quadratic estimate for p= 2, sharp extrapolation [8]
+then provides the right rate of growth for p/\e}atio\slash= 2. An example of C. Pérez shows the rates are sharp.
+Our method involves the use of the dyadic paraproduct πband its adjoint π∗
+b, both of which obey
+linear estimates in L2(w), see [2], like the Hilbert transform. It also uses Petermich l description of
+the Hilbert transform as an average of dyadic shift operator sS, [25] and reduces the estimate to
+obtaining corresponding estimates for the commutator [b,S]. After we decompose this commutator
+in three parts:
+[b,S]f= [πb,S]f+[π∗
+b,S]f+[λb,S]f,
+we estimate each commutator separately. This decompositio n has been used before to analyze the
+commutator, [25], [15], [16]. For precise definitions and de tail derivations, see Section 1.2. The first
+two commutators immediately give the desired quadratic est imates inL2(w)from the known linear
+bounds of the operators commuted. For the third commutator w e can prove a better than quadratic
+bound, in fact a linear bound. The following Theorem will be t he crucial part of the proof.
+Theorem 1.2. There exists a constant C >0, such that
+/ba∇dbl[λb,S]/ba∇dblL2(w)→L2(w)≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd, (1.2)
+for allb∈BMOdandw∈Ad
+2.
+This theorem is an immediate consequence of results in [14], [17] and [7], since the operator
+[λb,S]belongs to the class of Haar shift operators for which they ca n prove linear bounds. We
+present a different proof of this result and others, using Bell man function techniques and bilinear
+Carleson embedding theorems, very much in the spirit of [25] and [2]. These arguments were found
+independently by the author, and we think they can be of inter est.
+We then observe that for any Haar shift operator Tas defined in [17] the commutator [λb,T]
+is again a Haar shift operator, and therefore it obeys linear bounds in A2-characteristic of the
+weight as in Theorem 1.2. As a consequence, we obtain quadrat ic bounds for all commutators of
+Haar shift operators and BMO function b.In particular, this holds true for Haar shift operators
+inRnwhose averages recover the Riesz transforms [27] and for mar tingale transforms in R2whose
+averages recover the Beurling-Ahlfors operator [30], [9]. E xtrapolation will provide Lp(w)bounds
+which turn out to be sharp for the Riesz transforms and Beurlin g-Ahlfors operators as well. The
+following Theorem holds
+Theorem 1.3. LetTτbe the first class of Haar shift operators of index τ.Its convex hull include
+the Hilbert transform, Riesz transforms, the Beurling-Ahl fors operator and so on. Then, there exists
+a constantC(τ,n,p)which only depend on τ, nandpsuch that
+/ba∇dbl[b,Tτ]/ba∇dblLp(w)→Lp(w)≤C(τ,n,p)[w]2max{1,1
+p−1}
+Ap/ba∇dblb/ba∇dblBMO
+See Section 10 for definitions and precise statements of thes e results.
+31.2 The Hilbert transform case
+The bilinear operator
+fH(g)+H(f)g
+mapsL2×L2intoH1, hereHis the Hilbert transform and H1is the real Hardy space defined by
+H1(R) :={f∈L1(R) :Hf∈L1(R)}
+with norm
+/ba∇dblf/ba∇dblH1=/ba∇dblf/ba∇dblL1+/ba∇dblHf/ba∇dblL1.
+Because the dual of H1isBMO , we will pair with a BMO functionb.Using that H∗=−H, we
+obtain that
+/a\}b∇acketle{tfH(g)+H(f)g, b/a\}b∇acket∇i}ht=/a\}b∇acketle{tf, H(g)b−H(gb)/a\}b∇acket∇i}ht.
+Hence the operator g/ma√sto→H(g)b−H(gb)should beL2bounded. This expression H(g)b−H(gb)is
+called the commutator of Hwith theBMO functionb.More generally, we define as follows.
+Definition 1.4. The commutator of the Hilbert transform Hwith a function bis defined as
+[b,H](f) =bH(f)−H(bf).
+Our main concern in this paper is to prove that the commutator [b,H], as an operator from
+L2
+R(w)intoL2
+R(w)is bounded by the square of the A2-characteristic, [w]A2,of the weight times the
+BMO norm, ofb,/ba∇dblb/ba∇dblBMO,where
+/ba∇dblb/ba∇dblBMO:= sup
+I1
+|I|/integraldisplay
+I|b(x)−/a\}b∇acketle{tb/a\}b∇acket∇i}htI|dx.
+The supremum is taken over all intervals in R.Note that when we restrict the supremum to dyadic
+intervals this will define BMOdand we denote this dyadic BMO norm by /ba∇dbl·/ba∇dblBMOd.We now state
+our main results.
+Theorem 1.5. There exists Csuch that for all w∈A2,
+/ba∇dbl[b,H]/ba∇dblL2(w)→L2(w)≤C[w]2
+A2/ba∇dblb/ba∇dblBMO,
+for allb∈BMO.
+Once we have boundedness and sharpness for the crucial case p= 2, we can carry out the power
+of theAp-characteristic, for any 1< p <∞,using the sharp extrapolation theorem [8] to obtain
+Theorem 1.1. Furthermore, an example of C. Pérez [22] shows t his quadratic power is sharp. In
+[25], S. Petermichl showed that the norm of the commutator of the Hilbert transform is bounded by
+the supremum of the norms of the commutator of certain shift o perators. This result follows after
+writing the kernel of the Hilbert transform as a well chosen a verages of certain dyadic shift operators
+discovered by Petermichl. More precisely, S. Petermichl sh owed there is a non zero constant Csuch
+that
+/ba∇dbl[b,H]/ba∇dbl ≤Csup
+α,r/ba∇dbl[b,Sα,r]/ba∇dbl, (1.3)
+where the dyadic shift operator Sα,ris defined by
+Sα,rf=/summationdisplay
+I∈Dα,r/a\}b∇acketle{tf,hI/a\}b∇acket∇i}ht(hI−−hI+).
+4WherehIdenotes the Haar function associated to the interval I,I±denote the left and right
+halves ofI, andDα,ris a shifted and dilated system of dyadic intervals. Denote b ySthe shift op-
+erator corresponding to the standard dyadic intervals D.See the next section for a precise definition.
+Let us consider a compactly supported b∈BMOdandf∈L2.Expanding bandfin the Haar
+system associated to the dyadic intervals D,
+b(x) =/summationdisplay
+I∈D/a\}b∇acketle{tb,hI/a\}b∇acket∇i}hthI(x), f(x) =/summationdisplay
+J∈D/a\}b∇acketle{tf,hJ/a\}b∇acket∇i}hthJ(x);
+formally, we get the multiplication of bandfto be broken into three terms,
+bf=π∗
+b(f)+πb(f)+λb(f), (1.4)
+whereπbis the dyadic paraproduct, π∗
+bis its adjoint and λb(·) =π(·)b,defined as follows
+π∗
+b(f)(x) :=/summationdisplay
+I∈D/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tf,hI/a\}b∇acket∇i}hth2
+I(x),
+πb(f)(x) :=/summationdisplay
+I∈D/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tf/a\}b∇acket∇i}htIhI(x),
+λb(f)(x) :=/summationdisplay
+I∈D/a\}b∇acketle{tb/a\}b∇acket∇i}htI/a\}b∇acketle{tf,hI/a\}b∇acket∇i}hthI(x).
+Thus, we have
+[b,S] = [π∗
+b,S]+[πb,S]+[λb,S], (1.5)
+where
+S(f) =/summationdisplay
+I∈D/a\}b∇acketle{tf,hI/a\}b∇acket∇i}ht(hI−−hI+)
+and we can estimate each term separately. Notice that both πbandπ∗
+bare bounded operators in
+Lpforb∈BMO, despite the fact that multipication by bis not a bounded operator in L2unless
+bis bounded (L∞).Therefore,λbcan not be a bounded operator in Lp.However [λb,S]will be
+bounded on Lp(w)and will be better behaved than [b,S].Decomposition (1.5) was used to analyze
+the commutator with the shift operator first by Petermichl in [25], but also Lacey in [15] and authors
+in [16] to analyze the iterated commutators. Since all estim ates are independent on the dyadic grid,
+through out this paper we only deal with the dyadic shift oper atorSassociated to the standard
+dyadic grid D.For a single shift operator the hypothesis required on bandware that they belong
+to dyadicBMOdandAd
+2with respect to the underlying dyadic grid defining the opera tor. However
+since ultimately we want to average over all grids, we will ne edbandwbelonging to BMOdand
+Ad
+2for all shifted and scaled dyadic grids, that we will have if b∈BMO andw∈A2,non-dyadic
+BMO andA2.Beznosova has proved linear bounds for πbandπ∗
+b,[2], together with Petermichl’s
+[26] linear bounds for S,this immediately provides the quadratic bounds for [πb,S]and[π∗
+b,S].
+Theorem 1.5 will be proved once we show the quadratic estimat e holds for [λb,S]. We can actually
+obtain a better linear estimate as in Theorem 1.2. Some terms in (1.5) do also obey linear bounds.
+Theorem 1.6. There exists Csuch that
+/ba∇dblπ∗
+bS/ba∇dblL2(w)+/ba∇dblSπb/ba∇dblL2(w)≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd.
+for allb∈BMOd.
+5Note the three operators [λb,S], π∗
+bSandSπbare generalized Haar shift operators for which
+there are now two different proofs of linear bounds on L2(w)with respect to [w]Ad
+2,[17] and [7],
+and in this paper we present a third proof. We are now ready to e xplain the organization of this
+paper. In Section 2 we will introduce notation and discuss so me useful results about weighted Haar
+systems. In Section 3 we will start our discussion on how to fin d the linear bound for the term
+[λb,S],most of which will be very similar to calculations performed in [26]. In Section 4 we will
+introduce a number of Lemmas and Theorems that will be used. I n Section 5 we will finish the
+linear estimate for the term [λb,S],and prove Theorem 1.5. In Section 6 we prove the linear bound
+forπ∗
+bS.In Section 7 we reduce the proof of the linear bound for Sπbto verifying three embedding
+conditions, two are proved in this section, the third is prov ed in Section 8 using a Bellman function
+argument. In Section 9 we present the Lp(w)estimate of the commutator with a Haar shift operator,
+Theorem 1.3. Finally, in Section 10 we provide the sharpness for the commutators of Hilbert, Riesz
+transforms and Beurling-Ahlfors operators.
+2 Preliminaries and Notation
+Let us now introduce the notation which will be used frequent ly through this paper. Even though the
+Apconditions have already been introduced in (1.1), we will st ate the special case of this condition
+whenp= 2,namelyAd
+2since we will refer repeatedly to this. We say wbelongs toAd
+2class, if
+[w]Ad
+2:= sup
+I∈D/a\}b∇acketle{tw/a\}b∇acket∇i}htI/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI<∞. (2.1)
+Here we take the supremum over all dyadic interval in R.Note that if w∈A2thenw∈Ad
+2and
+[w]Ad
+2≤[w]A2.Intervals of the form [k2−j,(k+1)2−j)for integers j,kare called dyadic intervals.
+Again, let us denote Dthe collection of all dyadic intervals, and let us denote D(J)the collection
+of all dyadic subintervals of J.For any interval I∈ D, there is a Haar function defined by
+hI(x) =1
+|I|1/2(χI+(x)−χI−(x)),
+whereχIdenotes the characteristic function of the interval I, χI(x) = 1 ifx∈I, χI(x) = 0
+otherwise. It is a well known fact that the Haar system {hI}I∈Dis an orthonormal system in L2
+R.
+We also consider the different grids of dyadic intervals para metrized by α, r,defined by
+Dα,r={α+rI:I∈ D},
+forα∈Rand positive r.For each grid Dα,rof dyadic intervals, there are corresponding Haar func-
+tionshα,r
+I,I∈ Dα,rthat are an orthonormal system in L2
+R.Let us introduce a proper orthonormal
+system forL2
+R(w)defined by
+hw
+I:=1
+w(I)1/2/bracketleftbiggw(I−)1/2
+w(I+)1/2χI+−w(I+)1/2
+w(I−)1/2χI−/bracketrightbigg
+,
+wherew(I) =/integraltext
+Iw.We define the weighted inner product by /a\}b∇acketle{tf,g/a\}b∇acket∇i}htw=/integraltext
+Rfgw. Then, every function
+f∈L2(w)can be written as
+f=/summationdisplay
+I∈D/a\}b∇acketle{tf,hw
+I/a\}b∇acket∇i}htwhw
+I,
+where the sum converges a.e. in L2(w).Moreover,
+/ba∇dblf/ba∇dbl2
+L2(w)=/summationdisplay
+I∈D|/a\}b∇acketle{tf,hw
+I/a\}b∇acket∇i}htw|2. (2.2)
+6AgainDcan be replaced by Dα,rand the corresponding weighted Haar functions are an orthon ormal
+system inL2(w).For convenience we will observe basic properties of the disb alanced Haar system.
+First observe that /a\}b∇acketle{thK,hw
+I/a\}b∇acket∇i}htwcould be non-zero only if I⊇K, moreover, for any I⊇K,
+|/a\}b∇acketle{thK,hw
+I/a\}b∇acket∇i}htw| ≤ /a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+K. (2.3)
+Here is the the calculation that provides (2.3),
+|/a\}b∇acketle{thK,hw
+I/a\}b∇acket∇i}htw|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1
+|K|1/2w(I)1/2(χK+(x)−χK−(x))/bracketleftbiggw(I−)1/2
+w(I+)1/2χI+(x)−w(I+)1/2
+w(I−)1/2χI−(x)/bracketrightbigg
+w(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤1
+|K|1/2w(I)1/2/integraldisplay
+K/vextendsingle/vextendsingle/vextendsingle/vextendsinglew(I−)1/2
+w(I+)1/2χI+(x)+w(I+)1/2
+w(I−)1/2χI−(x)/vextendsingle/vextendsingle/vextendsingle/vextendsinglew(x)dx
+/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright
+A.
+IfK⊂I+, thenA≤w(I−)1/2w(I+)−1/2w(K). Thus
+|/a\}b∇acketle{thK,hw
+I/a\}b∇acket∇i}htw| ≤ /a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+K/radicalBigg
+w(K)w(I−)
+w(I+)w(I)≤ /a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+K.
+Similarly, if K⊂I−.IfK=I, thenA≤2w(K−)1/2w(K+)1/2.Thus
+|/a\}b∇acketle{thK,hw
+I/a\}b∇acket∇i}htw|=|/a\}b∇acketle{thK,hw
+K/a\}b∇acket∇i}htw| ≤ /a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+K2/radicalbig
+w(K−)w(K+)
+w(K)≤ /a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+K.
+Estimate (2.3) implies that |/a\}b∇acketle{thˆJ,hw−1
+ˆJ/a\}b∇acket∇i}htw−1/a\}b∇acketle{thJ,hw
+J/a\}b∇acket∇i}htw| ≤√
+2[w]1/2
+Ad
+2,whereˆJis the parent of J,
+|/a\}b∇acketle{thˆJ,hw−1
+ˆJ/a\}b∇acket∇i}htw−1/a\}b∇acketle{thJ,hw
+J/a\}b∇acket∇i}htw| ≤ /a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/2
+ˆJ/a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+J=/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/2
+ˆJ/parenleftbigg1
+|J|/integraldisplay
+Jw/parenrightbigg1/2
+=/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/2
+ˆJ/parenleftbigg2
+|ˆJ|/integraldisplay
+Jw/parenrightbigg1/2
+≤√
+2/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/2
+ˆJ/a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+ˆJ
+≤√
+2[w]1/2
+Ad
+2. (2.4)
+Also, one can deduce similarly the following estimate
+|/a\}b∇acketle{thJ,hw−1
+J/a\}b∇acket∇i}htw−1/a\}b∇acketle{thJ−,hw
+J/a\}b∇acket∇i}htw| ≤√
+2[w]1/2
+Ad
+2. (2.5)
+ForI/supersetnoteqlJ,hw
+Iis constant on J. We will denote this constant by hw
+I(J).Thenhw
+ˆJ(J)is the constant
+value ofhw
+ˆJonJand|hw
+ˆJ(J)| ≤w(J)−1/2,as can be seen by the following estimate,
+|hw
+ˆJ(J)|=/braceleftbiggw(ˆJ−)1/2/w(ˆJ)1/2w(ˆJ+)1/2≤w(J)−1/2ifJ=ˆJ+
+w(ˆJ+)1/2/w(ˆJ)1/2w(ˆJ−)1/2≤w(J)−1/2ifJ=ˆJ−.(2.6)
+Let us define the weighted averages, /a\}b∇acketle{tg/a\}b∇acket∇i}htJ,w:=w(J)−1/integraltext
+Jg(x)w(x)dx.As with the standard
+Haar system, we can write the weighted averages
+/a\}b∇acketle{tg/a\}b∇acket∇i}htJ,w=/summationdisplay
+I∈D:I/supersetnoteqlJ/a\}b∇acketle{tg,hw
+I/a\}b∇acket∇i}htwhw
+I(J). (2.7)
+7In fact, here is the derivation of (2.7).
+/a\}b∇acketle{tg/a\}b∇acket∇i}htJ,w=1
+w(J)/integraldisplay
+J/summationdisplay
+I∈D/a\}b∇acketle{tg,hw
+I/a\}b∇acket∇i}htwhw
+I(x)w(x)dx=1
+w(J)/integraldisplay
+J/summationdisplay
+I∈D/a\}b∇acketle{tg,hw
+I/a\}b∇acket∇i}htwhw
+I(J)w(x)dx
+=1
+w(J)/integraldisplay
+Jw(x)dx/summationdisplay
+I∈D:I/supersetnoteqlJ/a\}b∇acketle{tg,hw
+I/a\}b∇acket∇i}htwhw
+I(J) =/summationdisplay
+I∈D:I/supersetnoteqlJ/a\}b∇acketle{tg,hw
+I/a\}b∇acket∇i}htwhw
+I(J).
+Also, we will be using system of functions {Hw
+I}I∈Ddefined by
+Hw
+I=hI/radicalbig
+|I|−Aw
+IχIwhereAw
+I=/a\}b∇acketle{tw/a\}b∇acket∇i}htI+−/a\}b∇acketle{tw/a\}b∇acket∇i}htI−
+2/a\}b∇acketle{tw/a\}b∇acket∇i}htI. (2.8)
+Then,{w1/2Hw
+I}is orthogonal in L2with norms satisfying the inequality /ba∇dblw1/2Hw
+I/ba∇dblL2≤/radicalbig
+|I|/a\}b∇acketle{tw/a\}b∇acket∇i}htI,
+refer to [2]. Moreover, by Bessel’s inequality we have, for al lg∈L2,
+/summationdisplay
+I∈D1
+|I|/a\}b∇acketle{tw/a\}b∇acket∇i}htI/a\}b∇acketle{tg,w1/2Hw
+I/a\}b∇acket∇i}ht2≤ /ba∇dblg/ba∇dbl2
+L2. (2.9)
+3 Linear bound for [λb,S]part I
+In general, when we analyse commutator operators, a subtle c ancelation delivers the result one wants
+to find. In the analysis of the commutator [b,S], the part [λb,S]will allow for certain cancelation.
+First, let us rewrite [λb,S].
+[λb,S](f) =λb(Sf)−S(λbf)
+=/summationdisplay
+I∈D/a\}b∇acketle{tb/a\}b∇acket∇i}htI/a\}b∇acketle{tSf,hI/a\}b∇acket∇i}hthI−/summationdisplay
+J∈D/a\}b∇acketle{tλbf,hJ/a\}b∇acket∇i}ht(hJ−−hJ+)
+=/summationdisplay
+I∈D/summationdisplay
+J∈D/a\}b∇acketle{tb/a\}b∇acket∇i}htI/a\}b∇acketle{tf,hJ/a\}b∇acket∇i}ht/a\}b∇acketle{thJ−−hJ+,hI/a\}b∇acket∇i}hthI−/summationdisplay
+J∈D/summationdisplay
+I∈D/a\}b∇acketle{tb/a\}b∇acket∇i}htI/a\}b∇acketle{tf,hI/a\}b∇acket∇i}ht/a\}b∇acketle{thI,hJ/a\}b∇acket∇i}ht(hJ−−hJ+)
+=/summationdisplay
+J∈D/a\}b∇acketle{tb/a\}b∇acket∇i}htJ−/a\}b∇acketle{tf,hJ/a\}b∇acket∇i}hthJ−−/summationdisplay
+J∈D/a\}b∇acketle{tb/a\}b∇acket∇i}htJ+/a\}b∇acketle{tf,hJ/a\}b∇acket∇i}hthJ+−/summationdisplay
+J∈D/a\}b∇acketle{tb/a\}b∇acket∇i}htJ++/a\}b∇acketle{tb/a\}b∇acket∇i}htJ−
+2/a\}b∇acketle{tf,hJ/a\}b∇acket∇i}ht(hJ−−hJ+)
+=/summationdisplay
+J∈D/a\}b∇acketle{tb/a\}b∇acket∇i}htJ−−/a\}b∇acketle{tb/a\}b∇acket∇i}htJ+
+2/a\}b∇acketle{tf,hJ/a\}b∇acket∇i}hthJ−−/summationdisplay
+J∈D/a\}b∇acketle{tb/a\}b∇acket∇i}htJ+−/a\}b∇acketle{tb/a\}b∇acket∇i}htJ−
+2/a\}b∇acketle{tf,hJ/a\}b∇acket∇i}hthJ+
+=−/summationdisplay
+J∈D∆Jb/a\}b∇acketle{tf,hJ/a\}b∇acket∇i}ht(hJ++hJ−),
+recall the notation ∆Jb= (/a\}b∇acketle{tb/a\}b∇acket∇i}htJ+−/a\}b∇acketle{tb/a\}b∇acket∇i}htJ−)/2.To find the L2(w)operator norm of [λb,S], it is enough
+to deal with the operator
+Sb(f) =/summationdisplay
+I∈D∆Ib/a\}b∇acketle{tf,hI/a\}b∇acket∇i}hthI−.
+We shall state the weighted operator norm of Sbas a Theorem and give a detailed proof. Theorem
+1.2 is a direct consequence of the following Theorem. We will prove the following Theorem by the
+technique used in [26].
+Theorem 3.1. There exists a constant C >0, such that
+/ba∇dblSb/ba∇dblL2(w)→L2(w)≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd (3.1)
+for allb∈BMOdandw∈Ad
+2for allf∈L2(w).
+8One can easily check that choosing b=|I|1/2hI,yields/ba∇dblS/ba∇dblL2(w)→L2(w)≤C[w]Ad
+2.Inequality
+(3.1) is equivalent to the following inequality for any posi tive function f,g
+|/a\}b∇acketle{tSb,w−1f,g/a\}b∇acket∇i}htw| ≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/ba∇dblf/ba∇dblL2(w−1)/ba∇dblg/ba∇dblL2(w), (3.2)
+whereSb,w−1(f) =Sb(w−1f).Expanding fandgin the disbalanced Haar systems respectively for
+L2(w−1)andL2(w)yields for (3.2),
+|/a\}b∇acketle{tSb,w−1f,g/a\}b∇acket∇i}htw|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Sb,w−1/parenleftbig/summationdisplay
+I∈D/a\}b∇acketle{tf,hw−1
+I/a\}b∇acket∇i}htw−1hw−1
+I/parenrightbig/parenleftbig/summationdisplay
+J∈D/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htwhw
+J/parenrightbig
+wdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+I∈D/summationdisplay
+J∈D/a\}b∇acketle{tf,hw−1
+I/a\}b∇acket∇i}htw−1/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htw/integraldisplay
+hw
+JSb,w−1(hw−1
+I)wdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+I∈D/summationdisplay
+J∈D/a\}b∇acketle{tf,hw−1
+I/a\}b∇acket∇i}htw−1/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htw/a\}b∇acketle{tSb,w−1hw−1
+I,hw
+J/a\}b∇acket∇i}htw/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (3.3)
+In (3.3),
+/a\}b∇acketle{tSb,w−1hw−1
+I,hw
+J/a\}b∇acket∇i}htw=/angbracketleftBig/summationdisplay
+L∈D∆Lb/a\}b∇acketle{thL,w−1hw−1
+I/a\}b∇acket∇i}hthL−,hw
+J/angbracketrightBig
+w=/summationdisplay
+L∈D∆Lb/a\}b∇acketle{thL,hw−1
+I/a\}b∇acket∇i}htw−1/a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw
+Since/a\}b∇acketle{thL,hw−1
+I/a\}b∇acket∇i}htw−1/\e}atio\slash= 0,only whenL⊆Iand/a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw/\e}atio\slash= 0only whenL−⊆J,then we have
+non-zero terms if I⊆JorˆJ⊆Iin the sum of (3.3). Thus we can split the sum into four parts,/summationtext
+I=J,/summationtext
+I=ˆJ,/summationtext
+ˆJ/subsetnoteqlI,and/summationtext
+I/subsetnoteqlJ.Let us now introduce the truncated shift operator
+SI
+b(f) :=/summationdisplay
+L∈D(I)∆Lb/a\}b∇acketle{tf,hL/a\}b∇acket∇i}hthL−,
+and its composition with multiplication by w−1,
+SI
+b,w−1(f) :=/summationdisplay
+L∈D(I)∆Lb/a\}b∇acketle{tw−1f,hL/a\}b∇acket∇i}hthL−.
+We will see that the weighted norm /ba∇dblSI
+b,w−1χI/ba∇dblL2(w), plays a main role in our estimate for /a\}b∇acketle{tSb,w−1hw−1
+I,hw
+J/a\}b∇acket∇i}htw.
+4 Theorems and Lemmas
+To prove inequality (3.2) we need several theorems and lemma s. One can find the proof in the
+indicated references. First we recall some embedding theor ems. Another main result in [26] is a
+two-weighted bilinear embedding theorem, which was proved by a Bellman function argument.
+Theorem 4.1 (Petermichl’s Bilinear Embedding Theorem) .Letwandvbe weights so that /a\}b∇acketle{tw/a\}b∇acket∇i}htI/a\}b∇acketle{tv/a\}b∇acket∇i}htI≤
+Qfor all intervals Iand let{αI}be a non-negative sequence so that the three estimates below hold
+for allJ/summationdisplay
+I∈D(J)αI
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI≤Qv(J) (4.1)
+/summationdisplay
+I∈D(J)αI
+/a\}b∇acketle{tv/a\}b∇acket∇i}htI≤Qw(J) (4.2)
+9/summationdisplay
+I∈D(J)αI≤Q|J|. (4.3)
+Then there is csuch that for all f∈L2(w)andg∈L2(v)
+/summationdisplay
+I∈DαI/a\}b∇acketle{tf/a\}b∇acket∇i}htI,w/a\}b∇acketle{tg/a\}b∇acket∇i}htI,v≤cQ/ba∇dblf/ba∇dblL2(w)/ba∇dblg/ba∇dblL2(v).
+Corollary 4.2 (Bilinear Embedding Theorem) .Letwandvbe weights. Let {αI}be a sequence of
+nonnegative numbers such that for all dyadic intervals J∈ Dthe following three inequalities hold
+with some constant Q>0,/summationdisplay
+I∈D(J)αI/a\}b∇acketle{tv/a\}b∇acket∇i}htI|I| ≤Qv(J) (4.4)
+/summationdisplay
+I∈D(J)αI/a\}b∇acketle{tw/a\}b∇acket∇i}htI|I| ≤Qw(J) (4.5)
+/summationdisplay
+I∈D(J)αI/a\}b∇acketle{tw/a\}b∇acket∇i}htI/a\}b∇acketle{tv/a\}b∇acket∇i}htI≤Q|J|. (4.6)
+Then for any two nonnegative function f, g∈L2
+/summationdisplay
+I∈DαI/a\}b∇acketle{tfv1/2/a\}b∇acket∇i}htI/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI|I| ≤CQ/ba∇dblf/ba∇dblL2/ba∇dblg/ba∇dblL2
+holds with some constant C >0.
+Both Bilinear Embedding Theorems are key tools in our estimate . One version of such a theorem
+appeared in [19]. The original version of the next lemma also appeared in [26]. Using the fact that,
+for allI∈ D,|∆Ib| ≤2/ba∇dblb/ba∇dblBMOd,one can prove the following lemma similarly to its original p roof.
+Lemma 4.3. There is a constant csuch that /ba∇dblSI
+b,w−1χI/ba∇dblL2(w)≤c/ba∇dblb/ba∇dblBMOd[w]Ad
+2w−1(I)1/2for all
+intervalsIand weights w∈Ad
+2.
+We will also need the Weighted Carleson Embedding Theorem fr om [19], and some other in-
+equalities for weights.
+Theorem 4.4 (Weighted Carleson Embedding Theorem) .Let{αJ}be a non-negative sequence
+such that for all dyadic intervals I
+/summationdisplay
+J∈D(I)αJ≤Qw−1(I).
+Then for all f∈L2(w−1)/summationdisplay
+J∈DαJ/a\}b∇acketle{tf/a\}b∇acket∇i}ht2
+J,w−1≤4Q/ba∇dblf/ba∇dbl2
+L2(w−1).
+Theorem 4.5 (Wittwer’s sharp version of Buckley’s inequality) .There exist a positive constant C
+such that for any weight w∈Ad
+2and dyadic interval I∈ D,
+1
+|J|/summationdisplay
+I∈D(J)/parenleftbig
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI+−/a\}b∇acketle{tw/a\}b∇acket∇i}htI−/parenrightbig2
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI|I| ≤C[w]Ad
+2/a\}b∇acketle{tw/a\}b∇acket∇i}htJ.
+10We refer to [32] for the proof. You can find extended versions o f Theorem 4.4 and 4.5 to the
+doubling positive measure σin [24]. One can find the Bellman function proof of the followin g three
+Lemmas in [2].
+Lemma 4.6. For all dyadic interval J and all weights w.
+1
+|J|/summationdisplay
+I∈D(J)|I||∆Iw|21
+/a\}b∇acketle{tw/a\}b∇acket∇i}ht3
+I≤ /a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ.
+Lemma 4.7. Letwbe a weight and {αI}be a Carleson sequence of nonnegative numbers. If there
+exist a constant Q>0such that
+∀J∈ D,1
+|J|/summationdisplay
+I∈D(J)αI≤Q,
+then
+∀J∈ D,1
+|J|/summationdisplay
+I∈D(J)αI
+/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI≤4Q/a\}b∇acketle{tw/a\}b∇acket∇i}htJ
+and therefore if w∈Ad
+2the for any J∈ Dwe have
+1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tw/a\}b∇acket∇i}htIαI≤4Q[w]Ad
+2/a\}b∇acketle{tw/a\}b∇acket∇i}htJ.
+Lemma 4.8. Ifw∈Ad
+2then there exists a constant C >0such that
+∀J∈ D,1
+|J|/summationdisplay
+I∈D(J)/parenleftbigg/a\}b∇acketle{tw/a\}b∇acket∇i}htI+−/a\}b∇acketle{tw/a\}b∇acket∇i}htI−
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI/parenrightbigg2
+|I|/a\}b∇acketle{tw/a\}b∇acket∇i}htI/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI≤C[w]Ad
+2.
+5 Linear bound for [λb,S]part II
+We will continue to estimate the sum (3.3) in four parts.
+5.1/summationtext
+I=ˆJ
+For this case, it is sufficient to show that
+|/a\}b∇acketle{tSb,w−1hw−1
+ˆJ,hw
+J/a\}b∇acket∇i}htw| ≤c/ba∇dblb/ba∇dblBMOd[w]Ad
+2.
+Since/a\}b∇acketle{thk,hw
+I/a\}b∇acket∇i}htwcould be non-zero only if K⊆I,
+|/a\}b∇acketle{tSb,w−1hw−1
+ˆJ,hw
+J/a\}b∇acket∇i}htw|=/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig/summationdisplay
+L∈D∆Lb/a\}b∇acketle{tw−1hw−1
+L,hL/a\}b∇acket∇i}hthL−,hw
+J/angbracketrightBig
+w/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+L∈D∆Lb/a\}b∇acketle{thw−1
+ˆJ,hL/a\}b∇acket∇i}htw−1/a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw/vextendsingle/vextendsingle/vextendsingle
+has non-zero term only when L⊆ˆJ. Thus
+|/a\}b∇acketle{tSb,w−1hw−1
+L,hw
+J/a\}b∇acket∇i}htw|=/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+L∈D(ˆJ)∆Lb/a\}b∇acketle{thw−1
+ˆJ,hL/a\}b∇acket∇i}htw−1/a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw/vextendsingle/vextendsingle/vextendsingle
+=/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+L∈D(J)∆Lb/a\}b∇acketle{thw−1
+ˆJ,hL/a\}b∇acket∇i}htw−1/a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+L∈D(Js)∆Lb/a\}b∇acketle{thw−1
+ˆJ,hL/a\}b∇acket∇i}htw−1/a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw/vextendsingle/vextendsingle/vextendsingle
++|∆ˆJb/a\}b∇acketle{thw−1
+ˆJ,hˆJ/a\}b∇acket∇i}htw−1/a\}b∇acketle{thˆJ−,hw
+J/a\}b∇acket∇i}htw|
+≤ |/a\}b∇acketle{tSJ
+b,w−1hw−1
+ˆJ,hw
+J/a\}b∇acket∇i}htw|+|∆ˆJb/a\}b∇acketle{thw−1
+ˆJ,hˆJ/a\}b∇acket∇i}htw−1/a\}b∇acketle{thˆJ−,hw
+J/a\}b∇acket∇i}htw|
+11in the second equality, Jsdenotes the sibling of J, so for all L⊆Js,/a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw= 0.Then, by
+(2.4),
+|∆ˆJb/a\}b∇acketle{thw−1
+ˆJ,hˆJ/a\}b∇acket∇i}htw−1/a\}b∇acketle{thˆJ−,hw
+J/a\}b∇acket∇i}htw| ≤√
+2/ba∇dblb/ba∇dblBMOd[w]1/2
+Ad
+2. (5.1)
+So for the remaining part:
+|/a\}b∇acketle{tSJ
+b,w−1hw−1
+ˆJ,hw
+J/a\}b∇acket∇i}htw|=|hw−1
+ˆJ(J)/a\}b∇acketle{tSJ
+b,w−1χJ,hw
+J/a\}b∇acket∇i}htw| ≤c/ba∇dblb/ba∇dblBMOd[w]Ad
+2, (5.2)
+here the last inequality uses (2.6) and Lemma 4.3.
+5.2/summationtext
+I=J
+In this case, the argument is similar to the argument in Secti on 5.1. We have
+|/a\}b∇acketle{tSb,w−1hw−1
+J,hw
+J/a\}b∇acket∇i}htw|=|/summationdisplay
+L∈D∆Lb/a\}b∇acketle{thw−1
+J,hL/a\}b∇acket∇i}htw−1/a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw|
+here we have zero summands, unless L⊆J. Thus,
+|/a\}b∇acketle{tSb,w−1hw−1
+J,hw
+J/a\}b∇acket∇i}htw|=|/a\}b∇acketle{tSJ
+b,w−1hw−1
+J,hw
+J/a\}b∇acket∇i}htw|
+≤ |/a\}b∇acketle{tSJ+
+b,w−1hw−1
+J,hw
+J/a\}b∇acket∇i}htw|+|/a\}b∇acketle{tSJ−
+b,w−1hw−1
+J,hw
+J/a\}b∇acket∇i}htw|+|∆Jb/a\}b∇acketle{thw−1
+J,hJ/a\}b∇acket∇i}htw−1/a\}b∇acketle{thJ−,hw
+J/a\}b∇acket∇i}htw|
+≤c/ba∇dblb/ba∇dblBMOd[w]Ad
+2.
+In the last inequality, we use same arguments as in (5.2) for t he first two term, and (2.5) for the
+last term.
+5.3/summationtext
+ˆJ/subsetnoteqlIand/summationtext
+I/subsetnoteqlJ
+To obtain our desired results, we need to understand the supp orts ofSb(w−1hw−1
+I)andS∗
+b(whw
+J).
+Since
+Sb(w−1hw−1
+I) =/summationdisplay
+L∈D∆Lb/a\}b∇acketle{tw−1hw−1
+I,hL/a\}b∇acket∇i}hthL−=/summationdisplay
+L∈D∆Lb/a\}b∇acketle{thw−1
+I,hL/a\}b∇acket∇i}htw−1hL−,
+and/a\}b∇acketle{thw−1
+I,hL/a\}b∇acket∇i}htcan be non-zero only when L⊆I, Sb(w−1hw−1
+I)is supported by I.Also,
+/a\}b∇acketle{tSb(w−1hw−1
+I),hw
+J/a\}b∇acket∇i}htw=/a\}b∇acketle{thw−1
+I,S∗
+b(whw
+J)/a\}b∇acket∇i}htw−1, (5.3)
+12yield thatS∗
+b(whw
+J) =/summationtext
+L∈D∆Lb/a\}b∇acketle{twhw
+J,hL−/a\}b∇acket∇i}hthLis supported by ˆJ.Let us now consider the sum
+ˆJ/subsetnoteqlI.Then
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+I,J:ˆJ/subsetnoteqlI/a\}b∇acketle{tf,hw−1
+I/a\}b∇acket∇i}htw−1/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htw/a\}b∇acketle{tSb,w−1hw−1
+I,hw
+J/a\}b∇acket∇i}htw/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+I,J:ˆJ/subsetnoteqlI/a\}b∇acketle{tf,hw−1
+I/a\}b∇acket∇i}htw−1/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htw/a\}b∇acketle{thw−1
+I,S∗
+b(whw
+J)/a\}b∇acket∇i}htw−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+J∈D/summationdisplay
+I:I/supersetnoteqlˆJ/a\}b∇acketle{tf,hw−1
+I/a\}b∇acket∇i}htw−1/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htwhw−1
+I(ˆJ)/a\}b∇acketle{tSb,w−1χˆJ,hw
+J/a\}b∇acket∇i}htw/vextendsingle/vextendsingle/vextendsingle/vextendsingle(5.4)
+=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+J∈D/a\}b∇acketle{tf/a\}b∇acket∇i}htˆJ,w−1/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htw/a\}b∇acketle{tSb,w−1χˆJ,hw
+J/a\}b∇acket∇i}htw/vextendsingle/vextendsingle/vextendsingle/vextendsingle(5.5)
+≤ /ba∇dblg/ba∇dblL2(w)/parenleftbigg/summationdisplay
+J∈D/a\}b∇acketle{tf/a\}b∇acket∇i}ht2
+ˆJ,w−1/a\}b∇acketle{tSb,w−1χˆJ,hw
+J/a\}b∇acket∇i}ht2
+w/parenrightbigg1/2
+, (5.6)
+here (5.3) and the fact that S∗
+b(whw
+J)is supported by ˆJare used for equality (5.4), and (5.5) uses
+(2.7) and (5.3). If we show that
+/summationdisplay
+J∈D/a\}b∇acketle{tf/a\}b∇acket∇i}ht2
+ˆJ,w−1/a\}b∇acketle{tSb,w−1χˆJ,hw
+J/a\}b∇acket∇i}ht2
+w≤c/ba∇dblb/ba∇dbl2
+BMOd[w]2
+Ad
+2/ba∇dblf/ba∇dbl2
+L2(w−1), (5.7)
+then we have
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+I,J:ˆJ/subsetnoteqlI/a\}b∇acketle{tf,hw−1
+I/a\}b∇acket∇i}htw−1/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htw/a\}b∇acketle{tSb,w−1hw−1
+I,hw
+J/a\}b∇acket∇i}htw/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/ba∇dblb/ba∇dblBMOd[w]Ad
+2/ba∇dblf/ba∇dbl2
+L(w−1)/ba∇dblg/ba∇dblL2(w).
+To prove the inequality (5.7), we apply the Theorem 4.4. The e mbedding condition becomes
+/summationdisplay
+J∈D:J/subsetnoteqlI/a\}b∇acketle{tSb,w−1χˆJ,hw
+J/a\}b∇acket∇i}ht2
+w≤c/ba∇dblb/ba∇dbl2
+BMOd[w]2
+Ad
+2w−1(I)
+after shifting the indices. Since /a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw= 0unlessL⊆ˆJ,and we will sum over Jsuch that
+J/subsetnoteqlI, we can write
+/a\}b∇acketle{tSb,w−1χˆJ,hw
+J/a\}b∇acket∇i}htw=/summationdisplay
+L∈D∆Lb/a\}b∇acketle{tw−1χˆJ,hL/a\}b∇acket∇i}ht/a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw=/summationdisplay
+L∈D(ˆJ)∆Lb/a\}b∇acketle{tw−1χˆJ,hL/a\}b∇acket∇i}ht/a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw
+=/summationdisplay
+L∈D(I)∆Lb/a\}b∇acketle{tw−1χI,hL/a\}b∇acket∇i}ht/a\}b∇acketle{thL−,hw
+J/a\}b∇acket∇i}htw=/a\}b∇acketle{tSI
+b,w−1χI,hw
+J/a\}b∇acket∇i}htw.
+Thus,/summationdisplay
+J∈D:J/subsetnoteqlI/a\}b∇acketle{tSb,w−1χˆJ,hw
+J/a\}b∇acket∇i}ht2
+w=/summationdisplay
+J∈D:J/subsetnoteqlI/a\}b∇acketle{tSI
+b,w−1χI,hw
+J/a\}b∇acket∇i}ht2
+w≤ /ba∇dblSI
+b,w−1χI/ba∇dbl2
+L2(w),
+last inequality due to (2.2). By Lemma 4.3, the embedding cond ition holds. Hence we are done for
+the sum ˆJ/subsetnoteqlI. The part/summationtext
+I/subsetnoteqlJis similar to/summationtext
+ˆJ/subsetnoteqlI.One uses that Sb(w−1hw−1
+I)is supported by I
+and Theorem 4.4.
+135.4 Proof of Theorem 1.5
+We refer to [2] for following theorem.
+Theorem 5.1. The norm of dyadic paraproduct on the weighted Lebesgue spac eL2(w)is bounded
+from above by a constant multiple of the product of the Ad
+2characteristic of the weight wand the
+BMOdnorm ofb.
+To break [b,S]into three parts, as in (1.5), we assumed that b∈BMOdis compactly supported.
+However, we need to replace such a bwith a general BMOdfunction. In order to pass from a
+compactly supported bto generalb∈BMOd,we need the following lemma which is suggested in
+[10].
+Lemma 5.2. Supposeφ∈BMO. Let/tildewideIbe the interval concentric with Ihaving length |/tildewideI|= 3|I|.
+Then there is ψ∈BMO such thatψ=φonI,ψ= 0onR\/tildewideIand/ba∇dblψ/ba∇dblBMO≤c/ba∇dblφ/ba∇dblBMO.
+Proof. Without loss of generality, we assume /a\}b∇acketle{tφ/a\}b∇acket∇i}htI= 0.WriteI=/uniontext∞
+n=0Jnwheredist(Jn,∂I) =
+|Jn|, as in following figure.
+J0J1 J2 J3 J4
+/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright
+IK1 K2
+ThenJ0is the middle third of I.Forn >0, letKnbe the reflection of Jnacross the nearest
+endpoint of Iand set
+ψ(x) =
+
+φ(x)ifx∈I
+/a\}b∇acketle{tφ/a\}b∇acket∇i}htJnifx∈Kn
+0otherwise.
+This construction of ψsatisfies Lemma 5.2.
+By Theorem 3.1, Corollary 1.2, Theorem 5.1, Lemma 5.2 and usin g the fact /ba∇dblπb/ba∇dbl=/ba∇dblπ∗
+b/ba∇dbl, we
+can prove Theorem 1.5.
+Proof of Theorem 1.5. For any compactly supported b∈BMO,
+/ba∇dbl[b,H]/ba∇dblL2(w)→L2(w)≤Csup
+α,r/ba∇dbl[b,Sα,r]/ba∇dblL2(w)→L2(w)
+≤Csup
+α,r/parenleftbig
+/ba∇dbl[πb,Sα,r]/ba∇dblL2(w)→L2(w)+/ba∇dbl[π∗
+b,Sα,r]/ba∇dblL2(w)→L2(w)+/ba∇dbl[λb,Sα,r]/ba∇dblL2(w)→L2(w)/parenrightbig
+≤C/parenleftbig
+4/ba∇dblπb/ba∇dblL2(w)→L2(w)sup
+α,r/ba∇dblSα,r/ba∇dblL2(w)→L2(w)+C[w]A2/ba∇dblb/ba∇dblBMO/parenrightbig
+≤C[w]2
+A2/ba∇dblb/ba∇dblBMO.
+For fixedb, we consider the sequence of intervals Ik= [−k,k]and the sequence of BMO functions
+bkwhich are constructed as in Lemma 5.2. Then, there is a consta ntc, which does not depend on
+k,such that /ba∇dblbk/ba∇dblBMO≤c/ba∇dblb/ba∇dblBMO.Furthermore, there is a uniform constant Csuch that
+/ba∇dbl[bk,H]/ba∇dblL2(w)→L2(w)≤C[w]2
+A2/ba∇dblb/ba∇dblBMO. (5.8)
+Therefore, for some subsequence of integers kjandf∈L2(w),[bkj,H](f)converges to [b,H](f)
+almost everywhere. Letting j→ ∞ and using Fatou’s lemma, we deduce that (5.8) holds for all
+b∈BMO.
+146 Linear bound for π∗
+bS
+It might be useful to know what is the adjoint operator of S. Let us define sgn (I) =±1ifI=ˆI∓.
+Then, for any function f,g∈L2,
+/a\}b∇acketle{tSf,g/a\}b∇acket∇i}ht=/summationdisplay
+I∈D/summationdisplay
+J∈D/a\}b∇acketle{tf,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tg,hJ/a\}b∇acket∇i}ht/a\}b∇acketle{thI−−hI+,hJ/a\}b∇acket∇i}ht
+=/summationdisplay
+I∈D/a\}b∇acketle{tf,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tg,hI−/a\}b∇acket∇i}ht−/summationdisplay
+I∈D/a\}b∇acketle{tf,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tg,hI+/a\}b∇acket∇i}ht
+=/summationdisplay
+I∈D/a\}b∇acketle{tf,hI/a\}b∇acket∇i}ht/parenleftbig
+sgn(I−)/a\}b∇acketle{tg,hI−/a\}b∇acket∇i}ht+sgn(I+)/a\}b∇acketle{tg,hI+/a\}b∇acket∇i}ht/parenrightbig
+=/summationdisplay
+I∈D/a\}b∇acketle{tf,hˆI/a\}b∇acket∇i}htsgn(I)/a\}b∇acketle{tg,hI/a\}b∇acket∇i}ht
+=/angbracketleftBig
+f,/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tg,hI/a\}b∇acket∇i}hthˆI/angbracketrightBig
+=/a\}b∇acketle{tf,S∗g/a\}b∇acket∇i}ht.
+Now, we see the adjoint operator of dyadic shift operator Sis
+S∗f(x) =/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tf,hI/a\}b∇acket∇i}hthˆI(x).
+The following lemma provides the bound we are looking for the termπ∗
+bS.
+Lemma 6.1. Letw∈Ad
+2andb∈BMOd. Then, there exists Cso that
+/ba∇dblπ∗
+bS/ba∇dblL2(w)→L2(w)≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd.
+Proof. In order to prove Lemma 6.1 it is enough to show that for any pos itive square integrable
+functionf,g
+/a\}b∇acketle{tπ∗
+bS(fw−1/2),gw1/2/a\}b∇acket∇i}ht ≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/ba∇dblf/ba∇dblL2/ba∇dblg/ba∇dblL2. (6.1)
+Using the system of functions {Hw
+I}I∈Ddefined in (2.8), we can rewrite the left hand side of (6.1)
+/a\}b∇acketle{tπ∗
+bS(fw−1/2),gw1/2/a\}b∇acket∇i}ht=/a\}b∇acketle{tS(fw−1/2),πb(gw1/2)/a\}b∇acket∇i}ht=/summationdisplay
+I∈D/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tS(fw−1/2),hI/a\}b∇acket∇i}ht
+=/summationdisplay
+I∈D/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI/a\}b∇acketle{tb,hI/a\}b∇acket∇i}htsgn(I)/a\}b∇acketle{tfw−1/2,hˆI/a\}b∇acket∇i}ht
+=/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tfw−1/2,Hw−1
+ˆI/a\}b∇acket∇i}ht1/radicalBig
+|ˆI|
++/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tfw−1/2,Aw−1
+ˆIχˆI/a\}b∇acket∇i}ht1/radicalBig
+|ˆI|.(6.2)
+Our claim is that both sums in (6.2) are bounded by [w]Ad
+2/ba∇dblb/ba∇dblBMOd/ba∇dblf/ba∇dblL2/ba∇dblg/ba∇dblL2,i.e.
+/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tfw−1/2,Hw−1
+ˆI/a\}b∇acket∇i}ht1/radicalBig
+|ˆI|≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/ba∇dblf/ba∇dblL2/ba∇dblg/ba∇dblL2 (6.3)
+and
+/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tfw−1/2,Aw−1
+ˆIχˆI/a\}b∇acket∇i}ht1/radicalBig
+|ˆI|≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/ba∇dblf/ba∇dblL2/ba∇dblg/ba∇dblL2. (6.4)
+15First let us verify the bound for (6.3). Using Cauchy-Schwar z inequality,
+/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tfw−1/2,Hw−1
+ˆI/a\}b∇acket∇i}ht1/radicalBig
+|ˆI|
+≤/parenleftbigg/summationdisplay
+I∈D/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}ht2
+I/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/parenrightbigg1/2/parenleftbigg/summationdisplay
+I∈D1
+|ˆI|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tf,w−1/2Hw−1
+ˆI/a\}b∇acket∇i}ht2/parenrightbigg1/2
+≤ /ba∇dblf/ba∇dblL2/parenleftbigg/summationdisplay
+I∈D/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}ht2
+I/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/parenrightbigg1/2
+. (6.5)
+Thus, for (6.3), it is enough to show that
+/summationdisplay
+I∈D/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}ht2
+I/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI≤C[w]2
+Ad
+2/ba∇dblb/ba∇dbl2
+BMOd/ba∇dblg/ba∇dbl2
+L2. (6.6)
+It is clear that 2/a\}b∇acketle{tw/a\}b∇acket∇i}htˆI≥ /a\}b∇acketle{tw/a\}b∇acket∇i}htI,thus
+/summationdisplay
+I∈D/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}ht2
+I/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI=/summationdisplay
+I∈D/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}ht2
+I/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tw/a\}b∇acket∇i}htI/a\}b∇acketle{tw/a\}b∇acket∇i}ht−1
+I
+≤2[w]Ad
+2/summationdisplay
+I∈D/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}ht2
+I/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw/a\}b∇acket∇i}ht−1
+I.
+If we show for all J∈ D,
+1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw/a\}b∇acket∇i}ht−1
+I/a\}b∇acketle{tw/a\}b∇acket∇i}ht2
+I=1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw/a\}b∇acket∇i}htI≤[w]Ad
+2/ba∇dblb/ba∇dbl2
+BMOd/a\}b∇acketle{tw/a\}b∇acket∇i}htJ, (6.7)
+then by Weighted Carleson Embedding Theorem 4.4 with winstead ofw−1,we will have (6.6).
+Sinceb∈BMOd,{/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2}I∈Dis a Carleson sequence with constant /ba∇dblb/ba∇dbl2
+BMOdthat is
+1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2≤ /ba∇dblb/ba∇dbl2
+BMOd.
+Applying Lemma 4.7 with αI=/a\}b∇acketle{tb,hI/a\}b∇acket∇i}htwe have inequality (6.7). We now concentrate on the estimate
+(6.4), we can estimate the left hand side of (6.4) as follows.
+/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tfw−1/2,Aw−1
+ˆIχˆI/a\}b∇acket∇i}ht1/radicalBig
+|ˆI|
+=/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tfw−1/2/a\}b∇acket∇i}htˆIAw−1
+ˆI/radicalBig
+|ˆI|
+≤/summationdisplay
+I∈D/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|/a\}b∇acketle{tfw−1/2/a\}b∇acket∇i}htˆI|Aw−1
+ˆI|/radicalBig
+|ˆI|
+≤2/summationdisplay
+I∈D/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htˆI|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|/a\}b∇acketle{tfw−1/2/a\}b∇acket∇i}htˆI|Aw−1
+ˆI|/radicalBig
+|ˆI|
+= 2/summationdisplay
+I∈D/a\}b∇acketle{tgw1/2/a\}b∇acket∇i}htI/parenleftbig
+|/a\}b∇acketle{tb,hI−/a\}b∇acket∇i}ht|+|/a\}b∇acketle{tb,hI+/a\}b∇acket∇i}ht|/parenrightbig
+/a\}b∇acketle{tfw−1/2/a\}b∇acket∇i}htI|Aw−1
+I|/radicalbig
+|I|.
+16By Bilinear Embedding Theorem, inequality (6.4) holds provid ed the following three inequalities
+hold,
+∀J∈ D,1
+|J|/summationdisplay
+I∈D(J)/parenleftbig
+|/a\}b∇acketle{tb,hI−/a\}b∇acket∇i}ht|+|/a\}b∇acketle{tb,hI+/a\}b∇acket∇i}ht|/parenrightbig
+|Aw−1
+I|/radicalbig
+|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/a\}b∇acketle{tw/a\}b∇acket∇i}htI≤C/ba∇dblb/ba∇dblBMOd[w−1]Ad
+2,(6.8)
+∀J∈ D,1
+|J|/summationdisplay
+I∈D(J)/parenleftbig
+|/a\}b∇acketle{tb,hI−/a\}b∇acket∇i}ht|+|/a\}b∇acketle{tb,hI+/a\}b∇acket∇i}ht|/parenrightbig
+|Aw−1
+I|/radicalbig
+|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI≤C/ba∇dblb/ba∇dblBMOd[w−1]Ad
+2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ,(6.9)
+∀J∈ D,1
+|J|/summationdisplay
+I∈D(J)/parenleftbig
+|/a\}b∇acketle{tb,hI−/a\}b∇acket∇i}ht|+|/a\}b∇acketle{tb,hI+/a\}b∇acket∇i}ht|/parenrightbig
+|Aw−1
+I|/radicalbig
+|I|/a\}b∇acketle{tw/a\}b∇acket∇i}htI≤C/ba∇dblb/ba∇dblBMOd[w−1]Ad
+2/a\}b∇acketle{tw/a\}b∇acket∇i}htJ. (6.10)
+For (6.8), by Cauchy-Schwarz inequality
+1
+|J|/summationdisplay
+I∈D(J)/parenleftbig
+|/a\}b∇acketle{tb,hI−/a\}b∇acket∇i}ht|+|/a\}b∇acketle{tb,hI+/a\}b∇acket∇i}ht|/parenrightbig
+|Aw−1
+I|/radicalbig
+|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/a\}b∇acketle{tw/a\}b∇acket∇i}htI
+≤/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)/parenleftbig
+|/a\}b∇acketle{tb,hI−/a\}b∇acket∇i}ht|+|/a\}b∇acketle{tb,hI+/a\}b∇acket∇i}ht|/parenrightbig2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/a\}b∇acketle{tw/a\}b∇acket∇i}htI/parenrightbigg1/2/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)(Aw−1
+I)2|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/a\}b∇acketle{tw/a\}b∇acket∇i}htI/parenrightbigg1/2
+.
+Since /summationdisplay
+I∈D(|/a\}b∇acketle{tb,hI−/a\}b∇acket∇i}ht|+|/a\}b∇acketle{tb,hI+/a\}b∇acket∇i}ht|)2≤3/summationdisplay
+I∈D/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2, (6.11)
+1
+|J|/summationdisplay
+I∈D(J)(|/a\}b∇acketle{tb,hI−/a\}b∇acket∇i}ht|+|/a\}b∇acketle{tb,hI+/a\}b∇acket∇i}ht|)2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/a\}b∇acketle{tw/a\}b∇acket∇i}htI≤C[w−1]Ad
+21
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2≤C[w−1]Ad
+2/ba∇dblb/ba∇dbl2
+BMOd,
+and by Lemma 4.8,
+1
+|J|/summationdisplay
+I∈D(J)(Aw−1
+I)2|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/a\}b∇acketle{tw/a\}b∇acket∇i}htI≤C[w−1]Ad
+2.
+Thus embedding condition (6.8) holds. For (6.9), by Cauchy- Schwarz inequality and (6.11) we have
+1
+|J|/summationdisplay
+I∈D(J)(|/a\}b∇acketle{tb,hI−/a\}b∇acket∇i}ht|+|/a\}b∇acketle{tb,hI+/a\}b∇acket∇i}ht|)|Aw−1
+I|/radicalbig
+|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI
+≤C/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/parenrightbigg1/2/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)(Aw−1
+I)2|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/parenrightbigg1/2
+.
+By Theorem 4.5,
+1
+|J|/summationdisplay
+I∈D(J)(Aw−1
+I)2|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI=1
+|J|/summationdisplay
+I∈D(J)/parenleftbigg/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI+−/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI−
+2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/parenrightbigg2
+|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI≤C[w−1]Ad
+2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ.
+Similarly with (6.7), we have
+1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI≤[w−1]A2/ba∇dblb/ba∇dbl2
+BMOd/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ.
+To finish, we must estimate (6.10). In a similar way with (6.9), we need to estimate
+/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw/a\}b∇acket∇i}htI/parenrightbigg1/2/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)(Aw−1
+I)2|I|/a\}b∇acketle{tw/a\}b∇acket∇i}htI/parenrightbigg1/2
+.
+17By Lemma 4.6, we have
+1
+|J|/summationdisplay
+I∈D(J)(Aw−1
+I)2|I|/a\}b∇acketle{tw/a\}b∇acket∇i}htI≤[w−1]Ad
+21
+|J|/summationdisplay
+I∈D(J)(Aw−1
+I)2|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht−1
+I
+= [w−1]Ad
+21
+|J|/summationdisplay
+I∈D(J)/parenleftbigg/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI+−/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI−
+/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht3
+I/parenrightbigg2
+|I|
+≤C[w−1]Ad
+2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ.
+This completes the proof of Lemma 6.1.
+Due to the almost self adjoint property of the Hilbert transf orm, a certain bound for π∗
+bH
+immediately returns the same bound for Hπb. However we have to prove the boundedness of Sπb
+independently because Sis not self adjoint.
+7 Linear bound for Sπb
+Lemma 7.1. Letw∈Ad
+2andb∈BMOd. Then, there exists Cso that
+/ba∇dblSπb/ba∇dblL2(w)→L2(w)≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd.
+Proof. We are going to prove Lemma 7.1 by showing
+/a\}b∇acketle{tSπb(w−1f),g/a\}b∇acket∇i}htw≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/ba∇dblf/ba∇dblL2(w−1)/ba∇dblg/ba∇dblL2(w), (7.1)
+for any positive function f,g∈L2.Since/a\}b∇acketle{tSπb(f),hI/a\}b∇acket∇i}ht=sgn(I)/a\}b∇acketle{tπb(f),hˆI/a\}b∇acket∇i}ht=sgn(I)/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht,We
+have
+Sπb(f) =/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}hthI.
+By expanding gin the disbalanced Haar system for L2(w),
+/a\}b∇acketle{tSπb(w−1f),g/a\}b∇acket∇i}htw=/summationdisplay
+I∈D/a\}b∇acketle{tw−1f/a\}b∇acket∇i}htˆI/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}htsgn(I)/a\}b∇acketle{thI,g/a\}b∇acket∇i}htw
+=/summationdisplay
+I∈D/summationdisplay
+J∈Dsgn(I)/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI,w−1/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htw/a\}b∇acketle{thI,hw
+J/a\}b∇acket∇i}htw.
+Since/a\}b∇acketle{thI,hw
+J/a\}b∇acket∇i}htwcould be non zero only if J⊇I,we can split above sum into three parts,
+/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI,w−1/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht/a\}b∇acketle{tg,hw
+I/a\}b∇acket∇i}htw/a\}b∇acketle{thI,hw
+I/a\}b∇acket∇i}htw, (7.2)
+/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI,w−1/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht/a\}b∇acketle{tg,hw
+ˆI/a\}b∇acket∇i}htw/a\}b∇acketle{thI,hw
+ˆI/a\}b∇acket∇i}htw, (7.3)
+and /summationdisplay
+I∈D/summationdisplay
+J:J/supersetnoteqlˆIsgn(I)/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI,w−1/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htw/a\}b∇acketle{thI,hw
+J/a\}b∇acket∇i}htw. (7.4)
+18We claim that all sums, (7.2), (7.3), and (7.4), can be bounde d with a bound that depends on
+[w]Ad
+2/ba∇dblb/ba∇dblBMOdat most linearly. Since |/a\}b∇acketle{thI,hw
+I/a\}b∇acket∇i}htw| ≤ /a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+I,we can estimate (7.2)
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI,w−1/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht/a\}b∇acketle{tg,hw
+I/a\}b∇acket∇i}htw/a\}b∇acketle{thI,hw
+I/a\}b∇acket∇i}htw/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤/parenleftbigg/summationdisplay
+I∈D/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht2
+ˆI/a\}b∇acketle{tf/a\}b∇acket∇i}ht2
+ˆI,w−1/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw/a\}b∇acket∇i}htI/parenrightbigg1/2/parenleftbigg/summationdisplay
+I∈D/a\}b∇acketle{tg,hw
+I/a\}b∇acket∇i}ht2/parenrightbigg1/2
+≤C/ba∇dblg/ba∇dblL2(w)[w]1/2
+Ad
+2/parenleftbigg/summationdisplay
+I∈D/a\}b∇acketle{tf/a\}b∇acket∇i}ht2
+I,w−1/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/parenrightbigg1/2
+.
+By Weighted Carleson Embedding Theorem 4.4,
+/summationdisplay
+I∈D/a\}b∇acketle{tf/a\}b∇acket∇i}ht2
+I,w−1/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI≤C[w]Ad
+2/ba∇dblb/ba∇dbl2
+BMOd/ba∇dblf/ba∇dbl2
+L2(w−1)
+is provided by
+1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI≤[w]Ad
+2/ba∇dblb/ba∇dbl2
+BMOd/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ
+which we already have in (6.7). Thus, we have
+/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI,w−1/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht/a\}b∇acketle{tg,hw
+I/a\}b∇acket∇i}htw/a\}b∇acketle{thI,hw
+I/a\}b∇acket∇i}htw≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/ba∇dblf/ba∇dblL2(w−1)/ba∇dblg/ba∇dblL2(w).(7.5)
+Similarly to (7.2), we can estimate (7.3) using |/a\}b∇acketle{thI,hw
+ˆI/a\}b∇acket∇i}htw| ≤ /a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+I≤√
+2/a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+ˆI
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI,w−1/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht/a\}b∇acketle{tg,hw
+ˆI/a\}b∇acket∇i}htw/a\}b∇acketle{thI,hw
+ˆI/a\}b∇acket∇i}htw/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤√
+2/summationdisplay
+I∈D/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI,w−1|/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht|/a\}b∇acketle{tg,hw
+ˆI/a\}b∇acket∇i}htw/a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+ˆI
+= 2√
+2/summationdisplay
+I∈D/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/a\}b∇acketle{tf/a\}b∇acket∇i}htI,w−1|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|/a\}b∇acketle{tg,hw
+I/a\}b∇acket∇i}htw/a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+I
+≤2√
+2/parenleftbigg/summationdisplay
+I∈D/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht2
+I/a\}b∇acketle{tf/a\}b∇acket∇i}ht2
+I,w−1/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw/a\}b∇acket∇i}htI/parenrightbigg1/2/parenleftbigg/summationdisplay
+I∈D/a\}b∇acketle{tg,hw
+I/a\}b∇acket∇i}ht2/parenrightbigg1/2
+≤C/ba∇dblg/ba∇dblL2(w)[w]1/2
+Ad
+2/parenleftbigg/summationdisplay
+I∈D/a\}b∇acketle{tf/a\}b∇acket∇i}ht2
+I,w−1/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/parenrightbigg1/2
+≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/ba∇dblf/ba∇dblL2(w−1)/ba∇dblg/ba∇dblL2(w).
+Sincehw
+Jis constant on ˆI, forJ/supersetnoteqlˆIand we denote this constant by hw
+J(ˆI).Then we know by (2.7),
+/summationdisplay
+J:J/supersetnoteqlˆI/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htw/a\}b∇acketle{thI,hw
+J/a\}b∇acket∇i}htw=/summationdisplay
+J:J/supersetnoteqlˆI/a\}b∇acketle{tg,hw
+J/a\}b∇acket∇i}htwhw
+J(ˆI)/a\}b∇acketle{thI,w/a\}b∇acket∇i}ht=/a\}b∇acketle{tg/a\}b∇acket∇i}htˆI,w/a\}b∇acketle{thI,w/a\}b∇acket∇i}ht.
+19Thus, we can rewrite (7.4)
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+I∈Dsgn(I)/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI,w−1/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht/a\}b∇acketle{tg/a\}b∇acket∇i}htˆI,w/a\}b∇acketle{thI,w/a\}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤/summationdisplay
+I∈D/a\}b∇acketle{tw−1/a\}b∇acket∇i}htˆI/a\}b∇acketle{tf/a\}b∇acket∇i}htˆI,w−1|/a\}b∇acketle{tb,hˆI/a\}b∇acket∇i}ht|/a\}b∇acketle{tg/a\}b∇acket∇i}htˆI,w|/a\}b∇acketle{thI,w/a\}b∇acket∇i}ht| (7.6)
+=/summationdisplay
+I∈D/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|(|/a\}b∇acketle{thI−,w/a\}b∇acket∇i}ht|+|/a\}b∇acketle{thI+,w/a\}b∇acket∇i}ht|)/a\}b∇acketle{tf/a\}b∇acket∇i}htI,w−1/a\}b∇acketle{tg/a\}b∇acket∇i}htI,w. (7.7)
+We claim the sum (7.7) is bounded by [w]Ad
+2/ba∇dblb/ba∇dblBMOd/ba∇dblf/ba∇dblL2(w−1)/ba∇dblg/ba∇dblL2(w).We are going to prove
+it using Petermichl’s Bilinear Embedding Theorem 4.1. Thus, we need to show that the following
+three embedding conditions hold,
+∀J∈ D,1
+|J|/summationdisplay
+I∈D(J)|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI(|/a\}b∇acketle{thI−,w/a\}b∇acket∇i}ht|+|/a\}b∇acketle{thI+,w/a\}b∇acket∇i}ht|)1
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ,(7.8)
+∀J∈ D,1
+|J|/summationdisplay
+I∈D(J)|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI(|/a\}b∇acketle{thI−,w/a\}b∇acket∇i}ht|+|/a\}b∇acketle{thI+,w/a\}b∇acket∇i}ht|)1
+/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/a\}b∇acketle{tw/a\}b∇acket∇i}htJ,(7.9)
+∀J∈ D,1
+|J|/summationdisplay
+I∈D(J)|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI(|/a\}b∇acketle{thI−,w/a\}b∇acket∇i}ht|+|/a\}b∇acketle{thI+,w/a\}b∇acket∇i}ht|)≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd. (7.10)
+After we split the sum in (7.8):
+1
+|J|/summationdisplay
+I∈D(J)|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI|/a\}b∇acketle{thI−,w/a\}b∇acket∇i}ht|1
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI+1
+|J|/summationdisplay
+I∈D(J)|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI|/a\}b∇acketle{thI+,w/a\}b∇acket∇i}ht|1
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI,
+we start with Cauchy-Schwarz inequality to estimate the firs t sum of embedding condition (7.8),
+1
+|J|/summationdisplay
+I∈D(J)|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI|/a\}b∇acketle{thI−,w/a\}b∇acket∇i}ht|1
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI=1
+|J|/summationdisplay
+I∈D(J)|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/radicalbig
+|I−||∆I−w|
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI
+≤/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht2
+I/a\}b∇acketle{tw/a\}b∇acket∇i}htI/parenrightbigg1/2/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)|I−||∆I−w|21
+/a\}b∇acketle{tw/a\}b∇acket∇i}ht3
+I/parenrightbigg1/2
+≤C[w]1/2
+A2/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht/parenrightbigg1/2/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)|I−||∆I−w|21
+/a\}b∇acketle{tw/a\}b∇acket∇i}ht3
+I−/parenrightbigg1/2
+≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ. (7.11)
+Inequality (7.11) due to Lemma 4.6 and (6.7). Also, the other sum can be estimated by exactly the
+same method. Thus we have the embedding condition (7.8). To s ee the embedding condition (7.9),
+it is enough to show
+1
+|J|/summationdisplay
+I∈D(J)|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{thI−,w/a\}b∇acket∇i}ht| ≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/a\}b∇acketle{tw/a\}b∇acket∇i}htJ,
+20as we did above. We use Cauchy-Schwarz inequality for embedd ing condition (7.9), then
+1
+|J|/summationdisplay
+I∈D(J)|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{thI−,w/a\}b∇acket∇i}ht|=1
+|J|/summationdisplay
+I∈D(J)|/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht|/radicalbig
+|I−||∆I−w|
+≤/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht21
+/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/parenrightbigg1/2/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)|I−||∆I−w|2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/parenrightbigg1/2
+≤C/ba∇dblb/ba∇dblBMOd/a\}b∇acketle{tw/a\}b∇acket∇i}ht1/2
+J/parenleftbigg[w]A2
+|J|/summationdisplay
+I∈D(J)|I−||∆I−w|21
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI−/parenrightbigg1/2
+(7.12)
+≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/a\}b∇acketle{tw/a\}b∇acket∇i}htJ. (7.13)
+Here inequality (7.12) uses Lemma 4.7, and inequality (7.13 ) uses the fact that /a\}b∇acketle{tw/a\}b∇acket∇i}ht−1
+I≤2/a\}b∇acketle{tw/a\}b∇acket∇i}ht−1
+I−
+and Theorem 4.5 after shifting the indices.
+If we show the embedding condition (7.10), then we can immedi ately finish the estimate for (7.7)
+with bound C[w]Ad
+2/ba∇dblb/ba∇dblBMOd/ba∇dblf/ba∇dblL2(w−1)/ba∇dblg/ba∇dblL2(w).Combining this and (7.5) will give us our desire
+result.
+8 Proof for embedding condition (7.10)
+The following lemma lies at the heart of the matter for the pro of of the embedding condition (7.10).
+Lemma 8.1. There is a positive constant Cso that for all dyadic interval J∈ D
+1
+|J|/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw/a\}b∇acket∇i}ht1/4
+I/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/4
+I/parenleftbigg|∆I+w|+|∆I−w|
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI/parenrightbigg2
+≤C/a\}b∇acketle{tw/a\}b∇acket∇i}ht1/4
+J/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/4
+J, (8.1)
+wheneverwis a weight. Moreover, if w∈Ad
+2then for all J∈ D
+1
+|J|/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw/a\}b∇acket∇i}htI/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/parenleftbigg|∆I+w|+|∆I−w|
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI/parenrightbigg2
+≤C[w]Ad
+2.
+Proof of condition (7.10). By using Cauchy-Schwarz inequality and Lemma 8.1, we have:
+1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI(|/a\}b∇acketle{thI−,w/a\}b∇acket∇i}ht|+|/a\}b∇acketle{thI+,w/a\}b∇acket∇i}ht|)
+=1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/radicalbigg
+|I|
+2/parenleftbig
+|/a\}b∇acketle{tw/a\}b∇acket∇i}htI−+−/a\}b∇acketle{tw/a\}b∇acket∇i}htI−−|+|/a\}b∇acketle{tw/a\}b∇acket∇i}htI++−/a\}b∇acketle{tw/a\}b∇acket∇i}htI+−|/parenrightbig
+≤1√
+2/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)/a\}b∇acketle{tb,hI/a\}b∇acket∇i}ht2/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/a\}b∇acketle{tw/a\}b∇acket∇i}htI/parenrightbigg1/2/parenleftbigg1
+|J|/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI/a\}b∇acketle{tw/a\}b∇acket∇i}ht−1
+I(|∆I+w|+|∆I−w|)2/parenrightbigg1/2
+≤C[w]Ad
+2/ba∇dblb/ba∇dblBMOd.
+21We turn to the proof of Lemma 8.1. In the first place, we need to r evisit some properties of
+functionB(u,v) :=4√uvon the domain D0which is given by
+{(u,v)∈R2
++:uv≥1/2}.
+It is known, we refer to [2], that B(u,v)satisfies the following differential inequality in D0
+−(du,dv)d2B(u,v)(du,dv)t≥1
+8v1/4
+u7/4|du|2. (8.2)
+Furthermore, this implies the following convexity conditi on. For all (u,v),(u±,v±)∈D0,
+B(u,v)−B(u+,v+)+B(u−,v−)
+2≥C1v1/4
+u7/4(u+−u−)2, (8.3)
+whereu= (u++u−)/2andv= (v++v−)/2.Let us define
+A(u,v,∆u) :=aB(u,v)+B(u+∆u,v)+B(u−∆u,v),
+on the domain D1with some positive constant a >0.Here(u,v,∆u)∈D1means all pairs
+(u,v),(u+∆u,v),(u−∆u,v)∈D0.ThenAhas the size property and the convexity property:
+if(u,v,∆u)∈D1,then0≤A(u,v,∆u)≤(a+2)4√uv, (8.4)
+and
+A(u,v,∆u)−1
+2/bracketleftbig
+A(u+,v+,∆u1)+A(u−,v−,∆u2)/bracketrightbig
+≥C2v1/4
+u7/4(∆u2
+1+∆u2
+2), (8.5)
+whereu= (u++u−)/2,v= (v++v−)/2,and∆u= (u+−u+)/2.The property (8.5) is directly
+from the definition of function B(u,v).At the end, ∆uwill play the role of ∆Iw,∆u1is∆I+w,
+and∆u2is∆I−w.We can rewrite left hand side of the inequality (8.5) as follo ws
+A(u,v,∆u)−1
+2/bracketleftbig
+A(u+,v+,∆u1)+A(u−,v−,∆u2)/bracketrightbig
+=aB(u,v)+B(u+∆u,v)+B(u−∆u,v)−1
+2/bracketleftbig
+aB(u+,v+)+B(u++∆u1,v+)+B(u+−∆u1,v+)
++aB(u−,v−)+B(u−+∆u1,∆u−)+B(u−−∆u1,v−)/bracketrightbig
+=aB(u,v)−a
+2(B(u+,v+)+B(u−,v−))+B(u+,v)+B(u−,v)−1
+2/bracketleftBig
+B(u+∆u+∆u1,v+∆v)
++B(u+∆u−∆u1,v+∆v)+B(u−∆u+∆u2,v−∆v)+B(u−∆u−∆u2,v−∆v)/bracketrightBig
+.
+(8.6)
+Using Taylor’s theorem:
+B(u+u0,v+v0) =B(u,v)+∇B(u,v)(u0,v0)t+/integraldisplay1
+0(1−s)(u0,v0)d2B(u+su0,v+sv0)(u0,v0)tds,
+22and the convexity condition of B(u,v), we are going to estimate the lower bounds of (8.6).
+−1
+2B(u+∆u+∆u1,v+∆v)
+=−1
+2/parenleftBig
+B(u,v)+∇B(u,v)(∆u+∆u1,∆v)t/parenrightBig
+−/integraldisplay1
+0(1−s)(∆u+∆u1,∆v)d2B(u+s(∆u+∆u1),v+s∆v)(∆u+∆u1,∆v)tds
+≥ −1
+2/parenleftBig
+B(u,v)+∇B(u,v)(∆u+∆u1,∆v)t/parenrightBig
++1
+8/integraldisplay1
+0(1−s)(v+s∆v)1/4
+(u+s(∆u+∆u1))7/4(∆u+∆u1)2ds
+≥ −1
+2/parenleftBig
+B(u,v)+∇B(u,v)(∆u+∆u1,∆v)t/parenrightBig
++(∆u+∆u1)2
+8(4u)7/4/integraldisplay1
+0(1−s)(v+s∆v)1/4ds (8.7)
+≥ −1
+2/parenleftBig
+B(u,v)+∇B(u,v)(∆u+∆u1,∆v)t/parenrightBig
++(∆u+∆u1)2v1/4
+8(4u)7/4/integraldisplay1
+0(1−s)(1+s∆v
+v)1/4ds
+≥ −1
+2/parenleftBig
+B(u,v)+∇B(u,v)(∆u+∆u1,∆v)t/parenrightBig
++1
+72·43/4v1/4
+u7/4(∆u+∆u1)2. (8.8)
+Inequality (8.7) is due to the following inequalities
+|∆u|=|u+−u−|
+2≤|u++u−|
+2=uand|∆u1|=|u++−u+−|
+2≤u+≤2u.
+Since(1−s)1/4≤(1−|β|s)1/4≤(1+βs)1/4for any|β|<1, it is clear that
+/integraldisplay1
+0(1−s)(1+βs)1/4ds≥/integraldisplay1
+0(1−s)5/4ds=4
+9,
+and this allows the inequality (8.8). With the same argument s, we also estimate the following lower
+bounds:
+−1
+2/bracketleftBig
+B(u+∆u−∆u1,v+∆v)+B(u−∆u+∆u2,v−∆v)+B(u−∆u−∆u2,v−∆v)/bracketrightBig
+≥ −1
+2/parenleftBig
+B(u,v)+∇B(u,v)(∆u−∆u1,∆v)t/parenrightBig
++1
+72·43/4v1/4
+u7/4(∆u−∆u1)2
+−1
+2/parenleftBig
+B(u,v)+∇B(u,v)(−∆u+∆u2,−∆v)t/parenrightBig
++1
+72·43/4v1/4
+u7/4(−∆u+∆u2)2
+−1
+2/parenleftBig
+B(u,v)+∇B(u,v)(−∆u−∆u2,−∆v)t/parenrightBig
++1
+72·43/4v1/4
+u7/4(∆u+∆u2)2. (8.9)
+We can have the following inequality by combining (8.8), (8. 9) and (8.6),
+A(u,v,∆u)−1
+2/bracketleftbig
+A(u+,v+,∆u1)+A(u−,v−,∆u2)/bracketrightbig
+≥(a−2)B(u,v)−a
+2(B(u+,v+)+B(u−,v−))+B(u+,v)+B(u−,v)
++1
+36·43/4v1/4
+u7/4(2∆u2+∆u2
+1+∆u2
+2)
+≥1
+36·43/4v1/4
+u7/4(∆u2
+1+∆u2
+2). (8.10)
+23To see the inequality (8.10), using convexity condition of B(u,v) =4√uvand inequality: (1−s)u≤
+u−s∆u≤u+s∆u,
+(a−2)B(u,v)−a
+2/parenleftBig
+B(u+,v+)+B(u−,v−)/parenrightBig
++B(u+,v)+B(u−,v)
+=a/parenleftbigg
+B(u,v)−1
+2(B(u+,v+)+B(u−,v−)/parenrightbigg
+−/parenleftbigg3
+16∆u2/integraldisplay1
+0(1−s)v1/4(u+s∆u)−7/4ds+3
+16∆u2/integraldisplay1
+0(1−s)v1/4(u−s∆u)−7/4ds/parenrightbigg
+≥aC1v1/4
+u7/4∆u2−6
+16∆u2v1/4
+u7/4/integraldisplay1
+0(1−s)−3/4ds
+=aC1v1/4
+u7/4∆u2−3
+2∆u2v1/4
+u7/4
+=/parenleftbigg
+aC1−3
+2/parenrightbiggv1/4
+u7/4∆u2. (8.11)
+Choosing a constant asufficiently large so that aC1>3/2, quantity in (8.11) remains positive.
+This observation and discarding nonnegative terms yield in equality (8.10). Choosing the constant
+C2= 1/(36·43/4)in (8.5) completes the proof of the concavity property of A(u,v,∆u).We now
+turn to the proof of Lemma 8.1.
+Proof of Lemma 8.1. LetuI:=/a\}b∇acketle{tw/a\}b∇acket∇i}htI, vI:=/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI, u±=uI±, v±=vI±,∆uI= ∆Iw,∆u1=
+∆uI+,and∆u2= ∆uI−.Then by Hölder’s inequality (u,v,∆u),(u+,v+,∆u1),and(u−,v−,∆u2)
+belong to D1.FixJ∈ D,by properties (8.4) and (8.5)
+(a+2)|J|4/radicalbig
+/a\}b∇acketle{tw/a\}b∇acket∇i}htJ/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ≥ |J|A(uJ,vJ,∆uJ)
+≥1
+2/parenleftbigg
+|J+|A(u+,v+,∆u1)+|J−|A(u−,v−,∆u2)/parenrightbigg
++|J|C/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ
+/a\}b∇acketle{tw/a\}b∇acket∇i}ht7/4
+J(|∆J+u|2+|∆J−u|2).
+SinceA(u,v,∆u)≥0, iterating the above process will yield
+|J|4/radicalbig
+/a\}b∇acketle{tw/a\}b∇acket∇i}htJ/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ≥C/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/4
+I/a\}b∇acketle{tw/a\}b∇acket∇i}ht−7/4
+I(|∆I+w|2+|∆I−w|2). (8.12)
+Also, one can easily have
+|J|4/radicalbig
+/a\}b∇acketle{tw/a\}b∇acket∇i}htJ/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ≥C/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/4
+I/a\}b∇acketle{tw/a\}b∇acket∇i}ht−7/4
+I∆I+w2, (8.13)
+and
+|J|4/radicalbig
+/a\}b∇acketle{tw/a\}b∇acket∇i}htJ/a\}b∇acketle{tw−1/a\}b∇acket∇i}htJ≥C/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/4
+I/a\}b∇acketle{tw/a\}b∇acket∇i}ht−7/4
+I∆I−w2. (8.14)
+24Then,
+1
+|J|/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/4
+I/a\}b∇acketle{tw/a\}b∇acket∇i}ht−7/4
+I(|∆I+w|+|∆I−w|)2
+=1
+|J|/parenleftbigg/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/4
+I/a\}b∇acketle{tw/a\}b∇acket∇i}ht−7/4
+I(|∆I+w|2
++|∆I−w|2)+2/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/4
+I/a\}b∇acketle{tw/a\}b∇acket∇i}ht−7/4
+I(|∆I+w||∆I−w|)/parenrightbigg
+≤1
+|J|/parenleftBigg/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/4
+I/a\}b∇acketle{tw/a\}b∇acket∇i}ht−7/4
+I(|∆I+w|2+|∆I−w|2)
++2/parenleftbigg/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/4
+I/a\}b∇acketle{tw/a\}b∇acket∇i}ht−7/4
+I∆I+w2/parenrightbigg1/2/parenleftbigg/summationdisplay
+I∈D(J)|I|/a\}b∇acketle{tw−1/a\}b∇acket∇i}ht1/4
+I/a\}b∇acketle{tw/a\}b∇acket∇i}ht−7/4
+I∆I−w2/parenrightbigg1/2/parenrightBigg
+≤3
+C4/radicalbig
+/a\}b∇acketle{tw/a\}b∇acket∇i}htI/a\}b∇acketle{tw−1/a\}b∇acket∇i}htI.
+9 Recently developed tools and their applications
+The dyadic shift operator was first introduced in [20] to repl ace the weighted norm estimate for
+the Hilbert transform. It was also encountered in [29], so Ri esz transforms can be obtained as the
+result of averaging some dyadic shift operator. Recently, i n [17] and [7], a more general class of
+dyadic shift operators, so called the Haar shift operators w ere introduced. The Hilbert transform,
+Riesz transforms, and Beurling-Ahlfors operator are in the c onvex hull of this class, as they can
+be written as appropriate averages of Haar shift operators. LetDndenote the collection of dyadic
+cubes in Rn,Dn(Q)denotes dyadic subcubes of Q,and|Q|denotes the volume of the dyadic cube
+Q.We start with some definitions.
+Definition 9.1. A Haar function on a cube Q⊂Rnis a function HQsuch that
+(a)HQis supported on Q, and is constant on Dn(Q).
+(b)/ba∇dblHQ/ba∇dbl∞≤ |Q|−1/2.
+(c)HQhas a mean zero.
+Definition 9.2. Given an integer τ >0,we say an operator of the following form is in the first
+class of Haar shift operators of index τ
+Tτf(x) =/summationdisplay
+Q∈Dn/summationdisplay
+Q′,Q′′∈D(Q)
+2−τn|Q|≤|Q′|,|Q′′|aQ′,Q′′/a\}b∇acketle{tf,HQ′/a\}b∇acket∇i}htHQ′′(x),
+where the constant aQ′,Q′′satisfy the following size condition:
+|aQ′,Q′′| ≤C/parenleftbigg|Q′|
+|Q|·|Q′′|
+|Q|/parenrightbigg1/2
+. (9.1)
+25Note that once a choice of Haar functions has been made {HQ}Q∈Dn,then this is an orthogonal
+family, such that /ba∇dblHQ/ba∇dblL2≤1,so one could normalize in L2.Note that one can easily see that the
+dyadic shift operator Sbelongs to the first class of a Haar shift operator of index τ= 1with
+aI′,I′′=/braceleftbigg±1forI′=I, I′′=I∓
+0otherwise.
+One of the main result in [17] and [7] is the following
+Theorem 9.3 ([17], [7]) .LetTbe in the first class of Haar shift operators of index τ.Then for all
+w∈Ad
+2,there exists C(τ,n)which only depends on τandnsuch that
+/ba∇dblT/ba∇dblL2(w)→L2(w)≤C(τ,n)[w]Ad
+2.
+As a consequence of this Theorem, linear bounds for the Hilbe rt transform, Riesz transforms,
+and the Beurling-Ahlfors operator are recovered. There are n ow two different proofs of Theorem
+(9.3) in [17] and [7]. The commutator [λb,S]is also in the first class of Haar shift operators of index
+τ= 1.Recall the observation in the section 4,
+[λb,S](f) =−/summationdisplay
+I∈D∆Ib/a\}b∇acketle{tf,hI/a\}b∇acket∇i}ht(hI++hI−).
+Then we can see
+aI′,I′′=/braceleftbigg−∆IbforI′=I, I′′=I±
+0otherwise,
+moreover |aI′,I′′|=|∆Ib| ≤2/ba∇dblb/ba∇dblBMOdthis means the constant aI′,I′′satisfy the size condition (9.1)
+withC= 2√
+2/ba∇dblb/ba∇dblBMOd.These observations, Theorem 5.1, and Theorem 9.3 immediate ly recover
+the quadratic bound for the commutator of the Hilbert transf orm which was proved in this paper.
+We now define the second class of Haar shift operators of index τ.
+Definition 9.4. Given an integer τ >0,we say an operator Tof the form in Definition 9.2 is in the
+second class of Haar shift operators of index τ,ifTis bounded on L2and the function HQsatisfy
+the condition (a) and (b) in Definition 9.2.
+The second class of Haar shift operators is more general than the first class. One can easily
+observe that the operators πb,Sπbandπ∗
+bSdo not satisfy the condition (c) on Definition 9.1, however
+these operators satisfy the conditions of Definition 9.4. No te that the n-variable paraproduct is a
+sum of2n−1of the second class of Haar shift operator of index 1, the rest rictedn-variable dyadic
+paraproduct
+πbf=/summationdisplay
+Q∈Dn/a\}b∇acketle{tf/a\}b∇acket∇i}htQ/a\}b∇acketle{tb,HQ/a\}b∇acket∇i}htHQ.
+In [14], the linear estimate for the maximal truncations of t hese operators is presented. This also
+recovers our linear bound estimates for Sπbandπ∗
+bS. On the other hand, authors in [7] also
+reproduce the linear estimate for the dyadic paraproduct wi th different technique.
+Lemma 9.5. LetTτa Haar shift operator of the first class, then [λb,Tτ]is an operator of the same
+class.
+Proof. We are going to use the restricted multi-variable λboperator which is
+λbf=/summationdisplay
+Q∈Dn/a\}b∇acketle{tb/a\}b∇acket∇i}htQ/a\}b∇acketle{tf,HQ/a\}b∇acket∇i}htHQ.
+26One can get the n-variableλboperator by summing over 2n−1of restricted λboperator. Observe
+that,
+[λb,Tτ]f=λb(Tτf)−Tτ(λbf)
+=/summationdisplay
+Q∈Dn/summationdisplay
+Q′,Q′′∈D(Q)
+2−τn|Q|≤|Q′|,|Q′′|aQ′,Q′′/a\}b∇acketle{tb/a\}b∇acket∇i}htQ′′/a\}b∇acketle{tf,HQ′/a\}b∇acket∇i}htHQ′′−/summationdisplay
+Q∈Dn/summationdisplay
+Q′,Q′′∈D(Q)
+2−τn|Q|≤|Q′|,|Q′′|aQ′,Q′′/a\}b∇acketle{tb/a\}b∇acket∇i}htQ′/a\}b∇acketle{tf,HQ′/a\}b∇acket∇i}htHQ′′
+=/summationdisplay
+Q∈Dn/summationdisplay
+Q′,Q′′∈D(Q)
+2−τn|Q|≤|Q′|,|Q′′|aQ′,Q′′(/a\}b∇acketle{tb/a\}b∇acket∇i}htQ′′−/a\}b∇acketle{tb/a\}b∇acket∇i}htQ′)/a\}b∇acketle{tf,HQ′/a\}b∇acket∇i}htHQ′′.
+Since/vextendsingle/vextendsingleaQ′,Q′′(/a\}b∇acketle{tb/a\}b∇acket∇i}htQ′′−/a\}b∇acketle{tb/a\}b∇acket∇i}htQ′)/vextendsingle/vextendsingle≤ /ba∇dblb/ba∇dblBMO|aQ′,Q′′|,
+[λb,Tτ]remains in the same class of Tτ.
+Theorem 9.3 and Lemma 9.5 allow to extend our result to more ge neral class of commutators
+including the Riesz transforms and the Beurling-Ahlfors ope rator as in Theorem 1.3.
+10 Sharp bounds
+In this section, we start proving that the quadratic estimat e in Theorem 1.5 is sharp, by showing
+an example which returns quadratic bound. This example was d iscovered by C. Peréz [22] who
+is kindly allowing us to reproduce it in this paper. The same c alculations show that the bounds
+in Theorem 1.1 are also sharp for p/\e}atio\slash= 2and1< p <∞.Variations over this example will then
+show that the bounds in Theorem 1.3 are sharp for the Riesz tra nsforms and the Beurling-Ahlfors
+operator as well.
+10.1 The Hilbert transform
+Consider the weight, for 0<δ<1:
+w(x) =|x|1−δ.
+It is well known that wis anA2weight and
+[w]A2∼1
+δ.
+We now consider the function f(x) =x−1+δχ(0,1)(x)and BMO function b(x) = log|x|.We claim
+that
+/ba∇dbl[b,H]f(x)| ≥1
+δ2f(x).
+For0<x<1,we have
+[b,H]f(x) =/integraldisplay1
+0logx−logy
+x−yy−1+δdy=/integraldisplay1
+0log(x/y)
+x−yy−1+δdy
+=x−1+δ/integraldisplay1/x
+0log/parenleftBig
+1
+t/parenrightBig
+1−tt−1+δdt.
+27Now,/integraldisplay1/x
+0log(1/t)
+1−tt−1+δdt=/integraldisplay1
+0log(1/t)
+1−tt−1+δdt+/integraldisplay1/x
+1log(1/t)
+1−tt−1+δdt,
+and sincelog(1/t)
+1−tis positive for (0,1)∪(1,∞)we have for 0<x<1
+|[b,H]f(x)|>x−1+δ/integraldisplay1
+0log(1/t)
+1−tt−1+δdt. (10.1)
+But since /integraldisplay1
+0log(1/t)
+1−tt−1+δdt>/integraldisplay1
+0log(1/t)t−1+δdt=/integraldisplay∞
+0se−sδds=1
+δ2, (10.2)
+our claim follows and
+/ba∇dbl[b,H]f/ba∇dblL2(w)≥1
+δ2/ba∇dblf/ba∇dblL2(w)∼[w]2
+A2/ba∇dblf/ba∇dblL2(w).
+A first approximation of what the bounds in Lp(w)is given by an application of the sharp extrap-
+olation theorem for the upper bound, paired with the knowled ge of the sharp bound on L2(w)to
+obtain a lower bound.
+Proposition 10.1. For1<p<∞there exist constants candConly depending on psuch that
+c[w]2min{1,1
+p−1}
+Ap/ba∇dblb/ba∇dblBMO≤ /ba∇dbl[b,H]/ba∇dblLp(w)→Lp(w)≤C[w]2max{1,1
+p−1}
+Ap/ba∇dblb/ba∇dblBMO, (10.3)
+for allb∈BMO.
+Proof. Because the upper bound in (10.3) is the direct consequence of the quadratic bound in the
+Theorem 1.5 and sharp extrapolation theorem, we will only pr ove the lower bound. Let us assume
+that, for 1<r<2andα<1,
+/ba∇dbl[b,H]/ba∇dblLr(w)→Lr(w)≤C[w]2α
+Ar/ba∇dblb/ba∇dblBMO.
+This and the sharp extrapolation theorem return
+/ba∇dbl[b,H]/ba∇dblL2(w)→L2(w)≤C[w]2α
+A2/ba∇dblb/ba∇dblBMO.
+This contradicts to the sharpness (p= 2). Similarly, one can conclude for p>2.
+We now consider the weight w(x) =|x|(1−δ)(p−1)thenwis anApweight with [w]Ap∼δ1−p.By
+(10.1) and (10.2) we have
+/ba∇dbl[b,H]f/ba∇dblLp(w)≥1
+δ2/ba∇dblf/ba∇dblLp(w)= (δ1−p)2
+p−1/ba∇dblf/ba∇dblLp(w)∼[w]2
+p−1
+Ap/ba∇dblf/ba∇dblLp(w).
+This shows the upper bound in (10.3) is sharp for 1< p≤2.We use the duality argument to
+see the sharpness of the quadratic estimate for p >2.Note that the commutator is a self-adjoint
+operator:
+/a\}b∇acketle{tbH(f)−H(bf),g/a\}b∇acket∇i}ht=/a\}b∇acketle{tf,H∗(bg)/a\}b∇acket∇i}ht−/a\}b∇acketle{tf,bH∗(g)/a\}b∇acket∇i}ht=/a\}b∇acketle{tf,bH(g)−H(bg)/a\}b∇acket∇i}ht.
+Consider 1<p≤2and setu=w1−p′,then
+/ba∇dbl[b,H]/ba∇dblLp′(u)→Lp′(u)=/ba∇dbl[b,H]/ba∇dblLp′(w1−p′)→Lp′(w1−p′)=/ba∇dbl[b,H]∗/ba∇dblLp′(w1−p′)→Lp′(w1−p′)
+=/ba∇dbl[b,H]/ba∇dblLp(w)→Lp(w)≤C/ba∇dblb/ba∇dblBMOd[w]2
+1−p
+Ap(10.4)
+=C/ba∇dblb/ba∇dblBMO[w1−p′]2
+Ap′=C/ba∇dblb/ba∇dblBMO[u]2
+Ap′.
+Since the inequality in (10.4) is sharp, we can conclude that the result of Theorem 1.1 is also sharp
+forp>2.
+2810.2 Beurling-Ahlfors operator
+Recall the Beurling-Ahlfors operator Bis given by convolution with the distributional kernel p.v.1/z2:
+Bf(x,y) =p.v.1
+π/integraldisplay
+R2f(x−u,y−v)
+(u+iv)2dudv.
+Then the commutator of the Beurling-Ahlfors operator can be w ritten:
+[b,B]f(x,y) =p.v.1
+π/integraldisplay
+R2b(x,y)−b(s,t)
+((x−s)+i(y−t))2f(s,t)dsdt.
+It was observed, in [9], that the linear bound for the Beurling -Ahlfors operator is sharp in L2(w),
+with weights w(z) =|z|αand functions f(z) =|z|−αwhere|α|<2.Similarly, we consider weights
+w(z) =|z|2−δwhere0<δ<1. Note that w(z) =|z|2−δ:C→[0,∞)is aA2-weight with [w]A2∼
+δ−1.We also consider a BMO function b(x) = log|z|.LetE={(r,θ)|0< r <1,0< θ < π/ 2}
+andΩ ={(r,θ)|1<r <∞, π <θ < 3π/2}We are going to estimate |[b,B]f(z)|forz∈Ωwith a
+functionf(z) =|z|δ−2χE(z).Letz=x+iyandζ=s+ti.Then, forz∈Ω,
+|[b,B]f(z)|=1
+π/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+E(b(z)−b(ζ))f(ζ)
+(z−ζ)2dζ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=1
+π/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Eb(x,y)−b(s,t)
+((x−s)+i(y−t))2f(s,t)dsdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+=1
+π/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+E(log|z|−log|ζ|)|ζ|δ−2((x−s)2−(y−t)2)
+((x−s)2+(y−t)2)2dsdt
++i/integraldisplay
+E(log|z|−log|ζ|)|ζ|δ−2(2(x−s)(y−t))
+((x−s)2+(y−t)2)2dsdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
+Forz∈Ωandζ∈E,we have(x−s)(y−t)≥xyand by triangle inequality ((x−s)2+(y−t)2)2=
+|z−ζ|4≤(|z|+|ζ|)4.After neglecting the positive term (real part), we get
+|[b,B]f(z)|2≥4
+π2/parenleftbigg
+xy/integraldisplay
+Elog(|z|/|ζ|)|ζ|δ−2
+(|z|+|ζ|)4dsdt/parenrightbigg2
+=4
+π2/parenleftbigg
+xy/integraldisplayπ/2
+0/integraldisplay1
+0log(|z|/r)rδ−2r
+(|z|+r)4drdθ/parenrightbigg2
+=x2y2/parenleftbigg1
+|z|4/integraldisplay1
+0log(|z|/r)rδ−1
+(1+r/|z|)4dr/parenrightbigg2
+=x2y2/parenleftbigg1
+|z|4/integraldisplay1/|z|
+0log(1/t)(|z|t)δ−1
+(1+t)4|z|dt/parenrightbigg2
+=x2y2/parenleftbigg1
+|z|4−δ/integraldisplay1/|z|
+0log(1/t)tδ−1
+(1+t)4dt/parenrightbigg2
+.
+Since|z|/(|z|+1)≤1/(1+t),fort<1/|z|,we have
+|[b,B]f(z)|2≥x2y2/parenleftbigg1
+|z|−δ(|z|+1)4/integraldisplay1/|z|
+0log(1/t)tδ−1dt/parenrightbigg2
+=x2y2
+|z|−2δ(|z|+1)8/parenleftbigg|z|−δ(1+δlog|z|)
+δ2/parenrightbigg2
+=x2y2
+(|z|+1)8(1+δlog|z|)2
+δ4.
+29Then, we can estimate the L2(w)-norm as follows.
+/ba∇dbl[b,B]f/ba∇dbl2
+L2(w)≥1
+δ4/integraldisplay
+Ωx2y2(1+δlog|z|)2
+(|z|+1)8|z|2−δdxdy
+=1
+δ4/integraldisplay∞
+1/integraldisplay3π/2
+πr4cos2θsin2θ(1+δlogr)2
+(r+1)8r3−δdrdθ
+=π
+δ416/integraldisplay∞
+1r7−δ(1+δlogr)2
+(r+1)8dr≥π
+δ416/integraldisplay∞
+1r7−δ(1+δlogr)2
+(2r)8dr
+=π
+δ4212/integraldisplay∞
+1(1+δlogr)2r−1−δdr
+=π
+δ4212/parenleftbigg1
+δ+2δ
+δ2+2δ2
+δ3/parenrightbigg
+=5π
+212·1
+δ5.
+Combining with /ba∇dblf/ba∇dbl2
+L2(w)=π/2δ,we have that /ba∇dbl[b,B]f/ba∇dblL2(w)//ba∇dblf/ba∇dblL2(w)∼δ−2,which allows to
+conclude that the quadratic bound for the commutator with th e Beurling-Ahlfors operators is sharp
+inL2(w).Same calculations with weights w(z) =|z|(2−δ)(p−1)and functions f(z) =|z|(δ−2)(p−1)
+will provide the sharpness for 1<p≤2,and it is sufficient to conclude for all 1<p<∞because
+the Beurling-Ahlfors operator is essentially self adjoint o perator ( B∗=eiφB),so the commutator
+of the Beurling-Ahlfors operator is also self adjoint.
+10.3 Riesz transforms
+Consider weights w(x) =|x|n−δand functions f(x) =xδ−nχE(x)whereE={x|x∈(0,1)n∩
+B(0,1)},and a BMO function b(x) = log|x|.It was observed that |x|n−δis anA2-weight in Rn
+with[w]A2∼δ−1.We are going to estimate [b,Rj]fover the set Ω ={y∈B(0,1)c|yi<0for alli=
+1,2,...,n},whereRjstands for the j-th direction Riesz transform on Rnand is defined as follows:
+Rjf(x) =cnp.v./integraldisplay
+Rnyj
+|y|n+1f(x−y)dy,1≤j≤n,
+wherecn= Γ((n+1)/2)/π(n+1)/2.One can observe that, for all x∈Eand fixedy∈Ω,
+|yj−xj| ≥ |yj|and|y−x| ≤ |y|+|x|.
+Cndenotes a constant depending only on the dimension that may c hange from line to line. Then,
+|[b,Rj]f(y)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+E(yj−xj)(log|y|−log|x|)|x|δ−n
+|y−x|n+1dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥ |yj|/integraldisplay
+Elog(|y|/|x|)|x|δ−n
+(|y|+|x|)n+1dx
+≥ |yj|/integraldisplay
+E∩Sn−1/integraldisplay1
+0log(|y|/r)rδ−nrn−1
+(|y|+r)n+1drdσ=Cn|yj|/integraldisplay1/|y|
+0log(1/t)(t|y|)δ−1|y|
+(|y|+|y|t)n+1dt
+=Cn|yj|
+|y|n+1−δ/integraldisplay1/|y|
+0log(1/t)tδ−1
+(1+t)n+1dt≥Cn|yj|
+|y|n+1−δ/parenleftbigg|y|
+|y|+1/parenrightbiggn+1/integraldisplay1/|y|
+0log(1/t)tδ−1dt
+=Cn|yj|
+|y|−δ(|y|+1)n+1/parenleftbigg|y|−δ(1+δlog|y|)
+δ2/parenrightbigg
+=Cn|yj|
+(|y|+1)n+11+δlog|y|
+δ2.
+30We now can bound from below the L2(w)-norm as follows.
+/ba∇dbl[b,Rj]f(x)/ba∇dbl2
+L2(w)>Cn
+δ4/integraldisplay
+Ωy2
+j(1+δlog|y|)2
+(|y|+1)2n+2|y|n−δdy
+≥Cn
+δ4/integraldisplay
+Ω∩Sn−1/integraldisplay∞
+1γ2
+jr2(1+δlogr)2rn−δrn−1
+(r+1)2n+2drdσ(γ)
+≥Cn
+δ4/integraldisplay∞
+1(1+δlogr)2r2n−δ+1
+r2n+2dr
+=Cn
+δ4/integraldisplay∞
+1(1+δlogr)2r−δ−1dr=Cn
+δ5,
+which establishes sharpness for the commutator of the Riesz transform when p= 2.SinceR∗
+j=
+−Rj, one can easily check that the commutator of Riesz transform s are also self-adjoint operators.
+Furthermore, choosing weight w(x) =|x|(n−δ)(p−1),we will obtain the sharpness for 1<p<∞by
+the same argument we used in the case of the Hilbert transform .
+11 Acknowledgments
+This work is part of the author’s Ph.D dissertation [4]. The a uthor like to thank Professor Carlos
+Pérez for his comments and useful suggestions. Finally, the author is also very grateful to his
+graduate adviser María Cristina Pereyra for her suggestion s and helpful interaction.
+References
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+linear operators Studia Math. 104(2) (1993) 195-209.
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+Anal. 255 4 (2008), 994-1007.
+[3] S. Buckley, Summation condition on Weights, Michigan Math. J., 40(1) (1993) 153-170.
+[4] D. Chung, Ph.D Dissertation, University of New Mexico, (2010).
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+variables Ann. of Math. (2) 103 (1976), 611-635.
+[7] D. Cruz-Uribe, SFO, J. Martell and C. Pérez, Sharp weighted estimates for classical operators
+Preprint.
+[8] O. Dragicević, L. Grafakos, M. C. Pereyra, and S. Petermi chl,Extrapolation and sharp norm
+estimates for classcical operators on weighted Lebesgue sp acesPubl. Mat. 49 (2005), 73-91.
+[9] O. Dragicević and A. Volberg, Sharp estimate of the Ahlfor-Beurling operator via averagi ng
+Martingale transforms , Michigan Math. J. 51 (2003)
+[10] J. B. Garnett, Bounded analytic functions Acad. Press, NY, 1981.
+31[11] H. Helson and G. Szegö, A problem in prediction theory, Ann. Math. Pura. Appl. 51 (1960),
+107-138.
+[12] S. Hukovic, S. Treil, A. Volberg, The Bellman functions and the sharp weighted inequalities f or
+square functions Operator Theory : Advances and Applications, the volume in m emory of S.
+A. Vinogradov, v. 113, Birkhauser Verlag, (2000)
+[13] R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for conjugate function
+and the Hilbert transform Trans. Amer. Math. Soc. 176 (1973), 227-251.
+[14] T. Hytönen, M. Lacey, M. Reguera, and A. Vagharshakyan, Weak and Strong-type esti-
+mates for Haar Shift Operators: Sharp power on the Apcharacteristic (2009), available at
+http://arxiv.org/abs/0911.0713
+[15] M. Lacey, Haar shifts, Commutators, and Hankel operators CIMPA-UNESCO conference on
+Real Analysis and it Applications, La Falda Argentina, (200 8).
+[16] M. Lacey, S. Petermichl, J. Piper and B. Wick, Iterated Riesz Commutators: A simple proof of
+boundedness Proc. of El Escorial (2008).
+[17] M. Lacey, S. Petermichl and M. Reguera, SharpA2inequality for Haar shift operators Math.
+Ann. to appear.
+[18] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function Trans. Amer.
+Math. Soc. 165 (1972), 207-226.
+[19] F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for
+Haar multipliers J. Amer. Math. Soc. 12 (1999), 909-928.
+[20] C. Pérez, Endpoint estimates for commutators of singular integral op erators J. Funct. Anal.
+128 (1995), 163-185.
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+Littlewood maximal functions J. Fourier Anal. Appl. 3 (1997), 743-756.
+[22] C. Pérez, Personal communication (2009).
+[23] C. Pérez and G. Pradolini, Sharp weighted Endpoint Estimates for commutators of singu lar
+integrals Michigan Math. H. 49 (2001) 23-37.
+[24] M. C. Pereyra, Haar Multipliers meet Bellman Functions Revista Mat. Iber. 25 3 (2009), 799-
+840.
+[25] S. Petermichl, Dyadic shifts and a logarithmic estimate for Hankel operato rs with matrix symbol
+C. R. Acad. Sci. Paris 330 (2000), 455-460.
+[26] S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesg ue spaces in terms
+of the classical ApCharacteristic, Amer. J. of Math. 129 (2007) 1355-1375.
+[27] S. Petermichl, The sharp weighted bound for the Riesz transforms Proc. Amer. Math. Soc. 136
+(2008), 1237-1249.
+[28] S. Petermichl and S. Pott, An estimate for weighted Hilbert transform via square funct ions
+Trans. Amer. Math. Soc. 354 (2002), 1699-1703.
+32[29] S. Petermichl, S. Treil and A. Volberg, Why the Riesz transforms are averages of the dyadic
+shifts? Proc. of the 6th Inter. Conf. on Harmonic Analysis and Partia l Differential Equations
+(El Escorial, 2000) (2002) 209-228.
+[30] S. Petermichl and A. Volverg, Heating of the Ahlfors-Beurling operator: Weakly quasireg ular
+maps on the plane are quasiregular Duke Math J. 112 (2002), 281-305.
+[31] A. Vagharshakyan Recovering Singular Integrals from Haar Shifts (2009), available at
+http://arxiv.org/abs/0911.4968
+[32] J. Wittwer, A shap estimate on the norm of Martingale transform Math. Res. Lett. 7 (2000)
+1-12.
+Department of Mathematics and Statistics, 1 University of N ew Mexico, Albu-
+querque, NM 87131-0001 ,E-mail: dwchung@unm.edu and chdaewon@gmail.com
+33
\ No newline at end of file
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+arXiv:1001.0756v2  [gr-qc]  30 Jan 2011On particle collisionsin the gravitationalfield of the Kerr blackhole
+A.A. Griba,b, Yu.V.Pavlova,c
+aA.Friedmann Laboratory for Theoretical Physics, 30 /32 Griboedov can.,St.Petersburg 191023, Russia
+bTheoretical Physics and Astronomy Department, TheHerzen U niversity, 48, Moika, St.Petersburg 191186, Russia
+cInstitute ofMechanical Engineering, Russian Academy of Sc iences, 61 Bolshoy, V.O.,St.Petersburg 199178, Russia
+Abstract
+Scattering of particles in the gravitational field of Kerr bl ack holes is considered. It is shown that scattering energy o f particles in
+the centre of mass system can obtain very large values not onl y for extremal black holes but also for nonextremal ones exis ting
+in Nature. This can be used for explanation of still unresolv ed problem of the origin of ultrahigh energy cosmic rays obse rved in
+Augerexperiment. Extractionofenergyafterthe collision isinvestigated. Itisshownthatdueto thePenroseprocesst heenergyof
+theparticleescapingtheholeat infinitycanbelarge. Contr adictionsintheproblemofgettinghighenergeticparticle sescapingthe
+blackholeareresolved.
+Keywords: Black holes,Particle collisions,Cosmic rays
+1. Introduction
+In [1] we put the hypothesis that Active Galactic Nuclei
+(AGN) can be the sources of ultrahigh energy particles in cos -
+mic raysobserved recently by the AUGER group(see [2]) due
+to the processes of converting dark matter formed by super-
+heavy neutral particles into visible particles — quarks, le ptons
+(neutrinos), photons. Such processes as it was discussed pr e-
+viously in [3, 4] could lead to the origination of visible mat ter
+from the dark matter particles in the early Universe when the
+energy of particles was of the Grand Unification (GU) order
+(1013−1014GeV)i.e. attheendofinflationera.
+If AGN are rotating black holes then in [1] we discussed
+the idea that “This black hole acts as a cosmic supercollider
+in which superheavy particles of dark matter are accelerate d
+closetothehorizontotheGUenergiesandcanbescatteringi n
+collisions.” It was also shown that in Penrose process [5] da rk
+matter particle can decay on two particles, one with the nega -
+tiveenergy,theotherwiththepositiveoneandparticlesof very
+high energy of the GU order can escape the black hole. Then
+theseparticlesduetointeractionwithphotonsclosetothe black
+hole will loose energyanalogouslyupto the Greisen-Zatsep in-
+Kuzminlimit incosmology[6].
+Processes of converting dark matter particles into visible
+ones as well as collisions of dark matter particles close to t he
+horizonoftheblackholeintheAGNcanplayimportantrolein
+the solution of the problem of cusps as it was discussed in [7] .
+According to the hierarchical model the galaxies were forme d
+due to initial inhomogeneities of dark matter distribution . But
+after origination of rotating black holes at the centre of fu ture
+galaxies large part of the dark matter became redistributed so
+that it canbemainlypresentinthehaloofgalaxy[7].
+E-mailaddresses: andrei_grib@mail.ru (A.A.Grib),
+yuri.pavlov@mail.ru (Yu.V. Pavlov)The problem of the origin of ultrahigh energy cosmic rays
+(UHECR) is a major unsolved issue in astroparticle physics.
+Surely there can be di fferent mechanisms of getting UHECR
+from AGN. Our aim is full investigation of the role of pro-
+cesses of collisions and decays of elementary particles clo se
+to the horizonofthe Kerrblackholesin theAGN. Theseparti-
+clescanbesuperheavyparticlesofdarkmatterorusualprot ons,
+ironnuclei,etc. Inthelattercasesforusualparticlestog etlarge
+energyonemustconsidermultiplescattering.
+First calculations of the scattering of particles in the erg o-
+sphere of the rotating black hole, taking into account the Pe n-
+roseprocess,with theresult thatparticleswith highenerg ycan
+escape the black hole, were made in [8]. Recently in [9] it was
+shownthatfortherotatingblackhole(ifitisclosetothecr itical
+one)theenergyofscatteringisunlimited. Theresultof[9] was
+criticizedin[10,11]wheretheauthorsclaimedthatiftheb lack
+hole is not a critical rotating black hole so that its dimensi on-
+lessangularmomentum A/nequal1butA=0.998thentheenergyis
+limited.
+Inthispaperweshowthattheenergyofscatteringinthecen-
+treofmasssystemcanbestillunlimitedinthecasesofmulti ple
+scattering. InSection2,wecalculatethisenergy,reprodu cethe
+results of [9]–[11] and show that in some cases (multiple sca t-
+tering)theresultsof[10,11]onthelimitationsofthescat tering
+energyfornonextremalblackholesarenotvalid. InSection 2.1
+an evaluation of the time necessary for the freely falling pa rti-
+cletogettheunboundedlargeenergyincollisionwiththeot her
+particle in the centre of mass system is made. It is proved tha t
+in orderto have infinitely large energyof collision on the ho ri-
+zon infinitely large time from the beginning of free falling o f
+the particle on the rotating black hole is needed. Particle c olli-
+sionsinsidetherotatingblackholeareconsideredinSecti on3.
+In Section 4, we obtain the results for the extraction of the e n-
+ergy after collision in the field of the Kerr’s metric. It occu rsthat the Penrose process plays important role for getting la rger
+energies of particles at infinity. Our calculations show tha t the
+conclusion of [11] about the impossibility of getting at infi nity
+the energylargerthan the initial onein particlecollision sclose
+to theblackholeiswrong.
+Thesystem ofunits G=c=1isusedinthepaper.
+2. The energyofcollisionin the field ofblackholes
+Let us consider particles falling on the rotating chargeles s
+black hole. The Kerr’s metric of the rotating black hole in
+Boyer–Lindquistcoordinateshasthe form
+ds2=dt2−2Mr(dt−asin2θdϕ)2
+r2+a2cos2θ
+−(r2+a2cos2θ)/parenleftiggdr2
+∆+dθ2/parenrightigg
+−(r2+a2)sin2θdϕ2,(1)
+where
+∆=r2−2Mr+a2, (2)
+Misthemassoftheblackhole, J=aMisangularmomentum.
+In the case a=0 the metric (1) describes the static chargeless
+blackholeinSchwarzschildcoordinates. Theeventhorizon for
+the Kerr’sblackholecorrespondstothe value
+r=rH≡M+√
+M2−a2. (3)
+TheCauchyhorizonis
+r=rC≡M−√
+M2−a2. (4)
+Thesurfacecalled“thestaticlimit”isdefinedbytheexpres sion
+r=r0≡M+√
+M2−a2cos2θ. (5)
+The region of space-time between the horizon and the static
+limit isergosphere.
+For equatorial (θ=π/2) geodesics in Kerr’s metric (1) one
+obtains([13],§61)
+dt
+dτ=1
+∆/bracketleftigg/parenleftigg
+r2+a2+2Ma2
+r/parenrightigg
+ε−2Ma
+rL/bracketrightigg
+, (6)
+dϕ
+dτ=1
+∆/bracketleftigg2Ma
+rε+/parenleftigg
+1−2M
+r/parenrightigg
+L/bracketrightigg
+, (7)
+/parenleftiggdr
+dτ/parenrightigg2
+=ε2+2M
+r3(aε−L)2+a2ε2−L2
+r2−∆
+r2δ1,(8)
+whereδ1=1 for timelike geodesics ( δ1=0 for isotropic
+geodesics),τis the proper time of the moving particle, ε=
+const is the specific energy: the particle with rest mass mhas
+the energyεmin the gravitational field (1); Lm=const is the
+angular momentum of the particle relative to the axis orthog o-
+nal totheplaneofmovement.
+Let us find the energy Ec.m.in the centre of mass system of
+two colliding particles with the same rest min arbitrary gravi-
+tationalfield. It canbeobtainedfrom
+(Ec.m.,0,0,0)=mui
+(1)+mui
+(2), (9)whereui=dxi/ds. Taking the squared (9) and due to uiui=1
+oneobtains
+Ec.m.=m√
+2/radicalig
+1+ui
+(1)u(2)i. (10)
+The scalar product does not depend on the choice of the coor-
+dinate frame so (10) is valid in an arbitrary coordinate syst em
+andforarbitrarygravitationalfield.
+We denote x=r/M,xH=rH/M,xC=rC/M,A=a/M,
+ln=Ln/M. Apply the formula (10) for calculation of the en-
+ergyin thecentreofmassframeoftwo collidingparticleswi th
+angular momenta L1,L2, which are nonrelativistic at infinity
+(ε1=ε2=1) and are moving in Kerr’s metric. Using (1),
+(6)–(8)oneobtains[9]
+E2
+c.m.
+2m2=1
+x∆x/bracketleftigg
+2x2(x−1)+l1l2(2−x)
++2A2(x+1)−2A(l1+l2) (11)
+−/radicalig/parenleftig
+2x2+2(l1−A)2−l2
+1x/parenrightig/parenleftig
+2x2+2(l2−A)2−l2
+2x/parenrightig/bracketrightigg
+,
+where
+∆x=x2−2x+A2=(x−xH)(x−xC). (12)
+Tofindthelimit r→rHfortheblackholewithagivenangu-
+lar momentum Aone musttake in (11) x=xH+αwithα→0
+and do calculations up to the order α2. Taking into account
+A2=xHxC,xH+xC=2,afterresolutionofuncertaintiesinthe
+limitα→0oneobtains
+Ec.m.(r→rH)
+2m=/radicaligg
+1+(l1−l2)2
+2xC(l1−lH)(l2−lH),(13)
+where
+lH=2xH
+A=2
+A/parenleftig
+1+√
+1−A2/parenrightig
+. (14)
+In the limit r→rHfor the extremal black hole one obtains
+theexpression
+a=M⇒Ec.m.(r→rH)=√
+2m/radicalbigg
+l2−2
+l1−2+l1−2
+l2−2,(15)
+given first in [9], showing the unlimited increasing of the en -
+ergy of collision when the specific angular momentum of one
+of falling particles goes to the limiting possible value equ al to
+2Mnecessary for achieving the horizon of the extremal black
+hole.
+Formula(8)leadstolimitationsonthepossiblevaluesofth e
+angular momentum of falling particles: the massive particl e
+free falling in the black hole with dimensionless angular mo -
+mentumAbeingnonrelativisticatinfinity( ε=1)toachievethe
+horizon of the black hole must have angular momentum from
+theinterval
+−2/parenleftig
+1+√
+1+A/parenrightig
+=lL≤l≤lR=2/parenleftig
+1+√
+1−A/parenrightig
+.(16)
+Note that for the mentionedlimiting values the right hand si de
+oftheformula(8)iszerofor
+xR=2/parenleftig
+1+√
+1−A/parenrightig
+−A,xL=2/parenleftig
+1+√
+1+A/parenrightig
++A(17)
+2forl=lRandl=lLcorrespondingly.
+Thedependenceoftheenergyofcollision Ec.m.ontheradial
+coordinate for l1=lR,l2=lL(see (16)) is given on Fig. 1.
+Theboldfacelinecorrespondsto A=0.998,theordinarycurve
+1.1 1.2 1.3 1.4r/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtrH2.557.51012.51517.520Ec.m./Slash1m
+x1 x2
+Figure1: Thedependenceoftheenergyofparticlecollision onthecoordinate r.
+toA=0.9, the dotted to A=0.5, the line formed by points
+describes nonrotating black hole ( A=0). The fractures on
+dense line, denoted by dotted lines correspond to values x=
+xR(A)M/rH(A) (seexRin (17)) i.e. to zeroesof the expressions
+in rootsinformula(11).
+Puttingthelimitingvaluesofangularmomenta lL,lRintothe
+formula (13) one obtains the maximal values of the energy of
+collisionforparticlesfallingfrominfinity
+Emax
+c.m.(r→rH)=
+=2m
+4√
+1−A2/radicaltp/radicalvertex/radicalbt
+1−A2+/parenleftig
+1+√
+1+A+√
+1−A/parenrightig2
+1+√
+1−A2.(18)
+Thedependenceof Emax
+c.m.ontheangularmomentumoftheblack
+holeisshownonFig.2.
+0.2 0.4 0.6 0.8 1a/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtM481216Ec.m.max/Slash1m
+Figure 2: The dependence of the maximal energy of collision f or particles
+falling from infinity on the black hole angular momentum.
+For nonrotating black hole the maximal energy of collision
+dueto(18)is Emax
+c.m./m=2√
+5(firstfoundin[12]).
+ForA=1−ǫwithǫ→0formula(18)gives:
+Emax
+c.m.(r→rH)∼2/parenleftig
+21/4+2−1/4/parenrightigm
+ǫ1/4≈m·4,06
+ǫ1/4.(19)(see (8) in [11]). So even for values close to the extremal A=
+1 of the rotating black hole Emax
+c.m./mcan be not very large as
+it is mentioned in [10, 11]. So for Amax=0.998 considered
+as the maximal possible dimensionless angular momentum of
+the astrophysical black holes (see [14]), from (18) one obta ins
+Emax
+c.m./m≈18.97.
+Doesitmeanthatinrealprocessesofparticlescatteringin the
+vicinity of the rotating nonextremal black holes the scatte ring
+energy is limited so that no Grand Unification or even Planck-
+ean energies can be obtained? Let us show that the answer is
+no! If one takes into account the possibility of multiple sca t-
+tering so that the particle falling from the infinity on the bl ack
+hole with some fixed angular momentum changes its momen-
+tum in the result of interaction with particles in the accret ing
+disc and after this is again scattering close to the horizon t hen
+thescatteringenergycanbeunlimited.
+The limiting value of the angular momentum of the particle
+close to the horizonof the black hole can be obtainedfrom the
+condition of positive derivative in (6) dt/dτ >0, i.e. going
+“forward”intime. Soclosetothehorizononehastheconditi on
+l<ε2xH/Awhichforε=1 givesthe limitingvalue lH.
+From (8) one can obtain the permitted intervalin rfor parti-
+cleswithε=1andangularmomentum l=lH−δ. Inthesecond
+orderinδclosetothe horizononeobtains
+l=lH−δ⇒x/lessorsimilarxH+δ2x2
+C
+4xH√
+1−A2. (20)
+If the particle falling fromthe infinitywith l≤lRarrivesto the
+regiondefined by (20) and here it interacts with other partic les
+oftheaccretiondiscoritdecaysonmorelightparticlesoth atit
+getsthe largerangularmomentum l1=lH−δ, thendueto (13)
+thescatteringenergyinthe centreofmasssystemis
+Ec.m.≈m√
+δ/radicaligg
+2(lH−l2)
+1−√
+1−A2(21)
+and it increaseswithoutlimit for δ→0. ForAmax=0.998and
+l2=lL,Ec.m.≈3.85m/√
+δ.
+Note that for rapidly rotating black holes A=1−ǫthe dif-
+ferencebetween lHandlRisnotlarge
+lH−lR=2√
+1−A
+A/parenleftig√
+1−A+√
+1+A−A/parenrightig
+≈2(√
+2−1)√ǫ, ǫ→0. (22)
+ForAmax=0.998,lH−lR≈0.04 so the possibility of getting
+small additional angular momentum in interaction close to t he
+horizonseemsmuchprobable.
+Thatiswhywecometoaconclusionthatunboundedlylarge
+energycan be obtainedin the centre of mass system of scatter -
+ing particles. This means that physical processes not only o n
+the Grand Unification scale but even on the Planckean energy
+cangointhiscase.
+One can obtain the same result of getting unbounded large
+energy in the centre of mass frame for particles with di fferent
+massesm1/nequalm2andε1/nequalε2[15].
+Howeveran importantquestionis tolookhowparticleswith
+largeenergycanbeextractedfromtheblackhole? Someenerg y
+3surely will be loosed due to the red shift, but as we’ll show in
+thispaperultrarelativisticparticlesstillcanbeobserv edoutside
+theblackhole. Observationofthemcangiveusinsightintot he
+GrandUnificationandPlanckeanphysicsnearthehorizon.
+Beforeansweringonthisquestionletusinvestigatethepro b-
+lem aboutthe timeneededforthe energyto growunboundedly
+large.
+2.1. Thetimeofmovementbeforethecollisionwithunbounde d
+energy
+Let us show that in order to get the unboundedly growing
+energy one must have the time interval from the beginning of
+thefallinginsidetheblackholetothemomentofcollisiona lso
+growing infinitely. This is connected with the fact of infinit y
+of coordinate interval of time needed for a freely particle t o
+cross the horizon of the black hole (for Schwarzschild metri c
+thisquestionwasconsideredin[16]).
+From Fig. 1 one can see that the collision energy can get
+large values in the centre of mass system only if the collisio n
+occursclosetothehorizon. Unboundedlylargeenergyofcol li-
+sionsoutsideof a blackholeare possibleonlyforcollision son
+horizonx→xH(see (11),(13)).
+From equationof the equatorialgeodesic (6), (8) for a parti -
+clewithdimensionlessangularmomentum landspecificenergy
+ε=1 (i.e. the particle is non relativistic at infinity) falling on
+the black hole with dimensionless angular momentum Aone
+obtains
+dr
+dt=−(x−xH)(x−xC)√x/radicalbig
+2x2−l2x+2(A−l)2
+x3+A2x+2A(A−l).(23)
+So the coordinate time (proper time of the observer at rest fa r
+fromtheblackhole)oftheparticlefallingfromsomepoint r0=
+x0Mtothe point rf=xfM>rHisequalto
+∆t=Mx0/integraldisplay
+xf√x/parenleftig
+x3+A2x+2A(A−l)/parenrightig
+dx
+(x−xH)(x−xC)/radicalbig
+2x2−l2x+2(A−l)2.(24)
+Incaseoftheextremalrotatingblackhole( A=1,xR=xH=
+1) and the limiting value of the angular momentum l=2 the
+integral(24) isequalto
+∆t=M√
+2/parenleftigg2√x(x2+8x−15)
+3(x−1)+5ln√x−1√x+1/parenrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex0
+xf(25)
+andit divergesas( xf−1)−1forxf→1.
+In all other cases, when A≤1 and angular moment l<lH,
+integral(24) divergeslogarithmicallyif xf→xH. Thisfollows
+fromequality
+x3+A2x+2A(A−l)=(x−xH)(x2+xHx+2xH)+2A(lH−l).(26)
+Note that for the particle with ε=1 outside the horizon for
+A<1 the angular momentum l<lH(see (20)). So for all
+possible values of landAto get the collision with infinitely
+growing energy in the centre of mass system needs infinitely
+largetime.Fortheintervalofpropertimeofthefreefallingtotheblac k
+holeparticleoneobtainsfrom(8)
+∆τ=Mx0/integraldisplay
+xf√
+x3dx/radicalbig
+2x2−l2x+2(A−l)2. (27)
+Iftheangularmomentumoftheparticlefallinginsidethebl ack
+hole is such that lL<l<lRthen the proper time is finite for
+xf→xH. ForA=1,l=2the integral(27))is equalto
+∆τ=M
+3√
+2/parenleftigg
+2√x(3+x)+3ln√x−1√x+1/parenrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex0
+xf(28)
+and it diverges logarithmically when xf→1. So to get the
+collision with infinite energy one needs the infinite interva l of
+ascoordinateaspropertime ofthefreefallingparticle.
+Togetlargefiniteenergyonemusthavesomefinitetime[15].
+So to have the collision of two protons with the energy of the
+order of the Grand Unification one must wait for the extremal
+black hole of the star mass the time ∼1024s, which is larger
+thantheageoftheUniverse ≈1018s. Howeverforthecollision
+with the energy 103larger than that of the LHC one must wait
+only≈108s.
+From(7),(8)fortheangleoftheparticlefallinginequator ial
+planeoftheblackholeoneobtains
+∆ϕ=x0/integraldisplay
+xf√x(xl+2(A−l))dx
+(x−xH)(x−xC)/radicalbig
+2x2−l2x+2(A−l)2.(29)
+IfA/nequal0,thenintegral(29)is divergentfor xf→xH. Sobefore
+collision with infinitely large energy the particle must com mit
+infinitelylargenumberofrotationsaroundtheblackhole.
+3. Collisionofparticlesinside therotatingblack hole
+As one can see from formula (11) the infinite value of the
+collision energy in the centre of mass system can be obtained
+inside the horizonof the blackholeon the Cauchyhorizon(4) .
+Indeed, from (12) the zeroes of the denominator in (11): x=
+xH,x=xC,x=0.
+Let us find the expression for the collision energy for x→
+xC. Note that the Cauchy horizon can be crossed by the free
+falling from the infinity particle under the same conditions on
+the angular momentum (16) as in case of the event horizon.
+Denote
+lC=2xC
+A=2
+A/parenleftig
+1−√
+1−A2/parenrightig
+. (30)
+Notethat lL<lC≤lR≤lH.
+To find the limit r→rCfor the black hole with a given an-
+gular momentum Aone must take in (11) x=xC+αand do
+calculationswithα→0. Thelimitingenergyhastwodi fferent
+expressionsdependingonthe valuesofangularmomenta. If
+(l1−lC)(l2−lC)>0, (31)
+4i.e.l1,l2are either both larger than lC, or both smaller than lC,
+then
+Ec.m.(r→rC)
+2m=/radicaligg
+1+(l1−l2)2
+2xH(l1−lC)(l2−lC).(32)
+Thisformulais similarto(13)if everywhere H↔C. If
+(l1−lC)(l2−lC)<0, (33)
+i.e.l2∈(lL,lC),l1∈(lC,lR)(ortheopposite),then
+Ec.m.
+m=A
+1−√
+1−A2/radicaligg
+2(l1−lC)(lC−l2)√
+1−A2(x−xC),x→xC.(34)
+It is seen that the limit is infinite for all values of angular m o-
+mental1,l2(33). This could be interpreted [17] as the known
+instability of the internal Kerr’s solution (see [18, chap. 14]).
+However,fromEq.(6)we cansee
+dt
+dτ(r→rC+0)=/braceleftigg+∞,ifl>lC,
+−∞,ifl<lC.(35)
+That is why the collisions with infinite energy can not be real -
+ized (seealso[17]).
+4. The extraction of energy after the collision in
+Schwarzschild’sandKerr’smetrics
+Let us consider the case when interaction between particles
+with masses m, specific energies ε1,ε2, specific angular mo-
+mentaL1,L2fallingintoablackholeoccurredforsome r>rH.
+Let two new particles with rest masses µappeared, one of
+them (1)movedoutside theblack hole,the other(2)movedin-
+sideit. Denotethespecificenergiesofnewparticlesas ε1µ,ε2µ,
+their angular momenta (in units of µ) asL1µ,L2µ,vi=dxi/ds
+— their 4-velocities. Consider particle movement in the equ a-
+torialplaneoftherotatingblackhole.
+Conservation laws in inelastic particle collisions for the en-
+ergyandmomentumleadto
+m(ui
+(1)+ui
+(2))=µ(vi
+(1)+vi
+(2)). (36)
+Equations(36)for tandϕ-componentscanbewrittenas
+m(ε1+ε2)=µ(ε1µ+ε2µ), (37)
+m(L1+L2)=µ(L1µ+L2µ), (38)
+i.e. the sum of energiesand angularmomentaof colliding par -
+ticlesis conservedinthefield ofKerr’sblackhole.
+Forther-componentfrom(8) oneobtains
+−m/radicaligg
+ε2
+1+2M
+r3(aε1−L1)2+a2ε2
+1−L2
+1
+r2−∆
+r2
++/radicaligg
+ε2
+2+2M
+r3(aε2−L2)2+a2ε2
+2−L2
+2
+r2−∆
+r2=µ/radicaligg
+ε2
+1µ+2M
+r3(aε1µ−L1µ)2+a2ε2
+1µ−L2
+1µ
+r2−∆
+r2
+−/radicaligg
+ε2
+2µ+2M
+r3(aε2µ−L2µ)2+a2ε2
+2µ−L2
+2µ
+r2−∆
+r2.(39)
+Thesignsin(39)areputsothattheinitialparticlesandthe par-
+ticle(2)goinsidetheblackholewhileparticle(1)goesout side
+the black hole. The values ε1µ,ε2µare constants on geodesics
+([13],§61)sotheproblemofevaluationoftheenergyatinfinity
+extractedfrom the black hole in collision reducesto a probl em
+tofindthese values.
+Theinitialparticlesinourcaseweresupposedtobenonrela -
+tivistic atinfinity:ε1=ε2=1,so Eq.(39) becomes
+−m/radicaligg
+2M
+r3(a−L1)2+2M
+r−L2
+1
+r2
++/radicaligg
+2M
+r3(a−L2)2+2M
+r−L2
+2
+r2
+=µ/radicaligg
+ε2
+1µ+2M
+r3(aε1µ−L1µ)2+a2ε2
+1µ−L2
+1µ
+r2−∆
+r2
+−/radicaligg
+ε2
+2µ+2M
+r3(aε2µ−L2µ)2+a2ε2
+2µ−L2
+2µ
+r2−∆
+r2.(40)
+Consideratfirstthecase a=0ofthenonrotatingblackhole.
+Inthiscasethe Eq.(40)hastheform
+−m/radicaligg
+2M
+r−L2
+1
+r2/parenleftigg
+1−2M
+r/parenrightigg
++/radicaligg
+2M
+r−L2
+2
+r2/parenleftigg
+1−2M
+r/parenrightigg
+=µ/radicaltp/radicalbt
+ε2
+1µ−1+L2
+1µ
+r2/parenleftigg
+1−2M
+r/parenrightigg
+−/radicaltp/radicalbt
+ε2
+2µ−1+L2
+2µ
+r2/parenleftigg
+1−2M
+r/parenrightigg.(41)
+The maximal energyof collision is for L1=−L2,|L1|=|L2|=
+4M(see(16),(18)). FromEq.(41)onegetsfor |L1µ|=|L2µ|(in
+particular,for the radial movementof the productsof react ion)
+ε1µ≤ε2µ. The initial particles were supposed to be nonrela-
+tivistic on infinity and so from (37) one gets for the energy of
+theparticlemovingoutsidetheblackhole
+µε1µ≤m. (42)
+Forradialmovementtheenergyradiatedtoinfinityofthepar ti-
+cles collided in the Schwarzschild black hole can not be larg er
+thantherest energyofoneinitial particle!
+Notethatequalityin(42)isobtainedincase L1=−L2,|L1|=
+|L2|=4M,L1µ=L2µ=0 (the radial movement), if r=4M
+thenreactionoccursonthedoubledSchwarzschildradius.
+5For the case when the collision takes place on the horizon
+of the black hole ( r→rH) the system (37)–(40) can be solved
+exactly
+ε1µ=AL1µ
+2rH,ε2µ=2m
+µ−AL1µ
+2rH,L2µ=m
+µ(L1+L2)−L1µ.(43)
+In general case the system of three Eqs. (37)–(40) for
+four variablesε1µ, ε2µ,L1µ,L2µcan be solved numeri-
+cally for a fixed value of one variable (and fixed parameters
+m/µ,L1/M,L2/M,a/M,r/M). The example of numerical
+solution isµ/m=0.3,l1=2.2,l2=2.198,A=0.99,x=
+1.21,l1µ=16.35,l2µ=−1.69,ε1µ=7.215,ε2µ=−0.548.
+Note thatthe energyofthe secondfinal particleisnegativea nd
+theenergyofthefirstfinalparticlecontrarytothelimitobt ained
+in [11]is largerthanthe energyofinitial particlesasit mu stbe
+in the case of a Penrose process [8]. If the mass of the parti-
+cle is notverylargethisparticleis observedasultrarelat ivistic.
+What is the reason of this contradiction? Let us investigate the
+problemcarefully.
+Note that if one neglects the states with negative energy in
+ergosphere energy extracted in the considered process can n ot
+belargerthantheinitialenergyofthepairofparticlesati nfinity,
+i.e. 2m. The same limit 2 mfor the extracted energy for any
+(including Penrose process) scattering process in the vici nity
+of the black hole was obtained in [11]. Let us show why this
+conclusionisincorrect.
+If the angular momentumof the falling particles is the same
+then(see(40))onehasthesituationsimilartotheusualdec ayof
+the particle with mass 2 min two particles with mass µ. Due to
+thePenroseprocessinergosphereit ispossiblethatthepar ticle
+falling inside the blackholehas the negativerelative to in finity
+energy and then the extracted particle can have energy large r
+than2m.
+The main assumption made in [11] is the supposition of the
+collinearity of vectorsof 4-momentaof the particlesfalli ng in-
+side and outside of the black hole (see (9)–(11) in [11]). The
+authors of [11] say that these vectors are “asymptotically t an-
+gentto thehorizongenerator”.
+First note that from (8), (16) for A<1 andl≤lRorA=1,
+butl<2 the limit of dr/dτis not zero at the horizon. This
+derivativehasoppositesigns forthe falling and outgoingp arti-
+cles. Signsforothercomponentsofthe4-momentumareequal .
+In the limiting case ( A=1,l1=2) the expressions dt/dτ,
+dϕ/dτof the componentsof the 4-velocityof the infalling par-
+ticle(6),(7)gotoinfinitywhen r→rH,butdr/dτgoestozero.
+In spite of smallness of r-components in the expression of the
+squareofthe4-momentumvectortheyhavethefactor grrgoing
+to infinity at the horizon. So puttingthem to zero can lead to a
+mistake. Toseeif u(1)andv(1)arecollinearitisnecessarytoput
+the coordinate rof the collision point to the limit rH=Mand
+resolve the uncertainties ∞/∞and 0/0. For the falling particle
+ε=1,l=2 the expressions for components of the 4-vector u
+canbeeasilyfoundfrom(6)–(8). Fortheparticleoutgoingf rom
+theblackholeduetoexactsolutiononthehorizon(43)onepu ts
+ε1µ=l1µ/2+α,whereαissomefunctionof randl1µ,suchthat
+α→0 whenr→rH. Putting thisε1µinto (6)–(8) one gets forx=r/M→1
+vt
+(1)
+ut
+(1)=vϕ
+(1)
+uϕ
+(1)=α
+x−1+l1µ
+2, (44)
+vr
+(1)
+ur
+(1)=−/radicaligg
+2α2
+(x−1)2+2l1µα
+x−1+3
+8l2
+1µ−1
+2. (45)
+Duetothecondition dt/dτ>0(movementforwardintime)the
+necessary condition for collinearity is that both (44) and ( 45)
+mustbezero,whichisnottrue. Thisleadstotheconclusiont hat
+the considerations of the authors of [11] for scattering exa ctly
+on the horizon can not be used for the real situation of partic le
+scatteringclosetothe horizon.
+Allthisconfirmsourhypothesisin[1]thatthevicinityofth e
+rotatingblackholeisthearenaoftheGrandUnificationphys ics
+asitwasintheearlyUniverse. IntheearlyUniversegravita tion
+createdpairsofsuperheavy X,˜Xparticlesdecayingthenasshort
+living and long living XS,XLparticles on ordinary quarks and
+leptons. But then due to the breaking of the GU symmetry XL
+became metastable and survived up to our time as dark matter
+particles.
+In the vicinity of rotating black holes XLcan decay on or-
+dinary particles as it is possible for the GU energies as well
+as they can be converted into XSdecaying particles because
+/angbracketleftXL|XS/angbracketright/nequal0. These decays can go with baryon and CP-
+conservation breaking, so that di fferent hypotheseson the pro-
+cesses in the early Universe can be checked by astrophysical
+observationsofAGN.
+Acknowledgments
+The authors are indebted to Ted Jacobson and Thomas P.
+Sotiriou for putting their attention to a numerical mistake in
+thefirst versionofthepaper[19].
+References
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+7
\ No newline at end of file
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@@ -0,0 +1,219 @@
+arXiv:1001.0757v3  [hep-th]  20 Jun 2010On thenon-relativistic limit ofcharge
+conjugation inQED
+B.Carballo P ´erez∗andM. Socolovsky†
+Instituto de Ciencias Nucleares, Universidad Nacional Aut ´ onoma deM´ exico,
+Circuito exterior, Ciudad Universitaria, 04510, M´ exico D .F.,M´ exico
+Abstract
+Even if at the level ofthe non-relativisticlimitof full QED ,Cis not a symmetry,
+the limit of this operation does exist for the particular cas e when the electromag-
+neticfieldisconsideredaclassicalexternalobjectcouple dtotheDiracfield. This
+result extends the one obtained when fermions are described by the Schr¨ odinger-
+Pauliequation. We givetheexpressionsforboththe Cmatrixand the ˆCoperator
+for galilean electrons and positrons interacting with the e xternal electromagnetic
+field. Theresultisrelevantinrelation torecent experimen tswithantihydrogen.
+Keywords: charge conjugation;non-relativisticlimit;QED
+∗brendacp@nucleares.unam.mx
+†socolovs@nucleares.unam.mx1 Introduction 2
+1 Introduction
+It is very common to talk about P (parity or space inversion) a nd T (time
+inversion) symmetries, not only in the non-relativistic qu antum mechanics, but
+also in classical mechanics. Nevertheless, the existence o f the galilean limit of C
+(charge conjugation) is nowadays controversial ([1] vs [2] ). Despite the fact that
+some authors [1] consider that the C transformation does not have an equivalent
+in the non-relativistictheory, others [2] reach up to define theˆCoperator in non-
+relativisticquantummechanics,buttheydonotgiveanexpl icitexpressionforthis
+operator. It is generally believedthat the antiparticleco ncept can only be defined
+in the context of relativisticquantum mechanics, because i t is only in this regime
+that we obtain solutions for free particles with negative en ergy flying backward
+intime,whoseabsenceisinterpretedasparticlesofopposi temoment,chargeand
+energy, flying forward in time: antiparticles. As thecharge conjugationoperation
+makes the particle-antiparticle operation, is believed th at this operator only takes
+placeinthiscontext.
+What is clear is that the C invariance in relativistic theory should not neces-
+sarily remain in the galilean limit because the transition f rom Lorentz group to
+Galileangroup isnotcontinuous.
+The existence of the charge conjugationoperation in the non -relativisticlimit
+wasdemonstratedin2006[3,4],i.e.,inthecontextofgalil eanrelativity. Thisfact
+is different from the idea that C exists only in the context of relativistic physics,
+withtheLorentz-Poincar´ e groupas thesymmetryofspace-t ime.
+The Dirac equation interacting with a classical external el ectromagnetic field
+was considered in [3], as the first step for finding this limit. For this purpose, the
+authorstookintoaccount thefollowingfacts:
+•Cdoesnotbelongto theLorentzgroup O(3,1).
+•We can give ad hocdefinitions for C, not only in the non-relativisticquan-
+tum mechanics, but also in the classical lorentzian and gali lean mechanics
+[2].
+•Thereisasymmetrybetweenparticlesandantiparticlesint hequantumrel-
+ativistic theory. For the equations of both particles and an tiparticles, it is
+possibleto find a non-relativisticwell defined limit, witho utany contradic-
+tion between them. If we have electrons of low energies (slow electrons),
+wecanexpecttohavepositronsoflowenergies(slowpositro ns)[5],related
+bythechargeconjugationoperation.2 Onthe non-relativisticlimitofC inQED 3
+For consistency, any prescription in the non relativistica pproximation should
+be derived from the relativistic theory as the limiting case |β|<<1, whereβis
+theparticle”velocity”( c=1). ThatiswhythepositronSchr¨ odinger-Pauliequation
+was obtained in [3] from the positron Dirac equation, making the substitution
+ψC=eimt∼
+ψCand taking,then,thenon-relativisticlimitoftheresulti ngequation.
+It was found thus a natural definition for the charge conjugat ionmatrix in the
+non-relativisticlimit,
+/parenleftBigg∼
+ψ1
+∼
+ψ2/parenrightBigg
+C−→/parenleftBigg
+−∼
+ψ∗
+2∼
+ψ∗
+1/parenrightBigg
+:=Cnr/parenleftBigg∼
+ψ1
+∼
+ψ2/parenrightBigg
+, (1)
+where:
+Cnr=K/parenleftbigg0−1
+1 0/parenrightbigg
+, (2)
+withK, thecomplexconjugationoperation.
+Takingintoaccounttheworks[3,4],andalsosomepreviouss tudiesaboutthe
+non-relativisticlimitinquantumfieldtheory,likethenon -relativisticlimitof λΦ4
+theory [6], we are going to study the non-relativisticlimit of charge conjugation,
+getting a little closer to reality, that is, in the case of the quantum Dirac field
+coupledto anexternal electromagneticfield.
+2 Onthenon-relativisticlimitof Cin QED
+From the QEDlagrangiandensity:
+LQED=¯ψ(x)(iγµ∂µ−qγµAµ(x)−m)ψ(x)−1
+4Fµν(x)Fµν(x);(3)
+the equations for the coupled Dirac and electromagnetic fiel ds are deduced ([7],
+p. 84):
+(iγµ∂µ−qγµˆAµ(x)−m)ˆψ(x) = 0, (4)
+∂ˆFµν(x)
+∂xν=qˆ¯ψγµˆψ(x). (5)
+Inequations(4)and(5), ˆAµ(x)isnotconsideredexternalandispartofthedy-
+namicsofthesystem. Ifwemakeintheseequations ˆψ(x)−→ˆψC(x)(equivalent3 Thecaseofanexternal electromagnetic field 4
+heretochange qto−q),thenchanging ˆAµ(x)−→ −ˆAµ(x)thechargeconjugation
+operationiscompleted,showingtheCinvarianceofquantum electrodynamics.
+If we want now to take the non-relativistic limit of full QED, we must estab-
+lish this limit both for ˆψ(x)andˆAµ(x). The problem is that it is not possible to
+establisha non-relativisticlimitfor ˆAµ(x)because thephotonmassiszero.
+It isdiscardedfrom herethat,when weconsiderelectrons,p ositronsand pho-
+tons together, makes sense to talk about the charge conjugat ion operation in the
+non-relativistic limit. As we can not establish the non-rel ativistic limit for QED,
+fromthispointon,wewillforgettheequation(5)andwillas sumethattheelectro-
+magneticfield in (4) is now a classical external object, with perfectly determined
+valueˆAµ(x) =Aµ. That is, we are going to study the charge conjugation opera-
+tioninthecontextoftheDiracfieldcoupledtoaclassicalex ternalelectromagnetic
+field.
+3 Thecaseofan externalelectromagneticfield
+Whatwasdeterminedin[3]and[4]wastheexpressionforthe Cmatrixinthe
+non-relativisticlimitoftheDiracequationcoupledtoane xternalelectromagnetic
+field. It is also necessary to find now the ˆCoperator, which represents the charge
+conjugationoperationintheHilbertspaceanditisrelated totheCmatrixthrough
+Cˆ¯ψT(x) =ˆC†ˆψ(x)ˆC, in thecontextofthenon-relativisticlimitoftheDirac fie ld
+coupledtooto aclassicalexternal electromagneticfield.
+Thecreationandannihilationoperatorsintermsofthefield operatorsaregiven
+by:
+ˆb(p,r) =/integraldisplayd3x
+(2π)3/2/radicalbiggm
+Ep−u(p,r)ˆψ(x)eip·x, (6)
+ˆd†(p,r) =/integraldisplayd3x
+(2π)3/2/radicalbiggm
+Ep−v(p,r)ˆψ(x)e−ip·x, (7)
+ˆb†(p,r) =/integraldisplayd3x
+(2π)3/2/radicalbiggm
+Epˆ−
+ψ(x)u(p,r)e−ip·x, (8)
+ˆd(p,r) =/integraldisplayd3x
+(2π)3/2/radicalbiggm
+Epˆ−
+ψ(x)v(p,r)eip·x. (9)3 Thecaseofanexternal electromagnetic field 5
+At low speeds (|p|
+c−→0)the creation and annihilation operators ( ˆb†,ˆd†,ˆb,ˆd)
+willcontinuecreatingorannihilatingparticlesandantip articles. Thus,inthenon-
+relativistic limit, the form of the equations (6), (7), (8) a nd (9) will be the same;
+withthedifferencesthatinthiscasetheterm/radicalbiggm
+Epwillbeequalto 1,ˆψandˆ−
+ψwill
+besolutionsoftheSchr¨ odinger-Pauliequationsandthesp inorswillbereducedto:
+u(p,1) =/parenleftbigg1
+0/parenrightbigg
+, u(p,2) =/parenleftbigg0
+1/parenrightbigg
+,
+−u(p,1) =/parenleftbig
+1,0/parenrightbig
+,−u(p,2) =/parenleftbig
+0,1/parenrightbig
+, (10)
+and
+v(p,1) =/parenleftbigg
+1
+0/parenrightbigg
+, v(p,2) =/parenleftbigg
+0
+1/parenrightbigg
+,
+−v(p,1) =/parenleftbig
+−1,0/parenrightbig
+,−v(p,2) =/parenleftbig
+0,−1/parenrightbig
+. (11)
+TheˆCoperator corresponding to the Dirac field, as a function of th e creation
+and annihilationoperators isgivenby[8]:
+ˆC0=exp(iπ
+2/integraldisplay
+d3p2/summationdisplay
+r=1(ˆd†(p,r)ˆb(p,r)+ˆb†(p,r)ˆd(p,r)
+−ˆb†(p,r)ˆb(p,r)−ˆd†(p,r)ˆd(p,r))); (12)
+while in the presence of interactions, the charge conjugati on operator at time tis
+givenby ([7], p. 111):
+ˆC(t) =eiˆHTtˆC0e−iˆHTt, (13)
+whereˆHTisthetotalhamiltonian(Diracfieldcoupledtotheexternal electromag-
+neticfield)and ˆC0satisfies(12),keepingtheoperators ˆb†,ˆd†,ˆb,ˆdintheHeisenberg
+picture.
+By taking the non-relativistic limit, the form of the operat orˆC(t), given by
+(13), does not change; but then, the operators ˆb†,ˆd†,ˆb,ˆdwill be those for low
+speeds,and ˆHTwillbetheSchr¨ odinger-Pauli hamiltonian.
+Summarizing,inarelativisticcontextthechargeconjugat ionoperationisequiv-
+alenttochange ˆψ(x)−→ˆψC(x)andˆAµ(x)−→ −ˆAµ(x),bothfor ˆAµ(x)dynam-
+ics and for ˆAµ(x) =Aµclassical external. This is equivalent to say that, in a4 Conclusions 6
+relativistic context, the electron move in a field ˆAµ(x)like the positron does in a
+field−ˆAµ(x).
+Inthenon-relativisticlimit,thechargeconjugationoper ationonlymakessense
+forAµclassicalexternalanditisreducedonlytochange ˆψ(x)−→ˆψC(x). Inthis
+lastcase,tomakealsothechange Aµ−→ −AµdonotturntheSchr¨ odinger-Pauli
+equationforthepositronintotheSchr¨ odinger-Pauliequa tionfortheelectron.
+4 Conclusions
+Even if it is not possible to establish the charge conjugatio n operation in the
+non-relativistic limit of full quantum electrodynamics, a s is not possible to take
+this limit for the electromagnetic field because the photon m ass is zero, we can
+talkaboutthenon-relativisticlimitofchargeconjugatio nforthecaseoftheDirac
+field in the presence of a classical external electromagneti c field and give the ex-
+pressions,forboth the Cmatrixand the ˆCoperator, in thiscase.
+Regarding whether the process of renormalization could mod ify the above
+analysis,weshouldtakeintoaccountthatwearenotstudyin gtheelementsofthe
+dispersion matrix in each approximation of perturbation th eory. Instead, we are
+talkingaboutthenon-relativisticlimitofthehamiltonia n,withwhichispossibleto
+calculatethedispersionmatrix. Thus,renormalizationdo es not affect theprocess
+oftakingthelimit.
+5 Acknowledgment
+This work was partially support by the project PAPIIT IN 1186 09-2, DGAPA-
+UNAM, M´ exico. B. Carballo P´ erez also acknowledges financi al support from
+CONACyT,M´ exico.
+References
+[1] L. D. Landau y E. M. Lifshitz , 1997 , Quantum Electrodynamics, Volume
+IV(Butterworth-Heinemann).
+[2] I. I. Bigi,A. I. Sanda, 2000, CP Violation (Cambridg eUniversityPress).REFERENCES 7
+[3] A. Cabo, D.B. Cervantes, H. P´ erez Rojas y M. Socolovsky, 2006,Remark
+on charge conjugation in the non relativistic limit , International Journal of
+TheoreticalPhysics, 45,1989-2000;arXiv: hep-th/0504223.
+[4] M.Socolovsky,2006, Chargeconjugationinthegalileanlimit ,Electromag-
+neticPhenomena, 6,No. 10,132-135(2006); arXiv: hep-th/0603137.
+[5] M.Amorettietal., 2002,Nature 419,456-459.
+[6] M. A. B. B´ eg, R. C. Furlong , 1985 , λΦ4in the nonrelativistic limit , Phys.
+Rev.D,31,No. 6,1370-1373.
+[7] J. D. Bjorken and S. D. Drell, 1965, RelativisticQuantum Fields , McGraw-
+Hill,New York.
+[8] W.Greiner, J. Reinhardt, 1993, FieldQuantization (Springer), 314.
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+arXiv:1001.0758v1  [math.CT]  5 Jan 2010CLUSTER TILTING AND COMPLEXITY
+PETTER ANDREAS BERGH & STEFFEN OPPERMANN
+Abstract. We study the notion of positive and negative complexity of pa irs
+of objects in cluster categories. The first main result shows that the maximal
+complexity occurring is either one, two or infinite, dependi ng on the represen-
+tation type of the underlying hereditary algebra. In the the second result, we
+study the bounded derived category of a cluster tilted algeb ra, and show that
+the maximal complexity occurring is either zero or one whene ver the algebra
+is of finite or tame type.
+1.Introduction
+Cluster categories associated to finite dimensional hereditary alge bras were in-
+troduced in [BMRRT]. These 2-Calabi-Yau triangulated categories ar ise as orbit
+categories of derived categories, and provide a categorification o f the combinatorics
+of the cluster algebras introduced in [FoZ] by Fomin and Zelevinsky in t he acyclic
+case. They also provide a generalized framework for classical tilting theory, with
+the cluster tilting objects and their endomorphism rings, the cluste r tilted algebras.
+Given two objects in a triangulated category defined over a field, th eir total
+cohomology is a Z-graded vector space over the ground field. It therefore makes
+sense to study the rate of growth of the dimensions in both the “ne gative” and the
+“positive” direction, thus leading to the notion of negative and posit ive complexity.
+In this paper, we study the complexity of a cluster category, and s how that the
+maximal complexity occurring depends on the representationtype of the hereditary
+algebra we start with:
+Theorem. LetHbe a basic finite dimensional hereditary algebra over an alge -
+braically closed field, and let CHbe the corresponding cluster category. Then
+sup{cx∗
+CH(X,Y)|X,Y∈ CH}=
+
+1ifHhas finite type,
+2ifHhas tame type,
+∞ifHhas wild type.
+We also study the complexity of the derived category of a cluster tilt ed algebra,
+and show that in this case, the maximal complexity occurring depend s on the
+representation type of the algebra:
+Theorem. IfΛis a cluster tilted algebra of finite or tame representation t ype, then
+sup{cx∗
+Db(Λ)(X,Y)|X,Y∈ Db(Λ)}=/braceleftbigg
+0ifΛis hereditary,
+1otherwise
+We prove this by showing that a tame cluster tilted algebra has finitely many
+indecomposableCohen-Macaulaymodules. Finally, welookatsomeexa mplesshow-
+ing what can happen for wild cluster titled algebras.
+2000Mathematics Subject Classification. 16P90, 18E30, 18G15.
+Key words and phrases. Cluster categories, cluster tilted algebras, complexity.
+Both authors were supported by NFR Storforsk grant no. 16713 0.
+12 PETTER ANDREAS BERGH & STEFFEN OPPERMANN
+2.Preliminaries
+Throughoutthissection, wefixafield kandatriangulatedHom-finite k-category
+TwithsuspensionfunctorΣ. Thusforallobjects X,Y,ZinT, the setHom T(X,Y)
+is a finite dimensional k-vector space, and the composition
+HomT(Y,Z)×HomT(X,Y)→HomT(X,Z)
+isk-bilinear. Recall that a Serre functor onTis a triangle equivalence TS− → T,
+together with functorial isomorphisms
+HomT(X,Y)≃DHomT(Y,SX)
+of vector spaces for all objects X,Y∈ T, whereD= Hom k(−,k). By [BoK], such
+a functor is unique if it exists. For an integer d∈Z, the category Tis said to be
+d-Calabi-Yau if it admits a Serre functor which is isomorphic as a triangle functor
+to Σd.
+A subcategory of Tisthickif it is a full triangulated subcategory closed under
+direct summands. Now let CandDbe subcategories of T. We denote by thick1
+T(C)
+the full subcategory of Tconsisting of all the direct summands of finite direct sums
+of shifts of objects in C. Furthermore, we denote by C∗Dthe full subcategory of
+Tconsisting of objects Msuch that there exists a distinguished triangle
+C→M→D→ΣC
+inT, withC∈ CandD∈ D. Now for each n≥2, define inductively thickn
+T(C)
+to be thick1
+T/parenleftbig
+thickn−1
+T(C)∗thick1
+T(C)/parenrightbig
+, and denote/uniontext∞
+n=1thickn
+T(C) by thick T(C).
+This is the smallest thick subcategory of Tcontaining C.
+Given two objects XandYofT, we define the positive complexity ofthe ordered
+pair (X,Y) as
+cx+
+T(X,Y)def= inf{t∈N∪{0} | ∃a∈R: dimHom T(X,ΣnY)≤ant−1forn≫0}.
+Similarly, we define the negative complexity as
+cx−
+T(X,Y)def= inf{t∈N∪{0} | ∃a∈R: dimHom T(X,Σ−nY)≤ant−1forn≫0}.
+Wheneverwewritecx∗
+T(X,Y)andmakeastatement, itistobe understoodthatthe
+statement holds for both the positive and the negative complexity. By definition,
+the positive complexity is zero if and only if Hom T(X,ΣnY) = 0 for large n,
+whereas the negative complexity is zero if and only if Hom T(X,ΣnY) = 0 for
+smalln. Moreover, given integers a,b∈Z, there is an equality cx∗
+T(X,Y) =
+cx∗
+T(ΣaX,ΣbY). Note also that if Tisd-Calabi-Yau for some d, then cx+
+T(X,Y) =
+cx−
+T(Y,X); in particular the equality cx+
+T(X,X) = cx−
+T(X,X) holds in this case.
+The following elementary lemma shows that complexity in some sense be haves
+nicely on thick subcategories.
+Lemma 2.1. LetXandYbe objects of T. Thencx∗
+T(X′,Y)≤cx∗
+T(X,Y)for
+all objects X′∈thickT(X), andcx∗
+T(X,Y′)≤cx∗
+T(X,Y)for all objects Y′∈
+thickT(Y). In particular, the inequality cx∗
+T(X′,X′′)≤cx∗
+T(X,X)holds for all
+objectsX′,X′′∈thickT(X).
+Proof.We prove only the first inequality, by induction on the number nsuch that
+X′belongs to thickn
+T(X). If cx∗
+T(X,Y) =∞, then the inequality obviously holds.
+Hencewemayassumethatcx∗
+T(X,Y)isfinite, saycx∗
+T(X,Y) =c. Ifn= 1,then X′
+is a direct summand of finite direct sums of shifts of X, hence the inequality holds
+in this case. Next, suppose n >1 and that X′belongs to thickn−1
+T(X)∗thick1
+T(X).
+Then there exists a triangle
+X1→X′→X2→ΣX1CLUSTER TILTING AND COMPLEXITY 3
+in which X1∈thickn−1
+T(X) andX2∈thick1
+T(X). This triangle induces an exact
+sequence
+HomT(X2,ΣnY)→HomT(X′,ΣnY)→HomT(X1,ΣnY)
+of vector spaces for every n∈Z. By induction, both cx∗
+T(X1,Y) and cx∗
+T(X2,Y)
+are at most c. Therefore, there exist real numbers a1anda2such that
+dimHom T(X1,ΣnY)≤a1|n|c−1
+dimHom T(X2,ΣnY)≤a2|n|c−1
+for|n| ≫0. This gives
+dimHom T(X′,ΣnY)≤dimHom T(X1,ΣnY)+dimHom T(X2,ΣnY)
+≤(a1+a2)|n|c−1
+for|n| ≫0, showing that cx∗
+T(X′,Y) is at most c. The result now follows from the
+definition of thickn
+T(X). /square
+The aim of this paper is to determine the maximal complexity occurring in
+certain triangulated categories, via maximal orthogonal subcate gories. Recall that
+a subcategory CofTiscontravariantly finite inTif every object in Tadmits
+a rightC-approximation. Thus, for every object X∈ Tthere exists a morphism
+C→XwithC∈ C, such that everymorphism C′→XwithC′∈ Cfactorsthrough
+C. The following lemma provides a criterion under which a contravariant ly finite
+subcategory Cgenerates T(see [Iya] and [KeR, Section 5.5]). Consequently, we see
+from Lemma 2.1 that the maximal complexity of Tequals that of C.
+Lemma 2.2. LetCbe a contravariantly finite subcategory of T, and suppose there
+exists an integer n≥1such that the following are equivalent for every object X∈ T:
+(1)X∈ C,
+(2) Hom T(C,ΣiX) = 0for1≤i≤nand allC∈ C.
+Thenthickn+1
+T(C) =T.
+Proof.Choosentriangles
+K1/d47/d47C0f0/d47/d47X/d47/d47ΣK1
+............
+Kn−1/d47/d47Cn−2fn−2/d47/d47Kn−2/d47/d47ΣKn−1
+Kn/d47/d47Cn−1fn−1/d47/d47Kn−1/d47/d47ΣKn
+in which the morphisms fiare right C-approximations. Let Cbe any object in C.
+The triangles induce exact sequences
+··· →HomT(C,ΣjCi)(Σjfi)∗− −−−− →HomT(C,ΣjKi)→HomT(C,Σj+1Ki+1)→ ···
+for 0≤i≤n−1 (where we have denoted XbyK0). An induction argument shows
+that Hom T(C,ΣiKn) vanishes for 1 ≤i≤n, henceKnbelongs to C. Then another
+induction argument shows that Xbelongs to thickn+1
+T(C). /square
+Corollary 2.3. Given the assumptions from the previous lemma, the equality
+sup{cx∗
+T(X,Y)|X,Y∈ T }= sup{cx∗
+T(C,C′)|C,C′∈ C}
+holds.4 PETTER ANDREAS BERGH & STEFFEN OPPERMANN
+In the next section, we apply the above results to Calabi-Yau triang ulated cat-
+egories admitting subcategories with the properties displayed in the assumption of
+Lemma 2.2. Recall therefore that, if Tisd-Calabi-Yau for some d≥2, then a
+cluster tilting subcategory ofTis a contravariantly finite subcategory Csuch that
+the following are equivalent for any object X∈ T:
+(1)X∈ C,
+(2) Hom T(C,ΣiX) = 0 for 1 ≤i≤d−1 and all C∈ C.
+SinceTisd-Calabi-Yau, property (2) is equivalent to
+(3) Hom T(X,ΣiC) = 0 for 1 ≤i≤d−1 and all C∈ C.
+An object T∈ Tis acluster tilting object ofTif addTis a cluster tilting subcat-
+egory.
+Note that it follows directly from Corollary 2.3 that if C1andC2are cluster
+tilting subcategories of T, then
+sup{cx∗
+T(X,Y)|X,Y∈ T }= sup{cx∗
+T(C1,C′
+1)|C1,C′
+1∈ C1}
+= sup{cx∗
+T(C2,C′
+2)|C2,C′
+2∈ C2}.
+Therefore, in order to determine the maximal complexity of a Calabi- Yau triangu-
+lated category, any cluster tilting subcategory will do.
+3.Cluster categories
+Cluster categories associated to finite dimensional hereditary alge bras were in-
+troduced in [BMRRT] (and for hereditary algebras of Dynkin type Anin [CCS]).
+Letkbe an algebraically closed field and Ha basic finite dimensional heredi-
+taryk-algebra. Let Db(H) be the bounded derived category of finitely generated
+leftH-modules; this category is triangulated, its suspension functor Σ is just the
+shift of a complex. Finally, denote by τthe Auslander-Reiten translate in Db(H);
+this functor is induced by the usual Auslander-Reiten translate DTr on the non-
+projectiveindecomposable H-modules. Itwasshownin[Kel]thattheorbitcategory
+Db(H)/τ−1Σ is triangulated, with suspension functor induced by Σ. This is the
+cluster category CHassociatedto H. Its objectscoincide with the objectsin Db(H),
+and the functors Σ and τare equal. Given objects XandYofCH, the morphism
+space Hom CH(X,Y) is given by
+HomCH(X,Y)def=/circleplusdisplay
+i∈ZHomDb(H)(τ−iΣiX,Y),
+which is finite dimensional since His hereditary. We shall denote the suspension
+functor of CHby Σ as well. Moreover, given an H-module M, we shall also denote
+its image in CHbyM. By [BMRRT, Proposition 1.7(b)] the cluster category CH
+is 2-Calabi-Yau, that is, there is an isomorphism
+DHomCH(X,ΣY)≃HomCH(Y,ΣX)
+of vector spaces for all objects XandYinCH.
+In order to prove the main result, we need a result on the rate of gr owth of
+the sequence {dimτ−nH}∞
+n=1for a hereditary algebra H. Recall first that the
+representation type of a finite dimensional algebra (over an algebr aically closed
+field) iseither finite, tame orwild. An algebraisoffinite representationtypeifthere
+are only finitely many non-isomorphic indecomposable modules. Furth ermore, an
+algebra is of tame representation type if there exist infinitely many n on-isomorphic
+indecomposable modules, but they all belong to one-parameter fam ilies, and in
+each dimension there are finitely many such families. Finally, an algebra is of
+wild representation type if it is not of finite or tame type. In the latte r case, theCLUSTER TILTING AND COMPLEXITY 5
+representationtheory ofthe algebrais at least as complicated ast he classificationof
+finite dimensional vector spaces together with two non-commuting endomorphisms.
+Proposition 3.1. LetHbe a finite dimensional hereditary algebra of infinite rep-
+resentation type over an algebraically closed field. Define
+γ/parenleftbig
+τ−1
+H/parenrightbig
+:= inf{t∈N∪{0} | ∃a∈R: dimτ−nH≤ant−1forn≫0}.
+Then the following hold:
+(1)γ/parenleftbig
+τ−1
+H/parenrightbig
+= 2if(and only if )His tame.
+(2)γ/parenleftbig
+τ−1
+H/parenrightbig
+=∞if(and only if )His wild.
+Proof.(1) Suppose His tame. We may assume that His the path algebra of one of
+the Euclidean quivers /tildewideAn,/tildewideDn,/tildewideE6,/tildewideE7,/tildewideE8. We prove this case by using the theory
+of quadratic forms, and refer to [Ri1, Chapter 1] for unexplained n otation and
+terminology. Let nbe the number of vertices of the underlying Euclidean quiver.
+Letr∈Znbe a minimal positive radical vector. We claim that the set
+{x∈Zn|xroot,0≤x,r/notlessequalx}
+is finite. To see this, note that there exists an integer 1 ≤i≤nsuch that the ith
+entry in ris 1 (see [Ri1, table on page 8]). This implies that {r} ∪{ej|1≤j≤
+n,j/ne}ationslash=i}is aZ-basis for Zn, whereejdenotes the jth unit vector. With respect to
+this basis, the quadratic form is given by a matrix
+/parenleftbiggA0
+0 0/parenrightbigg
+in which Ais the matrix of the quadratic form of the corresponding Dynkin quiv er.
+Thus all the roots are of the form y+αen, whereα∈Zandyis a root of A. But
+then all the roots are also of the form y+αr, and since there are only finitely many
+choices for y, the claim follows.
+Now letPbe an indecomposable projective H-module. Then τ−mPis indecom-
+posable for all m≥0, and the roots {[τ−mP]}∞
+m=0are pairwise distinct. By the
+above, there exist integers i < jandvsuch that [ τ−iP]+vr= [τ−jP]. Denoting
+j−ibyl, we see that
+[τ−(αl+m)P] =αvr+[τ−mP]
+for anyα≥0 andm∈ {0,...,l−1}. This shows that the sequence {dimτ−nH}∞
+n=1
+grows linearly.
+(2) Suppose His wild, and let Pbe an indecomposable H-module. By [Tak,
+Theorem 2.4], there exists an integer msuch that
+lim
+n→∞dimτ−nP
+ρnnm−1
+is nonzero, where ρis the spectral radius of the Coxeter transformation of H. Now
+suppose that γ/parenleftbig
+τ−1
+H/parenrightbig
+is finite, so that there exist a t≥0 and an a∈Rsuch that
+dimτ−nP≤ant−1for large n. Then
+lim
+n→∞dimτ−nP
+ρnnm−1≤lim
+n→∞ant−1
+ρnnm−1
+= lim
+n→∞ant−m
+ρn
+= 0
+since, by [Ri2, Theorem], the spectral radius ρsatisfiesρ >1. This is a contradic-
+tion, hence γ/parenleftbig
+τ−1
+H/parenrightbig
+=∞. /square6 PETTER ANDREAS BERGH & STEFFEN OPPERMANN
+We now prove the main result. It shows that the maximal complexity in CH
+(positiveandnegative)iseitherone,twoorinfinite, dependingonth erepresentation
+type ofH.
+Theorem 3.2. LetHbe a basic finite dimensional hereditary algebra over an
+algebraically closed field, and let CHbe the corresponding cluster category. Then
+sup{cx∗
+CH(X,Y)|X,Y∈ CH}=
+
+1ifHhas finite type,
+2ifHhas tame type,
+∞ifHhas wild type.
+Proof.Consider the subcategory add HofCH. It is contravariantly finite since it
+contains only finitely many non-isomorphic indecomposable objects. Moreover, by
+[BMRRT, Theorem 3.3(b)], the following are equivalent for any object X∈ T:
+(1)X∈addH,
+(2) Hom T(H,ΣX) = 0.
+Thus the object His cluster tilting in CH, and so from Corollary 2.3 we see that
+sup{cx∗
+CH(X,Y)|X,Y∈ CH}= cx∗
+CH(H,H).
+SinceCHis Calabi-Yau, the maximal positive complexity equals the maximal neg-
+ative complexity. It therefore suffices to prove the result for neg ative complexity.
+By definition, the negative complexity cx−
+CH(H,H) equals the rate of growth of
+the dimensions of the vector spaces Hom CH(H,Σ−nH) asngrows. Since τ= Σ on
+CH, we obtain isomorphisms
+HomCH(H,Σ−nH)≃HomCH(H,τ−nH)
+≃/circleplusdisplay
+i∈ZHomDb(H)(τ−iΣiH,τ−nH)
+≃/circleplusdisplay
+i∈ZHomDb(H)(H,τi−nΣ−iH)
+of vector spaces. If His of finite representation type, then dimHom CH(H,τ−nH)
+is bounded as n→ ∞. Hence the result follows in this case.
+Suppose His of infinite representation type. Given integers iandj, the stalk
+complex τiΣjHinDb(H) is nonzero in degree j−1 wheni≥1, and in degree j
+wheni≤0. Thus when nis positive, the only nonzero term in the above direct
+sum appears when i= 0, that is, the term HomDb(H)(H,τ−nH). Therefore, for
+suchn, we obtain the isomorphisms
+HomCH(H,Σ−nH)≃HomDb(H)(H,τ−nH)
+≃HomH(H,τ−nH)
+≃τ−nH.
+Consequently, the negative complexity cx−
+CH(H,H) equals the rate of growth of the
+sequence {dimτ−nH}∞
+n=1. The result now follows from Proposition 3.1. /square
+4.Cluster tilted algebras
+LetHbe a basic finite dimensional hereditary algebra over some algebraica lly
+closed field, and Ta cluster tilting object in the cluster category CH. The corre-
+sponding cluster tilted algebra is the endomorphism ring End CH(T), itself a finite
+dimensional algebra. By [BMR], the functor
+Hom(T,−):CH/(τT)→mod(End CH(T))
+is an equivalence, hence one might suspect from Theorem 3.2 that ta me cluster
+tilted algebras have complexity two. However, this is notthe case: we show in this
+section that their complexity is at most one.CLUSTER TILTING AND COMPLEXITY 7
+In order to show this, we first recall some facts on Gorenstein alge bras. Let Γ be
+such an algebra, and denote by CM(Γ) the categoryof Cohen-Mac aulayΓ-modules,
+i.e.
+CM(Γ) = {M∈modΓ|Exti
+Γ(M,Γ) = 0 for all i >0}.
+It follows from general cotilting theory that this is a Frobenius exac t category, in
+whichtheprojectiveinjectiveobjectsaretheprojectiveΓ-modu les,andtheinjective
+envelopes are the left addΓ-approximations. Therefore the stab le category CM (Γ),
+which is obtained by factoring out all morphisms which factor throug h projective
+Γ-modules, is a triangulated category. Its shift functor is given by cokernels of left
+addΓ-approximations, the inverse shift is the usual syzygy funct or. Now let Db(Γ)
+be the bounded derived category of finitely generated Γ-modules. Furthermore,
+letDperf(Γ) be the thick subcategory of Db(Γ) consisting of objects isomorphic
+to bounded complexes of finitely generated projective Γ-modules. It follows from
+work by Buchweitz, Happel and Rickard (cf. [Buc], [Hap], [Ric]) that C M(Γ) and
+the quotient category Db(Γ)/Dperf(Γ) are equivalent as triangulated categories.
+The following lemma shows that if CM(Γ) is of finite type, i.e. contains on ly
+finitelymanynon-isomorphicindecomposableobjects, thenthemax imalcomplexity
+occurring in Db(Γ) is either one or zero.
+Lemma4.1. LetΓbe a finite dimensional Gorenstein algebra suchthat the cate gory
+CM(Γ)of Cohen-Macaulay Γ-modules has finitely many non-isomorphic indecom-
+posable objects. Then
+sup{cx+
+Db(Γ)(X,Y)|X,Y∈ Db(Γ)}=/braceleftbigg0ifΓhas finite global dimension,
+1otherwise
+Proof.LetXandYbe complexes in Db(Γ). As mentioned above, the categories
+CM(Γ) andDb(Γ)/Dperf(Γ) areequivalent, and sothere is a dense functor Db(Γ)→
+CM(Γ). If we denote by XandYthe images of XandYin CM(Γ), then it follows
+from [Buc, Corollary 6.3.4] that cx+
+Db(Γ)(X,Y) = cx+
+CM(Γ)(X,Y). Since CM(Γ) is
+of finite type, we see that cx+
+CM(Γ)(X,Y) is at most one. /square
+Recall from [KeR] that a cluster tilted algebra is Gorenstein of dimens ion one.
+The following result shows that if such an algebra Λ is tame, then CM(Λ ) is of
+finite type.
+Theorem 4.2. IfΛis a tame cluster tilted algebra, then the category CM(Λ)
+of Cohen-Macaulay Λ-modules has finitely many non-isomorphic indecomposable
+objects.
+Proof.By definition, there is a hereditary algebra Hand a cluster tilting object
+T∈ CHsuch that Λ = End CH(T). Moreover, by a theorem of Krause (cf. [Kra,
+Corollary 3.4]), the algebra His also tame. Therefore, at least two of the inde-
+composable direct summands of Tlie in the non-regular component. Let Tiby one
+such summand lying “as far to the right as possible”, that is, there is no path in
+this component from Tito any other summands of T. We may assume that τ−Ti
+comes from a projective H-module. Now for any X∈modHwe have
+X∈CM(Λ)⇐⇒Ext1
+Λ(X,Λ) = 0
+⇐⇒HomΛ(τ−Λ,X) = 0
+=⇒HomCH(τ−Ti,X)/(maps factoring through τT) = 0
+⇐⇒HommodH(τ−Ti,X)/(maps factoring through τT) = 0
+ForXpreprojective this is just Hom modH(τ−Ti,X), and this space only vanishes
+for finitely many X.8 PETTER ANDREAS BERGH & STEFFEN OPPERMANN
+W denote by Rthe direct sum of the regular summands of T. For almost all
+regular and preinjective H-modules Xwe have
+HommodH(τ−Ti,X)/(maps factoring through τT)
+= Hom modH(τ−Ti,X)/(maps factoring through τR).
+IfXlies in a homogeneous tube then the denominator vanishes, and henc e the
+space is non-zero.
+For any indecomposable regular Xthe dimension dimHom( τR,X) is at most
+the number of indecomposable summands of R. For all preinjective Xwe have
+dimHom( τR,X) = dimHom( τ1−ℓR,X) = dimHom( τR,τℓX), for some ℓonly
+depending on R, and hence there is a common bound for all the dimHom( τR,X)
+withXpreinjective. Now we have
+dimHom modH(τ−Ti,X)/(maps factoring through τR)
+/greaterorequalslantdimHom modH(τ−Ti,X)−dimHom modH(τ−Ti,τR)·dimHom modH(τR,X)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright
+bounded.
+Hence this space can only vanish if dimHom modH(τ−Ti,X) is sufficiently small.
+However this only happens for finitely many modules which are preinje ctive or lie
+in non-homogeneous regular tubes. /square
+Combining Lemma 4.1 and Theorem 4.2, we see that cluster tilted algebr as of
+finite or tame type are of complexity at most one.
+Theorem 4.3. IfΛis a cluster tilted algebra of finite or tame representation t ype,
+then
+sup{cx+
+Db(Λ)(X,Y)|X,Y∈ Db(Λ)}=/braceleftbigg
+0ifΛis hereditary,
+1otherwise
+Next, we look at three examples of wild cluster-tilted algebras. Thes e examples
+show that Theorem 4.3, and hence also Theorem 4.2, does not gener alize to wild
+cluster tilted algebras. For background on mutations of quivers wit h potentials, see
+[BIRS] and [DWZ].
+Examples. (1) Let Λ 0=k[1/d47/d47/d47/d472/d47/d473] be the path algebra of the
+wild quiver 1/d47/d47/d47/d472/d47/d473. This is a hereditary algebra, and therefore
+cx∗
+Db(Λ0)(X,Y) = 0 for any X,Y∈ Db(Λ0).
+(2) Let Λ 1be the cluster tilted algebra obtained from Λ 0by mutation at the
+vertex 2. Then Λ 1is the path algebra of the quiver
+2
+x2
+/d119/d119x1
+/d8/d8
+1z1/d42/d42z2/d52/d523y/d94/d94/d62/d62/d62/d62/d62/d62/d62
+subject to the relations given by the cyclic derivatives of the poten tial
+x1z1y+x2z2y, namely the relations {x1z1+x2z2,yx1,yx2,z1y,z2y}. Then
+sup{cx+
+Db(Λ1)(X,Y)|X,Y∈ Db(Λ1)}= 1.
+(3) Let Λ 2be the cluster tilted algebra obtained from Λ 1by mutation at the
+vertex 1. Then Λ 2is the path algebra of the quiver
+2
+y1
+/d22/d22y3
+/d39/d39y2
+/d30/d30/d62/d62/d62/d62/d62/d62/d62
+1x1/d55/d55
+x2/d72/d72
+3z2/d106/d106z1/d116/d116CLUSTER TILTING AND COMPLEXITY 9
+subject to the relations given by the cyclic derivatives of the poten tial
+x1y1z2+x2y2z1+x1y3z1−x2y3z2, namely the relations {x1y3+x2y2,x1y1−
+x2y3,y1z2+y3z1,y2z1−y3z2,z1x1,z2x2,z1x1−z2x2}. Then
+sup{cx+
+Db(Λ2)(X,Y)|X,Y∈ Db(Λ2)}= 2.
+Proof.The claims for Λ 0are clear. For Λ 1, the indecomposable projectives have
+the following composition structures:
+1
+x1
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x2
+/d30/d30/d62/d62/d62/d62/d62/d62/d62 2
+y
+/d15/d153
+z1
+/d0/d0/d0/d0/d0/d0/d0/d0/d0z2
+/d30/d30/d62/d62/d62/d62/d62/d62/d62
+2 2 3 1
+x2
+/d0/d0/d0/d0/d0/d0/d0/d0/d0
+x1/d30/d30/d62/d62/d62/d62/d62/d62/d62 1
+−x2/d0/d0/d0/d0/d0/d0/d0/d0/d0x1
+/d30/d30/d62/d62/d62/d62/d62/d62/d62
+2 2 2
+We see that the three simple modules satisfy
+Ω1
+Λ1(S1) =S2
+2 Ω1
+Λ1(S2) =S3 Ω2
+Λ1(S3) =S2.
+Consequently every simple Λ 1-module is eventually Ω-periodic, and therefore
+sup{cx+
+Db(Λ1)(X,Y)|X,Y∈ Db(Λ1)}= 1.
+For Λ 2, the indecomposable projective modules have the following composit ion
+structures:
+1
+z1
+/d0/d0/d0/d0/d0/d0/d0/d0/d0z2
+/d30/d30/d62/d62/d62/d62/d62/d62/d62
+3
+y1
+/d119/d119/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112
+y3
+/d15/d15y2
+/d39/d39/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78 3−y1
+/d119/d119/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112
+y3
+/d15/d15y2
+/d39/d39/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78
+2
+x2
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x1
+/d30/d30/d62/d62/d62/d62/d62/d62/d62 2
+x2
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x1
+/d30/d30/d62/d62/d62/d62/d62/d62/d62 2
+−x2
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x1
+/d30/d30/d62/d62/d62/d62/d62/d62/d62 2
+−x2
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x1
+/d30/d30/d62/d62/d62/d62/d62/d62/d62
+1 1 1 1 1
+2
+x1
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x2
+/d30/d30/d62/d62/d62/d62/d62/d62/d62
+1
+z1/d30/d30/d62/d62/d62/d62/d62/d62/d62 1
+z2/d0/d0/d0/d0/d0/d0/d0/d0/d0
+3
+3
+y1
+/d119/d119/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112/d112
+y3
+/d15/d15y2
+/d39/d39/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78
+2
+x2
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x1
+/d30/d30/d62/d62/d62/d62/d62/d62/d62 2
+x2
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x1
+/d30/d30/d62/d62/d62/d62/d62/d62/d62 2
+−x2
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x1
+/d30/d30/d62/d62/d62/d62/d62/d62/d62
+1 1 1 110 PETTER ANDREAS BERGH & STEFFEN OPPERMANN
+If we denote by Mnthe module with composition structure
+1
+x1
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x2
+/d30/d30/d62/d62/d62/d62/d62/d62/d62 1
+x1
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x2
+/d30/d30/d62/d62/d62/d62/d62/d62/d62··· 1
+x2
+/d0/d0/d0/d0/d0/d0/d0/d0/d0x1
+/d30/d30/d62/d62/d62/d62/d62/d62/d62
+3 3 3 ··· 3 3
+(withncomposition factors S1andn+ 1 composition factors S3), then one can
+show by direct calculation that
+Ω2
+Λ2(Mn) =Mn+2.
+Now note that
+S3=M0and Ω3
+Λ2(S1) =M1,
+so the rate of growth of the dimensions of the syzygies of these sim ple modules is
+linear. Finally, note that Ω1
+Λ2(S2) is an extension of S1⊕S1andS3, hence the rate
+of growth of the dimensions of its syzygies is at most linear. This show s that
+sup{cx+
+Db(Λ2)(X,Y)|X,Y∈ Db(Λ2)}= 2. /square
+Remark. As mentioned, the above example not only shows that Theorem 4.3 do es
+not generalize to wild cluster tilted algebras. It also shows that the s ame is true for
+Theorem 4.2, that is, there exist wild cluster tilted algebras with infinit ely many
+non-isomorphicindecomposableCohen-Macaulaymodules. Namely, b yLemma 4.1,
+the algebra in example (3) has this property.
+We conclude this paper with the following more general questions on t he com-
+plexity of wild cluster tilted algebras:
+Questions. (1) What numbers occur as
+sup{cx+
+Db(Λ)(X,Y)|X,Y∈ Db(Λ)}
+for Λ a cluster tilted algebra of wild type?
+(2) Given a wild hereditary algebra H, do all the numbers in (1) occur as
+sup{cx+
+Db(Λ)(X,Y)|X,Y∈ Db(Λ)}
+for some cluster tilted algebra Λ of type H?
+References
+[BoK] A. I. Bondal, M. M. Kapranov, Representable functors, Serre functors, and reconstruc-
+tions, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 11 83-1205, 1337;
+translation in Math. USSR-Izv. 35 (1990), no. 3, 519-541.
+[BMR] A. B. Buan, R. Marsh, I. Reiten, Cluster-tilted algebras , Trans. Amer. Math. Soc. 359
+(2007), no. 1, 323-332 (electronic).
+[BMRRT] A. B. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todoro v,Tilting theory and cluster
+combinatorics , Adv. Math. 204 (2006), no. 2, 572-618.
+[BIRS] A. B. Buan, O. Iyama, I. Reiten, D. Smith, Mutation of cluster-tilting objects and
+potentials , to appear in American J. Math.
+[Buc] R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-
+cohomology over Gorenstein rings , preprint, 1987, 155 pp; available at
+https://tspace.library.utoronto.ca/handle/1807/1668 2.
+[CCS] P. Caldero, F. Chapoton, R. Schiffler, Quivers with relations arising from clusters ( An
+case), Trans. Amer. Math. Soc. 358 (2006), no. 3, 1347-1364.
+[DWZ] H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potentials and their representa-
+tions I: Mutations , Selecta Math. (N.S.) 14 (2008), no. 1, 59-119.
+[FoZ] S. Fomin, A. Zelevinsky, Cluster algebras. I. Foundations , J. Amer. Math. Soc. 15
+(2002), no. 2, 497-529 (electronic).
+[Hap] D. Happel, On Gorenstein rings , inRepresentation Theory of Finite Groups and
+Finite-Dimensional Algebras (Bielefeld, 1991) , 389-404, Progr. Math. 95, Birkh¨ auser,
+1991.CLUSTER TILTING AND COMPLEXITY 11
+[Iya] O. Iyama, Maximal orthogonal subcategories of triangulated categor ies satisfying Serre
+duality, Oberwolfach Report (2005), 355-357.
+[Kel] B. Keller, On triangulated orbit categories , Doc. Math. 10 (2005), 551-581.
+[KeR] B. Keller, I. Reiten, Cluster-tilted algebras are Gorenstein and stably Calabi- Yau, Adv.
+Math. 211 (2007), no. 1, 123-151.
+[Kra] H. Krause, Stable equivalence preserves representation type , Comment. Math. Helv. 72
+(1997), no. 2, 266-284.
+[Ric] J. Rickard, Derived categories and stable equivalence , J. Pure Appl. Algebra 61 (1989),
+no. 3, 303-317.
+[Ri1] C. M. Ringel, Tame algebras and integral quadratic forms , Lecture Notes in Mathe-
+matics, 1099, Springer-Verlag, Berlin, 1984, xiii+376 pp.
+[Ri2] C. M. Ringel, The spectral radius of the Coxeter transformations for a gen eralized
+Cartan matrix Math. Ann. 300 (1994), no. 2, 331-339.
+[Tak] M. Takane, On the Coxeter transformation of a wild algebra , Arch. Math. (Basel) 63
+(1994), no. 2, 128-135.
+Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway
+E-mail address :bergh@math.ntnu.no
+E-mail address :Steffen.Oppermann@math.ntnu.no
\ No newline at end of file
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@@ -0,0 +1,872 @@
+arXiv:1001.0759v2  [astro-ph.CO]  2 Feb 2010Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 29 October 2018 (MN L ATEX style file v2.2)
+Cosmic shear requirements on the wavelength-dependence
+of telescope point spread functions
+E. S. Cypriano1,2, A. Amara3, L. M. Voigt2, S. L. Bridle2, F. B. Abdalla2,
+A. R´ efr´ egier4, M. Seiffert5,6, J. Rhodes5,6
+1Departamento de Astronomia, Instituto de Astronomia, Geof ´ ısica e Ciˆ encias Atmosf´ ericas, Universidade de S˜ ao Pau lo, Rua do Mat˜ ao 1226, Cidade Universit´ aria 05508-900, S˜ ao Paulo, SP, Brazil
+2Department of Physics and Astronomy, University College Lo ndon, Gower Street, London, WC1E 6BT, UK
+3Department of Physics, ETH Z¨ urich, Wolfgang-Pauli-Stras se 16, CH-8093 Z¨ urich, Switzerland.
+4Service d ´Astrophysique, CEA Saclay, 91191 Gif sur Yvette, France
+5Jet Propulsion Laboratory, California Institute of Techno logy, 4800 Oak Grove Drive, Pasadena, CA 91109
+6California Institute of Technology, 1201 E California Blvd ., Pasadena, CA 91125, USA
+Accepted . Received ; in original form
+ABSTRACT
+Cosmic shear requires high precision measurement of galaxy shapes in the presence
+of the observational Point Spread Function (PSF) that smears ou t the image. The
+PSF must therefore be known for each galaxy to a high accuracy. H owever, for several
+reasons, the PSF is usually wavelength dependent, therefore the differences between
+the spectral energy distribution of the observed objects introd uces further complexity.
+In this paper we investigate the effect of the wavelength-depende nce of the PSF,
+focusing on instruments in which the PSF size is dominated by the diffra ction-limit
+of the telescope and which use broad-band filters for shape measu rement.
+We first calculate biases on cosmological parameter estimation from cosmic shear
+when the stellar PSF is used uncorrected. Using realistic galaxy and s tar spectral
+energy distributions and populations and a simple three-component circular PSF we
+find that the colour-dependence must be taken into account for t he next generation of
+telescopes.Wethenconsidertwodifferentmethodsforremovingt heeffect(i)theuseof
+starsofthesamecolourasthe galaxiesand(ii)estimationofthegala xyspectralenergy
+distribution using multiple colours and using a telescope model for the PSF. We find
+that both of these methods correct the effect to levels below the t olerances required
+for per-cent level measurements of dark energy parameters. C omparison of the two
+methods favours the template-fitting method because its efficienc y is less dependent
+on galaxy redshift than the broad-band colour method and takes f ull advantage of
+deeper photometry.
+Key words: cosmology: observations - gravitational lensing - large-scale stru cture
+1 INTRODUCTION
+Measurements of the cosmic shear signal are expected to
+play a leading role in furthering our understanding of our
+Universe, in particular the nature of dark matter and dark
+energy or its possible alternatives such as modifications in
+gravity.The gravitational lensing oflight from distant ga lax-
+ies by intervening mass provides a powerful insight into the
+growthofstructureandtheexpansionhistoryoftheunivers e
+(for recent reviews see Hoekstra & Jain 2008; Munshi et al.
+2008; Refregier 2003).
+Several planned future dark energy missions are de-
+signed with weak lensing as a primary science driver,including ground-based projects: the KIlo-Degree Survey
+(KIDS)1; the Panoramic Survey Telescope and Rapid Re-
+sponse System (Pan-STARRS)2; the Dark Energy Survey
+(DES)3;andtheLargeSynopticSurveyTelescope (LSST)4,
+and space missions Euclid5(Laureijs 2009; Refregier et al.
+2010) and the Joint Dark Energy Mission (JDEM)6. The
+1http://www.astro-wise.org/projects/KIDS
+2http://pan-starrs.ifa.hawaii.edu
+3http://www.darkenergysurvey.org
+4http://www.lsst.org
+5http://www.euclid-imaging.net
+6http://jdem.gsfc.nasa.gov
+c/circlecopyrt0000 RAS2E. S. Cypriano et al.
+success of the method relies on accurate measurement of
+galaxy shapes (e.g. Heymans et al. 2006; Massey et al. 2007;
+Bridle et al. 2009).
+As light from galaxies passes through the atmosphere,
+telescope optics and measurement devices it is convolved
+with a kernel, referred to as the Point Spread Function
+(PSF). The size of the PSF is similar to the size of the galax-
+ies used to measure cosmic shear, making accurate determi-
+nation of the underlying galaxy shape a significant challeng e
+(e.g. Lewis 2009; Voigt & Bridle 2009). Precise modelling of
+the PSF is crucial (e.g. Paulin-Henriksson et al. 2008). Thi s
+can in principle be performed using accurate knowledge of
+the telescope optics, however in practice it is usual to make
+use of the point-like nature of stars by modelling the PSF
+based on the shapes of stellar images. The situation is com-
+plicated by (i) variations in the shape of the PSF across
+the detector and (ii) the wavelength-dependence of the PSF
+shape. In this paper we concentrate on the second of these
+two effects, focusing on a telescope which is nearly diffrac-
+tion limited and on a instrument that uses broad-band op-
+tical filters for shape measurement.
+Stars and galaxies have different spectral energy dis-
+tributions (SEDs), both intrinsic and observed, specially if
+we consider that galaxies are observed over a broad range
+of redshifts (See Figure 1). Therefore the PSF shape mea-
+sured from a stellar image will be different from the true
+PSF applied to the galaxy, leading to a bias on the mea-
+sured underlying galaxy shape. The wavelength-dependence
+of the PSF has a greater impact if observations are per-
+formed using broad-band filters. In this paper we quantify
+the effect of ignoring the wavelength-dependence of the PSF
+in a Euclid-like space mission and show the feasibility of re -
+ducing the effect by using colour information. We determine
+the wavelength-dependence of the PSF shape using a sim-
+ple model for the convolution kernel and applying realistic
+galaxy and stellar spectral energy distributions (SEDs).
+For the reasons we will discuss in the following sections
+this paper will focus on the wavelength-dependence of the
+PSF size. There are several measures of this quantity, such
+as the 50we adopt the full width at half-maximum intensity
+(FWHM) as our size parameter given that this is a more
+familiar quantity to the astronomical community and de-
+scribes well the simple PSF models we use in this paper.
+The paper is organised as follows: in Section 2 we set
+out a formalism for propagating a mis-estimation of the PSF
+shape through to biases on cosmological parameters. In Sec-
+tion 3 we describe the separate contributions to the PSF
+model and in Section 4 we predict the range in PSF sizes
+expected for two different broad-band filters using a realis-
+tic distribution of stellar and galaxy SEDs and investigate
+two different methods for reducing the bias. These involve
+using: (i) galaxy and stellar colours to correct the PSF size
+and (ii) template fitting. Also we place requirements on a
+simple model for the wavelength dependence for a Euclid-
+like survey. Finally, in Section 5 we discuss our findings.
+2 IMPLICATIONS OF USING AN
+INCORRECT PSF
+We first make a simple calculation of the shear bias induced
+by mis-estimating the size and ellipticity of the PSF andFigure 1. A sample of stellar and galaxy spectra showing the
+variety of SEDs among astronomical objects. An elliptical g alaxy
+atz= 0.9 (solid red), an irregular galaxy at z= 0.7 (dot-dashed
+blue), an A0V star (dashed green) and a K0III star (dotted ma-
+genta).
+compare it to the general systematics limit calculated in
+Amara & R´ efr´ egier (2007). We then consider the redshift
+dependence of the shear biases and propagate this through
+to biases on cosmological parameters.
+The shear measurement bias caused by an incorrect
+PSF will depend on the shear measurement method em-
+ployed. For simplicity we assume here that shears are cal-
+culated using unweighted quadrupole moments. Although
+this method is not feasible in practice due to the low signal-
+to-noise level in real images, it is related to the widely use d
+Kaiser, Squires and Broadhurst method (Kaiser et al. 1995).
+It has the advantage of being extremely easy to use for shear
+measurement bias calculations.
+Shear mis-estimates are often quantified by Taylor ex-
+panding the estimated two-component shear ˆ γi(i= 1,2) in
+terms of the true shear γias
+ˆγi=miγi+ci (1)
+wheremiis referred to as the multiplicative bias, cias the
+additive bias (Heymans et al. 2006) and it is often assumed
+m1≃m2≡mandc1≃c2≡c.
+Amara & R´ efr´ egier (2007) showed that, for a full-sky
+survey (2 ×104square degrees of extragalactic sky) with
+35 galaxies per square arcminute and a median redshift of
+0.9, the shear multiplicative error mmust be <1×10−3
+to keep systematic biases on cosmological parameters below
+random uncertainties, for a range of possible redshift evo-
+lution scenarios for m. Using the equations in Appendix A
+thistranslatestoarequirementonthePSFsize mis-estimat e
+δFPSFof
+δFPSF
+FPSF≃m/lessorequalslant1×10−3. (2)
+Amara & R´ efr´ egier (2007) also placed a requirement on the
+c/circlecopyrt0000 RAS, MNRAS 000, 000–0003
+mean square error σ2
+sys<10−7for the same survey. Inter-
+preting this as a requirement on the square of the shear
+measurement additive error c2, and neglecting the subdom-
+inant term in Eq. A8, this places a requirement on the PSF
+ellipticity mis-estimate δǫPSFiof
+δǫPSFi≃4c/lessorequalslant1×10−3. (3)
+Because theobservedSEDofagalaxy changes with red-
+shift, the appropriate PSF will also depend on galaxy red-
+shift. Therefore the PSF biases δFPSFandδǫPSFdepend on
+redshift and so do the multiplicative and additive biases m
+andc. If, for example, the impact of the multiplicative and
+additive biases as a function of redshift mimics a particu-
+lar cosmological parameter the requirements may be more
+stringent than the above more approximate calculation. We
+will therefore calculate PSF biases as a function of galaxy
+redshift and insert them into the more detailed calculation
+described below, which propagates the effect into biases on
+cosmological parameters.
+Use of the wrong PSF model will cause the measured
+cosmic shear cross power spectra between redshift bins iand
+j,ˆCκ
+ij(ℓ), to differ from the true cosmic shear power spectra
+Cκ
+ij(ℓ). If a particular systematic on the cosmic shear power
+spectrum ∆ Cκ
+ij(ℓ) =ˆCκ
+ij(ℓ)−Cκ
+ij(ℓ) is ignored then the bias
+on cosmological parameters δpαis given by
+δpα=F−1
+αβ/summationdisplay
+ℓ∆Cκ
+ij/parenleftbig
+Cov/bracketleftbig
+Cκ
+ij(ℓ),Cκ
+kl(ℓ)/bracketrightbig/parenrightbig−1∂Cκ
+kl(ℓ)
+∂pβ(4)
+wherei,j,k,landβare summed over, Cov/bracketleftbig
+Cκ
+ij(ℓ),Cκ
+kl(ℓ)/bracketrightbig
+is the two dimensional covariance matrix between the cross-
+spectra and Fis the Fisher matrix between the cosmological
+parameters (Huterer et al. 2006; Amara & R´ efr´ egier 2007).
+In the presence of a redshift dependent multiplicative
+bias, the measured lensing power spectrum can be given in
+terms of the true lensing power spectrum by
+ˆCκ
+ij(ℓ) =Cκ
+ij(ℓ)(1+mi+mj) (5)
+wheremiis the multiplicative bias for redshift bin i, aver-
+aged over all galaxies (Huterer et al. 2006, Eq. 16).
+The impact of additive errors depends to first order
+on the spatial variation of the additive errors. For the
+case of a wavelength dependent PSF this will induce power
+on the scale of the separation between stars of a typical
+colour, which may be propagated into cosmology (see also
+Guzik & Bernstein 2005). Here we focus on PSF size mis-
+estimates and therefore do not consider additive errors fur -
+ther.
+In this paper we compare the PSF sizes for stars with
+those for galaxies, without considering the galaxy morphol -
+ogy or profile. The equations derived above and in the
+Appendix make it possible to draw significant conclusions
+about the cosmology biases independent of considerations
+about the galaxy light distribution, if all parts of the gala xy
+have the same colour. Our main metric is the difference be-
+tween the FWHM of the stellar and galaxy PSFs. We av-
+erage this over populations of galaxies for various differen t
+PSF correction schemes.Figure 2. Contribution from each compoment of the PSF to the
+image quality (FWHM) as a function of wavelength. The dotted
+(blue) line represents the diffraction component, the long- dashed
+(red) line the CCD modulation transfer function component, the
+short-dashed (green) the achromatic component and the thic k
+solid line (magenta) shows the overall result taking into ac count
+all the previously described components. The solid (black) line
+shows the relative instrumental passband using a wide optic al
+filterF1 such as that proposed for Euclid. The dashed (black)
+vertical lines show an alternative, narrower, lensing filte r (F4).
+Thebackground shaded areasshow approximately the wavelen gth
+coverage of the grizyfilters.
+3 THE PSF MODEL
+We consider a simple instrument model in which the PSF is
+made up of three circular components, each with a different
+wavelength-dependence.This is reasonable for aspace-bas ed
+instrument composed mainly of reflective surfaces, such as
+Euclid. For such an instrument the PSF ellipticity is rel-
+atively insensitive to wavelength and the three main PSF
+componentsare (i) nearlydiffraction limited telescope opt ics
+giving rise to an Airy disk with size inversely proportional to
+the wavelength (ii) the CCD modulation transfer function
+(MTF) which tends to spread out higher energy photons
+more than lower energy photons and (iii) a wavelength in-
+dependent part such as telescope jitter. We assume for sim-
+plicity that each component is Gaussian with a wavelength-
+dependent size.
+We describe the total size of the PSF by its FWHM, F,
+which is given by the quadratic sum of the FWHM values
+of the three components
+F2
+PSF(λ) =F2
+D(λ)+F2
+MTF(λ)+F2
+J (6)
+Note that the addition in quadrature works reasonably well
+even if the diffraction limited component is not a Gaussian:
+we find adding FWHMs in quadrature works to better than
+5 per cent accuracy when an Airy disk is convolved with a
+Gaussian of the same FWHM, and improves to better than
+c/circlecopyrt0000 RAS, MNRAS 000, 000–0004E. S. Cypriano et al.
+2 per cent accuracy if the ratio of FWHMs is changed by a
+factor of 4 either way.
+The size of the diffraction limited image is given by
+FD(λ) = 0.154”/parenleftBigD
+1.2m/parenrightBig−1/parenleftBigλ
+7350˚A/parenrightBig
+(7)
+whereDis the diameter of the primary mirror. We take the
+contribution from the CCD MTF to be that measured em-
+pirically for an e2v CCD 231-84 (M. Cropper, priv. comm.),
+which is given approximately by
+FMTF(λ) =
+
+0.11”−0.027”/parenleftBig
+λ
+7000˚A/parenrightBig
+ifλ/lessorequalslant7000˚A
+0.09”−0.007”/parenleftBig
+λ
+7000˚A/parenrightBig
+ifλ >7000˚A(8)
+Finally, we take the contribution to the PSF size from the
+achromatic component to be
+FWHM J= 0.08” (9)
+as appropriate for a Euclid-like instrument.
+We plot the three contributions to the PSF image size
+in Fig. 2 for a 1.2 meter primary mirror. The image size
+is dominated by the diffraction limit of the instrument. As-
+suming the PSF contributions from different wavelengths all
+have the same centroid we can calculate the FWHM of the
+composite PSF from the FWHM of each component and
+the transmitted flux S(λ)T(λ), where S(λ) is the spectral
+energy distribution (SED) of the object and T(λ) is the in-
+strumental plus filter response, by
+F2
+PSF=/integraltext
+S(λ)T(λ)F2
+PSF(λ)dλ/integraltext
+S(λ)T(λ)dλ. (10)
+In this paper we assume T(λ) is the instrumental response
+for the case where a wide optical F1 filter (5500–9200 ˚A;
+see Fig. 2) is used for measuring the shapes of galaxies, as
+proposed for the Euclid satellite.
+4 PSF AND COSMOLOGY BIASES
+We use realistic galaxy and stellar populations to quan-
+tify the amount by which the PSF FWHM is likely to
+be mis-estimated. Galaxy SEDs are generated using mock
+catalogues designed to simulate the distribution of red-
+shifts, colours and magnitudes of galaxies in GOODS-N
+(Cowie et al. 2004; Wirth et al. 2004). Template spectra are
+taken from Coleman et al. (1980) and Kinney et al. (1996)
+and intermediate types obtained by linear interpolation of
+these templates (for further details see Abdalla et al. 2008 ).
+Inthis paper we include galaxies uptoaredshift of2. Galax-
+ies more distant than this will have small apparent sizes
+and are thus unlikely to be used to measure cosmic shear.
+For instance, for the Cosmos survey, much less than 10% of
+the lensing usable galaxies have z >2.0 (Leauthaud et al.
+2007). The PSF sizes for stars are estimated using stel-
+lar SEDs from the Bruzual-Persson-Gunn-Stryker (BPGS)
+Spectrophotometric Atlas7. The catalogue contains 175 dif-
+ferent SEDs covering a broad range of spectral types. The
+true PSF size for each of the star and galaxy types in the
+7http://www.stsci.edu/hst/observatory/cdbs/ astronomi -
+calcatalogs.html#bruzual-persson-gunn-stryker.Figure 3. Histogram of the predicted PSF sizes (FWHM) for the
+mock galaxy catalogue. The average value of the distributio n is
+marked on the plot with an arrow, as well as the sizes for typic al
+disk and halo stars G5IV and K0III.
+mock catalogues is then estimated by inserting its SED into
+Eq. 10.
+A histogram of the FWHM distribution for the galaxy
+population is shown in Fig. 3. The PSF size ranges from ap-
+proximately 0.175” to 0.220”, has an average of 0.1922” and
+a dispersion of 0.006”, or 3% of the mean PSF size. As an
+example, the FWHM of a G5 sub-giant star (taken here as a
+typical disk star) is typically 0.1884 arcsec, whereas a K0 g i-
+ant star (typical halo/bulge star) is typically 0.1897 arcs ec.
+In a conventional analysis which ignores the wavelength-
+dependence of the PSF, galaxies with small angular sepa-
+rations from the above two example stars will often have
+their shears underestimated. In fact, out of all the stars in
+the BPGS catalogue only a quarter of them (cold K and M
+stars) have PSF estimated-sizes larger than the average of
+the galaxies. The multiplicative shear mis-estimates for t he
+example G5 and K0 stars are of the order of m∼3×10−2
+relative totheaverage galaxy, inclear disagreement with o ur
+requirements on the PSF size error (Eq. 1) and therefore we
+need to explore methods for mitigating the effect. Below we
+describe two methods for correcting the PSF wavelength-
+dependence using colour information.
+Fluxes are obtained for each object in the mock cat-
+alogues for the filters F1,Y,JandHup to the limiting
+magnitudes 26.25, 24.0, 24.0 and 24.0 respectively (ABmag-
+nitudes, 5 σdetections). In addition, we consider different
+scenarios for the complementary ground based photometry.
+We assume here observations in the filters g,r,i,zandy
+with three different depths each, shallow, medium and deep
+(see Table 1). The fiducial optical depth used in this paper
+(medium)is chosen tocorrespond toaDESor Pan-STARRS
+type survey with two dedicated telescopes (PS2). Shallow
+and deep correspond to a Pan-STARRS type survey with
+one (PS1) or four (PS4) dedicated telecopes.
+c/circlecopyrt0000 RAS, MNRAS 000, 000–0005
+Figure 4. PSF FWHM versus colour for a realistic distribution of galax ies (black dots) and stars (magenta stars) of several differe nt
+spectral types. The continuous line is a second order polyno mial fit to the stellar sequence. The insets show the residual s between the
+galaxy and stellar polynomial FWHM values at a given colour, in units of 10−3arcsec. Results are shown using the default parameters
+for the mirror (1.2m) and survey depth (medium) for the colou rsF1−Y(top left), r−F1 (top right) and r−F4 (bottom right; F4 as
+the shape measuring filter). The effect of using a larger mirro r (1.5m) for r−F1 is also shown (bottom left). F1 and F4 correspond to
+the wavelength ranges 5500–9200 ˚A and 6120–8580 ˚A respectively.
+Table 1. Pan-STARRS (PS1, PS2 and PS4) and DES optical
+photometry depths. The values quoted here correspond to AB
+magnitudes of 5 σdetections.
+Band PS1 PS2 PS4 DES
+g 24.66 25.53 26.10 25.35
+r 24.11 24.96 25.80 24.85
+i 24.00 24.80 25.60 25.05
+z 22.98 23.54 24.10 24.65
+y 21.52 22.01 22.50 22.154.1 Broad-band colour method
+Here we investigate the possibility of estimating the galax y
+PSF from stars with the same broad-band colour. For this
+to work well two conditions must be met ( i) there must be a
+good correlation between a given broad-band colour and the
+PSF size and ( ii) this relation must be the same for both
+stars and galaxies, despite their potentially differingspe ctra.
+To assess the extent to which this applies for our fiducial
+instrument we have plotted the PSF sizes for the galaxy
+c/circlecopyrt0000 RAS, MNRAS 000, 000–0006E. S. Cypriano et al.
+andstellar populations against twodifferentcolours: F1−Y,
+which can be fully determined by an instrument containing
+a single optical filter plus the Yinfra-red band, such as the
+proposed Euclid design; and r−F1, which requires r-band
+observations which could come from the ground. The results
+are shown in the top two panels of Fig. 4.
+It is clear from Fig. 4 that there is a positive corre-
+lation between the PSF FWHM and the colour i.e. redder
+objects tend to be larger than bluer ones. This is unsur-
+prising given that the contribution to the PSF size from the
+diffraction limit of the telescope is the dominant component .
+It is also clear that the correlation between the PSF size and
+ther−F1 colour is tighter than with the F1−Ycolour. We
+expect this to occur because the r−F1 colour constrains
+the slope of the SED within the lensing measuring filter F1,
+whereas the F1−Ycolour constrains the SED slope at red-
+der wavelengths, which are less relevant. This interpretat ion
+is strengthened by the fact that other colours such as r−z,
+g−zandg−Y, which are also able to constrain the slope
+of the SED over the shape measuring filter, produce corre-
+lations almost as tight as those for r−F1.
+For the distribution of stellar SEDs used in this pa-
+per, we can see from Fig. 4 that the stars occupy a well
+defined locus in the FWHM versus colour space. We fit a
+parabola to this data and by doing so we can estimate the
+PSF size from a colour alone. From the figure we see that
+this line fits well through the stellar data points and thus
+we meet condition ( i) for the stars. It is fortunate that the
+wavelength-dependence of the PSF tends to be, to a first ap-
+proximation, relatively stable. This means that it is viabl e to
+use data from several high signal-to-noise stars from sever al
+different fields to empirically find this locus (provided that
+the wavelength independent effects on the PSF are properly
+dealt with). To test condition ( ii) we have to assess how
+well the PSF size-wavelength-dependence for galaxies fol-
+lows that of the stars. We therefore calculate the residuals
+between the FWHM of the galaxies and the fit to the stellar
+FWHM-colour relation (shown in the insets of the figure).
+The bias on the PSF size, /angbracketleftδFPSF/angbracketright, is reduced dra-
+matically by including ground-based photometry, decreas-
+ing in magnitude from 3 .6×10−3arcsec for F1−Yto
+0.16×10−3arcsec for r−F18. For a simple calculation
+in which the dependence of the multiplicative bias on red-
+shift is not taken into account, we find from Eq. 2 that
+δFPSFmust be <0.2×10−3arcsec for a typical PSF size
+ofF= 0.2 arcsec. The inclusion of ground-based photome-
+try thus allows us to meet our simple ‘back-of-the-envelope ’
+requirement (Eq. 2). Using a shallower r-band photometry
+depth (PS1-like) increases the bias by a factor of 2.5 from
+the value with the fiducial (medium) depth. However, using
+a deeper ground-based photometry depth (PS4-like) has no
+effect on the bias. This suggests that there is an intrinsic
+difference between the distribution of PSF sizes for typical
+star and galaxy SEDs (measured in the F1 band) with the
+samer−F1 colour, which is not reduced by increasing the
+photometry depth beyond ‘medium’.
+The wavelength-dependence of the PSF size can be re-
+8Unless otherwise stated, the photometry depths are for a
+‘medium-depth’ survey, corresponding to a PS2 survey (see T a-
+ble 1 for depths in the g,r,i,zandybands).duced by increasing the size of the primary mirror (since
+the telescope is diffraction-limited). We find that the bias i s
+reduced by a factor of 1.6 by increasing the mirror diame-
+ter from the fiducial value (1.2m) to 1.5m (Fig. 4 bottom-
+left panel)9. For instance, for a space telescope, such as
+the HST, with a 2.4m mirror, the diffraction component
+will contribute the same as the CCD MTF and the jitter
+at∼7500˚A. In this case the overall optics becomes much
+less chromatic, in particular for bluer wavelengths where t he
+CCD MTF partially compensates the diffraction effect. This
+is directly reflected in the bias we estimate as /angbracketleftδFPSF/angbracketrightgets
+to be as small as 0 .03×10−3arcsec (less than 5 times the
+value of our default configuration). On the other hand a
+mirror as small as 0.6m will induce a bias in the PSF size
+of 0.46×10−3arcsec, or 2.5 times larger than the bias we
+obtained with the fiducial configuration.
+Another way toreduce the chromaticity of the system is
+byusingashapemeasurmentfilterF4whose passbandis2/3
+narrower than F1 (See Fig. 2).By using the narrower shape
+measurement filter we increased to absoute bias value from
++0.16 to -0.18 miliarcsec. As we can see in Fig. 4 (bottom-
+right panel), this configuration indeed decreases the slope
+of the PSF FWHM–colour relation for stars and galaxies.
+However, for this particular colour of choice ( r−F4), the
+‘matching-up’ of stars and galaxies becomes poorer. If, for
+instance, we use the colour g−zinstead we get a bias of
+-0.11, that captures the improvement we expected by using
+a narrower filter.
+4.1.1 Redshift dependence
+Galaxies with the same intrinsic spectra will have different
+observed colours as a result of the range of galaxy redshifts .
+The bias on the PSF FWHM is thusredshift-dependent, and
+could potentially be very damaging to weak-lensing tomog-
+raphy. In the previous section we calculated the mean bias
+on the PSF FWHM for the galaxy SEDs averaged over all
+redshifts included in the catalogue ( z <2). In this section
+we investigate the effect of the redshift dependence of the
+galaxy colours.
+Fig 5 shows the redshift dependence of δFPSFfor the
+fiducial scenario (dashed red line). There is a large negativ e
+bias at high redshift which will affect a small fraction of
+galaxies. We also see clearly the biggest limtation of our
+global average method, since in that method positive and
+negative contributions will cancel.
+We propagate this through to dark energy related cos-
+mological parameters as explained in Section 2 and divide
+the biases on the parameter values by the statistical errors
+on the parameters found using the standard Fisher matrix
+approach, for a 20,000 square degree survey with 35 galaxies
+per square arcminute and a total uncertainty on each shear
+component of σγ= 0.35. The results are shown in Table 2.
+We see that, for all configurations all the dark energy
+parameter biases we get are smaller that the expected sta-
+tistical errors. In particular, for wa, the most demanding
+parameter those surveys are trying to determine, the bias
+9For these simulations the changes in the instrumental config u-
+ration do not change the signal-to-noise of the observation s, thus
+the same galaxies are observed in all cases.
+c/circlecopyrt0000 RAS, MNRAS 000, 000–0007
+Table 2. Biases on dark energy related cosmological parameters divi ded by the statistical uncertainty (0.035, 0.045 and 0.149 f or ΩDE,
+w0andwarespectively) on the cosmological parameter. Results are s hown for each correction method.
+Colour Information DLensing Photometry b(ΩDE)/σ(ΩDE)b(w0)/σ(w0)b(wa)/σ(wa)
+(m) filter depth
+Broad-band colour method
+r−F1 1.2 F1 Medium -0.05 0.17 -0.47
+r−F1 1.2 F1 Shallow -0.15 0.34 -0.79
+r−F1 1.2 F1 Deep -0.05 0.17 -0.47
+r−F1 1.5 F1 Medium -0.04 0.14 -0.36
+r−F4 1.2 F4 Medium -0.10 0.35 -0.49
+Template-fitting method
+F1,Y,J,H,g,r,i,z,y 1.2 F1 Medium 0.03 0.10 -0.43
+F1,Y,J,H,g,r,i,z,y 1.2 F1 Shallow 0.12 0.18 -0.72
+F1,Y,J,H,g,r,i,z,y 1.2 F1 Deep -0.02 0.06 -0.24
+F1,Y,J,H,g,r,i,z,y 1.5 F1 Medium 0.02 0.07 -0.31
+F4,Y,J,H,g,r,i,z,y 1.2 F4 Medium 0.01 0.07 -0.15
+Figure 5. Residual between the actual and the best estimate
+galaxy PSF FWHM as a function of redshift with the broad-band
+method using the r−F1 colour (red dashed) and the template-
+fitting method using the shape measurement filters F1(5500–
+9200˚A) (black solid) and F4 (6120–8580 ˚A) (blue dot-dashed).
+corrected by the broad-band colour method is less than a
+half of the statistical error level for our fiducial survey co n-
+figuration. The other trends follow those already discussed
+in the context of the global averaged biases discussed in the
+beginning of this Section.
+4.2 Template-fitting method
+The previous section assumes that we use a single colour to
+determine the correct PSF model to use for a given galaxy.
+In practice we will have more than two filters, which will
+be used to calculate photometric redshifts. We therefore
+also consider the use of a template fitting method to pre-
+dict the PSF FWHM of a galaxy by using all the colours
+available. Using ANNz (Collister & Lahav 2004) we trainedand validated a neural network using approximately one
+third of the simulated galaxies available, to predict the re d-
+shifts, spectral types and reddening of each galaxy, given
+the multi-colour information. With this information we can
+compute the SED of each object and use a model of the PSF
+wavelength-dependence to predict the PSF FWHM for this
+galaxy.
+A telescope model and stars will be used to build this
+model for the wavelength-dependence, and the accuracy of
+the model will depend on the stability of the wavelength-
+dependence with telescope properties, and the number of
+stars available to calibrate which model to use. In the case
+of the Hubble Space Telescope, a PSF model taken from
+the telescope design is routinely used in conjunction with
+calibration from any stars in the field to assess the telescop e
+configuration in a given observation. Therefore in this pape r
+we use the exact model as given in Eq. 10 and propagate the
+noisy and potentially biased galaxy SED estimates through
+to PSF biases and cosmological parameter biases.
+The comparison between this predicted PSF FWHM
+and the truth for the exact galaxy SED and redshift can be
+seen in Figure 6. We can also compare the global averaged
+bias to compare with the results we got from the broad-band
+method and understadn the trends. By using our standard
+configuration we obtain a bias of /angbracketleftδFPSF/angbracketright= +0.19 miliarc-
+sec, which is similar from what we got by using one single
+colour and also met our back-of-envelope requirment (See
+Fig. 6). The use of deeper photometry, a larger 1.5m mirror
+or the narrower lensing filter F4 reduced /angbracketleftδFPSF/angbracketrightto +0.18,
++0.13 and +0.12 respectively, following the expected trend s.
+The redshift dependence of the PSF FWHM bias is
+shown for the fiducial scenario (solid black line) and for a
+senario with the lensing filter F4 (dot-dashed blue line) in
+Fig. 5. Both cases show much less redshift evolution than
+the broad-band colour method, in particular for z >1.0.
+We propagate this (and all the other scenarios) through
+into biases on cosmological parameters and find the results
+given in the second line of Table 2. All the biases are smaller
+than the statistical errors. In Fig. 7 we show the statistica l
+confidence level uncertainties in the space w0−wafor our
+fiducial scenario. It can be seen in the figure that the resid-
+ual bias after the correction of the wavelength-dependence
+is well within the 68% CL. In comparison to the broad-
+band colour method the template fitting method produces
+c/circlecopyrt0000 RAS, MNRAS 000, 000–0008E. S. Cypriano et al.
+Figure 6. Comparison between the actual PSF sizes for objects
+with the SED of the galaxies of the mock catalogue and calcula ted
+PSF sizes given redshifts, reddening and spectral types obt ained
+through ANNz.
+smaller biases on cosmological parameters and takes more
+advantage of the deeper photometry. In this case by using
+the deepest photometry (PS4-like) we could see actual im-
+provement when compared to the default, medium depth
+(PS2/DES-like).
+4.3 Requirements on a simple
+wavelength-dependence model
+Now we consider requirements on the parameters of a
+simple model for PSF FWHM wavelength-dependence for
+the template fitting method. We approximate the FWHM-
+wavelength relation to be a simple linear function. The fidu-
+cial configuration is most closely approximated by a straigh t
+line whose slope is equal to 1 .63×10−5arcsec/˚A with
+FWHM(7350 ˚A) = 0.192”. We consider a range of slopes,
+from a pure wavelength independent PSF to a relation twice
+as steep as the fiducial configuration. In all the cases we kept
+the PSF FWHM constant at λ= 7350˚A, which are the cen-
+tral wavelenth of the lensing filters we are considering. In
+Figure 8 we show the variation of the systematic to statis-
+tical error for the dark energy parameters Ω DE,w0andwa
+for the case where the wavelength-dependence of the PSF
+has been corrected using the template-fitting method.
+In this Figure, as expected, one can see that the bias
+on cosmological parameters increases as the wavelength-
+dependence of the PSF gets stronger. For all dak energy pa-
+rameters thebiases lie within theregion where thestatisti cal
+errors are larger than the systematic ones |b(p)/σ(p)|<1.
+For the evolution of the dark energy equation of state pa-
+rameter the normalised bias |b(wa)/σ(wa)|approaches unity
+when the slope is double the value for the fiducial telescope
+mirror size and filter width.Figure 7. Bias in the dark energy equation of state parameters.
+The ellipse shows the region of the w0−waspace contained in
+the 68% confidence level contour for a weak-lensing Euclid-t ype
+survey, where the central values are respectively 0.0 and -0 .95
+(marked by a cross). The asterisk shows the effect of the resid ual
+bias induced by the wavelength-dependence effect.
+Figure 8. Variation of the systematic to statistical error ratio
+(b(p)/σ(p)) of dark energy parameters as a function of the slope
+of the PSF FWHM wavelength relation. The dashed (green), dot -
+dashed (blue) and solid (red) lines represent the error rati o for
+the parameters Ω DE,w0andwa, respectively.
+c/circlecopyrt0000 RAS, MNRAS 000, 000–0009
+5 DISCUSSION
+The wavelength-dependence of the point spread function is
+an effect that has to be carefully considered for the next
+generation of cosmic shear experiments, particularly if wi de
+bands are used for imaging. The different spectral energy
+distributions of stars and galaxies means that the point
+spread function obtained from stars is not the same as that
+for the galaxies and this can lead to a non-negligible bias in
+shear measurements.
+We have, for the first time, set out a formalism for test-
+ing the wavelength-dependence of the point spread function
+for diffraction limited imaging, using the parameters of a
+Euclid-like survey. Given these characteristics the domin ant
+wavelength-dependence comes from the PSF size, which we
+parameterise by the PSF full-width at half maximum inten-
+sity. We have shown that the fractional difference in PSF
+FWHM between the stars and the galaxies must be smaller
+than 1×10−3. We find the same fractional requirement on
+the PSF ellipticity difference. We have illustrated the for-
+malism on a fiducial Euclid-like telescope for which only the
+PSF size is significantly wavelength dependent.
+Weinvestigatedtwodifferentmethodsfor correctingthe
+effect and found that although they give similar results for
+the average PSF error, thedifferent dependencies of thePSF
+size bias on redshift leads to very different implications fo r
+biases on cosmological parameters. For this type of analysi s
+it is therefore necessary to take into account the redshift-
+ing of galaxy spectra and the cosmological parameters of
+interest.
+The first correction method we consider matches stars
+of a given colour to galaxies of the same colour. This is
+not expected to be a perfect correction method because
+the spectral energy distributions of two objects with the
+same colour are different. The stars and galaxies need to be
+matched up so they have similar SEDs within the imaging
+band used for cosmic shear galaxy shape measurement. This
+is best achieved if the colour considered matches well to the
+imaging band. We find that a telescope with a wide optical
+band plus infra-red bands cannot sufficiently self-correct f or
+PSF wavelength-dependence using the optical minus infra-
+red colour ( F1−Y). This is expected because the difference
+in luminosity between bands so widely spaced is not well
+correlated with the variation in luminosity within the opti -
+cal band itself. The PSF FWHM of the stars is on average
+3.6×10−3arcsec and a fractional error of 18 ×10−3, much
+greater than our requirement (1 ×10−3) .
+To realise the full potential of cosmic shear, all planned
+surveys will estimate the galaxy redshift using photometri c
+redshifts. This places stringent requirements on having ad -
+ditional photometry in multiple wavebands, which may be
+obtained from the ground or from space. This provides the
+ideal input into correction for the wavelength-dependence of
+the PSF. We consider the colour r −F1, which provides much
+more useful information about the spectral energy distribu -
+tion in the F1 optical band. The average difference in PSF
+FWHM between the stars and galaxies now meets our re-
+quirement. The redshift dependence shows some evolution
+and tends to present larger values for redshifts larger than
+1.0. However we find that the dark energy equation of state
+evolution parameter will not be biased by more than the sta-tistical error if this method is used, even in the case where
+we use a shallow ground based photometry.
+We therefore consider an additional method in which
+the full range of available wavebands are used, to match
+those used in photometric redshift estimation. We take ad-
+vantage of the fact that many methods for estimating the
+galaxy photometric redshift also provide an estimate of the
+galaxy spectral energy distribution. This allows a full mod el
+of the galaxy spectrum to be used in correcting for the PSF
+wavelength-dependence. The extent to which this is help-
+ful depends on how well the instrument PSF wavelength-
+dependence is already known. We do already have a reason-
+able model for the wavelength-dependence. Additionally we
+expect the wavelength dependence to be quite stable as a
+function of time, and furthermore it should be very well cal-
+ibratable using stars. In this work we assume that the model
+for the wavelength-dependence is therefore very well known ,
+and the limiting factor in the analysis comes from uncertain -
+ties in the galaxy photometry, which limit our knowledge of
+the galaxy redshift and spectral energy distribution. We fin d
+for the several cases we tested that the redshift dependence
+of the bias is smaller, when compared to the broad-band
+method and, as a consequence, the dark energy cosmologi-
+cal parameter biases are also smaller.
+Weconsider the effectofadifferenttelescope mirror size
+and imaging filter width. We find that the large mirror size
+does decrease the biases, as expected due to the smaller con-
+tribution to the wavelength-dependence from the diffractio n
+limit. The reduction is around 20 per cent for both correc-
+tion methods. Using a narrower filter reduces the scatter
+on the PSF FWHM but does not decrease the bias, for our
+particular configuration.
+Finally we consider a general linear PSF wavelength-
+dependence which matches well to the Euclid-like fidu-
+cial configuration. This allows a requirement on the linear
+slope of the PSF to be obtained for a given required ac-
+curacy on cosmological parameters. We find that, by us-
+ing the template fitting method, a survey in which the
+wavelength-dependence of the PSF is twice as strong as our
+fiducial Euclid-like survey just meets the requirement that
+|b(p)/σ(p)|<1.
+In this work we have studied the first order effects
+of a wavelength dependent PSF and have shown that this
+effect can be mitigated with the addition of photometric
+data. The next step is to consider higher order effects such
+as colour gradients and the spatial correlation function of
+galaxy colours. These issues will be tackled and discussed i n
+later work.
+ACKNOWLEDGEMENTS.
+Thispaperhas beentypesetfrom aT EX/LATEXfileprepared
+by the author.
+We are grateful to Peter Capak for providing the code
+used to generate the mock photometric catalogue. We thank
+Mark Cropper, Jerome Amiaux, Peter Doel, Ofer Lahav,
+James Kingston, Steve Kent, Michelle Antonik, Stephane
+Paulin-Henriksson, Gary Bernstein, Tom Kitching, Alexie
+Leauthaud and Gary Bernstein for helpful conversations.
+ESC acknowledges support from FAPESP (process num-
+c/circlecopyrt0000 RAS, MNRAS 000, 000–00010E. S. Cypriano et al.
+ber 2009/07154-8). LMV acknowledges support from STFC.
+SLB and FBA thanks the Royal Society for support in the
+form of a University Research Fellowship. The research de-
+scribed in this paper was performed in part at the Jet
+Propulsion Laboratory, California Institute of Technolog y,
+under a contract with the National Aeronautics and Space
+Administration.
+REFERENCES
+Abdalla F. B., Amara A., Capak P., Cypriano E. S., La-
+hav O., Rhodes J., 2008, Monthly Notices of the Royal
+Astronomical Society, 387, 969
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+Astronomical Society, 381, 1018
+Bridle S., et al., 2009, Annals of Applied Statistics, 3, 6
+Coleman G. D., Wu C.-C., Weedman D. W., 1980, Astro-
+physical Journal Supplement Series, 43, 393
+Collister A. A., Lahav O., 2004, Publications of the Astro-
+nomical Society of the Pacific, 116, 345
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+A., 2004, Astronomical Journal, 127, 3137
+Guzik J., Bernstein G., 2005, Physical Review D, 72,
+043503
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+Astronomical Society, 368, 1323
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+Particle Science, 58, 99
+Huterer D., Takada M., Bernstein G., Jain B., 2006,
+Monthly Notices of the Royal Astronomical Society, 366,
+101
+Kaiser N., Squires G., Broadhurst T., 1995, Astrophysical
+Journal, 449, 460
+Kinney A. L., Calzetti D., Bohlin R. C., McQuade K.,
+Storchi-Bergmann T., Schmitt H. R., 1996, Astrophysi-
+cal Journal, 467, 38
+Laureijs R., 2009, ArXiv e-prints 0912.0914
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+ment Series, 172, 219
+Lewis A., 2009, ArXiv e-prints 0901.0649
+Massey R., et al., 2007, Monthly Notices of the Royal As-
+tronomical Society, 376, 13
+Munshi D., Valageas P., van Waerbeke L., Heavens A.,
+2008, Physics Reports, 462, 67
+Paulin-Henriksson S., Amara A., Voigt L., Refregier A.,
+Bridle S. L., 2008, Astronomy and Astrophysics, 484, 67
+Refregier A., 2003, Ann. Rev. Astron. Astrophys., 41, 645
+Refregier A., Amara A., Kitching T. D., Rassat A.,
+Scaramella R., Weller J., Euclid Imaging Consortium f. t.,
+2010, ArXiv e-prints 1001.0061
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+Wirth G. D., et al., 2004, Astronomical Journal, 127, 3121
+APPENDIX A: MULTIPLICATIVE AND
+ADDITIVE SHEAR MEASUREMENT ERRORS
+FROM PSF MIS-ESTIMATES
+We follow Paulin-Henriksson et al. (2008), who define ob-
+ject size Rand two-component ellipticity ǫin terms of un-weighted quadrupole moments Qijas
+R2=Q11+Q22 (A1)
+ǫ1=Q11−Q22
+Q11+Q22(A2)
+ǫ2=2Q12
+Q11+Q22. (A3)
+It is shown in Paulin-Henriksson et al. (2008) that the sys-
+tematic bias on a galaxy ellipticity component δǫsys
+galican be
+approximated in terms of the mis-estimates of the PSF size
+δRPSFand PSF ellipticity δǫPSFias
+δǫsys
+gali≈/parenleftbigg
+RPSF
+Rgal/parenrightbigg2/parenleftBig
+2(ǫgali−ǫPSFi)δRPSF
+RPSF−δǫPSFi/parenrightBig
+(A4)
+whereǫgaliare the original (pre-PSF but post shear) galaxy
+ellipticity components and Rgalis the galaxy size. Similar
+definitions apply for the PSF ellipticity and size. This as-
+sumes the PSF and galaxy size and ellipticity measurements
+are made usingunweightedquadrupolemoments. This prop-
+agates into a bias on the shear estimate ˆ γi=γi+δγias
+δγi=δǫsys
+gali
+Pγ(A5)
+wherePγis the shear responsivitygiven by Pγ= 2−/angbracketleft|ǫ|2/angbracketright ∼
+1.8 for simple shear measurement methods. Similarly γi=
+ǫgali/Pγ.
+In this paper we quantify object sizes in terms of the
+slightly more intuitive quantity, the Full-Width at Half-
+Maximum (FWHM), Fwhere
+F= 2√
+ln2R (A6)
+for a Gaussian profile, which we use for the PSF component
+in this paper. We combine Eq. 1, Eq. A6 and Eq. A5 and
+compare with Eq. A4 to find
+m= 2/parenleftbigg
+RPSF
+Rgal/parenrightbigg2δFPSF
+FPSF(A7)
+ci=−1
+Pγ/parenleftbigg
+RPSF
+Rgal/parenrightbigg2/parenleftBig
+2ǫPSFiδFPSF
+FPSF+δǫPSFi/parenrightBig
+(A8)
+where the second term in cidominates for typical future sur-
+veys because ǫPSFiis small, in addition to the small value
+of the fractional uncertainty in the PSF FWHM. We have
+ignored the contribution to cifrom intrinsic galaxy elliptic-
+ities, as appropriate for the case where the average ciover
+randomlyorientedgalaxyellipticities isrequired.Where use-
+ful we have converted PSF sizes into FWHM values but we
+have left galaxy sizes written using Rsince the conversion
+fromRtoFis dependent on object profile, and is far from
+Gaussian for galaxies. As in Paulin-Henriksson et al. (2008 )
+we assume Rgal/greaterorequalslant1.5RPSFand thus RPSF/Rgal≃0.5. Thus,
+as expected, multiplicative errors arise from errors in the
+PSF size and additive errors arise from errors in the PSF
+ellipticity.
+c/circlecopyrt0000 RAS, MNRAS 000, 000–000
\ No newline at end of file
diff --git a/1001.0760.txt b/1001.0760.txt
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index 0000000000000000000000000000000000000000..de414876f7dc02cf3fb1a3f8b3e7c6243a309bb1
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+arXiv:1001.0760v2  [hep-ph]  7 Jan 20107 January 2010
+arXiv:1001.0760v2 [hep-ph]
+No Effect of Majorana Phases in Neutrino
+Oscillations
+C. Giunti
+INFN, Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Ita ly
+Abstract
+It is shown that the Majorana phases of the neutrino mixing ma trix cannot
+have any effect in neutrino oscillations, contrary to the argu ment presented in
+arXiv:0912.5266 . It is also shown that in a charged-current weak interaction pro-
+cess it is not possible to create a coherent superposition of different flavor neutrinos
+which is independent of the associated charged leptons.
+It has been argued in Ref. [1] that it is possible to observe the effect of the Majorana
+phases of the neutrino mixing matrix in a neutrino oscillation experimen t with an initial
+beam described by a superposition of different neutrino flavors. Un fortunately, the argu-
+ment presented in Ref. [1] is invalid, as shown in the following. It is also in contradiction
+with the well known proof that the Majorana phases do not have an y effect in neutrino
+oscillations in vacuum [2–4] as well as in matter [5].
+In the following we consider the simple νe-νµtwo-neutrino mixing framework con-
+sidered in Ref. [1]. In this framework, the left-handed flavor neutr ino fieldsνeL(x) and
+νµL(x) are unitary combinations of the left-handed massive neutrino field sν1L(x) and
+ν2L(x) with respective masses m1andm2(see Ref. [6]):
+νaL(x) =/summationdisplay
+k=1,2UakνkL(x) (a=e,µ), (1)
+whereUis the 2×2 unitary mixing matrix. Flavor neutrinos are produced and detecte d
+through charged-current (CC) weak-interaction processes de scribed by the CC weak-
+interaction Lagrangian
+LCC(x) =g√
+2/summationdisplay
+a=e,µaL(x)γρνaL(x)Wρ(x)+H.c.
+=g√
+2/summationdisplay
+a=e,µ/summationdisplay
+k=1,2aL(x)γρUakνkL(x)Wρ(x)+H.c., (2)
+wheregis the coupling constant. In general, the 2 ×2 unitary mixing matrix can be
+written as
+U=/parenleftbiggcosϑeiω1sinϑei(ω1+α)
+−sinϑei(ω2−α)cosϑeiω2/parenrightbigg
+. (3)
+1However, two of the three phases are unphysical, because they c an be eliminated by
+rephasing the two Dirac charged-lepton fields without affecting the respective kinetic
+and mass Lagrangians (the free Dirac Lagrangian is invariant under rephasing of the
+field) and the Lagrangians of the other interactions to which charg ed leptons take part
+(electromagnetic and weak neutral-current). On the other hand , a Majorana field cannot
+be rephased, because the Majorana mass term is not invariant und er rephasing of the
+field (see Ref. [6]). Therefore, if the massive neutrino fields are of M ajorana type, one
+of the three phases in Eq. (3), or a linear combination of them, is phy sical [2–4,7]. This
+phase is called “Majorana phase”. The choice of the Majorana phas e is arbitrary. Let us
+consider two possibilities:
+(1) Rephasing the charged-lepton fields by
+eL(x)→eiω1eL(x), µ L(x)→ei(ω2−α)µL(x), (4)
+we obtain
+U(1)=/parenleftbiggcosϑsinϑeiα
+−sinϑcosϑeiα/parenrightbigg
+. (5)
+This is the most convenient choice, because it allows to write the mixing matrix as
+U(1)=/parenleftbigg
+cosϑsinϑ
+−sinϑcosϑ/parenrightbigg/parenleftbigg
+1 0
+0eiα/parenrightbigg
+=R(ϑ)D(α). (6)
+In this form, the Majorana phase αis factorized in the diagonal matrix D(α) =
+diag(1,eiα) on the right of the mixing matrix.
+(2) Rephasing the charged-lepton fields by
+eL(x)→eiω1eL(x), µ L(x)→eiω2µL(x), (7)
+we obtain
+U(2)=/parenleftbiggcosϑsinϑeiα
+−sinϑe−iαcosϑ/parenrightbigg
+=D†(α)R(ϑ)D(α). (8)
+This is the choice adopted in Ref. [1].
+In order to investigate if there is any effect due to the Majorana ph aseαin neutrino
+oscillations, following Ref. [1] we consider an initial generic neutrino st ate which is a
+superposition of νeandνµ:
+|ν(0)∝angbracketright=Ae|νe∝angbracketright+Aµ|νµ∝angbracketright. (9)
+Since the mixing of states is given by
+|νa∝angbracketright=/summationdisplay
+k=1,2U∗
+ak|νk∝angbracketright(a=e,µ), (10)
+we have
+|ν(0)∝angbracketright=/parenleftbig
+AeU∗
+e1+AµU∗
+µ1/parenrightbig
+|ν1∝angbracketright+/parenleftbig
+AeU∗
+e2+AµU∗
+µ2/parenrightbig
+|ν2∝angbracketright. (11)
+The probability of νedetection at a time tis given by
+Pνe(t) =|∝angbracketleftνe|ν(t)∝angbracketright|2. (12)
+2Considering ultrarelativistic neutrinos, we obtain
+Pνe(t) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleAe|Ue1|2+AµUe1U∗
+µ1+/parenleftbig
+Ae|Ue2|2+AµUe2U∗
+µ2/parenrightbig
+exp/parenleftbigg
+−i∆m2t
+2E/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+,(13)
+where ∆m2=m2
+2−m2
+1andEis the neutrino energy.
+Using the choice of phases U(1)of the mixing matrix, it is clear that there is no effect
+of the Majorana phase in the oscillation probability, since the contrib ution ofαinUe2
+andU∗
+µ2cancels in the product:
+P(1)
+νe(t) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleAecos2ϑ−Aµcosϑsinϑ+/parenleftbig
+Aesin2ϑ+Aµcosϑsinϑ/parenrightbig
+exp/parenleftbigg
+−i∆m2t
+2E/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+.
+(14)
+This is enough to confute the argument presented in Ref. [1] that it is possible to
+observe the effect of the Majorana phases of the neutrino mixing m atrix in a neutrino
+oscillation experiment with an initial beam described by a superposition of different
+neutrino flavors. One can see it also by noting that the probability Pνe(t) in Eq. (13) is
+invariant under the rephasing
+Uak→eiφkUak(a=e,µ;k= 1,2), (15)
+which corresponds to the freedom to rephase arbitrarily the mass ive neutrino fields in
+the Dirac case, when the Majorana phase is unphysical. Indeed, a r ephasing of this type,
+U(1)
+a2→e−iαU(1)
+a2fora=e,µ, eliminates the Majorana phase in Eq. (5).
+It is however interesting to understand why the authors of Ref. [ 1] have been led to an
+incorrect conclusion as a consequence of their choice of phases U(2)of the mixing matrix.
+With this choice of phases the contribution of αdoes not cancel in Eq. (13):
+P(2)
+νe(t) =/vextendsingle/vextendsingle/vextendsingleAecos2ϑ−Aµeiαcosϑsinϑ
++/parenleftbig
+Aesin2ϑ+Aµeiαcosϑsinϑ/parenrightbig
+exp/parenleftbigg
+−i∆m2t
+2E/parenrightbigg/vextendsingle/vextendsingle/vextendsingle2
+. (16)
+The difference between P(1)
+νe(t) andP(2)
+νe(t) is an apparent paradox, because different
+choices of the phases of the mixing matrix cannot induce different ph ysical effects.
+A hint towards the solution of the paradox comes from the fact tha t in Eq. (16) the
+Majorana phase appears always in the product Aµeiα, i.e. as a change of phase of Aµ.
+Hence one is led to ask if the phases of the coefficients AeandAµin the initial neutrino
+state in Eq. (9) have physical meaning. The answer is that these ph ases have no physical
+meaning, because of the phase freedom of the charged-lepton fie lds, which induces a
+phase freedom in the corresponding charged-lepton states and in the associated flavor
+neutrino state. In order to clarify this statement, let us consider the complete leptonic
+state produced in a CC interaction process, for example in W+decay:
+|ψ(0)∝angbracketright=Ae|e+,νe∝angbracketright+Aµ|µ+,νµ∝angbracketright, (17)
+with
+Aa∝ ∝angbracketlefta+,νa|νaL(0)γρaL(0)W†
+ρ(0)|W+∝angbracketright(a=e,µ), (18)
+3at first order in perturbation theory, neglecting the effects of th e neutrino masses in
+the decay process. From Eq. (18) one can see that the phase fre edom of the charged-
+lepton fields implies that the phases of the initial amplitudes AeandAµare arbitrary
+and unphysical.
+This means that the state (9) cannot be realized in nature. In othe r words, it is not
+possible to create a coherent superposition of |νe∝angbracketrightand|νµ∝angbracketright. In fact, if the charged lepton
+is detected, the neutrino state is reduced to either |νe∝angbracketrightand|νµ∝angbracketright. On the other hand, if
+the charged lepton is not detected, one must sum all neutrino inter action probabilities
+over the different charged lepton contributions. This is equivalent o f considering the
+neutrino state as an incoherent superposition of |νe∝angbracketrightand|νµ∝angbracketright, which is better described
+by a density matrix. In this case we have
+Pνe(t) =/summationdisplay
+a=e,µ/vextendsingle/vextendsingle∝angbracketlefta+,νe|ψ(t)∝angbracketright/vextendsingle/vextendsingle2, (19)
+with|ψ(t)∝angbracketrightgiven by, from Eq. (17),
+|ψ(t)∝angbracketright=/parenleftbig
+AeU∗
+e1|e+,ν1∝angbracketright+AµU∗
+µ1|µ+,ν1∝angbracketright/parenrightbig
+e−iE1t
++/parenleftbig
+AeU∗
+e2|e+,ν2∝angbracketright+AµU∗
+µ2|µ+,ν2∝angbracketright/parenrightbig
+e−iE2t, (20)
+whereE1andE2are, respectively, the energies of ν1andν2, such that for the ultrarela-
+tivistic neutrinos under consideration E2−E1≃∆m2/2E. The amplitudes in Eq. (19)
+are given by
+∝angbracketlefte+,νe|ψ(t)∝angbracketright=Ae/parenleftbig
+|Ue1|2e−iE1t+|Ue2|2e−iE2t/parenrightbig
+, (21)
+∝angbracketleftµ+,νe|ψ(t)∝angbracketright=Aµ/parenleftbig
+Ue1U∗
+µ1e−iE1t+Ue2U∗
+µ2e−iE2t/parenrightbig
+. (22)
+Inserting these two expressions in Eq. (19), we obtain
+Pνe(t) =|Ae|2Pνe→νe(t)+|Aµ|2Pνµ→νe(t), (23)
+with the standard transition probabilities
+Pνe→νe(t) = 1−sin22ϑsin2/parenleftbigg∆m2t
+4E/parenrightbigg
+, P νµ→νe(t) = sin22ϑsin2/parenleftbigg∆m2t
+4E/parenrightbigg
+.(24)
+These probabilities areindependent oftheMajoranaphase αinboththechoices ofphases
+U(1)andU(2)of the mixing matrix, as well asin any other choice allowed by the repha sing
+freedom of the charged-lepton fields. Hence we see that the corr ect probability to detect
+aνeis independent of the Majorana phase and is independent of the pha ses of the initial
+amplitudes AeandAµ. Thus, the initial neutrino created in a charged-current interact ion
+process which can produce both νeandνµis not described correctly by the coherent state
+in Eq. (9), but can be described by the density matrix operator
+ˆρ(0) =|Ae|2|νe∝angbracketright∝angbracketleftνe|+|Aµ|2|νµ∝angbracketright∝angbracketleftνµ|. (25)
+Another way of seeing that the probability Pνe(t) in Eq. (13) is unobservable is to
+notice that it is not invariant under the rephasing
+Uak→eiφaUak(a=e,µ;k= 1,2), (26)
+4which corresponds to the freedom to rephase arbitrarily the char ged-lepton fields, as
+done, for example, in Eqs. (4) and (7). In fact, this lack of invarian ce is the origin of the
+difference between P(1)
+νe(t) in Eq. (14) and P(2)
+νe(t) in Eq. (16).
+Let us finally investigate if the Majorana phase has a physical effect in oscillations of
+a beam of neutrinos produced in a neutral-current interaction pro cess, which is the only
+possibility of neutrino production (and detection) besides the char ged-current processes
+considered above. In a neutral-current process, as for example Z0decay, neutrinos and
+antineutrinos are produced together and the different mass or fla vor eigenstates have
+the same initial amplitude, because neutral current interactions a re flavor-blind. In the
+two-neutrino framework the initial neutrino state is
+|ψ(0)∝angbracketright=1√
+2(|ν1,¯ν1∝angbracketright+|ν2,¯ν2∝angbracketright) =1√
+2(|νe,¯νe∝angbracketright+|νµ,¯νµ∝angbracketright). (27)
+It was shown in Ref. [8] that flavor transitions occur only if both the neutrino and the
+antineutrino are detected. Hence, we consider the case of detec tion of aνaat the time t
+and a ¯νbat the time ¯t, witha,b=e,µ. The relevant neutrino state is
+|ψ(t,¯t)∝angbracketright=1√
+2/parenleftBig
+|ν1,¯ν1∝angbracketrighte−i(E1t+¯E1¯t)+|ν2,¯ν2∝angbracketrighte−i(E2t+¯E2¯t)/parenrightBig
+, (28)
+where¯E1and¯E2are, respectively, the energies of ¯ ν1and ¯ν2, such that for the ultrarela-
+tivistic antineutrinos under consideration ¯E2−¯E1≃∆m2/2¯E. The detection probability
+is given by
+Pνa,¯νb(t,¯t) =|∝angbracketleftνa,¯νb|ψ(t,¯t)∝angbracketright|2
+=1
+2/parenleftbig
+|Ua1|2|Ub1|2+|Ua2|2|Ub2|2/parenrightbig
++Re/braceleftbigg
+Ua1U∗
+b1U∗
+a2Ub2exp/bracketleftbigg
+−i/parenleftbigg∆m2t
+2E+∆m2¯t
+2¯E/parenrightbigg/bracketrightbigg/bracerightbigg
+.(29)
+We confirm that flavor transitions occur only if both the neutrino an d the antineutrino
+are detected, as shown in Ref. [8], because when the neutrino or an tineutrino are not
+detected, the corresponding probabilities are
+P¯νb=/summationdisplay
+a=e,µPνa,¯νb(t,¯t) =1
+2, P νa=/summationdisplay
+b=e,µPνa,¯νb(t,¯t) =1
+2. (30)
+These are the same flavor probabilities as those in the initial state in E q. (27).
+Moreover, the probability Pνa,¯νb(t,¯t) is invariant under the rephasing in Eq. (15).
+Therefore, it is independent of the Majorana phase. One can also v erify it explicitly
+using the two choices of phases U(1)andU(2)of the mixing matrix.
+In conclusion, we have shown that the Majorana phases of the neu trino mixing matrix
+cannot have any effect in neutrino oscillations, in agreement with Ref s. [2–5] and contrary
+to the argument recently presented in Ref. [1]. We have also shown t hat in a charged-
+current weak interaction process it is not possible to create a cohe rent superposition of
+different flavor neutrinos which is independent of the associated ch arged leptons.
+5References
+[1] R. Adhikari and P. B. Pal, arXiv:0912.5266 .
+[2] S. M. Bilenky, J. Hosek and S. T. Petcov, Phys. Lett. B94, 495 (1980).
+[3] M. Doi, T. Kotani, H. Nishiura, K. Okuda and E. Takasugi, Phys. Le tt.B102, 323
+(1981).
+[4] J. Schechter and J. W. F. Valle, Phys. Rev. D23, 1666 (1981).
+[5] P. Langacker, S. T. Petcov, G. Steigman and S. Toshev, Nucl. P hys.B282, 589
+(1987).
+[6] C. Giunti and C. W. Kim, Fundamentals of Neutrino Physics and Astrophysics
+(Oxford University Press, Oxford, UK, 2007).
+[7] J. Schechter and J. W. F. Valle, Phys. Rev. D22, 2227 (1980).
+[8] A. Y. Smirnov and G. T. Zatsepin, Mod. Phys. Lett. A7, 1272 (1992).
+6
\ No newline at end of file
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+++ b/1001.0761.txt
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+ 
+An Accelerometer Based Instrumentation of the Golf Club:   
+Comparative Analysis of Golf Swings  
+ 
+Robert D. Grober 
+Department of Applied Physics 
+Yale University 
+New Haven, CT  06520 
+ 
+December 30, 2009 
+ 
+ The motion of the golf club is measur ed using two accelerometers mounted at 
+different points along the shaft of the golf club, both sensitive to acceleration along the 
+axis of the shaft.  The resu lting signals are resolved into differential and common mode 
+components.  The differential mode, a measure of the centripetal acceleration of the golf 
+club, is a reasonable proxy for club speed and can be used to unders tand details of tempo, 
+rhythm, and timing.  The common mode, related to the acceleration of the hands, allows 
+insight into the torques that generate speed in the golf swing.  This measurement scheme 
+is used in a comparative study of twenty-five golfers in which it is shown that club head 
+speed is generated in the downswing as a tw o step process.  The first phase involves 
+impulsive acceleration of the hands and club.  This is followed by a second phase where 
+the club is accelerated while th e hands decelerate.  This study serves to emphasize that 
+the measurement scheme yields a robust data set which provides deep insight into the 
+tempo, rhythm, timing and the torques that  generate power in the golf swing.   Introduction  
+ The use of electronics in the shaft or cl ub head of a golf club has been the subject 
+of considerable past work [ 1].  Modern implementations offer a large number of sensors 
+and computational power concealed within the sh aft.  Over time, the tendency has been to 
+make ever more sophisticated measurements in  an effort to obtain increasingly detailed 
+understanding of the golf swing.  This paper describes a relatively simple measurement 
+which yields a remarkably robust data set.    Measurement and Signal Analysis Summary
+ 
+ The measurement is described in detail in the companion paper, “ An 
+Accelerometer Based Measurement of the Golf Swing – Measurement and Signal 
+Analysis ” [2].  In summary, two accelerometers are mounted in the shaft of a golf club 
+with the sensing direction oriented along the axis of the shaft.  One accelerometer is 
+located under the grip, preferably at a point between the two hands.  The other is located 
+further down the shaft.  The output of th e accelerometers is dig itized and broadcast 
+wirelessly to a computer, enabling data storag e and signal analysis.  The data rate for the 
+entire wireless system is 4.4 ms/cycle, each cycle yielding data from both accelerometers.   
+ The data is analyzed within the contex t of the double pendulum model of the golf 
+swing, as discussed in Jorgensen’s The Physics of Golf  [3].  The model is shown 
+schematically in Fig. 1.  Our implementati on assumes no translational motion of the 
+center of the swing and assumes all motion is co nfined to a plane.  The upper arm of the 
+pendulum models the arms and body as a rigid rod of length  oriented at an angle 0l θ 
+with respect to the x -axis of the inertial, Cartesian, coor dinate system fixed in the plane of the swing.  The x-axis is aligned along the direction of gravitati onal acceleration.  The 
+golf club is modeled as a rigid rod of length lc at an angle φ with respect to the x -axis.  
+The orientation of the golf club is traditionally measured by the angle φθβ− =, which 
+roughly corresponds to the angle through which th e wrists are cocked.  Note here that β 
+is measured in the opposite orientation from θ and φ and is consistent with the definition 
+of Jorgensen.   
+ 
+xˆyˆ
+0l
+clβφθ
+ 
+ 
+Fig 1.  Geometry of the double pendulum.  The angle θ defines the angle of the upper 
+arm, l0 with respect to the x-axis and  defines the angle of the lower arm, lc, with respect 
+to the x -axis.  The angle β defines the angle of the lowe r arm with respect to the upper 
+arm, and is interpreted as the wrist cocking angle.   
+ 
+ The accelerometers are oriented along th e axis of the golf club, their positions 
+along the club measured from the hinged c oupling of the two rods and given by the 
+lengths r1 and r2.  Their position in space is given as  
+  () ( )y r l x r l rˆ sin sin ˆ cos cos 1 0 1 0 1 φθ φθ ++ +=r
+ (1a)   () ( )y r l x r l rˆ sin sin ˆ cos cos 2 0 2 0 2 φθ φθ ++ +=r (1b) 
+Taking two derivatives, the generali zed acceleration of the two points  and  is 1rr
+2rr
+  ( )
+() y r r l lx r r l l r
+ˆ cos sin cos sinˆ sin cos sin cos
+12
+1 02
+012
+1 02
+0 1
+φ φ φ φ θ θ θ θφ φ φ φ θ θ θ θ
+& & & & & && & & & & && &r
+− + − −+ + + − = (2a) 
+  ( )
+() y r r l lx r r l l r
+ˆ cos sin cos sinˆ sin cos sin cos
+22
+2 02
+022
+2 02
+0 2
+φ φ φ φ θ θ θ θφ φ φ φ θ θ θ θ
+& & & & & && & & & & && &r
+− + − −+ + + − = (2b) 
+Rewriting in the r-φ coordinate system a ttached to the golf club, 
+  ( )( )φ β θ β θ φ β θ β θ φˆ cos sin ˆ sin cos 02
+0 1 02
+02
+1 1& & & & & & & & & & &rl l r r l l r r + − + + + − =  (3a) 
+  ( )( )φ β θ β θ φ β θ β θ φˆ cos sin ˆ sin cos 02
+0 2 02
+02
+2 2& & & & & & & & & & &rl l r r l l r r + − + + + − =  (3b) 
+Projecting the acceleration along the negative rˆ-axis yields a positive centripetal 
+acceleration and the resulting sign als on each accelerometer are   
+  φ β θ β θ φcos sin cos ˆ*02021 1 1 g l l r r r S + + + = ⋅ − = & & & & & &r (4a) 
+  φ β θ β θ φcos sin cos ˆ*02022 2 2 g l l r r r S + + + = ⋅ − = & & & & & &r (4b) 
+Gravitational acceleration has been  added to these equations.  Note that the magnitude of 
+g* is the projection of the gravitational accele ration into the plane of motion, and is 
+therefore less than the gravitational acceleration, g = 9.8 m/s2.   
+ S1 and S 2 can be represented in term s of a common mode component, F(t), and 
+differential mode component, G(t), defined as  
+  ( )φ β θ β θcos sin cos ) (* 20 g l t F + + = & & &  (5a) 
+  ()22 1 ) ( φ&r r t G− =  (5b) 
+Using these definitions, S1 and S 2 can be written as,   ) ( ) (
+2 11
+1 t Gr rrt F S−+ =  (6a) 
+  ) ( ) (
+2 12
+2 t Gr rrt F S−+ =  (6b) 
+ The differential mode signal G(t) is recovered from S1 and S 2 by taking the 
+difference of the two signals, ()22 1 2 1 ) ( φ&r r t G S S− = = − .  For all the measurements 
+presented in this paper, r1-r2 = 0.546 m.  As will be discussed below, this signal is a 
+reasonable proxy for the speed of the club and can be used to provide insight into the 
+tempo, rhythm, and timing of the golf swing.  Its simplicity enables real-time biofeedback 
+[4].   
+ Determining F(t) from S1 and S 2 is more problematic.  Ref [ 2] proposes a solution 
+in which the function Si(t) is written as ) ( ) ( ) (t G a t F t S i i += and the quantity a i is 
+determined by minimizing .  This algorithm yields the definition 2)] (t G a i) ( [t S dti−∫
+  
+∫∫=2) () ( ) (
+t G dtt G t S dt
+ai
+i  (7) 
+and the resulting expression ) ( ) ( ) (t G a t S t F i i−= .  This method is used throughout this 
+paper.  As will be discussed below, F(t) is related to the acce leration of the hands and 
+provides insight into the torques that ge nerate speed in the golf swing.   
+ 
+  
+Fig 2(a).  The differential mode signal G(t), as described in the text, averaged over seven 
+swings of a professional golfe r.  Impact is centered at t = 0.  The y-axis is normalized to 
+units of g = 9.8 m/s2.  The separation between sensors, r1-r2 = 0.546 m.  The red circle is 
+the start of the swing; green ci rcle is the peak during the ba ckswing; blue circle is the 
+transition from backswing to downswing; and th e black circle indicates impact.  The red 
+and blue crosses are defined in Fig. 2(b).  Th e anomaly just to the right of impact is due 
+to shock induced oscillations in the out put of the sensor due to impact.   
+ 
+ 
+Fig 2(b):  The common mode signal, F(t), as described in the text, averaged over seven 
+swings of a professional golfe r.  Impact is centered at t = 0.  The y-axis is normalized to 
+units of g = 9.8 m/s2.  The red cross is the peak of F(t) and the blue cross is the minimum 
+of F(t).  The open circles are defined in Fig. 2( a).  We show the signal only until impact.  
+The signal immediately after impact is domin ated by shock induced oscillation of the 
+sensor signals, which is not re levant to this paper.    
+Differential Mode Data Interpretation  
+ Shown in Fig 2(a) is G (t) averaged over seven swings of a professional golfer.  
+The magnitude of the signal is norm alized to the acceleration of gravity, g.  Because the 
+differential mode signal is a di rect measure of the motion of the club, there are several 
+points in the swing which are easily identifie d.  They include 1) the beginning of the 
+swing (red circle), 2) point of maximum speed in the back swing (green circle), 3) 
+transition between backswing and downswing (blue circle), and 4) impact (black circle).  
+Note that impact results in a large shock to the system, which drives oscillations in the 
+sensor output.  When averaged over several swings, this shock appears as the anomaly 
+seen just to the right of the black circle in Fig. 2(a).    As an example of the usefulness of G(t), one can determine the duration of the 
+backswing and downswing for each individual sw ing, and then calculate the mean and 
+standard deviation of the entire data set.  Fo r the data of Fig. 2(a), the average duration of 
+the backswing is 731 ± 21 ms, and the averag e duration of the downswing is 258 ± 8 ms.  
+The resulting ratio of backswing to downswing time is 2.8:1.  This is very nearly the ratio 
+of 3:1 first discussed by John Novosel in the book Tour Tempo  [5], and subsequently by 
+Grober, et al . [6], in which Novosel’s observations were confirmed and a biomechanical 
+explanation for the ra tio was hypothesized.  
+ Common Mode Data Interpretation
+ 
+ Shown in Fig. 2(b) is F(t) calculated for the same data set shown in Fig. 2(a).  The 
+data is the average over seven swings of the player and is normalized to the acceleration of gravity, g .  The four points measured in G(t) (i.e. red, green, blue, and black circles) 
+are indicated in Fig 2(b) so that the key point s of the swing are easily identified.  The data 
+is only graphed thru impact, as shock induced oscillations dominate the region just after 
+impact and yield no particularly useful in formation.  An interesting feature in F(t) is the 
+structure observed during the downswing, characterized by the local maximum (red 
+cross), which occurs very near the beginni ng of the downswing, and the local minimum 
+(blue cross), which occurs when G(t) is about half way to its maximum.  The position of 
+these features relative to G(t) is indicated in Fig 2(a).  The following paragraphs address 
+the physical significance of  these features.   
+ As was discussed above, when interp reted in terms of  the double pendulum F(t) 
+has the definition ( )φ β θ β θcos sin cos ) (* 20 g l t F + + = & & &
+0 ~ cos.  Because F(t) has peaks that are 
+several times larger than g, and since g* is less than g , the g * term can safely be ignored 
+in the following analysis.  Additionally, for most of the downswing the hands are cocked, 
+β is of order π/2, β  and 1 ~ sinβ .  Therefore, F(t) can be approximated as 
+, which is a measure of th e acceleration of the point at  which the two arms of 
+the double pendulum are hinged.  This is essent ially the acceleration of the hands.  Thus, 
+the peak in F(t) near the red cross corresponds to a region of acceleration of the hands, 
+and the negative valued minimum near the blue  cross corresponds to a deceleration of the 
+hands.   θ& &0~ ) (l t F
+ To better understand why the hands acceler ate and decelerate as observed in Fig. 
+2(b), one needs to understand th e applied torque.  To understa nd the torque one needs to 
+consider the equations of motion of the double pendulum.  The Lagrangian of the double 
+pendulum is described in L.D. Landau and E.M. Lifschitz, Mechanics  [7] and was first applied to the golf swing by Jorgensen [3 ].  Written in terms of the coordinate system of 
+Fig. 1, the Lagrangian is 
+() (φ θ τ θ τ φ θ φ θ φ θ β θ− + + − > < + > < + + > < = ) cos(21
+21
+02 2222
+022
+0 1& & & &c c l l l m l m l m L ), (8) 
+where m 1 is the mass of the upper arm, l0 is the length of the upper arm,  is the 
+second moment of the upper arm, m2 is the mass of the lower arm, lc is the length of the 
+lower arm,  is the first moment of the lower arm, and  is the second moment 
+of the lower arm.  All moments are taken by averaging over the distribution of mass and 
+are measured relative to th eir most proximal point.  > <2
+0l
+> <cl > <2cl
+θτ is an applied torque that works to 
+increase the angle θ and βτ is an applied torque that  works to increase the angle 
+φθβ− =.  Using the shorthand notation, 
+  ,      and   2
+022
+0 1 l m l m I+ > < = θ > < =22 cl m I φ 0 2 l l m S c><= ,  (9) 
+the Lagrange equations of motion are  
+   (10a) β θ θ τ τ β φ β φ θ+ = + + sin cos2& & & & &S S I
+   (10b) β φ τ β θ φ β θ− = − + sin cos2& & & & &S I S
+Once again, we are concerned here with that region of the swing starting from the 
+transition and going into the downswing, dur ing which time the hands are cocked, β is of 
+order π/2, 0 ~ cosβ  and 1~ sinβ .  Additionally, the torque βτ is negative in the 
+downswing because it acts so as to un-cock the wrists, reducing β.  For convenience of 
+notation, we define the release torque βττ−=R .  The equations of motion are then 
+simplified to     (11a) 2φ τ τ θ θ θ& & & S I R− − =
+   (11b) 2θ τ φ φ& & &S I R+ =
+These two equations are particul arly insightful and their physi cal relevance is the subject 
+of the following paragraphs.   
+ As is indicated in Eq. 11(a), the only torque that can drive  positive is θ& &θτ, the 
+torque generated by the body.  The greater the applied torque, the greater is the 
+acceleration of the hands, and the larger is  this positive going region of the common 
+mode signal.  Because the angle β does not change appreciabl y during this region, both 
+the hands and the club must experience comparable acceleration.   and  are 
+comparable at low speeds if φ& &θ& &
+θ
+φ θφτ τI II
+R+~ .  Values of  are typically of order 0.2 kg 
+m2 while  is typically of order 1.2 kg m2 [φI
+θI 8], which means θττ<<R .   
+ Thus, the local maximum of the common m ode signal at the beginning of the 
+downswing is a measure of the acceleration of the hands and club in response to the 
+torque generated primarily by the body, θτ.   
+ The common mode signal increases, goes through a maximum, and then quickly 
+becomes negative.  During this region wher e the common mode signa l goes negative, the 
+differential mode signal, which is proportional to , increases dramatically, which 
+means  is large and positive.  As can be seen in Eq. 11(b), is driven both by the 
+torque 2φ&
+φ& &
+Rφ& &
+τ, which is applied by the golfer in an e ffort to release the club, and by the 
+centripetal term , which is associated with hand speed.  In contrast, and as is seen in 
+Eq. 11(a), the torque 2θ&S
+Rτ and the centripetal term  serve to decelerate the hands.  2φ&SWhen Rτ and  combine to be larger than 2φ&S θτ,  goes negative.  Thus, the negative 
+going region of the common mode signal is a direct measure of the deceleration of the 
+hands, which happens in respons e to the acceleration of th e club during the release.  The 
+greater the torque that releases the club and the faster the club moves, the more the 
+hands decelerate and the larger the nega tive going dip in the common mode signal .   θ& &
+ Eventually the common mode signal goe s through a local minimum.  This 
+generally occurs one-half to two-thirds of the way from the beginning of the downswing 
+to impact.  After this point, the common mode signal increases, eventually becoming 
+positive very near to impact.  The wrists fully un-cock during this later region of the 
+swing, β trends towards zero, 1 ~ cosβ  and 0~ sinβ .  Thus, near to impact, the 
+common mode signal ( )φ θ cos ) (* t F + = &
+20θ&lβsinβ θcos20l& g&+  is dominated by the 
+centripetal acceleration  and by the gravitational acceleration  both of which 
+are positive definite.   φcos*g
+ In summary, the common mode signal ty pical of the golf swing of professional 
+golfers has a distinctive structure during the downswing.  It exhibits a positive valued 
+maximum followed by a negative valued minimu m, and finally becomes slightly positive 
+just as the club gets to impact.  The positive maximum is associated with the impulsive 
+acceleration of the hands and club at the be ginning of the downswing.  The negative 
+minimum is associated with the release of the club, which serves to decelerate the hands.  
+The greater the positive maximum, the more pow erful is the acceleration at the start of 
+the downswing.  The deeper the negative minimum, the more powerful is the release of 
+the club .   
+ Detailed Comparison of Five Golfers  
+ Figs. 3 thru 7 are the differential and co mmon mode signals for five very different 
+golfers.  The first is an elderly, avid, amateu r golfer with a handicap of order twenty-five, 
+the second is an avid, amateur golfer with a handicap of order ten, the third is a 
+competitive collegiate golfer, the fourth is a PGA tour professional, and the fifth is a professional long drive competitor.   In each case, the peak of 
+G(t) provides an indication 
+of how fast the club is moving while F(t) provides some indication for how club head 
+speed is developed.  Commentary for each data set is provided in the figure captions.  These figures are meant to provide some insight as to how the data evolves from the case of an avid golfer who does not hit the ball very  far to the case of a golfer who hits the ball 
+just about as far as anybody.      
+ 
+Fig. 3:  Differential and co mmon mode data averaged over se ven swings of an elderly, 
+avid, amateur golfer with a handicap of order twenty-five.  The durat ion of the backswing 
+is 830 ± 18 ms and the duration of the downswing is 433 ± 18 ms, which is remarkably 
+consistent and characteristic of this beautiful ly rhythmic golf swing.  However, this golfer 
+is not particularly strong, as is clear from the peak of the differential mode G(t) and the 
+very small structure in the common mode F(t).  It is interesting that while the duration of 
+the backswing is only slightly slower than th at of the professional golfer (Fig. 6), the 
+duration of the downswing is much slower , suggesting significantly less physical 
+strength.     
+ 
+Fig. 4:  Differential and common mode data averaged over six swings of an amateur 
+golfer with a handicap of order ten.  The dur ation of the backswing is 817 ± 16 ms and 
+the duration of the downswing is 385 ± 20 ms.  It is particularly interesting to note that 
+while this golfer has a very respectable accel eration of the hands at  the beginning of the 
+downswing, he does not fully release the club.  This is generally done so as to obtain 
+better control over the club head, but at the cost of significant club head speed.   
+ 
+  
+ 
+Fig. 5:  Differential and co mmon mode data averaged over fo ur swings of a collegiate 
+golfer.  The duration of the backswing is 692 ± 48 ms and the duration of the downswing 
+is 201 ± 33 ms, which is extremely fast but not particularly beneficial.  Fluctuations in 
+tempo are of order 10%.  The club speed at im pact is less than that of the professional 
+golfer shown in the following figure.  However, the common mode F(t) data shows that 
+there is clearly the beginning of a very nice common mode structure.   
+ 
+  
+ 
+Fig. 6:  Differential and co mmon mode data averaged over eight swings of a PGA tour 
+professional.  The duration of the backswi ng is 724 ± 23 ms and the duration of the 
+downswing is 248 ± 4 ms.  Note that in comp arison with the collegiate golfer the max-
+min structure in the common mode F(t) is larger and wider.  This implies that larger 
+torques are sustained for longer periods of time over a larger swing arc.   
+ 
+ 
+  
+ 
+Fig. 7:  Differential and co mmon mode data averaged over ei ght swings of a professional 
+long drive competitor.  The duration of the ba ckswing is 750 ± 30 ms and the duration of 
+the downswing is 310 ± 11 ms, though it is clea r that the time over which the large 
+torques are applied in the downswing is considerably shorter.  Note that the vertical axes 
+have been increased by 50%  relative to the other golfers in order that that the entire data 
+set can be viewed.  This golfer hits the ball far because he is very strong, generating an 
+incredible amount of torque at the beginning of the downs wing, with a peak in the 
+common mode F(t) that is of order twice the valu e for the tour professional.   
+ Comparative Study of 25 Golfers  
+ The implication of the above analysis is that the greater the max-min structure of 
+F(t) during the downswing, the greater the resulti ng club head speed.  We have tested this 
+hypothesis by comparing the swings of 25 golfers of varying ability, from high-
+handicapper to tour professional.   
+ The test requires that we compare maximum club head speed with the size of the max-min structure of 
+F(t).  The following paragraphs outline  how this is accomplished.   
+ An expression for club head speed can be derived from the expression for the position of the club head,  
+  
+() ( )y l l x l l r c c c ˆ sin sin ˆ cos cos 0 0 φθ φθ ++ +=r, (12) 
+by first taking a derivative with respect to time, 
+  () ( )y l l x l l r c c c ˆ cos cos ˆ sin sin 0 0 φ φ θ θ φ φ θ θ& & & & &r+ + + = , (13) 
+and then taking the absolute value of this velocity, 
+  β φ θ θ φcos 202
+02 2 22
+c c c l l l l r& & & & &r+ + = .   (14) 
+In the vicinity of impact, 0~β , 1 ~ cosβ , and the above expression simplifies to 
+0l l r c cθ φ& & &r+ = .  Now the length of the lower arm of the pendulum (i.e. the club) is longer 
+than the length of the upper arm (i.e. the golfer’s arms), , and in the vicinity of 
+impact .  With these limits, 0l l c>
+θ φ& &> cl
+cc clll r φφθφ &&&&&r≈⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛+ =01 .  Thus, it is not completely 
+unreasonable to use  as a proxy indicator for relative club head speed 
+near to impact.  In particular, we will use the peak value of G(t) in the vicinity of impact.   (1r− =)22 ) ( φ&r t G 
+Fig. 8:  The signal F(t) for a golf swing.  As a measure of the size of F(t) during the 
+downswing, we integrate F(t) over two regions.  The first region (red shaded region) 
+starts at the beginning of the downswing (i.e blue circle), goes th rough the maximum, and 
+terminates at the zero crossing.  This integr ation yields a speed wh ich is interpreted as 
+approximating the maximum hand speed during the downswing.  The second region (blue 
+shaded region) spans th e negative going region of F(t).  This integration yields a speed 
+which is interpreted as approximating th e amount by which the hands slow down as 
+impact is approached.   
+ 
+ For most of the downswing 2 ~πβ , and therefore  is the acceleration 
+of the hands.  Integrating acceleration through tim e results in a change in velocity.  Our 
+proxy for the F(t) max-min structure in the downswi ng will involve an integration of F(t) 
+through two regions.  As is indicated in Fig. 8, Region I is the red shaded region.  It 
+begins at the start of the downswing, a point at which the entire sy stem is moving very 
+slowly.  Region I continues through the peak in F(t) and terminates at the zero crossing of 
+F(t).  This integral yields a speed, vI, which is approximately the maximum speed of the 
+hands during the downswing.  Region II is indicated as the blue shaded region and spans 
+the entire negative region of F(t).  This integration yields a negative number, vII , which is 
+interpreted as approximating the amount by which the hands slow down as the club is 
+released.  Finally, as a measure of the size of the max-min structure associated with F(t) θ& &0~ ) (l t Fon the downswing, we add v I to the negative of v II, yielding II I Fv v v−= .  While one can 
+conceive of a myriad of other indicators, vF is particularly simple.   
+  050100150
+05 15
+Integrated Common Mode SiPeak Differential Mode Signal ( g)
+10
+gnal (m/s)
+ 
+Fig. 9:  The peak value of G(t) as a function of vF for 200 golf swings sampled from 25 
+golfers.  As defined in the text, vF  is a proxy for the size of the max-min structure in F(t) 
+and the peak of G(t) is a proxy for the club head speed  at impact.  The golfers range 
+widely in capability from high-handicappers to PGA Tour professi onals and professional 
+long ball competitors.  As described in the text , the correlation coefficient for this data is 
+0.85, indicating a high degree of correlation.  Thus, we contend that the size of the 
+structure in F(t) correlates well with the speed  of the club at impact.   
+ 
+ In Fig. 9, the maximum value of G(t) is plotted relative vF for 200 golf swings 
+sampled from 25 golfers.  The golfers range widely in capability from high-handicappers 
+to PGA Tour professionals and professional lo ng ball competitors.  For each golfer, 5-10 
+swings are recorded while they are hitting either a 5 or 6 iron.  The data indicate a clear 
+trend:  the greater is the club speed proxy, th e larger is the max-min structure associated 
+with F(t).  For reference, the correlation coefficient is 0.85, calculated as    ()()
+()()∑ ∑∑
+− −− −=
+2 2y y x xy y x xcor  (15) 
+where x is vF, and x is the mean of vF , y is the peak of G(t) and y is the mean of the 
+peak of G(t).   
+ It is important to emphasize that whil e these measurements are precise, their 
+interpretation is approximate.  In particular the interpretation of the integrals vI and vII as 
+indicating a change in hand speed are only accura te in the limits that the upper and lower 
+arms of the pendulum move  in the same plane, the length of the upper arm, l0 does not 
+change, 2 ~πβ , etc.  Indeed, it is likely that none of these constraints are exact, varying 
+for every golfer and every golf swing, wh ich perhaps accounts fo r the width of the 
+distribution in Fig. 9.  For instance, if a gol fer lays the club off at the top of the swing, 
+making it flat relative to the swing plane, the sensors in the shaft are not perfectly aligned 
+with the direction of motion of the hands .  In this condition, the maximum in F(t) will be 
+suppressed.   
+ So, while there is no delusion here regarding perfect interpretation of 
+measurement, the trend in Fig.  9 is clear.  The maximum of G(t), which is a very 
+reasonable proxy for club head speed, scales with the size of the max-min structure of 
+F(t), which is a very reasonable proxy for the torques which accelerate the hands.   
+ In summary, the common mode signal F(t) provides deep insight into how torque 
+is applied to generate club head speed in the go lf swing.  It is a two step process.  Starting 
+at the beginning of the downswing, the firs t phase involves a rapi d acceleration of the 
+hands and club.  The height and width of the maximum of F(t) is a measure of this initial 
+acceleration.  This is then followed by a second phase, the release, in which the club accelerates while the hands decelerate.  The depth and width of the minimum of F(t) is a 
+measure of the intensity of the release.   
+ Conclusion
+ 
+ The motion of the golf club has been m easured using two accelerometers mounted 
+at different points along the shaft of the gol f club, both sensitive to acceleration along the 
+axis of the shaft.  Interpreted within the context of the double pendul um model of the golf 
+swing, the resulting signals ar e resolved into differential and common mode components.  
+The differential mode, a measure of the centr ipetal acceleration of the golf club, is a 
+reasonable proxy for club speed and can be used  to understand details of tempo, rhythm, 
+and timing.  The common mode, related to th e acceleration of the hands, allows insight 
+into the torques that generate speed in the golf swing.    A comparative study of twenty-five golf ers reveals that cl ub head speed in 
+accomplished golfers is generated as a two step  process.  Starting at the beginning of the 
+downswing, the first phase involves a rapid acce leration of the hands and club.  This is 
+then followed by a second phase, the release,  in which the club accelerates while the 
+hands decelerate.  This phenomenon is clearly  revealed in the common mode/differential 
+mode data analysis.    This paper demonstrates that this meas urement scheme yields a robust data set 
+which provides deep insight into tempo, rhythm, timing, and the torques that generate power in the golf swing.     
+Footnotes and References:  
+ 
+1)  See for instance, United States Pa tents 6,648,769; 6,402,634; 6,224,493; 6,638,175; 
+6,658,371; 6,611,792; 6,490,542; 6,385,559; and 6,192,323.     
+2)  R.D. Grober, An Accelerometer Based Instrumentation of the Golf Club: Measurement 
+and Signal Analysis , arXiv:1001.0956v1 [phys.ins-det].  
+ 
+3)  T.P. Jorgensen, The Physics of Golf  (AIP Press, American Institute of Physics, New 
+York, 1994).    
+4)  See United States Patent 7160200 and Sonic Golf, Inc., www.sonicgolf.com.   
+ 5)  J. Novosel and J. Garrity, 
+Tour Tempo , (Doubleday, New York, 2004).   
+ 6)  R.D. Grober, J. Cholewicki, 
+Towards a Biomechanical Understanding of Tempo in 
+the Golf Swing , arXiv:physics/0611291v1 .   
+ 7)  L.D. Landau, E.M. Lifshitz, 
+Mechanics , 3rd Edition, (Elsevier, New York, 1976), pg. 
+11.    8)  W.M. Pickering and G.T. Vickers, 
+On the double pendulum model of the golf swing , 
+Sports Engineering 2, 161-172 (1999).     
\ No newline at end of file
diff --git a/1001.0762.txt b/1001.0762.txt
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@@ -0,0 +1,851 @@
+arXiv:1001.0762v1  [astro-ph.SR]  6 Jan 2010Mid-Infrared Photometry of Cold Brown Dwarfs:
+Diversity in Age, Mass and Metallicity
+S K Leggett1
+sleggett@gemini.edu
+Ben Burningham2
+D Saumon3
+M S Marley4
+S J Warren5
+R L Smart6
+H R A Jones2
+P W Lucas2
+D J Pinfield2
+and
+Motohide Tamura7
+ABSTRACT
+1Gemini Observatory, Northern Operations Center, 670 N. A’ohok u Place, Hilo, HI 96720
+2Centre for Astrophysics Research, Science and Technology Rese arch Institute, University of Hertford-
+shire, Hatfield AL10 9AB
+3Los Alamos National Laboratory, PO Box 1663, MS F663, Los Alamos , NM 87545
+4NASA Ames Research Center, Mail Stop 245-3, Moffett Field, CA 940 35
+5Imperial College London, Blackett Laboratory, Prince Consort Ro ad, London SW7 2AZ
+6INAF/Osservatrio Astronomico di Torino, Strada Osservatrio 20 , 10025 Pino Torinese, Italy
+7National Astronomical Observatory of Japan, 2-21-1 Osawa, Mit aka, Tokyo 181-8588, Japan– 2 –
+We present new SpitzerIRAC photometry of twelve very late-type T dwarfs;
+ninehave [3.6], [4.5], [5.8]and[8.0]photometry andthree have [3.6]and[4 .5]pho-
+tometry only. Combining this with previously published photometry, w e investi-
+gate trends with type and color that are useful for both the plann ing and inter-
+pretation of infrared surveys designed to discover the coldest T o r Y dwarfs. The
+on-line Appendix provides a collation of MKO-system YJHKL′M′and IRAC
+photometry for a sample of M, L and T dwarfs. Brown dwarfs with eff ective
+temperature ( Teff) below 700 K emit more than half their flux at wavelengths
+longer than 3 µm, and the ratio of the mid-infrared flux to the near-infrared flux
+becomes very sensitive to Teffat these low temperatures. We confirm that the
+colorH(1.6µm)−[4.5] is a good indicator of Teffwith a relatively weak depen-
+dence on metallicity and gravity. Conversely, the colors H−K(2.2µm) and [4.5]
+−[5.8] are sensitive to metallicity and gravity. Thus near- and mid-infra red pho-
+tometry provide useful indicators of the fundamental propertie s of brown dwarfs,
+and if temperature and gravity are known, then mass and age can b e reliably
+determined from evolutionary models. There are twelve dwarfs cur rently known
+withH−[4.5]>3.0, and 500 ∼<TeffK∼<800, which we examine in detail. The
+ages of the dwarfs in the sample range from very young (0.1 – 1.0 Gyr ) to rela-
+tively old (3 – 12 Gyr). The mass range is possibly as low as 5 Jupiter mas ses to
+up to 70 Jupiter masses, i.e. near the hydrogen burning limit. The met allicities
+also span a large range, from [m/H]= −0.3 to [m/H]= +0 .3. The small number
+of T8 – T9 dwarfs found in the UKIRT Infrared Deep Sky Survey to d ate appear
+to be predominantly young low-mass dwarfs. Accurate mid-infrare d photometry
+of cold brown dwarfs is essentially impossible from the ground, and ex tensions to
+the mid-infrared space missions warm- Spitzerand WISE are desirable in order
+to obtain the vital mid-infrared data for cold brown dwarfs, and to discover more
+of these rare objects.
+Keywords : stars: low-mass, brown dwarfs — infrared: stars
+1. Introduction
+The last decade has seen a remarkable increase in our knowledge of t he bottom of the
+stellar main-sequence and of the low-mass stellar and sub-stellar (b rown dwarf) population
+of the solar neighbourhood. Two new classes have been added to th e spectral type sequence
+following M: L and T (e.g. Kirkpatrick 2005). T dwarfs with effective te mperatures ( Teff) as
+low as∼500 K are now known (Warren et al. 2007, Burningham et al. 2008, De lorme et al.– 3 –
+2008a, Leggett et al. 2009)andwe aretruly finding objects that p rovide the link between the
+low-mass stars and the giant planets. The very late-type T dwarf d iscoveries are primarily
+a result of the red and near-infrared wide-field surveys: the Two M icron All Sky Survey
+(2MASS; Skrutskie et al. 2006), the Sloan Digital Sky Survey (SDSS ; York et al. 2000),
+the UK Infrared Telescope (UKIRT) Infrared Deep Sky Survey (U KIDSS; Lawrence et al.
+2007), and the Canada France Hawaii Telescope’s Brown Dwarf Sur vey (CFBDS; Delorme
+et al. 2008b). Late-type T dwarfs have also been found as compan ions to stars in infrared
+imaging data (Mugrauer et al. 2006, Luhman et al. 2007).
+Interestingly, despite the great success of recent surveys, th ere are currently less than
+half the number of T dwarfs known than there are extrasolar plane ts;DwarfArchives.org
+lists 155 T dwarfs as opposed to the 407 exoplanets listed at exoplanet.eu , as of December
+2009. An increase in the number of known T dwarfs is expected, due to the ongoing CFBDS
+and UKIDSS surveys, and other imminent ground-based near-infr ared surveys, such as the
+Visible andInfrared Survey Telescope for Astronomy (VISTA, McP herson et al. 2006). Also,
+asTeffdecreases browndwarfsemitsignificant fluxatmid-infraredwavele ngths (e.g. Burrows
+et al. 2003, Leggett et al. 2009) and it is expected that the recent ly launched Wide-Field
+Infrared Survey Explorer (WISE, Liu et al. 2008) will detect a signifi cant number of cold
+brown dwarfs in its 2.8 – 3.8, 4.1 – 5.2 and 7.5 – 16.5 µm bands. Figure 1 demonstrates the
+rapidly increasing importance of the mid-infrared region for late-ty pe T dwarfs; for dwarfs
+cooler than 700 K more than half the flux is emitted at wavelengths lon ger than 3 µm.
+This paper investigates photometric trends seen in very late-type dwarfs at mid-infrared
+wavelengths, with the expectation that these longer wavelengths will be crucial for both the
+discovery and the understanding of the coolest T-type dwarfs, a nd even more so for the
+proposed cooler Y-type dwarfs (Kirkpatrick et al. 1999). We were awarded time on the
+Spitzer Space Telescope (Werner et al. 2004) to follow up at mid-infrared wavelengths late-
+type T dwarfs found by UKIDSS. In this paper we present new 3 – 8 µm photometry of
+UKIDSS dwarfs obtained with IRAC (Fazio et al. 2004), as well as pre viously unpublished
+IRACphotometryfromthe Spitzerarchive. Wecombinethesedatawithpreviouslypublished
+near- and mid-infrared photometry to examine trends in colors with type, which will be
+useful for the design and use of ongoing and planned infrared surv eys. We also examine in
+detail the colors of the 500 < TeffK<800 dwarfs for correlations with the photospheric
+parameters temperature, gravity and metallicity. We find that var ious colors do provide
+useful indicators of temperature and gravity, which in turn provid e estimates of mass and
+age (from evolutionary models) for these, usually isolated, very low mass objects.
+Figure 2 shows spectral energy distributions (SEDs) for a T8 dwar f withTeff≈750 K
+and a T9 dwarf with Teff≈600 K. Principal absorption features are indicated and it can be– 4 –
+seen that the flux from cool brown dwarfs emerges from windows b etween strong bands of,
+primarily, CH 4andH 2O absorption. As thetemperature dropsfrom750 Kto 600Kthera tio
+of the mid- to the near-infrared flux increases dramatically, and mo re flux emerges through
+the windows centered near 5 µm and 10 µm. Passbands are indicated for the near-infrared
+YJHKsystem used here, as well as the four IRAC bands and three short est-wavelength
+WISE bands. The IRAC and WISE filters sample regions of high flux, as well as regions of
+strong absorption where very little flux is emitted. Thus both camer as are sensitive to cold
+brown dwarfs, which can be identified by extreme colors in their phot ometric systems.
+2. Atmospheric and Evolutionary Models
+Atmospheric and evolutionary models used in this analysis were gener ated by members
+of our team (Marley et al. 2002, Saumon & Marley 2008). The atmosp heric models include
+all the significant sources of gas opacity (Freedman et al. 2008). H owever there are known
+deficiencies in the molecular opacity line lists. For CH 4, the list effectively contains no lines
+below 1.6 µm. For NH 3, there are no lines below 1.4 µm in our data base. The FeH line list
+is known to be incomplete for the F4∆–X4∆ transition in the 1.2 – 1.3 µm region (Cushing
+et al. 2003). This is not a concern for this study however as FeH ban ds disappear by the
+L7 spectral type (Cushing, Rayner & Vacca 2005) and are complet ely absent in late-type
+T dwarfs. Finally, our treatment of the red wing of the pressure br oadened K I resonance
+doublet (0.78 µm) does not properly reproduce the observations in the 0.8 - 1.1 µm range.
+Condensates of certain refractory elements are included in atmos pheric layers where
+physical conditions favor such condensation. Variations in conden sate cloud properties are
+treated in these models using a sedimentation parameter fsedwhich is the ratio of the sed-
+imentation velocity to the convective velocity (Ackerman & Marley 20 01). Small values of
+fsedrepresent thick clouds with small grain sizes, and large values repre sent thin clouds with
+largergrains. Theclouddecksimpactthefluxemittedinthenear-inf raredwavelength region
+for early-type L to early-type T dwarfs. Cloud-free models best fi t the spectra of later-type,
+cooler, objects and we employ such models here. In reality condens ates certainly form deep
+in the atmospheres of such cool T dwarfs but a combination of atmo spheric dynamics and
+cloud physics results in a negligible spectral influence from clouds at t hese effective temper-
+atures. We generally find that the colors and spectra of L dwarfs a re reproduced with fsed
+= 1 – 2, while those of later T dwarfs are reproduced with fsed= 4 or cloud-free ( fsed→ ∞)
+models (e.g. Knapp et al. 2004; Stephens et al. 2009).
+Our models also include vertical transport in the atmosphere, which significantly affects
+the chemical abundances of C-, N- and O-bearing species. The ver y stable molecules CO and– 5 –
+N2are dredged up from deep layers to the photosphere faster than they can reach chemical
+equilibrium with CH 4and NH 3, respectively, causing enhanced abundances of these species
+and decreased abundances of CH 4, H2O, and NH 3(e.g. Fegley & Lodders 1996, Hubeny
+& Burrows 2007, Saumon et al. 2007). Vertical transport is param eterized in our models
+by an eddy diffusion coefficient Kzz(cm2s−1) which is related to the mixing time scale.
+Generally, larger values of Kzzimply greater enhancement of CO and N 2over CH 4and NH 3,
+respectively. Values of log Kzz= 2 – 6, corresponding to mixing time scales of ∼10 yr
+to∼1 h respectively, reproduce the observations of T dwarfs (e.g. Le ggett et al. 2007b,
+Saumon et al. 2007). The mixing parameter Kzzhas a large influence on the 4.6 µm and
+9 – 15µm fluxes for both L and T dwarfs: increasing Kzzleads to increased CO and an
+increase in the CO absorption at 4.6 µm, and to decreased NH 3and so a decrease in the 9
+– 15µm NH 3absorption (e.g. Stephens et al. 2009). Models which do not include m ixing,
+usually referred to as chemical equilibrium models, do not reproduce observations, especially
+at these wavelengths.
+The evolutionary sequences of Saumon & Marley (2008) provide rad ius, mass, and
+age for a variety of atmospheric parameters. These have been co mputed with both cloudy
+(fsed= 2) and cloudless models as the surface-boundary condition.
+3. The Sample and Data Sources
+The primary sample for this paper consists of T dwarfs with spectra l types T7 and
+later, as defined by the Burgasser et al. (2006b) scheme. Very fe w of these are currently
+known — 26 are listed in the DwarfArchives.org compendium at the time of writing. The
+latest spectral type currently defined is T9, of which three are cu rrently known: ULAS
+J003402.77-005206.7 (hereafter ULAS 0034-00; Warren et al. 20 07), CFBDS J005910.90-
+011401.3 (CFBDS 0059-00; Delorme et al. 2008a) and ULAS J133553 .45+113005.2 (ULAS
+1335+11; Burningham et al. 2008). The discovery papers provide n ew spectral indices for
+typing such extreme T dwarfs, as the Burgasser et al. indices meas ure absorption regions
+that show little change beyond ∼T8 types. Figure 2 shows that very little flux remains to
+be absorbed at the near-infrared CH 4bands, for T8 and T9 dwarfs.
+TheSpitzerGeneral Observer programs 40449 and 60093, and the Director’s Time
+program 527, allowed us to obtain IRAC photometry of apparently v ery late-type T dwarfs
+discovered in the UKIDSSdata, that had anestimated brightness o f [4.5]<16.5 magnitudes.
+The Cycle 4 Target of Opportunity program and the Director’s Time p rogram provided
+photometry in all four IRAC bands, while the Cycle 6 warm mission prog ram provides only
+photometry at the shortest two wavelengths. All targets obser ved were found in the Large– 6 –
+Area Survey (LAS) component of the UKIDSS. The LAS will cover 38 00 deg2of the sky in
+theYJHKfilters to J= 19.6 (Vega magnitudes, 5 σ), with a large degree of overlap with
+existing SDSS optical data. The UKIDSS photometric system is desc ribed in Hewett et al.
+(2006) and is based on the Mauna Kea Observatories system (MKO; Tokunaga et al. 2002).
+Technical information on the survey data reduction and quality con trol is given in Dye et
+al. (2006). The identification and spectral classification methods f or the T dwarfs selected
+for theSpitzerprogram are described by Pinfield et al. (2008).
+We obtained IRAC photometry via GO 40449 for four T dwarfs, all of which were
+initially classified as having spectral types later than T7: ULAS J0857 15.96+091325.3(here-
+afterULAS0857+09),ULASJ101721.40+011817.9(ULAS1017+01 ),ULASJ123828.51+095351.3
+(ULAS1238+09)andULAS1335+11. Additional spectroscopy late r showed ULAS0857+09
+to have an earlier type of T6. The IRAC photometry for ULAS 1335+ 11 is published in
+Burningham et al. (2008). We were awarded SpitzerDirector’s Time (program 527) for
+photometry and spectroscopy of the T8.5 companion to an M dwarf , ULAS J214638.83-
+001038.7 or Wolf 940B. We present the photometry here, the spec troscopy will be pre-
+sented in a later paper. Three T7.5 dwarfs have been observed as p art of our warm
+Spitzerprogram GO 60093 to date, which provides the shorter wavelength IRAC pho-
+tometry: ULAS J01393977+0048138 (ULAS 0139+00), ULAS J015 02437+1359240 (ULAS
+0150+13) and ULAS J23212379+1354549 (ULAS 2321+13). Unpub lished four-band IRAC
+photometry for an additional five T7 and later type dwarfs were fo und in the Spitzer
+archive: 2MASS J00501994-3322402(2MASS 0050-33), 2MASS J0 3480772-6022270(2MASS
+0348-60), 2MASS J11145133-2618235 (2MASS 1114-26), SDSS J 150411.63+102718.4 (SDSS
+1504+10) and SDSS J162838.77+230821.1 (SDSS 1628+23). The ar chived data were ob-
+tained through GO programs 20544 and 40198, and GT 35. Table 1 list s the twelve dwarfs
+with previously unpublished photometry and gives the discovery ref erence, spectral type
+and associated Spitzerprogram number for each dwarf. Details of the observations and d ata
+reduction are given in §4.
+Previously published IRAC photometry for a reference sample of lat e-type M, L and
+T dwarfs was taken from Burgasser et al. (2008b); Burningham et al. (2008); Leggett et
+al. (2007b); Luhman et al. (2007); Patten et al. (2006); Warren e t al. (2007). IRAC
+photometry for the T3 dwarf 2MASS J12095613-1004008 is from S . Sonnett (priv. comm.
+2009). Uncertainties are 1 – 4 %, except for the [5.8] and [8.0] Luhma n et al. results, and
+the [8.0] Burningham et al. results, for which the uncertainties are 7 – 14%. To these should
+be added systematic uncertainties of 3%, which reflect both pipeline dependencies (Leggett
+et al. 2007b) and the uncertainty in the absolute calibration (Reach et al. 2005).
+YJHKphotometry on the UKIDSS/MKO system for ULAS 0150+13, ULAS 0 857+09– 7 –
+and ULAS 2321+13 were taken from Burningham et al. (2010). Othe r MKO-system YJHK
+for the dwarfs with new IRAC data and for a reference sample of dw arfs both with and
+without IRAC data were taken from previously published datasets: Chiu et al. (2006, 2008);
+Delorme et al. (2008a); Geballe et al. (2001); Golimowski et al. (2004 a); Knapp et al.
+(2004); Leggett et al (2000, 2002a, 2002b, 2009); Liu et al. (20 07); Lodieu et al. (2007);
+Luhman et al. (2007); Pinfield et al. (2008); Strauss et al. (1999); Tsvetanov et al (2000);
+Warren et al. (2007). For the T9 dwarf CFBDS 0059-00 only Ksis published, and we have
+calculated Kusing the Delorme et al. spectrum ( Kis 0.08 magnitudes fainter than Ks).
+The uncertainty in these data is typically 3%, but for the fainter obj ects it is 10 – 15%. The
+reference sample of objects were limited to dwarfs with known spec tral type, MKO-system
+photometry and σ(K)≤0.2 magnitudes.
+MKO-system JHKphotometryissynthesizedfromphotometricallyflux-calibratedsp ec-
+tra for late-type T dwarfs that do not have these measurements and do have IRAC data:
+2MASS 0050-33, 2MASS 0348-60 ( HKonly), 2MASS J093935482448279, 2MASS 1114-26
+and 2MASS J12373919+6526148. These spectra were taken from Burgasser et al. (2003b,
+2006a) and Liebert & Burgasser (2007). The Jphotometry for 2MASS 0348-60 was derived
+from the 2MASS photometry using the transformations based on t ype given by Stephens
+& Leggett (2004). Note that JHKphotometry on different systems can differ by several
+tenths of a magnitude for L and T spectral types (e.g. J2MASS−JMKO= 0.4 magnitudes
+for late-type T dwarfs, Stephens & Leggett) and that color tran sformations based on stars
+cannot be used for the T dwarfs. Synthetic YJHKphotometry was also included for the L9
+dwarf DENIS-P J0255-4700, calculated from the spectrum obtain ed by Cushing, Rayner &
+Vacca (2005). This dwarf has IRAC data and the photometry was c alculated by Stephens
+et al. (2009) but not published. The uncertainty in all these synthe sizedJHKis 5 – 10%,
+dominated by the uncertainty in the 2MASS photometry.
+Although the emphasis of this paper is on longer wavelengths, we hav e included the
+synthetic Y-band photometry for L and T dwarfs calculated by Hewett et al. (2 006), and
+have calculated other Y-bandphotometry where flux-calibrated spectra were availableac ross
+the 0.95 – 1.11 µmbandpass. These spectra were taken fromBurgasser et al. (20 03b, 2004b,
+2006a, 2006b, 2008a), Chiu et al. (2006), Cushing et al. (2008), K napp et al. (2004) and
+Liebert & Burgasser (2007). Primarily the Yphotometry was synthesized for T dwarfs
+to investigate trends in this bandpass for very late-type dwarfs, as these trends may have
+implications for use of the UKIDSS LAS YJHKdataset. The uncertainty in these Y
+magnitudes is ∼10%.
+Wehaveidentified knownbinariesinoursamplebyreferencetothelite ratureandknown
+current work: Allers et al. (2010); Bouy et al. (2003); Burgasser et al. (2003a, 2005, 2006c);– 8 –
+Freed et al. (2003); Golimowski et al. (2004b); Heintz (1972); Koe rner et al. (1999); Liu
+& Leggett (2005); Liu et al. (2006, 2010, and in preparation); Mar tin, Brandner and Basri
+(1999a); Reid et al. (2001); Stumpf et al. (2005).
+Finally, distance is a key parameter for luminosity investigations. Trig onometric paral-
+laxes for M, L and T dwarfs in the sample were taken from: Costa et a l. (2006), Dahn et al.
+(2002); the Hipparcos catalog (ESA 1997); Smart et al. (2010, an d private communication
+2009); Tinney (1996); Tinney et al. (1995, 2003); van Altena, Lee & Hoffleit (1995); Vrba
+et al. (2004).
+The on-line version of this paper includes an Appendix with a collation of the photom-
+etry for the 225 M, L and T dwarfs in our main and reference samples .JHKdata are given
+for all the dwarfs, and IRAC data for 75 of the dwarfs. Distance m oduli and binary status
+are also given where known. MKO-system L′M′photometry is included where available
+(although not used in this study), from Geballe et al. (2001); Golimow ski et al. (2004a);
+Jones et al. (1996); Leggett (1992); Leggett et al. (1998, 2001 , 2002b, 2007); Reid & Cruz
+(2002). All references are given in the on-line material; for complet eness we list here the dis-
+covery and spectral classification references: Becklin & Zuckerm an (1988); Bidelman (1985);
+Burgasser et al. (1999, 2000a, 2000b, 2002, 2003b, 2004b, 200 6b); Burningham et al. (2008,
+2009, 2010); Chiu et al. (2006, 2008); Cruz et al. (2003); Dahn et al. (2002); Delfosse et al.
+(1997); Delorme et al. (2008a); Fan et al. (2000); Geballe et al. (20 02); Giclas, Burnham &
+Thomas (1971); Gliese & Jahreiss (1979); Golimowski et al. (2004b) ; Hawley et al. (2002);
+Irwin et al. (1991); Kirkpatrick et al. (1997, 1999, 2000); Knapp e t al. (2004); Leggett
+(1992); Leggett et al. (2000); Lodieu et al. (2007); Luhman et al. (2007); Luyten (1944,
+1979); Martin et al. (1999b); Mugrauer et al. (2006); Nakajima et al. (1995); Pinfield et al.
+(2008); Reid et al. (2000); Ruiz, Leggett & Allard (1997); Strauss et al. (1999); Tinney et
+al. (1993; 2005); Tsvetanov et al (2000); Warren et al. (2007).
+4. New IRAC Photometry
+Table 1 gives the new IRAC photometry obtained as part of this stud y. The data were
+obtained through our Spitzerprograms GO 40449 and 60093, and DDT 527. Photometry
+was also extracted from IRAC images available in the archive taken as part of programs
+GO 20544 and 40198, and GT 35. In all cases individual frame times we re 30 seconds, with
+multiple position dithers and in some cases frame repeats. The total integration time for
+each object is given in the Table. Note that [3.6], [4.5], [5.8] and [8.0] are n ominal filter
+wavelengths and, as the photometry is not color-corrected for t he dwarfs’ spectral shapes,
+the results cannot be translated to a flux at the nominal wavelengt h (e.g. Cushing et al.– 9 –
+2008, Reach et al. 2005).
+For all but one filter for one dwarf, the post-basic-calibrated-da ta (pbcd) mosaics gen-
+erated by the Spitzerpipeline were used to obtain aperture photometry. For Wolf 940B
+some of the [5.8] images were badly affected by the bright primary and the mosaic was
+recreated excluding these frames. The photometry was derived u sing a 3 or 5 1.2-arcsecond
+pixel aperture radius (i.e. 7.2 or 12 arcsecond diameter), dependin g on the crowding of the
+field. Generally, concentric sky annuli could be used to determine th e sky levels. For ULAS
+1017+01 the field was crowded enough that separate sky apertur es were used, and for Wolf
+940B scattered light from the primary meant that separate sky ap ertures were required. For
+these two dwarfs the uncertainty in each measurement was taken to be the larger of that
+implied by the uncertainty images (noise pixel maps) provided by the Spitzerpipeline or
+the variation with sky aperture. For the other dwarfs the uncert ainties were determined
+from the uncertainty images. The aperture correction was taken from the IRAC handbook
+(http://ssc.spitzer.caltech.edu/irac/dh/ ).
+5. Observed Trends with Spectral Type
+In this Section we present and discuss observed trends with spect ral type. We use
+infrared spectral types for both the L dwarfs (as defined by Geb alle et al. 2002) and the T
+dwarfs (Burgasser et al. 2006b). Both optical (Kirkpatrick et al. 1999) and infrared L dwarf
+spectral types are commonly used. Generally the difference is not s ignificant, however for
+L dwarfs that are unusually blue or red in the near-infrared, indicat ing unusual condensate
+properties, there can be significant differences (e.g. Knapp et al. 2 004, Stephens et al. 2009).
+Data sources are given in §3. Interpretion of the observational data is based on the models
+described in §2; while these are advanced and do reproduce the data quite well, th ere are
+known deficiencies, and the conclusions reached here may evolve as the models improve.
+Figures3and4showabsolute YJHKandabsolute[3.6], [4.5], [5.8]and[8.0]magnitudes
+as a function of type. MYandMJbrighten from late-L to early-T as the condensate cloud
+decks clear from the photosphere; the clouds especially impact the 1µm region (Ackerman
+& Marley 2001). MHis an intermediate case and the brightness here is approximately
+constant from mid-L to mid-T. These trends have been known for s ome time (e.g. Dahn et
+al. 2002) and have now been confirmed, through studies of binaries , to be an intrinsic part
+of brown dwarf evolution, and not due to for example differences in g ravity or metallicity, or
+due to unresolved binarity (e.g. Liu et al. 2006, Looper et al. 2009). In the near-infrared
+the latest-type T dwarfs become rapidly fainter while the drop is muc h less pronounced in
+the mid-infrared, as more and more flux is distributed to the longer w avelengths at low– 10 –
+temperatures (Figures 1 and 2 and e.g. Burrows et al. 2003, Legge tt et al. 2009). There
+is a 4 – 5 magnitude increase in the near-infrared magnitudes betwee n T5 and T9 types,
+compared to only 1.5 – 2 magnitudes at the IRAC wavelengths.
+Figures 5 and 6 show various colors as a function of type, where aga in infrared spectral
+types are used for both the L and T dwarfs. The behavior of the Y−Jcolor is difficult
+to interpret; for early-type L to early-type T dwarfs both bands are impacted by the cloud
+decks. There is an apparent bluewards trend for the latest-type T dwarfs. This may be due
+to the first overtone band of pressure-induced H 2becoming significant at J(Borysow 2002),
+or to the brightening of Ydue to the weakening of the very broad 0.77 µm K I feature as K
+I condenses into KCl (Lodders 1999). The temperature depende nce of the latter is stronger
+than that of the former, and so this blueward trend in Y−Jis most likely due to K I
+condensation.
+The reddening and increased dispersion in the J−H,H−KandJ−Kcolors for
+around L3 to T2 types demonstrates the effect of the condensat e clouds, and the variability
+in the cloud properties. The less well-populated H−[4.5] plot also shows a wide dispersion
+for these types due to the impact of the clouds at H.
+T dwarfs are increasingly blue in J−H,H−KandJ−K, due to the loss of the
+cloud decks and the onset of CH 4absorption. However this blueward trend diminishes for
+the latest types, as very little flux remains to be absorbed by CH 4(Figure 2). For types
+later than T5 the H−KandJ−Kplots show more scatter — most likely this reflects
+variations in metallicity and gravity, as pressure-induced H 2opacity becomes important at
+these temperatures, which impacts the Kband, and which is very sensitive to metallicity
+and, to a lesser extent, gravity.
+At the IRAC bands, T5 and later dwarfs become rapidly redder in [3.6] −[4.5] and
+[5.8]−[8.0], but bluer in [4.5] −[5.8]. As temperature decreases, more flux emerges in the
+relatively clear 4 – 5 µm and>8µm windows, which lie between strong H 2O and CH 4
+bands (Figure2 and e.g. Leggett et al. 2009). The dispersion in the I RAC colors is relatively
+small compared to the photometric uncertainties. As previously me ntioned, variations in the
+condensate clouds lead to scatter in the H−[4.5] for L3 to T2 types. For the latest-type T
+dwarfs scatter is seen in [4.5] −[5.8] most likely due to variations in the CO absorption
+which impacts the [4.5] flux and is gravity- and metallicity-sensitive (se e§7).– 11 –
+6. Trends with Color
+6.1. Trends for M, L and T Dwarfs
+In this Section we show various trends with color for late-M, L and T d warfs. We
+compare the observations of the late-type T dwarfs to model seq uences calculated for 500
+≤TeffK≤1000, where all models are cloud-free (as appropriate for our late -type T dwarf
+sample) and also include significant but typical vertical mixing, with an eddy-diffusion pa-
+rameterKzz= 104(cm2s−1). Colors have been calculated for surface gravity g= 100, 300
+and 1000 ms−2, or logg= 4.0, 4.477 and 5.0 in cgsunits, and metallicity [m/H]= −0.3,
+[m/H]= +0 .0 and [m/H]= +0 .3. The evolutionary models show that for this temperature
+range log g= 5.0 corresponds to a ∼30 Jupiter-mass object aged between 1 and 10 Gyr,
+logg= 4.477 to a∼14 Jupiter-mass object aged between 0.2 and1.5 Gyr, andlog g= 4.0 to
+a very low-mass and young ∼6 Jupiter-mass object aged between 30 and 200 Myr. A gravity
+as high as log g= 5.5 is not relevant for a study of the coldest T dwarfs: at Teff= 500 K
+logg≤5.0 if age is ≤10 Gyr.
+Figure 7 shows absolute magnitudes as a function of various colors f or late-M, L and T
+dwarfs where symbols indicate spectral type. Values of Tefffrom the atmospheric models are
+indicated along the right axes for the log g= 4.477 and [m/H]= 0 case. Note again that the
+coolest dwarfs drop off much more rapidly in brightness at near-infr ared wavelengths than
+at longer wavelengths. In these plots the models sequences are us ed to illustrate the likely
+range of values only; we discuss the model implications for the atmos pheric parameters of
+the latest-type T dwarfs in §7. The dotted sequences are calculated from a solar metallicity
+logg= 4.477 model that does not include vertical mixing; the discrepancies w ith the
+observations confirm the conclusions of previous analyses that ve rtical mixing is significant
+in brown dwarf atmospheres ( §2).
+The model sequences that include mixing effectively encompass the r ange of datapoints
+quite well. The exceptions in the near-infrared are MY:Y−JandMJ:J−Hwhere the
+calculated Y, and for the latest-type T dwarfs J, both appear to be a few tenths of a
+magnitude too bright (see also Figure 2). This is presumably due to th e known opacity
+deficiencies and problems with modelling the wing of the strong K I reso nance doublet in
+thefar-red(see §2). Atlongerwavelengths, thereappearstobeasystematicoffse tincolorfor
+M[3.6]:[3.6]−[4.5] such that the calculated [3.6] fluxes are too faint. This is suppor ted by the
+model comparisons to the observed near- through mid-infrared s pectral energy distributions
+and IRAC magnitudes for 600 K brown dwarfs presented by Legget t et al. (2009) — the best
+fits all have the calculated [3.6] fluxes too low by 20–50%. It is not clea r if this is related to
+missing opacities in the near-infrared (which may lead to too high near -infrared fluxes and– 12 –
+too low mid-infrared fluxes), or if it is due to errors in the pressure- temperature structure of
+our non-equilibrium models.
+Figure 8 shows a selection of color:color plots. Sources with new IRAC photometry
+presented here are indicated, demonstrating that these data fill gaps in color sequences and
+also illustrate better the dispersion in some colors. The too-faint sy nthetic [3.6] magnitudes
+are again apparent. The use of a H−[4.5] color in Figures 7 and 8 is explained in §7.
+6.2. Survey Sensitivities to 500 K Brown Dwarfs
+We can use the absolute magnitudes to estimate the number of 500 K brown dwarfs
+(objects with masses between 7 and30 Jupiters for anagerange o f0.4 to 10Gyr) that will be
+found by various sky surveys. The UKIDSS LAS is expected to cove r 3800 deg2toJ= 19.6
+(Vega magnitudes, 5 σ); hence the LAS should detect 500 K brown dwarfs to 25 pc over a v ol-
+umeof6.4×103pc3. TheVISTAVIKINGsurvey( http://www.eso.org/sci/observing/policies/PublicSur veys/sciencePublicSurveys.html )
+is expected to reach J= 21.4 (Vega magnitudes, 5 σ) over 1500 deg2, implying a distance
+limit of 50 pc, and a volume of 1 .7×104pc3for detection of 500 K objects. The WISE
+sensitivities of 0.12, 0.16 and 0.65 mJy in bands 1, 2 and 3 respectively ( 2.8 – 3.8, 4.1 – 5.2
+and 7.5 – 16.5 µm; Mainzer et al. 2005) suggest that the mission will detect 500 K obj ects to
+10, 15 and 10 pc in each of these bands (based on our Spitzermeasurements and atmospheric
+models, e.g. Figure 6 of Leggett et al. 2009). Hence this all-sky missio n will sample a volume
+of 1.4×104pc3at band 2 for 500 K dwarfs, i.e. similar to the planned VISTA mission and
+about three times larger than the volume probed by the LAS. At Teff≈400 K however we
+estimate that the WISE volume should surpass the near-infrared V ISTA volume by about
+a factor of 6 — at this temperature VISTA should detect dwarfs ou t to around 17 pc and
+WISE out to around 10 pc. (At the faint limits these will be single-band detections and
+candidates will require follow up observations.)
+Metchev et al. (2008) and Pinfield et al. (2008) use the number of T d warfs found in the
+2MASS, SDSS and UKIDSS surveys to determine that the T dwarf ma ss function is most
+likely flat with α= 0 (or possibly declining to lower masses). The Monte Carlo simulations
+by Burgasser (2004a) then imply that the number of 500 K brown dw arfs is around 0.003 per
+100 K per pc3. Hence VISTA and WISE should find around forty-five 500 K brown d warfs
+compared to the fifteen expected in UKIDSS at the end of the LAS s urvey (and WISE
+should find around thirty cold 400 K brown dwarfs). Note that the James Webb Space
+Telescope will be able to obtain low-resolution spectroscopy at both near- and mid-infrared
+wavelengths for 500 K objects found by VISTA or WISE, given the e stimated NIRSpec and
+MIRI sensitivities of around 25 (Vega) magnitudes at Jand 20 (Vega) magnitudes at 5 –– 13 –
+10µm (http://www.stsci.edu/jwst/science/sensitivity/ , see also Marley & Leggett 2009).
+7. Dwarfs Cooler than 800 K and Temperature, Metallicity and Gravity
+Indicators
+We now examine observed and modelled trends for the coolest T dwar fs only, and iden-
+tify temperature, metallicity and gravity photometric indicators. F igures 1 and 2 demon-
+strate the rapidly increasing ratio of the mid-infrared flux to the ne ar-infrared flux for late-
+type T dwarfs. This means that a large wavelength baseline near-inf rared to mid-infrared
+color should be a good indicator of Teff. Warren et al. (2007), Leggett et al. (2009) and
+Stephens et al. (2009) show that the H−[4.5] color is optimum. This color is observed to be
+very sensitive to Teffat temperatures cooler than 1000 K (spectral types later than a round
+T6; Warren et al. and Stephens et al.), and this sensitivity is calculate d to increase for
+Teff<800 K (spectral types later than around T7; Leggett et al.). Step hens et al. (their
+Figure 14) show that the Teff:H−[4.5] relationship shows less scatter than colors using other
+near- or mid-infrared bandpasses. Figures 6, 7 and 8 demonstrat e the rapid reddening of
+H−[4.5] with type or decreasing Teffand the small dispersion in this color, for the late-type
+T dwarfs. H−[4.5] therefore shows relatively weak dependence on metallicity and g ravity.
+Leggett et al. (2009) show that plausible variations in gravity and me tallicity for a local
+disk sample, and variations in the chemical mixing efficiency, each impac tH−[4.5] by∼0.2
+magnitudes. This is small given the large degree of sensitivity to Teff: ∆(H−[4.5])≈0.7
+magnitudes for ∆ Teff= 100 K at Teff= 600 K for example.
+Hence we adopt H−[4.5] as our primary reference color for the very late-type T dwarf s,
+and investigate relationships between this color and others. We do n ot consider colors
+involving the [3.6] band, as the models systematically underestimate t he flux in this band
+(note that as very little flux is emitted in this spectral region the pro blem is not expected to
+affect the atmospheric structure). We look in detail at the known s ample of T dwarfs with
+IRAC photometry that have HMKO−[4.5]>3.0, which the models imply have Teff∼<800 K.
+Including the sample of T dwarfs with new IRAC photometry present ed here, twelve T
+dwarfs fall into this group. The dwarfs are listed in Table 2 together with some of their
+fundamental properties. Metallicity and age are reasonably well de termined for the three
+objects that are companions to stars, which means temperature and gravity can be well
+constrained. The parameters determined by model spectral fitt ing are less well constrained
+for the other dwarfs.
+Figure 6 shows that selecting HMKO−[4.5]>3.0 should provide a sample with spectral
+types later than T7. Two earlier-type dwarfs fall into our sample: t he very metal-poor– 14 –
+dwarfs 2MASS J0937347+293142 (T6p, 2MASS 0937+29, HMKO−[4.5] = 3.03) and 2MASS
+J12373919+6526148(T6.5e, 2MASS1237+65, HMKO−[4.5]= 3.01). Their inclusionisdueto
+the (small) dependency of H−[4.5] on metallicity. Four T7.5 dwarfs fall below our color cut-
+off. One is the metal-rich T7.5 dwarf 2MASS J1217110-031113 (2MAS S 1217-03, Saumon et
+al. 2007, HMKO−[4.5] = 2.75) whose exclusion is due to its enhanced metallicity. The three
+other excluded T7.5 dwarfs are objects observed as part of our w arm IRAC campaign, ULAS
+0139+00 ( HMKO−[4.5] = 2 .79), ULAS 0150+13 ( HMKO−[4.5] = 2 .99) and ULAS 2321+13
+(HMKO−[4.5] = 2.99). We discuss these dwarfs further below.
+Figure9showsabsolutemagnitudesasafunctionofcolorforthose dwarfsinTable2with
+measured distances; symbol type and color indicate the physical p roperties of each dwarf.
+Model sequences are also shown. We plot MHas a function of both H−KandH−[4.5],
+andM[4.5]as a function of [4.5] −[5.8]. These colors have been chosen as H−[4.5] has a
+small dependency on metallicity and gravity, while H−Kand [4.5] −[5.8] have significant
+dependencies on metallicity and gravity. Although the agreement is n ot perfect, the relative
+location of the eight objects in the plots is in agreement with our mode ls. The metal-poor
+T6 – T6.5 dwarfs 2MASS 0937+29 and 2MASS 1237+65 are brighter in MHandM[4.5]as
+predicted, and the models would imply Teff≥800 K for these dwarfs, consistent with earlier
+analyses. The locations of the 800 K log g∼5 approximately solar metallicity T7.5 dwarfs
+Gl 570D and HD 3651B are consistent with the models. The T8 2MASS 0 415-09 is slightly
+cooler than, and is probably more metal-rich, than Gl 570D, as implied from spectroscopic
+analyses. The T8.5 and T9 dwarfs Wolf 940B, ULAS 0034-00 and ULAS 1335+11 are still
+cooler, with Teff≈600 K, and ULAS 0034-00 and ULAS 1335+11 are lower gravity or mor e
+metal-rich than than Wolf 940B, as also indicated by spectral fitting .
+The metal-poor dwarf 2MASS 1237+65is noteworthy as it has very s trong Hαemission.
+Liebert and Burgasser (2007) explore and discount the possibility t hat it is young and low-
+gravity, which would also be inconsistent with our analysis. They sugg est that instead the
+object may be an interacting close double system. If the companion fills the Roche lobe then
+they calculate that the companion must be cooler than 650 K. As no s ignificant excess is
+seen at [4.5] in Figure 9, our models suggest the companion would have to be cooler than
+500 K. We conclude that the red H−[4.5] for the dwarf is due to its low metallicity and not
+to the detected presence of a cool companion.
+Figures 10 through 12 show H−[4.5] against observed and calculated colors. The ob-
+served near-infrared colors Y−JandJ−Hfor the sample show a small dispersion in
+Figure 10, and in the case of J−Hthe observed dispersion is smaller than modelled (most
+likely due to known inadequacies in the near-infrared opacities, §2). Both Figures 11 and 12
+however show a large dispersion in the observed H−Kand [4.5] −[5.8] colors. The models– 15 –
+indicate that the dispersion reflects metallicity and gravity variation s in the sample, as also
+seen in Figure 9. These colors also show significant scatter with spec tral type in Figures 5
+and 6. The observed dispersion in [4.5] −[5.8] is larger than calculated but the trends are
+in good agreement — the high-gravity and metal-poor dwarfs are blu er in both H−Kand
+[4.5]−[5.8]. In the case of H−Kthis is most likely due to the metallicity-sensitive (and
+to a lesser extent gravity-sensitive) pressure-induced H 2opacity in the K-band. For [4.5] −
+[5.8] an increase in gravity or decrease in metallicity leads to more flux a t [4.5] as the CO
+absorption is weakened (see Figure 3 in Leggett et al. 2009).
+Figures 11 and 12 suggest that ULAS 1017+01 and ULAS 1238+09 ar e low-gravity,
+logg= 4.0 – 4.5, possibly metal-rich objects, similar to the other LAS very la te-type T
+dwarfs ULAS 0034-00 and ULAS 1335+11 but not as cool. A gravity t his low for ULAS
+1017+01 disagrees with the value determined by Burningham et al. (2 008; Table 2) based on
+near-infrared data only. While including mid-infrared data is expecte d to provide a better
+estimate, the temperature and gravity determined here would imply a very young age for
+this dwarf of 0.1 – 0.4 Gyr (see the Notes to Table 2). This dwarf is spe ctrally peculiar in
+the near-infrared and is classified as T8p. If the object is instead a binary system composed
+of a 900 K and 700 K pair, or a 800 K and 600 K pair, then the calculated H−[4.5] color is
+reduced, and the H−Kand [4.5] −[5.8] colors increased, by 0.1 – 0.2 magnitudes, compared
+to the single dwarf solution. Hence the gravity could be higher (log g= 4.5 – 5.0) and the
+age more in line with what would be expected for a local disk dwarf (aro und 1 Gyr, see the
+Notes to Table 2). A parallax measurement would be helpful for purs uing this further.
+Figures 9 and 11 allow us to constrain the metallicity of the Wolf 940 sys tem: Burning-
+ham et al. (2009) give [Fe/H] = −0.06±0.20 based on the V−Kcolor of Wolf 940A but
+the photometric analysis presented here implies [m/H] = 0 to +0 .3 for the system. We can
+exclude the possibility that Wolf 940B is metal-poor.
+Although not part of our primary sample, we can also use Figure 11 to estimate the
+parameters of the three T7.5 dwarfs observed as part of our war m IRAC campaign (with
+therefore no [5.8] photometry): ULAS 0139+00, ULAS 0150+13 an d ULAS 2321+13. The
+{(H−K)MKO,HMKO−[4.5]}colors of these objects are {−0.11±0.08, 2.79±0.13},{0.27±
+0.16, 2.99±0.04}and{−0.01±0.03, 2.99±0.06}, respectively. Their position in Figure 11
+relativetothemodel sequences andotherdwarfssuggest that: ULAS0139+00haslog g≈5,
+[m/H]≈0 andTeff≈850 K; ULAS 0150+13 is a very metal-rich and/or low-gravity 750 –
+800 K object; ULAS 2321+13 has log g≈5, [m/H] >0 andTeff≈800 K. Hence the dwarfs
+have been excluded from the HMKO−[4.5]>3.0 sample due to their enhanced metallicity or,
+in the case of ULAS 0139+00, higher temperature.
+The 2MASS-selected dwarfs in our HMKO−[4.5]>3.0 sample tend to be high-gravity– 16 –
+and/or low-metallicity because they are selected for blue H−Kcolor. However the tendency
+for the LAS objects to be low-gravity (Table 2) is difficult to underst and. Figure 10 shows
+that the colors used to select the LAS dwarfs, YJH, should not impart any metallicity
+or gravity bias to the sample, as objects with a range in these param eters occupy very
+similar regions of the diagram. There should also be no gravity or age b ias introduced by a
+brightness selection, as field brown dwarfs with similar temperature s but different gravities
+have very similar luminosity, because the radius does not change sign ificantly after the first
+∼100 Myr (e.g. Burrows et al. 1997 their Figure 10; Saumon & Marley 20 08 their Figure 4.)
+The simulations of the mass function by Burgasser (2004a) does sh ow that the median field
+brown dwarf mass (and hence gravity) decreases to lower temper atures, because lower mass
+brown dwarfs start off cooler and remain cooler than higher mass br own dwarfs. However
+the typical mass and hence gravity is still calculated to be higher tha n what we observe
+for the LAS dwarfs: at Teff≈600 K Burgasser’s simulations indicate that, for a flat mass
+function and a constant birthrate, the median mass would be 30 – 40 MJupiter, with a median
+age older than 6 Gyr and log g≈5. The LAS T8 – T9 dwarf on the other hand appear
+to be generally younger than 2 Gyr and less massive than 20 M Jupiter. This puzzle will have
+to be re-examined when the sample of cold LAS brown dwarfs is larger and more significant
+conclusions can be drawn.
+8. Conclusions
+We have confirmed that H−[4.5] is a very sensitive temperature indicator for very
+late-T type brown dwarfs. Also H−Kand [4.5] −[5.8] show significant scatter for the
+late-type T dwarfs which the models indicate are due to metallicity and gravity variations.
+Unfortunately these effects are difficult to disentangle as an increa se in gravity has a similar
+effect to a decrease in metallicity. However the models and data sugg est that H−Kis
+more sensitive to metallicity and [4.5] −[5.8] to gravity, and so it is possible, as more cold
+brown dwarfs are found, and as the models continue to improve, th at we can photometrically
+distinguish between these changes.
+We provide new or improved estimates of temperature, gravity and metallicity, and
+hence mass and age, for the T8p ULAS 1017+01 and the T8.5 ULAS 12 38+09. Both dwarfs
+appear to be lowgravity, with log g= 4.0 – 4.5, and possibly metal-rich. ULAS 1017+01has
+Teff= 750 – 800 K, and ULAS 1238+09 is cooler with Teff= 575 – 625 K. These parameters
+imply low masses and young ages for the dwarfs: 8 – 12 M Jupiterand 0.1 – 0.4 Gyr, and
+6 – 10 M Jupiterand 0.2 – 1 Gyr, for ULAS 1017+01 and ULAS 1238+09, respectively. If
+the spectrally peculiar dwarf ULAS 1017+01 is a (900 + 700) K or (800 + 600) K binary– 17 –
+system, then the gravity of each component could be log g= 4.5 – 5.0 (and mass ∼20
+MJupiter) implying an age around 1 Gyr, more consistent with expectations fo r a local disk
+dwarf. We have also estimated the properties of three ∼800 K T7.5 dwarfs observed as part
+of our warm IRAC campaign: ULAS 0139+00, ULAS 0150+13 and ULAS 2321+13. ULAS
+0150+13 and ULAS 2321+13 are metal-rich, and ULAS 0150+13 may a lso have low gravity
+(and hence mass). Finally, the analysis presented here also constr ains the metallicity of the
+Wolf 940 M4+T8.5 system, as it excludes the possibility that the syste m is metal-poor.
+The T8 – T9 dwarfs found in the UKIDSS LAS appear to be predominan tly metal-rich,
+low gravity and therefore young low-mass dwarfs. The T7.5 – T8 2MA SS dwarfs on the
+other hand are mostly metal-poor and high gravity, and therefore older higher-mass dwarfs.
+While the 2MASS bias is understood as a color selection effect, the bias to lower gravities
+is not yet understood for the LAS dwarfs. The LAS sample is small, an d we look forward
+to the discovery of more 500 – 800 K brown dwarfs, for which both n ear- and mid-infrared
+data must be obtained.
+Photometric data at wavelengths longer than 3 µm are both important and useful for
+the latest-type T dwarfs with 500 ≤TeffK≤800. It is expected to be even more so for the
+cooler proposed Y-type objects, which are likely to have Teff≤400 K. It is only possible to
+get such ground-based data for very bright objects; the sensit ivity and robust performance
+of theSpitzerinstruments at 4 – 15 µm have revolutionized the study of cool brown dwarfs.
+We expect that WISE will provide a similarly dramatic advancement.
+Even with Spitzerwarm, the short wavelength IRAC photometry provides vital infor -
+mation that will be impossible to get fromthe ground, and which may be difficult to get with
+WISE. We hope the warm mission will be extended so that [3.6] and [4.5] p hotometry can be
+obtained for cold brown dwarfs that will be discovered in ongoing and planned near-infrared
+ground-based surveys.
+Also, although chemical mixing reduces the expected sensitivity of t he WISE 4.1 –
+5.2µm band to cold brown dwarfs, the same effect increases the sensitiv ity of its 7.5 –
+16.5µm band. WISE should detect 500 K brown dwarfs out to 15 pc, and ma y find the
+first 400 K object. Extending the WISE mission will provide a larger sa mple of these rare
+objects, which will fill the luminosity and temperature gap between t he cool brown dwarfs
+and the giant planets.
+This work is based on observations made with the Spitzer Space Telescope , which is
+operated by the Jet Propulsion Laboratory, California Institute o f Technology under a con-
+tract with NASA. Support for this work was provided by NASA throu gh an award issued
+by JPL/Caltech. Support for this work was also provided by the Spit zer Space Telescope– 18 –
+Theoretical Research Program, through NASA. SKL’s research is supported by the Gemini
+Observatory, which is operated by the Association of Universities f or Research in Astron-
+omy, Inc., on behalf of the international Gemini partnership of Arg entina, Australia, Brazil,
+Canada, Chile, the United Kingdom, and the United States of America . This research
+has benefitted from the SpeX Prism Spectral Libraries, maintained by Adam Burgasser at
+http://www.browndwarfs.org/spexprism. This research has also b enefitted from the M, L,
+and T dwarf compendium housed at DwarfArchives.org and maintaine d by Chris Gelino,
+Davy Kirkpatrick, and Adam Burgasser. Finally, we are grateful to John Stauffer for a very
+helpful referee’s report.– 19 –
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+Table 1. New IRAC Photometry
+Name Spec. Integ. [3.6] mag [4.5] mag [5.8] mag [8.0] mag Prog. Disc.
+(RA Dec)aType min (error mmag)bNum. Ref.c
+ULAS J085715.96+091325.3 T6 96.0 17.65(15) 16.73(16) 16.50(96) 1 6.36(308) 40449 1
+2MASS J00501994-3322402 T7 5.0 14.82(5) 13.57(3) 13.32(17) 13.0 0(22) 40198 2
+2MASS J03480772-6022270 T7 2.5 14.04(5) 12.51(4) 12.89(10) 12.2 8(14) 35 3
+SDSS J150411.63+102718.4 T7 7.5 15.44(7) 14.01(5) 14.37(27) 13.76 (59) 40198 4
+SDSS J162838.77+230821.1 T7 7.5 15.25(8) 13.86(6) 14.14(35) 13.55 (64) 40198 4
+2MASS J11145133-2618235 T7.5 2.5 14.01(5) 12.23(3) 13.22(17) 12 .25(22) 20544 2
+ULAS J013939.77+004813.8dT7.5 24.0 17.59(23) 16.33(12) 60093 5
+ULAS J015024.37+135924.0eT7.5 24.0 16.42(8) 15.12(4) 60093 1
+ULAS J232123.79+135454.9fT7.5 2.5 15.80(16) 14.16(6) 60093 1
+ULAS J101721.40+011817.9 T8pg64.0 17.29(18) 16.02(7) 15.90(33) 15.70(152) 40449 6
+ULAS J123828.51+095351.3 T8.5 24.0 17.09(17) 15.34(8) 15.32(51) 1 4.58 (91) 40449 6
+Wolf 940BhT8.5 18.5i16.44(16) 14.43(5) 15.38(150) 14.36(78) 527 7
+a2MASS names are given as HHMMSSSS ±DDMMSSS, SDSS and ULAS as HHMMSS.SS ±DDMMSS.S.
+bA 30 mmag (3%) error should be added in quadrature to the quoted r andom errors to account for systematics.
+cDiscovery references are (1) Burningham et al. (2010) (2) Tinney et al. 2005 (3) Burgasser et al. 2003b (4) Chiu et
+al. 2006 (5) Chiu et al. 2008 (6) Burningham et al. 2008 (7) Burningha m et al. 2009.
+dThis object’s {(H−K)MKO,HMKO−[4.5]}implies that it has log g≈5, [m/H] ≈0 andTeff≈850 K (see §7).
+eThis object’s {(H−K)MKO,HMKO−[4.5]}implies that it is a very metal-rich and/or low-gravity 750 – 800 K dwarf
+(see§7).
+fThis object’s {(H−K)MKO,HMKO−[4.5]}implies that it has log g≈5, [m/H] >0 andTeff≈800 K (see §7).
+gThis dwarf is spectrally a T8 in the J-band, but T6 in the H- andK-bands.
+hAlso known as ULAS J214638.83-001038.7.
+iEffective integration time for [5.8] was 9.5 minutes after removal of c orrupted frames.– 26 –
+Table 2. T Dwarf Sample with HMKO−[4.5]>3
+Name Spec. Previously Determined Parameters Referencesa
+Type TeffK log g[m/H] M JupAge Disc . πParam.
+Gyr
+2MASS J0937347+293142 T6p 925–975 5.2–5.5 −0.3 45–69 3–10 1 2 3
+2MASS J12373919+6526148 T6.5e 800-850 ∼5.0−0.2 20–40 1–4 4 2 5
+2MASS J11145133-2618235 T7.5 725–775 5.0–5.3 −0.3 30–50 3–8 6 7
+Gl 570D T7.5 800–820 5.1 0.00 31–47 2–5 8 9 10
+HD 3651B T7.5 780–840 5.3 +0 .15 40–72 3–12 11, 12 9 13
+2MASS J0415195-093506 T8 725–775 5.0–5.4 ∼>0 33–58 3–10 1 2 14
+2MASS J09393548-2448279bT8 500–700 5.0–5.3 −0.3 20–40 2–10 6 15
+ULAS J101721.40+011817.9cT8p [750–850] [5.0–5.5] [ ∼0] [33–70] [2–10] 16 16
+ULAS J123828.51+095351.3dT8.5 16
+Wolf 940BeT8.5 545–595 4.8–5.0 ∼0 20–32 3.5–6 17 9 17
+ULAS J003402.77-005206.7 T9 550–600 4.5 ∼>0 13–20 1–2 18 19 19
+ULAS J133553.45+113005.2 T9 500–550 4.0–4.5 ∼>0 5–20 0.1–2 15 20 15
+aDiscovery, trigonometric parallax and parameter references are :
+(1) Burgasser et al. 2002 (2) Vrba et al. 2004 (3) Geballe et al. 2009 (4) Burgasser et al. 1999 (5) Liebert &
+Burgasser 2007 (6) Tinney et al. 2005 (7) Leggett et al. 2007a (8) Burgasser et al. 2000a (9) ESA 1997 (10) Saumon
+et al. 2006 (11) Mugrauer et al. 2006 (12) Luhman et al. 2007 (13) L iu et al. 2007 (14) Saumon et al. 2007 (15)
+Leggett et al. 2009 (16) Burningham et al. 2008 (17) Burningham et al. 2009 (18) Warren et al. 2007 (19) Smart
+et al. 2010 (20) Smart et al. private communication 2009.
+bProbable binary, either a pair of 600 K brown dwarfs, or a 500 K and a 700 K brown dwarf pair (Burgasser et
+al. 2008b, Leggett et al. 2009).
+cTheparametersoriginallyderivedfromthenear-infrareddataonly forthisspectrallypeculiarobjectareuncertain.
+The photometric analysispresentedhere implies Teff= 750–800K, log g= 4.0–4.5, [m/H] ∼>0, mass8 – 12M Jupiter
+and age 0.1 – 0.4 Gyr. If this spectrally peculiar object is a binary syst em then the gravity, and hence mass and
+age, could be larger — log g= 4.5 – 5.0, masses ∼20 MJupiterand age∼1 Gyr.
+dThe photometric analysis presented here implies Teff= 575 – 625 K, log g= 4.0 – 4.5, [m/H] ∼>0, mass 6 –
+10 MJupiterand age 0.2 – 1.0 Gyr.
+eBurningham et al. (2009) give [Fe/H] = −0.06±0.20 based on the V−Kcolor of Wolf 940A; the photometric
+analysis presented here implies [m/H] = 0 .0 to +0.3 for the system.– 27 –
+Fig. 1.— The percentage of the total flux emitted at wavelengths lon ger than 3 µm, as a
+function of Teff, calculated from our models. Solar metallicity non-equilibrium models wit h
+Kzz= 104(cm2s−1) and log g= 5.0 were used. Cloudy models were used for the warmer
+temperatures: fsed= 3 atTeff= 1400 K and fsed= 4 atTeff= 1200 K. Cloud-free models
+were used for Teff≤1000 K.– 28 –
+Fig. 2.— Observed (black lines) and modelled (red lines) spectral ener gy distributions of
+the 750 K T8 dwarf 2MASS J0415195-093506 (lower spectrum; Sau mon et al. 2007) and the
+600 K T9 dwarf ULAS J003402.77-005206.7(upper spectrum; Legg ett et al. 2009). The zero
+flux level for the T9 dwarf is indicated by the dashed line at y= 2. Principal absorption
+features are identified. NH 3is also likely to be present at near-infrared wavelengths for the
+T9 dwarf. The MKO system YJHKbandpasses, as well as the IRAC and WISE filter
+bandpasses, are indicated.– 29 –
+Fig. 3.—Absolute YJHKmagnitudesontheMaunaKeaObservatoriesphotometricsystem,
+as a function of spectral type. Uncertainties in absolute magnitud es are smaller than the
+symbolsize. InfraredspectraltypesareusedforboththeLan dTdwarfs, andtheuncertainty
+in type is 0.5 – 1.0 subclasses. Green points are known binaries, and th e red point is Wolf
+940B for which we present new IRAC photometry in this paper.– 30 –
+Fig. 4.— Absolute IRAC [3.6], [4.5], [5.8] and [8.0] magnitudes as a function o f spectral type.
+Uncertainties in absolute magnitudes are similar to the symbol size. I nfrared spectral types
+are used for both the L and T dwarfs, and the uncertainty in type is 0.5 – 1.0 subclasses.
+Green points are known binaries, and the red point is Wolf 940B. Note the change of scale
+for the y axis compared to Figure 3.– 31 –
+Fig. 5.— Near-infrared colors on the Mauna Kea Observatories phot ometric system, as a
+function of spectral type. Infrared spectral types are used f or both the L and T dwarfs.
+Typical uncertainties are indicated by the error bar. Green points are known binaries, and
+the red points are dwarfs for which we present new IRAC photomet ry in this paper.– 32 –
+Fig. 6.— IRAC colors as a function of spectral type. Infrared spec tral types are used for
+both the L and T dwarfs. Typical uncertainties are indicated by the error bar. Symbols are
+as in Figure 5.– 33 –
+Fig. 7.— Color-Magnitude plots for late-M, L and T dwarfs where symb ol shape indicates
+spectral type: circles are M, triangles are L, squares are T0–T6.5 and asterisks are T7–T9
+types. Uncertainties in absolute magnitudes are smaller than or simila r to the symbol size.
+Uncertainty in color is 5 – 10%. Green symbols are known binaries, and the red point is Wolf
+940B. Model sequences are also shown for 500 ≤TeffK≤1000; values of Teffare shown on
+the right axes. Solid lines demonstrate the range in absolute magnitu de andTefffor a likely
+range in gravity, dashed lines for a plausible range in metallicity (see dis cussion in text and
+later Figures). All models are cloud-free and include vertical mixing w ithKzz= 104(cm2
+s−1), except for the dotted sequence for which Kzz= 0.– 34 –
+Fig. 8.— Color:color plots for late-M, L and T dwarfs where symbols and lines are as in
+Figure 7. Green points are known binaries, and the red points are dw arfs for which we
+present new IRAC photometry in this paper. Values of Tefffrom the models are shown on
+the right axes. Typical uncertainties are indicated by the error ba r.– 35 –
+Fig. 9.— Color-magnitude plots for late-type T dwarfs. Small dots ar e the generic sample,
+with known binaries shown as ringed symbols. Larger symbols are the dwarfs in Table 2
+and symbol size and color indicates gravity and metallicity respective ly. Largest to smallest
+filled circles correspond to log g≈5.4 (5.2 – 5.5), log g≈5.2 (5.0 – 5.4), log g≈5.0
+(4.8 – 5.1), and log g≈4.3 (4.0 – 4.5). Red symbols indicate metal-rich, green solar, and
+cyan metal-poor dwarfs; black are dwarfs with unconstrained met allicity. Model sequences
+with log g= 4.0, 4.5 and 5.0 and [m/H]=0 are shown as solid lines, and log g= 4.5
+with [m/H]= −0.3 and +0 .3 as dashed lines. In all three panels the metal-poor sequence
+is brightest, followed by the gravity sequences from high to low, follo wed by the metal-rich
+sequence. Crosses along the sequences indicate the 800K and 600 K points for each model
+set. The dwarfs listed in Table 2 are identified.– 36 –
+Fig. 10.— H−[4.5] color against Y−JandJ−H, theYJHare on the MKO system. Small
+dots are the generic sample, with known binaries shown as ringed sym bols. Larger symbols
+are the dwarfs in Table 2 and symbol size indicates gravity and colors indicate metallicity.
+Largest to smallest filled circles correspond to log g≈5.4 (5.2 – 5.5), log g≈5.2 (5.0 – 5.4),
+logg≈5.0 (4.8 – 5.1), and log g≈4.3 (4.0 – 4.5). Open circles are unconstrained gravity.
+Red symbols indicate metal-rich, green solar, and cyan metal-poor d warfs; black are dwarfs
+with unconstrained metallicity. Model sequences with a range of gra vity and metallicity are
+shown, with line types as in Figure 9 and as indicated on the right axes in this plot. The Teff
+values for the log g= 4.5 [m/H]= 0 model are indicated on the top axis, and crosses along
+the sequences indicate the 800K and 600K points for each model se t. The dwarfs listed in
+Table 2 are identified.– 37 –
+Fig. 11.— H−[4.5] color against H−Kwith symbols and sequences as in Figure 10.– 38 –
+Fig. 12.— H−[4.5] color against [4.5] −[5.8] with symbols and sequences as in Figure 10.
\ No newline at end of file
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+++ b/1001.0763.txt
@@ -0,0 +1,351 @@
+arXiv:1001.0763v2  [hep-th]  8 Aug 2010Topological Insulators and Superconductors from D-brane
+Shinsei Ryua1and Tadashi Takayanagib2
+aDepartment of Physics, University of California, Berkeley , CA 94720, USA
+bInstitute for the Physics and Mathematics of the Universe (I PMU),
+University of Tokyo, Kashiwa, Chiba 277-8582, Japan
+Abstract
+Realization of topological insulators (TIs) and supercond uctors (TSCs), such as
+thequantumspinHalleffectandthe Z2topological insulator, intermsofD-branesin
+string theory is proposed. We establish a one-to-one corres pondence between the K-
+theoryclassification ofTIs/TSCsandD-branecharges. Thes tringtheoryrealization
+of TIs and TSCs comes naturally with gauge interactions, and the Wess-Zumino
+term of the D-branes gives rise to a gauge field theory of topol ogical nature. This
+sheds light on TIs and TSCs beyond non-interacting systems, and the underlying
+topological field theory description thereof.
+1e-mail:sryu@berkeley.edu
+2e-mail:tadashi.takayanagi@ipmu.jp1 Introduction
+A gapped state of quantum condensed matter is called topological p hase when it supports
+stable gapless boundary modes, such as an edge or a surface stat e. The integer quantum
+Hall effect (QHE), which exists in d= 2 spatial dimensions and under a strong magnetic
+field, isthebest knownexample ofsuch aphase. The recent discove ry ofthequantumspin
+Hall effect (QSHE) in d= 2 and the Z2topological insulator in d= 3 [1–8] shows topo-
+logical phases can exist even in d>2 spatial dimensions, and can be protected by some
+discrete symmetries such as time-reversal symmetry (TRS, T), p article-hole symmetry
+(PHS, C), and chiral (or sublattice) symmetry (SLS, S).
+For non-interacting fermions, an exhaustive classification of topo logical insulators
+(TIs) and superconductors (TSCs) is proposed in Refs. [9,10]: TI s/TSCs are classified in
+terms of spatial dimensions dand the 10 = 2+8 symmetry classes (two “complex” and
+eight “real” classes) (Table 1). The ten symmetry classes are in one -to-one correspon-
+dence to the Riemannian symmetric spaces (without exceptional se ries) and, as pointed
+outin[10], theyareequivalent toK-theoryclassifying spaces [11]. Fo rexample, theIQHE,
+QSHE, and Z2TI are a topologically non-trivial state belonging to class A ( d= 2), AII
+(d= 2), and AII ( d= 3), respectively.
+The complete classification of non-interacting TIs and TSCs opens u p a number of fur-
+ther questions, most interesting among which are interaction effec ts: Do non-interacting
+topological phases continue to exist in the presence of interaction s? Can interactions give
+rise to novel topological phases other than non-interacting TIs/ TSCs? What is a topolog-
+ical field theory underlying TIs/TSCs, which can potentially describe TIs/TSCs beyond
+non-interacting examples?, etc.
+On the other hand, the ten-fold classification of TIs/TSCs reminds us of D-branes,
+which are fundamental objects in string theory, and are also class ified by K-theory [12]
+(Table 2) via the open string tachyon condensation [13]. It is then na tural to speculate
+a possible connection between TIs/TSCs and of D-branes. In this p aper, we propose a
+systematic construction of TIs/TSCs in terms of two D-branes (D p- and Dq-branes), pos-
+sibly with an orientifold plane (O-plane). Besides the appealing mathem atical similarity
+between TIs/TSCs and D-branes, realizing TIs/TSCs in string theo ry has a number of
+merits, since string theory and D-branesare believed to be rich eno ugh to reproduce many
+types of field theories and interactions in a fully consistent and UV co mplete way. Indeed,
+our string theory realizations of TIs/TSCs give rise to massive ferm ion spectra, which are
+in one-to-one correspondence with the ten-fold classification of T Is/TSCs, and come quite
+naturally with gauge interactions. These systems, while interacting , are all topologically
+stable, as protected by the K-theory charge of D-branes. We th us make a first step to-
+wardunderstanding interacting TIs/TSCs [14]. Wearealso separat ely preparing aregular
+paper with more details and expanded results in [15].
+In Dp-Dq-systems, massive fermions arise asan openstring excitation betw een thetwo
+D-branes. The distance between the branes corresponds to the mass of fermions. Open
+strings ending on the same D-branes give rise to a gauge field, which w e callAµ(Dp)
+and˜Aµ(Dq) with gauge group Gand˜G, respectively, and couple to the fermions. These
+1class\d0 1 2 3 4 5 6 7 T C S
+AZ0Z0Z0Z00 0 0
+AIII 0Z0Z0Z0Z0 0 1
+AIZ0 0 0 2 Z0Z2Z2+ 0 0
+BDIZ2Z0 0 0 2 Z0Z2+ + 1
+DZ2Z2Z0 0 0 2 Z00 + 0
+DIII 0Z2Z2Z0 0 0 2 Z−+ 1
+AII2Z0Z2Z2Z0 0 0 −0 0
+CII0 2Z0Z2Z2Z0 0 − −1
+C0 0 2 Z0Z2Z2Z00−0
+CI0 0 0 2 Z0Z2Z2Z+−1
+Table 1: Classification of topological insulators and superconducto rs [9,10];dis the space
+dimension; the left-most column (A, AIII, ..., CI) denotes the ten symmetry classes of
+fermionic Hamiltonians, which are characterized by the presence/a bsence of time-reversal
+(T), particle-hole (C), and chiral (or sublattice) (S) symmetries o f different types denoted
+by±1 in the right most three columns. The entries “ Z”, “Z2”, “2Z”, and “0” represent
+the presence/absence of topological insulators and supercondu ctors, and when they exist,
+types of these states (see Ref. [9] for detailed descriptions).
+two gauge fields play different roles in our construction: The gauge fi eldAµ“measures”
+K-theory charge of the D q-brane, and in that sense it can be interpreted as an “external”
+gauge field. In this picture, the D q-brane charge is identified with the topological (K-
+theory) charge of TIs/TSCs. On the other hand, ˜Aµis an internal degree of freedom on
+the Dq-brane. For example, in the integer/fractional QHE, the externa l gauge field is the
+electromagnetic U(1) gauge field, which measures the Hall conduct ivity, while the internal
+gauge field is the Chern-Simons (CS) gauge field describing the dynam ics of the droplet
+itself.
+The massive fermions can be integrated out, yielding the description of the topological
+phase in terms of the gauge fields. The resulting effective field theor y comes with terms of
+topological nature, such as the CS or the θ-terms. In our string theory setup, they can be
+readofffromtheWess-Zumino(WZ)actionoftheD-branes, bytak ingoneoftheD-branes
+as a background for the other. One can view these gauge-interac ting TIs/TSCs from D p-
+Dq-systems as ananalogue of the projective (parton) constructio n of the (fractional) QHE
+[16]. Our string theory realization of TIs/TSCs sheds light on extend ing the projective
+construction of the QHE to more generic TIs/TSCs; it tells us what t ype of gauge field is
+“natural” to couple with fermions in topological phases, and guaran tees the topological
+stability of the system.
+2D(−1) D0 D1 D2 D3 D4 D5 D6 D7 D8 D9
+type IIB Z0Z0Z0Z0Z0Z
+O9−(type I) Z2Z2Z0 0 0 Z0Z2Z2Z
+O9+0 0 Z0Z2Z2Z0 0 0 Z
+Table2: Dp-branecharges fromK-theory, classified by K( S9−p), KO(S9−p)andKSp( S9−p)
+[12]. AZ2charged Dp-brane with peven orpodd represents a non-BPS D p-brane or a
+bound state of a D pand an anti-D pbrane, respectively [13].
+Gclass\d0 1 2 3 4 5 6 7
+UAU - U - U - U -
+UAIII - U - U - U - U
+OAIO - - - Sp - U O
+OBDIO O - - - Sp - U
+ODU O O - - - Sp -
+ODIII - U O O - - - Sp
+SpAIISp - U O O - - -
+SpCII- Sp - U O O - -
+SpC- - Sp - U O O -
+SpCI- - - Sp - U O O
+Table 3: External G(left-most column) and internal ˜Ggauge groups for each spatial
+dimensiondand symmetry class; U, O, Sp, represents U(1), O(1) = Z2, and Sp(1) =
+SU(2), respectively.
+32 Complex case
+LetusstartwiththemostfamiliarexampleoftheQHE(classAin d= 2). Wefixthevalue
+ofpto bep= 5 by T-duality, and consider a D5-brane in type IIB string theory w hich
+extends in the x0,1,2,3,4,5directions in ten-dimensional space-time. We take the D q-brane
+withq= 5 in the x0,1,2,6,7,8directions (Table 4). By T-duality, this setup is equivalent
+to the D3-D7 system studied in [19–21]. Since the number of Neumann -Dirichlet (ND)
+directions is six, open string excitations between the D5-branes giv e rise to two Majorana
+fermions (Mj) [= one two-component Dirac fermion (Di), ψ] and no bosons. The distance
+between the D-branes in x9direction (∆ x9) is proportional to the mass mof the fermions.
+The low-energy effective theory is schematically summarized by the e ffective Lagrangian
+in the (2+1)-dimensional common direction of the two D5-branes,
+L=¯ψ[γµ(i∂µ−Aµ−˜Aµ)−m]ψ+···. (1)
+Integrating the massive fermions yields the CS termsk
+4π/integraltext
+A∧dAandk
+4π/integraltext˜A∧d˜Awith
+k=±1/2 (parity anomaly). The Hall conductivity is read off from the CS term forAµas
+σxy=k/(2π). Alternatively, the presence of the CS terms can be read off from the WZ
+action of either one of D5-branes, e.g.,
+SWZ
+D5∝/integraldisplay
+D5F∧F∧C2=/integraldisplay
+D5A∧F∧(dC)3 (2)
+for the external gauge field Aµ, whereC2is the RR 2-form from the D q-brane. When
+we change the sign of mby passing the D q-brane through the D p-brane, the value of k
+jumps from ±1/2 to∓1/2. If we instead put NfDq-branes, we have Nfcopies of massive
+Dirac fermions ψiwhich couple with U( Nf) gauge fields Aµand˜Aµ(when all D qare
+coincident).
+This brane construction can be extended to other even space dime nsionsd= 2nby
+considering D5-D qsystems with q= 5,7 (Table 4). This setup gives rise to the fermion
+spectrum consisting of one Dirac fermion per D q-brane, and the CS terms of the form
+∝k/integraltext
+A∧Fn. All of these brane configurations are identified with class A TIs, wh ich are
+characterized by the absence of any discrete symmetries.
+Now let us turn to AIII, which is characterized by the presence of S LS. We argue
+that SLS is equivalent to an invariance of the brane configurations u nder inversion of
+a coordinate in the Dirichlet-Dirichlet (DD) directions of open strings between the D5-
+and Dq-branes. One way to realize this is to assume two DD directions, say, x1and
+x9, and impose x1= 0. Indeed such a configuration is obtained by taking T-dual of
+class A configurations (Table 4). Again, the fermion spectrum cons ists of two Majorana
+fermions (=one Dirac fermion) for all dimensions, and the mass of th e fermion is, again,
+proportional to ∆ x9.
+In our setup in general, the number of Neumann-Neumann (NN) dire ctions #NN is
+equaltothespace-timedimensions d+1ofTIs/TSCs. Ontheotherhand, #DDrepresents
+the number of possible mass deformations: it is one if there is no SLS ( class A), while
+it is two in the presence of SLS. Finally, #ND is determined by #NN and #D D via the
+40 1 2 3 4 5 6 7 8 9 dA
+D5× × × × × ×
+D3× × × × 0Z(2 Mj)
+D5× × × × × × 2Z(2 Mj)
+D7× × × × × × × × 4Z(1 Di)
+0 1 2 3 4 5 6 7 8 9 dAIII
+D4× × × × ×
+D4× × × × × 1Z(2 Mj)
+D6× × × × × × × 3Z(2 Mj)
+Table 4: D p-Dqsystems for class A and AIII where p= 5 andq= 3,5,7 for A, and
+p= 4 andq= 4,6 for AIII. The D-branes extend in the µ-th direction denoted by “ ×” in
+the ten-dimensional space-time ( µ= 0,...,9);d+1 is the number of common directions
+of Dp-and Dq-branes; The last column shows the D q-brane charge, together with fermion
+spectra per D q-brane, where ” NfMj” or “NfDi” represents Nfflavor of Majorana and
+Dirac spinor, respectively.
+relation #ND = 10 −#NN−#DD. Note also that the T-dual in any ND directions does
+not change #NN, #DD and #ND and thus is a redundant operation.
+3 Real case
+To realize eight “real” symmetry classes in string theory, we need to implement TRS and
+PHS. To preserve PHS, we require the internal gauge field ˜Aµis not independent of its
+complex conjugate. This is the same as the orientation (Ω) project ion in string theory.
+To realize TRS, we recall that it can be viewed as a product of PHS and SLS [9]. As
+SLS can be imposed as a parity symmetry in string theory, we can inte rpret TRS as the
+orientifold projection.
+Let us start with class C and D, which are characterized by the pres ence of PHS but
+lack of TRS. We take an Ω projection of the class A setup. Note that there are two
+types of Ω projections, represented by two types of O9-plane, i.e ., O9−(orthogonal) and
+O9+(symplectic). While only O9−leads to supersymmetric type I string theory, here
+we consider both because the T-dual of O9+is equivalent to O p+planes (p≤8) in type
+II string theory. We realize class C and class D TSCs by considering a D 5-brane which
+extends in the x0,1,2,3,4,5directions, in the presence of an O9-plane. As before, we put
+a Dq-brane with q=d+ 3 so that there are d+ 1 common directions (Table 5). For
+class AII and AI, characterized by the presence of TRS but lack of PHS, we take the
+orientifold projection which leads to an O8-plane in type IIA theory [2 2]. By choosing
+(p,q) = (4,d+4), we obtain the brane configuration given in the third table in Table 5.
+Though the D-brane charges with an O p-plane forp≤8 are classified by KR-theory, the
+5same result can be obtained from KO-theory via T-duality for our pu rpose [22]. Finally,
+theremaining four classes, CII, BDI,CI andDIIIcanbeobtainedb y taking DDdirections
+to be two instead of one. SLS is imposed by requiring x9= 0 for all of these classes. These
+are O8- or O9-projection of the class AIII setup (Table 5).
+For these D-brane configurations, we chose a D p-brane to be a standard one with the
+integer K-theory charge so that it is regarded as the background (bulk) material itself.
+Then, we find that the K-theory charge of the D q-branes (shown in the last two columns
+in Table 5) agrees precisely with the corresponding classification of T Is/TSCs (Table 1).
+Moreover, the fermion content of these string theory realization s (denoted in the last two
+columns in Table 5 either by “ NfMj” or “NfDi” withNfan integer) can be compared
+with the Dirac representative of TIs/TSCs constructed in Ref. [9]. Indeed, they agree
+completely. It is also interesting to note that, for the Dirac repres entative of TIs/TSCs,
+themomentum dependence of the projection operator, which is one of key ingred ients in
+the classification of TIs/TSCs [9], looks quite similar to the spatial pro file of the tachyon
+field in string theory in realspace [12].
+Wenow describe thefield theory content of theD p-Dqsystems charges in moredetails.
+First, for the ZTIs/TSCs on the diagonal in Table 1 (“primary series”), the interna l
+gauge group is O(1) = Z2. In particular, for class D in d= 2, our string theory realization
+corresponds to (a proper supersymmetric generalization of) the honeycomb lattice Kitaev
+model in the weak pairing (non-Abelian) phase [17]. Similarly, for class D III ind= 3, it
+corresponds to an interacting bosonic model on the diamond lattice [18].
+ForZ2TIs/TSCs of the first descendant of the primary series, i.e, BDI ( d= 0), D
+(d= 1), DIII ( d= 2), and AII ( d= 3), the internal gauge group is O(1) = Z2(Table 3).
+ForZ2TIs/TSCs of the second descendant of the the primary series, i.e, D (d= 0),
+DIII (d= 1), AII (d= 2), and CII ( d= 3), the internal gauge group is U(1). For D and
+DIII, the U(1) unitary gauge field couples to two real fermions whic h can be combined
+into a single complex field. For AII and CII, the fermion spectrum con sists of 4 Mj. In
+this case, the external gauge field is Sp(1) = SU(2), which couples t o a doublet, ψ↑/↓.
+Each has 2 Mj (=1Di) degrees of freedom and couples to a U(1) inter nal gauge field as
+follows:
+L=¯ψ[γµ(i∂µ−Aµ−˜Aµ)−mM]ψ+···, (3)
+whereψ= (ψ↑,ψ↓)T, andMis a diagonal mass matrix whose eigenvalue is ±1 forψ↑/↓,
+respectively.
+Finally, for TIs/TSCs labeled by 2 Z, i.e., AII (d= 0), CII (d= 1), C (d= 2), and CI
+(d= 3), the gaugegroup is GטG= SU(2)×SU(2), with 4 Mj fermions in bi-fundamental.
+Even though the ten-dimensional string theories in the bulk are sup ersymmetric, our
+brane setups are not in general. When #ND = 4 with Zcharge, they exceptionally
+preserve a quarter of supersymmetries. In general, when #ND = 4 , there exist massive
+bosons in addition to the massive fermions. This happens when d= 4 for A, C, AI, AII,
+and whend= 3 for AIII, CII, CI and DIII. Since #NN >4 for all the other branes
+systems, we only have fermions from open strings between the D p- and Dq-branes and
+there are no tachyons.
+60 1 2 3 4 5 6 7 8 9 dC (O9−)D (O9+)
+D5× × × × × ×
+D3× × × × 00Z2(2 Mj)
+D4× × × × × 10Z2(1 Mj)
+D5× × × × × × 2Z(4 Mj) Z(1 Mj)
+D6× × × × × × × 30 0
+D7× × × × × × × × 4Z2(2 Di) 0
+0 1 2 3 4 5 6 7 8 9 dCI (O9−)DIII (O9+)
+D5× × × × × ×
+D2× × × 00 0
+D3× × × × 10Z2(2 Mj)
+D4× × × × × 20Z2(2 Mj)
+D5× × × × × × 3Z(4 Mj) Z(1 Mj)
+0 1 2 3 4 56 7 8 9 dAII (O8−)AI (O8+)
+D4× × × × ×
+D4× × × × × 0Z(4 Mj) Z(1 Mj)
+D5× × × × × × 10 0
+D6× × × × × × × 2Z2(4 Mj) 0
+D7× × × × × × × × 3Z2(2 Mj) 0
+D8× × × × × × × × × 4Z(1 Di) Z(1 Di)
+0 1 2 3 4 56 7 8 9 dCII (O8−)BDI (O8+)
+D4× × × × ×
+D3× × × × 00Z2(2 Mj)
+D4× × × × × 1Z(4 Mj) Z(1 Mj)
+D5× × × × × × 20 0
+D6× × × × × × × 3Z2(4 Mj) 0
+D7× × × × × × × × 4Z2(2 Di) 0
+Table 5: D p-Dqsystems for eight “real” symmetry classes, where p= 5 for classes C,
+D, CI, DIII, and p= 4 for classes AII, AI, CII, BDI. For classes AII, AI, CII, BDI, t he
+O8-plane extends except x5.
+7Edge 
+state Dq 
+Dp 
+Dp 
+Figure 1: A boundary of a TI/TSC as a brane intersection.
+#DD(O9−,O9+)(O8−,O8+)(O7−,O7+)(O6−,O6+)
+0Chiral
+1(C,D) (AII,AI)
+2(CI,DIII) (CII,BDI) (DIII,CI)
+3d≤2d≤2d≤2d≤2
+4d≤1d≤1d≤1d≤1
+Table 6: D p-Dqsystems in the presence of an O-plane; “ d≥2” means the constraint
+of possible spatial dimensions din the brane systems due to the existence of open string
+tachyons. “Chiral” denotes the existence of chiral fermions and is interpreted as boundary
+(edge) states. The horizontal direction is shifted by a T-duality in t he NN direction.
+We can take the T-duality further in NN directions. However, this lea d to theories
+withdifferent properties thanTIs/TSCs (Table 6). Notethat we ha ve succeeded to realize
+all TIs/TSCs in space dimensions d≤4.
+4 Boundary of TIs/TSCs
+A defining property of TIs/TSCs is the appearance of stable gaples s degrees of freedom,
+when the system is terminated by a ( d−1)-dimensional boundary. In our brane con-
+struction, the sample boundary can be constructed by bending th e Dp-brane toward the
+Dq-brane, tocreateanintersection between these branes(Fig.1) . This leadstoaposition-
+dependent fermion mass, which changes its sign at the intersection . This increases #ND
+by two and the correct number of massless fermions appears at th e intersection.
+85 Conclusions
+The main conclusion of this paper is that there is a one-to-one corre spondence between
+the tachyon-free D p-Dqsystems and the ten classes of topological insulators in d≤4 di-
+mensions. Indeed, we explicitly constructed the corresponding te n classes of D p-Dqbrane
+configurations in superstring theory. Two out of ten are realized in type II string theory
+without orientifolds, while the other eight require orientifolds. The K -theory charges of
+the Dq-branes agree with that of the topological insulators.
+One may wonder if there are other tachyon-free D p-Dqsystems which have not been
+considered in this paper. However, it turns out that their low-ener gy theories just cor-
+respond to multiple copies of the ten classes of topological insulator s (for details refer
+to [15]).
+Since a topological insulator has a mass gap in the bulk, the distance b etween the
+Dpand Dqis taken to be non-vanishing in its corresponding D-brane system. W hen we
+discuss the boundary of a TI/TSC, however, the D p-brane is bent toward the D q-brane
+and thus Dpand Dqare intersecting with each other. Therefore massless fermions ap pear
+at the intersection and they are identified with the boundary modes (or edge modes).
+We can also consider holographic descriptions of these systems by e xtending the con-
+structions in [20,21] in principle, though it is not possible to take the la rge-Nlimit ofZ2
+charged D-branes.
+Acknowledgments We acknowledge “Quantum Criticality and the AdS/CFT corre-
+spondence” miniprogram at KITP, “Quantum Theory and Symmetrie s” conference at
+University of Kentucky. We would like to thank K. Hori, P. Kraus, A. L udwig, J. Moore,
+M. Oshikawa, and S. Sugimoto for useful discussion, and A. Furusa ki for his clear lecture
+at“Development ofQuantumFieldTheoryandStringTheory”(YITP -W-09-04)atKyoto
+University. SR thanks Center for Condensed Matter Theory at Un iversity of California,
+Berkeley for its support. TT is supported in part by JSPS Grant-in- Aid for Scientific
+Research No. 20740132, and by JSPS Grant-in-Aid for Creative Sc ientific Research No.
+19GS0219.
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+[9] A. Schnyder, S. Ryu, A. Furusaki, A. Ludwig, Phys. Rev. B 78, 195125 (2008); AIP
+Conf. Proc. 1134, 10 (2009); S. Ryu, A. Schnyder, A. Furusaki, and A. Ludwig,
+arXiv:0912.2157 .
+[10] A. Kitaev, AIP Conf. Proc. 1134, 22 (2009).
+[11] The importance of K-theory in the physics of Fermi surfaces h as been first pointed
+out in P. Horava, Phys. Rev. Lett. 95, 016405 (2005).
+[12] E. Witten, JHEP 9812, 019 (1998); P. Horava, Adv. Theor. Math. Phys. 2, 1373
+(1999).
+[13] A. Sen, Int. J. Mod. Phys. A 20, 5513 (2005).
+[14] For a different approach to TIs from string theory, see, for e xample, A. Karch, Phys.
+Rev. Lett. 103, 171601 (2009).
+[15] S. Ryu and T. Takayanagi, arXiv:1007.4234 .
+[16] X. -G. Wen, Mod. Phys. Lett. B 5, 39 (1991); Phys. Rev. B 60, 8827 (1999).
+[17] Alexei Kitaev, Ann. of Phys. 321, 2 (2006).
+[18] Shinsei Ryu, Phys. Rev. B 79, 075124 (2009).
+[19] S. J. Rey, Prog. Theor. Phys. 177, 128 (2009).
+[20] J. L. Davis, P. Kraus and A. Shah, JHEP 0811, 020 (2008).
+[21] M. Fujita, W. Li, S. Ryu, and T. Takayanagi, JHEP 0906066, (2009).
+[22] For a KR theory analysis in similar brane systems, see T. Imoto, T . Sakai and
+S. Sugimoto, arXiv:0907.2968 , and references therein.
+10
\ No newline at end of file
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+arXiv:1001.0764v2  [cond-mat.str-el]  13 Jan 2010Optical Integral and Sum Rule Violation
+Saurabh Maiti, Andrey V. Chubukov
+Department of Physics, University of Wisconsin, Madison, W isconsin 53706, USA
+(Dated: November 9, 2018)
+The purpose of this work is to investigate the role of the latt ice in the optical Kubo sum rule in
+the cuprates. We compute conductivities, optical integral sW, and ∆Wbetween superconducting
+and normal states for 2-D systems with lattice dispersion ty pical of the cuprates for four different
+models – a dirty BCS model, a single Einstein boson model, a ma rginal Fermi liquid model, and a
+collective boson model with a feedback from superconductiv ity on a collective boson. The goal of
+the paper is two-fold. First, we analyze the dependence of Won the upper cut-off ( ωc) placed on
+the optical integral because in experiments Wis measured up to frequencies of order bandwidth.
+For a BCS model, the Kubo sum rule is almost fully reproduced a tωcequal to the bandwidth. But
+for other models only 70%-80% of Kubo sum rule is obtained up t o this scale and even less so for
+∆W, implying that the Kubo sum rule has to be applied with cautio n. Second, we analyze the sign
+of ∆W. In all models we studied ∆ Wis positive at small ωc, then crosses zero and approaches a
+negative value at large ωc, i.e. the optical integral in a superconductor is smaller th an in a normal
+state. The point of zero crossing, however, increases with t he interaction strength and in a collective
+boson model becomes comparable to the bandwidth at strong co upling. We argue that this model
+exhibits the behavior consistent with that in the cuprates.
+I. INTRODUCTION
+The analysisofsum rulesfor optical conductivityhas a
+long history. Kubo, in an extensive paper1in 1957, used
+a general formalism of a statistical theory of irreversible
+processes to investigate the behavior of the conductivity
+in electronic systems. For a system of interacting elec-
+trons,hederivedtheexpressionfortheintegralofthereal
+part of a (complex) electric conductivity σ(Ω) and found
+that it is independent on the nature of the interactions
+and reduces to
+/integraldisplay∞
+0Reσ(Ω)dΩ =π
+2ne2
+m(1)
+Herenis the density of the electrons in the system and
+mis the bare mass of the electron. This expression is
+exact provided that the integration extends truly up to
+infinity, and its derivation uses the obvious fact that at
+energies higher than the total bandwidth of a solid, elec-
+trons behave as free particles.
+The independence of the r.h.s. of Eq. (1) on temper-
+ature and the state of a solid (e.g., a normal or a super-
+conducting state – henceforth referred to as NS and SCS
+respectively) implies that, while the functional form of
+σ(Ω) changes with, e.g., temperature, the total spectral
+weight is conserved and only gets redistributed between
+different frequencies as temperature changes. This con-
+servation of the total weight of σ(Ω) is generally called a
+sum rule.
+One particular case, studied in detail for conventional
+superconductors, is the redistribution of the spectral
+weightbetween normaland superconductingstates. This
+is known as Ferrel-Glover-Tinkham (FGT) sum rule:2,3
+/integraldisplay∞
+0+ReσNS(Ω) =/integraldisplay∞
+0+Reσsc(Ω)+πnse2
+2m(2)
+wherensis the superfluid density, and πnse2/(2m) isthe spectral weight under the δ-functional piece of the
+conductivity in the superconducting state.
+In practice, the integration up to an infinite frequency
+is hardly possible, and more relevant issue for practical
+applicationsis whetherasum ruleis satisfied, at leastap-
+proximately,forasituationwhenthereisasingleelectron
+band which crosses the Fermi level and is well separated
+from other bands. Kubo considered this case in the same
+paper of 1957 and derived the expression for the “band”,
+or Kubo sum rule
+/integraldisplay‘∞′
+0Reσ(Ω)dΩ =WK=πe2
+2N/summationdisplay
+/vectork∇2
+/vectorkxε/vectorkn/vectork(3)
+wheren/vectorkis the electronic distribution function and ε/vectorkis
+the band dispersion. Prime in the upper limit ofthe inte-
+gration has the practical implication that the upper limit
+is much larger than the bandwidth of a given band which
+crosses the Fermi level, but smaller than the frequencies
+of interband transitions. Interactions with external ob-
+jects, e.g., phonons or impurities, and interactions be-
+tween fermions are indirectly present in the distribution
+function which is expressed via the full fermionic Green’s
+function as n/vectork=T/summationtext
+mG(/vectork,ωm). Forǫk=k2/2m,
+∇2
+/vectorkxε/vectork= 1/m,WK=πne2/(2m), and Kubo sum rule
+reduces to Eq. (1). In general, however, ε/vectorkis a lattice
+dispersion, and Eqs. (1) and (3) are different. Most im-
+portant, WKin Eq. (3) generally depends on Tand on
+the state of the system because of n/vectork. In this situation,
+the temperature evolutionofthe optical integraldoesnot
+reduce to a simple redistribution of the spectral weight
+– the whole spectral weight inside the conduction band
+changes with T. This issue was first studied in detail by
+Hirsch4who introduced the now-frequently-used nota-
+tion “violation of the conductivity sum rule”.
+In reality, as already pointed out by Hirsch, there is no
+true violation as the change of the total spectral weight2
+in a given band is compensated by an appropriatechange
+of the spectral weight in other bands such that the total
+spectral weight, integrated over all bands, is conserved,
+as in Eq. (1). Still, non-conservation of the spectral
+weight within a given band is an interesting phenomenon
+as the degree of non-conservation is an indicator of rele-
+vant energy scales in the problem. Indeed, when relevant
+energy scales are much smaller than the Fermi energy,
+i.e., changes in the conductivity are confined to a near
+vicinity of a Fermi surface (FS), one can expand εknear
+kFasεk=vF(k−kF)+(k−kF)2/(2mB)+O(k−kF)3
+and obtain ∇2
+/vectorkxε/vectork≈1/mB[this approximation is equiv-
+alent to approximating the density of states (DOS) by a
+constant]. Then WKbecomes πne2/(2mB) which does
+not depend on temperature. The scale of the tempera-
+ture dependence of WKis then an indicator how far in
+energy the changes in conductivity extend when, e.g., a
+system evolvesfrom a normal metal to a superconductor.
+Because relevant energy scales increase with the interac-
+tion strength, the temperature dependence of WKis also
+an indirect indicator of whether a system is in a weak,
+intermediate, or strong coupling regime.
+In a conventional BCS superconductor the only rele-
+vant scales are the superconducting gap ∆ and the impu-
+rity scattering rate Γ. Both are generally much smaller
+than the Fermi energy, so the optical integral should be
+almostT-independent, i.e., the spectral weight lost in a
+superconducting state at low frequencies because of gap
+opening is completely recovered by the zero-frequency δ-
+function. In a clean limit, the weight which goes into
+aδ−function is recovered within frequencies up to 4∆.
+This is the essence of FGT sum rule2,3. In a dirty limit,
+this scale is larger, O(Γ), but still WKisT-independent
+and there was no “violation of sum rule”.
+The issue of sum rule attracted substantial interest in
+the studies of high Tccuprates5–18,21–26in which pairing
+iswithout doubtsastrongcouplingphenomenon. Froma
+theoreticalperspective, the interestin this issue wasorig-
+inally triggered by a similarity between WKand the ki-
+netic energy K= 2/summationtextε/vectorkn/vectork.18–20For amodel with a sim-
+ple tight binding cosine dispersion εk∝(coskx+cosky),
+d2ε/vectork
+dk2x∼−ε/vectorkandWK=−K. For a more complex dis-
+persion there is no exact relation between WKandK,
+but several groups argued17,27,28thatWKcan still be
+regarded as a good monitor for the changes in the kinetic
+energy. Now, in a BCS superconductor, kinetic energy
+increases below Tcbecausenkextends to higher frequen-
+cies (see Fig.2). At strong coupling, Knot necessary
+increases because of opposite trend associated with the
+fermionic self-energy: fermions are more mobile in the
+SCS due to less space for scattering at low energies than
+they are in the NS. Model calculations show that above
+some coupling strength, the kinetic energy decreases be-
+lowTc29. While, as we said, there is no one-to-one cor-
+respondence between KandWK, it is still likely that,
+whenKdecreases, WKincreases.
+Agoodamountofexperimentalefforthasbeenputintoaddressingtheissueoftheopticalsumruleinthe c−axis7
+and in-plane conductivities8–16in overdoped, optimally
+doped, and underdoped cuprates. The experimental re-
+sults demonstrated, above all, outstanding achievements
+of experimental abilities as these groups managed to de-
+tect the value of the optical integral with the accuracy
+of a fraction of a percent. The analysis of the change
+of the optical integral between normal and SCS is even
+more complex because one has to (i) extend NS data to
+T < T cand (ii) measure superfluid density with the same
+accuracy as the optical integral itself.
+Theanalysisoftheopticalintegralshowedthatinover-
+doped cuprates it definitely decreases below Tc, in con-
+sistency with the expectations at weak coupling11. For
+underdoped cuprates, all experimental groups agree that
+a relative change of the optical integral below Tcgets
+much smaller. There is no agreement yet about the sign
+of the change of the optical integral : Molegraaf et al.8
+and Santander-Syro et al.9argued that the optical inte-
+gral increases below Tc, while Boris et al.10argued that
+it decreases.
+Theoretical analysis of these results21,22,25,28,30added
+one more degree of complexity to the issue. It is tempt-
+ing to analyze the temperature dependence of WKand
+relate it to the observed behavior of the optical integral,
+and some earlier works25,28,30followed this route. In the
+experiments, however, optical conductivity is integrated
+onlyup to a certainfrequency ωc, and the quantitywhich
+is actually measured is
+W(ωc) =/integraldisplayωc
+0Reσ(Ω)dΩ =WK+f(ωc)
+f(ωc) =−/integraldisplay′∞′
+ωcReσ(Ω)dΩ (4)
+The Kubo formula, Eq. (3) is obtained assuming that
+the second part is negligible. This is not guaranteed,
+however, as typical ωc∼1−2eVare comparable to the
+bandwidth.
+The differential sum rule ∆ Wis also a sum of two
+terms
+∆W(ωc) = ∆WK+∆f(ωc) (5)
+where ∆ WKis the variation of the r.h.s. of Eq. 3,
+and ∆f(ωc) is the variation of the cutoff term. Because
+conductivity changes with Tat all frequencies, ∆ f(ωc)
+also varies with temperature. It then becomes the issue
+whethertheexperimentallyobserved∆ W(ωc) ispredom-
+inantly due to “intrinsic” ∆ WK, or to ∆ f(ωc). [A third
+possibility is non-applicability of the Kubo formula be-
+cause of the close proximity of other bands, but we will
+not dwell on this.]
+For the NS, previous works21,22on particular models
+for the cuprates indicated that the origin of the temper-
+ature dependence of W(ωc) is likely the Tdependence
+of the cutoff term f(ωc). Specifically, Norman et. al.22
+approximated a fermionic DOS by a constant (in which3
+case, as we said, WKdoes not depend on temperature)
+and analyzed the Tdependence of W(ωc) due to the T
+dependenceofthecut-offterm. Theyfoundagoodagree-
+ment with the experiments. This still does not solve the
+problem fully as amount of the Tdependence of WKin
+the samemodelbut with alatticedispersionhasnotbeen
+analyzed. For a superconductor, which of the two terms
+contributes more, remains an open issue. At small fre-
+quencies, ∆ W(ωc) between a SCS and a NS is positive
+simply because σ(Ω) in a SCS has a δ−functional term.
+In the models with a constant DOS, for which ∆ WK= 0,
+previous calculations21show that ∆ W(ωc) changes sign
+atsomeωc, becomesnegativeatlarger ωcandapproaches
+zero from a negative side. The frequency when ∆ W(ωc)
+changessignis oforder∆ at weakcoupling, but increases
+as the coupling increases, and at large coupling becomes
+comparabletoabandwidth( ∼1eV). Atsuchfrequencies
+the approximation of a DOS by a constant is question-
+able at best, and the behavior of ∆ W(ωc) should gen-
+erally be influenced by a nonzero ∆ WK. In particular,
+the optical integral can either remain positive for all fre-
+quencies below interband transitions (for large enough
+positive ∆ WK), or change sign and remain negative (for
+negative ∆ WK). The first behavior would be consistent
+with Refs. 8,9, while the second would be consistent with
+Ref. 10. ∆ Wcan even show more exotic behavior with
+more than one sign change (for a small positive ∆ WK).
+We show various cases schematically in Fig.1.
+0
+ωc∆ W
+0
+ωc∆ W
+0
+ωc∆ W0
+ωc∆ W(a)(a)(a) (b)
+(c) (d)
+FIG. 1: Schematic behavior of ∆ Wvsωc, Eq. (4). The
+limiting value of ∆ Watωc=∞is ∆WKgiven by Eq. (3)
+Depending on the value of ∆ WK, there can be either one sign
+change of ∆ W(panels a and c), or no sign changes (panel b),
+or two sign changes (panel d).
+In our work, we perform direct numerical calculations
+of optical integrals at T= 0 for a lattice dispersion ex-
+tracted from ARPES of the cuprates. The goal of our
+work is two-fold. First, we perform calculations of the
+opticalintegralin the NS andanalyzehowrapidly W(ωc)
+approaches WK, in other words we check how much of
+the Kubo sum is recovered up to the scale of the band-
+width. Second, we analyze the difference between opticalintegral in the SCS at T= 0 and in the NS extrapolated
+toT= 0 and compare the cut off effect ∆ f(ωc) to ∆WK
+term. We also analyze the sign of ∆ W(ωc) at large fre-
+quencies and discuss under what conditions theoretical
+W(∞) increases in the SCS.
+We perform calculations for four models. First is a
+conventional BCS model with impurities (BCSI model).
+Second is an Einstein boson (EB) model of fermions in-
+teracting with a single Einstein boson whose propaga-
+tor does not change between NS and SCS. These two
+cases will illustrate a conventional idea of the spectral
+weight in SCS being less than in NS. Then we con-
+sidertwomoresophisticatedmodels: aphenomenological
+“marginal Fermi liquid with impurities” (MFLI) model
+of Norman and P´ epin30, and a microscopic collective bo-
+son (CB) model31in which in the NS fermions interact
+with a gapless continuum of bosonic excitations, but in a
+d−wave SCS a gapless continuum splits into a resonance
+and a gaped continuum. This model describes, in par-
+ticular, interaction of fermions with their own collective
+spin fluctuations32via
+Σ(k,Ω) = 3g2/integraldisplaydω
+2πd2q
+(2π)2χ(q,ω)G(k+q,ω+Ω) (6)
+wheregis the spin-fermion coupling, and χ(q,ω) is the
+spin susceptibility whose dynamics changes between NS
+and SCS.
+From our analysis we found that the introduction of
+a finite fermionic bandwidth by means of a lattice has
+generally a notable effect on both Wand ∆W. We
+found that for all models except for BCSI model, only
+70%−80% of the optical spectral weight is obtained by
+integrating up to the bandwidth. In these three models,
+there also exists a wide range of ωcin which the behavior
+of ∆W(ωc) is due to variation of ∆ f(ωc) which is domi-
+nant comparable to the ∆ WKterm. This dominance of
+the cut off term is consistent with the analysis in Refs.
+21,22,33.
+We also found that for all models except for the origi-
+nal version of the MFLI model the optical weight at the
+highest frequencies is greater in the NS than in the SCS
+(i.e., ∆W <0). This observation is consistent with the
+findings of Abanov and Chubukov32, Benfatto et. al.28,
+and Karakozov and Maksimov34. In the original ver-
+sion of the MFLI model30the spectral weight in SCS
+was found to be greater than in the NS (∆ W >0). We
+show that the behavior of ∆ W(ωc) in this model cru-
+cially depends on how the fermionic self-energy modeled
+to fit ARPES data in a NS is modified when a system
+becomes a superconductor and can be of either sign. We
+also found, however, that ωcat which ∆ Wbecomes neg-
+ative rapidly increases with the coupling strength and at
+strong coupling becomes comparable to the bandwidth.
+In the CB model, which, we believe, is most appropriate
+for the application to the cuprates, ∆ WK= ∆W(∞) is
+quite small, and at strong coupling a negative ∆ W(ωc)
+up toωc∼1eVis nearly compensated by the optical
+integral between ωcand “infinity”, which, in practice, is4
+an energy of interband transitions, which is roughly 2 eV.
+This would be consistent with Refs. 8,9.
+We begin with formulating our calculational basis in
+the next section. Then we take up the four cases and
+considerin eachcasethe extentto whichthe Kubosum is
+satisfied up to the order of bandwidth and the functional
+form and the sign of ∆ W(ωc). The last section presents
+our conclusions.
+II. OPTICAL INTEGRAL IN NORMAL AND
+SUPERCONDUCTING STATES
+The generic formalism of the computation of the op-
+tical conductivity and the optical integral has been dis-
+cussed several times in the literature21–23,26,29and wejust list the formulas that we used in our computations.
+The conductivity σ(Ω) and the optical integral W(ωc)
+are given by (see for example Ref. 35).
+σ′(Ω) =Im/bracketleftbigg
+−Π(Ω)
+Ω+iδ/bracketrightbigg
+=−Π′′(Ω)
+Ω+πδ(Ω)Π′(Ω)
+(7a)
+W(ωc) =/integraldisplayωc
+0σ′(Ω)dΩ =−/integraldisplayωc
+0+Π′′(Ω)
+ΩdΩ +π
+2Π′(0)
+(7b)
+where ‘X′’ and ‘X′′’ stand for real and imaginary parts
+ofX. We will restrict with T= 0. The polarization
+operator Π(Ω) is (see Ref. 36)
+Π(iΩ) =T/summationdisplay
+ω/summationdisplay
+/vectork(∇/vectorkε/vectork)2/parenleftBig
+G(iω,/vectork)G(iω+iΩ,/vectork) +F(iω,/vectork)F(iω+iΩ,/vectork)/parenrightBig
+(8a)
+Π′′(Ω) =−1
+π/summationdisplay
+/vectork(∇/vectorkε/vectork)2/integraldisplay0
+−Ωdω/parenleftBig
+G′′(ω,/vectork)G′′(ω+Ω,/vectork) +F′′(ω,/vectork)F′′(ω+Ω,/vectork)/parenrightBig
+(8b)
+Π′(Ω) =1
+π2/summationdisplay
+/vectork(∇/vectorkε/vectork)2/integraldisplay′/integraldisplay′
+dxdy/parenleftBig
+G′′(x,/vectork)G′′(y,/vectork) +F′′(x,/vectork)F′′(y,/vectork)/parenrightBignF(y)−nF(x)
+y−x(8c)
+where/integraltext′denotes the principal value of the integral,/summationtext
+/vectorkis understood to be1
+N/summationtext
+/vectork,(Nis the number of lat-
+tice sites), nF(x) is the Fermi function which is a step
+function at zero temperature, GandFare the normal
+and anomalous Greens functions. given by37
+For a NS, G(ω,/vectork) =1
+ω−Σ(k,ω)−ε/vectork+iδ(9a)
+For a SCS, G(ω,/vectork) =Zk,ωω+ε/vectork
+Z2
+k,ω(ω2−∆2
+k,ω)−ε2
+/vectork+iδsgn(ω)
+(9b)
+F(ω,/vectork) =Zk,ω∆k,ω
+Z2
+k,ω(ω2−∆2
+k,ω)−ε2
+/vectork+iδsgn(ω)
+(9c)
+whereZk,ω= 1−Σ(k,ω)
+ω, and ∆ k,ω, is the SC gap. Fol-
+lowing earlier works31,33, we assume that the fermionic
+self-energy Σ( k,ω) predominantly depends on frequency
+and approximate Σ( k,ω)≈Σ(ω) and also neglect the
+frequency dependence of the gap, i.e., approximate ∆ k,ω
+by ad−wave ∆ k. The lattice dispersion ε/vectorkis taken from
+Ref. 38. To calculate WK, one has to evaluate the Kubo
+term in Eq.3 wherein the distribution function n/vectork, is cal-
+culated from
+n(ε/vectork) =−2/integraldisplay0
+−∞dω
+2πG′′(ω,/vectork) (10)The 2 is due to the trace over spin indices. We show the
+distribution functions in the NS and SCS under different
+circumstances in Fig 2.
+The/vectork-summation is done overfirst Brillouin zone for a
+2-Dlattice witha62x62grid. Thefrequencyintegralsare
+done analyticallywhereverpossible, otherwise performed
+using Simpson’s rule for all regular parts. Contributions
+from the poles are computed separately using Cauchy’s
+theorem. For comparison, in all four cases we also calcu-
+lated FGT sum rule by replacing/integraltext
+d2k=dΩkdǫkνǫk,Ωk
+and keeping νconstant. We remind that the FGT is
+the result when one assumes that the integral in W(ωc)
+predominantly comes from a narrow region around the
+Fermi surface.
+Wewill firstuseEq3andcompute WKinNS andSCS.
+This will tell us about the magnitude of ∆ W(ωc=∞).
+We next compute the conductivity σ(ω) using the equa-
+tions listed above, find W(ωc) and ∆W(ωc) and compare
+∆f(ωc) and ∆WK.
+For simplicity and also for comparisons with earlier
+studies, for BCSI, EB, and MFLI models we assumed
+that the gap is just a constant along the FS. For CB
+model, we used a d−wave gap and included into consid-
+eration the fact that, if a CB is a spin fluctuation, its
+propagator develops a resonance when the pairing gap is
+d−wave.5
+FIG. 2: Distribution functions in four cases (a) BCSI model,
+where one can see that for ε >0, SC>NS implying KE in-
+creases in the SCS. (b) The original MFLI model of Ref. 30,
+where for ε >0, SC<NS, implying KE decreases in the SCS.
+(c) Our version of MFLI model (see text) and (d) the CB
+model. In both cases, SC >NS, implying KE increases in the
+SCS. Observe that in the impurity-free CB model there is no
+jump in n(ǫ) indicating lack of fermionic coherence. This is
+consistent with ARPES39
+A. The BCS case
+In BCS theory the quantity Z(ω) is given by
+ZBCSI(ω) = 1+Γ/radicalbig
+∆2−(ω+iδ)2(11)
+and
+ΣBCSI(ω) =ω(Z(ω)−1) =iΓω/radicalbig
+(ω+iδ)2−∆2(12)
+This is consistent with havingin the NS, Σ = iΓ in accor-
+dance with Eq 6. In the SCS, Σ( ω) is purely imaginary
+forω >∆ and purely real for ω <∆. The self-energy
+has a square-root singularity at ω= ∆.
+It is worth noting that Eq.12 is derived from the in-
+tegration over infinite band. If one uses Eq.6 for finite
+band, Eq.12acquiresanadditionalfrequencydependence
+at large frequencies of the order of bandwidth (the low
+frequency structure still remains the same as in Eq.12).
+In principle, in a fully self-consistent analysis, one should
+indeed evaluate the self-energy using a finite bandwidth.
+In practice, however, the self-energy at frequencies of or-
+der bandwidth is generally much smaller than ωand con-
+tribute very little to optical conductivity which predom-
+inantly comes from frequencies where the self-energy is
+comparable or even larger than ω. Keeping this in mind,
+below we will continue with the form of self-energy de-
+rived form infinite band. We use the same argument for
+all four models for the self-energy.
+For completeness, we first present some well known
+results about the conductivity and optical integral for aconstant DOS and then extend the discussion to the case
+where the same calculations are done in the presence of
+a particular lattice dispersion.
+0.10.20.30.4−4−20 
+ω in eVWSC−WNS∆ W (BCSI without lattice)
+Γ =70 meV
+Γ =50 meV
+Γ =3.5 meV
+FIG.3: TheBCSIcase withadispersionlinearized aroundthe
+Fermi surface. Evolution of the difference of optical integr als
+in the SCS and the NS with the upper cut-off ωcObserve
+that the zero crossing point increases with impurityscatte ring
+rate Γ and also the ‘dip’ spreads out with increasing Γ. ∆ =
+30meV
+For a constant DOS, ∆ W(ωc) =WSC(ωc)−WNS(ωc)
+is zero at ωc=∞and Kubo sum rule reduces to FGT
+sum rule. In Fig. 3 we plot for this case ∆ W(ωc) as a
+function of the cutoff ωcfor different Γ′s. The plot shows
+the two well known features: zero-crossing point is below
+2∆ in the clean limit Γ <<∆ and is roughly 2Γ in the
+dirtylimit21,40Themagnitudeofthe‘dip’decreasesquite
+rapidly with increasing Γ. Still, there is always a point
+of zero crossing and ∆ W(ωc) at large ωcapproaches zero
+from below.
+We now perform the same calculations in the presence
+of lattice dispersion. The results are summarized in Figs
+4,5, and 6.
+Fig 4 shows conductivities σ(ω) in the NS and the SCS
+and Kubo sums WKplotted against impurity scattering
+Γ. We see that the optical integral in the NS is always
+greater than in the SCS. The negative sign of ∆ WKis
+simply the consequence of the fact that nkis largerin the
+NS forǫk<0 and smaller for ǫk<0, and∇2ε/vectorkclosely
+follows−ε/vectorkfor our choice of dispersion38), Hence nkis
+larger in the NS for ∇2ε/vectork>0 and smaller for ∇2ε/vectork<
+0 and the Kubo sum rule, which is the integral of the
+product of nkand∇2ε/vectork(Eq. 3), is larger in the normal
+state.
+We also see from Fig. 4 that ∆ WKdecreases with Γ
+reflectingthefact that with toomuchimpurityscattering
+there is little difference in nkbetween NS and SCS.
+Fig 5 shows the optical sum in NS and SCS in clean
+and dirty limits (the parameters are stated in the fig-
+ure). This plot shows that the Kubo sums are almost
+completely recoveredby integrating up to the bandwidth
+of 1eV: the recoveryis 95% in the clean limit and ∼90%
+in the dirty limit. In Fig 6 we plot ∆ W(ωc) as a function
+ofωcin clean and dirty limits. ∆ W(∞) is now non-zero,
+in agreement with Fig. 4 and we also see that there is6
+0 0.5 100.51
+ω in eVσ(ω)Conductivities (BCSI)
+NS
+SC
+2∆
+0 50 100160180200
+Γ in meVWK in meVBCSI
+SC
+NS
+FIG. 4: Top - a conductivity plot for the BCSI case in the
+presence of a lattice. The parameters are ∆ = 30 meV, Γ =
+3.5meV. Bottom – the behavior of Kubosums. Note that (a)
+the spectral weight in the NS is always greater in the SCS, (b)
+the spectral weight decreases with Γ, and (c) the difference
+between NS and SCS decreases as Γ increases.
+little variation of ∆ W(ωc) at above 0 .1−0.3eVwhat
+implies that for larger ωc, ∆W(ωc)≈∆WK>>∆f(ωc).
+To make this more quantitative, we compare in Fig. 6
+∆W(ωc) obtained for a constant DOS, when ∆ W(ωc) =
+∆f(ωc), and for the actual lattice dispersion, when
+∆W(ωc) = ∆WK+ ∆f(ωc). In the clean limit there
+is obviously little cutoff dependence beyond 0 .1eV, i.e.,
+∆f(ωc) is truly small, and the difference between the
+two cases is just ∆ WK. In the dirty limit, the situation
+is similar, but there is obviously more variation with ωc,
+and ∆f(ωc) becomes truly small only above 0 .3eV. Note
+also that the position of the dip in ∆ W(ωc) in the clean
+limit is at a larger ωcin the presence of the lattice than
+in a continuum.
+B. The Einstein boson model
+We next consider the case of electrons interacting with
+a single boson mode which by itself is not affected by su-
+perconductivity. The primary candidate for such mode is
+an optical phonon. The imaginary part of the NS self en-
+ergy has been discussed numerous times in the literature.
+We make one simplifying assumption – approximate the
+DOS by a constant in calculating fermionic self-energy.
+We will, however, keep the full lattice dispersion in the
+calculationsof the optical integral. The advantageofthis0 0.5 10  0.51  
+ωc in eVW(ωc)/W(∞)Normal State Optical Sum (BCSI)
+Dirty Limit
+Clean Limit
+0 0.5 10  0.51  
+ωc in eVW(ωc)/W(∞)Superconducting State Optical Sum (BCSI)
+Dirty Limit
+Clean Limit
+FIG. 5: The evolution of optical integral in NS(top) and
+SCS(bottom) for BCSI case. Plots are made for clean limit
+(solid lines, Γ = 3 .5meV) and dirty limit (dashed lines,
+Γ = 150 meV) for ∆ = 30 meV. Observe that (a) W(0) = 0
+in the NS, but has a non-zero value in the SCS because of the
+δ-function (this value decreases in the dirty limit), and (b)
+the flat region in the SCS is due to the fact that σ′(ω) = 0 for
+Ω<2∆. Also note that ∼90−95% of the spectral weight is
+recovered up to 1 eV
+approximation is that the self-energy can be computed
+analytically. The full self-energy obtained with the lat-
+tice dispersion is more involved and can only be obtained
+numerically, but its structure is quite similar to the one
+obtained with a constant DOS.
+The self-energy for a constant DOS is given by
+Σ(iω) =−i
+2πλn/integraldisplay
+dǫkd(iΩ)χ(iΩ)G(ǫk,iω+iΩ) (13)
+where
+χ(iΩ) =ω2
+0
+ω2
+0−(iΩ)2(14)
+andλnis a dimensionless electron-boson coupling. Inte-
+grating and transforming to real frequencies, we obtain
+Σ′′(ω) =−π
+2λnωoΘ(|ω|−ωo)
+Σ′(ω) =−1
+2λnωolog/vextendsingle/vextendsingle/vextendsingle/vextendsingleω+ωo
+ω−ωo/vextendsingle/vextendsingle/vextendsingle/vextendsingle(15)
+In the SCS, we obtain for ω <0
+Σ′′(ω) =−π
+2λnωoRe/parenleftBigg
+ω+ωo/radicalbig
+(ω+ωo)2−∆2/parenrightBigg7
+0 0.1 0.2 0.30 2040
+ωc in eVWSC(ωc) − WNS(ωc)∆ W (BCSI−clean limit)
+ with lattice
+without lattice
+0.1 0.3 0.50 10
+ωc in eVWSC(ωc) − WNS(ωc)∆ W (BCSI−dirty limit)
+0.5 1−10
+ωc in eV∆ WLarger ωc
+FIG. 6: Evolution of ∆ Win the presence of a lattice (solid
+line) compared with the case of no lattice(a constant DOS,
+dashed line) for clean and dirty limits. ∆ = 30 meV, Γ =
+3.5meV(clean limit), Γ = 150 meV(dirty limit)
+Σ′(ω) =−1
+2λnωoRe/integraldisplay
+dω′1
+ω2o−ω′2−iδω+ω′
+/radicalbig
+(ω+ω′)2−∆2
+(16)
+Observe that Σ′′(ω) is no-zero only for ω <−ωo−∆.
+Also, although it does not straightforwardly follow from
+Eq. 16, but real and imaginary parts of the self-energy
+do satisfy Σ′(ω) =−Σ′(−ω) and Σ′′(ω) = Σ′′(−ω).
+Fig7 shows conductivities σ(ω) and Kubo sums WK
+as a function of the dimensionless coupling λ. We see
+that, like in the previous case, the Kubo sum in the NS
+is larger than that in the SCS. The difference ∆ WKis
+between 5 and 8 meV.
+Fig 8 shows the evolution of the optical integrals. Here
+we see the difference with the BCSI model – only about
+75% of the optical integral is recovered, both in the NS
+and SCS, when we integrate up to the bandwidth of 1 eV.
+The rest comes from higher frequencies.
+In Fig 9 we plot ∆ W(ωc) as a function of ωc. We see
+the same behavior as in the BCSI model in a clean limit
+– ∆W(ωc) is positive at small frequencies, crosses zero
+at some ωc, passes through a deep minimum at a larger
+frequency, and eventuallysaturatesat a negativevalue at
+the largest wc. However, in distinction to BCSI model,
+∆W(ωc) keeps varying with ωcup a much larger scale
+and saturates only at around 0 .8eV. In between the dip
+at0.1eVand0.8eV, thebehaviorofthe opticalintegralis
+predominantly determined by the variation of the cut-off
+term ∆f(ωc) as evidenced by a close similarity between
+the behavior of the actual ∆ Wand ∆Win the absence  0.5 1  0   0.1 0.2 
+Ω in eVσ(Ω)σ in NS and SCS(EB model)
+NS
+SC
+2∆+ωoωo
+00.20.40.6190200
+λ (the coupling constant)WK (meV)Kubo Sum (EB model)
+SC
+NS
+FIG. 7: Top- conductivities in the NS and the SCS for the EB
+model. The conductivity in the NS vanishes below ω0because
+of no phase space for scattering. Bottom - Kubo sums as a
+function of coupling. Observe that WKin the SCS is below
+that in the NS. We set ωo= 40meV, ∆ = 30 meV,λ=.5
+0 0.5 10.20.61  
+ωc in eVW(ωc)/W(∞)Normal State (EB model)
+0 0.5 10.20.61  
+ωc in eVW(ωc)/W(∞)Superconducting State (EB model)
+FIG. 8: Evolution of the optical integrals in the EB model.
+Note that W(0) has a non zero value at T= 0 in the NS
+because the self-energy at small frequencies is purely real and
+linear in ω, hence the polarization bubble Π(0) /negationslash= 0, as in an
+ideal Fermi gas. Parameters are the same as in fig. 78
+0.20.40.60.8−400
+ωc in eVWSC(ωc) − WNS(ωc)∆ W (EB model)
+with lattice
+without lattice∆ WK
+FIG. 9: ∆ Wvs the cut-off for the EB model. It remains neg-
+ative for larger cut-offs. Parameters are the same as before.
+The dot indicates the value of ∆ W(∞) = ∆WK
+of the lattice (the dashed line in Fig. 9).
+C. Marginal Fermi liquid model
+For their analysis of the optical integral, Norman and
+P´ epin30introducedaphenomenologicalmodelfortheself
+energy which fits normal state scattering rate measure-
+ments by ARPES41. It constructs the NS Σ′′(ω) out
+of two contributions - impurity scattering and electron-
+electronscatteringwhichtheyapproximatedphenomeno-
+logicallybythemarginalFermiliquid formof αωatsmall
+frequencies6(MFLI model). The total Σ′′is
+Σ′′(ω) = Γ + α|ω|f/parenleftbiggω
+ωsat/parenrightbigg
+(17)
+whereωsatis about ∼1
+2of the bandwidth, and f(x)≈1
+forx <1 and decreases for x >1. In Ref 30 f(x) was
+assumed to scale as 1 /xat largexsuch that Σ′′is flat at
+largeω. The real part of Σ( ω) is obtained from Kramers-
+Kr¨ onig relations. For the superconducting state, they
+obtainedΣ′′bycuttingofftheNS expressiononthelower
+end at some frequency ω1(the analog of ω0+∆ that we
+had for EB model):
+Σ′′(ω) = (Γ + α|ω|)Θ(|ω|−ω1) (18)
+where Θ( x) is the step function. In reality, Σ′′which fits
+ARPESintheNShassomeangulardependencealongthe
+Fermi surface42, but this was ignored for simplicity. This
+model had gained a lot of attention as it predicted the
+optical sum in the SCS to be larger than in the NS, i.e.,
+∆W >0 at large frequencies. This would be consistent
+with the experimental findings in Refs. 8,9 if, indeed, one
+identifies ∆ Wmeasured up to 1eV with ∆ WK.
+We will show below that the sign of ∆ Win the MFLI
+model actually depends on how the normal state results
+areextendedtothesuperconductingstateand, moreover,
+will argue that ∆ WKis actually negative if the extension
+is done such that at α= 0 the results are consistent withBCSI model. However, before that, we show in Figs 10-
+12 the conductivities and the optical integrals for the
+original MFLI model.
+0.2 0.6 100.10.2
+ω in eVσ(ω)Conductivities (Original MFLI)
+NS
+SC
+∆+ω1
+0 50 100120130140
+Γ (meV)Original MFLIWK (meV)SC
+NS
+α =0.75
+FIG. 10: Top –the conductivities in the NS and SCS in the
+original MFLImodel ofRef.30. Weset Γ = 70 meV,α= 0.75,
+∆ = 32 meV,ω1= 71meV. Note that σ′(ω) in the SCS
+begins at Ω = ∆+ ω1. Bottom – the behavior of WKwith Γ.
+In Fig 10 we plot the conductivities in the NS and the
+SCS and Kubo sums WKvs Γ atα= 0.75 showing that
+thespectralweightinthe SCSisindeedlargerthan inthe
+NS. In Fig 11 we show the behavior of the optical sums
+W(ωc) in NS and SCS. The observation here is that only
+∼75−80%oftheKubosumisrecovereduptothescaleof
+the bandwidth implying that there is indeed a significant
+spectral weight well beyond the bandwidth. And in Fig
+12 we show the behavior of ∆ W(wc). We see that it does
+not change sign and remain positive at all ωc, very much
+unlike the BCS case. Comparing the behavior of W(wc)
+with and without a lattice (solid and dashed lines in Fig.
+12) weseethatthe ‘finite bandwidth effect’ just shifts the
+curve in the positive direction. We also see that the solid
+line flattens above roughly half of the bandwidth, i.e., at
+these frequencies ∆ W(ωc)≈∆WK. Still, we found that
+∆Wcontinues going down even above the bandwidth
+and truly saturates only at about 2 eV(not shown in the
+figure) supporting the idea that there is ‘more’ left to
+recover from higher frequencies.
+The rationale for ∆ WK>0 in the original MFLI
+model has been provided in Ref. 30. They argued that
+this is closely linked to the absence of quasiparticle peaks
+in the NS and their restoration in the SCS state because
+the phase space for quasiparticle scattering at low ener-
+giesissmallerinasuperconductorthaninanormalstate.9
+0 0.5 100.20.40.60.81
+ωc in eVW(ωc)/W(∞) NS (Original MFLI)
+0 0.5 10.20.61  
+ωc in eVW(ωc)/W(∞)SCS (Original MFLI)
+FIG. 11: The evolution of the optical integral in the NS (top)
+and the SCS (bottom) in the original MFLI model. Parame-
+ters are the same as above. Note that only ∼75−80% of the
+spectral weight is recovered up to 1 eV.
+0.20.40.60.810 1020
+ωc in eVWSC(ωc) − WNS(ωc)NS and SCS ∆W (Original MFLI)
+with lattice
+without lattice
+∆ WK
+FIG. 12: Evolution of the difference of the optical integrals in
+theSCSandtheNSwith theuppercut-off ωc. Parameters are
+the same as before. Observe that the optical sum in the SCS
+is larger than in the NS and that ∆ Whas not yet reached
+∆WKup to the bandwidth. The dashed line is the FGT
+result.
+This clearly affects nkbecause it is expressed via the full
+Green’s function and competes with the conventional ef-
+fect of the gap opening. The distribution function from
+this model, which we show in Fig.2b brings this point
+out by showing that in a MFLI model, at ǫ <0,nkin a
+superconductor is larger than nkin the normal state, in
+clear difference with the BCSI case.
+We analyzed the original MFLI model for various pa-
+rameters and found that the behavior presented in Fig.
+12, where ∆ W(ωc)>0 for all frequencies, is typical but0 20 40175185195
+Γ in meVWK (meV)Original MFLI in BCS limit
+SC
+NS
+α =0.05
+FIG. 13: Behavior of WKwith Γ for the original MFLI model
+at very small α= 0.05. We set ω1= ∆ = 32 meV. Observe
+the inconsistency with WKin the BCSI model in Fig 4.
+0.20.40.60.8−0.400.4 
+ωc in eVWSC(ωc) − WNS(ωc)Original MFLI−two sign changes
+FIG. 14: The special case of α= 1.5,Γ = 5meV, other pa-
+rameters the same as in Fig. 10. These parameters are chosen
+to illustrate that two sign changes (indicated by arrows in t he
+figure) are also possible within the original MFLI model.
+not not a generic one. There exists a range of parame-
+tersαand Γ where ∆ WKis still positive, but ∆ W(ωc)
+changes the sign twice and is negative at intermediate
+frequencies. We show an example of such behavior in
+Fig14. Still, for most of the parameters, the behavior of
+∆W(ωc) is the same as in Fig. 12.
+Onmorecarefullookingwefoundtheproblemwiththe
+original MFLI model. We recall that in this model the
+self-energy in the SCS state was obtained by just cutting
+the NS self energy at ω1(see Eq.18). We argue that
+this phenomenological formalism is not fully consistent,
+at least for small α. Indeed, for α= 0, the MFLI model
+reduces to BCSI model for which the behavior of the self-
+energyis givenbyEq. (12). This self-energyevolveswith
+ωand Σ′′has a square-root singularity at ω= ∆ +ωo
+(withωo= 0). MeanwhileΣ′′intheoriginalMFLImodel
+in Eq. (18) simply jumps to zero at ω=ω1= ∆, and
+this happens for all values of αincluding α= 0 where the
+MFLI and BCSI model should merge. This inconsistency
+is reflected in Fig 13, where we plot the near-BCS limit
+of MFLI model by taking a very small α= 0.05. We
+see that the optical integral WKin the SCS still remains
+larger than in the NS over a wide range of Γ, in clear
+difference with the exactly known behavior in the BCSI10
+0 0.5 10  0.40.8Conductivities (Corrected MFLI)σ(ω)
+ω in eVNS
+SC
+2 ∆
+0 50 100100120WK (meV)
+Γ in meVCorrected MFLI
+SC
+NS
+FIG. 15: Top – σ(ω) in the NS and the SCS in the ‘corrected’
+MFLI model with the feedback from SC on the quasiparticle
+damping: iΓ term transforms intoΓ√
+−ω2+∆2. In the SCS σ
+now begins at Ω = 2∆. The parameters are same as in Fig.
+10. Bottom – the behavior of Kubo sum with Γ. Observe
+thatW(ωc) in the NS is larger than in the SCS.
+0.20.40.60.8−100  10 
+ωc in eVWSC(ωc) − WNS(ωc)Corrected MFLI
+without lattice
+with lattice
+∆ WK
+FIG. 16: Evolution of the difference of the optical integrals
+between the SCS and the NS with the upper cut-off ωcfor
+the “corrected” MFLI model. Now ∆ W(ωc) is negative above
+some frequency. Parameters are same as in the Fig 15.
+model, where WKis larger in the NS for all Γ (see Fig.
+4). In other words, the original MFLI model does not
+have the BCSI theory as its limiting case.
+We modified the MFLI model is a minimal way by
+changing the damping term in a SCS toΓ√
+−ω2+∆2to be
+consistent with BCSI model. We still use Eq. (18) for
+the MFL term simply because this term was introduced
+in the NS on phenomenological grounds and there is no
+way to guess how it gets modified in the SCS state with-out first deriving the normal state self-energy microscop-
+ically (this is what we will do in the next section). The
+results of the calculations for the modified MFLI model
+are presented in Figs. 15 and 16. We clearly see that the
+behavior is now different and ∆ WK<0 for all Γ. This
+is the same behavior as we previously found in BCSI
+and EB models. So we argue that the ‘unconventional’
+behavior exhibited by the original MFLI model is most
+likely the manifestation of a particular modeling incon-
+sistency. Still, Ref. 30 made a valid point that the fact
+that quasiparticles behave more close to free fermions in
+a SCS than in a NS, and this effect tends to reverse the
+signs of ∆ WKand of the kinetic energy43. It just hap-
+pens that in a modified MFLI model the optical integral
+is still larger in the NS.
+D. The collective boson model
+We now turn to a more microscopic model- the CB
+model. The model describes fermions interacting by ex-
+changing soft, overdamped collective bosons in a partic-
+ular, near-critical, spin or charge channel31,44,45. This
+interaction is responsible for the normal state self-energy
+and also gives rise to a superconductivity. A peculiar
+feature of the CB model is that the propagator of a col-
+lective boson changes below Tcbecause this boson is not
+an independent degree of freedom (as in EB model) but
+is made out of low-energy fermions which are affected by
+superconductivity32.
+The most relevant point for our discussion is that this
+model contains the physics which we identified above as
+a source of a potential sign change of ∆ WK. Namely,
+at strong coupling the fermionic self-energy in the NS
+is large because there exists strong scattering between
+low-energy fermions mediated by low-energy collective
+bosons. In the SCS, the density of low-energy fermions
+drops and a continuum collective excitations becomes
+gaped. Both effects reduce fermionic damping and lead
+to the increase of WKin a SCS. If this increase exceeds a
+conventional loss of WKdue to a gap opening, the total
+∆WKmay become positive.
+The CB model has been applied numerous times to the
+cuprates, most often under the assumption that near-
+critical collective excitations are spin fluctuations with
+momenta near Q= (π,π). This version of a CB bo-
+son is commonly known as a spin-fermion model. This
+model yields dx2−y2superconductivity and explains in a
+quantitative way a number of measured electronic fea-
+tures of the cuprates, in particular the near-absence of
+the quasiparticle peak in the NS of optimally doped and
+underdoped cuprates39and the peak-dip-hump structure
+in the ARPES profile in the SCS31,32,46,47. In our analy-
+sis we assume that a CB is a spin fluctuation.
+The results for the conductivity within a spin-fermion
+model depend in quantitative (but not qualitative) way
+on the assumption for the momentum dispersion of a col-
+lective boson. This momentum dependence comes from11
+high-energy fermions and is an input for the low-energy
+theory. Below we follow Refs. 31,33 and assume that
+the momentum dependence of a collective boson is flat
+near (π,π). The self energy within such model has been
+worked out consistently in Ref. 31,33. In the normal
+state
+Σ′′(ω) =−1
+2λnωsflog/parenleftBigg
+1+ω2
+ω2
+sf/parenrightBigg
+Σ′(ω) =−λnωsfarctanω
+ωsf(19)
+whereλnis the spin-fermion coupling constant, and ωsf
+is a typical spin relaxation frequency of overdamped spin
+collective excitations with a propagator
+χ(q∼Q,Ω) =χQ
+1−iΩ
+ωsf(20)
+whereχQis the uniform static susceptibility. If we use
+Ornstein-Zernike form of χ(q) and use either Eliashberg
+45or FLEX computational schemes48, we get rather sim-
+ilar behavior of Σ as a function of frequency and rather
+similar behavior of optical integrals.
+The collective nature of spin fluctuations is reflected in
+the fact that the coupling λand the bosonic frequency
+ωsfare related: λscales as ξ2, whereξis the bosonic
+mass (the distance to a bosonic instability), and ωsf∝
+ξ−2(see Ref. 49). For a flat χ(q∼Q) the product λωsf
+does not depend on ξand is the overall dimensional scale
+for boson-mediated interactions.
+In the SCS fermionic excitations acquire a gap. This
+gapaffectsfermionicself-energyintwoways: directly, via
+the change of the dispersion of an intermediate boson in
+the exchange process involving a CB, and indirectly, via
+the change of the propagator of a CB. We remind our-
+selves that the dynamics of a CB comes from a particle-
+hole bubble which is indeed affected by ∆.
+The effect of a d−wave pairing gap on a CB has been
+discussed in a number of papers, most recently in31. Ina SCS a gapless continuum described by Eq. (20) trans-
+forms into a gaped continuum, with a gap about 2∆ and
+a resonance at ω=ω0<2∆, where for a d−wave gap we
+define ∆ as a maximum of a d−wave gap.
+The spin susceptibility near ( π,π) in a superconductor
+can generally be written up as
+χ(q∼Q,Ω) =χQ
+1−iΠ(Ω)
+ωsf(21)
+where Π is evaluated by adding up the bubbles made
+out of two normal and two anomalous Green’s functions.
+Below 2∆, Π(Ω) is real ( ∼Ω2/∆ for small Ω), and the
+resonance emerges at Ω = ω0at which Π( ω0) =ωsf. At
+frequencies larger than 2∆, Π(Ω) has an imaginary part,
+and this gives rise to a gaped continuum in χ(Ω).
+The imaginary part of the spin susceptibility around
+the resonance frequency ω0is31
+χ′′(q,Ω) =πZoω0
+2δ(Ω−ω0) (22)
+whereZo∼2ωsfχ0/∂Π
+∂ω|Ω=ω0. The imaginary part
+of the spin susceptibility describing a gaped continuum
+exists for for Ω ≥2∆ and is
+χ′′(q,Ω) =Im/bracketleftBigg
+χ0
+1−1
+ωsf/parenleftbig4∆2
+ΩD(4∆2
+Ω2)+iΩK2(1−4∆2
+Ω2)/parenrightbig/bracketrightBigg
+≈Im/bracketleftBigg
+χ0
+1−1
+ωsf/parenleftbigπ∆2
+Ω+iπ
+2Ω/parenrightbig/bracketrightBigg
+forΩ>>2∆ (23)
+In Eq. (23) D(x) =K1(x)−K2(x)
+x, andK1(x) andK2(x)
+are Elliptic integrals of first and second kind. The real
+partofχis obtained by Kramers-Kr¨ onigtransformofthe
+imaginary part.
+Substituting Eq 6 for χ(q,Ω) into the formula for the
+self-energy one obtains Σ′′(ω) in a SCS state as a sum of
+two terms31
+Σ′′(ω) = Σ′′
+A(ω)+Σ′′
+B(ω) (24)
+where,
+Σ′′
+A(ω) =πZo
+2λnωoRe/parenleftBigg
+ω+ωo/radicalbig
+(ω+ωo)2−∆2/parenrightBigg
+comes from the interaction with the resonance and
+Σ′′
+B(ω) =−λn/integraldisplay|E|
+2∆dxReω+x/radicalbig
+(ω+x)2−∆2x
+ωsfK2/parenleftBig
+1−4∆2
+x2/parenrightBig
+/bracketleftBig
+1−4∆2
+xωsfD/parenleftbig4∆2
+x2/parenrightbig/bracketrightBig2
++/bracketleftBig
+x
+ωsfK2/parenleftbig
+1−4∆2
+x2/parenrightbig/bracketrightBig2(25)
+comes from the interaction with the gaped continuum. The real par t of Σ is obtained by Kramers-Kr¨ onig trans-12
+form of the imaginary part.
+0 0.5 10  0.20.4
+ω in eVσ(ω)Conductivities (CB model λ =1)
+NS
+SC
+2∆+ωo
+0 0.5 10.20.61  
+ω in eVσ(ω)Conductivities (CB model λ =10)
+NS
+SC
+2∆ + ωo
+FIG. 17: Conductivities and ∆ Wfor a fixed λωsf. Top –
+ωsf= 26meV,λ= 1,ωo= 40meV,Zo= 0.77 Bottom –
+ωsf= 2.6meV,λ= 10,ωo= 13.5meV,Zo= 1.22. The zero
+crossing for ∆ Wis not affected by a change in λbecause it
+is determined only by λωsf. We set ∆ = 30 meV.
+1 2 3120160200
+λ (coupling)CB model ( Ωo=40 meV)WK(meV)SCS
+NS
+FIG. 18: The behavior of Kubo sums in the CB model. Note
+that the spectral weight in the NS is always larger than in the
+SCS. We set ωsf= 26meV,λ= 1, and ∆ = 30 meV.
+We performed the same calculations of conductivities
+and optical integrals as in the previous three cases. The
+resultsaresummarizedinFigs. 17-22. Fig17showscon-
+ductivities in the NSand the SCSfortwocouplings λ= 1
+andλ= 10 (keeping λωsfconstant). Other parameters
+Zoandωoarecalculatedaccordingto the discussionafter
+Eq 21. for ωsf= 26meV,λ= 1, we find ωo= 40meV,
+Zo= 0.77. And for ωsf= 2.6meV,λ= 10, we find
+ωo= 13.5meV,Zo= 1.22. Note that the conductivity
+in the SCS starts at 2∆+ ωo(i.e. the resonance energy0 0.5 10.20.61  
+ωc in eVW(ωc)/W(∞)NS Optical Sums (CB model)
+0 0.5 10.20.61  
+ωc in eVW(ωc)/W(∞)SCS Optical Sums (CB model)
+FIG. 19: The evolution of the optical integrals in the NS
+and the SCS in the CB model. Note that about ∼75% of
+the spectral weight is recovered up to 1 eV. We set ωsf=
+26meV,λ= 1, and ∆ = 30 meV.
+0 0.5 1−200  20 
+ωc in eVWSC(ωc) − WNS(ωc)∆ W (CB model λ=1)
+with lattice
+without lattice
+∆ WK
+0.2 0.6 1  −200  20 
+ωc in eVWSC(ωc) − WNS(ωc)∆ W (CB model λ=10)
+with lattice
+without lattice
+∆ WK
+FIG. 20: ∆ W(in meV) for λ= 1(top) and λ= 10(bottom).
+We used ωsf= 26meV/λand ∆ = 30 meV. The zero crossing
+is not affected because we keep λωsfconstant. The notable
+difference is the widening of the dip at a larger λ.13
+000.51
+εn(ε)NS
+SC Jumpλ=1
+0   00.51
+εn(ε)NSSC λ =7
+FIG. 21: Distribution functions n(ǫ) for CB model for λ= 1
+andλ= 7andaconstant ωsf= 26meV. Weset ∆ = 30 meV.
+Forsmaller λ(top), quasiparticles neartheFSarewell defined
+as indicated by the well pronounced jump in n(ǫ). Forλ= 7,
+n(ǫ) is rather smooth implying that acoherence is almost lost.
+Some irregularities is the SCS distribution function are du e
+to finite sampling in the frequency domain. The irregulariti es
+disappear when finer mesh for frequencies is chosen.
+shows up in the optical gap), where as in the BCSI case
+it would have always begun from 2∆. In Fig 18 we plot
+the Kubo sums WKvs coupling λ. We see that for all λ,
+WKin the NS stays larger than in the SCS. Fig 19 shows
+the cutoff dependence of the optical integrals W(ωc) for
+λ= 1 separately in the NS and the SCS. We again see
+that only about 73% of the Kubo sum is recovered up
+to the bandwidth of 1 eVindicating that there is a sig-
+nificant amount left to recover beyond this energy scale.
+Fig 20 shows ∆ Wfor the two different couplings. We
+see that, for both λ’s, there is only one zero-crossing for
+the ∆Wcurve, and ∆ Wis negative at larger frequen-
+cies. The only difference between the two plots is that
+for larger coupling the dip in ∆ Wgets ‘shallower’. Ob-
+serve also that the solid line in Fig. 20 is rather far away
+fromthedashedlineat ωc>1meV, whichindicatesthat,
+although ∆ W(ωc) in this region has some dependence on
+ωc, still the largest part of ∆ W(ωc) is ∆WK, while the
+contribution from ∆ f(ωc) is smaller.0 0.5 10.10.2
+ω in eVσ(ω)Conductivities (CB model−larger λ ωsf)
+NS
+SC
+2 ∆ +ωo
+0.20.40.60.80  4 8 
+ωc in eVWSC(ωc) − WNS(ωc)  in meV∆ W (CB model−larger λ ωsf)
+with lattice
+without lattice
+FIG. 22: Top – conductivity at a larger value of ωsfλ(ωsf=
+26meV,λ= 7) consistent with the one used in Ref.33). Bot-
+tom – ∆ Wwith and without lattice. Observe that the fre-
+quencyof zero crossing of ∆ Wenhances compared to the case
+of a smaller λωsfand becomes comparable to the bandwidth.
+At energies smaller than the bandwidth, ∆ W >0, as in the
+Norman- P´ epin model.
+0 2 4 6103050
+Self Energy prefactorδKE in meVλ=10
+λ=1
+FIG. 23: Kinetic energy difference between the SCS and the
+NS,δKEWe setλto be either λ= 1 orλ= 10 and varied ωsf
+thuschangingthe overall prefactor inthe self-energy. Atw eak
+coupling ( λ= 1) the behavior is BCS-like – δKEis positive
+and increases with the overall factor in the self-energy. At
+strong coupling ( λ= 7),δKEshows a reverse trend at larger
+ωsf.
+The negative sign of ∆ W(ωc) above a relatively small
+ωc∼0.1−0.2eVimplies that the ‘compensating’ ef-
+fect from the fermionic self-energy on ∆ Wis not strong
+enough to overshadow the decrease of the optical inte-
+gral in the SCS due to gap opening. In other words,the
+CB model displays the same behavior as BCSI, EB, and14
+modified MFLI models. It is interesting that this holds
+despite the fact that for large λCB model displays the
+physics one apparently needs to reverse the sign of ∆ WK
+– the absence of the quasiparticle peak in the NS and its
+emergence in the SCS accompanied by the dip and the
+hump at larger energies. The absence of coherent quasi-
+particle in the NS at large λis also apparent form Fig
+21 where we show the normal state distribution functions
+for two different λ. For large λthe jump (which indicates
+the presence of quasiparticles) virtually disappears.
+On a more careful look, we found that indifference of
+δW(ωc) to the increase of λis merely the consequence of
+the fact that above we kept λωsfconstant. Indeed, at
+small frequencies, fermionic self-energy in the NS is Σ′=
+λω, Σ” = λ2ω2/(λωsf), and both Σ′and Σ′′increase
+withλifwekeep λωsfconstant. Butatfrequencieslarger
+thanωsf, which we actually probe by ∆ W(ωc), the self-
+energyessentiallydependsonlyon λωsf, andincreasing λ
+but keeping λωsfconstant does not bring us closer to the
+physicsassociatedwiththerecoveryofelectroncoherence
+in the SCS. To detect this physics, we need to see how
+things evolve when we increase λωsfabove the scale of
+∆ , i.e., consider a truly strong coupling when not only
+λ≫1 but also the normal state Σ NS(ω≥∆)>>∆.
+To address this issue, we took a larger λfor the same
+ωsfand re-did the calculation of the conductivities and
+optical integrals. The results for σ(ω) and ∆W(ωc) are
+presented in Fig. 22. We found the same behavior as be-
+fore, i.e., ∆ WKis negative. But we also found that the
+largeristhe overallscalefortheself-energy,thelargerisa
+frequency of zero-crossing of ∆ W(ωc). In particular, for
+thesame λandωsfthatwereusedinRef. 33tofittheNS
+conductivity data, the zero crossing is at ∼0.8eVwhich
+is quite close to the bandwidth. This implies that at a
+truly strong coupling the frequency at which ∆ W(ωc)
+changes sign can well be larger than the bandwidth of
+1eVin which case ∆ Wintegrated up to the bandwidth
+does indeed remain positive. Such behavior would be
+consistent with Refs.8,9. we also see from Fig. 22 that
+∆WKbecomes small at a truly strong coupling, and over
+a wide range of frequencies the behavior of ∆ W(ωc) is
+predominantly governed by ∆ f(ωc), i.e. by the cut-off
+term.50The implication is that, to first approximation,
+∆WKcan be neglected and positive ∆ W(wc) integrated
+to a frequency where it is still positive is almost compen-
+sated by the integral over larger frequencies. This again
+would be consistent with the experimental data in Refs.
+8,9.
+It is also instructive to understand the interplay be-
+tween the behavior of ∆ W(ωc) and the behavior of the
+difference of the kinetic energy between the SCS and the
+NS,δKE. We computed the kinetic energy as a function
+ofλωsfand present the results in Fig. 23 for λ= 1 and
+10. For a relatively weak λ= 1 the behavior is clearly
+BCS like- δKE>0 and increases with increasing λωsf.
+However, at large λ= 10, we see that the kinetic energy
+begin decreasing at large λωsfand eventually changes
+sign. The behavior of δKEat a truly strong coupling isconsistent with earlier calculation of the kinetic energy
+for Ornstein-Zernike form of the spin susceptibility43.
+We clearly see that the increase of the zero crossing
+frequency of ∆ W(ωc) at a truly strong coupling is cor-
+related with the non-BCS behavior of δKE. At the same
+time, the behavior of δW(ωc) is obviously not driven by
+the kinetic energyaseventually δW(ωc) changessignand
+become negative. Rather, the increase in the frequency
+range where ∆ W(ωc) remains positive and non-BCS be-
+havior of δKEare two indications of the same effect that
+fermions are incoherent in the NS but acquire coherence
+in the SCS.
+III. CONCLUSION
+In this work we analyzed the behavior of optical in-
+tegralsW(ωc)∝/integraltextωc
+oσ(ω)dωand Kubo sum rules in
+the normal and superconducting states of interacting
+fermionic systems on a lattice. Our key goal was to
+understand what sets the sign of ∆ WK= ∆W(∞) be-
+tween the normal and superconducting states and what
+is the behavior of W(ωc) and ∆W(ωc) at finite ωc. In a
+weak coupling BCS superconductor, ∆ W(ωc) is positive
+atωc<2∆ due to a contribution from superfluid den-
+sity, but becomes negative at larger ωc, and approach a
+negativevalue of∆ WK. Ourstudy wasmotivatedby fas-
+cinatingopticalexperimentsonthecuprates7–10. Inover-
+doped cuprates, there is clear indication11that ∆W(ωc)
+becomes negative above a few ∆, consistent with BCS
+behavior. In underdoped cuprates, two groups argued8,9
+that ∆Wintegrated up to the bandwidth remains posi-
+tive, while the other group argued10that it is negative.
+The reasoning why ∆ WKmay potentially change sign
+at strongcoupling involvesthe correlationbetween −WK
+and the kinetic energy. In the BCS limit, kinetic en-
+ergyobviouslyincreasesinaSCS becauseofgapopening,
+hence−WKincreases, and ∆ WKis negative. At strong
+coupling,thereisacountereffect–fermionsbecomemore
+mobile in a SCS due to a smaller self-energy.
+We considered four models: a BCS model with impu-
+rities, a model of fermions interacting with an Einstein
+boson, a phenomenological MFL model with impurities,
+and a model of fermions interacting with collective spin
+fluctuations. In all cases, we found that ∆ WKis neg-
+ative, but how it evolves with ωcand how much of the
+sum rule is recovered by integrating up to the bandwidth
+depends on the model.
+The result most relevant to the experiments on the
+cuprates is obtained for the spin fluctuation model.
+We found that at strong coupling, the zero-crossing of
+δW(ωc) occurs at a frequency which increases with the
+coupling strength and may become larger than the band-
+width at a truly strong coupling. Still, at even larger
+frequencies, ∆ W(ωc) is negative.15
+Acknowledgements
+We would like to thank M. Norman, Tom Timusk,
+Dmitri Basov, Chris Homes, Nicole Bontemps, Andres
+Santander-Syro, Ricardo Lobo, Dirk van der Marel, A.
+Boris, E. van Heumen, A. B. Kuzmenko, L. Benfato, andF. Marsigliofor many discussions concerningthe infrared
+conductivity and optical integrals and thank A. Boris, E.
+van Heumen, J. Hirsch, and F. Marsiglio for the com-
+ments on the manuscript. The work was supported by
+nsf-dmr 0906953.
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\ No newline at end of file
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+arXiv:1001.0765v2  [astro-ph.CO]  5 May 2010Mon. Not. R. Astron. Soc. 000, 1–10 (2010) Printed 12 November 2018 (MN L ATEX style file v2.2)
+Survival of Star-Forming Giant Clumps in
+High-Redshift Galaxies
+Mark R. Krumholz1⋆and Avishai Dekel2⋆
+1Department of Astronomy & Astrophysics, University of Cali fornia, Santa Cruz, CA 95060, USA
+2Racah Institute of Physics, The Hebrew University, Jerusal em 91904, Israel
+Accepted 12 March 2010
+ABSTRACT
+We investigate the effects of radiation pressure from stars on the survival of
+the star-forming giant clumps in high-redshift massive disc galaxies, during
+the most active phase of galaxy formation. The clumps, typically of m ass∼
+108−109M⊙andradius ∼0.5−1kpc,areformedintheturbulentgas-richdiscs
+by violent gravitational instability and then migrate into a central bu lge in
+∼10dynamicaltimes. We showthat the survivalordisruption ofthes e clumps
+under the influence of stellar feedback depends critically on the rat e at which
+they form stars. If they convert a few percent of their gas mass to stars per
+free-fall time, as observed for all local star-formingsystems an d implied by the
+Kennicutt-Schmidtlaw,theycannotbedisrupted.Onlyifclumpscon vertmost
+of their mass to stars in a few free-fall times can feedback produc e significant
+gas expulsion. We consider whether such rapid star formation is likely in high-
+redshift giant clumps.
+Key words: galaxies: formation — galaxies: ISM — galaxies: star clusters:
+general — galaxies: star formation — ISM: clouds — stars: formatio n
+1 INTRODUCTION
+A significant fraction of the massive galaxies, ∼
+1011M⊙in baryons, during the period from z=
+1.5−3 when star formation is at its peak and most
+stellar mass is assembled (Hopkins & Beacom 2006;
+Magnelli et al. 2009), form stars at high star forma-
+tion rates (SFR) of ∼100M⊙yr−1. Many of these are
+turbulent, gas-rich, extended rotating discs in which
+much of the star formation takes place in a few giant
+clumps(Cowie et al. 1995; van den Bergh et al. 1996;
+Elmegreen et al. 2004a, 2005, 2007). Based on this mor-
+phology, they were first termed “chain” or “clump-
+cluster” galaxies. In a typical galaxy of this type,
+10−40% of the UV rest-frame light is emitted from a
+few clumps of characteristic size ∼1kpc that form stars
+at tens of M⊙yr−1each (Elmegreen et al. 2004b, 2005;
+F¨ orster Schreiber et al. 2006; Genzel et al. 2008). These
+clumps are much more massive than the star-forming
+complexes in local galaxies. The star formation in these
+⋆Email: krumholz@ucolick.org (MRK);
+dekel@phys.huji.ac.il (AD)clumps, and their survival subject to stellar feedback,
+are central to our understanding of galaxy formation.
+Kinematically, most of the galaxies that host the
+clumps are thick rotating discs, with high velocity dis-
+persions of σ= 20−80kms−1(one dimensional),
+compared to σ≃10kms−1is present-day discs; they
+have rotation to dispersion ratios of V/σ∼1−7
+(Cresci et al. 2009). Estimates of the total gas fraction
+in star-forming galaxies, based on CO measurements,
+range from 0.2 to 0.8, with an average of ∼0.4−0.6
+(Tacconi et al. 2008; Daddi et al. 2008, 2009), system-
+atically higher than the typical gas fraction in today’s
+discs. These properties are generally incompatible with
+these systems being ongoing major mergers or remnants
+of such mergers (Shapiro et al. 2008; Bournaud et al.
+2008; Dekel et al. 2009a; Bournaud & Elmegreen 2009;
+Dekel et al. 2009b), though there may be counter exam-
+ples (Robertson & Bullock 2008).
+Instead, giant clumps form through a scenario,
+summarized by Dekel et al. (2009b), in which mas-
+sive galaxies at z∼2−3 are fed by a few nar-
+row and partly clumpy streams of cold gas ( ∼104K)
+that flow along the dark matter filaments of the cos-
+c/circleco√yrt2010 RAS2Mark R. Krumholz and Avishai Dekel
+mic web (e.g. Hahn et al. 2007) and penetrate deep
+into the centres of the massive dark matter haloes of
+∼1012M⊙(Birnboim & Dekel 2003; Kereˇ s et al. 2005;
+Dekel & Birnboim 2006; Ocvirk et al. 2008; Dekel et al.
+2009a). Indeed, the existence of these streams is an in-
+evitable prediction of the standard ΛCDM cosmology.
+They may produce the structures we observe as Lyman-
+alpha blobs (Furlanetto et al. 2005; Goerdt et al. 2009;
+Dijkstra & Loeb 2009). The angular momentum they
+carry leads the accreted material to form a disc of ra-
+diusRd∼10kpc. The continuous intense input of gas
+at the level of ∼100M⊙yr−1maintains a high gas sur-
+face density Σ, which drives a violent gravitational in-
+stability with a Toomre Q parameter below unity, Q≃
+σΩ/(πGΣ)<1, where σis the one-dimensional velocity
+dispersion and Ω is the angular velocity associated with
+the potential well (Toomre 1964). The disc instability is
+self-regulated at Q<∼1 by the gravitational interactions
+in the perturbed disc, which keep the disc thick and
+with a high velocity dispersion. The disc forms strong,
+transient spiral features that fragment to produce 5-10
+bound clumps, that together comprise ∼20% of the
+disc mass. The largest clumps have characteristic radii
+R≃7GΣ/Ω2∼1 kpc, and characteristic masses of a
+few percent of the disc mass, M∼109M⊙. A spectrum
+of smaller clumps with somewhat lower masses forms as
+well. The clumps’ large masses cause them to migrate to
+the centre of the disc on a short time scale of ∼10 disc
+crossing times, where they merge into a central bulge
+(Noguchi 1999; Immeli et al. 2004a,b; Bournaud et al.
+2007; Ceverino et al. 2009; Agertz et al. 2009). This vi-
+olent instability phase can last for more than a gigayear,
+during which the mass flow from the disc to the bulge is
+replenished by fresh accretion, keeping the mass within
+the disc radius divided quite evenly between disc, bulge
+and dark matter components.
+While this scenario of clump formation and migra-
+tion to build up bulges is appealing, it relies on the
+ability of clumps to survive for ∼10 disc dynamical
+times, i.e. a few hundred million years, while the gas
+in them turns into stars on a comparable timescale
+(Genzel et al. 2008; Dekel et al. 2009b; Ceverino et al.
+2009). However, at the SFR of a few tens of solar
+masses per year in the clumps, it is possible that they
+might be disrupted by stellar feedback on considerably
+shorter timescales. Murray et al. (2009) argue for ex-
+actly this scenario. In their models, clumps disrupt af-
+ter∼1 dynamical time, during which they turn only
+∼30% of their mass into stars. In this picture clumps
+would not survive long enough to migrate, and bulges
+would instead need to be built up by mergers. While
+this scenario seems difficult to reconcile with the es-
+timated ages of a few hundred Myr for the oldest stel-
+lar populations in some clumps (Elmegreen et al. 2009a;
+F¨ orster Schreiber et al. 2009), there is sufficient uncer-
+tainty in both the observational estimates of clump ages
+andthetheoretical modelingofclumpevolutionanddis-
+ruptiontomerit are-investigation oftheproblem,which
+we provide in this paper.The outline of the paper is as follows. In §2 we
+derive the expected gas ejection fraction as a function
+of SFR efficiency. In §3 we address the observational
+estimates of the SFR efficiency. In §4 we discuss some of
+the issues raised by our results and compare to previous
+work. We summarize our conclusions in §5.
+2 RADIATIVE FEEDBACK AND CLUMP
+SURVIVAL
+Consider a uniform-density giant gas clump of mass M,
+radiusR, and surface density Σ = M/(πR2). It forms
+stars at a rate ˙M∗, and the stars formed within it have a
+combined luminosity L. (We defer discussing the effect
+of sub-clumping within giant clumps to §3.3, since it
+does not change the qualitative result.) Characteristic
+numbers to keep in mind for the largest, best-observed
+clumps, found in galaxies with baryonic masses ∼1011
+M⊙, areM≃109M⊙,R≃1 kpc, and Σ ≃0.1 g cm−1.
+Clumpsin the ∼1010M⊙galaxies, whichare more com-
+mon, have masses ∼108M⊙and sizes that are at or be-
+low the resolution limit of present observations. We will
+assume that they have surface densities comparable to
+those their larger cousins; they cannot be much smaller,
+since the mean column densities of the galactic disks as
+a whole is Σ ∼0.05 g cm−2. We wish to evaluate the
+fraction eof clump mass that will be ejected by stellar
+feedback and the fraction E= 1−ethat is transformed
+into stars, because this is the critical parameter that
+determines whether the clump will form a bound stel-
+lar system. Both N-body simulations and analytic mod-
+els (e.g. Hills 1980; Kroupa 2001; Kroupa & Boily 2002;
+Baumgardt & Kroupa 2007) indicate that if E/greaterorsimilar0.5,
+then most of the stellar mass will remain bound, while
+ifE/lessorsimilar0.3 then no bound stellar system will be left.
+Small portions of the clump where Ewas locally higher
+may form bound clusters, but these will be orders of
+magnitude smaller than the initial clump.
+Several common feedback mechanisms are not im-
+portant for giant clumps in the relevant mass range.
+First, supernova feedback is unlikely to be effective
+in ejecting mass from these giant clumps, due to
+cooling and leakage of hot gas (Dekel et al. 2009b;
+Krumholz & Matzner 2009; Murray et al. 2009). Sec-
+ond, the pressure of warm ( ∼104K) ionized gas is in-
+effective because the escape velocity from the clump is
+larger than the gas sound speed of ∼10 km s−1. Third,
+protostellar outflows are unable to eject mass because
+they do not provide enough momentum (Fall et al.
+2010). Instead, the dominant feedback mechanism is
+likely to be radiation pressure from newly-formed stars,
+which creates a radiation-dominated H iiregion. The
+expansion of such a region follows a similarity solution
+(Krumholz & Matzner 2009), and Fall et al. (2010) use
+this solution to show that all the remaining gas will be
+ejected once the fraction of gas mass transformed into
+stars reaches a value
+c/circleco√yrt2010 RAS, MNRAS 000, 1–10Giant Clump Survival 3
+E=M∗
+M=Σ
+Σ+Σ crit, (1)
+where
+Σcrit=5ηftrap∝angbracketleftL/M∗∝angbracketright
+παcritGc, (2)
+ηis a constant of order unity that depends on the den-
+sitydistributionwithintheclump, ∝angbracketleftL/M∗∝angbracketrightisthelightto
+mass ratio of the stellar population, ftraprepresents the
+factor by which the radiation force is enhanced by trap-
+ping of re-radiated infrared light within the expanding
+shell, and αcritis a parameter of order unity that de-
+scribes the critical velocity required to eject mass from
+the cloud. Fall et al. (2010) assume fiducial values of
+η= 2/3 andftrap=αcrit= 2 for the local star-forming
+regions, and we adopt the same values here since there
+is no obvious reason for them to be systematically dif-
+ferent in the case of giant clumps at redshift 2. We refer
+toEas the final star fraction.1
+To determine the light to mass ratio, we must
+deal with the complication that the characteristic cross-
+ing time of a giant clump is rather long, tcr=
+R//radicalbig
+GM/R = 15M−1/2
+9R3/2
+1Myr, where M9=
+M/(109M⊙) andR1=R/(1 kpc). Depending on the
+exact values of MandR, this can be either greater
+than or less than the main-sequence lifetime of mas-
+sive stars. Thus we can neither assume a single burst,
+so that all stars are coeval, nor continuous star forma-
+tion, so that the population is in equilibrium between
+new stars forming and old ones evolving off the main
+sequence. However, we can treat these scenarios as two
+limiting cases, which must bracket anyreal stellar popu-
+lation. Krumholz & Tan (2007) point out that in stellar
+populations younger than 3 Myr, where no stars have
+left the main sequence yet, the light-to-mass ratio has
+a constant value, Ψ, for which we adopt Ψ ≈2200 erg
+s−1g−1(Fall et al. 2010). Once a stellar population is
+old enough to reach statistical equilibrium between star
+formation and star death, it instead has a nearly con-
+stant luminosity-to-star-formation rate ratio, Φ, which
+we take to be Φ ≈6.1×1017erg g−1(Krumholz & Tan
+2007). For any realistic stellar population whose light is
+dominated by young, massive stars as opposed to old
+ones, the luminosity is roughly equal to the smaller of
+these two limits, and for simplicity we simply take the
+light-to-mass ratio to be
+/angbracketleftBigL
+M∗/angbracketrightBig
+= min/parenleftbigg
+Ψ,Φ˙M∗
+M∗/parenrightbigg
+. (3)
+We refer to the first case as the “young stars” limit and
+the second as the “old stars” limit, since they repre-
+sent the opposite extremes of stellar populations that
+have undergone no evolution and populations that are
+1Note that this is sometimes referred to as star formation
+efficiency as well, but we avoid the term efficiency because it
+does not have a standard meaning, and different authors use
+it for different concepts.old enough to have reached equilibrium between star
+formation and stellar evolution.
+Substituting these two light-to-mass estimates into
+equation (1) gives
+E= max/bracketleftbigg/parenleftBig
+1+Σcrit,y
+Σ/parenrightBig−1
+,1−Σcrit,y
+Σ/parenleftbigg
+Φ
+Ψtdep/parenrightbigg/bracketrightbigg
+,(4)
+where
+Σcrit,y=5ηftrapΨ
+παcritGc≈1.2 g cm−2(5)
+is the critical density evaluated in the young-stars limit
+andtdep=M/˙M∗is the depletion time, i.e. the time
+that would be required to convert all of the gas into
+stars. The numerical evaluation in equation (5) is for
+the fiducial parameters of Fall et al. (2010). Note that
+this expression is a maximum rather than a minimum
+because Eis a decreasing function of ∝angbracketleftL/M∗∝angbracketright.
+Following Krumholz & McKee (2005), we define
+the dimensionless star-formation rate efficiency as the
+ratio between the free-fall time and the depletion time2,
+namely
+ǫff=˙M∗
+M/tff. (6)
+Krumholz & Tan (2007) show that ǫff≃0.01 across
+a very broad range of densities, size scales, and envi-
+ronments. Here we define the free-fall time as tff= /radicalbig
+3π/(32Gρ), where ρis the gas density. We discuss
+in§3 below whether this value of ǫffapplies in high-
+zgiant clumps. If we adopt it for now and make this
+substitution in equation (4), we find that
+E= max/bracketleftbigg/parenleftBig
+1+Σcrit,y
+Σ/parenrightBig−1
+,1−ǫff√
+8GΣcrit,yΦ
+(πΣM)1/4Ψ/bracketrightbigg
+(7)
+≃max/bracketleftBigg/parenleftbigg
+1+12
+Σ−1/parenrightbigg−1
+,1−0.086
+(Σ−1M9)1/4ǫff,−2/bracketrightBigg
+,(8)
+where Σ −1= Σ/(0.1 g cm−2),ǫff,−2=ǫff/100, and the
+numerical evaluation uses the fiducial parameters from
+Fall et al. (2010). We use Equation (8) to plot Eas a
+function of ǫffin Figure 1.
+What does this result imply for thesurvival ofhigh-
+zgiant clumps? We note that the second term in brack-
+ets in equation (8), corresponding to the case of a stel-
+lar population older than ∼3 Myr, is generally the one
+that applies for giant clumps. This reflects the fact that
+the crossing time for our fiducial values of the parame-
+ters is 15 Myr, which is significantly larger than 3 Myr.
+The young stellar population limit applies only if the
+2Note that the rate efficiency ǫffthat we have defined here
+is distinct both from the star formation rate, which has unit s
+ofM⊙yr−1and is not normalized by the ratio of mass over
+free-fall time, and the final stellar mass fraction E, which is
+dimensionless but does not carry any information about the
+star formation rate.
+c/circleco√yrt2010 RAS, MNRAS 000, 1–104Mark R. Krumholz and Avishai Dekel
+Figure 1. Final stellar fraction Eas a function of star for-
+mation rate efficiency ǫff, computed from Equation (7). We
+show results for giant clump surface densities Σ −1= 0.2,
+1.0, and 5.0 (green, red, and blue lines, as indicated), and for
+giant clump masses M9= 0.03, 0.3, and 3 .0 (dotted, solid,
+and dashed lines, as indicated). The arrows schematically i n-
+dicate the region around ǫff= 0.1 where we change from the
+oldstarslimit(low ǫff)to the young starslimit(high ǫff).The
+gray region from E= 0.3−0.5 indicates the rough boundary
+between stellar fractions E/lessorsimilar0.3, for which no bound stellar
+system will remain, and E/greaterorsimilar0.5, for which a majority of the
+stellar mass will remain bound.
+star formation rate efficiency is much higher than is ob-
+served in any star-forming systems anywhere in the lo-
+cal universe, ǫff,−2/greaterorsimilar10, or if the star-forming systems
+are significantly less massive or much more dense. This
+makes high- zgiant clumps very different from Galac-
+tic star clusters or even super-star clusters, such as
+those found in local starburst galaxies, e.g. the An-
+tennae or M82. These have lower masses and higher
+surface densities, with crossing times ∼0.1 Myr (e.g.
+McCrady & Graham 2007), placing them firmly in the
+young-star limit. Indeed, the first term in the brackets
+is identical to that derived by Fall et al. (2010) for local
+star-forming clumps.3
+We conclude that unless ǫff,−2≫1 for giant
+clumps, as opposed to ∼1for the local star-forming sys-
+tems, star-forming clumps with masses ∼108−109M⊙
+and surface densities ∼0.1 g cm−2cannot be disrupted
+by radiation pressure — so they end up converting most
+of their mass to stars. Thus, the expulsion fraction rel-
+evant for high- zgiant clumps, as derived in the old-star
+limit, is
+e= 1−E= 0.086Σ−1/4
+−1M−1/4
+9ǫff,−2. (9)
+SinceE/greaterorsimilar0.5, we expect the resulting stellar sys-
+3To get numerical agreement in the coefficient of Σ −1, we
+must adjust kby a factor of√
+0.4 to account for measuring
+the escape velocity at the surface instead of at the half-mas s
+radius.tems to remain gravitationally bound. Clumps with
+Σ−1= 1 have E<0.5 and suffer significant disrup-
+tion only if their masses are below ∼106M⊙, and they
+reachE<0.3 and undergo complete disruption only at
+masses/lessorsimilar2×105M⊙. Thus the observed ∼108−109
+M⊙giant clumps in high −zgalaxies should survive dis-
+ruption unless ǫff,−2≫1. Even if we are maximally
+conservative and assume that the smaller clumps have
+Σ−1= 0.5, i.e. that their surface densities do not exceed
+the mean surface densities of their host galaxies, our es-
+timated maximum mass for disruption only increases by
+a factor of 2.
+3 THE STAR FORMATION RATE
+EFFICIENCY IN HIGH-Z CLUMPS
+Equation (9) shows that the most important parame-
+ter in determining the survival of giant clumps is how
+quickly, normalized to their free-fall times, they turn
+themselves into stars. Only if they do so with a very
+high rate efficiency, ǫff,−2/greaterorsimilar10, do we expect signif-
+icant gas expulsion. In the local universe, one mea-
+suresǫffby determining the star formation rate of an
+object or a population of objects, and comparing this
+to the objects’ gas mass divided by the free-fall time
+computed for their density. This procedure was first
+applied by Zuckerman & Evans (1974) to the popula-
+tion of giant molecular clouds in the Milky Way, and
+has subsequently been extended to other objects by
+Krumholz & Tan (2007) and Evans et al. (2009). These
+measurements give ǫff,−2∼1 over a very wide range
+of star-forming environments, from small star clusters
+in the Milky Way to entire starburst galaxies. The re-
+sults are subject to considerable uncertainty, but values
+ǫff,−2/greaterorsimilar10 are strongly excluded by the data. However,
+we lack comparable data for star formation at high red-
+shifts. In this section, we therefore turn to the question
+of the likely value of ǫffin high-zgiant clumps.
+3.1 Estimates of ǫfffrom Observations of
+Giant Clumps
+Unfortunately, we cannot easily apply the direct mea-
+surement procedure used to determine ǫffin the local
+universe to the high- zclumps, because, while we can
+evaluate star-formation rates using H αluminosities, we
+are limited in our ability to measure the corresponding
+gas properties.
+For example, Elmegreen et al. (2009b) use stellar
+population synthesis to estimate masses and ages for
+the stellar populations seen in giant clumps in high- z
+galaxies, and they then compare the ages τto the clump
+dynamical times, defined as tdyn≡(Gρ)−1/2≈0.5tff,
+whereρis taken to be the stellarmass density. They
+find typical values τ/tdyn∼1−10, corresponding to
+τ/tff∼2−20, with a factor of ∼10 scatter. It is
+tempting to identify τwithtdepand simply estimate
+c/circleco√yrt2010 RAS, MNRAS 000, 1–10Giant Clump Survival 5
+ǫff∼tff/τ∼0.05−0.5, but this is likely to be a sig-
+nificant overestimate. Based on dynamical mass esti-
+mates, Genzel et al. (2008) estimate that gas comprises
+10-30% of the total mass within the disc radius. Esti-
+mates based on direct CO measurements indicate gas
+fractions of 45 −60% at z∼2 (Daddi et al. 2009;
+Tacconi et al. 2010). It is likely to be an even larger
+fraction of the mass within the dense, rapidly star-
+forming giant clumps, which are self-gravitating and
+lack a dark matter component. If the stars measured
+by Elmegreen et al. comprise only a small fraction f∗of
+the total mass in clump, then ǫffwill be reduced by a
+factor of roughly f1.5
+∗relative to the previous estimate –
+one power of f∗to account for the gas that has not yet
+formedstars, andanotherfactor of f0.5
+∗because thefree-
+fall time will be shorter than the value Elmegreen et al.
+estimate based on the stars alone. To give a sense of the
+possible magnitude of the error, note that Murray et al.
+(2009) estimate f∗= 0.2 for a giant clump in BX 482
+(and we argue in Section 4.2 that f∗is probably even
+smaller), and thisvalue of f∗would be sufficienttolower
+theestimate of ǫffbyafactor of10, to5 ×10−3−5×10−2.
+A secondary worry is that, as Elmegreen et al. point
+out, since the clumps are selected using rest-frame blue
+light there is a strong bias against selecting older, red-
+der clumps, causing an underestimate of τ. In general, τ
+is expected to be an underestimate of tdepbecause the
+former refers only to the stars that have already formed.
+Given this problem, many observers have at-
+tempted to estimate gas masses via the “inverse”
+Kennicutt (1998) law, namely by measuring the SFR
+surface density and then assuming that the gas surface
+density has the value required for the object to obey
+theKennicuttlaw (e.g. Genzel et al. 2006, 2008). Unfor-
+tunately this procedure does not yield an independent
+estimate of ǫff. The Kennicutt relation is
+˙Σ∗=AΣ1.4
+−1, (10)
+withA≃1.4M⊙yr−1kpc−2. Thus if the scale height
+of the galaxy is h, then the midplane gas density is ρ=
+Σ/h, the free-fall time is tff=/radicalbig
+3πh/(32GΣ), and we
+have
+ǫff,−2= 100˙Σ∗
+Σ/tff= 3.4A1.4h0.5
+0Σ−0.1
+−1, (11)
+whereh0=h/kpc and A1.4=A/(1.4M⊙yr−1
+kpc−2). The true value of ǫffis almost certainly a bit
+smaller than this, since star-forming clouds have densi-
+ties higher than the mean midplane density, and thus
+smaller free-fall times. Nonetheless, this calculation il -
+lustrates a crucial point: to the extent that galactic
+scale heights do not have a very large range of vari-
+ation (and the dependence is only to the 0.5 power),
+the statement that ǫff,−2∼1 is roughly equivalent to
+the Kennicutt law. (See Krumholz & McKee 2005 and
+Schaye & Dalla Vecchia2008 for amore detailed discus-
+sion of the relationship between volumetric and areal
+star formation laws.) Thus, no measurement of the gas
+surface density that assumes the Kennicutt law a priorican produce a value of ǫff,−2significantly different from
+this. Any measurement of ǫff,−2that did yield a signif-
+icantly larger value would necessarily place the galaxy
+well off the Kennicutt relation.
+3.2 Estimates of ǫfffrom Observations of
+Clump Host Galaxies
+Given the difficulties of estimating ǫffin giant clumps
+directly, we instead turn to indirect inferences, based
+on the more robust measurements of the overall disc
+properties. We first note that the observed correlation
+between total gas mass and total star-formation rate
+in high-zgalaxies (e.g. Carilli et al. 2005; Greve et al.
+2005; Gao et al. 2007) is consistent with these galaxies
+having the same value of ǫffas local star-forming sys-
+tems (Krumholz & Thompson 2007; Narayanan et al.
+2008a,b). Bothwell et al. (2009) have claimed to detect
+a deviation from a universal star formation law in spa-
+tially resolvedobservations ofthree z∼2systems. How-
+ever,Bothwell et al. obtain thisresult onlybecause they
+choose a non-standardconversion factor betweenCO lu-
+minosity and mass, which leads them to conclude that
+thegas fraction in thesesystems isonly ∼5−10%, much
+lower than in typical star forming galaxies at z∼2,
+and on the low side even for local star-forming galaxies.
+Tacconi et al. (2010), using a standard conversion fac-
+tor, conclude instead that these galaxies have molecular
+gas fractions of 35 −45%. This is consistent with the
+star formation law and the value of ǫffatz∼2 being
+the same as in the local universe.
+We can also approach the problem more theoreti-
+cally. The high- zclumps discs with which we are con-
+cerned have SFR ˙M∗∼100M⊙yr−1, baryonic mass
+Md∼1011M⊙, and disc radius Rd∼10kpc. We first
+expresstherelevantquantitiesinEquation (9)as afunc-
+tion of the clump mass M, surface density Σ, and SFR
+˙M∗. The free-fall time (which is√
+5/2 times the crossing
+timeR/V), expressed in 106yr, is
+tff,6≡/radicalbigg
+3π
+32Gρ= 16.7M−1/2
+9R3/2
+1= 12.3M1/4
+9Σ−3/4
+−1.(12)
+Then
+ǫff,−2≡˙M∗
+M/tff= 12.3˙M∗10Σ−3/4
+−1M−3/4
+9, (13)
+where˙M∗10is the SFR in a clump in 10 M⊙yr−1.
+Now let us express the clump quantities in terms of
+the disc mass, surface density, and total SFR. The disc
+surface density is
+Σd,−1= 0.663Md,11R−2
+d,10, (14)
+whereRd,10is the disc radius measured in units of 10
+kpc. The virialized clumps can be assumed to have col-
+lapsed by at least a factor 2 in radius, so the clump
+surface density is s≡4s4times larger than the disc’s,
+or
+Σ−1= 2.65s4Md,11R−2
+d,10, (15)
+c/circleco√yrt2010 RAS, MNRAS 000, 1–106Mark R. Krumholz and Avishai Dekel
+withs4/greaterorequalslant1.
+Assume that a fraction α(≃0.2) of the disc mass
+is inN(∼10) identical clumps and a fraction β(≃
+0.5) of the SFR is in the clumps (Dekel et al. 2009b;
+Ceverino et al. 2009). Then
+˙M∗,10= 0.5β0.5N−1
+10˙Md∗,100, (16)
+and
+M9= 2α0.2N−1
+10Md,11. (17)
+Substituting the last three expressions in eq. (13) we
+obtain in terms of the disc quantities
+ǫff,−2= 1.75s−3/4
+4β0.5α−3/4
+0.2N−1/4
+10˙Md∗,100M−3/2
+d,11R3/2
+d,10.(18)
+Note the very weak dependence on N. We see that
+the most straightforward observational estimates for the
+massive clumpy discs at z∼2 (e.g. the BzK galaxies,
+Genzel et al. 2008) yield ǫff,−2≃1.75 ande∼0.1, i.e.
+no significant expulsion.
+This estimate is based on the observed properties of
+galaxies over a relatively narrow range of galaxy masses
+and redshifts, but using simple theoretical arguments
+we can deduce how the results are likely to scale to
+other galaxies for which we currently lack direct obser-
+vations. For a self-gravitating disc, the circular velocit y
+is roughly V2
+d,200≃Md,11/Rd,10, soǫff,−2∝˙Md∗V−3
+d,
+namely
+ǫff,−2∝˙Md∗
+Mvir. (19)
+A constant ǫff,−2(∼1) is consistent with the SFR be-
+ing a constant fraction of the baryon accretion rate, be-
+cause the latter is roughly proportional to halo mass
+(Neistein et al. 2006; Birnboim et al. 2007):
+˙Md∗∼˙M∝Mvir. (20)
+The fact that the SFR follows the accretion rate is a
+natural result of the fact that the SFR is proportional
+to the mass of the available gas (Bouche et al. 2009;
+Dutton et al. 2009), and is consistent with the find-
+ing from simulations when compared to observed SFR
+(Dekel et al. 2009a). We learn that eis only weakly de-
+pendent on M.
+The redshift dependenceat agivenhalomass, using
+V3∝Mv(1+z)3/2and˙M∝(1+z)2, is
+ǫff,−2∝s−3/4(1+z)1/2. (21)
+The system can adjust the SFR to match the rate of
+gas supply by accretion with ǫff,−2∼1 at all times by
+slight variations in the contraction factor s. At higher
+redshift, the clumps should contract a bit further and
+form stars at a somewhat higher surface density.
+3.3 Sub-Resolution Clumping
+Our discussion of clump survival in the preceding sec-
+tions is based on the assumption that giant clumps rep-
+resent single star-forming molecular clouds, although weof course expect them to possess significant substruc-
+ture, as do local molecular clouds. These substructures
+are unresolved by current observations and by simula-
+tions. Here we discuss how their presence affects our
+conclusions.
+First note that, for this purpose, we do not care
+whether any sub-clumps within the giant clumps them-
+selves survive star formation feedback and form bound
+stellar clusters. To see why, consider an extreme case in
+which all the sub-clumps within the giant clump expel
+most of their gas, and thus do not leave behind bound
+remnants. This is what we might expect to happen if
+all the sub-clumps had surface densities similar to that
+of their parent giant clump, but had masses well below
+the∼105−106M⊙minimum survival mass that we
+computed in §2. In this case the sub-clumps would all
+form stars, expel their gas, and disperse, but both the
+stars and the expelled gas would still remain trapped
+within the much larger gravitational potential well of
+the giant clump. They could escape from this potential
+well only if the giant clump as a whole were disrupted
+by gas expulsion, which we have already shown in §2
+will happen only if ǫffis much larger than the expected
+value. Thus the end result of this scenario would be a
+boundgiant star cluster without any boundsub-clusters
+inside it.
+At the opposite extreme, suppose that all the sub-
+clumps were to remain bound and undergo negligible
+gas expulsion. We might expect this scenario if the
+sub-clumps all had surface densities much higher than
+that of their parent giant clump. In this case the sub-
+clumps would convert most of their mass to stars, form-
+ing bound clusters. All the bound clusters would irra-
+diate the remaining mass in the giant clump, imparting
+momentum to it. If the stars imparted enough momen-
+tum, this gas would be expelled. Assuming most of the
+mass were in the inter-clump medium, as is the case for
+local molecular clouds, this expulsion would unbind the
+giant clump, producing many small individually bound
+clusters that are not bound to one another. Conversely,
+if the imparted momentum were not sufficient to un-
+bind the giant clump, as we expect, the result would be
+a giant star cluster consisting of many smaller bound
+clusters, all gravitationally bound to one another.
+In either extreme scenario, whether or not sub-
+clumps survive does not make any difference to whether
+agiant clumpas awhole survives. This is dictated solely
+by the expulsion fraction from the giant clump. How-
+ever,sub-clumpingstill could makeadifference for giant
+clump survival by raising the value of ǫff. In this case
+the sub-clumps would still have ǫff,−2∼1, but the gi-
+ant clump would have ǫff,−2≫1 because it would have
+the same star formation rate but a much lower mean
+density, and thus a longer free-fall time.
+To see whether this is likely to happen, we note
+that the turbulent motions within a giant clump are
+likely to break it up into smaller sub-clumps, much a lo-
+cal molecular clouds are broken up into clump, filamen-
+tary structures by turbulence. In such a configuration,
+c/circleco√yrt2010 RAS, MNRAS 000, 1–10Giant Clump Survival 7
+a majority of the mass is at a density higher than the
+volumetric mean density ¯ ρthat we have computed, and
+wouldthereforehaveashorterfree-fall timeandahigher
+star formation rate. Quantitatively, we have computed
+the star formation rate as ˙M∗=ǫffM/tff(¯ρ), where M
+is the total mass of the giant clump, ¯ ρis its volume-
+averaged density, and tffis the free-fall time computed
+at that density. However, if most of the mass is at a
+densityρ >ρ, the appropriate mass might be the mass
+M(> ρ) above that higher density, and the appropri-
+ate timescale might be tff(ρ) computed for that density.
+While one might worry that this could be a significant
+effect, Krumholz & Thompson (2007) point out that it
+is in reality quite small. Turbulent systems generally
+have lognormal density distributions. For such a dis-
+tribution, the fraction of the cloud mass with density
+greater than ρis given by
+M(> ρ)
+M=1
+2/bracketleftbigg
+1+erf/parenleftbigg
+−2lnx+σ2
+ρ
+23/2σρ/parenrightbigg/bracketrightbigg
+, (22)
+wherex=ρ/¯ρ, ¯ρis the volumetric mean density, and
+σρis the dispersion of the density distribution. This
+is related to the Mach number Mof the turbulence
+byσ2
+ρ≈ln(1 +γM2), where γis a constant of or-
+der unity (Padoan & Nordlund 2002; Federrath et al.
+2008). For σρ= 2.5−3, the range of values expected for
+the Mach numbers found in giant clumps, the quantity
+M(> ρ)/tff(ρ) varies by only a factor of a few over a
+range of densities ρ/¯ρ≈10−1−105. Thus even if most
+of the mass is at a density vastly larger than the mean
+density we have used, as long as the mass distribution
+follows the lognormal form expected for supersonic tur-
+bulence, the star formation rate will not be modified sig-
+nificantly from our estimate using the volume-averaged
+density.
+4 DISCUSSION
+4.1 Turbulence and Energy Balance in Giant
+Clumps
+Our finding that the fraction of gas ejected from giant
+clumps depends critically on their dimensionless star-
+formation rate efficiency ǫffnaturally leads to the ques-
+tion of how this quantity is set, and whether the phys-
+ical processes responsible for setting ǫff,−2∼1 in the
+local universe might determine a different value in high-
+redshift clumps. Krumholz & McKee (2005) show that,
+as long as the gas in a molecular cloud is supersonically
+turbulent with a velocity dispersion comparable to the
+cloud’s virial velocity, as is observed to be the case in
+all molecular clouds in the local universe, ǫff,−2∼1 is
+the inevitable consequence. In contrast, in the absence
+of supersonic turbulence, simulations find that clouds
+undergo a rapid global collapse in which they convert
+all their mass into stars in roughly a dynamical time,
+i.e.ǫff,−2∼100 (e.g. Nakamura & Li 2007; Wang et al.2009).4Thus, a value of ǫff,−2∼1 may be expected in
+high-zgiant clumps only if they maintain the level of
+turbulence required to avoid rapid, global collapse.
+Whetherthe turbulencecan actually be maintained
+issomewhat less clear. Simulations showthatsupersonic
+turbulence decays in roughly one cloud-crossing time
+(e.g. Stone et al. 1998; Mac Low et al. 1998; Mac Low
+1999), so global collapse can be avoided only if this
+energy is replaced on a comparable timescale. In lo-
+cal, low-mass star-forming clouds ( /lessorsimilar104M⊙), observa-
+tions (Quillen et al. 2005), simulations (Nakamura & Li
+2007; Wang et al. 2009), and analytic theory (Matzner
+2007) all suggest that protostellar outflows can supply
+the necessary energy. In local giant molecular clouds
+with masses of ∼104−106M⊙, Hiiregions driven
+by the pressure of photoionized gas are likely to be
+able to supply the necessary energy (Matzner 2002;
+Krumholz et al. 2006).However,neitherofthesemecha-
+nismsareeffectiveforclumpswith M9∼1andΣ −1∼1,
+because they do not provide enough momentum input
+andbecause theyare overwhelmedbyradiation pressure
+(see Figure 2 of Fall et al. 2010).
+Supernova feedback (Dekel & Silk 1986) does not
+appear to be a likely candidate to drive the turbulence
+either.Supernovaedonotprovideenoughpowertodrive
+the observed level of turbulence (Dekel et al. 2009a),
+and analytic calculations (Harper-Clark & Murray
+2009; Krumholz & Matzner 2009), numerical simula-
+tions of isolated disk galaxies (Tasker & Bryan 2008;
+Joung et al. 2009), and numerical simulations of galax-
+ies in cosmological context (Ceverino & Klypin 2009)
+all indicate that supernova-heated gas is likely to escape
+through low-density holes in the molecular gas without
+driving much turbulence.
+Contrary to this conclusion, Lehnert et al. (2009)
+use the observed correlation between H αsurface bright-
+ness and linewidth in z∼2galaxies toargue thatsuper-
+nova feedback is responsible for driving the turbulence,
+based in part on simulations by Dib et al. (2006), who
+obtainascalingrelation betweenvelocitydispersion and
+supernova rate in numerical simulations. However, the
+efficiency with which supernova energy is coupled to the
+ISM is a free parameter in both Lehnert et al.’s analy-
+sis and in Dib et al.’s simulations, and their results are
+consistent with the data only if it is ∼25%, whereas in
+the dense environments found in high redshift galaxies
+it is expected to be far lower (Thompson et al. 2005).
+This conclusion is confirmed by the more recent simula-
+tions, which do not need toassume an efficiency because
+they have sufficient resolution to resolve the multiphase
+structure of the ISM. Finally, we note that the corre-
+lation between H αsurface brightness and linewidth ob-
+served by Lehnert et al. has a more prosaic explanation:
+4This can be avoided if clouds are magnetically subcritical
+(e.g. Nakamura & Li 2008), but magnetic fields in the early
+universe are likely to be weaker than those in the local uni-
+verse, and even in the local universe clouds do not appear to
+be subcritical in typical clouds (Crutcher et al. 2009).
+c/circleco√yrt2010 RAS, MNRAS 000, 1–108Mark R. Krumholz and Avishai Dekel
+in a marginally stable galactic disk of constant circular
+velocity, the velocity dispersion is proportional to the
+gas surface density, since Q=κσ/(πGΣ) = 1. Thus
+higher velocity dispersions correspond to higher surface
+densities, which in turn produce higher star formation
+rates in accordance with the standard Kennicutt (1998)
+relation. This naturally explains the observed correla-
+tion.
+Radiation pressure is another mechanism to con-
+sider. If radiation pressure is not able to drive mass out
+of the clump, as found above for ǫff∼0.01, this sug-
+gests that it might not be able to drive turbulence to
+the required virial level either, since the virial and es-
+cape velocities only differ by a factor of√
+2. However,
+it is unclear whether this conclusion is warranted. Ra-
+diation pressure cannot drive material out of a clump
+notbecause stars donotaccelerate material enough, but
+because they evolve off the main sequence before they
+are actually able to eject matter. As a result, they pro-
+duce expanding shells whose velocities greatly exceed
+the escape velocity. They simply fail to drive mass out
+because the clump because the driving sources turn off
+before the shells actually escape from the cluster. It is
+unclear if the expanding shells might provide enough
+energy to maintain the turbulence; this problem will re-
+quire further modeling.
+We are left with the possibility that the turbu-
+lence is driven by gravity. The driving source cannot
+be the collapse of the clump itself; although such a col-
+lapse does produce turbulence, it does so at the price
+of reducing the crossing time, raising the rate of en-
+ergy loss. Consequently, the collapse becomes a run-
+away process, and all the gas quickly converts to stars.
+However, as shown by Dekel et al. (2009a), the gravi-
+tational migration of the clumps through the galactic
+disc does provide enough power to maintain the turbu-
+lence within them. The main uncertainty in this model
+is how much of that power will go into driving internal
+motions within the clump, rather than motions in the
+external galactic disc. This depends on how the clumps
+are torqued by one another and by the disc. However,
+there is suggestive evidence from simulations of giant
+clumps that this mechanism might be viable. The sim-
+ulations of clumpy galaxies that have been done to
+date (e.g. Bournaud et al. 2007; Elmegreen et al. 2008;
+Agertz et al. 2009; Ceverino et al. 2009) either include
+no feedback or only supernova feedback (which is inef-
+fective). If the giant clumps formed in these simulations
+did lose their turbulence and undergo global collapse,
+then,dependingonthedetails ofthesimulationmethod,
+they would either convert all of their mass into stars, or
+all of the mass within them would collapse to the maxi-
+mumdensityallowed bythe imposed numerical pressure
+floor. This collapse would happen on the crossing time
+scale of a clump, which is much less than the time re-
+quired for the clumps to migrate to the galactic centre.
+However, such collapses are not observed in the simula-
+tions. This strongly suggests that, even in the absenceof feedback, gravitational power is sufficient to maintain
+the turbulence.
+We conclude that the generation of turbulence in
+the giant clumps is an important open issue, to be
+addressed by further studies including simulations of
+higher resolution.
+4.2 Comparison to Previous Work
+Our conclusion that giant clumps are not likely to be
+disrupted by feedback is in contrast with the findings of
+Murray et al. (2009) for the giant clump in the galaxy
+Q2346-BX 482 (Genzel et al. 2008), and this difference
+merits discussion. Based on the H αluminosity of the
+clump, Murray et al. estimate a total bolometric lumi-
+nosity of L= 4×1011L⊙, which corresponds to a star-
+formation rate of 34 M⊙yr−1in the old stars limit,
+or a stellar mass of 2 .6×108M⊙in the young stars
+limit, which Murray et al. assume. If these stars are in-
+deedyoung,thenthestar-formation ratecouldbehigher
+than 34 M⊙yr−1, but it could not be any lower.
+While the estimate of the star-formation rate is rel-
+atively straightforward, inferring the gas mass is much
+less so. Murray et al. (2009) take it to be 109M⊙based
+on an order-of-magnitude estimate for the Toomre mass
+in the galaxy. This choice is crucial to their result. The
+clump radius is r= 925 pc, so if we adopt this radius,
+the mean density-free fall time is 15 Myr, so ǫff,−2≃50.
+Using this value in equation (8), together with the cor-
+responding surface density and mass Σ −1= 0.8 and
+M9= 1, tells us that we are in the young stars limit
+andthat E= 0.06 – fully consistent with Murray et al.’s
+conclusion that only a relatively small fraction of the
+gas mass turns into stars, and that this is sufficient to
+expel the remaining gas. Murray et al. derive a slightly
+higher value E ∼1/3 because their criterion for ejection
+amounts to adopting αcrit∼10.
+However, this conclusion depends crucially on hav-
+ing a low estimate of the gas mass, and a correspond-
+ingly high estimate for ǫff,−2. Indeed, if the value of
+ǫff,−2≃50 were accurate, this clump would have the
+highest star formation rate efficiency of any known sys-
+tem. It is therefore useful to consider alternative meth-
+ods for estimating the mass. If we were to adopt the
+inverse-Kennicutt method, the observed star-formation
+rate per unit area ˙Σ∗≃13M⊙yr−1kpc−1, together
+with equation (10), gives a gas surface density of Σ −1≃
+5 (or 2300 M⊙pc−2). The corresponding gas mass is
+6×109M⊙, and recomputing ǫfffor this mass gives
+ǫff,−2≃3, consistent with the point we made earlier
+that the Kennicutt Law is in practice equivalent to hav-
+ingǫff,−2∼1. With this mass we would predict E ≃0.9,
+i.e. essentially no gas expulsion. One would expect sim-
+ilar results from Murray et al.’s models, because the ef-
+fect of this mass increase would be to increase the grav-
+itational force by a factor of 40 while leaving the radia-
+tiveforce unchanged.Thus while Murray et al.’s models
+suggest that the clump in BX 482 has stopped forming
+stars and all the remaining gas has just been expelled,
+c/circleco√yrt2010 RAS, MNRAS 000, 1–10Giant Clump Survival 9
+according to our estimate this clump can be only part
+of the way through its life and may continue to form
+stars.
+Murray et al. (2009) also suggest another method
+of estimating the gas mass. Based on the H αluminosity,
+if one assumes that the clump is filled with uniform-
+density gas, then ionization balance requires that this
+gas have a density of hydrogen nuclei
+nH=/radicalbigg
+3S
+4πr3α(B)/parenleftBig4X
+3X+1/parenrightBig
+, (23)
+whereXis the hydrogen mass fraction and this ex-
+pression assumes that He is singly-ionized. If, fol-
+lowing Murray et al. (2009), we adopt the young
+stars limit, then the ionizing luminosity is S=
+6.3×1046(M∗/M⊙) = 1 .6×1055photons s−1
+(Murray & Rahman 2009).5Combining this with the
+case B recombination coefficient α(B)= 3.46×10−13
+cm3s−1and the Solar hydrogen mass fraction X= 0.71
+(mean mass per H nucleus of 1 .4mH) givesnH= 21
+cm−3, corresponding to a mass of 2 .4×109M⊙and
+ǫff,−2= 14. Plugging this mass, surface density, and
+value of ǫff,−2into equation (8) gives E ≃0.2, i.e. the
+star fraction is more than three times what we would
+obtain using Murray et al.’s mass of M= 1×109
+M⊙. Adopting αcrit= 10 to shift our fiducial pa-
+rameters closer to those used in Murray et al. would
+giveE ≃0.8, no significant gas expulsion. We empha-
+size that these calculations are lower limits on the gas
+mass and upper limits on the fraction of mass ejected,
+because this mass estimate includes only ionized gas.
+However, models of both classical gas pressure-driven
+Hiiregions and ones driven by radiation pressure
+(Krumholz & Matzner 2009; Murray et al. 2009) sug-
+gest that the ionized gas mass in the H iiregion in-
+terior is significantly smaller than the mass of neutral
+gas swept up in the shell around it. Including this mass
+would lower ǫff,−2and increase Eeven further.
+In summary, our conclusions differ from those of
+Murray et al. (2009) not because of any difference in
+the physics of radiation feedback, but because they have
+used an estimated gas mass that produces an extraor-
+dinarily high value of ǫff.
+5 CONCLUSIONS
+Our main result in this paper, summarized in Equation
+(9) and Figure 1, is that the survival or disruption of
+giant star-forming clumps in high- zgalaxies depends
+critically on the rate at which they turn into stars. We
+find that as long as the high-redshift clumps convert
+their gas mass into stars at a rate of one to a few per-
+cent of the mass per free-fall time, ǫff∼0.01, as is ob-
+served in low-redshift star-forming systems from small
+5The ionizing luminosity is somewhat lower in the old stars
+limit:S= 2.5×1053(˙M∗/M⊙yr−1) = 8.5×1054photons
+s−1.galactic clusters to ultraluminous infrared galaxies, the
+clumps retain most of their gas and turn it into stars.
+As a result, they remain bound as they migrate into the
+galactic centre on timescales of ∼2 disc orbital times.
+A significant fraction of the clump gas could be ejected
+on a free-fall timescale before turning into stars only if
+clumps can convert /greaterorsimilar10% of their gas mass into stars
+in a free-fall time, forming stars much faster than any
+other star-forming system known. We argue the current
+high-zdata is consistent with the standard SFR rate
+efficiencies at the level of one to a few percent, with
+no significant evidence for a change in the star forma-
+tion process in high- zstar-forming galaxies. Neverthe-
+less, this is clearly an interesting issue to explore with
+more direct observational estimates of the gas mass in
+these galaxies.
+It is possible to check our theoretical arguments for
+clump survival with a number of possible observations.
+First, clumps can be disrupted by radiation pressure
+only if a majority of their gas is expelled, and the result-
+ing massive radiation-driven outflows from clumps may
+beobservable assystematic blueshifts attheclumploca-
+tions. A preliminary search for such a phenomenon have
+yielded a null result (K. Shapiro, private communica-
+tion, 2009), but further investigations of the kinematics
+in and around the clumps are worthwhile. It is possible
+that sufficient extinction could hide the blueshifted sig-
+nature, but we note that, if the gas surface densities are
+relatively low as, e.g., Murray et al. (2009) propose, the
+extinction is relatively mild.
+Second, stellar populations in high-redshift clumps
+can provide independent observational tests that could
+help distinguish between the two scenarios of clump
+survival or disruption. One such test involves the age
+spread of stars in actively star-forming clumps. If ra-
+diative feedback disrupts the clumps after one or a few
+free-fall times, the spread of stellar ages in each clump
+should not exceed ∼50 Myr. If clumps survive, on the
+other hand, then the age spreads may reach /greaterorsimilar100 Myr,
+with a high and roughly constant SFR during the life-
+time of the clump. There is preliminary observational
+evidence in favor of the latter (Elmegreen et al. 2009b;
+F¨ orster Schreiber et al. 2009), but this should be ex-
+plored further. In this case, migration toward the bulge
+will produce an age gradient, so that clumps closer to
+the bulge have systematically larger age spreads than
+those further out in the disc.
+A third test is associated with the properties of
+massive star clusters in the disc that are not actively
+forming stars. Clusters with masses /greaterorsimilar107M⊙can only
+be made in giant clumps, not in the rest of the disc,
+where molecular clouds are smaller. If clumps undergo
+rapid disruption by feedback, most of the clusters are
+likely to dissolve as is the case for local star formation
+(e.g. Fall et al. 2009), but the few clusters that may sur-
+vive will remain in the disc near their formation radii.
+None will migrate into the bulge, since the migration
+time varies as M2(Dekel et al. 2009b), and the masses
+of the clusters are much smaller than those of their par-
+c/circleco√yrt2010 RAS, MNRAS 000, 1–1010Mark R. Krumholz and Avishai Dekel
+ent gas clumps. In contrast, if gas clumps survive feed-
+back and convert most of their mass to stars, the re-
+sulting massive objects will rapidly migrate toward the
+galactic center. They will lose ∼50% of their mass due
+to tidal stripping (Bournaud et al. 2007), possibly in-
+cluding some massive sub-clusters, but they will deliver
+the rest of their mass to the bulge. Thus, if giant clumps
+do not survive, we expect to see a few massive clusters
+in the disk and none in the bulge. If, on the other hand,
+our model is correct, then there should be comparable
+masses of disk and bulge clusters. Those bulge clusters
+that survive today may correspond to the metal-rich
+globular cluster population seen in present-day galac-
+tic bulges (Brodie & Strader 2006; Shapiro et al. 2010;
+Romanowsky et al., in preparation).
+Even though we have concluded that the high- z
+giant clumps are expected to survive radiative stellar
+feedback, we do emphasize the important role that this
+process is likely to play in galaxy formation. For exam-
+ple, it can provide some of the pressure support needed
+in these giant clumps, it can drive non-negligible winds
+out of them, and it is likely to disrupt the less mas-
+sive clumps where stars form at later redshifts. In many
+circumstances, this mode of feedback is expected to be
+more important than supernova feedback, as it pushes
+away the cold dense gas while the latter mostly affects
+the hot dilute gas. We thus highlight the need for in-
+coroprating radiative stellar feedback in hydrodynam-
+ical simulations as well as in semi-analytic models of
+galaxy evolution.
+ACKNOWLEDGMENTS
+We acknowledge stimulating discussions with Andi
+Burkert, Daniel Ceverino, Reinhard Genzel, Norm Mur-
+ray, and Aaron Romanowsky, and helpful comments on
+the manuscript from Mark Dijkstra, Bruce Elmegreen,
+and the referee, Frederic Bournaud. This research has
+been supported by the Alfred P. Sloan Foundation
+(MRK); NASA through ATP grants NNX09AK31G
+(MRK), NAG5-8218 (AD), and as part of the Spitzer
+Theoretical Research Program, through a contract
+issued by the JPL (MRK); the National Science
+Foundation through grant AST-0807739 (MRK); the
+German-Israeli Foundation (GIF) through grant I-
+895-207.7/2005 (AD); the German Research Founda-
+tion (DFG) via German-Israeli Project Cooperation
+grant STE1869/1-1.GE625/15-1 (AD); the Israel Sci-
+ence Foundation (AD); and France-Israel Teamwork in
+Sciences (AD).
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+arXiv:1001.0766v2  [hep-th]  12 Mar 2010Tensor calculus on noncommutative spaces
+D V Vassilevich
+CMCC, Universidade Federal do ABC, Santo Andr´ e, SP, Brazil
+Department of Theoretical Physics, St Petersburg State Univer sity, St Petersburg,
+Russia
+E-mail:dvassil@gmail.com
+Abstract. It is well known that for a given Poisson structure one has infinitely
+many star products related through the Kontsevich gauge trans formations. These
+gauge transformations have an infinite functional dimension (i.e., co rrespond to an
+infinite number of degrees of freedom per point of the base manifold ). We show that
+on a symplectic manifold this freedom may be almost completely eliminate d if one
+extends the star product to all tensor fields in a covariant way and impose some
+natural conditions on the tensor algebra. The remaining ambiguity c orresponds either
+toconstantrenormalizationstothesymplecticstructure,ortom apsbetweenclassically
+equivalent field theory actions. We also discuss how one can introduc e the Riemannian
+metricinthis approachandtheconsequencesofourresultsforno ncommutativegravity
+theories.
+PACS numbers: 02.40.Gh, 04.60.Kz
+1. Introduction
+The deformation quantization program was formulated in its modern form in the papers
+[1, 2], see [3] for an overview. The main part of this program is the con struction of a
+star product, which is an associative deformation of the usual poin t wise product in the
+directionofa givenPoisson structure. Onsymplectic manifolds(non -degenerate Poisson
+structure) the problem was solved by Fedosov [4] in a covariant wa y. The existence of a
+starproduct foranarbitraryPoisson structurewasdemonstra ted byKontsevich [5], who
+also gave a closed formula for this product. To calculate higher orde rs of the star it is
+more convenient to use other methods [6]. The Kontsevich formula is written in a fixed
+coordinate frame, and, therefore, is not covariant. A globalizatio n of the Kontsevich
+product was done by Cattaneo and Felder [7], see also [8] for a covar iant version of
+the Kontsevich formality theorem. More recently, a manifestly cov ariant universal star
+product was constructed [9]. Further generalizations of the star product consist in its’
+extension to the exterior algebra of differential forms [10, 11, 12], Lie algebra valued
+differential forms [13], and tensor valued differential forms [14]. G auge theories with
+covariant star products were considered in [15].Tensor calculus on noncommutative spaces 2
+To a given Poisson structure one can associate infinitely many star p roducts related
+through the Kontsevich gauge transformations [5]. These gauge transformations depend
+on an arbitrary order (formal) differential operator. This means, that the star products
+are parametrized by an infinite number of fields, which are coefficient s in front of
+powers of the derivatives in this operator. In other words, the sp ace of star products
+has an infinite functional dimension, corresponding to an infinite num ber of degrees of
+freedom per point of the base manifold. Although the star product s related through the
+Kontsevich transformationsareequivalent inthesense ofdeform ationquantization, field
+theories based on such products are by no means equivalent. The r eason is very simple:
+the Kontsevich gauge transformations are not gauge symmetries in the field theory
+sense. To make them local symmetries of an action one has to add co rresponding gauge
+fields[16]. The number of such gauge fields is, in general, infinite. Some part of the
+gauge freedom may be used to extract physical fields [16], but the t otal ambiguity is
+enormous and must be reduced by imposing some natural restrictio ns on admissible star
+products.
+The problem of symmetries of a physical action on a noncommutative space is one
+of the central ones for noncommutative field theories [17]. Due to t he presence of the
+Poisson tensor, many symmetries are broken, but may be restore d by ”twisting”, which
+amounts to replacing the Lie algebra of symmetries by the correspo nding Hopf algebra
+with a twisted coproduct. The idea of twisting was discussed already in [18]. The first
+physical symmetry to be realized in this way was the Poincare one [19 ], which was then
+followed by other symmetries, as, e.g., the diffemorphism [20] andgau getransformations
+[21]. In this way one can define practically any symmetry on noncommu tative plane.
+There is, however, a drawback: twisted local symmetries are not b ona fide symmetries.
+One cannot use them for a gauge-fixing in order to remove non-phy sical degrees of
+freedom or to select a representative of a family of gauge-equivale nt configurations. It
+is desirable, therefore, to have the symmetries realized in the stan dard non-twisted way.
+It is natural to address both problems, namely the problem of symm etries and the
+problem of the abundance of free parameters, simultaneously. It is well known that
+the existence of a derivation satisfying the Leibniz rule is very restr ictive. But, the
+derivative ∇µmaps scalars to vectors, vectors to rank two tensors, etc. To d efine the
+derivative in a consistent way one has to extend the star product t o all tensor fields,
+thus adding more arbitrariness. Nevertheless, one gets a possibilit y to formulate some
+natural conditions which the algebra of tensors should satisfy. Th ese conditions appear
+to be restrictive enough to remove practically all ambiguities in the st ar product.
+This paper is organized as follows. In the next section we re-introdu ce a covariant
+star product [1] and remind some basic properties of its extension t o the tensor fields
+[22]. Then, in Sec. 3, we remind the structure of the Kontsevich gau ge transformations,
+impose some conditions on the star product, and show that the amb iguity is thus
+reduced to a space of a finite functional dimension. Next (Sec. 4), we show that all
+products satisfying the conditions of Sec. 3, can be realized throu gh a twist. In Sec. 5 we
+demonstrate thatforclosed star productsthe gaugefreedomis reduced further, andthatTensor calculus on noncommutative spaces 3
+practically all remaining gauge transformations are maps between c lassically equivalent
+actions. What remains, is just constant renormalization of the sym plectic structure
+(i.e., the freedom to add covariantly constant terms to the symplec tic structure with
+higher orders ofthe deformationparameter) andachoice ofa flat torsion-freesymplectic
+connection. In Sec. 6 we discuss some implications for noncommutat ive gravity theories.
+Finally, Sec. 7 contains concluding remarks. In this section we also ar gue that the set of
+the requirements that we impose on the star product is a natural r eplacement of locality
+in the noncommutative setting. We support these arguments by co nsidering the case of
+vanishing Poisson structure in Appendix Appendix A.
+2. The star product
+Consider asymplectic manifold Mwithasymplectic form ωµν(withthePoissonbivector
+ωµνbeing its’ inverse, ωµνωνρ=δρ
+µ). LetTMbe a tangent bundle, and T∗Mbe a
+cotangent bundle. Let αn,mbe a tensor field, αn,m∈TMn⊗T∗Mm≡Tn,m. This
+means,αn,mhasncontravariant and mcovariant indices.
+LetuschooseaChristoffelsymbolon Msuchthatthesymplectic formiscovariantly
+constant,
+∇µωνρ=∂µωνρ−Γσ
+µνωσρ−Γσ
+µρωνσ= 0. (1)
+Therefore, Mbecomes a Fedosov manifold [23]. Let us suppose that this connectio n is
+flat and torsion-free, i.e.,
+[∇µ,∇ν] = 0. (2)
+Locally, one can choose a coordinate system such that ωµν=const.and Γσ
+µν= 0. We
+shall call such coordinates the Darboux coordinates.
+One can then consider the algebra of formal power series T[[h]], where T≡/circleplustext
+n,mTn,m, withhbeing a deformation parameter, and define a covariant star produ ct
+[1] (it was recently used in [22] in the context of NC gravity)
+α⋆β=/summationdisplay
+khk
+k!ωµ1ν1...ωµkνk(∇µ1...∇µkα)·(∇ν1...∇νkβ). (3)
+We stress that this product respects the diffemorphism symmetry , i.e., it indeed maps
+tensors to tensors. Apart of this, it has the following obvious prop erties
+S1. α⋆β=αβ+∞/summationdisplay
+k=1hkCk(α,β),
+whereCkare bilinear differential operators.
+S2. Associativity:
+α⋆(β ⋆γ) = (α⋆β)⋆γ.
+S3. The order hterm is a Poisson bracket, C1(α,β) ={α,β}.
+S4. Stability on covariantly constant tensors: α⋆β=α·βif∇α= 0 or∇β= 0.Tensor calculus on noncommutative spaces 4
+S5. The Moyal symmetry:
+Ck(α,β) = (−1)kCk(β,α).
+S6. Derivation:
+∇α⋆β= (∇α)⋆β+α⋆(∇β).
+The list above is an extension of the requirements imposed on the sta r products of
+forms [12], except for S6, which was not requested in [12]. Instead o f S4 one usually
+considers a weaker property, namely, that the unit function is the unity of the algebra,
+1⋆ α=α ⋆1 =α. From the physical point of view, S4 means that for slowly varying
+fields the star product should look as the usual point wise product. The property S3
+definesa Poisson bracket on T
+{α,β}=ωµν∇µα·∇νβ (4)
+having the following properties.
+P1. Bracket degree:
+{αn1,m1,βn2,m2} ∈Tn1+n2,m1+m2.
+P2. Antisymmetry
+{α,β}=−{β,α}.
+P3. Product rule:
+{α,βγ}={α,β}γ+β{α,γ}.
+P4. There is a covariant derivative ∇:Tn,m→Tn,m+1such that [ ∇µ,∇ν] = 0 and
+∇{α,β}={∇α,β}+{α,∇β}.
+P5. Jacobi identity:
+{α,{β,γ}}+{γ,{α,β}}+{β,{γ,α}}= 0.
+The properties P1-P5 can be demonstrated by expanding corresp onding conditions
+S1-S6andpicking upappropriatepower of h, orbyusing theexplicit formula(4). Again,
+P1-P6 are extensions to arbitrary tensors of the requirements o n the Poisson structure
+of differential forms [24, 12].
+Let us make an important remark. It is known that the Leibniz rule S6 is very
+restrictive. However, since ∇maps a scalar to evector, imposing S6 ina way compatible
+with covariance requires first to extend the star product to tens or fields, which enlarges
+the freedom in the choice of the star product.
+In physical application the deformation parameter his imaginary, ¯h=−h. Then
+S5 yields that the star product is Hermitian,
+(α⋆β) =¯β ⋆¯α, (5)
+Since the symplectic structure is covariantly constant,
+ωµν⋆α=ωµν·α, (6)Tensor calculus on noncommutative spaces 5
+i.e.,ωµνis central in the corresponding commutator algebra.
+There is a natural integration measure (see [25] for a detailed disc ussion of traces
+in relation to star products)
+dµ(x) = (det( ωµν))−1
+2dx, (7)
+with respect to which the star product of tensors is closed provide d all indices are
+contracted in pairs,/integraldisplay
+Mdµ(x)αµν...ρ⋆βµν...ρ=/integraldisplay
+Mdµ(x)αµν...ρ·βµν...ρ. (8)
+In other words, the property (8) is valid when the integrand is diffeo morphism invariant.
+3. Gauge freedom
+3.1. Gauge transformations
+A star product corresponding to a given Poisson structure is not u nique. The
+arbitrariness in the choice of a star product is described by the Kon tsevich [5] gauge
+transformation ⋆→⋆′:
+α⋆′β=D−1(Dα⋆Dβ ), (9)
+where
+D= 1+hL. (10)
+HereLis a formal differential operator, i.e, it is sum of differential operato rs of arbitrary
+order
+L=∞/summationdisplay
+k=1Lµ1...µk∇µ1...∇µk, (11)
+and each Lµ1...µkcontains non-negative powers of the deformation parameter h. Eq. (11)
+is nothing else than a covariantization of corresponding formula in [5]. Here we also like
+to mention an interpretation of the gauge freedom in the framewor k of the Fedosov
+approach given in [26].
+We like to exclude from (9) the transformations which leave the star product
+invariant. They correspond to inner automorphisms of the algebra
+α→αϕ=eϕ
+⋆⋆α⋆e−ϕ
+⋆ (12)
+whereeϕ
+⋆= 1 +ϕ+ (1/2)ϕ ⋆ ϕ+...is the star-exponent. ϕis a scalar which can be
+expanded in non-negative powers of the deformation parameter h. In physical units his
+imaginary, and ϕis imaginary as well, so that we have a U(1)⋆gauge group (see [17]).
+It is easy to check that the transformations (12) indeed do not ch ange the star product,
+αϕ⋆βϕ= (α⋆β)ϕ, and that they can be represented in the form (9) with
+hL(ϕ) = 2hωµν(∇µϕ)∇ν+... , (13)
+where we omitted higher derivative terms. Therefore, in what follow s we exclude from
+the gauge transformations the terms whose linear part has the fo rm (13).Tensor calculus on noncommutative spaces 6
+3.2. Reduction of the gauge freedom
+Let us now study the following question. Suppose we have a Fedosov manifold with a
+symplectic structure ωµνand a flat torsionless symplectic connection ∇. What are the
+star products which preserve the rank of the tensors, Tn1,m1⋆ Tn2,m2⊂Tn1+n2,m1+m2,
+and satisfy the requirements S1-S6 with the Poisson bracket defin ed in Eq. (4)? Since
+the Poisson structure is given by (4), such products belong to the family consisting of
+the star product (3) and the products gauge-equivalent to (3).
+Interestingly, “physical uniqueness” of the star product (3) fo r scalars was
+demonstrated already in [1]. It was shown, that if one has only the sy mplectic structure
+and the flat covariant derivative, the star product must be a sum o f the same terms as
+on the right hand side of (3) but with possibly different coefficients. C orrect coefficients
+are then fixed by requiring associativity of the product.
+Here we shall allow for arbitrary structures to appear and then re duce the gauge
+freedom by using S1-S6. This will be done in three step. First, we sha ll show that L
+doesnot depend onthetensor degree. Second, we shall exclude t hefirst order terms in L
+modulo ”constant renormalizations” of the symplectic structure ( defined below). Third,
+we will show that the higher order terms in L(i.e., with two and more derivatives) have
+to be covariantly constant. Each step will include two sub-steps. W e shall begin with
+analyzing the linearized gauge transformations ⋆→⋆′
+1, where
+α⋆′
+1β=α⋆β−hL(α⋆β)+h(Lα⋆β)+h(α⋆Lβ). (14)
+Afterwards, the results will be extended to full non-linear gauge t ransformations (9).
+Perhaps, this is not the shortest way to prove the main result of th is section, but it is
+probably a more pedagogical one.
+A priori, each Lµ1...µkis a local linear map on the space of the tensors. According
+to our assumption, Lmust preserve the rank of the tensors, i.e., it maps Tn,mtoTn,m.
+Let us denote the restriction of Lµ1...µktoTm,nasLµ1...µk
+(n,m), and let us check under which
+conditions the property S4 holds. Take αn,mbeing covariantly constant, ∇αn,m= 0.
+Then
+0 =αn,m⋆′
+1β0,0−αn,m·β0,0 (15)
+=h/summationdisplay
+k/parenleftBig
+−Lµ1...µk
+(n,m)αn,m∇µ1...∇µkβ0,0+α(n,m)Lµ1...µk
+(0,0)∇µ1...∇µkβ0,0/parenrightBig
+as a consequence of S4 in the infinitesimal setting, Eq. (14). This las t equation is
+local and valid for an arbitrary scalar β0,0. Although α(n,m)above is restricted to being
+covariantly constant, it can have arbitrary value at a given point. T herefore, (15) gives
+enough conditions to conclude that
+Lµ1...µk
+(n,m)= I(n,m)Lµ1...µk
+(0,0), (16)
+where I (n,m)is the unit operator on T(n,m). To extend this result to full gauge
+transformations, let us fix a pair ( n,m) and consider the lowestnumberkfor which
+(16)does not hold. Then, for a covariantly constant αn,m, the terms with kderivativesTensor calculus on noncommutative spaces 7
+onβ0,0inαn,m⋆′β0,0−αn,m·β0,0(not restricting this difference to the linear order any
+more) look precisely as the expression in the brackets on the secon d line of Eq. (15).
+Due to the arbitrariness of β0,0this leads to a contradiction. We conclude, that (16)
+holds also if ⋆′is related to ⋆by a full gauge transformation (9).
+Next, let us study the restrictions on Lfollowing from the Leibniz rule S6. Locally,
+we can choose a Darboux coordinate system, so that ωµνis constant, and ∇=∂. We
+start with the first order part L=Lµ∂µ. Again, we start with the case when ⋆′and
+⋆are related by a linearized gauge transformation (14). The local ( h0) part of the
+product (3) is invariant under the gauge transformation with the fi rst-order L, which is
+a consequence of Eq. (16). The terms with lowest order derivative s in the transformed
+star product are coming from the h1part of (3):
+α⋆′
+1β−α⋆β
+=h2[−Lµ∂µ(∂ρα∂σβωρσ)+(∂ρLµ∂µα)(∂ρβ)ωρσ+(∂ρα)(∂ρLµ∂µβ)ωρσ]
+=h[∂ρα∂σβδωρσ], (17)
+where
+δωρσ=h[(∂µLρ)ωµσ+ωρµ(∂µLσ)] (18)
+is nothing else than the variation of ωρσunder infinitesimal diffeomorphisms
+generated by the vector field Lµ. The transformation which do not change ωρσ
+(symplectomorphisms) havealreadybeenexcluded, seesec.3.1, s othatfortheremaining
+Lµthe right hand side of (18) is non-zero. S6 immediately yields
+∂νδωρσ= 0. (19)
+This means that the allowed gaugetransformationaretheones whic h add tothe Poisson
+structure arbitrary constant terms containing at least one powe r of the deformation
+parameter h. It is natural to call these gauge transformations constant ren ormalizations
+of the Poisson structure.
+Let us extend this result beyond the linearized gauge transformat ions (14). To this
+end we write in the Darboux coordinates the product ⋆′in a form similar to S1
+α⋆′β=α·β+/summationdisplay
+j,lB(j),(l)(∂(j)α)·(∂(l)β), (20)
+where (j) and (l) are multi-indices, ∂(j)≡∂µ1...∂µj. Due to (16) the coefficient
+functions B(j),(l)do not depend on the tensor degree of αandβ. Moreover, because
+of S4,B(0),(l)=B(j),(0)= 0, and the sum in (20) starts with j=l= 1. Next we write
+∂ν(α⋆′β)−(∂να⋆′β)−(α⋆′∂νβ) =/summationdisplay
+j,l(∂νB(j),(l))(∂(j)α)·(∂(l)β) (21)
+for arbitrary αandβ. We conclude, that to satisfy S6, the coefficient functions B(j),(l)
+must be constant in the Darboux coordinates (or covariantly cons tant in an arbitrary
+coordinate system).Tensor calculus on noncommutative spaces 8
+For the future use it is convenient to introduce a “renormalized” st ar product
+α⋆Rβ=/summationdisplay
+khk
+k!ωµ1ν1
+R...ωµkνk
+R(∇µ1...∇µkα)·(∇ν1...∇νkβ),(22)
+which depends on a “renormalized” symplectic structure
+ωµν
+R=ωµν+h2ωµν
+1+h4ωµν
+2+... (23)
+It is supposed, that all correction terms ωµν
+jare covariantly constant,
+∇ρωµν
+j= 0. (24)
+Note, that odd powers of the deformation parameter hin (23) are excluded by the
+Moyal symmetry requirement S5. Since the h0part in (23) is unchanged, the product
+⋆Ris equivalent to ⋆in the sense of deformation quantization. Other equivalent star
+products ⋆′are obtained from ⋆Rby means of the gauge transformations (9). However,
+one has to exclude the transformations which do not change the an tisymmetric part of
+B(1),(1). This has to be done for the following reasons. First, one should avo id double
+countingofthedegreesoffreedom, whicharealreadyincludedin ωµν
+j, i.e. inthe constant
+antisymmetric part of B(1),(1). Second, a non-constant B(1),(1)contradicts S4, as we have
+just seen above. One can easily see, that this excludes the gauge t ransformations having
+a non-zero first-order part Lµ. To demonstrate this, let us expand Lµin a series of h
+and pick up the term with the lowest power of h. Then, to this order of h, thefullgauge
+transformation (including also arbitrary higher derivative terms) o f the antisymmetric
+part ofB(1),(1)is given by the linearized expression, c.f. right hand side of (17). The
+linearized transformations have already been analyzed above, yield ing that this lowest
+order part of Lµmust vanish. Consequently, Lµ= 0 to all orders.
+We have reduced the set of admissible star products to the produc ts, which are
+obtained from ⋆Rby means of the gauge transformations with vanishing first order p art
+Lµ. Moreover, we know that other Lµ1...µkare proportional to the unit operator on each
+Tn,mand do not depend on the tensor degree. The coefficient function B(j),(l)have been
+shown to be constants, which does not yet imply that Lµ1...µkmust be constants as well.
+Next, let us analyze the higher derivative terms in Lrestricting ourselves for a
+while to the linear order in L. Such terms change already the local, h0, part of the star
+product. For L=Lµ1...µn∇µ1...∇µnwe have in the Darboux coordinates
+α⋆′
+1β−α⋆β=−hn−1/summationdisplay
+k=1Ck
+nLµ1...µn(∂µ1...∂µkα)(∂µk+1...∂µnβ) (25)
+whereCk
+narebinomial coefficients. The terms with a larger number of derivativ es acting
+onαandβhave been omitted. Substitution of (25) in the Leibniz rule S6 yields
+∂ν(α⋆′
+1β)−(∂να⋆′
+1β)−(α⋆′
+1∂νβ)
+=−h(∂νLµ1...µn)n−1/summationdisplay
+k=1Ck
+n(∂µ1...∂µkα)(∂µk+1...∂µnβ) (26)
+up to higher derivative terms.Tensor calculus on noncommutative spaces 9
+Let us pick up the smallest nsuch that Lµ1...µnis not (covariantly) constant. Then,
+the terms with the lowest number of derivatives in ∇(α⋆′
+1β)−(∇α⋆′
+1β)−(α⋆′
+1∇β)
+are given in the Darboux coordinates precisely by the right hand side of (26). Since α
+andβare arbitrary, we conclude that
+∇νLµ1...µn= 0. (27)
+Next, we exclude non-constant part of Lµ1...µnµn+1, and so on.
+Note, that no cancellation can occur between the terms (17) and ( 25) since the
+former are antisymmetric in α↔β, while the latter are symmetric. According to the
+Moyalsymmetryrequirement, thesetwotypesofthetermscano nlyappearaccompanied
+by different powers of the deformation parameter h.
+Extension of (27) to full gauge transformations is straightforwa rd. It is enough to
+pick up for each nthe lowest power of the deformation parameter hat which (27) is not
+satisfied, and obtain a contradiction to S4.
+There is another, rather obvious, restriction onthe coefficients Lµ1...µnwhich follows
+from the Moyal symmetry requirement S5. Namely, these coefficien ts are allowed to
+contain odd powers of the deformation parameter honly.
+We have just demonstrated the following statement.
+Theorem 3.1. LetMbe a symplectic manifold with a symplectic structure ωµν
+and a flat torsionless connection ∇. Any covariant star product on the space of tensor
+fields over Msatisfying S1-S6 with the Poisson bracket (4) can be represe nted as
+α⋆Nβ=D−1(Dα⋆RDβ), (28)
+whereα,β∈T, the product ⋆Ris defined in (22), and the coefficients Lµ1...µnin the
+gauge operator Dare covariantly constant and contain odd powers of the defor mations
+parameter h. The coefficient with n=1 vanishes.
+We shall call the star product (28) the natural star product.
+Thefreedomremaininginthedefinitionof ⋆Nisthatofchoosingasinglescalarfield ‡
+depending also on h. Therefore, the space of natural star products for a given ωµνand
+∇has a functional dimension one. Let us remind, that the space of ar bitrary products
+⋆′corresponding to given ωµνand∇, eq. (9), has an infinitefunctional dimension.
+In Appendix Appendix A we explain how the arguments of this Section w ork in the
+commutative case ωµν= 0.
+4. Twist representation of the star product
+In this section, we are going to demonstrate that any natural sta r product (28) can
+be represented through a twist on a suitable Hopf algebra. A rathe r complete survey
+on the Hopf algebras can be found in the monographs [27] and [28]. A minimal set of
+necessary information is contained in [17].
+‡This can be seen by identifying constant coefficients Lµ1...µnwith coefficients in the Taylor expansion
+of a scalar field.Tensor calculus on noncommutative spaces 10
+Consider a Lie algebra A. One can make out of the universal enveloping algebra
+U(A) a Hopf algebra Hby introducing a primitive coproduct ∆ 0:H→H⊗Hsuch
+that ∆ 0(1) = 1⊗1 and ∆ 0(a) = 1⊗a+a⊗1, where ais a generator of A. A counit
+ε:H→Cis defined by the relations ε(a) = 0,ε(1) = 1, and an antipode S:H→H
+satisfiesS(1) = 1,S(a) =−a.
+Consider an invertible §elementF ∈H⊗H. If it satisfies the conditions
+(F ⊗1)(∆0⊗1)F= (1⊗F)(1⊗∆0)F, (29)
+(ε⊗1)F= 1 = (1 ⊗ε)F, (30)
+it is called a twist [30], and it defines a new twisted Hopf algebra.
+If the second equation (30) only is satisfied by F, the twisted algebra is quasi-
+Hopf, and not Hopf (see, e.g., [28], Theorem 2.4.2), meaning that the co-associativity
+is lost. Although quasi-Hopf algebras were recently considered in th e context of
+noncommutative quantum field theory [31], we do not consider such a possibility here
+and assume that Fsatisfies both the equations (29) and (30).
+Suppose that Hacts onT. There is a commutative point wise product µ0on
+T,µ0(α⊗β) =α·β. By using the twist, one can define a new associative product
+µF=µ0◦F−1, i.e., as
+µF(α⊗β) =µ0(F−1(α⊗β)). (31)
+We are going to demonstrate that each natural star product can be represented in this
+form
+α⋆Nβ=µ0(F−1
+N(α⊗β)) (32)
+for some twist FN.
+Let us pass to the Darboux coordinate system (transition to gene ral coordinates
+will be considered at the end of this section). Let Abe an abelian algebra generated by
+the partial derivatives ∂µ, andHbe the corresponding universal enveloping algebra
+with a primitive coproduct ∆ 0and the counit and antipode as defined above. In
+the Darboux coordinates the product ⋆Rcoincides with the Moyal product with a
+renormalized symplectic structure ωR. Therefore, it can be represented through the
+standard (Moyal) twist
+FR= exp(−hωµν
+R∂µ⊗∂ν) (33)
+To proceed with generic ⋆N, let us represent the admissible gauge transformations in an
+exponential form
+D= exp/parenleftBigg
+h∞/summationdisplay
+n=2lµ1...µn∂µ1...∂µn/parenrightBigg
+(34)
+§One can consider also non-invertible twist elements, see [29], though this leads to some technical
+complications.Tensor calculus on noncommutative spaces 11
+with constant coefficients lµ1...µn. Obviously, Dis an element of H. One can find
+FN∈H⊗Hsuch that (32) is satisfied:
+FN=FR˜F, (35)
+˜F−1= ∆0(D−1)(D⊗D). (36)
+Note that ˜FandFRcommute. Equation (35) follows from the fact that ⋆Nis a gauge
+transformation of ⋆R. To derive Eq. (36) one observes that the primitive coproduct
+corresponds to the usual Leibniz rule, i.e., ∂µ(α·β) =µ0(∆0(∂µ)(α⊗β)). Moreover,
+coproduct is an algebra morphism. Consequently, ∆ 0in (36) distributes the derivatives
+contained in D−1in the tensor product D⊗D. Explicitly,
+˜F= exp/parenleftBigg
+h∞/summationdisplay
+n=2lµ1...µnn−1/summationdisplay
+k=1Ck
+n∂µ1...∂µk⊗∂µk+1...∂µn/parenrightBigg
+(37)
+(cf Eq. (25)).
+Next, we have to show that FNsatisfies both (29) and (30). Equation (30) is
+straightforward. Since ε(∂µ) = 0, only the unit elements in ˜FandFRcontribute to
+(ε⊗1)Fand (1⊗ε)F. Eq. (30) follows immediately. By using certain commutativity
+properties and the fact that FRsatisfies Eq. (29), one can show, that FNsatisfies (29)
+if and only if ˜Fsatisfies the same equation with the primitive coproduct ∆ 0:
+(˜F ⊗1)(∆0⊗1)˜F= (1⊗˜F)(1⊗∆0)˜F. (38)
+Eq. (38) can be checked directly. Indeed, one can bring left and rig ht hand sides of (38)
+to the form
+exp/parenleftBigg
+h∞/summationdisplay
+n=2lµ1...µn/summationdisplay
+i+j+k=nn!
+i!j!k!∂µ1...∂µi⊗∂µi+1...∂µi+j⊗∂µi+j+1...∂µn/parenrightBigg
+Note, that the twist representation is not a universal property o f the star products.
+For generic star product such a representation is not known.
+Having defined the twist FN, we can define a new coproduct, ∆ = FN∆0F−1
+N, and,
+after suitably extending the algebra A, we can also define twisted symmetries (Poicare,
+diffeomorphism, Yang-Mills, etc.). Twisting the diffeomorphism seems u nnecessary,
+since we have these transformation realized in the standard way. H ere we repeat the
+interpretation oftwisted local symmetries suggested in [32, 22]: tw isted diffeomorphisms
+is what remains from the standard diffeomorphism symmetry when ωand∇are gauge
+fixed to some given values. It is also possible, that twisted symmetrie s are effective low-
+energysymmetries whenthenoncommutativity isdefined andfixed b ysomehigh-energy
+effects.
+In an arbitrary coordinate system the tensors ωµνandlµ1...µnare not constants.
+Therefore, (33) and (37) are not linear combinations of elements o f the tensor square
+ofU(A). To overcome this difficulty, one introduces a set of vector fields ξµ
+a, which
+are analogs of the vielbein fields of Riemannian geometry, that ωµν=ξµ
+aˆωabξν
+bwith
+a constant ˆ ωab. These fields locally describe the transition to a Darboux coordinate
+system. The rest is obvious. For example, ωµν∂µ⊗∂νis replaced by ˆ ωab(ξµ
+a∇µ)⊗(ξν
+b∇ν).
+The Hopf algebra is then generated by the fields ξµ
+a∇µ.Tensor calculus on noncommutative spaces 12
+5. Integration and closeness
+Let us check which of the natural star products are also closed, i.e . when the Equation
+(8) is satisfied by ⋆=⋆N. Letα∈Tm,nandβ∈Tn,m. The following chain of
+transformations is obvious in the Darboux coordinates. We suppos e that all indices of
+αandβare contracted in pairs, making the integrals below diffeomorphism inv ariant./integraldisplay
+Mdµ(x)α⋆Nβ=/integraldisplay
+Mdµ(x)D−1((Dα)⋆R(Dβ))
+=/integraldisplay
+Mdµ(x)(Dα)⋆R(Dβ) =/integraldisplay
+Mdµ(x)(Dα)·(Dβ)
+=/integraldisplay
+Mdµ(x)α·(D†Dβ), (39)
+whereDis given by (34) with (covariantly) constant coefficient. To obtain th e second
+line we neglect all total derivative terms coming from D−1and used closeness of ⋆R,
+which is obvious. The operator D†differs from Dby the signs in front of lµ1...µnwith
+oddn, i.e.
+D†D= exp/parenleftBigg
+2h∞/summationdisplay
+n=1lµ1...µ2n∂µ1...∂µ2n/parenrightBigg
+. (40)
+The integral (39) has to be equal to/integraltext
+dµ(x)α·βfor fairly arbitrary αandβ. This
+implies that Dhas to be unitary, D†D= 1. In other words, ⋆Nis closed iff
+lµ1...µn= 0 for n= 2k. (41)
+We see, that closeness provides further restrictions on the star product.
+An interesting property of the gauge transformations with const ant coefficients is
+that they are, in fact, rigidtransformations. Therefore, they do not require introduction
+of any new gauge fields to become symmetries of the action. Let us c onsider a classical
+action
+S=/integraldisplay
+dµ(x)P(fi,∇)⋆N, (42)
+wherefiare some fields, Pis a polynomial, where all products are ⋆Nproducts. We
+can rewrite Sas
+S=/integraldisplay
+dµ(x)D−1(P(Dfi,∇)⋆R) =/integraldisplay
+dµ(x)P(Dfi,∇)⋆R. (43)
+This means, that the replacement ⋆Nby⋆Ris compensated by the transformation
+fi→Dfi. Since the operator Dis invertible, the theories based on the two star
+products are classically equivalent.
+6. The metric
+Let us discuss how one can incorporate metric in the approach stud ied in this paper.
+The most obvious choice to request the metric to be consistent with the same symplectic
+connection, ∇µgνρ= 0, but this imposes too sever restrictions on the structure involv ed:Tensor calculus on noncommutative spaces 13
+because of the flatness of ∇it allows flat metrics only. A general discussion of the
+compatibility conditions on the Riemann and Poisson structures can b e found in [33].
+Let us reiterate the observation made in [22]: the use of diffeomorph ism covariant
+star products allows to construct gravity theories which are invar iant with respect to the
+standard diffeomorphisms transformations without the need to ma ke them twisted. In
+particular,noncommutativecounterpartsofall2Ddilatongravitie s[34]wereconstructed
+in [22] using the star product (3). These models contain a complex zw eibeineµ, a
+spin-connection, a dilaton, and auxiliary fields. Apart from the diffeo morphisms, these
+models are invariant under noncommutative U(1)⋆gauge transformations, which are
+deformations of the Euclidean Lorentz symmetries:
+δeµ=iλ⋆eµ, δ¯eµ=−i¯eµ⋆λ, (44)
+The metric
+gµν=1
+2(¯eµ⋆eν+ ¯eν⋆eµ), (45)
+is real and invariant under the U(1)⋆transformations.
+One of the models, corresponding to the conformally transformed string black hole,
+appeared to be integrable [22]. The solution for the zweibein in that mo del reads:
+eµ=u⋆∇µE,¯eµ=∇µ¯E ⋆u−1, (46)
+whereuis aU(1)⋆field which is canceled in the metric (45) and can be gauge-fixed
+tou= 1.Eis an arbitrary complex scalar field corresponding to the diffeomorph ism
+freedom. In the coordinates z1= ReE,z2= ImEthe zweiben is a constant unit
+matrix. (In the commutative case this led to trivialization of the geom etry). However,
+these coordinates ( z1,z2) need not be the Darboux coordinates of the symplectic
+structure. Therefore, the geometric structure on the noncom mutative model may be
+very nontrivial.
+Since the system possesses a full diffeomorphism invariance, the so lutions may be
+analyzed in any coordinate system. It is convenient to choose the D arboux coordinates,
+where the star product looks particularly simple (it coincides with the Moyal product):
+α⋆β= exp(iθ(∂x
+1∂y
+2−∂x
+2∂y
+1)α(x)β(y)|y=x,. (47)
+Hereθis a constant coefficient. We are using the physical units with an imagin ary
+deformation parameter ( h=iθ).
+To give an example of possible behavior of the metric, let us choose th e arbitrary
+function Ein the form
+E= sin(x1)+isin(x2). (48)
+This form is simple enough to allow explicit calculation of the star produc ts. On the
+other hand, it gives rise to a rich geometric structure. The metric is easy to calculate
+gµν=/parenleftBigg
+cos2x1−sinθsinx1sinx2
+−sinθsinx1sinx2cos2x2/parenrightBigg
+. (49)Tensor calculus on noncommutative spaces 14
+The determinant of this metric
+detgµν= cos2x1cos2x2−sin2θsin2x1sin2x2(50)
+is positive for small values of x1andx2, but changes the sign as x1andx2grow!
+To describe the geometry at small x1andx2it is convenient to introduce new
+coordinates z1= sinx1,z2= sinx2. In these coordinates the line element takes the
+form
+(ds)2= (dz1)2+(dz2)2−sinθf(z1)f(z2)dz1dz2, (51)
+wheref(z) =z/√
+1−z2. To identify the geometry corresponding to (51) one should
+calculate a diffeomorphism invariant corresponding to (51). The com mutative Riemann
+tensor reads
+R1
+212=−sinθf′(z1)f′(z2) (52)
+where we used the approximation of small θ, which is valid for small z1,z2. It is easy
+to see, that in this region, f′≃1, one has a constant curvature space. The sign of the
+curvature is defined by the sign of the noncommutativity paramete rθ.
+We see, that the geometry obtained is indeed very non-trivial. The s tructure of the
+solution is not described by the Cartan sector ( eµ) alone (which is gauge trivial), and
+not by the Poisson sector (the star product) alone (which is reduc ible to Moyal in some
+coordinates), but rather by a relation between these two sector s.
+A similar result has been obtained recently in [35], where it was shown th at even a
+very simple action (just a cosmological constant) leads to a large va riety of the solutions
+on a noncommutative plane.
+Therefore, to construct a noncommutative gravity theory it is no t enough to
+present an action for the gravity sector. One has to formulate an action principle
+which restricts /ba∇dblalso the symplectic structure ωµν
+Rand the symplectic connection ∇.
+Fortunately, since the Kontsevich gauge transformations with co nstantlµ1...µnlead to
+classically equivalent theories (see Sec. 5), there are no more relev ant parameters in
+the star product. Otherwise, one should have introduce a dynamic al equation for each
+parameter. This also implies, that a theory based on the generic sta r product with
+unrestricted gauge freedom requires an infinite number of equatio ns of motion, which
+makes such a theory meaningless.
+A similar problem exists also in theapproach to noncommutative gravit ies based on
+twisted diffeomorphism symmetries [20]. In this approach, one has to pre-fix a relation
+between the metric and noncommutativity, or between the star pr oduct and the Killing
+vectors of the metric, see [37].
+7. Conclusions
+In this paper, we have analyzed covariant star products on the sp ace of tensor fields
+over a Fedosov manifold with a given symplectic structure and a given flat torsionless
+/ba∇dblA relation between the metric and the symplectic structure can be d erived from the matrix models
+[36].Tensor calculus on noncommutative spaces 15
+symplectic connection. We have demonstrated, that although the space of star products
+has an infinite functional dimension, this space can be reduced to a s pace of functional
+dimension one after imposing some natural conditions (Theorem 3.1) . The products ⋆N
+satisfying these conditions, and, therefore, called natural star products, are constructed
+in the following way. First, take the symplectic structure ωµνand add arbitrary
+covariantly constant terms ωµν
+jmultiplied with h2j, see (23). Then, with this new
+symplectic structure ωµν
+Rone constructs a star product ⋆R, which is nothing else then
+a covariantization of the Moyal product. Physical interpretation of this procedure of
+constant renormalization of the symplectic structure is clear. Sinc e we are working in
+the framework of formal expansions, there is no way to ascribe a n umerical value to the
+deformation parameter h. In a more physical setup, h=iθ, whereθis measurable,
+at least in principle. One can sum up the expansion (23) and impose on ωµν
+Ra
+normalization condition, as is usually done in quantum field theory. For example, this
+may beωµν
+R=ωµνfor the physical value of the deformation parameter. The second step
+consists in the Kontsevich gauge transformation (28) of ⋆Rwith covariantly constant
+coefficients in front of the differential operators contained in D, which led us finally
+to⋆N. The geometrical meaning of this transformation and its physical in terpretation
+remain unclear.
+We have studied some basic properties of the natural star produc ts. We have shown
+thatallsuchproductscanberealizedthroughatwistonaHopfalge braandpresentedan
+explicitconstructionofthetwistelement. Wedemonstratedthatt heclosenessofthestar
+product imposes further restrictions on the parameters of the g auge transformations.
+Furthermore, we have shown that classically field theories based on ⋆Rand⋆Nare
+equivalent. This means that classically ωRand∇are the only relevant parameters in
+the star product.
+Let us now discuss the meaning of conditions S1-S6 (see Sec. 2) whic h we imposed
+on the star product. The conditions S1 – S3 (existence of the form al expansion,
+associativity, andrelationtothePoissonstructure)arerathers traightforwardextensions
+to all tensors of the conditions which are always imposed on the star products of scalar
+fields in the deformation quantization approach. Later on we reque st that the Poisson
+bracket is common for all tensors, which means that the noncommu tative structure
+of the manifold is universal and does not depend on the rank of the t ensor – an
+analog of universality of the metric in General Relativity. The conditio n S5 (the Moyal
+symmetry) ensures hermiticity of the star product for physics (im aginary) values of the
+deformation parameter h. The condition S4 (stability on covariantly constant tensors)
+means that slowly varying fields must not feel noncommutativity of t he manifold. From
+the mathematical point of view, S4 is a generalization of the condition of stability of
+unity. The condition S6 (the derivation), which includes derivatives, relates the star
+products calculated in nearby points (since ∇may be considered as a generator of
+infinitesimal translations). S4 and S6 are important properties of lo cal products which
+seem to be natural to keep in the nonlocal (noncommutative) case as well. These
+conditions appeared to be almost the same restrictive as locality, as we demonstratedTensor calculus on noncommutative spaces 16
+above.
+Covariant star product were also studied in [9, 10, 12, 13, 14, 15]. T he aim of
+these works was to make the star products as general as possible . Therefore, the
+restrictions considered above were not imposed. An interesting ma nifestly covariant
+construction of a star product of scalar functions was proposed in [38]. Finally, other
+types of noncommutativity, as the ones following from matrix algebr as, may also be
+covariantized and extended at least to differential forms [39].
+Wehavealsoconsidereda2Dgravitymodel[22]basedonthecovaria ntstarproduct.
+Despite the gauge triviality of the zweibein, the metric in this model ex hibits a very
+non-trivial (and, in fact, rather wild) behavior. This simple example h as demonstrated
+that the parameters in the star product should be fixed by indepen dent equations of
+motion – and this is only possible if the number of local parameters is re duced, as, for
+example, in the approach advocated in the present work.
+There is a number of possible further developments of the approac h reported here.
+The most important one seems to be an extension of the scheme to a rbitrary Poisson
+manifolds.
+Acknowledgments
+I am grateful to Shannon McCurdy and Markku Oksanen for corre spondence regarding
+covariant star products. This work was supported in part by CNPq and FAPESP.
+Appendix A. Commutative products
+Let us describe briefly commutative deformations of the point wise p roduct. In this case
+all calculations are considerably simpler. We put ωµν= 0 (at all orders of h), so that
+the manifold Mis no longer a symplectic one. Nevertheless, the product (3) exists and
+is just the usual point wise product α·β. Let us now fix a flat torsionless connection
+∇, which is no longer restricted to being consistent with any symplectic structure. We
+may define a new non-local covariant product as
+α∗β=D−1((Dα)·(Dβ)), (A.1)
+whereDis precisely as in (10). This product is obviously associative, so that S 1 and
+S2 are satisfied, and S3 defines a trivial Poisson structure. Next, we impose S4, which
+requires stability of the product on covariantly constant tensors . By repeating the
+calculations from the main text, we easily reproduce Eq. (16) which m eans that all
+Lµ1...µnare proportional to unit operators on the spaces of tensors of a fixed rank.
+Now we may conclude that the product (A.1) is commutative. Althoug h the inner
+automorphisms (12) act trivially if ⋆=·, the point wise product is diffeomorphism
+invariant, and we can exclude from the gauge transformations all Lhaving a non-zero
+first order part Lµ. The analysis of the higher order terms proceeds exactly as above
+yielding the condition (27).Tensor calculus on noncommutative spaces 17
+The integration measure (7) is not defined in the commutative case, but it is
+enough to take dµ(x) =dxin a coordinate system, where ∇=∂. Then we may
+repeat the analysis of Sec. 5 and conclude that all admissible produc ts obtained by
+gauge transformations from the ∗-product (A.1) correspond to classically equivalent
+actions. In other words, if one relaxes the locality assumption but im poses instead the
+set of ”natural” conditions described above, the resulting field the ories are classically
+equivalent to that with the local point wise product.
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\ No newline at end of file
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+Algorithms for Quantum Computers
+Jamie Smith and Michele Mosca
+1 Introduction
+Quantum computing is a new computational paradigm created by reformulating in-
+formation and computation in a quantum mechanical framework [30, 27]. Since the
+laws of physics appear to be quantum mechanical, this is the most relevant frame-
+work to consider when considering the fundamental limitations of information pro-
+cessing. Furthermore, in recent decades we have seen a major shift from just observ-
+ing quantum phenomena to actually controlling quantum mechanical systems. We
+have seen the communication of quantum information over long distances, the “tele-
+portation” of quantum information, and the encoding and manipulation of quantum
+information in many different physical media. We still appear to be a long way from
+the implementation of a large-scale quantum computer, however it is a serious goal
+of many of the world’s leading physicists, and progress continues at a fast pace.
+In parallel with the broad and aggressive program to control quantum mechan-
+ical systems with increased precision, and to control and interact a larger number
+of subsystems, researchers have also been aggressively pushing the boundaries of
+what useful tasks one could perform with quantum mechanical devices. These in-
+Jamie Smith
+Institute for Quantum Computing and Dept. of Combinatorics & Optimization
+University of Waterloo,
+with support from the Natural Sciences and Engineering Research Council of Canada
+e-mail: ja5smith@iqc.ca
+Michele Mosca
+Institute for Quantum Computing and Dept. of Combinatorics & Optimization
+University of Waterloo and St. Jerome’s University,
+and Perimeter Institute for Theoretical Physics,
+with support from the Government of Canada, Ontario-MRI, NSERC, QuantumWorks, MITACS,
+CIFAR, CRC, ORF, and DTO-ARO
+e-mail: mmosca@iqc.ca
+1arXiv:1001.0767v2  [quant-ph]  7 Jan 20102 Jamie Smith and Michele Mosca
+clude improved metrology, quantum communication and cryptography, and the im-
+plementation of large-scale quantum algorithms.
+It was known very early on [27] that quantum algorithms cannot compute func-
+tions that are not computable by classical computers, however they might be able to
+efficiently compute functions that are not efficiently computable on a classical com-
+puter. Or, at the very least, quantum algorithms might be able to provide some sort
+of speed-up over the best possible or best known classical algorithms for a specific
+problem.
+The purpose of this paper is to survey the field of quantum algorithms, which has
+grown tremendously since Shor’s breakthrough algorithms [57, 58] over 15 years
+ago. Much of the work in quantum algorithms is now textbook material (e.g. [54,
+39, 45, 43, 49] ), and we will only briefly mention these examples in order to provide
+a broad overview. Other parts of this survey, in particular, sections 4 and 5, give a
+more detailed description of some more recent work.
+We organized this survey according to an underlying tool or approach taken, and
+include some basic applications and specific examples, and relevant comparisons
+with classical algorithms.
+In section 2 we begin with algorithms one can naturally consider to be based
+on a quantum Fourier transform, which includes the famous factoring and discrete
+logarithm algorithms of Peter Shor [57, 58]. Since this topic is covered in several
+textbooks and recent surveys, we will only briefly survey this topic. We could have
+added several other sections on algorithms for generalizations of these problems,
+including several cases of the non-Abelian hidden subgroup problem, and hidden
+lattice problems over the reals (which have important applications in number the-
+ory), however these are covered in the recent survey [52] and in substantial detail in
+[22].
+We continue in section 3 with a brief review of classic results on quantum search-
+ing and counting, and more generally amplitude amplification and amplitude esti-
+mation.
+In section 4 we discuss algorithms based on quantum walks. We will not cover
+the related topic of adiabatic algorithms, which was briefly summarized in [52]; a
+broader survey of this and related techniques (“quantum annealing”) can be found
+in [26].
+We conclude with section 5 on algorithms based on the evaluation of the trace
+of an operator, also referred to as the evaluation of a tensor network, and which has
+applications such as the approximation of the Tutte polynomial.
+The field of quantum algorithms has grown tremendously since the seminal work
+in the mid-1990s, and a full detailed survey would simply be infeasible for one ar-
+ticle. We could have added several other sections. One major omission is the de-
+velopment of algorithms for simulating quantum mechanical systems, which was
+Feynman’s original motivation for proposing a quantum computer. This field was
+briefly surveyed in [52], with emphasis on the recent results in [13] (more recent
+developments can be found in [63]). This remains an area worthy of a comprehen-
+sive survey; like the many other areas we have tried to survey, it is difficult because
+it is still an active area of research. It is also an especially important area becauseAlgorithms for Quantum Computers 3
+these algorithms tend to offer a fully exponential speed-up over classical algorithms,
+and thus are likely to be among the first quantum algorithms to be implemented that
+will offer a speed-up over the fastest available classical computers.
+Lastly, one could also write a survey of quantum algorithms for intrinsically
+quantum information problems, like entanglement concentration, or quantum data
+compression. We do not cover this topic in this article, though there is a very brief
+survey in [52].
+One can find a rather comprehensive list of the known quantum algorithms (up to
+mid 2008) in Stephen Jordan’s PhD thesis [41]. We hope this survey complements
+some of the other recent surveys in providing a reasonably detailed overview of the
+current state of the art in quantum algorithms.
+2 Algorithms based on the Quantum Fourier transform
+The early line of quantum algorithms was developed in the “black-box” or “ora-
+cle” framework. In this framework part of the input is a black-box that implements
+a function f(x), and the only way to extract information about fis to evaluate it
+on inputs x. These early algorithms used a special case of quantum Fourier trans-
+form, the Hadamard gate, in order solve the given problem with fewer black-box
+evaluations of fthan a classical algorithm would require.
+Deutsch [27] formulated the problem of deciding whether a function f:f0;1g!
+f0;1gwas constant or not. Suppose one has access to a black-box that implements f
+reversibly by mapping x;07!x;f(x); let us further assume that the black box in fact
+implements a unitary transformation Ufthat mapsjxij0i7!jxijf(x)i. Deutsch’s
+problem is to output “constant” if f(0) =f(1)and to output “balanced” if f(0)6=
+f(1), given a black-box for evaluating f. In other words determine f(0)f(1)
+(wheredenotes addition modulo 2). Outcome “0” means fis constant and “1”
+means fis not constant.
+A classical algorithm would need to evaluate ftwice in order to solve this prob-
+lem. A quantum algorithm can apply Ufonly once to create
+1p
+2j0ijf(0)i+1p
+2j1ijf(1)i:
+Note that if f(0) =f(1), then applying the Hadamard gate to the first register
+yieldsj0iwith probability 1, and if f(0)6=f(1), then applying the Hadamard gate
+to the first register and ignoring the second register leaves the first register in the
+statej1iwith probability1
+2; thus a result ofj1icould only occur if f(0)6=f(1).
+As an aside, let us note that in general, given
+1p
+2j0ijy0i+1p
+2j1ijy1i4 Jamie Smith and Michele Mosca
+applying the Hadamard gate to the first qubit and measuring it will yield “0” with
+probability1
+2+Re(hy0jy1i); this “Hadamard test” is discussed in more detail in
+section 5.
+In Deutsch’s case, measuring a “1” meant f(0)6=f(1)with certainty, and a “0”
+was an inconclusive result. Even though it wasn’t perfect, it still was something that
+couldn’t be done with a classical algorithm. The algorithm can be made exact [24]
+(i.e. one that outputs the correct answer with probability 1) if one assumes further
+thatUfmapsjxijbi7!jxijbf(x)i, for b2f0;1g, and one sets the second qubit
+to1p
+2j0i1p
+2j1i. Then Ufmaps
+j0i+j1ip
+2j0ij1ip
+2
+7!(1)f(0) 
+j0i+(1)f(0)f(1)j1ip
+2!j0ij1ip
+2
+:
+Thus a Hadamard gate on the first qubit yields the result
+(1)f(0)jf(0)f(1)ij0ij1ip
+2
+and measuring the first register yields the correct answer with certainty.
+The general idea behind the early quantum algorithms was to compute a black-
+box function fon a superposition of inputs, and then extract a global property of f
+by applying a quantum transformation to the input register before measuring it. We
+usually assume we have access to a black-box that implements
+Uf:jxijbi7!jxijbf(x)i
+or in some other form where the input value xis kept intact and the second register
+is shifted by f(x)is some reversible way.
+Deutsch and Jozsa [28] used this approach to get an exact algorithm that decides
+whether f:f0;1gn7!f0;1gis constant or “balanced” (i.e. jf1(0)j=jf1(1)j),
+with a promise that one of these two cases holds. Their algorithm evaluated f
+only twice, while classically any exact algorithm would require 2n1+1 queries
+in the worst-case. Bernstein and Vazirani [12] defined a specific class of such
+functions fa:x7!a
x, for any a2f0;1gn, and showed how the same algorithm
+that solves the Deutsch-Jozsa problem allows one to determine awith two evalua-
+tions of fawhile a classical algorithm requires nevaluations. (Both of these algo-
+rithms can be done with one query if we have jxijbi7!jxijbf(x)i.) They further
+showed how a related “recursive Fourier sampling” problem could be solved super-
+polynomially faster on a quantum computer. Simon [59] later built on these tools to
+develop a black-box quantum algorithm that was exponentially faster than any clas-
+sical algorithm for finding a hidden string s2f0;1gnthat is encoded in a function
+f:f0;1gn!f0;1gnwith the property that f(x) =f(y)if and only if x=ys.
+Shor [57, 58] built on these black-box results to find an efficient algorithm for
+finding the order of an element in the multiplicative group of integers modulo N
+(which implies an efficient classical algorithm for factoring N) and for solving theAlgorithms for Quantum Computers 5
+discrete logarithm problem in the multiplicative group of integers modulo a large
+prime p. Since the most widely used public key cryptography schemes at the time
+relied on the difficulty of integer factorization, and others relied on the difficulty
+of the discrete logarithm problem, these results had very serious practical impli-
+cations. Shor’s algorithms straightforwardly apply to black-box groups, and thus
+permit finding orders and discrete logarithms in any group that is reasonably pre-
+sented, including the additive group of points on elliptic curves, which is currently
+one of the most widely used public key cryptography schemes (see e.g. [48]).
+Researchers tried to understand the full implications and applications of Shor’s
+technique, and a number of generalizations were soon formulated (e.g. [15, 36]).
+One can phrase Simon’s algorithm, Shor’s algorithm, and the various generaliza-
+tions that soon followed as special cases of the hidden subgroup problem . Consider
+a finitely generated Abelian group G, and a hidden subgroup Kthat is defined by
+a function f:G!X(for some finite set X) with the property that f(x) =f(y)if
+and only if xy2K(we use additive notation, without loss of generality). In other
+words, fis constant on cosets of Kand distinct on different cosets of G. In the case
+of Simon’s algorithm, G=Zn
+2andK=f0;sg. In the case of Shor’s order-finding
+algorithm, G=ZandK=rZwhere ris the unknown order of the element. Other
+examples and how they fit in the hidden subgroup paradigm are given in [51].
+Soon after, Kitaev [46] solved a problem he called the Abelian stabilizer prob-
+lemusing an approach that seemed different from Shor’s algorithm, one based in
+eigenvalue estimation. Eigenvalue estimation is in fact an algorithm of independent
+interest for the purpose of studying quantum mechanical systems. The Abelian sta-
+bilizer problem is also a special case of the hidden subgroup problem. Kitaev’s idea
+was to turn the problem into one of estimating eigenvalues of unitary operators. In
+the language of the hidden subgroup problem, the unitary operators were shift op-
+erators of the form f(x)7!f(x+y). By encoding the eigenvalues as relative phase
+shifts, he turned the problem into a phase estimation problem.
+The Simon/Shor approach for solving the hidden subgroup problem is to first
+compute åxjxijf(x)i. In the case of finite groups G, one can sum over all the el-
+ements of G, otherwise one can sum over a sufficiently large subset of G. For ex-
+ample, if G=Z, and f(x) =axmod Nfor some large integer N, we first compute
+å2n1
+x=0jxijaxi, where 2n>N2(we omit the “ mod N” for simplicity). If ris the order
+ofa(i.e.ris the smallest positive integer such that ar1) then every value xof the
+form x=y+krgets mapped to ay. Thus we can rewrite the above state as
+2n1
+å
+x=0jxijaxi=r1
+å
+y=0(å
+jjy+jri)jayi (1)
+where each value of ayin this range is distinct. Tracing out the second register we
+thus are left with a state of the form
+å
+jjy+jri6 Jamie Smith and Michele Mosca
+for a random yand where jgoes from 0 tob(2n1)=rc. We loosely refer to this state
+as a “periodic” state with period r. We can use the inverse of the quantum Fourier
+transform (or the quantum Fourier transform) to map this state to a state of the form
+åxaxjxi, where the amplitudes are biased towards values of xsuch that x=2nk=r.
+With probability at least4
+p2we obtain an xsuch thatjx=2nk=rj1=2n+1. One
+can then use the continued fractions algorithm to find k=r(in lowest terms) and
+thus find rwith high probability. It is important to note that the continued fractions
+algorithm is not needed for many of the other cases of the Abelian hidden subgroup
+considered, such as Simon’s algorithm or the discrete logarithm algorithm when the
+order of the group is already known.
+In contrast, Kitaev’s approach for this special case was to consider the map Ua:
+jbi7!jbaxi. It has eigenvalues of the form e2pik=r, and the statej1isatisfies
+j1i=1prr1
+å
+k=0jyki
+wherejykiis an eigenvector with eigenvalue e2pik=r:
+Ua:jyki7!e2pikx=rjyki:
+If we consider the controlled- Ua, denoted cUa, which mapsj0ijbi7!j0ijbiand
+j1ijbi7!j1ijbai, and if we apply it to the state (j0i+j1i)jykiwe get
+(j0i+j1i)jyki7!(j0i+e2pik=rj1i)jyki:
+In other words, the eigenvalue becomes a relative phase, and thus we can reduce
+eigenvalue estimation to phase estimation. Furthermore, since we can efficiently
+compute a2jby performing jmultiplications modulo N, one can also efficiently
+implement cUa2jand thus easily obtain the qubit j0i+e2pi2j(k=r)j1ifor integer
+values of jwithout performing cUaa total of 2jtimes. Kitaev developed an effi-
+cient ad hoc phase estimation scheme in order to estimate k=rto high precision, and
+this phase estimation scheme could be optimized further [24]. In particular, one can
+create the state
+(j0i+j1i)nj1i=2n1
+å
+x=0jxij1i=r1
+å
+k=02n1
+å
+x=0jxijyki
+(we use the standard binary encoding of the integers x2f0;1;:::; 2n1gas bits
+strings of length n) the apply the cUa2jusing the (nj)th bit as the control bit,
+and using the second register (initialized in j1i=åkjyki) as the target register, for
+j=0;1;2;:::; n1, to create
+åk(j0i+e2pi2n1k
+rj1i)


(j0i+e2pi2k
+rj1i)(j0i+e2pik
+rj1i)jyki (2)
+=å2n1
+x=0jxijaxi=år1
+k=0å2n1
+x=0e2pixk
+rjxijyki: (3)Algorithms for Quantum Computers 7
+If we ignore or discard the second register, we are left with a state of the form
+å2n1
+x=0e2pixk=rjxifor a random value of k2f0;1;:::; r1g. The inverse quantum
+Fourier transformation maps this state to a state åyayjyiwhere most of weight
+of the amplitudes is near values of ysuch that y=2nk=rfor some integer k.
+More specificallyjy=2nk=rj1=2n+1with probability at least 4 =p2; furthermore
+jy=2nk=rj1=2nwith probability at least 8 =p2. As in the case of Shor’s algo-
+rithm, one can use the continued fractions algorithm to determine k=r(in lowest
+terms) and thus determine rwith high probability.
+It was noted [24] that this modified eigenvalue estimation algorithm for order-
+finding was essentially equivalent to Shor’s period-finding algorithm for order-
+finding. This can be seen by noting that we have the same state in Equation 1 and
+Equation 2, and in both cases we discard the second register and apply an inverse
+Fourier transform to the first register. The only difference is the basis in which the
+second register is mathematically analyzed.
+The most obvious direction in which to try to generalized the Abelian hid-
+den subgroup algorithm is to solve instances of the hidden subgroup problem for
+non-Abelian groups. This includes, for example, the graph automorphism problem
+(which corresponds to finding a hidden subgroup of the symmetric group). There
+has been non-trivial, but limited, progress in this direction, using a variety of al-
+gorithmic tools, such as sieving, “pretty good measurements” and other group the-
+oretic approaches. Other generalizations include the hidden shift problem and its
+generalizations, hidden lattice problems on real lattices (which has important appli-
+cations in computational number theory and computationally secure cryptography),
+and hidden non-linear structures. These topics and techniques would take several
+dozen pages just to summarize, so we refer the reader to [52] or [22], and leave
+more room to summarize other important topics.
+3 Amplitude Amplification and Estimation
+A very general problem for which quantum algorithms offer an improvement is that
+of searching for a solution to a computational problem in the case that a solution x
+can be easily verified. We can phrase such a general problem as finding a solution
+xto the equation f(x) =1, given a means for evaluating the function f. We can
+further assume that f:f0;1gn7!f0;1g.
+The problems from the previous section can be rephrased in this form (or a small
+number of instances of problems of this form), and the quantum algorithms for these
+problems exploit some non-trivial algebraic structure in the function fin order to
+solve the problems superpolynomially faster than the best possible or best known
+classical algorithms. Quantum computers also allow a more modest speed-up (up to
+quadratic) for searching for a solution to a function fwithout any particular struc-
+ture. This includes, for example, searching for solutions to NP-complete problems.
+Note that classically, given a means for guessing a solution xwith probability p,
+one could amplify this success probability by repeating many times, and after a num-8 Jamie Smith and Michele Mosca
+ber of guess in O(1=p), the probability of finding a solution is in W(1). Note that the
+quantum implementation of an algorithm that produces a solution with probability
+pwill produce a solution with probability amplitudepp. The idea behind quan-
+tum searching is to somehow amplify this probability to be close to 1 using only
+O(1=pp)guesses and other steps.
+Lov Grover [37] found precisely such an algorithm, and this algorithm was
+analyzed in detail and generalized [16, 17, 38, 18] to what is known as “am-
+plitude amplification.” Any procedure for guessing a solution with probability p
+can be (with modest overhead) turned into a unitary operator Athat mapsj0i7!ppjy1i+p1pjy0i, where 0=00:::0,jy1iis a superposition of states encod-
+ing solutions xtof(x) =1 (the states could in general encode xfollowed by other
+“junk” information) and jy0iis a superposition of states encoding values of xthat
+are not solutions.
+One can then define the quantum search iterate (or “Grover iterate”) to be
+Q=AU0A1Uf
+where Uf:jxi7!(1)f(x)jxi, and U0=I2j0ih0j(in other words, maps j0i7!
+j0iandjxi7!jxifor any x6=0). Here we are for simplicity assuming there are
+no “junk” bits in the unitary computation of fbyA. Any such junk information can
+either be “uncomputed” and reset to all 0s, or even ignored (letting Ufact only on
+the bits encoding xand applying the identity to the junk bits).
+This algorithm is analyzed thoroughly in the literature and in textbooks, so we
+only summarize the main results and ideas that might help understand the later sec-
+tions on searching via quantum walk.
+If one applies Qa total of ktimes to the input state Aj00:::0i=sin(q)jy1i+
+cos(q)jy0i, wherepp=sin(q), then one obtains
+QkAj00:::0i=sin((2k+1)q)jy1i+cos((2k+1)q)jy0i:
+This implies that with kp
+4ppone obtainsjy1iwith probability amplitude close
+to 1, and thus measuring the register will yield a solution to f(x) =1 with high
+probability. This is quadratically better than what could be achieved by preparing
+and measuring Aj0iuntil a solution is found.
+One application of such a generic amplitude amplification method is for search-
+ing. One can also apply this technique to approximately count [19] the number of
+solutions to f(x) =1, and more generally to estimate the amplitude [18] with which
+a general operator Aproduces a solution to f(x) =1 (in other words, the transition
+amplitude from one recognizable subspace to another).
+There are a variety of other applications of amplitude amplification that cleverly
+incorporate amplitude amplification in a non-obvious way into a larger algorithm.
+Some of these examples are discussed in [52] and most are summarized at [41].
+Since there are some connections to some of the more recent tools developed in
+quantum algorithms, we will briefly explain how amplitude estimation works.Algorithms for Quantum Computers 9
+Consider any unitary operator Athat maps some known input state, say j0i=
+j00:::0i, to a superposition sin (q)jy1i+cos(q)jy0i, 0qp=2, wherejy1i
+is a normalized superposition of “good” states xsatisfying f(x) =1 andjy0iis a
+normalized superposition of “bad” states xsatisfying f(x) =0 (again, for simplic-
+ity we ignore extra junk bits). If we measure Aj0i, we would measure a “good” x
+with probability sin2(q). The goal of amplitude estimation is to approximate the
+amplitude sin (q). Let us assume for convenience that there are t>0 good states
+andnt>0 bad states, and that 0 <sin(q)<p=2.
+We note that the quantum search iterate Q=AU0A1Ufhas eigenvaluesjy+i=
+1p
+2(jy0i+ijy1i)andjyi=1p
+2(jy0iijy1i)with respective eigenvalues ei2qand
+ei2q. It also has 2n2 other eigenvectors; t1 of them have eigenvalue +1 and
+are the states orthogonal to jy1ithat have support on the states jxiwhere f(x) =1,
+andnt1 of them have eigenvalue 1 and are the states orthogonal to jy0ithat
+have support on the states jxiwhere f(x) =0. It is important to note that Ajyi=
+eiqp
+2jy+i+eiqp
+2jyihas its full support on the two dimensional subspace spanned
+byjy+iandjyi.
+It is worth noting that the quantum search iterate Q=AU0A1Ufcan also be
+thought of as two reflections
+UAj0iUf= (2Aj0ih0jA†I)(I2å
+xjf(x)=1jxihxj);
+one which flags the “good” subspace with a 1 phase shift, and then one that flags
+the subspace orthogonal to Aj0iwith a1 phase shift. In the two dimensional sub-
+space spanned by Aj0iandUfAj0i, these two reflections correspond to a rotation by
+angle 2 q. Thus it should not be surprising to find eigenvalues e2piqfor states in this
+two dimensional subspace. (In the section on quantum walks, we’ll consider a more
+general situation where we have an operator Qwith many non-trivial eigenspaces
+and eigenvalues.)
+Thus one can approximate sin (q)by estimating the eigenvalue of either jy+i
+orjyi. Performing the standard eigenvalue estimation algorithm on Qwith input
+Aj0i(as illustrated in Figure 1) gives a quadratic speed-up for estimating sin2(q)
+versus simply repeatedly measuring Aj0iand counting the frequency of 1s. In par-
+ticular, we can obtain an estimate ˜qofqsuch thatj˜qqj<ewith high (constant)
+probability using O(1=e)repetitions of Q, and thus O(1=e)repetitions of Uf,Aand
+A1. For fixed p, this implies that ˜ p=sin2(˜q)satisfiesjp˜pj2O(e)with high
+probability. Classically sampling requires O(1=e2)samples for the same precision.
+One application is speeding up the efficiency of the “Hadamard” test mentioned in
+section 5.2.3
+Another interesting observation is that increasingly precise eigenvalue estimation
+ofQon input Aj0ileaves the eigenvector register in a state that gets closer and closer
+to the mixture1
+2jy+ihy+j+1
+2jyihyjwhich equals1
+2jy1ihy1j+1
+2jy0ihy0j.
+Thus eigenvalue estimation will leave the eigenvector register in a state that contains
+a solution to f(x) =1 with probability approaching1
+2. One can in fact using this
+entire algorithm as subroutine in another quantum search algorithm, and obtain an10 Jamie Smith and Michele Mosca
+QFT M'&%$
+ !"#QFT1
+MrrrrLLLL
+rrrrLLLL
+j0i
+mrrrrLLLLy
+...rrrrLLLL
+A Qxj0i
+n
+...
+Fig. 1 The above circuit estimates the amplitude with which the unitary operator Amapsj0ito
+the subspace of solutions to f(x) =1. One uses m2W(log(1=e))qubits in the top register, and
+prepares it in a uniform superposition of the strings yrepresenting the integers 0 ;1;2;:::; 2m1
+(one can in fact optimize the amplitudes of each yto achieve a slightly better estimate [33]). The
+controlled- Qycircuit applies Qyto the bottom register when the value yis encoded in the top qubits.
+IfAj0i=sin(q)jy1i+cos(q)jy0i, wherejy1iis a superposition of solutions xtof(x) =1, and
+jy0iis a superposition of values xwith f(x) =0, then the value y=y1y2:::ymmeasured in the top
+register corresponds to the phase estimate 2 py=2mwhich is likely to be within2p
+2m(modulo 2 p) of
+either 2 qor2q. Thus the value of sin2py=2mis likely to satisfyjsin2py=2msin2qj2O(e).
+algorithm with success probability approaching 1 [50, 43], and the convergence
+rate can be improved further [61]. Another important observation is that for the
+purpose of searching, the eigenvalue estimate register is never used except in order
+to determine a random number of times in which to apply Qto the second register.
+This in fact gives the quantum searching algorithm of [16]. In other words, applying
+Qa random number of times decoheres or approximately projects the eigenvector
+register in the eigenbasis, which gives a solution with high probability. The method
+of approximating projections by using randomized evolutions was recently refined,
+generalized, and applied in [14].
+Quantum searching as discussed in this section has been further generalized in
+the quantum walk paradigm, which is the topic of the next section.Algorithms for Quantum Computers 11
+QFT M
+QFT1
+M j0i
+m
+...
+=
+QxrrrrLLLL
+rrrrLLLL
+j0i
+nrrrrLLLLy
+...rrrrLLLLQxrrrrLLLL
+rrrrLLLL
+j0i
+nrrrrLLLLy
+...rrrrLLLL
+forx2f1;2;:::;2m1g
+Fig. 2 The amplitude estimation circuit can also be used for searching for a solution to f(x) =1,
+as illustrated in the figure on the left. The second register starts off in the state Aj0i, and after
+a sufficiently precise eigenvalue estimation of the quantum search iterate Q, the second regis-
+ter is approximately projected in the eigenbasis. The idea projection would yield the mixed state
+1
+2jy+ihy+j+1
+2jyihyj=1
+2jy1ihy1j+1
+2jy0ihy0j, and thus yields a solution to f(x) =1 with
+probability approaching1
+2asmgets large (the probability is in W(1)once m2W(1=pp)). The
+same algorithm can in fact be achieved by discarding the top register and instead randomly picking
+a value y2f0;1;:::; 2m1gand applying QytoAj0i, as illustrated on the right.
+4 Quantum Walks
+Random walks are a common tool throughout classical computer science. Their
+applications include the simulation of biological, physical and social systems, as
+well as probabilistic algorithms such as Monte Carlo methods and the Metropolis
+algorithm. A classical random walk is described by a nnmatrix P, where the entry
+Pu;vis the probability of a transition from a vertex vto an adjacent vertex uin a graph
+G. In order to preserve normalization, we require that Pisstochastic — that is, the
+entries in each column must sum to 1. We denote the initial probability distribution
+on the vertices of Gby the column vector q. After nsteps of the random walk, the
+distribution is given by Pnq.
+Quantum walks were developed in analogy to classical random walks, but it was
+not initially obvious how to do this. Most generally, a quantum walk could begin in
+an initial state r0=jy0ihy0jand evolve according to any completely positive map
+Esuch that, after ttime steps, the system is in state r(t) =Et(r0). Such a quantum12 Jamie Smith and Michele Mosca
+walk is simultaneously using classical randomness and quantum superposition. We
+can focus on the power of quantum mechanics by restricting to unitary walk opera-
+tions, which maintain the system in a coherent quantum state. So, the state at time
+tcan be described by jy(t)i=Utjy0i, for some unitary operator U. However, it
+is not initially obvious how to define such a unitary operation. A natural idea is to
+define the state space Awith basisfjvi:v2V(G)gand walk operator ˜Pdefined by
+˜Pu;v=pPu;v. However, this will not generally yield a unitary operator ˜P, and a more
+complex approach is required. Some of the earliest formulations of unitary quantum
+walks appear in papers by Nayak and Vishwanath [53], Ambainis et al. [8], Kempe
+[44], and Aharonov et al. [2]. These early works focused mainly on quantum walks
+on the line or a cycle. In order to allow unitary evolution, the state space consisted
+of the vertex set of the graph, along with an extra “coin register.” The state of the
+coin register is a superposition of jLEFTiandjRIGHTi. The walk then proceeds by
+alternately taking a step in the direction dictated by the coin register and applying
+a unitary “coin tossing operator” to the coin register. The coin tossing operator is
+often chosen to be the Hadamard gate. It was shown in [2, 6, 8, 44, 53] that the mix-
+ing and propagation behaviour of these quantum walks was significantly different
+from their classical counterparts. These early constructions developed into the more
+general concept of a discrete time quantum walk, which will be defined in detail.
+We will describe two methods for defining a unitary walk operator. In a discrete
+time quantum walk, the state space has basis vectors fjui
+jvi:u;v2Vg. Roughly
+speaking, the walk operator alternately takes steps in the first and second registers.
+This is often described as a walk on the edges of the graph. In a continuous time
+quantum walk, we will restrict our attention to symmetric transition matrices P.
+We take Pto be the Hamiltonian for our system. Applying Schr ¨odinger’s equation,
+this will define continuous time unitary evolution in the state space spanned by
+fjui:u2Vg. Interestingly, these two types of walk are not known to be equivalent.
+We will give an overview of both types of walk as well as some of the algorithms
+that apply them.
+4.1 Discrete Time Quantum Walks
+LetPbe a stochastic matrix describing a classical random walk on a graph G. We
+would like the quantum walk to respect the structure of the graph G, and take into
+account the transition probabilities Pu;v. The quantum walk should be governed by
+a unitary operation, and is therefore reversible. However, a classical random walk is
+not, in general, a reversible process. Therefore, the quantum walk will necessarily
+behave differently than the classical walk. While the state space of the classical walk
+isV, the state quantum walk takes place in the space spanned by fju;vi:u;v2Vg.
+We can think of the first register as the current location of the walk, and the second
+register as a record of the previous location. To facilitate the quantum walk from a
+stateju;vi, we first mix the second register over the neighbours of u, and then swap
+the two registers. The method by which we mix over the neighbours of umust beAlgorithms for Quantum Computers 13
+chosen carefully to ensure that it is unitary. To describe this formally, we define the
+following states for each u2V:
+jyui=jui
+å
+v2Vp
+Pvujvi (4)
+jy
+ui=å
+v2Vp
+Pvujvi
+jui: (5)
+Furthermore, define the projections onto the space spanned by these states:
+P=å
+u2Vjyuihyuj (6)
+P=å
+u2Vjy
+uihy
+uj: (7)
+In order to mix the second register, we perform the reflection (2PI). Letting S
+denote the swap operation, this process can be written as
+S(2PI): (8)
+It turns out that we will get a more elegant expression for a single step of the quan-
+tum walk if we define the walk operator Wto be two iterations of this process:
+W=S(2PI)S(2PI) (9)
+=(2(SPS)I)(2PI) (10)
+=(2PI)(2PI): (11)
+So, the walk operator Wis equivalent to performing two reflections.
+Many of the useful properties of quantum walks can be understood in terms of
+the spectrum of the operator W. First, we define D, the nnmatrix with entries
+Du;v=pPu;vPv;u. This is called the discriminant matrix , and has eigenvalues in the
+interval [0;1]. In the theorem that follows, the eigenvalues of Dthat lie in the interval
+(0;1)will be expressed as cos (q1);:::;cos(qk). Letjq1i;:::;jqkibe the correspond-
+ing eigenvectors of D. Now, define the subspaces
+A=spanfjyuig (12)
+B=spanfjy
+uig: (13)
+Finally, define the operator
+Q=å
+v2Vjyvihvj (14)
+andfj
+=Qqj
+: (15)
+We can now state the following spectral theorem for quantum walks:
+Theorem 1 (Szegedy, [60]). The eigenvalues of W acting on the space A +B can be
+described as follows:14 Jamie Smith and Michele Mosca
+1. The eigenvalues of W with non-zero imaginary part are e2iq1;e2iq2;:::;e2iqk
+where cos(q1);:::;cos(qk)are the eigenvalues of D in the interval (0;1). The
+corresponding (un-normalized) eigenvectors of W (P)can be written asfj
+
+e2iqjSfj
+for j=1;:::;k.
+2. A\B and A?\B?span the +1eigenspace of W. There is a direct correspon-
+dence between this space and the +1eigenspace of D. In particular, the +1
+eigenspace of W has the same degeneracy as the +1eigenspace of D.
+3. A\B?and A?\B span the1eigenspace of W.
+We say that Pissymmetric ifPT=Pandergodic if it is aperiodic. Note that if Pis
+symmetric, then the eigenvalues of Dare just the absolute values of the eigenvalues
+ofP. It is well-known that if Pis ergodic, then it has exactly one stationary distri-
+bution (i.e. a unique +1 eigenvalue). Combining this fact with theorem (1) gives us
+the following corollary:
+Corollary 1. If P is ergodic and symmetric, then the corresponding walk operator
+W has unique +1eigenvector in Span (A;B):
+jyi=1pnå
+v2Vjyvi=1pnå
+v2Vjy
+vi (16)
+Moreover, if we measure the first register of jyi, we get a state corresponding to
+vertex u with probability
+Pr(u) =1
+nå
+v2VPu;v=1
+n: (17)
+This is the uniform distribution, which is the unique stationary distribution for the
+classical random walk.
+4.1.1 The Phase Gap and the Detection Problem
+In this section, we will give an example of a quadratic speedup for the problem of
+detecting whether there are any “marked” vertices in the graph G. First, we define
+the following:
+Definition 1. The phase gap of a quantum walk is defined as the smallest postive
+value 2 qsuch that e2iqare eigenvalues of the quantum walk operator. It is denoted
+byD(P).
+Definition 2. LetMVbe a set of marked vertices. In the detection problem , we
+are asked to decide whether Mis empty.
+In this problem, we assume that Pis symmetric and ergodic. We define the following
+modified walk P0:
+P0
+uv=8
+><
+>:Puvv=2M
+0 u6=v;v2M
+1 u=v;v2M:(18)Algorithms for Quantum Computers 15
+This walk resembles P, except that it acts as the identity on the set M. That is, if the
+walk reaches a marked vertex, it stays there. Let PMdenote the operator P0restricted
+toVnM. Then, arranging the rows and columns of P0, we can write
+P0=
+PM0
+P00I
+: (19)
+ByTheorem 1 , ifM=/ 0, then PM=P0=Pand

PM

=1. Otherwise, we have the
+strict inequality

PM

<1. The following theorem bounds

PM

away from 1:
+Theorem 2. If(1d)is the absolute value of the eigenvalue of P with second
+largest magnitude, and jMjejVj, then

PM

1de
+2.
+We will now show that the detection problem can be solved using eigenvalue esti-
+mation. Theorem 2 will allow us to bound the running time of this method. First,
+we describe the discriminant matrix for P0:
+D(P0)uv=8
+><
+>:Puvu;v=2M
+1 u=v;v2M
+0 otherwise :(20)
+Now, beginning with the state
+jyi=1pnå
+v2Vjyvi=1pnå
+v2Vp
+Pv;ujuijvi; (21)
+we measure whether or not we have a marked vertex; if so, we are done. Otherwise,
+we have the state
+jyMi=1p
+jVnMjå
+u;v2VnMp
+Pvujuijvi: (22)
+IfM=/ 0, then this is the state jyidefined in (16), and is the +1 eigenvector of
+W(P). Otherwise, by Theorem 1 , this state lies entirely in the space spanned by
+eigenvectors with values of the form e2iqj, where qjis an eigenvalue of PM. Ap-
+plying Theorem 2 , we know that
+qcos1(1de
+2)r
+de
+2: (23)
+So, the task of distinguishing between Mbeing empty or non-empty is equivalent
+to that of distinguishing between a phase parameter of 0 and a phase parameter of
+at leastq
+de
+2. Therefore, applying phase estimation to W(P0)on statejyMiwith
+precision O(p
+de)will decide whether Mis empty with constant probability. This
+requires time O(1p
+de).16 Jamie Smith and Michele Mosca
+By considering the modified walk operator P0, it can be shown that the detection
+problem requires O(1
+de)time in the classical setting. Therefore the quantum algo-
+rithm provides a quadratic speedup over the classical one for the detection problem.
+4.1.2 Quantum Hitting Time
+Classically, the first hitting time is denoted H(r;M). For a walk defined by P, start-
+ing from the probability distribution ronV,H(r;M)is the smallest value nsuch
+that the walk reaches a marked vertex v2Mat some time t2f0;:::;ngwith constant
+probability. This idea is captured by applying the modified operator P0and some n
+times, and then considering the probability that the walk is in some marked state
+v2M. LetrMbe any initial distribution restricted to the vertices VnM. Then, at
+time t, the probability that the walk is in an unmarked state is

Pt
+MrM

+1, where




+1
+denotes the L1norm. Assuming that Mis non-empty, we can see that

PM

<1. So,
+ast!¥, we have

Pt
+MrM

+1!0. So, as t!¥, the walk defined by P0is in a
+marked state with probability 1. As a result, if we begin in the uniform distribution
+ponV, and run the walk for some time t, we will “skew” the distribution towards
+M, and thus away from the unmarked vertices. So, we define the classical hitting
+time to be the minimum tsuch that
+

Pt
+MrM

+1<e (24)
+for any constant eof our choosing. Since the quantum walk is governed by a unitary
+operator, it doesn’t converge to a particular distribution the way that the classical
+walk does. We cannot simply wait an adequate number of time steps and then mea-
+sure the state of the walk; the walk might have already been in a state with high
+overlap on the marked vertices and then evolved away from this state! Quantum
+searching has the same problem when the number of solutions is unknown. We can
+get around this in a similar way by considering an expected value over a sufficiently
+long period of time. This will form the basis of our definition of hitting time. Define
+jpi=1p
+jVnMjå
+v2VnMjyvi: (25)
+Then, if Mis empty,jpiis a+1 eigenvector of W, and Wtjpi=jpifor all t.
+However, if Mis non-empty, then the spectral theorem tells us that jpilies in the
+space spanned by eigenvectors with eigenvalues e2iqjfor non-zero qj. As a result,
+it can be shown that, for some values of t, the state Wtjpiis “far” from the initial
+distributionjpi. We define the quantum hitting in the same way as Szegedy [60].
+The hitting time HQ(W)as the minimum value Tsuch that
+1
+T+1T
+å
+t=0

Wtjpijpi

21jMj
+jVj: (26)Algorithms for Quantum Computers 17
+This leads us to Szegedy’s hitting time theorem [60]:
+Theorem 3. The quantum hitting time H Q(W)is
+O0
+@1q
+1

PM

1
+A
+.
+Corollary 2. Applying Theorem 2 , if the second largest eigenvalue of P has mag-
+nitude (1d)andjMj=jVje, then H Q(W)2O(1p
+de)
+Notice that this corresponds to the running time for the algorithm for the detec-
+tion problem, as described above. Similarly, the classical hitting time is in O(1
+de),
+corresponding to the best classical algorithm for the detection problem.
+It is also worth noting that, if there no marked elements, then the system remains
+in the statejpi, and the algorithm never “hits.” This gives us an alternative way
+to approach the detection problem. We run the algorithm for a randomly selected
+number of steps t2f0;:::;Tgwith Tof size O(p
+1=de), and then measure whether
+the system is still in the state jpi; if there are any marked elements, then we can
+expect to find some other state with constant probability.
+4.1.3 The Element Distinctness Problem
+In the element distinctness problem, we are given a black box that computes the
+function
+f:f1;:::;ng! S (27)
+and we are asked to determine whether there exist x;y2f1;:::;ngwith x6=yand
+f(x) =f(y). We would like to minimize the number of queries made to the black
+box. There is a lower bound of W(n2=3)on the number of queries, due indirectly
+to Aaronson and Shi [1]. The algorithm of Ambainis [7] proves that this bound is
+tight. The algorithm uses a quantum walk on the Johnson graph J(n;p)has vertex
+set consisting of all subsets of f1;:::;ngof size p. Let S1andS2bep-subsets of
+f1;:::;ng. Then, S1is adjacent to S2if and only ifjS1\S2j=p1. The Johnson
+graph J(n;p)therefore hasn
+p

+vertices, each with degree p(np).
+The state corresponding to a vertex Sof the Johnson graph will not only record
+which subsetfs1;:::;spgf 1;:::;ngit represents, but the function values of those
+elements. That is,
+jSi=s1;s2:::;sp;f(s1);f(s2);:::;f(sp)
+: (28)
+Setting up such a state requires pqueries to the black box.
+The walk then proceeds for titeration, where tis chosen from the uniform dis-
+tribution onf0;:::;TgandT2O(1=p
+de). Each iteration has two parts to it. First,
+we need to check if there are distinct si;sjwith f(si) =f(sj)—that is, whether the18 Jamie Smith and Michele Mosca
+vertex Sis marked. This requires no calls to the black box, since the function values
+are stored in the state itself. Second, if the state is unmarked, we need to take a step
+of the walk. This involves replacing, say, siwith s0
+i, requiring one query to erase
+f(si)and another to insert the value f(s0
+i). So, each iteration requires a total of 2
+queries.
+We will now bound eandd. If only one pair x;yexists with f(x) =f(y), then
+there aren2
+p2

+marked vertices. This tells us that, if there are any such pairs at all,
+epsilon is W(p2=n2). Johnson graphs are very well-understood in graph theory. It is
+a well known result that the eigenvalues for the associated walk operator are given
+by:
+lj=1j(n+1j)
+p(np)(29)
+for 0jp. For a proof, see [20]. This give us d=n
+p(np). Putting this together,
+we find thatp
+1=deisO(npp). So, the number of queries required is O(p+npp).
+In order to minimize this quantity, we choose mto be of size Q(n2=3). The query
+complexity of this algorithm is O(n2=3), matching the lower bound of Aaronson and
+Shi [1].
+4.1.4 Unstructured Search as a Discrete Time Quantum Walk
+We will now consider unstructured search in terms of quantum walks. For unstruc-
+tured search, we are required to identify a marked element from a set of size n. Let
+Mdenote the set of marked elements, and Udenote the set of unmarked elements.
+Furthermore, let m=jMjandq=jUj. We assume that the number of marked el-
+ements, m, is very small in relation to the total number of elements n. If this were
+not the case, we could efficiently find a marked element with high probability by
+simply checking a randomly chosen set of vertices. Since the set lacks any structure
+or ordering, the corresponding walk takes place on the complete graph on nvertices.
+Let us define the following three states:
+jUUi=1
+qå
+u;v2Uju;vi (30)
+jUMi=1pnqå
+u2U
+v2Mju;vi (31)
+jMUi=1pnqå
+u2M
+v2Uju;vi: (32)
+Noting thatfjUUi;jUMi;jMUigis an orthonormal set, we will consider the action
+of the walk operator on the three dimensional space
+G=spanfjUUi;jUMi;jMUig: (33)Algorithms for Quantum Computers 19
+In order to do this, we will express the spaces AandB, defined in (12-13) in terms
+of a different basis. First we label the unmarked vertices:
+U=fu0;:::;uq1g: (34)
+Then, we define
+jgki=1pqq1
+å
+j=0e2pi jk
+qyuj
+(35)
+and
+jg
+ki=1pqq1
+å
+j=0e2pi jk
+qy
+ujE
+(36)
+with kranging from 0 to q1. Note thatjg0icorresponds to the definition of jpiin
+(25). We can then rewrite AandB:
+A=spanfjgkig (37)
+B=spanfjg
+kig: (38)
+Now, note that for k6=0, the space Gis orthogonal tojgkiandg
+k
+. Furthermore,
+jg0i=1pnpmjUMi+pqjUUi

+(39)
+and
+jg
+0i=1pnpmjMUi+pqjUUi

+: (40)
+Therefore, the walk operator
+W= (2PI)(2PI) (41)
+when restricted to Gis simply
+W0=
+2jg
+0ihg
+0jI
+2jg0ihg0jI
+: (42)
+Figure 4.1.4 illustrates the space Gwhich contains the vectors jg0iandg
+0
+.
+At this point, it is interesting to compare this algorithm to Grover’s search algo-
+rithm. Define
+jrti=WtjUUi: (43)
+Now, definer1
+t
+andr2
+t
+to be the projection of jrtionto spanfjUUi;jUMig
+andspanfjUUi;jMUigrespectively. We can think of these as the “shadow” cast by
+the vectorjrtion two-dimensional planes within the space G. Note thatjg0ilies in
+thespanfjUUi;jUMigand its projection onto spanfjUUi;jMUigisp
+q=njUUi.
+So, the walk operator acts onr1
+t
+by reflecting it around the vector jg0iand then
+aroundp
+q=njUUi. This is very similar to Grover search, except for the fact that20 Jamie Smith and Michele Mosca
+Fig. 3 The 3 dimensional space G. The walk operator acts on this space by alternately performing
+reflections in the vector jg0iandg
+0
+.
+the walk operator in this case does not preserve the magnitude of jrt
+1i. So, with each
+application of the walk operator,r1
+t
+is rotated by 2 q, where q=tan1(p
+m=q)is
+the angle between jUUiandjg0i. The case forr2
+t
+is exactly analogous, with the
+rotation taking place in the plane spanfjUUi;jMUig. So, we can think of the quan-
+tum walk as Grover search taking place simultaneously in two intersecting planes. It
+should not come as a surprise, then, that we can achieve a quadratic speedup similar
+to that of Grover search.
+We will now use a slightly modified definition of hitting time to show that the
+walk can indeed be used to find a marked vertex in O(pn)time. Rather than using
+the hitting time as defined in section 4.1.2, we will replace the state jpi=jg0iwith
+jUUi. Note that we can create jUUifromjpiby simply measuring whether the
+second register contains a marked vertex. This is only a small adjustment, since jg0i
+is close tojUUi. Furthermore, both lie in the space G, and the action of the walk
+operator is essentially identical for both starting states. It should not be surprising to
+the reader that the results of section 4.1.2 apply to this modified definition as well.
+In this case, the operator Pis defined by:
+Puv=(
+0 if u=v
+1
+n1ifu6=v:(44)Algorithms for Quantum Computers 21
+Letv0;:::;vn1be a labeling of the vertices of G. Let x0;:::;xn1denote the eigen-
+vectors of P, with xk
+vjdenoting the amplitude on vjinxk. Then, the eigenvalues of P
+are as follows:
+xk
+vj=1
+n
e2pi jk
+n: (45)
+Then, x0has eigenvalue 1, and is the stationary distribution. All the other xkhave
+eigenvalue1=(n1), giving a spectral gap of d=n2
+n1. Applying corollary 2, this
+gives us quantum hitting time for the corresponding quantum walk operator W
+HQ(W)r
+(n1)n
+n2=O(pn): (46)
+So, we run the walk for some randomly selected time t2f0;:::;T1gwith Tof size
+O(pn), then measure whether either the first or second register contains a marked
+vertex. Applying theorem 3, the probability that neither contains a marked vertex is
+1
+TT1
+å
+t=0jhrtjUUij21
+TT1
+å
+t=0jhrtjUUij
+=11
+2TT1
+å
+t=0

jrtijUUi

2
+1jMj
+2jVj
+Assuming thatjMjis small compared to jVj, we find a marked vertex in either the
+first or second register with high probability. Repeating this procedure a constant
+number of times, our success probability can be forced arbitrarily close to 1.
+4.1.5 The MNRS Algorithm
+Magniez, Nayak, Roland and Santha [47] developed an algorithm that generalizes
+the search algorithm described above to any graph. A brief overview of this al-
+gorithm and others is also given in the survey paper by Santha [56]. The MNRS
+algorithm employs similar principles to Grover’s algorithm; we apply a reflection in
+M, the space of unmarked states, followed by a reflection in jpi, a superposition of
+marked and unmarked states. This facilitates a rotation through an angle related to
+the number of marked states. In the general case considered by Magniez et al., jpi
+is the stationary distribution of the walk operator. It turns out to be quite difficult to
+implement a reflection in jpiexactly. Rather, the MNRS algorithm employs an ap-
+proximate version of this reflection. This algorithm requires O
+1p
+de
+applications
+of the walk operator, where dis the eigenvalue gap of the operator P, and eis the
+proportion vertices that are marked. In his survey paper, Santha [56] outlines some
+applications of the MNRS algorithm, including a version of the element distinctness
+problem where we are asked to find elements xandysuch that f(x) =f(y).22 Jamie Smith and Michele Mosca
+4.2 Continuous Time Quantum Walks
+To define a classical continuous time random walk for a graph with no loops, we
+define a matrix similar to the adjacency matrix of G, called the Laplacian :
+Luv=8
+><
+>:0 u6=v;uv=2E
+1 u6=v;uv2E
+deg(v)u=v:(47)
+Then, given a probability distribution p(t)on the vertices of G, the walk is defined
+by
+d
+dtp(t) =Lp(t): (48)
+Using the Laplacian rather than the adjacency matrix ensures that p(t)remains nor-
+malized. A continuous time quantum walk is defined in a similar way. For simplicity,
+we will assume that the Laplacian is symmetric, although it is still possible to define
+the walk in the asymmetric case. Then, since the Laplacian is Hermitian, we can
+simply take it to be the Hamiltonian of our system. Letting jr(t)ibe a normalized
+vector in CV, Schr ¨odinger’s equation gives us
+id
+dtjr(t)i=Ljr(t)i: (49)
+Solving this equation, we get an explicit expression for jr(t)i:
+jr(t)i=eiLtjr(0)i: (50)
+Letfjl1i:;;;jlnigbe the eigenvectors of Lwith corresponding values fl1;:::;lng.
+We can rewrite the expression for jr(t)iin terms of this basis as follows:
+jr(t)i=n
+å
+j=1eiljthljjr0ilj
+: (51)
+Clearly, the behaviour of a continuous time quantum walk is very closely related to
+the eigenvectors and spectrum of the Laplacian.
+Note that we are not required to take the Laplacian as the Hamiltonian for the
+walk. We could take any Hermitian matrix we like, including the adjacency matrix,
+or the transition matrix of a (symmetric) Markov chain.
+4.2.1 A Continuous Time Walk for Unstructured Search
+For unstructured search, the walk takes place on a complete graph. First, we define
+the following three states:Algorithms for Quantum Computers 23
+jVi=1p
+jVjå
+v2Vjvi (52)
+jMi=1p
+jMjå
+v2Mjvi: (53)
+The Hamiltonian we will use is a slightly modified version of the Laplacian for the
+complete graph, with an extra “marking term” added:
+H=jVihVj+jMihMj: (54)
+It is convenient to consider the action of this Hamiltonian in terms of the vectors
+jMiandM?
+, where
+M?E
+=1p
+jVnMjå
+v2VnMjvi=1p
+1hMjVi2(jSijMihMjSi): (55)
+As outlined in [21], we let a=hMjSi. We can then re-write the Hamiltonian in
+terms of the basisfjMi;M?
+g:
+H=
+a2ap
+1a2
+ap
+1a21a2
++
+1 0
+0 0
+(56)
+=
+1 0
+0 1
++a2
+1 0
+01
++ap
+1a2
+0 1
+1 0
+(57)
+=I+a2sZ+ap
+1a2sX (58)
+=I+a
+asZ+p
+1a2sX

+(59)
+where sXandsZare the Pauli X and Z operators. Note that the identity term in the
+sum simply introduces a global phase, and can be ignored. Note that the operator
+A=asZ+p
+1a2sX (60)
+has eigenvalues1, and is therefore Hermitian and unitary. Therefore, we can write
+eiHt=eiAat=cos(at)Iisin(at)A: (61)
+Also note that
+AjSi=
+asZ+p
+1a2sX

+ajMi+p
+1a2M?E

+(62)
+=jMi: (63)
+If we start with the state jSi, we can now calculate the state of the system at time t:
+jr(t)i=cos(at)jSiisin(at)AjSi (64)
+=cos(at)jSiisin(at)jMi: (65)24 Jamie Smith and Michele Mosca
+So, at time t, the probability of finding the system in a state corresponding to a
+marked vertex is
+å
+v2Mjhvjr(t)ij2=jMjcos2(at)
+jVj+sin2(at)
+jMj
+(66)
+=a2cos2(at)+sin2(at): (67)
+At time t=0, this is the same as sampling from the uniform distribution, as we
+would expect. However, at time t=p=2a=p=2
p
+N=M, we observe a marked
+vertex with probability 1. Therefore, this search algorithm runs in time O(p
+N=M),
+which coincides with the running time of Grover’s search algorithm and the related
+discrete time quantum walk algorithm.
+4.2.2 Mixing in Quantum Walks and the Glued Tree Traversal Problem
+While continuous time quantum walks give us a generic quadratic speedup over
+their classical counterparts, there are some graphs on which the quantum walk gives
+exponential speedup. One such example is the “glued trees” of Childs et al. [23]. In
+this example, the walk takes place on the following graph, obtained by taking two
+identical binary trees of depth dand joining their leaves using a random cycle, as
+illustrated in Figure 4. Beginning at the vertex ENTRANCE , we would like to know
+Fig. 4 A small glued tree graph.
+how long we need to run the algorithm to reach EXIT . It is straightforward to show
+that classically, this will take time exponential in d, the depth of the tree. Intuitively,
+this is because there are so many vertices and edges in the “middle” of the graph,
+that there is small probability of a classical walk “escaping” to the exit. We willAlgorithms for Quantum Computers 25
+prove that a continuous time quantum walk can achieve an overlap of W(P(d))with
+theEXIT vertex in time O(1
+Q(d)), where QandPare polynomials.
+LetVsdenote the set of vertices at depth s, so that V0=fENTRANCEgand
+V2d+1=fEXITg. Taking the adjacency matrix to be our Hamiltonian, the operator
+U(t) =eiHtacts identically on the vertices in Vsfor any s. Therefore, the states
+jsi=1p
+jVsjå
+v2Vsjvi (68)
+form a convenient basis. We can therefore think of the walk on Gas a walk on the
+statesfjsi: 0s2d+1g. We also note that, if Ais the adjacency matrix of G,
+Ajsi=8
+>>>>>><
+>>>>>>:p
+2js+1i s=0p
+2js1i s=2d+1p
+2js1i+2js+1i s=dp
+2js+1i+2js1i s=d+1p
+2js+1i+p
+2js1iotherwise :(69)
+Aside from the exceptions at ENTRANCE ,EXIT , and the vertices in the center, this
+walk looks exactly like the walk on a line with uniform transition probabilities.
+Continuous time classical random walks will eventually converge to a limiting
+distribution. The time that it takes for the classical random walk to get ‘close’ to
+this distribution is called the mixing time . More formally, we can take some small
+constant g, and take the mixing time to be the amount of time time it take to come
+within gof the stationary distribution, according to some metric. In general, we
+express the mixing time in terms of 1 =gandn, the number of vertices in the graph.
+Since quantum walks are governed by a unitary operator, we cannot expect the
+same convergent behaviour. However, we can define the limiting behaviour of quan-
+tum walks by taking an average over time. In order to do this, we define Pr (u;v;T).
+If we select t2[0;T]uniformly at random and run the walk for time t, beginning
+at vertex u, then Pr (u;v;T)is the probability that we find the system in state v. For-
+mally, we can write this as
+Pr(u;v;T) =1
+TZT
+0hvjeiAtjui2dt (70)
+where Ais the adjacency matrix of the graph. Now, if we take fjligto be the
+set of eigenvectors of Awith corresponding eigenvalues flg, then we can rewrite
+Pr(u;v;T)as follows:26 Jamie Smith and Michele Mosca
+Pr(u;v;T) =1
+TZT
+0å
+leilthvjlihljui2
+dt (71)
+=1
+TZT
+0å
+l;l0
+ei(ll0)thvjlihljuihvjl0ihl0jui
+dt (72)
+=å
+ljhvjlihljuij2(73)
++1
+Tå
+l6=l0hvjlihljuihvjl0ihl0juiZT
+0ei(ll0)tdt (74)
+=å
+ljhvjlihljuij2(75)
++å
+l6=l0hvjlihljuihvjl0ihl0jui1ei(ll0)T
+i(ll0)T: (76)
+In particular, we have
+lim
+T!¥Pr(u;v;T) =å
+ljhvjlihljuij2: (77)
+We will denote this value by Pr (u;v;¥). This is the quantum analogue of the
+limiting distribution for a classical random walk. We would now like to apply
+this to the specific case of the glued tree traversal problem. First, we will lower
+bound Pr (ENTRANCE ;EXIT ;¥). We will then show that Pr (ENTRANCE ;EXIT ;T)
+approaches this value rapidly as we increase T, implying that we can traverse the
+glued tree structure efficiently using a quantum walk.
+Define the reflection operator Qby
+Qjji=j2d1ji (78)
+This operator commutes with the adjacency matrix, and hence the walk operator
+because of the symmetry of the glued trees. This implies that Qcan be diagonalized
+in the eigenbasis of the walk operator eiAtfor any t. What is more, the eigenvalues
+ofQare1. As a result, ifjliis an eigenvalue of eiAt, then
+hljENTRANCEi=hljEXITi: (79)
+We can apply this to (77), yielding
+Pr(ENTRANCE ;EXIT ;¥) =å
+ljhENTRANCEjlij4(80)
+1
+2d+2å
+ljhENTRANCEjlij2(81)
+=1
+2d+2: (82)Algorithms for Quantum Computers 27
+Now, we need to determine how quickly Pr (ENTRANCE ;EXIT ;T)approaches Pr (ENTRANCE ;EXIT ;¥)
+as we increase T:
+jPr(ENTRANCE ;EXIT ;T)Pr(ENTRANCE ;EXIT ;¥)j (83)
+=å
+l6=l0hEXITjlihljENTRANCEihEXITjl0ihl0jENTRANCEi1ei(ll0)T
+i(ll0)T(84)
+å
+l6=l0jhljENTRANCEij2hl0jENTRANCEi21ei(ll0)T
+i(ll0)T(85)
+2
+Td(86)
+where dis the difference between the smallest gap between any two distinct eigen-
+values of A. As a result, we get
+Pr(ENTRANCE ;EXIT ;T)1
+2d12
+Td: (87)
+Childs et al. [23] show that disW(1=d3). Therefore, if we take Tof size O(d4), we
+get success probability O(1=d). Repeating this process, we can achieve an arbitrarily
+high probability of success in time polynomial in d— an exponential speedup over
+the classical random walk.
+4.2.3 AND-ORTree Evaluation
+AND-ORtrees arise naturally when evaluating the value of a two player combi-
+natorial game. We will call the players P0andP1. The positions in this game are
+represented by nodes on a tree. The game begins at the root, and players alternate,
+moving from the root towards the leaves of the tree. For simplicity, we assume that
+we are dealing with a binary tree; that is, for each move, the players have exactly
+two moves. While the algorithm can be generalizes to any approximately balanced
+tree, but we consider the binary case for simplicity. We also assume that every game
+lasts some fixed number of turns d, where a turn consists of one player making a
+move. We denote the total number of leaf nodes by n=2d. We can label the leaf
+nodes according to which player wins if the game reaches each node; they are la-
+beled with a 0 if P0wins, and a 1 if P1wins. We can then label each node in the
+graph by considering its children. If a node corresponds to P0’s turn, we take the
+AND of the children. For P1’s move, we take the ORof the children. The value at the
+root node tells us which player has a winning strategy, assuming perfect play.
+Now, since
+AND(x1;:::;xk) =NAND (NAND (x1;:::;xk)) (88)
+OR(x1;:::;xk) =NAND (NAND (x1);:::; NAND (xk)) (89)28 Jamie Smith and Michele Mosca
+we can rewrite the AND-ORtree using only NAND operations. Furthermore, since
+NOT(x1) = NAND (x1) (90)
+rather than label leaves with a 1, we can insert an extra node below the leaf and
+label it 0. Also, consecutive NAND operations cancel out, and will be removed. This
+is illustrated in figure 5. Note that the top-most node will be omitted from now on,
+and the shaded node will be referred to as the ROOT node. Now, the entire AND-OR
+tree is encoded in the structure of the NAND tree. We will write NAND (v)to denote
+theNAND of the value obtained by evaluating the tree up to a vertex v. The value of
+the tree is therefore NAND (ROOT ).
+Fig. 5 AnAND-ORtree and the equivalent NAND tree, with example values. Note that the top-most
+node will be omitted from now on, and the shaded node in the NAND tree will be referred to as the
+ROOT node.
+The idea of the algorithm is to use the adjacency matrix of the NAND tree as the
+Hamiltonian for a continuous time quantum walk. We will show that the eigenspec-
+trum of this walk operator is related to the value of NAND (ROOT ). This idea first
+appeared in a paper of Farhi, Goldman and Gutmann [29], and was later refined by
+Ambainis, Childs and Reichardt [9]. We begin with the following lemma:
+Lemma 1. Letjlibe an eigenvector of A with value l. Then, if v is a node in the
+NAND tree with parent node p and children C,
+hpjli=lhvjli+å
+c2Chcjli (91)
+Therefore, if Ahas an eigenvector with value 0, thenAlgorithms for Quantum Computers 29
+hpjl0i=å
+c2Chcjl0i (92)
+Using this fact and some inductive arguments, Ambainis, Childs and Reichardt [9]
+prove the following theorem:
+Theorem 4. IfNAND (ROOT ) =0, then there exists an eigenvector jl0iof A with
+eigenvalue 0such thatjhROOTjl0ij1p
+2. Otherwise, if NAND (ROOT ) =1, then for
+any eigenvectorjliof A withhljAjli<1
+2pn, we havehROOTjl0i=0.
+This result immediately leads to an algorithm. We perform phase estimation with
+precision O(1=pn)on the quantum walk, beginning in the state jROOTi. IfNAND (ROOT ) =
+0, get a phase of 0 with probability 1
+2. If NAND (ROOT ) =1, we will never get a
+phase of 0. This gives us a running time of O(pn). While we have restricted our
+attention to binary trees here, this result can be generalized to any m-ary tree.
+It is worth noting that unstructured search is equivalent to taking the ORofn
+variables. This is a AND-ORtree of depth 1, and nleaves. The O(pn)running time of
+Grover’s algorithm corresponds to the running time for the quantum walk algorithm.
+Classically, the running time depends not just on the number of leaves, but on the
+structure of the tree. If it is a balanced m-ary tree of depth d, then the running time
+is
+O0
+@ 
+m1+p
+m2+14m+1
+4!d1
+A: (93)
+In fact, the quantum speedup is maximal when we have an n-ary tree of depth 1.
+This is just unstructured search, and requires W(n)time classically.
+Reichardt and ˘Spalek [55] generalize the AND-OR tree problem to the evalu-
+ation of a broader class of logical formulas. Their approach uses span programs .
+A span program Pconsists of a set of target vector t, and a set of input vectors
+fv0
+1;v1
+1;:::;v0
+n;v1
+n;gcorresponding to logical literals fx1;x1;:::;xn;xng. The program
+corresponds to a boolean function fP:f0;1gn!f0;1gsuch that, for s2f0;1gn,
+f(s) =1 if and only if
+n
+å
+j=1vsj
+j=t: [42]
+Reichardt and ˘Spalek outline the connection between span programs and the evalu-
+ation of logical formulas. They show that finding s2f1(1)is equivalent to find-
+ing a zero eigenvector for a graph GPcorresponding to the span program P. In this
+sense, the span program approach is similar to the quantum walk approach of Childs
+et al.— both methods evaluate a formula by finding a zero eigenvector of a corre-
+sponding graph.30 Jamie Smith and Michele Mosca
+5 Tensor Networks and Their Applications
+A tensor network consists of an underlying graph G, with an algebraic object called
+atensor assigned to each vertex of G. The value of the tensor network is calculated
+by performing a series of operations on the associated tensors. The nature of these
+operations is dictated by the structure of G. At their simplest, tensor networks cap-
+ture basic algebraic operations, such as matrix multiplication and the scalar product
+of vectors. However, their underlying graph structure makes them powerful tools for
+describing combinatorial problems as well. We will explore two such examples—
+the Tutte polynomial of a planar graph, and the partition function of a statistical
+mechanical model defined on a graph. As a result, the approximation algorithm that
+we will describe below for the value of a tensor network is implicitly an algorithm
+for approximating the Tutte polynomial, as well as the partition function for these
+statistical mechanics models. We begin by defining the notion of a tensor. We then
+outline how these tensors and tensor operations are associated with an underlying
+graph structure. For a more detailed account of this algorithm, the reader is referred
+to [10]. Finally, we will describe the quantum approxiamtion algorithm for the value
+of a tensor network, as well as the applications mentioned above.
+5.0.4 Tensors: Basic Definitions
+Tensors are formally defined as follows:
+Definition 3. A tensor Mof rank mand dimension qis an element of Cqm. Its entries
+are denoted by Mj1;j2;:::;jm, where 0jkq1 for all jk.
+Based on this definition, a vector is simply a tensor of rank 1, while a square matrix
+is a tensor of rank 2. We will now define several operations on tensors, which will
+generalize many familiar operations from linear algebra.
+Definition 4. LetMandNbe two tensors of dimension qand rank mandnrespec-
+tively. Then, their product, denoted M
+N, is a rank m+ntensor with entries
+(M
+N)j1;:::;jm;k1;:::;kn=Mj1;:::;jm
Nk1;:::;kn: (94)
+This operation is simply the familiar tensor product. While the way that the entries
+are indexed is different, the resulting entries are the same.
+Definition 5. LetMbe a tensor of rank mand dimension q. Now, take aandbwith
+1a<bm. The contraction of Mwith respect to aandbis a rank m2 tensor
+Ndefined as follows:
+Nj1;:::;ja1;ja+1;:::jb1;jb+1;:::;jm=q1
+å
+k=0Mj1;:::;ja1;k;ja+1;:::jb1;k;jb+1;:::;jm: (95)Algorithms for Quantum Computers 31
+One way of describing this operation is that each entry in the contracted tensor is
+given by summing along the “diagonal” defined by aandb. This operation general-
+izes the partial trace of a density operator. The density operator of two qubit system
+can be thought of as a rank 4 tensor of dimension 2. Tracing out the second qubit is
+then just taking a contraction with respect to 3 and 4. It is also useful to consider the
+combination of these two operations:
+Definition 6. IfMandNare two tensors of dimension qand rank mandn, then for
+aandbwith 1amand 1bn, the contraction of MandNis the result of
+contracting the product M
+Nwith respect to aandm+b.
+We now have the tools to describe a number of familiar operations in terms of tensor
+operations. For example, the inner product of two vectors can be expressed as the
+contraction of two rank 1 tensors. Matrix multiplication is just the contraction of 2
+rank 2 tensors MandNwith respect to the second index of Mand the first index of
+N. Finally, if we take a Hilbert space H=Cq, then we can identify a tensor Mof
+dimension qand rank mwith a linear operator Ms;t:H
+t!H
+swhere s+t=m:
+Ms;t=å
+j1;:::;jmMj1;:::;jmjj1i
+:::
+jjsihjs+1j
+:::
+hjmj: (96)
+This correspondence with linear operators is essential to understanding tensor net-
+works and their evaluation.
+5.1 The Tensor Network Representation
+A tensor network T(G;M)consists of a graph G= (V;E)and a set of tensors M=
+fM[v]:v2Vg. A tensor is assigned to each vertex, and the structure of the graph
+Gencodes the operations to be performed on the tensors. We say that the value of
+a tensor network is the tensor that results from applying the operations encoded by
+Gto the set of tensors M. When the context is clear, we will let T(G;M)denote
+the value of the network. In addition to the typical features of a graph, we allow
+Gto have free edges — edges that are incident with a vertex on one end, but are
+unattached at the other. For simplicity, we will assume that all of the tensors in M
+have the same dimension q. We also require that deg(v), the degree of vertex v, is
+equal to the rank of the associated tensor M[v]. IfGconsists of a single vertex vwith
+mfree edges, then T(G;M)represents a single tensor of rank m. Each index of M
+is associated with a particular edge. It will often be convenient to refer to the edge
+associated with an index aasea. We now define a set of rules that will allow us to
+construct a tensor network corresponding to any sequence of tensor operations:
+IfMandNare the value of two tensor networks T(G;M)andT(H;N), then
+taking the product M
+Nis equivalent to taking the disjoint union of the two
+tensor networks, T(G[H;M[N)
+IfMis the value of a tensor network T(G;M), then contracting Mwith respect
+toaandbis equivalent to joining the free edges eaandebinG.32 Jamie Smith and Michele Mosca
+As a result, taking the contraction of MandNwith respect to aandbcorresponds
+to joining the associated edge eafrom T(G;M)andebfrom T(H;N)
+Some examples are illustrated in figure 6. Applying these simple operations itera-
+tively allows us to construct a network corresponding to any series of products and
+contractions of tensors. Note that the number of free edges in T(G;M)is always
+equal to the rank of the corresponding tensor M. We will be particularly interested
+in tensor networks that have no free edges, and whose value is therefore a scalar.
+Fig. 6 The network representation of a tensor Mof rank 4, and the contraction of two rank 4
+tensors, MandN.
+We can now consider the tensor network interpretation of a linear operator Ms;t
+defined in (96). Ms;tis associated with a rank m=s+ttensor M. An equivalent
+tensor network could consist of a single vertex with mfree edges (e1;:::;em). The
+operator Ms;tacts on elements of (Cq)
+t, which are rank ttensors, and can therefore
+be represented a single vertex with tfree edges (e0
+1;:::;e0
+t). The action of Ms;ton an
+element N2(Cq)
+tis therefore represented by connecting e0
+ktoes+kfork=1;:::;t.
+Figure 7 illustrates the operator M2;2acting on a rank 2 tensor. Note that the resulting
+network has two free edges, corresponding to the rank of the resulting tensor.
+Fig. 7 The network corresponding to the operator M2;2acting on a rank 2 tensor N.
+We now return to the tensor networks we will be most interested in— those with
+no free edges. Let us first consider the example of an inner product of two rank 1
+tensors. This corresponds to a graph consisting of two vertices uandvjoined by a
+single edge. The value of the tensor is thenAlgorithms for Quantum Computers 33
+T(G;M) =q
+å
+j=0(Mu)j
(M[v])j (97)
+That is, it is a sum of qterms, each of which corresponds to the assignment of an
+element off0;:::;q1gto the edge (u;v). We can refer to this assignment as an
+q-edge colouring of the graph. In the same way, the value of a more complex tensor
+network is given by a sum whose terms correspond to q-edge colourings of G. Given
+an colouring of G, letM[v]g=M[v]i1;:::;imwhere i1;:::;imare the values assigned by
+gto the edges incident at v. That is, a q-edge colouring specifies a particular entry of
+each tensor in the network. Then, we can rewrite the value of T(G;M)as follows:
+T(G;M) =å
+g 
+Õ
+v2VM[v]g!
+(98)
+where gruns over all q-edge colourings of G. Thinking of the value of a tensor
+network as a sum over all q-edge colourings of Gwill prove useful when we consider
+applications of tensor networks to statistical mechanics models.
+5.2 Evaluating Tensor Networks: the Algorithm
+In this section, we will first show how to interpret a tensor network as a series of
+linear operators. We will then show how to apply these operators using a quantum
+circuit. Finally, we will see how to approximate the value of the tensor network
+using the Hadamard test.
+5.2.1 Overview
+In order to evaluate a tensor network T(G;M), we first give it some structure, by
+imposing an ordering on the vertices of G,
+V(G) =fv1;:::;vng:
+We further define the sets Sj=fv1;:::;vjg. This gives us a natural way to depict the
+tensor network, which is shown in figure 8. Our evaluation algorithm will “move”
+from left to right, applying linear operators corresponding to the tensors Mvi. We
+can think of the edges in analogy to the “wires” of a quantum circuit diagram. To
+make this more precise, let vibe a vertex of degree d. Let tbe the number of vertices
+connecting vitoSi1andsbe the number of edges connecting vitoV(G)nSi. Then,
+the operator that is applied at viisMs;t
+vi. Letting jbe the number of edges from Si1
+toV(G)nSi, we apply the identity operator Ijon the jindices that correspond to
+edges not incident at vi. Combining these, we get an operator34 Jamie Smith and Michele Mosca
+Fig. 8 Part of the graph G, with an ordering on V(G).
+Ni=Ms;t
+vi
+Ij: (99)
+Note that, taking the product of these operators, Õn
+i=1Ni, gives the value of T(G;M).
+However, there are a few significant details remaining. Most notably, the operators
+Niare not, in general, unitary. In fact, they may not even correspond to square ma-
+trices. We will now outline a method for converting the Niinto equivalent unitary
+operators.
+5.2.2 Creating Unitary Operators
+Our first task is to convert the operators Niinto “square” operators— that is, op-
+erators whose domain and range have the same dimension. Referring to (99), the
+identity Ijis already square, so we need only modify Ms;t
+vi. In order to do this, we
+will add some extra qubits in the j0istate. We consider three cases:
+1. If s=t, we definegMs;t
+vi=Ms;t
+vi
+2. If s>t, then we add stextra qubits in thej0istate, and definegMs;t
+visuch that
+gMs;t
+vi
+jyi
+j0i
+(st)

+=Ms;t
+vijyi: (100)
+For completeness, we say that if the last stqubits are not in the j0istate,gMs;t
+vi
+takes them to the 0 vector.
+3. If s<t, then we define
+gMs;t
+vijyi= (Ms;t
+vijyi)
+j0i
+(ts): (101)
+Finally, we define
+˜Ni=gMs;t
+vi
+Ij; (102)
+which is a square operator. Now, to derive corresponding unitary operators, we make
+use of the following lemma:Algorithms for Quantum Computers 35
+Lemma 2. Let M be a linear map A :H
+t!H
+t, where H is a Hibert space H =
+Cq. Furthermore, let G =C2be a spanned byfj0i;j1ig. Then, there exists a unitary
+operator U M:(H
+t
+G)!(H
+t
+G)such that
+UM(jyi
+j0i) =1

M

Mjyi
+j0i+jfi
+j1i (103)
+UMcan be implemented on a quantum computer in poly (qt)time.
+A proof can be found in [10] as well as [4]. Applying this lemma, we can create n
+unitary operators U˜Njfor 1jn, and define
+U=n
+Õ
+j=1U˜Nj: (104)
+It is easily verified that
+h0j
+rUj0i
+r=T(G;M)
+Õj

Ms;t
+vj

(105)
+where ris dependent on the number of vertices nas well as the structure of Gand
+the ordering of the vertices v1;:::;vn.
+5.2.3 The Hadamard Test
+In order to approximate h0j
+rUj0i
+r, most authors suggest the Hadamard test —
+a well-known method for approximating the weighted trace of a unitary operator.
+First, we add an ancillary qubit that will act as a control register. We then apply the
+circuit outlined below, and measure the ancillary qubit in the computational basis.
+j0i H H
+jyi U
+Fig. 9 The quantum circuit for the Hadamard test.
+It is not difficult to show that we measure
+j0iwith probability1
+2
+1+RehfjUjyi

+j1iwith probability1
+2
+1RehyjUjyi

+:36 Jamie Smith and Michele Mosca
+So, if we assign a random variable Xsuch that X=1 when we measure j0iand
+X=1 when we measure j1i, then Xhas expected value Re hyjUjyi. So, in order
+to approximate Re hyjUjyito a precision of ewith constant probability p>1=2,
+we require O(e2)repetitions of this protocol. In order to calculate the imaginary
+portion, we apply the gate
+R=
+1 0
+0i
+to the ancillary qubit, right after the first Hadamard gate. It is important to note
+that the Hadamard test gives an additive approximation ofh0j
+rUj0i
+r. Additive
+approximations will be examined in section 5.2.5.
+5.2.4 Approximating h0j
+rUj0i
+rUsing Amplitude Estimation
+In order to estimate h0j
+rUj0i
+r, most of the literature applies the Hadamard test.
+However, it is worth noting that we can improve the running time by using the
+process of amplitude estimation , as outlined in section 3. Using the notation from
+section 3, we have Uf=U0andA=U, the unitary corresponding to the tensor
+network T(G;M). We begin in the state Uj0i
+r, and our search iterate is
+Q=UU 0U1U0: (106)
+In order to approximate h0j
+rUj0i
+rto a precision of e, our running time is in
+O(1=e), a quadratic improvement over the Hadamard test.
+5.2.5 Additive Approximation
+Letf:S!Cbe a function that we would like to evaluate. Then, an additive ap-
+proximation A with approximation scale D:S!Ris an algorithm that, on input
+x2Sand parameter e>0 , outputs A(x)such that
+Pr
+jA(x)f(x)jeD(x)
+c (107)
+for some constant cwith 0c<1=2. IfD(x)isO(f(x)), then Ais afully polynomial
+randomized approximation scheme , or FPRAS.
+We will now determine the approximation scale for the algorithm outlined above.
+Using amplitude estimation, we estimate
+h0j
+rUj0i
+r=T(G;M)
+Õj

Ms;t
+vj

(108)
+to within ewith time requirement in O(1=e). However, the quantity we actually
+want to evaluate is T(G;M), so we must multiply by Õj

Ms;t
+vj

. Therefore, our
+algorithm has approximation scaleAlgorithms for Quantum Computers 37
+D(G;M) =Õ
+j

Ms;t
+vj

 (109)
+We apply lemma 2 to each vertex in G, and require O(1=e)repetitions of the algo-
+rithm to approximate T(G;M)with the desired accuracy using amplitude estima-
+tion. This gives an overall running time
+O
+poly(qD(G))
jV(G)j
e
+(110)
+where D(G)is the maximum degree of any vertex in G. The algorithm is, therefore,
+an additive approximation of T(G;M).
+5.3 Approximating the Tutte Polynomial for Planar Graphs Using
+Tensor Networks
+In 2000, Kitaev, Freedman, Larsen and Wang [32, 31] demonstrated an efficient
+quantum simulation for topological quantum field theories. In doing so, they im-
+plied that the Jones polynomial can be efficiently approximated at e2pi=5. This sug-
+gests that efficient quantum approximation algorithms might exist for a wider range
+of knot invariants and values. Aharonov, Jones and Landau developed such a Jones
+polynomial for any complex value [5]. Yard and Wocjan’s developed a related algo-
+rithm for approximating the HOMFLYPT polynomial of a braid closure [64]. Both
+of these knot invariants are special cases of the Tutte polynomial. While the tensor
+network algorithm in section 5.3.2 follows directly from a later paper of Aharonov
+et al. [4], it owes a good deal to these earlier results for knot invariants.
+We will give a definition of the Tutte polynomial, as well as an overview of its
+relationship to tensor networks and the resulting approximation algorithm.
+5.3.1 The Tutte Polynomial
+The multi-variate Tutte Polynomial is defined for a graph G= (V;E)with edge
+weights w=fwegand variable qas follows:
+ZG(q;w) =å
+AEqk(A)Õ
+e2Awe (111)
+where k(A)denotes the number of connected components in the graph induced by
+A. The power of the Tutte polynomial arises from the fact that it captures nearly all
+functions on graphs defined by a skein relation . A skein relation is of the form
+f(G) =x
f(G=e)+y
f(Gne) (112)38 Jamie Smith and Michele Mosca
+where G=eis created by contracting an edge eandGneis created by deleting e.
+Oxley and Welsh [62] show that, with a few additional restrictions on f, iffis
+defined by a skein relation, then computing fcan be reduced to computing ZG. It
+turns out that many functions can be defined in terms of a skein relation, including
+1. The partition functions of the Ising and Potts models from statistical physics.
+2. The chromatic and flow polynomials of a graph G.
+3. The Jones Polynomial of an alternating link.
+The exact evaluation of the Tutte polynomial, even when restricted to planar graphs,
+turns out to be # P-hard for all but a handful of values of qandw. An exact algo-
+rithm is therefore believed to be impossible for a quantum computer. Indeed, even
+a polynomial time FPRAS seems very unlikely for the Tutte polynomial of a gen-
+eral graph and any parameters qandw. The interesting question seems to be which
+graphs and parameters admit an efficient and accurate approximation. By describ-
+ing the Tutte polynomial as the evaluation of a Tensor network, we immediately
+produce an additive approximation algorithm for the Tutte polynomial. However,
+we do not have a complete characterization of the graphs and parameters for which
+this approximation is non-trivial.
+5.3.2 The Tutte Polynomial as a Tensor Network
+Given a planar graph G, and an embedding of Gin the plane, we define the medial
+graph L G. The vertices of LGare placed on the edges of G. Two vertices of LGare
+joined by an edge if they are adjacent on the boundary of a face of G. IfGcontains
+vertices of degree 2, then LGwill have multiple edges between some pair of vertices.
+Also note that LGis a regular graph with valency 4. An example of a graph and its
+associated medial graph is depicted in figure 10.
+In order to describe the Tutte polynomial in terms of a tensor network, we make
+use of the generalized Temperley-Lieb algebra , as defined in [4]. The basis elements
+of the Temperley-Lieb algebra GT L(d)can be thought of as diagrams in which m
+upper pegs are joined to nlower pegs by a series of strands. The diagram must
+contain no crossings or loops. See figure 11 for an example of such an element. Two
+basis elements are considered equivalent if they are isotopic to each other or if one
+can be obtained from the other by ‘padding’ it on the right with a series of vertical
+strands. The algebra consists of all complex weighted sums of these basis elements.
+Given two elements of the algebra T1andT2, we can take their product T2
T1by
+simply placing T2on top of T1and joining the strands. Note that if the number of
+strands do not match, we can simply pad the appropriate element with some number
+of vertical strands. As a consequence, the identity element consists of any number of
+vertical strands. In composing two elements in this way, we may create a loop. Let
+us say that T1
T2contains one loop. Then, if T3is the element created by removing
+the loop, we define T1
T2=dT3, where dis the complex parameter of GT L(d).
+We would also like this algebra to accommodate drawings with crossings. Let
+T1be such a diagram, and T2andT3be the diagrams resulting from ‘opening’ theAlgorithms for Quantum Computers 39
+Fig. 10 A graph Gand the resulting medial graph LG.
+Fig. 11 An element of the Temperley-Lieb algebra with 6 upper pegs and 4 lower pegs.
+crossing in the two ways indicated in figure 12. Then, we define T1=aT2+bT3, for
+appropriately defined aandb.
+If we think of LGas the projection of a knot onto the plane, where the vertices
+ofLGare the crossings of the knot, we see that LGcan be expressed in terms of
+the generalized Temperly-Lieb algebra. We would like to take advantage of this in
+order to map LGto a series of tensors that will allow us to approximate the Tutte
+polynomial of G. Aharonov et. al. define such a representation rof the Temperley
+Lieb algebra. If T2GT L(d)is a basis element with mlower pegs and nupper pegs,40 Jamie Smith and Michele Mosca
+Fig. 12 An element that contains a crossing is equated to a weighted sum of the elements obtained
+by ‘opening’ the crossing.
+thenris a linear operator such that
+r(T):H
+m+1!H
+n+1(113)
+where H=Ckfor some k(which depends on the particular embedding of G) such
+that
+1.rpreserves multiplicative structure. That is, if T1hasmupper pegs and T2hasm
+lower pegs, then
+r(T2
T1) =r(T2)
r(T1): (114)
+2.ris linear. That is,
+r(aT1+bT2) =ar(T1)+br(T2): (115)
+For our purposes, kcan be upper bounded by the number of edges jE(LG)j=
+2jE(G)j, and corresponds to the value rin equation (105). To represent LG, each
+crossing (vertex) of LGis associated with a weighted sum of two basis elements,
+and therefore is represented by a weighted sum of the corresponding linear opera-
+tors,r(T1) =ar(T2) +br(T3). The minima and maxima of LGare also basis ele-
+ments of GT L(d), and can be represented as well. Finally, since LGis a closed loop,
+and has no ‘loose ends’, r(LG)must be a scalar multiple of the identity operator on
+H=Ck. This relates well to our notion that the value of a tensor network with no
+loose ends is just a scalar.
+We would like this scalar to be the Tutte polynomial of the graph G. In order to do
+this, we need to choose the correct values for a,bandd. Aharonov et. al. show how
+to choose these values so that the scalar is a graph invariant called the Kauffman
+bracket, from which we can calculate the Tutte polynomial for planar graphs. So,
+we can construct a tensor network whose value is the Kauffman bracket of LG, and
+thus gives us the Tutte polynomial of Gby the following procudure:
+1. Construct the medial graph LGand embed it in the plane.Algorithms for Quantum Computers 41
+2. Let L0
+Gbe constructed by adding a vertex at each local minimum and maximum
+ofLG. We need these vertices because we need to assign the linear operator
+associated with each minimum/maximum with a vertex in the tensor network.
+3. Each vertex vofL0
+Ghas a Temperley-Lieb element Tvassociated with it. For some
+vertices, this is a crossing; for others, it is a local minimum or maximum. Assign
+the tensor M[v] =r(Tv)to the vertex v.
+The tensor network we are interested in is then T(L0
+G;M), where M=fM[v]:v2
+V(L0
+G)g. Approximating the value of this tensor network gives us an approximation
+of the Kauffman bracket of LG, and therefore of the Tutte polynomial of G. Further-
+more, we know that L0
+Ghas maximum degree 4. Finally, we will assume that giving
+a running time of
+O
+poly(q4)
jE(G)j
e1

+(116)
+In this case, qdepends on the particular embedding f LG, but can be upper bounded
+byjEj. For more details on the representation rand quantum approximations of the
+Tutte polynomial as well as the related Jones polynomial, the reader is referred to
+the work of Aharonov et al., in particular [3] and [4].
+5.4 Tensor Networks and Statistical Mechanics Models
+Many models from statistical physics describe how simple local interactions be-
+tween microscopic particles give rise to macroscopic behaviour. See [11] for an
+excellent introduction to the combinatorial aspects of these models. The models we
+are concerned with here are described by a weighted graph G= (V;E). In this graph,
+the vertices represent particles, while edges represent an interaction between parti-
+cles. A configuration sof aq-state model is an assignment of a value from the set
+f0;:::;q1gto each vertex of G. We denote the value assigned to vbysv. For each
+edge e= (u;v), we define a local Hamiltonian he(su;sv)2C. The Hamiltonian for
+the entire system is then given taking the sum:
+H(s) =å
+e=(u;v)he(su;sv) (117)
+The sum runs over all the edges of G. The partition function is then defined as
+ZG(b) =å
+sebH(s)(118)
+where b=1=kTis referred to as the inverse temperature andkis Boltzmann’s con-
+stant. The partition function is critical to understanding the behaviour of the system.
+Firstly, the partition function allows us to determine the probability of finding the
+system in a particular configuration, given an inverse temperature b:
+Pr(s0) =ebH(s0)
+ZG(b)(119)42 Jamie Smith and Michele Mosca
+This probability distribution is known as the Boltmann distribution. Calculating the
+partition function also allows us to derive other properties of the system, such as
+entropy and free energy. An excellent discussion of the partition function from a
+combinatorial perspective can be found in [62].
+We will now construct a tensor network whose value is the partition function ZG.
+Given the graph G, we define the vertices of G0= (V0;E0)as follows:
+V0=V[fve:e2Eg (120)
+We are simply adding a vertex for each edge eofG, and identifying it by ve. We
+then define the edge set of G0:
+E0=f(x;ve);(ve;y):(x;y) =e2Eg (121)
+So, the end product G0resembles the original graph; we have simply added vertices
+in the middle of each edge of the original graph. The tensors that will be identified
+with each vertex will be defined separately for the vertex set Vof the original graph
+and the set VE=fvegof vertices we added to define G0. In each case, they will be
+dimension qtensors for a q-state model. For v2V,M[v]is an ‘identity’ operator;
+that is, it takes on the value 1 when all indices are equal, and 0 otherwise:
+(M[v])i1;::;im=(
+1 if i1=i2=:::=im
+0 otherwise(122)
+The vertices in VEencode the actual interactions between neighbouring particles.
+The tensor associated with veis
+(Mve)s;t=ebhe(s;t)(123)
+Now, let us consider the value of the tensor network in terms of q-edge colourings
+ofG:
+T(G0;M) =å
+g 
+Õ
+v2V0M[v]g!
+(124)
+where gruns over all q-edge colourings of G0. Based on the definitions of the tensors
+M[v]forv2V, we see that Õv2V0M[v]gis non-zero only when the edges incident at
+each vertex are all coloured identically. This restriction ensures that each non-zero
+term of the sum (124) corresponds to a configuration sof the q-state model. That is,
+a configuration scorresponds to a q-edge colouring gofG0where each g(e) =sv
+whenever eis incident with v2V. This gives us the following equality:Algorithms for Quantum Computers 43
+T(G0;M) =å
+g 
+Õ
+v2V0M[v]g!
+=å
+sÕ
+e=(u;v)ebhe(su;sv)
+=å
+sebH(s)
+=ZG(b)
+We now consider the time required to approximate the value of this tensor network.
+Recall that the running time is given by
+O
+poly(qD(G0))
V(G0)
e1
+(125)
+First, we observe that, if we assume that Gis connected, then jV(G0)jisO(jEj).
+We have not placed any restrictions on the maximum degree D(G0). The vertices
+ofG0of high degree must be in V, since all vertices in VEhave degree 2. Now,
+the tensors assigned to vertices of Vare just identity operators; they ensure that the
+terms of the sum from (124) are non-zero only when all the edges incident at vare
+coloured identically. As a result, we can replace each vertex v2Vwith deg (v)>3
+by(deg(v)2)vertices, each of degree 3. See figure 13 for an example. The tensor
+Fig. 13 Replacing a vertex vby(deg(v)2)vertices.
+assigned to each of these new vertices is the identity operator of rank 3. It is not
+difficult to show that this replacement does not affect the value of the tensor network.
+It does, however, affect the number of vertices in the graph, adding a multiplicative
+factor to the running time, which can now be written
+O
+poly(q)
D(G)
jEj
e1

+(126)
+For more detail on the approximation scale of this algorithm, the reader is referred
+to [10].
+Arad and Landau also discuss a useful restriction of q-state models called dif-
+ference models . In these models, the local Hamiltonians hedepend only on the dif-
+ference between the states of the vertices incident with e. That is, he(su;sv)is re-
+placed by he(jsusvj), wherejsusvjis calculated modulo q. Difference models44 Jamie Smith and Michele Mosca
+include the well-known Potts, Ising and Clock models. Arad and Landau show that
+the approximation scale can be improved when we restrict our attention to difference
+models.
+Van den Nest et al. [25, 40] show that the partition function for difference models
+can be described as the overlap between two quantum states,
+ZG=hyGj0
+@O
+e2E(G)jaei1
+A: (127)
+Van den Nest uses the state jyGito encode the structure of the graph, while each
+statejaeiencodes the strength of the local interaction at e. The actual computation
+of this overlap is best described as a tensor network. While this description does
+not yield a computational speedup over the direct method described above, it is
+an instructive way to deconstruct the problem. In the case of planar graphs, it also
+relates the partition function of a graph Gand its planar dual.
+Geraci and Lidar [34, 35] show that the Potts partition may be efficiently and
+exactly evaluated for a class of graphs related to irreducible cyclic cocycle codes.
+While their algorithm does not employ tensor networks, it is interesting to consider
+the implications of their results in the context of tensor networks. Is there a class of
+tensor networks whose value can be efficiently and exactly calculated using similar
+techniques?
+6 Conclusion
+In this chapter, we have reviewed quantum algorithms of several types: those based
+on the quantum Fourier transform, amplitude amplification, quantum walks, and
+evaluating tensor networks. This is by no means a complete survey of quantum al-
+gorithms; most notably absent are algorithms for simulating quantum systems. This
+is a particularly natural application of quantum computers, and these algorithms
+often achieve an exponential speedup over their classical counterparts.
+Algorithms based on the quantum Fourier transform, as reviewed in section 2
+include some of the earliest quantum algorithms that give a demonstrable speedup
+over classical algorithms, such as the algorithms for Deutsch’s problem and Simon’s
+problem. Many problems in this area can be described as hidden subgroup problems.
+Despite the fact that problems of this type have been well-studied, many questions
+remain open. For example, there is no efficient quantum algorithm known for most
+examples of non-Abelian groups. In particular, the quantum complexity of the graph
+isomorphism problem remains unknown.
+Similarly, the family of algorithms based on amplitude amplification and estima-
+tion have a relatively long history. While Grover’s original searching algorithm is
+the best-known example, this approach has been greatly generalized. For example,
+in [18] it is shown how the same principles applied by Grover to searching can beAlgorithms for Quantum Computers 45
+used to perform amplitude estimation and quantum counting. Amplitude amplifica-
+tion and estimation have become ubiquitous tools on quantum information process-
+ing, and are often employed as critical subroutines in other quantum algorithms.
+Quantum walks provide an interesting analogue to classical random walks. While
+they are inspired by classical walks, quantum walks have many properties that set
+them apart. Some applications of quantum walks are natural and somewhat unsur-
+prising; using a walk to search for a marked element is an intuitive idea. Others, such
+as evaluating AND-OR trees are much more surprising. Quantum walks remain an
+active area of research, with many open questions. For example, the relative compu-
+tational capabilities of discrete and continuous time quantum walks is still not fully
+understood.
+The approximation algorithm for tensor networks as described in this chapter
+[10] is relatively new. However, it is inspired by the earlier work of Aharonov et
+al. [5], as well as Yard and Wocjan [64]. The tensor network framework allows
+us to capture a wide range of problems, particularly problems of a combinatorial
+nature. Indeed, many # P-hard problems, such as evaluating the Tutte polynomial,
+can be captured as tensor networks. The difficulty arises from the additive nature
+of the approximation. One of the most important questions in this area is when this
+approximation is useful, and when the size of the approximation window renders
+the approximation trivial.
+Weve also briefly mentioned numerous other families of new quantum algorithms
+that we did not detail in this review article, and many others unfortunately go un-
+mentioned. We have tried to balance an overview of some of the classic techniques
+with a more in depth treatment of some of the more recent and novel approaches to
+finding new quantum algorithms.
+We expect and hope to see many more exciting developments in the upcoming
+years: novel applications of existing tools and techniques, new tools and techniques
+in the current paradigms, as well as new algorithmic paradigms.
+References
+1. Aaronson, S., Shi, Y .: Quantum lower bounds for the collision and the element distinctness
+problems. J. ACM 51(4), 595–605 (2004). DOI http://doi.acm.org/10.1145/1008731.1008735
+2. Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: STOC
+’01: Proceedings of the thirty-third annual ACM symposium on Theory of computing, pp.
+50–59. ACM, New York, NY , USA (2001). DOI http://doi.acm.org/10.1145/380752.380758
+3. Aharonov, D., Arad, I.: The bqp-hardness of approximating the jones polynomial (2006)
+4. Aharonov, D., Arad, I., Eban, E., Landau, Z.: Polynomial quantum algorithms for additive
+approximations of the potts model and other points of the tutte plane (2007)
+5. Aharonov, D., Jones, V ., Landau, Z.: A polynomial quantum algorithm for approximating the
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\ No newline at end of file
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+arXiv:1001.0768v3  [hep-ph]  6 May 2010UCRHEP-T483
+Gauged B−xiLorigin of Rparity and its implications
+Hye-Sung Lee
+Department of Physics, Brookhaven National Laboratory, Up ton, NY 11973, USA
+Ernest Ma
+Department of Physics and Astronomy, University of Califor nia, Riverside, CA 92521, USA
+(Dated: January, 2010)
+Gauged B−Lis a popular candidate for the origin of the conservation of Rparity, i.e. R=
+(−)3B+L+2j, in supersymmetry, but it fails to forbid the effective dimen sion-five terms arising from
+the superfield combinations QQQL,ucucdcec, anducdcdcNc, which allow the proton to decay.
+Changing it to B−xiL, where xe+xµ+xτ= 3 (with xi/negationslash= 1) for the three families, would
+forbid these terms while still serving as a gauge origin of Rparity. We show how this is achieved
+in two minimal models with realistic neutrino mass matrices , and discuss their phenomenological
+implications.
+Introduction
+In the Minimal Supersymmetric Standard Model
+(MSSM) ofparticle interactions, the imposition of Rpar-
+ity, i.e.R= (−)3B+L+2j, whereB,L, andjstand for
+baryon number, lepton number, and spin, respectively,
+serves two purposes. One is to establish a candidate for
+the dark-matter of the Universe, because the lightest su-
+persymmetric particle (LSP) is absolutely stable, being
+oddunder R. Theotheristoforbidtheotherwiseallowed
+renormalizable superfield terms LHu,LLec,LQdc, and
+ucdcdc, so that the proton does not decay as a result
+of these interactions. However, the higher-dimensional
+quadrilinear terms QQQLanducucdcecare still allowed,
+giving rise thus to effective dimension-five terms in the
+Lagrangian which would also induce fast proton decay
+[1]. This is a serious problem for grand unification, even
+at a scale as high as 1016GeV.
+With the addition of three neutral singlet superfields
+Nc, desirable for neutrino masses, gauged B−Lbecomes
+possible. In that case, Rparity is better understood the-
+oretically as a discrete remnant of B−Lbreaking. How-
+ever, the offending quadrilinear terms remain, together
+with the new term ucdcdcNc. To get rid of these terms,
+weproposeasimplesolution. Insteadofgauged B−L,we
+adopt a flavor-dependent variation, i.e. B−xiL, where
+the usual leptons still have L= 1, but xe,µ,τfor the
+three families are not equal to 1. In fact, the idea that
+there is just one Ncand that xe,µ,τ= (3,0,0) or (0,3,0)
+or (0,0,3) has already been explored in the past [2–6].
+Ifxi/negationslash= 1, then all the unwanted quadrilinear terms are
+forbidden by gauged B−xiLso that proton stability is
+assured as shown for (0 ,0,3) case in Ref. [3].
+Here we assume that there are three Nc, with each xi
+nonzero. Unlike the previous (0 ,0,3) type ofmodels, this
+opens up a new possibility that the spontaneousbreaking
+ofB−xiLby suitably chosen Higgs singlet superfields
+S1,2will result in an exact discrete residual symmetry
+which is just the usual Rparity. In the following, we
+will discuss the conditions on xiand construct two min-imal models with realistic neutrino mass matrices. We
+will examine their phenomenological constraints and the
+prognosis of their verification at the Large Hadron Col-
+lider (LHC).
+Model
+Our minimal model consists of the usual particle con-
+tent of the MSSM plus three singlet superfields Nc
+iwith
+L=−1 and two singlet superfields S1,2withL=±2.
+Under the gauged U(1)Xsymmetry of B−xiL, quarks
+have charges 1/3 and leptons have charges −xi. The
+two usual Higgs doublet superfields have charges 0 and
+the Higgs singlets S1,2have charges ∓2xS. The vari-
+ousxiandxSwill be determined by the requirements of
+anomaly cancellation and a realistic neutrino mass ma-
+trix.
+The anomaly-free conditions for the addition of U(1)X
+to the standard SU(3)C×SU(2)L×U(1)Ygauge group
+are easily written down. The [ SU(3)C]2U(1)Xand
+[U(1)X]3anomalies are automatically zero because of
+the vectorial nature of SU(3)CandU(1)X. The
+[gravity]2U(1)Xanomaly also vanishes for the same rea-
+son. The [ U(1)X]2U(1)Yanomaly is zero because the
+sum ofYcharges is zero separately for each family of
+quarksand foreach family ofleptons. The remainingtwo
+conditions, i.e. [ SU(2)L]2U(1)Xand [U(1)Y]2U(1)X, are
+given respectively by
+(3)(3)(1/3)−xe−xµ−xτ= 0, (1)
+(3)(3)[2(1 /6)2−(2/3)2−(−1/3)2](1/3)
++ [2(−1/2)2−(−1)2](−xe−xµ−xτ) = 0.(2)
+Both are satisfied if
+xe+xµ+xτ= 3. (3)
+The usual B−Lis recovered for xe=xµ=xτ= 1, and
+B−3Lτ[2] is obtained for xe=xµ= 0 and xτ= 3.2
+AsS1,2acquire vacuum expectation values, U(1)Xis
+broken. ( xe,xµ,xτ) as well as xSwill be chosen in such
+a way that a residual symmetry of B−xiLwill remain
+which is exactly Rparity, and that realistic neutrino
+masses and mixing are obtained via the canonical see-
+saw mechanism. We note that since the Higgs doublets
+do not transform under U(1)XandS1,2do not transform
+underSU(3)C×SU(2)L×U(1)Y, the resulting Zand
+Z′
+Xbosons do not mix. This avoids the stringent con-
+straint from precision electroweak measurements at the
+Zresonance. (See Ref. [7] and references therein.)
+To obtain Rparity (equivalently, matter parity) as a
+totalresidual symmetry from the breaking of B−xiL
+usingS1,2, these new singlet superfields (with L=±2
+andB= 0) should satisfy |xS|= 1/3. Since B= 1/3
+for quarks and L= 1 for leptons, we obtain the usual
+definition of Rparity for all particles if 3 xe,µ,τare odd
+integers. For a detailed discussion about conditions to
+get aZNout of aU(1) gauge symmetry, see Ref. [8] and
+references therein.
+Flavor-dependent U(1) models have been widely stud-
+ied to address many issues (e.g., see Ref. [9]). Our idea
+of having a particular discrete symmetry out of a flavor-
+dependent U(1) gauge symmetry can be viewed as a use-
+ful guide in constraining such models.
+Neutrino sector
+The general requirements discussed above would still
+allowaninfinite numberofpossiblemodels, untilrealistic
+neutrino masses and mixing are considered.
+If allxiare different for the three families of leptons,
+the charged lepton mass matrix and the Dirac mass ma-
+trix linking νiwithNc
+jare both constrained to be diag-
+onal. To obtain mixing among the three neutrinos, the
+3×3 Majorana matrix spanning Nc
+jmust have enough
+nonzero entries. However, the only sources of such terms
+areS1,2Nc
+jNc
+kandNc
+jNc
+kifxj+xk= 0. With three dif-
+ferentxe,µ,τand just ±xSto work with, this is clearly
+impossible.
+We now assume that xµ=xτ, then some simple alge-
+bra will show that there are only two solutions (recalling
+thatS1,2haveL=±2):
+(I)xS=−xe=xµ, (4)
+(II)xS=−xµ= (xe+xµ)/2. (5)
+Together with Eq. (3), this means that
+(I)xe,µ,τ= (−3,3,3), xS= 3, (6)
+(II)xe,µ,τ= (9,−3,−3), xS= 3.(7)
+In model (I), the 3 ×3 Majorana mass matrix spanning
+Nc
+jhas all nonzero entries: Nc
+eNc
+µ,τare invariant mass
+terms,Nc
+eNc
+ecomes from /angbracketleftS2/angbracketright, andNc
+µ,τNc
+µ,τcome from/angbracketleftS1/angbracketright. Inmodel(II), the Nc
+eNc
+eentryiszero, butallothers
+arenonzero: Nc
+µ,τNc
+µ,τcome from /angbracketleftS2/angbracketright, andNc
+eNc
+µ,τfrom
+/angbracketleftS1/angbracketright. Both are general enough for obtaining a realistic
+neutrinomassmatrix,withmixingamongallthreelepton
+families.
+Baryon triality
+As discussed above, the requirement of a realistic neu-
+trino mass matrix using only S1,2demands |xS|= 3 in-
+stead of |xS|= 1/3. This means that the totaldiscrete
+symmetry from B−xiLis not just Rparity any more.
+It has been extended to a larger symmetry. Following
+the general arguments in Ref. [8], we find that our total
+discrete symmetry is now Z6, which is a direct product
+ofRparity and baryon triality ( Z6=R2×B3). Un-
+der baryon triality ( B3=Z3) [10], baryons transform as
+ω= exp(2πi/3), so that the proton is absolutely stable,
+being the lightest particle with that charge. This result
+came as a pleasant surprise, because it was not our inten-
+tion to construct a model which contains B3. It points
+to a possible deep connection between neutrino mass and
+proton stability.
+In regard to baryon stability, there are other relevant
+anomaly-free U(1) gauge symmetry models [8, 11, 12],
+as well as discrete gauge symmetry models [10, 13, 14].
+Especially, we note that when model (I) is shifted by
+some hypercharge, it can reach the equivalent form of a
+model in Ref. [11].
+e−µ−τnonuniversality
+The salient prediction of our proposal is the existence
+of a new neutral gauge boson Z′
+X. It does not mix with
+the electroweak Zat tree level, but it couples to all
+quarks and leptons in a specified way. In particular, it
+breakse−µ−τuniversality. Thus it may be important
+as a one-loop effect [4] in the precision measurements of
+Z→ℓ+ℓ−. However, this effect is proportional to x2
+i,
+and its contribution to nonuniversality is zero for model
+(I).Asformodel(II), thepredictionisthatΓ( Z→e+e−)
+should be bigger than Γ( Z→µ+µ−) and Γ(Z→τ+τ−),
+and it is proportional to (81 −9)g2
+X. The present world
+averages are [15]
+Γe= 83.91±0.12 MeV, (8)
+Γµ= 83.99±0.18 MeV, (9)
+Γτ= 84.08±0.22 MeV. (10)
+After adding a kinematical correction of 0.19 MeV to Γ τ,
+we find the deviation of Γ efrom the average of Γ µand
+Γτto be bounded at 95% C.L. by
+∆Γe/Γµ,τ<0.002. (11)3
+Letr≡M2
+Z′
+X/M2
+Z, then the one-loopradiativecorrection
+toZ→ℓ+ℓ−fromZ′
+Xexchange in model (II) is given
+by [16]
+∆Γe
+Γµ,τ=(81−9)g2
+X
+8π2F2(r), (12)
+where
+F2(r) =−7
+2−2r−(2r+3)lnr+2(1+r)2×
+/bracketleftbiggπ2
+6−Li2/parenleftbiggr
+1+r/parenrightbigg
+−1
+2ln2/parenleftbiggr
+1+r/parenrightbigg/bracketrightbigg
+.(13)
+In the above, Li 2(x) =−/integraltextx
+0(dt/t)ln(1−t) is the Spence
+function. For r≫1 (i.e.M2
+Z′
+X≫M2
+Z) which will be
+required by Tevatron data in any case (as shown below),
+F2≃r−1[11/9+ (2/3)lnr] and the resulting numerical
+bound is
+g2
+X/M2
+Z′
+X∼<0.05 TeV−2, (14)
+to a good approximation. There is also an overall correc-
+tion to Γ( Z→hadrons) /Γ(Z→leptons), but that effect
+may be absorbed into the value of αSin QCD.
+LEP2 contact interaction
+The agreement of the SM and the LEP2 data provides
+the indirect bounds on the Z′
+Xmass. The strongest one
+for our models (I) and (II), where µandτhave the same
+chargesand ehas the chargeof oppositesign, comes from
+thee+e−→µ+µ−,τ+τ−channel with Λ−
+VV= 16 TeV
+[17]. (See Ref. [18] for some useful discussions.)
+TheZ′
+Xmass is bounded by
+M2
+Z′
+X∼>g2
+X
+4π|xexµ|(Λ−
+VV)2(15)
+for sufficiently large MZ′compared to the LEP2 energy.
+It results
+MZ′
+X∼>1.4 TeV in model (I) , (16)
+MZ′
+X∼>2.3 TeV in model (II) ,(17)
+forgX= 0.1.
+Phenomenology of Z′
+X
+The direct production of Z′
+Xis possible at hadron col-
+lidersthroughits couplingto quarks. Its decayto leptons
+is then a clear signature. At present, there is a Tevatron
+limit on the cross section σ(p¯p→Z′
+X→e+e−) at a
+center-of-mass energy Ecm= 1.96 TeV, based on an inte-
+grated luminosity of L= 2.5 fb−1[19], and similarly for
+dimuons [20]. To compare against these results, we takeFIG. 1: Tevatron bounds on Z′
+Xmass in models (I) and (II).
+gX= 0.1 for definiteness. We assume Γ Z′
+X= 0.01MZ′
+Xfor model (I) and Γ Z′
+X= 0.04MZ′
+Xfor model (II), which
+are approximately their true values if they only decay
+into particles of the Standard Model (SM), i.e. not their
+superpartners nor the additional singlets.
+For the numerical analysis, we use CompHEP/CalcHEP
+[21, 22] and the parton distribution functions of CTEQ6L
+[23]. Since the Tevatron bounds from dielectron and
+dimuon data are similar for the same coupling, we show
+only the dielectron result. We find MZ′
+X∼>830 GeV for
+model (I) and MZ′
+X∼>940 GeV for model (II) as shown
+in Fig. 1. (Since |xµ|<|xe|in model (II), the bound
+from the dimuon data is less stringent.) We note also
+that the e−µ−τnonuniversality constraint of Eq. (14)
+is easily satisfied.
+For the LHC discovery reach (with the design energy
+Ecm= 14 TeV) through the dilepton Z′
+Xresonance, we
+use cuts pT>20 GeV, |η|<2.4 (for each lepton) and
+|minv(ℓ+ℓ−)|<3ΓZ′
+X. SM background at the LHC with
+these cuts is negligible, and we just require 10 signal
+eventstoclaim its discoveryat the LHC forafixed flavor.
+Figs. 2(a) and 2(b) show the LHC discovery reach us-
+ing dileptons for models (I) and (II) respectively. From
+the dilepton resonance only, model (I) cannot be dis-
+tinguished from the flavor-independent case of B−L.
+In model (I), Z′
+Xwill be revealed by both e+e−and
+µ+µ−channels at the same luminosity ( L≃1 fb−1
+forMZ′
+X= 1.5 TeV). In model (II), the µ+µ−reso-
+nance will need a luminosity about an order of magni-
+tude larger than that for the e+e−resonance because
+σ(µ+µ−)/σ(e+e−)≃(−3)2/(9)2.
+Since Higgs doublets have zero charges under B−xiL,
+the channels which require nonzero charges such as the
+6-lepton resonance discussed in Ref. [24] will be absent.
+In the presence of Z′
+Xat the TeV scale, a (predomi-
+nantly right-handed) sneutrino as well as the usual neu-
+tralino can be a good LSP dark-matter candidate [25].4
+(a) (b)
+FIG. 2: The LHC discovery reach for (a) model (I) and (b) model (II). The required luminosity is the same for both e+e−
+andµ+µ−resonances in model (I). The required luminosity for the e+e−(solid line) resonance is about an order of magnitude
+smaller than that for the µ+µ−(dashed line) resonance in model (II).
+One-loop charged lepton processes
+There are two sources of lepton flavor violation. One
+comes from explicit interactions linking NcthroughS1,2;
+the other through the mismatch of lepton and slepton
+mass matrices (which is common to all supersymmetric
+models). Since we choose xe/negationslash=xµ=xτ, the latter ap-
+plies only to the µ−τsector, whereas the former applies
+to all leptons, but for one-loop processes such as µ→eγ,
+they are negligible because of the smallness of neutrino
+masses.
+The muon anomalous magnetic moment ( aµ) does not
+violate lepton flavor, so it has a contribution from Z′
+X,
+i.e. [26]
+∆aµ=3g2
+X
+4π2m2
+µ
+M2
+Z′
+X(18)
+for both models (I) and (II). Using the Tevatron bounds
+onMZ′
+XwithgX= 0.1, we find this to be at most of
+order 10−11, which is negligible compared to the present
+experimental accuracy. On the other hand, since we are
+considering a supersymmetric model, there are one-loop
+contributions to aµfrom the neutralinos and charginos,
+which may account for the possible deviation in its mea-
+surement at the Brookhaven Alternating Gradient Syn-
+chrotron [27].
+Conclusion
+We have proposed a new U(1) gauge symmetry B−
+xiL, withxe,µ,τ= (−3,3,3) [model (I)] or (9 ,−3,−3)
+[model (II)], in the context of supersymmetry with three
+neutral singlets Nc
+e,µ,τ. The spontaneous breaking of this
+U(1)Xby the addition of singlets S1,2withL=±2 andxS= 3 accomplishes three objectives. (i) The conven-
+tionalRparity survives as an exact discrete symmetry,
+desirable for having the LSP as a good dark-matter can-
+didate. (ii) Realistic neutrino masses and mixing are ob-
+tained. (iii) An exact residual Z3symmetry, i.e. baryon
+trialityB3, emerges which makes the proton absolutely
+stable.
+The neutral gauge boson of this new U(1)Xhas large
+couplings to leptons over quarks: three or nine times
+larger than in the case of B−L. Using present Tevatron
+bounds with gX= 0.1, we find MZ′
+Xto be greater than
+830 GeV in model (I), and 940 GeV in model (II). The
+indirect bounds from LEP2are 1 .4 TeV in model (I), and
+2.3TeVin model (II). It shouldbe accessibleat the LHC,
+possibly at a very early stage. To have a background-free
+signal of 10 dilepton events at the LHC, the Z′
+Xwith
+MZ′
+X= 1.5 TeV in model (I) may be discovered for an
+integratedluminosityofonlyabout1fb−1. Inmodel(II),
+because of the flavor-dependent charges, the event rate
+of thee+e−resonance is predicted to be nine times that
+ofµ+µ−.
+This work was supported by U. S. Department of En-
+ergy Grants No. DE-AC02-98CH10886 (HL) and No.
+DE-FG03-94ER40837 (EM).
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\ No newline at end of file
diff --git a/1001.0769.txt b/1001.0769.txt
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+arXiv:1001.0769v1  [astro-ph.CO]  5 Jan 2010TTK-10-01
+Tight constraints on F- and D-term hybrid inflation scenario s
+Richard Battye,1,∗Bj¨ orn Garbrecht,2,†and Adam Moss3,‡
+1Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy,
+University of Manchester, Manchester, M13 9PL UK
+2Institut f¨ ur Theoretische Physik, RWTH Aachen, 52056 Aach en, Germany
+3Department of Physics & Astronomy, University of British Co lumbia, Vancouver, BC, V6T 1Z1 Canada
+(Dated: November 3, 2018)
+We use present cosmological data from the cosmic microwave b ackground, large-scale structure
+and deuterium at high redshifts to constrain supersymmetri c F- and D-term hybrid inflation sce-
+narios including possible contributions to the CMB anisotr opies from cosmic strings. Using two
+different realizations of the cosmic string spectrum, we find that the minimal version of the D-term
+model is ruled out at high significance. F-term models are als o in tension with the data. We also
+discuss possible non-minimal variants of the models.
+PACS numbers: 98.80.Cq, 98.80.Jk
+Introduction: PresentobservationsoftheUniversepro-
+vide strong evidence for a topology which is close to spa-
+tially flat and a close to scale invariant spectrum of den-
+sity fluctuations. These are both predications of the sim-
+plest models of inflation based on the slow-roll approx-
+imation and hence provide prima-facie evidence for an
+inflationary epoch in the Early Universe. One of the ba-
+sic features of inflation is that the potential needs to be
+very shallow in order to give rise to perturbations with
+the observed amplitude. This can be difficult to arrange
+in many particle physics motivated scenariossince it typ-
+ically requires parametric fine-tuning in the theory or a
+trans-Planckian field-range for the inflaton.
+One class of models which naturally circumvents this
+issue are the Supersymmetric (SUSY) Hybrid Inflation
+scenarios which have F- and D-term variants [1–3]. In
+these models the tree level potential is completely flat
+and the slow-roll is induced by the radiative corrections.
+There are two key observational features of the minimal
+version of these scenarios: a spectral index of density
+fluctuations, nS, which is0.98–1.0,andcosmic stringsare
+often produced during the phase transition which ends
+inflation [4], forexample, if the brokensymmetry is U(1).
+For typical model parameters the amplitude of cosmic
+microwave background (CMB) anisotropies from strings
+will be∼10% of those produced by inflation [5].
+This is particularly interesting since the data from the
+Wilkinson Microwave Anisotropy Probe (WMAP) 3 year
+release suggested that nS= 0.958±0.016 which ap-
+pears to disfavor such models [6]. In subsequent work we
+pointed out that this discrepancy could be ameliorated
+by the inclusion of a component of the CMB anisotropies
+from cosmic strings [7]. In particular, we found that the
+minimal version of the F- and D-term scenarios, which
+have6freeparameters, fitted the dataaswellasorbetter
+∗Electronic address: rbattye@jb.man.ac.uk
+†Electronic address: garbrecht@physik.rwth-aachen.de
+‡Electronic address: adammoss@phas.ubc.cathan the standard 6 parameter fit. Similar results, based
+on a different cosmic string spectrum, were reported by
+Beviset al, although they did not specifically refer to the
+SUSY hybrid inflation models, rather they investigated a
+6 parameter model with the string tension, Gµ, allowed
+to vary and nS= 1 fixed [8].
+Since then cosmological data has moved on. WMAP
+have released their 5 year data [9], other CMB data
+on smaller angular scales has been published [10], the
+SDSS has been used to compute the matter power spec-
+trum [11] and tight constraints have been derived on
+the baryon density, quantified by the value relative to
+the critical density, Ω b, using Big Bang Nucleosynthe-
+sis (BBN) from measurements of deuterium at high red-
+shift [12]. The last two of these are particularly interest-
+ing in this context since we found in our earlier analysis
+that models which fitted the data due to the inclusion a
+string component with nS≈1 typically have values of
+h=H0/100kmsec−1Mpc−1and Ω bh2which are larger
+than the standard values [7].
+In thisletterwe will revisit the constraints on the
+SUSY hybrid inflation models focusing on the minimal
+F- and D-term variants and on those that are extended
+by a possible additional mass-square term for the infla-
+ton [13–16]. We will take into account the cosmic string
+component using two possible models. In model A we
+use the unconnected segment model [17] (USM) to pro-
+ducethestringspectrumusinganevolutionmodelforthe
+correlation length, rms string velocity and string wiggli-
+ness. This spectrum was used in our previous work [7]
+and also in refs. [18]. We also include results based on
+another spectrum created using the USM, which we will
+call model B. This spectrum, with different parameters
+from model A, was designed to match the results of the
+Abelian-Higgssimulationsreportedinref.[8](seeref.[19]
+for a discussion). It is our opinion that model A is likely
+to be closest to the true cosmic string spectrum, but the
+twospectraspanthedifferentpossibilitiesreportedinthe
+literature. We note that since the cosmic stringspectrum
+will be∼10% of the total, the level of accuracy required
+is much less than for the adiabaticspectrum, and the two2
+models give rise to qualitatively similar results.
+Methodology: Ourbasicmethods will be essentiallythe
+same as those described in ref. [7]. The F-term super-
+potential is given by W=κˆS(ˆ¯GˆG−M2) where ˆSis
+the singlet inflaton superfield and ˆG,ˆ¯Gare the waterfall
+fields responsible for the topological defects [4]. We in-
+clude the radiative corrections and minimal supergravity
+corrections, but ignore the curvature and tadpole terms.
+These terms were shown to have negligible effect in pre-
+vious work, where we assumed a positive sign for the tad-
+pole term. For the D-term model we use W=κˆSˆ¯GˆG.
+Spontaneous breaking of U(1) is induced by the D-term
+contribution g2(|G|2− |¯G|2+M2)2/8 to the potential.
+Without supergravity corrections, the predictions for in-
+flation and strings from the D-term model do not depend
+on the gauge coupling g. Taking account of minimal su-
+pergravity corrections, we find that for values of g>∼0.1,
+the spectrum receives an additional blue-tilt, which cor-
+responds to a less-good fit to the data [7]. When com-
+pared to small g, the hybrid inflation prediction for nS
+lies substantially above the central value from the stan-
+dard 6-parameter fit. In this work, we neglect minimal
+supergravity corrections, corresponding to a restriction
+of parameter space to g<∼0.1. Therefore in both the
+F- and D-term cases, the inflation model is specified by
+two parameters: the dimensionless coupling constant, κ,
+and the mass scale, M. It is possible to compute the
+three observable quantities nS,Gµand the scalar ampli-
+tudePRfrom these two parameters, and hence the fit
+is for 6 parameters (with the baryon density, Ω bh2, cold
+dark matter density Ω ch2, acoustic scale, θAand optical
+depth,τ, being the other 4). The reheat temperature is
+fixed to be TR= 109GeV but allowing this to vary only
+has a minor effect on our results.
+Werefertomoredetailsonthemodelstoref.[7], butin
+the presentcontext, it isuseful torecallthe basicfeatures
+distinguishing the F- and D-term models phenomenolog-
+ically. For κ2/g2<1/2 the string tension for the same
+value of Mis larger in the D-term model than the F-
+term. Hence, in order to comply with upper bounds on
+the string contribution to the CMB, D-term models pre-
+dict a smaller M, which also leads to a close to scale
+invariant spectrum, nS≈1.
+We use COSMOMC [20] to perform the analysis.
+We include data from WMAP5 [9], other CMB exper-
+iments [10] and the SDSS [11]. In addition we will also
+include constraints on Ω bh2from metal poor damped
+Lyman-αsystems. These have been used to compute
+the deuterium abundance at z≈2.6. One finds that
+log(D/H) =−4.55±0.03, which implies that Ω bh2=
+0.0213±0.0010 [12].
+Results: We find that the 4 cosmological parameters
+areclosetothe valuespreferredbythe standard6param-
+eter fit. In Fig. 1, we present 2D likelihoods for various
+parametersofthe F-andD-termmodelsforcosmicstring
+spectra A and B. The best-fitting values of Mandκare
+given in table I. We see that the values are not the samelog10 [κ]M (1016 GeV)
+−4−3−2−100.20.40.60.81
+−5−4−3−2−1log10 [κ]
+log10[κ]ns
+−5−4−3−2−10.9750.980.9850.990.99511.005
+log10[κ]−4−3−2−1
+FIG. 1: 2D likelihoods for D-Term (red, smaller values of κ)
+andF-term (blue, larger values of κ) for string model A (left
+set of two) and B (right set of two) using CMB+SDSS data.
+for the two string spectra. The central values for model
+A have changed from those reported in ref. [7] due to the
+inclusion of the new CMB and the SDSS data, although
+they remain within ∼1σ. The size of the errorbars have
+not changed significantly. The results for model B are
+somewhat different reflecting different ranges of Gµand
+nSwhich are favoured using the different string spec-
+tra [19]. Of particular importance is that model A re-
+strictsnS= 0.958±0.013 in a 7 parameter fit including
+Gµas the extra parameter, which is very similar to the 6
+parameter case. However, nS= 0.970±0.016 for model
+B, which is more compatible with nS= 1.
+One can define the relative likelihood of two models,
+labelled 1 and 2, by ∆ χ2=−2log(L1/L2). We will take
+model 1 to be the standard 6 parameter model using
+nSandASto define the initial power spectrum. Pos-
+itive values of this quantity indicate that model 2 fits
+the data better than the standard, whereas negative val-
+ues indicate a less good fit. This quantity was used in
+ref. [6] to quantify the level to which various variants
+of the standard cosmological model (for example, those
+with no reionization, or no dark matter) were poor fits to
+the data, We present the computed values of ∆ χ2for the
+F- and D-term models using the two cosmic string spec-
+tra in table II. We see that if only CMB data is included
+there is little to tell between the 6 parameter model and
+the F/D-term models, although the D-term model using
+string spectrum A is the least good fit. If one includes
+more data from SDSS and BBN, typically the fit of the
+hybrid inflation models become less good.
+Concentrating on the case of CMB+SDSS+BBN, it is
+clear that the D-term model using both string spectra
+is a much worse fit than the 6 parameter model with
+∆χ2=−17.6 for string spectrum A and -12.2 for B. This
+shows that the D-term model is severely in tension with3
+log10κM/1016GeV
+F(A)−2.79±0.300.450±0.062
+F(B)−1.94±0.340.563±0.038
+D(A)−4.55±0.210.196±0.030
+D(B)−3.84±0.230.330±0.047
+TABLE I: Constraints on minimal FandD-term parameters
+using string models A and B.
+the data. This is because the best-fitting D-term models
+and those with nS= 1, in the absence of the BBN prior,
+typically favour values of Ω bh2∼0.024−0.025. The
+situation ismorecomplicated in the F-termcase. In both
+cases the F-term model is disfavoured relative to the 6
+parameter model, but for string spectrum B the ∆ χ2is
+only−5.6. This would not be enough to exclude the
+model with high confidence, whereas for spectrum A the
+∆χ2=−12.2, which would. We have already alluded to
+the reason for this: spectrum B allows for larger values
+ofnSthan the spectrum A.
+In order to calibrate our interpretation of these values
+we have done two things. First, we computed ∆ χ2for
+a 5 parameter model with τ= 0. For each combination
+of data, we find ∆ χ2≈ −20, with roughly 5 σsignifi-
+cance that τ/negationslash= 0 (for example, CMB data alone gives
+τ= 0.090±0.017). We have also done the same for
+models with nS= 1, that is a Harrison-Zel’dovich (HZ)
+spectrum. For CMB+SDSS data we find ∆ χ2=−11.8,
+corresponding to the HZ model being excluded at 3 .4σ,
+and for CMB+SDSS+BBN data ∆ χ2=−16.8, corre-
+sponding to 4 .1σ(nS= 0.951±0.012). These values are
+presented in table II.
+Comparing the ∆ χ2for the F- and D-term models we
+can conclude that the D-term model is excluded at a
+level somewhere in the range of 3-4 σeven if strings are
+included, withthepreciseleveldependentonwhichstring
+spectrum is correct. Meanwhile the F-term model would
+be excluded at around 3 σif string spectrum A is correct,
+but cannot be excluded is spectrum B is correct.
+The statistical strength of our results is very strongly
+related to the value of D/Hmeasured in ref. [12] and
+the subsequent interpretation in terms of BBN. In order
+to explore the effects of systematic offsets in the value of
+Ωbh2on our results we have computed the value of ∆ χ2
+for a prior of ωb= Ωbh2= ¯ωb±0.001 (i.e. a different
+central value with the same statistical error as ref. [12]).
+For the D-term model with string spectrum A, the vari-
+ation inχ2for CMB+SDSS+BBN can be approximated
+by ∆χ2≈3160¯ωb−84.5. Even if the central value were
+to increase by 0.002, the model would still be in tension
+with the data, ∆ χ2≈ −11.
+Non-minimal models: Non-minimal F- and D-term
+models can be considered; these typically involve the in-
+clusion of a negative mass-square term in the potential
+and can allow for values of nSlower than 0.98 due to an
+enhanced negative curvature of the potential. In addi-
+tion, at the point of horizon exit, the potential can be
+effectively flattened. This leads to a lower energy-scaleCMB+SDSS+BBN+SDSS/BBN
+τ= 0-21.7-20.4-20.8 -20.2
+HZ-9.4-11.8-14.6 -16.8
+F(A)-1.2-4.4-10.2 -13.2
+F(B)1.20.2-4.0 -5.6
+D(A)-4.0-8.6-14.2 -17.6
+D(B)0.2-2.8-9.6 -12.2
+TABLE II: ∆ χ2=−2log(L1/L2) for minimal 6 parameter F
+andD-term scenarios. These are defined as model 2; model
+1 is the standard 6 parameter power law. We also present
+results relative to the 5 parameter τ= 0 and HZ models.
+of inflation and therefore to lower values for Mand a
+smaller string tension. In ref. [7] we considered a par-
+ticular class of F-term models [13] characterized by the
+addition of the potential term c2
+HH2S2, whereHdenotes
+the Hubble rate. This “Hubble-dependent” mass term
+genericallyarisesinthepresenceofanon-minimalK¨ ahler
+potential from the F-term that drives inflation.
+In the absence of F-terms, even for a non-minimal
+K¨ ahler potential, no such additional mass-square correc-
+tion can occur. This is why for the simplest realizations
+of D-term inflation, the exclusion of mass-square correc-
+tions is well-motivated. However, there are particular
+modelswheresub-dominantF-terms( e.g.fromcompact-
+ification in string models or from sneutrino potentials),
+thatdonotdriveinflation, maybepresent[14–16]. These
+can lead to mass-square corrections for the D-term po-
+tentials that may redden the spectrum and reduce the
+string contribution to the CMB.
+We aim to formulate the model in such a way that the
+number of parameters remains small ( e.gno more than
+7 parameters) and presume strings form. This can be
+achieved by the restriction to g<∼0.1 and adding the
+potential term in the form of g2c2
+HS2. This way, the
+predictions remain independent of the gauge coupling g.
+We note that in presence of a negative mass-squareterm,
+there may also be models with g>∼0.1, that are no longer
+disfavoured by the data. Unless one applies a restriction
+to more specific models, a systematic study of this region
+of parameter space would require additional free param-
+eters in the analysis and is therefore not performed here.
+In Fig. 2 we show 2D likelihoods for the non-minimal
+models using string spectrum A. In the F-term case, neg-
+ative values of c2
+Hlead to an increased red-tilt in nS, al-
+leviating the tension with data present for c2
+H= 0. The
+fit is slightly better than the 6 parameter model with
+∆χ2= 1.2. We find marginalized values of log10κ=
+−2.28±0.46,M= (0.346±0.065)×1016GeV and
+c2
+H=−0.027±0.011.
+For the D-term model the situation is more compli-
+cated, since for fixed PRandκthe potential can admit
+twosolutions, each with different Mand parameters nS
+andGµ. In the analysis we treat these independently,
+running separateMCMC chainsfor eachsolution branch.
+The results of the analysis are shown in Fig. 2. The4
+nsCH2
+−0.06−0.04−0.0200.02
+log10 [κ]
+ns− log10[1018 CH2/GeV2]
+0.920.940.960.981−2024
+log10[κ]−4−3−2−1
+FIG. 2: 2D likelihoods for non-minimal F- (top set of two)
+and D-term (bottom set of two) for string model A. In the
+D-term case the two sets of contours show the two solutions
+admitted by the potential, as explained in the text.
+solution for small κis essentially the same as the min-
+imal case. Here, Mis higher and a non-zero c2
+H= 0
+does not effect the inflation dynamics. We find log10κ=
+−4.45±0.22 andM= (0.203±0.032)×1016GeV, with
+∆χ2=−8.6.
+The other solution has a lower value of M, and a non-
+zeroc2
+Hflattens the potential near horizon exit. One gets
+sufficient e-foldings when the inflaton is placed close to
+the top of the potential and the resulting nSis redder,
+with the same slightly improved ∆ χ2= 1.2 as in the
+F-term model. For this branch larger values of κare al-lowed,sincethelower Mleadstoasmallerstringtension.
+We find log10κ=−3.78±0.44,M= (0.116±0.071)×
+1016GeV and −log10/parenleftbig
+1018c2
+H/GeV2/parenrightbig
+= 1.63±1.50. We
+note that the upper bound on κis related to the upper
+bound on Gµ.
+Conclusions: Minimal models of hybrid inflation rely
+only on two parameters ( κandM), and they do not suf-
+fer from the potential problem of trans-Planckian values
+of the inflaton field. Therefore they are predictive and it
+is possible to derive parametric constraints from cosmo-
+logical data. In Refs. [7, 8], it has been found that a close
+to scale invariant spectrum of adiabatic perturbations is
+in agreement with CMB data, when a sizable string com-
+ponent is present. In particular, minimal models of hy-
+brid inflation which predict these features appeared to
+be viable in that light, a finding that may open inter-
+esting directions in model building [16]. Using recent
+CMB data, SDSS and measurements of the deuterium
+abundance leads us to a revison of this picture: Mini-
+mal D-term models are ruled out at a level that can be
+estimated to be ≈4σ(the same is true for the HZ spec-
+trum) and the tension between minimal F-term models
+and the data is increasing. Non-minimal scenarios in-
+clude the negative mass-square term models discussed
+here and also the possibility of adding a negative-sign
+tadpole term [21]. This way, one can entirely remove
+tension with the data, but at the same time, also the
+very distinct predictions of minimal hybrid inflation for
+the range of the spectral index are lost.
+Acknowledgments: This researchwas supported by the
+Natural Sciences and Engineering Research Council of
+Canada and the British Columbia Knowledge Develop-
+ment Fund. We thank Douglas Scott for useful discus-
+sions.
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+arXiv:1001.0770v1  [astro-ph.HE]  5 Jan 20102009 Fermi Symposium, Washington, D.C., Nov. 2-5 1
+VERITAS Observations ofBlazars
+W. Benbow forthe VERITAS Collaboration
+Harvard-Smithsonian Center for Astrophysics, F.L. Whippl e Observatory, PO Box 6369, Amado, AZ 85645,
+USA
+The VERITAS array of four 12-m diameter imaging atmospheric -Cherenkov telescopes in southern Arizona is
+used to study very high energy (VHE; E >100 GeV) γ-ray emission from astrophysical objects. VERITAS is
+currently the most sensitive VHE γ-ray observatory in the world and one of the VERITAS collabor ation’s Key
+Science Projects (KSP) is the study of blazars. These active galactic nuclei (AGN) are the most numerous class
+of identified VHE sources, with ∼30 known to emit VHE photons. More than 70 AGN, almost all of wh ich
+are blazars, have been observed with the VERITAS array since 2007, in most cases with the deepest-ever VHE
+exposure. These observations have resulted in the detectio n of VHE γ-rays from 16 AGN (15 blazars), including
+8 for the first time at these energies. The VERITAS blazar KSP i s summarized in this proceeding and selected
+results are presented.
+1. Introduction
+Active galactic nuclei are the most numerous class
+of identified VHE γ-ray sources. These objects emit
+non-thermalradiationacross ∼20ordersofmagnitude
+in energy and rank among the most powerful particle
+accelerators in the universe. A small fraction of AGN
+possess strong collimated outflows (jets) powered by
+accretion onto a supermassive black hole (SMBH).
+VHEγ-ray emission can be generated in these jets,
+likely in a compact region very near the SMBH event
+horizon. Blazars, a class of AGN with jets pointed
+along the line-of-sight to the observer, are of par-
+ticular interest in the VHE regime. Approximately
+30 blazars, primarily high-frequency-peaked BLLacs
+(HBL), are identified as sources of VHE γ-rays, and
+some are spectacularly variable on time scales com-
+parable to the light crossing time of their SMBH ( ∼2
+min; [1]). VHE blazar studies probe the environment
+very near the central SMBH and address a wide range
+of physical phenomena, including the accretion and
+jet-formation processes. These studies also have cos-
+mological implications, as VHE blazar data can be
+used to strongly constrain primordial radiation fields
+(see the extragalactic background light (EBL) con-
+straints from, e.g., [2, 3]).
+VHE blazars have double-humped spectral energy
+distributions (SEDs), with one peak at UV/X-ray en-
+ergies and another at GeV/TeV energies. The ori-
+gin of the lower-energy peak is commonly explained
+as synchrotron emission from the relativistic electrons
+in the blazar jets. The origin of the higher-energy
+peak is controversial, but is widely believed to be the
+result of inverse-Compton scattering of seed photons
+off the same relativistic electrons. The origin of the
+seed photons in these leptonic scenarios could be the
+synchrotron photons themselves, or photons from an
+external source. Hadronic scenarios are also plausible
+explanations for the VHE emission, but generally are
+not favored.
+Contemporaneous multi-wavelength (MWL) obser-vations of VHE blazars, can measure both SED peaks
+and are crucial for extracting information from the
+observations of VHE blazars. They are used to con-
+strain the size, magnetic field and Doppler factor of
+the emission region, as well as to determine the origin
+(leptonic or hadronic) of the VHE γ-rays. In leptonic
+scenarios, such MWL observations are used to mea-
+sure the spectrum of high-energy electrons producing
+the emission, as well as to elucidate the nature of the
+seed photons. Additionally, an accurate measure of
+the cosmological EBL density requires accurate mod-
+eling of the blazar’s intrinsic VHE emission that can
+only be performed with contemporaneous MWL ob-
+servations.
+2.VERITAS
+VERITAS, a stereoscopic array of four 12-m
+atmospheric-Cherenkovtelescopes located in Arizona,
+is used to study VHE γ-rays from a variety of astro-
+physical sources [4]. VERITAS began scientific obser-
+vationswithapartialarrayinSeptember2006andhas
+routinely observed with the full array since Septem-
+ber 2007. The performance metrics of VERITAS in-
+clude an energy threshold of ∼100 GeV, an energy
+resolution of ∼15%, an angular resolution of ∼0.1◦,
+and a sensitivity yielding a 5 σdetection of a 1% Crab
+Nebula flux object in <30 hours1. VERITAS has an
+active maintenance program (e.g. frequent mirror re-
+coating and alignment) to ensure its continued high
+performance over time, and an upgrade improving
+both the camera (higher quantum-efficiency PMTs)
+and the trigger system has been proposed to the fund-
+ing agencies.
+1A VERITAS telescope was relocated during Summer 2009,
+increasing the array’s sensitivity by a factor ∼1.3.
+eConf C0911222 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
+3. VERITASBlazar KSP
+VERITAS observes for ∼750 h and ∼250 h each
+year during periods of astronomical darkness and par-
+tial moonlight, respectively. The moonlight observa-
+tionsarealmostexclusivelyusedforablazardiscovery
+program, and a large fraction of the dark time is used
+for the blazar KSP, which consists of:
+•A VHE blazar discoveryprogram ( ∼200 h / yr):
+Each year ∼10 targets are selected to receive
+∼10 h of observations each during astronomi-
+cal darkness. These data are supplemented by
+discovery observations during periods of partial
+moonlight.
+•A target-of-opportunity (ToO) observation pro-
+gram (∼50 h / yr): VERITAS blazar obser-
+vations can be triggered by either a VERI-
+TAS blazar discovery, a VHE flaring alert ( >2
+Crab) from the blazar monitoring program of
+the Whipple 10-m telescope or from another
+VHE instrument, or a lower-energy flaring alert
+(optical, X-ray or Fermi-LAT). Should the guar-
+anteedallocationbe exhausted, furthertime can
+be requested from a pool of director’s discre-
+tionary time.
+•Multi-wavelength (MWL) studies of VHE
+blazars ( ∼50 h / yr + ToO): Each year one
+blazar receives a deep exposure in a pre-planned
+campaign of extensive, simultaneous MWL (X-
+ray,optical, radio)measurements. ToOobserva-
+tion proposals for MWL measurements are also
+submitted to lower-energy observatories (e.g.
+Swift) and are triggered by a VERITAS discov-
+ery or flaring alert.
+•Distant VHE blazar studies to constrain the ex-
+tragalactic background light (EBL): Here dis-
+tant targets are given a higher priority in the
+blazar discovery program, as well as for the
+MWL observations of known VHE blazars, par-
+ticularly those with hard VHE spectra.
+4. BlazarDiscovery Program
+The blazars observed in the discovery program are
+largely high-frequency-peaked BL Lac objects. How-
+ever, the program also includes IBLs (intermediate-
+peaked) and LBLs (low-peaked), as well as flat spec-
+trum radio quasars (FSRQs), in an attempt to in-
+creasethe types ofblazarsknownto emit VHE γ-rays.
+The observed targets are drawn from a target list con-
+taining objects visible to the telescopes at reasonable
+zenith angles ( −8◦< δ <72◦), without a previously
+published VHE limit below 1.5% Crab, and with a
+measured redshift z <0.3. To further the study of theEBL a few objects having a large ( z >0.3) are also
+included in the target list. The target list includes:
+•All nearby ( z <0.3) HBL and IBL recom-
+mended as potential VHE emitters in [5, 6, 7].
+•The X-ray brightest HBL ( z <0.3) in the recent
+Sedentary [8] and ROXA [9] surveys.
+•Four distant ( z >0.3) BL Lac objects recom-
+mended by [5, 10].
+•Several FSRQ recommended as potential VHE
+emitters in [6, 11].
+•All nearby ( z <0.3) blazars detected by
+EGRET [12].
+•All nearby ( z <0.3) blazars contained in the
+Fermi-LAT Bright AGN Sample [13].
+•All sources ( |b|>10◦) detected by Fermi-LAT
+where extrapolations of their MeV-GeV γ-ray
+spectrum (including EBL absorption; assuming
+z = 0.3 if the redshift is unknown) indicates a
+possible VERITAS detection in less than 20 h.
+This criteria is the focus of the 2009-10 VERI-
+TAS blazar discovery program.
+5.VERITAS AGNDetections
+VERITAS has detected VHE γ-ray emission from
+16 AGN (15 blazars), including 8 VHE discoveries.
+These AGN are shown in Table I, and each has been
+detected by the Large Area Telescope (LAT) instru-
+ment aboard the Fermi Gamma-ray Space Telescope.
+Every blazar discovered by VERITAS was the sub-
+ject of ToO MWL observations to enable modeling of
+its simultaneously-measured SED. The known VHE
+blazars detected by VERITAS were similarly the tar-
+gets of MWL observations.
+5.1.Recent VERITAS BlazarDiscoveries
+Prior to the launch of Fermi VERITAS had discov-
+ered VHE emission from 2 blazars. These included
+the first VHE-detected IBL, WComae [14, 15], and
+the HBL 1ES0806+524 [16]. VERITAS has discov-
+ered 6 VHE blazars since the launch of Fermi. Three
+of these were initially observed by VERITAS prior to
+the release of Fermi-LAT results, due to the X-ray
+brightness of the synchrotron peaks of their SEDs.
+VHEemissionfrom3C66AwasdiscoveredbyVER-
+ITAS in September 2008 [17] during a flaring episode
+that was also observed by the Fermi-LAT [18]. The
+observedflux above200GeV was6% ofthe CrabNeb-
+ula flux and the measured VHE spectrum was very
+soft (Γ VHE∼4.1). RGBJ0710+591 was detected
+eConf C0911222009 Fermi Symposium, Washington, D.C., Nov. 2-5 3
+Table I VERITAS AGN Detections. The only non-blazar
+object is the radio galaxy M87. The blazars discovered
+at VHE by VERITAS are marked with a dagger.
+Object ClassRedshift
+M87 FR I 0.004
+Mkn421 HBL 0.030
+Mkn501 HBL 0.034
+1ES2344+514 HBL 0.044
+1ES1959+650 HBL 0.047
+WComae†IBL0.102
+RGBJ0710+591†HBL 0.125
+H1426+428 HBL 0.129
+1ES0806+524†HBL 0.138
+1ES0229+200 HBL 0.139
+1ES1218+304 HBL 0.182
+RBS0413†HBL 0.190
+1ES0502+675†HBL 0.341
+3C66A†IBL0.444?
+PKS1424+240†IBL ?
+VERJ0521+211†? ?
+(∼5.5σ; 3% Crab flux above 300 GeV; Γ VHE∼2.7)
+during VERITAS observations from December 2008
+to March 2009. The initial announcement of the VHE
+discovery [19] led to its discovery above 1 GeV in the
+Fermi-LAT data using a special analysis. RBS0413,
+a relatively distant HBL (z=0.19), was observed for
+16 h good-quality live time in 2008-092. These data
+resulted in the discoveryofVHE gamma-rays( >270γ,
+∼6σ) at a flux ( >200 GeV) of ∼2% of the Crab Neb-
+ula flux. The discovery[20] wasannounced simultane-
+ously with the LAT MeV-GeV detection. The VHE
+and other MWL observations, including Fermi-LAT
+data, for each of these three sources will be the sub-
+ject of a joint publication involving both the VERI-
+TAS and LAT collaborations.
+5.2. Discoveries Motivated byFermi-LAT
+The successful VHE discovery observations by
+VERITAS of three blazars was motivated primarily
+by results from the first year of LAT data taking. In
+particular, the VHE detections of PKS1424+240 [21]
+and 1ES0502+675 [22] were the result of VERITAS
+observationstriggeredbytheinclusionoftheseobjects
+in the Fermi-LAT Bright AGN List [13]. The former
+is only the third IBL known to emit VHE gamma-
+rays, and the latter is the most distant BL Lac object
+2RBS0413 was observed further by VERITAS in Fall 2009.(z= 0.341) detected in the VHE band. In addition,
+VERJ0521+211,likely associated with the radio-loud
+AGN RGBJ0521.8+2112, was detected by VERTAS
+in∼4 h of observations in October 2009 [23]. These
+observations were motivated by its identification as a
+>30 GeVγ-ray source in the public Fermi-LAT data.
+Its VHE flux is 5% of the Crab Nebula flux, placing it
+among the brightest VHE blazars detected in recent
+years. VERITAS later observed even brighter VHE
+flaring from VERJ0521+211 in November 2009 [24],
+leading to deeper VHE observations.
+6.BlazarsUpperLimits
+Morethan 50 VHE blazarcandidates wereobserved
+byVERITASbetweenSeptember2007andJune2009.
+The total exposure on the 49 non-detected candi-
+dates is ∼305 h live time (average of 6.2 h per can-
+didate). Approximately 55% of the total exposure is
+split amongst the 27 observed HBL. The remainder is
+divided amongst the 8 IBL (26%), 5 LBL (6%), and 9
+FSRQ (13%). There are no clearindications ofsignifi-
+cant VHE γ-ray emission from any of these 49 blazars
+[25]. However,theobservedsignificancedistributionis
+clearly skewed towards positive values (see Figure 1).
+A stacking analysis performed on the entire data sam-
+ple shows an overall excess of 430 γ-rays, correspond-
+ing to a statistical significance of 4.8 σ, observed from
+the directions of the candidate blazars. The IBL and
+HBLtargetsmakeup 96%ofthe observedexcess. Ob-
+servations of these objects also comprise ∼80% of the
+total exposure. An identical stacked analysis of all
+the extragalactic non-blazar targets observed, but not
+clearly detected ( >5σ), by VERITAS does not show
+a significant excess ( ∼120 h exposure). The stacked
+excess persists using alternate methods for estimating
+the background at each blazar location, and with dif-
+ferent event selection criteria (e.g. soft cuts optimized
+for sources with Γ VHE>4). The distribution of VHE
+flux upper limits is shown in Figure 1. These 49 VHE
+flux upper limits are generally the most-constraining
+ever reported for these objects.
+7.Multi-wavelength Studiesof VHE
+Blazars
+During the first three seasons of VERITAS obser-
+vations, pre-planned extensive MWL campaigns were
+organized for three blazars 1ES 2344+514 (2007-08),
+1ES 1218+304 (2008-09) and 1ES 0229+200 (2009-
+10 - ongoing). In addition, numerous ToO MWL-
+observationcampaignswereperformed. Theseinclude
+campaigns for every blazar/AGN discovered by VER-
+ITAS, and all include Swift (XRT and UVOT) data.
+All MWL campaigns on the VHE blazars discovered
+eConf C0911224 2009 Fermi Symposium, Washington, D.C., Nov. 2-5
+σ−5 −4 −3 −2 −1 0 1 2 3 4 5Entries
+024681012
+Crab Flux %0 2 4 6 8 10 12 14Entries
+024681012141618
+Figure 1: (Left) The preliminary significance measured from each of the 49 non-detected candidates using standard
+analysis cuts. The curve shows a Gaussian distribution, wit h mean zero and standard deviation one, normalized to the
+number of blazars. A similar result is obtained using analys is cuts optimized for soft-spectrum sources. (Right) The
+distribution of flux upper limits for the non-detected blaza rs in percentage of Crab Nebula flux above the observation
+threshold. The time-weighted average limit is less than ∼2% Crab flux.
+since the launch of Fermi include LAT detections. In
+addition, several MWL campaigns on the well-studied
+VHE blazars Mkn 421 and Mkn 501 (please see the
+contributions of D. Gall and A. Konopelko in these
+proceedings) were also performed. Highlights of these
+campaigns include:
+•1ES 2344+514: A major (50% Crab) VHE flare,
+along with correlations of the VHE and X-ray
+flux were observed from this HBL. The VHE
+and X-ray spectra harden during bright states,
+and a synchrotron self-Compton (SSC) model
+can explain the observed SED in both the high
+and low states [26].
+•1ES 1218+304: This HBL flared during VER-
+ITAS MWL observations. Its unusually hard
+VHE spectrum strongly constrains the EBL.
+Theobservedflaringrulesoutkpc-scalejetemis-
+sion as the explanation of the spectral hardness
+and places the EBL constraints on more solid-
+footing [27, 28].
+•1ES 0806+524: The observed SED of this new
+VHE HBL can be explained by an SSC model
+[16].
+•W Comae: This IBL, the first discovered at
+VHE, flared twice in 2008 [14, 15]. Modeling of
+the SED is improved by including an external-
+Compton (EC) component in an SSC interpre-
+tation.
+•3C 66A: This IBL flared at VHE and MeV-GeV
+energies in 2008[17, 18]. Similar to W Comae
+and PKS 1424+240, modeling of observed SED
+suggests a strong EC component in addition to
+an SSC component.
+•Mkn 421: This HBL exhibited major flaring be-
+havior for several months in 2008. Correlations
+of the VHE and X-ray flux were observed, along
+with spectral hardening with increased flux in
+both bands [29].•RGBJ0710+591: Modeling the SED of this
+HBL with an SSC model yields a good fit to
+the data. The inclusion of an external Compton
+component does not improve the fit.
+•PKS1424+240: ThebroadbandSEDofthisIBL
+(at unknown redshift) is well described by an
+SSC model favoring a redshift of less than 0.1
+[21]. Using the photon index measured with
+Fermi-LAT in combination with recent EBL ab-
+sorption models, the VERITAS data indicate
+that the redshift of PKS 1424+240 is less than
+0.66.
+8.Conclusions
+The first two years of the VERITAS blazar KSP
+were highly successful. Highlights include the detec-
+tion of more than a 16 VHE blazars with the obser-
+vations almost always having contemporaneous MWL
+data. Among these detections are 8 VHE blazar dis-
+coveries, including the first three IBLs known to emit
+VHEγ-rays. All but a handful of the blazars on the
+initial VERITAS discovery target list were observed,
+and the flux limits generated for those not VHE de-
+tected are generally the most-constraining ever. The
+excess seen in the stacked blazar analysis suggests
+that the initial direction of the VERITAS discovery
+program was well justified, and that follow-up obser-
+vations of many of these initial targets will result in
+VHE discoveries. In addition, the Fermi-LAT is iden-
+tifyingmanynewcompellingtargetsfortheVERITAS
+blazar discoveryprogram. These new candidates have
+already resulted in 3 VHE blazar discoveries. The
+future of the VERITAS blazar discovery program is
+clearly very bright.
+The MWL aspect of the VERITAS blazar KSP has
+also been highly successful. Every VERITAS obser-
+vation of a known, or newly discovered, VHE blazar
+has been accompanied by contemporaneous MWL ob-
+servations. These data have resulted in the identifica-
+eConf C0911222009 Fermi Symposium, Washington, D.C., Nov. 2-5 5
+tion of correlated VHE and X-ray flux variability, as
+well as correlated spectral hardening in both the VHE
+and X-ray bands. The VHE MWL observations were
+performed in both ”quiescent” and flaring states for
+some of the observed blazars. For the observed HBL
+objects, the SEDs can be well described by a simple
+SSC model in both high and low states. However, an
+additional external Compton component is necessary
+to adequately fit the SEDs of the IBL objects.
+The Fermi-LAT is already having a significant im-
+pact on the blazar KSP. In future seasons, the VER-
+ITAS blazar discovery program will focus its dis-
+covery program on hard-spectrum blazars detected
+by Fermi-LAT, and will likely have a greater focus
+on high-risk/high-reward objects at larger redshifts
+(0.3< z <0.7). In addition, the number of VHE
+blazars studied in pre-planned MWL campaigns will
+increase as data from the Fermi-LAT will be publicly
+available. In particular, the extensive pre-planned
+MWL campaigns will focus on objects that are note-
+worthy for the impact their data may have on under-
+standing the EBL. The simultaneous observations of
+blazars by VERITAS and Fermi-LAT will completely
+resolve the higher-energySED peak, often for the first
+time, enabling unprecedented constraints on the un-
+derlying blazar phenomena to be derived.
+Acknowledgments
+This research is supported by grants from the US
+Department of Energy, the US National Science Foun-
+dation,andtheSmithsonianInstitution, byNSERCin
+Canada, by Science Foundation Ireland, and by STFC
+in the UK. We acknowledge the excellent work of the
+technical support staff at the FLWO and the collab-orating institutions in the construction and operation
+of the instrument.
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+arXiv:1001.0771v3  [math.AT]  1 Apr 2011COMPLETION OF G-SPECTRA AND STABLE MAPS BETWEEN
+CLASSIFYING SPACES
+KÁRI RAGNARSSON
+Abstract. We prove structural theorems for computing the completion o f aG-spectrum at
+the augmentation ideal of the Burnside ring of a finite group G. First we show that a G-
+spectrum can be replaced by a spectrum obtained by allowing o nly isotropy groups of prime
+power order without changing the homotopy type of the comple tion. We then show that
+this completion can be computed as a homotopy colimit of comp letions of spectra obtained
+by further restricting isotropy to one prime at a time, and th at these completions can be
+computed in terms of completion at a prime.
+As an application, we show that the spectrum of stable maps fr omBGto the classifying
+space of a compact Lie group Ksplits non-equivariantly as a wedge sum of p-completed
+suspension spectra of classifying spaces of certain subquo tients of G×K. In particular this
+describes the dual of BG.
+1.Introduction
+Completions of G-spectra at ideals of the Burnside ring were introduced by Gr eenlees–May
+in [11] as a means to give a topological flavour to completion t heorems in stable equivariant
+homotopy theory. We say that the completion theorem holds fo r aG-spectrum Xif the natural
+map
+X∼=F(pt+,X)−→F(EG+,X),
+induced by the projection EG→ptis anI(G)-adic completion map, where I(G)is the aug-
+mentation ideal in the Burnside ring A(G). (In [11] Gcan be a compact Lie group, but we
+consider only finite groups in this paper.) The subscript +is used here to denote an added
+disjoint basepoint.
+Completion theorems are especially helpful in the case wher eXisG-split, in the sense of
+[17], for then one has a non-equivariant equivalence F(EG+,X)G≃F(BG+,i∗X),wherei∗X
+is the underlying non-equivariant spectrum of X. Thus, if the completion theorem holds for a
+G-splitG-spectrum X, then the natural map
+XG−→F(BG+,i∗X)
+is anI(G)-adic completion map. Taking homotopy groups, one deduces t hat the natural map
+X∗
+G(pt+)−→X∗(BG+)
+is anI(G)-adic completion map. This elucidates the importance of com pletion theorems: one
+is often interested in X∗(BG+), andX∗
+G(pt+)is usually easier to compute.
+The original, and most important, examples of completion th eorems are the Atiyah–Segal
+completion theorem [3, 4], where Xis real or complex periodic equivariant K-theory, and the
+Segal conjecture, proved by Carlsson in [6], where Xis the equivariant sphere spectrum SG.
+These were each originally proved as cohomological stateme nts and lifted to the spectrum level
+in [11]. The Atiyah–Segal completion theorem gives, in the c omplex case, an isomorphism
+ˆR[G]∼=−→KU0(BG)
+Date: March 17, 2011.
+12 KÁRI RAGNARSSON
+whereˆR[G]is the completion of the complex representation ring at its a ugmentation ideal1, and
+a similar statement in the real case. This inspired Segal to c onjecture an isomorphism
+A(G)∧
+I(G)∼=−→π0(BG+),
+which he later strengthened to a stronger from, involving hi gher cohomotopy groups. It is the
+stronger form which Carlsson proved.
+While completion theorems are very descriptive, their util ity is somewhat restricted by the
+fact that I(G)-adic completion has so far been very difficult to compute in pr actice. A notable
+exception is when Gis ap-group, for in this case results of May–McClure from [17] on c ompletion
+of Mackey functors suggest that the p-adic completion of Xfactors through the I(G)-adic
+completion, and the difference is controlled by the underlyi ng non-equivariant spectrum. In this
+paper we make this spectrum-level analogue of the May–McClu re result precise, and generalize
+it to an arbitrary finite group G, at the cost of restricting to G-spectra whose isotropy groups
+arep-groups (see Theorem C) below.
+More generally, we present an effective approach for computi ng theI(G)-adic completion of
+an arbitrary G-spectrum Xby restricting isotropy to p-groups, one prime at a time. The key
+to this is a result which allows us to replace Xby a spectrum whose I(G)-adic completion is
+easier to handle. For a G-spectrum X, we form a G-spectrum X[FP]which one can informally
+think of as the subspectrum of Xwhose isotropy groups are of prime power order; formally
+X[FP]def=EFP+∧X, whereEFPis the universal space for the family FPof subgroups in Gof
+prime power order (for any prime). The canonical map EFP→ptinduces a map X[FP]→X,
+and we have the following result.
+Theorem A (Replacement theorem) .For a bounded-below G-spectrum X, the natural map
+X[FP]→Xinduces a weak equivalence upon I(G)-adic completion.
+Of course, when Xis aG-CW spectrum, the replacement theorem gives an actual homot opy
+equivalence X[FP]∧
+I(G)≃X∧
+I(G).
+The next step in our process is to decompose X[FP]∧
+I(G)into component spectra X[Fp]∧
+I(G),
+for different primes p, whereFpis the family of p-subgroups in G. This decomposition is not
+quite a wedge sum splitting, as each component X[Fp]∧
+I(G)contains the spectrum X[F1]∧
+I(G),
+whereF1is the family containing only the trivial subgroup. Using a h omotopy colimit to keep
+track of this, we obtain the following result.
+Theorem B (Decomposition theorem) .For a bounded-below G-spectrum X,X[FP]∧
+I(G)is
+weakly equivalent to the homotopy colimit of the diagram con sisting of the natural maps
+X[F1]∧
+I(G)→X[Fp]∧
+I(G)for all primes p.
+It clearly suffices to take the homotopy colimit over primes th at divide the order of G, since the
+mapX[F1]∧
+I(G)→X[Fp]∧
+I(G)is a weak equivalence for other primes. However, it is theore tically
+useful to consider the diagram ranging over all primes, as th is allows us to deduce naturality in
+Gin certain circumstances.
+The completions of the component spectra can be computed as f ollows.
+Theorem C. LetXbe a bounded-below G-spectrum.
+(a)The completion map X[F1]→X[F1]∧
+I(G)is a weak equivalence.
+(b)For a prime p, theI(G)-adic completion of the cofibration sequence
+X[F1]−→X[Fp]−→X[Fp,F1]
+1A(G)acts onR[G]in an obvious way, and it is not hard to show that completion at I(G)is equivalent to
+completion at the augmentation ideal of R[G].COMPLETION OF G-SPECTRA AND STABLE MAPS BETWEEN CLASSIFYING SPACES 3
+is weakly equivalent to the cofibration sequence
+X[F1]−→X[Fp]∧
+I(G)−→X[Fp,F1]∧
+p.
+Theorem C makes precise the loosely worded statement above t hat thep-adic completion
+ofX[Fp]factors through the I(G)-adic completion, and the difference between them lies only
+in the subspectrum X[F1], which is unchanged by completion. When Gis ap-group one has
+X[Fp]≃X, and so one obtains the promised spectrum-level analogue of the May–McClure
+result.
+As an application of these results we can now give an explicit description of the spectrum of
+stable maps between classifying spaces. This description i s a generalization of the author’s work
+on homotopy classes of stable maps between classifying spac es in [20], and indeed this was the
+motivation for the work in this paper.
+Even before its proof, Lewis–May–McClure showed in [15] tha t the Segal conjecture implies
+the completion theorem for the suspension spectrum of B(G,K), the classifying space of prin-
+cipalK-bundles in the category of G-spaces. Their result was for finite groups K, and this
+was extended to compact Lie groups in [18]. The spectrum Σ∞
+GB(G,K)+isG-split, and there
+results a non-equivariant homotopy equivalence
+(Σ∞
+GB(G,K)G
++)∧
+I(G)≃F(BG+,Σ∞BK+).
+There is a non-equivariant splitting ([15, 21])
+Σ∞B(G,K)G
++≃/logicalordisplay
+[H,ϕ]Σ∞BW(H,ϕ)+,
+where the wedge sum runs over conjugacy classes of pairs (H,ϕ)consisting of a subgroup H≤G
+and a homomorphism ϕ:H→K, andW(H,ϕ)is a certain subquotient of G×K(see Section
+7 for details). Hence F(BG+,Σ∞BK+)is equivalent to the I(G)-adic completion of this wedge
+sum of classifying spaces, and this is usually considered th e best available description.
+The problem with this point of view is that, except when Gis ap-group, there has been no
+way to compute this completion; the action of A(G)on the wedge sum is not even understood.
+This problem can now be overcome: using the Theorem A one can r estrict the wedge sum to
+subgroups of prime power order, and using Theorems B and C, th e completion of each remaining
+classifying space can be described in terms of completion at the relevant prime, leading to the
+following result.
+Theorem D. For a finite group Gand a compact Lie group K, there is a homotopy equivalence
+F(BG+,Σ∞BK+)≃/logicalordisplay
+[H,ϕ]
+H∈FPΣ∞(BW(H,ϕ)∧
+p(H))+,
+where, for H∈FP\F1,p(H)is the prime that divides |H|, andp(1) = 0 .
+Applying the functor π0, one independently recovers [20, Theorem B]. One can also fo cus
+on a single prime pto obtain a wedge sum description of F(BG∧
+p+,Σ∞BK+)(see Corollary
+6.5), and applying π0to that description one independently recovers [20, Theore m A]. Taking
+K= 1in these results one obtains a description of the mapping dua ls ofBG+andBG∧
+p+(see
+Corollary 7.4.
+Outline. Following this introduction are three preliminary section s with mostly background
+material. In Section 2 we review families of subgroups and th eir universal spaces. Section 3
+contains a very brief overview of equivariant stable homoto py theory, as well as a proof of the
+known but unpublished result that weak equivalences can be d etected on geometric fixed-point
+spectra (Proposition 3.1). Completion at ideals of the Burn side ring is reviewed in Section 4.4 KÁRI RAGNARSSON
+The real work begins in Section 5, where we introduce the rest ricted isotropy spectra X[F],
+investigate their basic properties and prove Theorem A. The orems B and C are then proved
+in Section 6. Finally, the application to stable maps betwee n classifying spaces is treated in
+Section 7.
+Acknowledgments. The author is grateful to John Greenlees and Peter May for the ir per-
+mission to reproduce material from their overview article [ 13] in Section 3, and for their help
+regarding the proof and historical context of Proposition 3 .1. The author also thanks Tony
+Elmendorf for an explanation of his functor realizing syste ms of fixed points in [8], which allows
+the functorial construction of the restricted isotropy spe ctra. The method of proof is heavily
+dependent on the results of Greenlees from [10], and the auth or hopes this goes some way toward
+answering the question implicitly asked in the MathSciNet r eview of that paper.
+2.Families of subgroups and classifying spaces
+In this section, and throughout the paper, fix a finite group G. AfamilyFof subgroups
+ofGis a set of subgroups of Gthat is closed under conjugacy in Gand taking subgroups.
+The particular families that will be used in this paper are F1, the family consisting solely of
+the trivial subgroup; Fp, the family of p-subgroups for a prime p;FP=/uniontext
+pFp, the family of
+subgroups of prime power order; and Fall, the family of all subgroups.
+Given a familyF, aG-spaceXis said to be anF-space if the isotropy group of each point
+inXbelongs toF. There is a distinguished F-spaceEFwith the universal property that
+eachF-spaceXadmits a unique homotopy class of maps X→EF. The space EFcan be
+characterized by the fact that, for a subgroup H≤G, the fixed-point space EFHis contractible
+ifH∈F, and empty if H /∈F. Of course, these properties only determine EFup to homotopy,
+and to be explicit EFwill here be taken to mean the space constructed by Elmendorf in [8].
+Elmendorf describes a functor that, given a system of fixed po ints for subgroups of G, uses a
+categorical bar construction to construct a G-space realizing that system. The following lemma
+is a consequence of the construction.
+Lemma 2.1. For a familyF, the space EFis of finite type, and an inclusion of families
+Fa֒→Fbinduces an inclusion of G-CW complexes EFa→EFb, in particular a cofibration.
+The homotopy description of universal implies that EF1≃EGandEFall≃ptasG-spaces.
+The spaces EFpandFPare difficult to get a direct handle on, but they do relate to eac h other
+via a colimit description, which we will use throughout the p aper.
+Definition 2.2. LetDbe the category whose objects are the families F1andFp, for all primes
+p, and whose morphisms are the inclusions F1֒→Fp.
+Lemma 2.3. The natural map
+colim
+F∈DEF−→EFP
+is a homeomorphism.
+Proof. The system of fixed points for FPis the colimit of the systems of fixed points of the
+families in D. The point is now that the categorical bar construction is a b ig colimit construction,
+and colimits commute. /square
+Although a direct proof is easy in this case, it is fitting to at tribute Lemma 2.3 to Piacenza,
+who in [19] showed that Elmendorf’s functor in [8] is a Quille n equivalence from the systems of
+fixed points to G-spaces.COMPLETION OF G-SPECTRA AND STABLE MAPS BETWEEN CLASSIFYING SPACES 5
+The colimit in Lemma 2.3 is taken over the diagram
+EF1
+/d117/d117/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107
+/d123/d123/d119/d119/d119/d119/d119/d119/d119/d119
+/d35/d35/d71/d71/d71/d71/d71/d71/d71/d71
+EFp1EFp2··· EFpr···
+wherep1,p2,...is an enumeration of the primes. Thus EFPis very close to being the coproduct
+of the spaces EFp: the difference is that each EFpcontains EF1as a subspace, and the colimit
+construction is needed to keep track of this. Alternatively , one can describe EFPas a coproduct
+in the category of G-spaces under EF1, but there does not seem to be much to gain from this.
+We also observe that EFp=EF1for primes pthat do not divide the order of G, and so it
+suffices to take the colimit in Lemma 2.3 over primes that divid e the order of G. This is nice
+since one then obtains a finite colimit. For theoretical reas ons it is nevertheless convenient to
+include all primes, as that makes the construction natural i nG.
+3.Equivariant stable homotopy theory
+The subject of equivariant stable homotopy theory is much to o vast to cover in any detail in
+this article. An excellent overview of the subject is provid ed in [13], and a detailed account can
+be found in [16]. In this section we paraphrase passages from [13] with the aim that a reader
+familiar with the framework for non-equivariant stable hom otopy in [9] can follow the discussion
+in the paper, without going into much technical detail. The a uthor is grateful to John Greenlees
+and Peter May for the permission to reproduce their work here . In Proposition 3.1 we also fill
+a gap in the literature by presenting a proof that actual fixed points and geometric fixed points
+give rise to the same notion of weak equivalence.
+Fix acomplete G-universe U. This means Uis a real inner product space on which Gacts by
+linear isometries, and Udecomposes as a sum of countably many copies of each irreduci ble repre-
+sentations of G. AG-spectrum E(indexed on U) is then a collection of based G-spacesEV, one
+for each finite-dimensional G-subspace V⊆U, equipped with compatible G-homeomorphisms
+EV∼=− →ΩW−VEW−V= Map(SW−V,EW−V),
+one for each subspace inclusion V⊂W, whereW−Vis the orthogonal complement of Vin
+W, andSW−Vis the one-point compactification of W−V. (Observe that SW−Vis a sphere
+of dimension dimW−dimV, withG-action inherited from W.) A map E→E′ofG-spectra
+is a collection of G-space maps EV→E′
+V, compatible with the structure maps. Equivariant
+suspension spectra ( Σ∞
+GX), smash products ( X∧Y), and function spectra ( F(X,Y)) are defined,
+and behave, much as in the non-equivariant setting. Consequ ently one obtains mapping cones,
+mapping cylinders and homotopies in the standard way. For G-spectra EandF, we write
+[E,F]Gfor the set of homotopy classes of maps E→F.
+EveryG-spectrum has an underlying non-equivariant spectrum . For aG-spectrum E, this is
+obtained by first pulling Eback along the inclusion of fixed points i:UG֒→Uto obtain the
+naiveG-spectrum i∗Eindexed on the G-fixed universe UG, and then forgetting the G-action. A
+non-equivariant homotopy equivalence then means a homotopy equivalence between underlying
+non-equivariant spectra.
+In equivariant homotopy theory, G-orbits play the role of points, and therefore one must
+consider the generalized sphere spectra G/H+∧Sn
+G, whereSn
+Gis then-fold suspension of the G-
+equivariant sphere spectrum SGdef= Σ∞
+GS0. Consequently the homotopy groups of aG-spectrum
+Eare defined by setting
+πH
+n(E)def= [G/H+∧Sn
+G,E]6 KÁRI RAGNARSSON
+forH≤Gandn∈Z. This gives rise to a Mackey functor
+πn(E):H/mapsto→πH
+n(E).
+A spectrum Eisbounded below if there exists an integer Nsuch that πn(E) = 0 forn < N .
+A map between G-spectra is a weak equivalence if it induces an isomorphism on all homotopy
+groups. Cofibrations andfibrations are defined via the homotopy extension and homotopy
+lifting properties, respectively (here we ignore model str uctures and these terms are meant
+in the classical sense). As in the non-equivariant world, co fibration sequences coincide with
+fibration sequences up to sign and weak homotopy. It follows t hat fibre and cofibre sequences
+ofG-spectra both give rise to a long exact sequence of homotopy g roups, regarded as Mackey
+functors or (equivalently) one orbit at a time.
+Again, since G-orbits play the role of points, it is appropriate to regard t heG-spectra
+G/H+∧CSn
+G, where CSn
+G=I∧Sn
+Gis the cone on Sn
+G, as generalized cells. G-CW spectra
+are then constructed by attaching such cells along the bound ary sphere G-spectra G/H+∧Sn
+G.
+The standard results for CW-spectra carry over to the equiva riant world: We have an equivari-
+ant Whitehead theorem (a weak equivalence between G-CW spectra is a homotopy equivalence),
+cellular approximation of maps and homotopies, and approxi mation by G-CW spectra. The last
+result provides a functor Γthat assigns a G-CW spectrum ΓEto aG-spectrum E, and comes
+equipped with a natural weak equivalence γ: ΓE→E. We say that a G-spectrum is of finite
+typeif it is weakly equivalent to a G-CW spectrum with finitely many cells in each dimension.
+For a subgroup H≤G, one can regard the complete G-universe Uas a complete H-universe,
+and this gives rise to a forgetful functor from G-spectra to H-spectra. We denote this functor
+byi∗
+H, ori∗when there is no danger of confusion. The restriction functo rs preserve weak
+equivalences and cofibre sequences, and have left and right a djoint functors, which will not be
+discussed here.
+Fixed points of G-spectra have to be handled with some care. It is tempting to f orm fixed-
+point spectra by taking fixed points at the space level. The pr oblem is that this is not compatible
+with the structure maps in the spectrum, as Map(SW−V,(EW−V)G)is generally not homeomor-
+phic toMap(SW−V,EW−V)GwhenGacts non-trivially on W. One way around this problem is
+to restrict indexing to subspaces with trivial G-action and consider the naive G-spectrum i∗E.
+Taking space-level fixed points of i∗Eone gets a (non-equivariant) spectrum (i∗E)G, indexed on
+on theG-fixed universe UG, and the fixed-point spectrum EGis defined by EGdef= (i∗E)G. The
+fixed-point-spectrum functor satisfies the expected adjunc tion relations, and is used to define
+equivariant homology and cohomology groups. However it doe s not mirror the unstable fixed-
+point functor (or preserve smash products) as information a bout representations of Gis built
+intoΣ∞
+GX, even when Xhas trivial G-action. Instead, one has the homotopy equivalence
+(Σ∞
+GX+)G≃/logicalordisplay
+(H)Σ∞(EWG(H)+∧WG(H)XH), (1)
+originally due to tom Dieck [21], where the wedge sum runs ove r conjugacy classes of subgroups
+ofH, andWG(H) =NG(H)/His the Weyl group of HinG.
+An alternative approach to fixed-point spectra is taking spa ce-level fixed points of a G-
+spectrum, and then spectrifying the resulting prespectrum . This construction induces the
+geometric fixed-point spectrum functorΦG, so named because it satisfies ΦG(E∧E′)≃
+ΦG(E)∧ΦG(E′)forG-spectra E,E′, andΦG(Σ∞X)≃Σ∞(XG)for a based G-spaceX, in
+line with geometric intuition. It does not, however, satisf y the adjunction relation one demands
+of a fixed-point functor, and so plays a secondary, but import ant, role in equivariant stable
+homotopy theory. Formally, the geometric fixed-point spect rum is defined by a separation of
+isotropies. That is, one lets Pbe the family of proper subgroups in G, and defines /tildewideEPto be theCOMPLETION OF G-SPECTRA AND STABLE MAPS BETWEEN CLASSIFYING SPACES 7
+cofibre of the natural map EP+→EFall+≃S0. For aG-spectrum E, one can informally think
+of the smash product /tildewideEP∧Eas removing orbits whose isotropy groups are proper subgrou ps of
+G. Geometric intuition dictates that such orbits should have noG-fixed points (although this is
+false in equivariant stable homotopy theory), and this moti vates the definition of the geometric
+fixed-point spectrum as
+ΦG(E)def= (/tildewideEP∧E)G.
+For a subgroup H≤G, suitable adaptations of the above constructions yield a fix ed-point
+functor(−)Hand a geometric fixed-point functor ΦH, both taking values in WG(H)-spectra. We
+will only consider fixed-point spectra non-equivariantly, so it suffices to know that there are non-
+equivariant equivalences EH≃(i∗E)HandΦH(E)≃ΦH(i∗E), wherei∗Eis the restriction of
+Eto anH-spectrum. (A more elaborate construction is needed to obta in theWG(H)-spectrum
+structure.) The H-fixed-point spectra of a G-spectrum Esatisfy the adjunction
+πH
+∗(E)∼=π∗(EH).
+It follows immediately that a map between G-spectra is a weak equivalence if and only if it
+induces non-equivariant weak equivalences on all fixed-poi nt spectra. It is known to experts
+that this remains true if one looks at geometric fixed-point s pectra instead of actual fixed-
+point spectra, although no proof seems to appear in the liter ature. We remedy this in the
+following proposition, which was originally proved, but no t published, by Gaunce Lewis, and
+also independently discovered by John Greenlees. An outlin e of the proof presented here was
+communicated to the author by Peter May.
+Proposition 3.1. For a map f:X→YbetweenG-spectra, the following are equivalent.
+(i)fis a weak equivalence.
+(ii)For each H≤G, the map fHis a non-equivariant weak equivalence.
+(iii)For each H≤G, the map ΦH(f)is a non-equivariant weak equivalence.
+Proof. The equivalence between (i) and (ii) has already been establ ished. The implication (i)
+⇒(iii) is purely formal: A weak equivalence is an isomorphism in the homotopy category of
+G-spectra (obtained by formally inverting weak equivalence s). It follows that the restriction to
+H-spectra i∗f:i∗X→i∗Yis an isomorphism in the homotopy category of H-spectra. This
+remains true after smashing with /tildewideEP, wherePis the family of proper subgroups of H. Taking
+H-fixed points, we get an isomorphism in the homotopy category of spectra,
+(/tildewideEP∧i∗X)H(1∧i∗f)H
+−−−−−−→ (/tildewideEP∧i∗Y)H,
+which is the same as a non-equivariant weak equivalence ΦH(X)ΦH(f)−−−−→ΦH(Y).
+It now suffices to prove the implication (iii) ⇒(ii). By taking G-CW approximations if
+necessary, the implication (i) ⇒(iii) allows us to assume, without loss of generality, that Xand
+YareG-CW spectra. We now proceed by induction on G, noting that the claim is trivially true
+whenGis the trivial group. We may then inductively assume that the claim is true for all proper
+subgroups H < G . In particular this means that restricting to H-spectra yields non-equivariant
+weak equivalences
+(i∗
+HX)H(i∗
+Hf)H
+−−−−−→(i∗
+HY)H,
+which non-equivariantly is the same as weak equivalences
+XHfH
+−−→YH.
+Thus it remains only to prove that fGis a weak equivalence. Letting Zbe the cofibre of the
+mapf:X→Y, this is equivalent to showing that ZGis weakly contractible.8 KÁRI RAGNARSSON
+Consider the map Z∼=S0∧Zι∧1−−→/tildewideEP∧Zwhereιis the map in the cofibre sequence
+EP+→S0ι− →/tildewideEP. SinceZHis weakly contractible for every H∈P, [16, Proposition 9.2]
+implies that ι∧1is aG-equivalence, and in particular ZG≃(/tildewideEP∧Z)G= ΦG(Z). ButΦG(Z)
+is the cofibre of the map ΦG(X)ΦG(f)−−−−→ΦG(Y),which is a non-equivariant weak equivalence
+and hence ΦG(Z)is weakly contractible. It follows that ZGis weakly contractible, completing
+the induction and the proof. /square
+Finally, a G-spectrum EisG-split if the natural inclusion of EGinto the underlying non-
+equivariant spectrum of Esplits up to homotopy. In other words, EisG-split if there is a non-
+equivariant map i∗E→EG, whose composite with the non-equivariant inclusion EG֒→i∗E
+is homotopic to the identity. G-splitG-spectra play an important role in equivariant stable
+homotopy theory because of their interaction with freeG-spectra , meaning G-spectra that are
+equivalent to G-CW spectra built out of free G-cells. To keep things simple, we explain this
+interaction only for free G-CW complexes, in which case the result is due to Kosniowski [ 14],
+although a proof is also given in [17].
+Theorem 3.2. [14, 17] IfEis aG-splitG-spectrum and Xis a freeG-CW complex, then there
+is a natural isomorphism
+En
+G(X)∼=− →En(X/G),
+whereE∗denotes the cohomology theory represented by the underlying n on-equivariant spectrum
+ofE.
+A more general version of the theorem, where Xis allowed to be a free naive G-spectrum is
+proved in [16], but this will not be needed here.
+4.The Burnside ring and completion theorems
+TheBurnside ring of a finite group G, denoted A(G), is the Grothendieck group of isomor-
+phism classes of finite left G-sets. Its additive and multiplicative structures are indu ced by
+disjoint union and cartesian product, respectively. As a Z-module, the Burnside ring is gen-
+erated by the transitive G-sets. Thus, letting [X]denote the isomorphism class of a G-setX,
+A(G)has one basis element [G/H]for each conjugacy class of subgroups H≤G.
+4.1.Algebraic completion. For an ideal J⊆A(G)and anA(G)-module M, theJ-adic
+completion of Mis defined by
+M∧
+Jdef= lim←−/parenleftbig
+M/JM←M/J2M←M/J3M←···/parenrightbig
+As we are taking Gto be finite, A(G)is Noetherian, so when Mis finitely generated the Artin–
+Rees Lemma applies and we have M∧
+J∼=A(G)∧
+J⊗A(G)M. Moreover, A(G)∧
+Jis flat over A(G),
+and thus J-adic completion commutes with finite colimits for finitely g enerated A(G)-modules.
+We will primarily be interested in completion at the augment ation ideal of A(G), defined as
+follows. The augmentation ofA(G)is the homomorphism ǫdefined on isomorphism classes of
+G-sets by setting
+ǫ([X]) =|X|,
+where|X|denotes the cardinality of X, and then extending to a Z-linear homomorphism
+ǫ:A(G)→Z.
+As usual, the augmentation ideal I=I(G)is defined as the kernel of the augmentation:
+I=I(G)def= kerǫ.COMPLETION OF G-SPECTRA AND STABLE MAPS BETWEEN CLASSIFYING SPACES 9
+4.2.Cohomological completion theorems. The main motivation for considering completion
+at the augmentation ideal is the Segal Conjecture, which sti pulates that the natural map
+A(G)∧
+I∼=− →π0(BG+),
+whereπ∗denotes the cohomotopy functor, is an isomorphism. The conj ecture was settled by
+Carlsson in [6] where he bridged the gap between the results o btained in [1, 7] and [17]. Carlsson
+actually proved the “strong Segal conjecture”, describing also the higher cohomotopy groups of
+BG+as anI-adic completion. His theorem has since undergone a series o f generalizations in
+[2, 15, 18] to, amongst other things, give a description of th e homotopy groups of a mapping
+spectrum F(BG+,Σ∞BK+)for a finite group Gand a compact Lie group Kin terms of I-adic
+completions. This will be discussed in more detail in Sectio n 7.
+To understand the strong Segal conjecture and its generaliz ations, one must first recall Segal’s
+identification,
+A(G)∼=[SG,SG]G,
+whereSGdenotes the G-equivariant sphere spectrum. (See [21] for a proof and gene ralization
+to compact Lie groups.) For a G-spectrum E, the homotopy groups of the function spectrum
+F(SG,E)then receive an A(G)-module structure by pre-composition, and can hence be subj ected
+toI-adic completion. We say that the completion theorem holds for Eif the natural map
+E∗
+G(SG)→E∗
+G(EG+), induced by the map EG→pt, is anI-adic completion map. Carlsson’s
+theorem in [6] was the completion theorem for E=SG.
+The importance of completion theorems is most pronounced wh enEis aG-splitG-spectrum.
+In this case Theorem 3.2 gives an isomorphism E∗
+G(EG+)∼=E∗(EG+/G)∼=E∗(BG+), where
+E∗is the cohomology theory associated to the underlying non-e quivariant spectrum of E, and
+so the completion theorem for a G-split spectrum Ecan be interpreted as an isomorphism
+E∗(BG+)∼=E∗
+G(SG)∧
+I.
+The advantage here is that one is often interested in E∗(BG+), whileE∗
+G(SG)is known, or
+easy to calculate. The tricky thing is that I(G)-adic completions have so far been difficult to
+compute in practice, but in this paper we present methods to s implify this process.
+4.3.Spectrum-level completion. Completion theorems were given a more topological flavour
+by Greenlees–May in [11], where they defined a spectrum-leve l completion at ideals of the
+Burnside ring. This allowed them to rephrase the completion conjecture, asking, for a G-
+spectrum E, whether the natural map
+F(SG,E)−→F(EG+,E),
+induced by the map EG→pt, becomes an equivalence upon I-adic completion. We recall some
+basic properties of the spectrum-level completion here and refer the reader to [11] for details.
+First observe that one obtains a homotopy action of A(G)on aG-spectrum Eby letting
+α∈A(G)∼=[SG,SG]Gact via the composite
+E≃SG∧Eα∧1−−→SG∧E≃E.
+AG-spectrum Tisα-torsion if the homotopy class αn:T→Tis trivial for sufficiently large
+n. For an ideal J⊆A(G), we say that TisJ-torsion ifTisα-torsion for all α∈J. Define TJ
+to be the class of all J-torsionG-spectra. Greenlees–May then define the J-adic completion E∧
+J
+of aG-spectrum Eto be the Bousfield TJ-localization of E.
+To elaborate slightly on this, say that a spectrum WisTJ-acyclic ifW∧Tis contractible for
+everyT∈TJ. We say a spectrum XisTJ-local if[W,X] = 0for every TJ-acyclic spectrum W,
+and we say that a map f:E→Xis aTJ-equivalence if its cofibre is TJ-acyclic. A TJ-localization
+ofEis then a TJ-equivalence from Eto aTJ-local spectrum. Such a map is, up to homotopy,10 KÁRI RAGNARSSON
+initial among maps from EtoTJ-local spectra, and it follows that TJ-localization, and hence
+J-adic completion, is unique up to homotopy.
+Greenlees–May establish the existence of J-completions via an explicit construction E∧
+J≃
+F(M(J),E)for a certain spectrum M(J), but also point out that existence can be established
+formally by adapting Bousfield’s arguments from [5] to the eq uivariant setting.
+Greenlees–May describe the homotopy groups of a completed s pectrum in terms of the zeroth
+and first derived functors of the J-adic completion functor (the higher derived functors vani sh)
+in [11], and these derived functors are studied in [12]. For fi niteGthe Burnside ring A(G)is
+Noetherian, so when Eis of finite type, the first derived functor also vanishes, and there is a
+natural isomorphism of Mackey functors
+πq(E∧
+J)∼=πq(E)∧
+J.
+4.4.Non-equivariant effect of completion. Generalizing the augmentation, we consider for
+a subgroup H≤GtheH-fixed-point homomorphism
+φH:A(G)→Z
+defined by setting
+φH([X]) =|XH|
+for aG-setXand extending linearly. For a subgroup H≤Gand an ideal J⊆A(G), set
+φH(J)def=/angbracketleftbig
+φH(X)|X∈J/angbracketrightbig
+⊆Z.
+Greenlees proved the following useful theorem in [10].
+Theorem 4.1. [10]IfXis a bounded-below G-spectrum, then the map obtained from X→X∧
+J
+by taking ΦH-fixed points is non-equivariantly φH(J)-adic completion.
+5.The replacement theorem
+In this section we prove our first structural theorem for comp utingI(G)-adic completions,
+which is Theorem A of the introduction. The theorem says that whenXis a bounded-below G-
+spectrum, one can replace Xby a simpler spectrum X[FP]without changing the weak homotopy
+type of the I(G)-adic completion. The spectrum X[FP]admits a natural map X[FP]ι− →X, and
+is characterized by the fact that ΦH(ι)is a non-equivariant weak equivalence if H∈FP, and
+ΦH(X[FP])is weakly contractible if H /∈FP. (Recall from Section 2 that FPis the family of
+subgroups in Gof prime-power order for any prime.) The theorem therefore s ays in principle
+thatI(G)-adic completions are determined by prime-power fixed-poin t spectra.
+Geometric intuition would have one think of X[FP]as the subspectrum of Xconsisting of
+orbits with isotropy groups in FP. This is a useful analogy but the construction does not make
+sense in equivariant stable homotopy theory; instead X[FP]is defined as follows.
+Definition 5.1. LetFbe a family of subgroups in G. For a G-spectrum X, theF-isotropy
+restriction ofXis theG-spectrum
+X[F]def=EF+∧X.
+The functor RF:X/mapsto→X[F]is theF-isotropy restriction functor . Denote by η:RF⇒Idthe
+natural map induced by the projection EF→pt.
+The natural transformation map ηis induced by a trivial map, and to understand its proper
+role we make the following remark. Recall from Section 2 that Fall, the family consisting of
+all subgroups of Ghas a universal space homotopy equivalent to a point. For a G-spectrum
+Xwe therefore have a homotopy equivalence X[Fall]≃− →X.The inclusionF ⊆F allinduces
+an inclusion EF֒→EFalland hence a map X[F]→X[Fall], which we compose with theCOMPLETION OF G-SPECTRA AND STABLE MAPS BETWEEN CLASSIFYING SPACES 11
+equivalence X[Fall]≃− →Xto getηX. Thus the map EF→ptin the definition above should be
+thought of as the composite EF→EFall≃− →pt, but in the end projection to a point is trivial
+however one factors it.
+The next lemma records some elementary properties of the F-isotropy restriction.
+Lemma 5.2. LetFbe a family of subgroups in G.
+(a)RFpreserves the properties of being bounded below, of finite type or aG-CW spectrum.
+(b)RFcommutes with colimits.
+(c)LetXbe aG-spectrum, and let Hbe a subgroup of G. IfH /∈FthenΦH(X[F])is
+non-equivariantly weakly contractible. If H∈FthenΦH(X[F])ΦH(η)−−−−→ΦH(X)is a
+non-equivariant equivalence.
+(d)An inclusion of families Fa⊆Fbinduces a natural transformation RFa⇒RFbin which
+every map is a cofibration.
+Proof. Part (a) is immediate since EFis aG-CW complex of finite type, and (b) is a direct
+consequence of the smash product definition of RF.
+SinceΦHcommutes with suspension and smash product, one has a non-eq uivariant equiva-
+lence,
+ΦH(X[F])≃(EF+)H∧ΦH(X).
+Now,(EF+)His contractible for H /∈F, and equivalent to S0forH∈F. Part (c) follows.
+The natural transformation in part (d) is induced by the cofib rationEFa→EFbin the
+obvious way. /square
+A less obvious property of isotropy restriction is that it co mmutes with completion up to weak
+equivalence. To make sense of this, we first need a map compari ng the two ways of composing
+the isotropy restriction and completion functors. Let Xbe aG-spectrum. For a family of
+subgroupsFand an ideal J⊆A(G), a commutative diagram
+X∧
+J[F]
+/d15/d15X[F]/d59/d59/d118/d118/d118/d118/d118/d118/d118/d118/d118
+/d35/d35/d72/d72/d72/d72/d72/d72/d72/d72/d72
+X[F]∧
+J
+is obtained as the diagram
+F(SG,X)∧F(SG,EF+) /d47/d47F(M(I),X)∧F(SG,EF+)
+h
+/d15/d15X∧EF+∼=/d53/d53/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107/d107
+∼=
+/d41/d41/d83/d83/d83/d83/d83/d83/d83/d83/d83/d83/d83/d83/d83/d83
+F(SG∧SG,X∧EF+) /d47/d47F(M(I)∧SG,X∧EF+),
+where the horizontal maps are each induced by the natural map M(I)→SG, andhis the
+canonical map.
+Lemma 5.3. LetFbe a family of subgroups of Gand letJbe an ideal of A(G). IfXis a
+bounded-below G-spectrum, then the natural map h:X∧
+J[F]→X[F]∧
+Jis a weak equivalence.12 KÁRI RAGNARSSON
+Proof. By Proposition 3.1 it suffices to show that ΦH(f)is a non-equivariant weak equivalence
+for every subgroup HofG.
+WhenH /∈F, the spectrum ΦH(X∧
+J[F])is weakly contractible. By Theorem 4.1, ΦH(X[F]∧
+J)
+is non-equivariantly equivalent to the φH(J)-adic completion of the weakly contractible spectrum
+ΦH(X[F])and is therefore itself weakly contractible. It follows tri vially that ΦH(h)is a weak
+equivalence.
+ForH∈F, Theorem 4.1 gives non-equivariant equivalences
+ΦH(X∧
+J[F])≃ΦH(X∧
+J)≃ΦH(X)∧
+φH(J)
+and
+ΦH(X[F]∧
+J)≃ΦH(X[F])∧
+φH(J)≃ΦH(X)∧
+φH(J).
+In fact the theorem implies that the natural maps ΦH(X[F])→ΦH(X∧
+J[F])andΦH(X[F])→
+ΦH(X[F]∧
+J)are non-equivariantly φH(J)-adic completion maps. Since ΦH(h)respects these
+maps, it follows that it is a non-equivariant equivalence. /square
+Having defined the replacement spectra, we now prepare to pro ve the Replacement Theorem.
+Proposition 3.1 and Theorem 4.1 allow us to control the I(G)-adic completion of a spectrum if
+we can describe the ideals φH(I(G)). This is done in the following lemma.
+Lemma 5.4. For a subgroup H≤Gwe have
+φH(I(G)) =
+
+(0), ifH={1};
+(pk)for some k >0,ifHis ap-group; and
+Z, otherwise.
+Proof. The first claim, φ1(I(G)) =ǫ(I(G)) = 0 , is true by definition of I(G).
+For nontrivial Hwe observe that the ideal I(G)is generated by elements of the form
+[G/K]−|G:K|·[G/G]
+whereKruns through subgroups of G, and thus φH(I(G))is generated by the numbers
+(G/K)H−|G:K|=|NG(H,K)|
+|K|−|G:K|
+where the transporter NG(H,K)is the set of elements x∈Gsuch that xHx−1≤K.
+For a prime q, letGqbe a Sylow q-subgroup of G. Observe that if His not aq-group then
+NG(H,Gq)is empty so
+(G/Gq)H−|G:Gq|=−|G:Gq|.
+Suppose that the order of His divisible by two distinct primes. Then His not a q-group for
+any prime q, so, by the previous observation, |G:Gq|∈φH(I(G))for every prime qdividing G.
+As the greatest common divisor of the numbers |G/Gq|is1, this implies that φH(I(G)) =Z.
+Now suppose that His a nontrivial p-group. An argument similar to the previous paragraph
+shows that|Gp|, which is the greatest common divisor of the collection of nu mbers|G:Gq|for
+primesq/\e}atio\slash=p, is contained in φH(I(G)). Therefore the ideal φH(I(G))contains a power of p.
+On the other hand, for every finite G-setXthe setX\XHis a disjoint union of nontrivial H-
+orbits. As His ap-group each of these orbits has cardinality divisible by p. Since the numbers
+|XH|−|X|=−|X\XH|generate φH(I(G)), this implies that φH(I(G))is contained in the
+ideal(p). ThusφH(I(G))is of the form (pk)for some k >0. /square
+We can now prove the replacement theorem.COMPLETION OF G-SPECTRA AND STABLE MAPS BETWEEN CLASSIFYING SPACES 13
+Proof of Theorem A. For every subgroup HofG, the natural map f:X[FP]→Xinduces a
+non-equivariant commutative diagram
+ΦH(X[FP])ΦH(f)/d47/d47
+/d15/d15ΦH(X)
+/d15/d15
+ΦH(X[FP]∧
+I)ΦH(f∧
+I(G))/d47/d47ΦH(X∧
+I),
+where, by Theorem 4.1, the two vertical maps are φH(I(G))-adic completion maps, and the
+bottom map is the φH(I(G))-adic completion of the top map. When His a group of prime
+power order the top map is a weak equivalence and so the same is true of the bottom map. When
+Hhas order divisible by two primes the two vertical maps are co mpletion at φH(I(G)) =Z, so
+both spectra in the bottom row are weakly contractible and th e map between them is a weak
+equivalence by default. Since ΦH(f∧
+I(G))is a weak equivalence for every subgroup HofGit
+follows from Proposition 3.1 that f∧
+I(G)is a weak equivalence. /square
+6.The decomposition theorem
+In this section we explain the importance of Theorem A by show ing that the I(G)-adic
+completion of the replacement spectrum X[FP]can be computed in an efficient manner. That
+is, we show that the I(G)-adic completion of X[FP]decomposes as a colimit of I(G)-adic
+completions, the components of which can be computed as the c ompletion at a prime. The first
+step in this direction is the following observation.
+Lemma 6.1. For aG-spectrum X, the natural map
+colim
+F∈DX[F]−→X[FP]
+is a homotopy equivalence.
+Proof. This follows from Lemma 2.3 and the fact that smash products p reserve colimits. /square
+With a bit more work we can prove the following proposition, o f which Theorem B in the
+introduction is a special case.
+Proposition 6.2. LetJbe an ideal of A(G). For a bounded-below G-spectrum X, the natural
+map
+hocolim
+F∈D(X[F]∧
+J)−→X[FP]∧
+J
+is a weak equivalence.
+Proof. This map fits into a commutative diagram
+hocolim
+F∈D(X∧
+J[F])
+≃(5.3)
+/d15/d15≃/d47/d47colim
+F∈D(X∧
+J[F])≃
+(6.1)/d47/d47X∧
+J[FP]
+≃(5.3)
+/d15/d15hocolim
+F∈D(X[F]∧
+J) /d47/d47X[FP]∧
+J.
+As indicated, the maps in the upper right corner are equivale nces by Lemmas 6.1 and 5.3, respec-
+tively. Lemma 5.3 also implies for each F∈D, the map X[F]∧
+J→X∧
+J[F]is a weak equivalence,
+and hence the induced map of homotopy colimits on the left sid e is a weak equivalence. The14 KÁRI RAGNARSSON
+map from the homotopy colimit to the colimit along the top is a n equivalence since each map
+inX[F1]→X[Fp]in the diagram over which the colimit is taken is a cofibration (Lemma 2.1).
+It follows that the bottom map is a weak equivalence. /square
+Again we observe that in Lemma 6.1 and Proposition 6.2, the di agramD(defined in Section
+2) can be restricted to just include families Fpfor primes pthat divide the order of Gand the
+results still hold true. This is obviously more convenient f or computational purposes, since one
+then obtains a finite diagram. However, by including all prim es in the diagram, the result is
+natural in GwhenXis.
+For a bounded-below G-spectrum X, Theorem A and Proposition 6.2 combine to give a weak
+equivalence
+hocolim
+F∈D/parenleftig
+X[F]∧
+I(G)/parenrightig
+≃X∧
+I(G).
+We now show that the completions inside the homotopy colimit can be computed in terms of p-
+adic completions. That is, for a prime p, theI(G)-adic completion of the spectrum X[Fp]is very
+close to being a p-adic completion. The difference lies only in the cofibration X[F1]→X[Fp],
+the spectrum X[F1]being unchanged by I(G)-adic completion. To make this precise, we make
+the following definition.
+Definition 6.3. LetFa⊆Fbbe families of subgroups in G. For aG-spectrum X, denote by
+X[Fb,Fa]the cofibre of the map X[Fa]→X[Fb].
+Theorem C in the introduction is included in the following pr oposition as parts (a) and (c).
+Proposition 6.4. LetXbe a bounded-below G-spectrum.
+(a)The map X[F1]→X[F1]∧
+I(G)is a weak equivalence.
+(b)For a prime p, the map X[Fp,F1]→X[Fp,F1]∧
+I(G)is, up to weak equivalence, a p-adic
+completion map.
+(c)For a prime p, the completion X[Fp]∧
+I(G)fits, up to weak equivalence, into a cofibration
+sequence
+X[F1]−→X[Fp]∧
+I(G)−→X[Fp,F1]∧
+p.
+Proof. We prove part (a) by showing that the induced maps on geometri c fixed-point spectra
+are non-equivariant weak equivalences for all subgroups H≤Gand then applying Proposition
+3.1. For a nontrivial subgroup H≤G, the map ΦH(X[F1])→ΦH(X[F1]∧
+I(G))is a map of
+weakly contractible spectra, and hence a weak equivalence. For the trivial subgroup, the map
+Φ1(X)→Φ1(X[F1]∧
+I(G))is non-equivariantly completion at the ideal n1(I(G)) = 0 , which has
+no effect, so the map is a weak equivalence.
+Turning to part (b), we first note that for a subgroup H≤Gwe have, up to weak equivalence,
+a cofibration sequence
+ΦH(X[F1])−→ΦH(X[Fp])−→ΦH(X[Fp,F1]).
+We deduce that when H /∈FporH= 1, the geometric fixed-point spectrum ΦH(X[Fp,F1])
+is weakly contractible. Hence the map ΦH(X[Fp,F1])→ΦH(X[Fp,F1]∧
+I(G))can trivially
+be regarded as a p-adic completion map. For H∈ Fp\F1, the map ΦH(X[Fp,F1])→
+ΦH(X[Fp,F1]∧
+I(G))is non-equivariantly completion at (p)by Theorem 4.1 and Lemma 5.4.COMPLETION OF G-SPECTRA AND STABLE MAPS BETWEEN CLASSIFYING SPACES 15
+To complete the proof we consider the commutative square
+X[Fp,F1](−)∧
+I(G)/d47/d47
+(−)∧
+p
+/d15/d15X[Fp,F1]∧
+I(G)
+(−)∧
+I(G)+(p)
+/d15/d15
+X[Fp,F1]∧
+p(−)∧
+I(G)+(p)/d47/d47X[Fp,F1]∧
+I(G)+(p).(2)
+The preceding argument shows that taking geometric fixed poi nts at a subgroup H≤Gof this
+square one obtains the commutative square
+ΦH(X[Fp,F1])(−)∧
+p/d47/d47
+(−)∧
+p
+/d15/d15ΦH(X[Fp,F1]∧
+I(G))
+≃
+/d15/d15
+ΦH(X[Fp,F1]∧
+p)≃/d47/d47ΦH(X[Fp,F1]∧
+I(G)+(p)).
+From this we deduce that in the square (2), the lower and right side maps are weak equivalences,
+and thus the upper map is equivalent to the left map, up to weak equivalence, which is what we
+wanted to prove.
+Part (c) is an easy consequence of parts (a) and (b), using the fact that completions preserve
+cofibrations up to weak equivalences (they preserve fibratio n sequences by their construction as
+mapping spectra, and cofibration sequences are fibration seq uences up to weak equivalence). /square
+As promised, the combination of Theorems A, B and C provides a method for computing
+I(G)-adic completions of bounded-below G-spectra. The overall procedure may seem slightly
+complicated, so we permit ourselves a small departure from p recision to say that in essence these
+results allow us to compute X∧
+I(G)as a wedge sum/logicalortext
+pX[Fp]∧
+p. Enforcing precision again, things
+are a bit more complicated since the spectra X[Fp]are joined together at X[F1], necessitating
+the homotopy colimit construction instead of a simple wedge sum, and X[F1]is unaffected by
+completion, making the completions slightly harder to desc ribe. Some instances where the wedge
+sum description can be made precise are given in the followin g corollary.
+Corollary 6.5. LetXbe aG-spectrum that is bounded below.
+(a)IfX[F1]splits as a wedge sum of its p-completions, at different primes p, then there is
+a weak equivalence
+X∧
+I(G)≃/logicalordisplay
+pX[Fp]∧
+p.
+(b)If the cofibration X[F1]→Xsplits, then there is a weak equivalence
+X∧
+I(G)≃X[F1]∨/logicalordisplay
+pX[Fp,F1]∧
+p.
+Part (a) in Corollary 6.5 is particularly useful since it app lies when X[F1]is a torsion spec-
+trum, such as the classifying space of a group. Part (b) is use ful to deal with basepoint issues
+whenXis the suspension spectrum of a G-space with added, disjoint basepoint. Upon suspen-
+sion, the added basepoint becomes a sphere spectrum wedge su mmand that belongs to X[F1]
+and is unaffected by completion. In some cases the result allo ws us to set this sphere summand
+aside while completing and add it back at the end.16 KÁRI RAGNARSSON
+7.Stable maps between classifying spaces
+LetGbe a finite group. For a compact Lie group K, aprincipal (G,K)-bundle is a principal
+K-bundle in the category of G-spaces. Denote by B(G,K)the classifying space of principal
+(G,K)-bundles. Generalizing the Segal conjecture, May–Snaith– Zelewski proved the completion
+theorem for Σ∞
+GB(G,K)+in [18]. (This was originally proved for finite Kin [15].) Thus the
+natural map
+Σ∞
+GB(G,K)+∼=F(S0,Σ∞
+GB(G,K)+)−→F(EG+,Σ∞
+GB(G,K)+)
+is anI(G)-adic completion map.
+Now,Σ∞
+GB(G,K)+isG-split with underlying non-equivariant spectrum BKso by Theorem
+3.2 we have a non-equivariant equivalence
+F(EG+,Σ∞
+GB(G,K)+)G≃F(EG/G +,ι∗(Σ∞
+GB(G,K)+))≃F(BG+,Σ∞BK+).
+Thus the spectrum of stable maps between classifying spaces can be computed as the completion
+ofΣ∞
+GB(G,K)G
++. The tom Dieck splitting of this spectrum was further refined in [15] to obtain
+the non-equivariant splitting,
+Σ∞
+GB(G,K)G
++≃/logicalordisplay
+[H,ϕ]Σ∞BW(H,ϕ)+. (3)
+Here the wedge sum ranges over conjugacy classes of pairs (H,ϕ)whereH≤Gandϕis a group
+homomorphism H→K, andW(H,ϕ)is the Weyl group
+W(H,ϕ) =NG×K(∆(H,ϕ))/∆(H,ϕ),
+of the subgroup
+∆(H,ϕ) ={(h,ϕ(h))|h∈H}≤G×K.
+(Equivalently, W(H,ϕ)is the automorphism group of the principal (G,K)-bundle
+(G×K)/∆(H,ϕ)→G/H.)
+Collecting these facts, topologists commonly say that F(BG+,Σ∞BK+)is theI(G)-adic
+completion of the wedge sum of classifying spaces in (3). Thi s statement has up to now been
+rather vague and of limited practical use as the action of A(G)in terms of this non-equivariant
+splitting is not understood, and the effect of I-adic completion has so far been a mystery. Using
+the results from Sections 5 and 6 we can now make this a precise statement.
+First we note that, for a family Fof subgroups in G, we have
+Σ∞
+GB(G,K)+[F]≃Σ∞
+GB(F,K)+,
+whereB(F,K)is the classifying space of principal K-bundles in the category of F-spaces. The
+fixed-point spectrum Σ∞
+GB(F,K)G
++admits a non-equivariant splitting similar to the one for
+Σ∞B(G,K)G
++.
+Proposition 7.1 ([15]).LetGbe a finite group and Ka compact Lie group. For a family F
+of subgroups in G, there is a non-equivariant splitting
+Σ∞B(F,K)G
++≃/logicalordisplay
+[H,ϕ]
+H∈FΣ∞BW(H,ϕ)+.
+Proof. Lewis–May–McClure proved this for B(G,K) =B(Fall,K) in [15]. A minor adjustment
+generalizes their proof to any family of subgroups. /square
+This leads to the following description of stable maps betwe en classifying spaces, which is
+Theorem D of the introduction, and generalizes [20, Theorem B].COMPLETION OF G-SPECTRA AND STABLE MAPS BETWEEN CLASSIFYING SPACES 17
+Theorem 7.2. For a finite group Gand a compact Lie group K, there is a homotopy equivalence
+F(BG+,Σ∞BK+)≃/logicalordisplay
+[H,ϕ]
+H∈FPΣ∞(BW(H,ϕ)∧
+p(H))+,
+where, for H∈FP\F1,p(H)is the prime that divides |H|, andp(1) = 0 .
+Proof. Proposition 7.1 gives us
+Σ∞B(F1,K)G
++≃Σ∞BW(1,i)+
+whereiis the unique homomorphism 1→K, and
+Σ∞B(Fp,K)G
++≃/logicalordisplay
+[P,ϕ]
+P∈FpΣ∞BW(P,ϕ)+.
+ThusΣ∞B(F1,K)G
++is a summand of Σ∞B(Fp,K)G
++with cofibre
+/logicalordisplay
+[P,ϕ]
+1<P∈FpΣ∞BW(P,ϕ)+.
+Applying part (b) of Corollary 6.5 we therefore have a weak eq uivalence
+F(BG+,Σ∞BK+)≃Σ∞BW(1,i)+∨/logicalordisplay
+p/logicalordisplay
+[P,ϕ]
+1<P∈Fp(BW(P,ϕ)∧
+p)+,
+and this is just a different way to write the wedge sum in the sta tement of the theorem.
+What remains is to note that (by Morse theory) Khas the homotopy type of a CW-complex.
+Therefore B(G,K)also has the homotopy type of a CW-complex, and all the spectr a in sight
+have the homotopy type of G-CW spectra. Hence the weak equivalence is in fact a homotopy
+equivalence. /square
+We pause to note that
+BW(1,i)+=B(G×K)+≃BG+∧BK+.
+We have a stable splitting
+Σ∞BG+≃S∨/logicalordisplay
+pΣ∞BG∧
+p,
+where the sphere spectrum summand arises from the natural eq uivalence Σ∞X+≃S∨Σ∞X,
+and the wedge sum can be taken over all primes por just those primes that divide the order of
+G. It follows that we have a splitting
+Σ∞B(F1,K)G
++≃Σ∞BK+∨/logicalordisplay
+p/parenleftbig
+Σ∞BG∧
+p∧Σ∞BK+/parenrightbig
+,
+and we can rewrite the conclusion of Theorem 7.2 as a homotopy equivalence
+F(BG+,Σ∞BK+)≃Σ∞BK+∨/logicalordisplay
+p
+/parenleftbig
+Σ∞BG∧
+p∧Σ∞BK+/parenrightbig
+∨/logicalordisplay
+[P,ϕ]
+1<P∈Fp(BW(P,ϕ)∧
+p)+
+.
+The difference here is just a matter of bookkeeping, but it all ows us to easily focus attention on
+one prime at a time, obtaining the following corollary, whic h generalizes [20, Theorem A].18 KÁRI RAGNARSSON
+Corollary 7.3. For a finite group G, a compact Lie group Kand a prime p, there is a homotopy
+equivalence
+F(BG∧
+p+,BK+)≃/parenleftig
+Σ∞BG∧
+p+∧Σ∞BK+/parenrightig
+∨/logicalordisplay
+[P,ϕ]
+1<P∈Fp(BW(P,ϕ)∧
+p)+.
+Proof. For a prime q, letSqbe a Sylow q-subgroup of G. Standard arguments show that BG∧
+q+
+is a stable summand of both BG+andBSq+. Hence F(BG∧
+q+,BK+)is a stable summand of
+bothF(BG+,BK+)andF(BSq+,BK+). Furthermore, since BGsplits stably as a wedge sum
+of itsq-completions, every summand of F(BG+,BK+)is a summand of F(BSq+,BK+)for some
+primeq. Considering which summands of F(BG+,BK+)can belong to which F(BSq+,BK+)
+forces the desired result. For these purposes, the leading t erm in the wedge sum is rewritten as
+Σ∞BG∧
+p+∧Σ∞BK+≃(S∨Σ∞BG∧
+p)∧Σ∞BK+≃Σ∞BK+∨(BG∧
+p∧Σ∞BK+).
+/square
+A case of special interest is when K= 1and Theorem 7.2 and Corollary 7.3 describe the
+stable mapping duals of BG+andBG∧
+p+, respectively.
+Corollary 7.4. For a finite group G, there are non-equivariant equivalences
+D(BG+)≃/logicalordisplay
+H∈FPΣ∞(BWG(H)∧
+p(H))+,
+and
+D(BG∧
+p+)≃/logicalordisplay
+H∈FpΣ∞(BWG(H)∧
+p)+.
+References
+[1]Adams, J. F., Gunawardena, J. H., and Miller, H. The Segal conjecture for elementary abelian
+p-groups. Topology 24 , 4 (1985), 435–460.
+[2]Adams, J. F., Haeberly, J.-P., Jackowski, S., and May, J. P. A generalization of the Segal conjecture.
+Topology 27 , 1 (1988), 7–21.
+[3]Atiyah, M. F. Characters and cohomology of finite groups. Inst. Hautes Études Sci. Publ. Math. 9 (1961),
+23–64.
+[4]Atiyah, M. F., and Segal, G. B. Equivariant K-theory and completion. J. Differential Geometry 3
+(1969), 1–18.
+[5]Bousfield, A. K. The localization of spectra with respect to homology. Topology 18 , 4 (1979), 257–281.
+[6]Carlsson, G. Equivariant stable homotopy and Segal’s Burnside ring conj ecture. Ann. of Math. (2) 120 ,
+2 (1984), 189–224.
+[7]Caruso, J., May, J. P., and Priddy, S. B. The Segal conjecture for elementary abelian p-groups. II.
+p-adic completion in equivariant cohomology. Topology 26 , 4 (1987), 413–433.
+[8]Elmendorf, A. D. Systems of fixed point sets. Trans. Amer. Math. Soc. 277 , 1 (1983), 275–284.
+[9]Elmendorf, A. D., Kriz, I., Mandell, M. A., and May, J. P. Rings, modules, and algebras in
+stable homotopy theory , vol. 47 of Mathematical Surveys and Monographs . American Mathematical Society,
+Providence, RI, 1997. With an appendix by M. Cole.
+[10]Greenlees, J. P. C. Equivariant functional duals and completions at ideals of t he Burnside ring. Bull.
+London Math. Soc. 23 , 2 (1991), 163–168.
+[11]Greenlees, J. P. C., and May, J. P. Completions of G-spectra at ideals of the Burnside ring. In Adams
+Memorial Symposium on Algebraic Topology, 2 (Manchester, 19 90), vol. 176 of London Math. Soc. Lecture
+Note Ser. Cambridge Univ. Press, Cambridge, 1992, pp. 145–178.
+[12]Greenlees, J. P. C., and May, J. P. Derived functors of I-adic completion and local homology. J.
+Algebra 149 , 2 (1992), 438–453.
+[13]Greenlees, J. P. C., and May, J. P. Equivariant stable homotopy theory. In Handbook of algebraic
+topology . North-Holland, Amsterdam, 1995, pp. 277–323.
+[14]Kosniowski, C. Equivariant cohomology and stable cohomotopy. Math. Ann. 210 (1974), 83–104.COMPLETION OF G-SPECTRA AND STABLE MAPS BETWEEN CLASSIFYING SPACES 19
+[15]Lewis, L. G., May, J. P., and McClure, J. E. Classifying G-spaces and the Segal conjecture. In Current
+trends in algebraic topology, Part 2 (London, Ont., 1981) , vol. 2 of CMS Conf. Proc. Amer. Math. Soc.,
+Providence, R.I., 1982, pp. 165–179.
+[16]Lewis, Jr., L. G., May, J. P., Steinberger, M., and McClure, J . E.Equivariant stable homotopy
+theory , vol. 1213 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1986. With contributions by J.
+E. McClure.
+[17]May, J. P., and McClure, J. E. A reduction of the Segal conjecture. In Current trends in algebraic
+topology, Part 2 (London, Ont., 1981) , vol. 2 of CMS Conf. Proc. Amer. Math. Soc., Providence, R.I.,
+1982, pp. 209–222.
+[18]May, J. P., Snaith, V. P., and Zelewski, P. A further generalization of the Segal conjecture. Quart.
+J. Math. Oxford Ser. (2) 40 , 160 (1989), 457–473.
+[19]Piacenza, R. J. Homotopy theory of diagrams and CW-complexes over a categor y.Canad. J. Math. 43 , 4
+(1991), 814–824.
+[20]Ragnarsson, K. A Segal conjecture for p-completed classifying spaces. Adv. Math. 215 , 2 (2007), 540–568.
+[21]tom Dieck, T. Transformation groups , vol. 8 of de Gruyter Studies in Mathematics . Walter de Gruyter &
+Co., Berlin, 1987.
+Department of Mathematical Sciences, Depaul University, C hicago, IL, USA
+E-mail address :kragnars@math.depaul.edu
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+arXiv:1001.0772v2  [cond-mat.quant-gas]  23 Apr 2010Second sound and the density response function in uniform
+superfluid atomic gases
+H. Hu
+ACQAO and Centre for Atom Optics and Ultrafast Spectroscopy ,
+Swinburne University of Technology,
+Melbourne, Victoria 3122, Australia and
+Department of Physics, Renmin University of China, Beijing 100872, China
+E. Taylor
+Department of Physics, The Ohio State University, Columbus , Ohio, 43210, USA
+X.-J. Liu
+ACQAO and Centre for Atom Optics and Ultrafast Spectroscopy ,
+Swinburne University of Technology,
+Melbourne, Victoria 3122, Australia
+S. Stringari
+CNR-INFM BEC Centre and Dipartimento di Fisica,
+Universit` a di Trento, I-38050 Povo, Trento, Italy
+A. Griffin
+Department of Physics, University of Toronto,
+Toronto, Ontario, M5S 1A7, Canada
+(Dated: Apr. 26, 2010)
+1Abstract
+Recently there has been renewed interest in second sound in s uperfluid Bose and Fermi gases.
+By using two-fluid hydrodynamictheory, we review the densit y response χnn(q,ω) of these systems
+as a tool to identify second sound in experiments based on den sity probes. Our work generalizes
+the well-known studies of the dynamic structure factor S(q,ω) in superfluid4He in the critical
+region. We show that, in the unitary limit of uniform superflu id Fermi gases, the relative weight
+of second vs. first sound in the compressibility sum rule is gi ven by the Landau–Placzek ratio
+ǫLP≡(¯cp−¯cv)/¯cvfor all temperatures below Tc. In contrast to superfluid4He,ǫLPis much larger
+in strongly interacting Fermi gases, being already of order unity for T∼0.8Tc, thereby providing
+promising opportunities to excite second sound with densit y probes. The relative weights of first
+andsecondsoundarequitedifferentin S(q,ω)(measuredinpulsepropagationstudies)ascompared
+to Imχnn(q,ω) (measured in two-photon Bragg scattering). We show that fir st and second sound
+inS(q,ω) in a strongly interacting Bose-condensed gas are similar t o those in a Fermi gas at
+unitarity. However, in a weakly interacting Bose gas, first a nd second sound are mainly uncoupled
+oscillations of the thermal cloud and condensate, respecti vely, and second sound has most of the
+spectral weight in S(q,ω). We also discussthe behaviourof the superfluidandnormal fl uidvelocity
+fields involved in first and second sound.
+PACS numbers: 03.75.Kk, 03.75.Ss, 67.25.D-
+2I. INTRODUCTION
+The most dramatic effects related to superfluidity in liquid4He arise [1] when the dynam-
+ics of the two components are described by the two-fluid hydrodyn amics first discussed by
+Landau [2]. These equations only describe the dynamics when the non -equilibrium states
+are in local hydrodynamic equilibrium [3], which requires short collision t imes between the
+excitations forming the normal fluid. (This requirement is usually sum marized as ωτ≪1,
+whereωis the frequency of a collective mode and τis the appropriate relaxation rate.) The
+study of ultracold gases when they are in local equilibrium has been diffi cult because the
+density and the s-wave scattering length are typically not large enough. However, r ecent ex-
+perimental work on trapped Bose-condensed gases has reporte d some success, with evidence
+for a second sound mode in highly elongated (cigar-shaped) traps [4 ].
+Another approach to achieving conditions where the Landau two-fl uid description is cor-
+rect has been to consider a Fermi superfluid gas close to unitarity [ 5], where the s-wave
+scattering length between Fermi atoms in two different hyperfine s tates is infinite. Develop-
+ing earlier theoretical studies [6, 7], Taylor and co-workers [8] hav e recently given detailed
+predictions for the first and second sound breathing oscillations in a trapped Fermi gas at
+unitarity. The coupled differential equations of the Landau two-flu id description were solved
+variationally, with results which agreed with analytic predictions at T→0 andT→Tc. So
+far, only an isotropic trap has been studied.
+In [8], it was shown that (as in superfluid4He) the frequencies of first and second sound
+in unitary Fermi gases were quite well approximated by assuming tha t the solutions of the
+two-fluid equations corresponded to pure uncoupled density and t emperature waves. This
+is in sharp contrast to the situation in dilute, weakly interacting Bose gases, where first
+and second sound involve both density and temperature oscillations [9]. Na¨ ıvely, the first
+result above would lead one to believe that second sound in a unitary F ermi gas would have
+very small weight in the density response function. However, Arah ata and Nikuni [10] have
+shown that in a uniform Fermi gas at unitarity, a density disturbanc e can have first and
+second sound pulses of comparable magnitude at temperatures of order 0.8Tc.
+The purpose of this paper is to provide a systematic study of the de nsity response func-
+tionχnn(q,ω) of a uniform superfluid atomic gas in the hydrodynamic regime [11, 1 2],
+as described by the non-dissipative Landau two-fluid equations (se e sections II and III).
+3In this hydrodynamic region, the related dynamic structure facto r is given by S(q,ω)∝
+Imχnn(q,ω)/ω. In studies of superfluid4He,S(q,ω) is directly measured in Brillouin light
+scattering experiments [13, 14]. In dilute gases, density pulse prop agation experiments ef-
+fectively measure S(q,ω) as noted in [10]. In addition, experiments using two-photon Bragg
+scattering have a cross-section proportional to Im χnn(q,ω). While all these experiments
+probe the same density response function, Im χnn(q,ω), the relative weight of the low fre-
+quency resonances are strongest in S(q,ω) because of the extra factor of 1 /ωnoted above.
+This fact has significant implications for measuring second sound sinc e the frequency of
+second sound becomes much smaller than first sound as Tcis approached.
+We relate our analysis to classic discussions of the dynamic structur e factorS(q,ω) in
+the two-fluid region of superfluid4He in the critical region [13–17]. However, many new
+features arise when dealing with quantum gases.
+New aspects of our work include the use of the compressibility sum ru le (sections III
+and IV) in understanding the relative weights of first and second so und in the density
+response function. We also discuss the role played by the Landau–P laczek ratio ǫLP≡(¯cp−
+¯cv)/¯cvin determining these weights, with emphasis on new aspects that aris e in superfluid
+gases. Here, ¯ cpand ¯cvare the specific heats per unit mass at constant pressure and volu me,
+respectively. We discuss the differences between S(q,ω) of a unitary Fermi gas and that
+of a Bose-condensed gas (sections IV and V), and present result s for Bragg scattering with
+localized beams applied to the centre of a trapped unitary Fermi gas using the local density
+approximation (section VI). In B, we provide a detailed analysis of th e superfluid and
+normal fluid velocity fields associated with first and second sound.
+II. DENSITY RESPONSE FUNCTION
+Inthissection, wereview somebasicpropertiesofthedensity resp onsefunction χnn(q,ω),
+the related dynamic structure factor S(q,ω), and the experiments that measure these quan-
+tities. The density response function involves the correlation betw een the density at two
+different points at different times:
+χnn(r,t)≡ −iθ(t)∝angbracketleft[ˆρ(r,t),ˆρ(0,0)]∝angbracketright. (1)
+4Here,θ(t) is the step function and we have set the volume equal to unity. The Fourier
+transform Im χnn(q,ω) of the imaginary part of the density response function is related t o
+the dynamic structure factor S(q,ω) by
+Imχnn(q,ω+i0+) =−nπ[S(q,ω)−S(−q,−ω)]
+=−nπS(q,ω)(1−e−β/planckover2pi1ω), (2)
+wherenis the density, β= 1/Tis the inverse temperature, and in the last step we have
+made use of the “detailed balance” relation, S(q,−ω) =e−βωS(q,ω) and also the inversion
+symmetry of the system which gives S(−q,ω) =S(q,ω). Here and throughout this paper
+we set/planckover2pi1=kB= 1. Equation (2) can also be re-written as
+S(q,ω) =−1
+πn[N0(ω)+1]Imχnn(q,ω+i0+), (3)
+whereN0(ω) = (eβω−1)−1is the Bose distribution function. The latter arises in both
+Bose and Fermi fluids because density fluctuations always obey Bos e statistics. We see from
+(2) that Im χnn(q,ω) is the antisymmetric (frequency) part of the dynamic structure factor
+S(q,ω).
+The dynamic structure factor S(q,ω) is measured by using inelastic scattering (e.g., neu-
+tron and Brillouin light scattering) of probe particles, which transfe r momentum qand
+energyωto the system. S(q,ω) is discussed and calculated in all standard texts on many
+body physics (convenient reviews are given in [18, 19]). In contrast , the density response
+function χnn(q,ω) describes the density fluctuation induced by an external pertur bing po-
+tential. As an example, in atomic gases, the imaginary part of the den sity response function
+is probed in two-photon Bragg scattering and also describes the de nsity pulse produced by
+a blue-detuned laser. Brillouin and neutron scattering have been us ed extensively to study
+the collective modes in4He. These techniques cannot be applied to dilute atomic gases,
+however, since the gases are far too dilute to generate an apprec iable signal. Instead, spec-
+troscopic probes of dilute gases have successfully used Bragg sca ttering since this technique
+makes use of the stimulated light scattering processes induced by t wo counter-propagating
+laser beams, resulting in a strong enhancement of the signal.
+InBragg scattering experiments (see, e.g., [20]), by measuring th etotal momentum trans-
+ferred to the sample, one measures the imaginary part of the dens ity response function
+χnn(q,ω), where qandωare the momentum and energy transferred by the stimulated ab-
+sorption and emission of the photons [18]. If the Bragg pulse is short compared to 2 π/ωz,
+5Method System Quantity measured Response function
+Bragg Spectroscopy Gases Momentum transferred Imχnn(q,ω)
+Neutron Spectroscopy4HeNumber of scattered neutrons S(q,ω)
+Brillouin Spectroscopy ( ω≪T)4HeNumber of scattered photons S(q,ω)∼T
+ωImχnn(q,ω)
+Density pulse Gases Pulse amplitude1
+ωImχnn(q,ω)
+TABLE I: Summary of experimental probes. From left to right: the experimental method, the
+system(s) to which the method can be applied, the quantity be ing measured, and the density
+response function involved. In the two-fluid region of stron gly interacting Fermi gases and4He,
+second sound is weakly coupled to density fluctuations but ha s a small frequency ωat high temper-
+atures (below Tc). This means that, as a result of the extra factor of 1 /ωmultiplying the density
+response function, Brillouin spectroscopy and density pul ses are more sensitive to second sound
+than Bragg and neutron scattering.
+whereωzis the frequency of the harmonic trap along the axis of light propaga tion (q=ˆzq),
+the momentum transferred is
+∆Pz= 2qτ/parenleftbiggV
+2/parenrightbigg2
+Imχnn(q,ω), (4)
+whereVis the strength of the potential induced by the lasers and τis the pulse duration.
+As noted recently by Arahata and Nikuni [10], the density response function can also be
+measured by exciting a density pulse in a uniform gas. Within linear resp onse theory, the
+density fluctuation δn(r,t) induced by the application of an external perturbing potential
+δV(r,t) is
+δn(r,t) =/integraldisplaydq
+(2π)3/integraldisplaydω
+2πχnn(q,ω)δV(q,ω)eiq·r−iωt, (5)
+whereδV(q,ω) is the Fourier transform of the perturbing potential. Similar to the sound
+propagation experiments in [21, 22], consider a localized potential a pplied for a short du-
+rationτ < t < 0, and turned off at t= 0. Assuming for simplicity that the perturbing
+6potential only varies spatially along the z-axis (and q=ˆzq), fort >0, one obtains [10]
+δn(z,t) =1
+2π2/integraldisplay
+dq/integraldisplay
+dωδV(q)Imχnn(q,ω)
+ωeiqz−iωt. (6)
+Equation (6) reveals an important feature: in contrast to Bragg s cattering, where the re-
+sponse is proportional to Im χnn(q,ω), the response involved in the excitation of density
+pulses in proportional to Im χnn(q,ω)/ω. As we discuss in section III, for uniform super-
+fluids, this factor of 1 /ωleads to an enhancement in the signal of second sound in density
+pulse experiments as compared to its signal in Bragg spectroscopy . In table I, we review
+the quantities measured by the experimental probes of interest in both dilute gases and
+superfluid4He.
+Beforediscussing the formof the density response function inthe two-fluid hydrodynamic
+region in section III, we first review below some general properties of the dynamic structure
+factor that will be of use later on in understanding the contribution s from first and second
+sound in experimental probes of superfluid atomic gases.
+Quite generally, S(q,ω) satisfies various frequency moment sum rules. Two are of special
+particular interest in the two-fluid domain [23]. The f-sum rule (valid for all q) is
+/integraldisplay∞
+−∞ωS(q,ω) =q2
+2m. (7)
+The compressibility sum rule arises from the exact Kramers-Kronig id entity (also valid for
+allq)
+χnn(q,ω= 0) =/integraldisplay∞
+−∞dω′
+πImχnn(q,ω′)
+ω′, (8)
+which involves an inverse frequency moment. (Note that χnn(q,ω= 0) = Re χnn(q,ω= 0)
+is purely real.) Using (2), one can show that (8) is equivalent to
+χnn(q,ω= 0) =−2n/integraldisplay∞
+−∞dω′S(q,ω′)
+ω′. (9)
+The usefulness of (8) and (9) is due to the fact that in the long wave length limit ( q→0),the
+density response function χnn(q= 0,ω= 0) describes static thermodynamic fluctuations
+which can be related to the equilibrium isothermal compressibility [23, 2 4]
+lim
+q→0χnn(q,ω= 0) =−n∂n
+∂p/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+T. (10)
+Thus we arrive at the so-called compressibility sum rule for the dynam ic structure factor,
+lim
+q→0/integraldisplay∞
+−∞dω′S(q,ω′)
+ω′=1
+2m∂ρ
+∂p/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+T≡1
+2mv2
+T, (11)
+7where we have introduced the isothermal sound velocity vT.
+The sum rules in (7) and (11) are general and may be viewed as two co nstraints which
+S(q,ω) must always satisfy. The f−sum rule is used frequently in discussions of inelastic
+neutron scattering as a check on the experimental data. As we dis cuss in section III, the
+compressibility sumrulehasspecial significancewhenwediscuss thet wo-fluidhydrodynamic
+region, as first emphasized by Nozi` eres and Pines [23].
+III. DYNAMIC STRUCTURE FACTOR IN THE TWO-FLUID REGIME
+The Landau two-fluid equations for an isotropic (such that χnn(q,ω) =χnn(q,ω)) super-
+fluid (Bose or Fermi) describe the dynamics when collisions are strong enough to produce
+local equilibrium. Using the non-dissipative two-fluid equations, one fi nds the density re-
+sponse function (see Refs. [11, 12] and page 138 of [3]),
+χnn(q,ω) =nq2
+mω2−v2q2
+(ω2−u2
+1q2)(ω2−u2
+2q2), (12)
+where we have defined a new velocity
+v2≡T¯s2
+0
+¯cvρs0
+ρn0. (13)
+Here, ¯s0≡S0/Nmis the equilibrium entropy S0per unit mass and ¯ cv≡T(∂¯s/∂T)ρis
+the specific heat per unit mass at constant volume. The equilibrium su perfluid and normal
+fluid densities are denoted by ρs0andρn0, respectively. In (12), u1andu2are the well-
+known exact first and second sound velocities given by Landau’s hyd rodynamics. For later
+purposes, we note that these velocities satisfy the exact relation s
+u2
+1+u2
+2=v2+v2
+¯s, u2
+1u2
+2=v2
+Tv2=v2
+¯sv2
+γ, (14)
+wherevTis defined in (11) and the adiabatic sound velocity is defined as
+v2
+¯s≡∂P
+∂ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+¯s. (15)
+We note that thermodynamic relations [24] show that, quite gener ally, the ratio of the
+specific heats can be related to the isothermal and adiabatic sound velocities,
+γ≡¯cp
+¯cv=v2
+¯s
+v2
+T. (16)
+8The difference between v¯sandvTis a consequence of the finite thermal expansion.
+Equation (12) gives the density response for Bose and Fermi fluids described by the
+non-dissipative Landau two-fluid equations. The two-fluid density r esponse function in the
+presence of dissipation arising from transport coefficients has bee n given by Hohenberg
+and Martin [11, 12, 15]. (See also the discussion given by Vinen [14].) For our purposes
+in this paper, the primary effect of dissipation is to broaden the delta function peaks in
+the imaginary part of the non-dissipative density response functio n. It does not otherwise
+change the structure of the two-fluid response function and in pa rticular, does not affect in
+a significant way the weights of first and second sound in this functio n.
+From (12), it is straightforward to obtain the imaginary part of the density response
+function, which is the experimentally relevant quantity in dilute gases :
+Imχnn(q,ω) =−nπq2
+2m/braceleftbiggZ1
+ω[δ(ω−u1q)+δ(ω+u1q)]
++Z2
+ω[δ(ω−u2q)+δ(ω+u2q)]/bracerightbigg
+. (17)
+Here, we have defined the “traditional” amplitudes for first and sec ond sound, namely
+Z1≡u2
+1−v2
+u2
+1−u2
+2, Z2≡v2−u2
+2
+u2
+1−u2
+2. (18)
+Using (17) in (3), the two-fluid expression for the dynamic structu re factor is found to be
+S(q,ω)=q2
+2m[N0(ω)+1]/braceleftbiggZ1
+ω[δ(ω−u1q)+δ(ω+u1q)]
++Z2
+ω[δ(ω−u2q)+δ(ω+u2q)]/bracerightbigg
+. (19)
+The two-fluid dynamic structure factor in (19) turns out to satisf y both the f-sum and
+compressibility sum rules, given by (7) and (11) [23]. This makes use of the identity N0(ω)+
+1+N0(−ω)+1 = 1, satisfied by the Bose distribution. Substituting (19) into th e left-hand
+side of (7), one sees that the amplitudes Z1andZ2describe the weights of first and second
+sound, respectively, in the f-sum rule. That the sum of these two contributions saturate the
+f-sum rule is a consequence of the fact that Z1+Z2= 1. Similarly, substituting (19) into
+the left-hand side of (11), one sees the relative contributions of fi rst and second sound to
+the isothermal compressibility are given by Z1/u2
+1andZ2/u2
+2, respectively. We will denote
+these weights by Wi:
+W1≡Z1
+u2
+1, W2≡Z2
+u2
+2. (20)
+9To see that the compressibility sum rule is saturated by these contr ibutions from first and
+second sound, we note that Z1/u2
+1+Z2/u2
+2= 1/v2
+T, as can be seen from (14).
+Since we are working in the low frequency two-fluid hydrodynamic reg ion, we can use the
+fact that ω≪T. In this limit, the detailed balance factor in (19) can be approximated by
+N0(ω)+1≃T
+ω+1
+2+... (21)
+The first quantum correction, given by the 1 /2 term in (21) in this expansion, is needed to
+satisfy the sum rules previously discussed. This approximation ( ω≪T) is often referred to
+asthe “classical limit” intextbooks discussing the dynamic structur e factor, althoughin fact
+it describes both superfluid ( T < T c) and normal fluid ( T > T c) collisional hydrodynamics.
+Keeping only the leading order first term in (21), (19) simplifies to
+S(q,ω)≃T
+2m/braceleftbiggZ1
+u2
+1[δ(ω−u1q)+δ(ω+u1q)]
++Z2
+u2
+2[δ(ω−u2q)+δ(ω+u2q)]/bracerightbigg
+=−1
+πn/parenleftbiggT
+ω/parenrightbigg
+Imχnn(q,ω). (22)
+This expression is valid for Bose and Fermi superfluid gases as well as superfluid4He, that is,
+all superfluids described by an order parameter with a phase and am plitude. Equation (22)
+shows that inthe low-frequency two-fluid hydrodynamic region, Wi≡Zi/u2
+igives the weight
+of theithmode. In contrast, the weight of the ithmode in Bragg scattering [see (4) and
+(17)] is given by Bi, where
+B1≡Z1
+u1, B2≡Z2
+u2. (23)
+Comparing (22) with (17), we see that the hydrodynamic limit ( ω≪T) ofS(q,ω) involves
+an extra factor of 1 /ω.
+We see that while Im χnn(q,ω) and Im χnn(q,ω)/ωshare the same poles, the relative
+weights of first and second sound are quite different in these two fu nctions, with B2/B1=
+(W2/W1)(u2/u1) generally being much smaller than W2/W1due to the smallness of u2/u1
+(see Fig. 1).
+The two-fluid expression in (22) is known from earlier discussions of B rillouin scattering
+in superfluid4He (see section IV). In contrast to high-frequency inelastic neut ron scattering,
+Brillouin scattering is carried out with light in the visible part of the spec trum (ω≪T). As
+pointed out above [see (22)], this means that Brillouin scattering mea sures the imaginary
+10partofthedensityresponsefunctiondividedbythefrequency ω. Instudiesofsuperfluid4He,
+this factor of 1 /ωis crucial to probing second sound since, although only weakly couple d to
+density, the speed of second sound is small and hence, its weight in B rillouin light scattering
+can be significant.
+As noted in section II, a similar situation arises in studies of pulse prop agation in dilute
+gases, albeit for very different reasons. In this situation [see (6)], the amplitude of the
+density pulse induced by a localized (in time and space) density pertur bation is proportional
+to Imχnn(q,ω)/ω. The factor of 1 /ωhere results not from the detailed balance factor, but
+from the Fourier transform of the step-function used to model t he time dependence of the
+perturbing laser beams. Arahata and Nikuni [10] calculated the effe ct of producing a density
+fluctuation by a sudden perturbation in a superfluid Fermi gas at un itarity. The final result
+was two pulses moving with the speeds of first and second sound, wit h relative amplitudes
+given by Wias defined in (20).
+IV. SUPERFLUIDS WITH SMALL THERMAL EXPANSION
+In section III, we showed that the relative weights in S(q,ω) and Imχnn(q,ω) of first and
+second sound are given by Wi=Zi/u2
+iandBi=Zi/ui, respectively, where Ziis defined in
+(18). These results are valid for any type of superfluid, including we akly interacting atomic
+Bose gases and also strongly interacting Fermi gases. In this sect ion, we concentrate on
+strongly interacting Fermi gases where, as discussed above, the thermal expansion is rela-
+tively small, meaning that density and temperature fluctuations are not strongly coupled.
+Using the exact relations in (14), in this section we will show that to a v ery good approx-
+imation, W2/W1is given by (30) in this situation. In section V, we give a brief discussion
+of the dynamic structure factor in a dilute Bose gas, where the the rmal expansion can be
+much larger when the s-wave scattering length is small.
+Solving the coupled equations in (14), one obtains the expansions [14 , 15]
+u2
+1=v2
+¯s[1+(γ−1)x+···], u2
+2=v2
+γ[1−(γ−1)x+···], (24)
+in terms of the parameter defined by x≡v2/γv2
+¯s. These results show that to lowest order in
+the parameter x(γ−1)≡xǫLP, the first and second sound velocities are well approximated
+11/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54
+/s115/s101/s99/s111/s110/s100/s32/s115/s111/s117/s110/s100/s83/s111/s117/s110/s100/s32/s118/s101/s108/s111/s99/s105/s116/s105/s101/s115/s32/s91/s105/s110/s32/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s118
+/s70/s93
+/s84 /s47/s84
+/s67/s102/s105/s114/s115/s116/s32/s115/s111/s117/s110/s100
+FIG. 1: Speeds of first and second sound in a unitary superfluid Fermi gas. The dashed lines are
+the approximations c1=v¯sandc2=v/√γ, as given by the leading terms in (24) (from [8]).
+by
+c2
+1=v2
+¯s, c2
+2=v2
+γ=T¯s2
+0
+¯cpρs0
+ρn0. (25)
+In figure 1, we show the temperature dependence of u1andu2and compare these results
+with the leading order expressions c1andc2given by (25) (shown by the dashed lines).
+The thermodynamic functions needed to obtain these sound speed s are calculated using the
+microscopic theory of Nozi` eres and Schmitt-Rink (NSR) [25, 26]. W e see that c1andc2are
+extremely good approximations at all temperatures. At temperat uresT/greaterorsimilar0.4Tc,x=c2
+2/c2
+1
+is very small, even though ǫLP∼ O(1). At lower temperatures, while xis no longer small
+(x= 1/3 atT= 0),ǫLPbecomes extremely small. Thus in both limits, we find that the
+correction term xǫLPin (24) is negligible.
+The fact that u1≃c1andu2≃c2in a fluid with small thermal expansion such as the uni-
+taryFermi gasmeans that first andsecond soundpropagateate ssentially thesamevelocities
+as pure density and temperature oscillations, as would arise when ǫLP= 0. Surprisingly,
+this does not mean that first and second sound are uncoupled dens ity and temperature
+oscillations. We discuss this in detail in B.
+12Two-fluid hydrodynamics leads to the coupled equations [2, 3]
+u2δρ=δP, u2δ¯s= ¯s2
+0ρs0
+ρn0δT. (26)
+As noted above (see figure 1), the speeds of first and second sou nd are well approximated by
+c1andc2in (25). Physically, this means that to leading order, the pressure fl uctuations in
+first sound are at constant entropy per unit mass ¯ sand hence, δP≃(∂P/∂ρ)¯sδρ. Similarly,
+the temperature fluctuations in second sound are to leading order at constant pressure and
+hence,δT≃(∂T/∂¯s)pδ¯s. Using these in (26) leads to the leading order expressions for u1
+andu2given in (24).
+We remark that, qualitatively, our results in figure 1 for the temper ature dependence of
+u1andu2are in agreement with Arahata and Nikuni [10], who also based their w ork on
+NSR thermodynamics. The differences with [10] at high and low temper atures are easily
+understood. Direct numerical calculations based on NSR are difficult to do accurately for
+T <0.4Tcbecause of the extremely small value of the normal fluid density. Ou r results in
+figure 1 inthis regionare based onthe assumption that Goldstone ph onons arethe dominant
+thermal excitations at low Tat unitarity, which allows us to obtain the low temperature
+values of u1andu2analytically. In the opposite limit of high temperatures close to Tc,our
+calculations show that W2steadily increases, in contrast with that of [10]. For Tclose toTc,
+it is crucial to use a superfluid density with the correct critical beha viour [27], removing the
+spurious first-order behaviour predicted by NSR [28]. (In fact, th is behaviour is not unique
+to NSR and is symptomatic of any theory that treats phase fluctua tions in a perturbative
+way. See [29] for further discussion.)
+Using the values of u1,u2, andv(see figure 1), we can calculate Z2,W2andW1. The
+results are shown in figure 2. The relative weight of second and first sound in S(q,ω) is
+given by
+W2
+W1=Z2u2
+1
+Z1u2
+2=v2−u2
+2
+u2
+1−v2u2
+1
+u2
+2. (27)
+Using the exact two-fluid expressions for u1andu2gives the ratio W2/W1shown in figure 3.
+For comparison, we also plot the Landau-Placzek ratio ǫLP=γ−1, calculated directly from
+the same thermodynamic functions used to obtain the sound speed su1,u2, andv[7, 8].
+Infigure4, weplot S(q,ω)givenby(22). Thisshowsthefirstandsecondsoundresonances
+at a series of temperatures. The results shown in figures 2-4 show the remarkable feature
+that, even though the maximum value of Z2is∼0.05 atT∼0.8Tc, the relative weights of
+13/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48
+/s66
+/s50/s47/s40 /s66
+/s49/s43 /s66
+/s50/s41/s66
+/s49/s47/s40 /s66
+/s49/s43 /s66
+/s50/s41
+/s90
+/s50/s90
+/s49
+/s87
+/s50/s47/s40 /s87
+/s49/s43 /s87
+/s50/s41/s87
+/s49/s47/s40 /s87
+/s49/s43 /s87
+/s50/s41
+/s84 /s47/s84
+/s67/s83/s111/s117/s110/s100/s32/s119/s101/s105/s103/s104/s116/s115
+FIG. 2: The temperature dependence of the normalized amplit udesZi,Bi, andWiof first (i= 1)
+and second sound ( i= 2) in a Fermi gas at unitarity.
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53
+/s32/s87
+/s50/s47/s87
+/s49
+/s32/s76/s80/s32/s114/s97/s116/s105/s111 /s32
+/s76/s80
+/s84 /s47/s84
+/s67/s87
+/s50/s47/s87
+/s49/s32/s32/s32/s32 /s97/s110/s100 /s32/s32/s32/s32
+/s76/s80
+FIG. 3: Comparison between the ratio of the second and first so und amplitudes ( W2/W1) and the
+Landau-Placzek ratio ǫLP=γ−1 in a unitary Fermi gas.
+14/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s53/s48/s49/s48/s48
+/s32/s32/s83 /s40/s113 /s44 /s41/s32/s91/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s93
+/s47/s40 /s113/s118
+/s70/s41
+FIG. 4: The two-fluid dynamic structure factor given by (22) a t unitarity. The plot shows the
+first and second sound resonances as a function of the frequen cyω/qvF(where we take q= 0.1kF
+andvFis the Fermi velocity) for a series of temperatures from T= 0 toT=Tc, in steps of 0 .05Tc
+(each temperature is offset). For clarity, the delta function s in (22) are plotted as Lorentzians with
+a width ∆ = 0 .01qvF.
+first and second sound in S(q,ω) can be comparable in the temperature region T/greaterorsimilar0.8Tc.
+The relative weight of second sound is instead much smaller in Im χnn(see figure 5) for
+reasons we have discussed.
+Figure 3 also shows that W2/W1is quite well approximated by ǫLP. It is useful to derive
+this result more analytically. Using (14) in (27), we obtain after some algebra
+W2
+W1=1
+ǫLP/parenleftbiggu2
+1
+v2
+T−1/parenrightbigg2
+. (28)
+This expression was first given by Vinen [30]. Using the expansion for u2
+1in (24), one can
+reduce (28) to
+W2
+W1=ǫLP(1+γx+···)2. (29)
+15/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s53/s48/s49/s48/s48
+/s32/s32/s73/s109 /s40/s113 /s44 /s41/s32/s91/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s93
+/s47/s40 /s113/s118
+/s70/s41
+FIG. 5: The imaginary part of the two-fluid density response f unction at unitarity. In contrast to
+the results shown in figure 4, where the relative weights of se cond and first sound are determined
+byW2/W1, in Bragg scattering they are determined by the smaller rati oB2/B1[see (23)]. As a
+result, second sound has a much smaller weight in Bragg scatt ering that first sound.
+At high temperatures, where x=c2
+2/c2
+1≪1 (see figure 1), we have
+W2
+W1=ǫLP[1+2(1+ ǫLP)x+···]≃ǫLP=γ−1. (30)
+This result was first obtained almost 40 years ago [11, 14, 15, 31] in t he context of Brillouin
+light scattering [13] in superfluid4He near the critical region close to Tc. ForT/lessorsimilar0.4Tc, one
+can see from figure 1 that we can no longer assume x≪1 and hence expand (29).
+As noted above, the result in (30) was obtained in the classic literatu re on superfluid
+4He. The Landau-Placzek ratio ǫLPis extremely small (10−2−10−3) in superfluid4He and
+thus second sound has generally a very small amplitude in the dynamic structure factor.
+However, under pressure and close to Tc, the magnitude of ǫLPin liquid4He can be signif-
+icantly increased (to about ǫLP∼0.2) such that a second sound doublet in S(q,ω) can be
+measured using Brillouin light scattering techniques [13]. Such studies [16, 17] have verified
+the correctness of W2/W1=ǫLPin liquid4He at a pressure of P= 25 atmospheres in the
+16temperature region ( Tc−T)∼10−2−10−4K.
+TheLandau-Placzekratio ǫLP≡γ−1wasfirst derived in1934[32]todescribe therelative
+weight of the thermal diffusion central peak vs. sound waves which are exhibited by S(q,ω)
+describing the hydrodynamics of a normal fluid. Thus, (30) genera lizes the Landau-Placzek
+result [24, 32] to the case of two-fluid hydrodynamics. In a normal liquid above Tc, the
+dynamic structure factor given by (22) reduces to [24]
+S(q,ω) =T
+2m/braceleftBigg
+1
+v2
+¯s[δ(ω−v¯sq)+δ(ω+v¯sq)]+γ−1
+v2
+¯sδ(ω)/bracerightBigg
+. (31)
+Thus, above Tc, thesecondsounddoublet in(22)collapsesintoazerofrequencyc entral peak,
+withrelative weight γ−1compared toasoundwave at ω=v¯sq. When oneincludes damping
+duetotransportprocesses, thecentralpeakin(31)broadens , describingthethermaldiffusion
+relaxation mode of width DTq2(whereDTis the thermal diffusivity). The behaviour of
+S(q,ω) as one goes from below Tcto above Tcis smooth but complicated due to various
+singularities in the thermodynamic and transport coefficients (for f urther discussion, see
+[14].)
+V. DYNAMIC STRUCTURE FACTOR OF DILUTE BOSE GASES
+In this section, we discuss first and second sound in a uniform dilute B ose-condensed gas.
+The two-fluid hydrodynamics is again described by the results given in section III. However,
+we need to distinguish between the cases of weakly and strongly inte racting Bose gases. The
+latter case can be easily realized by considering a molecular Bose cond ensate on the BEC
+side of the BCS-BEC crossover in a two-component Fermi gas. In s uch a molecular Bose
+gas with density nB=n/2 and molecular mass mB= 2m, thes-wave molecular scattering
+length is given by aB≃0.6aF, whereaFis the atomic s-wave scattering length between the
+two Fermi components [33]. We will show that the behaviour of first a nd second sound in
+strongly interacting Bose gases is very similar to that in Fermi gas su perfluids near unitarity,
+but very different from that in weakly interacting Bose gases.
+Inadilute, weaklyinteracting Bosegas, thethermodynamicquantit iesthatenterthetwo-
+fluidequationscanbesolvedanalyticallyinpowersoftheinteractions trengthg≡4πaB/mB.
+To first order in g, the speeds of first and second sound are given by (see, for insta nce,
+17chapter 15 in [3])
+u2
+1=5T
+3mBg5/2(1)
+g3/2(1)+2g˜n0
+mB, u2
+2=gnc0
+mB. (32)
+Here,gn(z) =/summationtext∞
+l=1zl/lnis the usual Bose-Einstein function with fugacity z. In the limit
+of small g, the normal fluid density ρn0reduces to the density m˜n0≡ρ−mnc0of atoms
+thermally excited out of the condensate. The superfluid density ρs0is likewise given by the
+densitymnc0of condensate atoms. To lowest order in g,nc0is the condensate density of the
+ideal Bose gas.
+In figure 6, we show the temperature dependence of u1andu2, satisfying (14), in the case
+of weak interactions ( nBa3
+B= 10−5). The thermodynamic functions used to determine these
+sound velocities have been calculated using the Hartree–Fock–Pop ov (HFP) microscopic
+model for the thermal excitations [3]. The velocities are normalized in terms of a Fermi
+velocity defined by vF= (6π2nB)1/3/m(equivalent to the Fermi velocity of a two component
+Fermi gas through the BCS-BEC crossover). We also show for com parison the values of u1
+andu2given by the lowest order solutions described by (32). One sees tha t these uncoupled
+modes of oscillations are a good first approximation to the full solutio ns of the two-fluid
+equations in a weakly interacting BEC at all temperatures outside a s mall temperature
+region at low temperatures where first and second sound hybridize , exhibiting an avoided
+crossing. At this avoided crossing, the natures of first and secon d sound are interchanged.
+AsT→0, the first sound velocity coincides with the phonon velocity, as is th e case with
+strongly interacting Fermi gases and superfluid4He. For a dilute gas, this phonon velocity
+is the Bogoliubov–Popov phonon velocity/radicalbig
+gnc0/mB, and corresponds to an oscillation
+of the condensate component. Above the hybridization point, to a good approximation,
+second sound propagates with this velocity. Consequently, secon d sound in a dilute Bose
+gas (above the hybridization temperature) is essentially an oscillatio n of the condensate at
+all temperatures, witha staticthermal cloud. In contrast, first sound describes anoscillation
+of the thermal cloud in the presence of a stationary condensate.
+We note that HFP approximation we have used to calculate thermody namic quantities
+has the well-known problem near Tcof predicting a spurious first-order transition (see, for
+instance, [34]). As touched on in section IV, this is actually a very gen eral problem, and
+arises in any theory that treats bosonic phase fluctuations of the order parameter as a small
+parameter, including the HFP and NSR theories. Of course, this inclu des all perturbation
+18/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51
+/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50
+/s87
+/s50/s47/s87
+/s49
+/s32
+/s32/s32
+/s32
+/s76/s80/s83/s111/s117/s110/s100/s32/s118/s101/s108/s111/s99/s105/s116/s105/s101/s115/s32/s91/s105/s110/s32/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s118
+/s70/s93
+/s84 /s47/s84
+/s66/s69/s67/s110
+/s66/s97
+/s66/s51
+/s61/s49/s48/s45/s53
+FIG. 6: Speeds of first and second sound in a weakly interactin g molecular Bose gas with gas
+parameter nBa3
+B= 10−5. The blue dashed and red dotted lines are respectively the ap proximate
+speedsu1andu2given by (32). The inset shows the LP ratio ǫLP(red dashed line) and the weight
+ratioW2/W1(black solid line).
+theoriesthatgobeyondmean-fieldandonly ab initio QuantumMonte-Carlocalculationscan
+avoid this problem. (A competing fluctuation theory of the BCS-BEC crossover [35] avoids
+this problem by ignoring phase fluctuations.) The unphysical first-o rder transition leads to
+the result that the second sound velocity does not vanish at Tcas it should. In contrast to
+figure 1, we have not corrected the spurious behaviour of the sup erfluid (condensate) density
+close toTcin the results shown in figures 6 and 7. These figures also show the ex pected
+feature that the error becomes larger with increasing interaction strength. In spite of the
+shortcomings of HFP theory in the critical region close to Tc, we emphasize that it gives a
+goodestimate of thermodynamic quantities, outside thecritical re gion. All numerical results
+shown in figures (6)-(8) are obtained using this theory (apart fro m the uncoupled solutions
+shown in figure 6, which use the ideal gas expression for the conden sate density).
+In figure 6, the inset shows the much larger value of the LP ratio ǫLP(compared to a
+Fermi superfluid–see figure 3) and the ratio W2/W1of the amplitudes of second and first
+19sound in S(q,ω) as a function of the temperature. We note that ratio W2/W1is still given
+by the expressions in (27) and (28). However, now the parameter ǫLPis no longer small
+(≪1) and thus the expansion based on (24), which leads to (29) and (3 0), is no longer
+valid. We also explicitly note that W2/W1is not equal to ǫLPin either the weak or strong
+coupling limits.
+First and second sound in a dilute, weakly interacting Bose gas involve weakly coupled
+oscillations of the thermal cloud and condensate, respectively. Bo th modes contribute to the
+density response function. Consequently, bothfirst andsecond sound aresensitive todensity
+probes. This facthasbeenmadeuseofinarecent experiment involv ing trappedatomicBose
+gases by Meppelink et al. [4]. In this study, a density pulse was generated (as described in
+section II) and the velocity of the second sound pulse was measure d above the hybridization
+point by tracking the propagation of the density pulse. The inset in fi gure 6 shows that
+the amplitude of second sound in S(q,ω) and in density pulse propagation experiments
+is twenty times as strong as first sound for all temperatures abov e the crossing point at
+T∼0.1Tc. This result may explain the absence of a first sound pulse in the data of [4],
+which corresponded to nBa3
+B∼5×10−6.
+In figure 7, we plot first and second sound for a more strongly inter acting Bose gas (i.e.,
+with a larger value of g.) This situation is close to the unitarity Fermi gas results shown
+in figure 1 [36]. In particular, we see that the velocities of first and se cond sound never
+become degenerate (cross) and thus the hybridization shown in fig ure 6 does not occur. In
+this strongly interacting Bose superfluid, we also show the approxim ate velocities of u1and
+u2given by (25). These approximate values give a reasonable first ord er approximation to
+the full solutions of the first and second sound velocities, rather t han the expression in (32)
+that applies to a weakly interacting Bose gas as shown in figure 6. The inset shows the ratio
+ofW2/W1vs.T, as computed from (27). In this case, the amplitude of second sou nd in
+S(q,ω) is only 2-3 times larger than first sound for T/greaterorsimilar0.5Tc.
+In figure 8, we compare the values of W1/(W1+W2) andW2/(W1+W2) for a weakly
+interacting Bose gas ( nBa3
+B= 10−4) and a strongly interacting Bose gas ( nBa3
+B= 0.1). One
+sees that the latter case is qualitatively similar to the result in a Fermi superfluid gas at
+unitarity given in figure 2. In contrast, in a weakly interacting Bose g as, for the temperature
+range above the hybridization point, the amplitude of second sound is much larger than first
+sound in both S(q,ω) and density pulse propagation experiments.
+20/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54
+/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s50/s51
+/s76/s80
+/s32
+/s32/s32
+/s87
+/s50/s47/s87
+/s49/s83/s111/s117/s110/s100/s32/s118/s101/s108/s111/s99/s105/s116/s105/s101/s115/s32/s91/s105/s110/s32/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s118
+/s70/s93
+/s84 /s47/s84
+/s66/s69/s67/s110
+/s66/s97
+/s66/s51
+/s61/s49/s48/s45/s50
+FIG. 7: Speeds of first and second sound in a strongly interact ing molecular Bose gas with gas
+parameter nBa3
+B= 10−2. The blue dashed and red dotted lines are respectively the ap proximate
+speedsc1=υ¯sandc2=υ/√γgiven by (25). The inset shows the LP ratio ǫLP(dashed line)
+and the weight ratio W2/W1(black line), to be compared with the result in figure 3 for a st rongly
+interacting Fermi gas at unitarity.
+VI. BRAGG SCATTERING IN A TRAPPED ATOMIC GAS
+For illustration, we now consider the response of a trapped Fermi g as at unitarity to two-
+photon Bragg scattering due to first and second sound. We consid er an experimental setup
+similar to that used in [37], where a pair of laser beams are applied to the gas such that the
+region where they overlap is confined to the centre of the gas. As a first approximation to
+this configuration, we assume that the overlap region between the two beams is isotropic,
+with a Gaussian profile:
+δV(r,t) =V√
+2πσe−r2/2σ2cos(q·r−ωt). (33)
+Here,σis the width of the Bragg beams and qandωare the wavevector and frequency
+difference between the two beams.
+For a Bragg pulse of short duration ( τ≪2π/ω0, whereω0is the trap frequency), the
+21/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48
+/s110
+/s66/s97
+/s66/s51
+/s61/s32/s48/s46/s49/s87
+/s105/s32/s47/s32/s40 /s87
+/s49/s43/s87
+/s50/s41
+/s84 /s47/s84
+/s66/s69/s67/s110
+/s66/s97
+/s66/s51
+/s61/s49/s48/s45/s52
+FIG. 8: Temperature dependence of the normalized amplitude sW1/(W1+W2) (solid lines) and
+W2/(W1+W2) (dashed lines) for a weakly interacting Bose gas ( nBa3
+B= 10−4) and a strongly
+interacting Bose gas ( nBa3
+B= 0.1).
+momentum transferred to the gas is related to the imaginary part o f the density response
+function [see (4)], through a convolution of the atomic density:
+∆P(q,ω) =qV2τ
+4πNσ2/integraldisplay
+d3r n(r)e−r2/σ2Imχnn(q,ω;r) (34)
+In this expression, Im χnn(q,ω;r) is the imaginary part of the density response function,
+evaluated within a local density approximation (LDA). The imaginary p art of the two-fluid
+density response function is given by (17) and should be evaluated a t the local value of the
+density. The LDA is justified for wave vectors much larger than the inverse of the size of
+the trapped gas, so that the system can be considered, locally, as a uniform body.
+Results for the momentum transferred given by (34), as a functio n ofω/qvF, are shown
+in figure 9 for T= 0.75Tcand choosing σ= 0.5RTF, whereRTF≡/radicalbig
+/planckover2pi1/mω(24N)1/6is
+the Thomas–Fermi radius of a noninteracting two-component Fer mi gas. Although small,
+the second sound peak is well separated from the first sound peak , revealing that two-
+photon Bragg spectroscopy might become a practical way to meas ure second sound. Above
+and below this temperature, the contribution of second sound to t he momentum transfer
+22/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48
+/s32/s70/s105/s114/s115/s116/s32/s115/s111/s117/s110/s100
+/s32/s83/s101/s99/s111/s110/s100/s32/s115/s111/s117/s110/s100/s84/s114/s97/s110/s115/s102/s101/s114/s114/s101/s100/s32/s109/s111/s109/s101/s110/s116/s117/s109/s32/s91/s97/s114/s98/s46/s32/s117/s110/s105/s116/s93
+/s47/s40 /s113/s118
+/s70/s41/s84 /s32/s61/s32/s48/s46/s55/s53 /s84
+/s67/s44/s32 /s32/s61/s32/s48/s46/s53 /s82
+/s84/s70 
+FIG. 9: Momentum transferred to a trapped Fermi gas at unitar ity by two photon Bragg scattering
+dueto firstand second sound at T≃0.75Tc. Solid (blue) line shows the momentum transferred due
+to firstsoundas afunctionof the beamdetuning ωdividedby therelative wave vector q≡ |q1−q2|.
+vFis the Fermi velocity. Dashed (red) line shows the momentum t ransferred due to second sound.
+becomes significantly smaller, however.
+VII. CONCLUSIONS
+In this paper, we have given a detailed analysis of the density respon se function χnn(q,ω)
+and the related dynamic structure factor S(q,ω) for uniform superfluids in the two-fluid
+hydrodynamic limit. We have put special emphasis on the second soun d contribution to
+S(q,ω) in the case of a strongly interacting Fermi gas superfluid (at unita rity). While
+many features of the dynamic structure factor we discuss are kn own in the superfluid4He
+literature [13, 14], several new aspects arise in the case of ultraco ld superfluid gases.
+First and second sound both appear as resonances in the density r esponse function but
+their relative weights vary in different experimental probes. Two-p hoton Bragg scattering, a
+23well-developed toolinultracoldgasesusedtostudydensity fluctua tions(sofar, mainlyinthe
+collisionless region [20, 38, 39]), is directly proportional to Im χnn(q,ω). In contrast, the low-
+frequency limit of the dynamic structure factor S(q,ω) =−Imχnn(q,ω)/πn[1−exp(−βω)]
+measured in Brillouin light scattering, and also the amplitude of density pulses that can
+be generated in dilute gases [10], are proportional to Im χnn(q,ω)/ω. This extra factor of
+1/ωleads to a large enhancement of the weight of low frequency second sound in S(q,ω)
+compared to first sound.
+The relative spectral weights of first and second sound in χnn(q,ω) are very dependent
+on how strongly temperature fluctuations couple to density fluctu ations. For S(q,ω), this
+coupling is conveniently parametrized by the value of the dimensionles s Landau–Placzek
+ratioǫLP=γ−1, where γ, defined in (16), is the ratio of the specific heats (per units mass)
+at constant pressure and volume. At finite temperatures, the re lative weight of first and
+second sound ( W2/W1) is equal to ǫLPin both unitary Fermi gases and superfluid4He. The
+key difference between superfluid4He and a unitary Fermi superfluid lies in the value of ǫLP.
+Apart from a small region very close to the transition temperature Tc, one has ǫLP≪1 in
+superfluid4He [16, 17]. In contrast, for T/greaterorsimilar0.7Tc, one finds that ǫLPis of order unity in
+Fermi superfluids (see figures 3 and4)and hence thesecond soun dresonance hasappreciable
+weight in S(q,ω).
+Density pulse experiments as suggested in [10] seem to be a promisin g way of detecting
+both first and second sound pulses in Fermi gases, although more w ork is needed to clarify
+the nature of first and second sound in cylindrical geometries wher e such pulses would be
+generated. (See, for instance, recent work on this problem in [40].) The experimental results
+reported in [22], where a blue-detuned laser was used to generate a density pulse in a Fermi
+gas close to unitarity, revealed no clear signs of a second sound puls e. This experiment was
+carried out at very low temperatures, however, where second so und will not have appreciable
+weight in S(q,ω) (see figures 3 and 4). Alternately, even though the weight of sec ond sound
+is comparatively small in Bragg scattering, Bragg scattering has th e advantage that it can
+be used to probe in a very precise way the properties of uniform sup erfluid Fermi gases [37],
+including the velocity of second sound.
+In section V, we compare the two-fluid hydrodynamics in a uniform, d ilute, weakly in-
+teracting Bose condensed gas (figure 6) with a strongly interactin g Fermi superfluid gas.
+In such Bose superfluids, the thermal expansion (and hence, ǫLP) is much larger, with the
+24result that second sound is the dominant excitation in S(q,ω) and mainly involves a pure
+oscillation of the condensate in the presence of a static thermal co mponent [3, 4]. However,
+first and second sound in a strongly interacting Bose gas are similar t o those in a Fermi gas
+near unitarity (see figure 7).
+Acknowledgments
+We thank Lev Pitaevskii for useful comments. E.T. was supported by nsf-dmr 0907366
+and ARO W911NF-08-1-0338. H.H. and X.-J.L. were supported by AR C and NSFC. S.S.
+acknowledges the support of the EuroQUAM FerMix program. A.G. w as supported by
+NSERC and CIFAR.
+Appendix A: Phonon thermodynamics at low temperatures
+At low temperatures, Goldstone phonons determine the thermody namics of both super-
+fluid4He and Fermi gases. For phonons with velocity c, the free energy Fand normal fluid
+densityρn0are given by [41] (Recall that we have set /planckover2pi1=kB= 1)
+F=F0−Vπ2T4
+90c3(A1)
+and
+ρn0=2π2T4
+45c5. (A2)
+Using (A1), it is straightforward to show that
+¯s0=−1
+mN/parenleftbigg∂F
+∂T/parenrightbigg
+V,N=2π2T3
+45ρc3, (A3)
+P=−/parenleftbigg∂F
+∂V/parenrightbigg
+T,N=P0+π2T4
+90c3/bracketleftBigg
+1+3ρ
+c/parenleftbigg∂c
+∂ρ/parenrightbigg
+T,N/bracketrightBigg
+, (A4)
+and
+¯cv=2π2T3
+15ρc3. (A5)
+In arriving at these expressions, the temperature dependence o f the Goldstone phonon ve-
+locitychas been ignored. P0is the pressure in the ground state. These results can be
+combined to give
+¯cp= ¯cv−T/parenleftbigg∂¯s
+∂ρ/parenrightbigg
+T/parenleftbigg∂P
+∂T/parenrightbigg
+ρ/parenleftbigg∂P
+∂ρ/parenrightbigg−1
+T= ¯cv+4π4T7
+452ρ2c6v2
+T/bracketleftbigg
+1+3ρ
+c/parenleftbigg∂c
+∂ρ/parenrightbigg
+T/bracketrightbigg2
+,(A6)
+25and
+v2≡¯s2
+0T
+¯cvρs0
+ρn0=c2
+3ρs0
+ρ0. (A7)
+Combining the above results, the Landau–Placzek ratio at low tempe ratures is given by
+ǫLP=2π2T4
+135ρc3v2
+T/bracketleftbigg
+1+3ρ
+c/parenleftbigg∂c
+∂ρ/parenrightbigg
+T/bracketrightbigg2
+(A8)
+and hence, recalling that x=v2/γv2
+¯sandγ=v2
+¯s/v2
+T,
+/radicalbiggρs0
+ρn0ǫLPx=1
+3ρs0
+ρ0c2
+v2
+¯s/bracketleftbigg
+1+3ρ
+c/parenleftbigg∂c
+∂ρ/parenrightbigg
+T/bracketrightbigg
+. (A9)
+This result is valid for both the superfluid unitary Fermi gas and supe rfluid4He. Further
+simplifications are not possible without ab-initio or experimental information about c. At
+low temperatures, however, where ρs0≃ρ0andv¯s≃c(see figure 1), (A9) reduces to
+/radicalbiggρs0
+ρn0ǫLPx≃1
+3/bracketleftbigg
+1+3ρ
+c/parenleftbigg∂c
+∂ρ/parenrightbigg
+T/bracketrightbigg
+. (A10)
+Equations (A6) and (A7) also give us the result that γ→1 andx=v2/γv2
+¯s→1/3 as
+T→0. Finally, for a Fermi gas at unitarity, ( ρ/c)(∂c/∂ρ)T= 1/3 and (A10) is further
+reduced to the result given in (B10). In superfluid4He, the Gr¨ uneisen constant is almost a
+factor of 9 larger, ( ρ/c)(∂c/∂ρ)T≃2.84 [42].
+Appendix B: Superfluid and normal fluid velocity fields in second sound
+As Landau [2] originally noted, when the thermal expansion coefficien t can be neglected
+(i.e.,ǫLP= 0), one can show that second sound is a pure temperature oscillat ion, with no
+associated density fluctuations. This last feature follows from the fact that when ǫLP= 0,
+second sound involves no mass current ( j≡ρn0vn+ρs0vs= 0) and thus the continuity
+equation requires that δρ= 0. Likewise, first sound isa puredensity oscillation, andinvolves
+no fluctuations in the entropy per unit mass ¯ s. The non-dissipative two-fluid equations give
+the following equation for this quantity:
+∂δ¯s
+∂t=¯s0ρs0
+ρ0∇·(vs−vn). (B1)
+This shows that for pure adiabatic motion ( δ¯s=δT= 0) such as first sound, the superfluid
+andnormalfluidvelocitiesareidentical: vs=vn. Puttingtheseresultstogether, thevelocity
+26fields for first and second sound when ǫLP= 0 are given by
+v(1)
+s
+v(1)
+n= 1,ρs0v(2)
+s
+ρn0v(2)
+n=−1, (B2)
+where, for instance, v(1)
+sdenotes the superfluid velocity field associated with first sound
+(along the direction of propagation q). This Appendix discusses how these velocity ratios
+change when ǫLPis non-zero. Surprisingly, we find that changes are quite substant ial at all
+temperatures, including the low- Tregion where ǫLP≪1.
+Aswe discuss in the present paper and in[8], the Landau–Placzek ra tioǫLPincreases with
+temperature and in Fermi gases near unitarity, ǫLPis of order unity at high temperatures
+(see figure 3). As we have shown, this is large enough that second s ound has significant
+weight in the dynamic structure factor.
+To understand the effect that a non-zero value of ǫLPhas on the superfluid vsand normal
+fluidvnvelocity fields, one can solve the two-fluid equations for the values o f these fields
+associated with first and second sound ω=uq, whereu=u1oru2. Defining the variables
+j=ρs0vs+ρn0vnandw=ρs0(vs−vn), the plane-wave solutions of the linearized two-fluid
+equations are
+
+u2−v2−s0ρs0
+ρn0∂T
+∂ρ/vextendsingle/vextendsingle/vextendsingle
+¯s
+s0∂T
+∂ρ/vextendsingle/vextendsingle/vextendsingle
+¯su2−v2
+¯s
+
+w
+j
+= 0. (B3)
+Here,
+s0∂T
+∂ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+¯s=¯s0
+ρ0∂P
+∂¯s/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ρ=/radicalbiggρn0
+ρs0√ǫLPxv2
+¯s≡b2(B4)
+has units of velocity squared. Recall that xis defined below (24). Standard thermodynamic
+identities have been used to introduce ǫLP=γ−1. As the structure of (B3) makes clear, b2
+determinesthecouplingbetween density( ∝j)andentropyperunitmass( ∝w)fluctuations.
+[The latter is given by (B1).]
+Using (B3), it is straightforward to derive an expression for the ra tio of the velocity fields
+associated with first and second sound:
+vs
+vn=s0∂T
+∂ρ/vextendsingle/vextendsingle/vextendsingle
+¯s−ρn0
+ρs0(u2−v2
+¯s)
+s0∂T
+∂ρ/vextendsingle/vextendsingle/vextendsingle
+¯s+(u2−v2
+¯s). (B5)
+Substituting the expressions for the speeds of first and second s ound given by (24) into (B5),
+one finds
+v(1)
+s
+v(1)
+n≃v2
+¯s−b2
+v2
+¯s+ρs0
+ρn0b2(B6)
+27and
+ρs0v(2)
+s
+ρn0v(2)
+n≃ −v2
+¯s(1−x)+ρs0
+ρn0b2
+v2
+¯s(1−x)−b2. (B7)
+In section IV, we showed that c1andc2in (25), which are expansions in the small parameter
+ǫLPx, areexcellent approximationstothespeedsoffirstandsecondso undatalltemperatures
+(see figure 1). We thus expect (B6) and (B7) to be similarly good app roximations for the
+velocity fields at all temperatures. Using (B4) to remove thedepen dence ofthese expressions
+onv¯s, they reduce to
+v(1)
+s
+v(1)
+n≃1−/radicalBig
+ρn0
+ρs0ǫLPx
+1+/radicalBig
+ρs0
+ρn0ǫLPx(B8)
+and
+ρs0v(2)
+s
+ρn0v(2)
+n≃ −(1−x)+/radicalBig
+ρs0
+ρn0ǫLPx
+(1−x)−/radicalBig
+ρn0
+ρs0ǫLPx. (B9)
+WhenǫLPis set to zero, we recover the results in (B2), describing pure in-ph ase and out-
+of-phase oscillations of the superfluid and normal fluid components . However, we note that
+even though ǫLPvanishes as T→0, so does the normal fluid density ρn0, and it is not
+obvious that/radicalbig
+ρs0ǫLPx/ρn0is small at low temperatures. Conversely, at high temperatures,
+T→Tc, the factor of/radicalbig
+ρn0/ρs0multiplying√ǫLPxin (B8) and (B9) will enhance the small
+parameter√ǫLPx. It is thus crucial to know the temperature dependencies of ρs0/ρn0and
+ǫLPxin both limits.
+At low temperatures, the thermodynamics is dominated by Goldston e phonons. Using
+standard results in the phonon regime, one can show (see A) that in theT→0 limit,
+/radicalbiggρs0
+ρn0ǫLPx≃2
+3(B10)
+for a unitary Fermi gas. This non-zero limiting value as T→0 arises from the fact that both
+ǫLPxandρn0vanish as T4. For a Fermi gas at unitarity in the T→0 limit, using (B10) and
+the fact that x= 1/3 (see A), (B8) and (B9) give
+v(1)
+s
+v(1)
+n≃3
+5,ρs0v(2)
+s
+ρn0v(2)
+n≃ −2, (B11)
+even though ǫLPvanishes as T→0 (as shown in figure 3). If we had simply set ǫLP= 0, as
+often done in the superfluid literature, we would instead obtain the r esults given by (B2).
+In figure 10, we plot the velocity ratios in (B8) and (B9) for a superfl uid unitary Fermi
+gas. It is clear that the superfluid and normal fluid velocities which ar e involved in first
+28/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s50/s45/s49/s48/s49/s50
+/s84 /s47/s84
+/s67/s118
+/s115/s47/s118
+/s110/s102/s105/s114/s115/s116/s32/s115/s111/s117/s110/s100
+/s115/s101/s99/s111/s110/s100/s32/s115/s111/s117/s110/s100
+FIG. 10: Superfluid and normal fluid velocity fields associate d with first and second sound in a
+unitary Fermi superfluid. The solid (black) line shows the ra tiov(1)
+s/v(2)
+nof the superfluid and
+normal fluid velocity fields involved with first sound, (B8). T he dashed (red) line shows the ratio
+ρs0v(2)
+s/ρn0v(2)
+nof the superfluid and normal fluid currents associated with se cond sound, (B9). If
+we setǫLP= 0, these ratios would be 1 and −1, respectively, describing the velocity fields for pure
+density and temperature oscillations.
+and second sound are significantly different from those given in (B2) . This is a reflection of
+the fact that first and second sound are not pure uncoupled dens ity and temperature waves,
+respectively, which would be described by (B2) and correspond to ǫLP= 0 (and ρn0∝negationslash= 0).
+As we have shown in [8], such in-phase ( vs=vn) and out-of-phase zero current ( j= 0)
+solutions correspond to first and second sound velocities given by
+u2
+1=v2
+¯s, u2
+2=T¯s2
+0
+¯cvρs0
+ρn0≡v2. (B12)
+As we show in figure 2 of [8], these sound velocities are in good agreeme nt with a full
+calculation including the coupling associated with ǫLP. In this Appendix and section IV, we
+have carried out a careful analysis based on expanding to first ord er in the parameter ǫLPx,
+which is small ( ≪1) at all temperatures [14, 15]. The results in figure 10 show that th e
+29main effect of working with γnot equal to unity is simply that the second sound speed u2is
+given by v/√γas in (25), instead of vin (B12). In contrast, the first sound speed is always
+very well approximated by v¯s, even when γdeviates from unity.
+Initially, it seems surprising that the superfluid and normal fluid veloc ity fields shown in
+figure 10 are so different from those in (B2), since the first and sec ond sound velocities are
+fairly well approximated by (B12). (See figure 2 of [8].) This arises bec ause of two features
+evident in (B8) and (B9). First of all, we note that the first and seco nd sound speeds in
+(24) involve corrections of order ǫLPx≪1, while the related corrections are much larger in
+(B9) and (B11) since they enter as√ǫLPx. The second, and more important, reason is that
+the corrections involve the factors/radicalbig
+ρs0/ρn0and/radicalbig
+ρn0/ρs0, which are very large at T= 0
+andT=Tc, respectively.
+Substantial deviations fromthe ratiosgiven in(B2) also occur insup erfluid4He, although
+the magnitude will be different from the case of a Fermi superfluid at unitarity shown in
+figure 10. This feature was not commented on in the older superfluid4He literature [14,
+15, 31], which concentrated entirely on the frequency and damping of the first and second
+sound resonances appearing in the dynamic structure factor. Th e associated velocity fields
+calculated here were not discussed. These velocity fields may be dire ct experimental interest
+in future studies of two-fluid hydrodynamics in Fermi superfluids.
+Calculation shows that the various terms in (B8) and (B9) are not sm all compared to
+unity. However, if one formally expands these expressions to leadin g order in b2/v2
+¯s(=
+/radicalbig
+ρn0ǫLPx/ρs0), one finds
+v(1)
+s
+v(1)
+n= 1−δ,ρs0v(2)
+s
+ρn0v(2)
+n=−1−δ, (B13)
+where the leading correction to (B2) is found to be
+δ=ρ0
+ρn0b2
+v2
+¯s=ρ0
+ρn0/radicalbiggρn0
+ρs0ǫLPx. (B14)
+This value of δagrees with the expression originally worked out by Lifshitz (see pag e 71 in
+[43]) in the limit where u2
+2≪u2
+1, valid close to Tc. In the case of superfluid4He, the fact
+thatǫLP≪1 even close to Tc(it diverges very weakly at Tc) means that the first correction
+in (B13) is indeed small, δ≪1. In contrast, for a Fermi gas at unitarity, ǫLPis of order
+unity or larger near Tc(see figure 3) and as a result we find δ= 0.3 just below Tc[44]. In
+30this case, the expansion in (B13) is not valid, as can be seen by compa ring (B13) with the
+results in figure 10.
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+region below Tc.
+33
\ No newline at end of file
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+arXiv:1001.0773v1  [astro-ph.CO]  5 Jan 2010Simulating the universe on an intercontinental
+grid of supercomputers
+Simon Portegies Zwart1, Tomoaki Ishiyama2, Derek Groen1,3,4,
+Keigo Nitadori5, Junichiro Makino6, Cees de Laat7,
+Stephen McMillan8, Kei Hiraki9, Stefan Harfst1, Paola Grosso7,
+1Sterrewacht Leiden, P.O. Box 9513, 2300 RA Leiden, The
+Netherlands
+2Department of General and Sciences, University of Tokyo,
+Tokyo 153-8902, Japan
+3Astronomical Institute ”Anton Pannekoek” , University of
+Amsterdam, Amsterdam, The Netherlands
+4Section Computational Science, University of Amsterdam,
+Amsterdam, The Netherlands
+5Department of Astronomy, School of Science, University of
+Tokyo, Tokyo 113-8654, Japan
+6Center for Computational Astrophysics in Tokyo, Japan
+7Section System and Network Engineering Science, Universit y
+of Amsterdam, Amsterdam.
+8Department of Physics, Drexel University, Philadelphia, P A
+19104, USA;
+9Department of Creative Informatics, Graduate School of
+Information Science and Technology the University of Tokyo .
+Abstract
+Understanding the universe is hampered by the elusiveness o f its most common
+constituent, cold dark matter. Almost impossible to observ e, dark matter can be
+studied effectively by means of simulation and there is probab ly no other research
+field where simulation has led to so much progress in the last d ecade. Cosmological
+N-body simulations are an essential tool for evolving densi ty perturbations in the
+nonlinear regime. Simulating the formation of large-scale structures in the universe,
+however, is still a challenge due to the enormous dynamic ran ge in spatial and
+temporal coordinates, and due to the enormous computer reso urces required. The
+dynamic range is generally dealt with by the hybridization o f numerical techniques.
+Preprint submitted to Elsevier Preprint 10 September 2018Wedealwiththecomputationalrequirementsbyconnectingt wosupercomputersvia
+an optical network and make them operate as a single machine. This is challenging,
+if only for the fact that the supercomputers of our choice are separated by half the
+planet, as one is located in Amsterdam and the other is in Toky o. The co-scheduling
+of the two computers and the ’gridification’ of the code enabl es us to achieve a
+90% efficiency for this distributed intercontinental superc omputer. We conclude
+that running cosmological N-body simulations on a limited number of (<∼100)
+processors concurrently on more than 10 supercomputers in r ing topology with a
+high-bandwidth network would provide satisfactory perfor mance and is politicaly
+favorable regarding the acquisition of the resources.
+Keywords:
+Computer Applications: Physical Sciences and Engineering: Astron omy;
+ComputingMethodologies:Simulation,Modeling,andVisualization:Dist ributed
+1 Introduction
+Since the beginning the size and complexity of the universe have been in-
+creasing continuously. As a consequence we observe today large s tructures
+consisting of galaxies. The visible superstructure of the universe is made of
+baryonic material, e.g. stars and gas. But the majority of the mass is in
+the form of dark matter, which is affected by gravity but does not in teract
+electro-magnetically. The best model for the formation and evolut ion of this
+superstructure is called Λ cold dark-matter cosmology (ΛCDM)[3], a ccording
+to which the universe is about 13.7 billion years old and comprises of abo ut
+4% baryonic matter, ∼23% non-baryonic (dark) matter and 72% (dark) en-
+ergy, indicated by the letter Λ [10]. The nature of dark matter is unk nown and
+from an observational perspective it is hard to resolve this lacuna in our un-
+derstanding. Evidence for dark matter comes from a variety of ob servations,
+which includes the rapid rotation of the Milky-Way and other galaxies: its
+stars would be flung out in the absence of dark matter.
+Weakly-interacting massive particles form the most promising explan ation for
+dark matter [8], in which case the entire universe is filled with a finely gra ined
+substance that behaves like a gravitational fluid but is invisible other wise. The
+gravitational force in the Newtonian limit ∝1/r2, which is a long-range force
+in particular since the enclosed mass scale ∝r3. Together with the absence of
+a shielding mechanism, even very distant objects cannot be ignored . The way
+in which dark matter interacts is therefore rather simple and well un derstood.
+As a result, we can effectively study dark matter by simulation and us e these
+results to understand observations and to make predictions.
+2One of the most favorable techniques to study the formation of lar ge scale
+structure in the universe is by means of gravitational N-body simulations,
+in which each particle in the simulation represents the fluid of dark mat ter
+particles. The most widely used algorithm is the treePM method, in whic h
+the short range forces are resolved using a Barnes-Hut tree cod e, whereas the
+long range interactions are simulated using a Particle-Mesh method [1 1,4].
+This combination of methodologies provide good performance compa red to
+using only a tree code but still at a reasonable accuracy compared t o a pure
+particle-mesh technique, as in the treePM method we resolve short as well as
+long range forces reasonably well. In addition, the algorithm scales w ell to a
+large number of processors by optimized domain decomposition (Ish iyama et
+al. in press). Since simulating the entire universe is not really possible ( yet) we
+study a small part, using periodic boundary conditions to mimic ad infinitum .
+The computational demand for our large scale cosmological N-body simu-
+lation is enormous, and instead of running on a single supercomputer , we
+opted for running concurrently on two widely separated supercom puters. We
+demonstrate that that it can be efficient to run high-performance production
+simulations on a computational grid.
+2 The intercontinental grid
+The futureoflarge-scalescientific computing infrastructures aim s atdistribut-
+ing resources rather than concentrating supercomputers locally [5]. The higher
+cost effectiveness of distributing resources can be efficient for th ose applica-
+tions in which the compute time scales steeper than the communicatio n, or
+when the problem can be decomposed in domains [1].
+Instead of starting the simulation on one location and switch half way to
+another computer to continue the calculation, we run the simulation on two
+supercomputers concurrently. One of our computers is located in Amsterdam
+(the Netherlands) and the other is in Tokyo (Japan). The challenge is to
+efficiently use a large number of processors separated by half the p lanet with-
+out running in the inter-processor and intercontinental communic ation bot-
+tlenecks. If we can run successfully among two supercomputers s eparated by
+half the planet, we can be confident about upscaling our virtual org anization
+to include more than two supercomputers.
+Themanagement, politicalandtechnical issuesofcouplingthecomp uterswith
+an uncongested network and to schedule the resources proves t o be extremely
+challenging, and one can wonder if it is worth the effort. The prepara tions for
+our calculations lasted about a year. However, running on a single su percom-
+puter would have required its entire capacity for several months, which would
+3probably not have been granted. Acquiring relatively little time on a lar ge
+number of supercomputers proves to be considerably easier. With the exper-
+tise obtained in running on two supercomputers we can now extend t o more
+sites without much additional overhead. Of course, the political an d technical
+issues of coupling the computers with an efficient network and to sch edule the
+resources remain challenging, but many of these aspects will becom e easier
+once high-performance grid computing becomes mainstream. Addit ional com-
+plications arise by the required hybrid parallelization strategies, the diversity
+in topologies, scheduling, load balancing and the complications introdu ced by
+the different hardware architectures in particular if, as in our case , the code
+is machine dependent.
+One of the complications we encountered is related to the network t opology
+and internal hardware setup. Neither supercomputer is directly c onnected to
+the optical network, but communication is realized via a special node that
+is connected to the outside world with a 10GbE optical switch in Amste r-
+dam and Neterion NIC in Tokyo. Upon every communication step each pro-
+cessor: identifies the particles which need to be communicated, pac ks them
+and sends the particles to the supercomputer’s internal communic ation node,
+which subsequently sends the entire package of particles to the co mmunica-
+tion node outside the firewall. This package is subsequently transmit ted to the
+distant computer 9,400km away, as the bird flies. The data however , travels
+∼27,000km as the network links criss-crosses the Atlantic ocean, the U SA
+and the Pacific ocean. In Fig.1 we illustrate the network topology. W ith the
+speed of light in fiber this distance is covered in 0.138 seconds, result ing in a
+round-trip time of 0.277 seconds.
+ThenetworkcommunicationwasrealizedbyconfiguringtwovirtualL ANs(vLAN)
+to create two paths connecting both supercomputers. One vLAN was used
+for the production traffic and runtime synchronization, while the se cond was
+configured for testing purposes and for collecting the simulation da ta at the
+PByte storage array in Amsterdam. Both paths are dedicated to o ur experi-
+ments which prevents packet loss, packet reordering and latency changes due
+to congestion. In the secondary path we had to execute vLANs tr anslation
+in the JNG network, because the chosen vLANs numbers were unav ailable
+along the entire path length. The choice of an available vLAN number f or
+the end-to-end communication should have been trivial; but since ou r multi-
+domain light path lagged automatic configuration tools, human interv ention
+was necessary. The time required for solving problems depends on t he quick
+responses to email and phone calls, which is hindered by working over several
+time zones; debugging link failure at Layer2 in a multi-domain multi-vend or
+infrastructure is impractical and extremely time consuming. A self- healing or
+automatic setup would enormously help future experiments.
+The Research and Education Networks (REN) provided the links for free,
+4Fig. 1. Network topology for the ComoGrid experimental setu p. We used two
+main network connections between JGN2plus and StarLIGHT; o ne of then goes via
+L-LEX and Gigapop, the other connects directly from StarLIG HT to JGN2plus.
+Both paths have a network bandwidth of 10Gb/s, carried over 1 0 GE-LANPHY
+and 10 GE-WANPHY segments depending on the type of interface s present at
+the various switching points. Thick blue lines (LANPHY) and thick black lines
+(WANPHY).
+which is motivated by their vision to advertise and broaden their serv ices to
+the scientific community. In the last few years many RENs have adop ted a
+hybrid model for their architecture, expanding it to routed IP ser vices where
+users have access to dedicated light paths in which communication oc curs at
+lower layers of the open system-interconnection reference mode l. We settled
+for a flat Ethernet path, but still had to make the network reserv ations 3
+weeks in advance of each simulation restart, and then it would gener ally take
+about a week before it was fully operational. Our overall experience with the
+use of light paths however, is quite positive. We could not have perfo rmed
+our calculation without this type of network setup and certainly per ceived the
+advantage of the unrestricted and exclusive use of the links.
+52.1 Demand on the network
+While performing a cosmological N-body simulation in which the computa-
+tional domain is shared by two (or more) computers, the positions a nd ve-
+locities of all particles on both machines have to be synchronized thr oughout
+the simulation. This introduces an enormous demand on the network between
+the computers. Luckily only a small boundary layer between the two halves
+of the universe and the layer nearest the periodic boundary are co mmuni-
+cated between Amsterdam and Tokyo. The amount of data that ha s to be
+communicated per step is then
+Scomm= 144N2/3+4N3
+p+4N/SrByte. (1)
+HereNis the number of particles, Npis the resolution of the mesh in one
+dimension and Sris the sampling rate, which for the production simulation
+Sr= 5000. The first term is required for communicating the tree struc ture,
+the second term is for exchanging the particles in the border region , and the
+last term is for exchanging the mesh. The first term in this estimate d epends
+slightly on redshift ( z) and on the opening angle in the treecode ( θ), but is
+accurate for θ= 0.5.
+Whilerunning,densedark-matterclumpsmaynotbedistributedeve nlyacross
+thecomputationaldomainandloadbalancingisdonebyguaranteeing thatthe
+calculation time per step is the same on each computer, generating a variable
+boundary layer between the two computers. Where initially both com puter
+resolve exactly half the universe, in due time one of the two compute rs tends
+to deal with a larger volume.
+The communication within each of the supercomputers is realized by d omain
+decomposition, using theMessage Passing Interface ( mpich[2]).Towarrant ef-
+ficient andstabledata transfer we developed the parallel socket lib raryMPWide
+to facilitate the communication outside the local MPI domains (Groen etal
+2009, in preparation). This communication consists of data transf ers between
+the local compute nodes and the local communication node, and for the data
+transfer over the light-path between the supercomputers.
+The socket library is included in all processes, providing an interface simi-
+lar to regular MPI. A static communication topology is established at s tartup
+which uses multiple tcpconnection for each intra-cluster communication path
+and multiple tcpconnections for paths between supercomputers. The concur-
+rent use of multiple streams is realized by running a separate thread for each
+stream. For the smaller runs we used 16 tcpstreams, and 64 streams for the
+larger runs (see Tab.1).
+63 The Simulation environment
+We adopted the treePM code GreeMwhich was initially developed by [12] and
+rewritten by Ishiyama etal (in press) to run efficiently on the select ed hard-
+ware. The equations of motion are integrated in co-moving coordina tes using
+the leap-frog scheme with a shared but variable time-step. Good pe rformance
+at sufficient accuracy is then achieved when the time step is at least a n order
+of magnitude smaller than the crossing time in the densest halo [4]. Spu rious
+relaxation effects in the high-density clumps are prevented by intro ducing a
+Plummer softening to the force, which is comparable to the local inte r-particle
+distance.
+Since most simulation time is spent in calculating the gravitational forc es be-
+tween particles, we optimize this operation using the single precision X 86-64
+Streaming SIMD Extensions (SSE). The inverse square-root SSE in struction
+provides the best speedup in calculating Newtonian gravity. We furt her im-
+prove performance by minimizing RAM access through the 16 XMM reg isters
+for the force operations, and by operating on pairs of two 64-bit fl oating point
+words concurrently (Nitadori & Yoshikawa in preparation).
+With these optimizations the Power6 with symmetric multiprocessing is per
+core is about4.2% slower thanthe Intel based Cray XT4. This small d ifference
+in speed is achieved by adopting the x86 SSE version and the Power Alt iVec
+architectures, but for the former we use in-line assembly whereas for the latter
+we adopted the intrinsic functions (Nitadori & Yoshikawa in prepara tion).
+We measure the performance of the code using the Amsterdam and Tokyo
+machines separately with p= 1 top= 1024 processors and N= 2563toN=
+10243particles. The wall clock time tcpuis then fitted by tcpu≃14,500p−0.91
+seconds, with a ten fold increase of tcpufor every increase of the number of
+particles by 23. For relatively small Nwe lose scalability with respect to the
+number of processors for p≃103andN<∼2563.
+4 Simulating the Universe
+WeareinterestedinstructuresofkpctoMpcsize.Tominimizetheeff ectofthe
+periodic boundary conditions on such dark matter distributions we s elected
+our simulation box at z= 01to have sides of 30Mpc. The initial dark-matter
+distribution is generated at z≃65, assuming that the relation between the
+1Redshift is the standard measure of time in cosmology: the un iverse was born
+z→ ∞and today z≡0.
+7velocity and the potential in Zel’dovich approximation is the same as in t he
+linear theory [6]. The density field is then realized by multi-scale Gaussia n
+random fields, which is described in terms of its power spectrum and w as
+generated using MPGRAFIC [9].
+We further adopted cosmological parameters which are consisten t the 5-year
+WMAP results [7]. For clarity we opted for the nearest round values, which
+yields: matter (including dark) density Ω m= 0.3, dark energy density Ω Λ=
+0.7,theslopeforthescalarperturbationspectrum ns= 1.0,andtheamplitude
+of fluctuations σ8= 0.8. With these parameters the universe is about 13.7
+billion years old and the Hubble constant H0≃70km/s/Mpc.
+We ran several realizations at different resolution (in mass as well as spatially)
+to measure the performance before we completed a simulation to z= 0. An
+overview of the performed simulations is presented in Tab.1.
+NNpp (A+T) ttottcpuηp
+16777216 128 30+30 31.2 23.58 1.51
+134217728 256 30+30 116.8 105.7 1.81
+134217728 256 60+60 71.9 58.8 1.64
+134217728 256 120+120 53.0 36.8 1.39
+1073741824 256 500+250 60.0 40.0 —
+8589934592 256 500+250 380.0 310.0 —
+Table 1
+The performed simulations in a computational box of 30Mpc st arting at z≃65, a
+softening of 300pc and the opening angle in the tree-code θ= 0.3 forz >10 after
+whichθ= 0.5. The first column gives the number of particles in the simula tion
+followed by the number of mesh-cells in one dimension. Then w e give the number
+of processors in Amsterdam (A) and Tokyo (T). The next two col umns give the
+average wall-clock time for one step ( ttot) and the time spent computing the forces
+(tcpu), both in seconds. The last two columns gives the speed-up ηpwhich is defined
+as the wall-clock time of a single computer as fraction of the wall-clock time of
+the grid of supercomputers. Here we define ηpfor a grid where each computer has
+the same number of processors p. For the last two entries ηpis then not properly
+defined.
+The simulation with N= 16777216 took about 10 hours to complete from
+z≃65 toz= 0. In Fig.2 we show the wall-clock time of this run decomposed
+inthetimespendincalculation,communicationandstoringthedatato thefile
+system. About 25% of the time is spent in communication (see Tab.1) , which
+is roughly constant through the simulation. We encountered a few p roblems
+between z≃16 andz≃15, which are probably related to packet loss in the
+network transport protocol suite and resulted in an additional pe rformance
+loss of a few per-cent.
+8 0.0001 0.001 0.01 0.1 1 10
+ 0.01  0.1  1  10twall [hours]
+zttotaltcalctcommtstore
+Fig. 2. Wall-clock time as a function of redshift ( z) for the simulation with
+N= 16777216. The solid curve gives the wall-clock time, inclu ding the network
+wait stages, the dashed curve gives calculation time and the dotted curve indicates
+the time spend communicating. The lower thin dotted curve gi ves the time for
+storing data.
+Duringthecommunicationthespeedofdatatransferaverages21 .1Mbit/swith
+peaks of about 7Gbit/s and per step between 16MB and 26MB is tran sferred.
+For the larger runs the average throughput increases to about 2 08Mbit/s since
+the size of the packages increases with N(see Eq.1).
+The result of the simulation with N= 16777216 at z= 0 is presented in two
+panels in Fig.3. Each of the two panels show the universe as it is seen b y the
+two supercomputers, with the black parts on the right of the left p anel (left
+on the right image) indicate the part of the universe that resides on the other
+machine.
+5 Concluding remarks
+Theformationoflargescale structureintheuniverse canbestudie d effectively
+by means of simulation but requires phenomenal computer power. E ven with
+fully optimized numerical methods and approximations the wall-clock t ime
+9Fig. 3. Final snapshot of the simulation with N= 2563with Tokyo (left) and
+Amsterdam (right) . The black part of each figure represents t he memory share
+that is located at the other site.
+for such simulations can exceed several months. Our largest simula tion with
+N= 8589934592 has been running for about 1 million CPU hours (more th an
+1 month on 1024 processors) to reach z≃1.5 and we expect to spend another
+∼2 million CPU hours to reach z= 0.
+TherunsinTab.1areperformedonagridofsupercomputers. With ourimple-
+mentation thelatencyandthroughput ofthe communication poses a relatively
+small overhead to the calculation cost, in our largest simulation the c ommu-
+nication overhead<∼10%. We have therewith demonstrated that large-scale
+structure formation simulations are excellently suited for high-per formance
+grid computing.
+The adopted setup would in principle have reduced our wall-clock time b y
+almost a factor two compared to running on a single supercomputer , and our
+simulations in Tab.1 have indeed effectively benefitted from using the grid.
+However, we have spend more than a year preparing and optimizing t he code,
+and acquiring and scheduling the resources, none of which proved t o be trivial.
+On the other hand, if we would not have opted for running on a grid it w ould
+haveprovenextremelydifficulttoperformthesimulationsatallsince acquiring
+1024processorsonasupercomputer forhalf-yearviaa’normal’p roposalwould
+be challenging. Even acquiring half the resources on two supercomp uters with
+the promise to switch computers half-way the simulation would be diffic ult.
+Our strategy to run on a grid enabled us to secure the required CPU time on
+both supercomputers.
+During this project we all became very enthusiastic about the comp utational
+grid as a high-performance resource, despite the time we have spe nd with
+preparations. Thesuccess ofour gridisinpart a consequence oft herealization
+that latency forms no bottleneck, even if the computers are sepa rated by half
+10the planet. We cannot beat the speed of light, but bandwidth is likely t o
+improve with time, making high-performance grid computing very att ractive
+intheforeseeablefuture.Muchwork,however, isneededinimprov ingpractical
+matters, like scheduling issues, network acquisition and cooperatio n between
+supercomputer centers, each of which are major bottlenecks. W e seriously
+consider to perform our next run on a grid with many supercompute rs.
+ 1 10
+ 1  10  100ηp
+nscnp=128, ring
+np=128, star
+np=1024, ring
+np=1024, star
+Fig. 4. The speedup ( ηp) for a grid of nscsupercomputers each with pprocessors.
+The speedup is defined as the wall-clock time of the grid compu ter as fraction of
+the wall-clock time of one single supercomputer, assuming t hat each computer has
+pprocessors. The thick curves are calculated with p= 128 the thin curves with
+p= 1024. The solid curves are calculated assuming the ring top ology network,
+whereas for the dotted curves we adopted a star connected net work. We adopted
+an average bandwidth of b= 200Mbyte/s with a latency of νtlat= 0.828s for
+the external (grid) communication and 10Gbyte/s for intern al communication. The
+diagonal thin dashed curve indicates the ideal scaling.
+Based on our measurements for each of the two supercomputers and the in-
+tercontinental grid we constructed a performance model, which is composed
+of two main components: the calculation ( tcpu) and the data transfer over the
+gridtcomm=νtlat+Scomm/b, wheretlatis the network latency and bis the net-
+work throughput (see Eq.1). The parameter νis introduced to correct for the
+inefficiencies in our code which require several transmissions ( ν) to the other
+computer. In Fig.4 we present the results of the performance mo del based
+on the characteristics of the adopted computers, network and s oftware envi-
+ronment for N= 20483. The speedup is limited by the bandwidth, whereas
+latency, for which we adopted tlat= 0.138swith ν= 6, poses no limiting factor
+11to our simulations. Reducing the total latency will hardly improve the perfor-
+mance, but improving the throughput by a factor 10 would allow us to use an
+order of magnitude more processors per supercomputer while still acquiring
+acceptable speedup. Thetwo topologiesadoptedinFig.4areselect ed based on
+an ideal setup where the supercomputers are interconnected via a ring topol-
+ogy. A sub-optimal star topology leads to congestion as the commu nication
+tends to go through a single site. With the optimal topology (solid cur ves in
+Fig.4) running on a 10 supercomputers with<∼128 processors each results
+in a speed-up of a factor of ∼8, whereas running on 100 supercomputers the
+speedup would be only a factor of 10.
+The best strategy for performing cold dark matter simulations usin g a treePM
+code on a grid appears to be by acquiring p<∼100 processors on nsc∼10
+supercomputers in a ring network topology. The main reason for ad opting
+this strategy would be to be able to acquire the required compute re sources;
+it is considerably easier to obtain p∼100 processors for an extended period
+(∼1 year) from a dozen supercomputer centers, than to acquire th e required
+CPU hours on a single supercomputer. This strategy, however, wo uld require
+drastic changes in the co-scheduling policy of the supercomputer c enters.
+With our cosmological cold dark matter simulation we make the dream o f
+Foster and Kesselman [1] comes true, our calculation benefits from having
+multiple widely separated computers interconnected with a high-ban dwidth
+network, effectively operating as a single machine.
+Acknowledgments
+WearegratefultoHansBlom,MaxineBrown,AndreasBurkert, Ha ldenCohn,
+Jonathan Cole, Tom Defanti, Jan Eveleth, Katsuyuki Hasabe, Dou glas Heg-
+gie, Wouter Huisman, Piet Hut, Mary Inaba, Akira Kato, Walter Lioen , Kees
+Neggers, Breanndan ´O Nu´ aill´ an, Hanno Pet, Steven Rieder, Joaching Stadel,
+Huub Stoffers, Yoshino Takeshi, Thomas Tam, Jin Tanaka, Peter Ta venier,
+Mark van de Sanden, Ronald van der Pol Alan Verlo and Seiichi Yamamo to.
+We are grateful to Ben Moore for offering us a bottle of Pommery 1998 Cuv´ ee
+Louise Brut forthe first relevant scientific results coming fromthiscalculation.
+This work was supported by NWO (grants #643.200.503 and #639.073 .803),
+the NCF (project #SH-095-08), QosCosGrid (EU-FP6-IST-FET C ontract
+#033883), NSF IRNC, NAOJ, NOVA, LKBF and the JSPS. We thank th e
+network facilities of SURFnet, Masafumi Ooe; IEEAF; WIDE; North west Gi-
+gapop and the Global Lambda Integrated FAcility (GLIF) GOLE of Tr ans-
+Light Cisco on National LambdaRail, TransLight, StarLight, Nether Light,
+T-LEX, Pacific and Atlantic Wave. The calulations were performed on the
+supercomputers at Cray XT4 at the Center for Computational As trophysics
+12of National Astronomical Observatory of Japan and the national academic
+supercomputer center SARA in Amsterdam, the Netherlands.
+References
+[1] Foster, I., C. Kesselman, and S. Tuecke, “The Anatomy of t he Grid: Enabling
+Scalable Virtual Organizations,” ”International Journal of High Performance
+Computing Applications” 2001,15, 200.
+[2] Gropp, W., E. Lusk, N. Doss, and A. Skjellum, “A high-perf ormance, portable
+implementation of the MPI message passing interface standa rd,”Parallel
+Computing 1996,22, 6, 789–828.
+[3] Guth, A. H., “Inflationary universe: A possible solution to the horizon and
+flatness problems,” Phys. Rev. D 1981,23, 347–356.
+[4] Hockney, R. and J. Eastwood, Computer Simulation Using Particles , Adan
+Hilger Ltd., Bristol, UK, 1988, 540 pp.
+[5] Hoekstra, G., A., S. Portegies Zwart, M. Bubak, and P. Slo ot,Towards
+Distributed Petascale Computing , Petascale Computing: Algorithms and
+Applications, by David A. Bader (Ed.). Chapman & Hall/CRC co mputational
+science series 2008, 565pp.
+[6] Joyce, M. and B. Marcos (2007), “Quantification of discre teness effects in
+cosmological N-body simulations. II. Evolution up to shell crossing,” Phys.
+Rev. D 76 , 10, 103505.
+[7] Komatsu, E., J. Dunkley, M. R. Nolta, C. L., “Five-Year Wi lkinson Microwave
+Anisotropy Probe (WMAP) Observations: Cosmological Inter pretation,” 2008,
+ArXiv e-prints 803 .
+[8] Lee, B. W. and S. Weinberg (1977), “Cosmological lower bo und on heavy-
+neutrino masses,” Physical Review Letters 39 , 165–168.
+[9] Prunet, S., C. Pichon, D. Aubert, D. Pogosyan, R. Teyssie r, and S. Gottloeber,
+“Initial Conditions For Large Cosmological Simulations,” ApJS2008,178,
+179–188.
+[10] Spergel, D. N., R. Bean, O. Dor´ e, et la. “Three-Year Wil kinson Microwave
+Anisotropy Probe (WMAP) Observations: Implications for Co smology,” ApJS
+2007,170, 377–408.
+[11] Xu, G., “A New Parallel N-Body Gravity Solver: TPM,” ApJS1995,98, 355.
+[12] Yoshikawa, K. and T. Fukushige, “PPPM and TreePM Method s on GRAPE
+Systems for Cosmological N-Body Simulations,” Publ. Astr. Soc. Japan 2005,
+57, 849–860.
+13The authors
+Simon Portegies Zwart is professor in computational astrophysics at the
+Sterrewacht Leiden in the Netherlands. He received his Ph.D. in astr onomy
+at Utrecht University. His principal scientific interests are high-pe rformance
+computational astrophysics and the ecology of dense stallar syst ems.
+e-mail:spz@strw.leidenuniv.nl ,
+URL:http://www.strw.leidenuniv.nl/ ∼spz/
+Tomoaki Ishiyama and Derek Groen Are graduate students in the re-
+search groeps of Makino and Portegies Zwart, respectivly.
+Jun Makino is professor of astrophysics at the National Astronomical Ob-
+servatory of Japan. He recieved his PhD in astronomy at the Univer sity in
+Tokyo. His research interests are stellar dynamics, large-scale sc ientific simu-
+lation and high-performance computing.
+e-mail:makino@cfca.jp ,
+URL:http://www.artcompsci.org/ ∼makino
+Cees de Laat is associate professor of system and network engineering at
+the University of Amsterdam. He recieved his PhD in computer scienc e at the
+University of Amsterdam. His research interests are optical/switc hed Internet
+for data-transport in TeraScale eScience applications. e-mail: delaat@uva.nl ,
+URL:http://www.science.uva.nl/ ∼delaat
+Steve McMillan is professor of physics at Drexel University in Philadel-
+phia, Pennsylvania, U.S.A. He received his Ph.D. in astronomy from Har vard
+University. His principal scientific interests are high-performance computation
+and the astrophysics of star clusters and galactic nuclei.
+e-mail:steve@physics.drexel.edu ,
+URL:http://www.physics.drexel.edu/ ∼steve
+Kei Hiraki is professor in computer science at the University of Tokyo.
+He received his Ph.D. in phisics at the University of Tokyo. His researc h in-
+terests are parallel and distributed system, high-performance c omputing and
+networking.
+e-mail:hiraki@is.s.u-tokyo.ac.nl ,
+URL:http://www-hiraki.is.s.u-tokyo.ac.jp
+14Stefan Harfst is a postdoctoral fellow in the computational astrophysics
+group of Portegies Zwart. He received his Ph.D. in astrophysics at t he Uni-
+versity of Kiel, Germany. His research interests are stellar dynamic s and high-
+performance computing.
+e-mail:harfst@strw.leidenuniv.nl
+Keigo Nitadori Is a postdoctoral fellow at RIKEN, Japan. He received his
+Ph.D. in astrophysics at the University of Tokyo.
+e-mail:keigo@riken.jp
+Paola Grosso is a senior researcher at the system and network engineering
+group at the University of Amsterdam. She received her Ph.D. in Phy sics
+from the University of Turin in Italy. Her research interests are mo deling,
+programming and virtualization of networks for eScience application s.
+e-mail:p.grosso@uva.nl ,
+URL:http://www.science.uva.nl/ ∼grosso
+15
\ No newline at end of file
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@@ -0,0 +1,3879 @@
+arXiv:1001.0774v2  [cond-mat.quant-gas]  21 Dec 2018Exact relations for quantum-mechanical few-body and many- body problems
+with short-range interactions in two and three dimensions
+F´ elix Werner
+Department of Physics, University of Massachusetts, Amher st, MA 01003, USA
+Yvan Castin
+Laboratoire Kastler Brossel, ´Ecole Normale Sup´ erieure,
+UPMC and CNRS, 24 rue Lhomond, 75231 Paris Cedex 05, France
+(Dated: December 24, 2018)
+We derive relations between various observables for Nparticles with zero-range or short-range
+interactions, in continuous space or on a lattice, in two or t hree dimensions, in an arbitrary external
+potential. Some of our results generalize known relations b etween large-momentum behavior of
+the momentum distribution, short-distance behavior of the pair correlation function and of the
+one-body density matrix, derivative of the energy with resp ect to the scattering length or to time,
+and the norm of the regular part of the wavefunction; in the ca se of finite-range interactions, the
+interaction energy is also related to dE/da. The expression relating the energy to a functional of the
+momentum distribution is also generalized, and is found to b reak down for Efimov states with zero-
+range interactions, dueto a subleading oscillating tail in the momentum distribution. We also obtain
+new expressions for the derivative of the energy of a univers al state with respect to the effective
+range, the derivative of the energy of an efimovian state with respect to the three-body parameter,
+and the second order derivative of the energy with respect to the inverse (or the logarithm in the
+two-dimensional case) of the scattering length. The latter is negative at fixed entropy. We use
+exact relations to compute corrections to exactly solvable three-body problems and find agreement
+with available numerics. For the unitary gas, we compare exa ct relations to existing fixed-node
+Monte-Carlo data, and we test, with existing Quantum Monte C arlo results on different finite range
+models, our prediction that the leading deviation of the cri tical temperature from its zero range
+value is linear in the interaction effective range rewith a model independent numerical coefficient.
+PACS numbers:
+I. INTRODUCTION
+The experimental breakthroughs of 1995 having led to the first re alization of a Bose-Einstein condensate in an
+atomic vapor [1–3] have opened the era of experimental studies of ultracold gases with non-negligible or even strong
+interactions, in dimension lower or equal to three [4–8]. In these sys tems, the thermal de Broglie wavelength and the
+mean distance between atoms are much larger than the range of th e interaction potential. This so-called zero-range
+limit has interesting universal properties: Several quantities such as the thermodynamic functions of the gas depend
+on the interaction potential only through the scattering length a, a length characterizing the low-energy scattering
+amplitude of two atoms.
+ThisuniversalitypropertyholdsfortheweaklyrepulsiveBosegasint hreedimensions[9]uptotheorderofexpansion
+in (na3)1/2corresponding to Bogoliubov theory [10], nbeing the gas density. It is also true for the weakly repulsive
+Bose gas in two dimensions [11–13], even at the next order beyond Bo goliubov theory [14]. For amuch larger than
+the range of the interaction potential, the ground state of Nbosons in two dimensions is a universal N-body bound
+state [15–19]. In one dimension, the universality holds for any scatt ering length, and the Bose gas with zero-range
+interaction is exactly solvable by the Bethe ansatz both in the repuls ive case [20] and in the attractive case [21, 22].
+For spin 1/2 fermions, the universality properties are expected to be even stronger. The weakly interacting regimes
+in 3D [23–27] and in 2D [28] are universal, and the 1D case is also solvab le by Bethe ansatz for an arbitraryinteraction
+strength [29, 30]. Universality is expected to hold for an arbitrary s cattering length even in 3 D(see however [31]), as
+wasrecently tested by experimental studies on the BEC-BCS cros soverusing a Feshbach resonance, see e. g. [8, 32–39],
+and in agreement with unbiased Quantum Monte Carlo calculations [40 –45]. A similar universal crossover from BEC
+to BCS is expected in 2 Dwhen the parameter ln( kFa) varies from −∞to +∞[46–50]. Universality is also expected
+for mixtures in 2 D[50–52], and in 3 Dfor Fermi-Fermi mixtures below a critical mass ratio [51, 53, 54].
+In the zero-range regime, it is intuitive that the short-range or hig h-momenta properties of the gas are dominated
+by two-body physics. For example the pair distribution function g(2)(r12) of particles at distances r12much smaller
+than the de Broglie wavelength is expected to be proportional to th e modulus squared of the zero-energy two-body
+scattering wavefunction φ(r12), with a proportionality factor Λ gdepending on the many-body state of the gas.
+Similarly the large momentum tail of the momentum distribution n(k), at wavevectors much larger than the inverse2
+de Broglie wavelength, is expected to be proportional to the modulu s squared of the Fourier component of the zero
+energy scattering state ˜φ(k), with a proportionality factor Λ ndepending on the many-body state of the gas: Whereas
+two colliding atoms in the gas have a center of mass wavevector of th e order of the inverse de Broglie wavelength,
+their relative wavevector can access much larger values, up to the inverse of the interaction range, simply because the
+interaction potential has a width in the space of relative momenta of the order of the inverse of its range in real space.
+For these intuitive reasons, and with the notable exception of one- dimensional systems, one expects that the mean
+interaction energy Eintof the gas, being sensitive to the shape of g(2)at distances of the order of the interaction range,
+is not universal, but diverges in the zero-range limit; one also expect s that, apart from the 1D case, the mean kinetic
+energy, being dominated by the large-momentum tail of the moment um distribution, is not universal and diverges
+in the zero-range limit, a well known fact in the context of Bogoliubov theory for Bose gases and of BCS theory for
+Fermi gases. Since the total energy of the gas is universal, and Eintis proportional to Λ gwhileEkinis proportional
+to Λn, one expects that there exists a simple relation between Λ gand Λn.
+The precise link between the pair distribution function, the tail of th e momentum distribution and the energy of the
+gas was first established for one-dimensional systems. In [20] the value of the pair distribution function for r12= 0 was
+expressed in terms of the derivative of the gas energy with respec t to the one-dimensional scattering length, thanks
+to the Hellmann-Feynman theorem. In [55] the large momentum tail o fn(k) was also related to this derivative of the
+energy, by using a simple and general property of the Fourier tran sform of a function having discontinuous derivatives
+in isolated points.
+In three dimensions, results in these directions were first obtained for weakly interacting gases. For the weakly
+interacting Bose gas, Bogoliubov theory contains the expected pr operties, in particular on the short distance behavior
+of the pair distribution function [56–58] and the fact that the mome ntum distribution has a slowly decreasing tail.
+For the weakly interacting two-component Fermi gas, it was shown that the BCS anomalous average (or pairing field)
+∝an}b∇acketle{tˆψ↑(r1)ˆψ↓(r2)∝an}b∇acket∇i}htbehaves at short distances as the zero-energy two-body scatt ering wavefunction φ(r12) [59], resulting
+in ag(2)function indeed proportional to |φ(r12)|2at short distances. It was however understood later that the
+corresponding proportionality factor Λ gpredicted by BCS theory is incorrect [60], e.g. at zero temperature the BCS
+prediction drops exponentially with 1 /ain the non-interacting limit a→0−, whereas the correct result drops as a
+power law in a.
+More recently, in a series of two articles [61, 62], explicit expressions for the proportionality factors Λ gand Λnwere
+obtained in terms of the derivative of the gas energy with respect t o the inverse scattering length, for a two-component
+interacting Fermi gas in three dimensions, for an arbitrary value of the scattering length, that is, not restricting to
+the weakly interacting limit. Later on, these results were rederived in [63–65], and also in [66] with very elementary
+methods building on the intuition that g(2)∝ |φ(r12)|2at short distances and n(k)∝ |˜φ(k)|2at large momenta.
+These relations were recently tested by numerical four-body calc ulations [67]. An explicit relation between Λ gand
+the interaction energy was derived in [65]. Another fundamental re lation discovered in [61] and recently generalized
+in [68] to bosons, to Fermi-Bose mixtures and to fermions in 2D, expr esses the total energy as a functional of the
+momentum distribution and the spatial density.
+In the present work we derive generalizations of the relations of [2 0, 55, 61, 62, 65, 68] to two dimensional gases,
+to the general case of a mixture of an arbitrary number of atomic s pecies and spin component, and to the case of a
+small but non-zero interaction range (both on a lattice and in contin uous space). We also find entirely new results for
+the first order derivative of the energy with respect to the effect ive range and, in presence of the Efimov effect, with
+respect to the three-body parameter, as well as the second ord er derivative with respect to the scattering length.
+The article is organized as follows. In Section II we treat in detail the case of two-component Fermi gases. Relations
+holding for any system eigenstate for zero-range interactions ar e derived in Section IIB and summarized in Table II.
+We then consider lattice models (Tab. III, Sec. IIC) and finite-ran ge models in continuous space (Tab. IV, Sec. IID).
+In Section IIE we derive a model-independent expression for the co rrection to the energy due to a finite range or a
+finite effective range of the interaction. The generalization to ther modynamic equilibrium, where the system is in a
+statistical mixture of eigenstates, is discussed in Section IIF. In S ection III we turn to the case of spinless bosons. We
+focus on the case of zero-range interactions where, in 3 D, the Efimov effect leads to modifications or even breakdown
+of some relations, and to the appearance of a new relation. Then we show briefly in Section IV how to treat the case
+of an arbitrary mixture and present results for zero-range inter actions (Tab. VI). Finally we present applications of
+exact relations: For three particles we compute corrections to ex actly solvable cases and compare them to numerics
+(Sec. VA), and we check that exact relations are satisfied by exist ing fixed-node Monte-Carlo data for correlation
+functions of the unitary gas. We expect from our expression for t he leading finite-range correction to the energy that
+the leading finite-range correction to the critical temperature in t he BEC-BCS crossover depends only on the effective
+range of the interaction, an expectation that we test against the Quantum Monte Carlo calculations of [41, 44]. We
+conclude in Section VI.3
+II. TWO-COMPONENT FERMIONS
+In this Section we consider spin-1 /2 fermions. For a fixed number Nσof particles in each spin state σ=↑,↓, one
+can consider that particles 1 ,...,N ↑have a spin ↑and particles N↑+ 1,...,N ↑+N↓=Nhave a spin ↓, i.e. the
+wavefunction ψ(r1,...,rN) changes sign when one exchanges the positions of two particles ha ving the same spin [132].
+A. Models
+Here we introduce the three models used in this work to model interp article interactions.
+1. Zero-range model
+In this well-known model (see e.g. [69–76] and refs. therein) the in teraction potential is replaced by contact
+conditions on the many-body wavefunction: For any pair of particle si∝ne}ationslash=j, there exists a function Aij, hereafter
+called regular part of ψ, such that in 3 D
+ψ(r1,...,rN) =
+rij→0/parenleftbigg1
+rij−1
+a/parenrightbigg
+Aij(Rij,(rk)k/ne}ationslash=i,j)+O(rij), (1)
+and in 2D
+ψ(r1,...,rN) =
+rij→0ln(rij/a)Aij(Rij,(rk)k/ne}ationslash=i,j)+O(rij), (2)
+where the limit of vanishing distance rijbetween particles iandjis taken for a fixed position of their center of mass
+Rij= (ri+rj)/2 and fixed positions of the remaining particles ( rk)k/ne}ationslash=i,j. Fermionic symmetry of course imposes
+Aij= 0 if particles iandjhave the same spin. When none of the ri’s coincide, there is no interaction potential and
+Schr¨ odinger’s equation reads
+Hψ(r1,...,rN) =Eψ(r1,...,rN) (3)
+with
+H=N/summationdisplay
+i=1/bracketleftbigg
+−/planckover2pi12
+2m∆ri+U(ri)/bracketrightbigg
+ψ (4)
+wheremis the atomic mass and Uis an external potential. The crucial difference between the Hamilto nianHand
+the non-interacting Hamiltonian is the boundary condition (1,2).
+2. Lattice models
+These models were used for quantum Monte-Carlo calculations [40–4 3, 45, 77]. They can also be convenient for
+analytics, as used in [14, 78, 79] and in this work. Here particles live on a lattice, i. e. the coordinates are integer
+multiples of the lattice spacing b. The Hamiltonian reads
+H=H0+g0W (5)
+where
+H0=N/summationdisplay
+i=1/bracketleftbigg
+−/planckover2pi12
+2m∆ri+U(ri)/bracketrightbigg
+(6)
+W=/summationdisplay
+i<jδri,rjb−d(7)4
+Three dimensions Two dimensions
+ψ(r1,...,rN) =
+rij→0/parenleftbigg1
+rij−1
+a/parenrightbigg
+Aij(Rij,(rk)k/negationslash=i,j)+O(rij)ψ(r1,...,rN) =
+rij→0ln(rij/a)Aij(Rij,(rk)k/negationslash=i,j)+O(rij)
+(A(1),A(2))≡/summationdisplay
+i<j/integraldisplay/parenleftBig/productdisplay
+k/negationslash=i,jddrk/parenrightBig
+ddRijA(1)
+ij(Rij,(rk)k/negationslash=i,j)∗A(2)
+ij(Rij,(rk)k/negationslash=i,j)
+TABLE I: Notation for the regular part Aof the many-body wavefunction appearing in the contact cond itions (first line) and
+for the scalar product between such regular parts (second li ne).
+Three dimensions Two dimensions
+1dE
+d(−1/a)=4π/planckover2pi12
+m(A,A)dE
+d(lna)=2π/planckover2pi12
+m(A,A)
+2C≡lim
+k→+∞k4nσ(k) =4πm
+/planckover2pi12dE
+d(−1/a)C≡lim
+k→+∞k4nσ(k) =2πm
+/planckover2pi12dE
+d(lna)
+3/integraldisplay
+d3Rg(2)
+↑↓/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+∼
+r→0C
+(4π)21
+r2/integraldisplay
+d2Rg(2)
+↑↓/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+∼
+r→0C
+(2π)2ln2r
+4 E−Etrap=/planckover2pi12C
+4πmaE−Etrap= lim
+Λ→∞/bracketleftbigg
+−/planckover2pi12C
+2πmln/parenleftbiggaΛeγ
+2/parenrightbigg
++/summationdisplay
+σ/integraldisplayd3k
+(2π)3/planckover2pi12k2
+2m/bracketleftbigg
+nσ(k)−C
+k4/bracketrightbigg
++/summationdisplay
+σ/integraldisplay
+k<Λd2k
+(2π)2/planckover2pi12k2
+2mnσ(k)/bracketrightbigg
+5/integraldisplay
+d3Rg(1)
+σσ/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+=
+r→0Nσ−C
+8πr+O(r2)/integraldisplay
+d2Rg(1)
+σσ/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+=
+r→0Nσ+C
+4πr2lnr+O(r2)
+61
+33/summationdisplay
+i=1/summationdisplay
+σ/integraldisplay
+d3Rg(1)
+σσ/parenleftBig
+R+rui
+2,R−rui
+2/parenrightBig
+=
+r→0N1
+22/summationdisplay
+i=1/summationdisplay
+σ/integraldisplay
+d2Rg(1)
+σσ/parenleftBig
+R+rui
+2,R−rui
+2/parenrightBig
+=
+r→0N
+−C
+4πr−m
+3/planckover2pi12/parenleftbigg
+E−Etrap−/planckover2pi12C
+4πma/parenrightbigg
+r2+o(r2)+C
+4πr2/bracketleftbigg
+ln/parenleftBigr
+a/parenrightBig
++F
+32/bracketrightbigg
+−m
+2/planckover2pi12(E−Etrap)r2+o(r2)
+71
+2d2E
+d(−1/a)2=/parenleftbigg4π/planckover2pi12
+m/parenrightbigg2/summationdisplay
+n,En/negationslash=E|(A(n),A)|2
+E−En1
+2d2E
+d(lna)2=/parenleftbigg2π/planckover2pi12
+m/parenrightbigg2/summationdisplay
+n,En/negationslash=E|(A(n),A)|2
+E−En
+8/parenleftbiggd2F
+d(−1/a)2/parenrightbigg
+T<0,/parenleftbiggd2E
+d(−1/a)2/parenrightbigg
+S<0/parenleftbiggd2F
+d(lna)2/parenrightbigg
+T<0,/parenleftbiggd2E
+d(lna)2/parenrightbigg
+S<0
+9dE
+dt=/planckover2pi12C
+4πmd(−1/a)
+dt+/angbracketleftBigN/summationdisplay
+i=1∂tU(ri,t)/angbracketrightBigdE
+dt=/planckover2pi12C
+2πmd(lna)
+dt+/angbracketleftBigN/summationdisplay
+i=1∂tU(ri,t)/angbracketrightBig
+TABLE II: Relations for two-component fermions with zero-r ange interactions. The regular part Ais defined in Table I. Lines
+1-7 hold for any eigenstate, and can be generalized to finite t emperature by taking a thermal average in the canonical ense mble
+and by taking the derivatives of Ewith respect to aat constant entropy S. Line 8 holds in the canonical ensemble. Line 9
+holds for any time-dependence of scattering length and trap ping potential and any corresponding time-dependent stati stical
+mixture .5
+1 C≡4πm
+/planckover2pi12dE
+d(−1/a)C≡2πm
+/planckover2pi12dE
+d(lna)
+2dE
+d(−1/a)=4π/planckover2pi12
+m(A,A)dE
+d(lna)=2π/planckover2pi12
+m(A,A)
+3 Eint=/parenleftbigg/planckover2pi12
+m/parenrightbigg2C
+g0
+4 E−Etrap=/planckover2pi12C
+4πmaE−Etrap= lim
+q→0/braceleftBigg
+−/planckover2pi12C
+2πmln/parenleftbiggaqeγ
+2/parenrightbigg
++/summationdisplay
+σ/integraldisplay
+Dd3k
+(2π)3ǫk/bracketleftBigg
+nσ(k)−C/parenleftbigg/planckover2pi12
+2mǫk/parenrightbigg2/bracketrightBigg
++/summationdisplay
+σ/integraldisplay
+Dd2k
+(2π)2ǫk/bracketleftbigg
+nσ(k)−C/planckover2pi12
+2mǫkP/planckover2pi12
+2m(ǫk−ǫq)/bracketrightbigg/bracerightBigg
+51
+2d2E
+dg2
+0=|φ(0)|4/summationdisplay
+n,En/negationslash=E|(A(n),A)|2
+E−En
+6/parenleftbiggd2F
+dg2
+0/parenrightbigg
+T<0,/parenleftbiggd2E
+dg2
+0/parenrightbigg
+S<0
+7/summationdisplay
+Rb3g(2)
+↑↓(R,R) =C
+(4π)2|φ(0)|2/summationdisplay
+Rb2g(2)
+↑↓(R,R) =C
+(2π)2|φ(0)|2
+In the zero-range regime ktypb≪1
+8/summationdisplay
+Rb3g(2)
+↑↓/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+≃C
+(4π)2|φ(r)|2forr≪k−1
+typ/summationdisplay
+Rb2g(2)
+↑↓/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+≃C
+(2π)2|φ(r)|2forr≪k−1
+typ
+9 nσ(k)≃C/parenleftbigg/planckover2pi12
+2mǫk/parenrightbigg2
+fork≫ktyp
+TABLE III: Relations for two-component fermions in a lattic e model.Cis defined in line 1.
+in first quantization, i.e.
+H0=/summationdisplay
+σ/integraldisplay
+Dddk
+(2π)dǫkc†
+σ(k)cσ(k)+/summationdisplay
+r,σbdU(r)(ψ†
+σψσ)(r) (8)
+W=/summationdisplay
+rbd(ψ†
+↑ψ†
+↓ψ↓ψ↑)(r) (9)
+in second quantization. Here dis the space dimension, ǫkis the dispersion relation and c†
+σ(k) is creates a particle
+in the plane wave state |k∝an}b∇acket∇i}htdefined by ∝an}b∇acketle{tr|k∝an}b∇acket∇i}ht=eik·rfor anykbelonging to the first Brillouin zone D=/parenleftbig
+−π
+b,π
+b/bracketrightbigd.
+Accordingly the operator ∆ in (6) is the discrete representation of the Laplacian defined by −/planckover2pi12
+2m∝an}b∇acketle{tr|∆r|k∝an}b∇acket∇i}ht ≡ǫk∝an}b∇acketle{tr|k∝an}b∇acket∇i}ht.
+The simplest choice for the dispersion relation is ǫk=/planckover2pi12k2
+2m[14, 42, 45, 78, 79]. Another choice, used in [41, 77], is6
+Three dimensions Two dimensions
+1 C≡4πm
+/planckover2pi12dE
+d(−1/a)C≡2πm
+/planckover2pi12dE
+d(lna)
+2 Eint=C
+(4π)2/integraldisplay
+d3rV(r)|φ(r)|2Eint=C
+(2π)2/integraldisplay
+d2rV(r)|φ(r)|2
+3 E−Etrap=/planckover2pi12C
+4πmaE−Etrap= lim
+R→∞/braceleftBigg
+/planckover2pi12C
+2πmaln/parenleftbiggR
+a/parenrightbigg
++/summationdisplay
+σ/integraldisplayd3k
+(2π)3/planckover2pi12k2
+2m/bracketleftbigg
+nσ(k)−C
+(4π)2|˜φ′(k)|2/bracketrightbigg
++/summationdisplay
+σ/integraldisplayd2k
+(2π)2/planckover2pi12k2
+2m/bracketleftbigg
+nσ(k)−C
+(2π)2|˜φ′
+R(k)|2/bracketrightbigg/bracerightBigg
+In the zero-range regime ktypb≪1
+5/integraldisplay
+d3Rg(2)
+↑↓/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+≃C
+(4π)2|φ(r)|2forr≪k−1
+typ/integraldisplay
+d2Rg(2)
+↑↓/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+≃C
+(2π)2|φ(r)|2forr≪k−1
+typ
+6 nσ(k)≃C
+(4π)2|˜φ(k)|2fork≫ktyp nσ(k)≃C
+(2π)2|˜φ(k)|2fork≫ktyp
+TABLE IV: Relations for two-component fermions with a finite -range interaction potential V(r) in continuous space. Cis
+defined in line 1.
+the dispersion relation of the Hubbard model: ǫk=/planckover2pi12
+mb2d/summationdisplay
+i=1[1−cos(kib)]. More generally, what follows applies to
+anyǫksuch thatǫk→
+b→0/planckover2pi12k2
+2msufficiently rapidly and ǫ−k=ǫk.
+A key quantity is the zero-energy scattering state φ(r), defined by the two-body Schr¨ odinger equation (with the
+center of mass at rest)
+/parenleftbigg
+−/planckover2pi12
+m∆r+g0δr,0
+bd/parenrightbigg
+φ(r) = 0 (10)
+and by the normalization conditions
+φ(r)≃
+r≫b1
+r−1
+ain 3D (11)
+φ(r)≃
+r≫bln(r/a) in 2D. (12)
+A straightforward two-body analysis, detailed in App. A, yields the r elation between the scattering length and the
+bare coupling constant g0:
+1
+g0=m
+4π/planckover2pi12a−/integraldisplay
+Dd3k
+(2π)31
+2ǫkin 3D (13)
+1
+g0= lim
+q→0−m
+2π/planckover2pi12ln(aqeγ/2)+/integraldisplay
+Dd2k
+(2π)2P1
+2(ǫq−ǫk)in 2D (14)
+whereγ= 0.577216...is Euler’s constant and Pis the principal value. Other useful relations derived in App. A are
+φ(0) =−4π/planckover2pi12
+mg0in 3D (15)
+φ(0) =2π/planckover2pi12
+mg0in 2D (16)7
+Three dimensions Two dimensions
+/parenleftbigg∂E
+∂(−1/a)/parenrightbigg
+Rt=4π/planckover2pi12
+m(A,A)dE
+d(lna)=2π/planckover2pi12
+m(A,A)
+C≡lim
+k→+∞k4n(k) =8πm
+/planckover2pi12/parenleftbigg∂E
+∂(−1/a)/parenrightbigg
+RtC≡lim
+k→+∞k4n(k) =4πm
+/planckover2pi12dE
+d(lna)
+/integraldisplay
+d3Rg(2)/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+∼
+r→0C
+(4π)21
+r2/integraldisplay
+d2Rg(2)/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+∼
+r→0C
+(2π)2ln2r
+E−Etrapif∃lim=/planckover2pi12C
+8πmaE−Etrap= lim
+Λ→∞/bracketleftbigg
+−/planckover2pi12C
+4πmln/parenleftbiggaΛeγ
+2/parenrightbigg
++ lim
+Λ→+∞/integraldisplay
+k<Λd3k
+(2π)3/planckover2pi12k2
+2m/bracketleftbigg
+n(k)−C
+k4/bracketrightbigg
++/integraldisplay
+k<Λd2k
+(2π)2/planckover2pi12k2
+2mn(k)/bracketrightbigg
+1
+2/parenleftbigg∂2E
+∂(−1/a)2/parenrightbigg
+Rt=/parenleftbigg4π/planckover2pi12
+m/parenrightbigg2/summationdisplay
+n,En/negationslash=E|(A(n),A)|2
+E−En1
+2d2E
+d(lna)2=/parenleftbigg2π/planckover2pi12
+m/parenrightbigg2/summationdisplay
+n,En/negationslash=E|(A(n),A)|2
+E−En
+/parenleftbigg∂E
+∂lnRt/parenrightbigg
+a=/planckover2pi12
+m√
+3
+32|s0|2N(N−1)(N−2)/parenleftbiggd2F
+d(lna)2/parenrightbigg
+T<0
+×/integraldisplay
+dC/integraldisplay
+dr4...drN|B(C,r4,...,rN)|2/parenleftbiggd2E
+d(lna)2/parenrightbigg
+S<0
+TABLE V: Main results for spinless bosons in the limit of a zer o range interaction. In three dimensions, the derivatives a re taken
+for a fixed three-body parameter Rt. As discussed in the text, in three dimensions, the relation between energy and momentum
+distribution is valid if the large cut-off limit Λ →+∞exists, which is not the case for Efimovian states (i.e. eigen states whose
+energy depends on Rt). In the last relation in three-dimensions, Bis the three-body regular part defined in (152).
+and
+|φ(0)|2=4π/planckover2pi12
+md(−1/a)
+dg0in 3D (17)
+|φ(0)|2=2π/planckover2pi12
+md(lna)
+dg0in 2D. (18)
+In the zero-range limit ( b→0 withg0adjusted in such a way that aremains constant), the spectrum of the
+lattice model is expected to converge to the one of the zero-rang e model [41, 79], and any eigenfunction ψ(r1,...,rN)
+of the lattice model tends to the corresponding eigenfunction of t he zero-range model, provided all interparticle
+distances remain much larger than b. Let us denote by 1 /ktypthe typical length-scale on which the zero-rangemodel’s
+wavefunction varies: e.g. for the lowest eigenstates, it is on the or der of the mean interparticle distance, or on the
+order ofain the regime where ais small and positive and dimers are formed. The zero-range limit is the n reached if
+ktypb≪1.
+For lattice models, it will prove convenient to define the regular part Aby
+ψ(r1,...,ri=Rij,...,rj=Rij,...,rN) =φ(0)Aij(Rij,(rk)k/ne}ationslash=i,j). (19)8
+Three dimensions Two dimensions
+∂E
+∂(−1/aσσ′)=2π/planckover2pi12
+µσσ′(A,A)σσ′∂E
+∂(lnaσσ′)=π/planckover2pi12
+µσσ′(A,A)σσ′
+Cσ≡lim
+k→+∞k4nσ(k) =/summationdisplay
+σ′(1+δσσ′)8πµσσ′
+/planckover2pi12∂E
+∂(−1/aσσ′)Cσ≡lim
+k→+∞k4nσ(k) =/summationdisplay
+σ′(1+δσσ′)4πµσσ′
+/planckover2pi12∂E
+∂(lnaσσ′)
+/integraldisplay
+d3Rg(2)
+σσ′/parenleftbigg
+R+mσ′
+mσ+mσ′r,R−mσ
+mσ+mσ′r/parenrightbigg
+∼
+r→0(1+δσσ′)/integraldisplay
+d2Rg(2)
+σσ′/parenleftbigg
+R+mσ′
+mσ+mσ′r,R−mσ
+mσ+mσ′r/parenrightbigg
+∼
+r→0(1+δσσ′)
+×µσσ′
+2π/planckover2pi12∂E
+∂(−1/aσσ′)1
+r2×µσσ′
+π/planckover2pi12∂E
+∂(lnaσσ′)ln2r
+E−Etrap=/summationdisplay
+σ≤σ′1
+aσσ′∂E
+∂(−1/aσσ′)E−Etrap= lim
+Λ→∞
+−/summationdisplay
+σ≤σ′∂E
+∂(lnaσσ′)ln/parenleftbiggaσσ′Λeγ
+2/parenrightbigg
++/summationdisplay
+σ/integraldisplayd3k
+(2π)3/planckover2pi12k2
+2mσ/bracketleftbigg
+nσ(k)−Cσ
+k4/bracketrightbigg
++/summationdisplay
+σ/integraldisplay
+k<Λd2k
+(2π)2/planckover2pi12k2
+2mσnσ(k)/bracketrightBigg
+1
+2∂2En
+∂(−1/aσσ′)2=/parenleftbigg2π/planckover2pi12
+µσσ′/parenrightbigg2/summationdisplay
+n′,En′/negationslash=En|(A(n′),A(n))σσ′|2
+En−En′1
+2∂2En
+∂(lnaσσ′)2=/parenleftbiggπ/planckover2pi12
+µσσ′/parenrightbigg2/summationdisplay
+n′,En′/negationslash=En|(A(n′),A(n))σσ′|2
+En−En′
+/parenleftbigg∂2F
+∂(−1/aσσ′)2/parenrightbigg
+T<0/parenleftbigg∂2F
+∂(lnaσσ′)2/parenrightbigg
+T<0
+/parenleftbigg∂2E
+∂(−1/aσσ′)2/parenrightbigg
+S<0/parenleftbigg∂2E
+∂(lnaσσ′)2/parenrightbigg
+S<0
+TABLE VI: Main results for an arbitrary mixture with zero-ra nge interactions. In three dimensions, if the Efimov effect oc curs,
+the derivatives must be taken for fixed three-body parameter (s) and the expression for Ein line 4 breaks down.
+In the zero-range regime ktypb≪1, we expect that when the distance rijbetween two particles of opposite spin is
+≪1/ktypwhile all the other interparticle distances are much larger than band thanrij, the many-body wavefunction
+is proportional to φ(rij), with a proportionality constant given by (19):
+ψ(r1,...,rN)≃φ(rj−ri)Aij(Rij,(rk)k/ne}ationslash=i,j) (20)
+whereRij= (ri+rj)/2. If moreover rij≫b,φcan be replaced by its asymptotic form (11,12); since the contact
+conditions (1), (2) of the zero-range model must be recovered, we see that the lattice model’s regular part tends to
+the zero-range model’s regular part in the zero-range limit.9
+3. Finite-range continuous-space model
+Such models are used in numerical few-body correlated Gaussian an d many-body fixed-node Monte-Carlo calcula-
+tions (see e. g. [5, 67, 80–83] and refs. therein). They are also re levant to neutron matter [84]. The Hamiltonian
+reads
+H=H0+N↑/summationdisplay
+i=1N/summationdisplay
+j=N↑+1V(rij), (21)
+H0being defined by (6) where ∆ rinow stands for the usual Laplacian, and V(r) is an interaction potential between
+particles of opposite spin, which vanishes for r > bor at least decays quickly enough for r≫b. The two-body
+zero-energy scattering state φ(r) is again defined by the Schr¨ odinger equation −(/planckover2pi12/m)∆rφ+V(r)φ= 0 and the
+boundary condition (11,12). The zero-range regime is again reache d forktypb≪1 withktypthe typical relative
+wavevector [133]. Equation (20) again holds in the zero-range regim e, whereAnow simply stands for the zero-range
+model’s regular part.
+B. Relations in the zero-range limit
+1. First order derivative of the energy with respect to the sc attering length
+We now derive relations for the zero-range model. For some of the d erivations we will use a lattice model and take
+the zero-range limit in the end.
+Three dimensions:
+Let us consider a wavefunction ψ1satisfying the contact condition (1) for a scattering length a1. We denote by
+A(1)
+ijthe regular part of ψ1appearing in the contact condition (1). Similarly, ψ2satisfies the contact condition for a
+scattering length a2and a regular part A(2)
+ij. Then, as shown in Appendix B, the following lemma holds:
+∝an}b∇acketle{tψ1,Hψ2∝an}b∇acket∇i}ht−∝an}b∇acketle{tHψ1,ψ2∝an}b∇acket∇i}ht=4π/planckover2pi12
+m/parenleftbigg1
+a1−1
+a2/parenrightbigg
+(A(1),A(2)) (22)
+where the scalar product between regular parts is defined by
+(A(1),A(2))≡/summationdisplay
+i<j/integraldisplay/parenleftBig/productdisplay
+k/ne}ationslash=i,jddrk/parenrightBig/integraldisplay
+ddRijA(1)∗
+ij(Rij,(rk)k/ne}ationslash=i,j)A(2)
+ij(Rij,(rk)k/ne}ationslash=i,j). (23)
+We then apply (22) to the case where ψ1andψ2areN-body eigenstates of energy E1andE2. The left hand side of
+(22) then reduces to ( E2−E1)∝an}b∇acketle{tψ1|ψ2∝an}b∇acket∇i}ht. Taking the limit a2→a1gives the final result
+dE
+d(−1/a)=4π/planckover2pi12
+m(A,A) (24)
+for any eigenstate. This result is contained in the work of Tan [61, 62 ][134]. Note that, here and in what follows, we
+have assumed that the wavefunction is normalized: ∝an}b∇acketle{tψ|ψ∝an}b∇acket∇i}ht= 1.
+Two dimensions:
+The 2Dversion of the lemma (22) is
+∝an}b∇acketle{tψ1,Hψ2∝an}b∇acket∇i}ht−∝an}b∇acketle{tHψ1,ψ2∝an}b∇acket∇i}ht=2π/planckover2pi12
+mln(a2/a1) (A(1),A(2)), (25)
+as shown in Appendix B. As in 3 D, we deduce from the lemma the final result
+dE
+d(lna)=2π/planckover2pi12
+m(A,A). (26)10
+2. Large-momentum tail of the momentum distribution
+The momentum distribution is defined in second quantization by
+nσ(k) =∝an}b∇acketle{tˆc†
+σ(k)ˆcσ(k)∝an}b∇acket∇i}ht (27)
+where ˆcσ(k) annihilates a particle of spin σin the plane-wave state |k∝an}b∇acket∇i}htdefined by ∝an}b∇acketle{tr|k∝an}b∇acket∇i}ht=eik·r. This corresponds to
+the normalization
+/integraldisplayddk
+(2π)dnσ(k) =Nσ. (28)
+In first quantization,
+nσ(k) =/summationdisplay
+i:σ/integraldisplay/parenleftBig/productdisplay
+l/ne}ationslash=iddrl/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+ddrie−ik·riψ(r1,...,rN)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+(29)
+where the sum is taken over all particles of spin σ, i.e.iruns from 1 ,toN↑forσ=↑and fromN↑+1 toNforσ=↓.
+Three dimensions:
+The key point is that in the large- klimit, the Fourier transform with respect to riis dominated by the contribution
+of the short-distance divergence coming from the contact condit ion (1):
+/integraldisplay
+d3rie−ik·riψ(r1,...,rN)≃
+k→∞/integraldisplay
+d3rie−ik·ri/summationdisplay
+j,j/ne}ationslash=i1
+rijAij(rj,(rk)k/ne}ationslash=i,j). (30)
+From ∆(1/r) =−4πδ(r), we have the identity
+/integraldisplay
+d3re−ik·r1
+r=4π
+k2, (31)
+so that/integraldisplay
+d3rie−ik·riψ(r1,...,rN)≃
+k→∞4π
+k2/summationdisplay
+j,j/ne}ationslash=ie−ik·rjAij(rj,(rl)l/ne}ationslash=i,j). (32)
+Inserting this into (29) and expanding the modulus squared, the cr oss terms vanish in the large- klimit, so that
+C= (4π)2(A,A) (33)
+whereC≡limk→∞k4nσ(k). This can be rewritten using (24) as:
+C=4πm
+/planckover2pi12dE
+d(−1/a), (34)
+in agreement with Tan [62].
+Two dimensions:
+The 2Dcontact condition (2) now gives
+/integraldisplay
+d2rie−ik·riψ(r1,...,rN)≃
+k→∞/integraldisplay
+d2ri,e−ik·ri/summationdisplay
+j,j/ne}ationslash=iln(rij)Aij(rj,(rl)l/ne}ationslash=i,j). (35)
+From ∆(lnr) = 2πδ(r), we have the identity
+/integraldisplay
+d2re−ik·rlnr=−2π
+k2, (36)
+so that/integraldisplay
+d2rie−ik·riψ(r1,...,rN)≃
+k→∞−2π
+k2/summationdisplay
+j,j/ne}ationslash=ie−ik·rjAij(rj,(rl)l/ne}ationslash=i,j). (37)
+As in 3Dthis leads to
+C= (2π)2(A,A) (38)
+whereC≡limk→∞k4nσ(k), and thus from (24):
+C=2πm
+/planckover2pi12dE
+d(lna). (39)11
+3. Short-distance asymptotic behavior of the pair distribu tion function
+The pair distribution function, giving the probability density of finding a spin-↑particle at point R+r/2 and a
+spin-↓particle at point R−r/2, reads [135]
+g(2)
+↑↓/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+=/integraldisplay
+ddr1...ddrN|ψ(r1,...,rN)|2N↑/summationdisplay
+i=1N/summationdisplay
+j=N↑+1δ/parenleftBig
+R+r
+2−ri/parenrightBig
+δ/parenleftBig
+R−r
+2−rj/parenrightBig
+(40)
+=N↑/summationdisplay
+i=1N/summationdisplay
+j=N↑+1/integraldisplay/parenleftBig/productdisplay
+k/ne}ationslash=i,jddrk/parenrightBig/vextendsingle/vextendsingle/vextendsingleψ/parenleftBig
+r1,...,ri=R+r
+2,...,rj=R−r
+2,...,rN/parenrightBig/vextendsingle/vextendsingle/vextendsingle2
+.(41)
+In what follows we consider the spatially integrated pair distribution f unction
+G(2)
+↑↓(r)≡/integraldisplay
+d3Rg(2)
+↑↓/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+. (42)
+Three dimensions:
+Replacing the wavefunction in (41) by its asymptotic behavior given b y the contact condition (1) immediately yields:
+G(2)
+↑↓(r)∼
+r→0(A,A)1
+r2. (43)
+Expressing ( A,A) in terms of Ctrough (33) finally gives:
+G(2)
+↑↓(r)∼
+r→0C
+(4π)21
+r2. (44)
+In a measurement of all particle positions, the total number of pair s of particles of opposite spin which are separated
+by a distance smaller than sis
+Npair(s) =/integraldisplay
+r<sddrG(2)
+↑↓(r) (45)
+so that from (44)
+Npair(s)∼
+s→0C
+4πs, (46)
+as obtained in [61, 62].
+Two dimensions:
+The contact condition (2) similarly leads to
+G(2)
+↑↓(r)∼
+r→0C
+(2π)2ln2r. (47)
+After integration over the region r<sthis gives
+Npair(s)∼
+s→0C
+4πs2ln2s. (48)
+4. Expression of the energy in terms of the momentum distribu tion
+Three dimensions:
+AsshownbyTan[61],thetotalenergyofanyeigenstatehasasimple expressionintermsofthemomentumdistribution:
+E−Etrap=/planckover2pi12C
+4πma+/summationdisplay
+σ/integraldisplayd3k
+(2π)3/planckover2pi12k2
+2m/bracketleftbigg
+nσ(k)−C
+k4/bracketrightbigg
+(49)12
+or equivalently
+E−Etrap= lim
+Λ→∞/bracketleftBigg
+/planckover2pi12C
+4πm/parenleftbigg1
+a−2Λ
+π/parenrightbigg
++/summationdisplay
+σ/integraldisplay
+k<Λd3k
+(2π)3/planckover2pi12k2
+2mnσ(k)/bracketrightBigg
+(50)
+where
+C≡lim
+k→∞k4nσ(k), (51)
+and
+Etrap≡/angbracketleftBiggN/summationdisplay
+i=1U(ri)/angbracketrightBigg
+(52)
+is the trapping potential energy. A simple rederivation of this result is obtained using the lattice model (defined in
+Sec. IIA2): As shown in Section IIC3, one easily obtains Eq. (100), which yields (49) in the zero-range limit since
+D→R3andǫk→/planckover2pi12k2/(2m) forb→0.
+Two dimensions:
+The 2Dversion of (50) is
+E−Etrap= lim
+Λ→∞/bracketleftBigg
+−/planckover2pi12C
+2πmln/parenleftbiggaΛeγ
+2/parenrightbigg
++/summationdisplay
+σ/integraldisplay
+k<Λd2k
+(2π)2/planckover2pi12k2
+2mnσ(k)/bracketrightBigg
+(53)
+as was shown (for a homogeneous system) in [68]. This can easily be r ewritten in the following forms, which resem-
+ble (49):
+E−Etrap=−/planckover2pi12C
+2πmln/parenleftbiggaqeγ
+2/parenrightbigg
++/summationdisplay
+σ/integraldisplayd2k
+(2π)2/planckover2pi12k2
+2m/bracketleftbigg
+nσ(k)−C
+k4θ(k−q)/bracketrightbigg
+for anyq>0, (54)
+where the Heaviside function θensures that the integral converges at small k, or equivalently
+E−Etrap=−/planckover2pi12C
+2πmln/parenleftbiggaqeγ
+2/parenrightbigg
++/summationdisplay
+σ/integraldisplayd2k
+(2π)2/planckover2pi12k2
+2m/bracketleftbigg
+nσ(k)−C
+k2(k2+q2)/bracketrightbigg
+for anyq>0. (55)
+To derive this we again use the lattice model. We note that, if the limit q→0 is replaced by the limit b→0 taken for
+fixeda, Eq. (14) remains true (see App. A); repeating the reasoning of S ection IIC3 then shows that (101) remains
+true; taking the limit b→0 finally gives
+E−Etrap=−/planckover2pi12C
+2πmln/parenleftbiggaqeγ
+2/parenrightbigg
++/summationdisplay
+σ/integraldisplayd2k
+(2π)2/planckover2pi12k2
+2m/bracketleftbigg
+nσ(k)−C
+k2P1
+k2−q2/bracketrightbigg
+(56)
+whereq>0 is arbitrary; this can be rewritten as (53).
+5. Short-distance expansion of the one-body density matrix
+The one-body density matrix is defined as
+g(1)
+σσ/parenleftBig
+R−r
+2,R+r
+2/parenrightBig
+=/angbracketleftBig
+ˆψ†
+σ/parenleftBig
+R−r
+2/parenrightBig
+ˆψσ/parenleftBig
+R+r
+2/parenrightBig/angbracketrightBig
+(57)
+whereˆψσ(r) annihilates a particle of spin σat point r. Let us define a spatially integrated one-body density matrix
+G(1)
+σσ(r)≡/integraldisplay
+ddRg(1)
+σσ/parenleftBig
+R−r
+2,R+r
+2/parenrightBig
+. (58)
+This is related to the momentum distribution by Fourier transformat ion:
+G(1)
+σσ(r) =/integraldisplayddk
+(2π)deik·rnσ(k). (59)13
+Three dimensions:
+As shown below,
+G(1)
+σσ(r) =
+r→0Nσ−C
+8πr+O(r2), (60)
+and moreover the expansion can pushed to second order if one sum s over spin and averages over three orthogonal
+directions of r:
+1
+33/summationdisplay
+i=1/summationdisplay
+σG(1)
+σσ(rui) =
+r→0N−C
+4πr−m
+3/planckover2pi12/parenleftbigg
+E−Etrap−/planckover2pi12C
+4πma/parenrightbigg
+r2+o(r2) (61)
+where the ui’s are three orthogonal unit vectors. This last relation, as well as it s 2Dversion (65), also hold if one
+averages over all directions of runiformly on the unit sphere or unit circle. They generalize the result obtained in 1 D
+in [55], but the derivation is different from the 1 Dcase [136].
+To derive (60,61) we rewrite (59) as
+G(1)
+σσ(r) =Nσ+/integraldisplayd3k
+(2π)3/parenleftbig
+eik·r−1/parenrightbigC
+k4+/integraldisplayd3k
+(2π)3/parenleftbig
+eik·r−1/parenrightbig/parenleftbigg
+nσ(k)−C
+k4/parenrightbigg
+. (62)
+The first integral equals −(C/8π)r. In the second integral, we use
+eik·r−1 =
+r→0ik·r−(k·r)2
+2+o(r2). (63)
+The first term of this expansion gives a contribution to the integral proportional to the total momentum of the gas,
+which vanishes since the eigenfunctions are real. The second term is O(r2), which gives (60). Eq. (61) follows from
+the fact that the contribution of the second term, after averag ing over the directions of r, is given by the integral of
+k2[nσ(k)−C/k4], which is related to the total energy by (49).
+Two dimensions:
+As shown below,
+G(1)
+σσ(r) =
+r→0Nσ+C
+8πr2lnr+O(r2), (64)
+and for any pair of orthogonal unit vectors ( u1,u2)
+1
+22/summationdisplay
+i=1/summationdisplay
+σG(1)
+σσ(rui) =
+r→0N+C
+4πr2/bracketleftbigg
+ln/parenleftBigr
+a/parenrightBig
++F
+32/bracketrightbigg
+−m
+2/planckover2pi12(E−Etrap)r2+o(r2) (65)
+where
+F ≡2F3/parenleftbigg
+1,1;2,3,3;−1
+4/parenrightbigg
+=−16∞/summationdisplay
+i=11
+i[(i+1)!]2/parenleftbigg
+−1
+4/parenrightbiggi
+= 0.98625471.... (66)
+To derive (64,65) we rewrite (59) as
+G(1)
+σσ(r) =Nσ+I(r)+J(r) (67)
+with
+I(r) =/integraldisplayd2k
+(2π)2/parenleftbig
+eik·r−1/parenrightbigC
+k4θ(k−q) (68)
+and
+J(r) =/integraldisplayd2k
+(2π)2/parenleftbig
+eik·r−1/parenrightbig/parenleftbigg
+nσ(k)−C
+k4θ(k−q)/parenrightbigg
+(69)
+whereq>0 is arbitrary and the Heaviside function θensures that the integrals converge.14
+To evaluate I(r) we use standard manipulations to get
+I(r) =C
+8πr2[ln(qr)+4I]+O(r4) (70)
+where
+I=/integraldisplay∞
+1dxJ0(x)−1
+x3, (71)
+J0being a Bessel function. Evaluating this integral with Maple gives
+I=γ−1−ln2
+4+F
+128(72)
+whereFis the hypergeometric function defined in (66).
+Finally we evaluate J(r) using the same procedure as in 3 D: expanding the exponential [see (63)] yields an integral
+which can be related to the total energy thanks to (54).
+6. Second order derivative of the energy with respect to the s cattering length
+We denote by |ψn∝an}b∇acket∇i}htan orthonormal basis of N-body eigenstates which vary smoothly with 1 /a, and byEnthe
+corresponding eigenenergies. We will show that
+1
+2d2En
+d(−1/a)2=/parenleftbigg4π/planckover2pi12
+m/parenrightbigg2/summationdisplay
+n′,En′/ne}ationslash=En|(A(n′),A(n))|2
+En−En′in 3D (73)
+1
+2d2En
+d(lna)2=/parenleftbigg2π/planckover2pi12
+m/parenrightbigg2/summationdisplay
+n′,En′/ne}ationslash=En|(A(n′),A(n))|2
+En−En′in 2D (74)
+where the sum is taken on all values of n′such thatEn′∝ne}ationslash=En. This implies that for the ground state energy E0,
+d2E0
+d(−1/a)2<0 in 3D (75)
+d2E0
+d(lna)2<0 in 2D. (76)
+Eq.(75) was intuitively expected [85]: Eq. (46) shows that dE0/d(−1/a) is proportional to the probability of finding
+two particles very close to each other, and it is natural that this pr obability decreases when one goes from the BEC
+limit (−1/a→ −∞) to the BCS limit ( −1/a→+∞), i.e. when the interactions become less attractive [137]. Eq.(76)
+also agrees with intuition [138].
+For the derivation, it is convenient to use the lattice model (defined in Sec. IIA2): As shown in Sec.IIC4 one easily
+obtains (104), from which the result is deduced as follows. |φ(0)|2is eliminated using (17,18). Then, in 3 D, one uses
+d2En
+d(−1/a)2=d2En
+dg2
+0/parenleftbiggdg0
+d(−1/a)/parenrightbigg2
++dEn
+dg0d2g0
+d(−1/a)2(77)
+where the second term equals 2 g0dEn/d(−1/a)m/(4π/planckover2pi12) and thus vanishes in the zero-range limit. Similarly, in 2 D
+one uses the fact that
+d2En
+d(lna)2=d2En
+dg2
+0/parenleftbiggdg0
+d(lna)/parenrightbigg2
+(78)
+in the zero-range limit.15
+7. Time derivative of the energy
+We now consider the case where the scattering length a(t) and the trapping potential U(r,t) are varied with time.
+The time-dependent version of the zero-range model (see e.g. [86 ]) is given by Schr¨ odinger’s equation
+i/planckover2pi1∂
+∂tψ(r1,...,rN;t) =H(t)ψ(r1,...,rN;t) (79)
+when all particle positions are distinct, with
+H(t) =N/summationdisplay
+i=1/bracketleftbigg
+−/planckover2pi12
+2m∆ri+U(ri,t)/bracketrightbigg
+, (80)
+and by the contact condition (1) in 3 Dor (2) in 2Dfor the scattering length a=a(t). One then has the relations
+dE
+dt=/planckover2pi12C
+4πmd(−1/a)
+dt+∝an}b∇acketle{tψ(t)|N/summationdisplay
+i=1∂tU(ri,t)|ψ(t)∝an}b∇acket∇i}htin3D (81)
+dE
+dt=/planckover2pi12C
+2πmd(lna)
+dt+∝an}b∇acketle{tψ(t)|N/summationdisplay
+i=1∂tU(ri,t)|ψ(t)∝an}b∇acket∇i}htin2D, (82)
+whereE(t) =∝an}b∇acketle{tψ(t)|H(t)|ψ(t)∝an}b∇acket∇i}htis the total energy and Etrap(t) =∝an}b∇acketle{tψ(t)|/summationtextN
+i=1U(ri,t)|ψ(t)∝an}b∇acket∇i}htis the trapping potential
+energy. The relation (81) was first obtained by Tan [62]. A very simple derivation of these relations using the lattice
+model is given in Section IIC5. Here we give a derivation within the zero -range model.
+Three dimensions:
+We first note that the generalizationof the lemma (22) to the case o f two Hamiltonians H1andH2with corresponding
+trapping potentials U1(r) andU2(r) reads:
+∝an}b∇acketle{tψ1,H2ψ2∝an}b∇acket∇i}ht−∝an}b∇acketle{tH1ψ1,ψ2∝an}b∇acket∇i}ht=4π/planckover2pi12
+m/parenleftbigg1
+a1−1
+a2/parenrightbigg
+(A(1),A(2))+∝an}b∇acketle{tψ1|N/summationdisplay
+i=1[U2(ri,t)−U1(ri,t)]|ψ2∝an}b∇acket∇i}ht.(83)
+Applying this relation for |ψ1∝an}b∇acket∇i}ht=|ψ(t)∝an}b∇acket∇i}htand|ψ2∝an}b∇acket∇i}ht=|ψ(t+δt)∝an}b∇acket∇i}ht[and correspondingly a1=a(t),a2=a(t+δt) and
+H1=H(t),H2=H(t+δt)] gives:
+∝an}b∇acketle{tψ(t),H(t+δt)ψ(t+δt)∝an}b∇acket∇i}ht−∝an}b∇acketle{tH(t)ψ(t),ψ(t+δt)∝an}b∇acket∇i}ht=4π/planckover2pi12
+m/parenleftbigg1
+a(t)−1
+a(t+δt)/parenrightbigg
+(A(t),A(t+δt))
++∝an}b∇acketle{tψ(t)|N/summationdisplay
+i=1U(ri,t+δt)−U(ri,t)|ψ(t+δt)∝an}b∇acket∇i}ht.(84)
+Dividing by δt, taking the limit δt→0, and using the expression (33) of ( A,A) in terms of C, the right-hand-side of
+(84) reduces to the right-hand-side of (81). The left-hand-side of (84) can be rewritten using Schr¨ odinger’s equation
+asi/planckover2pi1d∝an}b∇acketle{tψ(t)|ψ(t+δt)∝an}b∇acket∇i}ht/dt. Using Schr¨ odinger’s equation again to Taylor expand |ψ(t+δt)∝an}b∇acket∇i}htin this last expression
+finally gives the result (81).
+Two dimensions:
+The relation (82) is derived similarly from the lemma
+∝an}b∇acketle{tψ1,H2ψ2∝an}b∇acket∇i}ht−∝an}b∇acketle{tH1ψ1,ψ2∝an}b∇acket∇i}ht=2π/planckover2pi12
+mln(a2/a1)(A(1),A(2))+∝an}b∇acketle{tψ1|N/summationdisplay
+i=1[U2(ri,t)−U1(ri,t)]|ψ2∝an}b∇acket∇i}ht. (85)
+C. Relations for the lattice model
+In this Section, as well as in Section IID, it will prove convenient to defineCby
+C≡4πm
+/planckover2pi12dE
+d(−1/a)in 3D (86)
+C≡2πm
+/planckover2pi12dE
+d(lna)in 2D. (87)16
+This new definition of Ccoincides with the identities (33) and (38) of Section IIB in the zero- range limit, as follows
+from (24,26).
+We will use the following lemma: For any wavefunctions ψ,ψ′,
+∝an}b∇acketle{tψ′|W|ψ∝an}b∇acket∇i}ht=|φ(0)|2(A′,A) (88)
+whereAandA′are the regular parts related to ψandψ′through (19), and the scalar product between regular parts
+is naturally defined as the discrete version of (23):
+(A′,A)≡/summationdisplay
+i<j/summationdisplay
+(rk)k/negationslash=i,j/summationdisplay
+Rijb(N−1)dA′∗
+ij(Rij,(rk)k/ne}ationslash=i,j)Aij(Rij,(rk)k/ne}ationslash=i,j). (89)
+The lemma simply follows from
+∝an}b∇acketle{tψ′|W|ψ∝an}b∇acket∇i}ht=/summationdisplay
+i<j/summationdisplay
+(rk)k/negationslash=i,jb(N−2)d(ψ′∗ψ)(r1,...,ri=rj,...,rj,...,rN). (90)
+1. First order derivative of the energy with respect to the sc attering length
+The Hellmann-Feynman theorem gives
+dE
+dg0=∝an}b∇acketle{tψ|dH
+dg0|ψ∝an}b∇acket∇i}ht=∝an}b∇acketle{tψ|W|ψ∝an}b∇acket∇i}ht. (91)
+But lemma (88) with ψ′=ψwrites
+∝an}b∇acketle{tψ|W|ψ∝an}b∇acket∇i}ht=|φ(0)|2(A,A). (92)
+Using the expressions (17,18) of |φ(0)|2, we conclude that
+dE
+d(−1/a)=4π/planckover2pi12
+m(A,A) in 3D (93)
+dE
+d(lna)=2π/planckover2pi12
+m(A,A) in 2D. (94)
+2. Interaction energy
+The left-hand-sideof(92) is obviouslyequaltothe mean interactio nenergyEintdivided byg0; in the right-hand-side
+of (92), (A,A) can be expressed in terms of Cusing (93,94) and the definition (86,87) of C:
+(A,A) =C
+(4π)2in 3D (95)
+(A,A) =C
+(2π)2in 2D. (96)
+This gives
+Eint
+g0=C
+(4π)2|φ(0)|2in 3D (97)
+Eint
+g0=C
+(2π)2|φ(0)|2in 2D. (98)
+Eliminating φ(0) thanks to (15,16) finally gives
+Eint=C/parenleftbigg/planckover2pi12
+m/parenrightbigg21
+g0(99)
+both in 3Dand 2D.17
+3. Relation between energy, momentum distribution and C
+Here we show that
+E−Etrap=/planckover2pi12C
+4πma+/summationdisplay
+σ/integraldisplay
+Dd3k
+(2π)3ǫk/bracketleftBigg
+nσ(k)−C/parenleftbigg/planckover2pi12
+2mǫk/parenrightbigg2/bracketrightBigg
+in 3D (100)
+E−Etrap= lim
+q→0−/planckover2pi12C
+2πmln/parenleftbiggaqeγ
+2/parenrightbigg
++/summationdisplay
+σ/integraldisplay
+Dd2k
+(2π)2ǫk/bracketleftbigg
+nσ(k)−C/planckover2pi12
+2mǫkP/planckover2pi12
+2m(ǫk−ǫq)/bracketrightbigg
+in 2D.(101)
+To derive this we start from the expression (99) of the interaction energy and eliminate 1 /g0thanks to (13,14). The
+desired quantity E−Etrap=Eint+Ekinis then obtained from
+Ekin=/summationdisplay
+σ/integraldisplay
+Dddk
+(2π)dǫknσ(k). (102)
+4. Second order derivative of the energy with respect to the c oupling constant
+We denote by |ψn∝an}b∇acket∇i}htan orthonormal basis of N-body eigenstates which vary smoothly with g0, and byEnthe
+corresponding eigenenergies. We apply second order perturbatio n theory to determine how an eigenenergy varies for
+an infinitesimal change of g0. This gives:
+1
+2d2En
+dg2
+0=/summationdisplay
+n′,En′/ne}ationslash=En|∝an}b∇acketle{tψn′|W|ψn∝an}b∇acket∇i}ht|2
+En−En′, (103)
+where the sum is taken over all values of n′such thatEn′∝ne}ationslash=En. Lemma (88) then yields:
+1
+2d2En
+dg2
+0=|φ(0)|4/summationdisplay
+n′,En′/ne}ationslash=En|(A(n′),A(n))|2
+En−En′. (104)
+5. Time derivative of the energy
+Equations (81,82) remain exact for the lattice model. Indeed, the H ellmann-Feynman theorem gives
+dE
+dt=∝an}b∇acketle{tdH
+dt∝an}b∇acket∇i}ht=dg0
+dt∝an}b∇acketle{tW∝an}b∇acket∇i}ht+∝an}b∇acketle{tN/summationdisplay
+i=1∂tU(ri,t)∝an}b∇acket∇i}ht. (105)
+The result then follows by using the lemma (88), the expressions (17 ,18) of|φ(0)|2, and the expressions (95,96) of
+(A,A) in terms of C.
+6. On-site pair distribution function
+We define the spatially integrated pair distribution function
+G(2)
+↑↓(r)≡/summationdisplay
+Rbdg(2)
+↑↓/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+. (106)
+Using (97,98) and expressing the interaction energy in terms of g(2)
+↑↓thanks to the second-quantized form (9) yields:
+G(2)
+↑↓(0) =C
+(4π)2|φ(0)|2in 3D (107)
+G(2)
+↑↓(0) =C
+(2π)2|φ(0)|2in 2D. (108)18
+7. Pair distribution function at short distances
+The last result can be generalized to finite but small r, assuming that we are in the zero-range regime ktypb≪1
+(introduced at the end of Sec. IIA2):
+G(2)
+↑↓(r)≃
+r≪1/ktypC
+(4π)2|φ(r)|2in 3D (109)
+G(2)
+↑↓(r)≃
+r≪1/ktypC
+(2π)2|φ(r)|2in 2D. (110)
+Indeed, the expression (41) of g(2)
+↑↓in terms of the wavefunction is valid for the lattice model with the obv ious
+replacement of the integrals by sums, so that
+G(2)
+↑↓(r) =/summationdisplay
+RbdN↑/summationdisplay
+i=1N/summationdisplay
+j=N↑+1/summationdisplay
+(rk)k/negationslash=i,jb(N−2)d/vextendsingle/vextendsingle/vextendsingleψ/parenleftBig
+r1,...,ri=R+r
+2,...,rj=R−r
+2,...,rN/parenrightBig/vextendsingle/vextendsingle/vextendsingle2
+.(111)
+Forr≪1/ktyp, we can replace ψby the short-distance expression (20), assuming that the multiple sum is dominated
+by the configurations where all the distances |rk−R|andrkk′are much larger than b:
+G(2)
+↑↓(r)≃(A,A)|φ(r)|2. (112)
+Expressing ( A,A) in terms of Cthanks to (95,96) gives (109,110).
+8. Momentum distribution at large momenta
+Assuming again that we are in the zero-range regime ktypb≪1, we will show that
+nσ(k)≃
+k≫ktypC/parenleftbigg/planckover2pi12
+2mǫk/parenrightbigg2
+(113)
+both in 3Dand in 2D. We start from
+nσ(k) =/summationdisplay
+i:σ/summationdisplay
+(rl)l/negationslash=ibd(N−1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
+ribde−ik·riψ(r1,...,rN)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+. (114)
+We are interested in the limit k≫ktyp. Sinceψ(r1,...,rN) is a function of riwhich varies on the scale of 1 /ktyp,
+except when riis close to another particle rjwhere it varies on the scale of b, we can replace ψby its short-distance
+form (20):
+/summationdisplay
+ribde−ik·riψ(r1,...,rN)≃˜φ(k)/summationdisplay
+j,j/ne}ationslash=ie−ik·rjAij(rj,(rl)l/ne}ationslash=i,j), (115)
+where˜φ(k) =∝an}b∇acketle{tr|φ∝an}b∇acket∇i}ht=/summationtext
+rbde−ik·rφ(r). Here we excluded the configurations where more than two partic les are at
+distances /lessorsimilarb, which are expected to have a negligible contribution to (114). Inse rting (115) into (114), expanding
+the modulus squared, and neglecting the cross-product terms in t he limitk≫ktyp, we obtain
+nσ(k)≃ |˜φ(k)|2(A,A). (116)
+Finally, ˜φ(k) is easily computed for the lattice model: for k∝ne}ationslash= 0, the two-body Schr¨ odinger equation (A1) directly
+gives˜φ(k) =−g0φ(0)/(2ǫk), andφ(0) is given by (15,16), which yields (113).
+D. Relations for a finite-range interaction in continuous sp ace
+We recall that in this Section, Cis again defined by (86,87).19
+1. Interaction energy
+As for the lattice model, we find that the interaction energy is propo rtional toC:
+Eint=C
+(4π)2/integraldisplay
+d3rV(r)|φ(r)|2in 3D (117)
+Eint=C
+(2π)2/integraldisplay
+d2rV(r)|φ(r)|2in 2D. (118)
+It was shown in [65] that this relation is asymptotically valid in the zero- range limit in 3 D. Here we show that it
+remains exact for any finite value of the range and we generalize it to 2D.
+For the derivation, we set
+V(r) =g0W(r) (119)
+whereg0is a dimensionless coupling constant which allows to tune a. The Hellmann-Feynman theorem then gives
+Eint=g0dE/dg 0. The result then follows by writing dE/dg 0=dE/d(−1/a)·d(−1/a)/dg0in 3DanddE/dg 0=
+dE/d(lna)·d(lna)/dg0in 2D, and by using the definition (86,87) of Cas well as the following lemmas:
+g0d(−1/a)
+dg0=m
+4π/planckover2pi12/integraldisplay
+d3rV(r)|φ(r)|2in 3D (120)
+g0d(lna)
+dg0=m
+2π/planckover2pi12/integraldisplay
+d2rV(r)|φ(r)|2in 2D. (121)
+To derive these lemmas, we consider two values of the scattering len gthai, i= 1,2, and the corresponding
+scattering states φiand coupling constants g0,i. The corresponding two-particle relative-motion Hamiltonians are
+Hi=−(/planckover2pi12/m)∆r+g0,iW(r). SinceHiφi= 0, we have
+lim
+R→∞/integraldisplay
+r<Rddr(φ1H2φ2−φ2H1φ1) = 0. (122)
+The contribution of the kinetic energies can be computed from Ostr ogradsky’s theorem and the large-distance form
+ofφ[139]. The contribution of the potential energies is proportional to g0,2−g0,1. Taking the limit a2→a1gives
+the results (120,121). Lemma (120) was also used in [65] and the abo ve derivation is essentially identical to the one
+of [65]. For this 3 Dlemma, there also exists an alternative derivation based on the two- body problem in a large
+box [140].
+2. Relation between energy and momentum distribution
+Three dimensions:
+E−Etrap=/planckover2pi12C
+4πma+/summationdisplay
+σ/integraldisplayd3k
+(2π)3/planckover2pi12k2
+2m/bracketleftbigg
+nσ(k)−C
+(4π)2|˜φ′(k)|2/bracketrightbigg
+(123)
+where˜φ′(k) =˜φ(k)+a−1(2π)3δ(k) =/integraltext
+d3re−ik·rφ′(r) with
+φ′(r) =φ(r)+1
+a. (124)
+This is simply obtained by adding the kinetic energy to (117) and by usin g the lemma:
+/integraldisplay
+d3rV(r)|φ(r)|2=4π/planckover2pi12
+ma−/integraldisplayd3k
+(2π)3/planckover2pi12k2
+m|˜φ′(k)|2. (125)
+To derive this lemma, we start from Schr¨ odinger’s equation −(/planckover2pi12/m)∆φ+V(r)φ= 0, which implies
+/integraldisplay
+d3rV(r)|φ(r)|2=/planckover2pi12
+m/integraldisplay
+d3rφ∆φ. (126)20
+On the other hand, applying Ostrogradsky’s theorem over the sph ere of radius R, using the asymptotic expression
+(11) ofφand taking the limit R→ ∞yields
+/integraldisplay
+d3rφ∆φ=4π
+a−/integraldisplay
+d3r(∇φ)2. (127)
+We then replace ∇φby∇φ′. Applying the Parseval-Plancherel relation to ∂iφ, and using the fact that φ′(r) vanishes
+at infinity, we get:
+/integraldisplay
+d3r(∇φ′)2=/integraldisplayd3k
+(2π)3k2|˜φ′(k)|2(128)
+The desired result (125) follows.
+Two dimensions:
+E−Etrap= lim
+R→∞/braceleftBigg
+/planckover2pi12C
+2πmln/parenleftbiggR
+a/parenrightbigg
++/summationdisplay
+σ/integraldisplayd2k
+(2π)2/planckover2pi12k2
+2m/bracketleftbigg
+nσ(k)−C
+(2π)2|˜φ′
+R(k)|2/bracketrightbigg/bracerightBigg
+(129)
+where˜φ′
+R(k) =/integraltext
+d3re−ik·rφ′
+R(r) with
+φ′
+R(r) = [φ(r)−ln(R/a)]θ(R−r). (130)
+This follows from (118) and from the lemma:
+/integraldisplay
+d2rV(r)|φ(r)|2= lim
+R→∞/braceleftbigg2π/planckover2pi12
+mln/parenleftbiggR
+a/parenrightbigg
+−/integraldisplayd2k
+(2π)2/planckover2pi12k2
+m|˜φ′
+R(k)|2/bracerightbigg
+. (131)
+The derivation of this lemma again starts with (126). Ostrogradsky ’s theorem then gives [139]
+/integraldisplay
+d2rφ∆φ= lim
+R→∞/braceleftbigg
+2πln/parenleftbiggR
+a/parenrightbigg
+−/integraldisplay
+r<Rd2r(∇φ)2/bracerightbigg
+. (132)
+We can then replace/integraltext
+r<Rd2r(∇φ)2by/integraltext
+d2r(∇φ′
+R)2, sinceφ′
+R(r) is continuous at r=R[139] so that ∇φ′
+Rdoes
+not contain any delta distribution. The Parseval-Plancherel relatio n can be applied to ∂iφ′
+R, since this function is
+square-integrable. Then, using the fact that φ′
+R(r) vanishes at infinity, we get
+/integraldisplay
+d2r(∇φ′
+R)2=/integraldisplayd2k
+(2π)2k2|˜φ′
+R(k)|2, (133)
+and the lemma (131) follows.
+3. Pair distribution function at short distances
+In the zero-range limit ktypb≪1, the short-distance behavior of the pair distribution function is g iven by the same
+expressions (109,110) as for the lattice model. Indeed, Eq.(112) is derived in the same way as for the lattice model;
+one can then use the zero-range model’s expressions (33,38) of ( A,A) in terms of C, since the finite range model’s
+quantitiesCandAtend to the zero-range model’s ones in the zero-range limit.
+4. Momentum distribution at large momenta
+In the zero-range regime ktypb≪1 the momentum distribution at large momenta k≫ktypis given by
+nσ(k)≃C
+(4π)2|˜φ(k)|2in 3D (134)
+nσ(k)≃C
+(2π)2|˜φ(k)|2in 2D. (135)
+Indeed, Eq.(116) is derived as for the lattice model, and ( A,A) can be expressed in terms of Cas in Subsec.IID3.21
+E. Derivative of the energy with respect to the effective rang e
+We now show that, in 3 D, the leading order finite-range correction to the zero-range mod el’s spectrum is given by
+the model-independent expression
+/parenleftbigg∂E
+∂re/parenrightbigg
+a= 2π/summationdisplay
+i<j/integraldisplay
+d3R/integraldisplay/parenleftBig/productdisplay
+k/ne}ationslash=i,jd3rk/parenrightBig
+Aij(R,(rk)k/ne}ationslash=i,j)
+E+/planckover2pi12
+4m∆R+/planckover2pi12
+2m/summationdisplay
+k/ne}ationslash=i,j∆rk−N/summationdisplay
+l=1U(rl)
+Aij(R,(rk)k/ne}ationslash=i,j)
+(136)
+wherereis the effective range of the interaction potential, the derivative is t aken inre= 0, the function Ais assumed
+to be real without loss of generality, and in the sum over lwe have set ri=rj=R. To obtain this result we use a
+modified version of the zero-range model, where the boundary con dition (1) is replaced by
+ψ(r1,...,rN) =
+rij→0/parenleftbigg1
+rij−1
+a+m
+2/planckover2pi12Ere/parenrightbigg
+Aij(Rij,(rk)k/ne}ationslash=i,j)+O(rij), (137)
+where
+E=E−2U(Rij)−
+/summationdisplay
+k/ne}ationslash=i,jU(rk)
++1
+Aij(Rij,(rk)k/ne}ationslash=i,j)
+/planckover2pi12
+4m∆R+/planckover2pi12
+2m/summationdisplay
+k/ne}ationslash=i,j∆rk
+Aij(Rij,(rk)k/ne}ationslash=i,j).(138)
+Equations (137,138) generalize the ones already used for 3 bosons in free space in [87, 88] (the predictions of [87] and
+[88] have been confirmed using different approaches, see [89] and R efs. therein, and [90, 91] respectively). Such a
+model was also used in the two-body case, see e.g. [92–94], and the modified scalar product that makes it hermitian
+was constructed in [95].
+For the derivation of (136), we consider an eigenstate ψ1of the zero-range model, satisfying the boundary condition
+(1) with a scattering length aand a regular part A(1), and the corresponding finite-range eigenstate ψ2satisfying
+(137,138) with the same scattering length aand a regular part A(2). As in App. B we get (B3), as well as (B6) with
+1/a1−1/a2replaced by mEre/(2/planckover2pi12). This yields (136).
+F. Generalization to statistical mixtures and to thermodyn amic equilibrium in the canonical ensemble
+The above results, summarized in Tables II, III and IV, hold for any eigenstate (apart from lines 8-9 of Tab. II
+and line 11 of Tab. III). Thus they can be generalized straightforw ardly to statistical mixtures of eigenstates. The
+relation for the time derivative of E(Tab. II line 9) holds for any time-evolving pure state, and thus also for any
+time-evolving statistical mixture.
+We turn to the case of thermal equilibrium in the canonical ensemble. We shall use the notation
+λ≡/braceleftBigg
+−1/ain 3D
+1
+2lnain 2D.(139)
+a. First order derivative of E.The thermal average in the canonical ensemble dE/dλcan be rewritten in the
+following more familiar way, as detailed in Appendix C:
+/parenleftbiggdE
+dλ/parenrightbigg
+=/parenleftbiggdF
+dλ/parenrightbigg
+T=/parenleftbiggdU
+dλ/parenrightbigg
+S(140)
+where(...) is the canonical thermal average, Fis the free energy, U=¯Eis the mean energy and Sis the entropy.
+Taking the thermal average of (24,26) thus gives
+/parenleftbiggdF
+dλ/parenrightbigg
+T=/parenleftbiggdU
+dλ/parenrightbigg
+S=4π/planckover2pi12
+m(A,A) (141)
+The other results (44,47,34,39,49,53) are generalized to finite tempe ratures in the same way.22
+b. Second order derivative of E.Taking a thermal average of the expression (73,74) we get after a simple ma-
+nipulation:
+1
+2/parenleftbiggd2E
+dλ2/parenrightbigg
+=/parenleftbigg4π/planckover2pi12
+m/parenrightbigg21
+2Z/summationdisplay
+n,n′;En/ne}ationslash=En′e−βEn−e−βEn′
+En−En′|(A(n′),A(n))|2(142)
+whereZ=/summationtext
+nexp(−βEn). This implies
+/parenleftbiggd2E
+dλ2/parenrightbigg
+<0. (143)
+Moreover one can check that
+/parenleftbiggd2F
+dλ2/parenrightbigg
+T−/parenleftbiggd2E
+dλ2/parenrightbigg
+=−β/bracketleftBigg/parenleftbiggdE
+dλ/parenrightbigg2
+−/parenleftbiggdE
+dλ/parenrightbigg2/bracketrightBigg
+<0, (144)
+which implies
+/parenleftbiggd2F
+dλ2/parenrightbigg
+T<0. (145)
+In usual cold atom experiments, however, there is no thermal res ervoir imposing a fixed temperature to the gas, one
+rather can achieve adiabatic transformations by a slow variation of the scattering length of the gas [96–100]. One also
+more directly accesses the mean energy Uof the gas rather than its free energy, even if the entropy is also m easurable
+[35]. The second order derivative of Uwith respect to λfor a fixed entropy is thus the relevant quantity to consider
+[101]. As shown in the appendix C one has in the canonical ensemble:
+/parenleftbiggd2U
+dλ2/parenrightbigg
+S=/parenleftbiggd2E
+dλ2/parenrightbigg
++/bracketleftbig
+Cov/parenleftbig
+E,dE
+dλ/parenrightbig/bracketrightbig2−Var(E)Var/parenleftbigdE
+dλ/parenrightbig
+kBTVar(E). (146)
+where Var(X) and Cov(X,Y) stand for the variance of the quantity Xand the covariance of the quantities Xand
+Yin the canonical ensemble, respectively. From the Cauchy-Schwar z inequality [Cov( X,Y)]2≤Var(X)Var(Y), and
+from the inequality (143), we thus conclude that
+/parenleftbiggd2U
+dλ2/parenrightbigg
+S<0. (147)
+To be complete, we also consider the process where λis varied so slowly that there is adiabaticity in the many-body
+quantum mechanical sense: The adiabatic theorem of quantum mec hanics [102] implies that in the limit where λis
+changed infinitely slowly, the occupation probabilitiesof each eigensp aceofthe many-body Hamiltonian do not change
+with time, even in presence of level crossings [103]. We note that this may require macroscopically long evolution
+times for a large system. For an initial equilibrium state in the canonica l ensemble, the mean energy then varies with
+λas
+Equant
+adiab(λ) =/summationdisplay
+ne−β0En(λ0)
+Z0En(λ) (148)
+where the subscript 0 refers to the initial state. Taking the secon d order derivative of (148) with respect to λin
+λ=λ0gives
+d2Equant
+adiab
+dλ2=/parenleftbiggd2E
+dλ2/parenrightbigg
+<0. (149)
+Finally, we compare the result of isentropic transformation (146) t o the one of the adiabatic transformation in the
+quantum sense (149). They differ by the second term in the right ha nd side of (146). A priori this term is extensive,
+and thus not negligible compared to the first term. We have explicitly c hecked this expectation for the Bogoliubov
+model Hamiltonian of a weakly interacting Bose gas. The discrepancy between (146) and (149) indicates that the
+limit of infinitely slow transformation does not commute with the therm odynamic limit. More explicitly, we see that
+ifλis varied so slowly that the quantum adiabaticity (148) is achieved, on e cannot assume any more that the system
+follows a sequence of thermal equilibrium states with a constant ent ropy; in practice this may require evolution times
+which grow exponentially with the system size.23
+III. SPINLESS BOSONS
+The wavefunction ψ(r1,...,rN) is now completely symmetric.
+Our results for bosons are shown in Table V. An obvious difference wit h the fermionic case is that there are no more
+spin indices in the pair distribution function g(2)and in the momentum distribution n(k). Accordingly, Eqs. (40,28)
+are replaced by [141]
+g(2)/parenleftBig
+R+r
+2,R−r
+2/parenrightBig
+=/integraldisplay
+dr1...drN|ψ(r1,...,rN)|2/summationdisplay
+i/ne}ationslash=jδ/parenleftBig
+R+r
+2−ri/parenrightBig
+δ/parenleftBig
+R−r
+2−rj/parenrightBig
+(150)
+and
+/integraldisplayddk
+(2π)dn(k) =N. (151)
+Animportantdifferencewiththefermioniccaseisthatin3 D, theEfimoveffectoccurs[71], andthezero-rangemodel
+is defined not only by the contact condition (1) and the Schr¨ odinge r equation (3), but also by a boundary condition
+in the limit where three particles approach each other: There exists a functionB, hereafter called three-body regular
+part, such that
+ψ(r1,...,rN)∼
+R→01
+R2sin/bracketleftbigg
+|s0|lnR
+Rt/bracketrightbigg
+Φ(Ω)B(C,r4,...,rN). (152)
+whereRtis the so-called three-body parameter and is an additional paramet er of the zero-range model; C= (r1+
+r2+r3)/3 is the center of mass of particles 1,2 and 3; RandΩare the hyperradius and the hyperangles associated
+with particles 1,2 and 3 [142] We recall the definition of RandΩ: The Jacobi coordinates are defined by r=r2−r1
+andρ= (2r3−r1−r2)/√
+3; then,R≡/radicalbig
+(r2+ρ2)/2, andΩ≡(α,ˆr,ˆρ) withα≡arctan(r/ρ), ˆr≡r/rand
+ˆρ≡ρ/ρ.s0=i·1.00624...is Efimov’s transcendental number, it is the imaginary solution of (19 2).Φ(Ω) is the
+normalized hyperangular part of an Efimov states’ wavefunction: Φ(Ω) =φs0(Ω)//radicalbig
+(φs0|φs0) [142]. We recall that, in
+the present case (bosons with zero total angular momentum), φs0(Ω)≡(1+Q)sin/bracketleftbig
+s0/parenleftbigπ
+2−α/parenrightbig/bracketrightbig
+/[√
+4πsin(2α)] where
+Q=P13+P23andPijexchangesparticles iandj. The hyperangularscalarproduct is defined by( φ|φ) =/integraltext
+dΩ|φ(Ω)|2
+with/integraltext
+dΩ≡2/integraltextπ/2
+0dαsin2(2α)/integraltext
+dˆr/integraltext
+dˆρ, wheredˆranddˆρarethe differential solid angles. Its value is givenin App. D.
+ForN= 3 it is well established that this model is self-adjoint and that it is the zero-range limit of finite-range
+models, see e.g. [75] and references therein. A recent numerical s tudy of several finite range models predicted for
+N= 4 the existence of tetramers of energies weakly depending on the model for a fixed three-body parameter [104],
+anexistence confirmedexperimentally[105]. This numericalstudy, t ogetherwith otherones[106, 107] claimthat there
+is no need to introduce a four-body parameter in the zero-range lim it, implying that the here considered zero-range
+model is self-adjoint for N= 4. Here we consider an arbitrary value of Nsuch that the model is self-adjoint.
+The main relations for zero-range interactions are displayed in Table V. Moreover we note that the relations for
+finite-range interactions, given in Tables III and IV for fermions, c an be easily generalized to the bosonic case.
+A. Relations which are analogous to the fermionic case
+The derivations of all relations of Table V are completely analogous to the fermionic case, except for lines 4 and 6
+of the left column (3D case).
+The first result in Table V was first obtained in [75] in the case N= 3. A simple way to derive it for any Nis to
+use the lattice model and to apply the reasoning of Sec.IIC1. The ke y point is that in the limit of a lattice spacing b
+much smaller than |a|, the three-body parameter corresponding to the lattice model is equal to a numerical constant
+timesb[143]. Thus, varying the coupling constant g0while keeping bfixed is equivalent to varying awhile keeping
+Rtfixed:
+dE
+dg0=/parenleftbiggdE
+d(−1/a)/parenrightbigg
+Rtd(−1/a)
+dg0(153)
+where the lattice model’s ( dE/d(−1/a))Rttends to the zero-range model’s one if one takes the zero-range lim it while
+keepingRtfixed [144]. The same reasoning explains why the second derivative in t he last line of Table V also has
+to be taken for a fixed Rt.24
+B. Derivative of the energy with respect to the three-body pa rameter
+The Efimov effect also gives rise to the following new relation between t he derivative of the energy with respect to
+the three-body parameter and the three-body regular part:
+/parenleftbigg∂E
+∂lnRt/parenrightbigg
+a=/planckover2pi12
+m√
+3
+32|s0|2N(N−1)(N−2)/integraldisplay
+dC/integraldisplay
+dr4...drN|B(C,r4,...,rN)|2. (154)
+This is similar to the relation (24) between the derivative with respect to the scattering length and the (two-body)
+regular part [145]. We will first derive this relation using the zero-ran ge model in the case N= 3, and then using a
+lattice model for any N.
+1. Derivation using the zero-range model for three particle s
+We consider two wavefunctions ψ1,ψ2, satisfying the two-body boundary condition (1) with the same sca ttering
+lengtha, and satisfying the three-body boundary condition with different t hree-body parameters Rt1,Rt2. The
+corresponding three-body regular parts are denoted by B1,B2. We show in the App. E that
+∝an}b∇acketle{tψ1,Hψ2∝an}b∇acket∇i}ht−∝an}b∇acketle{tHψ1,ψ2∝an}b∇acket∇i}ht=/planckover2pi12
+m3√
+3|s0|
+16sin/bracketleftbigg
+|s0|lnRt2
+Rt1/bracketrightbigg/integraldisplay
+dCB∗
+1(C)B2(C), (155)
+which yields (154) by choosing ψias an eigenstate of energy Eiand taking the limit Rt2→Rt1. We note that ψ1and
+ψ2do not satisfy lemma (22) because they are too singular for R→0.
+2. Derivation using a lattice model
+We now rederive (154) using a lattice model which is analogous to the m odel defined in Sec.IIA2, except that the
+Hamiltonian now contains a three-body interaction term:
+H=/integraldisplay
+Dd3k
+(2π)3ǫkc†(k)c(k) +/summationdisplay
+rb3U(r)(ψ†ψ)(r)+g0/summationdisplay
+rb3(ψ†ψ†ψψ)(r)+h0/summationdisplay
+rb3(ψ†ψ†ψ†ψψψ)(r).(156)
+The scattering length is related to g0as in Sec.IIA2. We define the zero-energy three-body scattering state
+φ0(r1,r2,r3) as the solution of H|φ0∝an}b∇acket∇i}ht= 0 fora=∞, with the boundary condition
+φ0(r1,r2,r3)∼1
+R2sin/bracketleftbigg
+|s0|lnR
+Rt/bracketrightbigg
+Φ(Ω) (157)
+in the limit where all interparticle distances tend to infinity. This define s the three-body parameter Rt(b,h0) for
+the lattice model. In order to derive (154) for any desired values of aandRt, we first choose bin such a way that
+Rt(b,h0= 0) is equal to the desired Rtdivided by enπ/|s0|, the zero-range limit n→ ∞being taken in the end. We
+then choose g0to reproduce the desired value of a. We then change h0to a small non-zero value, keeping fixed band
+g0(and thus also a). The Hellman-Feynman theorem writes:
+∂E
+∂h0=/summationdisplay
+rb3∝an}b∇acketle{t(ψ†ψ†ψ†ψψψ)(r)∝an}b∇acket∇i}ht=N(N−1)(N−2)/summationdisplay
+r4,...,rNb3(N−3)|ψ(r,r,r,r4,...,rN)|2.(158)
+For the lattice model we define the three-body regular part Bthrough:
+ψ(r,r,r,r4,...,rN) =φ0(0,0,0)B(r,r4,...,rN); (159)
+in the zero-range limit, we expect that this lattice model’s regular par t tends to the regular part of the zero-range
+model defined in (152), aswasdiscussedforthe two-bodyregular partAin Sec. IIA2. We thus have, in the zero-range
+limit:/parenleftbigg∂E
+∂(lnRt)/parenrightbigg
+a=N(N−1)(N−2)|φ0(0,0,0)|2/parenleftbigg∂h0
+∂(lnRt)/parenrightbigg
+b/integraldisplay
+d3rd3r4...d3rN|B(r,r4,...,rN)|2.(160)
+It remains to evaluate the expression between brackets: This is ac hieved by applying (160) to the case of an Efimov
+trimer in free space, where the regular part can be deduced from t he expression for the normalized wavefunction given
+in App. D.25
+C. Momentum distribution for an Efimov trimer
+For an Efimov trimer state at rest, for an infinite scattering length , we show in Appendix F that the atomic
+momentum distribution has the asymptotic expansion
+n(k) =
+k→∞C
+k4+D
+k5cos/bracketleftBig
+2|s0|ln(k√
+3/κ0)+ϕ/bracketrightBig
++... (161)
+wheres0=i·1.00624...solves
+scos(sπ/2)−8/√
+3sin(sπ/6) = 0, (162)
+the energy of the trimer Etrim=−/planckover2pi12κ2
+0/mdepends on the three-body parameter Rtas specified in (D1), and the
+quantitiesC,Dandϕare derived in the appendix F. The crucial point is that D∝ne}ationslash= 0: The momentum distribution
+has a slowly decaying oscillatory subleading tail.
+The calculations performed in appendix F also allow a straightforward numerical calculation of the atomic mo-
+mentum distribution for an Efimov trimer, both for low values of k, see Fig.1a, and for high values of k, see Fig.1b
+showing how n(k) approaches the asymptotic behavior (161). We have also derived in appendix F the exact value in
+k=0:
+n(k=0) =55.43379775608 ...
+κ3
+0. (163)
+0 1 2 3 4 5
+k [κ0]0102030405060n(k) [1/κ03](a)
+10 100
+k [κ0]-100-50050100k5 [n(k)-C/k4]   [κ02](b)
+FIG. 1: For a free space Efimov trimer at rest composed of three bosonic atoms of mass minteracting viaa zero range,
+infinite scattering length potential, atomic momentum dist ributionn(k) as a function of k. (a) Numerical calculation from the
+expressions derived in the appendix F. (b) Numerical calcul ation (solid line) and asymptotic behavior (161) (dashed li ne), with
+the horizontal axis in log scale. The unit of momentum is κ0, such that the trimer energy is −/planckover2pi12/mκ2
+0.
+D. Breakdown of the energy-momentum relation in the zero-ra nge model
+1. A non-converging integral
+As a consequence of (161), the integral/integraltext
+d3kk2[n(k)−C/k4] is not well-defined: After the change of variables
+x= lnk, the integrand behaves for x→ ∞as a linear superposition of ei|s0|xande−i|s0|x, that is as a periodic
+function of xoscillating around zero. This was overlooked in [68].
+2. Failure of a naive regularisation
+At first sight, however, this does not look too serious: one often a rgues, when one faces the integral of such an
+oscillating function of zero mean, that the oscillations at infinity simply average to zero. More precisely, let us define26
+the cut-off dependent energy of the Efimov trimer (here 1 /a= 0):
+E(Λ) =/integraldisplay
+k<Λd3k
+(2π)3/planckover2pi12k2
+2m/bracketleftbigg
+n(k)−C
+k4/bracketrightbigg
+. (164)
+For Λ→ ∞,E(Λ) is asymptotically an oscillating function of the logarithm of Λ, oscilla ting around a mean value ¯E.
+The naive expectation would be that the trimer energy Etrimequals¯E. This naive expectation is equivalent to the
+usual trick used to regularize oscillating integrals, consisting here in introducing a convergence factor e−ηln(k/κ0)in
+the integral without momentum cut-off and then taking the limit η→0+:
+lim
+η→0+/integraldisplay
+R3d3k
+(2π)3/planckover2pi12k2
+2m/bracketleftbigg
+n(k)−C
+k4/bracketrightbigg
+e−ηln(k/κ0)=¯E. (165)
+We have decided to really test this naive regularisation. As we now sho w, remarkably, it actually fails to give the
+correct energy of the Efimov trimer.
+a. Numerical evidence for E∝ne}ationslash=¯E.We first performed a numerical calculation of E(Λ), using the results of
+Appendix F to perform a very accurate numerical calculation of n(k). The result is shown as a solid line in Fig.2.
+We also developed a more direct technique allowing a numerical calculat ion ofE(Λ) without the knowledge of n(k),
+see Appendix G: The corresponding results are represented as + s ymbols in Fig.2 and are in perfect agreement with
+the solid line. As expected, E(Λ) is asymptotically an oscillating function of the logarithm of Λ, oscilla ting around a
+mean value ¯E. This may be formalized as follows. We introduce an arbitrary, non-z ero valuekminof the momentum,
+and we define
+δn(k)≡n(k)−C
+k4fork<kmin (166)
+δn(k)≡n(k)−/braceleftbiggC
+k4+D
+k5cos/bracketleftBig
+2|s0|ln(k√
+3/κ0)+ϕ/bracketrightBig/bracerightbigg
+fork>kmin. (167)
+The introduction of kminensures that the integral of k2δn(k) overkconverges around k=0. The subtraction of the
+asymptotic behavior of n(k) up to order O(1/k5) fork>kminensures that the integral of k2δn(k) overR3converges
+at infinity. As a consequence we get for Λ >kminthe splitting
+E(Λ) =/integraldisplay
+k<Λd3k
+(2π)3/planckover2pi12k2
+2mδn(k)+/integraldisplay
+kmin<k<Λd3k
+(2π)3/planckover2pi12k2
+2mD
+k5cos/bracketleftBig
+2|s0|ln(k√
+3/κ0)+ϕ/bracketrightBig
+. (168)
+With the change of variable x= ln(k√
+3/κ0), one can calculate the second integral explicitly. Since the first int egral
+converges in the limit Λ →+∞we obtain
+E(Λ) =¯E+/planckover2pi12D
+8π2m|s0|sin[2|s0|ln(Λ√
+3/κ0)+ϕ]+O(1/Λ), (169)
+with
+¯E=−/planckover2pi12D
+8π2m|s0|sin[2|s0|ln(kmin√
+3/κ0)+ϕ]+/integraldisplay+∞
+0dk/planckover2pi12k4
+4π2mδn(k). (170)
+From this last equation and the numerical calculations of n(k) first up to k= 1000κ0and then up to k≃5500κ0, we
+get two slightly different values of ¯Ewhich give an estimate with an error bar [146]:
+¯E≃0.89397(3)Etrim. (171)
+The key point is that ¯Esignificantly differs from Etrim: the naive regularisation does not give the correct value of the
+trimer energy!
+b. Physical explanation and expression of ¯E.We now show that an analytical expression for ¯Emay be obtained,
+whichconfirmsthenumericalconclusionandhasthecrucialadvant ageofexplainingthephysicsbehindthediscrepancy
+¯E∝ne}ationslash=Etrim.
+The firststep is to realizewhathappens in a finite rangeinteractionm odel, when onetakesthe zerorangelimit. E.g.
+in the lattice model, for a non-zero value of the lattice spacing b, one readily realizes that an exact energy-momentum
+relation holds for an arbitrary number of bosons even in three dimen sions. Repeating the reasoning of Sec. IIB4
+while keeping finite the lattice spacing b, we get
+E′−E′
+trap=/planckover2pi12C′
+8πma+/integraldisplay
+[−π/b;π/b]3d3k
+(2π)3/planckover2pi12k2
+2m/bracketleftbigg
+n′(k)−C′
+k4/bracketrightbigg
+(172)27
+0 1 2 3 4 5678 9
+ln(Λ/κ0)-2-1.5-1-0.500.5E(Λ) [ /h2κ02/m]E
+Etrim
+FIG. 2: Cut-off dependent energy E(Λ) as defined in (164) for a free space infinite scattering len gth Efimov trimer with a
+zero range interaction, as a function of the logarithm of the momentum cut-off Λ. Solid line: numerical result obtained viaa
+calculation of the momentum distribution n(k). Symbols +: direct numerical calculation of E(Λ) as exposed in the Appendix
+G. Dashed sinusoidal line: asymptotic oscillatory behavio r ofE(Λ) for large Λ, obtained in omitting O(1/Λ) in (169). Dashed
+horizontal line: mean value ¯Earound which E(Λ) oscillates at large Λ. The values of ¯Eobtained analytically (181) and
+numerically (171) are indistinguishable at the scale of the figure, and clearly deviate from the dotted line giving the tr ue energy
+Etrimof the trimer, exemplifying the failure of a at a first sight co nvincing application of an energy-momentum relation for
+bosons in three dimensions. The unit of momentum κ0is such that the true trimer energy is Etrim=−/planckover2pi12κ2
+0/m.
+where
+C′≡8πm
+/planckover2pi12dE′
+d(−1/a), (173)
+and the prime denotes quantities calculated within the lattice model. I f one then takes the zero-range limit by
+repeatedly dividing bby the discrete scaling factor [144], so as to ensure a well-defined va lue for the three-body
+parameterRt, the lattice model’s quantities E′,E′
+trap,C′andn′(k) converge to the zero-range model’s quantities E,
+Etrap,Candn(k); however one cannot simply remove the primes and replace the inte gration domain by R3in (172),
+because this would lead to an ill-defined integral, as we have seen. This paradox is due to the fact that the finite
+range interaction bin the lattice model has two effects, that conspire to ensure that t he energy-momentum relation
+(172) is correct: (i) it introduces a momentum cut-off of order 1 /b, and (ii) it changes the large momentum tails of
+the momentum distribution in a way that tends to zero when b→0 for fixed k, and that still can have a non-zero
+impact on the energy-momentum relation since the integral of k2/k5is UV divergent in 3D. The previous failure of
+the energy-momentum relation for the zero range model is thus du e to the fact that we have introduced a cut-off (164)
+or a regularisation (165) in the integral over kwithoutsubsequently modifying in an appropriate way the coefficient
+of the 1/k5part ofn(k).
+In a second step, we introduce a consistent way of performing a UV regularisation, that will both show up in the
+integral over kin the energy-momentum relation and affect the tail of the momentu m distribution. To be explicit, we
+turn back to the example of N= 3 bosons forming an Efimov trimer in free space, of energy Etrim=−/planckover2pi12κ2
+0/m, for an
+infinite scattering length. In the zero range model, when the positio nsr1andr2of the particles 1 and 2 symmetrically
+convergeto acommonpoint R12in space, the three-bodywavefunctiondivergesas B(2|r3−R12|/√
+3)/(−4πr12) where
+the function Bis known, see Appendix F. The one body momentum distribution has an explicit integral expression
+in terms of the Fourier transform ˜B(k) of this function B, as shown in that Appendix. The idea is then to introduce28
+a smooth regularisation simply replacing ˜B(k) with [147]
+˜Bη(k)≡˜B(k)e−ηln/bracketleftBig√
+1+k2/κ2
+0+k/κ0/bracketrightBig
+(174)
+and eventually take the limit η→0+. Performing the replacement (174) in the expression (F15,F16,F17 ,F18,F19) of
+the momentum distribution leads to a modified momentum distribution nη(k), with an asymptotic behavior nasymp
+η(k)
+fork→+∞that is modified as compared to (161): nη(k) =
+k→+∞nasymp
+η(k)+O(1/k6) with
+nasymp
+η(k) =Cη
+k4+e−2ηln(k√
+3/κ0)
+k5/braceleftBig
+¯Dη+Dηcos/bracketleftBig
+2|s0|ln(k√
+3/κ0)+ϕη/bracketrightBig/bracerightBig
+. (175)
+In the limit η→0+one has to recover (161) so that Cη→C,Dη→D,ϕη→ϕand¯Dη→0. The expressions of Cη
+and¯Dηare given in the Appendix I and confirm these requirements. What sh all play a crucial role in what follows
+is that, however, ¯Dηis not zero for η>0, it vanishes linearly with ηinη= 0. For this consistent regularisation, the
+energy-momentum relation holds in the limit of vanishing ηfor the Efimov trimer, as we prove in Appendix H
+Etrim= lim
+η→0+EηwithEη≡/integraldisplay
+R3d3k
+(2π)3/planckover2pi12k2
+2m/bracketleftbigg
+nη(k)−Cη
+k4/bracketrightbigg
+. (176)
+The definitions (166),(167) are then modified as
+δnη(k)≡nη(k)−C
+k4fork<kmin (177)
+δnη(k)≡nη(k)−nasymp
+η(k) fork>kmin. (178)
+This results in the splitting
+Eη=/integraldisplay
+R3d3k
+(2π)3/planckover2pi12k2
+2mδnη(k)+/planckover2pi12
+4π2m/integraldisplay+∞
+xmindxe−2ηx[¯Dη+Dηcos(2|s0|x+ϕη)] (179)
+where the change of variable x= ln(k√
+3/κ0) was used so that xmin= ln(kmin√
+3/κ0). Forη→0+, we can replace in
+the right hand side of (179) δnη(k) withδn(k) since the first integral converges absolutely, but we cannot exc hange
+the limit and the integration in the second integral. After explicit calcu lation of this second integral, we take η→0+
+and we recognize ¯Efrom (170) so that
+Etrim=¯E+/planckover2pi12
+8π2mlim
+η→0+¯Dη
+η. (180)
+As detailed in the Appendix I, ¯Dηand the above limit may be expressed as single integrals, which allows to evaluate
+¯Ewith a good precision:
+¯E= 0.8939667780883 ...Etrim (181)
+This confirms the numerical estimate (171).
+IV. ARBITRARY MIXTURE
+In this Section we consider a mixture of bosonic and/or fermionic ato ms with an arbitrary number of spin com-
+ponents. The Nparticles are thus divided into groups, each group corresponding t o a given chemical species and to
+a given spin state. We label these groups by an integer σ∈ {1,...,n}. Assuming that there are no spin-changing
+collisions, the number Nσof atoms in each group is fixed, and one can consider that particle ibelongs to the group σ
+ifi∈Iσ, where the Iσ’s are a fixed partition of {1,...,N}which can be chosen arbitrarily. For example, a possible
+choice isI1={1,...,N 1};I2={N1+ 1,...,N 1+N2}; etc. The wavefunction ψ(r1,...,rN) is then symmetric
+(resp. antisymmetric) with respect to the exchange of two partic les belonging to the same group Iσof bosonic (resp.
+fermionic) particles. Each atom has a mass miand is subject to a trapping potential Ui(ri), and the scattering length
+between atoms iandjisaij. We setmi=mσandaij=aσσ′fori∈Iσandj∈Iσ′. The reduced masses are29
+µσσ′=mσmσ′/(mσ+mσ′). We shall denote by Pσσ′the set of all pairs of particles with one particle in group σand
+the other one in group σ′, each pair being counted only once:
+Pσσ′≡ {(i,j)∈(Iσ×Iσ′)∪(Iσ′×Iσ)/ i<j}. (182)
+The definition of the zero-range model is modified as follows: In the c ontact conditions (1,2) the scattering length
+ais replaced by aij, and the limit rij→0 is taken for a fixed center of mass position Rij= (miri+mjrj)/(mi+mj);
+moreover Schr¨ odinger’s equation becomes
+N/summationdisplay
+i=1/bracketleftbigg
+−/planckover2pi12
+2mi∆ri+Ui(ri)/bracketrightbigg
+ψ=Eψ. (183)
+Our results are summarized in Table VI, where we introduced the not ation
+(A(1),A(2))σσ′≡/summationdisplay
+(i,j)∈Pσσ′/integraldisplay/parenleftBig/productdisplay
+k/ne}ationslash=i,jddrk/parenrightBig/integraldisplay
+ddRijA(1)
+ij(Rij,(rk)k/ne}ationslash=i,j)∗A(2)
+ij(Rij,(rk)k/ne}ationslash=i,j). (184)
+We note that since aσσ′=aσ′σthere are only n(n+1)/2 independent scattering length, and the partial derivatives
+with respect to one of these independent scattering lengths are t aken while keeping fixed the other independent
+scattering lengths.
+In 3Dthe Efimov effect can occur, e.g. if the mixture containsa bosonic gr oup, or at least three fermionic groups, or
+two fermionic groups with a mass ratio strictly larger than a critical v alue 13.6...[53]. In this case, as in the previous
+Section, the derivatives with respect to any scattering length hav e to be taken for fixed three-body parameter(s), and
+the relation between Eand the momentum distribution (line 4 of Table VI) breaks down [148]. T his relation was
+first obtained in [68] in 3 D, and in 2Dfor Fermi-Fermi mixtures. Here we express the first sum in a more e xplicit
+way in terms of the partial derivative of the energy, and we point ou t the breakdown of this relation in presence of
+the Efimov effect.
+The derivations are analogous to the ones of Sections II and III. T he lemmas (22,25) are replaced by
+∝an}b∇acketle{tψ1,Hψ2∝an}b∇acket∇i}ht−∝an}b∇acketle{tHψ1,ψ2∝an}b∇acket∇i}ht=
+
+2π/planckover2pi12
+µσσ′/parenleftbigg1
+a1−1
+a2/parenrightbigg
+(A(1),A(2))σσ′in 3D
+π/planckover2pi12
+µσσ′ln(a2/a1)(A(1),A(2))σσ′in 2D.(185)
+The pair distribution function is now defined by
+g(2)
+σσ′(u,v) =/integraldisplay
+dr1...drN|ψ(r1,...,rN)|2/summationdisplay
+i∈Iσ,j∈Iσ′,i/ne}ationslash=jδ(u−ri)δ(v−rj). (186)
+The Hamiltonian of the lattice model used in some of the derivations no w reads
+H=H0+/summationdisplay
+σ≤σ′g0,σσ′Wσσ′ (187)
+where
+H0=N/summationdisplay
+i=1/bracketleftbigg
+−/planckover2pi12
+2mi∆ri+Ui(ri)/bracketrightbigg
+(188)
+with∝an}b∇acketle{tr|∆r|k∝an}b∇acket∇i}ht ≡ −k2∝an}b∇acketle{tr|k∝an}b∇acket∇i}htand
+Wσσ′=/summationdisplay
+(i,j)∈Pσσ′δri,rjb−d. (189)
+In the formulas of Sec. II and App. A coming from the two-body sca ttering problem, one has to replace g0byg0,σσ′,
+abyaσσ′andmby 2µσσ′. Denoting the corresponding scattering state by φσσ′(r), the lemma (88) becomes
+∝an}b∇acketle{tψn′|Wσσ′|ψn∝an}b∇acket∇i}ht=|φσσ′(0)|2(A(n′),A(n))σσ′. (190)30
+V. APPLICATIONS
+In this Section, we apply some of the above relations, first to the th ree-body problem and then to the many-body
+problem. We consider the unitary limit a=∞in three dimensions.
+A. Three-body problem: corrections to exactly solvable cas es and comparison with numerics
+Here we use the known analytical expressions for the three-body wavefunctions to compute the corrections to the
+spectrum to first order in the inverse scattering length 1 /aand in the effective range re.
+1. Universal eigenstates in a trap
+The problem of three identical spinless bosons [108, 109] or two-c omponent fermions (say N↑= 2 andN↓=
+1) [108, 110] is exactly solvable in the unitary limit in an isotropic harmon ic trapU(r) = 1/2mω2r2. Here we restrict
+to zero total angular momentum, with a center of mass in its ground state, so that the normalization constants of the
+wavefunctions are also known analytically [75]. Moreover we restrict to universal eigenstates [149]. The spectrum is
+then given by
+E=Ecm+(s+1+2q)/planckover2pi1ω (191)
+whereEcmis the energy of the center of mass motion, sbelongs to the infinite set of real positive solutions of
+−scos/parenleftBig
+sπ
+2/parenrightBig
++η4√
+3sin/parenleftBig
+sπ
+6/parenrightBig
+= 0 (192)
+withη= +2 for bosons and −1 for fermions, and qis a non-negative integer quantum number describing the degree
+of excitation of an exactly decoupled bosonic breathing mode [86, 11 1]. We restrict for simplicity to states with q= 0.
+a. Derivative of the energy with respect to 1/a.Injecting the expression of the regular part Aof the normalized
+wavefunction [75] into the relation (24) or its bosonic version (Table V, line 1) we obtain
+∂E
+∂(−1/a)/vextendsingle/vextendsingle/vextendsingle
+a=∞=Γ(s+1/2)√
+2ssin/parenleftbig
+sπ
+2/parenrightbig
+Γ(s+1)/bracketleftBig
+−cos/parenleftbig
+sπ
+2/parenrightbig
++sπ
+2sin/parenleftbig
+sπ
+2/parenrightbig
++η2π
+3√
+3cos/parenleftbig
+sπ
+6/parenrightbig/bracketrightBig/radicalbigg
+/planckover2pi13ω
+m. (193)
+For the lowest fermionic state, this gives ( ∂E/∂(1/a))a=∞≃ −1.1980/radicalbig
+/planckover2pi13ω/m, in agreement with the value −1.19(2)
+which we extracted from the numerical solution of a finite-range mo del presented in Fig. 4a of [81], where the error
+bar comes from our simple way of extracting the derivative from the numerical data of [81].
+b. Derivative of the energy with respect to the effective rang e.Using relation (136), which holds not only for
+fermions but also for bosonic universal states, we obtain
+/parenleftbigg∂E
+∂re/parenrightbigg
+a=Γ(s−1/2)s(s2−1/2)sin(sπ/2)
+Γ(s+1)2√
+2/bracketleftbig
+−cos(sπ/2)+sπ/2·sin(sπ/2)+η2π/(3√
+3)·cos(sπ/6)/bracketrightbig√
+/planckover2pi1mω3.(194)
+For bosons, this result was derived previously using the method of [ 87] and found to agree with the numerical solution
+of a finite-range separable potential model for the lowest state [ 75]. For fermions, (194) agrees with the numerical
+data from Fig. 3 of [81] to ∼0.3% for the two lowest states and 5% for the third lowest state [150 ]; (194) also agrees
+to 3% with the numerical data from p. 21 of [75] for the lowest state of a finite-range separable potential model. All
+these deviations are compatible with the estimated numerical accur acy.
+2. Derivative of the energy of an Efimov trimer with respect to 1/a.
+The same relation (Table V, line 1) can be applied to Efimov trimers in fre e space. Using the expression of the
+normalized three-body wavefunction (see App. D) we get
+/parenleftbigg∂E
+∂(−1/a)/parenrightbigg
+Rt=/parenleftbigg
+−/planckover2pi12
+mE/parenrightbigg1/2πsinh(|s0|π/2)tanh(|s0|π)
+cosh(|s0|π/2)+π|s0|
+2sinh(|s0|π/2)−4π
+3√
+3cosh(|s0|π/6)(195)
+= 2.11267159347 ...×/parenleftbigg
+−/planckover2pi12
+mE/parenrightbigg1/2
+. (196)31
+0 0.1 0.2
+kF r00.001 g(2)(r)  r2 / kF4
+r=b
+FIG. 3: Pair distribution function g(2)
+↑↓(r) =
+∝angbracketleftˆψ†
+↑(r)ˆψ†
+↓(0)ˆψ↓(0)ˆψ↑(r)∝angbracketrightof the homogeneous unitary
+gas at zero temperature. Crosses: fixed-node Monte-Carlo
+results from Ref. [113]. Line: analytic expression (200),
+where the value ζ= 0.95 was taken to fit the Monte-Carlo
+results. The arrow indicates the range bof the square-well
+interaction potential.0 1 2 3 4
+kF r01g(1)
+σσ(r) / (n/2)
+FIG. 4: One-body density matrix g(1)
+σσ(r) =∝angbracketleftˆψ†
+σ(r)ˆψσ(0)∝angbracketright
+of the homogeneous unitary gas at zero temperature: com-
+parison between the fixed-node Monte-Carlo results from
+Ref. [114] (continuous curve) and the analytic expres-
+sion (199) for the small- kFrexpansion of g(1)
+σσup to first
+order (dashed straight line, red online) and second order
+(dotted parabola, blue online) where we took the value
+ζ= 0.95 extracted from the Monte-Carlo data for g(2)
+↑↓, see
+Fig. 3.
+This confirms and refines the value of the coefficient 2 .11 estimated numerically in [73]. We note that this numerical
+coefficient is simply ∆′(−π/2)/|s0|, where ∆(ξ) is Efimov’s universal function that was estimated numerically in [73]
+and computed very precisely in [112]. Our analytical calculation gives t he exact value of the derivative
+∆′(−π/2) = 2.125850069373 ..., (197)
+to be compared with the numerical estimate ∆′(−π/2)≃2.12 in [73].
+The expression (195) can also be obtained from the tail of the mome ntum distribution: The expression of the
+coefficientCfollows from Eq.(F1), and (195) is recovered using the relation on th e second line of the left column of
+Table V.
+B. Unitary Fermi gas: comparison with fixed-node Monte-Carl o
+For the homogeneous unitary gas (i.e. the two-component Fermi g as in 3Dwitha=∞) at zero temperature, we
+can compare our analytical expressions for the short-distance b ehavior of the one-body density matrix g(1)
+σσand the
+pair distribution function g(2)
+↑↓to the fixed-node Monte-Carlo results published by the Trento gro up in [82, 113, 114].
+In this case, g(1)
+σσ(R−r/2,R+r/2) andg(2)
+↑↓(R−r/2,R+r/2) depend only on rand not on σ,Rand the direction
+ofr. Expanding the energy to first order in 1 /(kFa) around the unitary limit yields:
+E=E0/parenleftbigg
+ξ−ζ
+kFa+.../parenrightbigg
+(198)
+whereE0is the ground state energy of the ideal gas, ξandζare universal dimensionless numbers, and the Fermi
+wavevector is related to the density through kF= (3π2n)1/3. Expressing Cin terms of ζthanks to (34,198) and
+inserting this into (61) gives
+g(1)
+σσ(r)≃n
+2/bracketleftbigg
+1−3ζ
+10kFr−ξ
+10(kFr)2+.../bracketrightbigg
+. (199)
+For a finite interaction range b, this expression is valid for b≪r≪k−1
+F[151]. Equation (109) yields
+g(2)
+↑↓(r)≃
+kFr≪1ζ
+40π3k4
+F|φ(r)|2. (200)32
+The interaction potential used in the Monte-Carlo simulations [82, 11 3, 114] is a square-well
+V(r) =/braceleftBigg
+−/parenleftbigπ
+2/parenrightbig2/planckover2pi12
+mb2forr<b
+0 for r>b(201)
+for which the zero-energy scattering state is
+φ(r) =/braceleftBigg
+sin/parenleftbigπr
+2b/parenrightbig
+/rforr<b
+1/r forr>b(202)
+and the range bwas taken such that nb3= 10−6i.e.kFb= 0.0309367.... Thus we can assume that we are in the
+zero-range limit kFb≪1, so that (199,200) are applicable.
+Figure 3 shows that the expression (200) for g(2)
+↑↓fits well the Monte-Carlo data of [113] if one adjusts the value of
+ζto 0.95. This value is close to the value ζ≃1.0 extracted from (198) and the E(1/a)-data of [82].
+Usingζ= 0.95 we can compare the expression (199) for g(1)
+σσwith Monte-Carlo data of [114] without adjustable
+parameters. Figure 4 shows that the first order derivatives agre e, while the second order derivatives are compatible
+within the statistical noise. This provides an interesting check of th e numerical results, even though any wavefunction
+satisfying the contact condition (1) would lead to g(1)
+σσandg(2)
+↑↓functions satisfying (44,60) with values of Ccompatible
+with each other.
+C. Dependence of the unitary gas critical temperature on the interaction range
+The key property underlying (136) is that the leading order change of each eigenstate energy of a spin 1 /2 Fermi
+gas due to a small but non-zero interaction range is linear in the effec tive range reof the interaction potential,
+which a coefficient which is model independent. As a consequence, th e leading order change of the thermodynamical
+potentials, and even of the critical temperature Tcof the Fermi gas, are also expected to be linear in re, with model
+independent coefficients.
+This expectation can be tested with the Quantum Monte Carlo data o f [41] and [44], where the critical temperature
+of the unitary gas was calculated for two different, finite range inte raction models [152]. In Fig.5, we plot the Monte
+Carlo data as a function of kFre, wherereis the effective range of the corresponding model. Our expectation is that
+all the data lie on the same straight line for low enough kF|re|, which is indeed essentially the case (considering the
+size of the error bars). We then conclude that the shift of Tcdue to the interaction range is of order
+δTc
+TF≃0.12kFre, (203)
+at lowreand in a model independent way. This results in a relative shift at the p ercent level for typical experiments
+on lithium, where kFre≈0.01.
+VI. CONCLUSION
+We derived relations between various observables for Nparticles of arbitrary masses and statistics in an external
+potential with zero-range or short-range interactions, in contin uous space or on a lattice, in two or three dimensions.
+Some of our results generalize the ones of [55, 61, 62, 65, 68]: The large-momentum behavior of the momentum
+distribution, the short-distance behavior of the pair correlation f unction and of the one-body density matrix, the
+derivativeoftheenergywithrespecttothescatteringlengthort otime,thenormoftheregularpartofthewavefunction
+(defined through the behavior of the wavefunction when two part icles approach each other), and, in the case of finite-
+range interactions, the interaction energy, are all related to the same quantity C; and the difference between the total
+energy and the trapping potential energy is related to Cand to a functional of the momentum distribution (which
+is also equal to the second order term in the short-distance expan sion of the one-body density matrix). For Efimov
+states with zero-range interactions, we found that this last relat ion breaks down, because the large-momentum tail of
+the momentum distribution contains a subleading oscillatory term.
+We also obtained entirely new relations: The second order derivative of the energy with respect to the inverse
+scattering length (or to the logarithm of the scattering length in tw o dimensions) is related to the regular part of
+the wavefunctions, and is negative at fixed entropy; the derivativ e of the energy of a universal state with respect to33
+-1 -0.5 0 0.5
+kF re00.050.10.150.20.25Tc / TF
+FIG. 5: Critical temperature Tcof the unitary Fermi gas as a function of the effective range reof the interaction potential, as
+given by the Quantum Monte Carlo results of [41] for the Hubba rd model (symbols with re<0) and of [44] for a continuous
+space model (symbols with re>0). The dashed line corresponds to a linear fit of the data over the interval kFre∈[−0.8,0.45].
+HerekBTF=/planckover2pi12k2
+F/(2m) is the Fermi energy of the ideal gas with the same density as t he unitary gas.
+the effective range of the interaction potential is also related to th e regular part; and the derivative of the energy
+of an efimovian state with respect to the three-body parameter is related to the a three-body analog of the regular
+part. Applications were presented in three dimensions for an infinite scattering length: the derivative of the energy
+with respect to the inverse scattering length was computed analyt ically and found to agree with numerics for Efimov
+trimers; the same was done for universal three-body states in a h armonic trap, not only for the derivative of the
+energy with respect to the inverse scattering length but also with r espect to the effective range; existing fixed-node
+Monte-Carlo data for the unitary Fermi gas were checked to satis fy exact relations. Also the derivative of the critical
+temperature of the unitary gas with respect to the effective rang e, expected from our results to be model-independent,
+is estimated from the Quantum Monte Carlo results of [41].
+The relations obtained here may be used in various other contexts. For example, the result (147) on the sign of the
+secondorderderivativeof Eat constantentropyis relevantto adiabaticrampexperiments [35 , 36, 98, 99, 115], and the
+relation (107) allows to directly compute Cusing diagrammatic Monte-Carlo [116]. Cis directly related to the closed-
+channel fraction in a two-channel model [64, 66], which allows to ex tract it [66] from experimental photoassociation
+measurements [32]. Calso plays an important role in the theory of radiofrequencyspectr a [65, 117–121] and in finite- a
+virial theorems [63, 122, 123].
+We can think of several generalizations of the relations presented here. All relations can be rederived in the case
+of periodic boundary conditions. The relations for finite-range mod els, obtained here for two-component fermions,
+can be generalized to arbitrary mixtures. The techniques used her e can be applied to the one-dimensional case to
+generalize the relations of [55]. For two-channel or multi-channel m odels one may derive relations other than the ones
+of [64–66]. In presence of the Efimov effect, the derivative of the e nergy with respect to the three-body parameter can
+easily be related to the short-distance behavior of the third order density correlation function thanks to (154,152);
+moreover the asymptotic behavior (161) of the momentum distribu tion is expected to hold for any state satisfying
+the three-body boundary condition (152), with a coefficient Dof the subleading tail in (161) related to ∂E/∂(lnRt)
+through a simple proportionality factor. Indeed, the singularities o f the wavefunction at short interparticle distances
+are generally related to large momentum tails.34
+Acknowledgments
+We thank S. Tan, M. Olshanii, E. Burovski, N. Prokof’ev and R. Ignat for useful discussions, as well as S. Giorgini,
+J. von Stecher and M. Thøgersen for sending numerical data from [81, 113, 114, 124]. F.W. is supported by NSF
+under Grant No. PHY-0653183. Y.C. is a member of IFRAF.
+Note
+While completing this work, we became aware of unpublished notes by T an where Eqs. (39,82) were obtained
+independently using the formalism of [125] [153].
+Appendix A: Two-body scattering for the lattice model
+For the lattice model defined in Sec. IIA2, we recall that φ(r) denotes the zero-energy two-body scattering state
+with the normalization (11,12). In this Appendix we derive the relation (13,14) between the coupling constant g0
+and the scattering length, as well as the expressions (15,16,17,18) ofφ(0). Some of the calculation resemble the ones
+in [78, 126].
+We consider a low-energy scattering state Φ q(r) of wavevector q≪b−1and energy E= 2ǫq≃/planckover2pi12q2/m, i.e. the
+solution of the two-body Schr¨ odinger equation (with the center o f mass at rest):
+(H0+V)|Φq∝an}b∇acket∇i}ht=E|Φq∝an}b∇acket∇i}ht (A1)
+whereH0=/integraltext
+Dddk/(2π)d2ǫk|k∝an}b∇acket∇i}ht∝an}b∇acketle{tk|andV=g0|r=0∝an}b∇acket∇i}ht∝an}b∇acketle{tr=0|, with the asymptotic behavior
+Φq(r) =
+r→∞eiqr+fqeiqr
+r+...in 3D (A2)
+Φq(r) =
+r→∞eiqr−fq/radicalBigg
+i
+8πqreiqr+...in 2D. (A3)
+Herefqis the scattering amplitude (in 2 Dthe present definition is from [127], and√
+i≡eiπ/4), which in the present
+case is independent of the direction of ras we will see. We then have the well-known expression
+|Φq∝an}b∇acket∇i}ht= (1+GV)|q∝an}b∇acket∇i}ht (A4)
+whereG= (E+i0+−H)−1, which, since
+G=G0+G0VG, (A5)
+is equivalent to
+|Φq∝an}b∇acket∇i}ht= (1+G0T)|q∝an}b∇acket∇i}ht (A6)
+where
+T=V+VGV. (A7)
+Indeed, (A4) clearly solves (A1), and one can check [using the fact that∝an}b∇acketle{tr|G0|r=0∝an}b∇acket∇i}htbehaves for r→ ∞as
+−m/(4π/planckover2pi12)eiqr/rin 3Dand−m//planckover2pi12/radicalbig
+i/(8πqr)eiqrin 2D] that (A6) satisfies (A2,A3) with
+fq=−m
+4π/planckover2pi12b3∝an}b∇acketle{tr=0|T|q∝an}b∇acket∇i}htin 3D (A8)
+fq=m
+/planckover2pi12b2∝an}b∇acketle{tr=0|T|q∝an}b∇acket∇i}htin 2D. (A9)
+Using (A7) and (A5) one gets
+∝an}b∇acketle{tr=0|T|q∝an}b∇acket∇i}ht=b−d/bracketleftbigg1
+g0−/integraldisplay
+Dddk
+(2π)d1
+E+i0+−2ǫk/bracketrightbigg−1
+. (A10)35
+In 3Dthe scattering length in defined by fq→
+q→0−a, which gives the relation (13) between aandg0. In 2D,
+fq=
+q→0−2π
+ln(qaeγ/2)−iπ/2+o(1)(A11)
+whereais by definition the 2 Dscattering length. Identifying the inverse of the right-hand-side s of Eqs.(A9) and
+(A11) and taking the real part gives the desired (14). We note tha t Eqs. (A11,14) remain true if q→0 is replaced by
+the limitb→0 taken for fixed a.
+To derive (15,16) we start from
+V|Φ∝an}b∇acket∇i}ht=T(E+i0+)|q∝an}b∇acket∇i}ht (A12)
+which directly follows from (A4). Applying ∝an}b∇acketle{tr=0|on the left and using (A8,A9) yields
+g0Φq(0) =−4π/planckover2pi12
+mfqin 3D (A13)
+g0Φq(0) =/planckover2pi12
+mfqin 2D. (A14)
+In 3D, we simply have φ=−a−1lim
+q→0Φq, and the result (15) follows. In 2 D, the situation is a bit more tricky because
+lim
+q→0Φq(0) = 0. We thus start with q >0, and we will take the limit q→0 later on. At finite q, we define φq(r) as
+being proportional to Φ q(r), and normalized by imposing the same condition (12) than at zero en ergy, but only for
+b≪r≪q−1. Inserting (A11) into (A14) gives an expression for Φ q(0). To deduce the value of φ(0), it remains to
+calculate the r-independent ratio φq(r)/Φq(r). But forr≫bwe can replace φq(r) and Φ q(r) by their values within
+the zero-rangemodel (since we also have b≪q−1) which we denote by φZR
+q(r) and ΦZR
+q(r). The two-body Schr¨ odinger
+equation
+−/planckover2pi12
+m∆ΦZR
+q=EΦZR
+q,∀r>0 (A15)
+implies that
+ΦZR
+q=eiq·r+NH(1)
+0(qr) (A16)
+whereNis a constant and H(1)
+0is an outgoing Hankel function. The contact condition
+∃A/ΦZR
+q(r) =
+r→0Aln(r/a)+O(r) (A17)
+together with the known short- rexpansion of the Hankel function [128] then gives
+A=−1
+ln(qaeγ/2)−iπ/2. (A18)
+Of course we also have ΦZR
+q/φZR
+q=A, which gives (16).
+Finally, Eqs.(17,18) are obtained from (15,16) using the relations d(m/(4π/planckover2pi12a))/d(1/g0) = 1 in 3 Dand
+d(1/g0)/d(lna) =−m/(2π/planckover2pi12) in 2D, which are direct consequences of the relations (13,14) between g0anda.
+Appendix B: Derivation of a lemma
+In this Appendix, we derive the lemma (22) in three dimensions, as well as its two-dimensional version (25).
+Three dimensions:
+By definition we have
+∝an}b∇acketle{tψ1,Hψ2∝an}b∇acket∇i}ht−∝an}b∇acketle{tHψ1,ψ2∝an}b∇acket∇i}ht=−/planckover2pi12
+2m/integraldisplay′
+d3r1...d3rNN/summationdisplay
+i=1[ψ∗
+1∆riψ2−ψ2∆riψ∗
+1]. (B1)36
+Here the notation/integraltext′means that the integral is restricted to the set where none of the particle positions coincide [154].
+We rewrite this as:
+∝an}b∇acketle{tψ1,Hψ2∝an}b∇acket∇i}ht−∝an}b∇acketle{tHψ1,ψ2∝an}b∇acket∇i}ht=−/planckover2pi12
+2mN/summationdisplay
+i=1/integraldisplay′/parenleftBig/productdisplay
+k/ne}ationslash=id3rk/parenrightBig
+lim
+ǫ→0/integraldisplay
+{ri/∀j/ne}ationslash=i,rij>ǫ}d3ri[ψ∗
+1∆riψ2−ψ2∆riψ∗
+1].(B2)
+We note that this step is not trivial to justify mathematically. The or der of integration has been changed and the
+limitǫ→0 has been exchanged with the integral over ri. We expect that this is valid in the presently considered
+case of equal mass fermions, and more generally provided the wave functions are sufficiently regular in the limit where
+several particles tend to each other.
+Since the integrand is the divergence of ψ∗
+1∇riψ2−ψ2∇riψ∗
+1, Ostrogradsky’s theorem gives
+∝an}b∇acketle{tψ1,Hψ2∝an}b∇acket∇i}ht−∝an}b∇acketle{tHψ1,ψ2∝an}b∇acket∇i}ht=/planckover2pi12
+2mN/summationdisplay
+i=1/integraldisplay′/parenleftBig/productdisplay
+k/ne}ationslash=id3rk/parenrightBig
+lim
+ǫ→0/summationdisplay
+j,j/ne}ationslash=i/dispoiint
+Sǫ(rj)[ψ∗
+1∇riψ2−ψ2∇riψ∗
+1]·dS (B3)
+where the surface integral is for ribelonging to the sphere Sǫ(rj) of center rjand radius ǫ, and the vector area dS
+points out of the sphere. We then expand the integrand by using th e contact condition, in the limit rij=ǫ→0 taken
+for fixed rjand fixed ( rk)k/ne}ationslash=i,j. UsingRij=rj+ǫu/2 withu≡(ri−rj)/rijwe get
+ψn=
+ǫ→0/parenleftbigg1
+ǫ−1
+an/parenrightbigg
+A(n)
+ij+1
+2u·∇RijA(n)
+ij+O(ǫ) (B4)
+∇riψn=
+ǫ→0−u
+ǫ2A(n)
+ij+1
+2ǫ/bracketleftBig
+∇RijA(n)
+ij−u/parenleftBig
+u·∇RijA(n)
+ij/parenrightBig/bracketrightBig
++O(1) (B5)
+wherenequals 1 or 2, and the functions A(n)
+ijand∇RijA(n)
+ijare taken at ( rj,(rk)k/ne}ationslash=i,j). This simply gives
+/dispoiint
+Sǫ(rj)[ψ∗
+1∇riψ2−ψ2∇riψ∗
+1]·dS=
+ǫ→04π/parenleftbigg1
+a1−1
+a2/parenrightbigg
+A(1)∗
+ijA(2)
+ij+O(ǫ) (B6)
+because the leading order term cancels and most angular integrals v anish. Inserting this into (B3) gives the desired
+lemma (22).
+Two dimensions:
+The derivation is analogous to the 3 Dcase. In (B3), the double integral on the sphere of course has to be replaced
+by a simple integral on the circle. Instead of (B4,B5), we now obtain, from the 2Dcontact condition (2),
+ψn=
+ǫ→0ln(ǫ/an)A(n)
+ij+O(ǫlnǫ) (B7)
+∇riψn=
+ǫ→0u
+ǫA(n)
+ij+O(lnǫ), (B8)
+which gives
+/contintegraldisplay
+Sǫ(rj)[ψ∗
+1∇riψ2−ψ2∇riψ∗
+1]·dS=
+ǫ→02πln(a2/a1)A(1)∗
+ijA(2)
+ij+O(ǫln2ǫ) (B9)
+and yields the lemma (25).
+Appendix C: First and second order isentropic derivatives o f the mean energy in the canonical ensemble
+One considers a system with a Hamiltonian H(λ) depending on some parameter λ, and at thermal equilibrium
+in the canonical ensemble at temperature T, with a density operator ρ= exp(−βH)/Z. In terms of the partition
+functionZ(T,λ) = Tre−βH(λ), withβ= 1/(kBT), one has the usual relations for the free energy F, the mean energy
+U= Tr(ρH) and the entropy S=−kBTr(ρlnρ):
+F(T,λ) =−kBTlnZ(T,λ) (C1)
+F(T,λ) =U(T,λ)−TS(T,λ) (C2)
+∂TF(T,λ) =−S(T,λ). (C3)37
+One now varies λfor a fixed entropy S. The temperature is thus a function T(λ) ofλsuch that
+S(T(λ),λ) = ct. (C4)
+The derivatives of the mean energy for fixed entropy are then:
+/parenleftbiggdU
+dλ/parenrightbigg
+S≡d
+dλ[U(T(λ),λ)] (C5)
+/parenleftbiggd2U
+dλ2/parenrightbigg
+S≡d2
+dλ2[U(T(λ),λ)]. (C6)
+Writing (C2) for T=T(λ) and taking the first order and the second order derivatives of th e resulting equation with
+respect toλ, one finds
+/parenleftbiggdU
+dλ/parenrightbigg
+S=∂λF(T(λ),λ) (C7)
+/parenleftbiggd2U
+dλ2/parenrightbigg
+S=∂2
+λF(T(λ),λ)−[∂T∂λF(T(λ),λ)]2
+∂2
+TF(T(λ),λ). (C8)
+It remains to use (C1) to obtain a microscopic expression of the abo ve partial derivatives of F, from the partition
+function expressed as a sum Z=/summationtext
+ne−βEnover the eigenenergies nof the Hamiltonian:
+∂λF(T,λ) =dE
+dλ(C9)
+∂2
+λF(T,λ) =d2E
+dλ2−βVar/parenleftbiggdE
+dλ/parenrightbigg
+(C10)
+∂2
+TF(T,λ) =−VarE
+kBT3(C11)
+∂T∂λF(T,λ) =Cov(E,dE/dλ )
+kBT2. (C12)
+The expectation value (...) stands for a sum over the eigenenergies with the canonical proba bility weights, and Var
+and Cov are the corresponding variance and covariance. E.g. E=Uand
+d2E
+dλ2≡/summationdisplay
+nd2En
+dλ2e−βEn
+Z(C13)
+Cov(E,dE/dλ )≡/summationdisplay
+nEndEn
+dλe−βEn
+Z−EdE
+dλ. (C14)
+Insertion of (C9) into (C7) gives (140). Insertion of (C10,C11,C12 ) into (C8) gives (146).
+Appendix D: Normalized wavefunction of an Efimov trimer
+In this Appendix we recall the wavefunction of an Efimov trimer and g ive the expression of its normalization
+constant. We consider an Efimov trimer state for three spinless bo sons of mass minteracting viaa zero range infinite
+scattering length potential. In order to avoid formal normalisability problems, we imagine that the Efimov trimer is
+trapped in an arbitrarilyweak harmonic potential, that is with a groun d state harmonic oscillatorlength a0arbitrarily
+larger than the trimer size. In this case, the energy of the trimer is essentially the free space energy Etrim=−/planckover2pi12κ2
+0
+m,
+κ0>0. According to Efimov’s theory [129]
+κ0=√
+2
+Rteπq/|s0|eArgΓ(1+s0)/|s0|(D1)
+whereRt>0 is a length known as the three-body parameter, the quantum num berqmay take all values in Zand
+the purely imaginary number s0=i|s0|is such that
+|s0|cosh(|s0|π/2) =8√
+3sinh(|s0|π/6), (D2)38
+so that|s0|= 1.00623782510 ...The corresponding three-body wavefunction Ψ may be written as
+Ψ(r1,r2,r3)≃ψCM(C)/bracketleftBig
+ψ(r12,|2r3−(r1+r2)|/√
+3)+ψ(r23,|2r1−(r2+r3)|/√
+3)+ψ(r31,|2r2−(r3+r1)|/√
+3)/bracketrightBig
+,
+(D3)
+whereC= (r1+r2+r3)/3 is the center of mass position of the three particles and the param eterization of ψis
+related to the Jacobi coordinates r=r2−r1andρ= [2r3−(r1+r2)]/√
+3. In our expression of Ψ, ψCMis the
+Gaussian wavefunction of the single particle ground state in the har monic trap, normalized to unity, and ψis a
+Faddeev component of the free space trimer wavefunction. The e xplicit expression of ψis known [129]:
+ψ(r,ρ) =Nψ√
+4πKs0(κ0/radicalbig
+r2+ρ2)
+(r2+ρ2)/2sin[s0(π
+2−α)]
+sin(2α)(D4)
+whereKs0is a Bessel function and α= atan(r/ρ). The normalization factor ensuring that ||Ψ||2= 1 may
+be calculated explicitly: One first performs the change of variables ( r1,r2,r3)→(C,r,ρ), whose Jacobian is
+D(r1,r2,r3)/D(C,ρ,r) = (−√
+3/2)3. To integrate over randρone introduces hyperspherical coordinates in which
+the wavefunction separates; one then faces known integrals on t he hyperradius [130] and on the hyperangles [87]. This
+leads to [75]:
+|Nψ|−2=/parenleftBigg√
+3
+2/parenrightBigg3
+3π2
+2κ2
+0cosh(|s0|π/2)/bracketleftbigg
+cosh(|s0|π/2)+|s0|π
+2sinh(|s0|π/2)−4π
+3√
+3cosh(|s0|π/6)/bracketrightbigg
+.(D5)
+We also recalled, as promised in the main text, the value of the hypera ngular scalar product derived in [75]:
+(φs0|φs0) =12π
+s0sin(s0π/2)/bracketleftbigg
+cos(s0π/2)−s0π
+2sin(s0π/2)−4π
+3√
+3cos(s0π/6)/bracketrightbigg
+. (D6)
+Appendix E: A lemma for three bosons in the zero-range model
+Here we prove the relation (155). The first step is to express the H amiltonian in hyperspherical coordinates [142]:
+∝an}b∇acketle{tψ1,Hψ2∝an}b∇acket∇i}ht−∝an}b∇acketle{tHψ1,ψ2∝an}b∇acket∇i}ht=−/planckover2pi12
+2m/parenleftBigg√
+3
+2/parenrightBigg3/integraldisplay∞
+0dRR5/integraldisplay
+dΩ/integraldisplay
+dC
+/braceleftbigg
+ψ∗
+1/parenleftbigg∂2
+∂R2+5
+R∂
+∂R+TΩ
+R2+1
+3∆C/parenrightbigg
+ψ2−[ψ∗
+1↔ψ2]/bracerightbigg
+(E1)
+=−/planckover2pi12
+2m/parenleftBigg√
+3
+2/parenrightBigg3/braceleftbigg/integraldisplay
+dRR5/integraldisplay
+dΩAC(R,Ω)+/integraldisplay
+dΩdCAR(Ω,C)
++/integraldisplay
+dRR5/integraldisplay
+dCAΩ(R,C)/bracerightbigg
+(E2)
+where
+AC(R,Ω)≡/integraldisplay
+dC/braceleftbigg
+ψ∗
+11
+3∆Cψ2−[ψ∗
+1↔ψ2]/bracerightbigg
+(E3)
+AR(Ω,C)≡/integraldisplay
+dRR5/braceleftbigg
+ψ∗
+1/parenleftbigg∂2
+∂R2+5
+R∂
+∂R/parenrightbigg
+ψ2−[ψ∗
+1↔ψ2]/bracerightbigg
+(E4)
+AΩ(R,C)≡/integraldisplay
+dΩ/braceleftbigg
+ψ∗
+1TΩ
+R2ψ2−ψ2TΩ
+R2ψ∗
+1/bracerightbigg
+, (E5)
+TΩbeing a differential operator acting on the hyperangles and called La placian on the hypersphere.
+ClearlyAC(R,Ω) =1
+3/integraltext
+dC∇C· {ψ∗
+1∇Cψ2−ψ2∇Cψ∗
+1}= 0, since the ψi’s are regular functions of Cfor every
+(R,Ω) except on a set of measure zero.
+In what follows we will use the following simple lemma: if Φ 1(R) and Φ 2(R) are functions which decay quickly at
+infinity and have no singularity except maybe at R= 0, then
+/integraldisplay
+dRR5/braceleftbigg
+Φ∗
+1/parenleftbigg∂2
+∂R2+5
+R∂
+∂R/parenrightbigg
+Φ2−[Φ∗
+1↔Φ2]/bracerightbigg
+=−lim
+R→0R/braceleftbigg
+F∗
+1∂F2
+∂R−F2∂F∗
+1
+∂R/bracerightbigg
+(E6)39
+whereFi(R)≡R2Φi(R).
+We now show that
+AΩ(R,C) = 0 for any CandR>0. (E7)
+We will use the fact that ψ1andψ2satisfy the two-body boundary condition with the same a, and apply lemma (22).
+More precisely, we will show that for any smooth function f(R,C) which vanishes in a neighborhood of R= 0,
+/integraldisplay
+dRR5/integraldisplay
+dCf(R,C)2AΩ(R,C) = 0; (E8)
+this clearly implies (E7). To show (E8) we note that
+−/planckover2pi12
+2m/parenleftBigg√
+3
+2/parenrightBigg3/integraldisplay
+dRR5/integraldisplay
+dCf(R,C)2AΩ(R,C) =−/planckover2pi12
+2m/parenleftBigg√
+3
+2/parenrightBigg3/integraldisplay
+dRR5/integraldisplay
+dΩ/integraldisplay
+dC/braceleftbigg
+(fψ∗
+1)TΩ
+R2(fψ2)−[ψ∗
+1↔ψ2]/bracerightbigg
+,
+(E9)
+which can be rewritten as
+/integraldisplay
+d3r1d3r2d3r3{(fψ∗
+1)H(fψ2)−[ψ∗
+1↔ψ2]}+/planckover2pi12
+2m/parenleftBigg√
+3
+2/parenrightBigg3/integraldisplay
+dRR5/integraldisplay
+dΩ/integraldisplay
+dC
+/braceleftbigg
+(fψ∗
+1)/parenleftbigg∂2
+∂R2+5
+R∂
+∂R+1
+3∆C/parenrightbigg
+(fψ2)−[ψ∗
+1↔ψ2]/bracerightbigg
+.(E10)
+The first integral in this expression vanishes, as a consequence of lemma (22). This lemma is indeed applicable to the
+wavefunctions fψi: They vanish in a neighborhood of R= 0 (see the discussion below (B2)), moreover they satisfy
+the two-body boundary condition for the same value of the scatte ring length a(as follows from the fact that Rvaries
+quadratically with rfor smallr). The second integral in (E10) vanishes as well: The contribution of the partial
+derivatives with respect to Rvanishes as a consequence of lemma (E6), and the contribution of ∆ Cvanishes because
+thefψi’s are regular functions of C.
+Finally,ARcan be computed using lemma (E6) and the boundary condition (152) , yielding (155).
+Appendix F: First two terms of the large- kexpansion of the momentum distribution of an Efimov trimer
+Here we show that the momentum distribution of an Efimov bosonic tr imer state of energy −/planckover2pi12κ2
+0/m(at rest and
+for an infinite scattering length) has the asymptotic expansion (16 1) with
+C/κ0=8π2sinh(|s0|π/2)tanh(|s0|π)
+cosh(|s0|π/2)+π|s0|
+2sinh(|s0|π/2)−4π
+3√
+3cosh(|s0|π/6)= 53.09722846003081 ... (F1)
+D/κ2
+0≃ −89.26260 (F2)
+ϕ≃ −0.8727976 (F3)
+1. Three-body state in momentum space
+We start from the three-body wavefunction in position space Ψ give n in Appendix D. To obtain the momentum
+distribution of the Efimov trimer, we need to evaluate the Fourier tr ansformation of Ψ. Rather than directly using
+(D4), we take advantage of the fact that the Faddeev componen tψobeys Schr¨ odinger’s equation with a source
+term. With the change to Jacobi coordinates, the Laplacian opera tor in the coordinate space of dimension nine reads/summationtext3
+i=1∆ri=1
+3∆C+2/bracketleftbig
+∆r+∆ρ/bracketrightbig
+so that
+−/bracketleftbig
+κ2
+0−∆r−∆ρ/bracketrightbig
+ψ(r,ρ) =δ(r)B(ρ). (F4)
+The source term in the right hand side originates from the fact that
+ψ(r,ρ)∼
+r→0−B(ρ)
+4πr(F5)40
+for a fixedρ, this 1/rdivergencecoming from the replacement of the interaction potent ial by the Bethe-Peierlscontact
+condition. Taking the Fourier transform of (F4) over randρleads to
+˜ψ(k,K) =−˜B(K)
+k2+K2+κ2
+0, (F6)
+where the Fourier transform is defined as ˜B(K)≡/integraltext
+d3ρe−iK·ρB(ρ).B(ρ) is readily obtained from (D4) by taking
+the limitr→0:
+B(ρ) =−Nψ(4π)1/2isinh(|s0|π/2)Ks0(ρ)
+ρ. (F7)
+The Fourier transform of this expression is known, see relation 6.67 1(5) in [130], so that
+˜B(K) =−Nψ2π5/2
+K(K2+κ2
+0)1/2/braceleftBigg/bracketleftbigg(K2+κ2
+0)1/2+K
+κ0/bracketrightbiggs0
+−/bracketleftbigg(K2+κ2
+0)1/2+K
+κ0/bracketrightbigg−s0/bracerightBigg
+. (F8)
+What we shall need is the large Kbehavior of ˜B(K). Expanding (F8) in powers of κ0/Kgives
+˜B(K) =Nψ2π5/2
+K2/bracketleftbig
+(2K/κ0)−s0−c.c./bracketrightbig
++O(1/K4). (F9)
+When necessary one may further use the relation
+(κ0/2)s0= (−1)q/parenleftBig
+Rt√
+2/parenrightBig−s0Γ(1+s0)
+|Γ(1+s0)|(F10)
+that can be deduced from (D1).
+The last step is to take the Fourier transform of (D3), using the ap propriate Jacobi coordinates for each Faddeev
+component (or simply by Fourier transforming the first Faddeev co mponent using the coordinates ( C,r,ρ) given
+above and by performing circular permutations on the particle labels ). This gives
+˜Ψ(k1,k2,k3) =/parenleftBigg√
+3
+2/parenrightBigg3
+˜ψCM(k1+k2+k3)/bracketleftBig
+˜ψ(|k2−k1|/2,√
+3|k3−(k1+k2)/2|/3)
++˜ψ(|k3−k2|/2,√
+3|k1−(k2+k3)/2|/3)+˜ψ(|k1−k3|/2,√
+3|k2−(k3+k1)/2|/3)/bracketrightBig
+.(F11)
+2. Formal expression of the momentum distribution
+To obtain the momentum distribution, it remains to integrate over k3andk2the modulus square of (F11). In the
+limitκ0aho→+∞, one can set
+|˜ψCM(k1+k2+k3)|2= (2π)3δ(k1+k2+k3). (F12)
+Integration over k3is then straightforward:
+n(k1) = 3(√
+3/2)6/integraldisplayd3k2
+(2π)3/vextendsingle/vextendsingle/vextendsingle˜ψ(|k2−k1|/2,√
+3|k1+k2|/2)+˜ψ(|k2+k1/2|,√
+3k1/2)+˜ψ(|k1+k2/2|,√
+3k2/2)/vextendsingle/vextendsingle/vextendsingle2
+.
+(F13)
+The factor 3 in the right hand side results from the fact that, in this article, we normalize the momentum distribution
+n(k) to the total number of particles (rather than to unity). One fur ther realizes that the sum of the squared moduli
+of the arguments of ˜ψis constant and equal to k2
+1+k2
+2+k1·k2for each term in the right hand side. One uses (F6),
+thus putting the denominator in (F6) as a common denominator, to o btain
+n(k1) = 3(√
+3/2)6/integraldisplayd3k2
+(2π)3/bracketleftBig
+˜B(√
+3|k1+k2|/2)+˜B(√
+3k1/2)+˜B(√
+3k2/2)/bracketrightBig2
+(k2
+1+k2
+2+k1·k2+κ2
+0)2. (F14)41
+For simplicity, we have assumed that the normalization factor Nψis purely imaginary, so that ˜B(K) is a real quantity.
+In the above writing of n(k1), the only “nasty” contribution is ˜B(√
+3|k1+k2|/2); the other contributions are “nice”
+since they only depend on the moduli k1andk2. Expanding the square in the numerator of (F14), one gets six ter ms,
+three squared terms and three crossed terms. The change of va riablek2=−(k′
+2+k1) allows, in one of the squared
+term and in one of the crossed term, to transform a nasty term int o a nice term. What remains is a nasty crossed
+term that cannot be turned into a nice one; in that term, as a compr omise, one performs the change of variable
+k2=−(k′
+2+k1/2). We finally obtain the momentum distribution as the sum of four con tributions,
+n(k1) =nI(k1)+nII(k1)+nIII(k1)+nIV(k1), (F15)
+with
+nI(k1) = 3(√
+3/2)6/integraldisplayd3k2
+(2π)3˜B2(√
+3k1/2)
+(k2
+1+k2
+2+k1·k2+κ2
+0)2(F16)
+nII(k1) = 3(√
+3/2)6/integraldisplayd3k2
+(2π)32˜B2(√
+3k2/2)
+(k2
+1+k2
+2+k1·k2+κ2
+0)2(F17)
+nIII(k1) = 3(√
+3/2)6/integraldisplayd3k2
+(2π)34˜B(√
+3k1/2)˜B(√
+3k2/2)
+(k2
+1+k2
+2+k1·k2+κ2
+0)2(F18)
+nIV(k1) = 3(√
+3/2)6/integraldisplayd3k2
+(2π)32˜B(√
+3|k2+k1/2|/2)˜B(√
+3|k2−k1/2|/2)
+(κ2
+0+k2
+2+3k2
+1/4)2. (F19)
+We shall now take the large k1limit, or equivalently formally the κ0→0 limit for a fixed k1. From the asymptotic
+behavior(F9) we see that ˜B(k1)2involvesa sum of “oscillating”terms involving k2s0
+1ork−2s0
+1, and of “non-oscillating”
+terms. We shall calculate first the resulting non-oscillating contribu tion, then the resulting oscillating one, up to order
+1/k5
+1included.
+3. Non-oscillating contribution up to O(1/k5
+1)
+We consider the small κ0limit successively for each of the four components of n(k1) in (F15).
+Contribution I:Taking directly κ0→0 in the integral defining nI, replacing ˜B(k1) by its asymptotic behavior (F9)
+and averaging out the oscillating terms k±2s0
+1gives the leading behavior
+∝an}b∇acketle{tnI(k1)∝an}b∇acket∇i}ht ≃3√
+3
+8π|Nψ|24π5
+k5
+1. (F20)
+Contribution II:In the integrand of (F17), we use the splitting
+(k2
+1+k2
+2+k1·k2+κ2
+0)−2=k−4
+1+/bracketleftbig
+(k2
+1+k2
+2+k1·k2+κ2
+0)−2−k−4
+1/bracketrightbig
+. (F21)
+The first term in the right hand side gives a contribution exactly scalin g as 1/k4
+1. In the contribution of the second
+term in the right hand side, one may take the limit κ0→0 and replace ˜B2(√
+3k2/2) by its asymptotic expression to
+get the subleading 1 /k5
+1contribution. Performing the change of variable k2=k1qin the integral and averaging out
+the oscillating terms k±2s0
+1gives
+∝an}b∇acketle{tnII(k1)∝an}b∇acket∇i}ht=C
+k4
+1−3√
+3
+2π|Nψ|24π5
+k5
+1+o(1/k5
+1), (F22)
+with
+C= 3(√
+3/2)6/integraldisplayd3k2
+(2π)32˜B2(√
+3k2/2). (F23)
+We calculate Cfrom the exact expression (F8) of ˜B: We integrate over solid angles and we use the change of variables√
+3
+2k2=κ0sinhα, whereαvaries from zero to + ∞, to take advantage of the fact that
+˜B(κ0sinhα) =−Nψ2π5/2
+κ2
+0sinhαcoshα/parenleftbig
+es0α−e−s0α/parenrightbig
+. (F24)42
+This leads to
+C= 12π3(√
+3/2)3|Nψ|2
+κ0/integraldisplay+∞
+0dα2−(e2s0α+c.c.)
+coshα, (F25)
+where we used the fact that N2
+ψ=−|Nψ|2. The resulting integral over αmay be extended over the whole real axis
+because the integrand is an even function of α; it may then be evaluated by using the general result (that we obta ined
+with contour integration)
+K(θ,s)≡/integraldisplay+∞
+−∞dαeisα
+coshα+cosθ=2π
+sinθsinh(sθ)
+sinh(sπ)(F26)
+wheresis a real number and θ∈]0,π[. One simply has to take θ=π/2,s= 0 ands=|s0|respectively. We get
+C=24π4
+κ0/parenleftBigg√
+3
+2/parenrightBigg3
+2sinh2(|s0|π/2)
+cosh(|s0|π)|Nψ|2. (F27)
+This, together with (D5), leads to the explicit expression (F1) for C.
+Contribution III:We directly take the limit κ0→0 and we replace the factors ˜Bby their asymptotic expressions
+in (F18). After the change of variable k2=k1q, angular integration and averaging out of the oscillating terms k±2s0
+1,
+this gives
+∝an}b∇acketle{tnIII(k1)∝an}b∇acket∇i}ht=9
+2π24π5|Nψ|2
+k5
+1/integraldisplay+∞
+0dqqs0+q−s0
+q4+q2+1+o(1/k5
+1). (F28)
+In this result, we change the integration variable setting q=eα, whereαvaries from −∞to +∞. The odd component
+of the integrand (involving sinh α) gives a vanishing contribution. The even component of the integra nd involves a
+rational fraction of cosh αto which we apply a partial fraction decomposition. Then we use (F26 ) to obtain
+nIII(k1) =4π5|Nψ|2
+k5
+13√
+3
+2πsinh(π|s0|/3)+sinh(2π|s0|/3)
+sinh(π|s0|)+o(1/k5
+1). (F29)
+Contribution IV:We directly take the limit κ0→0 and we replace the factors ˜Bby their asymptotic expressions
+in (F19). We perform the change of variable k2= (k1/2)q, we average out the oscillating terms k±2s0
+1. The angular
+integration in spherical coordinates of axis the direction of k1may be performed using
+/integraldisplay
+dv/parenleftbigg1+v
+1−v/parenrightbiggs0/2
+(1−v2)−1=/parenleftbigg1+v
+1−v/parenrightbiggs0/2
+/s0, (F30)
+where the variable vis restricted to the interval ( −1,1). This leads to
+∝an}b∇acketle{tnIV(k1)∝an}b∇acket∇i}ht=4π5|Nψ|2
+k5
+136
+π2/integraldisplay+∞
+0dqq
+q2+1(q2+3)−2/bracketleftbigg
+s−1
+0/parenleftbiggq+1
+|q−1|/parenrightbiggs0
++c.c./bracketrightbigg
++o(1/k5
+1). (F31)
+Calculating this integral directly is not straightforward because of the occurrence of the absolute value |q−1|. We
+thus split the integration domain in two intervals. For q∈[0,1] we setq= (X−1)/(X+1) (an increasing function
+ofX, whereXspans [1,+∞]). Forq∈[1,+∞] we setq= (X+1)/(X−1) (a decreasing function of X, whereX
+here also spans [1 ,+∞]). Then
+∝an}b∇acketle{tnIV(k1)∝an}b∇acket∇i}ht=4π5|Nψ|2
+k5
+19
+2π2/integraldisplay+∞
+1dX
+X(X2−1+X−2)(X−X−1)
+(X2+1+X−2)2/bracketleftbig
+s−1
+0Xs0−s−1
+0X−s0/bracketrightbig
++o(1/k5
+1).(F32)
+We then set X=eα, whereαranges from zero to + ∞, and we use the fact that the resulting integrand is an even
+function of αto extend the integral over the whole real axis. We integrate by pa rts, integrating the factor sin( α|s0|),
+and we perform a partial fraction decomposition of the resulting ra tional fraction of cosh α. Using (F26) and its
+derivatives with respect to θ, we get
+∝an}b∇acketle{tnIV(k1)∝an}b∇acket∇i}ht=−12π5|Nψ|2
+k5
+1×−6[cosh(2π|s0|/3)−cosh(π|s0|/3)]+√
+3|s0|[sinh(2π|s0|/3)+sinh(π|s0|/3)]
+2π|s0|sinh(π|s0|)+o(1/k5
+1).
+(F33)43
+Sum of the four contributions: Summing up the terms in 1 /k5
+1of the contributions nI,nII,nIIIandnIV, we
+obtain as a global prefactor the quantity
+S=−√
+3
+8+cosh(2π|s0|/3)−cosh(π|s0|/3)
+|s0|sinh(π|s0|). (F34)
+Multiplying (D2) on both sides by sinh( |s0|π/2) and using
+2sinhasinhb= cosh(a+b)−cosh(a−b),∀a,b (F35)
+we find that Sis exactly zero. As a consequence, the non-oscillating part of the m omentum distribution of an infinite
+scattering length Efimov trimer behaves at large kas
+∝an}b∇acketle{tn(k1)∝an}b∇acket∇i}ht=C
+k4
+1+o(1/k5
+1). (F36)
+4. Oscillating contribution at large k1
+In the large k1tail of the momentum distribution, we now include oscillating terms, having oscillating factors such
+ask±2s0
+1. The calculation techniques are the same of in the previous subsect ion, so that we give here directly the
+result. We find that the leading oscillating terms scale as 1 /k5
+1:
+n(k1)−∝an}b∇acketle{tn(k1)∝an}b∇acket∇i}ht=−12π5
+k5
+1|Nψ|2
+A/parenleftBigg
+k1√
+3
+κ0/parenrightBigg2s0
++c.c.
++o(1/k5
+1) (F37)
+where the complex amplitude Ais the sum of the contributions coming from each of the four compon ents
+(F16,F17,F18,F19) of the moment distribution,
+A=AI+AII+AIII+AIV. (F38)
+We successively find
+AI=3
+8π2/integraldisplay+∞
+0dqq2
+q4+q2+1=√
+3
+16π, (F39)
+AII=3
+4π2/integraldisplay+∞
+0dqq2s0
+q2/bracketleftbig
+(q4+q2+1)−1−1/bracketrightbig
+=−√
+3
+4πsinh(4π|s0|/3)+sinh(2π|s0|/3)
+sinh(2π|s0|), (F40)
+AIII=3
+2π2/integraldisplay+∞
+0dqqs0
+q4+q2+1=√
+3
+4πsinh(2π|s0|/3)+sinh(π|s0|/3)−i√
+3[cosh(2π|s0|/3)−cosh(π|s0|/3)]
+sinh(π|s0|),(F41)
+AIV=12
+π22−2s0/integraldisplay+∞
+0dqq(1+q2)s0
+(q2+3)2(q2+1)/integraldisplay2q/(1+q2)
+0dv(1−v2)s0/2
+1−v2≃0.0243657158 −0.0698680251 i.(F42)
+We have calculated analytically all these integrals, except for (F42) where the angular integration gives rise to the
+integral over vand thus to a difficult hypergeometric function. We used numerical in tegration for (F42). Finally
+A ≃0.1022397786 −0.1218775240 i. (F43)
+5. Momentum distribution at the origin
+The contribution nI(k1) is straightforward to calculate at all k1:
+nI(k1) =√
+3
+4πκ0/parenleftBigg√
+3
+2/parenrightBigg6˜B2(√
+3k1/2)
+(k2
+1+4κ2
+0/3)1/2. (F44)
+The contribution nII(k1) is also exactly calculable by performing the change of variable k2= (2/√
+3)sinhαand using
+the generalization of (F26):
+/integraldisplay+∞
+−∞dαeisα
+coshα−coshα0=2πsin[s(iπ−α0)]
+sinhα0sinh(sπ), (F45)44
+whereα0is a complex number with non-zero imaginary part. This allows to obtain an exact expression of nIII(k1)
+if one further applies integration by part, integrating the factor s in(|s0|α).
+We donot givehere the resulting expressions. Contrarilyto these fi rst three contributions to n(k1), the contribution
+nIV(k1) in (F19) indeed seems difficult to calculate analytically for an arbitrar yk1, and blocked our attempt to
+calculaten(k1) exactly. For k1=0however it becomes equal to the contribution nIIand may be evaluated exactly.
+We have thus calculated the value of n(k1= 0):
+nI(0) =3˜B2(0)
+8πκ0/parenleftBigg√
+3
+2/parenrightBigg6
+(F46)
+nII(0) =6√
+3˜B2(0)
+π|s0|2κ0/parenleftBigg√
+3
+2/parenrightBigg6/braceleftbigg
+1−1
+cosh(|s0|π)
++|s0|
+3cosh(|s0|2π/3)−cosh(|s0|4π/3)
+sinh(|s0|π)cosh(|s0|π)+5√
+3
+18/bracketleftbiggsinh(|s0|4π/3)+sinh( |s0|2π/3)
+sinh(|s0|π)cosh(|s0|π)−2/bracketrightbigg/bracerightBigg
+(F47)
+nIII(0) =6√
+3˜B2(0)
+π|s0|κ0/parenleftBigg√
+3
+2/parenrightBigg6
+cosh(|s0|π/3)−cosh(|s0|2π/3)+(|s0|/√
+3)[sinh(|s0|2π/3)+sinh( |s0|π/3)]
+sinh(|s0|π)(F48)
+nIV(0) =nII(0), (F49)
+with˜B(0) =−iNψ4π5/2|s0|/κ2
+0according to (F8). This leads to (163).
+Appendix G: A direct calculation of E(Λ)
+To calculate the cut-off dependent energy E(Λ) defined in (164) for an infinite scattering length Efimov trimer,
+the method consisting in calculating the momentum distribution n(k) and then integrating (164) is numerically
+demanding: a double integral has to be performed to obtain n(k), see (F19), so that the evaluation of E(Λ) results
+in a triple integral. A more direct formulation, involving only a double inte gration, is proposed here. One simply
+rewrites (164) as
+E(Λ) =/integraldisplay
+R3d3k
+(2π)3f(k)/planckover2pi12k2
+2m/bracketleftbigg
+n(k)−C
+k4/bracketrightbigg
+(G1)
+where the function f(k) is equal to unity for 0 ≤k≤Λ and is equal to zero otherwise. Then one plugs in (G1) the
+expression (F15) of n(k), also replacing Cwith its integral expression (F23). An integration over two vector s inR3
+appears:
+E(Λ) =Eeasy(Λ)+Ehard(Λ) (G2)
+Eeasy(Λ) = 3/parenleftBigg√
+3
+2/parenrightBigg6/integraldisplayd3k
+(2π)3f(k)/planckover2pi12k2
+2m/integraldisplayd3q
+(2π)3/bracketleftBigg˜B2(√
+3
+2k)+2˜B2(√
+3
+2q)+4˜B(√
+3
+2k)˜B(√
+3
+2q)
+(k2+q2+k·q+κ2
+0)2−2˜B2(√
+3
+2q)
+k4/bracketrightBigg
+(G3)
+Ehard(Λ) = 3/parenleftBigg√
+3
+2/parenrightBigg6/integraldisplayd3k
+(2π)3f(k)/planckover2pi12k2
+2m/integraldisplayd3q
+(2π)32˜B(√
+3
+2|q+k/2|)˜B(√
+3
+2|q−k/2|)
+(q2+3
+4k2+κ2
+0)2. (G4)
+The first part Eeasyof this expression originates from the bits nI,nII,nIIIof the momentum distribution and from
+C; angular integrations may be performed, one is left with a double inte gral over the moduli kandq. Takingκ0as a
+unit of momentum and /planckover2pi12κ2
+0/mas a unit of energy in what follows:
+Eeasy(Λ) = 3/parenleftBigg√
+3
+2/parenrightBigg6/parenleftbigg4π
+(2π)3/parenrightbigg2/integraldisplayΛ
+0dkk4
+2/integraldisplay+∞
+0dqq2/bracketleftBigg˜B2(√
+3
+2k)+2˜B2(√
+3
+2q)+4˜B(√
+3
+2k)˜B(√
+3
+2q)
+(k2+q2+1)2−k2q2−2˜B2(√
+3
+2q)
+k4/bracketrightBigg
+(G5)
+that we integrate numerically. The second part Ehard(Λ) in (G4) originates from the bit nIVof the momentum
+distribution. Performing the change of variables q= (k1−k2)/2 andk=k1+k2ensures that the factors ˜Bare now
+functions of the moduli k1andk2only,
+Ehard(Λ) = 3/parenleftBigg√
+3
+2/parenrightBigg6/integraldisplayd3k1
+(2π)3/integraldisplayd3k2
+(2π)31
+2(k1+k2)2f(|k1+k2|)2˜B(√
+3
+2k1)˜B(√
+3
+2k2)
+(k2
+1+k2
+2+k1·k2+1)2(G6)45
+so that angular integrations may again be performed, involving the in tegral
+I(k1,k2) =1
+2/integraldisplay1
+−1duk2
+1+k2
+2+2k1k2u
+(k2
+1+k2
+2+k1k2u+1)2f/parenleftbigg/radicalBig
+k2
+1+k2
+2+2k1k2u/parenrightbigg
+(G7)
+=1
+k1k2/bracketleftbigg
+ln(1+k2
+1+k2
+2+k1k2u)+1+(k2
+1+k2
+2)/2
+1+k2
+1+k2
+2+k1k2u/bracketrightbiggmax[−1,min(1,U)]
+−1(G8)
+whereuis the cosine of the angle between the vectors k1andk2,U= [Λ2−(k2
+1+k2
+2)]/(2k1k2), max(a,b) (resp.
+min(a,b)) is the largest (resp. smallest) of the two numbers aandb, and the notation [ F(u)]b
+astands forF(b)−F(a)
+for any function F(u). We also used the fact that |k1+k2| ≤Λ if and only if u≤U. This leads to
+Ehard(Λ) = 3/parenleftBigg√
+3
+2/parenrightBigg6/parenleftbigg4π
+(2π)3/parenrightbigg2/integraldisplay+∞
+0dk1k2
+1/integraldisplay+∞
+0dk2k2
+2I(k1,k2)˜B(√
+3
+2k1)˜B(√
+3
+2k2). (G9)
+Further simplifications may be performed. One can map the integrat ion to the domain k1≥k2since the integrand is
+a symmetric function of k1andk2. Then performing the change of variable k1=q+k/2 andk2=q−k/2, and using
+the fact that I(k1,k2) = 0 ifk1−k2>Λ, we obtain the useful form
+Ehard(Λ) = 3/parenleftBigg√
+3
+2/parenrightBigg6/parenleftbigg4π
+(2π)3/parenrightbigg2
+2/integraldisplayΛ
+0dk/integraldisplay+∞
+k/2dq(q2−k2/4)2I(q+k/2,q−k/2)˜B/bracketleftBigg√
+3
+2(q+k/2)/bracketrightBigg
+˜B/bracketleftBigg√
+3
+2(q−k/2)/bracketrightBigg
+,
+(G10)
+that we integrate numerically. A useful result to control the nume rical error due to the truncation of the integral over
+qto a value ≫Λ and≫1 is
+I(q+k/2,q−k/2)∼
+q→+∞k4−Λ4
+8q6. (G11)
+Appendix H: Validity of the energy-momentum relation for η→0+with a smooth regularisation
+Here we prove (176) for the skeptics. We take κ0as a unit of wavevector and /planckover2pi12κ2
+0/mas a unit of energy, so that
+the energy of the infinite scattering length bosonic Efimov trimer is Etrim=−1.
+First we obtain an integral expression for Eηfor a non-zero η, using the same technique as in Appendix G, after
+having replaced ˜Bby˜Bηin (F15). After angular integration we obtain
+Eη= 3/parenleftBigg√
+3
+2/parenrightBigg6/parenleftbigg4π
+(2π)3/parenrightbigg2/integraldisplay+∞
+0dkk2/integraldisplay+∞
+0dqq2/braceleftBigg
+k2
+2˜B2
+η(√
+3
+2k)+2˜B2
+η(√
+3
+2q)+4˜Bη(√
+3
+2k)˜Bη(√
+3
+2q)
+(k2+q2+1)2−k2q2−˜B2
+η(√
+3
+2q)
+k2
++/bracketleftbigg1
+kqln1+k2+q2+kq
+1+k2+q2−kq−2+k2+q2
+(k2+q2+1)2−k2q2/bracketrightbigg
+˜Bη/parenleftBigg√
+3
+2k/parenrightBigg
+˜Bη/parenleftBigg√
+3
+2q/parenrightBigg/bracerightBigg
+. (H1)
+We collect all the squared terms in ˜B2
+η, transforming ˜B2
+η/parenleftBig√
+3
+2q/parenrightBig
+into˜B2
+η/parenleftBig√
+3
+2k/parenrightBig
+by an exchange of the integration
+variablesqandk. As a consequence the integral over qcan be performed explicitly for these terms:
+/integraldisplay+∞
+0dqq2/bracketleftbiggq2+k2/2
+(k2+q2+1)2−k2q2−1
+q2/bracketrightbigg
+=−3π
+4k2+2
+(3k2+4)1/2. (H2)
+Using the same exchange trick we that some simplification occurs amo ng the crossed terms in ˜B2
+η/parenleftBig√
+3
+2q/parenrightBig
+˜B2
+η/parenleftBig√
+3
+2k/parenrightBig
+.
+For convenience we split the final result in three pieces:
+Eη=E(1)
+η+E(2)
+η+E(3)
+η (H3)46
+with
+E(1)
+η= 3/parenleftBigg√
+3
+2/parenrightBigg6/parenleftbigg4π
+(2π)3/parenrightbigg2/parenleftbigg
+−3π
+4/parenrightbigg/integraldisplay+∞
+0dkk2˜B2
+η/parenleftBigg√
+3
+2k/parenrightBigg
+k2+2
+(3k2+4)1/2(H4)
+(H5)
+E(2)
+η= 3/parenleftBigg√
+3
+2/parenrightBigg6/parenleftbigg4π
+(2π)3/parenrightbigg2/integraldisplay+∞
+0dkk2/integraldisplay+∞
+0dqq2˜Bη/parenleftBigg√
+3
+2k/parenrightBigg
+˜Bη/parenleftBigg√
+3
+2q/parenrightBigg
+1
+kqln1+k2+q2+kq
+1+k2+q2−kq(H6)
+E(3)
+η= 3/parenleftBigg√
+3
+2/parenrightBigg6/parenleftbigg4π
+(2π)3/parenrightbigg2
+(−2)/integraldisplay+∞
+0dkk2/integraldisplay+∞
+0dqq2˜Bη/parenleftBig√
+3
+2k/parenrightBig
+˜Bη/parenleftBig√
+3
+2q/parenrightBig
+(1+k2+q2+kq)(1+k2+q2−kq). (H7)
+In a second step we use the property
+˜Bη(sinhα) =˜B(0)
+|s0|sin(|s0|α)
+sinhαcoshαe−ηα. (H8)
+Hence we perform the change of variable k= (2/√
+3)sinhαandq= (2/√
+3)sinhβ. The first piece is transformed into
+E(1)
+η= 3/parenleftBigg√
+3
+2/parenrightBigg3/parenleftbigg4π
+(2π)3/parenrightbigg2/parenleftbigg
+−3π
+4/parenrightbigg/parenleftBigg˜B(0)
+|s0|/parenrightBigg2/integraldisplay+∞
+0dαsin2(|s0|α)e−2ηα/parenleftbigg2
+3+1
+3cosh2α/parenrightbigg
+. (H9)
+Taking the limit η→0+inE(1), we see that
+E(1)
+η= 3/parenleftBigg√
+3
+2/parenrightBigg3/parenleftbigg4π
+(2π)3/parenrightbigg2/parenleftbigg
+−3π
+4/parenrightbigg/parenleftBigg˜B(0)
+|s0|/parenrightBigg2/bracketleftbigg1
+6η+/integraldisplay+∞
+0dαsin2(|s0|α)
+3cosh2α+O(η)/bracketrightbigg
+. (H10)
+The second piece is transformed into
+E(2)
+η= 3/parenleftbigg4π
+(2π)3/parenrightbigg2/parenleftBigg˜B(0)
+|s0|/parenrightBigg2
+3
+4/integraldisplay+∞
+0dα/integraldisplay+∞
+0dβsin(|s0|α)sin(|s0|β)e−η(α+β)ln3
+4+sinh2α+sinh2β+sinhαsinhβ
+3
+4+sinh2α+sinh2β−sinhαsinhβ.
+(H11)
+Calculation of this double integral looks hopeless. However with the n atural change of variables
+α=y+x
+2(H12)
+β=y−x
+2, (H13)
+of Jacobian equal to 1 /2, one can use the magic identity
+3
+4+sinh2α+sinh2β+sinhαsinhβ=/parenleftbigg1
+2+coshx/parenrightbigg/parenleftbigg
+−1
+2+coshy/parenrightbigg
+. (H14)
+Using the fact that the integrand is a symmetric function of αandβ, we can restrict the integration domain to α≥β
+that isx≥0 so that, with the well known relation sin asinb= [cos(a−b)−cos(a+b)]/2, we obtain
+E(2)
+η= 3/parenleftbigg4π
+(2π)3/parenrightbigg2/parenleftBigg˜B(0)
+|s0|/parenrightBigg2
+3
+4/integraldisplay+∞
+0dx/integraldisplay+∞
+xdy1
+2[cos(|s0|x)−cos(|s0|y)]e−ηy/bracketleftbigg
+ln/parenleftbiggcoshx+1/2
+coshx−1/2/parenrightbigg
++ln/parenleftbiggcoshy−1/2
+coshy+1/2/parenrightbigg/bracketrightbigg
+.
+(H15)
+Since this integrand is now a sum of factorized terms, one of the inte grals may be calculated (in some cases, one needs
+to exchange the order of integration over xandy). We are left with a single integration, in which we may take the
+limitη→0+:
+E(2)
+η= 3/parenleftbigg4π
+(2π)3/parenrightbigg2/parenleftBigg˜B(0)
+|s0|/parenrightBigg2
+3
+8η/integraldisplay+∞
+0dxcos(|s0|x)ln/parenleftbiggcoshx+1/2
+coshx−1/2/parenrightbigg
++O(η). (H16)47
+The third piece is transformed into
+E(3)
+η= 3/parenleftbigg4π
+(2π)3/parenrightbigg2/parenleftBigg˜B(0)
+|s0|/parenrightBigg2
+(−2)×
+/integraldisplay+∞
+0dα/integraldisplay+∞
+0dβ9
+16sin(|s0|α)sin(|s0|β)sinhαsinhβe−η(α+β)
+(3
+4+sinh2α+sinh2β+sinhαsinhβ)(3
+4+sinh2α+sinh2β−sinhαsinhβ).(H17)
+Using the change of variables (H12), (H13) and the magic relation (H 14), we obtain the simpler form
+E(3)
+η= 3/parenleftbigg4π
+(2π)3/parenrightbigg2/parenleftBigg˜B(0)
+|s0|/parenrightBigg2
+(−2)/integraldisplay+∞
+0dx/integraldisplay+∞
+xdy9
+64[cos(|s0|x)−cos(|s0|y)](coshy−coshx)e−ηy
+(cosh2x−1/4)(cosh2y−1/4).(H18)
+We can take directly the limit η→0+without producing divergingterms in E(3)
+η. The integrand is a sum of factorized
+terms; when a term involves the factor cos( |s0|x), we calculate the integral over y; when a term involves the factor
+cos(|s0|y), we calculate the integral over x. We are thus left with single integration:
+E(3)
+η= 3/parenleftbigg4π
+(2π)3/parenrightbigg2/parenleftBigg˜B(0)
+|s0|/parenrightBigg2
+(−2)/integraldisplay+∞
+0dxπ√
+3
+16cos(|s0|x)3−2coshx
+4cosh2x−1+O(η). (H19)
+Finally, it remains to collect all the three pieces in Eη. The terms proportional to 1 /ηinE(1)
+ηandE(2)
+ηcan be
+checked to cancel exactly: one uses integration by part to eliminat e the logarithmic function in the integrand of the
+coefficient of 1 /ηinE(2)
+η, and the resulting integrals in E(1)
+ηandE(2)
+ηmay be calculated using (F26). What remains
+is
+lim
+η→0+Eη=−9√
+3
+128π3/parenleftBigg˜B(0)
+|s0|/parenrightBigg2/integraldisplay+∞
+0dx/bracketleftBigg
+sin2(|s0|x)
+cosh2x+cos(|s0|x)1−2
+3coshx
+cosh2x−1
+4/bracketrightBigg
+. (H20)
+This may be calculated using again (F26). From the value of ˜B(0) given in the Appendix F, we finally obtain the
+expected result
+lim
+η→0+Eη=−1. (H21)
+Appendix I: Momentum distribution asymptotics of an Efimov t rimer in presence of a smooth regularisation
+To understand the deviation between the true Efimov trimer energ yEtrim=−/planckover2pi12κ2
+0/mand the value ¯Epredicted
+by a at first sight convincing application of an energy-momentum rela tion, see (164), we suggested in the main text
+to apply the regularisation procedure (174) depending on a parame terηthat one eventually sets to 0+. Here we give
+the expressions of the coefficients Cηand¯Dηof the asymptotics (175) of the corresponding momentum distribu tion
+nη(k).
+The calculations are similar to the one of Appendix F. We shall need simp ly the asymptotic behavior of nη(k) after
+having averaged out the O(1/k5) contributions involving oscillating terms in k±2i|s0|:
+∝an}b∇acketle{tnη(k)∝an}b∇acket∇i}ht=Cη
+k4+¯Dη
+k5e−2ηln(√
+3k)+O(1/k6). (I1)
+The coefficient ¯Dηvanishes for η→0+but it is non zero for η>0:
+¯Dη=9
+2π2|Nψ|24π5/bracketleftbiggπ
+4√
+3+Iη+Jη+Kη/bracketrightbigg
+(I2)
+with
+Iη=/integraldisplay+∞
+0dq−(1+q2)
+1+q2+q4e−2ηlnq(I3)
+Jη=/integraldisplay+∞
+0dqqs0+q−s0
+1+q2+q4e−ηlnq(I4)
+Kη=/integraldisplay+∞
+0dq8q
+1+q2e−ηln/parenleftBig
+1+q2
+4/parenrightBig
+(q2+3)2/integraldisplay2q/(1+q2)
+0dve−ηln√
+1−v2
+1−v2/bracketleftBigg/parenleftbigg1+v
+1−v/parenrightbiggs0/2
++c.c./bracketrightBigg
+(I5)48
+The contributions Iη,JηandKηoriginate respectively from the bits nII,nIIIandnIVin (F15). Their expressions
+allow a numerical calculation of ¯Dηif desired. At first sight, the calculation of Kηis more difficult because it involves
+a double integration; since the inner integral is from 0 to a function o fq, it may however be advanced step by step
+in parallel with the evaluation of the outer integral, so that the comp lexity remains the same as for a single integral.
+Anyway, such a numerical calculation for a finite ηis not necessary, what matters is the knowledge of the derivative
+d¯Dη/dηinη= 0, see (180). We obtain for the derivatives:
+dIη
+dη|η=0= 0 (I6)
+dJη
+dη|η=0=/integraldisplay+∞
+0dq(−lnq)qs0+q−s0
+1+q2+q4(I7)
+=−0.2456950243427 ... (I8)
+dKη
+dη|η=0=−/integraldisplay+∞
+0dq16q
+1+q2(q2+3)−2ln/parenleftbigg1+q2
+4/parenrightbigg
+|s0|−1sin/bracketleftbigg
+|s0|ln/parenleftbigg1+q
+|1−q|/parenrightbigg/bracketrightbigg
+−/integraldisplay+∞
+0dq2ln/parenleftbigg1+q2
+|1−q2|/parenrightbigg/bracketleftbigg/parenleftbigg1+q
+|1−q|/parenrightbiggs0
++c.c./bracketrightbigg
+1
+2(q2+3)+ln/bracketleftBig
+2(1+q2)
+q2+3/bracketrightBig
+1−q2
+ (I9)
+= 0.04934911139697 ... (I10)
+Remarkably, in (I9) a single integral is obtained, after integration b y part, taking the derivative of the bit/integraltext2q/(1+q2)
+0dv.... All the integrals may be calculated numerically to a high precision with M aple, resulting in
+d¯Dη
+dη|η=0=−8.3720476291291 ...×κ2
+0 (I11)
+and (181).
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+[132] The corresponding state vector is |Ψ∝angbracketright= [N!/(N↑!N↓!)]1/2ˆA(| ↑,...,↑,↓,...,↓∝angbracketright⊗|ψ∝angbracketright) where there are N↑spins↑and
+N↓spins↓, and the operator ˆAantisymmetrizes with respect to all particles. The wavefun ctionψ(r1,...,rN) is then
+proportional to ( ∝angbracketleft↑,...,↑,↓,...,↓ |⊗∝angbracketleftr1,...,rN|)|Ψ∝angbracketright.
+[133] For purely attractive interaction potentials such as the square-well potential, above a critical particle numbe r, the ground
+state is a collapsed state and the zero-range regime can only be reached for certain excited states.
+[134] Our derivation is similar to the one given in the two-bo dy case and sketched in the many-body case in Section 3 of [62] .
+[135] In second quantization, (40) corresponds to g(2)
+↑↓(R+r/2,R−r/2) =∝angbracketleftˆψ†
+↑(R+r/2)ˆψ†
+↓(R−r/2)ˆψ↓(R−r/2)ˆψ↑(R+r/2)∝angbracketright.
+[136] In 2Dand 3D, our result does not follow from the well-known fact that, fo r a finite-range interaction potential in
+continuous space, −/planckover2pi12
+2m/summationtext
+σ∆G(1)
+σσ(r=0) equals the kinetic energy; indeed, the Laplacian does not c ommute with the
+zero-range limit in that case.
+[137] In the lattice model in 3 D, the coupling constant g0is always negative in the zero-range limit |a| ≫b, and is an increasing
+function of −1/a, as can be seen from (13).51
+[138] Eq. (48) shows that dE0/d(lna) is proportional to the probability of finding two particles very close to each other, and it
+is natural that this probability decreases when one goes fro m the BEC limit (ln a→ −∞) to the BCS limit (ln a→+∞),
+i.e. when the interactions become less attractive [in the la ttice model in 2 D, the coupling constant g0is always negative
+in the zero-range limit a≫b, and is an increasing function of ln a, as can be seen from (14)].
+[139] In 2Dwe assume, to facilitate the derivation, that V(r) = 0 forr > b, but the result is expected to hold for any V(r)
+which vanishes quickly enough at infinity.
+[140] We consider two particles of opposite spin in a cubic bo x of sideLwith periodic boundary conditions, and we work in
+the limit where Lis much larger than |a|andb. In this limit, there exists a “weakly interacting” eigenst ateψwhose
+energy is given by the “mean-field” shift E=g/L3. The Hellmann-Feynman theorem gives g0dE/dg 0=Eint[ψ]. But the
+wavefunction ψ(r1,r2)≃Φ(r12)/L3where Φ is the zero-energy scattering state normalized by Φ →1 at infinity. Thus
+Eint=/integraltext
+d3rV(r)|Φ(r)|2/L3. The desired Eq.(120) then follows, since Φ = −aφ.
+[141] In second quantization, (150) corresponds to g(2)(R+r/2,R−r/2) =∝angbracketleftˆψ†(R+r/2)ˆψ†(R−r/2)ˆψ(R−r/2)ˆψ(R+r/2)∝angbracketright.
+[142] See Chapter 3 in [75].
+[143] The value of this constant is irrelevant for what follo ws. It could be calculated e.g. by equating the energies of th e weakly
+bound Efimov trimers of the lattice model with the ones of the z ero-range model. This was done in [75, 108], not for the
+lattice model, but for a Gaussian separable potential model .
+[144] The zero-range limit for a fixed Rtcan be taken by repeatedly dividing bby the discrete scaling factor exp( π/|s0|) where
+s0is defined in (162) and by adjusting g0so thataremains fixed. In this limit the ground state energy tends to −∞as
+follows from the Thomas effect, but the restriction of the spe ctrum to any fixed energy window converges (see e.g. [75]).
+[145] We note that it was already speculated in [63] that, in p resence of the Efimov effect, “a three-body analog of the conta ct”
+may “play an important role”.
+[146] The following tricks are used in the numerical calcula tion of the integral appearing in (170). The integral is spli t in/integraltextkmax
+0+/integraltext+∞
+kmax. A linear scale is used to discretized k/κ0in between 0 and 10. For kin between 10 and kmax(kmax
+is either 1000 or 5500), ln( k/κ0) is discretized on a linear scale. The integral from kmaxto infinity is estimated with
+the formula δn(k)≃ −36/(k/κ0)6, an approximate asymptotic expression that was tested nume rically over the range
+100< k/κ 0<1000. A simple test of this overall procedure is to check the n ormalization of n(k). It is found that the
+numerical calculation gives the correct normalization fac tor within a ≃10−6relative error. We note that calculating δn(k)
+with a one percent error for k= 5500 requires a calculation of n(k) with a relative error ≃2×10−10.
+[147] The argument of the exponential in (174) is a regular fu nction ofkaroundk= 0, contrarily to the more primitive choice
+e−2ηln(k/κ0), and is quite natural considering (F8) and (F24).
+[148] This relation also breaks down for a Fermi-Fermi mixtu re with a mass ratio equal to the critical value 13 .6...above which
+the Efimov effect occurs. Indeed, the momentum distribution t hen has a subleading contribution δnσ(k)∝1/k5[131],
+leading toadivergentintegral inthis relation. For amass r atio strictly smaller thanthecritical ratio theintegral c onverges,
+becauseδnσ(k)∝1/k5+2swheres>0 is the scaling exponent of the three-body wavefunction, ψ(λr1,λr2,λr3)∝λs−2
+forλ→0 [62].
+[149] For Efimovian eigenstates, computing the derivative o f the energy with respect to the effective range would require to use
+a regularisation procedure similar to the one employed in fr ee space in [87, 89]. However the derivative with respect to
+1/acan be computed [75].
+[150] Here we used the value of the effective range re= 1.435r0[124] for the Gaussian interaction potential V(r) =−V0e−r2/r2
+0
+withV0equal to the value where the first two-body bound state appear s.
+[151] Forafinite-rangepotential one has g(1)
+σσ(r) =n/2−r2mEkin/(3/planckover2pi12V)+...whereVisthevolume; thekinetic energydiverges
+in the zero-range limit as Ekin∼ −Eint, thusEkin∼ −C/(4π)2/integraltext
+d3rV(r)|φ(r)|2from (99), so that Ekin∼Cπ/planckover2pi12/(32mb)
+for the square-well interaction. This behavior of g(1)(r) only holds at very short distance r≪band is below the resolution
+of the Monte-Carlo data.
+[152] Strictly speaking, (136) was derived for a parabolic k inetic energy dispersion relation, whereas the dispersion relation in
+the Quantum Monte Carlo Hubbard lattice model deviates from a parabola at large k. The deviations however scale as
+k4r2
+eat lowk, so are expected to give rise to a higher order, O(r2
+e) deviation to Tc.
+[153] Tan also derived Equation (54) independently from [68 ] using Ref. [125]’s Equation (17).
+[154] In other words, the Dirac distributions originating f rom the action of the Laplacian onto the 1 /rijdivergences can be
+ignored.
\ No newline at end of file
diff --git a/1001.0775.txt b/1001.0775.txt
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+arXiv:1001.0775v1  [math.GT]  5 Jan 2010Algebraic Structures Derived from Foams
+J. Scott Carter
+University of South AlabamaMasahico Saito
+University of South Florida
+September 12, 2018
+Abstract
+Foams are surfaces with branch lines at which three sheets merge. They have been used
+in the categorification of sl(3) quantum knot invariants and also in physics. The 2 D-TQFT of
+surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where
+saddle points correspond to multiplication and comultiplication. In this paper, we explore
+algebraic operations that branch lines derive under TQFT. In partic ular, we investigate Lie
+bracket and bialgebra structures. Relations to the original Frobe nius algebra structures are
+discussed both algebraically and diagrammatically.
+1 Introduction
+Frobenius algebras have been used extensively in the study o f categorification of the Jones poly-
+nomial [10], via 2-dimensional Topological Quantum Field T heory (2D-TQFT, [11]). For categori-
+fications of other knot invariants, 2-dimensional complexe s called foams have been used instead
+[9, 13]. Although 2 D-TQFT has been characterized [11] in terms of commutative Fr obenius alge-
+bras, foams have not been algebraically characterized in te rms of TQFT. Relations to Lie algebras,
+for example, have been suggested [9, 13] through their bound aries which are called webs and that
+are trivalent graphs. Foams have branch curves along which t hree sheets meet. Similar complexes
+appear as spines of 3-manifolds, and have been used for quant um invariants [4, 5, 7, 14].
+Herein we study the types of algebraic operations that appea r along the branch curves of foams
+in relation to 2 D-TQFT. Recall that a 2 D-TQFT is a functor from the category of 2-dimensional
+cobordisms to a category of R-modules (for some suitable ring R) that assigns an R-module to
+each connected component (circle) on the boundary of a surfa ce, and an R-module homomorphism
+to a surface. In the case of a foam, we examine the associated a lgebraic operations that might be
+associated to branching circles in relation to the Frobeniu s algebra structure that occurs on the
+unbranched surfaces. Specifically, we identify and study Li e algebra and bialgebra structures in
+relation to branch curves, and study their relations to the F robenius algebra structure.
+After reviewing necessary materials in Section 2, a Lie alge bra structure for the branch curves
+is studied in Section 3, and comultiplications of bialgebra s are examined in Section 4. The foam
+skein theory based on the bialgebra case is also defined in Sec tion 4.
+12 Preliminary
+Algebraic structures we investigate include Frobenius alg ebras, Lie algebras and bialgebras. We
+restrict to the following situations.
+ALiealgebra is a module Aover a unital commutative ring Rwith a binary operation [ ,] :
+A×A→Athat is bilinear, skew symmetric ([ x,y] =−[y,x] forx,y∈A) and satisfies the Jacobi
+identity ([ x,[y,z]]+[z,[x,y]]+[y,[z,x]] = 0 for x,y,z∈A).
+AFrobenius algebra is an algebra over R(that comes with associative linear multiplication
+µ:A⊗A→Aand unit η:R→A) with a non-degenerate form ǫ:A→Rthat is associative
+(ǫ(x⊗yz) =ǫ(xy⊗z) forx,y,z∈A). There is an induced co-associative comultiplication ∆ : A→
+A⊗A. See [3] for diagrams for Frobenius algebras which we will us e in this paper. A bialgebra
+is an algebra AoverRwith a comultiplication ∆ : A→A⊗Athat is an algebra homomorphism
+(∆(xy) = ∆(x)∆(y)) and a counit ǫ:A→Rsuch that ( ǫ⊗id)∆ = id = (id ⊗ǫ)∆. The following
+are typical examples.
+Example 2.1 LetA=ANbe the Frobenius algebra of truncated polynomial AN=R[X]/(XN)
+for a commutative unital ring R, with counit (Frobenius form) ǫdetermined by ǫ(xN−1) = 1 and
+ǫ(xi) = 0 for i/ne}ationslash=N−1. The comultiplication ∆ is determined by ∆(1) =/summationtextN−1
+i=0Xi⊗XN−1−i.
+Diagrammatically, this is represented by a “neck cutting” r elation [2], which we call a∆(1)-relation
+to distinguish the specific relation given in [2] for N= 2. See the right of Fig. 4 for a diagrammatic
+representation of the ∆(1)-relation in this case.
+In general, the ∆(1)-relation is also described as follows ( see [9, 11]). For a commutative
+Frobenius algebra Aover a unital ring Rof finite rank and with a non-degenerate Frobenius form
+ǫ, there is a basis {xi}and a dual basis {yi},i= 1,...,n, such that ǫ(xiyi) =δi,j, the Kronecker
+delta, and x=/summationtext
+iyiǫ(xix). This situation is depicted in Fig. 1, where the identity ma px/mapsto→xin
+the LHS corresponds to the annular cobordism in the left of th e figure, and the sum involving the
+Frobenius form ǫis depicted in the right of the figure.
+iΣy
+xii
+Figure 1: The ∆(1)-relation
+Example 2.2 The Frobenius algebra structure on A=Z[a,b,c][X]/(X3−aX2−bX−c) is pre-
+sented in [13] as follows. The multiplication and the unit ar e defined by those for polynomials,
+the Frobenius form (counit) ǫis defined by ǫ(1) =ǫ(X) = 0,ǫ(X2) =−1. The comultiplication is
+accordingly computed as
+∆(1) = −(1⊗X2+X⊗X+X2⊗1)+a(1⊗X+X⊗1)+b(1⊗1),
+∆(X) =−(X⊗X2+X2⊗X)+a(X⊗X)−c(1⊗1),
+∆(X2) =−(X2⊗X2)−b(X⊗X)−c(1⊗X+X⊗1).
+21
+A A A AA A
+23
+Figure 2: Operation on a branch circle
+We fix a 2 D-TQFT such that a connected circle corresponds to A. For TQFTs we refer to [11].
+We follow definitions of foams in [9, 13], except that facets o f foams are decorated by basis elements
+ofA, in a general way as in [6]. A foam without boundary is called c losed.
+We briefly summarize their definitions. FoamAis the category of formal linear combination
+overRof cobordisms of compact 2-dimensional complexes in 3-spac e with the following data. (1)
+Boundaries are planar graphs with trivalent rigid vertices . (2) For an interior point of a cobordism,
+the neighborhood of each point is homeomorphic to either Euc lidean 2-space (a facet) or a branch
+curvewhere three facets of half planes meet. (3) Each facet is orie nted, and the induced orientation
+on the branch curve is consistent among three facets that sha re the curve. (4) A cyclic order of
+facets are specified using the orientation of 3-space as depi cted in Fig. 2. (5) Each facet has a basis
+element of Aassigned. (6) An annular cobordism as depicted in the left of Fig. 1 is equivalent to
+the linear combination as depicted on the right. (7) Values θ(α,β,γ)∈Aof the theta foam, as
+depicted in Fig. 3 are specified.
+In [9, 13], it was shown that the values in Aof closed foams are well-defined for values of the
+theta foams, as long as the cyclic symmetry condition θ(α,β,γ) =θ(β,γ,α) =θ(γ,α,β) is satisfied.
+βα
+γ
+Figure 3: The theta foam
+By2D-TQFTforachosen A, thetwocircles ontheleftFig. 2aremappedtothefactors of A⊗A.
+For the cyclic order along the oriented branch circle as depi cted, make a correspondence between
+the facet labeled 1, 2, 3, respectively, to the first, second, and the target factor of A⊗A→A.
+Thus the cobordism near a branch circle as depicted in the figu re induces a linear map A⊗A→A
+under the chosen TQFT and the values of theta foams. Denote th is map by m:A⊗A→A. The
+goal of this paper is to investigate this map.
+In terms of maps among tensor products of As, we use planar graphs regularly used in knot
+theory, as well as Frobenius algebras as in [3]. In particula r, the Frobenius form (the counit)
+is depicted by a maximum, unit by a minimum, (co)multiplicat ions by trivalent vertices. In this
+convention, diagrams are read from bottom to top, correspon dingto the domain and range of maps.
+The map mcorresponding to theta foams has a specified cyclic order, as indicated on the right
+of Fig. 2. The map mis defined with this specific order, and the map with the opposi te order,
+depicted by a diagram with the opposite arrow, represent the mapm◦τ, whereτ:A⊗A→A⊗A
+3is the map induced from the transposition τ(x⊗y) =y⊗x.
+3 Lie algebras
+In this section we show that there are infinitely many TQFTs un der which Lie algebra structures
+are induced from the branch circle operation. Since our goal is to exhibit a Lie bracket, in this
+section we use the notation [ ,] :A×A→A, instead of m:A⊗A→A.
+Proposition 3.1 For any commutative unital ring Rand a positive odd integer N >1, there exist
+a Frobenius algebra AoverRand values of the theta foams in FoamAsuch that the branch circle
+operation minduces a non-trivial Lie algebra structure on A.
+Proof.LetA=R[X]/(XN) for an odd integer N >1. For simplicity we denote θ(Xa,Xb,Xc) by
+θ(a,b,c) in this proof.
+LetN >3. Define θ(a,b,c) = 1 if a= 0,b+c=Nand 1< b < c , as well as all cyclic
+permutations of such ( a,b,c). Define θ(a,b,c) =−1 ifa= 0,b+c=Nand 1< c < b, as well
+as all cyclic permutations of such ( a,b,c). Finally define θ(a,b,c) = 0 for all the other cases. For
+N= 3, replace the conditions 1 < b < cand 1< c < b, respectively, by b < candc < b. We show
+that these theta foam values induce Lie brackets as desired.
+0
+kXXj
+XjkX
+XjkXi1
+Σ
+XiXN 1 iN
+Figure 4: Evaluating bracket
+The operation [ Xj,Xk] is evaluated, using the ∆(1)-relation, by
+[Xj,Xk] =N−1/summationdisplay
+i=0θ(Xi,Xj,Xk)XN−1−i.
+This calculation is depicted in Fig. 4. Since θ(i,j,k) = 0 unless i+j+k=N, we have
+[Xj,Xk] =θ(i,j,k)XN−1−iwherei=N−(j+k) andN−1−i=j+k−1, so that [ Xj,Xk] =
+θ(N−(j+k),j,k)Xj+k−1. Note that if j+k > N, then the RHS is understood to be zero from
+the definition of θ. From the definition of θby cyclic ordering, the skew symmetry of [ ,] is clear.
+We show the Jacobi identity
+[Xj,[Xk,Xℓ]]+[Xℓ,[Xj,Xk]]+[Xk,[Xℓ,Xj]] = 0
+case by case. First we compute
+[Xj,[Xk,Xℓ]] =θ(N−(k+ℓ),k,ℓ)θ(N+1−(j+k+ℓ),j,k+ℓ−1),
+[Xℓ,[Xj,Xk]] =θ(N−(j+k),j,k)θ(N+1−(j+k+ℓ),ℓ,j+k−1),
+[Xk,[Xℓ,Xj]] =θ(N−(ℓ+j),ℓ,j)θ(N+1−(j+k+ℓ),k,ℓ+j−1),
+4hence it is sufficient to prove that the sum of the right-hand si des is zero.
+Case 1:j+k+ℓ > N+1.
+In this case, the second factors of the RHS are zero, so that al l terms are zero.
+Case 2:j+k+ℓ≤N+1 and k+ℓ > N.
+This case implies that j= 0 and k+ℓ=N+1. Since N+1 is even, kandℓhave the same
+parity. The first factor θ(N−(k+ℓ),k,ℓ) is 0 since N−(k+ℓ) =−1. (When the arguments of θ
+are out of range, then θ= 0. ) Suppose k=ℓ= (N+1)/2. Then the second and the third terms
+are
+θ(N−(j+k),j,k)θ(N+1−(j+k+ℓ),ℓ,j+k−1)
+=θ((N−1)/2,0,(N+1)/2)θ(0,(N+1)/2,(N−1)/2) = (−1)(−1) = 1,
+θ(N−(ℓ+j),ℓ,j)θ(N+1−(j+k+ℓ),k,ℓ+j−1)
+=θ((N−1)/2,(N+1)/2,0)θ(0,(N+1)/2,(N−1)/2) = (1)( −1) =−1,
+as desired. Hence assume k < ℓwithout loss of generality. For the second and third terms, w e have
+θ(N−(j+k)j,k)θ(N+1−(j+k+ℓ),ℓ,j+k−1)
+=θ(ℓ−1,0,k)θ(0,ℓ,k−1) = (1)( −1) =−1,
+θ(N−(ℓ+j),ℓ,j)θ(N+1−(j+k+ℓ),k,ℓ+j−1)
+=θ(k−1,ℓ,0)θ(0,k,ℓ−1) = (1)(1) = 1 ,
+as desired.
+Case 3:j+k+ℓ≤N+1 and k+ℓ≤N.
+First we check the case where two of j,k,ℓare the same. Suppose j=k. Then the second
+term is zero, as θ(N−(j+k),j,k) = 0. Furthermore, for the first and third terms, we have
+θ(N−(k+ℓ),k,ℓ) =−θ(N−(k+ℓ),ℓ,j) and
+θ(N+1−(j+k+ℓ),j,k+ℓ−1) =θ(N+1−(j+k+ℓ),k,ℓ+j−1)
+as desired. The other cases ( k=ℓ,j=ℓ) are checked similarly. Hence we may assume j < k < ℓ .
+Sinceθvanishes unless one of the entries is 0, the first factors of th e RHS are zero if N > k+ℓ
+and 0< j < k < ℓ . hence we may assume that k+ℓ=Norj= 0.
+We continue to examine specific subcases. Suppose that j= 0 and k+ℓ < N. The RHS
+becomes:
+θ(N−(k+ℓ),k,ℓ)θ(N+1−(k+ℓ),0,k+ℓ−1)
++θ(N−k,0,k)θ(N+1−(k+ℓ),ℓ,k−1)
++θ(N−ℓ,ℓ,0)θ(N+1−(k+ℓ),k,ℓ−1).
+Ifj= 0 and k+ℓ=N, we have
+θ(0,k,ℓ)θ(1,0,N−1)+θ(ℓ,0,k)θ(1,ℓ,k−1)+θ(k,ℓ,0)θ(1,k,ℓ−1).
+If 1< k, then the sum is 0 since θ(1,0,N−1) = 0 and the arguments of the second and the
+third factors are all non-zero. If k= 1, then θ(0,1,N−1) =θ(ℓ,0,1) =θ(1,ℓ,0), so the sum is 0.
+5Now suppose that k+ℓ < N. The first term is 0 and the second factors in the sum have
+arguments that are all non-zero unless k= 1. Ifk= 1, we have
+θ(N−1,0,1)θ(N−ℓ,ℓ,0)+θ(N−ℓ,ℓ,0)θ(N−ℓ,1,ℓ−1) = 0.
+Finally, suppose that j/ne}ationslash= 0, so that k+ℓ=N. The RHS becomes:
+θ(0,N−ℓ,ℓ)θ(1−j,j,N−1)+θ(0,j,N−ℓ)θ(1−j,ℓ,N−1)+θ(N−(ℓ+j),ℓ,j)θ(1−j,N−ℓ,ℓ+j−1).
+If 1< j, then the first argument of all the second factors is negative , so the sum is 0. If j= 1, then
+each term has a factor that is 0. /square
+Since the original motivation came from the foams in [9, 13], we examine the Frobenius algebra
+in [13] closely. In this case, the multiplication that is ind uced by branch circles also satisfies the
+Jacobi identity.
+Proposition 3.2 LetA=Z[a,b,c][X]/(X3−aX2−bX−c)with Frobenius structure defined as in
+Example 2.2 from [13]. The branch curve operation [,]is skew-symmetric and satisfies the Jacobi
+identity:
+[U,[V,W]]+[V,[W,U]] +[W,[U,V]]
+for anyU,V,W∈A.
+Proof.This is confirmed by calculations. From the axioms of Aand the theta foam values that are
+given in [13]:
+θ(1,X,X2) =θ(X2,1,X) =θ(X,X2,1) = 1 = −θ(1,X2,X) =−θ(X,1,X2) =−θ(X2,X,1)
+whileθ= 0 for any other arguments, we compute using the ∆(1) relatio n for Example 2.2:
+[1,X] =−1,
+[1,X2] =X−a,
+[X,X2] =−X2+aX+b.
+Then one computes
+[1,[X,X2]] =−X,
+[X,[X2,1]] =a,
+[X2,[1,X]] =X−a,
+as desired. In general, we consider cyclic permutations of Xj, Xk, andXℓin the expression
+[Xj,[Xk,Xℓ]]. Since the bracket is skew-symmetric, then we need only co nsider the cases in which
+j,k, andℓare distinct. The remaining case follows by skew-symmetry. /square
+We define the operation ∆:A→A⊗Athat is associated to the left of Fig. 5, a diagram that
+is up-side down of Fig. 2, in which one circle branches into tw o from bottom to top. A cyclic order
+is specified in the figure. If we specify the ordered tensor fac tors assigned to each sheet by Ai,
+63A AA A
+A2
+A1
+Figure 5: Upside-down operation
+i= 1,2,3, then the operation is defined as ∆:A1→A3⊗A2. A planar diagram representing this
+operation is depicted in the right of the figure. Imitating Sw eedler notation ∆( u) =/summationtextu(1)⊗u(2)
+for comultiplication, we denote ∆(u) =/summationtextu((1))⊗u((2)).The next lemma relates this operation to
+the unit map, and diagrammatic formulations are given in Fig . 6.
+Lemma 3.3 LetA=Z[a,b,c][X]/(X3−aX2−bX−c), with∆(1)-condition defined as in Exam-
+ple 2.2. The map ∆is computed as follows.
+∆(1) = 1 ⊗X−X⊗1,
+∆(X) =a(1⊗X−X⊗1)−(1⊗X2−X2⊗1),
+∆(X2) = (a2+b)(1⊗X−X⊗1)−a(1⊗X2−X2⊗1)+(X⊗X2−X2⊗X).
+Figure 6: ∆can be defined from left or right
+Direct calculations show
+Lemma 3.4 ∆ (V) =/summationtext[V,1(1)]⊗1(2)=/summationtext1(1)⊗[1(2),V].
+The diagram for this relation is depicted in Fig. 6. Other rel ations that follow are depicted in
+Fig. 7.
+Figure 7: Other symmetric relations
+The following relations hold for maps in Frobenius algebras and maps associated to branch
+circles. Here we used the notation minstead of [ ,] to formulate in tensor products. The equalities
+are diagrammatically represented in Fig. 8.
+72
+Figure 8: Web skein relations
+Proposition 3.5 ForA=Z[a,b,c]/(X3−aX2−bX−c)withθvalues as above, the map ∆:A→
+A⊗Asatisfies the following identities:
+(m⊗|)(|⊗∆) = ∆(1)( ǫµ)−τ,
+( (m⊗|)(|⊗∆) )2=|+∆(1)(ǫµ),
+m∆= 2|.
+Proof.The first and the third equalities are verified by calculation s on basis elements. For all Xi
+andXj, it is computed as [ Xi,Xj
+((1))]⊗Xj
+((2))=Xj⊗Xi+ǫ(Xi+j)∆(1). The second relation is
+diagrammatically computed as in Fig. 9. Note that the handle element ǫµ∆(1) is 3. /square
+(εµ∆)(1) 2
+Figure 9: Proof of the skein relation
+Remark 3.6 The skein relations stated in Proposition 3.5, as planar dia grams (instead of surface
+skein relation), coincide with those described in [9] as a de scription of Kuperberg’s invariant [12],
+with the choice of q= 1.
+Thus, the operation at branch curve of the foam used to catego rify the quantum sl(3) invariant
+satisfies the skein relations at the classical limit of the in variant.
+Figure 10: A surface skein relation in [9, 13]
+Remark 3.7 The second relation in Proposition 3.5 is related to the loca l surface skein relation
+in [9, 13] as follows. Their local relation is depicted in Fig . 10. Notice the negative signs, as well
+as resemblance to our relation. After performing their rela tions locally, move the holes of each
+term along the S1factor to the other side. Then one obtains a tube connecting t wo sheets. Then
+8perform the bamboo cutting relation , that is computed by applying ∆(1)-relation three times. In
+this case, one computes that it is the negative of the origina l bamboo segment. These negative
+signs cancel, and we obtain our equation. Thus, our relation follows from theirs, or algebraically
+as we have shown.
+We also point out that the second and the third relation in Pro position 3.5 have interpretations
+inFoamA. One simply takes the product of these diagrams with S1to obtain foams, and the
+equalities hold in FoamA. The first equality, however, is not realized in FoamA, as the intersection
+of surfaces are not allowed in FoamA.
+Figure 11: Cutting a bamboo segment
+4 Bialgebras
+In this section, we investigate functors whose image of bran ch curves induce bialgebra structure
+for group algebras. Let Gbe a group. Let A=R[G] be the group ring with a commutative unital
+ringR. It is well known that Ahas a commutative Hopf algebra structure defined as follows ( see,
+for example, [11]). Define ∆:A→A⊗Aby linearly extending ∆(x) =x⊗x. (This is different
+from the comultiplication as a Frobenius algebra ∆( x) =/summationtext
+x=yzy⊗z.) The unit map is defined as
+the same as the Frobenius unit map η(1) = 1 G, where 1 Gis the identity element of G. (The counit
+map as a Frobenius algebra is defined by ǫ(1G) = 1 and ǫ(x) = 0 for x/ne}ationslash= 1G.) The following shows
+that there is a strong requirement for group algebras to give bialgebra structures through branch
+curves.
+z
+z−1
+y−1
+xxx
+xy
+Figure 12: Comultiplication by theta foams
+Proposition 4.1 LetGbe an abelian group. For any unital ring R, the branch circle operation m
+induces a bialgebra structure on Aif and only if every non-identity element of Ghas order 2.
+9Proof.The ∆(1)-relation is written as ∆(1) =/summationtext
+y∈Gy⊗y−1, and the reducing ∆into the theta
+foam is depicted in Fig. 12. For ∆(x) =x⊗xto hold in the figure, we have y=z=x, and the
+value of the theta foam being θ(x,y−1,z−1) = 1 for y=z=xand 0 otherwise.
+Forθto satisfy the cyclic symmetry, this condition is satisfied i f and only if x−1=x(xhaving
+order 2) for any x∈G, and in this case, the theta foam values are determined by θ(x,x,x) = 1 for
+anyx∈Gand 0 otherwise. /square
+Figure 13: The compatibility condition of a bialgebra
+Remark 4.2 The condition of a bialgebra that the comultiplication is an algebra homomorphism
+(also called a compatibility condition) ∆(ab) =∆(a)∆(b) fora,b∈A, is represented by surfaces
+in Fig. 13.
+(Saddle)
+∆( (1))
+1 1 1
+(Closed foams)
+Figure 14: Surface skein relations
+Remark 4.3 In [1], skein modules for 3-manifolds based on embedded surf aces modulo the surface
+skein relations described in [2] were defined and studied. Su rface skein modules were generalized in
+[6] using general commutative Frobenius algebras. Such not ions are directly generalized to foams,
+with various skein relations at hand. Skein modules for sl(3) foams are analogously defined using
+the local skein relations given in [9, 13], for example.
+Here we propose local skein relations based on the foams in Pr oposition 4.1 with the bialgebra
+on branch curves for the group ring Z[x]/(x2−1). Considering that the move characteristic to
+bialgebras is the compatibility condition as depicted in Fi g. 13, we take a local change that happens
+at the saddle point of this move, as depicted in the top of Fig. 14 (labeled as saddle) as a local
+surface skein relation. Other relations in Fig. 14 are those coming from Frobenius algebra structure
+and the theta foam values as before.
+10Thus the skein module F(M) in this case can be defined to be the isotopy classes of foams i n
+a given 3-manifold Mmodulo the local surface skein relations in Fig. 14. Althoug h computations
+of this skein module in general is out of the scope of this pape r, it seems interesting to look into
+relations between foams and the topology of 3-manifolds.
+Acknowledgments
+JSC was supported in part by NSF Grant DMS #0603926. MS was sup ported in part by NSF
+Grants DMS #0603876 and #0900671.
+References
+[1] M.Asaeda; C.Frohman, A note on the Bar-Natan skein module , arXiv:math/0602262.
+[2] D.Bar-Natan, Khovanov’s homology for tangles and cobordisms , Geom. Topol. 9(2005) 1443–
+1499.
+[3] J.S.Carter, A.S.Crans, M.Elhamdadi, E.Karadayi, M.Sa ito,Cohomology of Frobenius Al-
+gebras and the Yang-Baxter Equation, Communications in Contemporary Mathematics 10,
+Suppl. 1 (2008), 791–814.
+[4] J.S.Carter, D.Flath, M.Saito, Classical and Quantum 6j Symbols, Mathematical notes, vol.
+43, Princeton University Press, 1995.
+[5] S.Chung, M.Fukuma, A.Shapere, Structures of topological lattice field theories in three di -
+mensions, Internat. J. Modern Phys. A 9(1994), 1305–1360.
+[6] U.Kaiser, Frobenius algebras and skein modules of surfaces in 3-manifolds , arXiv:0802.4068.
+[7] L.H.Kauffman, M.Saito, M.C.Sullivan, Quantum invariants of templates, J. Knot Theory
+Ramifications, 12(2003), 653-681.
+[8] M.Khovanov, A categorification of the Jones polynomial , Duke Math. J., 101(3)(1999), 359–
+426.
+[9] M.Khovanov, sl(3)link homology , Algebraic & Geometric Topology, 4(2004), 1045–1081.
+[10] M.Khovanov, Link homology and Frobenius extensions , Fundamenta Mathematicae, 190
+(2006), 179–190.
+[11] J.Kock, Frobenius algebras and 2D topological quantum field theorie s,London Mathematical
+Society Student Texts 59, Cambridge University Press, 2003.
+[12] G.Kuperberg, Spiders for rank 2Lie algebras, Comm. Math. Phys., 180(1)(1996), 109–151.
+[13] M.Mackaay; P.Vaz, The universal sl3-link homology , arXiv:math/0603307.
+[14] V.Turaev, O.Viro, State Sum Invariants of 3-Manifolds and Quantum 6J-Symbols , Topology
+31, No 4 (1992), 865–902.
+11
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+arXiv:1001.0776v2  [astro-ph.HE]  7 Jan 2010PROCEEDINGS OF THE 31stICRC, Ł´OD´Z 2009 1
+Atmospheric Variations as observed by IceCube
+Serap Tilav∗, Paolo Desiati†, Takao Kuwabara∗, Dominick Rocco†,
+Florian Rothmaier‡, Matt Simmons∗, Henrike Wissing§¶for the IceCube Collaboration/bardbl
+∗Bartol Research Institute and Dept. of Physics and Astronom y, University of Delaware, Newark, DE 19716, USA.
+†Dept. of Physics, University of Wisconsin, Madison, WI 5370 6, USA.
+‡Institute of Physics, University of Mainz, Staudinger Weg 7 , D-55099 Mainz, Germany.
+§III Physikalisches Institut, RWTH Aachen University, D-52 056 Aachen, Germany.
+¶Dept. of Physics, University of Maryland, College Park, MD 2 0742, USA.
+/bardblSee the special section of these proceedings
+Abstract.Wehavemeasuredthecorrelationofrates
+in IceCube with long and short term variations in
+the South Pole atmosphere. The yearly temperature
+variation in the middle stratosphere (30-60 hPa) is
+highly correlated with the high energy muon rate
+observed deepin the ice,andcausesa ±10%seasonal
+modulation in the event rate. The counting rates of
+the surface detectors, which are due to secondary
+particles of relatively low energy (muons, electrons
+and photons), have a negative correlation with tem-
+peratures in the lower layers of the stratosphere
+(40-80 hPa), and are modulated at a level of ±5%.
+The regionof the atmospherebetween pressure levels
+20-120 hPa, where the first cosmic ray interactions
+occurandtheproducedpions/kaonsinteractordecay
+to muons, is the Antarctic ozone layer. The anti-
+correlation between surface and deep ice trigger
+rates reflects the properties of pion/kaon decay and
+interaction as the density of the stratospheric ozone
+layer changes. Therefore, IceCube closely probes the
+ozone hole dynamics, and the temporal behavior of
+the stratospheric temperatures.
+Keywords : IceCube, IceTop, South Pole
+I. INTRODUCTION
+The IceCube Neutrino Observatory, located at the
+geographical South Pole (altitude 2835m), has been
+growing incrementally in size since 2005, surrounding
+its predecessor AMANDA[1]. As of March 2009, Ice-
+Cube consists of 59 strings in the Antarctic ice, and
+59 stations of the IceTop cosmic ray air shower array
+on the surface. Each IceCube string consists of 60
+Digital Optical Modules (DOMs) deployed at depths of
+1450-2450m, and each IceTop station comprises 2 ice
+Cherenkov tanks with 2 DOMs in each tank.
+IceCube records the rate of atmospheric muons with
+Eµ≥400 GeV. Muon events pass the IceCube Simple
+Majority Trigger (InIce SMT) if 8 or more DOMs are
+triggered in 5 µsec. The IceTop array recordsair showers
+with primary cosmic ray energies above 300 TeV. Air
+showers pass the IceTop Simple Majority Trigger (Ice-
+TopSMT) if 6 or more surface DOMs trigger within
+5µsec.As the Antarctic atmosphere goes through seasonal
+changes, the characteristics of the cosmic ray interac-
+tions in the atmosphere follow these variations. When
+the primary cosmic ray interacts with atmospheric nu-
+clei, the pions and kaons produced at the early interac-
+tions mainly determine the nature of the air shower. For
+thecosmicrayenergiesdiscussedhere,theseinteractions
+happen in the ozone layer (20-120 hPa pressure layer
+at 26 down to 14 km) of the South Pole stratosphere.
+During the austral winter, when the stratosphere is cold
+anddense,thechargedmesonsaremorelikelytointeract
+and produce secondary low energy particles. During the
+austral summer when the warm atmosphere expandsand
+becomes less dense, the mesons more often decay rather
+than interact.
+Figure 1 demonstrates the modulation of rates in
+relationto thetemporalchangesoftheSouthPole strato-
+sphere. The scaler rate of a single IceTop DOM on the
+surface is mostly due to low energy secondary particles
+(MeV electrons and gammas, ∼1 GeV muons)[2] pro-
+duced throughout the atmosphere, and therefore highly
+modulated by the atmospheric pressure. However, after
+correcting for the barometric effect, the IceTop DOM
+counting rate also reflects the initial pion interactions in
+the middle stratosphere. The high energy muon rate in
+the deep ice, on the other hand, directly traces the decay
+characteristics of high energy pions in higher layers
+of the stratosphere. The IceCube muon rate reaches its
+maximum at the end of January when the stratosphere
+is warmest and most tenuous. Around the same time
+Icetop DOMs measure the lowest rates on the surface
+as the pion interaction probability reaches its minimum.
+The high energy muon rate starts to decline as the
+stratosphere gets colder and denser. The pions will
+interact more often than decay, yielding the maximum
+rate in IceTop and the minimum muon rate in deep ice
+at the end of July.
+II. ATMOSPHERIC EFFECTS ON THE ICECUBERATES
+The Antarctic atmosphere is closely monitored by the
+NOAA Polar Orbiting Environmental Satellites (POES)
+and by the radiosonde balloon launches of the South
+Pole Meteorology Office. However, stratospheric data2 ATMOSPHERIC VARIATION
+ 180 200 220 240 260Tp [K]
+(a)p=20-30hPa
+30-40hPa
+40-50hPa
+50-60hPa
+60-70hPa
+70-80hPa
+80-100hPa
+ 1200 1300 1400 1500 1600 1700
+ 640 660 680 700 720 740IceTop DOM count. [Hz]
+Pressure [hPa]
+(b)Observed
+Pressure corrected
+Pressure
+ 900 1000 1100 1200
+05/01
+200707/01
+200709/01
+200711/01
+200701/01
+200803/01
+200805/01
+200807/01
+200809/01
+200811/01
+200801/01
+200903/01
+2009 180 200 220 240IceCube muon rate [Hz]
+Teff [K]
+(c)Observed
+Teff
+Fig. 1. The temporal behavior of the South Pole stratosphere from May 2007 to April 2009 is compared to IceTop DOM counting rate and
+the high energy muon rate in the deep ice. (a) The temperature profiles of the stratosphere at pressure layers from 20 hPa to 100 hPa where
+the first cosmic ray interactions happen. (b) The IceTop DOM c ounting rate (black -observed, blue -after barometric corr ection) and the surface
+pressure (orange). (c) The IceCube muon trigger rate and the calculated effective temperature (red).
+is sparse during the winter when the balloons do not
+reach high altitudes, and satellite based soundings fail
+to return reliable data. For such periods NOAA derives
+temperatures from their models. We utilize both the
+ground-based data and satellite measurements/models
+for our analysis.
+A. Barometric effect
+In first order approximation, the simple correlation
+between log of rate change ∆{lnR}and the surface
+pressure change ∆Pis
+∆{lnR}=β·∆P (1)
+whereβis the barometric coefficient.
+As shown by the black line in the Figure 1b, the
+observed IceTop DOM counting rate varies by ±10% in
+anti-correlationwithsurfacepressure,andthebarometri c
+coefficient is determined to be β=−0.42%/hPa. Using
+this value, the pressure corrected scaler rate is plotted
+as the smoother line (blue) in Figure 1b. The cosmic
+ray shower rate detected by the IceTop array also varies
+by±17% in anti-correlation with surface pressure, and
+can be corrected with a βvalue of −0.77%/hPa. As
+expected [3], the IceCube muon rate shown in Figure
+1c is not correlated with surface pressure. However,
+during exceptional stratospheric temperature changes,
+the secondordertemperatureeffect on pressurebecomes
+large enoughto cause anti-correlationof the high energy
+muon rate with the barometric pressure. During suchevents the effect directly reflects sudden stratospheric
+density changes, specifically in the ozone layer.
+B. Seasonal Temperature Modulation
+Figure 1 clearly demonstrates the seasonal temper-
+ature effect on the rates. The IceTop DOM counting
+rate, after barometric correction, shows ±5% negative
+temperature correlation. On the other hand, the IceCube
+muon rate is positively correlated with ±10% seasonal
+variation.
+Fromthephenomenologicalstudies[4][5],it isknown
+that correlation between temperature and muon intensity
+can be described by the effective temperature Teff,
+defined by the weighted average of temperatures from
+the surface to the top of the atmosphere. Teffapproxi-
+mates the atmosphere as an isothermal body, weighting
+each pressure layer according to its relevance to muon
+production in atmosphere [5][6].
+The variation of muon rate ∆Rµ/ < Rµ>is related
+to the effective temperature as
+∆Rµ
+< Rµ>=αT∆Teff
+< Teff>, (2)
+whereαTis the atmospheric temperature coefficient.
+Using balloon and satellite data for the South Pole
+atmosphere, we calculated the effective temperature as
+the red line in Figure 1c. We see that it traces the
+IceCube muon rate remarkably well. The calculated
+temperature coefficient αT= 0.9for the IceCube muonPROCEEDINGS OF THE 31stICRC, Ł´OD´Z 2009 3
+TABLE I
+Temperature and correlation coefficients of rates for diffe rent
+stratospheric layers of 20-100 hPa and Teff.
+IceCube Muon IceTop Count.
+P < Tp>αp
+Tγ αp
+Tγ
+(hPa) (K)
+20-30 214.0 0.512 0.953 -0.194 -0.834
+30-40 208.7 0.550 0.986 -0.216 -0.906
+40-50 207.3 0.591 0.993 -0.240 -0.946
+50-60 206.6 0.627 0.985 -0.261 -0.968
+60-70 206.3 0.656 0.971 -0.278 -0.975
+70-80 206.3 0.679 0.954 -0.292 -0.975
+80-100 206.5 0.708 0.927 -0.310 -0.971
+< Teff>αTγ α Tγ
+211.3 0.901 0.990 -0.360 -0.969
+rate agrees well with the expectations of models as well
+as with other experimental measurements[7][8].
+In this paper, we also study in detail the relation be-
+tween rates and stratospheric temperatures for different
+pressure layers from 20 hPa to 100 hPa as
+∆R
+< R >=αp
+T∆Tp
+< Tp>. (3)
+The temperature coefficients for each pressure layer
+αp
+Tandthe correlationcoefficient γare determinedfrom
+regression analysis. Pressure-corrected IceTop DOM
+counting rate, and IceCube muon rate are sorted in
+bins of∼10 days, and deviations ∆R/ < R > from
+the average values are compared with the deviation of
+temperatures at different depths ∆Tp/ < Tp>.
+We list the values of αp
+Tandγfor IceCube muon
+rate and IceTop DOM counting rate in Table 1. We
+find that the IceCube muon rate correlates best with
+the temperatures of 30-60 hPa pressure layers, while the
+IceTop DOM counting rate shows the best correlation
+with 60-80 hPa layers. In Figure 2. we plot the rate and
+temperature correlation for layers which yield the best
+correlation.
+III. EXCEPTIONAL STRATOSPHERIC EVENTS AND
+THEMUON RATES
+The South Pole atmosphere is unique because of
+the polar vortex. In winter a large-scale counter clock-
+wise flowing cyclone forms over the entire continent
+of Antarctica, isolating the Antarctic atmosphere from
+higher latitudes. Stable heat loss due to radiative cooling
+continues until August without much disruption, and the
+powerful Antarctic vortex persists until the sunrise in
+September. As warm air rushes in, the vortex loses its
+strength, shrinks in size, and sometimes completely dis-
+appears in austral summer. The density profile inside the
+vortex changes abruptly during the sudden stratospheric
+warming events, which eventually may cause the vortex
+collapse. The ozone depleted layer at 14-21 km altitude
+(ozone hole), observed in September/October period, is
+usually replaced with ozone rich layer at 18-30 km soon
+after the vortex breaks up.-15-10-5 0 5 10 15
+-20-15-10-5 0 5 10 15 20∆ R / < R > [%]
+∆ Tp / < Tp > [%]IceCube muon rate vs. Tp (p=40-50hPa)
+α = 0.591
+γ  = 0.993
+(a)
+-15-10-5 0 5 10 15
+-20-15-10-5 0 5 10 15 20∆ R / < R > [%]
+∆ Teff / < Teff > [%]IceCube muon rate vs. Teff
+α = 0.901
+γ  = 0.990
+(b)-15-10-5 0 5 10 15
+-20-15-10-5 0 5 10 15 20∆ R / < R > [%]
+∆ Tp / < Tp > [%]IceTop DOM count. vs. Tp (p=60-70hPa)
+α = -0.278
+γ  = -0.975
+(c)
+-15-10-5 0 5 10 15
+-20-15-10-5 0 5 10 15 20∆ R / < R > [%]
+∆ Teff / < Teff > [%]IceTop DOM count. vs. Teff
+α = -0.360
+γ = -0.969
+(d)
+Fig. 2. Correlation of IceCube muon and IceTop DOM counting r ates
+with stratospheric temperatures and Teff. (a) IceCube muon rate vs.
+temperature at 40-50 hPa pressure layer. (b) IceCube muon ra te vs.
+effective temperature. (c) IceTop DOMcounting ratevs.tem perature at
+70-80 hPa pressure layer. (d) IceTop DOM counting rate vs. ef fective
+temperature.
+Apart from the slow seasonal temperature variations,
+IceCube also probes the atmospheric density changes
+due to the polar vortex dynamics and vigorous strato-
+spheric temperature changes on time scales as short as
+days or even hours, which are of great meteorological
+interest.
+An exceptional and so far unique stratospheric event
+has already been observed in muon data taken with
+IceCube’s predecessor AMANDA-II.
+A. 2002AntarcticozoneholesplitdetectedbyAMANDA
+In late September 2002 the Antarctic stratosphere
+underwent its first recorded major Sudden Stratospheric
+Warming (SSW), during which the atmospheric temper-
+atures increased by 40 to 60 K in less than a week.
+This unprecedented event caused the polar vortex and
+the ozone hole, normally centered above the South Pole,
+to split intotwosmaller,separateoff-centerparts(Figur e
+3) [9].
+Figure4showsthestratospherictemperaturesbetween
+Septemberand October2002alongwith the AMANDA-
+II muon rate. The muon rate traces temperature varia-
+tions in the atmospherein great detail, with the strongest
+correlation observed for the 40-50hPa layer.
+B. South Pole atmosphere 2007-2008
+Unlike in 2002, the stratospheric conditions over
+Antarctica were closer to average in 2007 and 2008.
+In 2007 the polar vortex was off-center from the South
+Pole during most of September and October, resulting in
+greaterheatfluxinto thevortex,whichdecreasedrapidly
+in size. When it moved back over the colder Pole region
+in early November it gained strength and persisted until
+the beginning of December. The 2008 polar vortex was4 ATMOSPHERIC VARIATION
+Fig. 3. Ozone concentration over the southern hemisphere on
+September 20th 2002 (left) and September 25th 2002 (right) [ 10].
+Date 09/09 23/09 07/10 21/10 Rate [Hz] 
+64 66 68 70 72 74 T [K] 
+180 190 200 210 220 230 240 250  20 - 30  hPa 
+ 30 - 40  hPa 
+ 40 - 50  hPa 
+ 50 - 60  hPa 
+ 60 - 70  hPa 
+ 70 - 80  hPa 
+ 80 -100 hPa 
+Fig. 4. Average temperatures in various atmospheric layers over the
+South Pole (top) and deep ice muon rate recorded with the AMAN DA-
+IIdetector (bottom) during theAntarctic ozoneholesplito fSeptember-
+October 2002.
+one of the largest and strongest observed in the last 10
+years over the South Pole. Because of this, the heat flux
+entering the area was delayed by 20 days. The 2008
+vortex broke the record in longevity by persisting well
+into mid-December.
+In Figure 5 we overlay the IceCube muon rate over
+the temperatureprofilesoftheAntarcticatmospherepro-
+ducedbytheNOAAStratosphericanalysisteam[11].We
+note that the anomalous muon rates (see, for example,
+the sudden increase by 3% on 6 August 2008) observed
+by IceCube are in striking correlation with the middle
+and lower stratospheric temperature anomalies.
+We are establishing automated detection methods for
+anomalous events in the South Pole atmosphere as well
+as a detailed understanding and better modeling of
+cosmic ray interactions during such stratospheric events.
+IV. ACKNOWLEDGEMENTS
+We are grateful to the South Pole Meteorology Office
+and the Antarctic Meteorological Research Center of
+the University of Wisconsin-Madison for providing the
+Fig. 5. The temperature time series of the Antarctic atmosph ere
+produced by NOAA[11] are shown for 2007 and 2008. The pattern
+observed in the deep ice muon rate (black line) is superposed onto
+the plot to display the striking correlation with the strato spheric
+temperature anomalies.
+meteorological data. This work is supported by the
+National Science Foundation.
+REFERENCES
+[1] Karle, A. for the IceCube Collaboration (2008), IceCube : Con-
+struction Status and First Results, arXiv:0812.3981v1.
+[2] Clem, J. et al. (2008), Response of IceTop tanks to low-en ergy
+particles, Proceedings of the 30th ICRC, Vol. 1 (SH), p.237- 240.
+[3] Dorman, L. (2004), Cosmic Rays in the Earth’s Atmosphere and
+Underground, Springer Verlag.
+[4] Barrett P. et al. (1952), Interpretation of cosmic-ray m easurements
+far underground, Reviews of Modern Physics, Vol. 24, Issue 3 ,
+p.133-178.
+[5] Ambrosio,M.etal. (1997),Seasonal variations in theun derground
+muon intensity as seen by MACRO, Astroparticle Physics, Vol . 7,
+Issue 1-2, p.109-124.
+[6] Gaisser, T. (1990), Cosmic rays and particle physics. Ca mbridge,
+UK: Univ. Pr.
+[7] Grashorn, E. et al., Observation of the seasonal variati on in
+underground muon intensity, Proceedings of the 30th ICRC, V ol.
+5 (HE part 2), p.1233-1236.
+[8] Osprey, S., et al. (2009), Sudden stratospheric warming s seen in
+MINOS deep underground muon data, Geophys. Res. Lett., 36,
+L05809, doi:10.1029/2008GL036359.
+[9] Varotsos, C., et al. (2002), Environ. Sci. & Pollut. Res. , p.375.
+[10] http://ozonewatch.gsfc.nasa.gov
+[11] http://www.cpc.noaa.gov/products/stratosphere/s trat-trop/
\ No newline at end of file
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+arXiv:1001.0777v3  [math-ph]  13 May 2010Analytic functions of the annihilation operator
+Aleksandar Petrovi´ c∗
+M.Gorkog 27, 11000 Belgrade, Serbia
+A method for construction of an non-entire function fof the annihilation operator ais given for
+the first time. f(z) is analytic on some compact domain that does not separate th e complex plane.
+A new form of the identity is given, which is well suited for th e function’s domain. Using Runge’s
+polynomial approximation theorem, such a function fof the annihilation operator is defined on the
+whole domain. The constructed operators are given in terms o f dyads formed of Fock states.
+PACS numbers: 03.65.Ca, 03.65.Ta
+I. INTRODUCTION
+The annihilation operator in Hilbert space is operator awhich satisfies relation
+[a,a†] = 1. (1)
+A self-adjoint Number operator Nand the corresponding Fock states are constructed using a:
+N=a†a, N|n/an}bracketri}ht=n|n/an}bracketri}ht, n= 0,1,2,3,... (2)
+Eigenstates of aare called coherent states:
+a|α/an}bracketri}ht=α|α/an}bracketri}ht,|α/an}bracketri}ht=e−|α|
+22∞/summationdisplay
+n=0αn
+√
+n!|n/an}bracketri}ht, α∈C. (3)
+The coherentstates aredenoted with Greekletters ( |α/an}bracketri}ht,|β/an}bracketri}ht,|γ/an}bracketri}ht), and the Fock stateswith Latin letters ( |l/an}bracketri}ht,|k/an}bracketri}ht,|n/an}bracketri}ht,|m/an}bracketri}ht).
+The identity can be formed using eigenvectors |α/an}bracketri}ht:
+I=1
+π/integraldisplay
+d2α|α/an}bracketri}ht/an}bracketle{tα|. (4)
+ais not a normal operator, so The Spectral Theorem cannot be app lied to construct f(a). Nevertheless, every entire
+function of the operator acan be represented in a form analogous to the resolution of a norma l operator with respect
+to its projective measure:
+f(a) =1
+π/integraldisplay
+d2αf(α)|α/an}bracketri}ht/an}bracketle{tα|. (5)
+Contrary to the normal operator case, non-entire functions of acannot be represented in such a way [1],[2]. Previous
+unsuccessful attempts to use ln awere pointed out in [1]. Related questions concerning this issue were also studied by
+Perelomov[3]. The underlying problem is the fact that until now non-e ntire functions of the annihilation operatorhave
+not been studied in detail, and some straightforward assumptions a rising from resolution (5) have lead to erroneous
+conclusions. In this paper, we construct a function f(a) of the annihilation operator, where fis analytic on a compact
+domain that does not separate the complex plane. The operators c onstructed here are given in terms of dyads formed
+of Fock states.
+∗Electronic address: a.petrovic.phys@gmail.com2
+II. RESULTS
+A. New identity resolution
+Standard resolution of identity (4) is not suitable for non-entire fu nctions, so a new resolution of identity is given.
+To do that, non-normalized coherent states (NCS) are defined:
+/tildewider|α/an}bracketri}ht=eαa†|0/an}bracketri}ht=e|α|
+22
+|α/an}bracketri}ht. (6)
+Translation operator for NCS is:
+T(β) =eβa†, T(β)/tildewider|α/an}bracketri}ht=/tildewider|α+β/an}bracketri}ht, α,β∈C. (7)
+Now, an identity whose eigenstates are on the circle of radius R cent ered at the origin can be formed from NCS :
+I=−i/contintegraldisplay
+|γ|=Rdγ
+γ/tildewider|γ/an}bracketri}ht/tildewider/an}bracketle{tγ|J, (8)
+where
+J=1
+2π∞/summationdisplay
+n=0n!|n/an}bracketri}ht/an}bracketle{tn|. (9)
+B. Runge’s approximation theorem and functions of the annih ilation operator
+In this subsection, f(a) wherefis an analytic function defined on a compact domain Ω that does not se parate the
+complex plane, is constructed. Two classes of such functions are c onsidered. The first class consist of functions f
+defined on domains containing 0: 0 ∈Ω. The second class consists of functions which are not defined at 0 : 0/ne}ationslash∈Ω. A
+convenient theoretical tool for this is Runge’s polynomial approxim ation theorem.
+Theorem: Iffis an analytic function on a compact domain Ω that does not separate the complex plane, then there
+exists a sequence Pl(z) of polynomials such that converges uniformly to f(z) on Ω [4],[5],[6]:
+f(z) =∞/summationdisplay
+l=0Pl(z−z0) =∞/summationdisplay
+l=0dl/summationdisplay
+k=0c(l)
+k(z−z0)k,c(l)
+k∈C,z,z0∈Ω. (10)
+Let us consider the first class of analytic function (0 ∈Ω) and let
+f(z) =∞/summationdisplay
+l=0dl/summationdisplay
+k=0c(l)
+kzk,z∈Ω (11)
+be an expansion of fin a series of polynomials. Then (11) and (8) can be used to construc t the function f(a) of
+annihilation operator:
+f(a) =∞/summationdisplay
+l=0dl/summationdisplay
+k=0c(l)
+kak·I=−i∞/summationdisplay
+l=0dl/summationdisplay
+k=0c(l)
+k/contintegraldisplay
+|γ|=Rdγγk−1/tildewider|γ/an}bracketri}ht/tildewider/an}bracketle{tγ|J. (12)
+Using definition (6) of/tildewider|γ/an}bracketri}ht, and after performing integration, f(a) becomes:3
+f(a) =∞/summationdisplay
+l=0∞/summationdisplay
+n=0dl+n/summationdisplay
+m=nc(l)
+m−n/radicalbigg
+m!
+n!|n/an}bracketri}ht/an}bracketle{tm|, (13)
+which for α∈Ω gives:
+f(a)|α/an}bracketri}ht=f(α)|α/an}bracketri}ht. (14)
+Let us now examine the second class of f. Again we can use (10), but requiring z0/ne}ationslash= 0,z0∈Ω. However, a repetition
+of the above procedure using (8) leads to a divergence problem. To resolve this issue, a different resolution of identity,
+one involving translation (7) is chosen:
+I=ez0a†Ie−z0a†. (15)
+IfIin the right hand side of equation (15) is substituted with (8), and us ing (7) we get:
+I=−i/contintegraldisplay
+|γ|=Rdγ
+γ/tildewider|γ+z0/an}bracketri}ht/tildewider/an}bracketle{tγ|J e−z0a†. (16)
+Now we apply the operator, obtained by polynomial approximation, t o the identity (16):
+f(a) =∞/summationdisplay
+l=0dl/summationdisplay
+k=0c(l)
+k(a−z0)k·I=−i∞/summationdisplay
+l=0dl/summationdisplay
+k=0c(l)
+k/contintegraldisplay
+|γ|=Rdγγk−1/tildewider|γ+z0/an}bracketri}ht/tildewider/an}bracketle{tγ|J e−z0a†. (17)
+Again, using definition (6) of/tildewider|γ/an}bracketri}ht, and after performing integration, f(a) becomes:
+f(a) =∞/summationdisplay
+l=0dl/summationdisplay
+k=0c(l)
+k∞/summationdisplay
+n=0n+k/summationdisplay
+m=k/parenleftbiggn
+m−k/parenrightbigg/radicalbigg
+m!
+n!zn−m+k
+0|n/an}bracketri}ht/an}bracketle{tm|e−z0a†, (18)
+which for α∈Ω gives:
+f(a)|α/an}bracketri}ht=f(α)|α/an}bracketri}ht. (19)
+We can simplify (18), collecting coefficients at dyads, as
+f(a) =χe−z0a†, (20)
+χ=∞/summationdisplay
+l=0∞/summationdisplay
+n=0∞/summationdisplay
+m=0χ(l)
+nm|n/an}bracketri}ht/an}bracketle{tm|,
+χ(l)
+nm=/radicalbigg
+m!
+n!sl/summationdisplay
+k=pc(l)
+k/parenleftbiggn
+m−k/parenrightbigg
+zn−m+k
+0 ,p= max{0,m−n}, sl= min{m,dl}.4
+III. DISCUSSION AND CONCLUSION
+In this paper, for the first time an expression for a non-entire fun ctionf(a) of the annihilation operator is obtained.
+f(z) is an analytic function on an compact domain Ω ⊂Cthat does not separate the complex plane.
+A very significant application of our result is its use in a construction o f lna. Since ln zand 1/zcan be defined on
+domain which satisfies previous conditions, our method allows us to ob tain lnaand 1/a. Since, formally, [ln a,a†] =
+1/a, it follows
+[a†a,−ilna] =i. (21)
+Operator −ilnais conjugate to Number operator. It is not self-adjoint, but due t o commutation relation (21) it
+can serve as a good starting point for construction of the Phase O perator [1],[2]. Considering the significance of this
+result, it will be topic of a separate paper.
+Acknowledgments
+The autor would like to thank Drs M.Arsenovi´ c, D.Arsenovi´ c, D.Da vidovi´ c and J.Ajti´ c for their suggestions and
+help in preparation of this manuscript.
+[1] Lj. Davidovi´ c, D. Arsenovi´ c, M. Davidovi´ c and D.M. Da vidovi´ c, J.Phys.A: Math. Theor. 42, 235302 (2009).
+[2] M. Davidovi´ c, D. Arsenovi´ c and D.M. Davidovi´ c, J.Phy s.: Conf. Ser. 36, 46 (2006).
+[3] A.M. Perelomov, Theor. Math. Phys. 6(Engl.Transl.), 213 (1971).
+[4] E.B. Saff, ”Proceedings of Symposia in Aplied Mathematic Vol.36”, (1986).
+[5] A.I. Markushevich, ”Theory of function of complex varia ble”, (1977).
+[6] L.V. Ahlfors, ”Complex Analysis” (1979).
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+arXiv:1001.0778v1  [math.AP]  5 Jan 2010The Neumann Problem and Helmholtz Decomposition
+in Convex Domains
+Jun Geng and Zhongwei Shen∗
+Abstract
+We show that the Neumann problem for Laplace’s equation in a c onvex domain Ω
+with boundary data in Lp(∂Ω) is uniquely solvable for 1 < p <∞. As a consequence,
+we obtain the Helmholtz decomposition of vector fields in Lp(Ω,Rd).
+1 Introduction
+The main purpose of this paper is to prove the following.
+Theorem 1.1. LetΩbe a bounded convex domain in Rd,d≥2. Let1<p<∞. Then the
+LpNeumann problem for ∆u= 0inΩis uniquely solvable. That is, given any f∈Lp(∂Ω)
+with mean value zero, there exists a harmonic function uinΩ, unique up to constants, such
+that(∇u)∗∈Lp(∂Ω)and∂u
+∂n=fn.t. on∂Ω. Moreover, the solution satisfies the estimate
+/ba∇dbl(∇u)∗/ba∇dblp≤C/ba∇dblf/ba∇dblp, whereCdepends only on d,pand the Lipschitz character of Ω.
+Here and thereafter ( ∇u)∗denotes the nontangential maximal function of ∇uandn
+the unit outward normal to ∂Ω. By∂u
+∂n=fn.t. on∂Ω we mean that for a.e. P∈∂Ω,
+<∇u(x),n(P)>converges to f(P) asx→Pnontangentially. We remark that for a
+harmonic function uandp≥2, (∇u)∗∈Lp(∂Ω) implies that uis in the Sobolev space
+Lp
+1+1
+p(Ω) [8]. The solutions in Theorem 1.1 satisfy the estimate /ba∇dbl∇u/ba∇dblLp
+1/p(Ω)≤C/ba∇dblf/ba∇dblpfor
+2≤p<∞.
+It is known that the LpNeumann problem for ∆ u= 0 in a bounded C1domain is
+uniquely solvable for any p∈(1,∞) [2]. However, if Ω is a general Lipschitz domain, the
+sharp range of p’s, for which the LpNeumann problem in Ω is solvable, is 1 < p <2 +ε,
+whereε >0 depends on Ω (see [7, 14, 1]; also see [9] for references on related work on
+boundary value problems in Lipschitz domains). In [10], for any given L ipschitz domain Ω
+andp>2, Kim and Shen established a necessary and sufficient condition for t he solvability
+of theLpNeumann problem in Ω. More precisely, it is shown in [10] that the LpNeumann
+problem for ∆ u= 0 in Ω is solvable if and only if there exist positive constants C0andr0
+such that for any 0 < r < r 0andQ∈∂Ω, the following weak reverse H¨ older inequality on
+∂Ω,
+/braceleftbigg1
+rd−1/integraldisplay
+B(Q,r)∩∂Ω|(∇v)∗|pdσ/bracerightbigg1/p
+≤C0/braceleftbigg1
+rd−1/integraldisplay
+B(Q,2r)∩∂Ω|(∇v)∗|2dσ/bracerightbigg1/2
+,(1.1)
+∗Supported in part by NSF grant DMS-0855294.
+1holds for any harmonic function vin Ω satisfying ( ∇v)∗∈L2(∂Ω) and∂v
+∂n= 0 onB(Q,3r)∩
+∂Ω (see [10, Theorem 1.1]). Using this condition, Kim and Shen [10] obtain ed the solvability
+of theLpNeumann problem for ∆ u= 0 in bounded convex domains in Rdfor 1< p <∞
+ifd= 2; for 1< p <4 ifd= 3; and for 1 < p <3 +εifd≥4 (see [10, Theorem 1.2]).
+Theorem 1.1 extends the results in [10] in the case d≥3 and completely solves the Lp
+Neumann problem for Laplace’s equation in convex domains.
+Our approach to Theorem 1.1 follows the proof of Theorem 1.2 in [10]. T o establish
+the weak reverse H¨ older inequality (1.1), we use the square funct ion estimates for harmonic
+functions in Lipschitz domains and the local W2,2estimate in convex domains. This reduces
+the problem to the estimate of
+sup
+x∈B(P,r)|∇2v(x)|p−2[δ(x)]p−1−t, (1.2)
+wheret∈(0,1),δ(x) = dist(x,∂Ω), andvis a harmonic function in Ω such that∂v
+∂n= 0 on
+B(P,3r)∩∂Ω and (∇v)∗∈L2(∂Ω). In [10] the authors used the interior estimates and the
+localW2,2to obtain that for any x∈B(P,r)∩Ω,
+|∇2v(x)| ≤C
+r/bracketleftbiggr
+δ(x)/bracketrightbiggd
+2/braceleftbigg1
+rd/integraldisplay
+B(P,3r)∩Ω|∇v(y)|2dy/bracerightbigg1/2
+. (1.3)
+By a reflection argument the classical De Giorgi-Nash estimate implies that for any x∈
+B(P,r)∩Ω,
+|∇2v(x)| ≤C
+r/bracketleftbiggr
+δ(x)/bracketrightbigg2−α/braceleftbigg1
+rd/integraldisplay
+B(P,3r)∩Ω|∇v(y)|2dy/bracerightbigg1/2
+, (1.4)
+whereα >0 depends on Ω. Substituting (1.3) for d= 2,3 and (1.4) for d≥4 into (1.2)
+and choose tsufficiently close to 0, we see that the exponent of δ(x) would be positive for
+anyp >2 ifd= 2; forp <4 ifd= 3; and for p <3 +εifd≥4. This leads to the
+restriction of pford≥3 in [10, Theorem 1.2]. In this paper we will show that if d≥3, for
+anyx∈B(P,r)∩Ω,
+|∇2v(x)| ≤Cη
+r/bracketleftbiggr
+δ(x)/bracketrightbigg1+η/braceleftbigg1
+rd/integraldisplay
+B(P,3r)∩Ω|∇v(y)|2dy/bracerightbigg1/2
+(1.5)
+foranyη>0. Substituting(1.5)into(1.2),weseethattheexponentof δ(x)is−η(p−2)+1−t,
+which would be positive for any p>2 ifη>0 is sufficiently small.
+To show (1.5), we will prove that if Ω is a convex domain with smooth bou ndary, then
+for anyq >2,
+/braceleftbigg1
+rd/integraldisplay
+B(Q,r)∩Ω|∇v|qdx/bracerightbigg1/q
+≤C/braceleftbigg1
+rd/integraldisplay
+B(Q,2r)∩Ω|∇v|2dx/bracerightbigg1/2
+, (1.6)
+whenevervis harmonic in Ω and v∈C2(Ω),∂v
+∂n= 0 inB(Q,3r)∩∂Ω. The constant Cin
+(1.6) depends only on d,qand the Lipschitz character of Ω. Our proof of (1.6) is inspired
+by a recent paper of V. Maz’ya [11], in which he established the L∞gradient estimate for
+2solutions of the Neumann-Laplace problem in convex domains. More p recisely, it is proved
+in [11] that if q > dandf∈Lq(Ω) with mean value zero, then /ba∇dbl∇u/ba∇dblL∞(Ω)≤C/ba∇dblf/ba∇dblLq(Ω),
+where−∆u=fin Ω and∂u
+∂n= 0 on∂Ω. Although the proof of (1.6) does not rely on this
+estimate, the formulation of our main technical lemma, Lemma 2.2, as well as its proof, is
+motivated by [11].
+With Theorem 1.1 at our disposal, following the potential approach de veloped by Fabes,
+Mendez, Mitrea[3]inLipschitzdomains, wemaystudythesolvabilityof thePoissonequation
+withNeumann boundaryconditions inconvex domains. Inparticular, consider the boundary
+value problem
+
+∆u=f∈Lp
+−1,0(Ω),
+∂u
+∂n=g∈Bp
+−1/p(∂Ω),
+u∈W1,p(Ω).(1.7)
+HereLp
+−1,0(Ω) is the dual of Lq
+1(Ω) =W1,q(Ω) andBp
+−1/p(∂Ω) the dual of the Besov space
+Bq
+1/p(∂Ω) on∂Ω, whereq=p
+p−1. We will call u∈W1,p(Ω) a solution to (1.7) with data
+(f,g), if
+/integraldisplay
+Ω∇u·∇φdx=−<f,φ> Lp
+−1,0(Ω)×Lq
+1(Ω)+<g,Tr(φ)>Bp
+−1/p(∂Ω)×Bq
+1/p(∂Ω)(1.8)
+for anyφ∈W1,q(Ω), where Tr(φ) denotes the trace of φon∂Ω.
+Theorem 1.2. LetΩbe a bounded convex domain in Rd,d≥2. Let1< p <∞. Then
+for anyf∈Lp
+−1,0(Ω)andg∈Bp
+−1/p(∂Ω)satisfying the compatibility condition < f,1>=<
+g,1>, the Poisson problem (1.7) has a unique (up to constants) sol utionu. Moreover, the
+solutionusatisfies the estimate
+/ba∇dbl∇u/ba∇dblLp(Ω)≤C/braceleftBig
+/ba∇dblf/ba∇dblLp
+−1,0(Ω)+/ba∇dblg/ba∇dblBp
+−1/p(∂Ω)/bracerightBig
+, (1.9)
+whereCdepends only on d,pand the Lipschitz character of Ω.
+We remark that for bounded Lipschitz or C1domains, the inhomogeneous Neumann
+problem, ∆ u=f∈Lp
+1/p−s−1,0(Ω) in Ω,∂u
+∂n=g∈Bp
+−s(∂Ω) andu∈Lp
+1−s+1/p(Ω) with
+s∈(0,1) andp∈(1,∞), was studied in [3], where the authors obtained the solvability for
+the sharp ranges of pands. Analogous results for the inhomogeneous Dirichlet problem
+in Lipschitz or C1domains may be found in [8]. In particular, it follows from [3] that the
+boundary value problem (1.7) is solvable for p∈((3/2)−ε,3+ε) if Ω is Lipschitz; and for
+p∈(1,∞) if Ω isC1.
+LetLp
+σ(Ω) denote the subspace of functions vinLp(Ω,Rd) such that/integraltext
+Ωv·∇φdx= 0 for
+anyφ∈C1(Rd). As a corollary of Theorem 1.2, we establish the Helmholtz decompos ition
+ofLpvector fields on convex domains for 1 <p<∞.
+Theorem 1.3. LetΩbe a bounded convex domain in Rd,d≥2and1<p<∞. Then
+Lp(Ω,Rd) =gradW1,p(Ω)⊕Lp
+σ(Ω). (1.10)
+3That is, given any u∈Lp(Ω,Rd), there exist φ∈W1,p(Ω), unique up to a constant, and a
+uniquev∈Lp
+σ(Ω)such that u=∇φ+v.Moreover,
+max/braceleftbig
+/ba∇dbl∇φ/ba∇dblLp(Ω),/ba∇dblv/ba∇dblLp(Ω)/bracerightbig
+≤Cp/ba∇dblu/ba∇dblLp(Ω), (1.11)
+whereCpdepends only on d,pand the Lipschitz character of Ω.
+A useful tool in the study of the Navier-Stokes equations, the He lmholtz decomposition
+(1.10) is well known for smooth domains (see e.g. [4]). It was proved in [3] that (1.10)-(1.11)
+hold forp∈((3/2)−ε,3 +ε) if Ω is Lipschitz; and for p∈(1,∞) if Ω isC1. The range
+(3/2)−ε<p<3+εis known to be sharp for Lipschitz domains [3].
+2 Estimates on smooth convex domains
+The purpose of this section is to establish the following.
+Theorem 2.1. LetΩbe a bounded convex domain in Rd,d≥3withC2boundary. Let
+u∈C3(Ω). Suppose that ∆u= 0inΩand∂u
+∂n= 0onB(Q,3r)∩∂Ωfor someQ∈∂Ωand
+0<r<r 0. Then for any q>2,
+/braceleftbigg1
+rd/integraldisplay
+B(Q,r)∩Ω|∇u|qdx/bracerightbigg1/q
+≤C/braceleftbigg1
+rd/integraldisplay
+B(Q,2r)∩Ω|∇u|2dx/bracerightbigg1/2
+, (2.1)
+whereCdepends only on d,qand the Lipschitz character of Ω.
+The summation convention will be used in this section.
+TheproofofTheorem2.1reliesonthefollowinglemma. Aswementioned inIntroduction,
+the formulation of Lemma 2.2 as well as its proof is inspired by a paper o f Maz’ya [11].
+Lemma 2.2. LetΩbe a bounded convex domain with C2boundary. Suppose that v=
+(v1,...,vd)∈C2(Ω,Rd)andv·n= 0on∂Ω. Letg=|v|2. Then for a.e. t∈(0,∞),
+/integraldisplay
+{g=t}|∇g|dσ≤2√
+t/integraldisplay
+{g=t}
+
+/parenleftBigg/summationdisplay
+i,j/vextendsingle/vextendsingle∂vi
+∂xj−∂vj
+∂xi/vextendsingle/vextendsingle2/parenrightBigg1/2
++|div(v)|
+
+dσ
++2/integraldisplay
+{g>t}/braceleftbigg
+|div(v)|2−∂vi
+∂xj·∂vj
+∂xi/bracerightbigg
+dx,(2.2)
+whereσ=Hd−1denotes the d−1dimensional Hausdorff measure and {g=t}={x∈Ω :
+g(x) =t},{g>t}={x∈Ω :g(x)>t}.
+Proof.Let Ψ be a nonnegative Lipschitz function on [0 ,∞). It follows from integration by
+parts that
+4/integraldisplay
+ΩΨ(|v|2)∂vi
+∂xj·∂vj
+∂xidx=−2/integraldisplay
+ΩΨ′(|v|2)vk·∂vk
+∂xj·vi·∂vj
+∂xidx
+−/integraldisplay
+ΩΨ(|v|2)·vi·∂
+∂xi/braceleftbig
+div(v)/bracerightbig
+dx
++/integraldisplay
+∂ΩΨ(|v|2)vinj∂vj
+∂xidσ
+=−2/integraldisplay
+ΩΨ′(|v|2)vk·∂vk
+∂xj·vi·∂vj
+∂xidx
++2/integraldisplay
+ΩΨ′(|v|2)vk·∂vk
+∂xi·vi·div(v)dx
++/integraldisplay
+ΩΨ(|v|2)/braceleftbig
+div(v)/bracerightbig2dx
++/integraldisplay
+∂ΩΨ(|v|2)/braceleftbigg
+vinj∂vj
+∂xi−vinidiv(v)/bracerightbigg
+dσ.(2.3)
+This gives
+/integraldisplay
+ΩΨ(|v|2)/braceleftbigg/braceleftbig
+div(v)/bracerightbig2−∂vi
+∂xj·∂vj
+∂xi/bracerightbigg
+dx
+=/integraldisplay
+∂ΩΨ(|v|2)/braceleftbigg
+vinidiv(v)−vinj∂vj
+∂xi/bracerightbigg
+dσ
++2/integraldisplay
+ΩΨ′(|v|2)/braceleftbigg
+vk·∂vk
+∂xj·vi·∂vj
+∂xi−vk·∂vk
+∂xi·vi·div(v)/bracerightbigg
+dx.(2.4)
+Using the assumptions that v·n= 0 on∂Ω and Ω is a convex domain with C2boundary,
+we observe that
+vinidiv(v)−vinj∂vj
+∂xi=−β(vT;vT)≥0 on∂Ω,
+wherevT=v−(v·n)nis the tangential component of von∂Ω andβ(·,·) the second
+fundamental quadratic form of ∂Ω (see [6, p.137]). Hence,
+2/integraldisplay
+ΩΨ′(|v|2)·vk·∂vk
+∂xj·vi·∂vj
+∂xidx
+≤2/integraldisplay
+ΩΨ′(|v|2)·vk·∂vk
+∂xi·vi·div(v)dx
++/integraldisplay
+ΩΨ(|v|2)/braceleftbigg/braceleftbig
+div(v)/bracerightbig2−∂vi
+∂xj·∂vj
+∂xi/bracerightbigg
+dx.(2.5)
+5Letg=|v|2. Then|∇g|2= 4vk·∂vk
+∂xj·vi·∂vi
+∂xj. It follows from (2.5) that
+1
+2/integraldisplay
+ΩΨ′(g)/vextendsingle/vextendsingle∇g|2dx≤2/integraldisplay
+ΩΨ′(g)vk·∂vk
+∂xj·vi/braceleftbigg∂vi
+∂xj−∂vj
+∂xi/bracerightbigg
+dx
++2/integraldisplay
+ΩΨ′(g)·vk·∂vk
+∂xi·vi·div(v)dx
++/integraldisplay
+ΩΨ(g)/braceleftbigg/braceleftbig
+div(v)/bracerightbig2−∂vi
+∂xj·∂vj
+∂xi/bracerightbigg
+dx.(2.6)
+We now fix 0 <t <τ < ∞. Let Ψ be continuous so that Ψ( s) = 1 fors≥τ, Ψ(s) = 0
+fors≤t, and Ψ is linear on [ t,τ]. In view of (2.6), we obtain
+1
+2(τ−t)/integraldisplay
+t<g<τ|∇g|2dx≤1
+τ−t/integraldisplay
+t<g<τ|∇g||v|/braceleftBigg/summationdisplay
+i,j/vextendsingle/vextendsingle∂vi
+∂xj−∂vj
+∂xi/vextendsingle/vextendsingle2/bracerightBigg1/2
+dx
++1
+τ−t/integraldisplay
+t<g<τ|∇g||v||div(v)|dx
++/integraldisplay
+g>tΨ(g)/braceleftbigg/braceleftbig
+div(v)/bracerightbig2−∂vi
+∂xj·∂vj
+∂xi/bracerightbigg
+dx.(2.7)
+By the co-area formula, we may rewrite (2.7) as
+1
+2(τ−t)/integraldisplayτ
+t/integraldisplay
+g=s|∇g|dσds≤1
+τ−t/integraldisplayτ
+t/integraldisplay
+g=s|v|/braceleftBigg/summationdisplay
+i,j/vextendsingle/vextendsingle∂vi
+∂xj−∂vj
+∂xi/vextendsingle/vextendsingle2/bracerightBigg1/2
+dσds
++1
+τ−t/integraldisplayτ
+t/integraldisplay
+g=s|v||div(v)|dσds
++/integraldisplay
+g>tΨ(g)/braceleftbigg/braceleftbig
+div(v)/bracerightbig2−∂vi
+∂xj·∂vj
+∂xi/bracerightbigg
+dx.(2.8)
+Lettingτ→t+in (2.8), we obtain the desired estimate by the Lebesgue’s differentia tion
+theorem.
+Next we apply Lemma 2.2 to harmonic functions in Ω with normal derivat ives vanishing
+on part of the boundary.
+Lemma 2.3. LetΩbe a bounded convex domain with C2boundary and Q∈∂Ω. Let
+u∈C3(Ω). Suppose that ∆u= 0inΩand∂u
+∂n= 0onB(Q,2r)∩∂Ωfor somer >0. Then
+for a.e.t∈(0,∞),
+/integraldisplay
+g=t|∇g|dσ≤6√
+t/integraldisplay
+g=t|∇u||∇ϕ|dσ+2/integraldisplay
+g>t|∇u|2|∇ϕ|2dx, (2.9)
+whereg=|(∇u)ϕ|2andϕ∈C∞
+0(B(Q,2r)).
+6Proof.Letv= (∇u)ϕ. Thenv·n= 0 on∂Ω and
+∂vi
+∂xj=ϕ∂2u
+∂xi∂xj+∂u
+∂xi∂ϕ
+∂xj.
+It follows that div( v) = (∆u)ϕ+∇u·∇ϕ=∇u·∇ϕand
+∂vi
+∂xj−∂vj
+∂xi=∂u
+∂xi∂ϕ
+∂xj−∂u
+∂xj∂ϕ
+∂xi.
+Hence,/parenleftBigg/summationdisplay
+i,j/vextendsingle/vextendsingle∂vi
+∂xj−∂vj
+∂xi/vextendsingle/vextendsingle2/parenrightBigg1/2
++|div(v)|
+=/braceleftbig
+2|∇u|2|∇ϕ|2−2(∇u·∇ϕ)2/bracerightbig1/2+|∇u·∇ϕ|
+≤3|∇u||∇ϕ|.(2.10)
+Next note that
+|div(v)|2−∂vj
+∂xi∂vi
+∂xj=−ϕ2|∇2u|2−2ϕ·∂2u
+∂xi∂xj·∂u
+∂xi·∂ϕ
+∂xj
+=−/summationdisplay
+i,j/parenleftbigg
+ϕ∂2u
+∂xi∂xj−∂u
+∂xi∂ϕ
+∂xj/parenrightbigg2
++|∇u|2|∇ϕ|2
+≤ |∇u|2|∇ϕ|2.(2.11)
+Inviewof(2.10)-(2.11), estimate(2.9)inLemma2.3nowfollowsreadily fromLemma2.2.
+Lemma 2.4. LetΩbe a bounded convex domain in Rd,d≥3. Letf,gbe two nonnegative
+functions on Ω. Suppose that f∈C(Ω),g∈C1(Ω)and
+/integraldisplay
+g=t|∇g|dσ≤C0/braceleftbigg√
+t/integraldisplay
+g=t|f|dσ+/integraldisplay
+g>t|f|2dx/bracerightbigg
+(2.12)
+for a.e.t∈(0,∞). Then there exists Cdepending only on d,q,C0and the Lipschitz
+character of Ωsuch that
+/braceleftbigg/integraldisplay
+Ω|g|qdx/bracerightbigg1/q
+≤C/braceleftbigg/integraldisplay
+Ω|f|2pdx/bracerightbigg1/p
++C|Ω|1
+q−1/integraldisplay
+Ω|g|dx, (2.13)
+wherep>1and1
+q=1
+p−2
+d.
+Proof.By considering gδ=g+δand then letting δ→0+, we may assume that gis bounded
+from below by a positive constant. Using the co-area formula and (2 .12), we obtain
+/integraldisplay
+Ω|g|α|∇g|2dx=/integraldisplay∞
+0tα/integraldisplay
+g=t|∇g|dσdt
+≤C0/integraldisplay∞
+0tα/braceleftbigg
+t1
+2/integraldisplay
+g=t|f|dσ+/integraldisplay
+g>t|f|2dx/bracerightbigg
+dt
+≤C/integraldisplay
+Ω|g|α+1
+2|∇g||f|dx+C/integraldisplay
+Ω|g|α+1|f|2dx,(2.14)
+7whereα>−1. By the Cauchy inequality with an ε>0,
+/integraldisplay
+Ω|g|α+1
+2|∇g||f|dx≤ε/integraldisplay
+Ω|g|α|∇g|2dx+Cε/integraldisplay
+Ω|g|α+1|f|2dx.
+This, together with (2.14), implies that
+/integraldisplay
+Ω|g|α|∇g|2dx≤C/integraldisplay
+Ω|g|α+1|f|2dx. (2.15)
+Since Ω is convex, there exists a constant C, depending only on dand [diam(Ω)]d/|Ω|,
+such that/braceleftbigg/integraldisplay
+Ω|w−wΩ|2d
+d−2dx/bracerightbiggd−2
+d
+≤C/integraldisplay
+Ω|∇w|2dx, (2.16)
+wherew∈C1(Ω) andwΩdenotes the average of wover Ω. Let β >(1/2) andw=gβin
+(2.16). We obtain
+/braceleftbigg/integraldisplay
+Ω|g|2dβ
+d−2dx/bracerightbiggd−2
+d
+≤C/integraldisplay
+Ω|g|2β−2|∇g|2dx+C|Ω|−1−2
+d/braceleftbigg/integraldisplay
+Ω|g|βdx/bracerightbigg2
+. (2.17)
+Letα= 2β−2. It follows from (2.15) and (2.17) that
+/braceleftbigg/integraldisplay
+Ω|g|2dβ
+d−2dx/bracerightbiggd−2
+d
+≤C/integraldisplay
+Ω|g|2β−1|f|2dx+C|Ω|−1−2
+d/braceleftbigg/integraldisplay
+Ω|g|βdx/bracerightbigg2
+,(2.18)
+for anyβ >(1/2).
+We now choose p0>1 so that (2 β−1)p0=2dβ
+d−2. By H¨ older’s inequality,
+/integraldisplay
+Ω|g|2β−1|f|2dx≤/braceleftbigg/integraldisplay
+Ω|g|(2β−1)p0dx/bracerightbigg1/p0/braceleftbigg/integraldisplay
+Ω|f|2p′
+0dx/bracerightbigg1/p′
+0
+≤ε/braceleftbigg/integraldisplay
+Ω|g|(2β−1)p0dx/bracerightbiggp1
+p0+Cε/braceleftbigg/integraldisplay
+Ω|f|2p′
+0dx/bracerightbiggp′
+1
+p′
+0,(2.19)
+wherep1=2β
+2β−1. Note thatp1
+p0=d−2
+d. Also 2p′
+0=4dβ
+d−2+4βandp′
+1
+p′
+0=d−2+4β
+d. In view of
+(2.18)-(2.19), we obtain
+/braceleftbigg/integraldisplay
+Ω|g|2dβ
+d−2dx/bracerightbiggd−2
+d
+≤C/braceleftbigg/integraldisplay
+Ω|f|4dβ
+d−2+4βdx/bracerightbiggd−2+4β
+d
++C|Ω|−1−2
+d/braceleftbigg/integraldisplay
+Ω|g|βdx/bracerightbigg2
+.(2.20)
+Finally we let p=2dβ
+d−2+4βandq=2dβ
+d−2. It follows from (2.20) that
+/braceleftbigg/integraldisplay
+Ω|g|qdx/bracerightbigg1/q
+≤C/braceleftbigg/integraldisplay
+Ω|f|2pdx/bracerightbigg1/p
++C|Ω|−1
+2β−1
+dβ/braceleftbigg/integraldisplay
+Ω|g|βdx/bracerightbigg1/β
+. (2.21)
+Note that1
+q=1
+p−2
+dand 2β= (1−2
+d)q. Also−1
+2β−1
+dβ=1
+q−1
+β. Sinceβ <q, the desired
+estimate follows from (2.21) by H¨ older’s inequality.
+8We are now in a position to give the proof of Theorem 2.1.
+Proof of Theorem 2.1.
+Let 1< ρ < τ < 2. Choose ϕ∈C∞
+0(B(Q,τr)) such that ϕ= 1 inB(Q,ρr) and
+|∇ϕ| ≤C[(τ−ρ)r]−1. It follows from Lemmas 2.3 and 2.4 that
+/braceleftbigg/integraldisplay
+Ω|(∇u)ϕ|2qdx/bracerightbigg1/q
+≤C/braceleftbigg/integraldisplay
+Ω|∇u|2p|∇ϕ|2pdx/bracerightbigg1/p
++C|Ω|1
+q−1/integraldisplay
+Ω|(∇u)ϕ|2dx,(2.22)
+wherep>1 and1
+q=1
+p−2
+d. This yields that
+/braceleftbigg1
+rd/integraldisplay
+Ω∩B(Q,ρr)|∇u|2qdx/bracerightbigg1/(2q)
+≤C/braceleftbigg1
+rd/integraldisplay
+Ω∩B(Q,τr)|∇u|2pdx/bracerightbigg1/(2p)
+(2.23)
+for anyp >1 and1
+q=1
+p−2
+d, where 1< ρ < τ < 2. By a simple iteration argument, we
+obtain/braceleftbigg1
+rd/integraldisplay
+Ω∩B(Q,r)|∇u|qdx/bracerightbigg1/q
+≤C/braceleftbigg1
+rd/integraldisplay
+Ω∩B(Q,3r/2)|∇u|pdx/bracerightbigg1/p
+, (2.24)
+for any 2< p < q < ∞. Estimate (2.1) follows readily from (2.24) and the reverse H¨ older
+inequality,
+/braceleftbigg1
+rd/integraldisplay
+Ω∩B(Q,3r/2)|∇u|¯pdx/bracerightbigg1/¯p
+≤C/braceleftbigg1
+rd/integraldisplay
+Ω∩B(Q,2r)|∇u|2dx/bracerightbigg1/2
+, (2.25)
+where ¯p>2. Wemention that (2.25)holds even forsolutions of elliptic systems o f divergence
+form with bounded measurable coefficients. See e.g. [5, Chapter V] f or the interior case.
+The boundary case follows from the interior case by a reflection arg ument.
+Remark 2.5. Let Ω be a bounded Lipschitz domain in Rd,d≥2. Suppose that ∆ u= 0
+in Ω, (∇u)∗∈L2(∂Ω) and∂u
+∂n= 0 onB(Q,3r)∩∂Ω. Then the estimate (2.1) holds for
+2<q<3+εifd≥3; and for 2 <q<4+εifd= 2. To show this, one uses the fact that the
+L2Neumann problem in Lipschitz domains is solvable as well as the observa tion thatuisCα
+inB(Q,2r)∩Ω for some α>0 ifd≥3; and for some α>(1/2) ifd= 2. We refer the reader
+to [13, pp.188-189], where the same estimate was proved for a Lipsc hitz domain Ω, under the
+Dirichlet condition u= 0 onB(Q,3r)∩∂Ω. The proof in[13] extends easily to the case of the
+Neumann boundary condition. We point out that if d≥3, theCα(α>0) estimate follows
+from the De Giorgi-Nash estimate by a reflection argument. For the cased= 2, one may
+use the solvability of the LpNeumann problem in Lipschitz domains for some p= ¯p>2 and
+the square function estimates to show that ∇u∈L¯p
+1/¯p(B(Q,2r)∩Ω)⊂L2¯p(B(Q,2r)∩Ω).
+By Sobolev imbedding, this implies that uisCαonB(Q,2r)∩Ω for some α>(1/2). If Ω is
+C1, the estimate (2.1) holds for any d≥2 andq>2. This follows from the fact that the Lp
+Neumann problem in C1domains is solvable for any p >2. Since the results in this paper
+do not use the estimates mentioned above, we omit the details here.
+93 Weak reserve H¨ older inequality on the boundary
+The goal of this section is to prove the following.
+Theorem 3.1. Under the same conditions on Ωanduas in Theorem 2.1, we have
+/braceleftbigg1
+rd−1/integraldisplay
+B(Q,r)∩∂Ω|(∇u)∗|pdσ/bracerightbigg1/p
+≤C/braceleftbigg/integraldisplay
+B(Q,2r)∩∂Ω|(∇u)∗|2dσ/bracerightbigg1/2
+,(3.1)
+for anyp>2, whereCdepends only on d,pand the Lipschitz character of Ω.
+We begin with a local W2,2estimate.
+Lemma 3.2. Under the same conditions on Ωanduas in Theorem 2.1, we have
+/integraldisplay
+B(Q,r)∩Ω|∇2u|2dx≤C
+r2/integraldisplay
+B(Q,2r)∩Ω|∇u|2dx (3.2)
+whereCdepends only on d.
+Proof.See e.g. [10, p.1826].
+Letδ(x) = dist(x,∂Ω).
+Lemma 3.3. Letwbe a harmonic function in a bounded Lipschitz domain Ω. Letp >2.
+Fixx0∈Ωsuch thatδ(x0)≥c0diam(Ω). Then for any t∈(0,1),
+/integraldisplay
+∂Ω|(∇w)∗|pdσ≤C/braceleftbig
+diam(Ω)/bracerightbigtsup
+x∈Ω|∇2w(x)|p−2[δ(x)]p−1−t/integraldisplay
+Ω|∇2w|2dy
++C|∇w(x0)|p|∂Ω|,(3.3)
+whereCdepends only on d,p,t,c0and the Lipschitz character of Ω.
+Proof.See e.g. [10, p.1827].
+Proof of Theorem 3.1.
+Since Ω is a Lipschitz domain, by rotation and translation, we may assu me thatQ= 0
+and
+B(Q,C0r0)∩Ω =/braceleftbig
+(x′,xd) :xd>ψ(x′)/bracerightbig
+∩B(Q,C0r0)
+whereψ:Rd−1→Rsuch thatψ(0) = 0 and /ba∇dbl∇ψ/ba∇dbl∞≤M, andC0= 10√
+d(1+M). Let
+S(r) =/braceleftbig
+(x′,ψ(x′)) :|x′|<r/bracerightbig
+.
+We will show that if u∈C2(Ω) is harmonic in Ω and∂u
+∂n= 0 onS(8r), then
+/braceleftbigg1
+rd−1/integraldisplay
+S(r)|(∇u)∗|pdσ/bracerightbigg1/p
+≤C/braceleftbigg1
+rd−1/integraldisplay
+S(4r)|(∇u)∗|2dσ/bracerightbigg1/2
+, (3.4)
+whereCdepends only on d,pandM. Estimate (3.1) follows from (3.4) by a simple covering
+argument.
+10ForP∈∂Ω, define
+M1(∇u)(P) = sup/braceleftbig
+|∇u(x)|:x∈Ω,|x−P|<C0δ(x) and|x−P| ≤c0r/bracerightbig
+,
+M2(∇u)(P) = sup/braceleftbig
+|∇u(x)|:x∈Ω,|x−P|<C0δ(x) and|x−P|>c0r/bracerightbig(3.5)
+Note that ( ∇u)∗= max{M1(∇u),M2(∇u)}. The desired estimate for M2(∇u) follows
+readily from the interior estimates for harmonic functions. To hand leM1(∇u), we apply
+Lemma 3.3 to uon the Lipschitz domain Z(2r), where
+Z(ρ)(=/braceleftbig
+(x′,xd) :|x′|<ρandψ(x′)<xd<20√
+d(1+M)ρ/bracerightbig
+.
+This yields that
+1
+rd−1/integraldisplay
+S(r)|M1(∇u)|pdσ≤1
+rd−1/integraldisplay
+∂Z(2r)|(∇u)∗
+Z(2r)|pdσ
+≤Crt−d+1sup
+Z(2r)|∇2u(x)|p−2/bracketleftbig
+δ(x)/bracketrightbigp−1−t/integraldisplay
+Z(2r)|∇2u(y)|2dy
++C|∇v(x0)|p,(3.6)
+whereδ(x) = dist(x,Z(2r)) and (∇u)∗
+Z(2r)denotes the nontangential maximal function of
+∇uwith respect to the domain Z(2r). Note that the last term in the right-hand side of (3.6)
+may be treated easily, using the interior estimates.
+LetIdenote the first term in the right-hand side of (3.6). By Lemma 3.2,
+I≤Crt−d−1sup
+Z(2r)|∇2u(x)|p−2/bracketleftbig
+δ(x)/bracketrightbigp−1−t/integraldisplay
+Z(2r)|∇u(y)|2dy. (3.7)
+Letx∈Z(2r). It follows from the interior estimates that for any q>2,
+|∇2u(x)| ≤C
+δ(x)/braceleftbigg1
+[δ(x)]d/integraldisplay
+B(x,δ(x))|∇u|qdx/bracerightbigg1/q
+≤Crd
+q
+[δ(x)]1+d
+q/braceleftbigg1
+rd/integraldisplay
+Z(2r)|∇u|qdx/bracerightbigg1/q
+≤Crd
+q
+[δ(x)]1+d
+q/braceleftbigg1
+rd/integraldisplay
+Z(4r)|∇u|2dx/bracerightbigg1/2
+,(3.8)
+where we have used estimate (2.1) in the last step. This, together w ith (3.7), implies that
+I≤Crt−1+d
+q(p−2)sup
+x∈Z(2r)/bracketleftbig
+δ(x)/bracketrightbigp−1−t−(1+d
+q)(p−2)/braceleftbigg1
+rd/integraldisplay
+Z(4r)|∇u|2dy/bracerightbiggp/2
+.(3.9)
+Sincep−1−t−(1+d
+q)(p−2) = 1−t−d
+q(p−2), we may choose q >2 so large that the
+exponent of δ(x) in (3.9) is positive. Using δ(x)≤Cr, we then obtain
+I≤C/braceleftbigg1
+rd/integraldisplay
+Z(4r)|∇u|2dy/bracerightbiggp/2
+≤C/braceleftbigg1
+rd−1/integraldisplay
+S(4r)|(∇u)∗|2dσ/bracerightbiggp/2
+.(3.10)
+11Thus we have proved that
+1
+rd−1/integraldisplay
+S(r)|M1(∇u)|pdσ≤C/braceleftbigg1
+rd−1/integraldisplay
+S(4r)|(∇u)∗|2dσ/bracerightbiggp/2
+.
+This, together with the same estimate for M2(∇u), gives (3.4).
+Remark 3.4. Letp>2. It follows from Theorem 3.1 and [10, Theorem 1.1] that if Ω is a
+bounded convex domain with C2boundary, the LpNeumann problem for ∆ u= 0 in Ω is
+uniquely solvable. Moreover, the solution satisfies the estimate /ba∇dbl(∇u)∗/ba∇dblp≤C/ba∇dbl∂u
+∂n/ba∇dblp, where
+Cdepends only on d,pand the Lipschitz character of Ω.
+4 Proof of Theorem 1.1
+Let Ω be a bounded convex domain in Rd,d≥2. Letp>2. We need to show that the Lp
+Neumann problem for ∆ u= 0 in Ω is uniquely solvable. Since the case d= 2 is contained
+in [10], we will assume that d≥3.
+The uniqueness of the LpNeumann problem follows directly from the uniqueness of the
+L2Neumann problem. To establish the existence, it suffices to show tha t iff∈C∞
+0(Rd) and/integraltext
+∂Ωfdσ= 0, then the solution of the L2Neumann problem for ∆ u= 0 in Ω with boundary
+dataf|∂Ωsatisfies/ba∇dbl(∇u)∗/ba∇dblp≤C/ba∇dblf/ba∇dblp.
+To this end we approximate Ω from the outside by a sequence of conv ex domains {Ωj}
+with smooth boundaries and uniform Lipschitz characters. Let ujbe a solution to the L2
+Neumann problem for Laplace’s equation in Ω jwith datafj−αj, whereαjis the mean
+value offon∂Ωj. It follows from Remark 3.4 that
+/ba∇dbl(∇uj)∗/ba∇dblLp(∂Ωj)≤C/ba∇dblfj−αj/ba∇dblLp(∂Ωj), (4.1)
+whereCdepends only on d,pand the Lipschitz character of Ω. By a limiting argument (see
+e.g. [7]), there exists a subsequence, still denoted by {uj}, such that ∇uj→ ∇vuniformly
+on any compact subset of Ω, where vis a variational solution of the Neumann problem in Ω
+withdataf|∂Ω. Using thisand(4.1), we maydeduce that /ba∇dbl(∇v)∗/ba∇dblLp(∂Ω)≤C/ba∇dblf/ba∇dblLp(∂Ω). Since
+u−vis constant by the uniqueness of the variational solutions, we obta in/ba∇dbl(∇u)∗/ba∇dblLp(∂Ω)≤
+C/ba∇dblf/ba∇dblLp(∂Ω). This completes the proof.
+5 Proof of Theorem 1.2
+To establish the existence, we first reduce the problem to the case wheref= 0. The
+argument is standard. Let f∈Lp
+−1,0(Ω) andw= ΠΩ(f) =:RΩΠ(/tildewidef). Here RΩdenotes
+the operator restricting distributions in Rdto Ω, the map Π : E′(Rd)→ D′(Rd) is given by
+the convolution with the fundamental solution for ∆ in Rdwith pole at the origin, and /tildewidefis
+defined by</tildewidef,φ>=<f,RΩ(φ)>forφ∈C∞(Rd). Letq=p
+p−1. For anyφ∈Bq
+s(∂Ω) with
+s=1
+p, define
+<Λ(f),φ>=/integraldisplay
+Ω∇w·∇ψdx+<f,ψ>Lp
+−1,0(Ω)×Lq
+1(Ω), (5.1)
+12whereψis a function in Lq
+1(Ω) such that Tr(ψ) =φand/ba∇dblψ/ba∇dblLq
+1(Ω)≤C/ba∇dblφ/ba∇dblBq
+1/p(∂Ω). Here we
+have used the fact that the trace operator Tr:Lq
+1(Ω)→Bq
+1/p(∂Ω) is bounded and onto and
+that
+/ba∇dblφ/ba∇dblBq
+1/p(∂Ω)≈inf/braceleftbig
+/ba∇dblψ/ba∇dblLq
+1(Ω):Tr(ψ) =φ/bracerightbig
+.
+Since /integraldisplay
+Ω∇w·∇ψdx+<f,ψ>= 0 for any ψ∈C∞
+0(Ω)
+andC∞
+0(Ω) is dense in {u∈Lq
+1(Ω) :Tr(u) = 0}, it is easy to see that <Λ(f),φ >is well
+defined. Furthermore,
+|<Λ(f),φ>| ≤/braceleftbig
+/ba∇dblw/ba∇dblLp
+1(Ω)+/ba∇dblf/ba∇dblLp
+−1,0(Ω)/bracerightbig
+/ba∇dblψ/ba∇dblLq
+1(Ω)
+≤C/ba∇dblf/ba∇dblLp
+−1,0(Ω)/ba∇dblψ/ba∇dblLq
+1(Ω)
+≤C/ba∇dblf/ba∇dblLp
+−1,0(Ω)/ba∇dblφ/ba∇dblBq
+1/p(∂Ω),(5.2)
+where we have used the Calder´ on-Zygmund estimate /ba∇dblw/ba∇dblLp
+1(Ω)≤C/ba∇dblf/ba∇dblLp
+−1,0(Ω). It follows
+thatwis a weak solution to (1.7) with data ( f,Λ(f)) and/ba∇dblΛ(f)/ba∇dblBp
+−1/p(∂Ω)≤C/ba∇dblf/ba∇dblLp
+−1,0(Ω).
+Thus, by subtracting wfromu, we may always assume that f= 0.
+Next, we note that if Ω is a bounded Lipschitz domain, the solvability of the Neumann
+problem, 
+
+∆u= 0 in Ω,
+∂u
+∂n=g∈Bp
+−s(∂Ω) on∂Ω,
+u∈Lp
+1−s+1/p(Ω),(5.3)
+was established in [3] for ( s,1/p) in the (open) convex polygon Pformed by the vertices,
+(1−ε,0),(1,0),(1,(1+ε)/2),(ε,1),(0,1),(0,(1−ε)/2),
+whereε>0 depends on Ω. By interpolation, this, together with Theorem 1.1, im plies that
+the Neumann problem (5.3) in a convex domain is uniquely solvable if ( s,1/p) is the (open)
+convex polygon P1formed by the vertices
+(1−ε,0),(1,0),(1,(1+ε)/2),(ε,1),(0,1),(0,0).
+In particular, the Neumann problem (5.3) is solvable if s= 1/pand 2≤p<∞. As a result,
+we have proved Theorem 1.2 for 2 ≤p<∞. The case 1 <p<2 will be proved by a duality
+argument, given in the next section (see Remark 6.5).
+6 Proof of Theorem 1.3
+Note that
+Lp
+σ(Ω) =/braceleftbig
+u∈Lp(Ω,Rd) :/integraldisplay
+Ωu·∇ψdx= 0 for any ψ∈W1,q(Ω)/bracerightbig
+=/braceleftbig
+u∈Lp(Ω,Rd) : div(u) = 0 in Ω and u·n= 0 on∂Ω/bracerightbig
+,(6.1)
+13whereq=p
+p−1. Hereu·nis regarded as an element in Bp
+−1/p(∂Ω). LetXbe a normed vector
+space andSa subset of X. The set
+S⊥={ℓ∈X∗:<ℓ,f >= 0 for allf∈S}
+is called the set of annihilators of S. IfSis a closed subspace of a reflexive Banach space
+X, then (S⊥)⊥=S(see e.g. [12]). With this notation, we may write Lp
+σ(Ω) =S⊥⊂X=
+Lp(Ω,Rd), whereS= gradW1,q(Ω)⊂Lq(Ω,Rd). Thus the Lp-Helmholtz decomposition
+(1.10) may be written as
+Lp(Ω,Rd) = gradW1,p(Ω)⊕/braceleftbig
+gradW1,q(Ω)/bracerightbig⊥. (6.2)
+Lemma 6.1. LetΩbe a bounded Lipschitz domain in Rdand1<p<∞. ThenC∞
+σ(Ω) =
+{v∈C∞
+0(Ω,Rd) : div(v) = 0}is dense in Lp
+σ(Ω).
+Proof.LetXp(Ω) denote the closure of C∞
+σ(Ω) inLp(Ω,Rd). Clearly, grad W1,q(Ω)⊂
+(Xp(Ω))⊥. On the other hand, if u∈Lq(Ω,Rd) and/integraltext
+Ωu·vdx= 0 for any v∈C∞
+σ(Ω),
+thenu=−∇ψfor someψ∈L1
+loc(Ω) (see e.g. [4, pp.696-697] for a proof). This im-
+plies that u∈gradW1,q(Ω). Hence we obtain grad W1,q(Ω) = (Xp(Ω))⊥. It follows that
+Xp(Ω) = (grad W1,q(Ω))⊥=Lp
+σ(Ω) and thus C∞
+σ(Ω) is dense in Lp
+σ(Ω).
+Lemma 6.2. LetΩbe a bounded Lipschitz domain in Rdand1<p<∞. If the Helmholtz
+decomposition (1.10) with the estimate (1.11) holds for the exponentpand constant Cp, then
+it holds for the dual exponent q=p
+p−1and constant Cq=Cp.
+Proof.This follows from the fact that if X0,X1are closed subspaces of XandX=X0⊕X1,
+thenX∗=X⊥
+0⊕X⊥
+1. Note that if u∈Lp
+σ(Ω),v∈Lq
+σ(Ω),φ∈W1,p(Ω) andψ∈W1,q(Ω),
+then /integraldisplay
+Ωu·(v+∇ψ)dx=/integraldisplay
+Ω(u+∇φ)·vdx,
+/integraldisplay
+Ω∇φ·(v+∇ψ)dx=/integraldisplay
+Ω(u+∇φ)·∇ψdx.(6.3)
+By a simple duality argument, this shows that the estimate (1.11) hold s for the exponent q
+and constant Cq=Cp.
+Next we will show that the Lp-Helmholtz decomposition is equivalent to the solvability
+of (5.3) for s= 1/p.
+Theorem 6.3. LetΩbe a bounded Lipschitz domain in Rdand1<p <∞. Then the Lp-
+Helmholtz decomposition (1.10) with the estimate (1.11) ho lds if and only if the Neumann
+problem (5.3) is uniquely solvable for s= 1/p.
+Proof.Suppose that (5.3) is uniquely solvable for s= 1/p. The uniqueness of the solutions
+to (5.3) implies that Lp
+σ(Ω)∩gradW1,p(Ω) ={0}. Given any u= (u1,...,ud)∈Lp(Ω,Rd),
+let
+φ(x) =∂
+∂xi/integraldisplay
+ΩΓ(x−y)ui(y)dy, (6.4)
+14where Γ(x) denotes the fundamental solution for ∆ in Rdwith pole at the origin. By the
+Calder´ on-Zygmundestimate, φ∈W1,p(Ω)and/ba∇dbl∇φ/ba∇dblLp(Ω)≤C/ba∇dblu/ba∇dblLp(Ω). Sincediv( u−∇φ) =
+0 in Ω, it follows that Λ = ( u−∇φ)·n∈Bp
+−1/p(∂Ω) and/ba∇dblΛ/ba∇dblBp
+−1/p(∂Ω)≤C/ba∇dblu−∇φ/ba∇dblLp(Ω),
+where Λ may be defined by
+<Λ,ϕ>=/integraldisplay
+Ω(u−∇φ)·∇/tildewideϕdx
+forϕ∈Bq
+1/p(∂Ω) and/tildewideϕ∈W1,q(Ω) such that Tr(/tildewideϕ) =ϕon∂Ω. We now let v=u−∇(φ+
+ψ)∈Lp(Ω,Rd), whereψ∈W1,p(Ω) is a solution to (5.3) with boundary data Λ. Observe
+that for any ϕ∈C∞(Rd),
+/integraldisplay
+Ωv·∇ϕdx=<Λ,ϕ>−/integraldisplay
+Ω∇ψ·∇ϕ= 0.
+Thusv∈Lp
+σ(Ω). Also note that
+/ba∇dbl∇(φ+ψ)/ba∇dblLp(Ω)≤C/braceleftbig
+/ba∇dbl∇φ/ba∇dblLp(Ω)+/ba∇dblΛ/ba∇dblBp
+−1/p(∂Ω)/bracerightbig
+≤C/braceleftbig
+/ba∇dbl∇φ/ba∇dblLp(Ω)+/ba∇dblu−∇φ/ba∇dblLp(Ω)/bracerightbig
+≤C/ba∇dblu/ba∇dblLp(Ω).
+Sinceu=v+∇(φ+ψ), we obtain the Helmholtz decomposition (6.2).
+Next suppose that the Lp-Helmholtz decomposition (1.10) with estimate (1.11) holds.
+TheuniquenessfortheNeumannproblem(5.3)followsfromthefact thatLp
+σ(Ω)∩gradW1,p(Ω)
+={0}. To show the existence, let ψbe a solution of the L2Neumann problem in Ω with
+boundary data∂ψ
+∂n=g, whereg∈L∞(∂Ω) and/integraltext
+∂Ωgdσ= 0. Given u∈Lq(Ω,Rd)∩
+L2(Ω,Rd), writeu=v+∇φ, wherev∈Lq
+σ(Ω)∩L2
+σ(Ω) andφ∈W1,q(Ω)∩W1,2(Ω). This
+is possible since the Helmholtz decomposition holds for exponents qand 2. It follows that
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Ω∇ψ·udx/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Ω∇ψ·∇φdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+∂Ω∂ψ
+∂n·(φ−α)dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤ /ba∇dbl∂ψ
+∂n/ba∇dblBp
+−1/p(∂Ω)/ba∇dblφ−α/ba∇dblBq
+1/p(∂Ω)
+≤ /ba∇dbl∂ψ
+∂n/ba∇dblBp
+−1/p(∂Ω)/ba∇dblφ−α/ba∇dblW1,q(Ω),
+for anyα∈R. By Poincar´ e inequality, this yields that
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Ω∇ψ·udx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/ba∇dbl∂ψ
+∂n/ba∇dblBp
+−1/p(∂Ω)/ba∇dbl∇φ/ba∇dblLq(Ω).
+Using/ba∇dbl∇φ/ba∇dblLq(Ω)≤Cq/ba∇dblu/ba∇dblLq(Ω), we then obtain
+/ba∇dbl∇ψ/ba∇dblLp(Ω)≤C/ba∇dbl∂ψ
+∂n/ba∇dblBp
+−1/p(∂Ω) (6.5)
+by duality. Since L∞(∂Ω) is dense in Bp
+−1/p(∂Ω), the existence of solutions to (5.3) with
+data Λ, where Λ ∈Bp
+−1/p(∂Ω) and<Λ,1>= 0, follows from the estimate (6.5) by a simple
+limiting argument. This completes the proof.
+15Lemma 6.2 and Theorem 6.3 lead to the following.
+Theorem 6.4. LetΩbe a bounded Lipschitz domain in Rdand1< p <∞. Then the
+solvability of (5.3) for s= 1/pis equivalent to the solvability of (5.3) for s= 1/q, where
+q=p
+p−1.
+Remark 6.5. We show in Section 5 that if Ω is a bounded convex domain in Rd, then the
+Neumann problem (5.3) is solvable for s= 1/pand 2<p<∞. Thus, by Theorem 6.4, the
+Neumann problem (5.3) in convex domains with s= 1/pis solvable for any 1 < p <∞.
+This completes the proof of Theorem 1.2.
+Remark 6.6. Theorem 1.3 follows readily from Theorems 1.2 and 6.3.
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+16[12] M. Schechter, Principles of Functional Analysis , Graduate Studies in Math., vol. 36,
+AMS, Providence, RI, 2002.
+[13] Z. Shen, Bounds of Riesz transforms on Lpspaces for second order elliptic operators ,
+Ann. Inst. Fourier (Grenoble) 55(2005), no. 1, 173–197.
+[14] G. Verchota, Layer potentials and regularity for the Dirichlet problem f or Laplace’s
+equation in Lipschitz domains , J. Funct. Anal. 59(1984), 572–611.
+Department of Mathematics, University of Kentucky, Lexing ton, KY 40506
+E-mail address :jgeng@ms.uky.edu
+Department of Mathematics, University of Kentucky, Lexing ton, KY 40506
+E-mail address :zshen2@email.uky.edu
+November 18, 2018
+17
\ No newline at end of file
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+Shear Modes, Criticality and Extremal Black Holes
+Mohammad Edalati, Juan I. Jottar and Robert G. Leigh
+Department of Physics, University of Illinois at Urbana-Champaign,
+Urbana IL 61801, USA
+edalati,jjottar2,rgleigh@illinois.edu
+Abstract
+We consider a (2 + 1)-dimensional eld theory, assumed to be holographically dual to the extremal
+Reissner-Nordstr om AdS 4black hole background, and calculate the retarded correlators of charge
+(vector) current and energy-momentum (tensor) operators at nite momentum and frequency. We
+show that, similar to what was observed previously for the correlators of scalar and spinor opera-
+tors, these correlators exhibit emergent scaling behavior at low frequency. We numerically compute
+the electromagnetic and gravitational quasinormal frequencies (in the shear channel) of the extremal
+Reissner-Nordstr om AdS 4black hole corresponding to the spectrum of poles in the retarded correla-
+tors. The picture that emerges is quite simple: there is a branch cut along the negative imaginary
+frequency axis, and a series of isolated poles corresponding to damped excitations. All of these poles
+are always in the lower half complex frequency plane, indicating stability. We show that this analytic
+structure can be understood as the proper limit of nite temperature results as Tis taken to zero
+holding the chemical potential xed.arXiv:1001.0779v2  [hep-th]  10 Feb 20101 Introduction
+The AdS/CFT correspondence is emerging as a promising approach for analyzing strongly-coupled
+condensed matter systems.1Although the systems accessible through the duality need a lot of rene-
+ment in order to resemble actual condensed matter systems, (more than) a decade-long experience
+with the correspondence has taught us that it is extremely useful in extracting some universal features
+of strongly-interacting systems which would be hard, or even impossible, to obtain using eld theoretic
+methods. It is in this sense that applying the tools of the correspondence to study strongly-interacting
+condensed matter systems seems to be promising. Recent examples of such applications include mod-
+eling superconductivity and super
uidity [2, 3, 4, 5, 6], engineering systems with Schr odinger and
+Lifshitz symmetries [7, 8, 9, 10, 11, 12, 13, 14], or modeling a system which can exhibit non-Fermi
+liquid type behavior [15, 16, 17, 18, 19]. See [20, 21, 22] for reviews.
+Using the correspondence to study quantum critical regions of condensed matter systems is another
+example of such applications [23, 24, 25]. At the quantum critical point, the system is often a strongly-
+coupled conformal eld theory, and it is usually the case that a proper eld theoretic understanding
+is not known. The authors of [17] considered a simple gravitational background whose dual eld
+theory exhibits a variety of emergent quantum critical behaviors. They considered the background
+of extremal Reissner-Nordstr om AdS d+1black hole whose dual eld theory is supposed to be a d-
+dimensional strongly-coupled eld theory which is at zero temperature and nite U(1) charge density.
+Considering probe scalars as well as spinors in the geometry, it was shown that the retarded correlator
+of the dual operators show emergent quantum criticality at low frequency. The existence of such
+behaviors was attributed to the fact that the near horizon region of the background geometry (which
+encodes the IR physics of the boundary theory) is AdS 2R2, and the assumption that there exists
+some sort of IR CFT dual to the AdS 2region. It was shown that the behavior of the retarded Green
+functionGR(!;~k) at low frequency is dictated by GR(!= 0;~k) as well as the conformal dimension of
+some operators in the IR CFT. More specically, it was shown that GR(!;~k) can in general exhibit a
+scaling behavior for the spectral density, a log-periodic behavior, or in the case of fermionic operators
+indicate the existence of Fermi-like surfaces with quasi-particle excitations of non-Fermi liquid type.
+In [26] we examined the role played by the IR CFT in constraining the low frequency behavior of
+retarded correlators of charge vector and energy momentum tensor operators of the boundary the-
+ory. The main focus though was on extracting some transport coecients, such as conductivity and
+viscosity, of the boundary theory, and investigating the relationship between the IR CFT and the uni-
+versality of such coecients. As such, the analyses were limited to the case of zero spatial momentum.
+The boundary theory considered was a (2 + 1)-dimensional eld theory at zero temperature and nite
+charge density dual to an extremal Reissner-Nordstrom AdS 4black hole. Although temperature is
+zero, there is a scale in the problem set by the chemical potential for the charge density and moreover
+the entropy density is nite.2It was shown in [26] that, for example, the ratio of shear viscosity to
+the entropy density is 1 =4saturating the KSS bound [27]. There are a variety of reasons why such
+analyses at zero-temperature are interesting. First, calculating transport coecients involves taking
+the limit of small frequency, which at zero temperature becomes subtle and extra care is needed to deal
+with this subtlety. Secondly, they generalize the universality arguments, given for nite temperature
+1See [1] for a review of the AdS/CFT correspondence.
+2This is presumably a \large N" eect. Analysis of this has recently appeared in [19].
+2backgrounds [29, 28, 30], to backgrounds of extremal black holes. Third, the system studied may be
+in the universality class of some quantum critical points, hence understanding the universal behavior
+of the transport coecients of this system may shed light on the physics of quantum critical points.
+In this paper, we analyze the retarded correlators of the charge vector current and energy-momentum
+(tensor) operators of the aforementioned (2 + 1)-dimensional boundary theory for non-vanishing (spa-
+tial) momentum. Although we perform the analyses in the (2 + 1)-dimensional boundary theory,
+they can easily be carried out for higher dimensional boundary theories dual to extremal Reissner-
+Nordstrom AdS d+1black hole. Our focus here is on the retarded correlators of those operators which
+are in the so-called shear channel of the boundary theory. We nd that for generic momentum, similar
+to the cases of scalar and spinor operators studied in [17], the retarded correlators of such vector and
+tensor operators exhibit emergent scaling behavior at low frequency. These operators give rise in the
+IR to two sets of operators with dierent conformal dimensions. The low frequency scaling behaviors
+of the boundary theory retarded correlators, especially their scaling exponents, depend only on one of
+the two IR CFT operators. By taking linear combinations of the dual bulk modes, we construct the
+gauge invariant modes appropriate for extracting the quasinormal modes.
+We study the small frequency regime analytically for nite momentum, and we explore the ana-
+lytic structure of the shear channel retarded Green functions numerically for generic frequency and
+momentum. We present evidence for the existence of a branch cut in these Green functions at the
+origin, as well as a series of metastable modes corresponding to isolated poles in the lower half complex
+frequency plane. By turning on temperature we observe how the branch cut dissolves into a series
+of poles on the negative imaginary axis and determine the corresponding dispersion constant of the
+leading pole, in agreement with previous results.
+The organization of the paper is as follows. In section 2, we review the Reissner-Nordstr om AdS 4
+black hole background, its dual boundary eld theory, near horizon AdS 2R2geometry and the dual IR
+CFT. In section 3 we present the Linearized Einstein-Maxwell equations for the bulk modes dual to the
+(vector and tensor) operators in the shear channel of the boundary theory. Taking linear combinations
+of the modes, we construct the gauge-invariant combinations in terms of which the linearized Einstein-
+Maxwell equations reduce to a set of two coupled second-order dierential equations. By introducing
+\master variables" we then decouple these equations. In section 4 we solve the decoupled equations
+by matching the solutions in the outer region to the solutions in the inner AdS 2region. Having
+determined the solutions, we calculate the retarded correlators of the charge vector current and energy-
+momentum tensor operators of the boundary theory. For generic momentum, we then extract the low
+frequency emergent scaling behavior of their spectral functions. In section 5 we argue that the (shear-
+type) electromagnetic and gravitational perturbations of the extremal Reissner-Nordstr om AdS 4do
+not cause instability by showing that the associated quasinormal frequencies are all located in the
+lower half of the complex frequency plane. We then numerically compute the spectrum of the above-
+mentioned quasinormal frequencies and compare the numerical results with the analogous quasinormal
+frequencies of the non-extremal Reissner-Nordstr om AdS 4black hole background. In Appendix A we
+have summarized the nite-temperature equations we occasionally used in the bulk of the paper in
+order to extract the shear-type quasinormal frequencies of the non-extremal Reissner-Nordstr om AdS 4
+black hole.
+32 The Background and the Boundary Field Theory
+Consider the Einstein-Maxwell action in 3+1 spacetime dimensions with a negative cosmological con-
+stant  =3=L2
+S=1
+22
+4Z
+d4xpg
+R2L2FF

+; (2.1)
+whereLis the curvature radius of AdS 4. The background we consider is the Reissner-Nordstr om AdS 4
+black hole
+ds2=gdxdx=r2
+L2
+f(r)dt2+dx2+dy2
++L2
+r2f(r)dr2; (2.2)
+A=
+1r0
+r
+dt; (2.3)
+which is a solution to the Einstein-Maxwell equations obtained from (2.1), where
+f(r) = 1Mr0
+r3
++Q2r0
+r4
+;  =Qr0
+L2: (2.4)
+Here,r0is the horizon radius and is given by the largest real root of f(r0) = 0. Also, note that
+M= 1 +Q2. The temperature of the black hole takes the form
+T=r0
+4L2(3Q2) =
+43Q2
+Q; (2.5)
+while its entropy, charge and energy densities are given by
+s=2
+2
+4r0
+L2
+;  =2
+2
+4r0
+L2
+Q;  =r3
+0
+2
+4L4M; (2.6)
+respectively [31]. The background is extremal when Q2= 3, for which the temperature vanishes but
+the entropy density remains nite. The background is invariant under 
ipping the sign of At. We
+chooseto be positive resulting in Qto be positive. With this convention, Q=p
+3 at extremality.
+The background is dual to a (2 + 1)-dimensional strongly-coupled eld theory at nite temperature
+Tand nite charge density . The entropy and energy densities of the dual theory are given by sand
+in (2.6), respectively. Also, the asymptotic value of the bulk gauge eld At(1) =is interpreted in
+the dual theory as the chemical potential for the (electric) charge density. Little is known about the
+details of the dual theory from eld theory perspectives. On the other hand, using holography a lot
+has been learned (especially thermodynamical properties) regarding its strong-coupling behavior; see
+[31] and its citations.
+In this paper we work in the extremal limit, where the dual theory is at zero temperature but nite
+charge density. We will refer to this dual theory as the boundary eld theory. Although the black hole
+temperature vanishes at extremality, its horizon area remains nite whose dual interpretation is that
+the boundary theory has a nite entropy density at zero temperature.
+In the extremal limit, f(r) in the background metric (2.2) takes the form
+f(r) = 14r0
+r3
++ 3r0
+r4
+; (2.7)
+4which has a double zero at the horizon, and can be approximated near that region (to the leading
+order inrr0) by
+f(r)'6
+r2
+0(rr0)2: (2.8)
+The near horizon geometry is AdS 2R2. To see the emergence of this geometry, rst change the
+radial coordinate rtodened by
+rr0=L2
+6: (2.9)
+There is then a scaling limit [17] in which
+ds2=L2
+62
+dt2+d2
++r2
+0
+L2
+dx2+dy2
+; A =Q
+6dt: (2.10)
+The curvature radius of the AdS 2isL2=L=p
+6. The radial coordinate is interpreted holographically
+as the renormalization scale of the dual eld theory, and the near horizon region corresponds to the
+IR limit. This implies that the AdS 2R2geometry encodes the IR physics ( !!0) of the boundary
+theory.
+On general grounds of holography, one expects the gravity on the AdS 2space to be dual to a
+CFT 1. This led the authors of [17] to suggest that the (2 + 1)-dimensional boundary eld theory
+(which is dual to the extremal charged AdS 4black hole) 
ows in the IR to a xed point described
+by a CFT 1. Following [17] we will refer to this CFT 1as the IR CFT. The details of the AdS 2=CFT 1
+correspondence and how exactly the mapping works are poorly understood. In particular, it is not
+clear whether the CFT 1of the correspondence represents a conformal quantum mechanics or a chiral
+sector of a (1+1)-dimensional CFT. Nevertheless, not knowing the details of the IR CFT and just
+assuming its existence, the authors of [17] were able to show that the low frequency behavior of some
+observables in the boundary CFT is encoded in the IR CFT. In what follows, we consider the charge
+current and the energy-momentum tensor operators of the boundary eld theory and elucidate the
+role of the IR CFT in determining the low frequency behavior of the retarded Green functions of the
+two operators.
+3 Gauge Field and Metric Fluctuations
+The charge current operator in the boundary eld theory is dual to the 
uctuations of the bulk gauge
+eld in the transverse direction while the energy-momentum tensor operator is dual to the 
uctuations
+of the background metric. The calculation of the corresponding retarded Green functions starts with
+solving the linearized Einstein-Maxwell equations for these 
uctuations.
+3.1 Einstein-Maxwell Equations
+To obtain the linearized Einstein-Maxwell equations, we rst dene
+g= g+h; A=A+a; (3.1)
+5where gand Arepresent, respectively, the (extremal) metric and the gauge eld of the background,
+andhandaare the 
uctuations. We choose the so-called radial gauge
+ar= 0; hr= 0; (3.2)
+where=ft;x;y;rg. We proceed by Fourier transforming the 
uctuations
+h(t;x;r )ei!teikxh(r); a(t;x;r )ei!teikxa(r); (3.3)
+where, without loss of generality, we used the rotation invariance in the ( x;y) plane to set ky= 0 and
+denedkxk. The 
uctuations split into decoupled groups depending on whether they are even or
+odd with respect to parity, y!y. Accordingly, hty,hxy,ayhave odd parity while htt,htx,hxx,hyy,
+at,axall have even parity. In this paper, we study the odd parity modes which translates into the
+shear and (charge) diusion modes of the boundary theory. The analysis of the even parity modes
+(sound modes) will be given elsewhere.
+It is more convenient to raise the indices in htyandhxy(by the background metric  g) and work
+withhyt,hxy. It is also convenient to dene the dimensionless quantities
+u=r
+r0; w=!
+; q=k
+; (3.4)
+In this notation, the linearized Einstein-Maxwell equations for the odd parity modes hty,hxy,aythen
+read
+f(u)h
+u4hy00
+t(u) + 4u3hy0
+t(u) +12
+a0
+y(u)i
+3q[whxy(u) +qhyt(u)] = 0; (3.5)
+f(u)h
+u4f(u)hx00
+y(u) +
+u4f0(u) + 4u3f(u)
+hx0
+y(u)i
++ 3w[whxy(u) +qhyt(u)] = 0; (3.6)
+f(u)h
+u4f(u)a00
+y(u) +u2
+u2f0(u) + 2uf(u)
+a0
+y(u) +u2hy0
+t(u)i
++ 3
+w2f(u)q2
+ay(u) = 0;(3.7)
+wheref(u) is given in (2.7). There is also a constraint which comes from the yu-component of the
+linearized Einstein equations given by
+u4why0
+t(u) +u4f(u)qhx0
+y(u) +12
+way(u) = 0: (3.8)
+As we will see later in more detail, this constraint, evaluated asymptotically, encodes a Ward identity
+in the dual boundary theory, as one should expect.
+3.2 Gauge Invariant Modes
+Although working in the radial gauge (3.2) is convenient, it does not completely x the gauge freedom
+ofaandh. In the present case, it is easy to identify the gauge invariant combinations of elds.
+For the metric and gauge eld 
uctuations of our background, there is a U(1) gauge transformation
+A!~A=A+r, according to which f=@a@ais invariant. There are also gauge
+6transformations associated with dieomorphism under an arbitrary vector eld , implying that the
+
uctuationsf~h;~agandfh;agare equivalent if they are related as [32]
+~h=h
+rg+ gr+ gr
+=hr+r

+; (3.9)
+~a=a
+rA+Ar

+=a
+@A+A@

+; (3.10)
+whereris any torsionless connection and ris the connection associated with  g. It is important to
+notice that these gauge transformations are not changes of the coordinates. In particular, they aect
+only the elds handa, while all the other tensors, vectors, etc. remain invariant.
+In this paper, the 
uctuations of interest are hyt,hxyanday. Whileayis invariant under the residual
+gauge transformations, hytandhxytransform according to hyt!hyt+iwyandhxy!hxyiqy,
+respectively. Thus
+X(u) = qhyt(u) +whxy(u); (3.11)
+Y(u) =ay(u); (3.12)
+are invariant combinations.
+Note that at q= 0 the symmetries are enhanced, and hxy(u) is gauge invariant. For this special
+case, as can be seen in equation (3.6), hxy(u) decouples from the rest, while equations (3.7) and (3.8)
+together give a decoupled equation for ay(u). The decoupled equations read
+u4f(u)hx00
+y(u) +
+u4f0(u) + 4u3f(u)
+hx0
+y(u) +3w2
+f(u)hxy(u) = 0; (3.13)
+u2f(u)a00
+y(u) +
+u2f0(u) + 2uf(u)
+a0
+y(u) +1
+u23w2
+f(u)12
+u2
+= 0: (3.14)
+Equations (3.13) and (3.13) were studied in [26] and the retarded Green functions of the dual operators,
+TxyandJy, at q= 0 and small wwere calculated. For the most part in this paper, we consider qto
+be nite and non-zero.
+Even at nite qand w, the Einstein-Maxwell shear channel equations can be decoupled in terms
+of a pair of \master elds" [33]. This is remarkable, since in the boundary eld theory it corresponds
+to diagonalizing the entire renormalization group 
ow. The master elds are
+(u) =qf(u)u3
+w2f(u)q2X0(u)6
+u2f(u)q2
+w2f(u)q2+u
+1p
+1 + q2
+Y(u): (3.15)
+and they satisfy the decoupled equations
+
+u2f(u)0
+(u)0+
+uf0(u) +3
+u2f(u)
+w2f(u)q2

+6
+u3
+1p
+1 + q2
+(u) = 0: (3.16)
+As we will discuss in the next section, it is most convenient to organize numerical calculations in terms
+of these modes.
+74 Retarded Green Functions and Criticality
+In this section, we rst obtain formal expressions for the retarded Green functions of the vector current
+and energy momentum tensor operators of the boundary theory. By formal expressions we mean that
+we write formulas relating the above-mentioned retarded Green functions to ^, to be introduced
+below.
+Consider rst the solution of the equations (3.5), (3.6) and (3.7) asymptotically, for u!1 . For
+each of the elds in this problem, there is a constant asymptotic value that we denote by a hat. The
+solutions take the form
+hxy(u!1 ) =^hx
+y+3
+2w
+u2
+w^hx
+y+q^hy
+t
++xy
+u3+:::; (4.1)
+hyt(u!1 ) =^hy
+t3
+2q
+u2
+w^hx
+y+q^hy
+t
++yt
+u3+:::; (4.2)
+ay(u!1 ) = ^ay+y
+u3
+2(q2w2)
+u2
+w^hx
+y+q^hy
+t
++:::: (4.3)
+In these expressions, all quantities are understood to be functions of wand qeven if not explicitly
+indicated. Here, ^hx
+y;^hy
+tand ^ayare regarded, as usual, as the sources for the corresponding operators
+in the dual eld theory, while xy;ytandyare the corresponding one-point functions of those
+operators. The latter are understood to be functions of the sources, and that functional dependence,
+given the on-shell action, encodes the correlation functions of the dual eld theory.
+The renormalized action in our case is given by
+22
+4Sren=Z
+d4xpg
+R+6
+L2L2FF
+Z
+@Md3xp
+j
j2K4
+LZ
+@Md3xp
+j
jLZ
+@Md3xp
+j
j(3)R; (4.4)
+where
,Kand(3)Rare, respectively, the induced metric, the trace of the second fundamental
+form and the intrinsic curvature of the 3-dimensional boundary @M. The counterterm proportional to
+(3)Rremoves a linear divergence in the on-shell action for the Reissner-Nordstr om AdS 4background,
+with no contribution to its nite part. In order to extract Green functions that we are interested in
+holographically, it is enough to consider the (gravity) action up to quadratic terms in the 
uctuations.
+One nds
+Sren= lim
+u!1CZ
+d3xh
+u4hythy0
+tu4f(u)hxyhx0
+y+ 4u3
+1f(u)1=2
+(hythytf(u)hxyhxy)
+u4f0(u)hxyhxy12
+2
+ayhyt+u2f(u)aya0
+y

+3
+2uf(u)1=2(@thxy@xhyt)2i
+; (4.5)
+whereC=r02=(122
+4).
+The holographic prescription for evaluating the matrix of retarded Green functions when the op-
+8erators mix was given in [36] following [34, 35]3. Because we have rotational invariance in the spatial
+plane, we will always take the y-component of momentum to vanish, and hence it is most convenient to
+Fourier transform the x-direction only. Having imposed infalling boundary condition at the horizon,
+one writes the renormalized on-shell action in the form
+So:s:=Z
+dyZdwdq
+(2)2^i(w;q)Fij(w;q)^j(w;q); (4.6)
+where ^i=f^hy
+t;^hx
+y;^aygin our case. The prescription of [36] then reads
+GR(w;q) =2Fij(w;q)i=j
+Fij(w;q)i6=j:(4.7)
+In momentum space, (4.5) evaluates to
+Sren= 32CZ
+dyZdwdq
+(2)2h
+xy(w;q)^hx
+y(w;q)yt(w;q)^hy
+t(w;q)
++4
+2y(w;q)^ay(w;q) +:::i
+; (4.8)
+where the ellipsis contains terms which contribute at most to contact terms in two-point functions.
+Now, as we recalled above, the fxy;yt;ygare regarded as functions of f^hy
+t;^hx
+y;^ayg, and for
+present purposes, they can be regarded as linear functions. Also, by assumption, none of the cor-
+responding operators have one-point functions when the sources are removed. The vevs are not all
+independent in this case though. Indeed, the constraint (3.8) gives one more piece of information
+qxy+wyt=4
+w^ay (4.9)
+This equation is easily understood in the boundary theory. The dieomorphism Ward identity [21]
+takes the form
+g(x)rh^T(x)i+F(x)h^J(x)i= 0: (4.10)
+whosey-component evaluates to (4.9) at linearized order given the source ^ ay, and our notation.
+Thus, there are really only two independent vevs here, in accordance with the fact that the bulk
+problem can be reduced to the two gauge-invariant elds. What is more, the system is in fact diago-
+nalized by passing to the master elds. Asymptotically, we write the master elds as
+(u!1 )'^ 
+1 +^
+u+:::!
+; (4.11)
+where ^and^are functions of wand q, and we have in mind that the master elds satsify infalling
+boundary conditions at the horizon.
+3See also [37, 38] for a recent more systematic treatment of holographically computing Green functions in cases where
+the boundary theory operators mix.
+9By comparing asymptotic expansions, one easily deduces that
+^= 3q
+w^hx
+y+q^hy
+t
+6
+1p
+1 + q2
+^ay; (4.12)
+while
+^^= 3q
+wxy6
+1p
+1 + q2
+y; (4.13)
+where we have used (4.9) to simplify. It should be clear then that the on-shell action can be written
+as a function of the sources f^hy
+t;^hx
+y;^aygand ^. The latter two functions (of wand q) must be
+determined numerically. The matrix of retarded two-point functions in the shear channel are then
+determined by these two functions. This is in fact consistent with the requirements of the Ward
+identities.
+Returning to the on-shell action, using the holographic prescription given above, we deduce the
+retarded correlators of ^Jy,^Txyand ^Tyt
+Gyt;yt(w;q) =2q2G1(w;q); (4.14)
+Gyt;xy(w;q) =2qwG1(w;q); (4.15)
+Gxy;xy(w;q) =2w2G1(w;q); (4.16)
+Gyt;y(w;q) =1
+2q2[G1(w;q) +G2(w;q)]; (4.17)
+Gxy;y(w;q) =1
+2qw[G1(w;q) +G2(w;q)]; (4.18)
+Gy;y(w;q) =4G2(w;q); (4.19)
+where
+G1(w;q)3Cp
+1 + q2h
+1p
+1 + q2
+^+
+1 +p
+1 + q2
+^i
+; (4.20)
+G2(w;q)3Cp
+1 + q2h
+1 +p
+1 + q2
+^+
+1p
+1 + q2
+^i
+: (4.21)
+We have implicitly assumed here that the bulk equations of motion are to be solved by placing
+infalling boundary conditions on  (u). It can easily be veried that the prescription (4.7) implies
+thatGxy;yt(w;q) =Gyt;xy(w;q),Gy;xy(w;q) =Gxy;y(w;q), and up to a contact term Gy;yt(w;q) =
+Gyt;y(w;q).
+4.1 Frequency Expansions of ^and ^
+The expressions we derived above for the retarded Green functions are not very informative unless
+^(w;q) is known. One thing that can be done semi-analytically is to develop a series expansion in
+w. This is an extension of our earlier results to nite momentum. This analysis makes use of the fact
+that the near horizon geometry is AdS 2R2.
+Let us expand  (u) in a power series in w:
+O(u) = (0)
+O(u) +w(1)
+O(u) +w2(2)
+O(u) +:::; (4.22)
+10where the subscript denotes the outer region (see the discussion below). Since the black hole back-
+ground is extremal, taking the limit of w!0 becomes somewhat more subtle near the horizon. This
+is due to the fact that in the extremal case f(u) has a double zero at the horizon. A way to handle
+this issue was explained in [17]. Basically, one realizes that near the horizon the equations for  in
+(3.16) organize themselves as functions of !, where, using (2.9) and (3.4), we have
+u= 1 +wp
+12: (4.23)
+Since the coordinate is the suitable radial coordinate for the AdS 2part of the near horizon geometry,
+the wexpansion of  in that region can be written
+I() = (0)
+I() +w(1)
+I() +w2(2)
+I() +:::; (4.24)
+where the subscript Idenotes the inner region. Having implemented the (infalling) boundary condition
+at the horizon [34, 35] in the coordinate, one then matches the inner and outer expansions in the
+\matching region" where the !0 and w=!0 limits are taken. Since the dierential equations
+(3.16) are linear, and we also require that the solutions for the higher order terms in the expansions
+(4.22) and (4.24) do not include terms proportional to the zeroth-order solutions near the matching
+region, we just need to match (0)
+I() to (0)
+O(u) near that region [17] at leading order.
+4.1.1 Inner Region
+Substituting (4.24) in (3.16), we nd that the leading terms satisfy
+(0)00
+I() + 
+q2+ 22p
+1 + q2
+21!
+(0)
+I() = 0: (4.25)
+Equations (4.25) are identical to the equations of motion for massive scalar elds with eective AdS 2
+masses
+m2
+L2
+2= 1 +q2
+2p
+1 + q2: (4.26)
+Thus (0)
+I() source operators Oin the IR CFT with conformal dimensions =+1
+2where
+=1
+2q
+1 + 4m2
+L2
+2=1
+2q
+5 + 2 q24p
+1 + q2: (4.27)
+Having imposed the infalling boundary conditions at !1 , (0)
+I() take the following form near the
+matching region (where one takes the !0 and w=!0 limits)
+(0)
+I(u!1) =h
+(u1)1
+2++G(w)(u1)1
+2i
+: (4.28)
+Note that in writing (4.28) we used (4.23) to express in terms of u. Also, we chose a specic
+normalization for (0)
+I. Such a choice does not aect the calculation of the boundary theory Green
+11functions. In (4.28), G(w) denote the retarded Green functions of the aforementioned IR CFT
+operatorsO, and are given by [17, 26]
+G(w) =2ei(1)
+(1 +)w
+22: (4.29)
+The analytic structure of G(w) is the same as the retarded Green function of an IR CFT scalar
+operator discussed in [17]. Namely, for generic values of momentum q, there is a branch point at
+w= 0, and moreover the branch cut is chosen to extend along the negative imaginary axis in the
+complex w-plane.
+4.1.2 Outer Region and Matching
+In the outer region, the equations for (0)
+O(u) are obtained by setting w= 0 in (3.16). It is easy to
+show that near the matching region, the solutions for (0)
+O(u) are given by a linear combination of
+(u1)1
+2+and (u1)1
+2. For ease of notation, we dene
+(0)
+(u) = (u1)1
+2++:::; (0)
+(u) = (u1)1
+2+:::; u!1: (4.30)
+Then, matching (0)
+O(u) to (4.28), we obtain
+(0)
+O(u) =h
+(0)
+(u) +G(w)(0)
+(u)i
+: (4.31)
+The discussion here follows that of [17]. To higher orders, we can write
+(u) =(0)
+(u) +w(1)
+(u) +w2(2)
+(u) +:::; (4.32)
+(u) =(0)
+(u) +w(1)
+(u) +w2(2)
+(u) +:::; (4.33)
+where(n>0)
+ (u) and(n>0)
+ (u) are obtained demanding that in the u!1 limit, they are distinct from
+(0)
+(u) and(0)
+(u), respectively. Thus,
+O(u) = [(u) +G(w)(u)]: (4.34)
+Nearu!1 , one can expand (n)
+(u) and(n)
+(u) as follows (here n0)
+(n)
+(u!1 ) =a(n)
+
+1 +:::
++b(n)
+1
+u
+1 +:::
+; (4.35)
+(n)
+(u!1 ) =c(n)
+
+1 +:::
++d(n)
+1
+u
+1 +:::
+; (4.36)
+where the coecients a(n)
+;b(n)
+;c(n)
+andd(n)
+are all functions of q. So, asymptotically, we obtain
+^=h
+a(0)
++wa(1)
++O
+w2
i
++G(w)h
+c(0)
++wc(1)
++O
+w2
i
+; (4.37)
+^^=h
+b(0)
++wb(1)
++O
+w2
i
++G(w)h
+d(0)
++wd(1)
++O
+w2
i
+: (4.38)
+By virtue of (4.14){(4.19), these determine the frequency dependence of the retarded Green functions
+in the boundary theory, for small w.
+124.2 Criticality: Emergent IR Scaling
+Although the Green functions (4.14){(4.19) can be numerically computed by computing the coecients
+a(n)
+;b(n)
+;c(n)
+;d(n)
+, it turns out that one can analyze the low frequency behavior of the Green functions
+without knowing these coecients explicitly. For the cases of scalar and spinor operators, such analyses
+were performed in [17] where a variety of emergent IR phenomena were observed. In this subsection,
+we perform similar analyses of the retarded Green functions of the vector current and the energy-
+momentum tensor operators of the boundary theory at low frequency and determine their emergent
+IR behaviors. As we will see momentarily, one of the reasons which makes the low frequency analyses
+of the above-mentioned retarded Green functions non-trivial and interesting is the fact that there are
+now two IR CFT Green functions G(w) both of which can potentially play a role. Not surprisingly,
+the mixing ofG(w) in the expressions (4.14){(4.19) for the boundary theory Green functions is due
+to the coupled nature of the electromagnetic and gravitational perturbations we are considering in the
+charged (extremal) black hole background.
+As dened in (4.27), are real implying that a(n)
+;b(n)
+;c(n)
+andd(n)
+are all real. Thus the complex
+part of ^comes entirely from the IR CFT Green functions G(w). Suppose for generic momentum
+q, the product a(0)
++a(0)
+is non-vanishing. Expanding (4.20) and (4.21) for small w, and noticing that
+2+>3 while 2>1, one then deduces that Im G1(w;q) and ImG2(w;q) exhibit scaling behavior
+at small frequency:
+ImG1(w;q)'3Cp
+1 + q2
+1 +p
+1 + q2
+e0(q) ImG(w) [1 +:::]/w2; (4.39)
+ImG2(w;q)'3Cp
+1 + q2
+1p
+1 + q2
+e0(q) ImG(w) [1 +:::]/w2; (4.40)
+where
+e0(q) =b(0)
+
+a(0)
+ 
+c(0)
+
+a(0)
+d(0)
+
+b(0)
+!
+: (4.41)
+Substituting (4.39) and (4.40) into (4.14){(4.19), the spectral functions (the imaginary part of the
+aforementioned retarded Green functions) of the vector current ^Jyand the energy-momentum tensor
+operators ^Tytand ^Txyof the boundary eld theory show scaling behavior at small frequency
+ImGytyt(w;q)/w2; ImGxyxy(w;q)/w2+2; ImGytxy(w;q)/w1+2;
+ImGyty(w;q)/w2; ImGxyy(w;q)/w1+2; ImGyy(w;q)/w2:(4.42)
+Note that the low-frequency scaling behavior of the spectral functions above is emergent meaning
+that it is a consequence of the fact that the near horizon region of the background geometry (which
+translates into the IR physics of the boundary theory) contains an AdS 2factor. Although e0depends
+on details of the background geometry, the scaling behavior w2does not change by changing the
+geometry in the outer region as long as the AdS 2part of the near horizon geometry is kept intact.
+Also, note that because are strictly real, the retarded Green functions (4.14){(4.19) do not exhibit
+the log-periodicity behavior observed in [16, 17] for the cases of charged scalar and spinor operators.
+Of course, it is also of interest to consider the physical poles and branch points of the correlation
+functions at arbitrary qand w. We turn to a study of that problem in the following sections.
+135 Retarded Green Functions and Quasinormal Modes
+The expressions (4.14){(4.19) for the retarded Green functions of the boundary eld theory indicate
+that they generically have poles whenever ^(w;q) = 0. We denote by w
+n(q) the frequencies at which
+^(w;q) vanish. In order to obtain our boundary theory Green functions holographically, we imposed
+infalling boundary condition on  at the horizon. By further imposing  to vanish asymptotically
+we are essentially solving for their quasinormal modes (QNMs) in the extremal Reissner-Nordstr om
+AdS 4black hole background.
+In the context of the AdS/CFT correspondence, the connection between the QNMs of gravitational
+backgrounds and the singularities of the retarded Green functions of the dual boundary theories was
+rst suggested in [39]. It was then argued in [34] that the quasinormal frequencies of the decoupled
+
uctuations of a black hole background coincide with the poles in the retarded Green functions of
+the corresponding dual operators in the boundary theory. The authors of [40] then argued that
+in cases where the 
uctuations are coupled, in order for the quasinormal frequencies to coincide
+with the poles in the retarded Green functions of the dual operators, one should consider the gauge
+invariant combinations as the right set of variables and impose on them infalling boundary condition
+at the horizon together with an asymptotic normalizable boundary condition. In our case, as it is
+seen from the linearized Einstein-Maxwell equations (3.5){(3.8), the electromagnetic and gravitational
+perturbations of the background are certainly coupled. In section 3 we constructed the gauge invariant
+combinations X(u) andY(u) in terms of which the linearized Einstein-Maxwell equations are still
+coupled. We decoupled the equations for the gauge-invariant modes by introducing new variables
+(u). These variables are themselves gauge invariant by virtue of their denition (3.15) in terms
+ofX(u) andY(u). Thus, with  (u) being gauge invariant and satisfying decoupled second-order
+dierential equations, we can analyze their QNMs.
+To our understanding of the literature, the electromagnetic and gravitational QNMs of the extremal
+Reissner-Nordstr om AdS 4black hole have not been studied before4, which makes nding these QNMs
+an interesting problem in its own. Ref. [45] is a nice review of QNMs from both general relativity and
+the AdS/CFT perspectives.
+There are only very few backgrounds whose QNMs are known analytically. Although there are
+semi-analytic methods for calculating small or large (magnitudes of) quasinormal frequencies, one is
+usually forced to do numerics in order to extract the generic values of such frequencies. We refer
+the reader to [45] for an extensive list of analytic and numerical methods available in the literature
+on how to compute the QNMs of dierent backgrounds. In this section, we numerically compute
+the quasinormal frequencies of the (shear channel) electromagnetic and gravitational perturbations of
+the extremal Reissner-Nordstr om AdS 4black hole. Along the way, we nd it useful to compare our
+zero-temperature numerical results with the analogous QNMs of the non-extremal Reissner-Nordstr om
+AdS 4black hole background. Although, the results in the non-extremal case are not new, we present
+them here for the purpose of comparison and to study the T!0 limit (at xed ). By doing so, we
+4The QNMs of the four-dimensional asymptotically 
at extremal Reissner-Nordstr om black hole were studied in
+[42]. Ref. [43] analyzed the scalar, electromagnetic and gravitational QNMs of non-extremal Reissner-Nordstr om AdS 4
+black hole. Recently, the authors of [44] studied the QNMs of a massive, charged scalar eld in the background of the
+electrically, as well as the dyonic extremal Reissner-Nordstr om AdS 4black hole.
+14will see more clearly how the analytic structure at T= 0 is related to that of nite temperature.
+5.1 Generalities
+Before numerically computing the quasinormal frequencies for our problem, we discuss some general
+features that one expects to observe.
+5.1.1 Branch Cut
+For small w, the analytic structure is determined by the small frequency analysis that we presented
+earlier, pertaining to the near-horizon geometry. For generic values of q,G(w) are multi-valued and
+the boundary theory Green functions are then generically multi-valued as well. When q= 0, 2are
+integers: indeed 2 += 3 and 2= 1. In this case, the multi-valuedness of the Green functions is
+due to the existence of terms like wmlogwin their expressions, where mis a positive integer, in the
+extremal case.5For example, one can show that at small w
+Gxy;xy(w;q= 0)/iw
+1 +c1wlogw+c2iw; (5.1)
+wherec1andc2are numerical factors. Thus, the QNMs of  will also be multi-valued (in the
+extremal case). There is another, perhaps, more direct argument for the existence of the branch cut in
+the QNMs of  , which is based on analyzing the \tail" of the  
uctuations in the extremal black
+hole background [46]. An argument of this type was used in [44] to show the existence of the branch
+cut in the QNMs of a chargeless massless scalar in the extremal Reissner-Nordstr om AdS 4black hole
+background, and can easily be adopted to our case, as well. To that end, dene the (dimensionless)
+tortoise coordinate uvia
+du
+du=1
+u2f(u): (5.2)
+The asymptotic boundary of the background in the tortoise coordinate is obtained by taking u!0
+whereasu!1 is the near horizon region. Using the tortoise coordinate, we can write the equations
+in (3.16) in the form of Schr odinger equations (where we think of qas being xed)
+00
+(u) +
+3w2V(u)
+(u) = 0; (5.3)
+where the potentials V(u) are given by
+V(u) =6
+uf(u)
+1p
+1 + q2
++ 3q2f(u)u3f(u)f0(u): (5.4)
+It is understood that in the above expressions for V(u), one should use (5.2) to express uin terms
+ofu. Near the horizon, one easily nds
+V(u!1 )1
+u3: (5.5)
+5As we will recall below, such logs are absent at nite temperature, and the cut resolves into a series of Matsubara
+poles along the negative imaginary axis.
+15Such a power law behavior in the asymptotic form of the potentials is a characteristic of the background
+being extremal (as opposed to a non-extremal background for which the potential dies o exponentially
+asu!1 ). It is this power law behavior which gives rise to a branch cut [46] (for small frequencies),
+extended in the negative imaginary axis, for the QNMs of  . We will shortly see the appearance of
+this branch cut in our numerical plots of the QNMs.
+5.1.2 Stability
+Suppose are QNMs with frequencies w
+n. So, are ingoing at the horizon and vanish asymp-
+totically. Following [47], we will see shortly that, using the properties of the potentials V, these two
+boundary conditions constraint w
+n. Since are infalling at the horizon, one has eip
+3w
+n
+eip
+3w
+n(+u)where=r0t=L2is the dimensionless time coordinate and uis the tortoise coordinate
+dened in (5.2). Note that +uis the dimensionless ingoing Eddington coordinate. Dene
+(u) =eip
+3w
+nu	(u): (5.6)
+Substituting (5.6) into (5.3), and writing the result in terms of the coordinate u, we obtain
+
+u2f(u)	0
+(u)02ip
+3w
+n	0
+(u)U(u)	(u) = 0; (5.7)
+where the potentials U(u) are given by
+U(u) =6
+u3
+1p
+1 + q2
++3
+u2q2uf0(u): (5.8)
+We multiply (5.7) by 	(u) and integrate the result to obtain
+
+u2f(u)	(u)	0
+(u)1
+1Z1
+1duh
+u2f(u)	0
+(u)2+ 2ip
+3w
+n	(u)	0
+(u) +U(u)j	(u)j2i
+= 0
+(5.9)
+where we have also performed an integration by parts. Using (5.6) and (4.11) and the fact that  (u)
+are assumed to be QNMs, one obtains that as u!1 ,u2	0
+(u) is nite and 	(u) vanishes. At the
+horizon, both 	(u) and 	0
+(u) are nite and f(u) vanishes. Thus, all together, one concludes that
+u2f(u)	(u)	0
+(u)1
+1= 0. Take the complex conjugate of (5.9) and subtract the result from (5.9)
+to obtain
+Z1
+1du
+w
+n	(u)	0
+(u) +w
+n	(u)	0
+(u)
+= 0: (5.10)
+Integrating by parts the second term in (5.10), and noting that 	 (u) vanishes at u=1, we obtain
+
+Imw
+n
Z1
+1du	(u)	0
+(u) =i
+2w
+nj	(u= 1)j2: (5.11)
+Since 	(u= 1) is nite, if Re w
+n6= 0 equation (5.11) then implies that Im w
+n6= 0. Substituting
+(5.11) into (5.9), we obtain
+Z1
+1duh
+u2f(u)	0
+(u)2+U(u)j	(u)j2i
++p
+3
+Imw
+n
1w
+n2j	(u= 1)j2= 0: (5.12)
+16For all (real) values of q,U+(u) is always positive denite. This is because in order for U+(u) to be
+always positive denite, one has to have 3 q2+ 2p
+1 + q220, which is satised for all (real) values
+ofq. Thus, the integral (with subscript +) in (5.12) never becomes negative. From equation (5.12),
+we then reach the conclusion that Im w+
+n<0. This result indicates that the  +(u) 
uctuation does
+not cause instability, as eip
+3w+
+n+decays in time for all w+
+n. ForU(u), on the other hand, it can
+easily be shown that the condition for it being positive denite turns out to be 3 q22p
+1 + q220
+which is true only if q216
+9. Therefore, for jqj4
+3, the integral (with subscript ) in (5.12) is
+positive denite yielding Im w
+n(q)<0. Forjqj<4
+3, there are ranges of ufor whichU(u) becomes
+negative. As a result, when jqj<4
+3, without knowing the solution 	 (u), equation (5.12) cannot
+by itself determine the sign of Im w
+n(q). However, as we will see later on, it is easy to numerically
+conrm that even for jqj<4
+3one obtains Im w
+n(q)<0, so the (u) 
uctuation does not cause
+instability either.
+In the background geometry we are considering, the quasinormal frequencies of  are further
+constrained. Using equations (5.3) together with the infalling boundary condition (5.6), one easily
+concludes that if w
+n(q) is a quasinormal frequency of  , so isw
+n(q), that is to say that the
+quasinormal frequencies appear as pairs with the same imaginary part but opposite real part. On the
+eld theory side, the pairing of the poles of the retarded Green functions (4.14){(4.19) is due to the
+fact that the boundary theory is parity invariant.6
+5.2 Numerical Analysis: Matrix Method
+To compute the quasinormal frequencies of  (in the extremal case) numerically, we use the \matrix
+method" of [48]. Unlike the algorithm of [47] which is known not to work for extremal black hole
+backgrounds7, the matrix method of [48] is believed to be applicable here. The authors of [44] used
+this matrix method to nd the quasinormal frequencies of a charged minimally-coupled scalar eld in
+the extremal Reissner-Nordstr om AdS 4background. In what follows we rst present our numerical
+results for the quasinormal frequencies of  in the extremal case, then compare their general features
+with the analogous quasinormal frequencies when the Reissner-Nordstr om AdS 4black hole background
+is non-extremal.
+5.2.1 Zero Temperature
+For numerical purposes it is more convenient to switch to a new radial coordinate z= 1=u. In this
+coordinate, z= 1 represents the horizon while z= 0 is the asymptotic boundary. We write the
+equations (3.16) in terms of the new coordinate zand arrive at
+z2
+f(z)0
+(z)0+
+zf0(z) +3z2
+f(z)
+w2f(z)q2

+6z3
+1p
+1 + q2
+(z) = 0: (5.13)
+To apply the matrix method of [48], we rst isolate the leading behaviors of  (z) at the horizon and
+the boundary. The rest is then approximated by a power series in zaround a point z0such that the
+6This symmetry of the quasinormal modes would be broken, for example, in the presence of a magnetic eld.
+7This is because for extremal black holes, the horizon is an essential singularity.
+17radius of convergence of the series covers both the horizon and the boundary. That point in our case
+is taken to be z0=1
+2. Thus, we write
+(z) =eiwp
+12(1z)f(z)ip
+3w
+9zMX
+m=0a
+m(w;q)
+z1
+2m
+: (5.14)
+Substituting (5.14) into (5.13), we obtain a set of M+1 linear equations for M+1 unknowns ap(w;q)'s,
+and write them in a matrix form
+MX
+m=0A
+mp(w;q)a
+p(w;q) = 0: (5.15)
+Suppose now we choose a specic value for the spatial momentum q. The quasinormal frequencies w
+n
+for that particular value of qare solutions to the following equation
+detA
+mp(w
+n;q) = 0: (5.16)
+Since in (5.14) are approximated by power series, the more terms kept in the series, the more
+accurate results one obtains for the quasinormal frequencies of  .
+(a) ( b)
+-4-2024-8-6-4-20
+ReHwmLImHwmL
+-4-2024-10-8-6-4-20
+ReHwmLImHwmL
+Figure 1: Electromagnetic and gravitational quasinormal frequencies (in the shear channel) of extremal Reissner-
+Nordstr om AdS 4black hole. Plot (a) shows the quasinormal frequencies of  +and plot (b) shows the quasinormal
+frequencies of  . For both plots, q= 1 andM= 100. AsMis increased the poles on the negative imaginary axis get
+closer to one another and form a branch cut in the M! 1 limit.
+The plots (a) and (b) in Figure 1 show respectively the quasinormal frequencies of  +and for
+q= 1, where, due to space limitations, we have only shown a handful of them. We used M= 100
+for both plots. It can easily be veried numerically that, for both plots, as Mis increased the poles
+on the negative imaginary axis become closer to one another. This observation suggests that the
+sequence of discrete poles on the negative imaginary axis is due to approximating  in (5.14) by
+power series. In particular, although not numerically accessible, it should be the case that in the
+M!1 limit there is actually a branch cut [44]. This picture is consistent with the two arguments
+we presented earlier for the existence of the branch point at the origin in the extremal case. Notice
+18that the plots are symmetric under w
+n! w
+nas they should be. One also sees that the o-axis
+poles are approximately equally spaced and line up on almost straight lines. Aside from the poles on
+the negative imaginary axis, which coalesce as Mis increased, the qualitative behavior of the plots
+does not change by increasing M. In particular, we do not see any poles crossing into the real axis,
+indicating that these QNMs do not cause instabilities in the system. This observation is consistent
+with the argument we gave earlier in subsection 5.1.2. The argument there was not powerful enough
+to determine the sign of Im w
+n(q) for q2(4
+3;4
+3). The plot (b) of Figure 1 (for which q= 1) shows
+that the imaginary part of w
+nis negative. One can also plot the quasinormal frequencies of  for
+other values of qin the range (4
+3;4
+3) and observe that Im w
+nis always negative. Indeed we have
+shown one such plot in Figure 2(b).
+The plots in Figure 2 show the quasinormal frequencies of  forq= 1=p
+3. Although not shown,
+we considered other values of qand were able to verify that the qualitative patterns of the plots shown
+in Figures 1 and 2 stay the same as qis varied. Excluding the poles on the negative imaginary axis,
+which form a branch cut in the M!1 limit, we observed that independent of what value qtakes,
+there are no small quasinormal frequencies for  , while there are only two for  +(which are the
+ones closest to the real axis). Here, by small quasinormal frequencies we mean those which satisfy
+jw
+nj1. Therefore, one can conclude that at small frequencies jwj1, there are only two single
+poles in the boundary theory retarded Green functions (4.14){(4.19). Moreover, those poles do not
+approach zero as q!0.
+(a) ( b)
+-4-2024-10-8-6-4-20
+ReHwmLImHwmL
+-4-2024-10-8-6-4-20
+ReHwmLImHwmL
+Figure 2: Quasinormal frequencies of  (in the extremal case) for q= 1=p
+3. (a) Quasinormal frequencies of  +. (b)
+Quasinormal frequencies of  .M= 100 for both plots.
+The analysis at q= 0 is quite a bit simpler, but is qualitatively similar. In this case, the relevant
+equations are
+z2f(z)hx00
+y(z) +
+z2f0(z)2zf(z)
+hx0
+y(z) +3w2
+f(z)z2hxy(z) = 0; (5.17)
+f(z)a00
+y(z) +f0(z)a0
+y(z) +3w2
+f(z)12z2
+ay(z) = 0; (5.18)
+which are obtained from equations (3.13) and (3.14) by changing the radial coordinate to z= 1=u.
+19(Note that hxy(z) is gauge invariant at q= 0, while ay(z) is gauge invariant for all q.) We can nd
+the quasinormal frequencies of hxy(z) anday(z) similar to the steps we performed above to obtain the
+quasinormal frequencies of  . The only dierence is that the normalizable mode of hxy(z) falls o
+asymptotically as z3. Therefore in writing the analog of (5.14) for hxy(z) the factor of zshould be
+replaced by z3. The results are shown in Figures 3(a) and 3(b). Note that equation (5.17) is the same
+as the equation of motion for a minimally-coupled chargeless massless scalar in the background of the
+extremal Reissner-Nordstr om AdS 4black hole. The quasinormal frequencies of this scalar have been
+numerically computed in [44] using the same matrix method and a (density) plot resembling the one
+in Figure 3 (b) was generated there.
+(a) ( b)
+-4-2024-10-8-6-4-20
+ReHwmLImHwmL
+-4-2024-10-8-6-4-20
+ReHwmLImHwmL
+Figure 3: Quasinormal frequencies for q= 0 ofayandhx
+yin the extremal Reissner-Nordstr om AdS 4black hole
+background. Plot (a) shows the quasinormal frequencies of ayand plot (b) are the quasinormal frequencies of hx
+y. We
+setM= 100 for both plots.
+The pattern we observe in the two plots of Figure 3 is very much similar to the ones in Figures 1
+and 2. The plots were obtained for M= 100, but the pattern stays almost the same for higher values
+ofM. One still observes that the poles on the negative imaginary axis coalesce while the locations of
+the poles on each side of the imaginary axis do not change much. Also, we neither nd poles crossing
+into the upper half plane, nor do we nd poles which vanish. As it is seen from Figure 3(b), there are
+no (diagonally-oriented) quasinormal frequencies for hxyat smalljwj1 frequencies while there are
+two such frequencies for ayin that range. The quasinormal frequencies of ayin Figure 3(a) can be
+used to estimate, along the lines of [44], the UV behavior of a one-loop contribution to the free energy
+of the extremal background by the gauge eld ay. Knowing the one-loop corrections to the free energy
+enables one to compute a \1 =Ncorrection" to charge conductivity.
+5.2.2 Finite Temperature
+It is instructive to compare our results for the quasinormal frequencies of  in the extremal case to
+the quasinormal frequencies of  in the background of non-extremal Reissner-Nordstr om AdS 4black
+20hole. Written in terms of the z-coordinate, equations (A.9) take the form
+z2
+f(z)0
+(z)0+
+zf0(z) +Q2z2
+f(z)
+w2f(z)q2

+2Q2g(q)z3
+(z) = 0; (5.19)
+whereQ2<3 in the non-extremal case. The denition of  at nite temperature is given in (A.7).
+(a) ( b)
+-4-2024-10-8-6-4-20
+ReHwmLImHwmL
+-4-2024-8-6-4-20
+ReHwmLImHwmL
+Figure 4: Quasinormal frequencies of  in non-extremal Reissner-Nordstr om AdS 4black hole background. Plot (a)
+shows the quasinormal frequencies of  +while plot (b) shows the quasinormal frequencies of  . For both plots, q= 1,
+M= 100, and T= 0:09.
+In the non-extremal case, we also use w
+nto denote the quasinormal frequencies of  . In order to
+nd w
+nwe use the matrix method of [48] and follow the exact same steps as we did in the extremal
+case except that (5.14) is now replaced by
+(z) =f(z)iw=zMX
+m=0a
+m(w;q)
+z1
+2m
+; (5.20)
+where we have dened = 4T= , withTandbeing the temperature and chemical potential,
+respectively. The nite-temperature quasinormal frequencies of  have been plotted in Figure 4.
+Only a handful of w
+nhave been shown, though. To generate the plots, we chose a temperature of
+T= 0:09. Also, the plots were obtained for q= 1 andM= 100. The branch cut in Figures 1(a)
+and 1(b) are replaced at nite temperature by a sequence of almost equally-distanced poles along
+the negative imaginary axis. Note that the poles are all in the lower half of the frequency plane,
+indicating that they do not give rise to instabilities. As q!0, none of the quasinormal frequencies
+of  +approach zero. On the other hand, the lowest quasinormal mode of  , shown in Figure 4(b)
+by a larger red dot, approaches zero as q!0. This is the leading hydrodynamic pole. Thus, one
+concludes that the nite temperature generalization of (4.14){(4.19) all have the same hydrodynamic
+pole, namely the diusion constant Dis the same for all of them. In Figure 5, we plotted the lowest
+quasinormal frequency of  (shown in Figure 4(b) by a larger red dot) as a function of q, and tted
+the plot by a quadratic function. The solid black curve on the plot shows the t. From the slope of the
+tted curve, we obtain D' 0:17=, withT= 0:09. Substituting the numerically obtained diusion
+210.00.20.40.60.81.0-0.20-0.15-0.10-0.050.00
+kmImHwmLFigure 5: The lowest quasinormal frequency of  as function of q. The temperature is T= 0:09, andM= 100. The
+solid black curve is the quadratic t: iw'0:17q2.
+constant into the hydrodynamic equation = (Ts+)D, where,sandare respectively given in
+(2.4) and (2.6), one easily obtains =s'1=4.8
+(a) ( b)
+-4-2024-10-8-6-4-20
+ReHwmLImHwmL
+-4-2024-10-8-6-4-20
+ReHwmLImHwmL
+Figure 6: (a) Finite-temperature quasinormal frequencies of ayatq= 0. (b) Finite-temperature quasinormal frequen-
+cies ofhx
+y. The temperature is T= 0:09, andM= 100.
+We present the case of q= 0 separately. The relevant equations in this case read
+z2f(z)hx00
+y(z) +
+z2f0(z)2zf(z)
+hx0
+y(z) +Q2w2
+f(z)z2hxy(z) = 0; (5.21)
+f(z)a00
+y(z) +f0(z)a0
+y(z) +Q2w2
+f(z)4z2
+ay(z) = 0; (5.22)
+which are equations (A.5) and (A.6) expressed in terms of the coordinate z= 1=u. We nd the nite-
+temperature quasinormal frequencies of hxy(z) anday(z) following what we did above to obtain the
+8The sign 'is because of numerical uncertainties. One can of course show analytically that the ratio should be equal
+to 1=4[28, 29].
+22nite temperature quasinormal frequencies of  . The only dierence is that in writing the analog
+of (5.20) for hxy(z) we replace the factor of zbyz3. This is because asymptotically the normalizable
+mode ofhxy(z) falls o as z3.
+The quasinormal frequencies of ayandhxyhave been plotted in Figure 6 for a temperature of
+T= 0:09usingM= 100. We see that, as for q6= 0, there is no branch cut at nite temperature: it
+is replaced by a sequence of almost equally-distanced poles along the negative imaginary axis. Also,
+as it is seen from the plots, the nite-temperature quasinormal frequencies of ayandhxystay away
+from the real axis. To the extent that we have checked numerically, this behavior of the quasinormal
+frequencies is unchanged as Mis increased.
+There does not appear to be a qualitative dierence in the quasinormal spectrum between the
+T= 0 results and the nite temperature spectrum for small temperatures. To see that more clearly,
+we consider the limit T!0, holdingxed. For numerical simplicity, we have done this at q= 0, but
+do not expect an appreciable dierence at nite q. Referring to Figure 6, we consider the following
+three quantities. We denote by d1the dierence between the smallest quasinormal frequency on the
+negative imaginary axis and the origin, by d2the distance between the rst two poles on the imaginary
+axis and by d3, the distance between the two o-axis poles in the rst quadrant closest to the origin. In
+Figure 7(a), we demonstrate that as Tis lowered, holding xed, both d1andd2approach zero, while
+d3remains nite. Approaching T!0 requires ever-more numerical precision, but the plots suggest
+thatd1andd2extrapolate to zero at T= 0. This is consistent with the on-axis poles becoming the
+branch cut seen in the T= 0 theory, as one might expect.
+(a) ( b)
+0.000.050.100.150.200123456
+Tmdi
+0.000.050.100.150.200.00.51.01.52.02.53.03.5
+Tmdi
+Figure 7: (a) Plots of d1(red),d2(blue) and d3(green) as function of T= foray. (b) Plots of d1,d2andd3as
+function of T= forhx
+y. In the range of T=2[0:017;0:091], the red and blue curves almost overlap. The plots have
+been generated with M= 150.
+6 Conclusions
+In this paper, we considered a boundary eld theory dual to the extremal Reissner-Nordstr om AdS 4
+black hole, and analyzed the retarded two-point functions of the charge vector current and energy-
+23momentum (tensor) operators at nite frequency and momentum. The operators ^Jy,^Tytand^Txywhose
+two-point functions we analyzed were in the shear channel of the boundary eld theory, whose dual
+bulk modes ay,hytandhxywere odd under parity, y!y. We found that for generic momentum, the
+retarded correlators of the aforementioned vector and tensor operators exhibit emergent scaling behav-
+ior at low frequency. The operators ^Jy,^Tytand ^Txygave rise in the IR to two sets of scalar operators
+Owith conformal dimensions . The low frequency scaling exponents of the spectral densities in
+the boundary theory depended only on (or equivalently, ). We explored the analytic structure of
+the retarded correlators for nite momentum and small frequency, and argued that there should exist
+a branch cut in these correlators at the origin, as well as a series of metastable modes corresponding
+to isolated poles in the lower half of the complex frequency plane. Taking linear combinations of the
+dual bulk modes, we constructed the gauge invariant modes from which we were able to numerically
+compute the spectrum of the (shear-type) electromagnetic and gravitational quasinormal frequencies
+of the background. The numerical computations were in agreement with what we argued, analytically,
+for the existence of the branch cut in the retarded correlators as well as their poles being in the lower
+half of the complex frequency plane. Turning on temperature, we numerically observed how the branch
+cut dissolves into a series of poles on the negative imaginary axis, and determined the corresponding
+dispersion constant of the leading pole, in agreement with previous results. We performed the shear
+channel analyses in the (2+1)-dimensional boundary theory. One can straightforwardly generalize the
+same type of analyses for higher dimensional boundary theories dual to extremal Reissner-Nordstrom
+AdSd+1black hole, and, not surprisingly, reach the same conclusions as we did for the case of the
+(2 + 1)-dimensional boundary theory.
+Although more involved, analyzing the retarded correlators of the operators in the sound channel
+of the (2+1)-dimensional boundary theory can similarly be performed. In the bulk, one deals with the
+modes which are even under the parity, y!y:htt,htx,hxx,hyy,at,ax. Taking linear combinations
+of these modes, one can construct two gauge-invariant modes, and compute, either numerically or
+analytically for small wand q, the spectrum of their quasinormal frequencies (having diagonalized the
+system of equations). The sound wave dispersion relation w=w(q) can then be determined from the
+poles of, say, Gtt;tt(w;q) at small wand q. It is interesting to verify that w(q)!0 as q!0. The
+analysis for the sound channel of the (2 + 1)-dimensional boundary will be presented elsewhere.
+Acknowledgments
+We would like to thank T. Andrade, S. Baharian, A. Buchel, E. Fradkin, S. Hartnoll, A. Hashimoto,
+C. Hoyos-Badajoz, A. Karch, J. McGreevy, A. Sinha and D. Son for helpful discussions. M.E. would
+like to especially thank S. Hartnoll for correspondence. R.G.L. and M.E. are supported by DOE grant
+FG02-91-ER40709, J.I.J. is supported by a Fulbright-CONICYT fellowship.
+24A Finite Temperature Equations
+In the background of non-extremal Reissner-Nordstr om AdS 4black hole, the linearized Einstein-
+Maxwell equations for the (odd-parity) modes hyt,hxy,aytake the form
+f(r)h
+r4hy00
+t(r) + 4r3hy0
+t(r) + 4Qr2
+0L2a0
+y(r)i
+!kL4hxy(r)k2L4hyt(r) = 0; (A.1)
+f(r)h
+r4f(r)hx00
+y(r) +
+r4f0(r) + 4r3f(r)
+hx0
+y(r)i
++!2L4hxy(r) +!kL4hyt(r) = 0; (A.2)
+f(r)h
+r4f(r)L2a00
+y(r) +r2L2
+r2f0(r) + 2rf(r)
+a0
+y(r) +Qr2
+0r2hy0
+t(r)i
++L6
+!2f(r)k2
+ay(r) = 0; (A.3)
+Also, the constraint equation (the yr-component of the linearized Einstein equations) reads
+r4! hy0
+t(r) +r4f(r)k hx0
+y(r) + 4Qr2
+0L2! ay(r) = 0: (A.4)
+In the above expressions, f(r) is given in (2.4) and has a single zero at the (outer) horizon r0. Note
+thatM= 1 +Q2.
+In the non-extremal case, we choose to rescale !andkby. The dimensionless frequency and
+momentum, wand q, are dened in (3.4). Also, like the extremal case, we work with the dimensionless
+radial coordinate u=r=r0. The gauge-invariant combinations are given by the same expressions as in
+(3.11) and (3.12): X(u) = qhyt(u) +whxy(u) andY(u) =ay(u).
+Atq= 0,hxy(u) becomes gauge-invariant. Also, the linearized Einstein-Maxwell equations (A.1){
+(A.4) give, in this case, two decoupled equations for hxy(u) anday(u)
+u4f(u)hx00
+y(u) +
+u4f0(u) + 4u3f(u)
+hx0
+y(u) +Q2w2
+f(u)hxy(u) = 0; (A.5)
+u2f(u)a00
+y(u) +
+u2f0(u) + 2uf(u)
+a0
+y(u) +Q2
+u2w2
+f(u)4
+u2
+ay(u) = 0: (A.6)
+For nite (non-zero) q, on the other hand, it can easily be worked out that the linearized Einstein-
+Maxwell equations yield two coupled second-order dierential equations for X(u) andY(u). Similar
+to the extremal case studied in the bulk of the paper, these equations can be decoupled. We rst
+dene
+(u) =qf(u)u3
+w2f(u)q2X0(u)2Q2
+u2f(u)q2
+w2f(u)q2+ug(q)
+Y(u); (A.7)
+where
+g(q) =3
+4
+1 +1
+Q20
+@1s
+1 +16
+9
+1 +1
+Q22
+q21
+A: (A.8)
+The decoupled equations then read
+
+u2f(u)0
+(u)0+
+uf0(u) +Q2
+u2f(u)
+w2f(u)q2

+2Q2
+u3g(q)
+(u) = 0: (A.9)
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+28
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+arXiv:1001.0780v3  [astro-ph.CO]  23 Feb 2010Zhang et al. 2010, ApJ, 711, 1033-1043
+Preprint typeset using L ATEX style emulateapj v. 08/13/06
+LOCUSS: A COMPARISON OF CLUSTER MASS MEASUREMENTS FROM XMM-NEWTON AND SUBARU —
+TESTING DEVIATION FROM HYDROSTATIC EQUILIBRIUM AND NON-TH ERMAL PRESSURE SUPPORT*
+Yu-Ying Zhang1, Nobuhiro Okabe2,3, Alexis Finoguenov4,5, Graham P. Smith6, Rocco Piffaretti7, Riccardo
+Valdarnini8, Arif Babul9, August E. Evrard10,11, Pasquale Mazzotta12,13, Alastair J. R. Sanderson6, and
+Daniel P. Marrone14,**
+Zhang et al. 2010, ApJ, 711, 1033-1043
+ABSTRACT
+We compare X-ray hydrostatic and weak-lensing mass estimates fo r a sample of 12 clusters that
+have been observed with both XMM-Newton and Subaru. At an over-density of ∆ = 500, we obtain
+1−MX/MWL= 0.01±0.07 for the whole sample. We also divided the sample into undisturbed an d
+disturbed sub-samples based on quantitative X-ray morphologies u sing asymmetry and fluctuation
+parameters, obtaining 1 −MX/MWL= 0.09±0.06and−0.06±0.12for the undisturbed and disturbed
+clusters, respectively. In addition to non-thermal pressure sup port, there may be a competing effect
+associated with adiabatic compression and/or shock heating which le ads to overestimate of X-ray
+hydrostatic masses for disturbed clusters, for example, in the fa mous merging cluster A1914. Despite
+the modest statistical significance of the mass discrepancy, on av erage, in the undisturbed clusters, we
+detect a clear trend of improving agreement between MXandMWLas a function of increasing over-
+density,MX/MWL= (0.908±0.004)+(0 .187±0.010)·log10(∆/500). We also examine the gas mass
+fractions, fgas=Mgas/MWL, finding that they are an increasing function of cluster radius, with no
+dependence on dynamical state, in agreement with predictions fro m numerical simulations. Overall,
+our results demonstrate that XMM-Newton and Subaru are a powerful combination for calibrating
+systematic uncertainties in cluster mass measurements.
+Subject headings: cosmology: observations — galaxies: clusters: general — galaxies: clusters: indi-
+vidual (Abell 1914) — gravitational lensing: weak — surveys — X-ray s: galaxies:
+clusters
+1.INTRODUCTION
+Electronic address: yyzhang@astro.uni-bonn.de
+*This work is based on observations made with the XMM-
+Newton, an ESA science mission with instruments and contribu-
+tions directly funded by ESA member states and the USA (NASA) ,
+and data collected at Subaru Telescope and obtained from the
+SMOKA, which is operated by the Astronomy Data Center, Na-
+tional Astronomical Observatory of Japan.
+1Argelander-Institut f¨ ur Astronomie, Universit¨ at Bonn, Auf
+dem H¨ ugel 71, 53121 Bonn, Germany
+2Academia Sinica Institute of Astronomy and Astrophysics,
+P.O. Box 23-141, 10617 Taipei, Taiwan
+3Astronomical Institute, Tohoku University, Aramaki, Aoba -ku,
+Sendai, 980-8578, Japan
+4Max-Planck-Institut f¨ ur extraterrestrische Physik, Gie ssen-
+bachstraße, 85748 Garching, Germany
+5University of Maryland, Baltimore County, 1000 Hilltop Cir -
+cle, Baltimore, MD 21250, USA
+6School of Physics and Astronomy, University of Birmingham,
+Edgbaston, Birmingham, B152TT, UK
+7CEA, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
+8SISSA/ISAS, via Beirut 4, 34014 Trieste, Italy
+9Department of Physics and Astronomy, University of Victori a,
+3800 Finnerty Road, Victoria, BC Canada
+10Department of Physics and Michigan Center for Theoretical
+Physics, University of Michigan, Ann Arbor, MI 48109, USA
+11Departments of Physics and Astronomy, University of Cali-
+fornia, Berkeley, CA 94720, USA
+12Dipartimento di Fisica, Universita di Roma “Tor Vergata,”
+Via della Ricerca Scientifica 1, I-00133 Rome, Italy
+13Harvard-Smithsonian Center for Astrophysics, 60 Garden
+Street, Cambridge, MA 02138, USA
+14Kavli Institute for Cosmological Physics, Department of As -
+tronomy and Astrophysics, University of Chicago, Chicago, IL
+60637, USA
+**Hubble FellowThe mass function of galaxy clusters depends on both
+the matter density and the expansion history of the uni-
+verse. Indeed, clusters provided early evidence for a low
+density universe (White et al. 1993). Clusters, as pow-
+erful tools to constrain cosmological parameters (e.g.,
+Zhang & Wu 2003; Balogh et al. 2006; Henry et al.
+2009), currently receive much attention as a potential
+probe of the dark energy equation of state parameter
+(w=p/ρ, whereρis the energydensity and pis the pres-
+sure), through the evolution of the mass function. (e.g.,
+Vikhlinin et al. 2009a, 2009b). In addition, gas mass
+fractionmeasurementsalsopotentiallyprovideanimpor-
+tant cosmological probe, under the assumption that gas
+massfractionsdonot evolvewith redshift(e.g., Vikhlinin
+et al. 2002; Allen et al. 2004, 2008; Mantz et al.
+2010). Upcoming galaxy cluster surveys will shortly de-
+liver huge amounts of multi-wavelength data, e.g., from
+Subaru/Hyper-Suprime-Cam, eROSITA ,PLANCK , and
+South Pole Telescope (SPT). To achieve good control
+over systematic errors in cosmologicalmeasurements, for
+example of w, based on these surveys, it is crucial to
+understand cluster mass estimates.
+The density and temperature distributions of the hot
+X-ray emitting gas within galaxy clusters can be used
+to estimate the total mass of the cluster. Since the ac-
+celeration of the gas is still not well understood, it is
+assumed that the acceleration terms are negligible when
+estimating cluster masses from X-ray data; the resulting
+massestimatesthusinvoketheassumptionofhydrostatic
+equilibrium (H.E.), and are hereafter referred to as “X-
+ray hydrostatic mass estimates”. Such mass estimates
+also assume that the total pressure is dominated by the2 Zhang, Okabe, Finoguenov, et al.
+thermal pressure of the gas.
+Numerical simulations (e.g., Evrard 1990; Lewis et al.
+2000; Rasia et al. 2006; Nagai et al. 2007; Piffaretti &
+Valdarnini 2008, PV08, hereafter; Jeltema et al. 2008;
+Lau et al. 2009) have pointed out that X-ray hydrostatic
+massestimatesmayunderestimateclustermass,andthat
+this effect is most pronounced for clusters classified as
+disturbed, based on their X-ray morphology. There-
+fore, measurements of cosmological parameters based on
+the X-ray-measured cluster mass function or the X-ray-
+measured gas mass fraction may be biased. It is thus
+crucial, as a minimum, to calibrate observationally both
+the X-ray-measured cluster masses and gas mass frac-
+tions for a representative cluster sample. This opportu-
+nity is offered by measuring cluster masses using both
+X-ray and gravitational lensing data.
+X-ray and gravitational lensing mass measurements
+have complementaryadvantagesand disadvantages. The
+n2
+edependence of the X-ray emissivity of the intraclus-
+ter medium (ICM) helps guard against projection effects
+due to mass along the line of sight through the cluster.
+However, as discussed above, X-ray analysis requires as-
+sumptions to relate electromagnetic radiation to a model
+of the mass distribution. An ideal case is to use an ap-
+proach that is insensitive to the dynamical state to esti-
+mate the cluster mass. Gravitational lensing fulfills this
+requirement because the lensing signal is insensitive to
+the physical state and nature of the deflecting matter
+distribution. Lensing is, however, prone to projection
+effects because it is sensitive to all mass along the line
+of sight through the cluster (e.g., Hoekstra 2001). It
+is thus of paramount importance to combine these two
+techniques to develop a thorough understanding of clus-
+ter mass measurements. Since the 1990’s, the use of X-
+ray and lensing observations has been proposed to test
+deviations from H.E. and extra pressure support (e.g.,
+Miralda-Escude& Babul 1995; Wu & Fang 1996; Squires
+et al 1996; Allen 1998; Zhang et al. 2005, 2008; Mahdavi
+et al. 2008; Richard et al. 2010). Most earlier studies
+employed ”target of opportunity” mode — i.e., consid-
+ered clusters with available data — making the studies
+biased by design.
+The Local Cluster Substructure Survey (LoCuSS17; G.
+P. Smith et al. in preparation; Zhang et al. 2008; Haines
+et al. 2009a, 2009b; Sanderson et al. 2009; Marrone
+et al. 2009; Okabe et al. 2010a; Richard et al. 2010)
+is a systematic multi-wavelength survey of X-ray lumi-
+nous (LX,0.1−2.4keV≥2·1044erg s−1) galaxy clusters at
+0.15<∼z<∼0.3 selected from the ROSAT All-Sky Survey
+(RASS; Ebeling et al. 2000; B¨ ohringer et al. 2004) in a
+manner blind to the dynamical status. As a first step to-
+ward a comprehensive X-ray/lensingstudy we assembled
+a sample of 19 clusters with archival XMM-Newton ob-
+servations, and weak-lensing mass measurements in the
+literature (Zhang et al. 2008). The mean weak-lensing
+mass to X-ray hydrostatic mass ratio was found to be
+∝angbracketleftMWL/MX∝angbracketright= 1.09±0.08 at an over-density of ∆ = 500
+withrespecttothecriticaldensity. Mahdavietal. (2008)
+found the average X-ray to weak-lensing mass ratio to
+be 0.78±0.09 at ∆ = 500, being consistent with the
+results in Zhang et al. (2008). The uncertainties in the
+above analysis was dominated by measurement errors, in
+17http://www.sr.bham.ac.uk/locussparticular, the lensing masses drawn from the literature
+based on early lensing data (Bardeau et al. 2007; Dahle
+2006),withtypicalseeingofFWHM∼>0.′′8,usingfourdif-
+ferent camerason three different ≤3.6-m telescopes, with
+fields of view (FOVs) spanning r∼0.7−2Mpc at z∼0.2.
+The next step in our program is to employ uniform high-
+quality weak-lensing data from our own dedicated ob-
+serving program with Subaru/Suprime-CAM. Recently,
+the first batch of weak-lensing mass measurements have
+become available for 22 clusters based on data taken in
+good conditions (FWHM ≃0.′′6) through two-filters with
+detailsoutlinedinSection2.1(Okabeetal. 2010a;Okabe
+& Umetsu 2008). These clusters were selected based on
+observability from Mauna Kea on nights allocated to Lo-
+CuSS, and are thus unbiased with respect to their X-ray
+properties. Archival XMM-Newton data are availablefor
+12 of the 22 clusters — see Zhang et al. (2008) for de-
+tails. In this work, wethereforecompareX-ray(basedon
+theassumptionofH.E.)andweak-lensingmassestimates
+for these 12 clusters, and also compare our observational
+results with predictions from numerical simulations. It
+is worth noting that Subaru/Suprime-CAM provides an
+excellentmatch tothe XMM-Newton FOV,and alsothat
+these Subaru weak-lensing mass estimates have optimal
+precisioninthedensitycontrastrangeof500 ≤∆≤2000
+(Okabe et al. 2010a), again, well matched to the XMM-
+NewtonX-ray estimates.
+The outline of this paper is as follows. In Section 2
+we briefly describe the weak-lensing and X-ray analysis.
+The results are presented in Section 3, and discussed in
+Section 4. We summarize our conclusions in Section 5.
+Throughout the paper, we assume Ω m= 0.3, ΩΛ= 0.7,
+andH0= 70 kms−1Mpc−1. Confidence intervals cor-
+respond to the 68% confidence level. Unless explicitly
+stated otherwise, we apply the Orthogonal Distance Re-
+gression package (ODRPACK 2.0118, e.g., Boggs et al.
+1987) taking into account measurement errors on both
+variables to determine the parameters and their errors of
+the fitting.
+2.DATA ANALYSIS
+The 12 clusters in our sample are listed in Table 1.
+2.1.weak-lensing analysis
+The Subaru i′- andV-band imaging data and detailed
+weak-lensing analysis are described in Section 2 in Ok-
+abe et al. (2010a), in which the i′-band data are used
+for shape measurements, and the V-band data are used
+to remove foreground and cluster galaxies. Excluding
+unlensed galaxies from the background galaxy catalog is
+of prime importance for accurate weak-lensing mass esti-
+mates, especially at higher over-densities. This so called
+dilution bias increases as a function of the density con-
+trast ∆, and can cause MWL
+500andMWL
+2500to be biased
+low by∼20%−50% (Okabe et al. 2010a). Our two-
+filter lensing data allow us to construct secure red+blue
+backgroundgalaxysamples,definedasfaintgalaxieswith
+colors that are redder and bluer than the cluster red-
+sequence by a minimum color offset. This strategy typi-
+cally reduces the dilution bias to per cent level (Okabe et
+al. 2010a). The mean redshift for the background galaxy
+18http://www.netlib.org/odrpack and references thereinLoCuSS: A Comparison of Cluster Mass Measurements from XMM-Newton and Subaru 3
+catalogisestimatedbymatchingthemagnitudesandcol-
+ors of background galaxies to the COSMOS photometric
+redshift catalog (Ilbert et al. 2009). More precisely, the
+mean redshift is computed as a lens weighted average
+over the redshift distribution, dPWL/dz, which is defined
+by∝angbracketleftDls/Ds∝angbracketright=/integraltext
+zddzdPWL/dzDls/Ds, whereDsandDls
+arethe angulardiameterdistancestosourceandbetween
+lens and source, respectively.
+Galaxy cluster mass distributions are often modeled
+as Navarro, Frenk, and White (NFW; Navarro et al.
+1997)halosorsingularisothermalspheres(SIS).Inweak-
+lensing studies, the distortion signal is used to constrain
+the model parameters. Okabe et al. (2010a) have shown
+that the NFW model fits the lensing distortion profiles
+well in both statistical studies and their analysis of in-
+dividual clusters. The SIS model is statistically inade-
+quate to describe stacked tangential distortion profiles
+with pronounced radial curvatures, and is rejected at
+6σand 11σlevel, respectively, for their two sub-samples
+with the virial masses in the ranges of <6×1014h−1M⊙
+and≥6×1014h−1M⊙. It is also important to note that
+the mean ratio of masses obtained from SIS and NFW
+models is 0 .70±0.05atrWL
+500for our sample of12 clusters.
+The projected mass distribution can also be obtained
+using a model-independent approach, the so-called ζc-
+statistics method (Fahlman et al. 1994; Clowe et al.
+2000), which is complementaryto the tangential shear fit
+method. The ζc-statistics method measures the discrete
+integration of averaged tangential distortions of source
+galaxies outside given radii. The ζc-statistics method is
+thus less sensitive to the detailed structure of clusters on
+smallscalesthanusingmodelsfittedtothefulltangential
+shear data. Okabe et al.’s ζc-based model-independent
+projected masses are in good agreement with the pro-
+jected masses computed from the NFW-based models at
+∆ = 500, but not with the SIS-based models.
+We also compared the NFW-based spherical mass es-
+timates for our sample of 12 clusters with the spheri-
+cal mass estimates based on deprojecting the ζc-based
+model-independent masses. In the latter case, the spher-
+icalmasswasobtainedbyassuminganNFWprofile. The
+spherical masses using the ζc-statistics and NFW mod-
+els are statistically consistent, given their mean ratios of
+1.00±0.08, 0.96±0.07, and 1 .02±0.07, atrWL
+500,rWL
+1000,
+andrWL
+2500, respectively, weighted by the inverse square
+of the errors. For two clusters, A383 and A2390, the
+masses from the NFW models are lower than the de-
+projected masses from the ζc-statistics method due to
+substructures in the cluster cores.
+Based on the above tests, our joint analysis uses the
+NFWmassesobtainedfromthetangentialdistortionpro-
+files in Okabe et al. (2010a). The three-dimensional
+spherical cluster masses (see Table 2), MWL
+∆, within a
+sphere ofradius r∆, is derived following the NFW model,
+ρ∝r−1(1 +c∆r/r∆)−2, wherec∆is the concentration
+parameter.
+The weak-lensing analysis includes both statistical er-
+ror and errors of the photometric redshifts of source
+galaxies in mass measurements. The latter is estimated
+by bootstrap re-sampling to match the COSMOS cata-
+log. Leauthaudetal. (2010)found thatthe lattermainly
+matters for galaxies close to the lens. The typical 1 σ
+total uncertainty on the weak-lensing mass estimates is∼12%−21% at ∆ = 500 with four exceptions, i.e.,
+RXJ2129.6+0005, 39%; A68, 41%; A115 (south), 53%;
+and Z7160, 42%.
+Hoekstra (2001) pointed out that the uncertainty in
+weak-lensing mass estimates of clusters, caused by dis-
+tant large-scale structures (uncorrelated) along the line
+of sight is fairly small for deep observations (20 < R <
+26) of massive clusters at intermediate redshifts. The
+typical 1 σrelative uncertainty is about 6% if the lensing
+signal is measured out to 1 .5h−1
+50Mpc. All 12 clusters in
+our sample are massive clusters ( >5 keV) at intermedi-
+ate redshifts (0 .15≤z≤0.3) with deep Subaru obser-
+vations to ∼26 mag. The Subaru FOV covers the entire
+cluster up to a few Mpc for our sample. Therefore, the
+mean mass uncertainty caused by large-scale structures
+due to projection should be no more than 6%. Therefore
+we neglect this error in the mass estimates.
+2.2.X-ray analysis
+The X-rayanalysiswascarriedout independently from
+the weak-lensing analysis. The description of XMM-
+Newtondata and X-ray mass modeling is given in Sec-
+tion 2 in Zhang et al. (2008), and more details on the
+data reduction are given in Section 2 in Zhang et al.
+(2006, 2007). The radial temperature profile was mea-
+sured using spectral data including deprojection. The
+X-ray spectrum measuring the global temperature was
+used to calculate the cooling function for the conversion
+from X-ray surface brightness profile to electron num-
+ber density profile, in which we included both depro-
+jection and XMM-Newton point-spread-function correc-
+tions. The gas mass Mgas(≤rWL
+∆) was derived by inte-
+grating the electron number density, which was fitted by
+a double- βmodel, and assuming µe= 1.17. The X-ray
+hydrostatic mass MX(≤rWL
+∆) was measured from the
+temperature profile and electron number density profile
+assumingsphericalsymmetryandH.E.. Unlessexplicitly
+statedotherwise,both the X-raygasmassandthehydro-
+static mass are measured within the radius rWL
+∆obtained
+in the weak-lensing analysis. The typical 1 σuncertainty
+ontheX-rayhydrostaticmassis ∼18%−30%at∆ = 500
+with no exceptions (see Table 2).
+2.3.Selection of cluster centers
+The cluster centers were derived independently in the
+weak-lensing and X-ray analysis. In the weak-lensing
+analysis, we follow Okabe et al. (2010a, see their Sec-
+tion 3.2) and adopt the position of the brightest cluster
+galaxy (BCG) as the cluster center. This is motivated
+by the coincidence of the peak of the two-dimensional
+cluster mass distribution with the BCG position, and
+the strong gravitational lensed images centered close to
+the BCG position in some of the clusters (Richard et al.
+2010).
+In the X-ray analysis in Zhang et al. (2008), the clus-
+tercenterisdeterminedusingtheflatfieldedX-rayimage
+in the 0.7-2 keV band as follows. The procedure is ini-
+tiated by deriving the first flux-weighted center within
+a 1′aperture centered at the peak of the cluster X-ray
+emission. Iteratively, we re-derive the flux-weighted cen-
+ter but within the aperture, which is 1′larger than the
+previous one and centered at the previous flux-weighted
+center, till the coordinates of the flux-weighted center do
+not vary anymore. The iteration is less than 10 times to4 Zhang, Okabe, Finoguenov, et al.
+fulfill the goal. The final flux-weighted center is taken as
+the X-ray cluster center.
+We list the lensing and X-ray centers in Table 1; they
+agree to within 0 .14r2500for all except two clusters,
+namely A1914 and RXCJ2337.6+0016. We therefore
+test the sensitivity of the lensing and X-ray analysis to
+the choice of center, finding that the systematic error is
+small. For example, if we adopt the BCG as the cen-
+ter of the X-ray analysis of A1914 then the hydrostatic
+mass estimates change by 1%, 10%, and 3% at ∆ =500,
+1000, and 2500, respectively. Similarly, if we adopt the
+X-ray center as the center of the lensing analysis of the
+same cluster, then the lensing mass estimates change by
+0.1%, 0.2%, and 0.4% also at ∆ =500, 1000, and 2500,
+respectively. The X-ray analysis of the two clusters was
+thereforerevised using the weak-lensingdetermined clus-
+ter center.
+2.4.X-ray morphology
+The X-ray morphology of each cluster was determined
+by Okabe et al. (2010b) by calculating asymmetry ( A)
+and fluctuation parameters ( F; Conselice 2003) from the
+XMM-Newton X-ray images. In summary, the asymme-
+tryparameteris defined as A= (/summationtext
+ij|Iij−Rij|)//summationtext
+ijIij,
+the normalized sum of the absolute value of the flux
+residuals. Iijis the element of the XMM-Newton
+MOS1+MOS2 image in the 0.7-2.0 keV band, which is
+flat fielded, point source subtracted and re-filled assum-
+ing a Poisson distribution, and binned by 4′′×4′′.Rijis
+the element of the image derived by rotating the above
+image by 180◦. We take into account the position res-
+olution of XMM-Newton by allowing the cluster center
+falling into anyneighboringpixel of the r≤4′′circlecen-
+tered at the BCG and include this error in quadrature
+in calculating A. Dynamically immature clusters often
+show both an asymmetric X-ray morphology and an off-
+set betweenweak-lensingandX-raycenters. Therefore A
+is very sensitive to cluster dynamical state. The fluctua-
+tion parameter is given by F= (/summationtext
+ijIij−Bij)//summationtext
+ijIij,
+in which Bijis the element in the smoothed image. This
+parameter describes the degree of deviations from the
+smoothed distribution. We measure the errors of Aand
+Fassuming a Poisson noise computed within a radius of
+rWL
+500, excluding CCD gaps and bad pixels.
+TheFversusAplane is divided into four quadrants
+with cuts at A= 1.1 andF= 0.05 in Figure 1 in Ok-
+abe et al. (2010b). Our sample of 12 clusters occupy
+these quadrants as follows: (1) RXCJ2129, A209, A383,
+A1835,andA2390haveboth low AandlowF; (2) A2261
+and A1914 have high asymmetry parameters; (3) A68,
+RXCJ2337, A267, and Z7160 have high fluctuation pa-
+rameters; and (4) A115 (south) has both high Aand
+highF. The clusters falling into quadrant (1) are defined
+as dynamically undisturbed clusters, and the remaining
+clustersasdisturbed clustersbecausehigh Aand/orhigh
+Findicates that the cluster is still dynamically young.
+It is worth noting that the five undisturbed clusters have
+≤0.06r2500offset between the X-ray cluster center and
+the weak-lensing cluster center which is the BCG posi-
+tion.
+3.RESULTS
+3.1.X-ray hydrostatic mass versus weak-lensing massThe comparison between X-ray and weak-lensing
+masses for individual clusters is shown in the left panel
+of Figure 1. The X-ray to weak-lensing mass ratios vary
+in the range of ∼0.55−1.72. Undisturbed clusters have
+X-rayto weak-lensing mass ratios that generallyincrease
+towardsmallercluster-centricradii(higherover-density).
+In contrast, disturbed clusters show a much greater di-
+versity, with some disturbed clusters having mass ratios
+that increase toward larger cluster-centric radius. The
+distribution of ( MX−MWL)/MWLat ∆ = 500 (right
+panel of Figure 2) reveals that the well-known merging
+cluster A1914 is a ∼5σoutlier based on a naive calcula-
+tion of the mean mass ratio for the other 11 clusters. As
+pointed out in Section 2.2, A1914 also shows the largest
+offset(0.44r2500)between the X-rayand weak-lensingde-
+termined cluster centers.
+The average X-ray to weak-lensing mass ratios for
+the full sample and the undisturbed and disturbed sub-
+samples were calculated taking into account the errors in
+theX-rayandweak-lensingmassestimates—seeTable3
+and the right panel of Figure 1. MX/MWLis consistent
+with unity acrossthe full radialrangeprobed by the data
+for the full sample of 12 clusters. This also holds for the
+seven disturbed clusters, albeit with uncertainties ∼2×
+those for the full sample (as seen in simulations; e.g., Na-
+gai et al. 2007). In contrast, undisturbed clusters show a
+gentle decline in MX/MWLto larger cluster-centric radii
+— at ∆ = 500, the five undisturbed clusters show an
+average X-ray hydrostatic mass 9% ±6% lower than the
+average weak-lensing mass.
+Interestingly, given the apparent trend of MX/MWL
+with radius, MX/MWLis consistent with unity for the
+full sample at ∆ = 500. As noted above, A1914 — the
+mostextremeofthedisturbedclustershasamassratioof
+∼1.7 at ∆ = 500 (see Table 2 and Figure 1). We there-
+foreinvestigatetheextenttowhichA1914mightbedom-
+inating the results. We recalculated MX/MWLfor the
+full sample and the disturbed clusters excluding A1914
+(right panel of Figure 1 and Table 3), finding that the
+averageX-ray hydrostaticmass is now lowerthan the av-
+erageweak-lensingmassjust by6% ±5%. Within the un-
+certainties, our result that X-ray hydrostatic and weak-
+lensing masses agree for the full sample across the full
+radial range probed, is therefore insensitive to the inclu-
+sion/exclusion of A1914. However, there is evidence for
+shock heating of the ICM in the entropy map of A1914.
+In addition to non-thermal pressure support (likely the
+main reason for X-ray hydrostatic masses being under-
+estimated for undisturbed clusters), there may also be a
+competing effect associated with adiabatic compression
+and/or shock heating of the intracluster gas which leads
+to an overestimate of X-ray hydrostatic masses for some
+disturbed clusters. We therefore argue that robust con-
+straints on the bias of X-ray hydrostatic mass estimates
+for precision cluster cosmology requires both statistically
+large and complete (unbiased) samples in order to sam-
+ple the full range of physical processes at play within the
+underlying cluster population.
+We alsocalculatedthe cumulativeprobabilitydistribu-
+tion function of the X-ray to weak-lensing mass ratio as-
+sumingeachdatapointtobeaGaussiandistributedvari-
+able and taking into account the errors with 500 Monte
+Carlo simulations. The mean and its standard error areLoCuSS: A Comparison of Cluster Mass Measurements from XMM-Newton and Subaru 5
+listed in Table 4. Again, a clear trend is found that the
+average hydrostatic to weak-lensing mass ratio declines
+with cluster-centric radius for undisturbed clusters. The
+mass ratios for the full sample and disturbed clusters are
+consistent with unity across the full radial range.
+To further test our results on undisturbed and dis-
+turbed clusters based on our small sample, we applied
+the jackknife method to recalculate the average X-ray
+to weak-lensing mass ratios at ∆ = 500. We randomly
+removed one system from the five undisturbed clusters
+and obtained average X-ray to weak-lensing mass ratios
+in the range of 0.875-0.962. Therefore, our finding that
+the X-ray hydrostatic mass is on average lower than the
+weak-lensingmass for undisturbed clusters is robust. We
+applied the same procedure to the disturbed sub-sample
+and found that the average X-ray to weak-lensing mass
+ratios are in the range of 0.966-1.145. Given the large
+scatter and measurement errors, it is therefore unclear
+whether the average X-ray hydrostatic mass is lower or
+higher than the average weak-lensing mass for disturbed
+systems.
+Despite the modest statistical significance, the anal-
+ysis described above all points toward a trend of im-
+proving agreement between MXandMWLas a func-
+tion of increasing over-density. We therefore proceed
+a simple fit to the data, and obtain the following re-
+lation:MX/MWL= (0.908±0.004)+ (0 .187±0.010)·
+log10(∆/500). These results are consistent with those
+of Mahdavi et al. (2008), who found an X-ray to weak-
+lensing mass ratios of 1.06, 0.96, and 0.85 at ∆ = 2500,
+1000, and 500 respectively, with a typical error bar of
+10%. Our trend is slightly shallower than Mahdavi et
+al.’s result, but is in agreement within the uncertainties.
+Finally, we investigate the issue of scatter. The undis-
+turbed clusterspresentafactorof ∼2lessscatteraround
+their average X-ray to weak-lensing mass ratio than dis-
+turbed clusters (Table 3). A key question is whether
+this lower scatter implies lower intrinsic variance within
+the undisturbed sub-sample. We therefore calculated the
+real variance following Appendix A of Sanderson & Pon-
+man(2010)andfoundthattherealvariancefordisturbed
+clusters is ∼5−10×larger than that for undisturbed
+clusters at ∆ = 2500, 1000, and 500 (Table 4). This
+confirms that the smaller scatter measured for undis-
+turbed clusters reflects low intrinsic variance in this clus-
+ter population. This result is in agreement with studies
+based on numerical simulations, and is fully expected
+in both simulations and observations because of the non-
+smoothnessofthegasdistribution, complexthermal-and
+non-thermal-structure,deviationsfromsphericalsymme-
+try, etc., that are typical of dynamically immature clus-
+ters (e.g., Poole et al. 2006; Fabian et al. 2008; Zhang
+et al. 2009). We also stress that the fact that disturbed
+clusters are also typically less spherical than undisturbed
+clusters will also render the deprojection of the lensing
+signal via fitting a three-dimensional NFW model less
+valid in disturbed clusters than in undisturbed clusters.
+A thorough investigation of these issues requires a large
+complete sample of clusters with deep X-ray and lensing
+data.
+3.2.X-ray to weak-lensing mass ratio versus
+morphology indicatorsWe now investigate the dependence of the X-ray to
+weak-lensing mass ratio on the morphological parame-
+ters,A(asymmetry) and F(fluctuation), and the con-
+centration of the best-fit NFW halos from Okabe et al.
+(2010a, 2010b). As the mass estimates for individual
+clusters have large uncertainties, a detailed quantitative
+study of the relationship between X-ray to weak-lensing
+mass ratio and AandFis beyond the scope of that pos-
+sible with the current sample. We therefore restrict our
+attention in this section to general trends that will be
+worth following up with future larger samples.
+First, we plot the X-ray to weak-lensing mass ra-
+tio versus the asymmetry and fluctuation parameters
+in Figure 3. There is a general trend of decreasing
+MX/MWLwith increasing asymmetry in the sense that
+the most asymmetric clusters have the largest mass dis-
+crepancies with MX< MWL. A1914 is an obvious
+outlier from this apparent anti-correlation. We there-
+fore exclude A1914 and ignore the observational error
+bars in order to fit a “toy-model” to the data, obtain-
+ingMX/MWL= (1.408±0.257)−(0.454±0.263)·A.
+We also note that at fixed asymmetry undisturbed clus-
+ters generally have lower MX/MWLthan disturbed clus-
+ters, with undisturbed systems possibly tracing a steeper
+and tighter trend than the latter. However, we caution
+that these results are preliminary — for example, the
+straight-line fit discussed above is dominated by the left-
+most point in the left panel of Figure 3, namely A68. No
+trend is suggested between X-ray to weak-lensing mass
+ratio and fluctuation parameter in the middle panel of
+Figure 3.
+The mass and concentration ( c) of an individual NFW
+halo are anti-correlated. For example, the covariance
+forMWL
+500andc500isσ2
+M= 1.25,σMc=−0.44, and
+σ2
+c= 1.20 for the sample of 12 clusters. In practical
+terms, this meansthat clusters with higher weak-lensing-
+based values of c500/∝angbracketleftc500∝angbracketrightwill on average have higher
+values of MX/MWL, in which ∝angbracketleftc500∝angbracketrightis the average of the
+concentration parameters from Okabe et al. (2010a) for
+the sample of 12 clusters, weighted by the inverse square
+of the errors. Most significantly, the uncertainties on
+c500/∝angbracketleftc500∝angbracketrightandMX/MWLare correlated, which may in-
+duceacorrelationbetweenthequantitiesthemselves. We
+therefore plot MX/MWLversusc500/∝angbracketleftc500∝angbracketrightin the right
+panelofFigure 3— no obviousrelation isseen in this fig-
+ure, suggestingthat the effect issmalland/orthe effect is
+canceledoutbyothereffects. Thisresultdoesnotchange
+when we use the concentration parameter cvir=rvir/rs.
+A much larger sample and more detailed analysis are re-
+quired to investigate this issue further.
+Results from numerical simulations indicate that the
+cluster concentration parameter may be related to the
+cluster dynamical state (e.g., Neto et al. 2007). Using
+the Millennium Simulation, Neto et al. pointed out that
+the concentrations of out-of-equilibrium halos tend to be
+lower and have more scatter compared to their equilib-
+rium counterparts. This can also be investigated in the
+observed MX/MWLversusc500/∝angbracketleftc500∝angbracketrightplane, however,
+the current sample is too small.
+Finally, we note that the concentration measurements
+used in this article are based on weak-lensing constraints
+(Okabe et al. 2010a). Future papers in this series will use
+morepreciseconcentrationparameterestimatesbasedon6 Zhang, Okabe, Finoguenov, et al.
+combined strong- and weak-lensing models of the clus-
+ters. This will allow a more quantitative comparison be-
+tween observations and simulations.
+3.3.Gas mass fraction
+We define the gas mass fraction as Mgas(≤
+rWL
+∆)/MWL(≤rWL
+∆), and show the gas mass fractions
+for individual clusters at ∆ = 2500, 1000, and 500 in
+the left panel of Figure 4. There is a clear trend of in-
+creasing gas mass fraction toward larger cluster-centric
+radius. A similar trend is also common in both adiabatic
+simulations and in simulations with cooling (e.g., Evrard
+1990, Lewis et al. 2000, Kravtsov et al. 2005). We also
+calculate the average gas mass fractions for the full sam-
+ple and the undisturbed and disturbed sub-samples, at
+∆ = 2500, 1000, and 500 — right panel of Figure 4 and
+Table 5. The average gas mass fraction shows no depen-
+dence on cluster morphology, and approaches the cosmic
+mean baryon fraction Ω b/Ωm= 0.164±0.007 (Komatsu
+et al. 2009) at large cluster-centric radius. At ∆ ∼500,
+theaveragegasmassfractionsstandat90%ofthecosmic
+mean value, in good agreement with simulations (e.g.,
+Nagai et al. 2007).
+Our gas mass fractions are also in good agreement
+with Umetsu et al.’s (2009) joint Subaru weak-lensing
+and AMiBA Sunyaev–Zel’dovich effect (SZE) analysis
+of four clusters. Umetsu et al. obtained ∝angbracketleftfgas,2500∝angbracketright=
+0.105±0.015±0.012and∝angbracketleftfgas,500∝angbracketright= 0.126±0.019±0.016,
+where the first error is the statistical error, and the
+second one is the standard error from the average due
+to cluster-to-cluster variance. However, the gas mass
+fraction obtained by Mahdavi et al. (2008) using weak-
+lensing and X-ray data is slightly higher than our results:
+∝angbracketleftfgas,2500∝angbracketright= 0.119±0.006. A detailed understanding of
+this∼2σdiscrepancy awaits analysis of our complete
+volume-limited sample in a future article.
+Several X-ray-only studies have advocated gas mass
+fractions as a potential probe of the dark energy equa-
+tion of state parameter w(Vikhlinin et al. 2006; Allen
+et al. 2008). We therefore investigate how our weak-
+lensing-based gas mass fractions compare with X-ray-
+only gas mass fractions, defining the latter as Mgas(≤
+rX
+∆)/MX(≤rX
+∆) — i.e., the X-ray-only gas mass frac-
+tions are measured at radii determined from the X-ray
+data derived hydrostatic mass distribution. We cal-
+culated the average X-ray-only gas mass fractions at
+∆ = 2500, 1000, and 500 for the full sample and the
+undisturbed/disturbed sub-samples (Table 6). We found
+that the X-ray-only gas mass fractions are statistically
+indistinguishable from weak-lensing-basedgas mass frac-
+tions at all over-densities considered and for all three
+samples, except for disturbed clusters at ∆ = 500, for
+which the X-ray-only value appears to be biased low.
+The most significant result in the context of cluster cos-
+mology is the very close agreement at ∆ = 2500 between
+the respective gas mass fraction measurements for undis-
+turbed and disturbed clusters, regardless of whether the
+cluster mass measurement is based on X-ray or lensing
+data.
+Weexaminethisapparentmorphologicalindependence
+of gasmass fraction measurementsby looking at the pos-
+sible dependence of gas mass fraction measurements of
+individual clusters on the asymmetry and fluctuation pa-
+rameters. We plot gas mass fraction versus asymmetryand fluctuation parameters in Figure 5. The former re-
+sembles a scatter plot as might be expected based on
+the results above. However, undisturbed and disturbed
+clusters both follow the same trend (right panel of Fig-
+ure 5) of increasing gas mass fraction with fluctuation
+parameter. This may indicate a direct connection be-
+tween the gas mass fraction and substructure fraction,
+i.e., the fraction of cluster mass that resides in mas-
+sive sub-halos (substructures) within the cluster halo.
+Large substructure fractions are generally attributed to
+recent infall of galaxy groups/clusters into more mas-
+sive clusters — see Richard et al. (2010) for more de-
+tails. In a future article, we will investigate the rela-
+tionship between weak-lensing-based substructure frac-
+tions and cluster X-ray morphology, including the fluc-
+tuation parameter. For now, we fit a straight line to all
+12 data points in the right panel of Figure 5, obtaining:
+Mgas/MWL= (0.112±0.011) + (0 .502±0.197)·Fat
+∆ = 500.
+Finally, we compare the gas mass fraction with the
+weak-lensing mass. In this current narrow mass range,
+thereisnoobviousdependenceonthe weak-lensingmass.
+However, X-ray studies for samples with broad mass
+ranges have shown that there is a strong mass depen-
+dence (e.g., Pratt et al. 2009). This will be discussed
+later in Section 4.
+4.DISCUSSION
+In this section, we compare our results with recent nu-
+merical simulations of clusters. There is a broad consen-
+sus among numerical studies that total cluster masses
+derived from X-ray data underestimate the true cluster
+masses in both undisturbed and disturbed clusters for
+variety of reasons, most notably because of the deviation
+from H.E. and extra pressure support (e.g., turbulence)
+beside the thermal gas (e.g., Evrard 1990; Lewis et al.
+2000;Miniatietal. 2001; Miniati2003; Rasiaetal. 2006;
+Nagai et al. 2007; Pfrommer et al. 2007; PV08; Jeltema
+et al. 2008; Lau et al. 2009; Meneghetti et al. 2010).
+An obvious difference between numerical and observa-
+tional studies is that the true cluster mass is known in
+the former, and not in the latter. One solution is to con-
+struct fake weak-lensing data from the simulated cluster
+data (e.g., Meneghetti et al. 2008, 2010). However, large
+samples of such “observed” theoretical clusters are not
+yet available. We therefore rely on the insensitivity of
+the weak-lensing mass measurements to cluster thermo-
+dynamics to assume that our weak-lensing masses are,
+on average, well matched to “true” masses of the nu-
+merical studies. We therefore treat weak-lensing-based
+mass measurements as the “truth” in the comparisons
+described in this section.
+4.1.Summary of recent simulated cluster samples
+Before describing the comparison with simulations in
+the next section, we first summarize the key features of
+the simulated cluster samples against which we compare
+our results. We discuss in turn the samples of Nagai et
+al. (2007), Lau et al. (2009), PV08, and Jeltema et al.
+(2008).
+Nagai et al. (2007) studied 16 simulated clusters with
+T >2 keV in the mass range of M500∼(0.3−9)×
+1014M⊙h−1. They derived hydrostatic masses by ana-
+lyzingmock Chandra data (three projectionsper cluster)LoCuSS: A Comparison of Cluster Mass Measurements from XMM-Newton and Subaru 7
+following the method in Vikhlinin et al. (2006), which
+is similar to our reduction method (Zhang et al. 2008).
+We adopt Nagai et al.’s morphological classifications —
+their 48 simulated X-ray cluster observations comprise
+21 undisturbed (“relaxed” in their terminology) clusters
+and 27 disturbed (“unrelaxed”) clusters.
+Lau et al. (2009) studied the same simulated clusters
+asNagaiet al., but estimated hydrostaticmassesdirectly
+using the three-dimensional gas profiles. Importantly,
+Lau et al. used mass weighted temperatures ( TMW), in
+contrast to Nagai et al.’s temperatures that were recon-
+structed from mock observations. Lau et al. suggest that
+this difference explains the different average mass biases
+reported in the two papers. The reconstructed temper-
+ature profiles used by Nagai et al. are not spectroscopic
+ones, becausetheyarederivedbyweightingthe contribu-
+tion of temperature components along the line of sight to
+correct for the spectroscopic bias. Nevertheless, they are
+derivedfrom spectroscopicdata andsensitive tothe local
+multiphasestructureofthegas. IncontrasttoPVge2keV
+(see below) and Nagai et al., Lau et al. define a cluster
+as relaxed/undisturbed only if it appears so in all three
+projections, which gives 6 relaxed/undisturbed clusters
+and10unrelaxed/disturbedclustersintheirsample. The
+different fractions of undisturbed and disturbed clusters
+found by Nagai et al. and Lau et al. (based on the same
+simulated sample) arise from the fact that the perceived
+dynamical state can depend on viewing angle when using
+simple diagnostics.
+PV08 analyzed 100 simulated clusters (three pro-
+jections per cluster) with T > 2 keV (8 .2×
+1013M⊙h−1<∼M200<∼1.2×1015M⊙h−1) — this is to
+date the largest, complete volume-limited sample of sim-
+ulatedclustersforwhichthehydrostaticmassbiaseshave
+been investigated in detail. They applied different tech-
+niques to measure the X-ray hydrostatic mass in order to
+disentangle biases of different origin. Here, we compare
+our results with their average biases derived adopting an
+extended β-model to fit the gas density radial profile to
+estimate the hydrostatic mass, which is similar to the
+procedure in Vikhlinin et al. (2006), Nagai et al. (2007),
+and Zhang et al. (2008), and three-dimensional mass-
+weighted temperature profiles ( TMW) or spectroscopic-
+like temperature ( TSL; Mazzotta et al 2004).
+In order to improve the coverage at high mass end,
+we use an extended version of the PV08 sample, which
+is extracted from a larger cosmological simulation. The
+sample is constructed and analyzed exactly as the PV08
+sample and yields results fully consistent with the lat-
+ter. The new sample(PVge2keVsample, hereafter) com-
+prises∼120 clusters (3 projections per cluster) with
+T >2keV.Averagemassbiasesderivedforthe TMWcase
+are fully consistent with those derived using directly the
+three-dimensional gas profiles. The PVge2keV sample
+of 360 simulated X-ray cluster observations were divided
+into 180 undisturbed clusters and 180 disturbed clusters
+using their mock X-ray data; this matches the roughly
+50–50 split between disturbed and undisturbed clusters
+in the observed sample.
+Jeltema et al. (2008) analyzed 61 simulated clusters
+giving a 16% bias of the hydrostatic mass estimate on
+average at ∆ = 500. However, their analysis is restricted
+to just ∆ = 500, and their X-ray hydrostatic masses arederivedusingthetruethree-dimensionalgasprofilesonly.
+We therefore chose not to include a detailed comparison
+with their results. Nevertheless, notice that their results
+are fully consistent with those presented in this paper.
+In summary, the different hydrostaticmass reconstruc-
+tion methods, and in particular temperature definitions,
+adopted in the simulations discussed above imply that it
+is sensible to compare our observational results with (1)
+PVge2keV ( TMWcase) and Lau et al. (2009), and (2)
+PVge2keV ( TSLcase) and Nagai et al. (2007), with an
+emphasisonthe latterto matchouruse ofobservedspec-
+tral temperatures. We note that the simulated clusters
+extend to lower temperatures ( T∼>2keV) than the ob-
+served sample ( T≥5keV). However, the numerical stud-
+ies have shown that the dependence of hydrostatic mass
+biasonclustertemperature(mass)isnegligible. Theway
+to define the cluster dynamical state in PV08 and Nagai
+et al. (2007) should be more consistent with the one in
+our observations. However, for our comparison there is
+no point to correct the difference because the errors are
+tiny for the samples in simulations compared to the er-
+rors for the observational sample. Finally, we note that
+the cosmological parameters used in the simulations are
+slightly different from those used in this article; the im-
+pact of these small differences on our results/discussion
+is negligible.
+4.2.Comparing observational mass estimates to
+simulations
+We overplot the X-ray hydrostatic to weak-lensing
+mass ratios from the simulations in the right panel of
+Figure 1. We find that there is a general agreement be-
+tween observations and simulations in which the mass
+ratios are consistent with unity at small radii, e.g., at
+∆ = 2500, and the agreementdeterioratesat largerradii.
+Most strikingly, the simulations agree well with our re-
+sult that the mass ratio for undisturbed clusters declines
+gently to larger radii. We estimate that a firm detection
+of the discrepancy between simulations and observations
+for undisturbed clusters, would require a sample of ∼60
+clusters.
+For disturbed clusters, the simulations — particularly
+those using the spectroscopic-like temperatures — show
+a significant discrepancy between the X-ray hydrostatic
+andtruemassesatlargeradii, whilenodiscrepancyisde-
+tectedobservationallybetweentheX-rayhydrostaticand
+weak-lensing masses. The hydrostatic mass bias seen in
+simulations, when spectroscopic temperatures are used
+in the derivation of hydrostatic masses, might be ex-
+acerbated by over-cooling. Even when small-scale cold
+clumps of gas are masked in the mock analysis, over-
+coolingcanleadtounderestimatedspectraltemperatures
+because of the presence of a diffuse cold gas component.
+This may be the cause of the disagreement between the
+observedsample of12clustersand the simulated samples
+at large radii.
+4.3.Comparing observational gas mass fractions with
+simulations
+Within a given density contrast, the gas mass fraction
+tends to increase with the increasing cluster mass (e.g.,
+Vikhlinin et al. 2006, Gonzalez et al. 2007, Gastaldello
+et al. 2007, Pratt et al. 2009), with the trend being8 Zhang, Okabe, Finoguenov, et al.
+shallower at large radii. This behavior can be explained
+by the scale dependency introduced by cooling (Bryan
+2000), which regulates the amount of cool gas present in
+clusters. Numerical simulations can also reproduce the
+observed behavior of fgasin clusters (Valdarnini 2003,
+Kravtsov et al. 2005, Ettori et al.2006, Kay et al. 2007),
+and strongly support the cooling model.
+Due to the strong mass dependence of gas mass frac-
+tions seen in clusters, a proper comparison between sam-
+ple averagesrequiresthat the twosamplesapproximately
+cover the same mass range. This is accomplished by
+extracting a sub-sample of PVge2keV clusters that all
+have masses above the lowest mass of observed clus-
+ters:M500≥2.6·1014M⊙. This sub-sample contains
+Nsub= 68 clusters which are subsequently divided into
+34 undisturbed and 34 disturbed clusters based on the
+cluster dynamical state as explained in PV08. The av-
+erage gas mass fractions for all 68 simulated clusters
+andtheundisturbedanddisturbedsimulatedclustersare
+compared to the observed gas mass fractions in Figure 4.
+To illustrate the strong mass dependence of gas mass
+fractions, we also show in Figure 4 averagegasmass frac-
+tions for a more restrictive sub-sample of 32 PVge2keV
+clusters with M500≥5·1014M⊙.
+The average gas mass fractions of simulated clusters
+agree with those of observed clusters in the sense that
+they do not show any dependence on cluster morphol-
+ogy. The agreement between simulations and observa-
+tions is most striking at ∆ = 2500, where the observed
+and simulated gas mass fractions agree within the uncer-
+tainties when the mass range of the respective samples is
+matched. Overall there is a good agreement within the
+error bars, taking into account that the trend with mass
+is strong and that the mass function of the two samples
+(from simulations and observations) is not exactly the
+same.
+4.4.X-ray and lensing masses in simulated clusters
+As discussed at the beginning of Section 4, large sam-
+ples ofmock weak-lensingobservationsof simulated clus-
+ters are not yet available. The largest sample to date
+is that of Meneghetti et al. (2010), who studied three
+clusters, with three orthogonal projections per cluster,
+to generate a total set of nine mock observations. Their
+X-ray hydrostatic masses are typically biased low by 5%-
+20% due to the lack of H.E. in their simulated clusters,
+which is in agreement with our results. They also found
+the gas mass to be well reconstructed within the region
+where the X-ray surface brightness profile is extracted.
+Their gas mass measurements are independent of the dy-
+namical state of the cluster, with average deviations of
+1%±3% at ∆ = 2500 and 7% ±4% at 200 <∆<500.
+Although Meneghetti et al.’s sample is small, their re-
+sults agree well with our observational results that the
+gas mass to weak-lensing mass ratios are independent
+of the cluster dynamical state, and supports our finding
+that the radius with ∆ = 2500 appears to be the most
+robust radius at which to use cluster gas mass fractions
+for probing cosmological parameters.
+5.SUMMARY AND OUTLOOK
+We have presented a comparison of X-ray hydro-
+static and weak-lensing mass estimates for 12 clusters
+atz≃0.2 for which high-quality XMM-Newton andSubaru/Suprime-CAM data are available within the Lo-
+CuSS. Our main results are as follows:
+•For the full sample, we obtain 1 −MX/MWL=
+0.01±0.07 at an over-density of ∆ = 500. We
+also sub-divided the sample into undisturbed and
+disturbed sub-samples based on quantitative X-ray
+morphologies using asymmetry and fluctuation pa-
+rameters. We obtained 1 −MX/MWL= 0.09±0.06
+and−0.06±0.12for the undisturbed and disturbed
+clusters, respectively.
+•The scatter around the average X-ray hydrostatic
+to weak-lensing mass ratio for undisturbed clusters
+is half that of the disturbed clusters due to the
+lower intrinsic variance among the population of
+undisturbed clusters.
+•For disturbed clusters, it is unclear whether the X-
+ray hydrostatic mass is consistent with the weak-
+lensing mass, due to large scatter. In addition to
+non-thermalpressuresupport, there maybe a com-
+peting effect associated with adiabatic compression
+and/or shock heating of the intracluster gas which
+leads to overestimate of X-ray H.E. masses for dis-
+turbed clusters. The most prominent example of
+this in our sample is the famous merging cluster
+A1914.
+•Despite the modest statistical significance of the
+mass discrepancy in the undisturbed clusters, we
+detect a clear trend of improving agreement be-
+tweenMXandMWLas a function of increasing
+over-density, MX/MWL= (0.908±0.004)+(0.187±
+0.010)·log10(∆/500).
+•There is a general agreement between the X-ray
+hydrostatic to weak-lensing mass ratio in observed
+and simulated clusters in which the mass ratios are
+both consistent with unity at small radii (i.e., at
+∆ = 2500), with the agreement deteriorating at
+largerradii, i.e., outto∆ = 500. Thisdeterioration
+is dominated by disturbed clusters, with the undis-
+turbed simulated clusters reproducing well the ob-
+served gentle decline in MX/MWLas a function of
+increasing cluster-centric radius.
+•The weak-lensing mass-based cumulative gas mass
+fraction increases with radius, but still lies below
+the cosmic baryon fraction at the largest cluster-
+centric radii probed (i.e., ∆ = 500). An important
+findingistheabsenceofdependenceofthegasmass
+fractiononclusterdynamicalstate(i.e., X-raymor-
+phology). The X-ray-only gas mass fractions are
+also consistent with the weak-lensing mass-based
+gas mass fractions at ∆ = 2500, supporting the
+proposal to use this measurement as a probe of the
+dark energy equation of state parameter w.
+In summary, our results demonstrate that XMM-
+Newtonand Subaru are a powerful combination for cal-
+ibrating systematic uncertainties in cluster mass mea-
+surements and suggest an encouraging convergence be-
+tween X-ray, lensing, and numerical studies of cluster
+mass. Nevertheless, our observational sample remainsLoCuSS: A Comparison of Cluster Mass Measurements from XMM-Newton and Subaru 9
+very small at just 12 clusters. Our detailed results are
+therefore vulnerable to inclusion/exclusion of extreme
+clusters, as highlighted by our discussion of A1914. Ro-
+bust constraints on systematic uncertainties in cluster
+mass measurement therefore await a thorough investiga-
+tion of a large and complete volume-limited sample of
+clusters, such as that planned within LoCuSS. On the
+theoretical side, it is important to move as rapidly as
+possible toward large samples of mock weak-lensing and
+X-rayobservationsofsimulatedclusters,suchasthosere-
+cently pioneered by Meneghetti et al. (2008, 2010). With
+both of these observational and theoretical data sets in
+hand, a detailed and statistically robust comparison will
+be possible, helping to calibrate future cluster cosmology
+experiments.
+We acknowledge support from KICP in Chicago for
+hospitality, and thank our LoCuSS collaborators, espe-
+cially Masahiro Takada and Keiichi Umetsu, for help-
+ful comments on the manuscript. Y.Y.Z. thanks Mas-
+simo Meneghetti and Gabriel Pratt for useful discus-
+sion. Y.Y.Z. acknowledges support by the DFG through
+Emmy Noether Research Grant RE 1462/2, through
+Schwerpunkt Program 1177, and through project B6
+“Gravitational Lensing and X-ray Emission by Non-
+Linear Structures” of Transregional Collaborative Re-search Centre TRR 33 The Dark Universe, and support
+by the German BMBF through the Verbundforschung
+under grant 50OR0601. This work is supported by a
+Grant-in-Aid for the COE Program “Exploring New Sci-
+encebyBridgingParticle-MatterHierarchy”andG-COE
+Program “Weaving Science Web beyond Particle-Matter
+Hierarchy” in Tohoku University, funded by the Min-
+istryof Education, Science, Sports and Culture ofJapan.
+This work is, in part, supported by a Grant-in-Aid for
+Science Research in a Priority Area ”Probing the Dark
+Energy through an Extremely Wide and Deep Survey
+with Subaru Telescope” (18072001) from the Ministry
+of Education, Culture, Sports, Science, and Technology
+of Japan. N.O. is, in part, supported by a Grant-in-Aid
+fromtheMinistryofEducation,Culture, Sports, Science,
+and Technologyof Japan (20740099). A.F. acknowledges
+support from BMBF/DLR under grant 50OR0207 and
+MPG, and was partially supported by a NASA grant
+NNX08AX46G to UMBC. G.P.S. acknowledges support
+from the Royal Society and STFC. D.P.M. acknowledges
+support provided by NASA through Hubble Fellowship
+grant #HF-51259.01 awarded by the Space Telescope
+Science Institute, which is operatedby the Associationof
+Universities for Research in Astronomy, Inc., for NASA,
+under contract NAS 5-26555.
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+Zhang, Y.-Y., et al. 2008, A&A, 482, 451LoCuSS: A Comparison of Cluster Mass Measurements from XMM-Newton and Subaru 11
+Fig. 1.— Left panel : X-ray hydrostatic to weak-lensing mass ratios as a functio n of density contrast for individual clusters, in which red
+dashed and blue solid lines denote undisturbed and disturbe d clusters, respectively. Right panel : X-ray to weak-lensing mass ratios for all
+clusters (bottom), undisturbed clusters (middle), and dis turbed clusters (top) from our sample including (filled blac k boxes) and excluding
+(open black boxes) A1914, together with the X-ray hydrostat ic mass to true mass ratios from numerical simulations (PVge 2keV, red circles
+usingTMW, blue diamonds using TSL; Lau et al. 2009, green triangles using quasi- TMW; Nagai et al. 2007, light blue stars using TSL).
+The data points at each density contrast are off by 0.022 dex fo r clarity. We also show the best fit of the X-ray hydrostatic to weak-lensing
+mass ratio as a function of density contract for the five undis turbed clusters, MX/MWL= (0.908±0.004)+(0 .187±0.010)·log10(∆/500).
+(A color version of this figure is available in the online jour nal.)
+Fig. 2.— Normalized histograms of ( MX−MWL)/MWLfor all 12 clusters at ∆ = 2500 (left panel), ∆ = 1000 (middle pa nel), and
+∆ = 500 (right panel).12 Zhang, Okabe, Finoguenov, et al.
+Fig. 3.— X-ray hydrostatic to weak-lensing mass ratio at ∆ = 500 vs. as ymmetry parameter (left panel), fluctuation parameter (mid dle
+panel), and c500//angbracketleftc500/angbracketright(right panel), respectively. The colors and symbols are the same as shown in the left panel of Figure 1. The best
+fit with equal weighting of all data points except for A1914 (t he top rightmost point) is shown in the left panel. (A color ve rsion of this
+figure is available in the online journal.)
+Fig. 4.— Left panel: gas mass fractions using weak-lensing masses for individua l clusters. The colors, lines, and symbols are the same
+as shown in the left panel of Figure 1. Right panel: average gas mass fractions using weak-lensing masses (fille d symbols). Boxes, circles,
+and triangles denote all, undisturbed, and disturbed clust ers. The data sets are off by 0.048 dex at each density contrast for clarity. The
+gray horizontal band shows the 1 σrange of the cosmic mean baryon fraction from WMAP five-year data, Ω b/Ωm= 0.164±0.007, in
+Komatsu et al. (2009), and the black horizontal line corresp onds to 0 .9×the mean cosmic baryon fraction. Also shown are the gas mass
+fractions for the sub-samples drawn from the PVge2keV sampl e with mass cuts of M500≥5×1014M⊙(small green open symbols) and
+M500≥2.6×1014M⊙(big magenta open symbols). (A color version of this figure is available in the online journal.)LoCuSS: A Comparison of Cluster Mass Measurements from XMM-Newton and Subaru 13
+Fig. 5.— Gas mass to weak-lensing mass ratio at the radius with ∆ = 500 v s. asymmetry parameter (left panel) and fluctuation
+parameter (right panel). The best fit with equal weighting of each data point is shown in the right panel. The colors and sym bols are the
+same as shown in the left panel of Figure 1. (A color version of this figure is available in the online journal.)
+TABLE 1
+Cluster centers from weak-lensing and X-ray analysis
+Cluster Name X-ray Center [J2000] Weak-lensing Center [J2000] Offset X-r ay Morphology
+R.A. Decl. R.A. Decl. (arcmin) r2500
+A68 00 37 06.2 09 09 28.7 00 37 06.9 09 09 24.5 0.18 0.10 Disturbed
+A115 (south) 00 56 00.3 26 20 32.5 00 56 00.3 26 20 32.5 0.00 0.00 Disturbed
+A209 01 31 52.6 -13 36 35.5 01 31 52.5 -13 36 40.5 0.08 0.04 Undisturb ed
+A267 01 52 42.0 01 00 41.2 01 52 41.9 01 00 25.7 0.26 0.14 Disturbed
+A383 02 48 03.3 -03 31 43.6 02 48 03.4 -03 31 44.7 0.03 0.01 Undisturb ed
+A1835 14 01 01.9 02 52 35.5 14 01 02.1 02 52 42.8 0.13 0.06 Undisturbed
+A1914 14 26 00.9 37 49 38.8 14 25 56.7 37 48 59.2 1.22 0.44 Disturbed
+Z7160 14 57 15.2 22 20 31.2 14 57 15.1 22 20 35.3 0.07 0.05 Disturbed
+A2261 17 22 26.9 32 07 47.4 17 22 27.2 32 07 57.1 0.34 0.13 Disturbed
+A2390 21 53 37.1 17 41 46.4 21 53 36.8 17 41 43.3 0.08 0.03 Undisturbed
+RXCJ2129.6+0005 21 29 39.8 00 05 18.5 21 29 40.0 00 05 21.8 0.07 0.04 Undisturbed
+RXCJ2337.6+0016 23 37 37.8 00 16 15.5 23 37 39.7 00 16 17.0 0.48 0.25 Disturbed
+Note. Within the sample, A1914 and RXCJ2337.6+0016 show ext reme offsets between the X-ray and weak-lensing determined c luster
+central positions. Therefore, the X-ray analysis of the two clusters was revised using the weak-lensing determined clu ster centers.14 Zhang, Okabe, Finoguenov, et al.
+TABLE 2
+Cluster mass measurementsa
+Cluster Name rWL
+2500rWL
+1000rWL
+500
+MWLMXMgasMWLMXMgasMWLMXMgas
+A68 1.42+0.59
+−0.621.91±0.57 0.158±0.007 2.78+0.79
+−0.783.56±1.06 0.360±0.026 4.15+1.21
+−1.065.50±1.65 0.579±0.057
+A115 (south) 1.22+0.65
+−0.670.68±0.12 0.107±0.005 2.51+0.93
+−0.911.98±0.50 0.329±0.017 3.85+1.61
+−1.323.84±0.99 0.679±0.037
+A209 2.18+0.46
+−0.461.95±0.55 0.208±0.014 5.24+0.74
+−0.724.41±1.40 0.557±0.050 8.81+1.31
+−1.206.57±1.95 0.963±0.101
+A267 1.43+0.25
+−0.251.66±0.45 0.139±0.007 2.41+0.42
+−0.402.58±0.86 0.281±0.021 3.29+0.68
+−0.613.75±1.16 0.426±0.039
+A383 1.76+0.21
+−0.201.61±0.48 0.127±0.008 2.64+0.43
+−0.402.42±0.72 0.227±0.022 3.38+0.71
+−0.623.23±0.95 0.340±0.041
+A1835 2.88+0.57
+−0.582.98±0.89 0.423±0.023 6.15+0.95
+−0.905.66±1.68 0.844±0.080 9.65+1.70
+−1.518.59±2.50 1.261±0.154
+A1914 2.09+0.31
+−0.312.78±0.76 0.268±0.015 3.35+0.50
+−0.474.36±1.22 0.497±0.038 4.46+0.75
+−0.697.69±2.24 0.754±0.066
+Z7160 0.89+0.38
+−0.421.17±0.35 0.129±0.004 1.75+0.58
+−0.551.80±0.52 0.251±0.014 2.61+0.97
+−0.822.46±0.72 0.401±0.033
+A2261 3.55+0.44
+−0.442.77±0.75 0.266±0.026 5.95+0.74
+−0.714.54±1.20 0.516±0.065 8.12+1.23
+−1.126.24±1.65 0.786±0.114
+A2390 3.14+0.44
+−0.433.92±1.15 0.395±0.031 5.22+0.82
+−0.776.02±1.94 0.785±0.083 7.09+1.29
+−1.177.48±2.22 1.180±0.150
+RXCJ2129.6 1.38+0.53
+−0.541.75±0.52 0.166±0.009 2.97+0.69
+−0.713.07±0.93 0.364±0.031 4.68+1.09
+−0.974.46±1.29 0.581±0.065
+RXCJ2337.6 2.42+0.36
+−0.371.97±0.44 0.202±0.015 3.72+0.50
+−0.473.16±0.68 0.418±0.045 4.85+0.75
+−0.704.45±0.97 0.652±0.084
+Note.aThe values are in units of 1014solar mass.
+TABLE 3
+Comparison of X-ray and weak-lensing mass estimates
+Sample MX/MWL
+rWL
+2500rWL
+1000rWL
+500
+All 1.01±0.07 0.97±0.05 0.99±0.07
+Undisturbed 1.04±0.08 0.96±0.05 0.91±0.06
+Disturbed 0.98±0.12 0.97±0.09 1.06±0.12
+All-A1914 0.97±0.07 0.94±0.05 0.94±0.05
+Disturbed-A1914 0.92±0.11 0.91±0.07 0.97±0.08
+TABLE 4
+Results of Monte Carlo simulationsa
+Sample MX/MWL
+rWL
+2500rWL
+1000rWL
+500
+All 1.03±0.09±0.13 0.94±0.06±0.05 0.95±0.07±0.08
+Undisturbed 1.02±0.05±0.02 0.94±0.04±0.01 0.90±0.04±0.02
+Disturbed 1.05±0.11±0.22 0.94±0.07±0.08 1.02±0.08±0.11
+Note.aThe values quoted for each comparison are the mean, standard error, and real variance based on 500 Monte Carlo simulation s
+described in Section 3.1.
+TABLE 5
+Weak-lensing-based gas mass fractions
+Sample Mgas/MWL
+rWL
+2500rWL
+1000rWL
+500
+All 0.099±0.008 0.118±0.007 0.130±0.008
+Undisturbed 0.101±0.015 0.118±0.012 0.124±0.011
+Disturbed 0.096±0.009 0.119±0.009 0.136±0.011
+TABLE 6
+X-ray-only gas mass fractions
+Sample Mgas/MX
+rX
+2500rX
+1000rX
+500
+All 0.101±0.010 0.118±0.015 0.131±0.015
+Undisturbed 0.100±0.017 0.122±0.028 0.124±0.023
+Disturbed 0.101±0.018 0.115±0.024 0.110±0.026
\ No newline at end of file
diff --git a/1001.0781.txt b/1001.0781.txt
new file mode 100644
index 0000000000000000000000000000000000000000..30af7750d8c5dd41e5c68b398c57bf9ec6df0298
--- /dev/null
+++ b/1001.0781.txt
@@ -0,0 +1,425 @@
+arXiv:1001.0781v1  [cond-mat.str-el]  5 Jan 2010Tuning spin-orbit coupling and superconductivity at the Sr TiO3/LaAlO 3interface:
+a magneto-transport study
+M. Ben Shalom, M. Sachs, D. Rakhmilevitch, A. Palevski, and Y. Dagan,∗
+1Raymond and Beverly Sackler School of Physics and Astronomy , Tel-Aviv University, Tel Aviv, 69978, Israel
+(Dated: October 29, 2018)
+The superconducting transition temperature, T c, of the SrTiO 3/LaAlO 3interface was varied by
+the electric field effect. The anisotropy of the upper critica l field and the normal state magneto-
+transport were studiedas afunction of gate voltage. The spi n-orbit couplingenergy ǫSOis extracted.
+This tunable energy scale is used to explain the strong gate d ependence of the mobility and of the
+anomalous Hall signal observed. ǫSOfollows T cfor the electric field range under study.
+PACS numbers: 75.70.Cn, 73.40.-c
+Interfaces between strongly correlated oxides exhibit a
+variety of physical phenomena and are currently at the
+focus of intensive scientific research. An electronic recon-
+struction occurring at the interfaces may be at the origin
+of these phenomena.[1] It has been demonstrated that
+the interface between SrTiO 3(STO) and LaAlO 3(LAO)
+is highly conducting, having the properties of a two di-
+mensional electron (2DEG) gas. [2] At low temperatures
+the 2DEG has a superconducting ground state, whose
+critical temperature can be modified by an electric field
+effect.[3] The origin of the charge carriers and the thick-
+nessofthe conducting layersarestill underdebate. [1, 4–
+9] Recently, we have demonstrated that for carrier con-
+centrations at the range of 1013cm−2, a large, highly
+anisotropic magnetoresistance (MR) is observed.[10] Its
+strong anisotropy suggests that it stems from a strong
+magnetic scattering confined to the interface.
+Here we show that both superconducting and spin-
+orbit (SO) interaction can be modified by applying a
+gate voltage. The upper critical field applied parallel to
+the interface, H c2/bardbl, is approximately the weak coupling
+Clogston-Chandrasekhar paramagnetic limit: µ0Hc2∼=
+1.75kBTcforhighcarrierconcentrations. However,asthe
+density of the charge carrier is reduced, H c2/bardblbecomes as
+largeas five times this limit, suggestinga rapidly increas-
+ing SO coupling. This SO coupling energy ǫSOmanifests
+itself in many of the transport properties studied.
+We use a sample with 15 unit cells of LaAlO 3, de-
+posited by pulsed laser on an atomicallyflat SrTiO 3(100)
+substrate. The oxygen pressure during the deposition
+was maintained at 10−4Torr. The deposition was fol-
+lowed by a two-hour annealing stage at oxygen pressure
+of 0.2 Torr and a temperature of 400 C. The thickness
+of the LaAlO 3layer was monitored by reflection high en-
+ergy electron diffraction. Growth procedure is similar to
+the onedescribed elsewhere.[10] Alayerofgold wasevap-
+orated at the bottom of the sample and used as a gate
+when biased to ±50 V relative to the 2DEG. Contacts
+were made using a wire bonder in a Van Der Pauw ge-
+ometry. Wetookextracaretoensuretheabsenceofheat-
+ing by testing the resistance to be current-independent/s48/s46/s50 /s48/s46/s52/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57
+/s53 /s49/s48 /s49/s53/s49/s50/s51/s52/s53
+/s48/s46/s49 /s48/s46/s50/s48/s46/s48/s48/s46/s52/s48/s46/s56/s82/s32/s47/s32/s82 /s110/s40/s98/s41/s32/s32/s82/s32/s47/s32/s82 /s110
+/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s40/s97/s41
+/s32 /s32/s32
+/s48/s72 /s32/s40/s84/s41/s32 /s32/s84/s32/s61/s32/s50/s48/s32/s109/s75
+FIG. 1: Color on-line (a) Normalized sheet resistance versus
+temperature for the various gate voltages (from left to righ t)
+Vg=50 V, as-grown state, 10 V, -10 V, -50 V color and shape
+code applies to all graphs in the paper. (b) Resistivity versus
+theperpendicularmagnetic field. Inset: zoom onthelowfield
+region where superconductivity is suppressed.
+at the superconducting transition point, where current
+sensitivity is maximal. During measurements, the sam-
+ple was cooled down to a base temperature of 20 mK and
+a magnetic field of up to 18 T was applied. Samples ro-
+tation was done using a step motor with a resolution of
+0.015◦/step. All transport properties reported here were
+measuredin the as-grownstate (AG), and while applying
+Vg=50, 10, -10, -50 V. In order to accurately determine
+Tc, the remnant magnetic field was minimized by oscil-
+lating the field down to zero. In addition, the sample was
+kept parallel to the field where superconductivity is less
+sensitive to it.
+Fig.1a presents the normalized resistance R/R nas a
+function of temperature for the various V gvalues. R n
+is the resistance measured at zero field and at T=0.5 K,
+well above T c, for each V g. Tuning V gfrom +50 to -50
+V increases T cfrom 0.1 to 0.35 K.
+Fig.1b presents the resistance normalized with R n
+versusthe perpendicular magnetic field at T=20 mK.2
+/s48 /s53 /s49/s48 /s49/s53/s48/s52/s53/s48/s57/s48/s48/s48/s49/s53/s48/s51/s48/s48/s48/s54/s53/s49/s51/s48/s48/s50/s53/s53/s48
+/s110 /s32/s61/s32/s51/s46/s48 /s49/s48/s49/s51
+/s32/s99/s109/s45/s50
+/s32/s32
+/s48/s72 /s32/s40/s84/s41/s110 /s32/s61/s32/s52/s46/s52 /s49/s48/s49/s51
+/s32/s99/s109/s45/s50
+/s32/s32/s82/s32 /s40 /s41/s82
+/s110/s45/s32/s82
+/s115/s97/s116/s32/s45/s49
+/s115/s111/s72/s42/s110 /s32/s61/s32/s54/s46/s49 /s49/s48/s49/s51
+/s32/s99/s109/s45/s50
+/s32/s32
+/s72/s42
+/s32/s32
+/s110 /s32/s61/s32/s55/s46/s56 /s49/s48/s49/s51
+/s32/s99/s109/s45/s50
+FIG. 2: Sheet resistance versusthe magnetic field applied
+parallel to the interface and current for various charge den si-
+ties.
+A large, positive MR is observed for V g=50 V. Its mag-
+nitude gradually decreases as V gis reduced. The inset
+zooms on the superconducting low field region.
+Table 1summarizesthe superconductingpropertiesfor
+the various V g. We define H c2⊥(Tc) as the field (tem-
+perature) at which the resistance is R n/2. The Ginzburg
+Landau coherence length ξGL=/radicalbig
+Φ0/2πµ0Hc⊥is ex-
+tracted from H c2⊥, here Φ 0is the flux quantum.
+The charge carrier density n= 1/RHeis inferred from
+Hall measurements in the high field regime ( µ0H >14
+T). It is presented together with the calculated mobility
+µ=n/Rn. Alargevariationof µisobserved,indicatinga
+significant changein the scatteringrate with gate voltage
+as previously noted.[11] This behavior will be discussed
+later.
+Fig.2 presents the sheet resistance versusthe mag-
+netic field, H /bardbl, applied parallel to the interface and to
+the current. From the low field regime we extract the su-
+perconducting parallel critical field H c2/bardbl=H(R n/2). We
+note that while T cincreases merely to 0.35 K, H c2/bardblbe-
+comes as large as 2.5 T [table 1]. The high H c2/bardbland low
+Tcimplies that the paramagnetic limit is exceeded.[12]
+SinceξGLis rather large, 50-200 nm, it is reasonable to
+use the weak coupling BCS approximation. Accordingly,
+we expect superconductivity to exist in fields lower than
+gµBH≤3.5kBTc; here g is the gyromagnetic ratio and
+µBis the Bohr magneton.
+In Fig.3a µBHc2/bardblis plotted against kBTcfor differ-
+ent Vgvalues. The dashed straight line represents the
+paramagnetic limit using BCS weak coupling and g=2.
+As Tcincreases, the limit is exceeded by up to a fac-
+tor of∼5. This behavior suggests a strong SO coupling
+that relaxes the Clogston-Chandrasekharlimitations.[12]
+For a superconducting layer thickness d≪ξGLit is ex-pected that H c2/bardblbe much larger than H c2⊥. In this case
+the critical field is determined by the paramagnetic limit
+and the spin-orbit energy. From Fig.3a it appears that
+even for the lowest T c, superconductivity is quenched
+in this paramagnetic limit. We can therefore merely
+set an upper limit on the thickness of the conducting
+layer by analyzing the anisotropy of the critical field:
+d≤√
+3Φ0/πξGLµ0Hc2/bardbl. [13] The thickness upper limit
+is presented for the various V greaching a minimal value
+of 10 nm for n= 3×1013cm−2[table 1]. Previous esti-
+mations obtained similar values. They, however, relate d
+to the actual thickness.[14, 15]
+Let us describe the general behavior of R(H /bardbl) [Fig.2]
+using the n= 6.1×1013cm−2curve (green triangle)
+as an example: Above the superconducting transition,
+the sheet resistance reaches a roughly field-independent
+regime (0.9-1.8 T) in which the resistance approximately
+equals R n. We define H∗as the onset field where dR/dH
+becomes negative. For H >H∗the resistance drops to
+Rsat, the low resistance saturation regime (H /bardbl>10 T).
+Both H c2/bardbland H∗strongly depend on V g; they in-
+crease as V gchanges from 50 to -50 V, and H∗becomes
+unmeasurably high for V g=-50 V (black squares).
+TABLE I:
+Vg n×1013Mobility Tc µ 0Hc⊥µ0Hc/bardbldξGLǫSO
+Vcm−2cm/secV K T T nm nm meV
+-50 3.0 236 0.35 0.125 2.23 10 51 >2
+-10 4.4 392 0.28 0.098 1.37 14.4 58 0.58
+10 6.1 762 0.18 0.040 0.52 24.1 90 0.21
+50 7.8 1707 0.12 0.009 0.14 41.2 194 0.06
+AG 9.5 897 0.15 0.030 0.25 44 103 0.12
+Fig.3a suggests that SO coupling plays a major role in
+the system. We shall now describe the data presented
+so far using a single energy scale, ǫSO. We relate H∗to
+the breakdown field of the spin-orbit coupling energy:
+gµBH∗=ǫSO. TheǫSOvalues are presented assum-
+ing g=2 [table 1]. Above this field SO scattering is sup-
+pressed and completely vanishes at the high field satura-
+tion regime where all spins are aligned. In this scenario
+the high field saturation value R satis the remnant impu-
+rity scattering. The SO scattering rate can be evaluated
+from the difference between R nand R sat. Therefore R n-
+Rsatshould be proportional to h/τSO=ǫSO.
+Fig.3b presents R n−RsatandµBH∗versus k BTc.
+These two quantities scale as predicted. Furthermore,
+the saturation resistance R satroughly scales with the
+number of carriers. We further use the calculation of
+R. A. Klemm et al.which estimates the SO scattering
+time from H c2/bardbland T cfor a two-dimensional film in the
+paramagnetic limit.[16] As seen [Fig.3b, red circles], the
+general behavior is described by this (probably oversim-
+plified) model. We can therefore conclude that a single
+energy scale, tuned by V g, dominates the behavior of the3
+transport and affects superconductivity. For the carrier
+concentration range under study R sat≪Rn. Hence, the
+main contribution to the zero field resistance comes from
+SO scattering. This explains the strong dependence of
+the mobility on gate voltage.
+We note that for V g= 50 V the carrier concentra-
+tion is lower compared to the as-grown state. This is in
+contrast to a simple capacitor-like behavior with nega-
+tive charge carriers. Furthermore, despite the decrease
+in carrier concentration, R nunexpectedly decreases. We
+relate this peculiar behavior to the SO scattering pro-
+cesses dominating R n. We assume that static positive
+charges move to the interface when the highest positive
+voltage is applied and consequently nis reduced. This
+charge movement reduces the initial electric field at the
+interface and as a result decreases the SO scattering.
+Upon reducing the gate voltage from its maximal pos-
+itive value, electrons are drawn away from the interface
+in a reversible manner as expected, and an electric field
+is created. The initial electric field in the as-grown state
+results in enhanced SO scattering and higher resistance,
+despite the higher carrier concentration.
+The MR presented [Fig.1b, Fig.2] is highly anisotropic
+for high magnetic fields. For example, for n= 6.1×
+1013cm−2and H=18 T, the resistance vary from 0.12R n
+to 3.5R n(a factor of 30) for the in-plane and out-of-
+plane field configurations respectively. We study this
+anisotropy by rotating the sample around an in-plane
+axis which is perpendicular to the current, while keep-
+ing the total field constant, |/vectorH|=18 T. Fig.4a presents
+the normalized sheet resistance versusthe perpendicular
+field component. For the small angle range presented,
+the parallel field component is approximately constant
+≃18 T. Moreover, for this parallel field component, the
+resistance is insensitive to small changes in H /bardbl[Fig.2].
+As shown, the sheet resistance rapidly increases when a
+small perpendicular component is introduced. It reaches
+Rnfor H⊥≃1.5 T (an angle of about 4◦). This is sim-
+ilar to our previous observation at 2 K.[10] The curves
+n= 7.8,6.1×1013cm−2and AG (not shown) merge near
+1.5 T, while n= 4.4×1013cm−2departs form the gen-
+eral behavior. This can be easily explained since for this
+gate voltage the resistance does not saturate up to 18
+T [Fig.2]. For n= 3.0×1013cm−2, H=18 T is well be-
+low H∗and therefore the anisotropy cannot be observed
+at this too small a magnetic field. We relate the sud-
+den appearance of resistance with a small perpendicular
+field component to the orbital motion generated by this
+component. This motion is enhanced by the high mo-
+bility state (R ≃Rsat) induced by the presence of a large
+parallel magnetic field.
+Fig.4b presents the Hall resistivity ρxyat 20 mK
+versusthe magnetic field after subtracting out the lin-
+ear term obtained from a fit to the high field regime. A
+conspicuousdeviationfrom this linearpartin the Hall re-
+sistivity (anomalous Hall effect AHE), is observed. This/s48/s46/s48/s53/s48/s46/s49/s48
+/s48/s46/s48/s49 /s48/s46/s48/s50 /s48/s46/s48/s51/s48/s49/s50/s40/s98/s41/s40/s97/s41
+/s32/s32/s66/s72
+/s99 /s124/s124/s32/s40/s109/s101/s86/s41
+/s50 /s72
+/s99/s61/s49/s46/s55/s53/s107
+/s66/s84
+/s99
+/s32
+/s32/s50
+/s66/s72/s42
+/s32/s84/s104/s101/s111/s114/s121/s83/s79/s32/s40/s109/s101/s86/s41
+/s107
+/s66/s84
+/s99/s32/s40/s109/s101/s86/s41/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57
+/s82
+/s110/s45/s82
+/s115/s97/s116/s32/s40/s107 /s41
+/s32 /s82
+FIG. 3: (Color on-line) a. The parallel upper critical field H c/bardbl
+versuskBTcwhile V gis varied. The dashed line represents
+the expected behavior for the paramagnetic limit.
+b.ǫSOextracted from H∗[see Fig.2], and calculated from
+the measured superconducting properties (H c2/bardbland T c),[?]
+versuskBTc. The right axes presents ∆R, which scales with
+ǫSOas expected.
+AHE persists up to 100 K with a peculiar, roughly linear
+temperature dependence (not shown). As shown, the ap-
+plied V gvaries the AHE saturation field. This variation
+is inconsistent with a simple magnetic impurity scenario
+and with a simple ferromagnetic behavior. Furthermore,
+no hysteresis is observed for all carrier concentrations.
+Here we note that the AHE saturation field roughly fol-
+lows the behavior of ǫSOand may be related to it.
+Fig.4c compares the data in Fig.4b in the case where
+n= 4.4×1013cm−2(red circles), with data taken at
+constant magnetic field while rotating the sample (as in
+Fig.4a). For the rotation (brown pluses), the slope of
+the AHE is much steeper. The two measurements dif-
+fer at low perpendicular fields where the mobility is an
+order of magnitude higher for the rotation. The qual-
+itative variation between the measurements is expected
+since the AHE is usually proportional to some power of
+the resistivity. The AHE has been related to a multiple
+band structure (light and heavy)[11] and to incipient dis-
+ordered magnetism induced by the field[17]. The former
+is unlikely in view of the strong AHE slope in the case of
+the rotation experiment. For the case of light and heavy
+charge carriers, one would expect the former to domi-
+nate the resistivity and the latter to dominate the low
+field Hall. However, both the resistivity and the AHE
+are very susceptible to a small perpendicular field.
+Finally, we note that for the highest mobility state
+achieved (blue stars), quantum oscillations are observed
+in both MR and Hall measurements [Fig.1b, Fig.4b].
+These oscillations will be analyzed separately.[18]
+Spin-orbit interactions are expected to play a signif-4
+/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48
+/s45/s49/s48 /s48 /s49/s48/s45/s53/s48/s48/s53/s48
+/s48 /s49 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53
+/s40/s99/s41/s32/s110/s61/s51/s46/s48/s120/s49/s48/s49/s51
+/s32/s99/s109/s45/s50
+/s32/s110/s61/s52/s46/s52/s120/s49/s48/s49/s51
+/s32/s99/s109/s45/s50
+/s32/s110/s61/s54/s46/s49/s120/s49/s48/s49/s51
+/s32/s99/s109/s45/s50
+/s32/s110/s61/s55/s46/s56/s120/s49/s48/s49/s51
+/s32/s99/s109/s45/s50/s120/s121/s45/s82
+/s72/s72 /s32/s40 /s41/s32/s40/s98/s41
+/s32/s32/s120/s121/s45/s82
+/s72/s72/s32/s40 /s41/s40/s97/s41
+/s32
+/s83/s99/s97/s110
+/s82/s111/s116/s97/s116/s105/s111/s110
+/s32/s32
+/s48/s72 /s32/s40/s84/s41/s32/s32/s82/s47/s82/s110
+/s48/s72 /s40/s84/s41/s72 /s61/s49/s56/s32/s84
+FIG. 4: (Color on-line) a. Normalized resistance as a functi on
+of perpendicular magnetic field. Data is taken while rotatin g
+the sample in a constant magnetic field of 18 T. b. Hall re-
+sistivity after subtraction of a linear fit for high fields (AH E).
+c. The AHE for n= 4.4×1013cm−2while H is scanned, and
+while rotating the sample at a constant field.
+icant role at the interface due to the inversion symme-
+try breaking. The SO interaction term is of the form:
+ǫSO=p·σ×∇Vwherepis the carrier momentum, σis
+the Pauli spinor, and ∇Vis the electric field perpendic-
+ular to the interface in our system.[19] The strong gate
+dependence of ǫSOdirectly follows from this expression.
+As pointed out above, the initial local electric field is
+unknown. When a positive gate voltage was applied for
+the first time the number of charge carriers decreased,
+while the resistivity decreased. We believe that this is
+mainly due to a decrease in the total electric field near
+the interface, resulting in a decreased SO scattering and
+an enhanced mobility. This suggests that the initial field
+(AG state) is a consequence of the electronic reconstruc-
+tion. In this scenario, adding more LaAlO 3layers in-
+creases the as-grown field and hence the SO scattering.
+This explains the decrease in mobility while increasing
+the LaAlO 3thickness.[20]
+It seems that most scattering processes in the normal-
+state involvea spin flip, which stronglyimpede the trans-
+port. In a simple metal such processes do not contribute
+much to the resistivity, yet in our system they become
+important due to the strong SO coupling. This is clear in
+the case of in-plane spin flip processes; since SO coupling
+affects in-plane spin orientations, a spin flip process re-
+sults in momentum reversal and consequently in a strong
+contribution to the resistivity. These spins are strongly
+coupled to the in-plane momenta; therefore it is impos-
+sible to align them with H /bardblunless the energy associated
+with the field gµBH/bardblexceedsǫSO. For H>H∗, the spins
+gradually align along the field direction and so R nis re-
+duced. When all spins are aligned, the resistance reaches
+the saturation value R sat. For the out-of-plane field ori-
+entation, suppression of spin scattering is overwhelmed
+by the large positive MR.Tuning the gate voltage from 50 to -50 V increases the
+local electric field, resulting in an increase of ǫSOand
+H∗. Unlike R n(the zero field resistance), R satroughly
+scales with number of charge carriers deduced from the
+high field Hall measurements. This suggests that R sat
+is a result of standard impurity scattering. For Vg=-50
+V,ǫSOis extremely large and an in-plane field of 18 T
+is not enough to align the spins and suppress the spin-
+scattering resistivity. Upon applying a small perpendicu-
+lar field component and due to the large scattering time,
+orbital motion is immediately turned on along with the
+SO coupling. This results in a full recovery of the spin-
+MR.
+In summary, we study the phase diagram of the
+SrTiO 3/LaAlO 3interface in the region where T cin-
+creases, while reducing the carrier concentration by vari-
+ation of gate voltage. We demonstrate the important
+effect of spin-orbit (SO) interaction on both supercon-
+ductivity and on normal-state transport. The SO cou-
+pling energy ( ǫSO) is evaluated using two independent
+transport properties, and is also in agreement with theo-
+retical model given the superconducting parameters: T c
+and the upper critical parallel field. ǫSOfollows T cfor
+the electric field range studied. Our results suggest that
+SrTiO 3/LaAlO 3interfacesmaybeuseful forfutureoxide
+based devices controlling the orbital motion of electrons
+by acting on their spins. [21]
+We are indebted to G. Deutscher, A. Aharony, and
+Ora Entin-Wohlman for enlightening discussions. This
+research was partially supported by the Bikura program
+of the ISF grant No. 1543/08 by the ISF grant No.
+1421/08 and by the Wolfson Family Charitable Trust.
+A portion of this work was performed at the National
+High Magnetic Field Laboratory, which is supported by
+NSF Cooperative Agreement No. DMR-0654118, by the
+State of Florida, and by the DOE.
+∗yodagan@post.tau.ac.il
+[1] S. Okamoto and A. J. Millis, Nature (London) 428, 630
+(2004).
+[2] A. Ohtomo and H. Y. Hwang, Nature (London) 427, 423
+(2004).
+[3] N. Reyren et al., Science 317, 1196 (2007).
+[4] N. Nakagawa, H. Y. Hwang, and D. A. Muller, Nature
+Mater.5, 204 (2006).
+[5] R. Pentcheva and W. E. Pickett, Phys. Rev. Lett. 102,
+107602 (2009).
+[6] Z. S. Popovi´ c, S. Satpathy, and R. M. Martin, Phys. Rev.
+Lett.101, 256801 (2008).
+[7] W. Siemons et al., Phys. Rev. Lett. 98, 196802 (2007).
+[8] P. R. Willmott et al., Phys. Rev. Lett. 99, 155502 (2007).
+[9] M. Basletic et al., Nature Mater. 7, 621 (2008).
+[10] M. Ben Shalom et at, Phys. Rev. B 80, 140403 (2009).
+[11] C. Bell et al, Phys. Rev. Lett. 103, 226802 (2009).
+[12] A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962).5
+[13] J. P. Burger et al., Phys. Rev. 137, A853 (1965).
+[14] N. Reyren et al., Applied Physics Letters 94, 112506
+(2009).
+[15] Y. Kozuka et al., Nature 462(2009).
+[16] R. A. Klemm, A. Luther, and M. R. Beasley, Phys. Rev.
+B12, 877 (1975).[17] S. Seri and L. Klein, Phys. Rev. B 80, 180410 (2009).
+[18] M. Ben Shalom et al., To be published (????).
+[19] Y. A. Bychkov and ´E. I. Rashba, JETP Letters (1984).
+[20] C. Bell et al., Applied Physics Letters 94, 222111 (2009).
+[21] S. A. Wolf, Science 294, 1488 (2001).
\ No newline at end of file
diff --git a/1001.0782.txt b/1001.0782.txt
new file mode 100644
index 0000000000000000000000000000000000000000..93a45a268d5801e22ab19e7228bd918b8e27aa48
--- /dev/null
+++ b/1001.0782.txt
@@ -0,0 +1,986 @@
+Under consideration for publication in J. Fluid Mech. 1
+A Streamwise Constant Model of Turbulence
+in Plane Couette Flow
+D. F. G A Y M E1
+B. J. M c K E O N1, A. P A P A C H R I S T O D O U L O U2,
+B. B A M I E H3AND J. C. D O Y L E1
+1Division of Engineering and Applied Science, California Institute of Technology,
+Pasadena, CA 91106, U.S.A.
+2Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, U.K.
+3Department of Mechanical Engineering, University of California, Santa Barbara,
+CA 93106-5070, U.S.A.
+(Received ?? and in revised form ??)
+Streamwise and quasi-streamwise elongated structures have been shown to play a signif-
+icant role in turbulent shear 
ows. We model the mean behavior of fully turbulent plane
+Couette 
ow using a streamwise constant projection of the Navier Stokes equations. This
+results in a two-dimensional, three velocity component (2 D=3C) model. We rst use a
+steady state version of the model to demonstrate that its nonlinear coupling provides the
+mathematical mechanism that shapes the turbulent velocity prole. Simulations of the
+2D=3Cmodel under small amplitude Gaussian forcing of the cross-stream components
+are compared to DNS data. The results indicate that a streamwise constant projection
+of the Navier Stokes equations captures salient features of fully turbulent plane Couette
+
ow at low Reynolds numbers. A system theoretic approach is used to demonstrate the
+presence of large input-output amplication through the forced 2 D=3Cmodel. It is this
+amplication coupled with the appropriate nonlinearity that enables the 2 D=3Cmodel to
+generate turbulent behaviour under the small amplitude forcing employed in this study.
+1. Introduction
+The Navier Stokes (NS) equations provide a complete dynamical description of the
+three velocity components and pressure for simple canonical 
ows under the sole model-
+ing assumption that all important physical phenomena are captured by these equations.
+Unfortunately, these innite dimensional algebraically constrained equations are ana-
+lytically intractable. They have however, been extensively studied computationally and
+numerical solutions do exist. For plane Couette 
ow, the rst numerical solution was
+computed by Nagata (1990). Gibson et al. (2009) provide a detailed discussion of other
+work related to a full range of numerical plane Couette 
ow solutions. Ever increasing
+computing power will continue to allow progress toward understanding these local prop-
+erties. However, a full mathematical understanding of NS even in simple parallel 
ow
+congurations remains elusive, hence considerable eort has been applied to the search
+for more analytically-tractable 
ow models.
+In contrast, the Linearized Navier Stokes (LNS) equations can be analyzed using well
+developed tools from linear systems theory. For wall bounded shear 
ows, one particu-
+lar property of the LNS that has been extensively studied is disturbance amplication,
+e.g. Farrell (1988); Gustavsson (1991); Reddy & Henningson (1993); Farrell & IoannouarXiv:1001.0782v2  [physics.flu-dyn]  25 Nov 20102D. F. Gayme, B. J. McKeon, A. Papachristodoulou, B. Bamieh &J. C. Doyle
+(1993 b); Bamieh & Dahleh (2001); Jovanovi c & Bamieh (2005). Large disturbance am-
+plication is common in these 
ows because the linear operators governing them are
+non-normal, (i.e., the operator, A, is such that AA6=AA).
+The LNS are thought to capture the energy production of the full nonlinear system.
+Henningson & Reddy (1994) showed that non-normality and linear mechanisms are nec-
+essary conditions for subcritical transition to turbulence and it is widely believed that
+energy amplication is due to coupling terms that remain in linearized models, see for
+example (Trefethen et al. 1993). In smooth wall-bounded shear 
ows the linear coupling
+between the Orr-Sommerfeld and Squire equations associated with nonzero spanwise
+wave number has been shown to be required for the generation of the wall layer streaks
+that are necessary to maintain turbulence (Butler & Farrell 1993; Kim & Lim 2000). In
+this context, the term \streak" describes the \well-dened elongated region of spanwise
+alternating bands of low and high speed 
uid" (Walee et al. 1991). The LNS have also
+been used by Jovanovi c & Bamieh (2001) to predict certain second-order statistics of
+turbulent channel 
ow. The above results and a host of others illustrate the power of the
+LNS as a model for wall-bounded shear 
ows. There is however, one fundamental 
ow
+feature that linear models are unable to capture; the change in the mean velocity prole
+as the 
ow transitions from laminar to turbulent. In addition, linear analysis can only
+give local information regarding the full (nonlinear) system.
+Empirical models have been shown to be useful in capturing key aspects of many
+
ows. For example, Proper Orthogonal Decomposition (POD) has been successfully used
+to construct accurate low dimensional ordinary dierential equation models, e.g. (Lumley
+1967; Smith et al. 2005). However the preceding analysis utilizes existing experimental
+or numerical data, a limitation also applicable to eddy viscosity models. In general, data-
+driven or heuristic models can be said to suer from a lack of connection, of varying
+degree, to the governing equations of the problem.
+The model studied herein is an attempt to merge the benets of studying a physics-
+based set of equations, such as NS, with the analytical tractability of a simplied model,
+such as the LNS. It is developed based on the assumption that certain aspects of fully
+developed turbulent 
ow can be reasonably modeled as homogeneous in the streamwise
+direction, here denoted \streamwise constant". The idea that a streamwise constant
+model is sucient to capture mean prole changes from laminar to turbulent is strongly
+supported by the work of Reddy & Ioannou (2000), who showed that nonlinear interaction
+between the ( kx;kz) = (0;N) modes, where the k's are the streamwise and spanwise
+wave numbers, is the primary factor in determining the turbulent mean velocity prole
+in Couette 
ow. Further, as was discussed in Orlandi & Jim enez (1994), this type of
+model may be adequate to capture many of the eects associated with the generation of
+turbulent wall friction. A 21
+2D model along similar lines has also been developed for the
+viscous wall layer; Tullis & Pollard (1993) for example use such a model to study 
ow
+over riblets in this region. The physical and analytical basis for assuming homogeneity
+in the streamwise direction is discussed in the following section.
+1.1. Streamwise Coherence
+A growing body of work supports the notion that turbulence in wall-bounded shear 
ows
+is characterized by dynamically signicant coherent structures, particularly features with
+streamwise and quasi-streamwise alignment. Near-wall streaks (Kline et al. 1967), for
+example, have been shown to play a key role in energy production through the `near-wall
+autonomous cycle' discussed by Walee (1990); Hamilton et al. (1995); Walee (1997);
+Jim enez & Pinelli (1999). This cycle is generally agreed to be an important mechanismA Streamwise Constant Model of Turbulence in Plane Couette Flow 3
+in determining the low-order statistics of turbulent 
ows in the buer region and viscous
+sublayer, i.e. y+630 (Schoppa & Hussain 2002).
+More recent high Reynolds number studies have focused on the identication and
+characterization of streamwise coherence in the core, i.e. Kim & Adrian (1999); Morrison
+et al. (2004); Guala et al. (2006); Hutchins & Marusic (2007 a). These motions have been
+called large and very large scale motions (respectively LSM and VLSMs). They appear
+to have a similar signature to the near-wall streaks (Hutchins & Marusic 2007 b; Chung
+& McKeon 2010), but tend to be longer in extent, from one to ten times the outer
+length scale, . There is experimental evidence to suggest that at high Reynolds numbers
+(for example Re>7300), VLSMs contain more energy than the near-wall structures
+(Morrison et al. 2004; Hutchins & Marusic 2007 a,b). In turbulent boundary layers they
+have also been shown to modulate the near-wall turbulence, see for example Hutchins &
+Marusic (2007 b); Mathis et al. (2009), suggesting that they may play an important role
+in 
ow dynamics across a range of scales.
+In Couette 
ow, structures reminiscent of VLSMs have long been observed in the
+core through DNS of turbulent plane Couette 
ow (Lee & Kim 1991; Bech et al. 1995).
+Although some studies raised the concern that the structures were numerical artifacts,
+recent DNS at higher resolution and with longer box sizes (Komminaho et al. 1996;
+Tsukahara et al. 2006) have conrmed the existence of long streamwise alternating high
+and low speed streaky structures at the centerline. In experiments, VLSMs were rst
+identied through observations of a noticeable peak in the Fourier energy spectrum of
+the turbulence intensity at low frequencies (Komminaho et al. 1996; Kitoh & Umeki
+2008). The Couette 
ow experiments of Tillmark & Alfredsson (1998) found further
+evidence of very long structures in the form of long autocorrelations Ruu() and two-
+point correlations Ruu(
x) as well as periodic variation of spanwise correlations Ruu(
z)
+in the core. The streamwise extent of these correlations was longer than those generally
+seen in other wall-bounded 
ows. Komminaho et al. (1996) also found that in contrast
+to other 
ows, the streamwise correlations for Couette 
ow are larger at the center than
+near the wall. At channel center the zero cross distances of Ruu() andRuu(
x) have
+been observed to be three times that of the corresponding structures in Poiseuille 
ow
+(Kitoh et al. 2005). This makes Couette 
ow an ideal candidate to test the applicability
+of a streamwise constant model.
+Streamwise constant, kx= 0, perturbations to the LNS also produce the largest input-
+output response for both laminar (Bamieh & Dahleh 2001; Jovanovi c & Bamieh 2001;
+Farrell & Ioannou 1993 b, 1998) and turbulent (del Alamo & Jim enez 2006) base velocity
+proles. In addition, streaks of streamwise velocity naturally arise from the set of initial
+conditions that produce the largest energy growth (Butler & Farrell 1992; Farrell &
+Ioannou 1993 a), namely streamwise vortices. Even in linearly unstable 
ows, studies
+have shown that the amplitude of streamwise constant structures can exceed that of
+the linearly unstable modes (Jovanovi c & Bamieh 2004; Gustavsson 1991). For channel
+
ows, Bamieh & Dahleh (2001) explicitly showed that streamwise constant disturbances
+produce energy growth on the order of Re3whereas streamwise varying disturbances
+grow as a function of Re3
+2.
+In the present work we employ a streamwise constant model based on the previously
+discussed experimental and analytical evidence of the importance of streamwise homoge-
+nous features. This so-called two-dimensional, three (velocity) component, henceforth
+2D=3C, model for plane Couette 
ow is simulated under small amplitude Gaussian forc-
+ing. The results demonstrate the ability of this model to capture some important features
+of fully developed turbulent 
ow. In particular, it is demonstrated that: (1) the nonlinear
+terms in the 2 D=3Cmodel capture the momentum redistribution mechanism involved4D. F. Gayme, B. J. McKeon, A. Papachristodoulou, B. Bamieh &J. C. Doyle
+Figure 1. Flow geometry. Streamwise and spanwise boundaries are periodic, the bottom wall
+is stationary and the top wall moves in the x-direction with a velocity Uw. The channel half
+height is denoted and the full channel height is denoted h.
+in creating the shape of the turbulent velocity prole, (2) a stochastically forced 2 D=3C
+model can reproduce the appropriate turbulent mean velocity prole and Reynolds num-
+ber trends, and (3) this model produces amplication of small disturbances that is con-
+sistent with input-output studies of the LNS. The work is organized as follows: the next
+sections of the paper describe the model and simulation approach. Results and discus-
+sion follow, including a comparison between the model and a DNS dataset, before nal
+conclusions.
+2. The 2D=3CModel
+The 2D=3Cmodel discussed herein is obtained by setting streamwise ( x-direction)
+velocity derivatives in the full NS equations describing Couette 
ow to zero (Bobba
+2004). This can be thought of as a projection of the NS into the streamwise constant
+space. One can explicitly show that for Couette 
ow this 2 D=3Cformulation also results
+in a system with zero streamwise pressure gradient.
+The velocity eld is decomposed such that ~u= [U+u0
+sw;V+v0
+sw;W+w0
+sw]; where
+U=U(y) =y; V =W= 0 is the laminar Couette 
ow and ( u0
+sw;v0
+sw;w0
+sw) are the
+corresponding time-dependent deviations from laminar in the streamwise constant sense.
+The 
ow geometry is shown in Figure 1. The Reynolds number employed is Rew=Uwh
+,
+whereUwis the velocity of the top plate, his the channel height and is the kinematic
+viscosity of the 
uid. All distances and velocities are respectively normalized by hand
+Uw. In the sequel we will use ( u0
+sw;v0
+sw;w0
+sw) to denote
+u0
+sw
+Uw;v0
+sw
+Uw;w0
+sw
+Uw
+, and explicitly
+indicate the scaling only in the gure labels.
+A stream function
+v0
+sw=@ 
+@z;w0
+sw=@ 
+@y
+forces the appropriate 2 Dcontinuity. This results in the following model:
+@u0
+sw
+@t=@ 
+@z@u0
+sw
+@y@ 
+@z@U
+@y+@ 
+@y@u0
+sw
+@z+1
+Rew
u0
+sw
+@
 
+@t=@ 
+@z@
 
+@y+@ 
+@y@
 
+@z+1
+Rew
2 :(2.1)
+No slip boundary conditions at the wall and periodic boundary conditions in the spanwiseA Streamwise Constant Model of Turbulence in Plane Couette Flow 5
+direction are applied (without loss of generality they can also be used for the stream
+function equation since v0
+sw=w0
+sw= 0)@ 
+@z=@ 
+@y= 0) = const). This model
+retains many of the important 
ow features lost in a purely 2 Dmodel by maintaining
+all three velocity components. Equations (2.1) are an improvement over linear models
+because it is hypothesized that it is the nonlinearity in the u0
+sw(y;z;t ) equation that
+provides the mathematical mechanism for the redistribution of the 
uid momentum. This
+redistribution creates larger streamwise velocity gradients in the wall-normal direction
+and changes the plane Couette velocity prole from linear to its characteristic turbulent
+\S-shape". Meanwhile, the important features of the LNS are maintained. Linearization
+of (2.1) around the laminar prole produces a non-normal operator with a coupling term
+analogous to the one in the LNS.
+The laminar 
ow solution of Equation (2.1) was previously shown to be globally, that
+is nonlinearly, stable for all Reynolds numbers (Bobba et al. 2002), and therefore the
+laminar 
ow constitutes a unique solution. Consequently any transition mechanisms as-
+sociated with bifurcations, escape from the basin of attraction of the laminar solution or
+the like are not possible. So, any complications associated with these nonlinear phenom-
+ena can be eliminated from the analysis of these particular equations. Global asymptotic
+stability of the laminar solution also implies that without forcing, perturbations will
+eventually decay, in agreement with the results of Orlandi & Jim enez (1994) who found
+that after an initial perturbation a 2 D=3Cmodel decays (back to laminar) with time.
+The fact that one can analytically prove that the unforced 2 D=3Cmodel has a unique
+solution suggests that it is far more analytically tractable than NS. We do not pursue
+analytical studies of the 2 D=3Cmodel in the current work, but instead concern ourselves
+with showing the applicability of the model in describing important features of the 
ow
+eld. However, the fact that global statements about these equations can be made implies
+that future analytical studies are promising.
+As with any model, there are assumptions built into the 2 D=3Cmodel, and it is im-
+portant to understand how these relate to the physical phenomena associated with tur-
+bulent 
ows. Most obviously, small scale, three-dimensional turbulent activity, including
+the specics of several structures that are known to exist in the full 
ow, is not captured.
+While this makes appropriate scaling relationships more dicult to determine, it does
+not diminish the potential of the model for predicting and understanding key aspects of
+turbulence in plane Couette 
ow. The challenge lies in extending the 2 D=3Cmodel to
+incorporate aspects of the streamwise variation associated three-dimensional turbulent
+
ow.
+2.1. Modeling Framework
+No model is a perfect representation of reality. In addition to modeling assumptions,
+parameter errors or external in
uences on the system in question are often ignored.
+Inaccurate parameter estimates or linearization of a nonlinear system may change the
+model's ability to predict behavior. Environmental conditions that aect (or disturb) the
+system may also play an important role in its dynamics. This role is not captured by a
+typical model.
+Robust control theory has historically been used to analyze models in the presence of
+such modeling errors (`uncertainty') Doyle et al. (1991); Zhou et al. (1996). One typically
+represents all of the uncertainties using an uncertainty operator 
. The block diagram
+of Figure 2 is then used to depict a model subject to this uncertain set 
. Generally
+robust control tools provide a bound on k
k, below which a desired property can be
+maintained. Robust control tools do not require a detailed model of the particular un-
+certainty. This makes them appealing in situations where there are unknown (or hard to6D. F. Gayme, B. J. McKeon, A. Papachristodoulou, B. Bamieh &J. C. Doyle
+Figure 2. A robust control block diagram for a model
+subject to uncertainty. Generally a norm bound on 

+species the amount of uncertainty that a model can
+have before a desired property is lost (i.e., if the model
+is stable fork
k61, this implies robust stability).
+Figure 3. The approximation for il-
+lustrating the 2 D=3Cmodel's lack of
+robustness. The zero-mean noise as an
+approximation for the modeling errors
+and uncertainty. The noise acts as an
+additive \uncertainty" at each time
+step.
+model) environmental in
uences on the system, or when one can only specify the range
+on a parameter, rather than an exact value. However, since the uncertainty is generally
+specied through a bound that includes the worst case scenario, the results of this type
+of analysis may be very conservative. One way to mitigate this is to `structure' or shape
+the uncertainty, a process which relies on some understanding of the implicit modeling
+errors.
+In the context of a system comprising a wall-bounded shear 
ow, many disturbances
+can be modeled through the 
 block in Figure 2. These sources of modeling errors (uncer-
+tainties) can arise from assumptions on the boundary conditions or unmodeled dynamics.
+External sources of model uncertainty that are not captured in the NS equations, include
+phenomena such as acoustic noise and thermal 
uctuations in an experiment, or the
+build up of numerical error in simulations. In addition the uncertain set includes terms
+excluded by the modeling assumptions, namely the kx6= 0 modes in the 2 D=3Cmodel,
+or the nonlinear terms for the LNS model. See Bobba (2004) for a full characterization
+of the types of uncertainties present in shear 
ow problems. Obviously the latter class
+of perturbations are strictly bound to satisfy the NS equations, while the former are less
+constrained. Distributed wall roughness (i.e., surface imperfections present in any real
+surface), wall vibration, imperfect alignment of the walls or other parameter estimates
+may be captured through either stochastic or other forcing.
+In the present work, the framework of robust control is employed in a nontraditional
+manner. Instead of providing an upper bound on k
k, (i.e. a robustness guarantee)
+we describe the extent to which the laminar 
ow state is `fragile' (i.e. unable to be
+maintained in the face of innitesimal disturbances). One can think of this as an inverse
+robustness (or `fragility') problem, i.e. a discussion of a lack of robustness. In order to
+study the disturbance response of the 2 D=3Cmodel the system of Figure 2 is abstracted
+into the simplied setting of Figure 3. We further simplify by linearizing the 
  (y;z;t )
+equation which is equivalent to recognizing that advection terms in the stream function
+equation play a lesser role in redistributing momentum. The forcing and henceforth
+ are constrained to be small such that the nonlinear terms are at least an order of
+magnitude smaller than the linear ones in all cases studied here. This small amplitude
+noise assumption is very important in the development of this work because of the focus
+on the eect of small amplitude disturbances on a fragile system and because larger
+amplitude forcing can change the dynamics of the model.A Streamwise Constant Model of Turbulence in Plane Couette Flow 7
+For all of the numerical studies described herein we simulate
+@u0
+sw
+@t=@ 
+@z@u0
+sw
+@y@ 
+@z@U
+@y+@ 
+@y@u0
+sw
+@z+1
+Rew
u0
+sw+du
+@
 
+@t=1
+Rew
2 +d ;(2.2)
+with the same boundary conditions as in Equation (2.1). A short simulation study com-
+paring low-order streamwise velocity statistics supports the use of linearization in the  
+equation.
+Approximating the full 3 Dsystem using the interconnection in Figure 2 would involve
+nonlinear mixing modes. In order to approximate the range of frequencies associated
+with the full 3 Dsystem in the framework of Figure 3, zero mean stochastic forcing was
+applied to the 2 D=3Cmodel. In particular, the inputs du(y;z;t ) andd (y;z;t ) in (2.2)
+are small amplitude and Gaussian, as in Gayme et al. (2009). The input amplitudes are
+dened using the standard deviation, noise. Note that under these assumptions there is
+no coupling from the streamwise components ( u) back to the cross-stream components
+(the 
 equation). The plausibility of modeling the type of disturbances common to
+experimental conditions in this manner is conrmed by results from stochastic forcing of
+the LNS equations, which leads to 
ows dominated by streamwise elongated streaks and
+vortices that are strikingly similar to those observed in experiments (Farrell & Ioannou
+1993 b), as well as by the results of the simulation study discussed in Section 4.3. Further
+development of the model would be required to address this eective feedback mechanism.
+3. Approach
+Time-dependent simulations of the full coupled system (2.2) were carried out using a
+basic second-order central dierence scheme in both the spanwise ( z) and wall-normal ( y)
+directions. Periodic boundary conditions in zand no-slip boundary conditions in ywere
+applied. Simulations using the spectral methods of Weideman & Reddy (2000) were also
+performed for comparison. The pseudospectral simulations employ a Chebyshev inter-
+polant for the wall-normal direction and a Fourier method for the spanwise derivatives.
+The aspect ratio in all of the simulations was greater than 12 to 1 (spanwise to wall-
+normal) in order to eliminate box size eects; specically the usual computational box
+size wasLyLz=h12:8hwith 75100 grid points. A spanwise extent of 12 :8hwas
+selected to provide a direct comparison to the full eld DNS data from Tsukahara et al.
+(2006).
+In this study, the response of the streamwise velocity, u0
+sw, to forcing of the cross-
+stream velocity components, v0
+swandw0
+sw, was examined. A forcing input of zero mean
+small amplitude Gaussian noise evenly applied at each y{zplane grid point was selected
+ford . The other input forcing, du, was set to zero based on previous studies of the LNS,
+which showed that the response to streamwise body forcing is signicantly smaller than
+the response to spanwise/wall-normal plane forcing (Jovanovi c & Bamieh 2005). These
+studies used an order of magnitude argument to conclude that the dierence in response
+scales as1
+Re2. Furthermore, it is energy redistribution by streamwise vorticity (i.e. 
	)
+that is thought to be the primary eect governing the shape of the turbulent velocity
+prole (Hamilton et al. 1995). The response in the streamwise velocity component to this
+forcing may have a nonzero mean because of the nonlinearity in the uequation.
+The dierent discretization techniques naturally provide a comparison of dierent noise
+forcing distributions. For example, the Chebyshev grid results in a higher concentration
+of noise forcing near the walls. Throughout the present work it is assumed that signicant8D. F. Gayme, B. J. McKeon, A. Papachristodoulou, B. Bamieh &J. C. Doyle
+numerical errors are not introduced by the methods of discretization, i.e. the introduction
+of signicant noise arises only through the dterms of Equation (2.1).
+The time evolution of 
  in Equation (2.2) can be seen to be a stochastically forced
+heat equation, i.e. a linear stochastic partial dierential equation which can be solved
+analytically (see for example Swanson (2007) or Luo (2006) and the references therein).
+This is not pursued here because a simulation is a much simpler way to demonstrate
+the ecacy of the model. An exposition on It^ o calculus and Wiener chaos expansions is
+beyond the scope of this paper. Future work may involve pursuing analytical solutions
+to both the linear approximation to  and the full nonlinear system (2.1).
+4. Results and Discussion
+The results will be divided into three main sections. We begin by analyzing the DNS
+data of Tsukahara et al. (2006) in the light of the 2 D=3Cmodel and conrming the
+extent to which the assumptions of the 2 D=3Cmodel can be adduced through this
+data. Following that, a time independent version of Equation (2.1) is studied to verify
+the implicit model lter between  andu. Finally, results from full simulations of the
+time-dependent Equations (2.2) are presented and compared to the DNS data.
+4.1. Comparison of DNS data with 2D=3CModeling Assumptions
+Full details of the DNS dataset can be found in Tsukahara et al. (2006); a brief review
+of key aspects is given here. Three Reynolds numbers were considered, Rew= 3000;8600
+and 12800, all with computational domain size LxLyLz= 44:8hh12:8h,
+102496512 grid points, and a sampling time (tUw
+Lx) of 91. The fourth-order nite
+dierence scheme proposed in Morinishi (1995) was employed for the xandzdirections.
+A second-order nite dierence method was used for the ydirection.
+The friction coecient, Cf= 9:59103, is somewhat higher than in other studies,
+such as Robertson & Johnson (1970). Filling this friction factor into the relationship
+developed by Robertson (1959),
+r
+Cf
+2=G
+log10(1=4Rew)whereCf=w
+1=2(1=2Uw)2(4.1)
+withwused to denote shear stress at the wall, leads to an experimental constant G=
+0:199. Other values reported in the literature include G= 0:19 andG= 0:174 both
+from Robertson (1959) based on the data of Reichardt and Robertson respectively and
+G= 0:182 from the experimental study of El Telbany & Reynolds (1982).
+The turbulent mean velocity proles, turbulence intensities, Reynolds stresses and
+budgets of u0
+iu0
+jfrom this DNS show good agreement with the experimental results of
+Tillmark (1995) and the spectral DNS study of Komminaho et al. (1996) which used a
+larger box. The two-point correlations in uindicate that the box lengths used in both
+the streamwise, Ruu(
x), and spanwise, Ruu(
z), directions are sucient to eliminate
+any boundary condition-related spurious eects.
+In what follows, a streamwise constant projection of the DNS data is approximated
+through a streamwise ( x) average over the box length, which will highlight streamwise
+coherence of the order of the box length. The x-averaged DNS data is denoted ~uxave=
+(u0
+xave+U(y);v0
+xave;w0
+xave) to distinguish it from true streamwise constant data. Time
+averages are indicated by an overbar, (
).
+The ratio of the energy contained in the x-averaged DNS to that of the full eld provides
+a quantitative measure of the extent to which the DNS data can be approximated asA Streamwise Constant Model of Turbulence in Plane Couette Flow 9
+Component Total Energy Norm Percent of Total Energy
+k
k inx-averaged Norm
+u 0.5334 99.1
+uU 0.1686 90.2
+v 0.0279 19.0
+w 0.0412 15.0
+Table 1. Energy content in the x-averaged DNS velocity components at Rew= 3000.
+streamwise constant. For this comparison the squared 2-norm is used to approximate the
+energy in each 2-dimensional x-averaged velocity component
+kk2=Z
+ZZ1
+0(y;z)2dydz
+
z
+2LyLzNz1X
+k=10
+@Ny1X
+j=1
yj+1
+2
+2(yj+1;zk+1) +2(yj;zk+1)+
++2(yj+1;zk) +2(yj;zk)

+;(4.2)
+whereZis the spanwise extent, 
 z=z2z1is the spanwise distance between zgrid
+points and trapezoidal approximations are used for the inhomogeneous ygrid.
+Table 1 shows the total energy (based on the full DNS eld at Rew= 3000) and
+the percentage contained in each of the x-averaged velocity components ( u; v; w ) as
+well as in the deviation from laminar (denoted uU). This latter quantity is most
+representative of the energy associated with the dierences in the mean velocity prole
+for a turbulent versus a laminar 
ow. The computations show that x-averaged streamwise
+velocity contains 99% of the ( u) energy, whereas the corresponding deviation from laminar
+contains 90%. As expected, the x-averaging results in a larger loss of information in the
+spanwise and wall-normal velocity components.
+An examination of the DNS streamwise velocity eld at y+= 29, close to the outer edge
+of the region aected by the near-wall cycle, reveals the signature of streamwise elongated,
+large scale streaks in the streamwise/wall-normal plane of the full eld (Figure 4). These
+streaks are also visible in Figures 5(a) and 5(b) which depict contour plots of the deviation
+from laminar 
ow, u0
+xave=uxaveU, when averaged over 25% of the streamwise eld
+and the full eld respectively. Clearly, increasing the averaging length acts as a lter on
+structures of dierent streamwise extent. The average over the full box length retains
+strong evidence of structures across the entire spanwise/wall-normal plane. In particular,
+the strongest signature near the wall is in qualitative agreement with the near-wall model
+of energetic structures centered around y+15 with a statistical diameter of y+30.
+Another important feature of Figure 5(b) is that the maximum deviations from laminar
+
ow, which are out of spatial phase with one another, top to bottom, are associated with
+large-scale rolling motions which reach across the channel height.
+The above analysis shows that there is good agreement between the DNS data and our10D. F. Gayme, B. J. McKeon, A. Papachristodoulou, B. Bamieh &J. C. Doyle
+Figure 4. Anz{xplane contour plot of the streamwise velocity, u, from the DNS eld, (bottom
+up view) at y+= 29. Light colored contours denote regions of higher velocity and dark contours
+indicate lower velocity regions.
+(a) Average over 25% of streamwise box
+ (b) Average over full streamwise box
+Figure 5. y{zplane contour plots of the x-averaged (as an approximation for streamwise
+constant) DNS deviations from laminar ( u0
+xave) (a) averaged over 25% of the streamwise box
+length and (b) averaged over the full streamwise box length.
+assumptions. In the next section this data is used to suggest a time-independent model
+for (y;z) in order to study a steady state version of the streamwise velocity in (2.1).
+This is followed by simulation of the full system (2.2).
+4.2. Time Independent ux(y;z;t )Equation
+The response of Equation 2.1 to a stream function that is independent of time,  ss(y;z),
+is of interest for two reasons: (1) the forced solution for the streamwise velocity permits
+investigation of whether the 2 D=3Cmodel lters an appropriately shaped  ss(y;z) to-
+wards the expected shape of the turbulent velocity prole, and (2) it gives insight into the
+mathematical mechanisms that create the momentum (energy) transfer that generates
+the blunted prole. The analysis constitutes a weakly nonlinear analysis, in which the
+time-independent forcing takes the form  ss= ss0+" ss1+:::.
+In Barkley & Tuckerman (2007) it was shown that laminar-turbulent 
ow patterns in
+plane Couette 
ow could be reproduced using a stream function of the form  (y;z) =
+ 0(y)+ 1(y) cos(kzz)+ 2(y) sin(kzz). We use this study as guidance but set the zeroth-
+order term  0to zero because a nonzero  0produces a nonzero-mean spanwise 
ow w0
+ss,
+which is not representative of the velocity eld we are interested in studying. The DNS
+eld (Tsukahara et al. 2006) was also used as a guide to ensure that the rst-order
+term ss1as well as the corresponding wall-normal and spanwise velocities, respectivelyA Streamwise Constant Model of Turbulence in Plane Couette Flow 11
+v0
+ss1andw0
+ss1, contained representative features. For ease of computation and analysis
+a simple analytic model for  ss1(y;z) was selected, namely a doubly-harmonic model
+which matches the boundary conditions
+ ss=" ss1(y;z) ="sin2(y) cos2
+zz
+: (4.3)
+This corresponds to
+v0
+ss1(y;z) =2
+zsin2(y) sin2
+zz
+;andw0
+ss1(y;z) =sin (2y) cos2
+zz
+:
+The size of the perturbation, ", is a free variable to be explored, while the spanwise
+wavelength, z, is xed to a value determined using the DNS data.
+Streamwise averages of both v(x;y;z ) andw(x;y;z ) from the DNS data permit an
+estimate of  ss(y;z), (to within some constant), for that particular eld. A contour plot
+of the approximation based on w0
+xave(y;z) is shown in Figure 6( a). The value of z1:8h
+was chosen to match the results from a Fast Fourier Transform (FFT) of this data while
+maintaining the same box size (12 :8h) employed for the DNS. This value is also in the
+range of the spanwise wave number corresponding to maximum amplication of the
+linear operator (optimal spanwise spacing), kz2[2:8;4] (z2[1:6;2:2]h), reported in
+the literature (Farrell & Ioannou 1993 b; Butler & Farrell 1992; Gustavsson 1991). An
+initial perturbation amplitude of "= 0:00675 was selected based on the approximate
+values obtained by integrating v0
+ave(y;z) andw0
+ave(y;z). The estimated amplitude is very
+small, in agreement with the idea of using a nominal model plus an uncertainty d which
+is amplied through the coupling in the linear operator in a manner that is described
+and quantied in studies such as Trefethen et al. (1993) and Jovanovi c & Bamieh (2005).
+A contour plot re
ecting these parameter values is provided in Figure 6(b). It shows
+good qualitative agreement, in particular with the region of strongest signal in the DNS
+streamwise average (Figure 6(a)). The latter wall-normal variation is complicated (and
+Reynolds number-dependent), but a simple harmonic variation gives a reasonable rep-
+resentation. The velocity vector eld implied by Equation (4.3) is consistent with low
+speed 
uid being lifted up from the stationary wall and higher speed 
uid being pushed
+down from the moving wall, and as such supports the notion that the mechanisms of
+interest can be modeled using a single harmonic in both yandz.
+The stream function of Equation 4.3 with the selected "andzwas applied to the
+time-independent form of equation 2.2, yielding
+
+@ ss
+@z@
+@y+@ ss
+@y@
+@z+1
+Rew

+u0
+swss=@ ss
+@z@U
+@y: (4.4)
+A contour plot of the resulting u0
+swss(y;z) is depicted in Figure 7. This gure shows a
+u0
+swss(y;z) with near-wall rolls that are out of spanwise phase with one another similar to
+those seen in the x-averaged DNS data of Figure 5(b). The increased coherence associated
+with the steady state model relative to the DNS data manifests as an increased variation
+in the deviation from laminar (amplitude of the surface) particularly at the center of the
+channel. This eect is emphasized through comparison of the surface plots of Figures
+8(a) and 8(b). Note the dierent vertical axis scales for the two plots.
+Averages across the span of u0
+swss(y;z) for"= 0:00675 as well as for four additional "
+values are compared to a similar average of u0
+xavefrom the DNS in Figure 9(a). Clearly
+using ssfrom (4.3) as an input to (4.4) produces streamwise velocity proles whose
+shapes are consistent with u0
+xaveuUfrom the DNS. The peaks are, however located
+at dierent wall-normal positions. An amplitude that exactly matched both the magni-12D. F. Gayme, B. J. McKeon, A. Papachristodoulou, B. Bamieh &J. C. Doyle
+(a) xaveEstimated from w0
+xave
+ (b) ss(y;z) = 0:0675 sin2(y) cos2
+1:8z

+Figure 6. (a) Contour plot of the x-averaged DNS (as a streamwise constant,
+2D=3Capproximation) spanwise velocity deviations integrated to obtain the stream func-
+tion, xave(y;z) =@w0
+xave
+@y. (b) Contour plot of the simple harmonic model for
+ ss(y;z) = 0:00675 sin2(y) cos2
+1:8z

+with amplitude and wavelengths that approximate DNS
+data.
+(a) DNS Data
+ (b)u0
+swsspredicted from  ss
+Figure 7. Contour plots of (a) x-averaged DNS data and (b) the 2 D=3C(stream-
+wise constant) velocity deviations, u0
+avess, obtained using steady state estimate
+ ss(y;z) = 0:00675 sin2(y) cos2
+1:8143z

+, both plotted with the same contour levels.
+tude and location of the DNS peaks was not found even when dierent values of kzwere
+studied. This is not unexpected because of the simplicity of the wall-normal variation in
+the steady state model, as well as the streamwise constant and steady state assumptions
+(clearly the full turbulent eld is neither streamwise constant nor time-independent).
+However, this type of model claries the nonlinear role of cross-stream 
ow features in
+redistributing energy in the 
ow eld. These results suggest that the phenomenon that is
+responsible for blunting of the velocity prole in the mean sense is a direct result of the
+interaction between rolling motions caused by the y{zstream function and the laminar
+prole. In other words, this study provides strong evidence that the nonlinearity needed
+to generate the turbulent velocity prole comes from the nonlinear terms that are present
+in theu0
+sw(y;z;t ) equation of the 2 D=3Cmodel (2.1).
+The magnitude of forcing applied to the system is re
ected in the amplitude of  ss(y;z)
+which in turn aects the friction Reynolds number, Rethrough the skin friction arising
+due to the resultant mean velocity gradient at the wall. Increasing the amplitude ( ") inA Streamwise Constant Model of Turbulence in Plane Couette Flow 13
+(a)u0
+xaveDNS data
+ (b)u0
+swssestimated using  ss(y;z)
+Figure 8. Surface plot of velocity deviations (a) u0
+xave(y;z) from DNS Data at Rew= 3000 and
+(b)u0
+swss(y;z), obtained using steady state estimate  ss(y;z) = 0:00675 sin2(y) cos2
+1:8143z

+atRew= 3000, note the zscale dierence between (a) and (b)
+(a) Amplitude Variation
+ (b) Velocity Gradient at the Wall
+Figure 9. (a) Variation of the 2 D=3C(streamwise constant) velocity deviations, u0
+xavewith
+A; estimates are obtained using steady state estimate  (y;z) =Asin2(y) cos2
+1:8143z

+(b)
+Variation in the velocity gradient at the wall@u
+@y
+wallwith ssamplitude, A.
+Equation 4.3 is analogous to increasing the magnitude of the model uncertainty. These
+eects are emphasized in Figure 9(b) which provides a plot of "versus the velocity
+gradient at the wall. This behaviour will be discussed in more detail in Section 4.3.
+4.3. Time Dependent 2D=3CModel
+Time-dependent simulations of (2.2) were carried out using the basic second-order central
+dierence scheme and pseudo-spectral approaches described earlier. Table 2 lists the
+Reynolds number and forcing amplitude combinations considered. The window used for
+time averaging was 
 t= 100000h
+Uw.
+The initial simulation (Case 1, in Table 2) was carried out at Rew=Uwh
+= 3000 with
+d (x;y;t ) drawn from a zero mean Gaussian distribution with standard deviation (noise
+amplitude) noise = 0:01 applied at every point in the mesh.
+A comparison of Figures 5(b) and 10(a) shows that contours of constant streamwise
+velocity deviation from laminar from the DNS and the 2 D=3Csimulation are in good
+qualitative agreement. In particular, the spanwise oset in spatial phase between peaks14D. F. Gayme, B. J. McKeon, A. Papachristodoulou, B. Bamieh &J. C. Doyle
+(a)
+ (b)
+Figure 10. (a) Contour plot of u0sw(y;z;t ) obtained from the 2 D=3Cmodel for Case 1 with
+the same contour levels as in Figure 7. (b) The surface plot corresponding to (a).
+Table 2. Computation Details
+Case Reynolds Number noiseLyLzNyNzSquared Norm of
+the Noise Input
+1 3000 0.01 h12:8h75100 0 :0565
+2 3000 0.0125 h12:8h75100 0 :0882
+3 3000 0.004 h12:8h75100 0 :009
+4 8600 0.004 h12:8h75130 0 :0092
+5 12800 0.004 h 16:5h75130 0 :0092
+6 12800 0.001 h 16:5h75130 5:77e04
+Spec 1 3000 0.001 h 14:5h4081
+Spec 2 3000 0.002 h 14:5h4081
+from top to bottom is reproduced. While the dominant wavelength from the 2 D=3C
+simulation is somewhat longer than the z1:8 of the DNS (frequency analysis of u0sw
+indicates that most of the energy resides in wave lengths between 4 6z66:1) there is
+also a signicant contribution from z2. There is noticeably better agreement between
+the time-dependent model and the DNS (Figure 10(b) and 8(a)) than for the steady state
+analysis (Figure 8(b)), likely a consequence of the broadband stochastic, i.e. less coherent
+and time-dependent, forcing.
+4.3.1. Mean Velocity Prole
+Averaging the streamwise velocity eld obtained for Case 1 in Table 2 leads to the mean
+velocity prole (i.e. usw) shown in Figure 11(a). The mean prole can also be plotted inA Streamwise Constant Model of Turbulence in Plane Couette Flow 15
+(a) Mean Velocity Proles
+ (b) Mean Velocity Inner Units
+Figure 11. (a) Comparison of mean velocity prole from the 2 D=3Cmodel Case 1, usw(y;z;t ),
+withu(x;y;z;t ) from DNS. (b) Inner scaled velocity proles comparison of Cases 1 and 2 to the
+DNS data with Re52 for all data sets.
+inner units (Figure 11(b)) with the use of Equation 4.1 (with G= 0:1991 from Tsukahara
+et al. (2006)) to estimate the friction velocity, u. There is good agreement between the
+DNS and the Case 1 simulation, even with the assumption of a friction velocity that
+corresponds to the full 
ow. However, it is clear that below y+20 the 2D=3Cmodel
+underestimates the expected velocity prole (maximum error 7 :4%), and above that it
+overshoots it (maximum error 2 :4%). There are two obvious rst-order interpretations of
+these discrepancies. First, for cases 1{6, the noise is modeled as being evenly distributed
+across the grid while in reality the noise is likely higher in the buer region due to the
+proximity of the wall, and lower in the overlap layer. An improved noise model might
+improve the agreement. A second interpretation is that further from the wall the 
ow is
+better modeled by the streamwise constant approximation, while streamwise variation is
+more important in the dynamics of the near-wall region (in agreement with the known
+variation of the spectral distribution of streamwise energy in the full 
ow).
+A second (constant) noise amplitude at the same Reynolds number, Case 2, is also
+shown in Figure 11(b). The agreement with the DNS is certainly improved below y+= 20
+(maximum error 6 :19% aty+= 19), but at the expense of larger error further from the
+wall (56% between 20 < y+<30). These results further support the idea that
+a non-uniform noise forcing with increased noise near the wall versus that at channel
+center may more accurately re
ect the conditions in a real 
ow eld. This idea is further
+explored in Section 4.3.3.
+It should be noted that ucan also be computed directly from the velocity gradient at
+the wall. In both cases Reis underestimated by around 10% compared to the estimate
+from (4.1). Because of the limited number of points near the wall, the friction relationship
+from the full 
ow was preferred, with the understanding that this would only be correct
+if the 2D/3C model with noise exactly reproduced the mean 
ow behavior.
+4.3.2. Reynolds Number and Noise Amplitude Trends
+Four additional Reynolds number and noise amplitudes pairs were considered. The
+details for each of the cases 3{6, along with the computational domain and spatial res-
+olution, are provided in Table 2. Respective values of the norm k
k2, as computed in
+Equation (4.2), of the noise input computed over the box are also reported, since this is
+a more appropriate measure of the forcing when the box size varies.16D. F. Gayme, B. J. McKeon, A. Papachristodoulou, B. Bamieh &J. C. Doyle
+(a)
+ (b)
+Figure 12. (a)usw(y;z;t ) from 2D=3CModel for Cases 2 5 in Table 2 and (b) a comparison
+ofu+versusy+for Cases 4 and 5 with Recomputed based on the values used in Tsukahara
+et al. (2006).
+It is useful to introduce a normalized version of Equation (2.1) through the change of
+variables=t
+Reand 	 =Re  . This creates new expressions for the forcing in (2.2),
+Du=Redu(= 0) andD	=Re2d . The expression D	=Re2d indicates that an
+increase in noise produces a similar eect to an increase in Reynolds number (actuallyp
+Re), as observed in the increased deviation from laminar observed with increasing noise
+amplitude in 9(a). This is especially clear when considering the variation of Rebecause
+an increase in noise amplitude directly corresponds to increased velocity gradients at
+the wall due to the no-slip boundary conditions. Increased prole \blunting" with both
+increasingnoise (noise input energy) and Reynolds number in cases 2{5 can be observed
+in gure 12(a).
+For the higher Reynolds numbers (but constant noise amplitude) in Case 4 and 5,
+Figure 12(b) shows a worsening agreement in inner units with the DNS data from
+Tsukahara et al. (2006) atRe= 128:5 andRe= 181:3, respectively. The underes-
+timation below y+30 (in the buer layer) is more pronounced, but the agreement
+abovey+>30 remains of similar magnitude (max error 4:94% forRe= 128:5 and
+8:39% forRe= 181:3). We hypothesize that this worsened agreement may be repre-
+sentative of the increasing scale separation with increased Reynolds number. Near-wall
+motions that can eectively be modeled as streamwise constant at low Reynolds numbers
+have an increasingly short streamwise wavelength relative to the motions that scale with
+outer length scale . That the zero-error location consistently occurs around y+= 20{
+30, commonly thought to be the upper boundary of the buer layer, is consistent with
+this scale separation argument. For the same reason, the lack of model resolution in the
+near-wall region will be exacerbated with increasing Rew. In robust control terms, this
+points once again to an increase in the model uncertainty near the wall versus the channel
+center. Once again, a better uncertainty model could be accomplished through the use
+of a `structured uncertainty' which would include an increase in noise in the near-wall
+region.
+4.3.3. Varying Noise Amplitude and Distribution
+A preliminary eort to introduce a non-uniform distribution of noise was carried out
+by repeating the simulation using a pseudospectral scheme with a Chebyshev interpolant
+for the wall-normal direction. This scheme naturally produces increased noise near theA Streamwise Constant Model of Turbulence in Plane Couette Flow 17
+(a)
+ (b)
+Figure 13. (a) Comparison of u+versusy+from 2D=3CModel using Chebyshev spacing in y
+with DNS data at Rew= 3000 based on G= 0:1991 (Re= 52) (b) Comparison of usw(y;z;t )
+from 2D=3CModel at for Case 1 ( Rew= 3000 with noise = 0:01,energy = 0:0882), and Case
+6 (Rew= 12800 with noise = 0:001,energy = 5:77e04) same grid and box size.
+walls. Cases Spec 1 and Spec 2 in Table 2 are two such simulations, both at Rew= 3000,
+withnoise = 0:001 andnoise = 0:002 respectively. Figure 13(a) shows the resulting
+mean velocity proles. Clearly the noise level is too low for Spec 1, however for Spec 2
+the maximum error occurs in the buer layer and is of the order 5{6%. The results of
+the spectral simulations indicate that by further noise shaping one could improve the
+agreement throughout the prole and across a range of Reynolds numbers.
+As previously discussed there is a strong relationship between the friction Reynolds
+number and noise. As an illustration of this, Figure 13(b) shows that one can obtain
+similar mean velocity proles at two dierent Reynolds numbers simply by adjusting
+the noise amplitude, i.e. a higher Reynolds number requires a smaller (uniform) noise
+amplitude to develop a mean velocity prole that is similar to that of a lower Reynolds
+number case with higher noise amplitude. This result is consistent with observations
+of higher transitional Reynolds number associated with \quiet" experiments compared
+to ones with high background disturbance levels. Alternatively, a xed amplitude noise
+produces a larger response (more blunting) at higher Rethan lower ones because dis-
+turbance amplication increases with increasing Reynolds number. This example makes
+it clear that the noise amplitude and the friction Reynolds number are tightly coupled,
+while giving further evidence that Reynolds number dependent wall-normal shaping of
+the noise would be required to get a better model representation of the turbulent mean
+velocity proles.
+4.3.4. Characterization of the (small) disturbance amplication
+The results described herein indicate that a very small amount of stochastic noise
+forcing limited to the cross-stream components produces a very large response which
+corresponds to a behaviour that is not a solution of the unforced equations. The ability
+of our model (Equation 2.1), which has a unique solution in the unforced case, to produce
+a new 
ow condition due to such a forcing supports the notion that the model is not
+robust to small disturbances/uncertainty. The potential for disturbance amplication is
+not new, in fact it comes directly from the features of the LNS previously discussed,
+however the creation and maintenance of the new 
ow state is dierent and cannot
+come through the use of a linear model. A simple characterization of the amplication18D. F. Gayme, B. J. McKeon, A. Papachristodoulou, B. Bamieh &J. C. Doyle
+maintained through the forced response of (2.2), or the lack of robustness (`fragility') of
+the system, can be formulated as follows.
+Dening the squared 2-norm of the streamwise component of (2.2) (i.e. ku0swk2) to be
+the increase in streamwise energy from the base (laminar) 
ow, a so-called amplication
+factor is given by,
+u=ku0swk2
+knoisek2(4.5)
+which is a measure of the output energy for a given input (noise forcing amplitude).  u
+is a nonlinear analog of the `ensemble energy density' described in previous studies of
+the input-output response of the LNS, e.g. (Bamieh & Dahleh 2001; Jovanovi c & Bamieh
+2005). Those investigations showed that the coupling between the Orr-Sommerfeld and
+Squire modes enables very large, Reynolds number dependent disturbance amplication.
+The amplication factor for case 3{5, which all have approximately the same input energy,
+are respectively  u680, u2200, and  u2920. These trends are consistent with
+the low Reynolds number scaling trends based on the OSS equations. This agreement
+re
ects the eective restriction of the streamwise constant assumption to amplication
+of thekx= 0 modes (most amplied in the linear equation) as well as the source of the
+amplication in the 2 D=3Cmodel: a coupling in a similar linear operator, given by
+@
+@t 
+u0
+sw
+="
+
1
+1
+Rew


+0
+@U
+@y@
+@z1
+Rew
# 
+u0
+sw
+: (4.6)
+In this way computing  ufrom the simulation of (2.2) is analogous to studying the steady
+state nonlinear response to the most amplied 3 Dmode, (i.e. the kx= 0 mode).
+Equivalent amplication relationships between the cross-stream velocity components
+andnoise could similarly be investigated.
+Structures with long streamwise coherence have long been shown to have a signicant
+role in both transition and fully developed turbulent 
ows. Based on these observations
+we study a streamwise constant projection of the Navier-Stokes equations for plane Cou-
+ette 
ow, the 2 D=3Cmodel. Simulation of this model under small amplitude Gaussian
+forcing captures the turbulent mean velocity prole at low Reynolds numbers. Appropri-
+ate Reynolds number trends are also reproduced.
+A weakly nonlinear, steady state analysis demonstrates that 2 Dstream functions can
+produce appropriate mean velocity distributions when they are nonlinearly coupled to
+the 2D=3Cstreamwise velocity. This indicates that `swirling motions' in the y{zplane
+produce features consistent with the mean characteristics of fully developed turbulence.
+It also provides evidence that the nonlinear coupling in the 2 D=3Cmodel is responsible
+for creating the well known characteristic `S' shaped turbulent velocity prole.
+The use of small amplitude stochastic forcing as an input to the 2 D=3C(nominal)
+model is based on ideas from robust control. Experimental observations are used to
+simplify the NS equations to form this nominal model and the noise forcing is used to
+capture both uncertain parameter values and unmodeled eects. The resulting forced
+2D=3Cmodel allows one to isolate phenomena that can not be decoupled from a full
+simulation of NS while maintaining a suciently rich description of the physics that
+govern turbulent 
ow. Our physics-based model should provide greater insight into the
+dynamics of the system than an empirical technique. Such a model may also allow better
+control design.
+The linearized 2 D=3Cmodel (4.6) maintains the properties responsible for large dis-
+turbance amplication which have also been linked to subcritical transition. MaintenanceA Streamwise Constant Model of Turbulence in Plane Couette Flow 19
+of these linear mechanisms is critical to the success of this approach. It is the combina-
+tion of these linear processes along with the momentum transfer from the two nonlinear
+terms in the streamwise velocity equation that enable the model to capture the blunted
+turbulent velocity prole. This line of inquiry provides a complementary perspective to
+transient growth and structurally based models, in that the 2 D=3Cmodel oers some
+improvement in analytic tractability at the expense of streamwise detail. The results are
+especially promising because the computational and analytical tractability of this model
+makes it well suited to higher Reynolds number studies.
+A natural extension of the present work would be the development of a more appropri-
+ate model for the noise distribution. It is common in the controls literature for a system
+to have a so-called structured uncertainty which is based on the physics of a particular
+system. In this work the limitation of noise to only the 
  equation represents a rst
+level of such an approach. Knowledge of the physics, for example that the near-wall re-
+gion is under-resolved in the 2 D=3Cmodel, is a rst step. Numerical or experimental
+studies aimed at characterizing true spatial noise forcing patterns would further help in
+determining the correct model for noise distribution.
+5. Acknowledgements
+The authors would like to thank H. Kawamura and T. Tsukahara for providing us
+with their DNS data. This research is sponsored in part through a grant from the Boeing
+Corporation. B.J.M. gratefully acknowledges support from NSF-CAREER award number
+0747672 (program managers W. W. Schultz and H. H. Winter).
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\ No newline at end of file
diff --git a/1001.0783.txt b/1001.0783.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ac2aed549a279b2fd28f85f761654e734e9de7df
--- /dev/null
+++ b/1001.0783.txt
@@ -0,0 +1,443 @@
+Berd | Recovery Swaps   
+ 
+ 
+Recovery Swaps 
+We derive an arbitrage free relationship between recovery swap rates, digital default swap 
+spreads and conventional CDS spreads, and argue that the fair forward recovery rate used in 
+recovery swaps must contain a convexity prem ium over the expected recovery value.  Arthur M. Berd 
+Lehman Brothers Inc. 
+ 
+ 
+RECOVERY SWAP CONTRACTS 
+Recovery swaps are the most recent innovation in the credit derivatives market. In this 
+contract, two counterparties agree to exchange th e realized recovery vs. the preset recovery 
+value (recovery swap rate) in case of default, with no other payments being made in any other scenario. The typical recovery swaps have no running or upfront payments of any kind. In 
+keeping with the swaps nomenclature, the co unterparties are denoted  as the “payer” and 
+“receiver” of the realized recovery rate, correspondingly. From a payer’s perspective, the 
+payoff will be positive when the realized recovery is less than the preset swap rate level. 
+[1]      
+⎩⎨⎧ −=] [otherwise 0maturity]  prior to default  of  case [in R R  Payoffrealized rate  swap
+Recovery swaps allow investors to eliminate the uncertainty of the future recovery payment 
+which is present in the conventional (i.e. floating recovery) CDS, whose protection payment 
+in case of default depends on the post-default price of the refere nce obligation. In a sense, one 
+might also call these contracts “recovery forw ards” because they operate similarly to, for 
+example, an FX forward contr act which allows investors to eliminate the uncertainty about 
+the future FX rate. Such a name would also empha size the fact that their payoff is linear in 
+terms of the future realized (i.e. “spot”) re covery rate. However, we will follow the lingo 
+which is closer to the CDS market and cont inue calling these contracts “recovery swaps”. 
+The market for recovery swaps is driven primarily by the hedging needs of synthetic CDO investors and other structured credit market participants. The reason why they need recovery swaps is that the pricing and risk management of these more complex securities depends 
+separately on the default event risk and recovery risk. This is in contrast with the on-the-run 
+(par) conventional CDS contracts whose value depends on the combination of the two risk factors – the default loss risk. 
+Consider, for example, a synthetic  CDO tranche. Most models th at price these tranches must 
+rely on some assessment of the joint risk-neutral default probability between various 
+underlying names. This joint default probability is a non-linear function of individual default 
+probabilities of each name and some measure of their correlation, usually incorporated via a 
+copula-based framework.  
+The individual implied default probabilities must be calibrated to the observed conventional CDS spreads. It is this calibration which induces a strong dependence between the assumed 
+recovery rate and the single-name implied de fault probability. The relationship between the 
+two can be loosely summarized by th e so-called “credit triangle” formula: 
+[2]     
+()  Recovery - 1  y Probabilit Default  Risk  Loss Default   Spread   CDS × = ≈  
+November 2004 1 Berd | Recovery Swaps   
+ 
+ 
+When calibrated to a given observed spread level, the implied default probability will be 
+higher for higher values of assumed recovery rate. This implied default probability will then 
+become an input into complex correlated default calculations used to price the synthetic CDO 
+tranche. The same recovery value will be applied in the very end when figuring out the loss distribution on the tranche, but the dependence has been broken – the loss distribution on the 
+tranche will no longer depend just on the default loss ratio, but separately on each individual 
+default probability and each recovery rate. C onsequently, the tranche values and risk 
+measures also depend both explicitly, and implicitly, via calibration, on the recovery rate 
+assumptions for underlying names. 
+In fact, this dependence is not unique to the tranche products, but is generally the case for 
+most structured credit products and even of f-the-run conventional CDS with non-zero mark-
+to-market and non-par cash credit bonds. In all th ese cases there are so called “digital” default 
+risks which are characterized by a given fixed do llar loss in case of default. For example, in 
+case of an off-market CDS with positive mark-to- market the current present value on the CDS 
+represents a digital default risk because th is unrealized gain will disappear without 
+compensation if the underlying name defaults immediately. 
+For many years, there existed a small niche in the credit derivatives market which catered to 
+clients who needed to hedge such digital risks. The so-called digita l default swaps (DDS) 
+differ from the more conventional liquid CDS by pre-setting contractually the value of the 
+protection payment in case of default. This contractual protection payment is usually quoted 
+in terms of “contractual recovery” rate, which could be set at any number. If it was set at zero, 
+then a protection buyer would get a payment equal to the full notional in case of default. If it was set at 50%, then the protection buyer would get a payment equal to half of the notional.  
+One of the main uses of DDS was the risk ma nagement of corporate accounts receivables, 
+operational risks and other such exposures whose value is not naturally tied to the price of the 
+underlying issuer’s bonds or traded loans. Unlike asset managers or banks who need to hedge 
+the losses on a corporate bond or loan portfolios and thus find the conventional floating rate 
+CDS the most natural proxy, the corporate treas urers who need to he dge the specific dollar 
+exposure to a trading counterparty have found the DDS to be better suited (even though often 
+not liquid enough due to a niche nature of that market).  
+With the advent of the recovery swaps the toolkit for separate hedging of credit risks is 
+becoming more complete: 
+• Default loss risk (i.e. full credit ri sk with mark-to-market exposure):  hedged by 
+conventional (floating recovery) CDS 
+• Default event risk:  hedged by digital default swaps, i.e. DDS 
+• Recovery rate risk:  hedged by recovery swaps 
+• Credit risk without ma rk-to-market exposure:  hedged by constant maturity 
+default swaps (CMDS) 
+• Spread volatility risk:  hedged by (short-term) credit default swaptions 
+The conventional CDS still remain the linchpin of the credit derivatives market and continue 
+to account for the majority of all traded notionals, according to the most recent survey by the 
+British Bankers Association. But the existence of the expanded toolkit also implies certain 
+connections and (partial) substitution ability between various credit derivatives instruments.  
+November 2004 2 Berd | Recovery Swaps   
+ 
+ 
+ARBITRAGE-FREE PRICING OF RECOVERY SWAPS 
+Recovery swaps can be fully replicated by a combination of conventional and digital CDS. 
+This replication, as usual, implies an ar bitrage-free relationship between these three 
+instruments which we will derive below.  
+Consider a zero-investment composite trade depicted in Figure 1: a payer recovery swap with 
+the fixed swap rate R(swap), and a long-short position where an investor bought a digital 
+default protection with contractual recovery  and sold protection via conventional CDS.  DDSR
+The hedge ratios for the DDS and CDS should be chosen so that the net cash flows are zero in 
+all scenarios. Consider the case of default: 
+[3]     () ( )()realized CDS DDS DDS CDS swap default R H R H H R CF⋅ − +−⋅+−= 1 1  
+In order to guarantee that these cash flows are equal to zero regardless of the realized 
+recovery rate, one must have: 
+[4]     
+⎪⎩⎪⎨⎧
+−−==
+DDSswap
+DDSCDS
+RRHH
+111 
+Because the cash flows in case when the trades  mature without default are identically zero, 
+and because the replication requirement for th e case of default uniquely fixes both hedge 
+ratios, then the comparison of the premium cash  flows results in an arbitrage-free relationship 
+between the recovery swaps, CDS and DDS.  
+[5]     ()CDS
+swapDDS
+DDS DDS SRRR S ⋅−−=11 
+Indeed, if the quoted DDS spread were less (gr eater) than this value, then this trade would 
+have positive (negative) premium cash flows and zero cash flows otherwise,  i.e. there will be 
+an arbitrage.  
+Recalling that the spread for a DDS with zero co ntractual recovery encodes the term structure 
+of implied hazard rates and interest, we can interpret equation [5] as a fully consistent 
+generalization of the widely quoted “credit triangle” formula ()swap CDS R S h− = 1  for 
+arbitrary term structures of interest and hazard rates.  
+Figure 1. Recovery Swap replication using Digital and conventional CDS 
+ Notional Premium Payoff in case of 
+default Payoff when no 
+default 
+Payer Recovery Swap  1   -   R(swap)-R  -  
+Buy: Digital CDS  H(DDS)   -S(DDS)  1 - R(DDS)  - 
+Sell: Conventional CDS  H(CDS)   +S(CDS)  - (1-R)   -  
+Net Payments    H(CDS)*S(CDS)-
+H(DDS)*S(DDS)  R(swap)-R          
+-H(CDS)*(1 – R) + 
+H(DDS)(1 - R(DDS))  -  
+ 
+November 2004 3 Berd | Recovery Swaps   
+ 
+ 
+November 2004 4 Based on this understanding, it would be natural to think that the recovery swap rates should 
+be used for calibration of the implied hazard ra tes in the risk-neutral  framework. To prove 
+this, assume that we have observed the term  structure of CDS spre ads and recovery swap 
+rates for a range of maturities [0, T]. According to [5] we can obtain without any further assumptions the term structure of  DDS spreads with zero contractual recovery. The pricing of 
+such DDS depends solely on the term structure of risk-neutral survival probabilities (and possibly on the correlation of the default process and the risk-free discount factor). 
+In a very general form, the pricing of a zero contractual recovery DDS is given by: 
+[6] 
+()(){}{}{}{}∫ ∫ + ≤ < < ⋅ Ε = ⋅ Ε ⋅ ⋅ ⋅−T
+tdu u u u tT
+tu u t CDS
+swapI Z I Z du T t ST t Rτ τ ,, 11 
+Here, we have substituted the no-arbitrage DDS spread in terms of the market-observed CDS 
+spreads and recovery swap from eq. [5]. The variable τ denotes the (random) default time, 
+ denotes an indicator function for a random event X,  is the (random) credit risk-free 
+discount factor, and {}XIuZ
+{}•Εt denotes the expectation under th e risk-neutral measure at time t. 
+For convenience in notations, we have adopted an approximation of continuous DDS 
+premium payments (this approximation is irrelevant for the subsequent discussion). 
+Under the assumption of independence between the default process and the risk-free rates, the 
+expectations of the discount factor and the default indicator factor out. We have 
+, where () {u tZ u t ZΕ = , } ()u t Z,
+⎜⎝⎛−∫u
+th is the (non-random) riskless discount factor (i.e. the 
+prices of the zero-coupon credit risk-free bonds). Introducing the term structure of survival 
+probabilities and hazard rates accord ing to the standard reduced -form valuation methodology 
+, we get: (){}Ε =<u tI u t Q ,τ{} ()⎟⎠⎞⋅ = ds s exp
+[7]     ()() ()() ()( )() ∫ ∫⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅−T T
+CDS
+swapu t Q u h u t Z du u t Q u t Z du T t ST t R0 0, , , , ,, 11 
+One can easily calibrate the term structure of  implied hazard rates to the market-observed 
+CDS spreads and recovery swap rates from [7 ] without making any additional assumptions 
+about the recovery and default process. For example, in the case of flat CDS spreads, 
+recovery swap rates and hazard rates one immediately obtains the “credit triangle” 
+()swap CDS R S h− = 1 . 
+We see that the calibration procedure for the im plied hazard rates is completely independent 
+of our assumptions regarding the correlation between the recovery rates and default events. 
+While there is plenty of empiri cal evidence of a negative correlation between default rates and 
+recovery rates (see Altman et. al . [2002]), one can prove from [7] that the calibrated implied 
+hazard rate has the exact opposite behavior – it is a growing function of recovery swap rates, 
+given a level of observed CDS spreads. This is particularly obvious in the case when the 
+credit triangle formula applies.  
+Note also that in our derivation of the equations [6] and [7] we did not make an assumption 
+about the existence of a liquid DDS market – it is sufficient to assume that the CDS and 
+recovery swaps markets exist, and price the hypothetical DDS based on the known arbitrage-
+free relationship between the DDS and CDS spreads [5].  Berd | Recovery Swaps   
+ 
+ 
+As we explained, the recovery swaps are typica lly traded in a “par” format, i.e. with no 
+upfront or premium payments. The present value of such a swap is  zero at inception. 
+However, its present value will generally be not zero after the inception as the market 
+recovery rates fluctuate and m ove away from the contract fixe d rate. Consider a (receiver) 
+recovery swap with a contractual swap rate , and assume that the current market rate 
+for the recovery swap of the same maturity is . The present value of this swap is: swapR
+mktR
+[8]  (){} {}
+{}{}{}{}
+(){}{}∫∫ ∫∫
++ ≤ <+ ≤ < + ≤ <+ ≤ <
+⋅ Ε ⋅ − =⋅ Ε ⋅ − ⋅ ⋅ Ε =⋅ ⋅ − Ε =
+T
+tdu u u u t swap mktT
+tdu u u u t swapT
+tdu u u u u tT
+tdu u u u swap u t receiver
+I Z R RI Z R I Z RI Z R R PV
+ττ ττ
+where  is the (random) realized recovery rate in case of default. The last line in this 
+equation is obtained by noting that the first term in the second line is related to the current 
+market recovery rate, which is the rate that would have made the present value equal to zero. uR
+Comparing the last line in [8] with the equation [6] and recalling the definition of the risky PV01 for conventional CDS (see O’Kane and Turnbull [2003]), we finally obtain: 
+[9] 
+()mkt CDS CDS
+mktswap mkt
+receiver R S RPV SRR RPV , 011⋅ ⋅−−=  
+Here we have explicitly acknowledged that the RiskyPV01 is a function of the observed 
+current CDS spreads and recovery swap rates since it depends on the implied hazard rates 
+which in turn must be calibrated to  and  as discussed previously. The present 
+value of a payer recovery swap is obviously given by the same formula with an opposite sign. 
+Note that we have not made any assumptions about the dependence between the (risk-neutral) 
+interest rates, realized recovery rates and defaults in deriving the equation [9]. CDSSmktR
+THE CONVEXITY PREMIUM 
+Practitioners often use the empirical expected recovery rate (e.g. the long-term average rate of 40%) in calibration of the implied survival probabilities. We now know that it is the market-
+observed recovery swap rates that must play the role of the risk-neutral recovery parameter when relating the DDS and CDS spreads [5] and when calibrating the implied hazard rates to 
+CDS spreads [7]. However, in the absence of a liquid two-way recovery swap market the price discovery for recovery rates is still in its infancy and one might ask a question about how different should the “fair” recovery swap ra te be from the expected average (whether this 
+means expected under real or risk-neutral probability measure). 
+Equation [9] reveals an important feature of the recovery swaps that might shed some light on 
+the answer to this question. We see that the dependence on the market recovery rate is not 
+linear – the market rate enters not only the difference term in the numerator which reflects the 
+contract payoff structure, but also the denominator and the risky PV01 factor. The latter two dependencies reflect the calibration feedback, wh ich can also be interpreted as a convexity 
+effect if the recovery swap rates are assume d to be equal to expected recovery rates. 
+November 2004 5 Berd | Recovery Swaps   
+ 
+ 
+November 2004 6 Consider what would happen to the mark-to-market  of a receiver recovery swap if the market 
+recovery rate dropped while th e market CDS rates remained unchanged. The investor would 
+record a negative mark-to-market as she woul d expect that the (floating) recovery payment 
+will be now less than previously fixed contract  recovery rate which she would pay out in case 
+of default. At the same time, the implied default rate calibrated to the unchanged CDS spreads 
+will now be lower, and the expected likelihood of  default prior to maturity will also decrease, 
+partially offsetting the loss. Similarly, an upward revision to the expected  recovery rate would 
+produce a higher implied default probability and a greater likelihood of a positive payoff.1 
+Thus, the investor in a receiver recovery swap  benefits from both up and downward changes 
+in the market recovery rate co mpared to the contractual linear payoff estimate – which is a 
+manifestation of the recovery convexity. Being long such convexity has a positive value, therefore the fair value of the sw ap rate at which the investor can agree to deal is somewhat 
+higher than the expected future rate. In the rest of the paper we will try to estimate this 
+convexity correction in a heuristic manner. 
+Let us consider a receiver recovery swap with inception at time t, maturity at time T at some 
+intermediate time t<u<T  prior to expiry. The (zero) present value of the par recovery swap is 
+equal to the present value of the payoff scenarios when the default happens prior to the time u 
+plus the discounted present value of the remaining “live” recovery swap mark-to-market 
+values averaged over the ra ndom market recovery rates  and CDS spreads 
+ at time u. The first portion is simply equal to the present value of a recovery 
+swap with a shorter maturity u estimated at the initial date t, which leads us to:  (T u R,)
+)(T u SCDS ,
+[10]    
+() ()
+()() ()()()
+() ()
+()() () ()()
+⎭⎬⎫
+⎩⎨⎧< ⋅ ⋅−−⋅ Ε +⋅ ⋅−−=
+τu T u R T u S T u RPV T u ST u RT t R T u RZu t R u t S u t RPV u t Su t RT t R u t R
+CDS CDSswap
+u tCDS CDS
+swapswap swap
+, , , , 01 ,, 1, ,, , , , 01 ,, 1, ,0
+ 
+It is difficult to draw a specific conclusion from this relationship in full generality. However, 
+if we make certain simplifying assumptions we may be able to estimate the convexity effect. Let us assume that the interest rates are conditionally independent of the market recovery 
+swap rates and CDS spreads given the survival up to time u. Let us also ignore the 
+dependence of the risky PV01 on the level of spreads and recovery rates – both of these 
+dependencies are of second order of magnitude compared to the primar y dependence captured 
+by the pre-factors in front of the risky PV01. Finally, let us assume that the term structure of recovery swap rates is flat. Under these assumptions we get the following solution: 
+[11]  
+()()
+()()
+()()() (){}
+(){} ττ
+ττ
+< Ε< ⋅ Ε=
+⎭⎬⎫
+⎩⎨⎧< ⋅−Ε⎭⎬⎫
+⎩⎨⎧< ⋅−Ε
+=u T u Su T u S T u R
+u T u ST u Ru T u ST u RT u R
+T t RDDS tDDS t
+CDS tCDS t
+swap,, ,
+,, 11,, 1,
+,00
+ 
+                                                           
+1 As noted in the previous section,  these statements are not at all dependent  on the assumptions about the correlation 
+between the recovery rate and default events. Berd | Recovery Swaps   
+ 
+ 
+The second equation is expressed in terms of zero recovery DDS spreads, using [5]. 
+The equation [11] potentially depends on the time u on the right hand side, but not on the left 
+hand side – this is an artifact of our simplifying assumptions, ignoring small second order 
+effects. What this means is that we must estimate the equation [11] at a “typical” intermediate time before maturity and assume (or verify) that the time dependence is very mild indeed. 
+One can immediately see from [11] that the re lationship between the recovery swap rates and 
+risk-neutral expected recovery rates does in fact strongly depend on our assumptions about 
+the co-movement between the market recovery  rates and spreads under the risk-neutral 
+measure. For example, one could make an a ssumption that the zero-recovery DDS spreads 
+and recovery rates are conditionally independent given survival until u, and immediately 
+obtain a simple answer which would of course be fully consistent with the assumption of both 
+flat recovery rate term structure and of mild (or no) dependence on intermediate time u. 
+[12]     
+() (){ }τ< Ε = u T u R T t Rt swap , ,  
+However, such an assumption would go against the grain of the market behavior as we know 
+it. In fact, the DDS market is practically non-existent and if anything, the DDS spreads are set (not independently discovered) by market makers using the relationship [5] with the liquid 
+CDS spreads and the (estimates of or market values of) recovery rates as primary inputs. 
+Given this state of affairs, and also the f act the of the three types of securities, the 
+conventional CDS are by far the most liquid and the other two (DDS and recovery swaps) are 
+niche markets with limited price discovery, we believe that it is more meaningful to assume 
+that the CDS spreads are independent of the market recovery swap rates, conditionally on 
+survival until u. Under such an assumption, the relationship between the “fair” recovery swap 
+rates today and their expected values in the future is more complex: 
+[13]     
+()()
+()
+()()1
+, 111
+, 11, 1,
+,−
+⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+⎭⎬⎫
+⎩⎨⎧<−Ε − =
+⎭⎬⎫
+⎩⎨⎧<−Ε⎭⎬⎫
+⎩⎨⎧<−Ε
+=τ
+ττ
+uT u RuT u RuT u RT u R
+T t Rt
+tt
+swap  
+If we further assume a stationary  distribution of recovery swap rates (which could follow, for 
+example, from a mean-reverting dynamics and would be consistent with the requirement of 
+the mild time dependence), then we can calculate  the “fair” recovery swap rate based on the 
+parameters of such a distribution. For example, assuming that the (risk-neutral) distribution of 
+future market recovery rates is close to a normal with a mean R and a (relatively narrow) 
+standard deviation Rσ, we get: 
+[14]     ()
+()RR
+RRT t RR
+Rswap−+ ≈
+−+−− ≈1
+1111 ,2
+22σ
+σ 
+The second term on the right hand side is the amount by which the “fair” recovery swap rate 
+exceeds the expected recovery rate  – i.e. the convexity premium. 
+The convexity premium for the recovery swap rate  is intimately related with the DDS digital 
+premium discussed in Berd and Kapoor (2002) where we have argued that the fair DDS 
+November 2004 7 Berd | Recovery Swaps   
+ 
+ 
+November 2004 8 spread must be somewhat higher than what one would obtain using the expected (no-
+premium) recovery rates in [5]. Substituting [14]  into [5] we obtain the “fair” DDS spread, 
+which coincides with our estimate in the cited paper2: 
+[15]     ()()⎟⎟
+⎠⎞
+⎜⎜
+⎝⎛
+−+ ⋅−−⋅ =22
+1111
+R RRS R SR DDS
+CDS DDS DDSσ 
+The uncertainty about the recovery value can be quite high, in the 10-20% range, even when 
+conditioned on the current state of the economy (and much higher unconditionally over long 
+periods of time). For the typical levels of expected recovery rate of senior unsecured debt  (around 40%) and recovery uncertainty (around 15%), the estimate for the recovery swap rate 
+is 0.4+0.15
+2/(1-0.4)=43%, i.e. quite a bit higher than the 40% from which we started. Thus, 
+the convexity premium effect can have a significant impact on the pricing of both recovery 
+swaps and digital default swaps.  
+Market practitioners recognize the no-arbitrage relationship [5] between the recovery swaps, 
+CDS and DDS, as can be seen from the Figure 2. There, we show recently quoted mid-levels 
+for recovery swap rates, CDS and DDS (with contractual R=0) spreads. We also show the 
+“implied” recovery rate calculated from the comparison of the CDS and DDS spreads using 
+the equation [5]. We can see that the implied recovery swap rate is in line with the quoted 
+recovery swap rate (and is well inside the quoted bid-offer spread).  
+It is not possible to judge from the Figure 2 whether or not the ma rket recognizes the 
+convexity premium in recovery swap rates, b ecause the “expected recovery rate” is not an 
+observable quantity. However, investors who use such quotes in conjunction with their own 
+estimates of the expected recovery rate (and who agree with our assu mptions concerning the 
+dynamics of market recovery rates and CDS spreads) should take the convexity premium 
+effect into account when deciding whether th e recovery swap offers relative value. In 
+particular, a recovery swap rate that is higher than the investor’s projection of the expected rate may or may not be a sufficient indication of a relative value depending on the estimates 
+of the future uncertainty of the recovery rates,  in accordance with [14]. However, a recovery 
+swap rate that is lower than the investor’s pr ojection would, in our opinion, constitute such a 
+relative value because we know that the conv exity premium is positive. To monetize this 
+opinion, the investor would buy a receiver rec overy swap (i.e. pay a lower fixed rate). 
+Figure 2. Examples of quoted recovery swap, CDS and DDS rates 
+Ticker Recovery 
+Swap Rate (%) CDS   
+(bp) DDS(R=0) 
+(bp) Implied R 
+(%) 
+CA 34.5 80 122 34.8 
+FMC 39.5 174 289 39.7 
+GECC 38.0 26 42 38.6 
+T 37.0 242 386 37.4 
+                                                           
+2 The relative DDS spread convexity used in Berd and Kapoor (2002) can be estimated from the “no-premium” DDS 
+spread by taking its second derivative with respect to  mean recovery rate and leads precisely to [15]  
+()()2
+const221
+12
+11
+11
+R RRS
+R RRS R
+CDSSDDS
+CDSDDS
+CDS−= ⎟
+⎠⎞⎜
+⎝⎛
+−−⋅
+∂∂⋅⎟
+⎠⎞⎜
+⎝⎛
+−−⋅ ≈
+=−
+γ  Berd | Recovery Swaps   
+ 
+ 
+November 2004 9 In conclusion, we note that if or when recovery swaps become much more liquid, we would 
+regard the market-observed recovery swap ra tes to be the “true” implied recovery value 
+subsuming the convexity premium effects, and use it for calibration of implied default rates from market-observed CDS spreads – thus bringing the pricing framework for credit derivatives firmly into an arbitrage-free setting. There will be no need to worry about the 
+differences between the fair recovery swap  rate and the expected recovery rate.  
+In fact, when independent price discovery in recovery swap rates and CDS spreads becomes 
+reality, we can test the independence between these rates instead of simply assuming it, and 
+perhaps use the more general version of the e quation [11] to estimate the relationship between 
+the fair recovery swap rates and the parameters  of the joint distribution of future market 
+recovery rates and CDS spreads. The equation [5] would continue to govern the relationship between the CDS, DDS and recovery swaps. 
+Acknowledgments:  I would like to thank Roy Mashal, Lutz Schloegl and especially Marco 
+Naldi for very valuable comments and discussions. 
+REFERENCES 
+Altman, E. I., B. Brooks, A. Resti and A. Sironi (2002):  The Link between Default and 
+Recovery Rates: Theory, Empiri cal Evidence and Implications , working paper, NYU Stern 
+School of Business Berd, A. and V. Kapoor (2002):  Digital Premium , Journal of Derivatives, vol. 10 (3), p. 66 
+O’Kane, D., and S. M. Turnbull (2003):  Valuation of Credit Default Swaps , Quantitative 
+Credit Research Quarterly, vol.  2003-Q1/Q2, Lehman Brothers 
+ 
\ No newline at end of file
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new file mode 100644
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@@ -0,0 +1,1112 @@
+arXiv:1001.0784v2  [hep-th]  19 Jan 2010Faddeev-Popov spectra atthe Gribovhorizon
+J. Greensite
+Physics and Astronomy Dept., San Francisco State Universit y, San Francisco, CA 94132, USA
+(Dated: October 30, 2018)
+IpresentaperturbativecalculationofthespectrumoftheF addeev-PopovoperatorinCoulombgaugeinthree
+dimensions, and Landau gauge in two and three dimensions, wi th an ansatz for the gluon propagator as the
+non-perturbative input. The results show how the low-lying Faddeev-Popov eigenvalue spectrum ismodifiedas
+the firstGribovhorizon isapproached, and how the spectra ca n differinCoulomb and Landaugauges.
+I. INTRODUCTION
+The low-lying spectrum of the Faddeev-Popov(F-P) oper-
+ator, in Coulomb and covariant gauges, is a probe of the in-
+fraredpropertiesof non-abeliangaugetheories. Confineme nt
+in Coulomb gauge, in particular, is rather directly related to
+the F-P spectrum. The colorCoulombpotential,for example,
+involvesa productof inverse F-P operators, and the Coulom-
+bic self-energy of an isolated color charge, which is infrar ed
+divergentin a confiningtheory,dependscrucially on the den -
+sityoflow-lyingeigenvaluesoftheF-Poperator,asdiscus sed
+below. The connectionto confinement is less apparent in co-
+variant gauges, although the density of near-zero F-P eigen -
+modes is expected to be relevant to the infrared behavior of
+theghostpropagator.
+Coulomb and Laudau gauges are defined on the lattice as
+thesetofelements,oneachgaugeorbit,forwhichthequanti ty
+R[U]=−∑
+xd
+∑
+µ=1Tr[Uµ(x)] (1.1)
+is stationary with respect to infinitesimal gauge transform a-
+tions. Here ddenotes the number of space dimensions, in
+Coulomb gauge, and the number of spacetime dimensions in
+Landau gauge, which is a convention I will adopt from now
+on. In general,alongany gaugeorbit, there are manystation -
+ary points, known as Gribov copies, and at these points the
+F-P determinant may be positive or negative. This indefinite
+sign is closely to related to Neuberger’s Theorem [1], which
+demonstrates that BRST quantization of any non-abelian lat -
+tice gauge theory is ill-defined at the non-perturbative lev el.
+The picture is that in summing over all copies on a gauge or-
+bit,thecopieswithapositiveF-Pdeterminantareexactlyc an-
+celled by the copies with a negativeF-P determinant,and the
+functional integral vanishes. It is for this reason that a co n-
+straintof somekind,imposedonthe domainof functionalin-
+tegration,isnecessary. Ideallytherangeoffunctionalin tegra-
+tion should be a subspace (such as the FundamentalModular
+region)containingonlya singlegaugecopywithpositiveF- P
+determinantpergaugeorbit,butataminimumtheintegratio n
+range should lie within the Gribov region. This is the region
+which consists of all gauge copies in which the non-trivial
+eigenvaluesoftheF-Poperatorareallpositive;i.e.theGr ibov
+copieswhicharelocalminimaof R[U]. Theseare,infact,the
+configurationsobtainedbystandardlattice gauge-fixingal go-
+rithms. The Gribov region is completely bounded, and the
+FundamentalModularregionispartiallybounded,bythefirs tGribov horizon, where the lowest non-trivial F-P eigenvalu e
+vanishes. It has been argued by Zwanziger [2] that the vol-
+umeoftheGribovregionisconcentratedclosetothehorizon ,
+much as the volume of a sphere in a high dimensional Eu-
+clidean space is concentrated near the surface. Since the di -
+mensionality of the space of all lattice configurations is ve ry
+high indeed, the values of observables obtained at the Gri-
+bov horizon should dominate the expectation value. It would
+thenbeinterestingtounderstandexactlyhowproximitytot he
+Gribov horizon affects the behavior of various observables ,
+startingwiththespectrumoftheF-P operator.
+As a step in that direction, this article presents a pertur-
+bative calculation of the F-P spectra. Perturbation theory is
+not necessarily trustworthy when dealing with the low-lyin g
+eigenmodes, but something may still be learned from it. In
+particular, it would be interesting to see whether proximit y
+to the Gribov horizon changes the behavior of the low-lying
+spectra already at the perturbativelevel, and whether that be-
+havior is different, for some reason, in Coulomb and Landau
+gauges. The calculation is carried out for Landau gauge in
+two and three spacetime dimensions, and Coulomb gauge in
+three spacetime dimensions, to avoid the complicationsass o-
+ciated with renormalization in four dimensions.1The prox-
+imitytotheGribovhorizoniscontrolledbyamassparameter
+in the transverse gluon propagator, which is where the non-
+perturbativeinformation enters. I use an ansatz for the glu on
+propagator, motivated by Gribov’s expression [4], which al -
+lowsforanydesiredpowerbehaviorin theinfrared.
+II. F-PEIGENVALUESAND THECOULOMB
+SELF-ENERGY
+The Coulomb potential between a static quark-antiquark
+pairlocatedat points xandyisgivenbytheexpression
+VC(|x−y|)=−g2Cr
+dA/angbracketleftBig
+(M−1)ab
+xz(−∇2
+z)(M−1)ba
+zy/angbracketrightBig
+(2.1)
+1Yang-Mills theory in Coulomb gauge is trivial in two spaceti me dimen-
+sionsifthespacetimemanifold isflatandnon-compact, andf orthatreason
+Coulomb gauge in D=2 will not be considered here. Non-trivial features
+associated with the Coulomb gauge F-P operator do appear eve n in two
+dimensions, if the space direction is compactified to S1, and this case has
+been thoroughly discussed by Reinhardt and Schleifenbaum i n ref. [3].2
+whereCristhequadraticCasimirofquarksincolorrepresen-
+tationr,dAis the dimension of the adjoint representation of
+thegaugegroup,and MistheFaddeev-Popovoperator,which
+is
+Mab
+xy=/parenleftBig
+−δab∇2+gfabcAc
+i(x)∂i/parenrightBig
+δ3(x−y)(2.2)
+in the continuum. If VC(|x−y|)is confining, then this can
+only be attributed to an infrared singular behavior of M−1,
+which must be related somehow to the low-lying F-P eigen-
+valuespectrum.
+The perturbativeevaluationofthe F-P spectrumstarts with
+thefree-field, g2=0case,onafiniteperiodiclatticeofexten-
+sionL. TheeigenmodesofthecorrespondingF-Poperatorare
+simplytheplanewavestates
+φa(0)
+n,A=1√
+Veip·xχa
+A
+λ(0)
+n,A=2∑
+µ(1−cos(pµ))
+pi=2πni
+L,−L
+2<ni≤L
+2(2.3)
+and the/vectorχAare some set of orthonormal vectors spanning the
+dA-dimensional color space. The F-P eigenmodesand eigen-
+values{φn,A(x),λn,A}atg2>0 are also indexed by (n,A),
+denoted for brevity by n≡(n,A), and it is assumed that the
+eigenmodesandeigenvaluesarecontinuousanddifferentia ble
+functionsof g2,whichsmoothlyapproach(2.3)as g2→0. To
+connectthe eigenmodespectrumto the Coulombself-energy,
+we beginwiththeexpression
+Eself=g2Cr
+dAlim
+V→∞1
+V/angbracketleftBig
+(M−1)ab
+xz(−∇2
+z)(M−1)ba
+zx/angbracketrightBig
+(2.4)
+andinsertingthespectralrepresentation
+(M−1)ab
+xy=∑
+nφa
+n(x)φ∗b
+n(y)
+λn(2.5)
+thisbecomes[5]
+Eself=g2Cr
+dAlim
+V→∞1
+NcV∑
+n/angbracketleftbigg(φn|−∇2|φn)
+λn/angbracketrightbigg
+=g2Cr
+dA/integraldisplayλmax
+0dλ/angbracketleftbigg
+ρ(λ)(φλ|−∇2|φλ)
+λ2/angbracketrightbigg
+(2.6)
+whereρ(λ)isthenormalizedeigenvaluedensity
+ρ(λ)=lim
+V→∞1
+NcV∑
+nδ(λ−λn) (2.7)
+Ncis the number of colors, and V=Ld. In 2+1 dimensions,
+the integral in (2.6) is logarithmically divergent at the λ→0
+end of the integration even in an abelian theory, and this is
+because the Coulomb potential in an abelian theory confineswith a logarithmically rising potential. The criterion tha t the
+infrared divergence in the self-energy is stronger than log a-
+rithmicis
+lim
+λ→0/angbracketleftbiggρ(λ)(φλ|−∇2|φλ)
+λ1−ε/angbracketrightbigg
+>0 forsome ε>0 (2.8)
+Thisconditioninvolvesthenear-zeroF-Peigenmodes,aswe ll
+as the eigenvalues. However, assuming that the eigenvalue
+spectrum is non-degenerate (apart from some rather special
+casesinvolvingsymmetricgauge-fieldconfigurations),the nat
+fixedg2>0eachλnisassociatedwithaunique (n,A),which
+in turn determines p. Thenp2=p2(λ)in the infinite volume
+limitand
+lim
+λ→0/angbracketleftbiggρ(λ)(φλ|−∇2|φλ)
+λ1−ε/angbracketrightbigg
+≥lim
+λ→0/angbracketleftbiggρ(λ)p2(λ)
+λ1−ε/angbracketrightbigg
+(2.9)
+The proof of this inequality is given in the Appendix. It fol-
+lowsthat asufficientconditionforCoulombconfinementis
+lim
+λ→0/angbracketleftbiggρ(λ)p2(λ)
+λ1−ε/angbracketrightbigg
+>0 forsome ε>0 (2.10)
+III. THEAPPROACHTOTHEHORIZON
+It was stated above that numerical simulations find local
+minima of R, which means, strictly speaking, that all of the
+eigenvalues of the F-P operator are positive. This statemen t
+hastobequalifiedalittle. EvenapartfromGribovcopies,th e
+CoulombandLandaugaugeconditionsdonotentirelyfix the
+gauge,becauseif Uµ(x)satisfies the gaugecondition,so does
+GUµ(x)G†, whereG∈SU(N) is any position-independent
+groupelement. Thisisaremnantglobalgaugesymmetry,and
+it implies that at any stationary point of Rthere must be flat
+directionsalongthe gaugeorbitcorrespondingto zeromode s
+oftheF-P operator. Thesearethe trivialeigenmodes
+φa
+0,A(x)=1√
+Vχa
+A (3.1)
+The statement that the F-P determinant is positive in the Gri -
+bov region really refers to the determinant of the operator i n
+thesubspaceorthogonaltothese trivialzeromodes.
+Outside the Gribov region, some of the non-trivial F-P
+eigenvalues become negative, which means that for config-
+urations which lie exactly on the Gribov horizon there must
+be at least one non-trivial zero eigenvalue. However, in an
+infinite volume, the converse is not necessarily true: we can -
+not deduce, just from the fact that the spectrum of non-trivi al
+eigenvaluesbeginsatzero,thatthegaugefieldliesontheGr i-
+bov horizon. Even in an abelian theory, which has no Gribov
+horizon, the spectrum of the F-P operator −∇2in an infinite
+volumebeginsat λ=0.
+Let us begin with g2=0, i.e. a free-field theory, with the
+eigenvalues and eigenstates shown in (2.3). In this free cas e3
+     
+      λ
+pdH<0
+dH=0
+dH>0
+(a) TypeI scenario.     
+        λ
+pdH<0
+dH=0
+dH>0
+(b) TypeII scenario.
+FIG.1: Twoscenarios forthe behavior ofthe FPeigenvalue sp ectrum forgauge fieldconfigurations near the firstGribovhor izon. Justoutside
+the Gribov region ( dH<0), there is a small interval of negative eigenvalues, which shrinks to a single eigenvalue at the horizon ( dH=0). In
+the Type Iscenario, the interval of negative eigenvalues be gins atp=0;at the horizonthe nontrivial zero-mode isat p=0,and atsmall pthe
+eigenvalues growwithanon-standardpower λp∼p2+s. ForconfigurationsinsidetheGribovregion( dH>0)thegrowth λp∼p2isquadratic.
+In the Type II scenario, for configurations just outside the G ribov region, the interval of negative eigenvalues does not includep=0, and at
+the Gribovhorizon the non-trivial zero mode is at |p|>0.
+we have2
+ρ(λ)∝λ(d−2)/2,(φλ|−∇2|φλ)=λ(3.2)
+so that with an ultraviolet regulator,the Coulombself-ene rgy
+ind+1 dimensions is finite for all space dimensions d≥3,
+and marginally divergent (divergent as log (L)as extension
+L→∞) atd=2. The latter divergence is expected, since
+the Coulomb potential increases logarithmically in 2 +1 di-
+mensions, so the question in 2+1 dimensions is whether the
+conditionin eq.(2.8)is satisfiedforsome ε>0
+Outside the Gribov region, some of the non-trivial F-P
+eigenvalues become negative, and approaching the first Gri-
+bov horizon from the outside, the range of negativeeigenval -
+uesshouldshrinkaway. Rightonthehorizontheremustexist
+a non-trivial zero eigenvalue even for a finite spacetime vol -
+ume. Soletusimagineincreasing g2awayfromzero,andalso
+placingaconstraintinthefunctionalintegralbyintroduc inga
+dimensionful parameter dH, and requiring that if dH>0, the
+integrationisovergaugefieldsinsidetheGribovregion,ly ing
+a distance dHfrom the first Gribov horizon, while if dH<0,
+the integrationis overgaugefields outsidethe Gribovregion,
+at adistance |dH|fromthe horizon. Then
+/an}bracketle{tλp/an}bracketri}ht=λ(0)
+p+/an}bracketle{tΔλp/an}bracketri}ht
+=p2(1−F[g,p,dH]) (3.3)
+Weusepasanindexbecause,intheinfinitevolumelimit,itis
+bettertoreplacetheintegerindex nbythecontinuousindex p.
+2Tocompute ρ(λ),beginwiththevolumemeasureinmomentumspace,pro-
+portional to pd−1dp,and change variables to λ=λ(p)to arrive at ρ(λ)dλ.Also the expectation value of λp,Acan depend on neither the
+indexA, since this would violate global color symmetry, nor
+onthedirectionof p,whichwouldviolaterotationinvariance.
+Ifg=0 thenF=0, but we may speculate on the behavior
+ofF[g,p,dH]atg2>0asdHvaries. Suppose Fhastheform,
+nearp=0,
+F[g,p,dH] =a[g,dH]−b[g,dH]ps
++higherpowersof p(3.4)
+andb[g,dH]ispositiveforsmall |dH|. AtdH>0allnon-trivial
+eigenvalues are positive, so it must be that a[g,dH]<1 for
+smallp. Note that the eigenvaluespectrum in an infinite vol-
+umestillstartsat λ=0,eventhoughtheconfigurationsare,by
+definition, off the Gribov horizon. At dH<0 some eigenval-
+ues are negative, and if those are the eigenvaluesnear p=0,
+itmeansthat a[g,dH]>1. Thenegativeeigenvaluesmustjust
+disappearat dH=0,andthisisobtainedif a[g,0]=1exactly.
+Inthislast casethesubleadingpowerof pinF[g,p,dH]takes
+over,andwehave
+λp∼p2+s
+ρ(λ)∼λ(d−2−s)/(2+s)(3.5)
+Thisaqualitativechangeinthelow-lyingF-Pspectrum,com -
+pared to the behavior inside the Gribov region, and the suf-
+ficient condition (2.10) for Coulomb confinement is satisfied
+if
+2s+2>d (3.6)
+Inside the Gribov region, at p→0, the spectrum is simply a4
+rescalingofthe zeroth-orderspectrum
+λp=(1−a[g,dH])p2(3.7)
+and, in the case of Coulomb gauge, the Coulomb self-energy
+is finite. The conjecturedbehaviorof /an}bracketle{tλp/an}bracketri}htvs.p, fordHposi-
+tive,negative,andzero,issketchedinFig. 1(a).
+But the scenario just outlined is not the only possible be-
+havior near the horizon. Consider, in particular, the case t hat
+b[g,dH]is negativeforsmall |dH|. Thenwe have
+/an}bracketle{tλp/an}bracketri}ht=(1−a[g,dH])p2−/vextendsingle/vextendsingle/vextendsingleb[g,dH]/vextendsingle/vextendsingle/vextendsinglep2+s+higherpowersof p
+(3.8)
+anditispossiblethat /an}bracketle{tλp/an}bracketri}htispositivenear p=0wherethe p2
+termdominates,but negativein some finiteregionawayfrom
+p=0. The conjectured behavior in this case, for positive,
+negative, and vanishing dH, is indicated in Fig. 1(b), and in
+thiscasewewouldstill have λp∼p2atthehorizon,forsmall
+p2.
+Ofcourse,quantizationinCoulombandLandaugaugedoes
+not involve setting dHto some definite value. What isre-
+quired, however,is a constraint on the rangeof functionali n-
+tegrationtoliewithinthefirstGribovhorizon. Ifitistrue that
+entropydominatesdue to the high dimensionalityof the con-
+figuration space, and almost all of the volume of the Gribov
+regionisconcentratedat thehorizon,thenonlylattice con fig-
+urationsat orverynearthehorizonwill contributeto vacuu m
+expectation values in Coulomb and Landau gauge, just as if
+theconstraint dH=0wereimposed.
+IV. PERTURBATIVEEVALUATIONOF THEF-P
+SPECTRUM
+The possible spectra shown in Figs. 1(a) and 1(b) are pure
+speculationatthispoint,butitisinteresting,andsomewh atin
+thespirit ofGribov’soriginalwork[4], tosee howfarwe can
+goinunderstandingtheF-P spectrumwithordinaryperturba -
+tiontheory.
+LetusbeginwithlatticeSU(2)gaugetheoryineither d+1
+spacetime dimensions (Coulomb gauge) or dspacetime di-
+mensions (Landau gauge), starting on a finite d-dimensional
+volumeVand taking the infinite volume V→∞and lattice
+spacinga→0 limits at the end. The F-P operator on the lat-
+tice isgivenby[5]
+Mab
+xy= (K0)ab
+xy+(K1)ab
+xy+(M1)ab
+xy
+(K0)ab
+xy=δab∑
+i(2δxy−δx+ˆi,y−δx−ˆi,y)
+(K1)ab
+xy=1
+2gεacb∑
+i/bracketleftBig
+−Ac
+i(x)δx+ˆi,y+Ac
+i(y)δx−ˆi,y/bracketrightBig
+(M1)ab
+xy=−δab∑
+i/braceleftBig
+δxy/bracketleftbig
+(1−1
+2TrUi(x))+(1−1
+2TrUi(x−ˆi))/bracketrightbig
+−δx+ˆi,y(1−1
+2TrUi(x))−δx−ˆi,y(1−1
+2TrUi(y))/bracerightBig
+(4.1)where
+Aa
+j=1
+2igTr[σa(Uj(x)−U†
+j(x))] (4.2)
+The dimensionless lattice coupling gLis related to the gauge
+couplinggbyg2
+L=a4−Dg2, whereaisthelatticespacingand
+Dis the spacetime dimension. The eigenvalues and eigen-
+vectors of K0are those shown in eq. (2.3). The operator M1
+vanishesinthecontinuumlimit,soIwilljustignoreitinwh at
+follows, and treat K1as the only perturbation to K0. Lattice
+Fouriertransformswillbe definedsymmetrically
+Aa
+i(x) =1√
+V∑
+k/tildewideAa
+i(k)eikx
+/tildewideAa
+i(k) =1√
+V∑
+xAa
+i(x)e−ikx(4.3)
+Thefirst-ordercorrectionto λ(0)
+pis
+Δλ(1)
+p,A=/an}bracketle{tp,A|K1|p,A/an}bracketri}ht
+=1
+V∑
+x∑
+ye−ipxχa
+A(K1)ac
+xyeipyχc
+A
+=1
+2gχa
+Aεabcχb
+A∑
+i∑
+x1
+V/bracketleftBig
+−Ab
+i(x)eipi+Ab
+i(x−ˆi)e−ipi/bracketrightBig
+=−igχa
+Aεabcχb
+A1√
+V∑
+i/tildewideAb
+i(0)sin(pi) (4.4)
+Now, according to the above definition of the lattice Fourier
+transform,thelattice A-fieldatzeromomentumis
+/tildewideAb
+i(0)=1√
+V∑
+xAb
+i(x) (4.5)
+with−2/g<Ab
+i(x)<2/g. Then suppose that the lattice A-
+field in Coulomb or Landau gauge has a finite correlation
+lengthl. Thisimplies
+∑
+xAb
+i(x)∼±/radicalbigg
+V
+ldldA (4.6)
+where Ais the average value of Ab
+iin a hypercubic region
+of volume ld. Then, because of the factor of 1 /√
+Vin (4.4),
+thefirst-ordercorrectionto λ0
+pvanishesin theinfinite volume
+limit. Of course, the first-order contribution vanishes eve n
+in a finite volume upon taking the expectation value, since
+/an}bracketle{t/tildewideAb
+i(0)/an}bracketri}ht=0.
+At secondorder
+Δλp,A=∑
+k,B|(k,B|K1|p,A)|2
+λ0p−λ0
+k(4.7)5
+where
+(k,B|K1|p,A)
+=1
+2gχa
+Bεabcχc
+A∑
+i1
+V∑
+xe−ikx/parenleftBig
+−Ab
+i(x)eip(x+ˆi)
++Ab
+i(x−ˆi)eip(x−ˆi)/parenrightBig
+=1
+2gχa
+Bεabcχc
+A1√
+V∑
+iAb
+i(k−p)/parenleftBig
+−eipi+e−iki/parenrightBig
+(4.8)
+Then
+Δλp,A=1
+4g2∑
+B(χa
+Aεabcχc
+B)(χd
+Aεdefχf
+B)1
+V∑
+k1
+λ0p−λ0
+k
+×∑
+ij/tildewideAb
+i(k−p)/tildewideAe
+j(p−k)
+×/parenleftBig
+−eipi+e−iki/parenrightBig/parenleftBig
+−e−ipj+eikj/parenrightBig
+(4.9)
+In preparation for taking the continuum limit, we need to
+indicatepowersofthelattice spacingexplicitly. Let
+Δλ=a2Δλ′,p=ap′,Ac
+i(x)=aA′c
+i(x)(4.10)
+wheretheprimedquantitieshavethestandardengineeringd i-
+mensions of these quantities in the continuum formulation.
+We also have,using Δk′=2π/(La),
+1
+V∑
+k=1
+Ld1
+(Δk′)d∑
+k(Δk′)d
+=ad∑
+k/parenleftbiggΔk′
+2π/parenrightbiggd
+(4.11)
+Insertingtheseidentitiesinto(4.9)
+Δλ′
+p,A=1
+a2g2
+4∑
+B(χa
+Aεabcχc
+B)(χd
+Aεdefχf
+B)
+×ad∑
+k/parenleftbiggΔk′
+2π/parenrightbiggd1
+a2(λ′(0)
+p−λ′(0)
+k)
+∑
+ij/tildewideAb
+i(k−p)/tildewideAe
+j(p−k)
+×/parenleftBig
+−eip′
+ia+e−ik′
+ia/parenrightBig/parenleftBig
+−e−ip′
+ja+eik′
+ja/parenrightBig
+(4.12)
+Now we take the vacuum expectation value of Δλ′, noting
+that
+/an}bracketle{t/tildewideAb
+i(k)/tildewideAc(−k)/an}bracketri}ht=a2−dδbcDij(k′)(4.13)
+where,inLandaugauge, Dij(k′)isthefull(i.e.dressed)gluon
+propagator. In Coulomb gauge it is the spatial Fourier trans -formofthefull,equal-timesgluonpropagator. Thisgives
+/an}bracketle{tΔλ′
+p/an}bracketri}ht=1
+a2g2
+4∑
+B(χa
+Aεabcχc
+B)(χd
+Aεdbfχf
+B)
+×∑
+k/parenleftbiggΔk′
+2π/parenrightbiggd1
+λ′(0)
+p−λ′(0)
+k∑
+ijDij(p′−k′)
+×/parenleftBig
+−eip′
+ia+e−ik′
+ia/parenrightBig/parenleftBig
+−e−ip′
+ja+eik′
+ja/parenrightBig
+(4.14)
+At this point we can take the continuum limit, and make use
+of the transversality property qiDij(q)=0 of the gluon prop-
+agator,toobtain3
+/an}bracketle{tΔλ′
+p/an}bracketri}ht=g2∑
+B(χa
+Aεabcχc
+B)(χd
+Aεdbfχf
+B)
+×/integraldisplayddk′
+(2π)d1
+p′2−k′2p′
+ip′
+jDij(p′−k′)(4.15)
+The primes, having served their purpose, will now be
+dropped. It is understood that the unprimed quantities now
+havetheirstandardengineeringdimensions.
+Usingthecompetenessproperty
+∑
+Bχc
+Bχf
+B=δcf(4.16)
+we sum over the color indices, which just gives an overall
+factoroftwo. Theresultis
+/an}bracketle{tΔλp/an}bracketri}ht=−2g2pipj/integraldisplayddk
+(2π)d1
+k2−p2Dij(p−k)(4.17)
+(note the interchange of k2andp2in the denominator).
+Changingvariablesto q=p−k,andwriting
+Dij(q)=/parenleftbigg
+δij−qiqj
+q2/parenrightbigg
+D(q) (4.18)
+gives
+/an}bracketle{tΔλp/an}bracketri}ht=−2g2/integraldisplayddq
+(2π)dD(q)
+(p−q)2−p2
+×/parenleftbigg
+p2−(p·q)2
+q2/parenrightbigg
+(4.19)
+We nowgoto d-dimensionalsphericalcoordinates
+/integraldisplay
+ddq=Ad−1/integraldisplay∞
+0dqqd−1/integraldisplayπ
+0sind−2θ (4.20)
+3Wehaveignored, in lattice regularization, thecasethat λ′(0)
+p=λ′(0)
+k,which
+would have to be handled by degenerate perturbation theory. This case is
+zeromeasureinthecontinuumlimit,andwillnotrequirespe cialtreatment.6
+where
+Ad−1=2π(d−1)/2
+Γ/parenleftbigd−1
+2/parenrightbig (4.21)
+Define
+/tildewideD(q)=qd−2D(q) (4.22)and
+Rd=2Ad−1
+(2π)d=
+
+1/π2d=2
+1/(2π2)d=3
+1/(6π3)d=4(4.23)
+Then
+/an}bracketle{tΔλp/an}bracketri}ht=−g2Rd/integraldisplay∞
+0dqqd−1/integraldisplayπ
+0dθsind−2θ(1−cos2θ)1
+q2−2pqcosθD(q)p2
+=−g2Rd/integraldisplayπ
+0dθsind−2θ(1−cos2θ)/integraldisplay∞
+0dq1
+q−2pcosθqd−2D(q)p2
+=−g2Rdp2(I1+I2) (4.24)
+where
+I1=/integraldisplayπ/2
+0dθsind−2θ(1−cos2θ)/integraldisplay∞
+0dq1
+q−2pcosθ/tildewideD(q)
+I2=/integraldisplayπ
+π/2dθsind−2θ(1−cos2θ)/integraldisplay∞
+0dq1
+q−2pcosθ/tildewideD(q)
+(4.25)
+Makethechangeofvariables θ→π−θinI2
+I2=/integraldisplayπ/2
+0dθsind−2θ(1−cos2θ)/integraldisplay∞
+0dq1
+q+2pcosθ/tildewideD(q) (4.26)
+Then,inI1,it isusefultorewritetheintegralovermomenta q
+/integraldisplay∞
+0dq/tildewideD(q)
+q−2pcosθ=/braceleftbigg/integraldisplay2pcosθ
+0+/integraldisplay4pcosθ
+2pcosθ+/integraldisplay∞
+4pcosθ/bracerightbigg
+dq/tildewideD(q)
+q−2pcosθ
+=/integraldisplay2pcosθ
+0dq/tildewideD(q)
+q−2pcosθ+/integraldisplay2pcosθ
+0dq/tildewideD(4pcosθ−q)
+2pcosθ−q+/integraldisplay∞
+0dq/tildewideD(4pcosθ+q)
+2pcosθ+q
+=/integraldisplay2pcosθ
+0dq1
+2pcosθ−q[/tildewideD(4pcosθ−q)−/tildewideD(q)]+/integraldisplay∞
+0dq/tildewideD(4pcosθ+q)
+2pcosθ+q(4.27)
+Altogether,we haveto secondorder
+/an}bracketle{tλp/an}bracketri}ht=λ(0)
+p+/an}bracketle{tΔλp/an}bracketri}ht=p2/parenleftBig
+1−g2RdI[p,m,α]/parenrightBig
+(4.28)
+where
+I[p,m,α] =/integraldisplayπ/2
+0dθsind−2θ(1−cos2θ)/braceleftbigg/integraldisplay∞
+0dq1
+q+2pcosθ[/tildewideD(4pcosθ+q)+/tildewideD(q)]
++/integraldisplay2pcosθ
+0dq1
+2pcosθ−q[/tildewideD(4pcosθ−q)−/tildewideD(q)]/bracerightbigg
+(4.29)
+Them,αinI[p,m,α]are constants I will use to parametrize
+thetransversegluonpropagator D(q).V. AN ANSATZFORTHEGLUON PROPAGATOR
+Thegluonpropagators Dijin CoulombandLandaugauges
+are transverse with respect to spatial momenta qind+1 di-7
+mensions, and spacetime momenta qµindEuclidean dimen-
+sions,respectively. Thereforethesepropagatorshavethe form
+shownin (4.18). Inafreetheory
+D(q)=
+
+1/(2q)Coulombgauge
+1/q2Landaugauge(5.1)
+wherethepropagatorinCoulombgaugeisatequaltimes,with
+qthespace (ratherthanspacetime)momentum. The behavior
+(5.1) is expected at high momenta, but it is certainly not cor -
+rect at low momenta, as seen from lattice Monte Carlo sim-
+ulations. In Landau gauge, the current evidence is that D(0)
+is finite and non-zero at q=0 in three and four dimensions
+[7, 8], while D(q)→0 in two dimensions [9]. In Coulomb
+gaugeit appearsthat D(q)→0in fourdimensions[10].
+In order to allow for non-singular power behavior in the
+transverse gluon propagatoras p→0, I will adopt the ansatz
+that
+D(q)=1
+2/radicalbig
+q2+m2+α/qα(5.2)
+inCoulombgauge,and
+D(q)=1
+q2+m2+α/qα(5.3)
+in Landau gauge. Gribov’sproposalfor the gluonpropagator
+in these cases correspondsto α=2. The propagatorsgoover
+tofree-fieldbehavioras q→∞.
+-0.002 0 0.002 0.004 0.006 0.008 0.01
+ 0 2 4 6 8 10 12 14 16 18D(R)
+RL=24
+L=32
+L=50
+FIG.2: Equal-timesCoulomb-gaugegluonpropagatorin2+1d imen-
+sions, atβ=6andL3latticevolume, for L=24,32,50.
+I am not aware of any lattice Monte Carlo computation of
+the transverse gluon propagator in Coulomb gauge in d=3
+dimensions, in position space. In Fig. 2 I show data for D(R)
+obtainedfromtheequaltimescorrelator
+/an}bracketle{tTr[Aj(x,t)Aj(y,t)] (5.4)ofgluonfields
+Aj(x,t)=1
+2i(Uj(x,t)−U†
+j(x,t)) (5.5)
+on the lattice. The correlator is calculated via lattice Mon te
+CarlowithanSU(2)Wilsonaction,onan L3latticevolumeat
+couplingβ=6andL=24,32,50,withtheequal-timescorre-
+latorcomputedaftertransformingthegaugefieldstoCoulom b
+gauge. Note that as the lattice volume increases, the gluon
+propagatordevelopsa “dip”andactuallybecomesnegativea t
+the larger Rvalues. This behavior appears to rule out α=0,
+in whichthe propagatorshouldbe everywherepositive. A re-
+liable computation of D(q)asq→0 will probably require a
+large-scalelatticecalculation,ashasbeendonefortheLa ndau
+gauge.
+VI. RESULTSFOR F-PSPECTRA
+In section III I introduced a parameter dHto control the
+approach to the first Gribov horizon, and speculated on the
+low-pbehavior of λpas the horizon is approached. In the
+perturbative calculation, the mass parameter min the gluon
+propagator plays essentially the same role as dH. Note that
+in dimensionslower than 3+1, where I[p,m,α]is convergent,
+the coupling g2is dimensionful, and we may as well choose
+unitssuchthat g2=1. Then
+/an}bracketle{tλp/an}bracketri}ht=p2/parenleftBig
+(1−RdI[p,m,α]/parenrightBig
+(6.1)
+Expanding I[p,m,α]in leading powers of pnearp=0, we
+have
+RdI[p,m,α] =a[m,α]−b[m,α]ps
++higherpowersof p(6.2)
+inwhichcase
+/an}bracketle{tλp/an}bracketri}ht= (1−a[m,α])p2+b[m,α]p2+s
++higherpowersof p (6.3)
+Suppose, for a given α, it is possible to find a critical value
+m=mcsuchthat a[mc,α]=1andb[m,α]>0. Inthatcasewe
+havethe TypeI scenarioconjecturedinFig. 1(a)above;i.e.
+1.m<mcanda[m,α]>1: The low-lyingF-P eigenvalue
+spectrum has a range of negative eigenvalues, starting
+atp=0. We interpret this to mean that the trans-
+versegluonpropagator,whichdeterminesthespectrum
+atsecondorder,isdeterminedbyconfigurationsoutside
+the Gribovregion.
+2.m=mcanda[mc,α]=1: Theregionofnegativeeigen-
+values just disappears, and λp∼p2+s. This is the case
+of particularinterest,wherethegluonpropagatorisde-
+rivedfromconfigurationswhich mainlylie righton the
+Gribovhorizon.8
+_0 2 −1Type I Type II no solution Coulomb D=3no solution Type I Type II
+no solution Landau D=2Landau D=3
+ 1Type I
+αType II
+FIG.3: Summaryofthequalitativebehaviorofthelow-lying F-Pspectra,accordingto2ndorderperturbationtheory,fo rLandauandCoulomb
+gauges in D=2 and 3 dimensions. The sketch illustrates how th e behavior of the F-P spectra depends on the assumed infrared behavior of the
+gluon propagator, whichisparametrized bythe exponent α.
+3.m>mcanda[mc,α]<1. In this case the low-lying
+spectrum λp=(1−a[m,α])p2is just a rescalingof the
+free-fieldspectrum,andthegluonpropagatorisderived
+fromconfigurationsinsidetheGribovregion.
+It should be noted at this point that the Type I scenario is
+in some waysreminiscentofthe Dyson-Schwingerapproach,
+and indeed eq. (4.28) resembles the Dyson-Schwinger equa-
+tion for the ghost propagator in covariant gauges (see, e.g. ,
+Fischer [11]). Of course these equations are not the same;
+(6.1) is an equationforthe expectationvalueof F-P eigenva l-
+ues, not the inverse ghost propagator,and it is derivedfrom a
+perturbative expansion, not the Dyson-Schwinger equation s.
+Nevertheless, the scaling solution [12] is obtained from th e
+Dyson-Schwinger equation by tuning a coupling so that the
+bare inverse ghost propagatorin that equation is exactly ca n-
+celled by another term. In the absence of this tuning, the de-
+couplingsolution[13]isobtained. Similarly,inourappro ach,
+a massparameteris tunedtoexactlycancelthe p2termin the
+eigenvaluespectrum,resultinginanenhanceddensityofne ar-
+zeroeigenmodes. Themotivationforthetuningin ourcase is
+tostudytheF-PspectrumattheGribovhorizon,whichisonly
+relevant to physics if, in fact, the functional integral ove r the
+Gribovregionisdominatedbyhorizonconfigurations.
+TheTypeIIscenarioisobtainedif b[m,α]isnegativewhen
+a[m,α] =1, in which case there is still a range of negative
+eigenvalues, so this value of mis not the critical value. The
+critical value, corresponding to the horizon, is obtained a t a
+valuem=mcwherea[m,α]<1,suchthatthefunction
+/an}bracketle{tλp/an}bracketri}ht≈/parenleftBig
+1−a[mc,α]/parenrightBig
+p2−/vextendsingle/vextendsingle/vextendsingleb[mc,α]/vextendsingle/vextendsingle/vextendsinglepr+c[mc,α]pq(6.4)
+approximating /an}bracketle{tλp/an}bracketri}htat smallphas a zero value, but no nega-
+tive values, for one choice of p/ne}ationslash=0. In this case the horizon
+doesnotalterthepowerdependence λp∼p2nearp=0.BoththeTypeIandTypeIIscenariosassume that
+a[m,α]=RdI[p=0,m,α] (6.5)
+is finite. This is not necessarily the case, however, and it is
+easy to check that I[0,m,α]is divergentat all α≤0 for Lan-
+daugaugein twodimensionsandCoulombgaugeinthreedi-
+mensions,andisdivergentforall α≤−1forLandaugaugein
+three dimensions. There is no choice of m, for those choices
+ofα, which completely eliminates negative F-P eigenvalues.
+Thiswill bereferredtoasthe“nosolution”case.
+In order to determine which scenario is realized, at each
+choiceof αforwhich I[0,m,α]isfinite,it isnecessaryto cal-
+culateI[p,m,α]numerically. The result, for Coulomb gauge
+in three dimensions, and Landau gauge in two and three di-
+mensions, is indicated schematically in Fig. 3. To illustra te
+how these results are obtained, we consider in particular th e
+case ofα=1 for Coulomb and Landau gauges in three di-
+mensions(i.e. d=2forCoulomb,and d=3forLandau). We
+beginwith Coulombgauge(Figs. 4-7). Figure4(a)showsthe
+low-lyingF-Pspectrumat α=1andm=0.20<mc,andit is
+clear that there is a region of negative eigenvalues startin g at
+p=0. Asmis increased, the region of negative eigenvalues
+shrinksin size, untilata criticalvalue m=mc(α)the interval
+ofnegativeeigenvaluesjustvanishes. Figure4(b)display sthe
+low-lying spectrum just below, at, and just above the critic al
+massatα=1,whichis mc=0.2228. Atm/ne}ationslash=mc,λpispropor-
+tionaltop2nearp=0, witha proportionalityconstantwhich
+is positive or negative, depending on whether mis greater or
+lessthanmc. Butpreciselyat m=mc,wefindthat λp∝p2+s,
+withs=s(α)>0. Fig. 5 is a log-log plot of λpvs.pover
+a large range of p, atα=1 andmc=0.223. For the range
+0<p<1, we can determine that s=0.53 in this case, and
+λp≈1.21p2.53at smallp. At around p≡ |p|=1 (in units
+g2=1), the power behavior shifts to the free case, λp=p2,
+and continues that way for all higher p, as expected. This is9
+-1e-06-5e-07 0 5e-07 1e-06
+ 0  0.002  0.004  0.006  0.008  0.01λ
+pm=0.2
+(a)-1.5e-09-1e-09-5e-10 0 5e-10 1e-09 1.5e-09
+ 2e-05  4e-05  6e-05  8e-05  0.0001λ
+pm=0.20
+m=mc
+m=0.25
+(b)
+FIG.4: F-Pspectra at α=1. (a)m=0.20<mc. There is aninterval of negative eigenvalues inthe region 0 <p<0.009. (b)λpat lowp,for
+m=0.20<mc,m=mc=0.2228, and m=0.25>mc.
+10-1410-1210-1010-810-610-410-2100102104106
+ 1e-05 0.0001 0.001 0.01 0.1  1 10 100λ
+pfit   
+λ=p2
+FIG. 5: Log-log plot of the spectrum of the Fadeev-Popov oper ator,
+forα=1 at the critical mc=0.223. A best fit at p<1 yieldsλp=
+1.21p2.53.
+anexampleoftheTypeIscenario.
+Thenextquestionishow mcand2+schangeas αisvaried.
+As already noted, we must choose α>0 to reach the hori-
+zon, whichmeansthat D(0)=0, andthereforethe transverse
+gluonpropagatormustvanishatzeromomentumforCoulomb
+gauge in 2+1 dimensions, and for Landau gauge in 1+1 di-
+mensions. As α→0+, the increasingly singular behavior of
+the integrand in I[p,m,α]must be countered by an increas-
+inglylargevalueof mc,inordertosatisy a[mc],α]=1. Aplot
+ofmcvs.αis showninFig. 6.
+The power behavior λp=bp2+sin the low-lying spectrum
+is crucial for Coulomb confinement, and the exponent 2 +s
+vs.α, obtained at m=mcis shown in Fig. 7(a). In 2+1 di-
+mensionstheconditionforCoulombconfinement(beyondthe
+marginaldivergenceof the freetheory)is that s>0, whichis
+seento holdthroughouttherangeshown. 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
+ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2m
+α
+FIG. 6: Critical value mcfor the mass parameter in the transverse
+gluon propagator, vs. the power α.
+We also see that there is a sudden jump in sfrom roughly
+s=1tos=2atα=2. ThisiswherethetransitionfromType
+I to Type II behavior takes place. As α→2 the coefficient
+b[mc,α]approacheszero(cf.Fig. 7(b))andthenchangessign.
+Exactly at α=2, where b[mc,α]=0, the term which has the
+next higher power in ptakes over, accounting for the sudden
+jumpins.
+Landau gauge in three dimensions, at α=1, furnishes an
+exampleoftheTypeIIscenario. TheF-P spectrumat small p
+is shownin Fig. 8 forthe mass parameterabove( m=0.087),
+below(m=0.086),andequal m=mc=0.08644tothecritical
+value.
+VII. CONCLUSIONS
+If the integration over gauge fields is dominated by con-
+figurations on or near the first Gribov horizon, then the low-10
+ 0 1 2 3 4 5
+ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.22+s
+α
+(a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4
+ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2b[mc,α]
+α
+(b)
+FIG.7: (a)Exponent 2 +svs.α;and(b) coefficient bvsα; forthe power-law behavior λp=bp2+satthe criticalmass parameter m=mc(α),
+for the Coulomb gauge F-Pspectrum in2+1 dimensions. The sud den rise to2 +s=4atα=2is correlatedwith b→0
+-5e-06 0 5e-06 1e-05 1.5e-05 2e-05
+ 0  0.01  0.02  0.03  0.04λ
+pm=mc
+m>mc
+m<mc
+FIG. 8: The low-lying F-P eigenvalue spectra near the Gribov hori-
+zon, for Landau gauge in D=3 spacetime dimensions and α=1, ac-
+cording to second-order perturbation theory. This is an exa mple of
+the Type II scenario.
+est non-trivial F-P eigenvalue must be very close or equal to
+zero, even in a finite spacetime volume. The main finding of
+the perturbativetreatment presented here is that if there i s, in
+fact, a non-trivial zero mode, and the F-P eigenvalues are la -
+beled by the lattice momenta, then this non-trivialzero mod e
+mayoccurateitherzeromomentum(TypeIscenario)ornon-
+zero momenta (Type II scenario), depending on the infrared
+behavior of the gluon propagator. While the spectrum of F-
+P eigenvalues does not translate directly into a prediction for
+the behaviorof the ghostpropagator(becausethe momentum
+behavior of the F-P eigenmodes must also be taken into ac-
+count), it is natural to conjecture that the Type I scenario i s
+associated with an infrared singular ghost dressing functi on,
+as in Coulombgauge,while the Type II scenariocorresponds
+toafiniteghostdressingfunction,asappearstobethecasei nLandaugauge. Thiswouldmostlikelybethecaseif |φpA(k)|2
+isnarrowlypeakedaround k=p, whereφpA(k)isthe Fourier
+transformofanF-Peigenmode φpA(x)withalow-lyingeigen-
+valueλpA.
+Since the FP spectra at the Gribov horizon have been de-
+rived here from ordinary 2nd order perturbation theory (plu s
+anansatzforthegluonpropagator),thereisobviouslyaque s-
+tionofwhetherperturbationtheorycanbetrustedinthisco n-
+text. In D=3 spacetime dimensions the coupling g2has
+units of mass, so the expansion parameter at p→0 will be
+g2/m,whiletheexpansionparameteratlarge pwillbeg2/|p|.
+Theperturbativecalculationofthe FP eigenvaluespectrum at
+p→0 should therefore be trustworthy for large m/g2. Un-
+fortunately, we have seen that the critical mass parameter mc
+corresponding to the Gribov horizon is actually rather smal l,
+in units of g2, with, e.g., mc/g2=0.223 in Coulomb gauge,
+andmc/g2=0.0864 in Landau gauge in three spacetime di-
+mensions and α=1. Of course, the perturbative expansion
+may also involve some numerical factors, and without calcu-
+latingtohigherorders,orestimatingtheradiusofconverg ence
+insomeway,itisdifficulttojudgetheaccuracyofthesecond -
+order term in the series. But there is no particular reason fo r
+confidencein the second-orderresults at m=mcat the quan-
+titative level. It was argued however in section III, on rath er
+general grounds, that it is natural to expect either Type I or
+Type II behavior of the Faddeev-Popov spectrum at the Gri-
+bov horizon. The perturbative calculation, at this stage, s im-
+ply provides a concrete illustration in support of this rath er
+generalqualitativeargument.
+Acknowledgments
+ThisresearchwassupportedinpartbytheU.S.Department
+ofEnergyunderGrantNo.DE-FG03-92ER40711.11
+Appendix
+In order to derive the inequality (2.9) stated in section II,
+we beginwiththefollowing
+Theorem. Let Q be any Hermitian operator with a dis-
+crete set of eigenstates {|n/an}bracketri}ht}, whose corresponding eigen-
+valuespectrum {qn}isboundedfrombelow,andorderedsuch
+that qn≤qn+1. Here the index n runs from 1 up to the di-
+mension of the Hilbert space N H(which need not be finite).
+Let{|φn/an}bracketri}ht}be any other complete set of orthonormal states
+spanning the same Hilbert space as the {|n/an}bracketri}ht}. Then, for any
+N≤NH,
+N
+∑
+n=1/an}bracketle{tφn|Q|φn/an}bracketri}ht≥N
+∑
+n=1qn (A.1)
+This is a fairly trivial generalization of the inequality un der-
+lying the Rayleigh-Ritz variational method (the case N=1),
+andtheproofgoesasfollows: Define
+T≡N
+∑
+n=1/an}bracketle{tφn|Q|φn/an}bracketri}ht
+=N
+∑
+n=1∑
+k∑
+m/an}bracketle{tφn|k/an}bracketri}ht/an}bracketle{tk|Q|m/an}bracketri}ht/an}bracketle{tm|φn/an}bracketri}ht
+=∑
+mqmPN(m) (A.2)
+where
+PN(m)=N
+∑
+n=1/an}bracketle{tm|φn/an}bracketri}ht/an}bracketle{tφn|m/an}bracketri}ht (A.3)
+Observethat
+0≤PN(m)≤PNH(m)=1 (A.4)
+and
+NH
+∑
+m=1PN(m)=N (A.5)
+SincePN(m)≤1, and with regardto the constraint (A.5), the
+smallest possiblevalueof Tisobviouslyobtainedfor
+PN(m)=/braceleftbigg
+1m≤N
+0m>N(A.6)
+Substitutingthis optimalchoiceinto the last line of (A.2) , we
+findthat
+T≥Tmin=N
+∑
+m=1qm (A.7)andthe inequalitystated inthetheoremisestablished.
+Fromthistheoremit followsthat
+(N)
+∑
+n(φn|−∇2|φn)≥(N)
+∑
+nλ(0)
+n (A.8)
+where now the {φn}are the eigenstates of the F-P operator,
+andwherewe havedefined
+(N)
+∑
+n≡∑
+n,Aθ(N2−n·n) (A.9)
+Nowlet
+ρN(λ)=1
+NcolorsV(N)
+∑
+nδ(λ−λn) (A.10)
+Assuming non-degeneracy, each λdetermines n,Auniquely,
+ndetermines λ(0)
+n, andp2
+n=λ(0)
+nin the large-volumelimit. In
+thislimitwe canthereforewe canexpress(A.8)as
+/integraldisplay
+dλρN(λ)(φλ|−∇2|φλ)≥/integraldisplay
+dλ ρN(λ)p2(λ)(A.11)
+At smallλthereissomeleadingpowerbehavior
+/angbracketleftBig
+ρN(λ)(φλ|−∇2|φλ)/angbracketrightBig
+≈aλr
+/angbracketleftBig
+ρN(λ)p2
+λ/angbracketrightBig
+≈bλq(A.12)
+Then, since (A.11) must be true for any mode cutoff N, no
+matter how small, it follows that either r<q, orr=qand
+a>b. Eitherway,
+lim
+λ→0/angbracketleftbiggρN(λ)(φλ|−∇2|φλ)
+λ1−ε/angbracketrightbigg
+≥lim
+λ→0/angbracketleftbiggρN(λ)p2(λ)
+λ1−ε/angbracketrightbigg
+(A.13)
+Finally, giventhat all the near-zeromodesare includedin t he
+sum(A.9),we have
+ρN(λ)=ρ(λ)asλ→0 (A.14)
+andthe inequality(A.13)becomes
+lim
+λ→0/angbracketleftbiggρ(λ)(φλ|−∇2|φλ)
+λ1−ε/angbracketrightbigg
+≥lim
+λ→0/angbracketleftbiggρ(λ)p2(λ)
+λ1−ε/angbracketrightbigg
+(A.15)
+Thisestablisheseq.(2.9).
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+A. Cucchieri and D. Zwanziger, Nucl. Phys. Proc. Suppl. 119,
+727(2003) [arXiv:hep-lat/0209068];
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\ No newline at end of file
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+On the Origin of Gravity
+and the Laws of Newton
+Erik Verlinde1
+Institute for Theoretical Physics
+University of Amsterdam
+Valckenierstraat 65
+1018 XE, Amsterdam
+The Netherlands
+Abstract
+Starting from rst principles and general assumptions Newton's law of grav-
+itation is shown to arise naturally and unavoidably in a theory in which space
+is emergent through a holographic scenario. Gravity is explained as an entropic
+force caused by changes in the information associated with the positions of ma-
+terial bodies. A relativistic generalization of the presented arguments directly
+leads to the Einstein equations. When space is emergent even Newton's law of
+inertia needs to be explained. The equivalence principle leads us to conclude
+that it is actually this law of inertia whose origin is entropic.
+1e.p.verlinde@uva.nlarXiv:1001.0785v1  [hep-th]  6 Jan 20101 Introduction
+Of all forces of Nature gravity is clearly the most universal. Gravity in
uences and
+is in
uenced by everything that carries an energy, and is intimately connected with
+the structure of space-time. The universal nature of gravity is also demonstrated by
+the fact that its basic equations closely resemble the laws of thermodynamics and
+hydrodynamics2. So far, there has not been a clear explanation for this resemblance.
+Gravity dominates at large distances, but is very weak at small scales. In fact, its
+basic laws have only been tested up to distances of the order of a millimeter. Gravity is
+also considerably harder to combine with quantum mechanics than all the other forces.
+The quest for unication of gravity with these other forces of Nature, at a microscopic
+level, may therefore not be the right approach. It is known to lead to many problems,
+paradoxes and puzzles. String theory has to a certain extent solved some of these, but
+not all. And we still have to gure out what the string theoretic solution teaches us.
+Many physicists believe that gravity, and space-time geometry are emergent. Also
+string theory and its related developments have given several indications in this direc-
+tion. Particularly important clues come from the AdS/CFT, or more generally, the
+open/closed string correspondence. This correspondence leads to a duality between
+theories that contain gravity and those that don't. It therfore provides evidence for
+the fact that gravity can emerge from a microscopic description that doesn't know
+about its existence.
+The universality of gravity suggests that its emergence should be understood from
+general principles that are independent of the specic details of the underlying micro-
+scopic theory. In this paper we will argue that the central notion needed to derive
+gravity is information. More precisely, it is the amount of information associated with
+matter and its location, in whatever form the microscopic theory likes to have it, mea-
+sured in terms of entropy. Changes in this entropy when matter is displaced leads to
+an entropic force, which as we will show takes the form of gravity. Its origin therefore
+lies in the tendency of the microscopic theory to maximize its entropy.
+The most important assumption will be that the information associated with a
+part of space obeys the holographic principle [8, 9]. The strongest supporting evidence
+for the holographic principle comes from black hole physics [1, 3] and the AdS/CFT
+correspondence [10]. These theoretical developments indicate that at least part of
+the microscopic degrees of freedom can be represented holographically either on the
+boundary of space-time or on horizons.
+The concept of holography appears to be much more general, however. For instance,
+in the AdS/CFT correspondence one can move the boundary inwards by exploiting a
+holographic version of the renormalization group. Similarly, in black hole physics there
+exist ideas that the information can be stored on stretched horizons. Furthermore, by
+thinking about accelerated observers, one can in principle locate holographic screens
+2An incomplete list of references includes [1, 2, 3, 4, 5, 6, 7]
+2anywhere in space. In all these cases the emergence of the holographic direction is
+accompanied by redshifts, and related to some coarse graining procedure. If all these
+combined ideas are correct there should exist a general framework that describes how
+space emerges together with gravity.
+Usually holography is studied in relativistic contexts. However, the gravitational
+force is also present in our daily non-relativistic world. The origin of gravity, whatever
+it is, should therefore also naturally explain why this force appears the way it does,
+and obeys Newton law of gravitation. In fact, when space is emergent, also the other
+laws of Newton have to be re-derived, because standard concepts like position, velocity,
+acceleration, mass and force are far from obvious. Hence, in such a setting the laws of
+mechanics have to appear alongside with space itself. Even a basic concept like inertia
+is not given, and needs to be explained again.
+In this paper we present a holographic scenario for the emergence of space and
+address the origins of gravity and inertia, which are connected by the equivalence
+principle. Starting from rst principles, using only space independent concepts like
+energy, entropy and temperature, it is shown that Newton's laws appear naturally and
+practically unavoidably. Gravity is explained as an entropic force caused by a change
+in the amount of information associated with the positions of bodies of matter.
+A crucial ingredient is that only a nite number of degrees of freedom are associated
+with a given spatial volume, as dictated by the holographic principle. The energy, that
+is equivalent to the matter, is distributed evenly over the degrees of freedom, and thus
+leads to a temperature. The product of the temperature and the change in entropy due
+to the displacement of matter is shown to be equal to the work done by the gravitational
+force. In this way Newton's law of gravity emerges in a surprisingly simple fashion.
+The holographic principle has not been easy to extract from the laws of Newton
+and Einstein, and is deeply hidden within them. Conversely, starting from holography,
+we nd that these well known laws come out directly and unavoidably. By reversing
+the logic that lead people from the laws of gravity to holography, we will obtain a
+much sharper and even simpler picture of what gravity is. For instance, it claries why
+gravity allows an action at a distance even when there in no mediating force eld.
+The presented ideas are consistent with our knowledge of string theory, but if correct
+they should have important implications for this theory as well. In particular, the
+description of gravity as being due to the exchange of closed strings can no longer be
+valid. In fact, it appears that strings have to be emergent too.
+We start in section 2 with an exposition of the concept of entropic force. Section
+3 illustrates the main heuristic argument in a simple non relativistic setting. Its gen-
+eralization to arbitrary matter distributions is explained in section 4. In section 5 we
+extend these results to the relativistic case, and derive the Einstein equations. The
+conclusions are presented in section 7.
+32 Entropic Force.
+An entropic force is an eective macroscopic force that originates in a system with
+many degrees of freedom by the statistical tendency to increase its entropy. The force
+equation is expressed in terms of entropy dierences, and is independent of the details
+of the microscopic dynamics. In particular, there is no fundamental eld associated
+with an entropic force. Entropic forces occur typically in macroscopic systems such as
+in colloid or bio-physics. Big colloid molecules suspended in an thermal environment
+of smaller particles, for instance, experience entropic forces due to excluded volume
+eects. Osmosis is another phenomenon driven by an entropic force.
+Perhaps the best known example is the elasticity of a polymer molecule. A single
+polymer molecule can be modeled by joining together many monomers of xed length,
+where each monomer can freely rotate around the points of attachment and direct itself
+in any spatial direction. Each of these congurations has the same energy. When the
+polymer molecule is immersed into a heat bath, it likes to put itself into a randomly
+coiled conguration since these are entropically favored. There are many more such
+congurations when the molecule is short compared to when it is stretched into an
+extended conguration. The statistical tendency to return to a maximal entropy state
+translates into a macroscopic force, in this case the elastic force.
+By using tweezers one can pull the endpoints of the polymer apart, and bring it
+out of its equilibrium conguration by an external force F, as shown in gure 1. For
+deniteness, we keep one end xed, say at the origin, and move the other endpoint
+along thex-axis. The entropy equals
+S(E;x) =kBlog 
+(E;x) (2.1)
+wherekBis Boltzman's constant and 
+( E;x) denotes the volume of the conguration
+space for the entire system as a function of the total energy Eof the heat bath and
+the position xof the second endpoint. The xdependence is entirely a congurational
+eect: there is no microscopic contribution to the energy Ethat depends on x.
+€ F€ x
+€ T
+Figure 1: A free jointed polymer is immersed in a heat bath with temperature Tand pulled
+out of its equilibrium state by an external force F. The entropic force points the other way.
+4In the canonical ensemble the force Fis introduced in the partition function3
+Z(T;F) =Z
+dEdx 
+(E;x)e(E+Fx)=kBT: (2.2)
+as an external variable dual to the length xof the polymer. The force Frequired to
+keep the polymer at a xed length xfor a given temperature Ecan be deduced from
+the saddle point equations
+1
+T=@S
+@E;F
+T=@S
+@x: (2.3)
+By the balance of forces, the external force Fshould be equal to the entropic force, that
+tries to restore the polymer to its equilibrium position. An entropic force is recognized
+by the facts that it points in the direction of increasing entropy, and, secondly, that it
+is proportional to the temperature. For the polymer the force can be shown to obey
+Hooke's law
+Fpolymerconst
kBThxi:
+This example makes clear that at a macroscopic level an entropic force can be conser-
+vative, at least when the temperature is kept constant. The corresponding potential
+has no microscopic meaning, however, and is emergent.
+It is interesting to study the energy and entropy balance when one gradually lets
+the polymer return to its equilibrium position, while allowing the force to perform work
+on an external system. By energy conservation, this work must be equal to the energy
+that has been extracted from the heat bath. The entropy of the heat bath will hence be
+reduced. For an innite heat bath this would be by the same amount as the increase in
+entropy of the polymer. Hence, in this situation the total entropy will remain constant.
+This can be studied in more detail in the micro-canonical ensemble, because it
+takes the total energy into account including that of the heat bath. To determine the
+entropic force, one again introduces an external force Fand examines the balance of
+forces. Specically, one considers the micro canonical ensemble given by 
+( E+Fx;x ),
+and imposes that the entropy is extremal. This gives
+d
+dxS(E+Fx; x ) = 0 (2.4)
+One easily veries that this leads to the same equations (2.3). However, it illustrates
+that microcanonically the temperature is in general position dependent, and the force
+also energy dependent. The term Fxcan be viewed as the energy that was put in to
+the system by pulling the polymer out of its equilibrium position. This equation tells
+us therefore that the total energy gets reduced when the polymer is slowly allowed to
+return to its equilibrium position, but that the entropy in rst approximation stays
+the same. Our aim in the following sections is to argue that gravity is also an entropic
+force, and that the same kind of reasonings apply to it with only slight modications.
+3We like to thank B. Nienhuis and M. Shigemori for enlightning discussions on the following part.
+53 Emergence of the Laws of Newton
+Space is in the rst place a device introduced to describe the positions and movements
+of particles. Space is therefore literally just a storage space for information. This
+information is naturally associated with matter. Given that the maximal allowed in-
+formation is nite for each part of space, it is impossible to localize a particle with
+innite precision at a point of a continuum space. In fact, points and coordinates
+arise as derived concepts. One could assume that information is stored in points of a
+discretized space (like in a lattice model). But if all the associated information would
+be without duplication, one would not obtain a holographic description. In fact, one
+would not recover gravity.
+Thus we are going to assume that information is stored on surfaces, or screens.
+Screens separate points, and in this way are the natural place to store information
+about particles that move from one side to the other. Thus we imagine that this
+information about the location particles is stored in discrete bits on the screens. The
+dynamics on each screen is given by some unknown rules, which can be thought of
+as a way of processing the information that is stored on it. Hence, it does not have
+to be given by a local eld theory, or anything familiar. The microscopic details are
+irrelevant for us.
+Let us also assume that like in AdS/CFT, there is one special direction correspond-
+ing to scale or a coarse graining variable of the microscopic theory. This is the direction
+in which space is emergent. So the screens that store the information are like stretched
+horizons. On one side there is space, on the other side nothing yet. We will assume
+that the microscopic theory has a well dened notion of time, and its dynamics is time
+translation invariant. This allows one to dene energy, and by employing techniques
+of statistical physics, temperature. These will be the basic ingredients together with
+the entropy associated with the amount of information.
+3.1 Force and inertia.
+Our starting assumption is directly motivated by Bekenstein's original thought experi-
+ment from which he obtained is famous entropy formula. He considered a particle with
+massmattached to a ctitious "string" that is lowered towards a black hole. Just
+before the horizon the particle is dropped in. Due to the innite redshift the mass
+increase of the black hole can be made arbitrarily small, classically. If one would take a
+thermal gas of particles, this fact would lead to problems with the second law of ther-
+modynamics. Bekenstein solved this by arguing that when a particle is one Compton
+wavelength from the horizon, it is considered to be part of the black hole. Therefore,
+it increases the mass and horizon area by a small amount, which he identied with one
+bit of information. This lead him to his area law for the black hole entropy.
+We want to mimic this reasoning not near a black hole horizon, but in 
at non-
+6€ m€ Δx€ ΔS€ TFigure 2: A particle with mass approaches a part of the holographic screen. The screen
+bounds the emerged part of space, which contains the particle, and stores data that describe
+the part of space that has not yet emerged, as well as some part of the emerged space.
+relativistic space. So we consider a small piece of an holographic screen, and a particle
+of massmthat approaches it from the side at which space time has already emerged.
+Eventually the particle merge with the microscopic degrees of freedom on the screen,
+but before it does so, it already in
uences the amount of information that is stored on
+the screen. The situation is depicted in gure 2.
+Motivated by Bekenstein's argument, let us postulate that the change of entropy
+associated with the information on the boundary equals
+
S= 2kB when 
 x=~
+mc: (3.5)
+The reason for putting in the factor of 2 , will become apparent soon. Let us rewrite
+this formula in the slightly more general form by assuming that the change in entropy
+near the screen is linear in the displacement 
 x.
+
S= 2kBmc
+~
x: (3.6)
+To understand why it is also proportional to the mass m, let us imagine splitting the
+particle into two or more lighter sub-particles. Each sub-particle then carries its own
+associated change in entropy after a shift 
 x. Since entropy and mass are both additive,
+it is therefore natural that the entropy change is proportional to the mass. How does
+force arise? The basic idea is to use the analogy with osmosis across a semi-permeable
+membrane. When a particle has an entropic reason to be on one side of the membrane
+and the membrane carries a temperature, it will experience an eective force equal to
+F
x=T
S: (3.7)
+This is the entropic force. Thus, in order to have a non zero force, we need to have
+a non vanishing temperature. From Newton's law we know that a force leads to a
+7non zero acceleration. Of course, it is well known that acceleration and temperature
+are closely related. Namely, as Unruh showed, an observer in an accelerated frame
+experiences a temperature
+kBT=1
+2~a
+c; (3.8)
+whereadenotes the acceleration. Let us take this as the temperature associated with
+the bits on the screen. Now it is clear why the equation (3.6) for 
 Swas chosen to be
+of the given form, including the factor of 2 . It is picked precisely in such a way that
+one recovers the second law of Newton
+F=ma: (3.9)
+as is easily veried by combining (3.8) together with (3.6) and (3.7).
+Equation (3.8) should be read as a formula for the temperature Tthat is required
+to cause an acceleration equal to a. And not as usual, as the temperature caused by
+an acceleration.
+3.2 Newton's law of gravity.
+Now suppose our boundary is not innitely extended, but forms a closed surface. More
+specically, let us assume it is a sphere. For the following it is best to forget about
+the Unruh law (3.8), since we don't need it. It only served as a further motivation for
+(3.6). The key statement is simply that we need to have a temperature in order to
+have a force. Since we want to understand the origin of the force, we need to know
+where the temperature comes from.
+One can think about the boundary as a storage device for information. Assuming
+that the holographic principle holds, the maximal storage space, or total number of
+bits, is proportional to the area A. In fact, in an theory of emergent space this how
+area may be dened: each fundamental bit occupies by denition one unit cell.
+Let us denote the number of used bits by N. It is natural to assume that this
+number will be proportional to the area. So we write
+N=Ac3
+G~(3.10)
+where we introduced a new constant G. Eventually this constant is going to be iden-
+tied with Newton's constant, of course. But since we have not assumed anything
+yet about the existence a gravitational force, one can simply regard this equation as
+the denition of G. So, the only assumption made here is that the number of bits is
+proportional to the area. Nothing more.
+Suppose there is a total energy Epresent in the system. Let us now just make the
+simple assumption that the energy is divided evenly over the bits N. The temperature
+8€ m€ F€ M€ R€ TFigure 3: A particle with mass mnear a spherical holographic screen. The energy is evenly
+distributed over the occupied bits, and is equivalent to the mass Mthat would emerge in the
+part of space surrounded by the screen.
+is then determined by the equipartition rule
+E=1
+2NkBT (3.11)
+as the average energy per bit. After this we need only one more equation:
+E=Mc2: (3.12)
+HereMrepresents the mass that would emerge in the part of space enclosed by the
+screen, see gure 3. Even though the mass is not directly visible in the emerged space,
+its presence is noticed though its energy.
+The rest is straightforward: one eliminates Eand inserts the expression for the
+number of bits to determine T. Next one uses the postulate (3.6) for the change of
+entropy to determine the force. Finally one inserts
+A= 4R2:
+and one obtains the familiar law:
+F=GMm
+R2: (3.13)
+We have recovered Newton's law of gravitation, practically from rst principles!
+These equations do not just come out by accident. It had to work, partly for
+dimensional reasons, and also because the laws of Newton have been ingredients in the
+steps that lead to black hole thermodynamics and the holographic principle. In a sense
+we have reversed these arguments. But the logic is clearly dierent, and sheds new
+light on the origin of gravity: it is an entropic force! That is the main statement, which
+is new and has not been made before. If true, this should have profound consequences.
+93.3 Naturalness and robustness of the derivation.
+Our starting point was that space has one emergent holographic direction. The addi-
+tional ingredients were that (i) there is a change of entropy in the emergent direction
+(ii) the number of degrees of freedom are proportional to the area of the screen, and
+(iii) the energy is evenly distributed over these degrees of freedom. After that it is
+unavoidable that the resulting force takes the form of Newton's law. In fact, this rea-
+soning can be generalized to arbitrary dimensions4with the same conclusion. But how
+robust and natural are these heuristic arguments?
+Perhaps the least obvious assumption is equipartition, which in general holds only
+for free systems. But how essential is it? Energy usually spreads over the microscopic
+degrees of freedom according to some non trivial distribution function. When the lost
+bits are randomly chosen among all bits, one expects the energy change associated with
+
Sstill to be proportional to the energy per unit area E=A. This fact could therefore
+be true even when equipartition is not strictly obeyed.
+Why do we need the speed of light cin this non relativistic context? It was necessary
+to translate the mass Min to an energy, which provides the heat bath required for
+the entropic force. In the non-relativistic setting this heat bath is innite, but in
+principle one has take into account that the heat bath loses or gains energy when the
+particle changes its location under the in
uence of an external force. This will lead to
+relativistic redshifts, as we will see.
+Since the postulate (3.5) is the basic assumption from which everything else follows,
+let us discuss its meaning in more detail. Why does the entropy precisely change like
+this when one shifts by one Compton wave length? In fact, one may wonder why
+we needed to introduce Planck's constant in the rst place, since the only aim was
+to derive the classical laws of Newton. Indeed, ~eventually drops out of the most
+important formulas. So, in principle one could multiply it with any constant and still
+obtain the same result. Hence, ~just serves as an auxiliary variable that is needed
+for dimensional reasons. It can therefore be chosen at will, and dened so that (3.5)
+is exactly valid. The main content of this equation is therefore simply that there is
+an entropy change perpendicular to the screen proportional to the mass mand the
+displacement 
 x. That is all there is to it.
+If we would move further away from the screen, the change in entropy will in general
+no longer be given by the same rule. Suppose the particle stays at radius Rwhile the
+screen is moved to R0<R. The number of bits on the screen is multiplied by a factor
+(R0=R)2, while the temperature is divided by the same factor. Eectively, this means
+that only ~is multiplied by that factor, and since it drops out, the resulting force will
+stay the same. In this situation the information associated with the particle is no longer
+concentrated in a small area, but spreads over the screen. The next section contains a
+proposal for precisely how it is distributed, even for general matter congurations.
+4Inddimensions (3.10) includes a factor1
+2d2
+d3to get the right identication with Newton's constant.
+103.4 Inertia and the Newton potential.
+To complete the derivation of the laws of Newton we have to understand why the symbol
+a, that was basically introduced by hand in (3.8), is equal to the physical acceleration
+x. In fact, so far our discussion was quasi static, so we have not determined yet how
+to connect space at dierent times. In fact, it may appear somewhat counter-intuitive
+that the temperature Tis related to the vector quantity a, while in our identications
+the entropy gradient 
 S=
xis related to the scalar quantity m. In a certain way it
+seems more natural to have it the other way around.
+So let reconsider what happens to the particle with mass mwhen it approaches the
+screen. Here it should merge with the microscopic degrees of freedom on the screen,
+and hence it will be made up out of the same bits as those that live on the screen.
+Since each bit carries an energy1
+2kBT, the number of bits nfollows from
+mc2=1
+2nkBT: (3.14)
+When we insert this in to equation (3.6), and use (3.8), we can express the entropy
+change in terms of the acceleration as
+
S
+n=kBa
x
+2c2(3.15)
+By combining the above equations one of course again recovers F=maas the entropic
+force. But, by introducing the number of bits nassociated with the particle, we
+succeeded in making the identications more natural in terms of their scalar versus
+vector character. In fact, we have eliminated ~from the equations, which in view of
+our earlier comment is a good thing.
+Thus we conclude that acceleration is related to an entropy gradient. This will be
+one of our main principles: inertia is a consequence of the fact that a particle in rest
+will stay in rest because there are no entropy gradients. Given this fact it is natural to
+introduce the Newton potential  and write the acceleration as a gradient
+a=r:
+This allows us to express the change in entropy in the concise way
+
S
+n=kB

+2c2(3.16)
+We thus reach the important conclusion that the Newton potential  keeps track of
+the depletion of the entropy per bit. It is therefore natural to identify it with a coarse
+graining variable, like the (renormalization group) scale in AdS/CFT. Indeed, in the
+next section we propose a holographic scenario for the emergence of space in which the
+Newton potential precisely plays that role. This allows us to generalize our discussion
+to other mass distributions and arbitrary positions in a natural and rather beautiful
+way, and give additional support for the presented arguments.
+114 Emergent Gravity for General Matter Distributions.
+Space emerges at a macroscopic level only after coarse graining. Hence, there will be
+a nite entropy associated with each matter conguration. This entropy measures the
+amount of microscopic information that is invisible to the macroscopic observer. In
+general, this amount will depend on the distribution of the matter. The information is
+being processed by the microscopic dynamics, which looks random from a macroscopic
+point of view. But to determine the force we don't need the details of the information,
+nor the exact dynamics, only the amount of information given by the entropy, and the
+energy that is associated with it. If the entropy changes as a function of the location
+of the matter distribution, it will lead to an entropic force.
+Therefore, space can not just emerge by itself. It has to be endowed by a book
+keeping device that keeps track of the amount of information for a given energy dis-
+tribution. It turns out, that in a non relativistic situation this device is provided by
+Newton's potential . And the resulting entropic force is called gravity.
+We start from microscopic information. It is assumed to be stored on holographic
+screens. Note that information has a natural inclusion property: by forgetting certain
+bits, by coarse graining, one reduces the amount of information. This coarse graining
+can be achieved through averaging, a block spin transformation, integrating out, or
+some other renormalization group procedure. At each step one obtains a further coarse
+grained version of the original microscopic data. The gravitational or closed string side
+of these dualities is by many still believed to be independently dened. But in our
+view these are macroscopic theories, which by chance we already knew about before we
+understood they were the dual of a microscopic theory without gravity. We can't resist
+making the analogy with a situation in which we would have developed a theory for
+elasticity using stress tensors in a continuous medium half a century before knowing
+about atoms. We probably would have been equally resistant in accepting the obvious.
+Gravity and closed strings are not much dierent, but we just have not yet got used
+to the idea.
+The coarse grained data live on smaller screens obtained by moving the rst screen
+further into the interior of the space. The information that is removed by coarse
+graining is replaced by the emerged part of space between the two screens. In this way
+one gets a nested or foliated description of space by having surfaces contained within
+surfaces. In other words, just like in AdS/CFT, there is one emerging direction in
+space that corresponds to a "coarse graining" variable, something like the cut-o scale
+of the system on the screens.
+A priori there is no preferred holographic direction in 
at space. However, this is
+where we use our observation about the Newton potential. It is the natural variable
+that measures the amount of coarse graining on the screens. Therefore, the holographic
+direction is given by the gradient r of the Newton potential. In other words, the
+holographic screens correspond to equipotential surfaces. This leads to a well dened
+12Figure 4: The holographic screens are located at equipotential surfaces. The information on
+the screens is coarse grained in the direction of decreasing values of the Newton potential .
+The maximum coarse graining happens at black hole horizons, when  =2c2=1.
+foliation of space, except that screens may break up in to disconnected parts that each
+enclose dierent regions of space. This is depicted in gure 4.
+The amount of coarse graining is measured by the ratio =2c2, as can be seen
+from (3.16). Note that this is a dimensionless number that is always between zero and
+one. It is only equal to one on the horizon of a black hole. We interpret this as the
+point where all bits have been maximally coarse grained. Thus the foliation naturally
+stops at black hole horizons.
+4.1 The Poisson equation for general matter distributions.
+Consider a microscopic state, which after coarse graining corresponds to a given mass
+distribution in space. All microscopic states that lead to the same mass distribution
+belong to the same macroscopic state. The entropy for each of these state is dened
+as the number of microscopic states that 
ow to the same macroscopic state.
+We want to determine the gravitational force by using virtual displacements, and
+calculating the associated change in energy. So, let us freeze time and keep all the
+matter at xed locations. Hence, it is described by a static matter density (~ r). Our
+aim is to obtain the force that the matter distribution exerts on a collection of test
+particles with masses miand positions ~ ri.
+We choose a holographic screen Scorresponding to an equipotential surface with
+xed Newton potential  0. We assume that the entire mass distribution given by (x) is
+contained inside the volume enclosed by the screen, and all test particles are outside this
+volume. To explain the force on the particles, we again need to determine the work that
+is performed by the force and show that it is naturally written as the change in entropy
+multiplied by the temperature. The dierence with the spherically symmetric case is
+that the temperature on the screen is not necessarily constant. Indeed, the situation
+13is in general not in equilibrium. Nevertheless, one can locally dene temperature and
+entropy per unit area.
+First let us identify the temperature. We do this by taking a test particle and
+moving it close to the screen, and measuring the local acceleration. Thus, motivated
+by our earlier discussion we dene temperature analogous to (3.8), namely by
+kBT=1
+2~r
+kc: (4.17)
+Here the derivative is taken in the direction of the outward pointing normal to the
+screen. Note at this point  is just introduced as a device to describe the local accel-
+eration, but we don't know yet whether it satises an equation that relates it to the
+mass distribution.
+The next ingredient is the density of bits on the screen. We again assume that
+these bits are uniformly distributed, and so (3.10) is generalized to
+dN=c3
+G~dA: (4.18)
+Now let us impose the analogue of the equipartition relation (3.11). It is takes the
+form of an integral expression for the energy
+E=1
+2kBZ
+STdN: (4.19)
+It is an amusing exercise to work out the consequence of this relation. Of course, the
+energyEis again expressed in terms of the total enclosed mass M. After inserting our
+identications for the left hand side one obtains a familiar relation: Gauss's law!
+M=1
+4GZ
+Sr
dA: (4.20)
+This should hold for arbitrary screens given by equipotential surfaces. When a bit of
+mass is added to the region enclosed by the screen S, for example, by rst putting it
+close to the screen and then pushing it across, the mass Mshould change accordingly.
+This condition can only hold in general if the potential  satises the Poisson equation
+r2(~ r) = 4G(~ r): (4.21)
+We conclude that by making natural identications for the temperature and the in-
+formation density on the holographic screens, that the laws of gravity come out in a
+straightforward fashion.
+14€ m1  €  F 1  € δ r 1€ m2  €  F 2
+€ m3  €  F 3€ Φ=Φ0Figure 5: A general mass distribution inside the not yet emerged part of space enclosed by
+the screen. A collection of test particles with masses miare located at arbitrary points ~ riin
+the already emerged space outside the screen. The forces ~Fidue to gravity are determined
+by the virtual work done after innitesimal displacement ~ riof the particles.
+4.2 The gravitational force for arbitrary particle locations.
+The next issue is to obtain the force acting on matter particles that are located at
+arbitrary points outside the screen. For this we need a generalization of the rst pos-
+tulate (3.6) to this situation. What is the entropy change due to arbitrary innitesimal
+displacements ~ riof the particles? There is only one natural choice here. We want to
+nd the change sin the entropy density locally on the screen S. We noted in (3.16)
+that the Newton potential  keeps track of the changes of information per unit bit.
+Hence, the right identication for the change of entropy density is
+s=kB
+2c2dN (4.22)
+where is the response of the Newton potential due to the shifts ~ riof the positions
+of the particles. To be specic,  is determined by solving the variation of the Poisson
+equation
+r2(~ r) = 4GX
+imi~ ri
ri(~ r~ ri) (4.23)
+One can verify that with this identication one indeed reproduces the entropy shift
+(3.6) when one of the particles approaches the screen.
+Let us now determine the entropic forces on the particles. The combined work done
+by all of the forces on the test particles is determined by the rst law of thermody-
+namics. However, we need to express in terms of the local temperature and entropy
+15variation. Hence,X
+i~Fi
~ ri=Z
+STs (4.24)
+To see that this indeed gives the gravitational force in the most general case, one simply
+has to use the electrostatic analogy. Namely, one can redistribute the entire mass Mas
+a mass surface density over the screen Swithout changing the forces on the particles.
+The variation of the Newton potential can be obtained from the Greens function for
+the Laplacian. The rest of the proof is a straightforward application of electrostatics,
+but then applied to gravity. The basic identity one needs to prove is
+X
+i~Fi
~ ri=1
+4GZ
+S
+rr

+dA (4.25)
+which holds for any location of the screen outside the mass distribution. This is easily
+veried by using Stokes theorem and the Laplace equation. The second term vanishes
+when the screen is chosen at a equipotential surface. To see this, simply replace  by
+0and pull it out of the integral. Since  is sourced by only the particles outside the
+screen, the remaining integral just gives zero.
+The forces we obtained are independent of the choice of the location of the screen.
+We could have chosen any equipotential surface, and we would obtain the same values
+for~Fi, the ones described by the laws of Newton. That all of this works is not just
+a matter of dimensional analysis. The invariance under the choice of equipotential
+surface is very much consistent with idea that a particular location corresponds to an
+arbitrary choice of the scale that controls the coarse graining of the microscopic data.
+The macroscopic physics, in particular the forces, should be independent of that choice.
+5 The Equivalence Principle and the Einstein equations.
+Since we made use of the speed of light cin our arguments, it is a logical step to try
+and generalize our discussion to a relativistic situation. So let us assume that the mi-
+croscopic theory knows about Lorentz symmetry, or even has the Poincar e group as a
+global symmetry. This means we have to combine time and space in to one geometry.
+A scenario with emergent space-time quite naturally leads to general coordinate invari-
+ance and curved geometries, since a priori there are no preferred choices of coordinates,
+nor a reason why curvatures would not be present. Specically, we would like to see
+how Einstein's general relativity emerges from similar reasonings as in the previous
+section. We will indeed show that this is possible. But rst we study the origin of
+inertia and the equivalence principle.
+165.1 The law of inertia and the equivalence principle.
+Consider a static background with a global time like Killing vector a. To see the
+emergence of inertia and the equivalence principle, one has to relate the choice of this
+Killing vector eld with the temperature and the entropy gradients. In particular, we
+like to see that the usual geodesic motion of particles can be understood as being the
+result of an entropic force.
+In general relativity5the natural generalization of Newton's potential is [11],
+=1
+2log(aa): (5.26)
+Its exponent erepresents the redshift factor that relates the local time coordinate to
+that at a reference point with = 0, which we will take to be at innity.
+Just like in the non relativistic case, we like to use to dene a foliation of space,
+and put our holographic screens at surfaces of constant redshift. This is a natural
+choice, since in this case the entire screen uses the same time coordinate. So the
+processing of the microscopic data on the screen can be done using signals that travel
+without time delay.
+We want to show that the redshift perpendicular to the screen can be understood
+microscopically as originating from the entropy gradients6. To make this explicit, let
+us consider the force that acts on a particle of mass m. In a general relativistic setting
+force is less clearly dened, since it can be transformed away by a general coordinate
+transformation. But by using the time-like Killing vector one can given an invariant
+meaning to the concept of force [11].
+The four velocity uaof the particle and its acceleration abuaraubcan be expressed
+in terms of the Killing vector bas
+ub=eb; ab=e2arab:
+We can further rewrite the last equation by making use of the Killing equation
+rab+rba= 0
+and the denition of . One nds that the acceleration can again be simply expressed
+the gradient
+ab=rb: (5.27)
+Note that just like in the non relativistic situation the acceleration is perpendicular
+to screenS. So we can turn it in to a scalar quantity by contracting it with a unit
+outward pointing vector Nbnormal to the screen Sand tob.
+5In this subsection and the next we essentially follow Wald's book on general relativity (pg. 288-
+290). We use a notation in which candkBare put equal to one, but we will keep Gand~explicit.
+6In this entire section it will be very useful to keep the polymer example of section 2 in mind, since
+that will make the logic of our reasoning very clear.
+17The local temperature Ton the screen is now in analogy with the non relativistic
+situation dened by
+T=~
+2eNbrb: (5.28)
+Here we inserted a redshift factor e, because the temperature Tis measured with
+respect to the reference point at innity.
+To nd the force on a particle that is located very close to the screen, we rst use
+again the same postulate as in section two. Namely, we assume that the change of
+entropy at the screen is 2 for a displacement by one Compton wavelength normal to
+the screen. Hence,
+raS=2m
+~Na; (5.29)
+where the minus sign comes from the fact that the entropy increases when we cross
+from the outside to the inside. The comments made on the validity of this postulate
+in section 3.3 apply here as well. The entropic force now follows from (5.28)
+Fa=TraS=mera (5.30)
+This is indeed the correct gravitational force that is required to keep a particle at xed
+position near the screen, as measured from the reference point at innity. It is the
+relativistic analogue of Newton's law of inertia F=ma. The additional factor eis
+due to the redshift. Note that ~has again dropped out.
+It is instructive to rewrite the force equation (5.30) in a microcanonical form. Let
+S(E;xa) be the total entropy associated with a system with total energy Ethat contains
+a particle with mass mat position xa. HereEalso includes the energy of the particle.
+The entropy will in general also dependent on many other parameters, but we suppress
+these in this discussion.
+As we explained in the section 2, an entropic force can be determined micro-
+canonically by adding by hand an external force term, and impose that the entropy is
+extremal. For this situation this condition looks like
+d
+dxaS
+E+e(x)m; xa

+= 0: (5.31)
+One easily veries that this leads to the same equation (5.30). This xes the equilibrium
+point where the external force, parametrized by (x) and the entropic force statistically
+balance each other. Again we stress the point that there is no microscopic force acting
+here! The analogy with equation (2.4) for the polymer, discussed in section 2, should
+be obvious now.
+Equation (5.31) tells us that the entropy remains constant if we move the particle
+and simultaneously reduce its energy by the redshift factor. This is true only when the
+particle is very light, and does not disturb the other energy distributions. It simply
+18serves as a probe of the emergent geometry. This also means that redshift function
+(x) is entirely xed by the other matter in the system.
+We have arrived at equation (5.31) by making use of the identications of the
+temperature and entropy variations in space time. But actually we should have gone
+the other way. We should have started from the microscopics and dened the space
+dependent concepts in terms of them. We chose not to follow such a presentation, since
+it might have appeared somewhat contrived.
+But it is important to realize that the redshift must be seen as a consequence
+of the entropy gradient and not the other way around. The equivalence principle
+tells us that redshifts can be interpreted in the emergent space time as either due
+to a gravitational eld or due to the fact that one considers an accelerated frame.
+Both views are equivalent in the relativistic setting, but neither view is microscopic.
+Acceleration and gravity are both emergent phenomena.
+5.2 Derivation of the Einstein equations.
+We now like to extend our derivation of the laws of gravity to the relativistic case, and
+obtain the Einstein equations. This can indeed be done in natural and very analogous
+fashion. So we again consider a holographic screen on closed surface of constant redshift
+. We assume that it is enclosing a certain static mass conguration with total mass
+M. The bit density on the screen is again given by
+dN=dA
+G~(5.32)
+as in (4.18). Following the same logic as before, let us assume that the energy associated
+with the mass Mis distributed over all the bits. Again by equipartition each bit carries
+a mass unit equal to1
+2T. Hence
+M=1
+2Z
+STdN (5.33)
+After inserting the identications for TanddNwe obtain
+M=1
+4GZ
+Ser
dA (5.34)
+Note that again ~drops out, as expected. The equation (5.34) is indeed known to
+be the natural generalization of Gauss's law to General Relativity. Namely, the right
+hand side is precisely Komar's denition of the mass contained inside an arbitrary
+volume inside any static curved space time. It can be derived by assuming the Einstein
+equations. In our reasoning, however, we are coming from the other side. We made
+identications for the temperature and the number of bits on the screen. But we don't
+19know yet whether it satises any eld equations. The key question at this point is
+whether the equation (5.34) is sucient to derive the full Einstein equations.
+An analogous question was addressed by Jacobson for the case of null screens. By
+adapting his reasoning to this situation, and combining it with Wald's exposition of
+the Komar mass, it is straightforward to construct an argument that naturally leads
+to the Einstein equations. We will present a sketch of this.
+First we make use of the fact that the Komar mass can be re-expressed in terms of
+the Killing vector aas
+M=1
+8GZ
+Sdxa^dxbabcdrcd(5.35)
+The left hand side can be expressed as an integral over the enclosed volume of certain
+components of stress energy tensor Tab. For the right hand side one rst uses Stokes
+theorem and subsequently the relation
+rarab=Rb
+aa(5.36)
+which is implied by the Killing equation for a. This leads to the integral relation [11]
+2Z
+
+Tab1
+2Tgab
+nabdV=1
+4GZ
+RabnabdV (5.37)
+where  is the three dimensional volume bounded by the holographic screen Sand
+nais its normal. The particular combination of the stress energy tensor on the left
+hand side can presumably be xed by comparing properties on both sides, such as for
+instance the conservation laws of the tensors that occur in the integrals.
+This equation is derived in a general static background with a time like Killing
+vectora. By requiring that it holds for arbitrary screens would imply that also the
+integrands on both sides are equal. This gives us only a certain component of the
+Einstein equation. In fact, we can choose the surface  in many ways, as long as its
+boundary is given by S. This means that we can vary the normal na. But that still
+leaves a contraction with the Killing vector a.
+To get to the full Einstein we now use a similar reasoning as Jacobson [7], except now
+applied to time-like screens. Let us consider a very small region of space time and look
+also at very short time scales. Since locally every geometry looks approximately like
+Minkowski space, we can choose approximate time like Killing vectors. Now consider
+a small local part of the screen, and impose that when matter crosses it, the value of
+the Komar integral will jump precisely by the corresponding mass m. By following the
+steps described above then leads to (5.37) for all these Killing vectors, and for arbitrary
+screens. This is sucient to obtain the full Einstein equations.
+205.3 The force on a collection of particles at arbitrary locations.
+We close this section with explaining the way the entropic force acts on a collection
+of particles at arbitrary locations xiaway from the screen and the mass distribution.
+The Komar denition of the mass will again be useful for this purpose. The denition
+of the Komar mass depends on the choice of Killing vector a. In particular also in its
+norm, the redshift factor e. If one moves the particles by a virtual displacement xi,
+this will eect the denition of the Komar mass of the matter conguration inside the
+screen. In fact, the temperature on the screen will be directly aected by a change in
+the redshift factor.
+The virtual displacements can be performed quasi statically, which means that
+the Killing vector itself is still present. Its norm, or the redshift factor, may change,
+however. In fact, also the spatial metric may be aected by this displacement. We are
+not going to try to solve these dependences, since that would be impossible due to the
+non linearity of the Einstein equations. But we can simply use the fact that the Komar
+mass is going to be a function of the positions xiof the particles.
+Next let us assume that in addition to this xidependence of the Komar mass that
+the entropy of the entire system has also explicit xidependences, simply due to changes
+in the amount of information. These are de dierence that will lead to the entropic
+force that we wish to determine. We will now give a natural prescription for the entropy
+dependence that is based on a maximal entropy principle and indeed gives the right
+forces. Namely, assume that the entropy may be written as a function of the Komar
+massMand in addition on the xi. But since the Komar mass should be regarded
+as a function M(xi) of the positions xi, there will be an explicit and an implicit xi
+dependence in the entropy. The maximal entropy principle implies that these two
+dependences should cancel. So we impose
+S
+M(xi+xi); xi+xi

+=S
+M(xi);xi

+(5.38)
+By working out this condition and singling out the equations implied by each variation
+xione nds
+riM+TriS= 0 (5.39)
+The point is that the rst term represents the force that acts on the i-th particle due
+to the mass distribution inside the screen. This force is indeed equal to minus the
+derivative of the Komar mass, simply because of energy conservation. But again, this
+is not the microscopic reason for the force. In analogy with the polymer, the Komar
+mass represents the energy of the heat bath. Its dependence on the position of the
+other particles is caused by redshifts whose microscopic origin lies in the depletion of
+the energy in the heat bath due to entropic eects.
+Since the Komar integral is dened on the holographic screen, it is clear that like
+in the non relativistic case the force is expressed as an integral over this screen as well.
+We have not tried to make this representation more concrete. Finally, we note that
+21this argument was very general, and did not really use the precise form of the Komar
+integral, or the Einstein equations. So it should be straightforward to generalize this
+reasoning to higher derivative gravity theories by making use of Wald's Noether charge
+formalism [12], which is perfectly suitable for this purpose.
+6 Conclusion and Discussion
+The ideas and results presented in this paper lead to many questions. In this section
+we discuss and attempt to answer some of these. First we present our main conclusion.
+6.1 The end of gravity as a fundamental force.
+Gravity has given many hints of being an emergent phenomenon, yet up to this day
+it is still seen as a fundamental force. The similarities with other known emergent
+phenomena, such as thermodynamics and hydrodynamics, have been mostly regarded
+as just suggestive analogies. It is time we not only notice the analogy, and talk about
+the similarity, but nally do away with gravity as a fundamental force.
+Of course, Einstein's geometric description of gravity is beautiful, and in a certain
+way compelling. Geometry appeals to the visual part of our minds, and is amazingly
+powerful in summarizing many aspects of a physical problem. Presumably this explains
+why we, as a community, have been so reluctant to give up the geometric formulation
+of gravity as being fundamental. But it is inevitable we do so. If gravity is emergent,
+so is space time geometry. Einstein tied these two concepts together, and both have to
+be given up if we want to understand one or the other at a more fundamental level.
+The results of this paper suggest gravity arises as an entropic force, once space and
+time themselves have emerged. If the gravity and space time can indeed be explained
+as emergent phenomena, this should have important implications for many areas in
+which gravity plays a central role. It would be especially interesting to investigate
+the consequences for cosmology. For instance, the way redshifts arise from entropy
+gradients could lead to many new insights.
+The derivation of the Einstein equations presented in this paper is analogous to
+previous works, in particular [7]. Also other authors have proposed that gravity has
+an entropic or thermodynamic origin, see for instance [14]. But we have added an
+important element that is new. Instead of only focussing on the equations that govern
+the gravitational eld, we uncovered what is the origin of force and inertia in a context
+in which space is emerging. We identied a cause, a mechanism, for gravity. It is driven
+by dierences in entropy, in whatever way dened, and a consequence of the statistical
+averaged random dynamics at the microscopic level. The reason why gravity has to
+keep track of energies as well as entropy dierences is now clear. It has to, because
+this is what causes motion!
+The presented arguments have admittedly been rather heuristic. One can not expect
+22otherwise, given the fact that we are entering an unknown territory in which space does
+not exist to begin with. The profound nature of these questions in our view justies
+the heuristic level of reasoning. The assumptions we made have been natural: they t
+with existing ideas and are supported by several pieces of evidence. In the following
+we gather more supporting evidence from string theory, the AdS/CFT correspondence,
+and black hole physics.
+6.2 Implications for string theory and relation with AdS/CFT.
+If gravity is just an entropic force, then what does this say about string theory? Gravity
+is seen as an integral part of string theory, that can not be taken out just like that.
+But we do know about dualities between closed string theories that contain gravity
+and decoupled open string theories that don't. A particularly important example is
+the AdS/CFT correspondence.
+The open/closed string and AdS/CFT correspondences are manifestations of the
+UV/IR connection that is deeply engrained within string theory. This connection
+implies that short and long distance physics can not be seen as totally decoupled.
+Gravity is a long distance phenomenon that clearly knows about short distance physics,
+since it is evident that Newton's constant is a measure for the number of microscopic
+degrees of freedom. String theory invalidates the "general wisdom" underlying the
+Wilsonian eective eld theory, namely that integrating out short distance degrees
+of freedom only generate local terms in the eective action, most of which become
+irrelevant at low energies. If that were completely true, the macroscopic physics would
+be insensitive to the short distance physics.
+The reason why the Wilsonian argument fails, is that it makes a too conservative
+assumption about the asymptotic growth of the number of states at high energies. In
+string theory the number of high energy open string states is such that integrating them
+out indeed leads to long range eects. Their one loop amplitudes are equivalent to the
+tree level contributions due to the exchange of close string states, which among other
+are responsible for gravity. This interaction is, however, equivalently represented by
+the sum over all quantum contributions of the open string. In this sense the emergent
+nature of gravity is also supported by string theory.
+The AdS/CFT correspondence has an increasing number of applications to areas of
+physics in which gravity is not present at a fundamental level. Gravitational techniques
+are used as tools to calculate physical quantities in a regime where the microscopic
+description fails. The latest of these developments is the application to condensed
+matter theory. No one doubts that in these situations gravity emerges only as an
+eective description. It arises not in the same space as the microscopic theory, but in
+a holographic scenario with one extra dimension. No clear explanation exists of where
+this gravitational force comes from. The entropic mechanism described in this paper
+should be applicable to these physical systems, and explain the emergence of gravity.
+23Open  IR Closed 
+IR a) b) Figure 6: The microscopic theory in a) is eectively described by a string theory consisting
+of open and closed strings as shown in b). Both types of strings are cut o in the UV.
+The holographic scenario discussed in this paper has certainly be inspired by the
+way holography works in AdS/CFT and open closed string correspondences. In string
+language, the holographic screens can be identied with D-branes, and the microscopic
+degrees of freedom on these screens represented as open strings dened with a cut o
+in the UV. The emerged part of space is occupied by closed strings, which are also
+dened with a UV cut o, as shown in gure 6. The open and closed string cut os
+are related by the UV/IR correspondence: pushing the open string cut o to the UV
+forces the closed string cut o towards the IR, and vice versa. The value of the cut os
+is determined by the location of the screen. Integrating out the open strings produces
+the closed strings, and leads to the emergence of space and gravity. Note, however,
+that from our point of view the existence of gravity or closed strings is not assumed
+microscopically: they are emergent as an eective description.
+In this way, the open/closed string correspondence supports the interpretation of
+gravity as an entropic force. Yet, many still see the closed string side of these dualities
+is a well dened fundamental theory. But in our view gravity and closed strings are
+emergent and only present as macroscopic concept. It just happened that we already
+knew about gravity before we understood it could be obtained from a microscopic
+theory without it. We can't resist making the analogy with a situation in which we
+would have developed a theory for elasticity using stress tensors in a continuous medium
+half a century before knowing about atoms. We probably would have been equally
+resistant in accepting the obvious. Gravity and closed strings are not much dierent,
+but we just have to get used to the idea.
+246.3 Black hole horizons revisited
+We saved perhaps the clearest argument for the fact that gravity is an entropic force
+to the very last. The rst cracks in the fundamental nature of gravity appeared when
+Bekenstein, Hawking and others discovered the laws of black hole thermodynamics. In
+fact, the thought experiment mentioned in section 3 that led Bekenstein to his entropy
+law is surprisingly similar to the polymer problem. The black hole serves as the heat
+bath, while the particle can be thought of as the end point of the polymer that is
+gradually allowed to go back to its equilibrium situation.
+Of course, there is no polymer in the gravity system, and there appears to be no
+direct contact between the particle and the black hole. But here we are ignoring the
+fact that one of the dimensions is emergent. In the holographic description of this same
+process, the particle can be thought of as being immersed in the heat bath representing
+the black hole. This fact is particularly obvious in the context of AdS/CFT, in which a
+black hole is dual to a thermal state on the boundary, while the particle is represented
+as a delocalized operator that is gradually being thermalized. By the time that the
+particle reaches the horizon it has become part of the thermal state, just like the
+polymer. This phenomenon is clearly entropic in nature, and is the consequence of a
+statistical process that drives the system to its state of maximal entropy.
+Upon closer inspection Bekenstein's reasoning can be used to show that gravity
+becomes an entropic force near the horizon, and that the equations presented in section
+3 are exactly valid. He argued that one has to choose a location slightly away from the
+black hole horizon at a distance of about the order of the Compton wave length, where
+we declare that the particle and the black hole have become one system. Let us say
+this location corresponds to choosing a holographic screen. The precise location of this
+screen can not be important, however, since there is not a natural preferred distance
+that one can choose. The equations should therefore not depend on small variations of
+this distance.
+By pulling out the particle a bit further, one changes its energy by a small amount
+equal to the work done by the gravitational force. If one then drops the particle in
+to the black hole, the mass Mincreases by this same additional amount. Consistency
+of the laws of black hole thermodynamics implies that the additional change in the
+Bekenstein Hawking entropy, when multiplied with the Hawking temperature TH, must
+be precisely equal to the work done by gravity. Hence,
+Fgravity =TH@SBH
+@x: (6.40)
+The derivative of the entropy is dened as the response of SBHdue to a change in the
+distancexof the particle to the horizon. This fact is surely known, and probably just
+regarded as a consistency check. But let us take it one or two steps further.
+Suppose we take the screen even further away from the horizon. The same argument
+applies, but we can also choose to ignore the fact that the screen has moved, and lower
+25the particle to the location of the previous screen, the one closer to the horizon. This
+process would happen in the part of space behind the new screen location, and hence
+it should have a holographic representation on the screen. In this system there is
+force in the perpendicular direction has no microscopic meaning, nor acceleration. The
+coordinate xperpendicular to the screen is just some scale variable associated with
+the holographic image of the particle. Its interpretation as an coordinate is a derived
+concept: this is what it means have an emergent space.
+The mass is dened in terms the energy associated with the particle's holographic
+image, which presumably is a near thermal state. It is not exactly thermal, however,
+because it is still slightly away from the black hole horizon. We have pulled it out of
+equilibrium, just like the polymer. One may then ask: what is the cause of the change
+in energy that is holographically dual to the work done when in the emergent space we
+gradually lower the particle towards the location of the old screen behind the new one
+. Of course, this can be nothing else then an entropic eect, the force is simply due to
+the thermalization process. We must conclude that the only microscopic explanation
+is that there is an emergent entropic force acting. In fact, the correspondence rules
+between the scale variable and energy on the one side, and the emergent coordinate x
+the massmon the other, must be such that F=TrStranslates in to the gravitational
+force. It is straightforward to see that this indeed works and that the equations for the
+temperature and the entropy change are exactly as given in section 3.
+The horizon is only a special location for observers that stay outside the black
+hole. The black hole can be arbitrarily large and the gravitational force at its horizon
+arbitrarily weak. Therefore, this thought experiment is not just teaching us about
+black holes. It teaches us about the nature of space time, and the origin of gravity.
+Or more precisely, it tells us about the cause of inertia. We can do the same thought
+experiment for a Rindler horizon, and reach exactly the same conclusion. In this case
+the correspondence rules must be such that F=TrStranslates in to the inertial force
+F=ma. Again the formulas work out as in section 3.
+6.4 Final comments
+This brings us to a somewhat subtle and not yet fully understood aspect. Namely, the
+role of ~. The previous arguments make clear that near the horizon the equations are
+valid with ~identied with the actual Planck constant. However, we have no direct
+conrmation or proof that the same is true when we go away from the horizon, or
+eespecially when there is no horizon present at all. In fact, there are reasons to believe
+that the equations work slightly dierent there. The rst is that one is not exactly
+at thermal equilibrium. Horizons have well dened temperatures, and clearly are in
+thermal equilibrium. If one assumes that the screen at an equipotential surface with
+= 0is in equilibrium, the entropy needed to get the Unruh temperature(3.8) is given
+26by the Bekenstein-Hawking formula, including the factor 1 =4,
+S=c3
+4G~Z
+SdA: (6.41)
+This value for this entropy appears to be very high, and violates the Bekenstein bound
+[15] that states that a system contained in region with radius Rand total energy E
+can not have an entropy larger than ER. The reason for this discrepancy may be that
+Bekenstein's argument does not hold for the holographic degrees of freedom on the
+screen, or because of the fact that we are far from equilibrium.
+But there may also be other ways to reconcile these statements, for example by
+making use of the freedom to rescale the value of ~. This would not eect the nal
+outcome for the force, nor the fact that it is entropic. In fact, one can even multiply ~
+by a function f(0) of the Newton potential on the screen. This rescaling would aect
+the values of the entropy and the temperature in opposite directions: Tgets multiplied
+by a factor, while Swill be divided by the same factor. Since a priori we can not
+exclude this possibility, there is something to be understood. In fact, there are even
+other modications possible, like a description that uses weighted average over many
+screens with dierent temperatures. Even then the essence of our conclusions would
+not change, which is the fact that gravity and inertia are entropic forces.
+Does this view of gravity lead to predictions? The statistical average should give
+the usual laws, hence one has to study the 
uctuations in the gravitational force. Their
+size depends on the eective temperature, which may not be universal and depends on
+the eective value of ~. An interesting thought is that 
uctuations may turn out to
+be more pronounced for weak gravitational elds between small bodies of matter. But
+clearly, we need a better understanding of the theory to turn this in to a prediction.
+It is well known that Newton was criticized by his contemporaries, especially by
+Hooke, that his law of gravity acts at a distance and has no direct mechanical cause
+like the elastic force. Ironically, this is precisely the reason why Hooke's elastic force is
+nowadays not seen as fundamental, while Newton's gravitational force has maintained
+that status for more than three centuries. What Newton did not know, and certainly
+Hooke didn't, is that the universe is holographic. Holography is also an hypothesis, of
+course, and may appear just as absurd as an action at a distance.
+One of the main points of this paper is that the holographic hypothesis provides
+a natural mechanism for gravity to emerge. It allows direct "contact" interactions
+between degrees of freedom associated with one material body and another, since all
+bodies inside a volume can be mapped on the same holographic screen. Once this
+is done, the mechanisms for Newton's gravity and Hooke's elasticity are surprisingly
+similar. We suspect that neither of these rivals would have been happy with this
+conclusion.
+27Acknowledgements
+This work is partly supported by Stichting FOM. I like to thank J. de Boer, B.
+Chowdhuri, R. Dijkgraaf, P. McFadden, G. 't Hooft, B. Nienhuis, J.-P. van der Schaar,
+and especially M. Shigemori, K. Papadodimas, and H. Verlinde for discussions and
+comments.
+References
+[1] J. D. Bekenstein, \Black holes and entropy," Phys. Rev. D 7, 2333 (1973).
+[2] J. M. Bardeen, B. Carter and S. W. Hawking, \The Four laws of black hole
+mechanics," Commun. Math. Phys. 31, 161 (1973).
+[3] S . W. Hawking , \Particle Creation By Black Holes," Commun Math. Phys. 43,
+199-220, (1975)
+[4] P. C. W. Davies, "Scalar particle production in Schwarzschild and Rindler met-
+rics," J. Phys. A 8, 609 (1975)
+[5] W. G. Unruh, \Notes on black hole evaporation," Phys. Rev. D 14, 870 (1976).
+[6] T. Damour, "Surface eects in black hole physics, in Proceedings of the Sec-
+ond Marcel Grossmann Meeting on General Relativity", Ed. R. Runi, North
+Holland, p. 587, 1982
+[7] T. Jacobson, \Thermodynamics of space-time: The Einstein equation of state,"
+Phys. Rev. Lett. 75, 1260 (1995)
+[8] G. 't Hooft, \Dimensional reduction in quantum gravity," arXiv:gr-qc/9310026.
+[9] L. Susskind, \The World As A Hologram," J. Math. Phys. 36, 6377 (1995)
+[arXiv:hep-th/9409089].
+[10] J. M. Maldacena, \The large N limit of superconformal eld theories and super-
+gravity," Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113
+(1999)]
+[11] R. M. Wald, "General Relativity," The University of Chicago Press, 1984
+[12] R. M. Wald, \Black hole entropy is the Noether charge," Phys. Rev. D 48, 3427
+(1993) [arXiv:gr-qc/9307038].
+[13] L. Susskind, \The anthropic landscape of string theory," arXiv:hep-th/0302219.
+28[14] T. Padmanabhan, \Thermodynamical Aspects of Gravity: New insights,"
+arXiv:0911.5004 [gr-qc], and references therein.
+[15] J. D. Bekenstein, \A Universal Upper Bound On The Entropy To Energy Ratio
+For Bounded Systems," Phys. Rev. D 23(1981) 287.
+29
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+arXiv:1001.0789v1  [math.NT]  5 Jan 2010PERFECT FORMS AND THE COHOMOLOGY OF MODULAR
+GROUPS
+PHILIPPEELBAZ-VINCENT,HERBERTGANGL,ANDCHRISTOPHESOU LÉ
+Contents
+1. Introduction 1
+2. Voronoï’s reduction theory 3
+3. TheVoronoï complex 4
+4. TheVoronoï complex in dimensions 5,6 and 7 6
+5. Explicit homology classes 11
+6. Splitting offthe Voronoï complex Vor Nfrom Vor N+1for small N 13
+7. TheCohomology of modular groups 15
+References 17
+Abstract. ForN=5,6and7,usingtheclassificationofperfectquadraticform s,
+wecomputethehomologyoftheVoronoïcellcomplexesattach edtothemodular
+groupsSLN(Z) andGLN(Z). From this we deduce the rational cohomology of
+those groups.
+1. Introduction
+LetN/greaterorequalslant1 be an integer and let SLN(Z) be the modular group of integral ma-
+trices with determinant one. Our goal is to compute its cohom ology groups with
+trivial coefficients, i.e. Hq/parenleftbigSLN(Z),Z). The case N=2 is well-known and follows
+from the fact that SL2(Z) is the amalgamated product of two finite cyclic groups
+([19], [4], II.7, Ex.3, p.52). The case N=3 was done in [21]: for any q>0 the
+groupHq/parenleftbigSL3(Z),Z/parenrightbigis killed by 12. The case N=4 has been studied by Lee and
+Szczarbain[12]: modulo2,3and5–torsion, thecohomology g roupHq/parenleftbigSL4(Z),Z/parenrightbig
+is trivial whenever q>0, except that H3/parenleftbigSL4(Z),Z/parenrightbig=Z. In Theorem 7.3 below,
+wesolve the cases N=5, 6and 7.
+For these calculations we follow the method of [12], i.e. we u se the perfect
+forms of Voronoï. Recall from [22] and [13] that a perfect for m inNvariables is
+a positive definite real quadratic form honRNwhich is uniquely determined (up
+toascalar) byitsset ofintegral minimal vectors. Voronoï p roved in[22] that there
+are finitely manyperfect forms of rank N, modulo theaction of SLN(Z). These are
+known for N/lessorequalslant8(see §2below).
+Voronoï used perfect forms to define a cell decomposition of t he space X∗
+Nof
+positive real quadratic forms, the kernel of which is defined overQ. This cell
+decomposition (cf. §3) isinvariant under SLN(Z), hence it can be used to compute
+1991Mathematics Subject Classification. 11H55,11F75,11F06,11Y99,55N91, 20J06,57-04.
+Key words and phrases. Perfect forms, Voronoï complex, group cohomology, modular groups,
+machine calculations.
+12 P h. Elbaz-Vincent,H.Gangland C. Soul´e
+the equivariant homology of X∗
+Nmodulo its boundary. On the other hand, this
+equivariant homology turns out to be isomorphic to the group sHq/parenleftbigSLN(Z),StN/parenrightbig,
+whereStNis the Steinberg module (see [5] and §3.4 below). Finally, Bo rel–Serre
+duality [5] asserts that the homology H∗/parenleftbigSLN(Z),StN/parenrightbigis dual to the cohomology
+H∗/parenleftbigSLN(Z),Z/parenrightbig(modulo torsion).
+To perform these computations for N/lessorequalslant7, we needed the help of a computer.
+ThereasonisthattheVoronoïcelldecompositionof X∗
+Ngetssoonverycomplicated
+whenNincreases. Forinstance,when N=7,therearemorethantwomillionorbits
+of cells of dimension 18, modulo the action of SLN(Z) (see Figure 2 below). For
+this purpose, we have developed a C library [16], which uses P ARI [15] for some
+functionalities. The algorithms are based on exact methods . Asa result weget the
+full Voronoï cell decomposition of the spaces X∗
+NforN/lessorequalslant7 (with either GLN(Z)
+orSLN(Z)action). Thosedecompositions aresummarizedinthefigure sandtables
+below. Thecomputationsweredoneonseveralcomputersusin gdifferentprocessor
+architectures (which is useful for checking the results) an d forN=7 the overall
+computational timewasmore than ayear.
+Thepaper isorganized asfollows. In§2,werecall theVorono ï theory ofperfect
+forms. In §3, we introduce a complex of abelian groups that we call the “Voronoï
+complex” which computes the homology groups Hq/parenleftbigSLN(Z),StN/parenrightbig. In §4, we ex-
+plain how to get an explicit description of the Voronoï compl ex in rank N=5, 6
+or 7, starting from the description of perfect forms availab le in the literature (es-
+pecially in the work of Jaquet [11]). In Figures 1 and 2 we disp lay the rank of
+the groups in the Voronoï complex and in Tables 1–5 we give the elementary divi-
+sors of its differentials. The homology of the Voronoï complex (hence the gr oups
+Hq(SLN(Z),StN)) follows from this. It isgiven in Theorem 4.3.
+We found two methods to test whether our computations are cor rect. First,
+checking that the virtual Euler characteristic of SLN(Z) vanishes leads to a mass
+formula for the orders of the stabilizers of the cells of X∗
+N(cf. §4.5). Second, the
+identitydn−1◦dn=0 for the differentials in the Voronoï complex is a non-trivial
+equality whenthese di fferentials arewritten asexplicit (large) matrices.
+In §5 we give an explicit formula for the top homology group of the Voronoï
+complex (Theorem 5.1). In §6 we prove that the Voronoï comple x ofGLN(Z) is a
+direct factor of the Voronoï complex of GL6(Z) shifted by one. Finally, in §7 we
+explain how to compute the cohomology of SLN(Z) andGLN(Z) (modulo torsion)
+from our results on the homology of the Voronoï complex in §4. Our main result
+isstated in Theorem 7.3.
+Acknowledgments: Thefirst two authors are particularly indebted to the IHES
+for its hospitality. The second author thanks the Institute for Experimental Math-
+ematics in Essen, acknowledging financial support by the DFG and the European
+Commission as well as hospitality of the Newton Institute in Cambridge and the
+MPIfor Mathematics inBonn. Theauthors aregrateful toB.Al lombert, J.-G.Du-
+mas,D.-O.Jaquet, J.-C.König,J.Martinet, S.Morita,J-P. SerreandB.Souvignier
+for helpful discussions. The computations of the Voronoï ce ll decomposition were
+performed on the computers of the Institut de Mathématiques et Modélisation de
+Montpellier, theMPIBonn,theInstitut FourierinGrenoble andtheCentredeCal-
+cul Médicis and weare grateful to those institutions.
+Notation: For any positive integer nwe letSnbe the class of finite abelian
+groups the order of which has only prime factors less than or e qual ton.Perfectformsandthecohomologyofmodulargroups 3
+2. Vorono¨i’sreductiontheory
+2.1.Perfect forms. LetN/greaterorequalslant2 be an integer. We let CNbe the set of positive
+definite real quadratic forms in Nvariables. Given h∈CN, letm(h) be the finite
+set of minimal vectors of h, i.e. vectors v∈ZN,v/nequal0, such that h(v) is minimal.
+A formhis calledperfectwhenm(h) determines hup to scalar: if h′∈CNis such
+thatm(h′)=m(h), thenh′isproportional to h.
+Example 2.1.Theform h(x,y)=x2+y2has minimum 1 and precisely 4 minimal
+vectors±(1,0) and±(0,1). This form is not perfect, because there is an infinite
+number of positive definite quadratic forms having these min imal vectors, namely
+the forms h(x,y)=x2+axy+y2whereais a non-negative real number less than
+1. Bycontrast, theform h(x,y)=x2+xy+y2hasalso minimum1andhas exactly
+6 minimal vectors, viz. the ones above and ±(1,−1). This form is perfect, the
+associated lattice is the“honeycomb lattice”.
+Denote by C∗
+Nthe set of non-negative real quadratic forms on RNthe kernel of
+which is spanned by a proper linear subspace of QN, byX∗
+Nthe quotient of C∗
+Nby
+positive real homotheties, and by π:C∗
+N→X∗
+Nthe projection. Let XN=π(CN)
+and∂X∗
+N=X∗
+N−XN. LetΓbe eitherGLN(Z) orSLN(Z). The groupΓacts onC∗
+N
+andX∗
+Nonthe right by the formula
+h·γ=γthγ, γ∈Γ,h∈C∗
+N,
+wherehis viewed as a symmetric matrix and γtis the transpose of the matrix γ.
+Voronoï proved that there are only finitely many perfect form s modulo the action
+ofΓand multiplication bypositive real numbers ([22], Thm.p.1 10).
+The following table gives the current state of the art on the e numeration of perfect
+forms.
+rank12345678 9
+#classes 1112373310916/greaterorequalslant500000
+Theclassificationofperfectformsofrank8wasachievedbyD utour,Schürmann
+and Vallentin in 2005 [7], [18]. They have also shown that in r ank 9 there are at
+least 500000 classes of perfect forms. The corresponding cl assification for rank 7
+was completed by Jaquet in 1991 [11], for rank 6 by Barnes [2], and by Voronoï
+for theother dimensions. Werefer tothebook ofMartinet [13 ]for moredetails on
+the results upto rank 7.
+2.2.Acell complex. Givenv∈ZN−{0}welet ˆv∈C∗
+Nbe the form defined by
+ˆv(x)=(v|x)2,x∈RN,
+where (v|x) is the scalar product of vandx. Theconvex hull in X∗
+Nof a finite
+subsetB⊂ZN−{0}isthesubset of X∗
+Nwhichistheimageunder πofthequadratic
+forms/summationtext
+jλjˆvj∈C∗
+N, wherevj∈Bandλj/greaterorequalslant0. For any perfect form h, we let
+σ(h)⊂X∗
+Nbe the convex hull of the set m(h) of its minimal vectors. Voronoï
+proved in [22], §§8-15, that the cells σ(h) and their intersections, as hruns over
+all perfect forms, define a cell decomposition of X∗
+N, which is invariant under the
+action ofΓ. We endow X∗
+Nwith the corresponding CW-topology. Ifτis a closed
+cell inX∗
+Nandhaperfect form with τ⊂σ(h),weletm(τ)betheset ofvectors vin4 P h. Elbaz-Vincent,H.Gangland C. Soul´e
+m(h) such that ˆ vlies inτ. Anyclosed cellτisthe convex hull of m(τ), and for any
+twoclosed cellsτ,τ′inX∗
+Nwehavem(τ)∩m(τ′)=m(τ∩τ′).
+3. TheVorono¨icomplex
+3.1.Definition. Letd(N)=N(N+1)/2−1bethedimension of X∗
+Nandn/lessorequalslantd(N)
+a natural integer. We denote by Σ⋆
+n=Σ⋆
+n(Γ) a set of representatives, modulo the
+action ofΓ, of those cells of dimension ninX∗
+Nwhich meet XN, and byΣn=
+Σn(Γ)⊂Σ⋆
+n(Γ) the cellsσfor which any element of the stabilizer ΓσofσinΓ
+preserves the orientation. Let Vnbe the free abelian group generated by Σn. We
+define asfollows a map
+dn:Vn→Vn−1.
+For each closed cell σinX∗
+Nwe fix an orientation of σ, i.e. an orientation of
+the real vector space R(σ) of symmetric matrices spanned by the forms ˆ vwith
+v∈m(σ). Letσ∈Σnand letτ′be a face ofσwhich is equivalent under Γto
+an element inΣn−1(i.e.τ′neither lies on the boundary nor has elements in its
+stabilizer reversing the orientation). Given a positive ba sisB′ofR(τ′) we get a
+basisBofR(σ)⊃R(τ′) by appending to B′a vector ˆv, wherev∈m(σ)−m(τ′).
+Weletε(τ′,σ)=±1bethesignoftheorientation of Bintheoriented vector space
+R(σ) (this sign does not depend onthe choice of v).
+Next,letτ∈Σn−1bethe(unique) cell equivalent to τ′and letγ∈Γbesuchthat
+τ′=τ·γ. We defineη(τ,τ′)=1 (resp.η(τ,τ′)=−1) whenγis compatible (resp.
+incompatible) withthe chosen orientations of R(τ) andR(τ′).
+Finally wedefine
+(1) dn(σ)=/summationdisplay
+τ∈Σn−1/summationdisplay
+τ′η(τ,τ′)ε(τ′,σ)τ,
+whereτ′runs through theset of faces of σwhich are equivalent to τ.
+3.2.A spectral sequence. According to [4], VII.7, there is a spectral sequence
+Er
+pqconverging to the equivariant homology groups HΓ
+p+q(X∗
+N,∂X∗
+N;Z) of the ho-
+mology pair ( X∗
+N,∂X∗
+N), and such that
+E1
+pq=/circleplusdisplay
+σ∈Σ⋆pHq(Γσ,Zσ),
+whereZσis the orientation module of the cell σand, as above,Σ⋆
+pis a set of
+representatives, modulo Γ, of thep-cellsσinX∗
+Nwhich meet XN. Sinceσmeets
+XN, its stabilizerΓσis finite and, by Lemma 7.1 in §7 below, the order of Γσis
+divisible only by primes p/lessorequalslantN+1. Therefore, when qis positive, the group
+Hq(Γσ,Zσ) lies inSN+1.
+WhenΓσhappens tocontain anelement whichchanges theorientation ofσ,the
+groupH0(Γσ,Zσ) is killed by 2, otherwise H0(Γσ,Zσ)/simequalZσ). Therefore, modulo
+S2,wehave
+E1
+n0=/circleplusdisplay
+σ∈ΣnZσ,
+andthechoice ofanorientation foreachcell σgivesanisomorphism between E1
+n0
+andVn.Perfectformsandthecohomologyofmodulargroups 5
+3.3.Comparison. Weclaim that the di fferential
+d1
+n:E1
+n0→E1
+n−1,0
+coincides, up to sign, with the map dndefined in 3.1. According to [4], VII,
+Prop. (8.1), the differentiald1
+ncan be described asfollows.
+Letσ∈Σ⋆
+nand letτ′be a face ofσ. Consider the group Γστ′=Γσ∩Γτ′and
+denote by
+tστ′:H∗(Γσ,Zσ)→H∗(Γστ′,Zσ)
+the transfer map. Next,let
+uστ′:H∗(Γστ′,Zσ)→H∗(Γτ′,Zτ′)
+be the map induced by the natural map Zσ→Zτ′, together with the inclusion
+Γστ′⊂Γτ′. Finally, letτ∈Σ⋆
+n−1be the representative of the Γ-orbit ofτ′, letγ∈Γ
+besuch thatτ′=τ·γ,and let
+vτ′τ:H∗(Γτ′,Zτ′)→H∗(Γτ,Zτ)
+betheisomorphism inducedby γ. Thentherestrictionof d1
+ntoH∗(Γσ,Zσ)isequal,
+up tosign, tothe sum
+(2)/summationdisplay
+τ′vτ′τuστ′tστ′,
+whereτ′runs over aset of representatives of faces of σmoduloΓσ.
+Tocompare d1
+nwithdnwefirst note that, when τ∈Σn−1,
+vτ′τ:H0(Γτ′,Zτ′)=Z→H0(Γτ,Zτ)=Z
+isthe multiplication by η(τ,τ′), asdefined in§3.1. Next, when σ∈Σn,the map
+uστ′:H0(Γστ′,Zσ)=Zσ=Z→H0(Γτ′,Zτ′)=Z
+is the multiplication by ε(τ′,σ), up to a sign depending on nonly. Finally, the
+transfer map
+tστ′:H0(Γσ,Zσ)=Z→H0(Γστ′,Zσ)=Z
+is the multiplication by [ Γσ:Γστ′]. Multiplying the sum (2) by this number
+amounts to the same as taking the sum over all faces of σas in (1). This proves
+thatdncoincides, up tosign, with d1
+nonE1
+n0=Vn. /square
+In particular, we get that dn−1◦dn=0. Note that this identity will give us a
+non-trivial test of our explicit computations of the comple x.
+Notation: The resulting complex ( V•,d•) will be denoted by Vor Γ, and we call
+it theVoronoï complex .
+3.4.TheSteinbergmodule. LetTNbethespherical Titsbuilding of SLNoverQ,
+i.e. thesimplicialsetdefinedbytheorderedsetofnon-zero properlinearsubspaces
+ofQN. The reduced homology ˜Hq(TN,Z) ofTNwith integral coefficients is zero
+except when q=N−2,in which case
+˜HN−2(TN,Z)=StN
+is by definition the Steinberg module [5]. According to [20], Prop. 1, the relative
+homology groups Hq(X∗
+N,∂X∗
+N;Z) are zero except when q=N−1, and
+HN−1(X∗
+N,∂X∗
+N;Z)=StN.6 P h. Elbaz-Vincent,H.Gangland C. Soul´e
+From this it follows that, for all m∈N,
+HΓ
+m(X∗
+N,∂X∗
+N;Z)=Hm−N+1(Γ,StN)
+(see e.g. [20], §3.1). Combining this equality with the prev ious sections we con-
+clude that, modulo SN+1,
+(3) Hm−N+1(Γ,StN)=Hm(VorΓ).
+4. TheVorono¨icomplexindimensions 5, 6and7
+Inthis section, weexplain how to compute the Voronoï comple xes of rank N/lessorequalslant7.
+4.1.Checking the equivalence of cells. As a preliminary step, we develop an
+effective method to check whether two cells σandσ′of the same dimension are
+equivalent under the action of Γ. The cellσ(resp.σ′) is described by its set of
+minimal vectors m(σ) (resp.m(σ′)). We let b(resp.b′) be the sum of the forms ˆ v
+withv∈m(σ) (resp.m(σ′)). Ifσandσ′are equivalent under the action of Γthe
+same is true for bandb′, and the converse holds true since two cells of the same
+dimension are equal whenthey have aninterior point in commo n.
+Tocompare bandb′we first check whether or not they have the same determi-
+nant. Incasetheydo,welet M(resp.M′)bethesetofnumbers b(x)withx∈m(σ)
+(resp.b′(x) withx∈m(σ′)). Ifbandb′are equivalent, then the sets MandM′
+must be equal.
+Finally, if M=M′wecheckif bandb′areequivalent byapplying analgorithm
+of Plesken and Souvignier [17] (based on animplementation o f Souvignier).
+4.2.Finding generators of the Voronoï complex. In order to compute Σn(and
+Σ⋆
+n), we proceed as follows. Fix N/lessorequalslant7. LetPbe a set of representatives of the
+perfect forms of rank N. A choice ofPis provided by Jaquet [11]. Furthermore,
+for eachh∈P, Jaquet gives the list m(h) of its minimal vectors, and the list of all
+perfect forms h′γ(one for each orbit under Γσ(h)), whereh′∈Pandγ∈Γ, such
+thatσ(h) andσ(h′)γshare a face of codimension one. This provides a complete
+listC1
+hof representatives of codimension one faces in σ(h).
+From this, one deduces the full list F1
+hof faces of codimension one in σ(h)
+as follows: first list all the elements in the automorphism gr oupΓσ(h); this can
+be obtained by using a second procedure implemented by Souvi gnier [17] which
+gives generators for Γσ(h). We represent the latter generators as elements in the
+symmetric groupSM, whereMis the cardinality of m(h), acting on set m(h) of
+minimal vectors. Using those generators, we let GAP [9] list all the elements of
+Γσ(h), viewed aselements of the symmetric group above.
+The next step is to create a shortlist F2
+hof codimension 2 facets of σ(h) by
+intersecting allthetranslates under SMofcodimension 1facetswitheachmember
+ofC1
+hand only keeping those intersections with the correct rank ( =d(N)−2). The
+resulting shortlist is reasonably small and we apply the pro cedure of 4.1 to reduce
+the shortlist to aset of representatives C2
+hof codimension 2 facets.
+Wethen proceed by induction on the codimension to define alis tFp
+hof cells of
+codimension p>2 inσ(h). GivenFp
+h, we letCp
+h⊂Fp
+hbe a set of representatives
+for the action ofΓ. We then letFp+1
+hbe the set of cellsϕ∩τ, withϕ∈F2
+h, and
+τ∈Cp
+h.Perfectformsandthecohomologyofmodulargroups 7
+n 4567891011121314151617181920
+Σ⋆
+n/parenleftbigGL5(Z)/parenrightbig25101623252316943
+Σn/parenleftbigGL5(Z)/parenrightbig1761023
+Σ⋆
+n/parenleftbigGL6(Z)/parenrightbig31028711623295898741066103977542518157187
+Σn/parenleftbigGL6(Z)/parenrightbig346163340544636469200495
+Σ⋆
+n/parenleftbigSL6(Z)/parenrightbig3102871163347691115215321551113458522262187
+Σn/parenleftbigSL6(Z)/parenrightbig31018431694608151132127097043411427147
+Figure1. Cardinality ofΣnandΣ⋆
+nforN=5,6 (empty slots de-
+note zero).
+n 6 7 8 9 10 11 12 13 14 15 16
+Σ⋆
+n6 28 115 467 1882 7375 26885 87400 244029 569568 1089356
+Σn 1 60 1019 8899 47271 171375 460261 955128
+n17 18 19 20 21 22 23 24 25 26 27
+Σ⋆
+n1683368 2075982 2017914 1523376 876385 374826 115411 24623 3518 352 33
+Σn1548650 1955309 1911130 1437547 822922 349443 105054 21074 2798 305 33
+Figure2. Cardinality ofΣnandΣ⋆
+nforGL7(Z) .
+Next,weletΣ⋆
+nbeasystem of representatives modulo Γinthe union of thesets
+Cd(N)−n
+h,h∈P. We then compute generators of the stabilizer of each cell in Σ⋆
+n
+with the help of another algorithm developed by Plesken and S ouvignier in [17],
+andwecheckwhetherallgenerators preservetheorientatio n ofthecell. Thisgives
+usthe setΣnasthe set of those cells which pass that check.
+Proposition 4.1. The cardinality of ΣnandΣ⋆
+nis displayed in Figure 1 for rank
+N=5,6and in Figure2 forrank N =7.
+Remark4.2.ThefirstlineinFigure1hasalreadybeencomputedbyBatut(c f. [1],
+p.409, second column of Table2).
+4.3.Thedifferential. Thenext step isto compute the di fferentials of theVoronoï
+complexbyusingformula(1)above. InTable3,wegiveinform ation onthediffer-
+entialsintheVoronoï complexofrank5. Forinstancethesec ondline,denoted d11,
+isaboutthedifferential from V11toV10. InthebasesΣ11andΣ10,thisdifferentialis
+given by amatrix AwithΩ=513 non-zero entries, with m=46=card(Σ10) rows
+andn=163=card(Σ11) columns. The rank of Ais 42, and the rank of its kernel
+is121. Theelementary divisors of Aare1 (multiplicity 40) and 2(multiplicity 2).
+The cases of SL4(Z),GL6(Z) andSL6(Z) are treated in Table 1, Table 3 and Ta-
+ble 4, respectively.8 P h. Elbaz-Vincent,H.Gangland C. Soul´e
+AΩnmrankkerelementarydivisors
+d401001
+d511110 1(1)
+d601101
+d700100
+d801001
+d922111 2(1)
+Table1. Resultsontherankandelementarydivisors ofthedi ffer-
+entials for SL4(Z).
+AΩnmrankkerelementary divisors
+d801001
+d927116 1(1)
+d10186751 1(4), 2(1)
+d1151610 1(1)
+d1200100
+d1302002
+d1443221 5(1), 15(1)
+Table2. Resultsontherankandelementarydivisors ofthedi ffer-
+entials for GL5(Z).
+AΩnmrankkerelementary divisors
+d1017463343 1(3)
+d115131634642121 1(40), 2(2)
+d122053340163120220 1(120)
+d134349544340220324 1(217), 2(3)
+d1461536365443243121(320), 2(1), 6(2), 12(1)
+d155378469636312157 1(307), 2(3), 60(2)
+d16252620046915644 1(156)
+d17597492004451(41), 3(1), 6(1), 36(1)
+d184354950 1(5)
+Table3. Resultsontherankandelementarydivisors ofthedi ffer-
+entials for GL6(Z).
+Our results on the di fferentials in rank 7 are shown in Table 5. While the ma-
+trices are sparse, they are not sparse enough for e fficient computation. They have
+a poor conditioning with some dense columns or rows (this is a consequence of
+thefact that thecomplex isnot simplicial andnon-simplici al cellscan havealarge
+number ofnon-trivial intersections withthefaces). Wehav eobtained full informa-
+tionontherankofthedi fferentials. Forthecomputation oftheelementarydivisors
+complete results have been obtained in the case of matrices o fdnfor 10/lessorequalslantn/lessorequalslant14
+and 24/lessorequalslantn/lessorequalslant27 only. See[6] for adetailed description of the computatio n.
+4.4.Thehomology of theVoronoï complexes. Fromthe computation of thedif-
+ferentials, we can determine the homology of Voronoï comple x. Recall that if wePerfectformsandthecohomologyofmodulargroups 9
+AΩnmrankkerelementarydivisors
+d71210337 1(3)
+d8481810711 1(7)
+d914043181132 1(11)
+d106131694332137 1(32)
+d112952460169136324 1(129), 2(6), 6(1)
+d127614815460323492 1(318), 2(3), 4(2)
+d13123951132815491641 1(491)
+d141496612701132641629 1(637), 3(3), 12(1)
+d1512714 97012706293411(621), 2(5), 6(1), 60(2)
+d16649143497033995 1(338), 2(1)
+d1718321144349519 1(92), 3(2), 18(1)
+d1825727114198 1(17), 2(2)
+d1962142786 1(7), 10(1)
+d202871461 1(1), 3(4),504(1)
+Table4. Resultsontherankandelementarydivisors ofthedi ffer-
+entials for SL6(Z).
+AΩ n m rank ker elementary divisors
+d10 8 60 1 1 59 1
+d11 1513 1019 60 59 960 1(59)
+d1237519 8899 1019 960 7939 1(958), 2(2)
+d13356232 47271 8899 7938 39333 1(7937), 2(1)
+d141831183 171375 47271 39332 132043 1(39300), 2(29), 4(3)
+d156080381 460261 171375 132043 328218 ?
+d1614488881 955128 460261 328218 626910 ?
+d1725978098 1548650 955128 626910 921740 ?
+d1835590540 1955309 1548650 921740 1033569 ?
+d1937322725 1911130 1955309 1033568 877562 ?
+d2029893084 1437547 1911130 877562 559985 ?
+d2118174775 822922 1437547 559985 262937 ?
+d228251000 349443 822922 262937 86506 ?
+d232695430 105054 349443 86505 18549 ?
+d24593892 21074 105054 18549 2525 1(18544), 2(4), 4(1)
+d2581671 2798 21074 2525 273 1(2507), 2(18)
+d26 7412 305 2798 273 32 1(258), 2(7), 6(7),36 (1)
+d27 600 33 305 32 11(23), 2(4),28 (3),168 (1), 2016 (1)
+Table5. Resultsontherankandelementarydivisors ofthedi ffer-
+entials for GL7(Z).
+have acomplex of free abelian groups
+···→Zαf→Zβg→Zγ→···
+withfandgrepresented by matrices, then the homology is
+ker(g)/Im(f)/simequalZ/d1Z⊕···⊕Z/dℓZ⊕Zβ−rank(f)−rank(g),
+whered1,...,dℓare the elementary divisors of the matrix of f.
+WededucefromTables1–5thefollowingresultonthehomolog yoftheVoronoï
+complex.10 P h. Elbaz-Vincent,H.Gangland C. Soul´e
+Theorem 4.3. The non-trivial homology of the Voronoï complexes associat ed to
+GLN(Z)with N=5,6moduloS5isgiven by:
+Hn(VorGL5(Z))/simequalZ,if n=9,14,
+Hn(VorGL6(Z))/simequalZ,if n=10,11,15,
+while inthe case SL 6(Z)weget, moduloS7,that
+Hn(VorSL6(Z))/simequalZ,if n=10,11,12,20,
+Z2,if n=15.
+Furthermore, for N =7weget
+Hn(VorGL7(Z)⊗Q)/simequalQif n=12,13,18,22,27,
+0otherwise.
+Notice that, if Nis odd,SLN(Z) andGLN(Z) have the same homology modulo
+S2. Notice also that, for simplicity, in the statement of the th eorem wedid not use
+the full information given bythe list of elementary divisor s inTables 1–5.
+4.5.MassformulaefortheVoronoïcomplex. Letχ(SLN(Z))bethevirtualEuler
+characteristic ofthegroup SLN(Z). Itcanbecomputedintwoways. First,themass
+formula in [4] gives
+χ(SLN(Z))=/summationdisplay
+σ∈E(−1)dim(σ)1
+|Γσ|=d(N)/summationdisplay
+n=N(−1)n/summationdisplay
+σ∈Σ⋆n1
+|Γσ|,
+whereEis a family of representatives of the cells of the Voronoï com plex of rank
+Nmodulo the action of SLN(Z), andΓσis the stabilizer of σinSLN(Z). Second,
+by aresult of Harder [10], weknow that
+χ(SLN(Z))=N/productdisplay
+k=2ζ(1−k),
+henceχ(SLN(Z))=0 ifN/greaterorequalslant3.
+Anon-trivial check of our computations istotest the compat ibility of these two
+formulas, and the corresponding check for rank N=5 had been performed by
+Batut (cf. [1], where a proof of an analogous statement, for a nyN, but instead
+pertaining to well-rounded forms, which in our case are precisely the ones in Σ⋆
+•,
+isattributed toBavard [3]).
+If we add together the terms1
+|Γσ|for cellsσof the same dimension to a single
+term, then weget for N=6, starting withthe top dimension,
+45047
+1451520−10633
+11520+6425
+576−12541
+192
++7438673
+34560−3841271
+8640+9238
+15−266865
+448+14205227
+34560−14081573
+69120
++830183
+11520−205189
+11520+61213
+20736−1169
+3840+17
+1008−1
+2880
+=χ(SL6(Z))=0.
+ForN=7weobtain similarlyPerfectformsandthecohomologyofmodulargroups 11
+−290879
+107520+13994381
+103680−31815503
+13824+1362329683
+69120−6986939119
+69120
++7902421301
+23040−340039739981
+414720+174175928729
+120960−132108094091
+69120
++27016703389
+13824−13463035571
+8640+14977461287
+15360−22103821919
+46080
++8522164169
+46080−17886026827
+322560+1764066533
+138240−101908213
+46080+12961451
+46080
+−10538393
+414720+162617
+103680−721
+11520+43
+32256
+=χ(SL7(Z))=0.
+5. Explicithomologyclasses
+5.1.Equivariant fundamental classes.
+Theorem 5.1. The top homology group H d(N)/parenleftbigVorSLN(Z)⊗Q/parenrightbighas dimension 1.
+When N=4,5,6or7, it is represented bythe cycle
+/summationdisplay
+σ1
+|Γσ|[σ],
+whereσruns through the perfect forms of rank N and the orientation o f each cell
+isinherited from theone of X N/Γ.
+Proof.Thefirst assertion isclear since, by (3) above and (6) below w ehave
+Hd(N)/parenleftbigVorSLN(Z)⊗Q/parenrightbig/simequalHd(N)−N+1/parenleftbigSLN(Z),StN⊗Q/parenrightbig/simequalH0(SLN(Z),Q)/simequalQ.
+Inorder toprovethesecondclaim,writethedi fferential betweencodimension 0
+andcodimension1cellsasamatrix Aofsizen1×n0,withni=|Σd(N)−i(Γ)|denoting
+the number of codimension icells in the Voronoï cell complex. It can be checked
+thatineachofthe n1rowsofAthereareprecisely twonon-zero entries. Moreover,
+the absolute value of the ( i,j)-th entry of Ais equal to the quotient |Γσj|/|Γτi|(an
+integer), whereσj∈Σ0(Γ)andτi∈Σ1(Γ). Finally,onecanmultiplysomecolumns
+by−1 (which amounts to changing the orientation of the correspo nding codimen-
+sion 0 cell) in such a way that each row has exactly one positiv e and one negative
+entry. /square
+Example 5.2.ForN=5thedifferential matrix d14(cf. Table2)betweencodimen-
+sion 0and codimension 1isgiven by
+/parenleftigg40 0−15
+40−15 0/parenrightigg
+,
+sothekernel isgenerated by(3 ,8,8)=11520/parenleftbig1
+3840,1
+1440,1
+1440/parenrightbig,whiletheordersof
+the three automorphism groups are 3840, 1440 and 1440, respe ctively.12 P h. Elbaz-Vincent,H.Gangland C. Soul´e
+Example 5.3.Similarly, the differentiald20:V20→V19for rankN=6 (cf. Table
+3) is represented by the matrix
+0 0 96 0 0 0 −21
+3240 0 0 0 −21 0 0
+0 0 1440 0 0 −3 0
+0 0 0 18 0 −6 0
+−12960 0 0 0 0 12 0
+−3240 0 0 9 0 0 0
+0−360 0 1 0 0 0
+−4320 0 0 12 0 0 0
+0 0 960 −6 0 0 0
+0−216 96 0 0 0 0
+−45 45 0 0 0 0 0
+−2592 0 1152 0 0 0 0
+−3240 0 1440 0 0 0 0
+−432 0 192 0 0 0 0.
+Its kernel isgenerated by
+(28,28,63,10080,4320,30240,288)
+while theorders of the corresponding automorphism groups a re, respectively,
+103680,103680,46080,288,672,96,10080,
+andwenotethat28 ·103680=63·46080=10080·288=4320·672=30240·96.
+5.2.An explicit non-trivial homology class for rank N=5.The kernel of the
+6×7-matrixof d10forGL5(Z),displayed intheproof ofTheorem6.1,equation (4)
+below, is spanned by (0 ,0,0,0,0,1,−1) together with (5 ,1,−8,16,15,2,2). The
+latteroneprovidesanon-trivialhomologyclassin H10/parenleftbigVorGL5(Z)/parenrightbig/simequalH5(GL5(Z),Z)
+(moduloS5), given as a linear combination of cells (in terms of minimal vector
+indices) as follows:
+5ϕ/parenleftbig[1,15,4,16,10,11,17,18,3,5]/parenrightbig
++ϕ/parenleftbig[1,15,4,16,10,11,17,18,2,5]/parenrightbig
+−8ϕ/parenleftbig[1,15,4,10,11,17,18,3,2,5]/parenrightbig
++16ϕ/parenleftbig[1,6,15,4,16,10,11,17,18,2]/parenrightbig
++15ϕ/parenleftbig[1,6,15,4,16,10,11,17,18,5]/parenrightbig
++2ϕ/parenleftbig[1,6,7,4,16,10,11,17,2,5]/parenrightbig
++2ϕ/parenleftbig[1,6,7,19,13,20,15,10,11,2]/parenrightbig.
+Heretheindicesrefertothefollowingorderfortheset m(P1
+5)ofminimalvectors
+1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
+1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0
+0 1 1 0 0 −1 1 1 0 0 1 0 0 0 0 1 0 0 1 0
+0 0 0−1 0 0 0 0 0 0 1 −1−1 0 0 0 1 1 −1 1
+0 0 0−1 0 0 0 −1 1 1 0 0 0 −1−1 1 0 0 0 −1
+0−1−1 1−1 0 0 0 −1−1−1 0 0 0 0 −1−1−1 0 0
+andϕ(u) for a vector uof indices is the convex hull of the minimal vectors corre-
+sponding tothose indices, as in§2.2.Perfectformsandthecohomologyofmodulargroups 13
+6. Splittingoffthe Vorono¨icomplex VorNfromVorN+1forsmall N
+In this section, we will be concerned with Γ=GLN(Z) only and we adapt the
+notationΣn(N)=Σn(GLN(Z)) for the sets of representatives.
+6.1.Inflating well-rounded forms. LetAbe the symmetric matrix attached to
+a formhinC∗
+N. Suppose the cell associated to Aiswell-rounded , i.e., its set
+of minimal vectors S=S(A) spans the underlying vector space RN. Then we
+can associate to it a form ˜hwith matrix ˜A=/parenleftigg
+A0
+0m(A)/parenrightigg
+inC∗
+N+1, wherem(A)
+denotestheminimumpositivevalueof AonZN. Theset ˜Sofminimalvectorsof ˜A
+contains the ones from S, each vector being extended by an ( N+1)-th coordinate
+0. Furthermore, ˜Scontains the additional minimal vectors ±eN+1=±(0,...,0,1),
+and hence it spans RN+1, i.e.,˜Ais well-rounded as well. In the following, we will
+call forms like ˜Aas well as their associated cells inflated.
+Thestabilizer of hinGLN(Z)thereby embedsintotheoneof ˜hinsideGLN+1(Z)
+(at least modulo±Id) under the usual stabilization map.
+Note that, by iterating the same argument rtimes,Ainduces a well-rounded
+formalsoinΣ⋆
+•(N+r)which, for r/greaterorequalslant2,doesnotbelong to Σ•(N+r)sincethereis
+an obvious orientation-reversing automorphism of the infla ted form, given by the
+permutation which swaps the last twocoordinates.
+6.2.Thecase N=5.
+Theorem 6.1. Thecomplex VorGL5(Z)isisomorphic toadirect factor of VorGL6(Z),
+withdegrees shifted by1.
+Proof.TheVoronoïcomplexof GL5(Z)canberepresentedbythefollowingweighted
+graph withlevels
+0 : P1
+5
+−15/d61/d61/d61/d61/d61P2
+5
+40/d1/d1/d1/d1/d1
+40/d61/d61/d61/d61/d61P3
+5
+−15/d1/d1/d1/d1/d1
+1 : σ1
+1σ2
+1
+3 : σ1
+3
+−1
+/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106
+1
+/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111
+−2
+/d0/d0/d0/d0/d0/d0/d0/d0/d0/d0/d0
+−21
+/d62/d62/d62/d62/d62/d62/d62/d62/d62/d62/d62
+4 :σ1
+4
+/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79
+/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87/d87σ2
+4
+/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79
+/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84/d84σ3
+4
+/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111
+/d0/d0/d0/d0/d0/d0/d0/d0/d0/d0/d0σ4
+4
+/d0/d0/d0/d0/d0/d0/d0/d0/d0/d0/d0σ5
+4
+/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111/d111
+/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106/d106
+/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103/d103 σ6
+4
+/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102/d102
+−1
+/d12/d12/d12/d12/d12/d12/d12/d12/d12
+−1
+/d50/d50/d50/d50/d50/d50/d50/d50/d50
+5 :σ1
+5σ2
+5σ3
+5σ4
+5σ5
+5σ6
+5
+1
+/d49/d49/d49/d49/d49/d49/d49/d49/d49σ7
+5
+−1
+/d13/d13/d13/d13/d13/d13/d13/d13/d13
+6 : σ1
+614 P h. Elbaz-Vincent,H.Gangland C. Soul´e
+Herethenodesinline j(markedontheleft)representtheelementsin Σd(N)−j(5),
+i.e. we have 3, 2, 0, 1, 6, 7 and 1 cells in codimensions 0, 1, 2, 3 , 4, 5 and 6, re-
+spectively, and arrows show incidences of those cells, whil e numbers attached to
+arrows give the corresponding incidence multiplicities. S ince entering the multi-
+plicities relating codimensions 4 and 5 would make the graph rather unwieldy, we
+give them instead in terms of the matrix corresponding to the differentiald10con-
+necting dimension 10 to 9 (columns refer, in this order, to σ1
+5,...,σ7
+5, while rows
+refer toσ1
+4,...,σ6
+4)
+(4)−5 0−5 0−1 0 0
+0−2 0 2−2 0 0
+2−2 1 0 0 0 0
+0 0 2 1 0 0 0
+−1−2 1 0 1 0 0
+0 4 0 0 0 −1−1.
+As is apparent from the picture, there are two connected comp onents in that
+graph. The corresponding graph for GL6(Z) has three connected components, two
+of which are "isomorphic" (as weighted graphs with levels) t o the one above for
+GL5(Z), except for a shift in codimension by 5/parenleftbige.g. codimension 0 cells in Σ•(5)
+correspond tocodimension 5 cells in Σ•(6)/parenrightbig,i.e. ashift in dimension by 1.
+In fact, it is possible, after appropriate coordinate trans formations, to identify
+the minimal vectors (viewed up to sign) of any given cell in th e two inflated com-
+ponentsofΣ•(6)alludedtoabovewiththeminimalvectorsofanothercell whichis
+inflated from one in Σ•(5), except precisely oneminimal vector (up to sign) which
+isfixedunder the stabilizer of thecell.
+Letusillustratethiscorrespondence forthetop-dimensio nal cellσoftheperfect
+formP1
+5∈Σ14(5), also denoted P(5,1) in [11] and D5in [12], with the list m(P1
+5)
+of minimal vectors given already at the end of §5.2.
+Usingthealgorithmdescribedin§4.1,thecorrespondingin flatedcell /tildewideσinΣ15(6)
+can be found to be, in terms of its 21 minimal vectors of the per fect form P1
+6in
+Jaquet’s notation (see [11] and §5.2for the full list m(P1
+6)),
+v1v2v4v5v10v12v13v14v16v17v18v22v24v25v26v27v29v33v34v35v36
+1−1 0−1 0 0 0 −1 1 1 0 1 1 0 1 0 0 1 0 0 1
+0 1−1 0 0 0 −1 0 1 0 1 1 0 1 0 1 0 0 1 0 1
+0 0 1 1 0 −1 0 0−1 0 0−1 0 0−1−1 0−1−1 0−1
+0 0 0 0 1 0 0 0 −1−1−1 0 0 0 −1−1−1 0 0 0 −1
+0 0 0 0 0 1 1 1 −1−1−1−1−1−1 0 0 0 0 0 0 −1
+0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2
+Thetransformation
+γ=0−1−1 0 0 0
+0 0−1 0−1−1
+0 0 0 1 0 1
+0 0 0 1 0 0
+0 0 1−1 0 0
+−1−1−1 0−1 0
+sendsv1to(0,0,0,0,0,1) andsendseachoftheother vectors tothecorresponding
+one of the form ( v,0) where vis the corresponding minimal vector for P1
+5(in the
+order given above).Perfectformsandthecohomologyofmodulargroups 15
+One can verify that the other two perfect forms P2
+5andP3
+5(denoted by Voronoï
+A5andϕ2, respectively) give rise to a corresponding inflated cell in Σ15(6) in a
+similar way.
+Concerning the cells of positivecodimension inΣ•(5), it turns out that these all
+have a representative which is a facet in σ. Furthermore, the matrix γinduces an
+isomorphism from the subcomplex of Σ•(6) spanned by /tildewideσand all its facets to the
+complex obtained by inflation, as in §6.1 above, from the comp lex spanned byσ5
+and all its facets. Finally, one can verify that the cells att ached to P2
+5andP3
+5are
+conjugate, afterinflation,tocellsin Σ15(6),andthatthedifferentialsforVor GL5and
+VorGL6agree on these. Thisends the proof of the theorem. /square
+6.3.Othercases. A similar situation holds for Σ•(3) andΣ•(4), but asΣ•(3) con-
+sists of asingle cell only, the picture isfar less significan t.
+ForN=4, there is only one cell leftover in Σ•(4), in fact inΣ6(4), and it is
+already inflated from Σ5(3), as shown in §6.3. Hence its image in Σ⋆
+7(5) will allow
+an orientation reversing automorphism and hence will not sh ow up inΣ7(5). This
+illustrates the remark at theend of 6.1.
+Finally, for N=6, the cells in the third component of the incidence graph for
+GL6(Z) mentioned in the proof of Theorem 6.1 above appear, in inflat ed form, in
+the Voronoï complex for GL7(Z) which inherits the homology of that component,
+since in the weighted graph of GL7(Z), which is connected, there is only one in-
+cidence of an inflated cell with a non-inflated one. Therefore we do not have a
+splitting inthis case.
+7. TheCohomologyofmodulargroups
+7.1.Preliminaries. Recall the following simple fact:
+Lemma 7.1. Assume that p is a prime and g ∈GLN(R)has order p. Then p /lessorequalslant
+N+1.
+Proof.The minimal polynomial of gis the cyclotomic polynomial xp−1+xp−2+
+···+1. BytheCayley-Hamiltontheorem,thispolynomialdivides thecharacteristic
+polynomial of g. Therefore p−1/lessorequalslantN. /square
+Weshall also need the following result:
+Lemma 7.2. The action of GLN(R)on the symmetric space X Npreserves its ori-
+entation if and only if N isodd.
+Proof.The subgroup GL N(R)+⊂GLN(R) of elements with positive determinant
+is the connected component of the identity, therefore it pre serves the orientation
+ofXN. Anyg∈GLN(R) which is not in GL N(R)+is the product of an element
+of GLN(R)+with the diagonal matrix ε=diag(−1,1,...,1), so we just need to
+check whenεpreserves theorientation of XN. Thetangent space TXNofXNat the
+origin consists of real symmetric matrices m=(mij) of trace zero. The action of ε
+isgiven by m·ε=εtmε(cf. §2.1) and weget
+(m·ε)ij=mij
+unlessi=1orj=1andi/nequalj,inwhichcase( m·ε)ij=−mij. Letδijbethematrix
+withentry1inrow iandcolumn j,andzeroelsewhere. Abasisof TXNconsists of
+the matricesδij+δji,i/nequalj, together with N−1 diagonal matrices. For this basis,16 P h. Elbaz-Vincent,H.Gangland C. Soul´e
+the action ofεmapsN−1 vectors vto their opposite−vand fixes the other ones.
+Thelemmafollows. /square
+7.2.Borel–Serre duality. According to Borel and Serre ([5], Thm. 11.4.4 and
+Thm. 11.5.1), the group Γ=SLN(Z) orGLN(Z) is a virtual duality group with
+dualizing module
+Hv(N)(Γ,Z[Γ])=StN⊗˜Z,
+wherev(N)=N(N−1)/2isthevirtual cohomological dimension of Γand˜Zisthe
+orientation module of XN. It follows that there is along exact sequence
+(5)···→Hn(Γ,StN)→Hv(N)−n(Γ,˜Z)→ˆHv(N)−n(Γ,˜Z)→Hn−1(Γ,StN)→···
+whereˆH∗is the Farrell cohomology of Γ[8]. From Lemma 7.1 and the Brown
+spectral sequence ([4], X(4.1)) wededuce that ˆH∗(Γ,˜Z) lies inSN+1. Therefore
+(6) Hn(Γ,StN)≡Hv(N)−n(Γ,˜Z),moduloSN+1.
+WhenNisodd, then GL N(Z)is the product of SL N(Z)byZ/2, therefore
+Hm(GLN(Z),Z)≡Hm(SLN(Z),Z),moduloS2.
+WhenNis even, then the action of GL N(Z) on˜Zis given by the sign of the deter-
+minant (see Lemma7.2) and Shapiro’s lemmagives
+(7) Hm(SLN(Z),Z)=Hm(GLN(Z),M),
+with
+M=IndGLN(Z)
+SLN(Z)Z≡Z⊕˜Z,moduloS2.
+Tosummarize: when Γ=SLN(Z) orGLN(Z), whereN/lessorequalslant7, we know Hm(Γ,˜Z) by
+combining (3) (end of §3.4), Theorem 4.3 and (6). This allows us to compute the
+cohomology of GLN(Z). Theresults are given in Theorem 7.3below.
+7.3.Thecohomology of modulargroups.
+Theorem 7.3. (i)ModuloS5wehave
+Hm(SL5(Z),Z)=Zifm=0,5,
+0 otherwise.
+(ii)ModuloS7wehave
+Hm(GL6(Z),Z)=Zifm=0,5,8,
+0 otherwise,
+and
+Hm(SL6(Z),Z)=Z2ifm=5,
+Zifm=0,8,9,10,
+0 otherwise.
+(iii)For N=7, wehave,
+Hm(SL7(Z),Q)=Qifm=0,5,11,14,15,
+0 otherwise.
+Remark7.4.Moritaasksin[14]whethertheclassofinfiniteorderin H5(GL5(Z),Z)
+survives inthecohomology of the group of outer automorphis ms of thefree group
+of rank five.Perfectformsandthecohomologyofmodulargroups 17
+References
+[1]Batut,C.; Classificationofquinticeutacticforms, Math.Comput. 70,no.233(2001),395–417.
+[2]Barnes, E.S.; The complete enumeration of extreme senary forms, Phil. Trans. Roy. Soc. Lon-
+don249-A(1957), 461–506.
+[3]Bavard, C.; Classes minimales de réseaux et rétractions géométriques é quivariantes dans les
+espaces symétriques , J. London Math. Soc.(2) 64, no. 2(2001), 275–286.
+[4]Brown,K.; Cohomology of Groups, Graduate Textsin Mathematics 87, Springer-Verlag, New
+York,1994.
+[5]Borel, A.; Serre, J–P.; Corners and arithmetic groups, Comment. Math. Helv. 48(1973),
+436–491.
+[6]Dumas, J.–G.; Elbaz–Vincent, Ph.; Giorgi, P.; Urbanska, A. ;Parallel Computation of the
+Rank of Large Sparse Matrices from Algebraic K-theory. , PASCO 2007: Parallel Symbolic
+Computation ’07, 26–27 July, Waterloo, Canada. Proceeding s of the ACM(2007), 43–52.
+[7]DutourSikiric,M.;Schürmann,A.;Vallentin,F.; Classificationofeightdimensionalperfect
+forms,Electron. Res.Announc. Amer.Math. Soc. 13(2007), 21–32.
+[8]Farrell, F. T.; An extension of Tate cohomology to a class of infinite groups, J. Pure Appl.
+Algebra10, no. 2(1977/78), 153–161.
+[9]The GAP Group; GAP – Groups, Algorithms, and Programming, Version 4.4.12; 2008.
+(http://www.gap-system.org ).
+[10]Harder, G.; Die Kohomologie S-arithmetischer Gruppen über Funktionen körpern, Invent.
+Math.42(1977), 135–175.
+[11]Jaquet, D.-O.; Énumération complète des classes de formes parfaites en dim ension 7, Thèse
+de doctorat, Université de Neuchâtel (1991).
+[12]Lee, R.; Szczarba, R. H.; On the torsion in K 4(Z)and K5(Z), Duke Math. J. 45(1978), 101–
+129.
+[13]Martinet, J.; Perfect Lattices in Euclidean Spaces, Springer-Verlag, Grundlehren der Mathe-
+matischenWissenschaften 327, Heidelberg,2003.
+[14]Morita, S.; Cohomological structure of the mapping class group and beyo nd.in: Problems
+on mapping class groups and related topics, Proc. Sympos. Pu re Math., 74, Amer. Math. Soc.,
+Providence, RI,2006, 329–354.
+[15]The PARI–Group; PARI/GP, versions 2.1/rangedash2.4, Bordeaux,
+http://pari.math.u-bordeaux.fr/ .
+[16]PFPK:A C libraryfor computing Voronoï complexes, version 1.0.0, 2009.
+[17]Plesken, W.; Souvignier, B.; Computing isometries of lattices, J. Symb. Comput. 24(1997),
+327–334.
+[18]Schürmann,A.; Enumerating perfect forms. arXiv:0901.1587 (math.NT).
+[19]Serre,J–P.; Trees, Springer Monographs inMathematics. Springer-Verlag,Be rlin,2003.
+[20]Soulé,C.; Onthe3-torsion in K 4(Z).Topology 39, no.2, (2000), 259–265.
+[21]Soulé,C.; Thecohomology ofSL 3(Z).Topology 17, no. 1 (1978), 1–22.
+[22]Voronoï, G.: Nouvelles applications des paramètres continus à la théori e des formes quadra-
+tiques I,Crelle133 (1907), 97–178.
+InstitutFourier, UMR 5582 (CNRS-U niversit´eGrenoble1), 100ruedesMath´ematiques , Do-
+maineUniversitaire , BP74, 38402 SaintMartind’H`eres(France)
+E-mailaddress :Philippe.Elbaz-Vincent@ujf-grenoble.fr
+Departmentof Mathematical Sciences,SouthRoad, Universityof Durham(UnitedKingdom)
+E-mailaddress :herbert.gangl@durham.ac.uk
+IHES,LeBois-Marie35, RoutedeChartres,91440 Bures-sur-Yvette(France)
+E-mailaddress :soule@ihes.fr
\ No newline at end of file
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+arXiv:1001.0790v1  [hep-ex]  6 Jan 2010HARD COLLISIONS OF SPINNING PROTONS: HISTORY & FUTURE
+A.D. Krisch1†
+(1)Spin Physics Center, University of Michigan, Ann Arbor, MI 48 109-1040, USA
+†E-mail: krisch@umich.edu Opening Talk: DSPIN-09, JINR, Dubn a RU (1-5 Sept 2009)
+Abstract
+There will be a review of the history of polarized proton beam s, and a discussion
+of the unexpected and still unexplained large transverse sp in effects found in several
+high energy proton-proton spin experiments at the ZGS, AGS, Fermilab and RHIC.
+Next, there will be a discussion of possible future experime nts on the violent elastic
+collisionsofpolarizedprotonsat IHEP-Protvino’s70GeVU -70accelerator inRussia
+and the new high intensity 50 GeV J-PARC at Tokai in Japan.
+I will first discuss the violent elastic collisions of unpolarized protons. Fig. 1 shows
+the cross section for proton-proton elastic scattering plotted a gainst a scaled P2
+tvariable
+that was proposed in 1963 [1] and 1967 [2] following Serber’s [3] optica l model. This
+plot is from updates by Peter Hansen and me [4,5]. Notice that at sma llP2
+tthe cross-
+section drops off with a slope of about 10 (GeV/c)−2. Fourier transforming this slope
+gives the size and shape of the proton-proton interaction in the diff raction peak; it is a
+Gaussian with a radius of about 1 Fermi. At medium P2
+tthere is a component with a
+slope of about 3 (GeV/c)−2; however, this component disappears rapidly with increasing
+energy. At lab energies of a few TeV it has totally disappeared; then , there is a sharp
+destructive interference between the small- P2
+tdiffraction peak and the large- P2
+thard-
+scattering component. Since the diffraction peak is mostly diffractive , its amplitude must
+be mostly imaginary, as has been experimentally verified. Hence, the sharp destructive
+interference implies that the large- P2
+tcomponent is also mostly imaginary; thus, it is
+probably mostly diffractive . This large- P2
+tcomponent is probably the elastic diffractive
+scattering due to the directinteractions of the proton’s constituents; its slope of about
+1.5(GeV/c)−2impliesthatthese directinteractionsoccurwithinaGaussian-shapedregion
+of radius about 0.3 Fermi.
+Since the medium- P2
+tcomponent disappears at high energy, it is probably the direct
+elastic scattering of the two protons. This view is supported by the experimental fac t that
+proton-proton elastic scattering is the only exclusive process tha t still can be precisely
+measured at TeV energies. To understand this, note that directelastic scattering and all
+other exclusive processes must compete with each other for the t otal p-p cross-section,
+which is less than 100 millibarns. At TeV energies, there are certainly m ore than 105
+exclusive channels in this competition; thus, each channel has an av erage cross section of
+less than 1 microbarn. Moreover, since the medium- P2
+telastic component does not inter-
+fere strongly with either the large- P2
+tor small- P2
+tcomponents, its amplitude is probably
+real. Also note that the large- P2
+tcomponent intersects the cross section axis at about
+10−5below the small- P2
+tdiffractive component.
+An earlier version of Fig. 1 got me started in the spin business. In 196 6, we measured
+p-p elastic scattering at the ZGS at exactly 90◦
+cmfrom 5 to 12 GeV [6]; the sharp slope-
+change, shown by the stars, was apparently the first direct evide nce for constituents in the
+1proton. Dividing these 90◦
+cmp-p elastic cross sections by 4 (due to the protons’ particle
+identity)madeallthenexistingproton-protonelasticdata, above afewGeV,fitonasingle
+curve [2]. During a 1968 visit to Ann Arbor, Prof. Serber informed me that, by dividing
+the 90◦
+cmpoints by 4, I had made an assumption about the ratio of the spin sing let and
+triplet p-p elastic scattering amplitudes. I was astounded and said t hat I knew nothing
+about spin and certainly had not measured the spin of either proton . He said with a smile
+that both statements might be true; nevertheless, my nice fit req uired this assumption.
+Prof. Serber, as usual, spoke quietly; however, as a student, I h ad learned that he was
+almost always right. Thus, I looked for data on proton-proton elas tic scattering data, in
+the singlet or triplet spin states, above a few GeV. I found that non e existed and decided
+ 
+Figure 1. Proton-proton elastic cross-sections plot-
+ted vs. the scaled P2
+tvariable [4,5]. The 12 GeV/c
+Allabyet al.data were not corrected for 90◦
+cmpar-
+ticle identity effects.
+ 
+Figure 2. The proton-proton elastic
+cross section near 12 GeV in pure ini-
+tial spin states plotted vs. the scaled P2
+t-
+variable [11].
+ 
+Figure 3. Themeasuredspins-parallel/
+spins-antiparallel elastic cross-section ra-
+tio (σ↑↑/σ↑↓) plotted vs. P2
+t[12].
+2to try to polarize the protons in the ZGS.
+At the 1969 New York APS Meeting, I learned that EG&G was the repr esentative for
+a new polarized proton ion source made by ANAC in New Zealand. I discu ssed this with
+my long-time colleague, Larry Ratner, and then with Bruce Cork, Ar gonne’s Associate
+Director, and Robert Duffield, Argonne’s Director. They apparent ly decided it was a
+good idea; Duffield soon hired me as a consultant to Argonne at $100 p er month. In
+1973, after a lot of hard work by many people, the ZGS accelerated the world’s first high
+energy polarized proton beam [7].
+The ZGS needed some hardware to overcome both intrinsic and imper fection depolar-
+izing resonances. Fortunately, both types of resonances were f airly weak at the 12 GeV
+ZGS, which was the highest energy weak focusing accelerator ever built. All higher en-
+ergy accelerators wisely use strong focusing [8], which makes the de polarizing resonances
+much stronger. If we had first tried to accelerate polarized proto ns at a strong focusing
+accelerator, such as the AGS, we probably would have failed and aba ndoned the polarized
+proton beam business. Fortunately, it worked at the weak focusin g ZGS [7,9], and exper-
+iments [10] soon showed that the p-p total cross-section had sign ificant spin dependence;
+this surprised many people, including me.
+Figure 2 shows our perhaps most important result [11] from the ZGS polarized pro-
+ton beam. The 12 GeV proton-proton elastic cross section in pure in itial spin states is
+plotted against the scaled P2
+t-variable; in the diffraction peak the spin-parallel and spin-
+antiparallel cross-sections are essentially equal to each other an d to the unpolarized data
+from the CERN ISR at s= 2800GeV2; thus, in small-angle diffractive scattering, the
+protons in different spin states (and at different energies) all have about the same cross-
+section. The medium- P2
+tcomponent, which still exists near 12 GeV, has only a small spin
+dependence; again note that it has totally disappeared at 2800 GeV2. However, the be-
+havior of thelarge- P2
+thard-scattering component was a great surprise. When the prot ons’
+spins are parallel, they seem to have exactly the same behavior as th e much higher energy
+unpolarized ISR data; however, when their spins are antiparallel th eir cross-section drops
+with the medium- P2
+tcomponent’s steeper slope. When this data first appeared in 1977
+and 1978, people were totally astounded; most had thought that s pin effects would dis-
+appear at high energies. In the years following, many theoretical p apers tried to explain
+this unexpected behavior; none were fully successful. In particula r, the theory that is now
+called QCD, has been unable to deal with this data; Glashow once called this experiment
+”..the thorn in the side of QCD”. In his summary talk at Blois 2005, Stan Brodsky called
+this result ”..one of the unsolved mysteries of Hadronic Physics”.
+I learned something important from questions during two 1978 semin ars about this
+result. Two distinguished physicists, Prof. Weisskopf at CERN and t hen Prof. Bethe
+at Copenhagen a week later, asked the same question apparently in dependently. Each
+said that our big spin effect at large- P2
+twas quite interesting; but at 12 GeV, the spins-
+parallel/spins-antiparallel ratio was only large near 90◦
+cm, where particle identity was
+important for p-p scattering. They asked: How could one be sure t hat our large spin
+effect was due to hard-scattering at large- P2
+t, rather thanparticle identity near 90◦
+cm? One
+would be foolish to ignore the comments of two such distinguished the orists; moreover,
+they were related to Prof. Serber’s comment 10 years earlier.
+It seemed that their question could not be answered theoretically. Thus, we tried to
+answer it experimentally with a second ZGS experiment, which varied P2
+tby holding the
+3p-p scattering angle fixed at exactly 90◦
+cm, while varying the energy of the proton beam.
+This 90◦
+cmp-p elastic fixed-angle data [12] is plotted against P2
+tin Fig. 3, along with the
+fixed-energy data [11] of Fig. 2. There are large differences at sma llP2
+t, where the 90◦
+cm
+data are at very low energy; however, above P2
+tof about 1.5 (GeV/c)2, the two sets of
+data fall right on top of each other. The point at P2
+t= 2.5 (GeV/c)2, where the ratio is
+near 1, is just as much at 90◦
+cm, as the 5 (GeV/c)2point, where the ratio is 4. This data
+apparently convinced Profs. Bethe and Weisskopf that the large s pin effect was not due
+to 90◦
+cmparticle identity and it was a large- P2
+thard-scattering effect.
+ 
+Figure 4. Ann≡(σ↑↑−σ↑↓)/(σ↑↑+σ↑↓) is
+plotted against PLab. [5]Figure 4 shows Annfor p-p elastic scat-
+tering plotted against the lab momentum,
+PLab[5]; itincludes theZGSdatafromFig.3
+plus some lower energy data obtained from
+Willy Haeberli, who is an expert on low en-
+ergy p-p spin experiments. At the lowest
+momentum (near T= 10 MeV) Annis very
+close to−1; thus, two protons with paral-
+lel spins can never scatter at 90◦
+cm. Next
+Anngoes rapidly to +1; then protons with
+antiparallel spins can never scatter at 90◦
+cm.
+Then at medium energy, there are some os-
+cillations that were once thought to be due
+to dibaryon resonances, but may be due to
+the onset of N∗resonance production. In
+the ZGS region, Annfirst drops rapidly; it is
+next small and constant over a large range;
+it then rises rapidly to 0.6. These huge and
+sharp oscillations of Annseem impressive.
+I now turn to funding. In 1972 the AEC
+had agreed to shut down the ZGS in 1975 to get funding for PEP at SL AC. When
+the unique ZGS polarized beam started operating in 1973, the wisdom of this decision
+was questioned; AEC then set up a committee which extended ZGS op erations through
+1977. A second committee in 1976 extended operations of the ZGS p olarized beam until
+1979 [13]. Henry Bohm, the President of AUA, which operated Argon ne, asked ERDA,
+which had replaced AEC, to set up a third committee to again extend Z GS running.
+However, OMB objected, so there was no third committee; nevert heless, his efforts had
+some benefit. When James Kane, of ERDA, responded negatively to Dr. Bohm, his
+justification was that it might now be possible to accelerate polarized protons in a strong
+focusing accelerator such as the AGS; morover, Dr. Kaneofficially c opied me onhis letter.
+We had started interacting with Ernest Courant and others at Bro okhaven about
+polarizing the AGS, first at a 1977 Workshop in Ann Arbor [14]. Then at a 1978 Polarized
+AGS Workshop at Brookhaven [15], Brookhaven’s Associate Directo r, Ronald Rau, asked
+me for a copy of Dr. Kane’s letter. He used it to convince William Wallenme yer, the
+long-time Director of High Energy Physics at AEC, ERDA and DoE, to p rovide about
+$8 Million to Brookhaven, and about $2 Million split between Michigan, Arg onne, Rice
+and Yale, for the challenging project of accelerating polarized prot ons in the strong-
+focusing AGS, and later in the 400 GeV ISABELLE collider. ISABELLE w as canceled in
+41983, but was later reborn as RHIC, which is now colliding 250 GeV polar ized protons.
+It was far more difficult to accelerate polarized protons in the stron g focusing AGS
+than in the weak focusing ZGS. The strong focusing principal, invent ed by Courant,
+Livingston and Snyder [8], made possible all modern large circular acce lerators by using
+alternating quadrupole magnetic fields to strongly focus the beam a nd thus keep it small.
+Unfortunately, these strong quadrupole fields were very good at depolarizing protons.
+To accelerate polarized protons to 22 GeV at the AGS, one had to ov ercome 45 strong
+depolarizing resonances. This required: some very challenging hard ware; significantly
+upgrading the AGS controls; and spending lots of time individually over coming the 45
+depolarizing resonances. Michigan built 12 ferrite quadrupole magne ts to allow the AGS
+toovercomeits6intrinsicresonancesbyrapidlyjumpingitsvertical betatrontunethrough
+each resonance. Brookhaven was building their 12 power supplies; b ut each power supply
+had to provide 1500 Amps at 15,000 Volts (about 22 MW) during each q uadrupole’s
+1.6µsec rise-time. Overcoming the many imperfection depolarizing reson ances (occurring
+every 520 MeV) required programming the AGS’s 96 small correction dipole magnets
+to form a horizontal B-field wave of 4 oscillations at the instant when the proton energy
+passed through the Gγ= 4 inperfection resonance. Then, about 20 msec later in the AGS
+cycle, when Gγwas 5, the 96 magnets had a horizontal B-field wave with 5 oscillations ,
+etc. (G= 1.79285 is the proton’s anomalous magnetic moment, while γ=E/m.).
+After all this hardware was installed, an even larger problem was tun ing the AGS. In
+1988,whenweacceleratedpolarizedprotonsto22GeV,weneeded 7weeks ofexclusive use
+of the AGS; this was difficult and expensive. Once a week, Nicholas Sam ios, Brookhaven’s
+Director, would visit the AGS Control Room to politely ask how long the tuning would
+continue and to note that it was costing $1 Million a week. Moreover, it was soon clear
+that, except for Larry Ratner (then at Brookhaven) and me, no one could tune through
+these 45 resonances; thus, for some weeks, Larry and I worked 12-hour shifts 7-days
+each week. After 5 weeks Larry collapsed. While I was younger than Larry, I thought it
+unwise to try to work 24-hour shifts every day. Thus, I asked our Postdoc, Thomas Roser,
+who until then had worked mostly on polarized targets and scatter ing experiments, if he
+wanted to learn accelerator physics in a hands-on way for 12 hours every day. Apparently,
+he learned well, and now leads Brookhaven’s Collider-Accelerator Divis ion.
+One benefit from this difficult 7-week period [7,16] was learning that o ur method of
+individually overcoming each resonance, which had worked so well at t he ZGS [7,9], might
+work at the AGS, but would not be practical at higher energy accele rators. This lesson
+helped to launch our Siberian snake programs at IUCF [7,17] and the n SSC [18,19].
+In the 1980’s, a new proton collider, the SSC, was being planned; it wa s to have two
+20 TeV proton rings each about 80 km in circumference. Owen Chamb erlain and Ernest
+Courant encouraged me to form a collaboration to insure that polar ized protons would
+be possible in the SSC. We first organized a 1985 Workshop in Ann Arbo r, with Kent
+Terwilliger. This Workshop [18] concluded that it should be possible to a ccelerate and
+maintainthepolarizationof20TeVprotonsintheSSC, but onlyifthen ew Siberiansnake
+concept of Derbenev and Kondratenko [20] really worked; otherw ise, it would be totally
+impractical. Recall that it took 49 days to correct the 45 depolarizin g resonances at the
+AGS, about one per day. Each 20 TeV SSC ring would have about 36,00 0 depolarizing
+resonances to correct. Moreover, these higher energy resona nces would be much stronger
+and harder to correct; but even at one per day, this would require about 100 years of
+5tuning for each ring. The Workshop also concluded that one must pr ove experimentally
+that the too-good-to-be-true Siberian snakes really worked; otherwise, there would be no
+approval to install the 26 Siberian snakes needed in each SSC ring.
+Indiana’s IUCF was then building a new 500 MeV synchrotron Cooler Rin g [7]. Some
+of us Workshop participants then collaborated with Robert Pollock a nd others at IUCF
+to build and test the world’s first Siberian snake in the Cooler Ring. We b rought experi-
+ence with synchrotrons and high energy polarized beams, while the I UCF people brought
+experience with low energy polarized beams and the CE-01 detector , which was our po-
+larimeter. In 1989, we demonstrated that a Siberian snake could ea sily overcome a strong
+imperfection depolarizing resonance [7,17]. For 13 years we continu ed these experiments
+and learned many things about spin-manipulating polarized beams. Af ter the IUCF
+Cooler Ring shut down in 2002, this polarized proton and deuteron be am experiment
+program was continued at the 3 GeV COSY in Juelich, Germany from 20 02-2009 [7].
+In 1990 we formed the SPIN Collaboration and submitted to the SSC: Expression of
+Interest EOI-001 [19]. It proposed to accelerate and store polar ized protons at 20 TeV;
+and to study spin effects in 20 TeV p-p collisions. It was submitted a we ek before the
+deadline, which made it SSC EOI-001. Thus, we made the first presen tation to the SSC’s
+PAC before a huge audience that included many newspaper reporte rs and TV cameras.
+Perhaps partly due to this publicity, we were soon partlyapproved by SSC Director Roy
+Schwitters. By partlyI mean that he decided to add 26 empty spaces for Siberian snakes
+in each SSC Ring; each space was about 20 m long, which added about 0 .5 km to each
+Ring. Unfortunately, the SSC was canceled around 1993, before it was finished, but
+after $2.5 Billion was spent. Nevertheless, our detailed studies of th e behavior and spin-
+manipulation of polarized protons at IUCF and COSY helped in developin g polarized
+beams around the world: Brookhaven now has 250 GeV polarized pro tons in each RHIC
+ring [7,21]; perhaps someday CERN’s 7 TeV LHC might have polarized pr otons.
+We eventually accelerated polarized protonsto 22 GeV inthe AGS [7, 16] and obtained
+someAnndata [7,22,23] well above the ZGS energy of 12 GeV; but we never h ad enough
+polarized-beam data-time to get precise Anndata at high- P2
+t. But, during tune-up runs
+for theAnnexperiment, we used the unpolarized AGS proton beam to test our p olarized
+proton target and double-arm magnetic spectrometer by measur ingAnin 28 GeV proton-
+proton elastic scattering; this data resulted in an interesting surp rise. Despite QCD’s
+inabilitytoexplainthebig AnnfromtheZGS,ourQCDfriendshadmadeafirmprediction
+that the one-spin Anmust go to 0; they also said this prediction would become more firm
+at higher energies and in more violent collisions. But above P2
+t= 3 (GeV/c)2,Aninstead
+began to deviate from 0 and was quite large at P2
+t= 6 (GeV/c)2. This led to more
+controversy [23]; some QCD supporters said that our Andata must be wrong.
+Experimenters take such accusations seriously. Thus, we starte d preparing an exper-
+iment that could study Anat high-P2
+twith better precision. Our spectrometer worked
+well, but we could only use about 0.1% of the AGS beam intensity, becau se a higher in-
+tensity beam would heat our Polarized Proton Target (PPT) and dep olarize it. Thus, we
+started building a new PPT [7,24] that could operate with 20 times mo re beam intensity;
+this required4He evaporation cooling at 1 K, which has much more cooling power than
+our earlier3He evaporation PPT at 0.5 K. However, to maintain a target polarizat ion
+near 50% at 1 K required increasing the B-field from 2.5 to 5 Tesla. Thu s, we ordered
+a 5 T superconducting magnet from Oxford Instruments, with a B- field uniformity of a
+6few 10−5over the PPT’s 3 cm diameter volume. We also obtained a Varian 20 W at
+140 GHz Extended Interaction Oscillator; it was apparently the high est power 140 GHz
+microwave source available. As the PPT assembly started in 1989, we worried that, if the
+PPT model was wrong, the polarization might be only 10%; but we were very lucky; it
+was 96% [7,24].
+Moreover, the target polarization averaged 85% for a 3-month-lo ng run with high-
+intensity AGS beam in early 1990. As shown in Fig. 5, this let us precisely measure Anat
+even larger P2
+t. When these precise new data were published [25], some theorists se emed
+quite unhappy; they still believed the QCD prediction that Anmust go to 0, but they
+now refused to state at what P2
+tor energy this prediction would become valid. They
+also now said that QCD might not work for elastic scattering, which th ey now considered
+less fundamental than inelastic scattering, where they said QCD sh ould work. Thus,
+one result of our experiments was to make both elastic scattering e xperiments and spin
+experiments unpopular in some circles.
+Experimenters had also started doing inclusive polarization experime nts at Fermilab.
+Figure 6 shows the 400 GeV inclusive hyperon polarization experiment s from the 1970s
+and 1980s, led by Pondrum, Devlin, Heller and Bunce [26]; it clearly show s a small
+polarization at small Ptand a larger polarization at larger Pt. Moreover, their data is
+consistent with 12 GeV data from the KEK PS and with 2000 GeV data f rom the CERN
+ISR. These data do not support the QCD prediction that inelastic sp in effects disappear
+at high energy or high P2
+t.
+Another group at Fermilab, led by Yokosawa, developed a secondary polarized proton
+beam using the polarized protons from polarized hyperon decay. Th e beam’s intensity
+Figure 5. (Left)An≡(σ↑−σ↓)/(σ↑+σ↓) is
+plotted vs. P2
+tfor p-p elastic scattering [25].
+Figure 6. (Bottom) The inclusive Λ polariza-
+tion is plotted against Pt[26].
+7was only about 105per second, but its polarization was about 50% and its energy was
+about 200 GeV. They obtained some nice Andata on inclusive πmeson production [27],
+which are shown in Fig. 7. The Anvalues for π+andπ−mesons are both large but with
+opposite signs, while Anfor theπ0data is 50% smaller and is positive. These 200 GeV
+data provide little support for QCD.
+Figure 7. Anfor inclusive π-meson produc-
+tion is plotted vs. XF[27].We tried to measure spin effects in very
+high energy p-p scattering at UNK, which
+IHEP-Protvino started building around
+1986 [7]; IHEP and Michigan signed the
+NEPTUN-A Agreement in1989. Michigan’s
+main contribution was a 12 Tesla at 0.16 K
+Ultra-coldSpin-polarizedJet[7]. UNK’scir-
+cumference would be 21 km with 3 rings: a
+400 GeV warm ring and two 3 TeV super-
+conducting rings; its injector was IHEP’s ex-
+isting 70 GeV accelerator, U-70 [7]. By 1998
+the UNK tunnel and about 80% of its 2200
+warm magnets were finished, and 70 GeV
+protons were transferred into its tunnel with
+99% efficiency. However, progress became
+slower each year due to financial problems;
+in1998Russia’s MINATOM placed UNK on
+long-term standby [7].
+IHEP Director, A.A. Logunov, had ear-
+lier suggested moving our experiment to
+IHEP’s existing 70 GeV U-70 accelerator.
+By March 2002 the resulting SPIN@U-70
+Experiment on 70 GeV p-p elastic scatter-
+ing at high P2
+twas fully installed [7], except
+for our detectors and Polarized Proton Tar-
+get (PPT) [7,24]. However, just before our 4 tons of detectors, electronics and computers
+weretobeshipped, theUSGovernment suspended theUS-Russian PeacefulUseofAtomic
+Energy Agreement started by President Eisenhower in 1953. Neve rtheless, DoE asked us
+to send the shipment, since under the terms of the PUoAE Agreeme nt, the experiment
+should have been done exactly as planned; DoE faxed us a copy of th e Agreement. Thus,
+we sent the shipment. It arrived at Moscow airport on March 11, 20 02, where it was
+impounded for 8 months before it was returned to Michigan.
+Despite this problem, we remain friends with our IHEP colleagues and t here have
+been four SPIN@U-70 test runs using Russian detectors and an un polarized target; we
+participated in the November 2001 and April 2002 runs. We hope tha t the US-Russian
+PUoAE Agreement will soon be restarted so that the SPIN@U-70 ex periment can con-
+tinue and measure AnatP2
+tnear 12 (GeV/c)−2. However, if this international problem
+continues, we may try to do a similar experiment at Japan’s new high-in tensity 50 GeV
+proton accelerator, J-PARC [7] that is now starting to run. If J- PARC could accelerate
+polarized protons to 50 GeV, then one could study the large and mys terious elastic spin
+effects in both AnandAnnfor the first time in decades.
+8Figure 8. Inclusive pion asymmetry in proton-proton collisions [27] .
+Tosummarize, forthepast30yearsQCD-basedcalculationshavec ontinuedtodisagree
+with the ZGS 2-spin and AGS 1-spin elastic data, and the ZGS, AGS, Fe rmilab and now
+RHIC [28] inclusive data. To be specific:
+* These large spin effects do not go to zero at high-energy or high- Pt, as was predicted.
+* No QCD-based model can yet explain simultaneously all these large s pin effects.
+There is a BASIC PRINCIPLE OF SCIENCE:
+* If a theory disagrees with reproducible experimental data, then it must be modified.
+Precise spin experiments could provide experimental guidance for t he required modifica-
+tion of the theory of Strong Interactions. New experiments at hig her energy and higher
+Pton the proton-proton elastic cross-section’s: dσ/dt,AnnandAncould provide further
+guidance for these modifications, just as the RHIC inclusive Anexperiment [28] confirmed
+the earlier Fermilab experiments [27]. Elastic scattering is especially imp ortant because:
+* It is the only exclusive process large enough to be measured at TeV energy.
+This is probably because proton-proton elastic scattering is domina ted by the diffrac-
+tion due to the millions of inelastic channels that compete for the tota l cross-section of
+only about 100 milibarns at TeV energies. Many people may have forgo tten this simple
+but essential geometrical approach [1], which I learned from Prof. Serber’s optical model
+in 1963 [3]; perhaps it should now be learned or relearned by others.
+References
+[1] A.D. Krisch, Phys. Rev. Lett. 11, 217 (1963); Phys. Rev. 135, B1456 (1964).
+[2] A.D. Krisch, Phys. Rev. Lett. 19, 1149 (1967); Phys. Lett. 44, 71 (1973).
+[3] R. Serber, PRL 10, 357 (1963).
+[4] P.H. Hansen and A.D. Krisch, Phys. Rev. D15, 3287(1977).
+[5] A.D. Krisch, Z. Phys. C46, S113 (1990).
+[6] C.W. Akerlof et al., Phys. Rev. Lett. 17, 1105(1966); Phys. Rev. 159, 1138 (1967).
+[7] See http://theor.jinr.ru/ ∼spin/2009/table09.pdf for the talk’s figures and photos.
+9[8] E.D. Courant, M.S. Livingston, H.S. Snyder, Phys. Rev. 88, 1190 (1952).
+[9] T. Khoe et al., Particle Accelerators 6, 213 (1975).
+[10] E.F. Parker et al., Phys. Rev. Lett. 31, 783 (1973); W. deBoer et al., ibid ,34, 558
+(1975).
+[11] J.R. O’Fallon et al., P.R.L.39, 733 (1977); D.G. Crabb et al., ibid .41, 1257 (1978);
+A.D. Krisch, The Spin of the Proton , Scientific American, 240, 68 (May 1979).
+[12] A.M.T. Lin et al., P.L.74B, 273 (1978); E.A. Crosbie et al., P.R.D23, 600 (1981).
+[13] R.L. Walker, R.E. Diebold, G. Fox, J.D. Jackson, A.D. Krisch, T.A. O ’Halloran,
+D.D. Reeder, N.P. Samios, H.K. Ticho, Report: AEC Review Panel on ZGS (1976).
+[14] A.D. Krisch and A.J. Salthouse, eds., AIP Conf Proc. 42, (AIP, NY, 1978).
+[15] B. Cork, E.D.Courant, D.G.Crabb, A.Feltman, A.D.Krisch, E.F. Parker, L.G.Rat-
+ner, R.D. Ruth, K.M. Terwilliger, Accelerating Polarized Protons in AGS (1978).
+[16] F.Z. Khiari et al., Phys. Rev. D39, 45 (1989).
+[17] A.D. Krisch et al., Phys. Rev. Lett. 63, 1137(1989).
+[18] A.D. Krisch, A.M.T. Lin, O. Chamberlain, eds., AIP Conf Proc. 145, (AIP, 1985).
+[19]SSC-EOI-001 SPIN Collaboration , Michigan, Indiana, Protvino, Dubna, Moscow,
+KEK, Kyoto (1990).
+[20] Ya.S. Derbenev, A.M. Kondratenko, AIP Conf Proc. 51, 292, (AIP, 1979).
+[21] M. Bai et al., Phys. Rev. Lett. 96, 174801 (2006); AIP Conf. Proc. 915,22 (2007).
+[22] D.G. Crabb et al., Phys. Rev. Lett. 60, 2351 (1988).
+[23] A.D.Krisch, Collisionsof SpinningProtons ,ScientificAmerican, 257, 42(Aug1987).
+[24] D.G. Crabb et al., Phys. Rev. Lett. 64, 2627 (1990).
+[25] D.G. Crabb et al., Phys. Rev. Lett. 65, 3241 (1990).
+[26] G. Bunce et al., Phys. Rev. Lett. 36, 1113 (1976); K. Heller et al., ibid ,51, 2025
+(1983).
+[27] D.L. Adams et al., Z. Phys. C56, 181 (1992).
+[28] C. Aidala, AIP Conf. Proc. 1149, 124,(AIP, NY, 2009).
+10
\ No newline at end of file
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+arXiv:1001.0791v1  [astro-ph.CO]  5 Jan 2010Averaging in cosmological models
+A.A. Coley
+Department of Mathematics and Statistics
+Dalhousie University, Halifax, NS B3H 3J5, Canada
+aac@mathstat.dal.ca
+Abstract
+The averaging problem in cosmology is of considerable impor tance for the correct
+interpretation of cosmological data. We review cosmologic al observations and discuss
+someof theissuesregardingaveraging. Wepresentaprecise definitionofacosmological
+model and a rigorous mathematical definition of averaging, b ased entirely in terms of
+scalar invariants.
+1 Introduction
+Cosmological observations [1], based on the assumption of a spatially homogeneous and
+isotropic Friedmann-Lemaˆ ıtre-Robertson-Walker (FLRW) model plus small perturbations,
+are usually interpreted as implying that there exists dark energy, t he spatial geometry is flat,
+and that there is currently an accelerated expansion, giving rise to the so-called standard
+ΛCDM-concordance model. Although the concordance model is quit e remarkable, it does
+not convincingly fit all data (see below). Unfortunately, if the unde rlying cosmological model
+is not a perturbation of an exact flat FLRW solution, the convention al data analysis and
+their interpretation is not necessarily valid.
+For example, the standard analysis of type Ia supernovae (SNIa) and cosmic microwave
+1background (CMB) data in FLRW models cannot be applied directly whe n inhomogeneities
+or backreaction effects are present. However, supernovae dat a can be explained without dark
+energy in inhomogeneous models, where the full effects of general relativity (GR) come into
+play. In one approach exact inhomogeneous cosmological models ca n be utilised. Indeed,
+it has been shown that the Lemaˆ ıtre-Tolman-Bondi (LTB) solution can be used to fit the
+observed data without the need of dark energy, although it may be necessary to place the
+observer at a preferred location [2].
+A second approach, and the one of interest here, is backreaction s through averaging. The
+averaging problem in cosmology is of considerable importance for the correct interpretation
+of cosmological data. The correct governing equations on cosmolo gical scales are obtained by
+averagingtheEinsteinfieldequations(EFE)ofGR(plusatheoryofp hotonpropagation; i.e.,
+information on what trajectories actual particles follow). By assu ming spatial homogeneity
+and isotropy on the largest scales, the inhomogeneities affect the d ynamics though correction
+(backreaction) terms, which can lead to behaviour qualitatively and quantitatively different
+from the FLRW models; in particular, the expansion rate may be signifi cantly affected.
+2 Cosmological observations
+From the evidence of the CMB radiation, the universe was very smoo th at the time of
+last scattering. By the Copernican principle, the assumption of glob al isotropy and spatial
+homogeneity is then justified at the epoch of last scattering. Thus , the paradigm for our
+current standard model of the universe assumes the underlying g eometry is FLRW, with
+additional Newtonian perturbations, and in matching the cosmologic al observables that de-
+rive from such a geometry we have been led to the conclusion over th e past decade that the
+present–day universe is dominated by a cosmological constant, Λ, or other fluid–like “dark
+energy”, which violates the strong energy condition. In the case o f the ΛCDM paradigm,
+2dark energy only becomes appreciable at late epochs. Dark energy is widely described as
+the biggest problem in cosmology today.
+There are several problems regarding the ΛCDM model. First, it is diffi cult to under-
+stand the large value for Λ and why the contributions of ordinary ma tter and the repulsive
+component are roughly equal today, at around 10 billion years (the coincidence problem).
+Second, the universe is not perfectly homogeneous and isotropic ( or even perturbatively near
+homogeneity and isotropy). There are non-linear structures in th e real universe which are
+not described by perturbations around a smooth background, wit h a distribution that is sta-
+tistically homogeneous and isotropic above a scale of about 100 Mpc ( or, more precisely, 100
+h−1Mpc, but we shall omit the factor h−1for simplicity here) [3]. Linear perturbations are
+onlyvalid when boththe curvatureand thedensity contrasts rema in small, which is certainly
+not the case in the non-linear regime of structure formation when t he SNIe observations are
+made.
+Indeed, the largest structures so far detected are limited only by the size of the surveys
+that foundthem [4]. At the present epochthe distribution of matte r isfar fromhomogeneous
+onscales less than150–300Mpc. The actual universe has a sponge –like structure, dominated
+byhugevoidssurroundedbybubblewalls, andthreadedbyfilaments , withinwhichclustersof
+galaxies are located. Locally there two enormous voids, both 35 to 7 0 Mpc across, associated
+with the so-called velocity anomaly [5], a large filament known as the Sloa n great wall about
+400 Mpc long [6] and the Shapely supercluster with a core diameter of 40 Mpc at a distance
+of∼200 Mpc [7]. In addition, there has been detection of anomolously larg e local bulk flows
+[8] and evidence for a significant anisotropy in the local Hubble expan sion at distances of
+∼100 Mpc [9]. Recent surveys suggest [10] that some 40–50% of the p resent volume of the
+universeisinvoidsofacharacteristicscale30Mpc. Ifsmallerminivoids andlargersupervoids
+are included, then our observed universe is presently void–dominat ed by volume; thus within
+regions as small as 100 Mpc density contrasts ∼ −1 are observed leading to substantial
+gradients in the (local Ricci) spatial curvature [11]. Therefore, sp atial homogeneity is valid
+3only on scales larger than at least 100 Mpc [3], in contradiction with the predictions of the
+ΛCDM model in which the scale beyond which the distribution should bec ome uniform is
+about 10 Mpc [4].
+The present distribution of matter is clearly very complex, and since we cannot solve the
+EFE for this distribution of matter analytically, there is an important question as to how we
+operationally match the average geometry of this distribution to th e simple FLRW models.
+The mere fact that the universe is presently inhomogeneous means that the assumptions
+implicit in the FLRW approximation can no longer be justified at the pres ent epoch in the
+almost exact sense that they were justified at the epoch of last sc attering. The situation is
+further complicated by the fact that most data analysis based ont he standard model (FLRW
++ perturbations; ΛCDM ) is model- and prior- dependent [18].
+Consequently, spatial homogeneity only applies at the present day in an averaged sense.
+Given the observed inhomogeneities and that the nonlinear growth o f structure appears to
+be roughly correlated to the epoch when cosmic acceleration is infer red to begin, it has
+been suggested that the FLRW geometries are inadequate as a des cription of the universe
+at late times and the introduction of a smooth dark energy is a mistak en interpretation of
+the observations. A universe which is homogeneous and isotropic on ly statistically does not
+generally expand like an exactly homogeneous and isotropic universe , even on average. It is
+possible that there are large effects on the observed expansion ra te (and hence on other ob-
+servables) due to the backreaction of inhomogeneities in the univer se. Anything that affects
+the observed expansion history of the universe alters the determ ination of the parameters
+of dark energy; in the extreme it may remove the need for dark ene rgy. Indeed, it has been
+suggested in that inhomogeneities related to structure formation could be responsible for
+accelerated expansion [15].
+The effects of averaging can be signicant. Using perturbation theo ry, effects of order
+∼10−4are often quoted. However, these affects occur by averaging ov er the Hubble volumes
+4and not over regions of ∼100−200 Mpc. At best this is (only) a self-consistency analysis.
+In addition, there are highly non-Gaussian inhomogeneities in the late universe, and the
+coherence of structures causes small deviations in observations to sum to a large deviation,
+and there can be significant effects on observations from the back reaction of inhomogeneities
+[12].
+In [14] the hierarchy of the critical scales for large scale inhomogen eities (backreaction
+effects) were calculated, at which 10% effects show up from averag ing at different orders over
+a local domain in space-time. The dominant contribution comes from t he averaged spatial
+curvature, observable up to scales of ∼200 Mpc. The averaged spatial curvature typically
+leads to 10% (1%) effects up to ∼80 (240) Mpc. The cosmic variance of the local Hubble
+rate is 10% (5%) for spherical regions of radius 40 (60) Mpc. Below ∼40 Mpc, the cosmic
+variance of the Hubble rate is larger than 10%. At lower scales the kin ematical backreaction,
+due to second order perturbations caused by local inhomogeneitie s and anisotropies, are
+important. The crude estimates are comparable to the actual den sity variance determined
+from large scale structure surveys [3, 4, 11]. In addition, it has be en found that a matter
+model with discrete masses (rather than an idealised continuous flu id) leads to corrections
+for cosmological parameters ∼10−20% [13]. Indeed, it has been argued that the effects of
+averaging can theoretically be as large as ∼40% when the equivalence principle of GR is
+properly applied [11].
+There are also a number of other potential problems with the stand ard model. Apart
+from WMAP date ( z∼1100), the standard model is based on local observations ( z <2),
+and consequently it has been argued that the data does not convin cingly imply accelera-
+tion [14]. It is noteworthy that the quality of fit of the ΛCDM model ha s decreased with
+the introduction of each new SNIa data set, which may hint at inadeq uacy of the ΛCDM
+description [15]. Indeed, the standard model does not fit all data; there is tension between
+different SNIa data sets [16] and tension between different data se ts, especially between SNIa
+data and CMB data [16, 17], but also with nucleosynthesis and other la rge scale structure
+5data [11].
+2.0.1 Discussion: spatial curvature
+Clearly, backreaction (averaging) effects are real, but their relat ive importance still need to
+be determined. Within perturbation theory, the value of the norma lized spatial curvature,
+Ωk, is expected to be small. However, different authors have argued t hat Ωkcan be as large
+as 5−10% [11, 13, 15]. In particular, CMB data does not necessarily imply fla tness [15]; the
+position of the CMB peaks is consistent with significant spatial curva ture provided that the
+expansion history is sufficiently close to the spatially flat ΛCDM model. I ndeed, conclusions
+drawn about spatial curvature from the CMB are model- and prior- dependent; in a clumpy
+universe, the usual expression is inapplicable due to the non-trivial evolution of the spatial
+curvature as well as the fact that clumping contributes to the exp ansion rate, and there
+is no simple argument for obtaining the position of the CMB peaks. In a ddition, if the
+spatial curvature (parameter k) is allowed to be a function of position, then considerable
+spatial curvature (locally) is permissable (consistent with CMB obse rvations) [15, 13], since
+curvature can affect different observations at different scales in d ifferent ways (e.g., large
+scale structure, z <2, and CMB, z∼1100).
+Observational data perhaps suggests a normalized spatial curva ture|Ωk| ≈0.01−0.02
+(i.e., of about a percent). Combining these observations with large s cale structure obser-
+vations then puts stringent limits on the curvature parameter in th e context of adiabatic
+ΛCDM models; however, these data analyses are very model- and pr ior-dependent, and
+care is needed in the proper interpretation of the data. There is a h euristic argument that
+Ωk∼10−3−10−2(Ωk∼1%) [20], which is consistent with CMB observations [1] and agrees
+with estimates for intrinsic curvature fluctuations using realistically modelled clusters and
+voids in a Swiss-cheese model. In particular, the MG equations (see b elow) were explic-
+itly solved in a FLRW background geometry and it was found that the c orrelation tensor
+6(backreaction) is of the form of a spatial curvature [19]. Thus, th e averaged EFE for a flat
+spatially homogeneous, isotropic macroscopic space-time geometr y has the form of the EFE
+of GR for a non-flat spatially homogeneous, isotropic space-time ge ometry.
+Itmust beappreciatedthatsuchavalueforΩ k, atthe1%level, isrelativelylargeandmay
+have a significant dynamical effect on the evolution of the universe a nd the interpretation
+of cosmological observations. Indeed, in such a scenario the curr ent contribution from the
+spatial curvatureis comparable to the energy density inluminous ma tter. In addition, such a
+value cannot be naturally explained by inflation. From standard analy sis, depending on the
+initial conditions and the details of a specific model of inflation, |Ω−1|would be extremely
+small. Therefore, any value for Ω kat the 1 % level can only be naturally explained in terms
+of an averaging effect. In addition, such an effect would compromise any efforts to use data
+to constrain dark energy models (within the standard paradigm) wit h a variable equation of
+state [21].
+3 The averaging problem in cosmology
+3.1 General Approaches
+The Universe is not isotropic or spatially homogeneous on local scales . The gravitational
+FE on large scales are obtained by averaging the EFE of GR. It is nece ssary to use an exact
+covariant approach which gives a prescription for the correlation f unctions that emerge in
+an averaging of the full tensorial EFE.
+There are a number of approaches to the averaging problem. In th e approach of Buchert
+a 3+1 cosmological space-time splitting is employed (i.e., this procedur e is not generally
+covariant) and only scalar quantities are averaged (and thus the g overning equations are not
+7closed) [22]. The perturbative approach (backreaction about an F LRW background [12])
+involves averaging the perturbed EFE. However, a perturbation a nalysis cannot provide any
+information about an averaged geometry; thus perturbation the ory cannot be conclusive and
+provide a complete solution.
+To date the macroscopic gravity (MG) approach is the only approac h to the averaging
+problem in GR which gives a prescription for the correlation functions which emerge in an
+averaging of the non-linear FE (without which the averaging of the E FE simply amount
+to definitions of the new averaged terms) [23]. The MG space-time av eraging approach is a
+fully covariant, gaugeindependent andexact method, in which the a veraged EFE arewritten
+in the form of the EFE for the macroscopic metric tensor when the c orrelation terms are
+moved to the right-hand side of the averaged EFE to serve as the g eometric modification to
+the averaged (macroscopic) matter energy-momentum tensor. For the cosmological problem
+additional assumptions are required: with reasonable cosmological assumptions, the correla-
+tion tensor in Zalaletdinov’s scheme takes the form of a spatial curv ature [19], and Buchert’s
+scheme can be realized as a consistent limit [24].
+There are other approaches to averaging. The formal mathemat ical issues of averaging
+tensors on a differential manifold has recently been revisted. We no te that integrating
+scalars on spacetime regions is always well-defined and it may be possib le to avoid several
+of the technical problems of averaging by adopting an approach ba sed on scalar curvature
+invariants.
+4 Cosmological models
+A cosmological model is a mixedmodel, in that the matter is already assumed to be averaged
+but the geometry is not (necessarily). Therefore, we need a cons istent model for the matter,
+represented on the characteristic averaging scale, and its appro priate (averaged) physical
+8properties. It is known that the separation between the gravitat ional field and the matter is
+not scale invariant and the notion of a perfect fluid is not scale invaria nt; averaging (in the
+presence of a gravitational field) modifies the equation of state of the matter. In addition,
+since averaging does not conserve geodesics, we need further as sumptions in order to be able
+to compare the models with observational data.
+A precise definition of a cosmological model is necessary; i.e., a frame work in which
+to do averaging. The definition we shall adopt is given by the following c onditions C1
+– C5 [25]: C1. Spacetime Geometry: The spacetime geometry ( M,g) is defined by
+a smooth Lorentzian metric g(characterizing the macroscopic gravitational field) defined
+on a smooth differentiable manifold M. The macroscopic metric geometry is obtained by
+an appropriate spacetime averaging of the microgeometry; thus p art of the definition of a
+cosmological model consists of specifying the averaging scheme (which must be consistent
+with the physical assumptions of the model encapsulated in the con ditions C3 and C4 below)
+andthe cosmological scale over whichaveragingorthesmoothingoccurs(i.e., wemust specify
+the averaging scale ℓor averaging region).
+C2. Timelike Congruence: There exists a timelike congruence ( u) (in principle lo-
+cally), representing a family of fundamental observers. Mathema tically this means that the
+spacetime is I-non-degenerate andhence thespacetime isuniquely characterized byitsscalar
+curvature invariants [26]. In addition to the formal parts C1 and C2 of the definition of a
+cosmological model ( M,g,ℓ,u), we must also specify the physical relationship (interaction)
+between the macroscopic geometry and the matter fields, including how the matter responds
+to the macroscopic geometry.
+C3. Macroscopic FE: There exists an appropriate set of macroscopic FE relating the
+averaged matter and appropriately averaged (or macroscopic) g eometry. This is based on
+an underlying microscopic theory of gravity (such as, for example, GR), and an appropriate
+formalism to average the geometry and find corrections (correlat ions) due to averaging the
+9Einstein tensor in the resulting FE:
+˜Ga
+b+Ca
+b=Ta
+b, (4.1)
+where˜Ga
+b≡˜Ra
+b−1
+2δa
+b˜Rand˜Ra
+bis the Ricci tensor of the averaged macrogeometry, Ca
+bis
+the correlation tensor, and Ta
+bis the energy momentum tensor (already assumed averaged).
+C4. Equations of motion: We also need to know the trajectories along which the
+cosmological matter moves (and also the light trajectories, which d etermine observational
+relations). In principle, the average motion of a photon need not be a null geodesic in the
+averaged geometry [20]. C5. Observations: Finally, we need to be able to relate averaged
+quantities with physical observables, which ultimately must be consis tent with cosmological
+data.
+In the standard FLRW model there are a number of simplifications an d assumptions.
+The past approaches to averaging have been ideally suited to the FL RW models (with small,
+vanishing in the limit, perturbations). In these models, the macrome tricgis the FLRW
+metric (C1) and ualso has a geometric meaning (C2). In the usual point of view there
+are no correlations due to averaging (i.e., Ca
+b= 0) or, more precisely, they are negligible
+(C3). In this case it follows from the contracted Bianchi identities t hat energy-momentum is
+conserved: Tc
+b;c= 0, which relates the matter to the averaged geometry. All other effects are
+assumed negligible (C4). However, there is no formal argument tha t such assumptions arise
+froma rigorousaveraging scheme of someappropriate (physically m otivated) microgeometry.
+In addition, there are some important effects in the standard mode l which are not necessarily
+small perturbations.
+Since there are no scales explicitly specified in the model, in a sense the model is in-
+complete. Indeed, the model does not even have the ability to dete rmine whether there is
+a scale above which the geometry is exactly FLRW or whether at all sc ales the geometry is
+10only approximately FLRW(with a given perturbation scale). Further more, regarding C5, we
+can ask whether the model agrees with observations? If it does no t, then even if the model
+agrees in some approximate sense with most observations, there is no structure within which
+to discuss the potential small discrepancies with observed data, w hich is a deficiency of the
+model. If the model does, then it would be remarkable, although the re is still the need for a
+physical explanation for the dark energy. Finally, if observations in dicate that Ω k≈1−2%,
+then there is no physical mechanism within the model (particularly if t here is an inflationary
+period) to produce anintrinsic curvature parameter kof this magnitude, whereas aneffective
+curvature parameter ˆkof about a percent arises naturally from averaging.
+5 An approach to averaging using scalars
+Forany given spacetime ( M,g) we define the set of all scalar polynomial curvature invariants
+I ≡ {R,R1,R2,R3,R2,Rµ
+νRν
+µ,...,C2,...} (5.2)
+(where the Riare eigenvalues of the Ricci tensor, and C2≡CµναβCµναβ). Consider a
+spacetime ( M,g) with a set of invariants I. Then, if there does not exist a continuous
+metric deformation of ghaving the same set of invariants as g, the set of invariants will
+be called non-degenerate , and the spacetime metric, g,I-non-degenerate . This implies that
+for a metric which is I-non-degenerate the invariants characterize the spacetime uniq uely,
+at least locally. It was proven [26] that a 4D spacetime is either I-non-degenerate or the
+metric is a degenerate Kundt metric. This is a striking result because it tells us that the
+only metrics not locally determined by their scalar invariants must be o f Kundt form.
+Hence, in general, since we know how to average scalar quantities, w e can average all of
+the scalar curvature invariants that then represents an averag ed spacetime (with that set
+of averaged scalar invariants). In particular, we note that cosmo logical models (as defined
+above) belong to the set of spacetimes completely characterized b y their scalar invariants,
+11suggesting that we can average a cosmological model using scalar in variants. Therefore,
+we have a microgeometry completely characterized by its set of sca lar curvature invariants
+I. We then average these microgeometry scalar curvature invarian ts to obtain a new set
+of macrogeometry scalar curvature invariants ˜I, which now completely characterizes the
+macrogeometry [25].
+5.1 Averaging the geometry
+In the general mathematical context we want to describe the ave raged geometry (represented
+by the Riemann tensor and its covariant derivatives) and interpret the results. Let us
+consider, I, defined by (5.2), which is an ordered set of functions on M. Let us we write
+˜I ≡ {˜R,...,/tildewidestRµ
+νRνµ,...}, which is also an ordered set of functions. The question is then:
+does the ordered set of functions ˜Icorrespond to the associated scalar curvature invariants
+for some metric ˜ g(which could then serve to definethe macrometric ˜ g).
+It is certainly plausible that (some appropriately defined subset of) the ordered set of
+functions ˜Icorrespond to the associated scalar curvature invariants for so me macrometric ˜ g
+for the class of I-non-degenerate geometries that constitute the class of cosmo logical models
+defined. Since the geometries are I-non-degenerate and in 4D the properties of the geometry
+can be represented interms of scalars, andsince relations betwee n different terms (functions)
+in the set I(e.g.,RandR2are functionally dependent) and the corresponding terms in the
+set˜I(e.g.,˜Rand˜R2) are functionally related in exactly the same way and syzygies (e.g.,
+describing the algebraic type) are maintained under averaging, it fo llows that in general
+the set˜Igives rise to a macrometric ˜ g(which will have similar algebraic properties to the
+micrometric g).
+125.1.1 Proposal: Scalar Averaging Procedure
+Let us consider the ordered set of functions Iin the form of (5.2). First, let us omit any
+scalars from this set that are not algebraically independent (e.g., {R2,Rµ
+νRν
+µ...}) to obtain
+an (appropriate ‘independent’) subset IA. Second, for a particular spacetime, we omit any
+scalars from IAthat can be obtained from syzygies defining that particular spacet ime (e.g.,
+defining the algebraic type of the spacetime, such as the Segre typ e or the Petrov type).
+For example, for a Ricci tensor corresponding to the algebraic for m of a perfect fluid we
+could omit {R2,R3}(relative to {R,R1}). We consequently obtain the subset ISA: e.g.,
+ISA≡ {R,R1,...,C2,...}.For the spacetimes under consideration the microgeometry is
+then completely characterized by the (sub)set of scalar curvatu re invariants ISA.
+We now construct the new ordered set of functions ˜ISAby averaging the various scalar
+invariants of ISA:˜ISA≡ {˜R,/tildewidestR1,...,/tildewiderC2,...}, where all of the original scalar invariants
+omitted from the original set Iare replaced by a new set of functions obeying exactly the
+same algebraic properties (or syzygies) as ISA. Therefore, it is assumed that ˜ISAcomes
+equiped with these syzygies, so that we could construct the corre sponding set ˜Iconsisting
+of the members of ˜ISAand all of the corresponding syzygies. Consequently, the set ISAis an
+ordered set of functions (scalar curvature invariants) on Mwhich uniquely determines the
+macrogeometry with exactly the same algebraic properties as the o riginal microgeometry.
+5.1.2 Cosmological models
+In the case of a cosmological model, from C3 we have an effective set of FE and we only
+need to consider themacrogeometric Ricci tensor ˜Ra
+b(thecorrelation tensor isobtained from
+the averaging procedure). The microgeometric Ricci tensor Ra
+bis completely characterized
+by a set of scalar curvature invariants IR. Averaging these scalar curvature invariants we
+obtaintheset ˜I˜R, which completely characterizes themacrogeometric Ricci tensor ˜Ra
+b. Since
+13constructing the Ricci tensor from a set of scalar curvature inva riantsIRis relatively simple
+compared to the corresponding problem for the Riemann tensor, a nd since the reduced
+set of scalar curvature invariants IRis considerably smaller than I, we have reduced the
+complexity of the problem in this new averaging approach. Indeed, f or a Ricci tensor of the
+algebraic form of a perfect fluid, there are effectively (only) two ind ependent zeroth order
+scalar invariants, the Ricci scalar anda single Ricci eigenvalue (corr esponding to the effective
+energy density, ρ, andpressure, p, of the perfect fluid). Therefore, inthe context of the scalar
+averaging procedure, we have the set {˜R,/tildewidestR1}.
+It is necessary to determine whether the correlations due to aver aging alter the geometry
+or affect the effective energy-momentum tensor. This is partly a qu estion of interpretation,
+which must be done within the context of the underlying cosmological model. In particular,
+inthecosmologicalapplicationitmaybeappropriatetoreinterprett heaveragingcorrelations
+as corrections to the matter fields (and hence the effective equat ion of state) through the
+EFE.
+In [26] the specific example of a static spherically symmetric perfect fluid spacetime was
+considered. This is a simple and appropriate model for illustration sinc e it can include an
+arbitrary function of one variable, there is a non-vanishing pressu re, the averaging region
+doesnotchangewithtimeandthereareno gravitationalwaves. Th eaverage correlationscan
+be interpreted as contributing a small constant curvature term, arising from the averaging of
+local inhomogeneities in the micro-Ricci tensor to the smooth macro -Ricci tensor (consistent
+with the results of [19]).
+Acknowledgements . This work was supported by NSERC of Canada.
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+arXiv:1001.0792v1  [astro-ph.HE]  6 Jan 2010PSR J1907+0602: A Radio-Faint Gamma-Ray Pulsar Powering a1
+Bright TeV Pulsar Wind Nebula 2
+A. A. Abdo2,3,1, M. Ackermann4, M. Ajello4, L. Baldini5, J. Ballet6, G. Barbiellini7,8, 3
+D. Bastieri9,10, B. M. Baughman11, K. Bechtol4, R. Bellazzini5, B. Berenji4, 4
+R. D. Blandford4, E. D. Bloom4, E. Bonamente12,13, A. W. Borgland4, J. Bregeon5, 5
+A. Brez5, M. Brigida14,15, P. Bruel16, T. H. Burnett17, S. Buson10, G. A. Caliandro14,15, 6
+R. A. Cameron4, F. Camilo18, P. A. Caraveo19, J. M. Casandjian6, C. Cecchi12,13, 7
+¨O. C ¸elik20,21,22, A. Chekhtman2,23, C. C. Cheung20, J. Chiang4, S. Ciprini12,13, R. Claus4, 8
+I. Cognard24, J. Cohen-Tanugi25, L. R. Cominsky26, J. Conrad27,28,29, S. Cutini30, 9
+A. de Angelis31, F. de Palma14,15, S. W. Digel4, B. L. Dingus32, M. Dormody33, 10
+E. do Couto e Silva4, P. S. Drell4, R. Dubois4, D. Dumora34,35, C. Farnier25, C. Favuzzi14,15, 11
+S. J. Fegan16, W. B. Focke4, P. Fortin16, M. Frailis31, P. C. C. Freire36,63, Y. Fukazawa37, 12
+S. Funk4, P. Fusco14,15, F. Gargano15, D. Gasparrini30, N. Gehrels20,38, S. Germani12,13, 13
+G. Giavitto39, B. Giebels16, N. Giglietto14,15, F. Giordano14,15, T. Glanzman4, G. Godfrey4, 14
+I. A. Grenier6, M.-H. Grondin34,35, J. E. Grove2, L. Guillemot34,35, S. Guiriec40, 15
+Y. Hanabata37, A. K. Harding20, E. Hays20, R. E. Hughes11, M. S. Jackson27,28,41, 16
+G. J´ ohannesson4, A. S. Johnson4, T. J. Johnson20,38, W. N. Johnson2, S. Johnston42, 17
+T. Kamae4, H. Katagiri37, J. Kataoka43,44, N. Kawai43,45, M. Kerr17, J. Kn¨ odlseder46, 18
+M. L. Kocian4, M. Kuss5, J. Lande4, L. Latronico5, M. Lemoine-Goumard34,35, F. Longo7,8, 19
+F. Loparco14,15, B. Lott34,35, M. N. Lovellette2, P. Lubrano12,13, A. Makeev2,23, 20
+M. Marelli19, M. N. Mazziotta15, J. E. McEnery20, C. Meurer27,28, P. F. Michelson4, 21
+W. Mitthumsiri4, T. Mizuno37, A. A. Moiseev21,38, C. Monte14,15, M. E. Monzani4, 22
+A. Morselli47, I. V. Moskalenko4, S. Murgia4, P. L. Nolan4, J. P. Norris48, E. Nuss25, 23
+T. Ohsugi37, N. Omodei5, E. Orlando49, J. F. Ormes48, D. Paneque4, D. Parent34,35, 24
+V. Pelassa25, M. Pepe12,13, M. Pesce-Rollins5, F. Piron25, T. A. Porter33, S. Rain` o14,15, 25
+R. Rando9,10, P. S. Ray2, M. Razzano5, A. Reimer50,4, O. Reimer50,4, T. Reposeur34,35, 26
+S. Ritz33, M. S. E. Roberts2,23,51,1, L. S. Rochester4, A. Y. Rodriguez52, R. W. Romani4, 27
+M. Roth17, F. Ryde41,28, H. F.-W. Sadrozinski33, D. Sanchez16, A. Sander11, 28
+P. M. Saz Parkinson33,1, J. D. Scargle53, C. Sgr` o5, E. J. Siskind54, D. A. Smith34,35, 29
+P. D. Smith11, G. Spandre5, P. Spinelli14,15, M. S. Strickman2, D. J. Suson55, H. Tajima4, 30
+H. Takahashi37, T. Tanaka4, J. B. Thayer4, J. G. Thayer4, G. Theureau24, 31
+D. J. Thompson20, L. Tibaldo9,6,10, O. Tibolla56, D. F. Torres57,52, G. Tosti12,13, 32
+A. Tramacere4,58, Y. Uchiyama59,4, T. L. Usher4, A. Van Etten4, V. Vasileiou20,21,22, 33
+C. Venter20,60, N. Vilchez46, V. Vitale47,61, A. P. Waite4, P. Wang4, K. Watters4, 34
+B. L. Winer11, M. T. Wolff2, K. S. Wood2,1, T. Ylinen41,62,28, M. Ziegler3335– 2 –
+1Corresponding authors: A. A. Abdo, aous.abdo@nrl.navy.mil; M. S. E . Roberts, malloryr@gmail.com;
+P. M. Saz Parkinson, pablo@scipp.ucsc.edu; K. S. Wood, Kent.Wood@ nrl.navy.mil.
+2Space Science Division, Naval Research Laboratory, Washington, DC 20375, USA
+3National Research Council Research Associate, National Academ y of Sciences, Washington, DC 20001,
+USA
+4W. W. Hansen Experimental Physics Laboratory, Kavli Institute f or Particle Astrophysics and Cosmol-
+ogy, Department of Physics and SLAC National Accelerator Labor atory, Stanford University, Stanford, CA
+94305, USA
+5Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, I-56127 Pisa, Italy
+6Laboratoire AIM, CEA-IRFU/CNRS/Universit´ e Paris Diderot, Ser vice d’Astrophysique, CEA Saclay,
+91191 Gif sur Yvette, France
+7Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34127 T rieste, Italy
+8Dipartimento di Fisica, Universit` a di Trieste, I-34127 Trieste, It aly
+9Istituto Nazionale di Fisica Nucleare, Sezione di Padova, I-35131 P adova, Italy
+10Dipartimento di Fisica “G. Galilei”, Universit` a di Padova, I-35131 Pa dova, Italy
+11Department of Physics, Center for Cosmology and Astro-Particle Physics, The Ohio State University,
+Columbus, OH 43210, USA
+12Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, I-06123 P erugia, Italy
+13Dipartimento di Fisica, Universit` a degli Studi di Perugia, I-06123 Perugia, Italy
+14Dipartimento di Fisica “M. Merlin” dell’Universit` a e del Politecnico di Ba ri, I-70126 Bari, Italy
+15Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70126 Bari, It aly
+16Laboratoire Leprince-Ringuet, ´Ecole polytechnique, CNRS/IN2P3, Palaiseau, France
+17Department of Physics, University of Washington, Seattle, WA 981 95-1560, USA
+18Columbia Astrophysics Laboratory, Columbia University, New York, NY 10027, USA
+19INAF-Istituto di Astrofisica Spaziale e Fisica Cosmica, I-20133 Milan o, Italy
+20NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
+21CenterforResearchand ExplorationinSpaceScienceandTechnolo gy(CRESST), NASAGoddardSpace
+Flight Center, Greenbelt, MD 20771, USA
+22University of Maryland, Baltimore County, Baltimore, MD 21250, USA
+23George Mason University, Fairfax, VA 22030, USA
+24Laboratoire de Physique et Chemie de l’Environnement, LPCE UMR 611 5 CNRS, F-45071 Orl´ eans
+Cedex 02, and Station de radioastronomie de Nan¸ cay, Observato ire de Paris, CNRS/INSU, F-18330 Nan¸ cay,– 3 –
+France
+25Laboratoire de Physique Th´ eorique et Astroparticules, Universit ´ e Montpellier 2, CNRS/IN2P3, Mont-
+pellier, France
+26Department of Physics and Astronomy, Sonoma State University, Rohnert Park, CA 94928-3609, USA
+27Department of Physics, Stockholm University, AlbaNova, SE-106 9 1 Stockholm, Sweden
+28The Oskar Klein Centre for Cosmoparticle Physics, AlbaNova, SE-10 6 91 Stockholm, Sweden
+29Royal Swedish Academy of Sciences Research Fellow, funded by a gr ant from the K. A. Wallenberg
+Foundation
+30Agenzia Spaziale Italiana (ASI) Science Data Center, I-00044 Fras cati (Roma), Italy
+31Dipartimento di Fisica, Universit` a di Udine and Istituto Nazionale di Fisica Nucleare, Sezione di Trieste,
+Gruppo Collegato di Udine, I-33100 Udine, Italy
+32Los Alamos National Laboratory, Los Alamos, NM 87545, USA
+33Santa Cruz Institute for Particle Physics, Department of Physics and Department of Astronomy and
+Astrophysics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA
+34Universit´ edeBordeaux,Centred’ ´EtudesNucl´ eairesBordeauxGradignan,UMR5797,Gradignan,3 3175,
+France
+35CNRS/IN2P3, Centre d’ ´Etudes Nucl´ eaires Bordeaux Gradignan, UMR 5797, Gradignan, 3 3175, France
+36Arecibo Observatory, Arecibo, Puerto Rico 00612, USA
+37Department of Physical Sciences, Hiroshima University, Higashi-Hir oshima, Hiroshima 739-8526, Japan
+38University of Maryland, College Park, MD 20742, USA
+39Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, and Univer sit` a di Trieste, I-34127 Trieste, Italy
+40University of Alabama in Huntsville, Huntsville, AL 35899, USA
+41Department of Physics, Royal Institute of Technology (KTH), Alb aNova, SE-106 91 Stockholm, Sweden
+42Australia Telescope National Facility, CSIRO, Epping NSW 1710, Aust ralia
+43Department of Physics, Tokyo Institute of Technology, Meguro C ity, Tokyo 152-8551, Japan
+44Waseda University, 1-104 Totsukamachi, Shinjuku-ku, Tokyo, 16 9-8050, Japan
+45Cosmic Radiation Laboratory, Institute of Physical and Chemical R esearch (RIKEN), Wako, Saitama
+351-0198, Japan
+46Centre d’ ´Etude Spatiale des Rayonnements, CNRS/UPS, BP 44346, F-30128 Toulouse Cedex 4, France
+47Istituto Nazionale di Fisica Nucleare, Sezione di Roma “Tor Vergata ”, I-00133 Roma, Italy
+48Department of Physics and Astronomy, University of Denver, Den ver, CO 80208, USA
+49Max-Planck Institut f¨ ur extraterrestrische Physik, 85748 Gar ching, Germany– 4 –
+ABSTRACT 36
+37 We present multiwavelength studies of the 106.6 ms γ-ray pulsar PSR 38
+J1907+06 near the TeV source MGRO J1908+06. Timing observation s with 39
+Fermiresult in a precise position determination for the pulsar of R.A. = 40
+19h07m54.s7(2), decl. = +06◦02′16(2)′′placing the pulsar firmly within the TeV 41
+source extent, suggesting the TeV source is the pulsar wind nebula of PSR 42
+J1907+0602. Pulsed γ-ray emission is clearly visible at energies from 100 MeV to 43
+above 10 GeV. The phase-averaged power-law index in the energy r angeE >0.1 44
+GeV is Γ = 1 .76±0.05 with an exponential cutoff energy Ec= 3.6±0.5 GeV. 45
+We present the energy-dependent γ-ray pulsed light curve as well as limits on off- 46
+pulse emission associated with the TeV source. We also report the de tection of 47
+very faint (flux density of ≃3.4µJy) radio pulsations with the Arecibo telescope 48
+at 1.5 GHz having a dispersion measure DM = 82.1 ±1.1 cm−3pc. This indi- 49
+cates a distance of 3 .2±0.6 kpc and a pseudo-luminosity of L1400≃0.035 mJy 50
+kpc2. AChandra ACIS observation revealed an absorbed, possibly extended, 51
+compact (<∼4′′) X-ray source with significant non-thermal emission at R.A. = 52
+19h07m54.s76, decl. = +06◦02′14.6′′with a flux of 2 .3+0.6
+−1.4×10−14ergcm−2s−1. 53
+From archival ASCAobservations, we place upper limits on any arcminute scale 54
+2–10 keV X-ray emission of ∼1×10−13ergcm−2s−1. The implied distance to the 55
+pulsar is compatible with that of the supernova remnant G40.5 −0.5, located on 56
+the far side of the TeV nebula from PSR J1907+0602, and the S74 mo lecular 57
+cloud on the nearer side which we discuss as potential birth sites. 58
+Subject headings: pulsars: individual: PSR J1907+0602 — gamma rays: obser- 59
+vations 60
+1. Introduction 61
+The TeV source MGRO J1908+06 was discovered by the Milagro collabo ration at a me- 62
+dian energy of 20 TeV in their survey of the northern Galactic Plane ( Abdo et al. 2007) with 63
+a flux∼80% of the Crab at these energies. It was subsequently detected in the 300 GeV – 64
+20 TeVrangeby the HESS (Aharonian et al. 2009) andVERITAS (War d2008) experiments. 65
+The HESS observations show the source HESS J1908+063 to be clea rly extended, spanning 66
+∼0.3◦of a degree on the sky with hints of energy-dependent substruct ure. A decade earlier 67
+Lamb & Macomb (1997) cataloged a bright source of GeV emission fro m the EGRET data, 68
+GeV J1907+0557, which is positionally consistent with MGRO J1908+06 . It is near, but 69
+50Institut f¨ ur Astro- und Teilchenphysik and Institut f¨ ur Theore tische Physik, Leopold-Franzens-
+Universit¨ at Innsbruck, A-6020 Innsbruck, Austria
+51Eureka Scientific, Oakland, CA 94602, USA
+52Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB, 08193 Ba rcelona, Spain
+53Space Sciences Division, NASA Ames Research Center, Moffett Field, CA 94035-1000, USA
+54NYCB Real-Time Computing Inc., Lattingtown, NY 11560-1025, USA
+55Department of Chemistry and Physics, Purdue University Calumet, Hammond, IN 46323-2094, USA
+56Max-Planck-Institut f¨ ur Kernphysik, D-69029 Heidelberg, Germ any
+57Instituci´ o Catalana de Recerca i Estudis Avan¸ cats (ICREA), Ba rcelona, Spain
+58Consorzio Interuniversitario per la Fisica Spaziale (CIFS), I-10133 Torino, Italy
+59Institute ofSpace and AstronauticalScience, JAXA, 3-1-1Yosh inodai, Sagamihara,Kanagawa229-8510,
+Japan
+60North-West University, Potchefstroom Campus, Potchefstroo m 2520, South Africa
+61Dipartimento di Fisica, Universit` a di Roma “Tor Vergata”, I-0013 3 Roma, Italy
+62School of Pure and Applied Natural Sciences, University of Kalmar, SE-391 82 Kalmar, Sweden
+63Max-Planck-Institut f¨ ur Radioastronomie, Auf dem H¨ ugel 69, D -53121 Bonn, Germany– 5 –
+inconsistent with, the third EGRET catalog (Hartman et al. 1999) so urce 3EG J1903+0550 70
+(Roberts et al. 2001). The Large Area Telescope (LAT) (Atwood e t al. 2009) aboard the 71
+Fermi Gamma-Ray Space Telescope has been operating in survey mode since soon after its 72
+launch on 2008 June 11, carrying out continuous observations of t he GeV sky. The Fermi 73
+Bright Source List (Abdo et al. 2009b), based on 3 months of surve y data, contains 0FGL 74
+J1907.5+0602which is coincident with GeVJ1907+0557. The 3EG J190 3+0550source loca- 75
+tion confidence contour stretches between 0FGL J1907.5+0602 a nd the nearby source 0FGL 76
+J1900.0+0356, suggesting it was a conflation of the two sources.77
+TheFermiLAT collaboration recently reported the discovery of 16 previously -unknown 78
+pulsars by using a time differencing technique on the LAT photon data above 300 MeV 79
+(Abdo et al. 2009a). 0FGL J1907.5+0602 was found to pulse with a pe riod of 106.6 ms, 80
+have a spin-down energy of ∼2.8×1036erg s−1, and was given a preliminary designation 81
+of PSR J1907+06. In this paper we derive a coherent timing solution u sing 14 months 82
+of data which yields a more precise position for the source, allowing de tailed follow-up at 83
+other wavelengths, including the detection of radio pulsations using the Arecibo 305-m radio 84
+telescope. Energy resolved light curves, the pulsed spectrum, an d off-pulse emission limits at 85
+the positions of the pulsar and PWN centroid are presented. We the n report the detection of 86
+an X-ray counterpart with the Chandra X-ray Observatory and an upper limit from ASCA. 87
+Finally, we discuss the pulsar’s relationship to the TeV source and to t he potential birth 88
+sites SNR G40.5 −0.5 and the S74 H iiregion. 89
+2. Gamma-ray Pulsar Timing and Localization 90
+The discovery and initial pulse timing of PSR J1907+06 was reported b y Abdo et al. 91
+(2009a). The source position used in that analysis (R.A. = 286.965◦, Decl. = 6.022◦) was 92
+derived from an analysis of the measured directions of LAT-detect ed photons in the on-pulse 93
+phase interval from observations made from 2008 August 4 throu gh December 25. Here, we 94
+make use of a longer span of data and also apply improved analysis met hods to derive an 95
+improved timing ephemeris for the pulsar as well as a more accurate s ource position. 96
+For the timing and localization analysis, we selected “diffuse” class pho tons (events 97
+that passed the tightest background rejection criteria (Atwood et al. 2009)) with zenith 98
+angle<105◦as is standard practice and chose the minimum energy and extractio n radius 99
+to optimize the significance of pulsations. We accepted photons with E >200 MeV from 100
+within a radius of 0.7◦of the nominal source direction. We corrected these photon arriv al 101– 6 –
+times to terrestrial time (TT) at the geocenter using the LAT Scien ce Tool1gtbaryin its 102
+geocenter mode.103
+We fitted a timing model using Tempo2 (Hobbs et al. 2006) to 23 pulse times of arrival 104
+(TOAs) covering the interval 2008 June 30 to 2009 September 18. We note that during 105
+the on-orbit checkout period (before 2008 August 4) several ins trument configurations were 106
+tested that affected the energy resolution and event reconstru ction but had no effect on the 107
+LATtiming. TodeterminetheTOAs, wegeneratedpulseprofilesbyfo ldingthephotontimes 108
+according to a provisional ephemeris using polynomial coefficients ge nerated by Tempo2 in 109
+its predictive mode (assuming a fictitious observatory at the geoce nter). The TOAs were 110
+measured by cross correlating each pulse profile with a kernel dens ity template that was 111
+derived from fitting the full mission dataset (Ray et al. 2009). Finally , we fitted the TOAs 112
+to a timing model that included position, frequency, and frequency derivative. The resulting 113
+timing residuals are 0 .4msand are shown in Figure 1. The best-fit model is displayed in 114
+Table 1. The numbers in parentheses are the errors in the last digit o f the fitted parameters. 115
+The errors are statistical only, except for the position error, as described below. The derived 116
+parameters of ˙E,B, andτcare essentially unchanged with respect to those reported by 117
+Abdo et al. (2009a), but the position has moved by 1.2′. 118
+The statistical error on the position fit is <1′′; however, this is an underestimate of the 119
+true error. For example, with only one year of data, timing noise can perturb the position 120
+fit. We have performed a Monte Carlo analysis of these effects by sim ulating fake residuals 121
+using the fakeplugin for Tempo2 . We generated models with a range of frequency second 122
+derivatives ( ±2×10−22s−3, the allowed magnitude for ¨ νin our fits) to simulate the effects 123
+of timing noise and fitted them to timing models. Based on these simulat ions, we assigned 124
+an additional systematic error on the position of 2′′, which we added in quadrature to the 125
+statistical error in Table 1. As a result of the improved position estim ate provided by this 126
+timing analysis, we have adopted a more precise name for the pulsar o f PSR J1907+0602. 127
+3. Detection of Radio Pulsations 128
+To search for radio pulsations, we observed the timing position of PS R J1907+0602 129
+with the L-wide receiver on the Arecibo 305-m radio telescope. On 20 09 August 21 we made 130
+a 55-minute pointing with center frequency 1.51 GHz and total band width of 300 MHz, 131
+provided by three Wideband Arecibo Pulsar Processors (WAPPs, Do wd et al. (2000)), each 132
+individually capable of processing 100 MHz. We divided this band into 512 -channel spectra 133
+1http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/ index.html– 7 –
+accumulatedevery128 µs. ThesmallpositionaluncertaintyofPSRJ1907+0602derivedfro m 134
+the LAT timing means that a single Arecibo pointing covers the whole re gion of interest. 135
+After excising strong sources of radio-frequency interference withrfifind, one of the 136
+routinesofthePRESTOsignalanalysispackage(Ransom et al.2002 ), weperformedasearch 137
+by folding the raw data with the Fermitiming model into 128-bin pulse profiles. We then 138
+used the PRESTO routine prepfold to search trial dispersion measures between 0 and 1000 139
+pc cm−3. We found a pulsed signal with a signal-to-noise ratio S/N= 9.42and duty cycle 140
+of about 0.03 at a dispersion measure DM= 82.1±1.1 cm−3pc. This value was estimated 141
+by dividing the detection data into 3 sub-bands and making TOAs for e ach sub-band and 142
+fitting for the DM with tempo. 143
+We applied the same technique for 4 different time segments of 12.5 min utes each and 144
+created a time of arrival for each of them. We then estimated the b arycentric periodicity of 145
+the detected signal from these times of arrival. This differs from th e periodicity predicted 146
+by the LAT ephemeris for the time of the observation by ( −0.000005±0.000020) ms, i.e., 147
+the signals have the same periodicity.148
+Subsequent radio observations showed that the phase of the rad io pulses is exactly as 149
+predicted by the LAT ephemeris, apart from a constant phase offs et (depicted in Figure 2) 150
+A confirmation observation with twice the integration time (1.8 hr) wa s made on 2009 151
+September 4. The radio profile is shown in the bottom panel of Figure . 2 with an arbitrary 152
+intensity scale. The pulsar is again detected with S/N of 3.4, 5.1, 7.3 an d 8.6 at 1170, 1410, 153
+1510 and 1610 MHz. The higher S/N at the higher frequencies sugge st a positive spectral 154
+index, similartowhathasbeenobservedforPSRJ1928+1746(Cord es et al.2006). However, 155
+this might instead be due to scattering degrading the S/N at the lowe st frequencies— for 156
+the band centered at 1610 MHz the pulse profile is distinctively narro wer (about 2% at 50% 157
+power) than at 1410 or 1510 MHz (about 3%). At 1170 MHz the profi le is barely detectable 158
+but very broad. This suggests an anomalously large scattering time scale for the DM of the 159
+pulsar. Observations at higher frequencies will settle the issue of t he positive spectral index. 160
+For the 300 MHz centered at 1410 MHz, where the detection is clear , we obtain a total S/N 161
+of 12.4.162
+Withanantenna Tsys= 33K(givenbythefrequency-dependent antenna temperature of 163
+2This wasestimatedusinganothersoftwarepackage,SIGPROC(ap ackagedevelopedbyDuncanLorimer,
+see http://sigproc.sourceforge.net/), which processes the ban ds separately and produced S/N of 4.2, 6.3 and
+5.5 for the WAPPs centered at 1410, 1510 and 1610 MHz. Although S /N = 9.4 is close to the detection
+threshold for pulsars in a blind search, it is much more significant in this case because of the reduced number
+of trials in this search relative to a blind search.– 8 –
+25–27 K off the plane of the Galaxy plus 6 K of Galactic emission in the spe cific direction of 164
+the pulsar Haslam et al. 1982), Gain = 10.5 K Jy−1and 2 polarizations, and an inefficiency 165
+factor of 12% due to the 3-level sampling of the WAPP correlators, we obtain for the first 166
+detection a flux density at 1.4 GHz of S1400≃4.1µJy and for the second detection S1400≃ 167
+3.1µJy. Thesevaluesareconsistent giventhelargerelativeuncertaint iesintheS/Nestimates 168
+and the varying effect of radio frequency interference; at this DM , scintillation is not likely 169
+to cause a large variation in the flux density.170
+The time-averaged flux density is ≃3.4µJy. Using the NE2001 model for the electron 171
+distribution in the Galaxy (Cordes & Lazio 2002), we obtain from the p ulsar’s position and 172
+DM a distance of 3.2 kpc with a nominal error of 20%(Cordes & Lazio 20 02). The time- 173
+averaged flux density thus corresponds to a pseudo-luminosity L1400≃0.035 mJy kpc2. This 174
+is fainter than the least luminous young pulsar in the ATNF catalog (PS R J0205+6449, with 175
+a 1.4 GHz pseudo-luminosity of 0.5 mJy kpc2). It is, however, more luminous than the radio 176
+pulsations discovered through a deep search of another pulsar fir st discovered by Fermi, PSR 177
+J1741−2054 which has L1400∼0.025 mJy kpc2(Camilo et al. 2009). These two detections 178
+clearly demonstrate that some pulsars, as seen from the Earth, c an have extremely low 179
+apparent radio luminosities; i.e., similarly deep observations of other γ-ray selected pulsars 180
+might detect additional very faint radio pulsars. We note that thes e low luminosities, which 181
+may well be the result of only a faint section of the radio beam crossin g the Earth, are much 182
+lower than what has often been termed “radio quiet” in population sy nthesis models used to 183
+estimate the ratio of “radio-loud” to “radio quiet” γ-ray pulsars (eg. Gonthier et al. 2004). 184
+4. Energy-Dependent Gamma-ray Pulse Profiles 185
+Thepulse profileandspectral results reportedinthispaper useth esurvey datacollected 186
+with the LAT from 2008 August 4 through 2009 September 18. We se lected “diffuse” class 187
+photons (see §2) with energies E >100 MeV and, to limit contamination from photons from 188
+Earth’s limb, with zenith angle <105◦. 189
+Toexplorethedependenceofthepulseprofileonenergy, weselect edanenergy-dependent 190
+region of interest (ROI) with radius θ= 0.8×E−0.75degrees, but constrained not to be out- 191
+side the range [0 .35◦,1.5◦]. We chose the upper bound to minimize the contribution from 192
+nearby sources and Galactic diffuse emission. The lower bound was se lected in order to in- 193
+clude more photons from the wings of the point spread function (PS F) where the extraction 194
+region is small enough to make the diffuse contribution negligible. Figur e 2 shows folded 195
+light curves of the pulsar in 32 constant-width bins for different ene rgy bands. We use the 196
+centroid of the 1.4 GHz radio pulse profile to define phase 0.0. Two rot ations are shown in 197– 9 –
+each case. The top panel of the figure shows the folded light curve for photons with E >0.1 198
+GeV. The γ-ray light curve shows two peaks, P1 at phase 0.220 ±0.002 which determines 199
+the offset with the radio peak, δ. The second peak in the γ-ray, P2, occurs at phase 0.580 200
+±0.003. The phase separation between the two peaks is ∆ = 0 .360±0.004. The radio 201
+leadδand gamma peak separation ∆ values are in good agreement with the c orrelation 202
+predicted for outer magnetosphere models, (Romani & Yadigarog lu 1995) and observed for 203
+other young pulsars (Figure 3 of Abdo et al. (2009c)).204
+Pulsed emission from the pulsar is clearly visible for energies E >5 GeV with a chance 205
+probability of ∼4×10−8. Pulsed emission is detected for energies above 10 GeV with a 206
+confidence level of 99.8%. We have measured the integral and width s of the peaks as a 207
+function of energy and have found no evidence for significant evolu tion in shape or P1/P2 208
+ratio with energy. We note that the pulsar is at low Galactic latitude ( b∼ −0.89◦) where 209
+the Galactic γ-ray diffuse emission is bright ( it has not been subtracted fromthe lig ht curves 210
+shown.)211
+Figure 3 shows the observed LAT counts map of the region around P SR J1907+0602. 212
+We defined the “on” pulse as pulse phases 0.12 ≤φ≤0.68 and the “off” pulse as its 213
+complement (0.0 ≤φ <0.12 and 0.68 < φ≤1.0). We produced on-pulse (left panel) and 214
+off-pulse (right panel) images, scaling the off-pulse image by 1.27. The figure indicates the 215
+complexity of the region that must be treated in spectral fitting. B esides the pulsar there 216
+are multiple point sources, Galactic, and extragalactic diffuse contr ibutions. 217
+5. Energy Spectrum 218
+Thephase-averagedfluxofthepulsarwasobtainedbyperforming amaximumlikelihood 219
+spectral analysis using the FermiLAT science tool gtlike. Starting from the same data set 220
+described in §4, we selected photons from an ROI of 10 degrees around the pulsa r position. 221
+Sources from a preliminary version (based on 11 months of data) of the first FermiLAT 222
+γ−ray catalog (Abdo et al. 2009c) that are within 15 degree ROI aroun d the pulsar were 223
+modeled in this analysis. Spectra of sources farther away than 5◦from the pulsar were fixed 224
+at the cataloged values. Sources within 5◦degrees of the pulsar were modeled with a simple 225
+power law. For each of the sources in the 5◦degree region around the pulsar, we fixed the 226
+spectral index at the value in the catalog and fitted for the normaliz ation. Two sources 227
+that are at a distance >5◦showed strong emission and were treated the same way as the 228
+sources within 5◦. The Galactic diffuse emission (gll iemv02) and the extragalactic diffuse 229– 10 –
+background (isotropic iemv02) were modeled as well3. 230
+Theassumedspectralmodelforthepulsarisanexponentiallycut- offpowerlaw: dN/dE=
+No(E/Eo)−Γexp(−E/Ec).The resulting spectrum gives the total emission for the pulsar
+assuming that the γ-ray emission is 100% pulsed. The unbinned gtlikefit, using P6 v3
+instrument response functions (Atwood et al. 2009), for the ene rgy range E≥100 MeV
+gives a phase-averaged spectrum of the following form:
+dN
+dE= (7.06±0.43stat.+(+0.004
+−0.064)sys.)×10−11E−Γe−E/Ecγcm−2s−1MeV−1(1)
+where the photon index Γ = 1 .76±0.05stat.+ (+0.271
+−0.287)sys.and the cutoff energy Ec= 3.6± 231
+0.5stat.+(+0.72
+−0.36)sys.GeV. 232
+The integrated energy flux from the pulsar in the energy range E≥100 MeV is Fγ= 233
+(3.12±0.15stat.+ (+0.16
+−0.15)sys.)×10−10ergs cm−2s−1. This yields a γ-ray luminosity of Lγ= 234
+4πfΩFγd2= 3.8×1035fΩd2
+3.2ergs s−1above 100 MeV, where fΩis an effective beaming factor 235
+andd3.2=d/(3.2)kpc. This corresponds to an efficiency of η=Lγ/˙E= 0.13Fγd2
+3.2for 236
+conversion of spin-down power into γ−ray emission in this energy band. 237
+We set a 2 σflux upper limit on γ-ray emission from the pulsar in the off-pulse part 238
+ofFoff<8.31×10−8cm−2s−1. In addition to the γ-ray spectrum from the point-source 239
+pulsar PSR J1907+0602, we measured upper limits on γ-ray flux from the extended source 240
+HESS J1908+063 in the energy range 0.1–25 GeV. We performed binn ed likelihood analysis 241
+using the FermiScience Tool gtlike. In this analysis we assumed an extended source 242
+with gaussian width of 0.3◦andγ-ray spectral index of −2.1 at the location of the HESS 243
+source. The upper limits suggest that the spectrum of HESS J1908 +063 has a low energy 244
+turnover between 20 GeV and 300 GeV. Figure 4 shows the phase-a veraged spectral energy 245
+distribution for PSR J1907+0602 (green circles). On the same figur e we show data points 246
+from HESS for the TeV source HESS J1908+063 (blue circles) and th e 2σupper limits from 247
+Fermifor emission from this TeV source. Figure 5 shows an off pulse residua l map of the 248
+region around PSR J1907+0602. The timing position of the pulsar is ma rked by the green 249
+cross. The 5 σcontours from Milagro (outer) and HESS (inner) are overlaid. As ca n be seen 250
+from the residual map, there is no gamma-ray excess at the locatio n of either the pulsar or 251
+the PWN.252
+3Descriptions of the models are available at http://fermi.gsfc.nasa.go v/– 11 –
+6. X-ray Counterpart 253
+A 23 ks ASCAGIS exposure of the EGRET source GeV J1907+0557 revealed an ∼ 254
+8′×15′regionofpossibleextended hardemissionsurroundingtwopoint-like peakslying ∼15′255
+to the southwest of PSR J1907+0602 (Roberts et al. 2001) and no other significant sources 256
+in the 44′ASCAFOV. A 10 ks Chandra ACIS-I image of the ASCAemission (ObsID 7049) 257
+showed it to be dominated by a single hard point source, CXOU J19071 8.6+054858 with no 258
+compact nebular structure and just a hint of the several arcminu te-scale emission seen by 259
+ASCA.CXOUJ190718.6+054858seemed toturnofffor ∼2ksduringthe Chandra exposure, 260
+suggesting that it may be a binary of some sort or else a variable extr agalactic source. There 261
+is no obvious optical counterpart in the digital sky survey optical o r 2MASS near infrared 262
+images, nor in a Iband image taken with the 2.4m Hiltner telescope at MD M (Jules Halpern, 263
+private communication). This strongly suggests that it is not a near by source. An absorbed 264
+power law fits the spectrum of this source well, with absorption nH= 1.8+1.3
+−0.9×1022cm−2265
+(90% confidence region), a photon spectral index Γ = 0 .9+0.6
+−0.4, and an average 2–10 keV flux 266
+of 4.4+0.7
+−1.8×10−13ergcm−2s−1(68% confidence region). The fit absorption is similar to the 267
+estimated total Galactic absorption from the HEASARC nHtool of 1.6×1022cm−2based 268
+on the Dickey and Lockman (1990) HI survey (Dickey & Lockman 199 0), suggesting that 269
+annHof∼2×1022cm−2is a reasonably conservative estimate of interstellar absorption fo r 270
+sources deep in the plane along this line of sight.271
+The timing position of LAT PSR J1907+0602 is in the central 20′of theASCAGIS 272
+FOV (Figure 6). There is no obvious emission in the ASCAimage at the pulsar position. 273
+Using the methodology of Roberts et al. (2001), a 24 pixel radius ex traction region ( ∼6′), 274
+and assuming an absorbed power law spectrum with nH= 2×1022cm−2and Γ = 1 .5, we 275
+place a 90% confidence upper limit on the 2–10 keV flux Fx<5×10−14ergcm−2s−1. This 276
+suggests that for any reasonable absorption, the total unabso rbed X-ray flux from the pulsar 277
+plus any arcminute-scale nebula is less than 10−13ergcm−2s−1. 278
+PSR J1907+0602 was well outside of the FOV of the first Chandra observation, and 279
+so we proposed for an observation centered on the pulsar. We obt ained a 19 ks exposure 280
+with the ACIS-S detector (ObsID 11124). The time resolution of th e ACIS-S detector on 281
+boardChandra does not allow for pulse studies. The only source within an arcminute 282
+of the timing position and the brightest source in the FOV of the S3 ch ip is shown in 283
+Figure 7. It is well within errors of the timing position. Examination of t he X-ray image 284
+in different energy bands showed virtually no detected flux below ∼1keV and significant 285
+flux above 2.5 keV, suggesting a non-thermal emission mechanism fo r much of the flux. 286
+A comparison of the spatial distribution of counts between 0.75 keV and 2 keV to those 287
+between 2keV and 8keV shows some evidence for spatial extent be yond the point spread 288– 12 –
+function for the harder emission but not for the softer emission. T his would be consistent 289
+with an interpretation as predominantly absorbed but thermal emis sion from a neutron star 290
+surface surrounded by non-thermal emission from a compact puls ar wind nebula, which is 291
+the typical situation for young pulsars (see Kaspi et al. 2006, and references therein). We 292
+plot the Chandra 0.75-2keV, 2-8keV, and 0.75-8 keV images with an ellipse showing the 293
+timing position uncertainty, and a circle with a radius of 0 .8′′. From a modeled PSF, we 294
+estimate 80% of the counts should be contained within this circle. While this seems to be 295
+the case for the soft image, only roughly half the counts in the hard er image are contained 296
+within that radius. With only ∼12 source counts in the 0.75-2 keV image within 6′′and 297
+∼30 source counts in the 2-8 keV image, quantitative statements ab out source size and 298
+spectrum are difficult to make. We obtain a best fit position for the no minal point source of 299
+R.A. = 19h07m54.s76, decl. = +06◦02′14.6′′and estimated error of 0.7′′(an additional 0.1′′300
+centroid fitting uncertainty added to the nominal Chandra 0.6′′uncertainty). Using a 6′′301
+radius extraction region and an annulus between 6′′and 24′′for background, we extracted 302
+source and background spectra and fit them within XSPEC (Figure 8 ). A simple power law 303
+plus absorption model fit the data well in the energy range 2-10 KeV , with best fit values 304
+nH= 1.3×1022cm−2and Γ = 1 .6, with a total flux of 2 .3+0.6
+−1.4×10−14ergcm−2s−1. The 305
+low count rates and covariance between the absorption and photo n index meant the spectral 306
+parameters could not be simultaneously meaningfully constrained. F ixing the spectral index 307
+Γ = 1.6, a typical value for compact pulsar wind nebulae (Kaspi et al. 2006 ), we obtain a 308
+90% confidence region for the absorption of 0 .7−2.5×1022cm−2, consistent with a source a 309
+few kilo parsecs or more away and with CXOU J190718.6+054858 discu ssed above. We note 310
+that with such an absorption a significant thermal component in the below 2 keV emission 311
+is neither required nor ruled out by the spectral fitting.312
+7. Discussion 313
+The dispersion measure from the radio detection suggests a distan ce of 3.2 kpc, with 314
+a nominal error of 20%. However, there are many outliers to the DM error distribution, 315
+although the largest fractional errors tend to be from pulsars at high Galactic latitudes or 316
+very lowDMs(Deller et al.2009; Chatterjee et al.2009). ForPSRJ1 907+0602,atalatitude 317
+b=−0.9◦with a moderate DM, the distance estimate is likely to be reasonable. S ince the 318
+apparent γ-ray pulsed efficiency in the Fermipass-band is well above the median for other 319
+gamma-ray pulsars in Abdo et al. (2009d) (13% compared to 7.5%), it is worth checking 320
+secondary distance indicators to see if the DM measure could be a sig nificant overestimate 321
+of the true distance. We can use the X-ray observations of PSR J1 907+0602 to do this. 322
+Several authors have noted a correlation between the X-ray lumin osity of young pulsars and 323– 13 –
+their spin-down power (eg. Saito 1998, Possenti et al. 2002, Li, Lu and Li 2007). Most 324
+of these have the problem of using X-ray fluxes derived from the lite rature using a variety 325
+of instruments with no uniform way of choosing spectral extractio n regions. This can be 326
+especially problematic with Chandra data, since faint, arcminute scale emission can easily be 327
+overlooked. We compare our ASCAGIS upper limits to Figure 1 of Cheng, Taam and Wang 328
+(2004) who used only ASCAGIS data to derive their X-ray luminosity relationships. We see 329
+that the typical X-ray luminosity in the ASCAband for a pulsar with ˙E= 2×1036ergs−1330
+isLx∼1033−1034ergs−1with all of the pulsars used in their analysis with ˙E >1036ergs−1331
+havingLx>1032ergs−1. From these values and the ASCAupper limit, we derive a lower 332
+limit for the distance to LAT PSR J1907+0602 of ∼3 kpc. From Figure 2 of Li, Lu and 333
+Li (2007), who used XMM-Newton andChandra derived values, we see we can expect the 334
+luminosity to be between ∼1031.5−1034.5ergs−1. From our detection with Chandra, we 335
+again estimate a lower distance limit of ∼3 kpc. The “best guess” estimate from their 336
+relationship would result in a distance of ∼13kpc. We note that if we assume the pulsed 337
+emission to be apparently isotropic (i.e. fΩ= 1 as simple outer gap models suggest should 338
+approximately be the case, see Watters et al. (2009)), a distance of 9 kpc would result in 339
+100%γ-ray efficiency. 340
+ThederivedtimingpositionofPSRJ1907+0602iswellinsidetheextend edHESSsource, 341
+although ∼14′southwest of the centroid. The TeV source is therefore plausibly t he wind 342
+nebula of PSR J1907+0602. The physical size of this nebula is then>∼40 pc, and the 343
+integrated luminosity above 1 TeV is>∼40% that of the Crab, and in the MILAGRO band 344
+(∼20 TeV) at least twice that of the Crab. There is a hint of some spatia l dependence of 345
+the TeV spectrum in the HESS data, with the harder emission ( >2.5 TeV) peaking nearer 346
+the pulsar than the softer emission (Aharonian et al. 2009). If con firmed, this would be 347
+consistent withthehardeningoftheTeVemission observed toward sPSRB1823 −13, thought 348
+to be the pulsar powering HESS J1825 −137 (Aharonian et al. 2006). This latter pulsar has 349
+a spin period, characteristic age, and spin-down energy similar to PS R J1907+0602, and 350
+is also located near the edge of its corresponding TeV nebula. We also note that HESS 351
+J1825−137 subtends ∼1◦on the sky and has a flux level above 1 TeV of around 20% of the 352
+Crab. While the overall spectrum of HESS J1825 −137 is somewhat softer than the spectrum 353
+of HESS J1908+063, near the pulsar its spectrum is similarly hard. At a distance of ∼4 kpc, 354
+HESS J1825 −137 has a luminosity similar to the Crab TeV nebula, but with a much large r 355
+physical size of ∼70pc. Given the distance implied above and a flux above 1 TeV ∼17% of 356
+the Crab, HESS J1908+063 is similar in size and luminosity to HESS J1825 −137. 357
+At 20 TeV, HESS J1908+063 has a flux ∼80% of the Crab, and so at a distance>∼1.5 358
+times that of the Crab, is much more luminous at the highest energies . This is because there 359
+is no sign of a high-energy cutoff or break, as is seen in many other Te V nebulae. Aharonian 360– 14 –
+et al. (2009) place a lower limit of 19.1 TeV on any exponential cutoff to the spectrum. 361
+This implies that either the spectrum is uncooled due to a very low nebu lar magnetic field 362
+(<∼3µG, see, eg. de Jager (2008)), an age much less than the characte ristic age of 19.5 kyr, 363
+or else there is a synchrotron cooling break below the HESS band.364
+Our upper limits above a few GeV (Figure 4) requires there to be a low e nergy turnover 365
+between 20 GeV and 300 GeV. Given the nominal PWN spectrum, we co nstrain the overall 366
+PWN flux to be ≤25% of that of the pulsar. If only the HESS band is considered, and 367
+assuming the DM distance, the TeV luminosity LPWN= 5−8%˙E. However, since the TeV 368
+emission is generally thought to come from a relic population of electro ns the luminosity is 369
+likely a function of the spin-down history of the pulsar rather than t he current spin-down 370
+luminosity (eg. de Jager 2008). These numbers support consisten cy of the association of the 371
+TeV source with the pulsar, in the weak sense of not being discrepan t with other similar 372
+systems.373
+7.1. On the possible association with SNR G40.5 −0.5 374
+The bulk of HESS J1908+063 is between PSR J1907+0602 and the you ng radio SNR 375
+G40.5−0.5, suggesting a possible association. The distance estimate ( ∼3.4 kpc Yang et al. 376
+2006) and age (Downes et al. 1980) estimates of SNR G40.5 −0.5 are also consistent with 377
+those of PSR J1907+0602. If we use the usually assumed location fo r SNR G40.5-0.5 given 378
+by Langston et al. (2000) (RA=19h07m11.s9, Dec=6◦35′15′′), we get an angular separation of 379
+∼35′between the timing position for the pulsar and the SNR. However, th is position for the 380
+SNR is from single dish observations that were offset towards one br ight side of the nominal 381
+shell. WeusetheVLAGalacticPlaneSurvey 1420MHzimage(Stil et al.2 006)ofthisregion 382
+to estimate the SNR center to be RA=19h07m08.s6, Dec=6◦29′53′′(Figure 9) which, for an 383
+assumed distance of 3.2 kpc, would give a separation of ∼28 pc. Given the characteristic age 384
+of 19.5 kyr years, this would require an average transverse velocit y of∼1400 km/s. While 385
+velocities aboutthishighareseeninsomecases(eg. PSRB1508+55h asatransverse velocity 386
+of∼1100 km/s, Chatterjee et al. 2005 ), it is several times the averag e pulsar velocity and 387
+many times higher than the local sound speed. We note that pulsars with a braking index 388
+significantly less than n= 3 assumed in the derivation of the characteristic age could have 389
+ages as much as a factor of two greater (see eg. Kaspi et al. 2001 ), and thus a space velocity 390
+around half the above value may be all that is required. But with any r easonable assumption 391
+of birthplace, distance, and age, if the pulsar was born in SNR G40.5 −0.5, any associated 392
+X-ray or radio PWN should show a bow-shock and trail morphology, w ith the trail likely 393
+pointing back towards the SNR center. Unfortunately, the compa ctness and low number of 394– 15 –
+counts in our Chandra image precludes any definite statement about the PWN morphology. 395
+One arrives at a different, and lower, minimum velocity if one assumes t he pulsar was born 396
+at the center of the TeV PWN and moved to its present position, but the resulting velocity 397
+would still require a bow shock.398
+One can also get a pulsar offset towards the edge of a relic PWN if ther e is a significant 399
+density gradient in the surrounding ISM. A gradient will cause the su pernova blast wave to 400
+propagate asymmetrically. Where the density is higher, the revers e shock propagating back 401
+to the explosion center will also be asymmetric. This will tend to push t he PWN away from 402
+the region of higher density (Blondin et al. 2001; Ferreira & de Jager 2008). This has been 403
+invoked to explain the offsets in the Vela X and HESS J1825 −137 nebulae as well as several 404
+others. Infrared and radio imaging of the region shows that HESS J 1908+063 borders on 405
+a shell of material surrounding the S74 HII region, also known as th e Lynds Bright Nebula 406
+352. Russeil (2003) gives a kinematic distance of 3 .0±0.3 kpc for this star forming region, 407
+compatible with the pulsar distance. In this scenario, the pulsar wou ld not have to be highly 408
+supersonic to be at the edge of a relic nebula, and would not have to b e traveling away from 409
+the center of the TeV emission.410
+A third, hybrid possibility is that SNR G40.5 −0.5 is only a bright segment of a much 411
+larger remnant, whose emission from the side near the pulsar is conf used with that from the 412
+molecular cloud. The asymmetry would be explained by the difference in propagation speed 413
+in the lower density ISM away from the molecular cloud.414
+Our current Chandra data are insufficient to distinguish between the above scenarios. 415
+However, there is also the possibility of a compact cometary radio ne bula, such as is seen 416
+around PSR B1853+01 in SNR W44 (Frail et al. 1996) and PSR B0906 −49 (Gaensler et al. 417
+1998). In addition, sensitive long wavelength radio imaging could reve al any larger, faint 418
+SNR shells. Imaging with the EVLA and LOFAR of this region is therefor e highly desirable. 419
+The connection between the pulsar and the TeV nebula could be furt her strengthened by a 420
+confirmation of the spatio-spectral dependence of the nebula wh ere the spectrum hardens 421
+nearer to the pulsar.422
+TheFermiLAT Collaboration acknowledges the generous support of a number of agen- 423
+cies and institutes that have supported the FermiLAT Collaboration. These include the 424
+NationalAeronauticsandSpaceAdministrationandtheDepartmen t ofEnergyintheUnited 425
+States, the Commissariat ` a l’Energie Atomique and the Centre Natio nal de la Recherche Sci- 426
+entifique / Institut National de Physique Nucl´ eaire et de Physique des Particules in France, 427
+the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucle are in Italy, the 428
+Ministry of Education, Culture, Sports, Science and Technology (M EXT), High Energy Ac- 429– 16 –
+celerator Research Organization (KEK) and Japan Aerospace Exp loration Agency (JAXA) 430
+in Japan, and the K. A. Wallenberg Foundation and the Swedish Nation al Space Board in 431
+Sweden. The Arecibo Observatoryis part ofthe National Astrono my andIonosphere Center, 432
+which is operated by Cornell University under a cooperative agreem ent with the National 433
+Science Foundation. The National Radio Astronomy Observatory is a facility of the National 434
+Science Foundation Operated under cooperative agreement by As sociated Universities, Inc. 435
+Support for this work was provided by the National Aeronautics an d Space Administration 436
+through Chandra Award Number GO6-7136X issued by the Chandra X-Ray Observatory 437
+Center, which is operated by the Smithsonian Astrophysical Obser vatory for and on behalf 438
+of NASA under contract NAS8-03060. This research has made use of software provided 439
+by theChandra X-Ray Center in the application package CIAO. This research has ma de 440
+use of data obtained from the High Energy Astrophysics Science Ar chive Research Center 441
+(HEASARC), provided by NASA’s Goddard Space Flight Center.442
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+This preprint was prepared with the AAS L ATEX macros v5.2.– 19 –
+Table 1: Measured and Derived timing parameters of PSR J1907+060 2
+Fit and data-set
+Pulsar name............................... J1907+0602
+MJD range ................................ 54647–55074
+Number of TOAs .......................... 23
+Rms timing residual ( µs)................... 404
+Measured Quantities
+Right ascension, α......................... 19:07:54.71(14)
+Declination, δ.............................. +06:02:16.1(23)
+Pulse frequency, ν(s−1).................... 9.3780713067(19)
+First derivative of pulse frequency, ˙ ν(s−2)..−7.6382(4) ×10−12
+Second derivative of pulse frequency, ¨ ν(s−3) 2.5(6) ×10−22
+Epoch of frequency determination (MJD) .. 54800
+Dispersion measure, DM (cm−3pc)......... 82.1(11)
+Derived Quantities
+Characteristic age (kyr) ................... 19.5
+Surface magnetic field strength (G) ........ 3 .1×1012
+˙E(erg s−1) ................................ 2 .8×1036
+Assumptions
+Time units................................. TDB
+Solar system ephemeris model.............. DE405
+Note. — The numbers in parentheses are the errors in the last digit o f the fitted parameters. The errors
+are statistical only, except for the position error, as described in §2. The derived parameters of ˙E,B, andτc
+are essentially unchanged with respect to those reported by (Abd o et al. 2009a), but the position has moved
+by 1.2 arcmin.– 20 –
+Fig. 1.— Post-fit timing residuals for PSR J1907+0602. The reduced c hi-square of the fit is
+0.5.– 21 –
+300350400450500550Counts0.1-0.3 GeV400500600700800Counts0.3-1.0 GeV20406080100120140Counts1.0-3.0 GeV051015202530Counts> 3.0 GeV
+> 5.0 GeV700800900100011001200130014001500CountsP1 P2 P1 P2
+0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
+Pulse Phase0.00.20.40.60.81.0Amplitude (arb. units)
+Fig. 2.— Folded light curves of PSR J1907+0602 in 32 constant-width b ins for different
+energy bands and shown over two pulse periods with the 1.4 GHz radio pulse profile plotted
+in the bottom panel. The top panel of the figure shows the folded ligh t curve for photons
+withE >0.1 GeV. The other panels show the pulse profiles in exclusive energy ra nges:
+E >3.0 GeV (with E >5.0 GeV in black) in the second panel from the top; 1.0 to 3.0 GeV
+in the next panel; 0.3 to 1.0 GeV in the fourth panel; and 0.1 to 0.3 GeV in t he fifth panel.– 22 –
+Fig. 3.— The observed Fermi-LATcounts map of the region around PSR J1907+0602. Left:
+“on” pulse image, right: “off” pulse image. The open cross-hair mark s the location of the
+pulsar. Color scale shows the counts per pixel.– 23 –
+10-210-1100101102103104105
+E(GeV )10-1310-1210-1110-1010-9E2dN/dE (erg cm
+2s
+
1)
+Fig. 4.— Phase-averaged spectral energy distribution for PSR J19 07+0602 (green circles).
+Blue circles are data from HESS for HESS J1908+063 TeV source. 2 σupper limits from
+Fermi for emission from this TeV source are shown in blue. The black lin e shows the spectral
+model for the pulsar (equation 1). The upper limits suggest that th e spectrum of HESS
+J1908+063 has a low energy turnover between 20 GeV and 300 GeV.– 24 –
+-8 -6 -4 -2 0 2 4 6 8
+42.0 41.5 41.0 40.5 40.0 39.5 39.0 38.50.5
+0.0
+-0.5
+-1.0
+-1.5
+-2.0
+Galactic Latitude (deg.)
+Galactic Longitude (deg.)
+Fig. 5.— Residual map of the region around PSR J1907+0602 in the off- pulse. The timing
+position of the pulsar is marked by the cross. The 5 σcontours from Milagro (outer) and
+HESS (inner) are overlaid.– 25 –
+40.6 40.4 40.2 40.0 39.8 39.6 39.4-0.2
+-0.4
+-0.6
+-0.8
+-1.0
+-1.2
+-1.4
+Galactic Longitude (deg.)Galactic Latitude (deg.)PSR J1907+0602
+Fig. 6.— ASCAGIS 2-10 keV image of the region around PSR J1907+0602. The gree n
+contours are the 4-7 σsignificance contours from HESS.– 26 –
+0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
+R.A. (J2000)
+55.2 19:07:55.0 54.8 54.6 54.425.0
+6:02:20.0
+Decl. (J2000)15.0
+10.0
+05.0
+Fig. 7.— Chandra ACIS images of PSR J1907+0602. The blue ellipse shows the uncertain ty
+in the timing position. The green circle of radius 0.8′′is twice the FWHM of the 5keV PSF
+at this position, and should contain roughly 80% of the counts. The im age at 0.75-2 keV
+(Left), 2-8 keV (Center) and 0.75-8 keV (right) is shown. Color sc ale shows the counts per
+pixel.– 27 –! "#$! !%$! !&'()*+,-./012(3'4565/267/8 
+$ !9" : " ! "#$! !%)/5-03+,5 
+2;+''/,1/'/)<=1>7/8? 
+Fig. 8.— Chandra X-ray spectrum of PSR J1907+0602.– 28 –
+41.0 40.8 40.6 40.4 40.2 40.0 39.8-0.2
+-0.4
+-0.6
+-0.8
+-1.0
+-1.2
+-1.4S74 HESS J1908+063SNR G40.5-0.5This paper
+PSR J1907+0602Langston et al. position
+Fig. 9.— VGPS 1420 MHz image of region in Galactic coordinates showing r elationship
+between SNR G40.5 −0.5, HESS J1907+063 (blue contours representing the 4,5,6 and 7 σ
+significance levels), the star forming region S74, and PSR J1907+06 02.
\ No newline at end of file
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+arXiv:1001.0793v1  [cs.IT]  5 Jan 20101
+On the Vacationing CEO Problem: Achievable
+Rates and Outer Bounds
+Rajiv Soundararajan∗, Aaron B. Wagner†and Sriram Vishwanath∗
+∗Department of ECE, The University of Texas at Austin, Austin , TX 78712, USA
+†School of ECE, Cornell University, Ithaca, NY 14853, USA
+Abstract
+This paper studies a class of source coding problems that com bines elements of the CEO problem
+with the multiple description problem. In this setting, noi sy versions of one remote source are observed
+by two nodes with encoders (which is similar to the CEO proble m). However, it differs from the CEO
+problem in that each node must generate multiple descriptio ns of the source. This problem is of interest
+in multiple scenarios in efficient communication over netwo rks. In this paper, an achievable region and
+an outer bound are presented for this problem, which is shown to be sum rate optimal for a class of
+distortion constraints.
+I. INTRODUCTION
+Determining the theoretical limits of lossy compression sc hemes are of significant interest. Results
+in lossy source coding have applications in multiple domain s including multimedia communication &
+storage, image processing and distributed processing over sensor networks. A single representation for a
+single source is today a fairly well established field of rese arch [1]. When multiple representations and/or
+sources are involved, there are only a limited set of exact re sults known. The lossless compression of
+correlated sources, studied in [2] by Slepian and Wolf, is on e of the early success stories in this domain.
+Subsequently, the Gaussian two-terminal multiple descrip tion (MD) rate region was characterized in [3].
+More recently, many new results have emerged in the field of Ga ussian multiterminal source coding [4],
+[5]. In particular, the Gaussian CEO problem was studied in [ 6] and [7], where the sum rate and the
+entire rate region were characterized. [8] provides a simpl ified converse argument for the sum rate. The
+rate region of the Gaussian two-encoder problem was charact erized in [9].
+We consider a version of the CEO problem in which the CEO can “v acation.” The setup is described
+by Figure 1. We have a single source S, and two corrupted versions of the source X1=S+N1and
+November 15, 2018 DRAFT2
+X2=S+N2are available at the two encoders in the system. The encoders wish to communicate
+information about Sto a decoder, i.e., the CEO, which they accomplish by each sen ding a data packet at
+time 1 and another at time 2. The CEO may be on vacation during t ime 1, time 2, neither, or both, and
+she cannot receive data packets when she is vacationing. We a ssume that the CEO’s holiday schedule
+is unknown to the encoders. If the CEO works during time 1 and v acations during time 2, she expects
+to reproduce the source Sto distortion D1. Likewise, if she works during time 2 and vacations during
+time 1, she expects to reproduce Sto distortion D2. If the CEO eschews vacation and works during
+both periods, then she expects to reproduce Sto distortion D0. For convenience, we represent the three
+vacation states of the CEO by three separate receivers in Fig ure 1. Details on the system model and
+problem at hand are presented in Section II.
+This problem generalizes both the CEO problem, by omitting t he transmission at time 2, and the MD
+problem, by omitting the noises N1andN2. We note that a related problem is considered in [10], which
+also generalizesboth CEO and MD. However,unlike the other p roblem formulation, we are able to obtain
+a more conclusive sum rate result. The vacationing-CEO prob lem arises in multicast networks in which
+receivers enter and depart the systems at arbitrary times. S ee [11], [12] for additional discussion of this
+connection.
+The rest of this paper is organized as follows: In the next sec tion, we state the problem and describe the
+main result of the paper, which characterizes the sum rate of the Gaussian problem with CEO vacations.
+In Section III, we show that the sum rate described in Section II is achievable, and in Section IV, we
+provide a lower bound for the Gaussian vacationing-CEO prob lem. This lower bound combines converse
+techniques developed individually for the MD problem [3], [ 13] and the CEO problem [6], [8]. In fact,
+it is interesting to note that our lower bound requires the us e ofbothconverse techniques for the CEO
+problem, as neither alone is sufficient. In Section V, we esta blish the equivalence of the achievable sum
+rate and the lower bound on it, thus proving the main result of the paper.
+A. Notation
+We use capital letters to denote random variables and E[S]to denote the expected value of a random
+variableS.Alllogarithmsusedinthepaperarenaturallogarithms. Var(S|T)denotesES,T[(S−E[S|T])2].
+For¯S= (S1,S2),Cov(¯S|T)denotes the matrix E¯S,T[(¯S−E[¯S|T])(¯S−E[¯S|T])†], where ¯S†is the
+transpose of the vector ¯S.
+November 15, 2018 DRAFT3
+RECEIVER 0
+RECEIVER 2RECEIVER 1
+ENCODER 2 ENCODER 1
+R22X1=S+N1 X2=S+N2
+R12R11
+ˆS2ˆS1
+ˆS0R21
+Fig. 1. System Model
+II. PROBLEM STATEMENT AND MAINRESULT
+Let{X1i}n
+i=1and{X2i}n
+i=1be noisy observations of an underlying Gaussian source {Si}n
+i=1, observed
+by two different encoders. The observationsand the source a re assumed to be independentand identically
+distributed (i.i.d.) over i. For each time instant i, the observations are given by
+X1i=Si+N1i
+X2i=Si+N2i
+whereN1iandN2iare Gaussian distributed with mean zero and variance σ2
+N1andσ2
+N2.Sihas mean
+zero and variance σ2
+S. Encoder kobserves {Xki}n
+i=1fork= 1,2and sends two descriptions given by
+Ckl=fkl(Xn
+k), forl= 1,2to two receivers. Let Rkl≥1
+nlog|Ckl|. Receiver l, gets the messages
+f1l(Xn
+1)andf2l(Xn
+2), and applies decoding function ϕn
+l(f1l(Xn
+1),f2l(Xn
+2))to obtain an estimate of the
+sourceSn, denoted by ˆSn
+l.The central receiver gets all the four descriptions and app lies the function
+ϕn
+0(f11(Xn
+1),f21(Xn
+2),f12(Xn
+1),f22(Xn
+2))togetˆSn
+0.Wesaythatthetuple (R11,R12,R21,R22,D1,D2,D0)
+isachievable if there exist encoding and decoding functions such that
+Dl≥1
+nn/summationdisplay
+i=1E[(Si−ˆSli)2], l∈ {0,1,2}.
+November 15, 2018 DRAFT4
+We now state the main result of the paper. Let
+U={(U11,U12,U21,U22) :Ukl=Xk+Wklfork,l∈ {1,2},Wkl∼ N(0,σ2
+Wkl),
+(U11,U12)−X1−X2−(U21,U22),
+E[(S−E[S|U1l,U2l])2]≤Dlforl∈ {1,2}and
+E[(S−E[S|U11,U12,U21,U22])2]≤D0}.
+Theorem 1. The sum rate of the vacationing-CEO problem with distortion constraints (D1,D2,D0)such
+that1
+D1+1
+D2−max{1
+σ2
+N1,1
+σ2
+N2}−1
+σ2
+S≥1
+D0is given by
+inf
+(U11,U12,U21,U22)∈UI(X1,X2;U11,U12,U21,U22)+I(U11,U21;U12,U22).
+The technical condition on the distortions implies that the distortion constraint at the central receiver
+satisfiesmax{1
+D1,1
+D2} ≤1
+D0≤1
+D1+1
+D2−max{1
+σ2
+N1,1
+σ2
+N2}−1
+σ2
+S. This means that the central distortion
+constraint is comparable with (although lesser than) the di stortion constraints at the individual receivers.
+Note that the sum rate achieved here is a more general version of the achievable sum rate of the CEO
+problem with two sensors and the MD problem with two descript ions. In effect, the vacationing-CEO
+problem with just one sensor and encoder (or R21= 0andR22= 0) is a remote source version of
+the two description problem. In the absence of the central re ceiver (or D0≥σ2
+S), the vacationing-CEO
+problem reduces to two CEO problems corresponding to Receiv ers 1 and 2. We discuss the achievability
+in the following section.
+III. ACHIEVABILITY
+The achievable scheme discussed below is a Gaussian scheme. We define auxiliaries, U11,U12,U21and
+U22such that
+U11=X1+W11
+U12=X1+W12
+U21=X2+W21
+U22=X2+W22,
+November 15, 2018 DRAFT5
+where the vector W= (W11,W12,W21,W22)is Gaussian distributed with mean zero and covariance
+matrix
+Kw=
+σ2
+W11−a10 0
+−a1σ2
+W120 0
+0 0 σ2
+W21−a2
+0 0 −a2σ2
+W22
+.
+Kwis appropriately chosento meet the distortion constraints at Receivers1 and 2 and the central receiver.
+In effect Kwis chosen such that
+E/bracketleftBig
+(S−E[S|U1l,U2l])2/bracketrightBig
+≤Dl,l∈ {1,2}
+E/bracketleftBig
+(S−E[S|U11,U12,U21,U22])2/bracketrightBig
+≤D0.
+A. Codebook Generation
+Encoderk,k= 1,2, generates 2nR′
+k1Un
+k1and2nR′
+k2Un
+k2such that Uk1iandUk2iare generated i.i.d.
+according to the marginal of Uk1andUk2respectively. 2nR′
+k1Un
+k1and2nR′
+k2Un
+k2are binned into 2nRk1
+and2nRk2bins respectively.
+B. Encoding
+Encoderkchoosesthe pair (Un
+k1,Un
+k2)jointly typical with Xn
+kand transmits the respective bin indexes.
+There exists a pair (Un
+k1,Un
+k2)jointly typical with Xn
+kwith high probability if
+R′
+k1> I(Xk;Uk1)
+R′
+k2> I(Xk;Uk2) (1)
+R′
+k1+R′
+k2> I(Xk;Uk1,Uk2)+I(Uk1;Uk2).
+This multiple description encoding scheme is similar to the scheme in [14]. Since (Un
+11,Un
+12)−Xn
+1−
+Xn
+2−(Un
+21,Un
+22), by the Markov lemma (Lemma 14.8.1) in [1], we also have that (Un
+11,Un
+12,Un
+21,Un
+22)
+are jointly typical.
+C. Decoding at individual receivers
+Receiver l,l= 1,2, looks for Un
+1landUn
+2lthat are jointly typical in the bins corresponding to the bin
+indexes it receives. Receiver lwill be able to find unique codewords Un
+1landUn
+2lthat are jointly typical
+November 15, 2018 DRAFT6
+if
+R1l> R′
+1l−I(U1l;U2l)
+R2l> R′
+2l−I(U1l;U2l) (2)
+R1l+R2l> R′
+1l+R′
+2l−I(U1l;U2l).
+Receiver lgenerates an estimate of Sn, by constructing the minimum mean squared estimate (MMSE)
+E[Sn|Un
+1l,Un
+2l]. The decoding scheme resembles the decoding in the Berger-T ung scheme [15].
+D. Decoding at central receiver
+Receiver 0 mimics the decoding at Receiver 1 and 2 to find joint ly typical pairs (Un
+11,Un
+21)and
+(Un
+12,Un
+22)in the received bin indexes. Therefore, Receiver 0 will be ab le to find such unique codewords
+if the rates satisfy (2). Since (Un
+11,Un
+12,Un
+21,Un
+22)are jointly typical, Receiver 0 constructs the MMSE
+estimate of Sngiven by E[Sn|Un
+11,Un
+12,Un
+21,Un
+22].
+Note that the equations in (1) and (2) represent the entire ra te region achievable by the Gaussian
+scheme for the vacationing-CEO problem. We now consider the sum rate achievable by the Gaussian
+scheme.
+Lemma 1. The sum rate achievable by the Gaussian scheme is given by
+inf
+(U11,U12,U21,U22)∈UI(X1,X2;U11,U12,U21,U22)+I(U11,U21;U12,U22). (3)
+The lemma is proved in Appendix A. We present the lower bound o n the sum rate in the next section.
+IV. LOWERBOUND
+We now make a few definitions before presenting the lower boun d on the sum rate. Ckldenotes the
+code from Encoder kto Receiver lfork= 1,2andl= 1,2.
+Define,
+d11=1
+nn/summationdisplay
+i=1Var(X1i|C11,Sn) d21=1
+nn/summationdisplay
+i=1Var(X2i|C21,Sn)
+d12=1
+nn/summationdisplay
+i=1Var(X1i|C12,Sn) d22=1
+nn/summationdisplay
+i=1Var(X2i|C22,Sn) (4)
+t1=1
+nI(Xn
+1;C11,C12|Sn) t2=1
+nI(Xn
+2;C21,C22|Sn).
+November 15, 2018 DRAFT7
+We remark that in the following 0< D0<min{D1,D2}andmax{D1,D2}< σ2
+S. We now define for
+k= 1,2,
+Fk={(d1,d2,t) :d1,d2,t∈[0,∞)
+σ2
+Nke−2t≤min{d1,d2}max{d1,d2} ≤σ2
+Nk}.
+Further define,
+F={(d11,d12,d21,d22,t1,t2) : (dk1,dk2,tk)∈ Fk,k= 1,2
+1
+D1≤1
+σ2
+S+1
+σ2
+N1+1
+σ2
+N2−d11
+σ4
+N1−d21
+σ4
+N2(5)
+1
+D2≤1
+σ2
+S+1
+σ2
+N1+1
+σ2
+N2−d12
+σ4
+N1−d22
+σ4
+N2(6)
+1
+D0≤1
+σ2
+S+1−e−2t1
+σ2
+N1+1−e−2t2
+σ2
+N2}. (7)
+We have the following lemma which characterizes the paramet ers¯p= (d11,d12,d21,d22,t1,t2)defined
+above.
+Lemma 2. The parameters defined in (4) satisfy ¯p∈ F.
+Proof:The proof that
+1
+D0≤1
+σ2
+S+1−e−2t1
+σ2
+N1+1−e−2t2
+σ2
+N2
+follows directly from Lemma 3.1 in [7]. Also, in Theorem 1 in [ 8], it is shown that
+1
+Dl≤1
+σ2
+S+1
+σ2
+N1+1
+σ2
+N2−d1l
+σ4
+N1−d2l
+σ4
+N2
+forl= 1,2. By definition,
+ntk=I(Xn
+k,Ck1,Ck2|Sn) =h(Xn
+k|Sn)−h(Xn
+k|Ck1,Ck2,Sn)
+≥n
+2logσ2
+Nk−h(Xn
+k|Ckl,Sn),l= 1,2
+≥n
+2logσ2
+Nk−n
+2logdkl,l= 1,2.
+Therefore for k= 1,2,
+σ2
+Nke−2tk≤min{dk1,dk2}.
+Also, since E[Nn
+k|Ckl]achieves a smaller mean squared error in Nn
+kthan any other estimator,
+dkl=1
+nn/summationdisplay
+i=1Var(Xki|Ckl,Sn) =1
+nn/summationdisplay
+i=1Var(Nki|Ckl)≤σ2
+Nk
+November 15, 2018 DRAFT8
+fork= 1,2andl= 1,2. This concludes the proof of the lemma.
+Define,
+P1={(d11,d12,d21,d22,t1,t2) : (d11,d12,d21,d22,t1,t2)∈ F
+1
+D1=1
+σ2
+S+1
+σ2
+N1+1
+σ2
+N2−d11
+σ4
+N1−d21
+σ4
+N2(8)
+1
+D2=1
+σ2
+S+1
+σ2
+N1+1
+σ2
+N2−d12
+σ4
+N1−d22
+σ4
+N2(9)
+1
+D0=1
+σ2
+S+1−e−2t1
+σ2
+N1+1−e−2t2
+σ2
+N2} (10)
+P2={(d11,d12,d21,d22,t1,t2) : (d11,d12,d21,d22,t1,t2)∈ F
+1
+D1=1
+σ2
+S+1
+σ2
+N1+1
+σ2
+N2−d11
+σ4
+N1−d21
+σ4
+N2
+1
+D2=1
+σ2
+S+1
+σ2
+N1+1
+σ2
+N2−d12
+σ4
+N1−d22
+σ4
+N2
+1
+D0<1
+σ2
+S+1−e−2t1
+σ2
+N1+1−e−2t2
+σ2
+N2
+σ2
+N1e−2t1= min{d11,d12}σ2
+N2e−2t2= min{d21,d22}}.
+We denote
+P=P1∪P2.
+Note that the definition of Pimposes the restriction on the parameters to satisfy the ind ividual distortion
+constraints with equality. The central distortion constra intmay be satisfied with equality or the parameters
+satisfyσ2
+Nke−2tk= min{dk1,dk2}fork= 1,2. We also observe that P ⊂ F.
+Let¯p∈ F. Then∆F¯pis defined as
+∆F¯p={∆¯p= (∆d11,∆d12,∆d21,∆d22,−∆t1,−∆t2) : ∆d11,∆d12,∆d21,∆d22,∆t1,∆t2∈[0,∞)and
+(d11+∆d11,d12+∆d12,d21+∆d21,d22+∆d22,t1−∆t1,t2−∆t2)∈ P}.
+Lemma 3. ∆F¯p/ne}ationslash=φ∀¯p∈ F.
+Proof:The lemma is proved as follows. Consider ¯p∈ F. Then we increase d11andd12by∆d11and
+∆d12until we meet the distortion constraints at individual rece ivers with equality or d1l+∆d1l=σ2
+N1,
+l= 1,2. In the former case, we satisfy the individual distortion co nstraints with equality. In the latter
+case, we now increase d21andd22by∆d21and∆d22until we meet the individual distortion constraints
+with equality. We will be able to find such ∆d21and∆d22satisfying d2l+∆d2l≤σ2
+N2,l= 1,2, since
+November 15, 2018 DRAFT9
+Dl< σ2
+Sforl= 1,2. Now, we decrease t1by∆t1until the central distortion constraint is met with
+equality or σ2
+N1e−2(t1−∆t1)= min{d11+∆d11,d12+∆d12}. In the former case, we satisfy the central
+distortion with equality. In the latter case, we decrease t2by∆t2until the central distortion constraint is
+met with equality or σ2
+N2e−2(t2−∆t2)= min{d21+∆d21,d22+∆d22}. Therefore ∀¯p∈ F,∆F¯p/ne}ationslash=φ.
+Letk= 1,2andσ2
+Z≥0. Define, for (d1,d2,t)∈ Fk,
+rk(d1,d2,t,σ2
+Z) =t+1
+2log(σ2
+Nk+σ2
+Z)
+(d1+σ2
+Z)(d2+σ2
+Z)+1
+2log(σ2
+Nke−2t+σ2
+Z). (11)
+We now state the main result of this section.
+Lemma 4. The sum rate of the vacationing-CEO problem is lower bounded by
+inf
+¯p∈Psup
+σZ1,σZ2∈Rr1(d11,d12,t1,σ2
+Z1)+r2(d21,d22,t2,σ2
+Z2)+1
+2logσ4
+S
+D1D2. (12)
+Proof:By procedural steps, we have
+n(R11+R21+R12+R22)≥H(C11,C21)+H(C12,C22)
+≥H(C11,C21)+H(C12,C22)−H(C11,C21,C12,C22)
++H(C11,C21,C12,C22)−H(C11,C21,C12,C22|Xn
+1,Xn
+2)
+=I(Xn
+1,Xn
+2;C11,C21,C12,C22)+I(C11,C21;C12,C22)
+(a)=I(Sn;C11,C21,C12,C22)+I(Xn
+1,Xn
+2;C11,C21,C12,C22|Sn)
++I(C11,C21;C12,C22)
+(b)=I(Sn;C11,C21,C12,C22)+I(Xn
+1;C11,C12|Sn)+I(Xn
+2;C21,C22|Sn)
++I(C11,C21;C12,C22), (13)
+where (a) is true since Sn−(Xn
+1,Xn
+2)−(C11,C12,C21,C22)and (b) is true since (C11,C12)−Xn
+1−
+Sn−Xn
+2−(C21,C22).
+LetY1i=X1i+Z1iandY2i=X2i+Z2i, whereZ1iandZ2iare i.i.d Gaussians with mean zero and
+November 15, 2018 DRAFT10
+varianceσ2
+Z1andσ2
+Z2fori∈ {1,2,...,n}. Also,Z1iandZ2iare independent of Si,X1iandX2i. Now,
+I(C11,C21;C12,C22) =H(C11,C21)+H(C12,C22)−H(C11,C21,C12,C22)
+=H(C11,C21)+H(C12,C22)−H(C11,C21,C12,C22)
+−H(C11,C21|Sn,Yn
+1,Yn
+2)−H(C12,C22|Sn,Yn
+1,Yn
+2)
++H(C11,C21,C12,C22|Sn,Yn
+1,Yn
+2)+I(C11,C21;C12,C22|Yn
+1,Yn
+2,Sn)
+=I(Sn,Yn
+1,Yn
+2;C11,C21)+I(Sn,Yn
+1,Yn
+2;C12,C22)
+−I(Sn,Yn
+1,Yn
+2;C11,C12,C21,C22)+I(C11,C21;C12,C22|Yn
+1,Yn
+2,Sn)
+≥I(Sn,Yn
+1,Yn
+2;C11,C21)+I(Sn,Yn
+1,Yn
+2;C12,C22)
+−I(Sn,Yn
+1,Yn
+2;C11,C12,C21,C22). (14)
+Forl= 1,2,
+I(Sn,Yn
+1,Yn
+2;C1l,C2l) =I(Sn;C1l,C2l)+I(Yn
+1;C1l|Sn)+I(Yn
+2;C2l|Sn)
+since(Yn
+1,C1l)−Sn−(Yn
+2,C2l). By the definition of the rate distortion function for Gaussi an random
+variables, I(Sn;C1l,C2l)≥n
+2logσ2
+S
+DlandI(Yn
+k;Ckl|Sn)≥n
+2logσ2
+Nk+σ2
+Zk
+dkl+σ2
+Zkfork= 1,2. Therefore,
+I(Sn,Yn
+1,Yn
+2;C1l,C2l)≥n
+2logσ2
+S(σ2
+N1+σ2
+Z1)(σ2
+N2+σ2
+Z2)
+Dl(d1l+σ2
+Z1)(d2l+σ2
+Z2). (15)
+Observe that
+I(Sn,Yn
+1,Yn
+2;C11,C12,C21,C22) =I(Sn;C11,C21,C12,C22)+I(Yn
+1,Yn
+2;C11,C21,C12,C22|Sn)
+=I(Sn;C11,C21,C12,C22)+I(Yn
+1;C11,C12|Sn)
++I(Yn
+2;C21,C22|Sn), (16)
+where in the last step we used (Yn
+1,C11,C12)−Sn−(Yn
+2,C21,C22). Further, for k= 1,2
+I(Yn
+k;Ck1,Ck2|Sn) =−h(Yn
+k|Sn,Ck1,Ck2)+h(Yn
+k|Sn)
+(c)
+≤ −n
+2log(e2
+nh(Xn
+k|Sn,Ck1,Ck2)+e2
+nh(Zn
+k))+h(Yn
+k|Sn)
+=−n
+2log(e2
+n(h(Xn
+k|Sn)−I(Xn
+k;Ck1,Ck2|Sn))+e2
+nh(Zn
+k))+h(Yn
+k|Sn)
+=−n
+2log(σ2
+Nke−2tk+σ2
+Zk)+n
+2log(σ2
+Nk+σ2
+Zk), (17)
+November 15, 2018 DRAFT11
+where (c) follows from entropy power inequality (EPI). From (14), (15), (16) and (17),
+I(C11,C21;C12,C22)≥n
+2log(σ2
+N1+σ2
+Z1)(σ2
+N2+σ2
+Z2)σ4
+S
+(d12+σ2
+Z1)(d22+σ2
+Z2)(d11+σ2
+Z1)(d21+σ2
+Z2)D1D2
++n
+2log(σ2
+N1e−2t1+σ2
+Z1)(σ2
+N2e−2t2+σ2
+Z2)−I(Sn;C11,C21,C12,C22)
+Substituting the above in (13), we get
+R11+R21+R12+R22≥t1+t2+1
+2log(σ2
+N1+σ2
+Z1)(σ2
+N2+σ2
+Z2)σ4
+S
+(d12+σ2
+Z1)(d22+σ2
+Z2)(d11+σ2
+Z1)(d21+σ2
+Z2)D1D2
++1
+2log(σ2
+N1e−2t1+σ2
+Z1)(σ2
+N2e−2t2+σ2
+Z2)
+=r1(d11,d12,t1,σ2
+Z1)+r2(d21,d22,t2,σ2
+Z2)+1
+2logσ4
+S
+D1D2, (18)
+where the last equality is due to the definition in (11). From L emma 2, we have ¯p∈ F. By Lemma 3,
+∆F¯p/ne}ationslash=φ. Let∆¯p∈∆F¯p. Note that rk(dk1,dk2,tk,σ2
+Zk)is decreasing in dk1anddk2and increasing in
+tkfork= 1,2. This implies that
+rk(dk1,dk2,tk,σ2
+Zk)≥rk(dk1+∆dk1,dk2+∆dk2,tk−∆tk,σ2
+Zk)∀¯p∈ F.
+Therefore,
+R11+R21+R12+R22≥r1(d11+∆d11,d12+∆d12,t1−∆t1,σ2
+Z1)
++r2(d21+∆d21,d22+∆d22,t2−∆t2,σ2
+Z2)+1
+2logσ4
+S
+D1D2.
+By definition, ¯p+∆¯p∈ P. Therefore,
+R11+R21+R12+R22≥inf
+¯p∈Psup
+σZ1,σZ2∈Rr1(d11,d12,t1,σ2
+Z1)+r2(d21,d22,t2,σ2
+Z2)+1
+2logσ4
+S
+D1D2.
+In the following section, we show that the lower bound on the s um rate described above is achieved
+by the Gaussian scheme.
+V. EQUIVALENCE OF ACHIEVABLE SUM RATE AND LOWER BOUND
+Before we compare sum rate of the achievable scheme with the l ower bound, we present two lemmas
+about parameters introduced in the previous section which w ill be used in the comparison. We will use
+the notation ¯p= (d11,d12,d21,d22,t1,t2).
+Lemma 5. If1
+D1+1
+D2−max{1
+σ2
+N1,1
+σ2
+N2}−1
+σ2
+S≥1
+D0and¯p∈ P, then
+dk1+dk2−σ2
+Nke−2tk−σ2
+Nk≤0
+November 15, 2018 DRAFT12
+fork= 1,2.
+Proof:Let¯p∈ P1. Since
+1
+D1+1
+D2−max{1
+σ2
+N1,1
+σ2
+N2}−1
+σ2
+S≥1
+D0,
+substituting for1
+D1,1
+D2and1
+D0, from (8), (9) and (10) respectively, we get
+d11+d12−σ2
+N1−σ2
+N1e−2t1
+σ4
+N1+d21+d22−σ2
+N2−σ2
+N2e−2t2
+σ4
+N2+max{1
+σ2
+N1,1
+σ2
+N2} ≤0.
+Therefore either d11+d12−σ2
+N1−σ2
+N1e−2t1≤0ord21+d22−σ2
+N2−σ2
+N2e−2t2≤0. Letd11+d12−
+σ2
+N1−σ2
+N1e−2t1≤0. But since
+d11+d12−σ2
+N1−σ2
+N1e−2t1
+σ4
+N1+d21+d22−σ2
+N2−σ2
+N2e−2t2
+σ4
+N2+1
+σ2
+N1≤0
+⇒d11+d12−σ2
+N1e−2t1
+σ4
+N1+d21+d22−σ2
+N2−σ2
+N2e−2t2
+σ4
+N2≤0,
+andσ2
+N1e−2t1≤min{d11,d12}, it follows that d21+d22−σ2
+N2−σ2
+N2e−2t2≤0. Similarly, we can start
+withd21+d22−σ2
+N2−σ2
+N2e−2t2≤0, and use
+d11+d12−σ2
+N1−σ2
+N1e−2t1
+σ4
+N1+d21+d22−σ2
+N2−σ2
+N2e−2t2
+σ4
+N2+1
+σ2
+N2≤0,
+to show that d11+d12−σ2
+N1−σ2
+N1e−2t1≤0. Therefore we have now shown that if ¯p∈ P1, then
+dk1+dk2−σ2
+Nke−2tk−σ2
+Nk≤0fork= 1,2.
+Now, let ¯p∈ P2. Therefore, σ2
+Nke−2tk= min{dk1,dk2},k= 1,2. Sincemax{dk1,dk2} ≤σ2
+Nk, it
+follows that
+dk1+dk2−σ2
+Nk−σ2
+Nke−2tk= min{dk1,dk2}+max{dk1,dk2}−σ2
+Nk−σ2
+Nke−2tk
+= max{dk1,dk2}−σ2
+Nk
+≤0.
+Thus for all ¯p∈ P,dk1+dk2−σ2
+Nk−σ2
+Nke−2tk≤0,k= 1,2.
+We now state and prove the second lemma about the parameters. Let(dk1,dk2,tk)∈ Fkfork= 1,2.
+Define
+αk0=σ4
+Nke−2tk
+σ2
+Nk−σ2
+Nke−2tkαk1=σ2
+Nkdk1
+σ2
+Nk−dk1αk2=σ2
+Nkdk2
+σ2
+Nk−dk2(19)
+and
+gk(β) =1
+αk0+β−1
+αk1+β−1
+αk2+β.
+November 15, 2018 DRAFT13
+We use this function to partition the space of parameters (dk1,dk2,tk)∈ Fk. Define,
+Fk1={(dk1,dk2,tk)∈ Fk:gk(0)>0andgk(σ2
+Nk)≤0}
+Fk2={(dk1,dk2,tk)∈ Fk:gk(0)≤0}
+Fk3={(dk1,dk2,tk)∈ Fk:gk(σ2
+Nk)>0}.
+Lemma 6. Fork= 1,2,
+Fk=Fk1∪Fk2∪Fk3.
+Moreover, if1
+D1+1
+D2−max{1
+σ2
+N1,1
+σ2
+N2}−1
+σ2
+S≥1
+D0, then
+¯p∈ P ⇒(dk1,dk2,tk)∈ Fk1∪Fk2,k= 1,2.
+Proof:For every (dk1,dk2,tk)∈ Fk, one of either gk(0)>0andgk(σ2
+Nk)≤0orgk(0)≤0or
+gk(σ2
+Nk)>0is true and therefore Fk=Fk1∪Fk2∪Fk3.
+From Lemma5, ¯p∈ Pimpliesdk1+dk2−σ2
+Nke−2tk−σ2
+Nk≤0fork= 1,2. However, (dk1,dk2,tk)∈ Fk3
+impliesgk(σ2
+Nk)>0. This means that dk1+dk2−σ2
+Nke−2tk−σ2
+Nk>0. Therefore, ¯p∈ Pimplies,
+(dk1,dk2,tk)/∈ Fk3. Therefore,
+¯p∈ P ⇒(dk1,dk2,tk)∈ Fk1∪Fk2,k= 1,2.
+In order to show that the Gaussian scheme described in Sectio n III achieves the lower bound on the
+sum rate, we parametrize the achievable sum rate now. Define,
+d′
+k1= Var(Xk|Uk1,S) =σ2
+Nkσ2
+Wk1
+σ2
+Nk+σ2
+Wk1(20)
+d′
+k2= Var(Xk|Uk2,S) =σ2
+Nkσ2
+Wk2
+σ2
+Nk+σ2
+Wk2(21)
+t′
+k=I(Xk;Uk1,Uk2|S) =1
+2logσ2
+Nk(σ2
+Wk1+σ2
+Wk2+2ak)+σ2
+Wk1σ2
+Wk2−a2
+k
+σ2
+Wk1σ2
+Wk2−a2
+k. (22)
+We can rewrite the last equation above as
+1
+σ2
+Nke−2t′
+k
+1−e−2t′
+k+ak=1
+σ2
+Wk1+ak+1
+σ2
+Wk2+ak. (23)
+Let¯p′= (d′
+11,d′
+12,d′
+21,d′
+22,t′
+1,t′
+2)denote the parameters achieved by the Gaussian scheme. By de finition
+of(U11,U12,U21,U22)∈ U,¯p′∈ F. This means that the achievable parameters correspond to a G aussian
+scheme that satisfies the distortion constraints (5), (6) an d (7). We use the definition of functions in (11)
+November 15, 2018 DRAFT14
+and the parameters introduced above in the following lemma, relating them to the sum rate achievable
+by the Gaussian scheme.
+Lemma 7. For all(U11,U12,U21,U22)∈ Uandσ2
+Zk≥0,k∈ {1,2},
+I(X1,X2;U11,U12,U21,U22)+I(U11,U21;U12,U22)
+=r1(d′
+11,d′
+12,t′
+1,σ2
+Z1)+r2(d′
+21,d′
+22,t′
+2,σ2
+Z2)+I(U11;U12|S,Y1)+I(U21;U22|S,Y2)+1
+2logσ4
+S
+δ1δ2,
+(24)
+where
+1
+δ1=1
+σ2
+S+1
+σ2
+N1+1
+σ2
+N2−d′
+11
+σ4
+N1−d′
+21
+σ4
+N2
+1
+δ2=1
+σ2
+S+1
+σ2
+N1+1
+σ2
+N2−d′
+12
+σ4
+N1−d′
+22
+σ4
+N2,
+Y1=X1+Z1andY2=X2+Z2,Z1andZ2are independent of both X1andX2and Gaussian
+distributed with mean zero and variance σ2
+Z1andσ2
+Z2respectively.
+This lemma is proved in Appendix B. We now show that the Gaussi an scheme achieves the lower
+bound on the sum rate corresponding to every point ¯p∈ P. We prove this through the following lemma.
+Lemma 8. For every ¯p∈ P, there exists an achievable ¯p′∈ Fandσ2
+Zk≥0,k= 1,2, such that
+r1(d11,d12,t1,σ2
+Z1)+r2(d21,d22,t2,σ2
+Z2)+1
+2logσ4
+S
+D1D2
+=r1(d′
+11,d′
+12,t′
+1,σ2
+Z1)+r2(d′
+21,d′
+22,t′
+2,σ2
+Z2)+I(U11;U12|S,Y1)+I(U21;U22|S,Y2)+1
+2logσ4
+S
+δ2δ1.
+Proof:TheproofcloselyfollowsthediscussioninSection5in[13] .Let¯p= (d11,d12,d21,d22,t1,t2)∈
+P. Choosing d′
+kl=dklfork= 1,2andl= 1,2, from (8) and (9), we know that
+δ1=D1δ2=D2. (25)
+From Lemma 6, we know that (dk1,dk2,tk)∈ Fk1∪Fk2. By definition, Fk1∩Fk2=φ. We now consider
+two cases, (dk1,dk2,tk)∈ Fk1and(dk1,dk2,tk)∈ Fk2.
+A. Case 1: (dk1,dk2,tk)∈ Fk1
+Since(dk1,dk2,tk)∈ Pk1,gk(0)>0andgk(σ2
+Nk)≤0. Therefore, there exists an a∗
+k∈(0,σ2
+Nk]that
+solvesgk(ak) = 0. We set ak=a∗
+k. Further, d′
+k1=dk1andd′
+k2=dk2imply that σ2
+Wk1=αk1and
+November 15, 2018 DRAFT15
+σ2
+Wk2=αk2. Therefore, we conclude from (23) and gk(a∗
+k) = 0thatt′
+k=tk. We now need to show that
+this choice of a∗
+kis such that σ2
+Wk1σ2
+Wk2≥(a∗
+k)2. Sinceαk0≥0anda∗
+k∈(0,σ2
+Nk],
+αk0+a∗
+k≥a∗
+k⇒1
+αk0+a∗
+k≤1
+a∗
+k.
+Sinceg(a∗
+k) = 0,
+1
+σ2
+Wk1+a∗
+k+1
+σ2
+Wk2+a∗
+k≤1
+a∗
+k
+⇒1
+σ2
+Wk1+a∗
+k≤1
+a∗
+k−1
+σ2
+Wk2+a∗
+k
+⇒1
+σ2
+Wk1+a∗
+k≤σ2
+Wk2
+a∗
+k(σ2
+Wk2+a∗
+k)
+⇒σ2
+Wk1+a∗
+k≥(a∗
+k)2
+σ2
+Wk2+a∗
+k
+⇒σ2
+Wk1σ2
+Wk2≥(a∗
+k)2.
+Moreover, trivially,
+rk(dk1,dk2,tk,σ2
+Zk) =rk(d′
+k1,d′
+k2,t′
+k,σ2
+Zk)
+Also,
+Var(Ukl|S,Yk) =σ2
+Nk+σ2
+Wkl−σ4
+Nk
+σ2
+Nk+σ2
+Zk,l= 1,2
+Cov(Uk1,Uk2|S,Yk) =σ2
+Nk
+1 1
+1 1
++
+σ2
+Wk1−ak
+−akσ2
+Wk2
+−σ4
+Nk
+σ2
+Nk+σ2
+Zk
+1 1
+1 1
+
+The off diagonal entries in Cov(Uk1,Uk2|S,Yk)are zero if
+σ2
+Nk−ak=σ4
+Nk
+σ2
+Nk+σ2
+Zk.
+By choosing σ2
+Zk=akσ2
+Nk
+σ2
+Nk−akin this case,
+Var(Uk1|S,Yk)Var(Uk2|S,Yk) =|Cov(Uk1,Uk2|S,Yk)|
+andI(Uk1;Uk2|S,Yk) = 0. Note that we are allowed to choose σ2
+Zk=akσ2
+Nk
+σ2
+Nk−aksinceak∈(0,σ2
+Nk]in
+this case.
+November 15, 2018 DRAFT16
+B. Case 2: (dk1,dk2,tk)∈ Fk2
+In this case, we set ak= 0in (22) and achieve the corresponding t′
+k. Since,d′
+k1=dk1andd′
+k2=dk2,
+we haveσ2
+Wk1=αk1andσ2
+Wk2=αk2. It follows from (19) and (23) that
+1
+σ2
+Nk+1
+αk1+1
+αk2=1
+σ2
+Nke−2t′
+k.
+Sincegk(0)≤0, this implies that
+1
+αk0≤1
+αk1+1
+αk2
+⇒1
+αk0+1
+σ2
+Nk≤1
+σ2
+Nk+1
+αk1+1
+αk2
+⇒1
+σ2
+Nke−2tk≤1
+σ2
+Nk+1
+αk1+1
+αk2
+=1
+σ2
+Nke−2t′
+k.
+Therefore, we get that tk≤t′
+k. By achieving t′
+kinstead of tk, we still satisfy the central distortion
+constraint for the original problem and also ensure (d′
+k1,d′
+k2,t′
+k)∈ Fk. Further, we choose σ2
+Zk= 0in
+this case. Therefore
+rk(dk1,dk2,tk,0) =1
+2logσ4
+Nk
+dk1dk2=rk(dk1,dk2,t′
+k,0),
+Moreover, since σ2
+Zk= 0andak= 0
+I(Uk1;Uk2|S,Yk) =I(Uk1;Uk2|S,Xk) = 0.
+The lemma follows from the cases considered above.
+Therefore, it follows from Lemma 4 and Lemma 8 that for every ¯p∈ P, there exists an achievable
+¯p′∈ Fsuch that the sum rate achievable by the Gaussian scheme is eq ual to the lower bound on the
+sum rate. This proves the optimality of the Gaussian scheme f or the sum rate of the vacationing-CEO
+problem.
+VI. CONCLUSION
+We introduced the vacationing-CEO problem which in essence , is a CEO problem with multiple
+descriptions. We described a Gaussian achievable scheme an d presented a lower bound for the sum
+rate as an optimization problem over the code parameters. We also showed that the Gaussian scheme
+is optimal in terms of sum rate for a class of distortion const raints. Future work includes extending the
+result to other distortion regimes and considering a two ter minal source coding problem with multiple
+descriptions.
+November 15, 2018 DRAFT17
+APPENDIX A
+PROOF OF LEMMA1
+In order to prove Lemma 1, we need to show that ∀δ >0, there exist (R′
+11,R′
+12,R′
+21,R′
+22)and
+(R11,R12,R21,R22)that satisfy (1) and (2) such that
+|R11+R21+R12+R22−I(X1,X2;U11,U12,U21,U22)−I(U11,U21;U12,U22)| ≤δ.
+Letǫ=δ
+8and(U11,U12,U21,U22)∈ U. We choose
+R′
+11=I(X1;U11)+ǫ R′
+12=I(X1;U12|U11)+I(U11;U12)+ǫ
+R′
+21=I(X2;U21)+ǫ R′
+22=I(X2;U22|U21)+I(U21;U22)+ǫ
+R11=R′
+11−I(U11;U21)+ǫ R 21=R′
+21+ǫ
+R12=R′
+12−I(U12;U22)+ǫ R 22=R′
+22+ǫ.
+Note that (R′
+11,R′
+12,R′
+21,R′
+22)satisfy (1) and (R11,R12,R21,R22)satisfy (2). Therefore,
+R11+R21+R12+R22=R′
+11+R′
+12+R′
+21+R′
+22−I(U11;U21)−I(U12;U22)+4ǫ
+=I(X1;U11,U12)+I(U11;U12)+I(X2;U21,U22)+I(U21;U22)
+−I(U11;U21)−I(U12;U22)+8ǫ
+=I(X1,X2;U11,U12,U21,U22)+I(U11,U21;U12,U22)+δ.
+Allowing δ→0, we see that the Gaussian scheme achieves the sum rate
+inf
+(U11,U12,U21,U22)∈UI(X1,X2;U11,U12,U21,U22)+I(U11,U21;U12,U22).
+APPENDIX B
+PROOF OF LEMMA7
+By procedural steps, we have
+I(X1,X2;U11,U21,U12,U22)+I(U11,U21;U12,U22)
+=I(S;U11,U21,U12,U22)+I(X1,X2;U11,U21,U12,U22|S)
++I(U11,U21;U12,U22)
+=I(S;U11,U21,U12,U22)+I(X1;U11,U12|S)+I(X2;U21,U22|S)
++I(U11,U21;U12,U22)
+=I(S;U11,U21,U12,U22)+t′
+1+t′
+2+I(U11,U21;U12,U22).
+November 15, 2018 DRAFT18
+Recall that Y1=X1+Z1andY2=X2+Z2whereZ1andZ2are Gaussians with mean zero and
+varianceσ2
+Z1andσ2
+Z2and independent of S,X1andX2. Now,
+I(U11,U21;U12,U22) =h(U11,U21)+h(U12,U22)−h(U11,U21,U12,U22)
+=h(U11,U21)+h(U12,U22)−h(U11,U21,U12,U22)
+−h(U11,U21|S,Y1,Y2)−h(U12,U22|S,Y1,Y2)
++h(U11,U21,U12,U22|S,Y1,Y2)+I(U11,U21;U12,U22|Y1,Y2,S)
+=I(S,Y1,Y2;U11,U21)+I(S,Y1,Y2;U12,U22)
+−I(S,Y1,Y2;U11,U12,U21,U22)+I(U11;U12|S,Y1)+I(U21;U22|S,Y2).
+(26)
+Forl= 1,2, letδl=σ2
+Se−2I(S;U1l,U2l). Now, we can compute mutual information expressions betwee n
+Gaussian random variables or use the fact that Gaussian rand om variables satisfy Lemma 3.1 in [7] with
+equality to conclude that,
+1
+σ2
+Se2I(S;U1l,U2l)=1
+σ2
+S+1−e−2I(X1;U1l|S)
+σ2
+N1+1−e−2I(X2;U2l|S)
+σ2
+N2
+⇒1
+δl=1
+σ2
+S+1
+σ2
+N1+1
+σ2
+N2−d′
+1l
+σ4
+N1−d′
+2l
+σ4
+N2.
+Therefore,
+I(S,Y1,Y2;U1l,U2l) =I(S;U1l,U2l)+I(Y1;U1l|S)+I(Y2;U2l|S)
+=1
+2logσ2
+S(σ2
+N1+σ2
+Z1)(σ2
+N2+σ2
+Z2)
+δl(d′
+1l+σ2
+Z1)(d′
+2l+σ2
+Z2). (27)
+Observe that
+I(S,Y1,Y2;U11,U12,U21,U22) =I(S;U11,U21,U12,U22)+I(Y1,Y2;U11,U21,U12,U22|S)
+=I(S;U11,U21,U12,U22)+I(Y1;U11,U12|S)+I(Y2;U21,U22|S),
+(28)
+and fork= 1,2
+I(Yk;Uk1,Uk2|S) =−h(Yk|S,Uk1,Uk2)+h(Yk|S)
+(a)=−1
+2log(e2
+nh(Xk|S,Uk1,Uk2)+e2
+nh(Zk))+h(Yk|S)
+=−1
+2log(e2
+n(h(Xk|S)−I(Xk;Uk1,Uk2|S))+e2
+nh(Zk))+h(Yk|S)
+=−1
+2log(σ2
+Nke−2t′
+k+σ2
+Zk)+1
+2log(σ2
+Nk+σ2
+Zk), (29)
+November 15, 2018 DRAFT19
+where (a) follows from EPI for Gaussians. From (26), (27), (2 8) and (29),
+I(U11,U21;U12,U22) =1
+2log(σ2
+N1+σ2
+Z1)(σ2
+N2+σ2
+Z2)σ4
+S
+(d′
+12+σ2
+Z1)(d′
+22+σ2
+Z2)(d′
+11+σ2
+Z1)(d′
+21+σ2
+Z2)δ1δ2
++1
+2log(σ2
+N1e−2t′
+1+σ2
+Z1)(σ2
+N2e−2t′
+2+σ2
+Z2)−I(S;U11,U21,U12,U22)
++I(U11;U12|S,Y1)+I(U21;U22|S,Y2)
+and
+I(X1,X2;U11,U12,U21,U22)+I(U11,U21;U12,U22)
+=r1(d′
+11,d′
+12,t′
+1,σ2
+Z1)+r2(d′
+21,d′
+22,t′
+2,σ2
+Z2)+I(U11;U12|S,Y1)+I(U21;U22|S,Y2)+1
+2logσ4
+S
+δ1δ2.
+REFERENCES
+[1] T. M. Cover and J. A. Thomas, Elements of Information Theory . New York: John Wiley & Sons, 1999.
+[2] D. Slepian and J. K. Wolf, “Noiseless coding of correlate d information sources,” IEEE Trans. Inf. Theory , vol. 19, pp.
+471–480, Jul. 1973.
+[3] L. Ozarow, “On a source coding problem with two channels a nd three receivers,” Bell Syst. Tech. J. , vol. 59, pp. 1909–1921,
+Dec. 1980.
+[4] Y. Oohama, “Gaussian multiterminal source coding,” IEEE Trans. Inf. Theory , vol. 43, pp. 1912–1923, Nov. 1997.
+[5] D. Krithivasan and S. Pradhan, “Lattices for distribute d source coding: Jointly Gaussian sources and reconstructi on of a
+linear function,” IEEE Trans. Inf. Theory , vol. 55, pp. 5628–5651, Dec. 2009.
+[6] Y. Oohama, “The rate distortion function for the quadrat ic Gaussian CEO problem,” IEEE Trans. Inf. Theory , vol. 44, pp.
+55–67, May 1998.
+[7] V. Prabhakaran, K. Ramachandran, and D. Tse, “Rate regio n of the quadratic Gaussian CEO problem,” in Proc. IEEE Int
+Symp Info Theory , Chicago, USA, 2004.
+[8] J. Wang, J. Chen, and X. Wu, “On the minimum sum rate of Gaus sian multiterminal source coding: new proofs,” in Proc.
+IEEE Int Symp Info Theory , Seoul, Korea, 2009.
+[9] A.B.Wagner,S.Tavildar,andP.Viswanath,“Rateregion oftheQuadratic GaussianTwo-Encoder Source-CodingProbl em,”
+IEEE Trans. Inf. Theory , vol. 54, pp. 1938–1961, May 2008.
+[10] J. Chen and T. Berger, “Robust distributed source codin g,”IEEE Trans. Inf. Theory , vol. 54, pp. 3385–3398, Aug. 2008.
+[11] E. Ahmed and A. B. Wagner, “Binary erasure multiple desc riptions: Worst-case distortion,” in Proc. IEEE Int Symp Info
+Theory, 2009, pp. 55–59.
+[12] ——, “Binary erasure multiple descriptions: Average-c ase distortion,” in Proc. Information Theory Workshop , 2009, pp.
+166–170.
+[13] H. Wang and P. Viswanath, “Vector Gaussian multiple des criptions with individual and centralized receivers,” IEEE Trans.
+Inf. Theory , vol. 53, pp. 2133–2153, Jun. 2007.
+[14] T. Cover and A. El Gamal, “Achievable rates for multiple descriptions,” IEEE Trans. Inf. Theory , vol. 28, pp. 851–857,
+Nov. 1982.
+[15] S. Y. Tung, Multiterminal Source Coding . PhD thesis, School of Electrical and Computer Engineering , Cornell University,
+Ithaca, NY, 1978.
+November 15, 2018 DRAFT
\ No newline at end of file
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--- /dev/null
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@@ -0,0 +1,291 @@
+arXiv:1001.0794v1  [astro-ph.IM]  6 Jan 2010Consequences of spectrograph illumination for the accuracy of
+radial-velocimetry
+Boisse, I.1,a, Bouchy,F.1,2,Chazelas,B.3,Perruchot,S.2,Pepe,F.3,Lovis,C.3,andH´ ebrard,G.1
+1Institut d’Astrophysique de Paris, Universit´ e Pierre et M arie Curie, UMR7095 CNRS, 98bis bd. Arago, 75014 Paris,
+France
+2ObservatoiredeHauteProvence,CNRS /OAMP, 04870St Michell’Observatoire,France
+3ObservatoiredeGen` eve,Universit´ ede Gen` eve,51Ch. des Maillettes, 1290Sauverny,Switzerland
+Abstract. Forfiber-fedspectrographs witha stable external waveleng th source, scrambling properties of opti-
+cal fibers and, homogeneity and stabilityof the instrument i llumination are important for the accuracy of radial-
+velocimetry. Optical cylindric fibers are known to have good azimuthal scrambling. In contrast, the radial one
+is not perfect. In order to improve the scrambling ability of the fiber and to stabilize the illumination, optical
+doublescramblerareusuallycoupledtothefibers.Despitet hat,ourexperience onSOPHIEandHARPShaslead
+to identified remaining radial-velocity limitations due to the non-uniform illumination of the spectrograph. We
+conductedtestsonSOPHIEwithtelescopevignetting,seein gvariationandcenteringerrorsonthefiberentrance.
+We simulated the light path through the instrument in order t o explain the radial velocity variation obtained
+with our tests. We then identified the illumination stabilit y and uniformity has a critical point for the extremely
+high-precisionradialvelocityinstruments(ESPRESSO@VL T,CODEX@E-ELT).Testsonsquareandoctagonal
+section fibers are now under development and SOPHIE will be us ed as a bench test to validate these new feed
+optics.
+1 Introduction
+High-precisionradial-velocimetry(RV)isstillamaintec h-
+nique in the search and characterization of planetary sys-
+tems.Uptodate,mostoftheknownextrasolarplanetshave
+been detected with this method. It will remain at the fore-
+front of exoplanet science for the coming years thanks to
+itsefficiencyinfindinglow-massplanetssinceHARPShas
+reached an RV accuracy less than 1ms−1(ref. [7]). In the
+aim to reach Earth-type planets and, also measure the ex-
+pansion of the universe, new spectrographs in the visible
+are nowin studyto reacha precisionofonlyfew cms−1in
+the near future (ESPRESSO@VLT [8], CODEX@E-ELT
+[9]).
+To reach this level of accuracy, we must avoid a cer-
+tain number of limitations. Among these, one can quote
+the stellar noise (activity, pulsation, surface granulati on),
+thecontaminationbyexternalsources(moon,close-byob-
+jects,backgroundcontinuum),andwhatlightmayencounter
+duringitspath(e.g.atmosphericdispersion).Aspartofin -
+strumental factors, CCD cosmetics, charge transfer ine ffi-
+ciency(ref.[2]),wavelengthcalibration(refs.[5]and[1 2])
+shouldbementioned.Foranexhaustivediscussionofthese
+limitations,we refer to[10]. Theauthorshighlightthe im-
+pact of the spectrograph illumination as one of the main
+limiting factors, and the need to analyze and quantify it.
+In this proceeding, we then focus on the limitation due to
+ae-mail:iboisse@iap.frthe injection of light in the fiber-fed spectrograph. This
+work is based on our experiment on two high-resolution,
+high-precision fiber-fed spectrographs, SOPHIE@1.93m,
+Observatoire de Haute-Provence (OHP), France (refs. [3]
+and [11]) and HARPS@3.6m, EuropeanSouthernObser-
+vatory(ESO)-LaSilla, Chile (ref.[7]).
+Up to day, SOPHIE has a Doppler accuracy of about
+5 ms−1, as most of high-accuracy spectrographs involved
+in exoplanet survey. In service since November 2006, we
+identified that instrumental limitations mainly come from
+theoldCassegrainFiberAdaptater(developedforELODIE
+spectrograph).In order to reach the so far unique HARPS
+level of accuracy,we conductedtests that illustrate the in -
+complete fiber scrambling. Detected with SOPHIE, these
+effectswerealsocharacterizedonHARPSatalowerlevel.
+Inthedevelopmentofthefutureinstruments,theoptimiza-
+tion ofSOPHIEis a benchtest forthe improvementofthe
+spectrographilluminationuniformityandstability.
+The proceedingis structured as followed. In a the sec-
+ondsection,weexposethepropertiesofthefiber-fedspec-
+trographsandthelightinjection.Thethirdsectionreview s
+the different effects that are detected in the RV due to the
+incompleteandimperfectscramblingobtainedwiththestan -
+dard fibers. In the last section, we describe solutions pro-
+posedtooptimizethefibersscrambling.EPJWeb ofConferences
+2 Fiber-fedspectrographs
+HARPS and SOPHIE are fiber-fed spectrographs with si-
+multaneousreferencecalibrations.Thestellarlightcoll ected
+by the telescope are lead to the instrument througha stan-
+dard step-index multi-mode cylindrical optical fiber. SO-
+PHIE has two observing mode using two di fferent fibers,
+the High-Resolution (HR) and the High-E fficiency mode
+(HE). In HR mode, the spectrograph is fed by a 40.5- µm
+slitsuperimposedontheoutputofthe100- µmfiber,reach-
+ing a spectral resolution of λ/∆λ=75,000. In HE mode,
+thespectrographisdirectlyfedbythe100- µmfiberwith a
+resolution power of 40,000. Both SOPHIE fibers have an
+sky acceptance of 3-arcsec. HARPS has 70- µm fiber with
+a sky acceptance of 1-arcsec and a spectral resolution of
+110,000(seeTable1).
+Non-uniform illumination of the slit or output fiber at
+thespectrographentrancedecreasestheradial-velocityp re-
+cision.Indeed,variationsinseeing,focusandimageshape
+at the fiber entrance may induced non-uniform illumina-
+tioninside the pupilofthe spectrograph.Theoptical aber-
+rationsleadtovariationsinthecentroidsofthestellarli nes
+onthefocalplane.
+Optical fibers are used to lead the light from the tele-
+scope to the entrance of the spectrograph. They have the
+propertiestoscramblertheatmospherice ffectsandguiding
+and centering errors discussed previously. But the scram-
+blingabilityofonemultimodefiberisnotperfectasshown
+inFig.1.Withaninputo ffaxesimage,theazimuthalscram-
+bling is good whereas it remains some e ffects in the ra-
+dial one. This pattern observed is bigger in far field than
+in near field. Near field and far field are respectively de-
+fined as, the brightness distribution across the output face
+ofthefiber,and,astheangulardistributionoflightoffiber
+output beam. Therefore, changes in the input beam cause
+onlycircularsymmetricchangesinthefiberoutputpattern.
+Incontrast,imperfectradialscramblingallowssmallzona l
+errorsto remain.
+HARPS and the SOPHIE HR modeare equippedwith
+optical double scramblers. It is used to increase and im-
+prove the uniformity and stabilization of the illumination
+of the spectrograph entrance (refs. [4] and [6]). The near
+field of a circular fiber is observed to be better scramble
+than the far field. In spite of only one fiber guiding the
+light from the telescope to the instrument, double scram-
+bler is composedof two fibers coupledwith two doublets.
+The system is designed to inverse near field and far field
+in order to induce a better radial scrambling, although it
+causesfluxlosses(ref.[1]).
+Table 1. Comparison of parameters of fiber-fed spectrographs
+SOPHIEandHARPS .
+Instrument Doppler prec. 1 pixel Resol. ∆λ/λ
+HARPS/lessorequalslant1ms−1800 ms−1110,000
+SOPHIE(HR) ∼4ms−11400 ms−175,000
+SOPHIE(HE) ∼10ms−11400 ms−140,000
+Fig. 1.Illumination at the output of one cylindrical optical fiber
+in the near and the far field. Sources are o ffaxes. A high degree
+ofazimuthalscramblingisobserved.Incontrast,itremain ssome
+effects inthe radial one.
+3 RVeffectsof fiberimperfect scrambling
+3.1 Centering/Guiding onthe fiber entrance
+The degree of radial scrambling describes the stability of
+theoutputbeamastheinputimageismovedfromthecen-
+ter to the edge of the fiber. It is possible to detect in RV
+some variations due to the insu fficiency of radial scram-
+bling. The light from a stable star is movedwith the guid-
+ing/centeringsystemfromanedgetotheotherofthefiber.
+We compute the RV with the data reduction pipeline for
+different positions of the star. The test was done for the
+two modes of SOPHIE and for HARPS. The results are
+computedinTable2.
+Because the scrambling of the fiber is not perfect, the
+movementofthe inputimagecorrespondstoa shiftonthe
+Table 2. Variation observed in RV as the input image (star) is
+moved from the center to the edge of the fiber. HARPS and SO-
+PHIEwithHigh-Resolutionfiberusedadoublescramblerinst ead
+of the High-Efficiency mode of SOPHIE.
+Instrument fib.diam. RVe ffect [ms−1]
+HARPS 1” ∼3
+SOPHIE(HR) 3” ∼13
+SOPHIE(HE) 3” ∼36Newtechnologiesforprobingthe diversityofbrowndwarfsa ndexoplanets
+Fig. 2.Schema illustrating how the vignetting of the telescope
+is translated at the output of the fiber HE in the far-field, i.e . at
+theentranceofthespectrograph.Forexternaloccultation ,theRV
+decrease whereas, for internal occultationthe RV increase .
+detector.Weobservedfirstthat,wideristhefiber,moreim-
+portant is the RV variations. Moreover,the value of meter
+per second per pixel vary as a function of the resolution
+power.Movingtheinputimagefromthecentertotheedge
+isthenexpectedtohavealowere ffectforahigherspectral
+resolution.Inaddition,weexpectthatthedoublescramble r
+improvetheradialscrambling,i.e. decreasetheRV e ffect.
+3.2 Vignetting telescope
+Thepupilofthetelescope,i.e.theentranceofthetelescop e
+orthefarfieldoftheinputlightisknowntobemorestable
+than the telescope image. We would like to test the e ffect
+in RV of variations of the far field. For that, we vignette
+the telescope onday skywith the dome.We computedthe
+RV with the pipeline SOPHIE, estimating that the varia-
+tion of Barycentric Earth RV is not significant during our
+measurements.We madethistest with the HE modein or-
+der that the variation of the far-field of the telescope are
+projected on the grating (i.e. without double scrambler).
+WeremarkedthatthewayoftheRVvariationsdependson
+exteriororinteriorvignetting(Fig.2).
+3.3 Optical pathsimulations
+WesimulatedtheopticalpathintheSOPHIEspectrograph.
+We observed a displacement of the spectrum in function
+of wavelength on the focal plane when the input beam at
+the entrance of the spectrograph is center-illuminated or
+center-darked,correspondingrespectivelywhenvignetti ng
+external and internal part of the pupil in HE mode (cf.
+Sect.3.2). Variation of the slit or the optical fiber illumi-
+nationaredirectlytranslatedonthespectruminFig.3.We
+observed that the shift is more significant for an external
+occultation. Because the e ffect is not symmetric along an
+order and not monitored by the calibration lamp, the final
+computed RV vary. So, variations of the far-field pattern
+projectedontothegratingcanintroduceradial-velocitys hifts
+at thedetectorloweringthefinalprecisioninRV.
+Fig. 3.Displacement of the position of a nine wavelength as a
+function of a internal (green) or an external (red) occultat ion of
+the entrance of the spectrograph onthe CCDdetector.
+Fig.4.Whentheseeingisgood,thetelescopeimageislesswide.
+With the double scrambler in HR mode, near field becomes far-
+fieldandisprojectedonthecollimatorattheentranceofthe spec-
+trograph. Thepatterniscomparable tothat observed inHEwh en
+vignetting telescope (Fig.2)
+3.4 RVeffects due toseeingvariations
+Variations of the far-field of the input beam in HE mode
+is equivalent to a variation of the telescope image in HR
+mode,duetothedoublescrambler.Variationsinseeingin-
+duce a variation of the input image as illustrate in Fig. 4.
+Theinputbeamattheentranceofthespectrographiscenter-
+illuminated(externaloccultation)whenatargetisobserv ed
+with a good seeing in HR mode. The previous tests and
+simulations explain a systematic e ffect observedwith SO-
+PHIE named as ”seeing e ffect”. For a sample of stars ob-
+servedathighsignaltonoiseratio(SNR)inHRmode,we
+observedadecreaseinRVcorrelatedwithanestimationof
+seeing. The value of the seeing is not monitored with SO-
+PHIE.We estimatedit with calculatingthe relativefluxby
+unitofexposuretime :
+S=SNR2
+Texp10−MV/2.5(1)
+with SNR, the signal to noise ratio of the spectra, T expthe
+time exposure of the measurement, M Vthe visual magni-
+tude of the target. The seeing decrease when the seeing
+estimation Sincrease. With a lower seeing, the input im-
+age is smaller than the fiber width, and as shown in theEPJWeb ofConferences
+Seeing Estimation [arb.unit]
+Fig. 5.Radial-velocity measurements as a function of a seeing
+index (cf. text). The seeing decrease when the seeing estima tion
+Sincrease. As the seeingis lower,the RV decreases.
+simulations,inducedavariationofthefarfieldinputinthe
+spectrograph,due to the double scrambler. We expect and
+observed a decrease of RV when the value of the seeing
+decreaseasshowninFig.5.
+Moreover,whentheinputimageissmallerthanthedi-
+ameter of the fiber, we are more sensible to e ffect of guid-
+ing and centering system. The guiding and centering sys-
+tem may inducedvariationof the position of the inputim-
+ageontheslit ortheentranceofthefiber,andsointroduce
+RV variations. These RV variation are quite random, and
+addsomenoise.
+Whereas the seeing e ffect is directly related to the op-
+ticalpathinthespectrograph,it maybequantified.We are
+modeling this variations with our data in order to remove
+thisnoisewitha softwaretool.
+We note that the e ffect on HARPS is expected to be
+lower. The 1” width of the fibers is smaller than the SO-
+PHIEandroughlyequivalenttothetypicalseeingatLaSilla .
+Furthemore the quality image provided by the 3.6-m tele-
+scopeiscloseto0.7arcsec.Hence,HARPSislesssensible
+tothevariationsofseeing.Moreover,theresolutionpower
+of the instrument is better and then the incomplete radial
+scramblinghasalowerimpactinRVasshowninSect.3.1.
+FurthermoretheopticalaberrationsonHARPSaresmaller
+thaninSOPHIE.
+4 Perspectivesandconclusions
+The CODEX experiment calls for a Doppler precision as
+low as 1 cms−1. This work shows that the future instru-
+ments need for better scrambling to reach this accuracy.
+New types of fibers for feeding the telescope or new dou-
+ble scrambler are needed to improve the precision and to
+minimize the noise due to variationsof the illuminationat
+the entrance of the spectrograph. To remove the sphericalsymmetry of cylindrical fibers, octagonal and square sec-
+tionopticalfibersaretheninstudyatGeneva,OHP-LAM,
+and ESO Garching. As first results, very good scrambling
+properties are observed in the near field, whereas strange
+patternsinfarfieldarenotwellunderstoodatthemoment.
+Itneedsfutureteststoimprovethemeasurements,tostudy
+its best used, on all the path or in a double scrambling
+mode. SOPHIE is then going to be a bench test for the
+futureinstruments.
+We note that additional solution is being installed on
+HARPS toreducetheguidingandcenteringlimitations.A
+tip-tilt mirror generate small-amplitude and high frequen -
+ciesvariationsin ordertoinducea spreadofthelight.
+AnewguidingcameraisnowinoperationonSOPHIE
+and allows a better guiding and centering and also allows
+to monitorthe exactseeing.
+References
+1. Bouchy&ConnesA&A,1999,Volume136,193
+2. Bouchy, F., Isambert, J., Lovis, C. Boisse, I. et al. EAS
+PublicationsSeries , Volume37,2009,pp.247-253
+3. Bouchy,F.,H´ ebrard,G.,Udry,S.,Delfosse,X.,Boisse,
+I.etal. A&A,Volume505,2009,pp.853
+4. Brown,T.M.1990,ASPC, 8,335B
+5. Lovis,C.& Pepe,F. 2007,A&A,468,1115L
+6. Hunter,T.D.&Ramsey,L.W. 1992,PASP, 104,1244
+7. Mayor, M., Udry, S., Lovis, C. et al. 2009, A&A, 493,
+639
+8. Pasquini, L., Manescau, A., Avila, G. et al. 2009,
+svlt.conf,411
+9. Pasquini, L., Avila, G., Dekker, H. et al. 2008,
+SPIE.7014E,51P
+10. Pepe,F. &Lovis,C., Phys.Scr. T130,(2008)014007
+11. Perruchot, S., Kohler, D., Bouchy, F. et al. 2008,
+SPIE.7014E,17P
+12. Wildi,F., Pepe,F., Lovis,C., Chazelas,B. et al. 2009,
+SPIE.7440E,19W
\ No newline at end of file
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+ 1 On Unconventional Electron Pairing In a Periodic Potential  Valentin Voroshilov, Physics Department, Boston University, Boston, MA, 02215, USA, valbu@bu.edu  On the assumption that two electrons with the same group velocity effectively attract each other a simple model Hamiltonian is proposed to question the existence of unconventional electron pairs formed by electrons in a strong periodic potential.  71.10.Li  Despite intensive experimental and theoretical research the mechanism responsible for high temperature superconductivity1 is yet not clear. The goal of this letter is to offer some speculations on the matter as a step toward the future deeper theoretical analysis.  Among different theoretical approaches there is a view that formation of electron pairs is a natural property of interacting electrons immersed in a strong periodic potential2,3. In particular, as one of the possibilities a hole-electron paring mechanism is suggested, where an electron is paired with a hole traveling in space in opposite direction4. As a possible extension of that hole-electron paring mechanism we assume that electron pairs are formed by electrons traveling with the same group velocity (at this point we do not impose limits on the energy of interacting electrons).  Based on the assumption above, we write a model Hamiltonian in the following form (Eq. 1, 2): € H=(εp−µ)apσ+pσ∑apσ+U,εp=−t(cospx+cospy).                          (1) In Eq. (1) the first term represents kinetic energy of electrons in a 2D square lattice in tight binding approximation; t > 0 is a hopping integral. Here and below we assume the lattice constant, the Plank’s constant, and the Boltzmann’s constant are equal to unity. For all electrons momenta lie in the first Brillouin zone € −π<px,y<π; µ is a chemical potential introduced in the Hamiltonian as a Lagrange multiplier; € apσ+ and € apσ are creation and annihilation operators for an electron with momentum € p=(px,py) and spin component σ (€ σ=+,−). Term U in Eq. (1) describes an effective attraction between electrons with the same group velocity.   2 For an electron with momentum € p=(px,py) and kinetic energy € εp=−t(cospx+cospy) there might exists three more electrons with the same group velocity € vp=∇εp, namely, € p/=(px,π⋅sgn(py)−py), € p//=(π⋅sgn(px)−px,py), € p*=(π⋅sgn(px)−px,π⋅sgn(py)−py) (€ sgn(px,y) is the signum function). Thus, in general, there might be formed a cluster of four electrons with the same group velocity. In this letter we limit ourselves to the simplest interaction term (Eq. 2) € U=−WNV+V,                     € V+=ap↑+ap*↓+p∑.                                         (2) In Eq. (2) N is the number of sites, –W < 0 is a coupling constant for paired electrons, and the summation runs over the first Brillouin zone. Let us introduce a Bogolyubov canonical transformation (Eq. 3) € ap↑=upbp↑−vpbp*↓+,   € ap↓=upbp↓+vpbp*↑+,    € up=up*>0,   € vp=vp*>0,   € up2+vp2=1.  (3) With the means of transformation (3) the standard procedure5 gives (Eq. 4) for the energy of the system as a function of the occupation numbers € H|E>=E|E>,bpσ+bpσ|E>=npσ|E>,npσ=0,1,                                        (4a) € E=−2µvp2p∑+(εp−µ(up2−vp2))npσp,σ∑−WNupvp(1−p∑np+−np−)⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2.                           (4b) Equation (5) € −2µupvp=(up2−vp2)WNupvpp∑(1−<np+>−<np−>)                                                (5) insures minimization of the free energy of the system. This equation defines parameters up and vp of canonical transformation (3). Chemical potential can be related to the number of electrons as € Ne=−∂<E>∂µ which gives (Eq. 6) € Ne=2vp2p∑+(up2−vp2)(<np+>+<np->)p∑.                                                   (6)   3 After elementary transformations we obtain (Eq. 7) € up=u=121−2µΔ,               € vp=v=121+2µΔ,         € Δ=WN(1−<np+>−<np−>)p∑      (7) Solution (7) exist only when € 2|µ||Δ|≤1 (this is the only regime to be discussed below). Combining Eq. (6) and (7) we find the chemical potential (Eq. 8) € µ=Ne−NN*W2.                                                        (8) For elementary excitations we obtain (Eq. 9) € np=<bpσ+bpσ>=[1+exp(ωpT)]−1,                                               (9a) € ωp=εp−µ(u2−v2)+2WNu2v2(1−<np+>−<np−>)p∑=εp+Δ2.                         (9) When for all momenta € εp+W2>0 (i.e. when € W4t>1, the only case we consider here) there are no excitations in the ground state (€ np≡0 at T = 0); expression (9) shows the existence of a gap in the excitation spectrum, and pairing function € <ap++ap*++>=−uv≠0 shows the existence of electron pairs (even at a half filling when Ne = N, and µ = 0). From the point of view of the BCS theory6, one would conclude on the existence of superconductivity. However, at T = 0 electron density € <apσ+apσ>=Ne2N is a constant, which would mean that the Fermi see fills up the whole 1st Brillouin zone for all numbers of electrons. Correcting interaction term (2) (for example, by limiting attractive interactions being only between electrons with energy above the Fermi level), or/and including in the analysis four-electron clusters might lead to a physically more adequate picture. There is also an interesting outlook, related to the notion that HTSC are doped Mott insulators7. The linkage between a parent insulating phase and a superconductive one is yet elusive and leaves a room for speculations.     4 The most general form of a Hamiltonian for a system of interacting non-relativistic electrons in an external field is given by a standard first quantization expression (Eq. 10)   € H= p α22mα=1Ne∑+U( r α)+12V(β=1β≠αNe∑α=1Ne∑ r α, r β)α=1Ne∑,                   € V( r α, r β)=V( r β, r α).                 (10)  This expression can be rewritten in a different form (Eq. 11) € H=H0+Hint,            € H0= p α22mα=1Ne∑+Ueff( r α)α=1Ne∑,              € Hint=12Veff(β=1β≠αNe∑α=1Ne∑ r α, r β),         (11a)   € Ueff( r α)=U( r α)+W( r α),                       € Veff( r α, r β)=V( r α, r β)−W( r α)+W( r β)Ne−1.             (11b) For any function   € W( r α) Hamiltonian (11) is exactly equal to Hamiltonian (10). However, if one intends on using a perturbation theory, the difference may arise from different sets of eigenfunctions for one-particle operators using to write the Hamiltonian in a second quantization form (and the use of   € Veff( r α, r β) can provide a better convergence of the perturbation series). In a mean-field approach function  € W( r α) can be seen as an effective energy of a single electron in the filed of all other electrons.  When electrons are localized because of strong interactions, the inclusion of function   € W( r α) might make a significant difference. If function  € W( r α) has the same symmetry as lattice potential   € U( r α) has, the effective potential   € Ueff( r α) should lead to the band structure similar to the one provided by a “naked” lattice potential   € U( r α). Although, the parameters of the system can be modulated, for example, “potential”   € W( r α) can increase the potential barriers of potential   € U( r α), which effectively changes hopping constant t (i.e. the width of the band). If the symmetry of term   € W( r α) is different from the symmetry of lattice potential   € U( r α)(i.e. the spatial symmetry of the electron system is different from the spatial symmetry of the lattice), that difference might lead to important changes in the structure of energy bands.  In particular, if the effective field of electrons, which is also strong, is of a different symmetry than the lattice field, in result, even if formally the lattice is at a half filling, effectively electrons can occupy “pseudo sites” by two on each such a site (a “pseudo lattice”). Such a dynamic symmetry braking may lead to an insulating phase where the band theory predicts a metallic one.  5 Under an assumption that copper-oxides might experience such a symmetry breaking, one could argue that in model (1) at a formal half-filling when Ne = N, the effective number of sites Neff is to be equal to a half the number of actual sites N; and, hence, in all expressions (2) – (9) N should be replaced with € Neff=Ne2. This leads (at T = 0) to€ µ=Δ2=W2, the energy gap still exists, but the pairing function € <ap++ap*++>=0 (since u = 0). This might be seen as a sign for the existence of a pseudo-gap.  In the end we briefly examine the existence of pairs when attractive interactions are limited between only some of the electrons.  Let us assume that at T = 0 electron distribution in momentum space is of simplest form (Eq. 12) € <apσ+apσ>=1,εp<εF0,εp>εF⎧ ⎨ ⎩ ,               -2t  < εF  <  2t.                                       (12)  We also assume that in term (2) two electrons experience attraction only when they both have energy above Fermi level εF. Under these assumptions Fermi level has to be negative εF < 0; and we have to modify the Hamiltonian and write it in the following form (Eq. 13)  € H=(εp−µ)apσ+pσεp<εF−εF<εp∑apσ+(εp−µ)apσ+pσεF<εp<−εF∑apσ−WNV+V,V+=ap↑+ap*↓+pεF<εp<−εF∑.          (13)  Canonical transformation (3) remains applied to the operators with momenta within region εF < εp  <  -εF, and outside this region we can set € apσ=bpσ. Calculations show that solution (7) remains correct for u and v, but the summation in the expression for Δ should be limited by momenta for which εF < εp  <  -εF. Expression (8) for the chemical potential stays but now it is also equal to the Fermi level, so € εF=µ=Ne−NN*W2; which means Ne < N, and € N−NeN*W4t<1.     6 Expression (9) is to be replaced by 
+€ np=<bpσ+bpσ>=[1+exp(εp−µT)]−1,εp<εF,−εF<εp[1+exp(εp+Δ2T)]−1,εp<εp<−εF⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ .                               (14) Since by the assumptions there are no excitations in the ground state, hence € −2t+Δ(T=0)2>0, which provides an additional condition on the parameters of the system; € W4tN1pεF<εp<−εF∑>1.   When € −µ<Δ(T=0)2 the excitation spectrum has an energy gap. What differs this model from a BCS picture is that parameter Δ is never equal to zero (for all the values of the parameters when solution (14) exists and consistent with condition (12)). However, when the temperature of the system is such that€ Δ=−µ2 solution (7) gives u = 1, v = 0, and pairing function becomes zero € <ap++ap*++>=0. What kind of a transition is happening at that temperature is a matter of a subsequent study.  We see that a model based on Hamiltonian (1), (2), or (13) shows unusual and interesting properties. At this point there is no solid theoretical basis to support the speculations on relationships between the model and properties of HTSC. The future work will show if such a basis exists. Appendix Let us write Schrödinger equation for two non-relativistic electrons with anti-parallel spins (all the constants a set to unity, terms are rearranged)   € ∂2∂ r 12+∂2∂ r 22+2(E−V( r 1)−V( r 2))⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Ψ( r 1, r 2)=2U( r 1, r 2)Ψ( r 1, r 2), where   € U( r 2, r 2) describes the Coulomb’s interaction and   € V( r )describes an external field. The standard approach is to seek the solution with the means of a Green’s function   € Ψ( r 1, r 2)=Φ( r 1, r 2)+2d∫∫ r 1'd r 2'G( r 1, r 2; r 1', r 2')U( r 1', r 2')Ψ( r 1', r 2')   7 where “free” solution   € Φ( r 1, r 2) and Green’s function   € G( r 1, r 2; r 1', r 2') satisfy equations   € ∂2∂ r 12+∂2∂ r 22+2(E−V( r 1)−V( r 2))⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Φ( r 1, r 2)=0   € ∂2∂ r 12+∂2∂ r 22+2(E−V( r 1)−V( r 2))⎛ ⎝ ⎜ ⎞ ⎠ ⎟ G( r 1, r 2; r 1', r 2')=δ( r 1− r 1')δ( r 2− r 2'). “Free” solution   € Φ( r 1, r 2)can be represented as   € Φ( r 1, r 2)=ϕ p ( r 1)ϕ q ( r 2)+ϕ q ( r 1)ϕ p ( r 2) where   € ∂2∂ r +2(E p −V( r ))⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ϕ p ( r )=0    and      € E=E p +E q . Functions   € ϕ p ( r ) can be normalized as   € ϕ p *( r )ϕ p ( r ')=δ( r − r ') p ∑. In that case the Green’s function can be written as   € G( r 1, r 2; r 1', r 2')=12ϕ p 1( r 1)ϕ p 1*( r 1')ϕ p 2( r 2)ϕ p 2*( r 2')E p +E q −E p 1−E p 2 p 1, p 2∑. For a strong periodic external field the approximate solution (for example tight binding approximation) gives   € E p ≈α−t(cos(px)+cos(py))=α+ε p  (we consider only 2D square lattice). In that case the Green’s function can be rewritten as   € G( r 1, r 2; r 1', r 2')=12ϕ p 1( r 1)ϕ p 1*( r 1')ϕ p 2( r 2)ϕ p 2*( r 2')ε p +ε q −ε p 1−ε p 2 p 1, p 2∑. It is clear that for this solution the Green’s function exhibits irregular behavior (which is unique for “energy spectrum”   € ε p =−t(cos(px)+cos(py))), namely, there are momenta   €  p  and   €  q  for which   € ε p +ε q =0. It happens, in particular, when   €  q = p *€ =(π⋅sgn(px)−px,π⋅sgn(py)−py), i.e. when two “free” electrons have the same group velocity. Further investigation is needed to examine connection between this irregularity and electron pairing in HTSC.  8 It might be plausible to offer a “naïve” picture for two electrons paired in a strong periodic potential. When two electrons are localized at neighboring sites (having opposite spins), their exchange interaction (which takes much longer time than for almost free electrons) results in strong correlations between the electrons, and the paired electrons have a tendency traveling together by hopping simultaneously unto neighboring sites in the same direction. This correlation might make electrons with the same group velocity being special, and might have a significant influence on properties of the system at the optimal electron density (i.e. enough electrons to pair, and enough empty sites for hopping).                                                 1  J.G. Bednorz and K.A. Muller, Z. Physik B 64, 189 (1986).  2  J. Bergou, S. Varro, and M. V. Fedorov, J. Phys A: Math. Gen. 14, 2305-2315 (1981).  3  S.M. Mahajan and A. Thyagaraja, DOE/ET-53088-642, IFSR # 642, January (1994).  4  A. Brinkman, H. Hilgenkamp, Physica C 422, 71-75 (2005); doi:10.1016/j.physc.2005.03.011. 5 N. N. Bogoliubov; V. V. Tolmachev, D. V. Shirkov (1958). A New Method in the Theory of Superconductivity. Moscow: Academy of Sciences Press. — New York, Consultants Bureau, 1959. 6 J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Microscopic Theory of Superconductivity", Phys. Rev. 106, 162 - 164 (1957). 7 Patrick A. Lee, Naoto Nagaosa, Xiao-Gang Wen; http://arxiv.org/abs/cond-mat/0410445. 
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+arXiv:1001.0797v1  [math.CO]  6 Jan 2010ON THE EXISTENCE PROBLEM OF THE TOTAL DOMINATION
+VERTEX CRITICAL GRAPHS
+MOO YOUNG SOHN, DONGSEOK KIM, YOUNG SOO KWON, AND JAEUN LEE
+Abstract. The existence problem of the total domination vertex critical grap hs has been
+studied in a series of articles. The aim of the present article is twofold . First, we settle the
+existence problem with respect to the parities of the total dominat ion number mand the
+maximum degree ∆ : for even mexceptm= 4, there is no m-γt-critical graph regardless
+of the parity of ∆; for m= 4 or odd m≥3 and for even ∆, an m-γt-critical graph exists if
+and only if ∆ ≥2⌊m−1
+2⌋; form= 4 or odd m≥3 and for odd ∆, if ∆ ≥2⌊m−1
+2⌋+7, then
+m-γt-critical graphs exist, if ∆ <2⌊m−1
+2⌋, thenm-γt-critical graphs do not exist. The only
+remaining open cases are ∆ = 2 ⌊m−1
+2⌋+k,k= 1,3,5. Second, we study these remaining
+open cases when m= 4 or odd m≥9. As the previously known result for m= 3 [1,2],
+we also show that for ∆( G) = 3,5,7, there is no 4- γt-critical graph of order ∆( G) + 4.
+On the contrary, it is shown that for odd m≥9 there exists an m-γt-critical graph for all
+∆≥m−1.
+1.Introduction
+A domination and its variations in graphtheory have been studied wide ly and extensively
+because of its rich applications [2,6,8,11]. Two books by Haynes, H edetniemi and Slater
+provide a well written survey on this subject [4,5]. We refer to [4] fo r notation and general
+terminology.
+LetG= (V(G),E(G)) be a simple graph of order n(G). The minimum degree and the
+maximum degree of a graph Gare denoted by δ(G) and ∆( G), respectively. A subset
+S⊆Visa dominating set ofGif every vertex not in Sis adjacent to a vertex in S. The
+domination number ofG, denoted by γ(G), is the minimum cardinality of dominating sets.
+A subset S⊆Visa total dominating set ofGif every vertex of Gis adjacent to a vertex
+inS. Thetotal domination number ofG, denoted by γt(G), is the minimum cardinality of
+total dominating sets. A total dominating set of cardinality γt(G) is called a γt(G)-set.
+Goddard et al. introduced the concept of total domination critical graphs [2]. A graph G
+with no isolated vertex is total domination vertex critical if for any vertex vofGthat is not
+adjacent to a leaf, a vertex of degree one, the total domination n umber of G−vis less than
+the total domination number of G. Such a graph is said to be γt-criticalorm-γt-criticalif
+its total domination number is m. It is well known that the order of m-γt-critical graph G
+is at least ∆( G) +m. So, they suggested the following classification problem of the tota l
+domination critical graphs.
+2000Mathematics Subject Classification. Primary 05C50.
+Key words and phrases. total domination numbers, total domination vertex critical graph s, maximal
+degree.
+This work was supported by Basic Science Research Program throu gh the National Research Foundation
+of Korea(NRF) grant funded by the Korea government(MEST)(2 009-0073714).
+12 MOO YOUNG SOHN, DONGSEOK KIM, YOUNG SOO KWON, AND JAEUN LEE
+Problem 1 ( [2]).Characterize m-γt-critical graphs Gwith order ∆(G)+m.
+There have been a series of articles regarding this problem. Mojdeh and Rad found 3- γt-
+critical graphs of order 3+∆( G) for any even ∆( G) and showed that there is no 3- γt-critical
+graphGof order 3+∆( G) for ∆(G) = 3,5 [11]. In [1], Chen and Sohn proved that there is
+no 3-γt-critical graph of order ∆( G)+3 with ∆( G) = 7 and δ(G)≥2. Furthermore, they
+gave a family of 3- γt-critical graphs of order ∆( G)+3 with odd ∆( G)≥9 andδ(G)≥2.
+Hassankhani and Rad proved that there is no 4- γt-critical graph of order ∆( G) + 4 with
+δ(G)≥2 for ∆(G) = 3,5 [3]. There have been several partial results on the existence
+problem of the total domination vertex critical graphs from differe nt point of views.
+The aim of the present article is twofold. First, we settle the existen ce problem with
+respect to the parities of the total domination number mand the maximum degree ∆ in
+Theorem 2.
+Theorem 2. If there exists an m-γt-critical graph of order ∆+mfor some ∆thenm= 4
+orm≥3is odd. Conversely, for any m= 4or oddm≥3,
+(1)if∆<2⌊m−1
+2⌋, then there exists no m-γt-critical graph of order ∆+m.
+(2)For any even ∆≥2⌊m−1
+2⌋, there exists an m-γt-critical graph of order ∆+m.
+(3)For any odd ∆≥2⌊m−1
+2⌋+7, there exists an m-γt-critical graphs of order ∆+m.
+Theorem 2 implies that the only remaining cases are ∆ = 2 ⌊m−1
+2⌋+k,k= 1,3,5.
+Second, we study these remaining open cases when m= 4 or odd m≥9. When m= 4, we
+show that there is a 4- γt-critical graph Gof order ∆( G) +4 with δ(G)≥2 if and only if
+∆(G) = 2,4,6,8 or ∆(G)≥9. For odd m≥9, it is shown that there exists an m-γt-critical
+graphGof order ∆( G)+mwithδ(G)≥2 if and only if ∆( G)≥m−1.
+The outline of this paper is as follows. In section 2, we review some defi nitions and
+previous results. In section 1, some properties of m-γt-critical graph of order ∆+ mwill be
+given. In section 4, we provide the proof of the Theorem 2. In sect ion 5, we deal with the
+remaining open cases for m= 4 and m≥9.
+2.Preliminaries
+Inthissection, wereviewsomedefinitionsandpreviousresults. The degree, neighborhood
+and closed neighborhood of a vertex vin a graph Gare denoted by d(v),N(v) andN[v] =
+N(v)∪ {v}, respectively. For a subset SofV, we set N(S) =/uniontext
+v∈SN(v) andN[S] =
+N(S)∪S. The graph induced by S⊆Vis denoted by G[S]. The cycle, path and complete
+graph on nvertices are denoted by Cn,PnandKn, respectively. A vertex of degree one is
+called aleaf. A vertex vofGis called a support vertex if it is adjacent to a leaf. Let S(G)
+be the set of all support vertices of G. Thecoronaof a graph H, denoted by cor(H), is the
+graph obtained from Hby adding a leaf adjacent to each vertex of H.
+For two graphs G1andG2and for two vertices v1∈V(G1) andv2∈V(G2), avertex
+amalgamation ofG1andG2with two vertices v1andv2is a graph whose vertex set is
+(V(G1)−v1)∪(V(G2)−v2)∪{v}and edge set is
+E(G1−v1)∪E(G2−v2)∪{vu|v1u∈E(G1)}∪{vw|v2w∈E(G2)}.
+The vertex amalgamation method is useful to construct a new γt-critical graph by the
+following proposition.TOTAL DOMINATION VERTEX CRITICAL GRAPHS 3
+Proposition 3 ( [2]).LetFandHbej-γt-critical and k-γt-critical graphs, respectively,
+with minimum degrees at least two and let Gbe a graph formed by identifying a vertex of
+Fwith a vertex of H. Ifγt(G) =j+k−1thenGis alsoγt-critical.
+Lemma 4. For anyi= 1,2, letGibe anmi-γt-critical graph Giof order∆(Gi)+miwith
+δ(Gi)≥2and letvi∈V(Gi)be a vertex of maximum degree in Gi. If each component of
+G[V(Gi)−N[vi]]is aP2then the vertex amalgamation GofG1andG2withv1andv2is an
+(m1+m2−1)-γt-critical graph of order ∆(G)+m1+m2−1, where∆(G) = ∆(G1)+∆(G2).
+Proof.Letvbe the vertex of Gwhose degree is ∆( G) = ∆(G1) +∆(G2), namely, vis an
+amalgamated vertex. For any u∈N(v), (V(G)−N[v])∪{u}is a total dominating set of G
+and whose cardinality is m1+m2−1. Hence γt(G)≤m1+m2−1. LetSbe aγt(G)-set of
+G. Suppose v∈S. Then,vis adjacent to a vertex u∈S−{v}. Without loss of generality,
+we may assume that u∈V(G1). Then, ( V(G1)∩(S−{v}))∪{v1}) is a total dominating
+set ofG1. Furthermore, for Sto dominate G2−N[v], we have |V(G2)∩(S−{v})| ≥m2−1.
+Hence,|S| ≥m1+m2−1, which means that γt(G) =m1+m2−1. By Proposition 3, Gis
+an (m1+m2−1)-γt-critical graph of order ∆( G)+m1+m2−1. /square
+The following two lemmas are known results in [2] which will be used in this p aper.
+Lemma 5 ( [2]).IfGis aγt-critical graph, then γt(G−v) =γt(G)−1for every v∈
+V−S(G). Furthermore, a γt(G−v)-set contains no neighbor of v.
+Lemma 6 ( [2]).If a graph Ghas nonadjacent vertices uandvsuch that v /∈S(G)and
+N(u)⊆N(v), thenGis notγt-critical.
+Mojdeh and Rad [11] found the following lemma about a total dominatio n vertex critical
+graphGof order ∆( G)+γt(G) withδ(G)≥2.
+Lemma 7 ( [11]).There is no 3-γt-critical graph Gof order ∆(G)+3with∆(G) = 3,5
+andδ(G)≥2.
+3.Some properties of γt-critical graph Gwithγt(G) =n−∆(G)
+In this section, we find some properties of γt-critical graph Gwithγt(G) =n−∆(G).
+Throughout the section, we assume the following notation for γt-critical graph Gwith
+γt(G) =n−∆(G) andδ(G)≥2 unless stated otherwise. Let vbe a vertex whose degree is
+the maximum degree ∆( G). SinceGisγt-critical, it follows that γt(G−v) =γt(G)−1 =
+n−∆(G)−1. LetSbe aγt-set ofG−v. Then, S=V(G)−N[v] by Lemma 5. Let
+H1,H2,···,Htbe the components of G[S] and let V(Hi) =Sifor alli= 1,2,...,t. We
+find the following two lemmas regarding the γt-critical graph Gwithγt(G) =n−∆(G) and
+δ(G)≥2.
+Lemma 8. Everyγt-critical graph Gwithγt(G) =n−∆(G)andδ(G)≥2is connected.
+Proof.Suppose that Gis not connected. Then, at least one of H1,H2,···,Htis also a
+connected component of G, sayHiissuch acomponent. Since δ(G)≥2,|V(Hi)|=|Si| ≥3.
+Choose a spanning tree TofHiand one end vertex uofT. Then, Si−uis a total
+dominating set of Hiand furthermore S−uis a total dominating set of G−v, which is a
+contradiction. /square4 MOO YOUNG SOHN, DONGSEOK KIM, YOUNG SOO KWON, AND JAEUN LEE
+Lemma 9. LetGbe aγt-critical graph with γt(G) =n−∆(G)andδ(G)≥2. Then,
+(1)Hiis aP2or aP3fori= 1,2,···,t.
+(2)IfG[S]containsa P3component, then G[S] =P3. Furthermore, for the P3=u1u2u3,
+N(u2)∩N(v) =∅andN(v)is a disjoint union of nonempty sets N(u1)−u2and
+N(u3)−u2.
+(3)IfHiis aP2for alli= 1,2,...,t,i.e., H i=uiwi, then for any u∈S,N(u)∩N(v)/n⌉}ationslash=
+∅andN(v)is a disjoint union of N(u1)−w1,N(w1)−u1,...,N(ut)−wt,N(wt)−ut.
+Proof.(1)First, we aimtoshowthat∆( Hi)≤2fori= 1,2,···,t. Supposethat∆( Hj)≥3
+for some 1 ≤j≤t. Letube a vertex of Hjwhose degree in Hjis at least 3. Choose a
+spanning tree TofHjcontaining all edges incident to u. Then,Thas at least three leaves.
+Letu1,u2,u3be three leaves in T. For any x∈N(v), letS′= (S− {u2,u3})∪ {v,x}.
+Then,S′is a total dominating set of Gand hence γt(G)≤ |S′|=|S|=γt(G)−1,which is
+a contradiction. Therefore, ∆( Hi)≤2 fori= 1,2,···,t. It implies that Hiis a path or a
+cycle for i= 1,2,···,t.
+Suppose that there exists jsuch that Hjis a cycle u1u2···uku1fork≥3. Then, there is
+uℓsuch that N(uℓ)∩N(v)/n⌉}ationslash=∅. Without loss of generality, we assume N(u1)∩N(v)/n⌉}ationslash=∅and
+pick a vertex x∈N(u1)∩N(v). Then, S′′= (S−{u2,u3})∪{v,x}is a total dominating
+set ofG,which is a contradiction. So, for all i= 1,2,...,t,Hiis a path.
+Suppose that there exists a path Hi=u1u2···ukfork≥4. Then, ( S−{u1,uk})∪{v,x}
+for some x∈N(v) is a total dominating set of G,which is a contradiction. Therefore, Hi
+is aP2or aP3for alli= 1,2,···,t.
+(2) LetG[S] contains a P3component, say u1u2u3. IfG[S] contains another component
+w1w2w3which is isomorphic to P3, then for some x∈N(v), (S− {u3,w3})∪ {v,x}is a
+total dominating set of G, which is a contradiction. Next if we suppose G[S] contains a
+P3and at least one P2, sayw1w2. Then,N(v)∩N(w1)/n⌉}ationslash=∅becauseδ(G)≥2. For some
+x∈N(v)∩N(w1), (S− {u3,w2})∪ {v,x}is a total dominating set of G, it leads us a
+contradiction. Therefore, G[S] =P3=u1u2u3.
+Sinceδ(G)≥2, (N(ui)−u2)∩N(v)/n⌉}ationslash=∅for anyi= 1 or 3. If N(u2)∩N(v)/n⌉}ationslash=∅then for
+anyx∈N(u2)∩N(v),{v,x,u 2}is a total dominating set of G, which is a contradiction.
+Hence,N(v) is a disjoint union of nonempty sets N(u1)−u2andN(u3)−u2.
+(3) Since δ(G)≥2,N(u)∩N(v)/n⌉}ationslash=∅for anyu∈S. Furthermore, for any x∈N(v),
+N(x)∩S/n⌉}ationslash=∅because Sis a total dominating set of G−v. We want to show that
+|N(x)∩S|= 1 for any x∈N(v). Suppose that there exists an x∈N(v) such that
+ui,wi∈N(x) for some i= 1,2,...,t. Then, S′= (S− {ui,wi})∪ {v,x}is a total
+dominating set of G, which is a contradiction.
+For the next case, suppose that there exists an x∈N(v) such that ui,uj∈N(x) for
+some different i,j. Choose yi∈N(v)∩N(wi) andyj∈N(v)∩N(wj). Then, one can easily
+check that ( S− {ui,uj,wi,wj})∪ {v,x,yi,yj}is a total dominating set of G, which is a
+contradiction. Similarly, one can show that a contradiction occurs if |N(x)∩S| ≥2 for some
+x∈N(v). It implies that N(v) is a disjoint union of N(u1)−w1,N(w1)−u1,...,N(ut)−
+wt,N(wt)−ut. /square
+These results can be summarized to obtain general figures of γt-critical graph Gwith
+γt(G) =n−∆(G) andδ(G)≥2 as in Figure 1.TOTAL DOMINATION VERTEX CRITICAL GRAPHS 5
+......
+a).....................
+b)
+Figure 1. Figuresof γt-criticalgraph Gwithγt(G) =n−∆(G)andδ(G)≥2
+where all vertices in the boxes are adjacent to vertices connecte d to boxes by
+thick lines and there could be edges between vertices in different box es or in
+the same box. This convention will be used for other figures.
+4.Proof of Theorem 2
+In this section, we shall give a proof of Theorem 2. Suppose that Gis anm-γt-critical
+graph of order ∆( G) +m. Letvbe a vertex for which d(v) = ∆(G). By Lemma 9, each
+connected component of G[V(G)−N[v]] is aP2or aP3and if there exists a component P3
+thenG[V(G)−N[v]] =P3. Hence, m−1 =|G[V(G)−N[v]]|is 3 or even. It implies that
+m= 4 orm≥3 is odd.
+Ifm= 4, then G[V(G)−N[v]] is aP3and ∆(G)≥2 by Lemma 9 (2). If m≥3 is odd
+then each component of G[V(G)−N[v]] is aP2and ∆(G)≥m−1 by Lemma 9 (3). Hence,
+form= 4 or odd m≥3 if ∆<2⌊m−1
+2⌋, then there exists no m-γt-critical graph of order
+∆+m.
+Form= 4 and for even ∆ ≥2, letGbe a graph whose vertex set is {v} ∪(U∪W)∪
+{u1,u2,u3}with|U|=|W|= ∆/2 and whose edge set is composed of {vx,vy,u 1x,u3y|x∈
+U, y∈W} ∪ {u1u2,u2u3}as in Figure 1 a) and the subgraph induced by the vertices in
+between UandWisK∆/2,∆/2−E(M), where Mis an 1-factor of K∆/2,∆/2. Then, one can
+show that Gis a 4-γt-critical graph of order ∆( G)+4.
+For odd m≥3 and for even ∆ ≥m−1, letG1be a graph whose vertex set is
+{v1} ∪(U1∪W1)∪ {u1,w1}with|U1|=|W1|= (∆−m+ 3)/2 and whose edge set is
+composed of {v1x,v1y,u1x,w1y|x∈U1, y∈W1} ∪ {u1w1}and the subgraph induced by
+the vertices in between U1andW1isK(∆−m+3)/2,(∆−m+3)/2−E(M), where Mis an 1-
+factor of K(∆−m+3)/2,(∆−m+3)/2. Then, one can show that G1is a 3-γt-critical graph of order
+∆(G1) + 3 = ∆ −m+ 6. Note that C5is a 3-γt-critical graph of order 5. So, the vertex
+amalgamation GofG1and (m−3)/2 5-cycles with v1and any vertex in each ( m−3)/2
+5-cycles as in Figure 2 is an m-γt-critical graph of order ∆ −m+6+4·m−3
+2= ∆+mby
+Proposition 4. Hence, for m= 4 or odd m≥3 and for any even ∆ ≥2⌊m−1
+2⌋, there exists
+anm-γt-critical graph of order ∆( G)+m.6 MOO YOUNG SOHN, DONGSEOK KIM, YOUNG SOO KWON, AND JAEUN LEE
+.........
+Figure 2. Figures of m-γt-critical graph of order ∆ + m, where each box
+contains∆−m+3
+2vertices and the subgraph induced by the vertices in two
+boxes is K(∆−m+3)/2,(∆−m+3)/2−E(M), where Mis a 1-factor of
+K(∆−m+3)/2,(∆−m+3)/2.
+Now, we want to consider odd ∆. In the paper [11], Mojdeh and Rad s howed that there
+is no 3-γt-critical graph Gof order ∆( G) + 3 for ∆( G) = 3,5. In [1], Chen and Sohn
+proved that there is no 3- γt-critical graph of order ∆( G)+3 with ∆( G) = 7. Furthermore,
+they gave a family of 3- γt-critical graphs of order ∆( G) + 3 with ∆( G)≥9. For any
+oddm≥3 and for any odd ∆ ≥m+ 6, letG2be a 3-γt-critical graph of order 12 with
+∆(G2) = 9 and δ(G2)≥2 and let G3be an (m−2)-γt-critical graph of order ∆+ m−11
+with ∆(G3) = ∆−9≥m−3 andδ(G3)≥2. Letvi∈V(Gi) be a vertex such that
+d(vi) = ∆(Gi) for each i= 2,3. Then, the vertex amalgamation GofG2andG3with the
+verticesv2andv3is anm-γt-critical graph of order ∆+ mwith ∆(G) = ∆ and δ(G3)≥2
+by Proposition 4. In the next section, we construct a 4- γt-critical graph of order ∆+4 for
+any odd ∆ ≥9. Hence, for any m= 4 or odd m≥3 and for any odd ∆ ≥2⌊m−1
+2⌋+ 7,
+there exists an m-γt-critical graph of order ∆+ m.
+5.m= 4or odd m≥9
+The only remaining open cases are ∆ = 2 ⌈m−1
+3⌉+k,k= 1,3,5. In this section, we prove
+that there is no 4- γt-critical graph of order ∆ + 4 with δ(G)≥2 for ∆ = 3 ,5 or 7. For
+oddm≥9, it will be shown that there exists an m-γt-critical graph of order ∆+ mwith
+δ(G)≥2 for any odd ∆ ≥2⌈m−1
+3⌉+1.
+Theorem 10. There is no 4-γt-critical graph Gof order∆(G)+4with∆(G) = 3,5,7and
+δ(G)≥2.
+Proof.LetGbe aγt-critical graph with γt=n−∆(G) andδ(G)≥2. For any vertex
+u∈V(G), letSube aγt(G−u)-set. Choose v∈V(G) such that d(v) = ∆(G). Since
+n(G) = ∆(G) + 4, we can assume that V(G)−N[v] ={u,z,w}. SinceGis 4-γt-critical,
+by Lemma 5, it follows that Sv={u,z,w}andN(u)∪N(w)−{z}=N(v).Furthermore,
+N(u)∩N(w) ={z}. Otherwise, say x∈N(u)∩N(w), then{v,x,u}is aγt(G)-set, which
+is a contradiction.
+Suppose that |N(u)∩N(v)| ≥2 and|N(w)∩N(v)| ≥2. Then, for any x∈N(u)∩N(v),
+Sx={z,w,y}or{w,y,x 1}for some y∈N(w)∩N(v) andx1∈N(u)∩N(v). IfSx=
+{z,w,y}thenydominates all elements in N(u)∩N(v)−{x}and hence, {w,y,x 2}is also a
+total dominating set of G−xfor anyx2∈N(u)∩N(v)−{x}. Therefore, we assume that
+for anyt∈N(v),|St∩N(v)| ≥2 in the case |N(u)∩N(v)| ≥2 and|N(w)∩N(v)| ≥2.TOTAL DOMINATION VERTEX CRITICAL GRAPHS 7
+It divides into three cases depending on ∆( G).
+Case 1. ∆(G) = 3. We assume that N(u)∩N(v) ={x1}andN(w)∩N(v) ={y1,y2}.
+SinceG−y2is the cycle C6which has a total domination number 4. It is a contradiction.
+Case 2. ∆(G) = 5. It divides into two cases depending on |N(u)∩N(v)|.
+Case 2.1. We assume that N(u)∩N(v) ={x1}andN(w)∩N(v) ={y1,y2,y3,y4}. It is
+obvious that there is no edges x1yj(j= 1,2,3,4) inG. If we delete y1, there is the cycle
+C6inGwhich have a total domination number 4. It is a contradiction.
+Case 2.2. We assume that N(u)∩N(v) ={x1,x2}andN(w)∩N(v) ={y1,y2,y3}. It is
+obvious that for any i= 1,2,3,yicannot be adjacent to both x1andx2. Without loss
+of generality, we can assume that x1y1,x1y2/∈E(G). It implies that Sx2={x1,y3,w},
+x1y3∈E(G) andx2y3/∈E(G). By considering Sy3, one can show that x2y1∈E(G) or
+x2y2∈E(G). Letx2y1∈E(G). Then, Sy1={x1,y3,u}andy2y3∈E(G). Furthermore,
+Sy2={x2,y1,u}andy1y3∈E(G). In this case, {x,y3,u}is a total dominating set of G, a
+contradiction.
+Case 3. ∆(G) = 7. It divides into three cases depending on |N(u)∩N(v)|.
+Case 3.1. We assume that N(u)∩N(v) ={x1}andN(w)∩N(v) ={y1,y2,y3,y4,y5,y6}.
+It is obvious that Gis not 4-γt-critical graph.
+Case 3.2. We assume that N(u)∩N(v) ={x1,x2}andN(w)∩N(v) ={y1,y2,y3,y4,y5}.
+By the Pigeonhole Principle, we can assume that x1∈Sy1∩Sy2∩Sy3. Forj= 1,2,3,
+Syj∩{y4,y5} /n⌉}ationslash=∅. By the Pigeonhole Principle, we can assume that Sy1=Sy2={x1,y4,u}.
+Since{x1,y4,u}is aγt(G−y1)-set and x1y2/∈E(G),y2y4∈E(G).Therefore {x1,y4,u}is
+not aγt(G−y2)-set. It is a contradiction.
+Case 3.3. We assume that N(u)∩N(v) ={x1,x2,x3}andN(w)∩N(v) ={y1,y2,y3,y4}.
+It divides into four cases depending on existing edges between {x1,x2,x3}. Suppose that
+there is no edges in {x1,x2,x3}. Without loss of generality, let Sx1={x2,y1,w}. Then,
+x2y1,x3y1∈E(G) andx1y1/∈E(G). By the similar way, we can assume that x1y2,x3y2∈
+E(G) andx1y3,x2y3∈E(G). Furthermore, x2y2,x3y3/∈E(G). Considering Sy4, we may
+assume that Sy4={x1,y2,u}. Then,y1y2∈E(G) andx1y4,y2y4/∈E(G). Ifx2y4∈E(G)
+orx3y4∈E(G) then{x2,y1,u}or{x3,y1,u}is aγt(G)-set, which is a contradiction.
+Hence, we may assume that x2y4,x3y4/∈E(G). It implies that Sy2={x2,y3,u}and hence
+y3y4∈E(G). Let us consider Sy3. Sincey2y4/∈E(G),Sy3={x3,y1,u}. It implies that
+y1y4∈E(G). Then, {x2,y1,u}is aγt(G)-set, a contradiction.
+If there is one edges in {x1,x2,x3}, we assume that x2x3∈E(G). Without loss of
+generality, let Sx1={x2,y1,w}. Then, x2y1∈E(G) andx1y1/∈E(G). Also, without
+loss of generality, we may assume that Sx2={x1,y2,w}. It implies that x1y2∈E(G) and
+x3y2∈E(G). In this case, {x3,y2,w}is aγt(G)-set. It is a contradiction.
+If there is two or three edges in {x1,x2,x3}, one can similarly get a contradiction as the
+case that there is one edge in {x1,x2,x3}. /square
+Lemma 11. LetGbe a connected graph with ∆(G) = 9or∆(G)≥11. Then there are
+positive integers 3,2 =s1,s2=s3satisfying the following two conditions;
+(1) 3+2+ s2+s3= ∆(G)
+(2) 2 =s1≤s2=s3.8 MOO YOUNG SOHN, DONGSEOK KIM, YOUNG SOO KWON, AND JAEUN LEE
+Figure 3. Figures of 4- γt-critical graphs with ∆( G) = 9,11.
+Now we construct a family of 4- γt-critical graphs of order ∆( G) +4 with δ(G)≥2 and
+∆(G) = 9 or ∆( G)≥11.
+LetHbe a copy of the complement graph K3of the complete graph K3. LetV(H) =
+{x1,x2,x3}. LetHibe a graph with a vertex set V(Hi) ={yi1,yi2,···,yisi}fori= 1,2,3.
+Suppose that 2 = s1≤s2=s3. LetFbe the graph obtained from H1∪H2∪H3by adding
+edgesy1jy2k,y2ky3ℓ,y1jy3ℓforj= 1,2,k= 1,2,···,s2, andℓ= 1,2,···,s3,j/n⌉}ationslash=k,j/n⌉}ationslash=ℓ,
+k/n⌉}ationslash=ℓ. LetGbe the graph obtained from H∪Fand four new vertices v,u,z,w by adding
+edgesxiyjkfor 1≤i,j≤3,i/n⌉}ationslash=jand 1≤k≤sj, and then joining vto every vertex in
+H∪F, joining uto every vertex in Hand joining wto every vertex in F, and adding the
+edgesuzandzw. Then ∆( G) = 3+2+ s2+s3. Two figures in Figure 3 are examples of
+4-γt-critical graphs with ∆( G) = 9,11.
+Theorem 12. The graph Gin Figure 3 is 4-γt-critical.
+Proof.It is obvious that γt(G) = 4. So we only prove that Gisγt-critical graph. First,
+{v,y11,w},{v,x1,u},{v,x1,y11}and{u,w,z}is a total dominating set of G−u,G−w,
+G−zandG−vrespectively. For any vertex xi∈V(G),{w,yi1,z}is a total dominating set
+ofG−xi. For any vertex yjk∈V(G), It is easy to choose a total dominating set of G−yjk,
+In general, for any vertex a∈V(G),γt(G−a) = 3. So Gis a 4-γt-critical graph. /square
+By Theorems 2, 10 and 12, we have the following corollary.
+Corollary 13. There is a 4-γt-critical graph Gof order ∆(G) + 4withδ(G)≥2if and
+only if∆(G) = 2,4,6,8or∆(G)≥9.TOTAL DOMINATION VERTEX CRITICAL GRAPHS 9
+Figure 4. Figures of m γt-critical graph Gwithγt(G) =n−∆(G) and
+δ(G)≥2 form= 9.
+......
+Figure 5. Figures of m γt-critical graph Gwithγt(G) =n−∆(G) and
+δ(G)≥2 form≥9, where each box contains∆−7
+2vertices and the subgraph
+induced by the vertices in two boxes is K∆−7
+2,∆−7
+2−E(M), where Mis a
+1-factor of K∆−7
+2,∆−7
+2.
+From now on, we aim to consider an m-γt-critical graph Gof order ∆( G) +mwith
+δ(G)≥2 for any odd m≥9 and odd ∆( G)≥m.
+Theorem 14. For any odd m≥9and for any odd ∆≥m, there exists an m-γt-critical
+graphGof order∆+mwith∆(G) = ∆andδ(G)≥2.
+Proof.Assume that there exists a 9- γt-critical graph G1of order ∆ 1+9 with ∆( G1) = ∆ 1
+andδ(G1)≥2 for any odd ∆ 1≥9. Then for odd m≥9 and for any odd ∆ ≥m, one
+can construct m-γt-critical graph Gof order ∆+ mwith ∆(G) = ∆ and δ(G)≥2 using a
+vertex amalgamation of G1and several C5’s. Hence, it suffices to show that there exists a
+9-γt-critical graph Gof order ∆+9 with ∆( G) = ∆ and δ(G)≥2 for any odd ∆ ≥9.
+Forany∆ ≥9,letG= (V,E)beagraphwhosevertexsetis {v}∪/uniontext4
+i=1(Ui∪Wi∪{ui,wi}),
+where
+Ui={xi}fori= 1,2, U3={x31,x32},U4={x41,x42,...,x4∆−7
+2},
+Wi={yi}fori= 1,2,3, W4={y41,y42,...,y4∆−7
+2}
+and its edge set is composed of
+{vx,vy,xu i,ywi,uiwi|x∈Ui, y∈Wi, i= 1,2,3,4}
+∪ {xix3i,yix3i|xi∈Ui, yi∈Wi, i= 1,2}
+∪ {y3x,y3y|y3∈W3, x∈U4, y∈W4}10 MOO YOUNG SOHN, DONGSEOK KIM, YOUNG SOO KWON, AND JAEUN LEE
+as in Figure 4 and the subgraph induced by the vertices in U4andW4isK∆−7
+2,∆−7
+2−E(M),
+whereMis a 1-factor of K∆−7
+2,∆−7
+2. For our convenience, let Ni=Ui∪Wi∪ {ui,wi}for
+i= 1,2,3,4. We want to show that Gis a 9-γt-critical graph of order ∆ + 9. Let S
+be a total dominating set of G. Then, one can check that γt(G) =|S| ≥8 because for
+i= 1,2,3,4,|S∩Ni| ≥2 forSto dominate uiandwj. Suppose that γt(G) = 8. Then,
+|S∩Ni|= 2 for any i= 1,2,3,4. Especially, |S∩N3|= 2. IfS∩N3={x31,u3}then
+forSto dominate y3,S∩N3is{x4j,u4}or{y4j,w4}for some j= 1,2,...,∆−7
+2. In either
+cases,W4orU4is not dominated. For other choices of S∩N3, one can similarly show
+thatV(G) is not totally dominated by Sif|S∩N3|= 2. So, γt(G) =|S| ≥9. For
+S1={ui,wi|i= 1,2,4}∪{v,x31,u3},S1is total dominating set of G. Hence, γt(G) = 9.
+If we delete ujfor some j= 1,2,3,4, then for some y∈Wj,{ui,wi|i= 1,2,3,4, i/n⌉}ationslash=
+j}∪{v,y}is a total dominating set of G−uj. Hence, γt(G−uj) = 8. Similarly, one can
+show that γt(G−wj) = 8. If we delete x1fromGthen{ui,wi|i= 2,3,4}∪{y1,w1}is
+a total dominating set of G−ujand hence γt(G−x1) = 8. If we delete x3,1fromGthen
+{u1,w1,x2,y2,x32,y3,x41,y41}isatotaldominatingset of G−x3,1andhence γt(G−x31) = 8.
+Similarly, one can show that for any z∈V(G),γt(G−z) = 8. Therefore, Gis a 9-γt-critical
+graph of order ∆+9. /square
+By Theorems 2 and 14, we have the following corollary.
+Corollary 15. For any odd m≥9, there exists an m-γt-critical graph Gof order∆(G)+m
+withδ(G)≥2if and only if ∆(G)≥m−1.
+Remark: We settled the existence problem with respect to the parities of the total domi-
+nation number mand the maximum degree ∆ except some cases. The only remaining ope n
+cases are ∆ = 5 ,7,9 form= 5 and ∆ = 7 ,9,11 form= 7.
+References
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+critical graphs , Discrete Math. 286(2004), 255–261.
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+Dekker, New York, 1997.
+[6] T. W. Haynes, L. C. Merwe and C. M. Mynhardt, Total domination edge critical graphs with minimum
+diameter , Ars Combinatoria. 66(2003), 79–96.
+[7] M. Henning, T. H. Lucas and L. C. Merwe, Total domination critical graphs with respect to relative
+complements , Ars Combinatoria. 64(2002), 169–179.
+[8] M. A. Henning and C. M. Mynhardt, The diameter of paired-domination vertex critical graphs ,
+Czechoslovak Math. J. 58(4)(2008), 887–897.
+[9] M. A. Henning and N. J. Rad, On total domination vertex critical graphs of high connecti vity, Discrete
+Applied Mathematics, 157(8)(2009), 1969–1973.
+[10] M. Loizeaux and L. C. Merwe, A total domination vertex critical graph with diameter two , Bulletin of
+the Institute of Combinatorics and its Applications. 48(2006), 63–65.TOTAL DOMINATION VERTEX CRITICAL GRAPHS 11
+[11] D. A. Mojdeh and N. J. Rad, On the total domination critical graphs , Electronic Notes in Discrete
+Mathematics, 24(2006), 89–92.
+[12] D. A. Mojdeh and N. J. Rad, On an open problem concerning total domination critical gra phs, Expo.
+Math.25(2007), 175–179.
+[13] N. J. Rad and D. A. Mojdeh, A note on k-γt-critical graphs , Preprint.
+[14] N. J. Rad and S. R. Sharebaf, A remark on total domination critical graphs , Applied Mathematical
+Sciences, 2(41), (2008), 2025–2028.
+Mathematics, Changwon National University Changwon 641-7 73, Korea
+E-mail address :mysohn@changwon.ac.kr
+Department of Mathematics, Kyonggi University, Suwon, 443 -760 Korea
+E-mail address :dongseok@kgu.ac.kr
+Department of Mathematics, Yeungnam University, Kyongsan , 712-749, Korea
+E-mail address :ysookwon@yu.ac.kr
+Department of Mathematics, Yeungnam University, Kyongsan , 712-749, Korea
+E-mail address :julee@yu.ac.kr
\ No newline at end of file
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+Topical Bias in Generalist Mathematics Journals
+Joseph F. Grcary
+Generalist mathematics journals exhibit bias toward the branches of
+mathematics by publishing articles about some subjects in quantities far
+disproportionate to the production of papers in those areas within all of
+mathematics. Bias is used here because it is the shortest English word with
+Webster's meaning of \a tendency of a statistical estimate to deviate in one
+direction from a true value." This paper quanties the bias, which seems
+not to be discussed previously, and suggests some consequences of it.
+The mathematical topics that are generally agreed to be the major
+branches of mathematics and the production of papers about them can
+be determined from the Mathematical Subject Classication and from two
+databases based on the classication [3]. The classication dates in some
+form to 1931, when Zentralblatt f ur Mathematik und ihre Grenzgebiete began
+publishing annual reviews of papers grouped into broad subject areas; Math-
+ematical Reviews started publishing similar material in 1940. The American
+Mathematical Society created hierarchical codes to classify papers for its de-
+funct Mathematical Oprint Service in the late 1960s [8]. This AMS (MOS)
+Classication quickly became the de facto index for mathematical literature.
+Both Zentralblatt andMathematical Reviews maintain historical databases
+of indexed papers that can be accessed now through the world wide web.
+The data presented here for the decade 2000{2009 were gathered from
+theZentralblatt database in January, 2010. The 854,547 items that were
+available at the time will increase as Zentralblatt completely assimilates pa-
+pers from the recent past. All the records can be retrieved by Mathematical
+Subject Classication.
+Figure 1 displays the percent of all mathematics papers that address
+each of the major branches of mathematics enumerated by Subject Classi-
+cation. The present 5-digit codes begin with the 2-digit numbers that re
ect
+the coarsest level of dierentiation, namely, the major branches of mathe-
+Please read and cite the corrected, published article in Notices of the AMS ,
+57(11):1421{1424, 2010.
+y6059 Castlebrook Drive, Castro Valley, CA 94552 USA (jfgrcar@comcast.net).
+1arXiv:1001.0798v6  [math.HO]  30 Nov 2010 0  1  2  3  4  5  6  7  8  9  10
+Computer science  68
+Partial differential eq.  35
+Numerical analysis  65
+Operations research  90
+Fluid mechanics  76
+Statistics  62
+Quantum theory  81
+Ordinary differential eq.  34
+Mech. of deform. solids  74
+Probability theory  60
+Game theory, economics  91
+Combinatorics  05
+Systems theory, control  93
+Dynamical systems  37
+Operator theory  47
+Biology  92
+Number theory  11
+Information and comm.  94
+Functional analysis  46
+Statistical mechanics  82
+Differential geometry  53
+Logic and foundations  03
+Group theory  20
+Relativity theory  83
+Algebraic geometry  14
+Calculus of variations  49
+One complex variable  30
+General topology  54
+Global analysis  58
+Mech. of particles and sys.  70
+Associative rings  16
+Linear, multilinear algebra  15
+Fourier analysis  42
+Real functions  26
+Classical thermodynamics  80
+Manifolds, cell complexes  57
+Optics  78
+Several complex variables  32
+Approximations  41
+Nonassociative rings  17
+Functional equations  39
+Special functions  33
+Commutative rings  13
+Measure and integration  28
+Convex & discrete geom.  52
+Topological groups  22
+Ordered alg. structures  06
+Integral equations  45
+Geophysics  86
+Geometry  51
+Algebraic topology  55
+Category theory  18
+Astronomy  85
+Field theory, polynomials  12
+Potential theory  31
+Abstract harmonic anal.  43
+Integral transforms  44
+General alg. systems  08
+K-theory  19
+Sequences and series  40Percent of All Mathematics Papers 2000-2009extend to 11.9 and 14.5
+Primary Subject
+Secondary OnlyFigure 1: The distribution of eort in mathematical research is indicated by
+the percent of all mathematics papers addressing a particular Mathematical
+Subject Classication. Three subject classes are omitted: general 00, history
+01, and education 97. Papers for which a subject is primary are indicated
+by red, white, or blue shading (the shading is explained elsewhere). Papers
+for which a subject is only secondary are indicated by grey shading.
+2matics. Currently, sixty-three of the 2-digit numbers are assigned. Each
+paper so catalogued receives one primary code and optionally any number
+of secondary codes. For example, when the data were gathered, 22,443 pa-
+pers from 2000{2009 in the Zentralblatt database had either a primary or
+a secondary code in the \group theory" classication, 20. Thus, approxi-
+mately 22443 =854547 or 2.63 percent of all mathematics papers discussed
+group theory during 2000{2009. This value is recorded in gure 1.
+The grey shading in gure 1 indicates papers for which the associated
+classication is not the primary subject. Such papers are prevalent across
+mathematics, so they are included in the count of papers for each subject.
+Since a paper may contribute to several subjects, the percentages for all
+classes sum to 156.5. The red, white, and blue shades in the gure distin-
+guish subjects in a manner to be explained.
+The American Mathematical Society publishes three print journals \de-
+voted to research articles in all areas of mathematics." The extent to which
+these journals fulll the Society's promise of inclusiveness can be exam-
+ined by calculating percentages similar to those in gure 1 but specic to
+the journals in question. The data presented here are for Proceedings of
+the American Mathematical Society ; similar observations can be made for
+theJournal and the Transactions . The Zentralblatt database holds 4,758
+papers from the Proceedings during the past decade, of which 309 had a pri-
+mary or secondary code beginning with 20. Therefore, of all papers in the
+Proceedings , roughly 309 =4758 or 6 :49 percent discuss group theory. This
+percentage for the generalist journal, 6 :49, contrasts with the 2 :63 percent
+among all mathematics papers. The fraction of papers about group theory
+in the journal is over twice the fraction in mathematics as a whole.
+The bias of a journal for or against a given branch of mathematics may
+be dened as the ratio of the fraction of papers about the subject in the
+journal to the fraction of papers about the subject in all of mathematics.
+It is convenient to represent the ratio as a base 2 logarithm, so a bias in
+favor has a positive value and a bias against has a negative value. The
+Proceedings of the American Mathematical Society thus has a positive bias
+for group theory of 6 :49=2:63 or 2 :47 = 2+1 :31.
+Figure 2 displays the biases of the Proceedings toward all the branches
+of mathematics. The wide range of values indicates that the journal is un-
+representative of mathematics research. It has a strong bias (2+1to over
+2+2) in favor of twenty-ve subjects that are colored red in gures 1 and
+2. Comparing the gures reveals a strong positive bias for three subjects
+that are of interest to relatively few mathematicians, in that each subject
+accounts for less than 1 percent of all mathematics papers: commutative
+3Positive Bias of Proc. AMS
+by Primary or Secondary Subject for 2000-2009
+Negative Bias of Proc. AMS1 2 3
+-5 -4 -3 -2 -1log2 = 0
+0 = log2Commutative rings  13
+Abstract harmonic anal.  43
+Functional analysis  46
+Algebraic topology  55
+Manifolds, cell complexes  57
+Fourier analysis  42
+Operator theory  47
+Potential theory  31
+Several complex variables  32
+Measure and integration  28
+General topology  54
+One complex variable  30
+Field theory, polynomials  12
+Topological groups  22
+Algebraic geometry  14
+Associative rings  16
+Group theory  20
+Number theory  11
+K-theory  19
+Special functions  33
+Global analysis  58
+Convex & discrete geom.  52
+Differential geometry  53
+Real functions  26
+Integral transforms  44
+Sequences and series  40
+Logic and foundations  03
+Category theory  18
+Nonassociative rings  17
+Approximations  41
+Ordered alg. structures  06
+Dynamical systems  37
+Functional equations  39
+Linear, multilinear algebra  15
+35  Partial differential eq.
+51  Geometry
+34  Ordinary differential eq.
+60  Probability theory
+49  Calculus of variations
+05  Combinatorics
+45  Integral equations
+08  General alg. system s
+81  Quantum theory
+94  Information and comm.
+70  Mech. of particles and sys.
+76  Fluid mechanics
+78  Optics
+86  Geophysics
+65  Numerical analysi s
+92  Biology
+91  Game theory, economic s
+62  Statistics
+82  Statistical mechanics
+93  Systems theory, control
+80  Classical thermodynamic s
+83  Relativity theory
+90  Operations research
+74  Mech. of deform. solid s
+68  Computer science
+85  Astronom yFigure 2: Topical bias in Proceedings of the American Mathematical Society .
+Bias is the ratio of the fraction of publications in the journal for a given
+subject to the fraction of publications in all of mathematics for that subject.
+Note the scale is logarithmic to the base 2. The journal had no papers about
+the one subject at bottom. Subjects with strong positive or negative bias
+have red or blue shading, respectively, to ease their identication in other
+gures.
+4rings 13 (bias 2+2 :58), abstract harmonic analysis 43 (bias 2+2 :52), and alge-
+braic topology 55 (bias 2+2 :17). In contrast, the Proceedings has a neutral
+to slightly negative bias for three subjects that are of interest to many more
+mathematicians, in that each of them constitutes over 4 percent of all math-
+ematics papers: ordinary dierential equations 34 (bias 20:39), probability
+theory 60 (bias 20:51), and combinatorics 05 (bias 20:54).
+The ratio of biases for two subjects is easily seen to equal the ratio of
+the conditional probabilities that papers about the respective subjects would
+appear in the journal. Viewed in this way, the biases of the Proceedings are
+remarkable. For example, the journal is roughly 6 :52+2 :17=20:54times
+more likely to be the publisher of a paper about algebraic topology than
+about combinatorics.
+The journal has a strong negative bias (21to much below) against eigh-
+teen subjects that are colored blue in gure 2. The same shading in gure
+1 reveals the strong negative bias occurs for many branches of mathematics
+about which most papers are written. Indeed, the Proceedings is strongly bi-
+ased against seven of the ten most heavily published subjects. Consequently,
+the generalist journal neglects subjects to which many mathematicians con-
+tribute, or whose very creation is associated with some mathematicians. To
+cite a few examples: Tukey [1] for statistics 62, Lax [11] for numerical anal-
+ysis 65, Moser [9] for mechanics 70, Ladyzhenskaya [4] for 
uid mechanics
+76, Dobrushin [10] for statistical mechanics 82, Dantzig [2] for operations
+research 90, Shannon [5] for information theory 94, and of course, Wiener
+[6] for systems theory 93, and von Neumann [7] for game theory 91.
+TheTransactions has roughly the same biases as the Proceedings (Figure
+3). Among heavily published subjects that appear in quantity in either
+journal, the Proceedings has stronger positive biases for functional analysis
+46 and for operator theory 47.
+Explanations for the biases are in a sense circular, in that authors submit
+where history suggests acceptance is likely, or that editorial boards more ac-
+curately evaluate the familiar. For example, not all branches of mathematics
+employ just the telegraphic style that is prevalent in these journals. Thus
+biases are self-perpetuating, although surely procedures could be found so
+submissions that are exceptions to what is usually published do not prove
+the rule.
+The signicant question raised by the data is not how biases occur or
+how to manage them, but rather, whether the present topical distribution in
+generalist journals best serves mathematics. Biases in professional journals
+impart an illusory picture of a eld that can be dangerous if it becomes so
+pervasive as to aect the evolution of the underlying subject matter. In
+5Positive Bias of Trans. AMS
+by Primary or Secondary Subject for 2000-2009
+Negative Bias of Trans. AMS1 2 3
+Algebraic topology  55
+Manifolds, cell complexes  57
+Commutative rings  13
+K-theory  19
+Algebraic geometry  14
+Topological groups  22
+Several complex variables  32
+Category theory  18
+Group theory  20
+Abstract harmonic anal.  43
+Associative rings  16
+Global analysis  58
+Potential theory  31
+Field theory, polynomials  12
+Convex & discrete geom.  52
+Fourier analysis  42
+Differential geometry  53
+Measure and integration  28
+Nonassociative rings  17
+Integral transforms  44
+Number theory  11
+Functional analysis  46
+One complex variable  30
+Special functions  33
+Dynamical systems  37
+General topology  54
+Operator theory  47
+Logic and foundations  03
+General alg. systems  08
+Ordered alg. structures  06
+Real functions  26
+Partial differential eq.  35
+Combinatorics  05
+Probability theory  60
+Calculus of variations  49
+41  Approximations
+51  Geometry
+15  Linear, multilinear algebra
+40  Sequences and series
+34  Ordinary differential eq.
+45  Integral equations
+39  Functional equations
+70  Mech. of particles and sys.
+82  Statistical mechanics
+81  Quantum theory
+76  Fluid mechanics
+92  Biology
+80  Classical thermodynamics
+94  Information and comm.
+78  Optics
+74  Mech. of deform. solids
+91  Game theory, economics
+93  Systems theory, control
+68  Computer science
+65  Numerical analysis
+62  Statistics
+90  Operations research
+83  Relativity theory
+85  Astronomy
+86  Geophysics
+-5 -4 -3 -2 -1log2 = 0
+0 = log2Figure 3: Topical bias in Transactions of the American Mathematical Soci-
+ety. The scale is logarithmic to the base 2. Subjects are shaded as in gure 2
+to ease comparison with the Proceedings . The journal had no papers about
+the two subjects at bottom.
+6the present case, since the sponsoring Society does not explain the topical
+selectivity of its generalist journals, and since the literature does not examine
+the issue, in the absence of clarifying public discussion, readers may assume
+the contents of the 
agship journals conrm prejudices about the branches
+of the eld. If unchecked over many years, these opinions may in
uence
+decisions about curricula, publications, and stang that can fragment the
+research community.
+Such dissolution of mathematics may be occurring already as evidenced
+by the proliferation and growth of elds, particularly since the middle of the
+last century, that have considerable mathematical content but whose fac-
+ulties and professional societies have little overlapping membership. When
+dierent elds sponsor dierent branches of mathematics, then the subelds
+may adopt the cultures of their sponsors, so the branches of mathematics
+eventually may come to disagree over acceptable idiom, notation, rigor, and
+terminology. Barriers among the branches of mathematics entail a large
+opportunity cost, so to speak, because possible collaborations and synergies
+may not be realized. Moreover, as governments increasingly view scientic
+research as a component of national wealth, disparities may be expected
+to grow in the allocation of resources to the elds that encompass dierent
+branches of mathematics. In this way the fragmentation of mathematical
+research cedes to non-mathematicians a greater degree of responsibility to
+choose which branches of mathematics to encourage and how they should
+develop.
+Acknowledgments
+The author is grateful to B. Wegner for clarifying the history of the MSC,
+and to the editor and the referees for advice that improved this article.
+References
+[1] D. R. Brillinger. John Wilder Tukey (1915{2000). Notices of the
+AMS , 49(2):193{201, 2002.
+[2] R. Cottle, E. Johnson, and R. Wets. George B. Dantzig (1914{2005).
+Notices of the AMS , 54(3):344{362, 2007.
+[3] G. Fairweather and B. Wegner. Mathematics Subject Classication
+2010. Notices of the AMS , 56(7):848, 2009.
+7[4] S. Friedlander and others. Olga Alexandrovna Ladyzhenskaya
+(1922{2004). Notices of the AMS , 51(11):1320{1331, 2005.
+[5] S. W. Golomb and others. Claude Elwood Shannon (1916{2001).
+Notices of the AMS , 49(1):8{16, 2002.
+[6] D. Jerison and D. Stroock. Norbert Wiener. Notices of the AMS ,
+42(4):430{438, 1995.
+[7] R. J. Leonard. From Parlor Games to Social Science: von Neumann,
+Morgenstern, and the Creation of Game Theory 1928{1944. Journal
+of Economic Literature , 33(2):730{761, 1995.
+[8] W. LeVeque. The AMS | Then, Now, and Soon. Notices of the
+AMS , 35(6):785{789, 1988. Available at www.ams.org/ams/lev2.pdf .
+[9] J. N. Mather and others. J urgen K. Moser (1928{1999). Notices of the
+AMS , 47(11):1392{1393, 2000.
+[10] R. Minlos, S. Shlosman, and N. Vvedenskaya. Memories of Roland
+Dobrushin. Notices of the AMS , 43(4):428{430, 1996.
+[11] Norwegian Academy of Science and Letters. Lax Receives Abel Prize.
+Notices of the AMS , 52(6):632{633, 2005.
+8
\ No newline at end of file
diff --git a/1001.0799.txt b/1001.0799.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3b2be47c79a56a6e75a9f7eb702563690f5a897a
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+++ b/1001.0799.txt
@@ -0,0 +1,2127 @@
+arXiv:1001.0799v1  [astro-ph.CO]  6 Jan 2010Draft version October 3, 2018
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+ORBITAL STRUCTURE OF MERGER REMNANTS: TRENDS WITH GAS FRACT ION IN 1:1 MERGERS
+Loren Hoffman1, Thomas J. Cox2,3, Suvendra Dutta2,4, Lars Hernquist2
+Draft version October 3, 2018
+ABSTRACT
+Since the violent relaxation in hierarchical merging is incomplete, elliptic al galaxies retain a wealth
+of information about their formation pathwaysin their present-da yorbital structure. Recent advances
+in integral field spectroscopy, multi-slit infrared spectroscopy, a nd triaxial dynamical modeling tech-
+niques have greatly improved our ability to harvest this information. A variety of observational and
+theoretical evidence indicates that gas-rich major mergers play a n important role in the formation of
+elliptical galaxies. We simulate 1:1 disk mergers at seven different initial gas fractions ( fgas) ranging
+from 0 to 40%, using a versionof the TreeSPHcode Gadget-2that in cludes radiativeheating and cool-
+ing, star formation, and feedback from supernovae and active ga lactic nuclei. We classify the stellar
+orbits in each remnant and construct radial profiles of the orbital content, intrinsic shape, and ori-
+entation. The dissipationless remnants are typically prolate-triaxia l, dominated by box orbits within
+rc∼1.5Re, and by tube orbits in their outer parts. As fgasincreases, the box orbits within rcare
+increasingly replaced by a population of short axis tubes ( z−tubes) with near zero net rotation, and
+the remnants become progressively more oblate and round. The lon g axis tube ( x−tube) orbits are
+highly streaming and relatively insensitive to fgas, implying that their angular momentum is retained
+from the dynamically cold initial conditions. Outside rc, the orbital structure is essentially unchanged
+by the gas. For fgas/greaterorsimilar15%, gas that retains its angular momentum during the merger re-f orms a
+disk, that appears in the remnants as a highly streaming z−tube population superimposed on the hot
+z−tube distribution formed by the old stars. In the 15-20%gasremna nts, this population appearsasa
+kinematically distinct core (KDC) within a system that is slowly rotating or dominated by minor-axis
+rotation. These remnants show an interesting resemblance, in bot h their velocity maps and intrinsic
+orbital structure, to the KDC galaxy NGC4365 (van den Bosch et a l. 2008). At 30-40% gas, the rem-
+nants are rapidly rotating, with sharp embedded disks on ∼1Rescales. We predict a characteristic,
+physically intuitive orbital structure for 1:1 disk mergerremnants, with a distinct transition between 1
+and 3Rethat will be readily observable with combined data from the 2D kinemat ics surveys SAURON
+and SMEAGOL. Our results illustrate the power of direct comparison s between N−body simulations
+and dynamical models of observed systems to constrain theories o f galaxy formation.
+Subject headings: stellar dynamics – methods: n-body simulations – galaxies: elliptical an d lenticu-
+lar, cD – galaxies: formation – galaxies: interactions – galaxies: kinem atics and
+dynamics
+1.INTRODUCTION
+1.1.Motivation
+In the standard ΛCDM concordance cosmology
+(Ostriker & Steinhardt 1995; Dodelson et al. 1996;
+Spergel et al. 2007), structure in the Universe grows
+hierarchically, through a progression of smaller bodies
+accreting material and merging to form larger systems
+(e.g. White & Rees 1978). Cosmological N-body
+simulations starting from a Gaussian random field
+of linear density fluctuations at high redshift (e.g.
+Springel et al. 2005b), together with analytic models
+of gravitational collapse (e.g. Bertschinger 1985), have
+given us a fairly clear picture of how dark matter (DM)
+assembles in the Universe. Extended Press-Schechter
+l-hoffman@northwestern.edu
+1Department of Physics and Astronomy, Northwestern
+University, Dearborn Observatory, 2131 Tech Drive, Evanst on,
+IL 60208
+2Department of Astronomy, Harvard University, 60 Garden
+Street, Cambridge, MA 02138
+3Carnegie Observatories, 813 Santa Barbara Street,
+Pasadena, CA 91101
+4Faculty of Arts and Sciences, Harvard University, Universi ty
+Hall, Cambridge, MA 02138theory (Press & Schechter 1974; Bond et al. 1991;
+Lacey & Cole 1993) provides an analytic formalism for
+computing halo merger rates and assembly histories,
+which matches N-body simulations remarkably well
+(Lacey & Cole 1994; Genel et al. 2009b), and has been
+used in a variety of semianalytic models of structure for-
+mation (e.g. Cole et al. 2000; Manrique & Salvador-Sole
+1996; Kauffmann et al. 1999; Somerville & Primack
+1999; Croton et al. 2006; Somerville et al. 2008;
+Stewart et al. 2009b). Simulated DM halos also appear
+to share a universal internal morphology, with density
+and velocity anisotropy profiles similar to the generic
+outcome of violent relaxation following dissipationless
+collapse or strong tidal shocking (Dubinski & Carlberg
+1991; Navarro et al. 1997; Bullock et al. 2001a,b;
+Navarro et al. 2008; Miller & Smith 1979; van Albada
+1982; McGlynn 1984, 1990; Spergel & Hernquist 1992;
+Huss et al. 1999; MacMillan et al. 2006; Bellovary et al.
+2008).
+There is no such simple theory of hierarchical galaxy
+formation, because the luminous components of galax-
+ies are formed through complex baryonic physics. For
+instance radiative heating and cooling, star formation2
+(SF) and gas expulsion through stellar winds, energy
+and momentum feedback from supernovae and active
+galactic nuclei (AGN), ram pressure stripping, and res-
+onant effects in dynamically cold systems, are all im-
+portant ingredients in determining the baryonic struc-
+ture of galaxies (e.g. Croton et al. 2006; Springel
+2000; Springel et al. 2005a; Best et al. 2007; Cox et al.
+2006b; Ceverino & Klypin 2009; Martig & Bournaud
+2009; D’Onghia et al. 2009). Cosmological models of
+galaxy formation must therefore be calibrated with high-
+resolution simulations on galactic scales that study these
+effects in isolation, coupled with direct constraints from
+observations.
+Violent relaxation in mergers tends to drive galaxies
+toward a universal, fully mixed structure (Alladin 1965;
+Lynden-Bell 1967; Syer & White 1998; Thomas et al.
+2009), while gas accretion and stellar outflows produce
+new dynamically cold components (Rix & White 1990;
+Block et al. 2002; Genel et al. 2009a; Dekel et al. 2009;
+Bournaud & Elmegreen 2009; Martig & Bournaud 2009;
+Daddi et al. 2009). From the hot stellar distributions
+of elliptical galaxies, we deduce that they are the most
+merger-dominated systems. Ellipticals and bulges con-
+tain the majority of the stellar mass in the local Uni-
+verse (e.g. Gadotti 2009), and often serve as a testing
+ground for theory since they are the most evolved under
+the complex combination of processes driving galaxy for-
+mation. Since the violent relaxation in mergers is incom-
+plete, elliptical galaxies retain a wealth of information
+about their formation histories in their present-day dis-
+tribution functions (e.g. Eggen et al. 1962; Lynden-Bell
+1967; White 1980; Valluri et al. 2007).
+Recent advances in integral field spectroscopy (IFS;
+Bacon et al. 1995, 2001; Hill et al. 2006; Weijmans et al.
+2009), multi-slit infrared spectroscopy (Norris et al.
+2008; Proctor et al. 2009), and triaxial dynami-
+cal modeling techniques (van de Ven et al. 2008;
+van den Bosch et al. 2008; de Lorenzi et al. 2006, 2009)
+have greatly improved our ability to harvest this in-
+formation. The SAURON project (Bacon et al. 2001;
+Emsellem et al. 2004) will produce high resolution,
+2D kinematic maps within ∼1 effective radius ( Re)
+for a representative sample of ∼100 nearby elliptical
+galaxies and spiral bulges, using a panoramic integral
+field spectrograph mounted on the William Herschel
+Telescope. The data on 48 early-type galaxies released
+to date has revealed an unexpectedly rich variety of
+kinematic structures, which poses a new challenge
+for galaxy formation simulations (Jesseit et al. 2007;
+Burkert et al. 2008). Dynamical modeling studies have
+shown that, in practice, 2D maps of the first four mo-
+ments (h1−h4) of the line-of-sight velocity distribution
+(LOSVD) provided by SAURON are typically sufficient
+to uniquely reconstruct the 3D stellar orbital distri-
+bution (van de Ven et al. 2008; van den Bosch et al.
+2008). Complex features present in many systems, such
+as embedded disks and kinematically distinct cores
+(KDCs), can provide especially strong constraints on the
+intrinsic structure (van den Bosch & van de Ven 2008).
+Agoodexampleofthepowerofthesenewtechniquesis
+providedbythe caseofNGC4365. Thismassiveoldellip-
+tical is known for its minor axis rotation (Wagner et al.
+1988; Bender et al. 1994) and KDC (Davies et al. 2001),
+and is therefore a natural candidate for dynamical mod-eling. Statler et al. (2004) modeled this galaxy using
+a velocity field fitting method (Statler 1994a,b) that
+made use of the surface brightness and full 2D velocity
+map from SAURON, but not the higher moments of the
+LOSVDs. They found that the system was nearly max-
+imally triaxial, ruling out axisymmetry at >95% confi-
+dence.
+A few years later, van den Bosch et al. (2008) mod-
+eled the same galaxy using an advanced new triaxial
+Schwarzschild modeling (Schwarzschild 1979) code that
+can incorporate all of the LOSVD moments up to h4.
+They reached a qualitatively different conclusion - that
+the system was nearly oblate axisymmetric. The pre-
+dominance of the minor-axis rotation in the outer parts
+of the map owed to a high degree of cancellation of the
+orbits rotating in a prograde and retrograde sense about
+the short axis, and streaming of the smaller population
+of orbits rotating about the intrinsic long axis. They
+found no major transition in the orbital structure at the
+boundary of the kinematically “decoupled” core, mak-
+ing it unlikely that it formed in a separate infall event.
+The orbital structure of the 15-20% gas merger remnants
+in this paper bears a tantalizing resemblance to that of
+NGC4365 (compare e.g. our Figures 9 - 10 and 17 - 18
+with Figures 7, 11 and 12 of van den Bosch et al. 2008).
+However a limitation of the SAURON spectrograph
+is its small field of view, corresponding to ∼1Reon a
+typical elliptical target. The outer parts of galaxies are
+less relaxed than their inner parts, and bar-like modes
+in mergers efficiently transport angular momentum out-
+ward (Ostriker & Peebles 1973; Fall 1979; Hernquist
+1993; Barnes & Hernquist 1996; Hopkins et al. 2008b;
+Hoffman et al. 2009c). A gas-rich merger between two
+spiral galaxies with halos might be expected to produce
+a remnant with three distinct components in its distri-
+bution function: (i) an inner part formed through dissi-
+pation, (ii) a middle part reflecting the dynamically cold
+distribution of the disk stars, and (iii) an outer part aris-
+ing from the pre-existing stellar halo populations. Ob-
+servations with about four times the spatial coverage of
+SAURON would be able to detect these dynamical sub-
+components,andprobethepartsofgalaxiesretainingthe
+most memory of their progenitors’ angular momentum
+and internal structure. Hints of increased complexity in
+the angularmomentum profilesat largeradiihaveindeed
+been observed in a few elliptical galaxies (Coccato et al.
+2009; Proctor et al. 2009).
+A number of projects designed to extend SAURON-
+style dynamical modeling out to larger radii are cur-
+rently underway. The SMEAGOL survey (Proctor et al.
+2009; Foster et al. 2009) will obtain smoothed 2D maps
+ofh1−h4out to∼3Refor a representative sample
+of 25 nearby ellipticals, using the new stellar kinemat-
+ics with multiple slits (SKiMS) technique (Norris et al.
+2008; Proctor et al. 2008, 2009) with the DEIMOS
+spectrograph on the 10-meter Keck-II telescope. The
+data analysis will include triaxial dynamical model-
+ing with the advanced particle-based method NMAGIC
+(Syer & Tremaine 1996; de Lorenzi et al. 2006, 2008,
+2009). The wide-field IFS VIRUS-P has been used
+to obtain 2D stellar kinematics out to 3 −4Refor
+several giant ellipticals (Hill et al. 2006; Blanc et al.
+2009; Murphy et al. 2009), and multiple pointings of
+the SAURON spectrograph have been used to measure3
+the stellar LOSVDs out to ∼4Rein NGC3379 and
+NGC821 (Weijmans et al. 2009). Surveys using globular
+clusters (GCs) and planetary nebulae (PNe) as discrete
+tracers of the distribution function (Romanowsky et al.
+2009a; Douglas et al. 2002, 2007; Napolitano et al. 2009;
+Nantais & Huchra 2009; Schuberth et al. 2009) can
+probe much larger radii (out to ∼10Re) and place fur-
+ther constraints on dynamical models (Chaname et al.
+2008).
+Combined with simulations aimed at establishing the
+characteristic orbital structure arising from various for-
+mation pathways, these observational programs will pro-
+vide unprecedented insight into the physics of galaxyfor-
+mation.
+1.2.Orbits in galactic potentials
+Todeducetheformationhistoriesofgalaxiesfromtheir
+orbital structure, we must identify macroscopic groups
+of orbits that are confined to a well-defined neighbor-
+hood of phase space because they followed similar evo-
+lutionary pathways. The isolating integrals of motion
+(or quasi-isolating integrals in the case of perturbed po-
+tentials; Contopoulos 1963b; Goodman & Schwarzschild
+1981; Siopis & Kandrup 2000; Kandrup & Siopis 2003)
+parameterize the phase space region in which a star re-
+mains localized, and encode whatever information about
+its initial conditions (ICs) is preserved once the sys-
+tem is fully phase-mixed (e.g. Binney & Tremaine 2008;
+Binney & Spergel 1984; Gomez & Helmi 2009). Orbits
+which conserve at least one isolating integral per degree
+of freedom are called regular. It is the ubiquity of reg-
+ular orbits in galaxies that permits the rich variety in
+their structure, mirroring their varied formation path-
+ways (Schwarzschild 1979).
+The regularorbits in a static potential can be classified
+into families that conserve qualitatively similar integrals
+of the motion, and therefore have similar morphologies.
+Whichorbitalclassagivenstarwilloccupyisdetermined
+by the available phase space for different kinds of orbits
+in the potential, and its ICs - the phase space need not
+be uniformly populated. In a time-varying (or otherwise
+non-ideal) potential such as an ongoing merger, stars dif-
+fuse in the space of their conserved integrals, but not
+completely. If they do not cross boundaries between the
+orbital families, then their qualitative character may be
+preservedfrom the ICs. Crossingbetween orbital bound-
+aries can serve as a collective relaxation mechanism (e.g.
+Barnes & Hernquist 1996). An intuitive grasp of the or-
+bital classes is therefore essential to understanding how
+dynamical systems evolve and relax.
+We begin with a brief overview of the types of regular
+orbits that are possible in various idealized potentials,
+leading up to a classification of the orbits in triaxial sys-
+tems into families that conserve similar integrals. For
+a more thorough and rigorous presentation we refer the
+reader to Binney & Tremaine (2008).
+The simplest conceivable model is a spherical poten-
+tial. In this case the symmetry about all three Carte-
+sian axes implies conservation of the angular momentum
+vector, so every orbit is confined to a plane. The star
+oscillates in radius with frequency ωrwhile precessing in
+azimuth with frequency ωθ. If these two frequencies are
+commensurate ( mωr+nωθ= 0 for some integers mand
+n) then the orbit closes on itself, as in a Kepler potential.More generally, the orbits form rosettes that eventually
+fill an annulus between the minimum and maximum of
+the radial oscillations (pericenter and apocenter).
+Very few galaxies are spherical, but many are consis-
+tent with axisymmetry. In an axisymmetric potential
+the angular momentum component about the symme-
+try axis, Lz, is conserved. The direction of /vectorLprecesses
+about the zaxis asLxandLyvary. The radial oscilla-
+tions no longer return the star to the exact same peri-
+center and apocenter every cycle, but are still bounded
+between some p=rminanda=rmax.
+Twoorbitswith the same energyand Lzcanlook quite
+different from eachother, rangingfrom orbits nearlycon-
+fined to the x−yplane, resembling eccentric orbits in
+a thin disk, to puffed-up orbits nearly filling a spherical
+annulus over long times. This suggests that the orbits
+are constrained by another integral in addition to Eand
+Lz, related to how the energyis apportioned into vertical
+and radial motion. Though it cannot be expressed ana-
+lytically for a general axisymmetric potential, this third
+integral ( I3) allows an assortment of stable axisymmetric
+systems, from thin disks formed by quiescent accretion,
+to nearly spherical systems heated by persistent pertur-
+bations or discrete encounters (e.g. Contopoulos 1960,
+1963a; Henon & Heiles 1964; Saaf 1968; Richstone 1982;
+Binney & Tremaine 2008).
+Fig. 1.— Toy model of the box-tube boundary: Elliptical
+ring in a frictionless, concave trough. The ring is initially
+in equilibrium, with its long axis aligned with the long axis of the
+trough. If an angular velocity /vector ωis imparted to the ring, it must
+go up and over a barrier to spin all the way around. Above some
+criticalωc, the ring will make it over the barrier and continue spin-
+ning; below ωcit will just librate about the equilibrium position.
+This analogue owes to Binney & Spergel (1982).
+However some galaxies show clear evidence of nonax-
+isymmetric shapes (e.g. Franx et al. 1991), and sim-
+ulations of dissipationless violent relaxation in merg-
+ers or collapses generically produce triaxial remnants
+(e.g. van Albada 1982). There is a unique den-
+sity distribution stratified on similar triaxial ellipsoids
+whose potential is separable in ellipsoidal coordinates
+(de Zeeuw & Lynden-Bell 1985), known as the “per-
+fect ellipsoid”. Its distribution function and integrals
+of motion can be expressed analytically, and its or-
+bital structure has been studied extensively (St¨ ackel
+1890; Eddington 1915; de Zeeuw & Lynden-Bell 1985;
+de Zeeuw 1985; Statler 1987). The orbits in the per-
+fect ellipsoid were classified into four major families by4
+de Zeeuw (1985): short-axis tubes ( z−tubes) which ro-
+tateabouttheshort( z−)axisofthepotential, twoclasses
+of long-axis tubes ( x−tubes) which revolve about the
+long (x−)axis, and box orbits which behave like per-
+turbed simple harmonic oscillators. Because it is ana-
+lytic, this model has been widely used as a starting point
+for understanding the orbital structure of more general
+triaxial potentials.
+Tube orbits resemble the orbits in axisymmetric po-
+tentials, and may be thought of loosely as precessing
+ellipses driven by the bar-like potential of the triax-
+ial ellipsoid (Binney & Spergel 1982). They conserve
+angular momentum-like integrals and therefore avoid
+the origin of the potential and the zero-velocity sur-
+face. It can be shown that y−tube orbits rotating about
+the intermediate axis in the perfect ellipsoid are unsta-
+bletoverticalperturbations(Heiligman & Schwarzschild
+1979; de Zeeuw 1985), but both x−tube and z−tube
+orbits are allowed. Observations of both major- and
+minor-axis rotation in some elliptical galaxies there-
+fore strongly suggests that these systems are triaxial
+(Illingworth 1977; Schechter & Gunn 1979; Binney 1985;
+Wagner et al. 1988; Franx et al. 1991). x−tube orbits
+are most prevalent in prolate potentials, and are popu-
+lated by stars with large initial angular momenta about
+the long axis. z−tubes are the dominant type of orbit
+in oblate systems. Tube orbits oscillate in radius within
+some bounds panda, and one component of /vectorLnever
+switches sign. Any net angular momentum of a triax-
+ial system must be carried by the tube orbits, so only
+these orbits can retain information about a galaxy’s ini-
+tial sense of rotation.
+Just as tube orbits may be thought of as precessing
+ellipses, box orbits may be regarded loosely as axial or-
+bits (or elongated ellipses) librating about the x−axis.
+Binney & Spergel (1982) illustrated the transition be-
+tween box and tube orbits using an intuitive toy model.
+Imagine an elliptical ring lying in a concave, frictionless
+trough,with itslongaxisinitiallyparalleltothelongaxis
+of the trough, as shown in Figure 1. This configuration
+allows the ring to lie as low in the trough as possible.
+Now imagine trying to spin the ring in the trough, by
+applying an impulsive kick of energy1
+2Iω2to one of its
+ends. To spin all the wayaroundthe ringmust goup and
+overabarrier, sincethe curvatureofthe troughalongthe
+perpendicular direction is greater. There is some criti-
+cal precession frequency ωc, above which the ring will
+get over the barrier. Below ωcit will just librate about
+the equilibrium configuration, with no definite sense of
+rotation.
+The trough’s different curvature along the two axes is
+analogous to triaxiality of a gravitational potential, and
+the librating mode is analogous to box orbits. If the cur-
+vature is the same along both directions perpendicular
+to the rotation axis (as in an axisymmetric potential),
+then the energy barrier is zero. Note also that the effec-
+tive barrier in a triaxial potential is greatest for a star
+rotating about the y−axis, since in this case the differ-
+ence in curvature along the two perpendicular directions
+is greatest, making y−tubes the most susceptible to in-
+stability.
+Box orbits are prevalent in triaxial systems with shal-
+low inner density profiles, and conserve integrals simi-lar to the energies of independent harmonic oscillations
+about each Cartesian axis. Stars on box orbits have no
+definite sense of rotation, and can therefore pass arbi-
+trarily close to both the origin (they are “centrophilic”)
+and the zero-velocity surface. Over time, they densely
+fill a 3-dimensional box-like region centered on the origin
+(de Zeeuw1985; Statler 1987; Binney & Tremaine2008).
+Powerfulnuclear processes such as gas inflow, starbursts,
+and black hole growth are thought to be major drivers of
+galaxy evolution, and box orbits may be responsible for
+conveying information about the rapidly varying central
+potential to large radii.
+Fig. 2.— Orbital composition of a typical merger rem-
+nant.Remnant i(see Table 1), 15% gas, viewed along a sight-line
+near the y−axis.First column: Projected surface brightness, Σ,
+colored on a logarithmic scale. All four maps use the same col or
+scaling to show the relative contribution of each orbital cl ass.Sec-
+ond column: LOS velocity v, with overplotted isophotes. The size
+of the maps is 1 Re×1Re, and the color scale runs from −vmaxto
+vmax, where vmaxis the maximum value of |v|in any pixel. The
+number in the upper right corner gives the value of vmaxin km/s.
+The two right panels show the smoothed LOSVDs at the location s
+specified by the white letters on the Σ and vmaps.Top:LOSVD
+at point A, near the major axis. Bottom: LOSVD at point B, near
+the minor axis. The heavy black line is the full LOSVD, while t he
+thin lines with markers break it down by orbital class.
+Figure 2 shows an example of how the three orbital
+classescontributeto the projectedsurfacebrightnessand
+kinematicsinoneofoursimulatedmergerremnants. The
+remnant is shown in projection along the y−axis, which
+maximally separates the orbital classes in 2D space. The
+boxes and z−tubes are elongated along the major axis,
+while the x−tubes are elongated along the minor axis
+of the projection. All four surface brightness maps are
+plotted on the same color scale, to show the relative or-
+bital populations at different locations in the sky plane.
+Note that the isophotes of each individual orbital class
+appear boxier than the combined isophotes, which are
+nearly elliptical in shape.5
+Thex−tube orbits are responsible for the minor-axis
+rotation in the remnant, and the z−tubes produce the
+major-axis rotation. There is a striking difference in the
+amount of streaming of the two classes of tubes: the
+velocity scale on the x−tube map is ±145 km/s, while
+it is only ±53 km/s on the z−tube map. The remnant
+therefore appears dominated by minor-axis rotation in
+the velocity maps, even though z−tube orbits dominate
+its stellar mass.
+Twolocal LOSVDs areshown on the right, at the loca-
+tions indicated on the maps: one at a point near the cen-
+teralongthemajoraxis(pointA)andoneneartheminor
+axis farther out (point B). At point A the z−tubes have
+a flat-topped distribution with a high velocity dispersion
+and negative kurtosis, indicative of canceling streams ro-
+tating in opposite directions. The box orbits are strongly
+peaked at v= 0, with a large excess in the tails giving
+the distribution a high positive kurtosis. The combi-
+nation of the z−tube and box orbits yields a combined
+LOSVD that is closer to a Gaussian. At point B the
+x−tube orbits make a larger relative contribution. Their
+LOSVD is peaked at a high negative velocity and skewed
+right, indicative of a heated streaming population. The
+combined distribution has substantial net streaming mo-
+tion and is slightly skewed in the direction opposite the
+mean velocity.
+An important idealization in the perfect ellipsoid
+model is that it asymptotes to a flat core. Most
+elliptical galaxies actually have central density cusps,
+with slopes ranging from dlnρ/dr∼0.5−2.5 (e.g.
+Crane et al. 1993; Faber et al. 1997; Lauer et al. 2005,
+2007; Kormendy et al. 2009), which may be thought
+of as perturbations to the perfect ellipsoid poten-
+tial. These perturbations lead to trapping of box
+orbits by resonances (e.g. Binney & Tremaine 2008;
+Contopoulos & Mertzanides 1977; Sridhar & Touma
+1996; Collett et al. 1997; Merritt & Valluri 1999;
+Tremaine & Yu 2000), /vector n·/vector ω= 0 for integer /vector n(e.g.
+z−tube orbits are trapped by the 1:1 resonance between
+ωxandωy). Stars in an island of phase space around
+a stable resonance librate around the resonant orbit;
+these trapped box orbits, called “boxlets,” generally
+avoid the origin (e.g. Carpintero & Aguilar 1998;
+Merritt & Valluri 1999; Fulton & Barnes 2001). As
+the perturbation grows larger (the cusp gets steeper),
+more phase space is occupied by resonant islands
+until they overlap, producing regions populated by
+ergodic orbits that eventually fill the entire portion of
+their 5D energy surface not occupied by regular orbits
+(e.g. Merritt & Valluri 1996; Valluri & Merritt 1998;
+Kandrup & Siopis 2003; Kalapotharakos et al. 2004).
+For simplicity we do not distinguish between regular
+box orbits, boxlets, and ergodic orbits in this paper, and
+will loosely use the term “box” to refer to any orbit with
+no definite sense of rotation. This will be sufficient for
+the global comparison with observations desired in this
+study. Spectralanalysisofselectedorbits(Hoffman2007;
+Clozel 2008) reveals that the orbits classified as boxes
+are generically centrophilic (at least in the inner parts)
+and stochastic, but they remain fairly localized in phase
+space over a Hubble time. We defer a detailed study of
+the nature of the box orbits in the remnants to future
+work.
+The effect of an inner cusp on the orbital pop-ulations can be understood by means of a simple
+experiment - adding a central point mass to a flat-cored,
+triaxial potential with a large population of box or-
+bits. This “experiment” has been studied extensively,
+since it has direct astrophysical relevance to black
+holes in the nuclei of galaxies (e.g. Norman et al.
+1985; Gerhard & Binney 1985; Quinlan et al. 1995;
+Sigurdsson et al. 1995; Merritt & Valluri 1997;
+Merritt & Quinlan 1998; Valluri & Merritt 1998;
+Poon & Merritt 2001; Holley-Bockelmann et al. 2002;
+Kalapotharakos et al. 2004). Since box orbits pass
+arbitrarily close to the origin, they are deflected by the
+point mass at pericenter passage, causing the orbits to
+diffuse within box phase space. The velocity changes are
+primarily along the axis of approach (Chandraskehar
+1943), the x−axis for box orbits librating about the
+long axial orbit, so the angular momentum diffusion
+is mostly in the y−zplane. Eventually the star may
+wander into z−tube phase space (there is no space for
+y−tubes since they are unstable). Since box orbits also
+strike the zero-velocity surface, they convey information
+about their pericentric evolution to large radii.
+As box orbits diffuse across the z−tube boundary,
+the shape of the potential also becomes more oblate,
+shrinking the phase space available for boxes and ex-
+panding that for z−tubes (e.g. Kalapotharakos et al.
+2004). A point mass as small as 2-3% of the total mass
+can seed this transformation in a flat-cored, triaxial sys-
+tem (Gerhard & Binney 1985; Merritt & Valluri 1997;
+Merritt & Quinlan 1998; Valluri & Merritt 1998). When
+the central mass concentration (CMC) is not an ideal
+point mass, the diffusion timescale varies with the degree
+of central concentration - in an r−1cusp the timescales
+are typically longer than the lifetime of a galaxy,
+while in an r−2cusp they are short (Merritt & Valluri
+1996; Merritt & Fridman 1996; Fridman & Merritt 1997;
+Holley-Bockelmann et al. 2001). In hierarchical struc-
+ture formation, there is an ongoing exchange between
+processes that induce triaxiality (e.g. violent relaxation
+in mergers) and gas inflows that deepen the central po-
+tential well (e.g. Dubinski 1994). This process therefore
+undoubtedly playsan important role in galaxyevolution.
+1.3.Gas-rich mergers
+In current semi-analytic models, ∼70% of the stellar
+mass in present-day ellipticals and classical bulges as-
+sembles through major mergers (Hopkins et al. 2009a,d;
+Fakhouri & Ma 2008; Conroy & Wechsler 2009). An
+elliptical galaxy-sized halo has on average undergone
+∼1 major merger since z∼2−3, the epoch dur-
+ing which most of its stellar mass formed (Kauffmann
+1996; De Lucia et al. 2006; Hopkins et al. 2008a, 2009a;
+Stewart et al. 2008). The last major merger was typ-
+ically between two spiral galaxies, with gas fractions
+ranging from ∼10% for systems with stellar masses
+around 3 ×1011M⊙, to∼50% for 1010M⊙systems
+(Stewart et al. 2009a; Erb et al. 2006 and references
+therein).
+The hypothesis that elliptical galaxies form through
+mergers between spirals (Toomre & Toomre 1972) ac-
+tually preceded the acceptance of the concordance cos-
+mology by about two decades, based on the proper-
+ties of the galaxies themselves. Gas-rich tidal tails,
+and rings and shells indicative of the recent disrup-6
+tion of a dynamically cold system, often surround
+galaxies otherwise resembling ordinary giant ellipti-
+cals (Arp 1966; van Dokkum 2005). Early simula-
+tions of mergers between disk galaxies could explain a
+wide variety of the properties of observed ellipticals,
+including their r1/4law density profiles (van Albada
+1982; McGlynn 1984; Hernquist 1992; Naab & Trujillo
+2006), slow rotation and anisotropic velocity distribu-
+tions (Ostriker & Peebles 1973; Aarseth & Binney 1978;
+White 1978; Gerhard 1981), flat rotation curves (White
+1978; Farouki & Shapiro 1982; Efstathiou et al. 1982),
+fine structure (Hernquist & Spergel 1992), and ap-
+parently triaxial shapes (Gerhard 1981; Barnes 1988;
+Franx et al. 1991). A simple counting argument based
+on the numbers of observed interacting pairs and ellipti-
+cal galaxies in the local universe made the prospect that
+these pairs turn into ellipticals quite plausible (Toomre
+1977).
+It was apparent that dissipation must play a
+large role in mergers long before hydrodynamic
+simulations with realistic gas fractions became fea-
+sible (Negroponte & White 1983; Hernquist 1989;
+Nieto et al. 1991; Barnes & Hernquist 1991; Lutz 1991;
+Ashman & Zepf 1992; Hernquist et al. 1993). The
+measured phase space densities at the centers of el-
+liptical galaxies far exceed the maximum densities in
+observed spirals, implying a violation of Liouville’s
+theorem unless the initial disks contain /greaterorsimilar25-30% of
+their mass in gas (Carlberg 1986; Hernquist et al. 1993;
+Robertson et al. 2006a). Dynamically cold components,
+such as embedded disks and KDCs, are often observed
+in elliptical galaxies (e.g. Franx & Illingworth 1988;
+Jedrzejewski & Schechter 1988; Hernquist & Barnes
+1991; Rix & White 1990; Scorza & Bender 1995;
+Jesseit et al. 2007). The high specific frequencies of
+GCs in ellipticals relative to spirals imply that mergers
+must trigger the formation of many new clusters from
+the available gas (Ashman & Zepf 1992; Fall & Zhang
+2001).
+More recent simulations have shown that merg-
+ers with ∼30% gas produce remnants that fall
+on the observed fundamental plane scaling rela-
+tion (Djorgovski & Davis 1987; Dressler et al. 1987;
+Gonzalez-Garcia & van Albada 2003; Robertson et al.
+2006a; Hopkins et al. 2008c), and that the remnants
+of 1:1 mergers between 40% gas disks match the 1D
+kinematic properties of observed ellipticals far bet-
+ter than dissipationless merger remnants (Cox et al.
+2006a). 2D kinematic maps of gas-rich merger rem-
+nants display many of the intriguing features seen in
+real galaxies, including misaligned rotation, central ve-
+locity dispersion dips, counter-rotating disks, and KDCs
+(Jesseit et al. 2007; Barnes 1992; Hernquist & Barnes
+1991). The shapes of the LOSVDs of simulated
+gas-rich merger remnants display the same trends
+as ellipticals in the SAURON sample (Bender et al.
+1994; Bendo & Barnes 2000; Naab & Burkert 2001;
+Bournaud et al. 2005b; Gonzalez-Garcia et al. 2006;
+Naab et al. 2006a; Hoffman et al. 2009a), and they
+occupy essentially the same part of the anisotropy-
+ellipticity ( δ−ǫ) plane as the SAURON galaxies (with
+thenotableexceptionofsomegiantslowrotatorsthatare
+very round, anisotropic, and featureless; Burkert et al.2008). These results motivate studies of the intrinsic
+orbital structure of gas-rich merger remnants, to see to
+what extent the 3D distribution functions of elliptical
+galaxies can indeed be explained with binary mergers,
+and shed light on how the observable features arise phys-
+ically.
+To our knowledge, Barnes (1992) was the first to apply
+orbital analysis in the style of de Zeeuw (1985); Statler
+(1987) to simulated merger remnants. He ran a series
+of dissipationless disk galaxy mergers with varying disk
+orientations and impact parameters, and classified the
+stellar orbits in the remnants based on the sign changes
+in their angular momentum. He found a wide variety
+in the remnant shapes and orbital structure, depending
+on the encounter parameters. Some of the remnants dis-
+played substantial orientation twists (see also Gerhard
+1983). The mergers often produced both x−tube and
+z−tube populations with significant net rotation, result-
+ing in large kinematic misalignments. Since the rotation
+ofthe majorityof observedellipticals is well-alignedwith
+the major axis, he argued that dissipationless disk merg-
+ers cannot be the generic way to form elliptical galaxies.
+Barnes & Hernquist (1996) classified the stellar or-
+bits in mergers with 10% gas, and found that even
+this small gas component has a dramatic effect on the
+remnant shapes and orbital structure. Gravitational
+torques during the merger drain much of the gas of its
+angular momentum, causing it to collapse inward and
+form a dense CMC (Hernquist 1989; Barnes & Hernquist
+1991; Mihos & Hernquist 1994a,b, 1996; Hopkins et al.
+2009c), essentially a point mass to stars at 1 Re. The
+CMC destabilizes box orbits, leading to a global trans-
+formation of the remnant to a more oblate shape.
+This result is not surprising in light of the studies
+by e.g. Gerhard & Binney (1985); Merritt & Quinlan
+(1998), showing that a central point mass with just 2%
+of the total mass can globally transform the structure of
+a triaxial system.
+Jesseit et al. (2005); Naab et al. (2006a) performed a
+detailedstudyoftheorbitalstructureofalargesampleof
+equalandunequalmassmergerremnantswith0and10%
+gas (Naab & Burkert 2003), with an emphasis on relat-
+ing the intrinsic structure to photometric and 1D kine-
+matic observables. They studied the relation between
+the orbital structure and isophotal shapes in further de-
+tail in Jesseit et al. (2008). Using spectral classification
+(Carpintero & Aguilar 1998), they found that box orbits
+typically dominate the inner parts of the dissipationless
+remnants, while x−tubes and z−tubes become dominant
+at larger radii. The box to z−tube ratio was the primary
+determinant of kinematic properties such as the location
+of the remnants in the h3,4−v/σplanes. When a gas
+component was added, the box population was highly
+suppressed and the remnants became z−tube dominated
+and oblate. The shape of the z−tube orbits in the more
+axisymmetric dissipative remnants made the isophotes
+less boxy. The 1:1 mergerremnants were slowlyrotating,
+while the 3:1 remnants were found to be rapidly rotating
+and disky. They concluded that observed rapidly rotat-
+ing ellipticals could form from dissipative 3:1 mergers,
+but boxy, slowly rotating systems (Kormendy & Bender
+1996) could not have formed through dissipative disk
+mergers.
+In this work we analyze the orbital structure of 1:17
+TABLE 1
+Merger orbits
+Orbit θ1φ1θ2φ2
+i0 0 71 30
+j-109 90 71 90
+k-109 -30 71 -30
+l-109 30 180 0
+m 0 0 71 90
+n-109 -30 71 30
+o-109 30 71 -30
+p-109 90 180 0
+merger remnants in simulations including SF and feed-
+back, enabling us to consider the high gas fractions char-
+acteristicofspiralgalaxiesat z∼2(Stewart et al.2009a;
+Daddi et al. 2009). At fixed fgasthe effect on the rem-
+nant structure may be diminished by including SF and
+the dissipational features may become more spatially ex-
+tended, since the gas can be converted to collisionless
+material early on in the merger and undergo subsequent
+violent relaxation. We quantify the variation of the or-
+bital structure and intrinsic shape with gas fraction for
+fgasranging from 0 to 40%, and discuss the physical
+mechanisms that may be driving the orbital transforma-
+tions, with direct comparisons against dynamical mod-
+els of observed systems in mind. We relate the orbital
+structureofthe remnantsto theirappearancein 2Dkine-
+matic maps on 1 Reand 3Rescales, and show that a wide
+range of different kinematic structures can be accounted
+for simply by varying fgasin 1:1 disk mergers.
+1.4.Outline
+In§2 we describe our merger simulations and remnant
+analysis methods, and in §3 we present our results. The
+intrinsic structure of our eight dissipationless remnants
+is presented in §3.1, as a baseline for understanding the
+effect of adding a gas component. We show how the or-
+bitalstructureandshapeoftheremnantsvarieswith fgas
+in§3.2, and discuss the KDCs and embedded disks that
+arise in the gas-rich remnants in §3.3. In§3.4 we briefly
+describethestructureoftheremnantsbeyond ∼1Re, de-
+ferring a more detailed and quantitative presentation to
+Hoffman et al. (2009b). In §4 we summarize our results
+and conclude. The radial profiles of the orbital struc-
+ture, intrinsic shape, and orientation of all 48 dissipative
+remnants are presented in the Appendix.
+2.SIMULATIONS AND METHODS
+2.1.Simulations
+Our galaxy merger simulations were performed us-
+ing the publicly available TreeSPH (Smoothed Par-
+ticle Hydrodynamics) code Gadget-2 (Springel 2005;
+Springel et al. 2001), which uses an advanced formula-
+tion of SPH that explicitly conserves both energy and
+entropy when appropriate (Springel & Hernquist 2002).
+In addition to the standard features, our version of the
+software includes sub-resolution prescriptions for radia-
+tive cooling (Katz et al. 1996; Dav´ e et al. 1999), SF, andfeedback from supernovae and AGN, as described in de-
+tail in Springel et al. (2005a).
+SFandsupernovafeedbackaretreatedusingthe multi-
+phase prescription of Springel & Hernquist (2003) and
+Springel et al. (2005a). The interstellar medium (ISM)
+consists of a cold cloud phase in which stars form, and
+a hot phase that provides pressure support for the disk.
+The density dependence of the SF rate is calibrated to
+match the observed Schmidt-Kennicutt Law (Schmidt
+1959; Kennicutt 1998). Radiative cooling drives gas
+transferfromthe hot phaseto the coldphase, while heat-
+ing from supernovae feeds gas back into the hot phase.
+The parameter qEOSsmoothly varies the effective equa-
+tion of state (EOS) between isothermal ( qEOS= 0) and
+the pure multi-phase model ( qEOS= 1); higher values
+ofqEOScorrespond to a stiffer EOS and permit stable
+disks at higher gas fractions. In all of the simulations in
+this paper we used qEOS= 0.25.
+The simulations also include sink particles repre-
+senting black holes, which accrete gas at a rate
+˙MBH= min( ˙MBondi,˙MEdd), where ˙MBondi∝
+M2
+BHρgas//radicalbig
+c2s+v2is a rate based on the Bondi-Hoyle-
+Lyttleton formula (Bondi 1952; Bondi & Hoyle 1944;
+Hoyle & Lyttleton 1939), and ˙MEddis the Eddington
+rate. A fraction ǫfǫrof the accretion energy is re-
+leased thermally and isotropically into the surrounding
+gas, where ǫr= 0.1 is the assumed radiative efficiency,
+andǫf= 0.05 is the fraction of the radiated luminosity
+that can couple thermally to the gas.
+The gravitational softening length was ǫ= 140 pc for
+the stellar and gas components in our simulations, and
+ǫ= 290 pc for the halo component. We decreased the
+parameter ηcontrolling the Gadget-2 timestep criterion
+(∆t=/radicalbig
+2ηǫ/|/vector a|, where/vector ais the particle acceleration)
+from the default value of 0.025 to 0.0025, to ensure that
+the steep central cusp that formed in the gas-rich simu-
+lations was preserved for several Gyrs after the merger.
+We set up our merger ICs using standard tech-
+niques presented in Springel et al. (2005a), which have
+been employed in a variety of previous applica-
+tions (e.g. Di Matteo et al. 2005; Naab et al. 2006a;
+Hopkins et al. 2006; Robertson et al. 2006a; Cox et al.
+2006a;Johansson et al.2009;Hopkins et al.2009b). The
+galaxy models consisted of exponential disks embedded
+in dark matter halos; stellar bulges were not included
+in this study. The total mass was specified through the
+virial velocity, v200= 160 km/s for an approximately
+Milky Way-sized galaxy, by Mtot=v3
+200/(10GH0). The
+halo was modeled as a Hernquist (1990) profile with spin
+parameter λ= 0.033, and scale radius chosen to match
+an NFW profile with concentration c= 0.9 in the inner
+parts.
+The disk mass was set to Md=fdMtot, withfd=
+0.041. Its scale length of rd= 3.9 kpc was determined by
+the requirement that it be centrifugally supported with
+the same specific angular momentum as the halo, Jd=
+fdJtot. The stellar component of the disk was assigned a
+radially constant vertical scale height of 0.2 rd, while the
+vertical structure of the gas component was set by the
+requirement of hydrostatic equilibrium. The disk models
+were realized with 80000 equal-mass particles, a fraction
+1−fgasof them in collisionless stars and the other fgas
+in SPH particles. The halo was comprised of 1200008
+collisionless dark matter particles.
+For each simulation two identical disk galaxies were
+constructed following this procedure, and placed on a
+parabolic orbit with an impact parameter of 7.1 kpc. At
+their initial separationof140kpc, the center-of-masstra-
+jectories were well-approximated by point mass orbits.
+At least in the inner parts, the final structure is not
+highly sensitive to our choice of small impact parame-
+ter, since the most bound particles (the stars) lose most
+of their orbital angular momentum to the halo by the
+time the cores merge even in wide encounters (Barnes
+1998; Cox et al. 2006a). The effect of the merger impact
+parameter on the stellar halo structure will be explored
+in future work (Hoffman et al. 2009b).
+The orientation of the disks relative to the merger
+(x−y) planewasparametrizedbythetwoangles θandφ,
+the polar angle of the spin vector relative to the z−axis
+and its azimuthal angle relative to the axis of approach,
+as shown in Figure 6 of Toomre & Toomre (1972). We
+usedtheschemeproposedbyBarnes(1992)tosamplethe
+spaceofpossiblediskorientationsinarelativelyunbiased
+way, including prograde, retrograde, and polar orbits.
+The spin of the first disk coincides with each of the four
+symmetryaxesofaregulartetrahedronpointingupward,
+while that of the second disk points along the axes of the
+corresponding downward-pointing tetrahedron. The re-
+sulting eight merger orbits are listed in Table 1. This set
+of encounter orbits has been used in a number of pre-
+vious studies (e.g. Barnes 1992; Naab & Burkert 2003;
+Naab et al. 2006a; Cox et al. 2006a), which may be used
+asareferencepointforcomparisonwithourresultswhere
+appropriate. The eight encounters listed in Table 1 were
+repeated at seven different gas fractions, fgas= 0, 5, 10,
+15, 20, 30, and 40%, for a total of 56 merger simulations.
+Once on their orbits, the galaxies reach their first peri-
+center passage at t≈0.35 Gyrs, at which point their
+morphologies become strongly distorted by tidal forces
+from the other galaxy, and the stars and gas each form a
+bar. The stellar bar slightly trails the gas bar, torquing
+back on the gas and draining its angular momentum
+so that the gas flows inward and undergoes a burst of
+star SF (Mihos & Hernquist 1994a; Barnes & Hernquist
+1996; Escala et al. 2004; Hopkins et al. 2009b). At t≈
+1.8 Gyrs the galaxies reach second passage, and their
+cores merge shortly thereafter. The strong and persis-
+tent gravitational torques during the second passage and
+final merger typically induce a stronger central starburst
+than at first passage. By about 0.5 Gyrs after the merger
+the size, shape, and velocity dispersion of the remnant
+(measuredwithin ∼1Re) reachasteadystate(Cox et al.
+2006a), and the system may be considered relaxed.
+2.2.Remnant analysis
+We freeze the potential at t= 4.3 Gyrs (about
+2.5 Gyrs after the merger is complete), and represent
+it using the bi-orthogonal “designer” basis expansion
+(Clutton-Brock 1972, 1973; Binney & Tremaine 2008)
+presented in Hernquist & Ostriker (1992),
+ρ(r,θ,φ)=/summationdisplay
+nlmAnlmρnl(r)Ylm(θ,φ) (1)
+Φ(r,θ,φ)=/summationdisplay
+nlmBnlmΦnl(r)Ylm(θ,φ).(2)Thetermsinthis expansionindividuallysatisfyPoisson’s
+equation,andthelowest-ordertermisaHernquist(1990)
+profile, which should be a good first approximation to
+the density profile of a remnant resembling an elliptical
+galaxy. Before computing the expansion coefficients we
+symmetrize the particle distribution by reflecting all of
+the positions and velocities about the origin, effectively
+increasing the particle statistics by a factor of two and
+reducing global fluctuations. The orbit of each stellar
+particle is then integrated through ∼300 radial turning
+points (peri/apocenter passages) in the static potential
+usingaBulirsch-Stoerintegrator(Bulirsch & Stoer1966;
+Press et al. 1992). This technique allows all of the stellar
+orbitstobeefficientlyfollowedfor ∼150dynamicaltimes,
+including halo orbits for which this corresponds to many
+Hubble times. It also lessens spurious relaxation and
+simplifies the orbital analysis by fixing the principle axes
+of the system.
+Each stellar orbit was classified as a box, x−tube, or
+z−tube orbit using a simple algorithm based on the sign
+changes in the star’s angular momentum, introduced by
+Barnes (1992). At each timestep we computed the angu-
+lar momentum vector /vectorj, and checked whether each com-
+ponent of /vectorjhad changed sign since the last step. At
+the end of the integration we constructed the vector /vectork
+such that ki= 1 ifjinever changed sign, and ki= 0
+otherwise. The orbit was assigned the classification code
+C=kx+2ky+4kz, which is 0 for a box orbit, 1 for an
+x−tube orbit, and 4 for a z−tube orbit. For a perfect
+triaxial ellipsoid, other values of Care unphysical.
+This classification scheme depends on a correct speci-
+fication of the system orientation, and many of the rem-
+nants have intrinsic orientation twists. We therefore di-
+agonalized the inertia tensor, in the form Iij=/summationtextxixj
+(e.g. Aguilar & Merritt 1990), in ∼50 cumulative en-
+ergy bins, and used the local orientation based on the
+star’s energy for the classification. The eigenvectors of
+Iijare the principal axes of the remnant’s figure, and
+the square roots of the eigenvalues give the relative scale
+lengths along the three principal axes, a,b, andc. All
+particles - gas, stars, and dark matter - were included in
+Iij, since a star’s orbit is determined by the sum of all
+three components. Note that since we used the local ori-
+entation for the classification, e.g. the position angle of
+thez−tuberotationmayappeartovarywith galactocen-
+tric radius in the remnants with significant orientation
+twists.
+Only∼1% of the stellar orbits were not assigned C
+values of 0, 1, or 4, except in rare instances of sud-
+den orientation twists or locations where the potential
+was very nearly spherical. The results of our simple
+classification scheme were consistent with spectral clas-
+sification (Carpintero & Aguilar 1998; Binney & Spergel
+1982, 1984; Hoffman 2007), and we determined that the
+simpler algorithm was better suited for the noisy poten-
+tials and gross analysis desired in this work. However
+note that this algorithm does not distinguish between
+boxes, resonant boxlets, and ergodic orbits.
+For each remnant, we present radial profiles of the or-
+bital structure and intrinsic shape as shown for one ex-
+ample in Figure 3. First we order the stellar particles
+by energy, and divide them into ∼20 energy bins with
+equal numbers of particles. The mass fraction of stars9
+Fig. 3.— Radial profiles of the orbital structure and intrinsic shape of remnant i, 40% gas. Left:Fraction of the stellar mass
+in each of the three major orbital classes vs. percentile in b inding energy. The horizontal axis is labeled by the mean rad ius (in units of
+Re) of the stars in the current energy bin (see text for details) .Left center: Rotation bias, β= (N+−N−)/(N++N−), of the x−tube
+andz−tube orbits. β= 0 corresponds to perfect cancellation of the rotation, whi leβ=±1 means that all of the stars are streaming in
+the same direction. We choose the overall sign so that βis positive at the 40th percentile in binding energy (1 Re).Right center: Intrinsic
+shapes of the remnants. The heavy green line is the triaxiali ty parameter, T, for all particles in the current energy bin and more bound.
+The black line is the intrinsic ellipticity ǫ=c/a, and the red dashed line is the smaller of the other two axis ra tios. The horizontal line
+atT= 0.5 marks the boundary between oblate and prolate shapes. Both stars and DM particles are included in the evaluation of the
+intrinsic shapes. Right:Intrinsic orientations ( θsymm,φsymm) and kinematic misalignments (Ψ kin), in degrees. The red solid and dashed
+lines are the polar ( θsymm) and azimuthal ( φsymm) angles of the symmetry axis, in this case the short axis sinc e the remnant is oblate. In
+general we will plot θ,φof the short axis in red if the remnant is oblate ( T <0.5) at 1Re, andθ,φof the long axis in blue if the remnant
+is prolate ( T >0.5) at 1Re.θandφare evaluated in a coordinate system where the disks’ initia l orbital angular momentum points in
+the ˆzdirection, and the galaxies initially approach one another along the x-axis. To evaluate Ψ kin(equation 3; black line), we return to
+the local principal axes coordinate system. In all panels, t he 1Rescale probed by SAURON and the 3 Rescale typical of SKiMS data are
+marked with vertical lines.
+in box,x−tube, and z−tube orbits is computed for each
+energy bin, and plotted vs. percentile in binding energy
+(by mass), as in the left-hand panel of the figure. For
+ease of comparison with observational results, we also
+compute the mean galactocentric radius of the particles
+in each energy bin, and label the bins corresponding to
+0.3, 1, 3, and the largest integer number of Re.
+Forunderstandingkinematicmaps, itisusefultoquan-
+tify not only the populations of the major orbital classes,
+but also their degree of streaming. For this purpose we
+plot profiles of
+β= (N+−N−)/(N++N−) (3)
+for thex−tube and z−tube orbits, as shown in the left
+center panel. Here N+andN−are the numbers of par-
+ticles rotating in either direction; β= 0 corresponds to
+perfect cancellation of the rotation, while β=±1 means
+that all of the stars are streaming in the same direction.
+The sign conventionwas chosen by requiring βto be pos-
+itive at the 40th percentile in binding energy (1 Re).
+The intrinsic shape and orientation vector are com-
+puted by diagonalizing the inertia tensor as described
+previously. We plot the maximum and minimum axis
+ratios,ǫ=c/aandrmin= min(b/a,c/b), and the triaxi-
+ality parameter,
+T= (a2−b2)/(a2−c2), (4)
+in the right center panel of Figure 3. Tranges from 0 for
+a perfectly oblate spheroid to 1 for a prolate spheroid. In
+the right-hand panel we plot the polar and azimuthal an-
+gles (θsymmandφsymm) of the symmetry axis - the long
+axis ifT >0.5 within 1 Re, or the short axis if T <0.5.
+If the short axis orientation is plotted then we color the
+lines red; otherwise we color them blue. The shape of a
+remnant that is oblate within 1 Remay become prolatefarther out; in this case we still plot the orientation of
+the short axis at all radii, which does not make the an-
+gles ill-defined since the remnants are never too close to
+axisymmetricat largeradii. Note that θsymmandφsymm
+represent angles between a vector and an axis(whose di-
+rection is defined only up to a sign), and therefore range
+from 0 to 90◦. When plotting the orientation profiles, we
+use a fixed coordinate system where the z−axis points
+alongthe disks’ initial orbitalangularmomentum vector,
+and thex−axis is along the initial direction of approach.
+On the same axes we plot the intrinsic kinematic mis-
+alignment,
+Ψkin= arctan |jz/jx|, (5)
+or the angle between the short axis and the net angular
+momentum vector. To compute Ψ kinwe revert back to
+the local principal axes system, so that Ψ kin= 0 always
+corresponds to rotation about the intrinsic short axis.
+To place our work in an observational context, it is
+also useful to show how the intrinsic orbital structure
+imprints itself on SAURON-like 2D kinematic maps. We
+constructed histograms of the LOS velocity in 40 ×40
+spatial bins within 1 Re, using 80 velocity bins within
+±3.5σ. For each LOSVD we performed a least-squares
+fit to the 5-parameter function
+F(y) =Ae−w2/2[1+h3H3(w)+h4H4(w)],(6)
+wherew= (y−v)/σ, andvandσrepresent the mean
+velocity and velocity dispersion. Here H3andH4are
+Gauss-Hermite (GH) polynomials, and the parameters
+h3andh4measure the skewness and kurtosis of the dis-
+tribution (van der Marel & Franx 1993; Gerhard 1993;
+Binney & Merrifield 1998). The Gadget particles were
+smoothedoveraradius hsmooth= max(hsee,hngb), where
+hsee= 150 pc corresponds to a seeing of 1.5” at 20Mpc,10
+andhngbis 1.7 times the distance to the 128th nearest
+neighbor.
+We focus primarily on the velocity fields in this work.
+Notethat v, asderivedfromthefitofequation4,deviates
+systematically from the true mean of the distribution for
+nonzeroh3andh4(see van der Marel & Franx 1993 for
+the relevant correction terms). However we follow the
+convention of previous authors (e.g. Bender et al. 1994)
+andsimplyuse vfromtheGHfittorepresentthevelocity
+in this paper.
+When displayingallofthe remnantsat agiven fgas, we
+choose the viewing angles randomly (from an isotropic
+distribution) to give a fair representation of their ap-
+pearance in kinematic maps. We label each map with
+the LOS angles θandφ, whereθis the polar angle be-
+tween the LOS and the z−axis, and φis the azimuthal
+angle between the LOS and the y−axis. The apparent
+major-axis rotation is maximized when θ= 90◦, and the
+minor-axis rotation is maximal at φ= 0 for a given θ.
+3.RESULTS AND DISCUSSION
+3.1.The dissipationless remnants
+The structure of the dissipationless remnants is pre-
+sented in Figure 4. There is substantial variation in
+the orbital distribution and shape over the eight merger
+orbits - for instance remnant jis uniformly prolate
+and dominated by x−tube orbits, while remnant ois
+oblate-triaxial and composed primarily of box orbits and
+z−tubes. Overallthe remnantsaredominated by boxor-
+bits in their inner parts ( r/lessorsimilar1.5Re), and tube orbits in
+their outskirts. This is not surprisingsince the Hernquist
+(1990)profile,whichgenericallyprovidesagoodfittothe
+density profiles of dissipationless merger remnants, has a
+shallow inner cusp ( ρ∝r−1) and much steeper outer
+profile (ρ∝r−4)1.
+The inner shapes of the remnants aretypically prolate-
+triaxial and highly flattened ( T∼0.8,ǫ∼0.6), consis-
+tent with previous results for head-on mergers of stellar
+systems that have shed most of their orbital angular mo-
+mentum to the halo (Miller & Smith 1980; White 1983;
+Barnes 1992; Cox et al. 2006a). Farther out the rem-
+nants become rounder ( ǫ∼0.8) and closer to maximally
+triaxial. See also Dubinski (1994); Kazantzidis et al.
+(2004); Debattista et al. (2008); Valluri et al. (2009);
+Abadi et al. (2009); Romano-Diaz et al. (2009), who
+studied halo shapes in a cosmological context and found
+that the response of the DM to baryonic condensation
+into galaxies drives the halos to rounder and more oblate
+shapes.
+There is substantial cancellation of the prograde and
+retrograde rotation of the z−tube orbits ( βnear zero),
+especially in the inner parts. On the other hand the
+x−tube populations are highly streaming, with βtyp-
+ically approaching one at large radii. Remnant jis a
+notable exception to this rule, with the rotation of its
+dominant x−tube orbits canceling nearly perfectly over
+the full range of radii plotted. Upon closer examination
+of this remnant, it turns out that this owes to the sym-
+1A star in what we are calling the “outer parts” ( ∼3−6Re) still
+occupies the innerDMhalo(which alsofollowsroughly a Hernquist
+1990 profile but with a much larger scale radius). The total de nsity
+(stars + DM) at these radii scales roughly like r−2- still steep
+compared to r−1, but not nearly as steep as r−4.metry of the ICs. Both disks on orbit jare substantially
+inclined, but their spins point in opposite directions. If
+thex−tube orbits in the remnant are separated by their
+disk of origin, nearly all of those from disk 1 rotate in
+one sense about the long axis, while those from disk 2
+rotate in the opposite sense, giving the net cancellation
+seen in Figure 4.
+The dissipationless remnants do not show strong ori-
+entation twists, and their long axis (which is typically
+the axis of symmetry, since the remnants are prolate)
+lies in the merger plane ( θsymm= 90◦), along the axis of
+the final coalescence of the stellar components. This is
+consistent with the interpretation of Novak et al. (2006):
+the time-varying tidal field along the merger axis cou-
+ples coherently to the stellar orbits, producing a bounce
+that elongates the remnant along the direction of tidal
+compression (Miller 1978; White 1983). The kinematic
+misalignments are large since the net rotation tends to
+favor the long axis, and noisy since the magnitude of the
+rotation is often small.
+In the 1Rekinematic maps, the dissipationless rem-
+nants are generally slowly rotating (e.g. remnants jand
+p) or are dominated by minor-axis rotation (e.g. rem-
+nantsmandn). Some of the remnants (e.g. pand
+l) definitely show both major and minor axis rotation,
+though small in magnitude ( v/σ∼0.15).
+Figure 4 is a baseline for understanding how a gas
+component affects the remnants - it may be compared
+with Figures 15 - 20 in the Appendix to see how the or-
+bital structure evolves for each merger orbit as fgasis
+increased from 0 to 40%.
+3.2.Variation with gas fraction
+In Figure 5 we illustrate the effect of dissipation on
+the remnant structure by varying the gas fraction for a
+fixed merger orbit, i, whose structural transformations
+are representative of the overall trends. Case iis an
+encounter between a prograde disk in the merger plane
+(θ1=φ1= 0), and a highly inclined disk that is also
+slightly prograde ( θ2= 71◦,φ2= 30◦). The orbital
+structure, shape, and orientation profile of this remnant
+are shown for seven different gas fractions in Figure 5.
+Unlike in Figure 4, the velocitymaps areall in projection
+alongthe y−axis, toclearlyshowthe rotationofboth the
+x−tube and z−tube orbits and how it varies with fgas.
+Asfgasincreases from 0 to 40%, the population of box
+orbits within ∼1.5Redeclines, and the z−tube popu-
+lation increases. The most rapid change in the orbital
+populations occurs between 0 and 20% gas. The outer
+orbital structure ( r/greaterorsimilar2Re) is relatively unaffected by
+the gas, and the x−tube population does not display an
+obvious trend.
+The variation in the intrinsic shape closely follows the
+change in the orbital populations. As fgasincreases, the
+inner parts become progressively more oblate; the 30-
+40% gas remnants are nearly oblate axisymmetric within
+1Re(T/lessorsimilar0.1). The remnants also become rounder with
+higherfgas:ǫincreases from ∼0.6 for the dissipationless
+remnant to ∼0.8 at the highest gas fractions.
+Though the mass in z−tube orbits within ∼2Rerises
+substantially as fgasincreases from 0 to 15%, the rem-
+nants show little change in their major-axisrotation over
+this range, since the mass in prograde and retrograde or-
+bits is nearly equal ( β∼0). The 15% gas remnant,11
+Fig. 4.— Intrinsic structure of the dissipationless remnants. The first four columns follow the format of Figure 3. The merge r
+orbit (see Table 1) is specified on the left for each row. In the right-hand column we show a 1 Revelocity map of the remnant, viewed along
+a randomly (isotropically) chosen LOS. The two angles speci fying the LOS are indicated by the labels on the right. θis the polar angle
+between the LOS and the short axis (this axis was chosen as the reference throughout the paper since most of the remnants wi th gas are
+oblate), and φis the azimuthal angle between the LOS and the intermediate a xis. The major-axis rotation is maximal at θ= 90◦, and the
+minor-axis rotation is maximized at φ= 0 for fixed θ. The velocity scale runs from −vmaxtovmax, wherevmaxis the maximum value of
+|v|. The value of vmax(in km/s) is indicated by the label in the upper right corner o f each map.12
+Fig. 5.— Evolution of the intrinsic structure of remnant iwith gas fraction. The format is the same as in Figure 4, except
+that now each row corresponds to a different fgas. In this figure all of the velocity maps are shown in projectio n along the intermediate
+axis, (θ,φ) = (0,0), to allow the major- and minor-axis rotation in the remnan ts at different fgasto be compared directly.13
+Fig. 6.— Cancellation of the z−tubes and streaming of
+thex−tube orbits. Upper left: Velocity map of remnant m, 10%
+gas, viewed in a near-intermediate axis projection. Upper right:
+Same map with x−tube orbits removed. Lower left: LOSVD at
+location A, near the major axis. Lower right: LOSVD at location
+B, near the minor axis. The velocity scale is the same on both
+maps.
+though dominated by z−tubes within 1 Re, still appears
+as a minor-axis rotator in the projected velocity map.
+In the 20% gas remnant, the innermost z−tube orbits
+begin to show substantial streaming, in a sense opposite
+the net rotation farther out. This inner streaming shows
+up as a prominent KDC in the velocity map. Note that
+the minor-axis rotation present at lower gas fractions is
+still there in this map, but it is dwarfed on the veloc-
+ity scale by the rapid rotation within the KDC. In the
+30-40%gas remnants, the z−tube orbits show significant
+streaming ( β∼0.3) over the full range of radii plotted,
+and the edge-on kinematic maps display rapid, disk-like
+major-axis rotation. This rapid rotation about the in-
+trinsicshortaxisgiveskinematicmisalignmentsnearzero
+in the inner parts.
+Although the rotation bias of the z−tube orbits in the
+30% gas remnant appearsfairly uniform, the cause of the
+streaming in the inner and outer parts is quite different.
+In the outer parts the net angular momentum is retained
+from the ICs, and is therefore present at all gas fractions.
+The inner streaming is a direct consequence of dissipa-
+tion - the gas that retains its angular momentum during
+the merger re-forms a cold disk as it dissipates energy,
+which forms new stars primarily after the violent relax-
+ation of the merger is complete. Thus the inner part of
+the gas-rich remnants is a superposition of a hot, slowly-
+rotating component consisting of the stars present before
+the merger (hereafter referred to as “old stars”), and a
+maximally rotatingdisk composed of“new stars”formed
+through dissipation. This is why βisonlyaround 0.3 de-
+spite the high maximum velocities in the maps. We will
+elaborate on this point in section 3.4.The inner parts of the gas-rich remnants are often
+rapidly tumbling, which produces large intrinsic orien-
+tation twists between ∼1 and 3Re, as in the bottom two
+rows of in Figure 5. Note that the orientations in the
+inner parts of these remnants are an arbitrary snapshot
+in time; had the simulation been frozen earlier or later
+θsymmandφsymmcould have quite different values.
+In the preceding discussion we have demonstrated that
+the low-fgasremnantsshowlittle major-axisrotationnot
+because they lack z−tube orbits, but because there is a
+largedegreeofcancellationbetweenthe z−tubesrotating
+in opposite directions. An observer who measured the
+major-and minor-axisrotationspeeds of the typical 10%
+gas remnant might naively think that the system was
+prolate in shape and dominated by x−tube orbits, when
+in fact the predominance of the minor-axis rotation is
+caused by a sub-dominant population of x−tubes that
+are highly streaming.
+We highlight this point in Figure 6 showing remnant
+mat 10% gas, which has comparable masses in x−tubes
+andz−tubes within 1 Re, but somewhat more z−tubes
+(see Figure 16). The upper left panel is a velocity map
+of the remnant in a nearly edge-on projection (maximiz-
+ing both the major- and minor-axis rotation), and the
+second panel shows the same map with the x−tube or-
+bits removed. Without the x−tubes, the remnant shows
+almost no rotation at all. The LOSVDs at the locations
+marked by the white letters are plotted in the lower two
+panels. The lower left panel clearly shows the canceling
+streams of z−tube orbits along the major axis, while the
+lower right panel depicts the highly streaming x−tubes
+along the minor axis, peaked at nearly 200 km/s. In the
+lower left panel, the large population of box orbits fills
+in the gap in the bimodal z−tube distribution, yielding
+a total LOSVD that is nearly Gaussian in shape.
+Figure 7 provides insight into the physical explanation
+forthis orbitalstructurebymappingoutthe locations, in
+the original disks, of the stars that end up in each of the
+three orbital classes in remnant i, withfgas=0, 15, and
+30%. Disk 1 is the direct disk, and disk 2 is the inclined
+one. Only old stars are included in this plot, so the dissi-
+pativediskcomponentinthe30%gasremnantisleft out.
+Each disk was partitioned into cylindrical bins contain-
+ing 300 stars apiece, and the bins were labeled accord-
+ing to which orbital class formed the plurality of stars
+among those 300 in the final remnant - black for boxes,
+green for z−tubes, and red for x−tubes. The bins fa-
+voring tube orbits were further labeled based on whether
+they tended to stream in one direction in the final rem-
+nant, or to have canceling rotation. Bins with β >0.3
+are indicated with upward-pointing arrows and labeled
+as “counterclockwise,” while those with β <−0.3 are
+indicated with downward-pointing arrows and labeled as
+“clockwise.” Bins with |β|<0.3 are labeled as “cancel-
+ing” and indicated with filled circles. The blue circles
+mark the half-mass radii of the original disks.
+In the dissipationless simulation, most of the stars
+within the half-mass radius become box orbits in the fi-
+nal remnant. When a dissipative component is added
+(second and third columns), the set of stars that be-
+came box orbits in the collisionless case instead form
+a population of z−tube orbits with largely canceling
+rotation, especially in disk 2. It is apparent that
+this cancellation does notarise primarily from the ICs,14
+Fig. 7.— Spatial distribution in the original disks of stars that end up on box, x−tube, and z−tube orbits, for merger
+orbiti.Disk 1 (top row) is on a direct, prograde orbit ( θ1=φ1= 0), while disk 2 (bottom row) is inclined and prograde ( θ2= 71◦,
+φ2= 30◦). Each disk was partitioned into cylindrical bins containi ng 300 stars each, and we identified the dominant orbital clas s of the
+stars from each bin in the final remnant. If tubes were dominan t, then we also computed βfor the bin. Bins with |β|<0.3 are labeled
+with circles, while those with |β|>0.3 are labeled with triangles pointing up or down according to the sense of rotation (clockwise or
+counterclockwise, where the overall sign is arbitrary). Th e key for different orbital classes is given in the legend. The three columns show
+the remnants at 0, 15, and 30% gas. Only old stars are included (the gas particles in the initial disks are excluded). The bl ue circles mark
+the half-mass radii of the original disks.
+since it occurs for stars in the same disk. Note that
+the spatial boundary of the canceling z−tube popu-
+lation changes little between 15 and 30% gas, sug-
+gesting that the boundary is set by a bar forma-
+tion criterion that determines which orbits “would have
+been” boxes (e.g. Miller & Smith 1979; Norman & Silk
+1983; Barnes & Hernquist 1996; Bournaud et al. 2005a;
+MacMillan et al. 2006; Hopkins et al. 2009b), rather
+than by the direct gravity of the dissipative compo-
+nent. The box population might be fully transformed
+intoz−tubes so long as a critical gas mass is exceeded.
+It should also be noted that the efficiency with which the
+gas is torqued inward decreases with increasing fgas, so
+the CMC mass comprises a smaller fraction of the ini-
+tial gas mass in higher- fgasremnants (see Figure 7 of
+Hopkins et al. 2009b).
+Stars outside the half-mass radius in disk 1 generally
+becomez−tube orbits in the remnant, with a higher de-
+gree of streaming than in the inner parts. The fate of
+these stars is relatively insensitive to fgas, though some
+box orbits from this region in the dissipationless rem-
+nant do become z−tubes in the simulations with gas
+(compare the top right and top middle panels). The
+stars originating in the outer part of disk 2 end up in a
+uniformly streaming x−tube population, irrespective of
+fgas. Their streamingabout the long axis of the remnant
+can be explained simply in terms of the merger geome-try - see e.g. Figure 3 of Novak et al. (2006). The long
+axis lies in the merger plane, so stars on nearly circular
+orbits in the outskirts of an inclined disk start out with
+largetangentialvelocitycomponents aboutthis axis, and
+are therefore liable to occupy the x−tube “start space”
+(e.g. Schwarzschild 1979, 1993). The uniform rotation
+of these orbits in the final remnant suggests that they do
+not cross orbital boundaries during the merger; their an-
+gular momentum is retained from the initial spin of the
+inclined disk, and is a direct signature ofthe dynamically
+cold nature of the progenitor galaxy. Figure 7 provides
+a good example of how orbital classification can isolate
+sets of stars with similar physical histories.
+So far we have focused the discussion on individual
+remnants; Figure 8 shows the dependence of the orbital
+structure on fgas, averaged over all eight merger orbits.
+We compute the structural parameters on ∼1Rescales
+since this radius has been the main focus of triaxial dy-
+namical modeling efforts to date (e.g. van de Ven et al.
+2008; van den Bosch et al. 2008; de Lorenzi et al. 2006;
+van den Bosch 2008), and defer further discussion of the
+outer orbital structure to section 3.5.
+As expected fromthe preceding discussion, the z−tube
+population increases, the mass in box orbits decreases,
+and the shapes become rounder and more oblate with
+increasing fgas. This transformation occurs primarily
+between 0 and 15% gas. At higher gas fractions the av-15
+Fig. 8.— Inner structure ( /lessorsimilar1Re), averaged over eight merger orbits. Upper left: Fraction of the stellar mass in each orbital class
+vs.fgas, for the 25-45th percentile in binding energy ( ∼0.5−1.3Re).Upper right: Rotation bias ( β) of thex−tube and z−tube orbits, as
+defined in equation 1. Lower left: Intrinsic shape of the 40% most bound particles. The filled di amonds are for all matter (including DM
+particles), while the asterisks are for stars only. Lower right: Intrinsic orientation twist between the 25th and 50th perce ntile in binding
+energy, and kinematic misalignment (Ψ kin) for the 40% most bound particles. Ψ kinis defined as in Figure 3. The orientation twist is
+computed as max( θij),i25<(i,j)< i50, the maximum angle between the symmetry axes computed withi n any two cumulative energy
+bins between the 25th and 50th percentiles.
+erage shapes, box, and z−tube fractions level off, con-
+sistent with the “critical mass” interpretation of the box
+orbit suppression. The apparent break in the trend at
+30% gas arises because these remnants are very round.
+In aremnantthat is verynearlyspherical, the distinction
+between the two classes of tubes is blurred, and the tri-
+axiality parameter and orientation vector are ill-defined
+(see e.g. remnant min Figure 19). The 40% gas rem-
+nants are slightly more flattened than those at 30% gas,
+presumably owing to the rotation of the stronger em-
+bedded disks. The stellar component is slightly more
+flattened than the mass including the DM, particularly
+at lowfgas. There is a mild downward trend in the pop-
+ulation of x−tube orbits with fgasbetween zeroand 15%
+as the shape of the potential becomes more oblate, but
+the round potentials at higher gas fractions are slightly
+more amenable to x−tubes.
+At lowfgasthex−tube orbits tend to stream far more
+than the z−tube orbits. This trend reverses at the high-
+est gas fractions owing to the rapid rotation of the em-
+bedded disks, and the direct modification of the gravita-
+tional potential by the large CMC, which may blur thelines between x−tube and z−tube orbits as the potential
+becomes very round. The streaming of the x−tube or-
+bits might also fall off because of the figure rotation; the
+long axis no longer necessarily lies in the merger plane,
+so more stars may be randomly scattered onto tangen-
+tial orbits about the long axis. The exact reason for
+this trend in the x−tube rotation needs further investi-
+gation. The cancellation of the z−tube rotation is most
+complete around 10% gas, perhaps because the stronger
+CMC more efficiently disrupts box orbits than at lower
+fgas, while at higher fgasthe dissipation begins to play
+a direct role.
+In the high- fgasremnants, the fraction of box orbits
+often turns upward in the very center. This effect is
+partially hidden in e.g. Figure 20 since we started the
+profiles at the 15th percentile in binding energy. It can
+be naturally explained since only the innermost compo-
+nent (the CMC itself) contains no smaller CMC, and so
+can support stable box orbits. The violent relaxation
+during the final merger of the two cores provides a nat-
+ural setting for the onset of triaxiality. However the box
+population computed within the central starburst is ex-16
+Fig. 9.— A variety of disk-like KDCs at 15-20% gas. Top row: 1Revelocity maps. Bottom row: Corresponding h3maps. The
+color scaling and labels are as in previous figures. Each colu mn is labeled at the top with the merger orbit and gas fraction , and at the
+bottom with the LOS.
+pected to be highly sensitive to numerical effects, includ-
+ing artificial flattening of the core density profile owing
+to the gravitational softening, and imprecise centering in
+the orbital classification, so we chose to omit the very
+innermost scales from the structural profiles.
+The remnants do not typically show large orientation
+twists within the 50th percentile in binding energy, or
+∼1.5Re(excluding the KDCs, since the shape may not
+be well-resolved on such small scales). The low fgas
+remnants typically have large kinematic misalignments,
+while the angular momentum of the 40% gas remnants
+is well aligned with the short axis of the potential.
+Profiles of the orbital structure, shape, and orientation
+of all 48 dissipative remnants (5-40% gas) are shown in
+Figures 15 to 20 of the Appendix. We have outlined the
+main trends with fgasin this section, but there is sub-
+stantial variation in the structural transformations from
+one merger orbit to the next. Some of the remnants ( l
+andp) are alreadytransformedinto oblate, z−tube dom-
+inated systems at 5% gas, while others (e.g. iandk) are
+still box-dominated and substantially prolate at this gas
+fraction. Remnant oisoblateat0%gasandbecomespro-
+late at 5% gas. A small gas component transforms rem-
+nantjfrom a nearly pure prolate spheroid dominated by
+x−tube orbits, to a triaxial system with a large z−tube
+population. As the CMC alters the shape of the poten-
+tial, both directly and indirectly through its interactions
+with the stars, the phase space boundaries between the
+orbital classes move. This shifting of the orbital bound-
+aries can induce further changes in the potential, leading
+to a cascade effect. The causes of the more complex or-
+bital transformations in some of the individual remnants
+are an interesting topic for future study.
+3.3.Kinematically distinct cores and embedded disks
+The 15-40% gas remnants often contain a cold z−tube
+population that appears as a KDC or embedded disk in
+∼1Revelocity maps, similar to the sub-components ob-
+servedinsomeoftheSAURONgalaxies(Krajnovic et al.
+Fig. 10.— Orbital structure of a typical KDC remnant.
+Upper left: Velocity map of remnant i, 20% gas, viewed in an
+oblique projection. Upper right: LOSVD at location A, near the
+majoraxis withinthe KDC. Lower left: LOSVD at location B, near
+the major axis outsidethe KDC. Lower right: LOSVD at location
+C, near the minor axis. Color scaling and labels are as in prev ious
+figures.
+2008).
+This component typically appears as a KDC in the 15-
+20% gas remnants, with a large kinematic twist relative
+to the stars around 1 Re. Figure 9 depicts five selected
+KDC systems, from a variety of different remnants and
+viewing angles. 1 Revelocity maps are presented in the17
+top row, while the bottom row shows the correspond-
+ing maps of h3to establish the disk-like character of the
+KDCs. In some cases there are also large photometric
+twists on the KDC scale, as in column 4 of Figure 9.
+The KDCs are visible from nearly all viewing angles in
+the∼50% of the remnants that have them (see Fig-
+ures 17 - 18).
+Theouterpartsofthe KDC remnantsshowavarietyof
+different types of kinematics, depending of the remnant
+and projection - some show no rotation (e.g. column 1 of
+Figure 9), some minor-axis or oblique rotation (columns
+2 and 3), and some a significant amount of major-axis
+rotation either aligned with or counter to the KDC (but
+always less disk-like than the KDC; e.g. columns 2 and
+4).h3andvare always tightly anticorrelated within
+the KDC, but may have a positive (columns 2 and 5) or
+weaker negative (column 4) correlation outside the core.
+Some of the remnants show multiple kinematically dis-
+tinctparts,e.g. remnant iat20%gasdisplaysminor-axis
+rotation with h3andvcorrelated, and major-axis rota-
+tion counter to the KDC, with h3andvanticorrelated,
+at the outer edge of the map.
+The cores of the KDC remnants are not nearly as dis-
+tinct in their underlying orbital structure as they are
+in their kinematics, as van den Bosch et al. (2008) ob-
+served in the dynamical modeling of NGC4365. This is
+because the mass of the streaming population produc-
+ing the KDC is generally small, and the KDCs consist of
+an oblate z−tube population on top of a system already
+oblate and dominated by z−tube orbits because of the
+strong CMC. Figure 10 illustrates this point by showing
+Fig. 11.— Embedded disk formed by dissipation. Upper
+left:Velocity map of remnant i, 40% ga s, viewed at ( θ,φ) =
+(72,77)◦.Upper right: Same map with the new stars, formed fro
+m gas during the merger, removed. Lower left: Same map with
+only the 15% of the new stars that form ed lastremoved. Lower
+right:LOSVD at location A, broken down by orbital class and
+stellar ag e. Color scaling and labels are as in previous figur es.the orbital breakdown of the LOSVDs at three different
+locations in the KDC remnant depicted in the second
+column of Figure 9: one within the KDC (point A), one
+along the major axis outside the KDC, in a region show-
+ing no rotation (point B), and one near 1 Realong the
+minor axis, in the part dominated by minor-axis rota-
+tion (point C). The orbital population at all three loca-
+tions is (perhaps surprisingly) comprised of roughly the
+same mixture of z−tube,x−tube, and box orbits, with
+z−tube orbits always in the majority. The very differ-
+ent appearanceof the three regionsin the kinematic map
+owes to large differences in the degree of orbital stream-
+ing - at point A the z−tubes are highly streaming; the
+x−tubes are highly streaming at point C; and at point B
+bothclassesoftubes aredistributed symmetricallyabout
+v= 0.
+In the 30-40% gas remnants, the scale of the disk com-
+ponent is typically large enough to fill most or all of the
+SAURON-scale kinematic maps. The embedded disk in
+these remnants is typically quite cold, because the disk
+stars form in a slow trickle extending over ∼2 Gyrs af-
+ter the merger (Robertson et al. 2006b; Hoffman 2007)
+and are therefore subject to very little violent relaxation.
+Figure 11 shows remnant iat 40% gas, which has one of
+the strongest embedded disks in our sample, at a view-
+ing angle about 30◦from edge on. The contribution
+from old stars, present as collisionless particles in the
+initial disks, is shown in the upper right panel. When
+only old stars are included, the maximum velocity on
+the map drops from 135 to 66 km/s, and the rotation
+is no longer well-aligned with the major axis. The old
+stars do still display more major-axis rotation than most
+remnants at low fgas, since they also contract and spin
+up in response to the rapid condensation of the gas (e.g.
+Eggen et al. 1962; Blumenthal et al. 1986; Gnedin et al.
+2004; Villalobos et al. 2009).
+The lower right panel shows what is left when only the
+15%ofthe new starsthat formed latest(∼5% ofthe total
+stellar mass) are removed. The maximum velocity still
+plummets to 72 km/s, and what remains appears similar
+to a KDC system, since the most recently-formed stars
+dominate the outer portions of the embedded disk. The
+contributions of old and recently-formed stars to a typi-
+cal LOSVD near 1 Rein the embedded disk is shown in
+the lower right panel. The stars that formed late have
+a much smaller velocity dispersion, and when superim-
+posed on the hot old population they yield a velocity
+distribution strongly skewed opposite the mean velocity
+(see also Hoffman et al. 2009a).
+3.4.Outer structure
+We have shown that the outer parts of the remnants
+(outside ∼1.5Re) are less relaxed than the inner parts,
+and are largely unaffected by the gas content of the disks
+(see Figures 5 and 7). While kinematic maps of gas-rich
+remnantsaredominated by dissipationalrotationon 1 Re
+scales,theirkinematicsfartheroutreflectsthe“dry”part
+of the merger remnant that is left essentially untouched
+by the gas, resulting in large kinematic twists between
+around 1 and 3 Re.
+This point is illustrated clearly in Figure 12, which
+compares velocity maps of remnant mat 0 and 40%
+gas on three different scales, 1, 3, and 7 Re, in the same
+obliqueprojection. Profilesofthe orbitalpopulations are18
+Fig. 12.— Kinematic transitions in the outer parts. The top row shows dissipationless remnant m, vie wed at ( θ,φ) = (68,35)◦,
+while the bottom row shows the same remnant at 40% gas, along t h e same LOS. Left column: 1Revelocity maps. Left center: 3Revelocity
+maps, with th e 1 Remaps superimposed inside the boxes. Right center: 7Revelocity maps, with the 1 and 3 Remaps inside the boxes.
+Right:Radial profiles of the orbital content. All maps are plotted o n the same velocity scale. Gaussian (rather than GH) fitting w as used
+in this figure because of the nois y LOSVDs at large radii.
+reproduced in the right-hand panel for comparison. On
+1Rescales the dissipationless and gas-rich remnants look
+entirely different, the former being a minor-axis rotator
+while the latter is dominated by rapid disk-like rotation.
+However when we zoom out to 3 Re, the similarity in
+their outer structure becomes apparent. The gas has
+thoroughly transformed the inner part of the 40% gas
+remnant, while the outer parts are left as they were in
+the dissipationless case. The same observation can be
+made from the orbital profiles - the orbital populations
+outside∼1.5Reare similar in the 0 and 40% gas rem-
+nants, but are radically different within 1 Re, where the
+dissipationless remnant is comprised mostly of box or-
+bits and the 40% gas remnant is dominated by z−tubes.
+The similarity in the outer kinematic maps is even more
+apparent on 7 Rescales.
+Sharp kinematic and orbital transitions such as those
+in Figure 12 will be clearly observable in new sur-
+veys such as SMEAGOL, probing the stellar orbital
+structure out to ∼3Rescales (Proctor et al. 2009;
+Foster et al. 2009), or the PN.S survey, probing the
+kinematics out to ∼7Rescales using planetary nebu-
+lae as tracers (Douglas et al. 2002; Romanowsky et al.
+2003; Coccato et al. 2009). Predictions for the statistics
+of kinematic twists and abrupt changes in the rotation
+parameter, λR(Emsellem et al. 2007), in such surveys
+will be further discussed and quantified in Hoffman et al.
+(2009b).
+The intrinsic outer structure of the remnants is sum-
+marized in Figure 13, which is the same as Figure 8 but
+for the structure between ∼3 and 7Re. The orbitalpopulations, shapes, and orbital streaming are far less
+sensitive to fgasthan in the inner parts. The outer stel-
+lar population is dominated by tube orbits, as expected
+given the steep effective density profile at these radii,
+with a roughly even mix of x−tubes and z−tubes. The
+exchangebetween box and z−tube orbits with increasing
+fgasis still noticeable, though diminished. From visual
+inspection of some randomly selected orbits, box orbits
+in the halo tend to be more stochastic and spherical in
+shape (Clozel 2008), and we will further investigate the
+nature of the outer box orbits in future work.
+Both classes of tubes have a higher rotation bias than
+in the inner parts. The x−tube orbits are especially
+highly streaming, which is a signature of the stars’ dy-
+namically cold origin. To highlight this point, we have
+also plotted the streaming fraction of the x−tube and
+z−tube orbits over the same range of relative binding
+energies for a series of re-mergers of the 20 and 40% gas
+remnants (see Hoffman et al. 2009a), intended to repre-
+sent “dry” mergers between elliptical galaxies that have
+already exhausted their gas in a previous major merger
+(e.g. van Dokkum 2005; Bell et al. 2006; Naab et al.
+2006b; Kormendy et al. 2009). The streaming of the
+z−tubes is not that much different in the remnants of
+mergers between dynamically hot ellipticals than in the
+disk mergers, but the x−tube orbits show far less or-
+dered rotation in the dry merger remnants. In a merger
+between hot systems, the random motions of the stars
+provide a source of initial angular momentum about the
+long axis, with no preferred sense of rotation. The uni-
+form streaming of the outer x−tube orbits might be a19
+Fig. 13.— Outer structure ( ∼3−7Re), averaged over eight merger orbits. Same as Figure 8, except that the orbital structure
+is averaged over the 75-95th percentiles in binding energy, and the shapes and kinematic misalignment are evaluated wit hin the 80th
+percentile. The intrinsic orientation twist is computed as max(θij), withiandjrunning from the 25th to the 85th percentile in binding
+energy ( ∼0.5−5Re). On the tube rotation plot (upper right panel), we have adde d data points for re-mergers of the 20 and 40% gas
+remnants (labeled as “dry” mergers - see text).
+tell-tale sign of a cold (late-type) progenitor in dynami-
+cal models of observed systems.
+The shapes of the matter distribution between 3 and
+7Reare round ( ǫ∼0.8 on average), and nearly max-
+imally triaxial. The stars are substantially more flat-
+tened (ǫ∼0.75) and oblate ( T∼0.4) than the matter
+as a whole (including DM). Large kinematic misalign-
+ments are common in the outer parts, producing large
+kinematic twists between ∼1 and 3Rein the most gas-
+rich remnants, as shown previously in Figure 12. The
+30-40% gas remnants also typically have large instrin-
+sic orientation twists between ∼1 and 5Re, owing to the
+rapid figure rotation of the inner component.
+4.SUMMARY AND CONCLUSIONS
+We have shown that a variety of observed kinematic
+structures can be accounted for just by varying the
+gas fraction in binary 1:1 mergers. The projected
+kinematics within 1 Retends to fall into one of four
+categories: (i) rapid disk-like rotation about the major
+axis; (ii) a prominent, disk-like KDC with slow or
+minor-axis rotation farther out; (iii) uniform slow
+rotation, with vmax/lessorsimilar40 km/s; and (iv) prominent
+minor axis rotation, with little or no rotation aboutthe major axis. In Figure 14 we classify the eight
+remnants at each fgasinto these four categories
+based on visual inspection of their velocity maps in
+projection along the y−axis (intrinsic variations far
+outweigh projection effects in the global appearance
+of the kinematic maps). The 0-10% gas remnants are
+generally slowly rotating or dominated by minor-axis
+rotation, in agreement with the previous results of
+Naab et al. (2006a); Jesseit et al. (2007); Cox et al.
+(2006a). The 15-20% gas remnants often look similar
+to the lower- fgasremnants but with disk-like KDCs
+at their centers, and bear an interesting resemblance
+to observed galaxies such as NGC4365 (Statler et al.
+2004; van den Bosch et al. 2008; Hoffman et al. 2009c),
+NGC5813, and NGC4458 (Emsellem et al. 2004;
+Cappellari et al. 2007; van den Bosch 2008).
+The 30-40%gasremnants generically show rapid, disk-
+like rotation with v/σup to∼1 (see also Rix & White
+1990; Scorza & Bender 1995; Robertson et al. 2006b;
+Krajnovic et al. 2008). None of them display the ex-
+treme rotation ( v/σ∼2) of the most rapid rota-
+tors in the SAURON sample, although other stud-
+ies (Robertson et al. 2006b; Robertson & Bullock 2008)
+have shown that 1:1 mergers between 60-80% gas disks20
+can yield higher values of v/σ. Highv/σvalues are al-
+ways accompanied by high, anticorrelated h3values in
+our remnants, so the subset of SAURON rapid rotators
+with a shallow h3−v/σrelation may require a different
+formation mechanism.
+The variety of features that can be produced with
+1:1 mergers alone suggests that there are degeneracies
+in the formation scenarios leading to a given gross
+outcome. For instance cold embedded disks can arise
+either from gas-rich merging or from cosmological
+gas inflow and secular evolution (Naab & Burkert
+2001; Robertson et al. 2004; Dekel & Birnboim 2006;
+Robertson et al. 2006b; Bournaud et al. 2007b;
+Robertson & Bullock 2008; Elmegreen et al. 2008;
+Genzel et al. 2008; Shapiro et al. 2008; Dekel et al.
+2009; Naab et al. 2009b; Keres & Hernquist 2009;
+Ceverino et al. 2009). Lower progenitor mass ra-
+tios (∼3-4:1 instead of 1:1 mergers) can produce
+remnants resembling many SAURON rapid rota-
+tors even in dissipationless simulations (Naab et al.
+1999; Cretton et al. 2001; Naab & Burkert 2003;
+Bournaud et al. 2004, 2005b; Burkert & Naab 2005;
+Jesseit et al. 2007; Hopkins et al. 2009b; Jesseit et al.
+2009; Johansson et al. 2009). Sequential minor mergers
+can produce slowly rotating ellipticals as well as major
+mergers with low fgas, that are in better agreement
+with observed round, isotropic, and featureless sys-
+tems such as NGC4486, NGC4552, and NGC 5846
+(Weil & Hernquist 1994, 1995; Bournaud et al. 2007a;
+Cappellari et al. 2007; Burkert et al. 2008; Naab et al.
+2009a; Gonzalez-Garcia et al. 2009). These scenarios
+must be distinguished with more detailed structural and
+dynamical analysis (e.g. Naab et al. 2006b; Jesseit et al.
+2007; Burkert et al. 2008; de Lorenzi et al. 2008;
+van den Bosch et al. 2008; Robertson & Bullock 2008;
+Shapiro et al. 2008; Jeong et al. 2008; Hoffman et al.
+2009a; Thomas et al. 2009; Romanowsky et al. 2009b;
+Romano-Diaz et al. 2009).
+Our orbital analysis suggests a rather simple picture
+of how the structure of the remnants arises. Non-
+axisymmetric torques in the merger trigger the forma-
+tion of a triaxial structure with a large population of
+box orbits. When gas is present, the strong CMC formed
+throughdissipation(e.g. Mihos & Hernquist1994a) con-
+verts the majority of the box population into z−tube or-
+bits, which have nearly canceling rotation since box dif-
+fusion has no preferred direction. Stars in inclined disks
+begin with a large polar angular momentum component
+(jθ) that is retained throughout the merger, so they do
+not cross orbital boundaries. This produces a streaming
+x−tube population in the remnants, that is most pro-
+nounced at large radii. This picture of the x−tube pop-
+ulation is supported by the facts that (i) co-planar merg-
+ers produce remnants almost entirely devoid of x−tube
+orbits; (ii) the phase space available to x−tubes in the
+remnant potentials is generally underpopulated (Barnes
+1992; Hoffman 2007); and (iii) in some observed ellipti-
+cals the velocity ellipsoid is flattened along the polar axis
+(van den Bosch 2008).
+Ingas-richmergerremnants, thereisalsoapronounced
+streaming z−tube population on small scales owing to
+the gasthat dissipatesenergyand retainsits angularmo-
+mentum during the merger (Hopkins et al. 2009b). Out-
+side a fairly well-defined boundary (see e.g. Figures 7
+Fig. 14.— Classification of the remnant kinematics. The
+eight remnants of each gas fraction were categorized as (i) s howing
+rapidmajor-axisrotation, (ii)slowlyrotating witha disk -likeKDC,
+(iii) slowly rotating throughout, or (iv) showing rapid min or-axis
+rotation, based on visual inspections of their velocity map s (see
+text).
+and 12), the orbital structure is largely unaffected by the
+gas. This boundary corresponds to the radius where box
+orbits cease to dominate in the dissipationless remnants,
+implying that these regions may be left relatively intact
+because they have never had a large population of radial
+orbits to convey information about the galactic center to
+their location. Time-dependent orbital classification and
+idealized evolving models are needed to verify this in-
+terpretation (e.g. Hoffman et al. 2009c; Villalobos et al.
+2009).
+The characteristic orbital structure arising from this
+picture, and in particularthe sharpkinematic transitions
+predictedbetween ∼1and3Re, shouldbereadilyobserv-
+able with surveys in progress such as SMEAGOL. Some
+pronounced kinematic transitions have been observed
+around these radii in survey pilot studies (Proctor et al.
+2009; Coccato et al. 2009), but more data will be needed
+to determine whether these transitions are of the same
+nature as those in the merger simulations. Note that
+some features, e.g. the uniform streaming of the x−tube
+orbits, can only be captured with dynamical modeling -
+substantial minor-axis rotation alone could be produced
+by either a dominant population of x−tubes with a small
+rotation bias, or a smaller, highly streaming population.
+We do not necessarily expect most real galaxies to dis-
+play the sharp features in their orbital structure found
+in this paper, since cosmological galaxy formation his-
+tories are far more complex than isolated 1:1 mergers
+between pure disks. However the results of our analy-
+sis illustrate how orbital analysis can isolate subsets of
+the stellar population with similar histories and place in-
+tuitive constraints on galaxy formation mechanisms. In
+future work we hope to extend this type of analysis to a
+broader range of merger parameters and more complex
+cosmological formation scenarios.
+We thank Glenn van de Ven, Remco van den Bosch,
+Aaron Romanowsky, and Phil Hopkins for enlightening
+discussions, especially on relating our work to observa-21
+tions. We are also grateful to Bart Willems and Jaczek
+Braden for technical help. This work was supported in
+part by a Lindheimer Postdoctoral Fellowship at North-
+western University. Computations were performed ontheFugucomputer cluster at Northwestern, funded by
+NSF MRI grant PHY-0619274 to Vicky Kalogera, and
+theSauronclusteratthe parallelcomputingcenterofthe
+Institute for Theory and Computation at the Harvard-
+Smithsonian Center for Astrophysics.
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+APPENDIX
+In Figures 15 - 20 we present radial profiles of the orbital populatio ns, intrinsic shapes, and orientations of all eight
+merger remnants with each initial gas fraction (5, 10, 15, 20, 30, a nd 40%). The format of the figures is identical to
+that of Figure 4 in the body of the paper (instrinsic structure of th e dissipationless remnants). The figures may be
+viewed one-at-a-time to get a feel for the variation with merger or bit at fixed fgas, or compared row-for-row to follow
+the evolution of the structure with fgasfor a fixed merger orbit.24
+Fig. 15.— Intrinsic structure of the 5% gas remnants.25
+Fig. 16.— Intrinsic structure of the 10% gas remnants.26
+Fig. 17.— Intrinsic structure of the 15% gas remnants.27
+Fig. 18.— Intrinsic structure of the 20% gas remnants.28
+Fig. 19.— Intrinsic structure of the 30% gas remnants.29
+Fig. 20.— Intrinsic structure of the 40% gas remnants.
\ No newline at end of file
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+arXiv:1001.0800v1  [astro-ph.EP]  6 Jan 2010Discovery of Diffuse Hard X-ray Emission around Jupiter
+withSuzaku
+Y. Ezoe, K. Ishikawa and T. Ohashi
+Department of Physics, Tokyo Metropolitan University, 1-1 Minami- Osawa, Hachioji,
+Tokyo 192-0397, JAPAN
+ezoe@phys.metro-u.ac.jp
+Y. Miyoshi
+Solar-Terrestrial Environment Laboratory, Nagoya University, Furo-cho, Chikusa-ku,
+Nagoya 464-8601, JAPAN
+N. Terada
+Department of Geophysics, Tohoku University, 6-3 Aoba, Aramak i, Aoba-ku, Sendai,
+Miyagi 980-8578, JAPAN
+Y. Uchiyama
+Stanford Linear Accelerator Center, Standford University, 257 5 Sand Hill Road, Menlo
+Park, CA 94025, USA
+and
+H. Negoro
+Department of Physics, Nihon University, 1-8-14 Kanda Surugada i, Chiyoda, Tokyo
+101-8308, JAPAN
+Received ; accepted– 2 –
+ABSTRACT
+We report the discovery of diffuse hard (1–5 keV) X-ray emission ar ound
+Jupiter in a deep 160 ks SuzakuXIS data. The emission is distributed over
+∼16×8Jovianradiusandspatially associatedwiththeradiationbelts andth eIo
+Plasma Torus. Itshows aflatpower-law spectrumwithaphotoninde x of1.4±0.2
+with the 1–5 keV X-ray luminosity of (3.3 ±0.5)×1015erg s−1. We discussed its
+origin and concluded that it seems to be truly diffuse, although a poss ibility of
+multiple background point sources can not be completely rejected w ith a limited
+angular resolution. If it is diffuse, the flat continuum indicates that X -rays arise
+by the non-thermal electrons in the radiation belts and/or the Io P lasma Torus.
+The synchrotron and bremsstrahlung models can be rejected fro m the necessary
+electron energy and X-ray spectral shape, respectively. The inv erse-Compton
+scattering off solar photons by ultra-relativistic (several tens Me V) electrons can
+explain the energy and the spectrum but the necessary electron d ensity is /greaterorsimilar10
+times larger than the value estimated from the empirical model of Jo vian charge
+particles.
+Subject headings: planets and satellites: individual (Jupiter, Io) — X-rays: general– 3 –
+1. Introduction
+In situ measurements and ground-based observations have show n the existence of
+Jupiter’s high-energy electron Van Allen radiation belts with evidence for electrons at
+energies of keV to 50 MeV (Berge & Gulkis 1976; Van Allen et al. 1975; F ischer et al. 1996;
+Bolton et al. 2002). Theoretically, electrons are energized by the b etatron acceleration
+via the inward radial diffusion (Goertz et al. 1979; Miyoshi et al. 1999 ) and recently non-
+adiabatic acceleration via wave-particle interactions has been discu ssed (Horne et al. 2008).
+The trapped electrons lose their energy through synchrotron ra dio emission and some
+fraction are absorbed by Jovian moons and rings (Bagenal et al. 20 04). However, there
+remains much debate over the precise energy spectrum and spatia l distribution of the
+energetic electrons in the radiation belts.
+Jupiter is the most luminous planet in the solar system at X-ray wavele ngth
+(Metzger et al. 1983; Waite et al. 1997; Bhardwaj et al. 2007). It e mits X-rays from mag-
+netic poles by the chargeexchange interaction with the solar wind an d magnetospheric heavy
+ions and also by the bremsstrahlung emission via energetic electrons (Gladstone et al. 2002;
+Branduardi-Raymont et al. 2004; Branduardi-Raymont et al. 2007 a). A typical X-ray
+luminosity of the auroral emission is ∼1016erg s−1. The low-latitude atmosphere also
+exhibits scattered radiation of solar X-rays with a typical luminosity of∼3×1015erg
+s−1(Branduardi-Raymont et al. 2007b). Besides Jupiter itself, faint s oft (<1 keV) X-rays
+from the Io Plasma Torus (IPT) have been detected ( ∼1×1014erg s−1, Elsner et al. 2002).
+It may arise from bremsstrahlung emission by warm electrons with en ergies of few hundred
+to few thousand eV. Although we can expect hard X-rays ( >1 keV) from energetic keV to
+MeV electrons in the Jupiter’s radiation belts, there has been no rep ort on such an emission
+because past instruments lacked a necessary sensitivity to detec t the extended hard X-ray
+emission. This indication motivated us to perform a decisive analysis us ing a deep 160 ks– 4 –
+observation data of Jupiter with Suzaku.
+2. Observations
+Suzaku(Mitsuda et al. 2007) observed Jupiter on 2006 February 24-28 wit h the
+X-ray Imaging Spectrometer (XIS) (Koyama et al. 2007). The XIS consists of three
+front-illuminated (FI) CCDs (XIS0, 2 and 3) and one back-illuminated (BI) CCD (XIS1).
+Due to the low-earth orbit of Suzakuand the large effective area, the XIS-FI has the
+lowest particle background among all X-ray CCDs in currently availab le X-ray observatories.
+Figure 5 in Mitsuda et al. (2007) summarizes the background normaliz ed by the effective
+area and the field of view, which is a good measure of sensitivity deter mined by background
+for spatially extended emission. The background of the XIS FI is the lowest among X-ray
+CCDs onboard Suzaku,Chandra andXMM-Newton . It is 2 and 5 times lower than the
+XMM-Newton MOS and pn, and Chandra ACIS-I in 2–5 keV, respectively. For this reason,
+we use only the XIS data in this paper.
+During the observations, the spacecraft was repointed four time s to allow for the
+planet’s motion (0.7 ∼1.8 arcmin per day). The XIS was operated in the normal mode. The
+data reduction was performed on the version 2.0.6.3 screened data provided by the Suzaku
+processing facility, using the HEAsoft analysis package ver 6.3.1. Th e net exposure of each
+FI and BI chip was 159 ks. For spectral fits, we generated respon se matrices and auxiliary
+files with xisrmfgen andxissimarfgen .
+Although an optical loading from Jupiter (visual magnitude mV=−2 during
+Suzaku observations) often becomes a problem in X-ray observat ions (Elsner et al. 2002),
+we concluded that it is negligible in this case, because optical loading eff ects such as
+degradation of the energy resolution, changes in the energy scale s and/or the detection– 5 –
+efficiency were not seen in XIS spectra. For confirmation, we estima ted the optical blocking
+power of the XIS compared to CCDs onboard XMM-Newton (Lumb 2000). We found
+that the SuzakuXIS has almost the same blocking power as the XMM-Newton MOS
+with a thick filter, with which the observation of Jupiter was success fully conducted
+(Branduardi-Raymont et al. 2004).
+3. Extended X-ray Emission
+3.1. Imaging analysis
+To search for any emission from Jupiter, we have to remove Suzaku’s orbital motion
+and transform the data into Jupiter’s comoving frame using an ephe meris obtained from
+the Jet Propulsion Laboratory (JPL). Before conducting this pro cedure, we checked X-ray
+images without these corrections. We created two-band mosaic ima ges using XIS1 (BI) and
+XIS0+2+3 (FI) data as shown in figure 1. We corrected the data fo r positional difference in
+exposure times by using the exposure map generator xisexpmapgen . An extended emission
+is detected in both the 0.2–1 keV and 1–5 keV bands along the path of Jupiter.
+Beside the extended emission, we found signatures of emission from point sources near
+Jupiter in the 1–5 keV band. Since we found no known point sources in the X-ray source
+catalogue compiled by NASA1, we decided to detect sources from the Suzakuimage. We
+utilized the wavelet function program wavdetect in the CIAO package2. We created four
+1–5 keV images with different bin sizes (1, 2, 4, and 8) and ran the poin t search program by
+setting the significant threshold above 1 ×10−6, which corresponds to one spurious source
+in a 1024 ×1024 pixel map (XIS field of view). We summarized source lists and rem oved
+1http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/high energysource/high energysource.pl
+2http://cxc.harvard.edu/ciao/– 6 –
+possible spurious detections by visual inspection.
+The obtained source positions are shown in figure 1 (b) (green circle s with red lines).
+Twenty sources are detected with the 1–5 keV X-ray flux of 1 ∼3×10−14erg s−1cm−2.
+Here we converted a XIS count rate into flux, assuming a power-law spectrum with an
+photon index of 1.5 and an average absorption column toward this fie ld3, which represents a
+typical spectrum of the background active galactic nucleus. This n umber is consistent with
+the canonical cosmic X-ray background model (Giacconi et al. 200 1), which predicts 27
+background sources above the detection limit. Although the sourc e list contains marginally
+detected point sources, we decided to remove X-ray photons fro m the 1–5 keV image
+analysis afterward, for safety. We excluded circular regions with a diameter of 2 arcmin
+centered at individual sources. The diameter is equal to the beam s ize or the half power
+diameter of the XIS4. Since all the sources are faint, contamination from the sources is
+thus negligible. Because there are no significant point sources arou nd Jupiter in the 0.2–1
+keV image (fig. 1 a), we conducted this exclusion process only for th e 1–5 keV image.
+We then corrected the two-band images for Suzaku’s orbital motion and Jupiter’s
+ephemeris. In case of the 1–5 keV image, to take into account the e xcluded regions, we
+created the exposure map for each observation and excluded the point source regions. Then
+we transformed the exposure map considering the Jupiter’s orbit a nd divided the orbital
+motion corrected image by the corrected map. In case of the 0.2–1 keV image, we made the
+same analysis except for excluding the point source regions.
+The results obtained are shown in figure 2. We detected a significant hard X-ray
+emission extended over ∼16×8Rjwhich is positionally associated with the Jupiter’s
+3http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl
+4http://www.astro.isas.jaxa.jp/suzaku/doc/suzaku td– 7 –
+radiation belts, and Io’s orbit. Because Io’s orbit coincides with the I PT, the emission
+is also spatially associated with the IPT. The X-ray luminosity of the ex tended emission
+arises from (3.6 ±0.4)×1015erg s−1and (3.3±0.5)×1015erg s−1in 0.2–1 and 1–5 keV,
+respectively. Here and below the uncertainties are 68 % confidence intervals. We assumed a
+distance to Jupiter of 5.0 AU on 24-28 Feb 2006. An estimated conta mination of the point
+sources outside the excluded regions to the extended emission is ∼20 %. Hence, most of the
+emission is due to the extended component.
+To check the distribution of the emission, we created projection pr ofiles along the
+horizontal axis as shown in figure 3. The hard X-ray emission is signific antly extended
+more than the point spread function (green line). We found that th e profile can be roughly
+represented by a simulated model using xissimarfgen , which is a sum of a point source
+and a elliptical uniform emission with radii of 4 and 1.3 arcmin (12 and 4 Rj). The emission
+is thus extended over a wide region ( >6Rj). On the other hand, the soft emission is
+marginally consistent with the point source. There is an excess on th e right side of the soft
+X-ray peak, which suggests emission from the IPT that has been re ported by Elsner et al.
+(2002).
+The hard X-rays are hence unlikely to originate only from a single point source or
+Jupiter. Furthermore, an artificial effect on the extended morph ology due to the analysis
+can be excluded because of the different morphology from the soft X-ray image. The
+emission thus seems to contain truly diffuse component related to Jo vian magnetospheric
+processes.
+We note that the faintness of the emission should have hindered its d etection in
+the past observations. Considering the angular resolution of XMM-Newton , its surface
+brightness will be 1/30 ∼1/40 of Jupiter’s aurora, From figure 5 in Branduardi-Raymont
+et al. (2007a), we can roughly estimate the CCD background level a s 1/10 of the auroral– 8 –
+emission. Therefore the detection of the diffuse emission shuold be q uite difficult. If the
+emission is related to Jovian radiation belts, there can be also time var iation (see §4).
+3.2. Spectral analysis
+To investigate characteristics of the emission, we analyzed its X-ra y spectrum by
+extracting photons around the extended emission region. For simp licity, we excluded the
+point sources from the event files without orbital motion correctio ns and extracted events
+from a circular region with a radius of 3 arcmin centered at the path o f Jupiter. The
+surrounding region with an outer radius of 6 arcmin after subtract ing the point sources
+was utilized as a background. Consequently, this extended emission region includes both
+Jupiter and the extended emission, because the limited angular reso lution hindered us to
+spatially separate these two components.
+Figure 4 shows the spectrum obtained. It consists of a flat continu um extending up
+to 5 keV and signs of lines around 0.25 keV and 0.56 keV. Hence, we fitt ed the spectrum
+with a power-law plus two Gaussian model. The best-fit model repres ents data well with
+χ2/ν∼0.5. No absorbing column was required, as expected from the sourc e’s proximity.
+The obtained photon index was 1.4 ±0.2, while the central energies of the two Gaussian
+were 0.24 ±0.1 keV and 0.56 ±0.1 keV. Fluxes for the 0.24 keV and 0.56 keV lines were
+(3.9±0.6)×10−5and (2.0±0.5)×10−5photons cm−2s−1, respectively.
+Characteristics of the two lines such as a central energy and a flux were similar to
+those of the Jupiter’s auroral emission observed with XMM-Newton in Nov 2003 (see
+table 3 in Branduardi-Raymont et al. 2007a). The two lines at 0.24 keV and 0.56 keV
+are presumably a line complex from C5+and ionized Mg, Si, S and a K αemission from
+O6+, respectively. Such line emissions are distinct features of the char ge exchange reaction– 9 –
+(Cravens et al. 2003). Taken together with the fact that the 0.2– 1 keV X-rays arise from a
+compact region near Jupiter, most of soft X-rays likely originate fr om Jupiter’s aurora via
+the charge exchange.
+In contrast, the 1–5 keV X-ray emission was represented by a simp le flat continuum
+with a photon index of 1.4. Since the peak of the X-rays coincides with Jupiter (fig. 2 b),
+some fractions of the emission can arise from Jupiter.
+The power-law spectrum was in fact seen in the past XMM-Newton observations
+of Jupiter’s aurora (Branduardi-Raymont et al. 2004). For compa rison, we plotted
+auroral emission models in three past XMM-Newton observations (tables 2 and 3 in
+Branduardi-Raymont et al. 2007a) in figure 4. The 1–5 keV fluxes of these models are
+smaller than that of the Suzakubest-fit model by a factor of 2 ∼3. This is consistent with
+an independent estimation from the projection profile that most of the extended emission
+comes from diffuse emission other than Jupiter.
+4. Discussion
+We have discovered extended hard X-ray emission around Jupiter. Although its
+spatial association with Jupiter strongly suggests a diffuse compon ent related to Jovian
+magnetospheric processes, we first have to examine serendipitou s background sources.
+Because we corrected the image for the Jupiter’s orbital motion, a ny faint background
+source on the Jupiter’s path, that is missed in our point source ident ification procedure, can
+be seen as extended emission in the orbital motion corrected image. Considering the almost
+symmetric projection profile (fig. 3) centered at Jupiter, the spa tial distribution of the
+background sources should be symmetric about Jupiter. The most plausible case is that the
+emission is composed of Jupiter itself and a single background point so urce near the center– 10 –
+of the Jupiter’s orbit. To test this hypothesis, we excluded the cen tral part of the Jupiter’s
+orbit with a circle region in the same way as figure 1 (b), and created t he orbital motion
+corrected image and projection profile in 1–5 keV. We still observed a significant extended
+emission around Jupiter, although the total count rate decrease d by∼15 %. Hence, a single
+background source cannot fully explain the observed emission. Sinc e multiple point sources
+with symmetric spatial distribution against Jupiter are less probable , below we discuss the
+origin of the emission, considering that the emission is truly diffuse.
+Theflatpower-law continuum oftheemission suggestsanon-therm al mechanism. There
+are three candidates for the emission mechanism; synchrotron em ission, bremsstrahlung
+emission, and inverse-Compton scattering. The synchrotron inte rpretation can be easily
+rejected in terms of the required electron energy. From the X-ra y image, we can observe
+the hard X-rays at the Io’s orbital plane (a distance from Jupiter, r=5.9Rj) or even more
+distant area. At r∼6Rj, the magnetic field of the Jovian magnetosphere is quite weak on
+the order of 0.01 G. With such a weak magnetic field, if we want to gene rate synchrotron
+emission in the X-ray wavelength range, TeV electrons are necessa ry. This energy is more
+than 104times higher than the observed maximum electron energy in the radia tion belts
+(∼50 MeV) (Bolton et al. 2002). Even if TeV electrons exist, their Larm or radius (3 ×1011
+cm for 1 TeV electron at r=6Rj= 4×1010cm) will be too large, so that TeV electrons
+are not trapped in the magnetic field line and will escape from the Jupit er’s radiation belts.
+The bremsstrahlung emission is possible in terms of the electron ener gy, because
+we need only keV electrons which have been observed in the past in-s itu measurements
+(Bagenal et al. 2004). Such electrons will interact in an ambient med ium and produce not
+only continuum but also X-ray lines by collisional excitation and inner-s hell ionization (see
+§3.1 in Tatischeff 2002). From in-situ measurements, the plasma at r∼6Rjis known to be
+mainly composed of heavy ions such as S+and O+(see fig. 23.2 in Bagenal et al. 2004).– 11 –
+We estimated an equivalent width of S K αline (2.3 keV). We used cross sections for K-shell
+ionization by electron impact from Santos et al. (2003) and the brem sstrahlung loss rate
+of electrons given by Skibo et al. (1996), to estimate the line and con tinuum intensities,
+respectively. The calculated equivalent width of the S K αline was as large as ∼10 keV,
+which should be observed in the XIS spectrum with the good energy r esolution ( ∼130 eV
+at 6 keV, Koyama et al. 2007). As shown in figure 4, no such line exists around 2 keV.
+Hence, the bremsstrahlung interpretation can be rejected.
+The remaining possibility is inverse-Compton scattering. We consider the scattering of
+solar light by ultra-relativistic electrons in the radiation belts. In this process, the scattered
+photon energy will be
+∼8 keV(Eph/1.4 eV) (Ee/50 MeV)2, (1)
+whereEphis an energy of solar optical photons and Eeis that of relativistic electrons
+(Rybiki & Lightman 1979). Considering the maximum observed electr on energy of 50 MeV
+in the inner radiation belts (Bolton et al. 2002), the inverse-Compto n scattering is possible
+in terms of the electron energy. Although 50 MeV electrons have no t been observationally
+well confirmed at outer regions ( r >severalRj), there may exist such ultra-relativistic
+electrons in these regions as expected from the empirical model (e .g., Divine & Garret
+1983). Also, since this mechanism does not need the emission line, the observed spectral
+shape can be safely explained.
+Assuming the inverse Compton scattering, we can estimate the inde x of the
+electron number density spectrum as ∼2, from the photon index of the X-ray spectrum
+(Rybiki & Lightman 1979). This is consistent with the Divine-Garrett (DG) empirical
+charged particle model of Jovian magnetosphere (Divine & Garret 1 983) (∼3 at 6Rj) and
+another model based on the Galileodata (Garret et al. 2003) ( ∼3 at 8Rj).
+We then proceeded to estimate the necessary electron number de nsity. For simplicity,– 12 –
+we assumed that the emission region is an oblate spheroid with radii of 8, 8, and 4 Rj,
+and the monotonic electron energy of 50 MeV. We here considered a nisotropic angular
+distribution of the inverse-Compton scattering (Brunetti et al. 2 001). During the Suzaku
+observation, the Sun-Jupiter-Earth angle was 10◦. Because of the larger cross section of the
+back scattering, the inverse-Compton flux will increase by a facto r of∼3. The estimated
+average electron number density to explain 80% of the hard X-rays was∼0.005 cm−3.
+For comparison, we utilized the DG empirical model, which provides a th eoretical
+electron spectrum as a function of the distance from Jupiter. The DG model is based
+on the data taken with PioneerandVoyager. To date, it is the best model to estimate
+the MeV electron distribution in the inner radiation belts ( <8Rj). We found that
+the>20 MeV electron number density is only ∼0.0007 and 0.0001 cm−3at 4 and 6 Rj,
+respectively. The density will decrease as the distance increases. Hence, there is a factor of
+7∼50 discrepancy. Although the situation can be somewhat relaxed by an uncertainty of
+the DG model (Bolton et al. 2001) and/or time variations of the trap ped MeV electrons
+(Miyoshi et al. 1999; Bolton et al. 2002; Santos-Costa et al. 2008) , this interpretation
+needs further investigations.
+In conclusion, further X-ray, radio and in-situ observations are n ecessary to understand
+the origin of this diffuse hard X-ray emission, to identify the emission r egion with the
+Jupiter’s magnetosphere, the IPT and/or else, and to clearly distin guish the emission from
+background point sources.– 13 –
+REFERENCES
+Bagenal, F., Dowling, T.E., & McKinnon, W.B. 2004, Jupiter: The Planet, Satellites and
+Magnetosphere (Cambridge University Press).
+Berge, G. L. & Gulkis, S. 1976, Jupiter (ed. Gehrels, T.; Univ. Arizon a Press).
+Bhardwaj, A., et al. 2007, Planetary Space Sci., 55, 1135.
+Bolton, S. J., et al., 2001, Geophys. Res. Lett., 28, 907.
+Bolton, S. J. et al. 2002,. Nature 415, 98).
+Branduardi-Raymont, G. et al. 2004, A&A, 424, 331.
+Branduardi-Raymont, G. et al. 2007a, A&A,. 463, 761.
+Branduardi-Raymont, G. et al. 2007b, Planetary Space Sci., 55, 11 26.
+Brunetti, G., Cappi, M., Setti, G., Feretti, L., & Harris, D. E. 2001, A& A, 372, 755
+Creavens, T.E., 2003, J. Geophys. Res., 108, 1465.
+Divine, N., & Garret, H.B. 1983, J. Geophys. Res., 88, 6889.
+Elsner, R.F. et al. 2002, ApJ, 572, 1077.
+Fischer, H. M., Pehlke, E., Wibberenz, G., Lanzerotti, L. J. & Mihalov, J. D. 1996, Science
+272, 856.
+Giacconi, R. et al., 2001, ApJ, 551, 624.
+Garret, H.B. et al, 2003, NASA/JPL publication 03-006.
+Gladstone, G.R. et al. 2002, Nature, 415, 1000.– 14 –
+Goertz, C. K., J. A. Van Allen, & M. F. Thomsen, 1979, J. Geophys. R es., 84, 87.
+Horne, R.B. et al. 2008, Nature Physics 4, 301.
+Koyama, K. et al. 2007, PASJ, 59, S23.
+Lumb, D. 2000, XMM PHS Tools - EPIC Optical loading.
+Metzger, A.E. et al. 1983, J. Geophys. Res., 88, 7731.
+Mitsuda, K. et al. 2007, PASJ, 59, S1.
+Miyoshi, Y., et al. 1999, Geophys. Res. Lett., 26, 9.
+Rybiki, G.B., & Lightman, A.R. 1979, Radiative Processes in Astrophys ics (A Wiley-
+Interscience Publication).
+Santos, J.P., Parente, F., & Kim, Y-K. 2003, J. Phys. B: At. Mol. Opt o. Phys., 36, 4211.
+Santos-Costa, D., et al., 2008, J. Geophys. Res., 113, A01204.
+Serlemitsos, D., et al., 2007, PASJ, 59, S9.
+Skibo, J.G., Ramaty, R., & Purcell, W.R. 1996, A&AS, 120, 403.
+Tatischeff, V., 2002, astro-ph/020839.
+Uchiyama, Y. et al., 2008, PASJ, 60, S35.
+Van Allen, J. A., Baker, D. N., Randall, B. A. & Sentman, D. D. 1975, Sc ience, 188, 459.
+Waite Jr., J.H. et al. 1997, Science, 276, 104.
+This manuscript was prepared with the AAS L ATEX macros v5.2.– 15 –
+Fig. 1.— SuzakuXIS mosaic images of the vicinity of Jupiter in the (a) 0.2–1 keV (BI)
+and (b) 1–5 keV (FI) bands, displayed on the J2000.0 coordinates. Exposures are corrected
+and the count unit is counts s−1binned pixel−1. For clarity, images are binned by a factor
+of 8 and smoothed by a Gaussian of σ= 5 pixels. Black circles show the size and posi-
+tion of Jupiter during the observations. Considering known pointing uncertainty of Suzaku
+(Uchiyama et al. 2008), positions of the circles are slightly shifted by +25 and −25 arcsec in
+the Ra and Dec directions, respectively, in order that the extende d emission coincides with
+the Jupiter’s path. Green circles with red lines mark excluded point so urce regions when we
+create the 1–5 keV image in comoving frame (see text).– 16 –
+Fig. 2.— SuzakuXIS images after correcting for the satellite’s orbital motion and Ju piter’s
+ephemeris in the (a) 0.2–1 keV (BI) and (b) 1–5 keV (FI) bands. The images are binned
+and smoothed in the same way as figure 1. In the panel (a), a circle in dicates the expected
+position and size of Jupiter whose diameter is 39 arcsec. In the pane l (b), grey lines indicate
+the equatorial crossing of magnetic field lines at 2, 4, 6, and 8 Jovian radius (Rj). A black
+line is the path traced by Io. A photograph of Jupiter by Cassinitaken from the JPL web
+site is overlaid. An arrow indicates the direction of the Sun. Boxes ar e used to obtain the
+projection profiles in figure 3.– 17 –
+Fig. 3.— Projection profiles along the horizontal axis, extracted fr om box regions in figure 2.
+Black and red points show the 0.2–1 and 1–5 keV data, respectively. Errors are 1 σstatistical
+ones. Green lines are a simplified model of Jupiter’s X-rays (42 arcse c diameter) in the two
+energy bands, whose normalization and offset are tuned to coincide with the profile. The
+point spread function is almost the same at different energies (Serle mitsos et al. 2007). A
+dashed blue line indicates the expected profile of an uniform emission e xtended over an
+elliptical region with radii of 4 and 1.3 arcmin. A solid black line is the sum of the dashed
+green and blue lines. Dashed black lines are background levels estimat ed by fitting the data
+above the projected distance >14 arcmin with a constant.– 18 –
+Fig. 4.— Background subtracted BI (red) and FI (black) spectrum of the extended emission
+region, compared with the best-fit power-law plus two Gaussian mod els (solid line). Two
+Gaussians are shown in dashed and dash-dotted lines. Purple, gree n and blue dashed lines
+plottedfortheFIspectrumareJupiter’sauroralcontinuumemiss ionmodelsin XMM-Newton
+observations (Branduardi-Raymont et al. 2007a).
\ No newline at end of file
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+arXiv:1001.0801v1  [cond-mat.str-el]  6 Jan 2010Typeset with jpsj2.cls <ver.1.2.2> Full Paper
+Quantum Kagome antiferromagnet ZnCu 3(OH) 6Cl2.
+Philippe Mendels1∗andFabrice Bert1
+1Laboratoire de Physique des Solides, Universit´ e Paris-Su d XI Orsay, UMR 8502 CNRS, 91405
+Orsay Cedex, France
+The frustration of antiferromagnetic interactions on the loosely c onnected kagome lattice
+associated to the enhancement of quantum fluctuations for S=1/ 2 spins was acknowledged
+long ago as a keypoint to stabilize novel ground states of magnetic m atter. Only very re-
+cently, the model compound Herbersmithite, ZnCu 3(OH)6Cl2, a structurally perfect kagome
+antiferromagnet, could be synthesized and enables a close compar ison to theories. We review
+and classify various experimental results obtained over the past y ears and underline some of
+the pending issues.
+KEYWORDS: frustration, kagome, spin liquids, quantum magn etism
+1. Introduction
+After the initial papers by Anderson and Fazekas in 1973-741)and the revival of their
+seminal mainstream proposal of a resonating valence bond gr ound state in the context of High
+Temperature Superconductors,2)kagome antiferromagnets have been recognized since the
+early 90’s as the corner stone of the search for a quantum spin liquid state. Strong indications
+for very original physics had been found in some experimenta l systems with a kagome based
+geometry, such as magnetic freezing occuring only at very lo w temperatures on the scale of
+the exchange energy and fluctuating ground states, two featu res recognized as the landmarks
+of frustration.3,4)Only recently, a ”New Candidate emerged for a quantum spin li quid”.5)
+Namely, Herbertsmithite, a rare mineral6)which has been synthesized for the first time in
+20057)has brought a tremendous excitement to the field. Altogether withκ(ET)2Cu2(CN)3, it
+combineslowdimensionality, aquantumcharacterassociat edwithspins1/2andtheabsenceof
+magneticfreezing.Thiswascelebratedas”AnEndtotheDrou ghtofQuantumSpinLiquids”8)
+in the context of a grail quest for the synthesis of a kagome mo del system to compare with
+theories. Although theoretical models have been developed over the last 20 years, there is still
+no consensus about the expected ground state. Most of the mod els point at an exotic ground
+state and two broad classes of spin liquids have been explore d theoretically, in particular for
+thekagome antiferromagnet, (i) topological spin liquidsw hich feature aspin gap, a topological
+order and fractional excitations;9)(ii) algebraic spin liquids with gapless excitations in all spin
+sectors.10)
+∗E-mail: mendels@lps.u-psud.fr
+1/22J. Phys. Soc. Jpn. Full Paper
+Here, we focus on the review of the experimental properties o f Herbertsmithite and discuss
+the deviations to the ideal case and their implications with regards to the important issue of
+the existence of a gap.
+2. Structure, interactions and phase diagram of the generic family of parata-
+camites, Zn xCu4−x(OH)6Cl2, 0< x <1
+Herbertsmithite ZnCu 3(OH)6Cl2is thex= 1 end compound of the Zn-paratacamite fam-
+ily ZnxCu4−x(OH)6Cl2.6,7)It can be viewed as a double variant of the parent clinoatacam ite
+compound ( x= 0), situated at the other end of the family. Starting from th e latter, a Jahn-
+Teller distorted S= 1/2 pyrochlore, the symmetry first relaxes from monoclinic ( P21/n) to
+rhombohedral ( R3m) around x= 0.33, leading to a perfect kagome lattice in the a-bplane
+with isotropic planar interactions; then, in the c- elongated x >0.33 pyrochlore structure,
+the magnetic bridge along c-axis between a-bkagome planes is progressively suppressed by
+replacing the apical Cu2+by a diamagnetic Zn2+. Due to a more favorable electrostatic en-
+vironment, Cu2+is expected to preferentially occupy the distorted octahed ral kagome sites
+only. When x= 1 theS= 1/2 ions should therefore form structurally perfect kagome la yers
+that are themselves well separated by diamagnetic Zn2+(Fig. 1).
+2.1 Clinoatacamite
+In clinoatacamite ( x= 0) two successive transitions to long-range ordered phase s are
+observed at 18 K and 6 K. They are signalled by the development of well defined internal
+fields in µSR,11)by peaks in specific heat and accidents in magnetic susceptib ility.12)At
+the 6 K transition a ferromagnetic component along the b-axis develops which likely relates
+to some Dzyaloshinskii-Moriya anisotropy and/or weak ferr omagnetic interactions between
+the planes.13,14)While in the low- Tphase a combination of three planar interactions, as
+expected from the low symmetry of the structure, ( Jplanar∼53 K) and anisotropy ∼0.08J
+are necessary to explain the inelastic neutron data,13)there are still inconsistencies between
+the results from different techniques. On one hand, µSR indicates that the order parameter
+fully develops at the upper transition (18 K). Below 6 K, besi des some rearrangement of the
+magnetic structure which might explain the change in the int ernal field at the muon sites,
+an unexpected enhancement of fluctuations is observed.11)On the other hand, only a weak
+amount of entropy is released at 18 K and a clear signature of l ong-range ordering appears in
+neutron diffraction only below 6 K, suggesting a highly frustr ated state with weak moments
+between 6 and 18 K in strong contradiction with µSR results.12,14)
+2.2 Phase diagram
+Owing to its extreme sensitivity to static frozen phases, ze ro fieldµSR experiments enable
+totrack inminutedetails themagnetism oftheentireparata camite family.15)For intermediate
+2/22J. Phys. Soc. Jpn. Full Paper
+Clinoatacamite
+Cu2(OH)3ClHerbertsmithite
+ZnCu3(OH)6Cl2Zn-paratacamite
+ZnxCu4-x(OH)6Cl2Zn/Cu
+Zn
+Cu I Cu I Cu IZn/Cu II
+Cu I Cu I Cu I Cu III
+Cu I Cu ICu II
+Cu
+Zn/CuZn/Cu
+x=0 x=0.33 x=1P21/n R-3m
+Fig. 1. (Color online) Top left: Structure of Herbertsmithite from.7)Top right: Simple sketch of the
+structure where Zn and Cu sites only have been represented. Bot tom: Evolution of the structure
+of paratacamites when Zn content is increased. A structural tra nsition occurs around x= 0.33
+which yields well defined perfect kagome planes, assumed to be filled w ith Cu only.
+Zn contents 0 .33< x <0.66, the low temperature phase appears to be locally inhomoge neous
+and yields two components in the µSR signal; one from a frozen magnetic environment rem-
+iniscent of the ordered phase in clinoatacamite which gets m ore and more disordered as x
+increases, and one from a dynamical environment of the muon ( Fig.2). The evolution of this
+frozen fraction detected in µSR yields the phase diagram plotted in Fig. 4. Interestingly for
+x >0.66, the frozen fraction gets lower than the experimental acc uracy (a few percent) and
+a fully ”liquid” phase appears although 1/3rdof the inter-layer 3D coupling paths are still
+active. Accordingly, the small ferromagnetic component fo und in macroscopic magnetization
+measurements disappears progressively when going from x= 0 to 1 (Fig.3).
+The large domain of stability of a ”liquid” phase in the phase diagram of paratacamites
+likely points at the weakness of the interplane coupling as w ell as at the shortness of the in
+plane correlation length. No magnetic freezing has ever bee n observed in the x= 1 compound
+down to the lowest probed temperature of 20 mK.15)The upper boundvalue of a hypothetical
+3/22J. Phys. Soc. Jpn. Full Paper
+/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s48/s46/s51/s48/s46/s54/s48/s46/s57
+/s32/s84/s105/s109/s101/s32/s40 /s115/s41/s32/s77/s117/s111/s110/s32/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110
+/s32/s32
+/s120/s32/s61/s32/s48/s120/s32/s61/s32/s48/s46/s51/s51/s120/s32/s61/s48/s46/s53
+/s120/s32/s61/s32/s48/s46/s54/s54/s120/s32/s61/s32/s49/s46/s48
+/s120/s32/s61/s32/s48/s46/s49/s53
+/s84/s61/s49/s46/s53/s75
+Fig. 2. (Color online) Evolution of the low- T µSR asymmetry when Zn content is progressively
+increased. The oscillations due to the existence of well defined stat ic fields progressively disappear
+and, forx >0.5, the frozen magnetism disappears. (Adapted from15))
+frozen moment would then be less than 6 10−4µB(µSR). This is completely consistent with ac
+susceptibility results for x= 1, which behaves monotonously down to the lowest temperatu re
+(50 mK) and with neutron scattering where no magnetic diffract ion Bragg peak was detected
+at variance with clinoatacamite.16)
+Herbertsmithite therefore appears as the first structurall y perfect kagome antiferromagnet
+not displaying any magnetic transition and opens wide the fie ld of experimental investigations
+in quantum kagome antiferromagnets.
+2.3 Interactions
+In Herbertsmithite, the 119◦Cu-OH-Cu bond-angle yields a moderate planar interaction
+J≃180(10) K in comparison with other cuprates. Note that this v alue is consistent with that
+observedin cubaneswherea similar OHbridgelinksadjacent coppers.14)From asimpleCurie-
+Weiss analysis of the variation of the high- Tsusceptibility with x, the interaction between
+apical and planar Cu can be estimated to be weakly ferromagne tic,J′∼0.1J, in agreement
+with neutron low- Tdata on the x= 0 compound. The paratacamite family therefore offers
+an interesting possibility to explore weakly ferromagneti cally coupled kagome layers7)for
+0.33< x <1.
+4/22J. Phys. Soc. Jpn. Full Paper
+Fig. 3. (Color online) Top panel: Susceptibility versus Tforx= 0, 0.5, 0.66 and 1.0 in the parat-
+acamite family. The corresponding symbols can be read from the bot tom panel. The T= 0
+ferromagnetic component decreases progressively from x= 0 (inset) to x= 1.0. Bottom panel:
+The apparent Curie-Weiss temperatures are extracted from linea r fits of the inverse susceptibility
+up to 300 K. These values have to be corrected in order to extract the exact value of the exchange
+interaction, (see text). Adapted from.7)
+3. Magnetism of the kagome planes
+3.1 Static susceptibility
+Probing the susceptibility is a crucial issue regarding the comparison and discrimination
+between models. It has represented a major challenge since t he discovery of the compound.
+Various results are now reported through macroscopic susce ptibility, and local techniques, by
+shift NMR measurements through35Cl17)and17O NMR18)orµSR.19)Among all these local
+probes,17O NMR is certainly the most sensitive to the physics of the kag ome planes since
+oxygen from the OH group mediates the superexchange couplin g between two adjacent Cu.
+The hyperfine coupling of Cl to Cu is found more than one order o f magnitude smaller, itself
+5/22J. Phys. Soc. Jpn. Full Paper
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48
+/s80/s50
+/s49/s47/s110
+/s32/s32/s32
+/s90/s110/s32/s99/s111/s110/s116/s101/s110/s116/s83/s116/s97/s116/s105/s99
+/s68/s121/s110/s97/s109/s105/s99/s70/s114/s111/s122/s101/s110/s32/s102/s114/s97/s99/s116/s105/s111/s110
+/s82/s51/s109/s48 /s49/s48 /s50/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48
+/s32/s32/s70/s114/s111/s122/s101/s110/s32/s70/s114/s97/s99/s116/s105/s111/s110
+/s84/s32/s40/s75/s41/s120/s32/s61/s32/s48
+/s120/s32/s61/s32/s48/s46/s49/s53
+/s120/s32/s61/s32/s48/s46/s51/s51
+/s120/s32/s61/s32/s48/s46/s53
+Fig. 4. (Color online) Phase diagram of the paratacamite family as ded uced from the T→0 frozen
+fractions found in zero field µSR experiments15)(inset).
+much larger than the muon one. We will focus first on17O NMR.
+Typical17O NMR spectra are displayed in Fig. 5. From a detailed analysi s of the marked
+singularities of the powder lineshape which positions are g overned by the magnetic shift and
+quadrupolar effects, two different sites can be identified. The m ost intense line at high T,
+named main(M), is associated with O sites linked to two Cu sit es in the kagome planes.
+Tracking its position gives the shift, hence one can extract the hereafter called ”intrinsic”
+susceptibility from 300 K down to 0.45 K. The other line (D) wh ich is very sharp at low- Tis
+characteristic of O sites linked with one Cu and one Zn in the k agome planes. The existence
+of this line points at the existence of defects resulting mai nly from Cu-Zn site disorder. This
+will be further discussed in the next sections as well as the c orresponding details of the17O
+NMR spectra such as the broadening observed at low Tfor the (M) line.
+Cl NMR shift could be as well followed on partially oriented p owders but due to a large
+broadening when Tis decreased, it becomes impossible to determine the suscep tibility below
+15 K.
+It is worth noting that no quantitative derivation of the cou plings of the probes to the
+various Cu sites has been made up to now but the diversity of th e observed behaviors at low
+Tmight likely be associated to the ratios of couplings streng th of each probe to the various
+Cu locations including the non-kagome Zn site.20)
+One can divide the behavior of the susceptibility into 3 T-ranges (Fig. 6):
+6/22J. Phys. Soc. Jpn. Full Paper
+/s54/s46/s52 /s54/s46/s53 /s54/s46/s54 /s54/s46/s55 /s54/s46/s56 /s54/s46/s57 /s55/s46/s48
+/s32/s32
+/s56/s53/s32/s75
+/s72/s32/s40/s84/s101/s115/s108/s97/s41
+/s32/s32
+/s54/s48/s32/s75
+/s32/s32
+/s53/s48/s32/s75/s51/s48/s32/s75/s49/s53/s32/s75
+/s32/s32
+/s50/s48/s32/s75
+/s32/s32
+/s49/s48/s32/s75
+/s32/s32
+/s53/s32/s75
+/s32/s32/s32/s49/s46/s51/s32/s75/s114/s101/s102/s101/s114/s101/s110/s99/s101
+/s48/s46/s52/s55/s32/s75
+/s32
+/s52/s48/s32/s75
+/s68/s77
+/s49/s55/s53/s32/s75/s32
+Fig. 5. (Color online)17O NMR spectra from 175 K to 0.47 K from ref.18)The shift reference is
+indicated by the vertical line. (M) and (D) refer to the two oxygen s ites in the kagome planes as
+described in the text. Large full green circles indicate the position o f the center of the line that
+wouldbe expected from χmacro.At 175K,adetailedanalysiswasperformedandtwocontributions
+are evident from the displayed fits (see text and section 4). The ha tched area represents the
+contribution from (D) oxygens sitting nearby one Zn defect in the k agome plane.
+3.1.1 High T:T >150K
+In macroscopic susceptibility ( χmacro) experiments as well as for17O NMR, or35Cl NMR,
+an effective Curie-Weiss behavior is observed at high T. From a correct fit using high temper-
+ature series expansions, one can extract a value J= 170(10) K from .21,22)
+3.1.2 Intermediate T:10< T <150K
+Below 150 K, macroscopic and local susceptibilities as meas ured from35Cl and17O NMR,
+start to markedly differ as seen directly from Fig. 5 and 6. A cle ar upturn is observed in
+χmacrowhereas the variation of the oxygen shift goes the opposite d irection.35Cl NMR data
+also show a deviation from χmacro. This is certainly the best evidence that the compound is
+not free from defects. Would it be, the only contribution to17O andχmacromeasurements
+would come from fully occupied Cu2+kagome planes and would be identical provided that no
+staggered component dominates the O signal. Through17O shift measured from 300 K ∼1.7
+Jdown to 0.45 K ∼J/400, a broad crossover with a smooth maximum from the Curie-W eiss
+7/22J. Phys. Soc. Jpn. Full Paper
+/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s49/s50/s51/s52
+/s48/s46/s48/s49/s46/s48/s120/s49/s48/s45/s51/s50/s46/s48/s120/s49/s48/s45/s51/s51/s46/s48/s120/s49/s48/s45/s51/s52/s46/s48/s120/s49/s48/s45/s51
+/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s45/s48/s46/s48/s50/s45/s48/s46/s48/s52/s45/s48/s46/s48/s54/s45/s48/s46/s48/s56/s83/s117/s115/s99/s101/s112/s116/s105/s98/s105/s108/s105/s116/s121/s32/s40/s99/s109/s51
+/s47/s109/s111/s108/s32/s67/s117/s41/s32/s78/s77/s82/s32/s76/s105/s110/s101/s115/s104/s105/s102/s116/s32/s40/s37/s41
+/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s49/s55
+/s79/s51/s53
+/s67/s108
+Fig. 6. (Color online) Thermal variationof the susceptibility ofthe ka gomeplanes measured from the
+shift of the main17O NMR line (red open symbols) compared to the macroscopic suscept ibility
+( green solid line). The red dashed line is a guide to the eye for the low- Tshift. Inset:35Cl data
+show the same trend as17O data although the accuracy is much reduced below 50 K. Adapted
+from ref.17,18)
+to the low- Tregime is observed between ∼2J/3∼120 K and ∼J/3∼60 K, see Fig. 6.
+Such maxima are also observed in other corner-sharing kagom e based lattices23–26)but theT
+dependence of the shift differs from one compound to another. A rapid decrease of the shift
+is further observed at lower T, between J/3 andJ/20∼10 K.
+3.1.3 Low T:T <5−10K
+ForT <10 K∼J/20, the shift finally levels off at a finitevalue. For instance, no difference
+can be found between the17O spectra taken at 0.45 and 1.3 K. The non-zero value and the
+Tdependence of the shift lead to the conclusion that the kagom e plane susceptibility of
+Herbertsmithite has a finitevalue at T→0 and the system is probably gapless.
+3.2 Dynamical susceptibility
+On the dynamical side, inelastic neutron scattering (INS) d ata taken at 35 mK (Fig. 7),
+indicate an increase of χ′′(ω) with decreasing ωand clearly supports the absence of a spin gap
+at least larger than 0.1 meV. The apparently divergent power law behavior χ′′(ω)∼ω−0.7(3)
+is also quite unusual. Moreover χ′′is only weakly Qdependent and points at the absence of
+a characteristic length to describe the correlations.16)
+T1relaxation measurements complement the insight into the st ructure of the excitation
+8/22J. Phys. Soc. Jpn. Full Paper
+/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53/s48/s50/s52/s32/s84/s32/s61/s32/s51/s53/s32/s109/s75
+/s32/s84/s32/s61/s32/s49/s48/s32/s75
+/s32
+/s32/s40/s109/s101/s86/s41/s34 /s40 /s41 /s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
+Fig. 7. (Color on line) Imaginary part of the dynamical susceptibility in tegrated over momentum
+transfer. The apparently divergent behavior is indicated by a powe r law fit, as represented by the
+full line on the 35 mK data. The exponent -0.5(3) is in line with the -0.7(3 ) obtained from earlier
+data, in ref.16)
+spectrum.Thethermaldependenceoftherelaxation rateobt ained for17O,18) 35Cland63Cu17)
+NMR are plotted in Fig. 8. Quite strikingly, in the low Tregime (T <30 K∼J/6), the high
+field relaxation rate of O, Cl and Cu nuclei follow the same rat her unusual variation, namely
+a power law with the sublinear exponent T−1
+1∼T0.71(5).17,18)Given the filtering form factors
+which are quite different for all probes, this indicates that e xcitations are likely weakly Q-
+dependent. The T-dependence clearly demonstrates the absence of a gap in the excitations
+and confirm the neutron results. Note that a very different T-behavior, namely a moderate yet
+observable increase of the relaxation followed by a saturat ion is observed in µSR experiments
+forT <1 K but this might be related to weak interactions between mag netic defects.15)
+4. Magnetic defects and their contribution to macroscopic measurements
+In addition to the previously reported discrepancy between17O shift and macroscopic
+susceptibility, there are several types of arguments which consistently fit with a scenario
+of dilution of the kagome magnetic lattice of Herbertsmithi te. This comes out from inter-site
+mixingdefects of CuandZnspecies even for compoundswithth eideal stoechiometry. Namely,
+part of the non-magnetic Zn2+substitutes copper on the kagome plane, and respectively, t o
+preserve stoichiometry, the same amount of copper substitu tes zinc on the inter-plane site.
+One should note that samples of different origins yield very si milar results in macroscopic
+measurements,35Cl NMR and µSR, so that the various interpretations which have been
+9/22J. Phys. Soc. Jpn. Full Paper
+/s48/s46/s49/s49/s49/s48
+/s48/s46/s49/s49
+/s48 /s53 /s49/s48 /s49/s53/s48/s46/s48/s48/s46/s52/s48/s46/s56
+/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48/s32/s49/s55
+/s40/s49/s47/s84
+/s49/s41/s32/s40/s109/s115/s45/s49
+/s41
+/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s51/s53
+/s67/s108
+/s54/s51
+/s40/s49/s47/s84
+/s49/s41/s32/s40/s109/s115/s45/s49
+/s41/s44/s32/s51/s53
+/s40/s49/s47/s84
+/s49/s41/s32/s40/s115/s45/s49
+/s41/s32
+/s32
+/s32/s54/s51
+/s67/s117/s32
+/s32/s49/s55
+/s79
+/s32
+/s84/s45/s49
+/s49/s126 /s84/s48/s46/s55/s49/s40/s53/s41
+Fig. 8. (Color on line) Semi-log plot of the relaxation rates obtained fo r all probe nuclei,17O,63Cu
+and35Cl. Inset: Linear plot of T−1
+1with a power law fit for the low T17O data. Adapted from
+ref.17,18)
+successively proposed relate rather with progress in the un derstanding of Herbertsmithite
+physics than with differences between samples.
+It is fair to point that a different picture of perfectly filled C u2+kagome planes was
+initially favored based on a quite funded Jahn-Teller argum ent.7)Indeed, since Cl−are quite
+far from Cu2+located in the kagome planes, the quasi-planar configuratio n is thought to
+be most favorable to Cu occupation through Jahn-Teller stab ilization. This statement was
+reinforced by the interpretation of early shift measuremen ts through µSR which was not then
+confirmed by other local probes. Yet, as pointed above, it is t oday quite hard to explain all the
+experimental findings within the initial framework of what w as initially hailed as a ”perfect”
+compound. The Jahn-Teller energy stabilization might not b e large enough to completely
+prevent inter-site mixing and it would be interesting to kno w more about the Cu Jahn-Teller
+energy in this compound.
+4.1 Direct evidence of intersite mixing defects
+Although Zn and Cu have close neutron scattering cross-sect ions, structural refinements
+are sensitive enough, as compared to x-ray ones, to suggest a large amount 7-10% of such
+inter-site mixing defects,27,28)depending on the origin of the samples.
+NMR spectra taken at high T, in the uncorrelated regime, see Fig. 5, give a local evi-
+dence for this. Indeed, the NMR powder average spectrum disp lays many singularities which
+10/22J. Phys. Soc. Jpn. Full Paper
+correspond to singular orientations of shift and quadrupol ar tensors. Although oxygen occu-
+pies only one crystallographic site in Herbertsmithite, it was found impossible to fit such a
+powder lineshape29)with only one O site. The additional site is well explained by oxygens
+surrounded by one Cu2+contributing to its shift and one Zn2+, ie a spin vacancy which does
+not contribute to the shift as sketched in Fig. 9. A quantitat ive estimate of a 5% magnetic
+dilution was calculated from the ratio of intensities of thi s ”defect” (D) line to that of the
+main (M) line. The shift of this second site is reported in Fig . 9 and compared to half the shift
+of the main line. These two quantities match well in the high- Tregime where correlations are
+negligible but differences occur at low- T. In addition, the relative sharpness of the defect-line
+clearly point at some different low- Tphysics for the defect sites. We postpone the related
+discussion to the next section but, interestingly, the NMR l ineshape can be satisfyingly repro-
+duced in exact diagonalization of small spin clusters on the kagome lattice with random spin
+vacancies.30)
+Finally, the low energy magnetic spectral weight in inelast ic neutron scattering is found
+to shift linearly with field towards higher energies, which i s commonly interpreted as a Zee-
+man splitting of paramagnetic-like S= 1/2 centers. Yet, because of linewidth and scattering
+intensity arguments developed in16,28)they cannot be interpreted as arising from strictly
+non-interacting spins. Either weakly coupled coppers occu pying the Zn site due to intersite
+mixing, or paramagnetic centers due to spinless defects (se e below) could yield a qualitative
+interpretation for this.
+4.2 Macroscopic measurements: quasi-free Cu2+spins on the Zn2+site
+4.2.1 Susceptibility
+The intersite mixing is responsible for two kinds of magneti c defects that can show up in
+the susceptibility of Herbertsmithite.
+First, the misplaced Zn2+in the kagome planeact as spinless impurities which are like ly to
+induce a staggered magnetization of the nearby Cu2+spins as commonly observed in strongly
+correlated systems.31)This could well relate to the magnetic broadening of the17O line at
+lowT, see Fig.5. Probing the response to such defects is a ”pertur b to reveal” strategy which
+might prove to be rewarding. A special section is devoted to t his in the discussion part. As
+observed in the kagome-like S=1/2 Volborthite compound, wh ere non-magnetic impurities
+can be introduced in a controlled manner, one can then expect that the whole extended
+defect around the spin vacancy will behave as a paramagnetic center with a small effective
+moment.32)The corresponding contribution is often omitted in the cont ext of Herbertsmithite
+and ought to be considered.
+Second, the misplaced Cu2+on the Zn interlayer site are weakly coupled to the kagome
+planes if one refers to the weak interkagome plane coupling i n the paratacamite family (see
+section 2.3), and can be considered as quasi-free S=1/2 spin s. While the first type of defects
+11/22J. Phys. Soc. Jpn. Full Paper
+/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53
+/s40/s68/s41/s40/s77/s41/s79
+/s67/s117/s72/s79
+/s67/s117/s72
+/s67/s117 /s79
+/s90/s110/s67/s117 /s79
+/s90/s110/s90/s110 /s67/s117/s79
+/s90/s110 /s67/s117/s79/s32/s76/s105/s110/s101/s115/s104/s105/s102/s116/s32/s40/s37/s41
+/s84/s32/s40/s75/s41
+Fig. 9. (Color on line) Comparison between the shift of the main line (M: open red squares) and
+that from the defect line(D: full blue circles (adapted from ref.18)). The corresponding oxygen
+environment are displayed at the right of the figure. The red line rep resents half the M shift which
+isacorrectapproximationoftheD shift downto J/4.Lowerin temperature,the Dshift isfound to
+decrease more rapidly. This indicates a depressed susceptibility nea r a spin vacancy. The sketch at
+the bottom left of the figure represents what is expected in an idea l case where singlet dimers are
+formed next to a spin vacancy.46)Dzyaloshinskii Moriya anisotropy slightly corrects this singlet
+picture.45)
+is well studied by17O NMR, the second type is a classical case of weakly coupled pa ramag-
+netic centers, which give the dominant contribution, at low temperature, to the macroscopic
+susceptibility. This is indeed what is observed in these mac roscopic measurements. From a
+simple Curie analysis of the low- TCurie-like tail of the susceptibility, one gets an estimate of
+5-10% of such Zn-defects in the kagome planes.
+4.2.2 Low Tand High Field Magnetization
+In lowTmagnetization measurements under strong magnetic fields (s ee Fig. 10),33)the
+magnetization is linear at low field and shows a pronounced do wnward curvature above ∼2 T
+as expected for the Brillouin type saturation of free spins, although a saturation plateau at
+higher field is not observed. Instead, the magnetization for H/greaterorsimilar8 T increases rather linearly
+with the applied field. As a first approximation, one can split the total magnetization as the
+sum of two terms.
+12/22J. Phys. Soc. Jpn. Full Paper
+First, a linear term that is likely to arise from the suscepti bility of the Cu2+spins of the
+kagome planes. Due to their strong antiferromagnetic inter actions, their magnetization is not
+expected to deviate from linearity up to very high fields gµBH∼J. As shown in the inset
+of fig. 10, this kagome plane susceptibility is much lower tha n the macroscopic susceptibility
+at the same T= 1.7 K but significantly higher than the local susceptibility ob tained from
+17O NMR. However if the effective moment of the extended paramagn etic defects induced by
+spin vacancies in the kagome lattice is small, they may not be saturated in the maximum
+applied field of 14 T and they then also contribute to the linea r magnetization. As a result
+the extracted susceptibility must be considered as an upper bound to the intrinsic kagome
+plane susceptibility.
+Second, a saturated magnetization which can be isolated by s ubtraction of the high field
+linear contribution (discussed above) from the measured to tal magnetization (see red curve in
+Fig. 10). This part of the magnetization resembles a Brillou in function for a spin 1/2 although
+the saturation is reached at a slightly higher field suggesti ng some residual antiferromagnetic
+interactions of ∼1 K.15,27,33)This second contribution can be safely attributed to the in-
+terlayer Cu2+defects. From the value of the saturated magnetization one g ets about 7% of
+such defects out of the total number of Cu, in perfect agreeme nt with the direct and indirect
+estimates from other studies.
+4.2.3 Specific heat
+The specific heat of Herbertsmithite was measured down to 0.1 K under various applied
+fields up to 14 T.16,27)Since no isostructural non-magnetic compound is available , reliable
+conclusions about the magnetic contribution cannot be extr acted above 5-10 K because of the
+dominant contribution from phonons. The evolution of the Tdependence of the specific heat
+under an applied field can be fitted satisfactorily with a Scho ttky model of quasi-free spins,
+which value is quite accurately S= 1/2, assuming an amount of 6 % of the total number of
+Cu. Including interactions between the ”quasi-free” Cu on t he Zn site might indicate that the
+analysis developed in22)could be corrected through the addition of extra terms. Whet her such
+corrections to a pure Schottky analysis would correct the sm all 2 K value measured for ∆ E
+(fig . 11) or whether the latter reveals some intrinsic energy scale in Herbertsmithite is still an
+open issue. From the 0 .66< x≤1.0 wide range of compositions in the paratacamite family
+where no ordering occurs, and the weakness of interplanar in teractions that they generate, it
+seemssafetoassessthat theintersiteCucannot influencedr amatically thesusceptibility ofthe
+kagome planes. Using such a scenario and assuming in additio n that the kagome susceptibility
+is unaffected by spinless Zn defects, a very detailed analysis , using series expansions and Pade
+approximants for the pure kagome lattice22)show that both low- Tsusceptibility and specific
+heat data in Herbertsmithite can be fairly reproduced with a few % weakly interacting spins.
+There are some discrepancies, especially for the deduced en tropy which is found smaller than
+13/22J. Phys. Soc. Jpn. Full Paper
+/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s48/s50/s52/s54/s56/s49/s48
+/s49/s48/s45/s51/s49/s48/s45/s50/s48/s44/s49 /s49 /s49/s48 /s49/s48/s48/s40/s99/s109/s51
+/s47/s109/s111/s108/s32/s67/s117 /s41
+/s32/s32/s77/s47/s77
+/s115/s97/s116/s32/s40/s37/s41
+/s72/s32/s40/s84/s41/s49/s46/s55/s32/s75
+/s32/s84/s32/s40/s75/s41
+/s49/s55
+/s79/s108/s111/s119/s32/s72
+Fig. 10. (Color on line) Field dependence of the 1.7 K macroscopic magn etization (black squares)
+of Herbertsmithite normalized to the saturated magnetization of 1 mole of Cu2+S=1/2 spins.
+The magnetization of the defective inter-layer Cu2+spins (red open squares) is estimated by
+subtracting the linear contribution of the kagome planes (blue line) f rom the total magnetization.
+Inset : the kagome plane susceptibility at 1.7 K, i.e. the slope of the line ar contribution, is shown
+as the blue circle with large downward error bar on the plot of the mac roscopic susceptibility
+measured in 0.1 T (black curve) and17O NMR lineshift (open circles) reproduced from Fig. 6.
+Adapted from ref.33)
+that of the pure kagome lattice. This might point at the need t o include the response of the
+spinless defects or other extra terms (see discussion secti on) in the interpretation of these
+results.
+Overall, due to the dominant contribution from these free ce nters, it is quite hard to
+conclude about the analytic dependence of the specific heat w hich behavior depends strongly
+on the selected T-range.
+5. Discussion
+As emphasized in the introduction, the discovery of Herbert smithite has triggered many
+new ideas about the ground state of quantum kagome antiferro magnets.34–36)Exact diagonal-
+izations of finite clusters in the S= 1/2 quantum limit had earlier suggested a picture of a
+fluctuating ground state with an unusual continuum of single t excitations between a singlet
+ground state and the first excited triplet, at variance with t he triangular HAF (Fig.12)37,38)
+which features a more connected edge-sharing lattice. The c entral question of whether the
+small spin-gap /lessorsimilarJ/20 survives or vanishes at the thermodynamic limit is still p ending,39)see
+fig. 13. A remarkable result is that an effective dimer-model yi elded a similar energy landscape
+14/22J. Phys. Soc. Jpn. Full Paper
+0.5 1 2 4 8 16 32
+T (K)0.040.21525Cv (J mol-1K-1)0 T
+1 T
+2 T
+3 T
+5 T
+7 T
+9 T
+0 2 4 6 8
+T (K)-0.200.20.40.6
+∆CV/T  (JK-2mol-1)1 T - 3 T
+1 T - 9 T
+6%    S = 1/2
+1.5% S = 3/2
+02 4 6 8H (T)
+051015∆E/kB (K)
+Fig. 11. (Coloronline) ThelowtemperatureheatcapacityofHerber tsmithiteasafunctionofapplied
+field. Inset top: The temperature dependence of Cv(1 T) - Cv(3 T ) and Cv(1 T) - Cv(9 T), fitted
+to two models, S= 1/2 andS= 3/2. The latter cannot be fitted simultaneously with the Cv(1
+T) - Cv(3 T) and Cv(1 T) - Cv(9 T) curves. Even for S= 1 the fit is less good than for S= 1/2.
+Inset bottom: field dependence of the S= 1/2 doublet. Adapted from ref.27)
+with the same entropy.40)These are quite strong indications that rather than condens ing into
+an ordered state, the system breaks into singlets which reso nate to lower the total energy,
+possibly pointing at a RVB state. Quite recently, more propo sals have been made for the
+ground state of the KHAF, but it is not the aim of this paper to r eview in detail the various
+models.
+From an experimentalist point of view, extracting the exist ence and value of the gap is
+of crucial importance. The dynamical susceptibility which could be extracted from neutron
+scattering, its Qdependence, the NMR relaxation rates and the specific heat ar e also stringent
+testsof theories althoughtherearelittle predictions att hepresentstage aboutthesequantities
+and their exact T-dependence.
+Two main features are emerging from the experimental studie s on Herbertsmithite, which
+are reported in the previous sections. The ground state poss esses a finite magnetic suscepti-
+bility. Hence it is not a pure singlet state and, concomitant ly, no spin-gap is observed in the
+excitations. One is therefore led to the central question of how near (or far) Herbertsmithite
+is from the ideal S=1/2 kagome Heisenberg antiferromagnet.
+Various additional terms need to be incorporated into the He isenberg Hamiltonian. We
+review them sequentially below but they likely have to be com bined in order to account
+adequately for the present experimental data in Herbertsmi thite. This might have a positive
+counterpart in the future since varying them through more ad vanced materials engineering
+15/22J. Phys. Soc. Jpn. Full Paper
+Fig. 12. Energyspectrum using exactdiagonalizationstechniques, from ref.37)The striking difference
+betweenthetriangularandkagom´ eantiferromagnets,labeledTA HandKHAF, isthecontinuumof
+excitations in the singlet channel. The gap between the first singlet a nd triplet states is estimated
+at ∆< J/20, but might vanish at the thermodynamic limit.39)
+might prove a rewardingway to isolate their influenceon thei ntrinsicproperties of thekagome
+lattice and could help discriminate between models.
+5.1 Applied field
+In the context where macroscopic susceptibility is dominat ed by the defects contribution
+at lowT, the main measurement of the susceptibility relies upon loc al NMR measurements
+which are all performed in fields of the order of a few Teslas. T his represents a few K energy
+for spins 1/2. The non-zero value of the susceptibility and t he shape of its T-variation both
+indicate that in a scenario of a singlet-triplet gap, the val ue of the gap under an applied field,
+should be less than J/200∼1 K. There is no calculation at present of how the field could
+decrease the value of the gap and this is an issue which should be addressed in the future.
+5.2 Dzyaloshinskii-Moriya (DM) anisotropy
+Initially introduced in the context wherethe macroscopic s usceptibility was assumed to be
+intrinsic, the DM anisotropy has been incorporated into exa ct diagonalization calculations in
+ordertoaccount fortheupturnof thesusceptibility below ≃J, afeature whichis not expected
+in the unperturbed case.41)The existence of a DM anisotropy− →Dij.− →Si×− →Sjis granted since
+there is no inversion symmetry between two adjacent Cu.− →Dhas two components, one along
+z,Dz, and one planar Dpwhich act differently on the susceptibility and the specific he at (see
+Fig. 11). Inorder to explain the low- Tupturnof the susceptibility, one would need Dp>|Dz|,
+16/22J. Phys. Soc. Jpn. Full Paper
+Fig. 13. (Color on line) Gap determined from exact diagonalizations te chniques versus the inverse
+size of the sample. The sample size was doubled within 10 years. The ga p might vanish at the
+thermodynamic limit39)as shown by the least square fit with a 3% J accuracy. Unpublished da ta,
+courtesy from P. Sindzingre and C. Lhuillier.
+withDp∼0.3J. Yet, these values do not match those which were later extrac ted from the
+broad ESR line at room- T(Fig. 14), iea value of Dpwhich does not exceed 0.05-0.1 J42)
+(Fig. 15). Therefore only 15-30 % of the upturncan be account ed for by DM anisotropy, which
+validates the defect approach. Within the latter approach, DM anisotropy induces a decrease
+of entropy which might solve one of the problems of the defect -only approach developed in
+ref.22)
+OneofthemostinterestingoutcomefromthestudyoftheDMan isotropyasaperturbation
+oftheHeisenbergHamiltonian is thatan unexpectedphasedi agram appearsFig. 15). Whereas
+in the classical case, any minute amount of DM anisotropy wil l lead to long-range order,43)for
+spinsS= 1/2 a quantum critical point appears which is governed by the va lue ofD/J∼0.1
+where the system switches from a liquid to a N´ eel state.44,45)Noticeably, Herbertsmithite
+appears to be close to this quantum critical point.
+The absence of a gap in the susceptibility is then well explai ned by a mixing of singlet
+and triplet states due to the DM interaction. Indeed with a va lue of the order of 0.1 J, DM
+interactions likely smear out a gap which is predicted to be /lessorsimilarJ/20.
+17/22J. Phys. Soc. Jpn. Full Paper
+/s57 /s49/s48 /s49/s49 /s49/s50 /s49/s51/s45/s53/s48/s53
+/s49/s48/s32/s75/s32/s69/s83/s82/s32/s83/s105/s103/s110/s97/s108/s32/s40/s97/s46/s117/s46/s41
+/s32
+/s72 /s32/s40/s84/s41/s68
+/s112/s68
+/s122
+/s51/s48/s48/s32/s75
+Fig. 14. (Color on line) ESR spectra taken at 300 and 10 K. The broad linewidth is due to a large
+anisotropy.The dashed line shows the anisotropy of the g- shift tensor, hence the minute deviation
+from a pure Heisenberg character of the spins, quite common for C u2+ions. In the inset the
+Dzyaloshinskii vectors are represented. The Cu and O are respec tively represented by small and
+large full circles (red and blue online). Adapted from ref.42)
+5.3 In-plane magnetic defects
+Asdevelopedintheprevioussection,somequasi-freeCures ideontheZnsiteanddominate
+the low-Tmacroscopic measurements, yet they do not fully account for the experimental
+observations. DM interactions can complete the view but spi n vacancies in the kagome planes
+alsogeneratealocal magneticresponse.Thiscomesintopla yinthemacroscopicmeasurements
+but yield specific signatures for local techniques. It is the aim of this paragraph to review
+shortly this topic with respect to the experimental results .
+Can the presence of 5 - 10% dilution defects alter drasticall y the intrinsic physics? Down
+to 0.3J, this does not seem to be much the case if one refers to numeric al calculations
+of the macroscopic susceptibility and the specific heat.41)More insights on the local and
+extended response around spinless defects come out from exa ct diagonalization studies of
+the effect of non-magnetic impurities in kagome antiferromag nets, including the case where
+DM interactions are present. For D= 0, a non-magnetic impurity is found to induce a dimer
+freezingin thetwo adjacent triangles it belongs to, as expe cted from therelief of frustration.46)
+Further from the impurity, a staggered response is generate d like in all antiferromagnets.31)
+The major difference with the unfrustrated case is that the res ponse is not peaked on the sites
+18/22J. Phys. Soc. Jpn. Full Paper
+/s48/s46/s48/s51 /s48/s46/s48/s54 /s48/s46/s48/s57 /s48/s46/s49/s50/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57
+/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48
+/s49/s48/s51
+/s49/s48/s50
+/s49/s48/s49
+/s49/s48/s48/s32/s68
+/s112/s47/s74
+/s32
+/s68
+/s122/s47/s74/s50
+/s49/s48/s52
+/s72/s101/s114/s98/s101/s114/s116/s115/s109/s105/s116/s104/s105/s116/s101/s78/s233/s101/s108/s32/s112/s104/s97/s115/s101/s109/s50 /s65/s70
+/s68/s47/s74/s32
+/s32
+/s77/s111/s109/s101/s110/s116/s45/s102/s114/s101/s101/s32/s112/s104/s97/s115/s101
+Fig. 15. (Color on line) Inset: Contour plot of the value obtained for DzandDpfrom the analysis
+of the high- TESR spectrum. Main panel: Phase diagram adapted from.44)The Herbertsmithite
+lies near the QCP where a N´ eel ordered phase would be observed. A dapted from ref.42)
+next to the impurity. Both the tendency to dimer freezing and the staggered response hold for
+as large values of D/Jas 1, while the critical point at D/J= 0.1 does not seem much affected
+by the presence of these spinless defects45)(Fig. 16). Since for Herbertsmithite D/J/lessorsimilar0.1, one
+expects in17O NMR a well defined susceptibility for the oxygens next to a va cancy, reflected
+in the sharp ”defect line” (D) whereas, combining the stagge red response to the large number
+of defects, the susceptibility on further sites is expected to be more distributed as observed
+below 50 K on the main line (M). The experimental observation s agree with that trend and
+the non-zero value of the T→0 susceptibility for the (D) line argue in favor of not being f ar
+from the quantum critical point, 0 .06< D/J < 0.1, in agreement with ESR analysis.42,45)
+6. Conclusions
+Herbertsmithite is the only kagome compound not showing any magnetic transition at
+least down to temperatures of the order of 10−4J. In this respect, it has opened wide the
+field of experimental investigations of the kagome physics, which has been at the center of
+theoretical researches on highly frustrated magnets for mo re than 20 years. Its ground state
+is not a simple singlet state and excitations are gapless. Co nsiderable progress can be now
+anticipated in filling the gap between experiments and theor y and overall in unveiling the
+ground state of the pure kagome antiferromagnet, especiall y through the investigation of
+the thermal dependence of the excitations. Models should ta ke into account both the DM
+19/22J. Phys. Soc. Jpn. Full Paper
+Fig. 16. (Color on line) Evolution of the in-plane magnetizations aroun d a spin vacancy when D
+increases. The calculations are performed under a field B=J/20, similar to the applied field
+in NMR experiments and an angle of 30◦from the zaxis. The bond spin correlation values are
+represented by the thickness of the lines. The susceptibility remain s small for the sites adjacent
+to a spin vacancy. Adapte from ref.45)
+interaction ( ∼0.1J) which, due to low symmetry, might unfortunately not be avoi ded in any
+kagome based compounds and the existence of spinless defect s (∼a few %). Extensive exact
+diagonalization studies combining most of the experimenta l parameters have recently paved
+the way. Working out the consequences of these perturbation s might also help to understand
+and probe the kagome physics.
+Certainly, the synthesis of single crystals and the control led variation of both DM
+anisotropy and the amount of defects through new synthesis r outes are highly desirable for
+further progress in the understanding of the ground state of the kagome antiferromagnet. We
+are still at the early era of a new story! More quantum frustra ted compounds are being and
+will be synthesized.
+7. Acknowledgments
+We thank Pr. H. Kawamura and Pr. S. Maegawa for offering us the op portunity to present
+this review in this special issue.
+We thank A. Olariu, M. de Vries and A. Zorko for their particip ation at various stages of
+the work performed by our group and for their help in the prepa ration of the manuscript.
+We thank C. Lhuillier for discussions and a critical reading of our manuscript.
+We deeply thank J. Helton, C. Lhuillier, F. Mila, I. Rousocha tzakis, P. Sindzingre, M. de
+Vries and A. Zorko for providing original figures of their wor k for this review.
+The work on Highly Frustrated Magnets at LPS has been support ed by an ANR grant
+”OxyFonda” No. NT05-4-41913 and by the ESF research network ”Highly Frustrated Mag-
+netism”. It has benefitted from the access to the Large Scale M uon Facilities PSI and ISIS,
+20/22J. Phys. Soc. Jpn. Full Paper
+supported by the EC FP 6 program, Contract No. RII3-CT-2003- 505925.
+21/22J. Phys. Soc. Jpn. Full Paper
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+arXiv:1001.0802v1  [astro-ph.GA]  6 Jan 2010Draft version November 6, 2018
+Preprint typeset using L ATEX style emulateapj v. 11/10/09
+MOLECULAR CLOUD EVOLUTION III. ACCRETION VS.STELLAR FEEDBACK
+Enrique V ´azquez-Semadeni, Pedro Col ´ın, Gilberto C. G ´omez, and Alan W. Watson1
+Centro de Radioastronom´ ıa y Astrof´ ısica, Universidad Na cional Aut´ onoma de M´ exico, Campus Morelia, Apdo. Postal 3 -72, Morelia,
+58089, M´ exico
+Draft version November 6, 2018
+ABSTRACT
+We numerically investigate the effect of feedback from the ionizing ra diation heating from massive
+stars on the evolution of giant molecular clouds (GMCs) and their sta r formation efficiency (SFE),
+which we treat as a time-dependent quantity. We consider the GMCs ’ evolution from their formation
+by colliding warm neutral medium (WNM) streams up to advanced star -forming stages. We find,
+in agreement with our previous studies, that the star-forming reg ions (“clouds”) within the GMCs
+are invariably formed by gravitational contraction, so that their in ternal non-thermal motions must
+contain a significant component of global convergence, which does not oppose gravity. After an initial
+period of contraction, the collapsing clouds begin forming stars, wh ose feedback evaporates part of the
+clouds’ mass, opposing the continuing accretion from the infalling ga s. The competition of accretion
+against dense gas consumption by star formation (SF) and evapor ation by the feedback, regulates
+the clouds’ mass and energy balance, as well as their SFE. We find th at, in the presence of feedback,
+the clouds attain levels of the SFE that are consistent at all times with observational determinations
+for regions of comparable SF rates (SFRs). However, we observe that the dense gas mass is larger
+in general in the presence of feedback, and that the total (dens e gas + stars) is nearly insensitive to
+the presence of feedback, suggesting that the total mass is det ermined by the accretion, while the
+feedback inhibits mainly the conversion of dense gas to stars, beca use it acts directly to reheat and
+disperse the gas that is directly on its way to forming stars.
+We find that the factor by which the SFE is reduced upon the inclusion of feedback is a decreasing
+function of the cloud’s mass, for clouds of size ∼10 pc. This naturally explains the larger observed
+SFEs of massive-star forming regions. We also find that the clouds m ay attain a pseudo-virialized
+state, with a value of the virial mass very similar to the actual cloud m ass. However, this state differs
+from true virializationin that the clouds are the center of a large-sc alecollapse, continuously accreting
+mass, rather than being equilibrium entities. Finally, we also calculate t he density probability density
+functions of the clouds, finding that they in general exhibit the bimo dal shape characteristic of ther-
+mally bistable flows, rather than a lognormal form, which is characte ristic of isothermal flows. This
+supports suggestions that low density, atomic gas pervades molec ular clouds. We conclude that the
+general state of star-forming regions is likely to be one of gravitat ional collapse, although most of the
+mass in a GMC may be not participating of the instantaneous star-fo rming activity, as recently sug-
+gested by Elmegreen (2007); that is, that SF is a spatially and tempo rally intermittent phenomenon,
+with strong, localized bursts interspersed within much more quiesce nt gas.
+Subject headings: interstellar matter – stars: formation – turbulence
+1.INTRODUCTION
+The evolution of giant molecular clouds (GMCs) is a
+keyingredient in ourunderstanding ofthe starformation
+process. In particular, the low observed star formation
+efficiency (SFE) at the scale of whole GMCs ( ∼2%;
+Myers et al. 1986) remains a topic of strong debate, with
+there being two main competing scenarios that attempt
+to explain it. These scenarios refer essentially to the
+effect of the stellar feedback (mainly from massive stars)
+on the star-forming clouds. One is the scenario that the
+stars quickly disrupt their parent clouds by dispersal
+and/or photoionization, before the gaseous mass of the
+cloud is completely converted to stars (e.g. Whitworth
+1979; Elmegreen 1983; Franco, Shore & Tenorio-Tagle
+1994; Williams & McKee 1997;
+Hartmann, Ballesteros-Paredes & Bergin 2001). In
+e.vazquez@crya.unam.mx, p.colin@crya.unam.mx, g.gomez @crya.unam.mx, alan@astro.unam.mx
+1Also: Instituto de Astronom´ ıa, Universidad Nacional
+Aut´ onoma de M´ exico, A.P. 70-264, 04510, M´ exico, D.F., M´ exicothis case, the SFR of active star-forming sites may be
+large for brief periods, and then halted by the very stars
+that have just been formed.
+In the other scenario, the role of stellar feedback
+is to drive turbulent motions within the GMC, which
+oppose its self-gravity, allowing it to remain in near
+hydrostatic equilibrium for times significantly longer
+than its free-fall time ( tff) (Krumholz & McKee 2005;
+Krumholz, Matzner, & McKee 2006; Li & Nakamura
+2006). In this case, the low efficiency of star formation
+would be due to the dual role of supersonic turbulence in
+self-gravitating clouds, of opposing global collapse of the
+cloud while promoting local collapse of turbulent density
+enhancements, which involve small fractions of the
+total cloud mass (Klessen, Heitsch & Mac Low 2000;
+V´ azquez-Semadeni, Ballesteros-Paredes, & Klessen
+2003, see also the reviews by Mac Low & Klessen 2004;
+Ballesteros-Paredes et al. 2007).
+Another controversy, related to the control of the
+SFE, refers to the nature of the motions originating the2 V´ azquez-Semadeni et al.
+linewidths observed in GMCs and their substructure.
+The latter were initially proposed to correspond to
+gravitational contraction by Goldreich & Kwan (1974),
+but this suggestion was quickly deemed untenable by
+Zuckerman & Palmer (1974), who noted that it would
+imply totalGalacticSFRs ofthe orderofthe totalmolec-
+ular gas mass in the Galaxy ( ∼109M⊙) divided by the
+typical free-fall time for a GMC ( ∼4 Myr), or ∼250M⊙
+yr−1, an estimate roughlytwoordersofmagnitude larger
+than the observed Galactic SFR. Zuckerman & Evans
+(1974) then suggested that the observedlinewidths could
+correspond instead to random, small-scale2turbulent
+motions, a notion that has prevailed until the present.
+However, a number of workers have recently advocated
+a return to the gravitational contraction picture, noting
+that various observational properties of clouds and
+clumps can be well matched by models dominated by
+gravitational contraction (e.g., Hartmann & Burkert
+2007; Peretto, Hennebelle & Andr´ e 2007;
+V´ azquez-Semadeni et al. 2009). Moreover, the no-
+tion of completely random, small-scale turbulent
+motions appears difficult to reconcile with the recent
+realization that the principal component of the velocity
+differences within clouds and clumps at all scalesappears
+to be “dipolar”, indicative of coherent motions at the
+scale of the whole cloud or clump (Heyer & Brunt 2007;
+Brunt et al. 2009). In general, several studies comparing
+simulations and observations have concluded that the
+motions in molecular clouds are consistent with large-,
+rather than small-scale driving (Ossenkopf & Mac Low
+2002; Brunt 2003; Padoan et al. 2009).
+If a return to the collapsing scenario is to be
+considered, it is necessary to somehow avoid the
+Zuckerman & Palmer (1974) criticism of it. This is ac-
+tually not so difficult because that criticism neglects
+the internal structure of the GMCs. Recent numeri-
+cal studies of cloud formation by converging streams of
+warm neutral gas in the interstellar medium show that
+the clouds are born turbulent, due to one or more of
+thethermal, thin-shell, andKelvin-Helmholzinstabilities
+(Hennebelle & P´ erault 1999;Koyama & Inutsuka 2000;
+Koyama & Inutsuka 2002; Audit & Hennebelle 2005;
+Heitsch et al. 2005; V´ azquez-Semadeni et al. 2006). The
+turbulence is subsonic with respect to the warm gas,
+but supersonic with respect to the cold phase, imply-
+ing large density fluctuations in the latter. In fact, it has
+recently been proposed that molecular clouds may actu-
+ally contain a warmer, atomic substrate in which colder
+molecular clumps are embedded (Hennebelle & Inutsuka
+2006). In either case, the molecular cloud contains large,
+nonlinear density enhancements in which the local free-
+fall time is significantly shorter than the cloud’s average.
+Thus, once the global collapse begins, the local clumps
+may complete their collapses earlier than the bulk of the
+cloud. They can thus begin forming stars that can begin
+their feedback action on the GMC before it completes
+the bulk collapse.
+In this paper, we present numerical simulations
+aimed at investigating this scenario, in which we use
+the same cloud-formation setup of previous papers
+2Zuckerman & Evans (1974) referred to these motions as “lo-
+cal”, and explicitly discarded large-scale coherent motio ns such as
+gravitational contraction.(V´ azquez-Semadeni et al. 2007; Banerjee et al. 2009),
+but including a prescription for stellar feedback mimick-
+ingionizationheatingfrommassivestars. With thistool,
+we investigate the effect of the feedback on the global
+SFE of the evolving GMC, as well as the nature of the
+motions in the cloud, in a first effort to shed light on
+these issues. As we shall see, it turns out that the phys-
+ical conditions in the clouds differ significantly from the
+“normal” picture, since accretion of gas from the warm
+diffusemediumisanintegralpartoftheclouds’dynamics
+and evolution, and thus the clouds cannot be considered
+as isolated.
+The plan of the paper is as follows. In §2 we describe
+the numerical code, and the implementation of our star
+formation and stellar feedback prescriptions. In §3 we
+describethesimulations,andin §4wedescribetheresults
+concerningthe controlofthe SFEby stellarfeedbackand
+the nature of the “clouds” themselves. Finally, in §6 we
+present a summary and some conclusions.
+2.THE NUMERICAL MODEL
+2.1.Heating and cooling
+The numerical simulations used in this work were per-
+formedusingthehydrodynamics+N-bodyAdaptiveRe-
+finement Tree code ART (Kravtsov et al. 1997; Kravtsov
+2003). Among the physical processes implemented in
+ART, relevant for our physical problem, are the radia-
+tive heating and cooling of the gas, its conversion into
+stars, ionization-like heating from stellar feedback, and
+self-gravity, from both the stars and the gas.
+We use heating (Γ) and cooling (Λ) functions of the
+form
+Γ=2.0×10−26erg s−1(1)
+Λ(T)
+Γ=107exp/parenleftbigg−1.184×105
+T+1000/parenrightbigg
++1.4×10−2√
+Texp/parenleftbigg−92
+T/parenrightbigg
+cm3.(2)
+Thesefunctionsarefitstothevariousheatingandcooling
+processes considered by Koyama & Inutsuka (2000), as
+given by equation (4) of Koyama & Inutsuka (2002). As
+noted in V´ azquez-Semadeni et al. (2007, hereafter Paper
+I), eq. (4) in Koyama & Inutsuka (2002) contains two
+typographical errors. The form used here incorporates
+the necesary corrections, kindly provided by H. Koyama
+(2007, private communication). With these heating and
+cooling laws, the gas is thermally unstable in the density
+range 1/lessorsimilarn/lessorsimilar10 cm−3(cf. Paper I).
+2.2.Star formation and stellar feedback prescriptions
+In ART, star formation is modeled as taking place in
+the coldest and densest regions, defined by T < T SFand
+n > n SF, whereTandnare the local temperature and
+number density of the gas, respectively, and nSFandTSF
+are respectively a density and a temperature threshold.
+We setTSF= 9000 K, which is easily satisfied by all
+cells with density nSF, so in practice our SF condition
+depends on density only.
+A stellar particle of mass m∗is placed in a grid cell
+where these conditions are simultaneously satisfied, andMolecular Cloud Evolution III. Accretion vs.Stellar Feedback 3
+thismassisremovedfromthegasmassinthecell. There-
+after, the particle is treated as non-collisional, and fol-
+lows N-body dynamics. No other criteria are imposed.
+In each gas cell that satisfies the above criteria a stel-
+lar particle is formed with a mass equal to 50% of the
+gas mass contained in the cell. Since the stellar particles
+are more massive than a single star, each stellar parti-
+cle should be considered as a small cluster, within which
+the individual stellar masses are distributed according to
+some stellar initial mass function (IMF).
+Stellar particles inject thermal energy at a rate ˙Eerg
+Myr−1per star with mass greater than 8 M⊙contained
+in the stellar particle. We assume a Miller & Scalo
+(1979) IMF, implying that each stellar particle of mass
+133M⊙produces one 8- M⊙star. The energy is de-
+positedin the cellin whichthe stellarparticleisinstanta-
+neously located, over a typical OB stellar lifetime, which
+we assume to be 10 Myr.
+It is important to note that, although initially we ex-
+perimented with realistic values of ˙Ebased on the Ly-
+man continuum fluxes of stars with masses between 10
+and 20M⊙(e.g., Diaz-Miller et al. 1998), we found that,
+because all the energy is deposited in a single cell, and
+the neighboring cells are heated exclusively by conduc-
+tion, rather than by radiative heating, the resulting HII
+regions were not so realistic. Thus, we opted instead for
+taking˙Eas a free parameter, and adjusting it until we
+obtained realistic HII regions, with temperatures ∼104
+K, diameters of a few parsecs, and expansion velocities
+of a few tens of km s−1.
+Note also that we resort to the common strategy of
+turning off the cooling in the cell where a stellar par-
+ticle is located, so that the cell can reach realistically
+high temperatures. Otherwise, the cooling can dissipate
+most of the thermal energy deposited in very dense cells.
+In the real ISM this does not occur because the stellar
+heating is applied through ionization, so that the tem-
+perature reached in the star’s immediate environment is
+independent of the medium’s local density. Instead, in
+the simulations, the cooling depends on the density, and
+the temperature resulting from the balance between the
+stellar heating and the cooling does depend on the den-
+sity. This problem is avoided by turning off the cooling
+in the cell where the stellar particle is located. Note
+that this contradicts claims that the need to turn off the
+cooling can be alleviated simply by increasing the resolu-
+tion (e.g., Ceverino & Klypin 2009). We argue that this
+problem can only be alleviated by performing radiation-
+hydrodynamics simulations. In their absence, we con-
+sider that turning off the cooling is actually a better
+model of the effect of stellar feedback, because it allows
+mimicking the fact that the gas temperature in the HII
+regions is independent of the local density.
+Finally, in order to further constrain the physical con-
+ditions in the HII regions, we also impose a “ceiling” to
+the temperature in the cell containing the stellar particle
+because otherwise, with the cooling off, the temperature
+in the cell might diverge. We set this “ceiling” to 106K.
+Although this procedure is mostly one of trial-and-
+error, we consider it to be the most adequate one for our
+purposes, since it is the HII regions that drive the tur-
+bulent motions in the dense gas in our simulations, and
+so it is them that must have realistic properties, even atthe expense of a somewhat ad-hoc SF prescription. We
+show a typical HII region in our simulation in Fig. 1.
+2.3.Refinement
+The numerical box is initially covered by a grid of 1283
+(zeroth level) cells. The mesh is subsequently refined as
+the matter distribution evolves. The maximum allowed
+refinement level was set to four, so that high-density re-
+gions have an effective resolution of 20483cells, with a
+minimum cell size of 0.125 pc. Cells are refined (halved
+in linear size) if the gas mass within the cell is greater
+than 0.32 M⊙. That is, the cell is refined by a factor of 2
+when the density increasesby a factor of 8, so that, while
+refinement is active, the grid cell size ∆ xscales with den-
+sitynas ∆x∝n−1/3. Once the maximum refinement
+level is reached, no further refinement is performed, and
+the cell’s mass can reach much larger values. In par-
+ticular, a stellar particle is formed when a fourth-level
+cell reaches a density nSF= 4×106cm−3, or a mass of
+243.5M⊙, again assuming µ= 1.27 (we use this value be-
+cause we do not follow the actual chemistry, and thus we
+assume the entire box to be filled with atomic hydrogen.)
+Thus, a stellar particle typically has a mass /greaterorsimilar122M⊙.
+Note that, because we use only four levels of re-
+finement, the largest densities arising in the simula-
+tion are by far not sufficiently resolved according to the
+“Jeans criterion” proposed by Truelove et al. (1997) for
+adaptive-mesh codes. Specifically, at our stellar-particle
+formation threshold density of 4 ×106cm−3, and assum-
+ing a gas temperature of T∼15 K at that density, we
+find that the Jeans length (using the adiabatic sound
+speed) is ∼0.031 pc, while the minimum cell size, at
+0.125 pc, is roughly 4 times larger. Thus, according to
+those authors, one should expect artificial fragmentation
+to occur in our simulations. However, we do not con-
+sider this to be a problem because we are not concerned
+here with the numbers and masses of the stellar parti-
+cles formed in the simulation, but simply with the total
+amount of mass that goes into stars.
+3.THE SIMULATIONS
+We consider four simulations using the same setup as
+in Paper I, which represents the evolution of a region of
+256pcperside, initiallyfilledwithwarmgasatauniform
+density of n0= 1 cm−3and a temperature T0= 5000 K,
+implying an adiabatic sound speed cs= 7.4 km s−1(as-
+suming a mean particle mass µ= 1.27). The whole nu-
+mericalbox thus contains5 .25×105M⊙. In this medium,
+we set up two streams moving with the same speed
+vinf= 5.9 km s−1(corresponding to a Mach number of
+0.8 with respect to the unperturbed medium) in opposite
+senses along the x-direction. The streams have a radius
+of 32 pc and a length of 112 pc each, so that the total
+mass in the two inflows is 2 .25×104M⊙. The flows col-
+lide head on at the box’s center (see Fig. 1 of Paper I).
+To the inflow velocity field we superpose a field of initial
+low-amplitudeturbulent velocityfluctuations, inorderto
+trigger the instabilities in the compressed layer. We cre-
+ate this initial velocity fluctuation field with a new ver-
+sionofthe spectralcode used in Vazquez-Semadeni et al.
+(1995) and Passot et al. (1995), modified to run in paral-
+lel in shared-memory architectures. The simulations are
+evolved for about 40 Myr.4 V´ azquez-Semadeni et al.
+Fig. 1.— Cross sections of the density ( left panel ), temperature ( middle panel ) andy-speed (right panel ), shown on the x-zplane, of a
+typical isolated HII region. The scale bar near the top indic ates length in parsecs.
+TABLE 1
+Run parameters
+Run vrms Feedback
+name [km s−1]
+LAF0 1.7 off
+LAF1 1.7 on
+SAF0 0.1 off
+SAF1 0.1 on
+The collision nonlinearly triggers a transition to
+the cold phase, forming a turbulent, cold, dense
+cloud (Audit & Hennebelle 2005; Heitsch et al. 2005;
+Heitsch et al 2006; V´ azquez-Semadeni et al. 2006), con-
+sisting of a complex network of sheets, filaments, and
+clumps of cold gas embedded in a warm diffuse substrate
+(Audit & Hennebelle 2005; Hennebelle & Inutsuka 2006;
+Hennebelle & Audit 2007). The largest cold structures
+may become gravitationally unstable and begin to col-
+lapse. Eventually, they proceed to forming stars, which
+then heat their environment, forming expanding “HII re-
+gions” that tend to disperse the clumps.
+In the simulations reported here, we vary only two
+parameters: the amplitude of the initial velocity fluc-
+tuations and whether the stellar feedback is on or off.
+We consider a “large-amplitude” (LA) and a “small-
+amplitude” (SA) case for the initial velocity fluctuations,
+for which the three-dimensional velocity dispersions are
+vrms∼1.7 km s−1andvrms∼0.1 km s−1, respectively.
+We thus employ a mnemonic nomenclature for the runs
+using the acronyms LA or SA, followed by F0 or F1, in-
+dicating that feedback is off or on, respectively. Table 1
+summarizes the runs considered in the paper.
+4.RESULTS
+4.1.Evolution of the simulations
+The simulations performed here behave very similarly
+to previous simulations with similar setups, performed
+with other codes. In particular, our SA runs are very
+similar to run L256∆ v0.17 in Paper I and the run pre-
+sented by Banerjee et al. (2009). The main feature of
+these runs is that, because the initial fluctuations are
+very mild, the flow collision creates a large, coherent
+“pancake” of cold, dense gas, which is able to undergo
+gravitational collapse as a whole. This results in the
+formation of a dense, massive, and turbulent region at
+the site where the global collapse finally converges, with
+physical properties similar to those of high-mass star
+forming regions (V´ azquez-Semadeni et al. 2009). How-
+ever, a recent study varying the parameters of the flow
+Fig. 2.— View in projection of the whole numerical box for sim-
+ulation LAF1 at t≈31.64 Myr. The box size is 256 pc. The green
+dots indicate stellar particles. The light yellow spots are transient
+dense cores, highlighted by saturation of the color table. I n the
+electronic version, this figure shows an animation of the ent ire evo-
+lution of the simulation up to t= 40 Myr. Records are spaced by
+a time interval ∆ t≈0.14 Myr.
+collision (Rosas-Guevara et al. 2010) shows that the co-
+herence of the collapse may be lost in the presence of
+stronger initial velocity fluctuations, and the SFE is de-
+creased. In such cases, smaller clouds appeared to be
+less strongly gravitationally bound, with the effect of de-
+creasing the SFE. This feature also happens in our LA
+simulations, in which the cloud formed by the initial flow
+was much more fragmented and scattered over the simu-
+lationvolume. Asaresult, SFalsooccursinamuchmore
+scattered manner, and the SFEs are in general smaller
+in the LA runs than in their SA counterparts.
+However, in general a common pattern is followed by
+all simulations: the transonic converging flows in the dif-
+fuse gas induce a phase transition to the cold phase of
+the atomic gas, which is highly prone to gravitational
+instability. This can be seen as follows. The thermal
+pressure at our initial conditions is 5000 K cm−3. From
+Fig. 2 of Paper I, it can be seen that the thermal bal-
+ance conditions of the cold medium at that pressure are
+n∼130 cm−3,T∼40 K. At these values, the Jeans
+length and mass are ∼7 pc and ∼640M⊙, respectively.
+These sizes and masses are easily achievable by a large
+fraction of the cold gas structures, which can then pro-
+ceed to gravitational collapse and form stars. Moreover,
+the ensemble of these clumps may also be gravitationally
+unstable as a whole, the likelihood of this being larger
+for greater coherence of the large-scale pattern.
+Regions of active star formation form in both sets of
+simulations by the gravitational merging of pre-existingMolecular Cloud Evolution III. Accretion vs.Stellar Feedback 5
+Fig. 3.— Cross-section view through the Central Cloud in run
+SAF1 at t≈33 Myr, at which time it has grown to a mass of ∼
+3×104M⊙. The plane of the image isshown froman inclined line of
+sight for better perspective. The cloud is seen to contain nu merous
+HII regions mixed withdense regions. In the electronic vers ion, this
+figure shows an animation of the build-up of this cloud, illus trating
+how it forms by the continued accretion of infalling materia l from
+the globally collapsing GMC. In the animation, note that man y
+of the infalling clumps form stellar particles before reach ing the
+center, and are disrupted by the local stellar feedback. How ever,
+once the Central Cloud is fully assembled, it resists disper sal, and
+forms stars at a high rate.
+smaller-scale clumps, which, altogether, form a larger-
+scale GMC. Figure 2 shows a whole-box image of the
+density field in run LAF1, in projection. In the elec-
+tronic version of the paper, this figure shows an anima-
+tion of this run from t= 0 tot≈40 Myr, illustrating
+the entire evolution of the simulation, from the assembly
+of the cloud, to its advanced star-forming epochs. In the
+animation, subsequent “records” are separated by time
+intervals ∆ t≈0.14 Myr.
+In theSAruns, the largeststar-formingregionformsin
+the center of the simulation, due to the coherent collapse
+oftheentiresheet-likecloudformedbythecollision. This
+was the region shown in V´ azquez-Semadeni et al. (2009)
+to exhibit physical conditions typical of actual high-mass
+star-formingregions. We referto this regionas“theCen-
+tral Cloud”. Figure 3 shows a view of this region at
+t≈33 Myr, a time at which the central cloud has grown
+to a mass of nearly 3 ×104M⊙in run SAF1 (cf. Fig. 5).
+In the case ofthe LAruns, because star formation occurs
+in a much more scattered fashion, we study two of the
+regions exhibiting the strongest star formation activity,
+neither of which is located at the center of the numerical
+box. These are shown in Fig. 4, and we refer to them as
+Cloud 1 and Cloud 2.
+4.2.Effect of stellar feedback on the SFE and on the
+clouds’ evolution
+Our main interest in this contribution is the effect of
+the feedback on the efficiency of the star formation pro-
+cess, and the identification of the mechanism through
+which this effect is accomplished. Figure 5 shows the
+evolution of the dense gas mass and the stellar mass in
+these simulations, with ( right panels ) and without ( left
+panels) feedback. The solidlines refer to the total masses
+in the computational box, while the dottedlines refer to
+the masses in the Central Cloud. Figures 6 and 7 show
+the corresponding plots for Cloud 1 and Cloud 2. Here,
+thesolid lines representthe massesforthe full simulation
+Fig. 4.— Cross-section view through Clouds 1 ( top panel ) and
+2 (bottom panel ) att∼35 Myr in simulation LAF1. The plane
+of view is located at x= 100 pc for Cloud 1 and at x= 150
+pc for Cloud 2. The dots represent the stellar particles. In t he
+electronic edition, these figures show animations of the evo lution
+of both clouds from t= 23 to t= 40 Myr.
+1001000
+SAF0
+t(Myr)0 10 20 30 401001000SAF1
+t(Myr)0 10 20 30 40
+Fig. 5.— Evolution of the dense gas mass and the stellar mass
+for the SA simulations, without ( left panels , run SAF0) and with
+(right panels , run SAF1) feedback. The solidlines refer to the total
+masses in the computational box, while the dottedlines refer to the
+masses in a cylinder with length and diameter of 10 parsecs wi th its
+axis along the x-direction, and which contains the Central Cloud.
+box, and the dotted, short-dashed, and long-dashed lines
+represent the masses contained in cylinders of length and
+diameter 10, 20, and 30 pc, respectively, enclosing the
+clouds. We do this because the clouds have very compli-
+cated morphologies, with filaments that extend out over
+tens of parsecs and connecting with other clouds (Fig.
+4), thus making it virtually impossible to fully enclose
+the “clouds” in any given cylindrical volume.
+It is seen from Figs. 5-7 that the inclusion of feedback
+(right panels ) causes the dense gas mass to be largerand
+the stellar mass to be smaller than in the case without
+feedback in general, even though the total cloud mass
+(dense gas + stars) in the simulations is nearly the same
+in both the cases with and without feedback (Fig. 8). As
+a consequence, the instantaneous SFE, defined as
+SFE(t) =M∗(t)
+Mdense(t)+M∗(t), (3)
+whereMdense(t) is the mass of the dense ( n/greaterorsimilar50 cm−3)6 V´ azquez-Semadeni et al.
+1001000
+LAF0
+t(Myr)0 10 20 30 401001000LAF1      
+t(Myr)0 10 20 30 40
+Fig. 6.— Evolution of the dense gas mass and the stellar mass
+for Cloud 1 in the LA simulations, without ( left panels ) and with
+(right panels ) feedback. The solidlines refer to the total masses
+in the computational box. The other lines refer to the masses in
+cylinders of length and diameter 10 pc ( dotted lines ), 20 pc ( short-
+dashed lines ), and 30 pc ( long-dashed lines ).
+gas in the simulation, and M∗(t) is the stellar mass, nat-
+urally decreases upon the inclusion of feedback in both
+sets of simulations (Fig. 9). Note that in eq. (3) we have
+explicitly written out the time dependence of the quan-
+tities involved.
+The SFE is seen to be reduced by a larger factor ( ∼
+10×) in the case of the LA runs, in which the collapse is
+less focused and less massive, than in the case of the SA
+ones (∼3×), in which the opposite is true. In addition,
+inFig. 10weshowtheevolutionofthe SFEat thelevelof
+the clouds. The left panel showsthe evolution ofthe SFE
+for the Central Cloud in the SA runs. The middle panels
+show the corresponding plots for Cloud 1 and Cloud 2 in
+the LAF0 run (without feedback), and the right panels
+show the SFEs in the LAF1 run (with feedback). Again
+we see a trend for the less massive cloud (Cloud 1) to
+suffer a greater reduction of its SFE (by a factor of ∼20,
+from∼60-90% to ∼3-4%) than the more massive one
+(Cloud2, byafactorof ∼10, from ∼70-80%to ∼7-8%).
+We discuss the possible causes for this mass-dependent
+effect of the feedback in Sec. 4.3.2.
+ThefactorbywhichtheSFEisreducedupontheinclu-
+sion of the feedback in the simulations is plotted versus
+the system’s mass in Fig. 11, both for the full amount of
+densegasinthesimulations,andforeachofthecloudswe
+have been considering. We see that two sets of points are
+clearly defined in this plot, one for the clouds, and one
+for the simulations. In both cases, however, the trend
+of larger reduction factor at smaller dense gas mass is
+clearly observed, although, at the level of simulations,
+we see that their masses are not very different. Thus,
+in this case the different reduction factors must include
+a contribution from the larger degree of fragmentation
+occurring in the LA simulations due to the larger am-1001000
+LAF0
+t(Myr)0 10 20 30 401001000LAF1      
+t(Myr)0 10 20 30 40
+Fig. 7.— Evolution of the dense gas mass and the stellar mass
+for Cloud 2 in the LA simulations, without ( left panels ) and with
+(right panels ) feedback. The solidlines refer to the total masses
+in the computational box. The other lines refer to the masses in
+cylinders of length and diameter 10 pc ( dotted lines ), 20 pc ( short-
+dashed lines ), and 30 pc ( long-dashed lines ).
+plitude of the initial turbulent fluctuations. beginfigure
+In order to compare the SFEs in our simulated clouds
+with those of real molecular clouds, it is convenient to
+note that regions of low-mass star formation, such as
+Taurus (Goldsmith et al. 2008) or the Chamaeleon II
+dark cloud (Spezzi et al. 2008) generally have low SFEs
+(∼1-5%), while cluster forming cores have SFE ∼30-
+50% (Lada & Lada 2003). Thus, we can check whether
+the SFEs and SFRs of our three clouds follow the same
+trend. Figure 12 shows the evolution of the SFRs for our
+three clouds. From these plots, we find that the average
+SFR of the Central Cloud starting from t= 32 Myr (the
+time at which a large, roughly stationary SFR sets in) is
+∝angbracketleftSFR∝angbracketright ∼1450M⊙Myr−1, while for Clouds 1 and 2, we
+find∝angbracketleftSFR∝angbracketright ∼30 and 60 M⊙Myr−1during their entire
+star-forming stages, respectively. From here, and using
+the instantaneous SFE at t= 40 Myr from Fig. 10, we
+can then plot the SFE versusthe SFR. This is shown
+in Fig. 13, in which both the SFR and the SFE have
+been multiplied by an extra factor of 0.5, to respresent
+the fact that the efficiency within the stellar particles,
+which are created at a density of n= 4×106cm−3in
+our simulations, is still smaller than unity. We take 0.5
+as a representative value.
+We see that the Central Cloud has values of the
+SFR and the SFE comparable to those of cluster-
+forming cores (Lada & Lada 2003). Specifically,
+V´ azquez-Semadeni et al. (2009) estimated an SFR /greaterorsimilar
+250M⊙Myr−1for the Orion Nebula Cluster (ONC).
+This calculation used an estimated age spread of the
+ONC of /lessorsimilar2 Myr (Hillenbrand 1997), and Tobin et al.
+(2009)’s result of there being 1613 stars in the ONC.Molecular Cloud Evolution III. Accretion vs.Stellar Feedback 7
+Fig. 8.— Evolution of the total (dense gas + stars) mass for the
+four simulations. The SA simulations are shown in the top panel ,
+while the LA runs are shown in the bottom panel . Theblack, solid
+lines refer to simulations with feedback and the red, dotted lines
+represent runs without it. The colors are shown in the electr onic
+version only.
+LAF0
+LAF1
+SAF0
+SAF1
+t(Myr)20 25 30 35 400.0010.010.11
+Fig. 9.— Evolution of the instantaneous star formation efficiency
+(SFE), as defined in eq. (3), in the full simulation box in the f our
+runs.Taking this number as a proxy for the total stellar
+production of this region, and a mean stellar mass of
+0.3M⊙(Hillenbrand & Carpenter 2000), this implied a
+total stellar mass of ∼500M⊙. On the other hand,
+Krumholz & Tan (2007) quote a total stellar mass in the
+ONC of∼4600M⊙(Hillenbrand & Hartmann 1998) and
+an age spread of ∼3 Myr (Tan et al. 2006), implying an
+SFR∼1530M⊙Myr−1. These values bracket the SFR
+we measured for our Central Cloud.
+Concerning the SFE, Fig. 13 shows that the Central
+Cloud has SFE ∼10%, which is comparable to that of
+the Orion A cloud (Carpenter 2000), in which the OMC-
+1 clump is contained. Thus, our Central Cloud may be
+compared to the Orion A cloud, and its dense core, dis-
+cussed in V´ azquez-Semadeni et al. (2009), is comparable
+to the OMC-1 clump.
+Ontheotherhand, Clouds1and2areseeninFig.13to
+have SFRs comparable to those of low-mass star-forming
+clouds,∼10-100M⊙Myr−1, such as Taurus, Cha II or
+Lupus, in which the SFE is known to be lower, ∼1-4%
+(Spezzi et al. 2008). This compares very well with the
+SFEswe reportforourClouds1 and 2in the same figure.
+We conclude that Clouds 1 and 2 are good models of
+low-mass star-forming regions, while the Central Cloud
+is a good model of a massive SF region, as discussed in
+V´ azquez-Semadeni et al. (2009).
+4.3.The physical processes acting on the clouds
+4.3.1.Cloud “destruction”
+Up to the 40-Myr time to which we have evolved our
+simulations, the three large clouds (the Central Cloud in
+run SAF1 and Clouds 1 and 2 in run LAF1) do not show
+any instances of the dense gas mass reversing its increas-
+ing trend and beginning to decrease due to the feedback,
+as can be seen in Figs. 5 through 7. Apparently, the
+HII region-like feedback we use is unable to overwhelm
+the large gravitational potential well of these clouds and
+their enveloping “atomic” gas reservoirs.
+Instead,completedestructionseemstobeabletooccur
+in small clumps. This can be seen in the animation cor-
+responding to Fig. 3, in which various small clumps are
+seentobe destroyedbytheir stellarproducts. Threepar-
+ticularly conspicuous ones are, first, the one that forms
+a stellar particle at the very starting frame (record 179)
+of the animation, slightly above and to the right of the
+screen’s center. Next, another stellar particle appears in
+the clump almost at the left border of the frame, about
+2/3 of the way from bottom to top, at record 187. Fi-
+nally, a third particle appears at record 189, slightly be-
+lowand to the right ofthe screen’scenter, as the result of
+the collision of two clumps. In all of these cases, a single
+stellar particle is formed ( ∼120M⊙), and the clump is
+destroyed. Itis worthnotingthat actually, the expansion
+of the HII regions formed produces new clumps from the
+materialcollected around it, but these new clumps either
+disperse, or simply do not form new stars.
+Thus, we conclude, similarly to
+Krumholz, Matzner, & McKee (2006), that small
+clouds (“clumps”) are rapidly destroyed, while large
+clouds may survive for longer times. However, our
+clouds exhibit a fundamental difference with respect to
+the model considered by those authors, namely that the
+clouds are accreting in general. In the next section we8 V´ azquez-Semadeni et al.
+Fig. 10.— Evolution of the instantaneous star formation efficiency (SF E) in the dense clouds in the simulations. The left panel shows
+the SFE for the Central Cloud of the SA runs, with and without f eedback. The measurements refer to a cylindrical box with a d iameter
+and a length of 10 pc. The middle panels show the SFE for Clouds 1 and 2 in the LAF0 simulation (without feedback), for three different
+cylindrical boxes, of length and diameter indicated in the l abels. The gaps in the curves for the smaller cylindrical box es correspond to
+times when the stellar particles migrate out of them, and no n ew particles have been formed. The right panels show the SFE for Clouds 1
+and 2 in the LAF1 simulation (with feedback).
+Fig. 11.— Reduction factor of the SFE at 40 Myr upon inclu-
+sion of the feedback in the simulations, both for the three cl ouds
+as well as for the entire mass of dense gas in the simulations. The
+error bars in the masses indicate the range of values taken by the
+systems once their initial assembly has finished (see Figs. 5 -7).
+The error bars in the reduction factor for Clouds 1 and 2 denot e
+the maximum and minimum values of the ratio SFE(w/o feed-
+back)/SFE(w/feedback), using the SFE data for the three cyl inder
+sizes of 10, 20 and 30 pc.
+now discuss this feature.
+4.3.2.Accretion vs. feedback
+One crucial feature in all our simulations is that the
+clouds are accreting material from the surrounding dif-
+fuse medium. This is fundamentally different from mod-
+els in which the clouds are isolated entities, in rough bal-
+ance between their self-gravity and the turbulent pres-
+sure, possibly driven by the stellar feedback. The accre-
+tion competes with SF and stellar feedback in regulating
+the cloud’s mass and coherence, with important conse-
+quences. First of all, this implies that simple observa-
+tional estimates of the SFE in GMCs based on measur-
+ing the stellar mass and dividing it by the cloud’s mass
+may be failing to take into account the additional “raw
+material” for SF contained in the part (or the whole) of
+the atomic envelope of the clouds that will eventually be
+incorporated into the GMC.
+Second, the competition between feedback and accre-Fig. 12.— Evolution of the stellar particle formation rate, aver-
+aged over 2-Myr intervals, of the Central Cloud ( top panel ), Cloud
+1 (middle panel ) and Cloud 2 ( bottom panel ). For this plot we
+only use the 10-pc cylindrical volumes for Clouds 1 and 2. Bec ause
+our stellar particles form at a density of n= 4×106cm−3, actual
+star formation rates should be a factor of 2-3 lower. Gaps ind icate
+periods over which no new stars are formed.Molecular Cloud Evolution III. Accretion vs.Stellar Feedback 9
+Fig. 13.— SFEversusstellar-particle formation rate for the three
+clouds in the simulations, using the data from Figs. 10 and 12 . The
+values of both the SFR and the SFE have been multiplied by a
+factor of 0.5, representative of the still-lower-than-uni ty efficiency
+within our stellar particles.
+tion may explain our observation from Sec. 4.2 that
+cases with feedback are characterized by largerdense
+gas masses and smaller stellar masses than their coun-
+terparts without feedback. The smaller stellar mass is
+not surprising, as the obvious effect of stellar feedback
+is to reheat the cold, collapsing star-forming gas, thus
+reducing the SFR. However, the larger cold gas mass in
+the presence of feedback is indeed surprising, since both
+gas consumption by SF and the “ionization” by stellar
+feedback act to reduce the dense gas mass. Our result
+implies that the rate of dense gas consumption by SF
+far outweighs its rate of destruction by stellar feedback,
+so that the net effect of reducing the SFR is to allow a
+larger amount of dense gas to be collected by the accre-
+tion. This scenario is supported by Fig. 8, which shows
+the total (dense gas + stars) in the clouds in the two
+sets of simulations. It is seen that the total cloud mass is
+nearly the same with and without feedback, suggesting
+thatthe total cloud mass is mainly determined by the ac-
+cretion, while the ratio of dense gas to stellar mass seems
+to be mainly determined by the feedback.
+These results moreover support the suggestions by
+Elmegreen (2007) and V´ azquez-Semadeni et al. (2009)
+that the majority of the gas in a GMC is not partici-
+pating of the SF process at any given time. The latter
+authors suggested that this is how local regions of SF
+mayhavea veryhigh specific SFR ( ∼(10Myr)−1), while
+that of their whole parent GMCs may be much smaller
+(∼(300 Myr)−1; McKee & Williams 1997). Because it
+isinjected bythe newlyformed stars, the stellarfeedback
+actspreferentiallyongasthatisabouttoformstarsnext.
+This allows an efficient suppression of SF, through only
+a modest fraction of the total available dense gas being
+destroyed. That is, if SF is a highly localized process,
+then it is possible to achieve significant reductions of the
+SFR with only a modest reduction of the total amount of
+dense gas, by targeting the destruction precisely at the
+star-forming gas.
+This mechanism may also explain the trend observed
+in Sec. 4.2, that the SFE is more strongly reduced by
+the feedback in cases where the collapsing gas mass is
+smaller. This may be understood as a consequence of thefact that stellar feedback is localized, while the accretion
+is extended, and more so for greater mass of the globally
+gravitationally unstable region that will form the cloud.
+Thus, as we observe in the animation corresponding to
+Fig. 3, the low-mass fragments that are undergoing SF
+whileontheirwaytothecentralsiteoftheglobalcollapse
+in run SAF1 are easily destroyed by their stellar activity.
+Instead, the massiveCentralCloud is not destroyed, asit
+continues to accrete mass from large distances at a pace
+that outweighs the local destruction by stellar feedback.
+Clouds 1 and 2 in run LAF1, which do not involve such
+an extended collapse, are intermediate cases, in which
+the clouds are not destroyed by the feedback, but the
+latter is more efficient in reducing the SFE.
+Thus, we can conclude that the efficacy of the feed-
+back in destroying the cloud is maximal when its region
+of influence is comparable to the spatial extent of the
+infall, and is progressively reduced as the latter involves
+progressively larger coherence lengths.
+4.4.Physical conditions in the clouds
+Here we consider the global physical properties of the
+clouds. We postpone a discussion of the properties of
+individual clumps within the clouds for a future study,
+to be performed at higher resolution. One such study,
+including magnetic fields, although not including stellar
+feedback, has recently been presented by Banerjee et al.
+(2009).
+4.4.1.Density PDFs
+It is important to determine the physical conditions
+in our clouds in order to assess their degree of realism.
+One basic diagnostic is the probability density function
+(PDF)ofthedensityfield. Althoughitiswellestablished
+that in isothermal flows the density PDF takes a lognor-
+mal form (V´ azquez-Semadeni 1994; Padoan et al. 1997;
+Passot & V´ azquez-Semadeni 1998; Nordlund & Padoan
+1999; Ostriker et al. 1999, 2001; Federrath et al. 2008),
+for non-isothermal flows the expectation is in gen-
+eral different, with a near-power-law tail develop-
+ing at high densities for flows softer than isother-
+mal (Passot & V´ azquez-Semadeni 1998; Scalo et al.
+1998; Nordlund & Padoan 1999; Gazol et al. 2005;
+de Avillez & Breitschwerdt 2004, although see, e.g.,
+Wada & Norman 2001, 2007), and a bimodal form aris-
+ingforthermallybistableflows(V´ azquez-Semadeni et al.
+2000; Gazol et al. 2005; Audit & Hennebelle 2005, see
+also the review by V´ azquez-Semadeni 2009).
+Figure 14 shows the density PDFs for the whole sim-
+ulation box of the SA runs top panels and of the LA
+runsbottom panels . Theright panels show the entire
+density range, while the left panels show the PDF for
+the dense gas ( n≥100 cm−3) only. The whole-range
+PDFsexhibitthebimodalitytypicalofthermallybistable
+flows, although the high-density tail is seen to exhibit
+an excess over the power-law in both the cases with
+and without feedback. This is probably due to the ac-
+tion of self-gravity (Klessen 2000; Dib & Burkert 2005;
+V´ azquez-Semadeni et al. 2008). In addition, the cases
+with feedback show a slight excess over the cases with-
+out it at densities 103/lessorsimilarn/lessorsimilar105cm−3, probably due
+to the formation of compressed regions by the expand-
+ing “HII regions”. The high-density PDFs, being just10 V´ azquez-Semadeni et al.
+Fig. 14.— Density PDFs for the entire simulation boxes of the
+SA runs ( top panels ) and the LA runs ( bottom panels ). Theright
+panelsshow the PDFs forthe entire range ofdensities, while the left
+panelsshow the PDFs of the dense gas only ( n≥100 cm−3). The
+red, dotted lines refer to the simulations without feedback, while
+theblack, solid lines show represent the simulations with feedback.
+then >100 cm−3tail of the whole-range PDFs, only
+show more detail of the very-high density gas, up to the
+star-forming density of 4 ×106cm−3, but with the same
+shape of the curves as in the whole-range PDFs.
+In order to see the PDFs at the locations of the
+three main clouds we have studied (the Central Cloud
+in the SA runs and Clouds 1 and 2 in the LA
+runs), we show in Fig. 15 their corresponding density
+PDFs. It is noteworthy that the PDFs again show a
+roughly power-law shape at high densities over three
+to four orders of magnitude in density, in agreement
+with the expectation for softer-than-isothermal flows
+(Passot & V´ azquez-Semadeni 1998; Nordlund & Padoan
+1999). This is at odds with results from numerical sim-
+ulations of turbulent isothermal gas in closed boxes, but
+then again our clouds are notisothermal. Instead, they
+are characterized by an effective polytropic equation of
+stateP∝ργ
+eff, withγeffvarying from 0.8 to nearly unity
+forn/greaterorsimilar100 cm−3, as can be seen in Fig. 16. Real in-
+terstellar molecular gas is not expected to be exactly
+isothermal, either(Scalo et al.1998;Spaans & Silk 2000;
+Jappsen et al. 2005). Moreover,our clouds areimmersed
+in a diffuse, warmer medium which, as proposed, e.g.,
+by Li & Goldsmith (2003) and Hennebelle & Inutsuka
+(2006), infiltrates the clouds to some degree. Thus, the
+density PDFs for the volumes containing our clouds also
+exhibit the bimodal shape characteristic of thermally
+bistable flows, and have low-density tails that extend
+down to the WNM regime, contrary to what happens
+in isothermal simulations.
+4.4.2.Velocity dispersions and virial masses
+Finally, we investigate the global velocity dispersion
+(∆v) in the Central Cloud and Clouds 1 and 2. ThisFig. 15.— Density PDFs for the three main clouds in the simu-
+lations: the Central Cloud in the SA runs ( left panel ), and Clouds
+1 (middle panel ) and 2 ( right panel ) in the LA runs. The PDFs
+for the Central Cloud are computed in a cylinder with length a nd
+diameter equal to 10 pc, while the PDFs for Clouds 1 and 2 are
+computed in cylinders of length and diameter equal to 10, 20 a nd
+30 pc. Red lines indicate cases without feedback, and are dis -
+placed downwards by a factor of 10 for better viewing in the ca ses
+of Clouds 1 and 2. Black lines indicate cases with feedback.
+is shown in Fig. 17. The solidlines show the density-
+weighted value, while the dottedlines show the volume
+weighted value. Blacklines denote cases with feedback,
+andredlines correspond to cases without it.
+For the Central Cloud and Cloud 2, which are the two
+most massive ones, the density-weighted velocity disper-
+sion, which highlights the dense gas, decreases upon the
+inclusion of feedback. Without feedback, in the Cen-
+tral Cloud, it reaches very high values at the end of
+the run ( ∼15 km s−1, corresponding to Mach numbers
+Ms∼50-75), which in fact are significantly larger than
+typical values for cloud complexes of comparable mass
+(M/greaterorsimilar104M⊙; e.g., Dame et al. 1986; Rathborne et al.
+2009). Instead, in therunincluding feedback, ∆ vreaches
+values∼5-6 km s−1(rms Mach number, Ms∼20), in
+much better agreement with typically observed values.
+On the other hand, the volume-weighted velocity disper-
+sion, which highlights the less dense gas, is seen in Fig.
+17 toincrease upon the inclusion of feedback. In Clouds
+1 and 2, which are less massive and more scattered than
+the CentralCloud, ∆ vdoes not achieveexceedinglylarge
+values in any case.
+These results can be understood as a consequence of
+the fact that the dense gas acquires its largest velocities
+in the case of free-fall collapse. However, the collapse
+flow is dismantled in its final (fastest) stages by the stel-
+lar feedback, so that it is the high-velocity dense gas that
+is preferentially destroyed by the feedback. On the other
+hand, this gas becomes warm, diffuse, high-velocity ex-
+pandinggas, which is the one highlighted by the volume
+weighting. These results again reinforce the notion thatMolecular Cloud Evolution III. Accretion vs.Stellar Feedback 11
+Fig. 16.— Phase-space diagrams of temperature vs. density for
+the Central Cloud ( left panels ), Cloud 1 ( middle panels ) and Cloud
+2 (right panels ). Thetop panels show the cases without feedback,
+and the bottom panels show the cases with feedback. It can be seen
+that, for the majority of the points, the temperature only va ries
+between ∼40 and 10 K for 100 < n <106cm−3. The exceptions
+are locations recently heated by new stars.
+the feedback is mostly applied on the gas that is closest
+(but not quite there yet) to forming stars.
+Once we havedeterminations of the velocity dispersion
+in the cylinders containing the clouds, it is natural to
+measurethe virialmass( Mvir) ofthe cloudsandcompare
+it with their real mass M, in order to check whether they
+look “virialized”. We compute the virial mass through
+the standard formula
+Mvir≡210/parenleftbiggR
+pc/parenrightbigg/parenleftbigg∆v
+km s−1/parenrightbigg2
+M⊙,(4)
+(see, e.g., Caselli et al. 2002; Tachihara et al. 2002;
+Klessen et al. 2005). For the real cloud mass we con-
+sider the sum of the gas (dense + diffuse) and stellar
+masses in the volume being considered. It is impor-
+tant to note that, with this prescription, variations in
+the value of the ratio come exclusively from the estimate
+of the virial mass, because it doesdepend on the weight-
+ing used, while the actual cloud mass is independent of
+the method of weighting.
+Figure 18 shows the evolution of the ratio M/Mvirfor
+the three clouds, showing the cases with and without
+feedback, and using volume- and density-weighting for
+the calculation of ∆ v. In all cases we observe that, at
+advanced stages of the evolution, when the clouds have
+been fully assembled ( t/greaterorsimilar25 Myr), this ratio is closest to
+unity when feedback is included and density-weighting is
+usedinthecalculationof∆ v. Thecaseswithnofeedback
+anddensityweightingaresystematicallylowerthanunity
+by factors ∼2-10. The cases with volume weighting are
+more strongly fluctuating, but it is noteworthy that the
+ratio often takes values close to unity anyway.Fig. 17.— Evolution of the velocity dispersion σin the three
+clouds. The solidlines show the density-weighted value, while the
+dottedlines show the volume weighted value. Redlines correspond
+to cases without feedback, and blacklines correspond to cases with
+feedback.
+From these results, we conclude that, in the case with
+feedback, the clouds appear to be in a pseudo-virialized
+state with respect to the velocity dispersion of the dense
+gas (highlighted by the density weighting). In this
+state, there is an approximateforce balance between self-
+gravity and feedback driving. However, it differs from
+true virialization because of the presence of mass sources
+(the accretion from the diffuse environment) and sinks
+(the consumption by SF and the destruction by stellar
+feedback) in the system. These seem to self-regulate, so
+as to be capable of maintaining an approximately con-
+stant cloud mass while accretion persists. On the other
+hand, when the velocity dispersion of the warm gas is
+taken into account (still in the case with feedback), the
+clouds appear to be slightly sub-virial, suggesting that
+the newly formed warm gas in the HII regions is capable
+of escaping the cloud.
+In the cases without feedback, the clouds also appear
+tobe sub-virialevenwith respecttothe density-weighted
+virialmassestimate. Inthiscasethisappearstobeacon-
+sequence of the dense gas being in free-fall collapse, and
+into a potential well produced not only by its own self-
+gravity,but alsobythepreviously-formedstarsthathave
+fallen there too (see V´ azquez-Semadeni et al. 2009).12 V´ azquez-Semadeni et al.
+Central Cloud
+0.010.11
+Cloud 1
+0.010.11
+Cloud 2
+t(Myr)0 10 20 30 400.010.11
+Fig. 18.— Evolution of the ratio of actual to virial mass for the
+three clouds. Redlines correspond to the runs without feedback.
+Blacklines correspond to the runs with feedback. Solidlines cor-
+respond to the density-weighted velocity dispersion, whil edotted
+lines correspond to the volume-weighted one.
+5.CAVEATS AND LIMITATIONS
+Of course, our simulations are not free of caveats and
+limitations. Two that stand out are the resolution and
+the nature of the feedback that we have included. Con-
+cerning the resolution, in this paper we have used a rela-
+tively limited one, in order to speed up the simulations,
+at the expense of not fulfilling the Jeans criterion pro-
+posed by Truelove et al. (1997). However, as explained
+in Sec. 2.3, we do not consider this a problem for our
+study, since we have restricted it to the global properties
+oftheclouds, ratherthantheirdetailedstructure, andwe
+have avoided addressing issues related with the fragmen-
+tation of the final stages of collapse in the clouds, such
+as the mass distribution of the cores and stellar prod-
+ucts. We plan to perform a higher-resolution study in a
+forthcoming paper, in which we can address these issues.
+The second limitation is the nature of the stel-
+lar feedback that we have considered, which in this
+study has been restricted to local heating represent-
+ing the ionizing radiation from massive stars, simi-
+larly to the approach used by, e.g., Yorke et al. (1982);
+Vazquez-Semadeni et al. (1995); Passot et al. (1995), ne-
+glecting other sources such as outflows from stars of all
+masses (e.g., Norman & Silk 1980; Li & Nakamura 2006;Matzner 2007; Nakamura & Li 2007; Carroll et al. 2009;
+Wang et al. 2009). We do this in part for the numerical
+simplicity of the approach, and in part because here we
+have been mainly interested in the SFE and and cloud
+evolution at the scale of GMCs, for which the expansion
+of HII regions is likely the main driver (Matzner 2002).
+However, the neglect of bipolar outflows may introduce
+a non-negligible bias in our finding that smaller-scale
+clumps are destroyed by the feedback, and this should
+be confirmed by future simulations that can better re-
+solve these objects and include bipolar outflows.
+6.SUMMARY AND CONCLUSIONS
+In this paper we have presented a numerical investiga-
+tionofthe evolutionofdense(“molecular”)clouds, start-
+ingfromtheirformationbytransoniccompressionsinthe
+warm neutral medium, followed by a phase transition to
+the cold neutral medium, the onset of gravitational col-
+lapse, star formation, and, finally, energy feedback from
+massive-star ionizing radiation and the formation of ex-
+panding HII regions.
+A crucial difference between the results from
+our simulations and other models and theories of
+the self-regulation of star formation and its ef-
+ficiency (e.g., Whitworth 1979; Elmegreen 1983;
+Franco, Shore & Tenorio-Tagle 1994; McKee & Tan
+2003; Krumholz & McKee 2005) is that the clouds in
+our simulations are in general accreting material at high
+rates from the surrounding diffuse medium, rather than
+having a fixed mass. This implies that the material
+making up a cloud is constantly being renewed over time
+because, on the one hand it is accreting fresh gas, and on
+the other it is losing mass to SF and stellar ionization.
+The star-forming regions can be considered to be not
+objects, but rather the lociwhere the gas is just “passing
+through”, from the diffuse-gas state to the “star” state,
+similarly to the nature of a flame, which is the locus of
+the gas undergoing combustion in a candle, with fresh
+air entering at its base, burning while it transits through
+the flame, and the exhaust gases leaving at the top.
+Witin this framework, we have found that the SFE
+in the clouds is readily decreased by feedback to levels
+consistent with observational determinations at all times
+during the clouds’ evolution up to the maximum integra-
+tion time of 40 Myr that we have considered. This is a
+significant improvement over our previous non-magnetic
+studies of the SFE in the context of cloud evolution
+without feedback, in which the SFE at late times is of-
+ten found to be excessive (e.g., V´ azquez-Semadeni et al.
+2007; Rosas-Guevara et al. 2010). However, we have
+found that the reduction factor upon the inclusion of
+feedback is an inverse function of the dense gas mass in
+the system, and of the degree of coherence of the global
+collapse, as illustrated in Fig. 11. This result is com-
+plemented by the additional observation that low-mass
+clumps are fully destroyed by the feedback. Together
+they imply that at some point, below a certain critical
+collapsingmass,theappearanceofastationarystatemay
+not occuranymore. This is in qualitative agreementwith
+the results of Krumholz, Matzner, & McKee (2006), al-
+though their model does not include mass accretion onto
+the cloud. We plan to present one such model in a future
+paper.
+In general, our results may provide an explanationMolecular Cloud Evolution III. Accretion vs.Stellar Feedback 13
+for the observational fact that regions of massive-star
+formation, which are themselves more massive than re-
+gions of low mass-star formation, appear to have higher
+SFEs (Lada & Lada 2003) than regionsforming low-and
+intermediate-mass stars (Evans et al. 2009). Essentially,
+our results imply that the more massive the region, the
+less effective the feedback is in reducing the SFE. Specif-
+ically, these results indicate that the role of the feedback
+is not the same in all clouds, but rather depends on the
+initialconditionsofthelarge-scalecollapsethat produces
+each individual region. Regions in which the amount of
+mass involved in the collapse overwhelms the destruc-
+tive action of the feedback may reach a stationary state
+that appears virialized, but in which there is actually
+a countinuous processing of material, since on the one
+hand the cloud is accreting mass from the outer infalling
+material, and on the other hand it is losing mass through
+consumption by SF and evaporation by the feedback. In
+this case, termination of the SF episode probably re-
+quires the termination of the gas reservoir involved in
+the global collapse, and is independent of the feedback.
+However, this need not be in contradiction with the low
+observedglobalefficiencyofSF,ifonlyasmallfractionof
+a GMC’s mass is involved in the collapse at any moment
+intime, with therestbeingeithersupported(by, e.g., the
+magnetic field), or in the process of dispersal (Elmegreen
+2007). Alternatively, the SF episode may be terminated
+upon the initiation of supernova events, which we have
+not included in the present study.
+An additional consequence of the mechanisms de-
+scribed in this paper is that the most massive clouds
+appear to actually contain more mass when feedback is
+included than when it is not. This suggests that, first,
+unimpeded SF is more efficient at removing mass from
+the dense gasphase to deposit into stars, than the evapo-ration of dense gas by the stellar feedback, so that, when
+the latter inhibits SF, the accretion onto the cloud ac-
+cumulates larger amounts of dense gas than otherwise.
+Second, this indicates that the deposition of the feedback
+energy into the clouds is accurately targeted to gas that
+is already on the verge of forming stars. If this gas is a
+minor fraction of the total dense gas in a cloud (i.e, SF is
+a highly spatially intermittent phenomenon in molecular
+clouds; V´ azquez-Semadeni et al. 2009), then its destruc-
+tion is a highly efficient way to reduce the SFE without
+destroying a large fraction of the cloud.
+Finally, we have also investigated the density PDF in
+the volumes containing the clouds in our simulations,
+finding that in general they retain the bimodal shape
+characteristic of thermally bistable flows. The PDF
+of the dense gas exclusively shows no turnover at low
+densities, indicating that low densities are most abun-
+dant. The high-density tails of the PDFs have power-law
+shapes, as expected for softer-than-isothermal flows, al-
+though this last result may be biased by our usage of a
+singlecoolinglawappropriateforatomicgas,andextrap-
+olating it to molecular-gas densities, rather than using a
+specific molecular cooling law. We expect to address this
+shortcoming in future works.
+ACKNOWLEDGMENTS
+We are grateful to A. Kravtsov for providing us with
+the numerical code, and to N. Gnedin for providing us
+with the useful analysis and graphics package IFRIT.
+The calculations were performed on the 8-core server at
+CRyA-UNAM, acquired through UNAM-PAPIIT grant
+IN112806to P.C. This work hasreceived partialfinancial
+support from CONACYT grants U47366-F to E.V.-S.
+and J50402-F to G.C.G., and UNAM-PAPIIT IN106809
+to G.C.G.
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@@ -0,0 +1,276 @@
+arXiv:1001.0803v1  [cond-mat.mtrl-sci]  6 Jan 2010Electronic and magnetic properties of bilayer graphene wit h
+intercalated adsorption atoms C, N and O
+S. J. Gong1[*], W. Sheng2, Z. Q. Yang2, J. H. Chu1,3
+1. National Laboratory for Infrared Physics,
+Shanghai Institute of Technical Physics,
+Chinese Academy of Sciences, Shanghai 200083, China
+2. National Laboratory for Surface Physics,
+Fudan University, Shanghai 200433, China. and
+3. Key Laboratory of Polar Materials and Devices, Ministry o f Education,
+East China Normal University, Shanghai 200241, China
+Abstract
+We present an ab-initiodensity function theory to investigate the electronic and m agnetic struc-
+tures of the bilayer graphene with intercalated atoms C, N, a nd O. The intercalated atom although
+initially positioned at the middle site of the bilayer inter val will finally beadsorbed to one graphene
+layer. Both N and O atoms favor the bridge site (i.e. above the carbon-carbon bondingof the lower
+graphene layer), while the C atom prefers the hollow site (i. e. just above a carbon atom of the
+lower graphene layer and simultaneously below the center of a carbon hexagon of the upper layer).
+Concerning the magnetic property, both C and N adatoms can in duce itinerant Stoner magnetism
+by introducing extended or quasilocalized states around th e Fermi level. Full spin polarization can
+be obtained in N-intercalated system and the magnetic momen t mainly focuses on the N atom. In
+C-intercalated system, both the foreign C atom and some carb on atoms of the bilayer graphene are
+induced to be spin-polarized. N and O atoms can easily get ele ctrons from carbon atoms of bilayer
+graphene, which leads to Fermi level shifting downward to va lence band and thus producing the
+metallic behavior in bilayer graphene.
+PACS Numbers: 71.15.Mb, 73.43.Cd, 81.05. Uw
+Typeset by REVT EX 1Introduction: Since the discovery of monolayer graphene in the year 2004 [2], this
+two-dimensional material remains in the focus of active research m otivated by its novel
+physical properties and a promising potential for applications[3–5]. Experimental groups
+have enabled preparation and study of systems with one or a small n umber of graphene
+layers [6]. Bilayer graphene, which is made of two stacked graphene layers, is considered to
+be particularly importance for electronics applications because of it s special band structure
+[7]. Coupling of the two monolayer graphene sheets in the usual A-B s tacking of bilayer
+graphene yields pairs of hyperbolically dispersing valence and conduc tion bands that are
+split from one another by the interlayer interaction[8]. The band gap of the bilayer can
+be easily opened when a difference between the electrostatic poten tial of the two layer is
+introduced, either by chemical doping or by applying gate voltage [9 –17], which makes
+graphene channels have a high resistance for the OFF state. Angle -resolved photoemission
+(ARPES) measurements indicatesuch agapinpotassium dopedbilaye r grapheneepitaxially
+grown on SiC [14], and infrared spectroscopy measurements also de tect the similar gap in
+the electrostatically gated bilayer graphene [10–12].
+In this work, we carried out first-principle calculations and theoret ical analysis to explore
+the electronic and magnetic properties of bilayer graphene with inte rcalated atoms C, N and
+O. The calculations were performed using the projector augmente d wave (PAW) formalism
+of density functional theory (DFT) as implemented in the Vienna abinitio simulation pack-
+age (VASP) [18]. Because generalized gradient approximation (GGA) [19] gives essentially
+no bonding between graphene planes and leads to excessively large v alues of bilayer distance
+[13], we performed the calculations within the localized density approx imation (LDA). We
+find that LDA gives rise to a bilayer distance of 3.34 ˚A, in good agreement with the ex-
+periment value. Some previous investigations used LDA to optimize th e structure to get a
+reasonable bilayer distance, and sequently used GGA to calculate th e electronic structure
+[9]. To keep consistent, we use LDA in all our calculations for the inves tigated system. An
+energy cutoff of 400 eV for plane-wave expansion of the PAWs is use d. The model system
+here consists of a 4 ×4 supercell with a foreign atom intercalated between the two couplin g
+graphene layers. The supercell parameters are set to be the sam e asa=b= 9.84˚Ain
+thexyplane (aandbindicate the crystal lattice constants). The Brillouin zone is sampled
+using a 11 ×11×1 Γ centered k-point grid. For geometry optimization, all the intern al
+coordinates are relaxed until the Hellmann-Feynman forces are les s than 0.01 eV/ ˚A.The
+2vacuum thickness along the z axis is 16 ˚Ato avoid the interaction between graphene layers
+of adjacent supercells.
+We considered the /tildewideA-B Bernal stacking structure for bilayer graphene. Top-view for the
+bilayer graphene is shown in Fig. 1(a), in which violet color is for the lowe r layer, and gray
+color is for the upper layer. Seen from the top-view, ‘ /tildewideA’ position in the lower layer coincides
+with ‘B’ position in the upper layer, ‘A’ position in the upper layer is exac tly the center of
+the hexagon of the lower graphene layer from the top-view, and ‘ /tildewideB’ position in the lower
+layer is right the center of the hexagon of the upper graphene laye r. The foreign atom is
+intercalated between the upper and lower graphene layers and is init ially positioned in the
+middle position of the bilayer distance. Three kinds of initial positions a re considered for
+each intercalated atom: bridge, top andhollow positions, which aren oted inthe following by
+using subscript“Bri”, “Top”, and “Hol” when needed, for example, we use N Brito indicates
+the bilayer system with N atom positioned at the bridge site. Bridge sit e is above the
+carbon-carbon bonding in the lower graphene layer, top site indicat es the middle position
+between the coinciding positions ‘ /tildewideA’ and ‘B’, and hollow site is just above a carbon atom
+of the lower graphene layer and simultaneously below the center of a carbon hexagon of the
+upper layer. The structure relaxation calculations show that, if th e intercalated atom is
+initially positioned at the bridge or hollow site, it will finally be adsorbed to one graphene
+layer and away from the other layer. While for the top site, the fore ign atom will keep being
+located in the middle position of the bilayer interval.
+The binding energy is defined as: dE=Egraphene+Eatom−Etotal,whereEgrapheneis the
+energy of the clean bilayer graphene, Eatomstands for the energy of the single foreign atom,
+andEtotalisthetotalenergy ofthebilayer graphenewith theintercalated at om. The binding
+energy, the distance between the foreign atom and its nearest C a tom, and the interlayer
+distance of the doped bilayer graphene are illustrated in Table I. Com paring the data in
+Table I, we find both N and O atoms favor the bridge site, while the C at om prefers the
+hollow site. Nointercalated atomisstable at thetopsite. At the topp osition, therelaxation
+calculations show that nearest C-N bonding is about 1.83 ˚A, and C-O bonding is about 1.74
+˚A, which is much larger than the typical lengths of C-N (1.47 ˚A)and C-O (1.42 ˚A) [20],
+implying the physisorption rather than the chemisorption. For the h ollow site, O atom is
+adsorbed to the layer which provides the nearest C atom, and away from the hexagonal
+3cental of the other graphene layer. O atom and the nearest C ato m form the C=O bonding
+with the length of 1.39 ˚A, smaller than the the length of C-O bonding in the O Bristructure
+(1.44˚A). The calculation shows N Holstructure doesn’t exist because of the negative binding
+energy. Concerning the bilayer distance, with the foreign atom inte rcalated in the bilayer
+space, the distance is enlarged for all the investigated systems.
+Figure 1 displays the ground states for C, N and O configurations, w here Fig. 1(b),
+(c) and (d) are the C Hol, NBri, and O Bristructures, respectively. In the C Holsystem, the
+optimized interlayer distance is about 3.39 ˚A, which is a little larger than the value of 3.34
+˚Afor the pure bilayer graphene. Calculations show that C-C bonding in the upper layer
+nearly keeps unchanged. The carbon atom which is right below the fo reign C is pushed
+down, forming a “dumbbell” at the saddle point. The foreign C bonds w ith the adjacent
+three carbon atoms with bonding length being 1.54 ˚A, which is a standard length for sp3
+hybridization [20]. In N Brisystem, the bilayer distance is enlarged to the value of 3.87 ˚A,
+and the length of C-N is 1.43 ˚A, indicating the chemically adsorption not the physically
+adsorption. The two carbon atoms bonded with N atom are drawn ou t of the graphene
+layer, and the C-C bonding is 1.55 ˚A, which implies sp3hybridization of the two carbon
+atoms. In O Brisystem, bilayer distance is about 3.76 ˚A, length of C-O bonding is 1.44 ˚A,
+and the two carbon atoms bonded with O atom are also drawn out of t he graphene layer
+with the C-C bonding is 1.51 ˚A. Obviously, N and O atoms have the similar ground state
+structure.
+Side-view of the charge distributions for C Bri, NBri, and O Brisystems are displayed in Fig.
+2. Although C Bristructure is not the ground state for C-intercalated system, we show its
+charge contour together with those of N Briand O Brisystems, to provide a clear comparison.
+We know O atom is lack of two electrons, when it is adsorbed in the bilaye r graphene, it
+strongly interacts with its adjacent carbon atoms andget electro ns fromthem (see the upper
+panel of Fig. 2). N atom lacks three electrons, it also gets electron s from its adjacent carbon
+atoms (see the middle panel of Fig.2). C atom lacks four electrons to get saturated, and it
+tends to share electrons with its adjacent carbon atoms in bilayer g raphene (see the lower
+panel of Fig. 2). All these three atoms show strong interaction wit h lower graphene layer.
+Concerning their interactions with the upper graphene layer, judg ing from Fig. 2, C atom
+has the strongest interaction with the upper layer, N is the second , and O is the third. We
+compare the stable positions for C, N and O atoms in monolayer and bila yer graphene, as
+4illustrated in table II. It is clear that N atom has the same stable posit ion in both monolayer
+andbilayer graphene, so doesthe O atom. Cadatomis stable at theb ridge site inmonolayer
+graphene, and Hollow site in bilayer graphene. We believe the different stable positions for
+C atom results from its interaction with the upper graphene layer.
+The C Holstructure is favored as the ground state for C atom intercalated in the bilayer
+graphene. The spin-resolved band structures and density of sta tes (DOS) for C Holsystem
+are shown in Fig. 3. Fig. 3(a) and (b) are the band structures for t he majority spin and
+minority spin, respectively. It is clearly seen that impurity bands for the majority and
+minority spin components lie, respectively, lower and higher than the Fermi level. In the
+spin-resolved total DOS (see Fig. 3(c)), two narrow peaks at the opposite side of the Fermi
+level are observed, indicating the itinerant magnetism triggered by the intercalated C atom.
+In the following, we will find the itinerant magnetism comes from not on ly the foreign C
+atom, but also the carbon atoms of the bilayer graphene. In the or bital-resolved PDOS
+of the foreign C atom, sstate and pstate have peaks in the same energy range, which is
+indicative of the sp3hybridization. The C-C bonding between the foreign C and the neare st
+carbon atoms with the value of 1.54 ˚A, is also a proof of the sp3hybridization, which has
+been pointed in the previous analysis. When the spin degree of freed om is neglected, our
+calculation from first principle predicts a twofold degenerate peak a t the Fermi level (Fig.
+3(e)), but the spin unpolarized state is not the ground state. Inc luding the spin degree of
+freedom, the balance between the majority and minority spin compo nents will be destroyed.
+The spin density distribution of the lower graphene layer of the C Holsystem are shown in
+Fig. 3(f). Both the foreign C atom and the pushed down carbon ato m are magnetic, yet
+we cannot see their magnetic moment distribution in Fig. 3(f), becau se they are not in the
+lower graphene plane. Seen from the top-view, these two carbon a toms occupy coinciding
+positions, which are noted in Fig. 3(f) by the letter ‘C’. The three nea rest carbon atoms
+bonded with the foreign C atom are nearly nonmagnetic and the next -near-neighbors are
+magnetic. On the whole, the total magnetic moment of the C Holsystem is about 1.32 µB,
+and the magnetic moment distributions show threefold symmetry, w hich is similar with the
+hydrogen adsorption on the graphene plane [23].
+Spin-resolved band structure and density of states for N Brisystem are shown in Fig. 4,
+where Fig. 4(a) and (b) show the band structure for majority and minority spin, respec-
+5tively, and Fig. 4(c) and (d) display the total DOS of N Brisystem and PDOS of N atom,
+respectively. In the total DOS, very narrow and sharp peak at th e Fermi level is observed.
+We draw the PDOS of N atom and find the peak arises from from the N a dsorption and
+the N atom is nearly 100% spin polarized. In the band structure, we fi nd the very localized
+states at the Fermi level in the minority spin band structure. Such quasilocalized states give
+rise to the strong Stoner magnetism with magnetic moment of 0.65 µB located at the N
+atom. In the PDOS of N atom, between the energy -0.5 eV and -1 eV, another narrow peak
+is obtained. Both in the majority and minority spin band structures, the corresponding flat
+impurity bands in the energy range from -0.5 eV to -1 eV are observe d. In addition, both in
+the majority and minority band structures, the characteristic co nical point at K point still
+can be clearly identified, implying the bilayer graphene is not strongly p urturbed by the N
+atom. The critical difference is that, in freestanding graphene the Fermi level coincides with
+the conical point, while the Fermi level obtained in N-intercalated sy stem is shifted down-
+ward and becomes below the conic point. A shift downwards (upward s) means the holes
+(electrons) are donated by the adsorption atom. The manganese doping (electron donor)
+results in the upward shift of the Fermi level, reported by previous investigations [9].
+Band structure and density of states of O Brisystem are displayed in Fig. 5, where Fig.
+5(a), (b) and (c) are the band structure, total density of stat es, and partial density of states
+of O atom, respectively. The calculation shows the O Brisystem is nonmagnetic. From the
+above analysis about N Briand C Holsystems, we see the impurity states localized around the
+Fermi level play an important role in the magnetic properties, and th e intercalated atom
+itself is spin-polarized. In O Brisystem, we do not get such localized states around the Fermi
+level, and the O atom is nonmagnetic. In the PDOS of O atom, we find th e peak nearest
+to the Fermi level is around the energy of 0.75 eV. From the band st ructure, we can also
+see that the impurity state nearest to the Fermi level is at the ene rgy of 0.75 eV. Like N Bri
+system, the O Brisystem also undergoes a change from semimetal to metal, because O atom
+get electrons form graphene, thus the Fermi level shifts downwa rd.
+Insummary, we calculated thestructure, electronic andmagnetic properties ofthebilayer
+graphene with intercalated atoms C, N, and O by the ab-initiodensity function theory.
+Impuritiesevenatverylowdensitymaybringfruitfulphysical prop ertiesforbilayergraphene
+system. Structure relaxation shows that the intercalated atom a lthough initially positioned
+6at the middle position of the bilayer distance, will finally be adsorbed to one graphene layer
+and away from the other layer. The bilayer distance is enlarged for a ll the investigated
+adsorption systems. Both N and O atoms are stable at the bridge sit e, while C atom
+is stable at the hollow site. Concerning the magnetic property, O-int ercalated system is
+nonmagnetic, while N and C atoms can induce stoner magnetism by intr oducing extended
+or quasilocalized states around the Fermi level. Nearly 100% spin pola rization is obtained in
+N-intercalated system and the magnetic moment focuses on the N a tom. For C-intercalated
+system, the magnetic moment distributes on the foreign C atom and certain carbon atoms
+of the bilayer graphene. Both in N-intercalated and O-intercalated systems, the Fermi level
+is obviously shift downward, inducing the metallic behavior of the bilaye r graphene.
+Acknowledgements: This work was supported by the the Ministry of Sciences and
+Technology through the 973-Project (No. 2007CB924901), the National Natural Science
+Foundation of China (Grant Nos. 60221502 and 1067027), the Gra nd Foundation of Shang-
+hai Science and Technology (05DJ14003).
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+8I. FIGURE CAPTIONS
+Fig. 1(a)/tildewideA-B Bernal stacking structure for bilayer graphene. (b) C Holsystem, where the
+atom in violet is the foreign C atom. (c) N Brisystem, where the atom in blue is the
+foreign N atom. (d) O Brisystem, where the atom in red is the foreign O atom.
+Fig. 2Side-view of charge contours for C Bri, NBriand O Brisystems, respectively.
+Fig. 3Electronic and magnetic structures of the C Holsystem. (a) Band structure of ma-
+jority spin, (b) band structure of minority spin, (c) total density of states, (d)the
+orbital-resolved DOS for the foreign C atom, (e) density of states for spin unpolarized
+CHolsystem, (f) distribution of the spin density on the lower graphene la yer..
+Fig. 4Spin-resolvedbandstructureanddensityofstatesofN Brisystem. (a)Bandstructure
+ofthe majorityspin; (b) bandstructure ofthe minorityspin, (c) t otaldensity ofstates,
+(d) partial density of states of N atom.
+Fig. 5Band structure and density of states of O Brisystem. (a) Band structure, (b) total
+density of states, (c) partial density of states of O atom.
+Table I The binding energy ( dE), the length of bonding between the adsorption atom and
+its nearest C atom ( aC-atom), and the bilayer distance of the adsorption system ( dis.).
+The stable position for each intercalated atom is noted by box.
+Table II The stable positions for adsorption atoms C, N, and O in monolayer an d bilayer
+graphene.
+9This figure "fig1.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0803v1---/g4521.741.39 --- 1.83 1.49dE (eV)Top         Hol           Bri Top         Hol            BriC N O
+Position Top             Hol         Bri
+1.45 2.80 1.93 --- 1.31 0.37 1.68 1.86 2.10
+aC-atom(A)1.46/g452
+1.54 1.43 1.44
+dis.(A)3.533.39 3.87 3.62 3.87 3.613.85 3.76This figure "fig2.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0803v1areference  20.
+bb
+Bilayer    Bridge               BridgeC N O
+Monolayer              Bridge
+Bridge    Hollow Bridgea b
+ 
+  reference  21.
+ This figure "fig3.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0803v1This figure "fig4.jpg" is available in "jpg"
+ format from:
+http://arxiv.org/ps/1001.0803v1-3 -2 -1 0 1 2 30.00.40.81.2
+  
+(c)Energy (eV)PDOS (arb.units) DOS (arb.units)
+Energy (eV)Energy (eV)
+-3 -2 -1 0 1 2 30369
+  
+(b)-3-2-10123
+  
+K Γ M(a)
\ No newline at end of file
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+arXiv:1001.0804v1  [math.DG]  6 Jan 2010AFFINE IMAGES OF RIEMANNIAN MANIFOLDS
+ALEXANDER LYTCHAK
+Abstract. We describe all affine maps from a Riemannian man-
+ifold to a metric space and all possible image spaces.
+1.Introduction
+A mapf:X→Ybetween metric spaces is called affineif it
+preserves the class of linearly parametrized minimizing geodesics. If
+Xis a geodesic metric space then a continuous map fis affine if and
+only if it sends midpoints to midpoints. A map is called locally affine
+(anotherwordis totally geodesic )iftherestrictionofthemaptosome
+neighborhood of any point is affine. Basic examples of (locally) affine
+maps are (locally) isometric embeddings, rescalings and projections to
+a factor in a direct product decomposition.
+Affine maps arise naturally in questions related to super-rigidity (see
+the explanations and the literature list in [Oht03]), in the study of
+Berwald spaces in Finsler geometry ([Sza81]), in the study of produc t
+decompositions ([FL08]) and in the study of isometric actions on non-
+positively curved spaces ([AB98]). It seems to me to be of independe nt
+interest, to understand to what extent the geodesics determine the
+metric of a given space.
+We give a complete description of all affine maps f:M→Ywhere
+Mis a smooth Riemannian manifold and Yan arbitrary metric space.
+First, an important special case of the main result:
+Theorem 1.1. LetMbe a connected complete smooth Riemannian
+manifold. There is a locally affine map f:M→Yto some metric
+space that is not a local homothety if and only if the universa l covering
+ofMis a product or a symmetric space of higher rank.
+Here, we saythatamap f:X→Yisalocal homothety if, forany
+pointx∈X, there is some neighborhood Uofxand a non-negative
+1991Mathematics Subject Classification. 53C20.
+The author was supported in part by the SFB 611 Singul¨ are Ph¨ anomene und
+Skalierung in mathematischen Modellen , by the MPI for mathematics in Bonn and
+by the Heisenberg grant from the DFG.
+1numbera, such that for all x1,x2∈Uthe equality d(f(x1),f(x2)) =
+a·d(x1,x2) holds true. If Xis connected then the number adoes not
+depend on the point x. Thus, up to rescaling by a, such map locally is
+an isometric embedding.
+Fortheformulationofthemaintheoremwe will need two definitions.
+Definition 1.2. A Riemannian submersion f:M→M1between
+smooth Riemannian manifolds will be called flatif the fibers are totally
+geodesic and the horizontal distribution is integrable.
+A Riemannian submersion f:M→M1between complete Riemann-
+ian manifolds is flat if and only if the lift to the universal coverings
+˜f:˜M→˜M1is the projection map of a direct product decomposition
+˜M=˜M1×M2→˜M1. Another useful formulationisthata Riemannian
+submersion f:M→M1is flat if and only if the tangent distribution
+of the fibers is a parallel distribution ([Vil70] or [Sak96]).
+Definition 1.3. Let(M,g)be a Riemannian manifold. We will call a
+function |·|:TM→Ra holonomy invariant Finsler structure if the
+restriction of |·|to each tangent space TpMis a (possibly not smooth
+or not strictly convex) norm and if |v1|=|v2|for any two vectors
+related by the holonomy along some piece-wise smooth curve. For a
+holonomy invariant Finsler structure |·|we will call the identity map
+Id: (M,g)→(M,|·|)an admissible change of metric.
+Note that an admissible change is a bi-Lipschitz map. Clearly, ho-
+lonomy invariant Finsler structures are in one-to-one correspond ence
+with norms |·|on a fixed tangent space TpMthat are invariant under
+the action of the holonomy group Holp. We refer to Subsection 4.3 for
+more information about such norms.
+Now we can state the main result:
+Theorem 1.4. LetMbe a complete Riemannian manifold. Then any
+locally affine map f:M→Yto a metric space Yis a composition
+f=fi◦fa◦fpof the following factors:
+(1)The map fp:M→M1is flat Riemannian submersion onto a
+smooth complete Riemannian manifold M1;
+(2)The map fa: (M1,g)→(M1,| · |)is an admissible change of
+metric;
+(3)The map fiis a locally isometric embedding.
+Moreover, any map fof this kind is locally affine.
+This global theorem is a consequence of the following local version
+of this result. To state it we recall the following notation from [Oht03 ].
+Letf:M→Ybe a locally affine map defined on a Riemannian
+2manifold M. For a tangent vector vtoMwe denote by γvthe geodesic
+inthedirection of vandby|v|fthevelocity oftheimagegeodesic f◦γv.
+Now the local result reads as follows:
+Theorem 1.5. LetMbe a smooth Riemannian manifold and let f:
+M→Ybe a locally affine map. Then | ·|fis a continuous family of
+semi-norms on the tangent bundle of Mthat is invariant under the
+parallel translation.
+Remark 1.1.It is important to stress that we consider all geodesics
+to be parametrized affinely. If one parametrize all geodesics by the ar-
+clengthit is immediately clear that they determine themetric uniquely.
+If, on the other hand, one allows the geodesics to be parametrized arbi-
+trary, then thequestion becomes much harder. Forinstance, if Xisflat
+andYa Finsler manifold this question is essentially Hilbert’s fourth
+problem. If XandYare Riemannian manifolds then such projective
+mappings have recently been investigated by Matveev (cf. [Mat07]) .
+In the setting of metric spaces it seems to be almost impossible to say
+something meaningful about such maps.
+The results of this paper generalize previous works [Vil70], [Sza81],
+[Oht03] and [HL07]. Vilms classifies in [Vil70] all affine maps between
+smoothRiemannianmanifolds. Szaboextended in[Sza81]thisresult t o
+bijective maps between smooth Riemannian and smooth Finsler man-
+ifolds providing a characterization of all Berwald spaces. On the oth er
+hand, Ohta ([Oht03]) studied maps from a Riemannian manifold to
+(locally uniquely geodesic) metric spaces and proved that such maps
+are compositions of affine maps to continuous Finsler manifolds and
+isometric immersions. However, one cannot combine results of [Oht0 3]
+and [Sza81], since the Finsler metric obtained is only continuous (even
+point-wise) and the Finsler geometric method used in [Sza81] cannot
+work.
+Our proof is a self-contained combination and simplification of ideas
+used in [Oht03] and [HL07]. We hope, that our proof may lead to an
+understanding of affine maps on singular non-positively curved spac es.
+The paper is structured as follows. In Section 2 we recall the proof
+of the semi-continuity of the the semi-Finsler structure |·|f. In Section
+3, the core of the paper, we prove Theorem 1.5. In Section 4 we rec all
+basics about invariant norms andFinsler structures. Finally, inSect ion
+5 we prove Theorem 1.1 and Theorem 1.4.
+32.Basics
+2.1.Setting. LetMbe a Riemannian manifold and let f:M→Y
+be a locally affine map to a metric space Y. We will denote by dthe
+distance in Mand by¯dthe distance in Y. In this and in the next
+section we are going to deal with local questions only. Thus we may
+replaceMby a small strictly convex open neighborhood Uof a given
+pointx. IfUis sufficiently small then f:U→Yis affine. By the
+definition of | · |f, for any geodesic γv: [a,b]→Vand alls,t∈[a,b]
+we have ¯d(f(γv(s)),f(γv(t))) =|s−t| · |v|f. From this we deduce
+|λv|f=|λ|·|v|ffor allλ∈R.
+In this section we are going to prove that fis locally Lipschitz and
+that|·|fis a continuous family of semi-norms. With a minor additional
+assumption it has been proved in [Oht03]. For the convenience of the
+reader, we recall and slightly simplify Ohta’s proof.
+2.2.Lipschitz continuity. We start with
+Lemma 2.1. The map fis continuous.
+Proof.We fixp∈Uand are going to prove the continuity at the point
+p. Inordertodoso, itisenoughtoshowthatthehomogeneousfunc tion
+|·|fis bounded on the unit sphere in T1
+pM⊂TpM.
+Choosevi∈T1
+pMto be the vertices of a regular simplex ∆. Set
+xi= expp(ǫvi), forsufficiently small ǫ >0. LetA0(ǫ)betheunionofall
+xi. LetAk(ǫ) be the set of all points that lie on a shortest geodesic be-
+tween some pair of points in Ak−1(ǫ). The sets Kǫ:=1
+ǫexp−1
+p(An−1(ǫ))
+converge in the Hausdorff topology to the boundary ∂∆ of ∆. Thus Kǫ
+does not contain the origin for small ǫ. On the other hand, Kǫcontains
+a small continuous perturbation of ∂∆. Thus Kǫcarries a non-trivial
+homology class of Hn−1(TpM\{0}). Therefore, Kǫintersects all rays
+starting from the origin. Thus, for sufficiently small ǫ, the compact
+setAn−1:=An−1(ǫ) does not contain p, but it intersects any geodesic
+starting from p.
+Setl= max{¯d(f(p),f(xi))}. By induction on kand the triangle
+inequality, ¯d(f(p),f(x))≤2klfor allx∈Ak. SinceAn−1is compact,
+the number ρ=d(p,An−1) is positive. Thus for any x∈An−1we
+get¯d(f(p),f(x))≤Cd(p,x) withC= 2k·l/ρ. SinceAn−1intersects
+any geodesic starting in p, we deduce that | · |fis bounded by Con
+T1
+pM. /square
+Lemma 2.2. The map | · |fis continuous and fis locally Lipschitz
+continuous.
+4Proof.Choose vectors vn∈TpnUconverging to v∈TpU. Setxn=
+exp(vn);x= exp(v). Then xnconverges to xand by continuity of f,
+the images f(pn) converge to f(p) and the images f(xn) converge to
+f(x). Thus|vn|f=¯d(f(pn),f(xn))→¯d(f(p),f(x)) =|v|f.
+The Lipschitz constant of fatpis bounded by the supremum of |·|f
+on the unit sphere in TpU. Thus, the continuity of | · |fimplies the
+Lipschitz continuity of f. /square
+2.3.Finsler structure. Now we claim:
+Lemma 2.3. The function |·|frestricted to any tangent space TpUis
+a semi-norm.
+Proof.One can follow the proof in ([Oht03]). Another short proof goes
+as follows. The function | · |fcoincides with the metric differential
+defined by Kirhcheim ([Kir94]) for any locally Lipschitz continuous
+map from a manifold to a metric space. Kirchheim proved that this
+metric differential is a semi-norm at almost all points. By continuity
+of|·|fonTU, in our case this is a semi-norm at all points. /square
+3.Main argument
+We are going to use the notations and assumptions from the last
+section. In this section we are going to prove that the semi-norms |·|f
+are invariant under parallel translations along arbitrary piecewise C1
+curves in U. Approximating such a curve by a geodesic polygon in
+the theC1-topology and using the fact that parallel transport behaves
+continuously with respect to such approximations, we deduce that it
+is enough to prove that the semi-norms are parallel along any geode sic
+polygon. Therefore, it is enough to prove that |·|fis parallel along any
+geodesic γinU.
+Thus let us fix a geodesic γ: [−a,a]→Uparametrized by the
+arclength. Set p=γ(0) and denote by Pt:TpM→Tγ(t)Mthe parallel
+transport along γ.
+We call a vector h∈TxUregular if for all v∈TxU
+lim
+t→0|h+tv|f+|h−tv|f−2|h|f
+t= 0
+Avector h∈TxUisregular ifandonly if thesemi-norm |·|f:TxU→
+Ris differentiable at h. By the Theorem of Rademacher almost all
+vectors in TxMare regular. Note that his regular if and only if rhis
+regular, for all real non-zero r.
+We call a vector h∈TpUgoodif, for almost all t∈[−a,a], the
+vectorht=Pt(h) is a regular vector (in Tγ(t)U). Applying Fubini’s
+5theorem we deduce that almost all h∈TpUare good. By continuity,
+it suffices to show that for any good vector h, the function l(t) =|ht|f
+is constant. Since Ptand| · |fare positively homogeneous, we may
+assume after rescaling that exp( ht)∈Uexists for all t∈[−a,a].
+Lemma 3.1. The function l(t)is Lipschitz continuous.
+Proof.For allt,s∈(−a,a), by the triangle inequality ||ht|f−|hs|f|=
+|¯d(f(γ(t)),f(exp(ht))−¯d(f(γ(s),f(exp(hs)))| ≤¯d(f(γ(t),f(γ(s)) +
+¯d(f(exp(ht),f(exp(hs)).
+The parallel transport and the exponential map are smooth and f
+is Lipschitz continuous, thus we can estimate ||ht|f−|hs|f|from above
+byA·|t−s|, for some A >0. /square
+Sincel(t) is Lipschitz continuous, it is enough to prove that the
+derivative of l(t) is 0 almost everywhere. Thus, it suffices to prove that
+ifht0is regular and if the derivative l′(t) exists at t0, this derivative
+must be zero. To prove this claim, we may assume without loss of
+generality (reparametrizing γ) thatt0= 0 and that this derivative is
+non-negative. Thus, we have reduced our task to proving the follo wing
+claim:
+Proposition 3.2. Letγbe a geodesic starting in p. Lethbe a regular
+vector in TpU. Lethtbe the parallel translates of halongγand let the
+function l(t) =|ht|fbe differentiable at 0. Thenl′(0)≤0.
+Proof.Again, we may assume that exp( ht) exists for all t. Assume
+l′(0) = 2δ >0. Then, for all small positive t, we have |ht|f≥ |h|f+δt.
+Denote by ηtthe geodesic ηt(r) = exp( rht). Setxt,r:=ηt(r) and
+p=x0,0=γ(0). Finally, set µt,r:= exp−1
+p(xt,r)∈TpU. The triangle
+inequality gives us
+¯d(f(xt,r),f(xt,−r) = 2r|ht|f≤¯d(f(p),f(xt,r)+¯d(f(p)),f(xt,−r) =|µt,r|f+|µt,−r|f
+Therefore
+2r|h|f+2rδt≤ |µt,r|f+|µt,−r|f
+Denote by vr∈TpMthe derivative vr:=d
+dt|t=0µt,r. The proof of the
+following differential geometric lemma will be given below.
+Lemma 3.3. In the above notations one has limr→0(||vr−v−r||/r) = 0.
+Given this lemma, it is easy to derive a contradiction: We find and
+fix a small rwith|vr−v−r|f≤δr/4. By definition, µt,±r= (±r)·h+
+t·v±r+o(t). Thus|µt,−r|f≤ |−rh+t·vr|f+δrt/2, for small t. Hence
+|µt,r|f+|µt,−r|f≤ |r·h+t·vr|f+|rh−t·vr|f+δrt.
+By assumption, his regular. Thus |rh+tvr|f+|rh−tvr|f= 2|rh|f+
+o(t). For small t, this contradicts to |µt,r|f+|µt,−r|f≥2r+2rδt./square
+6It remains to provide:
+Proof of Lemma 3.3. Letη=η0. Denote by Ythe Jacobi field Y(r) =
+d
+dtηt(r) alongη. LetVrbe the Jacobi field Vr(s) :=d
+dt|t=0expp(µt,r
+rs).
+By construction, Y′(0) = 0, Vr(0) = 0 and Vr(r) =Y(r). Finally,
+V′
+r(0) =d
+dtµt,r
+r=vr
+r. Thus, we only need to prove that |V′
+r(0)+V′
+−r(0)|
+converges to 0, as rgoes to 0. Since the Jacobi field ( Vr+V−r) has
+value 0 at the point 0, it suffices to prove that |(Vr+V−r)(r)|=o(r).
+In fact we have ||Y(t)−Y(0)|| ≤Ct2and||Vr(t)−tV′
+r(0)|| ≤Ct2
+for all|t| ≤rand some Cindependent of r. Together this gives us
+||V−r(r)+V−r(−r)|| ≤Cr2and||Y(r)−Y(−r)|| ≤Cr2. Together this
+gives us||V−r(r)+Vr(r)|| ≤Cr2. This finishes the proof of Lemma 3.3,
+Proposition 3.2 and Theorem 1.5. /square
+4.Remarks on invariant norms
+In this section we collect some elementary observation, probably we ll
+known to experts, for which we could not find a reference.
+4.1.Invariant norms. We recall a few definitions. A semi-norm on
+a finite-dimensional vector space Vis a non-negative, homogeneous,
+convex function q:V→[0,∞). It is called a normifq(v)>0 for all
+v/\e}atio\slash= 0. It is called a Minkowski norm, if qis smooth outside the origin
+and the Hessian D2
+xqis positive definite at all points x/\e}atio\slash= 0. The second
+condition is equivalent to the requirement that qcan be expressed as
+q=q0+q1, whereq0is a scalar product and q1is a norm.
+For two norms q1,q2onVthere is some L≥1 withq1≤L·q2and
+q2≤L·q1. The distance |q1−q2|between q1andq2is defined to be
+the infimum of log( L) taken over the set of all such L. The topology
+on the set of norms defined by this distance function coincides with t he
+Hausdorff topology on the set of convex centrally-symmetric bodie s. It
+is well known that the set of Minkowski norms is dense in the set of all
+norms.
+Lemma 4.1. LetVbe a Euclidean space and let Gbe a closed subgroup
+of the orthogonal group. Then the set of G-invariant Minkowski norms
+is dense in the set of all G-invariant norms.
+Proof.Letqbe aG-invariant norm. Choose some small ǫ >0. We find
+a Minkowski norm q1with|q1−q| ≤ǫ. Letq2be the norm obtained
+fromq1by the averaging procedure, i.e., q2(x) =/integraltext
+Gq1(gx)dµ(g), where
+µis the unit volume Haar measure on G. Thenq2is aG-invariant
+Minkowski normandwestillhave |q−q2| ≤ǫ, sinceqisG-invariant. /square
+7Lemma 4.2. LetV,Gbe as above. If Gacts transitively on the unit
+sphere then the only G-invariant norms are multiples of the given Eu-
+clidean norm. If Gdoes not act transitively on the unit sphere then
+there are non-Euclidean G-invariant Minkowski norms.
+Proof.The first statement is clear. To prove the second statement we
+proceed as follows. If Gacts irreducibly, consider any orbit Gpand
+letCbe its convex hull. If Gacts reducibly, we consider orthogonal
+G-invariant subspaces V1,V2⊂VwithV1⊕V2=V, and let Cbe the
+convex hull of the union of the unit spheres S1⊂V1andS2⊂V2. In
+both cases we obtain a G-invariant convex body that is not strictly
+convex. Its symmetrization around the origin and the norm qdefined
+by the symmetrized body is still G-invariant and not strictly convex.
+Due to the previous lemma, we find a G-invariant Minkowski norm qn
+arbitrary close to q. Sinceqis non-Euclidean, qnis non-Euclidean as
+well, at least for large n. /square
+4.2.Holonomy group. For simplicity, we will restrict ourselves to
+the complete case. Thus let ( M,g) be a complete smooth Riemannian
+manifold. Let p∈Mbeapointandlet HandH0denote theholonomy
+groupHolpand its identity component respectively. The tangent space
+V=TpMsplits under the action of H0in a uniquely defined way as
+V=V0⊕V1⊕...⊕Vi, wheretheactionof HonV0istrivialandtheaction
+onVi,i≥1 is irreducible and not trivial. Moreover, this decomposition
+determines the unique direct product decomposition of the univers al
+covering ˜MofM([Sak96]). By the uniqueness of this decomposition
+up to permutation, the action of Hpreserves this decomposition of V,
+possibly up to a permutation of summands. In particular, if H0acts
+reducibly then the closure ¯HofHacts non-transitively on the unit
+sphere.
+Iftheactionof H0ontheunitsphereisirreducible, theneitheritacts
+transitively on the unit sphere or the space Mis a locally symmetric
+space of rank at least two, with irreducible universal covering, due to
+the theorem of Berger-Simons ([Sim62]). In this case, each element of
+Hacts as the differential of a local isometry, thus it preserves the t ype
+of a vector in its Weyl chamber. Hence, the action of ¯Hon the unit
+sphere is non-transitive as well.
+We conclude:
+Lemma 4.3. LetMbe a complete Riemannian manifold. Let p∈M
+be a point, V=TpMand letHbe the holonomy group at the point p.
+Then the following are equivalent:
+(1)Hdoes not act transitively on the unit sphere V1ofV;
+8(2)The closure of Hdoes not act transitively on V1;
+(3)There are non-Euclidean Minkowski norms on Vinvariant un-
+derH;
+(4)The universal covering of Mis either a direct product or a
+symmetric space of higher rank.
+4.3.Description of invariant norms. In this subsection we are go-
+ing to provide a description of (Minkowski) norms invariant under the
+action of a continuous holonomy group. Since these statements ar e not
+used in the rest of the paper we will omit some details.
+ThusletMbeaRiemannianmanifoldandassumethattheholonomy
+groupH=Holpis connected (this happens, for instance, if Mis
+complete and simply connected). Then (due to the theorems of Ber ger-
+Simons an de Rham) the action of HonV=TpMis polar, i.e., there
+is a linear subspace Σ of Vthat intersects all orbits of Hwith all
+intersection being orthogonal. The stabilizer N(Σ) acts on Σ as a
+finite Coxeter group W. Then each H-invariant norm qonVrestricts
+to aW-invariant norm ¯ qon Σ. We have:
+Proposition 4.4. The restriction q→¯qis a bijection between the set
+ofH-invariant norms on VandW-invariant norms on Σ. Moreover,
+qis a Minkowski norm if and only if ¯qis a Minkowski norm.
+Proof.Ifqis a (Minkowski) norm then so is its restriction to any sub-
+space, in particular ¯ q. On the other hand, each W-invariant function ¯ q
+extends to a unique H-invariant function qonV. Now, if ¯ qis a norm
+then so is q, due to [HT01], p. 107. Moreover, if one can represent ¯ q
+as a sum ¯ q0+ ¯q1of a scalar product and a norm, then, averaging, we
+may assume that ¯ q0and ¯q1areW-invariant as well. Thus they extend
+to anH-invariant scalar product and norm q0andq1withq=q0+q1.
+Finally, an H-invariant function is smooth if and only if its restriction
+toWis smooth ([Dad82]). Smoothing ¯ qat the origin, we deduce, that
+¯qis smooth in Σ \{0}if and only if qis smooth in V\{0}. Thusqis
+a Minkowski norm if and only if ¯ qis. /square
+We illustrate this result by two examples:
+Example 4.1.LetMis a symmetric space and let Fbe a maximal
+flat through a point p. LetWbe the group of all isometries of M
+that fixpand leaves Finvariant. Then the set of parallel (smooth)
+Finsler structures is in one-to-one correspondence with the set o f all
+W-invariant (Minkowski) norms on TpM.
+Example 4.2.LetM=M1×M2be the product of two irreducible
+not locally symmetric spaces. For p∈M, choose any unit vector e1in
+9TpM1ande2inTpM2. ThenonecanchooseΣtobetheplanegenerated
+bye1ande2. The group Whas 4 elements and is generated by the the
+reflections ei→ −ei.
+5.The conclusion
+5.1.The if part. We are going to prove Theorem 1.4. First we would
+like to see that each map as described in the theorem is locally affine.
+A composition of continuous locally affine map is locally affine. A
+locallyisometricembeddingislocallyaffine. Aflatsubmersionislocally
+given by a projection onto a direct factor, thus it is locally affine as
+well. Therefore, it suffices to prove that an admissible change of a
+metric is locally affine. We are going to reduce the statement to the
+smooth case, where the result is known (cf. [Sza81]).
+Thus let ( M,g) be a smooth complete Riemannian manifold and let
+| · |be a holonomy invariant Finsler structure. Fix a point x∈M.
+Aproximate the norm |·|onTpMby Minkowski norms |·|nnorms on
+TpMinvariant under the holonomy group Holp(Lemma 4.1). Extend
+each norm | · |nto a smooth holonomy invariant Finsler structure on
+M. Due to [Sza81], the identity map Id: (M,g)→(M,|·|n) is affine.
+Since minimizing geodesics in the smooth Finsler manifold ( M,| · |n)
+arelocally unique, all ( M,|·|n) geodesics are images of ( M,g) geodesics
+(i.e., the map Id: (M,|·|n)→(M,g) is affine as well). We find some
+Lsuch that the identity Id: (M,g)→(M,| · |n) is L-bilipschitz, for
+alln. We find some strictly convex ball Vof radius raroundx, such
+that any geodesic between points of Vthat is not contained in Vhas
+length at least 4 Lr. It follows, that any ( M,g)-geodesic contained in
+Vis a minimizing geodesic in ( M,| · |n), for all n. Since minimizing
+(!) (M,| · |n) geodesics converge to minimizing ( M,| · |) geodesics,
+we deduce that Id: (V,g)→(V,| · |) sends minimizing geodesics to
+minimizing geodesics. The linear parametrization of the geodesics is
+clear byapproximation (orbythefact thatgeodesics areauto-pa rallel).
+5.2.Decomposition of f.LetMbe a complete Riemannian mani-
+fold and let f:M→Ybe a locally affine map. Due to Theorem 1.5,
+the function | · |fdefines a parallel family of semi-norms. For each
+q∈M, we set Vq={v∈TqM||v|f= 0}. Since the semi-norms are
+parallel this is a parallel distribution.
+Assume first that Mis simply connected. Then by the theorem
+of de Rham this distribution q→Vqis the vertical distribution of a
+projection fp:M=M1×M2→M1onto a direct factor. The map
+ffactors through fpasf=ˆf◦fp. Moreover, ˆfcoincides with the
+restriction of fto any horizontal slice M1× {x2}. Thus ˆfis again
+10affine, and the semi-norm |·|ˆfcoincides with the restriction of |·|f. By
+the definition of Vq, this semi-norm is a norm. Thus on M1we have an
+admissible change of metric ( M1,g)→(M1,|·|f) and the induced map
+fi: (M1,|·|f)→Yis by definition of ·|fa local isometry.
+Assume now that Mis not simply connected. Consider the lift ˜fof
+fto the universal covering ˜M. Consider a horizontal slice M′
+1in˜M.
+The restriction of ˜fto this slice is locally bi-Lipschitz. On the other
+hand this restriction factors through the canonical projection o f the
+sliceM′
+1toM. This implies that the leaves of the integral distribution
+VqinMare closed submanifolds that define a Riemannian submersion
+fp:M→M1for some manifold M1covered by M′
+1. It is clear that
+the Riemannian submersion fpis flat. The rest follows from the simply
+connected case.
+5.3.Proof of Theorem 1.1. To prove Theorem 1.1, first let Mbe
+locally irreducible and not locally symmetric of higher rank. Let f:
+M→Ybe a locally affine map. Due to Theorem 1.5 and Lemma 4.3,
+at each point p∈M, the semi-norm |·|fonTpMis either constant 0 or
+a positive multiple of the metric g, the multiple not depending on the
+point. Thus the map feither sends the whole manifold Mto a point
+or rescales distances in any small neighborhood of any given point by
+the constant number a.
+If, on the other hand, Mis either locally reducible or locally sym-
+metric of higher rank, then there is an admissible change of metric to
+a non-Euclidean (smooth) Finsler structure. This change is certain ly
+not a local homothety, but it is a locally affine map by Theorem 1.4.
+References
+[AB98] S. Adams and W. Ballmann. Amenable isometry groups of Hadam ard
+spaces.Math. Ann. , 312:183–195, 1998.
+[Dad82] J. Dadok. On the C∞chevalley theorem. Adv. in Math. , 44:121–131,1982.
+[FL08] Th. Foertsch and A. Lytchak. The de Rham decomposition th eorem for
+metric spaces. Geom. Funct. Anal. , 18:120–134, 2008.
+[HL07] P. Hitzelberger and A. Lytchak. Spaces with many affine func tions.Proc.
+Amer. Math. Soc. , 235:2263–2271, 2007.
+[HT01] W. Hill and T. Tam. On g-invariantnorms. Linear Algebra Appl. , 331:101–
+112, 2001.
+[Kir94] B. Kirchheim. Rectifiable metric spaces: local structure and the regularity
+of the Hausdorff measure. Proc. Amer. Math. Soc. , 121:113–123, 1994.
+[Mat07] V. Matveev. Proof of the projective lichnerowicz-obatac onjecture. J. Diff.
+Geom., 75:459–502, 2007.
+[Oht03] S. Ohta. Totally geodesic maps into metric spaces. Math. Z. , 244(1):47–65,
+2003.
+11[Sak96] T. Sakai. Riemannian geometry . Translations of Mathematical Mono-
+graphs, 149. American Mathematical Society, Providence, RI, 19 96.
+[Sim62] J. Simons. On transitivity of holonomy systems. Annals of Math. , 76:213–
+234, 1962.
+[Sza81] Z. I. Szabo. Positive definite Berwald spaces. Tensor N.S. , 38:25–39, 1981.
+[Vil70] J. Vilms. Totally geodesic maps. J. Differential Geometry , 4:73–79, 1970.
+12
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+arXiv:1001.0805v2  [nucl-th]  26 Apr 2010Possible Σ(1
+2−)under the Σ∗(1385)peak in KΣ∗photoproduction
+Puze Gao, Jia-Jun Wu and B.S. Zou
+Institute of High Energy Physics, CAS, P.O. Box 918(4), Beij ing 100049, China and
+Theoretical Physics Center for Science Facilities, CAS, Be ijing 100049, China
+The LEPS collaboration has recently reported a measurement of the reaction γn→K+Σ∗−(1385)
+with linearly polarized photon beam at resonance region. Th e observed beam asymmetry is sizably
+negative at Eγ= 1.8−2.4GeV, in contrast to the presented theoretical prediction. In this paper, we
+calculate this process in the framework of the effective Lagr angian approach. By including a newly
+proposed Σ( JP=1
+2−) state with mass around 1380 MeV, the experimental data for b othγnand
+γpexperiments can be well reproduced. It is found that the Σ(1
+2−) and/or the contact term may
+play important role and deserve further investigation.
+PACS numbers: 14.20.Jn, 25.20.Lj, 13.60.Le, 13.60.Rj
+I. INTRODUCTION
+With the development of accelerator facilities,
+strangeness production from photon-nucleon scattering
+has been extensively studied at resonance region [1–9].
+These experiments can provide not only more informa-
+tion on the properties and interactions of the known res-
+onances, but also clues for the existence of some new res-
+onances. For KΣ∗(1385) photoproduction, high statistic
+datahavebeen availableonlyrecently. TheCLAScollab-
+oration has studied the reaction γ+p→K++Σ∗0(1385)
+with unpolarized photon beam at energy Eγ= 1.5−
+4 GeV [2]. The LEPS collaboration has reported the
+reaction γ+n→K++Σ∗−(1385), with a linearly polar-
+ized photon beam at Eγ= 1.5−2.4 GeV [9]. The high
+statistic data and the polarized observables provide more
+information and challenges for theoretical studies.
+Theoreticalinvestigationson KΣ∗(1385)photoproduc-
+tion include the work by Lutz and Soyeur [10], which
+mainly studies the t-channel processes, the work by
+D¨ oring, Oset and Strottman [11], where the role of
+∆(1700) is addressed, and the work by Oh, Ko, and
+Nakayama [12], where the roles of Nand ∆ resonances
+in s-channel are stressed and compared with the CLAS
+data [2].
+For the approach by Oh, Ko, and Nakayama [12], the
+total cross section for γ+p→K++ Σ∗0(1385) is cal-
+culated in the framework of gauge-invariant effective La-
+grangians. The results are in reasonable agreement with
+the CLAS data [2], showing that the s-channel Nand
+∆ resonances above KΣ∗threshold may give important
+contributions to cross sections. Although this theory can
+well describethe totalcrosssectionof KΣ∗photoproduc-
+tion from CLAS aswell asfrom LEPSexperiment [9], the
+theoretical prediction deviates greatly from the data for
+linear beam asymmetry measured by LEPS with polar-
+ized photon beam. This is an urgent problem for theo-
+retical studies.
+From studies of baryon spectroscopy and structures,
+five quark qqqqqcomponents are proposed to play im-
+portant roles in some baryons [13, 14]. A few years
+ago, JaffeandWilczekhavepromotedadiquark-diquark-antiquarkpictureforthepentaquarkbaryons[15]. Zhang
+et al.then studied the JP=1
+2−pentaquark baryons
+based on this picture and predicted a Σ(1
+2−) state with
+mass around 1360 MeV [16]. A more general pentaquark
+model[13]withoutintroducingexplicitlydiquarkclusters
+predictsthatΣ(1
+2−)hasamasssimilartoΛ(1
+2−), whichis
+around1405MeV. From these two models, one would ex-
+pect a Σ(1
+2−) state with mass around 1380 MeV. Recent
+studies on K−p→Λπ+π−process have shown some evi-
+dence for the existence of the Σ(1
+2−) near 1380 MeV [17].
+In this work, we study the KΣ∗(1385) photoproduction
+processes with the consideration of the case that a por-
+tion ofKΣ(1
+2−) photoproduction is mixed in.
+This paper is organized as follows. In section II,
+the theoretical framework is presented for the KΣ∗and
+KΣ(1
+2−) photoproduction from the nucleons. In section
+III, the numerical results for cross sections and the beam
+asymmetry are presented and compared with the exper-
+imental data, with some discussions. In section IV, we
+give the summery of this work.
+II. THEORETICAL FRAMEWORK
+The effective Lagrangian method is an important the-
+oretical approach in describing the various processes at
+resonance region. We use the effective Lagrangians of
+Ref. [12] for KΣ∗photoproduction, where the contact
+term is derived from Ref. [18] to keep the amplitude
+gauge invariant. In the following equations we use Σ∗
+and Σ denoting the Σ∗(3
+2+) at 1385 MeV and the Σ(1
+2−)
+near 1380 MeV, respectively.
+A.KΣ∗(3
+2+)photoproduction
+For the reaction γN→KΣ∗(3
+2+), the Feynman dia-
+grams are shown in Fig. 1, where the incoming momenta
+arekandpfor photon and nucleon, respectively, and
+the outgoing momenta are qandp′forKmeson and the2
+FIG. 1: Feynman diagrams for γ+N→K+ Σ∗(3
+2+). (a)
+t-channel; (b) s-channel; (c) u-channel; (d) contact term.
+Σ∗, respectively. From the investigation of Ref. [12], the
+main contributions comefrom the t-channel Kmeson ex-
+change,thes-channel Nand∆aswellastheirresonances
+exchange, the u-channel Λ (for neutral propagator only)
+and Σ∗(3
+2+) exchange and the contact term.
+For the t-channel Kmeson exchange, the relevant ef-
+fective Lagrangians are
+LγKK=ieAµ(K−∂µK+−∂µK−K+),(1)
+LKNΣ∗=fKNΣ∗
+mK∂µKΣ∗µ·τN+H.c., (2)
+with the isospin structure of KΣ∗Ncoupling
+K= (K−,K0),Σ·τ=/parenleftBigg
+Σ0√
+2Σ+
+√
+2Σ−−Σ0/parenrightBigg
+,N=/parenleftbigg
+p
+n/parenrightbigg
+,
+(3)
+wherefKNΣ∗is the coupling constant and is taken as
+fKNΣ∗=−3.22 from the SU(3) flavor symmetry relation
+as in Ref. [12]; mKis the mass for Kmeson.
+For the s-channel nucleon exchange, the effective La-
+grangians for the interactions contain Eq. (2) as well as
+LγNN=−eN(γµAµQN−κN
+2MNσµν∂νAµ)N ,(4)
+whereQNis the electric charge (in unite of e), andκN
+denotes the magnetic moment of the nucleon.
+For the s-channel spin-3
+2and spin-5
+2resonances ex-
+change, the effective Lagrangians for the interactions are
+LγNR(3
+2±
+) =−ief1
+2MNNΓ(±)
+νFµνRµ
+−ef2
+(2MN)2∂νNΓ(±)FµνRµ+H.c.,(5)
+LγNR(5
+2±
+) =ef1
+(2MN)2NΓ(∓)
+ν∂αFµνRµα
+−ief2
+(2MN)3∂νNΓ(∓)∂αFµνRµα+H.c.,(6)and
+LRKΣ∗(3
+2±
+) =−h1
+mK∂αKΣ∗µΓ(±)
+αRµ
++ih2
+(mK)2∂µ∂αKΣ∗
+αΓ(±)Rµ+H.c.,(7)
+LRKΣ∗(5
+2±
+) =ih1
+m2
+K∂µ∂βKΣ∗αΓ(∓)
+µRαβ
+−h2
+(mK)3∂µ∂α∂βKΣ∗
+µΓ(∓)Rαβ+H.c.,(8)
+whereFµν=∂µAν−∂νAµ;RµandRµαdenote the
+spin-3
+2and spin-5
+2fields, respectively, and
+Γ(±)
+µ=/parenleftbigg
+γµγ5
+γµ/parenrightbigg
+,Γ(±)=/parenleftbigg
+γ5
+1/parenrightbigg
+. (9)
+For ∆ and ∆ resonances of isospin-3/2, the effective La-
+grangians have the isospin structure
+KΣ∗·T(1
+2,3
+2)∆ =√
+3K−Σ∗+∆++−√
+2K−Σ∗0∆+
+−K−Σ∗−∆0+K0Σ∗+∆+−√
+2K0Σ∗0∆0
+−√
+3K0Σ∗−∆−.(10)
+We consider 3 PDG resonances in s-channel, namely,
+N3
+2−(2080), ∆ 3
+2−(1940) and ∆ 5
+2+(2000), which are the
+most prominent ones as stated in Ref. [12]. The cou-
+pling constants f1andf2can be calculated from the
+helicity amplitudes in PDG [19] or model predictions
+with Eq.(B3) of Ref. [12]. For γN∆ coupling, we have
+f1= 4.04 andf2= 3.87. From the predicted helicity am-
+plitudes in Ref. [20], one gets f1=−1.25 andf2= 1.21
+forγpN∗(2080) coupling; f1= 0.381 and f2=−0.256
+forγnN∗(2080) coupling; f1= 0.39 andf2=−0.57
+forγN∆(1940) coupling, and f1=−0.68,f2=−0.062
+forγN∆(2000) coupling. For ∆ KΣ∗coupling, h1= 2.0
+andh2= 0 are used from h1=−fK∆Σ∗/√
+3 with
+fK∆Σ∗=−3.46 [21]. For the resonances coupling to
+KΣ∗, the coupling constants h1andh2can be calculated
+from Eq.(B11)-(B18) of Ref. [12] from the model pre-
+dicted amplitudes G(l) [22]. In our concrete calculation,
+h1= 0.24 andh2=−0.54 forN∗(2080)KΣ∗coupling,
+h1=−0.68 andh2= 1.0 for ∆(1940) KΣ∗coupling, and
+h1=−1.1 andh2= 0.21 for ∆(2000) KΣ∗coupling are
+taken.
+The reaction γp→K+Σ∗0contains the u-channel
+Λ(1116) exchange. The effective Lagrangians are
+LγΛΣ∗=−ief1
+2MΛΛγνγ5FµνΣ∗
+µ
+−ef2
+(2MΛ)2∂νΛγ5FµνΣ∗
+µ+H.c.,(11)
+LKNΛ=gKNΛ
+MN+MΛNγµγ5Λ∂µK+H.c.,(12)3
+wheref1= 4.52,f2= 5.63 are obtained from decay
+width Γ(Σ∗→Λγ) and model predicted helicity am-
+plitudes; and gKNΛ=−13.24 is estimated from flavor
+SU(3) symmetry relation as in Ref. [12].
+For u-channel Σ∗exchange, the effective Lagrangian
+forγΣ∗Σ∗is
+LγΣ∗Σ∗=eΣ∗
+µAαΓα,µν
+γΣ∗Σ∗
+ν, (13)
+with
+AαΓα,µν
+γΣ∗=QΣ∗Aα/parenleftbig
+gµνγα−1
+2(γµγνγα+γαγµγν)/parenrightbig
+−κΣ∗
+2MNσαβ∂βAαgµν, (14)
+whereQΣ∗denotes the electric charge of Σ∗in unit of e
+andκΣ∗denotes the anomalous magnetic moment of Σ∗;
+κΣ∗0= 0.36 andκΣ∗−=−2.43 are taken from the quark
+model prediction [23].
+For each vertex of these channels, a form factor is at-
+tached to describe the off-shell properties of the ampli-
+tudes. For t-channel Kmeson exchange, we adopt the
+form factor [12]
+FM=Λ2
+M−m2
+K
+Λ2
+M−q2
+t, (15)
+whereqt=k−qisthe4-momentumtransferint-channel.
+For the s-channel and u-channel processes, we adopt the
+form factor
+FB(q2
+ex,Mex) =/parenleftbiggnΛ4
+B
+nΛ4
+B+(q2ex−M2ex)2/parenrightbiggn
+,(16)
+wherethe qexandMexarethe4-momentumandthemass
+of the exchanged hadron, respectively. For s-channel N
+and ∆ exchange and the u-channel processes, we take
+n= 1 as in Ref. [12]; for s-channel resonances exchange,
+n→ ∞is taken to obtain the Gaussian form for the form
+factors [12]
+FB(q2
+s,MR)|n→∞= exp/parenleftBig
+−(q2
+s−M2
+R)2
+Λ4
+B/parenrightBig
+.(17)
+The cutoff parameters Λ Mand ΛBwere taken to be 0.83
+and 1.0 GeV, respectively, in Refs. [9, 12], by fitting to
+theγp→K+Σ∗0data.
+The contact term illustrated with Fig. 1(d) serves to
+keep the full amplitude gauge invariant. The contact
+currents for KΣ∗photoproduction are related to the La-
+grangian of Eq.(2). For the process γp→K+Σ∗0, we
+adopt the contact current [12, 18]
+Mµν
+c=iefKNΣ∗
+mK(gµνft−qµCν),(18)
+whereCνis expressed as
+Cν=−(2q−k)νft−1
+t−m2
+K/parenleftbig
+1−h(1−fs)/parenrightbig
+−(2p+k)νfs−1
+s−M2
+N/parenleftbig
+1−h(1−ft)/parenrightbig
+.(19)Here the Lorenz index µandνcouple to that of Σ∗and
+the photon respectively; ft=F2
+Mandfs=F2
+B(s,MN)
+are form factors squared, and t=q2
+tands=q2
+sare
+squared momentum transfer for t- and s-channel; his a
+parameter to be fitted to experiments, and h= 1 is used
+in Ref.[12]. From this contact term, we can check that
+the total amplitude is gauge invariant. For the process
+γn→K+Σ∗−, the contact current is [18]
+Mµν
+c=ie√
+2fKNΣ∗
+mK(gµνft−qµCν),(20)
+with
+Cν=−(2q−k)νft−1
+t−m2
+K/parenleftbig
+1−h(1−fu)/parenrightbig
++(2p′−k)νfu−1
+u−M2
+Σ∗/parenleftbig
+1−h(1−ft)/parenrightbig
+.(21)
+Wherefu=F2
+B(u,M∗
+Σ) is form factor squared, and u=
+q2
+uis squared momentum transfer for u-channel.
+For the propagators, we use 1 /(q2
+t−m2
+K) for t-channel
+Kmeson exchange. For the propagator of a baryon with
+massmand 4-momentum p, we use/negationslashp+m
+p2−m2for spin-1/2
+propagator;
+/negationslashp+m
+p2−m2/parenleftBig
+−gµν+γµγν
+3+γµpν−γνpµ
+3m+2pµpν
+3m2/parenrightBig
+(22)
+for spin-3/2 propagator; and
+/negationslashp+m
+p2−m2Sαβµν(p,m) (23)
+for spin-5/2 propagator, where
+Sαβµν(p,m) =1
+2(gαµgβν+gανgβµ)−1
+5gαβgµν
+−1
+10(γαγµgβν+γαγνgβµ+γβγµgαν+γβγνgαµ),(24)
+with
+gµν=gµν−pµpν
+m2,
+γµ=γµ−pµ
+m2/negationslashp. (25)
+For the s-channel resonances with sizeable width Γ, we
+replace the denominator1
+p2−m2in the propagators by
+1
+p2−m2+imΓ, and replace min the rest of the propagators
+by/radicalbig
+p2.
+B.KΣ(1
+2−)photoproduction
+Since five quark components for baryons may exist,
+there may be a Σ(1
+2−) that has a large probability of five
+quark structure with mass near 1380 MeV as models pre-
+dict [13, 16]. Both Σ∗(3
+2+) and Σ(1
+2−) decay strongly to4
+Λπ, which are detected by experiments. Thus we con-
+sider that KΣ(1
+2−) photoproduction may also contribute
+to the measured cross sections. We constrain our study
+to processes with sandpwave hadronic vertices, since
+the contributions from higher waves are relatively sup-
+pressed. Wealsoneglectthecontributionsfromsomeres-
+onances either for lack of information on the couplings or
+the couplings are small. Thus the main contributions to
+KΣ(1
+2−) photoproduction are from the t-channel Kme-
+son exchange, the s-channel Nexchange, the u-channel
+Σ(1
+2−) exchange (and Λ exchange for γp→K+Σ0(1
+2−))
+and the contact term. The Feynman diagrams are the
+same as Fig. 1. The effective Lagrangian for KNΣ(1
+2−)
+coupling can be expressed as [21]
+LKNΣ=−igKNΣKΣ·τN+H.c.,(26)
+wherethecouplingconstant gKNΣistobefittedtoexper-
+iments; The isospin structure is the same as Eq.(3). The
+effective Lagrangians for γKKandγNNare described
+in Eq.(1) and Eq.(4), respectively. For γΣ(1
+2−)Σ(1
+2−)
+vertex, the effective Lagrangian can be expressed as [21]
+LγΣΣ=−eΣ(γµAµQΣ−κΣ
+2MNσµν∂νAµ)Σ,(27)
+whereQΣis the electric charge(in unite of e), andκΣde-
+notestheanomalousmagneticmomentforΣ(1
+2−), andwe
+take the predicted values κΣ0=−0.43 andκΣ−=−1.74
+from the diquark model [16]. For the γΛΣ0(1
+2−) coupling
+in theγp→K+Σ0process, the effective Lagrangian is
+expressed as [21]
+LγΛΣ=egγΛΣ
+4(MΛ+MΣ)Σγ5σµνΛFνµ+H.c.,(28)
+where we take gγΛΣ= 1.16. Note that although this
+coupling has some uncertainty, it is largely suppressed
+and has very small effect in this process.
+ForeachvertexintheFeynmandiagrams,aformfactor
+is attached. Similar to the KΣ∗(3
+2+) photoproduction
+processes, we adopt the form factor as in Eq.(15) for t-
+channel Kmeson exchange, and Eq.(16) for s-channel
+and u-channel processes.
+The contact term for KΣ(1
+2−) photoproduction is re-
+lated to the Lagrangian of Eq.(26). Following Refs. [12,
+18], we adopt the contact current
+Mν
+c=iegKNΣCν(29)
+forγp→K+Σ0(1
+2−), where the Lorenz index νcouples
+to that of the photon, and Cνis expressed as Eq.(19),
+whereh= 1 is taken. For γn→K+Σ−(1
+2−) process, we
+adopt the contact current:
+Mν
+c=ie√
+2gKNΣCν, (30)
+whereCνis expressed as Eq.(21), where h= 1 is taken.
+With these contact currents, one can check that the total
+amplitudes are gauge invariant.Sincepreviousevidenceforthe newΣ(1
+2−) indicatesits
+masstobearoundΣ∗(1385)[17], hereweassumeitsmass
+be the same as Σ∗(1385). The coupling constant gKNΣ
+and the relevant cut-off parameter Λ Mare unknown pa-
+rameters, and will be tuned to fit the data.
+III. RESULTS AND DISCUSSIONS
+The effective Lagrangian methods employ hadronic
+fields as basic degrees of freedom for the interactions,
+which introduce many parameters such as the coupling
+constants and cutoff parameters. Many of the parame-
+ters are not well constrained, and approximations such
+as flavor SU(3) symmetry relation or model predictions
+are used. These may bring some uncertainty in the
+calculation. For example, in this paper, we consider
+3 PDG resonances in s-channel as in Ref. [12]. Their
+masses, widths and coupling constants are not well con-
+strained by experiments. Our choices of the parameters
+from the allowed range of PDG [19] and model calcu-
+lations [20, 22] certainly have big uncertainties, similar
+as discussed in Ref. [24] for the study of γn→K+Σ−.
+However, for the cross sections and beam asymmetry
+of the reaction γn→K+Σ∗−(1385)→K+Λπ−at
+Eγ= 1.5−2.4 GeV reported by the LEPS collabora-
+tion [9], these 3 resonances mainly contribute to cross
+sections around Eγ= 1.8 GeV, and their contributions
+aresmallfor Eγ>2.1GeV. Evenwithout constraintfrom
+other sources, adjusting these parameters cannot repro-
+duce the measured beam asymmetry qualitatively. We
+need to find other ingredients to describe the cross sec-
+tions and beam asymmetry data simultaneously.
+The measurement by the LEPS Collaboration is lim-
+ited to forward angles with cos θc.m.≥0.6, where θc.m.is
+the forward angle of the produced Kmeson in the c.m.
+frame. The theoretical predictions by the original set
+of parameters ( h= 1 for the contact term) [12] deviate
+fromthe LEPSresultsforthe linearbeam asymmetry[9],
+which can be interpreted as
+Abeam=σ⊥−σ/bardbl
+σ⊥+σ/bardbl, (31)
+whereσ⊥andσ/bardbldenote the cross sections for beam po-
+larization vertical and parallel to the reaction plane, re-
+spectively.
+Inthiswork,wefindtwopossibleingredientstoexplain
+the observed beam asymmetry. One is by including the
+possible contribution of KΣ(1
+2−) photoproduction, and
+another is to assume a different h parameter for the γn
+reaction from the γpreaction. To incorporate the new
+ingredients to fit both cross sections and beam asymme-
+try, the minimum set of parameters to be adjusted from
+those used in Ref. [9, 12] are listed in Table I. For both
+schemes, the narrower widths for the three N∗and ∆∗
+resonances are used, but still within the PDG uncertain-
+ties [19]. The reached χ2of the two schemes are also5
+scheme h ΛM ΓN∗(2080) Γ∆(1940) Γ∆(2000) gKNΣ(ΛM)χ2
+I 1.0 (fixed) 0.8 GeV 0.25 GeV 0.15 GeV 0.15 GeV 1.34 (1.6 GeV) 97
+II 1.11 0 (fixed) 102
+[9, 12] (PDG [19]) 1.0 (fixed) 0.830.3 (0.12∼0.63)0.3 (0.10∼0.78)0.3 (0.07∼0.52)0 (fixed) ∼180
+TABLE I: Adjusted parameters for γn→K+Σ∗−with two schemes compared with original ones in Refs. [9, 12] and PDG
+range [19], and corresponding χ2for 39 data points in Figs. 3 & 4.
+/s49/s46/s54 /s49/s46/s56 /s50/s46/s48 /s50/s46/s50 /s50/s46/s52/s45/s49/s46/s50/s45/s49/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48
+/s110 /s62 /s99/s111/s115
+/s99/s109
+/s32/s32/s32/s66/s101/s97/s109/s32/s65/s115/s121/s109/s109/s101/s116/s114/s121
+/s69 /s40/s71/s101/s86/s41
+FIG. 2: Linear beam asymmetry for γ+n→K++Σ−(1
+2−)
+with cos θc.m.= 0.6−1.
+listed. In scheme I, we keep h= 1 fixed for the contact
+term to be identical to the γp→K+Σ∗0process, and
+then include the contribution of KΣ(1
+2−) photoproduc-
+tion for the process. The coupling constant of Σ(1
+2−) to
+KNand the corresponding cut-off parameter are tuned
+to describe the experimental data. In scheme II, we do
+not consider Σ(1
+2−) production, and tune parameter hof
+the contact term to describe the data.
+In Fig. 2 we show the result of the linear beam asym-
+metry for γn→K+Σ−(1
+2−) with cos θc.m.≥0.6,i.e.,
+the cross sections in Eq. (31) are integrated results for
+cosθc.m.= 0.6−1. The process has a large negative
+beam asymmetry that approaches −1. This result comes
+from the fact that K+Σ−(1
+2−) is produced mainly from
+the t-channel kaon exchange process. For the t-channel
+kaonexchangeprocess, the photon spin isnecessaryto be
+vertical to the reaction plane due to angular momentum
+conservation, and such photon corresponds to the beam
+(electromagnetic field) with polarization parallel to the
+reaction plane, i.e.,σ⊥= 0. Thus the K+Σ−(1
+2−) pro-
+duction contributes negatively to the beam asymmetry.
+If a portion of the detected Λ πstems from Σ(1
+2−), the
+measured beam asymmetry can be pulled to the negative
+side, and the observed negative beam asymmetry may be
+explained.
+In Fig. 3, the differential cross sections for the process
+γn→K+Σ∗−with respect to cos θc.m.are shown and
+compared with the LEPS data [9]. The solid lines are the
+results of scheme I, and the dashed lines are the results/s49/s46/s54 /s49/s46/s56 /s50/s46/s48 /s50/s46/s50 /s50/s46/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48
+/s69 /s40 /s71/s101/s86 /s41/s99/s111/s115
+/s99/s46/s109/s46/s61/s48/s46/s57/s45/s49/s46/s48
+/s32/s32/s32
+/s32/s49/s46/s54 /s49/s46/s56 /s50/s46/s48 /s50/s46/s50 /s50/s46/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48/s32
+/s32/s99/s111/s115
+/s99/s46/s109/s46/s61/s48/s46/s55/s45/s48/s46/s56
+/s32/s32
+/s49/s46/s54 /s49/s46/s56 /s50/s46/s48 /s50/s46/s50 /s50/s46/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48/s32
+/s32/s100 /s100/s99/s111/s115
+/s99/s46/s109/s46/s40 /s98 /s41
+/s99/s111/s115
+/s99/s46/s109/s46/s61/s48/s46/s56/s45/s48/s46/s57
+/s69 /s40 /s71/s101/s86 /s41
+/s32/s32/s49/s46/s54 /s49/s46/s56 /s50/s46/s48 /s50/s46/s50 /s50/s46/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48
+/s32/s32/s100 /s100/s99/s111/s115
+/s99/s46/s109/s46/s40 /s98 /s41
+/s99/s111/s115
+/s99/s46/s109/s46/s61/s48/s46/s54/s45/s48/s46/s55/s32
+/s32
+FIG. 3: Differential cross sections for γ+n→K++ Σ(∗)−
+with four cos θc.m.intervals. Theoretical results with scheme
+I (solid curves) and scheme II (dashed curves) are compared
+with the LEPS data [9]. The dot-dashed curves demonstrate
+the contributions from Σ(1
+2−) production in scheme I.
+of scheme II. The contributions from Σ(1
+2−) production
+in scheme I are also shown by the dash-dotted lines.
+In Fig. 4 and Fig. 5, the results for the linear beam
+asymmetry and integrated cross sections of the reaction
+γn→K+Σ∗−are shown and compared with the LEPS
+data [9], respectively. The cross sections are intergrated
+resultsinphasespacecos θc.m.= 0.6−1.0. Thesolidlines
+are the results of scheme I, i.e., including both Σ∗(3
+2+)
+production with h= 1 and the contribution from Σ(1
+2−).
+The dashed lines are the results of scheme II, i.e., for
+Σ∗(3
+2+) production alone with h= 1.11. The dotted
+lines are the results with the h= 1 and without the
+contribution from Σ(1
+2−). The contribution from Σ(1
+2−)
+production to the cross section alone is shown by the
+dash-dotted line in Fig. 5.
+From Fig.3-5, one can see that both schemes can well
+describe the LEPS data. For scheme I, since the Σ(1
+2−)
+alone gives a large negative beam asymmetry approach-
+ing−1 as shown in Fig. 2, a portion of Σ(1
+2−) in this pro-
+cess helps to explain the observed negative beam asym-
+metry. At the same time, we find that tuning the pa-6
+/s49/s46/s54 /s49/s46/s56 /s50/s46/s48 /s50/s46/s50 /s50/s46/s52 /s50/s46/s54 /s50/s46/s56 /s51/s46/s48 /s51/s46/s50 /s51/s46/s52 /s51/s46/s54 /s51/s46/s56 /s52/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56
+/s99/s111/s115
+/s99/s109
+/s32/s32/s66/s101/s97/s109/s32/s65/s115/s121/s109/s109/s101/s116/s114/s121
+/s69 /s40/s71/s101/s86/s41
+FIG. 4: The linear beam asymmetry for γ+n→K++Σ(∗)−
+with cos θc.m.integrated from 0.6 to 1. Results of scheme
+I (solid curve), and scheme II (dashed curve) are compared
+with the LEPS data [9]. The dotted curve demonstrate the
+result with h= 1 and without Σ(1
+2−) contribution.
+/s49/s46/s54 /s49/s46/s56 /s50/s46/s48 /s50/s46/s50 /s50/s46/s52 /s50/s46/s54 /s50/s46/s56 /s51/s46/s48 /s51/s46/s50 /s51/s46/s52 /s51/s46/s54 /s51/s46/s56 /s52/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54
+/s32/s32/s40 /s110/s45 /s62 /s32 /s41/s40 /s98/s41
+/s69 /s40/s71/s101/s86/s41/s99/s111/s115
+/s99/s46/s109/s46
+FIG. 5: Integrated cross sections for γ+n→K++Σ(∗)−with
+cosθc.m.= 0.6−1.0 of scheme I (solid curve) and scheme
+II (dashed curve), compared with the LEPS data [9]. The
+dotted and dot-dashed curves demonstrate the contribution s
+from Σ∗(3
+2+) and Σ(1
+2−), respectively, in the scheme I.
+rameter hto a larger value as scheme II can also have
+such an effect. The χ2for the two schemes are 97 and
+102, respectively, compared to the 39 experimental data
+points in Figs. 3 & 4. As a comparison, the correspond-
+ing previous theoretical prediction [9] without including
+the two new ingredients give a χ2value about 180.
+From the comparison of the results with the LEPS
+data, one can not tell for sure which one of the two
+schemes is better. In fact, some other combinations of h
+andgKNΣ, suchas h= 1.06combinedwith gKNΣ= 1.04,
+may also describe the present data. However, one can/s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s49/s46/s49/s49/s46/s50
+/s32/s32/s67/s114/s111/s115/s115/s32/s83/s101/s99/s116/s105/s111/s110/s32/s40 /s98/s41
+/s69 /s40/s71/s101/s86/s41
+FIG. 6: Total cross sections for γ+p→K++ Σ(∗)0of
+scheme I (solid curve) and scheme II (dashed curve), com-
+pared with the CLAS preliminary data [2]. The dotted
+and dot-dashed curves demonstrate the contributions from
+Σ∗(3
+2+) and Σ(1
+2−), respectively, in the scheme I.
+see from Fig. 5 that for higher incident energy, the
+two schemes give very distinctive predictions. Thus the
+schemes and mechanisms can be distinguished when data
+on reaction γn→K+Σ∗−at higher incident energies are
+available.
+The above results show that while the original theoret-
+ical prediction with Σ∗(3
+2+) production alone with h= 1
+fails to reproduce the data for γn→K+Σ∗−[9, 12],
+the data can be well reproduced either by increasing h
+to 1.11 or by including an additional Σ(1
+2−) resonance
+around 1380 MeV. Since the parameters for the original
+theoretical prediction come from the fits to the data on
+theγp→K+Σ∗0process, we need to check how the new
+sets of parameters fit to the γp→K+Σ∗0data. The
+results are shown in Fig. 6 for the total cross sections
+of the reaction γp→K+Σ∗0with the above sets of pa-
+rameters, comparing with the CLAS data [2]. Again,
+the solid line is the result for Σ∗(3
+2+) production with
+h= 1 combined with the contribution from Σ(1
+2−) with
+gKNΣ= 1.34; the dotted line and the dashed line are
+the results for Σ∗(3
+2+) production alone with h= 1 and
+h= 1.11, respectively. The Σ(1
+2−) contribution is also
+shown by the dash-dotted line with gKNΣ= 1.34. From
+Fig. 6 one sees that the combined result of Σ∗(3
+2+) pro-
+duction with h= 1 and the Σ(1
+2−) production (scheme
+I, solid line) can be compatible with the CLAS data on
+the whole, while the results with h= 1.11 (dashed line)
+deviate from the data for larger incident energy.
+Our results show that the combined results of Σ∗(3
+2+)
+andΣ(1
+2−)production(solidline)canbecompatiblewith
+both the γn→K+Σ∗−reaction data and the γp→
+K+Σ∗0reaction data with the same set of parameters7
+(scheme I). Without the Σ(1
+2−) contribution, different
+parameter hneed to be used for γn→K+Σ∗−andγp→
+K+Σ∗0processes.
+The properties and couplings for Σ(1
+2−), the contact
+term, and the s-channel resonances still have some un-
+certainties and need further investigation. More data of
+broader energy range on these processes will be helpful
+to clarify the roles of the contact term and the Σ(1
+2−)
+resonance.
+IV. SUMMARY
+In summery, we study the process of KΣ∗(1385) pho-
+toproduction from the nucleons in the framework of the
+effective Lagrangian method. From recent studies, there
+is possibly a Σ state with Jp=1
+2−near 1380 MeV. We
+consider the case that Σ(1
+2−) may contribute to the ob-
+servables of KΣ∗(1385) photoproduction in experiments.
+Our results show that the Σ(1
+2−) production can give
+large negative contribution to beam asymmetry, which
+helps to explain the large negative linear beam asymme-try observed by the LEPS experiment. With a portion of
+Σ(1
+2−), the same set of parameters can reproduce both
+the data of γn→K+Σ∗−from the LEPS experiment
+and the data of γp→K+Σ∗0from the CLAS experi-
+ment. On the other hand, without including Σ(1
+2−), the
+presentdataon γn→K+Σ∗−canbedescribedbytuning
+a parameter hin the contact term, which means differ-
+ent choices of parameter hin the contact term for γn
+andγpreactions are needed. To distinguish the roles of
+Σ(1
+2−) and the contact term, different predictions of the
+two schemes are presented for future experimental study.
+Acknowledgments
+We thank Kanzo Nakayama and Yongseok Oh for the
+helpful discussions. This project is supported by the Na-
+tional Natural Science Foundation of China under Grant
+10905059, 10875133, 10821063, and by China Postdoc-
+toralScience Foundation and the Ministry of Science and
+Technology of China (2009CB825200).
+[1] R. Bradford et al.(CLAS Collaboration), Phys. Rev. C
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+[2] L.GuoandD.P.Weygand(CLASCollaboration), in Pro-
+ceedings of the International Workshop on the Physics of
+Excited Baryons (NSTAR05) , edited by S. Capstick, V.
+Crede, andP.Eugenio(WorldScientific, Singapore, 2006,
+pp. 306-309.
+[3] I.Hleiqawi et al.(CLASCollaboration), Phys.Rev.C 75,
+042201(R) (2007); Phys. Rev. C 76, 039905(E) (2007).
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\ No newline at end of file
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@@ -0,0 +1,478 @@
+arXiv:1001.0807v2  [astro-ph.CO]  16 Apr 20101001.0807
+Probing the cosmic acceleration from combinations of differ ent
+data sets
+Yungui Gong,1,∗Bin Wang,1,2,†and Rong-Gen Cai1,3,‡
+1College of Mathematics and Physics,
+Chongqing University of Posts and Telecommunications, Cho ngqing 400065, China
+2Department of Physics, Fudan University, Shanghai 200433, China
+3Institute of Theoretical Physics, Chinese Academy of Scien ces,
+Beijing 100190, China
+Abstract
+We examine in some detail the influence of the systematics in d ifferent data sets including
+type Ia supernova sample, baryon acoustic oscillation data and the cosmic microwave background
+information on the fitting results of the Chevallier-Polars ki-Linder parametrization. We find that
+the systematics in the data sets does influence the fitting res ults and leads to different evolutional
+behavior of dark energy. To check the versatility of Chevall ier-Polarski-Linder parametrization, we
+also perform the analysis on the Wetterich parametrization of dark energy. The results show that
+both the parametrization of dark energy and the systematics in data sets influence the evolutional
+behavior of dark energy.
+PACS numbers: 95.36.+x, 98.80.Es
+∗Electronic address: gongyg@cqupt.edu.cn
+†Electronic address: wangb@fudan.edu.cn
+‡Electronic address: cairg@itp.ac.cn
+1I. INTRODUCTION
+The convincing fact that our universe is experiencing accelerated e xpansion [1, 2] has
+become one of the most important and mysterious issues of modern cosmology. In the
+framework of general relativity, the cosmic acceleration is attribu ted to an exotic energy
+called dark energy (DE). The simplest candidate of the DE is the cosm ological constant
+with equation of state (EOS) w=−1. Despite some severe problems such as the cosmolog-
+ical constant problem and the coincidence problem, the cosmologica l constant is favored by
+current astronomical observations. Besides, there aresome ot her DE models with EOS vary-
+ing with cosmic time either above −1 or below −1. Interestingly, some current observations
+even give the hint that EOS of DE has crossed −1 at least once [3, 4]. To narrow down the
+DE candidates, one has to examine the EOS carefully by confronting different observational
+data sets including the type Ia supernova (SNIa) luminosity distanc e, the cosmic microwave
+background (CMB) temperature anisotropy and the baryon acou stic oscillations (BAO) in
+the galaxy power spectrum, and this could be the best what we can d o to approach the
+truth of DE.
+In addition to the EOS, a new diagnostic of DE was introduced in [5] whic h is called
+Omdiagnostic. It is a combination of the Hubble parameter and the cosm ological redshift,
+which depends upon the first derivative of the luminosity distance an d is less sensitive to
+observational errors than the EOS. If the value of Omis the same at different redshifts, then
+the DE is the cosmological constant. The slope of Omcan differentiate between different
+DE models with w >−1 orw <−1 even if the value of the matter density is not accurately
+known.
+Analyzing the Constitution SNIa sample [6] together with the BAO dat a [7, 8] by using
+the popular Chevallier-Polarski-Linder (CPL) parametrization [9, 10 ], it was revealed that
+there appears the increase of Omandwat redshifts z <0.3 [11]. This suggests that
+the cosmic acceleration may have already been over the peak and no w the acceleration is
+slowing down. However, including the CMB shift parameter which is ano ther independent
+observable at high redshift, it was found that the result changes d ramatically and the value
+ofOmbecomes un-evolving which is consistent with the ΛCDM model. Furthe r check shows
+that the CPL ansatz is unable to fit the data simultaneously at low and high redshifts. It
+was argued that this could either due to the systematics in some of t he data sets which is not
+2sufficiently understood or because the CPL parametrization is not v ersatile to accommodate
+the cosmological evolution of DE suggested by the data [11]. The dat a sets used in the
+analysis of [11] are limited to the Constitution set of 397 SNIa in combin ation with BAO
+distance ratio of the distance measurements obtained at z= 0.2 andz= 0.35 and the CMB
+shift parameter. It isexpected thatif morecombinations of new da tasets areincluded, these
+arguments can be clarified. In [4], it was showed that the result of th e analysis on the CPL
+model [11] heavily depends on the choice of BAO data. The result obt ained by using the
+BAO distance ratio data was found not consistent with that by using other observational
+data, and this inconsistency can be overcome if the BAO Aparameter [12] is employed
+instead [4]. In this work we are going to investigate this problem furth er by comparing
+different data set combinations among SNIa, BAO and CMB.
+II. OBSERVATIONAL DATA
+For the SNIa data, we use the Constitution sample [6] and the first y ear Sloan digital
+sky survey-II (SDSS-II) SNIa (hereafter Sdss2) [13]. The Cons titution sample consists of
+the Union sample [14] together with 185 CfA3 SNIa data, which totally contains 397 SNIa.
+The CfA3 addition makes the cosmologically useful sample of nearby S NIa much larger than
+before, which reduces the statistical uncertainty to the point wh ere systematics plays the
+largest role. To test the systematic differences and consistencies , in [6] four light curve
+fitters,SALT,SALT2,MLCS2k2 withRV= 3.1 (MLCS31), MLCS2k2 withRV= 1.7
+(MLCS17), have been used. For the Constitution SNIa data using t he template SALT
+(hereafter Csta), the intrinsic uncertainty of 0.138 mag for each CfA3 SNIa, the peculiar
+velocity uncertainty of 400km/s, and the redshift uncertainty ha ve been considered [6].
+These data were suggested by observers to be the best data for model independent analysis
+of the expansion history. The Constitution SNIa data using the tem plate SALT2 (hereafter
+Cstb), excludes the SNIa with z <0.01 ort1st>10d, so it has 351 SNIa data. Using the
+template MLCS17 on the Constitution data (hereafter Cstc), the SNIa with Av≥1.5 and
+t1st>10dhas been cut out and it excludes those SNIa whose MLCS17 fit has a r educed
+χ2
+νbeing 1.6 or higher, thus Cstc has only 372 SNIa data. The Cstd samp le is formed by
+using the template MLCS31 on the Constitution sample and cutting ou t the SNIa with
+Av≥1.5 andt1st>10d. It excludes any SNIa whose MLCS31 fit has a reduced χ2
+ν
+3being 1.5 or higher, so that it contains 366 SNIa data. We will examine t he systematic
+differences brought by the Constitution sample with different light cu rve fitters. In addition
+we will also consider Sdss2 sample and investigate the systematic diffe rence it brought, to
+compare with the Constitution sample. The Sdss2 consists of 103 ne w SNIa with redshifts
+0.04< z <0.42, discovered during the first season of the SDSS-II supernova survey, 33
+nearby SNIa with redshfits 0 .02< z <0.1, 56 SNIa with redshifts 0 .16< z <0.69 from
+ESSENCE [15, 16], 62 with redshifts 0 .25< z <1.01 SNIa from SNLS [17], and 34 SNIa
+with redshifts 0 .21< z <1.55 from HST. For simplicity, we only use the Sdss2 data with the
+templateMLCS2K2. Forthe288Sdss2 SNIadata, wealsoconsider t heredshiftuncertainties
+from spectroscopic measurements and from peculiar motions of th e host galaxy. For the
+redshift uncertainty, σz,pec, arises from peculiar velocities of and within host galaxies, we
+takeσz,pec= 0.0012. In addition, the intrinsic error of 0.16 mag is added to the unce rtainty
+of the distance modulus [13].
+To fit the SNIa data, we define
+χ2(p,H0) =/summationdisplay
+i=1[µobs(zi)−µ(zi,p,H0)]2
+σ2
+i, (1)
+where the extinction-corrected distance modulus µ(z,p,H0) = 5log10[dL(z)/Mpc] + 25, σi
+is the total uncertainty in the SNIa data, and the luminosity distanc e for a flat universe is
+dL(z,p,H0) =1+z
+H0/integraldisplayz
+0dx
+E(x), (2)
+where the dimensionless Hubble parameter E(z) =H(z)/H0. Because the normalization
+of the luminosity distance is unknown, the nuisance parameter H0in the SNIa data is not
+the observed Hubble constant. We marginalize over the nuisance pa rameterH0with a flat
+prior, which leads to [18]
+χ2
+sn(p) =/summationdisplay
+i=1α2
+i
+σ2
+i−(/summationtext
+iαi/σ2
+i−ln10/5)2
+/summationtext
+i1/σ2
+i−2ln/parenleftBigg
+ln10
+5/radicalBigg
+2π/summationtext
+i1/σ2
+i/parenrightBigg
+, (3)
+whereαi=µobs(zi)−25−5log10[H0dL(zi)].
+Besides considering these SNIa data sets individually, our analysis will also investigate
+the combination of SNIa data with the BAO data. For the BAO data, w e first use the BAO
+distance measurements obtained at z= 0.2 andz= 0.35 from joint analysis of the 2dFGRS
+and SDSS data [8]. Defining dz(p,H0) =rs(zd)/DV(z), where the comoving sound horizon
+4and effective distance are
+rs(z,p) =/integraldisplay∞
+zdx
+cs(x)E(x), (4)
+DV(z,p,H0) =/bracketleftbiggd2
+L(z)
+(1+z)2z
+H(z)/bracketrightbigg1/3
+=H−1
+0/bracketleftBigg
+z
+E(z)/parenleftbigg/integraldisplayz
+0dx
+E(x)/parenrightbigg2/bracketrightBigg1/3
+. (5)
+The redshift zdat the baryon drag epoch is fitted with the formula [19]
+zd=1291(Ω m0h2)0.251
+1+0.659(Ω m0h2)0.828[1+b1(Ωbh2)b2], (6)
+b1= 0.313(Ω m0h2)−0.419[1+0.607(Ω m0h2)0.674], b2= 0.238(Ω m0h2)0.223,(7)
+where Ω m0is the current value of the dimensionless matter energy density, Ω bis the di-
+mensionless baryon matter energy density, and h=H0/100. The sound speed cs(z) =
+1//radicalbig
+3[1+¯Rb/(1+z)], where ¯Rb= 315000Ω bh2(Tcmb/2.7K)−4andTcmb= 2.726K. In [8], two
+pointsd0.2andd0.35andtheircovariancematrixaregiven. Wewill use d0.2= 0.1905±0.0061,
+d0.35= 0.1097±0.0036 and their covariance matrix (hereafter Bao2) as the second sample of
+BAO data in the data fitting. When we use these data, we add two mor e parameters Ω bh2
+andh. Based on these two BAO distance measurements, we can derive th e BAO distance
+ratioDV(0.35)/DV(0.2) = 1.736±0.065 (hereafter BaoR), which is relatively model inde-
+pendent quantity. This BAO distance ratio was employed in the analys is in [11]. In our case
+we will further use another two radial BAO data at redshifts z= 0.24 andz= 0.43 [20] by
+using ∆z(z) =H(z)rs(zd)/c(hereafter BaoZ). Combining BAO data sets Bao2 and BaoZ,
+we have the data set called Bao4. Therefore, for different BAO dat a sets, we can perform
+χ2statistics for the model parameter pas follows
+χ2
+Baor(p) =[DV(0.35)/DV(0.2)−1.736]2
+0.0652, (8)
+χ2
+BaoZ(p,h,Ωbh2) =[∆z(0.24)−0.0407]2
+0.00112+[∆z(0.43)−0.0442]2
+0.00152, (9)
+and
+χ2
+Bao2(p,h,Ωbh2) =/summationdisplay
+ij∆diCov−1(di,dj)∆dj, (10)
+wherei,j= 0.2,0.35, ∆d0.2=d0.2−0.1905, ∆d0.35=d0.35−0.1097.
+Since the SNIa and BAO data contain information about the universe at relatively low
+redshifts, we will include the CMB information by implementing the Wilkins on microwave
+anisotropy probe 5 year (WMAP5) data to probe the entire expans ion history up to the last
+5scattering surface. In addition to employing the CMB shift paramet erRby defining the
+reduced distance in the flat universe to the last scattering surfac ez∗as done in [11]
+R(p) =/radicalbig
+Ωm0/integraldisplayz∗
+0dz
+E(z)= 1.71±0.019, (11)
+we will add the acoustic scale lain the data analysis. The acoustic scale lais defined by
+la=πdL(z∗)
+(1+z∗)rs(z∗)= 302.1±0.86, (12)
+where the redshift z∗is given by [21]
+z∗= 1048[1+0 .00124(Ω bh2)−0.738][1+g1(Ωm0h2)g2] = 1090.04±0.93,(13)
+g1=0.0783(Ω bh2)−0.238
+1+39.5(Ωbh2)0.763, g2=0.560
+1+21.1(Ωbh2)1.81. (14)
+We have three fitting parameters xi= (R, la, z∗) to include the CMB information now.
+Thus we have χ2
+cmb(p,h,Ωbh2) =/summationtext
+ij∆xiCov−1(xi,xj)∆xj, ∆xi=xi−xobs
+iand Cov( xi,xj)
+is the covariance matrix for the three parameters [22]. The CMB inf ormation provides a
+systematic check of the DE model by combining with the low redshift S NIa and BAO data
+sets.
+III. FITTING RESULTS
+We employ the Monte-Carlo Markov Chain (MCMC) method to explore t he parameter
+spacepand the nuisance parameters hand Ω bh2in the data analysis. Our MCMC code
+[18] is based on the publicly available package COSMOMC [23].
+We first explore the widely used CPL parametrization [9, 10]
+w(z) =w0+waz
+1+z. (15)
+In this model, the dimensionless Hubble parameter for a flat universe is
+E2(z) =H2(z)
+H2
+0= Ωm0(1+z)3+(1−Ωm0)(1+z)3(1+w0+wa)exp(−3waz/(1+z)).(16)
+In this model, we have three parameters p= (Ωm0, w0, wa). Knowing the expansion history
+of the universe, we can construct the Omdiagnostic by defining
+Om(z) =E2(z)−1
+(1+z)3−1. (17)
+6−1.5 −1 −0.5 0 0.5 1−25−20−15−10−50
+w0waSN
+SN+BaoR
+SN+Bao2
+SN+BaoZ
+SN+Bao4
+SN+Bao4+wmap
+(a)−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−40−35−30−25−20−15−10−505
+w0waSN
+SN+BaoR
+SN+Bao2
+SN+BaoZ
+SN+Bao4
+SN+Bao4+wmap
+(b)
+−1.4−1.2 −1−0.8−0.6−0.4−0.2 00.20.40.6−20−15−10−50
+w0waSN
+SN+BaoR
+SN+Bao2
+SN+BaoZ
+SN+Bao4
+SN+Bao4+wmap
+(c)−1.5 −1 −0.5−10−8−6−4−202
+w0wa
+SN
+SN+BaoR
+SN+Bao2
+SN+BaoZ
+SN+Bao4
+SN+Bao4+wmap
+(d)
+FIG. 1: The marginalized 1 σcontours of w0andwain the CPL model. The green line is for SNIa,
+the dashed black line is for SNIa+BaoR, the cyan line is for SN Ia+Bao2, the magenta line is for
+SNIa+BaoZ, the blue line is for SNIa+Bao4, and the red line is for SNIa+Bao4+WMAP5. The
+SNIa used in (a)-(d) are Csta, Cstb, Cstc, and Cstd, respecti vely.
+Omdiagnostic is useful in establishing the properties of DE at low redshif ts. The constant
+Omindicates that the DE is the cosmological constant and the bigger va lue ofOmshows
+thatwis bigger [5]. Om(z) is less sensitive to observational errors than EOS w(z). Due to
+the degeneracy between Ω m0andw, the property of w(z) depends on the current value of
+matter energy density Ω m0. However, Omdiagnostic provides a null test of the cosmological
+constant without knowing the exact value of Ω m0. To reconstruct Om(z), we need to apply
+the specific model and model parameters. Following [11], we consider the uncertainties of
+Ωm0,w0andwawhen we we reconstruct Om(z).
+7−3 −2.5 −2 −1.5 −1 −0.5 0−20−15−10−505
+w0wa
+SN
+SN+BaoR
+SN+Bao2
+SN+BaoZ
+SN+Bao4
+SN+Bao4+wmap
+(a)−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.500.20.40.60.811.21.41.61.82
+w0awaSN
+SN+BaoR
+SN+Bao2
+SN+BaoZ
+SN+Bao4
+SN+Bao4+wmap
+−20 −15 −10 −5 000.511.52
+(b)
+−5−4.5 −4−3.5 −3−2.5 −2−1.5 −1−0.5 000.511.522.5
+w0awaSN
+SN+BaoR
+SN+Bao2
+SN+BaoZ
+SN+Bao4
+SN+Bao4+wmap
+−20 −15 −10 −5 000.511.522.5
+(c)−5−4.5 −4−3.5 −3−2.5 −2−1.5 −1−0.5 000.511.522.533.544.55
+w0awaSN
+SN+BaoR
+SN+Bao2
+SN+BaoZ
+SN+Bao4
+SN+Bao4+wmap
+−20 −15 −10 −5 0012345
+(d)
+FIG. 2: The marginalized 1 σcontours of w0(wa
+0) andwa. The label of contours is the same as
+Fig. 1. (a) is for the CPL model with Sdss2 SNIa data. (b)-(d) a re for the Wetterich model. The
+SNIa used in (b)-(d) are Csta, Cstd and Sdss2, respectively.
+Fig. 1 shows contours of the fitting results for the CPL parametriz ation by combining
+different SNIa data together with different BAO data and the combin ation of the WMAP5
+data. The SNIa data used in Figs. 1a-1d and 2a are Csta, Cstb, Cst c, Cstd, and Sdss2,
+respectively. The greenlines areforusing just SNIa data alonewith different templates. The
+dashedblacklines areforSN+BaoR,thecyanlinesareforSN+Bao2. SincetheBaoRdatais
+derived from Bao2 data, so the result using BaoR data is compatible w ith that of Bao2. Also
+we can see that the constraint is alittle better with Bao2thanthat w ith BaoR. The magenta
+lines are for SN+BaoZ, we see that the constraint from BaoZ is in gen eral consistent with
+that from Bao2 although the former gives a little tighter constraint . These behaviors keep
+800.20.40.60.8 100.20.40.60.81
+(a)          zOmz
+00.20.40.60.8 100.20.40.60.81
+(b)          zOmz
+00.20.40.60.8 100.20.40.60.81
+(c)          zOmz
+00.20.40.60.8 100.20.40.60.81
+(d)          zOmz
+FIG. 3: The marginalized 1 σand 2σerrors of Om(z) reconstructed in the CPL model by using the
+Csta SNIa data. (a) uses the combination of SNIa+BaoR, (b) us es the combination of SNIa+BaoZ,
+(c) uses the combination of SNIa+Bao4 and (d) uses the combin ation of SNIa+Bao4+WMAP5.
+the same for SNIa data with different templates. The blue lines are fo r SN+Bao4, where
+Bao4 is the combination of Bao2+BaoZ. The combination of the SN+Ba o4+WMAP5 is
+shown in the red solid lines. The observation that the CPL ansatz is st rained to describe
+the DE behavior suggested by data at low and high redshifts by comp aring Csta+BaoR and
+Csta+BaoR+CMB shift parameter [11] has been reduced by using th e same SN data (Csta)
+with BaoZ or Bao4, or other SN data sets (Cstb-Cstd, Sdss2) with arbitrary combinations
+of BAO data. This confirms that the systematics in some of the data sets really matters the
+fitting results.
+For the Csta data, the reconstructed Om(z) for SNIa data in combination with BAO and
+CMB data is shown in Fig. 3. Since Om(z) is not a constant at 1 σlevel if we combine
+900.20.40.60.8 100.20.40.60.81
+(a)          zOmz
+00.20.40.60.8 100.20.40.60.81
+(b)          zOmz
+00.20.40.60.8 100.20.40.60.81
+(c)          zOmz
+00.20.40.60.8 100.20.40.60.81
+(d)          zOmz
+FIG. 4: The marginalized 1 σand 2σerrors of Om(z) reconstructed in the CPL model with
+SNIa+BaoR. The SNIa data used in (a)-(d) are Cstb, Cstc, Cstd , and Sdss2, respectively.
+Cata SNIa with BaoR, BaoZ or Bao4, so the ΛCDM model is excluded at 1σlevel if use the
+combination of Csta SNIa and BAO data. However, when the WMAP5 d ata is added, the
+ΛCDM model is consistent at 1 σlevel. The growth in the value of Om(z) at low redshifts
+becomes smaller when we change the BAO data from BaoR to BaoZ or B ao4. In Fig. 4,
+we show the marginalized 1 σand 2σerrors of Om(z) reconstructed using the BaoR and
+different SNIa data sets. The growth in the value of Om(z) at low redshifts in Fig. 3a by
+using Csta+BaoR as observed in [11] gets flattened if we change SNI a data sets from Csta
+to Cstb, Cstc, Cstd and Sdss2. This shows that the finding in [11] is n ot a general behavior
+even at low z. The evolutional behavior of Om(z) at low redshifts is changed for different
+selections of the SNIa data. Thus the striking observation that ou r universe is slowing down
+[11] based on the fitting at low redshifts for CPL ansatz is caused by the systematics of
+the specially chosen SNIa and BAO data sets. By choosing some othe r data sets, the CPL
+10parametrization is in compatible with combinations of data sets at low a nd high redshifts.
+The un-evolving Om(z) is also allowed even at low redshifts. From Figs. 3a and 4, we see
+that the ΛCDM model is not allowed at 1 σlevel for the combination of BaoR and Csta,
+Cstb or Cstc, although the cosmological constant is inside the 1 σcontours of w0andwain
+Figs. 1b and 1c. This shows the advantage of Omdiagnostic because the uncertainty of
+Ωm0is taken accounted for.
+In the above discussions we have focused on the commonly used fun ctional form for DE,
+the CPL parametrization, andarguedthat the systematics inSNIa andBAO data sets really
+affects the evolution of the DE and the acceleration of the universe . Besides the systematics
+ofdatasets, whethertheinfluencesontheevolutionoftheDEand theaccelerationarecaused
+by the specific choice of the parametrization of the DE is another int eresting question to
+ask. In [11], the incompatibility of the CPL ansatz fitting to the data s imultaneously at low
+and high redshifts was attributed to the versatility of the CPL para metrization. Choosing
+another ansatz of DE parametrization, it was found significantly be tter than that of CPL,
+that ansatz can provide a good fit to the combination of Csta+BaoR and the CMB shift
+data. In the rest of this work we are going to further examine the v ersatility of the CPL
+parametrization by considering a different ansatz of DE parametriz ation and confronting it
+with different combinations of various data sets as used above.
+What we are going to study is the Wetterich parametrization of the f orm [24]
+w(z) =w0
+1+waln(1+z). (18)
+Whenz= 0,w(z) =w0, and at large redshift z≫1,w(z)≈0. The Wetterich parametriza-
+tion becomes the constant parametrization w(z) =w0whenwa= 0.
+In this model, the dimensionless Hubble parameter for a flat universe is
+E2(z) = Ωm0(1+z)3+(1−Ωm0)(1+z)3[1+waln(1+z)]3w0/wa. (19)
+To ensure the positivity of DE, we require that wa≥0. For the convenience of numerical
+calculation, we take the independent parameters as wa
+0=w0/waandwa. The parameters
+in this model are p= (Ωm0, wa
+0, wa), the number is the same as those in the case of CPL
+ansatz.
+The compatibility checks in confronting with different SNIa, BAO data sets together with
+CMB data are shown in Figs. 2b-2d. In Figs. 2b-2d, we have used the Csta, Cstd and Sdss2
+1100.20.40.60.8 100.10.20.30.40.5
+(a)          zOmz
+00.20.40.60.8 100.10.20.30.40.5
+(b)          zOmz
+00.20.40.60.8 100.10.20.30.40.5
+(c)          zOmz
+00.20.40.60.8 100.10.20.30.40.5
+(d)          zOmz
+FIG. 5: The marginalized 1 σand 2σerrors of Om(z) reconstructed in the Wetterich model. (a)
+uses the combination of Csta SNIa+BaoR, (b) uses the combina tion of Csta SNIa+BaoZ, (c) uses
+the combination of Sdss2 SNIa+BaoR, and (d) uses the combina tion of Sdss2 SNIa+BaoZ.
+SNIa data sets, respectively. As indicated in Fig. 1 and Fig. 2, the gr een lines are just
+for SNIa data alone, the dashed black lines are for SN+BaoR, the cy an lines are results
+for SNIa+Bao2, the magenta lines indicate SNIa+BaoZ, the blue lines are for SNIa+Bao4,
+and the solid red lines are for the combination of SNIa+Bao4+WMAP5. It is interesting
+to see that different from the CPL ansatz, this parametrization allo ws the compatibility of
+differentSNIa, BaoandCMBdatasets, evenforcomparingthecom binationsofCsta+BaoR,
+Csta+BaoZ and Csta+Bao4+WMAP5. The Wetterich ansatz is found able to fit different
+data sets both at low and high redshifts. In Fig. 5, we present the b ehaviors of the Om(z)
+reconstructed bycomparing theinfluence ofdifferent SNIa andBA O datasets. Interestingly,
+the evolution of Om(z) is not affected much by different SNIa and BAO data sets. The
+12Om(z) is perfectly consistent with ΛCDM for different data sets.
+Forthecomparisonofdifferentdatasetsanddifferentparametriz ations, wesummarizethe
+minimum value of χ2in Table I. From Table I, we observe the reliability of parametrizations
+when using different data sets. Comparing two different parametriz ations, we see that both
+of them can fit well of the data, while the CPL model is a little better th an the Wetterich
+parametrization for the Csta SNIa and the combination of Csta SNI a with BAO data.
+TABLE I: The minimum value of χ2for different combinations of data sets and models.
+Data CPL (χ2/DOF)Wetterich
+Csta 462.06/394 466.33/394
+Csta+BaoR 462.43/395 467.62/395
+Csta+Bao2 462.45/394 467.73/394
+Csta+BaoZ 462.10/394 466.60/394
+Csta+Bao4 464.11/396 468.57/396
+Csta+Bao4+WMAP5 468.73/399 468.69/399
+Sdss2 227.55/285 227.09/285
+Sdss2+BaoR 229.77/286 229.45/286
+Sdss2+Bao2 229.93/285 229.65/285
+Sdss2+BaoZ 228.77/285 228.57/285
+Sdss2+Bao4 231.15/287 231.02/287
+Sdss2+Bao4+WMAP5 231.71/290 231.94/290
+IV. CONCLUSION
+To summarize, we have examined the influence of the systematics in d ifferent data sets in
+SNIa and BAO on the fitting results of the CPL parametrization. We f ound that the tension
+observed in [11] between low z(Csta+BaoR) and the high z(CMB) data is not a general
+behavior. By using SNIa with other templates and other BAO data se ts, the incompatibility
+of the CPL parametrization will disappear. This result supports the speculation that the
+systematics inthedatasetscanaffectthefittingresultsandleads todifferent evolutionofthe
+DE model [11]. However this answer is still not definite. The different e volutions in Om(z)
+13as we observed by using different combinations of variousSNIa and B AO data sets disappear
+when we use the Wetterich parametrization to replace the CPL para metrization. This again
+brings the problem as raised in [11] that the CPL parametrization migh t be blamed to be
+not so versatile. In order to disclose the exact answer, we have to examine more models
+of DE. This work offers the attempt of using different combinations o f various data sets at
+low and high redshifts to examine the effects of data systematics on the evolution of the DE
+and the acceleration of the universe. Further investigation along t his line by including more
+data sets and examining more DE models are called for and we will repor t our progress in
+this direction in the future work.
+Acknowledgments
+RGCandBWthankChongqingUniversityofPostsandTelecommunicat ions forthewarm
+hospitality during their visits. This work was partially supported by NN SF of China (Nos.
+10821504, 10878001, 10975168 and 10935013) and the National Basic Research Program of
+China under grant No. 2010CB833004. YG was partially supported b y the Natural Science
+Foundation Project of CQ CSTC under grant No. 2009BA4050.
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+arXiv:1001.0808v1  [cond-mat.str-el]  6 Jan 2010Metal-insulatortransitionofthe Kagom ´elatticefermions at1/3filling
+Satoshi Nishimoto,1Masaaki Nakamura,2Aroon O’Brien,3and Peter Fulde3,4
+1Leibniz-Institut f¨ ur Festk¨ orper- und Werkstoffforschu ng Dresden, D-01171 Dresden, Germany
+2Department of Physics, Tokyo Institute of Technology, Toky o 152-8551, Japan
+3Max-Planck-Institut f¨ ur Physik komplexer Systeme, N¨ oth nitzer Straße 38, D-01187 Dresden, Germany
+4Asia Pacific Center for Theoretical Physics, Pohang, Korea
+(Dated: October 28, 2018)
+We discuss the metal-insulator transition of the spinless f ermion model on the Kagom´ e lattice at 1/3-filling.
+The system is analyzed using exact diagonalization, the den sity-matrix renormalization group methods and the
+random phase approximation. In the strong-coupling region , the charge-ordered ground state is consistent with
+thepredictionsoftheeffectivemodelwithaplaquetteorde r. Wefindthatthequalitativepropertiesofthemetal-
+insulator transition are totallydifferent depending on th e sign of the hopping integrals, reflecting the difference
+of bandstructure atthe Fermilevel.
+PACS numbers: 71.10.Hf, 71.27.+a, 71.10.-w
+Introduction — For a long time, the exotic phenomena
+emergent in geometrical frustration have been fascinating
+but challenging subjects of research in condensed matter
+physics [1]. An essential difficulty of frustrated systems i s
+attributedtotheirhighly-degenerategroundstatesin the clas-
+sical sense. It is common practice to determine how the
+degeneracy is removed or how the frustration is minimized
+by adding quantum fluctuations. On a related issue, one of
+the most intensively studied systems is quantum spins on
+the Kagom´ e lattice, which consists of corner-sharing tria n-
+gles. Very recently, so-called herbertsmithite has been re -
+vealed as a promising candidate for an experimental real-
+ization of an S= 1/2Kagom´ e antiferromagnet. Both ex-
+perimentally [2, 3] and theoretically [4], the low-tempera ture
+phaseofthissystemischaracterizedbyaspinliquidstatew ith
+a smallorpossiblyevennoenergygap.
+When we turn our attention to itinerant systems on the
+Kagom´ e lattice, a greater variety of physical phenomena ca n
+be observed. Of particular interest is the competition of di s-
+tinctorderedstatesassociatedwiththechargedegreesoff ree-
+dom. From this point of view, an intriguing system is that
+of1/3-filledKagom´ elattice fermions. If long-rangerepul sive
+interactions are taken into account, there exists a huge num -
+ber of (nearly) degenerate charge-ordered (CO) states in th e
+strong-coupling regime; a melting of the CO may take place
+assomeratiooftheinteractionandhopping.
+An experimental candidate to realize this situation is
+hydrogen-bondedcrystals of alkali-hydrosulfatesor hydr ose-
+lenates M 3H(XO4)2(M=Cs,Rb,K,Tl, X=S,Se) [5]. These
+materials exhibit an abrupt change in the electrical conduc -
+tivity accompanied with a ferroelastic deformation around
+Tc= 400K: above Tcthe conductivity within the a-bplane
+is very high and it seems to be related to the hopping motion
+of protons, i.e., super-protonconductivity;below Tc, the pro-
+tons appear to be localized in some ordering pattern which
+satisfies the condition that one tetrahedron of XO−
+4has only
+one hydrogen bond (see Fig. 1). In the regime where only
+proton ordering is taken into account, an effective model ca n
+provideadescriptionoftheprotonsformingtheKagom´ enet -work, as discussed later in the paper. Since the number of
+corner-sharing triangles is equivalent to the number of tet ra-
+hedra,thecriticalbehaviorinthesematerialscanbedescr ibed
+bya metal-insulatortransition(MIT)in theKagom´ elattic e at
+1/3filling.
+Another possible experimental realization may be given
+by recently developed laser-cooling techniques. It was re-
+ported [6] that a two-dimensional optical trimerized Kagom´ e
+lattice was constructed and that the coupling constants cou ld
+becontrolledusingatriplelaserbeamdesign. Furthermore ,a
+schemeproposedthisyearwouldconstructanopticalKagom´ e
+latticeat1/3fillingwiththeuseofjusttwostandingwavesa nd
+morecontrollableinteractions[7].
+Motivated by the optical lattice systems, a model of hard-
+core bosons on the Kagom´ e lattice was studied with large
+scale quantumMonte Carlo simulations and phenomenologi-
+cal dual vortex theory [8, 9]. At 1/3 and 2/3 fillings a weak
+first-order superfluid-solid transition induced by the near est-
+neighbor repulsion was found. Thus, in our work presented
+here it is natural to consider how this transition occurs in
+fermionsystems. Althoughthe effectivemodel canbe shown
+to be equivalent for hard core bosons and spinless fermions
+in the strong-couplinglimit, features for each, in the weak to
+intermediate coupling regimes, may differ completely due t o
+the absence of a band picture in boson systems; here the dis-
+/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle
+/Bullet/Circle/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle
+/Bullet/Circle/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle
+/Bullet/Circle/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle
+/Bullet/Circle/CP/D1
+/CQ/D1
+/CG /C7/BG
+/B4/D9/D4 /DB /CP/D6/CS/B5
+/CG /C7/BG
+/B4/CS/D3 /DB/D2 /DB /CP/D6/CS/B5/Bullet/Circle/C5/Bullet/Circle/D4/D6/D3/D8/D3/D2 /D7/CX/D8/CT/D7/C5/BP/CA/CQ/B8 /BV/D7 /CG/BP/CB/B8 /CB/CT/CY /C3/CP/CV/D3/D1 /AJ /CT /D0/CP/D8/D8/CX
/CT/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle/Bullet/Circle /Bullet/Circle
+/Bullet/Circle
+/Bullet/Circle /Bullet/Circle
+FIG.1: (Color online) Latticestructure of hydrogen-bonde d crystals
+M3H(XO 4)2where the proton sites form the Kagom´ e lattice (right
+figure). The left figure presents an example of protons order b elow
+thecriticaltemperature. Thesolid(red)linesdenotehydr ogenbonds.2
+t>0
+kxky-4-2 0 2 4t<0
+kxky-4-2 0 2 4
+FIG.2: (Coloronline)BandstructureoftheKagom´ elattice fermions
+for positive (left)and negative (right) hopping energies.
+tinctionbetweenthetwo casesis non-trivial.
+In this Letter, we therefore study the MIT of the spin-
+less fermion model on the Kagom´ e lattice at 1/3 filling
+with the nearest-neighbor repulsive interactions using th e
+exact-diagonalization (ED) and density-matrix renormali za-
+tion group (DMRG) techniques. Assuming a CO state in the
+strong-couplingregime,wedemonstratethatthenatureoft he
+MIT strongly dependson the sign of hoppingintegral (inver-
+sionofthe signcorrespondsto aswitch between 1/3and2/3
+fillings). Lastly, the random phase approximation (RPA) is
+employedto confirmthenumericalresults.
+Model—TheHamiltonianis
+H=t/summationdisplay
+/angbracketlefti,j/angbracketright(c†
+icj+H.c.)+V/summationdisplay
+/angbracketlefti,j/angbracketrightninj,(1)
+wherec†
+j(cj)isacreation(annihilation)operatorofaspinless
+fermion and nj(=c†
+jcj) is the correspondingnumber opera-
+tor. The repulsive interaction V(>0) is assumed to act only
+between neighboring sites ∝angbracketlefti,j∝angbracketright. In the non-interacting case
+V= 0,thedispersionrelationisgivenbythreebands:
+ε(k) =−2t,
+t/bracketleftBigg
+1±/radicalBigg
+1+8coskx
+2cosky
+2cos/parenleftbiggkx−ky
+2/parenrightbigg/bracketrightBigg
+(2)
+(see Fig. 2). It is particularly worth noting that the flat ban d
+at 1/3 filling is only filled when t >0. Since the density of
+statesattheFermileveldiverges,thephysicalproperties inthe
+weak-tointermediate-couplingregimeareexpectedtogrea tly
+differfromthosefor t <0. Wenotethatourmodelwith t >0
+at1/3filling is equivalentto that with t <0at2/3filling via
+theparticle-holetransformation.
+Let us now consider the case of t= 0withV >0. The
+nearest-neighbor repulsions Vare minimized when each the
+corner-sharing triangle is occupied by exactly one fermion .
+This means that the system is in macroscopically degenerate
+COstate. However,thedegeneracymaybeliftedwhenafinite
+hoppingisintroduced,andthena CO state withsome period-
+icity must have the lowest energy. The candidates of the CO
+patterns are quoted in Fig. 3. By looking at the pattern III,
+onerealizesthekineticenergycan begainedthrougha cycli c
+hopping process of three fermions on each hexagon. Thus,
+theliftingtakesplacewith processesoforder |t|3/V2(lowerorder contributions, e.g. t2/V, are the same for all configu-
+rations and therefore contribute a constant energyshift ra ther
+thanlifting the degeneracy). The effectiveHamiltonianis de-
+rivedas[10–12]
+Heff=12|t|3
+V2/summationdisplay
+hexagon(c†
+1c†
+3c†
+5c2c4c6+H.c.),(3)
+where the sum is over all hexagons on the lattice and the six
+sites on each hexagonare labeled by 1-6in a clockwise man-
+ner. This is equivalent to the quantum dimer model on the
+honeycomblatticewhichhasaplaquette-orderedgroundsta te
+with three-fold degeneracy [13]. The effective model would
+beatleastqualitativelyvalidaslongasthesystemisaCOin -
+sulator. We note that here the sign of hoppingintegral has no
+influence on the physicsbecause the honeycomblattice has a
+bipartitestructure.
+Numerical analysis — First, we compare the lowest ener-
+gies of each CO pattern to determine if pattern III is indeed
+stabilized for V≫ |t|>0. We apply the ED technique with
+periodic boundary conditions. A 12- (18-)site cluster is us ed
+to calculate an energy difference EI−EII(EII−EIII) [see
+Fig. 4], where EI,EII, andEIIIare the energies of the CO
+patterns I, II, and III, respectively. The periodicity of wa ve
+functions is checked in order to identify one CO state from
+another. As shown in Fig. 5, for both signs of t,EIIIhas
+thelowestground-stateenergywhile EIIisthesecond-lowest
+in the large- Vregime; the energy differences are scaled like
+EII−EIII∼ O(V−2)andEI−EII∼ O(V−3). Since
+the second-order contributions O(|t|/V2)cancel in the en-
+ergydifferences,the behaviorof EII−EIIIseems to be con-
+sistent with the existence of an energy gain O(|t|3/V2)from
+theringexchangeprocessintheeffectiveHamiltonian(3). As
+expected, EII−EIIIhasnodependenceonthesignof tinthe
+large-Vregionasseenin Fig.5(b).
+Next, we show that the system indeed exhibits the CO-
+melting MIT at a finite critical interaction V=Vc. To find
+theMITcriticalpoint,thesingle-particlegapiscalculat edus-
+ingtheDMRG method. TheDMRG calculationisperformed
+on aLx×Lycluster shown in Fig. 4(b), where the periodic
+boundaryconditionsareimposedinthe ydirectionwiththree
+unitcellsandthe openboundaryconditionsin the xdirection
+/G49 /G49 /G49 /G49 /G49 /G49
+FIG. 3: (Color online) Charge-ordering patterns in the 1/3- filled
+Kagom´ e lattice characterized by wave vectors I (qa,qb) = (0,0),
+II(0,π),andIII(2π/3,2π/3). ThecircleonpatternIIIcorresponds
+tothe cyclic hopping process byEq.(3).3
+withLxunitcells. Westudyseverallengthsoftheclusterwith
+up toLx= 8(78sites) and extrapolate the finite-size results
+to the thermodynamic limit Lx→ ∞. The single-particle
+gap is thus given by ∆ = lim Lx→∞∆(Lx)with∆(Lx) =
+E(Nf+1,Lx,Ly)+E(Nf−1,Lx,Ly)−2E(Nf,Lx,Ly),
+whereE(Nf,Lx,Ly)is the ground-state energy of the cor-
+responding cluster with Nffermions. The upper panels of
+Figure6showtheextrapolatedvaluesof ∆asafunctionof V
+in units of |t|. The critical points are estimated as Vc∼2.6
+and4.0fort >0andt <0, respectively. The behaviors
+of the gap opening are quite different: the gap for t >0in-
+creases almost linearly, while it rises gradually for t <0like
+theBerezinskii-Kosterlitz-Thoulesstransition.
+In order to gain further insight, we also calculate the or-
+der parameter corresponding to the CO pattern III. The CO
+phenomenon is observed as a state with a broken transla-
+tional symmetry: actually, there are two or more degenerate
+ground states and oneconfiguration of the degenerate states
+is picked out as the ground state by an initial condition of
+the DMRG calculation, as a consequence of the cylindri-
+cal boundary conditions [14]. The order parameter is now
+defined as the amplitude of the charge-density modulation
+η(Lx) =n1−1
+2(n2+n3)for aLx×3cluster, where n1
+andn2(=n3) are the charge densities of fermion-rich and
+fermion-poor sites on a triangle at the center of the cluster .
+In the lower panels of Figure 6, we show the extrapolatedre-
+sults of order parameter ηto the thermodynamic limit, i.e.,
+η= limLx→∞η(Lx), as a function of V. We find that the
+estimatedcriticalpointsarealmostthesameasthoseobtai ned
+from∆for both signs of t. As seen similarly for ∆., the be-
+havior of ηdiffers markedly depending on the sign of t. For
+t >0,thetransitionappearstobenearlydiscontinuous,which
+mightbeanalogoustothatinameltingMITobservedinhard-
+core bosons on the triangular lattice [15]; for t <0, it looks
+continuous, which looks like a weakly first-order phase tran -
+sitionforhard-corebosonsontheKagom´ elattice [8]. Henc e,
+we conclude that the criticality for positive tis in stark con-
+trast tothatfornegative t.
+Now, we comment on the difference in values of the MIT
+criticalpointsbetween t >0andt <0. Asmentionedabove,
+all the triangles are only allowed to be singly occupied in th e
+/G28 /G61 /G29 /G28 /G62 /G29
+/G78/G79
+FIG. 4: (Color online) (a) Finite clusters of the Kagom´ e lat tice ap-
+pliedfortheEDmethod. The12-and18-siteclustersareused toob-
+tainEI−EIIandEII−EIII,respectively. (b) Lx×3cluster(shaded
+regime)for theDMRG method, where theopen andperiodic boun d-
+aryconditions are taken for the xandydirections, respectively./G70 /G61 /G74 /G74 /G65 /G72 /G6E /G20 /G49 /G49 /G49/G70 /G61 /G74 /G74 /G65 /G72 /G6E /G20 /G49 /G49/G70 /G61 /G74 /G74 /G65 /G72 /G6E /G20 /G49
+/G30 /G30 /G2E /G30 /G30 /G30 /G35 /G30 /G2E /G30 /G30 /G31 /G30 /G2E /G30 /G30 /G31 /G35/G30/G30 /G2E /G30 /G30 /G35/G30 /G2E /G30 /G31/G30 /G2E /G30 /G31 /G35
+/G30 /G30 /G2E /G30 /G30 /G30 /G31 /G30 /G2E /G30 /G30 /G30 /G32/G30/G30 /G2E /G30 /G30 /G32/G30 /G2E /G30 /G30 /G34/G30 /G2E /G30 /G30 /G36/BX
+/C1 /C1/A0 /BX
+/C1 /C1 /C1/BX
+/C1/A0 /BX
+/C1 /C1
+/BD /BP/CE
+/BE/BD /BP/CE
+/BF
+/D3 /B4/BD /BP/CE
+/BF/B5/D3 /B4/BD /BP/CE
+/BE/B5
+/D8 /BQ /BC/D8 /BO /BC/D8 /BQ /BC/D8 /BO /BC
+/B4/CP/B5 /B4
/B5/B4/CQ/B5
+FIG.5: (Coloronline)(a)Energydifferences EII−EIIIasafunction
+of1/V2in units of |t|, and (c)EI−EIIas a function of 1/V3. (c)
+Relationship of the energies EI,EII, andEIIIcorresponding to the
+COpatterns I - IIIinFig.3.
+/G30 /G30 /G2E /G31 /G30 /G2E /G32 /G30 /G2E /G33/G30/G31/G32
+/G30 /G30 /G2E /G31 /G30 /G2E /G32/G30/G30 /G2E /G31/G30 /G2E /G32/G30 /G2E /G33/G30/G32/G34/G36/G38
+/G30 /G35 /G31 /G30 /G31 /G35/G30/G30 /G2E /G31/G30 /G2E /G32/G30 /G2E /G33/G30 /G30 /G2E /G31 /G30 /G2E /G32 /G30 /G2E /G33/G30/G31/G32/G33
+/G30 /G30 /G2E /G31 /G30 /G2E /G32 /G30 /G2E /G33/G30/G30 /G2E /G31/G30 /G2E /G32/G30 /G2E /G33/G30 /G2E /G34/G30 /G2E /G35/G30/G35/G31 /G30
+/G30 /G35 /G31 /G30 /G31 /G35/G30/G30 /G2E /G31/G30 /G2E /G32/G30 /G2E /G33/G30 /G2E /G34/A1
+/A1/AH
+/AH/CE /CE/A1/B4 /C4
+/DC/B5/BD /BP/C4/DC
+/AH /B4 /C4
+/DC/B5/BD /BP/C4/DC
+/A1/B4 /C4
+/DC/B5/BD /BP/C4/DC/AH /B4 /C4
+/DC/B5/BD /BP/C4/DC
+/B4/CP/B5 /B4/CQ/B5
+/AW /A0 /AW /A0
+/CE
+Ꜵ /BE /BM /BI /CE
+Ꜵ /BG /BM /BC
+FIG. 6: (Color online) Single-particle gap ∆and order parameter η
+asafunctionofthenearest-neighbor repulsion Vfor(a)positiveand
+(b) negative tvalues. Insets: Finite-size scaling of each quantity for
+(a)V= 2,3,4, and5; and (b)V= 2,3,4,5, and6, from bottom to
+top.
+CO state. Therefore,oursystemcan bemappedintoa single-
+band half-filled Hubbard model on honeycomb lattice with
+on-site Coulomb interaction O(V)if we regard each trian-
+gle as a single site. The single-particle gap is then expecte d
+to behave like ∆≈V−Weffin the large- Vregion, where
+Weffis the effective bandwidth. Actually, as seen in Fig. 6,
+thegapbehaveslike ∆∼V−5tand∆∼V−8|t|fort >0
+andt <0, respectively. In general, the MIT critical strength
+is roughly scaled by the bandwidth, i.e., Vc∝Weff. The as-
+sumptionoftheeffectivebandwidths Weff= 5tfort >0and
+8|t|fort <0arecompatiblewiththe obtainedcriticalpoints.
+The narrower bandwidth for t >0might be interpreted as a
+remnantoftheflat bandinthenoninteractingcase[16].
+Randomphaseapproximation —Inaddition,theMITisin-
+vestigated analytically using the RPA [17]. The charge sus-
+ceptibilityisgivenbytheKuboformulaas
+Xαγ(q,iωl) =1
+Nuc/integraldisplayβ¯h
+0dτeiωlτ∝angbracketleftnq,α(τ)n−q,γ(0)∝angbracketright(4)4
+0 2 4 6 80123
+   CO Ins.
+(pattern III)Metal/CZ
+/BU/CC/CE
+/D8 /BQ /BC/D8 /BO /BC
+FIG.7: (Color online) Phase diagram of the 1/3-filled t-Vmodel on
+the Kagom´ e latticeobtained by the random phase approximat ion.
+where∝angbracketleft···∝angbracketrightistheensembleaverage, nq,αisanumberopera-
+tor in momentum space, β=kBTis inverse temperature, ωl
+is the Matsubarafrequencyof bosons,and Nucis the number
+of unit cells. Then, applyingthe RPA, the charge susceptibi l-
+ityis obtainedin thefollowingmatrixform,
+X(q,ıωl) =/bracketleftBig
+1+2v(q)X(0)(q,ıωl))/bracketrightBig−1
+X(0)(q,ıωl),(5)
+wherethematrixelementsof v(q)aregivenby
+v12(q) =Vcos(qx/2), v13(q) =Vcos(qy/2)(6)
+v23(q) =Vcos((qx−qy)/2), (7)
+with the relation vji=vijandvii= 0. The bare susceptibil-
+ityis definedas
+X(q,ıωl)(0)
+αγ≡
+1
+Nu/summationdisplay
+k/summationdisplay
+µνu∗
+k,αµuk+q,ανu∗
+k+q,γνuk,γµΓ(k;q)µν
+ωl,(8)
+where
+Γ(k;q)µν
+ωl=fk+q,ν−fk,µ
+ı¯hωl−εk+q,ν+εk,µ, (9)
+withfbeing the Fermi distribution function. ukis a matrix
+whichdiagonalizesthekineticpartofeq.(1)inthemomentu m
+spacek.
+TheMIT criticalpointat finite temperatureisfoundbyde-
+terminingthedivergenceofthediagonalpartofeq.(5)fort he
+three sets of CO patterns, I, II, and III. We foundthat the CO
+pattern III is always most stable, in agreement with the ED
+result. The obtained phase diagram is shown in Fig. 7. At
+high temperature( kBT/|t| ≫1), the MIT phase boundaryis
+almost independent of the sign of t; however, for decreasing
+temperature the boundaries gradually move apart from each
+other. The critical points are estimated as Vc∼3and4.5for
+t >0andt <0, respectively. These values seem to be not
+onlyqualitativelybutalsoquantitativelyconsistentwit hthoseobtainedby the DMRG method. It is also interesting that the
+boundariesshowreentrantbehaviors.
+Conclusion —Westudythemetal-insulatortransitionofthe
+spinlessfermionsontheKagom´ elatticeat1/3-fillingusin gthe
+exact diagonalization, density-matrix renormalization g roup,
+andrandomphaseapproximationtechniques. Intheregionof
+largeV, the CO pattern in the ground-stateis consistent with
+what is predictedby the ring exchangemodel. The behaviors
+of the single-particle gap opening for positive and negativ e
+hopping integrals are qualitatively quite different; this is fur-
+ther reflectedin the differing values of the critical point Vc.
+These differences may arise from the distinct nature of the
+density of states at the Fermi level in each case. A possible
+further extension of this work is a consideration of spin de-
+grees of freedom. This is of interest because spins in the CO
+phasearealsofrustratedandtheconfigurationis nontrivia l.
+Acknowledgment — We thank T. Aonuma, G. Fiete,
+Y. Ohta, and F. Pollmann for very useful discussions. M. N.
+acknowledgesthevisitorsprogramattheMax-Planck-Insti tut
+f¨ ur Physik komplexer Systeme, Global Center of Excellence
+Program “Nanoscience and Quantum Physics” of the Tokyo
+Institute of Technology by MEXT, and Industrial Technol-
+ogy Research Grant Program in 2005-2008 from the New
+EnergyandIndustrialTechnologyDevelopmentOrganizatio n
+(NEDO)ofJapan.
+[1] R.MoessnerandA.P.RamirezPhysicsToday,pp.24,Febru ary
+2006.
+[2] J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko,
+B. M. Bartlett, Y. Yoshida, Y. Takano, A. Suslov, Y. Qiu, J.-
+H. Chung, D. G. Nocera, and Y. S. Lee, Phys. Rev. Lett. 98,
+107204 (2007).
+[3] P. Mendels, F. Bert, M. A. de Vries, A. Olariu, A. Harrison ,
+F. Duc, J. C. Trombe, J. S. Lord, A. Amato, and C. Baines,
+Phys. Rev. Lett. 98, 077204 (2007).
+[4] P.Sindzingre andC.Lhuillier,arXiv:0907.4164.
+[5] H. Kamimura, Y. Matsuo, S. Ikehata, T. Ito, M. Komukae, an d
+T. Osaka, Phys.Stat.Sol.(b) 241, 61 (2004).
+[6] L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts,
+H. Fehrmann, and M. Lewenstein, Phys. Rev. Lett. 93, 030601
+(2004).
+[7] J. Ruostekoski, Phys.Rev. Lett. 103, 080406 (2009).
+[8] S. V. Isakov, S. Wessel, R. G. Melko, K. Sengupta, and
+Y. B.Kim, Phys.Rev. Lett. 97, 147202 (2006).
+[9] K. Sengupta, S. V. Isakov, and Y. B. Kim, Phys. Rev. B 73,
+245103 (2006).
+[10] E.Runge and P.Fulde, Phys.Rev. B 70, 245113 (2004).
+[11] P.FuldeandF.Pollmann,Ann.Phys.(Berlin) 17,7,441(2008).
+[12] A. O’Brien,F.Pollmannand P.Fulde, tobe published.
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+144416 (2001).
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+[16] D. Poilblanc andH. Tsunetsugu, arXiv:0911.2413v1 (20 09).
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+Science (2006).
\ No newline at end of file
diff --git a/1001.0809.txt b/1001.0809.txt
new file mode 100644
index 0000000000000000000000000000000000000000..56d49138d35b2f1e14faa7e63b3fdc66e22037f0
--- /dev/null
+++ b/1001.0809.txt
@@ -0,0 +1,533 @@
+arXiv:1001.0809v3  [physics.chem-ph]  30 Nov 2010Excitation energy transfer: Study with non-Markovian dyna mics
+Xian-Ting Liang∗
+Department of Physics and Institute of Modern Physics,
+Ningbo University, Ningbo, 315211, China
+In this paper, we investigate the non-Markovian dynamics of a model to mimic the excitation en-
+ergy transfer (EET) between chromophores in photosynthesi s systems. The numerical path integral
+method is used. This method includes the non-Markovian effec ts of the environmental affects and
+it does not need the perturbation approximation in solving t he dynamics of systems of interest. It
+implies that the coherence helps the EET between chromophor es through lasting the transfer time
+rather than enhances the transfer rate of the EET. In particu lar, the non-Markovian environment
+greatly increase the efficiency of the EET in the photosynthes is systems.
+Keywords: Photosynthesis, Excitation energy transfer; No n-Markovian; Decoherence.
+PACS numbers: 33.15.Hp, 03.65.Yz, 82.50.-m
+I. INTRODUCTION
+Photosynthesis provides almost all the energy for life
+on Earth. Therefore, many interests have been attracted
+on this topic in past decades not only in biology but
+also in chemistry and physics [1]. The typical antenna of
+harvestingphotons in photosynthesis systems are protein
+complexes that hold pigments [chlorophyll (Chl), bacte-
+riochtorophyll (BChl), phycobilin, carotenoid, bacterio-
+pheophytin (BPhy), etc.]. For example, in a green sulfur
+bacteria, the Fenna-Matthews-Olson (FMO) protein is a
+trimer made of identical subunits, each of which contains
+sevenBChlmolecules[2]. Anotherpurplebacterium, Rh.
+sphaeroides[3] includesaspecial pairChls (P)in the cen-
+ter, accessory Bchls (B Land B M) on the each side of P,
+and BPhys (H Land H M) next to the each BChls.
+The photosynthesis starts with the absorption of pho-
+ton of sunlight by the light-harvestingpigments, followed
+by the excitation energy transfer (EET) to the reaction
+center, where charge separation is initiated and physi-
+cal energy is converted into chemical energy [4]. At low
+light intensities, the quantum efficiency of the transfer
+of energy is very high, with a near unity quantum yield.
+Although this process has been investigated extensively
+the details and the mechanism of the energy transfer has
+not been fully understood today.
+There is a hypothesis that the high efficiency of near
+unity EET attributes to coherent superpositions between
+excitation states that belong to different chromophores
+of the photosynthesis systems. It is deduced from the
+following analogous facts. Engel et al.[5] found that in
+FMO protein the coherence between the two excitation
+states clearly lasts for a longer time which is similar to
+the time scale of the EET. Lee et al.[6] also found that in
+Rh. sphaeroides the coherence of the excitations created
+∗Email: xtliang@ustc.edufrom laser pulses on adjacent B Land H Lcan prolong for
+longer time which is long enough for the coherent trans-
+fer of energies between chromophores B Land H L. These
+kindsofexperimentalresultsimplythatelectronicexcita-
+tions may move coherently through these photosynthesis
+systems ratherthan by incoherent hopping motion as has
+usually been assumed.
+In the aspect of theory, more than fifty years earlier,
+F¨ orster[7] suggested a formula of the efficiency of the
+resonance EET. This theory give a picture that excita-
+tion energies are transferred in the photosynthesis sys-
+tems through incoherent hopping from the higher levels
+to the lower levels. However, this theory only describes
+the cases that the couplings between chromophores are
+weak. The master equation of Redfield form [8] has also
+been used in the investigations of the EET. However,
+this method of the original form is limited because it
+derived from the Markovian and perturbation approxi-
+mations. A modified Redfield theory, first suggested by
+Zhang and Fleming [9], and later elaborated by Yang
+and Fleming [10], has then been used in the investiga-
+tions of the EET. It partly overcomes the shortcomings
+of the conventional one. But, it cannot take into ac-
+count any processes related to quantum coherence and
+decoherence. Jang et al. [11] developed the F ¨ orster-
+Dexter theory for the EET. In their method the weak
+couplings between chromophores are not needed, but it
+asks the couplings between the system and the environ-
+ment be weak. Ishizaki and Fleming [12] recently used
+thereducedhierarchyequationapproachdealingwiththe
+quantum coherent and incoherent dynamics of the EET.
+It avoids the usage of perturbative truncation, and can
+describe the quantum coherent wavelike motion and in-
+coherent hopping in the same framework. Mohseni et
+al. [13] developed a quantum walk approach, based on a
+quantum trajectory picture in Born-Markovian and sec-
+ular approximations, as a natural frameworkfor incorpo-
+ratingquantum dynamicaleffects in theEET.It isshown
+that the interplay between the coherent dynamics of the
+system and the incoherent action of the environment can2
+lead to significantly greater transport efficiency than the
+coherent dynamics on its own. Similar results have also
+been obtained independently by Plenio et al.[14]. Much
+othereffortbasbeencontributedtothefieldsinlastyears
+[15–19].
+However, in the existing investigations as being listed
+above, two important aspects paid not enough attention
+to. One is how the non-Markovian [20, 21] effects influ-
+ences the dynamics of the EET, and the other is how to
+describe the evolution rate of the EET. In this paper, we
+shall use the numerical path integral method developed
+by Makri’s group investigating the EET, where we will
+pay particular attention to above two problems.
+This article is organized as follows. In Sec. II, we
+introduce the model and compare the numerical path in-
+tegral method to the master equation method through
+calculating the time evolution of the population of sites.
+In Sec. III we investigate the evolutions of the coherent
+term of the reduced density matrix and the transfer rate
+of the EET in many kinds of conditions. It will be shown
+that the coherence helps the energy transfer in the model
+system through lasting the transfer time rather than in-
+creasing the transfer rate of the EET, and as we consider
+the non-Markovian effects the energy transfer will be in-
+creased greatly. Through the model study we can obtain
+some important implications for energy transfer in pho-
+tosynthesis systems. Finally, Sec. IV is devoted to the
+concluding remarks.
+II. MODEL AND TIME EVOLUTION OF THE
+POPULATION OF SITES
+The Hamiltonian ofthe EETmodel canbe represented
+by Frenkel exciton Hamiltonian [10, 12].
+Htot=Hel+Hph+Hre+Hel−ph,(1)
+where
+Hel=2/summationdisplay
+j=1|j/angbracketrightε0
+j/angbracketleftj|+∆(|1/angbracketright/angbracketleft2|+|2/angbracketright/angbracketleft1|),(2)
+Hph=2/summationdisplay
+j=1Hph
+j, (3)
+Hre=2/summationdisplay
+j=1|j/angbracketrightλj/angbracketleftj|, (4)
+Hel−ph=2/summationdisplay
+j=1Hel−ph
+j=2/summationdisplay
+j=1Vjuj. (5)
+Here,|j/angbracketrightrepresents the state where only the jth site is
+in its excitation electronic state |ϕje/angbracketrightand another is in
+its ground electronic state |ϕkg/angbracketright, i.e.,|j/angbracketright ≡ |ϕje/angbracketright|ϕkg/angbracketright.
+While we define the ground state |0/angbracketright ≡ |ϕ1g/angbracketright|ϕ2g/angbracketright.
+Eq.(2) is the electronic Hamiltonian in which the Hamil-
+tonian of the ground electronic state is set to be zero.ε0
+jis the excitation electronic energy of the jth site in
+the absence of phonon. ∆ is the electronic coupling be-
+tween the two sites, which responsible for the EET be-
+tween the individual sites. Hph
+j=/summationtext
+ξ¯hωξ/parenleftBig
+p2
+ξ+q2
+ξ/parenrightBig
+/2
+is the phonon Hamiltonian associated with the jth site,
+whereqξ, pξ,andωξare the dimensionless coordinate,
+conjugate momentum, and frequency of the ξth phonon
+mode, respectively. λj=/summationtext
+ξ¯hωξd2
+jξ/2 is the reorganiza-
+tion energy of the jth site, where djξis the dimensionless
+displacement of the equilibrium configuration of the ξth
+phonon mode between the ground and excitation states
+of thejth site.Hel−phis the coupling Hamiltonian be-
+tween thejth site and phonon modes and the Vjand
+ujare defined as Vj=|j/angbracketright/angbracketleftj|anduj=/summationtext
+ξ¯hωξdjξqξ.
+Here, we assume the phonon modes associated with one
+site are uncorrelated with those of another site. If we
+setλ1=λ2=λ,namely, suppose that the reorganiza-
+tion energy of the 1-th and 2-th sites are equal, and reset
+the zero point energy at ( ε0
+2−ε0
+1+u2−u1)/2,and set
+cξ= (d2ξ−d1ξ)/2,we can obtain the total Hamiltonian
+presented with Pauli matrix as
+H=ǫ
+2σz+∆σx−σz/summationdisplay
+ξcξ¯hωξqξ
++/summationdisplay
+ξ¯hωξ/parenleftbig
+p2
+ξ+q2
+ξ/parenrightbig
+/2, (6)
+whereǫ=ε0
+2−ε0
+1.Eq.(6) is the standardform ofthe spin-
+boson model. Our following investigations start from
+this Hamiltonian. Thus, we can investigate the model
+with numerical path integral scheme which is developed
+by Makri group [22], and was widely used by Thorwart
+group [23] and many other groups [24] in last years.
+In order to compare numerical path integral scheme
+to master equations approach of Redfield form we firstly
+investigate the time evolution of the population of sites
+for the model. The spectral distribution function is also
+employedastheDrude-Lorentzdensity(theover-damped
+Brownian oscillator model):
+J(ω) = 2λωγ
+ω2+γ2, (7)
+as Ref.[12]. The parameters are taken as the same val-
+ues as Ref.[12], namely, ǫ= 100 cm−1,∆ = 100 cm−1,
+γ= 53.08 cm−1(γ−1= 100 fs),andT= 300 k. To
+implement the numerical path integral scheme, we use
+the iterative tensor multiplication (ITM) algorithm de-
+rived from the the quasiadiabatic propagator path inte-
+gral (QUAPI) technique. We use the time step δt= 10 fs
+which isshorterthan the correlationtime ofthe bath and
+the characteristic time of the two-level system. In order
+to include as much non-Markovian effect of the bath as
+possible in the ITM algorithm, one should choose δkmax
+as large as possible so that the dominant non-Markovian
+effect of the bath is included. Here, δkmaxis the maxi-
+malnumberoftimesteps, andthe δkmaxδtistheincluded
+memory time in the calculations. In another word, the
+memory effects of the bath within time δkmaxδtare con-
+sidered in the algorithm. If set δkmax= 0, this method3
+/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48
+/s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50
+/s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50
+/s32/s82/s101/s100/s102/s105/s101/s108/s100
+/s32
+/s116/s40/s102/s115/s41/s32/s80/s73/s40 /s107
+/s109/s97/s120/s61/s48/s41
+/s116/s40/s102/s115/s41/s32/s80/s73/s40 /s107
+/s109/s97/s120/s61/s51/s41/s32
+/s32/s32
+/s32
+FIG. 1: (Color online) Time evolutions of the population of
+site 1, obtained from master equations of Redfield form (blac k
+line connects black squares), and from ITM algorithm based
+on QUAPI of δkmax= 0 (blue dashed line connects blue
+stars), and δkmax= 3 (red dotted line connects red circles)
+for different λ. Here,ǫ= 100 cm−1,∆ = 100 cm−1,γ= 53.08
+cm−1(γ−1= 100 fs),andT= 300 k. Initially, the system is
+populated on the site 1 with state |ψ0/angbracketright=|1/angbracketright.
+then reduce to the Markovian approximation. But, the
+Markovian results are not same to the ones from master
+equations of Redfield form, because the latter is not only
+Markovian but also perturbative. The time evolutions of
+the population of site 1 (suppose at the initial time the
+system populated on the site 1) obtained from ITM and
+master equations of Redfield form are plotted in Fig.1.
+In Fig.1, we plot the time evolutions of the population
+of site 1 by using the master equations of Redfield form
+and the ITM algorithm as λ= ∆/50,∆/5,∆,and 5∆.
+The initial state is |ψ0/angbracketright=|1/angbracketright, where|1/angbracketrightis the basis state
+of the system. It is shown that the evolutions of the pop-
+ulation of site 1 obtained from the master equations of
+Redfield form are in good agreement with the ones ob-
+tained from the ITM algorithm of δkmax= 0. We shall
+useδkmax= 3,andδt= 10 fs to investigate the dynam-
+ics under non-Markovianapproximation in this paper, so
+in this case, the memory effects of the bath within 30 fs
+are included. In fact, the memory time of the bath with
+Drude-Lorentz density function is longer than this value,
+the memory effects beyond 30 fs will not be included in
+this paper. The time evolutions of the population of site
+1 from ITM of δkmax= 3,are also plotted in Fig.1 which
+is quite distinct from the Markovian results. It shows
+that when we investigate this kind of system the non-
+Markovianeffects are non-ignorable. In the following, we
+shall use the ITM algorithm with δkmax= 3 to study the
+decoherence and the transfer rate of the EET. The mem-
+ory effects of the bath have not been completely included
+in our calculations but the dominating affects have been
+included, because when we increase δkmaxfrom 3 to 4 the
+results have little change and this change even cannot be
+distinguished by eyes.III. TIME EVOLUTIONS OF COHERENCE
+AND EET
+It is reported[5, 6] that the decoherencetime ofthe co-
+herent superposition state produced by laser pulses has
+the same time scale of the EET between two adjacent
+chromophores. People thus suppose that coherence may
+help the EET in photosynthesis. On the other hand, the
+theoretical analysis shows that the environment is not a
+hindrance but a help to the EET [13, 14]. So, we want to
+know further that how the coherence helps the EET and
+how about non-Markovian environment compares to the
+Markovian one in the helps of the EET. In this section
+we shall discuss these kinds of problems through investi-
+gating above dynamical model to mimic photosynthesis
+systems.
+Similar to Sec.II we can easily plot the evolution of the
+coherence, namely, the the evolution of the off-diagonal
+term in the reduced density matrix. It is also convenient
+toplot the evolutionofthe changingrateofpopulationof
+site2. Thechangingrateofthe populationofthesite2in
+fact reflects the transfer rate of the EET from the site 1
+to the site 2. Comparing the evolutions of the coherence
+and the transfer rate (denoted by kin this paper) of
+the EET, we can judge whether the transfer rate of the
+EET increase or not as the coherence increase, and as
+the system in a non-Markovian environment instead of
+Markovian one.
+In Figs.2 and 3 we plot the evolutions of the ρ12(t) and
+kwith time in different λ. Here we set λvary from ∆ /5,
+to ∆/4,∆/3,∆/2, andǫ= 100 cm−1,∆ = 100 cm−1.
+In this paper we always set γ= 53.08 cm−1(γ−1= 100
+fs),andT= 300 k. The initial state is set in the max-
+imal coherent superposition state |ψ0/angbracketright= (|1/angbracketright+|2/angbracketright)/√
+2
+in Fig.2, and in the basis state |ψ0/angbracketright=|1/angbracketrightin Fig.3. Figs.4
+and 5 plot the evolutions of the ρ12(t) andkwith time t
+in different ∆. Here, we set ǫ= 100 cm−1,λ= 20 cm−1,
+and ∆ vary from ǫ, toǫ/2,ǫ/3, andǫ/4, the initial state
+is|ψ0/angbracketright= (|1/angbracketright+|2/angbracketright)/√
+2 in Fig.4, and |ψ0/angbracketright=|1/angbracketrightin Fig.5.
+In the Figs.2-5, the time evolutions of the coherence are
+plotted in the upper panels (a) and (c) and transfer rate
+kof the EET are plotted in the lower panels (b) and (d).
+In the left panels (a) and (b) are the results of Marko-
+vian approximation δkmax= 0, and in right panels (c)
+and (d) are the results of non-Markovian approximation
+δkmax= 3.
+The Table 1 is made by using the data of Figs.2 and
+3, and the Table 2 is made from data of Figs.4 and 5. In
+the Tables 1 and 2 we numerically show the decoherence
+times and the net area of curve kin different conditions.
+The positivevalue oftransferrateofthe EETmeansthat
+the energy transfers from the site 1 to the site 2, and the
+negative value of the kmeans that it transfers back to
+the site 1 from the site 2. The net transfer quantity of
+energy from the site 1 to the site 2 is proportional to the
+netarea under the curve k(where the net area denote al-
+gebraicsum of the area). From Figs.2-5and Tables1 and
+2 we can obtain the following results. (1) The transfer4
+/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54
+/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52
+/s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54
+/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s32
+/s32/s32/s84/s114/s97/s110/s115/s102/s101/s114/s32/s114/s97/s116/s101/s32/s111/s102/s32/s101/s110/s101/s114/s103/s121/s40/s97/s41/s32/s32 /s32/s40/s99/s41/s68/s101/s99/s111/s104/s101/s114/s101/s110/s99/s101
+/s116/s40/s102/s115/s41/s40/s98/s41/s40/s100/s41/s32
+/s116/s40/s102/s115/s41/s32
+/s32
+/s32
+FIG. 2: (Color online) Time evolutions of the coherence [up-
+per panels (a) and (c)] and transfer rate of the EET [lower
+panels (b) and (d)], in Markovian approximation δkmax= 0
+[left panels (a) and (b)], and in non-Markovian approximati on
+δkmax= 3 [right panels (c) and (d)], for different values of λ.
+The initial state is |ψ0/angbracketright= (|1/angbracketright+|2/angbracketright)/√
+2. The values of other
+parameters are the same as in Fig.1.
+/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48
+/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48
+/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52
+/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51
+/s48 /s52/s48/s48 /s56/s48/s48 /s49/s50/s48/s48 /s49/s54/s48/s48/s32/s84/s114/s97/s110/s115/s102/s101/s114/s32/s114/s97/s116/s101/s32/s111/s102/s32/s101/s110/s101/s114/s103/s121/s40/s97/s41/s32
+/s32
+/s40/s99/s41/s68/s101/s99/s111/s104/s101/s114/s101/s110/s99/s101
+/s40/s98/s41
+/s32
+/s116/s40/s102/s115/s41/s40/s100/s41/s32/s32/s32
+/s32
+/s116/s40/s102/s115/s41/s32
+/s32
+FIG. 3: (Color online) Time evolutions of the coherence and
+transfer rate of the EET in Markovian and non-Markovian
+approximations. The parameters in Fig.3 are similar to them
+in Fig.2 except the initial state |ψ0/angbracketright=|1/angbracketright.
+rate of energy kfrom the site 1 to the site 2 are oscilla-
+tions, the coherent term ρ12(t) are also oscillations. The
+two kinds ofevolution curvesalmost havethe samedecay
+times. It means that the system is transferring the en-
+ergyin all of the coherence time. However, the maximum
+ofkis nether at the maximal points nor at the minimal
+points of the curve ρ12(t). This means that the increase
+of coherence does not result in the increase of the trans-
+fer rate of the EET. Thus, we obtain the first conclusion
+in this paper that coherence helps the energy transfer
+throughprolongingthetransfertimeratherthan increase
+the transfer rate. As seen in the figures and tables that
+although the time of energy transfer is extended due to
+the increase of coherence its efficiency may decrease be-
+cause the energy will be transferred between the site 1
+and the site 2 back and forth. The breath-like transfer/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52
+/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51
+/s48 /s52/s48/s48 /s56/s48/s48 /s49/s50/s48/s48 /s49/s54/s48/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54
+/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s32/s32/s84/s114/s97/s110/s115/s102/s101/s114/s32/s114/s97/s116/s101/s32/s111/s102/s32/s101/s110/s101/s114/s103/s121/s40/s97/s41/s32
+/s32
+/s40/s99/s41/s68/s101/s99/s111/s104/s101/s114/s101/s110/s99/s101
+/s116/s40/s102/s115/s41/s40/s98/s41
+/s32
+/s116/s40/s102/s115/s41/s40/s100/s41/s32/s32
+/s32
+FIG. 4: (Color online) Time evolutions of the coherence [up-
+per panels (a) and (c)] and transfer rate of the EET [lower
+panels (b) and (d)], in Markovian approximation δkmax= 0
+[left panels (a) and (b)], and in non-Markovian approximati on
+δkmax= 3 [right panels (c) and (d)], for different values of ∆.
+The initial state is |ψ0/angbracketright= (|1/angbracketright+|2/angbracketright)/√
+2. The values of other
+parameters are the same as in Fig.1.
+/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50
+/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48
+/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53
+/s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s32/s32/s84/s114/s97/s110/s115/s102/s101/s114/s32/s114/s97/s116/s101/s32/s111/s102/s32/s101/s110/s101/s114/s103/s121/s40/s97/s41/s32
+/s32
+/s40/s99/s41/s68/s101/s99/s111/s104/s101/s114/s101/s110/s99/s101
+/s116/s40/s102/s115/s41/s40/s98/s41
+/s32
+/s116/s40/s102/s115/s41/s40/s100/s41/s32/s32
+/s32
+FIG. 5: (Color online) Time evolutions of the coherence and
+transfer rate of the EET in Markovian and non-Markovian
+approximations. The parameters in Fig.5 are similar to them
+in Fig.4 except the initial state |ψ0/angbracketright=|1/angbracketright.
+of energy in fact is ineffective. (2) From the Tables 1
+and 2 we see that the environment really helps the en-
+ergy transfer. It is clearly shown that an environment
+with stronger damping may stop the breath-like trans-
+fer of energy, which increase the transfer quality of the
+EET. In particular, as the non-Markovian effects are in-
+cluded the net area under the curve k, namely, the trans-
+fer quality of the EET is increased greatly in all cases. It
+means that the environment with non-Markovian effects
+will help the EET. This is the second conclusion in this
+paper. It is worthwhile to point out that as the initial
+state is the coherent superposition state the energy may
+be transferred from site 2 to 1 then the net area of the k
+curve is negative as shown at 410 fs in Table 2.5
+TABLE I: (Obtained from Figs.2 and 3) Transfer quantity of
+energy and decoherence times in different λas ∆ =ǫ= 100
+cm−1.
+|ψ0/angbracketright= (|1/angbracketright+|2/angbracketright)/√
+2 |ψ0/angbracketright=|1/angbracketright
+kmax=0 kmax=3 kmax=0 kmax=3
+λτareaτareaτareaτarea
+∆/53590.16098
+almost same0.44934 2742.92073
+almost same3.44756
+∆/42650.08718 0.45430 3902.96761 3.45388
+∆/32070.03219 0.46212 2602.99283 3.46209
+∆/21490.00473 0.47830 2402.99969 3.47830
+TABLE II: (Obtained from Figs. 4 and 5) Transfer quantity
+of energy and decoherence times in different ∆ as ǫ= 100
+cm−1,λ=ǫ/5.
+|ψ0/angbracketright= (|1/angbracketright+|2/angbracketright)/√
+2 |ψ0/angbracketright=|1/angbracketright
+kmax=0 kmax=3 kmax=0 kmax=3
+∆τareaτareaτareaτarea
+ǫ/2410-0.02476
+almost same0.45031 8002.92073
+almost same3.45018
+ǫ/38700.09477 0.41073 8302.90845 3.41011
+ǫ/413200.25661 0.41204 13802.62069 3.37671
+ǫ/519000.39017 0.43823 17002.18892 3.19348
+IV. SUMMARY
+To summary, in this paper we used a model introduced
+in Ref.[12] investigating the time evolutions of popula-
+tion by using the numerical path integral, which is qual-itatively identical to the results obtained from reduced
+hierarchy equation approach in Ref.[12]. The path inte-
+gral method includes the non-Markovian effects of bath.
+By use of the method we furthermore investigated the
+evolutions of the coherence and the transfer rate of the
+EET between two sites. We clearly show that there exist
+the energy transfer in all decoherence time. The increase
+ofcoherencedoes not directly result in the increaseofthe
+transfer rate of the EET. The helps of coherence to the
+energy transfer results from the increase of transfer time
+rather than the transfer rate. The energy transfer be-
+tween two sites within first severalhundred femtoseconds
+islikeabreathingmotioninMarkovianenvironment. We
+have also obtained that the non-Markovian effects play
+important role to the dynamics of the energy transfer in
+the model systems in the initial several hundreds fem-
+toseconds and it results in high efficiency of the EET.
+Our findings from spin-boson model study have impor-
+tant implications for understanding the energy transfer
+in photosynthesis systems.
+Acknowledgments
+I like to thank Profs. Wei-Min Zhang and Yi-Zhong
+Zhuo for helpful discussions. This project was sponsored
+by National Natural Science Foundation of China (Grant
+No. 61078065), and K.C.Wong Magna Foundation in
+Ningbo University.
+[1] May and O. Kuhn, Charge and Energy Transfer Dynam-
+ics in Molecular Systems, (Wiley, Weinheim, 2004); J.
+Cogdell, A. Gall, and J. K¨ ohler, Q. Rev. Biophys. 39,
+227 (2006); H. van Amerongen, L. Valkunas, and R.
+van Grondelle, Photosynthetic Excitons, (World Scien-
+tific, Singapore, 2000); J. Gilmore and R. H. MacKenzie,
+Chem. Phys. Lett. 421, 266 (2006).
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\ No newline at end of file
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+arXiv:1001.0810v1  [astro-ph.CO]  6 Jan 2010Relaxing constraints on dark matter annihilation
+Leonid Chuzhoy
+Department of Astronomy and Astrophysics, The University o f Chicago, 5640 S. Ellis, Chicago, IL 60637,
+USA; chuzhoy@oddjob.uchicago.edu
+ABSTRACT
+The relic abundance of thermal dark matter particles is generally as sumed to be inversely
+proportional to their annihilation rate, which is therefore constra ined by the present matter
+density,∝angbracketleftσannv∝angbracketright ∼10−26Ω−1
+dmcm3·sec−1. Here we point out that much lower values of ∝angbracketleftσannv∝angbracketright
+arepossibleforheavydarkmattercandidates( mX/greaterorsimilar10TeV)thatcoupletootherparticlespecies
+through the electroweak force. With heavy dark matter particles present the early universe may
+evolveaccordingtothefollowingscenario. Afteranearlyentryinto matter-dominatedphase,dark
+matter particles form self-gravitating microhalos. Collisional intera ction between dark matter
+particles and the surroundingradiation field eventually leads to micro halosgravothermalcollapse
+and annihilation of most dark matter particles. For sufficiently heavy dark matter candidates
+(mX/greaterorsimilar10 TeV) the universe can return to radiation-dominated phase bef ore the nucleosynthesis
+and thereafter follow the ”standard” scenario.
+Subject headings: cosmology: theory – early universe
+1. Introduction
+While there is a strong evidence for the ex-
+istence of non-baryonic dark matter, little is
+known about its origins. In the currently popu-
+lar scenario, dark matter particles and antipar-
+ticles start in thermal equilibrium with radia-
+tion. When temperature drops below the par-
+ticle mass, mX, their comoving density begins an
+exponential fall until the annihilation time-scale,
+tann= (nX∝angbracketleftσannv∝angbracketright)−1, becomes larger than the
+age of the Universe. From this point the comoving
+density of dark matter particles is assumed frozen.
+Forthe relicabundance ofdarkmatter particlesto
+be consistent with the present value of Ω dmtheir
+annihilation rate at the time of freeze-out must
+equal (Kolb & Turner 1989)
+∝angbracketleftσannv∝angbracketright ∼3·10−26cm3·sec−1.(1)
+Unless dark matter is of non-thermal origin
+(Chung, Kolb & Riotto 1998), lower values of
+∝angbracketleftσannv∝angbracketrightwould overclose the universe and there-
+fore can be excluded.
+If dark matter particles interact through any
+known force, during the decoupling epoch, whenT∼mX, their annihilation cross-section can not
+exceedσann/lessorsimilarα/T2∼α/m2
+X. Using this fact,
+Griest & Kamionkowski (1990) obtained from (1)
+anupperlimitonthemassofdarkmatterparticles
+mX/lessorsimilar300 TeV. (2)
+However, in this paper we point out that dark
+matter candidates with higher mass and lower an-
+nihilation cross-section are not necessarily incon-
+sistent with the present value of Ω dm. Unlike the
+standard model, which assumes the dark matter
+comoving density freezes out soon after the tem-
+peratureoftheUniversedropsbelow mX, weshow
+that it is possible for it to drop again later, after
+dark matter particles form virialized halos.
+2. Dark matter annihilation inside halos
+Unlike the standard model, where the matter
+becomes dominant only around z∼104, in a Uni-
+verse with low ∝angbracketleftσannv∝angbracketrightand high mXthe initial
+transition to matter domination happens much
+earlier. Consequently the growth of density per-
+turbations and the formation of the first gravita-
+1tionally bound halos would be likewise shifted to
+a much earlier epoch.
+The collapse of dark matter particles into halos
+gives a strong boost to the dark matter annihi-
+lation rate. Whereas previously the time-scale for
+particleannihilationwasincreasingfasterthanthe
+age of the Universe, freeze-out of the local density
+inside gravitationally bound halos causes the re-
+versal of this trend. Once the Hubble time begins
+to catch up with the annihilation time-scale, the
+dark matter comoving density is no longer con-
+stant.
+Subsequent evolution of the Universe depends
+on the nature of the dark matter particles. For
+purely collisionless dark matter it can be shown
+(see Appendix A) that the comoving matter den-
+sity eventually approaches an asymptote, ρm(1+
+z)−3∝t−1/4. This moderate rate of decline turns
+out to be insufficient for the Universe to return to
+radiation dominated phase before the nucleosyn-
+thesis. However, annihilation inside halos can be
+greatly sped up if dark matter particles interact
+with the surrounding radiation (made up mostly
+of relativistic quarks and leptons).
+At present weakly interacting massive parti-
+cles (WIMPs) are the most popular candidates
+for the role of dark matter. The collisional cou-
+pling between WIMPs and other species is usu-
+ally assumed to be negligible. At our epoch this
+assumption is generally correct, as the Universe
+temperature is far below the mass scale of W±
+andZbosonswhich mediate the weakforce. How-
+ever, in the early universe, when T/greaterorsimilar300 GeV,
+the coupling becomes very efficient. At high tem-
+peratures WIMPs scattering cross-section rises to
+σelastic∼α2/T2, making the ratio between the
+relaxation time and the age of the Universe
+τrel
+tHubble∼mX+T
+α2mPl, (3)
+wheremPlis the Planck mass. For mX/lessorsimilar1013
+TeV and 0 .1/lessorsimilarT/lessorsimilar1013TeV this ratio drops
+below unity, implying that WIMPs kinetic energy
+is tightly coupled to the radiation.
+Collisional coupling between dark matter and
+radiation has a two-fold effect on the formation
+of self-gravitating dark matter halos, which is ex-
+pected to begin after the matter-radiation equal-
+ity. On small scales, Tvirial< Tγ, the energy
+transfer from radiation to dark matter preventsthecollapseofthelow-masshalos. Onlargescales,
+Tvirial/greaterorsimilarTγ, the effect is the opposite. Since
+self-gravitating objects have negative heat capac-
+ity, the energy loss to the surrounding radiation
+field would lead dark matter particles to collapse
+deeper into the gravitational well, thereby raising
+the virial temperature. Thus the continuous en-
+ergy loss to the background radiation results in a
+gravothermal collapse.1
+The time-scale for particle diffusion in self-
+gravitating objects, which in our case also sets the
+time-scale for gravothermal collapse of larger ha-
+los, is
+τcoll∼∝angbracketleftσelv∝angbracketright
+Gmx, (4)
+whereσel∼α2/T2is the scattering cross-section
+for the weakly interacting particles. If, as gener-
+ally predicted by the inflation models, the density
+perturbation spectrum is scale-invariant, then the
+velocity dispersion of dark matter particles in ha-
+los with a size above the damping scales should
+be of order v∼0.1cand the virial temperature
+T∼0.01mX. Thus in large halos the gravother-
+mal collapse proceeds on a timescale
+τcoll∼103/parenleftBigmX
+1TeV/parenrightBig−3
+sec. (5)
+If halos consist equally of dark matter particles
+and anti-particles (i.e. there is no dark matter
+asymmetry) at some point during the gravother-
+mal collapsenearlyall ofthem mayannihilate into
+radiation. Provided the mass of the particle is
+greater than ∼10 TeV, halos collapse and the
+resulting conversion of dark matter to radiation
+would end before nucleosynthesis and thereafter
+the evolution of the Universe can proceed accord-
+ing to the ”standard” scenario.
+3. Discussion
+Itfollowsfromourscenariothattheexistenceof
+stable particles with high mass and low annihila-
+tion cross-section does not violate the constraints
+on the present matter density. If several species of
+1During the epoch of reionization, 6 /lessorsimilarz/lessorsimilar20, inverse
+Compton scatterings between electrons and the CMB pho-
+tons have a similareffect on baryonic halos. However, there
+the gravothermal collapse is generally avoided either by en -
+ergy injection from stars and miniquasars or by hydrogen
+recombination, which depletes the free electron abundance .
+2such particles existed in the early universe, each
+could have been eliminated in separate periods of
+matter domination followed by return to radiation
+domination.
+To satisfy the constraints on abundance of the
+light elements the last reversal to radiation dom-
+ination must have happened prior to the nucle-
+osynthesis. Nevertheless it is possible for the tran-
+sitional matter dominated phase to leave its im-
+print on the presently observed universe. Besides
+leaving a small remnant of heavy particles, which
+at present compose the dark matter, the transi-
+tional matter dominated phase might affect later
+universe by creating a population of primordial
+black holes and baryon asymmetry.
+The violation of CP-symmetry in the early uni-
+verse (Sakharov 1967) makes it natural to expect
+a small differential between the coupling strengths
+of particles and anti-particles. Inside collapsing
+halos the drag force produced on dark matter
+particles through elastic scatterings with the es-
+caping radiation may be different from the force
+on dark matter anti-particles. As a result, spa-
+tial segregation of dark matter particles and anti-
+particles would be produced. Similarly, a dif-
+ferential drag by the infalling dark matter par-
+ticles on the outflowing quarks and anti-quarks
+(or leptons and anti-leptons) would produce lo-
+cal over/underabundancesof baryons and leptons.
+During halos gravothermal collapse the existence
+of such segregation would make it impossible for
+the entirehalo to annihilate intoradiation. There-
+fore the collapse of particles (or anti-particles)
+dominating at the halo center should lead to the
+formation of a black hole.
+Since black holes do not conserve baryon num-
+bers (Hawking 1974; Zeldovich 1976; Carr 1976),
+formation of a central black hole would convert a
+local particle asymmetry into a global one. De-
+pending on their initial mass, these primordial
+black holes may have evaporated long before the
+present epoch, possibly further amplifying the
+matter-antimatterasymmetryin the process(Dol-
+gov 1982). Alternatively, if their mass was suffi-
+ciently large, significant population of primordial
+black holes may still be found in our Universe.
+TheauthorthanksE.W.Kolb,M.S.TurnerandM. Beltran for stimulating discussions.
+Appendix A: Annihilation of collisionless
+particles in virialized halos
+We consider the evolution of matter den-
+sity field, assuming the annihilation cross-section
+scales as ∝angbracketleftσannv∝angbracketright ∝v−xat non-relativistic veloci-
+ties. While it is common to assume x= 0, many
+new particle candidates with x= 1 have been
+proposed (e.g. Hisano et al. 2004; Profumo 2005;
+Lattanzi & Silk 2008). Here we shall assume that
+xcan take any value in the physically plausible
+range 0≤x <2.
+After decoupling from radiation matter den-
+sity drops to a point when the annihilation time-
+scale,tann= (nx∝angbracketleftσannv∝angbracketright)−1, becomes longer than
+the Hubble time, so that the comoving density is
+frozen. As long as the density field remains nearly
+homogeneous with the density and velocity dis-
+persion falling respectively as a−3anda−2with
+the increasing scale factor, a,tann/tHubblecontin-
+ues to rise. However, this trend is reversed af-
+ter matter collapses into halos. Inside an isolated
+gravitationally bound halos both the local density
+and velocity dispersion remain roughly constant,
+freezing the growth of tannuntil it becomes com-
+parable with tHubble. At this point a halo would
+start losing mass at a significant rate via parti-
+cle annihilation. The subsequent halo evolution of
+can be described as following.
+The mass loss from particle annihilation and
+the resulting decrease of binding energy would
+cause the halo to expand, so that the annihila-
+tion time-scale tracks the age of the halo. Because
+the dynamic time is much shorter than tHubble,
+the halo always remains close to dynamical equi-
+librium. Since the gravitational binding energy is
+proportional to Eg∝M2/R, whereMandRare
+the total massandthe radiusofthe halo, changing
+the mass and radius by dManddRrespectively,
+changesEgby
+dEg=Eg/parenleftbigg
+2dM
+M−dR
+R/parenrightbigg
+. (6)
+The change of halo kinetic energy, Ek, in the an-
+nihilation process depends on the velocity depen-
+dence of σannand the particle velocity distribu-
+tion. Assuming Maxwell-Boltzmann distribution,
+we find that the average kinetic energy of an an-
+3nihilating particle is (1 −x/2)(mX/angbracketleftbig
+v2/angbracketrightbig
+/2), where/radicalbig
+∝angbracketleftv2∝angbracketrightis the local velocity dispersion. Thus
+dEk
+Ek=/parenleftBig
+1−x
+2/parenrightBigdM
+M. (7)
+Combining the virial theorem, Eg=−2Ek, with
+equations (6) and (7), we find that the radius
+of the halo increases with the falling mass as
+R∝M−1+x/2. Further, by using the scalings
+ρ∝M/R3,v∝/radicalbig
+M/Rand (ρv−x)∝t−1, we
+find that the total mass of the halo falls with time
+asM∝tµ,µ=−4/(16+2x−x2). For the entire
+range 0≤x <2,µis confined to a narrow range
+−0.25≤x <−0.235.
+Modeling the Universe as an ensemble of iso-
+lated halos, we can follow the global evolution of
+matter and radiation density fields
+d2a
+adt2=−4πG
+3(ρm+2ργ),(8)
+dρm
+dt=−/parenleftbigg
+3da
+adt−µ
+t/parenrightbigg
+ρm,(9)
+dργ
+dt=−4da
+adt−µ
+tρm, (10)
+whereργandρmare respectively, the mean densi-
+ties ofradiation and matter and ais the expansion
+factor. Solving the equations, we find the asymp-
+totic solutions
+a∝t(2+µ)/3, (11)
+ρm=2+5µ+2µ2
+12πGt2, ργ=−(2+µ)µ
+8πGt2,(12)
+ρm
+ργ=−2(1+2µ)
+3µ. (13)
+Forµ≈ −1/4 we get Ω m≈4/7 and Ω γ≈3/7.
+REFERENCES
+Carr B.J. 1976, ApJ, 206, 8
+ChungD. J.H., KolbE.W., RiottoA. 1998,Phys.
+Rev. Lett. 81, 4048
+Dolgov A.D. 1992, Phys. Rept. 222, 309Griest K., Kamionkowski M. 1990, Phys. Rev.
+Lett., 64, 615
+Hawking S. W. 1974, Nature 248, 30
+Hisano J., Matsumoto S., & Nojiri M. M. 2004,
+Phys. Rev. Lett., 92, 031303
+Kolb E.W., Turner M.S., ”The Early Eniverse”
+(Addison-Wesley, Redwood City)
+Lattanzi M., & Silk J. 2009, Phys. Rev. D., 79,
+3523
+Profumo S., 2005, Phys. Rev. D 72, 103521
+Sakharov A.D. 1967, JETP, 5, 24
+Zeldovich Ya. B., 1976, JETP Lett. 24, 25
+This 2-column preprint was prepared with the AAS L ATEX
+macros v5.2.
+4
\ No newline at end of file
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+arXiv:1001.0811v1  [math.GR]  6 Jan 2010ON TYPES AND CLASSES OF COMMUTING MATRICES
+OVER FINITE FIELDS
+JOHN R. BRITNELL AND MARK WILDON
+Abstract. This paper addresses various questions about pairs of simi-
+larity classes of matrices which contain commuting element s. In the case
+of matrices over finite fields, we show that the problem of dete rmining
+such pairs reduces to a question about nilpotent classes; th is reduction
+makes use of class types in the sense of Steinberg and Green. W e in-
+vestigate the set of scalars that arise as determinants of el ements of the
+centralizer algebra of a matrix, providing a complete descr iption of this
+set in terms of the class type of the matrix.
+Several results are established concerning the commuting o f nilpotent
+classes. Classes which are represented in the centralizer o f every nilpo-
+tent matrix are classified—this result holds over any field. N ilpotent
+classes are parametrized by partitions; we find pairs of part itions whose
+corresponding nilpotent classes commute over some finite fie lds, but not
+over others. We conclude by classifying all pairs of classes , parametrized
+by two-part partitions, that commute. Our results on nilpot ent classes
+complement work of Koˇ sir and Oblak.
+1.General introduction
+LetFqbe a finite field, and let CandDbe classes of similar matrices in
+Matn(Fq). We say that CandDcommute if thereexistcommuting matrices
+XandYsuch that X∈CandY∈D. In this paper we are concerned with
+the problem of deciding which similarity classes commute.
+A matrix is determined up to similarity by its rational canon ical form.
+This however is usually too sharp a tool for our purposes, and many of
+our results are instead stated in terms of the class type of a matrix. This
+notion, which seems first to have appeared in the work of Stein berg [14], is
+important in Green’s influential paper [8] on the characters of finite general
+linear groups. Lemma 2.1 of that paper implies that the type o f a matrix
+determines its centralizer up to isomorphism; this fact is a lso implied by our
+Theorem 2.7, which says that two matrices with the same class type have
+conjugate centralizers.
+Date: 15 December 2009.
+12 JOHN R. BRITNELL AND MARK WILDON
+The main body of this paper is divided into three sections. In §2 we
+develop a theory of commuting class types; the results of thi s section re-
+duce the general problem of determining commuting classes t o the case of
+nilpotent classes. A key step in this reduction is Theorem 2. 8, which states
+that if similarity classes CandDcommute, then any class of the type of C
+commutes with any class of the type of D.
+Relationships between class types and determinants are dis cussed in §3.
+We provide a complete account of those scalars which appear a s determi-
+nants in the centralizer of a matrix of a given type; this resu lt, stated as
+Theorem 3.1, has appeared without proof in [2, §3.4], and as we promised
+there, we present the proofhere. We also discuss theproblem of determining
+which scalars appear as the determinant of a matrix of a given type. This
+problem appears intractable in general, and we provide only a very partial
+answer. But we identify a special case of the problem which le ads to a dif-
+ficult but highly interesting combinatorial problem, to whi ch we formulate
+Conjecture 3.3 as a plausible solution.
+In§4 we make several observations concerning the problem of com muting
+nilpotent classes; this is a problem which has attracted att ention in several
+different contexts over the years, and there is every reason to suppose that
+it is hard. Among other results, we determine in Theorem 4.6 t he nilpotent
+classes which commute with every other nilpotent class of th e same dimen-
+sion, andinTheorem4.10weclassify allpairsofcommutingn ilpotentclasses
+of matrices whose nullities are at most 2. We describe a const ruction on ma-
+trices which produces interesting and non-obvious example s of commuting
+nilpotent classes. This construction motivates Theorem 4. 8, which says that
+for every prime pand positive integer r, there exists a pair of classes of
+nilpotent matrices which commute over the field Fpaif and only if a > r.
+As far as the authors are aware, it has not previously been obs erved that the
+commuting of nilpotent classes, as parameterized by partit ions, is dependent
+on the field of definition.
+More detailed outlines of the results of §2,§3 and§4 are to be found at
+the beginnings of those sections.
+1.1.Background definitions. We collect here the main prerequisite defi-
+nitions concerning partitions, classes and class types tha t we require.
+Partitions. Wedefinea partition tobeaweakly decreasingsequenceof finite
+length whose terms are positive integers; these terms are ca lled thepartsofTYPES AND CLASSES OF COMMUTING MATRICES 3
+the partition. We shall denote the j-th part of a partition λbyλ(j). The
+sum of the parts of λis written as |λ|.
+Given partitions λandµ, we write λ+µfor the partition of |λ|+|µ|
+whose multiset of parts is the union of the multisets of parts ofλand ofµ.
+We shall write 2 λforλ+λ, and similarly we shall define tλfor all integers
+t∈N0. A partition µwill be said to be t-divisible if it is expressible as tλ
+for some partition λ; ifsλ=tµthen we may write µ=s
+tλ.
+We shall require the dominance order /trianglerightequalon partitions. For two partitions
+λandµwe say that λdominates µ, and write λ/trianglerightequalµ(orµ/triangleleftequalλ) if
+j/summationdisplay
+i=1λ(i)≥j/summationdisplay
+i=1µ(i)
+for allj∈N. (Ifiexceeds the number of parts in a partition, then the
+corresponding part is taken to be 0.)
+Letλbe a partition with largest part λ(1) =a. Theconjugate partition
+λis defined to be ( λ(1),...,λ(a)), where λ(j) is the number of parts of λof
+size at least j. It is a well-known fact (see for instance [11, 1.11]) that th e
+conjugation operation on partitions reverses the dominanc e order; that is,
+λ/trianglerightequalµif and only if µ/trianglerightequalλ.
+A geometric interpretation of the dominance order is develo ped by Ger-
+stenhaber in [5] and [6]; the issues with which the latter pap er is concerned
+are similar in many respects to those considered in §4 of the present paper,
+although Gerstenhaber’s approach using algebraic varieti es is very different.
+Similarity classes. LetKbe a field. A class of similar matrices in Mat n(K)
+is determined by the following data: a finite set Fof irreduciblepolynomials
+overK, and for each f∈ Fa partition λfof a positive integer, such that
+n=/summationdisplay
+f∈F|λf|degf.
+The characteristic polynomial of a matrix Min this class is/producttext
+ff|λf|. There
+is a decomposition of Vgiven by
+V=/circleplusdisplay
+f/circleplusdisplay
+jVf(j),
+whereMacts indecomposably on the subspace Vf(j) with characteristic
+polynomial fλf(j). This decomposition is, in general, not unique. By a
+change of basis, we may express Mas/circleplustext
+f/circleplustext
+jPf(j), where Pf(j) is a matrix
+representing the action of MonVf(j); we say that Pf(j) is acyclic block
+ofM.4 JOHN R. BRITNELL AND MARK WILDON
+IfF={f1,...,ft}and the associated partitions are λ1,...,λ trespec-
+tively, then we shall define the cycle type ofMto be the formal expression
+cyc(M) =fλ1
+1···fλt
+t.
+The order in which the polynomials appear in this expression is, of course,
+unimportant.
+Nilpotent classes. We shall denote by N(λ) the similarity class of nilpotent
+matrices withcycle type fλ
+0, wheref0(x) =x. We denoteby J(λ) theunique
+matrix in upper-triangular Jordan form in the similarity cl assN(λ).
+Ifλ= (λ(1),...,λ(k)) we shall omit unnecessary brackets by writing
+N(λ(1),...,λ(k)) forN(λ) andJ(λ(1),...,λ(k)) forJ(λ).
+Class types. More general than the notion of similarity class is that of cl ass
+type. If Mis a matrix of cycle type fλ1
+1···fλt
+t, where for each ithe poly-
+nomialfihas degree di, then the class type ofMis the formal string
+ty(M) =dλ1
+1···dλt
+t.
+Here too, the order of the terms is unimportant.
+Any string of this form will be called a type. Thedimension of the type
+dλ1
+1···dλt
+tis defined to be d1|λ1|+···+dt|λt|. We shall say that the type T
+isrepresentable over a field Kif there exists a matrix of class type Twith
+entries in K; the dimension of such a matrix is the same as the dimension of
+the type. Clearly not all types are representable over all fie lds; for instance
+the type1λ1µ1νis not representable over F2since there are only two distinct
+linear polynomials over this field; similarly 3λis not representable over R
+since there are no irreducible cubics over R.
+Similar matrices have the same cycle type and the same class t ype, and
+so we may meaningfully attribute types of either kind to simi larity classes.
+We shall say that a class type Tisprimary if it isdλfor some dandλ.
+Otherwise Tiscompound . Ifdλappears as a term in the type T, we say that
+dλis aprimary component ofT. We may also say that a matrix, a similarity
+class of matrices, or a cycle type is primary or compound, acc ording to its
+class type, and we may refer to its primary components.
+We have already defined what it means for two similarity class es to com-
+mute. We generalise this idea to types, as follows.
+Definition. LetSandTbe class types. We say that SandTcommute
+over a field Kif there are matrices XandYoverKsuch that Xhas class
+typeS, andYhas class type T, andXandYcommute.TYPES AND CLASSES OF COMMUTING MATRICES 5
+The field Kwill not always be mentioned explicitly if it is clear from th e
+context.
+2.Commuting types of matrices
+This section proceeds as follows. In §2.1 we prove several results relating
+the class type of a polynomial in a matrix Mto the class type of M, leading
+up to Theorem 2.6: that two similarity classes have the same c lass type if
+andonlyiftheycontain representatives whicharepolynomi alinoneanother.
+This result is then used in the proof of Theorem 2.8, which sta tes that two
+similarity classes commute if and only if their class types c ommute.
+Using Theorem 2.8, we proceed to reduce our original problem of deciding
+which similarity classes commute, first to the case of primar y types in §2.2,
+and thence to the case of nilpotent classes in §2.3. At the end of §2.3 we
+give examples illustrating both steps of this reduction.
+2.1.Polynomials and commuting types. IfMis a matrix of primary
+class type dλthen it has associated with it a single irreducible polynomi al
+fsuch that its cycle type is fλ. It is clear that f(M) is nilpotent. The
+following lemma and proposition describe its associated pa rtition.
+Lemma 2.1. LetMbe a matrix of cycle type fλ, wheredegf=d. For
+eachj, letmjbe the number of parts of λof sizej. Then
+dmj= (nullf(M)j−nullf(M)j−1)−(nullf(M)j+1−nullf(M)j).
+Proof.LetPbe a cyclic block of M. If the dimension of Pisdhthen
+the characteristic polynomial of Pisfh. Ifj≥h, then null f(P)j=dh;
+otherwise null f(P)j=dj.
+SinceMis a direct sum of cyclic blocks of dimensions dλ(1),dλ(2),..., it
+follows that
+nullf(M)j=/summationdisplay
+h≤jdhmh+/summationdisplay
+h>jdjmh,
+and hence
+nullf(M)j+1−nullf(M)j=/summationdisplay
+h>jdmh.
+This implies the lemma. /square
+Proposition 2.2. LetMbe a matrix of primary type dλ. If the cycle type
+ofMisfλthenf(M)is nilpotent of type 1dλ.
+Proof.Sincef(M) is nilpotent, it is primary and its associated polynomial,
+f0(x) =x, is linear. The result is now immediate from Lemma 2.1. /square6 JOHN R. BRITNELL AND MARK WILDON
+We use the preceding proposition to give some information ab out the
+type ofF(M), where Mis a primary matrix and Fis any polynomial. The
+following lemma will be required.
+Lemma 2.3. LetMandNbe nilpotent matrices with associated partitions
+µandνrespectively. Then µ/triangleleftequalνif and only if rankMj≤rankNjfor all
+j∈N.
+Proof.The rank of Mjis equal to the sum of the jsmallest parts of the
+conjugate partition µ. The rank of Njcan be calculated similarly in terms
+ofν. It follows easily that rank Mj≤rankNjfor alljif and only if µ/trianglerightequalν.
+Thelemma now follows from thefact that thedominanceorder /trianglerightequalis reversed
+by conjugation of partitions. /square
+Proposition 2.4. LetXbe a primary matrix of class type dλwith entries
+from a field K, and let F∈K[x]be any polynomial. The type of F(X)is
+eµfor some edividing d, and some partition µsuch that e|µ|=d|λ|and
+eµ/triangleleftequaldλ.
+Proof.Let the cycle type of Xbefλwherefis an irreducible polynomial
+of degree d. Ifαis a root of fin a splitting field, then the eigenvalues of
+F(X) are the conjugates over KofF(α). Hence F(X) is of primary type,
+and ifg∈K[x] is the irreducible polynomial associated with F(X), then
+the degree of gdividesd. Leteµbe the type of F(X).
+LetY=F(X). We observe that g(Y) = (g◦F)(X) is a nilpotent matrix,
+and hence fdividesg◦F; letg◦F=kf. By Proposition 2.2, f(X) has type
+1dλ, whileg(Y) has type 1eµ. For each i∈N0we haveg(Y)i=k(X)if(X)i
+and hence im g(Y)i⊆imf(X)i. It follows that rank g(Y)i≤rankf(X)ifor
+everyi∈N. Now from Lemma 2.3 we see that eµ/triangleleftequaldλ, as required. /square
+WhenKisafinitefield, Proposition2.4hasthefollowingpartial co nverse.
+Proposition 2.5. IfXis a primary matrix of class type dλwith entries
+fromFq, andDis a similarity class of matrices also of this class type, the n
+there is a polynomial F∈Fq[x]such that F(X)∈D.
+Proof.Letf∈Fq[x] be the irreducible polynomial associated with X. Sup-
+posethattheadditiveJordan–Chevalleydecompositionof XisX+N, where
+Xis semisimple and Nis nilpotent; recall that XandNcan be expressed
+as polynomials in X. Without loss of generality, we may suppose that
+X= diag(P,...,P),TYPES AND CLASSES OF COMMUTING MATRICES 7
+where the cyclic block Phas minimum polynomial f.
+Letgbe the irreducible polynomial associated with the similari ty class
+D, and let αandβbe roots of fandgrespectively in Fqd. There exists a
+polynomial G∈Fq[x], coprime with f, such that G(α) =β. If we define
+Q=G(P),
+thenQhas minimum polynomial g. Let
+Y= diag(Q,...,Q ).
+ThenY=G(X), and since Xis polynomial in X, it follows that Yis too.
+Moreover, if we set Y=Y+N, thenYis polynomial in X, and it is clear
+thatYlies in the similarity class D. /square
+LetCandDbe similarity classes of Mat n(Fq). We say that Dispoly-
+nomial in Cif there exists a polynomial Fwith coefficients in Fqsuch that
+F(X)∈Dfor allX∈C.
+Theorem 2.6. LetCandDbe similarity classes of Matn(Fq). The classes
+CandDhave the same type if and only if CandDare polynomial in one
+another.
+Proof.We observe that applying a polynomial to a matrix cannot incr ease
+its number of primary components. So if CandDare polynomial in one
+another, then they have the same number of components. Moreo ver there is
+a pairing between the primary components C1,...,C tofCandD1,...,D t
+ofDsuch that CiandDiare polynomial in one another for all i. It will
+therefore be sufficient to prove the result in the case that bot hCandDare
+primary. Suppose that ty( C) =dλfor some d∈Nand some partition λ. It
+follows from Proposition 2.4 that Dhas class type eµwhereedividesdand
+eµ/triangleleftequaldλ. By symmetry we see that e=dandλ=µ, as required.
+For the converse, suppose that ty( C) = ty(D). LetT1,T2,...,T tbe the
+primary components of ty( C), and let X= diag(X1,...,X t) be an element
+ofCsuch that ty( Xi) =Tifor alli. Let the minimum polynomial of the
+blockXibefai
+i, where fiis irreducible. By Proposition 2.5, there exist
+polynomials F1,...,F t∈Fq[x] such that diag( F1(X1),...,F t(Xt))∈D. By
+the Chinese Remainder Theorem, there exists a polynomial F∈Fq[x] such
+that
+F(x)≡Fi(x) modfai
+i(x) for all i.
+And now we see that F(X)∈D, as required. /square8 JOHN R. BRITNELL AND MARK WILDON
+It was proved by Green [8, Lemma 2.1] that the type of a matrix d eter-
+mines its centralizer up to isomorphism. Using Theorem 2.6 w e may prove
+the following stronger result.
+Theorem 2.7. LetXandYbe matrices in Matn(Fq)with the same class
+type. Let CentXandCentYbe the centralizers in Matn(Fq)ofXand
+Yrespectively. Then CentXandCentYare conjugate by an element of
+GLn(Fq).
+Proof.By Theorem 2.6 there exist polynomials FandGsuch that F(X) is
+conjugate to YandG(Y) is conjugate to X. Now the centralizer Cent F(X)
+is a subalgebra of Cent Xwhich is conjugate to Cent Y; similarly the cen-
+tralizer Cent G(Y) is a subalgebra of Cent Ywhich is conjugate to Cent X.
+Since Cent Xand Cent Yare finite, it is clear that Cent X= CentF(X)
+and that Cent Y= CentG(Y), which suffices to prove the theorem. /square
+An obvious corollary of Theorem 2.6, which has been stated in [3,§3.2], is
+that classes of thesametypecommute. We arenow in aposition toestablish
+a stronger result. Recall that types SandTare said to commute if there
+exist commuting matrices XandYwith types SandTrespectively.
+Theorem 2.8. LetCandDbe similarity classes of matrices over Fq. Then
+CandDcommute if and only if ty(C)andty(D)commute.
+Proof.One half of the double implication is trivial, since if the si milarity
+classes commute then by definition the class types do. For the other half,
+notice that if ty( C) and ty( D) commute then there exist commuting simi-
+larity classes C′andD′such that ty( C′) = ty(C) and ty( D′) = ty(D). Let
+X′andY′be commuting matrices from C′andD′respectively. Then there
+exist polynomials FandGsuch that F(X′)∈CandG(Y′)∈D, and clearly
+F(X′) andG(Y′) commute. /square
+We remark that Theorems 2.6, 2.7 and 2.8 do not hold for matric es over
+an arbitrary field. There are counterexamples in Mat 2(R), for instance. Let
+R(α) andR(β) be distinct quadratic extensions of R. LetCandDbe the
+similarity classes of rational matrices with characterist ic polynomials x2−α
+andx2−βrespectively; then ty( C) = ty(D) = 2(1). Since the eigenvalues α
+andβare not polynomial in one another, it is clear that neither ar eCand
+D. Moreover, the classes CandDdo not commute. It is for this reason
+that our consideration of commuting types is for the most par t restricted to
+matrices with entries from a finite field.TYPES AND CLASSES OF COMMUTING MATRICES 9
+2.2.Reduction to primary types. The next step in our strategy is to
+reducethequestion of whichclass typescommute to thecorre spondingques-
+tion about primary types. This is accomplished in Propositi on 2.9 below.
+We shall need the following two definitions.
+Definition. Aseparation operation on a type Tis the replacement of a
+primary component dλofTbydµdν, whereλ=µ+ν. Aseparation ofTis
+a type obtained from Tby repeated applications of separation operations.
+Definition. LetSandTbe types. We shall say that SandTcommute
+componentwise over a field Kif the primary components of SandTcan
+be ordered so that S=cλ1
+1···cλt
+tandT=dµ1
+1···dµk
+t, wherecλi
+icommutes
+withdµi
+ioverKfor each i.
+This definition, it should be noted, does not preclude the pos sibility that
+typesSandTcommute componentwise, even if one or both of them cannot
+be represented over the field K. For example, 1(1,1,1)commutes componen-
+twise with 1(1)1(1)1(1)overF2according to the definition, even though the
+latter type is not representable. The examples at the end of §2.3 illustrate
+why this freedom is desirable.
+Proposition 2.9. LetSandTbe types which are representable over a finite
+fieldFq. ThenSandTcommute over Fqif and only if there exist separa-
+tionsS⋆ofSandT⋆ofTsuch that S⋆andT⋆commute componentwise.
+Proof.LetXandYbe commuting matrices with entries from Fq, whose
+types are SandTrespectively. It is well known and easy to show that there
+exists a decomposition V=V1⊕···⊕Vtsuch that both XandYact as
+transformations of primary type on each of the summands Vi. Suppose that
+the action of XonVihas type cλi
+i, and the action of Yhas type dµi
+i. Then
+it is clear that the primary types cλi
+ianddµi
+icommute, that cλ1
+1···cλt
+tis a
+separation of Sand that dµ1
+1···dµt
+tis a separation of T.
+For the converse, suppose that the primary types cλi
+ianddµi
+icommute,
+thatcλ1
+1···cλt
+tis a separation of Sand that dµ1
+1···dµt
+tis a separation of T.
+Then, from Theorem 2.8, it follows that for any choice of irre ducible poly-
+nomialsfiof degree ciandgiof degree di, the classes fλi
+iandgµi
+icommute.
+IfXiandYiare commuting representatives of these respective classes , then
+thematrices X= diag(X1,...,X t)andY= diag(Y1,...,Y t)commute. Now
+each primary type cλi
+iderives (under separation operations) from a partic-
+ular component of S. If we select our polynomials fiin such a way that
+blocks deriving from the same component of Shave the same polynomial,10 JOHN R. BRITNELL AND MARK WILDON
+then we find that ty( X) =S. Similarly we can choose the polynomials giso
+that ty(Y) =T, and it follows that SandTcommute. /square
+2.3.Reduction to nilpotent classes. We now complete the reduction of
+our general problem of commuting classes to the case of nilpo tent classes.
+Recall that we denote by N(λ) the similarity class of nilpotent matrices with
+cycle type fλ
+0, wheref0(x) =x. Recall also that a partition is said to be
+t-divisible if it is tνfor some partition ν.
+Theorem 2.10. LetS=cλandT=dµbe primary types of the same
+dimension. Let h= hcf(c,d)andℓ= lcm(c,d). ThenSandTcommute
+overFqif and only if λisd
+h-divisible, µisc
+h-divisible, and the nilpotent
+classesN(h
+dλ)andN(h
+cµ)commute over Fqℓ.
+Proof.Supposethat SandTcommute over Fq. LetXandYbecommuting
+elements of Mat n(Fq) withcycle types fλandgµrespectively, wheredeg f=
+cand degg=d. Letα1,...,α cbe the roots of fandβ1,...,β dthe roots of
+gin the extension field Fqℓ. Over this extension field, it is easy to see that
+the cycle types of XandYare given by
+cyc(X) = (x−α1)λ···(x−αc)λ,
+cyc(Y) = (x−β1)µ···(x−βd)µ.
+LetW=Fn
+qℓ, and let Wijdenote the maximal subspace of Won which
+X−αiIandY−βjIare both nilpotent. (So W=/circleplustext
+ijWij.) Letλijandµij
+be the partitions such that the type of XonWijis 1λijand the type of Y
+onWijis 1µij. Then clearly/summationtextd
+j=1λij=λfor alli, while/summationtextc
+i=1µij=µfor
+allj.
+SinceXandYhave entries in Fq, it follows that the Frobenius automor-
+phismξ/mapsto→ξqofFqℓinducesanisomorphismbetweenthe Fqℓ/a\}bracketle{tX,Y/a\}bracketri}ht-modules
+VijandVi′j′whenever i−j≡i′−j′modh. Hence
+λij=λi′j′andµij=µi′j′whenever i−j≡i′−j′modh.
+Therefore the partitions λijfori∈ {1,...,c}andj∈ {1,...,d}are deter-
+mined by the partitions λ1kfork∈ {1,...,h}, and since
+λ=d
+hh/summationdisplay
+k=1λ1k,
+it follows that λisd
+h-divisible. Similarly, µisc
+h-divisible.
+Now clearly the actions of XandYon the subspace/circleplustexth
+k=1V1kcommute.
+The type of Xon this submodule (defined over Fqℓ) is 1λ11···1λ1h, which isTYPES AND CLASSES OF COMMUTING MATRICES 11
+aseparation of1h
+dλ. Similarlythetypeof Yonthesubmoduleisaseparation
+of 1h
+cµ. Hence, by the ‘if’ direction of Proposition 2.9, the types 1h
+dλand
+1h
+cµcommute over Fqℓ. In particular, it follows from Theorem 2.8 that the
+nilpotent classes N(h
+dλ) andN(h
+cµ) commute over this field.
+For the converse, let λ′=h
+dλandµ′=h
+cµ, and supposethat the nilpotent
+classesN(λ′)andN(µ′)commuteover Fqℓ. Weshalldenoteby mtheinteger
+|λ′|, which of course is equal to |µ′|. Letαandβbe elements of Fqℓwhose
+degrees over Fqarecanddrespectively. Since N(λ′) andN(µ′) commute
+overFqℓ, so do the classes with cycle types ( x−α)λ′and (x−β)µ′. LetX
+andYbe commuting elements of these respective classes. Let φbe an
+embedding of the matrix algebra Mat m(Fqℓ) into Mat ℓm(Fq); then it is not
+hard to see that φ(X) has class type cλandφ(Y) has class type dµ. It
+follows that these types commute over Fq. /square
+It is worth noting that Theorem 4.8 below implies that the ref erences
+to particular fields in the statement of Theorem 2.10 are esse ntial. The
+following special case of the theorem, however, does not dep end on the field
+of definition.
+Proposition 2.11. Letd,k∈N. The types d(k)and1(k,...,k)commute over
+any field.
+Proof.Ifthefieldinquestion isfinite, thenthepropositionfollow s fromThe-
+orem 2.10. For it suffices to show that the type1
+d(k,...,k) = (k) commutes
+with itself over Fqd, and certainly this is the case.
+A straightforward modification of the last paragraph of the p roof of The-
+orem 2.10 would allow us to deal with arbitrary fields; howeve r we prefer the
+following short argument involving tensor products. Let fbe an irreducible
+polynomial of degree dand letPbe the companion matrix of f. The type
+d(k)is represented by the dk×dkmatrix
+P(k)=
+P I
+P I
+......
+P
+.
+LetJ=J(k) be the k-dimensional Jordan block with eigenvalue 1. It is
+clear that P(k)commutes with the tensor product I⊗J(which is obtained
+from the matrix above by substituting Ifor each occurrence of P). And
+I⊗Jis conjugate to J⊗I= diag(J,...,J), which has type 1(k,...,k). Hence
+the types d(k)and 1(k,...,k)commute. /square12 JOHN R. BRITNELL AND MARK WILDON
+We end this section with two examples of how the steps in our re duction
+can be carried out, which illustrate the various results of t his section.
+Example. Letp,q,r,sandtbe the following irreducible polynomials
+overF2:
+linear: p(x) =x, q(x) =x+1;
+quadratic: r(x) =x2+x+1;
+cubic: s(x) =x3+x+1, t(x) =x3+x2+1.
+LetCbethesimilarityclassofmatricesover F2withcycletype p(12,12)q(2,2,2)r(3)s(1)
+and letDbe the similarity class with cycle type r(7,5)t(2,2,1). We shall prove
+thatCcommutes with D.
+By Theorem 2.8, this is equivalent to showing that the types
+S= 1(12,12)1(2,2,2)2(3)3(1),
+T= 2(7,5)3(2,2,1)
+commute. This, in turn, will follow from Proposition 2.9, if we can show
+thatScommutes componentwise with the separation T⋆= 2(7,5)3(2)3(2)3(1)
+ofT. (This example was chosen to make the point that it is not nece ssary
+that the separated types can be represented over F2.) By Theorem 2.10 we
+see that 1(12,12)commutes with 2(7,5)overF2if and only if 1(6,6)commutes
+with 1(7,5)overF4; that this is the case follows from Proposition 4.7 below,
+which implies that the nilpotent classes N(6,6) andN(7,5) commute over
+F4. It is immediate from Theorem 2.10 that 1(2,2,2)commutes with 3(2), and
+that 2(3)commutes with3(2). HenceSandT⋆commutecomponentwise, and
+soCandDcommute.
+TheconversedirectionsofProposition2.9andTheorem2.10 caninprinci-
+plebeusedaspartofaargumentthat twosimilarity classes d onotcommute;
+again, results about commuting of nilpotent classes will ge nerally be needed
+to complete such an argument. The following example is illus trative.
+Example. Let the polynomials p,q,r,sandt, the class C, and the type
+Sbe as in the previous example. Let Dbe the similarity class over F2with
+cycle type r(8,4)t(2,2,1). The class type of Dis
+T= 2(8,4)3(2,2,1).
+Suppose that a separation T∗ofTcommutes componentwise with a sepa-
+rationS∗ofS; then one of 2(8)or 2(8,4)is a component of T∗. The first
+possibility is ruled out since S∗can have no component of dimension 16.TYPES AND CLASSES OF COMMUTING MATRICES 13
+The only possible component of S∗of dimension 24 is 1(12,12), and so if
+our supposition is correct, then the primary types 2(8,4)and 1(12,12)must
+commute over F2. By Theorem 2.10, this is the case only if 1(8,4)and 1(6,6)
+commute. But by Proposition 4.9 below, the nilpotent classe sN(8,4) and
+N(6,6) do not commute over any field. It follows that CandDdo not
+commute.
+3.Types and determinants
+The main object of this section is to establish Theorem 3.1, c oncerning
+determinants of elements of centralizer algebras. The foll owing definition is
+key.
+Definition. LetMbe a matrix with class type dλ1
+1···dλt
+t. Thepart-size
+invariant ofMis defined to be the highest common factor of all of the parts
+of the partitions λ1,...,λ t.
+Theorem 3.1. LetM∈Matn(Fq)have part-size invariant k. The deter-
+minants which occur in the centralizer of MinMatn(Fq)are precisely the
+k-th powers in Fq.
+Part of the motivation for this investigation comes from the authors’ pa-
+per [2] on the distribution of conjugacy classes of a group Gacross the cosets
+of a normal subgroup H, whereG/His abelian. The centralizing subgroup
+of a class Cwith respect to Hwas defined to be the subgroup Cent G(g)·H,
+whereg∈Cmay be chosen arbitrarily. It was proved that if Gis finite and
+G/His cyclic, then the classes with centralizing subgroup Kare uniformly
+distributed across the cosets of HinK.
+Theorem 3.1 treats the case where G= GLn(Fq) andH= SLn(Fq). It is
+clear that the subgroups Klying in the range H≤K≤Gmay bedefined in
+terms of the determinants of their elements; specifically, t he index |K:H|is
+equal to the order of the subgroup of F×
+qgenerated by the determinants of
+the matrices in K. Hence, in order to calculate the centralizing subgroup of
+a matrix, we must decide which determinants occur in its cent ralizer. The
+following corollary of Theorem 3.1 shows that the answer to t his question
+depends only on the class type of the matrix concerned.
+Corollary 3.2. LetM∈GLn(Fq)have part-size invariant k, and let c=
+hcf(q−1,k). The centralizing subgroup of the conjugacy class of Mis the
+unique index csubgroup of GLn(Fq)containing SLn(Fq).14 JOHN R. BRITNELL AND MARK WILDON
+In§3.1 below we prove a special case of Theorem 3.1, namely that t he
+determinants in the centralizer of a nilpotent matrix are k-th powers, where
+kis the part-size invariant. The proof of Theorem 3.1 is compl eted in§3.2.
+We end in §3.3 by discussing the natural—but surprisingly hard—quest ion
+of which scalars can appear as the determinant of a matrix of a given type.
+3.1.Determinants in the centralizer of a nilpotent matrix. In this
+section we let M∈Matn(Fq) be a nilpotent matrix lying in the similarity
+classN(λ). LetA= CentMbe the subalgebra of Mat n(Fq) consisting of
+the matrices that centralize M. We shall find the composition factors of
+V=Fn
+qas a right A-module; using this result we describe the determinants
+of the matrices of A. For some related results on the lattice of A-submodules
+ofV, the reader is referred to [7, Chapter 14].
+Definition. Forv∈Vwe define the heightofv, written ht( v), to be the
+least integer hsuch that v∈kerMh.
+Definition. We shall say that a vector u∈Vis acyclic vector forMifu
+is not in the image of M.
+The proof of the following well-known lemma is straightforw ard, and is
+omitted.
+Lemma 3.3. An element Y∈Ais uniquely determined by its effect on the
+cyclic vectors of M. Ifu1,...,u tare linearly independent cyclic vectors and
+v1,...,vtare any vectors such that ht(vi)≤ht(ui)for every i, then there is
+an element Y∈Asuch that uiY=vifor each i.
+As in Lemma 2.1, we let mhbe the number of parts of λof sizeh. For
+h∈N0, we shall write Vhfor kerMh.
+Proposition 3.4. For each h∈N, the subspace Vhis anA-submodule of V
+containing Vh+1M+Vh−1as anA-submodule. Moreover if mh/\e}atio\slash= 0then
+Vh/ (Vh+1M+Vh−1)
+is a simple A-module of dimension mh.
+Proof.The proof of the first statement is straightforward, and we om it it;
+we shall outline a proof of the second statement.
+Letu1,...,u mhbe a maximal set of linearly independent cyclic vectors
+each of height h. It is not hard to see that u1,...,u mhspan a complement
+inVhtoVh+1M+Vh−1. By the previous lemma, for any vectors v1,...,vmhTYPES AND CLASSES OF COMMUTING MATRICES 15
+inVh, there exists Y∈Asuch that uiY=vifor each i. This implies
+thatAacts as a full matrix algebra in its action on the quotient mod ule
+Vh/(Vh+1M+Vh−1). Hence the quotient module is simple. /square
+Forhsuch that mh/\e}atio\slash= 0, let Sh=Vh/(Vh+1M+Vh−1) be the simple
+A-module constructed in Proposition 3.4. If h/\e}atio\slash=h′and both ShandSh′are
+defined, then by Lemma 3.3, it is possible to define a matrix Y∈Asuch
+thatYacts as the identity on the cyclic vectors spanning Sh, and as the
+zero map on the cyclic vectors spanning Sh′. The simple modules Shand
+Sh′are therefore non-isomorphic as A-modules.
+Proposition 3.5. TheA-module Vhas a composition series in which the
+simpleA-module Shappears with multiplicity h.
+Proof.The action of the nilpotent matrix MonVhinduces a non-zero ho-
+momorphism of simple A-modules
+VhMi−1
+Vh+1Mi+Vh−1Mi−1−→VhMi
+Vh+1Mi+1+Vh−1Mi
+for each isuch that 1 ≤i≤h−1. This gives us hdistinct composition
+factors of Vh, each isomorphic to Sh. It now follows from the Jordan–H¨ older
+theorem that in any composition series of V, the simple module Shappears
+at least with multiplicity h. Finally, by comparing dimensions using the
+equation
+dimV=n=/summationdisplay
+hhmh=/summationdisplay
+hhdimSh,
+we see that equality holds for each h, and that the A-module Vhas no other
+composition factors. /square
+Proposition 3.6. IfMis nilpotent, and has part-size invariant k, then the
+determinants that appear in CentMarek-th powers in Fq.
+Proof.GivenY∈CentMletYhdenote the matrix in Mat mh(Fq) which
+gives the action of Yon the simple A-module Sh. Using the composition
+series given by the previous theorem to compute det Ywe get
+detY=/productdisplay
+h
+mh/negationslash=0(detYh)h.
+Since the part-size invariant of mis the highest common factor of the set
+{h|mh/\e}atio\slash= 0}, we see that det Yis ak-th power. /square16 JOHN R. BRITNELL AND MARK WILDON
+It is worth remarking that it is also possible to prove Propos ition 3.5 in a
+way that gives the required composition series in an explici t form. We have
+avoided this approach in order to keep the notation as simple as possible.
+The following example indicates how to construct a suitable basis ofVin a
+small case.
+Example. LetM∈Mat5(Fq) be a nilpotent matrix in the similarity class
+N(2,2,1). Letu1,u2be cyclic vectors of Mof height 2, and let vbe a cyclic
+vector of Mof height 1. Then with respect to the basis u1,u2,v,u1M,u2M
+ofF5
+q, the centralizer of Mconsists of all matrices of the form
+
+α β ⋆ ⋆ ⋆
+γ δ ⋆ ⋆ ⋆
+ζ ⋆ ⋆
+α β
+γ δ
+
+where gaps denote zero entries, and ⋆is used to denote an entry we have no
+need to specify explicitly. The key to obtaining this matrix in the required
+form is to order the elements of the basis correctly. The foll owing principles
+determine a suitable ordering on the basis: elements come in decreasing
+order of height; cyclic vectors come first among elements of t he same height;
+ifbicomes before bjthenbiMcomes before bjM.
+3.2.Proof of Theorem 3.1. The proof has two steps. We first show that
+ifMis a matrix with entries in Fqand part-size invariant k, then every
+k-th power in Fqappears as the determinant of a matrix in Cent M. In the
+second, we use Proposition 3.6 to show that no other powers ca n appear.
+We begin with the following lemma.
+Lemma 3.7. LetS(k)be the set of k-th powers in Fq. Letd∈Nand
+θ∈F×
+q. Then the number of irreducible polynomials of degree doverFq
+with constant term θis
+1
+d(q−1)/summationdisplay
+k|d
+S(k)∋θµ(k)hcf(q−1,k)(qd/k−1).
+This number is non-zero for all choices of dandθand for all q.
+Proof.We give an elementary proof of the existence of a polynomial w ith
+degreedand constant term θ. For the number of polynomials, see for
+instance [1, §5.2].TYPES AND CLASSES OF COMMUTING MATRICES 17
+Letαbe a generator of the multiplicative group F×
+qd, and let β=n(α)
+wheren:F×
+qd→F×
+qis the norm homomorphism. It is clear that βgenerates
+F×
+q. Letcbe such that 0 < c < qand (−1)dθ=βc. SinceFqdhas no proper
+subfield of index less than q, and since the multiplicative order of αcis at
+least (qd−1)/c, it is easy to see that αccannot lie in a proper subfield
+ofFqd. It follows that the minimum polynomial of αcoverFqhas degree d
+and constant term θ, as required. /square
+Proposition 3.8. LetPbe a matrix with class type d(j). Then for any
+θ∈Fq, there exists a matrix in CentPwith determinant θj.
+Proof.We may assume that θis non-zero. By Lemma 3.7 there exists an
+irreducible polynomial foverFqwith degree dand constant term ( −1)dθ.
+LetCbe the similarity class containing P, and let Dbe the class of matrices
+with cycle type f(j). SinceCandDhave the same class type, it follows from
+Theorem 2.6 that they commute. Therefore Pcommutes with an element of
+D. It is clear from the construction of Dthat its elements have determinant
+θj, as required. /square
+We now extend Proposition 3.8 to a general matrix.
+Proposition 3.9. IfMis a matrix with part-size invariant k, then for any
+ζ∈Fq, there exists a matrix in CentMwith determinant ζk.
+Proof.LetP1,...,P sbe the distinct cyclic blocks of M; soMis conjugate
+to/circleplustext
+iPi. For each ilet the class type of the block Bibedhi
+i. By Proposi-
+tion 3.8, for any scalars θithat we choose, there exist matrices X1,...,X s
+such that Xi∈CentBifor alli, and det Xi=θhi
+i. ThusMcommutes with
+a conjugate of the matrix diag( X1,...,X s), which has determinant/producttext
+iθhi
+i.
+It will therefore be enough to show that there exist non-zero scalars
+θ1,...,θssuch that/producttext
+iθhi
+i=ζk. But we know that k= hcf(h1,...,h s),
+and so there exist integers aisuch that k=/summationtext
+iaihi; it follows that we can
+simply take θi=ζaifor alli. /square
+We now turn to the second step in the proof of Theorem 3.1.
+Proposition 3.10. LetMbe a matrix with part-size invariant k. The
+determinant of an element of CentMis ak-th power in Fq.
+Proof.LetMact onV=Fn
+q. For each irreducible polynomial foverFq
+which divides the minimal polynomial of M, letVfbe the largest subspace18 JOHN R. BRITNELL AND MARK WILDON
+ofVon which f(M) acts nilpotently. Then V=/circleplustextVf, and each sum-
+mandVfis invariant under Cent M. It follows that if Y∈CentMthen
+detY=/producttextdetYf, whereYfis the restriction of YtoVf. Therefore, it will
+be sufficient to show that det Yis ak-th power for each f.
+Letλ= (h1,...,h s) be the partition associated with a given fin the
+rational canonical form of M. From the definition of the part-size invariant,
+each of the parts hiis divisible by k. LetMfbe the restriction of MtoVf,
+and letYf∈CentMf.
+ByProposition2.2, f(Mf)isnilpotentwithassociatedpartition dλ, where
+dis the degree of f. It is clear, then, that the part-size invariant of f(Mf)
+isk. SinceYfis in the centralizer of f(Mf), it follows from Proposition 3.6
+that detYfis ak-th power in Fq, as required. /square
+Combining the results of Propositions 3.9 and 3.10 gives The orem 3.1.
+3.3.Determinants in classes of a given type. It is natural to ask which
+determinants are represented among matrices of a given type . This question
+leads to a hard problem in arithmetic combinatorics, to whic h we have been
+able to find only a partial solution.
+It is clear that if Tis a type representable over the field Fq, then there is a
+matrix of type Twith zero determinant if and only if Thas a primary com-
+ponent 1λfor some λ. This leaves us to decide which non-zero determinants
+can arise. For primary types this question is easily answere d.
+Lemma 3.11. Letλbe a partition of k∈N, letd∈N, and let θ∈F×
+q.
+There is an invertible matrix over Fqwith type dλand determinant θif and
+only ifθis ak-th power in F×
+q.
+Proof.IfMis a matrix of type dλthenMhas characteristic polynomial fk.
+The determinant of Mis therefore a k-th power. That every k-th power in
+F×
+qis obtained in this way follows easily from Lemma 3.7. /square
+The following pair of propositions establish a sufficient con dition on a
+type for it to represent all non-zero determinants.
+Proposition 3.12. Letd∈Nbe coprime with q−1, and let T=dλ1···dλt
+be a type representable over Fq. IfL=|λ1|+···+|λt|is also coprime to
+q−1, then every element of F×
+qis the determinant of a matrix of type T.
+Proof.It is an easy consequence of Lemma 3.7 that if dis coprime with q−1,
+then there are the same number of irreducible polynomials of degreedwithTYPES AND CLASSES OF COMMUTING MATRICES 19
+any non-zero constant term. It follows that, for a generator θof the cyclic
+groupF×
+q, there exists a permutation σof the set of irreducible polynomials
+of degree d, such that fσ(0) =θf(0) for all f.
+LetCbe a similarity class of type T, whose members have determinant
+α. Consider the class C′obtained from Cby applying the permutation σ
+to the irreducible polynomials which appear in its cycle typ e. It is easy
+to see that C′has the same type as C, and that the members of C′have
+determinant αθL, whereLis as in the statement of the proposition. Now
+θLis a generator of F×
+qsinceLis coprime with q−1, and so it is clear that
+by repeated applications of the permutation σwe can obtain any non-zero
+determinant of our choice. /square
+Proposition 3.13. LetTbe a type representable over a finite field Fq. For
+eachdletLdbe the sum of the sizes of the partitions associated with the
+components of degree dinT. IfdLdis coprime with q−1for anyd, then
+every element of F×
+qis a determinant of a matrix of type T.
+Proof.This follows immediately from Proposition 3.12. /square
+It should be noted that Proposition 3.13 does not come close t o giving a
+necessary condition for a type to contain all non-zero deter minants. Finding
+conditions which are both necessary and sufficient appears to be a highly
+intractable problem.
+A special case of considerable interest is that of lineartypes, of the form
+1λ1···1λt. (These are precisely the types of triangular matrices over Fq.)
+We make use of the following definition.
+Definition. LetAbe an abelian group of order m(written multiplica-
+tively) and let π= (π1,...,π m)∈Zm. We say that an element x∈Ais
+π-expressible if there exists an ordering g1,...,gmof the elements of Gsuch
+thatx=gπ1
+1···gπmm.
+The relevance of this definition to our problem is easily expl ained. Let T
+be the linear type 1λ1···1λtwheret≤q−1. Letπ∈Zq−1be defined by
+π= (|λ1|,...,|λt|,0,...,0).
+Then we observe that the non-zero determinants represented inTare pre-
+cisely the π-expressible elements of F×
+q.
+IfAis an abelian group of exponent nthen we observe that adding mul-
+tiples of nto the entries of πdoes not affect π-expressibility in A; we may20 JOHN R. BRITNELL AND MARK WILDON
+therefore assume that all of the entries of πsatisfy 0 ≤πi≤n−1. Sim-
+ilarly, reordering the entries of πcannot affect π-expressibility, and so we
+may suppose that they appear in decreasing order.
+Numerical evidence obtained by the authors supports the fol lowing con-
+jecture.
+Conjecture. LetAbe a cyclic group of order m. Letπ= (π1,...,π m)∈
+(Z/mZ)m, whereπ1≥ ··· ≥ πm. Letπ′be the partition obtained from π
+by subtracting πmfrom each part (thereby ensuring that the last part is 0).
+Then every element of Aisπ-expressible unless one of the following holds:
+(1)π′= (m−r,r,0,...,0) for some r, or
+(2) There exists an integer p >1 which divides each part of π′, and
+which also divides m.
+This conjecture is known to be true in the case that mis a prime (see [4,
+Theorem 1.2]). For our purposes, we would like it to be true fo rA=F×
+qfor
+allq; that is, whenever m+1 is a power of a prime. This would provide a
+complete classification of the determinants occurring in li near types. In the
+very special case when q= 2rand|F×
+q|= 2r−1 is a Mersenne prime, the
+result of [4] already gives such a classification.
+4.Commuting nilpotent classes
+In§2 the question of which similarity classes of matrices over a finite
+field commute was reduced to the analogous problem for nilpot ent classes.
+The question of which nilpotent classes commute with a given nilpotent
+classN(λ) appears to be a very hard problem, and we shall not attempt
+to answer it in any generality. We shall, however, treat a var iety of special
+cases, and make a number of observations which, so far as we ha ve been
+able to determine, do not appear in the existing literature. Our approach is
+elementary, and leads to results which, for the most part, ap ply to matrices
+definedover anarbitraryfield. (For someotherrecent result sontheproblem
+of commuting nilpotent classes over algebraically closed fi elds, obtained by
+the methods of Lie theory, the reader is referred to [12] and [ 13].)
+Our results may be summarized as follows. Proposition 4.1 de scribes the
+nilpotent classes that commute with N(λ) whenλhas a single part. This
+result has appeared previously in [12]; our Proposition 4.2 is similar to, but
+slightly stronger than, the result which appears there as Pr oposition 2.
+Similarly, we deal in Proposition 4.4 with the case that λ= (n−1,1)
+for some n, and in Proposition 4.5 with the case that λ= (2,...,2). UsingTYPES AND CLASSES OF COMMUTING MATRICES 21
+these results we are able to classify those nilpotent classe s that commute
+with every nilpotent class of the same dimension; this is The orem 4.6.
+We next establish a condition for the nilpotent classes N(n,n) andN(n+
+1,n−1) to commute; these classes are found to commute over any infi nite
+field, and over the finite field Fprprovided that p(p2r−1)/edoes not divide
+n, where e= 1 ifp= 2 and e= 2 otherwise. As well as augmenting
+our list of commuting classes, this result is particularly s ignificant, since it
+demonstrates that commuting of classes is in some cases depe ndent on the
+field of definition. Finally, we use the results just mentione d to classify those
+commuting nilpotent classes whose associated partitions h ave no more than
+two parts; this result, stated as Theorem 4.10, is valid over any field.
+The following definition will be useful in what follows.
+Definition. LetMbe a nilpotent transformation of a space V. Acyclic
+basisforMis a basis BofVwith the property that for each v∈B, either
+vM= 0, or else vM∈B.
+Earlier in §3.1 we defined a cyclic vector for Mto be a vector which is
+not in the image of M. LetM∈N(h1,...,h k), and let Bbe a cyclic basis
+forM. ThenBcontains cyclic vectors v1,...,vk, where ht vi=hifor alli;
+in fact
+B={viMj|1≤i≤k,0≤j < hi}.
+ByLemma3.3, anelementofCent Misdeterminedbyitsactionon v1,...,vk.
+4.1.Cyclic nilpotent classes and partition refinements. Recall that
+J(λ), orJ(λ(1),...,λ(k)), is the unique upper-triangular matrix in Jordan
+form in the similarity class N(λ). The next proposition is concerned with
+the case where λ= (n) for some n. It is well known that the elements of the
+centralizer algebra Cent J(n) are the polynomials in J(n)—see for example
+[9, Ch. III, Corollary to Theorem 17].
+Proposition 4.1. Letλ= (h1,...,h k). ThenJ(n)commutes with a con-
+jugate of J(λ)if and only if h1−hk≤1.
+Proof.WriteEifor the matrix whose ( x,y)-th entry is 1 if k=y−x, and
+0 otherwise. The matrices E0,E1,...,E n−1form a basis for the centralizer
+algebra of J(n). LetMbe a non-zero nilpotent element of this algebra; then
+for some din the range 0 < d≤n−1 we can write
+M=/summationdisplay
+i≥dαiEi,22 JOHN R. BRITNELL AND MARK WILDON
+for scalars αi, withαd/\e}atio\slash= 0.
+It is easy to check that null Ms= min(sd,n) for all integers s. Lethbe
+the least integer such that hd≥n. Then it follows from Lemma 2.1 that M
+is conjugate to J(λ), where
+λ= (h,...,h,h −1,...,h−1)
+is the partition with n−hdparts of size h−1 and (h+ 1)d−nparts of
+sizeh. This establishes the proposition. /square
+The terminology in the first of the following definitions is bo rrowed from
+[10,§3].
+Definition. A partition is almost rectangular if its largest part differs from
+its smallest part by at most 1.
+Definition. Letλandµbe partitions. We say that µis arefinement ofλ
+ifµis the disjoint union of subpartitions whose sizes are the pa rts ofλ. We
+say that a refinement of λisalmost rectangular if all of the subpartitions
+involved are almost rectangular.
+For instance, (5 ,3,1) = (3 + 2 ,2 + 1,1) has (3 ,2,2,1,1) as an almost-
+rectangular refinement. It is worth noting that while the rel ation given by
+“µis a refinement of λ” is clearly transitive, the relation given by “ µis an
+almost rectangular refinement of λ” is not.
+Proposition 4.2. Letµ1andµ2be partitions of n. If there exists a parti-
+tionλwhich has both µ1andµ2as almost rectangular refinements, then the
+conjugacy classes represented by the Jordan blocks J(µ1)andJ(µ2)com-
+mute.
+Proof.Consider the subpartitions ν1ofµ1andν2ofµ2whose parts combine
+to create a single part of λof sizeh. Sinceν1andν2are almost rectangular,
+they yield Jordan blocks whose classes commute with that of J(h). But
+the centralizer of J(h) consists of polynomials in J(h), and it follows that
+the classes of J(ν1) andJ(ν2) have representatives which are polynomials
+inJ(h). So these representatives commute, and hence J(µ1) andJ(µ2) have
+conjugates which commute. /square
+The preceding proposition is slightly more general than [12 , Proposi-
+tion 2], which states that the nilpotent classes N(λ) andN(µ) commute
+ifµis an almost rectangular refinement of λ. It is noted in [12] that there
+exist examples of classes commuting that cannot be explaine d in this way.TYPES AND CLASSES OF COMMUTING MATRICES 23
+We remark that our Proposition 4.2 does not account for all co mmuting
+between classes, either. We illustrate this fact with the ex ample and the
+proposition below; other examples will be seen in subsequen t sections.
+Example. There is no partition which has both (2 ,2) and (3 ,1) as an
+almost rectangular refinement, but the classes N(2,2) andN(3,1) commute
+over any field. We leave the proof of this to the reader, while r emarking
+that it is a special case of any one of Propositions 4.4, 4.5 an d 4.7 below.
+Proposition 4.3. Letλbe a partition, and let λbe its conjugate partition.
+Then the nilpotent classes with partitions λandλcommute.
+Proof.Letλ= (h1,...,h k), where h1≥ ··· ≥ hk. LetNbe nilpotent of
+typeλ, and let u1,...,u kbe cyclic vectors for N, such that uihas height
+hifor alli. By Lemma 3.3 there is a unique matrix M∈CentNsuch that
+uiM=ui+1for alli, withukM= 0.
+Ifλ= (5,5,3,2), for instance, then the actions of NandMon the cyclic
+basis can be represented as follows:
+N M
+u1• /d47/d47•/d47/d47•/d47/d47•/d47/d47•
+u2• /d47/d47•/d47/d47•/d47/d47•/d47/d47•
+u3• /d47/d47•/d47/d47•
+u4• /d47/d47•u1•
+/d15/d15•
+/d15/d15•
+/d15/d15•
+/d15/d15•
+/d15/d15u2•
+/d15/d15•
+/d15/d15•
+/d15/d15• •
+u3•
+/d15/d15•
+/d15/d15•
+u4••
+It is easy to check that Mis nilpotent, with associated partition λ./square
+In general there does not exist a partition which has both λandλas
+almost rectangular refinements, as is shown by the example il lustrating the
+proof above, or by the case λ= (4,1,1).
+4.2.Universally commuting classes. The object of this section is to
+classify, in Theorem 4.6, the partitions to which the follow ing definition
+refers.
+Definition. A partition λofnisuniversal with respect to a field Kif
+N(λ) commutes with N(µ) overKfor every partition µofn.
+The reference to the field in this definition is in fact redunda nt; it is a
+consequence of Theorem 4.6 that a partition which is univers al with respect24 JOHN R. BRITNELL AND MARK WILDON
+to one field is universal with respect to any field. To prove the theorem, we
+shall require the following two propositions.
+Proposition 4.4. Letλbe a partition of n. The matrix J(n−1,1)commutes
+with a conjugate of J(λ)if and only if one of the following holds:
+(1)λhas a part of size 1, and ifλ−is obtained from λby removing this
+part, then J(n−1)commutes with a conjugate of J(λ−); Proposition
+4.1 provides a classification in this case.
+(2)nis even, and all of the parts of λare of size 2.
+(3)λhas a part of size 3, and its other parts are of size 1or2, with at
+least one part of size 1.
+(4)n= 3andλ= (3).
+Proof.The centralizer algebra of J(n−1,1) has the basis
+{Ei|0≤i≤n−2}∪{F,G,H},
+where
+/summationdisplay
+iαiEi+βF+γG+δH=
+α0α1α2... α n−2β
+0α0α1αn−30
+0 0α0αn−40
+.........
+0 0 0 α00
+0 0 0 ... γ δ
+.
+A nilpotent element of this algebra must have α0=δ= 0. We suppose that
+Mis such an element, and that Mis non-zero. By Lemma 2.1 the partition
+λassociated with Mis determined by the sequence of ranks of powers of
+M.
+Ifαi= 0 for all i < n−2, thenαn−2,βandγare the only entries that
+are possibly non-zero. It is easy to see that the rank sequenc e
+(rankI,rankM,rankM2,rankM3)
+must be either ( n,2,1,0) or (n,1,0,0). In the first case the partition asso-
+ciated with Mis (3,1n−3), which is covered by either part (iii) or part (iv)
+of the lemma. In the second case the partition is (2 ,1n−2), which is covered
+by part (i) or part (ii).
+Now suppose that there exists i < n−2 such that αi/\e}atio\slash= 0. Let mbe
+the least such i. Ifm <(n−2)/2 then it is not hard to see that the rankTYPES AND CLASSES OF COMMUTING MATRICES 25
+sequence is
+(n,n−m−1,n−2m−1,...,0).
+The partition λgiven by this data has one more part of size 1 than the
+partition λ−given by the data
+(n−1,n−m−1,n−2m−1,...,0).
+Butλ−corresponds to the rank sequence for an element of the centra lizer
+algebra of J(n−1), and so this case is covered by case (i) of the lemma. If
+m >(n−2)/2 then the same situation occurs if βγ= 0. But if βandγ
+are both non-zero then the rank sequence obtained is ( n−m−1,1,0). The
+corresponding partition λis covered by part (iii) of the lemma.
+The final case to analyse occurs when nis even and m= (n−2)/2. If
+M2/\e}atio\slash= 0 then the situation of the previous paragraph applies. Oth erwise the
+rank sequence is ( n,n/2,0) and all of the parts of λhave size 2, as in part
+(ii) of the lemma. /square
+Proposition 4.5. Letλbe the partition of 2swhich has sparts of size 2,
+and letµbe any partition of 2s. ThenJ(λ)commutes with a conjugate of
+J(µ).
+Proof.By a straightforward inductive argument, we may supposetha tµhas
+no subpartition of even size. If µhas only one part then the result follows
+from Proposition 4.1; so we may assume that µhas exactly two parts, s+t
+ands−t.
+A cyclic basis for N=J(λ) has the form B={e1,...,es,f1,...,fs},
+where the vectors fiare in the kernel of N, andeiN=fifor alli. LetMbe
+the matrix whose action is defined by eiM=ei+1,fiM=fi+1for 1≤i < s,
+and
+esM=
+
+fs−t+1ift >0,
+0 otherwise,
+fsM= 0.
+It is easy to see that Mcommutes with N, hence it suffices to show that
+M∈N(µ). A basis for ker Mis given by {fs,es−fs−t}, so null M= 2.
+It follows that the partition associated with Mhas two parts, and since e1
+is a cyclic vector of height s+t, this partition must be ( s+t,s−t), as
+required. /square26 JOHN R. BRITNELL AND MARK WILDON
+Theorem 4.6. The universal partitions are precisely those with no part
+greater than 2, together with λ= (3).
+Proof.Supposethat λhas no part of size greater than 2. If all of theparts of
+λhave size 2, then J(λ) commutes with all nilpotent classes, by Proposition
+4.5. Otherwise λhas a subpartition λmofmfor every m≤n. Letµbe a
+partition of nwith largest part m. Then since λmis an almost rectangular
+refinement of m, it follows from Proposition 4.1 that J(λm) commutes with
+a conjugate of J(m). Now if λ′denotes the partition obtained by deleting
+theparts of λmfromλ, and ifµ′is obtained by deleting a part of size mfrom
+µ, then we may suppose inductively that J(λ′) commutes with a conjugate
+ofJ(µ′). It follows that J(λ) commutes with a conjugate of J(µ).
+Conversely, supposethat λhas largest part h >2. IfJ(λ) commutes with
+J(n) then by Proposition 4.1 all of its parts have size horh−1. Then we
+see from Proposition 4.4 that J(λ) does not commute with a conjugate of
+J(n−1,1), except in the single case that λ= (3). /square
+4.3.Commuting of classes N(n,n)andN(n+1,n−1).Themain object
+of this section is to prove Proposition 4.7 below, which give s a necessary and
+sufficient condition for the classes N(n,n) andN(n+1,n−1) to commute.
+This case is of particular interest because the field enters i n an essential way.
+In Theorem 4.8 we use this proposition to show that for every p rimepand
+positive integer r, there exists a pair of classes of nilpotent matrices which
+commute over the field Fprif and only if s > r.
+Proposition 4.7 is motivated by a natural construction on ma trices. Sup-
+pose that XandYare commuting matrices over a field K, and let
+D=/parenleftigg
+X0
+0X/parenrightigg
+, E=/parenleftigg
+Y I
+0Y/parenrightigg
+.
+Clearly the matrices DandEcommute. We may assume that XandY
+(andhence DandE) arenilpotent; then thisconstruction (andother similar
+ones) may in principal be used to find new cases of commuting ni lpotent
+classes. The partition labelling the class of Dis clearly 2 λ, whereλlabels
+the class of X. The partition labelling the class of Eis harder to calculate,
+and depends on the characteristic of K.
+We have no occasion to make systematic use of this constructi on in the
+present paper, but the following example is illustrative. L etX=Y=J(n).
+ThenD∈N(n,n). The partition labelling the class of Eis (n+1,n−1)TYPES AND CLASSES OF COMMUTING MATRICES 27
+except in the case that char Kdividesn, in which case it is ( n,n). It follows
+thatN(n,n) andN(n+ 1,n−1) commute over fields of all but finitely
+many characteristics, the exceptions being the prime divis ors ofn. We note,
+however, that the present method gives no information about whether the
+classes commute in fields of these exceptional characterist ics; this gives an
+indication that the following proposition is non-trivial.
+Proposition 4.7. Letpbe a prime, and let
+e=/braceleftigg
+1ifp= 2,
+2otherwise.
+Then the nilpotent types (n,n)and(n+1,n−1)commute over Fprif and
+only ifnis not divisible by p(p2r−1)/e.
+Proof.LetMbe nilpotent of type ( n+1,n−1), acting on a space Vover
+Fpr. Take a cyclic basis {ui,wj|0≤i≤n,1≤j≤n−1}forV, with
+uiM=ui−1andwjM=wj−1for alliandj. LetUkandWkdenote the
+subspaces /a\}bracketle{tuj|0≤j≤k/a\}bracketri}htand/a\}bracketle{twj|1≤j≤k/a\}bracketri}htrespectively—we take
+W0={0}andWn=Wn−1. LetVkdenoteUk⊕Wkfor allk. For each pair
+(x,y) withx∈Vn−1andy∈Vn−2, there is an unique nilpotent element Y
+of CentMsuch that unY=xandwn−1Y=y; it follows from Lemma 3.3
+that all of the nilpotent elements of Cent Mcan be obtained in this way.
+LetY∈CentMbe nilpotent, and define α,β,γ,δ by
+unY∈αun−1+γwn−1+Vn−2,
+wn−1Y∈βun−2+δwn−2+Vn−3.
+Thereader may findhelpfulthe following diagrammatic repre sentation of Y.
+un
+•α/d47/d47
+γ
+/d32/d32/d65/d65/d65/d65/d65/d65/d65/d65/d65/d65un−1
+•α/d47/d47
+γ
+/d34/d34/d69/d69/d69/d69/d69/d69/d69/d69/d69un−2
+•α/d47/d47
+γ
+/d34/d34/d69/d69/d69/d69/d69/d69/d69/d69/d69un−3
+•u3
+•α/d47/d47
+γ
+/d31/d31/d63/d63/d63/d63/d63/d63/d63/d63/d63u2
+•α/d47/d47
+γ
+/d31/d31/d63/d63/d63/d63/d63/d63/d63/d63/d63u1
+•α/d47/d47u0
+•
+•
+wn−1δ/d47/d47β/d60/d60/d121/d121/d121/d121/d121/d121/d121/d121/d121
+•
+wn−2δ/d47/d47β/d60/d60/d121/d121/d121/d121/d121/d121/d121/d121/d121
+•
+wn−3•
+w3δ/d47/d47β/d63/d63/d127/d127/d127/d127/d127/d127/d127/d127/d127
+•
+w2δ/d47/d47β/d63/d63/d127/d127/d127/d127/d127/d127/d127/d127/d127
+•
+w1β/d64/d64/d1/d1/d1/d1/d1/d1/d1/d1/d1
+The matrix A=/parenleftigg
+α β
+γ δ/parenrightigg
+describes the maps induced by Y,
+Yk:Vk
+Vk−1−→Vk−1
+Vk−2,
+wherekis in the range 1 < k < n . Outside of this range, the map Yn
+has domain /a\}bracketle{tun+Vn−1/a\}bracketri}htof dimension 1, while Y1has codomain /a\}bracketle{tu0/a\}bracketri}htof28 JOHN R. BRITNELL AND MARK WILDON
+dimension 1. These maps are represented by the first row and th e first
+column of Arespectively.
+Claim. The kernel of Yhas dimension 2 if and only if Ais non-singular.
+Proof of Claim. IfAis invertible, then the maps Ykare injective for k >1.
+It follows easily that if v∈kerYthenv∈V1. It is now easy to check that
+v∈ /a\}bracketle{tu1,βu2−αw1/a\}bracketri}htand so null Y= 2 in this case.
+Conversely, suppose that Ais singular. If α=β= 0 then V1⊆kerY,
+and so null Y≥3. So let us suppose that αandβare not both 0. Then
+there exists z∈V1such that zY=u0. SinceAis singular, the map Y2
+has a non-trivial kernel, and it follows that there exists v∈V2\V1such
+thatvY∈V0=/a\}bracketle{tu0/a\}bracketri}ht. Say that vY=σu0; now we have a set of three
+kernel vectors, {u0, βu1−αw1, v−σz}, which is linearly independent
+sincev−σz /∈V1. So null Y≥3 in this case as well. /square
+The dimension of ker Ytells us the number of parts in the partition asso-
+ciated with the class of Y. This partition therefore has two parts if and only
+if the matrix Ais non-singular. Note that since Yn+1= 0, no part can be
+larger than n+1, and therefore the only possible partitions are ( n+1,n−1)
+and (n,n). The former corresponds to the class of Mitself, while the latter
+case occurs when Yn= 0, which is the case if and only if un∈kerYn.
+Now we observe that
+unYn=Y1◦Y2◦···◦Yn(vn+Vn−1)
+=u0R1An−2C1,
+whereR1andC1are, respectively, thefirstrow andthefirstcolumn of A. So
+the partition of Yis (n,n) precisely when R1An−2C1= (0), or equivalently,
+when the matrix Anhas a zero for its top left-hand entry.
+Claim. Every element of GL 2(Fpr) is either a scalar matrix, or else is
+conjugate to a matrix with a zero for its top left-hand entry.
+Proof of Claim. Every quadratic polynomial over Fpris the characteristic
+polynomial of a unique similarity class of non-scalar matri ces. Thus if Xis
+a non-scalar matrix with characteristic polynomial x2+σx+τ, thenXis
+conjugate to
+
+0 1
+−τ−σ
+,
+as required. /squareTYPES AND CLASSES OF COMMUTING MATRICES 29
+Now suppose that GL 2(Fpr) contains a non-scalar element Xwhich is an
+n-th power in the group. Then Xhas a conjugate X′with a zero for its top
+left-hand entry. Clearly X′is also an n-th power; by choosing a,b,c,δto be
+the entries of an n-th root of X′, we can construct a matrix Yin CentM
+whose type is ( n,n).
+There exist non-scalar n-th powers in GL 2(Fpr) provided that nis not
+divisible by the exponent of PGL 2(Fpr). This exponent is p(p2r−1)/e, and
+the proof of Proposition 4.7 is complete. /square
+Remark. This argument also goes to show that the nilpotent types ( n,n)
+and (n−1,n+1) commute over any infinite field K, since the exponent of
+PGL2(K) is infinite.
+Theorem 4.8. Letpbe a prime, and r≥1. There exist partitions λand
+µ, such that N(λ)commutes with N(µ)over the fields Fpafora > r, but
+not fora≤r.
+Proof.We use a famous theorem of Zsigmondy [15] which states that if
+k≥2,t≥3, and (k,t)/\e}atio\slash= (2,6), then there is a prime divisor of kt−1 which
+does not divide ks−1 for any ssuch that 1 ≤s < t.
+LetL= lcm({p2s−1|1≤s≤r}), and let n=pL/e. We observe that
+p(p2a−1)/edividesnwhenever a≤r. Whena > rwe invoke Zsigmondy’s
+Theorem with ( k,t) = (p,2a), or with ( k,t) = (4,3) ifp= 2 and t= 3; this
+tells us that p2a−1 has a prime divisor qwhich does not divide p2s−1 for
+s < a. Clearly qdoes not divide n, and sop(p2a−1)/edoes not divide n. It
+now follows from Proposition 4.7 that the partitions ( n,n) and (n+1,n−1)
+have the property stated in the theorem. /square
+Remark. The authors have found no case where the commuting of nilpo-
+tent classes dependson the field of definition in dimension le ss than 12. This
+is the dimension of the smallest example given by Propositio n 4.7: that of
+N(6,6) andN(7,5), which commute over every field except F2.
+4.4.Classes corresponding to two-part partitions. We end by estab-
+lishing a result which, together with results already prese nted, will allow us
+to classify, over any field K, pairs of partitions ( λ,µ) with at most two parts,
+such that N(λ) andN(µ) commute over K. We note that classes with at
+most 2 parts are precisely those whose elements have nullity at most 2.30 JOHN R. BRITNELL AND MARK WILDON
+Proposition 4.9. Letλ= (a,b)andµ= (c,d), wherea+b=c+dand
+a > c≥d > b. IfN(λ)andN(µ)commute over a field Kthenc=dand
+a−b= 2.
+Proof.The case that c=danda−b= 2 has been dealt with in Proposition
+4.7 and the ensuing remark. We may therefore suppose that a−b >2.
+LetM∈N(λ), and let {v,vM,...,vMa−1,w,wM,...,wMb−1}be a cyclic
+basis for M. LetW= kerMa−2; soWis the span of all the basis vectors
+apart from vandvM. Suppose that Yis nilpotent and commutes with M;
+then it is not hard to see that W⊆kerYa−2. SinceYis nilpotent we have
+vY∈αvM+Wfor some α∈K.
+Suppose first that α/\e}atio\slash= 0; then we see that vYa−1=αa−1vMa−1, while
+vYa= 0. Hence vis a cyclic vector for Yof height a. It follows that if the
+partition associated with Yhas only 2 parts then it must be λ.
+Suppose alternatively that α= 0, so vY∈W. We shall show that
+nullY≥3, and so the partition associated with Yhas more than 2 parts.
+Firstobservethat vMa−2andvMa−1areinker Y, sincevMa−2Y=vYMa−2∈
+WMa−2={0}. Furthermore it is easy to show that vMa−3YandwMb−1Y
+both lie in /a\}bracketle{tvMa−1/a\}bracketri}ht, and hence a non-zero linear combination of these two
+vectors lies in ker Y. We have therefore found three linearly independent
+vectors in ker Y, as required. /square
+The following theorem simply collects together elements of Propositions
+4.1, 4.7 and 4.9; it requires no further proof.
+Theorem 4.10. Suppose that λandµare partitions of nwith at most two
+parts, and that N(λ)andN(µ)commute over a field K. Assume without
+loss of generality that the largest part of λis at least as large as the largest
+part ofµ. Then one of the following holds.
+(1)λ=µ.
+(2)n= 2m,λ= (n)andµ= (m,m).
+(3)n= 2m,λ= (m+1,m−1),µ= (m,m)and, ifKis finite then the
+exponent of PGL2(K)does not divide m.
+(4)n= 2m+1,λ= (n)andµ= (m+1,m).
+References
+[1]John R. Britnell , ‘Cyclic, separable and semisimple matrices in the special linear
+groups over a finite field’, J. London Math. Soc. (2) 66 (2002) 605–622.TYPES AND CLASSES OF COMMUTING MATRICES 31
+[2]John R. Britnell andMark Wildon , ‘On the distribution of conjugacy classes
+between the cosets of a finite group in a cyclic extension’, Bull. London Math. Soc.
+40 (5) (2008) 897–906.
+[3]John R. Britnell andMark Wildon , ‘Commuting elements in conjugacy classes:
+An application of Hall’s Marriage Theorem’, J. Group Theory , to appear.
+[4]Andr´as G´acs,Tam´as H´eger,Zolt´an L´or´ant Nagy andD¨om¨ot¨or P´alv¨olgyi,
+‘Permutations, hyperplanes and polynomials over finite fiel ds’, preprint.
+[5]Murray Gerstenhaber , ‘On nilalgebras and linear varieties of nilpotent matrice s
+III’,Ann. of Math. 70 (1) (1959) 167–205.
+[6]Murray Gerstenhaber , ‘Ondominance andvarieties ofcommutingmatrices’, Ann.
+of Math. 73 (2) (1961) 324–348.
+[7]Israel Gohberg ,Peter Lancaster andLeiba Rodman ,Invariant subspaces of
+matrices and applications (Wiley, New York, 1986).
+[8]J. A. Green , ‘The characters of the finite general linear groups’, Trans. Amer. Math.
+Soc.80 (2) 1955 402–447.
+[9]Nathan Jacobson ,Lectures in abstract algebra II: Linear Algebra (Van Nostrand,
+1953)
+[10]Tomaˇz KoˇsirandPolona Oblak , ‘On pairs of commuting nilpotent matrices’,
+Transformation Groups 14 (1) (2009) 175–182.
+[11]I. G. Macdonald ,Symmetric functions and Hall polynomials (Second Edition, Ox-
+ford University Press, Oxford, 1995).
+[12]Polona Oblak , ‘The upper bound for the index of nilpotency for a matrix com -
+muting with a given nilpotent matrix’, Linear and Multilinear Algebra 56 (6) (2008)
+701–711.
+[13]Dmitri I. Panyushev , ‘Two results on centralisers of nilpotent elements’, J. Pure
+Appl. Algebra 212 (2008) 774–779.
+[14]R. Steinberg , ‘A geometric approach to the representations of the full li near group
+over a Galois field’, Trans. Amer. Math. Soc. 71 (1951) 274–282.
+[15]K. Zsigmondy , ‘Zur Theorie der Potenzreste’, Monatsh. f¨ ur Math. u. Phys. 3 (1892)
+265–284.
\ No newline at end of file
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+arXiv:1001.0812v1  [nucl-th]  6 Jan 2010EffectsoftheΛ(1405)ontheStructureofMulti-AntikaonicNuclei
+TakumiMutoa,ToshikiMaruyamab,ToshitakaTatsumic
+aDepartment of Physics, Chiba Institute ofTechnology, 2-1- 1 Shibazono,Narashino, Chiba 275-0023, Japan
+bAdvanced Science Research Center, Japan Atomic EnergyAgen cy, Tokai, Ibaraki 319-1195, Japan
+cDepartment ofPhysics, Kyoto University, Kyoto 606-8502, J apan
+Abstract
+TheeffectsoftheΛ(1405)(Λ∗)onthestructureofthemulti-antikaonicnucleus(MKN),in which
+severalK−mesons are embedded to form deeply bound states, are conside red based on chiral
+symmetry combined with a relativistic mean-field theory. It is shown that additional attraction
+resulting from the Λ∗pole has a sizable contribution to not only the density profil es for the
+nucleonsand K−mesonsbut also the groundstate energyof the K−mesonsandbindingenergy
+oftheMKN asthenumberoftheembedded K−mesonsincreases.
+Key words: multi-antikaonicnuclei,chiralsymmetry,kaoncondensat ion,subthreshold
+resonanceΛ(1405)
+PACS:21.85.+d,11.30.Rd,21.65.Jk,26.60.-c
+1. Introduction
+Exploringmulti-strangenesssystemsisanimportantaspec tofunderstandinghadrondynam-
+icsindensematter. Kaoncondensationinneutronstarsmaye xistasastrangeness-nonconserving
+system, where kaon condensatesare spontaneouslyproduced fromnormal matter throughweak
+processes, N+n→N+p+K−,N+e−→N+K−+νe(N=p,n)[1]. Recentlymulti-antikaonic
+nuclei(abbreviatedasMKN),whereseveralantikaons( K−mesons)areboundinthegroundstate
+of the nucleus,have beeninvestigated[2, 3], stimulatedby the proposalto exploredeeplybound
+kaonic nuclear states and subsequent theoretical and exper imental studies[4]. The MKN is a
+strangeness-conservingsystem and should be formedby embe ddingaK−meson in the nucleus
+through strong processes. Both the kaon-condensed state in neutron-star matter and the MKN
+formed in experiments are cold, dense objects originating f rom the common ¯K−Nand¯K−¯K
+interactionsin densematter,sothat theymaybecloselyrel atedwitheachother.
+WehaveconsideredpropertiesoftheMKNwithintheframewor kofarelativisticmean-field
+theory (RMF) coupledwith the nonlineare ffectivechiral Lagrangian[2]. It has been shown that
+thelowest K−energy,ωK−,increasesasthenumberofembedded K−mesons,|S|,becomeslarge
+andthatitentersintothesubthresholdresonanceregionof theΛ(1405)(Λ∗),whereωK−≃mΛ∗−
+mN=467MeV.Thisisbecausethecontributiontotheenergyfromt herepulsive ¯K−¯Kinteraction
+becomes sizable with the increase in |S|as compared with the attractive ¯K−Ninteraction. In
+this paper,we take into account the Λ∗-pole contributionas well as range termsand study these
+effectsonthestructureoftheMKN.
+Preprint submitted to Nuclear Physics A December 14, 20182. Formulation
+A spherical symmetry is assumed for the MKN, and the mass numb erA, the number of
+protonsZ, and the number|S|of embedded K−mesons with the lowest energy ωK−are kept
+fixed. We start with the e ffective chiral Lagrangian, which incorporates s-wave interactions
+between the (nonlinear) ¯Kmesons and nucleons of the scalar type simulated by the KNsigma
+term,ΣKN,andofthevectortype(Tomozawa-Weinbergterm). Thenonli nearK−fieldΣisgiven
+asΣ=exp[2i(K+T4+i5+K−T4−i5)/f], whereT4±i5(≡T4±iT5) is the SU(3) generatorand f(=
+93 MeV) the meson decayconstant. The K−field is representedas K−(r)=fθ(r)/√
+2 withθ(r)
+beingthechiralangleinthecondensateapproximation[2]. These¯K−Ninteractionsarereplaced
+bythosegeneratedbythe σandω,ρmesons-exchanges,respectively,withintheRMF[2].
+Thethermodynamicpotential ΩfortheMKNisderivedunderalocaldensityapproximation
+for the nucleons[2]. The correction to the energy density, ∆ǫ(r), from theΛ∗is introduced
+throughthesecond-orderperturbationwithrespecttothea xialcurrentofhadrons, ˆAµ
+5=f∂µK−+
+···+(gΛ∗/2)(¯Λ∗γµp+h.c.)+···withgΛ∗beingthecouplingconstantfor K−pΛ∗vertex:
+∆ǫ=−i/integraldisplay
+d4z∝angb∇acketleftx|T/tildewideωK−ˆA0
+5(z)/tildewideωK−ˆA0
+5(0)|x∝angb∇acket∇ight×/parenleftigg
+−1
+2sin2θ/parenrightigg
+real part⇒ −1
+2f2/tildewideω2
+K−sin2θ/bracketleftigg
+ρs
+p/braceleftigg
+dp+g2
+Λ∗
+2f2mΛ∗−mN−ωK−
+(mΛ∗−mN−ωK−)2+γ2
+Λ∗/bracerightigg
++dnρs
+n/bracketrightigg
+,(1)
+where the smooth parts ∝dpρs
+p,dnρs
+nare the range terms with ρs
+p(r) (ρs
+n(r)) being the scalar
+density of the proton (neutron)and the pole contributionco mes from theΛ∗withγΛ∗being the
+width. Thesetermsareabsorbedintothee ffectivenucleonmasses. Wecallthesecontributionsto
+theenergythesecond-ordere ffects(SOE)[1]. InEq.(1), /tildewideωK−(r)[≡ωK−−VCoul(r)]isthelowest
+energyof the K−shifted in the presence of the Coulomb potential. The parame ters,dp,dn,gΛ∗,
+andγΛ∗are determinedso as to reproducethe on-shell s-waveK−Nscatteringlengths[5]. The
+classicalK−field equationisgivenfrom δΩ/δθ=0 as
+∇2θ(r)=sinθ(r)/bracketleftigg
+m∗2
+K(r)−2/tildewideωK−(r)X0(r)−/tildewideω2
+K−(r)cosθ(r)
+−/tildewideω2
+K−(r)cosθ(r)/braceleftigg
+ρs
+p(r)dp+g2
+Λ∗
+2f2mΛ∗−mN−ωK−
+(mΛ∗−mN−ωK−)2+γ2
+Λ∗+dnρs
+n(r)/bracerightigg/bracketrightigg
+,(2)
+wherem∗2
+K(r) (=m2
+K−2gσKmKσ(r)) is the square of the e ffective mass of the K−, andX0(r) (=
+gωKω0(r)+gρKR0(r))representsthe ¯K−Nvectorinteraction. Inthesequantities, giK(i=σ,ω,ρ)
+are the coupling constants, while σ(r),ω0(r), andR0(r) are the mean fields of the σmeson and
+the time componentsof the ωandρmesons,respectively. Togetherwith Eq.(2) oneobtainsthe
+coupled equationsof motion (EOM) for the other mesons σ,ω,ρ, and the Poisson equation for
+the Coulombpotential VCoul(r):
+−∇2σ(r)+m2
+σσ(r)=−dU
+dσ(r)+gσN(ρs
+p(r)+ρs
+n(r))+2gσKmKf2(1−cosθ(r)),(3a)
+−∇2ω0(r)+m2
+ωω0(r)=gωN(ρp(r)+ρn(r))−2gωK/tildewideωK−(r)f2(1−cosθ(r)), (3b)
+−∇2R0(r)+m2
+ρR0(r)=gρN(ρp(r)−ρn(r))−2gρK/tildewideωK−(r)f2(1−cosθ(r)), (3c)
+∇2VCoul(r)=4πe2(ρp(r)−ρK−(r)), (3d)
+2whereρi(r) (i=p,n,K−) are the number densities and giN(i=σ,ω,ρ) the coupling constants.
+The coupled equations (2) and (3a) −(3d) are solved self-consistently, and the density distrib u-
+tionsρi(r)andotherquantitiesareobtainedasfunctionsofthe radia ldistance r.
+3. NumericalResults
+We take the15
+8O (A=15,Z=8) as a reference nucleus. The K−optical potential depth UKis
+chosentobe UK=−80MeV.
+3.1. Density profiles
+The density distributions of the protons, neutrons, and the distribution of the strangeness
+density[=−ρK−(r)]areshownfor|S|=4and8inFig.1. Thesolidlinesareforthepreviousresult
+without the SOE[2], and the dashed-dottedlines for the pres ent result with the SOE. Due to the
+0 1 2 3 4–0.50.00.5ρi ( fm−3 )|S|=4
+ρp
+ρn
+−ρK
+r (fm)0 1 2 3 4–0.50.00.5ρi ( fm−3 )|S|=8
+ρp
+ρn
+−ρK
+r (fm)
+Figure1: Thedensitydistributionsofprotons,neutrons,a ndthestrangenessdensity[ =−ρK−(r)]fortheMKNwith A=15,
+Z=8, and|S|=4, 8in thecase of UK=−80 MeV.
+SOE, the K−mesons and the protonsare attracted more to each other than t he case without the
+SOE,sinceintheformerthe K−liesbelowtheresonanceregionofthe Λ∗andfeelsanadditional
+attractionthroughcouplingwiththe Λ∗pole. Asaresult,thecentraldensitiesoftheprotonsand
+K−mesons become larger. On the other hand, neutrons are pushed outward from the center of
+the MKN due to the weakly repulsive e ffect from the range term ( ∝dnρs
+n,dn<0 in Eq. (2) ).
+These features become remarkable for a large value of |S|(Compare the cases of |S|=4 and 8).
+The central baryon density ρ(0)
+B(=ρp(r=0)+ρn(r=0)) becomesρ(0)
+B∼3.5ρ0withρ0=0.153
+fm−3for|S|∼8. Onecanseea “neutronskin”structurewitha thickness(1 −2)fmfor|S|∼8. In
+addition, for a larger |S|, the proton and K−density distributions tend to be more uniform near
+the center.
+3.2.|S|-dependenceof thelowest K−energyandbindingenergy
+In Fig. 2, the lowest energyof the K−,ωK−, is shownas a functionof |S|. The energydiffer-
+enceperunitofstrangeness,[ E(A,Z,|S|)−E(A,Z,0)]/|S|(=mK−B(A,Z,|S|)/|S|withB(A,Z,|S|)
+beingthebindingenergyoftheMKN),isshownasafunctionof |S|inFig.3. Inthesefiguresthe
+solid lines are for the result without the SOE[2], and the das hed-dotted lines for the result with
+the SOE. FromFig. 2, the ωK−is shownto be loweredby ∼40 MeV fromthat withoutthe SOE
+duetotheadditionalattractionbroughtaboutfromthe Λ∗pole. Nevertheless, ωK−increaseswith
+30 5 10 15350400450500ωK−  ( MeV )mΛ(1405)−mNmK
+|S|unbound
+Figure 2: The lowest energy of the K−,ωK−, for the MKN
+withA=15,Z=8, and|S|=2,8 in thecase of UK=−80 MeV.0 5 10 15100200300400mK
+mΛ(1116)−mN
+|S|with 
+without range terms 
+and Λ(1405) ( MeV )
+B /|S|B /|S|
+Figure 3: The energy di fference per|S|, [E(A,Z,|S|)−
+E(A,Z,0)]/|S|as functions of|S|.B(A,Z,|S|) is the binding
+energy of the MKN.
+anincreasein|S|since therepulsive ¯K−¯Kinteractionoverwhelmstheattractive ¯K−Ninterac-
+tions at large|S|. For|S|≥12 (in the case of UK=−80 MeV), K−mesons become unbound,
+whereωK−/greaterorsimilarmΛ∗−mNabovetheΛ∗-resonanceregion.
+From Fig. 3, the B/|S|steadily increases with |S|in the case in which the SOE is included,
+whileitshowslittledependenceupon |S|withouttheSOE.Onefindsthat mK−B/|S|>mΛ(1116)−
+mN, wheremΛ(1116)is the free mass of the lightest hyperon Λ(1116). Hence the MKN decays
+through strong processes such as K−NN→Λ(1116)N, so that it is not stable as a self-bound
+object. Thisresultqualitativelyagreeswiththatin Gazda et al.[3].
+4. Concludingremarks
+With regardto creating self-boundobjects for the MKN, hype ron-mixingeffects may be re-
+sponsibleforformationofmorestronglyboundstates. Itha sbeenshowninaliquid-droppicture
+that coexistenceofantikaonsandhyperonsleadstohighlyd enseself-boundobjects,whichmay
+decay only through weak processes[7]. There is a controvers y about the possible existence of
+such objects depending on the adopted models and approximat ions[6]. A realistic framework
+including antikaons and hyperons as well as nucleons beyond the local density approximation
+forbaryonsis necessaryforfurtherinvestigation.
+Acknowledgments
+ThisworkissupportedinpartbytheGrant-in-AidforScient ificResearch(No.20028009).
+References
+[1] H. Fujii, T.Maruyama, T.Muto and T.Tatsumi, Nucl. Phys. A 597(1996) 645.
+[2] T.Muto, T.Maruyama and T.Tatsumi, Phys.Rev. C 79(2009) 035207.
+[3] D. Gazda, E.Friedman, A.Gal, and J.Mareˇ s, Phys.Rev. C 76(2007) 055204; Phys.Rev. C 77(2008) 045206.
+[4] For arecent review, A.Galand R.S.Hayano (Eds.),Nucl Ph ys.A 804(2008) 1.
+[5] A. D.Martin, Nucl. Phys. B 179(1981) 33.
+[6] D. Gazda, E.Friedman, A.Gal, and J.Mareˇ s, Phys.Rev. C 80(2009) 035205.
+[7] T.Muto, Nucl Phys. A 804(2008) 322.
+4
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+arXiv:1001.0813v1  [astro-ph.HE]  6 Jan 2010Centaurus A at Ultra-High Energies
+Roger W. ClayA,C, Benjamin J. WhelanA, and Philip G. EdwardsB
+ASchool of Chemistry and Physics, University of Adelaide, Ad elaide SA 5005
+BCSIRO ATNF, Narrabri Observatory, Locked Bag 194, Narrabri NSW 2390
+Cemail: roger.clay@adelaide.edu.au
+Abstract: We review the importance of Centaurus A in high ene rgy astrophysics as a
+nearby object with many of the properties expected of a major source of very high energy
+cosmic rays and gamma-rays. We examine observational techn iques and the results so far
+obtained in the energy range from 200GeV to above 100EeV and a ttempt to fit those
+data with expectations of Centaurus A as an astrophysical so urce from VHE to UHE
+energies.
+Keywords: acceleration of particles — galaxies: active — gamma rays: o bservations — cosmic rays
+1 Introduction
+The field of very high energy astrophysics deals with
+processes associated with the acceleration and inter-
+actions of particles at energies above those accessible
+with spacecraft observatories, characteristically above
+a few 100 GeV, up to the highest particle energies found
+in nature, above 100 EeV. The massive particles at
+these energies are known as cosmic rays and at the
+top of the energy range are referred to as ultra high
+energy (UHE). The observed energetic photons, which
+are seen at lower energies, are known as very high en-
+ergy (VHE) gamma-rays.
+The all sky cosmic ray spectrum exhibits a very
+steep dependence of flux against energy. It extends
+over 30 orders of magnitude of flux and ten orders of
+magnitude in energy to above 100 EeV with rather lit-
+tle deviation from a featureless power law relationship.
+There is a steepening at PeV energies, known as the
+“knee” (Hillas 1984). At energies in the EeV range,
+there is then a flattening known as the “ankle”. The
+knee is thought to represent either an energy limit to
+the acceleration ability of most galactic sources or a
+limit to the ability of our Milky Way galaxy to securely
+contain and build up an internal cosmic ray flux. The
+ankle is thought to represent a change from predomi-
+nantly galactic sourced cosmic rays to an extragalactic
+flux (Gaisser & Stanev 2006). It is thus most reason-
+able to look at energies above a few EeV for direct
+observational evidence of Centaurus A (Cen A) as a
+cosmic ray source.
+The origins of cosmic rays are not securely known.
+It is thought that supernovae or supernova remnants
+(SNR) are the most probable origins of cosmic rays
+which originate in the Milky Way and that such par-
+ticles are energised through diffusive shock accelera-
+tion (Protheroe & Clay 2004). There appear to be
+severe limitations to energies accessible through this
+process and the highest energy cosmic rays are postu-
+lated to originate in some different environment out-
+side our galaxy (Hillas 1984). Cen A, our closest ac-
+tive galaxy, is a relatively local extragalactic objectwhich may contain regions such as its extended radio
+lobes, or supermassive central black hole, with physi-
+cal properties which enable cosmic ray acceleration to
+exceed energy limitations which apply in galaxies like
+the Milky Way. For this reason, over almost 40 years,
+Cen A has been the target of observational searches for
+evidence that it is a significant VHE or UHE source.
+We briefly review the techniques used to study Cen A,
+review the reasons why Cen A is an attractive observa-
+tional target, and examine the observational progress
+which has been made.
+2 Particle Acceleration
+Processes and Sites
+There is a general expectation that cosmic ray particles
+primarily receive energy through diffusive shock ac-
+celeration (Berezinskii et al. 1990; Protheroe & Clay
+2004). This is a process whereby charged particles
+diffusively cross an astrophysical shock front multiple
+times, receiving a boost in energy with each crossing.
+This concept has been shown to have the happy char-
+acteristic that a power law cosmic ray energy spec-
+trum results. Such acceleration processes were first
+proposed by Fermi who noted that head on collisions
+with moving magnetic clouds resulted in a transfer of
+energy to already energetic particles, and that head
+on collisions were statistically preferred over others
+(Berezinskii et al. 1990). Fermi’s original process proved
+to be very slow and the realisation that multiple (sta-
+tistical) crossings of a shock front provided a much
+faster (“first order”) process appeared to provide a
+practical acceleration model. In such a picture, there
+is a clear requirement that (a subset of) the accelerat-
+ing particles are repeatedly scattered across the shock.
+An energy upper limit of the process results from prop-
+erties of the source region which finally fail to provide
+sufficient scattering at the highest energies.
+An alternative non-stochastic scenario is that ac-
+celeration is associated with the voltage drop created
+12 Publications of the Astronomical Society of Australia
+by a rapidly spinning supermassive black hole threaded
+by magnetic fields induced by currents flowing in a sur-
+rounding disk or torus (Levinson 2000). In this case,
+the maximum achievable energy is apparently in the
+UHE region although more detailed modelling will be
+required to clearly determine limits imposed by energy
+loss mechanisms such as curvature radiation.
+Cen A contains a supermassive black hole and also
+exhibits evidence of substantial shocks with evidence
+for particle acceleration associated with their related
+jets (Hardcastle et al. 2007). We shall see below that
+it has regions which are capable of scattering parti-
+cles magnetically as required. Whether those neces-
+sary conditions are sufficient for the acceleration of
+particles to ultra high energies is the question to be
+answered observationally.
+3 Particle Propagation andAt-
+tenuation
+At very high energies, astrophysical particles are ca-
+pable of having inelastic and elastic collisions with
+particles and fields in the source, and between the
+source and our observatories within the Milky Way
+galaxy. These interactions are important to under-
+standing the astrophysics of the particles which are
+observed. Although limited in size, source regions can
+contain strong magnetic fields, intense photon fields
+over a great energy range, and a high plasma den-
+sity. In intergalactic space, our knowledge of fields
+and energy densities is limited but we expect, at least,
+that there will be some magnetic fields, starlight, infra-
+red radiation, and the cosmic microwave background
+(Driver et al. 2008). Closer to home, messenger par-
+ticles will transit whatever fields are contained in our
+local group of galaxies, our galactic halo and the plane
+of the Milky Way. In the latter context, it is expected
+that the known magnetic fields of our galaxy (with an
+underlying regular field at levels of a few µG but up to
+10µG if a random component is included (Sun et al.
+2008)) will have deflected all charged cosmic rays by
+significant amounts (Stanev 1997). Astrophysical an-
+gular uncertainties then exceed instrumental ones for
+charged cosmic rays.
+It is possible for accelerated protons to interact
+(most likely in a source region) and convert to neu-
+trons. This, for instance, is an argument for a possible
+cosmic ray excess in the direction of our galactic centre
+since the neutrons will not suffer deflection in galac-
+tic plane magnetic fields which would certainly scatter
+protons out of a directional beam (Clay 2000). Iso-
+lated neutrons decay within a few minutes when at
+rest but cosmic ray neutrons with a high relativistic
+gamma factor will survive long distances in the lab-
+oratory frame. However, at the distance of Cen A,
+neutrons with energies below 400 EeV (just above the
+highest energy cosmic ray recorded from any direc-
+tion (Bird et al. 1995)) would decay before reaching
+us. We note that such decay is statistical and some
+neutrons might be observable even at lower energies
+but the resulting flux would be greatly attenuated be-
+low 100 EeV. It seems that neutrons generated withina Cen A source will not provide a direct undeflected
+beam, although protons could interact in some inter-
+mediate matter, resulting in a “halo” around the di-
+rection of a source.
+Cosmic ray interactions can additionally produce a
+flux of high energy neutrinos. These could be through
+interactions with the CMB or with particles and fields
+close to the source. In the latter case, LUNASKA
+(James et al. 2009) or northern UHE neutrino detec-
+tors such as ANTARES (Brown et al. 2009) might
+search for signals from the direction of Cen A.
+4 Interactions with Photon
+Fields
+Our Universe is known to be pervaded by the cos-
+mic microwave background (CMB) having a photon
+number density a thousand times that of characteris-
+tic plasma densities. Despite the low energies of mi-
+crowave photons, VHE photons and UHE nuclei see
+them as significant targets over modest astrophysical
+distances. Photons from Cen A are expected to be
+severely attenuated (Protheroe 1986a,b) over a range
+of energies, with a spectral cut-off beginning in the en-
+ergy range 130 to 200 TeV depending on the strength of
+the intergalactic magnetic field (Clay et al. 1994). In
+this absorption feature, attenuation lengths of a few
+kpc are expected for its deepest point at a little over
+1 PeV. The absorption feature progressively weakens at
+higher energies and, for sources within our galaxy, at
+much higher energies the absorption dip may be passed
+and photon attenuation may be reduced (Protheroe
+1986a). However, for photons from the more distant
+Cen A with a much greater path length, the absorption
+will be strong up to 10 EeV (Protheroe 1986b). The
+Pierre Auger Observatory (see Section 11) is capable
+of selecting photons from its overall detected flux at
+such energies and a search for photons from Cen A
+above 10 EeV would seem worthwhile.
+Cosmic ray nuclei will also interact with the CMB
+but this is not important until much higher energies
+than for photons, and the attenuation length is much
+greater. This attenuation phenomenon is convention-
+ally named the GZK effect after the people (Greisen,
+Zatsepin and Kuzmin) who proposed it in 1966 for cos-
+mic ray protons interacting on the CMB (Berezinskii et al.
+1990). The interaction has a proton energy thresh-
+old of about 60 EeV, a factor of 100,000 times greater
+than energies associated with the photon attenuation.
+There are also interaction processes for other nuclei
+on the CMB which become important at about this
+energy. However, whilst the characteristic attenuation
+length due to the GZK effect is of the order of 100 Mpc,
+the interaction mean free path is of the order of the
+distance to Cen A. This is due to the modest energy
+loss per interaction. Thus both VHE gamma-rays and
+UHE cosmic rays sourced from Cen A are expected to
+show evidence of interactions with the CMB.
+It appears that attenuation compatible with the
+GZK effect is evident in the all sky cosmic ray spec-
+trum of the Pierre Auger Observatory (Abraham et al.
+2008a), although evidence for a propagation cut-off re-www.publish.csiro.au/journals/pasa 3
+quires an assumption that the cosmic ray source spec-
+trum does not contain a similar feature. The current
+dataset at these energies is small and it is questionable
+if any presently observed events from the direction of
+Cen A could show a statistically convincing evidence
+for the existence of, or a lack of, GZK attenuation (al-
+though see Section 12).
+5 Magnetic Fields
+Charged cosmic ray particles will have their propaga-
+tion directions changed in their passage through as-
+trophysical magnetic fields. That deflection will de-
+pend on the particle rigidity, the ratio of momentum
+and charge. At the energies of interest here, this is
+effectively the ratio of the energy and the charge. A
+convenient rule of thumb is that the radius of curva-
+ture of a 1 PeV proton trajectory perpendicular to a
+uniform 1 microgauss magnetic field is 1 pc. Charac-
+teristic galactic fields are at these levels or just above,
+but their structure and, particularly, their extension
+out of the galactic plane are poorly known. Also,
+their strength within our local group of galaxies and
+the remaining intergalactic space between ourselves
+and Cen A is largely unconstrained by observation
+(Beck 2008). There is evidence that richer groups
+may be pervaded by multiple µG level magnetic fields
+(Clarke et al. 2001; Feretti & Johnston-Hollitt 2004)
+but this may not apply in our local region. This lack
+of knowledge is a major problem since, even at pro-
+ton energies of 100 EeV, we are still dealing with a
+radius of curvature of only 100 kpc in a characteristic
+magnetic field. As a result, we are unable to specify
+whether charged cosmic ray propagation over a dis-
+tance of 3.8 Mpc from Cen A is diffusive (with an un-
+certain magnetic turbulence scale size) or whether we
+can assume roughly linear propagation. A clear scat-
+tered cosmic ray signal from Cen A would give us in-
+valuable information regarding our local extragalactic
+magnetic fields (though of course this requires a knowl-
+edge of the intrinsic source size).
+6 The Air Shower Technique
+Our atmosphere is opaque to primary radiation at en-
+ergies with which very high energy astrophysics deals.
+Also, the flux of astronomical particles becomes suffi-
+ciently low at those energies and above such that direct
+satellite observation ceases to be effective for reason-
+able spacecraft detector collecting areas. The effective
+way of working at higher energies is through the use of
+our atmosphere as a target and observing the cascades
+of particles, known as “air showers” or “extensive air
+showers (EAS)”, which are produced as incident par-
+ticles deposit their energy, first as conversion to sec-
+ondary particle mass and kinetic energy and then into
+atmospheric gas excitation and ionisation. A good in-
+troduction to the physical processes in air showers can
+be found in Allan (1971).
+Primary gamma-rays initiate cascades which, to a
+first approximation, develop through successive pro-
+cesses of pair production and then bremsstrahlung ofthe daughter electrons and positrons. The mean free
+paths for those processes are related and are between
+30 and 40 g cm−2(compared to the thickness of the ver-
+tical atmosphere of about 1000 g cm−2). This rather
+simple cascade develops “exponentially” in particle
+number until ionisation energy losses begin to inhibit
+further development. The cascade thus reaches a max-
+imum particle number (often stated as a number of
+“electrons”, Ne). Such cascades are statistical in char-
+acter but the short interaction mean free paths com-
+pared to the total atmospheric depth result in a rather
+smooth development profile.
+Primary nuclei also initiate cascades but these are
+more complex and irregularly structured. They begin
+with a strong interaction which produces pions. The
+neutral pions decay to a pair of gamma-rays which then
+cascade as we have just seen. However, the charged
+pions are likely to decay (they may interact again if
+the atmospheric conditions are conducive) to muons.
+Those muons will most likely continue to traverse the
+atmosphere without further major interactions, just
+suffering a continuous energy loss from their ionisa-
+tion and excitation of atmospheric gases. This cascade
+now has three components. They are: the remnants
+of the original particle which only loses a fraction of
+its initial energy at each interaction (the nuclear core),
+the muons, and the electromagnetic cascades. A key
+point is that the nuclear core continues to inject en-
+ergy through further interactions, resulting in the initi-
+ation of superimposed electromagnetic cascades. The
+overall “shower” particle number is then a superposi-
+tion of electron numbers in successive electromagnetic
+cascades, building and decaying, plus the integrated
+numbers of muons. This picture is further compli-
+cated by the fact that the cosmic ray beam (at least
+at the lower energies) is a mixture of nuclear compo-
+nents (Gaisser & Stanev 2006). These various nuclei
+have their own interaction mean free paths for initiat-
+ing cascades (longest for protons — 80 g cm−2— and
+shorter for more massive nuclei) and, though difficult,
+and probably not possible on an event by event basis,
+this offers a means for studying the beam composition,
+or its change with energy.
+All cascades contain charged particles which scat-
+ter through interacting with atmospheric gas. As a re-
+sult, the electromagnetic cascades spread laterally with
+a characteristic distance of below 100 m for the nu-
+merically dominant electromagnetic component. How-
+ever, some electromagnetic particles can scatter to very
+large “core distances” and ground-based detecting ar-
+rays such as the Pierre Auger Observatory record par-
+ticles at kilometres from a lateral extension of the orig-
+inal cosmic ray trajectory. The muons scatter rather
+little but retain their direction from their initiating
+interaction which results in a characteristic spread of
+hundreds of metres at sea level. Again, a very few can
+be found kilometres from the core.
+Air shower cascades are studied by sampling a se-
+lection of their components. This is efficient in terms
+of enabling the detection of rare events (at the high-
+est energies the flux may be measured in terms of
+km−2century−1). This sampling can be accomplished
+with a sparse array of ground-based charged particle4 Publications of the Astronomical Society of Australia
+detectors which sample the cascade at a single devel-
+opment level. The requirement that particles reach the
+ground limits this technique to energies above about
+100 TeV at sea level or to detector arrays located at
+very high altitudes (Amenomori et al. 2000). Alter-
+natively, that observational energy threshold can be
+reduced by detecting the bright beam of forward di-
+rected ˇCerenkov light produced in the atmosphere us-
+ing large optical photon collecting telescopes which
+“image” those photons onto a photomultiplier “cam-
+era”. The VHE gamma-ray telescopes such as H.E.S.S.
+(Aharonian et al. 2005) fall into this category. At the
+highest energies, where a low level of light emission
+per shower particle is not a limiting factor, isotropic
+nitrogen fluorescence light, produced by the cascade
+exciting atmospheric gas, can be successfully recorded.
+This enables a large collection area to be achieved with
+large mirrors viewing the cascade from the side. The
+Pierre Auger Observatory employs this technique to-
+gether with a large array of ground-based large area
+particle detectors.
+7 Differentiating between
+Gamma-Rays and Cosmic
+Rays
+In recent years, the study of very high energy gamma-
+rays has become an important component of astro-
+physics. This has become possible partly through im-
+provements in instrumental sensitivity and angular res-
+olution but, also, through significant improvements
+in the software discrimination between gamma-ray in-
+duced extensive air showers and the numerically domi-
+nant cosmic ray showers. The gamma-ray showers are
+predominantly electromagnetic with interaction pro-
+cesses (pair production and bremsstrahlung) which oc-
+cur at rather small intervals in the atmosphere. The
+resulting cascades are rather simple and smooth. On
+the other hand, cosmic rays initiate and feed cascades
+through a nuclear process with a much longer mean
+free path and their interactions produce muons in ad-
+dition to electromagnetic particles. The cascade de-
+velopment is then rather irregular and also has an ir-
+regular geometrical spread due to the muons, which
+can travel, with rather little scattering, at significant
+angles away from the central cascade core. Differen-
+tiation between gamma-ray and cosmic ray initiated
+cascades conventionally depends on vetoing cosmic ray
+cascades. This is achieved either by detecting an irreg-
+ular shower development (or irregular ˇCerenkov im-
+age), a development which peaks at an atmospheric
+depth characteristic of nuclei for a given total energy
+(or particle content), or a muon content greater than
+expected for a gamma-ray cascade.
+VHE gamma-ray astronomy using the atmospheric
+ˇCerenkov technique has proved to be very efficient in
+producing images which have good gamma-ray to cos-
+mic ray discrimination. This is due to careful Monte
+Carlo modelling of the cascade and imaging processes
+to develop suitable image cuts (Aharonian et al. 2005).
+These are broadly based on vetoing the larger, less con-strained, cosmic ray images, together, in some cases,
+with a requirement that point sources under study are
+at known positions in the image.
+PeV gamma-ray studies have more commonly ex-
+plicitly used a muon veto in which cascades with sig-
+nificant muon numbers have been rejected. This ap-
+proach has had mixed success. At even higher energies
+in the EeV range, gamma-ray initiated showers are ex-
+pected to reach maximum development deeper in the
+atmosphere than cosmic ray showers and work is on-
+going to select potential gamma-ray cascades on this
+basis. Presently, the Pierre Auger Observatory claims
+upper limits to the UHE photon fraction using this
+method (Abraham et al. 2009).
+8 Searches at TeV Energies
+The first searches for VHE gamma-ray emission from
+Cen A were made with the Narrabri Stellar Inten-
+sity Interferometer. The Intensity Interferometer con-
+sisted of two 6.5 m diameter segmented optical reflec-
+tors mounted on a 188 m diameter circular track. The
+interferometer took advantage of the Bose-Einstein sta-
+tistical nature of light, with optical photons from stars
+tending to arrive in clumps. The correlations in inten-
+sity between the two reflectors enabled the diameters
+of bright stars to be inferred. The light pool from
+ˇCerenkov radiation produced by VHE cosmic rays has
+a similar extent to the track diameter, and so calcu-
+lations and tests were performed to confirm that the
+ˇCerenkov light signal was not contaminating the mea-
+surements of steller diameters (Hanbury Brown et al.
+1969). It was recognised, however, that the Inten-
+sity Interferometer could also be put into service as
+aˇCerenkov detector. Grindlay et al. (1975b) used a
+120 m separation between reflectors and operated the
+two as a stereo detector. They also employed a novel
+background rejection scheme, using off-axis photomul-
+tipliers to detect ˇCerenkov light from the penetrat-
+ing muon component of cosmic-ray initiated cascades,
+which allowed ∼30% of recorded events to be rejected.
+Between 1972 and 1974, a sample of 11 sources, in-
+cluding pulsars, X-ray binaries, the Galactic Centre,
+and AGN, were observed, with source selection based
+on X-ray and SAS-2 (30 MeV to 100 MeV) gamma-ray
+results. A time-averaged 4.5 σexcess was detected in a
+total observing time of 51 hr on Cen A (Grindlay et al.
+1975a), corresponding to a integral flux above 300 GeV
+of (4.4±1)×10−11cm−2s−1. As the outer radio lobes
+were well outside the beam, these were excluded as
+the source. Possible variability of the gamma-ray sig-
+nal (Grindlay et al. 1975a), coupled with theoretical
+modelling, also excluded the inner radio lobes as the
+gamma-ray source, with a proposed model of the gamma-
+ray flux arising from inverse Compton scattering in the
+nucleus of Cen A being favoured (Grindlay 1975).
+Subsequent VHE observations over the next 35 years
+yielded negative results. The Durham group, based
+at Narrabri, observed between March 1987 and April
+1988 with their MkIII telescope for a total of 44 hr of
+good on-source data. The 3 σflux upper limit above
+300 GeV was 7.8 ×10−11cm−2s−1(Carrami˜ nana et al.www.publish.csiro.au/journals/pasa 5
+1990). In March 1997, 6.75 hours of observations were
+made with the Durham Mk6 telescope, which provided
+better discrimination against the cosmic ray background,
+with a 3σflux upper limit above 300 GeV of
+5.2×10−11cm−2s−1(Chadwick et al. 1999).
+The JANZOS group observed Cen A from New
+Zealand for 56.9 hr between April 1988 and June 1989,
+reporting a 95% confidence level upper limit on the
+flux above 1 TeV of 2.2 ×10−11cm−2s−1(Allen et al.
+1993a).
+Interest in VHE emission from Cen A was rekin-
+dled by the EGRET detection of Cen A in the 30 MeV
+to 30 GeV range (Steinle et al. 1998; Sreekumar et al.
+1999). The CANGAROO 3.8 m telescope was used in
+March and April 1999 to record a total of 45 hr of on-
+and off-source data. The resulting 3 σflux upper lim-
+its above 1.5 TeV were 5.5 ×10−12cm−2s−1for a point
+source at the core of the galaxy, and 1.3 ×10−11cm−2s−1
+for an extended region of radius 14′centred on the
+core (Rowell et al. 1999). In March and April 2004,
+further observations were made with three 10 m tele-
+scopes of the CANGAROO-III array. From 10.6 hours
+of on-source data, upper limits were set for the sev-
+eral regions of interest: for the core of Cen A the 2 σ
+upper limit above 424 GeV was 4.9 ×10−12cm−2s−1
+(Kabuki et al. 2007).
+The H.E.S.S. group achieved a detection of Cen A
+with over 120 hr of observation between April 2004
+and July 2008. Their measured integral flux above
+250 GeV was (1.56 ±0.67)×10−12cm−2s−1. The de-
+tection is concentrated on the central galaxy region,
+not the lobes. This excess “only matches the position
+of the core, the pc/kpc inner jets and the inner radio
+lobes” (Aharonian et al. 2009).
+The H.E.S.S. flux is a factor of almost 30 below the
+original report of Grindlay et al. (1975a). However,
+the Intensity Interferometer observations were made
+during an extended period of enhanced X-ray emis-
+sion in the early 1970s. Although X-ray monitoring
+was infrequent during the 1980s, it appears Cen A has
+been in a relatively quiescent state for most of the last
+30 years (Bond et al. 1996; Turner et al. 1997; Steinle
+2006). A low X-ray state would plausibly result in a
+low flux of inverse Compton scattered VHE gamma-
+rays.
+TeV detections and upper limits are plotted in Fig-
+ure 1.
+9 Searches at PeV Energies
+Searches at PeV energies are made using air shower
+arrays of particle detectors. The Buckland Park air
+shower array was used between 1978 and 1981 to study
+the anisotropy of cosmic rays above an energy of 1 PeV.
+The directional accuracy of the array was 3◦×sec(θ),
+whereθis the zenith angle. The study confirmed that
+the overall cosmic ray flux shows no strong sidereal
+isotropy at these energies, and the work was then ex-
+tended to search for more localized excesses. The least
+isotropic declination band was that between −40◦and
+−45◦, and the largest excess in bins of 1 hr in right
+ascension coincided with Cen A. The overall signif-icance was 2.7 σ, a value not unexpected by chance
+given the number of bins examined, but which encour-
+aged further investigation. There was some evidence
+in the binned data for excess event numbers toward
+both outer radio lobes (Clay et al. 1984). Some sup-
+porting evidence was also noted in two other South-
+ern Hemisphere experiments (Farley and Storey 1954;
+Kamata et al. 1968), though with differing energy thresh-
+olds, angular resolutions and years of operation. It
+was also acknowledged that, at PeV energies, the sig-
+nal could not be due to gamma-rays from Cen A as the
+path length for interactions of PeV gamma-rays with
+CMB photons is only 10 kpc (Clay et al. 1984).
+The JANZOS experiment combined TeV telescopes
+with a PeV scintillator array, and a search for gamma-
+rays above 100 TeV was conducted with data taken
+between October 1987 and January 1992. No signifi-
+cant excess was found over this complete time range,
+but an excess was observed over 48 days from April
+to June 1990, with the excess concentrated in events
+with enegies below 200 TeV, consistent with the effects
+of the expected absorption at higher energies. Monte
+Carlo simulations were used to derive a probability of
+2% for the observed 3.8 σexcess to arise by chance
+(Allen et al. 1993b). A contour map of the signifi-
+cance showed a peak that coincided, within the an-
+gular resolution of the array, with the core of Cen A.
+Buckland Park data taken between March 1988 and
+February 1989 was examined and no significant excess
+found from Cen A (Bird & Clay 1990) — this being
+consistent with the JANZOS result for this period.
+A larger Buckland Park data-set, from 1984 to
+1989, was split, a priori , into three event size bins, and
+a excess found in the lowest size bin, corresponding to
+energies below 150 TeV (Clay et al. 1994) (hence be-
+low the CMB absorption feature). A confidence level
+of 99.4% was claimed for the excess. A Kolmogorov-
+Smirnov test indicated there was no significant evi-
+dence for enhanced periods of emission over this five-
+year period. A contour map of the excess showed a
+peak suggestive of a point source compatible with the
+core of Cen A.
+PeV detections and upper limits are also plotted
+in Figure 1.
+10 SUGAR
+The Sydney University Giant Air Shower Recorder
+(SUGAR) (Winn et al. 1986), apart from having a cre-
+ative acronym, was notable and important in pioneer-
+ing the start of a new era of cosmic ray study. Like the
+huge Pierre Auger Observatory (below), with an en-
+closed area of 70 km2SUGAR’s design recognised that
+the flux of cosmic rays at the highest energies is so low
+that its ground detectors required large separations
+and could not realistically be connected by cable to
+their direct neighbours. They required a measure of lo-
+cal autonomy. In the case of SUGAR, this was through
+the realisation of a local coincidence between two muon
+detectors at each station and the tape recording of
+their data plus time stamping for later global array
+analysis. The detector sites were also autonomous in6 Publications of the Astronomical Society of Australia
+A
+BCDE
+F
+GHI
+IJ
+K
+Figure 1: Reported detections (solid circles) and
+upper limits (arrows) to the flux from Cen A at
+TeV and PeV energies. (The labels are as follows:
+A, H.E.S.S., Aharonian et al. 2005; B, H.E.S.S.,
+Aharonian et al. 2009; C, Narrabri, Grindlay et
+al. 1975a; D, Durham, Carraminana et al. 1990
+E, Durham, Chadwick et al. 1999; F, JANZOS,
+Allen et al. 1993a; G, CANGAROO, Rowell et al.
+1999; H, CANGAROO-III, Kabuki et al. 2007; I,
+JANZOS, Allen et al. 1993b; J, Buckland Park,
+Clay et al. 1994; K, Buckland Park, Clay et al.
+1984.) The Chadwick et al. 1999 point (E) has
+been moved horizontally from the actual 300GeV
+energy threshold for clarity.
+terms of power, generating their own power thermo-
+electrically. Previous arrays had detected real time
+coincidences between spaced detectors to initiate data
+acquisition following the arrival of a suitably energetic
+shower. SUGAR operated for long enough to detect a
+significant number of showers with energies above the
+GZK cut-off energy and, until the commissioning of
+the Pierre Auger Observatory, was the only array at
+the highest energies with Cen A in its field of view.
+SUGAR was shown to have a problem with photo-
+multiplier afterpulsing which could make some energy
+assignments uncertain, although its direction determi-
+nations would not have been affected by that. Data
+from this array was used for studying a possible as-
+sociation of the highest energy cosmic rays with the
+direction of our galactic centre (Bellido et al. 2001).
+No excess from the direction of Cen A was evident in
+that analysis. Figure 2 shows the SUGAR highest en-
+ergy events (SUGAR Catalogue 1986) in the vicinity
+of Cen A (for comparison with Figure 3 for the Pierre
+Auger Observatory).
+11 The Pierre Auger Obser-
+vatory
+The Pierre Auger Observatory (PAO) is the largest
+cosmic ray detector ever built. It currently consists
+of a Southern Hemisphere site located near the town°l=-40 °l=-80°b=0°b=3079
+12664
+Figure 2: SUGAR events with energies
+above 60EeV in the vicinity of Centaurus A
+(SUGAR Catalogue 1986). The events are
+labelled by their energy in EeV. The red star
+indicates the position of Cen A.
+of Malarg¨ ue, Argentina, with planning of a North-
+ern Site in Colorado, USA, underway (Harton et al.
+2009). The southern site covers an area of approxi-
+mately 3000 km2(Suomij¨ arvi et al. 2009). The PAO
+employs two cosmic ray detection methods through
+the Surface Detector (SD) and the Fluorescence De-
+tector (FD) (Abraham et al. 2004; Bellido et al. 2005;
+Allekotte et al. 2008).
+The SD consists of more than 1600 autonomous
+particle detector stations which employ the water-
+ˇCerenkov detection technique. The particle detectors
+operate by recording ˇCerenkov light emitted when rela-
+tivistic charged particles in an air shower pass through
+1.2 m deep purified water enclosed in large-area (10 m2)
+tanks. These stations are arranged on a triangular
+grid, with 1.5 km spacing. This spacing results in the
+SD being fully efficient for detecting showers with a
+primary energy of above 3 ×1018EeV at zenith angles
+of 60◦or less (Suomij¨ arvi et al. 2009). Statistical en-
+ergy uncertainties from the SD are approximately 17%,
+with an additional systematic uncertainty of 7% at
+1019eV (increasing to 15% at 1020eV) arising from cal-
+ibration with FD energies (see below) (Di Guilio et al.
+2009). Directional uncertainties are approximately 1 .5◦
+at energies around 3 EeV, reducing to less than 1◦for
+energies above approximately 10 EeV (Bonifazi et al.
+2009). It has a duty cycle of slightly less than 100%
+giving a current integrated exposure of more than 12,000 km2sr yr,
+increasing by approximately 350 km2sr yr per month
+(Sch¨ ussler et al. 2009).
+The FD makes use of the air fluorescence method.
+Twenty-four telescopes are separated into 4 groups of
+6 telescopes (each group being termed a FD ‘site’),
+which overlook different sections of the SD. Each tele-
+scope views 30◦in azimuth, giving each site a 180◦
+azimuthal field of view, and 28.6◦in elevation. In each
+telescope a camera consisting of an array of photomul-
+tiplier tubes, viewing separate regions of sky, collects
+the light emitted by nitrogen molecules excited by the
+EAS. Using pulse timing information from triggered
+pixels, the axis along which the shower front propa-www.publish.csiro.au/journals/pasa 7
+gates can be reconstructed. The energy of the shower,
+apart from a small amount of ‘invisible energy’ carried
+by neutrinos and muons which are not visible to the
+FD, is proportional to the integrated light flux along
+the shower’s path. This allows an effectively calori-
+metric measurement of particle energy (Bellido et al.
+2005). The statistical energy uncertainties are approx-
+imately 9%, with a systematic uncertainty of approx-
+imately 22% arising from factors such as limitations
+of knowledge of the instantaneous atmospheric pro-
+file and aerosol content (Di Guilio et al. 2009). The
+requirement of clear, moonless nights for the opera-
+tion of the FD means that its duty cycle is about 13%
+(Sch¨ ussler et al. 2009).
+The colocation of the FD and SD allows events to
+be observed by both detectors. The events for which
+this occurs are termed ‘hybrids’. For these events,
+the timing information from a triggered SD station
+is added to that from the FD trace to reconstruct the
+arrival direction (Bellido et al. 2005). This allows a
+greater accuracy in determining the shower axis than
+is possible with either method alone, and the average
+hybrid directional uncertainty is 0.6◦(Bonifazi et al.
+2009). An additional advantage of the hybrid method
+is that it allows the SD energy scale to be determined.
+By a comparison of independent SD and FD recon-
+structions of the same events, SD energies can be cali-
+brated against the essentially calorimetric values from
+the FD (Di Guilio et al. 2009).
+Composition studies are performed primarily with
+the FD through measurements of the position of shower
+maximum, Xmax. This value indicates the slant depth
+in the atmosphere, in g cm−2, at which the flux of fluo-
+rescent light from the EAS reaches its maximum. From
+shower to shower the value of Xmax fluctuates due
+to the statistical nature of shower initiation and de-
+velopment. On average, however, nuclei are expected
+to have smaller Xmaxvalues than protons at a given
+energy, and fluctuations in Xmaxare expected to be
+smaller. Consequently, FD measurements are used to
+study the behaviour, as a function of energy, of both
+/angbracketleftXmax/angbracketright(the energy dependence of which is termed
+the ‘elongation rate’) and the RMS of Xmaxto look
+for possible changes in the composition of the primary
+particles (Bellido et al. 2009). It should be noted, how-
+ever, that the interpretation of these results is not clear
+due to uncertainties in hadronic interaction physics at
+such high energies (Bellido et al. 2009; D’Urso et al.
+2009).
+The large duty cycle of the SD makes the prospect
+of utilizing it to determine CR composition highly de-
+sirable. This is a somewhat more difficult task than
+with the FD, however, as a direct measurement of
+Xmax is not possible with the SD. Methods such as
+studying the risetime of the particle signal in the SD
+stations, the shower front radius of curvature, the ratio
+of the muonic to electromagnetic contributions to the
+signal and the azimuthal asymmetry in station signal
+around the shower axis are currently being investigated
+for their suitability in composition determination with
+the SD (Wahlberg et al. 2009).12 Observational Results at the
+Highest Energies
+The Pierre Auger Observatory is the only system presently
+recording data at EeV energies from the direction of
+Centaurus A. It presented its first skymap in 2007
+(Abraham et al. 2007) displaying the 27 highest en-
+ergy events at that time. This map appears to show
+event clustering in the general direction of Cen A. Data
+in that skymap which are in the vicinity of Cen A are
+shown in Figure 3 (Abraham et al. 2008b) which il-
+lustrates apparent clustering around the direction of
+Cen A.
+°l=-40 °l=-80°b=0°b=30
+84
+6663 58
+7969148
+70
+6480
+Figure 3: Auger events above 57EeV in the vicin-
+ity of Centaurus A (Abraham et al. 2008b). The
+events are labelled by their energy in EeV.
+The Auger paper interprets the dataset as a whole
+as being statistically associated with the directions of
+local AGNs. This was on the basis of an a priori
+search “prescription”. The Pierre Auger Collabora-
+tion has not yet developed a discovery prescription for
+Cen A and, as a result, no a priori statistical analysis
+is presently possible. However, one can comment on
+the properties of the dataset.
+Hillas (2009) has independently examined this Auger
+data set and confirms the conclusion that the highest
+energy events are associated with rather typical Seyfert
+galaxies in clusters at distances of typically ∼50 Mpc.
+The clustering of events near Cen A is confirmed and
+an origin in Cen A, or alternatively, NGC 5090 consid-
+ered. The close proximity of Cen A, however, led Hillas
+to conclude that Cen A is probably an inactive 1020eV
+accelerator as more distant galaxies play a larger role
+than might have been expected on the basis of a simple
+inverse square law flux dependence.
+More recently (Hague et al. 2009), the Pierre Auger
+Collaboration has shown continuing evidence for a con-
+centration of the highest energy events in the direction
+of Cen A. Those data show an excess of events above
+55 EeV within 18◦of Cen A which is well above the
+68% confidence interval for a sample from an isotropic
+distribution. In that range, 12 events are found where
+2.7 are expected on the basis of an isotropic flux. Ap-
+proximately 30% of the Auger events show some evi-
+dence of being members of such clustering out to 30◦
+from Cen A, approximately the same fraction (10/27)8 Publications of the Astronomical Society of Australia
+as found in the original Auger skymap.
+On the basis of this evidence, one might speculate
+that Cen A is the source of a substantial fraction of
+the extragalactic cosmic rays at energies above the an-
+kle of the cosmic ray energy spectrum. In that case,
+one notes that the angular size of the cosmic ray “im-
+age” is appreciably greater than the known physical di-
+mensions of the astronomical source on the sky. That
+cannot be explained by instrumental errors since, as
+noted above, the known angular resolution of the ob-
+servatory at these energies is below 1◦(Bonifazi et al.
+2009) and one naturally invokes magnetic scattering
+in intergalactic space or within our galactic region. If
+the scattering occurs in intergalactic space, one might
+plausibly assume a 10 kpc turbulence cell size, leading
+to a total scattering deflection of the order of 20 times
+the scattering in an individual cell (after the passage
+of approximately 400 individual cells). To fit the ob-
+served excess around Cen A, this requires a turbulent
+intergalactic field of strength 0.1 µG.
+At these energies, any deflections from the direc-
+tion of Cen A in known regular galactic plane magnetic
+fields are likely to be modest (a few degrees (Stanev
+1997)) but the extent and strength of magnetic fields in
+any galactic halo around the Milky Way are unknown
+(Sun et al. 2008) and could plausibly be substantial if
+the dimensions of the halo are large. The Auger ex-
+cess appears to limit the possibility of such effects from
+regular fields since there seems to be no great asymme-
+try perpendicular to the galactic plane although there
+may be an oval axis in that direction for the excess
+just discussed.
+In a scenario in which the object Cen A is the origin
+of the Pierre Auger “Centaurus A” excess, one must
+explain the apparent similarity between the distribu-
+tion of energies within the excess when compared with
+the totality of Auger events. Using the published event
+data (Abraham et al. 2008b), there is no significant
+difference between the two spectra (Figure 4) on the
+basis of a Kolmogorov Smirnov test (at >99% level),
+admittedly with a very limited event list. If that con-
+tinues to be the case, one might have to abandon the
+GZK cut-off as the source of a deficit of events above
+60 EeV and argue for a source acceleration limitation.
+An examination of event data presented by the
+Pierre Auger Collaboration with their 2007 sky map
+of the highest energy events shows a possible system-
+atic variation in particle energy across the Cen A ex-
+cess, with the highest energy events being at the high-
+est (positive) Galactic latitudes. Such an effect could
+be due to chance, but it could also count as evidence
+against the excess being associated with Cen A since
+there is not now a symmetry in the event energies
+about the central region. An explanation could be that
+the cosmic ray source for the highest energy events
+is in the northern lobe, or there could be contamina-
+tion from another source in the supergalactic plane, or
+simply that the combination of intergalactic scatter-
+ing and the passage through structured regular fields
+combine to produce the effect.
+The energy variation with direction could also sug-
+gest that there is another possible approach to un-
+derstanding the “Centaurus A” excess. This is thatEnergy (EeV)60 80 100 120 140 160 E≤ Fraction of Events 
+00.20.40.60.81
+Figure 4: Energy distribution for events in-
+side and outside a 25◦circle centred on Cen A
+(Abraham et al. 2008b). The dotted red line is for
+the events more than 25◦from Cen A.
+the propagation is dominated by regular intergalac-
+tic fields and that the particles are deflected as in a
+magnetic spectrometer, with the true source being in
+the direction from which an “infinite” energy particle
+would have been seen. Since the excess has its high-
+est energy particles furthest north from the galactic
+plane, one would invoke a magnetic field parallel to
+that plane (perpendicular to the supergalactic plane)
+with a “true” source region some distance further to
+the north such that the angular deflection is inversely
+proportional to the particle rigidity. In this scenario,
+it could be that the deflecting magnetic field is a halo
+field of the Milky Way. This would require a prod-
+uct of magnetic field strength for the regular compo-
+nent of the field and its spatial extent of the order of
+50µG kpc. This may not be incompatible with mod-
+els of magnetic fields in groups of galaxies or extended
+galactic halos.
+13 Centaurus A as a Cosmic
+Ray Source
+As we saw, it is usual to think of cosmic rays being
+accelerated to their observed energy in a rather slow
+statistical process. There are alternative possibilities ,
+such as a single acceleration through a very large po-
+tential step or a “top down” model in which an ultra-
+energetic particle is the result of the decay of an exotic
+highly massive initial particle. If the process is some-
+thing like diffusive shock acceleration, the acceleration
+site must be such that its scattering fields are capable
+of returning accelerating particles many times across a
+shock front. This would appear to require local mag-
+netic fields with products of strengths and physical
+dimensions such that a particle radius of gyration at
+the highest energies can be contained within the phys-
+ical boundaries of the field. This is often expressed
+through one of the “Hillas diagrams” (Hillas 1984).
+When it comes to considering Cen A as a source,
+possible extremes of the spectrum of sites would bewww.publish.csiro.au/journals/pasa 9
+within the modest strength magnetic fields enclosed in
+one or other of its giant radio lobes, or (at the other
+extreme) within very strong fields close to the cen-
+tral engine. Somewhere within a jet, or the southern
+shock, could also be candidate sites specific to Cen A.
+The radius of gyration of a cosmic ray proton (in kpc)
+is numerically close to its energy (in units of EeV)
+divided by the magnetic field strength (in units of mi-
+crogauss). Containment within an acceleration region
+will require that the region is significantly larger than
+that radius of gyration. Since it is the particle rigid-
+ity which is relevant, this requirement would be eased
+in proportion to the nuclear charge for heavier nuclei.
+This would mean that the acceleration of protons in
+200 kpc lobes of Cen A (the whole of a lobe) would re-
+quire magnetic fields filling a lobe at microgauss levels
+in order to accelerate particles to the measured Auger
+limit of about 200 EeV.
+Under such conditions ((Protheroe & Clay 2004)
+equation 41), a time of the order of 100 million years is
+required for the acceleration process. This would seem
+to be approximately the limit of possible acceleration
+both under a 108year estimate of AGN lifetimes and
+an estimate of microgauss strength fields in the lobes.
+The cosmic ray energy spectrum extends over 30 orders
+of magnitude in flux and very few accelerated cosmic
+rays are required to reach the highest energies - they
+are statistical anomalies. The source magnetic field
+must be strong enough and large enough in scale for
+the highest energy particles to be capable of one last
+diffusive scattering across the shock front.
+An alternative extreme of the spectrum of possi-
+ble Cen A acceleration sites might be within the most
+central volume of the AGN, close to the black hole.
+The majority of TeV and PeV detections are consistent
+with an excess concentrated on the central galaxy re-
+gion, and not the outer lobes. A central region with di-
+mensions of, say, 10 pc would require a shock contain-
+ment field strength approaching 1 Gauss. This would
+be substantial but not unreasonable. A difficulty with
+such a region would be to accelerate particles to UHE
+energies over a substantial period of time within a
+dense photon field containing photons with energies
+substantially above those of the CMB. This is because
+cosmic ray energy loss interactions would be significant
+from at least EeV energies. This attenuation at ener-
+gies below the ankle of the cosmic ray spectrum makes
+it difficult to see how the Auger Cen A spectrum could
+bear similarity with the conventional spectrum from
+other directions.
+Cen A is a radio galaxy with highly extended jets
+and lobes. As we noted, it could be that such lobes
+play a key part in accelerating particles to the high-
+est observed energies. Nagar & Matulich (2008) have
+discussed the possible role of objects with that mor-
+phology as sources of the Auger highest energy events.
+They note that there is a number of such sources (5)
+in the general vicinity of Cen A out of a total of 10
+in the “field of view” of the PAO. They also note that
+such objects seem to be statistically closely related to
+the directions of the highest energy PAO events. This
+proposition seems to be arguable but, if this is the case,
+the contribution of Cen A to the total flux must bebelow a level proportional to its radio emission since,
+including all such objects, Cen A dominates the total
+sum of the radio fluxes typically by at least an order
+of magnitude (Nagar & Matulich 2008). This is due
+to its proximity to us, and it could be that some of the
+other objects are more effective at accelerating par-
+ticles to the highest energies. We noted that Cen A
+may be a variable source at high energies, which may
+support this idea (Hillas 2009).
+Recently, Rieger & Aharonian (2009) have consid-
+ered Cen A as a VHE gamma-ray and UHE cosmic-
+ray source, and conclude that advection dominated ac-
+cretion disk models can account for the production of
+the TeV emission close to the core via inverse Comp-
+ton scattering of sub-mm disk photons by accelerated
+electrons. As it is unlikely that protons could be ac-
+celerated to EeV energies in this region, they propose
+shear acceleration along the kpc-scale jet as the origin
+for these particles.
+14 Conclusions
+Centaurus A has been a popular potential source of
+cosmic rays for close to half a century. A number of
+cosmic ray and VHE gamma-ray searches for excesses
+from that region have been made. An early search by
+Grindlay et al. (1975a,b) with VHE gamma-rays was
+encouraging, showing evidence for a positive observa-
+tion at a time of a large X-ray flare, and, very re-
+cently, H.E.S.S. has also provided evidence that Cen A
+is a VHE gamma-ray source. The Buckland Park air
+shower array found a signal with appropriate spectral
+characteristics which included evidence for CMB ab-
+sorption at sub-PeV energies. Also, recently, the Pierre
+Auger Observatory has shown evidence of a cluster-
+ing of UHE cosmic ray events around Cen A although
+without evidence for a different spectrum to that of
+other directions. Taken with observations which have
+produced upper limits and recognising that none of
+these observations had well defined a priori statistical
+analysis procedures, one cannot say with confidence
+that Centaurus A is a major source at UHE energies.
+However, it is the nearest example to us of one of the
+few classes of objects which have been identified as
+having a structure possibly capable of accelerating par-
+ticles to the highest observed energies. Acceleration of
+cosmic ray particles to the highest measured energies
+in Cen A would be at the limit of parameters associ-
+ated with the acceleration process. A reduction in the
+cosmic ray flux (from the general direction of Cen A)
+above about 60 EeV from the power law at lower ener-
+gies could be a source effect rather than a GZK cut-off,
+which would (and may well) otherwise apply to more
+distant sources.
+For the future, we clearly require more data. Fur-
+ther work at VHE (H.E.S.S.) energies to better define
+the source region and any possible extended structure
+would be great progress. An extension of the gamma-
+ray spectrum upwards towards the CMB absorption
+feature using the ˇCerenkov technique, with better an-
+gular uncertainty than Buckland Park, such as the
+TenTen concept (Rowell et al. 2008), would be a ma-10 Publications of the Astronomical Society of Australia
+jor asset. Also, solid confirmation and understanding
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+arXiv:1001.0817v1  [physics.atom-ph]  6 Jan 2010Preliminary studies for anapole moment
+measurements in rubidium and francium.
+D. Sheng and L. A. Orozco
+Joint Quantum Institute, Department of Physics, University of Ma ryland and
+National Institute of Standards and Technology, College Park, MD 20742-4111, USA
+E-mail:lorozco@umd.edu
+E. Gomez
+Instituto de F´ ısica, Universidad Aut´ onoma de San Luis Potos´ ı, Sa n Luis Potos´ ı, SLP
+78290, M´ exico
+Abstract. Preparations for the anapole measurement in Fr indicate the possib ility
+of performing a similar measurement in a chain of Rb. The sensitivity an alysis based
+on a single nucleon model shows the potential for placing strong limits on the nucleon
+weak interaction parameters. There are values of the magnetic fie lds at much lower
+values than found before that are insensitive to first order chang es in the field. The
+anapole moment effect in Rb corresponds to an equivalent electric fie ld that is eighty
+times smaller than Fr, but the stability of the isotopes and the curre nt performance
+of the dipole trap in the apparatus, presented here, are encoura ging for pursuing the
+measurment.
+PACS numbers: 31.30.jp, 32.10.FN, 21.60-n
+Keywords : Parity non conservation, anapole moment, blue detuned dipole tra p.Preliminary study for anapole moment measurements in rubid ium and francium 2
+1. Introduction
+The constraints obtained from Atomic Parity Non-Conservation (P NC) on the weak
+interaction and its manifestation both at low energy and in hadronic e nvironments are
+unique [1]. The information it provides is complementary to that obtain ed with high
+energyexperiments. Thelast twenty yearshave seensteadypro gressintheexperimental
+advances [2, 3, 4, 5] together with atomic theoretical calculations [6, 7, 8, 9, 10] having
+a precision better than 1% [11, 12, 13].
+As we prepare for a new generation of PNC experiments with radioac tive isotopes
+[14, 15], we continue to study the measurement strategy and find a dvances both in the
+understanding of the parameters and on the specific experimenta l approaches that we
+are taking to achieve the ultimate goal. This paper presents progre ss on both fronts.
+We explore the possibility of measurements in Rb and Fr in our current apparatus
+and show the calculated anapole moments using current single partic le nuclear models.
+We find the possible constraints in the nuclear weak interaction para meters that such
+measurements can bring and present the current performance o f our atomic trap.
+The general approach for the PNC experiments under considerat ion is the
+interference between an allowed transiton and the weak interactin g PNC transition
+[16, 17]. These experiments are going to take place in atomic traps an d will require
+access to accelerators that can deliver the different isotopes.
+2. The anapole moment in atoms
+We start reviewing the basics of the anapole moment following very clo sely the work
+we have done in planning the experiment in Fr [15]. Parity nonconserva tion in atoms
+appears through two types of weak interaction: Nuclear spin indep endent and nuclear
+spin dependent [18]. Nuclear spin dependent PNC occurs in three way s [19, 20, 13]:
+An electron interacts weakly with a single valence nucleon (nucleon ax ial-vector current
+AnVe), thenuclearchiralcurrentcreatedbyweakinteractionsbetwe ennucleons(anapole
+moment), andthecombinedactionofthehyperfineinteractionand thespin-independent
+Z0exchange interaction from nucleon vector currents ( VnAe). [21, 22, 23].
+Assuming an infinitely heavy nucleon without radiative corrections, t he
+Hamiltonian is [24]:
+H=G√
+2(κ1iγ5−κnsd,iσn·α)δ(r), (1)
+whereG= 10−5m−2
+pis the Fermi constant, m pis the proton mass, γ5andαare Dirac
+matrices, σnarePauli matrices, and κ1iandκnsd,i(nuclear spin dependent) with i=p,n
+for a proton or a neutron are constants of the interaction. At tr ee levelκnsd,i=κ2i, and
+in the standard model these constants are given by
+κ1p=1
+2(1−4sin2θW),κ1n=−1
+2,Preliminary study for anapole moment measurements in rubid ium and francium 3
+κ2p=−κ2n=κ2=−1
+2(1−4sin2θW)η, (2)
+with sin2θW∼0.23 the Weinberg angle and η= 1.25.κ1i(κ2i) represents the coupling
+between nucleon and electron currents when an electron (nucleon ) is the axial vector.
+The first term of Eq. 1 gives a contribution that is independent of th e nuclear
+spin and proportional to the weak charge ( QW) in the approximation of the shell model
+with a single valence nucleon of unpaired spin. For the standard mode l values, the
+weak charge is almost equal to minus the number of neutrons Nthat we take to be
+proportional to the number of protons Z. The second term is nuclear spin dependent
+and due to the pairing of nucleons its contribution has a weaker depe ndence on Z[25]:
+Hnsd
+PNC=G√
+2KI·α
+I(I+1)κnsdδ(r), (3)
+where
+K= (I+1/2)(−1)I+1/2−l, (4)
+withlthe nucleon orbital angular momentum, Iis the nuclear spin. The terms
+proportional to the anomalous magnetic moment of the nucleons an d the electrons are
+neglected.
+The interaction constant is then:
+κnsd=κa−K−1/2
+Kκ2+I+1
+KκQW, (5)
+whereκ2∼ −0.05 from Eq. 2 is the tree level approximation, and we have two
+corrections, the effective constant of the anapole moment κa, andκQWthat is generated
+by the nuclear spin independent part of the electron nucleon intera ction together with
+the hyperfine interaction.
+The contributions to the interaction constant can be estimated by [20, 25, 26]:
+κa=9
+10gαµ
+mp˜r0A2/3,
+κQW=−1
+3QWαµN
+mp˜r0AA2/3, (6)
+whereαisthefinestructureconstant, µandµNarethemagneticmomentoftheexternal
+nucleon and of the nucleus respectively in nuclear magnetons, ˜ r0= 1.2 fm is the nucleon
+radius,A=Z+N, andggives the strength of the weak nucleon-nucleon potential
+withgp∼4 for protons and 0 .2< gn<1 for neutrons [24]. For a heavy nucleus like
+francium, the anapole moment contribution is the dominant one ( κa,p/κQW= 14 and
+κa,p/κ2,n= 9). Rubidium is heavy enough that the anapole moment contribution still
+dominates ( κa,p/κQW= 20 and κa,p/κ2,n= 5).
+Vetteret al.[2] set an upper limit on the anapole moment of Thallium and Wood
+et al.[3, 27] measured with an uncertainty of 15% the anapole moment of133Cs by
+extracting the dependence of atomic PNC on the hyperfine levels inv olved. The results
+form atomic PNC and other measurements in nuclear physics have sim ilar uncertainty,
+but do not completely agree with each other [21]. It is desirable to hav e other atomicPreliminary study for anapole moment measurements in rubid ium and francium 4
+PNC measurements to resolve the discrepancy. In particular, in th e method proposed
+the anapole moment dominates over the spin independent PNC contr ibution. The
+projected signal to noise is 60 times higher than that of the Cs meas urement [15].
+Measurements in ions have also been proposed [28, 29]. Following Khr iplovich [24] the
+anapole moment is:
+a=−π/integraldisplay
+d3rr2J(r), (7)
+withJthe electromagnetic current density.
+Flambaum, Khriplovich and Sushkov [20] by including weak interaction s between
+nucleons intheircalculationofthenuclear current density, estimat etheanapolemoment
+from Eq. 7 of a single valence nucleon to be
+a=1
+eG√
+2Kj
+j(j+1)κa=Canj, (8)
+wherejis the nucleon angular momentum. The calculation is based on the shell model
+for the nucleus, under the assumption of homogeneous nuclear de nsity and a core with
+zeroangularmomentumleavingthevalencenucleoncarryingallthea ngularmomentum.
+Dimitrev and Telitsin [30, 31] have looked into many body effects in anap ole moments
+and find strong compensations among many-body contributions, s till there is about a
+factor of two difference with the single particle result (Eq. 8) [23].
+We estimate with Eqs. 6 and 8 the anapole moments of five francium iso topes on
+the neutron deficient side with approximately one minute lifetimes and five rubidium
+isotopes that lie on both sides of the stability region. Eq. 8 is still a goo d choice to
+give qualitative andquantative guideline of the anapolemoment measu rement. We have
+studied the Fr isotopes extensively in Ref. [32]. In even-neutron is otopes, the unpaired
+valence protongenerates the anapolemoment, whereas intheodd -neutronisotopesboth
+the unpaired valence proton and neutron participate. In the latte r case, one must add
+vectorially thecontributionsfromboththeprotonandtheneutro ntoobtaintheanapole
+moment as follows.
+a=Can
+pjp·I+Can
+njn·I
+I2 I, (9)
+withCan
+ijitheanapolemomentforasinglevalencenucleon iinashell modeldescription
+as given by Eq. 8, with the appropriate values of jpandjndepending on the isotope
+and using gp=4 andgn=1. Then we can use as an operational definition for the anapole
+moment constant the following equation:
+a=1
+eG√
+2(I+1/2)
+I(I+1)κaI, (10)
+This way of defining the anapole moment absorbs the angular moment um constant K
+from Eq. 4 in κa.
+Figure 1a shows the effective constant for the anapole moment for five different
+isotopes of francium (triangles) and rubidium (open squares). Fr h as an unpaired πh9/2
+proton for all the isotopes considered; the odd neutron in208,210Fr is an νf5/2orbit,Preliminary study for anapole moment measurements in rubid ium and francium 5
+while in212Fr the extra neutron is on a νp1/2orbital. There is a clear even-odd neutron
+number alternation in Fr due to the pairing of neutrons. For Rb, the alternation is no
+longer evident due to changes in the orbitals for the valence nucleon s. In particular the
+value ofκahas a different sign for the two stable isotopes of rubidium (85Rb and87Rb).
+The nucleon orbitals used for rubidium are πf5/2for isotopes 84 and 85, πp3/2for 86-88,
+νg9/2for 84 and 86 and νf5/2for 88 [33]. The two neutron holes in85Rb deform the
+nucleus very slightly and change the order of the proton orbitals fr omπp3/2in87Rb
+toπf5/2in85Rb. The result is that the spin and orbital contributions to the angu lar
+momentum are anti-aligned in85Rb and they are aligned in87Rb. The alignment is
+responsible for the sign change in κa. The authors of Ref. [34] use Eq. 2 to calculate
+the anapole moment constant and find no sign change between87Rb and85Rb. We
+consider even and odd isotopes with the vector sum of Eq. 9. The sig n change that
+we get comes from the contribution of K(Eq. 4) in our operational definition of the
+anapole moment Eq. 10. The quantity measured experimentally, the amplitude of the
+E1 PNC transition, also contains the sign change.
+Figure 1b presents a sensitivity analysis of the effective anapole con stant for the
+Rb isotopes to the change in gp=4 andgn=1 by fifty percent up and down around the
+values used in Fig.1a. The range of values still preserves the basic st ructure of the plot,
+and should allow a study of the gn/gpratio. There still remains the question of the
+sensitivity to the configuration used for the particular nucleus. Th e calculation of the
+anapole constant uses the orbital expected to be the dominant on e. The actual orbital
+may be a different one or even a superposition of different orbitals. U sing a proton
+orbitalπp3/2for86Rb changes κafrom 0.45 to -0.13, while using a proton orbital pid5/2
+for88Rb gives a smaller change from -0.06 to 0.01. Rb is a tractable nucleus a s it is
+around the neutron magic number of 50. This is not the case for Cs w here the nuclear
+structure calculations are more complicated.
+3. Constraints to weak meson-nucleon interaction constant s from anapole
+measurements
+The anapole moment constant ( κa) depends on the strength of the weak nucleon-
+nucleus potential, characterized by gpfor a proton and gnfor a neutron. Equation
+18 of Ref. [25] gives a relation between the weak nucleon-nucleus co nstants ( gpandgn)
+that appear in the expression for the anapole moment (Eq. 6) and t he meson-nucleon
+parity nonconserving interaction constants formulated by Despla nques, Donoghue, and
+Holstein (DDH) [35]. Evaluating the relations with the DDH “best” value s for the weak
+meson-nucleon constants we arrive at the following expressions
+Y= 3.61(−X+1.77gp+0.26), (11)
+Y= 2.5(X+2.65gn−0.29), (12)
+withX= (fπ−0.12h1
+ρ−0.18h1
+ω)×107andY=−(h0
+ρ+0.7h0
+ω)×107combinations of
+weak meson-nucleon constants. Figure 2 shows the expected con straints on the weakPreliminary study for anapole moment measurements in rubid ium and francium 6
+Figure 1. (Color online.) Anapole moment effective constant for different isoto pes. a)
+francium (triangles) and rubidium (open squares) with gp= 4, gn= 1. b) Sensitivity
+analysis for the Anapole moment effective constant for different Rb isotopes. The
+limits come from varying gpandgnby fifty percent from the values in a). The lines
+are only to guide the eye.
+meson-nucleon constants from an anapole moment measurement u sing Eqs. 11 and 12.
+The figure is analogous to Fig. 8 in Ref. [21] that shows the constrain ts obtained from
+different experiments. This figure complements the one that appea rs in the review of
+Behr and Gwinner [1] as it adds the rubidium numbers to the constrain ts obtained from
+the anapole moment measurement in Cs considering only the experime ntal uncertainty
+[3, 27] and the calculations for Fr.
+Figure 2 shows the expected constraints for a 3% measurement in f rancium and
+rubidium. The main contribution to the anapole moment in neutron eve n alkali atoms
+comes from the valence proton, and the experiment provides a mea surement of gp.
+Using Eq. 11 we obtain constraints in the direction of the Cs result of Fig. 2a. The
+fractional experimental uncertainty in gpis the same as the measurement one. ThePreliminary study for anapole moment measurements in rubid ium and francium 7
+actual final uncertainty for gpis considerably higher and depends on the accuracy of
+the theoretical calculations [25]. A measurement in any other neutr on even alkali atom
+generates constraints in the same direction as the Cs result, with a confidence band size
+proportional to the measurement uncertainty. Fig. 2a shows the expected constraint
+from a 3% measurement in a neutron even atom (rubidium or francium ). The neutron
+odd alkali atoms have contributions from the valence neutron. The constraints obtained
+(Eq. 12) are not colinear to those of Cs and generate a crossing re gion in Fig. 2.
+The size of the error band depends on the relative contributions of the valence proton
+and neutron to the anapole moment which depends on the atom. All t he calculations
+assumegp= 4 and gn= 1. Fig. 2b shows constraints obtained from measurements in
+rubidium if the values of gnandgpvary. The precise crossing region changes, as well as
+the constrained area (set in the figure by the error bars). This re sult shows in a different
+way the robustness of the proposed measurement. It may be able to give us more than
+just the DDH coupling constant constraints, but some information of the ratio of gn/gp.
+4. Experimental requirements
+This section presents the experimental requirements to measure the anapole moment
+in chains of Rb and Fr isotopes. Most of the details are in Ref [15], but h ere we focus
+on the differences for Rb and new ways that we have to perform the measurement.
+The measurement relies on driving an anapole allowed electric dipole ( E1) transition
+between hyperfine levels of the ground state. The transition prob ability is very small
+and is enhanced by interfering it with an allowed Raman (or magnetic dip ole (M1))
+transition. The excitation is coherent and allows for long interaction times. The signal
+to noise ratio is linear with time (up to the coherence time). We report on the progress
+towards an anapole measurement and the possibility of making the me asurement in
+rubidium.
+4.1. Source of atoms
+The work with radioactive atoms requires on-line trapping with an acc elerator to have
+access to reasonably short lifetime isotopes. We take the numbers for the production
+of unstable isotopes from what is available at TRIUMF in the Isotope S eparator and
+Accelerator (ISAC), where we are an approved experiment. A 500 MeV proton beam
+collides with an uranium carbide target to produce francium as fission fragments. A
+voltage up to 60 kV extracts the atoms as ions from the target. Th e beam goes through
+a mass separator and into the trapping area. The yield is up to 2 ×1011s−1for Rb, and
+2×106s−1for Fr, but it is expected to reach 108-109for Fr once the accelerator runs
+at full capacity.
+Our current apparatus to go on-line with the accelerator has a vac uum chamber
+to capture atoms in a high efficiency magneto-otpical trap (MOT) op erating in batch
+mode with a neutralizer, as described in Ref. [36]. A second science c hamber, withPreliminary study for anapole moment measurements in rubid ium and francium 8
+Figure 2. (Color online.) a) Constraints to DDH weak meson-nucleon interactio n
+constants from an anapole moment measurement in Rb and Fr. The is otopes with
+even number of neutrons give constraints aligned with the 14% Cs re sult [3], while
+those with odd number of neutrons give constraints in an different d irection. Cs result
+(small dashed line), Fr 3% measurement (solid line) and Rb 3% measure ment (long
+dashed line). b) Sensitivity analysis to the changes in the values of gn, gpby fifty
+percent. The Error bars mark the 3 % limits given a set of values. The color lines
+follow iso- gnand iso-gp.Preliminary study for anapole moment measurements in rubid ium and francium 9
+controlled electric and magnetic environments for the PNC experime nts, connects with
+the first through differential pumping. Tests for the transfer of atoms (Rb) between the
+two show efficiencies above 50% that allows accumulation of atoms for a longer time.
+4.2. Measurement
+The anapole moment constant scales with the atomic mass number as A2/3(Eq. 6).
+The anapole induced electric dipole ( E1) transition amplitude scales faster due to
+additional enhancement factors [15]. It is about 83 times larger in Fr than in Rb.
+The magnetic dipole ( M1) transition amplitude between hyperfine levels, on the other
+hand, has about the same value for both species. The M1 transition gives the main
+systematic error contribution, and the figure of merit is the ratio o f the two transition
+amplitudes |AE1/AM1| ∼1×10−9for francium and |AE1/AM1| ∼1×10−11for rubidium.
+In order to do a measurement in rubidium it becomes more important t o understand
+and suppress the M1 contribution.
+The experiment starts by optical pumping all the atoms to a particu lar level
+|F1,m1/angbracketright. The atoms interact with two transitions for a fixed time. One corre sponds to
+a Raman transition (parity conserving) and the other to the anapo le induced electric
+dipole (E1) transition (parity violating). Both are resonant with the ground state
+hyperfine transition |F1,m1/angbracketright → |F2,m2/angbracketrightin the presence of a static magnetic field. This
+triad of vectors defines the coordinate axis. The interference of both transitions gives a
+signal linear in the anapole moment [15]. The field driving the E1 transition is inside
+a microwave Fabry-Perot cavity. The signal to measure is the popu lation in |F2,m2/angbracketrightat
+the end of the interaction with a given handedness of the system.
+The specific magnetic field and transition levels reduce the sensitivity to
+fluctuations. There is an operating point where the transition freq uency varies
+quadratically with the magnetic field for a |∆m|= 1 transition. The operating field
+grows with the hyperfine separation and it is larger in Fr than in Rb. Ta ble 1 shows the
+magneticfieldfordifferent isotopesofFrandRb. Thetransition m1= 0.5→m2=−0.5
+in odd neutron isotopes has a small magnetic field. The electronic con tribution to the
+linear Zeeman effect cancels for the two levels at low magnetic fields, b ut the nuclear
+magnetic contribution remains since the two states belong to differe nt hyperfine levels.
+The magnetic field for odd neutron Fr isotopes is smaller than previos ly reported ( ∼
+2000 G) [15].
+The dimensions of the microwave cavity scale with the wavelength of t he transition
+(λm∼6 mm for Fr and λm∼6 cm for Rb). The mirror separation of the Fabry-Perot
+cavity should be at least 20 cm for Rb. The anapole signal remains unc hanged between
+Fr and Rb by putting more power in the microwave cavity to compensa te for the loss
+in the nuclear size.
+We hold the atoms in place for the measurement using a far off resona nce trap
+(FORT) [37]. Changes in the FORT wavelength allow its use for both rub idium and
+francium. The dipole trap causes an ac Stark shift that is different f or the two hyperfinePreliminary study for anapole moment measurements in rubid ium and francium 10
+Table 1. Operating point for the magnetic field ( B0), resonant frequency ( νm) and
+Zeeman sublevels ( m1,m2) for the transition.
+Atom Isotope Spin m1m2B0(G) νm(Mhz)
+Rb 84 2 -0.5 0.5 0.2 3084
+85 5/2 0 -1 186.1 2992
+86 2 0.5 -0.5 0.3 3947
+87 3/2 0 -1 654.2 6602
+88 2 0.5 -0.5 0.03 1191
+Fr 208 7 0.5 -0.5 3.3 49880
+210 6 0.5 -0.5 3.4 46768
+212 5 0.5 -0.5 4.5 49853
+levels. The differential shift causes a change of the resonant freq uency and eventually
+leads to decoherence. The Stark shift in a FORT is in the same directio n for both
+hyperfine levels but of different size due to the different detuning. T he differential shift
+is proportional to the hyperfine splitting, and it is reduced by an ord er of magnitude in
+rubidum.
+The measurement in both species depends on the effectiveness of t he suppression
+mechanisms [15]. The first suppression mechanism works by having th e atoms in the
+magnetic field node (electric field antinode). The reduction depends on the magnitude
+of the field at the edges of the atomic cloud. Since the wavelength inc reases by an
+order of magnitude in rubidium, the suppression improves by the sam e amount. The
+second suppression mechanism works by forcing the M1 transition to have the wrong
+polarization for the levels considered. It remains unchanged in rubid ium. The atoms
+oscillate around the magnetic field node for the third suppression me chanism. The
+suppression is proportional to the M1 field, and since it gets reduced because of the
+better positioning in the node, we can gain an order of magnitude (as suming no increase
+in the driving field power). The suppression mechanisms work better in rubidium than
+in francium by two orders of magnitude because of better positionin g to the magnetic
+field node. The improvement compensates the two orders of magnit ude loss in the figure
+of merit ( |AE1/AM1|).
+We compare the requirements in rubidium to those established on Tab le III of Ref.
+[15] for francium. We assume an increase in the microwave power to k eep the same
+E1 transition amplitude. The magnetic field stability is still about 10−5but since now
+the magnetic field is smaller this means that the field has to be controlle d to about 10
+µG. The requirements on all the systematic effects that depend on t heM1 transition
+produced by the microwave cavity increase by two orders of magnit ude. This is because
+by increasing the microwave field we increase the E1 andM1 transition amplitude by
+thesame amount. The systematic effects introduced by thedipolet rapor Ramanbeams
+remain the same.Preliminary study for anapole moment measurements in rubid ium and francium 11
+5. Optical Dipole trap
+We report on the experimental implementation of the optical dipole t rap in the science
+chamber. The dipole trap design aims to decrease the photon scatt ering and differential
+ac Stark shift introduced by the laser forming the trap [38, 39]. We u se a FORT to
+reduce the photon scattering rate. The ac Stark shift depends o n the position in the
+trap and the atomic state. The shift changes with time as the atoms move in the trap.
+We choose a blue detuned trap where the atoms are confined on the dark region of the
+trap.
+Weusearotatingdipoletrapbecausewecancontroltheshapeand sizedynamically.
+A laser rotating faster than the motion of the atoms creates a time averaged potential
+equivalenttoahollowbeampotential[40]. Thelaserbeampropagating inthezdirection
+goes through two acousto-optical modulators (AOMs) placed bac k-to-back in the xand
+ydirections respectively. We use the beam that corresponds to the first diffraction
+order in both directions, the (1,1) mode. We scan the modulation fre quency of both
+AOMswithtwophase-lockedfunctiongenerators(StanfordRese archSR345)togenerate
+different hollow beam shapes.
+The general expression of the time averaged potential U(ρ,z) in the radial direction
+for linearly polarized light and a detuning larger than the hyperfine st ructure splitting,
+but smaller than the fine structure splitting is [41]:
+U(ρ,z) =¯hγ
+24IS/bracketleftBigg1
+δ1/2+2
+δ3/2/bracketrightBigg/contintegraltext
+ρ′∈lI(ρ−ρ′,z)dl
+/contintegraltextdl(13)
+whereγis the natural linewidth, ISis the saturation intensity defined as IS=
+2π2¯hcγ/(3λ3), andI(ρ,z) is the Gaussian beam intensity at position ( ρ,z). The integral
+over the contour of the rotating laser beam lgives the time averaged potential. The
+detunings δ1/2andδ3/2in units of γ.
+Tightly focusing the laser at the position of the atoms confines them along the
+beam axis. Fig. 3 shows the shape of the potential both along the ra dial and axial
+directions for a circular shaped trap.
+We study the lifetime andspin relaxation rateofsuch a circular trapw ith a beam of
+400 mW and blue detuned 2.5 nm from the87RbD2line. The spin relaxation is critical
+for the anapole measurement. The beam rotating frequency is 50 k Hz, which is much
+faster than the oscillation frequency of the trap ( ≤1 kHz). The trap has a transverse
+diameter of 150 µm, an axial diameter of 24 mm, and a potential depth of 60 µK
+(normalized potential value of 0.4 in Fig 3). We measure the atom numb er in the dipole
+trap after a pre-set hold time by shinning a 100 µs long resonant pulse and imaging
+the fluorescence into a photomultiplier tube (Fig. 4a). We image the fl uorescence from
+a region of radius of 2 mm. We see a rapid decay during the first 100 ms from fast
+atoms that can not be confined in the radial direction. The rapid dec ay is followed by
+a slower decay (2.5 s lifetime). The slow decay is shorter than the MOT lifetime (30
+s) and corresponds to the continous diffusion of the atoms out of t he imaging region.
+This is supported by the calculation shown in Fig. 4a that gives the rem aining numberPreliminary study for anapole moment measurements in rubid ium and francium 12
+Figure 3. (Color online) Time averagedpotential along the axial and radial dire ctions
+for a cigar shaped trap. The aspect ratio is not to scale in the figure .
+of atoms in the imaging area using the expected temperature of the atoms. We follow
+the method of Ref. [42] to measure the spin relaxation rate. We load the atoms from a
+magneto-optical trap (MOT) to the dipole trap, turn off the magne tic field and MOT
+beams and pump the atoms to the F= 1 ground state. We get the relaxation rate due
+to the interaction of the atoms with the dipole trap laser by comparin g the populations
+of the atoms in both hyperfine levels as a function of time. Figure 4b s hows the fraction
+of atoms in the F= 2 ground state as a function of time. An exponential fit to the dat a
+gives a spin relaxation time of 840 ms, similar to previous measurement s [42]. This is
+a first step towards the 20 ms coherent interaction of the propos ed data taking cycle in
+Ref. [15]
+We reduce the diffusion of the atoms in the axial direction by adding a o ne-
+dimensional (1D) blue denuned standing wave with a frequency differ ent from the one
+usedfortherotatingtrap. Thecombinationoftightradialconfine ment fromtherotating
+trapandconfinement intheaxialdirectionfromthestandingwave g ives ahigher density
+dipole trap (Fig. 5). It also opens the possibility to study the motion o f atoms in 2D
+billiards with arbitrary transverse shape [43].
+The symmetry of the trap is an important point for the anapole mome nt
+measurement. As we scan the beam, there will be diffraction power c hanges on the
+AOMs. This has been pointed out in the study with Bose-Einstein cond ensates where
+the uniformity is required to avoid parametric excitation [44]. We fee dforward on the
+RF power to reduce the diffraction variations [45]. Fig. 6 shows the inc reased stability
+in the diffraction power as we rotate the beams using this method.Preliminary study for anapole moment measurements in rubid ium and francium 13
+Figure 4. a) Lifetime measurement of the atoms in the dipole trap, filled square s
+experimental data, dashed line atoms escaping model, continuous lin e exponential fit.
+b) Measurement of the spin relaxation time by plotting the fraction o f the atoms in
+theF= 2 state in the dipole trap. The continuous line is the exponential fit ( lifetime
+840 ms).Preliminary study for anapole moment measurements in rubid ium and francium 14
+Figure 5. Fluorescence image of the atoms 35 ms after turning off the magnet ic field
+and MOT beams. a) rotating dipole trap. b) rotating dipole trap and 1 D blue detuned
+standing wave. Gravity (g) goes into the paper in the figure.
+Figure 6. Intensity profile of the diffracted light showing a few oscilation period s that
+generate the trap with (solid line) and without (dashed line) feedfor ward on the RF
+power to the AOMs.
+6. Conclusions
+The measurement ofthe anapolemoment in achain of isotopes canco nstraint thevalues
+oftheDDHweakmeson-nucleon interactionconstants. Themeasu rement oftheanapole
+moment is possible in any of the heavy alkali atoms (rubidium, cesium or francium) but
+it becomes increasingly difficult with decreasing atomic number. The an apole moment
+for the two Rb naturally available isotopes has the opposite sign which can be useful
+for the study of systematic effects. Neutron odd isotopes have t ransitions insensitive
+to magnetic field fluctuations at small static values of the magnetic fi eld. We have a
+working rotating blue detuned dipole trap necessary to hold the ato ms for the duration
+of the anapole moment measurement. The trap shows a spin relaxat ion time of 840 ms.
+The dipole trap will be used in future measurements in both francium a nd rubidium.
+Acknowledgments
+Work supported by NSF. E. G. acknowledges support from CONACY T (cooperacion
+bilateral). We would like to thank J. A. Behr, B. A. Brown, N. Davidson , S. Schnelle, G.Preliminary study for anapole moment measurements in rubid ium and francium 15
+D. Sprouse, for helpful discussions and A. Perez Galvan for his wor k on the apparatus
+and interest on this project.
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\ No newline at end of file
diff --git a/1001.0818.txt b/1001.0818.txt
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+arXiv:1001.0818v1  [astro-ph.GA]  6 Jan 2010The Solar Nebula on Fire: A Solution to the Carbon Deficit in
+the Inner Solar System
+Jeong-Eun Lee1, Edwin A. Bergin2, Hideko Nomura3
+ABSTRACT
+Despite a surface dominated by carbon-based life, the bulk compos ition of the
+Earth is dramatically carbon poor when compared to the material av ailable at
+formation. Bulk carbon deficiency extends into the asteroid belt re presenting a
+fossil record of the conditions under which planets are born. The in itial steps of
+planet formation involve the growth of primitive sub-micron silicate an d carbon
+grains in the Solar Nebula. We present a solution wherein primordial ca rbon
+grains are preferentially destroyed by oxygen atoms ignited by hea ting due to
+stellar accretion at radii <5 AU. This solution can account for the bulk carbon
+deficiency in the Earth and meteorites, the compositional gradient within the
+asteroid belt, and for growing evidence for similar carbon deficiency in rocks
+surrounding other stars.
+Subject headings: ISM: abundances – Stars: formation – Physical Data and
+Processes: astrochemistry
+1. Introduction
+Theearlystagesofplanetformationrelyonthegrowthofsmall gra insintolargerbodies.
+Locally, the relative chemical composition of planets, comets and as teroids within the solar
+system provides indirect clues to the environment and processes t hat existed and operated
+billions of years ago. In this regard, the bulk elemental composition r epresents a key fossil
+record, as it can be compared to the distribution of elements availab le at birth as represented
+by the Sun, interstellar clouds, or extra-solar protoplanetary dis ks.
+1Department of Astronomy and Space Science, Astrophysical Res earch Center for the Structure and
+Evolution of the Cosmos, Sejong University, Seoul 143-747, Kore a; jelee@sejong.ac.kr
+2Department of Astronomy, The University of Michigan, 500 Church Street, Ann Arbor, Michigan 48109-
+1042; ebergin@umich.edu
+33Department of Astronomy, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan;
+nomura@kusastro.kyoto-u.ac.jp– 2 –
+Figure 1 illustrates the dramatic gradient in the amount of carbon re lative to silicon
+incorporated into planets, comets, and asteroids, represented by meteorites. In this figure
+the H/Si ratio serves to separate the various bodies with the prov iso that the Sun/ISM
+represents the total amount of material available at the start. M aterial that formed far from
+the Sun (comets) incorporate water and other hydrogen rich ices , but do not include the H 2
+gas that is the major component of the Sun and ISM. In contrast r ocks within the terrestrial
+planet region (r <5 AU) incorporate successively less H-rich ices, a well known result d ue
+to the snow-line lying inside the asteroid belt. Icy bodies have an abun dance of carbon in
+the form of ices and amorphous carbon grains. In contrast the Ea rth has three orders of
+magnitude less carbon (relative to Si) than was available during forma tion as represented
+by the Sun/ISM. While the Earth measurement does have uncertain ty, as we do not know
+the amount of carbon sequestered in the core, estimates of the a mount of hidden carbon still
+lead to significant carbon deficit (All´ egre et al. 2001). Moreover, primitive carbonaceous
+meteorites, as tracers of the starting materials for the Earth, a lso exhibit carbon deficiency.
+In ISM∼60% of the cosmic carbon is locked in some form of graphite or amorph ous
+carbon grains (Savage & Sembach 1996). These primordial grains n eed to be destroyed
+prior to planet formation to account for the composition of the Ear th and carbonaceous
+chondrites. That carbon grains have been detected in comets with near-solar abundances
+(Fig. 1) suggests that any destruction mechanism was inactive in th e cometary formation
+zone near the giant planets. Another hint lies in the variation of spec tral classes within the
+asteroid belt, which may be indicative of compositional gradients in th e relative amounts of
+carbonaceous material (Gradie & Tedesco 1982). Thus there are hints in the solar system of
+processes that were active in the hot inner nebula that kept primor dial carbon grains from
+being incorporated into rocks. However, the same process was ina ctive in the colder outer
+nebula.
+In addition, the detection of carbon and iron in the atmosphere of s ome white dwarfs
+(Jura 2006) is an unexpected result as the strong gravitational fi eld of a white dwarf is
+anticipated to lead to settling of heavy elements in a photosphere do minated by either
+hydrogen or helium (Paquette et al. 1986). The measured C/Fe abu ndance ratio in the
+atmosphere of these white dwarfs is well below solar and consistent with external supply by
+rocky bodies that are deficient in carbon (Jura 2008). In sum, the solution to the carbon
+deficit in the inner solar system is a natural outcome of star and plan et formation, perhaps
+throughout the Galaxy.
+The standard model for the Earth’s composition is that the planet f ormed from material
+less volatile thansodium, but with anadditional contribution ofvolatile s (Waenke &Dreibus
+1988). In this approach, carbon is considered a volatile element sinc e, in thermodynamic– 3 –
+equilibrium, in a mixture with a solar composition, it forms molecules such as CO and CH 4.
+Therefore, the standard cosmochemical model presumes that t he inner solar system was
+entirely vaporized at a temperature greater than 1,800 K (Grossm an 1972). However, the
+computeddiskmidplanetemperatureat1AUislessthan1,000Kinmost modelsofaccreting
+pre-main-sequence stars (D’Alessio et al. 2005). Thus, primordial carbon grains should have
+been incorporated into the Earth. Carbon must have been presen t in the nebular gas in a
+volatile form (CO, CH 4, or CO 2). At 1 AU this gaswould not be assimilated into silicate-rich
+planetesimals, but would be accreted by the Sun and gas giants or los t to space. Beyond the
+nebular snow-line large bodies could incorporate the carbon grains a nd ices, which can be
+contributed to the young Earth by impacts. Some previous solution s to this problem include
+ultraviolet heating and vaporization of infalling grains (Simonelli et al. 1 997), reactions with
+OH (Gail 2002), and impacts (Bond et al. 2009). However, the infall phase is short and
+models suggest that either none or all grains, including silicates, are vaporized (Simonelli et
+al. 1997) and in the warm inner disk OH may not be abundant enough (G lassgold et al.
+2004; Glassgold et al. 2009; Woitke et al. 2009; Gorti & Hollenbach 20 04).
+In this letter we meld measurements of relative amounts of carbon a nd silicon found
+within solar system bodies with our current understanding of the ph ysics and chemistry in
+extra-solar protoplanetary systems to demonstrate that the in nermost regions of the Solar
+Nebula was on fire and rocks forming within the terrestrial planet zo ne for the Sun and,
+likely other stars, are carbon-poor.
+2. Model Calculations and Results
+The paradigm of terrestrial planet formation involves the growth o f solids in the solar
+nebula, which is a protoplanetary disk. Grains initially present in the dis k upper layers settle
+to a dust-rich midplane where they coagulate and grow into planetes imals (Weidenschilling
+& Cuzzi 1993). Depending upon their size, grains can be suspended in the disk atmosphere
+via interactions with the turbulent gas (Weidenschilling & Cuzzi 1993) . In this theory sub-
+micron sized grainsareheld in theupper layers, by turbulent stirring , andcoagulateto larger
+sizes where they eventually settle to the midplane. Grains settle to a vertical layer, zsett,
+that depends on the disk physical structure, the strength of tu rbulence, parameterized by
+anαviscosity (Hartmann 2001), and the grain mass to surface area ra tio. This height can
+be determined by setting the timescale of settling, tsett, equal to the timescale of diffusive
+turbulent stirring of the disk, tstir(Dullemond & Dominik 2004):
+ξtsett=tstir (1)– 4 –
+ξ/bracketleftbigg4
+3σ
+mcs(z)
+Ω2
+Kρ(z)/bracketrightbigg
+=Sc(z)
+αcs(z)z2
+Hp(2)
+Scis the Schmidt number which relates the decoupling of grains from tur bulence.m/σis
+the grain mass-to-cross section ratio, which is m/σ= 4ρda/3, whereρdis the grain density
+andais the grain radius. Here, ρdis calculated from graphite, whose density is 2.24 g cm−3,
+with assumed porosities (Ormel et al. 2007). Hpis the pressure scale height which is given
+byHp=cs,0/Ωk. ΩK=/radicalbig
+G M⋆/R3denotes the local Kepler frequency and the sound
+speed at the midplane, cs,0=/radicalBig
+kTgas
+µgasmp. For perfectly coupled particles Sc= 1, which is the
+case for grains smaller than a= 100µm (Dullemond & Dominik 2004). When ξ= 1 the
+settling layer ( zsett), where grains are suspended by turbulence, is reached. The dep letion
+layer (zdepl), above which the disk is entirely depleted (by settling) of grains of t hat size,
+can be determined by setting ξ= 100 (Dullemond & Dominik 2004). We adopt α= 0.01,
+which seems appropriate according to the Balbus-Hawley instability m echanism to generate
+viscosity (Hartmann 2001).
+The gas and dust temperature and density in the disk are calculated by using the
+parameters listed in the caption of Figure 2 (see Nomura et al. 2007 f or details).1For a
+dust model, we use a dust size distributions of dn/da∝a−3.5with grain radii ranging from
+0.01µm to 10µm, which are larger than those of the ISM grain model (Mathis et al. 1 977),
+assuming the possibility of grain growth in dark clouds. In the calculat ion of stirring depths,
+we adopt various porosities for different grain sizes according to Or mel et al. (2007).
+Figure 2 presents the estimates of zsettandzdepland the gas and dust temperature
+profiles at 1 AU in our disk model. We also show the chemical abundance profiles of key
+oxygen reservoirs2. Disk accretion onto the star and an active chromosphere power X -ray
+and ultraviolet radiation fields that heat the disk surface to high tem peratures at the depth
+where grainsaresuspended by turbulent stirring (Glassgoldet al. 2 004; Nomura et al. 2007);
+in contrast the disk midplane is cooler. A key facet of disk thermal mo dels is that gas and
+dust temperatures are decoupled in the upper layers. Figure 2 sho ws thatTgasbetweenzsett
+andzdeplcan be quite high ( >500K) for grains ≤10µm. In addition, within this depth range
+oxygen-bearing molecules are predominantly photodissociated with nO= 1.8×10−4nHat<
+1The gas heating via PAHs is included in the model (e.g., Aikawa & Nomura 2 006), but we note that the
+temperature structure may have some dependency on models (e.g ., Gorti & Hollenbach 2004).
+2The chemical/thermal model did not include the chemistry of water a nd OH formation. For these
+abundances we used the analytical approximations of Bethell & Ber gin (2009), which rely on knowledge of
+the local radiation field and gas temperature and density. The resu lt is consistent with other self-consistent
+model results (e.g. Gorti & Hollenbach 2004).– 5 –
+5 AU. At 10 and 20 AU, however, the dust temperature is lower than the water desorption
+temperature (110 K) at all depths so that a significant amount of o xygen is locked on grain
+surfaces, as expected beyond the snowline.
+Hot oxygen atoms erode carbon grains and release the carbon into the gas. The physical
+and chemical conditions between zsettandzdeplat the terrestrial planet forming zone in our
+model are such that the chemical sputtering of carbon grains by h ot oxygen atoms will be
+operative. Thus, forcarbon, theneedtoincreasethedusttemp eratureabovethesublimation
+point, as in the standard cosmochemical model, is evaded. Silicates w ill not be subject to
+similar chemisputtering (Barlow & Silk 1977) and remain the seeds for t errestrial worlds.
+The oxidation rate of carbon-based grains is: koxy=nOσvOY(s−1).nOandvOare the
+numberdensityandthethermalvelocityofoxygenatoms, respec tively.Yistheyield(carbon
+atoms removed per incident oxygen atoms). We adopt laboratory s tudies for the yield as
+given in Table 2 of Draine (1979), which covers the appropriate temp erature range. For
+Tgas= 500 K and 1000 K, Y∼0.02 and 0.2, respectively. A summary of recent experiments
+atTgas>1000 K can be found in Vierbaum and Roth (2002); these results are consistent
+with our assumptions.
+The key question is the destruction timescale, tdest, as grains cascade to smaller sizes
+when carbon atoms are removed. If tdestis shorter than tstir, any carbon grain that is lifted
+above the settling layer will be destroyed. To determine tdestwe have constructed a kinetic
+chemical network, such as gn+1+O→gn+C+O(with n = # of carbon atoms) with the
+rate specified above. This network is then extended down to carbo n atoms and we solve
+the system of ordinary differential equations to calculate tdestof the biggest grain (0.002
+µm) feasible to the computational memory and time. We then extend t hese results to larger
+grains by extrapolating this timescale (see Eq. (4) below).
+As shown above, the reaction rate ( koxy) is proportional to the grain surface area ( σ=
+πa2) in a given temperature and abundance of atomic oxygen. For σ, we have related the
+grain volume to its mass:
+4
+3πa3=mC
+ρd, (3)
+HeremCis the total mass of carbon atoms contained in the given size grain. B ased on this
+model with no porosity, a 0.002 µm grain (g3000) contains 3000 carbon atoms. For the initial
+abundance of each grain ( g#) in this kinetic calculation, we adopted the total number of
+carbon atoms (per H atom) derived from the MRN interstellar grain s ize distribution for
+graphite (see Figure 6.19 of Tielens 2005) and extrapolated to very small grains.– 6 –
+Figure 3 shows the abundance variation of atoms in carbon grains as a function of time
+at 1 AU. The physical conditions for the kinetic calculation are obtain ed from our disk model
+at the depth (0.122 AU) where Tgas= 500 K. These results demonstrate that, assuming no
+source term, there is a rapid decay of carbon grains to reduced ca rbon content and size.
+We definetdestof a certain size of grain when X(g1) decreases by 4 orders of magnitude in
+the reservoir. Therefore, 0.002 µm grain (g3000) is destroyed within 2 years at the depth of
+Tgas= 500 K at 1 AU. tdestof 0.002µm grain at 3, 10, and 20 AU are 14, 100, and 400
+years, respectively (see Table 1). In the calculation of tdestat 10 and 20 AU, we assume all
+oxygen is atomic although a significant amount is likely locked on grain su rfaces as water
+ice. Therefore, the calculated tdestat 10 and 20 AU is the lower limit. In this disk model,
+the same size grains are destroyed at the depth where Tgas= 1000 K about 3 times faster
+compared to the depth where Tgas= 500 K. Our estimates of destruction timescales are very
+conservative as we have examined the timescale at 500 K and defined the timescale after a
+several orders of magnitude decrease in abundance.
+As grains grow in size by colliding aggregation, the porosity, the meas urement of volume
+that is not occupied by carbon atoms, increases (Ormel et al. 2007 ). The enlargement
+parameterψis defined by ψ=V/V∗, whereV∗is the volume that carbon grains occupy
+in its compacted state while Vis the total spatial extent of the given size grain. Thus, the
+porosity is defined as [( V−V∗)/V]×100 = [1−1/ψ]×100. Therefore, the same size of grain
+of a higher porosity contains a lower number of carbon atoms to red ucetdest.ψ=1 does
+not necessarily mean no porosity, but we assume no porosity for ψ=1 for the conservative
+calculation of tdest(i.e., the longest destruction timescale) of a given size grain. Accord ing
+to the results of Ormel et al. (2007), we adopt ψ=1, 8, and 27 for 0.1, 1, and 10 µm
+grains, respectively. Therefore, the porosity of 0.1, 1, and 10 µm grains are 0, 88, and 96%,
+respectively.
+tdestat a given condition is proportional to 1 /koxy×#(C)grain, where #(C)grainis the
+total number of carbon atoms contained in the grain, and thus, pr oportional to the size of
+grain (1/σ×a3∝a). However, the porosity reduces tdestsince #(C)grainof a given size
+decreases. Therefore, tdestof a given size grain can be calculated by:
+tdest,a= (a
+0.002µm)ψ−1
+atdest,0.002, (4)
+wheretdest,0.002andtdest,aaretdestof 0.002µm grain and a grain with size a.ψais the
+enlargement factor for grains of a certain size. The results of the combination of grain
+size and porosity are presented in Table 2. At the depth of Tgas= 500 K,tdest< tstir
+for grains smaller than 10 µm at 1 and 3 AU while at 10 and 20 AU, only very small– 7 –
+grains (0.01 µm) satisfy the condition. Therefore, all primordial carbon grains a re destroyed
+within the terrestrial planet forming zone by turbulent stirring abo vezsettwithintstir. This
+effectively stops incorporating carbon grains in the initial stages of planetesimal formation
+– the coagulation of small grains in the upper layers and the subsequ ent settling to lower
+heights. Moreover, this mechanism will decrease in efficiency within th e asteroid belt, which
+could produce the observed asteroid compositional gradients.
+3. Discussion
+Carbon grain destruction in the dense UV free midplane has been exp lored by Finocchi
+et al. 1997, Bauer et al. 1997, and Gail (2001, 2002). In this work O H is the primary agent
+for carbon grain destruction. However, the amount of OH in the dis k midplane has some
+uncertainty as these authors determine an OH abundance of ∼10−10−10−7relative to H 2
+(Finocchi et al. 1997, Bauer et al. 1997), while Glassgold et al. (2009 ) find an abundance
+well below 10−12. Our model relies on an abundance of oxygen atoms on the disk surf ace
+(e.g. Fig. 2) in a region that is clearly exposed to UV radiation, ensurin g a ready supply of
+atoms with O/OH ≫1. However, our model does not preclude destruction of carbon g rains
+in the midplane by OH, if the abundance is high enough.
+As a result of chemical destruction of carbon grains all carbon in th e inner solar system
+will reside in the gas, which will accrete into the star or dissipate. Car bon grains that reach
+sizes ofa>50−100µm in the outer disk will settle below the layer where chemisputtering
+dominates. This brings up an important question: if the Earth did not incorporate carbon
+grains or accrete sufficient amounts from the gas, how did the Eart h assimilate any carbon?
+We suggest that maybe the problem of the Earth’s carbon is related to the origin of the
+Earth’s oceans. At 1 AU it is likely that grains were too hot to be sufficie ntly coated with
+icesandinsteadtheEarthpotentiallyreceived itswaterfromimpact sofbodiesthatoriginate
+beyond the nebular snow-line near 2.5 AU (e.g. Morbidelli et al. 2000, b ut see also Drake
+2004). These bodies are likely also carbon rich (e.g. Gradie & Tedesco 1982) and could have
+brought carbon with water ice. The mass of carbon in the Earth’s ma ntle is∼5.5×1023g
+with perhaps up to an order of magnitude additional carbon seques tered in the core (All´ egre
+et al. 2001). In the study of terrestrial planet formation perfor med by O’Brien et al. (2006)
+they estimate that ∼15% (median value of simulations) of the Earth was delivered by bodies
+that originated beyond 2.5 AU. If we assume that these bodies cont ained the same amount
+of carbon as in carbonaceous chondrites of ∼10 mg/g (Waenke & Dreibus 1988) then this
+corresponds to a delivery of roughly 1025g of carbon, which is in the exact range of that
+estimated for the Earth (see Bond et al. 2009 for an additional look at this issue).– 8 –
+We are grateful to M. Jura for motivating this work and J. Cuzzi fo r helpful discussions.
+This work was supported by the National Research Foundation of K orea(NRF) grant funded
+by the Korea government(MEST) (No.2009-0062865).
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+This preprint was prepared with the AAS L ATEX macros v5.2.– 10 –
+Fig. 1.— Carbon abundances relative to Silicon in the Solar System. Abu ndance ratios
+of carbon (listed as C) relative to silicon for representative bodies in the solar system as
+a function of their H/Si content. Shaded areas denote specific ty pes of bodies (2 Oort
+cloud comets, 4 classes of chondritic meteorites, Earth) or instan ces with strong similarities
+(Sun and ISM). Abundance estimates are taken from the literatur e (Geiss 1987, Wasson &
+Kallemeyn 1988, Sofia et al. 1994, All´ egre et al. 2001, Min et al. 2005 , Grevesse et al.
+2007). The amount of hydrogen on the Earth is estimated assuming a hydrosphere mass of
+1.4×1024g. Figure adapted and expanded from Geiss 1987.– 11 –
+10-710-610-510-4Abundance Relative to Total H
+0.300.250.200.150.10
+Z [AU]O 
+OHCO
+H2O 
+102461002461000246Temperature (K)
+0.3 0.2 0.1 0.0
+Z (AU)10-510-410-310-210-1100101102Visual Extinction  (mag) Tgas
+Tdust
+0.1 µm
+a=0.01 µm1 µm10 µm
+zsett zdeplAV
+Fig. 2.— Left:Temperatures and zsettandzdeplin the protoplanetary disk atmosphere.
+Gas and dust temperatures as a function of vertical height above the midplane at 1 AU.
+Also shown are the estimated depletion and settling depths for vario us grain sizes. Right:
+Chemical abundances of key oxygen carriers as a function of vert ical depth. The chemical
+calculation is stopped at z∼0.1 AU when the gas/dust temperatures reach equilibrium.
+The model assumes an axisymmetric α-viscous disk (e.g., Hartmann 2001) with α= 0.01
+and˙M= 10−8M⊙yr−1, which is surrounding a central star with parameters of M∗= 1.0M⊙,
+R∗= 2R⊙, andT∗= 4000K. Both UV and X-ray spectral profiles from the central st ar are
+assumed to be the same as those of TW Hydra, and they are normaliz ed so that the total
+FUV luminosity is LFUV= 1×1031ergs s−1(which corresponds to GFUV= 300G0at the
+disk radius of 100 AU, where G0= 1.6×10−3ergs s−1cm−2) and the X-ray luminosity is
+LX= 1×1029ergs s−1.– 12 –
+Fig. 3.— Kineticsofcarbongraindestruction at1AU.Plot ofgrainabu ndances –maximum
+grain size ( a(g3000)) is∼0.002µm – with variable number of carbon atoms as a function
+of time using conditions at the depth where Tgas= 500 K (see Table 1). Various line types
+represent grains containing different carbon atoms. The gray arr ow indicates the timescale
+(tdestof 0.002µm grain) in which the abundance of g1decreases by 4 orders of magnitude.– 13 –
+Table 1: Destruction timescale of 20 ˚A grains at the depth of Tgas= 500 K
+1 AU 3 AU 10 AU 20 AU
+Depth (AU) 0.122 0.494 2.047 5.180
+Hp(AU) 0.033 0.101 0.460 1.164
+n(H2) (cm−3) 1.42×1092.03×1082.50×1077.50×106
+tstir(yr) 160 840 3200 8400
+tdest(yr) 2 14 100 400
+Table 2: Destruction timescale of various size of grains at the depth ofTgas= 500 K
+Grain Size ( µm) Porosity (%) 1 AU 3 AU 10 AU 20 AU
+(160 yr)a(840 yr)a(3200 yr)a(8400 yr)a
+0.01 0 10 yr 70 yr 500 yr 2000 yr
+0.1 0 100 yr 700 yr 5000 yr 20000 yr
+1 88 125 yr 875 yr 6250 yr 25000 yr
+10 96 370 yr 2593 yr 18520 yr 74000 yr
+atstirat each radius as given in Table 1.
\ No newline at end of file
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+arXiv:1001.0820v1  [cs.AI]  6 Jan 20101
+Abstract Answer Set Solvers with Backjumping
+and Learning
+(long version)
+YULIYA LIERLER
+Department of Computer Science
+University of Texas at Austin
+1 University Station C0500
+Taylor Hall 2.124
+Austin, USA
+E-mail: yuliya@cs.utexas.edu
+Abstract
+Nieuwenhuis, Oliveras, and Tinelli (2006) showed how to des cribe enhancements of the
+Davis-Putnam-Logemann-Loveland algorithm using transit ion systems, instead of pseu-
+docode. We design a similar framework for several algorithm s that generate answer sets
+for logic programs: smodels ,smodels cc,asp-sat with Learning ( cmodels ), and a newly
+designed and implemented algorithm sup. This approach to describing answer set solvers
+makes it easier to prove their correctness, to compare them, and to design new systems.
+KEYWORDS : answer set programming, inference, learning
+1 Introduction
+Answer Set Programming (ASP) is a methodology commonly used for s olving com-
+binatorial search problems (Lifschitz 2008). In the development o f ASP solvers,
+computational ideas behind SAT solvers (Gomes et al. 2008) play an im portant
+role. Influence of SAT solvers development on ASP systems is twofo ld. On the one
+hand, such ASP solvers as assat1andcmodels2follow the so called SAT-based
+approach where a SAT solver is invoked for search, possibly multiple t imes. On
+the other hand, “native” ASP solvers that implement search proce dures specifically
+suited for logic programs often adopt computational techniques f rom SAT solvers.
+For instance, dlv3implements backjumping (Ricca et al. 2006), and smodels cc4(Ward and Schlipf 2004)
+extends the answer set solver smodels5by introducing restarts, conflict-driven
+1http://assat.cs.ust.hk/ .
+2http://www.cs.utexas.edu/users/tag/cmodels .
+3http://www.dbai.tuwien.ac.at/proj/dlv/ .
+4http://www.nku.edu/ ∼wardj1/Research/smodels cc.html .
+5http://www.tcs.hut.fi/Software/smodels/ .2 Y. Lierler
+backjumping, learning, and forgetting – techniques widely used in SA T solvers.
+The ASP solver sup6(Lierler 2008) implements these features also.
+In this paper our main goal is to show how the “abstract” approach to describ-
+ing SAT solvers proposed in (Nieuwenhuis et al. 2006) can be extende d to ASP
+solvers that use these sophisticated features. Usually computat ion procedures are
+described in terms of pseudocode. In (Nieuwenhuis et al. 2006), th e authors pro-
+posed an alternative approach to describing dpll -like procedures. They introduced
+an abstract framework that captures what ”states of computa tion” are, and what
+transitions between states are allowed. In this way, it defines a dire cted graph such
+that every execution of the dpll procedure corresponds to a path in this graph.
+Some edges may correspond to unit propagation steps, some to br anching, some to
+backtracking. This allows the authors to model a dpll -like algorithm by a mathe-
+matically simple and elegant object, graph, rather than a collection o f pseudocode
+statements. In (Lierler 2008), we extended this framework for d escribing such ASP
+algorithms as smodels ,asp-sat with Backtracking, and supwithout Learning . In
+this paper, we expand our previous work on abstract answer set s olvers to cover
+such features as backjumping and learning (and also forgetting an d restart). We
+start by introducing an abstract framework that captures a gen eral mechanism of
+these sophisticated features in ASP solvers. For instance, this fr amework provides
+the transition underlying the process of learning a clause, but it doe s not suggest
+which clause shall be learned. Similarly, it provides a general descript ion of back-
+jumping but it does not supply the means for computing a “backjump clause”
+necessary for an answer set solver to perform backjumping. We t hen enhance this
+abstract framework to capture enough information about a stat e of computation
+for deriving a backjump clause.
+Usually, dpll -like procedures implement conflict-driven backjumping and learn-
+ing where a particular learning schema such as, for instance, Decision orFirs-
+tUIP (Mitchell 2005) is applied for computing a special kind of a backjump c lause.
+There are two common methods for describing a backjump clause co nstruction.
+One employs the implication graph (Marques-Silva and Sakallah 1996) a nd the
+other employs resolution (Mitchell 2005). Ward and Schlipf (2004) e xtended the
+notion of an implication graph to the smodels algorithm. They then defined an
+algorithm for computing FirstUIP backjump clauses utilized by smodels ccto im-
+plement conflict-driven backjumping and learning. In this paper we in troduce the
+algorithms BackjumpClause andBackjumpClauseFirstUIP based on resolution and
+the enhanced abstract framework that compute Decision andFirstUIP7backjump
+clauses respectively.
+In (Lierler 2008), we introduced the basic algorithm underlining the s ystemsup
+but neglected some of its features: conflict-driven backjumping, learning, forgetting,
+and restarts. Here we account for these techniques and use an a bstract framework
+designed in this paper for describing system sup. We emphasize that the work
+on this abstract framework helped us to develop ASP solver sup, to incorporate
+6http://www.cs.utexas.edu/users/tag/sup .
+7The names of the backjump clauses follow (Mitchell 2005).Abstract Answer Set Solvers with Backjumping and Learning ( long version) 3
+learning into its algorithm, and to prove its correctness. We analyze d performance
+ofsup against such answer set solvers as cmodels ,smodels ,smodels cc, and
+clasp8. Overall, supperforms well against these rival systems.
+We start the paper with Section 2 that reviews the abstract DPLL f ramework
+introduced in (Nieuwenhuis et al. 2006) and some logic programming co ncepts. In
+Section 3, we define a graph representing the application of the algo rithm for finding
+supporting models of a logic program. This paves the way to defining a graph rep-
+resenting the application of the smodels algorithm to a program in Section 4. Sec-
+tion 4.2 elaborates on the relationship between previously defined ab stract frame-
+works. Section 5 extends the abstract DPLL framework by introd ucing an addi-
+tional inference rule so that the generate and test algorithm of the SAT-based ASP
+systemcmodels may be characterized by this graph. In Section 6, we review the
+abstract framework that describes DPLL enhanced by backjump ing and learning.
+In Section 7, we define a general abstract framework for describ ing ASP algorithms
+that implement such phenomena as backjumping and learning. In Sec tion 7.2 we
+describe the algorithms of systems smodels ccandsupby means of this framework.
+In Section 8 we extend the abstract generate and test framework to accommodate
+backjumping and learning, and in Section 8.2 we use these findings to d escribe the
+cmodels algorithm. Section 9 extends the framework to capture additional infor-
+mation about a computation state of a solver, states the correct ness results, and
+describes how the frameworks are related to each other. Section 10 provides the
+proofs for these results. In Section 10.3 and 11 we introduce the a lgorithms based
+on the extended framework for computing a backjump clause that are important in
+implementing conflict-driven backjumping and learning. In Section 12 we introduce
+the concept of an extended graph for the generate and test abstract framework and
+state the correctness results. Section 13 provides the proofs f or these results. At last,
+in Section 14 we provide the experimental analysis that compares pe rformance of
+supwith other answer set solvers.
+2 Review: Abstract DPLL and Logic Programs
+2.1 Abstract Classical DPLL
+For a set σof atoms, a recordMrelative to σis a list of literals over σwhere
+(i) some literals in Mare annotated by ∆ that marks them as decision literals,
+(ii)Mcontains no repetitions.
+The concatenation of two such lists is denoted by juxtaposition. Fr equently, we
+consider a record as a set of literals, ignoring both the annotations and the or-
+der between its elements. A literal lisunassigned by a record if neither lnor its
+complement lbelongs to it.
+8http://www.cs.uni-potsdam.de/clasp/ .4 Y. Lierler
+Unit Propagate :
+M=⇒Ml if/braceleftbiggC∨l∈Fand
+C⊆M
+Decide :
+M=⇒Ml∆if/braceleftbiggMis consistent and
+lis unassigned by M
+Fail:
+M=⇒FailState if/braceleftbiggMis inconsistent and
+Mcontains no decision literals
+Backtrack :
+Pl∆Q=⇒Plif/braceleftbiggPl∆Qis inconsistent, and
+Qcontains no decision literals
+Fig. 1. The transition rules of the graph dpF.
+Astate relative to σis either a distinguished state FailState or a record relative
+toσ. For instance, the states relative to a singleton set {a}of atoms are
+FailState ,∅,a,¬a,a∆,¬a∆,a¬a,a∆¬a,
+a¬a∆,a∆¬a∆,¬aa,¬a∆a,¬aa∆,¬a∆a∆,
+where by∅we denote the empty list.
+IfCis a disjunction (conjunction) of literals then by Cwe understand the con-
+junction (disjunction) of the complements of the literals occurring inC. We will
+sometimes identify Cwith the multi-set of its elements.
+For any CNF formula F(a finite set of clauses), we will define its DPLL graph
+dpF. The nodes of dpFare the states relative to the set of atoms occurring in F.
+We use the terms “state” and “node” interchangeably. Recall tha t a node is called
+terminal in a graph if there is no edge leaving this node in the graph. If a state is
+consistent and complete then it represents a truth assignment fo rF.
+The set of edges of dpFis described by a set of “transition rules.” Each transition
+rule is an expression M=⇒M′followed by a condition, where MandM′are nodes
+ofdpF. Whenever the condition is satisfied, the graph contains an edge fr om node M
+toM′. Generally, an edge in the graph may be justified by several transit ion rules.
+Figure 1 presents four transition rules that characterize the edg es ofdpF.
+This graph can be used for deciding the satisfiability of a formula Fsimply
+by constructing an arbitrary path leading from node ∅until a terminal node M
+is reached. The following proposition shows that this process always terminates,
+thatFis unsatisfiable if MisFailState , and that Mis a model of Fotherwise.
+Proposition 1
+For any CNF formula F,
+(a) graph dpFis finite and acyclic,Abstract Answer Set Solvers with Backjumping and Learning ( long version) 5
+(b) any terminal state of dpFother than FailState is a model of F,
+(c)FailState is reachable from ∅indpFif and only if Fis unsatisfiable.
+For instance, let Fbe the set consisting of the clauses
+a∨b
+¬a∨c.
+Here is a path in dpF:
+∅ =⇒(Decide )
+a∆=⇒(Unit Propagate )
+a∆c =⇒(Decide )
+a∆cb∆(1)
+The name of the transition rule after each = ⇒shows which rule justifies the presence
+of this edge in the graph. Since the state a∆cb∆is terminal, Proposition 1(b)
+asserts that{a,c,b}is a model of F. Here is another path in dpFfrom∅to the
+same terminal node:
+∅ =⇒(Decide )
+a∆=⇒(Decide )
+a∆¬c∆=⇒(Unit Propagate )
+a∆¬c∆c=⇒(Backtrack )
+a∆c =⇒(Decide )
+a∆cb∆(2)
+Path (1) corresponds to an execution of dpll in the sense of (Davis et al. 1962);
+path (2) does not, because it applies Decide toa∆even though Unit Propagate
+could be applied in this state.
+Note that the graph dpFis a modification of the classical DPLL graph defined
+in (Nieuwenhuis et al. 2006, Section 2.3). It is different in three ways. First, its
+states are pairs M||Ffor all CNF formulas F. For the purposes of this section,
+it is not necessary to include F. Second, the description of the classical DPLL
+graph involves a “PureLiteral” transition rule. We dropped this rule b ecause it
+does not correspond to any of the propagation rules used in answe r set solvers
+whose algorithms we will model in this paper. Third, in the definition of t hat
+graph, each Mis required to be consistent. In case of DPLL, due to the simple
+structure of a clause, it is possible to characterize the applicability o fBacktrack in
+a simple manner: when some of the clauses become inconsistent with t he current
+partial assignment, Backtrack is applicable. In ASP, it is not easy to describe the
+applicability of Backtrack if only consistent states are taken into account. We
+introduced inconsistent states in the graph dpFto facilitate our work on extending
+this graph to model algorithms of answer set solvers.
+In the rest of this section we give a proof of Proposition 1.
+Lemma 1
+For any CNF formula Fand any state l1...lnreachable from∅indpF, every
+modelXofFsatisfiesliif it satisfies all decision literals l∆
+jwithj≤i.6 Y. Lierler
+Proof
+By induction on the path from ∅tol1...ln. The property of Xthat we need to
+prove trivially holds in the initial state ∅, and we will prove that all transition rules
+ofdpFpreserve it.
+Take a model XofF, and consider an edge M=⇒M′whereMis a listl1...lk
+such that Xsatisfiesliif it satisfies all decision literals l∆
+jwithj≤i.
+It is clear that the rule justifying the transition from MtoM′is different from
+Fail. For each of the other three rules, M′is obtained from a prefix of Mby append-
+ing a list of literals containing at most one decision literal. Due to the indu ctive
+hypothesis, it is sufficient to show that if Xsatisfies all decision literals in M′
+thenXsatisfies all M′.
+Unit Propagate :M′isMl. By the inductive hypothesis, for every literal in M
+the property in question holds. We need to show that X|=l. From the definition of
+Unit Propagate , for some clause C∨l∈F,C⊆M. Consequently, M|=¬C. From
+the inductive hypothesis and the assumption that Xsatisfies all decision literals
+inM′and hence in M, it follows that X|=M. SinceXis a model of F, we conclude
+thatX|=l.
+Decide :M′isMl∆. Obvious.
+Backtrack :Mhas the form Pl∆QwhereQcontains no decision literals. M′
+isPl. By the inductive hypothesis, it trivially follows that for every literal inP
+the property in question holds. We need to show that X|=l. Assume that X|=l.
+SinceQdoes not contain decision literals, and the assumption that Xsatisfies all
+decision literals in M′and hence in P,Xsatisfies all decision literals in Pl∆Q, that
+isM. By the inductive hypothesis, it follows that XsatisfiesM. This is impossible
+becauseMis inconsistent.
+ıProof of Proposition 1
+(a) The finiteness of dpFis obvious. For any list Nof literals by|N|we denote the
+length of N. Any state Mother than FailState has the form M0l∆
+1M1...l∆
+pMp,
+wherel∆
+1...l∆
+pare all decision literals of M; we define α(M) as the sequence of
+nonnegative integers |M0|,|M1|,...,|Mp|, andα(FailState ) =∞. By the definition
+of the transition rules defining the edges of dpF, if there is an edge from a state M
+toM′indpFthenα(M)< α(M′), where <is understood as the lexicographical
+order. It follows that if a state M′is reachable from Mthenα(M)< α(M′).
+Consequently the graph is acyclic.
+(b) Consider any terminal state Mother than FailState . From the fact that Decide
+is not applicable, we conclude that Mhas no unassigned literals. Since neither
+Backtrack norFailis applicable, Mis consistent. Consequently Mis an assignment.
+It follows that for any clause C∨l∈FifC/\e}atio\slash⊆MthenC∩M/\e}atio\slash=∅. Furthermore,
+sinceUnit Propagate is not applicable, we conclude that if C⊆Mthenl∈M.
+Consequently, M|=C∨l. HenceMis a model of F.
+(c) Left-to-right: Since FailState is reachable from∅, there is an inconsistent state M
+without decision literals that is reachable from ∅. By Lemma 1, any model of F
+satisfiesM. SinceMis inconsistent we conclude that Fhas no models.Abstract Answer Set Solvers with Backjumping and Learning ( long version) 7
+Right-to-left: From (a) it follows that there is a path from ∅to some terminal
+state. By (b), this state cannot be different from FailState , because Fis unsatisfi-
+able.
+2.2 Logic Programs
+We consider programs consisting of finitely many rules of the form
+a←b1,...,bl,not bl+1,...,not bm (3)
+whereais an atom or symbol ⊥, and each bi(1≤i≤m) is an atom. We will
+identify the body of (3) with the conjunction
+b1∧...∧bl∧¬bl+1∧...¬∧bm (4)
+and also with the set of its conjunctive terms. If the head aof a rule (3) is an atom
+then we will identify (3) with the clause
+a∨¬b1∨...∨¬bl∨bl+1∨...∨bm. (5)
+Ifais⊥then we call rule (3) a constraint and identify (3) with the clause
+¬b1∨...∨¬bl∨bl+1∨...∨bm. (6)
+We will often omit the symbol ⊥when referring to a constraint.
+We will use two abbreviated forms for a rule (3): The first is
+a←B
+whereBstands for b1,...,bl,not bl+1,...,not bm. The second abbreviation is
+a←D,F (7)
+whereDstands for the positive part of the body b1,...,bl, andFstands for the
+negative part of the body not bl+1,...,not bm.
+Thereduct ΠXof a program Π with respect to a set Xof atoms is obtained
+from Π by
+•removing each rule (7) such that F∩X/\e}atio\slash=∅, and
+•replacing each remaining rule (7) by a←D.
+A setXof atoms is an answerset for a program Π if Xis minimal (with respect to
+set inclusion) among the sets of atoms that satisfy the reduct ΠX(Gelfond and Lifschitz 1988).
+For example, let Π be the program
+a←not b
+b←not a
+c←a
+d←d.(8)
+Consider set{a,c}. Reduct Π{a,c}is
+a←
+c←a
+d←d.(9)8 Y. Lierler
+Set{a,c}satisfies the reduct and is minimal, hence {a,c}is an answer set of Π.
+Consider set{a,c,d}. The reduct Π{a,c,d}is (9). Set{a,c,d}satisfies the reduct
+but is not minimal and hence it is not an answer set of Π.
+ByBodies (Π,a) we denote the set of the bodies of all rules of Π with head a.
+For any set Mof literals, by M+we denote the set of positive literals from M. For
+any consistent and complete set Mof literals (that is, an assignment), if M+is an
+answer set for a program Π, then Mis a model of Π. Moreover, in this case Mis
+asupported model of Π, in the sense that for every atom a∈M,M|=Bfor some
+B∈Bodies (Π,a).
+A setUof atoms occurring in a program Π is said to be unfounded (Van Gelder et al. 1991)
+on a consistent set Mof literals w.r.t. Π if for every a∈Uand every B∈
+Bodies (Π,a),B∩M/\e}atio\slash=∅orU∩B+/\e}atio\slash=∅. There is a tight relation between un-
+founded sets and answer sets: For any model Mof a program Π, M+is an answer
+set for Π if and only if Mcontains no non-empty subsets unfounded on Mw.r.t. Π
+(Corollary 2 from (Sacc´ a and Zaniolo 1990)9).
+For instance, let Π be program (8) and let Mbe a consistent set {a,¬b,c,d}of
+literals. We already demonstrated that M+={a,c,d}is not an answer set of Π.
+Accordingly, its subset {d}is unfounded on{a,¬b,c,d}w.r.t. Π, because the only
+rule in Π with din the head
+d←d
+is such that U∩B+={d}∩{d}/\e}atio\slash=∅.
+We say that a program Π entails a formula Fwhen for any consistent and com-
+plete set Mof literals, if M+is an answer set for Π, then M|=F. For instance,
+any program Π entails each rule occurring in Π.
+3 Generating Supported Models
+In Section 4 we will define, for an arbitrary program Π, a graph smΠrepresenting
+the application of the smodels algorithm to Π; the terminal nodes of smΠare
+answer sets of Π. As a step in this direction, we describe here a simple r graph
+atleast Π.
+3.1 Graph atleast Π
+The terminal nodes of atleast Πare supported models of Π. The transition rules
+defining atleast Πare closely related to procedure Atleast (Simons 2000, Sec-
+tions 4.1), which is one of the core procedures of the smodels algorithm.
+The nodes of atleast Πare the states relative to the set of atoms occurring in Π.
+The edges of the graph atleast Πare described by the transition rules Decide ,Fail,
+Backtrack introduced in Section 2.1 and the additional transition rules10presented
+9The Corollary 2 from (Sacc´ a and Zaniolo 1990) refers to ”ass umption sets” rather than un-
+founded sets. But as the authors noted, in the context of this corollary the two concepts are
+equivalent.
+10The names of some of these rules follow (Ward 2004).Abstract Answer Set Solvers with Backjumping and Learning ( long version) 9
+Unit Propagate LP :
+M=⇒M a if/braceleftbigga←B∈Π and
+B⊆M
+All Rules Cancelled :
+M=⇒M¬aifB∩M/\e}atio\slash=∅for allB∈Bodies (Π,a)
+Backchain True :
+M=⇒Ml if
+
+a←B∈Π,
+a∈M,
+B′∩M/\e}atio\slash=∅for allB′∈Bodies (Π,a)\{B},
+l∈B
+Backchain False :
+M=⇒Mlif
+
+a←l,B∈Π,
+¬a∈Mora=⊥,
+B⊆M
+Fig. 2. The additional transition rules of the graph atleast Π.
+in Figure 2. Note that each of the rules Unit Propagate LP andBackchain False
+is similar to Unit Propagate : the former corresponds to Unit Propagate onC∨l
+wherelis the head of the rule, and the latter corresponds to Unit Propagate on
+C∨lwherelis an element of the body of the rule.
+This graph can be used for deciding whether program Π has a suppor ted model
+by constructing a path from ∅to a terminal node:
+Proposition 2
+For any program Π,
+(a) graph atleast Πis finite and acyclic,
+(b) any terminal state of atleast Πother than FailState is a supported model
+of Π,
+(c)FailState is reachable from ∅inatleast Πif and only if Π has no supported
+models.
+For instance, let Π be program (8). Here is a path in atleast Π:
+∅ =⇒(Decide )
+a∆=⇒(Unit Propagate LP )
+a∆c =⇒(All Rules Cancelled )
+a∆c¬b =⇒(Decide )
+a∆c¬bd∆(10)
+Since the state a∆c¬bd∆is terminal, Proposition 2(b) asserts that {a,c,¬b,d}is
+a supported model of Π.
+The assertion of Proposition 2 will remain true if we drop the transitio n rules
+Backchain True andBackchain False from the definition of atleast Π.
+In the rest of this section we give a proof of Proposition 2.10 Y. Lierler
+Lemma 2
+For any program Π and any state l1...lnreachable from∅inatleast Π, every
+supported model Xfor Π satisfies liif it satisfies all decision literals l∆
+jwithj≤i.
+Proof
+By induction on the path from ∅tol1...ln. Similar to the proof of Lemma 1. We
+will show that the property in question is preserved when the trans ition from M
+toM′is justified by any of the four new rules.
+Take a supported model Xfor Π, and consider an edge M=⇒M′whereMis a
+listl1...lksuch that Xsatisfiesliif it satisfies all decision literals l∆
+jwithj≤i.
+Assume that Xsatisfies all decision literals in M′.
+Unit Propagate LP :M′isMa. By the inductive hypothesis, for every literal in M
+the property in question holds. We need to show that X|=a. By the definition of
+Unit Propagate LP ,B⊆Mfor some rule a←B. Consequently, M|=B. From the
+inductive hypothesis and the assumption that Xsatisfies all decision literals in M′
+and hence in M, it follows that X|=M. SinceXis a model of Π we conclude
+thatX|=a.
+All Rules Cancelled :M′isM¬aandB∩M/\e}atio\slash=∅for every B∈Bodies (Π,a).
+Consequently, M|=¬Bfor every B∈Bodies (Π,a). By the inductive hypothesis,
+for every literal in Mthe property in question holds. We need to show that X|=¬a.
+By contradiction. Assume that X|=a. From the inductive hypothesis and the
+assumption that Xsatisfies all decision literals in M′and hence in M, it follows
+thatX|=M. SinceM|=¬Bfor every B∈Bodies (Π,a), it follows that X|=¬B.
+We conclude that Xis not a supported model of Π.
+Backchain True :M′isMl. By the inductive hypothesis, for every literal in M
+the property in question holds. We need to show that X|=l. By contradiction.
+Assume X|=l. Consider the rule a←Bcorresponding to this application of
+Backchain True . Sincel∈B,X|=¬B. By the definition of Backchain True ,
+B′∩M/\e}atio\slash=∅for every B′inBodies (Π,a)\B. Consequently, M|=¬B′for every B′
+inBodies (Π,a)\B. From the inductive hypothesis and the assumption that X
+satisfies all decision literals in M′and hence in M, it follows that X|=M. We
+conclude that X|=¬B′for every B′inBodies (Π,a)\B. HenceXis not supported
+by Π.
+Backchain False :M′isMl. By the inductive hypothesis, for every literal in Mthe
+property in question holds. We need to show that X|=l. By contradiction. Assume
+thatX|=l. By the definition of Backchain False there exists a rule a←l,Bin Π
+such that¬a∈MandB⊆M. Consequently, M|=¬aandM|=B. From the
+inductive hypothesis and the assumption that Xsatisfies all decision literals in M′
+and hence in M, it follows that X|=M. We conclude that X|=¬aandX|=B.
+From the fact that X|=l, it follows that Xdoes not satisfy the rule a←l,B, so
+that it is not a model of Π.
+ıProof of Proposition 2
+Parts (a) and (c) are proved as in the proof of Proposition 1, using Lemma 2.
+(b) LetMbe a terminal state so that none of the rules are applicable. From th eAbstract Answer Set Solvers with Backjumping and Learning ( long version) 11
+fact that Decide is not applicable, we conclude that Massigns all literals. Since
+neitherBacktrack norFail is applicable, Mis consistent. Consequently, Mis an
+assignment. Since Unit Propagate LP is not applicable, it follows that for every
+rulea←B∈Π, ifB⊆Mthena∈M. Consequently, if M|=BthenM|=a.
+We conclude that Mis a model of Π. We will now show that Mis a supported
+model of Π. By contradiction. Suppose that Mis not a supported model. Then,
+there is an atom a∈Msuch that M/\e}atio\slash|=Bfor every B∈Bodies (Π,a). Since M
+is consistent, B∩M/\e}atio\slash=∅for every B∈Bodies (Π,a). Consequently, All Rules
+Cancelled is applicable. This contradicts the assumption that Mis terminal.
+The fact that the assertion of Proposition 2 remains true if we drop the transition
+rulesBackchain True andBackchain False from the definition of atleast Πfollows
+from the proof of Proposition 2 (b) that does not refer to those r ules.
+3.2 Relation between dpFandatleast Π
+It is well known that the supported models of a program can be char acterized as
+models of program’s completion in the sense of (Clark 1978). It turn s out that the
+graphatleast Πis identical to the graph dpF, whereFis the (clausified) comple-
+tion of Π. To make this claim precise, we first review the notion of comp letion.
+For any program Π, its completion consists of Π and the formulas tha t can be
+written as
+¬a∨/logicalordisplay
+B∈Bodies(Π,a)B (11)
+for every atom ain Π. ıCNF−Comp (Π) is the completion converted to CNF us-
+ing straightforward equivalent transformations. In other words , ıCNF−Comp (Π)
+consists of clauses of two kinds:
+1. the rules a←Bof the program written as clauses
+a∨B, (12)
+2. formulas (11) converted to CNF using the distributivity of disjun ction over
+conjunction11.
+Proposition 3
+For any program Π, the graphs atleast ΠanddpCNF-Comp (Π)are equal.
+For instance, let Π be the program
+a←b,not c
+b.(13)
+Its completion is
+(a↔b∧¬c)∧b∧¬c, (14)
+11It is essential that repetitions are not removed in the proce ss of clausification. For instance,
+ıCNF−Comp(a←not a) is the formula ( a∨a)∧(¬a∨¬a).12 Y. Lierler
+and ıCNF−Comp (Π) is
+(a∨¬b∨c)∧(¬a∨b)∧(¬a∨¬c)∧b∧¬c. (15)
+Proposition 3 asserts that atleast Πcoincides with dpCNF-Comp (Π).
+From Proposition 3, it follows that applying the Atleast algorithm to a program
+essentially amounts to applying dpll to its completion.
+In the rest of this section we give a proof of Proposition 3.
+It is easy to see that the states of the graphs atleast ΠanddpCNF-Comp (Π)
+coincide. We will now show that the edges of atleast ΠanddpCNF-Comp (Π)coincide
+also.
+It is clear that there is an edge M=⇒M′inatleast Πjustified by the rule
+Decide if and only if there is an edge M=⇒M′indpCNF-Comp (Π)justified by
+Decide . The same holds for the transition rules Fail andBacktrack .
+We will now show that if there is an edge from a state Mto a state M′in the
+graphdpCNF-Comp (Π)justified by the transition rule Unit Propagate then there is
+an edge from MtoM′inatleast Π. Consider a clause C∨l∈ıCNF−Comp (Π)
+such that C⊆M. We will consider two cases, depending on whether C∨lcomes
+from (12) or from the CNF of (11).
+Case 1.C∨lisa∨Bcorresponding to a rule a←B.
+Case 1.1. lisa. Then there is an edge from MtoM′inatleast Πjustified by
+the transition rule Unit Propagate LP .
+Case 1.2. lis an element of B. ThenBhas the form l,DandCisa∨D. From
+C⊆Mwe conclude that D⊆Mand¬a∈M. There is an edge from MtoM′in
+the graph atleast Πjustified by the following instance of Backchain False :
+M=⇒M l if
+
+a←l,D∈Π,
+¬a∈M,
+D⊆M.
+Case 2.C∨lhas the form¬a∨D, whereDis one of the clauses of the CNF of
+/logicalordisplay
+B∈Bodies(Π,a)B.
+ThenDhas the form/logicalordisplay
+B∈Bodies(Π,a)f(B)
+wherefis a function that maps every B∈Bodies (Π,a) to an element of B.
+Case 2.1. lis¬a. ThenCisD, so that D⊆M. Consequently f(B)∈B∩D⊆
+B∩M, so that B∩M/\e}atio\slash=∅for every B∈Bodies (Π,a). There is an edge from M
+toM′inatleast Πjustified by All Rules Cancelled .
+Case 2.2. lis an element of D. From the construction of D, it follows that l=
+f(B)∈Bfor some rule a←B. ThenCis
+¬a∨/logicalordisplay
+B′∈Bodies(Π,a)\Bf(B′).
+FromC⊆Mwe conclude that a∈Mand that f(B′)∈Mfor every B′∈Abstract Answer Set Solvers with Backjumping and Learning ( long version) 13
+Bodies (Π,a)\B. Sincef(B′) is a conjunctive term of B′, it follows that B′∩M/\e}atio\slash=∅.
+Then there is an edge from MtoM′inatleast Πjustified by Backchain True .
+We will now show that if there is an edge from a state Mto a state M′in
+the graph atleast Πjustified by one of the transition rules Unit Propagate LP ,
+All Rules Cancelled ,Backchain True , andBackchain False then there is an edge
+fromMtoM′indpCNF-Comp (Π).
+Case 1. The edge is justified by Unit Propagate LP . Then there is a rule a←B∈Π
+whereB⊆M, andM′isMa. By the construction of ı CNF−Comp (Π),a∨B∈
+ıCNF−Comp (Π). There is an edge from MtoM′indpCNF-Comp (Π)justified by
+the following instance of Unit Propagate :
+M=⇒Ma if/braceleftbiggB∨a∈ıCNF−Comp (Π) and
+B⊆M.
+Case 2. The edge is justified by All Rules Cancelled . By the definition of All
+Rules Cancelled , there is an atom asuch that for all B∈Bodies (Π,a),B∩M/\e}atio\slash=∅;
+andM′isM¬a. Consequently, Mcontains the complement of some literal in B.
+Denote one of such literals by f(B), so that f(B)∈M. From the construction of
+ıCNF−Comp (Π),
+¬a∨/logicalordisplay
+B∈Bodies(Π,a)f(B)
+belongs to ı CNF−Comp (Π). By the choice of f,
+/logicalordisplay
+B∈Bodies(Π,a)f(B)⊆M.
+There is an edge from MtoM′indpCNF-Comp (Π)justified by the following instance
+ofUnit Propagate :
+M=⇒M¬aif
+
+/logicalordisplay
+B∈Bodies(Π,a)f(B)∨¬a∈ıCNF−Comp (Π),
+/logicalordisplay
+B∈Bodies(Π,a)f(B)⊆M.
+Case 3. The edge is justified by Backchain True . By the definition of Backchain
+True , there is a rule a←B∈Π and a literal l∈Bsuch that a∈M; for all
+B′∈Bodies (Π,a)\B,B′∩M/\e}atio\slash=∅; andM′isMl. Letf(B′) be an element of B′
+such that f(B′)∈M. From the construction of ı CNF−Comp (Π),
+¬a∨l∨/logicalordisplay
+B′∈Bodies(Π,a)\Bf(B′)
+belongs to ı CNF−Comp (Π). By the choice of f,
+/logicalordisplay
+B′∈Bodies(Π,a)\Bf(B′)⊆M.
+There is an edge from MtoM′indpCNF-Comp (Π)justified by the following instance14 Y. Lierler
+ofUnit Propagate :
+M=⇒Ml if
+
+¬a∨l∨/logicalordisplay
+B′∈Bodies(Π,a)\Bf(B′)∈ıCNF−Comp (Π),
+(¬a∨/logicalordisplay
+B′∈Bodies(Π,a)\Bf(B′))⊆M.
+Case 4. The edge is justified by Backchain False . By the definition of Backchain
+False , there is a rule a←l,B∈Π such that¬a∈M,B⊆M, andM′isMl.
+By the construction of ı CNF−Comp (Π),a∨B∨l∈ıCNF−Comp (Π). There
+is an edge from MtoM′indpCNF-Comp (Π)justified by the following instance of
+Unit Propagate :
+M=⇒Mlif/braceleftBigg
+a∨B∨l∈ıCNF−Comp (Π) and
+a∨B⊆M.
+4 Answer Set Solver Smodels
+4.1 Abstract Smodels
+We now describe the graph smΠthat represents the application of the smodels
+algorithm to program Π. smΠis a graph whose nodes are the same as the nodes
+of the graph atleast Π. The edges of smΠare described by the transition rules of
+atleast Πand the additional transition rule:
+Unfounded :
+M=⇒M¬aif/braceleftbiggMis consistent, and
+a∈Ufor a set Uunfounded on Mw.r.t. Π.
+This transition rule of smΠis closely related to procedure Atmost (Simons 2000,
+Sections 4.2), which together with the procedure Atleast forms the core of the
+smodels algorithm.
+The graph smΠcan be used for deciding whether program Π has an answer set
+by constructing a path from ∅to a terminal node:
+Proposition 4
+For any program Π,
+(a) graph smΠis finite and acyclic,
+(b) for any terminal state MofsmΠother than FailState ,M+is an answer set
+of Π,
+(c)FailState is reachable from ∅insmΠif and only if Π has no answer sets.
+To illustrate the difference between smΠandatleast Π, assume again that Π is
+program (8). Path (10) in the graph atleast Πis also a path in smΠ. But state
+a∆c¬bd∆, which is terminal in atleast Π, is not terminal in smΠ. This is notAbstract Answer Set Solvers with Backjumping and Learning ( long version) 15
+surprising, since{a,c,¬b,d}+={a,c,d}is not an answer set of Π. To get to a
+state that is terminal in smΠ, we need two more steps:
+...
+a∆c¬bd∆=⇒(Unfounded ,U={d})
+a∆c¬bd∆¬d=⇒(Backtrack )
+a∆c¬b¬d(16)
+Proposition 4(b) asserts that {a,c}is an answer set of Π.
+The assertion of Proposition 4 will remain true if we drop the transitio n rulesAll
+Rules Cancelled ,Backchain True , andBackchain False from the definition of smΠ.
+In the rest of this section we give a proof of Proposition 4.
+We say that a model Mof a program Π is unfounded-free if no non-empty subset
+ofMis an unfounded set on Mw.r.t. Π.
+Lemma 3 (Corollary 2 from (Sacc´ a and Zaniolo 1990) )
+For any model Mof a program Π, M+is an answer set for Π if and only if Mis
+unfounded-free.
+Lemma 4
+For any unfounded set Uon a consistent set Mof literals w.r.t. a program Π, and
+any assignment X, ifX|=MandX∩U/\e}atio\slash=∅, thenX+is not an answer set for Π.
+Proof
+Assume that X+is an answer set for Π. Then Xis a model of Π. By Lemma 3,
+it follows that Xis unfounded-free. Hence any non-empty subset of Xincluding
+X∩Uis not unfounded on X. This means that for some rule a←Bin Π such
+thata∈X∩U,B∩X=∅andX∩U∩B+=∅. FromX|=M(M⊆X) and
+B∩X=∅we conclude that B∩M=∅. SinceB∩X=∅andXis an assignment,
+B⊆X. It follows that B+⊆X. Consequently U∩B+=X∩U∩B+=∅. This
+contradicts the assumption that Uis an unfounded set on M.
+Lemma 5
+For any program Π, any state l1...lnreachable from∅insmΠ, and any assign-
+mentX, ifX+is an answer set for Π then Xsatisfiesliif it satisfies all decision
+literalsl∆
+jwithj≤i.
+Proof
+By induction on the path from ∅tol1...ln. Recall that for any assignment X, ifX+
+is an answer set for Π, then Xis a supported model of Π, and that the transition
+systemsmΠextendsatleast Πonly by the transition rule Unfounded . Given our
+proof of Lemma 2, we only need to demonstrate that application of Unfounded
+preserves the property.
+Consider a transition M=⇒M′justified by Unfounded , whereMis a sequence
+l1...lk.M′isM¬a, such that a∈U, whereUis an unfounded set on Mw.r.t Π.
+Take any assignment Xsuch that X+is an answer set for Π and Xsatisfies all16 Y. Lierler
+decision literals l∆
+jwithj≤k. By the inductive hypothesis, X|=M. ThenX|=¬a.
+Indeed, otherwise awould be a common element of XandU, andX∩Uwould
+be non-empty, which contradicts Lemma 4.
+ıProof of Proposition 4
+Parts (a) and (c) are proved as in the proof of Proposition 1, using Lemma 5.
+(b) As in the proof of Proposition 2(b) we conclude that Mis a model of Π. Assume
+thatM+is not an answer set. Then, by Lemma 3, there is a non-empty unfou nded
+setUonMw.r.t. Π such that U⊆M. It follows that Unfounded is applicable
+(with an arbitrary a∈U). This contradicts the assumption that Mis terminal.
+The fact that the assertion of Proposition 4 remains true if we drop the transition
+rulesAll Rules Cancelled ,Backchain True , andBackchain False from the definition
+ofsmΠfollows from the proof of Proposition 4 (b) that does not refer to t hose rules.
+4.2 Smodels Algorithm
+We can view a path in the graph smΠas a description of a process of search
+for an answer set for a program Π by applying inference rules. Ther efore, we can
+characterize the algorithm of an answer set solver that utilizes the inference rules of
+smΠby describing a strategy for choosing a path in smΠ. A strategy can be based,
+in particular, on assigning priorities to some or all inference rules of smΠ, so that
+a solver will never apply a transition rule in a state if a rule with higher pr iority is
+applicable to the same state.
+We use this method to describe the smodels algorithm. System smodels assigns
+priorities to the inference rules of smΠas follows:
+Backtrack ,Fail≫
+Unit Propagate LP ,All Rules Cancelled ,Backchain True ,Backchain False≫
+Unfounded≫
+Decide .
+For example, let Π be program (8). The smodels algorithm may follow a path
+∅ =⇒(Decide )
+a∆=⇒(Unit Propagate LP )
+a∆c =⇒(All Rules Cancelled )
+a∆c¬b =⇒(Unfounded )
+a∆c¬b¬d
+in the graph smΠ, whereas it may never follow path (10), because Unfounded has
+a higher priority than Decide .
+4.3 Tight Programs
+We will now review the definitions of a positive dependency graph and a tight
+program. The positive dependency graph of a program Π is the directed graph G
+such thatAbstract Answer Set Solvers with Backjumping and Learning ( long version) 17
+•the nodes of Gare the atoms occurring in Π, and
+•Gcontains the edges from atobi(1≤i≤l) for each rule
+a←b1,...,bl,not bl+1,...,not bm
+in Π where ais an atom.
+A program is tight if its positive dependency graph is acyclic. For instance, pro-
+gram (8) is not tight since its positive dependency graph has a cycle d ue to the rule
+d←d. On the other hand, the program constructed from (8) by remov ing this rule
+is tight.
+Recall that for any program Π and any assignment M, ifM+is an answer set of Π
+thenMis a supported model of Π. For the case of tight programs, the con verse
+holds also: M+is an answer set for Π if and only if Mis a supported model
+of Π (Fages 1994) or, in other words, is a model of the completion of Π.
+It turns out that for tight programs the graph smΠis “almost identical” to the
+graphdpF, whereFis the clausified completion of Π. To make this claim precise,
+we need the following terminology.
+We say that an edge M=⇒M′in the graph smΠissingular if
+•the only transition rule justifying this edge is Unfounded , and
+•some edge M=⇒M′′can be justified by a transition rule other than Un-
+founded orDecide .
+For instance, let Π be the program
+a←b
+b←c.
+The edge
+a∆b∆¬c∆=⇒(Unfounded ,U={a,b})
+a∆b∆¬c∆¬a
+in the graph smΠis singular, because the edge
+a∆b∆¬c∆=⇒(All Rules Cancelled )
+a∆b∆¬c∆¬b
+belongs to smΠalso.
+With respect to the actual smodels algorithm (Simons 2000), singular edges of
+the graph smΠare inessential: in view of priorities for choosing a path in smΠde-
+scribed in Section 4.2 smodels never follows a singular edge. Indeed, the transition
+ruleUnfounded has the lower priority than any other transition rule but Decide .
+Bysm−
+Πwe denote the graph obtained from smΠby removing all singular edges.
+Proposition 5
+For any tight program Π, the graph sm−
+Πis equal to each of the graphs atleast Π
+anddpCNF-Comp (Π).
+For instance, let Π be the program (13). This program is tight, its co mpletion is (14),
+and ıCNF−Comp (Π) is formula (15). Proposition 5 asserts that, sm−
+Πcoincides
+withdpCNF-Comp (Π)and with atleast Π.18 Y. Lierler
+From Proposition 5, it follows that applying the smodels algorithm to a tight
+program essentially amounts to applying dpll to its completion. A similar relation-
+ship, in terms of pseudocode representations of smodels anddpll , is established
+in (Giunchiglia and Maratea 2005).
+In the rest of this section we give a proof of Proposition 5.
+Lemma 6
+For any tight program Π and any non-empty unfounded set Uon a consistent set
+Mof literals w.r.t. Π there is an atom a∈Usuch that for every B∈Bodies (Π,a),
+B∩M/\e}atio\slash=∅.
+Proof
+By contradiction. Assume that, for every a∈Uthere exists B∈Bodies (Π,a) such
+thatB∩M=∅. By the definition of an unfounded set it follows that for every atom
+a∈Uthere isB∈Bodies (Π,a) such that U∩B+/\e}atio\slash=∅. Consequently the subgraph
+of the positive dependency graph of Π induced by Uhas no terminal nodes. Then,
+the program Π is not tight.
+ıProof of Proposition 5
+In view of Proposition 3, it is sufficient to prove that sm−
+Πequalsatleast Π; or,
+in other words, that every edge of smΠjustified by the rule Unfounded only is
+singular. Consider such an edge M=⇒M′. We need to show that some transi-
+tion rule other than Unfounded orDecide is applicable to M. By the definition of
+Unfounded ,Mis consistent and there exists a non-empty set Uunfounded on M
+w.r.t. Π. By Lemma 6, it follows that there is an atom a∈Usuch that for every
+B∈Bodies (Π,a),B∩M/\e}atio\slash=∅. Therefore, the transition rule All Rules Cancelled is
+applicable to M.
+5 Generate and Test
+In this section, we present a modification of the graph dpF(Section 2.1) that
+includes testing “partial” assignments of Ffound by dpll .
+LetFbe a CNF formula, and let Gbe a formula formed from atoms occur-
+ring inF. The terminal nodes of the graph gtF,Gdefined below are models of
+formulaF∧G.
+This modification of the graph dpFis of interest, for example, in connection
+with the fact that answer sets of a program Π can be characterize d as models of
+its completion extended by so called loop formulas of Π (Lin and Zhao 20 02). If
+ıCNF−Comp (Π), as above, is the completion converted to CNF, and LF(Π) is
+the conjunction of all loop formulas of Π, then for any assignment M,M+is an
+answer set of Π iff Mis a model of ı CNF−Comp (Π)∧LF(Π). Hence, the terminal
+nodes of the graph gtCNF-Comp (Π),LF(Π)will correspond to answer sets of Π.
+The nodes of the graph gtF,Gare the same as the nodes of the graph dpF. The
+edges of gtF,Gare described by the transition rules of dpFand the additionalAbstract Answer Set Solvers with Backjumping and Learning ( long version) 19
+transition rule:
+Test:
+M=⇒Mlif
+
+Mis consistent ,
+G|=M,
+l∈M
+It is easy to see that the graph dpFis a subgraph of gtF,G. The latter graph
+can be used for deciding whether a formula F∧Ghas a model by constructing a
+path from∅to a terminal node:
+Proposition 6
+For any CNF formula Fand a formula Gformed from atoms occurring in F,
+(a) graph gtF,Gis finite and acyclic,
+(b) any terminal state of gtF,Gother than FailState is a model of F∧G,
+(c)FailState is reachable from ∅ingtF,Gif and only if F∧Gis unsatisfiable.
+Note that to verify the applicability of the new transition rule Test we need a pro-
+cedure for testing whether Gentails a clause, but there is no need to explicitly write
+outG. This is important because LF(Π) can be very long (Lin and Zhao 2002).
+For instance, let Π be the nontight program
+d←d.
+Its completion is
+d↔d,
+and ıCNF−Comp (Π) is
+(d∨¬d).
+This program has one loop formula
+d→⊥.
+Proposition 6 asserts that a terminal state ¬dofgtCNF-Comp (Π),d→⊥is a model of
+ıCNF−Comp (Π)∧LF(Π). It follows that {¬d}+=∅is an answer set of Π. To com-
+pare with the graph dpCNF-Comp (Π): statedis a terminal state in dpCNF-Comp (Π)
+whereasdis not a terminal state in gtCNF-Comp (Π),d→⊥because the transition rule
+Test is applicable to this state.
+asp-sat with Backtracking (Giunchiglia et al. 2006) is a procedure that com-
+putes models of the completion of the given program using dpll , and tests them
+until an answer set is found. The application of this procedure to a p rogram Π
+can be viewed as constructing a path from ∅to a terminal node in the graph
+gtCNF-Comp (Π),LF(Π)by adopting a strategy that Test is applied to a state Monly
+whenMis an assignment.
+In the rest of this section we give a proof of Proposition 6.
+Lemma 7
+For any CNF formula F, a formula Gformed from atoms occurring in F, and a
+path from∅to a state l1...lningtF,G, any model XofF∧Gsatisfiesliif it
+satisfies all decision literals l∆
+jwithj≤i.20 Y. Lierler
+Proof
+By induction on the path from ∅tol1...ln. Similar to the proof of Lemma 1. We
+will show that the property in question is preserved by the transitio n ruleTest.
+Take a model XofF∧Gand consider an edge M=⇒M′whereMis a list
+l1...lksuch that Xsatisfiesliif it satisfies all decision literals l∆
+jwithj≤i.
+Assume that Xsatisfies all decision literals from M. By the inductive hypothesis,
+X|=M. We will show that the rule justifying the transition from MtoM′is
+different from Test. By contradiction. M′isMl. By the definition of Test,G|=M.
+SinceXis a model of F∧Git follows that X|=M. This contradicts the fact that
+X|=M.
+ıProof of Proposition 6
+Part (a) and part (c) Right-to-left are proved as in the proof of P roposition 1.
+(b) LetMbe any terminal state other than FailState . As in the proof of Proposi-
+tion 1(b) it follows that Mis a model of F. The transition rule Test is not applicable.
+HenceG/\e}atio\slash|=M. In other words Mis a model of G. We conclude that Mis a model
+ofF∧G
+(c) Left-to-right: Since FailState is reachable from ∅, there is a state Mwithout
+decision literals such that Mis reachable from ∅and the transition rule Fail is
+applicable in M. Then, Mis inconsistent. By Lemma 7, any model of F∧G
+satisfiesM. SinceMis inconsistent we conclude F∧Gis unsatisfiable.
+6 Review: Abstract DPLL with Learning
+Most modern SAT solvers implement such sophisticated techniques a s backjumping
+and learning:
+Backjumping: Chronological Backtracking (used in classical dpll ) can be
+seen as a prototype of Backjumping. Unlike Backtracking that und oes only the
+previously made decision, Backjumping is generally able to backtrack further
+in the search tree by undoing several decisions at once.
+Learning : Most modern SAT solvers implement so called conflict-driven
+backjumping and learning : whenever backjumping is performed they add
+(learn) a “backjump clause” to the clause database of a solver. Le arning back-
+jump clauses prevents a solver from reaching “similar“ inconsistent states.
+In this section we will extend the graph dpFto capture the ideas behind back-
+jumping and learning. The new graph will be closely related to the DPLL System
+with Learning graph introduced in (Nieuwenhuis et al. 2006, Section 2.4).
+We first note that the graph dpFis not adequate to capture such technique as
+learning since it is incapable to reflect a change in a state of computat ion related
+to newly learned clauses. We start by redefining a state so that it inc orporates
+information about changes performed on a clause database.
+For a CNF formula F, anaugmented state relative to Fis either a distinguished
+stateFailState or a pair M||Γ where Mis a record relative to the set of atoms
+occurring in F, and Γ is a (multi-)set of clauses over atoms of Fthat are entailed
+byF.Abstract Answer Set Solvers with Backjumping and Learning ( long version) 21
+Unit Propagate λ:
+M||Γ =⇒Ml||Γ if/braceleftbiggC∨l∈F∪Γ and
+C⊆M
+Backjump :
+Pl∆Q||Γ =⇒Pl′||Γ if/braceleftbiggPl∆Qis inconsistent and
+F|=l′∨P
+Learn :
+M||Γ =⇒M||C,Γ if/braceleftbiggevery atom in Coccurs in Fand
+F|=C
+Fig. 3. The additional transition rules of the graph dplF.
+We now define a graph dplFfor any CNF formula F. Its nodes are the augmented
+states relative to F. The transition rules Decide andFail ofdpFare extended to
+dplFas follows: M||Γ =⇒M′||Γ (M||Γ =⇒FailState ) is an edge in dplFjustified
+byDecide (Fail) if and only if M=⇒M′(M=⇒FailState ) is an edge in dpF
+justified by Decide (Fail). Figure 3 presents the other transition rules of dplF. We
+refer to the transition rules Unit Propagate λ,Backjump ,Decide , andFail of the
+graphdplFasBasic . We say that a node in the graph is semi-terminal if no rule
+other than Learn is applicable to it.
+We will omit the word “augmented” before “state” when this is clear f rom a
+context.
+The graph dplFcan be used for deciding the satisfiability of a formula Fsimply
+by constructing an arbitrary path from node ∅||∅to a semi-terminal node:
+Proposition 7
+For any CNF formula F,
+(a) every path in dplFcontains only finitely many edges justified by Basic tran-
+sition rules,
+(b) for any semi-terminal state M||Γ ofdplFreachable from∅||∅,Mis a model
+ofF,
+(c)FailState is reachable from ∅||∅indplFif and only if Fis unsatisfiable.
+On the one hand, Proposition 7 (a) asserts that if we construct a p ath from∅||∅so
+that Basic transition rules periodically appear in it then some semi-ter minal state
+will be eventually reached. On the other hand, Proposition 7 (b) and (c) assert
+that as soon as a semi-terminal state is reached the problem of dec iding whether
+formula Fis satisfiable is solved. The proof of this proposition is similar to the
+proof of Theorem 2.12 from (Nieuwenhuis et al. 2006).
+For instance, let Fbe the formula
+a∨b
+¬a∨c.22 Y. Lierler
+Here is a path in dplF:
+∅||∅ =⇒(Learn )
+∅||b∨c =⇒(Decide )
+¬b∆||b∨c =⇒(Unit Propagate λ)
+¬b∆c||b∨c =⇒(Unit Propagate λ)
+¬b∆ca||b∨c(17)
+Since the state¬b∆cais semi-terminal, Proposition 7 (b) asserts that {¬b,c,a}is
+a model of F.
+Recall that the transition rule Backtrack of the graph dpF– a prototype of Back-
+jump – is applicable in any inconsistent state with a decision literal in dpF. The
+transition rule Backjump , on the other hand, is applicable in any inconsistent state
+with a decision literal that is reachable from ∅||∅(the proof of this statement is
+similar to the proof of Lemma 2.8 from (Nieuwenhuis et al. 2006)). The application
+ofBackjump wherel∆is the last decision literal and l′islcan be seen as an appli-
+cation of Backtrack . This fact shows that Backjump is essentially a generalization of
+Backtrack . The subgraph of dpFinduced by the nodes reachable from ∅is basically
+a subgraph of dplF.
+7 Answer Set Solver with Learning
+In this section we will extend the graph smΠto capture backjumping and learning.
+As a result we will be able to model the algorithms of systems smodels ccandsup.
+7.1 Graph smlΠ
+An(augmented) state relative to a program Π is either a distinguished state Fail-
+State or a pair of the form M||Γ where Mis a record relative to the set of atoms
+occurring in Π, and Γ is a (multi-)set of constraints formed from ato ms occurring
+in Π that are entailed by Π.
+For any program Π, we will define a graph smlΠ. Its nodes are the augmented
+states relative to Π. The transition rules Unit Propagate LP ,All Rules Cancelled ,
+Backchain True ,Unfounded ,Decide andFail ofsmΠare extended to smlΠas
+follows:M||Γ =⇒M′||Γ (M||Γ =⇒FailState ) is an edge in smlΠjustified by a
+transition rule Tif and only if M=⇒M′(M=⇒FailState ) is an edge in smΠ
+justified by T. Figure 4 presents the other transition rules of smlΠ.
+We refer to the transition rules UnitPropagate LP ,All Rules Cancelled ,Backchain
+True,Backchain False λ,Unfounded ,Backjump LP ,Decide , andFail of the graph
+smlΠasBasic . We say that a node in the graph is semi-terminal if no rule other
+thanLearn LP is applicable to it.
+The graph smlΠcan be used for deciding whether a program Π has an answer
+set by constructing a path from ∅||∅to a semi-terminal node:
+Proposition 8
+For any program Π,Abstract Answer Set Solvers with Backjumping and Learning ( long version) 23
+Backchain False λ:
+M||Γ =⇒Ml||Γ if
+
+a←l,B∈Π∪Γ,
+¬a∈Mora=⊥,
+B⊆M
+Backjump LP :
+Pl∆Q||Γ =⇒Pl′||Γ if/braceleftbiggPl∆Qis inconsistent and
+Π entails l′∨P
+Learn LP :
+M||Γ =⇒M||←B,Γ if Π entails B
+Fig. 4. The additional transition rules of the graph smlΠ.
+(a) every path in smlΠcontains only finitely many edges labeled by Basic tran-
+sition rules,
+(b) for any semi-terminal state M||Γ ofsmlΠreachable from∅||∅,M+is an
+answer set of Π,
+(c)FailState is reachable from ∅||∅insmlΠif and only if Π has no answer sets.
+Thus if we construct a path from ∅||∅so that Basic transition rules periodically
+appear in it then some semi-terminal state will be eventually reached ; as soon as a
+semi-terminal state is reached the problem of finding an answer set is solved.
+For instance, let Π be program (8). Here is a path in smlΠwith every edge
+annotated by the name of a transition rule that justifies the prese nce of this edge
+in the graph :
+∅||∅ =⇒(Decide )
+a∆||∅ =⇒(Unit Propagate LP )
+a∆c||∅ =⇒(All Rules Cancelled )
+a∆c¬b||∅ =⇒(Decide )
+a∆c¬bd∆||∅ =⇒(Unfounded )
+a∆c¬bd∆¬d||∅ =⇒(Backjump LP )
+a∆c¬b¬d||∅ =⇒(Learn LP )
+a∆c¬b¬d||¬a∨¬c∨b∨¬d(18)
+Since the state a∆c¬b¬dis semi-terminal, Proposition 8 (b) asserts that
+{a,c,¬b,¬d}+={a,c}
+is an answer set for Π.
+Proof of Proposition 8 is in Section 10.
+As in case of the graphs dpFanddplF,Backjump LP is applicable in any
+inconsistent state with a decision literal that is reachable from ∅||∅(Proposition 11
+from Section 9), and is essentially a generalization of the transition r uleBacktrack
+of the graph smΠ.
+Modern SAT solvers often implement such sophisticated techniques as restart
+and forgetting in addition to backjumping and learning:24 Y. Lierler
+Restart : A solver restarts the dpll procedure whenever the search is not
+making “enough” progress. The idea is that upon a restart a solver will explore
+a new part of the search space using the clauses that have been lea rned.
+Forgetting : This technique is usually implemented in relation with conflict-
+driven backjumping and learning. When a solver “notes” that earlier learned
+clauses are not helpful anymore it removes (forgets) them from t he clause
+database. Forgetting allows a solver to avoid a possible exponential space
+blow-up introduced by learning.
+We may extend the graph smlΠwith the following transition rules that capture
+the ideas behind these technique:
+Restart :
+M||Γ =⇒∅|| Γ
+Forget LP :
+M||←B,Γ =⇒M||Γ.
+The transition rules Restart andForget LP are similar to the analogous rules
+in (Nieuwenhuis et al. 2006) for extending dpll procedure with restart and for-
+getting techniques. It is easy to prove a result similar to Proposition 8 for the
+graphsmlΠwithRestart andForget LP (for such graph a state is semi-terminal if
+no rule other than Learn LP ,Restart ,Forget LP is applicable to it.)
+7.2 Smodels ccand Sup Algorithms
+In Section 4.2 we demonstrated a method for specifying the algorith m of an answer
+set solver by means of the graph smΠ. In particular, we described the smodels al-
+gorithm by assigning priorities to transition rules of smΠ. In this section we use this
+method to describe the smodels cc(Ward and Schlipf 2004) and sup(Lierler 2008)
+algorithms by means of smlΠ.
+Systemsmodels ccenhances the smodels algorithm with conflict-driven back-
+jumping and learning. Its strategy for choosing a path in the graph smlΠis similar
+to that of smodels . System smodels ccassigns priorities to inference rules of smlΠ
+as follows:
+Backjump LP ,Fail≫
+Unit Propagate LP ,All Rules Cancelled ,Backchain True ,Backchain False λ≫
+Unfounded≫
+Decide .
+Also,smodels ccalways applies the transition rule Learn LP in a non-semi-
+terminal state reached by an application of Backjump LP , because it implements
+conflict-driven backjumping and learning.12In Section 11 we discuss details on
+which clause is being learned during the application of Learn LP .
+12Systemsmodels cc(sup) also implements restarts and forgetting that may be modele d by the
+transition rules Restart andForget LP . An application of these transition rules in smlΠrelies
+on particular heuristics implemented by the solver.Abstract Answer Set Solvers with Backjumping and Learning ( long version) 25
+In (Lierler 2008), we introduced the simplified supalgorithm that relies on back-
+tracking rather than conflict-driven backjumping and learning tha t are actually
+implemented in the system. We now present the supalgorithm that takes these
+sophisticated techniques into account.
+Systemsupassigns priorities to inference rules of smlΠas follows:
+Backjump LP ,Fail≫
+Unit Propagate LP ,All Rules Cancelled ,Backchain True ,Backchain False λ≫
+Decide≫
+Unfounded .
+Similarly to smodels cc,supalways applies the transition rule Learn LP in a
+non-semi-terminal state reached by an application of Backjump LP .
+For example, let Π be program (8). Path (18) corresponds to an ex ecution of
+systemsup, but does not correspond to any execution of smodels ccbecause for
+the latter Unfounded is a rule of higher priority than Decide . Here is another path
+insmlΠfrom∅||∅to the same semi-terminal node:
+∅||∅ =⇒(Decide )
+a∆||∅ =⇒(Unit Propagate LP )
+a∆c||∅ =⇒(All Rules Cancelled )
+a∆c¬b||∅ =⇒(Unfounded )
+a∆c¬b¬d||∅(19)
+Path (19) corresponds to an execution of system smodels cc, but does not corre-
+spond to any execution of system supbecause for the latter Decide is a rule of
+higher priority than Unfounded .
+The strategy of supof assigning the transition rule Unfounded the lowest priority
+may be reasonable for many problems. For instance, it is easy to see that transi-
+tion rule Unfounded is redundant for tight programs. The supalgorithm is similar
+to SAT-based answer set solvers such as assat (Lin and Zhao 2004) and cmod-
+els(Giunchiglia et al. 2006) (see Section 8.2) in the fact that it will first co mpute
+a supported model of a program and only then will test whether this model is
+indeed an answer set, i.e., whether Unfounded is applicable in this state.
+8 Generate and Test with Learning
+In this section we model backjumping and learning for the generate and test pro-
+cedure by defining a graph gtlF,Gthat extends gtF,G(Section 5) in a similar
+manner as dplF(Section 6) extends dpF.
+8.1 Graph gtlF,G
+An(augmented) state relative to a CNF formula Fand a formula Gformed from
+atoms occurring in Fis either a distinguished state FailState or a pair of the
+formM||Γ, where Mis a record (Section 2.1) relative to the set of atoms occurring
+inF, and Γ is a (multi-)set of clauses formed from atoms occurring in Fthat are
+entailed by F∧G.26 Y. Lierler
+The nodes of the graph gtlF,Gare the augmented states relative to a CNF
+formulaFand a formula Gformed from atoms occurring in F. The edges of gtlF,G
+are described by the transition rules Unit Propagate λ,Decide ,Fail ofdplF, the
+transition rules
+Backjump GT :
+Pl∆Q||Γ =⇒Pl′||Γ if/braceleftbiggPl∆Qis inconsistent and
+F∧G|=l′∨P
+Learn GT :
+M||Γ =⇒M||C,Γ if/braceleftbiggevery atom in Coccurs in Fand
+F∧G|=C
+and the transition rule Test ofgtF,Gthat is extended to gtlF,Gas follows:
+M||Γ =⇒M′||Γ is an edge in gtlF,Gjustified by Test if and only if M=⇒M′is
+an edge in gtF,Gjustified by Test.
+We refer to the transition rules Unit Propagate λ,Test,Decide ,Fail,Back-
+jump GT of the graph gtlF,GasBasic . We say that a node in the graph is
+semi-terminal if no rule other than Learn GT is applicable to it.
+The graph gtlF,Gcan be used for deciding whether a formula F∧Ghas a model
+by constructing a path from ∅||∅to a terminal node:
+Proposition 9
+For any CNF formula Fand a formula Gformed from atoms occurring in F,
+(a) every path in gtlF,Gcontains only finitely many edges labeled by Basic
+transition rules,
+(b) for any semi-terminal state M||Γ ofgtlF,Greachable from∅||∅,Mis a model
+ofF∧G,
+(c)FailState is reachable from ∅||∅ingtlF,Gif and only if F∧Gis unsatisfiable.
+As in case of the graph dplF, the transition rule Backjump GT is applicable in
+any inconsistent state with a decision literal that is reachable from ∅||∅. We call
+such states backjump states.
+Proposition 10
+For any CNF formula Fand a formula Gformed from atoms occurring in F, the
+transition rule Backjump GT is applicable in any backjump state in gtlF,G.
+Proofs of Propositions 9 and 10 are given in Section 13.
+8.2 Cmodels Algorithm
+Systemcmodels implements an algorithm called asp-sat with Learning (Giunchiglia et al. 2006)
+that extends asp-sat with Backtracking by backjumping and learning.
+The application of cmodels to a program Π can be viewed as constructing a
+path from∅||∅to a terminal node in the graph gtlF,G, where
+•Fis the completion of Π converted to conjunctive normal form, and
+•GisLF(Π).Abstract Answer Set Solvers with Backjumping and Learning ( long version) 27
+In Sections 4.2 we demonstrated a method for specifying the algorit hm of an
+answer set solver by means of the graph smΠ. We use this method to describe the
+cmodels algorithm using the graph gtlF,G. System cmodels assigns priorities
+to the inference rules of gtlF,Gas follows:
+Backjump GT ,Fail≫
+Unit Propagate λ≫
+Decide≫
+Test.
+Also,cmodels always applies the transition rule Learn GT in a non-semi-
+terminal state reached by an application of Backjump GT .
+The priorities imposed on the rules by cmodels guarantee that the transition
+ruleTest is applied to a model of F∪Γ (clausified completion Fextended by learned
+clauses Γ). This allows cmodels to proceed with its search in case if a found model
+is not an answer set. Furthermore, the cmodels strategy guarantees that in a state
+reached by an application of Test, firstBackjump GT will be applied and then in the
+resulting state Learn GT will be applied. The clause learned due to this application
+ofLearn GT is derived by means of loop formulas (see (Giunchiglia et al. 2006)).
+In this sense cmodels uses loop formulas to guide its search.
+Systems sag (Lin et al. 2006) and clasp (Gebser et al. 2007) are answer set
+solvers that are enhancements of cmodels . First, they compute and clausify pro-
+gram’s completion and then use unit propagate on resulting proposit ional formula
+as an inference mechanism. Second, they guide their search by mea ns of loop for-
+mulas. Third, they implement conflict-driven backjumping and learnin g. Also,sag
+uses SAT solvers for search. The systems differ from cmodels in the following:
+•they maintain the data structure representing an input logic progr am through
+out the whole computation,
+•in addition to implementing inference rules of the graph gtlF,Gthey also
+implement the inference rule Unfounded ofsmΠ. A hybrid graph combining
+the inference rule Unfounded ofsmΠand the inference rules of gtlF,Gmay
+be used to describe the sagandclasp algorithms.
+Systemsagassigns the same priorities to the inference rules of the hybrid grap h
+ascmodels . Also,sag at random decides whether to apply the inference rule
+Unfounded in a state.
+On the other hand, system clasp assigns priorities to the inference rules of the
+hybrid graph as follows:
+Backjump GT ,Fail≫
+Unit Propagate λ,Unfounded≫
+Decide .
+Likecmodels , bothsagandclasp always apply the transition rule Learn GT
+in a non-semi-terminal state reached by an application of Backjump GT .28 Y. Lierler
+9 Backjumping and Extended Graph
+Recall the transition rule Backjump LP ofsmlΠ
+Backjump LP :
+Pl∆Q||Γ =⇒Pl′||Γ if/braceleftbiggPl∆Qis inconsistent and
+Π entails l′∨P.
+A state in the graph smlΠis abackjump state if it is inconsistent, contains a
+decision literal, and is reachable from ∅||∅. Note that it may be not clear a priori
+whether Backjump LP is applicable to a backjump state and if so to which state the
+edge due to the application of Backjump LP leads. These questions are important
+if we want to base an algorithm on this framework. It turns out that Backjump LP
+is always applicable to a backjump state:
+Proposition 11
+For a program Π, the transition rule Backjump LP is applicable to any backjump
+state insmlΠ.
+Proposition 11 guarantees that a backjump state in smlΠis never semi-terminal. In
+the end of this section we show how Proposition 11 can be derived fro m the results
+proved later in this paper. Next question to answer is how to continu e choosing
+a path in the graph after reaching a backjump state. To answer th is question we
+introduce the notions of reason and extended graph.
+For a program Π, we say that a clause l∨Cis areason forlto be in a list of
+literalsPlQ w.r.t Π if Π entails l∨CandC⊆P. We can equivalently restate
+the second condition of Backjump LP “Π entails l′∨P” as “there exists a reason
+forl′to be in Pl′w.r.t. Π” (note that l′∨Pis a reason for l′to be in Pl′). We
+call a reason for l′to be inPl′abackjump clause . Note that Proposition 11 asserts
+that a backjump clause always exists for a backjump state. It is cle ar that we may
+continue choosing a path in the graph after reaching a backjump st ate if we know
+how to compute a backjump clause for this state. We now define a gr aphsml↑
+Π
+that shares many properties of smlΠbut allows us to give a simpler procedure for
+computing a backjump clause.
+Anextended record Mrelative to a program Π is a list of literals over the set of
+atoms occurring in Π where
+(i) each literal linMis annotated either by ∆ or by a reason for lto be in M
+w.r.t. Π,
+(ii)Mcontains no repetitions,
+(iii) for any inconsistent prefix of Mits last literal is annotated by a reason.
+For instance, let Π be the program
+a←not b
+c.
+The list of literals
+b∆a∆¬b¬b∨¬aAbstract Answer Set Solvers with Backjumping and Learning ( long version) 29
+is an extended record relative to Π. On the other hand, the lists of lit erals
+a∆¬a∆a∆¬b¬b∨¬ab∆b∆a∆¬b¬b∨¬ac∆
+are not extended records.
+Anextended state relative to a program Π is either a distinguished state FailState
+or a pair of the form M||Γ where Mis an extended record relative to Π, and Γ
+is the same as in the definition of an augmented state (i.e., Γ is a (multi-) set of
+constraints formed from atoms occurring in Π that are entailed by Π .) It is easy to
+see that for any extended state Srelative to a program Π, the result of removing
+annotations from all nondecision literals of Sis a state of smlΠ: we will denote this
+state by S↓.
+For instance, consider program a←not b . All pairs
+FailState∅||∅a∆¬b¬b∨¬a||∅ ¬a∆bb∨a||∅
+are among valid extended states relative to this program. The corr esponding states S↓
+are
+FailState∅||∅a∆¬b||∅ ¬a∆b||∅.
+We now define a graph sml↑
+Πfor any program Π. Its nodes are the extended
+states relative to Π. The transition rules of smlΠare extended to sml↑
+Πas follows:
+S1=⇒S2is an edge in sml↑
+Πjustified by a transition rule Tif and only if S↓
+1=⇒S↓
+2
+is an edge in smlΠjustified by T.
+We will omit the word “extended” before “record” and “state” whe n this is clear
+from a context.
+The following lemma formally states the relationship between nodes of the graphs
+smlΠandsml↑
+Π:
+Lemma 8
+For any program Π, if S′is a state reachable from ∅||∅in the graph smlΠthen
+there is a state Sin the graph sml↑
+Πsuch that S↓=S′.
+The definitions of Basic transition rules and semi-terminal states in sml↑
+Πare
+similar to their definitions for smlΠ.
+Proposition 8↑
+For any program Π,
+(a) every path in sml↑
+Πcontains only finitely many edges labeled by Basic tran-
+sition rules,
+(b) for any semi-terminal state M||Γ ofsml↑
+Π,M+is an answer set of Π,
+(c)sml↑
+Πcontains an edge leading to FailState if and only if Π has no answer
+sets.
+Note that Proposition 8↑(b), unlike Proposition 8 (b), is not limited to semi-
+terminal states that are reachable from ∅||∅. As in the case of the graph smlΠ,
+sml↑
+Πcan be used for deciding whether a program Π has an answer set. Fu rther-
+more, the new graph provides the means for computing a backjump clause that
+permits practical application of the transition rule Backjump LP : Sections 10.330 Y. Lierler
+and 11 describe the BackjumpClause (Algorithm 1) and BackjumpClauseFirstUIP
+(Algorithm 2) procedures that compute Decision andFirstUIP backjump clauses
+respectively.
+We say that a state in the graph sml↑
+Πis abackjump state if its record is incon-
+sistent and contains a decision literal. Unlike the definition of a backju mp state in
+smlΠ, this definition does not require a backjump state to be reachable f rom∅||∅
+insml↑
+Π. As in case of the graph smlΠ, any backjump state in sml↑
+Πis not semi-
+terminal:
+Proposition 11↑
+For a program Π, the transition rule Backjump LP is applicable to any backjump
+state insml↑
+Π.
+Proposition 8 (b), (c) and Proposition 11 easily follow from Lemma 8 an d Propo-
+sition 8↑(b), (c) and Proposition 11↑respectively. Proof of Proposition 8 (a) is
+similar to the proof of Proposition 8↑(a).
+Next section will present the proofs for Proposition 8↑, Lemma 8, and Proposi-
+tion 11↑. It is interesting to note that the proofs of Lemma 8 and Propositio n 11↑
+implicitly provide the means for choosing a path in the graph sml↑
+Π:
+•given a state M||Γ and a transition rule Unit Propagate LP ,All Rules Can-
+celled ,Backchain True ,Backchain False λ, orUnfounded applicable to M||Γ,
+the proof of Lemma 8 describes a clause that may be used to constr uct a
+recordM′so that there is an edge M||Γ =⇒M′||Γ due to this transition
+rule,
+•given a backjump state M||Γ, the proof of Proposition 11↑describes a back-
+jump clause that can be used to construct a record M′so that there is an
+edgeM||Γ =⇒M′||Γ due to Backjump LP .
+Furthermore, the construction of the proof of Proposition 11↑paves the way for
+procedure BackjumpClause presented in Algorithm 1.
+10 Proofs of Proposition 8↑, Lemma 8, Proposition 11↑
+10.1 Proof of Proposition 8↑
+Lemma 9
+For any program Π, an extended record Mrelative to Π, and every assignment X
+such that X+is an answer set for Π, if Xsatisfies all decision literals in Mthen
+X|=M.
+Proof
+By induction on the length of M. The property trivially holds for ∅. We assume that
+the property holds for any state with nelements. Consider any state Mwithn+ 1
+elements. Let Xbe an assignment such that X+is an answer set for Π and X
+satisfies all decision literals in M. We will now show that X|=M.
+Case 1.Mhas the form Pl∆. By the inductive hypothesis, X|=P. SinceX
+satisfies all decision literals in M,X|=l.Abstract Answer Set Solvers with Backjumping and Learning ( long version) 31
+Case 2.Mhas the form Pll∨C. By the inductive hypothesis, X|=P. By the
+definition of a reason, (i) Π entails l∨Cand (ii)C⊆P. From (ii) it follows that
+P|=¬C. Consequently, X|=¬C. From (i) it follows that for any assignment X
+such that X+is an answer set, X|=l∨C. Consequently, X|=l.
+The proof of Proposition 8↑assumes the correctness of Proposition 11↑that we
+demonstrate in Section 10.3.
+Proposition 8↑
+For any program Π,
+(a) every path in sml↑
+Πcontains only finitely many edges labeled by Basic tran-
+sition rules,
+(b) for any semi-terminal state M||Γ ofsml↑
+Π,M+is an answer set of Π,
+(c)sml↑
+Πcontains an edge leading to FailState if and only if Π has no answer
+sets.
+Proof
+(a) For any list Nof literals by|N|we denote the length of N. Any state M||Γ
+has the form M0l∆
+1M1...l∆
+pMp||Γ, where l∆
+1...l∆
+pare all decision literals of M;
+we define α(M||Γ) as the sequence of nonnegative integers |M0|,|M1|,...,|Mp|, and
+α(FailState ) =∞. For any states SandS′ofsml↑
+Π, we understand α(S)< α(S′)
+as the lexicographical order. We first note that for any state M||Γ, value of αis
+based only on the first component Mof the state. Second, there is a finite number
+of distinct values of αdue to the fact that there is a finite number of distinct Ms
+over Π. We conclude that there is a finite number of distinct values of αfor the
+states of sml↑
+Π, even though the number of distinct states in sml↑
+Πis infinite.
+By the definition of the transition rules of sml↑
+Π, if there is an edge from M||Γ
+toM′||Γ′insml↑
+Πformed by any Basic transition rule then α(M||Γ)< α(M′||Γ′).
+Then, due to the fact that there is a finite number of distinct values ofα, it follows
+that there is only a finite number of edges due to the application of Ba sic rules
+possible in any path.
+(b) LetM||Γ be a semi-terminal state so that none of the Basic rules are applica ble.
+From the fact that Decide is not applicable, we conclude that Massigns all literals.
+Furthermore, Mis consistent. Indeed, assume that Mis inconsistent. Then, since
+Failis not applicable, Mcontains a decision literal. Consequently, M||Γ is a back-
+jump state. By Proposition 11↑, the transition rule Backjump LP is applicable
+inM||Γ. This contradicts our assumption that M||Γ is semi-terminal.
+Also,Mis a model of Π: since Unit Propagate LP is not applicable in M||Γ, it
+follows that for every rule a←B∈Π, ifB⊆Mthena∈M.
+Assume that M+is not an answer set. Then, by Lemma 3, there is a non-empty
+unfounded set UonMw.r.t. Π such that U⊆M. It follows that Unfounded is
+applicable (with an arbitrary a∈U) inM||Γ. This contradicts the assumption
+thatM||Γ is semi-terminal.
+(c) Left-to-right: There is a state M||Γ insml↑
+Πsuch that there is an edge be-
+tweenM||Γ andFailState . By the definition of sml↑
+Π, this edge is due to the transi-
+tion rule Fail. Consequently, state M||Γ is such that Mis inconsistent and contains32 Y. Lierler
+no decision literals. By Lemma 9, for every assignment Xsuch that X+is an an-
+swer set for Π, XsatisfiesM. SinceMis inconsistent we conclude that Π has no
+answer sets.
+Right-to-left: Consider the process of constructing a path cons isting only of edges
+due to Basic transition rules. By (a), it follows that this path will even tually reach
+a semi-terminal state. By (b), this semi-terminal state cannot be different from
+FailState , because Π has no answer sets. We conclude that there is an edge le ading
+toFailState .
+10.2 Proof of Lemma 8
+The proof uses the notion of loop formula (Lin and Zhao 2004).
+Given a set Aof atoms by Bodies (Π,A) we denote the set that consists of the
+elements of Bodies (Π,a) for allainA. Let Π be a program. For any set Yof atoms,
+theexternal support formula (Lee 2005) for Yis
+/logicalordisplay
+B∈Bodies(Π,Y),B+∩Y=∅B. (20)
+We will denote the external support formula by ESΠ,Y. For any set Yof atoms,
+theloop formula forYis the implication
+/logicalordisplay
+a∈Ya→ESΠ,Y.
+We can rewrite this formula as the disjunction
+/logicalanddisplay
+a∈Y¬a∨ESΠ,Y. (21)
+From the Main Theorem in (Lee 2005) we conclude:
+Lemma on Loop Formulas
+For any program Π, Π entails loop formulas (21) for all sets Yof atoms that occur
+in Π.
+For a state Sin the graph sml↑
+Π, we say that S↓insmlΠis theimage ofS.
+Lemma 8
+For any program Π, if S′is a state reachable from ∅||∅in the graph smlΠthen
+there is a state Sin the graph sml↑
+Πsuch that S↓=S′.
+Proof
+Since the property trivially holds for the initial state ∅||∅, we only need to prove
+that all transition rules of smlΠpreserve it.
+Consider an edge M||Γ =⇒M′||Γ′in the graph smlΠsuch that there is a state
+M1||Γ in the graph sml↑
+Πsatisfying the condition ( M1||Γ)↓=M||Γ. We need to
+show that there is a state in the graph sml↑
+Πsuch that M′||Γ′is its image in smlΠ.
+Consider several cases that correspond to a transition rule leadin g fromM||Γ to
+M′||Γ′:Abstract Answer Set Solvers with Backjumping and Learning ( long version) 33
+Unit Propagate LP :
+M||Γ =⇒M a||Γ if/braceleftbigga←B∈Π and
+B⊆M.
+M′||Γ′isMa||Γ. It is sufficient to prove that M1aa∨B||Γ is a state of sml↑
+Π. It is
+enough to show that a clause a∨Bis a reason for ato be inMa. By applicability
+conditions of Unit Propagate LP ,B⊆M. Since Π entails its rule a←B, Π entails
+a∨B.
+All Rules Cancelled :
+M||Γ =⇒M¬a||Γ ifB∩M/\e}atio\slash=∅for allB∈Bodies (Π,a).
+M′||Γ′isM¬a||Γ. Consider any B∈Bodies (Π,a). SinceB∩M/\e}atio\slash=∅,Bcontains a
+literal from M: call itf(B). It is sufficient to show that
+¬a∨/logicalordisplay
+B∈Bodies(Π,a)f(B) (22)
+is a reason for¬ato be inM¬a.
+First, by the choice of f(B),f(B)∈M; consequently,
+/logicalordisplay
+B∈Bodies(Π,a)f(B)⊆M.
+Second, since f(B)∈B, the loop formula ¬a∨ESΠ,{a}entails (22). By Lemma
+on Loop Formulas , it follows that Π entails (22).
+Backchain True :
+M||Γ =⇒Ml||Γ if
+
+a←B∈Π,
+a∈M,
+B′∩M/\e}atio\slash=∅for allB′∈Bodies (Π,a)\{B},
+l∈B.
+M′||Γ′isMl||Γ. Consider any B′∈Bodies (Π,a)\B. SinceB′∩M/\e}atio\slash=∅,B′contains
+a literal from M: call itf(B′). A clause
+l∨¬a∨/logicalordisplay
+B′∈Bodies(Π,a)\Bf(B′)· (23)
+is a reason for lto be in Ml. The proof of this statement is similar to the case of
+All Rules Cancelled .
+Backchain False λ:
+M||Γ =⇒Ml||Γ if
+
+a←l,B∈Π∪Γ,
+¬a∈Mora=⊥,
+B⊆M.
+M′||Γ′isMl||Γ. A clause l∨B∨ais a reason for lto be inMl. The proof of this
+statement is similar to the case of Unit Propagate LP .
+Unfounded :
+M||Γ =⇒M¬a||Γ if/braceleftbiggMis consistent and
+a∈Ufor a set Uunfounded on Mw.r.t. Π.34 Y. Lierler
+M′||Γ′isM¬a||Γ. Consider any B∈Bodies (Π,U) such that U∩B+=∅. By the
+definition of an unfounded set, it follows that B∩M/\e}atio\slash=∅. Consequently, Bcontains
+a literal from M: call itf(B). The clause
+¬a∨/logicalordisplay
+Bodies(Π,U),B+∩U=∅f(B) (24)
+is a reason for¬ato be inM¬a. The proof of this statement is similar to the case
+ofAll Rules Cancelled .
+Backjump LP ,Decide ,Fail, andLearn LP : obvious.
+The process of turning a state of smlΠreachable from∅||∅into a corresponding
+state ofsml↑
+Πcan be illustrated by the following example: Consider a program Π
+a←not b
+b←not a,not c
+c←not f
+←k,d
+k←l,not b
+←m,not l,not b
+m←not k,not l(25)
+and a path in smlΠ
+∅||∅=⇒(Decide )
+a∆||∅=⇒(All Rules Cancelled )
+a∆¬b||∅=⇒(Decide )
+a∆¬bc∆||∅=⇒(Backchain True )
+a∆¬bc∆¬f||∅=⇒(Decide )
+a∆¬bc∆¬f d∆||∅=⇒(Backchain False λ)
+a∆¬bc∆¬f d∆¬k||∅=⇒(Backchain False λ)
+a∆¬bc∆¬f d∆¬k¬l||∅=⇒(Backchain False λ)
+a∆¬bc∆¬f d∆¬k¬l¬m||∅=⇒(Unit Propagate LP )
+a∆¬bc∆¬f d∆¬k¬l¬mm||∅(26)
+The construction in the proof of Lemma 8 applied to the nodes in this p ath gives
+following states of sml↑
+Π:
+∅||∅
+a∆||∅
+a∆¬b¬b∨¬a||∅
+a∆¬b¬b∨¬ac∆||∅
+a∆¬b¬b∨¬ac∆¬f¬f∨¬c||∅
+a∆¬b¬b∨¬ac∆¬f¬f∨¬cd∆||∅
+a∆¬b¬b∨¬ac∆¬f¬f∨¬cd∆¬k¬k∨¬d||∅
+a∆¬b¬b∨¬ac∆¬f¬f∨¬cd∆¬k¬k∨¬d¬l¬l∨b∨k||∅
+a∆¬b¬b∨¬ac∆¬f¬f∨¬cd∆¬k¬k∨¬d¬l¬l∨b∨k¬m¬m∨l∨b||∅
+a∆¬b¬b∨¬ac∆¬f¬f∨¬cd∆¬k¬k∨¬d¬l¬l∨b∨k¬m¬m∨l∨bmm∨k∨l||∅(27)
+It is clear that these nodes form a path in sml↑
+Πwith every edge justified by the
+same transition rule as the corresponding edge in path (26) in smlΠ.Abstract Answer Set Solvers with Backjumping and Learning ( long version) 35
+10.3 Proof of Proposition 11↑
+In this section Π is an arbitrary and fixed logic program.
+For a record M, bylcp(M) we denote its largest consistent prefix. We say that a
+clauseCisconflicting on a list Mof literals if Π entails C, andC⊆lcp(M). For
+example, let Mbe the first component of the last state in (27):
+a∆¬b¬b∨¬ac∆¬f¬f∨¬cd∆¬k¬k∨¬d¬l¬l∨b∨k¬m¬m∨l∨bmm∨k∨l. (28)
+Then,lcp(M) is obtained by dropping the last element mm∨k∨lofM. It is clear
+that the reason m∨k∨lformto be inMis a conflicting clause on M.
+Lemma 10
+The literal that immediately follows lcp(M) in an inconsistent record M, has the
+formlCwhereCis a conflicting clause on M.
+Proof
+By the requirement (iii) of the definition of an extended record, the literal that
+immediately follows lcp(M) may not be annotated by ∆. Consequently, the literal
+has the form lC. We now show that Cis a conflicting clause on M. SinceCis a
+reason for lto be in lcp(M)lC, it immediately follows that Π entails C,Ccan be
+written as l∨C′, andC′⊆lcp(M). Sincelimmediately follows the largest consistent
+prefix of M,l∈lcp(M). Consequently, C⊆lcp(M). We conclude that Cis indeed
+a conflicting clause on M.
+For any inconsistent record l1···lnand any conflicting clause Con this record,
+byβl1···ln(C) we denote the set of numbers isuch that li∈C. (It is clear that
+every element from Cequals to one of the literals in l1···ln.) The relation I<J
+between subsets I,Jof{1···n}is understood here as the lexicographical order
+between IandJsorted in descending order. For instance, {2 6 7}<{6 7 8}
+because{7 6 2}<{8 7 6}in lexicographical order.
+Recall that the resolution rule can be applied to clauses C∨landC′∨¬land
+produces the clause C∨C′, called the resolvent ofC∨landC′∨¬lonl.
+Lemma 11
+LetMbe a record and let lBbe a nondecision literal from lcp(M). If clause Dis
+the resolvent of Band a clause Cconflicting on Mthen
+(i)Dis a clause conflicting on M,
+(ii)βM(D)< βM(C).
+For instance, let Mbe (28), let reason ¬m∨l∨bfor¬minlcp(M) beB, and let
+conflicting clause m∨k∨lonMbeC. ThenD, the result of resolving Btogether
+withC, is clause k∨l∨b. Lemma 11 asserts that k∨l∨bis a conflicting clause
+onMand that βM(D)< βM(C). Indeed, βM(D) ={2 6 7}andβM(C) ={6 7 8}.36 Y. Lierler
+Proof
+(i) Clause Dis a resolvent of BandCon some literal l′. Then, for some literal
+l′∈B,l′∈C. The clause Ccan be written as l′∨C′.
+In order to demonstrate that Dis a conflicting clause we need to show that
+D⊆lcp(M) and Π entails D.
+SinceBis a reason for lto be inlcp(M), Π entails BandBhas the form l∨B′
+whereB′⊆lcp(M). Since Cis a conflicting clause on M,C⊆lcp(M) and Π
+entailsC. From the fact that lcp(M) is consistent, it follows that there is no literal
+inB′such that its complement occurs in C. Consequently, l′/\e}atio\slash∈B′so thatl′isl
+andDisB′∨C′. We conclude that D⊆lcp(M). From the fact that Π entails B, Π
+entailsC, and the construction of D, it follows that Π entails D.
+(ii) From the proof of (i) it follows that Dis a resolvent of BandConl
+whereBhas the form l∨B′. SinceBis a reason for lto be in lcp(M), every
+literal in B′precedes linlcp(M). SinceDis derived by replacing linCwithB′,
+βM(D)< βM(B).
+Let record Mbel1···li···ln, thedecision level of a literal liis the number of
+decision literals in l1···li: we denote it by decM(li). We will also use this notation
+to denote the decision level of a set of literals: For a set P⊆Mof literals, decM(P)
+is the decision level of the literal in Pthat occurs latest in M. For record Mand a
+decision level jbyMjwe denote the prefix of Mthat consists of the literals in M
+that belong to decision level less than jand byMj]we denote the prefix of Mthat
+consists of the literals in Mthat belong to decision level less than or equal to j.
+For instance, let Mbe record (28) then decM(¬k) = 3,decM(¬b c¬k) = 3,M3is
+a∆¬b¬b∨¬ac∆¬f¬f∨¬c, andM3]isMitself.
+Lemma 12
+For an inconsistent record Mand a conflicting clause l∨ConM, ifdecM(l)>
+decM(c) for allc∈Cthenlcp(M)decM(C)]ll∨Cis a record.
+Proof
+We need to show that (i) l/\e}atio\slash∈lcp(M)dec(C)]and (ii)l∨Cis a reason for lto be
+inlcp(M)dec(C)]l, i.e, Π entails l∨CandC⊆lcp(M)dec(C)].
+Sincel∨Cis conflicting on M,l∨C⊆lcp(M). From the consistency of
+lcp(M) and the fact that l∈lcp(M), it follows that l/\e}atio\slash∈lcp(M). Consequently,
+l/\e}atio\slash∈lcp(M)dec(C)].
+Sincel∨Cis conflicting on M, Π entails l∨Candl∨C⊆lcp(M). Consequently,
+C⊆lcp(M). From the definition of decM(C), it follows that decM(C) is the decision
+level of the literal in Cthat occurs latest in lcp(M). By the definition of a decision
+level,C⊆lcp(M)decM(C)].
+Proposition 11↑
+For a program Π, the transition rule Backjump LP is applicable to any backjump
+state insml↑
+Π.Abstract Answer Set Solvers with Backjumping and Learning ( long version) 37
+Proof
+LetM||Γ be a backjump state in sml↑
+Π. LetRbe the list of reasons that are assigned
+to the nondecision literals in lcp(M).
+Consider the process of building a sequence C1,C2,...of clauses so that
+•C1is the reason of the member of Mthat immediately follows lcp(M), and
+•Cj(j>1) is a resolvent of Cj−1and some clause in R
+while derivation of new clauses is possible. From Lemma 11 (i) and the ch oice ofC1
+andR, it follows that any clause in C1,C2...is conflicting. By Lemma 11 (ii)
+we conclude that βM(Cj)< βM(Cj−1) (j>1). It is clear that this process will
+terminate after deriving some clause Cm, since the number of conflicting clauses
+onMis finite. It is clear that clause Cmcannot be resolved against any clause in R.
+Case 1.Cmis the empty clause. Since M||Γ is a backjump state, Mcontains a
+decision literal l∆. By part (iii) of the definition of a record, lbelongs to lcp(M).
+Consequently, Mcan be represented in the form lcp(M)decM(l)l∆Q.
+By the choice of C1,C1is a reason and must consist of at least one literal.
+Consequently, m>1. Clause Cmis derived from clauses Cm−1and some clause
+inR. SinceCmis empty, Cm−1is a unit clause l′. We will show that
+lcp(M)decM(l)l∆Q||Γ =⇒lcp(M)decM(l)l′l′||Γ
+is an application of Backjump LP . It is sufficient to demonstrate that lcp(M)decM(l)l′l′
+is a record. Since lcp(M)decM(l)l∆Qis a record, we only need to show that l′/\e}atio\slash∈
+lcp(M)decM(l)and clause l′is a reason for l′to be in lcp(M)decM(l)l′. Recall that
+Cm−1, i.e.,l′, is a conflicting clause. Consequently, Π entails l′andl′∈lcp(M).
+Sincelcp(M) is consistent, l′/\e}atio\slash∈lcp(M) so that l′/\e}atio\slash∈lcp(M)decM(l). On the other
+hand, from the fact that Π entails l′it immediately follows that clause l′is a reason
+forl′to be in lcp(M)decM(l)l′.
+Case 2.Cmis not empty. Since Cmis a conflicting clause on M, the complement
+of any literal in Cmbelongs to lcp(M). Furthermore, every such complement is a
+decision literal in lcp(M). Indeed, if this complement is ll∨B∈lcp(M) thenl∨B
+is one of the clauses Bi, and it can be resolved against Cm.
+By the definition of a decision level, there is at most one decision literal that
+belongs to any decision level. It follows that Cmcan be written as l∨C′
+mso
+thatdecM(l)>decM(c) for any c∈C′
+m. Consequently, Mcan be written as
+lcp(M)decM(l)l∆Q. Note that
+lcp(M)decM(l)l∆Q||Γ =⇒lcp(M)decM(C′m)]lCm||Γ
+is an application of Backjump LP . Indeed, by Lemma 12 lcp(M)decM(C′
+m)]lCmis a
+record.
+Algorithm 1 presents procedure BackjumpClause that computes a backjump
+clause for any backjump state in the graph sml↑
+Π. The algorithm follows from the
+construction of the proof of Proposition 11↑. It is based on the iterative application
+of the resolution rule on reasons of the smallest inconsistent prefix of a state. The38 Y. Lierler
+BackjumpClause (M||Γ);
+Arguments :M||Γ is a backjump state in sml↑
+Π
+Return Value :Cis a backjump clause
+begin
+C←the reason of the member of Mthat immediately follows lcp(M);
+N←the list of the nondecision literals in lcp(M);
+R←the list of the reasons that are assigned to the literals in N;
+whileC∩N/negationslash=∅do
+l←a literal in C∩N;
+B←the clause in Rthat contains l;
+C′←the resolvent of CandBonl;
+ifC′=∅then
+returnC
+C←C′
+returnC;
+Algorithm 1: A procedure for generating a backjump clause.
+proof of Proposition 11↑allows to conclude the termination of BackjumpClause and
+asserts that a clause returned by the procedure is a backjump cla use on a backjump
+state.
+For instance, let Π be (25). Consider an execution of BackjumpClause on Π and
+backjump state (28). The table below gives the values of lcp(M),C,N, andR
+during the execution of the BackjumpClause algorithm. By Ciwe denote a value
+ofCbefore the i-th iteration of the while loop.
+lcp(M)a∆¬b¬b∨¬ac∆¬f¬f∨¬cd∆¬k¬k∨¬d¬l¬l∨b∨k¬m¬m∨l∨b
+C1m∨k∨l
+N¬b¬b∨¬a¬f¬f∨¬c¬k¬k∨¬d¬l¬l∨b∨k¬m¬m∨l∨b
+R¬b∨¬a,¬f∨¬c,¬k∨¬d,¬l∨b∨k,¬m∨l∨b
+C2k∨l∨bis the resolvent of C1and¬m∨l∨b
+C3k∨bis the resolvent of C2and¬l∨b∨k
+C4¬d∨bis the resolvent of C3and¬k∨¬d
+C5¬d∨¬ais the resolvent of C4and¬b∨¬a(29)
+The algorithm will terminate with the clause ¬d∨¬a. Proof of Proposition 11↑
+asserts that (i) this clause is a backjump clause such that dandaare decision
+literals in Mand (ii) the transition
+a∆¬b¬b∨¬ac∆¬f¬f∨¬cd∆¬k¬k∨¬d¬l¬l∨b∨k¬m¬m∨l∨bmm∨k∨l||∅=⇒
+a∆¬b¬b∨¬a¬d¬d∨¬a||∅(30)
+insml↑Πis an application of Backjump LP . Indeed, by Lemma 12 lcp(M)decM(¬a)]¬d¬d∨¬a,
+in other words a∆¬b¬b∨¬a¬d¬d∨¬a, is a record.
+Note that a backjump clause may be derived in other ways than capt ured by
+BackjumpClause algorithm: the transition rule Backjump LP is applicable with
+an arbitrary backjump clause. Usually, dpll -like procedures implement conflict-
+driven backjumping and learning where a particular learning schema s uch as, forAbstract Answer Set Solvers with Backjumping and Learning ( long version) 39
+instance, Decision orFirstUIP (Mitchell 2005) is applied for computing a special
+kind of a backjump clause. It turns out that the BackjumpClause algorithm captures
+theDecision learning schema for ASP. Typically, SAT solvers impose an order for
+resolving the literals during the process of Decision backjump clause derivation. We
+can impose similar order by replacing the line
+l←a literal in C∩N
+in the algorithm BackjumpClause with
+l←a literal in C∩Nthat occurs latest in lcp(M).
+In fact, the sample application of BackjumpClause algorithm described in (29)
+follows this ordering.
+This section introduced BackjumpClause algorithm that derives a Decision back-
+jump clause for an arbitrary backjump state. In the next section we will introduce
+an algorithm that will compute an ASP counterpart of FirstUIP backjump clause.
+11 FirstUIP Conflict-Driven Backjumping and Learning
+Conflict-driven backjumping and learning proved to be a highly succe ssful tech-
+nique in modern SAT solving. Furthermore, in (Zhang et al. 2001) the authors
+investigated the performance of various learning schemes and est ablished exper-
+imentally that FirstUIP clause is the most useful single clause to learn. Success of
+conflict-driven learning led to the implementation of its ASP counterp art in systems
+smodels cc,clasp , andsup. There are two common methods for describing a back-
+jump clause construction in the SAT literature. The first one employ es the implica-
+tion graph (Marques-Silva and Sakallah 1996) and the second one em ployes resolu-
+tion (Mitchell 2005). Ward and Schlipf (Ward and Schlipf 2004) exten ded the defi-
+nition of an implication graph to the smodels algorithm and implemented FirstUIP
+learning schema in answer set solver smodels cc. In the previous section we used
+sml↑
+Πformalism and resolution to describe the BackjumpClause algorithm for com-
+puting an ASP counterpart of a Decision backjump clause. In (Gebser et al. 2007)
+the authors used the concepts from constraint processing to imp lementFirstUIP
+learning schema in answer set solver clasp .
+This section presents the BackjumpClauseFirstUIP algorithm for computing an
+ASP counterpart of a FirstUIP backjump clause by means of sml↑
+Πformalism and
+resolution. The BackjumpClauseFirstUIP algorithm is employed by the system sup
+in its implementation of conflict-driven backjumping and learning.
+The Algorithm 2 presents procedure BackjumpClauseFirstUIP that computes a
+FirstUIP backjump clause for any backjump state in the graph sml↑
+Π.
+We now state the correctness of the algorithm BackjumpClauseFirstUIP . We
+start by showing its termination. By C1we will denote the initial value assigned to
+clauseC. From Lemma 11 (i) and the choice of C1we conclude that at any point of
+computation clause Cis conflicting on M. By Lemma 11 (ii), the value of βM(C)
+decreases with each new assignment of clause Cin thewhile loop. It follows that
+thewhile loop will terminate since the number of conflicting clauses ConMsuch40 Y. Lierler
+BackjumpClauseFirstUIP (M||Γ);
+Arguments :M||Γ is a backjump state in sml↑
+Π
+Return Value :Cis a backjump clause
+begin
+C←the reason of the member of Mthat immediately follows lcp(M);
+l←the literal in Cthat occurs latest in lcp(M);
+P←the sublist of lcp(M) that consists of the literals that belong to the decision
+leveldec(l);
+R←the list of the reasons that are assigned to the literals in P;
+while|C∩P|>1do
+l←the literal in Cthat occurs latest in P;
+B←the clause in Rthat contains l;
+C←the resolvent of CandBonl;
+returnC;
+Algorithm 2: A procedure for generating a FirstUIP backjump clause.
+that|C∩P|>1 is finite. By Cmwe will denote the clause Cwith which the
+while loop terminates. In other words BackjumpClauseFirstUIP returnsCm. We
+now show that Cmis indeed a backjump clause. We already concluded that Cm
+is a conflicting clause on M. Furthermore, from the termination condition of the
+while loop|Cm∩P|≤1. From the choice of C1andPit follows that|Cm∩P|= 1.
+Consequently, Cmcan be written as l∨C′
+mwherelis in singleton Cm∩P. By
+Lemma 11 (ii), β(Cm)≤β(C1). From the definition of βand the choice of Pit
+follows that decM(l)>decM(c) for allc∈C′
+m. By Lemma 12, lcp(M)decM(C′m)]lCm
+is a record. In other words, transition
+M||Γ =⇒lcp(M)decM(C′m)]lCm||Γ
+is an application of Backjump LP . Consequently, Cmis a backjump clause.
+For instance, let Π be (25). Consider an execution of BackjumpClauseFirstUIP
+on Π and a backjump state (28). The table below gives the values of lcp(M),C,P,
+andRduring the execution of BackjumpClauseFirstUIP . ByCiwe denote a value
+ofCbefore the i-th iteration of the while loop.
+lcp(M)a∆¬b¬b∨¬ac∆¬f¬f∨¬cd∆¬k¬k∨¬d¬l¬l∨b∨k¬m¬m∨l∨b
+C1m∨k∨l
+P d∆¬k¬k∨¬d¬l¬l∨b∨k¬m¬m∨l∨b
+R¬k∨¬d,¬l∨b∨k,¬m∨l∨b
+C2k∨l∨bis the resolvent of C1and¬m∨l∨b
+C3k∨bis the resolvent of C2and¬l∨b∨k.
+TheBackjumpClauseFirstUIP algorithm will terminate with the clause k∨b. The
+proof of the correctness of BackjumpClauseFirstUIP asserts that k∨bis a backjump
+clause and the transition
+a∆¬b¬b∨¬ac∆¬f¬f∨¬cd∆¬k¬k∨¬d¬l¬l∨b∨k¬m¬m∨l∨bmm∨k∨l=⇒
+a∆¬b¬b∨¬akk∨b||∅(31)Abstract Answer Set Solvers with Backjumping and Learning ( long version) 41
+insml↑Πis an application of Backjump LP .
+12 Extended Graph: Generate and Test
+In this section we introduce an extended graph gtl↑
+F,Gfor thegenerate and test
+abstract framework gtlF,Gsimilar as in Section 9 we introduced sml↑
+ΠforsmlΠ.
+For a formula H, we say that a clause l∨Cis areason forlto be in a list PlQ
+of literals w.r.t. HifH|=l∨CandC⊆P.
+An(extended) record Mrelative to a formula His a list of literals over the set
+of atoms occurring in Hwhere
+(i) each literal linMis annotated either by ∆ or by a reason for lto be in M
+w.r.t.H,
+(ii)Mcontains no repetitions,
+(iii) for any inconsistent prefix of Mits last literal is annotated by a reason.
+An(extended) state relative to a CNF formula F, and a formula Gformed from
+atoms occurring in Fis either a distinguished state FailState or a pair of the
+formM||Γ, where Mis an extended record relative to F∧G, and Γ is the same
+as in the definition of an augmented state (i.e., Γ is a (multi-)set of clau ses formed
+from atoms occurring in Fthat are entailed by F∧G.) For any extended state S
+relative to FandG, the result of removing annotations from all nondecision literals
+ofSis a state of gtlF,G: we will denote this state by S↓.
+For a CNF formula Fand a formula Gformed from atoms occurring in F, we
+will define a graph gtl↑
+F,G. The set of the nodes of gtl↑
+F,Gconsists of the extended
+states relative to FandG. The transition rules of gtlF,Gare extended to gtl↑
+F,G
+as follows: S1=⇒S2is an edge in gtl↑
+F,Gjustified by a transition rule Tif and
+only ifS↓
+1=⇒S↓
+2is an edge in gtlF,Gjustified by T.
+The lemma below formally states the relationship between nodes of th e graphs
+gtlF,Gandgtl↑
+F,G:
+Lemma 13
+For any CNF formula Fand a formula Gformed from atoms occurring in F, ifS′
+is a state reachable from ∅||∅in the graph gtlF,Gthen there is a state Sin the
+graphgtl↑
+F,Gsuch that S↓=S′.
+The definitions of Basic transition rules and semi-terminal states in gtl↑
+F,Gare
+similar to their definitions for gtlF,G.
+Proposition 9↑
+For any CNF formula Fand a formula Gformed from atoms occurring in F,
+(a) every path in gtl↑
+F,Gcontains only finitely many edges labeled by Basic
+transition rules,
+(b) for any semi-terminal state M||Γ ofgtl↑
+F,G,Mis a model of F∧G,
+(c)gtl↑
+F,Gcontains an edge leading to FailState if and only if F∧Gis unsatis-
+fiable.42 Y. Lierler
+We say that a state in the graph gtl↑
+F,Gis abackjump state if its record is
+inconsistent and contains a decision literal. As in case of the graph gtlF,G, any
+backjump state in gtl↑
+F,Gis not semi-terminal:
+Proposition 10↑
+For any CNF formula Fand a formula Gformed from atoms occurring in F, the
+transition rule Backjump GT is applicable in any backjump state in gtl↑
+F,G.
+Proposition 9 (b), (c) and Proposition 10 easily follow from Lemma 13 a nd Propo-
+sition 9↑(b), (c) and Proposition 10↑respectively. Proof of Proposition 9 (a) is
+similar to the proof of Proposition 9↑(a).
+13 Proofs of Proposition 9↑, Lemma 13, Proposition 10↑
+13.1 Proof of Proposition 9↑
+Lemma 14
+For any CNF formula F, a formula Gformed from atoms occurring in F, an ex-
+tended record Mrelative to F∧G, and any model XofF∧G, ifXsatisfies all
+decision literals in MthenX|=M.
+Proof
+By induction on the length of M. The property trivially holds for ∅. We assume that
+the property holds for any state with nelements. Consider any state Mwithn+ 1
+elements. Let Xbe a model of F∧Gsuch that Xsatisfies all decision literals in M.
+Case 1.Mhas the form Pl∆. By the inductive hypothesis, X|=P. SinceX
+satisfies all decision literals in M,X|=l∆.
+Case 2.Mhas the form Pll∨C. By the inductive hypothesis, X|=P. By the
+definition of a reason (i) F∧Gentailsl∨Cand (ii)C⊆P. From (ii) it follows
+thatP|=¬C. Consequently, X|=¬C. From (i) it follows that X|=l∨C. We
+conclude that X|=l.
+The proof of Proposition 9↑assumes the correctness of Proposition 10↑that we
+demonstrate in Section 13.3.
+ıProof of Proposition 9↑
+Parts (a) and (c) are proved as in the proof of Proposition 8↑, using Lemma 14.
+(b) LetM||Γ be a semi-terminal state so that none of the Basic rules are applica ble.
+From the fact that Decide is not applicable, we conclude that Massigns all literals.
+Furthermore, Mis consistent. Indeed, assume that Mis inconsistent. Then, since
+Failis not applicable, Mcontains a decision literal. Consequently, M||Γ is a back-
+jump state. By Proposition 10↑, the transition rule Backjump GT is applicable in
+M||Γ. This contradicts our assumption that M||Γ is semi-terminal.
+Also,Mis a model of F: sinceUnit Propagate λis not applicable, it follows that
+for every clause C∨l∈F∪Γ ifC⊆Mthenl∈M. Consequently, M|=C∨l.
+Furthermore, Mis a model of G: sinceTest is not applicable, then G/\e}atio\slash|=M. We
+conclude that M|=G. Consequently, Mis a model of F∧G.Abstract Answer Set Solvers with Backjumping and Learning ( long version) 43
+13.2 Proof of Lemma 13
+For a state Sin the graph gtl↑
+F,G, we say that S↓ingtlF,Gis theimage ofS.
+Lemma 13
+For any CNF formula Fand a formula Gformed from atoms occurring in F, ifS′
+is a state reachable from ∅||∅in the graph gtlF,Gthen there is a state Sin the
+graphgtl↑
+F,Gsuch that S↓=S′.
+Proof
+Since the property trivially holds for the initial state ∅||∅, we only need to prove
+that all transition rules of gtlF,Gpreserve it.
+Consider an edge M||Γ =⇒M′||Γ′in the graph gtlF,Gsuch that there is a
+stateM1||Γ in the graph gtl↑
+F,Gsatisfying the condition ( M1||Γ)↓=M||Γ. We
+need to show that there is a state in the graph gtl↑
+F,Gsuch that M′||Γ′is its
+image in gtlF,G. Consider several cases that correspond to a transition rule lead ing
+fromM||Γ toM′||Γ′:
+Unit Propagate λ:
+M||Γ =⇒Ml||Γ if/braceleftbiggC∨l∈F∪Γ and
+C⊆M.
+M′||Γ′isMl||Γ. It is sufficient to prove that M1lC∨l||Γ is a state of gtl↑
+F,G. It is
+enough to show that a clause C∨lis a reason for lto be in Mlw.r.t.F∧G, i.e,
+F∧G|=C∨landC⊆M. By applicability conditions of Unit Propagate λ,C⊆M.
+By the definition of a state F∧Gentails Γ. Since C∨l∈F∩Γ,F∧G|=C∨l.
+Test:
+M||Γ =⇒Ml||Γ if
+
+Mis consistent ,
+G|=M,
+l∈M.
+M′||Γ′isMl||Γ. It is sufficient to prove that M1lM||Γ is a state of gtl↑
+F,G.Mhas
+the form l∨C. It is enough to show that a clause l∨Cis a reason for lto be in
+Mlw.r.t.F∧G. It is trivial that C⊆M. By applicability condition of the rule,
+G|=l∨C.
+Backjump GT ,Decide ,Fail, andLearn GT : obvious.
+13.3 Proof of Proposition 10↑
+For a state MlC||Γ, we say that a reason Cis abackjump clause if there is a
+transition Backjump GT leading to MlC||Γ ingtlF,G.
+In this section Fis an arbitrary and fixed CNF formula and Gis an arbitrary
+and fixed formula formed from atoms occurring in F.
+For a record M, bylcp(M) we denote its largest consistent prefix. We say that a
+clauseCisconflicting on a list Mof literals if F∧GentailsC, andC⊆lcp(M).
+Lemmas 10, 11, 12 hold for the case of extended record relative to a formula. The
+proofs of the lemmas have to be modified only by replacing Π with F∧G.
+Proposition 10↑is proved as Proposition 11↑.44 Y. Lierler
+Algorithms BackjumpClause andBackjumpClauseFirstUIP are applicable to the
+backjump states of the graph gtl↑
+F,G.
+14 Experiments with Sup
+Here we present experimental analysis that compares performan ce of the system
+supversuscmodels ,clasp ,smodels , andsmodels cc. We start by describing the
+implementation details of sup.
+The implementation of suputilizes
+•the interface of SAT-solver minisat (v1.12b) that supports non-clausal con-
+straints described in (Een and S¨ orensson 2003) in order to introd uce addi-
+tional inference possibilities, but unit propagation. In particular, sup im-
+plements Backchain True andAll Rules Cancelled by means of non-clausal
+constraints and it uses the unit propagate of minisat to capture Unit Propa-
+gate LP andBackchain False .
+•parts ofcmodels code that eliminate weight and choice rules; perform model
+verification; and compute loop formulas. In particular, supuses the latter two
+parts ofcmodels code to capture Unfounded .
+In the experiments we used the following versions of the systems: sup v. 0.1,
+supv. 0.2,cmodels v. 3.77 using minisat v. 1.12b, clasp v. 1.0.5,smodels v. 2.32,
+smodels ccv. 1.08 (implemented on top of smodels v. 2.26). System sup(v. 0.1
+and v. 0.2) extends the implementation of minisat v. 1.12b. Therefore, we compare
+supperformance against cmodels that uses minisat 1.12b for its inference. Sys-
+temsupv. 0.1 stands for a version of supthat implements Unit Propagate LP ,All
+Rules Cancelled , andBackchain False λpropagation rules, and does not implement
+Backchain True . System supv 0.2, on the other hand, also implements Backchain
+True .
+All considered solvers use preprocessor lparse (see Footnote 5) to ground the
+problems so that the systems are run on identical ground instance s. Grounding time
+is not accounted for in solving time. All times are reported in seconds . Symbol tout
+stands for the fact that a system did not terminate with a solution a fter 10 minutes.
+Sup 0.1, Sup, Cm, Cl, Sm cc,andSmstand for supv. 0.1,supv. 0.2,cmodels ,
+clasp ,smodels cc, andsmodels respectively. The symbol −tabbreviates the flag
+−temp that allows supto forget learnt clauses due to loop formulas (by default sup
+adds these clauses into permanent clause database). The symbol −aabbreviates
+the flag−atomreason that forces cmodels , likesup, to add only a clause implied
+by some loop formula and unsatisfied by a current model rather tha n the complete
+loop formula unsatisfied by the model. By default, cmodels adds a complete loop
+formula unsatisfied by the model. All experiments were run on Intel( R) Pentium(R)
+D CPU 3.00GHz, 2 cpu cores, cache size 1024 KB, running Linux.
+Table 1 presents the experiments run on tight programs. Recall th at for tight
+programs (i) the transition rule Unfounded ofsupis never used for inference and
+(ii) the transition rule Test ofcmodels is never used for inference.Abstract Answer Set Solvers with Backjumping and Learning ( long version) 45
+Instance Sup 0.1 Sup Cm Cl SmccSm
+Towers of Hanoi: http://asparagus.cs.uni-potsdam.de
+towers-hanoi.35 643.68 92.96 74.03 23.86 115.83 62.19
+towers-hanoi.36 6183.05 70.71 117.56 38.98 112.44 86.01
+towers-hanoi.37 635.71 30.75 290.65 34.25 84.75 120.53
+towers-hanoi.38 6243.41 233.82 37.03 70.94 99.87 168.88
+towers-hanoi.39 696.59 tout tout 123.40 384.24 237.90
+towers-hanoi.40 6tout 113.15 30.21 114.39 124.73 329.08
+towers-hanoi.41 6123.00 69.51 103.61 169.43 168.74 466.15
+towers-hanoi.42 6tout 389.88 91.28 182.10 tout tout
+towers-hanoi.43 6tout 42.40 353.74 228.37 204.89 tout
+towers-hanoi.44 6501.80 438.78 498.89 tout tout tout
+Pigeon Holes with 10 holes: pgh#pigeons
+pgh7 0.18 0.22 0.14 0.10 20.98 1.67
+pgh8 0.75 1.10 0.68 1.00 tout 8.87
+pgh9 7.60 9.30 4.03 5.73 tout 47.31
+Queens Normal Encoding: q.lp.#queens
+q.lp.18 0.14 0.14 0.14 0.08 155.62 7.84
+q.lp.22 0.28 0.27 0.30 0.14 tout tout
+q.lp.24 0.35 0.35 0.38 0.20 tout tout
+q.lp.30 0.70 0.71 0.75 0.44 tout tout
+Queens Cardinality Constraint Encoding: q.lp2.#queens
+q.lp2.18 0.06 0.05 0.06 0.04 147.88 2.24
+q.lp2.22 0.11 0.10 0.12 0.08 tout 191.76
+q.lp2.24 0.15 0.14 0.17 0.11 tout 267.27
+q.lp2.30 0.30 0.28 0.32 0.21 tout tout
+TOAST: http://asparagus.cs.uni-potsdam.de
+sequence3-ss3-Plain 138.08 49.69 78.45 10.51 444.41 tout
+sequence4-ss2-Plain 14.86 14.68 24.50 9.56 99.85 71.42
+sequence4-ss3-Plain 408.99 468.25 tout 294.27 tout tout
+sequence3-ss2 8.11 9.00 8.34 5.75 79.23 46.07
+sequence3-ss3 137.37 49.84 78.45 10.39 444.16 tout
+sequence4-ss2 16.13 16.88 13.38 8.96 102.98 74.85
+sequence4-ss3 103.33 207.21 16.74 233.09 tout tout
+Vertex Cover vcx.# minimum size vertex cover:
+http://www.cs.engr.uky.edu/ai/benchmarks.html
+vc1.53 8.30 3.23 19.81 tout tout 218.59
+vc2.50 11.55 6.76 13.32 94.43 tout 11.73
+vc3.55 0.65 4.09 64.44 512.56 tout 10.59
+vc4.54 6.49 10.19 23.05 tout tout tout
+Table 1: Experiments: Tight problems.
+Table 2 presents the experiments run on nontight programs:46 Y. Lierler
+Instance Sup Sup -t Cm Cm -a Cl SmccSm
+Deterministic Automaton: http://www.fmi.uni-stuttgart.de/szs/
+research/projects/synthesis/benchmarks030923.html
+mutex3Morin 15.30 15.25 15.68 15.68 15.45 306.30 153.60
+mutex4IDFD 0.80 0.74 0.66 0.64 0.80 35.53 13.46
+phi3Morin 0.96 0.95 0.62 0.72 0.37 2.50 2.80
+phi4IDFD 12.11 14.85 0.39 2.98 0.02 0.31 0.16
+phi4Morin tout 448.82 105.54 tout 95.50 tout tout
+phi5IDFD tout tout 67.14 tout 138.56 tout tout
+Bounded Model Checking:
+http://www.tcs.hut.fi/ ∼kepa/experiments/boundsmodels/
+dp10.i.O2.b12 30.88 3.51 10.14 22.29 0.20 tout 63.11
+dp10.s.O2.b9 0.54 0.42 0.78 0.68 0.14 26.18 13.04
+dp12.i.O2.b14 254.79 88.18 188.89 272.55 7.17 tout tout
+dp12.s.O2.b10 5.89 2.01 0.76 2.05 0.72 tout 337.28
+dp6.i.O2.b8 0.20 0.20 0.18 0.16 0.02 0.93 0.32
+dp8.i.O2.b10 0.98 0.77 1.53 2.60 0.03 12.97 4.12
+dp8.s.O2.b8 0.12 0.20 0.15 0.10 0.02 2.58 1.18
+Hamiltonian Cycle: http://www.cs.engr.uky.edu/ai/benchmarks.html
+hc1Stout 2.82 tout tout tout tout tout
+hc2S0.29 5.37 13.50 8.60 0.38 153.44 tout
+hc3S1.28 8.15 5.94 3.10 tout tout tout
+hc4S7.08 2.81 tout 0.94 2.18 14.92 tout
+Table 2: Experiments: Nontight problems.
+Overall the results demonstrated by supplace the system in the class of highly
+efficient answer set solvers.
+15 Related Work
+Simons (2000) and Ward (2004) described the smodels andsmodels ccalgorithms,
+respectively, by means of pseudocode and demonstrated their co rrectness. In this
+paper we designed an abstract framework that was used as an alte rnative method
+for describing these algorithms and demonstrating their correctn ess.
+Gebser and Schaub (2006) provided a deductive system for descr ibing inferences
+involved in computing answer sets by tableaux methods. The abstra ct framework
+presented here can be viewed as a deductive system also, but of a v ery different
+kind. First, it accounts for phenomena such as backjumping and lea rning (and also
+forgetting and restart) whereas the Gebser-Schaub system do es not. Second, we
+describe backtracking by an inference rule, and the Gebser-Scha ub system does
+not. Accordingly, the derivations considered in this paper describe search process,
+and derivations in the Gebser-Schaub system do not. Also, the abs tract frameworkAbstract Answer Set Solvers with Backjumping and Learning ( long version) 47
+discussed here does not have any inference rule similar to Cut; this is why its
+derivations are paths, rather than trees.
+16 Conclusions
+In this paper we showed how to model advanced algorithms for comp uting answer
+sets of a program by means of simple mathematical objects, graph s. We extended
+the abstract frameworks proposed in (Lierler 2008) for describin g native and SAT-
+based ASP algorithms to capture such sophisticated features as b ackjumping and
+learning. We characterized the algorithms of systems smodels cc,sup, andcmod-
+elsthat implement these features. We note that the work on this abst ract frame-
+work suggested the implementation of answer set solver supand the experimental
+analysis presented here demonstrates that supis a competitive representative in
+the family of answer set solvers. The abstract framework simplifies the analysis of
+the correctness of algorithms and allows us to study the relationsh ip between vari-
+ous algorithms by analyzing the differences in strategies of choosing a path in the
+graph. For example, the description of the smodels ccandsupalgorithms in this
+framework reflects their differences in a simple manner via distinct as signments of
+priorities to edges of the graph that characterize these systems . Also we used this
+framework to describe two algorithms for computing Decision andFirstUIP back-
+jump clauses for the implementation of conflict-driven backjumping and learning.
+This formalism provided the transparent means for specifying thes e algorithms.
+We believe that the development of this abstract framework power ful enough to
+describe advanced features of answer set solvers in a simple manne r will promote
+the use of these sophisticated features in more solvers. This work helped us design
+the new solver sup, and we hope that in the future it will suggest designs of other
+systems for computing answer sets.
+Acknowledgments
+We are grateful to Marco Maratea for bringing to our attention th e work by
+Nieuwenhuis et al. (2006), to Vladimir Lifschitz for the numerous disc ussions, to
+Martin Gebser, Michael Gelfond, and Miros/suppress law Truszczy´ nski for valuable ideas and
+comments, to anonymous referees for their suggestions. The au thor was supported
+by the National Science Foundation under Grant IIS-0712113.
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+arXiv:1001.0821v1  [cs.DS]  6 Jan 2010BeyondBidimensionality: ParameterizedSubexponential
+AlgorithmsonDirectedGraphs
+FredericDorn∗Fedor V. Fomin∗Daniel Lokshtanov∗VenkateshRaman†
+Saket Saurabh∗
+Abstract
+In 2000Alber et al. [ SWAT 2000 ] obtained the first parameterizedsubexponentialalgorith m on
+undirected planar graphs by showing that k-DOMINATING SETis solvable in time 2O(√
+k)nO(1),
+wherenis the input size. This result triggered an extensive study o f parameterized problems on
+planarandmoregeneralclassesofsparsegraphsandculmina tedinthecreationofBidimensionality
+TheorybyDemaineetal.[ J.ACM2005 ]. ThetheoryutilizesdeeptheoremsfromGraphMinorThe-
+oryofRobertsonandSeymour,andprovidesasimplecriteria forcheckingwhetheraparameterized
+problemissolvablein subexponentialtimeonsparsegraphs .
+While bidimensionality theory is an algorithmic framework on undirected graphs, it remains
+unclearhowtoapplyittoproblemsondirectedgraphs. Thema inreasonisthatGraphMinorTheory
+for directed graphs is still in a nascent stage and there are n o suitable obstruction theorems so far.
+Even the analogue of treewidth for directed graphs is not uni que and several alternative definitions
+havebeenproposed.
+In this paper we make the first step beyond bidimensionality b y obtaining subexponential time
+algorithms for problems on directed graphs. We develop two d ifferent methods to achieve subex-
+ponential time parameterized algorithms for problems on sp arse directed graphs. We exemplify
+our approacheswith two well studied problems. For the first p roblem,k-LEAFOUT-BRANCHING ,
+which is to find an oriented spanningtree with at least kleaves, we obtain an algorithmsolving the
+problem in time 2O(√
+klogk)n+nO(1)on directed graphs whose underlying undirected graph ex-
+cludessome fixedgraph Has a minor. Forthe special case when the inputdirectedgraph is planar,
+the running time can be improvedto 2O(√
+k)n+nO(1). The second example is a generalization of
+the DIRECTED HAMILTONIAN PATHproblem, namely k-INTERNAL OUT-BRANCHING , which is
+tofindanorientedspanningtreewithatleast kinternalvertices. We obtainanalgorithmsolvingthe
+problemintime 2O(√
+klogk)+nO(1)ondirectedgraphswhoseunderlyingundirectedgraphexclu des
+somefixedapexgraph Hasaminor. Finally,weobservethatforany ε >0,thek-DIRECTED PATH
+problemissolvablein time O((1+ε)knf(ε)), wherefissomefunctionof ε.
+Ourmethodsarebasedonnon-trivialcombinationsofobstru ctiontheoremsforundirectedgraphs,
+kernelization, problem specific combinatorialstructures and a layering techniquesimilar to the one
+employedbyBaker toobtainPTAS forplanargraphs.
+1 Introduction
+Parameterized complexity theory is a framework for a refined analysis of hard ( NP-hard) problems.
+Here,everyinput instance Iofaproblem Πisaccompanied withaninteger parameter kandΠissaidto
+be fixed parameter tractable (FPT) if there is an algorithm ru nning in time f(k)·nO(1), wheren=|I|
+andfis a computable function. A central problem in parameterize d algorithms is to obtain algorithms
+∗Department of Informatics, Universityof Bergen, Norway.
+{dorn|fedor.fomin|daniello|saket.saurabh }@ii.uib.no .
+†The Institute of Mathematical Sciences, Chennai, India.
+vraman@imsc.res.in
+1with running time f(k)·nO(1)such that fis as slow growing function as possible. This has led to
+the development of various graph algorithms with running ti me2O(k)nO(1)— notable ones include k-
+FEEDBACK VERTEXSET[7],k-LEAFSPANNING TREE[28],k-ODDCYCLETRANSVERSAL [31],
+k-PATH[4], andk-VERTEXCOVER[8] in undirected graphs. A natural question was whether we c an
+getsubexponential time algorithms for these problems, that is, can we have algorith ms with running
+time2o(k)nO(1). It is now possible to show that these problems do not admit al gorithms with running
+time2o(k)nO(1)unless the Exponential Time Hypothesis (ETH) [22, 27] fails . Finding algorithms with
+subexponential running time on general undirected graphs i s a trait uncommon to parameterized algo-
+rithms.
+However,thesituation changes completelywhenweconsider problemsontopological graphclasses
+like planar graphs or graphs of bounded genus. In 2000, Alber et al. [1] obtained the first parameterized
+subexponential algorithm onundirected planar graphs by sh owing that k-DOMINATING SETissolvable
+in time2O(√
+k)nO(1). This result triggered an extensive study of parameterized problems on planar
+and more general classes of sparse graphs like graphs of boun ded genus, apex minor-free graphs and
+H-minor free graphs. All this work led to subexponential time algorithms for several fundamental
+problems like k-FEEDBACK VERTEXSET,k-EDGEDOMINATING SET,k-LEAFSPANNING TREE,
+k-PATH,k-r-DOMINATING SET,k-VERTEXCOVERto name a few on planar graphs [1, 12, 25], and
+more generally, on H-minor-free graphs [13, 15, 16]. These algorithms are obtai ned by showing a
+combinatorial relation between the parameter and the struc ture of the input graph and proofs require
+strong graph theoretic arguments. This graph-theoretic an d combinatorial component in the design of
+subexponential timeparameterized algorithms makes it of a n independent interest.
+Demaine et al. [13] abstracted out the “common theme” among t he parameterized subexponential
+time algorithms on sparse graphs and created the meta-algor ithmic theory of Bidimensionality. The
+bidimensionality theory unifies and improves almost all kno wn previous subexponential algorithms on
+sparegraphs. Thetheory isbasedonalgorithmic andcombina torial extensions tovariouspartsofGraph
+Minors Theory of Robertson and Seymour [32] and provides a si mple criteria for checking whether a
+parameterized problem issolvable insubexponential timeo nsparse graphs. Thetheory applies tograph
+problems that are bidimensional in the sense that the value of the solution for the problem in q uestion
+onk×kgridor “grid likegraph” isatleast Ω(k2)andthevalueof solution decreases whilecontracting
+or sometime deleting the edges. Problems that are bidimensi onal include k-FEEDBACK VERTEXSET,
+k-EDGEDOMINATING SET,k-LEAFSPANNING TREE,k-PATH,k-r-DOMINATING SET,k-VERTEX
+COVERand many others. In most cases we obtain subexponential time algorithms for a problem using
+bidimensionality theory in following steps. Given an insta nce(G,k)to a bidimensional problem Π,
+in polynomial time we either decide that it is an yes instance toΠor the treewidth of GisO(√
+k).
+In the second case, using known constant factor approximati on algorithm for the treewidth, we find a
+tree decomposition of width O(√
+k)forGand then solve the problem by doing dynamic programming
+over the obtained tree decomposition. This approach combin ed with Catalan structure based dynamic
+programmingovergraphsofboundedtreewidthhasledto 2O(√
+k)nO(1)timealgorithmfor k-FEEDBACK
+VERTEXSET,k-EDGEDOMINATING SET,k-LEAFSPANNING TREE,k-PATH,k-r-DOMINATING
+SET,k-VERTEXCOVERand many others on planar graphs [12, 13, 20] and in some cases likek-
+DOMINATING SETandk-PATHonH-minor free graphs [13, 18]. We refer to surveys by Demaine
+and Hajiaghayi [15] and Dorn et al. [19] for further details o n bidimensionality and subexponential
+parameterized algorithms.
+Whilebidimensionality theoryisapowerfulalgorithmicfr ameworkonundirectedgraphs, itremains
+unclear how to apply it to problems on directed graphs (or dig raphs). The main reason is that Graph
+Minor Theory for digraphs is still in a nascent stage and ther e are no suitable obstruction theorems so
+far. For an example, even the first step of the framework does n ot work easily on digraphs, as there is
+no unique notion of directed k×kgrid. Given a k×kundirected grid we can make 2O(k2)distinct
+directed grids by choosing orientations for the edges. Henc e, unless we can guarantee a lower bound of
+Ω(k2)on thesize of solution of aproblem for anydirectedk×kgrid, thebidimensionality theory does
+2not look applicable for problems ondigraphs. Eventheanalo gue oftreewidth for digraphs isnot unique
+andseveral alternative definitions havebeenproposed. Onl yrecently thefirstnon-trivial subexponential
+parameterized algorithms on digraphs wasobtained. Alon et al. [3] introduced the method of chromatic
+coding,avariantofcolorcoding[4],andcombineditwithdi videandconquertoobtain 2O(√
+klogk)nO(1)
+fork-FEEDBACK ARCSETintournaments.
+Our contribution. In this paper we make the first step beyond bidimensionality b y obtaining subexpo-
+nential time algorithms for problems on sparse digraphs. We develop two different methods to achieve
+subexponential time parameterized algorithms for digraph problems when the input graph can be em-
+bedded on some surface or the underlying undirected graph ex cludes some fixedgraph Hasaminor.
+Quasi-bidimensionality. Our first technique can be thought of as “bidimensionality in disguise”. We
+observe that given a digraph D, whose underlying undirected graph UG(D)excludes some fixed graph
+Has a minor, if we can remove o(k2)vertices from the given digraph to obtain a digraph whose un-
+derlying undirected graph has a constant treewidth, then th e treewidth of UG(D)iso(k). So given an
+instance(D,k)to a problem Π, in polynomial time we either decide that it is an yes instanc e toΠor
+the treewidth of UG(D)iso(k). In the second case, as in the framework based on bidimension ality,
+we solve the problem by doing dynamic programming over the tr ee decomposition of UG(D). The
+dynamic programming part of the framework is problem-speci fic and runs in time 2o(k)+nO(1). We
+exemplify this technique on awell studied problem of k-LEAFOUT-BRANCHING .
+We say that a subdigraph Ton vertex set V(T)of a digraph Don vertex set V(D)is anout-tree
+ifTis an oriented tree with only one vertex rof in-degree zero (called the root). The vertices of Tof
+out-degree zero are called leavesand every other vertex is called an internal vertex . IfTis a spanning
+out-tree, that is, V(T) =V(D), thenTis called an out-branching ofD. Now we are in position to
+define the problem formally.
+k-LEAFOUT-BRANCHING (k-LOB): Givenadigraph Dwiththevertexset V(D)andthe
+arc setA(D)and a positive integer k, check whether there exists an out-branching with at
+leastkleaves.
+The study of k-LEAFOUT-BRANCHING has been at forefront of research in parameterized algo-
+rithms in the last few years. Alon et al. [2] showed that the pr oblem is fixed parameter tractable by
+giving an algorithm that decides in time O(f(k)n)whether a strongly connected digraph has an out-
+branching withatleast kleaves. BonsmaandDorn[6]extendedthisresulttoalldigra phs, andimproved
+therunningtimeofthealgorithm. Recently,Kneisetal.[28 ]providedaparameterizedalgorithmsolving
+the problem in time 4knO(1). This result was further improved to 3.72knO(1)by Daligaut et al. [10].
+Fernauet al.[21]showedthat for therooted version ofthepr oblem, whereapart from theinput instance
+we are also given a root rand one asks for a k-leaf out-branching rooted at r, admits a O(k3)kernel.
+Furthermore theyalso show that k-LOB does not admit polynomial kernel unless polynomial hie rarchy
+collapses to third level. Finally, Daligault and Thomass´ e [11] obtained a O(k2)kernel for the rooted
+version of the k-LOB problem and gave aconstant factor approximation algor ithm fork-LOB.
+Using our new technique in combination with kernelization r esult of [21], we get an algorithm for
+k-LOB that runs in time 2O(√
+klogk)n+nO(1)for digraphs whose underlying undirected graph is H-
+minor-free. Forplanar digraphs our algorithm runs in 2O(√
+k)n+nO(1)time.
+Kernelization and Divide & Conquer. Our second technique is a combination of divide and conquer,
+kernelization and dynamic programming over graphs of bound ed treewidth. Here, using a combina-
+tion of kernelization and a Baker style layering technique f or obtaining polynomial time approximation
+schemes [5], we reduce the instance of a given problem to 2o(k)nO(1)many new instances of the same
+problem. These new instances have the following properties : (a) the treewidth of the underlying undi-
+rected graph of these instances is bounded by o(k); and (b) the original input is an yes instance if and
+only if at least one of the newly generated instance is. We exh ibit this technique on the k-INTERNAL
+OUT-BRANCHING problem, a parameterized version of a generalization of D IRECTED HAMILTONIAN
+PATH.
+3k-INTERNAL OUT-BRANCHING (k-IOB): Given a digraph Dwith the vertex set V(D)
+and the arc set A(D)and a positive integer k, check whether there exists an out-branching
+with at least kinternal vertices.
+PrietoandSloper[30]studiedthe undirected versionofthisproblemandgaveanalgorithmwithrunning
+time24klogknO(1)and obtained a kernel of size O(k2). Recently, Fomin et al. [23] obtained a vertex
+kernel of size 3kand gave an algorithm for the undirected version of k-IOB running in time 8knO(1).
+Gutin et al. [26] obtained an algorithm of running time 2O(klogk)nO(1)fork-IOB and gave a kernel
+of size of O(k2)using the well known method of crown-decomposition. Cohen e t al. [9] improved the
+algorithm for k-IOB and gave an algorithm with running time 49.4knO(1). Here, we obtain a subexpo-
+nential time algorithm for k-IOB with running time 2O(√
+klogk)+nO(1)on directed planar graphs and
+digraphs whose underlying undirected graphs are apex minor -free.
+Finally, we also observe that for any ε >0, there is an algorithm finding in time O((1+ε)knf(ε))
+a directed path of length at least k(thek-DIRECTED PATHproblem) in a digraph which underlying
+undirected graphexcludesafixedapexgraphasaminor. Theex istence ofsubexponential parameterized
+algorithm for this problem remains open.
+2 Preliminaries
+LetDbe a digraph. By V(D)andA(D)we represent the vertex set and arc set of D, respectively.
+Givenasubset V′⊆V(D)ofadigraph D,letD[V′]denote thedigraph induced by V′. Theunderlying
+graphUG(D)ofDis obtained from Dby omitting all orientations of arcs and by deleting one edge
+from each resulting pair of parallel edges. A vertex uofDis anin-neighbor (out-neighbor ) of a vertex
+vifuv∈A(D)(vu∈A(D),respectively). The in-degree d−(v)(out-degree d+(v)) ofavertex visthe
+number of its in-neighbors (out-neighbors). We say that a su bdigraph Tof a digraph Dis anout-tree
+ifTis an oriented tree with only one vertex rof in-degree zero (called the root). The vertices of Tof
+out-degree zero are called leavesand every other vertex is called an internal vertex . IfTis a spanning
+out-tree, thatis, V(T) =V(D),thenTiscalledan out-branching ofD. Anout-branching (respectively.
+out-tree) rooted at ris calledr-out-branching (respectively. r-out-tree). We define the operation of a
+contraction ofadirected arc asfollows. Anarc uviscontracted asfollows: addanewvertex u′,andfor
+each arcwvorwuadd the arc wu′and for an arc vworuwadd the arc u′w, remove all arcs incident
+touandvand the vertices uandv. We call a loopless digraph Drooted, if there exists a pre-specified
+vertexrof in-degree 0as a root randd+(r)≥2. The rooted digraph Dis calledconnected if every
+vertex in V(D)isreachable from rby adirected path.
+LetGbe an undirected graph with the vertex set V(G)and the edge set E(G). For a subset V′⊆
+V(G), byG[V′]we mean the subgraph of Ginduced by V′. ByN(u)we denote (open) neighborhood
+ofuthat is the set of all vertices adjacent to uand byN[u] =N(u)∪ {u}. Similarly, for a subset
+D⊆V, wedefine N[D] =∪v∈DN[v]. Thediameter of a graph G, denoted by diam(G), is defined to
+be themaximum length of ashortest path between anypair of ve rtices ofV(G).
+Given an edge e=uvof a graph G, the graph G/eis obtained by contracting the edge uv; that is,
+wegetG/ebyidentifying thevertices uandvand removing alltheloops andduplicate edges. A minor
+of a graph Gis a graph Hthat can be obtained from a subgraph of Gby contracting edges. A graph
+classCisminor closed if any minor of any graph in Cis also an element of C. A minor closed graph
+classCisH-minor-free or simply H-freeifH /∈ C. Agraph Hiscalled anapexgraph if theremoval of
+one vertex makes it aplanar graph.
+Atree decomposition of a (undirected) graph Gis a pair(X,T)whereTis a tree whose vertices
+we will call nodesandX= ({Xi|i∈V(T)})is a collection of subsets of V(G)such that (a)/uniontext
+i∈V(T)Xi=V(G),(b)for each edge vw∈E(G), there is an i∈V(T)such that v,w∈Xi, and(c)
+foreachv∈V(G)thesetofnodes {i|v∈Xi}formsasubtreeof T. Thewidthofatreedecomposition
+4({Xi|i∈V(T)},T)equalsmaxi∈V(T){|Xi|−1}. Thetreewidth of agraph Gis the minimum width
+over all tree decompositions of G. Weuse notation tw(G)to denote the treewidth of agraph G.
+Aparameterized problemissaidtoadmita polynomial kernel ifthereisapolynomial timealgorithm
+(where the degree of the polynomial is independent of k), called a kernelization algorithm, that reduces
+the input instance down to an instance with size bounded by a p olynomial p(k)ink, while preserving
+the answer. This reduced instance is called a p(k)kernelfor the problem. See [29] for an introduction
+tokernelization.
+3 Method I– QuasiBidimensionality
+Inthissectionwepresent ourfirstapproach. Ingeneral, asu bexponential timealgorithm usingbidimen-
+sionality is obtained by showing that the solution for a prob lem in question is at least Ω(k2)onk×k
+(contraction) grid minor. Using this we reduce the problem t o a question on graph with treewidth o(k).
+Westartwithalemmawhichenablesustousetheframeworkofb idimensionality fordigraph problems,
+though not as directly as for undirected graph problems.
+Lemma1. LetDbeadigraphsuchthat UG(D)excludes afixedgraph Hasaminor. Foranyconstant
+c≥1, if there exists a subset S⊆V(D)with|S|=ssuch that tw(UG(D[V(D)\S]))≤c, then
+tw(UG(D)) =O(√s).
+Proof.By[15],forany H-minor-freegraph Gwithtreewidthmorethan r,thereisaconstant δ >1only
+dependent on Hsuchthat Ghasar
+δ×r
+δgridminor. Suppose tw(UG(D))> δ(c+1)√sthenUG(D)
+contains a (c+1)√s×(c+1)√sgrid as a minor. Notice that this grid minor can not be destroy ed by
+any vertex set Sof size at most s. That is, if we delete any vertex set Swith|S|=sfrom this grid, it
+will still contain a (c+ 1)×(c+ 1)subgrid. Thus, UG(D[V(D)\S])contains a (c+ 1)×(c+ 1)
+grid minor and hence by [22, Exercise 11.6] we have that tw(UG(D[V(D)\S]))> c. This shows
+that we need to delete more than svertices from UG(D)to obtain a graph with treewidth at most c, a
+contradiction.
+Using Lemma 1, we show that k-LEAF-OUT-BRANCHING problem has a subexponential time al-
+gorithm on digraphs Dsuch that UG(D)exclude a fixed graph Has a minor. For our purpose a rooted
+versionof k-LOB willalsobeuseful whichwedefinenow. Inthe R OOTEDk-LEAF-OUT-BRANCHING
+(R-k-LOB) problemapart from Dandktherootrofthetreesearchedforisalsoapartoftheinput and
+theobjectiveistocheckwhetherthereexistsan r-out-branching withatleast kleaves. Wenowstateour
+main combinatorial lemmaand postpone its proof for a while.
+Lemma 2. LetDbe a digraph such that UG(D)excludes a fixed graph Has a minor, kbe a positive
+integer and r∈V(D)be the root. Thenin polynomial time either wecan construct a nr-out-branching
+withat least kleaves in Dor find adigraph D′such that following holds.
+•UG(D′)excludes the fixed graph Hasaminor;
+•Dhas anr-out-branching with at least kleaves if and only if D′has anr-out-branching with at
+leastkleaves;
+•there exists a subset S⊆V(D′)such that |S|=O(k)andtw(U(D′[V(D′)\S])≤c,ca
+constant.
+Combining Lemmata1and 2weobtain the following result.
+Lemma 3. LetDbe a digraph such that UG(D)excludes a fixed graph Has a minor, kbe a positive
+integer and r∈V(D)be a root. Then in polynomial time either we can construct an r-out-branching
+withatleast kleavesinDorfindadigraph D′suchthat Dhasanr-out-branching withatleast kleaves
+if and only if D′has anr-out-branching with at least kleaves. Furthermore tw(UG(D′)) =O(√
+k).
+5When a tree decomposition of UG(D)is given, dynamic programming methods can be used to
+decide whether Dhas an out-branching with at least kleaves, see [26]. The time complexity of such a
+procedure is 2O(wlogw)n,wheren=|V(D)|andwisthe widthof thetree decomposition. Nowweare
+ready toprove the maintheorem of this section assuming the c ombinatorial Lemma2.
+Theorem1. Thek-LOBproblemcanbesolvedintime 2O(√
+klogk)n+nO(1)ondigraphswith nvertices
+such that the underlying undirected graph excludes afixed gr aphHas aminor.
+Proof.LetDbe a digraph where UG(D)excludes a fixed graph Has a minor. We guess a vertex
+r∈V(D)asaroot. Thisonlyaddsafactor of ntoouralgorithm. ByLemma3,wecaneither compute,
+in polynomial time, an r-out-branching with at least kleaves in Dor find a digraph D′withUG(D′)
+excluding a fixedgraph Has aminor and tw(UG(D′)) =O(√
+k). In the later case, using the constant
+factor approximation algorithm of Demaine et al. [17] for co mputing the treewidth of a H-minor free
+graph, we find a tree decomposition of width O(√
+k)forUG(D′)in timenO(1). With the previous
+observation thatwecanfindan r-out-branching withatleast kleaves, ifexistsone,intime 2O(√
+klogk)n
+usingdynamic programming over graphs ofbounded treewidth , wehavethat wecansolve R- k-LOB in
+time2O(√
+klogk)nO(1). Hence, weneed 2O(√
+klogk)nO(1)tosolve the k-LOB problem.
+To obtain the claimed running time bound we use the known kern elization algorithm after we have
+guessed the root r. Fernau et al. [21] gave an O(k3)kernel for R- k-LOB which preserves the graph
+class. That is, given an instance (D,k)of R-k-LOB, in polynomial time they output an equivalent
+instance(D′′,k)of R-k-LOB such that (a) if UG(D)isH-minor free then so is UG(D′′); and (b)
+|V(D′′)|=O(k3). We will use this kernel for our algorithm rather than the O(k2)kernel for R- k-
+LOB obtained by Daligault and Thomass´ e [11], as they do not p reserve the graph class. So after we
+have guessed the root r,weobtain an equivalent instance (D′′,k)for R-k-LOB using the kernelization
+procedure described in[21]. Thenusing thealgorithm descr ibed intheprevious paragraph wecansolve
+R-k-LOB in time 2O(√
+klogk)+nO(1). Hence, weneed 2O(√
+klogk)n+nO(1)to solvek-LOB.
+Given a tree decomposition of width wofUG(D)for a planar digraph D, we can solve k-LOB
+using dynamic programming methods intime 2O(w)n. Thisbrings us tothe following theorem.
+Theorem2. [⋆]1Thek-LOBproblemcanbesolvedintime 2O(√
+k)n+nO(1)ondigraphswith nvertices
+when theunderlying undirected graph isplanar.
+3.1 ProofofLemma 2
+To prove the combinatorial lemma we need a few recent results from the literature on out-branching
+problems. We start with some definitions given in[11]. A cutofDis asubset Ssuch that there exists a
+vertexz∈V(D)\Ssuchthat zisnot reachable from rinD[V(D)\S]. Wesay that Dis2-connected
+if there exists no cut of size one in Dor equivalently there are at least two vertex disjoint paths fromr
+toevery vertex in D.
+Lemma 4 ([11]).LetDbe a rooted 2-connected digraph with rbeing its root. Let αbe the number of
+verticesin Dwithin-degree atleast 3. ThenDhasanout-branching rootedat rwithatleast α/6leaves
+and such an out-branching can befound in polynomial time.
+A vertexv∈V(D)is called a nice vertex ifvhas anin-neighbor which is not its out-neighbor. The
+following lemmais proved in[11].
+Lemma 5 ([11]).LetDbe a rooted 2-connected digraph rooted at a vertex r. Letβbe the number
+of nice vertices in D. ThenDhas an out-branching rooted at rwith at least β/24leaves and such an
+out-branching can be found inpolynomial time.
+1The proofs marked with[ ⋆] have been moved tothe appendix due tospace restrictions.
+6Proof of Lemma2. Toprovethecombinatorial lemma, weconsider twocases base d onwhether or not
+Dis2-connected.
+Case 1)Disarooted 2-connected digraph.
+Weprove this case inthe following claim.
+Claim1. LetDbearooted 2-connecteddigraphwithroot randapositiveinteger k. Theninpolynomial
+time,wecanfindanout-branchingrootedat rwithatleast kleavesorfindaset Sofatmost 30kvertices
+whose removal results in adigraph whose underlying undirec ted graph has treewidth one.
+Proof.Ifα≥6k, then weare done by Lemma4. Similarly if β≥24k, then weare done by Lemma5.
+Hence we assume that α <6kandβ <24k. LetSbe the set of nice vertices and vertices of in-degree
+at least3inG. Then|S|< α+β≤30k. Observe that D[V(D)\S]is simply a collection of directed
+paths where every edge of the path is a directed 2-cycle. This is because D[V(D)\S]contains only
+those vertices which are not nice (that is, those vertices wh ose in-neighbors are also out-neighbors) and
+are of in-degree at most two. Hence, if there is an arc xyinD[V(D)\S], then the arc yxalso exists in
+D[V(D)\S]. Nextwenotethat D[V(D)\S]doesnotcontainadirectedcycleoflengthmorethantwo.
+Weprovethelastassertionasfollows. Let Cbeadirectedcyclein D[V(D)\S]oflengthatleast 3. Since
+Dis a rooted 2-connected digraph, we have a vertex von the cycle Csuch that there is a path from rto
+vwithout using any other vertex from the cycle C. This implies that the in-degree of vis at least 3inD
+andhence v∈S,contrarytoourassumption that v /∈S. Thisprovesthat D[V(D)\S]doesnotcontain
+a directed cycle of length more than two. Hence the underlyin g undirected graph UG(D[V(D)\S])is
+just acollection of paths and hence tw(UG(D[V(D)\S]))is one.
+Case 2)Disnot2-connected.
+SinceDisnot2-connected,ithascutvertices,thoseverticesthatsepara terfromsomeothervertices.
+Wedeal withthe cut vertices in three cases. Let xbea cut vertex of D. Thethree cases weconsider are
+following.
+Case 2a) There exists anarc xythat disconnects at least twovertices from r.
+In this case, we contract the arc xy. After repeatedly applying Case 2a), we obtain a digraph D′
+such that any arc out of a cut vertex xofD′disconnects at most 1vertex. The resulting digraph D′is
+the one mentioned in the Lemma. Since we have only contracted some arcs iteratively to obtain D′, it
+is clear that UG(D′)also excludes Has aminor. Theproof that such contraction does not decrease the
+number of leaves follows from areduction rule given in[21]. Weprovide aproof for completion.
+Claim 2. [⋆]LetDbe a rooted connected digraph with root r, letxybe an arc that disconnects at
+least two vertices from randD′be the digraph obtained after contracting the arc xy. ThenDhas an
+r-out-branching withat least kleaves if and only if D′has anr-out-branching with at least kleaves.
+Now we handle the remaining cut-vertices of D′as follows. Let Sbe the set of cut vertices in D′.
+For every vertex x∈ S, we associate a cut-neighborhood C(x), which is the set of out-neighbors of
+xsuch that there is no path from rto any vertex in C(x)inD′[V(D′)\ {x}]. ByC[x]we denote
+C(x)∪{x}. Thefollowing observation isused to handle other cases.
+Claim 3. LetSbe the set of cut vertices in D′. Then for every pair of vertices x,y∈ Sandx/\⌉}atio\slash=y, we
+have that C[x]∩C[y] =∅.
+Proof.To the contrary let us assume that C[x]∩C[y]/\⌉}atio\slash=∅. We note that C[x]∩C[y]can only have
+a vertexv∈ {x,y}. To prove this, assume to the contrary that we have a vertex v∈C[x]∩C[y]and
+v /∈ {x,y}. But then it contradicts the fact that v∈C[x], asxdoesn’t separate vfromrdue to the path
+betweenrandvthroughy. Thus,either x∈C(y)ory∈C(x). Withoutlossofgenerality let y∈C(x).
+This implies that we have an arc xyand there exists a vertex z∈C(y)such that z /∈C(x). But then
+the arcxydisconnects at least twovertices yandzfromrand hence Case 2awould have applied. This
+proves the claim.
+7Now we distinguish cases based on cut vertices having cut-ne ighborhood of size at least 2or1.
+LetS≥2andS=1be the subset of cut-vertices of D′having at least two cut-neighbors and exactly one
+neighbor respectively.
+Case 2b) S≥2/\⌉}atio\slash=∅.
+We first bound |S≥2|. LetAc={xy|x∈ S≥2,y∈C(x)}be the set of out-arcs emanating from
+thecut vertices in S≥2toitscut neighbors. Wenowprove thefollowing structural c laim whichisuseful
+for bounding the size of S≥2.
+Claim 4. [⋆]IfD′has anr-out-branching T′with at least kleaves then D′has anr-out-branching
+Twith at least kleaves and containing all the arcs of Ac, that is,Ac⊆A(T). Furthermore such an
+out-branching can be found inpolynomial time.
+We know that in any out-tree, the number of internal vertices of out-degree at least 2is bounded by
+the number of leaves. Hence if |S≥2| ≥kthen we obtain an r-out-branching TofD′with at least k
+leaves using Claim 4 and weare done. Sofrom now onwards weass ume that|S≥2|=ℓ≤k−1.
+We now do a transformation to the given digraph D′. For every vertex x∈ S≥2, we introduce an
+imaginary vertexxiand add an arc uxiif there is an arc ux∈A(D′)and add an arc xivif there is an
+arcxv∈A(D′). Basically we duplicate the vertices in S≥2. Let the transformed graph be called Ddup.
+Wehavethefollowingtwoproperties about Ddup. First,novertexin S≥2∪{xi|x∈ S≥2}isacutvertex
+inDdup. Wesum up thesecond property in the following claim.
+Claim 5. The digraph D′has anr-out-branching Twith at least kleaves if and only if Dduphas an
+r-out-branching T′withat least k+ℓleaves.
+Proof.Givenan r-out-branching TofD′withat least kleaves, weobtain an out-branching T′ofDdup
+with at least k+ℓleaves by adding an arc xxitoTfor every x∈ S≥2. Since every vertex of S≥2is an
+internal vertex in T, this process only adds {xi|x∈ S≥2}as leaves and hence we have at least k+ℓ
+leaves inT′.
+For the converse, assume that Dduphas anr-out-branching T′with at least k+ℓleaves. First, we
+modify the out-branching so that not both of xandxiare internal vertices and we do not lose any leaf.
+This can be done easily by making all out arcs in the out-branc hing from xand making xia leaf. That
+is, ifN+
+T′(xi)is the set of out-neighbors of xiinT′then we delete the arcs xiz,z∈N+
+T′(xi)and add
+xzfor allz∈N+
+T′(xi). Thisprocess cannotdecrease thenumber ofleaves. Further more wecanalways
+assumethatifexactlyoneof xandxiisaninternal vertex,then xistheinternalvertexin T′. Nowdelete
+all the vertices of {xi|x∈ S≥2}fromT′and obtain T. Since the vertices in the set {xi|x∈ S≥2}
+are leaves of T′, we have that Tis anr-out-branching in D′. Since in the whole process we have only
+deletedℓvertices wehave that Thas at least kleaves.
+Nowwemoveon tothe last case.
+Case 2c) S=1/\⌉}atio\slash=∅.
+Consider the arc set Ap={xy|x∈ S=1,y∈C(x)}. Observe that Ap⊆A(D′)⊆A(Ddup)
+andApforms amatching inDdupbecause of Claim 3. Let Ddup
+cbe the digraph obtained from Ddupby
+contractingthearcsof Ap. Thatis,foreveryarc uv∈Ap,thecontractedgraphisobtainedbyidentifying
+the vertices uandvasuvand removing all the loops and duplicate arcs.
+Claim 6. LetDdup
+cbe the digraph obtained by contracting the arcs of ApinDdup. Then the following
+holds.
+1. Thedigraph Ddup
+cis2-connected;
+2. IfDdup
+chas anr-out-branching Twith at least k+ℓleaves then Dduphas anr-out-branching
+with at least k+ℓleaves.
+8Proof.The digraph Ddup
+cis2-connected by the construction as we have iteratively remov ed all cut-
+vertices. If Ddup
+chasanr-out-branching Twithatleast k+ℓleavesthenwecanobtaina r-out-branching
+withat least k+ℓleaves for Ddupbyexpanding each of the contracted vertices toarcs in Ap.
+We are now ready to combine the above claims to complete the pr oof of the lemma. We first apply
+Claim 1 on Ddup
+cwithk+ℓ. Either we get an r-out-branching T′with at least k+ℓleaves or a
+setS′of size at most 30(k+ℓ)such that tw(UG(Ddup
+c[V(Ddup
+c)\S]))is one. In the first case, by
+Claims 5 and 6 we get an r-out-branching Twith at least kleaves in D′. In the second case we know
+thatthereisavertexset S′ofsizeatmost 30(k+ℓ)suchthat tw(UG(Ddup
+c[V(Ddup
+c)\S′]))isone. Let
+S∗={u|uv∈S′,vu∈S′,u∈S′}bethesetofverticesobtainedfrom S′byexpandingthecontracted
+vertices in S′. Clearly the size of |S∗| ≤2|S′| ≤60(k+ℓ)≤120k=O(k).We now show that the
+treewidth of the underlying undirected graph of Ddup[V(Ddup)\S∗]is at most 3. This follows from
+the observation that tw(UG(Ddup
+c[V(Ddup
+c)\S′]))is one. Hence given a tree-decomposition of width
+one forUG(Ddup
+c[V(Ddup
+c)\S′])we can obtain a tree-decomposition for UG(Ddup[V(Ddup)\S∗])
+by expanding the contracted vertices. This can only double t he bag size and hence the treewidth of
+UG(Ddup[V(Ddup)\S∗])isat most 3,asthebagsizecan atmost be 4. Nowwetake S=S∗∩V(D′)
+and since V(D′)⊆V(Ddup), we have that tw(UG(D[V(D)\S]))≤3. This concludes the proof of
+the lemma.
+4 Method II-Kernelization andDivide & Conquer
+In this section we exhibit our second method of designing sub exponential time algorithms for digraph
+problems through the k-INTERNAL OUT-BRANCHING problem. In this method we utilize the known
+polynomial kernel for theproblem and obtain acollection of 2o(k)instances such that theinput instance
+isan “yes” instance if and only if one of the instances inour c ollection is. The property of the instances
+in the collection which we make use of is that the treewidth of the underlying undirected graph of these
+instances is o(k). The last property brings dynamic programming on graphs of b ounded treewidth into
+picture as thefinal step of the algorithm.
+Here, we will solve a rooted version of the k-IOB problem, called R OOTEDk-INTERNAL OUT-
+BRANCHING (R-k-IOB), where apart from Dandkwe are also given a root r∈V(D), and the
+objectiveistofindan r-out-branching, ifexistsone,withatleast kinternalvertices. The k-IOB problem
+can be reduced to R- k-IOB by guessing the root rat the additional cost of |V(D)|in the running time
+of the R-k-IOB problem. Henceforth, we will only consider R- k-IOB. We call an r-out-treeTwithk
+internal vertices minimalifdeleting any leaf results inan r-out-tree withat most k−1internal vertices.
+A well known result relating minimal r-out-treeTwithkinternal vertices with a solution to R- k-IOB
+isas follows.
+Lemma 6 ([9]).LetDbe a rooted connected digraph with root r. ThenDhas anr-out-branching T′
+withat least kinternal vertices if andonly if Dhas aminimal r-out-treeTwithkinternal vertices with
+|V(T)| ≤2k−1. Furthermore, given a minimal r-out-treeT, we can find an r-out-branching T′with
+at leastkinternal vertices inpolynomial time.
+Wealso need another known result about kernelization for k-IOB.
+Lemma7 ([26]).k-INTERNAL OUT-BRANCHING admits apolynomial kernel of size 8k2+6k.
+In fact, the kernelization algorithm presented in [26] work s for all digraphs and has a unique re-
+duction rule which only deletes vertices . This implies that if we start with a graph G∈Gwhere G
+excludes a fixed graph Has a minor, then the graph G′obtained after applying kernelization algorithm
+still belongs to G.
+Our algorithm tries to find a minimal r-out-tree Twithkinternal vertices with |V(T)| ≤2k−1
+recursively. As the first step of the algorithm we obtain a set of2o(k)digraphs such that the underlying
+9undirected graphs have treewidth O(√
+k), and the original problem is a “yes” instance if and only at
+least one of the 2o(k)instances is a“yes” instance. More formally, weprove the fo llowing lemma.
+Lemma 8. LetHbe a fixed apex graph and Gbe a minor closed graph class excluding Has a minor.
+Let(D,k)be an instance to k-INTERNAL OUT-BRANCHING such that UG(D)∈G. Then there exists
+acollection
+C=/braceleftBig
+(Di,k′,r)|Diis asubgraph of D,k′≤k,r∈V(D),1≤i≤/parenleftbigg8k2+6k√
+k/parenrightbigg/bracerightBig
+,
+of instances such that tw(UG(Di)) =O(√
+k)foralliand(D,k)has an out-branching withat least k
+internal verticesifandonlyifthereexistsan i,randk′≤ksuchthat (Di,k′,r)hasanr-out-branching
+withat least k′internal vertices.
+Proof.Theideaof theproof istodoBaker style layering technique [ 5] combined withkernelization. In
+thefirststepweapplythekernelization algorithm givenbyL emma7on (D,k)andobtain anequivalent
+instance(D′,k′)where|D′| ≤8k2+ 6kandk′≤kfork-IOB. From now onwards we will confine
+ourselves to (D′,k′). Observe that since the kernelization algorithm only delet es vertices to obtain the
+reduced instance from theinput digraph, wehave that UG(D′)∈G.
+Now we reduce the k-IOB problem to the R- k-IOB problem by guessing a vertex r∈V(D′)as a
+root. Furthermore wetrytofindaminimal r-out-treeTwithk′internal vertices with |V(T)| ≤2k′−1.
+This suffices for our purpose if we know that every vertex in V(D′)is reachable from the root r, as in
+this case Lemma6isapplicable.
+We start with a BFS starting at the vertex rinUG(D′). Let the layers created by doing BFS on r
+beLr
+0,Lr
+1,...,Lr
+t. Ift≤ ⌈√
+k⌉, then the collection Crconsists of (D′,k′,r). Fort≤√
+k, the fact
+thattw(D′) =O(√
+k)follows from the comments later in the proof. Hence from now o nwards we
+assume that t >⌈√
+k⌉. Now we partition the vertex set into (⌈√
+k⌉) + 1parts where the q-th part
+contains all vertices which are at a distance of q+i(⌈√
+k⌉)fromrfor various values of i. That is, let
+V(D′) =∪qPq,q∈ {0,...,⌈√
+k⌉}. We define Pq=/uniontextLr
+q+i(⌈√
+k⌉+1), i∈/braceleftBig
+0,...,/floorleftBig
+t−√
+k
+⌈√
+k⌉+1/floorrightBig/bracerightBig
+.It is
+clear from the definition of Pqthat it partitions the vertex set V(D′). If the input is an “yes” instance
+then there exists a partition Pasuch that it contains at most/ceilingleftBig
+2k′−1
+⌈√
+k⌉/ceilingrightBig
+≤2√
+kvertices of the minimal
+r-out-treeTweare seeking for. Weguess the partition Paand obtain the collection
+Cr(Pa) =/braceleftBig
+(D′[V(D′)\Pa∪Z],k′,r)/vextendsingle/vextendsingle/vextendsingleZ⊆Pa,|Z| ≤2√
+k/bracerightBig
+.
+We now claim that for every Z⊆Pa,|Z| ≤2√
+k,tw(UG(D′[V(D′)\Pa∪Z])) =O(√
+k). Let
+V′=V(D′)\Pabe the set of vertices after removal of Pafrom the vertex set of D′. Let the resultant
+underlying undirected graph be G′=UG(D′[V′])with connected components C1,...,C ℓ. We show
+that each connected component CiofG′hasO(√
+k)treewidth. More precisely, every connected com-
+ponentCiofG′is a subset of at most ⌈√
+k⌉+1consecutive layers of the BFS starting at r. If we start
+withUG(D′),anddeleteallBFSlayersaftertheselayersandcontract al lBFSlayersbeforetheselayers
+into asingle vertex v, weobtain a minor MofUG(D′). This minor Mhas diameter at most ⌈√
+k⌉+2
+and contains Cias an induced subgraph. Since UG(D′)∈G′, we have that M∈G. Furthermore,
+Demaine and Hajiaghayi [14] have shown that for any fixed apex graphH, everyH-minor-free graph
+of diameter dhas treewidth O(d). This implies that the tw(Ci)≤tw(M)≤ O(√
+k). Notice that
+since every connected component of G′has treewidth O(√
+k),G′itself has O(√
+k)treewidth. Given
+a tree-decomposition of width O(√
+k)forG′, we can obtain a tree-decomposition of width O(√
+k)for
+UG(D′[V(D′)\Pa∪Z])byadding Ztoeverybag. Thecollection Crisgivenby ∪⌈√
+k⌉
+a=0Cr(Pa). Finally
+the collection C=∪r∈V(D′)Cr.
+By the pigeon hole principle we know that if (D′,k′)is an yes instance then there exists a Pa
+containing atmost 2√
+kverticesoftheminimaltree Twearelookingfor. Sincewehaverunthrough all
+10r∈V(D′)as a potential root as well as all subsets of size at most 2√
+kas the possible intersection of
+V(T)withPa, we know that (D′,k′)has an out-branching with at least kinternal vertices if and only
+if there exists an i,randk′≤ksuch that (Di,k′,r)∈ Chas ar-out-branching with at least k′internal
+vertices. This concludes the proof of the lemma.
+Given a tree decomposition of width wforUG(D), one can solve R- k-IOB in time 2O(wlogw)n
+using a dynamic programming over graphs of bounded treewidt h as described in [26]. This brings us to
+the maintheorem of this section.
+Theorem 3. Thek-IOBproblem can besolved intime 2O(√
+klogk)+nO(1)ondigraphs with nvertices
+such that the underlying undirected graph excludes afixed ap ex graph Hasaminor.
+Proof.As the first step of the algorithm we apply Lemma 8 and obtain co llectionCsuch that for every
+(D,k,r)∈ C,tw(UG(D))∈ O(√
+k). Then using the constant factor approximation algorithm of
+Demaine et al. [17] for computing the treewidth of a H-minor free graph, we find a tree decomposition
+of widthO(√
+k)forUG(D)in timekO(1). Finally, weapply dynamic programming algorithm running
+in time(√
+k)O(√
+k)= 2O(√
+klogk)on each instance in C. If for any of them we get an yes answer we
+return “yes”, else wereturn “no”. Therunning timeof the alg orithm isbounded by
+|C|·2O(√
+klogk)+nO(1)= 2O(√
+klogk)·2O(√
+klogk)+nO(1)= 2O(√
+klogk)+nO(1).
+We have an additive term of nO(1)as we apply the algorithm only on the O(k2)size kernel. This
+completes the proof.
+5 Conclusion andDiscussions
+We have given the first subexponential parameterized algori thms on planar digraphs and on the class of
+digraphs whose underlying undirected graph excludes afixed graphHor an apex graph as a minor. We
+haveoutlinedtwogeneraltechniques, andhaveillustrated themontwowellstudiedproblemsconcerning
+oriented spanning trees (out branching)— one that maximize s the number of leaves and the other that
+maximizes the number of internal vertices. One of our techni ques uses the grid theorem on H-minor
+graphs, albeit in a different way than how it is used on undire cted graphs. The other uses Baker type
+layering technique combined with kernelization and solves the problem on a subexponential number of
+problems whose instances have sublinear treewidth.
+We believe that our techniques will be widely applicable and it would be interesting to find other
+problemswheresuchsubexponential algorithmsarepossibl e. Twofamousopenproblemsinthiscontext
+are whether the k-DIRECTED PATHproblem (does a digraph contains a directed path of length at least
+k) and the k-DIRECTED FEEDBACK VERTEXSETproblem (does a digraph can be turned into acyclic
+digraph by removing at most kvertices) have subexponential algorithms (at least) on pla nar digraphs.
+However, for the k-DIRECTED PATHproblem, we can reach “almost” subexponential running time .
+More precisely, wehave the following theorem.
+Theorem 4. [⋆]For anyε >0, there is δsuch that the k-DIRECTED PATHproblem is solvable in time
+O((1+ε)k·nδ)ondigraphs with nvertices such that the underlying undirected graph exclude s afixed
+apex graph Hasaminor.
+Let use remark that similar O((1+ε)knf(ε))results can also be obtained for many other problems
+including P LANARSTEINERTREE.
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+126 Appendix
+6.1 ProofofTheorem 2
+Proof.We only give an outline of dynamic programming algorithm for planar digraphs that given a
+tree-decomposition of width wdecides whether Dhas an out-branching with at least kleaves in time
+2O(w)n. Therest of the proof is sameas Theorem 1.
+Tree collections. LetGbe an undirected graph with edge set E(G)and letE′⊆E(G). Let
+S⊆V(G)be a vertex set separating E′fromE(G)\E′, that is,Scontains all vertices incident to at
+least one edge of Eand at least one edge of E(G)\E′. We consider a forest Fwith disjoint trees on
+edgesofE′andeachintersecting atleastonevertexof S. Letusdenotethecollection ofallsuchforests
+Fbyforests E′(S).
+We define an equivalence relation ∼onforests E′(S)as: for two forests F1,F2∈ F,F1∼ F2if
+there is a bijection µ:F1→ F2such that for every tree T∈Fwe have that T∩S=µ(T)∩S.
+Letq-forests(S)denote the cardinality of both, the quotient set of forests E′(S)plus the quotient set
+offorests E′\E(G)(S)by relation ∼. In general, q-forests(S)≤ |S|!. In [20], the authors show for a
+planar graph Gof treewidth whowtodecompose Gbyseparators of size O(w), such that for eachsuch
+separator S,q-forests(S)is bounded by 2O(w). Thesebranch decompositions are very closely related
+to tree decompositions with width parameters bounding each other by constants. Thus, we can simply
+talk about tree decompositions with someadditional struct ure.
+In this case we use standard dynamic programming on such tree decompositions (X,T)(see e.g.
+[25]) At every step of dynamic programming for each node of T, we keep track of all the ways the
+required out-branching can cross the separator Srepresented by X. In other words, we count all the
+ways parts of the out-branching can be routed through E. In the underlying undirected graph, this is
+proportional to q-forests(S). Since an out-branching is rooted, every subtree is rooted, too. Thus, the
+only overhead in the directed case compared to the undirecte d is that we have to guess for each tree
+TFinFif its root is in S. In this case, we guess which of the vertices of TF∩Sis the root. The
+number of guesses is bounded by 2O(w)and hence the dynamic programming algorithm runs in time
+O(2O(w)n).
+6.2 ProofofClaim2
+Proof.Letthearc xydisconnect atleast twovertices yandwfromrandletD′bethedigraph obtained
+fromDby contracting the arc xy. LetTbe anr-out-branching of Dwith at least kleaves. Since every
+path from rtowcontains the arc xy,Tcontainsxyaswell and neither xnoryisaleaf of T. LetT′be
+the tree obtained from Tbycontracting xy.T′is anr-out-branching of D′withat least kleaves.
+Fortheconverse, let T′beanr-out-branching of D′withatleast kleaves. Let x′bethevertexin D′
+obtained by contracting the arc xy, and letube the parent of x′inT′. Notice that the arc ux′inT′was
+initially the arc uxbefore the contraction of xy, since there is no path from rtoyavoidingxinD. We
+obtain an r-out-branching TofDfromT′,byreplacing thevertex x′bythevertices xandyandadding
+the arcsux,xyand arc sets {yz:x′z∈A(T′)∧yz∈A(D)}and{xz:x′z∈A(T′)∧yz /∈A(D)}.
+All these arcs belong to A(D)because all the out-neighbors of x′inD′are out-neighbors either of xor
+ofyinD. Finally, x′must be an internal vertex of T′sincex′disconnects wfromr. HenceThas at
+least as manyleaves as T′.
+6.3 ProofofClaim4
+Proof.LetT∗be anr-out-branching of D′with at least kleaves and containing the maximum number
+of arcs from the set Ac. IfAc⊆A(T∗), then we are through. So let us assume that there is an arc
+e=xy∈Acsuchthat e /∈A(T∗). Noticethat sincetheverticesof S≥2arecutvertices, theyarealways
+internal vertices in any out-branching rooted at rinD. In particular, the vertices of S≥2are internal
+13vertices in T∗. Furthermore by Claim 3 we know that yis an end-point of exactly one arc in Ac. Letz
+betheparent of yinT∗. Nowobtain T∗
+e=T∗\{zy}∪{xy}. Observethat T∗
+econtains atleast kleaves
+and has morearcs from ActhanT∗. Thisis contrary to our assumption that T∗isanr-out-branching of
+D′with at least kleaves and containing the maximum number of arcs from the set Ac. This proves that
+D′has anr-out-branching Twith at least kleaves and containing all the arcs of Ac.
+Observe that starting from any r-out-branching T′ofD′wecan obtain the desired Tin polynomial
+timeby simple arc exchange operations described in theprev ious paragraph.
+6.4 ProofofTheorem 4
+Proof.LetPbe a path of length kin a digraph D. The vertex set of Pcan be covered by at most b
+balls of radius k/bin the metric of UG(D). LetFbe a subgraph of UG(D)induced by the vertices
+contained in bballs of radius k/b. We claim that there is a constant c(depending only on the size of
+the apex graph H), such that tw(F)≤c·k/√
+b. Indeed, because Fis apex minor-free, it contains a
+partially triangulated (d·tw(F)×d·tw(F))-grid as a contraction for some d >0[24]. One needs
+Ω((tw(F)b/k)2)balls of radius k/bto cover such a grid, and hence to cover F[12]. But on the other
+hand,Fis covered by at most bballs of radius k/b, and the claim follows. Byan easy adaptation of the
+algorithm from[18]forundirected H-minor-freegraphs, itispossibletofindintime 2O(tw(F)·nO(1),if
+the subdigraph of Dwith the underlying undirected graph Fcontains a directed path of length k. Thus
+these computations can be done in time 2cH·k/√
+b·nO(1)for some constant cH>0depending only on
+the size of H.
+Putting things together, to check if Dcontains a path of length k(and if yes, to construct such a
+path), we try all possible sets of bverticesBand for each such set we construct a graph Finduced
+by vertices at distance at most k/bfrom vertices of B. IfDcontain a k-path, then this path should be
+covered by at least one such set of bballs. For each such graph, wecheck, if the corresponding di rected
+subgraph contains a k-path. Thetotal running timeof thealgorithm is
+O(/parenleftbiggn
+b/parenrightbigg
+2c·k/√
+b·nc) =O( 2c·k/√
+b·nb+c)
+for some constant c. By putting b= (c/(log(1 + ε))2andδ=b+c, we complete the proof of the
+theorem.
+14
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@@ -0,0 +1,1033 @@
+arXiv:1001.0822v5  [cond-mat.mes-hall]  20 Jan 2011Atomic-scale Field-effect Transistor as a Thermoelectric P ower Generator and
+Self-powered Device.
+Yu-Shen Liu, Hsuan-Te Yao, and Yu-Chang Chen∗
+Department of Electrophysics, National Chiao Tung Univers ity, 1001 Ta Hush Road, Hsinchu 30010, Taiwan
+Using first-principles approaches, we have investigated th e thermoelectric properties and the en-
+ergy conversion efficiency of the paired metal-Br-Al junctio n. Owing to the narrow states in the
+vicinity of the chemical potential, the nanojunction has la rge Seebeck coefficients such that it can
+be considered an efficient thermoelectric power generator. W e also consider the nanojunction in a
+three-terminal geometry, where the current, voltage, powe r, and efficiency can be efficiently mod-
+ulated by the gate voltages. Such current-voltage characte ristics could be useful in the design of
+nano-scale electronic devices such, as a transistor or swit ch. Notably, the nanojunction as a transis-
+tor with a fixed finite temperature difference between electro des can power itself using the Seebeck
+effect.
+I. INTRODUCTION
+In the past decade, considerable concern has arisen
+regarding the transport properties of atomic-scale junc-
+tions, which are the basic building blocks for molecular
+electronics1,2. This concern is motivated by the aspira-
+tion to develop new forms of electronic devices based on
+subminiature structures and by the desire to understand
+the fundamental properties of electron transport under
+non-equilibrium3. A growing number of research studies
+are now available to diversify the scope of electron trans-
+port properties in molecular/atomic junctions, including
+current-voltage characteristics4–6, inelastic electron tun-
+neling spectroscopy (IETS)7–16, shot noise17–20, count-
+ing statistics21, local heating22,23, and gate-controlled ef-
+fects24–28. Substantialprogressin experiment and theory
+has been achieved29–31.
+Recently, growing attention has been paid to the ther-
+mopower of nanojunctions. Pioneering experiments have
+been conducted to measure the Seebeck coefficients at
+the atomic and molecular level32–35. The Seebeck co-
+efficient is related not only to the magnitude, but also
+to the slope of the transmission function in the vicin-
+ity of chemical potentials. Thus, it can provide richer
+information than the current-voltage characteristics re-
+garding the electronic structures of the molecule bridg-
+ing the electrodes36. The thermoelectric effect hybridizes
+the electron and energy transport, which complicates
+the fundamental understanding for quantum transport
+of electron and energy under non-equilibrium conditions.
+This has spurred rapid developments in the fundamental
+thermoelectrictheoryinnanojunctions37–47includingthe
+effects of electron-vibration interactions48–50. The See-
+beck coefficient is typically measured under equilibrium
+condition, non-equilibrium current and inelastic effects
+potentially offer new possibilities of engineering systems
+leading to enhanced the thermopower51,52. The emer-
+genceofthermoelectricnanojunctionsmayalsohavepro-
+found implications on the design of subminiature energy-
+conversion devices, such as nano-refrigerators53–55.
+Thermoelectric power generators in bulk systems em-
+ploy electron gas as a working fluid. It directly converts
+thermal energy into electric energy using the Seebeck ef-fect. Provided a temperature difference is maintained
+across the device, it can generate electric power con-
+vertedfromthe thermal energy. In this paper, weexplore
+the energy conversion mechanism of nanojunctions. We
+present a parameter-free first-principles calculations for
+a thermoelectric power generator in a truly atomic-scale
+system. Togainfurther insightintothemechanismofen-
+ergy conversion, we also developed an analytical theory
+for it. As a specific example, we consider an atomic junc-
+tion depicted in Fig. 1(a), where the left and right elec-
+trodes serve as independent cold- and hot-temperature
+reservoirs, respectively. This is not just an academic
+example since recent studies have demonstrated the ca-
+pability to assemble one magnetic atom on a thin layer
+of atoms on the surface of STM56. A similar technique
+might be applicable to a single Al atom adsorbed onto a
+layerofBratomsonthemetalsurface. Inthisregard,the
+atomic junction could be formed by bringing two iden-
+tical pieces of metal-Br-Al surfaces close together before
+the reconstruction of Br and Al atoms occurs.
+AsitwasfoundinRef.57, thepairedmetal-Br-Aljunc-
+tion shows interesting device properties, such as negative
+differential resistance, that utilize the relatively narrow
+density of states (DOSs) near the chemical potentials.
+The narrow DOSs are due to the weak coupling between
+the Al atoms and the electrodes via the “spacer” Br
+atoms58,59. This junction also serves as an efficient field-
+effect transistor because the narrow DOSs near chemi-
+cal potentials can be easily shifted by the gate voltages.
+As the Seebeck coefficients are relevant to the slope of
+DOSs, the sharp DOSs result in a large magnitude in
+the Seebeck coefficient52. When a temperature differ-
+ence is maintained between electrodes in a closed cir-
+cuit, the Seebeck effect generates an electromotive force
+(emf), which drives a current flowing through the junc-
+tion, hence the nanojunction can also be considered as
+an efficient thermoelectric power generator60.
+We investigate further the possibility of powering an
+atomic-scale device as a field-effect transistor using heat
+instead of electricity. To illustrate this point, we con-
+sider the paired metal-Br-Al junction in a three-terminal
+geometry. When a finite temperature difference is main-
+tainedbetweenelectrodes,theSeebeckeffectconvertsthe2
+FIG. 1: (color online) (a) Schematics of the thermoelectric
+junction. The Br-jellium, Al-Br, and Al-Al distance are 1 .8,
+4.9, and 8 .6 a.u., respectively. The Al-Al distance has been
+significantly enlarged. (b) The directions of electric curr entI
+and thermal current JforS <0. Inside the power generator,
+current travels from the lower to the higher potential, whic h
+is opposite of what occurs when the nanojunction is a passive
+element in a circuit.
+electron’s thermal current into the electric current flow-
+ing through the nanojunction. We observe that the cur-
+rent’s magnitude, polarity, and power on-off are control-
+lable by the gate field. Such current-voltage characteris-
+tics could be useful in the design of nanoscale electronic
+devices, such as a transistor or switch.The nanojunction,
+therefore, canbe consideredatransistorthatemploysthe
+Seebeck effect to power itself by converting the thermal
+energy into electric energy. The results of this study may
+be of interest to researchers attempting to develop new
+styles of thermoelectric nano-devices.
+The flow of the discussion in this paper is as follows.
+In Sec. II, we describe density functional theory, theory
+of thermoelectricity, and theory of thermoelectric power
+generator as a self-powered electronic device. In Sec. III,
+wediscussthe thermoelectricpropertiesofnanojunctions
+and the thermoelectric power generatoras a self-powered
+device. Finally, we conclude our findings in Sec. IV.
+II. THEORETICAL METHODS
+We briefly present an introduction of the Lippmann-
+Schwinger (LS) equation and density-functional theory
+(DFT) in subsection A. In subsection B, we present
+the theory of the Seebeck coefficient, electric conduc-
+tance and thermal conductance. In subsection C, we
+present the theory to calculate the Seebeck-induced elec-
+tric current, voltage, power, and efficiency of energy con-
+version. These quantities are calculated in the trulyatomicscalejunctionintermsofthenonequilibriumscat-
+tering wave-functions obtained self-consistently in the
+framework of DFT+LS calculations. Both DFT+LS and
+DFT+Keldysh nonequilibrium Green’s function (NEGF)
+havebeenappliedextensivelytoawiderangeofproblems
+of nonequilibrium quantum transport of device physics
+problems under finite biases from first-principles ap-
+proaches. The formal connection between DFT+LS and
+DFT+NEGF can be found in Ref. 61.
+A. Density Functional Theory and
+Lippmann-Schwinger Equation
+We model a nanoscale junction as a passive element
+formed by two semi-infinite metal with planar surfaces
+held a fixed distance apart connecting to an external
+battery of bias VB, with a nano-structured object bridg-
+ing the gap between them. The full Hamiltonian of the
+system is H=H0+V, where H0is the Hamiltonian
+due to the biased bimetallic electrodes that we model
+as ideal metal (jellium model), and Vis the scattering
+potential of a group of atoms bridging the gap. We as-
+sume that the left electrode is positively biased such that
+VB= (µR−µL)/e, whereµL=µandµR=µ+eVBare
+the chemical potential deep in the left and right elec-
+trodes, respectively.
+The unperturbed wavefunctions of the biased bimetal-
+lic electrodes have the form, Ψ0,L(R)
+EK/bardbl(r) =eiK/bardbl·r⊥·
+uL(R)
+EK/bardbl(z), where r⊥is the coordinate parallel to the sur-
+faces and zis the coordinate normal to them. Electrons
+are free to move in the plane perpendicular to the z
+direction, and K/bardblis the momentum of electron in the
+plane parallel to the electrode surfaces. Partial charges
+spilled from the electrode positive-background edges into
+the vacuum region. This causes an electrostatic poten-
+tial barrier between two electrodes. The charge density
+distribution and effective single-particle wave functions
+uL(R)
+EK/bardbl(z) are calculated in the framework of the density
+functional formalism by solving the coupled Shr¨ odinger
+and Poisson equations iteratively until self-consistency is
+obtained62. The wave function describes the electrons
+incident from the right electrode, satisfying the following
+boundary condition :
+uR
+EK/bardbl(z) = (2π)−3
+2×/braceleftbigg1√kR(e−ikRz+R eikRz), z→ ∞,
+1√kLT e−ikLz, z→ −∞.
+(1)
+The group of atoms is considered in the scattering
+approaches. Corresponding to each of the unperturbed
+wave functions, a Lippmann-Schwinger equation involv-
+ing a Green’s function for the biased bimetallic junction
+is solved in the plane wave basis, where a basis of ap-
+proximately 2300 plane waves has been chosen for this
+study63:3
+ΨL(R)
+EK/bardbl(r) = Ψ0,L(R)
+EK/bardbl(r)+/integraldisplay
+d3r1/integraldisplay
+d3r2G0
+E(r,r1)V(r1,r2)ΨL(R)
+EK/bardbl(r2), (2)
+where ΨL(R)
+EK/bardbl(r) represents the nonequilibrium scattering
+wave function of the electrons with energy Eincident
+from the left (right) electrode. The quantity G0
+Eis theGreen’s function for the biased bimetallic electrodes and
+V(r1,r2) is the scattering potential electrons experience:
+V(r1,r2) =Vps(r1,r2)+/braceleftbigg
+(Vxc[n(r1)]−Vxc[n0(r1)])+/integraldisplay
+dr3δn(r3)
+|r1−r3|/bracerightbigg
+δ(r1−r2), (3)
+whereVps(r1,r2) is the electron-ion interaction poten-
+tial represented with pseudopotential; Vxc[n(r)] is the
+exchange-correlation potential calculated at the level of
+the local-density approximation; n0(r) is the electron
+density for the pair of biased bare electrodes; n(r) is the
+electron density for the total system; and δn(r) is theirdifference. The wave functions of the entire system are
+calculated iteratively until self-consistency is obtained63.
+These right- and left-moving wave functions, weighed
+with the Fermi-Dirac distribution function according to
+their energies and temperatures, are applied to calculate
+the electric current as:
+I(µL,TL;µR,TR) =e/planckover2pi1
+mi/integraldisplay
+dE/integraldisplay
+dr⊥/integraldisplay
+dK/bardbl/bracketleftBig
+fE(µR,TR)IRR
+EE,K||(r)−fE(µL,TL)ILL
+EE,K||(r)/bracketrightBig
+, (4)
+whereIRR(LL)
+EE′,K||(r) =/bracketleftBig
+ΨR(L)
+E,K||(r)/bracketrightBig∗
+∇ΨR(L)
+E′,K||(r)−
+∇/bracketleftBig
+ΨR(L)
+E,K||(r)/bracketrightBig∗
+ΨR(L)
+E′,K||(r) anddr⊥represents an ele-
+ment of the electrode surface. Here, we assume that
+the left and right electrodes are independent electron
+reservoirs, with the electron population described by the
+Fermi-Dirac distribution function, fE(µL(R),TL(R)) =
+1/{exp[/parenleftbig
+E−µL(R)/parenrightbig
+/(kBTL(R))] + 1}, whereµL(R)and
+TL(R)are the chemical potential and the temperature
+in the left (right) electrode, respectively. More detailed
+descriptions of theory can be found in Refs. 5,63,64.
+The above expression can be cast in a Landauer-
+B¨ uttiker formalism:
+I(µL,TL;µR,TR) =2e
+h/integraldisplay
+dEτ(E)[fE(µR,TR)−fE(µL,TL)],
+(5)whereτ(E) =τR(E) =τL(E) is a direct consequence of
+the time-reversal symmetry, and τR(L)(E) is the trans-
+mission function of the electron with energy Eincident
+from the right (left) electrode,
+τR(L)(E) =π/planckover2pi12
+mi/integraldisplay
+dr⊥/integraldisplay
+dK||IRR(LL)
+EE,K||(r).(6)
+Note that eis positive in the definition of Eq. (5).
+The electron’s thermal current, defined as the rate at
+which thermal energy flows from the right (into the left)
+electrode, is
+JR(L)
+el(µL,TL;µR,TR) =2
+h/integraldisplay
+dE(E−µR(L))τ(E)[fE(µR,TR)−fE(µL,TL)], (7)
+respectively. B. Theory of the Thermoelectric Properties in the
+Nanoscale Junctions
+We assume that the nanojunction is not connected to
+external circuit such that µR=µL=µandTR=TL=4
+T, and then weconsideran infinitesimal current[( dI)T=
+I(µ,T;µ,T+dT)]induce byan infinitesimal temperature
+difference dTraisedin the rightelectrode[i.e., TR=TL+
+dT]. The Seebeck effect generates a voltage difference
+dVin the right electrode [i.e., µR=µL+dV] which
+drives current [( dI)V=I(µ,T;µ+edV,T)]. The current
+cannot actually flow, thus, ( dI)Tcounterbalances ( dI)V
+[i.e.,dI= (dI)T+ (dI)V= 0]. The Seebeck coefficient
+(defined as S=dV
+dT) can be obtained by expanding the
+Fermi-Dirac distribution functions in ( dI)Tand (dI)Vto
+the first order in dTanddV:
+S=−1
+eTK1(µ,T)
+K0(µ,T), (8)
+where
+Kn(µ,T) =−/integraldisplay
+dEτ(E)(E−µ)n∂fE(µ,T)
+∂E.(9)
+The Seebeck coefficient up to the lowest order in temper-
+atures is,
+S≈αT, (10)
+whereα=−π2k2
+B∂τ(µ)
+∂E/(3eτ(µ))47. Here, we have ex-
+pandedKn(µ,T) to the lowest order in temperatures
+by using Sommerfeld expansion: K0≈τ(µ),K1≈
+[π2k2
+Bτ′(µ)/3]T2, andK2≈[π2k2
+Bτ(µ)/3]T2. The See-
+beck coefficient is positive (negative) when the slope
+of transmission function is negative (positive) near the
+chemical potential.
+When a finite temperature difference ∆ T=TR−TL>
+0 is raised in the right electrode, the Seebeck effect gen-
+erates a finite electromotive force, emf=|∆V(TL,TR)|,raising the potential energy of the charge. We assume
+that the voltage difference ∆ V(TL,TR) is applied to the
+right electrode such that µ=µLandµR=µL+
+e∆V(TL,TR). The voltage difference ∆ V(TL,TR) gener-
+ated by the temperature difference ∆ Tis approximately
+given by,
+∆V(TL,TR)≈/integraldisplayTR
+TLS(µL,T)dT, (11)
+where the Seebeck coefficient S(µ,T) is given by Eq. (8)
+withµ=µL. Equation (11) becomes exact when ∆ T→
+0. The sign of ∆ V(TL,TR) depends on the sign of the
+Seebeck coefficient S(µ,T). For example, ∆ V(TL,TR)<
+0 forS <0, as shown in Fig. 1(b).
+Equation (11) is an approximation when ∆ Tis finite.
+We have to remind ourselves of the fact that the left
+chemical potential µLdiffers from the right chemical po-
+tentialµRwhen the temperature difference ∆ Tis large.
+In this case, the Seebeck coefficient becomes relevant to
+bothµLandµRwith a more complicated form52,
+S(µL,TL;µR,TR) =−1
+eK1(µL,TL)/TL+K1(µR,TR)/TR
+K0(µL,TL)+K0(µR,TR),
+(12)
+whereK0(µL(R),TL(R)) andK1(µL(R),TL(R)) are given
+by Eq. (9). A more elaborate evaluation of the voltage
+difference ∆ V(TL,TR) can be performed using,
+∆V(TL,TR)≈/integraldisplay∆T/2
+0S(µ−e∆V(T−T,T+T)
+2,T−T;µ+e∆V(T−T,T+T)
+2,T+T)dT, (13)
+whereT= (TL+TR)/2, ∆T=TR−TLandµ=EF.
+The induced voltage by the Seebeck effect using
+Eq.(11)and(13)havebeen numericallyverified, showing
+thatEq.(11)isagoodapproximationwhenthetransmis-
+sion function τ(E) does not change rapidly around the
+chemical potentials and when ∆ Tis not too large. Ac-
+cordingly,weevaluate∆ V(TL,TR)throughoutthisstudy
+using Eq. (11) instead of Eq. (13) for simplicity.
+Corresponding to the electric current, the extra elec-
+tron’s thermal current induced by an infinitesimal tem-
+perature ( dT) and voltage ( dV) across the junctions is,
+dJel= (dJel)T+(dJel)V. (14)
+where (dJel)T=Jel(µ,T;µ,T+dT) and (dJel)V=
+Jel(µ,T;µ+edV,T) can be calculated by Eq. (7).
+The electron’s thermal conductance (defined as kel=dJel/dT) can be decomposed into two components:
+κel(µ,T) =κT
+el(µ,T)+κV
+el(µ,T),(15)
+whereκT
+el= (dJel)T/dTandκV
+el= (dJel)V/dT. Using
+Sommerfeld expansion, they can be can be written as,
+κT
+el(µ,T) =2
+hK2(µ,T)
+T, (16)
+and
+κV
+el(µ,T) =2e
+hK1(µ,T)S(µ,T), (17)
+whereK1(µ,T) andK2(µ,T) are given by Eq. (9). One
+should note that κV
+el= 0 if the Seebeck coefficient of the
+system is zero.5
+FIG. 2: (color online) The circuit diagram of the thermoelec -
+tric power generator depicted in Fig. 1(b) as a self-powered
+electronic device. The thermoelectric nanojunction can be
+represented as an ideal battery of emf=VABgenerated by
+the Seebeck effect and an internal resistance rfor the nano-
+junction as an electronic device. We assume that the externa l
+resistorR→0 in this study for simplicity.
+Byexpanding Kn(µ,T)andS(µ,T)tothelowestorder
+in temperatures, Eqs. (16) and (17) become
+κV
+el≈βVT3andκT
+el≈βTT, (18)
+whereβV=−2π4k4
+B[τ′(µ)]2/[9hτ(µ)] and βT=
+2π2k2
+Bτ(µ)/(3h). In the above expansions, we have
+applied the Sommerfeld expansions: K1(µ,T)≈
+[π2k2
+Bτ′(µ)/3]T2,K2(µ,T)≈[π2k2
+Bτ(µ)/3]T2, and
+Eq. (10).
+In the limit of zero bias, Eq. (5) yields the electric
+conductance (defined as σ(T) =dI/dV),
+σ(T) =2e2
+h/integraldisplay
+dEfE(µ,T)[1−fE(µ,T)]τ(E)/(kBT),
+(19)
+which is relatively insensitive to temperatures if tunnel-
+ing is the major transport mechanism.
+C. Theory of the Thermoelectric Power Generator
+as a Self-powered Electronic Device
+In this subsection, we present a closed-circuit theory
+of the thermoelectric nanojunction as a power genera-
+tor and a self-powered electronic device by itself. We
+consider the nanojunction with the Seebeck coefficientS <0 and ∆T=TR−TL>0 such that the open-circuit
+Seebeck-effect induced emfis|∆V(TL,TR)|, as depicted
+in Fig. 1(b). Figure 1(b) can be represented as a cir-
+cuit diagram shown in Fig. 2, where the thermoelectric
+junction is considered to consist an ideal battery of emf
+plusaninternalresistance r. Thethermoelectricjunction
+employs the Seebeck effect to generate an emfand con-
+verts thermal energy into electric energy. Concurrently,
+the nanojunction itself serves as a passive element with
+resistance rwhich consumes electric energy.
+We choose clockwise as positive such that the induced
+currentI >0 forS <0. Beginning at point Aas the
+current traverses the circuit in the positive direction, we
+obtain from Kirchhoff’s loop rule,
+VAB−Ir−IR= 0, (20)
+where the open-circuit emfisVAB=−∆V(TL,TR)>0
+forS <0,ris the internal resistance of the nanojunc-
+tion, and Ris the resistance of the external resistor, as
+depicted in Fig. 2. Note that the current Iencounters
+a potential increase due to the source of emfbetween
+pointsAandB, a potential drop [ VBC=Ir] due to
+the internal resistance between points BandC, and a
+potential drop [ VCA=IR] as the current traverses the
+external resistor with resistance Rbetween points Cand
+A, respectively.
+In particular, we carry out this research considering
+R→0 such that the voltage drop between points Cand
+AisVCA=IR= 0. It yields
+VAB=Ir. (21)
+It implies that external circuit does not consume electric
+energy because the potential difference across two ter-
+minals of the thermoelectric power generators (i.e., the
+terminal voltage) is zero due to R→0. This simplified
+scenario helps us to analyze the detailed mechanism of
+energy conversion between the thermal current and elec-
+tric current. Such analysis will provide us an insight into
+the device physics of the thermoelectric junctions as self-
+powered electronic devices.
+The current I=σVABcan be evaluated by Eq. (21),
+whereVABis the Seebeck-effect induced emf=
+|∆VTL,TR|[which is evaluated by Eq. (12)] and σ≈1/ris
+the conductance of nanojunction [which is evaluated by
+Eq. (19)]. We have numerically verified and confirmed
+thatI= (I)∆V= (I)∆T, where
+(I)∆V=−2e
+h/integraldisplay
+dEτ(E)[fE(µL+e∆V(TL,TR),TL)−fE(µL,TL)], (22)
+and
+(I)∆T=2e
+h/integraldisplay
+dEτ(E)[fE(µL,TL+∆T)−fE(µL,TL)].
+(23)The above two equationshave the sign conventioncon-
+formed to the direction of current depicted Fig. 1(b) and6
+Fig. 3 for S <0. We may interpret ( I)∆Tas the cur-
+rent which traverses the thermoelectric nanojunction as
+a ”pure” source of emfgenerated by the temperature
+difference ∆ Tusing the Seebeck effect. Similarly, ( I)∆V
+is the current which traverses the nanojunction as an in-
+ternal resistor. Equations (22) and (23) describe the flow
+of electrons associated with the probability flux.The flow of probability density can also transport en-
+ergy as electrons travel between two electrodes. The
+electrons that travel with energy Efrom the right (left)
+electrode carry an amount of energy ( E−µR(L)). Cor-
+respondingly, the ∆ V-induced energy current carried by
+the probability current flowing out of the right electrode
+(into to the left electrode) is,
+(JR(L)
+el)∆V=−2
+h/integraldisplay
+dE(E−µR(L))τ(E)[fE(µL+e∆V(TL,TR),TL)−fE(µL,TL)], (24)
+whereµR=µL+e∆V(TL,TR) and ∆V(TL,TR) is given
+by Eq. (11). Similarly, the ∆ T-induced electron’s ther-
+malcurrentcarriedbytheprobabilitycurrentflowingout
+of the right electrode (into to the left electrode) is,
+(Jel)∆T=2
+h/integraldisplay
+dE(E−µL)τ(E)[fE(µL,TL+∆T)−fE(µL,TL)].
+(25)
+Note that ( JR
+el)∆Vand (JL
+el)∆Vare positive and
+(JR
+el)∆V>(JL
+el)∆VforS <0. The sign convention de-
+finedinEqs.(22)-(25)complywiththe directiondepicted
+in Fig. 1(b) and Fig. 2 for S <0.
+Subtracting ( JL
+el)∆Vfrom (JR
+el)∆V, we obtain
+∆P= (JR
+el)∆V−(JL
+el)∆V>0, (26)
+where ∆Pis the electric power delivered by the thermo-
+electric junction served as a ”pure” battery which em-
+ploys the Seebeck effect to convert the electron’s ther-
+mal current into electric energy. If the external resistor
+with resistance R→0, then the external circuit does
+not consume electric energy. In this case, the nanojunc-
+tion simultaneously serves as an electronic device which
+consumes the entire electric power generated by itself,
+∆P≈σ(∆V)2≈/integraldisplayTR
+TLSdT/integraldisplayTR
+TLSσdT. (27)
+Note that Eq. (27) can alternatively be expressed as,
+∆P=−∆V(TL,TR)·I(TL,TR)>0.(28)
+Eqs. (26)-(28) have been numerically verified to be con-
+sistent with each other. If the system is n-type (i.e.,
+S <0), then ∆ V(TL,TR)<0 andI(TL,TR)>0, as
+shown in Fig. 1(b). Similarly, if the system is p-type
+(i.e.,S >0), then ∆ V(TL,TR)>0 andI(TL,TR)<0.
+Both cases render ∆ P >0.
+III. RESULTS AND DISCUSSION
+Provided that a finite temperature difference is raised
+between electrodes, the Seebeck effect will generate an
+electric current. It is observed that the nanojunction it-
+self can be deemed a field-effect transistor, where gatefield can control the current. Thus and so, the atomic
+junction in three-terminal geometry can be considered
+as an electronic device which can be self-powered by pro-
+viding a temperature difference across the junction.
+As an a specific example, we consider an atomic-scale
+junction in a three-terminal geometry as a thermoelec-
+tric power generator, depicted in Fig. 6(a). In the paired
+metal-Br-Al junction, we assume that the distance be-
+tween two Al atoms is sufficiently large. The interac-
+tion between two pieces of metal-Br-Al is minimal, and
+in this fashion, no rearrangement of atom positions oc-
+curs. Accordingly, the phonon’sthermalcurrentcould be
+suppressed because of poor mechanical coupling between
+two pieces of metal-Br-Al is minimal. We neglect it for
+simplicity in this study.
+A. Thermoelectric Properties of the Junction
+In an open circuit, an infinitesimal temperature dT
+raised in the right electrode induces ( dI)Tand (dI)V
+which counterbalance each other. This renders the See-
+beck coefficient defined in Eq. (8), as described in sub-
+section IIB.
+Correspondingly, the electron’s thermal conductance
+is defined as κel=dJel/dT, as given by Eq. (15). The
+electron’s thermal conductance can be decomposed into
+two components, κel=κT
+el+κV
+el, whereκT
+elandκV
+elare
+given by Eqs. (16) and (17), respectively.
+Fig. 3(a) shows that the system is characterized by a
+sharp transmission function corresponding to a narrow
+DOSs near the chemical potential. As it was found in
+Ref. 57, the DOSs become narrower when the distance
+betweentwoAlatomsincreases. ThenarrowDOSsresult
+in substantial Seebeck coefficients, as shown in Fig. 3(b).
+It shows that the thermoelectric junction is n-type ( S <
+0) since the narrow DOSs are slightly above the chem-
+ical potential and the slope of the transmission τ′(µ) is
+positive. At low temperatures (about T <100 K), the
+Seebeck coefficients remain linear ( S≈αT), as described
+in Eq. (10). We present the absolute values of κT
+eland
+κV
+elas functions of temperatures in Fig. 3(c) and (d),
+respectively. At low temperatures, κV
+el≈ −βVT3and
+κT
+el≈βTT, as shown in Eq. (18). The inset of Fig. 3(c)7
+/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s49/s52/s48/s45/s55/s48/s48
+/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s50/s120 /s49/s48/s45/s51
+/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s55/s120 /s49/s48/s45/s52
+/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48/s48/s49/s48/s48/s50/s48/s48
+/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48/s49/s69/s45/s53/s49/s69/s45/s51/s48/s46/s49
+/s83/s40 /s86/s47/s75/s41
+/s84/s40/s75/s41/s86 /s101/s108/s40/s69/s114/s103/s83/s45/s49
+/s75/s45/s49
+/s41
+/s84 /s101/s108/s40/s69/s114/s103/s83/s45/s49
+/s75/s45/s49
+/s41
+/s84/s40/s75/s41/s84/s40/s75/s41/s40/s50/s101/s50
+/s47/s104/s41
+/s84/s40/s75/s41/s40/s97/s41/s40/s98/s41
+/s40/s99/s41/s40/s100/s41/s69/s110 /s101/s114/s103/s121 /s40/s101/s86/s41/s68/s79/s83/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
+/s69/s110 /s101/s114/s103/s121/s40/s101/s86/s41
+FIG. 3: (color online) (a) The DOSs (inset: transmission
+functions) as afunction of energies; (b) theSeebeck coeffici ent
+S; (c) the thermal conductance induced by the temperature
+difference ( κV
+el) (inset: conductance σ); and (d) the thermal
+conductance induced by the voltage difference ( κT
+el) as a func-
+tion of temperatures.
+shows the zero-bias electric conductance as a function of
+temperatures calculated from Eq. (19).
+B. The Junction as a Thermoelectric Power
+Generator and a Self-powered Electronic Device
+Suppose that the junction is not connected to battery
+(i.e.,µR=µL=µ) andTR=TL=T. Provided that
+a finite temperature difference ∆ T=TR−TLis raised
+in the right electrode (i.e., TL=TandTR=T+∆T),
+the Seebeck effect generates a finite electromotive force
+emf=|∆V(TL,TR)|, where ∆ V(TL,TR) is given by
+Eq. (11). This voltage difference induces a net elec-
+tric current I(TL,TR) which travels from lower to higher
+potential inside the power generator, as described in
+Eq. (22) and Fig. 1(b). Correspondingly, the electron’s
+thermalcurrent( JR
+el)∆V[Eq.(24)]absorbsenergyin heat
+from the hot (right) electrode, produces electric power
+∆P[Eq. (26)] using the Seebeck effect, and then gives
+up energy in heat to the cold (left) electrode via ( JL
+el)∆V
+[Eq. (24)], as described in Fig. 1(b) for S <0.
+To gain further insight into the mechanism of energy
+conversion, we theoretically analyze the electron’s ther-
+malcurrent. ApplyingSommerfeld expansionto/parenleftbig
+JR
+el/parenrightbig
+∆V
+and/parenleftbig
+JL
+el/parenrightbig
+∆Vin Eq. (24), we obtain
+/parenleftbig
+JR
+el/parenrightbig
+∆V≈σ(∆V)2T/∆T+σ(∆V)2/2,(29)
+and
+/parenleftbig
+JL
+el/parenrightbig
+∆V≈σ(∆V)2T/∆T−σ(∆V)2/2,(30)
+where we have simplified Eqs. (29) and (30) us-
+ing the following approximations: τ(µ+e∆V)≈FIG. 4: (color online) (a) and (b) are the schematic rep-
+resentation of the power generator for ( JL
+el)∆V>0 and
+(JL
+el)∆V<0, respectively. Each electrode provides one half of
+the electric power ∆ P= (JR
+el)∆V−(JL
+el)∆Vusing the Seebeck
+effect. The thermal currents JR
+phand (Jel)∆Tare driven by
+the temperature difference ∆ Tand both are not converted
+into electric energy.
+τ(µ),τ′(µ+e∆V)≈τ′(µ),σ≈2e2τ(µ)/h,S≈
+−π2k2
+B∂τ(µ)
+∂ET/(3eτ(µ)), and ∆ V≈S∆T.
+Equations. (29) and (30) immediately provides the law
+of energy conservation as further described below. The
+thermoelectric junction serves as an ideal source of emf
+which delivers electric power ∆ P=/parenleftbig
+JR
+el/parenrightbig
+∆V−/parenleftbig
+JL
+el/parenrightbig
+∆Vconverted from the electron’s thermal currents using the
+Seebeck effect. Concurrently, the nanojunction itself
+servesasapassiveelementwhichconsumeselectricpower
+dissipated by the internal resistor at a rate of σ(∆V2).
+Equations (29) and (30) imply that each electrode ap-
+proximately provides one half of the electric power us-
+ing the Seebeck effect and the electric power is mainly
+converted from ( Jel)∆V. No energy conversion is pos-
+sible when the Seebeck coefficients is vanishing because/parenleftbig
+JR
+el/parenrightbig
+∆V= 0. We note that JR
+el= (JR
+el)∆V+(Jel)∆Tre-
+moves heat energy from the hot electrode, converts heat
+energy into electric energy, and rejects waste heat to the
+cold electrode by JL
+el. Both (Jel)∆TandJR
+phdo not play
+active role in the energy conversion. The above discus-
+sions are summarized in Fig. 4(a).
+The energy current/parenleftbig
+JL
+el/parenrightbig
+∆Vflowing into the cold elec-
+trode could be negative when ∆ Tis sufficiently large, as
+shown in Fig. 4(b). To show this, we integrate Eq. (11)
+using Eq. (10), K1(µ,T)≈[π2k2
+Bτ′(µ)/3]T2,K2(µ,T)≈
+[π2k2
+Bτ(µ)/3]T2. We obtain ∆ V≈αT2[(∆T/T)2+8
+2(∆T/T)]/2, where α=−[π2k2
+B∂τ(µ)
+∂E/(3eτ(µ))]. To-
+gether with Eq. (30), we arrive at
+/parenleftbig
+JL
+el/parenrightbig
+∆V≈σST∆V{1−[(∆T/T)+(∆T/T)2]/4},(31)
+from which we observe that/parenleftbig
+JL
+el/parenrightbig
+∆Vchanges sign at
+∆T≈1.236T(which is universal), and/parenleftbig
+JL
+el/parenrightbig
+∆V<0
+when ∆T/greaterorapproxeql1.236T. Fig. 4(a) and 4(b) illustrate the
+schematicrepresentationofenergyconversionforpositive
+and negative/parenleftbig
+JL
+el/parenrightbig
+∆V, respectively. Fig. 4(a) and 4(b)
+illustrate the schematic representation of energy conver-
+sion for positive and negative/parenleftbig
+JL
+el/parenrightbig
+∆V, respectively.
+Let us focus on the energy conversion efficiency ηel
+defined as the ratio of the electric power ∆ Pgenerated
+by the Seebeck effect to the electron’s thermal current
+JR
+el= (JR
+el)∆V+ (Jel)∆Twhich removes thermal energy
+from the high temperature reservoir:
+ηel=∆P
+JR
+el, (32)
+where ∆ P, (JR
+el)∆V, and (Jel)∆Tare given by Eq. (28),
+(24), and (25), respectively. In the above definition, we
+have neglect other effects [ e.g.the phonon’s thermal cur-
+rentJR
+phand possible photon radiation] which transfer
+energy from the hot to cold electrodes. From this view-
+point, the energy conversion efficiency presented in this
+study may be overestimated. We note that the size of
+the paired metal-Br-Al system is small. This may lead
+to suppression of the photon radiation because photon
+radiation is proportional the surface area. Moreover, the
+paired metal-Br-Al system has a poor mechanical link
+between electrodes when the distance of two Al atoms
+is sufficiently far apart. The poor mechanical coupling
+between two phonon reservoir could lead to suppression
+of the phonon’s thermal current. These two features are
+helpful to increase the energy conversion efficiency. Note
+that the phonon’s thermal current and photon radiation
+do not affect the magnitudes of the current and electric
+power generated by the Seebeck effect.
+To show that the atomic junction can be considered
+as a field-effect transistor which can be self-powered, we
+consider the nanojunction in a three-terminal geometry
+with a finite temperature ∆ T= 100 K maintained be-
+tween electrodes (where TL= 200 K and TR= 300 K).
+The gate field is introduced as a capacitor composed of
+two parallel circular charged disks separated at a cer-
+tain distance from each other. The axis of the capacitor
+is perpendicular to the transport direction. One plate
+is placed close to the nano-object while the other plate,
+placed far away from the nano-object, is set to be the
+zero reference field24–27.
+The nanojunction can be considered as a field-effect
+transistor which is powered by itself via the tempera-
+ture difference maintained between electrodes using the
+Seebeck effect. We observe that the gate voltages can
+shift the narrow DOSs and transmission function near
+the chemical potential, as shown in Fig. 5. Figure 6
+shows the induced potential ∆ V(TL,TR) [upper panel;
+given by Eq. (11)] and current I(TL,TR) [lower panel;/s45/s49/s48 /s45/s53 /s48/s86
+/s71/s61/s32/s45/s55/s46/s57/s56/s32/s86/s86
+/s71/s61/s32/s45/s51/s46/s57/s57/s32/s86/s86
+/s71/s61/s32/s45/s50/s46/s51/s57/s32/s86/s86
+/s71/s61/s32/s48/s32/s86/s86
+/s71/s61/s32/s48/s46/s56/s32/s86/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
+/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s86
+/s71/s61/s32/s49/s46/s53/s57/s32/s86
+/s32
+FIG. 5: (color online) Transmission functions as functions of
+energies for various gate voltages ( VG= -7.98, -3.99, -2.39, 0,
+0.8, 1.59 V). The Fermi level is set to be zero of energy.
+given by Eq. (22)] for TL= 200 K and TR= 300 K
+as functions of gate voltages. We observe that the ex-
+tremal of ∆ V(TL,TR) andI(TL,TR) do not occurs at
+the same value of VGbecause the gate voltage also
+modulates the conductance of the junction [note that
+σ=I(TL,TR)/∆V(TL,TR)]. Figure 6 demonstrates that
+thegatefield canefficientlycontrolthe magnitude, polar-
+ity, and power on-offof the voltage and current induced
+by the Seebeck effect. Such current-voltage characteris-
+tics could be useful in the design of nanoscale electronic
+devices such as a transistor or switch. Note that the in-
+duced voltage and induced current described in Fig. 6
+comply with the sign convention described in Fig. 1(b).
+Figure 7 shows the electric power ∆ P[upper panel;
+givenbyEq.(26)]deliveredbythebatteryandtheenergy
+conversion efficiency ηel[lower panel; given by Eq. (32)]
+as functions of the gate voltages. It illustrates that the
+gate field is capable of controlling and optimizing the
+electric power and energy conversion efficiency ηel. The
+energy conversion efficiency ηel≈0.09 if optimized by
+the gate field. The optimized power is about 3 .5 nW at
+VG≈ −2 V. We also observe that the maximum values of
+electric power and energy conversion efficiency occur at
+different voltages, similar to the case of induced voltage
+and current. If a macroscopic system can be formed by
+twoidenticalpiecesofsurfaces,eachofwhichisformedby
+a layer of Al atoms adsorbing onto metal surface coated
+with a layer of Br atoms as spacer. The macroscopic sys-
+tem could be equivalent to more than 1012single paired
+metal-Br-Al-Al-Br-M junctions which are connected in9
+/s45/s48/s46/s48/s49/s48/s45/s48/s46/s48/s48/s53/s48/s46/s48/s48/s48
+/s45/s53 /s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52
+/s32/s73/s110/s100/s117/s99/s101/s100/s32/s86/s111/s108/s116/s97/s103/s101/s32/s40/s86/s41/s65/s108
+/s66/s114
+/s40/s98/s41
+/s40/s99/s41/s84
+/s82
+/s86
+/s71/s32/s40/s86/s41/s67/s117/s114/s114/s101/s110/s116/s40 /s65/s41/s84
+/s76/s86
+/s71/s40/s97/s41
+/s66/s114/s65/s108
+FIG. 6: (color online) (a) Schematic of the three-terminal
+junction. The Seebeck-effect induced (b) voltage V(TL,TR)
+and (c) current I(TL,TR) as functions of gate voltages via the
+temperature difference ∆ T=TR−TL, whereTL= 200 K and
+TR= 300 K.
+parallel on 1 cm2surface. We conjecture that the power
+could be substantial in such a macroscopic system.
+Finally, we investigate the efficiency of energy conver-
+sionηelas a function of TC(=TL) and ∆T=TR−TL
+forVG= 0 , as shown in Fig. 8. The paired metal-Br-
+Al junction shows sufficiently largeefficiency around 0 .05
+whenTC= 300 K and ∆ T= 60 K. Such a high efficiency
+is attributed to the enhanced the Seebeck coefficients.
+IV. CONCLUSIONS
+In summary, we developed a theory for thermoelec-
+tric power generator in the truly atomic scale system.
+The theory is general to any atomic/molecular junction
+where electron tunneling is the major transport mecha-
+nism. As an example, we investigated the thermoelec-
+tric properties and the efficiency of energy conversion of
+the paired metal-Br-Al junction from first-principles ap-
+proaches. Owing to the narrow states near the chemi-
+cal potentials, the nanojunction has large Seebeck coef-
+ficients; thus, it can be considered as an efficient ther-
+moelectric power generator. Provided that a finite tem-/s45/s53 /s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s50/s52
+/s40/s98/s41
+/s69/s102/s102/s105/s99/s105/s101/s110/s99/s121 /s32
+/s101/s108/s32
+/s32
+/s86
+/s71/s32/s40/s86/s41/s69/s108/s101/s99/s116/s114/s105/s99/s32/s112/s111/s119/s101/s114/s32/s40/s110/s87/s41/s40/s97/s41
+FIG. 7: (color online) (a) The electric power ∆ P, converted
+from the thermal energy using the Seebeck effect, as function s
+of gate voltages. (b) the corresponding efficiency of energy
+conversion ηelvs.VG. The temperatures of electrodes are
+maintained at TL= 200 K and TR= 300 K
+/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48
+/s49/s48/s48/s50/s48/s48/s51/s48/s48
+/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52
+/s32
+/s84
+/s67/s40/s75/s41/s101/s108
+/s84/s40/s75/s41/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52
+FIG. 8: (color online) Efficiency of energy conversion ηelas
+a function of TL=TCand ∆T=TR−TL.
+perature difference is maintained between electrodes, the
+thermoelectric power generator converts thermal energy
+into electric energy. The optimized electric power gener-
+atedbytheSeebeckeffectisabout3 .5nWatTL(R)= 200
+(300) K at VG≈ −2 V. To gain further insight into the
+mechanism of energy conversion, we investigate the elec-
+tron’s thermal currents analytically. The electron’s ther-
+mal current which removes heat from the hot tempera-
+ture reservoir can be decomposed into two components,
+JR
+el=/parenleftbig
+JR
+el/parenrightbig
+∆V+ (Jel)∆T. Only/parenleftbig
+JR
+el/parenrightbig
+∆Vis capable of10
+converting energies. The electron’s thermal current re-
+movesheatfromthehotreservoirvia/parenleftbig
+JR
+el/parenrightbig
+∆Vandrejects
+waste heat into the cold reservoir via/parenleftbig
+JL
+el/parenrightbig
+∆V. No en-
+ergy conversion is possible when the Seebeck coefficients
+is vanishing.
+We also consider the nanojunction in a three-terminal
+geometry, where the current, voltage, and electric power
+can be modulated by the gate voltages, which shift the
+states of the junction. We observe that the gate field can
+control the magnitude, power on-off, and polarity of the
+induced current and voltages generated by the Seebeck
+effect. Such current-voltage characteristics could be use-
+ful in the design of nanoscale electronic devices such asa transistor or switch. Notably, the nanojunction as a
+transistor with a fixed finite temperature difference be-
+tweenelectrodescanpoweritselfusingthe Seebeckeffect.
+The results of this study may be of interest to researchers
+attempting to develop new forms of thermoelectric nano-
+devices.
+The authors thank Ministry of Education, Aiming
+for Top University Plan (MOE ATU), National Center
+for Theoretical Sciences (South), and National Science
+Council(Taiwan)forsupportunderGrantsNSC97-2112-
+M-009-011-MY3, 098-2811-M-009-021, and 97-2120-M-
+009-005.
+∗Electronic address: yuchangchen@mail.nctu.edu.tw
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\ No newline at end of file
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+arXiv:1001.0823v3  [math.GT]  8 Feb 2012CONNECTIVITY OF COMPLEXES OF SEPARATING CURVES
+EDUARD LOOIJENGA
+In memory of Fritz Gr¨ unewald (1949-2010)
+ABSTRACT . We prove that the separating curve complex of a closed ori-
+entable surface of genus gis(g−3)-connected. We also obtain a connec-
+tivity property for a separating curve complex of the open su rface that is
+obtained by removing a finite set from a closed one, where it is assumed
+that the removed set is endowed with a partition and that the s eparating
+curves respect that partition. These connectivity stateme nts have implica-
+tions for the algebraic topology of the moduli space of curve s.
+1. S TATEMENTS OF THE RESULTS
+LetSbe a connected oriented surface of genus gwith finite first Betti
+number2g+n(i.e., a closed surface with npunctures) and make the cus-
+tomary assumption that Shas negative Euler characteristic: if g=0, then
+n≥3and ifg=1, thenn≥1. We recall that the curve complex C(S)of
+Sis the simplicial complex whose vertex set consists of the isotopy c lasses
+of embedded (unoriented) circles in Swhich do not bound in Sa disk or a
+cylinder. A finite set of vertices spans a simplex precisely whe n its elements
+can be represented by embedded circles that are pairwise disjoin t. Thus, a
+closed 1-dimensional submanifold AofSwithk+1connected components
+such that every connected component of its complement has negativ e Euler
+characteristic defines a k-simplex σAofC(S)and every simplex of C(S)is
+thus obtained.
+This complex has proven to be quite useful in the study of the mappi ng
+class group of S. For the purposes of studying the Torelli group of Sa sub-
+complex Csep(S)ofC(S)can render a similar service. It is defined as the full
+subcomplex of C(S)spanned by the separating vertices of C(S), where a ver-
+tex is called separating if a representative embedded circle separates Sinto
+two components. Our main result for the case when Sis closed is contained
+in the following theorem.
+Theorem 1.1 (Ag).Ifn≤1, then the simplicial complex Csep(S)is(g−3)-
+connected.
+2010 Mathematics Subject Classification. 57N05, 55U15.
+Key words and phrases. separating curve complex.
+Support by the Mathematical Research Center at Tsinghua Uni versity is gratefully
+acknowledged.
+12 EDUARD LOOIJENGA
+Previous work on this topic that we are aware of concerns the case n=0.
+Farb and Ivanov announced in 2005 [1, Thm. 4] that Csep(S)is connected for
+g≥3. Putman gave in [5, Thm. 1.4] another proof of this and showed that
+Csep(S)is simply connected for g≥4(op. cit. , Thm. 1.11). In that paper
+he also mentions that Hatcher and Vogtmann have proved that Csep(S)is
+⌊1
+2(g−3)⌋-connected for all g(unpublished).
+Remark 1.2.Possibly the connectivity bound in Theorem 1.1 is the best pos-
+sible for every positive genus. In a paper with Van der Kallen [3] we showed
+that the quotient of Csep(S)by the action of the Torelli group of Shas the ho-
+motopy type of a bouquet of (g−2)-spheres. In the situation of the theorem,
+Csep(S)has dimension 2g−4+nso that its connectivity is half the dimension
+(asn≤1). In particular, we cannot conclude that Csep(S)is spherical.
+Before we state a version for the case n≥2, we point out a consequence
+that pertains to the moduli space of curves. Consider the Teichm ¨ uller space
+T(S)ofS, a contractible manifold on which acts the mapping class group
+Γ(S). The action is proper and the orbit space may be identified with the
+moduli space Mgof curves of genus g. The Harvey bordification ofT(S),
+here denoted by T(S)+⊃ T(S), is a (noncompact) manifold with boundary
+with corners to which the action of Γ(S)naturally extends. This action is
+also proper and according to [4] the orbit space M+
+g:=Γ(S)\T(S)+is a
+compactification of Mgthat can also be obtained from the Deligne-Mumford
+compactification Mg⊃ Mgas a ‘real oriented blowup’ of its boundary ∆g:=
+Mg−Mg. The walls of T(S)+define a closed covering of the boundary
+∂T(S)+and any nonempty corner closure is an intersection of walls. As is
+well-known, the curve complex C(S)can be identified with the nerve of this
+covering of ∂T(S)+. Since the corner closures are contractible, Weil’s nerve
+theorem implies that ∂T(S)+has the same homotopy type as C(S).
+Let∆g,0⊂∆gdenote the irreducible component of the Deligne-Mumford
+boundary whose generic point parameterizes irreducible curves w ith one
+singular point. We may understand Mc
+g:=Mg−∆g,0as the moduli space of
+stable genus gcurves all of whose nodes are separating (which is equivalent
+to the irreducible components of the curve being smooth and with thei r
+genera summing up to g) and∆c
+g:=∆g−∆g,0as the locus in Mc
+gthat
+parameterizes the singular ones among them. If Γis a subgroup of Γ(S)
+with the property that every Dehn twist along a separating curve inShas a
+positive power lying in Γ, then this defines a (not necessarily finite) cover
+˜Mc
+g→Mc
+g.
+Corollary 1.3. SupposeΓ⊂Γ(S)is as above and is in addition torsion free. If
+we denote by ˜∆c
+g⊂˜Mc
+gthe preimage of ∆c
+g, then the pair (˜Mc
+g,˜∆c
+g)is(g−2)-
+connected. Moreover, Hk(Mc
+g,∆c
+g;Q) =0fork≤g−2.
+Proof. LetT(S)+
+sepbe obtained from T(S)+by removing the walls that cor-
+respond to the nonseparating vertices of C(S). ThenT(S)+
+sepis the preim-
+age ofMc
+ginT(S)+. The same reasoning as above shows that ∂T(S)+
+sepCONNECTIVITY OF COMPLEXES OF SEPARATING CURVES 3
+is homotopy equivalent to C(S)sepand so∂T(S)+
+sepis(g−3)-connected.
+It follows that we can construct a relative CW complex (Z,∂T(S)+
+sep)ob-
+tained from ∂T(S)+
+sepby attaching cells of dimension ≥g−1in aΓ(S)-
+equivariant manner as to ensure that Zis contractible and no nontrivial
+element of Γ(S)fixes a cell. Then Γacts freely on Z(as it does on the
+contractible space T(S)+
+sep) and so there is a Γ-equivariant homotopy equiv-
+alenceZ→T(S)+
+seprelative to ∂T(S)+
+sep. It follows that we also have a
+homotopy equivalence Γ\Z→˜Mc
+grelative to ˜∆c
+gand we conclude that
+(˜Mc
+g,˜∆c
+g)is(g−2)-connected.
+The last assertion follows from the existence of a normal subgroup
+Γ⊂Γ(S)of finite index that is torsion free. For if Γis such a group, then
+Hk(Mc
+g,∆c
+g;Q)∼=Hk(˜Mc
+g,˜∆c
+g;Q)Γ(S)/Γ=0fork≤g−2. /square
+A similar statement holds for the universal curve Mg,1.
+Whenn > 1 , we need to come to terms with the fact that the separability
+notion has no good heriditary properties: if Tis a closed surface, A⊂Ta
+compact 1-dimensional submanifold representing a simplex of C(T)andSa
+connected component of T−A, then a vertex of C(S)may split S, but not T.
+This happens precisely when the vertex in question separates two boundary
+components of ∂Sthat lie on the same connected component of T−S. So the
+basic object should be, what Andy Putman calls in [6], a partitioned surface:
+a closed surface minus a finite set, for which the removed set comes with a
+partition. This leads to the following definition.
+Definition 1.4. LetNbe the set of points of Sat infinity (the cusps) and let
+Pbe a partition of N. We call a vertex of C(S)separating relative to Pif a
+representative embedded circle α⊂Shas the property that S−αhas two
+connected components each of which meets Nin a union of parts of P. We
+denote by C(S,P)the full subcomplex of C(S)spanned by such vertices.
+One might also understand C(S,P)as the full subcomplex of C(S)spanned
+by the isotopy classes of embedded cycles which are separating on the sur-
+faceSPthat is obtained by capping off for each part of Pthe corresponding
+set of cusps by a sphere with that many holes. Notice that C(S,P)⊂ C sep(S)
+and that we have equality when Pis discrete or Nis empty.
+We shall prove Theorem 1.1 by induction and simultaneously with
+Theorem 1.5 (Ag,n).Supposeg > 0 andn=|N|> 1. LetPbe a partition of
+N. ThenC(S,P)is(g−2)-connected.
+To be precise, the induction starts with g=0, where the statements ( Ag)
+and (A0,n) are trivially true and the induction strategy will be to show th at
+(i) (Ah,n) forh < g implies ( Ag) and
+(ii) (Ag) and (Ah,k) for(h,k)<(g,n)(for the lexicographic ordering)
+imply (Ag,n).
+I am indebted to Allen Hatcher for pointing out that the stronger ve rsion
+of Theorem 1.5 that I stated in a previous version was incorrect. Ye t it4 EDUARD LOOIJENGA
+may be that some such statement might hold. For instance, if r(P)denotes
+the number of nonempty parts of Pands(P)the number of parts with at
+least two elements, is it true that C(S,P)is(g+r(P) +s(P) −4)-connected
+wheng > 0 (as I claimed in the earlier version)? In case g=0,C(S,P)is a
+complex of dimension r(P) +s(P) −4. Is this(r(P) +s(P) −5)-connected?
+In other words, is this complex spherical?
+I am grateful to the referee, whose meticulous job helped to impr ove the
+paper. The proof of Lemma 2.3 follows a suggestion by the referee and
+simplifies my original one.
+I also gratefully acknowledge support by the Mathematical Scien ces Cen-
+ter of Tsinghua University at Beijing, where some of this work was done.
+2. P ROOFS
+Before we start off, we mention the following elementary fact that we will
+frequently use.
+Lemma 2.1. LetXibe adi-connected space ( di= −1meansXi/nega⊔ionslash=∅), where
+i=1,...,k . Then the iterated join X1∗···∗Xkis(−2+/summationtextk
+i=1(di+2))-connected.
+Proof that ( Ah,n) forh < g , alln, implies ( Ag).So here n≤1. We must
+show that Csep(S)is(g−3)-connected. For g < 2 , there is nothing to show
+and so we may assume that g≥2. A theorem of Harer [2, Thm. 1.2] asserts
+thatC(S)is(2g−3)-connected. So it is certainly (g−3)-connected. Let Ck
+be the subcomplex of C(S)that is the union of Csep(S)and thek-skeleton of
+C(S). SoC−1=Csep(S)andCk=C(S)forklarge. Notice that a finite set
+of vertices of C(S)spans a simplex of Ckif and only if no more than k+1
+of these are nonseparating. Hence a minimal simplex of Ck−Ck−1is repre-
+sented by a compact 1-dimensional submanifold A⊂Swithk+1connected
+components, each of which is nonseparating. We prove that the bounda ry
+of the star of such a simplex in Ckis a(g−3)-connected subcomplex of Ck−1.
+This property implies that the pair (C(S),Csep(S))is(g−2)-connected. Since
+C(S)is(g−3)-connected, it then follows that Csep(S)is. Let{Si}i∈Ibe the set
+of connected components of S−A. Notice that if giis the genus of Si, then
+gi< g. An Euler characteristic argument shows that
+g−1=k+1+/summationdisplay
+i∈I(gi−1).
+We denote by Nithe set of ‘cusps’ of Si, i.e., the finitely many points needed
+to make Sia closed surface. So an element of Niis given by possibly a cusp
+ofS(if it exists and if it is also a cusp of Si) or by a connected component of
+Ain the boundary of Siendowed with the orientation it receives as such . The
+setNicomes with an evident partition Pi: ifShas a cusp and Nicontains it,
+then this cusp makes up a singleton part of Piand any other two elements
+ofNibelong to the same part of Piif and only if they come from connected
+components of Athat lie on the same connected component of S−Si. (NB:CONNECTIVITY OF COMPLEXES OF SEPARATING CURVES 5
+beware that a connected component of S−Sicould be simply a connected
+component AoofA; then its two orientations define a 2-element part of Pi.)
+connected components of Athat bound Si. Note that since the connected
+components of Aare nonseparating, we always have |Ni|≥2. By our in-
+duction hypothesis C(Si,Pi)is then(gi−2)-connected. The boundary of the
+star of the k-simplex σAdefined by AinCklies inCk−1and can be identified
+with the (|I|+1)-fold join
+∂σA∗/parenleftbig
+∗i∈IC(Si,Pi)/parenrightbig
+.
+Since∂σAis a combinatorial (k−1)-sphere, this join has by Lemma 2.1
+connectivity at least −2+k+/summationtext
+i∈Igi. By the displayed formula above this
+is equal to g−4+|I|and is therefore ≥g−3. /square
+The proof of Ag,nbegins with a discussion. We now assume that g > 0
+andn≥2. Letx∈N. Notice that S′:=S∪{x}has still negative Euler charac-
+teristic. We put N′:=N−{x}andP′:=P|N′. The goal is to compare C(S′,P′)
+withC(S,P). There is in general no forgetful map C(S,P)→C(S′,P′)be-
+cause there will be vertices of C(S,P)that do not give vertices of C(S′,P′).
+Let us first identify this set of vertices.
+Denote by Σx⊂N−{x}the set of y∈N−{x}for which {x,y}is a union of
+parts ofP. In other words, if Pxdenotes the part of Pthat contains x, thenΣx
+is empty if Pxhas more than 2 elements, equals Px−{x}ifPxis a 2-element
+set, and equals the set of y/nega⊔ionslash=xfor which Pyis a singleton in case Px={x}.
+Then the vertices of C(S,P)that have no image in C(S′,P′)are precisely the
+verticesαofCsep(S)which for some y∈Σxbound a disk neighborhood of
+{x,y}inS∪{x,y}. Such a disk neighborhood can be thought of as a regular
+neighborhood of an arc in S∪{x,y}connecting the two added cusps; this
+may help to explain why we have chosen to denote this set of vertic es by
+arc(S,P)(x). Denote by C(S,P)xthe full subcomplex of C(S,P)spanned by the
+vertices not in arc (S,P)(x).
+Observe that arc (S,P)(x)is empty (so that C(S,P)x=C(S,P)) ifΣxis.
+Lemma 2.2. The link in C(S,P)of every vertex of arc(S,P)(x)is a subcomplex
+ofC(S,P)xthat projects isomorphically onto C(S′,P′).
+Proof. A vertex of arc (S,P)(x)defines a y∈Σxand (up to isotopy) a closed
+diskDinS∪{x,y}that is a neighborhood of {x,y}. The inclusion S−D⊂S′
+identifies the link in question with C(S′,P′). /square
+Denote by ˜Pthe refinement of Pwhich coincides with PonN−Pxand
+partitions Pxfurther into {x}andPx−{x}. So˜P′=P′. It is clear that C(S,P)
+is a subcomplex of C(S,˜P). Notice that arc (S,P)(x) =C(S,P)∩arc(S,˜P)(x)
+(we have arc (S,P)(x) =arc(S,˜P)(x)unless|Px|=2) andC(S,P)x=C(S,P)∩
+C(S,˜P)x. We denote by fthe forgetful simplicial map C(S,˜P)x→C(S′,P′)so6 EDUARD LOOIJENGA
+that we have the diagram
+C(S,P)x⊂C(S,˜P)xf−→C(S′,P′).
+∩ ∩
+C(S,P)⊂C(S,˜P)
+Lemma 2.3. The map f:C(S,˜P)x→C(S′,P′)is a homotopy equivalence.
+Proof. Choose an arc γwhich connects xwith another point of Nand defines
+a vertex of arc (S,P)(x)and observe that the full subcomplex K⊂ C(S,˜P)x
+spanned by vertices that avoid γdefines a section of f(the inclusion S−γ⊂
+S′is isotopic to a homeomorphism). We shall prove that C(S,˜P)xadmitsK
+as a deformation retract. (The proof will in fact show that each fiber of|f|
+is a tree and essentially produces for every element of |C(S,˜P)x|the unique
+path in its |f|-fibre that connects it to the point of |K|.)
+Denote by Kr⊂ C(S,˜P)xthe subcomplex whose simplices can be repre-
+sented by a closed submanifold A⊂Swhich meets γtransversally in at
+mostrpoints. This defines a filtration K=K0⊂K1⊂K2⊂ ··· whose
+union isC(S,˜P)x. Although this filtration is infinite, it is enough to construct
+for every r≥0a deformation retraction of |Kr+1|onto|Kr|, for in the sim-
+plicial setting an infinite sequence of deformation retractions still gives a
+deformation retraction.
+We do this per simplex: if σis a simplex of Kr+1that is not in Krand
+is minimal for this property, then its link in Kr+1lies inKrand so it suf-
+fices to define for such a σa deformation retraction hσof|StarKr+1(σ)|onto
+|LinkKr+1(σ)|.
+The simplex σis represented by a closed submanifold A⊂Sof which
+every connected component meets γtransversally and is such that A∩γhas
+cardinality r+1(a number that cannot be made smaller in its isotopy class).
+Letx0be the point of A∩γclosest to x. Denote by α0the connected compo-
+nent ofAwhich contains x0and choose in S′a thin regular neighborhood of
+the union of α0and the subarc of γwhich connects x0withx. The boundary
+of that neighborhood has two connected components. Both lie in Sand only
+one of them is isotopic to α0. Denote by α′
+0the other boundary component.
+Ifτis a simplex of Kr+1which contains σ, then adding α′
+0toτgives also a
+simplexτ′ofKr+1and the codimension one face τ′′ofτ′obtained by re-
+movingα0is contained in Kr. So if we regard |StarKr+1(σ)|as the cone over
+|LinkKr+1(σ)|with the barycenter of σas its vertex, then there is a simplicial
+map from this cone to its base which sends the barycenter to α′
+0and is the
+identity on the base. Its geometric realization yields the desire dhσ./square
+Corollary 2.4. The complex C(S,P)∪C(S,˜P)xis canonically homotopy equiv-
+alent to the join arc(S,P)(x)∗C(S′,P′)(where arc(S,P)(x)is discrete).
+Proof. The set of vertices of C(S,P)∪C(S,˜P)xnot inC(S,˜P)xis arc(S,P)(x). The
+link of any such vertex is contained in C(S,˜P)xand by Lemma 2.2 that link
+projects isomorphically onto C(S′,P′). In view of Lemma 2.3 this impliesCONNECTIVITY OF COMPLEXES OF SEPARATING CURVES 7
+that the inclusion of this link in C(S,˜P)xis also a homotopy equivalence.
+Hence the natural inclusion C(S,P)∪ C(S,˜P)x⊂arc(S,P)(x)∗ C(S,˜P)xis a
+homotopy equivalence. The corollary follows. /square
+From now on we assume that Agholds and that Ah,kholds for all (h,k)
+smaller than (g,n)for the lexicographic ordering. Our goal is to prove Ag,n.
+Lemma 2.5. The pair (C(S,P)∪C(S,˜P)x,C(S,P))is(g−1)-connected.
+Proof. IfPx={x}, then ˜P=Pand there is nothing to show. We therefore
+assume that Pxhas more than one element. Denote by Ckthe subcomplex of
+C(S,P)∪C(S,˜P)xthat is the union of C(S,P)and thek-skeleton of C(S,P)∪
+C(S,˜P)x: a finite set of vertices of C(S,P)∪C(S,˜P)xspans a simplex of Ckif
+and only if no more than k+1of these separate xfromPx−{x}. Notice that
+C−1=C(S,P)andCk=C(S,P)∪ C(S,˜P)xforklarge. A minimal simplex
+ofCk−Ck−1is represented by a compact 1-dimensional submanifold A⊂S
+withk+1connected components, each of which separates xfromPx−{x}
+(the graph that is associated to Ais then a string with k+2nodes). We prove
+that the boundary of the star of such a simplex in Ck−1is(g−2)-connected if
+g > 0 . We enumerate the connected components of Aasα0,...,α kand the
+connected components of S−AasS0,...,S k+1such that αiis a boundary
+component of SiandSi+1and so that S0resp.Sk+1is punctured by xresp.
+Px−{x}. The cusps of S−Aare naturally indexed by ^N:=N⊔{i±}k
+i=0,
+wherei−resp.i+corresponds to the cusp defined by αionSiresp.Si+1.
+Let^Ni⊂^Nindex the set of cusps on Si. Denote by Pithe partition of
+(N−Px)∩Sithat is simply the restriction of Pand denote by ^Pithe partition
+of^Nithat on(N−Px)∩Siis equal to Piand has what remains of ^Nias a
+single part. So this new part is {x,0+}fori=0,{(i−1)−,i+}for0 < i < k +1
+and(Px−{x})∪{k−}fori=k+1.
+The reason for introducing these partitions is that we can now observ e
+that the boundary of the star of the k-simplex σAdefined by AinCklies in
+Ck−1and can be identified with the iterated join
+∂σA∗C(S0,^P0)∗···∗C(Sk+1,^Pk+1).
+It is then enough to show that this join is (g−2)-connected for g > 0 . Since
+∂σAis a(k−1)-sphere, it is (k−2)-connected. The connectivity of a factor
+C(Si,^Pi)withgi> 0is at least gi−2. So by Lemma 2.1 the connectivity of
+the above join is at least −2+k+/summationtext
+{i:gi>0}gi=g+k−2≥g−2./square
+Proof of ( Ag,n).We must show that C(S,P)is(g−2)-connected. In view of
+Lemma 2.5 it suffices to show that C(S,P)∪C(S,˜P)xhas that property.
+If arc(S,P)(x) =∅, thenn > 2 and so our induction hypothesis implies that
+C(S′,P′)is(g−2)-connected by Ag,n−1. It follows from Corollary 2.4 that
+C(S,P)∪C(S,˜P)xis homotopy equivalent to C(S′,P′)and hence is (g−2)-
+connected.8 EDUARD LOOIJENGA
+If arc(S,P)(x)/nega⊔ionslash=∅, then we may have n=2. At least we know that C(S′,P′)
+is(g−3)-connected (invoke Agifn=2). But since C(S,P)∪ C(S,˜P)xis
+homotopy equivalent to arc (S,P)(x)∗C(S′,P′)(by Corollary 2.4), it is (g−2)-
+connected. /square
+REFERENCES
+[1] B. Farb, N.V. Ivanov: The Torelli geometry and its applications: research announ cement,
+Math. Res. Lett. 12 (2005), 293–301.
+[2] J. L. Harer: Stability of the homology of the mapping class groups of orie ntable surfaces,
+Ann. of Math. (2) 121 (1985), 215–249.
+[3] W. van der Kallen, E. Looijenga: Spherical complexes attached to symplectic lattices,
+Geom. Dedicata, 152 (2011), 197-211.
+[4] E. Looijenga: Cellular decompositions of compactified moduli spaces of po inted curves,
+In: The moduli space of curves (Texel Island, 1994), 369–400 , Progr. Math. 129,
+Birkh¨ auser, Boston, MA, 1995.
+[5] A. Putman: A note on the connectivity of certain complexes associated t o surfaces, Enseign.
+Math. (2) 54 (2008), 287–301.
+[6] A. Putman: Cutting and pasting in the Torelli group, Geom. Topol. 11 (2007) 829–865.
+E-mail address :E.J.N.Looijenga@uu.nl
+MATHEMATISCH INSTITUUT , UNIVERSITEIT UTRECHT , P.O. B OX80.010, NL-3508 TA
+UTRECHT (NEDERLAND )
\ No newline at end of file
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+arXiv:1001.0824v2  [cs.DS]  3 Feb 2010Symposium on TheoreticalAspects of Computer Science 2010( Nancy, France), pp. 513-524
+www.stacs-conf.org
+APPROXIMATE SHORTEST PATHS AVOIDING A FAILED VERTEX :
+OPTIMAL SIZE DATA STRUCTURES FOR UNWEIGHTED GRAPHS
+NEELESH KHANNA1AND SURENDER BASWANA2
+1Oracle India Pvt. Ltd, Bangalore-560029, India.
+E-mail address :neelesh.khanna@gmail.com
+2Indian Institute of Technology Kanpur, India.
+E-mail address :sbaswana@cse.iitk.ac.in
+Abstract. LetG= (V,E) be any undirected graph on Vvertices and Eedges. A path P
+between any two vertices u,v∈Vis said to be t-approximate shortest path if its length is
+at mostttimes the length of the shortest path between uandv. We consider the problem
+of building a compact data structure for a given graph Gwhich is capable of answering
+the following query for any u,v,z∈Vandt >1.
+reportt-approximate shortest path between uandvwhen vertex zfails
+Wepresentdatastructures for thesingle source as well all- pairs versions of this problem.
+Our data structures guarantee optimal query time. Most impr essive feature of our data
+structures is that their size nearlymatch the size of their best static counterparts.
+1. Introduction
+The shortest paths problem is a classical and well studied al gorithmic problem of com-
+puter science. This problem requires processing of a given g raphG= (V,E) onn=|V|
+vertices and m=|E|edges to compute a data structure using which shortest path o r dis-
+tance between any two vertices can be efficiently reported. Tw o famous and thoroughly
+studied versions of this problem are single source shortest paths (SSSP) problem and all-
+pairs shortest paths (APSP) problem.
+Most of the applications of the shortest paths problem invol ve real life graphs and
+networks which areproneto failureof nodes(vertices) andl inks(edges). Thishas motivated
+researchers to design dynamic solution for the shortest paths problem. For this purpose, o ne
+has to first develop a suitable model for the shortest paths pr oblem in dynamic networks.
+In fact two such models exists, and each of them has its own alg orithmic objectives.
+The shortest paths problem in the first model is described as f ollows : There is an
+initial graph followed by an on-line sequence of insertion a nd deletion of edges interspersed
+1998 ACM Subject Classification: E.1[Data Structures] :Graphs andNetworks; G.2.2 [Discrete Math-
+ematics] :Graph Theory - Graph Algorithms .
+Key words and phrases: Shortest path, distance, distance queries, oracle.
+Part of this work was done while the authors were at Max-Planc k Institute for Computer Science, Saar-
+bruecken, Germany during the period May-July 2009.
+c/circlecopyrtN.KHANNA andS. BASWANA
+CC/circlecopyrtCreativeCommonsAttribution-NoDerivsLicense514 N. KHANNA AND S. BASWANA
+with shortest path (or distance) queries. Each query has to b e answered with respect to the
+graph which exists at that moment (incorporating all the upd ates preceding the query on
+the initial graph). A trivial solution of this problem is to r ecompute all-pairs shortest paths
+from scratch after each update. This is certainly a wasteful approach since a single update
+usually does not cause a huge change in the all-pairs distanc e information. Therefore, the
+algorithmic objective here is to maintain a data structure w hich can answer distance query
+efficiently and can be updated after any edge insertion or dele tion in an efficient manner. In
+particular, the time required to update the data structure h as to be substantially less than
+the running time of the best static algorithm. Many novel alg orithms have been designed
+in the last ten years for this problem and its variants (see [6 ] and the references therein).
+On one hand the first model is important since it captures the w orst possible hardness
+of any dynamicgraph problem. Ontheother hand, it can also be considered as a pessimistic
+model for real life networks. It is true that the networks are never immune to failures. But
+in addition to it, it is also rareto have networks which may ha ve arbitrary numberof failures
+in normal circumstances. It is essential for network design ers to choose suitable technology
+to make sure that the failures are quite infrequent in the net work. In addition, when a
+vertex or edge fails (goes down), it does not remain failed/d own indefinitely. Instead, it
+comes up after some finite time due to simultaneous repair mec hanism going on in the
+network. These aspects can be captured in the second model wh ich takes as input a graph
+and a number ℓ≪n. This model assumes that there will be at most ℓvertices or edges
+which may be inactive at any time, though the corresponding s et of failed vertices or edges
+may keep changing as the time progresses : the old failed vert ices become active while some
+new active vertices may fail. The algorithmic objective in t his model is to preprocess the
+given graph to construct a compact data structure which for a ny subset Sof at most ℓ
+vertices may answer the following query for any u,v∈V.
+Report the shortest-path (or distance) from utovinG\S.
+It is desired that each query gets answered in optimal time : retrieval of distance in
+O(1) time and the shortest path in time which is of the order of t he number of its edges.
+The ultimate research goal would be to understand the comple xity of the above problem
+for any given value ℓ. In this pursuit, the first natural step would be to efficiently solve
+and thoroughly understand the complexity of the problem for the case ℓ= 1, that is, the
+shortest paths problem avoiding any failed vertex. Interes tingly, this problem appears as a
+sub problem in many other related problems, namely, Vickrey pricing of networks [9], most
+vital node of a shortest path [11], the replacement path prob lem [12], and shortest paths
+avoiding forbidden subpaths [1].
+The first nontrivial and quite significant breakthrough on th e all-pairs version of this
+problem was made by Demetrescu et al. [7]. They designed an O(n2logn) space data
+structure, namely distance sensitivity oracle , which is capable of reporting the shortest path
+between any two vertices avoiding any single failed vertex. The preprocessing time of this
+data structure is O(mn2). Recently, Bernstein and Karger [4] improved the preproce ssing
+time toO(mnlogn). Though Θ( n2logn) space bound of this all-pairs distance sensitivity
+oracle is optimal up to logarithmic factors, it is too large f or many real life graphs which
+appear in various large scale applications [13]. In most of t hese graphs usually m≪n2,
+henceatableofΘ( n2)sizemay betoolargeforpractical purposes. However, itis alsoknown
+[7] that even a data structure which reports exact distances from a fixed source avoiding a
+single failed vertex will require Ω( n2) space in the worst case. So approximation seems to
+be the only way to design a small space compact data structure for the problem of shortestAPPROXIMATE SHORTEST PATHS AVOIDING A FAILED VERTEX 515
+paths avoiding a failed vertex. A path between u,v∈Vis said to be t-approximate shortest
+path if its length is at most ttimes that of the shortest path between the two. The factor
+tis usually called the stretch. We would like to state here tha t many algorithms and data
+structures have been designed in the last fifteen years for th e static all-pairs approximate
+shortest paths (see [2, 13] and references therein). The pri me motivation underlying these
+algorithms has been to achieve sub-quadratic space and/or s ub-cubic preprocessing time
+for the static APSP problem. However, no data structure was d esigned in the past for
+approximate shortest paths avoiding any failed vertex.
+In this paper, we present really compact data structures whi ch are capable of reporting
+approximate shortest paths between two vertices avoiding a ny failed vertex in undirected
+graphs. The most impressive feature of our data structures i s their nearly optimal size. In
+fact their size almost matches the size of their best static c ounterparts.
+1.1. New Results
+Single source approximate shortest paths avoiding any failed vertex.
+First we address weighted graphs. For the weighted graphs, w e present an O(mlogn) time
+constructible data structure of size O(nlogn) which can report 3-approximate shortest
+path from the source to any vertex v∈Vavoiding any x∈V. We then consider the case
+of undirected unweighted graphs. For these graphs, we prese nt anO(nlogn
+ǫ3) space data
+structure which can even report (1+ ǫ)-approximate shortest path for any ǫ >0.
+All-pairs approximate shortest paths avoiding any failed vertex.
+Among the existing data structures for static all-pairs app roximate shortest paths, the
+approximate distance oracle ofThorupandZwick[13]standsoutduetoitsamazingfeature s.
+Thorup and Zwick [13] showed that an undirected graph can be p reprocessed in sub-cubic
+time to build a data structure of size O(kn1+1/k) for any k >1. This data structure, despite
+of its sub-quadratic size, is capable of reporting (2 k−1)-approximate distance between any
+two vertices in O(k) time (and the corresponding approximate shortest path in o ptimal
+time), and hence the name oracle. Moreover, the size-stretch trade off achieved by this data
+structureis essentially optimal. Itis a very natural quest ion to explore whether it is possible
+to design all-pairs approximate distance oracle which may h andle single vertex failure. We
+show that it is indeed possible for unweighted graphs. For th is purpose, we suitably modify
+the approximate distance oracle of Thorup and Zwick [13] usi ng some new insights and our
+single source data structure mentioned above. These modific ations make the approximate
+shortest-paths oracle of Thorup and Zwick handle vertex fai lure easily, and (surprisingly)
+still preserving the old (optimal) trade-off between the spa ce and the stretch. For precise
+details, see Theorem 5.3.
+For the algorithmic details missing in this extended abstra ct due to page limitations,
+we suggest the reader to refer to the journal version [10]. Ou r data structures can be easily
+adapted for handling edge failure as well without any increa se in space or time complexity.
+2. Preliminaries
+We use the following notations and definitions in the context of a given undirected
+graphG= (V,E) withn=|V|,m=|E|and a weight function ω:E→R+.
+•Tr: single source shortest path tree rooted at r.
+•P(x,y) : the shortest path between xandy.516 N. KHANNA AND S. BASWANA
+•δ(x,y) : the length of the shortest path between xandy.
+•P(x,y,z) : the shortest path between xandyavoiding vertex z.
+•δ(x,y,z) : the length of the shortest path between xandyavoiding vertex z.
+•Tr(x) : the subtree of Trrooted at x.
+•Gr(x) : the subgraph induced by the vertices of set Tr(x) and augmented by vertex
+rand edges from ras follows. For each v∈Tr(x) with neighbors outside Tr(x),
+keep an edge ( r,v) of weight = min (u,v)∈E,u/∈Tr(x)(δ(r,u) +ω(u,v)).
+•P::Q: a path formed by concatenating path Qat the end of path Pwith an edge
+(u,v)∈E, whereuis the last vertex of Pandvis the first vertex of Q.
+•E(X) : the set of edges from Ewith at least one endpoint in X.
+Our algorithms will also use a data structure for answering l owest common ancestor (LCA)
+queries on Tr. There exists an O(n) time computable data structure which occupies O(n)
+space and can answer any LCA query in O(1) time (see [3] and references therein).
+3. Single source 3-approximate shortest paths avoiding a fa iled vertex
+We shall first solve a simpler sub-problem where the vertex wh ich may fail belong to
+a given path P∈Tr. Then we use divide and conquer strategy wherein we decompos eTr
+into a set of disjoint paths and for each such path, we solve th is sub-problem.
+3.1. Solving the Sub-Problem : the failures of a vertex from a given path P (r,t)
+Given the shortest path tree Tr, letP(r,t) =/a\}⌊ra⌋ketle{tr(=x0),x1,...,xk(=t)/a\}⌊ra⌋ketri}htbe any shortest path
+present in Tr. We shall design an O(n) space data structure which will support retrieval of a
+3-approximate shortest path from rto anyv∈Vwhen some vertex from P(r,t) fails. The
+preprocessing time of our algorithm will be O(m+nlogn) which matches that of Dijkstra’s
+algorithm. The algorithm is inspired by the algorithm of Nar delli et al. [11] for computing
+the most vital vertex on a shortest path. Consider vertex xilying on the path P(r,t). We
+PSfrag replacementsr
+xi−1
+xi
+xi+1
+OiUi
+DiP
+Figure 1: Partitioning of the shortest path tree Tratxi∈P
+partition the tree Tr\{xi}into the following 3 parts (see Figure 1).
+(1)Ui: the tree Trafter removing the subtree Tr(xi)
+(2)Di: the subtree of Trrooted at xi+1
+(3)Oi: the portion of Trleft after removing Ui,xi, andDi.
+Note that a vertex of the tree Tris either a vertex of the path P(r,t) or it belongs to some
+Oifor some i. We build the following two data-structures of total O(n) size.APPROXIMATE SHORTEST PATHS AVOIDING A FAILED VERTEX 517
+(1) a data structure to retrieve 3-approximate shortest pat h fromrto anyv∈Di.
+(2) a data structure to retrieve 3-approximate shortest pat h fromrto anyv∈Oi.
+3.1.1.Data structure for 3-approximate shortest paths to vertice s ofDiwhenxihas failed.
+Considerthevertex xi+1andanyothervertex y∈Di. Notethattheshortestpath P(xi+1,y)
+remains intact even after removal of xi, and its length is certainly less than δ(r,y). Based
+on this simple observation one can intuitively see that in or der to travel from rtoywhen
+xifails, we may travel along shortest route to xi+1(that isP(r,xi+1,xi)) and then along
+P(xi+1,y). Using triangle inequality and the fact that the graph is un directed, the length
+of this path P(r,xi+1,xi) ::P(xi+1,y) can be approximated as follows.
+δ(r,xi+1,xi)+δ(xi+1,y)≤δ(r,y,xi)+δ(y,xi+1,xi)+δ(xi+1,y)
+≤δ(r,y,xi)+2δ(xi+1,y)
+≤δ(r,y,xi)+2δ(r,y)≤3δ(r,y,xi)
+Therefore, in order to support retrieval of 3-approximate s hortest path to any v∈Diin
+optimal time, it suffices to store the path P(r,xi+1,xi).
+In order to devise ways of efficient computation and compact st orage of P(r,xi+1,xi)
+for a given i, we use the following lemma about the structure of the path P(r,xi+1,xi).
+Lemma 3.1. The shortest path P(r,xi+1,xi)is of the form P1::P2whereP1is a shortest
+path from rin the subgraph induced by Ui∪Oi, andP2is a path present in Di.
+Itfollows thatinordertocompute P(r,xi+1,xi), firstweneedtocomputeshortestpaths
+fromrin the subgraph induced by Ui∪Oi. Letδi(r,v) denote the distance from rtov∈
+Ui∪Oiin this subgraph. Note that δi(r,v) forv∈Uiand the correspondingshortest path is
+thesameas in theoriginal graph, andis already present in Tr. For computingshortestpaths
+fromrto vertices of Oi, we build a shortest path tree (denoted as Tr(Oi)) fromrin the
+subgraph induced by vertices Oi∪{r}and the following additional edges. For each z∈Oi
+withatleastoneneighborin Ui, weaddanedge( r,z)withweight= min(u,z)∈E,u∈Ui(δ(r,u)+
+ω(u,z)). Applying Lemma 3.1, let ( yi,zi) be the edge of P(r,xi+1,xi) joining the sub
+path present in Ui∪Oiwith the sub path present in Di. This edge can be identified
+using the fact that this is the edge which minimizes δi(r,y) +ω(y,z) +δ(xi+1,z) over all
+z∈Di,y∈Ui∪Oi,(y,z)∈E. The vertex xi+1stores the path P(r,xi+1,xi) implicitly
+by keeping the edge ( yi,zi) and the tree Tr(Oi). The shortest path P(r,xi+1,xi) can be
+retrieved in optimal time using the trees Tr,Tr(Oi), and the edge ( yi+1,zi+1). Due to
+mutual disjointness of Oi’s, the overall space requirement of the data structurefor r etrieving
+P(r,xi+1,xi) for alli≤kwill beO(n).
+3.1.2.Data structure for 3-approximate shortest paths to vertice s ofOiwhenxihas failed.
+In order to compute 3-approximate shortest path to Oiupon failure of xi, we shall use the
+approximateshortestpathsto Diascomputedabove. Hereweuseaninterestingobservation
+which states that if we have a data structure to retrieve α-approximate shortest paths from
+rto vertices of Diwhenxifails, then we can use it to have a data-structure to retrieve
+α-approximate shortest paths to vertices of Oias well. To prove this result, this is how
+we proceed. Consider the subgraph induced by Oiand augmented with vertex rand some
+extra edges which are defined as follows.518 N. KHANNA AND S. BASWANA
+•For each o∈Oihaving neighbors from Ui, keep an edge ( r,o) and assign it weight
+= min (u,o)∈E,u∈Ui(δ(r,u) +ω(u,o)).
+•For each o∈Oihaving neighbors from Di, keep an edge ( r,o) and assign it weight =
+min(u,o)∈E,u∈Di(ˆδ(r,u,xi)+ω(u,o)), where ˆδ(r,u,xi) is theα-approximate distance
+touupon failure of xi. (In the present situation we have α= 3.)
+Let us denote this graph as Gr(Oi). Observation 3.3 is based on the following lemma which
+is easy to prove.
+Lemma 3.2. The Dijkstra’s algorithm from rin the graph Gr(Oi)computes α-approximate
+shortest paths from rto allv∈Oiavoiding xi.
+Observation 3.3. If we can design a data structure for retrieving (1 + ǫ)-approximate
+shortest paths from rto vertices of Diupon failure of xi, then it can also be used to design
+a data structure which can support retrieval of (1 + ǫ)-approximate shortest paths to all
+vertices of the graph upon failure of xi.
+We compute and store the shortest path tree rooted at rin the graph Gr(Oi). This tree
+along with the tree Trand the data structure described in the previous sub-sectio n suffice
+for retrieval of 3-approximate shortest paths to o∈Oiupon failure of xi.
+Query answering: Suppose the oracle receives a query asking for approximate s hortest
+path from rtovavoiding xi∈P(r,t). It first invokes lowest common ancestor (LCA)
+query between vandxionTr. IfLCA(v,xi)/\e}atio\slash=xi, the shortest path from rtovremains
+unaffected and so it reports the path P(r,t). Otherwise, it determines if v∈Diorv∈Oi.
+Depending upon the two cases, it reports the approximate sho rtest path between randvi
+using one of the two data structures described above.
+Theorem 3.4. An undirected weighted graph G= (V,E), a source r∈V, and a shortest
+pathP∈Trcan be processed in O(m+nlogn)time to build a data structure of O(n)space
+which can report 3-approximate shortest path from rto anyv∈Vavoiding any single failed
+vertex from P.
+3.2. Handling the failure of any vertex in Tr
+We follow divide and conquer strategy based on the following simple lemma.
+Lemma 3.5. There exists an O(n)time algorithm to compute a path PinTrwhose removal
+splitsTrinto a collection of disjoint subtrees Tr(v1),...Tr(vj)such that
+• |Tr(vi)|< n/2for each i≤j.
+•P∪iTr(vi) =TandP∩Tr(vi) =∅ ∀i.
+First we compute the path P∈Tras mentioned in Lemma 3.5. We build the data
+structure for handling failure of any vertex from Pby executing the algorithm of Theorem
+3.4. Let v1,...,vjbe the roots of the sub trees of Trconnected to the path Pwith an edge.
+For each 1 ≤i≤j, we solve the problem recursively on the subgraph Gr(vi), and build
+the corresponding data structures. Lemma 3.5 and Theorem 3. 4 can be used in straight
+forward manner to prove the following theorem.
+Theorem 3.6. An undirected weighted graph G= (V,E)can be processed in O(mlogn+
+nlog2n)time to build a data structure of size O(nlogn)which can answer, in optimal time,
+any 3-approximate shortest path query from a given source rto any vertex v∈Vavoiding
+any single failed vertex.APPROXIMATE SHORTEST PATHS AVOIDING A FAILED VERTEX 519
+4. Single source (1+ ǫ)-approximate shortest paths avoiding a failed vertex
+In this section, we shall present a compact data structure fo r single source (1 + ǫ)-
+approximate shortest paths avoiding a failed vertex in an un weighted graph. Let level(v)
+denote the level (or distance from r) of vertex vin the tree Tr. LetUx,Dx,Oxdenote
+the partitions of the tree Trformed by deletion of vertex x, with the same meaning as
+that ofUi,Di,Oidefined for xiin the previous section. On the basis of Observation 3.3,
+our objective is to build a compact data structure which will support retrieval of (1+ ǫ)-
+approximate shortest-paths to vertices of Dxuponfailureof xfor anyx∈V. Letuchild(x)
+denote the root of the subtree corresponding to Dx(it is similar to xi+1in case of Di). For
+reporting approximate distance between randv∈Dxwhenxfails, the data structure of
+previoussection reportspathoflength δ(r,uchild(x),x)+δ(uchild(x),v) whichisbounded
+byδ(r,v,x)+2δ(uchild(x),v). It should be noted that the approximation factor associat ed
+with it is already bounded by (1+ ǫ) for any ǫ >0 if the following condition holds.
+C:uchild(x) isclosetov, that is, δ(uchild(x),v)≤ǫ
+2δ(r,v).
+We shall build a supplementary data structure which will ens ure that whenever the
+condition Cdoes not hold, there will be some ancestor wofvlying on P(x,v), called a
+specialvertex, satisfying the following two properties.
+(1)δ(w,v)≪δ(r,v), that is wis much closer to vthanr.
+(2) vertex wstores approximate shortest path to ravoiding x(with the approximation
+factor arbitrarily close to 1).
+We shall refer to such vertices was special-vertices.
+4.1. Constructing the set of special vertices
+Lethbe the height of BFS tree rooted at r. LetLbe a set of integers such that
+L={i|⌊(1+ǫ)i⌋< h}. For a given i∈L, we define a subset Siof special vertices as
+Si={u∈V|level(u) =⌊(1+ǫ)i⌋ ∧ |Tr(u)| ≥ǫlevel(u)}. We define the set of special
+vertices as S=∪∀i∈LSi. In addition, we also introduce the following terminologie s.
+•S(v): the nearest ancestor of vwhich belongs to set S.
+•V(u): For a vertex u∈S,V(u) denotes the set of vertices v∈VwithS(v) =u.
+In essence, the vertex uwill serve as the special vertex for each vertex from V(u).
+For failure of any vertex x∈P(r,u), each vertex of set V(u) will query the data
+structure stored at ufor retrieval of approximate shortest path/distance from t he
+source.
+We now state two simple lemmas based on the above constructio n.
+Lemma 4.1. Letv∈V\S, thenδ(v,S(v))≤/parenleftbig2ǫ
+1+ǫ/parenrightbig
+level(v)ifǫ <1
+Lemma 4.2. Letube a vertex at level ℓandu∈S. ThenV(u)≥ǫℓ.
+Ifwecanensurethatthedatastructureforaspecialvertex u(forretrievingapproximate
+shortest paths from rupon failure of any x∈P(r,u)) is of size O(level(u)), then it would
+follow from Lemma 4.2 that the space required by our suppleme ntary data structure will
+be linear in n.520 N. KHANNA AND S. BASWANA
+4.2. The data structure for a special vertex
+Consider a special vertex vwithlevel(v) =⌊(1+ǫ)i⌋We shall now describe a compact
+data structure stored at vwhich will facilitate retrieval of approximate shortest pa th from
+rtovupon failure of any vertex x∈P(r,v).
+Letv′be the special vertex which is present at level ⌊(1+ǫ)i−1⌋and is ancestor of v.
+The data structure stored at vwill be defined in a way that will prevent it from storing
+information that is already present in the data structure of some special vertex lying on
+P(r,v′).
+Ifx∈P(v′,v), then the data structure described in the previous section itself stores a
+path which is (1+2 ǫ)-approximation of P(r,v,x).
+Let us now consider the nontrivial case when x∈P(r,v′),x/\e}atio\slash=v′. In order to discuss
+this case, we would like to introduce the notion of detour. To understand it, let us visualize
+the paths P(r,v,x) andP(r,v) simultaneously. Since P(r,v,x) andP(r,v) have the same
+end-points and xdoesn’t lie on P(r,v,x), there must be a middleportion of P(r,v,x) which
+intersects P(r,v) at exactly two vertices, and the remaining portion of P(r,v,x) overlaps
+withP(r,v). Thismiddleportion is called a detour. We now define it more formally. Let
+aandbbe two vertices on the shortest path P(r,v). We use a≺bto denote that vertex a
+is closer to rthan vertex b. The notation a/pre⌋edesequalbwould mean that either a≺bora=b. .
+So here is the definition of detour (and the underlying observ ation).
+Definition 4.3. Letx∈P(r,y). When xfails, the path P(r,y,x) will be of the form
+ofP(r,a) ::pa,b::P(b,y), where r/pre⌋edesequala≺x≺b/pre⌋edesequalyand the path pa,bis such that
+pa,b∩P(a,b) ={a,b}. In other words, pa,bmeetsP(a,b) only at the end points. We shall
+callpa,bas the detour associated with the shortest path P(r,y,x).
+Letpa,brepresent the detour w.r.t. to P(r,v,x). The handling of failure of vertices
+x∈P(r,v) which lie above v′would depend upon the detour pa,b. This detour can be of
+any of the following types (see Figure 2 for illustration).
+•I :b/pre⌋edesequalv′.
+•II :v′≺b.PSfrag replacements
+ℓ0
+x xa a
+b
+bv′ v′
+v vr r
+⌊(1+ǫ)i⌋ ⌊(1+ǫ)i⌋pa,b pa,b
+(i) (ii)
+Figure 2: pa,bis shortest detour of P(r,v,x). (i) : detour of type I, (ii) : detour of type II
+Handling detours of type I is relatively easy. Let wbe the farthest ancestor of vsuch
+thatw∈Sand level of wis greater or equal to the level of b. In this case, vstores the
+corresponding detour implicitly by just keeping a pointer t o the vertex w.
+Handling detours of type II is slightly tricky since we can’t afford to store each of them
+explicitly. However, we shall employ the following observa tion associated with the detours
+of type II to guarantee low space requirement.APPROXIMATE SHORTEST PATHS AVOIDING A FAILED VERTEX 521
+Observation 4.4. Letα1,α2,...,αtbe the vertices on P(r,v) (in the increasing order of
+their levels) such that the shortest detour corresponding t oP(r,v,αi) is of type II ∀i, then
+δ(r,v,α1)≥δ(r,v,α2)≥ ··· ≥δ(r,v,αt)
+It follows from the above observation that if δ(r,v,αi)≤(1+ǫ)δ(r,v,αj) for any i < j,
+thenP(r,v,αi) may as well serve as (1+ ǫ)-approximate shortest path from rtovavoiding
+αj. Inotherwords,weneednotstorethedetourassociatedwith P(r,v,αj)insuchsituation.
+Using this observation, we shall have to explicitly store on lyO(log1+ǫn) detours of type II.
+Moreover, we do not store explicitly detours of type II whose length is much larger than
+level(v). Specifically, if P(r,v,x)≥1
+ǫlevel(v), thenvwill merely store pointer to the path
+P(r,uchild(x),x) ::P(uchild(x),v). This ensures that each detour of type II which vhas
+to explicitly store will have length O(1
+ǫlevel(v)).
+Itfollows fromtheabove description that foraspecial vert exvandx∈P(r,v), thedata
+structure associated with vstores (1+2 ǫ)-approximation of the path P(r,v,x). Moreover,
+the total space required by the data structure associated wi th all the special vertices will be
+O(nlogn
+ǫ3). This supplementary data structure combined with the data structure of previous
+section can report (1+6 ǫ)-approximation of P(r,z,x) for any z,x∈V.
+Theorem 4.5. Given an undirected unweighted graph G= (V,E), source r∈V, and any
+ǫ >0, we can build a data structure of size O(nlogn
+ǫ3)that can report (1 +ǫ)-approximate
+shortest path from rto anyz∈Vavoiding any failed vertex in optimal time.
+5. All-pairs (2k−1)(1+ǫ)-approx. distance oracle avoiding a failed vertex
+We start with a brief description of the approximate distanc e oracle of Thorup and
+Zwick [13]. The key idea to achieve sub-quadratic space is to store distance from each
+vertex to only a small set of vertices. For retrieving approx imate distance between any two
+verticesu,v∈V, it is ensured that there is a third vertex wwhich is closeto both of them,
+and whose distance from both of them is known. To realize this idea, Thorup and Zwick
+[13] introduced two novel structures called ballandclusterwhich are defined for any two
+subsetsA,Bof vertices as follows. (here δ(v,B) denotes the distance between vand its
+nearest vertex from B).
+Ball(v,A,B) ={w∈A|δ(v,w)< δ(v,B)}C(w,A,B) ={v∈V|δ(v,w)< δ(v,B)}
+Construction of (2 k−1)-approximate distance oracle of Thorup and Zwick [13] emp loys a
+k-level hierarchy Ak=/a\}⌊ra⌋ketle{tA0⊇A1⊇A2...⊇Ak−1⊃Ak/a\}⌊ra⌋ketri}htof subsets of vertices as follows.
+A0=V,Ak=∅, andAi+1for anyi < k−1 is formed by selecting each vertex from Ai
+independently with probability n−1/k.
+The data structure associated with the (2 k−1)-approximate distance oracle of Thorup
+and Zwick [13] stores for each vertex v∈Vthe following information :
+•the vertices of set ∪i<kBall(v,Ai,Ai+1) (and their distances).
+•the vertex from Ainearest to v(to be denoted as pi(v)).
+Due to randomization underlying the construction of Ak, the expected size of
+Ball(v,Ai,Ai+1) isO(n1/k), and hence the space required by the oracle is O(kn1+1/k).
+We shall now outline the ideas in extending the (2 k−1)-approximate distance oracle to
+handle single vertex failure. Kindly refer to the extended v ersion [10] of this paper for
+complete details.522 N. KHANNA AND S. BASWANA
+5.1. Overview of all-pairs approx. distance oracles avoiding a failed vertex
+Firstly thenotations usedby thestatic approximate distan ce oracle of [13], in particular
+ball and cluster, get extended for single vertex failure in a natural manner as follows. (here
+δ(v,B,x) is the distance between vand its nearest vertex from BinG\{x}).
+Ballx(v,A,B) ={w∈A|δ(v,w,x)< δ(v,B,x)}
+Cx(w,A,B) ={v∈V|δ(v,w,x)< δ(v,B,x)}
+Letpx
+i(v) denote the vertex from Aiwhich is nearest to vinG\{x}. Along the lines of the
+static approximate distance oracle of Thorup and Zwick [13] , the basic operation which the
+approximate distance oracle avoiding a failed vertex shoul d support is the following :
+Report distance (exact or approximate) between vandw∈Aiifw∈Ballx(v,Ai,Ai+1)
+for any given v,x∈V.
+However, it can be observed that we would have to support this operation implicitly in-
+stead of explicitly keeping Ballx(v,Ai,Ai+1) for each v,x,i. Ourstarting point is the simple
+observation that clusters andballsareinverses of each oth ers, that is, w∈Ballx(v,Ai,Ai+1)
+is equivalent to v∈Cx(w,Ai,Ai+1). Now we make an important observation. Consider
+the subgraph Gi(w) induced by the vertices of set ∪x∈VCx(w,Ai,Ai+1). This subgraph
+preserves the path P(w,v,x) for each x,v∈Vifw∈Ballx(v,Ai,Ai+1). So it suffices
+to keep a single source (approximate) shortest paths oracle onGi(w) withwas the root.
+Keeping this data structure for each w∈Aiprovides an implicit compact data structure for
+supporting the basic operation mentioned above. Using Theo rem 4.5, it can be seen that
+the space required at a level iwill be of the order of/summationtext
+w∈Ai| ∪x∈VCx(w,Ai,Ai+1)|, but
+it is not clear whether we can get an upper bound of the order of n1+1/kon this quantity.
+Here, as a new tool, we introduce the notion of ǫ-truncated balls and clusters.
+Definition 5.1. Given a vertex x, any subsets A,B, andǫ >0
+Ballx(v,A,B,ǫ ) =/braceleftbigg
+w∈A|δ(v,w,x)<δ(v,B,x)
+1+ǫ/bracerightbigg
+Instead of dealing with the usual balls (and clusters) under deletion of single vertex,
+we deal with ǫ-truncated balls (and clusters) under deletion of single ve rtex. We note that
+the inverse relationship between clusters and balls gets se amlessly extended to ǫ-truncated
+balls and clusters under single vertex failure as well. That is,
+/summationdisplay
+w∈Ai|∪x∈VCx(w,Ai,Ai+1,ǫ)|=/summationdisplay
+v∈V|∪x∈VBallx(v,Ai,Ai+1,ǫ)|
+So it suffices to get an upper bound on the size of the set ∪x∈VBallx(v,Ai,Ai+1,ǫ) for any
+vertexv∈V. The following lemma states a very crucial property of ǫ-truncated balls which
+leads to prove the existence of a small set SofO(1
+ǫ2logn) vertices such that
+∪x∈VBallx(v,Ai,Ai+1,ǫ)⊆ ∪x∈SBallx(v,Ai,Ai+1)∪Ball(v,Ai,Ai+1) (5.1)
+Lemma 5.2. In a given graph G= (V,E), letvbe any vertex and let u=pi+1(v). Letx1
+andx2be any two vertices on the P(v,u)path with x1appearing closer to von this path
+andδ(v,Ai+1,x1)≤(1+ǫ)δ(v,Ai+1,x2). Then
+Ballx1(v,Ai,Ai+1,ǫ)⊆Ball(v,Ai,Ai+1)∪Ballx2(v,Ai,Ai+1)APPROXIMATE SHORTEST PATHS AVOIDING A FAILED VERTEX 523
+Proof.Letwbe any vertex in Ai. It suffices to show the following. If wdoes not belong
+toBall(v,Ai,Ai+1)∪Ballx2(v,Ai,Ai+1), thenwdoes not belong to Ballx1(v,Ai,Ai+1,ǫ).
+The proof is based on the analysis of the following two cases.
+Case 1 : The vertex x2is present in P (v,w,x1).
+Since,w /∈Ball(v,Ai,Ai+1), therefore, δ(v,w) is at least δ(v,u). Hence using triangle
+inequality, δ(v,x2) +δ(x2,w)≥δ(v,u). Nowδ(v,u) =δ(v,x2) +δ(x2,u) (sincex2lies on
+P(v,u)). Hence δ(x2,w)≥δ(x2,u). Moreover, since x1does not appear on P(x2,u), so
+δ(x2,u) =δ(x2,u,x1). So
+δ(x2,w,x1)≥δ(x2,u,x1) (5.2)
+Now it is given that x2∈P(v,w,x1), soP(v,w,x1) must be of the form P(v,x2,x1) ::
+P(x2,w,x1), the length of which is at least δ(v,x2,x1) +δ(x2,u,x1) using Equation 5.2.
+The latter quantity is at least δ(v,u,x1) which by definition is at least δ(v,Ai+1,x1). Hence
+w /∈Ballx1(v,Ai,Ai+1), and therefore, w /∈Ballx1(v,Ai,Ai+1,ǫ).
+Case 2 : The vertex x2is not present in P (v,w,x1).
+In this case, δ(v,w,x1) =δ(v,w,{x1,x2})≥δ(v,w,x2). The value δ(v,w,x2) is in turn
+at least δ(v,Ai+1,x2) sincew /∈Ballx2(v,Ai,Ai+1). It is given that δ(v,Ai+1,x2)≥
+δ(v,Ai+1,x1)
+1+ǫ, hence conclude that δ(v,w,x1)≥δ(v,Ai+1,x1)
+1+ǫ. Sow /∈Ballx1(v,Ai,Ai+1,ǫ).
+We shall now outline the construction of a small set Sof vertices which will satisfy
+Equation 5.1. Let u=pi+1(v) and let P(v,u) =v(=x0),x1,...,xℓ(=u). Observe that
+∪x∈VBallx(v,Ai,Ai+1,ǫ) =∪1≤j≤ℓBallxj(v,Ai,Ai+1,ǫ). For any node x∈P(u,v), let
+value(x) =δ(v,Ai+1,x), and let hbe the maximum valueof any node on this path. The
+setSis initially empty.
+Letα(1) be the largest index from [1 ,ℓ] such that value(xi)≥h/(1 +ǫ). It can
+be seen that for all j < α(1),δ(v,Ai+1,xj)≤(1 +ǫ)δ(v,Ai+1,xα(1)). Therefore, it
+follows from Lemma 5.2 that for each vertex x∈ {x1,...,xα(1)},Ballx(v,Ai,Ai+1,ǫ)⊆
+Ballxα(1)(v,Ai,Ai+1)∪Ball(v,Ai,Ai+1). So we insert xα(1)toS. Similarly α(2)∈
+[α(1) + 1,ℓ] be the greatest integer such that value(xα(2))≥h/(1 +ǫ)2. We add xα(2)
+toS, and so on. It can be seen that the set Sconstructed in this manner will satisfy
+Equation 5.1 and its size will be O(log1+ǫh) =O(logn
+ǫ).
+It can be shown using elementary probability theory that for eachx∈V, the set
+Ballx(v,Ai,Ai+1) has size O(n1/klogn) with high probability. Therefore, the construction
+of the set Soutlined above implies the following crucial bound for each v∈V,i < k −1
+which helps us design all-pairs approximate distance oracl e avoiding a failed vertex.
+|∪x∈VBallx(v,Ai,Ai+1,ǫ)|=O/parenleftbigg
+n1/klog2n
+ǫ/parenrightbigg
+Using this equation, and owing to inverse relationship betw een clusters and balls, it fol-
+lows that/summationtext
+w∈Ai| ∪x∈VCx(w,Ai,Ai+1,ǫ)|=O/parenleftBig
+n1+1/klog2n
+ǫ/parenrightBig
+. Our all-pairs approximate
+distance oracle avoiding any failed vertex will keep the fol lowing data structures.
+•Letpx
+i(v,ǫ) denoteavertex wfromAiwithδ(v,w,x)≤(1+ǫ)δ(v,px
+i(v),x). Wekeep
+a data structure Ni∀i < k, usingwhich we can retrieve px
+i(v,ǫ). Thisdata-structure
+is obtained by suitable augmentation of our single source (1 +ǫ)-approximate oracle.
+•For each w∈Ai, we keep our single source (1+ ǫ)-approximate oracle in Gi(w,ǫ)
+which is the subgraph induced by ∪x∈VCx(w,Ai,Ai+1,ǫ).524 N. KHANNA AND S. BASWANA
+It follows that the overall space required by the data struct ure will be O(kn1+1/klog3n
+ǫ4).
+The query algorithm and the analysis on the stretch of the app roximate distance reported
+by the oracle are similar in spirit to that of Thorup and Zwick [13] (see [10] for details).
+Theorem 5.3. Given an integer k >1and a fraction ǫ >0, an unweighted graph G=
+(V,E)can be processed to construct a data structure which can answ er(2k−1)(1 +ǫ)-
+approximate distance query between any two nodes u∈Vandv∈Vavoiding any single
+failed vertex in O(k)time. The total size of the data structure is O(kn1+1/klog3n
+ǫ4).
+Future work. (i)Can we design a data structure for single source (1+ ǫ)-approx. shortest
+paths avoiding a failed vertex for weighted graphs ? Such a da ta structure will immediately
+extend our all-pairs approx. distance oracle avoiding a fai led vertex to weighted graphs.
+(ii)How to design approx. distance oracles avoiding two or more f ailed vertices ? Recent
+work of Duan and Pettie [8], and Chechik et al. [5] provides ad ditional motivation for this.
+References
+[1] M. Ahmed and A. Lubiw. Shortest paths avoiding forbidden subpaths. In STACS ’09: Proceedings
+of 26th International Symposium on Theoretical Aspects of C omputer Science , pages 63–74, Freiburg,
+Germany, 2009. Schloss Dagstuhl - Leibniz-Zentrum fuer Inf ormatik, Germany.
+[2] S. Baswana and T. Kavitha. Faster algorithms for approxi mate distance oracles and all-pairs small
+stretch paths. In FOCS ’06: Proceedings of the 47th Annual IEEE Symposium on Fo undations of
+Computer Science , pages 591–602, Washington, DC, USA, 2006. IEEE Computer So ciety.
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+Nineteenth Annual ACM -SIAM Symposium on Discrete Algorith ms, pages 506–515, Philadelphia, PA,
+USA, 2009. Society for Industrial and Applied Mathematics.
+[9] J. Hershberger and S. Suri. Vickrey prices and shortest p aths: What is an edge worth? In FOCS ’01:
+Proceedings of the 42nd IEEE symposium on Foundations of Com puter Science , page 252, Washington,
+DC, USA, 2001. IEEE Computer Society.
+[10] N. Khanna and S. Baswana. Approximate shortest paths av oiding a failed vertex : optimal data struc-
+tures for unweighted graphs. http://www.cse.iitk.ac.in/ ∼sbaswana/publications/algorithmica-09.pdf .
+[11] E. Nardelli, G. Proietti, and P. Widmayer. Finding the m ost vital node of a shortest path. Theor.
+Comput. Sci. , 296(1):167–177, 2003.
+[12] L. Roditty. On the k-simple shortest paths problem in we ighted directed graphs. In SODA ’07: Proceed-
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+phia, PA, USA, 2007. Society for Industrial and Applied Math ematics.
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+This work is licensed under the Creative Commons Attributio n-NoDerivsLicense. To view a
+copyofthislicense,visit http://creativecommons.org/licenses/by-nd/3.0/ .
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+arXiv:1001.0825v1  [physics.soc-ph]  6 Jan 2010The Morphology of Urban Agglomerations for Developing Coun tries: A
+Case Study with China
+Kausik Gangopadhyay∗
+Indian Institute of Management Kozhikode,
+IIMK Campus P.O., Kozhikode 673570, India
+B. Basu†
+Physics and Applied Mathematics Unit
+Indian Statistical Institute
+Kolkata-700108, India
+Abstract
+In this article, the relationship between two well-accepte d empirical propositions regarding the dis-
+tribution of population in cities, namely, Gibrat’s law and Zipf’s law, are rigorously examined using
+the Chinese census data. Our findings are quite in contrast wi th the most of the previous studies
+performed exclusively for developed countries. This motiv ates us to build a general environment
+to explain the morphology of urban agglomerations both in de veloped and developing countries. A
+dynamic process of job creation generates a particular dist ribution for the urban agglomerations
+and introduction of Special Economic Zones (SEZ) in this abs tract environment shows that the
+empirical observations are in good agreement with the propo sed model.
+I. INTRODUCTION
+Social phenomenon is a pertinent topic of discussion among the Econ omists and Econophysicists -
+partly because, human behavior can be explained in terms of Econom icmotives as well as a manifestation
+of a complex natural system. One of the interesting observation is distribution of dwellers in different
+urban agglomerations. A simple empirical law, namely Zipf’s law [1], is ofte n successful in describing the
+distribution of populations for various cities [2] in a nation.
+In Economics, there is a body of literature devoted to explain morph ology of cities. The survey paper
+by Gabaix and Ioannides [3] enlists most of them. Krugman [4] have loo ked at the top 135 U.S. cities and
+have found that the log-rank of a city bears a linear relation to the lo g-size of the same in a significant
+way. The slope of the linear relation is also found to be quite close to on e as expected from the Zipf’s
+law.
+Gabaix [5] investigate into the growth of cities and their adherence t o the Zipf’s law. This is because
+Zipf’s law is not a static phenomenon, but is the outcome of a dynamic p rocess. Different cities have
+presumablydifferent growthprocesses. We can expressthe expe cted growthrateofa city with population
+Sas a random variable, µ(S). The standard deviation in the growth rate of cities with population Sare
+denoted by σ(S). If either µ(S) orσ(S) is a non-trivial function of Sat least in the upper tail of the
+distribution of S, there would be violations of Zipf’s law. This is a consequence of the Gib rat’s law being
+followed in the upper tail of the city distribution. Gibrat’s law propose s that the growth rate process of
+a city is independent of the size of the city. Therefore the mean gro wth rate and the standard deviation
+of the growth rate for a city is independent of its size. It must be cla rified that Gibrat law does not say
+that the growth rate of any city follows the same stochastic proce ss. It only says that there is no relation
+between growth rate of a city and its size.
+Gibrat law and its relation to Zipf’s law is particularly pertinent for a nat ion experiencing growth
+in urban inhabitants. A developing country is very different compare d to its developed counterparts in
+terms of economic and social structures. Therefore, the inter r elationship between this two empirical
+conjectures might be particularly interesting. A pertinent case st udy is the People’s Republic of China,
+∗Electronic address: kausik@iimk.ac.in
+†Electronic address: banasri@isical.ac.in, Fax:91+(033) 2577-30262
+where urbanization is taking place in a fast pace. We investigate into t he occurrence of these laws in case
+of China. The next section discusses our empirical analysis with the fi ndings. A model is proposed in
+Section III along with an appropriate simulation study. The concludin g remarks are noted in Section IV.
+II. DATA TREATMENT
+The People’s Republic of China conducted censuses in 1953, 1964, an d 1982. At the 2000 census, the
+total population stood at approximately 1.29533 billion, which is about 22% of total population in the
+world. 36% of the Chinese population used to reside in urban agglomer ations in 2000. We use the data
+[6] from 1990 and 2000 census (plotted in Fig. 1).
+10410510610710−410−310−210−1100
+City−size: x −−−−−−−−−−−−−>Complemenatry CDF: PC(x) −−−−−−−−−−−−−>
+(a)Census year 1990:
+rank of a city plotted
+against its size10410510610710810−410−310−210−1100
+City−size: x −−−−−−−−−−−−−>Complemenatry CDF: PC(x) −−−−−−−−−−−−−>
+(b)Census year 2000:
+rank of a city plotted
+against its size78910111213141516−202468101214
+City−size (ln Scale) −−−−−−−−−−−−−>Growth Rate −−−−−−−−−−−−−>
+(c)Scatter plot of city
+growth against city size
+(1990-2000)
+FIG. 1: Chinese Cities: 1990-2000
+A. Verification of Zipf’s Law
+Letp(·) be a probability density function of the city-size distribution. The c orresponding cumulative
+distribution function (CDF) and the complementary cumulative distr ibution function (CCDF) are given
+byP(·) andPC(·), respectively. By definition,
+P(x) =/integraldisplayx
+0p(x′)dx′;PC(x) = 1−P(x)
+In case of city-size distribution following the Zipf’s law,
+pα(x) =Cx−αandPC
+α(x) =C
+α−1x−(α−1)(1)
+whereαandCare constants. αis called the exponent of the power law. This family of power law
+distributions for α>1 are known as the Pareto distribution. From equation (1), it is obvio us thatpα(x)
+diverges to infinity for any value of α >1 asx→0. Therefore, some minimum value, xmin, is usually
+considered for the support of the Pareto distribution.
+TABLE I: Data Description : Values of city-population are reported in units of thousan ds. The left truncation
+of the data is determined through the value of xmin. The numbers in parenthesis represent the standard errors
+for the estimates Source: [6]
+Census n Min Max Mean Median First Third Estimate of α
+Year Value Value Quartile Quartile Linear Fit MLE
+2000 1462 50.08 14230.99 298.27 136.63 80.86 265.42 1.7544 2.2975
+(0.0018) (0.0572)
+1990 1345 25.02 7821.79 156.33 68.71 44.23 128.96 1.7701 2.2308
+(0.0032) (0.0736)
+The slope of the plot, in which log of the rank of a city, log( Rx), is plotted against the log of its
+population, log( x), has been used to estimate the exponent of the power law in almost all the previous3
+studies. It has been shown [5, 7] that this produces a biased estima te of the power law exponent.
+Alternatively the Maximum Likelihood Estimator[8]The MLE is given by th e expression, /hatwideαMLE= 1 +
+n/bracketleftBig/summationtextn
+i=1log/parenleftBig
+xi
+xmin/parenrightBig/bracketrightBig−1
+(MLE) produces the most efficient estimate. We find [9] the estimate ofαto be
+significantly bigger than 2 as a departure from the Zipf’s law (see Tab le I).
+B. Verification of Gibrat’s Law
+Thecitiesin theuppertail ofthesizedistributionfollowaconstantra teofgrowthforvariousdeveloped
+countries [10]. It is interesting to repeat this exercise for a develop ing nation, where urbanization is
+happening fast to notice any discrepancy among cities in terms of gr owth regarding size. We perform
+various non-parametric as well as parametric exercises on the dat a to find out the relationship between
+the size of of a city and its growth rate.
+8 9 10 11 12 13 14 15−10123456E(Growth | City Size)
+City Size (log scale)
+(a)Epanechnikov Kernel8 9 10 11 12 13 14 15−1−0.500.511.522.5E(Growth | City Size)
+City Size (log scale)
+(b)Gaussian Kernel
+FIG. 2: Kernel estimates of population growth against city- size (dotted line represents the 95% bootstrapped
+confidence interval)
+We plot the growth rate of population in all available urban agglomerat ionsfor the period of 1990-2000
+against the population of the corresponding urban agglomeration in 1990. The standard non-parametric
+measure is to use the Kernel estimates of local mean. Suppose, th e growth rate of a city, gi, bears some
+relation with the size of the city, Si, modeled as:
+gi=m(Si)+ǫi
+for alli= 1,2,...,n,nbeing the total number of cities with available data. The objective is t o find
+a smooth estimate of local means of growth rate over size and to ve rify whether there is any visible
+relationship between growth and size based on this estimate m(·).giis the growth rate of the ith city
+over 1990-2000. We perform a Kernel density regression in the su pport ofSi.[11] The local average
+smooths around the point s, and the smoothing is done using a kernel, i.e. a continuous weight fun ction
+symmetric around s. The bandwidth hof a kernel determines the scale of smoothing. The Nadaraya-
+Watson estimate [12] of m(·) is given by the following expression,
+ˆm(s) =n−1/summationtextn
+i=1Kh(s−Si)gi
+n−1/summationtextn
+i=1Kh(s−Si)
+We use two most popular Kernels, Gaussian and Epanechnikov. For G aussian Kernel, K(ψ) =
+(2π)−1/2exp/bracketleftbig
+−1
+2(ψ)2/bracketrightbig
+, and for the Epanechnikov Kernel, K(x) =3
+4/parenleftbig
+1−ψ2/parenrightbig
+·1|ψ|≤1. For both the
+kernels, we find that m(·) does depend on the size. The visual observation is verified throug h the fol-
+lowing regression, where the growth rate of a city is regressed on t he size of the city [13]. We find a
+significant [14] negative coefficient for the variable of city-size.
+gi= 2.635−4.681×10−7·S90+S00
+2
+(0.039) (8.982×10−8)4
+8 9 10 11 12 13 14 1500.511.522.533.544.5E(Variance in Growth | City Size)
+City Size (log scale)
+(a)Epanechnikov Kernel8 9 10 11 12 13 14 150123456E(Variance in Growth | City Size)
+City Size (log scale)
+(b)Gaussian Kernel
+FIG. 3: Kernel estimates of variances in population growth a gainst city-size (dotted line represents the 95%
+bootstrapped confidence interval)
+We conclude that there is a definite variation among cities in terms of g rowth process and the overall
+evidence indicates that the growth process is negatively biased aga inst the cities of higher sizes at least
+at the upper tail of the distribution.
+III. A MIGRATION BASED MODEL
+To illustrate the empirical anomalies found in the context of distribut ion of urban agglomerations in
+China, we can motivate our findings with a mathematical model of city formation. There are several
+recent attempts [15–17] to model urban growth. It uses the idea that the growth of cities resembles to
+that of the two-dimensional aggregates of particles. There are r esults in the the statistical physics of
+clusters regarding the growth of the two-dimensional aggregate s of particles. These results are applied in
+the context of modeling the population distribution of urban agglome rations. In particular, the model of
+diffusion limited aggregation(DLA) predicted the existence of only on e large fractal cluster that is almost
+perfectly screened from incoming development units so that almost all the cluster growth occurs in the
+extreme peripheral tips. The morphology of cities is also explained us ing a percolation model[18], where
+the scaling of the urban perimeter of individual cities and the distribu tion of system of cities are tested.
+The intermittency mechanism[19] is used to model[20] a large scale city formation and understand the
+universal properties of the social phenomenon of city formation a nd global demographic development. In
+a different approach[21], the laws of population growth is explained us ing theCity Clustering Algorithm
+(CCA). The CCA is used to examine Gibrat’s law of proportional growt h and finds that the mean growth
+rate of a cluster exhibits deviations from the Gibrat’s law.
+For China, we need a model that is consistent with the empirical phen omenons observed and yet
+models the violations of the power law as found in the data. However, it must be taken into account that
+in the developed countries, this empirical observations are often r eversed as we have found out from the
+literature. We introduce the aspect of Special Economic Zones in my model and explain the empirical
+anomalies in contrast to the developed countries in terms of Special Economic Zones. We construct a
+baseline environment without any Special Economic Zones. Then we a dd Special Economic Zones to that
+environment to observe any effect due to introduction of SEZ.[23]
+There areklocations in a country. Jobs are spawn one at a time. The probability o f a job being
+spawn in a location is a function number of already existing jobs in that location. More particularly,
+the probability of an additional job being created at the ithlocation is proportional to nγ
+i, whereniis
+the number of already existing jobs at the ithlocation. We let jobs spawn at different location until
+total number of jobs becomes N. The parameter γis an important parameter of scale. If γis 1, the
+growth rate of a city is independent of its size. On the other hand, if γis less than unity, larger cities
+are discriminated against regarding growth. A value of γbeing more than one means that the growth
+process favours the large cities to growth against the smaller cities .
+We introduce a migration based Special Economic Zones in this model. T he government introduce
+the feature of Special Economic Zones by giving special privileges to some cities. The privileged urban
+agglomerations are chosen in such a way that they are not from the most populous cities. A number of
+new jobs are created in the locations of the SEZs. These new jobs r equire higher skill levels compared
+to the previously existing jobs. A worker matched with these jobs le ave their old locations of work and5
+move to the new location. Also higher skilled workers are primarily from the top ranking cities.
+A. A Simulation Study
+To evaluate the performance of our economically tenable model, we r esort to the widely used technique
+of simulation. We choose 3,000 locations ( k) and one million agents ( N). Jobs are spawn randomly in
+various locations are defined in our framework until the total numb er of spawned jobs is equal to total
+number of agents. We choose the value of γto be 0.9 so that there is a negative bias towards the
+growth of top ranking cities as observed as observed in the data. W e consider the top 2,500 locations and
+estimate the power law coefficient using the maximum likelihood method. we find ˆαMLEto be 1.0419
+with standard error of the estimate being 0.0208.This baseline study is devoid of any SEZ and is quite
+in accordance with the Zipf’s Law.
+10010110210310410−410−310−210−1100
+City−size (log scale) −−−−−−−−−−−−−>Complemenatry CDF: PC(x) (log scale)−−−−−−−−−−−−−>
+(a)Beforeintroduction of SEZs,
+city sizes plotted against
+corresponding ranks10010110210310410−410−310−210−1100
+City−size (log scale) −−−−−−−−−−−−−>Complemenatry CDF: PC(x) (log scale)−−−−−−−−−−−−−>
+(b)Afterintroduction of SEZs, city
+sizes plotted against corresponding
+ranks
+FIG. 4: Simulation study for the model
+To introduce SEZ in this model, we randomly select 270 locations outsid e the top 300 locations and
+introduceanumberofnewjobsinthoselocationsequaling20%ofalre adyexistingjobsintheeconomy.[24]
+Workersfrom the top 300locations arerandomly matched with the n ewly created jobs and once matched,
+theymigratetothelocationoftheirnewjobs. Wecompute ˆ αMLEinthesamewayconsideringtopranking
+2,500 locations and find it to be 1.2667 with 0.0259 to be the standard e rror of the estimate. This is
+demonstrative of the high value of αestimated using the data for China. Moreover, αestimated for
+the census year of 2000 is higher than that for the census year of 1990. It is associated with the rising
+importance of SEZs in the Chinese economy.
+IV. DISCUSSION
+Economists often surmise[3] that Zipf’s law is the consequence of Gib rat’s law as far as city-size
+distribution isconcerned. Asimultaneousviolationofboth isnatural. However,Gibrat’slawis associated
+with the free market economy[22]. A breech in Gibrat’s law implies a wedg e in the free market. A
+possible source of this wedge is debatable. We focus on government ’s intervention on the natural process
+of morphologyof cities. The cities under SEZ are subject to very diff erent economic regulationscompared
+to their counterparts in the rest of the country. This is analogous to a wedge in a perfectly competitive
+economic system.
+It has been pointed out[10] that the Zipf’s exponent does depend o n the cut-off in the upper tail of
+the city size distribution. The difference in socio-economic structur e may give rise to different values of
+the Zipf’s exponent with the same minimum cut-off. It is observed tha t in case of China, the exponent of
+Zipf’s law augments for the year of 2000 compared to the value in the year of 1990. However, number of
+locations above the minimum cut-off are quite close (see Table I). This phenomenon cannot be explained
+by a static process as modeled in [10]. Nevertheless, our model rec onciles this empirical scenario with6
+the gradual importance of SEZs in China.
+[1] G.K.Zipf, Human Behavior and the Principle of Least Effort (Addison-Wesley, Cambridge, MA, 1949)
+[2] In this article, “Urban Agglomeration” and “City” have t hroughout been used interchangeably. The literature
+starting from the Zipf’s law have historically looked into t he population distribution of the urban areas
+formally denoted as “Cities”. However the more general noti on of urban agglomerations have been used in
+the relatively recent literature.
+[3] The Evolution of City Size Distributions, Xavier Gabaix and Yannis Ioannides, Handbook of Regional and
+Urban Economics 4, V. Henderson and J-F. Thisse eds, 2004, North-Holland, 234 1-2378.
+[4] The Self-Organizing Economy, (Blackwell Publishers Ox ford, UK and Cambridge, MA).
+[5] Xavier Gabaix, Zipfs Law for Cities: An Explanation, Qua rterly Journal of Economics 114 (3), August 1999,
+p.739-67.
+[6] Chinese Census data uploaded in http://www.citypopulation.de/China.html .
+[7] A. Clauset, C. R. Shalizi, and M. J. Newman, Power-law distributions in empirical data , arXiv:0706.1062v1.
+[8] The MLE is given by the expression, /hatwideαMLE= 1+n/bracketleftBig/summationtextn
+i=1log/parenleftBig
+xi
+xmin/parenrightBig/bracketrightBig−1
+[9] Kausik Gangopadhyay and B. Basu, City Size Distribution for India and China, Physica A 388(2009),
+2682-2688.
+[10] Jan Eeckhout, Gibrats Law for (all) Cities, American Ec onomic Review 94(5), (2004), 1429-1451.
+[11] We have chosen the interval [1 .1·min
+iSi,0.9·max
+iSi] to exclude the effect of the boundaries to some extent.
+[12] Adrian Pagan and Aman Ullah, Nonparametric Econometri cs, Cambridge University Press, 1999.
+[13] We choose the average of the populations in 1990 and 2000 at a city, respectively denoted as S90andS00, as
+the population of the city. We can however choose the populat ion of the city in 1990 as the regression. The
+result is quite similar even quantitatively.
+[14] The standard errors of the estimates are shown in the par enthesis below.
+[15] M. Batty and P. Longley, Fractal Cities (Academic Press , San Diego, 1994)
+[16] L. Benguigui, A new aggregation model. Application to t own growth, Physica A 219,(1995) 13
+[17] T.M. Witten and L.M. Sander, Diffusion-Limited Aggrega tion, a Kinetic Critical Phenomenon, Phys. Rev.
+Lett47(1981) 1400
+[18] H A. Makse et al. Modeling Urban Growth Patterns with Cor related Percolation, arXiv:cond-mat/9809431v1
+[cond-mat.dis-nn], Phys. Rev. E (1 December 1998); Nature 3 77, 608 (1995)
+[19] Y.B. Zeldovich et al. The Almighty Chance (World Scient ific, Singapore 1990)
+[20] D. H. Zanette and S.C. Manrubi, Role of Intermittency in Urban Development: A Model of Large-Scale City
+Formation, Phys. Rev. Lett. 79(1997) 523
+[21] H.D. Rozenfeld et. al.,Laws of population growth, arXi v:08082202
+[22] Juan Carlos Cordoba, On the Distribution of City Sizes, Journal of Urban Economics, January 2008.
+[23] A Special Economic Zone (SEZ) is a geographical region t hat has economic laws that are more liberal than
+a country’s typical economic laws. The category ’SEZ’ cover s a broad range of more specific zone types,
+including Free Trade Zones (FTZ), Export Processing Zones ( EPZ), Free Zones (FZ), Industrial Estates
+(IE), Free Ports, Urban Enterprize Zones and others. Usuall y the goal of a structure is to increase foreign
+investment. One of the earliest and the most famous Special E conomic Zones were found by the government
+of the People’s Republic of China under Deng Xiaoping in the e arly 1980s. The most successful Special
+Economic Zone in China, Shenzhen, has developed from a small village into a city with a population over
+ten million within 20 years.
+[24] There is nothing special about the numbers used in this c onstructed model. A numerical experiment with
+different values for the parameters would qualitatively yie ld the same response.
\ No newline at end of file
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+arXiv:1001.0826v1  [math.RT]  6 Jan 2010La conjecture locale de Gross-Prasad pour les groupes
+sp´ eciaux orthogonaux : le cas g´ en´ eral
+C. Moeglin, J.-L. Waldspurger
+31 d´ ecembre 2009
+Introduction
+SoitFun corps local non archim´ edien de caract´ eristique nulle. On appelle espace
+quadratique un couple ( V,q) form´ e d’un espace vectoriel VsurFde dimension finie et
+d’une forme bilin´ eaire qsurV, sym´ etrique et non d´ eg´ en´ er´ ee. Soient ( V,q) et (V′,q′) deux
+espaces quadratiques. On note detd′leurs dimensions et GetG′leurs groupes sp´ eciaux
+orthogonaux. On suppose d>d′etdetd′de parit´ es distinctes. On suppose donn´ ee une
+d´ ecomposition orthogonale de ( V,q) en la somme de ( V′,q′) et d’un espace quadratique
+qui est lui-mˆ eme somme orthogonale de plans hyperboliques et d’une droite quadratique
+(D,qD). Le groupe G′est alors un sous-groupe de G. On d´ efinit ν0∈F×/F×,2en fixant
+un ´ el´ ement non nul v0∈Det en posant
+ν0= (−1)dqD(v0,v0)/2.
+Soientσ, resp.σ′, une repr´ esentation admissible irr´ eductible de G(F), resp.G′(F). Gross
+et Prasad ont d´ efini une multiplicit´ e m(σ,σ′). Par exemple, dans le cas o` u d=d′+ 1,
+introduisons des espaces EσetEσ′dans lesquels se r´ ealisent σetσ′. Alorsm(σ,σ′) est
+la dimension de l’espace complexe des applications lin´ eaires l:Eσ→Eσ′telles que
+l◦σ(g′) =σ′(g′)◦lpour toutg′∈G′(F). La d´ efinition g´ en´ erale est rappel´ ee en 1.2.
+D’apr` es [AGRS] et [GGP] corollaire 15.2, on a toujours m(σ,σ′)≤1.
+Rappelons la classification conjecturale des repr´ esentations adm issibles irr´ eductibles
+deG(F) dans le cas dimpair. Pour tout entier naturel pair N, on fixe une forme sym-
+plectique sur CNet on note Sp(N,C) son groupe symplectique. Notons WFle groupe
+de Weil absolu de FetWDF=WF×SL(2,C) le groupe de Weil-Deligne. Notons
+Φ(G) l’ensemble des classes de conjugaison par Sp(d−1,C) d’homomorphismes conti-
+nusϕ:WDF→Sp(d−1,C) qui sont semi-simples et dont la restriction ` a SL(2,C)
+est alg´ ebrique. On conjecture que l’ensemble des classes de repr´ esentations admissibles
+irr´ eductibles de G(F) est union disjointe de L-paquets ΠG(ϕ) index´ es par les ϕ∈Φ(G).
+Cette classification se ram` ene ` a celle des repr´ esentations temp ´ er´ ees de la fa¸ con suivante.
+Consid´ erons les donn´ ees suivantes :
+•un L´ eviˆL=GL(d1,C)×...×GL(dt,C)×Sp(d0−1,C) deSp(d−1,C);
+•des homomorphismes ϕ0:WDF→Sp(d0−1,C) etϕj:WDF→GL(dj,C) pour
+j= 1,...,t, v´ erifiant les mˆ emes conditions que ϕet qui sont temp´ er´ es, c’est-` a-dire que
+les images de WFpar ces homomorphismes sont relativement compactes;
+•des r´ eelsb1>b2>...>b t>0.
+Notons|.|Fla valeur absolue usuelle de F×, que l’on identifie par la th´ eorie du corps
+de classes ` a un caract` ere de WF, puis deWDF. Introduisons l’homomorphisme
+ϕˆL= (ϕ1⊗|.|b1
+F)⊗...⊗(ϕt⊗|.|bt
+F)⊗ϕ0
+1deWDFdansˆL. En le poussant par l’inclusion de ˆLdansSp(d−1,C), il devient un
+´ el´ ement de Φ( G). Inversement, soit ϕ∈Φ(G). Alors il existe des donn´ ees comme ci-
+dessus, uniques ` a conjugaison pr` es, de sorte que ϕ=ϕˆL. Il peut ne correspondre ` a ˆL
+aucun L´ evi de G(c’est le cas si Gn’est pas d´ eploy´ e et d0= 1). Dans ce cas, on pose
+ΠG(ϕ) =∅et, pour unifier les notations, S(ϕ)/S(ϕ)0={1},EG(ϕ) =∅. Supposons qu’` a
+ˆLcorresponde un L´ evi LdeG. On a
+L=GL(d1)×...×GL(dt)×G0,
+o` uG0est un groupe de mˆ eme type que G. Pour tout j= 1,...,t, notonsπ(ϕj) la
+repr´ esentation admissible irr´ eductible de GL(dj,F) d´ etermin´ ee par ϕjpar la corres-
+pondance de Langlands, prouv´ ee par Harris-Taylor et Henniart. Admettons la conjec-
+ture pour les repr´ esentations temp´ er´ ees. A ϕ0correspond un L-paquet ΠG0(ϕ0) de
+repr´ esentations temp´ er´ ees de G0(F). SoitPle sous-groupe parabolique de G, de com-
+posante de L´ evi L, pour lequel la suite ( b1,...,bt) est positive dans un sens usuel. Pour
+σ0∈ΠG0(ϕ0), notonsσle quotient de Langlands de la repr´ esentation induite
+IndG
+P((π(ϕ1)⊗|det|b1
+F)⊗...⊗(π(ϕt)⊗|det|bt
+F)⊗σ0),
+c’est-` a-diresonuniquequotientirr´ eductible.OnnoteΠG(ϕ)l’ensembledecesrepr´ esentations
+σquandσ0d´ ecrit ΠG0(ϕ0). Rappelons que le paquet ΠG0(ϕ0) est param´ etr´ e (conjectu-
+ralement) par un ensemble EG0(ϕ0) de caract` eres d’un groupe S(ϕ0)/S(ϕ0)0isomorphe
+` a un produit de facteurs Z/2Z. On a rappel´ e ce param´ etrage en [W1] 4.2, on le note
+ǫ/maps⊔o→σ(ϕ0,ǫ). On pose S(ϕ)/S(ϕ)0=S(ϕ0)/S(ϕ0)0,EG(ϕ) =EG0(ϕ0) et, pour un
+´ el´ ementǫ∈ EG(ϕ), on noteσ(ϕ,ǫ) la repr´ esentation d´ eduite de σ0=σ(ϕ0,ǫ).
+Les repr´ esentations de G(F) se param` etrent de fa¸ con similaire dans le cas o` u dest
+pair. Pour tout entier naturel N, on fixe une forme bilin´ eaire sym´ etrique non d´ eg´ en´ er´ ee
+surCNetonnoteO(N,C),resp.SO(N,C),songroupeorthogonal,resp. sp´ ecial orthogo-
+nal. Consid´ erons un homomorphisme ϕ:WDF→O(N,C) v´ erifiant les mˆ emes conditions
+que pr´ ec´ edemment. Par composition avec le d´ eterminant, on ob tient un caract` ere qua-
+dratique de WDF, forc´ ement trivial sur SL(2,C), donc un caract` ere quadratique de WF.
+Par la th´ eorie du corps de classes, il d´ etermine un ´ el´ ement δ(ϕ)∈F×/F×,2. D´ efinissons
+un autre ´ el´ ement de ce groupe par δ(q) = (−1)d/2det(q). On note Φ( G) l’ensemble des
+classes de conjugaison par SO(d,C) d’homomorphismes ϕ:WDF→O(d,C) v´ erifiant
+les conditions pr´ ec´ edentes et tels que δ(ϕ) =δ(q). Pourϕ∈Φ(G), on d´ efinit comme
+ci-dessus le L-paquet ΠG(ϕ) : ou bien il est vide, ou bien c’est l’ensemble des quotients de
+Langlandsissusderepr´ esentations π(ϕj)⊗|det|bj
+FdegroupesGL(dj,F)etdes´ el´ ements σ0
+d’unL-paquet ΠG0(ϕ0). LeL-paquet est param´ etr´ e par un ensemble EG(ϕ) =EG0(ϕ0)
+de caract` eres d’un groupe S(ϕ)/S(ϕ)0=S(ϕ0)/S(ϕ0)0produit de facteurs Z/2Z. Si-
+gnalons toutefois que, pour associer un L´ evi LdeG` a un L´ evi ˆLdeO(d,C), il faut
+consid´ ererO(d,C) comme le L-groupe deG, ce qui sous-entend que l’on fixe des donn´ ees
+suppl´ ementaires cach´ ees (pr´ ecis´ ement, on identifie un sous -tore maximal de SO(d,C) au
+L-groupe d’un sous-tore maximal de G).
+On admet la validit´ e des conjectures pour les repr´ esentations te mp´ er´ ees, telles qu’on
+les a formul´ ees en [W1] 4.2 et 4.3, compl´ et´ ees comme en 2.1 ci-des sous dans le cas
+dimpair. Dans [W1] 4.8, on a pr´ ecis´ e les param´ etrages des L-paquets temp´ er´ es et ce
+sont ces param´ etrages que nous utilisons. Les param´ etrages d esL-paquets de G(F) ne
+d´ ependent d’aucune donn´ ee auxiliaire dans le cas o` u dest impair. Ils d´ ependent par
+contre de l’´ el´ ement ν0d´ efini ci-dessus dans le cas o` u dest pair. Celui-ci sert ` a normaliser
+les facteurs de transfert.
+2Soitϕ∈Φ(G).SiGest quasi-d´ eploy´ e, ondit que ϕest g´ en´ erique s’il existe σ∈ΠG(ϕ)
+et un type de mod` eles de Whittaker de sorte que σadmette un mod` ele de ce type. Rap-
+pelons que si dest impair, il n’y a qu’un seul type de mod` eles de Whittaker tandis que ,
+sidest pair, il y a plusieurs types de tels mod` eles, autant que d’orbites unipotentes
+r´ eguli` eres de G(F). SiGn’est pas quasi-d´ eploy´ e, on introduit un espace quadratique
+(V,q) de mˆ emes dimension et d´ eterminant que ( V,q), mais dont le groupe sp´ ecial or-
+thogonalGest quasi-d´ eploy´ e. A ϕest associ´ e un L-paquet ΠG(ϕ) de repr´ esentations de
+G(F). On dit que ϕest g´ en´ erique s’il existe un ´ el´ ement σde ce paquet et un type de
+mod` eles de Whittaker de sorte que σadmette un mod` ele de ce type.
+On peut ´ evidemment remplacer GparG′dans les consid´ erations ci-dessus. Soient
+ϕ∈Φ(G) etϕ′∈Φ(G′). En suivant Gross et Prasad, on a d´ efini en [W1] 4.9 un signe
+E(ϕ,ϕ′) et des caract` eres ǫdeS(ϕ)/S(ϕ)0etǫ′deS(ϕ′)/S(ϕ′)0. On pose
+µ(G,G′) =/braceleftbigg1,siGetG′sont quasi-d´ eploy´ es,
+−1, sinon.
+On sait que si E(ϕ,ϕ′) =µ(G,G′),ǫappartient ` a EG(ϕ) etǫ′appartient ` a EG′(ϕ′). Si
+au contraire E(ϕ,ϕ′) =−µ(G,G′),ǫn’appartient pas ` a EG(ϕ) etǫ′n’appartient pas ` a
+EG′(ϕ′).
+En plus des conjectures de classification, on doit admettre certain s r´ esultats issus de
+la formule des traces locale tordue, afin d’assurer la validit´ e du th´ e or` eme de [W2].
+Notre principal r´ esultat est le th´ eor` eme suivant, qui sera d´ e montr´ e dans la section 3.
+Th´ eor` eme .Soientϕ∈Φ(G)etϕ′∈Φ(G′). On suppose ϕetϕ′g´ en´ eriques. Alors :
+(i) toutes les repr´ esentations induites dont les ´ el´ ements de ΠG(ϕ)et deΠG′(ϕ′)sont
+les quotients de Langlands sont irr´ eductibles;
+(ii) siE(ϕ,ϕ′) =−µ(G,G′), on am(σ,σ′) = 0pour tousσ∈ΠG(ϕ),σ′∈ΠG′(ϕ′);
+(iii) siE(ϕ,ϕ′) =µ(G,G′), on a
+m(σ(ϕ,ǫ),σ′(ϕ′,ǫ′)) = 1
+etm(σ,σ′) = 0pour tousσ∈ΠG(ϕ),σ′∈ΠG′(ϕ′)tels que(σ,σ′)/\e}a⊔io\slash= (σ(ϕ,ǫ),σ′(ϕ′,ǫ′)).
+Pourdimpair,lesconjectures declassificationdesrepr´ esentations de G(F)fontpartie
+des r´ esultats annonc´ es par Arthur, au moins dans le cas d’un gro upe quasi-d´ eploy´ e ([A1]
+th´ eor` eme 30.1). Dans le cas dpair, Arthur annonce un r´ esultat un peu plus faible, o` u on
+ne distingue pas deux repr´ esentations conjugu´ ees par le group e orthogonal tout entier,
+cf. [W1] 4.4. En admettant seulement ces conjectures plus faibles, on peut ´ enoncer un
+th´ eor` eme similaire ` a celui ci-dessus, o` u on regroupe les repr´ e sentations conjugu´ ees par le
+groupe orthogonal.
+Il y a trois ingr´ edients dans la preuve du th´ eor` eme. En pr´ ecisa nt un raisonnement
+dˆ u ` a Gan, Gross et Prasad, on prouve dans la premi` ere section que les multiplicit´ es
+sont compatibles ` a l’induction, sous des hypoth` eses convenables (proposition 1.3). Cela
+ram` ene les assertions (ii) et (iii) du th´ eor` eme au cas temp´ er´ e, pourvu que toutes les in-
+duites intervenant soient irr´ eductibles, c’est-` a-dire pourvu qu e l’assertion (i) soit v´ erifi´ ee.
+Dans la deuxi` eme section, on ´ etablit un crit` ere g´ en´ eral d’irr´ eductibilit´ e pour ces in-
+duites (th´ eor` eme 2.13). Signalons qu’il est ´ egalement vrai pour les groupes symplec-
+tiques. La cons´ equence de ce crit` ere est que, parmi ces induite s, il y en a qui sont ”les
+plus r´ eductibles” qu’il est possible. C’est-` a-dire que si l’une des ind uites est r´ eductible,
+3celles-ci le sont aussi. Ce sont celles pour lesquelles la repr´ esentat ion que l’on induit ad-
+met un mod` ele de Whittaker (il y en a plusieurs s’il y a plusieurs types d e tels mod` eles).
+Pour obtenir le (i) du th´ eor` eme, il reste ` a utiliser une conjectur e de Shahidi d´ emontr´ ee
+par Muic qui affirme que si un quotient de Langlands est g´ en´ erique (ce qui fait partie
+de nos hypoth` eses : les paquets sont g´ en´ eriques), alors l’induit e dont il est quotient est
+irr´ eductible.
+1 Induction et multiplicit´ es
+1.1 Notations
+Selon le cas, on d´ esigne une repr´ esentation d’un groupe soit par la repr´ esentation elle-
+mˆ eme,disons π,soitparunespace Eπdanslequelelleser´ ealise.Onintroduiraparfoisune
+repr´ esentation σd’un groupe sp´ ecial orthogonal sans pr´ eciser au d´ epart de qu el groupe
+il s’agit. Dans ce cas, on notera dσla dimension de l’espace quadratique ( V,q) du groupe
+sp´ ecial orthogonal duquel σest une repr´ esentation. De mˆ eme, on introduira parfois une
+repr´ esentation πd’ungroupelin´ eaire(onentendparl` aungroupe GL(d,F))sanspr´ eciser
+ce groupe. On notera dπl’entier tel que πest une repr´ esentation de GL(dπ,F). On note
+ˇσla contragr´ ediente d’une repr´ esentation σ.
+Soient(V,q)unespacequadratiquedegroupesp´ ecialorthogonal Getσunerepr´ esentation
+lisse deG(F). La notation
+σ=π1×...×πt×σ0,
+ou encore
+σ= (×i=1,...,tπi)×σ0
+signifie ce qui suit. On fixe une d´ ecomposition
+V=X1⊕...⊕Xt⊕V0⊕Yt⊕...⊕Y1.
+On suppose que les espaces XietYisont totalement isotropes et non nuls. En posant
+Vi=Xi⊕Yipouri= 1,...,t, on suppose que les espaces Vi, pouri= 0,...,tsont
+orthogonaux. On note dila dimension de Xi(ou encore celle de Yi) etd0celle deV0. On
+notePle sous-groupe parabolique de Gqui conserve le drapeau
+X1⊂X1⊕X2⊂...⊂X1⊕...⊕Xt,
+Uson radical unipotent et Msa composante de L´ evi form´ ee des ´ el´ ements qui conserven t
+tous les espaces XietYi. On noteG0le groupe sp´ ecial orthogonal de V0et on fixe une
+base deXiqui permet de noter GL(di) le groupe lin´ eaire de l’espace Xi. On a l’´ egalit´ e
+M=GL(d1)×...×GL(dt)×G0.
+Le termeσ0est une repr´ esentation lisse de G0(F) et, pour tout i= 1,...,t,πiest une
+repr´ esentation lisse de GL(di,F). Alorsσest la repr´ esentation
+IndG
+P(π1⊗...⊗πt⊗σ0)
+deG(F).
+4Dans le cas o` u la dimension ddeVest paire, cette notation est ambig¨ ue car cette
+repr´ esentation induite peut d´ ependre du choix de la d´ ecomposit ion deV. En pratique,
+l’important sera que ces d´ ecompositions soient choisies de fa¸ con c oh´ erentes.
+On utilise une notation similaire
+π=π1×...×πt=×i=1,...,tπi
+pour des repr´ esentations de groupes lin´ eaires.
+Soitπune repr´ esentation admissible irr´ eductible d’un groupe lin´ eaire. E lle poss` ede
+un caract` ere central ωπ, qui est un caract` ere de F×. Il existe un unique r´ eel eπtel que
+|ωπ(λ)|=|λ|eπdπ
+Fpour toutλ∈F×. On appelle eπl’exposant de π. Soitb∈R. On
+noteπ|.|b
+Fla repr´ esentation π⊗ |det|b
+F. Siπest unitaire, on a eπ|.|b
+F=b. On appelle
+support cuspidal de πl’ensemble avec multiplicit´ es {ρ1,..,ρk}form´ e de repr´ esentations
+irr´ eductibles et cuspidales tel que πsoit un sous-quotient de l’induite ×i=1,...,kρi. On
+utilise ici, ` a la suite de Zelevinsky, la notion d’ensembles avec multiplicit´ e s. Ils sont
+toujours finis et peuvent ˆ etre consid´ er´ es comme des familles fin ies prises ` a l’ordre pr` es.
+Soienteetfdeux r´ eels tels que e−f∈N. Par abus de notation, on d´ esigne par
+[e,f] l’ensemble des r´ eels xtels quee≥x≥fete−x∈N. On ordonne cet ensemble
+par l’ordre oppos´ e de l’ordre ordinaire. Soit ρune repr´ esentation admissible irr´ eductible
+et cuspidale d’un groupe lin´ eaire. On note < e,f > ρl’unique sous-module irr´ eductible
+de la repr´ esentation induite ×x∈[e,f]π|.|x
+F, le produit ´ etant pris dans l’ordre que l’on vient
+de pr´ eciser. Dans le cas particulier o` u ρest unitaire et o` u e= (a−1)/2 etf= (1−a)/2
+pour un entier a≥1, on note aussi St(ρ,a) cette repr´ esentation (une repr´ esentation de
+Steinberg g´ en´ eralis´ ee). En g´ en´ eral, < e,f > ρest une telle repr´ esentation de Steinberg
+g´ en´ eralis´ ee tordue par le caract` ere |det|(e+f)/2
+F. Pour simplifier la notation, on utilisera
+aussi la notation [ e,f] dans le cas o` u f=e+1. Par convention, [ e,f] =∅dans ce cas.
+SoitN≥1unentier.Pourtout k= 1,...,NnotonsQN−k,klesous-groupeparabolique
+triangulaire sup´ erieur de GLNde composante de L´ evi
+LN−k,k=GLN−k×GL(1)×...×GL(1),
+avecktermesGL(1). Notons UN−k,kson radical unipotent et PN−k,kle sous-groupe de
+QN−k,kform´ e des ´ el´ ements dont les projections dans les kderniers facteurs GL(1) de
+LN−k,kvalent 1. En particulier, PN−1,1est le groupe appel´ e ”mirabolique”. On fixe un
+caract` ere continu et non trivial ψFdeFet on d´ efinit un caract` ere ψkdePN−k,k(F) par
+ψk(p) =ψF(/summationdisplay
+i=N−k+1,...,N−1pi,i+1),
+o` u lespi,i+1sont les coefficients de p.
+1.2 D´ efinition des multiplicit´ es
+On fixe pour la suite de la section deux espaces quadratiques ( V,q) et (V′,q′) comme
+dans l’introduction. On pose r= (d−d′−1)/2 et on introduit un espace Zde dimension
+2r+1 surF, muni d’une base ( vk)k=−r,...,r. On d´ efinit une forme bilin´ eaire sym´ etrique
+qZsurZpar
+qZ(/summationdisplay
+k=−r,...,rλkvk,/summationdisplay
+k=−r,...,rλ′
+kvk) = 2(−1)dν0λ0λ′
+0+/summationdisplay
+k=1,...,r(λ−kλ′
+k+λkλ′
+−k).
+5On fixe un isomorphisme de ( V,q) sur la somme orhogonale ( V′,q′)⊕(Z,qZ). On note
+Qle sous-groupe parabolique de Gqui conserve le drapeau de sous-espaces
+Fvr⊂Fvr⊕Fvr−1⊂...⊂Fvr⊕...⊕Fv1.
+On noteNson radical unipotent. On d´ efinit un caract` ere ψNdeN(F) par la formule
+ψN(n) =ψF(/summationdisplay
+k=0,...,r−1q(nvk,v−k−1)).
+Le groupeG′est inclus dans QetψNest invariant par conjugaison par G′(F). Soient
+σetσ′des repr´ esentations lisses de G(F) etG′(F). On noteHomG′(F),ψF(σ,σ′) l’espace
+des applications lin´ eaires l:Eσ′→Eσtelles que
+l(σ(ng′)e) =ψN(n)σ(g′)l(e)
+pour tousn∈N(F),g′∈G′(F),e∈Eσ. On notem(σ,σ′) oum(σ′,σ) la dimension
+de cet espace HomG′(F),ψF(σ′,σ). Cette dimension ne d´ epend pas des choix effectu´ es
+(l’espaceZ, sa base, le caract` ere ψF). Siσetσ′sont irr´ eductibles, on a m(σ,σ′)≤1.
+1.3 Induction
+Consid´ erons une repr´ esentation induite
+σ=π1|.|b1
+F×...×πt|.|bt
+F×σ0
+deG(F), o` u
+•pouri= 1,...,t,πiest une repr´ esentation admissible irr´ eductible et temp´ er´ ee d’u n
+groupe lin´ eaire GL(Ni,F);
+•σ0est une repr´ esentation admissible irr´ eductible et temp´ er´ ee de G0(F), o` uG0est
+le groupe sp´ ecial orthogonal d’un certain sous-espace V0deV, cf. 1.1;
+•lesbisont des nombres r´ eels tels que b1≥b2≥...≥bt≥0.
+On consid` ere une repr´ esentation
+σ′=π′
+1|.|b′
+1
+F×...×π′
+t′|.|b′
+t′
+F×σ′
+0
+deG′(F) v´ erifiant des hypoth` eses similaires.
+On a d´ efini la multiplicit´ e m(σ,ˇσ′) dans le paragraphe pr´ ec´ edent . On remarque qu’` a
+conjugaison pr` es, on peut supposer que le plus petit des deux esp acesV0etV′
+0est inclus
+dans le plus grand et qu’alors, ces deux espaces verifient les mˆ emes hypoth` eses que Vet
+V′, avec la mˆ eme constante ν0. On peut donc d´ efinir la multiplicit´ e m(σ0,σ′
+0). La section
+est consacr´ ee ` a la preuve de la proposition suivante.
+Proposition .Sous ces hypoth` eses, on a l’´ egalit´ e
+m(σ,ˇσ′) =m(σ0,σ′
+0).
+Remarque. Onpourraitremplacer σ0ouσ′
+0parleurscontragr´ edientesdanslesecond
+membre de cette ´ egalit´ e, cf. [W3] 7.1(1).
+Convention. Dans la suite interviendront des in´ egalit´ es portant sur b1etb′
+1. Dans le
+cas o` ut= 0, auquel cas b1n’est pas d´ efini, on consid` ere que b1= 0 dans ces in´ egalit´ es.
+De mˆ eme pour b′
+1.
+61.4 Une premi` ere in´ egalit´ e
+On suppose d=d′+1. On consid` ere les donn´ ees du paragraphe pr´ ec´ edent en ce qui
+concerne le groupe G′. On consid` ere une repr´ esentation induite
+σ=π|.|b
+F×σ0
+deG(F),o` uπestunerepr´ esentationadmissibleirr´ eductibled’ungroupelin´ eair eGL(N,F),
+unitaire et de la s´ erie discr` ete, best un r´ eel et σ0est une repr´ esentation admissible, pas
+forc´ ement irr´ eductible, d’un groupe G0(F).
+Lemme.Supposons b≥b′
+1. Alorsm(σ,ˇσ′)≤m(σ′,ˇσ0).
+Conform´ ement ` a notre convention, l’in´ egalit´ e b≥b′
+1signifieb≥0 sit′= 0. La
+d´ emonstration ci-dessous est inspir´ ee de celle du th´ eor` eme 15 .1 de [GGP].
+Preuve. R´ ealisons σcomme en 1.1. On fixe une d´ ecomposition
+V=X⊕V0⊕Y
+comme dans ce paragraphe, dont on d´ eduit des groupes Pet
+M=GL(N)×G0.
+On a
+σ=IndG
+P(π|.|b
+F⊗σ0).
+A toutg∈P(F)\G(F), associons le sous-espace g−1(X) deV. Il est totalement isotrope
+etdedimension N.Notons Ulesous-ensemble des g∈P(F)\G(F)telsquedim(g−1(X)∩
+V′) =N−1 etXcelui desgtels quedim(g−1(X)∩V′) =N. AlorsUest un ouvert de
+compl´ ementaire le ferm´ e X. On v´ erifie que Uest form´ e d’une seule orbite sous l’action
+deG′(F). L’ensemble Xest en g´ en´ eral form´ e d’une seule orbite, sauf dans le cas o` u d
+est impair et N=d′/2, o` u il y en a deux.
+NotonsEσl’espace de la repr´ esentation induite σform´ e de fonctions sur G(F) ` a
+valeurs dans Eπ⊗CEσ0v´ erifiant la condition habituelle de transformation ` a gauche
+parP(F). NotonsEσ,Ule sous-espace form´ e des fonctions ` a support dans UetEσ,Xle
+quotientEσ/Eσ,U. Ces espaces sont stables par G′(F) et on a une suite exacte
+0→HomG′(F)(Eσ,X,Eˇσ′)→HomG′(F)(Eσ,Eˇσ′)→HomG′(F)(Eσ,U,Eˇσ′).(1)
+Montrons que
+HomG′(F)(Eσ,X,Eˇσ′) ={0}. (2)
+Supposons que Xsoit form´ e d’une seule orbite pour l’action de G′(F). Quitte ` a
+effectuer une conjugaison par un ´ el´ ement de G(F), on peut supposer X⊂V′etY⊂V′.
+En posantV′
+0=V0∩V′, on a
+V′=X⊕V′
+0⊕Y
+NotonsP′etM′le sous-groupe parabolique de G′et sa composante de L´ evi associ´ es
+` a cette d´ ecomposition. On a P′=P∩G′et l’application P′(F)g′/maps⊔o→P(F)g′est un
+hom´ eomorphisme de P′(F)\G′(F) surX. On en d´ eduit que la repr´ esentation de G′(F)
+dansEσ,Xest une repr´ esentation induite ` a partir d’une repr´ esentation d eM′(F). Pour
+d´ ecrire celle-ci, on doit remarquer que, pour m′= (x,g′
+0)∈GL(N,F)×G′
+0(F), on a
+7δP(m′)1/2=|det(x)|(d0+N−1)/2
+F tandis que δP′(m′)1/2=|det(x)|(d0+N−2)/2
+F . On en d´ eduit
+qu’en notant σ0|G′la restriction de σ0` aG′(F), la repr´ esentation de G′(F) dansEσ,Xest
+´ egale ` a
+IndG′
+P′((π⊗|det|b+1/2
+F)⊗σ0|G′),
+c’est-` a-dire ` a
+π|.|b+1/2
+F×σ0|G′,
+avec les notations de 1.1. Par le th´ eor` eme de seconde r´ eciprocit ´ e de Bernstein, on a
+HomG′(F)(Eσ,X,Eˇσ′) =HomM′(F)((π⊗|det|b+1/2)⊗σ0|G′,(ˇσ′)¯P′),
+le terme (ˇσ′)¯P′d´ esignant le module de Jacquet normalis´ e. La repr´ esentation π´ etant
+irr´ eductible et unitaire, l’exposant de π|.|b+1/2est ´ egal ` ab+1/2. La repr´ esentation (ˇ σ′)¯P′
+est de longueur finie. Pour d´ emontrer (2),il suffit de prouver que , pour toutsous-quotient
+irr´ eductible cette repr´ esentation, de la forme π♯⊗σ′
+♯, l’exposant e♯deπ♯ne peut pas ˆ etre
+´ egal ` ab+1/2. Par le calcul habituel du module de Jacquet d’une induite, on sait qu ’une
+telle repr´ esentation π♯intervient dans une induite
+π1,♭×...×πt′,♭×π0,♯×πt′,♮×...×π1,♮
+o` u :
+•pouri= 1,...,t′, il existe une repr´ esentation irr´ eductible τi,♭d’un groupe lin´ eaire de
+sorte queπi,♭⊗τi,♭intervienne comme sous-quotient d’un module de Jacquet de ˇ π′
+i|.|−b′
+i
+relatif ` a un sous-groupe parabolique triangulaire inf´ erieur (parc e qu’on calcule le module
+de Jacquet de σ′relativement ` a ¯P′);
+•pouri= 1,...,t′, il existe une repr´ esentation irr´ eductible τi,♮d’un groupe lin´ eaire
+de sorte que πi,♮⊗τi,♮intervienne comme sous-quotient d’un module de Jacquet de π′
+i|.|b′
+i
+relatif ` a un sous-groupe parabolique triangulaire inf´ erieur;
+•il existe une repr´ esentation irr´ eductible σ′
+0,♭d’un groupe sp´ ecial orthogonal de sorte
+queπ0,♯×σ′
+0,♯intervienne dans unmoduledeJacquet de ˇ σ′
+0relativement ` a unsous-groupe
+parabolique ”triangulaire inf´ erieur”.
+NotonsNi,♭,Ni,♮etN0,♯les rangs des groupes lin´ eaires dont πi,♭,πi,♮etπ0,♯sont des
+repr´ esentations et ei,♭,ei,♮ete0,♯les exposants de ces repr´ esentations. Alors on a les
+´ egalit´ es
+N= (/summationdisplay
+i=1,...,t′Ni,♭)+N0,♯+(/summationdisplay
+i=1,...,t′Ni,♮),
+Ne♯= (/summationdisplay
+i=1,...,t′Ni,♭(ei,♭−b′
+i))+N0,♯e0,♯+(/summationdisplay
+i=1,...,t′Ni,♮(ei,♮+b′
+i)).
+Parce que les repr´ esentations π′
+iet ˇσ′
+0sont temp´ er´ ees et qu’on consid` ere leurs modules
+de Jacquet relatifs ` a des sous-groupes paraboliques inf´ erieurs , les nombres ei,♭,ei,♮ete0,♯
+sont n´ egatifs ou nuls. On en d´ eduit l’in´ egalit´ e e♯≤b′
+1. Sous l’hypoth` ese de l’´ enonc´ e, on
+a donce♯≤b, a fortiorie♯/\e}a⊔io\slash=b+1/2, et cela prouve (2) dans le cas o` u Xest form´ e d’une
+seule orbite. Quand il y a deux orbites, Eσ,Xest somme de deux sous-repr´ esentations
+auxquelles on peut appliquer le mˆ eme raisonnement. D’o` u (2).
+Etudions la repr´ esentation Eσ,UdeG′(F). Quitte ` a effectuer une conjugaison par un
+´ el´ ement de G(F), on se ram` ene ` a la situation suivante. On a une d´ ecomposition
+V′=X′⊕D′⊕V0⊕Y′,
+8o` uX′etY′sont totalement isotropes de dimension N−1 et les espaces X′⊕Y′,D′
+etV0sont orthogonaux. L’espace D′est une droite port´ ee par un vecteur v′
+0tel que
+q′(v′
+0,v′
+0) = 2(−1)d′ν0. Rappelons que V=V′⊕D, o` uDest une droite port´ ee par
+un vecteur v0tel queq(v0;v0) = 2(−1)dν0. On a les ´ egalit´ es X=X′⊕F(v0+v′
+0),
+Y=Y′⊕F(v0−v′
+0). PosonsV′
+∞=D′⊕V0, notonsG′
+∞son groupe sp´ ecial orthogonal,
+P′
+∞lesous-groupeparaboliquede G′form´ edes´ el´ ements quiconservent X′,P′
+∞=M′
+∞U′
+∞
+sa d´ ecomposition habituelle, o` u M′
+∞=GL(N−1)×G′
+∞. On aG0⊂G′
+∞et on v´ erifie
+queP∩G′= (GL(N−1)×G0)U′
+∞. La repr´ esentation Eσ,UdeG′(F) est une induite
+compacte ` a partir de ce groupe P∩G′. Ici, il n’y a pas de d´ ecalage sur les modules : la
+restriction de δP` aP∩G′est le module usuel de ce groupe. Par contre, la repr´ esentation
+que l’on induit n’est pas triviale sur U′
+∞(F). Ce groupe U′
+∞(F) admet une filtration ` a
+deux crans
+0→ ∧2(X′)→U′
+∞(F)→(V′
+∞⊗FX′)→0.
+Par projection orthogonale de V′
+∞surD′, on obtient une projection de U′
+∞(F) surD′⊗F
+X′,dontonnote U′
+♭(F)lenoyau,puisuneprojectionde( GL(N−1,F)×G0(F))U′
+∞(F)sur
+leproduitsemi-direct GL(N−1,F)⋉(D′⊗FX′).Enchoisissant desbasesconvenables, on
+peut identifier cedernier groupeausous-groupe mirabolique PN−1,1(F) deGL(N,F). On
+a aussi une projection de ( GL(N−1,F)×G0(F))U′
+∞(F) surG0(F), d’o` u une projection
+(GL(N−1,F)×G0(F))U′
+∞(F)→PN−1,1(F)×G0(F).
+Notons (π|.|b
+F)|PN−1,1la restriction de π|.|b
+Fen une repr´ esentation de PN−1,1(F). Par la
+projection ci-dessus, on peut consid´ erer ( π|.|b
+F)|PN−1,1⊗σ0comme une repr´ esentation de
+(GL(N−1,F)×G0(F))U′
+∞(F). On a alors
+Eσ,U=indG′
+(GL(N−1)×G0)U′∞((π|.|b
+F)|PN−1,1⊗σ0),
+o` uindd´ esigne l’induction ` a supports compacts. D’apr` es [BZ] 3.5, la repr ´ esentation
+(π|.|b
+F)|PN−1,1poss` ede une filtration
+{0}=τN+1⊂τN⊂τN−1⊂...⊂τ1= (π|.|b
+F)|PN−1,1.
+Les quotients de cette filtration v´ erifient
+τk/τk+1≃indPN−1,1
+PN−k,k(∆k(π|.|b
+F)⊗ψk)
+o` u ∆k(π|.|b
+F) est lak-i` eme d´ eriv´ ee de π|.|b
+F, cf. [BZ] 4.3. L’induction ` a supports compacts
+´ etant un foncteur exact, on en d´ eduit une filtration
+{0}=µN+1⊂µN⊂µN−1⊂...⊂µ1=Eσ,U
+dont les quotients v´ erifient
+µk/µk+1≃indG′
+PN−k,kG0U′
+♭((∆k(π|.|b
+F)⊗ψk⊗σ0). (3)
+Pour toutk= 1,...,N, on a une suite exacte
+1→HomG′(F)(µk/µk+1,ˇσ′)→HomG′(F)(µk,ˇσ′)→HomG′(F)(µk+1,ˇσ′).(4)
+Montrons que
+(5) pourk= 1,...,N−1,HomG′(F)(µk/µk+1,ˇσ′) ={0}.
+9Introduisonslesous-groupeparabolique Q′
+N−kdeG′form´ edes´ el´ ementsquiconservent
+le sous-espace engendr´ e par les N−kpremiers vecteurs de base de X′. Sa composante de
+L´ eviL′
+N−ks’identifie ` a GL(N−k)×G′
+k, o` uG′
+kest un groupe sp´ ecial orthogonal de mˆ eme
+type queG′. On aPN−k,kG0U′
+♭⊂Q′
+N−ket, dans la formule (3), la repr´ esentation que l’on
+induit est triviale sur le radical unipotent de Q′
+N−k(F). Introduisons la repr´ esentation µ′
+k
+deG′
+k(F) d´ efinie par
+µ′
+k=indG′
+k
+(UN−k,kG0U′
+♭)∩G′
+k(ψk⊗σ0).
+Alors
+µk/µk+1≃IndG′
+Q′
+N−k(∆k(π|.|b
+F)⊗µ′
+k) = ∆k(π|.|b
+F)×µ′
+k.
+Par le th´ eor` eme de seconde adjonction de Bernstein, on a
+HomG′(F)(µk/µk+1,σ′) =HomL′
+N−k(F)(∆k(π|.|b
+F)⊗µ′
+k,σ′
+¯Q′
+N−k).
+Puisqueπest unitaire et de la s´ erie discr` ete, elle est de la forme St(ρ,a), pour une
+repr´ esentation irr´ eductible et cuspidale ρet un entier a≥1. D’apr` es [Z] proposition 9.6,
+∆k(π|.|b
+F) est nulle si kn’est pas un multiple de dρ. Sik=ldρpour un entier l≥1, on
+a ∆k(π|.|b
+F) =<(a−1)/2+b,(1−a)/2+b+l >ρ. L’exposant de cette repr´ esentation
+est strictement sup´ erieur ` a b. La preuve de (2) montre que, pour tout sous-quotient
+irr´ eductible π♯⊗σ′
+♯de (ˇσ′)¯Q′
+N−k, l’exposant de π♯est inf´ erieur ou ´ egal ` a b. L’espace
+d’homomorphismes ci-dessus est donc nul, ce qui prouve (5).
+En utilisant (4) et (5) successivement pour k= 1,...,N−1, on obtient une injection
+HomG′(F)(Eσ,U,Eˇσ′)→HomG′(F)(µN,ˇσ′). (6)
+Pourk=N,larepr´ esentation∆k(π|.|b
+F)disparaˆ ıtdelad´ efinitionde µN.Lacontragr´ ediente
+del’induite ` asupports compactsd’une repr´ esentation admissible´ etant l’induite ordinaire
+de la contragr´ ediente, on a
+HomG′(F)(µN,ˇσ′) =HomG′(F)(σ′,IndG′
+P0,NG0U′
+♭(ψ−1
+N⊗ˇσ0))
+=HomP0,N(F)G0(F)U′
+♭(F)(σ′,ψ−1
+N⊗ˇσ0).
+Remarquonsque P0,NU′
+♭n’estautrequeleradicalunipotentd’unsous-groupeparabolique
+deG′contenu dans P′
+∞, de composante de L´ evi GL(1)N−1×G′
+∞. Alors le dernier espace
+ci-dessus n’est autre que celui not´ e HomG0(F),ψ′
+F(σ′,ˇσ0) en 1.2, pour un choix convenable
+de caract` ere ψ′
+F. Sa dimension est m(σ′,ˇσ0). En utilisant (1), (2) et (6), on obtient le
+lemme./square
+1.5 Rappel d’un r´ esultat de Gan, Gross et Prasad
+On conserve la situation du paragraphe pr´ ec´ edent. On suppose πcuspidale et on fait
+l’hypoth` ese suivante :
+(H) soitπ♯une repr´ esentation irr´ eductible d’un groupe lin´ eaire telle qu’il exis te une
+repr´ esentation irr´ eductible τ♯d’un groupe lin´ eaire ou d’un groupe sp´ ecial orthogonal de
+sorte queπ♯⊗τ♯intervienne comme sous-quotient d’un module de Jacquet de l’une des
+repr´ esentations π′
+i, ˇπ′
+iou ˇσ′
+0; soit de plus β∈R; alors aucun´ el´ ement du support cuspidal
+deπ♯n’est ´ egal ` a π|.|β
+F.
+10Lemme.Sous ces hypoth` eses, on a l’´ egalit´ e
+m(σ,ˇσ′) =m(σ′,ˇσ0).
+Cf. [GGP] th´ eor` eme 15.1, dont on reproduit la d´ emonstration.
+Preuve. Bernstein a d´ ecompos´ e la cat´ egorie des repr´ esenta tions lisses de G′(F) en
+sommedirectedesous-cat´ egoriesindex´ eespardesclassesd’ine rtiedesupportscuspidaux.
+Notonsprojla projection sur la sous-cat´ egorie contenant ˇ σ′. On a la suite exacte
+0→proj(Eσ,U)→proj(Eσ)→proj(Eσ,X)→0.
+D’apr` es la description de Eσ,Xdonn´ ee en 1.4, l’hypoth` ese (H) implique que proj(Eσ,X) =
+{0}. Donc
+proj(Eσ,U) =proj(Eσ).
+Mais, pour toute repr´ esentation lisse τdeG′(F), on a
+HomG′(F)(τ,ˇσ′) =HomG′(F)(proj(τ),ˇσ′).
+Donc
+HomG′(F)(Eσ,Eˇσ′) =HomG′(F)(Eσ,U,Eˇσ′).
+On d´ ecritEσ,Ucomme en 1.4. Puisque πest cuspidale, la filtration de ( π|.|b
+F)PN−1,1n’a
+qu’un quotient non nul : on a ( π|.|b
+F)PN−1,1=τN. L’injection (6) de 1.4 devient une
+bijection et le r´ esultat s’ensuit. /square
+1.6 L’in´ egalit´ e m(σ,ˇσ′)≤m(σ0,σ′
+0)
+Dans la situation de 1.3, on va d´ emontrer l’in´ egalit´ e
+m(σ,ˇσ′)≤m(σ0,σ′
+0). (1)
+Remarquonsd’abordque,puisquelesrepr´ esentations πietπ′
+isontirr´ eductiblesettemp´ er´ ees,
+ce sont des induites de s´ eries discr` etes. En les ´ ecrivant comme d e telles induites, on ob-
+tient denouvelles expressions de σetσ′, similaires auxexpressions primitives, maiso` ules
+nouvelles repr´ esentations πietπ′
+isont de la s´ erie discr` ete. En oubliant cette construction,
+on peut supposer que les πietπ′
+isont de la s´ erie discr` ete.
+On utilisera plusieurs fois la construction suivante. Choisissons une r epr´ esentation
+admissible irr´ eductible ρdeGL((d+ 1−d′)/2,F), unitaire et cuspidale, v´ erifiant les
+analogues de l’hypoth` ese (H) du paragraphe pr´ ec´ edent o` u σ′est remplac´ e par ˇ σ′ouσ.
+Unetellerepr´ esentation existe. Consid´ erons larepr´ esentatio nρ×σ′d’ungroupedemˆ eme
+type queG′. On a
+ρ×σ′=ρ×π′
+1|.|b′
+1
+F×...×π′
+t′|.|b′
+t′
+F×σ′
+0.
+Mais les hypoth` eses sur ρnous permettent de permuter les premiers facteurs :
+ρ×σ′=π′
+1|.|b′
+1
+F×...×π′
+t′|.|b′
+t′
+F×ρ×σ′
+0.
+C’estuneexpressiondumˆ emetypequecellede σ′:t′estdevenut′+1,ler´ eelsuppl´ ementaire
+b′
+t′+1est nul et on n’a pas chang´ e σ′
+0. Remarquons que dρ×σ′=d+1. On a l’´ egalit´ e
+11m(σ,ˇσ′) =m(ρ×σ′,ˇσ). (2)
+En effet, cela r´ esulte de 1.5, o` u l’on remplace respectivement π,σ0etσ′parρ,σ′et
+σ.
+Revenons ` a notre probl` eme et supposons d’abord que tous les bietb′
+isont nuls. Dans
+ce cas, prouvons :
+(3) on a l’´ egalit´ e m(σ,ˇσ′) =m(σ0,σ′
+0).
+Il s’agit ici d’induites unitaires, donc σetσ′sont semi-simples. D´ ecomposons σ′en
+composantes irr´ eductibles : σ′=⊕j=1,...,kσ′
+j. On a
+m(σ,ˇσ′) =/summationdisplay
+j=1,...,km(σ,ˇσ′
+j).
+Il s’agit ici de repr´ esentations temp´ er´ ees. En utilisant la propo sition 5.7 et les lemmes
+5.3 et 5.4 de [W3], on a, pour tout j= 1,...,k,
+m(σ,ˇσ′
+j) =m(σ0,ˇσ′
+j).
+Sidσ0<d′, on a
+/summationdisplay
+j=1,...,km(σ0,ˇσ′
+j) =/summationdisplay
+j=1,...,km(ˇσ′
+j,σ0) =m(ˇσ′,σ0).
+Les mˆ emes r´ esultats de [W3] entraˆ ınent que cette derni` ere m ultiplicit´ e vaut m(ˇσ′
+0,σ0).
+Mais, pour les repr´ esentations irr´ eductibles, le changement d’un e repr´ esentation en sa
+contragr´ ediente ne modifie pas la multiplicit´ e ([W3] 7.1(1)). Le dern ier nombre est donc
+´ egal ` am(σ0,σ′
+0), d’o` u (3) dans ce cas. Supposons maintenant dσ0> d′. On choisit ρ
+comme ci-dessus, relativement aux repr´ esentations σ0etσ′. Remarquons que ρv´ erifie
+alors pour tout jles mˆ emes hypoth` eses relativement aux repr´ esentations σ0etσ′
+j. Donc,
+d’apr` es (2), m(σ0,ˇσ′
+j) =m(ρ×σ′
+j,ˇσ0), puis
+/summationdisplay
+j=1,...,km(σ0,ˇσ′
+j) =m(ρ×σ′,ˇσ0).
+On peut maintenant appliquer les mˆ emes r´ esultats de [W3]. Ils entra ˆ ınent que ce dernier
+terme vaut m(σ0,σ′
+0). Cela prouve (3).
+Revenons au cas g´ en´ eral. On d´ efinit un invariant
+N(σ,σ′) = (/summationdisplay
+i=1,...,t;bi/ne}ationslash=0Ni)+(/summationdisplay
+i=1,...,t′;b′
+i/ne}ationslash=0N′
+i).
+SoitNun entier naturel. On va d´ emontrer par r´ ecurrence sur Nque l’in´ egalit´ e (1) est
+v´ erifi´ ee pour toutes donn´ ees G,G′,σ,σ′v´ erifiant les conditions requises et telles que
+N(σ,σ′) =N. Le casN= 0 est couvert par (3). On suppose maintenant N >0 et on
+fixe des donn´ ees avec N(σ,σ′) =N.
+1ercas. On suppose d=d′+1,t≥1 etb1≥b′
+1. Posons
+σ1=π2|.|b2
+F×...×πt|.|bt
+F×σ0.
+C’est une repr´ esentation d’un groupe G1de mˆ eme type que Get on aσ=π1|.|b1
+F×σ1.
+La situation permet d’appliquer le lemme 1.4 : on a
+m(σ,ˇσ′)≤m(σ′,ˇσ1).
+12On applique l’hypoth` ese de r´ ecurrence aux groupes G′etG1et ` a leurs repr´ esentations
+σ′etσ1. C’est loisible puisque N(σ′,σ1) =N(σ,σ′)−dπ1<N. On en d´ eduit m(σ′,ˇσ1)≤
+m(σ0,σ′
+0), puis (1).
+2`emecas. On suppose t′≥1 etb′
+1≥b1. Choisissons une repr´ esentation ρv´ erifiant les
+hypoth` eses permettant d’appliquer (2). Grˆ ace ` a cette relatio n, on am(σ,ˇσ′) =m(ρ×
+σ′,ˇσ). Mais les repr´ esentations ρ×σ′etσv´ erifient les hypoth` eses du premier cas. Donc
+m(ρ×σ′,ˇσ)≤m(σ0,σ′
+0).
+3`emecas. On suppose t≥1 etb1≥b′
+1. On raisonne comme dans le deuxi` eme cas,
+` a ceci pr` es que les repr´ esentations ρ×σ′etσv´ erifient maintenant les hypoth` eses du
+deuxi` eme cas.
+PourN >0, on est forc´ ement dans l’un des deuxi` eme ou troisi` eme cas et cela prouve
+(1)./square
+1.7 Produit multilin´ eaire
+On suppose d=d′+1. Fixons un sous-tore d´ eploy´ e maximal A′deG′et un sous-tore
+d´ eploy´ emaximal AdeGquicontient A′.Fixonsdessous-groupesparaboliquesminimaux
+P′
+mindeG′etPmindeG, contenant respectivement A′etA. On peut les choisir de sorte
+que :
+•soientn′≤nles entiers tels que A≃GL(1)netA′≃GL(1)n′; alors le plongement
+deA′dansAs’identifie ` a ( a1,...,an′)/maps⊔o→(a1,...,an′,1,...,1);
+•l’ensembledesracinessimplesde a′relatif` aP′
+minestform´ edescaract` eres( a1,...,an′)/maps⊔o→
+aia−1
+i+1pouri= 1,...,n′−1 et de (a1,...,an′)/maps⊔o→an′, sauf dans le cas o` u d′= 2n′, auquel
+cas la derni` ere racine est remplac´ ee par ( a1,...,an′)/maps⊔o→an′−1an′; de mˆ eme pour l’ensemble
+des racines simples de Trelatif ` aPmin.
+Onpeutsupposerquelessous-groupesparaboliquesservant` ad ´ efinirlesrepr´ esentations
+σetσ′de 1.3 contiennent les sous-groupes paraboliques Pmin, resp.P′
+min. On fixe des
+sous-groupes compacts sp´ eciaux K′deG′(F) etKdeG(F), en bonne position relative-
+ment ` aP′
+minetPmin.
+En 1.3, on a suppos´ e que les bietb′
+i´ etaient r´ eels et v´ erifiaient certaines in´ egalit´ es.
+Oublions ces conditions en prenant pour bietb′
+ides nombres complexes quelconques. On
+introduit les param` etres z= (z1,...,zt) etz′= (z′
+1,...,z′
+t′), aveczi=q−bietz′
+i=q−b′
+i, o` u
+qest le nombre d’´ el´ ements du corps r´ esiduel de F. On note plutˆ ot nos repr´ esentations
+σzetσ′
+z′. Suivant Bernstein, on peut consid´ erer σzetσ′
+z′comme les sp´ ecialisations pour
+ces valeurs des param` etres de repr´ esentations ` a valeurs dan s l’alg` ebre
+R=C[z±1
+1,...,z±1
+t,(z′
+1)±1,...,(z′
+t′)±1].
+La repr´ esentation σzse r´ ealise dans un espace Ede fonctions
+e:K→Eπ1⊗C...⊗CEπt⊗Eσ0
+et cet espace est ind´ ependant de z. On note ˇσzla contragr´ ediente de σz. Elle se r´ ealise
+de mˆ eme dans un espace ˇEde fonctions
+ˇe:K→Eˇπ1⊗C...⊗CEˇπt⊗Eˇσ0.
+On introduit le produit bilin´ eaire naturel sur E סE:
+<e,ˇe>=/integraldisplay
+K<e(k),ˇe(k)>dk,
+13o` u le produit int´ erieur est l’accouplement naturel sur
+(Eπ1⊗C...⊗CEπt⊗Eσ0)×(Eˇπ1⊗C...⊗CEˇπt⊗Eˇσ0).
+Les mˆ emes consid´ erations valent pour σ′
+z′et on introduit des espaces E′etˇE′, munis d’un
+produit bilin´ eaire. Pour e∈ E, ˇe∈ˇE,e′∈ E′et ˇe′∈ˇE′, posons
+L(z,z′;e′,ˇe′,e,ˇe) =/integraldisplay
+G′(F)<σ′
+z′(x)e′,ˇe′><σz(x)e,ˇe>dx.
+NotonsDle domaine de ( C×)n×(C×)n′d´ efini par les relations
+q−1/2<|zi|,|zj|,|ziz′
+j|,|ziz′−1
+j|<q1/2,
+lesi,jparcourant tous les entiers possibles.
+Lemme.(i) L’int´ egrale L(z,z′;e′,ˇe′,e,ˇe)est absolument convergente pour (z,z′)∈ D.
+(ii) Il existe un polynˆ ome non nul D∈ Ret, pour tous e,ˇe,e′,ˇe′, il existe un
+polynˆ omeL(z,z′;e′,ˇe′,e,ˇe)∈ Rde sorte que
+D(z,z′)L(z,z′;e′,ˇe′,e,ˇe) =L(z,z′;e′,ˇe′,e,ˇe)
+pour tout (z,z′)∈ D.
+Preuve. Pour simplifier, on suppose d/\e}a⊔io\slash= 2netd′/\e}a⊔io\slash= 2n′. On indiquera plus loin
+comment adapter la preuve si l’une de ces conditions n’est pas v´ erifi ´ ee. Posons A=
+X∗(A′) etAR=X∗(A′)⊗ZRn′. L’ensemble des racines simples de A′relatif ` aP′
+min
+s’identifie ` a un ensemble ∆′de formes lin´ eaires sur AR. On introduit l’ensemble des poids
+{̟α;α∈∆′}. Fixons une uniformisante ̟FdeFet identifions A` a un sous-groupe de
+A′(F) par
+m= (m1,...,mn′)/maps⊔o→(̟m1
+F,...,̟mt′
+F).
+Introduisons le sous-ensemble A+form´ e des m∈ Atels quem1≥m2≥...≥mt′≥0.
+D’apr` es la d´ ecomposition de Cartan, il existe un sous-ensemble fin i Γ′deG′(F), contenu
+dans le commutant de A′, de sorte que G′(F) soit union disjointe des ensembles K′mγ′K′
+pourm∈ A+etγ′∈Γ′. On a
+L(z,z′;e′,ˇe′,e,ˇe) =/integraldisplay
+K′×K′/summationdisplay
+γ′∈Γ′/summationdisplay
+m∈A+mes(mγ′)<σ′
+z′(k1mγ′k2)e′,ˇe′>
+<σz(k1mγ′k2)e,ˇe>dk 1dk2,
+o` umes(mγ′) =mes(K′mγ′K′)mes(K′)−2. Nos repr´ esentations ´ etant lisses, cela nous
+permet de fixer γ′∈Γ′et de remplacer l’int´ egrale L(z,z′;e′,ˇe′,e,ˇe) par la s´ erie
+S(z,z′;e′,ˇe′,e,ˇe) =/summationdisplay
+m∈A+mes(mγ′)<σ′
+z′(m)e′,ˇe′><σz(m)e,ˇe>
+Consid´ erons un ´ el´ ement T= (T1,...,Tt′)∈ Atel queT1>... >T t′>0. Un tel ´ el´ ement
+permet de d´ ecomposer A+en union disjointe de sous-ensembles A+(Q′), o` uQ′parcourt
+les sous-groupes paraboliques de G′qui contiennent P′
+min, cf. [A2] 3.9. A Q′sont associ´ es
+un sous-ensemble ∆′(Q′)⊂∆′et une d´ ecomposition AR=AQ′,R⊕ AQ′
+R. Le premier
+14sous-espace est l’intersection des annulateurs des α∈∆′(Q′) et le second est engendr´ e
+par les coracines ˇ αpourα∈∆′(Q′). On´ ecrit tout m∈ ARsous la forme m=mQ′+mQ′
+conform´ ement ` a cette d´ ecomposition. L’ensemble A+
+Q′est celui des m∈ A+tels que
+•α(mQ′−TQ′)>0 pourα∈∆\∆′(Q′);
+•̟α(mQ′−TQ′)≤0 pourα∈∆′(Q′).
+On peut fixer Q′et remplacer S(z,z′;e′,ˇe′,e,ˇe) par la s´ erie S(Q′;z,z′;e′,ˇe′,e,ˇe) o` u
+l’on restreint la somme aux m∈ A+(Q′). NotonsM′la composante de L´ evi de Q′qui
+contientA′. On ´ ecritM′=GL(N1)×...×GL(Nk)×G′
+1, o` uG′
+1est un groupe de mˆ eme
+type queG′. Les ´ el´ ements e′et ˇe′´ etant fix´ es, les r´ esultats de Casselman nous disent
+que, siα(T) est assez grand pour tout α∈∆, la propri´ et´ e suivante est v´ erifi´ ee. Notons
+pQ′:Eσ′
+z′→Eσ′
+z′,Q′et ˇp¯Q′:Eˇσ′
+z′→E(ˇσ′
+z′)¯Q′les projections sur les modules de Jacquet.
+Alors il existe un produit bilin´ eaire M′(F) invariant sur ces modules de sorte que, pour
+toutm∈ A+(Q′), on ait l’´ egalit´ e
+<σ′
+z′(m)e′,ˇe′>=δQ′(m)1/2<σ′
+z′,Q′(m)pQ′(e′),ˇp¯Q′(ˇe′)>.
+Remarquons qu’` a Q′est naturellement associ´ e un sous-groupe parabolique QdeGconte-
+nantPmin: le L´ eviMdeQestGL(N1)×...×GL(Nk)×G1, o` uG1est de mˆ eme type
+queG. Un ´ el´ ement de A+(Q′) v´ erifie les mˆ emes conditions relativement ` a Qque celles
+indiqu´ ees ci-dessus et on peut appliquer le r´ esultat de Casselman. Les ´ el´ ements eet ˇe
+´ etant fix´ es, si α(T) est assez grand pour tout α∈∆, on a une ´ egalit´ e
+<σz(m)e,ˇe>=δQ(m)1/2<σz,Q(m)pQ(e),p¯Q(ˇe)>
+pour tout m∈ A+(Q′). Notons C(M′) l’intersection de Aavec le centre de M′(F). On
+aC(M′) =Zk. Remarquons que C(M′) est aussi contenu dans le centre de M(F). La
+repr´ esentation σz,Qest de longueur finie, la longueur ´ etant born´ ee ind´ ependamment de
+z. D’apr` es le mˆ eme calcul qu’en 1.4(2), les restrictions ` a C(M′) =Zkdes caract` eres
+centraux de ses sous-quotients irr´ eductibles sont de la forme
+c= (c1,...,ck)/maps⊔o→χz(c) =χ(c)/productdisplay
+i=1,...,t;j=1,...,kz(fi,j−f−i,j)cj
+i, (1)
+o` u :
+•lesfi,jetf−i,jsont des entiers naturels v´ erifiant
+/summationdisplay
+i=1,...,t(fi,j+f−i,j) =Nj;
+•χest le caract` ere central d’un sous-quotient irr´ eductible de σ1,Q, o` u1= (1,...,1)∈
+(C×)n.
+Cela entraˆ ıne qu’il existe un entier l≥0 de sorte que, pour tous e, ˇeet toutm∈ A,
+la fonction
+c/maps⊔o→<σz,Q(cm)pQ(e),p¯Q(ˇe)> (2)
+surC(M′) est combinaison lin´ eaire de fonctions
+c/maps⊔o→χz(c)/productdisplay
+j=1,...,kclj
+j, (3)
+o` u lesljsont des entiers naturels inf´ erieurs ou ´ egaux ` a l. Un r´ esultat analogue vaut pour
+la fonction
+c/maps⊔o→<σ′
+z′,Q′(m)pQ′(e′),ˇp¯Q′(ˇe′)>.
+15Le groupe C(M′) agit par translations sur A. On v´ erifie que l’ensemble A+(Q′) est
+contenu dans la r´ eunion d’un nombre fini d’orbites. On peut d´ ecom poser notre s´ erie
+S(Q′;z,z′;e′,ˇe′,e,ˇe) en somme finie de s´ eries o` u l’on restreint l’ensemble de sommation
+` aune intersection ( mC(M′))∩A+(Q′). Consid´ erons une telles´ erie. L’application c/maps⊔o→mc
+identifiel’intersectionpr´ ec´ edente ` auncˆ one C(M′)+d´ efinipardesin´ egalit´ es cj−cj+1≥Cj
+pourj= 1,...,k−1 etck≥Ck, o` u lesCjsont certaines constantes. On v´ erifie qu’il y a
+une constante m>0 telle que mes(K′mcγ′K′) =mδQ′(c)−1pour tout cdans ce cˆ one.
+Alors notre s´ erie est born´ ee par une somme de s´ eries de la forme
+/summationdisplay
+c∈C(M′)+δQ′(c)−1/2δQ(c)1/2χ′
+z′(c)χz(c)/productdisplay
+j=1,...,k|cj|lj+l′
+j. (4)
+On calcule
+δQ′(c)−1/2δQ(c)1/2=q−/summationtext
+j=1,...,kNjcj/2.
+Consid´ erons l’expression (1). La repr´ esentation σ1est temp´ er´ ee, donc χest born´ ee sur
+C(M′)+. Si on note zle plus grand des nombres |zi|±1,χz(c) est essentiellement born´ e
+surC(M′)+par/producttext
+j=1,...,kz/summationtext
+j=1,...,kNjcj. De mˆ eme, χ′
+z′(c) est essentiellement born´ e par/producttext
+j=1,...,k(z′)/summationtext
+j=1,...,kNjcj. La s´ erie (4) est donc essentiellement born´ ee par
+/summationdisplay
+c∈C(M′)+/productdisplay
+j=1,...,t(zz′q−1/2)Njcj|cj|lj+l′
+j.
+Cettederni` eres´ erieestconvergentesousleshypoth` esesd u(i)del’´ enonc´ eetcelad´ emontre
+cette assertion.
+On v´ erifie que la somme de la s´ erie (4) est une fraction rationnelle en z,z′. Remar-
+quons que les termes qui y interviennent parcourent des ensemble s finis ind´ ependants des
+vecteurse, ˇe,e′, ˇe′. En reprenant le calcul ci-dessus, on voit que, pour d´ emontrer le (ii)
+de l’´ enonc´ e, il suffit de prouver l’assertion suivante. Consid´ ero ns les diff´ erentes fonctions
+de la forme (3) qui peuvent intervenir. Elles sont d´ etermin´ ees pa r un caract` ere χet des
+familles d’entiers fi,j−f−i,jetlj, ces donn´ ees parcourant des ensembles finis. Notons
+(fh,z)h∈Hcette famille de fonctions, l’ensemble d’indices H´ etant donc fini. Notons fzla
+fonction (2). Ecrivons
+fz=/summationdisplay
+h∈HCh(z)fh,z.
+On doit prouver que les diff´ erents coefficients Ch(z) sont des fractions rationnelles en
+z, de d´ enominateur born´ e ind´ ependamment de eet ˇe. On doit aussi prouver l’assertion
+similaire relative au groupe G′, mais elle se prouve ´ evidemment de la mˆ eme fa¸ con. Pour
+zen position g´ en´ erale, la famille de fonctions ( fh,z)h∈Hest lin´ eairement ind´ ependante.
+On peut donc fixer une famille ( ch)h∈Hd’´ el´ ements de C(M′) telle que le d´ eterminant de
+la matrice ( fh,z(ch′))h,h′∈Hsoit non nul pour au moins une valeur de z. Ce d´ eterminant
+est donc un ´ el´ ement non nul de R. Les coefficients Ch(z) sont d´ etermin´ es par le syst` eme
+d’´ equations
+fz(ch′) =/summationdisplay
+h∈HCh(z)fh,z(ch′)
+pour touth′∈H. Le membre de gauche de cette ´ equation appartient ` a R. Il en r´ esulte
+que lesCh(z) sont des fractions rationnelles, de d´ enominateur divisant le d´ et erminant
+ci-dessus, lequel ne d´ epend pas des vecteurs eet ˇe. Cela d´ emontre l’assertion requise et
+le lemme, sous les hypoth` eses d/\e}a⊔io\slash= 2n,d′/\e}a⊔io\slash= 2n′.
+16Supposons d= 2n. Cela implique que Gest d´ eploy´ e et l’hypoth` ese sur le plongement
+deV′dansVimplique que G′est lui-aussi d´ eploy´ e. Donc n′=n−1. La seule chose
+qui change dans le raisonnement ci-dessus est que, si N1+...+Nk=n′, le L´ evi du
+parabolique Qassoci´ e` aQ′estGL(N1)×...×GL(Nk)×GL(1).Cela n’aaucune incidence.
+Supposons maintenant d′= 2n′. Les deux groupes GetG′sont d´ eploy´ es et on a n=n′.
+Dans la d´ efinition de A+, la derni` ere in´ egalit´ e mn′≥0 doit a priori ˆ etre remplac´ ee
+parmn′−1+mn′≥0. Mais on peut d´ ecomposer cet ensemble en deux, l’un sur lequel
+mn′≥0 et l’autre sur lequel mn′<0. En changeant l’identification de A′avecGL(1)n′
+en inversant la derni` ere coordonn´ ee, et en changeant en cons ´ equence le groupe Pmin,
+le deuxi` eme ensemble se ram` ene ` a un ensemble du premier type (la condition sur mn′
+devientmn′>0aulieude mn′≥0,maisc’est sansimportance). Onpeut doncconsid´ erer
+queA+est d´ efini par les mˆ emes in´ egalit´ es que pr´ ec´ edemment. Pour un sous-groupe
+parabolique Q′tel queN1+...+Nk<n′, rien ne change.
+SoitQ′tel queN1+...+Nk=n′etNk≥2. Si ∆′(Q′) contient la racine m/maps⊔o→
+mn′−1−mn′, de nouveau rien ne change. Supposons que ∆′(Q′) contienne la racine
+m/maps⊔o→mn′−1+mn′. On d´ efinit Qde sorte que sa composante de L´ evi Msoit ´ egale ` a
+GL(N1)×...×GL(Nk−1)×G1. On v´ erifie les deux propri´ et´ es suivantes :
+(5) notons ∆ l’ensemble des racines simples de Aassoci´ e ` aPminet ∆(Q) le sous-
+ensemble associ´ e ` a Q; pourm∈ A+(Q′) etα∈∆\∆(Q), on aα(m)>α(T);
+(6) il existe C∈Ntel quemi≤Cpour tout m∈ A+(Q′) et touti=N1+...+
+Nk−1+1,...,n′.
+L’assertion (5) vient de l’inclusion ∆ \∆(Q)⊂∆′\∆′(Q′). D´ emontrons (6). Posons
+e=N1+...+Nk−1. Soitm∈ A+(Q′). Cette hypoth` ese implique que l’on peut ´ ecrire
+(me+1−te+1,...,mn′−tn′) = (z,...,z,−z)+(p1,...,pNk−1,−pNk),
+avecz >0,p1+...+pf≤0 pour tout f= 1,...,Nk−1,p1+...+pNk= 0. Cela entraˆ ıne
+pNk≥0. Alorsz=tn′−mn′−pNkest major´ e (puisque l’on a suppos´ e mn′≥0). On a
+aussip1≤0, doncme+1=te+1+z+p1est major´ e. Puisque me+1≥me+2≥...≥mn′≥0,
+(6) s’ensuit.
+La relation (5) nous permet d’appliquer les r´ esultats de Casselman a ux termes pro-
+venant du groupe G, pour le sous-groupe parabolique Q. On remplace dans les raisonne-
+ments ci-dessus le groupe C(M′) par son analogue C(M) pour le groupe M. Grˆ ace ` a la
+relation (6), A+(Q′) est inclus dans un nombre fini d’orbites pour l’action de ce groupe.
+La preuve se poursuit alors comme pr´ ec´ edemment.
+Soit enfinQ′tel queN1+...+Nk=n′etNk= 1. On d´ ecompose notre s´ erie en une
+sommesurles mtelsquemn′>tn′etd’unesommesurles mn′≤tn′.Lapremi` eresetraite
+comme dans le cas g´ en´ eral, en prenant Qde composante de L´ evi GL(N1)×...×GL(Nk).
+La seconde se traite comme dans le cas particulier ci-dessus en pren antQde composante
+de L´ eviGL(N1)×...×GL(Nk−1)×G1. Cela ach` eve la preuve. /square
+1.8 Preuve de l’in´ egalit´ e m(σ0,σ′
+0)≤m(σ,ˇσ′)
+On consid` ere la situation de 1.3 et on suppose d’abord d=d′+ 1. Il n’y a rien ` a
+prouver sim(σ0,σ′
+0) = 0. On suppose donc m(σ0,σ′
+0) = 1. Introduisons un polynˆ ome
+D(z,z′) v´ erifiant les conditions du (ii) du lemme 1.7. Fixons d’abord des familles z
+etz′, telles que D(z,z′)/\e}a⊔io\slash= 0 et dont toutes les coordonn´ ees sont de module 1. Les
+17repr´ esentations σzetσ′
+z′sont temp´ er´ ees. La proposition 5.7 et le lemme 5.3 de [W3] nous
+disent qu’il existe e∈ E, ˇe∈ˇE,e′∈ E′et ˇe′∈ˇE′tels queL(z,z′;e′,ˇe′,e,ˇe)/\e}a⊔io\slash= 0.
+Remarque. Dans [W3], on a consid´ er´ e des produits hermitiens plutˆ ot que des p ro-
+duits bilin´ eaires, mais la traduction est facile puisque les repr´ esent ations sont unitaires.
+Onadoncaussi L(z,z′;e′,ˇe′,e,ˇe)/\e}a⊔io\slash= 0.Ecrivons lescoordonn´ ees zietz′
+idenosfamilles
+sous la forme zi=q−βi,z′
+i=q−β′
+i. Pours∈C, introduisons les familles z(s) etz′(s) de
+coordonn´ ees zi(s) =q−sβi+(s−1)bi,z′
+i(s) =q−sβ′
+i+(s−1)b′
+i. Pour tous e, ˇe,e′, ˇe′, l’application
+s/maps⊔o→L(z(s),z′(s);e′,ˇe′,e,ˇe) est holomorphe. Elle est non nulle pour au moins un choix de
+e, ˇe,e′, ˇe′. NotonsNle plus grand entier tel que, pour tous e, ˇe,e′, ˇe′, cette application
+soit divisible par sN. Posons
+Lσ,σ′(e′,ˇe′,e,ˇe) =lims→0s−NL(z(s),z′(s);e′,ˇe′,e,ˇe).
+Il y a au moins un choix de vecteurs e, ˇe,e′, ˇe′tel queLσ,σ′(e′,ˇe′,e,ˇe)/\e}a⊔io\slash= 0. Dans le
+domaine D, on v´ erifie ais´ ement l’´ egalit´ e
+L(z,z′;σ′
+z′(g′)e′,ˇe′,σz(g′)e,ˇe) =L(z,z′;e′,ˇe′,e,ˇe)
+pour toutg′∈G′(F). On a une mˆ eme relation pour l’application L(z,z′;e′,ˇe′,e,ˇe) et
+cette ´ egalit´ e se prolonge alg´ ebriquement ` a toutes familles z,z′. Il en r´ esulte que l’appli-
+cationLσ,σ′ci-dessus v´ erifie la relation
+Lσ,σ′(σ′(g′)e′,ˇe′,σ(g′)e,ˇe) =Lσ,σ′(e′,ˇe′,e,ˇe).
+Fixons alors ˇ eet ˇe′et d´ efinissons une application lin´ eaire l:Eσ→Eˇσ′par l’´ egalit´ e
+<e′,l(e)>=Lσ,σ′(e′,ˇe′,e,ˇe).
+Elle appartient ` a HomG′(σ,ˇσ′). Pour un bon choix de ˇ e, ˇe′, elle est non nulle. Donc
+m(σ,ˇσ′)≥1.
+Supposons maintenant d > d′+ 1. On choisit ρcomme en 1.6(2). On a m(σ,ˇσ′) =
+m(ρ×σ′,ˇσ).D’apr` escequel’onvientdeprouver,cettederni` eremultiplicit´ eestsup´ erieure
+ou ´ egale ` a 1.
+L’in´ egalit´ e que l’on vient de prouver et celle de 1.6 d´ emontrent la pro position 1.3.
+2 Irr´ eductibilit´ e et repr´ esentations g´ en´ eriques
+2.1 Rappels sur les param´ etrages
+Dans cette section, Gest un groupe sp´ ecial orthogonal ou symplectique d´ efini sur F.
+Pr´ ecis´ ement, cela signifie que l’on fixe un espace vectoriel VsurFde dimension finie et
+une forme bilin´ eaire qnon d´ eg´ en´ er´ ee sur Vqui est soit sym´ etrique, soit antisym´ etrique.
+AlorsGest la composante neutre du groupe d’automorphismes de ( V,q). On notedGla
+dimension de V. Un groupe similaire G′est dit de mˆ eme type que Gs’il est associ´ e ` a
+un couple ( V′,q′) satisfaisant les conditions suivantes : q′v´ erifie la mˆ eme condition de
+sym´ etrie que q; le plus grand des espaces ( V,q) et (V′,q′) est isomorphe ` a la somme
+orthogonale du plus petit et de plans hyperboliques.
+On consid` ere que le L-groupe ˆGdeGest :
+18•le groupe sp´ ecial orthogonal complexe SO(ˆdG,C), o` uˆdG=dG+ 1, siGest sym-
+plectique;
+•le groupe symplectique complexe Sp(ˆdG,C), o` uˆdG=dG−1, siGest sp´ ecial
+orthogonal ”impair” (c’est-` a-dire que dGest impair);
+•le groupe orthogonal complexe O(ˆdG,C), o` uˆdG=dG, siGest sp´ ecial orthogonal
+”pair” (c’est-` a-dire que dGest pair).
+Toutes les repr´ esentations de groupes r´ eductifs que l’on consid ´ erera seront suppos´ ees
+admissibles et de longueur finie. Dans les cas symplectique ou sp´ ecial orthogonal im-
+pair, on conjecture (et nous admettons cette conjecture) que l’ensemble des classes de
+conjugaison de repr´ esentations irr´ eductibles temp´ er´ ees de G(F) se d´ ecompose en union
+disjointe de L-paquets ΠG(ϕ), o` uϕparcourt les classes de conjugaison par ˆGd’homo-
+morphismes ϕ:WDF→ˆGqui v´ erifient quelques conditions usuelles de continuit´ e et
+de semi-simplicit´ e et qui sont temp´ er´ es, c’est-` a-dire que l’image deWFparϕest re-
+lativement compacte. Un tel homomorphisme ϕse pousse en un homomorphisme de
+WDFdansGL(ˆdG,C). Via la correspondance de Langlands (th´ eor` eme de Harris-Tay lor
+et Henniart), on associe ` a ϕune repr´ esentation irr´ eductible π(ϕ) deGL(ˆdG,F). Elle est
+temp´ er´ eeetautoduale.Onpeutlaprolongerenunerepr´ esent ationduproduitsemi-direct
+GL(ˆdG,F)⋊{1,θ}, o` uθest l’automorphisme ext´ erieur habituel d´ efini par θ(g) =tg−1.
+Notons ˜π(ϕ) sa restriction ` a la composante connexe non neutre GL(ˆdG,F)θ. D’autre
+part, pour toute repr´ esentation σ, notons Θ σson caract` ere. Les propri´ et´ es essentielles
+duL-paquet ΠG(ϕ) sont les suivantes :
+(1) la somme Θ ΠG(ϕ)=/summationtext
+π∈ΠG(ϕ)Θπest une distribution stable;
+(2)siGestquasi-d´ eploy´ e, ilexiste c∈C×telqueladistributionΘ ˜π(ϕ)soitletransfert,
+par endoscopie tordue, de cΘΠG(ϕ);
+(3) siGn’est pas quasi-d´ eploy´ e, introduisons sa forme quasi-d´ eploy´ e eGet leL-
+paquet ΠG(ϕ) de repr´ esentations de G(F); alors il existe c∈C×tel que Θ ΠG(ϕ)soit le
+transfert, par endoscopie ordinaire, de cΘΠG(ϕ).
+Ici, les correspondances endoscopiques entre classes de conjug aison stable sont in-
+jectives. Les conditions (2) et (3) se traduisent concr` etement de la fa¸ con suivante. Soit
+g∈G(F) un ´ el´ ement semi-simple fortement r´ egulier. Alors on a une ´ ega lit´ e
+ΘΠG(ϕ)(g) =c/summationdisplay
+˜x∆(g,˜x)Θ˜π(ϕ)(˜x), (4)
+o` u ˜xparcourt un certain sous-ensemble fini de GL(ˆdG,F)θassoci´ e ` aget ∆(g,˜x) est un
+facteur de transfert (c’est l’inverse du facteur de Kottwitz-She lstad). Par ind´ ependance
+lin´ eaire des caract` eres, ces ´ egalit´ es d´ eterminent uniquemen t le paquet ΠG(ϕ).
+Remarque. Dans le cas d’un groupe orthogonal impair, les conjectures pos´ ee s en
+[W1] 4.2 faisaient intervenir l’endoscopie tordue entre GetGL(ˆdG+1)θ. Ici, on utilise
+l’endoscopie tordue plus habituelle entre GetGL(ˆdG)θ. Pour la validit´ e des r´ esultats
+de cette section, celle-ci suffit. Mais, pour le reste de l’article, on do it admettre aussi la
+validit´ e des conjectures telles qu’on les a formul´ ees en [W1].
+Dans le cas d’un groupe sp´ ecial orthogonal pair, il faut imposer ` a ϕune condition
+portant sur le d´ eterminant det◦ϕ(celui-ci doit correspondre au discriminant de la forme
+q, cf. [M1] paragraphe 2.1 et l’introduction ci-dessus), et on consid` ere les classes de
+conjugaison de tels ϕparSO(ˆdG,C) et non par O(ˆdG,C). Le paquet ΠG(ϕ) v´ erifie les
+mˆ emes conditions que ci-dessus. Mais les correspondances entre classes de conjugaison
+stable ne sont plus injectives : deux classes conjugu´ ees par le gro upe orthogonal tout
+19entier ne sont pas discernables par endoscopie tordue. Notons G+ce groupe orthogonal
+et fixons un ´ el´ ement w∈G+(F)\G(F). La relation (4) devient
+ΘΠG(ϕ)(g)+ΘΠG(ϕ)(wgw−1) =c/summationdisplay
+˜x∆(g,˜x)Θ˜π(ϕ)(˜x).
+Cesrelationsned´ eterminent plusΠG(ϕ).Toutefois, pourtouterepr´ esentation σdeG(F),
+notonsσwla repr´ esentation g/maps⊔o→σ(wgw−1) et posons ΠG(ϕ)w={πw;π∈ΠG(ϕ)}. Alors
+l’ensemble avec multiplicit´ es ¯Π(ϕ) = ΠG(ϕ)⊔ΠG(ϕ)west uniquement d´ etermin´ e.
+Comme on l’a expliqu´ e dans l’introduction, la d´ efinition des L-paquets s’´ etend au cas
+non temp´ er´ e en utilisant la classification de Langlands.
+Notation. Soitπune repr´ esentation irr´ eductible de G(F). On pose πGL=π(ϕ),
+o` uϕest le param` etre de Langlands tel que π∈ΠG(ϕ). C’est une repr´ esentation de
+GL(ˆdG,F).
+2.2 Induction et modules de Jacquet
+Soitπune repr´ esentation irr´ eductible temp´ er´ ee d’un groupe lin´ eair eGL(dπ,F). Le
+supportcuspidalde πalaformesuivante:ilexisteunensemblefiniavecmultiplicit´ es, que
+l’on noteJord(π), de couples ( ρ,a), o` uρest une repr´ esentation irr´ eductible cuspidale et
+unitaire d’un groupe lin´ eaire et a≥1 est un entier, de telle sorte que le support cuspidal
+deπest exactement
+∪(ρ,a)∈Jord(π)∪x∈[(a−1)/2,−(a−1)/2]{ρ|.|x
+F}.
+On a donc d´ efini Jord(π) pour toute repr´ esentation irr´ eductible temp´ er´ ee πd’un
+groupelin´ eaire. Ontranspose cela en uned´ efinition de Jord(π)pourtouterepr´ esentation
+irr´ eductible temp´ er´ ee πdeG(F), on posant Jord(π) =Jord(πGL), avec la notation
+introduite en 2.1. Remarquons que Jord(π) ne d´ epend que du L-paquet Π contenant π,
+ce qui permet de d´ efinir Jord(Π) pour un tel L-paquet.
+Soitπune repr´ esentation irr´ eductible temp´ er´ ee de G(F). L’ensemble Jord(π) a les
+propri´ et´ es suivantes. Pour tout ( ρ,a)∈Jord(π), ou bien la mutliplicit´ e de ( ρ,a) dans
+Jord(π) est paire, ou bien la repr´ esentation de WDFassoci´ ee ` a la repr´ esentation de
+SteinbergSt(ρ,a) est ` a valeurs dans un groupe classique de mˆ eme type que le group e
+dual deG. Dans ce dernier cas, on dira que ( ρ,a) a bonne parit´ e. Dans les autres cas,
+on dira que ( ρ,a) n’a pas bonne parit´ e. Ce dernier cas couvre ` a la fois celui o` u ρn’est
+pas autodual et celui o` u le param` etre de Langlands de St(ρ,a), bien qu’autodual, ne se
+factorise pas par le bon groupe classique.
+Le facteur de transfert de Kottwitz et Shelstad est compatible ` a l’induction. Cela
+signifie la chose suivante. Supposons Gquasi-d´ eploy´ e et fixons un tel facteur ∆ pour
+l’endoscopie tordue entre GetGL(ˆdG)θ. SoitLun L´ evi de G. Il lui correspond un L´ evi
+θ-stableLdeGL(ˆdG).Soientg∈L(F)et ˜x∈L(F)θdes´ el´ ementssuffisamment r´ eguliers.
+D´ efinissons un facteur ∆ L,Lθ(g,˜x) par
+∆L,Lθ(g,˜x) = ∆(g,˜x) si les classes de conjugaison stable de gdansL(F) et de ˜xdans
+L(F)θse correspondent;
+∆L,Lθ(g,˜x) = 0 sinon.
+Alors ∆L,Lθest un facteur de transfert pour le couple ( L,Lθ). Cela entraˆ ıne que le
+transfert est compatible ` a l’induction et cela a la cons´ equence suiv ante. Soit Π un paquet
+20de repr´ esentations temp´ er´ ees de G(F). Supposons donn´ ee une d´ ecomposition en union
+disjointe (au sens des ensembles avec multiplicit´ es) :
+Jord(Π) =E ⊔F ⊔ ˇE,
+o` uˇE={(ˇρ,a);(ρ,a)∈ E}. Il existe un L-paquet temp´ er´ e Π′d’un groupe de mˆ eme type
+queGtel queF=Jord(Π′). Alors
+(1) siGest symplectique ou sp´ ecial orthogonal impair, Π est exactement form´ e des
+composantes irr´ eductibles de toutes les induites
+(×(ρ,a)∈ESt(ρ,a))×π′
+quandπ′d´ ecrit Π′.
+Dans le cas o` u Gest sp´ ecial orthogonal pair, on a une assertion analogue. On doit
+remplacer Π et Π′par les paquets ¯Π et¯Π′d´ efinis en 2.1. Dans le cas o` u le sous-groupe
+parabolique Pservant ` a d´ efinir l’induite ci-dessus n’est pas semblable ` a wPw−1, il faut
+sommer sur les deux induites possibles.
+Notonsϕetϕ′les param` etres de Langlands de Π et Π′. L’assertion (1) r´ esulte sim-
+plement du fait que la somme des caract` eres des composantes en q uestion est stable et
+se transf` ere en le caract` ere de l’induite tordue
+(×(ρ,a)∈ESt(ρ,a))טπ(ϕ′).
+Or cette derni` ere n’est autre que ˜ π(ϕ)./square
+En particulier, un paquet temp´ er´ e Π est form´ e de repr´ esenta tions de la s´ erie discr` ete
+si et seulement si tous les ´ el´ ements de Jord(Π) sont de bonne parit´ e et interviennent
+avec multiplicit´ e 1.
+Une autre cons´ equence concerne les modules de Jacquet. Fixons un sous-groupe pa-
+raboliquePdeGde L´ eviGL(d1)×...×GL(dm)×G′. Pouri= 1,...,m, soientρi
+une repr´ esentation irr´ eductible cuspidale de GL(di,F), pas forc´ ement unitaire. Pour
+une repr´ esentation irr´ eductible πdeG(F), notonsJacρ1,...,ρm(π) la repr´ esentation semi-
+simple deG′(F) telle que le semi-simplifi´ e du module de Jacquet πPsoit la somme de
+ρ1⊗...⊗ρm⊗Jacρ1,...,ρm(π) et de repr´ esentations dont les premi` eres composantes ne so nt
+pas ´ egales ` a ρ1⊗...⊗ρm. Soitπune repr´ esentation irr´ eductible temp´ er´ ee de G(F). On a
+(2) supposons Jacρ1,...,ρm(π)/\e}a⊔io\slash={0}; alors pour tout ( ρ,a)∈Jord(π), il existe deux
+´ el´ ementsbρ,aetˇbρ,ade [(a+1)/2,−(a+1)/2], avecbρ,a>ˇbρ,atels que :
+•la famille (ρi)i=1,...,ms’obtienne en m´ elangeant les familles ( ρ|.|x
+F)x∈[(a−1)/2,bρ,a], pour
+(ρ,a)∈Jord(π), sans changer l’ordre dans chacune d’elles;
+•lafamille(ˇρi)i=m,...,1s’obtienne enm´ elangeantlesfamilles( ρ|.|x
+F)x∈[ˇbρ,a,−(a−1)/2],pour
+(ρ,a)∈Jord(π), sans changer l’ordre dans chacune d’elles.
+En effet, supposons Gsymplectique ou sp´ ecial orthogonal impair. Soit Π le L-paquet
+contenantπ. D’apr` es un lemme analogue ` a [MW] lemme 4.2.1, la somme des caract` e res
+desJacρ1,...,ρm(π1), pourπ1∈Π, est stable et a pour transfert un certain module de
+Jacquet ”tordu” Jacθ
+ρ1,...,ρm(πGL). En tant que repr´ esentation d’un groupe lin´ eaire non
+tordu, celui-ci se construit de fa¸ con analogue ` a ci-dessus. On n otePun sous-groupe
+parabolique de GL(ˆdG) de L´ evi
+GL(d1)×...×GL(dm)×GL(d0)×GL(dm)×...×GL(d1).
+AlorsJacθ
+ρ1,...,ρm(πGL) est la repr´ esentation semi-simple de GL(d0,F) telle que le module
+de Jacquet ( πGL)Psoit la somme de
+ρ1⊗...⊗ρm⊗Jacθ
+ρ1,...,ρm(πGL))⊗ˇρm⊗...⊗ˇρ1,
+21et de repr´ esentations dont les composantes non centrales ne so nt pas celles ci-dessus.
+L’hypoth` ese que Jacρ1,...,ρm(π)/\e}a⊔io\slash={0}impose que ce module de Jacquet tordu n’est pas
+nul. Mais on sait bien calculer ( πGL)Pet la conclusion s’ensuit. Dans le cas o` u Gest
+sp´ ecial orthogonal pair, la d´ emonstration s’adapte en consid´ e rant le paquet ¯Π./square
+On a un r´ esultat plus pr´ ecis pour des familles particuli` eres. Par e xemple, fixons une
+repr´ esentation irr´ eductible ρcuspidale et unitaire d’un groupe lin´ eaire et un r´ eel x>0
+et supposons ρi=ρ|.|x
+Fpour touti= 1,...,m. On a
+(3) supposons Jacρ1,...,ρm(π)/\e}a⊔io\slash={0}; alorsxest un demi-entier et ( ρ,2x+1) intervient
+avec multiplicit´ e au moins mdansJord(π).
+Cela r´ esulte imm´ ediatement de (2). /square
+Soit Π un paquet de repr´ esentations temp´ er´ ees de G(F). Soit (ρ,a)∈Jord(Π) avec
+a≥2, notonsmsa multiplicit´ e. Notons Π−le paquet temp´ er´ e tel que Jord(Π−) se
+d´ eduise de Jord(Π) en rempla¸ cant les mcopies de ( ρ,a) parmcopies de ( ρ,a−2).
+Prenonsρi=ρ|.|(a−1)/2
+Fpouri= 1,...,m. Soitπ∈Π. Alors
+(4) toutes les composantes irr´ eductibles de Jacρ1,...,ρm(π) appartiennent ` a Π−dans le
+cas symplectique ou orthogonal impair, ` a ¯Π−dans le cas orthogonal pair.
+En notant ϕetϕ−les param` etres de Langlands des paquets Π et Π−, on calcule
+facilement
+Jacθ
+ρ1,...,ρm(π(ϕ)) =π(ϕ−).
+Alors la mˆ eme preuve que celle de (2) entraˆ ıne (4). /square
+2.3 Support cuspidal ´ etendu
+Dans la suite de la section, on suppose Gsymplectique ou sp´ ecial orthogonal impair.
+On indiquera dans le dernier paragraphe 2.15 comment adapter les ar guments au cas
+d’un groupe sp´ ecial orthogonal pair.
+Soitπune repr´ esentation irr´ eductible de G(F). On appelle support cuspidal ´ etendu
+deπle support cuspidal ordinaire de πGL. C’est un ensemble avec multiplicit´ es de
+repr´ esentationsirr´ eductiblescuspidalesnonn´ ecessairement unitairesdegroupeslin´ eaires.
+Onremarquequeparsaconstruction,cetensembleeststablepar passage` alacontragr´ ediente.
+Lemme.Soitπune repr´ esentation de G(F)de la forme π=σ×π′, o` uσest une
+repr´ esentation irr´ eductible d’un groupe lin´ eaire et π′une repr´ esentation irr´ eductible d’un
+groupe de mˆ eme type que G. Alors tout sous-quotient irr´ eductible de πa pour support
+cuspidal ´ etendu l’union disjointe du support cuspidal ´ etendu de π′et des supports cus-
+pidaux ordinaires de σetˇσ.
+Ceci permet de parler du support cuspidal ´ etendu d’une repr´ es entation induite mˆ eme
+si celle-ci n’est pas irr´ eductible : c’est le support cuspidal ´ etend u de n’importe lequel de
+ses sous-quotients irr´ eductibles.
+Preuve. Soit πune repr´ esentation irr´ eductible de G(F). On peut la r´ ealiser comme
+sous-quotient d’une induite
+(×i=1,...,vρi)×πcusp,
+o` uπcuspest une repr´ esentation irr´ eductible cuspidale d’un groupe de mˆ e me type que G
+et, pour tout i= 1,...,v,ρiest une repr´ esentation irr´ eductible cuspidale d’un groupe
+lin´ eaire, pas forc´ ement unitaire. Appelons support cuspidal ord inaire deπl’ensemble
+22avec multiplicit´ es {ρi;i= 1,...,v}⊔ {ˇρi;i= 1,...,v} ⊔{πcusp}. On sait qu’il ne d´ epend
+pas de l’induite (1) choisie. Il est form´ e de repr´ esentations de gr oupes lin´ eaires et d’au
+plus une repr´ esentation πcuspd’un groupe de mˆ eme type que G(cette repr´ esentation
+disparaˆ ıt ici et dans la suite quand ce groupe est r´ eduit ` a 1). On v a montrer
+(1) le support cuspidal ´ etendu de πest l’union disjointe de celui de πcuspet du
+compl´ ementaire de {πcusp}dans le support cuspidal ordinaire de π.
+Cette assertion entraˆ ınelelemme car, danssous leshypoth` eses del’´ enonc´ e, lesupport
+cuspidal ordinaire de tout sous-quotient irr´ eductible de σ×π′est l’union disjointe de
+celui deπ′et de ceux de σet ˇσ.
+Pour prouver (1), on utilise la remarque suivante :
+(2) supposons que πapparaisse comme sous-quotient d’une induite σ1×...×σv×π′,
+o` uπ′et lesσisont irr´ eductibles et o` u v≥1, et que l’on sache que le support cuspidal
+´ etendu soit r´ eunion de celui de π′et des supports cuspidaux ordinaires des σiet des ˇσi
+pouri= 1,...,v; alors (1) est vrai pour π.
+En effet, le support cuspidal ordinaire de πest forc´ ement r´ eunion de celui de π′et de
+ceux desσiet ˇσipouri= 1,...,v. En raisonnant par r´ ecurrence sur dG, on peut supposer
+que le support cuspidal ´ etendu de π′se d´ eduit de son support cuspidal ´ etendu de la fa¸ con
+prescrite par (1) et on en d´ eduit que le support cuspidal ´ etendu deπse d´ eduit de la
+mˆ eme fa¸ con de son support cuspidal ordinaire.
+Siπn’est pas temp´ er´ ee, on r´ ealise πcomme quotient de Langlands d’une induite
+comme en (2), o` u π′est temp´ er´ ee et les σisont des repr´ esentations temp´ er´ ees tordues
+par un caract` ere. Par d´ efinition du support cuspidal ´ etendu d eπ, les hypoth` eses de (2)
+sont satisfaites. Donc (1) est v´ erifi´ ee pour π.
+Supposons que πsoit temp´ er´ ee et qu’il existe un´ el´ ement ( ρ,a) deJord(π) tel que, ou
+bien (ρ,a) ne soit pas la bonne parit´ e, ou bien ( ρ,a) soit de bonne parit´ emais intervienne
+avecmultiplicit´ e aumoins2.Alors2.2(1)permet der´ ealiser πcommesous-quotient d’une
+induite de sorte que les hypoth` eses de (2) soient satisfaites. D’o` u la conclusion dans ce
+cas.
+Supposons que πsoit temp´ er´ ee, que Jord(π) soit form´ e de couples ( ρ,a) de bonne
+parit´ e intervenant avec multiplicit´ e 1, et que πne soit pas cuspidale. Cette derni` ere
+hypoth` ese entraˆ ıne que l’on peut r´ ealiser πcomme sous-module d’un induite ρ′×π′,
+o` uπ′est irr´ eductible et ρ′est irr´ eductible et cuspidale, pas forc´ ement unitaire. D’apr` es
+l’hypoth` ese sur Jord(π), 2.2(2) entraˆ ıne qu’il existe ( ρ,a)∈Jord(π) tel quea≥2 et
+ρ′=ρ|.|(a−1)/2
+F. La relation 2.2(3) entraˆ ıne alors que le support cuspidal ´ etendu deπest
+r´ eunion de celui de π′et de{ρ′,ˇρ′}. Autrement dit, les hypoth` eses de (2) sont v´ erifi´ ees
+et on conclut.
+On est ramen´ e au cas o` u πest cuspidale. Mais alors (1) est tautologique. /square
+Remarque. Soientπunerepr´ esentationirr´ eductibletemp´ er´ eede G(F),ρunerepr´ esentation
+irr´ eductible cuspidale et unitaire d’un groupe lin´ eaire et s∈R. Supposons que l’in-
+duiteρ|.|s
+F×πait un sous-quotient irr´ eductible ayant mˆ eme support cuspidal q u’une
+repr´ esentation temp´ er´ ee. Alors l’une des conditions suivantes est v´ erifi´ ee :
+•s= 0;
+•sest un demi-entier avec |s| ≥1,ρ≃ˇρetJord(π) contient le couple ( ρ,2|s|−1)
+•s=±1/2 et (ρ,2) a bonne parit´ e.
+Cela r´ esulte du lemme ci-dessus et des propri´ et´ es du support c uspidal ´ etendu d’une
+repr´ esentation temp´ er´ ee.
+232.4 Support cuspidal ´ etendu et induction dans le cas temp´ e r´ e
+Lemme.Soitπune repr´ esentation irr´ eductible temp´ er´ ee de G(F). On suppose qu’elle
+est sous-quotient d’une induite
+<e,f > ρ×π′
+o` u[e,f]est un segment tel que e≥0≥f,ρest une rep´ esentation irr´ eductible cuspidale
+et unitaire d’un groupe lin´ eaire et π′est une repr´ esentation irr´ eductible temp´ er´ ee d’un
+groupe de mˆ eme type que G. Alors l’une des propri´ et´ es suivantes est v´ erifi´ ee :
+•ρ≃ˇρ,eetfsontdesdemi-entierset Jord(π) =Jord(π′)∪{(ρ,2e+1),(ρ,−2f+1)};
+•ρ/\e}a⊔io\slash≃ˇρ,e=−fetJord(π) =Jord(π′)∪{(ρ,2e+1),(ˇρ,2e+1)}.
+Preuve. Le support cuspidal ´ etendu de πest, d’apr` es le lemme pr´ ec´ edent, l’union de
+celui deπ′avec∪x∈[e,f]{ρ|.|x
+F} ∪x∈[−f,−e]{ˇρ|.|x
+F}. Les conditions sur ρ,e,fr´ esultent des
+propri´ et´ es rappel´ ees en 2.2 du support cuspidal ´ etendu de s repr´ esentations temp´ er´ ees. /square
+2.5 Un lemme technique
+Lemme.Consid´ erons une induite
+σ|.|s×π′,
+o` uπ′est une repr´ esentation irr´ eductible temp´ er´ ee d’un groupe de mˆ eme type que G,
+σ=St(ρ,a)estunerepr´ esentationdeSteinbergg´ en´ eralis´ eetemp´ er´ e ed’ungroupelin´ eaire
+ets >0est un r´ eel. Alors le quotient de Langlands de cette induite est l’uniqu e sous-
+quotient irr´ eductible qui poss` ede une inclusion dans une induite de la forme
+(×x∈[(a−1)/2,−(a−1)/2]ˇρ|.|x−s
+F)×τ,
+o` uτest une repr´ esentation non n´ ecessairement irr´ eductible d’un g roupe de mˆ eme type
+queG. En particulier si
+σ|.|s
+F×π′֒→(×x∈[(a−1)/2,−(a−1)/2]ˇρ|.|x−s
+F)×τ
+avecτcomme ci-dessus, l’induite σ|.|s
+F×π′est irr´ eductible.
+Preuve. Il est clair que le quotient de Langlands de l’induite σ|.|s
+F×π′a la pro-
+pri´ et´ e requise car il est inclus dans ˇ σ|.|−s
+F×π′et cette induite est elle-mˆ eme incluse dans
+(×x∈[(a−1)/2,−(a−1)/2]ˇρ|.|x−s
+F)×π′. Soit donc πun sous-quotient irr´ eductible de l’induite
+ayant une inclusion comme dans l’´ enonc´ e. Comme πest irr´ eductible, une telle inclu-
+sion en donne une de mˆ eme type mais avec τirr´ eductible. On suppose donc que τest
+irr´ eductible. Par r´ eciprocit´ e de Frobenius un module de Jacquet convenable de π′admet
+(⊗x∈[(a−1)/2,−(a−1)/2]ˇρ|.|x−s
+F)⊗τcomme quotient.
+On sait calculer tous les termes du module de Jacquet de l’induite σ|.|s
+F×π′. Les
+termes d’un module de Jacquet cuspidal de cette induite peuvent se regrouper en sous-
+ensembles param´ etr´ es par un demi-entier entier x∈[(a+1)/2,−(a−1)/2] et un terme
+24cuspidal, ( ⊗j=1,...,kρj|.|yj
+F)×π′
+cuspdu module de Jacquet de π′. Les termes correspondants,
+´ ecrits sous la forme
+(⊗(ρ′,z′)∈Eρ′|.|z′
+F)⊗π′
+cusp
+sont tels que le l’ensemble ordonn´ e Es’obtient en m´ elangeant les ensembles ordonn´ es
+´ ecrits ci-dessous sans permuter l’ordre interne ` a chacun de ces sous-ensembles :
+{ρ|.|s+i
+F;i∈[(a−1)/2,x]}(c’est l’ensemble vide si x= (a+1)/2 );
+{ˇρ|.|i′−s
+F;i′∈[(a−1)/2,−x[}(c’est l’ensemble vide si x=−(a−1)/2) ;
+{ρj|.|yj
+F;j= 1,...,k}.
+Ainsi lesapremiers exposants d’un tel terme, ´ ecrits z′
+ipouri= 1,...,ase d´ ecomposent
+ena1termes du premier ensemble, a2termes du deuxi` eme et a3termes du troisi` eme.
+Donc on a :
+/summationdisplay
+i=1,...,az′
+i=/summationdisplay
+ℓ∈[(a−1)/2,(a+1)/2−a1](s+ℓ)+/summationdisplay
+ℓ∈[(a−1)/2,(a+1)/2−a2](−s+ℓ)+/summationdisplay
+j=1,...,a3yj.
+Puisqueπ′est une repr´ esentation temp´ er´ ee, on a sˆ urement/summationtext
+j=1,...,a3yj≥0 et donc
+/summationdisplay
+i=1,...,az′
+i≥a1s−a2s≥ −a2s.
+Pour le terme consid´ er´ e du module de Jacquet de π, cette somme vaut −as. Comme
+a2≤a, on doit avoir ´ egalit´ e a2=v, d’o` ua1=a3= 0. Notons Ple sous-groupe
+parabolique qui sert ` a d´ efinir l’induite σ|.|s
+F×π′. Les termes du module de Jacquet de
+l’induiteσ|.|s
+F×π′qui v´ erifient les conditions pr´ ec´ edentes sont ceux qui provien nent du
+sous-quotient ˇ σ|.|−s
+F⊗π′du module de Jacquet ( σ|.|s
+F×π′)P. Ils n’interviennent que dans
+le quotient de Langlands de notre induite. Donc πest ce quotient de Langlands. Cela
+d´ emontre la premi` ere assertion de l’´ enonc´ e. Sous l’hypoth` e se de la seconde assertion,
+l’unique sous-module irr´ eductible de σ|.|s
+F×π′est le quotient de Langlands d’apr` es ce
+que l’on vient de prouver. Puisque l’unique sous-module irr´ eductible e t aussi l’unique
+quotient irr´ eductible, la repr´ esentation est irr´ eductible. /square
+2.6 Induction et L-paquets temp´ er´ es
+Ci-dessous, on utilise ` a plusieurs reprises la remarque ´ el´ ementa ire suivante. Soit xun
+nombre r´ eel nonnul et ρunerepr´ esentation irr´ eductible cuspidale et unitaired’ungroupe
+lin´ eaire. Soient aussi m≥1 un entier et σune repr´ esentation de G(F). On suppose
+qu’il n’existe pas d’inclusion de la forme σ ֒→ρ|.|x
+F×σ′o` uσ′est une repr´ esentation
+quelconque. Alors l’induite ρ|.|x
+F×···ρ|.|x
+F×σ,, o` u il y amcopies deρ|.|x
+F, a un unique
+sous-module irr´ eductible. C’est un calcul de module de Jacquet et d e r´ eciprocit´ e de
+Frobenius pour le parabolique standard de sous-groupe de Levi GL(mdρ)×G. En effet,
+les sous-quotients irr´ eductibles de ce module de Jacquet de la form eτ⊗τ′v´ erifient
+soit que le support cuspidal de τn’est pasmcopies deρ|.|x
+F, soit queτest l’induite
+irr´ eductible ρ|.|x
+F× ··· ×ρ|.|x
+F. Dans ce dernier cas, on a n´ ecessairement τ′≃σet un
+tel terme n’intervient qu’avec multiplicit´ e au plus 1 comme sous-quot ient irr´ eductible
+du module de Jacquet. Par r´ eciprocit´ e de Frobenius, tout sous- module irr´ eductible de
+l’induite´ ecrite asonmoduledeJacquet qui admet ( ρ|.|x
+F×···×ρ|.|x
+F)⊗σcomme quotient
+25irr´ eductible. Comme le foncteur de Jacquet est exact, cela force l’unicit´ e d’un tel sous-
+module irr´ eductible. Remarquons que, d’apr` es 2.1(2), l’hypoth` e se surσest v´ erifi´ ee si les
+deux conditions suivantes le sont :
+•σest une repr´ esentation irr´ eductible temp´ er´ ee;
+•xn’est pas un demi-entier positif, ou xest un tel demi-entier mais Jord(σ) ne
+contient pas ( ρ,2x+1).
+Lemme.SoitΠun paquet de repr´ esentations temp´ er´ ees. Soit x >0un demi-entier
+tel que(ρ,2x+ 1)∈Jord(Π). On notemla multiplicit´ e de (ρ,2x+ 1)dansJord(Π)
+etΠ−le paquet de repr´ esentations temp´ er´ ees qui se d´ eduit de Πen rempla¸ cant les m
+copies de (ρ,2x+1)parmcopies de (ρ,2x−1). Pour tout π−∈Π−, on noteπl’unique
+sous-module irr´ eductible de l’induite ρ|.|x
+F×···ρ|.|x
+F×π−. Alorsπest une repr´ esentation
+temp´ er´ ee de Πet l’application ainsi d´ efinie de Π−dansΠest une injection.
+Preuve. Soit π∈Π. Supposons qu’il existe une inclusion de la forme π֒→ρ|.|x
+F×···×
+ρ|.|x
+F×σ, avecmcopies deρ|.|x
+F. On v´ erifie que l’on n’a certainement aucune inclusion de
+la formeσ֒→ρ|.|x
+F×σ′. Sinon, on aurait une inclusion π ֒→ρ|.|x
+F×···×ρ|.|x
+F×σ, avec
+m+1 copies de ρ|.|x
+FetJord(π) contiendrait m+1 copies de ( ρ,a), cf. 2.1(3). Les calculs
+de modules de Jacquet expliqu´ es ci-dessus montrent alors que si u ne telle inclusion se
+produit avec σque l’on suppose irr´ eductible, tout sous-quotient irr´ eductible du module
+de Jacquet de πde la forme ρ|.|x
+F× ···ρ|.|x
+F⊗σ′′, o` u il y a mcopies deρ|.|x
+F, v´ erifie
+σ′′≃σ. Un peu plus g´ en´ eralement supposons que π∈Π et que le module de Jacquet de
+πcontient un sous-quotient irr´ eductible de la forme ρ|.|x
+F×···ρ|.|x
+F⊗σ′′; alors quitte ` a
+changerσ′′on peut supposer que ce sous-quotient est en fait un quotient du m odule de
+Jacquet de πet, par r´ eciprocit´ e de Frobenius que πest un sous-module comme ci-dessus.
+Quand on applique le module de Jacquet pour le parabolique GL(mdρ,F)⊗G′(pour
+G′convenable) ` a la distribution stable form´ ee de la somme des ´ el´ em ents deπet que
+l’on projette sur le caract` ere de la repr´ esentation ρ|.|x
+F×···×ρ|.|x
+F(avecm-copies), on
+obtient unedistribution stable associ´ ee ` aΠ−. Ainsi d’apr` es ceque l’ona vuci-dessus, une
+repr´ esentation π∈Π soit disparaˆ ıt dans cette proc´ edure, soit contribue par le cara ct` ere
+deσ(avec les notations ci-dessus). Ainsi n´ ecessairement σ∈Π−et toute repr´ esentation
+de Π−est obtenue par cette proc´ edure. On a donc d´ efini sur un sous -ensemble de Π un
+inverse surjectif de l’application de l’´ enonc´ e. Cela montre que cet te derni` ere application
+est bien une injection de Π−dans Π./square
+2.7 Propri´ et´ e des repr´ esentations temp´ er´ ees ayant un mod` ele
+de Whittaker
+Supposons Gquasi-d´ eploy´ e. On d´ efinit de la fa¸ con habituelle la notion de mod` e le de
+Whittaker. Il y a plusieurs types de tels mod` eles, autant que de cla sses de conjugaison
+d’´ el´ ements unipotents r´ eguliers dans G(F). On utilise la propri´ et´ e suivante :
+soitπune repr´ esentationirr´ eductibletemp´ er´ eede G(F); il existe uneunique repr´ esentation
+temp´ er´ ee irr´ eductible π0deG(F)ayant un mod` ele de Whittaker d’un type fix´ e et telle
+queJord(π0) =Jord(π).
+Cf. [K] th´ eor` eme 3.4, [W1] th´ eor` eme 4.9.
+26Lemme.Supposons Gquasi-d´ eploy´ e, soit πune repr´ esentation irr´ eductible de G(F)
+ayant un mod` ele de Whittaker. On suppose que le support cuspidal ´ etendu deπest celui
+d’une repr´ esentation temp´ er´ ee. Alors πest temp´ er´ ee.
+Preuve. On d´ emontre ce lemme par r´ ecurrence sur le rang de Get pour cela on a
+besoin d’une cons´ equence du lemme. Pour fixer les notations, on no te Π un paquet de
+repr´ esentations temp´ er´ ees et on suppose que le support cus pidal ´ etendu de πest le mˆ eme
+que celui des ´ el´ ements de Π. Le lemme dit alors que πappartient ` a Π. Ceci montre que le
+support cuspidal ordinaire de πest bien d´ etermin´ e. On peut d´ ecrire ce support cuspidal.
+En effet, notons ϕle morphisme de WDFdans leL-groupe deGparam´ etrisant Π. Notons
+ϕLlemorphisme de WDFdans ceL-groupequi est trivial sur SL(2,C) et qui, sur WF, est
+le compos´ e de l’inclusion : WF֒→WDF=WF×SL(2,C);w/maps⊔o→/parenleftbigg
+w,/parenleftBigg
+|w|1/2
+F0
+0|w|−1/2
+F/parenrightBigg/parenrightbigg
+avecϕ. AinsiϕLd´ etermine un paquet de Langlands assez particulier. Le support c us-
+pidal ´ etendu de tout ´ el´ ement de ce paquet est le mˆ eme que celu i des ´ el´ ements de Π.
+Il existe exactement une repr´ esentation de ce paquet qui soit le quotient de Langlands
+d’une induite ayant un mod` ele de Whittaker. Notons π0le sous-quotient irr´ eductible
+de cette induite ayant un mod` ele de Whittaker. Le lemme dit que π0est temp´ er´ ee et
+isomorphe ` a π. On pourrait montrer que π=π0est la repr´ esentation duale au sens
+d’Aubert, Schneider-Stuhler du quotient de Langlands. Ce qui nou s importe est que si π
+est cuspidal, alors ϕL=ϕ, doncJord(π) est un ensemble de couples ( ρ,1).
+Prouvons le lemme. On fixe πcomme dans l’´ enonc´ e et Π comme ci-dessus. Si πest
+cuspidale, elle est a fortiori temp´ er´ ee, il n’y a rien ` a d´ emontrer . Sinon, on ´ ecrit πcomme
+sous-module d’une induite de repr´ esentations cuspidales
+π֒→(×i=1,...,vρi|.|si
+F)×πcusp, (1)
+o` uπcuspest une repr´ esentation irr´ eductible cuspidale d’un groupe de mˆ e me type que G,
+v≥1 est un entier et, pour tout i= 1,...,v,ρiest une repr´ esentation cuspidale unitaire
+irr´ eductible d’un groupe lin´ eaire et siest un nombre r´ eel. Comme πcuspa n´ ecessairement
+un mod` ele de Whittaker et est une repr´ esentation d’un groupe de rang plus petit que
+G, on sait, par r´ ecurrence, que Jord(πcusp) est un ensemble de couples ( ρ,1). Puisque
+πcuspest de la s´ erie discr` ete, cet ensemble est sans multiplicit´ es et pou r tout (ρ,1) y
+intervenant, ( ρ,1) est de bonne parit´ e, en particulier ρest autodual.
+Soit (ρ,a)∈Jord(Π). On suppose d’abord que ρn’est pas autoduale. Ainsi (ˇ ρ,a)∈
+Jord(Π) et il existe un sous-ensemble Ede{1,...,v}tel que∪i∈E{ρi|.|si
+F,ˇρi|.|−si
+F}=
+∪x∈[(a−1)/2,−(a−1)/2]{ρ|.|x
+F,ˇρ|.|x
+F}. Un tel ensemble n’est pas unique, en g´ en´ eral, et on en
+fixe un. Ainsi πest certainement un sous-quotient irr´ eductible de l’induite :
+(×x∈[(a−1)/2,−(a−1)/2]ρ|.|x
+F)×(×i=1,...,v;i/∈Eρi|.|si
+F)×πcusp.
+Il existe donc un sous-quotient irr´ eductible σde l’induite ×x∈[(a−1)/2,−(a−1)/2]ρ|.|x
+Fet un
+sous-quotient irr´ eductible π′de l’induite ( ×i=1,...,v;i/∈Eρi|.|si
+F)×πcusptel queπsoit un sous-
+quotient irr´ eductible de l’induite σ×π′. N´ ecessairement σa un mod` ele de Whittaker au
+sensusuel et π′enaundumˆ emetypeque π.Ainsiσ≃<(a−1)/2,−(a−1)/2>ρ.Enfait,
+on connaˆ ıt aussi π′. En effet le support cuspidal ´ etendu de l’induite ( ×i=1,...,v;i/∈Eρi|.|si
+F)×
+πcuspest celui des repr´ esentations temp´ er´ ees dans le paquet Π′, qui se d´ eduit de Π en
+enlevant (ρ,a) et (ˇρ,a). En appliquant le lemme par r´ ecurrence, on sait que π′est une
+27repr´ esentation temp´ er´ ee. Il en est donc de mˆ eme de tout so us-quotient de σ×π′, donc
+πest temp´ er´ ee. Cela prouve le lemme dans ce cas.
+Supposons maintenant que Jord(Π) contient un ´ el´ ement ( ρ,a) tel queρsoit auto-
+duale, avec une multiplicit´ e not´ ee mv´ erifiant l’une des conditions suivantes :
+•m>2;
+•m= 2 etaest pair;
+•m= 2 etaest impair mais ( ρ,1) n’intervient pas dans Jord(πcusp).
+Ces conditionssont parexemple automatiques si( ρ,a) estdemauvaiseparit´ e. Alors E
+comme ci-dessus existe encore et π′d´ efini comme ci-dessus a encore son support cuspidal
+´ etendu qui se d´ eduit de celui de πen enlevant deux copies de ( ρ,a). On conclut comme
+ci-dessus.
+D’autre part, si a= 1 pour tout ( ρ,a)∈Jord(Π), lessisont tous nuls et σest certai-
+nementunerepr´ esentationtemp´ er´ eed’apr` esl’inclusion(1).I lnoussuffitdoncmaintenant
+de d´ emontrer le lemme dans le cas o` u Jord(Π) contient un´ el´ ement ( ρ,a) de bonne parit´ e
+aveca≥2. On peut se limiter au cas o` u la multiplicit´ e de ( ρ,a) est inf´ erieure ou ´ egale
+` a 2. On fixe un tel ( ρ,a) et on suppose que aest maximal avec cette propri´ et´ e.
+On fait d’abord la d´ emonstration dans le cas o` u la multiplicit´ e de ( ρ,a) dansJord(π)
+est 1 pour clarifier la m´ ethode. On fixe une inclusion (1) et on sait qu’il existei0∈
+{1,...,v}tel queρi0|.|si0
+F≃ρ|.|ζ(a−1)/2
+Fpour un signe convenable ζ. On fixe une telle
+inclusion avec l’hypoth` ese suppl´ ementaire que pour tous les choix possibles,i0est mi-
+nimum. On suppose d’abord que i0= 1 et on conclut : il existe un sous-quotient π′de
+l’induite ( ×i=2,...,vρi|.|si
+F)×πcusptel queπsoit un sous-module irr´ eductible de l’induite
+ρ|.|ζ(a−1)/2
+F×π′. Le support cuspidal ´ etendu de π′est celui des repr´ esentations temp´ er´ ees
+dans le paquet qui se d´ eduit de Π en rempla¸ cant ( ρ,a) par (ρ,a−2). De plus π′admet
+un mod` ele de Whittaker. Par l’hypoth` ese de r´ ecurrence, on sait queπ′est temp´ er´ ee.
+Ainsi siζ= +,πest un sous-module irr´ eductible de l’induite ρ|.|(a−1)/2
+F×π′. On applique
+le lemme 2.6 avec ici m= 1,π−=π′. Ce lemme nous dit que πappartient ` a Π, donc
+est une repr´ esentation temp´ er´ ee. Consid´ erons le cas o` u ζ=−. Sous cette hypoth` ese, π
+est le sous-module de Langlands de l’induite ρ|.|−(a−1)/2
+F×π′. Commeπa un mod` ele de
+Whittaker, le th´ eor` eme 1.1 de [Mu] montre que cette induite est irr ´ eductible et πest
+donc aussi un sous-module irr´ eductible de l’induite ρ|.|(a−1)/2
+F×π′. On conclut comme
+pr´ ec´ edemment. Au passage d’ailleurs cette conclusion montre qu e l’induite ne peut pas
+ˆ etre irr´ eductible et que ce cas ne peut se produire.
+Il suffit donc de d´ emontrer que i0= 1. Supposons qu’il n’en soit pas ainsi mais que
+ζ= +. Dans ce cas ρi0−1|.|si0−1
+F×ρ|.|(a−1)/2
+Fest soit irr´ eductible soit de longueur deux.
+Dans le premier cas on peut commuter les facteurs ce qui contredit la minimalit´ e de i0.
+Dans le deuxi` eme cas, ρi0−1|.|si0−1
+F≃ρ|.|(a−3)/2
+Fet on peut remplacer ρ|.|(a−3)/2
+F×ρ|.|(a−1)/2
+F
+par son unique sous-quotient ayant un mod` ele de Whittaker. Ce so us-quotient est <
+(a−1)/2,(a−3)/2>ρ. Or<(a−1)/2,(a−3)/2>ρ֒→ρ|.|(a−1)/2
+F×ρ|.|(a−3)/2
+Fet on peut
+donc encore ´ echanger i0eti0−1 ce qui contredit la minimalit´ e de i0. Donc siζ= +,
+i0= 1.
+On suppose que ζ=−. Ici la deuxi` eme partie de l’argument ci-dessus est diff´ erente.
+Onpeutremplacer ρ|.|−(a−3)/2
+F×ρ|.|−(a−1)/2
+Fpar<−(a−3)/2,−(a−1)/2>ρ.Enproc´ edant
+ainsi de proche ne proche, on montre qu’il existe un segment d´ ecr oissant [e,−(a−1)/2]
+et une inclusion
+π ֒→<e,−(a−1)/2>ρ×π′, (2)
+o` uπ′est une repr´ esentation irr´ eductible convenable. On va montrer que n´ ecessairement
+28e= (a−3)/2.De(1)ontireque πestunsous-quotientirr´ eductibledel’induite ρ|.|(a−1)/2
+F×
+π1o` uπ1a pour support cuspidal ´ etendu celui de πo` u on a remplac´ e ( ρ,a) par (ρ,a−2).
+Commeπ1a n´ ecessairement un mod` ele de Whittaker, π1est temp´ er´ ee. On sait calculer
+les modules de Jacquet de l’induite ρ|.|(a−1)/2
+F×π1. On voit que (2) force l’existence d’une
+inclusion :
+π1֒→ ×j∈[e,−(a−3)/2]ρ|.|j
+F×π2,
+o` uπ2ne nous int´ eresse pas. Comme π1est une repr´ esentation temp´ er´ ee, n´ ecessairement
+e= (a−3)/2 comme annonc´ e. Le support cuspidal ´ etendu de π1contient donc
+∪x∈[(a−3)/2,−(a−3)/2{ρ|.|x
+F}
+avec multiplicit´ e au moins 2. Ceci est donc aussi vrai pour le suppor t cuspidal de π
+et il existe des entiers b,b′sup´ erieurs ou ´ egaux ` a a−2 tel que (ρ,b) et (ρ,b′) soient
+dansJord(Π). Par maximalit´ e de a, on ab,b′≤aet comme ( ρ,a) n’intervient qu’avec
+multiplicit´ e 1 dans Jord(Π), l’un des deux vaut a−2. AinsiJord(Π) contient ( ρ,a) et
+(ρ,a−2) et le support cuspidal ´ etendu de π′se d´ eduit de celui de Jord(π) en enlevant
+(ρ,a) et (ρ,a−2). On applique l’hypoth` ese de r´ ecurrence ` a π′qui a n´ ecessairement
+un mod` ele de Whittaker et on sait que π′est temp´ er´ ee. Ainsi πest le sous-module de
+Langlands de l’induite <(a−3)/2,−(a−1)/2>ρ×π′. D’apr` es [Mu] thorme 1.1, cette
+induite doit ˆ etre irr´ eductible, elle est donc isomorphe ` a <(a−1)/2,−(a−3)/2>ρ×π′
+on trouve encore une inclusion avec i0= 1 ce qui contredit la minimalit´ e de i0.
+Il nous reste donc ` a voir le cas o` u la multiplicit´ e de ( ρ,a) dansJord(π) est ´ egale ` a 2.
+On proc` ede comme ci-dessus mais ici, i0et remplac´ e par i1<i2o` upourj= 1,2, il existe
+unsigneζjtel queρij|.|sij
+F=ρ|.|ζj(a−1)/2
+F. Onfixe une telle inclusion. Enproc´ edant comme
+ci-dessus, on pousse d’abord vers la gauche les ρ|.|ζij(a−1)/2
+F o` uζj= +. Puis on pousse les
+autres. On montre que pour tout j= 1,2 tel queζj=−, il existe un demi-entier ejtel
+que [ej,ζj(a−1)/2] soit un segment d´ ecroissant et une inclusion
+π ֒→(×j=1,2;ζj=+ρ|.|(a−1)/2
+F)×(×j=1,2;ζj=−<ej,ζj(a−1)/2>ρ)×π′,
+pourπ′irr´ eductible convenable. Comme ci-dessus on v´ erifie que ou bien ζj= +, ou bien
+ej≥(a−3)/2 et que l’´ egalit´ e est n´ ecessaire par maximalit´ e de a(ici on utilise le fait
+que l’on a d´ ej` a pouss´ e tous les ρ|.|(a−1)/2
+Fen premi` ere position). On note m+le nombre
+dej= 1,2 tels queζj= + etm−= 2−m+. On pose
+τ=ρ|.|(a−1)/2
+F×...×ρ|.|(a−1)/2
+F×<(a−3)/2,−(a−1)/2>ρ×...×<(a−3)/2,−(a−1)/2>ρ,
+avecm+copies de la premi` ere repr´ esentation et m−copies de la seconde. Ainsi π′est
+une repr´ esentation irr´ eductible ayant un mod` ele de Whittaker e t dont le support cus-
+pidal ´ etendu s’obtient ` a partir Jord(Π) en enlevant le support cuspidal de τet celui
+de ˇτ. Autrement dit, le support cuspidal ´ etendu de π′s’obtient en rempla¸ cant d’abord
+dansJord(Π) les 2 copies de ( ρ,a) par 2 copies de ( ρ,a−2) puis en enlevant 2 m−co-
+pies de (ρ,a−2); il se peut que 2 m−>2 mais comme dans la d´ emonstration du cas
+m= 1 cela ne gˆ ene pas. Ainsi le support cuspidal ´ etendu de π′est le support cuspi-
+dal d’une repr´ esentation temp´ er´ ee et π′est donc une repr´ esentation temp´ er´ ee. On pose
+m=inf(m+,m−).
+On suppose que m/\e}a⊔io\slash= 0, doncm=m+=m−= 1, et on remarque que l’inclusion
+π ֒→τ×π′se factorise necessairement par le sous-module irr´ eductble de τayant un
+29mod` ele de Whittaker. Celui-ci est <(a−1)/2,−(a−1)/2>ρ. La repr´ esentation π′a
+pour support cuspidal ´ etendu celui de πdont on a enlev´ e les deux copies de ( ρ,a). C’est
+donc bien le support cuspidal ´ etendu d’une repr´ esentation temp ´ er´ ee et par hypoth` ese de
+r´ ecurrence, on sait que π′est temp´ er´ ee. Ainsi πest un sous-module d’une repr´ esentation
+induite temp´ er´ ee et est donc temp´ er´ ee.
+On suppose donc que m= 0. On a donc soit m+= 2 etm−= 0, soitm−= 2
+etm+= 0. Dans le cas o` u m+= 2, on sait que πest une repr´ esentation temp´ er´ ee en
+appliquant le lemme 2.6. Si m−= 2, on sait que πest le sous-module de Langlands de
+l’induiteτ×π′. Cette induite est donc irr´ eductible d’apr` es [Mu] thorme 1.1. D’o` u encore
+π≃ˇτ×π′
+֒→ρ|.|(a−1)/2
+F×ρ|.|(a−1)/2
+F×<(a−3)/2,−(a−3)/2>ρ×<(a−3)/2,−(a−3)/2>ρ×π′.
+On trouve encore une inclusion
+π֒→ρ|.|(a−1)/2
+F×ρ|.|(a−1)/2
+F×π′′,
+o` uπ′′est une repr´ esentation temp´ er´ ee. L’ensemble Jord(π′′) se d´ eduit de Jord(Π) en
+rempla¸ cant les 2 copies de ( ρ,a) par 2 copies de ( ρ,a−2). En particulier Jord(π′′) ne
+contient pas ( ρ,a) et contient au moins 2 copies de ( ρ,a−2) (on peut avoir a= 2 bien
+que le cas o` u aest pair ait d´ ej` a ´ et´ e d´ emontr´ e). Il suffit d’appliquer le lemme 2 .6 pour
+conclure que σest une repr´ esentation temp´ er´ ee. Cela termine la preuve. /square
+2.8 D´ efinition des points de r´ eductibilit´ e possible pour une
+repr´ esentation temp´ er´ ee
+Soitπune repr´ esentation irr´ eductible temp´ er´ ee de G(F). On note RP(π) (pour
+”r´ eductibilit´ epossible”)l’ensembledescouples( ρ,x),o` uρunerepr´ esentationirr´ eductible
+cuspidale unitaire d’un groupe lin´ eaire et xun nombre r´ eel, tel que :
+•ρ|.|x
+F×πait pour support cuspidal ´ et´ endu le support cuspidal d’une repr ´ esentation
+temp´ er´ ee de GL(ˆdG+2dρ,F);
+•de plus, six= 0, (ρ,1) est de bonne parit´ e mais n’appartient pas ` a Jord(π).
+Ce sont exactement les couples d´ ecrits dans la remarque 2.3 sauf q uandx= 0 o` u
+on a restreint les possibilit´ es. Remarquons que, pour ( ρ,x)∈RP(π), on a ˇρ=ρet
+(ρ,−x)∈RP(π).
+Proposition .Soientπunerepr´ esentationirr´ eductibletemp´ er´ eede G(F),ρunerepr´ esentation
+cuspidale unitaire irr´ eductible d’un groupe lin´ eaire et xun r´ eel non nul. On suppose que
+(ρ,x)/∈RP(π). Alors l’induite ρ|.|x
+F×πest irr´ eductible.
+Preuve. Le cas x= 0 r´ esulte de la classification des repr´ esentations temp´ er´ ees . Dans
+tout ce qui suit, on suppose x/\e}a⊔io\slash= 0 et, par sym´ etrie, on peut supposer x>0.
+On appelle s´ erie discr` ete strictement positive une s´ erie discr` et e telle que tous les
+termes de son module de Jacquet cuspidal soient de la forme ( ⊗i=1,...,vρi|.|si
+F)⊗πcuspo` u
+πcuspest une repr´ esentation irr´ eductible cuspidale d’un groupe de mˆ e me type que G,
+vest un entier convenable et, pour i= 1,...,v,ρiest une repr´ esentation irr´ eductiible
+cuspidale unitaire d’un groupe lin´ eaire et siest un r´ eel strictement positif. Remarquons
+que lessisont forc´ ement des demi-entiers.
+30On suppose d’abord que πest une s´ erie discr` ete strictement positive. On va alors
+montrer la propri´ et´ e suivante : l’induite ρ|.|x
+F×πest irr´ eductible ou elle contient un
+sous-quotient irr´ eductible qui est une repr´ esentation temp´ e r´ ee. En effet, soit σun sous-
+quotient irr´ eductible de l’induite ρ|.|x
+F×π.
+On regarde les termes d’un module de Jacquet cuspidal de σc’est-` a-dire les inclusions
+σ֒→(×i=1,...,v+1ρi|.|si
+F)×πcusp. (1)
+D’apr` es la propri´ et´ e de positivit´ e de πtous lessiqui interviennent sont strictement
+positifs sauf ´ eventuellement un, pour i=i0disons, tel que ρi0≃ˇρetsi0=−x. Sii0= 1,
+alorsσest le quotient de Langlands de l’induite ρ|.|x
+F×πd’apr` es le lemme 2.5. Supposons
+quei0>1 et supposons d’abord que x/\e}a⊔io\slash= 1/2. Cette hypoth` ese assure que, pour tout
+i= 1...i0−1, les induites ρi|.|si
+Fסρ|.|−x
+Fsont irr´ eductibles car si∈1/2Netsi+xest
+un nombre r´ eel positif qui n’est pas 1. Ainsi l’inclusion ci-dessus don ne une inclusion
+analogue mais o` u ˇ ρ|.|−x
+Fa ´ et´ e pouss´ e ` a la premi` ere place. Dans ce cas σest le quotient de
+Langlands. Donc si x/\e}a⊔io\slash= 1/2,σest soit une s´ erie discr` ete positive soit est le quotient de
+Langlands. Ainsi si l’induite ρ|.|x
+F×πest r´ eductible, elle contient un sous-quotient qui
+est une s´ erie discr` ete et x∈RP(π) comme annonc´ e.
+Lecaso` ux= 1/2est demˆ eme nature; onnepeut pas”pousser” ˇ ρ|.|−1/2
+F` alapremi` ere
+place s’il existe i= 1,...,i0−1 tel queρi≃ˇρetsi= 1/2. Mais dans ce cas, le terme du
+moduledeJacquet quel’onconsid` erev´ erifielapropri´ et´ edepos itivit´ elargequicaract´ erise
+les repr´ esentations temp´ er´ ees. Donc ici, soit σest une repr´ esentation temp´ er´ ee soit σest
+le quotient de Langlands de l’induite ρ|.|x
+F×π. Et on conclut comme ci-dessus.
+On suppose maintenant que πest une repr´ esentation temp´ er´ ee quelconque et on
+prouve la proposition par r´ ecurrence sur le rang de G. On suppose que x /∈RP(π) et on
+montre que l’induite ρ|.|x
+F×πest irr´ eductible.
+On suppose que πn’est pas une s´ erie discr` ete strictement positive puisque ce cas a
+d´ ej` a ´ et´ e vu. Alors il existe :
+•une repr´ esentation irr´ eductible cuspidale unitaire ρ0d’un groupe lin´ eaire;
+•un segment [ e0,f0] form´ e de demi-entiers tel que e0≥0≥f0;
+•une repr´ esentation irr´ eductible temp´ er´ ee π′d’un groupe de mˆ eme type que G;
+de sorte que l’on ait une inclusion :
+π ֒→<e0,f0>ρ0×π′.
+Siπn’est pas de la s´ erie discr` ete, cela r´ esulte de 2.2(1). Si πest de la s´ erie discr` ete, c’est
+le lemme 3.1 de [M2]. Si ρ0/\e}a⊔io\slash≃ˇρ0n´ ecessairement e0=−f0et d’apr` es le lemme 1.4 :
+RP(π) =/braceleftBigg
+RP(π′)∪(ρ0,e0+1)∪(ρ0,−f0+1) siρ0≃ˇρ0,
+RP(π′)∪(ρ0,e0+1)∪(ˇρ0,e0+1) siρ0/\e}a⊔io\slash≃ˇρ0.
+Ainsi si (ρ,x)/∈RP(π), l’induite ρ|.|x
+F×<e0,f0>ρ0est irr´ eductible car soit ρ/\e}a⊔io\slash≃ρ0, soit
+ρ≃ρ0etx/\e}a⊔io\slash=d0+1, (onrappelle que x>0 cequi force x/\e}a⊔io\slash=f0−1). Demˆ eme soit ρ/\e}a⊔io\slash≃ˇρ0,
+soitρ≃ˇρ0etx/\e}a⊔io\slash=−f0+1, d’o` ul’irr´ eductibilit´ e de l’induite <−f0,−e0>ˇρ0×ρ|.|x
+Fet par
+dualit´ e celle de <e0,f0>ρ0סρ|.|−x
+F. Commex /∈RP(π′), par l’hypoth` ese de r´ ecurrence,
+on sait aussi que ρ|.|x
+F×π′est irr´ eductible. On a donc une suite de morphismes :
+ρ|.|x
+F×π֒→ρ|.|x
+F×<e0,f0>ρ0×π′≃<e0,f0>ρ0×ρ|.|x
+F×π′
+≃<e0,f0>ρ0סρ|.|−x
+F×π′≃ˇρ|.|−x
+F×<e0,f0>ρ0×π′.
+L’irr´ eductibilit´ e cherch´ ee r´ esulte alors du lemme 2.5 et cela termin e la preuve. /square
+312.9 R´ eductibilit´ e et g´ en´ ericit´ e
+Lemme.Soitρune repr´ esentation irr´ eductible d’un groupe lin´ eaire, que l’on sup pose
+cuspidale, unitaire et autoduale. Soient e,fdes demi-entiers tels que e≥0ete+f/\e}a⊔io\slash= 0.
+Onsuppose que (ρ,2e+1)et(ρ,2|f|+1)ont labonne parit´ e. Supposons Gquasi-d´ eploy´ e,
+soitπune repr´ esentation temp´ er´ ee de G(F), irr´ eductible, ayant un mod` ele de Whittaker.
+Alors
+(i) sif≤0, l’induite<e,f > ρ×πest r´ eductible;
+(ii) supposons que f >0et que(ρ,f)∈RP(π); alors l’induite < e,f > ρ×πest
+r´ eductible.
+Preuve.Leshypoth` esesde(i)commede(ii)assurentquetoutso us-quotientirr´ eductible
+de l’induite ´ ecrite a mˆ eme support cuspidal ´ etendu qu’une repr´ e sentation temp´ er´ ee π′:
+dansle cas(i), Jord(π′) =Jord(π)∪{(ρ,2d+1),(ρ,−2f+1)}et dans lecas (ii), Jord(π′)
+se d´ eduit de Jord(π) en rempla¸ cant ( ρ,2f−1) par (ρ,2d+ 1). Or ces induites ont un
+sous-quotient irr´ eductible ayant un mod` ele de Whittaker. On app lique alors le lemme
+2.7 pour montrer que ce sous-quotient irr´ eductible est temp´ er´ e. Puisque l’induite n’est
+pas temp´ er´ ee, elle est r´ eductible. /square
+2.10 La notion de liaison
+On fixe une repr´ esentation irr´ eductible cuspidale et unitaire ρd’un groupe lin´ eaire et
+un segment [ e,f] de nombres r´ eels. Soit πune repr´ esentation irr´ eductible temp´ er´ ee de
+G(F). On dit que ( ρ,e,f) etJord(π) sont li´ es si les conditions suivantes sont satisfaites :
+eest un demi-entier et
+(1) si (ρ,2|e|+1) n’est pas de bonne parit´ e, il existe un entier a≥1 de mˆ eme parit´ e
+que 2e+1 tel que ( ρ,a)∈Jord(π) et les segments [ e,f] et [(a−1)/2,−(a−1)/2] sont
+li´ es au sens de Zelevinsky;
+(2) si (ρ,2|e|+1) est de bonne parit´ e, alors soit e≥ −1/2 etf≤1/2, soit il existe
+(ρ,a)∈Jord(π) avec (a+1)/2∈[d,f]∪[−f,−d].
+La notion de liaison ne d´ epend que de Jord(π) et non de π. De plus, ( ρ,e,f) et
+Jord(π) sont li´ es si et seulement si (ˇ ρ,−f,−e) etJord(π) le sont.
+Proposition .On suppose Gquasi-d´ eploy´ e. Soit (ρ,e,f)comme ci-dessus et soit π
+une repr´ esentation irr´ eductible temp´ er´ ee de G(F)ayant un mod` ele de Whittaker. On
+suppose que (ρ,e,f)etJord(π)sont li´ es et que e+f/\e}a⊔io\slash= 0. Alors l’induite <e,f > ρ×π
+est r´ eductible.
+Preuve. Par sym´ etrie, on suppose e+f >0, a fortiori e >0. On suppose d’abord
+que (ρ,e,f) satisfait la propri´ et´ e (1) de la condition de liaison. On fixe acomme dans
+cette propri´ et´ e; comme ( ρ,a) n’a pas bonne parit´ e (puisque ( ρ,2e+1) ne l’a pas) πest
+une induite de la forme <(a−1)/2,−(a−1)/2>ρ×π′, o` uπ′est une repr´ esentation
+irr´ eductible temp´ er´ ee convenable. On a donc un isomorphisme :
+<e,f > ρ×π≃<e,f > ρ×<(a−1)/2,−(a−1)/2>ρ×π′.
+32Or l’induite < e,f > ρ×<(a−1)/2,−(a−1)/2>ρdans leGLconvenable n’est
+pas irr´ eductible d’o` u la r´ eductibilit´ e de l’induite de gauche comme an nonc´ e. Ici on n’a
+d’ailleurs pas utilis´ e le fait que πa un mod` ele de Whittaker.
+On suppose que c’est (2) de la d´ efinition de la liaison qui est satisfait. En particulier
+(ρ,2e+ 1) et (ρ,2|f|+ 1) sont de bonne parit´ e. Si f≤1/2 le lemme 2.9 montre que
+l’induite est r´ eductible. On suppose donc que f >1/2 et qu’il existe un entier a≥1
+tel que (ρ,a)∈Jord(π) et (a+1)/2∈[e,f]. On noteσle sous quotient irr´ eductible de
+l’induite<e,f > ρ×πayant un mod` ele de Whittaker. Ainsi σest aussi un sous-quotient
+irr´ eductible de l’induite
+<(a−1)/2,f >ρ×<e,(a+1)/2>ρ×π.
+Onnoteπ′lesous-quotient irr´ eductible del’induite <e,(a+1)/2>ρ×πayant unmod` ele
+de Whittaker et on sait que π′est une repr´ esentation temp´ er´ ee, cf. lemme 2.7. Ainsi σ
+est un sous-quotient de l’induite <(a−1)/2,f >ρ×π′. Si l’induite < e,f > ρ×πest
+irr´ eductible, elle co¨ ıncide avec σetσest un sous-module de l’induite <−f,−e >ρ×π.
+Donc un module de Jacquet cuspidal de σcontient un terme ρ|.|−f
+F⊗...⊗ρ|.|−e
+F⊗.... On
+sait calculer les modules de Jacquet cuspidaux de l’induite <(a−1)/2,f >ρ×π′. Pour
+qu’ils contiennent le terme pr´ ec´ edent, il faut n´ ecessairement q u’il existex∈[−f,−e] tel
+que le module de Jacquet de π′contienne un terme de la forme ρ|.|−x
+F⊗···car le facteur
+<(a−1)/2,f >ρne peut contribuer au mieux qu’au sous-segment [ −f,(a−1)/2] de
+[−f,−e]. Ceci contredit le fait que π′est temp´ er´ ee et termine la preuve. /square
+2.11 Un r´ esultat d’irr´ eductibilit´ e
+Lemme.Soient(ρ,e,f)comme en 2.10et πune repr´ esentation irr´ eductible et temp´ er´ ee
+deG(F). On suppose que (ρ,e,f)etJord(π)ne sont pas li´ es, que e+f/\e}a⊔io\slash= 0et que,
+ou bienen’est pas un demi-entier, ou bien eest demi-entier et (ρ,2|e|+1)n’est pas de
+bonne parit´ e. Alors l’induite <e,f > ρ×πest irr´ eductible.
+Preuve. On suppose, par sym´ etrie, que e+f >0. On d´ emontre d’abord le lemme
+dans le cas o` u soit ρ/\e}a⊔io\slash≃ˇρ, soit 0/∈[e,f]. On gagne le fait que pour tout x∈[e,f[,
+l’induite (dans un groupe lin´ eaire convenable)
+<e,x> ρסρ|.|−x+1
+F
+est irr´ eductible. Supposons de plus pour le moment que πest une s´ erie discr` ete. Alors
+pour toutx∈[e,f], (ρ,x)/∈RP(π) (puisque ( ρ,x) n’a pas la bonne parit´ e) et l’induite
+ρ|.|x
+F×πest irr´ eductible donc isomorphe ` a ˇ ρ|.|−x
+F×π. De proche en proche, on montre
+alors que l’on a une inclusion
+<e,f > ρ×π ֒→ˇρ|.|−f
+F×···ˇρ|.|−x+1
+F×<e,x> ρ×π
+֒→ˇρ|.|−f
+F×···×ˇρ|.|−e
+F×π.
+Et cela donne l’irr´ eductibilit´ e annonc´ ee dans l’´ enonc´ e d’apr` es le lemme 2.5.
+On enl` eve l’hypoth` ese que πest une s´ erie discr` ete. S’il n’en est pas ainsi, πest une
+sous-repr´ esentation d’une induite de la forme <(a−1)/2,−(a−1)/2>ρ′×π′, o` uπ′
+33est une repr´ esentation irr´ eductible temp´ er´ ee. Il est imm´ ed iat que (ρ,e,f) etJord(π′) ne
+sont pas li´ es car Jord(π′) est un sous-ensemble de Jord(π). De plus puisque ( ρ,e,f) et
+Jord(π) ne sont pas li´ es, on sait que si ρ≃ρ′, les segment [( a−1)/2,−(a−1)/2] et
+[e,f] ne sont pas li´ es. Il en est de mˆ eme si ρ≃ˇρ′car si (ρ,a)∈Jord(π) alors (ˇρ,a)
+aussi. Par sym´ etrie par rapport ` a 0, les segments [( a−1)/2,−(a−1)/2] et [−f,−e]
+ne sont li´ es que si les segments [( a−1)/2,−(a−1)/2] et [e,f] sont li´ es. Ainsi l’induite
+<e,f > ρ×<(a−1)/2,−(a−1)/2>ρ′est irr´ eductible et il en est de mˆ eme de l’induite
+<(a−1)/2,−(a−1)/2>ρ′×<−f,−e >ˇρ. On admet par r´ ecurrence, que l’induite
+<e,f > ρ×π′est irr´ eductible donc isomorphe ` a l’induite <−f,−d>ˇρ×π′et on a alors
+une s´ erie d’isomorphismes :
+<e,f > ρ×π ֒→<e,f > ρ×<(a−1)/2,−(a−1)/2>ρ′×π′
+≃<(a−1)/2,−(a−1)/2>ρ′×<e,f > ρ×π′
+≃<(a−1)/2,−(a−1)/2>ρ′×<−f,−e>ˇρ×π′
+≃<−f,−e>ˇρ×<(a−1)/2,−(a−1)/2>ρ′×π′.
+Et l’irr´ eductibilit´ e annonc´ ee r´ esulte du lemme 2.5.
+Le cas restant est plus d´ elicat. On suppose donc que ρ≃ˇρet que 0 ∈[e,f], c’est-` a-
+dire queeetfsont des entiers tels que e≥0≥f. Pour ´ eviter les confusions de signes,
+on posef+=−f≥0.
+On traite d’abord le cas o` u πest une s´ erie discr` ete strictement positive. Soit σun
+sous-quotient irr´ eductible de l’induite < e,f > ρ×π. On va montrer que soit σest une
+repr´ esentation temp´ er´ ee, soit σest le quotient de Langlands de l’induite <e,f > ρ×π.
+On consid` ere les termes constants cuspidaux de σc’est-` a-dire les inclusions
+σ֒→(×i=1,...,v′ρ′
+i|.|s′
+i
+F)×πcusp, (1)
+o` uπcuspest irr´ eductible et cuspidale et, pour tout i= 1,...,v′,ρ′
+iest une repr´ esentation
+irr´ eductible cuspidale et unitaire d’un groupe lin´ eaire et s′
+i∈R. Les termes constants
+cuspidaux de toute l’induite sont index´ es par le choix d’un entier x∈[e+ 1,−f+] et
+d’un terme constant cuspidal pour π, c’est-` a-dire d’une inclusion :
+π ֒→(×i=1,...,vρi|.|si
+F)×πcusp
+et s’obtiennent en m´ elangeant les 3 ensembles ordonn´ ees suivant s (en gardant l’ordre
+dans chaque ensemble)
+∪i∈[e,x]{ρ|.|i
+F};∪i∈[f+,−x[{ρ|.|i
+F};∪i=1,...,v{ρi|.|si
+F}.
+Untel terme v´ erifie laconditionde positivit´ edes repr´ esentation s temp´ er´ ees si x≤f++1.
+Le point est donc de d´ emontrer que si σcontient un terme comme ci-dessus avec x >
+f++1 alorsσest le quotient de Langlands de l’induite <e,−f+>ρ×π.
+On fixe donc σet un terme comme ci-dessus avec x > f++ 1. On utilise le fait
+que pour tout i= 1,...,v, siρi≃ρalorssiest un demi-entier non entier : ( ρ,2si+ 1)
+doit avoir bonne parit´ e, or eest entier et ( ρ,2e+ 1) n’a pas bonne parit´ e. Donc pour
+tout entier y∈[e,−f+]∪[f+,−e] et pour tout i= 1,...,v, l’induiteρ|.|y
+F×ρi|.|si
+Fest
+irr´ eductible. On peut donc commuter ces facteurs. De mˆ eme pou r touty∈[e,x] et tout
+y′∈[f+,−x[ l’induiteρ|.|y
+F×ρ|.|y′
+Fest irr´ eductible car y−y′≥x−f+>1. On peut donc
+34aussi commuter de tels facteurs et finalement l’assertion sur σse traduit par l’existence
+d’une inclusion :
+σ֒→(×j∈[f+,−x]ρ|.|j
+F)×(×i=1,...,vρi|.|si
+F)×(×y∈[e,x[ρ|.|y
+F)×πcusp.
+Pour touty∈[e,x[,ρ|.|y
+F×πcuspest irr´ eductible (proposition 2.8) et donc isomorphe ` a
+ρ|.|−y
+F×πcusp. D’o` u finalement une inclusion
+σ֒→(×j∈[f+,−x]ρ|.|j
+F)×(×i=1,...,vρi|.|si
+F)×(×y∈[x,e]ρ|.|−y
+F)×πcusp
+≃(×j∈[f+,−e]ρ|.|j
+F)×(×i=1,...,vρi|.|si
+F)×πcusp.
+Etσest le quotient de Langlands de l’induite comme annonc´ e, cf. lemme 2.5 . Ainsi
+< e,f > ρ×πa un seul sous-quotient irr´ eductible qui n’est pas temp´ er´ e, c’e st le sous-
+quotient de Langlands. Mais une telle induite ne peut avoir de sous-qu otient irr´ eductible
+qui sont des repr´ esentations temp´ er´ ees car son support cu spidal ´ etendu est l’union de
+celui deπavec les 2 segments [ e,−e]∪[f+,−f+] bas´ es sur ρ. Ceci n’est pas le support
+cuspidal ´ etendu d’une repr´ esentation temp´ er´ ee car ( ρ,2e+ 1) n’a pas bonne parit´ e, ni
+d’ailleurs (ρ,2f++1), et que e/\e}a⊔io\slash=f+par hypoth` ese. D’o` u l’irr´ eductibilit´ e.
+On consid` ere maintenant une repr´ esentation irr´ eductible temp ´ er´ eeπquelconque et
+on prouve l’irr´ eductibilit´ e par r´ ecurrence. Si πn’est pas une s´ erie discr` ete strictement
+positive, alors, comme on l’a dit en 2.8, il existe une repr´ esentation c uspidaleρ′(unitaire
+irr´ eductible), un segment [ e′,−f′] de demi-entiers avec e′,f′≥0, et une repr´ esentation
+irr´ eductible temp´ er´ ee π′avec une inclusion :
+π֒→<e′,−f′>ρ′×π′.
+De plus si ρ′≃ρsoit (ρ,2e′+ 1) et (ρ,2f′+ 1) sont de bonne parit´ e ce qui entraˆ ıne
+quee′−eetf′+fsont des demi-entiers non entiers, soit ( ρ,2e′+1) et (ρ,2f′+1) ne
+sont pas de bonne parit´ e ce qui force e′=f′et, par l’hypoth` ese de non liaison, que les
+segments [e,f] et [e′,−f′] ne sont pas li´ es. Ainsi l’induite <e,f > ρ×<e′,−f′>ρ′est
+irr´ eductible. On v´ erifie de la mˆ eme fa¸ con que l’induite et < e′,−f′>ρ′×<−f,−e >ρ
+est irr´ eductible. On a donc :
+<e,f > ρ×π ֒→<e,f > ρ×<e′,−f′>ρ′×π′≃
+<e,−f′>ρ′×<e,f > ρ×π′.
+On sait d’apr` es le lemme 2.4 que Jord(π) =Jord(π′)∪{(ρ,2e′+1),(ρ,2f′+1)}et ainsi
+(ρ,e,f) n’est pas li´ e ` a Jord(π′). Par l’hypoth` ese de r´ ecurrence < e,f > ρ×π′est donc
+aussi irr´ eductible et on a encore une inclusion
+<e,f > ρ×π ֒→<e′,−f′>ρ′×<−f,−e>ρ×π′
+≃<−f,−e>ρ×<d′,−f′>ρ′×π′.
+Avec le lemme 2.5, cette inclusion montre l’irr´ eductibilit´ e de l’induite <e,f > ρ×π. Cela
+termine la preuve. /square
+352.12 Un second r´ esultat d’irr´ eductibilit´ e
+Proposition .Soient(ρ,e,f)commeen2.10et πunerepr´ esentationirr´ eductibletemp´ er´ ee
+deG(F).Onsupposeque (ρ,e,f)etJord(π)nesontpasli´ es.Alorsl’induite <e,f > ρ×π
+est irr´ eductible.
+Preuve. Entenantcomptedulemmepr´ ec´ edent,ilnousreste` a voirlecaso` u eestdemi-
+entier et (ρ,2|e|+1) est de bonne parit´ e. On peut encore supposer e+f >0. L’hypoth` ese
+de non liaison assure alors que f >1/2 et que pour tout x∈[e,f],ρ|.|x
+F/∈RP(π). En
+particulier, pour un tel x, on sait que ρ|.|x
+F×πest irr´ eductible (proposition 2.8), donc
+isomorphe ` a ρ|.|−x
+F×π. De plus pour tous y,y′∈[d,f],y+y′≥2f >1 et les induites
+ρ|.|y
+F×ρ|.|−y′
+Fsont donc aussi irr´ eductibles. On a donc un isomorphisme :
+(×x∈[e,f]ρ|.|x
+F)×π≃(×x∈[−f,−e]ρ|.|x
+F)×π.
+D’o` u
+<e,f > ρ×π ֒→(×x∈[e,f]ρ|.|x
+F)×π≃(×x∈[−f,−e]ρ|.|x
+F)×π.
+Et on conclut, avec le lemme 2.5, ` a l’irr´ eductibilit´ e de l’induite. /square
+2.13 Crit` ere d’irr´ eductibilit´ e
+G´ en´ eralisons la notion de liaison. Soit k∈Net pour tout i= 1,...,ksoit (ρi,ei,fi)
+un triplet form´ e d’une repr´ esentation irr´ eductible ρicuspidale et unitaire d’un groupe
+lin´ eaireetd’unsegment[ ei,fi]denombresr´ eels.Soitaussi πunerepr´ esentationirr´ eductible
+temp´ er´ ee de G(F). On dit que {(ρi,ei,fi)i=1,...,k}etJord(π) ne sont pas li´ es si pour tout
+i= 1,...,k, (ρi,ei,fi) etJord(π) ne sont pas li´ es et si, pour tout j= 1,...,k,j/\e}a⊔io\slash=i, les
+induites<ei,fi>ρi×<ej,fj>ρjet<ei,fi>ρi×<−fj,−ej>ˇρjsont irr´ eductibles.
+Th´ eor` eme .Soient{(ρi,ei,fi)i=1,...,k}commeci-dessuset πunerepr´ esentationirr´ eductible
+temp´ er´ ee de G(F). On suppose que pour tout i= 1,...,k,ei+fi/\e}a⊔io\slash= 0.
+(i) On suppose que Gest quasi-d´ eploy´ e et que πa un mod` ele de Whittaker. Alors
+l’induite (×i=1,...,k< ei,fi>ρi)×πest irr´ eductible seulement si {(ρi,ei,fi)i=1,...,k}et
+Jord(π)ne sont pas li´ es.
+(ii) On suppose que {(ρi,ei,fi)i=1,...,k}etJord(π)ne sont pas li´ es. Alors l’induite
+(×i=1,...,k<ei,fi>ρi)×πest irr´ eductible.
+En d’autres termes, la condition de non liaison est n´ ecessaire et suffi sante pour avoir
+l’irr´ eductibilit´ e de l’induite si πa un mod` ele de Whittaker et est seulement suffisante
+sans cette hypoth` ese de mod` ele de Whittaker.
+Preuve. Prouvons (i). On suppose que l’induite figurant dans cette assertion est
+irr´ eductible. Il faut certainement que pour tout i,j= 1,...,k,i/\e}a⊔io\slash=j, les induites <
+ei,fi>ρi×<ej,fj>ρjsoient irr´ eductibles. Pour un choix de j= 1,...,k, l’irr´ eductibilit´ e
+de l’induite de l’assertion est aussi ´ equivalente ` a l’irr´ eductibilit´ e de la mˆ eme induite
+mais o` u on remplace < ej,fj>ρjpar sa contragr´ ediente <−fj,−ej>ˇρj; on doit donc
+avoir aussi l’irr´ eductibilit´ e pour tout i,j= 1,...,k,i/\e}a⊔io\slash=j, des induites < ei,fi>ρi
+×<−fj,−ej>ˇρj. De plus pour tout i= 1,...,k, l’induite< ei,fi>ρi×πdoit aussi
+36ˆ etre irr´ eductible et donc, d’apr` es la proposition 2.10, ( ρi,ei,fi) etJord(π) ne sont pas
+li´ es. Cela montre que les conditions de non liaison sont bien n´ ecessa ires pour avoir
+l’irr´ eductibilit´ e quand πa un mod` ele de Whittaker.
+Prouvons (ii). On suppose que, pour tout i= 1,...,k,ei+fi>0, sinon on remplace
+<ei,fi>ρipar<−fi,−ei>ˇρi.
+En permutant, comme on en a le droit, on suppose aussi que e1+f1≥ ··· ≥ek+fk.
+Avec ces choix, l’induite
+(×i=1,...,k<ei,fi>ρi)×π (1)
+a un unique quotient irr´ eductible, σ. De plusσintervient avec multiplicit´ e 1 comme
+sous-quotient irr´ eductible et l’induite
+(×i=1,...,k<−fi,−ei>ˇρi)×π≃(×i=k,...,1<−fi,−ei>ˇρi)×π (2)
+aununiquesous-moduleirr´ eductibleetilestisomorphe` a σ.Pourd´ emontrerl’irr´ eductibilit´ e
+de l’induite (1), il suffit donc de montrer que (1) est un sous-module d e (2) et on va mˆ eme
+v´ erifier que (1) et (2) sont isomorphes. On a les isomorphismes
+(×i=1,...,k<ei,fi>ρi)×π≃(×i=1,...,k−1<ei,fi>ρi)×<−fk,−ek>ˇρk×π
+≃<−fk,−ek>ˇρk×(×i=1,...,k−1<ei,fi>ρi)×π.
+Et de proche en proche on construit donc un isomorphisme de (1) et (2). Ceci termine
+la preuve. /square
+2.14 Irr´ eductibilit´ e et L-paquets g´ en´ eriques
+SoitGlaformeint´ erieurequasi-d´ eploy´ eede G.PourunL-paquetΠderepr´ esentations
+temp´ er´ ees de G(F), on note Π leL-paquet de repr´ esentations temp´ er´ ees de G(F) qui
+correspond` aΠ,autrementditdontleparam` etredeLanglandse stlemˆ emequeceluideΠ.
+On aJord(Π) =Jord(Π). Dans l’´ enonc´ e ci-dessous, l’induite d´ esigne une repr´ esenta tion
+deG(F) ou deG(F) selon que πappartient ` a Π ou ` a Π .
+Corollaire .Soitk∈N. Pour tout i= 1,...,k, soitσiune repr´ esentation irr´ eductible
+d’un groupe lin´ eaire, unitaire et de la s´ erie discr` ete, et soit siun nombre r´ eel non nul.
+SoitΠunL-paquet de repr´ esentations irr´ eductibles temp´ er´ ees de G(F). Alors l’induite
+(×i=1,...,kσi|.|si
+F)×π
+est irr´ eductible pour tout π∈Π⊔Πsi et seulement s’il existe une repr´ esentation π0∈Π
+ayant un mod` ele de Whittaker et tel que l’induite pr´ ec´ edente soit irr´ eductible pour
+π=π0.
+Preuve. Il est ´ evident que si l’induite de l’´ enonc´ e est irr´ educt ible pour tout π∈Π⊔Π,
+elle l’est en particulier pour π=π0ayant un mod` ele de Whittaker. R´ eciproquement on
+suppose que cette induite est irr´ eductible pour une repr´ esenta tionπ=π0ayant un
+mod` ele de Whittaker. On ´ ecrit, pour tout i= 1,...,k,σi|.|si
+Fsour la forme < ei,fi>ρi
+et on asi=ei+fi/\e}a⊔io\slash= 0. Le (i) du th´ eor` eme pr´ ec´ edent dit que les {(ρi,ei,fi)i=1,...,k}
+etJord(Π) ne sont pas li´ es. Le (ii) du mˆ eme th´ eor` eme donne alors l’irr´ e ductibilit´ e de
+l’induite pour tout π∈Π⊔Π. Cela termine la preuve. /square
+372.15 Le cas des groupes sp´ eciaux orthogonaux pairs
+On suppose que Gest sp´ ecial orthogonal pair. La principale diff´ erence avec les cas
+symplectique etsp´ ecial orthogonalimpairestquenotrenotation simplifi´ eeσ1×...×σm×π
+pour les induites est trop impr´ ecise car d’une part, il peut y avoir de ux sous-groupes
+paraboliques non conjugu´ es de mˆ eme L´ evi GL(d1)×...×GL(dt)×G′, d’autre part,
+l’identificationdugroupecentral G′n’estcanoniquequ’` aconjugaisonpr` esparun´ el´ ement
+du groupe orthogonal tout entier (ce qui fait que selon l’identificat ion choisie, πest
+remplac´ epar πw).Onpeutrem´ edier` aceladelafa¸ consuivante.Fixonsuned´ eco mposition
+V=Fv1⊕...⊕Fvt⊕Van⊕Fv−t⊕...⊕Fv−1
+de sorte que
+•les vecteurs visont isotropes et q(vi,v−i) = 1 pour i=±1,...,±t;
+•les espaces Fv1⊕Fv−1,...,Fvt⊕Fv−tetVansont deux ` a deux orthogonaux;
+•la restriction de q` aVanest anisotrope.
+Appelonsfamilleparaboliqueunefamille( Ij)j=1,...,mv´ erifiantlesconditionssuivantes:
+•chaqueIjest un sous-ensemble non vide de {±1,...,±t};
+•en posant ˇIj={−i;i∈Ij}, les ensembles I1,...,Im,ˇI1,...,ˇImsont deux ` a deux
+disjoints.
+Pour une telle famille et pour j= 1,...,t, notonsXjle sous-espace de Vengendr´ e
+par les vecteurs vipouri∈ ∪k=1,...,jIk. NotonsPI1,...,Imle sous-groupe parabolique de G
+form´ e des ´ el´ ements qui conservent le drapeau de sous-espa ces
+X1⊂X2⊂...⊂Xm.
+La composante de L´ evi de PI1,...,Ims’identifie ` a GL(d1)×...×GLdm×G′o` u, pour tout
+j,djest le nombre d’´ el´ ements de IjetG′est le groupe sp´ ecial orthogonal du sous-
+espace deVengendr´ e par Vanet lesvipouri/\e}a⊔io\slash∈/uniontext
+j=1,....,m(Ij∪ˇIj). Cette identification
+est canonique (` a automorphismes int´ erieurs pr` es pour les blocs GL(dj), mais cela n’a
+pas d’importance). Supposons d’abord Van/\e}a⊔io\slash={0}. Alors l’application qui, ` a une famille
+parabolique ( Ij)j=1,...,m, associePI1,...,Imest injective. La classe de conjugaison de ce
+parabolique est d´ etermin´ ee par la suite d’entiers d1,...,dm. Supposons maintenant Van=
+{0}. L’application n’est plus injective. Le d´ efaut d’injectivit´ e est le suiv ant. Consid´ erons
+une famille parabolique ( Ij)j=1,...,mtelle qu’il existe i0∈ {1,...,t}de sorte que
+(/uniondisplay
+j=1,...,m(Ij∪ˇIj)) ={±1,...,±t}\{i0,−i0}.
+On peut compl´ eter cette famille en deux autres, en lui ajoutant u n ´ el´ ementIm+1´ egal
+soit ` a{i0}, soit ` a{−i0}. On a
+PI1,...,Im=PI1,...,Im,{i0}=PI1,...,Im,{−i0}.
+Remarquons que les L´ evi de ces paraboliques ne sont pas exactem ent les mˆ emes. Le
+premier a un bloc central G′qui est un groupe SO(2) d´ eploy´ e, les deux autres n’ont pas
+de bloc central, mais ont un facteur GL(1) suppl´ ementaire. Cela correspond aux deux
+identifications possibles de SO(2) d´ eploy´ e avec GL(1). Par ailleurs, la suite d1,...,dmne
+d´ etermine plus toujours la classe de conjugaison du parabolique.
+Dans les constructions des paragraphes pr´ ec´ edents, il convie nt maintenant, ` a chaque
+fois que l’on se donne une induite σ1×...×σm×π, de fixer auparavant une famille
+38parabolique ( Ij)j=1,...,met de pr´ eciser que l’induite est IndG
+PI1,...,Im(σ1⊗...⊗σm⊗π). Au
+cours d’une d´ emonstration, la famille parabolique peut changer. Pa r exemple, dans le
+cas o` um= 2, dire que l’on peut permuter σ1etσ2signifie que l’on a
+IndG
+PI1,I2(σ1⊗σ2⊗π) =IndG
+PI2,I1(σ2⊗σ1⊗π).
+De mˆ eme, quand IndG
+PI1(σ⊗π) admet un quotient de Langlands, celui-ci est une sous-
+repr´ esentation de IndG
+PˇI1(ˇσ⊗π).
+Comme on le voit, le groupe SO(2) d´ eploy´ e peut intervenir dans nos raisonnements.
+Il convient de consid´ erer que, pour ce groupe, aucune repr´ es entation n’est de la s´ erie
+discr` ete, a fortiori aucune n’est cuspidale. Les repr´ esentatio ns temp´ er´ ees πsont les ca-
+ract` eres unitaires. Pour une telle repr´ esentation, on a Jord(π) ={ρ,ρ−1}, o` uρest un
+caract` ere unitaire de GL(1,F). L’´ el´ ement ρa bonne parit´ e si et seulement si ρ=ρ−1,
+c’est-` a-dire ρest un caract` ere quadratique.
+Certains raisonnements utilisent la caract´ erisation des repr´ ese ntations temp´ er´ ees par
+la positivit´ e de leurs exposants cuspidaux. La notion de positivit´ e e st a priori diff´ erente
+pour un groupe sp´ ecial orthogonal pair de ce qu’elle est pour un gr oupe symplectique ou
+sp´ ecial orthogonal impair. Par exemple, pour un exposant cuspid al (b1,...,bt) relatif ` a un
+sous-groupe parabolique minimal, la condition de positivit´ e s’exprime p ar les in´ egalit´ es
+b1≥0, b1+b2≥0,...,b1+...+bt≥0 (1)
+dans les cas symplectique ou sp´ ecial orthogonal impair, tandis qu’il faut ajouter la condi-
+tion
+b1+...+bt−1−bt≥0 (2)
+dans le cas sp´ ecial orthogonal pair. Mais l’utilisation des familles para boliques ci-dessus
+´ elimine cette diff´ erence. En effet, posons Ij={j}pour toutj= 1,...,t. Si (b1,...,bt) est
+un exposant relatif au parabolique PI1,...,It, alors (b1,...,bt−1,−bt) est un exposant relatif
+` aPI1,...,It−1,ˇIt. Si on impose les conditions (1) pour chacune des familles parabolique s, la
+condition (2) en r´ esulte.
+Dans certains raisonnements, par exemple la preuve du lemme 2.6, il c onvient de
+remplacer les paquets Π par les paquets plus grossiers ¯Π d´ efinis en 2.1. Enfin, la premi` ere
+propri´ et´ e de 2.7 n’est plus vraie. Il peut y avoir deux repr´ esen tationsπ1etπ2deG(F)
+ayant un mod` ele de Whittaker d’un type fix´ e et v´ erifiant Jord(π1) =Jord(π2). Mais
+alorsπ2=πw
+1et cette unicit´ e ` a conjugaison pr` es par le groupe orthogonal t out entier
+suffit pour assurer la validit´ e de nos raisonnements.
+Avec ces adaptations, on v´ erifie que les r´ esultats des paragrap hes pr´ ec´ edents et leurs
+preuves restent valables.
+Remarque. On a pr´ esent´ e en 2.1 et admis la forme fine des conjectures. On pe ut se
+contenter d’un forme un peu plus faible o` u l’on ne param` etre plus qu e les paquets ¯Π, cf.
+[W1] 4.4 pour des ´ enonc´ es pr´ ecis. Nos raisonnements restent v alables. L’int´ erˆ et de cette
+forme plus faible des conjectures est qu’elle est un r´ esultat annon c´ e par Arthur, ainsi
+qu’on l’a dit dans l’introduction.
+3 Preuve du th´ eor` eme
+SoientGetG′comme dans l’introduction. Rappelons le th´ eor` eme principal que no us
+allons maintenant d´ emontrer.
+39Th´ eor` eme .Soientϕ∈Φ(G)etϕ′∈Φ(G′). On suppose ϕetϕ′g´ en´ eriques. Alors :
+(i) toutes les repr´ esentations induites dont les ´ el´ ements de ΠG(ϕ)et deΠG′(ϕ′)sont
+les quotients de Langlands sont irr´ eductibles;
+(ii) siE(ϕ,ϕ′) =−µ(G,G′), on am(σ,σ′) = 0pour tousσ∈ΠG(ϕ),σ′∈ΠG′(ϕ′);
+(iii) siE(ϕ,ϕ′) =µ(G,G′), on a
+m(σ(ϕ,ǫ),σ′(ϕ′,ǫ′)) = 1
+etm(σ,σ′) = 0pour tousσ∈ΠG(ϕ),σ′∈ΠG′(ϕ′)tels que(σ,σ′)/\e}a⊔io\slash= (σ(ϕ,ǫ),σ′(ϕ′,ǫ′)).
+Preuve. Introduisons la forme quasi-d´ eploy´ ee GdeG. Il y a un L´ evi
+L=GL(d1)×...×GL(dm)×G0
+deG, des repr´ esentations irr´ eductibles temp´ er´ ees σjdeGL(dj,F) et des r´ eels bjpour
+j= 1,...,m, et un param` etre de Langlands temp´ er´ e ϕ0∈Φ(G0) de sorte que la suite
+(b1,...,bm) soit strictement positive relativement ` a un sous-groupe parabo lique de com-
+posante de L´ evi Let que ΠG(ϕ) soit l’ensemble des quotients de Langlands des induites
+σ1|.|b1
+F×...×σm|.|bm
+F×π0 (1)
+pourπ0∈ΠG0(ϕ0).
+Remarque. Ici, on peut supposer et on suppose que groupe G0n’est pas un groupe
+SO(2) d´ eploy´ e : on peut remplacer un tel groupe par un facteur GL(1).
+SiLne correspond ` a aucun L´ evi de G(cela se produit quand G0={1}etGn’est
+pas quasi-d´ eploy´ e), le paquet ΠG(ϕ) est vide. Sinon, Lcorrespond ` a un L´ evi
+L=GL(d1)×...×GL(dm)×G0
+deGet ΠG(ϕ) est form´ e des quotients de Langlands des induites (1) pour π0∈ΠG0(ϕ0).
+Par hypoth` ese, il existe π∈ΠG(ϕ) qui admet un mod` ele de Whittaker d’un cer-
+tain type. C’est le quotient de Langlands de l’induite (1) pour une repr ´ esentationπ0∈
+ΠG0(ϕ0) qui admet elle-aussi un mod` ele de Whittaker. D’apr` es [Mu] thorm e 1.1, le quo-
+tient de Langlands de l’induite admet un mod` ele de Whittaker si et seu lement si l’induite
+est irr´ eductible. Donc cette induite est irr´ eductible. En appliquan t le corollaire 2.14 au
+paquet ΠG0(ϕ0) et, quand il existe, au paquet ΠG0(ϕ0), on voit que toutes les induites
+(1) sont irr´ eductibles pour π0∈ΠG0(ϕ0) ouπ0∈ΠG0(ϕ0). Cela d´ emontre l’assertion
+concernant ΠG(ϕ) du (i) du th´ eor` eme.
+La mˆ eme chose vaut du cˆ ot´ e de G′. On introduit les objets similaires, auquels on
+ajoute des′. L’assertion concernant ΠG′(ϕ′) du (i) du th´ eor` eme s’obtient comme ci-
+dessus. Supposons E(ϕ,ϕ′) =−µ(G,G′). Si l’un des paquets ΠG(ϕ) ou ΠG′(ϕ′) est vide,
+l’assertion (ii) l’est aussi. Supposons ces deux paquets non vides, a f ortiori, les L´ evi L
+etL′existent. Soient ǫ∈ EG(ϕ) etǫ′∈ EG′(ϕ′). Alorsσ(ϕ,ǫ) est l’induite (1) pour
+π0=σ(ϕ0,ǫ). On a une assertion analogue pour σ′(ϕ′,ǫ′). D’apr` es la proposition 1.3, on
+a
+m(σ(ϕ,ǫ),σ′(ϕ′,ǫ′)) =m(σ(ϕ0,ǫ),σ′(ϕ′
+0,ǫ′)). (2)
+On aE(ϕ,ϕ′) =E(ϕ0,ϕ′
+0) etµ(G,G′) =µ(G0,G′
+0), doncE(ϕ0,ϕ′
+0) =−µ(G0,G′
+0).
+D’apr` es [W1] th´ eor` eme 4.9, on a m(σ(ϕ0,ǫ),σ′(ϕ′
+0,ǫ′)) = 0, d’o` u l’´ egalit´ e cherch´ ee
+m(σ(ϕ,ǫ),σ′(ϕ′,ǫ′)) = 0.
+40Supposons maintenant E(ϕ,ϕ′) =µ(G,G′). Montrons qu’alors, les L´ evi LetL′
+existent bel et bien. Si ce n’est pas le cas, l’un des groupes G0ouG′
+0est r´ eduit ` a
+{1}. Le param` etre ϕ0ouϕ′
+0correspondant est vide et on a E(ϕ,ϕ′) =E(ϕ0,ϕ′
+0) = 1.
+Alorsµ(G,G′) = 1 et les deux groupes GetG′sont quasi-d´ eploy´ es. Donc les L´ evi LetL′
+existent, contrairement ` a l’hypoth` ese. De plus, l’hypoth` ese E(ϕ,ϕ′) =µ(G,G′) entraˆ ıne
+que le couple ( ǫ,ǫ′) param` etre un ´ el´ ement du produit ΠG(ϕ)×ΠG′(ϕ′). Ces paquets sont
+donc non vides. La preuve du (iii) du th´ eor` eme est alors la mˆ eme qu e celle de (ii) : pour
+ǫ∈ EG(ϕ) etǫ′∈ EG′(ϕ′), on a l’´ egalit´ e (2); en appliquant le th´ eor` eme 4.9 de [W1] au
+membre de droite de cette ´ egalit´ e, on obtient l’assertion cherch´ ee./square
+Remarque. Supposons que Fsoit le compl´ et´ e d’un corps de nombres ken une place
+vde ce corps et que Gsoit la composante en cette place vd’un groupe sp´ ecial orthogonal
+Gd´ efini surk. Notons Al’anneau des ad` eles de k. Consid´ erons une repr´ esentation au-
+tomorphe irr´ eductible πdeG(A) intervenant dans le spectre discret. Conjecturalement,
+il lui correspond une famille ( πi)i=1,...,kde repr´ esentations automorphes irr´ eductibles de
+groupes lin´ eaires intervenant dans le spectre discret. En notant dil’entier tel que πi
+soit une repr´ esentation automorphe de GL(di,A), on aˆdG=/summationtext
+i=1,...,kdi. Ce r´ esultat
+conjectural est annonc´ e par Arthur, sous les mˆ emes r´ eserv es que dans le cas local, cf.
+l’introduction. Supposons que toutes les πisoient cuspidales. Notons π=πvla compo-
+sante de πen la place v. Alorsπappartient ` a un paquet Π auquel on peut appliquer
+nos r´ esultats, c’est-` a-dire que c’est un paquet g´ en´ erique. C ela r´ esulte de [M3] proposition
+5.1.
+Bibliographie
+[AGRS] A. Aizenbud, D. Gourevitch, S. Rallis, G. Schiffmann : Multiplicity one theo-
+rems, pr´ epublication 2007
+[A1] J. Arthur : An introduction to the trace formula , Clay Math. Proc. 4 (2005),
+p.1-253
+[A2] ———– : A local trace formula , Publ. Sc. IHES 73 (1991), p.5-96
+[BZ] I. Bernstein, A. Zelevinsky : Induced representations of reductive p-adic groups
+I, Ann. Sc. ENS 10 (1977), p.441-472
+[GGP] W. Gan, B. Gross, D. Prasad : Symplectic local root numbers, central cri-
+ticalL-values and restriction problems in the representation theory of c lassical groups,
+pr´ epublication 2009
+[K] T. Konno : Twisted endoscopy implies the generic packet conjecture , Isra¨ el J. of
+Math. 129 (2002), p.253-289
+[M1] C. Moeglin : Repr´ esentations quadratiques unipotentes des groupes cl assiques
+p-adiques, Duke Math. J. 84 (1996), p.267-332
+[M2] ————– : Sur la classification des s´ eries discr` etes des groupes cla ssiquesp-
+adiques : param` etres de Langlands et exhaustivit´ e , J. Eur. Math. Soc. 4 (2002), p.143-200
+[M3]————–: Image des op´ erateurs d’entrelacements normalis´ eset pˆ o les des s´ eries
+d’Eisenstein , pr´ epublication 2009
+[MW] C.Moeglin, J.-L.Waldspurger : Sur le transfert des traces d’un groupe classique
+p-adique ` a un groupe lin´ eaire tordu , Selecta math. 12 (2006), p.433-515
+[Mu] G. Muic : A proof of Casselman-Shahidi’s conjecture for quasi-split classical
+groups, Can. Math. Bull. 43 (2000), p.90-99
+[W1]J.-L.Waldspurger: La conjecture localedeGross-Prasadpour les repr´ esentat ions
+temp´ er´ ees des groupes sp´ eciaux orthogonaux , pr´ epublication 2009
+41[W2] ———————–: Calcul d’une valeur d’un facteur ǫpar une formule int´ egrale ,
+pr´ epublication 2009
+[W3] ———————– : Une formule int´ egrale reli´ ee ` a la conjecture locale de Gr oss-
+Prasad,2` emepartie : extension aux repr´ esentations temp´ er´ ees , pr´ epublication 2009
+[Z] A. Zelevinsky : Induced representations of reductive p-adic groups II. On irredu-
+cible representations of GL(n), Ann. Sc. ENS 13 (1980), p.165-210
+CNRS Institut de math´ ematiques de Jussieu
+175 rue du Chevaleret
+75013 Paris
+moeglin@math.jussieu.fr, waldspur@math.jussieu.fr
+42
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+arXiv:1001.0827v1  [cs.IR]  6 Jan 2010Document Clustering with K-tree
+Christopher M. De Vries and Shlomo Geva
+Faculty of Science and Technology,
+Queensland University of Technology, Brisbane, Australia
+chris@de-vries.id.au s.geva@qut.edu.au
+Abstract. This paper describes the approach taken to the XML Min-
+ing track at INEX 2008 by a group at the Queensland University of
+Technology. We introduce the K-tree clustering algorithm i n an Infor-
+mation Retrieval context by adapting it for document cluste ring. Many
+large scale problems exist in document clustering. K-tree s cales well with
+large inputs due to its low complexity. It offers promising re sults both in
+terms of efficiency and quality. Document classification was c ompleted
+using Support Vector Machines.
+Key words: INEX,XMLMining,Clustering, K-tree,Tree,VectorQuan-
+tization, Text Classification, Support Vector Machine
+1 Introduction
+The XML Mining track consists of two tasks, classification and cluste ring. Clas-
+sification labels documents in known categories. Clustering groups s imilar doc-
+uments without any knowledge of categories. The corpus consiste d of 114,366
+documents and 636,187 document-to-document links. It is a subse t of the XML
+Wikipedia corpus [1]. Submissions were made for both tasks using seve ral tech-
+niques.
+We introduce K-tree in the Information Retrieval context. K-tre e is a tree
+structured clustering algorithm introduced by Geva [2] in the conte xt of signal
+processing. It is particularly suitable for large collections due to its lo w com-
+plexity. Non-negative Matrix Factorization (NMF) was also used to s olve the
+clustering task. Applying NMF to document clustering was first desc ribed by
+Xu et. al. at SIGIR 2003 [3]. Negentropy has been used to measure c lustering
+performance using the labels provided for documents. Entropy ha s been used by
+many researchers [4–6] to measure clustering results. Negentro py differs slightly
+but is fundamentally measuring the same system property.
+The classification task was solved using a multi-class Support Vector Ma-
+chine (SVM). Similar approaches have been taken by Joachims [7] and Tong and
+Koller[8]. We introduce a representationfor links named Link Frequen cy Inverse
+Document Frequency (LF-IDF) and make several extensions to it .
+Sections 2, 3, 4, 5, 6 and 7 discuss document representation, clas sification,
+cluster quality, K-tree, NMF and clustering respectively. The pape r ends with a
+discussion of future research and a conclusion in Sects. 8 and 9.2 Document Representation
+Document content was represented with TF-IDF [9] and BM25 [10]. S top words
+wereremovedand the remainingterms werestemmed usingthe Port eralgorithm
+[11]. TF-IDF is determined by term distributions within each document and the
+entire collection. Term frequencies in TF-IDF were normalized for do cument
+length. BM25 works with the same concepts as TF-IDF except that is has two
+tuning parameters. The BM25 tuning parameters were set to the s ame values
+as used for TREC [10], K1 = 2 and b= 0.75.K1 influences the effect of term
+frequency and binfluences document length.
+Links were represented as a vector of LF-IDF weighted link freque ncies. This
+resulted in a document-to-document link matrix. The row indicates t he origin
+and the column indicates the destination of a link. Eachrow vector of the matrix
+represents a document as a vector of link frequencies to other do cuments. The
+motivation behind this representation is that documents with similar c ontent
+will link to similar documents. For example, in the current Wikipedia both car
+manufacturers BMW and Jaguar link to the Automotive Industry do cument.
+Termfrequencies weresimply replacedwith link frequenciesresulting in LF-IDF.
+Link frequencies were normalized by the total number of links in a doc ument.
+All representations were culled to reduce the dimensionality of the d ata. This
+is necessary to fit the representations in memory when using a dens e representa-
+tion. K-tree will be extended to work with sparse representations in the future.
+A feature’s rank is calculated by summation of its associated column v ector.
+This is the sum of all weights for each feature in all documents. Only t he top n
+features are kept in the matrix and the rest are discarded. TF-ID F was culled
+to the top 2000 and 8000 features. The selection of 2000 and 8000 features is
+arbitrary. BM25 and LF-IDF were only culled to the top 8000 featur es.
+3 Classification Task
+The classification task was completed using an SVM and content and lin k infor-
+mation. This approach allowed evaluation of the different document r epresenta-
+tions. It allowed the most effective representation to be chosen fo r the clustering
+task.
+SVMmulticlass[12] was trained with TF-IDF, BM25 and LF-IDF represen-
+tations of the corpus. BM25 and LF-IDF feature vectors were co ncatenated to
+train on both content and link information simultaneously. Submission s were
+made only using BM25, LF-IDF or both because BM25 out performed TF-IDF.
+3.1 Classification Results
+Table 1 lists the results for the classification task. They are sorted in order of de-
+creasing recall. Recall is simply the accuracy of predicting labels for d ocuments
+not in the training set. Concatenating the link and content represe ntations did
+not drastically improve performance. Further work has been subs equently per-
+formed to improve classification accuracy.Name Recall Name Recall
+Expe 5 tf idf T5 10000 0.7876 Expe 4 tf idf T5 100 0.7231
+Expe 3 tf idf T4 10000 0.7874 Kaptein 2008NBscoresv02 0.6981
+Expe 1 tf idf TA 0.7874 Kaptein 2008run 0.6979
+Vries text and links 0.7849 Romero nave bayes 0.6767
+Vries text only 0.7798 Expe 2.tf idf T4 100 0.6771
+Boris inex tfidf1 sim 0.38.3 0.7347 Romero nave bayes links 0. 6814
+Boris inex tfidf sim 037 it3 0.7340 Vries links only 0.6233
+Boris inex tfidf sim 034 it2 0.7310
+Table 1. Classification Results
+3.2 Improving Representations
+0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 105,00010,00015,00020,00025,00030,00035,00040,00045,00050,000
+cosine similarity between linked documentslink count636,187 INEX XML Mining Links
+Fig.1.Text Similarity of LinksSeveral approaches have been carried out
+to improve classification performance.
+They were completed after the end of of-
+ficial submissions for INEX. The same
+train and test splits were used. All fea-
+tures were used for text and links, where
+earlier representations were culled to the
+top 8000 features. Links were classi-
+fied without LF-IDF weighting. This was
+to confirm LF-IDF was improving the
+results. Document length normalization
+wasremovedfromLF-IDF.Itwasnoticed
+that many vectors in the link representa-
+tion contained no features. Therefore, inbound links were added t o the repre-
+sentation. For i, the source document and j, the destination docu ment, a weight
+of one is added to the i, j position in the document-to-document link m atrix.
+This represents an outbound link. To represent an inbound link, i is t he destina-
+tion document and j is the source document. Thus, if a pair of docum ents both
+link to each other they receive a weight of two in the corresponding c olumns
+in their feature vectors. Links from the entire Wikipedia were insert ed into this
+matrix. This allows similarity to be associated on inbound and outbound links
+outside the XML Mining subset. This extends the 114,366 ×114,366 document-
+to-document link matrix to a 114,366 ×486,886 matrix. Classifying links in this
+waycorrespondstotheideaofhubsandauthoritiesinHITS[13].Ove rlaponout-
+bound links indicates the document is a hub. Overlap on inbound links ind icates
+the document is an authority. The text forms a 114,366 ×206,868document term
+matrix when all terms are used. The link and text representation we re combined
+using twomethods. In the first approachtext and links wereclassifi edseparately.
+The ranking output of the SVM was used to choose the most approp riate la-
+bel. We call this SVM by committee. Secondly, both text and link featu res were
+converted to unit vectors and concatenated forming a 114,366 ×693,754 matrix.
+Table 2 highlights the performance of these improvements.Dimensions Type Representation Recall
+114,366 Links (XML Mining subset) unweighted 0.6874
+114,366 Links (XML Mining subset) LF-IDF 0.6906
+114,366 Links (XML Mining subset) LF-IDF no normalization 0 .7095
+486,886 Links (Whole Wikipedia) unweighted 0.7480
+486,886 Links (Whole Wikipedia) LF-IDF 0.7527
+486,886 Links (Whole Wikipedia) LF-IDF no normalization 0. 7920
+206,868 Text BM25 0.7917
+693,754 Text, Links (Whole Wikipedia) BM25 + LF-IDF committ ee 0.8287
+693,754 Text, Links (Whole Wikipedia) BM25 + LF-IDF concate nation 0.8372
+Table 2. Classification Improvements
+The new representation for links has drastically improved performa nce from
+a recall of 0.62 to 0.79. It is now performing as well as text based clas sification.
+However, the BM25 parameters have not been optimized. This could further
+increase performance of text classification. Interestingly, 97 pe rcent of the cor-
+rectly labeled documents for text and link classification agree.To fu rther explain
+this phenomenon, a histogram of cosine similarity of text between link ed docu-
+ments was created. Figure 1 shows this distribution for the links in XM L Mining
+subset. Most linked documents have a high degree of similarity based on their
+text content. Therefore, it is valid to assume that linked document s are highly
+semantically related. By combining text and link representations we c an disam-
+biguate many more cases. This leads to an increase in performance f rom 0.7920
+to 0.8372 recall. The best results for text, links and both combined, performed
+the same under 10 fold cross validation using a randomized 10% train a nd 90%
+test split.
+LF-IDF link weighting is motivated by similar heuristics to TF-IDF term
+weighting. In LF-IDF the link inverse document frequency reduces the weight
+of common links that associate documents poorly and increases the weight of
+links that associate documents well. This leads to the concept of sto p-links that
+are not useful in classification. Stop-links bare little semantic inform ation and
+occur in many unrelated documents. Considerfor instance a docum ent collection
+of the periodic table of the elements, where each document corres ponds to an
+element. In such a collection a link to the “Periodic Table” master docu ment
+would provide no information on how to group the elements. Noble gas es, alkali
+metals and every other category of elements would all link to the “Pe riodic
+Table” document. However, links that exist exclusively in noble gases or alkali
+metals would be excellent indicators of category. Year links in the Wikip edia
+are a good example of a stop-link as they occur with relatively high fre quency
+and convey no information about the semantic content of pages in w hich they
+appear.4 Document Cluster Quality
+The purity measure for the track is calculated by taking the most fr equently oc-
+curring label in each cluster. Micro purity is the mean purity weighted by cluster
+size and macro is the unweighted arithmetic mean. Taking the most fr equently
+occurring label in a cluster discards the rest of the information rep resented by
+the other labels. Due to this fact negentropy was defined. It is the opposite of
+information entropy [14]. If entropy is a measure of uncertainty as sociated with
+a random variable then negentropy is a measure of certainty. Thus , it is better
+when more labels of the same class occur together. When all labels ar e evenly
+distributed across all clusters the lowest possible negentropy is ac hieved.
+Negentropy is defined in Equations (1), (2) and (3). Dis the set of all doc-
+uments in a cluster. Xis the set of all possible labels. l(d) is the function that
+maps a document dto its label x.p(x) is the probability for label x.H(D) is
+the negentropy for document cluster D. The negentropy for a cluster falls in
+the range 0 ≤H(D)≤1 for any number of labels in X. Figure 2 shows the
+difference between entropy and negentropy. While they are exact opposites for
+a two class problem, this property does not hold for more than two c lasses. Ne-
+gentropy always falls between zero and one because it is normalized. Entropy is
+bounded by the number of classes. The difference between the max imum value
+for negentropy and entropy increase when the number of classes increase.
+l(d) ={(d1,x1),(d2,x2),...,(d|D|,x|D|)} (1)
+p(x) =|{d∈D:x=l(d)}|
+|D|(2)
+H(D) = 1+1
+log2|X|/summationdisplay
+x∈X
+p(x)/negationslash=0p(x)log2p(x) (3)
+0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.912 Class Problem
+p(class 1)f(x, 1−x)
+  
+negentropy
+entropy
+Fig.2.Entropy Versus NegentropyThe difference between purity andne-
+gentropycaneasilybedemonstratedwith
+an artificial four class problem. There are
+six of each of the labels A, B, C and D.
+For each cluster in Solution 1 purity and
+negentropy is 0.5. For each cluster in So-
+lution 2 the purity is 0.5 and the negen-
+tropy is 0.1038.Purity makes no differen-
+tiation between the two solutions. If the
+goal of document clustering is to group
+similardocumentstogetherthen Solution
+1 is clearly better because each label oc-
+curs in two clusters instead of four. The
+grouping of labels is better defined because they are less spread. F igures 3 and
+4 show Solutions 1 and 2.Cluster Label Counts
+1 A=3, B=3
+2 A=3, C=3
+3 B=3, D=3
+4 C=3, D=3
+A B A C B D C D
+Fig.3.Solution 1Cluster Label Counts
+1 A=3, B=1, C=1, D=1
+2 B=3, C=1, D=1, A=1
+3 C=3, D=1, A=1, B=1
+4 D=3, A=1, B=1, C=1
+ABCD BCDA CDAB DABC
+Fig.4.Solution 2
+5 K-tree
+The K-tree algorithm is a height balanced cluster tree. It can be dow nloaded
+from http://ktree.sf.net. It is inspired by the B+-tree where all data records are
+stored in the leaves at the lowest level in the tree and the internal n odes form a
+nearest neighbour search tree. The k-means algorithm is used to p erform splits
+when nodes become full. The constraints placed on the tree are rela xed in com-
+parison to a B+-tree. This is due to the fact that vectors do not have a total
+order like real numbers.
+B+-tree of order m
+1. All leaves are on the same level.
+2. Internal nodes, except the root, contain between ⌈m
+2⌉andmchildren.
+3. Internal nodes with nchildren contain n−1 keys, partitioning the children
+into a search tree.
+4. The root node contains between 2 and mchildren. If the root is also a leaf
+then it can contain a minimum of 0.
+K-tree of order m
+1. All leaves are on the same level.
+2. Internal nodes contain between one and mchildren. The root can be empty
+when the tree contains no vectors.
+3. Codebook vectors (cluster representatives) act as search k eys.
+4. Internal nodes with nchildren contain nkeys, partitioning the children into
+a nearest neighbour search tree.
+5. The level immediately above the leaves form the codebook level co ntaining
+the codebook vectors.
+6. Leaf nodes contain data vectors.
+The leaf nodes of a K-tree contain real valued vectors. The searc h path in
+the tree is determined by a nearest neighbour search. It follows th e child node
+associatedwithnearestvector.Thisfollowsthesamerecursivede finitionofaB+-
+treewhereeachtreeismadeupofasmallersubtree.Thecurrentim plementation
+of K-tree uses Euclidean distance for all measures of similarity. Fut ure versions
+will have the ability to specify any distance measure.5.1 Building a K-tree
+The K-tree is constructed dynamically as data vectors arrive. Init ially the tree
+contains a single empty root node at the leaf level. Vectors are inser ted via a
+nearest neighbour search, terminating at the leaf level. The root o f an empty
+tree is a leaf, so the nearest neighbour search terminates immediat ely, placing
+the vector in the root. When m+1 vectors arrive the root node can not contain
+any more keys. It is split using k-means where k= 2 using all m+ 1 vectors.
+The two centroids that result from k-means become the keys in a ne w parent.
+New root and child nodes are constructed and each centroid is asso ciated with a
+child.The vectorsassociatedwith eachcentroidfromk-meansare placedintothe
+associated child. This process has created a new root for the tree . It is now two
+levelsdeep.Theroothastwokeysandtwochildren,makingatotalo fthreenodes
+in the tree. Now that the tree is two levels deep, the nearest neighb our search
+finds the closest centroid in the root and inserts it in the associated child. When
+a new vector is inserted the centroids are updated along the neare st neighbour
+searchpath. They areweighted by the number ofdatavectorsco ntained beneath
+them. This process continues splitting leaves until the root node be comes full.
+K-means is run on the root node containing centroids. The keys in th e new root
+node become centroids of centroids. As the tree grows internal a nd leaf nodes
+are split in the same manner. The process can potentially propagate to a full
+root node and cause construction of a new root. Figure 6 shows th is construction
+process for a K-tree of order three ( m= 3).
+0 1 2 3 4 5 6 7 8 910 11 12
+x 104050100150200250300350400
+number of documentstime (seconds)K−tree and k−means Performance Comparison
+INEX XML Mining Collection
+  
+k−means
+K−tree
+Fig.5.K-tree PerformanceThe time complexity of building a K-
+tree fornvectors is O( nlogn). An in-
+sertion of a single vector has the time
+complexity of O(log n). These proper-
+ties are inherent to the tree based algo-
+rithm. This allows the K-tree to scale ef-
+ficiently with the number of input vec-
+tors. When a node is split, k-means is
+always restricted to m+ 1 vectors and
+two centroids ( k= 2). Figure 5 compares
+k-means performance with K-tree where
+kfor k-means is determined by the num-
+ber of codebook vectors. This means that
+both algorithms produce the same number of document clusters an d this is nec-
+essary for a meaningful comparison. The order, m, for K-tree was 50. Each
+algorithm was run on the 8000 dimension BM25 vectors from the XML m ining
+track.
+5.2 K-tree Submissions
+K-tree was used to create clusters using the Wikipedia corpus. Doc uments were
+representedas8000dimensionBM25weightedvectors.Thus,clus terswereformed
+using text only. This representation was used because it was most e ffective textnode vector child link k-means performed on enclosed vectors
+the dashed parts represent the nearest neighbour searchlevel 1
+level 2
+level 3root node
+nodes above the leaves contain codebook vectors
+leaf nodes contain the data vectorsnodes above the codebook level are clusters of clusters
+Fig.6.K-tree Construction
+representation in the classification task. The K-tree was constru cted using the
+entire collection. Cluster membership was determined by comparing e ach doc-
+ument to all centroids using cosine similarity. The track required a su bmission
+with 15 clusters but K-tree does not produce a fixed number of clus ters. There-
+fore, the codebook vectors were clustered using k-means++ whe rek= 15. The
+codebook vectors are the cluster centroids that exist above the leaf level. This
+reduces the number of vectors used for k-means++, making it quic k and inex-
+pensive. As k-means++ uses a randomised seeding process, it was r un 20 times
+to find the solution with the lowest distortion. The k-means++ algorit hm [15]
+improves k-means by using the D2weighting for seeding. Two other submission
+were made representing different levels of a K-tree. A tree of orde r 100 had 42
+clusters in the first level and a tree of order 20 had 147 clusters in t he second
+level. This made for a total of three submissions for K-tree.
+Negentropy was used to determine the optimal tree order. K-tre e was built
+using the documents in the 10% training set from the classification ta sk. A tree
+was constructed with an order of 800 and it was halved each time. Ne gentropywasmeasuredintheclustersrepresentedbytheleafnodes.Asth eorderdecreases
+the size of the nodes shrinks and the purity increases. If all cluste rs became
+pure at a certain size then decreasing the tree order further wou ld not improve
+negentropy.However, this was not the case and negentropy con tinued to increase
+as the tree order decreased. This is expected because there will u sually be some
+imperfection in the clustering with respect to the labels. Therefore , the sharp
+increaseinnegentropyin aK-treebelowanorderof100suggestst hatthe natural
+cluster size has been observed. This can be seen in Fig. 7. The “left a s is” line
+represents the K-tree as it is built initially. The “rearranged” line rep resents the
+K-tree when all the leaf nodes have been reinserted to their neare st neighbours
+without modifying the internal nodes.
+0 100 200 300 400 500 600 700 8000.40.50.60.70.80.91
+K−tree ordernegentropyINEX XML Mining Collection
+Labeled documents only
+  
+left as is
+rearranged
+Fig.7.K-tree NegentropyNegentropy was calculated using the
+10% training set labels provided on clus-
+ters for the whole collection. This was
+used to determine which order of 10, 20
+or 35 fed into k-means++ with k= 15
+was best. A tree of order 20 provided the
+best negentropy.
+6 Non-negative Matrix
+Factorization
+NMF factorizes a matrix into two matri-
+ces where all the elements are ≥0. IfV
+is an×mmatrix and ris a positive integer where r < min(n,m), NMF finds
+two non-negativematrices Wn×randHr×msuch that V≈WH. When applying
+this process to document clustering Vis a term document matrix. Each column
+inHrepresents a document and the largest value represents its clust er. Each
+row inHis a cluster and each column is a document.
+The projected gradient method was used to solve the NMF problem [1 6].V
+was a 8000 ×114366 term document matrix of BM25 weighted terms. The algo-
+rithm ran for a maximum of 70 iterations. It produced the WandHmatrices.
+Clusters membership was determined by the maximum value in the colum ns of
+H. NMF was run with rat 15, 42 and 147 to match the submissions made with
+K-tree.
+7 Clustering Task
+Every team submitted at least one solution with 15 clusters. This allow s for a
+direct comparison between different approaches. It only makes se nse to compare
+results where the number of clusters are the same. The K-tree pe rformed well
+according to the macro and micro purity measures in comparison to t he rest of
+the field. The difference in macro and micro purity for the K-tree sub missions
+can be explained by the uneven distribution of cluster sizes. Figure 9 shows thatName Size Micro Macro Name Size Micro Macro
+K-tree 15 0.4949 0.5890 QUT LSK 1 15 0.4518 0.5594
+QUT LSK 3 15 0.4928 0.5307 QUT LSK 4 15 0.4476 0.4948
+QUT Entire collection 15 15 0.4880 0.5051 QUT LSK 2 15 0.4442 0 .5201
+NMF 15 0.4732 0.5371 Hagenbuchner 15 0.3774 0.2586
+Table 3. Clustering Results Sorted by Micro Purity
+Method Clusters Micro Macro Clusters Micro Macro Clusters M icro Macro
+left as is 17 0.4018 0.5945 397 0.5683 0.6996 7384 0.6913 0.72 02
+rearranged 17 0.4306 0.6216 371 0.6056 0.7281 5917 0.7174 0. 7792
+cosine 17 0.4626 0.6059 397 0.6584 0.7240 7384 0.7437 0.7286
+Table 4. Comparison of Different K-tree Methods
+0 5 10 150.10.20.30.40.50.60.70.80.91
+clusters sorted by puritycluster purityINEX 2008 XML Mining Submissions (15 clusters)
+  QUT Entire 15
+QUT LSK 1
+QUT LSK 2
+QUT LSK 3
+QUT LSK 4
+K−tree
+NMF
+Hagenbuchner
+0 5 10 1500.050.10.150.20.25
+clusters sorted by sizecluster sizeINEX 2008 XML Mining Submissions (15 clusters)
+  
+QUT Entire 15
+QUT LSK 1
+QUT LSK 2
+QUT LSK 3
+QUT LSK 4
+K−tree
+NMF
+Hagenbuchner
+Fig.8.All Submissions with 15 Clusters
+0 2 4 6 8 10 12 14 1600.10.20.30.40.50.60.70.80.91K−tree (15 clusters)
+clusters sorted by puritycluster property
+  
+purity
+size
+Fig.9.K-tree Breakdownmany of the higher purity clusters are small. Macro purity is simply the average
+purity for all clusters. It does not take cluster size into account. Micro purity
+does take size into account by weighting purity in the average by the cluster
+size. Three types of clusters appear when splitting the x-axis in Fig. 9 in thirds.
+There are very high purity clusters that are easy to find. In the mid dle there are
+some smaller clusters that have varying purity. The larger, lower pu rity clusters
+in the last third are hard to distinguish. Figure 8 shows clusters sort ed by purity
+and size.K-treeconsistentlyfound higherpurity clustersthan ot hersubmissions.
+Even with many small high purity clusters, K-tree achieved a high micr o purity
+score. The distribution of cluster size in K-tree was less uniform tha n other
+submissions. This can be seen in Figure 8. It found many large cluster s and
+many small clusters, with very few in between.
+The K-tree submissions were determined by cosine similarity with the c en-
+troids produced by K-tree. The tree has an internal ordering of c lusters as well.
+A comparison between the internal arrangement and cosine similarit y is listed in
+Table 4. This data is based on a K-tree of order 40. Levels 1, 2 and 3 p roduced
+17, 397 and 7384 clusters respectively. Level 3 is the above leaf or codebook
+vector level. The “left as is” method uses the K-tree as it is initially built . The
+rearranged method uses the K-tree when all vectors are reinser ted into the tree
+to their nearest neighbour. The cosine method determines cluster membership
+by cosine similarity with the centroids produced. Nodes can become e mpty when
+reinserting vectors. This explains why levels 2 and 3 in the rearrange d K-tree
+contain less clusters. Using cosine similarity with the centroids improv ed purity
+in almost all cases.
+8 Future Work
+The work in this area falls into two categories, XML mining and K-tree. Further
+work in the XML mining area involves better representation of struc ture. For
+example, link information can be included into clustering via a modified sim ilar-
+ity measure for documents. Future work with the K-tree algorithm will involve
+different strategies to improve quality of clustering results. This will require ex-
+tra maintenance of the K-tree as it is constructed. For example, r einsertion of
+all data records can happen every time a new root is constructed.
+9 Conclusion
+In this paperanapproachtothe XML mining trackwaspresented,d iscussedand
+analyzed.A new representationfor links wasintroduced, extende d and analyzed.
+It was combined with text to further improve classification perform ance. The K-
+tree algorithm was applied to document clustering for the first time. The results
+show that it is well suited for the task. It produces good quality clus tering
+solutions and provides excellent performance.References
+1. Denoyer, L., Gallinari, P.: The Wikipedia XML Corpus. SIG IR Forum (2006)
+2. Geva, S.: K-tree: a height balanced tree structured vecto r quantizer. Proceedings
+of the 2000 IEEE Signal Processing Society Workshop Neural N etworks for Signal
+Processing X, 2000. 1(2000) 271–280 vol.1
+3. Xu, W., Liu, X., Gong, Y.: Document clustering based on non -negative matrix
+factorization. In: SIGIR ’03: Proceedings of the 26th annua l international ACM
+SIGIR conference on Research and development in informaion retrieval, New York,
+NY, USA, ACM (2003) 267–273
+4. Surdeanu,M., Turmo,J., Ageno,A.: Ahybridunsupervised approachfordocument
+clustering. In: KDD ’05: Proceedings of the eleventh ACM SIG KDD international
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+(2005) 685–690
+5. Hotho, A., Staab, S., Stumme, G.: Ontologies improve text document clustering.
+Third IEEE International Conference on Data Mining, 2003. I CDM 2003. (Nov.
+2003) 541–544
+6. Steinbach, M., Karypis, G., Kumar, V.: A comparison of doc ument clustering
+techniques. 34(2000) 35
+7. Joachims, T.: Text categorization with Support Vector Ma chines: Learning with
+many relevant features. In: Text categorization with Suppo rt Vector Machines:
+Learning with many relevant features. (1998) 137–142
+8. Tong, S., Koller, D.: Support vector machine active learn ing with applications to
+text classification. Journal of Machine Learning Research 2(2002) 45–66
+9. Salton, G., Fox, E.A., Wu, H.: Extended boolean informati on retrieval. Commu-
+nications of the ACM 26(11) (1983) 1022–1036
+10. Robertson, S., Jones, K.: Simple, proven approaches to t ext retrieval. Update
+(1997)
+11. Porter, M.: An algorithm for suffix stripping. Program: El ectronic Library and
+Information Systems 40(3) (2006) 211–218
+12. Tsochantaridis, I., Joachims, T., Hofmann, T., Altun, Y .: Large Margin Methods
+for Structured and Interdependent Output Variables. Journ al of Machine Learning
+Research 6(2005) 1453–1484
+13. Kleinberg, J.M.: Authoritative sources in a hyperlinke d environment. J. ACM
+46(5) (1999) 604–632
+14. Shannon, C., Weaver, W.: The mathematical theory of comm unication. University
+of Illinois Press (1949)
+15. Arthur, D., Vassilvitskii, S.: k-means++: the advantag es of careful seeding. In:
+SODA ’07: Proceedings of the eighteenth annual ACM-SIAM sym posium on Dis-
+crete algorithms, Philadelphia, PA, USA, Society for Indus trial and Applied Math-
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+Computation 19(10) (2007) 2756–2779
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+arXiv:1001.0828v1  [physics.soc-ph]  6 Jan 2010pacs: 89.75.Da, 89.65.-s, 89.65.Gh
+City Size Distributions For India and China
+Kausik Gangopadhyay∗
+Indian Institute of Management Kozhikode,
+IIMK Campus P.O., Kozhikode 673570, India
+B. Basu†
+Physics and Applied Mathematics Unit
+Indian Statistical Institute
+Kolkata-700108, India
+Abstract
+This paper studies the size distributions of urban agglomer ations for India and China. We have
+estimated the scaling exponent for the Zipf’s law with the In dian census data for the years of 1981-
+2001 and the Chinese census data for 1990 and 2000. Along with the biased linear fit estimate,
+the maximum likelihood estimate for the Pareto and Tsallis q-exponential distribution has been
+computed. For India, the scaling exponent is in the range of [ 1.88, 2.06] and for China, it is in the
+interval [1.82, 2.29]. The goodness-of-fit tests of the esti mated distributions are performed using the
+Kolmogorov-Smirnov statistic.
+I. INTRODUCTION
+Nature, in spite of its complex character, often displays macrosco picregularity, which can be described
+broadly by simple laws at different scales. Examples include the size dist ribution of islands [1] and lunar
+craters[2], occurrence offorest fires [3] and solarflares [4], web surfings[5], wealth and income distribution
+in societies [6], and also football goal distribution [7]. However, the s ize-distributions of populations in
+cities aces them all in arresting our longest attention [8].
+It isconjectured that the citysizes obeyasurprisinglysimple law, kn ownasZipf’s law[9] (alternatively
+known as Pareto distribution or simply power law), which states that the population-wise rank, Rx, of
+a city with xnumber of inhabitants is proportional to x−α, withαbeing close to one. This law has
+received empirical endorsement from different studies using data f rom USA, Switzerland [10], Brazil [11]
+etc. However, fewer analysis have been conducted for urban agg lomerations with comparatively lower
+populations. For example, a recent work [12] shows that the size dis tribution of towns and villages are
+completely different from that ofcities. Another interesting study [13, 14] looks at the spatial distribution
+of city population and observes deviation from the usual power law. Some other studies reveal that a
+deviation of the power law may be observed if all the urban agglomera tions of an urbanized nation is
+considered[15,16]. Somealternativestatisticaldistributions[16–1 8]aresuggestedtoincludethedeviation
+from the power law. In general, the distribution of urban agglomera tions in USA can be well-described
+by using a Tsallis q-exponential distribution [16], which is an extension of the standar d Zipf-Mandelbrot
+law [17] proposed in the context of generalized statistical mechanic s [19].
+The empirical phenomenon of the power law receives the theoretica l support from some mathematical
+models [20–22]. These theories model the evolution of the distributio n of the city-size either as a time
+dependent process, or as a result of interactions among individuals . In both cases, reality is an approxi-
+mation of the theoretical prediction of the limiting distribution and th e empirically observed distribution
+converges to its limiting value depending on either the time-length of t he process, or the total popula-
+tion of the considered society. This motivates us to analyze empirica lly the population distribution of
+cities for the two most populous countries of the world such as India and China, which are arguably the
+foremost ancient civilizations as well. These two countries are compa ratively less urbanized and possess
+∗Electronic address: kausik@iimk.ac.in
+†Electronic address: banasri@isical.ac.in, Fax:91+(033) 2577-30262
+remarkably heterogeneous socio-economic structures. In any c ase, the data from India and China should
+be ideal to test the theoretical limiting predictions.
+In this work, we carry out an empirical investigation with census dat a of India for the time period of
+1981-2001as well as with the Chinese census data for the years of 1990and 2000. We estimate the scaling
+exponents for the Pareto distribution and also for a more generaliz ed Tsallis q-exponential distribution.
+Also, goodness-of-fit tests for these hypothesized distribution have been performed. The methodology is
+described in Section II; whereas the data and the results are discu ssed in Section III. We conclude our
+study in the final section.
+II. METHODOLOGY
+Letp(·) be a probability density function of the city-size distribution. The c orresponding cumulative
+distribution function (CDF) and the complementary cumulative distr ibution function (CCDF) are given
+byP(·) andPC(·), respectively. The CDF, P(x) is the probability that a city has a population less than
+or equal to x and the CCDF, PC(x), is the probability that a city has a population greater than x. By
+definition,
+P(x) =/integraldisplayx
+0p(x′)dx′;PC(x) = 1−P(x) (1)
+In case of city-size distribution following the Zipf’s law,
+pα(x) =Cx−αandPC
+α(x) =C
+α−1x−(α−1)(2)
+whereαandCare constants. αis called the exponent of the power law. This family of power law
+distributions for α >1 are known as the Pareto distribution. From equation (2), it is obvio us thatpα(x)
+diverges to infinity for any value of α >1 asx→0. Therefore, some minimum value, xmin, is usually
+considered for the support of the Pareto distribution. The corre sponding probability density function,
+the CDF and the CCDF are given by:
+pα(x) =(α−1)·xα−1
+min
+xα;Pα(x) = 1−/parenleftBigxmin
+x/parenrightBigα−1
+;PC
+α(x) =/parenleftBigxmin
+x/parenrightBigα−1
+.(3)
+A more general distribution, namely the Tsallis q-exponential distribution, has been proposed in [16].
+The probability density function, the CDF and the CCDF of this distrib ution, as given in [23], are noted
+below:
+pθ,σ(x) =θ
+σ/parenleftBig
+1+x
+σ/parenrightBig−θ−1
+;Pθ,σ(x) = 1−/parenleftBig
+1+x
+σ/parenrightBig−θ
+;PC
+θ,σ(x) =/parenleftBig
+1+x
+σ/parenrightBig−θ
+(4)
+From the equations (3) and (4), it is evident that the two distributio ns of Pareto and Tsallis q-
+Exponential are approximately identical for large values of x, when we set θto (α−1) andσtoxmin.
+The slope of the plot, in which log of the rank of a city, log( Rx), is plotted against the log of its
+population, log( x), has been used to estimate the exponent of the power law in almost all the previous
+studies. It has been shown [24] that this produces a biased estimat e of the power law exponent. Alter-
+natively the Maximum Likelihood Estimator (MLE) produces the most e fficient estimate. For a sample
+consisting cities with populations x1,x2, ...,xn, the log-likelihood of the sample is described by the
+following expression:
+l(x;ν) =n/summationdisplay
+i=1log(p(xi;ν)) (5)
+wherep(xi;ν)istheprobabilitydensityfunctionofthe ithobservation xidrawnfromacertaindistribution
+with parameter ν. The function l(x;ν) is maximized with respect to the parameter νto derive the
+maximum likelihood estimate /hatwideνof the parameter ν. Mathematically,
+/hatwideν= argmax
+νl(x;ν). (6)3
+In particular, the MLE for the Pareto distribution with xminas the minimum value is given by:
+/hatwideαMLE= 1+n/bracketleftBiggn/summationdisplay
+i=1log/parenleftbiggxi
+xmin/parenrightbigg/bracketrightBigg−1
+(7)
+The solution of the following system of simultaneous equations [23] r epresents the MLE for the Tsallis
+q-exponential distribution with parameters ( θ,σ) andxminas the minimum value:
+/hatwideθMLE=n/bracketleftBig/summationtextn
+i=1log/parenleftBig
+1+xi//hatwideσMLE
+1+xmin//hatwideσMLE/parenrightBig/bracketrightBig−1
+/hatwideσMLE=−/hatwideθMLExmin
+1+xmin//hatwideσMLE+/hatwideθMLE+1
+n/summationtextn
+i=1xi
+1+xi//hatwideσMLE(8)
+The Fisher’s Information matrix [25] gives the asymptotic variance o f the MLE. We can also compute
+the standard error in our estimate by the technique of Bootstrap ping [26]. In this method, we draw sub-
+samples from our original sample and compute the maximum likelihood es timates for those sub-samples.
+The standard error in the estimates obtained from different sub-s amples is our estimate for the standard
+error of the MLE.
+So far, we have treated xminas an exogenous parameter in estimating the scaling exponent. To d erive
+[24] an endogenous value for xmin, it is required to minimize the distance between the two CDFs, one
+obtained from the data and the other arising out of the best-fitte d power law model, contingent on the
+value ofxmin. In general, if we choose ˆ xminhigher than the true value of xmin, then the size of the data
+set is effectively reduced. Due to statistical fluctuations, a reduc ed data-set augments the error level for
+the empirical distribution, when compared with the fitted theoretic al distribution. On the other hand,
+if ˆxminis smaller than the true value of xmin, the distribution will differ because of the fundamental
+difference between the data and the fitted model. Kolmogorov-Smir nov (KS) statistic [27] is a standard
+measure to quantify this distance, D, between the two probability distributions with CDFs F1(·) and
+F2(·). Mathematically speaking,
+D= sup
+x|F1(x)−F2(x)| (9)
+It may be worth noting that the CDF of the best-fitted power law de pends on the choice of xminand
+Dis minimized with respect to this xmin. This leads to the optimal model, which is the closest one to
+the empirical distribution among the class of best fitted models. Simu ltaneously, we obtain ˆ xmin, the
+optimal estimate for xmin.
+Goodness-of-fit Tests
+It might be interesting to test the null hypothesis [25] of empirical d istribution following our estimated
+distribution (Pareto or Tsallis q-exponential). It should be mentioned here that even if we estimate
+the parameters of our distribution using the empirical observation s, it is not anyway imperative for the
+observations to be actually from that particular distribution. We re quire a rigorous procedure [24] to test
+the validity of our sample following the specified distribution.
+After hypothesizing the empirical distribution from the optimal pow erlawmodel, we simulate asimilar
+sample from that particular distribution. The optimal power law mode l for the simulated sample is
+estimated by minimization of the relevant KS statistic over the values ofxmin. Thereby, we obtain the
+optimal value for the KS statistic for this particular sample. If this v alue is greater than or equal to the
+corresponding value obtained from the actual data, it is an evidenc e in favour of the real data being from
+the best fitted power-law distribution. Otherwise, it is rather unlike ly that the data is actually from the
+hypothesized power law distribution.
+We generatea largenumber ofsamples with the same size as that of t he data and calculate the fraction
+of samples, where the optimal KS statistic exceeds the one for the real data. We denote this fraction as
+thep-value of our test statistic. If this p-value is large, say close to one, then evidently the real data is
+from the best fitted power law distribution. On the other hand, if th isp-value is close to zero, we fail to
+accept our hypothesis of data being drawn from a power law distribu tion. In terms of the level of the
+test, if the p-value is less than the specified level of the test, the null hypothes is of the data following a
+power law distribution is rejected.4
+Study No. Data-set nxminxmaxMeanMedian Quartile 1 Quartile 3
+IIndia (2001) 330710.0016434.39 84.3023.42 15.39 48.23
+IIIndia (2001) 174203.3816434.39 956.38430.50 267.66 865.55
+IIIIndia (1991) 162160.5012596.24 759.45365.31 219.75 654.49
+IVIndia (1981) 152120.429194.02 576.86277.22 171.68 519.37
+VChina (2000) 146250.0814230.99 298.27136.63 80.86 265.42
+VIChina (2000) 514200.1014230.99 658.95358.81 254.67 578.36
+VIIChina (1990) 134525.027821.79 156.33 68.71 44.23 128.96
+VIIIChina (1990) 280151.587871.79 503.05274.58 189.83 456.18
+TABLE I: Data Description : Values of x (city-population) are reported in units of thou sands. The left trunca-
+tion of the data is determined through the value of xmin.
+Source: [28] and [29]
+We repeat this entire exercise for the null hypothesis of data follow ing a Tsallis q-exponential distri-
+bution as well.
+III. DATA ANALYSIS AND RESULTS
+Data Description
+The Indian Census is conducted once in a decade. We have detailed da ta [28] for the year of 2001.
+According to the census conducted on the first day of March, 200 1, the population of India stood at
+1,027,015,247 persons. In that census data, there is a complete en umeration for the population of 4378
+Indian urban agglomerations, 35 of which, have a population greate r than a million. The data shows that
+only 27.86 percent of Indians live in these urban agglomerations; the rest of the Indians live in numerous
+rural agglomerations.
+The People’s Republic of China conducted censuses in 1953, 1964, an d 1982. In 1987 the government
+announced that the fourth national census would take place in 199 0 and that there would be one every
+ten years thereafter. The 1982 census, which reported a total population of 1,008,180,738, is generally
+accepted as significantlymore reliable, accurate, and thoroughth an the previoustwo. At the 2000census,
+the total population stood at approximately 1.29533 billion, which is ab out 22% of total population in
+the world. 36% of the Chinese population used to reside in urban agglo merations in 2000. We use the
+data [29] from 1990 and 2000 census.
+In general, the theories for modeling the city-size distribution does not differentiate between an urban
+agglomeration and a rural one. However, the census data does no t disclose the complete enumeration of
+the sizes for the rural agglomerations. Therefore, we analyze th e size-distribution of the urban agglom-
+erations alone. In the data, there are many towns with lesser numb er of inhabitants compared to that of
+many villages. If we consider the data for urban agglomerationsin its entirety, this would lead to a biased
+data set for the population agglomeratesas a whole and hence, to a biased set of estimates for the studied
+statistical models. This suggests us to consider the urban agglome rates over certain minimum value. We
+decide to set it at 10,000 for the Indian census data for the year of 2001. There is a trade-off involved in
+finding this minimum cut off for a town’s population size. The choice of a r ather high value (say 20,000)
+causes us to loose a large fraction (nearly half) of our data set; wh ereas choice of a lower value would
+accentuate the problem of biased data-set. Most importantly, a s mall movement of the minimum value
+to either direction would not alter our estimates even quantitatively .
+For the censuses of 1981 and 1991, we do not have the complete en umeration of the population figures
+in the Indian cities. However, we find the individual data for the popu lation of Indian cities with a
+certain minimum number of inhabitants. For example, in 1991, we have individual data regarding 185
+urban agglomeration with the total population of 125,457,068 perso ns; while the total urban population
+of India in 1991 was 217,611,012. Moreover, individual figures of all t he cities above the population of
+160,000 are included in these data. Therefore, we set the minimum va lue to 160,000 to left-censor our
+data-set. Further, to have a comparative study among the data from 1981, 1991 and 2001, we also work
+with all the Indian cities in 2001 with at least 200,000 dwellers.
+We also use the individual data [29] on the population of urban agglome rations in China for the years5
+of 1990 and 2000. We work with two different values for xminin both of these data-sets. The lower
+cut-offs (cases V and VII for the years 2000 and 1990) are chose n such as to include the data-set in
+its entirety baring a few outliers. The relatively higher cut-offs (as in cases VI and VIII for the years
+2000 and 1990) are selected to compare the corresponding figure s for the top ranking cities alone. It
+may be mentioned here that the distributions of Tsallis q-exponential and Pareto differ only at the lower
+level. We tabulate the descriptive statistics regarding all the data- sets in Table I for all the eight cases
+considered.
+Results
+We report the estimates corresponding to all the eight different ca ses studied in this paper in Table
+II. We have elaborated the estimation procedure in Section II. The usual linear fit estimate using the
+Pareto Distribution is given by /hatwideαlf; whereas the maximum likelihood estimate of the same is denoted
+as/hatwideαMLE. It is emphasized here that the linear fit estimate, /hatwideαlf, is not suitable for a proper estimation
+of the scaling parameter. As most of the prevalent literature has b ased their conclusion based on this
+measure, we compute it merely to measure the quantity and directio n of the bias in this estimate. We
+find a considerable bias in the linear fit estimate compared to the corr esponding one obtained using
+the technique of MLE. Also, the MLE of the parameters for the Tsa llisq-exponential distribution are
+expressed as /hatwideθMLEand/hatwideσMLEin Table II. For each study, we plot the corresponding data-set an d the
+fitted CCDFs. The graphical representation shows that in all the c ases studied the estimated Pareto
+distribution and the fitted Tsallis q-exponential distribution are almost identical. Therefore, we rest rict
+our attention to Pareto distribution alone for further investigatio n.
+Pareto Tsallis Minimized KS Distance Estimate with
+Study No. Distribution Distribution Pareto Dist. Tsallis Dist.
+/hatwideαLF/hatwideαMLE/hatwideθMLE/hatwideσMLEstatistic p-valuestatistic p-value
+I1.9923 1.8827 0.8827 0.0084 0.0348 0.0000.0348 0.000
+(0.0010) (0.0153) (0.0002) (0.0132)
+II1.9133 2.0320 1.0320 0.0530 0.0420 0.2900.0420 0.286
+(0.0073) (0.0782) (0.0164) (0.0709)
+III 1.8946 2.0601 1.0601 0.0192 0.0648 0.0050.0648 0.006
+(0.0083) (0.0838) (0.0025) (0.0692)
+IV1.8893 1.9909 0.9909 0.0224 0.0557 0.0520.0557 0.053
+(0.0085) (0.0804) (0.0027) (0.0675)
+V1.8976 1.8480 0.8480 0.0120 0.0568 0.0000.0568 0.000
+(0.0036) (0.0222) (0.0005) (0.0167)
+VI1.7544 2.2975 1.2975 -0.1056 0.0217 0.5310.0217 0.871
+(0.0018) (0.0572) (0.0424) (0.0550)
+VII 1.8967 1.8241 0.8241 0.0076 0.0682 0.0000.0682 0.000
+(0.0031) (0.0225) (0.0001) (0.0167)
+VIII 1.7701 2.2308 1.2308 0.0666 0.0229 0.9130.0229 0.913
+(0.0032) (0.0736) (0.0299) (0.0675)
+TABLE II: Estimates from various data-sets considered: The standard errors are in parenthesis
+In case of India, the estimated exponent ( /hatwideαMLE) is within the range of [1 .8827,2.0601] for different
+cases considered. The value is a good approximation of the theoret ical predicted value of 2 for α. For
+China,/hatwideαMLEdepends on the chosen value of xmin. It is in the range of [1 .8241,2.2975]contingent on the
+choice of xmin. For higher values of xmin, the estimate is bigger. It might be interesting to compare it
+with other studies. In case of cities of Brazil [11] with xminas 30,000 the estimated value of αis found to
+be 2.41 for 1970 and 2.36 for 1980-2000. The corresponding linear fi t estimate for 2400 U.S. cities [10] is
+2.1 for the year of 2000 and that for Switzerland stands at 2.0. The estimates using the data from Japan
+[15] is rather interesting. It is shown that the Zipf’s law ( α= 2) holds for the period of 1970-2000. Before
+and after this time period, αis significantly greater than 2. Using the KS statistic, we have compu ted
+the optimal estimate in the class of best fitted models and the endog enous value for xmin. In all the
+cases, it is almost same as the exogenously fixed value of xminin the data.
+However, the data-set may contain a lot of non-sampling errors. T he scatter plot reveals that the6
+percentage of variation in our estimates is quite large, if we consider the slope of the fitted line neglecting
+a small number of observations. This is indicative of a poor quality of t he data. So, we consider only two
+digits after the decimal place for all our estimates as significant. Th e interpretations for all the computed
+estimates in Table II should be modified accordingly.
+A natural question may arise whether there is any other suggeste d distribution from the exponential
+family that explains the data better. There is an indication [18] that t he Weibull distribution provides
+us a satisfactory adjustment for some ranges of the data. We ca rry out a likelihood maximization (LM)
+test [25] with our data-set, where the null hypothesis of data bein g from the Pareto distribution is tested
+againstthealternativeofvariousotherstatisticaldistributions, namelyWeibull, Exponential,Exponential
+with a cutoff and log-normal. The p-value of the test statistic is always equal to one, which indicates
+that the null hypothesis is strongly accepted against the specified various alternatives. However, there is
+a word of caveat regarding this observation. The result does not im ply that the assumption of the data
+following the Pareto distribution is justified. It is only a better descr iption of the data over the other
+specified alternatives.
+To test the validity of Pareto distribution, we compute the KS statis tic as the distance between the
+empirical CDF and that of a fitted distribution as elaborated in Sectio n II. The relevant p-values in Table
+II reveals that when we analyse the data comprising the sample of In dian cities with a higher xmin, the
+null hypothesis of Pareto distribution is accepted at the 5% critical level for all the years. But the null
+hypothesis can not be accepted if we consider the full sample for th e year of 2001. In case of Chinese
+cities, the null hypothesis is again rejected with the full sample. How ever, it is well-accepted at any
+critical level, if the short sample with higher ranked cities is considere d.
+Finally, the graphical representation discloses that for both the c ountries of India and China, the
+comparativelyhigherrankedcitieshavedisproportionatelymoredw ellerscomparedtoratherlowerranked
+ones. In general, the theories for size distribution implying Zipf’s law d oes not take into account any
+rural to urban migration, while modeling the distribution of urban agg lomerations. However, it is an
+important phenomenon in the developing countries like India and China . In these countries, various
+Economic opportunities drive [30] people from rural agglomeration s to urban ones and also from smaller
+towns to larger cities. A theory, taking into account this factor, c an explain the size distribution of the
+entire sample for the Indian (or Chinese) urban agglomerations.
+IV. CONCLUSION
+In this work, we have shown that the city-size distribution for both the countries of India and China
+follow the Zipf’s law, if only we work with a more trimmed sample keeping th exminquite high. We
+have estimated the scaling exponent, by the linear fit method as well as by a more accurate technique of
+Maximum Likelihood Estimator, which is found to be nearly 2 as predicte d. The maximum likelihood
+estimation with Tsallis q-exponential distribution is also performed, although the estimate d CDF of this
+distribution is identical with its Pareto counterpart. The novelty of our work lies in the goodness-of-fit
+tests. The Kolmogorov-Smirnov statistic for the sample with the co mputedp-value implies that the full
+sample does not follow a Pareto or q-exponential Tsallis distribution too well. However, it gives a good
+approximation for a restricted sample with top ranking cities.
+Acknowledgement: The authors thank Soumyasree Bandyapadh yay for compilation of data for this
+project.7
+10310410510610710810−410−310−210−1100101
+City−size: x −−−−−−−−−−−−−>Complemenatry CDF: PC(x) −−−−−−−−−−−−−>
+  
+2001 INDIAN CITIES OVER 10k
+  Linear Fit
+  Pareto CDF
+  Tsallis q−exponential CDF
+(a)Indian Cities with population over 10,000 in 200110510610710810−410−310−210−1100101
+City−size: x −−−−−−−−−−−−−>Complemenatry CDF: PC(x) −−−−−−−−−−−−−>
+  
+2001 INDIAN CITIES OVER 200k
+  Linear Fit
+  Pareto CDF
+  Tsallis q−exponential CDF
+(b)Indian Cities with population over 200,000 in 2001
+10510610710810−410−310−210−1100101
+City−size: x −−−−−−−−−−−−−>Complemenatry CDF: PC(x) −−−−−−−−−−−−−>
+  
+1991 INDIAN CITIES OVER 160k
+  Linear Fit
+  Pareto CDF
+  Tsallis q−exponential CDF
+(c)Indian Cities with population over 160,000 in 199110510610710810−410−310−210−1100101
+City−size: x −−−−−−−−−−−−−>Complemenatry CDF: PC(x) −−−−−−−−−−−−−>
+  
+1981 INDIAN CITIES OVER 120k
+  Linear Fit
+  Pareto CDF
+  Tsallis q−exponential CDF
+(d)Indian Cities with population over 120,000 in 1981
+FIG. 1: Indian Cities: 1981-20018
+10510610710810−410−310−210−1100101
+City−size: x −−−−−−−−−−−−−>Complemenatry CDF: PC(x) −−−−−−−−−−−−−>
+  
+2000 CHINESE CITIES OVER 50k
+  Linear Fit
+  Pareto CDF
+  Tsallis q−exponential CDF
+(a)Chinese Cities with population over 50,000 in 200010510610710810−410−310−210−1100101
+City−size: x −−−−−−−−−−−−−>Complemenatry CDF: PC(x) −−−−−−−−−−−−−>
+  
+2000 CHINESE CITIES OVER 200k
+  Linear Fit
+  Pareto CDF
+  Tsallis q−exponential CDF
+(b)Chinese Cities with population over 200,000 in 2000
+10410510610710810−410−310−210−1100101
+City−size: x −−−−−−−−−−−−−>Complemenatry CDF: PC(x) −−−−−−−−−−−−−>
+  
+1990 CHINESE CITIES OVER 25k
+  Linear Fit
+  Pareto CDF
+  Tsallis q−exponential CDF
+(c)Chinese Cities with population over 25,000 in 199010510610710810−410−310−210−1100101
+City−size: x −−−−−−−−−−−−−>Complemenatry CDF: PC(x) −−−−−−−−−−−−−>
+  
+1990 CHINESE CITIES OVER 150k
+  Linear Fit
+  Pareto CDF
+  Tsallis q−exponential CDF
+(d)Chinese Cities with population over 150,000 in 1990
+FIG. 2: Chinese Cities: 1990 and 2000
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+Growth Proc. Nat. Acad. Sci. 105(2008) 18702
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+[21] Damian H. Zanette , Zipf’s law and city sizes: A short tutorial review on multipli cative processes in urban
+growth, (To appear in Advances in Complex Systems), arXiv:0704.31 70v1
+[22] Y. Dover, Physica A 334(2004) 591
+[23] C. R. Shalizi, Maximum Likelihood Estimation for q-Exponential (Tsallis ) Distribution , arxiv:math/0701854
+[24] A. Clauset, C. R. Shalizi, and M. J. Newman, Power-law distributions in empirical data , arXiv:0706.1062v1
+[25] C. R. Rao, Linear Statistical Inference and Its Applications , John Wiley and Sons, Inc (2002).
+[26] B. Efron and R. J. Tibsirani, An Introduction to the Bootstrap , Chapman and Hall, Boca Raton (1998).
+[27] I. M. Chakravarti, R. G. Laha, and J. Roy, Handbook of Methods of Applied Statistics , Volume I, John Wiley
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\ No newline at end of file
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+arXiv:1001.0830v1  [cs.IR]  6 Jan 2010K-tree: Large ScaleDocument Clustering
+Christopher M DeVries
+Faculty ofScienceand Technology
+Queensland Universityof Technology
+Brisbane, Australia
+chris@de-vries.id.auShlomo Geva
+Facultyof Scienceand Technology
+Queensland Universityof Technology
+Brisbane, Australia
+s.geva@qut.edu.au
+ABSTRACT
+We introduce K-tree in an information retrieval context. It
+is an efficient approximation of the k-means clustering algo-
+rithm. Unlike k-means it forms a hierarchy of clusters. It
+has been extended to address issues with sparse represen-
+tations. We compare performance and quality to CLUTO
+using document collections. The K-tree has a low time com-
+plexity that is suitable for large document collections. Th is
+tree structureallows for efficient disk based implementatio ns
+where space requirements exceed that of main memory.
+Categories andSubject Descriptors
+H.3.3[Information Storage and Retrieval ]: Information
+Search and Retrieval— Clustering
+General Terms
+Algorithms, Performance
+Keywords
+Document Clustering, K-tree, B-tree, Search Tree, k-means
+1. K-TREE
+The K-treeis aheightbalancedcluster tree. AK-treecon-
+tains nodes of order m, restricting the maximum number of
+vectors in any node. There are two types of nodes. Leaf
+nodes contain 1 to mvectors. Internal nodes contain 1 to m
+(vector, child node) pairs. A nearest neighboursearch tree is
+built bottom-up by splitting full nodes with k-means. Once
+the tree increases in depthit forms a hierarchy of“clusters of
+clusters”until the above leaf level is reached. The above le af
+level contains cluster centres and pointers to the leaves co n-
+taining vectors inserted into the tree. For more informatio n
+see Geva [3] and the K-tree Project Page [1].
+The algorithm shares many similarities with BIRCH [7] as
+both are inspired by the B+-tree data structure. However,
+BIRCH does not keep the inserted vectors in the tree. As
+a result, it can not be used for a nearest neighbour search
+tree. This makes precise removal of vectors from the tree
+impossible.
+The K-tree algorithm was initially designed to work with
+dense vectors. This causes space and efficiency problems
+Copyright is held by theauthor/owner(s).
+SIGIR'09, July 19–23, 2009, Boston, Massachusetts, USA.
+ACM 978-1-60558-483-6/09/07.when dealing with sparse document vectors. We will high-
+light the issue using the INEX 2008 XML Mining collection
+[2]. It contains 114,366 documents and 206,868 stopped and
+stemmed terms. TF-IDF culling is performed by ranking
+terms. A rank is calculated by summing all weights for each
+term. The 8000 terms with the highest rank are selected.
+This reduced matrix contains 10,229,913 non-zero entries. A
+document term matrix using a 4 byte float requires 3.4 GB
+of storage. A sparsely encoded matrix with 2 bytes for the
+term index and 4 bytes for the term weighting requires 58.54
+MB of storage. It is expected that sparse representation wil l
+also improve performance of a disk based implementation.
+More data will fit into cache and disk read times will be
+reduced.
+Unintuitively,asparse representation performs worse wit h
+K-tree. The root of the K-tree contains means of the whole
+collection. Therefore, the vectors contain all terms in the
+collection. This makes them dense. When buildinga K-tree,
+most of the time is spent near the root of tree performing
+nearest neighbour searches. This explains why the sparse
+representation is slower. The most frequently accessed vec -
+tors are not sparse at all.
+2. MEDOID K-TREE
+We propose an extension to K-tree where all cluster cen-
+tres are document exemplars. This is inspired by the k-
+medoids algorithm [5]. Exemplar documents are selected
+by choosing the nearest documents to the cluster centres
+produced by k-means clustering. Internal nodes now con-
+tain pointers to document vectors. This reduces memory
+requirements because no new memory is allocated for clus-
+ter centres. Run time performance is increased because all
+vectors are sparse. The trade-off for this increased perfor-
+mance is a drop in cluster quality. This is expected because
+we are throwing away information. Cluster centres in the
+tree are no longer weighted and are not updated upon inser-
+tion.
+3. EXPERIMENTAL SETUP
+Performance and quality is compared between CLUTO [4]
+and K-tree. The k-means and repeated bisecting k-means
+algorithms were chosen from CLUTO. The medoid K-tree
+was also used to select 10% of the corpus for sampling. This
+sample was used to construct a K-tree. The resulting K-
+tree was used to perform a nearest neighbour search and
+produce a clustering solution. In all cases the K-tree or-
+der was adjusted to alter the number of clusters at the leaflevel. CLUTO was then run tomatch the number of clusters
+produced. All algorithms were single threaded.
+The INEX2008 XML Miningcorpus and aselection of the
+RCV1 corpus used by [6] were clustered. The INEX collec-
+tion consists of 114,366 documents from Wikipedia with 15
+labels. The subset of the RCV1 corpus consists of 193,844
+documents with 103 industry labels. Both corpora were re-
+stricted to the most significant 8000 terms. This is required
+to fit the data in memory when using K-tree with a dense
+representation.
+Micro averaged purity and entropy are compared. Micro
+averaging weights the score of a cluster by its size. Purity
+and entropy are calculated by comparing the clustering so-
+lution to the labels provided. Run-times are also compared.
+4. EXPERIMENTAL RESULTS
+ CLUTO k−means
+CLUTO repeated bisecting k−means
+K−tree
+medoid K−tree
+hybrid sampled K−tree
+0 2000 4000 6000 8000 10000 120000.40.450.50.550.60.650.70.750.80.850.9
+number of clusterspurity
+  
+0 2000 4000 6000 8000 10000 1200000.511.522.53
+number of clustersentropy
+  
+0 2000 4000 6000 8000 10000 12000050010001500200025003000
+number of clustersrun−time in seconds
+  
+Figure 1: INEX 200801000200030004000500060007000800090001000011000120000.20.250.30.350.40.450.50.550.60.65
+number of clusterspurity
+  
+0 2000 4000 6000 8000 10000 120001.522.533.544.555.5
+number of clustersentropy
+  
+0 2000 4000 6000 8000 10000 120000500100015002000250030003500400045005000
+number of clustersrun−time in seconds
+  
+Figure 2: RCV1
+K-tree produces large numbers of clusters in significantly
+less time. The graphs show an early poor run-time perfor-
+mance of K-tree. This has been artificially inflatedby choos-
+ing a large tree order. One can choose a smaller tree order
+and find a smaller number of clusters higher in the tree. The
+computational cost of k-means dominates when choosing a
+large K-tree order. Note that the k-means algorithms in K-
+tree and CLUTO differ. K-tree runs k-means to convergence
+using dense vectors. CLUTO stops after a specified numberof iterations and uses sparse vectors. Using the medoid K-
+tree further increases run-time performance and the combi-
+nation of both K-tree variants falls somewhere in between.
+These solutions provide significant performance gains when
+large numbers of clusters are required. This leaves a trade
+off between performance and quality. Figures 1 and 2 show
+quality relationships and run-time performance.
+5. APPLICATIONSININFORMATIONRE-
+TRIEVAL
+The performance gain of K-tree is most pronounced when
+many small clusters are required. Many practical applica-
+tions require this. For example, many clusters are useful fo r
+collection selection when deciding how to spread a collec-
+tion across many machines. When using clustering for link
+discovery, a large number of small clusters are useful. A
+small number of highly semantically related documents are
+candidates for linking.
+Due to the dynamic nature of the algorithm, clusters can
+beproducedincrementallywithouthavingtoprocessalldoc -
+uments at once. This also allows for easy updates as new
+documents arrive.
+The tree structure can be used for interactive collection
+exploration. A user can start browsing from any point in
+the tree and generalise or specialise what they are viewing
+by traversing up or down the tree. A list of documents
+ranked against the current cluster centre can be displayed.
+This allows the user to conceptualise clusters as they move
+through the tree.
+6. REFERENCES
+[1] K-tree project page, http://ktree.sourceforge.net. 2 009.
+[2] Ludovic Denoyer and Patrick Gallinari. The Wikipedia
+XML Corpus. SIGIR Forum , 2006.
+[3] S. Geva. K-tree: a height balanced tree structured
+vector quantizer. Neural Networks for Signal Processing
+X, 2000. Proceedings of the 2000 IEEE Signal
+Processing Society Workshop , 1:271–280 vol.1, 2000.
+[4] G. Karypis. CLUTO-A Clustering Toolkit. 2002.
+[5] L. Kaufman, P. Rousseeuw, Delft(Netherlands). Dept.
+of Mathematics Technische Hogeschool, and
+Informatics. Clustering by means of medoids. Technical
+report, Technische Hogeschool, Delft(Netherlands).
+Dept. of Mathematics and Informatics., 1987.
+[6] Y. Song, W.Y. Chen, H. Bai, C.J. Lin, and E.Y.
+Chang. Parallel Spectral Clustering. 2008.
+[7] T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH: A
+New Data Clustering Algorithm and Its Applications.
+Data Mining and Knowledge Discovery , 1(2):141–182,
+1997.
\ No newline at end of file
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+arXiv:1001.0832v1  [physics.atm-clus]  6 Jan 2010Oscillations in the total photodetachment cross sections o f a
+triatomic anion
+B.C. Yang and M.L. Du∗
+Institute of Theoretical Physics, Academia Sinica, Beijin g 100190, China
+(Dated: November 15, 2018)
+Abstract
+The total photodetachment cross section of a linear triatom ic anion is derived for arbitrary
+laser polarization direction. The cross section is shown to be strongly oscillatory when the laser
+polarization direction is parallel to the axis of the system ; the oscillation amplitude decreases and
+vanishes as the angle between the laser polarization and the anion axis increases and becomes
+perpendicular to the axis. The average cross section over th e orientations of the triatomic system
+is also obtained. The cross section of the triatomic anion is compared with the cross section of a
+two-center system. We find there are two oscillation frequen cies in the triatomic anion in contrast
+to only one oscillation frequency in the two-center case. Cl osed-orbit theory is used to explain the
+oscillations.
+PACS numbers: 32.80.Fb,32.80.Qk,33.70.Ca
+∗Electronic address: duml@itp.ac.cn
+1I. INTRODUCTION
+Photodetachment of negative ions in the presence of a static elect ric field has been an
+active research area in the last decades [1–25]. The most interestin g feature in the cross
+section is the induced oscillations above photodetachment thresho ld by the static electric
+field. The oscillations in the total cross sections can be understood using closed-orbit theory
+[26–28]. In an effort to understand the oscillations in various proces ses involving two-center
+system such as the photoionization cross section of diatomic molecu le [29], the scattering
+ofD2molecule by fast electron [30], a molecule in a strong laser field [31], abov e-threshold
+ionization[32]andharmonicgeneration[33], AfaqandDuextendedth eone-center H−model
+for photodetachment and developed a two-center model for pho todetachment[34–36]. They
+demonstrated the cross sections in the two-center system show strong oscillation which can
+be explained using closed-orbit theory. In particular, a detached e lectron orbit connecting
+thetwo centers isidentified to beresponsible forthe oscillation inthe total photodetachment
+cross sections of the two-center system.
+To understand the structural information on linear triatomic nega tive anions such as
+BeCl−
+2,HCN−,CS−
+2and CO−
+2[37], Afaq et al.[38] recently studied the photodetachment of a
+triatomic anion with three centers when the axis of the triatomic ion is perpendicular to the
+laser polarization direction. Interference patterns for detache d-electron on a screen placed
+at a large distance from the system were demonstrated, but the t otal cross section was found
+to be smooth and no oscillation was observed for this configuration.
+Here we extend the study of the total photodetachment cross s ection of the above tri-
+atomic anion system to the general case with an arbitrary laser pola rization direction. We
+will derive analytic formulas for the total cross section which depen ds on photon energy,
+laser polarization direction and other parameters characterizing t he triatomic anion. It will
+be shown that the cross section shows strong oscillations when the laser polarization is par-
+allel to the system axis and the oscillation amplitudes gradually decrea se to zero as the laser
+polarization is changed to be perpendicular to the axis. We also obtain ed the cross section
+averaged over the orientations of the system. We compare the cr oss section of the triatomic
+anion with that of the two-center system. The oscillations in photod etachment cross sec-
+tions for the triatomic anion appear much enhanced compared to th e two-center case. We
+find there are two oscillation frequencies in the triatomic anion. The t wo oscillations are
+2explained using closed-orbit theory. Atomic units will be used unless s pecially noted.
+II. FORMULAS FOR TOTAL PHOTODETACHMENT CROSS SECTION
+The linear triatomic anion interacting with a laser is shown schematically in Fig.1. Sym-
+bols 1,0 and 2 represent the three atomic centers in the system. It is convenient to choose
+the z-axis in the direction of the three-center axis and the middle ce nter denoted by 0 as
+the origin of coordinates. Let dbe the distance between two adjacent centers. The laser
+polarization direction is denoted as ( θL,φL) with respect to the zaxis.
+In the triatomic anion, one active electron is assumed. This is an exte nsion of the two-
+center model for photodetachment[34–36] to a three-center m odel by Afaq et al.[38]. In the
+photodetachment process, there are two steps: in the first ste p, the active electron absorbs
+one photon energy Ephand escapes from the negative anion as an electron wave from each
+center; in the second step, the outgoing waves from each center propagate out to large
+distances. The interference of the outgoing waves from each cen ter produces oscillatory
+cross section.
+For the general case, as illustrated in Fig. 1, Let Ψ+
+1, Ψ+
+0and Ψ+
+2be the detached-waves
+from center 1, 0 and 2 respectively. Following the previous approac h in the two-center
+case[36], the outgoing detached-electron wave Ψ+
+Mfrom the triatomic anion can be written
+as a linear combination given by[36]
+Ψ+
+M=1√
+3(Ψ+
+1+Ψ+
+0+Ψ+
+2). (1)
+Let(r1,θ1,φ1), (r0,θ0,φ0)and(r2,θ2,φ2)represent thespherical coordinatesofthedetached-
+electron relative to the three centers respectively. The detache d-electron wave generated
+from each center has been worked previously[25]. They can be writt en as
+Ψ+
+1=4Bk2i
+(k2
+b+k2)2f(θ1,φ1;θL,φL)exp(ikr1)
+kr1,
+Ψ+
+0=4Bk2i
+(k2
+b+k2)2f(θ0,φ0;θL,φL)exp(ikr0)
+kr0,
+Ψ+
+2=4Bk2i
+(k2
+b+k2)2f(θ2,φ2;θL,φL)exp(ikr2)
+kr2, (2)
+wherek=√
+2Eand E is the detached-electron energy; kbis related to the binding energy
+Ebof H−byEb=k2
+b
+2. The photon energy is given by Eph=E+Eb.Bis a normalization
+3constant[3], cis the speed of light and is approximately equal to 137a.u.. The angular factor
+such asf(θ0,φ0;θL,φL) represents the dependence of the detached-electron wave fu nction
+on the outgoing direction ( θ0,φ0) for center 0, and it can be written as [25]
+f(θ0,φ0;θL,φL) = cosθ0cos(θL)+sinθ0sin(θL)cos(φ0−φL). (3)
+The angular factors f(θ1,φ1;θL,φL) andf(θ2,φ2;θL,φL) can be written out in a similar way.
+After substituting Eqs.(2) and Eqs.(3) in Eq.(1), we can get the exp licit expression for
+the detached-electron wave function Ψ+
+M. The resulting formula for Ψ+
+Mcan be simplified
+because the calculation of the photodetachment cross section re quires the knowledge of
+Ψ+
+Mat large distances where r1,r0andr2are much greater than the distance dbetween
+two neighboring centers. Let ( r,θ,φ) be the spherical coordinates of the detached-electron
+relative to the origin. Then we approximate the phase terms using r1≈r−dcosθ,r0=r,
+r2≈r+dcosθ; in other places we can set r1≈r2≈r0=r,θ1≈θ2≈θ0=θ. With these
+approximations, Ψ+
+Mbecomes
+Ψ+
+M(r,θ,φ) =4Bk2i
+(k2
+b+k2)21√
+3f(θ,φ;θL,φL)[1+2cos( kdcosθ)]exp(ikr)
+kr,r→ ∞.(4)
+Eq.(4) describes the detached-electron wave of thelinear triatom ic anionwhen the detached-
+electron is far away from the anion.
+The differential cross section is [5]
+dσ(q)
+ds=2πEph
+cj·n, (5)
+j=i
+2(Ψ+
+M∇Ψ+∗
+M−Ψ+∗
+M∇Ψ+
+M),
+whereσis the photodetachment cross section, dsis the differential area on a surface Γ such
+as the surface of a large sphere enclosing the anion, qis the coordinate on the surface Γ, n
+is the exterior norm vector at qandjis the detached-electron flux.
+The detached-electron flux in the radial direction can be evaluated as
+jr(r,θ,φ) =j·ˆr=i
+2(Ψ+
+M∂Ψ+∗
+M
+∂r−Ψ+∗
+M∂Ψ+
+M
+∂r). (6)
+When Eq.(4) is substituted in Eq.(6), we get
+j·ˆr=16B2k4
+3k(k2
+b+k2)4f2(θ,φ;θL,φL)
+r2[1+4cos( kdcosθ)+4cos2(kdcosθ)].(7)
+4To get the total photodetachment cross section, we integrate t he differential photode-
+tachment cross section using Eq.(5) and Eq.(7) over θandφ,
+σ(E,d,θ L) =2πEph
+c/integraldisplay
+(j·ˆr)r2sinθdθdφ. (8)
+After a straightforward integration, we find the total photodet achment cross section can be
+written as a product form
+σ(E,d,θ L) =σ0(E)A3(kd,θL), (9)
+where
+σ0(E) =16√
+2B2π2E3
+2
+3c(Eb+E)3, (10)
+A3(kd,θL) = 1+4
+3I(kd)+2
+3I(2kd). (11)
+In the above equations, σ0(E) is the smooth total photodetachment cross section of H−[3]
+andA3(kd,θL)isamodulationfunctionforthetriatomicanion. Thefunction I(S)appearing
+in Eq.(11) is given by
+I(S) = 3cos2θL[sinS
+S+3cosS
+S2−3sinS
+S3]+3[sinS
+S3−cosS
+S2]. (12)
+Thefunction I(S)alsoappearsinthemodulationfunctionforthetwo-centerproble mstudied
+previously[36].
+III. OSCILLATIONS AND LIMITS OF CROSS SECTIONS
+First we show the cross section obtained by Afaq et al.[38] for the perpendicular config-
+uration can be obtained from the general formulas. When the laser polarization direction is
+perpendicular to the direction of the axis, θL=π
+2. From Eqs.(9)-(12) we immediately have
+σ(E,d,π
+2) =σ0(E)[1+4[sin(kd)
+(kd)3−cos(kd)
+(kd)2]+4[sin(2kd)
+(2kd)3−cos(2kd)
+(2kd)2]].(13)
+The result in Eq.(13) is identical to that given by Afaq et al.[38]. In Fig.2 we compare the
+photodetachment cross section in Eq.(13) with the cases θL= 0 and θL=π
+4in Eqs.(9)-
+(12) when d= 4a0. Indeed we do not find any oscillation in the the cross section for
+θL=π
+2in agreement with Afaq et al.[38]. But when the direction of laser polarization is
+5not perpendicular to the axis, we observe a hint of oscillation as illustr ated in the insert. In
+fact, the oscillations will become strong and obvious as the paramet erS=kdis increased
+to be comparable or greater than π. In Fig.3 we show the photodetachment cross sections in
+Eqs.(9)-(12) for several dandθLvalues. The following points can be derived directly from
+Fig.3. First, we do not observe obvious oscillation in the photodetach ment cross sections for
+the perpendicular laser polarization ( θL=π
+2) even when dis increased. But when the laser
+polarization direction is not perpendicular to the anion axis, there ar e strong oscillations.
+Second, the oscillation amplitude depends on the laser polarization dir ection. In fact, the
+oscillation amplitude increases and reaches maximum as the laser polar ization direction
+is turned from perpendicular to parallel direction with respect to th e anion axis ( θL=
+0). However, the oscillation frequency is not sensitive to the laser p olarization direction.
+The modulation function A3suggests the oscillation frequency increases but the oscillation
+amplitude decreases as the parameter dis increased. The oscillations are easily observed
+whendis approximately around 30 −100a0in our system.
+The limits of the photodetachment cross sections of the anion can b e obtained from the
+limits of I(S). Using Taylor series expansion,one can show
+lim
+S→0I(S,θL) = 1,
+lim
+S→∞I(S,θL) = 0. (14)
+The above results are independent of the value of θL. Therefore we conclude from Eqs.(9)-
+(12) that in the low energy limit the total photodetachment cross s ection of the three-center
+system is three times of the photodetachment cross section of a s ingle-center system and in
+the large photon energy limit the photodetachment cross section o f the three-center system
+approaches the photodetachment cross section of a single-cent er system.
+It is also straightforward to get the photodetachment cross sec tion averaged over the
+orientations of the anion. Let us assume the direction of the anion is random with respect
+to the laser polarization direction. Defining the averaging by
+¯σ(E,d) =/integraldisplayπ
+0sinθLdθL/integraldisplay2π
+0σ(E,d,θ L)dφL//integraldisplayπ
+0sinθLdθL/integraldisplay2π
+0dφL. (15)
+The integrals can be evaluated to give the following results,
+¯σ(E,d) =σ0(E)¯A3(kd), (16)
+¯A3(kd) = 1+4
+3sin(kd)
+kd+2
+3sin(2kd)
+2kd. (17)
+6It is interesting that the average cross section is equal to the lase r polarization depen-
+dent cross section in Eqs.(9)-(12) evaluated at a special angle sat isfyingθL= arccos(1√
+3)
+(approximately 54 .74◦). The same angle was noticed for the two-center system earlier[36 ].
+IV. FOURIER ANALYSIS
+The averaging process does not change the basic oscillatory struc ture of the cross section.
+However, the oscillation amplitudes in the average cross section are reduced to one third of
+the values obtained for the laser polarization dependent cross sec tion atθL= 0. In Fig.4(a)
+we compare the average cross section and the laser polarization de pendent cross section at
+θL= 0.
+The oscillations are better analyzed using a transformation. For th e cross section σ(E)
+we define F(x) using the following integral
+F(x) =/integraldisplayE2
+E1[σ(E)−σ0(E)]sin[π(E−E1)
+(E2−E1)]kexp(−ikx)dE. (18)
+In Fig.4(b) we present the corresponding transformations of the two cross sections in
+Fig.4(a). Several points can be derived from Fig.4 regarding the osc illations. First, there
+are two peaks in each transformation. The peaks correspond to t wo oscillations in each cross
+section. Second, the oscillation frequencies are not changed by th e averaging procedure.
+Third, the oscillation amplitudes in the average cross section are con siderable reduced.
+We now compare the average photodetachment cross section of t he triatomic anion with
+that of the two-center negative ion studied earlier[36]. In Fig.5(a) we show the two cross
+sections. One can see that the oscillation in the photodetachment c ross section for the
+triatomic anion is enhanced compared to that for the two-center s ystem. In Fig.5(b) we
+show the corresponding transformations of the two cross sectio ns in Fig.5(a). It is clear
+from Fig.5(b) that there are two oscillations for the triatomic anion b ut only one oscillation
+for the two-center system.
+Closed-orbit theory was extended previously to explain the oscillatio n in the cross sec-
+tion of the two-center system. The oscillation was identified as an int erference between
+the detached-electron wave emitted at one center and the sourc e of the wave at another
+center[36]. A similar explanation can be made here. For example, the d etached-electron
+wave produced at center 1 will reach center 0 and 2 as it propagate s out. The overlap of this
+7detached-electron wave fromcenter 1 with the source at center 0 and the source at 2 produce
+the sin(kd) and sin(2 kd) oscillations in the total cross section. Detached-electron wave f rom
+center 2 also have similar interference terms. The final oscillation am plitudes include all the
+interference terms. We will not repeat the detail of such derivatio n which is quite similar to
+the two-center case[36]. We emphasize that such a derivation base d on closed-orbit theory
+establishes that kdis the action/integraltext
+p·dqof the detached-electron propagating from one cen-
+ter to a neighboring center and 2 kdis the action of the detached-electron propagating from
+center 1 to center 2 or from center 2 to center 1. The two oscillatio ns are directly associated
+with the detached-electron orbits from one center to a neighborin g center having action kd
+and the detached-electron orbits connecting center 1 and 2 havin g action 2 kd.
+V. CONCLUSIONS
+We have studied the photodetachment of the recent triatomic anio n model[38] in the
+general case. We have derived the total photodetachment cros s section for arbitrary laser
+polarization direction. It is demonstrated there are two oscillation f requencies in the cross
+sections. The amplitudes of the oscillations can be varied by changing the laser polarization
+direction. Theamplitudes arelargest whenthelaserpolarizationispa ralleltotheanionaxis.
+As the laser polarization direction is turned to be perpendicular to th e axis, the oscillation
+amplitudes decrease and vanish. We also obtained the average cros s section for random
+orientations of anion. The averaging procedure modifies the oscillat ion amplitudes but it
+does not change the oscillation frequencies. The two oscillations in th e present three-center
+system can be explained using closed-orbit theory as interference between detached-electron
+waves produced from one center and the sources at other cente rs. Two types of detached-
+electron orbits are responsible for the two oscillations.
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+[31] S. X. Hu and L. A. Collins, Phys. Rev. A 94, 073004 (2005).
+[32] Xibin Zhou, Robynne Lock, Wen Li, Nick Wagner, Margaret M. Murnane,and Henry C.
+Kapteyn, Physical Review Letters 100, 073902 (2008).
+[33] H. Hetzheim, C. Figueira de Morisson Faria, and W. Becke r, Phys. Rev. A 76,023418 (2007).
+9[34] A.Afaq and M. L. Du, Commun.Theor.Phys. 46119 (2006).
+[35] A.Afaq and M. L. Du, Commun.Theor.Phys. 501401 (2008).
+[36] A.Afaq and M. L. Du, J. Phys. B 42105101 (2009).
+[37] V. R. Bhardwaj, D. Mathur, and F. A. Rajgara, Phys. Rev. L ett.80, 3220 (1998).
+[38] A.Afaq et al., Appl. Phys. Lett. 94, 041125 (2009).
+10d
+d2dZψ1+
+ψ0+
+ψ2+1
+oθL
+2++
+−+
+−
+−
+FIG. 1: The schematic representation for the photodetachme nt of the linear triatomic anion. The
+dotted arrows point to the laser polarization direction. Th e plus and minus lobes represent the
+angular amplitude of the detached-electron from each cente r.
+110 1 2 3 4 5 6 7 8 9 1000.511.522.533.54
+Photon energy (eV)Cross section (a.u.)
+  
+d=4a0
+1020304050607000.050.10.150.20.25
+Photon energy (eV)Cross section (a.u.)
+  
+θL=0
+θL=π/4
+θL=π/2
+FIG. 2: The photodetachment cross section of the triatomic a nion at d= 4a0and the laser
+polarization direction θLare respectively equal to 0,π
+4,π
+2. The inset provides a hint of oscillation
+forθL= 0 and θL=π
+4.
+120 2 4 6 800.511.522.53
+Photon energy (eV)Cross section (a.u.)
+  
+θL=0
+θL=π/4
+θL=π/2
+0 2 4 6 800.511.52
+Photon energy (eV)Cross section (a.u.)
+  
+θL=0
+θL=π/4
+θL=π/2
+0 2 4 6 800.511.52
+Photon energy (eV)Cross section (a.u.)
+  
+θL=0
+θL=π/4
+θL=π/2
+0 2 4 6 800.511.52
+Photon energy (eV)Cross section (a.u.)
+  
+θL=0
+θL=π/4
+θL=π/2(a) d=10a0
+(d) d=500a0(c) d=100a0(b) d=50a0
+FIG. 3: The photodetachment cross sections of the triatomic anion with different values of dand
+θL.
+130.511.522.533.544.5500.511.52
+Photon energy (eV)Cross section (a.u.)
+  
+0 50 100 150 200 250 300   1   2   3   4   5   6   7
+x/a0|F(x)| (arb. units)average cross section
+cross section for θL=0(a)
+(b)
+FIG. 4: (a) The total photodetachment cross section for d= 100a0andθL= 0 (solid line) and the
+average cross section (dotted line). (b) The transformatio ns defined in Eq.(18) for the above two
+cross sections.
+140.511.522.533.544.5500.511.52
+Photon energy (eV)Cross section (a.u.)
+  
+0 50 100 150 200 250 30000.511.522.5
+x/a0|F(x)| (arb. units)
+  two−center system
+triatomic system(a)
+(b)
+FIG. 5: (a) The orientation average photodetachment cross s ections of the two-center system and
+theorientation average triatomicanionsystem. Theparame terd= 100a0. (b)Thetransformations
+defined in Eq.(18) for the above two cross sections.
+15
\ No newline at end of file
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+arXiv:1001.0833v2  [cs.IR]  2 Feb 2010Random Indexing K-tree
+ChristopherM. DeVries Lance DeVine ShlomoGeva
+Faculty ofScienceand Technology
+QueenslandUniversityofTechnology
+Brisbane, Australia
+chris@de-vries.id.au l.devine@qut.edu.au s.geva@qut.e du.au
+Abstract Random Indexing (RI) K-tree is the combi-
+nation of two algorithms for clustering. Many large
+scale problems exist in document clustering. RI K-tree
+scales well with large inputs due to its low complexity.
+It also exhibits features that are useful for managing
+a changing collection. Furthermore, it solves previ-
+ousissueswithsparsedocumentvectorswhenusingK-
+tree. The algorithms and data structures are defined,
+explained and motivated. Specific modifications to K-
+tree are made for use with RI. Experiments have been
+executed to measure quality. The results indicate that
+RI K-tree improves document cluster quality over the
+originalK-tree algorithm.
+Keywords RandomIndexing,K-tree,Dimensionality
+Reduction, B-tree, Search Tree, Clustering, Document
+Clustering,VectorQuantization,k-means
+1 Introduction
+The purpose of this paper is to present and analyse
+the combination of Random Indexing (RI) with
+the K-tree algorithm. Both RI and K-tree adapt to
+changingdataanddecreasethecostofcomputationally
+intensive vector based applications. This combination
+is particularly suitable to the representation and
+clustering of very large document collections.
+Documents are typically represented in vector
+space as very sparse high dimensional vectors. RI
+can reduce the dimensionality and sparsity of this
+representation. In turn, the condensedrepresentationis
+highly effective when working with K-tree. The paper
+is focused on determining the effectiveness of using
+RI with K-tree through experiments and comparative
+analysisofresults.
+Sections 2 to 6 discuss K-tree, Random Indexing,
+DocumentRepresentation,ExperimentalSetupandEx-
+perimental results respectively. The paper ends with a
+conclusionin Section7.
+2 K-tree
+K-tree[6,1]isaheightbalancedclustertree. Itwasfirst
+introduced in the context of signal processing by Geva
+Proceedings of the 14th Australasian Document Comput-
+ing Symposium, Sydney, Australia, 4 December 2009.
+Copyright for thisarticle remainswiththe authors.[10]. Thealgorithmisparticularlysuitabletoclustering
+of large collections due to its low complexity. It is
+a hybrid of the B+-tree and k-means algorithm. The
+B+-tree algorithm is modified to work with multi di-
+mensionalvectorsandk-meansisusedtoperformnode
+splits in the tree. K-tree is also related to Tree Struc-
+tured Vector Quantization (TSVQ) [9]. TSVQ recur-
+sively splits the data set, in a top-down fashion, using
+k-means. TSVQ does not generally produce balanced
+trees.
+K-tree achieves its efficiency through execution of
+the high cost k-means step over very small subsets of
+the data. The number of vectors clustered during any
+step in the K-tree algorithm is determined by the tree
+order(usually ≪1000)andit is independentof collec-
+tion size. It is efficient in updatingthe collection while
+maintaining clustering properties through the use of a
+nearest neighbour search tree that directs new vectors
+totheappropriateleafnode.
+The K-tree forms a hierarchy of clusters. This hi-
+erarchy supports multi-granular clustering where gen-
+eralisation or specialisation is observed as the tree is
+traversed from a leaf towards the root or vice versa.
+The granularity of clusters can be decided at run-time
+byselectingclustersthatmeetcriteriasuchasdistortion
+orclustersize.
+2.1 K-tree andDocument Clustering
+The K-tree algorithm is well suited to clustering large
+document collections due to its low time complexity.
+The time complexity of building K-tree is O(n log n)
+where n is the number of bytes of data to cluster. This
+is due to the divide and conquer properties inherent to
+thesearchtree. DeVriesandGeva[5,6]investigatethe
+run-timeperformanceandqualityofK-treebycompar-
+ing results with other INEX submissions and CLUTO
+[13]. CLUTOisapopularclusteringtoolkitusedinthe
+informationretrievalcommunity. K-treehasbeencom-
+pared to k-means, including the CLUTO implementa-
+tion,andprovidescomparablequalityandamarkedin-
+creaseinrun-timeperformance. However,K-treeforms
+a hierarchy of clusters and k-means does not. Com-
+parison of the quality of the tree structure will be un-
+dertakeninfurtherresearch. Therun-timeperformance
+increase of K-tree is most noted when a large number
+of clusters are required. This is useful in terms of doc-ument clustering because there are a huge number of
+topics in a typical collection. The on-line and incre-
+mental nature of the algorithm is useful for managing
+changing document collections. Most clustering algo-
+rithms are one shot and must be re-run when new data
+arrives. K-tree adapts as new data arrives and has the
+lowtimecomplexityofO(logn)forinsertionofasingle
+document. Additionally, the tree structure also allows
+for efficient disk based implementations when the size
+ofdatasetsexceedsthatofmainmemory.
+2.2 K-tree Definition
+K-tree buildsa nearest neighboursearch tree overa set
+ofrealvaluedvectors Vinddimensionalspace.
+∀v∈V:v∈Rd(1)
+It is inspired by the B+-tree where all data records are
+stored in leaf nodes. Tree nodes, N, consist of a se-
+quence of (vector, child node) pairs of length l. The
+tree order, m, restricts the number of vectors stored in
+anynodetobetweenoneand m.
+1≤l≤m (2)
+N=/angbracketleft(v1,c1),...,(vl,cl)/angbracketright (3)
+The tree consists of two types of nodes. Leaf nodes
+containthedatavectorsthatwereinsertedintothetree.
+Internal nodes contain clusters. A cluster vector is
+the mean of all data vectors contained in the leaves of
+all descendant nodes (i.e. the entire cluster sub-tree).
+This followsthe same recursivedefinitionof a B+-tree
+whereeachtreeismadeupofasetofsmallersub-trees.
+Upon construction of the tree, a nearest neighbour
+search tree is built in a bottom-up manner by splitting
+full nodes using k-means [14] where k= 2. As the
+tree depth increases it forms a hierarchy of “clusters
+of clusters” from the root to the above-leaf level.
+The above-leaf level contains the finest granularity
+cluster vectors. Each leaf node stores the data vectors
+pointed to by the above-leaf level. The efficiency of
+K-tree stems from the low complexity of the B+-tree
+algorithm,combinedwithonlyeverexecutingk-means
+on a relatively small number of vectors, defined by the
+treeorder,andbyusingasmall valueof k.
+2.3 Modifications to K-tree
+The K-tree algorithm was modified for use with RI.
+This modified version will be referred to as “Modified
+K-tree” and the original K-tree will be referred to as
+“UnmodifiedK-tree”.
+All the document vectors created by RI are of unit
+length in the modified K-tree. Therefore, all centroids
+are normalisedto unit lengthat all times. The k-means
+used for node splits in K-tree was changed to use ran-
+domisedseedingandrestartifitdidnotconvergewithin
+sixiterations. Theprocessalwaysconvergedquicklyin
+ourexperiments;althoughitispossibletoconstrainthe
+numberofrestartswedidnotfindthistobenecessary.0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.20.40.60.811.21.41.61.8
+Figure1: Level1
+The original K-tree algorithm does not modify any
+of the centroids. They are simply the means of the
+vectors they represent. The k-means implementation
+runs to complete convergence and seeds centroids via
+perturbation of the global mean. To create two seeds
+the global mean is calculated and then the two seeds
+are createdbymovingawayfromthe meanin opposite
+directions.
+2.4 K-tree Example
+Figures 1 to 3 are K-tree clusters in two dimensions.
+1000 points were drawn from a random normal distri-
+butionwithameanof1.0andstandarddeviationof0.3.
+TheorderoftheK-tree, m,was11. Thegreydotsrepre-
+sent the data set, the black dots represent the centroids
+and the lines represent the Voronoi tessellation of the
+centroids. Eachofthedatapointscontainedwithineach
+tile of the tessellation are the nearest neighboursof the
+centroid and belong to the same cluster. It can be seen
+that the probabilitydistributionis modelledat different
+granularities. The top level of the tree is level 1. It is
+thecoarsest grainedclustering. Inthisexampleit splits
+the distribution in three. Level 2 is more granular and
+splitsthecollectioninto19sub-clusters. Theindividual
+clustersinlevel2canonlybearrivedatthroughanear-
+est neighbour association with a parent cluster in level
+1 of the tree. Level 3 is the deepest level in the tree
+consistingofclustercentroids. The4thlevelisthedata
+set ofvectorsthatwereinsertedintothetree.
+2.5 Building K-tree
+The K-tree is constructed dynamically as data vectors
+arrive. Initially the tree contains a single empty root
+nodeat theleaf level. Vectorsare insertedviaa nearest
+neighboursearch,terminatingattheleaflevel. Theroot
+ofanemptytreeisaleaf,sothefirst mdatavectorsare
+storedintheroot,atwhichpointthenodebecomesfull.
+When the m+ 1vector arrives the root is split using
+k-means where k= 2, clustering all m+ 1vectors
+into two clusters. The two centroids that result from
+k-means are then promoted to become the centroids in
+a new root. The vectors associated with each centroid0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.20.40.60.811.21.41.61.8
+Figure2: Level2
+0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.20.40.60.811.21.41.61.8
+Figure3: Level3
+are placed into a child node. This promotion process
+has created a new root and two leaf nodes in the tree.
+The tree is now two levels deep. Insertion of a new
+data vector follows a nearest neighbour search to find
+the closest centroid in the root. The vector is inserted
+into the associated child. When a new vector is in-
+serted the centroids are updated recursively along the
+nearest neighbour search path, all the way back to the
+root node. The propagated means are weighted by the
+number of data vectors contained beneath them. This
+ensuresthatanycentroidinK-treeisthemeanvectorof
+allthedatavectorscontainedintheassociatedsubtree.
+This insertion process continues, splitting leaves when
+they become full, until the root node itself becomes
+full. K-means is then run on the root node containing
+centroids. The vectors in the new root node become
+centroids of centroids. As the tree grows, internal and
+leaf nodesaresplit in thesame manner. Theprocessof
+promotioncanpotentiallypropagatetocauseafullroot
+node at which point the construction of a new root fol-
+lowsandthetreedepthisincreasedbyone. Atalltimes
+the tree is guaranteed to be height balanced. Although
+the tree is always heightbalancednodescan contain as
+little as one vector. In this case the tree will contain
+many more levels than a tree where each node is halffull. Section 4 showsthis constructionprocessfora K-
+treeoforderthree (m= 3).
+2.6 Sparsity and K-tree
+K-tree was originally designed to operate with dense
+vectors. When a sparse representation is used perfor-
+mance degrades even though there is significantly less
+datatoprocess. Theclustersinthetoplevelsofthetree
+aremeansofmostofthetermsinthecollectionandare
+not sparse at all. The algorithm updatescluster centres
+along the insertion path in the tree. Since document
+vectors have very high dimensionality this becomes a
+veryexpensiveprocess.
+The medoid K-tree [6] extended the algorithm to
+use a sparse representation and replace centroids with
+document examples. This improved run-time perfor-
+mance and decreased memory usage. Unfortunately it
+decreasedqualitywhenusingsparsedocumentvectors.
+The documentexamplesin the rootof the tree were al-
+mostorthogonaltonew documentsbeinginserted. The
+documentswereunlikelytohavemeaningfuloverlapin
+vocabulary.
+TheapproachtakenbyDe Vries andGevaat INEX
+2008[5] is a simple approachto dimensionalityreduc-
+tion or feature selection. It is called TF-IDF culling
+and it is performed by ranking terms. A rank is cal-
+culated by summing all weights for each term. The
+weights are the BM25 weight for each term in each
+document. Thiscanalsobeexplainedasthesumofthe
+columnvectorinthedocumentbytermmatrix. Thetop
+n terms with the highest rank are selected, where n is
+thedesireddimensionality. Thisworksparticularlywell
+with term occurrences due to the Zipf law distribution
+of terms [19]. The collection frequency of a term is
+inverselyproportionaltoitsrankaccordingtocollection
+frequency. Most of the term weights are contained in
+themostfrequentterms.
+3 Random Indexing
+RandomIndexing(RI)[18]isan efficient,scalable and
+incremental approach to the word space model. Word
+spacemodelsusethedistributionoftermstocreatehigh
+dimensionaldocumentvectors. The directionsof these
+documentvectorsrepresentvarioussemanticmeanings
+andcontexts.
+Latent Semantic Analysis (LSA) [7] is a popular
+word space model. LSA creates context vectors from
+adocumenttermoccurrencematrixbyperformingSin-
+gular Value Decomposition (SVD). Dimensionality re-
+ductionisachievedthroughprojectionofthedocument
+term occurrence vectors onto the subspace spanned by
+the vectorswith the largest Eigen valuesin the decom-
+position. This projection is optimal in the sense that it
+minimisesthevariancebetweentheoriginalmatrixand
+theprojectedmatrix. Incontrast,RandomIndexingfirst
+createsrandomcontextvectorsoflowerdimensionality,
+and then combines them to create a term occurrence
+matrix in the dimensionally reduced space. Each termnode vector child link k-means performed on enclosed vectors
+the dashed parts represent the nearest neighbour searchlevel 1
+level 2
+level 3root node
+nodes above the leaves contain codebook vectors
+leaf nodes contain the data vectorsnodes above the codebook level are clusters of clusters
+Figure4: K-treeConstruction
+in the collection is assigned a random vector, and the
+document term occurrence vector is then a superposi-
+tion of all the term randomvectors. There is no matrix
+decompositionandhencetheprocessisefficient.
+The RI process is conceptually very different from
+LSA and doesnot have the same optimalityproperties.
+The context vectors used by RI should optimally be
+orthogonal. Nearlyorthogonalvectorscan be used and
+have been found to perform similarly [4]. These vec-
+torscanbedrawnfromarandomGaussiandistribution.
+TheJohnsonandLinden-Strausslemma[11]statesthat
+if points are projected into a randomly selected sub-
+space of sufficiently high dimensionality, then the dis-
+tancesbetweenthepointsareapproximatelypreserved.
+Thesametopologythatexistsinthehigherdimensional
+spaceisreflectedinthelowerdimensionalrandomlyse-
+lectedsubspace. Consequently,RI offerslowcomplex-
+ity dimensionality reduction while still preserving the
+topologicalrelationshipsamongstdocumentvectors.
+3.1 Random Indexing Definition
+In RI, each dimension in the original space is given a
+randomly generated index vector. The index vectors
+are high dimensional, sparse and ternary. Sparsity is
+controlledviaaseedlengththatspecifiesthenumberof
+randomly selected non-zero dimensions. Ternary vec-
+torsconsistofrandomlydistributed+1and-1valuesin
+thenon-zerodimensions.
+In the context of document clustering, RI can be
+viewed as a matrix multiplication of a document bytermmatrix Dandatermbyindex-vectormatrix I. Al-
+ternatively, Ican be referredto as a randomprojection
+matrix. Each row vector in Drepresents a document,
+eachrowvectorin Iisanindexvector, nisthenumber
+of documents, tis the number of terms and ris the
+dimensionalityof thereducedspaced. Risthe reduced
+matrixwhereeachrowvectorrepresentsadocument.
+Dn×tIt×r=Rn×r (4)
+RI has several advantages. It can be performed in-
+crementallyand on-line as data arrives. Any document
+can be indexed (i.e. encoded as an RI vector) inde-
+pendently from all other documents in the collection.
+This eliminates the need to build and store the entire
+documentbytermmatrix. Additionally,newlyencoun-
+tereddimensions(terms)inthedocumentcollectionare
+easily accommodatedwithouthavingto recalculatethe
+projection of previously encoded documents. In con-
+trast, SVD requires global analysis where the number
+ofdocumentsandtermsarefixed. Thetimecomplexity
+ofRIisalsoveryattractive. Itislinearinthenumberof
+termsinadocumentandindependentofcollectionsize.
+3.2 ChoiceofIndex Vectors
+The index vectors used in RI were chosen to be
+sparse and ternary. Ternary index vectors for RI were
+introduced by Achlioptas [2] as being well suited for
+database environments. The primaryconcernof sparse
+index vectors is reducing time and space complexity.
+Bingham and Mannila [4] run experiments indicatingFigure5: RandomIndexingExample
+that sparse index vectors do not affect the quality of
+results. This is not the only choice when creating
+index vectors. Kanerva [12] introduces binary spatter
+codes. Plate [15] explores Holographic Reduced
+Representations that consist of dense vectors with
+floatingpointvalues.
+3.3 Random Indexing Example
+In practice, to construct a document vector, the docu-
+ment vector is initially set to zero, and then the sparse
+index vector for each term in the document is added
+to the document vector. The weight of the added term
+index vector may be determined by TF-IDF or another
+weighting scheme. When all terms have been added,
+thedocumentvectorisnormalisedtounitlength. There
+isnoneedtoexplicitlyformtherandomprojectionma-
+trix in Equation 4 up-front. The random index vectors
+for each term can be generated and stored as they are
+first encountered. The fact that each index vector is
+sparse means that the vectors use less memory to store
+andarefastertoadd.
+The effect of this approach is that each document
+will have a particular signature that can be compared
+with other documents via cosine similarity. The docu-
+mentsignatureisthusavectorontheunithyper-sphere.
+In the simple scenario in Figure 5 the indexvectors
+forthefourwordstravel,mars,spaceandtelescope,are
+added to the document vector as they are encountered
+in the text of the document. Afterwards, the document
+shouldbenormalised.
+Thesparseindexvectorscanbeefficientlystoredby
+simplystoringthe positionofthe non-zeroentrieswith
+the sign of the position indicating whether it is one or
+negativeone.
+3.4 Random Indexing K-tree
+The time complexity of K-tree depends on the length
+of the document vectors. K-tree insertion incurs two
+costs,findingtheappropriateleafnodeforinsertionand
+k-means invocation during node splits. It is therefore
+desirabletooperatewithlowerdimensionalvectorrep-
+resentation.
+The combination of RI with K-tree is a good fit.
+Both algorithms operate in an on-line and incremental
+mode. This allows it to track the distribution of data
+as it arrives and changes over time. K-tree insertions
+and deletions allow flexibility when tracking data in
+volatile and changing collections. Furthermore, K-treeperforms best with dense vectors, such as those pro-
+ducedbyRI.
+4 Document Representation
+The INEX 2008 XML Mining collection was used to
+complete the experiments. It contains 114,366 docu-
+ments that are a subset of the XML Wikipedia corpus
+[8]. 15differentcategorieswereprovidedforthedocu-
+ments.
+Document content was represented with BM25
+[17]. Stop words were removed and the remaining
+terms were stemmed using the Porter algorithm [16].
+BM25 is determined by term distributions within each
+document and the entire collection. BM25 works
+with similar concepts as TF-IDF except that is has
+two tuning parameters. The BM25 tuning parameters
+were set to the same values as used for TREC [17],
+K1 = 2andb= 0.75.K1influencestheeffectofterm
+frequencyand binfluencesdocumentlength.
+Links were represented using LF-IDF [5]. This re-
+sulted in a document-to-documentlink matrix. If there
+is a link between documents iandjthen a value of
+one is added to position i,jandj,iin the matrix. If
+two documents both link to each other a value of two
+isrecordedin their respectivevectors. Eachrowvector
+of the matrix represents a documentas a vector of link
+frequenciesto andfromotherdocuments.
+The motivation behind this representation is that
+documents with similar content will link to similar
+documents. For example, in the current Wikipedia
+both car manufacturers BMW and Jaguar link to the
+AutomotiveIndustrydocument. Linkfrequencieswere
+weighted with the same Inverse Document Frequency
+heuristic from TF-IDF. The idea is to decrease the
+weight of highly frequent links and increase the
+weight of less frequent links. Links to year documents
+in the Wikipedia are examples of “stop links” that
+are weighted down by this heuristic. Unlike term
+frequencies in TF-IDF the link frequencies in LF-IDF
+are not normalised. De Vries and Geva [5] found that
+normalising link frequencies decreased classification
+performance.
+When document and link representations are com-
+bined they are both converted to unit vectors and con-
+catenated. Converting each representation to unit vec-
+tors ensures that the weights of one representation do
+not dominate the other. De Vries and Geva [5] found
+thistobeeffectiveforclassification.
+5 Experimental Setup
+Experiments have been run to measure the quality dif-
+ference between variousconfigurationsof K-tree. Sec-
+tion2.3describesthemodificationsmadetoK-tree. Ta-
+ble1lists all theconfigurationstested.
+The following conditions were used when running
+theexperiments.1. Each K-tree configuration was run a total of 20
+times.
+2. Thedocumentswereinsertedinadifferentrandom
+ordereachtimeK-treeisbuilt.
+3. If RI was used, the index vectors were generated
+statistically independently each time K-tree was
+built.
+4. For each K-tree built, k-means++ [3] was run 20
+times on the codebook vectors to create 15 clus-
+ters.
+5. Alldocumentvectorswereunitisedafterperform-
+ingdimensionalityreduction.
+The conditions listed above resulted in 400 mea-
+surements for each K-tree configuration. For each of
+the 20 K-treesbuilt, k-means++was run20 times. The
+repetitionoftheexperimentsistomeasurethevariance
+caused by the random insertion order into K-tree, the
+randomised seeding process in k-means in the modi-
+fied K-tree and the randomised seeding process of k-
+means++.
+Assessment of clustering quality is based on
+the INEX XML Mining track. The set of 114,366
+documents, belonging to 15 classes were used to
+evaluate clustering quality of INEX submissions. The
+cluster labels are taken from the Wikipedia itself.
+K-tree generates clusters in an unsupervised manner,
+and it is not necessarily going to produce 15 clusters
+at a particular level in the tree. In order to re-use
+the INEX test collection, it was necessary to post
+process the K-tree and to reduce a cluster level in the
+tree to 15 clusters by using k-means++. Note that
+this is a low cost operation involving only a small
+numberofvectors,whichisnotrequiredinanordinary
+application. Itisdoneforthesolepurposeofproducing
+comparable results with the INEX benchmark data.
+The same approach was taken at INEX 2008 by De
+Vries and Geva [5]. For a comparison of entropy and
+purity to be meaningful they have to be measured on
+thesame numberofclusters.
+Micro averaged purity and entropy are compared.
+Micro averaging weights the score of a cluster by its
+size. Purity and entropy are calculated by comparing
+the clustering solution to the labels provided. A higher
+purityscore indicatesa higherquality solutionbecause
+the clusters are more pure with respect to the ground
+truth. A lower entropy score indicates a higher quality
+solutionbecausethereismoreorderwith respecttothe
+groundtruth.
+6 Experimental Results
+Tables 3 to 7 contain results for the K-tree configura-
+tions tested listed in Table 1. Table 2 lists the meaning
+of the symbols used. Figures 6 and 7 are graphical
+representationsoftheaveragemicropurityandentropy.The unmodified K-tree using TF-IDF culling and
+BM25 had unexpected results as seen in Table 3. The
+averagemicropurityandentropypeakedat400dimen-
+sions. Performingthisdimensionalityreductionatthese
+lowerdimensionshadnot beenperformedbefore. This
+isaninterestingandunexpectedresultandfutureexper-
+imentswillneedtodetermineifthephenomenonoccurs
+indifferentcorpora.
+Improvementsin micro purity have been tested for
+significancevia t-tests. Thenullhypothesisisthat both
+results come from the same distribution with the same
+mean. In this case they are not significantly different.
+If the null hypothesis is rejected then the difference is
+statistically significant.
+The modifications made to K-tree for use with RI
+had a significant impact. The unmodified K-tree and
+modified K-tree were compared. Specifically, config-
+urations B and D, and configurations C and E were
+tested against each other. All dimensions were com-
+paredagainsteachother. Theimprovedperformanceof
+the modified K-tree was statistically significant for all
+dimensions (100 vs 100, 200 vs 200 and so on) with a
+p-valueof0 orextremelycloseto 0( p <1×10−100).
+The modified K-tree using RI was tested with two
+representations. Configurations D and E were tested
+at all dimensions. The null hypothesis was rejected
+at all dimensions except 10000. This means that
+BM25 performed significantly better than the BM25
++ LF-IDF representation at all dimensions except
+10000. At 10000 dimensions the difference was not
+considered statistically significant with a p-value of
+0.3. The increased performance of this representation
+in classification did not apply to clustering when using
+RI. The LF-IDF representation may be interfering
+with the BM25 representation and approaches such as
+reducing the weight of LF-IDF in the RI process or
+performing RI separately on each representation and
+then concatenating the reduced vectors may improve
+performance. Running k-means on the full sparse
+vectors will also indicate if RI is responsible for this.
+Further experimentation is required to provide more
+evidenceforthisresult.
+The unexpected results in configuration A were
+tested against the best RI configuration,E. The highest
+average at 400 dimensions in configuration A was
+testedagainstalldimensionsinconfigurationE(400vs
+100, 400 vs 200, 400 vs 400, 400 vs 1000 and so on).
+The RI K-tree, configuration E, became statistically
+more significant at 2000 dimensions with a p-value
+of1.48×10−6and thus rejected the null hypothesis.
+For dimensions 4000 through 10000, the performance
+difference was statistically significant, with a p-value
+of0inallcases. Thus,RIK-treeimprovesresults,even
+over the unexpected high results of configuration A,
+by embedding the original 200,000 dimensional term
+spaceintoat least a2000dimensionreducedspace.ID K-tree Representation
+A Unmodified TF-IDFCulling,BM25
+B Unmodified RI, BM25+ LF-IDF
+C Unmodified RI,BM25
+D Modified RI, BM25+ LF-IDF
+E Modified RI,BM25
+Table 1: K-treeTest Configurations
+01000 2000 3000 4000 5000 6000 7000 8000 9000 100000.320.340.360.380.40.420.440.460.480.5
+dimensionsaverage micro purity
+  
+A
+B
+C
+D
+E
+Figure6: PurityVersusDimensions
+01000 2000 3000 4000 5000 6000 7000 8000 9000 1000022.22.42.62.833.23.4
+dimensionsaverage micro entropy
+  
+A
+B
+C
+D
+E
+Figure7: EntropyVersusDimensions
+6.1 INEX Results
+The INEX XML Mining track is a collaborativeevalu-
+ation forum where research teams improveapproaches
+in supervised and unsupervised machine learning with
+XML documents. Participants make submissions and
+theevaluationresultsarelater released.
+The RI K-tree in configurationE performson aver-
+age at a comparable level to the best results submitted
+to the INEX 2008 XML Mining track. The top two
+results from the track had a micro purity of 0.49 and
+0.50. These are not average scores for the approaches
+butthebestresultsparticipantsfound. TheRIK-treein
+configurationEhadamaximummicroentropyof0.55.
+Thisis10%greaterthanthe INEXsubmissions.Symbol Meaning
+αAverageMicroEntropy
+βStandardDeviationof α
+γAverageMicroPurity
+δStandardDeviationof γ
+Table2: SymbolsforResults
+Dimensions α β γ δ
+100 2.6299 0.0194 0.3981 0.0067
+200 2.4018 0.0207 0.4590 0.0085
+400 2.2762 0.0263 0.4814 0.0093
+800 2.2680 0.0481 0.4768 0.0155
+1000 2.2911 0.0600 0.4703 0.0192
+2000 2.3302 0.0821 0.4569 0.0254
+4000 2.3751 0.1103 0.4401 0.0331
+8000 2.3868 0.1068 0.4402 0.0300
+10000 2.3735 0.1062 0.4431 0.0306
+Table3: A:UnmodifiedK-tree,TF-IDFCulling,BM25
+Dimensions α β γ δ
+100 3.0307 0.0149 0.3093 0.0045
+200 2.9295 0.0206 0.3300 0.0079
+400 2.7962 0.0379 0.3648 0.0143
+800 2.6781 0.0718 0.3921 0.0236
+1000 2.6509 0.0842 0.3959 0.0260
+2000 2.6315 0.1262 0.3908 0.0345
+4000 2.6380 0.1451 0.3860 0.0356
+8000 2.6371 0.1571 0.3844 0.0382
+10000 2.6302 0.1540 0.3876 0.0385
+Table 4: B: Unmodified K-tree, Random Indexing,
+BM25+LF-IDF
+Dimensions α β γ δ
+100 2.9308 0.0213 0.3337 0.0089
+200 2.7902 0.0335 0.3724 0.0126
+400 2.6151 0.0417 0.4089 0.0116
+800 2.5170 0.0703 0.4238 0.0197
+1000 2.5066 0.0858 0.4234 0.0240
+2000 2.4701 0.0938 0.4275 0.0258
+4000 2.4581 0.0979 0.4261 0.0271
+8000 2.4530 0.1139 0.4260 0.0318
+10000 2.4417 0.1019 0.4283 0.0283
+Table 5: C: Unmodified K-tree, Random Indexing,
+BM25Dimensions α β γ δ
+100 3.1527 0.0227 0.3105 0.0047
+200 3.0589 0.0266 0.3312 0.0065
+400 2.9014 0.0259 0.3726 0.0065
+800 2.6690 0.0336 0.4204 0.0085
+1000 2.5890 0.0319 0.4349 0.0090
+2000 2.3882 0.0428 0.4700 0.0129
+4000 2.2558 0.0443 0.4879 0.0144
+8000 2.1933 0.0473 0.4935 0.0162
+10000 2.1735 0.0496 0.4969 0.0171
+Table 6: D: ModifiedK-tree, RandomIndexing,BM25
++ LF-IDF
+Dimensions α β γ δ
+100 3.0717 0.0263 0.3269 0.0074
+200 2.9078 0.0291 0.3706 0.0087
+400 2.6832 0.0293 0.4191 0.0077
+800 2.4696 0.0350 0.4555 0.0106
+1000 2.4093 0.0399 0.4673 0.0115
+2000 2.2826 0.0422 0.4853 0.0137
+4000 2.2094 0.0416 0.4937 0.0141
+8000 2.1764 0.0429 0.4975 0.0149
+10000 2.1686 0.0440 0.4981 0.0161
+Table7: E:ModifiedK-tree,RandomIndexing,BM25
+7 Conclusion
+RI K-tree was introduced as an attractive approach for
+large scale document clustering. This is the first time
+RI and K-tree have been combined. The results show
+that RI K-tree improves quality of clustering results,
+evenovertheunexpectedresultsfoundwhenusingTF-
+IDF culling. Furtherexperimentsare requiredto deter-
+mine if the unexpectedeffect of TF-IDF culling at low
+dimensions is an anomaly or actually exists in many
+collections. Additionally, RI K-tree is an efficient and
+high quality approach to overcome previous problems
+with sparse representations when using K-tree. Unfor-
+tunately the combination of BM25 and LF-IDF repre-
+sentations did not improve results in clustering as they
+didinearlierclassificationresults.
+References
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+arXiv:1001.0834v1  [math.LO]  6 Jan 2010A TRICHOTOMY FOR A CLASS OF
+EQUIVALENCE RELATIONS
+LONGYUN DING
+Abstract. LetXn, n∈Nbe a sequence of non-empty sets, ψn:X2
+n→
+R+. We consider the relation E((Xn,ψn)n∈N) on/producttext
+n∈NXnby (x,y)∈
+E((Xn,ψn)n∈N)⇔/summationtext
+n∈Nψn(x(n),y(n))<+∞. IfE((Xn,ψn)n∈N) is a
+Borel equivalence relation, we show a trichotomy that eithe rRN/ℓ1≤B
+E,E1≤BE, orE≤BE0.
+We also prove that, for a rather general case, E((Xn,ψn)n∈N) is an
+equivalence relation iff it is an ℓp-like equivalence relation.
+1.Introduction
+Atopological spaceiscalleda Polish space ifitisseparableandcompletely
+metrizable. Let X,Ybe Polish spaces and E,Fequivalence relations on
+X,Yrespectively. A Borel reduction ofEtoFis aBorel function θ:X→Y
+such that (x,y)∈Eiff (θ(x),θ(y))∈F, for allx,y∈X. We say that Eis
+Borel reducible toF, denotedE≤BF, if there is a Borel reduction of Eto
+F. IfE≤BFandF≤BE, we say that EandFareBorel bireducible and
+denoteE∼BF. We refer to [4] and [8] for background on Borel reducibility .
+There are several famous dichotomy theorems on Borel reduci bility. The
+first one is the Silver’s dichotomy theorem [13].
+Theorem 1.1 (Silver).LetEbe aΠ1
+1equivalence relation. Then Ehas
+either at most countably many or perfectly many equivalence classes, i.e.
+E≤Bid(N)orid(R)≤BE.
+There are three dichotomy theorems concerning E0. Before introducing
+these theorems, we recall definitions of equivalence relati onsE0,E1,Eω
+0.
+(a) Forx,y∈2N, (x,y)∈E0⇔ ∃m∀n≥m(x(n) =y(n)).
+(b) Forx,y∈2N×N, (x,y)∈E1⇔ ∃m∀n≥m∀k(x(n,k) =y(n,k)).
+(c) Forx,y∈2N×N, (x,y)∈Eω
+0⇔ ∀k∃m∀n≥m(x(n,k) =y(n,k)).
+Theorem 1.2. LetEbe a Borel equivalence relation. Then
+(a) (Harrington-Kechris-Louveau [5]) eitherE≤Bid(R)orE0≤BE;
+(b) (Kechris-Louveau [9]) ifE≤BE1, thenE≤BE0orE∼BE1;
+(c) (Hjorth-Kechris [7]) ifE≤BEω
+0, thenE≤BE0orE∼BEω
+0.
+Date: November 18, 2018.
+2000Mathematics Subject Classification. Primary 03E15, 54E35, 46A45.
+Research partially supported by the National Natural Scien ce Foundation of China
+(Grant No. 10701044).
+12 LONGYUN DING
+AnotherclassofinterestingBorelequivalencerelationsc omefromclassical
+Banach sequence spaces. Let p≥1. Forx,y∈RN, (x,y)∈RN/ℓp⇔
+x−y∈ℓp. It was shown by G. Hjorth [6] that every Borel equivalence
+relationE≤BRN/ℓ1is either essentially countable or satisfies E∼BRN/ℓ1.
+Kanovei asked whether the position of RN/ℓ1in the≤B-structure is similar
+withE1andEω
+0(see [8], Question 5.7.5).
+Question 1.3 (Kanovei) .Does every Borel equivalence relation E≤RN/ℓ1
+satisfy either E≤BE0orE∼BRN/ℓ1?
+Two kinds of ℓp-like equivalence relations were introduced by T. M´ atrai
+[11] and the author [1]. (1) Let f: [0,1]→R+. Forx,y∈[0,1]N, (x,y)∈
+Ef⇔/summationtext
+n∈Nf(|x(n)−y(n)|)<+∞. (2) Let (Xn,dn), n∈Nbe a sequence
+of metric spaces, p≥1. For (x,y)∈/producttext
+n∈NXn, (x,y)∈E((Xn,dn)n∈N;p)⇔/summationtext
+n∈Ndn(x(n),y(n))p<+∞.
+In this paper, we introduce a notion surpassing both (1) and ( 2). Let
+Xn, n∈Nbe a sequence of non-empty sets, ψn:X2
+n→R+. Forx,y∈/producttext
+n∈NXn, (x,y)∈E((Xn,ψn)n∈N)⇔/summationtext
+n∈Nψn(x(n),y(n))<+∞. Though
+wedidnotfindanaturalnecessaryandsufficientconditiontha tE((Xn,ψn)n∈N)
+be an equivalence relation, we establish the following tric hotomy.
+Theorem 1.4. IfE=E((Xn,ψn)n∈N)is a Borel equivalence relation, then
+eitherRN/ℓ1≤BE,E1≤BE, orE≤BE0.
+From this trichotomy, we can see that Kanovei’s problem is va lid within
+equivalence relations of the form E((Xn,ψn)n∈N).
+It was shown by R. Dougherty and G. Hjorth [2] that, for p,q≥1,
+RN/ℓp≤BRN/ℓqiffp≤q. We complete a different picture for 0 <p≤1 by
+showing that RN/ℓp∼BRN/ℓ1.
+Via a process of metrization, we prove that, for a rather gene ral case,
+equivalence relations E((Xn,ψn)n∈N) coincide with ℓp-like equivalence rela-
+tionsE((Xn,dn)n∈N;p).
+2.A trichotomy for sum-like equivalence relations
+We denote the set of all non-negative real numbers by R+.
+Definition 2.1. LetXn, n∈Nbe a sequence of non-empty sets, ψn:X2
+n→
+R+. We define a relation E((Xn,ψn)n∈N) on/producttext
+n∈NXnby
+(x,y)∈E((Xn,ψn)n∈N)⇐⇒/summationdisplay
+n∈Nψn(x(n),y(n))<+∞
+forx,y∈/producttext
+n∈NXn. If (Xn,ψn) = (X,ψ) for every n∈N, we write
+E(X,ψ) =E((Xn,ψn)n∈N) for the sake of brevity.
+It is hard to find a natural necessary and sufficient condition t o determine
+whenE((Xn,ψn)n∈N) is an equivalence relation. We need the following
+definition:A TRICHOTOMY FOR A CLASS OF EQUIVALENCE RELATIONS 3
+Definition 2.2. IfE((Xn,ψn)n∈N) is an equivalence relation, we call it a
+sum-like equivalence relation . Furthermore, if Xn, n∈Nis a sequence of
+Polish spaces and every ψnis Borel function, it is called a sum-like Borel
+equivalence relation .
+The following easy lemma is very useful for the study of sum-l ike equiva-
+lence relations.
+Lemma 2.3. LetE=E((Xn,ψn)n∈N)be a sum-like equivalence relation.
+Forx,y∈/producttext
+n∈NXn, we have
+(x,y)∈E⇐⇒/summationdisplay
+x(n)/negationslash=y(n)ψn(x(n),y(n))<+∞.
+Proof.SinceEis an equivalence relation, ( x,x)∈E. It follows that
+/summationdisplay
+x(n)=y(n)ψn(x(n),y(n))≤/summationdisplay
+n∈Nψn(x(n),x(n))<+∞.
+Then the lemma follows. /square
+Definition 2.4. Let (Xn,dn), n∈Nbe a sequence of pseudo-metric spaces.
+Forp≥1, anℓp-like equivalence relation E((Xn,dn)n∈N;p)isE((Xn,ψn)n∈N)
+whereψn(u,v) =dn(u,v)pforu,v∈Xn. If (Xn,dn) = (X,d) for every
+n∈N, we writeE(X,d;p) =E((Xn,dn)n∈N;p) for the sake of brevity.
+Proposition 2.5. LetE((Xn,dn)n∈N;p)be anℓp-like equivalence relation.
+Then there exists metric d′
+nonXnfor eachnsuch thatE((Xn,d′
+n)n∈N;p) =
+E((Xn,dn)n∈N;p).
+Proof.We can see that following d′
+n’s meet the requirements.
+d′
+n(u,v) =
+
+0, u =v,
+2−n, u /ne}ationslash=v,dn(u,v)≤2−n,
+dn(u,v), dn(u,v)>2−n,
+foru,v∈Xn. /square
+Proposition 2.6. LetEi, i∈Nbe a sequence of Borel equivalence relations.
+Then for every p≥1there is an ℓp-like Borel equivalence relation E=
+E((Xn,dn)n∈N;p)such thatEi≤BEforn∈N.
+Proof.Fori∈N, letEibe a Borel equivalence relation on a Polish space
+Yi. We define χi:Y2
+i→R+by
+χi(u,v) =/braceleftbigg
+0,(u,v)∈Ei,
+1,(u,v)/∈Ei.
+Now fix a bijection /an}bracketle{t·,·/an}bracketri}ht:N2→N. For every i,j∈N, ifn=/an}bracketle{ti,j/an}bracketri}ht, we denote
+Xn=Yianddn=χi. It is easy to verify that E=E((Xn,dn)n∈N;p) is an
+ℓp-like Borel equivalence relation. For i∈N, defineθi:Yi→/producttext
+n∈NXnby
+θi(u)(/an}bracketle{tk,j/an}bracketri}ht) =/braceleftbigg
+u, k=i,
+ak, k/ne}ationslash=i,4 LONGYUN DING
+whereak∈Ykis independent of u. Clearlyθiis a Borel reduction of Eito
+E. /square
+Now suppose that E=E((Xn,ψn)n∈N) is a sum-like Borel equivalence
+relation. Its position in the ≤B-structure of Borel equivalence relations is
+determined by the following condition:
+(ℓ1)∀c >0∃x,y∈/producttext
+n∈NXnsuch that ∃m∀n > m(ψn(x(n),y(n))< c)
+and/summationdisplay
+n∈Nψn(x(n),y(n)) = +∞.
+Lemma 2.7. If(ℓ1)holds, then RN/ℓ1≤BE.
+Proof.Firstly, we show a well known fact: RN/ℓ1≤B[0,1]N/ℓ1. Fix a
+bijection /an}bracketle{t·,·/an}bracketri}ht′:N×Z→N. Forz∈RN, we define θ′(z)∈[0,1]Nas
+θ′(z)(/an}bracketle{tm,k/an}bracketri}ht′) =
+
+0, z (m)<k,
+z(m)−k, k≤z(m)<k+1,
+1, k +1≤z(m).
+Thenθ′witness that RN/ℓ1≤B[0,1]N/ℓ1.
+Secondly, we construct a reduction of [0 ,1]N/ℓ1toE.
+Forl∈N, by condition ( ℓ1), there exist xl,yl∈/producttext
+n∈NXnsuch that
+∃m∀n>m(ψn(xl(n),yl(n))<2−l) and/summationtext
+n∈Nψn(xl(n),yl(n)) = +∞. Then
+wecan select two sequences ofnatural numbers( il)l∈N,(jl)l∈Nsatisfyingthat
+(i)il<jl<il+1forl∈N;
+(ii)ψn(xl(n),yl(n))<2−lforil≤n≤jl;
+(iii) 1≤/summationtext
+il≤n≤jlψn(xl(n),yl(n))<1+2−l.
+Fix an element an∈Xnfor everyn∈N. Forz∈[0,1]N, we define
+ϑ(z)∈/producttext
+n∈NXnby
+ϑ(z)(n) =
+
+xl(n), il≤n≤jl,/summationtext
+il≤m≤nψm(xl(m),yl(m))≤z(l);
+yl(n), il≤n≤jl,/summationtext
+il≤m≤nψm(xl(m),yl(m))>z(l);
+an,otherwise.
+Note that, for z,w∈[0,1]Nandl∈N, we have
+|z(l)−w(l)|−2−l</summationdisplay
+il≤n≤jl
+ϑ(z)(n)/negationslash=ϑ(w)(n)ψn(ϑ(z)(n),ϑ(w)(n))<|z(l)−w(l)|+2−l.
+Therefore, by Lemma 2.3,
+(ϑ(z),ϑ(w))∈E⇐⇒/summationtext
+ϑ(z)(n)/negationslash=ϑ(w)(n)ψn(ϑ(z)(n),ϑ(w)(n))<+∞
+⇐⇒/summationtext
+l∈N|z(l)−w(l)|<+∞
+⇐⇒z−w∈ℓ1.
+It follows that [0 ,1]N/ℓ1≤BE. /squareA TRICHOTOMY FOR A CLASS OF EQUIVALENCE RELATIONS 5
+If (ℓ1) fails, then there exists c>0 such that
+∃m∀n>m(ψn(x(n),y(n))<c)⇒/summationdisplay
+n∈Nψn(x(n),y(n))<+∞
+forx,y∈/producttext
+n∈NXn. Now we denote
+Fn={(u,v)∈X2
+n:ψn(u,v)<c}.
+Lemma 2.8. If(ℓ1)fails, letc >0andFn, n∈Nbe defined as above.
+Then
+(1)forx,y∈/producttext
+n∈NXn,
+(x,y)∈E⇐⇒ ∃m∀n>m((x(n),y(n))∈Fn);
+(2)there exists N0such that, for n > N 0,Fnis a Borel equivalence
+relation on Xn.
+Proof.Since (ℓ1) fails, clause (1) is trivial.
+(2) Firstly, assume for contradiction that, there exist a st rictly increasing
+sequence of natural numbers ( nk)k∈Nanduk∈Xnksuch thatψnk(uk,uk)≥
+c. Select an x∈/producttext
+n∈NXnwithx(nk) =ukfork∈N. Then we have/summationtext
+n∈Nψn(x(n),x(n)) = +∞. This is impossible, since Eis an equiva-
+lence relation. So there exists N1such that, for n > N 1,u∈Xn, we have
+ψn(u,u)<c, i.e. (u,u)∈Fn.
+Secondly, assume for contradiction that, there exist a stri ctly increasing
+sequenceofnaturalnumbers( nk)k∈Nanduk,vk∈Xnksuchthatψnk(uk,vk)<
+c,ψnk(vk,uk)≥c. Selectx,y∈/producttext
+n∈NXnsuch thatx(nk) =uk,y(nk) =vk
+fork∈Nandx(n) =y(n) for other n. We have ψn(x(n),y(n))< cfor
+n > N 1. Since (ℓ1) fails,/summationtext
+n∈Nψn(x(n),y(n))<+∞, i.e. (x,y)∈E.
+Clearly/summationtext
+n∈Nψn(y(n),x(n)) = +∞, so (y,x)/∈E. A contradiction! Hence
+there exists N2such that, for n > N 2,u,v∈Xn, we haveψn(u,v)< c⇒
+ψn(v,u)<c, i.e. (u,v)∈Fn⇒(v,u)∈Fn.
+With a similar argument, we can prove that there exists N3such that,
+forn>N3,u,v,r∈Xn, we have (u,v),(v,r)∈Fn⇒(u,r)∈Fn.
+In summary, for n > N 0= max{N1,N2,N3},Fnis an equivalence rela-
+tion. Since ψnis Borel, so is Fn. /square
+Recall that E0(N) is an equivalence relation on NNsimilar toE0on 2N.
+Forx,y∈NN, (x,y)∈E0(N)⇔ ∃m∀n>m(x(n) =y(n)). It is well known
+thatE0∼BE0(N) (see Proposition 6.1.2 of [4]).
+Lemma 2.9. For any equivalence relation Eon/producttext
+n∈NXn, if there is a
+sequenceFn⊆X2
+n, n∈Nsuch that (1) and (2) of Lemma 2.8 hold, then
+eitherE1≤BE,E∼BE0, orEis trivial, i.e. all elements in/producttext
+n∈NXnare
+equivalent.
+Proof.Forn > N 0, sinceFnis a Borel equivalence relation on Xn, from
+the Silver dichotomy theorem [13], either there are at most c ountably many
+Fn-equivalence classes or there are perfectly many Fn-equivalence classes.6 LONGYUN DING
+Case 1. There exists a strictly increasing sequence of natural numb ers
+(nk)k∈Nsuch that there are perfectly many Fnk-equivalence classes. Then
+there is a continuous embedding hk: 2N→Xnkfor everyk∈Nsuch that
+(hk(z),hk(w))∈Fnkiffz=w. Defineθ: 2N×N→/producttext
+n∈NXnby
+θ(x)(n) =/braceleftbigg
+hk(x(k,·)), n=nk,
+an, otherwise,
+wherean∈Xnis independent of x. By Lemma 2.8.(1), it is straightforward
+to check that θis a reduction of E1toE.
+Case 2.There exists Nsuch that, for n>N,Fnhas only one equivalence
+class. From Lemma 2.8.(1), we see that Eis trivial.
+Case 3.If case 1 fails, then there exists N′such that, for n>N′,Fnhas
+at most countably many equivalence classes. So E≤BE0(N). If case 2 fails,
+then there exists a strictly increasing sequence of natural numbers (nk)k∈N
+such thatFnkhas more than one equivalence class. Thus E0≤BE. Since
+E0∼BE0(N), we haveE∼BE0. /square
+Now we have already completed the proof of the following tric hotomy.
+Theorem 2.10. LetE=E((Xn,ψn)n∈N)be a sum-like Borel equivalence
+relation. Then either RN/ℓ1≤BE,E1≤BE, orE≤BE0.
+Corollary 2.11. LetE=E((Xn,ψn)n∈N)be a sum-like Borel equivalence
+relation. If E≤BRN/ℓ1, then either E≤BE0orE∼BRN/ℓ1.
+Proof.It is well known that E1/ne}ationslash≤BRN/ℓ1(see [9] Theorem 4.2), so if E≤B
+RN/ℓ1, thenE1/ne}ationslash≤BE. Hence the corollary follows. /square
+It was shown by R. Dougherty and G. Hjorth [2] that, for p,q≥1,
+RN/ℓp≤BRN/ℓq⇐⇒p≤q.
+The following corollary shows that, for p≤1, the situation is different.
+Corollary 2.12. For0<p≤1, we have RN/ℓp∼BRN/ℓ1.
+Proof.Note that RN/ℓp=E(R,ψ) whereψ(u,v) =|u−v|pforu,v∈R. It
+is easy to see that ( ℓ1) holds for E(R,ψ), soRN/ℓ1≤BRN/ℓp.
+For the other direction, we claim that RN/ℓp≤BRN/ℓqfor 0<p<q≤1.
+We sketch the proof for Theorem 1.1 of [2], that RN/ℓp≤BRN/ℓqfor
+1≤p<q, and check that it is also valid for 0 <p<q≤1.
+It will suffice to prove for the case 0 <q
+2<p<q ≤1. We denote ρ=p
+q
+andr= 4−ρ. Then1
+2< ρ <1 and1
+4< r <1
+2. In [2] p. 1838, the authors
+constructed a continuous function ¯Kr:R→R2, and proved that there are
+m′,M′>0 such that
+m′|s−t|ρ≤ /bardbl¯Kr(s)−¯Kr(t)/bardbl2≤M′|s−t|ρ,
+fors,t∈[i−1,i+1],i∈Z. And/bardbl¯Kr(s)−¯Kr(t)/bardbl2≥1 ifs,tare not in the
+same interval [ i−1,i+1],i∈Z.A TRICHOTOMY FOR A CLASS OF EQUIVALENCE RELATIONS 7
+Define a mapping θ:RN→RNsuch that, for x∈RNandk∈N,
+¯Kr(x(k)) = (θ(x)(2k),θ(x)(2k+1)).
+Forw= (s,t)∈R2, denote /bardblw/bardblq= (|s|q+|t|q)1
+q. Note that
+1√
+2/bardblw/bardbl2≤ /bardblw/bardbl∞≤ /bardblw/bardblq≤21
+q/bardblw/bardbl∞≤21
+q/bardblw/bardbl2.
+Forx,y∈RN, we have
+θ(x)−θ(y)∈ℓq⇐⇒/summationtext
+k∈N/bardbl¯Kr(x(k))−¯Kr(y(k))/bardblq
+q<+∞
+⇐⇒/summationtext
+k∈N/bardbl¯Kr(x(k))−¯Kr(y(k))/bardblq
+2<+∞
+⇐⇒/summationtext
+k∈N|x(k)−y(k)|p<+∞
+⇐⇒x−y∈ℓp.
+Thus,θis a reduction of RN/ℓptoRN/ℓq. /square
+3.Metrization
+Inthissection, weshowthatasum-likeequivalencerelatio nE((Xn,ψn)n∈N)
+coincides with an ℓp-like equivalence relation if thefollowing conditions hol d.
+(m1)DenoteX=/uniontext
+n∈NXn. There is a unique function ψ:X2→R+
+such thatψn=ψ↾X2
+nandψ(u,v) = 1if noXncontains both u,v.
+(m2)For anyu,v,r∈X, ifu,v,r∈Xn, then there exists m > nsuch
+thatu,v,r∈Xm.
+Letψn= min{ψn,1}. Wecanseethat E((Xn,ψn)n∈N) =E((Xn,ψn)n∈N).
+Therefore, we may assume ψn≤1 if needed.
+Lemma 3.1. Let(Xn,ψn)n∈Nsatisfy(m1)and(m2). Ifψn≤1, then
+E((Xn,ψn)n∈N)is an equivalence relation iff the following conditions hold :
+(i)ψ(u,u) = 0foru∈X;
+(ii)there is aC≥1such that for u,v,r∈X,
+ψ(v,u)≤Cψ(u,v);ψ(u,r)≤C(ψ(u,v)+ψ(v,r)).
+Proof.If conditions (i) and (ii) hold, it is trivial that E((Xn,ψn)n∈N) is an
+equivalence relation. We only need to prove the other direct ion.
+Now assume that E((Xn,ψn)n∈N) is an equivalence relation.
+Firstly, for any u∈X, by (m2), there is an infinite set I⊆Nsuch that
+u∈Xnforn∈I. Letx∈/producttext
+n∈NXnwithx(n) =uforn∈I. Since
+(x,x)∈E((Xn,ψn)n∈N), we haveψ(u,u) = 0.
+Secondly, for u,v∈X, ifψ(u,v) = 0, we claim that ψ(v,u) = 0. By (m2),
+there is an infinite set I⊆Nsuch thatu,v∈Xnforn∈I. Therefore, for
+anyx,y∈/producttext
+n∈NXn, ifx(n) =u,y(n) =vforn∈Iandx(n) =y(n) for
+n /∈I, we have (x,y)∈E((Xn,ψn)n∈N). Hence (y,x)∈E((Xn,ψn)n∈N). It
+follows that/summationtext
+n∈Iψ(v,u) =/summationtext
+n∈Iψ(y(n),x(n))<+∞. Thusψ(v,u) = 0.
+Now assume for contradiction that, for every k∈Nthere areuk,vk∈X
+such thatψ(vk,uk)>2kψ(uk,vk)>0. Sinceψn≤1, 0<ψ(uk,vk)<2−k.
+By (m2), there are infinitely many nsuch thatuk,vk∈Xn.
+Select a finite set Ik⊆Nfor everyksatisfying that8 LONGYUN DING
+(i)uk,vk∈Xnforn∈Ik;
+(ii) 2−k≤ |Ik|ψ(uk,vk)≤2−(k−1);
+(iii) ifk1<k2, then max Ik1<minIk2.
+Now we define x,y∈/producttext
+n∈NXnby
+/braceleftbigg
+x(n) =uk,y(n) =vk, n∈Ik,k∈N,
+x(n) =y(n) =an,otherwise,
+wherean∈Xnis independent of xandy. Then we have
+/summationdisplay
+n∈Nψ(x(n),y(n)) =/summationdisplay
+k∈N|Ik|ψ(uk,vk)≤/summationdisplay
+k∈N2−(k−1)<+∞,
+so (x,y)∈E((Xn,ψn)n∈N). On the other hand, we have
+/summationdisplay
+n∈Nψ(y(n),x(n)) =/summationdisplay
+k∈N|Ik|ψ(vk,uk)>/summationdisplay
+k∈N2k|Ik|ψ(uk,vk)≥/summationdisplay
+k∈N1 = +∞,
+so (y,x)/∈E((Xn,ψn)n∈N). A Contradiction! We complete the proof of
+that there is C1≥1 such that ψ(u,v)≤C1ψ(v,u) foru,v∈X.
+With a similar argument, we can prove that there is C2≥1 such that
+ψ(u,r)≤C2(ψ(u,v) +ψ(v,r)) foru,v,r∈X. /square
+Lemma 3.2. LetE((Xn,ψn)n∈N),E((Xn,ϕn)n∈N)be two sum-like equiva-
+lence relations, both satisfying (m1)and(m2). Ifϕn≤1, then
+E((Xn,ψn)n∈N)⊆E((Xn,ϕn)n∈N)
+⇐⇒ ∃A≥1∀u,v∈X(ϕ(u,v)≤Aψ(u,v)).
+Proof.“⇐” is trivial. “ ⇒” follows similarly as the proof of Lemma 3.1. /square
+Corollary 3.3. LetE((Xn,ψn)n∈N),E((Xn,ϕn)n∈N)be two sum-like equiv-
+alence relations, both satisfying (m1)and(m2). Ifψn,ϕn≤1, then
+E((Xn,ψn)n∈N) =E((Xn,ϕn)n∈N)
+⇐⇒ ∃A≥1∀u,v∈X(A−1ψ(u,v)≤ϕ(u,v)≤Aψ(u,v)).
+Before introducing the Metrization lemma, we recall severa l basic notions
+onrelations. Let Xbea non-empty set, wedenote ∆( X) ={(u,u) :u∈X}.
+A subsetU⊆X2is called symmetricif ( u,v)∈U⇒(v,u)∈Uforu,v∈X.
+ForU,V⊆X2we defineU◦V⊆X2by
+(u,r)∈U◦V⇐⇒ ∃v((u,v)∈U,(v,r)∈V) (∀u,v,r∈X).
+Lemma 3.4 (Metrization lemma [10], p. 185) .LetUn, n∈Nbe a sequence
+of subsets of X2such that
+(i)U0=X2;
+(ii)eachUnis symmetric and ∆(X)⊆Un;
+(iii)Un+1◦Un+1◦Un+1⊆Unfor eachn.
+Then there is a pseudo-metric donXsatisfying that
+Un⊆ {(u,v) :d(u,v)<2−n} ⊆Un−1(∀n≥1).A TRICHOTOMY FOR A CLASS OF EQUIVALENCE RELATIONS 9
+Theorem 3.5. Let(Xn,ψn)n∈Nsatisfy(m1)and(m2). ThenE((Xn,ψn)n∈N)
+is an equivalence relation iff it is an ℓp-like equivalence relation.
+Proof.Assumethat E((Xn,ψn)n∈N) is an equivalence relation. Without loss
+of generality, we may assume that ψn≤1. From Lemma 3.1, there is C≥1
+such that, for u,v,r∈X,
+ψ(v,u)≤Cψ(u,v) andψ(u,r)≤C(ψ(u,v)+ψ(v,r)).
+Therefore, if ψ(u,v),ψ(v,r),ψ(r,s)<ε, then
+ψ(u,s)≤C(ψ(u,v)+C(ψ(v,r)+ψ(r,s)))<(2C2+C)ε.
+Now denote B= 2C2+C. We define U0=X2and
+Un={(u,v) :ψ(u,v)<B−n,ψ(v,u)<B−n}(n≥1).
+It follows that, for each n,Unis symmetric and Un+1◦Un+1◦Un+1⊆Un. By
+Lemma 3.1.(i), ∆( X)⊆Un. Then the metrization lemma gives a pseudo-
+metricdonXsuch thatUn⊆ {(u,v) :d(u,v)<2−n} ⊆Un−1.
+It is easy to check that, d(u,v) = 0 iffψ(u,v) = 0. Ifψ(u,v)≥B−2, then
+(u,v)/∈U2,d(u,v)≥2−3.
+Denotep= log2B≥1. If 0< ψ(u,v)< B−2, assume that B−(n+1)≤
+ψ(u,v)<B−nfor somen≥2. Thenψ(v,u)<CB−n<B−(n−1). It follows
+that (u,v)∈Un−1,(u,v)/∈Un+1, 2−(n+2)≤d(u,v)<2−(n−1). So
+B−2d(u,v)p<B−2(2−(n−1))p≤ψ(u,v)<B2(2−(n+2))p≤B2d(u,v)p.
+Therefore, we have E((Xn,ψn)n∈N) =E((Xn,d↾Xn);p). /square
+Corollary 3.6. Let(Xn,ψn)n∈Nsatisfy that, for n < m,Xn⊆Xmand
+ψn=ψm↾X2
+n. ThenE((Xn,ψn)n∈N)is an equivalence relation iff it is an
+ℓp-like equivalence relation.
+In particular, E(X,ψ)is an equivalence relation iff there are pseudo-
+metricdonXandp≥1such thatE(X,ψ) =E(X,d;p).
+4.Further remarks
+Letusconsideraspecialcaseofsum-likeequivalencerelat ion. Letψf(u,v) =
+f(|u−v|) wheref:R+→R+. Then conditions (i) and (ii) for E(R,ψf) in
+Lemma 3.1 read as (see also [11], Proposition 2).
+(i)f(0) = 0;
+(ii)there is aC≥1such that for s,t∈R+,
+f(s+t)≤C(f(s)+f(t)), f(s)≤C(f(s+t)+f(t)).
+DenoteNf={x∈RN:/summationtext
+n∈Nf(|x(n)|)<+∞}. ThenE(R,ψf) is an
+equivalence relation iff Nfis a subgroup of ( RN,+). Furthermore, another
+interesting problem is to determine when Nfis a linear subspace of RN.
+This problem was studied by S. Mazur and W. Orlicz (see [12], 1 .7). It was
+also considered in [12] that when Nf’s are Banach spaces.10 LONGYUN DING
+Theorem 4.1 (Mazur-Orlicz) .Letf:R+→R+satisfy that, as n→ ∞,
+f(tn)→0ifftn→0. The necessary and sufficient condition for Nfto be a
+linear space is that
+(a)there exist constants C >0,ε>0such thatf(s+t)≤C(f(s)+f(t))
+fors,t<ε;
+(b)for everyρ >0there are constants D >0,δ >0such thatf(s)≤
+Df(t)fort<δ,s<ρt .
+Note that for tn≥min{ε,δ}>0, n∈N, we havef(tn)/ne}ationslash→0. Thus there
+isc>0 such that f(t)≥cfort≥min{ε,δ}. If we assume that f≤1, then
+conditions (a) and (b) in this theorem turn to
+(a)’there exists a constant C′>0such thatf(2s)≤C′f(s)fors∈R+;
+(b)’there exists a constant D′>0such thatf(s)≤D′f(t)fors<t.
+For almost all knownexamples of sum-likeequivalence relat ionsE(R,ψf),
+Nf’s are linear spaces. In the end, we present an example in whic hNfis
+not linear as follows.
+Example 4.2.Letg:R+→R+beanincreasingfunctionwith g(0) = 0, such
+thatg′(t) is decreasing with lim t→0g′(t) = +∞. For example, g(x) =√x
+is such a function. Let ( an)n∈Nbe a strictly decreasing sequence of positive
+numbers with lim n→∞an= 0.
+Denotekn=g(an)
+an. Thenkn< kn+1. We consider equations y=knx
+andy−g(an+1) =−kn+1(x−an+1). Their solution is ( bn,knbn) where
+bn=2kn+1an+1
+kn+kn+1. We can see that an+1<bn<an.
+Now define f:R+→R+by
+f(t) =
+
+0, t = 0,
+−kn+1(t−an+1)+g(an+1), an+1≤t<bn,
+knt, b n≤t<an,
+g(a0), a 0≤t.
+It is easy to check that, for s,t∈R+,
+f(s+t)≤f(s)+f(t), f(s)≤f(s+t)+f(t).
+Note thatf(an+1)
+f(bn)=1
+2/parenleftBig
+1+kn+1
+kn/parenrightBig
+. Since lim t→0g(t)
+t= limt→0g′(t) = +∞,
+we can find a sequence ( an)n∈Nsuch thatkn+1
+kn→+∞,f(an+1)
+f(bn)→+∞as
+n→ ∞. Then condition (b) in Theorem 4.1 fails.
+References
+[1] L. Ding, Borel reducibility and H¨ older( α) embeddability between Banach spaces ,
+preprint, available at http://arxiv.org/abs/0912.1912 .
+[2] R. Dougherty and G. Hjorth, Reducibility and nonreducibility between ℓpequivalence
+relations , Trans. Amer. Math. Soc. 351 (1999) 1835–1844.
+[3] I. Farah, Basis problem for turbulent actions II: c0-equalities , Proc. London Math.
+Soc. 82 (2001), no. 3, 1–30.
+[4] S. Gao, Invariant Descriptive Set Theory, Monographs an d Textbooks in Pure and
+Applied Mathematics, vol. 293, CRC Press, 2008.A TRICHOTOMY FOR A CLASS OF EQUIVALENCE RELATIONS 11
+[5] L. A. Harrington, A. S. Kechris, and A. Louveau, A Glimm-Effros dichotomy for
+Borel equivalence relations , J. Amer. Math. Soc. 3 (1990), no. 4, 903–928.
+[6] G. Hjorth, Actions by the classical Banach spaces , J. Symbolic Logic 65 (2000), no.
+1, 392–420.
+[7] G. Hjorth and A. S. Kechris, New dichotomies for Borel equivalence relations , Bull.
+Symbolic Logic 3 (1997), no. 3, 329–346.
+[8] V. Kanovei, Borel Equivalence Relations: Structure and Classification, University
+Lecture Series, vol. 44, A. M. S., 2008.
+[9] A. S. Kechris and A. Louveau, The classification of hypersmooth Borel equivalence
+relations , J. Amer. Math. Soc. 10 (1997), no. 1, 215–242.
+[10] J. L. Kelley, General Topology, D. Van Nostrand Company , Inc., 1955.
+[11] T. M´ atrai, Onℓp-like equivalence relations , Real Anal. Exchange 34 (2008/09), no.
+2, 377–412.
+[12] S. Mazur and W. Orlicz, On some classes of linear spaces , Studia Math., 17 (1958),
+97–119.
+[13] J. H. Silver, Counting the number of equivalence classes of Borel and coan alytic equiv-
+alence relations , Ann. Math. Logic 18 (1980), no. 1, 1–28.
+School of Mathematical Sciences and LPMC, Nankai Universit y, Tianjin,
+300071, P.R.China
+E-mail address :dinglongyun@gmail.com
\ No newline at end of file
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+ 1  
+ 
+On the application of Canonical Perturbation Theory  
+up to the dissociation threshold 
+ 
+ 
+ 
+ 
+Sahin BUYUKDAGLI and Marc JOYEUX (a)  
+Laboratoire de Spectrométrie Physique (CNRS UMR 558 8), 
+Université Joseph Fourier Grenoble 1, 
+BP 87, 38402 St Martin d'Hères Cedex, France  
+ 
+ 
+ 
+Submitted to :  Chemical Physics Letters 
+ 
+ 
+Abstract : We investigate a model system consisting of a Morse  oscillator strongly coupled 
+to a doubly-degenerate bending degree of freedom an d show that Canonical Perturbation 
+Theory is able to provide a fairly precise, though not exact, approximation of the Hamiltonian 
+up to the dissociation threshold. Quantum mechanica l results and classical ones are discussed 
+in this Letter.  
+ 
+(a) email : Marc.JOYEUX@ujf-grenoble.fr  2  We have recently shown how canonical perturbation theory (CPT) can be adapted to 
+investigate the highly excited dynamics of semi-rig id molecules, floppy molecules (i.e. 
+molecules with several equilibrium positions), and non-adiabatic systems (i.e. molecules with 
+several crossing diabatic electronic surfaces) : Fo r a recent review, see for example Ref. [1] 
+and references therein. Whether based on classical mechanics, like the Birkhoff-Gustavson 
+method [2] and the Lie transform one [3], or quantu m mechanics, like the Van-Vleck 
+procedure [4], the purpose of CPT is always to rewr ite the Hamiltonian of the investigated 
+system in terms of as complete as possible a set of  classical constants of the motion or good 
+quantum numbers. The transformed Hamiltonian, whose  dynamics can usually be analyzed in 
+great detail thanks to the conserved quantities, th en provides the keys for understanding the 
+principal properties of the initial system. For exa mple, we have used CPT to characterize the 
+bifurcations of semi-rigid molecules and the isomer ization dynamics of floppy molecules 
+[5,6], to detect monodromy, a rather newly describe d mathematical concept [7], in floppy 
+molecules [8,9] as well as in CO 2 [10], to understand the effect of the X 2A1-A2B2 conical 
+intersection on the vibronic spectrum of NO 2 and imagine an efficient method to adjust the 
+coefficients of an effective Hamiltonian which desc ribes this intersection [11], and to define 
+meaningful local modes for the study of intramolecu lar vibrational energy redistribution in 
+floppy molecules like H 2O2 [12]. The purpose of the present Letter is to answ er the following 
+question : Up to what energy is CPT expected to des cribe correctly a molecule with a 
+dissociation threshold ? Indeed, it is well known, that a Morse oscillator 
+2)( 2
+21) 1 (0RR
+RMeDpH−−−+ =α is formally equivalent, up to the dissociation thr eshold, to 
+an anharmonic oscillator 2xI IH+=ω, where I is the action integral of the harmonic 
+oscillator ( )(21+=nIh in the semiclassical limit), 2/ 12)/2 (MDαω=  its fundamental 
+frequency, and )4/( 2D xω−=  the first anharmonicity. This result is obtained w hen solving  3 analytically Schrödinger's equation [13], but also when performing second order CPT [14] 
+(higher orders lead to null corrections). A Morse o scillator is therefore perfectly well 
+described by CPT up to the threshold. But what abou t a Morse oscillator coupled to another 
+degree of freedom ? To our knowledge, this point ha s not been studied in detail and it is the 
+goal of this Letter to provide an indication of wha t can be expected with this respect. More 
+precisely, we investigated a model system consistin g of a Morse oscillator strongly coupled to 
+a doubly-degenerate bending degree of freedom and s howed that CPT is still able to provide a 
+fairly precise, though not exact, approximation of the Hamiltonian up to the dissociation 
+threshold. Quantum mechanical results and classical  ones will successively be briefly 
+discussed in the remainder of this article. 
+ The Hamiltonian we studied describes the vibration s of a linear triatomic molecule 
+expressed in Jacobi coordinates, with the non-disso ciating stretching degree of freedom 
+frozen at equilibrium. It writes 
+)( 2)(
+22
+2
+2
+022 0 0)cos 1 () 1 ()
+sin )( 11(21
+21 RR RR
+Re C eDp
+p
+mr MR pMH−− −−−+ −+ + + + =α α ϕ
+γγ
+γ
+            (1) 
+where R is the Morse oscillator coordinate and Rp its conjugate momentum, γ the Jacobi 
+bending angle and γp its conjugate momentum, ϕp the vibrational angular momentum 
+arising from the degeneracy of the bending motion, and 2MR  and 2
+0mr  the moments of 
+inertia associated with the Morse oscillator and th e frozen stretch degree of freedom, 
+respectively. The last term in the right-hand side of Eq. (1) describes a realistic stretch-bend 
+interaction potential, in the sense that the bend b ehaves like an harmonic oscillator close to 
+the bottom ( R,γ)=( 0R,0) of the potential energy surface (PES), while th e potential no longer 
+depends on γ at very large R stretches (cf. Fig. 1). Moreover, the geometry of the PES 
+depends crucially on the relative values of C and D : Indeed, if DC≤, there exists a saddle at  4 πγ=, which is located below the dissociation threshold , while there is no such saddle in the 
+case where DC> (cf. Fig. 1). Numerical values used throughout thi s work are α=2.5 Å -1, 
+0R=2.5 Å, 2
+0mr =1.6948 10 -27  kg.Å 2, 2
+0MR =8.0370 10 -27  kg.Å 2, D=15000 cm -1, and C=5625 
+cm -1 or C=18750 cm -1. These values result in a stretch fundamental freq uency of about 2700 
+cm -1 and a bend frequency of about 450 cm -1 ( C=5625 cm -1) or 830 cm -1 ( C=18750 cm -1). 
+ Exact eigenvalues and eigenvectors of the Hamilton ian of Eq. (1) were first obtained 
+by expanding H up to very high order (300) with respect to the R coordinate in the 
+neighbourhood of R=0R and diagonalizing the Hamiltonian matrix of size 3 000*3000 built in 
+the direct product basis of the rotating Morse osci llator and the free rotor. 98 rotationless 
+(0==lJ ) bound states with up to 40 quanta of excitation i n the bend degree of freedom and 
+8 quanta in the stretch were obtained for C=5625 cm -1, and 48 rotationless bound states with 
+up to 20 quanta of excitation in the bend and 7 qua nta in the stretch for C=18750 cm -1. All 
+states are converged to better than 0.01 cm -1. The list of assigned states can be obtained by 
+sending an email to the authors. 
+ CPT calculations were then performed, in order to compare exact and perturbative 
+results. The procedure to use, as well as the form of the transformed Hamiltonian, actually 
+depend on the relative values of C and D. Indeed, if DC≤, wave functions may be 
+delocalized over the whole range πγ≤≤0 , because of the saddle at πγ=. In this case, the 
+procedure we derived for floppy molecules [1,6] mus t be used. The first step of this procedure 
+consists in expanding the Hamiltonian in the neighb ourhood of a Minimum Energy Path 
+(MEP) in Taylor series with respect to the stretch coordinate and in Fourier series with respect 
+to the bend coordinate. After transformation of the  stretch coordinates into dimensionless 
+creation and annihilation operators ),(11aa+, the expanded Hamiltonian writes  5 ∑+=
+NPMmkNPM mk
+kmMPN J aa h H
+,,,,2
+11)()(cos )( σγ       (2) 
+where the kmMPN h are real coefficients, and P is equal to 0 or 1. The operator γγσ∂∂=sin  
+arises from the non-commutativity of γcos  and 22
+2sin 1
+sin 12sin 
+ϕγγ γγγ
+∂∂
+∂∂
+∂∂− −=J . A series 
+of canonical transformations )exp( )exp( SHS H − →  is then applied, where S is chosen such 
+as to cancel the terms with mk≠ up to increasingly higher values of mk+. After a certain 
+number of transformations, called the "order" of th e perturbation series, the remaining terms 
+with mk≠, which have hopefully become very small in the inv estigated region of the phase 
+space, are finally neglected, so that one is left w ith a perturbative Hamiltonian of the form 
+∑=
+NPMkNPM k
+kMPN J nk H
+,,,2
+1)()(cos σγ        (3) 
+where the kMPN k are real coefficients and 111aan+= denotes the stretch quantum number. 
+The eigenvalues and eigenvectors of the Hamiltonian  of Eq. (3) are easily obtained from the 
+diagonalization of matrices of approximate size 60* 60 built in the basis of the free rotor. It is 
+reminded, that application of CPT leads to so-calle d "asymptotic" series, which means that the 
+eigenvalues and eigenvectors of the Hamiltonian of Eq. (3) converge towards those of the 
+exact Hamiltonian up to a certain order and then di verge again. The order at which the series 
+diverges is connected to the famous small divisor p roblem and cannot be determined a priori . 
+As can be seen in Fig. 1, there exist two MEPs in t he case where DC≤ : the first one is 
+0=γ (the horizontal axis), while the second one, shown  as a dot-dashed line in Fig. 1, can be 
+expressed as )(γMEP MEP RR= . It turns out that, for C=5625 cm -1, use of the two MEP 
+results in very similar accuracies : expansion arou nd 0=γ leads to an absolute average error 
+between the 98 exact and perturbative eigenvalues sm aller than 10 cm -1 for all perturbation 
+orders comprised between (or equal to) 4 and 13, wit h a minimum error of 6.75 cm -1 at order  6 10, while expansion around ) (γMEP MEP RR=  leads to an average error smaller than 10 cm -1 
+for all orders comprised between 6 and 11, with a mi nimum error of 6.25 cm -1 at order 9. 
+ In contrast, the range of values, which γ can reach, is narrower and localized above the 
+minimum of the PES in the case where DC>, so that one can use the usual procedure for 
+semi-rigid molecules [1,14]. Accordingly, the Hamilt onian is first expanded in Taylor series 
+around the minimum of the PES and rewritten in terms  of the ladder operators associated with 
+dimensionless normal coordinates. The series of can onical transformations 
+)exp( )exp( SHS H − →  is then applied in order to keep in the perturbati ve Hamiltonian only 
+the terms which depend on the stretch quantum numbe r 111aan+= and the bend quantum 
+number yyxxaaaan22222+ ++ = , where 2 x and 2 y denote the two components of the 
+degenerate bending motion. The obtained polynomes i n 1n and 2n are called Dunham 
+expansions (see Eq. (4) below). For C=18750 cm -1, the convergence of the Dunham 
+asymptotic series is however exceedingly slow. Examin ation of the differences between exact 
+and perturbative energies shows that this slow conver gence is due to a non-negligible 1:2 
+Fermi resonance between the two degrees of freedom. W hen taking this resonance into 
+account in the choice of the successive operators S, the perturbative Hamiltonian is obtained 
+in the form 
+c.c. ) )( (2
+22
+2
+,21 1
+,21++ + = ∑ ∑+
+yx
+mkmk
+km 
+mkmk
+km aannfannhH      (4) 
+where the km h and km f are real coefficients. The first term in the right -hand side of Eq. (4) is 
+the polynomial Dunham expansion and the second one t he Fermi resonance. Again, the 
+eigenvalues and eigenvectors of the Hamiltonian of E q. (4) are very easily obtained from the 
+diagonalization of small matrices of size less than  11*11 built in the direct product basis of 
+the non-degenerate and doubly-degenerate harmonic o scillators. The average absolute error  7 between the exact and perturbative energies of the 4 8 bound states is 12.8 cm -1 at 10th order 
+of CPT, but it increases again rapidly at orders la rger than 13. To illustrate more clearly the 
+agreement between exact and perturbative results, th e wave functions for 3 states are shown in 
+Fig. 2. These states (#42, #43 and #47) have respec tive energies 14578, 14667 and 14929 cm -1 
+above the minimum of the PES, that is, they are loc ated close to the dissociation threshold at 
+15000 cm -1. Comparison of the bottom row (exact calculations) and the top one (perturbative 
+results) shows that the wave functions are indeed ver y similar. 
+ Conclusion of the first, quantum mechanical part o f this study is therefore that, when 
+the geometry of the PES is not too complex, CPT is able to provide approximations of 
+dissociating systems with several degrees of freedo m, which are simple and nevertheless 
+precise up to the threshold. In the remainder of th is Letter, we next present a few results of the 
+classical analysis, which give a clear illustration  of the simplifications brought to the 
+investigated system by the CPT procedure. 
+ The left column of Fig. 3 shows the ) ,(RpR Poincaré surfaces of section (SOS) at γ=0 
+and the ),(γγp SOS at R=2.5 Å for the exact Hamiltonian of Eq. (1) with C=5625 cm -1 at an 
+energy E=13000 cm -1 above the minimum of the PES, that is, 2000 cm -1 below the threshold. 
+The ),(RpR SOS displays the usual elongated triangular form of  dissociating coordinates, 
+while the ),(γγp SOS looks essentially like that of the pendulum. Howe ver, at this energy, 
+many tori have already split into resonance islands  and a thin macroscopic chaotic region has 
+appeared around the separatrix between rotation-like  and vibration-like motions, the size of 
+which increases rapidly when energy approaches the d issociation limit. The right column of 
+Fig. 3 shows the corresponding surfaces of section a t the same energy for the transformed 
+Hamiltonian of Eq. (3), that is, the ) ,(11pq SOS at γ=0 and the ),(γγp SOS at 1q=0. The 
+transformed Hamiltonian being separable, one should not be surprised to notice that all the  8 subtleties (resonance islands, chaos) of the preced ing SOS have disappeared. Moreover, as we 
+already stated, the canonical transformation maps t he Morse oscillator on a slightly 
+anharmonic one. Therefore, the two SOS on the right- hand side of Fig. 3 can hardly be 
+distinguished from those of the harmonic oscillator  and the pendulum, respectively. 
+Nonetheless, it is emphasized that the essential geo metrical features of the original 
+Hamiltonian are preserved. For example, the periods and action integrals of the periodic orbits 
+[R], [ γ] and [SN], which organize the classical phase space and also act as backbones for the 
+quantum mechanical wave functions [5,6] (see below), agree almost perfectly for the two 
+systems. 
+ Fig. 4 displays the same information for the the e xact Hamiltonian of Eq. (1), with C 
+however equal to 18750 cm -1, and the perturbative Hamiltonian of Eq. (4). More precisely, the 
+right column of Fig. 3 shows the ) ,(11pq SOS at 2q=0 and the ) ,(22pq SOS at 1q=0. In 
+addition to the fact that γ remains localized in the well centred around γ=0, the essential 
+difference between Figs. 3 and 4 results from the pe riod-doubling bifurcation, which the [R] 
+periodic orbit (PO) undergoes around 10000 cm -1. At this bifurcation, the [R] PO becomes 
+unstable - we therefore label it [R*] above the bifu rcation's energy - while the same PO 
+covered twice, which is labelled [2R], remains stable  and progressively acquires, with 
+increasing energies, a pronounced ⊃ shape in the ( R,γ) plane. This bifurcation is due to the 1:2 
+Fermi resonance between the stretching and bending d egrees of freedom and has been 
+observed, for example, in the spectra of CO 2 and CS 2. Macroscopic chaos first appears around 
+the separatrix encompassing [R*] and develops more rapidly than in Fig. 3, because of the 
+higher value of the coupling coefficient C. It is again observed, in the right column of Fig.  4, 
+that CPT drastically simplifies the appearance of t he SOS, although it preserves the essential 
+organization of the classical phase space around th e [R*], [2R] and [ γ] POs. Let us finally 
+note that these three POs have been plotted in Fig. 2 on top of the quantum mechanical wave  9 functions for states #42, #43 and #47. One can chec k, as usual, that the POs, in addition to 
+organizing the classical phase space, also play the  role of backbones for the quantum wave 
+functions. The agreement between exact and perturbat ive quantum results suggests that the 
+quantum world is much more sensitive to the properti es of these POs than to the other details 
+(resonance islands, chaos, etc...) of the classical  mechanics. 
+ In conclusion, we have shown that CPT is able to pro vide a fairly precise, though not 
+exact, approximation of the Hamiltonian of a vibrati ng molecular system with several degrees 
+of freedom up to the dissociation threshold. The me thod probably fails for too complex PES 
+and/or too strong couplings but it is still expecte d to be very useful for studying the highly 
+excited dynamics of a large variety of molecules.  10 REFERENCES 
+ 
+[1] M. Joyeux and D. Sugny, Can. J. Phys. 80 (2002) 1459 
+[2] F. Gustavson, Astron. J. 71 (1966) 670 
+[3] A.J. Dragt and J.M. Finn, J. Math. Phys. 17 (197 6) 2215 
+[4] E.C. Kemble, The fundamental principles of quan tum mechanics, McGraw-Hill, New-
+York, 1937 
+[5] M. Joyeux, S.C. Farantos and R. Schinke, J. Phy s. A 106 (2002) 5407 
+[6] M. Joyeux, S. Yu. Grebenshchikov, J. Bredenbeck, R. Schinke and S.C. Farantos, Adv. 
+Chem. Phys. 130 (2005) 267 
+[7] J.J. Duistermaat, Commun. Pure Appl. Math. 33 (1 980) 687 
+[8] M. Joyeux, D.A. Sadovskii and J. Tennyson, Chem.  Phys. Lett. 382 (2003) 439 
+[9] K. Efstathiou, M. Joyeux and D.A. Sadovskii, Phy s. Rev. A 69 (2004) 032504 
+[10] R.H. Cushman, H.R. Dullin, A. Giacobbe, D.D. Holm, M. Joyeux, P. Lynch, D.A. 
+Sadovskii and B.I. Zhilinskii, Phys. Rev. Lett. 93 (2004) 024302 
+[11] M. Joyeux, R. Jost and M. Lombardi, J. Chem. P hys. 119 (2003) 5923 
+[12] M. Joyeux, J. Chem. Phys. 122 (2005) 074303 
+[13] S. Flügge, Practical quantum mechanics, Spring er, New York, 1974 
+[14] E.L. Sibert, J. Chem. Phys. 88 (1988) 4378 
+  11 FIGURE CAPTIONS 
+ 
+Figure 1  : Contour plots of the potential energy surface of  Eq. (1) for coupling constants 
+C=5625 cm -1 (top plot) and C=18750 cm -1 (bottom plot). Contour lines range from 1000 cm -1 
+to 16000 cm -1 above the minimum of the PES, with increments of 10 00 cm -1. Filled circles 
+indicate the positions of the minimum of the PES (o n the γ=0 axis), as well as the saddle (on 
+the γ=π axis) for C=5625 cm -1. In this later plot, the dot-dashed line shows the 
+)(γMEP MEP RR=  Minimum Energy Path between the two equilibria. The dissociation 
+threshold is located at 15000 cm -1 above the bottom of the PES and the saddle at arou nd 9140 
+cm -1. 
+ 
+Figure 2  : Bottom : Contour plots in the ( R,γ) plane of the wave functions of states #42 
+(E=14578.26 cm -1), #43 ( E=14667.04 cm -1) and #47 ( E=14928.51 cm -1) for the Hamiltonian 
+of Eq. (1) with C=18750 cm -1. The thick lines indicate the positions of the [ γ], [R*] and [2R] 
+periodic orbits at corresponding energies (the [R*]  PO lies on top of the R axis) and the dot-
+dashed lines the contours of the PES at E=5000, 10000 and 14900 cm -1. Top : Same plots, in 
+the ),(21qq plane, for the Hamiltonian of Eq. (4) obtained from  10th order CPT. 
+ 
+Figure 3  : Left column : ) ,(RpR SOS at γ=0 and ),(γγp SOS at R=2.5 Å for the 
+Hamiltonian of Eq. (1) with C=5625 cm -1 at E=13000 cm -1 (that is, 2000 cm -1 below the 
+threshold). Filled circles indicate the positions o f some periodic orbits. Right column : 
+),(11pq SOS at γ=0 and ),(γγp SOS at 1q=0 for the 8th order perturbative Hamiltonian of 
+Eq. (3) at the same energy. 
+  12 Figure 4  : Left column : ) ,(RpR SOS at γ=0 and ),(γγp SOS at R=2.5 Å for the 
+Hamiltonian of Eq. (1) with C=18750 cm -1 at E=13000 cm -1 (that is, 2000 cm -1 below the 
+threshold). Filled circles indicate the positions o f some periodic orbits. Right column : 
+),(11pq SOS at 2q=0 and ) ,(22pq SOS at 1q=0 for the 10th order perturbative Hamiltonian 
+of Eq. (4) at the same energy.  13 FIGURE 1 
+ 
+ 
+ 
+ 
+ 
+  14 FIGURE 2 
+ 
+ 
+ 
+ 
+ 
+  15 FIGURE 3 
+ 
+ 
+ 
+ 
+ 
+  16 FIGURE 4 
+ 
+ 
+ 
+ 
+ 
+ 
\ No newline at end of file
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@@ -0,0 +1,240 @@
+arXiv:1001.0836v3  [quant-ph]  5 Jul 2010Quantum Annealing with JarzynskiEquality
+MasayukiOhzekia,HidetoshiNishimorib
+aDepartment ofSystems Science, Kyoto University, Yoshida- Honmachi, Sakyo-ku, Kyoto 606-8501, Japan
+bDepartment ofPhysics, Tokyo Institute ofTechnology, Oh-o kayama, Meguro-ku, Tokyo 152-8551, Japan
+Abstract
+We show a practical application of an well-known nonequilib riumrelation, the Jarzynski equality, in quantum com-
+putation. Its implementation may open a way to solve combina torial optimization problems, minimization of a real
+single-valued function, cost function, with many argument s. It has been disclosed that the ordinary quantum com-
+putational algorithm to solve a kind of hard optimization pr oblems, has a bottleneck that its computational time is
+restrictedtobeextremelyslowwithoutrelevanterrors. Ho wever,byournovelstrategyshowninthepresentstudy,we
+mightovercomesuchadi fficulty.
+Keywords: Jarzynskiequality,quantumannealing,optimizationprob lem
+1. Introduction: QuantumAnnealing
+To reduce power loss in electric circuits, we have to
+minimize the circuit length. This kind of problems are
+formulatedintoamoregenerictasktominimizeormax-
+imize a real single-valued function of multivariables,
+cost function. This is called optimization problem [1].
+To solve these problems is one of the most important
+tasks and has broad applications in science and engi-
+neering. Well-known examples with discrete variables
+are satisfiability problems, exact cover, maximum cut,
+Hamiltongraph,andtravelingsalesmanproblem.
+Most of the above exemplified cases are mapped
+into a generic problem to find the ground state for a
+kindof systemsseen in statistical physics, spin glasses.
+Its Hamiltonian is denoted as H0in the present study.
+One of the generic algorithms to solve optimization
+problems in reasonable time by exploiting resources in
+physicsisquantumannealing(QA)[2,3,4]. InQA,we
+introduce artificial degrees of freedom of quantum na-
+ture,noncommutativeoperatorsinordertoinducequan-
+tumfluctuations.
+H(t)=f(t)H0+{1−f(t)}H1, (1)
+wheref(t) isassumed to bea monotonicallyincreasing
+functionsatisfying f(0)=0 andf(τ)=1. The anneal-
+ingtime is denotedby τ. Thetranseverse-fieldoperator
+for spin glasses is often used as a quantum fluctuation,
+H1=−/summationtext
+iσx
+i. The quantum annealing starts from the
+groundstate of H1, a uniform linear combinationof allspin configurations on the basis of σz
+i. The quantum
+system is driven by gradually decreasing the quantum
+fluctuation according to f(t). The adiabatic theorem
+guarantees that we can reach a nontrivial ground state
+ofH0aftersufficientlyslow quantumsweep τc∼1/∆2,
+where∆istheenergygapoftheinstantaneousquantum
+system (1) [5]. If we consider the cases in which the
+quantum system as in Eq. (1) has a minimum energy
+gap vanishing as∆∼exp(−αN) for increasing the sys-
+tem sizeN, QA does not work well in reasonable time
+[6, 7].
+2. JarzynskiEquality
+Toovercometheabovedi fficultyinhardoptimization
+problems, we propose a novel method in conjunction
+with a theoretical piece from non-equilibrium statisti-
+cal physics, the Jarzynski equality (JE), in the present
+study [8, 9]. The Jarzynski equality is written by an
+well-knownexpressionas,
+/angbracketleftBig
+e−βW/angbracketrightBig
+=Zτ(β)
+Z0(β), (2)
+where the angular brackets denote the average over all
+realizationsinapredeterminedprocessstartingfroman
+initial equilibrium state and Wis the work done during
+theprocess. Thepartitionfunctionsfortheinitialandfi-
+nalHamiltoniansarewrittenas Z0(β)andZτ(β)within-
+versetemperatureβ,respectively. Wehereshortlyrecall
+the formulationof JE for the classical system on a heat
+Preprintsubmitted to Computer Physics Communications November 18,2018bath. Let us considera thermal nonequilibriumprocess
+in a finite-time schedule t0=0≤t≤tn=nδt. Ther-
+mal fluctuations can be simulated by the master equa-
+tion. The probability that the system is in a state σkat
+timetkis denoted as P(σk;tk). The transition probabil-
+ity per unit timeδtis defined as M(σk+1|σk;tk). In the
+original formulation of JE, the work is defined as the
+energydifferencemerelyattributedtothe changeofthe
+Hamiltonian,butwecanconstructJEalsointhecaseof
+changing the inverse temperature by defining the work
+as−βW(tk)=−(β(tk+1)−β(tk))H0for the stateσ. The
+left-handside ofJE canthenbeexpressedas
+/angbracketleftBig
+e−βW/angbracketrightBig
+=/summationdisplay
+{σk}n−1/productdisplay
+k=0/braceleftBig
+e−βW(tk)eδtM(σk+1|σk;tk)/bracerightBig
+טP(σ0;t0), (3)
+where˜P(σ0;t0) denotesthe initial equilibriumdistribu-
+tion. Even if the transition term exp( δtM(σk+1|σk;tk))
+is removed in this equation, JE is trivially satisfied as
+one can simply confirm. A non-trivial feature of JE is
+in the insertion of the transition term. From Eq. (3),
+it is straightforward to prove JE by use of the detailed-
+balance condition. An observant reader may think the
+above formulation without any consideration of quan-
+tum nature is not available for the application to QA.
+NeverthelesswecanapplytheclassicalJEtoQAbyaid
+oftheclassical-quantummapping[10].
+3. Classical-quantummapping
+The classical-quantum mapping leads us to a special
+quantum system, in which the (instantaneous) equilib-
+rium state of the above stochastic dynamics can be ex-
+pressed as the ground state. A general form of such a
+special quantum Hamiltonian is given as Hq(σ′|σ;t)=
+I−eβ(t)
+2H0M(σ′|σ;t)e−β(t)
+2H0. This Hamiltonian has the
+ground state as|Ψeq(t)∝an}bracketri}ht=/summationtext
+σe−β(t)H0/2|σ∝an}bracketri}ht/√Z(t). The
+ground state energy is 0, which can be explicitly
+shown by the detailed-balance condition. On the other
+hand, the excited states have positive-definite eigenval-
+ues, which can be confirmed by the application of the
+Perron-Frobeniustheorem.
+In the above special quantum system, we can deal
+with a quasi-equilibrium stochastic process as an adi-
+abatic quantum-mechanical dynamics in QA. Let us
+consider QA for the above special quantum system by
+setting the parameter corresponding to the temperature
+T→∞(β→0). Thisconditiongivesthetrivialground
+state for H1in the preceding section with an uniformlinear combination,similarly to the ordinaryQA. If we
+changeT→0 very slowly, one can obtain the ground
+state ofHq, which expresses the very low-temperature
+equilibrium state for H0, the cost function of the opti-
+mizationproblemthatwewishsolve. We howevercon-
+sider to construct a protocol with the same spirit as JE
+by using the special quantum system to overcome the
+bottleneck of the ordinary QA as proposed in the fol-
+lowingsection.
+4. QuantumJarzynskiannealinganditsapplication
+We prepare a trivial ground state with a uni-
+form linear combination as the initial condition in
+the ordinary QA. This initial state corresponds to
+the high-temperature equilibrium state |Ψeq(t0)∝an}bracketri}ht ∝
+exp(−β(t0)H0/2)|σ∝an}bracketri}htwithβ(t0)≪1. We introduce the
+exponentiatedworkoperator Wexp(tk)=exp(−(β(tk+1)−
+β(tk))H0/2). It looks like a non-unitary operator,
+but we can construct this operation by considering
+an extended quantum system [11, 12]. If we apply
+Wexp(tk) to the preceding quantum state |Ψeq(tk)∝an}bracketri}ht, it is
+changed into a state corresponding to the equilibrium
+distribution with the inverse temperature β(tk+1). Af-
+ter then, the time-evolution operator U(σ′|σ;tk+1)=
+exp(−iδtHq(σ′|σ;tk+1)//planckover2pi1) also does not alter this state,
+sinceit isthegroundstate of Hq(σ′|σ;tk+1). Theresult-
+ingstate aftertherepetitionoftheaboveprocedureis
+|Ψ(tn)∝an}bracketri}ht∝n−1/productdisplay
+k=0/braceleftBig
+Wexp(tk)Uk+1(σk+1|σk;tk)/bracerightBig
+.
+×|Ψeq(t0)∝an}bracketri}ht (4)
+This is essentially of the same form as Eq. (3). We
+measure the obtained state by the projection onto a
+specified stateσ′. The probability is then given by
+|∝an}bracketle{tσ′|Ψ(tn)∝an}bracketri}ht|2,whichmeansthatthegroundstatewewish
+to find is obtained with the probability proportional to
+exp(−β(tn)H0). If we continue the above procedure up
+toβ(tn)≫1, we can efficiently obtain the ground state
+ofH0. This is called the quantum Jarzynski anneal-
+ing (QJA) in the present study. Most of the readers,
+who are familiar with the ordinary computation, have
+considered that it may seem unnecessary to apply the
+time-evolutionoperator U(σk+1|σk;tk),whichexpresses
+change between states by quantum fluctuations, at the
+middle step between the operations of the exponenti-
+ated work operators Wexp(tk). The time-evolution op-
+erator does not mean an artificial control but describes
+the change by quantum nature during quantum com-
+putation, which is inherent property in quantum com-
+putation. However we here remember the nontrivial
+20.020.040.060.080.10t0.20.40.60.81.0PGS
+Figure 1: The performance of QA (triangles), and QJA (cir-
+cles). The reference curve describes the instantaneous Gib bs-
+Boltzmannfactor.
+point of JE. Even if quantum nature a ffects the instan-
+taneous quantum state, the property of JE guarantees
+that we keep the quantum state expressing the instan-
+taneous equilibrium state. If one considers to simu-
+late this procedure in “classical” computers, we have
+to need the repetition of the pre-determined process to
+deal with all fluctuationsin the nonequilibrium-process
+average, since we assume an ensemble in equilibrium
+intheformulationofJE.However,whenwe implement
+QJA in “quantum” computation, we operate QJA to a
+single quantum system in principal since the classical
+ensembleismappedtothe quantumwave function. We
+do not need the repetitionof the same proceduredi ffer-
+entlyfromtheclassical case[12].
+Let us take a simple instance to search the minimum
+fromaone-dimensionalrandompotential,whichisfor-
+mulated as the Hamiltonian H0=−/summationtextN
+i=1Vi|i∝an}bracketri}ht∝an}bracketle{ti|. Here
+Videnotesthe potentialenergyat site iand chosenran-
+domly. We employ a linear schedule for tuning the pa-
+rameterβfrom 0 to 100. Figure 1 shows the plot for
+QJA (upper circles), which are fixed along the refer-
+encecurves(solidcurve)representingtheinstantaneous
+Gibbs-Boltzmann factor. In contrast, QA (lower tri-
+angles) can not sufficiently find the ground state since
+we considera veryshortannealingisconsideredin this
+case.
+5. Summary
+We consider an application of JE to quantum com-
+putation as QA to solve the optimization problems by
+using the classical-quantummapping. As we expected,
+this protocol keeps the quantum system to express the
+equilibrium state for the instantaneous inverse temper-
+ature. The result by QJA shown here gives the groundstateinashortannealingandimpliesthatwemayover-
+comethe difficultiesin hardoptimizationproblemsand
+solve them in a reasonable time. The present result is
+nothing but preliminary one. We should address the
+problem on practical e fficiency for several interesting
+hardproblemswewishtosolveinthefuturestudy[13].
+Acknowledgement
+ThisworkwassupportedbyCREST, JST.
+References
+[1] M.R.GareyandD.S.Johnson,ComputersandIntractabili ty: A
+Guide to the Theory of NP-Completeness Freeman, (San Fran-
+cisco, 1979).
+[2] A. B. Finnila, M. A. Gomez, C. Sebenik, S. Stenson, and J. D .
+Doll, Chem. Phys.Lett. 219,343 (1994).
+[3] T.Kadowaki and H.Nishimori, Phys.Rev. E 58, 5355 (1998).
+[4] S.Morita, and H.Nishimori, J.Math. Phys. 49, 125210 (2008).
+[5] S.Suzuki and M.Okada, J.Phys.Soc. Jpn. 74, 1649 (2005).
+[6] T. Jorg, F. Krzakala, J. Kurchan, A. C. Maggs, Phys. Rev. L ett.
+101, 147204 (2008).
+[7] A. P.Young, S. Knysh, and V. N,Smelyanskiy, Phys. Rev. Le tt.
+104, 020502 (2010).
+[8] C. Jarzynski, Phys.Rev. Lett. 78, 2690 (1997).
+[9] C. Jarzynski, Phys.Rev. E 56, 5018 (1997).
+[10] R. D. Somma, C. D. Batista, and G. Ortiz, Phys. Rev. Lett. 99,
+030603 (2007).
+[11] P.Wocjan,C.Chiang,D.Nagaj,andA.Abeyesinghe,Phys .Rev.
+A.80, 022340 (2009).
+[12] M.Ohzeki, work in progress.
+[13] M.Ohzeki and S.Tanaka, work in progress.
+3
\ No newline at end of file
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+arXiv:1001.0837v1  [cond-mat.str-el]  6 Jan 2010Freezing of spin dynamics and ω/Tscaling in
+underdoped cuprates
+Igor Sega1and Peter Prelovˇ sek1,2
+1Joˇ zef Stefan Institute, Ljubljana, Slovenia
+2Faculty of Mathematics and Physics, University of Ljubljan a, Ljubljana, Slovenia
+E-mail:igor.sega@ijs.si
+Abstract. The memory function approach to spin dynamics in doped antif erromagnetic
+insulator combined with the assumption of temperature inde pendent static spin correlations
+and constant collective mode damping leads to ω/Tscaling in a broad range. The theory
+involving a nonuniversal scaling parameter is used to analy ze recent inelastic neutron scattering
+results for underdoped cuprates. Adopting modified damping function also the emerging central
+peak in low-doped cuprates at low temperatures can be explai ned within the same framework.
+1. Introduction
+It is by now experimentally well established that magnetic s tatic and dynamical properties of
+high-Tccuprates are quite anomalous. Early inelastic neutron scat tering (INS) experiments
+on low-doped La 2−xSrxCuO4(LSCO) [1, 2] revealed that local, i.e., q-integrated dynamic
+spin response in the normal state (NS) exhibits anomalous ω/Tscaling, not reflected in
+instantaneous spin-spin correlation length ξTwhich shows no significant T-dependence below
+room temperature. Subsequently similar behaviour has been found in a number of other
+compounds, i.e., in underdoped YBaCu 3O6+x(YBCO) and in Zn-doped YBCO [2]. More
+recent INS experiments on heavily underdoped (UD) cuprates , including Li-doped LSCO [3],
+YBCO [4, 5, 6, 7], and Pr 1−xLaCexCuO4−δ(PLCCO) [8], confirm the universal features of
+anomalous NS spin dynamics so that ω/Tscaling is found in a broad range both in q-integrated
+susceptibility χ′′
+L(ω) [3, 5, 7] and in χ′′
+q(ω) at the commensurate AFM q=Q= (π,π) [3, 5, 8].
+Typically, χ′′
+Q(ω) is a Lorentzian with the characteristic relaxation rate sc aling as Γ = αT,
+but with a nonuniversal α[3, 5, 8]. Similarly, χ′′
+L(ω,T) =χ′′
+L(ω,0)f(ω/T) has been used [2, 8, 6],
+withf(x) = 2/πarctan[A1x+A2x3] and material dependent A1,2. It has been also observed
+that at low T < Tgsome intensity is gradually transfered into a central peak ( CP) [3, 4] whereas
+the inelastic response saturates. This freezingmechanism appears to be entirely dynamical in
+origin since ξTas well as the integrated intensity are unaffected by the cross over.
+The present authors introduced a theory of spin dynamics in d oped AFM [9] which describes
+the scaling behavior as a dynamical phenomenon based on two e xperimental observations: a)
+ξTis (almost) independent of T, and b) the system is metallic with finite spin collective-mo de
+damping. Then the system close to AFM naturally exhibits ω/Tscaling in a wide energy range,
+with saturation at low-enough T.
+Our starting point in the analysis of recent INS experiments is the dynamical spin
+susceptibilty [9]χq(ω) =−ηq
+ω2+ωMq(ω)−ω2q, (1)
+where the spin stiffness ηq∼2J(∼240 meV) is only weakly q-dependent ( Jis the exchange
+coupling), ωq= (ηq/χ0
+q)1/2is an effective collective mode frequency, χ0
+q=χq(ω= 0) is the
+static susceptibility and Mqis (the complex) memory function containing information on col-
+lective mode damping γq=M′′
+q(ω). In the NS of cuprates low-frequency collective modes
+atq∼Qare generally overdamped so that γq> ωq. UD cuprates close to the AFM phase
+0 50 100 150 200 250 300
+T [K]01020304050ΓQ(T) [meV]ζ=1
+1.5
+2
+2.5
+3
+4
+6
+8
+2 4 68
+ζ0123
+αγ=40 meV
+γ=100 meV
+α=π/(2ζ)
+Figure 1. Relaxation rate Γ Qvs.Tfor different
+parameters ζandγ=100 meV. Inset: dependence
+of slope parameter αonζfor two different γ. For
+comparison, an estimate for αis included.have low charge-carrier concentration
+but pronounced spin fluctuations whose
+dynamics is quite generally governed by
+the sum rule
+/integraldisplay∞
+0dω
+πχ′′
+q(ω)cothω
+2T=Cq,(2)
+whereCqis strongly peaked at Qwith
+a characteristic width κT= 1/ξT.
+Moreover, the total sum rule is for a
+system with local magnetic moments
+(spin 1/2) given by (1 /N)/summationtext
+qCq= (1−
+ch)/4, where chis an effective (hole)
+doping.
+While the formalism so far is very
+general, we now introduce approxi-
+mations specific to UD cuprates [9].
+INS experiments listed above indicate
+that within the NS the effective q
+width of χ′′
+q(ω), i.e., dynamical κ(ω),
+is only weakly T- andω-dependent,
+even on entering the regime with the
+CP response [4]. Within further analysis we assume the comme nsurate AFM response at Qand
+the double-Lorentzian form Cq=C/[(q−Q)2+κ2
+T]2although qualitative results at low ωdo
+not depend on a particular form of Cq.
+2. Paramagnetic metal:
+We also assume that the damping γq(ω) is dominated by particle-hole excitations being only
+weaklyqandωdependent. Hence γq(ω)∼γ, withγa phenomenological parameter. The
+assumption of constant γin Eq. (1) then leads to
+χ′′
+q(ω)∼χ0
+qωΓq
+ω2+Γ2q,Γq=η
+γχ0q. (3)
+Note that recent INS data are fully consistent with this form which has been used to extract
+ΓQ(T) [5].
+We next exploit the sum rule, Eq.(2), to determine Γ q. As shown elsewhere [9, 10] Γ Qis
+mainly determined by the parameter
+ζ=πγCQ/(2η), (4)
+subjectto T≪γwhichis experimentally relevant. Theresults arepresente dinFig. 1for arange
+ofζ= 1−8. While Γ Q(0) = Γ0
+Q∼γexp(−2ζ) [9, 10], for T >Γ0
+Qthe variation is nearly linear00.51χ"(QAF,ω,T)/χ"(QAF,ω,0)exp., ω=3 meV
+theory5 meV 8 meV
+050100150200250
+T [K]00.51 12 meV
+050100150200250
+T [K]32.5 meV
+Γ(0)=8 meV
+α=0.18 meV/K
+050100150200250
+T [K]01020304050[meV]ωQ(T)
+Γ(T)
+Figure 2. Temperature evolution of χ′′
+Q(ω,T) for
+UD YBCO with x= 0.45 [7] (symbols) compared
+with theoretical result, Eq. (3), with ζ= 1.8 and
+γ=60 meV (lines).Figure 3. Scaling of theoretical
+normalised χ′′
+L(ω,T) based on ζandγas
+in Fig. (2). For comparison, the scaling
+function f(x) as frequently used in fits
+to experiments is also plotted.
+0 2 4 6 8
+ω/T00.20.40.60.81χ"L(ω,T)/χ"L(ω,T=0)
+ω=3 meV
+5
+8
+12
+16
+20
+32.5
+f(x)=2/π tan-1[0.56x+0.09x3]
+0 100 200 300
+T [K]050100150200κ2χQ [a.u.]ζ=1.5
+2.0
+2.5
+3.0
+0 0.1 0.2 0.3 0.4 0.5
+q [r.l.u.]00.20.40.60.81χ(q)/χ(0)T~ 0 K
+25
+50
+100
+300
+C(q)/C(0)
+Figure 4. Left pannel: the temperature
+dependence of κ2χQfor several ζas a function of
+T. Right pannel: T-dependence of χqrelative to a
+Lorentzian Cq.ΓQ∼αTbeing a manifestation of the
+ω/Tscaling. But αis not universal and
+depends on ζ(and weakly on γ). Then
+to leading order
+ΓQ(T)∼=max{πT
+2ζ,γexp(−2ζ)},(5)
+from which α=π/(2ζ) is identified [10].
+Note that the above Γ Qleads to χ′′
+L(ω)
+consistent with the “marginal Fermi
+liquid” model introduced by Varma et
+al. [11]. However, contrary to the usual
+assumption of proximity to a quantum
+critical point where ξT∝1/T, hereξT∼
+const.
+Recently a number of INS experi-
+ments have been reported where χ′′
+Q(ω)
+can be well described by Eq.3, includ-
+ing linear-in- Tbehaviour of Γ Qwhere,
+depending on material α∼0.18−0.75 [3, 5, 8]. All these αrequire rather large ζ, which implies
+very low saturation Γ0
+Q, whereas INS data for YBCO and LSCO indicate quite substanti al Γ0
+Q.
+However, saturation of Γ Q, setting in for T < Tg, is accompanied by simultaneous appearance
+of the CP which absorbs the ‘missing’ sum rule.
+In Fig. 2 INS measurements by Hinkov et al. [6, 7] on UD YBCO wit hx= 0.45 [7] are
+presented together with theoretical curves were the only re levant parameter is ζ= 1.8, while
+in Fig. 3 theoretical scaling function χ′′
+Q(ω,T)/χ′′
+Q(ω,0) for the same ζ(andγ) is plotted. The
+overall agreement between theory and experiment (Fig. 2) is quite satisfactory. The agreement
+is less satisfactory for ω= 32.5meV and should be attributed to the breakdown of scaling sin ce
+ωQ> γ/2, as also evident in Fig. 3. Note that the ad hocansatz for f(x) commonly used
+(withA2= 0) can be easily obtained assuming a Lorentzian dependence ofχ0
+qonq. A simplecalculation yields arctan( A1ω/T) withA1∼1/α, provided that κ≪1 butκ2χ0
+Q∼const (see
+Fig. 4).
+3. CP response:
+The advantage of the memory-function formalism is that the e mergence of the CP at T < Tgin
+the spin response can be as well treated within the same frame work. One has to assume that
+unlike in a paramagnet the mode damping γqis not constant but may acquire an additional low
+frequncy contribution. In particular we can take ˜Mq∼iγ−δ2/(ω+iλ), withT-dependent δ
+andλ, which leads to χ′′
+q(ω) of the form used also to analyse experimental INS data for YB CO
+withx= 0.35 [4, 14].
+Forλ→0 butδ2/λ≫γthe modified ˜Mqleads to two distinct energy scales and hence to
+two contributions to spin dynamics, i.e., the CP part χc
+q(ω) and the regular contribution χr
+q(ω),
+χc
+q(ω)∼χ0
+qΓc
+Γc−iω, χr
+q(ω)∼χr0
+qΓr
+Γr−iω,q∼Q, (6)
+valid for ω < λandλ < ω≪γ, respectively. Thus, below Tgnew scales are set by Γ r= Ω2
+q/γ
+and Γc= (η/δ2)λ/χ0
+q, withχr0
+q=η/Ω2
+qand Ω2
+q=ω2
+q+δ2. If one assumes that δsaturates
+at lowT, as is manifest by saturation of Γ r[5], Γc∝λ/χ0
+qbecomes the smallest energy scale,
+resulting in a quasielastic peak of width Γ c. The saturation of Γ r, although at present unclear
+physically, is responsible for the transfer of spectral wei ght, since Cr
+q∼Cq−πT/(γΓr) [10],
+which is again consistent with experiment on x= 0.35 YBCO [5].
+4. Conclusions
+The approach presented gives a consistent explanation of th eω/Tscaling both in χ′′
+q(ω) as well
+as inχ′′
+L(ω). It is based on two well established experimental facts: th e overdamped nature of
+the response and the saturation of κTat lowωandT. The appearance of the CP for T < Tgis
+easily incorporated into the formalism via the (almost) sin gularω- andT-dependent damping
+˜Mq(ω). However, the question as to the origin of CP remains to be se ttled.
+References
+[1] B Keimer, R J Birgeneau, A Cassanho, Y Endoh, R W Erwin, M A K astner, and G Shirane 1991 Phys. Rev.
+Lett.67, 1930
+[2] for a review see M A Kastner, R J Birgeneau, G Shirane, and Y Endoh 1998 Rev. Mod. Phys. 70, 897
+[3] W Bao, Y Chen, Y Qiu, and J L Sarrao 2003 Phys. Rev. Lett. 91, 127005
+[4] C Stock, W J L Buyers, Z Yamani, C L Broholm, J-H Chung, Z Tun , R Liang, D Bonn, W N Hardy, and R
+J Birgeneau 2006 Phys. Rev. B73, 100504(R)
+[5] C Stock, W J L Buyers, Z Yamani, Z Tun, R J Birgeneau, R Liang , D Bonn, and W N Hardy 2008 Phys.
+Rev.B77, 104513
+[6] V Hinkov, D Haug, B Fauque, P Bourges, Y Sidis, A Ivanov, C B ernhard, C T Lin, and B Keimer 2008
+Science319, 597
+[7] V Hinkov 2007 In-plane anisotropy of the spin-excitation spectrum in twi n-freeYBa2Cu3O6−x(Ph. D. thesis,
+University of Stuttgart)
+[8] S D Wilson, S Li, H Woo, P Dai, H A Mook, C D Frost, S Komiya, an d Y Ando 2006 Phys. Rev. Lett. 96,
+157001
+[9] P Prelovˇ sek, I Sega, and J Bonˇ ca 2004 Phys. Rev. Lett. 92, 027002
+[10] I Sega and P Prelovˇ sek 2009 Phys. Rev. B79, 140504(R)
+[11] C M Varma, P B Littlewood, S Schmitt-Rink, E Abrahams, an d A E Ruckenstein 1989 Phys. Rev. Lett. 63,
+1996
+[12] G Aeppli, T E Mason, S M Hayden, H A Mook, and J Kulda 1997 Science278, 1432
+[13] S Chakravarty, B I Halperin, and D R Nelson 1988 Phys. Rev. Lett. 60, 1057
+[14] B I Halperin and C M Varma 1976 Phys. Rev. B14, 4030
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+arXiv:1001.0838v2  [math.AG]  31 Dec 2010PROJECTIVIZED RANK TWO TORIC VECTOR BUNDLES ARE
+MORI DREAM SPACES
+JOS´E LUIS GONZ ´ALEZ∗
+Abstract. We prove that the Cox ring of the projectivization P(E) of a rank two
+toric vector bundle E, over a toric variety X, is a finitely generated k-algebra. As
+a consequence, P(E) is a Mori dream space if the toric variety Xis projective and
+simplicial.
+1.Introduction
+Mori dream spaces were introduced by Hu and Keel in [10] as a class o f varieties with
+interesting features from the point of view of Mori theory. For ins tance, their pseudoef-
+fective and nef cones are both polyhedral, and the Mori program c an be carried out for
+any pseudoeffective divisor on these varieties. Additionally, their ps eudoeffective cones
+can be decomposed into finitely many closed convex chambers that a re in correspondence
+with the birational contractions of XhavingQ-factorial image (see Proposition 1.11 [10]).
+A Mori dream space can be defined as a normal projective Q-factorial variety X, with
+Pic(X)Q=N1(X)Qand a finitely generated Cox ring (see Definition 2.2). One basic
+example is that of toric varieties, where the Cox ring is a polynomial rin g in finitely many
+variables (see [4]). Thus, a projective simplicial toric variety is a Mori dream space. One
+more example is given by the (log) Fano varieties, which have recently been proven to be
+Mori dream spaces (see [2]). Other references for recent work o n Cox rings of particular
+Mori dream spaces are [1], [3] and [15].
+In [9], Hering, Mustat ¸˘ a and Payne ask whether projectivizations of toric vector bundles
+are Mori dream spaces, and give some evidence for a positive answe r. For example, the
+Mori cones of these varieties are rational polyhedral, and they ar e known to be Mori
+dream spaces for toric vector bundles that split as a sum of line bund les. Toric vector
+bundles were classified by Klyachko (see [11]) via an equivalence of ca tegories under which
+each toric vector bundle corresponds to a collection of filtrations o f a suitable vector space
+(see Theorem 2.1).
+In this paper, we prove that the Cox ring of the projectivization P(E) of a rank two
+toric vector bundle Eover an arbitrary toric variety Xis finitely generated as a k-algebra.
+2000Mathematics Subject Classification . Primary 14M25; Secondary 14E30, 14F05.
+Key words and phrases . Toric vector bundles, Mori dream spaces, toric varieties, Cox rin gs, Klyachko
+filtrations.
+∗The author was partially supported by the NSF under grant DMS-05 02170 and by the David and
+Lucile Packard Foundation under M. Mustat ¸˘ a’s fellowship.
+12 JOS´E LUIS GONZ ´ALEZ
+As a consequence, P(E) is a Mori dream space, if Xis assumed to be projective and
+simplicial. In the proof, we consider a particular set of generators o f the group N1(P(E)),
+and describe a finite set of generators for the associated Cox ring . This is done by
+considering a finer grading of the Cox ring induced by the torus actio n, and describing
+the graded pieces and the multiplication map in terms of data arising fr om Klyachko’s
+filtrations of E.
+Alternative proofs of our main result can be found in [12] and [8]. In fact, in these
+articles the authors establish the finite generation of Cox rings in th e more general setting
+ofT-varieties of complexity one. We recall that a normal variety with an algebraic action
+of a torus Tis called a T-variety, and that its complexity is defined to be the lowest
+codimension of a T-orbit on the variety. Hence projectivizations of rank two toric ve ctor
+bundles are examples of T-varieties of complexity one. On the other hand, we hope that
+our methods will provide new ideas for treating the analogous proble m in the case of
+projectivizations of higher rank toric vector bundles.
+Acknowledgments. I am grateful to Milena Hering, Mircea Mustat ¸˘ a and Sam Payne
+from whom I learned the questions that motivated this work. I than k M. Hering for
+bringing the paper [12] to my attention, and I especially thank Mircea Mustat ¸˘ a for many
+inspiring conversations.
+2.Toric Vector Bundles and Cox rings
+All our varieties are defined over a fixed algebraically closed field k. By aline bundle
+on a variety Z, we mean a locally free sheaf of rank one on Z. The groups of line
+bundles on Zmodulo linear equivalence andmodulo numerical equivalence aredenot ed by
+Pic(Z)andN1(Z), andtheir Q-extensions ofscalarsaredenotedbyPic( Z)QandN1(Z)Q,
+respectively. We follow the convention that the geometric vector bundle associated to the
+locally free sheaf Eis the variety V(E) =Spec/circleplustext
+m≥0SymmE∨, whose sheaf of sections
+isE. Also, when we consider the fiber of Eover a point z∈Z, we mean the fiber over z
+of the projection f:V(E)→Z. Lastly, by the projectivization P(E) ofE, we mean the
+projective bundle Proj/circleplustext
+m≥0SymmEoverZ. This bundle is endowed with a projection
+π:P(E)→Zand an invertible sheaf OP(E)(1) (see II.7 in [7]).
+LetXbe ann-dimensional toric variety corresponding to the nondegenerate f an ∆ in
+the lattice N∼=Zn. We denote the algebraic torus acting on XbyT, and we designate its
+unit element by t0. We denote the character lattice Hom( N,Z)∼=ZnofTbyM. Thus,
+T= Speck[M] = Spec k[χu|u∈M] andXhas an open covering given by the affine
+toric varieties Uσ= Speck[σ∨∩M] corresponding to the cones σ∈∆. We denote the
+rays in ∆ by ρ1,ρ2,...,ρ d. For each ray ρj, we denote its primitive lattice generator by vj
+and its associated torus invariant prime divisor by Dj. For a detailed treatment of toric
+varieties we refer to [5].PROJECTIVIZED RANK TWO TORIC VECTOR BUNDLES ARE MORI DREAM S PACES 3
+Atoric vector bundle on the toric variety Xis a locally free sheaf Etogether with an
+action of the torus Ton the variety V(E), such that the projection f:V(E)→Xis
+equivariant and Tacts linearly on the fibers of f. Given any T-invariant open subset U
+ofX, there is an induced action of TonH0(U,E), defined by the equation
+(t·s)(x) =deft(s(t−1x)),
+for anys∈H0(U,E),t∈Tandx∈X. This action induces a direct sum decomposition
+(1) H0(U,E) =/circleplusdisplay
+u∈MH0(U,E)u,
+whereH0(U,E)u=def{s∈H0(U,E)|t·s=χu(t)sfor eacht∈T}. We refer to the
+summands in the decomposition from (1) as the isotypical componen ts ofH0(U,E) with
+respect to the torus action. Let us denote the fiber of Eovert0byE, and for any T-
+invariant open subset U, letevt0:H0(U,E)→Ebe the evaluation mapat t0. Foreach ray
+ρj∈∆ and each u∈M, the evaluation map evt0gives an inclusion H0(Uρj,E)u֒−→E.
+Ifu,u′∈Msatisfy/an}brack⌉tl⌉{tu,vj/an}brack⌉tri}ht ≥ /an}brack⌉tl⌉{tu′,vj/an}brack⌉tri}ht, thenevt0(H0(Uρ,E)u)⊆evt0(H0(Uρ,E)u′). In
+particular, evt0(H0(Uρ,E)u) depends only on /an}brack⌉tl⌉{tu,vj/an}brack⌉tri}ht, or equivalently, only on the class
+ofuinM/ρ⊥
+j∩M∼=Z. Hence we may simply denote this space by Eρj(/an}brack⌉tl⌉{tu,vj/an}brack⌉tri}ht). The
+ordered collection Eρj={Eρj(i)|i∈Z}gives a decreasing filtration of E, and the set
+of these filtrations {Eρj|1≤j≤d}are known as the Klyachko filtrations associated
+toE. Klyachko proved in [11] that these filtrations of Esatisfy a certain compatibility
+condition, and that they completely describe E. More precisely,
+Theorem 2.1 (Klyachko) .The category of toric vector bundles on the toric variety X
+is equivalent to the category of finite dimensional k-vector spaces Ewith collections of de-
+creasing filtrations {Eρ(i)|i∈Z}, indexed by the rays of ∆, satisfying the following com-
+patibility condition: For each cone σ∈∆, there is a decomposition E=/circleplustext
+¯u∈M/σ⊥∩ME¯u
+such that
+Eρ(i) =/summationdisplay
+/an}brack⌉tl⌉{t¯u,vρ/an}brack⌉tri}ht≥iE¯u,
+for every ray ρ⊆σand every i∈Z.
+Some references for recent work on toric vector bundles are [6], [9 ], [13] and [14]. Next,
+we give the definition of the algebras that we will call Cox rings of varie ties, and we briefly
+compare this definition with the one used in [10].
+Definition 2.2.LetXbe a variety such that Pic( X)Qis finite dimensional. Any
+k-algebra of the form
+Cox/parenleftbig
+X,(L1,L2,...,Ls)/parenrightbig
+=def/circleplusdisplay
+(m1,m2,...,ms)∈ZsH0(X,L⊗m1
+1⊗L⊗m2
+2⊗···⊗L⊗ms
+s)
+whereL1,L2,...,Lsare line bundles on Xwhose classes span Pic( X)Q, will be called a
+Cox ring ofX.4 JOS´E LUIS GONZ ´ALEZ
+Inthe setting of Definition2.2, it iseasy to see that the finite genera tionof oneCox ring
+ofXisequivalent tothe finitegenerationof every Cox ring of X. In[10], HuandKeel give
+a definition of Cox rings for any projective variety Xthat satisfies Pic( X)Q=N1(X)Q.
+TheirdefinitionissimilartoDefinition2.2, buttheyrequirethelinebundle sL1,L2,...,Ls
+to satisfy some extra conditions. Namely, the classes of L1,L2,...,Lsarerequired to form
+a basis for Pic( X)Qand their affine hull must contain the pseudoeffective cone of X. For
+projective varieties that satisfy Pic( X)Q=N1(X)Q, it is easy to see that a particular
+(equivalently every) Cox ring of Xin the sense of Hu and Keel is finitely generated if and
+only if a particular (equivalently every) Cox Ring of Xin the sense of Definition 2.2 is
+finitely generated. Since our interest lies in the finite generation of t hesek-algebras, and
+this does not depend on the definition of Cox ring that we use, we will u se Definition 2.2
+since it applies in the case of the projectivization of a toric vector bu ndle over anarbitrary
+toric variety.
+Remark 2.3.Given a toric variety Xthere exists a toric resolution of singularities,
+i.e. a smooth toric variety X′and a proper birational toric morphism f:X′→X. Given
+a toric vector bundle EonX, the induced map f′:P(f∗E)→P(E) is also proper and
+birational. In this case f∗Eis a toric vector bundle on X′and the finite generation
+of a Cox ring of P(f∗E) implies the finite generation of any Cox ring of P(E). To see
+this we consider line bundles L1,L2,...,LsonP(E) whose classes span Pic( P(E))Qand
+line bundles L′
+1,L′
+2,...,L′
+s′onP(f∗E) whose classes span Pic( P(f∗E))Q. The finite gen-
+eration of the algebra Cox/parenleftbig
+P(f∗E),(L′
+1,L′
+2,...,L′
+s′,f′∗L1,f′∗L2,...,f′∗Ls)/parenrightbig
+implies the
+finite generation of the subalgebra
+/circleplusdisplay
+(m1,m2,...,ms)∈ZsH0(P(f∗E),f′∗L⊗m1
+1⊗f′∗L⊗m2
+2⊗···⊗f′∗L⊗ms
+s) = Cox/parenleftbig
+P(E),(L1,L2,...,Ls)/parenrightbig
+.
+Therefore, to prove the finite generation of the Cox rings of proj ectivizations of rank two
+toric vector bundles over arbitrary toric varieties, it is enough to c onsider the case when
+the base is a smooth toric variety.
+Now, let Xbe a smooth toric variety and let π:P(E)→Xbe the projectiviza-
+tion of the rank two toric vector bundle EoverX. The classes of the line bundles
+OX(D1),OX(D2),...,OX(Dd) span Pic( X)Q=N1(X)Q. Since Pic( P(E)) = Pic( X)⊕Z·
+[OP(E)(1)], the classes of the line bundles OP(E)(1),π∗OX(D1),π∗OX(D2),...,π∗OX(Dd)
+span Pic( P(E))Q=N1(P(E))Q. Therefore the following algebra is a Cox ring of P(E) in
+the sense of Definition 2.2:
+C=/circleplusdisplay
+(m,m1,...,md)∈Zd+1H0(P(E),OP(E)(m)⊗π∗OX(m1D1)⊗···⊗π∗OX(mdDd)).PROJECTIVIZED RANK TWO TORIC VECTOR BUNDLES ARE MORI DREAM S PACES 5
+By the projection formula, Cis isomorphic to the algebra
+R=/circleplusdisplay
+(m,m1,...,md)∈Zd+1H0(X,SymmE ⊗O X(m1D1)⊗···⊗O X(mdDd)),
+whereSymmE= 0 form <0. For each m= (m1,...,m d)∈Zd, let us denote the toric
+line bundle OX(m1D1)⊗···⊗O X(mdDd) onXbyLm, and the fiber of Lmovert0by
+Lm. Note that for each m,m′∈Z, eachm,m′∈Zdand each u,u′∈M, the product
+ofH0(X,SymmE ⊗ Lm)uandH0(X,Symm′E ⊗Lm′)u′in the algebra Ris contained in
+H0(X,Symm+m′E ⊗Lm+m′)u+u′. Therefore we get a finer grading for Rgiven by
+R=/circleplusdisplay
+(u,m,m)∈M×Z×ZdR(u,m,m)=/circleplusdisplay
+(u,m,m)∈M×Z×ZdH0(X,SymmE ⊗Lm)u.
+Note that each of the homogeneous components R(u,m,m)=H0(X,SymmE ⊗ Lm)uis a
+finite dimensional k-vector space. For each l∈Z+, letR(l)be the Veronese subalgebra of
+Rgiven by
+(2)R(l)=/circleplusdisplay
+(u,m,m)∈M×Z×ZdR(lu,lm,lm)=/circleplusdisplay
+(u,m,m)∈M×Z×ZdH0(X,SymlmE ⊗Llm)lu.
+SinceRis a domain and H0(X,OX) is a finitely generated k-algebra, it follows from
+general considerations that the finite generation of Ris equivalent to the finite generation
+ofR(l)for anyl∈Z+.
+3.Preliminary lemmas
+LetW1,W2,...,W lbe subspaces of a vector space W. For every collection of nonneg-
+ative integers m,c1,c2,...,c l, we denote by Symm
+W(Wc1
+1,Wc2
+2,···,Wcl
+l) the subspace of
+SymmWequal to the image of the composition of the natural maps
+W⊗c1
+1⊗W⊗c2
+2⊗···⊗W⊗cl
+l⊗W⊗(m−/summationtextl
+i=1ci)−→W⊗m−→SymmW,
+ifm≥/summationtextl
+i=1ci, or the subspace 0 of SymmW, otherwise.
+Lemma3.1.LetW1,W2,...,W qbe distinct subspaces of a vector space W. Letm,m′,
+c1,c2,...,c q,c′
+1,c′
+2,...,c′
+qbe nonnegative integers.
+(a)If/summationtextq
+i=1ci≤mand/summationtextq
+i=1c′
+i≤m′, then
+µ/parenleftbigg
+Symm
+W/parenleftbig
+Wc1
+1,...,Wcq
+q/parenrightbig
+⊗Symm′
+W/parenleftbig
+Wc′
+1
+1,...,Wc′
+q
+q/parenrightbig/parenrightbigg
+=Symm+m′
+W/parenleftbig
+Wc1+c′
+1
+1,...,Wcq+c′
+q
+q/parenrightbig
+,
+whereµ:SymmW⊗Symm′W→Symm+m′Wis the multiplication map.
+(b)IfW1,W2,...,W qare one-dimensional, and Wis two-dimensional, then
+Symm
+W(Wc1
+1)∩Symm
+W(Wc2
+2)∩···∩Symm
+W(Wcq
+q) =Symm
+W(Wc1
+1,Wc2
+2,···,Wcq
+q).
+Furthermore, this subspace of SymmWis nonzero precisely when m≥/summationtextq
+i=1ci, and in
+that case its dimension is m+1−/summationtextq
+i=1ci.6 JOS´E LUIS GONZ ´ALEZ
+Proof.
+(a)The conclusion follows at once from the commutativity of the diagram
+/parenleftbig
+(/circlemultiplytext
+iW⊗ci
+i)⊗W⊗(m−/summationtextci)/parenrightbig
+⊗/parenleftbig
+(/circlemultiplytext
+iW⊗c′
+i
+i)⊗W⊗(m′−/summationtextc′
+i)/parenrightbig
+≀
+/d15/d15/d47/d47W⊗m⊗W⊗m′ /d47/d47SymmW⊗Symm′W
+µ
+/d15/d15
+(/circlemultiplytext
+iW⊗(ci+c′
+i)
+i)⊗W⊗((m+m′)−/summationtext(ci+c′
+i)) /d47/d47W⊗(m+m′) /d47/d47Symm+m′W.
+(b)We fix an isomorphism of k-algebras between ⊕h≥0SymhWand the polynomial ring
+in two variables k[x,y]. The subspaces W1,...,W qofWcorrespond to the linear spans of
+somedistinct linearforms l1,...,lq. Thesubspaces ∩q
+i=1Symm
+W(Wci
+i)andSymm
+W(Wc1
+1,···,
+Wcq
+q) ofSymmWboth correspond to the homogeneous polynomials of degree mdivisible
+bylc1
+1···lcq
+q. From this observation the conclusion follows. /square
+LetEbe a toric vector bundle of rank two on X. LetEbe the fiber of Eover the
+unit of the torus, and let Eρ1,Eρ2,...,Eρdbe the Klyachko filtrations associated to E. Let
+V1,V2,...,V pbe the distinct one-dimensional subspaces of Ethat appear in the Klyachko
+filtrations of E, i.e. each Vlis equal to Eρj(i) for some j∈ {1,2,...,d}and some i∈
+Z. We now define the subsets A0,A1,...,A pof{1,2,...,d}, which intuitively classify
+the filtrations according to their one-dimensional subspace, as fo llows. For each l∈
+{1,2,...,p}, we define
+Al={j∈ {1,2,...,d} | Eρj(i) =Vlfor some i∈Z},
+and we also define
+A0={1,2,...,d}/integerdivide/uniondisplay
+1≤l≤pAl.
+For each j∈ {1,2,...,d}letl(j) be the unique element of {0,1,2,...,p}such that j∈
+Al(j). We next introduce the following collection ofpossibly empty subsets of{1,2,...,d}:
+J={J|J⊆/uniondisplay
+1≤l≤pAl,andJ∩Alhas at most one element, for each l= 1,2,...,p}.
+Note that Jis a finite set. For each j∈ {1,2,...,d}, we define the integers ajandbjby
+aj= max{i∈Z| Eρj(i) =E}and
+bj=
+
+max{i∈Z|dimkEρj(i) = 1}ifj∈/uniontext
+1≤l≤pAl
+aj+1 if j∈A0.
+To each filtration Eρjwe associate the linear functional λjonMR×Rd+1defined by
+λj:MR×Rd+1−→ R
+(u,m,m 1,...,m d)/mapst≀−→/an}brack⌉tl⌉{tu,vj/an}brack⌉tri}ht−ajm−mj
+bj−aj
+for anyu∈MRand any m,m1,m2,...,m d∈R.PROJECTIVIZED RANK TWO TORIC VECTOR BUNDLES ARE MORI DREAM S PACES 7
+Remark 3.2.To motivate the preceding definitions, we note that for each j∈ {1,2,
+...,d}, and for each m∈Z≥0,m∈Zdandu∈M, we have
+((SymmE)⊗Lm)ρj(/an}brack⌉tl⌉{tu,vj/an}brack⌉tri}ht) =Symm
+E/parenleftbig
+(Eρj(bj))max{0,⌈λj(u,m,m)⌉}/parenrightbig
+⊗Lm,
+where⌈x⌉= min{l∈Z|l≥x}for anyx∈R. This equality is a reformulation of
+Example 3.6 in [6], and it can also be verified through direct computation .
+We now define a rational polyhedral cone QJinMR×Rd+1for each J∈ J, as
+follows. Given such a set J={j1,j2,...,j q}, the cone QJis defined as the set of elements
+(x,w,w 1,...,w d) satisfying the linear inequalities
+w≥0, (3)
+q/summationdisplay
+h=1λjh(x,w,w 1,...,w d)≤w, (4)
+λj(x,w,w 1,...,w d)≤0 for each j∈ {1,2,...,d}/integerdivide/uniondisplay
+1≤h≤qAl(jh), (5)
+λjh(x,w,w 1,...,w d)≥0 for each h∈ {1,2,...,q}, (6)
+λjh(x,w,w 1,...,w d)≥λj(x,w,w 1,...,w d) for each h∈ {1,2,...,q} (7)
+and each j∈Al(jh),
+where(x,w,w 1,...,w d)arethecoordinatesin MR×Rd+1. Thefollowinglemmamotivates
+the definition of these cones.
+Lemma3.3.For each g= (u,m,m)∈M×Z×Zdwe have:
+(a)If(u,m,m)belongs to the cone QJ, for some J={j1,j2,...,j q} ∈ J, then
+evt0/parenleftbig
+H0(X,SymmE ⊗Lm)u/parenrightbig
+=Symm
+E/parenleftbigg
+V⌈λj1(g)⌉
+l(j1),V⌈λj2(g)⌉
+l(j2),...,V⌈λjq(g)⌉
+l(jq)/parenrightbigg
+⊗Lm.
+(b)IfH0(X,SymmE ⊗Lm)u/n⌉}ati≀nslash= 0, then(u,m,m)belongs to the cone QJfor some J∈ J.
+(c)Assume that (u,m,m)∈c(M×Z×Zd)belongs to the cone QJ, for some J={j1,j2,
+...,jq} ∈ J, wherec= lcm{bj−aj|j= 1,2,...,d}. ThenH0(X,SymmE ⊗Lm)u/n⌉}ati≀nslash= 0.
+Proof.
+(a)From the definition of the Klyachko filtrations of SymmE ⊗Lm, we have that
+evt0/parenleftbig
+H0(X,SymmE ⊗Lm)u/parenrightbig
+=/intersectiondisplay
+1≤j≤d(SymmE ⊗Lm)ρj(/an}brack⌉tl⌉{tu,vj/an}brack⌉tri}ht).
+For each j∈ {1,2,...,d}/integerdivide/uniontext
+1≤h≤qAl(jh), the inequality λj(g)≤0 implies that ( SymmE⊗
+Lm)ρj(/an}brack⌉tl⌉{tu,vj/an}brack⌉tri}ht) =SymmE⊗Lm. For each h∈ {1,2,...,q}, the inequality λjh(g)≥0
+implies that ( SymmE ⊗ Lm)ρjh(/an}brack⌉tl⌉{tu,vjh/an}brack⌉tri}ht) =Symm
+E(V⌈λjh(g)⌉
+l(jh))⊗Lm. Similarly, for each
+h∈ {1,2,...,q}and each j∈Al(jh)the inequality λjh(g)≥λj(g) implies that
+(SymmE ⊗Lm)ρj(/an}brack⌉tl⌉{tu,vj/an}brack⌉tri}ht)⊇(SymmE ⊗Lm)ρjh(/an}brack⌉tl⌉{tu,vjh/an}brack⌉tri}ht).8 JOS´E LUIS GONZ ´ALEZ
+Therefore,
+evt0/parenleftbig
+H0(X,SymmE ⊗Lm)u/parenrightbig
+=/intersectiondisplay
+0≤l≤p/intersectiondisplay
+j∈Al(SymmE ⊗Lm)ρj(/an}brack⌉tl⌉{tu,vj/an}brack⌉tri}ht) =/intersectiondisplay
+1≤h≤q/intersectiondisplay
+j∈Al(jh)(SymmE ⊗Lm)ρj(/an}brack⌉tl⌉{tu,vj/an}brack⌉tri}ht)
+=/intersectiondisplay
+1≤h≤q(SymmE ⊗Lm)ρjh(/an}brack⌉tl⌉{tu,vjh/an}brack⌉tri}ht) =/intersectiondisplay
+1≤h≤qSymm
+E(V⌈λjh(g)⌉
+l(jh))⊗Lm
+=Symm
+E/parenleftbigg
+V⌈λj1(g)⌉
+l(j1),V⌈λj2(g)⌉
+l(j2),...,V⌈λjq(g)⌉
+l(jq)/parenrightbigg
+⊗Lm.
+(b)We define
+I=/braceleftbig
+l∈ {1,2,...,p} |max{λj(u,m,m)|j∈Al} ≥0/bracerightbig
+.
+We can assume that I/n⌉}ati≀nslash=φ, since otherwise ( u,m,m) belongs to QJforJ=φ∈ J.
+Letl1,l2,...,lqbe the distinct elements of I, and for each h∈ {1,2,...,q}let us choose
+jh∈Alhsuch that λjh(u,m,m) = max{λj(u,m,m)|j∈Alh}. Clearly, the set J=def
+{jh|h= 1,2,...,q}belongs to J. Then (u,m,m) satisfies (3) and (4) since
+/intersectiondisplay
+1≤h≤qSymm
+E/parenleftbig
+V⌈λjh(u,m,m)⌉
+l(jh)/parenrightbig
+⊗Lm=evt0/parenleftbig
+H0(X,SymmE ⊗Lm)u/parenrightbig
+/n⌉}ati≀nslash= 0,
+and it satisfies (5)-(7) by the definitions of IandJ. Thus, ( u,m,m)∈QJ.
+(c)By (a), it suffices to show that Symm
+E/parenleftbig
+Vλj1(u,m,m)
+l(j1),Vλj2(u,m,m)
+l(j2),...,Vλjq(u,m,m)
+l(jq)/parenrightbig
+is non-
+zero, which is true by (4) and Lemma 3.1 (b). /square
+4.Finite generation of the Cox ring of P(E)
+In the next theorem we prove that any Cox ring of P(E), in the sense of Definition 2.2,
+is finitely generated. As a corollary we obtain that P(E) is a Mori dream space as defined
+by Hu and Keel in [10], if the toric variety Xis projective and ∆ is simplicial (i.e. each
+cone in ∆ is spanned by as many vectors as its dimension). Throughou t this section we
+use the notation from §2-3.
+Theorem 4.1.Any Cox ring of the projectivization P(E)of a rank two toric vector
+bundleEover an arbitrary toric variety Xis finitely generated as a k-algebra.
+Proof. By Remark 2.3 we can assume that Xis smooth. It suffices to find a finite
+set of generators for the k-algebra R(c)from (2), where c= lcm{bj−aj|j= 1,2,...,d}.
+For each set J∈ J, letGJ⊆c(M×Zd+1) be a finite set of generators for the semigroup
+QJ∩c(M×Zd+1). For each g∈M×Zd+1, letβgbe ak-basis of Rg. We claim that the
+finite set
+β=def/uniondisplay
+J∈J/uniondisplay
+g∈GJβgPROJECTIVIZED RANK TWO TORIC VECTOR BUNDLES ARE MORI DREAM S PACES 9
+generates R(c)as ak-algebra. In order to prove the claim, consider ( u,m,m)∈c(M×
+Z×Zd) such that H0(X,SymmE ⊗ Lm)u/n⌉}ati≀nslash= 0. By Lemma 3.3 (a) and (b), there exist
+J={j1,j2,...,j q} ∈ J, such that ( u,m,m)∈QJand
+evt0/parenleftbig
+H0(X,SymmE ⊗Lm)u/parenrightbig
+=Symm
+E/parenleftBig
+Vλj1(u,m,m)
+l(j1),Vλj2(u,m,m)
+l(j2),...,Vλjq(u,m,m)
+l(jq)/parenrightBig
+⊗Lm.
+Now, fix an expression ( u,m,m) =/summationtext
+g∈GJcgg, wherecg∈Z≥0for each g∈GJ. Let
+g= (ug,mg,mg) for each g∈GJ, be the corresponding coordinates in M×Z×Zd. From
+Lemma 3.3 (a) applied in the cone QJ, we get that for each g∈GJ,
+evt0/parenleftbig
+H0(X,SymmgE ⊗Lmg)ug/parenrightbig
+=Symmg
+E/parenleftBig
+Vλj1(g)
+l(j1),Vλj2(g)
+l(j2),...,Vλjq(g)
+l(jq)/parenrightBig
+⊗Lmg.
+Since/summationtext
+g∈GJcgλjh(g) =λjh(u,m,m) for each h∈ {1,2,...,q}, it follows by Lemma 3.3
+(c) and Lemma 3.1 that in the commutative diagram
+/circlemultiplytext
+g∈GJ(Rg)⊗cg
+/d15/d15/d47/d47R(u,m,m)/d127/d95
+evt0
+/d15/d15 /circlemultiplytext
+g∈GJ(SymmgE⊗Lmg)⊗cg /d47/d47SymmE⊗Lm
+the images of/circlemultiplytext
+g∈GJ(Rg)⊗cgandR(u,m,m)inSymmE⊗Lmcoincide. The injectivity of
+evt0implies that/circlemultiplytext
+g∈GJ(Rg)⊗cgsurjects onto R(u,m,m), and this completes the proof. /square
+Corollary 4.2.The projectivization P(E)of a rank two toric vector bundle Eover
+the projective simplicial toric variety Xis a Mori dream space.
+Proof. The finite generation of any Cox ring of P(E) in the sense of Hu and Keel is a
+consequence of Theorem 4.1. Any simplicial toric variety is Q-factorial, and a projective
+bundle over a Q-factorial variety is again Q-factorial, hence the additional conditions
+follow at once from the hypotheses. /square
+The aim of Theorem 4.1, and of our work in this article, is to give furthe r steps toward
+investigating the finite generation of the Cox rings of projectivizat ions of toric vector
+bundles of arbitrary rank (see Question 7.2 in [9]). In the articles [12] and [8], the authors
+prove the finite generation of the Cox rings of T-varieties of complexity one. In particular,
+they provide alternative proofs of Theorem 4.1, since the project ivization of a rank two
+vector bundle has complexity one as a T-variety. We expect that our proof will provide
+new insights for treating this question in the case of toric vector bu ndles of arbitrary rank.
+References
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+Jos´e Luis Gonz ´alez, Department of Mathematics, University of Michigan.
+E-mail address :jgonza@umich.edu
\ No newline at end of file
diff --git a/1001.0840.txt b/1001.0840.txt
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+On the stability of a very dense deuterium -tritium plasma ball
+Yuri Kornyushin
+Maître Jean Brunschvig Research Unit, Chalet Shalva, Randogne, 3975 -CH
+Email: jacqie@bluewin.ch
+General conditions of a stability of a very dense deuterium -tritium
+plasma ball are discussed. It is shown that the decrease in the size of a
+plasma ball (increase in the plasma density) can be achieved somewhat
+easier when the temperature and the pressure in the plasma ball are kept
+high en ough for all the particles, the nuclei and the delocalized
+electrons, to be described by classical statistics.
+PACS: 52.27.Aj; 52.25.Kn; 25.60.Pj
+Let us consider a very dense neutral deuterium -tritium plasma ball of a radius R, consisting
+ofNnuclei and Ndelocalized electrons. The Gibbs free energy of the plasma ball is
+ΦPVKUTS, (1)
+where Pis the external pressure, V(4/3)R3is the volume of a ball, Kis the kinetic
+energy of the particles, Uis the potential electrostatic energy, Tis the absolute
+temp erature, and Sis the entropy. When the delocalized electrons are degenerate, the
+kinetic energy of them in the ball of a radius RisKe1.105 N5/32/mR2[1]. The potential
+electrostatic energy of a neutral plasma ball consists of two terms. One of them,
+0.75e2Ng, describes a negative contribution of the screening to the potential electrostatic
+energy [1] (here eis elementary charge, and 1/ gis the screening radius). The origin of this
+contribution is a relaxation of the delocalized electrons to lower en ergy state of screening
+of the ions. When delocalized electrons relax to more equilibrium state, screening the ions,
+the electrostatic energy of a sample decreases. Screening cuts off the long -range
+electrostatic field of the ions, that is, leads to the de crease in the electrostatic energy of a
+sample. This problem was considered in [2]. In [2] the ionization energy of impurities in
+semiconductors was discussed. The effective ionization energy Wewas calculated in [2] for
+the case when the radius of a groun d state is much smaller than the screening radius. The
+result was as follows: We=W0e2Ng(here W0is the initial ionization energy). This
+corresponds to taking into account only the first term in Eq. (29) in [1].
+The other term of the electrostatic ene rgy,1.2e2N/R, describes (in the model of
+homogeneously distributed in a ball charge of every particle) a negative contribution,
+coming from the absence of the self -action in quantum mechanics. Potential energy U(r) in
+Schrödinger equation for elementary charged particle does not include the potential energy
+of the interaction of the considered elementary charged particle with the field produced by
+the same particle. This means that in quantum mechanics the action of the elementary
+charged particle on itse lf, that is a self -action, is not taken into account. Averaging
+Schrödinger equation quantum -mechanically, one can obtain that the energy of a particle
+consists of its kinetic energy and its potential energy in the external (relative to the
+regarded partic le) potential field. So one can see that the energy of the regarded particle do
+not include the potential energy of the field created by the particle. This means that any2elementary charge ein any state has no electrostatic potential energy of its own in quantum
+mechanics, see e.g. [3]. From this follows that the electrostatic potential energy of the
+regarded neutral plasma ball is 1.2e2N/R.
+For degenerate electrons gis proportional to 1/ R1/2[1,4]. For the classical ones gis
+proportional to 1/ R3/2[4]. For Nclassical particles, as is well known, the negative
+contribution of the entropy to Gibbs free energy is 3kTNln(R/R0), where R0is some
+parameter.
+So, when electrons are degenerate, the negative part of the Gibbs free energy (one term
+is proportional to 1/R, the other one is proportional to 1/ R1/2, the third one is proportional to
+ln(R/R0)) cannot compensate for the positive term of the kinetic energy, proportional to
+1/R2, in the case of a decrease in the ball radius R. So when temperature Tand the plasma
+density N/Vcorrespond to the delocalized electrons being degenerate, the further decrease
+in the plasma ball size is somewhat difficult.
+On the contrary, when the temperature is high enough for all 2 Nparticles (the nuclei
+and the delocalized elect rons) to be classical, the kinetic energy of all the particles is 3 kTN.
+It does not depend on the ball radius R. In this case further decrease in the plasma ball size
+can be somewhat easier. Decrease in the plasma ball size, however, leads to the increase in
+the plasma density and brings the plasma ball out of the classical statistics, when the
+plasma density increases. This makes the further decrease in the plasma ball size more
+difficult. To keep the process of the plasma ball size decreasing easier, the temperature and
+the pressure should be kept high enough, when classical statistics is relevant. The process
+of decreasing the plasma ball size can result in the beginning of nuclear fusion, when the
+plasma density becomes high enough. However this requires extremely high temperature
+and pressure.
+It should be noted that the process of decreasing of the regarded plasma ball size, when
+the particles are essentially degenerate, is possible [5,6]. Here it was discussed how to keep
+this process of the plasma den sification easier.
+References
+1.Y. Kornyushin, Low Temp. Physics, 34, 838 (2008).
+2.G. F. Neumark, Phys. Rev. B5, 408 (1972).
+3.M. Ya. Amusia and Y. Kornyushin, Contemp. Phys., 41, 219 (2000).
+4.C. Kittel, Quantum Theory of Solids , Wiley, New York (1963).
+5.J. Lin dl, Phys. Plasmas, 2, 3933 (1995).
+6.S. Eliezer et al, Laser Part. Beams, 21, 599 (2003).
+О стабильности очень плотной дейтериево-тритиевой плазмы
+Ю. В. Корнюшин
+Maître Jean Brunschvig Research Unit, Chalet Shalva, Randogne, 3975 -CH
+Email: jacqie@bluewin.ch
+Обсуждаются общие условия стабильности очень плотной 
+дейтериево-тритиевой плазмы. Показано, что уменьшение размера 
+плазменного шара (увеличение плотности плазмы) можно 3достигнуть легче при обеспечении достаточно высоких температур 
+и давлений, при которых все частицы (как ядра, так и 
+делокализованные электроны) описываются классической 
+статистикой.
+PACS: 52.27.Aj; 52.25.Kn; 25.60.Pj
+Рассмотрим очень плотный шар нейтральной дейтериево-тритиевой плазмы радиуса 
+R, содержащий N ядер и N делокализованных электронов. Термодинамический 
+потенциал рассматриваемого шара
+ΦPVKUTS. (1)
+Здесь P обозначает внешнее давление, объём шара V(4/3)R3,K– кинетическая 
+энергия частиц, U– потенциальная электростатическая энергия, T– абсолютная 
+температура и S– энтропия. Если делокализованные электроны вырождены, их 
+кинетическая энергия в шаре радиуса R есть Ke1.105 N5/32/mR2 [1]. Потенциальная 
+электростатическая энергия нейтральной плазмы состоит из двух слагаемых. Одно 
+из них, 0.75e2Ng, представляет отрицательный вклад экранирования в 
+потенциальную электростатическую энергию [1] (здесь e– элементарный заряд, а 1/g
+– радиус экранирования). Это слагаемое происходит от релаксации 
+делокализованных электронов к более равновесному экранирующему ионы 
+состоянию. При этом электростатическая энергия образца уменьшается. 
+Экранирование обрезает дальнодействующее электростатическое поле ионов и 
+обусловливает уменьшение электростатической энергии образца. Эта задача была 
+рассмотрена в [2]. В [2] обсуждалась энергия ионизации примесей в 
+полупроводнике. В [2] была вычислена эффективная энергия ионизации примеси We
+для случая когда радиус основного состояния значительно меньше радиуса 
+экранирования. Было получено что We=W0e2ng (здесь W0– энергия ионизации без 
+учёта экранирования). Этот результат соответствует учёту только первого 
+слагаемого в уравнении (29) в [1].
+Другое слагаемое электростатической энергии, 1.2e2N/R, описывает (в модели 
+однородно распределённых в шаре элементарных электрических зарядов всех 
+частиц) отрицательный вклад, происходящий от отсутсвия самодействия 
+элементарных зарядов в квантовой механике. Потенциальная энергия U(r) в 
+уравнении Шредингера для заряженной элементарным зарядом частицы не включает 
+потенциальную энергию взаимодействия рассматриваемой элементарной частицы с 
+полем, которое она сама производит. Это значит что в квантовой механике дейстие 
+частицы, заряженной элементарным зарядом, на саму себя исключается. Усредняя 
+уравнение Шредингера квантово-механически получаем, что энергия частицы 
+состоит из кинетической энергии и потенциальной энергии частицы во внешнем по 
+отношению к ней поле. Так что видим, что энергия рассматриваемой частицы не 
+содержит потенциальную энергию поля, создаваемого самой частицей, то есть в 
+квантовой механике элементарный заряд e в любом состоянии не обладает 
+собственной потенциальной электростатической энергией (смотри, например [3]). 
+Отсюда следует, что электростатическая энергия рассматриваемого нейтрального 
+плазменного шара есть 1,2e2N/R.4Для вырожденных электронов g пропорционально 1/R1/2[1,4]. Для классических 
+электронов g пропорционально 1/R3/2 [4]. Для N частиц, как известно, отрицательный 
+вклад энтропии в термодинамический потенциал есть 3kTNln(R/R0), где R0–
+некоторый параметр. 
+Для вырожденных электронов отрицательные слагаемые термодинамического 
+потециала (одно из них  пропорционально 1/R, а другое пропорционально 1/R1/2,
+третье пропорционально ln(R/R0)) не могут скомпенсировать положительное 
+слагаемое, кинетическую энергию, пропорциональную 1/R2, при уменьшении 
+радиуса шара R. Так что, когда температура T и плотность плазмы N/V
+соответствуют вырождению делокализованных электронов, дальнейшее уменьшение 
+размеров плазменного шара затруднительно.
+Когда же температура достаточно высока и все частицы, как ядра так и 
+делокализованные электроны, являются классическими, кинетичесая энергия 2N
+частиц K3kTN; она не зависит от радиуса шара R. В этом случае дальнейшее 
+уменьшение размера плазменного шара облегчается. Уменьшение размера 
+плазменного шара приводит к увеличению плотности плазмы, что в свою очередь 
+выводит плазму из области классической статистики, когда плотность плазмы 
+возрастает. При этом дальнейшее уменьшение размера плазменного шара становится 
+затруднительным. Для продолжения уменьшения размера плазменного шара 
+(увеличения плотности плазмы) температуру и давление следует держать на 
+должном уровне, когда имеет место классическая статистика. Этот процесс может 
+привести к началу термоядерной реакции. Однако для такого развития событий 
+требуются очень высокие температура и давление.
+Следует заметить, что процесс уменьшения рамера плазменного шара в случае 
+существенно вырожденных частиц возможен [5,6]. Здесь обсуждается возможность 
+облегчения процесса уплотнения плазмы. 
+Литература
+1.Y. Kornyushin, Low Temp. Physics, 34, 838 (2008).
+2.G. F. Neumark, Phys. Rev. B5, 408 (1972).
+3.M. Ya. Amusia and Y. Kornyushin, Contemp. Phys., 41, 219 (2000).
+4.C. Kittel, Quantum The ory of Solids , Wiley, New York (1963).
+5.J. Lindl, Phys. Plasmas, 2, 3933 (1995).
+6.S. Eliezer et al, Laser Part. Beams, 21, 599 (2003).
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+arXiv:1001.0841v1  [cond-mat.dis-nn]  6 Jan 2010Anomalous diffusion in disordered multi-channel
+systems
+R´ obert Juh´ asz
+Research Institute for Solid State Physics and Optics, H-1525 Bud apest, P.O.Box 49,
+Hungary
+E-mail:juhasz@szfki.hu
+Ferenc Igl´ oi
+Research Institute for Solid State Physics and Optics, H-1525 Bud apest, P.O.Box 49,
+Hungary
+Institute of Theoretical Physics, Szeged University, H-6720 Sze ged, Hungary
+E-mail:igloi@szfki.hu
+Abstract. We study diffusion of a particle in a system composed of Kparallel
+channels, wherethe transitionrateswithin the channelsarequenc hed randomvariables
+whereas the inter-channel transition rate vis homogeneous. A variant of the
+strong disorder renormalization group method and Monte Carlo simu lations are used.
+Generally, we observe anomalous diffusion, where the average dista nce travelled by
+the particle, [ /an}bracketle{tx(t)/an}bracketri}ht]av, has a power-law time-dependence [ /an}bracketle{tx(t)/an}bracketri}ht]av∼tµK(v), with
+a diffusion exponent 0 ≤µK(v)≤1. In the presence of left-right symmetry of
+the distribution of random rates the recurrent point of the multi-c hannel system is
+independent of K, and the diffusion exponent is found to increase with Kand decrease
+withv. In the absence of this symmetry, the recurrent point may be shif ted with K
+and the current can be reversed by varying the lane change rate v.Anomalous diffusion in disordered multi-channel systems 2
+1. Introduction
+Brownian motion is one of the most studied and the best understood stochastic
+processes, both in homogeneous and in random environments [1, 2, 3, 4]. Physical
+motivation to study diffusion in disordered media is provided by transp ort processes
+(molecular diffusion, electric conduction, biological transport in a ce ll) on the one hand
+and by relaxation in complex systems (in spin glasses and random ferr omagnets) on
+the other. For asymmetric hoppings, when the microscopic transit ion rates depend
+on the direction, too, disorder can strongly modify the properties of the transport in
+dimensions, d <2 [4].
+The effect of disorder is particularly strong in 1 d, in which case several exact results
+are known [5, 6, 7, 8, 9, 10]. Many of those are obtained in a mathema tically rigorous
+way, such as Sinai-scaling of the average square displacement, [ /an}bracketle{tx2(t)/an}bracketri}ht]avin time,t[7].
+This behaves at the recurrent point as:
+[/an}bracketle{tx2(t)/an}bracketri}ht]av∼ln4(t), (1)
+where/an}bracketle{t.../an}bracketri}htstands for the thermal average for a given realization of disorder and [...]av
+is used to denote averaging over quenched disorder. Some presum ably exact results are
+alsocalculatedbyaphysically motivatedmethodusingastrongdisord errenormalization
+group approach [11]. In this method sites of the lattice having the sm allest barrier in
+the random potential landscape are consecutively eliminated and ne w transition rates
+are perturbatively calculated between remaining sites. Thus during renormalization the
+fastestratesareeliminatedandatthefixedpointonlytheslowexcit ationsremain, which
+govern the cooperative dynamics of the system. The fixed point of the transformation
+is a so called infinite disorder fixed point, at which the ratio of transitio n rates between
+neighbouring sites goes either to zero or to infinity, so that the tra nsformation becomes
+asymptotically exact. The infinite disorder fixed point of the 1 drandom random walk
+is isomorphic with the fixed point of some 1 drandom quantum magnets[12], such as
+the random transverse-field Ising spin chain[13] and some critical p roperties of the two
+models are interrelated [14].
+It is convenient to introduce a control parameter of the random r andom walk, δ,
+which has the value δ= 0 at the recurrent point, which corresponds to the fixed point
+of the strong disorder renormalization group transformation. In the recurrent point the
+correlation length of the system is divergent. On the contrary for δ/ne}ationslash= 0 when the walk is
+driftedtooneoranotherdirectionthecorrelationlengthisfinitean dthisregimeiscalled
+the transient phase. In the transient phase the average travelle d distance, [ /an}bracketle{tx(t)/an}bracketri}ht]av, is
+non-zero and has a power-low time-dependence:
+[/an}bracketle{tx(t)/an}bracketri}ht]av∼tµ1, (2)
+where the diffusion exponent, µ1=µ1(δ)≤1, is a continuous function of the control
+parameter. According to renormalization group analysis in the vicinit y of the critical
+point it behaves as[11, 14]:
+µ1(δ) =δ+O(δ2). (3)Anomalous diffusion in disordered multi-channel systems 3
+In reality, the channel in which the transport takes place is often n ot strictly
+one-dimensional. For example, we may think of vehicular traffic in multi- lane roads
+[15] or the active transport in cells, which is realized by molecular moto rs moving on
+filaments that are composed of several proto-filaments [16]. Bes ides, the problem of one
+dimensional random walks with long (but bounded) jump lengths is also equivalent to
+random walks with nearest neighbour jumps in a strip of finite width. T his problem
+arises also in the context of disordered iterated maps [18] or for ra ndom walks with
+inertia [19]. Comparatively, less results are known for random walks o n strips [17, 20],
+which can be considered to be in a way between dimensions one and two . Only very
+recently, it has been shown that at the recurrent point of the mult i-channel system,
+Sinai scaling in Eq.(1) is valid [21]. The same conclusion is obtained by one o f us by
+using a variant ofthestrong disorder renormalizationgroupmetho d[22]. Indeed, during
+renormalization the strip is renormalized to a random chain, thus res ults quoted above
+in Eqs. (2) and (3) are expected to hold in this case, too.
+In this work we are going to study the properties of a random walk in a disordered
+multi-channel system, we pay particular attention to the transien t phase. In this
+study we consider systems composed of Kchannels with quenched random transition
+rates within channels and allow symmetric transitions between chann els with a site-
+independent lane change rate v. We measure the diffusion exponent in the system,
+µK(v), and study its dependence on Kandv. In the strong coupling limit, v→ ∞, and
+inthevicinityoftherecurrent pointweobtainanalyticalresults. In thegeneralsituation
+we use numerical methods, either Monte-Carlo simulations or numer ical application of
+the strong disorder renormalization group method.
+The rest of the paper is organized as follows. The model is presente d in Sec. 2
+and its properties are analyzed in the strong coupling limit in Sec.3. The essence of
+the renormalization procedure and phenomenological scaling consid erations are given in
+Sec.4. Numerical analysis for K= 2 and for varying lane change rate is given in Sec.5.
+Our results are discussed in the final Section.
+2. Model
+2.1.1dsystems
+We define the model first in one-dimension, on the discrete points, l= 1,2,...,L, with
+periodic boundaries. A continuous time stochastic process is consid ered in which the
+particle jumps from site lto sitel+1 with a transition rate, pl, which may be different
+from the transition rate of the reverse jump, ( l+1→l), denoted by ql+1. Here the pairs
+of rates ( pl,ql) are independent and identically distributed random variables drawn from
+the distribution P(p,q). We generally consider the limit L→ ∞and are interested in
+the asymptotic (long time) behavior.
+The control parameter of the 1 drandom walk is defined as[11, 14]:
+δ=[lnp]av−[lnq]av
+var[lnp]+var[ln q], (4)Anomalous diffusion in disordered multi-channel systems 4
+where var[ y] stands for the variance of y. In the 1 dproblem, the mean velocity and the
+diffusion constant of a particle on a finite ring is expressed as a Keste n random variable
+and therefore the diffusion exponent, µ1, can be calculated exactly even for not small δ
+[5, 6, 9]. It is given by the positive root of the equation:
+/bracketleftbigg/parenleftbiggq
+p/parenrightbiggµ1/bracketrightbigg
+av= 1, (5)
+provided it is smaller than one. Otherwise the particle moves with a finit e mean
+velocity and µ1= 1. Indeed for small δEq(5) gives back the leading behavior of
+the renormalization group result in Eq.(3).
+In this paper we consider two types of random environments. For type A (or
+symmetric) randomness the transition rates satisfy the relation that the product, plql,
+does not depend on the position and the distribution of the rates is s ymmetric at the
+recurrent point, i.e. P(p,q) =P(q,p). Here we use a bimodal disorder:
+pl=1
+ql=/braceleftBigg
+λwith probability c,
+1/λwith probability 1 −c,(6)
+for which the control parameter is given by:
+δA=2c−1
+4c(1−c)1
+lnλ, (7)
+and the diffusion exponent is:
+µ1(A) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleln(c−1−1)
+2lnλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (8)
+Fortype B (or asymmetric) randomness the distribution of rates is non-symmetric at
+the recurrent point either. We have used the following distribution:
+pl= 1, ql=/braceleftBigg
+λwith probability c,
+1/λwith probability 1 −c,(9)
+the control parameter is given by:
+δB=−2c−1
+4c(1−c)1
+lnλ, (10)
+and the diffusion exponent is:
+µ1(B) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleln(c−1−1)
+lnλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (11)
+2.2. Multi-channel systems
+The multi-channel system consists of i= 1,2,...,Kchannels in which the transition
+rates are drawn independently from each other but from the same distribution P(p,q).
+Lane changes occur with rate vbetween nearest neighbor lanes (also between lane i=K
+andi= 1), which does not depend on the position. This means that the tra nsition rate,
+w[(l,i)→(k,j)], for a jump from site ( l,i) to site ( k,j) is given by:
+w[(l,i)→(k,j)] =δi,j/parenleftBig
+p(i)
+lδl+1,k+q(i)
+lδl,k/parenrightBig
++δl,k(δi+1,j+δi−1,j)v .(12)Anomalous diffusion in disordered multi-channel systems 5
+Similarly to the one dimensional model, the multi-channel system has a recurrent
+point at, otherwise it is transient in the positive or negative direction s. The transient
+regions consist of two phases: in the sublinear transient phase µK<1, while in the
+ballistic phase µK= 1.
+For symmetric randomness, it is easy to see that the recurrent po int is not shifted
+forK >1 compared to that of K= 1. It is due to the fact that here recurrence is
+connected with the left-right symmetry of the distribution of the t ransition rates, which
+stays invariant after connecting the channels, too. On the contr ary, for asymmetric
+randomness, the condition of recurrence depends on Kandv.
+3. Strong-coupling limit
+First, we discuss the strong coupling limit ( v→ ∞), which is simple enough to obtain
+analyticalresults. Inthefollowing, thesetof Ksiteswiththehorizontalcoordinate lwill
+be called the lth layer. In the limit v→ ∞, the particle being in the lth layer, visits all
+sites of the layer infinitely many times before jumping to another laye r. Consequently,
+the effective transition rate between layers is just the average of the interlayer jump
+rates over the vertical coordinate:
+˜pl=/summationtext
+ip(i)
+l
+K,˜ql=/summationtext
+iq(i)
+l
+K, (13)
+and the problem is reduced to an effective one-dimensional random w alk. The
+corresponding 1 drandom environment is given by the transition rates, ˜ pland ˜ql, the
+distributions of which follow fromthe distributions of the original cha in, respectively. In
+the following we analyse the properties of the strong coupling limit for the two different
+types of randomness.
+3.1. Symmetric randomness
+Let us consider a general symmetric randomness, where p(i)
+lq(i)
+l= 1 at all ( l,i) sites
+and the probability density of p(i)
+lis denoted by ρ(p). As we have already mentioned,
+ifδA= 0 then the system is recurrent for any K >1, due to the symmetry of the
+distribution P(p,q).
+Specially for K= 2, thediffusionexponent µ2canbegiven intermsof µ1asfollows.
+For the effective one-channel system with rates given in Eq. (13)) , equation (5) can be
+written as:/integraldisplay
+dp1ρ(p1)/integraldisplay
+dp2ρ(p2)/parenleftbigg1/p1+1/p2
+p1+p2/parenrightbiggµ2
+=
+/integraldisplay
+dp1ρ(p1)(p1)−µ2/integraldisplay
+dp2ρ(p2)(p2)−µ2. (14)
+The diffusion exponent µ1for the case K= 1 is determined by/integraldisplay
+dpρ(p)(p)−2µ1= 1. (15)Anomalous diffusion in disordered multi-channel systems 6
+Comparing the latter two equations, we obtain the simple relation:
+µ2= 2µ1, v→ ∞. (16)
+ForK >2 a similar relation holds only for weak disorder and in the vicinity of the
+recurrent point. Here for weak disorder we use the parametrizat ionpl= 1 +ǫl,
+with|ǫl| ≪1 in which case in the K-channel system the control parameter, δ(K)
+Ais
+simply related to the control parameter in the one lane system, δA, asδ(K)
+A≃KδA.
+Consequently one obtains for the diffusion exponent from eq.(3):
+µK≃Kµ1, µ1≪1 and v→ ∞. (17)
+Thus with increasing number of channels, the extension of the sub- linear transient
+regime, measured by δAis decreasing and, for any fixed δA>0 and sufficiently large K,
+the transport becomes ballistic.
+For large values of Kthe effective transition rates in Eq.(13) will approach a
+Gaussian distribution with a relative mean deviation, which is decreasin g withK.
+Consequently the disorder of the effective couplings will be weak for largeK, thus one
+expects that µK/K→const. We have checked this expectation by using the bimodal
+distribution in Eq.(6), when for general K, the diffusion exponent has been obtained by
+solving numerically the equation:
+K/summationdisplay
+n=0K!
+n!(K−n)!cK−n(1−c)n/parenleftbiggnλ2+K−n
+n+(K−n)λ2/parenrightbiggµK
+= 1. (18)
+The obtained values of µKfor different number of channels are shown in Fig.1 for two
+different bimodal parameters. Indeed, µKis found to be approximately linear in Kfor
+0204060
+10 20µK
+Kc=1/3, λ2=1/8
+c=1/5, λ2=1/2
+Figure 1. Diffusion exponent, µK, plotted against the number of channels, K, in the
+large coupling limit for symmetric randomness in Eq.(6) calculated from Eq.(18).
+largeKin both cases.Anomalous diffusion in disordered multi-channel systems 7
+3.2. Asymmetric randomness
+Next, we consider the bimodal disorder in Eq.(9). The probability c=c∗
+Kfor which the
+system is recurrent is the root of the following equation:
+K/summationdisplay
+n=0K!
+n!(K−n)!(c∗
+K)K−n(1−c∗
+K)nln(K−n)λ+nλ−1
+K= 0. (19)
+ForK= 2 it is explicitly given as:
+c∗
+2(λ) =1
+2(1+f−/radicalbig
+1+f2), f= lnλ/ln[(λ+λ−1)/2]. (20)
+For weak disorder we have lim λ→1c∗
+2(λ) =c∗
+1= 1/2, which is shifted to lim λ→∞c∗
+2(λ) =
+1−√
+2/2 in the strong disorder limit. For general K, this strong disorder limiting value
+is given by: lim λ→∞c∗
+K(λ) = 1−2−1/K.
+The dynamical exponent, µK, is given as the positive root of the equation:
+K/summationdisplay
+n=0K!
+n!(K−n)!(c)K−n(1−c)n/parenleftbigg(K−n)λ+nλ−1
+K/parenrightbiggµK
+= 1. (21)
+It is easy to see that the boundary between the sub-linear and the ballistic phase where
+µ1= 1 is independent of K, sinceµK= 1 for any K >1 ifµ1= 1.
+4. Renormalization and phenomenological scaling
+We are interested in the time dependence of the displacement of the particle in the
+random environment defined in the previous section. To obtain infor mation on the
+dynamics in an infinite system, we shall follow an indirect way in which finit e systems
+ofsizeK×Lareconsidered andthedynamical exponent isestimated fromthe fi nite-size
+scaling ofthestationaryprobability current. Fortheestimationof thelatter quantity we
+shall apply a real space renormalization group method which is a varia nt of the strong
+disorder renormalization group method and has been used to study disordered quantum
+spin chains [12, 13, 14], random walks in one-dimensional random envir onments[11] and
+random nonequilibrium processes[23, 24]. For higher dimensional dyn amical systems,
+just as the present problem, the procedure has been formulated in terms of transition
+rates [22, 27, 26]: states with the largest exit rate are gradually re moved from the
+configuration space andone can infer the dynamics fromthe finite- size scaling of the last
+remaining exit rate. This method has been applied to a series of proble ms [22, 27, 26].
+A common difficulty of these methods in higher dimensions is that the to pology of the
+network of transitions between configurations does not remain inv ariant but it becomes
+more and more complicated in the course of the procedure. In the p resent problem,
+one can however keep the topology invariant by eliminating complete la yers, see Fig. 2.
+The rates of “diagonal” transitions ( l,i)→(l±1,j) withi/ne}ationslash=jare initially zero (see
+Eq.(12)), but during the procedure they become positive. Thus, t he invariant topology
+of the network of transitions for K= 2 is a ladder with both vertical anddiagonal edges.
+In the followings, the precise definition of the renormalization metho d is given.Anomalous diffusion in disordered multi-channel systems 8
+l
+l+1l+1 l−1 l−2
+l−2 l−1l+2
+l+2
+Figure 2. Illustration of a renormalization step for K= 2. Layer lis decimated and
+new transitions are created (indicated by dashed lines), the effect ive rates of which are
+calculated from the rates of transitions starting from or ending in t he eliminated sites
+(shown in red).
+In the steady state, the probability that the particle resides on sit e (l,i) will be
+denoted by πl(i). Introducing the row vectors πl= (πl(i))1≤i≤K, for a fixed random
+environment (i.e. for a fixed set of transition probabilities), the sta tionary probabilities
+satisfy the following system of linear equations which express the co nservation of
+probability[22]:
+πlSl=πl−1Pl−1+πl+1Ql+1,1≤l≤L. (22)
+Here, the K×Kmatrices Pl,QlandSlare composed of the jump rates as follows:
+Pl(i,j) =w[(l,i)→(l+1,j)]
+Ql(i,j) =w[(l,i)→(l−1,j)]
+Sl(i,j) =−w[(l,i)→(l,j)]i/ne}ationslash=j
+Sl(i,i)≡/summationdisplay
+j[Pl(i,j)+Ql(i,j)]−/summationdisplay
+j/negationslash=iSl(i,j). (23)
+Let us introduce the exit rate of the lth layer as Ω l:= 1//bardblS−1
+l/bardbl, where the matrix
+norm/bardbl · /bardblis defined as /bardblA/bardbl:= max i/summationtext
+j|A(i,j)|. In a renormalization step, layer l
+is decimated, which results in a one layer shorter system with effectiv e rates in the
+adjacent layers given by
+˜Pl−1=Pl−1S−1
+lPl
+˜Ql+1=Ql+1S−1
+lQl
+˜Sl−1=Sl−1−Pl−1S−1
+lQl
+˜Sl+1=Sl+1−Ql+1S−1
+lPl. (24)
+This transformation leaves the sojourn probability at the remaining sites, as well as the
+probability current Jin the horizontal direction invariant. Following these rules, L−1
+layers are decimated so that a single one remains. One can show that the remainingAnomalous diffusion in disordered multi-channel systems 9
+effective rates at the last remaining layer are independent of the or der of elimination of
+the other L−1 layers. There are therefore only Ldifferent states the procedure can
+end up with, depending on the layer which is not decimated. The magnit ude of the
+current that is invariant under the renormalization is related to the smallest among the
+Lpossible last exit rates Ω ˜l≡minl{˜Ωl}as|J|=|π˜l(˜P˜l−˜Q˜l)| ≃/summationtext
+iπ˜l(i)Ω˜lfor large L.
+An efficient algorithm for finding the rate Ω ˜lis the following. For the sake of
+convenience, let us assume that Lis an integer power of 2. First, the system is divided
+into two equal parts. The sites in the first part are decimated (in an arbitrary order).
+This modifies the two surface layers of the other part. Then, star ting again from the
+initial system, the second part is eliminated. This modifies the two sur face layers of the
+first part. This procedure is then applied recursively to each part w ith a halved length
+until the procedure ends up with Lparts of length 1. The exit rates of these Llayers
+are calculated and the smallest one is selected.
+The relation to strong disorder renormalization methods is more tra nsparent if the
+smallest remaining exit rate Ω ˜lis found in the way that the layers with the actually
+largest exit rates are eliminated one by one. Then the last remaining la yer is expected
+to be layer ˜lwith high probability. It has been shown that, in the course of the st rong
+disorder renormalization, the vertical jump ratesare non-decre asing while the horizontal
+and diagonal jump rates tend to zero, so that the multi-channel s ystem renormalizes to
+an effective one-channel system [22]. Nevertheless, the previous algorithm needs only
+O(LlnL)operationspersampleincontrast withthe O(L2)operationsofthesecond one.
+Another advantage of the former implementation is that it applies wit hout difficulties
+also to discrete randomness, such as for symmetric distribution wh ere all exit rates are
+initially equal. In the numerical calculations we have therefore impleme nted the former
+algorithm.
+Inthefollowing, theminimal exit rateΩ ˜lwill bedenotedbyΩ L, asweareinterested
+the scaling of this quantity with the system size Lfor fixedK. For a one-channel system
+(K= 1) it is known from the analytical solution of the renormalization equ ations that
+ΩL, intypical samples, scales withthesystem size asΩ L∼L−1/µ1inthetransient phase,
+whereµ1is the unique positive root of Eq. (5) [25]. On the other hand, from ex act
+results on the time-dependence of displacement [5, 6, 9], one obtain s that the current
+scales as J∼L−1/µ1forµ1<1 andJ∼L−1forµ1>1. Here, µ1is again the root of
+Eq. (5). Comparing the finite-size behavior of Jand Ω Lone concludes that the typical
+value of the probability at the last site ˜lmust tend to an L-independent constant for
+largeLifµ1<1. This means that the walker typically spends a finite fraction of the
+time at a single site (favourite site). For K >1, we have no exact expression of µK
+at our disposal but the multi-channel system renormalizes to an eff ective one-channel
+system. Therefore we expect that the typical value of the proba bility of finding the
+particle in the favourite layer is still asymptotically L-independent. This implies that
+J∼ΩL∼L−1/µK, (25)
+where the first proportionality holds if the diffusion exponent µKis less than one. TheAnomalous diffusion in disordered multi-channel systems 10
+importance ofthis relationis thatthe diffusion exponent can beestim ated by computing
+ΩLwhich is a much less numerical effort than the direct solution of Eqs. ( 22).
+In the frame of a phenomenological random trap picture of the sys tem, the
+renormalization procedure can be naturally interpreted. The rand om environment in
+which the particle moves can be thought of as a collection of consecu tive, spatially
+localized trapping regions. In some regions, the particle may spend v ery long times and
+obviously the trapping times in these regions determine predominant ly the traveling
+time of the particle. The renormalization procedure means a kind of c oarse-graining
+of the environment in the course of which trapping regions are step by step contracted
+and, simultaneously, trapping regions that had been contracted t o a single site are
+completely eliminated one by one in the order of increasing trapping tim e. For late
+times, when the latter process dominates, the inverse of the effec tive exit rate gives
+roughly the release time from the corresponding trapping region. T he inverse of the last
+exit rate Ω Lcan thus be interpreted as the mean release time from the “deepes t” trap
+of the environment. Close to the fixed-point of the transformatio n (i.e. when the largest
+effective exit rate is very small) the system can be treated as an effe ctive one-channel
+system. Fromtheanalytical treatment oftherenormalizationgro upequations for K= 1
+it is known[25] that the distribution of exit rates has a power-law tail P>(˜Ω)∼˜Ω−µK,
+which isexpected toholdfor K >1(see thenumerical results inFig. 5). Themeantime
+needed for the particle to make a complete tour on a finite ring is given approximately
+ast= 1/J∼/summationtext
+i1/˜Ωi. IfµK<1 (i.e. in the zero-velocity phase), the above sum of
+random variables is dominated by the largest one, i.e./summationtext
+i1/˜Ωi∼1/ΩLand we obtain
+J∼ΩL. We see, that this simple phenomenological theory corroborates r elation (25).
+5. Numerical Results
+We have applied two numerical methods for the investigation of the m odel: Monte Carlo
+simulations and the renormalization group method. In case of Monte Carlo simulations,
+we have generated 104independent random environments with L= 107,K= 2 and
+simulated the process in each sample with 100 different (equidistantly distributed)
+starting positions for times up to t= 222−228and computed the average displacement,
+[/an}bracketle{tx(t)/an}bracketri}ht]av. Besides, we have have considered finite systems of size L= 24−214,
+K= 2, generated 104−106independent random environments for each system size
+and computed the average of lnΩ Lby the renormalization group method. Regarding
+the efficiency, the two methods complement each other. The renor malization is more
+efficient in the vicinity of the recurrent point, whereas by the simulat ion one can treat
+longer time-scales far from the recurrent point.
+We start with the asymmetric randomness, where we choose c= 0.4 andλ= 0.1
+in the bimodal distribution in Eq.(9). The average displacements calcu lated by Monte
+Carlo simulations are shown in Fig.3 for various values of the lane chang e ratev.
+As seen in this figure, the direction of the walk has changed by varyin gv. For
+small lane change rate, v < vc≈1.66 the walker goes ahead, whereas for large values ofAnomalous diffusion in disordered multi-channel systems 11
+ 0
+-1
+-2
+-3
+-4
+ 16  12  8  4  0[x(t)]av
+ln[t]v=1.6  
+v=1.65
+v=1.66
+v=1.7  
+Figure 3. Left: Average position (horizontal coordinate) of a walker in time f or
+different values of vfor the bimodal distribution in Eq.(9) with c= 0.4 andλ= 0.1
+measured in Monte Carlo simulation. The direction of the motion is reve rsed at
+v≈1.66.
+v > vc, it moves backwards. At v=vcthe average displacement tends to a constant in
+the limit t→ ∞. We have also studied the typical time-scale in the problem, tL∼Ω−1
+L,
+by the renormalization method. According to phenomenological sca ling at the recurrent
+point we have lnΩ L∼L1/2(see Eq.(1)) whereas, in the transient phase, the dependence
+is algebraic: Ω L∼L−1/µ2(see Eq.(25)). In order to check both type of behavior, we
+have plotted [lnΩ L]avas a function of L1/2(Fig.4, left panel) and as a function of ln L
+(Fig.4, right panel).
+-200-150-100-50 0
+ 120  80  40  0[lnΩL]av
+L1/2v=0.5  
+v=1     
+v=1.66
+v=2     
+v=3      0
+-50
+-100
+-150
+ 10  8  6  4  2[lnΩL]av
+lnLv=0.5  
+v=1     
+v=1.66
+v=2     
+v=3     
+Figure 4. The average of the logarithm of the minimal exit rate, [lnΩ L]av, plotted
+againstL1/2(left panel) and against ln L(right panel) for the bimodal distribution
+used in Fig.3. At the recurrent point, v=vc, the linear dependence in the plot in
+the left panel corresponds to Sinai-scaling in Eq.(1). In the transie nt phase, linear
+dependence is seen in the plot in the right panel, and the asymptotic v alues of the
+slopes identified with 1 /µ2(v) depend on v(see Eq.(25)).
+The results in Fig.4 are in complete agreement with the scaling theory. In
+the transient phase, the diffusion exponent µ2(v), which can be extracted from the
+asymptotic slopes of the curves in the right panel, are found to dep end onv. We are
+going to study this issue in details for the symmetric disorder, where the condition of
+recurrence is exactly known.Anomalous diffusion in disordered multi-channel systems 12
+For symmetric disorder we consider also two-lane systems and use t he bimodal
+randomness in Eq.(6). In our first example we have c= 1/8 andλ2= 1/3, which,
+according to Eq.(8), results in a diffusion exponent µ1= 1/3 forK= 1. By application
+of the renormalization group procedure we have calculated sample d ependent minimal
+exit rates Ω Land studied their distribution. For different lengths, L, the appropriate
+scaling combination is: Ω LL1/µ2, see Eq.(25), and the distributions in terms of this
+reduced variable show a nice scaling collapse. This is illustrated in Fig.5 f or the lane
+change rate, v= 0.1, using a diffusion exponent µ2= 0.726. The scaling curve is very
+well described by the Fr´ echet-distribution[28]:
+PFr(u) =µ2uµ2−1exp(−uµ2), (26)
+withu=u0ΩLL1/µ2. Here the non-universal constant, u0, which depends on the
+amplitude of the tail of P>(Ω) is the only free parameter.
+The distribution of the low-energy excitations in strongly disordere d systems has
+been studied previously in Ref.[29]. Using the strong disorder renorm alization group
+method it has been argued that in several models the smallest excita tions, the energy
+of which follow an asymptotic power-law behavior, can be considered asymptotically
+independent and therefore their distribution is in the Fr´ echet for m. Our result in Fig.5
+indicates that the disordered multi-channel problem satisfies this c onjecture.
+-12-8-4
+-5 0 5 10 15 20ln[P(x)]
+x=-ln(ΩLL1/µ2)L=28
+L=29
+L=210
+L=211
+L=212
+L=213
+L=214
+Figure 5. Distribution of the scaling variable Ω LL1/µ2, calculated for K= 2 with
+symmetric bimodal randomness ( c= 1/8, λ2= 1/3 andv= 0.1). The scaling collapse
+is obtained with a diffusion exponent µ2= 0.726. The solid line is the Fr´ echet-
+distribution given in Eq.(26) with the parameter u0= 0.392.
+Wehaverepeatedthepreviousrenormalizationgroupcalculationfo rdifferentvalues
+ofvand estimated the diffusion exponent by comparing the average valu e of [lnΩ L]av
+and that of [lnΩ L/2]av. From this we have obtained effective exponents:
+µeff
+2(L) =[lnΩL/2]av−[lnΩL]av
+ln2, (27)
+which are shown in the left panel of Fig.6 as a function of τ=L1/µ2. These exponents
+are compared with those calculated by the Monte Carlo simulations. I n this case theAnomalous diffusion in disordered multi-channel systems 13
+average position of the walker, [ x(t)]av, is calculated at time, t, and from the results at
+two different times, tandt·∆, we have obtained:
+µeff
+2(t) =ln[x(t·∆)]av−ln[x(t)]av
+ln∆. (28)
+These are also shown in the left panel of Fig.6 as a function of τ=t.
+ 0.9
+ 0.8
+ 0.7
+ 0.6
+ 16  12  8  4µeff(τ)
+ln[τ]v=10-3 MC
+v=10-2 MC
+v=10-1 MC
+v=1     MC
+v=10-3 RG
+v=10-2 RG
+v=10-1 RG
+v=1     RG 1.1
+ 1
+ 0.9
+ 0.8
+ 0.7
+ 16  12  8  4µeff(τ)
+ln[τ]v=10-3 MC
+v=10-2 MC
+v=10-1 MC
+v=1     MC
+v=10-3 RG
+v=10-2 RG
+v=10-1 RG
+v=1     RG
+Figure 6. Effective diffusion exponent of the random walk in the two-channelp roblem
+for various values of the lane change rate, v, calculated either by the Monte Carlo
+simulations ( τ=t) or by the renormalization method ( τ=Lµ2). We used symmetric
+bimodal randomness. Left panel: c= 1/8, λ2= 1/3, so that µ1= 1/3. Right panel:
+c= 1/3,λ2= 0.1768, so that µ1= 0.4.
+As can be seen in this figure the calculated effective exponents have strongτ
+dependenceforsmallsystems(renormalizationmethod)andfors horttimes(simulation).
+However, the true exponents obtained by the two methods, which should be seen in the
+limitτ→ ∞seem to agree well. As expected, µ2is a continuous function of the lane
+change rate and it is found to decrease monotonically with v. In the large v-limit it
+approaches the known exact limiting value: lim v→∞µ2= 2µ1, see Eq.(16). Similar
+tendency is seen in the right panel of Fig.6 for different parameters of the distribution:
+c= 1/3,λ2= 0.1768, so that µ1= 0.4. In both distributions the diffusion exponent
+remains smaller than one, even for very small value of v, thus the system is always in
+the anomalous diffusion regime.
+This type of behavior is going to change if we use such type of distribu tions, for
+whichµ1reaches or exceeds the value of 0 .5. In this case 2 µ1≥1, thus the transport in
+the two-lane system is ballistic. This type of behavior is illustrated in Fig .7 forc= 1/4,
+λ2= 1/9, i.e.µ1= 0.5 (left panel) and for c= 1/4,λ2= 1/5, i.e.µ1= 0.683 (right
+panel).
+Indeed the effective exponents calculated by the Monte Carlo meth od approach
+one, which is a clear signal of the ballistic behavior. (We note that in th e borderline
+case,µK= 1, there may be logarithmic corrections to the ballistic behavior jus t as for
+K= 1 atµ1= 1 [4])
+In this case the exponent extracted from the finite size scaling of Ω Lis greater than
+1 and thus differs from the true diffusion exponent, which is equal to 1. The exponent
+obtained by the renormalization in this phase appears in the correct ions to scaling toAnomalous diffusion in disordered multi-channel systems 14
+ 1.4
+ 1.2
+ 1
+ 0.8
+ 16  12  8  4µeff(τ)
+ln[τ]v=10-1 MC
+v=1     MC
+v=10-3 RG
+v=10-2 RG
+v=10-1 RG
+v=1     RG 2.2
+ 1.8
+ 1.4
+ 1
+ 16  12  8  4µeff(τ)
+ln[τ]v=10-2 MC
+v=10-1 MC
+v=1     MC
+v=10-2 var
+v=10-3 RG
+v=10-2 RG
+v=10-1 RG
+v=1     RG
+Figure 7. The same as in Fig.6 for ballistic transport. Here the µeffcalculated by
+the renormalization group method are the disorder induced diffusion exponents, which
+represent correction to scaling effects. Left panel: c= 1/4,λ2= 1/9, so that µ1= 0.5.
+Right panel: c= 1/4,λ2= 1/5, thusµ1= 0.683. Here we also present the exponent
+3−σ(indicated by ”var”), in which σis the effective exponent of the second moment
+of the displacement measured in the simulations.
+the ballistic behavior. To be concrete, this exponent governs the s caling of the second
+moment ofthedisplacement, Σ L, which isexpected toscaleasΣ L∼Lσ, withσ= 3−µ2.
+In the right panel of Fig.7 the renormalization estimates for µ2are compared with the
+estimates for 3 −σ, which are obtained in Monte Carlo simulations and good agreement
+is found.
+Finally, we mention that, for symmetric randomness, it is possible to d rive the
+system from the sublinear transient phase to the ballistic phase by v arying the lane
+change rate.
+6. Discussion
+Inthispaperwehavestudiedthediffusionofaparticleinaquasi-one- dimensionalsystem
+which consists of Kdisordered channels. We have investigated the transient phase by
+MonteCarlosimulationsandbyavariantofthestrongdisorderreno rmalizationmethod.
+In this phase, the diffusion exponent is found to depend on Kand the lane change rate.
+For asymmetric disorder, the recurrent point is shifted compared to that of the one-
+lane model and the interaction between the lanes is able to cause a co unter-intuitive
+reversal ofthedirectionofmotionwhenthelanechangerateisvar ied. Thisphenomenon
+is analogous to the ratchet effect in a saw-tooth potential. For sym metric disorder the
+recurrent point stays invariant and thediffusion exponent inthe tr ansient phase is found
+to increase with Kand decrease with the lane change rate.
+The numerical value of the diffusion exponent for K >1 and for intermediate
+lane change rates 0 < v <∞is non-trivial. It is easy to see, that a naive random
+trap description, when applied to the lanes separately, gives a wron g result. Indeed,
+identifying the trapping regions in each lane, then regarding them po int-like traps
+which are concentrated on a single site would lead after simple argume ntation on theAnomalous diffusion in disordered multi-channel systems 15
+probability of simultaneous occurrence of traps with a waiting time at leasttin all lanes
+to the diffusion exponent µK=Kµ1, independent of vand the type of randomness.
+The problem with this argument is that, although, the finite extensio n of trapping
+regions does not play a role in one dimension, this circumstance canno t be disregarded
+in a coupled multi-channel system. After long times the particle may e ncounter long
+enough overlapping trapping regions in all lanes, where it cannot pas s even in the most
+favourable lane without changing lanes. Such overlapping regions mu st be treated as
+a whole and they lead to a non-trivial dependence of the diffusion exp onent on the
+distribution of transition rates.
+The choice that the lane change rate is site-independent allows an alt ernative
+interpretation of the model. The particle can be thought to move in o ne-dimension
+but in a time-dependent potential, which fluctuates stochastically w ith ratevbetween
+Kdifferent random potentials each of which is determined by the jump r ates in the K
+lanes. A similar but spatially non-random problem, the first passage t ime of a particle
+over a homogeneous barrier the height of which fluctuates betwee n two different values
+has been thoroughly studied [30, 31, 32] In case of very low flipping f requency the
+mean first passage time is dominated by first passage time over the h igher barrier. For
+very high flipping frequency the particle feels an average potential and the first passage
+time is determined by the mean height of the potential. For intermedia te frequencies
+the passing of the particle occurs primarily when the height of the ba rrier is small.
+Therefore, when varying the frequency, the mean first passage time exhibits a minimum.
+This phenomenon is called resonant activation. The optimal frequen cy is a known to
+decrease exponentially with the minimal height of the barrier [32].
+Our result are qualitatively in agreement with the properties of reso nant activation.
+For long times, the particle encounters higher and higher barriers, therefore the optimal
+lane change rateat which therelease time is minimal becomes smaller an dsmaller as the
+particle proceeds. Consequently, for late times ( t→ ∞), the diffusion exponent assumes
+its maximum in the limit v→0, in agreement with the numerical results. Note that
+these two limits can not be interchanged, therefore the (asympto tic) diffusion exponent
+of the model as a function of the lane change rate is non-analytical atv= 0.
+Acknowledgments
+This work has been supported by the Hungarian National Research Fund under grant
+No OTKAK62588, K75324andK77629andbya German-Hungarianex change program
+(DFG-MTA).
+References
+[1] Alexander S, Bernasconi J, Schneider W R and Orbach R 1981 Rev. Mod. Phys. 53175
+[2] Havlin S and Ben-Avraham D 1987 Adv. Phys. 36695
+[3] Haus W and Kehr K W 1987 Phys. Rep. 150263
+[4] Bouchaud J-P and Georges A 1990 Phys. Rep. 195127Anomalous diffusion in disordered multi-channel systems 16
+[5] Solomon F 1975 Ann. Prob. 31
+[6] Kesten H, Kozlov M V and Spitzer F 1975 Compositio Math. 30145
+[7] Sinai Ya G 1982 Theor. Prob. Appl. 27247
+[8] Golosov A O 1984 Commun. Math. Phys. 92491
+[9] Derrida B and Pomeau Y 1982 Phys. Rev. Lett. 48627; Derrida B 1983 J. Stat. Phys. 31433
+[10] Zeitouni O 2006 J. Phys. A: Math. Gen. 39R433
+[11] Fisher D S, Le Doussal P and Monthus C 1998 Phys. Rev. Lett. 803539
+Le Doussal P, Monthus C, and Fisher D S 1999 Phys. Rev. E 594795
+[12] Ma S-K, Dasgupta C and Hu C-K 1979 Phys. Rev. Lett. 431434
+[13] Fisher D S 1992 Phys. Rev. Lett. 69534; 1995 Phys. Rev. B 516411
+[14] For a review, see: Igl´ oi F and Monthus C 2005 Phys. Rep. 412277
+[15] Chowdhury D, Santen L, and Schadschneider A, 2000 Phys. Rep. 329199
+[16] J. Howard, Mechanics of Motor Proteins and the Cytoskeleton (Sinauer, Sunderland, 2001).
+[17] Key E S 1984 Ann. Prob. 12529
+[18] Radons G 2004 Physica D 1873
+Fichtner A and Radons G 2005 New J. Phys. 730
+[19] Sz´ asz D and T´ oth B 1984 J. Stat. Phys. 3727
+Alili S 1999 J. Stat. Phys. 94469
+[20] Bolthausen E and Goldsheid I 2000 Commun. Math. Phys. 214429
+[21] Bolthausen E and Goldsheid I 2008 Commun. Math. Phys. 278253
+[22] Juh´ asz R 2008 J. Phys. A: Math. Theor. 41315001
+[23] J. Hooyberghs, F. Igl´ oi and C. Vanderzande 2003 Phys. Rev. Lett. 90100601; 2004 Phys. Rev. E
+69066140
+[24] Juh´ asz R, Santen L and Igl´ oi F 2005 Phys. Rev. Lett. 94, 010601; 2005 Phys. Rev. E 72046129
+[25] Igl´ oi F, Juh´ asz R and Lajk´ o P 2001 Phys. Rev. Lett. 861343
+Igl´ oi F 2002 Phys. Rev. B 65064416
+[26] Pigolotti S and Vulpiani A 2008 J. Chem. Phys. 128154114
+[27] Monthus C and Garel T 2008 J. Phys. A: Math. Theor. 41255002; 2008 J. Stat. Mech. P07002;
+2008J. Phys. A: Math. Theor. 41375005
+[28] J. Galambos, 1978 The Asymptotic Theory of Extreme Order Statistics Wiley, New York
+[29] R. Juh´ asz, Y.-C. Lin, and F. Igl´ oi 2006 Phys. Rev. B 73224206
+[30] Doering C R and Gadoua J C 1992 Phys. Rev. Lett. 692318
+[31] Bier A and Astumian R D 1993 Phys. Rev. Lett. 711649
+[32] Bogu˜ n´ a M, Porr` a J M, Masoliver J and Lindenberg K 1998 Phys. Rev. E 573990
\ No newline at end of file
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+arXiv:1001.0842v1  [cond-mat.soft]  6 Jan 2010Fluctuating Nematodynamics using the Stochastic Method of Lines
+A. K. Bhattacharjee, Gautam I. Menon, and R. Adhikari
+The Institute of Mathematical Sciences, C.I.T. Campus, Tar amani, Chennai 600013, India
+(Dated: November 16, 2018)
+We construct Langevin equations describing the fluctuation s of the tensor order parameter Qαβ
+in nematic liquid crystals by adding noise terms to time-dep endent variational equations that follow
+from the Ginzburg-Landau-de Gennes free energy. The noise i s required to preserve the symmetry
+and tracelessness of the tensor order parameter and must sat isfy a fluctuation-dissipation relation
+at thermal equilibrium. We construct a noise with these prop erties in a basis of symmetric traceless
+matrices and show that the Langevin equations can be solved n umerically in this basis using a
+stochastic version of the method of lines. The numerical met hod is validated by comparing equi-
+librium probability distributions, structure factors and dynamic correlations obtained from these
+numerical solutions with analytic predictions. We demonst rate excellent agreement between numer-
+ics and theory. This methodology can be applied to the study o f phenomena where fluctuations in
+both the magnitude and direction of nematic order are import ant, as for instance in the nematic
+swarms which produce enhanced opalescence near the isotrop ic-nematic transition or the problem
+of nucleation of the nematic from the isotropic phase.
+PACS numbers: 05.10.Gg, 02.50.Ey, 02.60.Cb, 64.70.mf
+I. INTRODUCTION
+Fluctuation phenomena in nematic liquid crystals are
+typically studied within Ericksen-Leslietheory, which as-
+sumes that the orientation of the normalized nematic di-
+rector is the only fluctuating variable[1]. This approxi-
+mation is adequatedeep within the nematic phase, where
+the strength of nematic order is not significantly affected
+by thermal fluctuations. However, in the vicinity of
+the weakly first-order isotropic-nematic transition, sig-
+nificant fluctuations in the nematic order are observed,
+suggesting that the phase-only approximation embodied
+in Leslie-Ericksen theory is inadequate [2]. The study of
+nucleation in quenches from the isotropic to the nematic
+phase involves the growth of one phase within another,
+mandating the use of descriptions capable of describing
+both isotropic and nematic phases on the same foot-
+ing. In these and similar situations, a tensorial descrip-
+tion of nematic order which uses the symmetric, trace-
+less quadrupole moment tensor Qαβ, is appropriate, as
+first clarified by de Gennes in his Ginzburg-Landau the-
+ory of the isotropic-nematic transition[3]. The Ginzburg-
+Landau-de Gennes (GLdG) approach provides a simple,
+but accurate phenomenological description of nematic
+fluctuations in the static case[4].
+The description of the fluctuating dynamics of the ori-
+entation tensor within GLdG theory has received consid-
+erably less attention[5, 6]. Understanding the results of
+inelastic scattering experiments on nematic systems[7],
+the description of the rate of nucleation into the ne-
+matic phase[8], the modeling of nontrivial stresses aris-
+ing from Casimir interactions[9] and the calculation of
+the spectrum of capillary waves on the isotropic-nematic
+interface[10] are all problems which require a dynami-
+cal theory of fluctuations in the orientation tensor. This
+problem is addressed in this paper, in which we present
+and solve the Langevin equations for dynamical fluctu-ations at equilibrium for the nematic orientation ten-
+sor. These are stochastic non-linear partial differential
+equations for the five components of the orientation ten-
+sor. Analytical solutions can be obtained when these
+equations are linearized. For the solution of the gen-
+eral non-linear equations, we propose an efficient numer-
+ical method, based on a stochastic generalization of the
+method of lines. We compare our results with analytic
+results where such calculations are possible, finding ex-
+cellent agreement.
+II. FLUCTUATING NEMATODYNAMICS
+Orientational order in the nematic phase is described
+by a second-rank, symmetric traceless tensor Qαβ(x,t).
+This is the second moment of the microscopic orienta-
+tional distribution function. The tensor can be expanded
+as
+Qαβ=3
+2S(nαnβ−1
+3δαβ)+1
+2T(lαlβ−mαmβ).(1)
+The three principal axes of this tensor, obtained by di-
+agonalizing Qαβin a local frame, specify the direction of
+nematic ordering n, the codirector land the joint nor-
+mal to these, labeled by m. The principal values Sand
+Trepresent the strength of ordering in the direction of n
+andm, quantifying, respectively, the degree of uniaxial
+and biaxial nematic order.
+The static fluctuations of Qαβcan be calculated from
+a Ginzburg-Landau functional, first proposed by de
+Gennes, based on an expansion in rotationally invariant
+combinations of Qαβand its gradients. The Ginzburg-
+Landau-de Gennes functional Fis
+F=/integraldisplay
+d3x[1
+2ATrQ2+1
+3BTrQ3+1
+4C(TrQ2)2(2)
++E′(TrQ3)2+1
+2L1(∂αQβγ)(∂αQβγ)].2
+Here,A=A0(1−T/T∗)T∗denoting the supercooling
+transition temperature, A0a constant, L1is an elas-
+tic constant and α,β,γdenote the Cartesian directions.
+From the inequality1
+6(TrQ2)3≥(TrQ3)2, higher pow-
+ers ofTrQ3can be excluded for the description of the
+uniaxial phase. Uniaxial phases are described by E′=
+0 while biaxial phases require E′/negationslash= 0. For the nematic
+phase (rod-like molecules) B <0 whereas for the dis-
+cotic phase (plate-like molecules) B >0. The quantities
+C andE′must always be positive to ensure bounded-
+ness and stability of the free energy in all phases. We
+omit other symmetry-allowed gradient terms in this pa-
+per, thus working in the limit where all three Frank con-
+stants are assumed to be equal. Such symmetry-allowed
+terms, as also total derivative surface terms, can be ac-
+countedfor without essentialchange, usingthe numerical
+method described below.
+In the limit that hydrodynamic interactions may be
+neglected, i.e.the Rouse or free-draining limit, the dy-
+namical fluctuations of Qαβare not coupled to other hy-
+drodynamic variables. The Langevin equations are those
+appropriate to a non-conserved order parameter with an
+overdamped, relaxational dynamics of the form
+∂tQαβ=−ΓαβµνδF
+δQµν+ξαβ, (3)
+Here the kinetic coefficients Γ αβµν, defined as Γ αβµν=
+Γ[δαµδβν+δανδβµ−2
+3δαβδµν], ensure that the dynam-
+ics preserves the symmetry and tracelessness property of
+the order parameter. In the absence of long-range forces,
+a local approximation for the kinetic coefficients is ad-
+equate and Γ can be taken as constant. The ξαβare
+symmetric, traceless Gaussian white noises, which sat-
+isfy a fluctuation-dissipation relation at equilibrium of
+the form
+/angb∇acketleftξαβ(x,t)/angb∇acket∇ight= 0, (4)
+/angb∇acketleftξαβ(x,t)ξµν(x′,t′)/angb∇acket∇ight= 2kBTΓαβµνδ(x−x′)δ(t−t′).(5)
+HerekBis the Boltzmann constant, Tthe temperature
+and/angb∇acketleft/angb∇acket∇ightdenotes the average over the probability distri-
+bution of the noise. These Langevin equations, together
+with the fluctuation-dissipationrelation for the noise, en-
+sure that the stationary one-point probability distribu-
+tionofQαβ,P[Qαβ], convergestoBoltzmannequilibrium
+withP[Qαβ]∼exp(−F/kBT).
+The equations above are five coupled, non-linear
+stochasticpartialdifferentialequations,withanoiseterm
+which has a tensorial structure. A numerical method
+of solution must maintain the symmetry and traceless
+ofQαβ. To ensure equilibrium dynamics, it must also
+maintain the balance between fluctuation and dissipa-
+tion. These two stringent requirements may be satisfied
+by transforming to a basis in which Qαβis traceless and
+symmetric by construction. Symmetry and tracelessness
+ofQαβis automatic. As we show below, the noise can be
+constructed out of independent Gaussian noises.We expand the orientational tensor in a basis of sym-
+metric traceless matrices Ti
+αβas
+Qαβ(x,t) =5/summationdisplay
+i=1ai(x,t)Ti
+αβ (6)
+withT1=/radicalbig
+3/2ˆ zˆ z,T2=/radicalbig
+1/2(ˆ x ˆ x−ˆ y ˆ y),T3=√
+2ˆ x ˆ y,T4=√
+2ˆ xˆ zandT5=√
+2ˆ y ˆ z. The com-
+plete basis of matrices is orthogonal in the sense that
+Ti
+αβTj
+αβ=δij. Inpreviousworkwehavepresentedexplic-
+itly the equations for the basis coefficients ai(x,t) that
+follow from the deterministic part of the relaxational ki-
+netics∂tQαβ=−ΓαβµνδF/δQ µν[11]. (These differ from
+the equations derived by others in that we include all
+non-linearities as well as an additional symmetry-allowed
+gradient ( L2) term.) Here we focus on how an explicit
+construction of the noise can be implemented by expand-
+ing in the same basis.
+We expand the noise as
+ξαβ(x,t) =5/summationdisplay
+i=1ξi(x,t)Ti
+αβ, (7)
+where each ξi(x,t) is a zero-mean Gaussian white noise.
+From the orthogonality of the basis the inverse relation
+is
+ξi(x,t) =/summationdisplay
+α,βξαβ(x,t)Ti
+αβ. (8)
+From this, and the fluctuation-dissipation relation it fol-
+lows that
+/angb∇acketleftξi(x,t)ξj(x′,t′)/angb∇acket∇ight=/summationdisplay
+αβµν/angb∇acketleftξαβ(x,t)ξµν(x′,t′)/angb∇acket∇ightTi
+αβTj
+µν,(9)
+=/summationdisplay
+αβµν2kBTΓαβµνTi
+αβTj
+µνδ(x−x′)δ(t−t′),
+= 2kBTΓδijδ(x−x′)δ(t−t′).
+Thisshowsthat the non-triviallycorrelatednoise ξαβcan
+be constructed from uncorrelated noises ξi. Thus, by
+construction, the noise ξαβis symmetric, traceless and
+satisfies the fluctuation-dissipation relation.
+When anharmonic terms are ignored in the Ginzburg-
+Landau-de Gennes functional, the Langevin equations
+are linear and correlation functions can be calculated ex-
+plicitly in the Tbasis. Then, the Langevin equations of
+motion in terms of
+ai(q,t) =/integraldisplay
+d3xexp(−iq·x)ai(x,t),(10)
+are
+∂tai(q,t) =−Γ(A+L1q2)ai(q,t)+ξi(q,t).(11)
+From this, the static and dynamic correlations follow im-
+mediately,
+Cij(q) =/angb∇acketleftai(q)aj(−q)/angb∇acket∇ight=kBT
+A+L1q2δij,(12)3
+0 5 10 15 200.10.20.30.40.50.60.70.80.91
+τ<v(t)v(t+τ)>
+  
+Numeric
+Analytic−1−0.5 00.5102468x 104
+vFrequency
+FIG. 1: (Color online) Autocorrelation function for the
+Ornstein-Uhlenbeck process, showing /angbracketleftv(t)v(t+τ)/angbracketrightas a func-
+tionofthetime increment τ. Theinsetshows thehistogram of
+fluctuations. It is Gaussian with the expected variance. The
+numerical parameters chosen are Γ = kBT= 0.1,dt= 1.0.
+The average is taken over 10 independent realizations while
+the integration is performed for 106SRK4 steps.
+and
+Cij(q,τ) =/angb∇acketleftai(q,t)aj(−q,t+τ)/angb∇acket∇ight (13)
+=Cij(q)exp[−Γ(A+L1q2)τ],
+whereCij(q) is the static structure factor. The static
+and dynamic correlations for Qαβare then obtained by
+returning to the original basis. The stationary probabil-
+ity distribution generated by the Langevin dynamics is
+Gaussian with zero mean and variance Cij(q), consistent
+with Boltzmann equilibrium.
+III. STOCHASTIC METHOD OF LINES
+The fluctuating nematodynamics equations contained
+in Eq. (3) are five non-linear stochastic partial differen-
+tial equations. In general these have no analytical so-
+lutions and reliable numerical methods are therefore es-
+sential for their study. Here we combine the method of
+lines for solving initial-value partial differential equations
+with a stochastic Runge-Kutta integrator for systems of
+stochastic ordinary differential equations. This enables
+us to construct an accurate and efficient solver for the
+equations of fluctuating nematodynamics. Our results
+here build on previous work [11], where a method of
+lines approach was used to solve the deterministic time-
+dependent Ginzburg-Landau equations numerically. The
+methodology here can thus be thought of as a general-
+ization of the method of lines to stochastic partial differ-
+ential equations.
+The method of lines is based on the idea of semidis-
+cretisation, where an initial-value partial differentialequation in space and time is discretised only in the spa-
+tial variable [12]. This yields a (possibly large) system
+of ordinary differential equations which is then solved by
+standard numerical integrators. To apply this method to
+stochastic partial differential equations, we must account
+forthefactthatintegratorsforordinarydifferentialequa-
+tions do not automatically provide efficient and accurate
+solutions of stochastic differential equations. Qualita-
+tively, the noise term in a stochastic differential equation
+is a rapidly varying function and hence must be inte-
+grated with some care. At a more technical level, the
+noise is a Wiener process and the theory of stochastic
+integration must be used to evaluate it correctly [13].
+Common stochastic integrators include those due to
+Maryuama [14] and Milstein [15]. In this work, we use
+anintegratorproposedrecentlybyWilkie[16], basedona
+multi-step Runge-Kutta strategy. The integratoris accu-
+rate and easy to implement by making small changes to
+a deterministic Runge-Kutta integrator. Further, since
+it is an explicit integrator, no matrix inversions are in-
+volved. This makes it attractive when the method of
+lines discretisation produces a large system of ordinary
+differential equations, as in our case.
+To test Wilkie’s algorithm for a stochastic Runge-
+Kutta integrator, henceforth denoted as SRK4, we first
+check that the fluctuation-dissipation is obeyed. We
+performed a simple benchmark test on the Ornstein-
+Uhlenbeck process, represented as the Ito differential
+equation,
+dv(t) =−Γv(t)dt+dW(t), (14)
+wheredW(t) =√2kBTΓdtN(0,1) is the increment of
+the stochastic variable in the interval dt, andN(0,1) is
+a zero-mean unit-variance normal deviate. By construc-
+tion, the increments of this stochastic variable are inde-
+pendent and normally distributed with mean /angb∇acketleftdW(t)/angb∇acket∇ight=
+0. The particular choice of the variance ensures that the
+equilibrium distribution of vis a Gaussian with variance
+kBT. The stationary two-point autocorrelation of the
+velocity from Eqn.(14) is,
+/angb∇acketleftv(t)v(t+τ)/angb∇acket∇ight=kBTexp(−Γτ). (15)
+Fig.(1) shows the autocorrelation as a function of time
+and the histogram of equal-time fluctuations of v. The
+variance /angb∇acketleftv2/angb∇acket∇ight=kBTis correctly reproduced, as is the
+exponential decay of the autocorrelation function. We
+conclude that SRK4 is suitable as an integrator for prob-
+lems where the fluctuation-dissipation relation must be
+maintained.
+IV. NUMERICAL METHOD
+We now apply the method of lines together with
+SRK4 to obtain a stochastic method lines discretisa-
+tion (SMOL) for the equations of fluctuating nemato-
+dynamics. We benchmark our numerical results by com-
+paring autocorrelations within a harmonic theory which4
+−0.6 −0.4 −0.2 0 0.2 0.4 0.6050100150200250300350400450
+a1 (q)Frequency
+FIG. 2: (Color online) Histogram of the real part of ai(q) for
+nx=ny= 6 in a box of dimension Lx=Ly= 16, using a
+harmonic free energy, with kBT=A= 0.05 andL1= 0.5,
+Γ = 1.0. The fluctuations are Gaussian with the expected
+variance. The histogram is obtained from 20 independent
+realizations, each realization contributing 4000 time ste ps.
+accurately describes fluctuations about the isotropic
+phase. We then consider expansions about the ordered
+state, comparing static correlationsobtained analytically
+within the Frank approximation with our numerical re-
+sults.
+We use a finite-difference discretisation with nearest-
+neighbour stencils for gradients and the Laplacian. We
+implement periodic boundary conditions. Specifically, in
+three dimensions, we consider a box of dimension Lx,Ly
+andLzalong the Cartesian directions, and grid these
+lengths with equal grid spacing ∆ x= ∆y= ∆z= 1.
+The latter defines lattice units for the spatial coordi-
+nate. We define corresponding discrete time units for
+the temporal variables by choosing ∆ t= 1. Fourier
+modes are labelled by the wave-vector q= (qx,qy,qz),
+where each component is of the form qα= 2πnα/Lα,
+withnα= 0,1,2,...,(Lα−1).
+With this discretisation, theLaplacianin Fourierspace
+is given by
+L(q) = 2[cos( qx)+cos(qy)+cos(qz)−3].(16)
+The nearest-neighbour finite difference stencil suffers
+from lack of isotropy at high wavenumbers. This can be
+improvedthroughtheuseofhigher-pointstencils[17–19].
+Applying the method of lines discretisation to Eq. (3)
+reduces it to a system of stochastic ordinary differential
+equations, whose Fourier representation in the harmonic
+approximation of Eq. (11) is
+∂tai(q,t) =−ΓDai(q,t)+ξi(q,t).(17)
+The Fourier representation of the drift-diffusion dynam-
+ics is encoded in the linear operator D,
+D=A−L1L(q). (18)nxny
+  
+−15 −10 −5 0 5 10 15−15−10−5051015 Numeric
+Analytic
+(a)
+0 0.5 1 1.5 2 2.500.10.20.30.40.50.60.70.80.91
+qC(q)
+  
+Numeric
+Analytic
+(b)
+FIG. 3: (Color online) (a) Contour plot of the structure fact or
+Cij(q) and (b) angular average of Cij(q). The parameters are
+kBT=A= 0.05,L1= 0.5,Γ = 1.0. The time averaging is
+over 5×104time steps and ensemble averaging is over 20
+independent realizations in a box with Lx=Ly= 64. The
+relaxation time scale is τ= (ΓA)−1= 20, the diffusion time
+scale of the smallest Fourier mode is τd=L2
+x/(4π2L1Γ) =
+207.51 and the correlation length is λ=/radicalbig
+L1/A= 3.16.
+Fourier representations of the one and two-dimensional
+method of lines discretisations are obtained by setting
+the corresponding wavenumbers to zero. The static and
+dynamic autocorrelations in Fourier space follow in a
+straightforward manner though the replacement of q2by
+its discrete Laplacian representation. The results are
+Cij(q) =kBT
+Dδij, (19)
+Cij(q,τ) =Cij(q)exp(−ΓDτ). (20)
+It is also useful to define an angle-averagedstructure fac-5
+0 5 10 15 2000.20.40.60.81
+τ< ai(q, t) ai(−q, t+τ) >
+  
+qx = qy = 2
+  
+ Analytic
+  
+qx = qy = 3
+  
+qx = 2, qy = 3
+FIG. 4: (Color online) Autocorrelation Cij(q,τ) for the linear
+Langevin equation, calculated for small wavenumbers. Nu-
+merical parameters are kBT=A= 0.05,L1= 0.5,Γ = 1.0.
+The integration is performed for 4 .2×103time steps on a 162
+grid. The time average is taken over 4 ×103time steps and
+an ensembleaverage is takenover40 independentrealizatio ns.
+The relaxation time scale is τ= (ΓA)−1= 20, the diffusion
+time scale of the shortest Fourier mode τd=L2
+x/(4π2L1Γ)∼
+13 and the correlation length λ=/radicalbig
+L1/A= 3.16.
+0 1 2 3 4−0.500.511.522.533.54x 10−3
+qCθ (q)
+  
+Numeric
+Analytic
+FIG. 5: (Color online) Angularly averaged structure factor
+of the nematic phase. The numerical parameters are kBT=
+0.05,A=−3.5,B=−10kBT,C= 2.67,L1= 32.0,Γ = 0.01
+on a 642grid. The relaxation time scale is τ= (ΓA)−1=
+28.57, the diffusion time scale of the shortest Fourier mode i s
+τd=L2
+x/(4π2L1Γ) = 324 .23 and the correlation length is λ=/radicalbig
+L1/A= 3.02. The time average is taken over 5 ×104time
+steps and an ensemble average is taken over 20 independent
+realizations.torC(q) =/summationtext
+|q|=qCij(q/angb∇acket∇ightfor comparison with the nu-
+merical simulation.
+We now compare theoretical and numerical results: In
+Fig. (2) we show the histogram of the ai(q) for a par-
+ticular Fourier mode. This is normally distributed, as
+expected, with zero mean and variance as required by
+thermal equilibrium. Similarly, all Fourier modes exam-
+ined have correct normal distributions. The variances
+obtained are compared in Fig. 3(a) with the analytical
+values by plotting contours of Cij(q). There is excel-
+lent agreement. A close inspection reveals some degree
+ofanisotropyin both the analyticaland numericalresults
+at high wavenumbers. This is attributed to the lack of
+isotropy of the nearest-neighbour finite-difference Lapla-
+cian mentioned earlier. However, the anisotropies are
+removed upon angular averaging, as shown in Fig. 3(b).
+Thus, the present discretisation should be adequate in
+most cases, unless highly accurate isotropies are required
+from the simulation. From these results, we conclude
+that correlations in thermal equilibrium are accurately
+captured by the stochastic method of lines approach.
+We next compare the dynamics of fluctuations at equi-
+librium, by comparing two-point autocorrelation func-
+tions calculated analytically and numerically. Fig.(4)
+showsCij(q,τ) for three sets of Fourier modes. The ex-
+ponential decay of the autocorrelation function is repro-
+duced accurately within the numerics and fit the theo-
+retical curve very closely. We conclude, therefore, that
+thestochasticmethodoflinesaccuratelyreproducesboth
+static and dynamic fluctuations in a harmonic theory.
+Finally, we compare theory and simulation in a situ-
+ation where a linearization of the Qαβequations about
+Qαβ= 0 is inapplicable, that of director fluctuations
+within the nematic phase. In the Tbasis, the equations
+of motion are
+∂tai=−Γ [(A+CTrQ2)ai (21)
++ (B+6E′TrQ3)Ti
+αβQ2
+αβ−L1∇2ai]+ξi.
+where,Q2
+αβis the traceless symmetric projection of
+Q2
+αβ.These equations of motion are anharmonic, and the
+analytical solutions obtained earlier within the harmonic
+expansion are no longer available for comparison. We
+therefore extract the fluctuations of the angular displace-
+ments from the uniform nematic ground state using an
+approach based on the Frank free energy.
+Consider an uniform uniaxial nematic with director n0
+with small fluctuations δn(x). Decomposing the fluctu-
+ations into parts parallel and perpendicular to n0and
+imposing the normalization of the director, we find from
+the Frank free energy that,
+/angb∇acketleft|δn⊥(q)|2/angb∇acket∇ight=kBT
+Kq2, (22)
+whereK= (9S2/2)L1is a Frank constant [4]. Since fluc-
+tuations in the plane perpendicular to n0can be charac-
+terized through a single angle θ, an equivalent result is6
+/angb∇acketleftθ(q)θ(−q)/angb∇acket∇ight=kBT/Kq2. In the semidiscrete represen-
+tation, we obtain
+/angb∇acketleftθ(q)θ(−q)/angb∇acket∇ight=−kBT
+KL(q). (23)
+Fig.(5) shows the angular average of the static
+correlations of director fluctuations Cθ(q) =/summationtext
+|q|=q/angb∇acketleftθ(q)θ(−q)/angb∇acket∇ight. The formally divergent q= 0
+mode is excluded both from the numerical data and
+analytical result. Given that the analytical result is
+obtained from a linearization about the aligned state
+whereas the numerical solution is calculated from the
+equations of motion arising from the full non-linear free
+energy, the agreement between theory and simulation
+is satisfactory. The stochastic method of lines thus
+accurately captures equilibrium fluctuations in the
+ordered state as well as in the disordered one.
+V. DISCUSSION AND CONCLUSION
+The numerical method of solution presented in this
+paper can be applied to a variety of problems in ne-
+matodynamics where accounting for fluctuations in ne-
+matic orderareimportant. To the best ofour knowledge,
+no systematic study of anharmonic fluctuations existswithin the time-dependent Ginzburg-Landau framework.
+In previous work, Stratonovich [5] presented fluctuating
+equations of motion for harmonic fluctuations in terms
+of Langevin equations, analyzing these equations within
+thetracelesssymmetricbasisdescribedhereinastraight-
+forward manner[6]. Our equations of motion contain the
+necessary nonlinearities and our numerical methodology
+accounts for them in a computationally straightforward
+way.
+We also present, to the best of our knowledge for the
+first time, a systematic stochastic integration scheme ca-
+pable of yielding highly accurate solutions of the non-
+linear equations of nematodynamics. These equations
+can be applied to study fluctuations of nematic order at
+the isotropic-nematic interface, pseudo-Casimir interac-
+tions close to the isotropic-nematic transition, as well as
+nucleationphenomenainnematogens. Thestudyofthese
+andsimilarproblemsisongoingandwill be reportedelse-
+where.
+VI. ACKNOWLEDGEMENTS
+GIM thanks the DST(India) and the Indo-French Cen-
+tre for the Promotionof Advanced Research(CEFIPRA)
+for support.
+[1] P. G. de Gennes and J. Prost, The Physics of Liquid
+Crystals (Clarendon Press, Oxford, 1993), 2nd ed.
+[2] T. Moses, J. Reeves, and P. Pirondi, Americal Journal of
+Physics75, 220 (2007).
+[3] P. G. de Gennes, Molecular Crystals and Liquid Crystals
+12, 193 (1971).
+[4] E. F. Gramsbergen, L. Longa, and W. H. de Jeu, Physics
+Reports 135, 195 (1986).
+[5] R. Stratonovich, Sov.Phys.JETP 43, 672 (1976).
+[6] P. D. Olmsted and P. Goldbart, Phys. Rev. A 41, 4578
+(1990).
+[7] B. J. Berne and R. Pecora, Dynamic Light Scattering:
+With Applications to Chemistry, Biology, and Physics
+(Dover, New York, 2000).
+[8] A. Cuetos and M. Dijkstra, Phys. Rev. Lett. 98, 095701
+(2007).
+[9] A. Ajdari, L. Peliti, and J. Prost, Phys. Rev. Lett. 66,
+1481 (1991).
+[10] F. Schmid, G. Germano, S. Wolfsheimer, andT. Schilling, Macromolecular Symposia 252, 110 (2007).
+[11] A.K.Bhattacharjee, G.I.Menon, andR.Adhikari,Phys.
+Rev. E78, 026707 (2008).
+[12] O. A. Liskovets, J. Diff. Eqs. 1, 1308 (1965).
+[13] C. W. Gardiner, Handbook of Stochastic Methods
+(Springer, Berlin, 1983).
+[14] G. Maruyama, Rend. Circolo. Math. Palermo 4, 48
+(1955).
+[15] G. N. Milstein, Numerical integration of stochastic dif-
+ferential equations (Kluwer, London, 1995).
+[16] J. Wilkie, Phys. Rev. E 70, 017701 (2004).
+[17] M. Abramowitz and I. A. Stegun, Handbook of Mathe-
+matical Functions: with Formulas, Graphs, and Mathe-
+matical Tables (Dover, 1964).
+[18] A. Shinozaki and Y. Oono, Phys. Rev. E 48, 2622 (1993).
+[19] M. Patra and M. Karttunen, Numerical Methods for Par-
+tial Differential Equations 22, 936 (2005).
\ No newline at end of file
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--- /dev/null
+++ b/1001.0843.txt
@@ -0,0 +1,269 @@
+arXiv:1001.0843v1  [astro-ph.SR]  6 Jan 2010Comm. in Asteroseismology
+Contribution to the Proceedings of the 38thLIAC/HELAS-ESTA/BAG, 2008
+Overshooting
+B. Dintrans
+LATT, Universit´ e de Toulouse, CNRS, 14 av. Edouard Belin, 3 1400 Toulouse, France
+Abstract
+Overshooting occurs in stars when convective elements pene trate into adjacent
+radiative zones. In the Sun, it leads to the so-called ‘tachocl ine’ at the base
+of the outer convection zone and this region is becoming a key ingredient of
+the standard solar dynamo model as strong toroidal magnetic fields may be
+generated there. However this overshoot is not predicted by the mixing-length
+theory of convection where convective elements must stop at the border of a
+convectively unstable region. I will review the main proper ties of this convective
+overshoot in stellar interiors, with in particular its subt le dependence with the
+thermal diffusion, and will present the most recent results o btained from 2-D
+and 3-D direct numerical simulations (DNS) of penetrative c onvection.
+Session: Physics and uncertainties in massive stars on the M S and close to it
+Introduction
+In the standard model of thermal convection based on the mixin g-length phe-
+nomenology (B¨ ohm-Vitense 1958), the penetration of conve ctive elements into
+surrounding stable layers is not possible. Indeed, the mixin g-length velocity v
+of a convective blob is related to its temperature contrast δTby
+v2∝δT
+TgℓwithδT∝ ∇−∇ ad, (1)
+wheregandℓdenote the gravity and the mixing length, respectively, and ∇
+and∇adthe usual temperature ‘nabla’. Following Schwarzschild’s cri terion,
+the interface that separates a convective zone from a radiat ive one corresponds
+to∇=∇adtherefore the blob must stop here. However, the mixing-leng th
+formalism is based on an acceleration budget and setting the blob acceleration
+to zero does not mean that its velocity is itself zero, i.e., the mixing-length2 Overshooting
+theory neglects inertia. In fact, the blob penetrates into the stable layer over a
+small distance, then it is sloweddown bythe buoyancybrakin gand finallystops.
+This small penetration is not anecdotal because it has impor tant consequences
+in stellar physics:
+- it leads to different evolutionary tracks in the HR diagram a s massive stars
+live longer on the main sequence due to the additional mixing induced by the
+overshootingabovetheconvectivecore(Rosvick&Vandenbe rg1998,Perryman
+et al. 1998).
+- in the standard α−Ωmodel of the solar dynamo, the regeneration of poloidal
+magnetic fields into toroidal ones takes place in the tachocl ine where vertical
+and latitudinal differential rotation is expected to be most efficient (Parker
+1955, Brandenburg & Subramanian 2005).
+- heliosismology showed that the solar core rotation is almo st rigid (Brown et
+al. 1989). Internal gravity waves propagating in stellar rad iative zones are
+suspected to play the main role in this angular momentum redi stribution (Zahn
+et al. 1997, Talon & Charbonnel 2003). The most wide-spread e xcitation
+model involves penetrative convection from neighboring co nvection zones as
+strong downdrafts extend a substantial distance into the ad jacent stable zones
+sothatinternalgravitywavescanberandomlygenerated(Di ntransetal. 2005).
+Modelling of overshooting
+Assuming that the temperature gradient in the overshoot lay er is close to the
+adiabatic one of the next convection zone, Roxburgh (1978) i nvestigated the
+overshoot of convective cores and derived what it is now call ed the ‘Roxburgh’s
+integral constraint’:
+/integraldisplayrc
+0(L−Lrad)d/parenleftbigg1
+T/parenrightbigg
+>0, (2)
+where the radius rccorresponds to the edge of the overshoot region, while L
+andLraddenote the total and radiative luminosity, respectively. T he overshoot
+layer corresponds in this relation to the region where L < L rad(i.e., negative
+contribution) while the convection zone means L > L rad(i.e., positive contri-
+bution). However, this relation gives an upper limit of the o vershoot extent
+because it does not take into account the dissipation acting on the penetrating
+elements. Indeed, the viscous dissipation Dνenters in the heat equation and
+affects the blob dynamics such that the correct relation read s now (Roxburgh
+1989):
+/integraldisplayrc
+0(L−Lrad)d/parenleftbigg1
+T/parenrightbigg
+=/integraldisplayrc
+04πr2Dν
+TdrwithDν∼ν/parenleftbigg∂ui
+∂xj/parenrightbigg2
+,(3)B. Dintrans 3
+whereνdenotes the kinematic viscosity. As the viscous dissipatio n is always
+positive, this term acts to decrease the negative part of the integrand in the
+left-hand side and thus the radius rc.
+Following Roxburgh’s investigation, Zahn (1991) showed tha t the penetra-
+tion extent also depends on the value of the Peclet number Peassociated with
+the convective elements, that is,
+Pe=vℓ
+χ, (4)
+whereχdenotes the thermal diffusivity. Small Peclet’s numbers mean that
+the radiative diffusion is stronger than the advection and th e convective blob
+rapidly loses its identity compared to the surrounding medi um. As a conse-
+quence, it does not feel any more the buoyancy braking and tra vels over long
+distances in the radiative zone (Zahn called this regime ‘ove rshoot’). On the
+other hand, a blob with a large Peclet number does not thermal ize rapidly with
+its surrounding (i.e., no radiative losses)and keepits ident ity: it is then strongly
+slowed down by the buoyant force and its penetration is weake r (the so-called
+‘penetration’ regime). Penetrationleads to an adiabatic st ratificationbelow the
+convection zone because of the induced entropy mixing, wher eas overshoot has
+no significant influence on the local stratification.
+Assuming that the velocity Wof the penetrating motions obeys the usual
+mixing-length scaling of thermal convection, Zahn derived the following rela-
+tions for both the subadiabatic extent of a convective regio n and the width of
+the thermal boundary layer:
+Lp∝f1/2W3/2andLbound∝/parenleftbiggχHp
+g/parenrightbigg1/2
+, (5)
+wherefdenotes a filling factor of convective plumes (i.e., fraction o f the area
+occupied by the plumes) and Hpthe pressure scale-height at the bottom of
+the convection zone. Applied to the Sun, these two scalings l ead to a pene-
+tration extent of about 0.2−0.3Hp(∼15 000km) and an overshoot layer
+≃1km. These scalings seem however to overestimate the amount o f con-
+vective penetration inferred by helioseismology as the stu dy of the oscillations
+in the asymptotic laws of solar acoustic modes rather predic ts an upper limit
+of∼0.1Hpfor the extent of the whole convective overshooting (Roxbur gh &
+Vorontsov 1994; Christensen-Dalsgaard et al. 1995; see als o Aerts et al. 2003
+for the case of a massive star).4 Overshooting
+Numerical simulations of overshooting
+The bad news
+The rapid development of supercomputers has lead several gr oups to under-
+take direct numerical simulations of convective overshoot ing. However, many
+numerical difficulties appear when one tries to simulate stel lar convection:
+- the molecular viscosity in stellar interiors is small, typ icallyν∼1cm2/s
+(Parker 1979). Granulation cells in the upper convection zo ne of the Sun
+correspond to a typical size and velocity of about L= 1000km andU= 1
+km/s, respectively. It leads to Reynolds’ number of about Re=UL/ν∼1011,
+meaning that solar convection is a highly turbulent and nonl inear phenomenon.
+- we know from Kolmogorov’s theory of turbulence (Kolmogorov 1941) that
+the ratio between the largest scale (i.e., the injection scale L) and the smallest
+one (i.e., the dissipation scale ℓd) behaves as L/ℓd∼Re3/4such that the
+solar dissipation scale is about one centimeter. In other wor ds, one needs
+(at least) 108gridpoints in each spatial direction to well reproduce both the
+injection of energy in the largest structures and its viscou s dissipation in the
+smallest ones (the turbulent cascade scenario). As this num ber of gridpoints is
+clearly beyond the scope oftoday supercomputers, several n umerical techniques
+have been developed to overcome this restriction: (i) inste ad of doing Direct
+Numerical Simulations (DNS), one can perform Large-Eddy Si mulations (LES)
+where the small-scale dynamics is modelled using a sub-grid scale approach
+(Lesieur 2008); (ii) one can also artificially increase the d issipation scale by
+adding a large-scale viscosity in the whole computational d omain, the price to
+pay being that motions are essentially laminar ( Re∼102···104).
+- finally, another problem concerns the different time scales that coexist in a
+convection zone. Indeed, the local turnover time scale of con vection is roughly
+given by:
+τ∼Hp/URMS, (6)
+whereURMSis the typical (turbulent) RMS velocity of convective eddie s. Ap-
+plied to the solar convection zone, it leads to turnover time scales of about the
+day at the surface to several months at the bottom (Spruit 197 4). As a con-
+sequence, simulations should reproduce motions whose dyna mical time scales
+can vary by two orders of magnitude in the computational doma in. Moreover,
+the thermal time scale is also much more longerthan these tur nover time scales
+by orders of magnitude as, e.g., the solar thermal flux expresse d in dynamical
+units is very small at the bottom of the convection zone ( F⊙/(ρc3
+s)∼10−11).
+This ratio is in essence equal to the ratio of the dynamical to thermal time
+scales, meaning that realistic simulations should be integ rated on a very longB. Dintrans 5
+Figure 1: Penetration extent ∆v.s. stability parameter S.Left: in the 2-D case
+where both the penetration and overshoot regimes exist (fro m Hurlburt et al. 1994).
+Right: in the 3-D case where only the overshoot regime remains (fro m Brummell et
+al. 2002).
+time to make sure that a thermal-relaxed state is reached. It i s not possible in
+practice to do that with a decent spatial resolution and the t hermal fluxes are
+commonly overestimated by orders of magnitude in numerical simulations.
+DNS of convective overshooting in local cartesian boxes
+FollowingHurlburt et al. (1984), the majorityofDNS doneso faruse polytropic
+solutions for the internal structure of the star. Indeed, pol ytropes with ρ∝
+Tm(mbeing the polytropic index) allow to easily specify the stre ngth of the
+convective instability as the hydrostatic equilibrium imp lies in that case
+dT
+dz=−g
+(m+1)R∗, (7)
+wheregis the downward-directed gravity, zthe altitude and R∗the perfect gas
+constant. For a monatomic gas with an adiabatic index γ= 5/3, this relation
+shows that the temperature gradient is equal to the adiabati c one(=−g/cp)
+when the polytropic index is mad= 1/(γ−1) = 3/2. As a consequence,
+a stable layer corresponds to m > m adand a convectively-unstable one to
+m < m ad.
+Hurlburt et al. (1986, 1994) used this model to study the over shooting in
+2-D simulations of compressible convection for a solar case , that is, a radiative
+zone with polytropic index m3>3/2located above a convective zone with
+polytropic index m2<3/2. In particular, they studied the influence of the6 Overshooting
+relative stabilityparameter Son the penetration extent ∆, whereS=−(mad−
+m3)/(mad−m2), and found the following scalings (Fig. 1, left):
+
+
+∆∝S−1for small S-values: penetration regime ,
+∆∝S−1/4for large S-values: overshoot regime.(8)
+However, these scaling laws have not been fully confirmed in t he 3-D case by
+Brummell et al. (2002) as only the overshoot regime ∆∝S−1/4is recovered
+(Fig.1, right). The reason simply lies in the smaller filling f actorffor the con-
+vective plumes compared to the 2-D case (Eq. 5): the spaced do wndrafts that
+penetrate into the stably stratified layer are not enough to m aintain an adia-
+batic stratification below the convection zone and only the t hermal boundary
+layer exists.
+t=1000.0
+Figure 2: Vorticity isocontours in a 2-D star-in-a-box simu lation of penetrative con-
+vection for a solar-like star (Dintrans & Brandenburg 2009) .
+Another interesting 2-D and 3-D simulations of penetrative convection are
+the ones donebyRoxburgh& Simmons(1993)and Singhet al. (19 98). Indeed,
+a more realistic model of the star interior has been consider ed in the 2-D case
+through a temperature-dependent radiative conductivity. The main result is
+that the viscous dissipation contribution in the RHS of Roxb urgh’s integral
+constraint (3) decreases with the Prandtl number. It suggest s that in the
+astrophysical limit of very small Prandtl numbers ( Pr∼10−6), the stable and
+unstable contributions in the LHS of Roxburgh’s integral str ictly balance each
+other and are sufficient to obtain a good estimate for the overs hoot extentB. Dintrans 7
+(Roxburgh & Simmons 1993). In the 3-D case (Singh et al. 1998), a sub-
+grid scale modelling of the small scales led to the confirmati on of the scaling
+relationships between this extent and both the imposed bott om flux and the
+vertical velocity of convective downdrafts (see Eq. 5).
+Global DNS
+In the last decade, global direct simulations of overshootin g have greatly im-
+proved following two numerical approaches:
+- the spherical shape of the star is reproduced using a pseudo -spectral code
+where physical fields are projected onto spherical harmonic s. Browning et al.
+(2004) used this method with the 3-D anelastic ASH code to stu dy the core
+overshooting in a massive A-type star of 2M⊙, where only the inner 30% by
+radiusofthestarhasbeenconsidered(convectivecoreands omeofitssurround-
+ing radiative envelope). Although the achievement of the th ermal relaxation
+in these simulations remains questioning, they found that t he convective core
+adopts a prolate shape while overshooting is larger at the eq uator than at the
+poles, yielding an overall spherical shape to the whole cent ral region.
+- the star is embedded in a topologically Cartesian domain an d the sphericity
+is reproduced through radial-dependent forcing or damping profiles. This novel
+approach has been developed by Dintrans & Brandenburg (2009 ) and first
+results in 2-D are promising as both the penetration and over shoot regimes are
+observed, whereas internal gravity waves are efficiently exc ited in the central
+radiative zone (Fig. 2).
+Conclusion
+Convective overshooting is today included in stellar evolu tion codes and evo-
+lutionary tracks of massive stars reproduce satisfactoril y the observations of
+clusters when an overshooting above convective cores is ass umed. Theoreti-
+cal studies showed that the overshoot region is composed of t wo nested layers
+whose the size depends on the value of the Peclet number of pen etrating mo-
+tions. Numerical studies in 2-D and 3-D of penetrative conve ction confirmed
+this picture, except that only the overshoot layer (i.e., the t hermal boundary
+one) exists in 3-D due to the weak filling factor of convective plumes. These
+local and global direct numerical simulations are however s till quite far from
+realistic values of stellar interiors. Flows are less turbu lent than in reality and
+the overshoot regime dominates. Further progress are clear ly needed in the
+numerical side of this problem (e.g., development of large-ed dy simulations), in
+its theoretical modelling (e.g., turbulent closure models; K upka & Montgomery
+2002) and in the observational signatures of overshooting f rom asteroseismol-
+ogy.8 Overshooting
+References
+Aerts, C., Thoul, A., Daszy´ nska, J., et al. 2003, Sci, 300, 1 926
+B¨ ohm-Vitense, E. 1958, ZA, 46, 108
+Brandenburg, A., & Subramanian, K. 2005, PhR, 417, 1
+Brown, T. M., Christensen-Dalsgaard, J., Dziembowski, W.A ., et al. 1989, SoPh,
+343, 526
+Browning, M. K., Brun, A. S., & Toomre, J. 2004, ApJ, 601, 512
+Brummell, N. H., Clune, T. L., & Toomre, J. 2002, ApJ, 570, 825
+Christensen-Dalsgaard, J., Monteiro, M. J. P. F. G., & Thomp son, M. J. 1995,
+MNRAS, 276, 283
+Dintrans, B., Brandenburg, A., Nordlund, ˚A., et al. 2005, A&A, 438, 365
+Dintrans, B., & Brandenburg, A. 2009, ˚A., in preparation
+Hurlburt, N. E., Toomre, J., & Massaguer, J. M. 1984, ApJ, 282 , 557
+Hurlburt, N. E., Toomre, J., & Massaguer, J. M. 1986, ApJ, 311 , 563
+Hurlburt, N. E., Toomre, J., Massaguer, J. M., & Zahn, J.-P. 1 994, ApJ, 421, 245
+Kolmogorov, A. N. 1941, DoSSR, 30, 301
+Kupka, F., & Montgomery, M. H. 2002, MNRAS, 330, L6
+Lesieur, M. 2008, Turbulence in Fluids (Springer-Verlag)
+Parker, E. N. 1955, ApJ, 122, 293
+Parker, E. N., 1979, Cosmical Magnetic Fields, (Oxford Univ ersity Press)
+Perryman, M. A. C., Brown, A. G. A., Lebreton, Y., et al. 1998, A&A, 331, 81
+Rosvick, J. M., & Vandenberg, D. A. 1998, AJ, 115, 1516
+Roxburgh, I. W. 1978, A&A, 65, 281
+Roxburgh, I. W. 1989, A&A, 211, 361
+Roxburgh, I. W. & Simmons, J. 1993, A&A, 277, 93
+Roxburgh, I. W., & Vorontsov, S. V. 1994, MNRAS, 268, 880
+Singh, H. P., Roxburgh, I. W., & Chan, K. L. 1998, A&A, 340, 178
+Spruit, H. C. 1974, SoPh, 34, 277
+Talon, S., & Charbonnel, C. 2003, A&A, 405, 1025
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\ No newline at end of file
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+arXiv:1001.0844v1  [quant-ph]  6 Jan 2010Dynamics of entanglement in the
+transverse Ising model
+Zhe Chang∗and Ning Wu†
+Institute of High Energy Physics
+Chinese Academy of Sciences
+P. O. Box 918(4), 100049 Beijing, China
+Abstract
+We study the evolution of nearest-neighbor entanglement in the one dimensional
+Ising model with an external transverse field. The system is i nitialized as the so
+called “thermal ground state” of the pure Ising model. We ana lyze properties of
+generation of entanglement fordifferentregions ofexternal transversefields. Wefind
+that the derivation of the time at which the entanglement rea ches its first maximum
+with respect to the reciprocal transverse field has a minimum at the critical point.
+This is a new indicator of quantum phase transition.
+PACS numbers: 03.67.Bg, 75.10.Pq
+1 Introduction
+Entanglement has been recognized as an important resource for q uantum information and
+computation [1], and has been the subject of intense research ove r the past few years [2].
+Recently, there has been extensive analysis of entanglement in qua ntum spin models [3].
+Various models have been considered for entanglement generation , and their static [4, 5]
+as well as dynamical properties [6, 7, 8] have been investigated. Th e reason for this is
+that, many of these models can be realized in cold atomic gas in an optic al lattice [9, 10].
+A special one among these models is the anisotropic XY model. It can b e solved
+exactly by means of the Jordan-Wigner transformation [11]. A lot of works have been
+done on the dynamics of entanglement in this model. In Ref.[6], the time evolution of
+initial Bell states was studied. Ref.[7] investigated the dynamics of e ntanglement of a
+system prepared in a thermal equilibrium state, and the system sta rts evolving after the
+transverse field was turned off.
+∗e-mail:changz@mail.ihep.ac.cn
+†e-mail:wun@mail.ihep.ac.cn
+1In this paper, we analyze a spin-1/2 transverse Ising chain with a pa rticular separable
+initial state, the so-called “thermal ground state” [4] for the cas e of zero field. We study
+the generation of nearest-neighbor entanglement as a function o f the external transverse
+field, and find the time at which the entanglement reaches its first ma ximum. It turns
+out that the derivation of this time with respect to the reciprocal t ransverse field has a
+minimum at the critical point. This is a new indicator of quantum phase t ransition.
+2 Static correlations
+The anisotropic XY model is described by the Hamiltonian
+H=−λ
+4N/summationdisplay
+i=1[(1+γ)σx
+iσx
+i+1+(1−γ)σy
+iσy
+i+1]−1
+2N/summationdisplay
+i=1σz
+i, (1)
+whereσα
+iare the Pauli matrix of the spin at site iand we assume periodic boundary
+conditions. We have neglected a common factor and absorbed the e xternal field into the
+reciprocal field λ. The anisotropy parameter γconnects the quantum Ising model for
+γ= 1. In the range 0 < γ≤1, the model belongs to the Ising universality class and
+undergoes a quantum phase transition at the critical point λc= 1. We will consider the
+case ofγ= 1 in this paper, namely the transverse Ising model
+H=−1
+2N/summationdisplay
+i=1(λσx
+iσx
+i+1+σz
+i). (2)
+The Hamiltonian defined in Eq.(2) can be mapped into a one-dimensional spinless
+fermion system with creation and annihilation operators c†
+iandcivia the Jordan-Wigner
+transformationσx
+i−iσy
+i
+2=/producttext
+j<i(1−2c†
+jcj)ci,σx
+i+iσy
+i
+2=c†
+i/producttext
+j<i(1−2c†
+jcj),σz
+i
+2=c†
+ici−1
+2. By
+making use of the transformation
+ηk=1√
+N/summationdisplay
+leikl(αkcl+iβkc†
+l) (3)
+further, with αk=Λk−(1+λcosk)√
+2[Λ2
+k−(1+λcosk)Λk], βk=λsink√
+2[Λ2
+k−(1+λcosk)Λk], we get the fermionic
+Hamiltonian finally
+H=/summationdisplay
+kΛk/parenleftbigg
+η†
+kηk−1
+2/parenrightbigg
+, (4)
+where the spectrum is Λ k=√
+1+λ2+2λcosk.
+The evolution of the system is governed by Eq.(2). In the Heisenber g picture, the time
+evolution of the spinless fermion operator ci(t) is [6]
+ci(t) =1
+N/summationdisplay
+k,l[cosk(l−i)A(k,t)cl+sink(l−i)B(k,t)c†
+l], (5)
+2whereA(k,t) =eiΛkt−2iβ2
+ksinΛkt,B(k,t) =−2iαkβksinΛkt.
+The dynamics of the system also depends on the initial state. We cho ose the “thermal
+ground state” ρ0=1
+2(|N+/an}bracketri}ht/an}bracketle{tN+|+|N−/an}bracketri}ht/an}bracketle{tN−|) of the pure Ising model (no transverse field
+is applied) as the initial state. Here |N+/an}bracketri}ht=|→/an}bracketri}ht1...|→/an}bracketri}htN, and|N−/an}bracketri}ht=|←/an}bracketri}ht1...|←/an}bracketri}htN
+are the two degenerate ground states of the Ising model with all s pins pointing to the
+positive (negative) xdirection. Note that |N+/an}bracketri}htand|N−/an}bracketri}htare the ground states of the
+Hamiltonian (2) with λ→∞. When λis finite, they are even not eigenstates of the
+Hamiltonian. Thus, the evolution from this initial state must be nontr ivial for finite λ.
+Physically, this can be viewed as an instantaneous, i.e., idealized sudde n quench [8] in the
+reciprocal field λ1→λ2withλ1→∞andλ2=λ.
+Due to the translational symmetry, we need only to consider avera ges of the form
+/an}bracketle{tc1.../an}bracketri}ht0. Namely, we choose site-1 as “the first site”, where /an}bracketle{t.../an}bracketri}ht0denote the average over
+the initial state ρ0. We rewrite|N±/an}bracketri}htin the fermionic representation,
+|N±/an}bracketri}ht=/parenleftbigg1√
+2/parenrightbiggN
+(1±c†
+1)(1±c†
+2)...(1±c†
+N)|01,02,...,0N/an}bracketri}ht, (6)
+where|01,02,...,0N/an}bracketri}htdenotes the vacuum of the fermions, or the state for all spins dow n.
+Here we have to pay attention to the order of operators in this equ ation. We write (1 ±c†
+1)
+before operators on all other sites in order to be compatible with th e correlation/an}bracketle{tc1.../an}bracketri}ht0.
+In other words, if another site iis chosen as “the first site”, we should write the state
+as|N±/an}bracketri}ht= (1√
+2)N(1±c†
+i)(1±c†
+i+1)...(1±c†
+i−1)|0i,0i+1,...,0i−1/an}bracketri}htto calculate averages like
+/an}bracketle{tci.../an}bracketri}ht0, due to the periodic boundary conditions.
+The only single-site averages are /an}bracketle{tc1/an}bracketri}ht0and/an}bracketle{tc†
+1c1/an}bracketri}ht0(or their complex conjugate). It is
+easy to see that/an}bracketle{tN+|c1|N+/an}bracketri}ht=1
+2=−/an}bracketle{tN−|c1|N−/an}bracketri}ht, so/an}bracketle{tc1/an}bracketri}ht0= 0. As to/an}bracketle{tc†
+1c1/an}bracketri}ht0, we have
+/an}bracketle{tN+|c†
+1c1|N+/an}bracketri}ht=1
+2/an}bracketle{t01|(1+c1)c†
+1c1(1+c†
+1)|01/an}bracketri}ht=1
+2. Similarly,/an}bracketle{tN−|c†
+1c1|N−/an}bracketri}ht=1
+2. So that,
+/an}bracketle{tc†
+1c1/an}bracketri}ht0=1
+2. We can also see this point from 0 = /an}bracketle{tσz
+1
+2/an}bracketri}ht0=/an}bracketle{tc†
+1c1−1
+2/an}bracketri}ht0.
+Next we consider two-point correlations of the form /an}bracketle{tc1cl/an}bracketri}ht0, l≥2. Forl= 2, we have
+/an}bracketle{tN+|c1c2|N+/an}bracketri}ht=/parenleftbigg1
+2/parenrightbigg2
+/an}bracketle{t01,02|(1+c2)(1+c1)c1c2(1+c†
+1)(1+c†
+2)|01,02/an}bracketri}ht
+=/parenleftbigg1
+2/parenrightbigg2
+/an}bracketle{t01|(1+c1)c1(1−c†
+1)|01/an}bracketri}ht·/an}bracketle{t02|(1+c2)c2(1+c†
+2)|02/an}bracketri}ht
+=−1
+4.
+Similarly/an}bracketle{tN−|c1c2|N−/an}bracketri}ht=−1
+4. So that/an}bracketle{tc1c2/an}bracketri}ht0=−1
+4. We can show that /an}bracketle{tc†
+1c2/an}bracketri}ht0=1
+4in
+the same manner. For l >2, whenclorc†
+lcrosses (1±c†
+2) to its right side, this will make
+it be (1∓c†
+2). But/an}bracketle{t02|(1±c2)(1∓c†
+2)|02/an}bracketri}ht= 0. So we have obtained all the two-point
+correlations
+/an}bracketle{tc†
+icj/an}bracketri}ht0=1
+4(δi+1,j+δi−1,j+2δi,j),
+/an}bracketle{tcicj/an}bracketri}ht0=1
+4(−δi+1,j+δi−1,j),
+/an}bracketle{tc†
+ic†
+j/an}bracketri}ht0=1
+4(δi+1,j−δi−1,j). (7)
+3By the same means one can check that the averages involving three fermion operators
+all vanish.
+3 Reduced density matrix and evolution of concur-
+rence
+Making use of the periodic boundary conditions, we can focus on the reduced density
+matrixρ12(t) at time tof the first and second spins. In the {|↑↑/an}bracketri}ht,|↑↓/an}bracketri}ht,|↓↑/an}bracketri}ht,|↓↓/an}bracketri}ht}basis,
+the matrix elements reads,
+ρ12
+11(t) =/an}bracketle{t↑↑|ρ12(t)|↑↑/an}bracketri}ht
+=/an}bracketle{t↑↑|ρ12(t)S+
+1S−
+1S+
+2S−
+2|↑↑/an}bracketri}ht+/an}bracketle{t↑↓|ρ12(t)S+
+1S−
+1S+
+2S−
+2|↑↓/an}bracketri}ht
++/an}bracketle{t↓↑|ρ12(t)S+
+1S−
+1S+
+2S−
+2|↓↑/an}bracketri}ht+/an}bracketle{t↓↓|ρ12(t)S+
+1S−
+1S+
+2S−
+2|↓↓/an}bracketri}ht
+=tr12(ρ12(t)S+
+1S−
+1S+
+2S−
+2)
+=tr(ρ(t)S+
+1S−
+1S+
+2S−
+2)
+=/an}bracketle{tc†
+1c1c†
+2c2/an}bracketri}htt.
+Other matrix elements can be obtained similarly. Thus, we have
+ρ12(t) =
+/an}bracketle{tc†
+1c1c†
+2c2/an}bracketri}htt−/an}bracketle{tc†
+1c1c2/an}bracketri}htt/an}bracketle{tc†
+2c2c1/an}bracketri}htt/an}bracketle{tc2c1/an}bracketri}htt
+−/an}bracketle{tc†
+2c†
+1c1/an}bracketri}htt/an}bracketle{tc†
+1c1c2c†
+2/an}bracketri}htt−/an}bracketle{tc1c†
+2/an}bracketri}htt/an}bracketle{tc1c2c†
+2/an}bracketri}htt
+/an}bracketle{tc†
+1c†
+2c2/an}bracketri}htt−/an}bracketle{tc2c†
+1/an}bracketri}htt/an}bracketle{tc1c†
+1c†
+2c2/an}bracketri}htt/an}bracketle{tc1c†
+1c2/an}bracketri}htt
+/an}bracketle{tc†
+1c†
+2/an}bracketri}htt/an}bracketle{tc2c†
+2c†
+1/an}bracketri}htt/an}bracketle{tc†
+2c1c†
+1/an}bracketri}htt/an}bracketle{tc1c†
+1c2c†
+2/an}bracketri}htt
+, (8)
+All the elements can be equally evaluated in the Heisenberg picture, e .g.,ρ12
+11(t) =
+/an}bracketle{tc†
+1c1c†
+2c2/an}bracketri}htt=/an}bracketle{tc†
+1(t)c1(t)c†
+2(t)c2(t)/an}bracketri}ht0. We have mentioned that all the static averages in-
+volving three fermion operators equal zero. From Eq.(5), the fer mion operators in the
+Heisenberg picture are merely some linear superpositions of the sam e set of operators in
+the Schrodinger picture. So all matrix elements involving three oper ators in Eq.(8) also
+vanish. Using Wick’s theorem, one gets all non-vanishing elements of ρ12(t) as following
+ρ12
+11(t) =/an}bracketle{tc†
+1(t)c1(t)/an}bracketri}ht0/an}bracketle{tc†
+2(t)c2(t)/an}bracketri}ht0−/an}bracketle{tc†
+1(t)c†
+2(t)/an}bracketri}ht0/an}bracketle{tc1(t)c2(t)/an}bracketri}ht0−/an}bracketle{tc†
+1(t)c2(t)/an}bracketri}ht/an}bracketle{tc†
+2(t)c1(t)/an}bracketri}ht0,
+ρ12
+22(t) =ρ12
+33(t) =/an}bracketle{tc†
+1(t)c1(t)/an}bracketri}ht0−ρ12
+11(t),
+ρ12
+44(t) = 1−ρ12
+11(t)−2ρ12
+22(t),
+ρ12
+14(t) =ρ12∗
+41(t) =/an}bracketle{tc2(t)c1(t)/an}bracketri}ht0,
+ρ12
+23(t) =ρ12∗
+32(t) =−/an}bracketle{tc1(t)c†
+2(t)/an}bracketri}ht0. (9)
+4We see that if we can calculate /an}bracketle{tc†
+1(t)c1(t)/an}bracketri}ht0,/an}bracketle{tc†
+1(t)c2(t)/an}bracketri}ht0and/an}bracketle{tc1(t)c2(t)/an}bracketri}ht0, then the re-
+duced density matrix is totally determined.
+Using Eq.(5) and (7), we find, in the thermodynamic limit, that
+/an}bracketle{tc†
+1(t)c1(t)/an}bracketri}ht=1
+8π/integraldisplayπ
+−πdksink(A∗(k)−A(k))(B(k)−B∗(k))+1
+4π/integraldisplayπ
+−πdk(|A(k)|2+|B(k)|2)
++1
+4π/integraldisplayπ
+−πdkcosk(|A(k)|2−|B(k)|2)
+=1
+π/integraldisplayπ
+0dkλsin2ksin2Λkt
+Λ2
+k+1
+2,
+/an}bracketle{tc†
+1(t)c2(t)/an}bracketri}ht=1
+8π/integraldisplayπ
+−πdk(1+cos2 k)(|A(k)|2−|B(k)|2)+1
+4π/integraldisplayπ
+−πdkcosk(|A(k)|2+|B(k)|2)
++1
+8π/integraldisplayπ
+−πdksin2k[A(k)−A∗(k)]B∗(k)
+=1
+π/integraldisplayπ
+0dkλcosksin2ksin2Λkt
+Λ2
+k+1
+4,
+/an}bracketle{tc1(t)c2(t)/an}bracketri}ht=1
+4π/integraldisplayπ
+−πdksin2kA(k)B(k)+1
+8π/integraldisplayπ
+−πdk[B(k)2−A(k)2](1−cos2k)
+=1
+2π/integraldisplayπ
+0dksin2k[isin2Λ kt
+Λk−cos2Λ kt−2λsin2Λkt
+Λ2
+k(cosk+λ)] (10)
+Note that/an}bracketle{tc†
+1(t)c2(t)/an}bracketri}htis indeed real, so that we have ρ12
+23(t) =ρ12
+32(t).
+For bipartite entanglement, a commonly used measure for arbitrar y states of two
+qubits is the so-called concurrence [12]. The concurrence is defined as
+C(t) = max{0,2λmax(t)−tr/radicalbig
+ρ12(t)˜ρ12(t)}, (11)
+˜ρ12(t) =σy⊗σyρ12∗(t)σy⊗σy, (12)
+whereλmaxis the largest eigenvalue of the matrix/radicalbig
+ρ12(t)˜ρ12(t).
+In the present case,
+ρ12(t) =
+ρ12
+11(t) 0 0 ρ12
+14(t)
+0ρ12
+22(t)ρ12
+23(t) 0
+0ρ12
+23(t)ρ12
+22(t) 0
+ρ12∗
+14(t) 0 0 ρ12
+44(t)
+, (13)
+5Figure 1: The evolution of bipartite entanglement for the initial state being th e “thermal ground state”
+of the Ising model.
+the four eigenvalues of/radicalbig
+ρ12(t)˜ρ12(t) are of the form
+λ1(t) =||ρ12
+14(t)|+/radicalig
+ρ12
+11(t)ρ12
+44(t)|,
+λ2(t) =||ρ12
+14(t)|−/radicalig
+ρ12
+11(t)ρ12
+44(t)|,
+λ3(t) =|ρ12
+22(t)+ρ12
+23(t)|,
+λ4(t) =|ρ12
+22(t)−ρ12
+23(t)|. (14)
+In Fig.1, we give a plot of the evolution of entanglement as a function o f timetand
+parameter λ. At the beginning, the reduced density matrix is
+ρ12(0) =1
+4
+1 0 0 1
+0 1 1 0
+0 1 1 0
+1 0 0 1
+. (15)
+So that the concurrence equals zero. At any later fixed instant, t he entanglement as a
+function of λmay have different behaviors in different time intervals. For times bef ore
+6Figure 2: (a): The concurrence at different (b): The concurrence at differ ent times
+timest= 0.5,1.0,1.5,2.0 as a function of λ, t= 4.0,5.0,6.0,7.0 as a function of λ,
+in the region of t <2.5, in the region of t >3.0.
+aboutt= 2.5 (Fig.2 (a)), the entanglement maintains its magnitude for large λwith
+slow oscillations. As λincreases, the height of each peak deceases gradually, and finally
+vanishes as λ→∞. This is easy to understand. For λ→∞, the Hamiltonian reduces to
+the pure Ising model, and the initial state is just the eigenstate of t he Ising model. Thus,
+only a phase factor contributes to the times evolving state, which d oes not change the
+entanglement. Physically, this corresponds to λ1=λ2=∞, hence no quench takes place.
+For the narrow band 2 .47< t <2.69, there is no entanglement at all for all λ. For times
+greater than around t= 3.0 (Fig.2 (b)), the entanglement emerges abruptly, but only has
+non-vanishing values for very short interval of λ. As the time increases, the generations of
+entanglement are different fromeach other for different λ. Forλis less than about λ= 0.8
+(Fig.3 (a)), the concurrence will encounter more than one maximum s. In a time region
+between the first two maximums, no entanglement appear. Althoug h weak oscillations
+emerge after this region, the entanglement can last for a fairly long time with a not too
+small magnitude. For 0 .80< λ <1.07, there is only one maximum as the time increases.
+After a short time, the entanglement vanishes. For λ >1 (Fig. 3 (b)), strong oscillations
+emerge. The entanglement reaches its maximum fast, then sudden ly decreases to zero.
+Such a behavior will repeat again at later times, and finally the entang lement vanishes
+completely.
+For anyλ, there is a time Tm(λ) at which the entanglement reaches its first maximum
+(Fig.4). This time is a monotonically decreasing function of λ. Taking its derivative
+with respect to λ, we find that the derivativedTm(λ)
+dλhas a minimum at the critical point
+λ= 1. This interesting property maybe an indicator of the quantum ph ase transition at
+the critical point.
+7Figure 3: (a): The dynamics of entanglement (b): The dynamics of entanglem ent for
+for different reciprocal fields λ= 0.2,0.4,0.6,0.8, different reciprocal fields λ= 1.5,3.0,
+in the region of λ <0.8, in the region of λ >1.07.
+Figure 4: The time at which the entanglement reaches its first maximum and its d erivative.
+84 Brief discussions and conclusions
+In this paper, we have studied the dynamics of the transverse Isin g chain prepared in
+the “thermal ground state” of the pure Ising model. For different values of the reciprocal
+field, the evolution of the entanglement has dramatic distinctions. W e also obtained the
+times at which the entanglement reaches its first maximum, which is a m onotonically
+decreasing function of the parameter λ. The derivative of this quantity with respect to
+the reciprocal field gets its minimum at the critical field. This is a new ind icator of the
+quantum phase transition. Let us finally remark that the initial stat e we choose is simple
+to be prepared experimentally. One can first cool the pure Ising sy stem down to near
+absolute zero, then turn on some desired transverse field at t= 0 to generate the wanted
+entanglement.
+Acknowledgments
+We would like to thank X. Li for useful discussions. The work was sup ported partially by
+the NSF of China under Grant No. 10875129.
+References
+[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information
+(Cambridage University Press, Cambridge, 2000).
+[2] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Re v. Mod. Phys. 81,
+865 (2009).
+[3] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).
+[4] T. J. Osborne and M. A. Nielsen, Phys, Rev. A 66, 032110 (2002).
+[5] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature (London) 416, 608 (2002).
+[6] L. Amico, A. Osterloh, F. Plastina, R. Fazio, and G. M. Palma, Phys , Rev. A 69,
+022304 (2004).
+[7] A. Sen(De), U. Sen, and M. Lewenstein, Phys, Rev. A 72, 052319 (2005).
+[8] H. Wichterich and S. Bose, Phys, Rev. A 79, 060302(R) (2009).
+[9] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys, Rev. Lett.
+81, 3108 (1998).
+[10] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, and I. B loch, Nature
+(London) 425, 937 (2003).
+[11] E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961).
+[12] W. K. Wootters, Phys, Rev. Lett. 80, 2245.
+9
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+arXiv:1001.0845v1  [math-ph]  6 Jan 2010PLASMA WAVES REFLECTION FROM A BOUNDARY
+WITH SPECULAR ACCOMMODATIVE BOUNDARY
+CONDITIONS
+N. V. Gritsienko, A. V. Latyshev, A. A. Yushkanov
+Moscow State Regional University, Radio st.,10a, Moscow 10 5005, Russia
+e-mail: natafmf@yandex.ru, avlatyshev@mail.ru, yushkan ov@inbox.ru
+Abstract
+In the present work the linearized problem of plasma wave refl ection from a
+boundary of a half–space is solved analytically. Specular a ccommodative conditions
+of plasma wave reflection from plasma boundary are taken into consideration. Wave
+reflectance is found as function of the given parameters of th e problem, and its
+dependence on the normal electron momentum accommodation c oefficient is shown
+by the authors. The case of resonance when the frequency of se lf-consistent electric
+field oscillations is close to the proper (Langmuir) plasma o scillations frequency,
+namely, the case of long wave limit is analyzed in the present paper. Refs. 17. Figs.
+6.
+Keywords: degenerate plasma, half-space, normal electron momentum a ccommodation
+coefficient, specular accommodative boundary condition, lo ng wave limit, wave ref-
+lectance.
+PACS numbers: 52.35.-g 52.90.+z 02.60.Nm
+1. BASIC EQUATIONS
+Research of degenerate electron plasma behaviour, process es which take
+place in plasma under electric field, plasma waves becomes mo re and more
+actual at the present time in connection with the problems of such intensively
+developing areas as microelectronics and nanotechnologie s [1] – [6].
+This work continues the research of the electron plasma beha viour in
+external longitudinal alternating electric field [7] – [11] .
+In the present work linearized problem of plasma wave reflect ion from
+a boundary of a half-space of conductive medium is solved ana lytically.
+Specular accommodative conditions of electron reflection f rom plasma boun-
+dary are taken into consideration. The diffuse boundary cond itions were con-
+siderated in [8]–[10].2
+Expression for wave reflectance is obtained and it is shown th at in the
+case when normal electron momentum accommodation coefficien t takes on a
+value of zero the wave reflectance is expressed by the formula obtained earlier
+in [8], [11].
+Let us consider degenerate plasma which is situated in a half -spacex >0.
+We assume that self-consistent electric field E(r,t)inside plasma has one
+x–component and varies along the axis xonly:E={Ex(x,t),0,0}.
+In this case the electric field is perpendicular to the plasma boundary
+which is situated in the plane x= 0.
+Hereωpis Langmuir (proper) plasma oscillation frequency, ωp=4πe2N
+m,
+Nis the electron numerical density (concentration), mis the mass of the
+electron.
+Let us take the system of equations which describes plasma be haviour. As
+the kinetic equation we take τ–model Vlasov /emdash.cyr Boltzmann kinetic equation:
+∂f
+∂t+v∂f
+∂r+eE∂f
+∂p=feq(r,t)−f(r,v,t)
+τ. (1.1)
+Heref=f(r,v,t)is the electron distribution function, eis the electron
+charge,p=mvis the electron momentum, mis the electron mass, τis the
+characteristic time period between two collisions, feq=feq(r,t)is the local
+equilibrium distribution function of Fermi and Dirac, feq= Θ(EF(t,x)−E),
+whereΘ(x)is the function of Heaviside,
+Θ(x) =/braceleftBigg
+1, x >0,
+0, x <0,
+EF(t,x) =1
+2mv2
+F(t,x)is the disturbed kinetic energy of Fermi, E=1
+2mv2is
+the kinetic energy of the eletron.
+Let us consider the Maxwell equation for the electric field
+divE(r,t) = 4πe/integraldisplay
+(f(r,v,t)−f0(v))dΩF,. (1.2)
+where
+dΩF=2d3p
+(2π/planckover2pi1)3, d3p=dpxdpydpz.
+Heref0is the undisturbed electron distribution function of Fermi and
+Dirac,f0(E) = Θ(EF−E),/planckover2pi1is the Planck constant, EF=1
+2mv2
+Fis the3
+undisturbed kinetic energy of Fermi, vFis the electron velocity on the Fermi
+surface which is considered as spherical.
+Let us search for the solution of the system (1.1) and (1.2) in the following
+form
+f=f0(E)+EFδ(EF−E)H(x,µ,t), µ=vx
+v.
+We obtain (see [9]) linearized system of equations of Vlasov /emdash.cyr Maxwell:
+∂H
+∂t1+µ∂H
+∂x1+H(x1,µ,t1) =µe(x1,t1)+1
+21/integraldisplay
+−1H(x1,µ′,t1)dµ′,(1.3)
+∂e(x1,t1)
+∂x1=3ω2
+p
+2ν21/integraldisplay
+−1H(x1,µ′,t1)dµ′. (1.4)
+Heree(x1,t1)is dimensionless function
+e(x,t) =evF
+νEfEx(x,t),
+x1=x/lis the dimensionless coordinate, where l=vFτis the average
+free path of electrons, t1=νtis the dimensionless time , νis the effective
+frequency of electron scattering, ν= 1/τ.
+Supposing that kis the dimensional wave number, we introduce the dimen-
+sionless wave number k1=kvF
+ωp, then we have kx=k1x1
+ε, whereε=ν
+ωp.
+Let us introduce the quantity ω1=ωτ=ω/ν.
+2. BOUNDARY CONDITIONS STATEMENT
+Let the plasma wave move to the plasma boundary situated in th e plane
+x1= 0. The electric field of the wave changes according to the follo wing law
+e+(x1,t1) =E1exp(−i(k1x1
+ε+ω1t1)). (2.1)
+The amplitude of this wave E1we assume to be given. On the plasma
+boundary this wave reflects and the electric field of the reflec ted wave has
+the following form
+e−(x1,t1) =E2exp(i(k1x1
+ε−ω1t1)). (2.2)4
+The amplitude E2is unknown and is to be found from the problem
+solution. The quantities ω1andk1are not independent, the following depen-
+denceω1=ω1(k1)is determined from the solution of the dispersion equation
+which will be introduced below.
+It is required to determine what part of the wave energy is abs orbed under
+the wave reflection from the plasma boundary, and what part of the energy
+is reflected, and also to find the phase shift of the wave. It mea ns we have
+to calculate the reflectance which is determined as square of module of the
+ratio of reflected and incoming waves amplitudes
+R(k,ω,ε) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleE2
+E1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
+(2.3)
+and to find the argument of the amplitudes ratio
+φ(k,ω,ε) = arg/parenleftBigE2
+E1/parenrightBig
+= argE2−argE1. (2.4)
+Let us outline the time variable of the functions H(x1,µ,t1)ande(x1,t1),
+assuming
+H(x1,µ,t1) =e−iω1t1h(x1,µ), e(x1,t1) =e−iω1t1e(x1).(2.5)
+The system of equations (1.3) and (1.4) in this case will be tr ansformed
+to the following form:
+µ∂h
+∂x1+(1−iω1)h(x1,µ) =µe(x1)+1
+21/integraldisplay
+−1h(x1,µ′)dµ′,(2.6)
+de(x1)
+dx1=3ω2
+p
+2ν21/integraldisplay
+−1h(x1,µ′)dµ′. (2.7)
+Further instead of x1,t1we write x,t. We rewrite the system of equations
+(2.6) and (2.7) in the form:
+µ∂h
+∂x+z0h(x,µ) =µe(x)+1
+21/integraldisplay
+−1h(x,µ′)dµ′, z0= 1−iω1.(2.8)
+de(x)
+dx=3
+2ε21/integraldisplay
+−1h(x,µ′)dµ′. (2.9)5
+We consider the external electric field outside the plasma li mit is absent.
+This means that for the field inside plasma on the plasma bound ary the
+following condition is satisfied:
+e(0) = 0. (2.10)
+The non-flowing condition for the particle (electric curren t) flow through
+the plasma boundary means that
+1/integraldisplay
+−1µh(0,µ)dµ= 0. (2.11)
+In the kinetic theory for the description of the surface prop erties the
+accommodation coefficients are used often. Tangential momen tum and energy
+accommodation coefficients are the most–used. For the proble m considered
+the normal electron momentum accommodation under the scatt ering on the
+surface has the most important significance.
+The normal momentum accommodation coefficient is defined by th e follo-
+wing relation:
+αp=Pi−Pr
+Pi−Ps,0/lessorequalslantαp/lessorequalslant1, (2.12)
+wherePiandPrare the flows of normal to the surface momentum of incoming
+to the boundary and reflected from it electrons,
+Pi=0/integraldisplay
+−1µ2h(0,µ)dµ, P r=1/integraldisplay
+0µ2h(0,µ)dµ, (2.13)
+quantity Psis the normal momentum flow for electrons reflected from the
+surface which are in thermodynamic equilibrium with the wal l,
+Ps=1/integraldisplay
+0µ2hs(µ)dµ, гдеhs(µ) =As,0< µ <1.(2.14)
+The function hs(µ)is the equilibrium distribution function of the correspon-
+ding electrons. This function is to satisfy the condition si milar to the non-
+flowing condition:
+0/integraldisplay
+−1µh(0,µ)dµ+1/integraldisplay
+0µhs(µ)dµ= 0. (2.15)6
+We are going to consider the relation between the normal mome ntum
+accommodation coefficient αpand the diffuseness coefficient qfor the case of
+specular and diffuse boundary conditions which are written i n the following
+form:
+h(0,µ) = (1−q)h(0,−µ)+as,0< µ <1.
+Hereqis the diffuseness coefficient ( 0/lessorequalslantq/lessorequalslant1),asis the quantity
+determined from the non-flowing condition.
+From the non-flowing condition we derive
+as=−2q0/integraldisplay
+−1µh(0,µ)dµ=qAs.
+After that we find the difference between the flows
+Pi−Pr=q0/integraldisplay
+−1µ2h(0,µ)dµ−1/integraldisplay
+0µ2asdµ=
+=q0/integraldisplay
+−1µ2h(0,µ)dµ−q1/integraldisplay
+0µ2Asdµ=qPi−qPs.
+Substituting the expressions obtained to the definition of t he normal momen-
+tum accommodation coefficient we obtain that αp=q.
+Thus, for the specular and diffuse boundary conditions the no rmal momen-
+tum accommodation coefficient αpcoincides with the diffusion coefficient q.
+Together with the specular and diffuse boundary conditions o ther variants
+of boundary conditions are used in the kinetic theory.
+In particular, accommodative boundary conditions are wide ly used. They
+are divided into two forms: diffuse accommodative and specul ar accommo-
+dative boundary conditions (see [12]).
+We consider specular accommodative boundary conditions. F or the func-
+tionhthese conditions will be written in the following form:
+h(0,µ) =h(0,−µ)+A1+A2µ,0< µ <1. (2.16)
+Coefficients A1andA2can be derived from the non-flowing condition and
+the definition of the normal electron momentum accommodatio n coefficient.7
+The problem statement is completed. Now the problem consist s in finding
+of such solution of the system of equations (2.8) and (2.9), w hich saatisfies
+the boundary conditions (2.10)–(2.16). Further, with use o f the amplitudes
+of reflected and incoming waves found it is required to find the reflectance
+of the incoming wave energy (2.3) and the argument of the ampl itudes ratio
+(2.4).
+3. THE RELATION BETWEEN FLOWS AND BOUNDARY
+CONDITIONS
+First of all let us find expression which relates the constant sA0, A1from
+the boundary condition (2.13).
+To carry this out we will use the condition of non-flowing (2.1 2) of the
+particle flow through the plasma boundary, which we will writ e as a sum of
+two flows:
+N0≡1/integraldisplay
+0µh(0,µ)dµ+0/integraldisplay
+−1µh(0,µ)dµ= 0.
+After evident substitution of the variable in the second int egral we obtain:
+N0≡1/integraldisplay
+0µ/bracketleftBig
+h(0,µ)−h(0,−µ)/bracketrightBig
+dµ= 0.
+Taking into account the relation (2.16), we obtain that A0=−2A1/3.
+With the help of this relation we can rewrite the condition (2. 16) in the
+following form:
+h(0,µ) =h(0,−µ)+A1(µ−2
+3),0< µ <1. (3.1)
+We consider the momentum flow of the electrons which are movin g to the
+boundary. According to (3.1) we have:
+Pi=Pr−1
+36A1. (3.2)
+It is easy to see further that
+Ps=As/3. (3.3)8
+With the help of the formula (3.2) we will rewrite the definitio n of the
+accommodation coefficient (2.12) in the form:
+αpPr−αpAs
+3+A1
+36(1−αp) = 0. (3.4)
+Let us consider the condition (2.15). From this condition we obtain that
+As=−20/integraldisplay
+−1µh(0,µ)dµ= 21/integraldisplay
+0µh(0,−µ)dµ.
+Using the condition (3.1), we then get
+As= 21/integraldisplay
+0µh(0,µ)dµ. (3.5)
+Now with the help of the second equality from (2.13) and (3.4) we rewrite
+the relation (3.3) in the integral form:
+αp1/integraldisplay
+0/parenleftBig
+µ2−2
+3/parenrightBig
+h(0,µ)dµ=−1
+36(1−αp)A1. (3.6)
+Now the boundary problem consists of the equations (2.8) and (2.9) and
+boundary conditions (2.10), (3.1) and (3.6).
+4. SEPARATION OF VARIABLES AND
+CHARACTERISTICAL SYSTEM
+Application of the general Fourier method of the separation of variables
+in several steps results in the following substitution:
+hη(x,µ) = exp(−z0x
+η)Φ(η,µ), e η(x) = exp(−z0x
+η)E(η),(4.1)
+whereηis the spectrum parameter or the parameter of separation, wh ich is
+complex in general.
+We substitute the equalities (4.1) into the equations (2.8) and (2.9). We
+obtain the following characteristic system of equations:
+z0(η−µ)Φ(η,µ) =ηµE(η)+η
+21/integraldisplay
+−1Φ(η,µ′)dµ′, (4.2)9
+z0E(η) =−3
+ε2·η
+21/integraldisplay
+−1Φ(η,µ′)dµ′. (4.3)
+Let us introduce the designations:
+γ=ω
+ωp−1, η2
+1=ε2z0
+3, z0= 1−i1+γ
+ε, c= 2η2
+1z0.
+Substituting the integral from the equation (4.3) into (4.2 ), we come to
+the following system of equations:
+(η−µ)Φ(η,µ) =E(η)
+z0(ηµ−η2
+1),−η2
+1E(η) =η
+21/integraldisplay
+−1Φ(η,µ′)dµ′.(4.4)
+Solution of the system (4.4) depends essentially on the cond ition if the
+spectrum parameter ηbelongs to the interval −1< η <1. In connection
+with this the interval −1< η <1we will call as continuous spectrum of the
+characteristic system.
+Let the parameter η∈(−1,1). Then from the equations (4.4) in the
+class of general functions we will find eigenfunction corres ponding to the
+continuous spectrum:
+Φ(η,µ) =F(η,µ)E(η)
+z0, (4.5)
+where
+F(η,µ) =Pµη−η2
+1
+η−µ−cλ(η)
+ηδ(η−µ). (4.6)
+In the equation (4.6) δ(x)is the delta–function of Dirac, the symbol Px−1
+means the principal value of the integral under integrating of the expression
+x−1, the fuction λ(z)is called as dispersion function of the problem,
+λ(z) = 1+z
+c1/integraldisplay
+−1η2
+1−zµ
+µ−zdµ. (4.7)
+The function (4.5) is called eigenfunction of the continuou s spectrum,
+since the spectrum parameter ηfills out the continuum (−1,+1)compactly.
+The eigensolutions of the given problem can be found from the equalities10
+(4.7). The dispersion function λ(z)we express in the terms of the Case [13]
+dispersion function:
+λ(z) = 1−1
+z0+1
+z0/parenleftBig
+1−z2
+η2
+1/parenrightBig
+λc(z),
+where
+λc(z) = 1+z
+21/integraldisplay
+−1dτ
+τ−z=1
+21/integraldisplay
+−1τ dτ
+τ−z
+is the Case dispersion function [13].
+The boundary values of the dispersion function from above an d below the
+contour are calculated according to the Sokhotsky formulas
+λ±(µ) =λ(µ)±iπµ
+2η2
+1z0(η2
+1−µ2),−1< µ <1,
+from where we have
+λ+(µ)−λ−(µ) =iπ
+η2
+1z0µ(η2
+1−µ2),
+λ+(µ)+λ−(µ)
+2=λ(µ),−1< µ <1,
+where
+λ(µ) = 1+µ
+2η2
+1z01/integraldisplay
+−1η2
+1−η2
+η−µdη,
+and the integral in this equality is understood as singular i n terms of the
+principal value by Cauchy. Besides that, the function λ(µ)can be represented
+in the following form:
+λ(µ) = 1−1
+z0+1
+z0/parenleftBig
+1−µ2
+η2
+1/parenrightBig
+λc(µ), λc(µ) = 1+µ
+2ln1−µ
+1+µ.
+5. EIGENFUNCTIONS OF THE DISCRETE SPECTRUM AND
+PLASMA WAVES11
+According to the definition, the discrete spectrum of the cha racteristic
+equation is a set of zeroes of the dispersion equation
+λ(z)/z= 0. (5.1)
+We start to search zeroes of this equation. Let us expand take Laurent
+series of the dispersion function:
+λ(z) =λ∞+λ2
+z2+λ4
+z4+···,|z|>1. (5.1)
+Here
+λ∞≡λ(∞) = 1−1
+z0+1
+3z0η2
+1=2γ+iε+γ(γ+iε)
+(1+γ+iε)2,
+λ2=−1
+z0/parenleftBig1
+3−1
+5η2
+1/parenrightBig
+=−9+5iε(1+γ+iε)
+15(1+γ+iε)2,
+λ4=−1
+z0/parenleftBig1
+5−1
+7η2
+1/parenrightBig
+=−15+7iε(1+γ+iε)
+35(1+γ+iε)2.
+One can easily see that in collisional plasma (i.e. when ε >0) the coefficient
+λ∞/ne}ationslash= 0. Consequently, the dispersion equation has infinity as a zer oηi=
+∞, to which the discrete eigensolutions of the given system co rrespond:
+h∞(x,µ) =µ/z0, e∞(x) = 1.
+This solution is naturally called as mode of Drude. It descri bes the volume
+conductivity of metal, considered by Drude (see, for exampl e, [14]).
+Let us consider the question of the plasma mode existance in d etails. We
+find finite complex zeroes of the dispersion function. We use t he principle of
+argument. We take the contour Γ+
+ε= ΓR∪γε, (see Fig. 1), which is passed
+in the positive direction and which bounds the biconnected d omainDR. This
+contour consists of the circumference {ΓR:|z|=R, R= 1/ε,ε >0}, and
+the contour γε, which includes the cut [−1,+1], and stands at the distance
+ofεfrom it.
+According to the principle of argument the number [15] of zer oesNin the
+areaDεequals to:
+N=1
+2πi/contintegraldisplay
+Γεdlnλ(z).12
+xy
+R=1
+εγε
+γεDRΓR
+ΓRz
+•
+1 O −1• •
+Fig. 1.•
+Considering the limit in this equality when ε→0and taking into account
+that the dispersion function is analytic in the neighbourho od of the infinity,
+we obtain that
+N=1
+2πi1/integraldisplay
+−1dlnλ+(τ)
+λ−(τ)=1
+πi1/integraldisplay
+0dlnλ+(τ)
+λ−(τ)=1
+πargλ+(τ)
+λ−(τ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
+0.(5.2)
+Consider the curve
+γ=/braceleftbig
+z:z=G(τ),0/lessorequalslantτ/lessorequalslant+1/bracerightbig
+,
+whereG(τ) =λ+(τ)/λ−(τ).It is obvious that G(0) = 1,lim
+τ→1G(τ) = 1 .
+Hence, according to (5.3), the number of zeroes Nequals to the double
+number of turns of the curve γround the point of origin, i.e. N= 2κ(G),
+κ(G) = Ind [0,+1]G(τ).13
+/s48 /s49 /s50 /s51/s48/s51/s54
+/s32/s32/s68/s45
+/s68/s43/s76/s76
+Fig. 2.
+Thus, the number of zeroes of the dispersion function belong ing to the
+complex plane out of the segment [−1,1]of the real axis equals to the double
+index of the function G(τ), calculated on the semi-axis [0,+1].
+We take on the plane (γ,ε)(see Fig. 2) the curve L, determined by
+equations: γ=−1+/radicalbig
+L1(τ), ε=/radicalbig
+L2(τ),0/lessorequalslantτ/lessorequalslant1,where
+L1(τ) =−3µ2[s2+λ0(1+λ0)]2
+λ0[s2+(1+λ0)2], L2(τ) =−3τ2s2
+λ0[s2+(1+λ0)2].
+Here
+λ0(τ) = 1+τ
+2ln1−τ
+1+τ, s(τ) =π
+2τ.
+In the same way, as was shown in the work [16], we can prove that if
+(γ,ε)∈D+, then the index of the problem equals to unity, i.e. N= 2is the
+number of zeroes which equals to two, and if (γ,ε)∈D−, then the index of
+the problem equals to zero, i.e. N= 0.
+We would like to notice, that in the work [17] (see also [16]) t he method
+of examination of the boundary mode in the case when (γ,ε)∈Lwas
+developed.
+Since the dispersion function is even its zeroes differ from e ach other by
+sign. We designate these zeroes as following ±η0, byη0we take the zero which14
+satisfies the condition Reη0>0. The following solution corresponds to the
+zeroη0:
+hη0(x,µ) = exp(−z0x
+η0)E2
+z0η0µ−η2
+1
+η0−µ, e η0(x) = exp(−z0x
+η0)E2.(5.2)
+This solution is naturally called as mode of Debay (this is pl asma mode).
+In the case of low frequencies it describes well-known scree ning of Debay
+[3]. The external field penetrates into plasma on the depth of rD, rDis the
+raduis of Debay. When the external field frequancies are close to Langmuir
+frequencies, the mode of Debay describes plasma oscillatio ns (see, for instance,
+[3, 14]).
+From the equalities (2.2) for the wave e−(x,t)with the help of (2.6) and
+the equality (4.1) follows the relation between the wave num berkand the
+zero of the dispersion function η0(ω,ν):ikx
+ε=−z0x
+η0,henceη0≡η0(γ,ε) =
+1+γ
+k+iε
+k.
+The equalities (2.5) and (2.6) jointly with (5.2) mean that t he reflected
+wave corresponds to the zero η0:
+Hη0=η2
+1−η0µ
+z0(µ−η0)exp/bracketleftBig
+i/parenleftBigkx
+ε−ωt/parenrightBig/bracketrightBig
+, eη0= exp/bracketleftBig
+i/parenleftBigkx
+ε−ωt/parenrightBig/bracketrightBig
+,
+and the wave incoming to the plasma boundary corresponds to t he symmetric
+zero−η0:
+H−η0=η2
+1+η0µ
+z0(µ+η0)exp/bracketleftBig
+−i/parenleftBigkx
+ε+ωt/parenrightBig/bracketrightBig
+, e−η0= exp/bracketleftBig
+−i/parenleftBigkx
+ε+ωt/parenrightBig/bracketrightBig
+,
+6. EXPANSION IN THE TERMS OF EIGENFUNCTIONS
+In the work [11] it was shown that from the non–flowing conditi on (2.11)
+and the condition on the electric field (2.10) it results that the trivial (equal
+to zero) solution of the present problem corresponds to the p ointηi=∞.
+We will show that the system of equations (2.8) and (2.9) with the boundary
+conditions (3.1), (3.5) and (2.10) has the solution which ca n be represented15
+as an expansion by the eigenfunctions of the characteristic system:
+h(x,µ) =E2
+z0η0µ−η2
+1
+η0−µexp(ikx
+ε)+E1
+z0η0µ+η2
+1
+η0+µexp(−ikx
+ε)+
++1
+z01/integraldisplay
+0exp/parenleftBig
+−z0x
+η/parenrightBig
+F(η,µ)E(η)dη, (6.1)
+e(x) =E2exp(ikx
+ε)+E1exp(−ikx
+ε)+1/integraldisplay
+0exp/parenleftBig
+−z0x
+η/parenrightBig
+E(η)dη.(6.2)
+HereE1is given, and E2is unknown coefficient. Both of the variables
+(amplitudes of Debay) correspond to the discrete spectrum, E(η)is unknown
+function, which is called eigenfunction of continuous spec trum.
+Our purpose is to find the coefficient of the continuous spectru m and the
+relation which connects the coefficients of the discrete spec trum.
+Let us substitute the expansion (6.1) into the boundary cond ition (3.1).
+We get the following equation in the interval 0< µ <1:
+1/integraldisplay
+0/bracketleftBig
+F(η,µ)−F(η,−µ)/bracketrightBig
+E(η)dη+(E1+E2)ϕ0(µ) =−2
+3z0A1+z0A1µ.(6.3)
+Here
+ϕ0(µ) =η2
+1−η0µ
+µ−η0+η2
+1+η0µ
+µ+η0.
+Extending the function E(η)into the interval (−1,0)evenly we transform
+the equation (6.3) to the following form:
+1/integraldisplay
+−1F(η,µ)E(η)dη+(E1+E2)ϕ0(µ)−z0A1µ=−2
+3z0A1signµ.(6.4)
+Let us substitute the eigenfunctions of the continuous spec trum into the
+equation (6.4). We obtain singular integral equation with C auchy kernel in
+the interval (−1,1):
+(E1+E2)ϕ0(µ)+1/integraldisplay
+−1ηµ−η2
+1
+η−µE(η)dη−cλ(µ)
+µE(µ)−z0A1µ=16
+=−2
+3z0A1signµ. (6.5)
+7. SOLUTION OF THE SINGULAR EQUATION
+We introduce the auxiliary function
+M(z) =1/integraldisplay
+−1ηz−η2
+1
+η−zE(η)dη, (7.1)
+the boundary values of which on the real axis above and below i t are related
+by the Sokhotsky formulas:
+M+(µ)−M−(µ) = 2πi(µ2−η2
+1)E(µ). (7.2)
+M+(µ)+M−(µ)
+2=M(µ),−1< µ <+1, (7.3)
+where
+M(µ) =1/integraldisplay
+−1ηµ−η2
+1
+η−µE(η)dη,
+and the singular integral in this equality is understood as s ingular in the
+sense of principal value of Cauchy.
+With the help of the Sokhotsky formulas for the dispersion and auxiliary
+function we reduce the equation (6.5) to the boundary condit ion of the
+problem of determination of analytic function by its jump on the contour:
+λ+(µ)[M+(µ)+(E1+E2)ϕ0(µ)−z0A1µ]−λ−(µ)[M−(µ)+(E1+E2)ϕ0(µ)−
+−z0A1µ] =iπ
+3η2
+1A1µ(η2
+1−µ)signµ,−1< µ <1.
+This equation has general solution (see [15]):
+λ(z)[ϕ(z)(E1+E2)+M(z)−z0A1z] =2z0A1
+3c1/integraldisplay
+−1µ(µ2−η2
+1)signµ
+µ−zdµ+C1z,
+whereC1is an arbitrary constant.17
+Let us introduce auxiliary function
+T(z) =1
+c1/integraldisplay
+−1µ(µ2−η2
+1)signµ
+µ−zdµ.
+Then from the general solution we can easy find M(z):
+M(z) =−(E1+E2)ϕ(z)+z0A1z+2
+3z0A1T(z)
+λ(z)+C1z
+λ(z).(7.4)
+Let us eliminate the pole of the solution (7.4) in the infinity . We get that
+C1=−z0A1λ∞.
+Poles in the points z=±η0can be eliminated with the help of one equality
+since the functions constituting the general solution are u neven:
+z0A1=(E1+E2)λ′(η0)(η2
+1−η2
+0)
+(2/3)T(η0)−λ∞η0. (7.5)
+We substitute the expansion (6.1) for the function h(x,µ)to the integral
+boundary condition (3.5). We get the following equation:
+E1m(−η0)+E2m(η0)+1/integraldisplay
+0m(η)E(η)dη=−1
+36z0A11−αp
+αp.(7.6)
+In (7.6) the following designations were introduced:
+m(±η0) =1/integraldisplay
+0/parenleftBig
+µ2−2
+3/parenrightBig
+F(±η0,µ)dµ, m(η) =1/integraldisplay
+0/parenleftBig
+µ2−2
+3/parenrightBig
+F(η,µ)dµ.
+The coefficient of the continuous spectrum we will find from the Sokhotsky
+formula (7.1) after the substitution of the general solutio n (7.4) into it:
+E(η) =1
+2πi(η2−η2
+1)/bracketleftBigg
+2
+3/bracketleftBigT+(η)
+λ+(η)−T−(η)
+λ−(η)/bracketrightBig
+−λ∞η/bracketleftBig1
+λ+(η)−1
+λ−(η)/bracketrightBig/bracketrightBigg
+.(7.7)
+Let us notice that under the transition through the positive part of the
+cut(0,1)functions T(z)andλ(z)make jumps, which differ only by sign.
+Indeed, let us represent the formula for T(z)in the following form:
+T(z) =z
+c1/integraldisplay
+0(µ2−η2
+1)/bracketleftBigg
+1
+µ−z+1
+µ+z/bracketrightBigg
+dµ.18
+This integral can be calculated easily in explicit form. On t he cut this
+integral is calculated according to the following formula :
+T(η) =η
+c/bracketleftBig
+1+(η2−η2
+1)ln/parenleftBig1
+η2−1/parenrightBig/bracketrightBig
+,−1< η <+1.
+Now from the Sokhotsky formula for the difference of boundary values we
+obtain that under condition 0< η <1the following equality takes place:
+λ+(η)−λ−(η) =λ(η)±iπη(η2
+1−η2)
+c,
+T+(η)−T−(η) =T(η)±iπη(η2−η2
+1)
+c.
+Now one can naturally find that
+T+(η)λ−(η)−T−(η)λ+(η) = 2(T(η)+λ(η))·iπη(η2−η2
+1)
+c,
+λ−(η)−λ+(η) = 2iπη(η2−η2
+1)
+c.
+With the help of the last relations we find the coefficient of the c ontinuous
+spectrum from (7.5):
+E(η) =z0A1Q(η),whereQ(η) =2
+3η[T(η)+λ(η)]−λ∞η2
+cλ+(η)λ−(η).(7.8)
+We introduce the integral
+T0(z) =1
+c1/integraldisplay
+0η2−η2
+1
+η−zdη.
+It is evident that in the complex plane this integral is calcu lated by the
+following formula:
+T0(z) =z
+c/bracketleftBig1
+2+z+(z2−η2
+1)ln/parenleftBig1
+z−1/parenrightBig/bracketrightBig
+.
+With the help of this function we represent the dispersion fun ction in the
+form:λ(z) = 1−zT0(z)+zT0(−z),the function T(z)we also express in terms
+of this integral: T(z) =zT0(z)+zT0(−z).The sum of two last expressions
+equals to: λ(z)+T(z) = 1+2zT0(−z).Let us note that the integral T(−z)is19
+not singular on the cut 0< η <1. The sum λ(η)+T(η)on the cut 0< η <1
+is calculated in explicit form without applying to integral s:
+λ(η)+T(η) = 1+1
+c/bracketleftBig
+η−2η2+2η(η2−η2
+1)ln(1/η+1)/bracketrightBig
+.
+We calculate the integrals m(±η0)andm(η)in explicit form. The integrals
+m(±η0)can be determined easily:
+m(±η0) =−(η2
+0−η2
+1)/bracketleftBig
+−1
+6+η0+η0(η0−2
+3)ln(1
+η0−1)/bracketrightBig
+.
+Let us find the integral m(η). We have:
+m(η) =1/integraldisplay
+0/parenleftBig
+µ2−2
+3µ/parenrightBig
+(η2
+1−ηµ)dµ
+µ−η−2η2
+1z0(η−2
+3)λ(η)θ+(η).
+Hereθ+(η)is the characteristic function of the interval 0< η <1, i.e.
+θ+(η) =/braceleftBig1,0< η <1,
+0,−1< η <0.
+Computating the integral in the preceding equality we obtai n that the
+integralm(η)is calculated by the formula:
+m(η) = (1
+6−η)(η2−η2
+1)+(η−2
+3)/bracketleftBig
+−c+2η2−η(η2−η2
+1)f+(η)/bracketrightBig
+,
+where
+f+(η) =
+
+ln1+η
+η,0< η <1,
+ln1−η
+η,−1< η <0.
+Now the equation (7.6) with the help (7.8) we rewrite in the fo rm:
+E1m(−η0)+E2m(η0) =−z0A1/parenleftBig1−αp
+36αp+Qm/parenrightBig
+, (7.9)
+where
+Qm=1/integraldisplay
+0m(η)Q(η)dη.
+After that substituting the variable z0A1into the equation (7.9) according
+to (7.5), we derive the equation:
+E1m(−η0)+E2m(η0) =−/parenleftBig1−αp
+36αp+Qm/parenrightBigE1+E2
+2
+3T(η0)−λ∞η0,20
+from which we can find the amplitude required E2:
+E2=−αpm(−η0)A(η0)+B(η0)C(αp)
+αpm(η0)A(η0)+B(η0)C(αp)E1. (7.10)
+We introduced the following designations in the formula (7. 10):
+A(η0) =2
+3T(η0)−λ∞η0, C(αp) =1−αp
+36+αpQm,
+B(η0) = (η2
+1−η2
+0)λ′(η0).
+Thus, all the coefficients of the expansions (6.1) and (6.2) ar e determined
+unambiguously, and this completes the proof of these expans ions.
+It is seen from the equality (7.10) that under condition αp= 0we have:
+E2=−E1, i.e. under condition of pure specular reflection of electro ns from
+the boundary the wave reflectance equals to unity: R= 1, and the phase
+shift of the incoming and reflected waves is equal to 180◦, i.e.φ=π,from
+where we see that argE2= argE1+π.
+Let us represent the formula (7.10) in the form which is more c onvenient
+for numerical analysis. Let us designate the ratio of amplit udes asK, K=
+E2/E1, the
+K=−1+αp[m(η0)−m(−η0)]
+αpm(η0)+C(αp)D(η0), (7.9)
+where
+D(η0) =B(η0)
+A(η0)=(η2
+1−η2
+0)λ′(η0)
+(2/3)T(η0)−λ∞η0.
+8. LONG WAVE LIMIT
+For study of the incoming wave reflectance R=|K|2and the phase shift
+φ= argKwe will use the formula (7.11).
+We consider the dispersion equation with small values of the wave number
+k:
+λ/parenleftBig
+iεz0
+k/parenrightBig
+=λ∞−λ2k2
+ε2z2
+0= 0. (8.1)
+We assume the frequency ωis complex: ω=ω0+iω1. Then the quantity
+γis complex also: γ=γ0+iγ1. Hereγ0=ω0/ωp−1, γ1=ω1/ωp. From21
+the equation (8.1) we find that when kis smallγ0= 0.3k2, γ1=−0.5ε. We
+express the parameters of the problem in terms of kandε:λ∞= 0.6k2(1−iε),
+z0=1
+2−1+0.3k2
+ε, η2
+1=−iε
+3(1+0.3k2), η0=1+0.3k2+i0.5ε
+k.
+With the help of these parameters let us carry out the study of t he
+reflectance and the phase shift in long wave limit (when kis small by magni-
+tude).
+On the Fig. 3 one can see the dependence of the reflectance Ron the
+wave number kfor the case when ε= 10−2. The curves 1,2,3correspond to
+the following values of the accommodation coefficient αp= 0.1,0.5,1.0. The
+curve4corresponds to the diffuse boundary conditions (see [10]). F rom the
+graph it is seen that when ktakes on small values the curve 3(corresponding
+toαp= 1) coincides practically with the curve 4(to which corresponds
+the case q= 1), which was obtained by the linearization of the reflectance
+value by k. Taking into account that under αp= 0the reflectance is equal
+to reflectance for specular boundary conditions (i.e. when q= 0) we can
+conclude that specular accommodative boundary conditions approximate
+specular and diffuse boundary conditions under αp=qvery well.
+On the Fig. 4 the dependence of the reflectance Ron the quantity εfor the
+caseαp= 1is represented. The curves 1,2,3,4,5correspond to the following
+values of the wave number k= 0.001,0.01,0.05,0.1,0.2. The more accurate
+analysis shows that with the growth of the accommodation coe fficient the
+value of the reflectance decreases.
+On the Fig. 5 the dependence of the reflectance Ron the value of the
+accommodation coefficient αpfor the case ε= 10−3is presented. The curves
+1,2,3,4correspond to the following values of the wave number k= 0.05,
+0.1,0.15,0.25.
+On the Fig. 6 the dependence of the angle φ= argK(of the phase
+shift) on the quantity εfor the case k= 0.2is represented. The curves
+1,2,3correspond to the following values of the accommodation coe fficient
+αp= 0.1,0.5,1. The analysis shows that the dependence between the values
+of the angle φand the wave number and the accommodation coefficient as22
+/s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s48/s46/s57/s50/s48/s46/s57/s54/s49/s46/s48/s48
+/s32/s32/s49
+/s50
+/s51
+/s52
+/s107/s82
+/s48/s46/s48/s53 /s48/s46/s49/s48/s48/s46/s57/s50/s48/s46/s57/s54/s49/s46/s48/s48
+/s32/s32/s82
+/s49
+/s50
+/s51
+/s52
+/s53
+Fig. 3. Fig. 4.
+/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s57/s50/s48/s46/s57/s54/s49/s46/s48/s48
+/s32/s32/s49
+/s50
+/s51
+/s52
+/s112/s82
+/s48/s46/s48/s53 /s48/s46/s49/s48/s51/s46/s49/s51/s53/s51/s46/s49/s51/s56/s51/s46/s49/s52/s49
+/s32/s32
+/s107/s97/s114/s103/s32/s75
+/s49
+/s50
+/s51
+Fig. 5. Fig. 6.
+well is small.
+9. CONCLUSION
+In the present work new boundary conditions for the question s of plasma
+wave reflection from the plane boundary of a half–space of deg enerate plasma
+were proposed. These boundary conditions are naturally cal led as specular
+accommodative conditions. Such boundary conditions are mo st adequate for
+the problems of normal propagation of plasma waves (perpend icular to the
+boundary), since accommodation coefficient under such bound ary conditions
+is normal electron momentum accommodation coefficient.23
+In the present paper the analytical solution of the problem o f plasma wave
+reflection from a boundary with normal electron momentum acc ommodation
+is obtained. The analysis of the main parameters of the probl em in long
+wave limit is carried out. This analysis shows that the bound ary conditions
+proposed are intermediate between pure specular and pure di ffuse boundary
+conditions. Indeed, from the Fig. 3 it is seen that all the gra phs showing
+the dependence between the reflectance and the wave number ar e located
+between graphs corresponding to specular and diffuse bounda ry conditions.
+REFERENCES
+1.Fortov V. E. (ed.) The encyclopaedia of low temperature plasmas//
+1997–2009. Volums I–VII. Moscow, Nauka (in Russian).
+2.Vedenyapin V. V. Boltzmann and Vlasov kinetic equations. Moscow,
+Fizmatlit, 2001 (in Russian).
+3.Abrikosov A. A. Fundamentals of the Theory of Metals, Nauka, Moscow,
+1977; North Holland, Amsterdam, 1988.
+4. Dressel M., Gr¨ uner G. Electrodynamics of Solids. Optical Properties of
+Electrons in Matter . Cambridge. Univ. Press. 2003. 487 p.
+5.Boyd T.J.N., Sanderson J.J. The Physics of Plasmas// Cambridge Univ.
+Press. 2003. 546 pp.
+6.Liboff R.L. Kinetic theory: classical, quantum, and relativistic
+description// 2003. Springer Verlag. New York, Inc. 587 pp.
+7.Latyshev A. V. and Yushkanov A. A. Analytical solution of the problem
+on behaviour the degenerate electronic plasmas// – Chapter 10 in
+"Encyclopaedia of low temperature plasma". Vol. VII-I. P. 1 59–177.
+Moscow, 2008 (in Russian).24
+8.Latyshev A. V. and Yushkanov A. A. Reflection and Transmission
+of Plasma Waves at the Interface of Crystallities. – Computa tional
+Mathematics and Mathematical Physics, 2007, Vol. 47, No 7, p p. 1179–
+1196.
+9.Latyshev A. V. and Yushkanov A. A. Reflection of plasma waves from a
+plane boundary. – Theor. Math. Phys. V. 150 (3), 425 – 435 (200 7).
+10.Latyshev A. V. and Yushkanov A. A. Reflection of a Plasma Wave from
+the Flat Boundary of a Degenerate Plasma. – Technical Physic s. 2007.
+Vol. 52. No. 3, pp. 306 – 312.
+11.Latyshev A. V. and Yushkanov A. A. Boundary value problems for
+degenerate electronic plasmas// Monograph. Moscow State R egional
+University, 2006 (in Russian). 274 P.
+12.Latyshev A. V. and Yushkanov A. A. The Method of Singular Equations
+in Boundary Value Problems in Kinetic Theory. – Theor. Math. Phys.
+2005. 143(3). P. 855–870.
+13.Case K. M. and Zweifel P. F. Linear Transport Theory. Addison–Wesley
+Publ. Comp. Reading, 1967.
+14.Platzman P. M. and Wolf P. A. Waves and interactions in solid state
+plasmas. Academic Press. New York and London. 1973.
+15.Gakhov F. D. , Boundary Value Problems [in Russian], Nauka, Moscow
+(1977); English transl., Dover, New York (1990).
+16.Latyshev A. V. and Yushkanov A. A. Analytic solution of boundary
+– value problems for nonstationary model kinetic equations . – Theor.
+Math. Phys. 1992. V. 92. №1, p.p. 782–790.
+17.Latyshev A. V. and Yushkanov A. A. Nonstationary boundary problem
+for model kinetic equations at critical parameters. – Teor. and Math.
+Phys. 1998. V. 116. №. 2. P. 978 – 989.
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+arXiv:1001.0847v1  [gr-qc]  6 Jan 2010The post-Minkowskian limit of f(R)-gravity
+Salvatore Capozziello⋄,Arturo Stabile♮, Antonio Troisi‡
+⋄Dipartimento di Scienze Fisiche, Universit` a di Napoli ”Fe derico II” and INFN sez. di
+Napoli Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthi a, I-80126 - Napoli, Italy
+♮Dipartimento di Ingegneria, Universita’ del Sannio and INF N sez. di Napoli,
+gruppo collegato di Salerno, C-so Garibaldi, I - 80125 Benev ento, Italy and
+‡Dipartimento di Ingegneria Meccanica, Universita’ di Sale rno,
+Via Ponte Don Melillo, I - 84084 Fisciano (SA), Italy
+(Dated: September 27, 2018)
+Weformally discussthepost-Minkowskianlimitof f(R)-gravitywithoutadoptingconformal trans-
+formations but developing all the calculations in the origi nal Jordan frame. It is shown that such
+an approach gives rise, in general, together with the standa rd massless graviton, to massive scalar
+modes whose masses are directly related to the analytic para meters of the theory. In this sense, the
+presence of massless gravitons only is a peculiar feature of General Relativity. This fact is never
+stressed enough and could have dramatic consequences in det ection of gravitational waves. Finally
+the role of curvature stress-energy tensor of f(R)-gravity is discussed showing that it generalizes the
+so called Landau-Lifshitz tensor of General Relativity. Th e further degrees of freedom, giving rise
+to the massive modes, are directly related to the structure o f such a tensor.
+PACS numbers: 04.50.Kd, 04.25.Nx, 04.30.-w
+Keywords: alternative theories of gravity, weak field limit , gravitational waves
+I. INTRODUCTION
+Astrophysical observations of the last decade suggests the intr oduction of new ingredients in order to achieve a
+self-consistent picture of cosmos. In particular, the observatio n that Hubble flow is currently experiencing a speeding
+up has completely changed the approach to standard cosmology ind ucing to take into account theoretical approaches
+more general than the standard lore of General Relativity (GR).
+The simplest explanation of such a cosmic acceleration requires to inc lude the cosmological constant in the
+Friedmann-Robertson-Walker cosmology (Concordance Model). T his ingredient gives rise to a negative pressure
+contribution needed to balance the standard matter attractive in teraction. Although the Concordance Model repre-
+sents the best fit model with respect to all samples of data coming f rom supernovae, large-scale structure and cosmic
+microwave radiation [1], several conceptual problems come out to t heoretically define and to give an explanation to
+the observed value of the cosmological constant . Furthermore, assuming the existence of both dark energy and dark
+matter, we should find out new fundamental ingredients capable of giving account to almost 95% of the total amount
+of cosmic matter-energy.
+Due to these difficulties, people have considered alternative approa ches to GR that could be able to frame the
+observed late time acceleration and missing matter without introduc ing new ingredients. In this sense, higher order
+gravity [2] and, in particular, fourth order gravity represent an in teresting scheme which could, potentially, address
+the problems.
+Up to now, these theories have been investigated both at cosmolog ical scale and in the weak field limit with
+significant results [3–6]. It has been shown that an accelerating late time behaviour can be easily recovered [7] and,
+in addition, it can be coherently related to an early time inflationary ex pansion [8]. Furthermore, such an approach
+seems to deserve attention even at smaller scales. In fact, modify ing the gravity action in favor of a non-linear
+Lagrangian in the Ricci scalar implies, in the Newtonian limit, correction s to the gravitational potential which can
+induce an astrophysical phenomenology interesting at galactic sca les. In particular one can fit the rotation curves
+of spiral galaxies and the haloes of galactic clusters without the intr oduction of dark matter [9]. Besides, several of
+these extended models evade Solar System tests so they are not in conflict with positive experimental results of GR
+[10, 11].
+A relevant aspect of higher order gravity theories is that, in the po st-Minkowskian limit (i.e. small fields and no
+prescriptions on the propagation velocity), the propagation of th e gravitational fields turns out to be characterized
+by waves with both tensorial and scalar modes [12, 13]. This issue re presents a striking difference between GR and
+extended gravity since, in the standard Einstein scheme, only tens orial degrees of freedom are allowed. As matter of
+facts, the gravitational waves can represent a fundamental to ol to discriminate between GR and alternative gravities
+[15, 16].
+In this paper, we want to develop, formally, the post-Minkowkian limit of analytic f(R)-gravity models which, in
+our opinion, has never been pursued with accuracy stressing enou gh some peculiar points. As shown by the same2
+authors for the Newtonian limit, we will show that it is different from th e same limit of GR since massive modes
+naturally come out in the gravitational radiation. This occurrence h as a deep meaning since points out that the
+presence of massless modes only is nothing else but the particular ca se of GR while massive and ghost modes are
+present in general [13].
+The layout of the paper is the following. In Sec. II, we discuss the po st-Minkowskian limit of f(R)-gravity.
+Considerations on gravitational wave massive modes are developed in Sec.III. Sec.IV is devoted to the discussion of
+the role of the stress-energy tensor in f(R)-gravity. Concluding remarks are drawn in Sec. V.
+II. THE POST-MINKOWSKIAN LIMIT OF f(R)- GRAVITY
+Any theory of gravity has to be discussed in the weak field limit approx imation. This ”prescription” is needed to
+test if the given theory is consistent with the well-established Newto nian Theory and with the Special Relativity as
+soon as the the gravitational field is weak or is almost null. Both requir ements are fulfilled by GR and then they can
+be considered two possible paradigms to confront a given theory, a t least in the weak field limit, with GR itself. In
+[17, 18], the Newtonian limit of f(R)-gravity is investigated always remaining in the Jordan frame [14]. Fr om our
+point of view, this is important since, by perturbatively approximatin g a field, some conformal features could be lost.
+Here we want to derive, formally, the post-Minkowskian limit of f(R)-gravity.
+The post-Minkowskian limit of any theory of gravity arises when the r egime of small field is considered without any
+prescription on the propagation of the field. This case has to be clea rly distinguished with respect to the Newtonian
+limit which, differently, requires both the small velocityand the weak fi eld approximations. Often, in literature, such a
+distinction is not clearly remarked and several cases of pathologica l analysis can be accounted. The post-Minkowskian
+limit of GR gives rise to massless gravitational waves. An analogous st udy can be pursued considering, instead of the
+Hilbert-Einstein Lagrangian linear in the Ricci scalar R, a general function f(R). The only assumption that we are
+going to do is that f(R) is an analytic function. The gravitational action is then
+A=/integraldisplay
+d4x√−g/bracketleftbigg
+f(R)+XLm/bracketrightbigg
+, (1)
+whereX=16πG
+c4is the coupling, Lmis the standard matter Lagrangian and gis the determinant of the metric. The
+field equations, in metric formalism, read1
+f′Rµν−1
+2fgµν−f′
+;µν+gµν/squaregf′=X
+2Tµν (2)
+3/squaref′+f′R−2f=X
+2T , (3)
+withTµν=−2√−gδ(√−gLm)
+δgµνthe energy momentum tensor of matter ( Tis the trace), f′=df(R)
+dRand/squareg=;σ;σ. We
+adopt a (+ ,−,−,−) signature, while the conventions for Ricci’s tensor is Rµν=RσµσνandRαβµν= Γα
+βν,µ+...for
+the Riemann tensor, where
+Γµ
+αβ=1
+2gµσ(gασ,β+gβσ,α−gαβ,σ) (4)
+aretheChristoffel symbolsofthe gµνmetric. Actually, in orderto performapost-Minkowskianlimit offield e quations,
+one has to perturb Eqs. (2) on the Minkowski background ηµν. In such a case the invariant metric element becomes
+ds2=gστdxσdxτ= (ηστ+hστ)dxσdxτ(5)
+withhµνsmall (O(h)2≪1). We assume that the f(R)-Lagrangian is analytic (i.e. Taylor expandable) in term of the
+Ricci scalar, which means that
+1All considerations are developed here in metric formalism.3
+f(R) =/summationdisplay
+nfn(R0)
+n!(R−R0)n≃f0+f′
+0R+1
+2f′′
+0R2+.... (6)
+The flat-Minkowski background is recovered for R=R0≃0.
+Field equations (2), at the first order of approximation in term of th e perturbation [19], become:
+f′
+0/bracketleftbigg
+R(1)
+µν−R(1)
+2ηµν/bracketrightbigg
+−f′′
+0/bracketleftbigg
+R(1)
+,µν−ηµν/squareR(1)/bracketrightbigg
+=X
+2T(0)
+µν (7)
+wheref′
+0=df
+dR/vextendsingle/vextendsingle/vextendsingle
+R=0,f′′
+0=d2f
+dR2/vextendsingle/vextendsingle/vextendsingle
+R=0and/square=,σ,σthat is now the standard d’Alembert operator of flat space-time.
+From the zero-order of Eqs.(2), one gets f(0) = 0, while Tµνis fixed at zero-order in Eq.(7) since, in this perturbation
+scheme, the first order on Minkowski space has to be connected w ith the zero order of the standard matter energy
+momentumtensor2. TheexplicitexpressionsoftheRiccitensorandscalar,atthefirs torderinthemetricperturbation,
+read
+
+
+R(1)
+µν=hσ
+(µ,ν)σ−1
+2/squarehµν−1
+2h,µν
+R(1)=hστ,στ−/squareh(8)
+withh=hσσ. Eqs. (7) can be written in a more suitable form by introducing the co nstantξ=−f′′
+0
+f′
+0, that is
+hσ
+(µ,ν)σ−1
+2/squarehµν−1
+2h,µν−1
+2(hστ,στ−/squareh)ηµν+ξ(∂2
+µν−ηµν/square)(hστ,στ−/squareh) =X
+2f′
+0T(0)
+µν. (9)
+By choosing the transformation ˜hµν=hµν−h
+2ηµνand the gauge condition ˜hµν
+,µ= 0, one obtains that field equations
+and the trace equation, respectively, read3
+
+
+/square˜hµν+ξ(ηµν/square−∂2
+µν)/square˜h=−X
+f′
+0T(0)
+µν
+/square˜h+3ξ/square2˜h=−X
+f′
+0T(0). (10)
+In orderto derive the analytic solutionsof Eqs. (10), we can adopt a momentum- description. This approachsimplifies
+the equations and allows to fix the physical properties of the proble m. In such a scheme, we have:
+
+
+k2˜hµν(k)+ξ(kµkν−k2ηµν)k2˜h(k) =X
+f′
+0T(0)
+µν(k)
+k2˜h(k)(1−3ξk2) =X
+f′
+0T(0)(k)(11)
+where
+
+
+˜hµν(k) =/integraltextd4x
+(2π)2˜hµν(x)e−ikx
+T(0)
+µν(k) =/integraltextd4x
+(2π)2T(0)
+µν(x)e−ikx(12)
+2This formalism descends from the theoretical setting of New tonian mechanics which requires the appropriate scheme of a pproximation
+when obtained from a more general relativistic theory. This scheme coincides with a gravity theory analyzed at the first o rder of
+perturbation in the curved spacetime metric.
+3The gauge transformation is h′
+µν=hµν−ζµ,ν−ζν,µwhen we perform a coordinate transformation as x′µ=xµ+ζµwith O(ζ2)≪1.
+To obtain the gauge and the validity of the field equations for both perturbation hµνand˜hµν, theζµhave to satisfy the harmonic
+condition /squareζµ= 0.4
+are the Fourier transforms of the perturbation ˜hµν(x) and of the matter tensor T(0)
+µν. We have defined, as usual,
+kx=ωt−k·xandk2=ω2−k2;˜h(k) andT(0)(k) are the traces of ˜hµν(k) andT(0)
+µν(k). In the momentum space,
+one can easily recognize the solutions of Eqs.(11); ˜hµν(k) turns out to be
+˜hµν(k) =X
+f′
+0T(0)
+µν(k)
+k2+ξX
+f′
+0k2ηµν−kµkν
+k2(1−3ξk2)T(0)(k), (13)
+which fulfils the condition ˜hµν
+,µ= 0 (that is ˜hµν(k)kµ= 0). The perturbation variable hµν(k) can be obtained by
+inverting the relation with the tilded variables. In particular, by inser ting the new stress-energy tensor S(0)
+µν(k) =
+T(0)
+µν(k)−1
+2ηµνT(0)(k) with the trace S(0)(k) =ηµνS(0)
+µν(k), one obtains:
+hµν(k) =X
+f′
+0S(0)
+µν(k)
+k2+ξX
+2f′
+0k2ηµν+2kµkν
+k2(1−3ξk2)S(0)(k), (14)
+which represents a wave-like solution, in the momentum space, with a massless and a massive contributions. The
+massive term is due to the pole in the denominator of the second term : themassis directly related with the physical
+properties of the pole itself and, thanks to the parameter ξ, depends on the analytic form of the model (i.e. f′
+0and
+f′′
+0). The wavelike solution in the configuration space is obtained by the in verse Fourier transform of hµν(k).
+III. MASSIVE MODES IN GRAVITATIONAL WAVES
+The presence of the massive term is a feature emerging from the int rinsic non-linearity of f(R)-gravity. Specifically,
+it is related to the fact that f′′
+0∝ne}ationslash= 0, which is zero in GR where f(R) =R. This means that massless states are nothing
+else but a particular case among the gravitational theories. A similar situation emerges also in the Newtonian limit:
+the Newton potential is recovered only as the weak field limit of GR. In general, Yukawa-like corrections, and then
+characteristic interaction lengths, are present [17]
+Some considerations are in order at this point. It is worth noticing th at field Eqs. (2) can be written putting in
+evidence the Einstein tensor in the l.h.s. [3]. In such a case, higher th an second order differential contributions can
+be considered as a sources in the r.h.s. as well as the energy-momen tum tensor of standard matter:
+Gµν=Rµν−1
+2Rgµν=T(curv)
+µν+T(m)
+µν, (15)
+where
+
+
+T(m)
+µν=Tµν
+f′(R)
+T(curv)
+µν=1
+2gµνf(R)−f′(R)R
+f′(R)+f′(R);µν−gµν/squaregf′(R)
+f′(R).(16)
+Actually, if we consider the perturbed metric (5) and develop the Ein stein tensor up to the first order in perturbations,
+we have
+Gµν∼G(1)
+µν=hσ
+(µ,ν)σ−1
+2/squarehµν−1
+2h,µν−1
+2(hστ,στ−/squareh)ηµν (17)
+while the curvature stress-energy tensor gives the contribution s
+T(curv)
+µν∼ξ(∂2
+µν−ηµν/square)(hστ,στ−/squareh). (18)
+This expression allows to recognize that, in the space of momenta, s uch a quantity will be responsible of the pole-like
+term which implies the introduction of a massive degree of freedom int o the particle spectrum of gravity. In fact,
+inserting these two expressions into the the field Eqs. (15) and con sidering Eqs.(8), we obtain the solution:5
+/squarehµν(x) =−X
+f′
+0/bracketleftbigg
+S(0)
+µν(x)+Σξ
+µν(x)/bracketrightbigg
+(19)
+where Σξ
+µν(x) is related to the curvature stress-energy tensor and is defined as
+Σξ
+µν(x) =ξ
+2/integraldisplayd4k
+(2π)2k2ηµν+2kµkν
+1−3ξk2S(0)(k)eikx. (20)
+The general solution for the metric perturbation hµν(x), when the field equations are (15), can be rewritten as
+hµν(x) =X
+f′
+0/integraldisplayd4k
+(2π)2S(0)
+µν(k)
+k2eikx+ξX
+2f′
+0/integraldisplayd4k
+(2π)2k2ηµν+2kµkν
+k2(1−3ξk2)S(0)(k)eikx, (21)
+where the second pole-like term is present. In vacuum (i.e. T(m)
+µν= 0), Eqs. (10) become
+
+
+k2[˜hµν(k)+ξ(kµkν−k2ηµν)˜h(k)] = 0
+k2˜h(k)(1−3ξk2) = 0(22)
+showing that allowed solutions are of two types, i.e.:
+
+
+ω=±|k|
+hµν(x) =/integraltextd4k
+(2π)2hµν(k)eikxwithh(k) = 0, (23)
+and
+
+
+ω=±/radicalBig
+k2+1
+3ξ
+hµν(x) =−/integraltextd4k
+(2π)2/bracketleftbigg
+ηµν+6ξkµkν
+6/bracketrightbigg
+h(k)eikxwithh(k)∝ne}ationslash= 0. (24)
+It is evident, that the first solution represents a massless gravito n according to the standard prescriptions of GR while
+the second one gives a massive degree of freedom with m2= (3ξ)−1=−f′
+0
+3f′′
+0. Thanks to this propertiy, we can rewrite
+Eqs.(10) introducing a scalar field φ=/square˜hso that the general system can be rearranged in the following way
+
+
+/square˜hµν=−X
+f′
+0T(0)
+µν+/bracketleftbigg
+∂2
+µν−ηµν/square
+3m2/bracketrightbigg
+φ
+(/square+m2)φ=−X
+f′
+0m2T(0)(25)
+which suggests that the higher order contributions act, in the pos t-Minkowskian limit, as a massive scalar field whose
+mass depends on the derivatives f′(R) andf′′(R), calculated on the unperturbed background metric.
+The massive mode is directly related to the coefficients of the Taylor e xpansion and it is interesting to note that
+they dermine also the value of the Yukawa correction in the Newtonia n approximation [17, 18]. On the other hand, it
+is straightforward to see that massive modes are directly related t o the non-trivial structure of the trace equation as
+it is easy to see from Eq.(3). In GR, the Ricci scalar is univocally fixed b eingR= 0 in vacuum and R∝ρin presence
+of matter, where ρis the matter-energy density.6
+IV. THE STRESS-ENERGY TENSOR IN f(R)-GRAVITY AND THE GRAVITATIONAL RADIATION
+As we have seen, higher order theories of gravity introduce furth er degrees of fredom which can be taken into
+account by defining an additional ”curvature source term” in the r .h.s. of field equations. This quantity behaves as
+an effective stress-energy tensor that can characterize the en ergy loss due to the gravitational radiation. Although
+the procedure to calculate the stress-energy tensor of the gra vitational field in GR is often debated, one can extend
+the formalism to more general theories and obtain this quantity by v arying the gravitational Lagrangian. In GR, this
+quantity is a pseudo-tensor and is tipically referred to as the Landa u-Lifshitz energy-momentum tensor [20].
+The calculations of GR need to be extended when dealing with higher or der gravity. In the case of f(R)-gravity,
+we have
+δ/integraldisplay
+d4x√−gf(R) =δ/integraldisplay
+d4xL(gµν,gµν,ρ,gµν,ρσ)≈/integraldisplay
+d4x/parenleftbigg∂L
+∂gρσ−∂λ∂L
+∂gρσ,λ+∂2
+λξ∂L
+∂gρσ,λξ/parenrightbigg
+δgρσ= (26)
+.=/integraldisplay
+d4x√−gHρσδgρσ= 0.
+The Euler-Lagrange equations are then
+∂L
+∂gρσ−∂λ∂L
+∂gρσ,λ+∂2
+λξ∂L
+∂gρσ,λξ= 0, (27)
+which coincide with the field Eqs. (2) in vacuum. Actually, even in the ca se of more general theories, it is possible to
+define an energy-momentum tensor that turns out to be defined a s follows:
+tλ
+α=1√−g/bracketleftbigg/parenleftbigg∂L
+∂gρσ,λ−∂ξ∂L
+∂gρσ,λξ/parenrightbigg
+gρσ,α+∂L
+∂gρσ,λξgρσ,ξα−δλ
+αL/bracketrightbigg
+. (28)
+This quantity, together with the energy-momentum tensor of mat terTµν, satisfies a conservation law as required by
+the Bianchi identities. In fact, in presence of matter, one has Hµν=χ
+2Tµν, and then
+(√−gtλ
+α),λ=−√−gHρσgρσ,α=−X
+2√−gTρσgρσ,α=−X(√−gTλ
+α),λ, (29)
+and, as a consequence,
+[√−g(tλ
+α+XTλ
+α)],λ= 0 (30)
+that is the conservationlaw given by the Bianchi identities. We can no w write the expression ofthe energy-momentum
+tensortλ
+αin term of the gravity action f(R) and its derivatives:
+tλ
+α=f′/braceleftbigg/bracketleftbigg∂R
+∂gρσ,λ−1√−g∂ξ/parenleftbigg√−g∂R
+∂gρσ,λξ/parenrightbigg/bracketrightbigg
+gρσ,α+∂R
+∂gρσ,λξgρσ,ξα/bracerightbigg
+−f′′R,ξ∂R
+∂gρσ,λξgρσ,α−δλ
+αf , (31)
+It is worth noticing that tλ
+αis a non-covariant quantity in GR while its generalization, in fourth ord er gravity, turns
+out to satisfy the covariance prescription of standard tensors ( see also [2]). On the other hand, such an expression
+reduces to the Landau-Lifshitz energy-momentum tensor of GR a s soon as f(R) =R, that is
+tλ
+α|GR=1√−g/parenleftbigg∂LGR
+∂gρσ,λgρσ,α−δλ
+αLGR/parenrightbigg
+(32)
+where the GR Lagrangian has been considered in its effective form, i.e . the symmetric part of the Ricci tensor, which
+effectively leads to the equations of motion, that is
+LGR=√−ggµν(Γρ
+µσΓσ
+ρν−Γσ
+µνΓρ
+σρ). (33)7
+It is important to stress that the definition of the energy-moment um tensor in GR and in f(R)-gravity are different.
+This discrepancy is due to the presence, in the second case, of high er than second order differential terms that cannot
+be discarded by means of a boundary integration as it is done in GR. We have noticed that the effective Lagrangian
+of GR turns out to be the symmetric part of the Ricci scalar since th e second order terms, present in the definition
+ofR, can be removed by means of integration by parts.
+On the other hand, an analytic f(R)-Lagrangian can be recast, at linear order, as f∼f′
+0R+F(R), where the
+function Fsatisfies the condition: lim R→0F →R2. As a consequence, one can rewrite the explicit expression of tλ
+α
+as:
+tλ
+α=f′
+0tλ
+α|GR+F′/braceleftbigg/bracketleftbigg∂R
+∂gρσ,λ−1√−g∂ξ/parenleftbigg√−g∂R
+∂gρσ,λξ/parenrightbigg/bracketrightbigg
+gρσ,α+∂R
+∂gρσ,λξgρσ,ξα/bracerightbigg
+−F′′R,ξ∂R
+∂gρσ,λξgρσ,α−δλ
+αF.(34)
+The general expression of the Ricci scalar, obtained by splitting its linear (R∗) and quadratic ( ¯R) parts once a
+perturbed metric (5) is considered, is
+R=gµν(Γρ
+µν,ρ−Γρ
+µρ,ν)+gµν(Γρ
+σρΓσ
+µν−Γσ
+ρµΓρ
+νσ) =R∗+¯R, (35)
+(notice that LGR=−√−g¯R). In the case of GR tλ
+α|GR, the Landau-Lifshitz tensor presents a first non-vanishing term
+at order h2. A similar result can be obtained in the case of f(R)-gravity. In fact, taking into account Eq.(34), one
+obtains that, at the lower order, tλ
+αreads:
+tλ
+α∼tλ
+α|h2=f′
+0tλ
+α|GR+f′′
+0R∗/bracketleftbigg/parenleftbigg
+−∂ξ∂R∗
+∂gρσ,λξ/parenrightbigg
+gρσ,α+∂R∗
+∂gρσ,λξgρσ,ξα/bracketrightbigg
+−f′′
+0R∗
+,ξ∂R∗
+∂gρσ,λξgρσ,α−1
+2f′′
+0δλ
+αR∗2=
+=f′
+0tλ
+α|GR+f′′
+0/bracketleftbigg
+R∗/parenleftbigg∂R∗
+∂gρσ,λξgρσ,ξα−1
+2R∗δλ
+α/parenrightbigg
+−∂ξ/parenleftbigg
+R∗∂R∗
+∂gρσ,λξ/parenrightbigg
+gρσ,α/bracketrightbigg
+. (36)
+Considering the perturbed metric (5), we have R∗∼R(1), whereR(1)is defined as in (8). In terms of handη, we get
+
+
+∂R∗
+∂gρσ,λξ∼∂R(1)
+∂hρσ,λξ=ηρλησξ−ηλξηρσ
+∂R∗
+∂gρσ,λξgρσ,ξα∼hλξ
+,ξα−h,λ
+α. (37)
+Clearly, the first significant term in Eq. (36) is of second order in the perturbation expansion. We can now write the
+expression of the energy-momentum tensor explicitly in term of the perturbation h; it is
+tλ
+α∼f′
+0tλ
+α|GR+f′′
+0{(hρσ
+,ρσ−/squareh)[hλξ
+,ξα−h,λ
+α−1
+2δλ
+α(hρσ
+,ρσ−/squareh)]
+−hρσ
+,ρσξhλξ
+,α+hρσ λ
+,ρσh,α+hλξ
+,α/squareh,ξ−/squareh,λh,α}. (38)
+Considering the tilded perturbation metric ˜hµν, the more compact form
+tλ
+α|f=1
+2/bracketleftbigg1
+2˜h,λ
+α/square˜h−1
+2˜h,α/square˜h,λ−˜hλ
+σ,α/square˜h,σ−1
+4(/square˜h)2δλ
+α/bracketrightbigg
+, (39)
+is achieved. As matter of facts, the energy-momentum tensor of the gravitational field, which expresses the energy
+transport during the propagation, has a natural generalization in the case of f(R)-gravity. We have adopted here the
+Landau-Lifshitz definition but other approaches can be taken into account [21]. The general definition of tλ
+α, obtained
+above, consists of a sum of a GR contribution plus a term coming from f(R)-gravity:
+tλ
+α=f′
+0tλ
+α|GR+f′′
+0tλ
+α|f. (40)
+However, as soon as f(R) =R, we obtains tλ
+α=tλ
+α|GR. As a final remark, it is worth noticing that massive modes of
+gravitational field come out from tλ
+α|fsince/square˜hcan be considered an effective scalar field moving in a potential: tλ
+α,
+in this case, represents a transport tensor.8
+V. CONCLUDING REMARKS
+In this paper, we have formally studied the post-Minkowskian limit of f(R)-gravity developing all the calculations
+in the Jordan frame. The main result is that, beside standard massle ss modes of GR, further massive modes emerge
+and they are directly determined by the analytic parameters of f(R)-gravity, that is the coefficients f′
+0andf′′
+0of the
+Taylor expansion. This fact is extremely relevant since it does not de pend on the considered f(R)-model but it is
+a general feature that can be enounciated in the following way: Massless gravitons are a peculiar characteristic of
+GR while extended or alternative theories have, in general, further massive or ghost states [13]. It is worth noticing
+that several indications in this sense are present in literature [12, 2 2, 23] but their relevance, from an experimental
+viewpoint, has never been stressed enough.
+On the other hand, a similar result comes out also in the Newtonian limit o f the same theories: Yukawa-like
+correctionsto the gravitationalpotential emerge in general and they are absent only in the case of GR. It is interesting
+to note that also the characteristic lengths of such corrections a re related to f′
+0andf′′
+0as shown in [17]. Also in this
+case, the Newtonian potential, coming from the weak field limit of GR, is only a particular case.
+These results pose interesting problems related to the validity of GR at all scales. It seems that it works very well at
+local scales(Solar System) where effects of further gravitationa ldegreesoffreedom cannot be detected. As soon as one
+is investigating larger scales, as those of galaxies, clusters of galax ies, etc., further corrections have to be introduced
+in order to explain both astrophysical large-scale dynamics [7, 9] an d cosmic evolution [3, 10]. Alternatively, huge
+amounts of dark matter and dark energy have to be invoked to exp lain the phenomenology, but, up today there are
+no final answer for these new constituents at fundamental level. Furthermore, the fact that, up to now, only massless
+gravitationalwaves have been investigated could be a shortcoming preventing the possibility to find out other forms of
+gravitational radiation. Tests in this sense could come, for example , from the stochastic background of gravitational
+waves where massive modes could play a crucial role in the cosmic back ground spectrum [24, 25].
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+arXiv:1001.0848v2  [hep-ph]  15 Feb 2011Two-Photon, Two-gluon and Radiative Decays of Heavy
+Flavoured Mesons
+Arpit Parmar∗1, Bhavin Patel†2and P C Vinodkumar∗3
+∗Department of Physics, Sardar Patel University, Vallabh Vi dyanagar-388120, Gujarat,
+INDIA.
+†LDRP Institute of Technology and Research, Gandhinagar-38 2015, Gujarat, INDIA.
+Abstract
+Here we present the two-photon and two-gluon decay widths of th e S-wave
+(ηQ∈c,b) and P-wave ( χQ∈c,bJ) charmonium and bottonium states and the ra-
+diative transition decay widths of c¯c,b¯bandc¯bsystems based on Coulomb
+plus power form of the inter-quark potential ( CPPν) with exponent ν. The
+Schr¨odinger equation is solved numerically for different choices of the ex-
+ponentν. We employ the masses of different states and their radial wave
+functions obtained from the study to compute the two-photon an d two-gluon
+decay widths and the E1 and M1 radiative transitions. It is found tha t the
+quarkonia mass spectra and the E1 transition can be described by t he same
+interquark model potential of the CPPνwithν= 1.0 forc¯candν= 0.7
+forb¯bsystems, while the M1 transition (at which the spin of the system
+changes) and the decay rates in the annihilation channel of quarko nia are
+better estimated by a shallow potential with ν <1.0.
+Keywords: Quarkonia, Annihilation, Radiative Transition
+PACS numbers: 12.39.Pn, 12.39.Jh, 14.40.Pq, 13.20.Gd
+1. Introduction
+The renaissance in the hadron spectroscopy particularly in the hea vy
+flavour sector associated with the large number of charm and beau ty states
+1arpitspu@yahoo.co.in
+2azadpatel2003@yahoo.co.in
+3pothodivinod@yahoo.com
+Preprint submitted to Nuclear Physics A October 25, 2018observed in recent experiments [1, 2, 3, 4] has created renewed in terest in
+the study of hadron spectroscopy. Particularly the discovery of ηb(1S) state,
+ηc(2S) state and other orbitally excited states in the charmonia and bott onia
+systems [5] has stimulated the theoretical phenomenologists to ta ke a new
+look and refine their parameters that describe the nature of quar k-antiquark
+interaction at the hadronic scale. The goodness of the spectrosc opic parame-
+ters like the interquark potential and its parameters that describ e the masses
+of the bound states and the corresponding wave functions obtain ed from the
+phenomenology, in the descriptions of other properties like the dec ay (in the
+annihilation channel) and transition properties now become the pros pects
+for a detail investigation. With the recent CLEO measurements [4, 5] of
+the two-photon decay rates of the even-parity, P-wave 0++(χQ0) and 2++
+(χQ2) states and with renewed interest in radiative decays of heavy qua rko-
+nium states, it seems appropriate to have another look at the two- photon
+decay of heavy quarkonia from the view point of quark-antiquark in teraction
+[6, 7]. Though the physics of quarkonium decay seems to be better u nder-
+stood within the conventional framework of QCD [8], unlike the two- photon
+width of S-wave ( ηQ) which can be predicted from the corresponding J/ψ
+and Υ leptonic widths, the similar prediction for the P-wave ( χQ) states is
+not conclusive. Similarly the radiative transitions in heavy quarkonia ( c¯c,
+b¯bandc¯b) states have drawn much theoretical interest [9, 10], as they ca n
+provide direct information on the nature of Q¯Qinteraction.
+Most of the existing theoretical values for the decay rates areba sed onpoten-
+tial model calculations that employ different types of interquark po tentials
+[11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. We consider the conven-
+tional nonrelativistic formalism for computations of the decay prop erties of
+the heavy flavour systems. In the traditional non-relativistic bou nd state
+calculation, the two-photon and two-gluon widths of quarkonium st ates de-
+pend on the ℓthderivatives of the radial wave function at the origin [11].
+As considered by many authors [24, 25, 26, 27, 28], the contribut ion from
+the radiative corrections to these decays are also incorporated in the present
+study.
+The paper is organized as follows. In Section 2, we describe the phen omeno-
+logical quark-antiquark interaction potential and extract the pa rameters that
+describe the ground state masses of c¯c,b¯bandc¯bsystems. We also compute
+the low lying orbital excited states of these systems. In section 3 w e employ
+the spectroscopic parameters of the c¯candb¯bsystems to study the two pho-
+ton and two-gluon decay widths. The radiative transitions (E1 and M 1) of
+2thec¯c,b¯bandc¯bsystems are described in section 4. In section 5 we present,
+analyze and discuss our results to draw important conclusions.
+2. The phenomenology and extraction of the spectroscopic pa ram-
+eters
+For the description of the quarkonium bound states, we adopt the phe-
+nomenological Coulomb plus power potential ( CPPν) expressed as [24, 25]
+V(r) =−4
+3αs
+r+Arν(1)
+Here,Ais the confinement strength of the potential and αsis the running
+strong coupling constant which is computed as,
+αs(µ2) =4π
+(11−2
+3nf) ln(µ2/Λ2)(2)
+Where,nfis the number of flavors, µis the renormalization scale related to
+the constituent quark mass and Λ is the QCD scale which is taken as 0.1 50
+GeV by fixing αs= 0.118 at theZ−boson mass (91 Ge V) [5].
+The potential parameter, Aof Eqn.1 is similar to the string strength σof
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56 /s50/s46/s48 /s50/s46/s50/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48/s48/s46/s51/s53/s48/s46/s52/s48/s48/s46/s52/s53/s48/s46/s53/s48/s48/s46/s53/s53/s48/s46/s54/s48/s48/s46/s54/s53/s48/s46/s55/s48/s48/s46/s55/s53/s48/s46/s56/s48/s48/s46/s56/s53/s65/s32/s105/s110/s32/s71/s101/s118/s43/s49
+/s80/s111/s116/s101/s110/s116/s105/s97/s108/s32/s83/s116/s114/s101/s110/s103/s116/s104/s32/s32/s99 /s99
+/s32/s98 /s99
+/s32/s98 /s98
+Figure 1: Potential strength A(inGeVν+1) obtained from the ground state spin-average
+mass against the choices of potential index, ν(0.1≤ν≤2.0)
+the Cornell potential. We particularly chose to vary νin our study as very
+3different interquark potentialscan provide fairlygooddescription o fthe mass
+spectra, while the transitions and other decay properties are ver y sensitive
+to the choice of interquark potential. Thus the present study on t he decay
+properties of heavy flavour mesons based on the CPPνmodel by varying
+the exponent ν(0.1≤ν≤2.0) can provide significant understanding of the
+quark-antiquark interaction in the mesonic states while they under go a tran-
+sition or decay through annihilation channels. The different choices o fνhere
+then correspond to different potential forms. So, the potential parameterA
+expressed in Ge Vν+1can be different for each choices of ν. The model poten-
+tial parameter Aand the mass parameter of the quark/antiquark ( m1,m2)
+are fixed using the known ground state center of weight (spin aver age) mass
+and the hyperfine splitting ( M3S1−M1S0) of the ground state c¯candb¯bsys-
+tems respectively. The spin average mass for the ground state is c omputed
+for the different choices of νin the range, 0 .1≤ν≤2.0. The spin average
+or the center of weight mass, MCWis calculated from the known experimen-
+tal/theoretical values ofthe pseudoscalar ( J= 0)andvector ( J= 1)mesonic
+mass as
+Mn,CW=/summationtext
+J(2J+1)MnJ
+/summationtext
+J(2J+1)(3)
+The Schr¨odinger equation is numerically solved using the mathematica note-
+book [ver. 3.0] of the Runge-Kutta method [29]. For computing the m ass
+difference between different spin degenerate mesonic states, we c onsider the
+spin dependent part of the usual one gluonexchange potential (O GEP) given
+by [30, 31, 32, 33, 34]. Accordingly, the spin-dependent part, VSD(r) contains
+three types of interaction terms, such as the spin-spin, the spin- orbit and the
+tensor part as
+VSD(r) =VSS(r)/bracketleftbigg
+S(S+1)−3
+2/bracketrightbigg
++VLS(r)/parenleftBig
+/vectorL·/vectorS/parenrightBig
++
+VT(r)/bracketleftBigg
+S(S+1)−3(/vectorS·/vector r)(/vectorS·/vector r)
+r2/bracketrightBigg
+(4)
+The spin-orbit term containing VLS(r) and the tensor term containing VT(r)
+describe the fine structure of the meson states, while the spin-sp in term
+containing VSS(r) proportional to 2( /vector sq·/vector s¯q) =S(S+ 1)−3
+2gives the spin
+singlet-triplet hyperfine splitting. The coefficient of these spin-dep endent
+4terms of Eqn.4 can be written in terms of the vector ( VV) and scalar ( VS)
+parts of the static potential, V(r) described in Eqn. 1 as [32]
+VLS(r) =1
+2m1m2r/parenleftbigg
+3dVV
+dr−dVS
+dr/parenrightbigg
+(5)
+VT(r) =1
+6m1m2/parenleftbigg
+3d2VV
+dr2−1
+rdVV
+dr/parenrightbigg
+(6)
+VSS(r) =1
+3m1m2∇2VV=16παs
+9m1m2δ(3)(/vector r) (7)
+HereVVis the coulumb part (1stterm) and VSis the confining part (2nd
+term) of Eqn. 1. The computed masses of the Q¯Q(2s+1LJ) states are listed
+in Table 1, 2 and 3 for different combinations of Q∈b,c. The spectroscopic
+parametersthuscorrespondtothefittedquarkmasses, thepo tential strength
+A, the potential exponent, νand the corresponding radial wave functions.
+The fitted mass parameters are mc= 1.28GeV/c2,mb= 4.4GeV/c2while
+the potential strength Afor each choices of νare shown in Fig. 1. The ℓth
+derivative of the corresponding radial wave functions at the origin obtained
+for each choices of the exponent νare plotted in Fig. 2.
+/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s124/s82/s40/s108/s41
+/s40/s48/s41/s124/s50/s32/s49/s83
+/s32/s49/s80
+/s99 /s99
+/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s50/s52/s54/s56/s49/s48/s124/s82/s40/s108/s41
+/s40/s48/s41/s124/s50/s32/s49/s83
+/s32/s49/s80
+/s98 /s98
+Figure 2: Square of ℓthderivative of radial wave function at origin in GeV2ℓ+3
+3. Two-photon and two-gluon decay widths for quarkonium sta tes
+The extracted model parameters and the radial wavefunctions a re be-
+ing employed here to compute the two-photon (Γ γγ(Q¯Q)) and two gluon
+5Table 1: Masses (in GeV) of charmonium states ( n2S+1LJ) in the CPPνmode with
+different choices of ν
+ν Expt.
+State 0.1 0.5 0.7 0.9 1.0 1.1 1.5 2.0 [5]
+13S13.076 3.088 3.093 3.097 3.099 3.100 3.106 3.111 3.097
+11S03.045 3.008 2.994 2.981 2.976 2.971 2.955 2.940 2.980
+13P23.174 3.357 3.430 3.495 3.524 3.552 3.647 3.737 3.556
+13P13.171 3.347 3.419 3.484 3.514 3.542 3.640 3.736 3.511
+13P03.167 3.324 3.386 3.441 3.466 3.489 3.570 3.647 3.415
+11P13.172 3.350 3.422 3.485 3.514 3.542 3.636 3.727 3.526
+13D33.215 3.520 3.653 3.774 3.830 3.830 4.068 4.249
+13D23.214 3.523 3.662 3.792 3.854 3.914 4.129 4.354
+13D13.214 3.521 3.663 3.796 3.860 3.923 4.152 4.399 3.770
+11D23.214 3.521 3.658 3.785 3.844 3.901 4.105 4.314
+23P23.121 3.461 3.633 3.803 3.870 3.970 4.292 4.661 3.929
+23P13.120 3.455 3.624 3.792 3.875 3.958 4.277 4.647
+23P03.118 3.439 3.599 3.757 3.835 3.912 4.206 4.541
+21P13.120 3.457 3.626 3.740 3.877 3.960 4.277 4.643
+23S13.087 3.326 3.444 3.558 3.615 3.670 3.881 4.119 3.686
+21S03.079 3.290 3.390 3.487 3.533 3.579 3.749 3.936 3.638
+6Table 2: Masses (in GeV) ofBcstates (n2S+1LJ) in theCPPνmode with different choices
+ofν
+ν Others
+State 0.1 0.5 0.7 1.0 1.5 2.0 [41] [42] [43] [26] [44]
+13S16.326 6.335 6.339 6.343 6.348 6.351 6.416 6.337 6.317 6.332 6.338
+11S06.305 6.279 6.268 6.256 6.241 6.231 6.380 6.264 6.253 6.270 6.271
+13P26.453 6.679 6.771 6.889 7.043 7.156 6.837 6.747 6.743 6.762 6.768
+13P16.451 6.673 6.764 6.882 7.038 7.155 6.772 6.730 6.717 6.734 6.741
+13P06.448 6.656 6.740 6.848 6.989 7.092 6.693 6.700 6.683 6.699 6.706
+11P16.452 6.675 6.765 6.882 7.035 7.148 6.775 6.736 6.729 6.749 6.750
+13D36.505 6.889 7.059 7.286 7.599 7.840 7.003 7.005 7.007 7.081 7.045
+13D26.504 6.891 7.065 7.304 7.643 7.915 7.000 7.012 7.001 7.077 7.041
+13D16.504 6.890 7.066 7.308 7.659 7.947 6.959 7.012 7.008 7.720 7.028
+11D26.505 6.890 7.062 7.297 7.626 7.886 7.001 7.009 7.016 7.079 7.036
+23P26.387 6.812 7.027 7.345 7.852 8.313 7.186 7.153 7.134 7.156 7.164
+23P16.386 6.808 7.021 7.337 7.843 8.303 7.136 7.135 7.113 7.126 7.145
+23P06.384 6.797 7.003 7.309 7.792 8.229 7.081 7.108 7.088 7.091 7.122
+21P16.386 6.809 7.022 7.339 7.842 8.301 7.139 7.142 7.124 7.145 7.150
+23S16.343 6.640 6.785 6.997 7.326 7.617 6.896 6.899 6.902 6.881 6.887
+21S06.338 6.615 6.748 6.939 7.232 7.487 6.875 6.856 6.867 6.835 6.855
+7Table 3: Masses (in GeV) of bottonia states ( n2S+1LJ) in theCPPνmode with different
+choices of ν
+ν Expt.
+State 0.1 0.5 0.7 0.9 1.0 1.5 2.0 [5]
+13S19.447 9.455 9.458 9.460 9.461 9.466 9.468 9.460
+11S09.427 9.404 9.395 9.388 9.385 9.372 9.363 9.389
+13P29.595 9.838 9.938 10.025 10.065 10.231 10.356 9.912
+13P19.593 9.831 9.929 10.016 10.056 10.223 10.348 9.893
+13P09.590 9.817 9.909 9.989 10.026 10.179 10.292 9.859
+11P19.594 9.834 9.932 10.018 10.057 10.223 10.346
+13D39.653 10.066 10.250 10.419 10.498 10.842 11.111
+13D29.653 10.067 10.254 10.427 10.509 10.871 11.163 10.162
+13D19.652 10.066 10.253 10.428 10.511 10.881 11.182
+11D29.653 10.060 10.252 10.423 10.504 10.859 11.143
+23P29.530 9.984 10.214 10.443 10.556 11.100 11.599 10.269
+23P19.529 9.970 10.208 10.434 10.547 11.088 11.583 10.255
+23P09.528 9.970 10.130 10.413 10.522 11.044 11.518 10.232
+21P19.530 9.981 10.210 10.437 10.549 11.090 11.585
+23S19.481 9.796 9.950 10.101 10.175 10.524 10.835 10.023
+21S09.476 9.773 9.918 10.057 10.125 10.444 10.724
+8(Γgg(Q¯Q))decay widths. Two photonwidthsoforbitallyexcited quarkonium
+statesχQJ=0,2→γγare suppressed by the mass of the heavy quark while
+the two-photon decays of the spin one state χQ,1is forbidden by the Landau-
+Yang theorem [35, 36]. Most of the quark model predictions [5, 21, 3 7] for
+theηQ→γγwidth are comparable with the experimental value. However
+the theoretical predictions for the P-wave (χQ0,2→γγ) widths differ largely
+from the experimental observations [5]. This warranted to incorpo rate con-
+tribution from QCD corrections. Thus with the one-loop QCD radiativ e
+corrections the decay widths of1S0(ηQ),3P0(χQ0) and3P2(χQ2) states into
+two photons are computed according to the non relativistic expres sion given
+by [11, 14, 24, 25, 38, 39, 40]
+Γγγ(ηQ) =3Q4
+eα2
+emMηQ|R0(0)|2
+2m3
+Q/bracketleftbigg
+1−αs
+π(20−π2)
+3/bracketrightbigg
+(8)
+Γγγ(χQ0) =27Q4
+eα2
+emMχQ0|R′
+1(0)|2
+2m5
+Q/bracketleftBig
+1+B0αs
+π/bracketrightBig
+(9)
+Γγγ(χQ2) =4
+1527Q4
+eα2
+emMχQ2|R′
+1(0)|2
+2m5
+Q/bracketleftBig
+1+B2αs
+π/bracketrightBig
+(10)
+Here,B0=π2/3−28/9 andB2=−16/3 are the next to leading order (NLO)
+QCD radiative corrections [40, 45, 46] and are considered to be the same for
+charmonium and bottonium systems.
+Similarly, the two-gluon decay width of ηQ,χQ0andχQ2states are given by
+[47],
+Γgg(ηQ) =α2
+sMηc|R0(0)|2
+3m3
+Q[1+CQ(αs/π)] (11)
+Γgg(χQ0) =3α2
+sMχQ0|R′
+1(0)|2
+m5
+Q[1+C0Q(αs/π)] (12)
+Γgg(χQ2) =/parenleftbigg4
+15/parenrightbigg3α2
+sMχQ2|R′
+1(0)|2
+m5
+Q[1+C2Q(αs/π)] (13)
+Here, the quantities in the square brackets are the NLO QCD radiat ive cor-
+rections [40, 45, 46] and the coefficients, CQ,C0QandC2Qfor the charmonia
+(Q=c) and bottonia ( Q=b) are listed in Table 4.
+The value of the radial wave functions |R(ℓ)
+Q¯Q(0)|2of Eqns. 8-10 are obtained
+9tbp
+Table 4: NLO QCD Correction coefficients for the two-gluon decays
+Q=c Q=b
+CQ4.8 4.4
+C0Q8.77 10.0
+C2Q−4.827−0.1
+from the numerical solution of the schrodinger equation correspo nding to the
+CPPνpotential model description of the Q¯Qbound states described in sec-
+tion 2. The computed digamma widths of the c¯candb¯bstates are listed in
+Table 5 and 6 while the digluon widths of c¯candb¯bstates are listed against
+the exponent, νin Table 7 and 8 respectively.
+4. Radiative transitions
+As the mass spectra of the quarkonia states are well known, the in vestiga-
+tion of radiative transitions become important and such study could help us
+to understand the theory of strong interaction in the nonpertur bative regime
+of QCD. The motivation for the present work is to study the radiativ e transi-
+tions (E1 and M1 transitions) of the Q¯Q(Q∈b,c) states and to investigate
+the interquark potentials that provides the right description of th e quarkonia
+states. The nonrelativistic treatment adopted for the study of t hese heavy
+flavour mesonic systems allows as to apply the usual multipole expans ion in
+electrodynamics tocompute thetransitionbetween the quarkonia states with
+the emission of a photon. The leading terms in this expansion corresp ond
+to the E1 and M1 transitions. The electric dipole term (E1) is respons ible
+for the transition between the SandPstates without changing the spin of
+the quark-antiquark pair, while the magnetic dipole term (M1) descr ibes the
+transition between S= 1 andS= 0 states without changing relative orbital
+momentum ( L) of the quark-antiquark pair. Accordingly, electric transitions
+do not change quark spin. These transitions have ∆ L=±1 and ∆S= 0.
+The E1 radiative transition width between initial state, ( n2s+1LJ) to final
+state, (n′2s+1L′′
+J) is given by [48],
+Γ(i→f+γ) =4α/angb∇acketlefteQ/angb∇acket∇ight2
+3(2J′+1)SE
+ifω3|εif|2(14)
+10Table 5: Two-photon decay widths (in keV) ofc¯cstates with different choices of ν
+νΓγγ(ηc) Γ γγ(χc0) Γ γγ(χc2)
+0.5 6.375 2.539 0.309
+0.7 7.875 4.839 0.529
+1.0 9.684 9.357 1.162
+[4] - 2 .36±0.35±0.22 0.66±0.07±0.06
+[5] 6.41+3.657
+−3.123 2.397±0.4 0.493±0.06
+[14] - 3.72 0.49
+[15] - 2.50 0.28
+[16] - 1.39 0.44
+[17] - 3.34 0.43
+[18] - 3.78 -
+[19] - 1.99 0.30
+[20] - 1.29 0.46
+[21] - 2.90 0.50
+[22] - 1.56 0.56
+[26] 4.33 - -
+[45] - 3.50 0.93
+[49] 4.252 - -
+11Table 6: Two-photon decay widths (in keV) ofb¯bstates with different choices of ν
+ν Γγγ(ηb) Γγγ(χb0) Γγγ(χb2)
+0.5 0.410 0.064 0.011
+0.7 0.504 0.097 0.016
+1.0 0.618 0.185 0.030
+[13] - 0.080 0.008
+[15] 0.460 - -
+[16] 0 .220±0.040 - -
+[19]pert. 0.30 0.032 0.007
+[19]nonpert. 0.32 0.094 0.005
+[20] 0.214 - -
+[27] 0.350 - -
+[28] 0 .384±0.047 - -
+[31] 0.230 - -
+[49] 0.313 - -
+[50] 0 .466±0.101 - -
+[51] 0.170 - -
+[52] 0.520 - -
+[53] 0.560 - -
+12Table 7: Two-gluon decay widths (in MeV) ofc¯cstates with different choices of ν
+ν Γgg(ηc) Γ gg(χc0) Γ gg(χc2)
+0.5 32.209 10.467 1.169
+0.7 39.790 19.949 2.003
+1.0 48.927 38.574 4.396
+[5] 26 .7±3.0 10.2±0.7 2.03±0.12
+[19]pert. 15.70 4.68 1.72
+[19]nonpert. 10.57 4.88 0.69
+Table 8: Two-gluon decay widths (in MeV) ofb¯bstates with different choices of ν
+ν Γgg(ηb) Γgg(χb0) Γgg(χb2)
+0.5 11.921 1.826 0.285
+0.7 14.644 2.745 0.429
+1.0 17.945 5.250 0.822
+[13] 12.46 2.15 0.22
+[19]pert. 11.49 0.96 0.33
+[19]nonpert. 12.39 2.74 0.25
+Here,α= 1/137,ωis the photon energy and /angb∇acketlefteQ/angb∇acket∇ight, the mean charge content
+of theQ¯Qsystem are expressed as
+ω=M2
+i−M2
+f
+2Mi(15)
+and
+/angb∇acketlefteQ/angb∇acket∇ight=/vextendsingle/vextendsingle/vextendsingle/vextendsinglem¯QeQ−mQe¯Q
+mQ+m¯Q/vextendsingle/vextendsingle/vextendsingle/vextendsingle(16)
+respectively. Here, MiandMfare the initial and final state mass of the
+quarkonia respectively. The statistical factor SE
+if=SE
+fiis given by,
+SE
+if=max(ℓ,ℓ′)/braceleftbiggJ1J′
+ℓ′s ℓ/bracerightbigg2
+(17)
+The overlap integral εifis given by
+εif=3
+ω/integraldisplay∞
+0drunℓ(r)un′ℓ′(r)/bracketleftBigωr
+2j0/parenleftBigωr
+2/parenrightBig
+−j1/parenleftBigωr
+2/parenrightBig/bracketrightBig
+(18)
+13Table 9: E1 transition widths (in keV) ofc¯cstates against potential index ν
+Initial Final ν Others
+State State 0.5 0.7 1.0 [5] [55]
+13P213S1163.98 250.22 383.48 406.00 315
+13P113S1148.31 229.62 361.21 320.40 41
+13P013S1115.53 173.42 263.62 130.56 120
+11P111S0302.44 451.50 671.05 482
+13D313P2137.86 245.87 432.36 402
+13D213P236.23 68.29 131.01 69.5
+13D213P1127.47 231.43 423.43 31.3
+13D113P23.89 7.68 15.23 3.88
+13D113P168.65 129.97 245.70 99
+13D113P0127.71 240.03 447.72 299
+11D211P1157.16 285.76 523.89 389
+23P213S170.34 140.10 252.68
+23P223S136.97 83.93 164.41
+23P113S166.96 132.92 258.15
+23P123S132.65 73.81 172.63
+23P013S158.48 114.28 216.53
+23P023S122.63 49.36 112.39
+23P111S0122.85 226.91 415.25
+21P121S065.04 146.99 332.99
+The E1 transition rates of charmonia, bottonia and Bcsystems are listed in
+Table 9, 10 and 11.
+The magnetic dipole transitions (M1) flip the quark spin, so their ampli-
+tudes are proportional to the quark magnetic moments and thus r elated in-
+versely tothe quarkmass. Thus, M1 transitions areweaker thant heE1 tran-
+sitions and this causes difficulties in the experimental observations. Along
+with other exclusive processes, the magnetic dipole (M1) transition s fromthe
+spin-triplet S-wave vector (V) state to the spin-singlet S-wave ps eudoscalar
+(P) state have also been considered as a valuable testing ground to further
+constrain the phenomenological quark model of hadrons. The M1 t ransi-
+tions have ∆ L= 0 and the n2s+1LJ→n′2s′+1LJ′+γtransition rate for Q¯Q
+14Table 10: E1 transition widths (in keV) ofb¯bstates against potential index ν
+Initial Final ν Others
+State State 0.5 0.7 1.0 [5] [55]
+13P213S129.06 44.28 70.29 11.88 31.6
+13P113S127.58 42.00 67.43 18.91 27.8
+13P013S124.75 37.18 58.42 3.24 22.2
+11P111S040.29 60.45 94.05
+13D313P217.28 41.00 76.82 22.6
+13D213P24.37 10.63 20.60
+13D213P114.32 34.53 65.35
+13D113P20.48 1.17 2.32 0.50
+13D113P17.86 19.02 36.75 10.7
+13D113P012.38 30.04 58.47 20.1
+11D211P116.84 44.06 83.96
+23P213S111.05 20.42 35.84 12.7
+23P223S15.54 13.34 32.92 14.5
+23P113S110.19 19.92 34.86 12.0
+23P123S14.43 12.50 30.82 12.4
+23P013S110.19 14.14 32.24 10.9
+23P023S14.43 4.44 25.40 9.17
+21P111S014.39 25.84 43.99
+15Table 11: E1 transition widths (in keV) ofBcstates against potential index ν
+Initial Final ν Others
+State State 0.5 0.7 1.0 [44] [26] [41] [42]
+13P213S182.04 128.18 200.07 83 107 109.8 112.6
+13P113S178.22 122.73 193.48 11 78.9 67.8 99.5
+13P013S167.93 105.03 163.13 60 67.2 32.1 79.2
+11P111S0119.47 184.66 282.73 81.8 56.4
+13D313P268.56 125.28 225.588 18.7 98.7
+13D213P217.6 33.13 63.37 4.4 24.7
+13D213P157.06 105.92 198.57 35.8 88.8
+13D113P21.93 3.71 7.22 0.2 2.7
+13D113P131.3 59.38 113.06 11.4 49.3
+13D113P051.4 97.17 183.53 43.9 88.6
+11D211P173.20 136.21 325.46 120.8 92.5
+23P213S134.78 68.56 134.25 9.1 25.8
+23P223S117.88 41.63 97.24 91.3 73.8
+23P113S133.89 66.66 130.62 6.3 22.1
+23P123S116.74 38.91 91.62 53.5 54.3
+23P013S131.53 61.18 118.43 5.9 21.9
+23P023S113.86 31.36 73.25 24.3 41.2
+21P111S048.08 91.81 174.50
+21P121S024.87 57.80 137.46
+16mesonic system is given by [54, 55, 56]
+Γ(i→f+γ) =α
+3µ2ω3SM
+if(2J′+1)M2(19)
+Where
+µ=/vextendsingle/vextendsingle/vextendsingle/vextendsingleeQ
+mQ−e¯Q
+m¯Q/vextendsingle/vextendsingle/vextendsingle/vextendsingle(20)
+and,mQ/¯Qis the mass of the heavy quark/antiquark. The statistical factor
+SM
+if=SM
+fiis
+SM
+if= 6(2s+1)(2s′+1)/braceleftbiggJ1J′
+s′ℓ s/bracerightbigg2/braceleftbigg11
+21
+21
+2s′s/bracerightbigg2
+(21)
+The matrix element Mcontains the integral related to the static term ( I1)
+and the term related to the contribution from the two quark confin ing ex-
+change current ( Ic) in the non-relativistic limit as
+M=I1+Ic (22)
+Where
+I1=/angb∇acketleftf|j0(ωr/2)|i/angb∇acket∇ight (23)
+and
+Ic=−1
+mQ/angb∇acketleftf|Arν|i/angb∇acket∇ight (24)
+Here,Arνis the confining part of the quark-antiquark potential described b y
+CPPνmodel.
+The magnetic dipole transition rate (allowed M1 transitions) corresp onds to
+triplet-singlet transitions between S- wave states, Γ( n3S1→n1S0+γ) of the
+c¯c,b¯bandc¯bsystems are listed in Table 12, 13 and 14 respectively. Other
+allowed M1 transitions n3S1→n′1S0+γwithn>n′are nonrelativistically
+forbidden [58] and require relativistic treatment which is beyond the scope of
+the present study. The values within the brackets are the results computed
+by incorporating the two-quark exchange current contribution a s given by
+[56, 57]. Other theoretical model predictions and available experime ntal
+results are also listed for comparison.
+17Table 12: M1 transition widths (in keV) ofc¯cstates against potential index ν
+Initial Final ν Others
+State State 0.5 0.7 0.8 1.0 [55] [26] [57]
+13S111S01.29(0.47) 2.41(0.96) 3.12(1.28) 4.57(2.01) 2.0 1.05 1.05
+23S121S00.12(0.02) 0.40(0.07) 0.61(0.10) 1.39(0.20) 1.26 0.14
+33S131S00.04(0.004) 0.16(0.009) 0.28(0.011) 0.76(0.012) 0.0124
+Expt [5]: Γ(13S1→11S0γ) = 1.21±0.37
+Table 13: M1 transition widths (in eV) ofb¯bstates against potential index ν
+Initial Final ν Others
+State State 0.5 0.7 0.8 1.0 [55] [26] [57]
+13S111S07.28(5.26) 13.70(10.11) 16.47(12.22) 24.01(18.20) 8.95 9.7 18.9
+23S121S00.67(0.41) 1.80(1.10) 3.02(1.80) 6.87(4.01) 1.51 1.6 2.77
+33S131S00.19(0.10) 0.76(0.38) 1.34(0.64) 1.98(0.86) 0.83 0.9 1.32
+Table 14: M1 transition widths (in keV) ofBcstates against potential index ν
+Initial Final ν Others
+State State 0.5 0.7 0.8 1.0 [26] [59] [44] [57]
+13S111S00.15(0.02) 0.30(0.06) 0.37(0.08) 0.55(0.13) 0.07 0.01 0.08 0.05
+23S121S00.01(0.01) 0.04(0.06) 0.07(0.15) 0.17(0.48) 0.03 0.03 0.01 0.004
+33S131S00.005(0.01) 0.02(0.07) 0.03(0.18) 0.09(0.79) 0.08 eV
+185. Results and Discussions
+The computed masses of charmonium, Bcmeson andbottoniumlow lying
+states using the CPPνmodel are shown in Table 1, 2 and 3 respectively
+with different choices of exponent ν. Their respective experimental values
+(PDG average) are also listed for comparison. The best possible cho ice of
+the exponent ( ν) for the description of quarkonia ( b¯b&c¯c) spectra would be
+the one with minimum statistical deviations of the predicted mass spe ctra
+with the corresponding experimental values. For this we consider a bout
+ten experimentally known states ( n2S+1LJ) ofc¯candb¯bsystems as given in
+Tables1and3, andcomputetherootmeansquaredeviationsofthe predicted
+masses of these states for each choices of νas
+SDmass (ν) =/radicaltp/radicalvertex/radicalvertex/radicalbt1
+NN/summationdisplay
+i=1/bracketleftbiggMCPPν(n2S+1LJ)−Mexp(n2S+1LJ)
+Mexp(n2S+1LJ)/bracketrightbigg2
+i(25)
+The computed values of the SDmass( ν) forc¯candb¯bsystems are plotted
+againstνin Fig. 3. The figure shows distinct minima around ν= 1.0 in
+the case of c¯cand around ν= 0.7 in the case of b¯bsystems. Thus, we
+conclude that charmonia spectra are better described with the co rnell-like
+potential with ν= 1.0, while the bottonia spectra are better described by a
+relatively flat potential with ν= 0.7. In the absence of known experimental
+excited states for Bcsystem, we consider that better predictions for the Bc
+spectra must lie within the range of the exponent 0 .7≤ν≤1.0. And it is
+found that the predicted masses of the Bcstates for the choices of νin the
+range, 0.7≤ν≤0.9 areingoodagreement with other theoretical predictions
+[26, 41, 42, 43].
+The computed two-photon and two-gluon decay widths of ηQ,χQ0andχQ2
+states (Q∈c,b) using the spectroscopic parameters of CPPνmodel are com-
+pared with other theoretical predictions in Tables 5, 6, 7 and 8 resp ectively.
+The experimentally known two-photon and two-gluon widths of the c¯cstates
+arealsocompared inTable5 and7 respectively. Thetwo-gluondecay process
+accountsforsubstantial partoftheirhadronicdecays andhenc ewehavecom-
+paredourresults ofthe c¯csystemwiththeirmeasuredhadronicdecaywidths.
+Thoughtherearemanymodel predictions forthetwo-photondec ay widths of
+χc0andχc2states, only very few predictions for the ηc→γγstate exist. Our
+predictions for Γ γγ(c¯c) with the exponent lying in the range, 0 .4< ν <0.7
+and that for Γ gg(c¯c) in the range, 0 .3<ν <0.5 are in accordance with other
+19/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50/s48/s46/s48/s48/s52/s48/s46/s48/s48/s54/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50/s48/s46/s48/s49/s52/s48/s46/s48/s49/s54/s83/s68/s109/s97/s115/s115/s99 /s99
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56 /s50/s46/s48 /s50/s46/s50/s45/s48/s46/s48/s48/s49/s48/s46/s48/s48/s48/s48/s46/s48/s48/s49/s48/s46/s48/s48/s50/s48/s46/s48/s48/s51/s48/s46/s48/s48/s52/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s83/s68/s109/s97/s115/s115/s98 /s98
+Figure 3: rms deviation in mass for quarkonia states
+model predictions. Fig. 4 shows the trend lines for the two-photon and two-
+gluon decay widths of the charmonia states with exponent, ν. The horizontal
+lines with error bar represents the respective experimental value s obtained
+from the PDG data [5]. Similar to the rms deviations computed for the m ass
+predictions, the rms deviations of the predicted two-photon width s of the
+charmonia states with their respective experimental values show a minimum
+aroundν= 0.5 (see Fig. 6(a)) while that in the case of two-gluon widths
+occur around ν= 0.4 (See Fig. 6(b)). The shaded regions in Fig. 4 with
+left aligned lines show the neighborhood region of the exponent νat which
+SDmass is minimum for c¯cstates. And the right aligned shaded region show
+the region of ν(0.4<ν <0.7) around which the minimum rms deviations of
+the two-photon widths ( SD−γγ) occur. The present analysis thus clearly
+indicates that the mass predictions and the annihilation widths studie d here
+cannot be explained by any single potential choice.
+In the case of b¯bsystems, Fig. 5 shows the trend lines for the two-photon
+and two-gluon decay widths with ν. These decay widths of ηb,χb0andχb2
+states have not been measured. However many other theoretica l models pre-
+dicted the two-photon decay of ηbstate, while very few model predictions
+exist forχb0andχb2states. Our two-photon and two-gluon widths of the
+b¯bsystems within the exponent, 0 .3<ν≤0.7 are in accordance with other
+theoretical predictions. New experimental results for these dec ay widths of
+b¯bsystem are awaited. Present widths in the range of exponent 0 .3<ν <0.7
+for two-photon and two-gluon decays are marked in Fig. 5(a) as rig ht aligned
+shaded region.
+Further, the present study clearly indicates that the Q¯Qsystem in its anni-
+20/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52/s48/s49/s50/s51/s52/s53/s54/s55/s56/s57/s49/s48/s32
+/s99
+/s32
+/s99/s40/s69/s120/s112/s116/s46/s41
+/s32
+/s99/s48
+/s32
+/s99/s48/s40/s69/s120/s112/s116/s41
+/s32
+/s99/s50
+/s32
+/s99/s48/s40/s69/s120/s112/s116/s46/s41/s105/s110/s32/s107/s101/s86
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48/s103/s103/s32/s105/s110/s32/s77/s101/s86/s32
+/s99
+/s32
+/s99/s40/s69/s120/s112/s116/s46/s41
+/s32
+/s99/s48
+/s32
+/s99/s48/s40/s69/s120/s112/s116/s41
+/s32
+/s99/s50
+/s32
+/s99/s48/s40/s69/s120/s112/s116/s46/s41
+Figure 4: Two-photon and two-gluon decay width of charmonia stat es against potential
+indexν
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s32
+/s98
+/s32
+/s98/s48
+/s32
+/s98/s50/s105/s110/s32/s107/s101/s86
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s103/s103/s32/s105/s110/s32/s77/s101/s86/s32
+/s98
+/s32
+/s98/s48
+/s32
+/s98/s50
+Figure 5: Two-photon and two-gluon decay widths of bottonia stat es against potential
+indexν
+/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s48/s50/s52/s83/s68
+/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s48/s49/s48/s50/s48/s83/s68/s103/s103
+Figure 6: rms deviation in two-photon and two-gluon decay of charm onia
+21hilation channel (digamma/digluon decays) interact more weakly com pared
+to their bound state spectral excitations.
+The low lying electric dipole transition rates of the c¯c,b¯bandc¯bsystems
+are listed in Table 9, 10 and 11 respectively, with other theoretical m odel
+predictions and with available experimental results. Their trendlines drawn
+againstνare shown in Fig. 7. The horizontal lines that cut the E1-curves
+correspond to the respective experimental values. The shaded r egion corre-
+spond to the neighbourhood of νfor which the SDmass is minimum. The
+present results for the E1-transition rates agree with the exper imental data
+[5] in the neighbourhood region of ν= 1.0 in the case of charmonium systems
+and in the neighbourhoodregion of ν= 0.7 in the case of bottonium systems.
+These values of νare the same at which SDmass becomes minimum.
+The predicted M1 transition rates with the correction due to the co nfining
+exchange current contribution for Q¯Qstates are drawn in Fig. 8 with re-
+spect to the exponent ν. Present result in the case of J/ψ→ηcγis in
+agreement with the known experimental result of 1 .21±0.37keVat the
+exponentν≈0.8 with the confining exchange current contribution while it
+agrees with the prediction at the exponent ν≈0.5 without the exchange
+current contribution. Our results for c¯csystem in the range, 0 .7≤ν≤1.0 of
+the exponent, νare in accordance with the values predicted by other models.
+From the present study, the importance of the two -quark confin ing exchange
+current contribution in the M1 transition widths of c¯csystem becomes very
+clear. In the case of b¯bandc¯bstates, our predictions with and without the
+exchange current contribution in the range 0 .6<ν <1.0 of the exponent, ν
+are in accordance with other model predictions. However lack of ex perimen-
+tal data for these decay widths does not allow to draw conclusion in f avour
+of a particular choice of the exponent ν.
+To summarize, we find that the description of the quarkonia mass sp ectra
+and the E1 transition can be described by the same interquark mode l poten-
+tial of theCPPνwithν= 1.0 forc¯candν= 0.7 forb¯bsystems, while the
+M1 transition (at which the spin of the system changes) and the dec ay rates
+in the annihilation channel of quarkonia are better estimated by a sh allow
+potential with ν <1.0. We look forward to future experimental data related
+to the decay properties of c¯bandb¯bsystems.
+Acknowledgment
+Part of this work is done with a financial support from DST, Governm ent of
+India, under a Major Research Project SR/S2/HEP-20/2006 .
+22/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56 /s50/s46/s48/s48/s49/s51/s48/s50/s54/s48/s51/s57/s48/s53/s50/s48/s54/s53/s48/s55/s56/s48/s68/s101/s99/s97/s121/s32/s87/s105/s100/s116/s104/s32/s49/s51
+/s80
+/s50/s49/s51
+/s83
+/s49
+/s49/s51
+/s80
+/s49/s49/s51
+/s83
+/s49
+/s32/s49/s51
+/s80
+/s48/s49/s51
+/s83
+/s49
+/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s68/s101/s99/s97/s121/s32/s87/s105/s100/s116/s104/s32/s49/s51
+/s80
+/s50/s49/s51
+/s83
+/s49
+/s32/s49/s49
+/s68
+/s50/s49/s49
+/s80
+/s49
+/s32/s50/s51
+/s80
+/s48/s51
+/s83
+/s49
+/s32/s49/s49
+/s80
+/s49/s49/s49
+/s83
+/s48
+/s32/s49/s51
+/s68
+/s49/s49/s51
+/s80
+/s48/s99 /s98
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s68/s101/s99/s97/s121/s32/s87/s105/s100/s116/s104/s32/s49/s51
+/s80
+/s50/s49/s51
+/s83
+/s49
+/s32/s49/s51
+/s80
+/s49/s49/s51
+/s83
+/s49
+/s32/s49/s51
+/s68
+/s49/s49/s51
+/s80
+/s48
+/s32/s50/s51
+/s80
+/s50/s50/s51
+/s83
+/s49
+/s32/s50/s51
+/s80
+/s48/s50/s51
+/s83
+/s49
+Figure 7: E1 transition decay widths for quarkonium states against potential index ν
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s49/s50/s68/s101/s99/s97/s121/s32/s87/s105/s100/s116/s104/s32/s105/s110/s32/s107/s101/s86/s32/s49/s51
+/s83
+/s49/s49/s49
+/s83
+/s48
+/s32/s50/s51
+/s83
+/s49/s49
+/s83
+/s48
+/s32/s49/s51
+/s83
+/s49/s49/s49
+/s83
+/s48/s40/s69/s120/s112/s116/s41
+/s32/s51/s51
+/s83
+/s49/s49
+/s83
+/s48/s99 /s99
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56
+/s32/s49/s51
+/s83
+/s49/s49/s49
+/s83
+/s48
+/s32/s50/s51
+/s83
+/s49/s49
+/s83
+/s48
+/s51/s51
+/s83
+/s49/s49
+/s83
+/s48/s68/s101/s99/s97/s121/s32/s87/s105/s100/s116/s104/s105/s110/s32/s107/s101/s86/s99 /s98
+/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s68/s101/s99/s97/s121/s32/s87/s105/s100/s116/s104/s105/s110/s32/s101/s86/s32/s49/s51
+/s83
+/s49/s49/s49
+/s83
+/s48
+/s32/s50/s51
+/s83
+/s49/s49
+/s83
+/s48
+/s51/s51
+/s83
+/s49/s49
+/s83
+/s48/s98 /s98
+Figure 8: M1 transition decay widths for quarkonium states against potential index ν
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\ No newline at end of file
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+arXiv:1001.0849v2  [hep-ph]  31 May 2011Properties of Light Flavour Baryons in
+Hypercentral quark model
+Kaushal Thakkar∗, Bhavin Patel†, Ajay Majethiya‡and
+P.C.Vinodkumar∗
+∗Department of Physics, Sardar Patel University, Vallabh Vidyanag ar- 388 120,
+†LDRP, Gandhinagar, Gujarat,‡KITRC, Kalol, Gujarat, INDIA.
+E-mail:kaushal21185@yahoo.co.in
+Abstract.
+The light flavour baryons are studied within the quark model using th e hyper
+central description of the three-body system. The confinement potential is assumed as
+hypercentral coulomb plus power potential ( hCPP ν) with power index ν. The masses
+and magnetic moments of light flavourbaryonsare computed for diff erent power index,
+νstarting from 0.5 to 1.5. The predicted masses and magnetic moment s are found to
+attain a saturated value with respect to variation in νbeyond the power index ν >
+1.0. Further we computed transition magnetic moments and radiativ e decay width of
+light flavour baryons. The results are in good agreement with known experimental as
+well as other theoretical models.
+PACS numbers: 12.39.Jh, 12.39.Pn, 13.40.Em, 13.40.HqProperties of Light Flavour Baryons in Hypercentral quark m odel 2
+1. Introduction
+Baryons are not only the interesting systems to study the quark d ynamics and their
+properties but are also interesting from the point of view of simple sy stems to study
+three body interactions. In the last two decades, there has been great advancement
+in the study of baryon properties. The ground state masses and m agnetic moments of
+many low lying baryons have been measured experimentally. The magn etic moments of
+all octet baryons ( JP=1
+2+) are known accurately except for Σ0which has a life time
+too short. For the decuplet baryons ( JP=3
+2+), the experimental measurements are
+poor as they have very short life times due to available strong intera ction decay chan-
+nels. The Ω−is an exception as it is composed of three s quarks which decays via we ak
+interaction causing longer life time for it [1]. The ∆ particles are produ ced in scattering
+the pion, photon, or electron beams off a nucleon target. High prec ision measurements
+of the N →∆ transition by means of electromagnetic probes became possible wit h the
+advent of the new generation of electron beam facilities such as LEG S, BATES, ELSA,
+MAMI, and those at the Jefferson Lab. Many such experimental pr ograms devoted to
+the study of electromagnetic properties of the ∆ have been repor ted in the past few
+years [2, 3, 4]. The electromagnetic transition of ∆ →Nγhave been the subject of
+intense study [5, 6, 7, 8, 9, 10]. The experimental information prov ides new incentives
+for theoretical study of these observables.
+Theoretically, there exist serious discrepancies between the quar k model and exper-
+imental results particularly in the predictions of their magnetic mome nts [11, 12, 13].
+Prediction of transition magnetic moments between the decuplet to octet (3
+2+→1
+2+)
+is as important as the prediction of the masses and magnetic moment s of the baryons
+(octet and decuplet) for testing of any model hypothesis and und erstanding the dynam-
+ics of quarks and meaning of the constituent mass of the quarks in t he hadronic scale.
+Various attempts including lattice QCD (Latt) [14, 15, 16], chiral pe rturbation theory
+(χPT) [17, 18, 19, 20, 21], relativistic quark model (RQM) [22, 23], non r elativistic
+quark model (NRQM) [24], QCD sum rules (QCDSR) [12, 13, 25, 26] , ch iral quark
+soliton model ( χQSM) [27, 28], chiral constituent quark model ( χCQM) [29], chiral bag
+model (χB) [30], cloudy bag model [31], quenched lattice gauge theory [32] et c., have
+been tried, but with partial success.
+The importance of three body interaction in the description of bary on was felt in
+many cases. In this context it is found that the six dimensional hype r central model
+with coulomb plus power potential ( hCPP ν) is successful in predicting the masses and
+magneticmoments ofheavyflavour baryons(baryoncontainingch armorbeautyquarks)
+[33, 34]. Unlike in the case of many other potential models, in the hCPP νmodel, the
+confinement potential expressed in terms of the three body hype r spherical co-ordinate
+is able to account for the three body effects.Properties of Light Flavour Baryons in Hypercentral quark m odel 3
+Accordingly, in this paper we extend the hCPP νmodel in the light flavour baryonic
+sector to compute the masses, magnetic moments of octet and de cuplet baryons. We
+also study the electromagnetic transition and radiative decay width of those baryons. In
+section 2 the hypercentral scheme anda brief introduction of hCPP νpotential employed
+for the present study are described. Section 3 describes the com putational details of
+the magnetic moments of octet and decuplet baryons and the3
+2+→1
+2+transition
+magnetic moments. Section 4 describes the radiative decay widths f or those transition.
+In section 5, we discuss our results while comparing with other theor etical predictions
+and experimental results and draw important conclusions.
+2. Hyper Central Scheme for Baryons
+Quark model description of baryons is a simple three body system of interest. Generally
+the phenomenological interactions among the three quarks are st udied using the two-
+body quark potentials such as the Isgur Karl Model [35], the Capst ick and Isgur
+relativistic model [36, 37], the Chiral quark model [38], the Harmonic Oscillator model
+[39, 40] etc. The three-body effects are incorporated in such mod els through two-body
+and three-body spin-orbit terms [33, 41]. The Jacobi Co-ordinate s to describe baryon
+as a bound state of three different constituent quarks is given by [4 2]
+/vector ρ=1√
+2(/vector r1−/vector r2) ;/vectorλ=(m1/vector r1+m2/vector r2−(m1+m2)/vector r3)/radicalbig
+m2
+1+m2
+2+(m1+m2)2(1)
+Such that
+mρ=2m1m2
+m1+m2;mλ=2m3(m2
+1+m2
+2+m1m2)
+(m1+m2)(m1+m2+m3)(2)
+Herem1,m2andm3are the constituent quark mass parameters.
+In the hypercentral model, we introduce the hyper spherical coo rdinates which are
+given by the angles
+Ωρ= (θρ,φρ) ; Ωλ= (θλ,φλ) (3)
+together with the hyper radius, xand hyper angle ξrespectively as,
+x=/radicalbig
+ρ2+λ2;ξ= arctan/parenleftBigρ
+λ/parenrightBig
+(4)
+The model Hamiltonian for baryons can now be expressed as
+H=P2
+ρ
+2mρ+P2
+λ
+2mλ+P2
+R
+2M+V(ρ,λ) =P2
+x
+2m+V(x) (5)
+Here the potential V(x) is not purely a two body interaction but it contains three-body
+effects also. The three body effects are desirable in the study of ha drons since the non-
+abelian nature of QCD leads to gluon-gluon couplings which produce th ree-body forcesProperties of Light Flavour Baryons in Hypercentral quark m odel 4
+[43]. Using hyperspherical coordinates, thekinetic energy operat orP2
+x
+2mofthe three-body
+system can be written as
+P2
+x
+2m=−1
+2m/parenleftbigg∂2
+∂x2+5
+x∂
+∂x−L2(Ωρ,Ωλ,ξ)
+x2/parenrightbigg
+(6)
+WhereL2(Ωρ,Ωλ,ξ) is the quadratic Casimir operator of the six dimensional rotational
+groupO(6) and its eigen functions are the hyperspherical harmonics, Y[γ]lρlλ(Ωρ,Ωλ,ξ)
+satisfying the eigenvalue relation
+L2Y[γ]lρlλ(Ωρ,Ωλ,ξ) =γ(γ+4)Y[γ]lρlλ(Ωρ,Ωλ,ξ) (7)
+Hereγis the grand angular quantum number and it is given by γ= 2ν+lρ+lλ,
+andν= 0,1,...andlρandlλbeing the angular momenta associated with the ρandλ
+variables.
+If the interaction potential is hyper spherical such that the pote ntial depends only
+on the hyper radius x, then the hyper radial schrodinger equation corresponds to the
+hamiltonian given by Eqn.(5) can be written as
+/bracketleftbiggd2
+dx2+5
+xd
+dx−γ(γ+4)
+x2/bracketrightbigg
+φγ(x) =−2m[E−V(x)]φγ(x) (8)
+whereγis the grand angular quantum number.
+For the present study we consider the hyper central potential V(x) as the hyper
+coulomb plus power (hCPP ν) form given by [33, 34, 44]
+V(x) =−τ
+x+βxν+κ+Vspin (9)
+In the above equation the spin independent terms correspond to c onfinement potential
+in the hyperspherical co-ordinates. The form of the potential th ough hyper central,
+belong to a generality of potentials of the form −Arα+krǫ+V0whereA,k,αandǫare
+non negative constants where as V0can have either sign. There are many attempts with
+different choices of αandǫto study the hadron properties [45]. For example, Cornell
+potential has α=ǫ= 1, Lichtenberg potential has α=ǫ= 0.75. Song-Lin potential has
+α=ǫ= 0.5 and the Logarithmic potential of Quigg and Rosner corresponds t oα= 0,
+ǫ→0 [45]. Martin potential corresponds to α=0,ǫ= 0.1 [45] while Grant, Rosner and
+Rynes potential corresponds to α= 0.045,ǫ= 0; Heikkil¨ a, T¨ ornqusit and Ono potential
+corresponds to α= 1,ǫ= 2/3 [46]. Potentials in the region 0 ≤α≤1.2, 0≤ǫ≤1.1
+ofα−ǫvalues are also been explored [47]. So it is important to study the beha vior
+of different potential scheme with different choices of αandǫto know the dependence
+of their parameters to the hadron properties. The spin independe nt part of potential
+defined by Eqn.(9) corresponds to α= 1 andǫ=ν. Hereτof the hyper-coulomb, βof
+the confining term and κare the model parameters. The parameter τis related to the
+strong running coupling constant αsas [33, 34]
+τ=2
+3bαs (10)Properties of Light Flavour Baryons in Hypercentral quark m odel 5
+where b is the model parameter,2
+3is the color factor for the baryon. The potential
+parameters treated here are similar to the one employed for the st udy of heavy flavour
+baryons [33, 34]. The strong running coupling constant is computed using the relation
+αs=αs(µ0)
+1+33−2nf
+12παs(µ0)ln(µ
+µ0)(11)
+whereαs(µ0= 1GeV)≈0.6 is considered in the present study. The spin dependent
+part of the three body interaction potential of Eqn.(9) is taken as [33, 41]
+Vspin(x) =−1
+4αse−x
+x0
+xx2
+0/summationdisplay
+i<j/vector σi·/vector σj
+6mimj/vectorλi·/vectorλj (12)
+where,x0is the hyperfine parameter of the model.
+The six dimensional radial Schrodinger equation described by Eqn.(8 ) has been
+solved in the variational scheme with the hyper coloumb trial radial w ave function given
+by [43]
+ψωγ=/bracketleftbigg(ω−γ)!(2g)6
+(2ω+5)(ω+γ+4)!/bracketrightbigg1
+2
+(2gx)γe−gxL2γ+4
+ω−γ(2gx) (13)
+The wave function parameter g and hence the energy eigen value ar e obtained by ap-
+plying virial theorem for a chosen potential index ν.
+The baryon masses are determined by the sum of the model quark m asses plus
+kinetic energy, potential energy and the spin hyperfine interactio n as
+MB=/summationdisplay
+imi+/angbracketleftH/angbracketright (14)
+For the present calculations, we have employed the same mass para meters of the light
+flavour quarks ( mu= 338 MeV, md= 350 MeV, ms= 500 MeV) as used in [33]. We
+fix other parameters (b of Eqn.(10) and x0of Eqn.(12)) of the model for each choice of
+νusing the experimental center of weight (spin-average) mass and hyper fine splitting
+of the octet decuplet baryons. The procedure is repeated for diff erent choices of νand
+the computed masses of octet and decuplet baryons are listed in Ta ble 1 and Table 2
+respectively.
+3. Magnetic moments of light baryons
+Now the magnetic moment of the baryons are computed in terms of it s quarks spin-
+flavour wave function of the constituent quarks as
+µB=/summationdisplay
+i/angbracketleftφsf|µi/vector σi|φsf/angbracketright (15)
+where
+µi=ei
+2mi(16)Properties of Light Flavour Baryons in Hypercentral quark m odel 6
+Hereeiandσirepresents the charge and the spin of the quark constituting the baryonic
+stateand |φsf/angbracketrightrepresents thespin-flavour wave functionoftherespective bar yonic state
+as listed in [48]. Here, mithe mass of the ithquark in the three body baryon is taken
+as an effective mass of the constituting quarks as their motions are governed by the
+three body force described through the hCPP νpotential appeared in the hamiltonian
+5. Accordingly, within the baryon the mass of the quarks may get mo dified due to its
+binding interactions with other two quarks. We account for this bou nd state effect by
+replacing the mass parameter miof Eqn.(16) by defining an effective mass to the bound
+quarks,meff
+ias given by [33, 34, 44]
+meff
+i=mi
+1+/angbracketleftH/angbracketright/summationtext
+imi
+ (17)
+such thatMB=3/summationtext
+i=1meff
+i.The computations are repeated for the different choices of
+the flavour combinations of qqq (q = u, d, s). The computed magnet ic moments of the
+octet and decuplet baryons are listed in Table 3 and 4 respectively.
+4. Radiative Decay Width
+The radiative decays of baryons provide much better understand ing of the underlying
+structure of baryons and the dependence on the constituent qu ark mass. Though
+the non-relativistic model of Isgur and Karl successfully predicte d the electromagnetic
+properties of the low lying octet baryons but it fails to provide a good description of
+the radiative decay of the decuplet baryons [35, 50]. Thus, the suc cessful prediction of
+the electromagnetic properties of octet baryons as well as the de cuplet baryons become
+detrimental for all the phenomenological models. The radiative dec ay width of the
+baryons can be computed using the relation given by [44]
+ΓR=q3
+4π2
+2J+1e2
+m2p|µ3
+2+→1
+2+|2(18)
+wherempis the proton mass, µ3
+2+→1
+2+is the radiative transition magnetic moments, q
+is the photon energy and is given by M3
+2+−M1
+2+.
+The transition magnetic moments for3
+2+→1
+2+are computed as
+µ3
+2+→1
+2+=/summationdisplay
+i/angbracketleftφ3
+2+
+sf|µi/vector σi|φ1
+2+
+sf/angbracketright (19)
+|φ3
+2+
+sf/angbracketrightrepresentthespinflavourwavefunctionofthequarkcomposition fortherespective
+decuplet baryons while |φ1
+2+
+sf/angbracketrightrepresent the spin flavour wave function of the quark
+composition for the octet baryons. The value of µiis given by Eqn.(16) and the meff
+iProperties of Light Flavour Baryons in Hypercentral quark m odel 7
+/s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56/s57/s48/s48/s57/s53/s48/s49/s48/s48/s48/s49/s48/s53/s48/s49/s49/s48/s48/s49/s49/s53/s48/s49/s50/s48/s48/s49/s50/s53/s48/s49/s51/s48/s48/s49/s51/s53/s48/s49/s52/s48/s48/s49/s52/s53/s48/s77/s97/s115/s115/s32/s105/s110/s32/s77/s101/s86
+/s80/s111/s116/s101/s110/s116/s105/s97/s108/s32/s105/s110/s100/s101/s120/s32 /s32/s32/s32/s112
+/s32/s32/s110
+/s32/s32
+/s32/s32
+/s32/s32
+/s32/s32
+/s32/s32
+/s32/s32/s40/s97/s41
+Figure 1. Variation of octet baryon masses with respect to potential index ν.
+Experimental masses are shown with error bar. The shaded region show minimum
+root mean square deviation with experimental results.
+forthetransitioniscalculatedusinggeometricmeanofeffectivequa rkmassesofdecuplet
+and octet baryons as given by [49]
+meff
+i=/radicalbigg
+meff
+i(3
+2+)meff
+i(1
+2+)(20)
+The calculated transition magnetic moments are listed in Table 5
+We also calculate the branching ratioΓR
+Γ(Baryon)using the experimental total decay
+width Γ(Baryon) of the respective decuplet baryons. The computed values of rad iative
+decay width and the branching ratio for different choices of the pot ential power indices
+are listed in Table 6.
+5. Results and Discussion
+The masses of octet and decuplet baryons in the hypercentral co ulomb plus power po-
+tential (hCPP ν) model with the different choices of potential index νhave been studied.
+Fig.(1) and Fig.(2) show the behaviour of the predicted masses of th e octet anddecuplet
+baryons with the potential index νin the range, 0.5 ≤ν≤1.5. The trend lines here
+show saturation of the masses beyond ν≥1.0. The shaded regions in Fig.(1) and in
+Fig.(2) show the neighbour hood region of νat which the predicted masses are having
+minimum root mean square deviation with the experimental masses.Properties of Light Flavour Baryons in Hypercentral quark m odel 8
+/s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56/s49/s50/s48/s48/s49/s51/s48/s48/s49/s52/s48/s48/s49/s53/s48/s48/s49/s54/s48/s48/s49/s55/s48/s48/s49/s56/s48/s48/s77/s97/s115/s115/s32/s105/s110/s32/s77/s101/s86
+/s80/s111/s116/s101/s110/s116/s105/s97/s108/s32/s105/s110/s100/s101/s120/s32 /s32/s32
+/s32
+/s32
+/s32
+/s32
+/s32
+/s32
+/s32/s32
+/s32
+/s32/s40/s98/s41
+Figure 2. Variation of decuplet baryon masses with respect to potential inde xν.
+Experimental masses are shown with error bar. The shaded region show minimum
+root mean square deviation with experimental results.
+/s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56/s45/s50/s45/s49/s48/s49/s50/s51/s77/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s115/s32/s105/s110/s32
+/s80/s111/s116/s101/s110/s116/s105/s97/s108/s32/s105/s110/s100/s101/s120/s32 /s32/s32/s32/s112
+/s32/s32/s110
+/s32
+/s32
+/s32
+/s32
+/s32/s40/s97/s41
+Figure 3. Behaviour of the magnetic moments of octet baryons with respect to
+potential index ν. The known experimental values are shown with error bar. The
+shaded region show minimum root mean square deviation with experime ntal results.Properties of Light Flavour Baryons in Hypercentral quark m odel 9
+/s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56/s45/s50/s48/s50/s52/s54/s77/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s115/s32/s105/s110/s32
+/s80/s111/s116/s101/s110/s116/s105/s97/s108/s32/s105/s110/s100/s101/s120/s32 /s32/s32
+/s32
+/s32/s32/s32
+/s32/s32/s32
+/s32/s32/s32
+/s32/s32
+/s32/s32
+/s32/s32
+/s32/s32/s32/s40/s98/s41
+Figure 4. Behaviour of the magnetic moments of decuplet baryons with respe ct to
+potential index ν. The known experimental values are shown with error bar.
+Table 1. Masses of octet baryons ( JP=1
+2+)
+hCPP ν Octet Mass(MeV) Octet Mass(MeV)
+Model Baryon Our Others Baryon Our Others
+0.5 uud(p) 1065.68 939.00[11] uds(Σ0) 1344.67 1193.00[11]
+0.7 967.41 938.27[51] 1239.94 1192.64[51]
+1.0 931.08 866.00[52] 1203.29 1022.00[52]
+1.5 924.27 938.27[1] 1195.98 1192.64[1]
+0.5 ddu(n) 1067.24 939.00[11] dds(Σ−) 1345.46 1197.00[11]
+0.7 971.74 939.57[51] 1243.71 1197.45[51]
+1.0 935.77 866.00[52] 1205.99 1022.00[52]
+1.5 929.04 939.56[1] 1199.06 1197.45[1]
+0.5 uds(Λ0) 1289.26 1116.00[11] ssu(Ξ0) 1420.19 1315.00[11]
+0.7 1183.59 1115.68[51] 1331.65 1314.64[51]
+1.0 1147.34 1022.00[52] 1297.09 1215.00[52]
+1.5 1139.88 1115.68[1] 1291.43 1314.86[1]
+0.5 uus(Σ+) 1339.95 1189.00[11] ssd(Ξ−) 1428.97 1321.00[11]
+0.7 1235.98 1189.39[51] 1340.44 1321.39[51]
+1.0 1198.84 1022.00[52] 1306.55 1215.00[52]
+1.5 1191.50 1189.37[1] 1299.61 1321.71[1]Properties of Light Flavour Baryons in Hypercentral quark m odel 10
+Table 2. Masses of decuplet baryons ( JP=3
+2+)
+hCPP ν Decuplet Mass(MeV) Decuplet Mass(MeV)
+Model Baryon Our Others Baryon Our Others
+0.5 uuu(∆++) 1361.68 1232.00[11] uds(Σ∗0) 1530.40 1384.00[11]
+0.7 1264.17 1230.82[51] 1428.43 1384.18[51]
+1.0 1228.63 1344.00[52] 1390.47 1447.00[52]
+1.5 1221.21 1232.00[1] 1383.66 1383.70[1]
+0.5 uud(∆+) 1358.3 1232.00[11] dds(Σ∗−) 1534.72 1387.00[11]
+0.7 1260.78 1230.57[51] 1431.66 1387.18[51]
+1.0 1223.74 1344.00[52] 1395.44 1447.00[52]
+1.5 1216.33 1232.00[1] 1387.45 1387.20[1]
+0.5 ddu(∆0) 1360.22 1232.00[11] ssu(Ξ∗0) 1640.49 1532.00[11]
+0.7 1263.77 1231.87[51] 1549.05 1531.81[51]
+1.0 1228.69 1344.00[52] 1516.19 1583.00[52]
+1.5 1221.25 1232.00[1] 1508.03 1531.80[1]
+0.5 ddd(∆−) 1356.79 1232.00[11] ssd(Ξ∗−) 1641.69 1535.00[11]
+0.7 1260.32 1234.73[51] 1553.1 1534.95[51]
+1.0 1223.65 1344.00[52] 1519.35 1583.00[52]
+1.5 1217.80 1232.00[1] 1512.23 1535.00[1]
+0.5 uus(Σ∗+) 1534.60 1383.00[11] sss(Ω−) 1804.12 1672.00[11]
+0.7 1430.54 1382.74[51] 1718.22 1672.45[51]
+1.0 1392.93 1447.00[52] 1678.7 1701.00[52]
+1.5 1386.16 1382.80[1] 1668.16 1672.45[1]
+The computed magnetic moments of the octet and decuplet baryon s are compared
+with the known experimental results as well as with other model pre dictions in Table 3
+and 4 respectively. Present results for the choice of ν≈0.7 are found to be in agreement
+with the known experimental values as well as with other model pred ictions. Here, it
+should be noted that the better agreement occur for the choice o fν(0.6≤ν≤0.7)
+slightly below the saturation region ( ν≥1) (See Fig.1 - 4). However the experimental
+measurements of decuplet states are difficult and the known values for the ∆ - baryons
+carry large errors [1, 2, 3].
+The available experimental results for the ∆++are 4.5±0.95 and 3.5-7.5 are in very
+good agreement with our calculated magnetic moment 4.52 at ν= 0.7. The calculated
+magnetic moments for ∆+, ∆0, and Ω−are also in good agreement with experimental
+result while comparing with other theoretical models.
+The behavior of the predicted magnetic moments of octet and decu plet baryons
+with potential index νare shown in Fig.(3) and Fig.(4) respectively. The same satura-Properties of Light Flavour Baryons in Hypercentral quark m odel 11
+Table 3. Magnetic moments of octet baryons ( in µN)
+Various models p n Λ Σ+Σ0Σ−Ξ0Ξ−
+hCPP ν
+ν= 1.5 3.07 -2.04 -0.65 2.64 0.84 -0.98 -1.50 -0.55
+ν= 1.0 3.04 -2.07 -0.64 2.63 0.83 -0.98 -1.49 -0.55
+ν= 0.7 2.93 -1.93 -0.62 2.54 0.81 -0.95 -1.46 -0.54
+ν= 0.5 2.66 -1.76 -0.57 2.35 0.74 -0.88 -1.37 -0.51
+EXPT. [1] 2.79 -1.91 -0.61 2.46 -1.16 -1.25 -0.65
+QCDSR [26] 2.82 -1.97 -0.56 2.31 0.69 -1.16 -1.15 -0.64
+χCQM [29] 2.80 -2.11 -0.66 2.39 0.54 -1.32 -1.24 -0.50
+χPT [17] 2.58 -2.10 -0.66 2.43 0.66 -1.10 -1.27 -0.95
+Latt [14] 2.79 -1.60 -0.50 2.37 0.65 -1.08 -1.17 -0.51
+CDM [54] 2.79 -2.07 -0.71 2.47 -1.01 -1.52 -0.61
+QM [55] 2.79 -1.91 -0.59 2.67 0.78 -1.10 -1.41 -0.47
+QM+T [55] 2.79 -1.91 -0.61 2.39 0.63 -1.12 -1.24 -0.69
+BAGCHI [11] 2.88 -1.91 -0.71 2.59 0.83 -0.92 -1.45 -0.62
+Dai fit A [56] 2.84 -1.87 2.46 -1.06 -1.28 -0.61
+Dai fit B [56] 2.80 -1.92 2.46 -1.23 -1.26 -0.63
+SIMON [57] 2.54 -1.69 -0.69 2.48 0.80 -0.90 -1.49 -0.63
+SU(3)BR. [58] 2.79 -1.97 -0.60 2.48 0.66 -1.16 -1.27 -0.65
+PQM [53] 2.68 -1.99 -0.56 2.52 -1.17 -1.27 -0.59
+tion trends towards saturation beyond the potential index ν >1.0 are observed. The
+shaded region in Fig.(3) corresponds to the region of ν(0.6< ν <0.7) for which the
+predicted octet baryon magnetic moments show minimum root mean s quare deviation
+with the experiments. The predicted magnetic moments of the decu plet baryons in the
+same region of ν(0.6< ν <0.7) are found to be closer to the existing experimental
+values of ∆ and Ω baryons.
+As the octet magnetic moments are known experimentally, we calcula te the per-
+centage variations of the different model predictions with respect to the experimental
+values and are given in Table 7 for comparison. The present hCPP ν≈0.7prediction for p,
+n, Λ, Σ+baryons are much better with lesser percentage error compared to other model
+predictions. And the average percentage variations from proton to Ξ−obtained from
+the Table 7 is about 8 % only, while that for the lattice predictions and t hat ofχPT
+predictions is about 10 and 11 % respectively. It can also be seen tha t the predictions
+of QCDSR and PQM are having lower variations of about 4 % only.
+The transition magnetic moments obtained from the present study (hCPP νmodel)
+are in accordance with other theoretical predictions with much less variations with theProperties of Light Flavour Baryons in Hypercentral quark m odel 12
+Table 4. Magnetic moments of decuplet baryons (in µN)
+Various
+models ∆++∆+∆0∆−Σ∗+Σ∗0Σ∗−Ξ∗0Ξ∗−Ω−
+hCPP ν
+ν= 1.5 4.69 2.37 0.05 -2.34 2.61 0.29 -2.42 0.53 -1.92 -1.68
+ν= 1.0 4.66 2.35 0.05 -2.33 2.60 0.28 -2.40 0.53 -1.91 -1.67
+ν= 0.7 4.52 2.29 0.05 -2.25 2.53 0.27 -2.32 0.52 -1.87 -1.63
+ν= 0.5 4.19 2.12 0.05 -2.08 2.35 0.26 -2.15 0.49 -1.77 -1.56
+Expt. 4.5 ±0.95 2.70+1.0
+−1.3≈0 -2.02±0.06
+[1, 2, 3] 3.5-7.5
+LCQCD [12] 4.40 2.20 0.00 -2.20 2.70 0.20 -2.28 0.40 -2.00 -1.56
+QCDSR [13] 4.39 2.19 0.00 -2.19 2.13 0.32 -1.66 -0.69 -1.51 -1.49
+Latt [14] 4.91 2.46 0.00 -2.46 2.55 0.27 2.02 0.46 -1.68 -1.40
+χPT [17] 6.04 2.84 -0.36 -3.56 3.07 0.00 -3.07 0.36 -2.56 -2.02
+χPT [18] 4.00 2.10 -0.17 -2.25 2.00 -0.07 -2.20 0.10 -2.00 input
+RQM [22] 4.76 2.38 0.00 -2.38 1.82 -0.27 -2.36 -0.60 -2.41 -2.48
+NRQM [24] 5.56 2.73 -0.09 -2.92 3.09 0.27 -2.56 0.63 -2.20 -1.81
+χQSM [27] 4.73 2.19 -0.35 -2.9 2.52 -0.08 -2.69 0.19 -2.48 -2.27
+χCQM [29] 4.51 2.00 -0.51 -3.02 2.69 0.02 -2.64 0.54 -1.84 -1.71
+χB [30] 3.59 0.75 -2.09 -1.93 2.35 -0.79 -3.87 0.58 -2.81 -1.75
+EMS [49] 4.56 2.28 0.00 -2.28 2.56 0.23 -2.10 0.48 -1.90 -1.67
+LCQCDSR[59] 6.34 3.17 0.00 -3.17
+choices ofν. However the experimentally known value for the transition magnet ic mo-
+ments of (3.23 ±0.1) ∆0→nγis higher than theoretical model predictions (see Table 5).
+The parameter free predictions of the radiative decay width for ∆ →Nγ(N=n,p)
+transitions obtained here are in very good agreement with experime nt compared to
+other model predictions (see Table 6). Prediction for other decup let to octet radiative
+transitions are well within the experimental limits.
+At the end, we like to point out important feature of the hCPP νmodel is the
+saturation behaviour of the predicted properties of the baryons withν >1. Similar
+saturation behaviour was also observed in the mass predictions of t hehCPP νmodel in
+the heavy flavour sector [33].
+It thus suggests that hCPP ν≥1model can adequately represents the three body
+interactions among the quarks constituting the baryons.
+Acknowledgement: The authorsacknowledge thefinancial support fromtheUni-Properties of Light Flavour Baryons in Hypercentral quark m odel 13
+Table 5. Magnitude of the transition Magnetic moments ( |µ3
+2+→1
+2+|) inµN
+Decay Mode Transition( |3
+2+→1
+2+|) Magnetic moments( µN)
+Expression hCPP νothers Expt. [3]
+∆+→pγ |2√
+2
+3(µu−µd)|ν= 0.5 2.20 2.57 [60]
+ν= 0.7 2.40 2.76 [61]
+ν= 1.0 2.49 2.48 [49]
+ν= 1.5 2.50 2.50 [9]
+∆0→nγ |−2√
+2
+3(µd−µu)|ν= 0.5 2.23 2.57 [60]
+ν= 0.7 2.42 2.76 [61] 3.23 ±0.1
+ν= 1.0 2.51 2.58 [49]
+ν= 1.5 2.52 2.50 [9]
+Σ∗+→Σ+γ |2√
+2
+3(µu−µs)|ν= 0.5 1.91 2.21 [60]
+ν= 0.7 2.06 2.24 [61]
+ν= 1.0 2.13 2.13 [49]
+ν= 1.5 2.14 2.10 [9]
+Σ∗0→Σ0γ|√
+2
+3(2µs−µu−µd)|ν= 0.5 0.89 0.88 [60]
+ν= 0.7 0.97 1.01 [61]
+ν= 1.0 1.00 0.96 [49]
+ν= 1.5 1.01 0.89 [9]
+Σ∗0→Λ0γ |√
+2√
+3(µu−µd)|ν= 0.5 1.93 2.24 [60]
+ν= 0.7 2.09 2.46 [61]
+ν= 1.0 2.15 2.25 [49]
+ν= 1.5 2.16 2.30 [9]
+Σ∗−→Σ−γ |2√
+2
+3(µs−µd)|ν= 0.5 0.21 0.44 [60]
+ν= 0.7 0.22 0.22 [61]
+ν= 1.0 0.23 0.22 [49]
+ν= 1.5 0.23 0.31 [9]
+Ξ∗0→Ξ0γ |2√
+2
+3(µu−µs)|ν= 0.5 2.05 2.22 [60]
+ν= 0.7 2.17 2.46 [61]
+ν= 1.0 2.23 2.27 [49]
+ν= 1.5 2.24 2.20 [9]
+Ξ∗−→Ξ−γ|2√
+2
+3(2µs−µd)|ν= 0.5 0.22 0.44 [60]
+ν= 0.7 0.23 0.27 [61]
+ν= 1.0 0.24 0.32 [49]
+ν= 1.5 0.24 0.31 [9]Properties of Light Flavour Baryons in Hypercentral quark m odel 14
+Table 6. Radiative decay widths (Γ Rin MeV) and branching ratio
+Decay Mode hCPP νRadiative Decay Width( ΓR) in MeV Branching Ratio in %
+Our Others Expt. Symbol our Expt. [1]
+∆+→pγ ν = 0.5 0.501 0.363 [60] 0.424
+ν= 0.7 0.601 0.430 [5] 0.510
+ν= 1.0 0.643 0.900 [9] 0.64 [62]ΓR
+Γ(∆)0.545 0.52-0.60
+ν= 1.5 0.644 0.546
+∆0→nγ ν = 0.5 0.517 0.363 [60] 0.438
+ν= 0.7 0.603 0.430 [5] 0.511
+ν= 1.0 0.655 0.900 [9] 0.64 [62]ΓR
+Γ(∆)0.555 0.52-0.60
+ν= 1.5 0.655 0.555
+Σ∗+→Σ+γ ν= 0.5 0.111 0.100 [60] 0.310
+ν= 0.7 0.129 0.100 [5] 0.361
+ν= 1.0 0.137 0.11 [9]ΓR
+Γ(Σ∗+)0.383 -
+ν= 1.5 0.139 0.390
+Σ∗0→Σ0γ ν = 0.5 0.021 0.016 [60] 0.058
+ν= 0.7 0.026 0.017 [5] 0.072
+ν= 1.0 0.027 0.021 [9] <1.750 [63]ΓR
+Γ(Σ∗0)0.075
+ν= 1.5 0.027 0.022 [6] 0.077
+Σ∗0→Λ0γ ν = 0.5 0.216 0.241 [60] 0.600
+ν= 0.7 0.265 0.730
+ν= 1.0 0.274 0.470 [9] <2.100 [64]ΓR
+Γ(Λ∗0)0.763 1.3 ±0.4
+ν= 1.5 0.279 0.275 [6] 0.776
+Σ∗−→Σ−γ ν= 0.5 0.001 0.004 [60] 0.003
+ν= 0.7 0.001 0.003 [5] 0.003
+ν= 1.0 0.001 0.002 [9] <0.009 [65]ΓR
+Γ(Σ∗−)0.003<0.024
+ν= 1.5 0.001 0.003
+Ξ∗0→Ξ0γ ν = 0.5 0.185 0.131 [60] 2.043
+ν= 0.7 0.200 0.129 [5] 2.197
+ν= 1.0 0.216 0.140 [9]ΓR
+Γ(Ξ∗0)2.378<4.0
+ν= 1.5 0.211 2.319
+Ξ∗−→Ξ−γ ν= 0.5 0.001 0.005 [60] 0.019
+ν= 0.7 0.002 0.003 [5] 0.021
+ν= 1.0 0.002 0.003 [9]ΓR
+Γ(Ξ∗−)0.023<4.0
+ν= 1.5 0.002 0.023Properties of Light Flavour Baryons in Hypercentral quark m odel 15
+Table 7. Percentagevariationinthe predictionsofmagneticmomentsofoct etbaryons
+system hCPP ν QCDSRχCQMχPT BAGCHI PQM LATTICE
+(In shaded region) [26] [29] [17] [11] [53] [14]
+p 0.75 1.07 0.35 7.52 3.22 3.94 0.00
+n 2.14 3.14 10.4 9.54 0.00 4.18 16.23
+Λ 1.63 8.20 9.83 8.10 16.3 8.19 18.0
+Σ+0.40 6.09 2.84 1.21 5.28 2.43 6.30
+Σ−20.7 0.00 13.7 5.17 20.6 0.86 6.80
+Ξ013.6 8.00 0.80 0.00 16.0 1.60 6.40
+Ξ−18.7 1.53 23.0 46.1 4.61 9.23 21.5
+Average 8.20 4.00 8.70 11.00 9.43 4.34 10.74
+versity Grant commission, Government of India under a Major rese arch project F. 32-
+31/2006 (SR).
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\ No newline at end of file
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+arXiv:1001.0850v1  [nucl-th]  6 Jan 2010Suppression of Charge Equilibration leading to the Synthes is of Exotic Nuclei
+Yoritaka Iwata1, Takaharu Otsuka2,3, Joachim A. Maruhn4, and Naoyuki Itagaki2
+1GSI Helmholtzzentrum f¨ ur Schwerionenforschung, D-64291 Darmstadt, Germany
+2Department of Physics, University of Tokyo, Hongo, Tokyo 11 3-0033, Japan
+3Center for Nuclear Study, University of Tokyo, Hongo, Tokyo 113-0033, Japan and
+4Institut f¨ ur Theoretische Physik, Universit¨ at Frankfur t, D-60325 Frankfurt, Germany
+(Dated: June 15, 2018)
+Charge equilibration between two colliding nuclei can take place in the early stage of heavy-ion
+collisions. A basic mechanism of charge equilibration is pr esented in terms of the extension of single-
+particle motion from one nucleus to the other, from which the upper energy-limit of the bombarding
+energy is introduced for significant charge equilibration a t the early stage of the collision. The for-
+mulafor this limit is presented, and is compared to various e xperimental data. It is examined also by
+comparison to three-dimensional time-dependentdensity f unctional calculations. The suppression of
+charge equilibration, which appears in collisions at the en ergies beyond the upper energy-limit, gives
+rise to remarkable effects on the synthesis of exotic nuclei w ith extreme proton-neutron asymmetry.
+PACS numbers: 21.10.Ft, 25.70.-z, 25.70.Hi
+Chargeequilibrationisarapidprocessduringthe early
+stageofheavy-ioncollisionswithatime scaleof10−22sec
+(for a review, see [1]). Despite many theoretical at-
+tempts, its mechanism has remained an open problem,
+where only the relation with the isovector giant dipole
+resonance (iv-GDR) was discussed in certain cases (for
+example, see [2–8]). The charge equilibration is quite
+important, because it naturally prevents the synthesis of
+exotic nuclei with extreme proton-neutron asymmetry.
+As there are and will be the 3rd generation RI-beam fa-
+cilities, it is an urgent question whether such synthesis
+can be enhanced with higher beam energies or not.
+In this Letter, we first point out that there is an upper
+limitofthebombardingenergyforthefastandsignificant
+charge equilibration in the initial stage of the collision.
+Microscopic three-dimensional (time-dependent) density
+functional calculations are systematically performed for
+the purpose of examination. We actually employ Skyrme
+time-dependent Hartree-Fock (TDHF) theory, which is
+a rather unique presently feasible method for the treat-
+ment ofnon-perturbativeprocessessuchasmulti-nucleon
+transfers in a realistic framework.
+The suppression of charge equilibration brings about a
+favorable situation for the synthesis of exotic nuclei. The
+evaluation of the upper limit thus can have crucial sig-
+nificance for experiments on nuclei with extreme proton-
+neutron asymmetry. We shall consider the collision of a
+target nucleus with mass number A1, neutron (proton)
+numberN1(Z1), with a projectile nucleus with A2,N2,
+andZ2. The total mass, neutron, and proton numbers
+are denoted by A,N,Z, respectively. We look at this
+problem from an intuitive and basic viewpoint. We begin
+with a picture that the charge equilibration takes place
+as wave functions of nucleons propagate from their orig-
+inal nucleus to the other nucleus in the initial stage of
+the collision. Namely, the regime of individual single-
+particle motion spreads out following the lowering of po-
+tential barrier between the two nuclei after the touching.Because this spreading occurs as a consequence of un-
+blocked single-particle motion, the process can be very
+fast; a particle travels into the other side within ∼10−22
+sec at a quarter of light velocity, which roughly corre-
+sponds to the Fermi energy of normal nuclear matter.
+On the other hand, this propagation can be prohibited
+by the Pauli principle, while the neutron (or proton) ex-
+cess weakens this effect. Another important factor is the
+difference of velocities of the two nuclei. Because this
+spreading needs a certain time, it does not lead to charge
+equilibration in the initial stage if the relative velocity of
+the colliding nuclei is too large. In the charge equilibra-
+tion, protons and neutrons from both nuclei, particularly
+those near the Fermi levels, are mixed within a certain
+time after the touching.
+We can introduce an ansatz that, in order for the fast
+charge equilibration to occur, the relative velocity vrof
+thetwonucleiatthecollisionmustbebelowthevelocities
+corresponding to the proton or neutron Fermi momenta
+of both nuclei. By denoting the minimum of these four
+velocities as vmin
+F, the present effect can be summarized
+by the statement that the upper limit of the bombarding
+energy for charge equilibration is determined by vr=
+vmin
+F. The upper limit of the energy in the laboratory
+frame for charge equilibration is expressed as the sum
+of the kinetic energy for velocity vmin
+Fand the Coulomb
+energy at touching:
+ECE,lab/A[MeV]
+=/planckover2pi12(3π2ρmin)2/3
+2m+e2Z1Z2
+4πǫ0r0A1+A2
+A1A2(A1/3
+1+A1/3
+2),(1)
+ρmin= min
+i/parenleftbig
+ρni,ρzi/parenrightbig
+= min
+i/parenleftBigg
+Ni/parenleftBig4πr0
+3A1/3
+i/parenrightBig−1
+(1−3¯ǫ)(1+¯δ),Zi/parenleftBig4πr0
+3A1/3
+i/parenrightBig−1
+(1−3¯ǫ)(1−¯δ)/parenrightBigg
+,(2)
+wherem,e,ǫ0, andr0are the nucleon mass, the charge2
+1.5 1.6 1.7 1.8 1.9 2.0 2.10255075
+1.5 1.6 1.7 1.8 1.9 2.0 2.1 1.5 1.6 1.7 1.8 1.9 2.0 2.1 1.5 1.6 1.7 1.8 1.9 2.0 2.1
+N/ZYields [%]
+5.0 MeV E     /A = cm   6.0 MeV E     /A = cm   7.0 MeV E     /A = cm   8.0 MeV E     /A = cm   
+FIG. 1: Yield distribution of final fragments as a function of theirN/Zratios for the collisions of208Pb +132Sn (N/Zratio
+is discretized by 0.05) based on TDHF calculation (SLy4d). F our cases with different Ecm/Avalues are presented. Columns
+corresponding to the equilibrium value of N/Z=1.58 are colored in blue, and connected by red lines. For ref erence, the N/Z
+ratios for208Pb and132Sn are 1.54 and 1.64, respectively.
+TABLE I: Ecm/Avalues [MeV] of the upper limit of charge
+equilibration obtained by TDHF calculations with SLy4d and
+SkM* parameters, compared to those by the proposed for-
+mula, eq. (1). For reference, the values obtained by the Ferm i
+gas model with standard parameters are shown.
+Collision TDHF TDHFEq. (1) Fermi
+(SLy4d) (SkM*) gas
+(i)208Pb +238U6.5±0.56.5±0.56.919.46
+(ii)208Pb +132Xe6.5±0.56.5±0.56.509.03
+(iii)208Pb +132Sn6.5±0.56.5±0.56.369.03
+(iv)208Pb +40Ca3.5±0.53.5±0.53.665.14
+(v)208Pb +24Mg2.5±0.52.5±0.52.363.52
+(vi)208Pb +24O2.5±0.52.5±0.52.183.52
+(vii)208Pb +16O1.5±0.51.5±0.51.752.50
+(viii)208Pb +4He<1.0<1.0 0.480.70
+(ix)24Mg +24O5.5±1.05.5±1.05.999.5
+unit, the vacuum permittivity, and the usual nuclear ra-
+dius parameter (1.2 fm), respectively. Here we express
+the minimum velocity by the corresponding minimum of
+the proton or neutron density of the two nuclei using the
+formula vmin
+F=/planckover2pi1(3π2ρmin)1/3/m. In Eq. (2), i= 1,2
+distinguishes the two initial nuclei, and ¯ ǫand¯δ(func-
+tions ofAi) are introduced based on the droplet model
+[9, 10] to take into account the effect of neutron and pro-
+ton skins.
+In order to confirm the validity of this picture, we
+perform three-dimensional TDHF calculations systemat-
+ically. Many TDHF events were obtained with different
+values of impact parameter being incremented by 0.25
+fm. These events are summed over impact parameters
+with weights of geometric cross section, and elastic cases
+are discarded. By collecting TDHF events, we sort them
+in terms of N/Zratio with N/Zbeing discretized into
+bins of width 0.05. Figure 1 shows the yield distribu-
+tion of final fragments for208Pb +132Sn reaction. Go-
+ing from low to high Ecm(total kinetic energy in the
+center-of-mass frame) of the collision, the peak energy
+of the yield distribution as a function of N/Zis almostconstant at the beginning, and is shifted later with the
+lowering of peak height. A clear decrease of the yield
+of charge-equilibrated fragments for Ecm/A≥7.0 MeV
+is noticed. On the other hand, very neutron-rich nuclei
+withN/Z∼2.0 simultaneously start to be produced. By
+takingE1as the energy at which the peak height starts
+to be lowered (6.0 MeV in this case), and E2as the en-
+ergy at which the yield of the equilibrated N/Zbecomes
+about50%(7.0MeVinthiscase),theupperenergy-limit
+is defined in TDHF as ECE= (E1+E2)/2 (6.5 MeV in
+this case). The uncertainty from the energy bin value is
+(E2−E1)/2 (0.5 MeV in this case). Such TDHF results
+are summarized in the third and forth columns of Table
+I. TDHF calculations with two different parameter-sets
+result in the same upper energy-limit. Note that a large
+value is obtained for24Mg +24O, implying that a sim-
+ple mass dependence (small values for reactions between
+light nuclei, and vice versa) cannot explain this, while
+the mass asymmetry plays a certain role.
+Let us have a comparison between the present upper-
+limit and the TDHF results. Figure 2 depicts how the
+upper limit given by Eq. (1) changes as functions of
+N/ZandA1/A2(orA2/A1). The upper energy-limit
+comes down significantly low for higher mass asymme-
+try, while it depends only weakly on the total mass. Al-
+though charge equilibration can compete with Coulomb
+excitation particularly in collisions involving nuclei with
+largeZ, no evidence of a major change due to large Z
+is found in the results of Eq. (1) or those of TDHF cal-
+culations. However, non-negligible decrease of the upper
+energy-limit due to the proton-neutron asymmetry is no-
+ticed. The corresponding values obtained by the formula
+in Eq. (1) is shown in the fifth column of Table I. For
+reference, the values obtained by the simple Fermi gas
+model (kF= 1.36 fm−1for both protons and neutrons)
+are shown in its sixth column. By comparing the re-
+sults of Eq. (1) with the TDHF results shown in the
+third and forth columns, the agreement is remarkable,
+including a high value in the last row. Comparison be-
+tween the results of Eq.(1) (blue circles) and the TDHF
+results (blue bars) is also made in Fig. 2. Consequently,3
+ N / Z  㻜㻡㻝㻜 A 1 / A2 =  1 
+ 10
+ 100
+㻜㻚㻡 㻝 㻝㻚㻡 㻞(i)
+(ix)
+(viii)(vii)(ii)(iii)
+(iv)
+(v)(vi)E   /A [MeV] FP 
+FIG. 2:N/Zdependence of the upper energy-limit of charge
+equilibration based on Eq. (1) in the center-of-mass frame,
+whereA1> A2is assumed without loss of generality. Values
+with different total masses A= 100, 200, 300, 400, and 500
+are plotted for each A1/A2, which correspond to black lines
+from bottom to top in each group (values are too crowded to
+distinguish the total mass difference, for the cases of A1/A2=
+10 and 100), and lines are drawn to guide eyes. Each TDHF
+calculation is shown as a blue bar, where the central points
+correspond to the value obtained by Eq. (1) for each reaction ,
+and Roman numbers distinguish reactions shown in Table I.
+the upper-limit of charge equilibration depends largely
+on the Fermi energy, and the proton-neutron asymmetry
+can contribute to the shift of the upper-limit, where the
+dependence of the upper-limit on the mass-asymmetry is
+remarkable.
+Let us move on to the comparison to the experiments.
+Itcanbeseenthatexistingexperimentaldataagreeswith
+the present upper-limit formula. For instance, the fol-
+lowing experiments show charge equilibration:40Ar +
+58Ni atElab/A=7.0 MeV [11], and56Fe +165Ho (209Bi)
+atElab/A= 8.3 MeV [12]. The following experiment
+shows the disappearance of charge equilibration:112Sn
++124Sn atElab/A=50 MeV [13, 14]. Recently, experi-
+ments:124,112Sn +124,112Sn atElab/A= 35 and 50 MeV
+respectively was performed at Michigan State University
+[16]. One remarkable result here is that the final frag-
+ments arenot so closeto chargeequilibrium even in lower
+energy collisions with Elab/A=35 MeV. The disappear-
+ance of charge equilibration for this experiment has not
+been explained. Because the upper limit of charge equi-
+libration is calculated to be Elab/A=27.6 MeV from Eq.
+(1), this question can be understood now.
+Thesuppressionofchargeequilibrationcontributenat-
+urally to the production of exotic fragment; more ex-
+otic nuclei far from the equilibrated N/Zratio are to be
+synthesized above the present upper-limit. Because thePb  +   Ca 208   40
+3.0 x10       s -226.0 x10       s -229.0 x10       s -22
+10 fm
+9.0 x10       s -226.0 x10       s -223.0 x10       s -22
+10 fmE      /A
+ = 3 MeVcm 
+E      /A
+ = 4 MeVcm 
+FIG. 3: Real-time dynamics of charge distribution for208Pb
++40Ca (SLy4d) is shown for a fixed impact parameter 7.5
+fm. The upper and the lower panels show cases with Ecm/A
+= 3.0 MeV and 4.0 MeV, respectively. For both cases,208Pb
+is coming from the left, and40Ca from the right. The colored
+regions show the distribution of charge (proton-rich part) ,
+and the contours being incremented by 0.02 fm−3show the
+density. Domains with the proton-to-neutron density ratio
+greater than 0.78 and 0.72 are colored in red and yellow, re-
+spectively.
+bombardingenergiesofthe 3rd generationRI-beam facil-
+ities are sufficiently high to exceed the upper-limit, the
+present upper-limit ensures more production of further
+exotic isotopes by the latest and the future RI-beam fa-
+cilities. In fact, in the experiment of Ref. [15], the yield
+of exotic fragments was increased simply by putting the
+beam energy higher than the present upper-limit. It is
+alsoofmuchinteresttoexplorethenovelpossibilityofthe
+synthesis of exotic nuclei by collisions including only β-
+stable nuclei. Such possibilities have not attracted much
+attention, but may emerge with various feasibilities of
+the production of exotic spieces, if the experiment is set
+for energies beyond the present upper-limit. As an ex-
+ample, Fig. 3 shows the real-time dynamics of208Pb +
+40Ca around the energy where the production of exotic
+nuclei starts; the upper energy limit is 3.66 MeV (See
+Table I), fusion appears at Ecm/A= 3.0 MeV (upper
+panels), and break-up is seen at 4.0 MeV (lower panels).
+β-unstable fragments are emitted only for the bombard-
+ing energy higher than the upper-limit; for instance, a
+small fragment in the lower-right panel of Fig. 3 is62Mn
+(numbersofnucleonareroundedtobeinteger), forwhich
+the stable isotope is55Mn.
+We now point out some interesting details of charge-
+equilibration dynamics for collisions below the present
+upper energy limit. We shall first investigate it with a
+focus on the iv-GDR in a collision between light nuclei.
+Figure 4 (a) shows the time-evolution of charge distri-
+bution for24O +24Mg. We see the appearance of the
+iv-GDR and the oscillation of well-localized charge (4.5,
+6.0×10−22sec). The charge equilibration is synchro-
+nized with the iv-GDR. This is, however, seen only for4
+time10 fm4.5 x10      s-226.0 x10      s-227.5 x10      s-22Mg +   O24   24
+Mg +    Pb24   208
+time10 fm4.5 x10      s7.5 x10      s10.5 x10      s(a)
+(b)
+FIG. 4: Real-time dynamics of charge distribution for24Mg
++24O and24Mg +208Pb (SLy4d) are shown for Ecm/A= 4.5
+MeV and 2.0 MeV with the impact parameters 5.0 fm and 7.5
+fm, respectively. For both cases,24Mg is incoming from the
+left, and the states are evolving into fusion. The descripti on
+manner is the same as Fig. 3, where parts with the proton to
+neutron density ratio larger than 1.00 (0.75) and 0.80 (0.70 )
+are colored in red and yellow in case of24Mg +24O (24Mg +
+208Pb), respectively.
+collisions between light nuclei. Similar results are ob-
+tained in [2–8]. Next, we move on to collisions involving
+a heavier nucleus; e.g.24Mg +208Pb, as shown in Fig. 4
+(b). Charge-equilibration dynamics is not similar to the
+motion of the iv-GDR in this case, because no significant
+oscillationsofthelocalizedchargeappear. Insteadaradi-
+ally layered structure of the composite nucleus is formed
+(7.5×10−22sec), in which a relatively neutron-rich core
+appears in the center, a proton-rich layer surrounds it,
+and a neutron-rich skin is in the surface. This seems to
+be due to the Coulomb repulsion, hence the radial distri-
+bution of charge is formed. Here one can find analogous
+situation of isovectormonopole excitation. It means that
+the radial flow plays a prominent role in such low energy
+collisions. Similar layered structures are obtained in the
+TDHF calculations (with both Skyrme parameter-sets)
+listed in Table I except for24Mg +24O. Consequently,
+chargeequilibration, which isdependentonthe mass, has
+been clarified to be related not only with the isovectorgi-
+ant dipole excitations, but also with the isovector giant
+monopole excitations.
+Finally, let us comment on the diffusion-type mech-
+anism towards charge equilibrium. The mechanism of
+charge equilibration discussed in this Letter is valid at
+lower energies (lower than the upper energy-limit). At
+much higher energies, collisions between nucleons and
+the diffusion contribute mostly to charge equilibration
+(for example, see [17, 18]). As the diffusion is a slow pro-
+cess, it is unlikely to attain the chargeequilibrium withinthe initial stage. Energy-dependent equilibration mecha-
+nisms of a similar kind have been reported in condensed
+matter physics (see [19–21]), where the equilibration is
+fast for lower energies (temperatures), and quite slow for
+higher energies. Such a similarity between completely
+different physical systems is interesting.
+In this Letter the mechanism of charge equilibration,
+in which nucleons with the Fermi velocity play a primary
+role, has been presented, and its validity is examined by
+comparisonto virtuallyall existingrelevant experimental
+data. This concept has been further analyzed in terms of
+a three-dimensional time-dependent density function for-
+malism. The upper energy-limit of charge equilibration
+hasacrucialimpact onthenuclearsynthesisofexoticnu-
+clei. Properties presented in this Letter will give a sound
+motivation for the production of further exotic isotopes
+by the latest and the future RI-beam facilities.
+This work was supported in part by EMMI project
+of the Helmholtz Alliance, by JSPS Core-to-Core pro-
+gram EFES and by grant-in-aid for Scientific Research
+(A) 20244022. Y.I. thanks for the JSPS fellowship, and
+express his gratitude to Dr. C. Simenel. T.O. thanks
+Prof. P. Bonche for long-term valuable collaboration.
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+395.
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\ No newline at end of file
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+arXiv:1001.0851v1  [math.RA]  6 Jan 2010Left ideals in an enveloping algebra, prelie products and
+applications to simple complex Lie algebras
+L. Foissy
+Laboratoire de Mathématiques, Université de Reims
+Moulin de la Housse - BP 1039 - 51687 REIMS Cedex 2, France
+e-mail : loic.foissy@univ-reims.fr
+ABSTRACT. We characterize prelie algebras in words of left id eals of the enveloping algebras
+and in words of modules, and use this result to prove that a sim ple complex finite-dimensional
+Lie algebra is not prelie, with the possible exception of f4.
+Keywords. Prelie algebras, simple complex finite dimension al algebras.
+AMS classification. 17B20; 16S30; 17D25.
+Contents
+1 Prelie products on a Lie algebra 3
+1.1 Preliminaries and recalls on symmetric coalgebras . . . . . . . . . . . . . . . . . . 3
+1.2 Extension of a prelie product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
+1.3 Prelie product associated to a left ideal . . . . . . . . . . . . . . . . . . . . . . . . 8
+1.4 Prelie products on a Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
+1.5 Good and very good pointed modules . . . . . . . . . . . . . . . . . . . . . . . . 10
+2 Are the complex simple Lie algebras prelie? 12
+2.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
+2.2 Representations of small dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 12
+2.3 Proof in the generic cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
+2.4 Case of so2n+1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
+2.5 Proof for sl6andg2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
+3 Dendriform products on cofree coalgebra 17
+3.1 Preliminaries and results on tensor coalgebras . . . . . . . . . . . . . . . . . . . . 17
+3.2 Left ideal associated to a dendriform Hopf algebra . . . . . . . . . . . . . . . . . 18
+3.3 Dendriform products on a tensorial coalgebra . . . . . . . . . . . . . . . . . . . . 19
+3.4 Dendriform structures on a cofree coalgebra . . . . . . . . . . . . . . . . . . . . . 21
+4 MuPAD computations 22
+4.1 Dimensions of simple modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
+4.2 Computations for sl6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
+4.3 Computations for g2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
+1Introduction
+A (left) prelie algebra, or equivalently a left Vinberg alge bra, or a left-symmetric algebra [1, 2, 13]
+is a couple (V,⋆), whereVis a vector space and ⋆a bilinear product on Vsuch that for all
+x,y,z∈V:
+(x⋆y)⋆z−x⋆(y ⋆z) = (y ⋆x)⋆z−y ⋆(x⋆z).
+This axiom implies that the bracket defined by [x,y] =x⋆y−y⋆xsatisfies the Jacobi identity,
+soVis a Lie algebra; moreover, (V,⋆)is a left module on V. The free prelie algebra is described
+in [1] in terms of rooted trees, making a link with the Connes- Kreimer Hopf algebra of Renor-
+malization [3, 6].
+The aim of this text is to give examples of Lie algebras which a re not prelie, namely the
+simple complex Lie algebras of finite dimension. For this, we use the construction of [11] of the
+extension of the prelie product of a prelie algebra (g,⋆)which gives to the symmetric coalgebra
+S(g)an associative product ∗, making it isomorphic to U(g). This construction is also used in
+[9] to classify right-sided cocommutative graded and conne cted Hopf algebras. A remarkable
+corollary of this construction is that S(g)≥2=/circleplustext
+n≥2Sn(g)is a left ideal of (S(g),∗). As a
+consequence, there exists a left ideal IofU(g), such that the augmentation ideal U(g)+ofU(g)
+can be decomposed as U(g)+=g⊕I. We prove here the converse result: more precisely, given
+any Lie algebra g, we prove in theorem 11 that there exists a bijection into the se two sets:
+• PL(g) ={⋆|⋆is a prelie product on ginducing the bracket of g}.
+• LI(g) ={Ileft ideal of U(g)| U+(g) =g⊕I}.
+Let then a prelie algebra g, andIthe ideal corresponding to the prelie product of g. The
+g-module U(g)/Icontains a submodule of codimension 1, isomorphic to (g,⋆), and the special
+element1 +I. This leads to the definition of good pointed modules (definit ion 13). The sets
+PL(g)andLI(g)are in bijection with the set GM(g)of isoclasses of good pointed modules, as
+proved in theorem 14.
+Ifgis semisimple, then any good pointed module is isomorphic to K⊕(g,⋆)as ag-module.
+This leads to the definition of very good pointed module, and w e prove that if gis a semisimple
+Lie algebra, then it is prelie if, and only if ghas a very good pointed module. If gis a simple
+complex lie algebra, with the possible exception of f4, we prove that it has no very good pointed
+module, so gis not prelie. The proof is separated into three cases: first, the generic cases, then
+so2n+1, and finally sl6andg2, which are proved by direct computations using the computer al-
+gebra system MuPAD pro 4. We conjecture that this is also true forf4(the computations would
+be similar, though very longer).
+We also give in this text a non cocommutative version of theor em 11, replacing enveloping
+algebras (which are symmetric coalgebras, as the base field i s of characteristic zero) by cofree
+coalgebras, and prelie algebras by dendriform algebras [8, 10]. IfAis a cofree coalgebra, we
+prove in theorem 32 that there is a bijection between these tw o sets:
+• DD(A) ={(≺,≻)|(A,≺,≻,∆)is a dendriform Hopf algebra }.
+• LI(A) =/braceleftbigg
+(∗,I)|(A,∗,∆)is a Hopf algebra and
+Iis a left ideal of Asuch that A+=Prim(A)⊕I/bracerightbigg
+.
+The text is organized as follows: the first section deals with the general results on prelie
+algebras: after preliminaries on symmetric coalgebras, th eorem 11 is proved and good pointed
+modules are introduced. The second section is devoted to the study of good pointed modules
+over simple complex Lie algebras. The results on dendriform algebras are exposed in the third
+section and the MuPAD procedures used in this text are writte n in the last section.
+2Notations.
+1.Kis a commutative field of characteristic zero. Any algebra, c oalgebra, bialgebra, etc, of
+this text will be taken over K.
+2. Letgbe a Lie algebra. We denote by U(g)its enveloping algebra and by U+(g)the
+augmentation ideal of U(g).
+1 Prelie products on a Lie algebra
+1.1 Preliminaries and recalls on symmetric coalgebras
+LetVbe a vector space. The algebra S(V)is given a coproduct ∆, defined as the unique algebra
+morphism from S(V)toS(V)⊗S(V), such that ∆(v) =v⊗1+1⊗vfor allv∈V. Let us recall
+the following facts:
+•Let us fix a basis (vi)i∈ΛofV. We define SΛas the set of sequences a= (ai)i∈Λof elements
+ofN, with a finite support. For an element a∈SΛ, we put l(a) =/summationdisplay
+i∈Λai. For all a∈ SΛ,
+we put:
+va=/productdisplay
+i∈Λvai
+i
+ai!.
+Then(va)a∈SΛis a basis of S(V), and the coproduct is given by:
+∆(va) =/summationdisplay
+b+c=avb⊗vc.
+•LetS+(V)be the augmentation ideal of S(V). It is given a coassociative, non counitary
+coproduct ˜∆defined by ˜∆(x) = ∆(x)−x⊗1−1⊗xfor allx∈S+(V). In other terms,
+puttingS+Λ =SΛ−{(0)},(va)a∈S+Λis a basis of S+(V)and:
+˜∆(va) =/summationdisplay
+b+c=a
+b,c∈S+Λvb⊗vc.
+•Let˜∆(n):S+(V)−→S+(V)⊗(n+1)be then-th iterated coproduct of S+(V). Then:
+Ker/parenleftig
+˜∆(n)/parenrightig
+=n/circleplusdisplay
+k=1Sk(V).
+In particular, Prim(S(V)) =V.
+•(Sn(V))n∈Nis a gradation of the coalgebra S(V).
+Lemma 1 InS+(V)⊗S+(V),Ker(˜∆⊗Id−Id⊗˜∆) =Im(˜∆)+V⊗V.
+Proof.⊇. Indeed, if y=˜∆(x)+v1⊗v2∈Im(˜∆)+V⊗V, as˜∆is coassociative:
+(˜∆⊗Id)(y)−(Id⊗˜∆)(y) = (˜∆⊗Id)◦˜∆(x)−(Id⊗˜∆)◦˜∆(x)
++˜∆(v1)⊗v2−v1⊗˜∆(v2)
+= 0.
+⊆. LetX∈Ker(˜∆⊗Id−Id⊗˜∆). Choosing a basis (vi)i∈ΛofV, we put:
+X=/summationdisplay
+a,b∈S+Λxa,bva⊗vb.
+3By homogeneity, we can suppose that Xis homogeneous of a certain degree n≥2inS+(V)⊗
+S+(V). Ifn= 2, thenX∈V⊗V. Let us assume that n≥3. Then:
+(˜∆⊗Id)◦˜∆(X) =/summationdisplay
+a,b,c∈S+Λxa+b,cva⊗vb⊗vc= (Id⊗˜∆)◦˜∆(X) =/summationdisplay
+a,b,c∈S+Λxa,b+cva⊗vb⊗vc.
+So, for all a,c∈ S+Λ,b∈ SΛ,xa+b,c=xa,b+c.
+Leta,b,a′,b′∈ S+Λ, such that a+b=a′+b′. Let us show that xa,b=xa′,b′.
+First case. Let us assume that the support of aanda′are not disjoint. For all i∈Λ, we put:
+ci=/braceleftbigga′
+i−aiifa′
+i−ai>0,
+0ifa′
+i−ai≤0,
+c′
+i=/braceleftbiggai−a′
+iifai−a′
+i>0,
+0ifai−a′
+i≤0.
+Thencandc′belong to SΛ. Moreover, for all i∈Λ,ai+ci=a′
+i+c′
+i, soa+c=a′+c′, or
+a′=a+c−c′. Asa+b=a′+b′,b′=b−c+c′.
+For alli∈Λ:
+ai−c′
+i=/braceleftbigga′
+iifai−a′
+i>0,
+aiifai−a′
+i≤0.
+Soa−c′∈ SΛ. Moreover, if a−c′= 0, then if ai> a′
+i,a′
+i= 0, so the support of a′is included
+in{i /ai≤a′
+i}. Ifai≤a′
+i, thenai= 0, so the support of ais included in {i /ai> a′
+i}. As a
+consequence, the supports of aanda′are disjoint: this is a contradiction. So a−c′∈ S+Λ. As
+a−c′andb′∈ S+Λ:
+xa′,b′=xa−c′+c,b′=xa−c′,b′+c=xa−c′,b+c′.
+Asa−c′andb∈ S+Λ:
+xa−c′,b+c′=xa,b.
+The proof is similar if the support of bandb′are not disjoint, permuting the roles of (a,a′)and
+(b,b′).
+Second case. Let us assume that the supports of aanda′, and the supports of bandb′are
+disjoint. We then denote, up to a permutation of the index set Λ:
+a= (a1,...,ak,0,...,0,...),
+a′= (0,...,0,a′
+k+1,...,a′
+k+l,0,...),
+where the ai’s and the a′
+j’s are non-zero. As a+b=a′+b′, necessarily bk+1/ne}ationslash= 0.
+First subcase. l(a)orl(a′)>1. We can suppose that l(a)>1. Then aand(a1−
+1,a2,...,a k,0,...)are elements of S+Λbecause l(a)>1and their support are non disjoint.
+By the first case:
+xa,b=x(a1−1,a2,...,ak,0,...),b+(1,0,...).
+Moreover, the supports of (a1−1,a2,...,a k,0,...)and(a1−1,a2,...,ak,1,0,...)are not disjoint.
+Asbk+1>0,b+(1,0,...)−(0,...,0,1,0,...)belongs to S+Λ. By the first case:
+x(a1−1,a2,...,ak,0,...),b+(1,0,...)=x(a1−1,a2,...,ak,1,0,...),b+(1,0,...,0,−1,0,...).
+Finally, the support of (a1−1,a2,...,ak,1,0,...)anda′are not disjoint, so:
+x(a1−1,a2,...,ak,1,0,...),b+(1,0,...,0,−1,0,...)=xa′,b′.
+4Ifl(b)orl(b′)>1, the proof is similar, permuting the roles of (a,a′)and(b,b′).
+Second subcase. l(a) =l(a′) =l(b) =l(b′) = 1. Thenva⊗vb,va′⊗vb′∈V⊗V. By the
+homogeneity condition, xa,b=xa′,b′= 0.
+Hence, we put xa+b=xa,bfor alla,b∈ S+Λ: this does not depend of the choice of aandb.
+So:
+X=/summationdisplay
+c∈S+Λxc/summationdisplay
+a+b=c
+a,b∈S+Λva⊗vb=/summationdisplay
+c∈S+Λxc˜∆(vc),
+soX∈Im(˜∆). ✷
+Lemma 2 InS(V),Im(˜∆)∩(V⊗V) =S2(V).
+Proof.⊆. By cocommutativity of ˜∆.⊇. Ifv,w∈V, thenv⊗w+w⊗v=˜∆(vw).✷
+Lemma 3 LetWbe a subspace of S+(V), such that S+(V) =V⊕W. There exists a unique
+coalgebra endomorphism φofS(V), such that φ|V=IdVandφ
+/circleplusdisplay
+n≥2Sn(V)
+=W. Moreover,
+φis an automorphism.
+We shall denote from now:
+S≥2(V) =∞/circleplusdisplay
+n=2Sn(V).
+Proof. Existence. We denote by πWthe projection on Win the direct sum S(V) = (1)⊕V⊕
+W. We define inductively φ|Sn(V). Ifn= 0, it is defined by φ(1) = 1 . Ifn= 1, it is defined by
+φ|V=IdV. Let is assume that φis defined on the subcoalgebra Cn−1=K⊕V⊕...⊕Sn−1(V)
+withn≥2and let us define φonSn(V). Letva∈Sn(V). Then ˜∆(va)∈Cn−1⊗Cn−1, so
+(φ⊗φ)◦˜∆(va)is already defined. Moreover:
+(˜∆⊗Id)◦(φ⊗φ)◦˜∆(va) = (φ⊗φ⊗φ)◦(˜∆⊗Id)◦˜∆(va)
+= (φ⊗φ⊗φ)◦(Id⊗˜∆)◦˜∆(va)
+= (Id⊗˜∆)◦(φ⊗φ)◦˜∆(va).
+By lemma 1, (φ⊗φ)◦˜∆(va)∈Im(˜∆)+V⊗V. Moreover, as ˜∆is cocommutative, using lemma
+2:
+(φ⊗φ)◦˜∆(va)∈(Im(˜∆)+V⊗V)∩S2(S(V))
+∈Im(˜∆)+(V⊗V)∩S2(S(V))
+∈Im(˜∆)+S2(V)
+∈Im(˜∆).
+Letwa∈S+(V), such that (φ⊗φ)◦˜∆(va) =˜∆(wa). We put then φ(va) =πW(wa). So
+φ(va)∈W. Moreover, wa−πW(wa)∈V⊆Ker(˜∆), so:
+(φ⊗φ)◦∆(va) =φ(va)⊗φ(1)+φ(1)⊗φ(va)+(φ⊗φ)◦˜∆(va)
+=φ(va)⊗1+1⊗φ(va)+˜∆(wa)
+=φ(va)⊗1+1⊗φ(va)+˜∆(πW(wa))
+= ∆(φ(va)).
+Soφ|Cnis a coalgebra morphism.
+5Unicity. Let˜φbe another coalgebra endomorphism satisfying the required properties. Let
+us show that φ(va) =˜φ(va)by induction on n=l(a). Ifn= 0or1, this is immediate. Let us
+assume the result at all rank < n,n≥2. Then:
+˜∆(φ(va)−˜φ(va)) = (φ⊗φ−˜φ⊗˜φ)
+/summationdisplay
+b+c=a∈S+Λvb⊗vc
+= 0,
+by the induction hypothesis. So φ(va)−˜φ(va)∈Prim(S(V)) =V. Moreover, it belongs to
+φ(S≥2(V))+˜φ(S≥2(V))⊆W. AsV∩W= (0),φ(v1...vn) =˜φ(v1...vn).
+We have defined in this way an endomorphism φ, such that φ|V=IdVandφ(Sn(V))⊆W
+for alln≥2. We now show that φis an automorphism. Let us suppose that Ker(φ)/ne}ationslash= (0). Asφ
+is a coalgebra morphism, Ker(φ)is a non-zero coideal, so it contains primitive elements, th at is
+to say elements of V: impossible, as φ|Vis monic. So φis monic. Let us prove that va∈Im(φ)
+by induction on l(a). Ifl(a) = 0 or1, thenφ(va) =va. We can suppose that va=λv1...vk,
+whereλis a non-zero scalar. Then:
+˜∆k−1(va) =λ/summationdisplay
+σ∈Skvσ(1)⊗...⊗vσ(k)=λ/summationdisplay
+σ∈Skφ(vσ(1))⊗...⊗φ(vσ(k)) =˜∆k−1(φ(va)).
+Sova−φ(va)∈Ker(˜∆k−1) =K⊕V⊕...⊕Sk−1(V)⊆Im(φ)by the induction hypothesis. So
+va∈Im(φ)andφis epic. As a consequence, φ(S≥2(V)) =W. ✷
+1.2 Extension of a prelie product
+Definition 4 A (left) prelie algebra is a vector space g, with a product ⋆satisfying the
+following property: for all x,y,z∈g,
+(x⋆y)⋆z−x⋆(y ⋆z) = (y ⋆x)⋆z−y ⋆(x⋆z).
+Remark. A prelie algebra gis also a Lie algebra, with the bracket given by:
+[x,y] =x⋆y−y ⋆x.
+This bracket will be called the Lie bracket induced by the pre lie product.
+Let(g,⋆)be a prelie algebra. By [5] and [11], permuting left and right, the prelie product ⋆
+can be extended to the coalgebra S(g)in the following way: for all x,y∈g,P,Q,R∈S(g),
+
+
+1⋆P=P,
+(xP)⋆y=x⋆(P ⋆y)−(x⋆P)⋆y,
+P ⋆(QR) =/summationtext/parenleftbig
+P(1)⋆Q/parenrightbig/parenleftbig
+P(2)⋆R/parenrightbig
+.
+ThenS(g)is given an associative product ∗defined by P∗Q=/summationtextP(1)/parenleftbig
+P(2)⋆Q/parenrightbig
+. Moreover,
+(S(g),∗,∆)is isomorphic, as a Hopf algebra, to U(g). There exists a unique isomorphism ξ⋆of
+Hopf algebras:
+ξ⋆:/braceleftbiggU(g)−→(S(g),∗,∆)
+x∈g−→x∈g.
+Lemma 5 For alln≥1,g⋆Sn(g)⊆Sn(g).
+Proof. By induction on n. This is immediate for n= 1. Let us assume the result at rank
+n−1. LetP=yQ∈Sn(g), withy∈gandQ∈Sn−1(g). For all x∈g, asxis primitive:
+x⋆P= (x⋆y)Q+y(x⋆Q).
+Note that x⋆y∈gandx⋆Q∈Sn−1(g)by the induction hypothesis. So x⋆P∈Sn(g).✷
+6Proposition 6 Letgbe a prelie algebra. We denote S≥2(g) =/circleplusdisplay
+n≥2Sn(g). Then:
+1.S≥2(g)is a left ideal for ∗.
+2.S≥2(g)is a bilateral ideal for ∗if, and only if, ⋆is associative on g.
+Proof. 1. Letx∈gandQ∈Sn(g), withn≥2. Then, by lemma 5:
+x∗Q=xQ+x⋆Q∈Sn+1(V)+Sn(V).
+As a consequence, S≥2(g)is stable by left multiplication by an element of g. Asggenerates
+(S(g),∗)(because it is isomorphic to U(g)),S≥2(g)is a left ideal.
+2,⇐=. Let us first show that Sn(g)⋆g⊆S≥2(g)for alln≥2by induction on n. For
+n= 2, takex,y,z∈g. Then (xy)⋆ z=x ⋆(y ⋆ z)−(x ⋆ y)⋆ z= 0, as⋆is associative
+ong. Let us assume the result at rank n−1 (n≥3). Letx∈g,P∈Sn−1(V),y∈g.
+Then(xP)⋆ z=x ⋆(P ⋆ z)−(x ⋆ P)⋆ z. By the induction hypothesis, P ⋆ z∈S≥2(g). By
+lemma 5, x ⋆(P ⋆ z)∈S≥2(g). By lemma 5, x ⋆ P∈Sn−1(g). By the induction hypothesis,
+(x⋆P)⋆z∈S≥2(g). SoSn(g)⋆g⊆S≥2(g)for alln≥2.
+Let us now prove that S≥2(g)∗g⊆S≥2(g). LetP∈S≥2(g)andy∈g. We put ∆(P) =
+P⊗1+1⊗P+/summationtextP′⊗P′′. Then:
+P∗y=Py+P ⋆y+/summationdisplay
+P′(P′′⋆y).
+Note that Pyand/summationtextP′(p′′⋆ y)belong to S≥2(g). We already proved P ⋆ y∈S≥2(g). Asg
+generates (S(g),∗),S≥2(g)is a right ideal. By the first point, it is a bilateral ideal.
+2,=⇒. Let us assume that ⋆is not associative on g. There exists x,y,z∈g, such that
+x⋆(y ⋆z)−(x⋆y)⋆z/ne}ationslash= 0. Then:
+(xy)∗z=xyz+x(y ⋆z)+y(x⋆z)+(xy)⋆z
+=xyz+x(y ⋆z)+y(x⋆z)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+∈S≥2(g)+x⋆(y ⋆z)−(x⋆y)⋆z/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+∈V−{0},
+so(xy)∗z /∈S≥2(g), which is not a right ideal. ✷
+Definition 7 Letgbe a Lie algebra. We define:
+1.PL(g) ={⋆|⋆is a prelie product on ginducing the bracket of g}.
+2.LI(g) ={Ileft ideal of U(g)| U+(g) =g⊕I}.
+Proposition 8 There exists an application:
+Φg:/braceleftbiggPL(g)−→ LI (g)
+⋆−→ξ−1
+⋆(S≥2(g))
+Proof.Φg(⋆)is indeed an element of LI(g), asS≥2(g)is a left ideal of (S(g),∗)andξ⋆is an
+isomorphism of algebras. ✷
+71.3 Prelie product associated to a left ideal
+Proposition 9 LetI∈ LI(g). We denote by ̟gthe projection on gin the direct sum
+U(g) = (1)⊕g⊕I. Then the product ⋆defined by x⋆y=̟g(xy)is an element of PL(g). This
+defines an application Ψg:LI(g)−→ PL(g).
+Proof. Letx,y∈g. InU(g),xy−yx= [x,y]∈g, so:
+x⋆y−y ⋆x=̟g(xy−yx) =̟g([x,y]) = [x,y],
+so⋆induces the Lie bracket of g. It remains to prove that it is prelie. Let us fix a basis (vi)i∈Λ
+ofg. By the Poincaré-Birkhoff-Witt theorem, U(g)is isomorphic to S(g)as a coalgebra. Using
+lemma 3, there exists an isomorphism of coalgebras:
+φI:/braceleftbiggS(g)−→ U(g)
+v∈g−→g,
+such that φI(S≥2(g)) =I. We denote by (va)a∈SΛthe basis of U(g), image of the basis (va)a∈SΛ
+ofS(g). As a consequence:
+•A basis of Iis given by (va)a∈SΛ,l(a)≥2.
+•For alla∈ SΛ,∆(va) =/summationdisplay
+b+c=avb⊗vc.
+We putδj= (δi,j)i∈Ifor allj∈I(sovi=vδifor alli∈I) and, for all a,b∈ SΛ:
+vavb=/summationdisplay
+c∈SΛxc
+a,bvc.
+By definition, for all i,j∈I:
+vi⋆vj=/summationdisplay
+k∈Ixδk
+δi,δjvk.
+Let us prove the prelie relation for vi,vj,vk. It is obvious if i=j: let us assume that i/ne}ationslash=j.
+Then, in U(g):
+∆/parenleftig
+vivj−vδi+δj/parenrightig
+=/parenleftig
+vivj−vδi+δj/parenrightig
+⊗1+1⊗/parenleftig
+vivj−vδi+δj/parenrightig
++vi⊗vj+vj⊗vi−vδi⊗vδj−vδj⊗vδi
+=/parenleftig
+vivj−vδi+δj/parenrightig
+⊗1+1⊗/parenleftig
+vivj−vδi+δj/parenrightig
+.
+Sovivj−vδi+δj∈Prim(S(g)) =g. So:
+vivj=vδi+δj+/summationdisplay
+k∈Ivδk
+δi,δjvδk=vδi+δj+vi⋆vj.
+Hence:
+̟g(vi(vjvk)) =̟g/parenleftigg/summationdisplay
+c∈SΛcc
+δj,δkvivc/parenrightigg
+=̟g
+/summationdisplay
+c∈SΛ
+l(c)≥2cc
+δj,δkvivc
+/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+∈I,left ideal
++̟g(vi(vj⋆vk))
+= 0+vi⋆(vj⋆vk)
+=̟g((vivj)vk)
+=̟g(vδi+δjvk)+̟g((vi⋆vj)vk)
+=̟g(vδi+δjvk)+(vi⋆vj)⋆vk.
+8So:
+vi⋆(vj⋆vk)−(vi⋆vj)⋆vk=̟g(vδi+δjvk) =vj⋆(vi⋆vk)−(vj⋆vi)⋆vk.
+Hence, the product ⋆is prelie. ✷
+1.4 Prelie products on a Lie algebra
+Proposition 10 The applications ΦgandΨgare inverse bijections.
+Proof. Let⋆∈ PL(g). We put I= Φg(⋆)and•= Ψg(I). Letπgbe the canonical surjection
+onginS(g). Then, as ξ⋆(I) =S≥2(g),ξ⋆◦̟g=πg◦ξ⋆. So, for all x,y∈g:
+ξ⋆(x•y) =ξ⋆◦̟g(xy) =πg◦ξ⋆(xy) =πg(x∗y) =πg(xy+x⋆y) =x⋆y=ξ⋆(x⋆y).
+Asξ⋆is monic, x•y=x⋆x, soΨg◦Φg(⋆) =⋆.
+LetI∈ LI(g). We put ⋆= Ψg(I). We have to prove that ξ−1
+⋆(S≥2(g)) =I, that is to say
+ξ⋆(I) =S≥2(g). Because they are both complements of (1)⊕gandξ⋆is bijective, it is enough to
+proveξ⋆(I)⊆S≥2(g). With the preceding notations, let va∈I,l(a)≥2, and let us prove that
+ξ⋆(va)∈S≥2(g). Ifl(a) = 2, we put a=δi+δj. Then we saw that va=vivj−vi⋆vjifi/ne}ationslash=j. If
+i=j, in the same way, 2va=vivj−vi⋆vj. So, up to a non-zero mutiplicative constant λ:
+ξ⋆(va) =λ(vi∗vj−vi⋆vj) =λ(vivj+vi⋆vj−vi⋆vj) =λvivj∈S≥2(g).
+Ifl(a)≥3, we puta=a′+δifor a well-chosen i, withl(a′)≥2. Then, there exists a non-zero
+constant λsuch that, in U(g),˜∆(va) =λ(viva′). So, asIis a left ideal, va−λviva′∈g∩I= (0),
+sova=λviva′. Thenξ⋆(va) =vi∗φ(va′). By the induction hypothesis, φ(va′)belongs to S≥2(g),
+left ideal for ∗, sovabelongs to S≥2(g). ✷
+As a conclusion:
+Theorem 11 Letgbe a Lie algebra. There exists a prelie product on ginducing its Lie
+bracket, if, and only if, there exists a left ideal IofU(g)such that U+(g) =g⊕I. More precisely,
+there exists a bijection between the sets:
+• PL(g) ={⋆|⋆is a prelie product on ginducing the bracket of g}.
+• LI(g) ={Ileft ideal of U(g)| U+(g) =g⊕I}.
+It is given by Ψg:LI(g)−→ PL(g), associating to a left ideal Ithe prelie product defined by
+x⋆y=̟g(xy), where̟gis the canonical projection on gin the direct sum U+(g) =g⊕I. The
+inverse bijection is given by Φg:PL(g)−→ LI(g), associating to a prelie product ⋆the left ideal
+U(g)Vect(xy−x⋆y, x,y ∈g).
+Proof. It only remains to prove that Φg(⋆)is generated by the elements xy−x⋆y,x,y∈g.
+Using the isomorphism ξ⋆, it is equivalent to prove that the left ideal S≥2(V)of(S(V),∗)is
+generated by the elements x∗y−x⋆y,x,y∈g. By definition of ∗, for allx,y∈g:
+x∗y−x⋆y=xy+x⋆y−x⋆y=xy,
+so it is equivalent to prove that the left ideal S≥2(V)is generated by S2(V). Let us denote by
+Jthe left ideal generated by S2(V). AsS≥2(V)is a left ideal, J⊆S≥2(V). Letv1,...,vn∈g,
+withn≥2. Let us prove that v1...vn∈Jby induction on n. This is obvious if n= 2. Ifn≥3:
+v1∗(v2...vn) =v1...vn+v1⋆(v2...vn).
+By lemma 5, v1⋆(v2...vn)∈Sn−1(V). By the induction hypothesis, as n≥3,v1∗(v2...vn)
+andv1⋆(v2...vn)belong to J, sov1...vn∈J. As a conclusion, S≥2(V) =J. ✷
+9Corollary 12 Letgbe a Lie algebra. There exists an associative product on ginducing its
+Lie bracket, if, and only if, there exists a bilateral ideal IofU(g)such that U+(g) =g⊕I. More
+precisely, there exists a bijection between the sets:
+1.AS(g) ={⋆|⋆is an associative product on ginducing the bracket of g}.
+2.BI(g) ={Ibilateral ideal of U(g)/U+(g) =g⊕I}.
+It is given by Ψg:LI(g)−→ PL(g), associating to a left ideal Ithe associative product defined
+byx⋆y=̟g(xy), where̟gis the canonical projection on gin the direct sum U+(g) =g⊕I.
+Proof. It is enough to verify that Ψg(BI(g))⊆ AS(g)andΦg(AS(g))⊆ BI(g). Let
+⋆∈ AS(g). By proposition 6, S≥2(g)is a bilateral ideal, so is Φg(⋆) =ξ−1
+⋆(S≥2(g)). Let
+I∈ ASgr(g)). Theng≈ U+(g)/Iinherits an associative product, which is ⋆= Ψg(I). ✷
+1.5 Good and very good pointed modules
+Definition 13 Letgbe a Lie algebra.
+1. Apointedg-module is a couple (M,m), whereMis ag-module and m∈M.
+2. Let(M,m)be a pointed g-module. The application Υmis defined by:
+Υm:/braceleftbiggg−→M
+x−→x.m.
+3. A pointed g-module (M,m)isgoodif the following assertions hold:
+•Υmis injective.
+•Im(Υm)is a submodule of Mwhich does not contains m.
+•M/Im(Υm)≈K(trivialg-module) as a g-module.
+4. A pointed g-module (M,m)isvery good ifΥmis bijective.
+Note that all good pointed g-modules have the same dimension, so we can consider the set
+GM(g)of isomorphism classes of good pointed g-modules. Similarly, we can consider the set
+VGM(g)of very good pointed g-modules.
+Remark. If(M,m)is good, then it is cyclic, generated by m. Moreover, M/Im(Υm)is
+one-dimensional, trivial, generated by m=m+Im(Υm).
+Theorem 14 Letgbe a Lie algebra. The following application is a bijection:
+Θ :/braceleftbiggLI(g)−→ GM (g)
+I−→(U(g)/I,1).
+Proof. Let us first proove that Θis well-defined. As I∈ LI(g), as a vector space
+U(g)/I= (1)⊕g. For all x,y∈g,xy∈ U+(g), sox.y∈ U+(g)/I=ginU(g)/I, as
+I⊆ U+(g). As a conclusion, the subspace gofU(g)/Iis a submodule. Moreover, Υ1(x) =xfor
+allx∈g. SoΥ1is injective, and its image is the submodule g, so does not contain 1. Finally,
+(U(g)/I)/Im(Υ1)≈ U(g)/U+(g)≈Kas ag-module. So (U(g)/I,1)is a good pointed g-module.
+Let us consider:
+Θ′:/braceleftbiggGM(g)−→ LI (g)
+(M,m)−→Ann(m) ={x∈ U(g)|x.m= 0}.
+10Let us prove that Θ′is well-defined. Let (M,m)be a good pointed g-module. Then Ann(m)
+is a left-ideal of U(g). Letx∈Ann(m). Then x.m= 0, sox.m= 0inM/Im(Υm), so
+x.m=ε(x)m= 0. As a conclusion, ε(x) = 0 , soAnn(m)⊆ U+(g). Let us now show that
+U+(g) =g⊕Ann(m). First, if x∈g∩Ann(m), thenΥm(x) =x.m= 0. AsΥmis monic,
+x= 0. Ify∈ U+(g), theny.m=ε(y)m= 0inM/Im(Υm), soy.m∈Im(Υm): there exists
+x∈g, such that y.m=x.m. Hence, y−x∈Ann(m), soy=x+(y−x)∈g+Ann(m). Finally,
+Ann(m)∈ LI(g).
+Moreover, if (M,m)and(M′,m′)are isomorphic (that is to say there is an isomorphism of
+g-modules from MtoM′sendingmtom′), thenAnn(m) =Ann(m′). SoΘ′is well-defined.
+Let(M,m)be a good pointed g-module. Then Θ◦Θ′(M,m) = (U(g)/Ann(m),1)≈(M,m),
+asMis cyclic, generated by m. Let now I∈ LI(g). ThenΘ′◦Θ(I)is the annihilator of 1in
+U(g)/I, so is equal to I. As a conclusion, ΘandΘ′are inverse bijections. ✷
+Definition 15 The set GM′(g)is the set of isomorphism classes of couple ((M,m),V),
+where:
+•(M,m)is a good pointed g-module.
+•Vis a submodule of Msuch that M=V⊕Im(Υm).
+Proposition 16 For any Lie algebra g,GM′(g)is in bijection with VGM(g).
+Proof. Let((M,m),V)be an element of GM′(g). AsM/Im(Υm)≈K, the complement V
+ofIm(Υm)is trivial and one-dimensional. As m /∈Im(Υm),Vadmits a unique element m′=
+m+x.m, wherex∈g. Let us show that (Im(Υm),−x.m)is very good. Let m′′∈Im(Υm). There
+existsy∈g, such that y.m=m′′. Theny.m′=ε(y)m′= 0 =y.m+y.(x.m) =m′′+y.(x.m),
+som′′=y.(−x.m) = Υ−x.m(y). Hence, Υ−x.mis epic.
+Let us assume that Υ−x.m(y) = 0. Theny.m′=ε(y)m′= 0 =y.m+y.(x.m) =y.m= Υm(x).
+AsΥmis monic, y= 0, soΥ−x.mis also monic.
+Moreover, if ((M,m),V)≈((M′,m′),V′), then(Im(Υm),−x.m)and(Im(Υm′,−x′.m′)are
+isomorphic, so the following application is well-defined:
+Λ :/braceleftbiggGM′(g)−→ VGM (g)
+((M,m),V)−→(Im(Υm),−x.m).
+Let now (M,m)be a very good pointed g-module. Let us prove that ((K⊕M,1+m),K)
+is an element of GM′(g). First, for all x∈g,Υ1+m(x) =ε(x)1 +x+m= 0 + Υ m(x), so
+Υ1+m= Υm. As a consequence, Im(Υ1+m) =Mis a submodule which does not contains 1+m,
+Υ1+mis monic, and Kis a complement of Im(Υ1+m). Hence, ((K⊕M,1+m),K)∈ GM′(g).
+Moreover, if (M,m)≈(M′,m′), then((K⊕M,1 +m),K)≈((K⊕M′,1 +m′),K), so the
+following application is well-defined:
+Λ′:/braceleftbiggVGM(g)−→ GM′(g)
+(M,m)−→((K⊕M,1+m),K).
+Let(M,m)∈ VGM(g). ThenΛ◦Λ′((M,m)) = (M,m). Let((M,m),V)∈ GM′(g). Then
+the following application is an isomorphism of pointed g-modules and sends KtoV:
+/braceleftbigg(K⊕Im(Υm),1−x.m)−→M
+λ+m′′−→λm′+m′′.
+SoΛandΛ′are inverse bijections. ✷
+112 Are the complex simple Lie algebras prelie?
+The aim of this section is to prove that the complex finite-dim ensional simple Lie algebras are
+not prelie. In this section, the base field is the field of compl exC. We shall use the notations
+of [4] in the whole section. In particular, if gis a simple complex finite-dimensional Lie algebra,
+Γ(a1,...,an)is the simple g-module of hightest weight (a1,...,an).
+2.1 General results
+Proposition 17 Letgbe a finite-dimensional semi-simple Lie algebra. Then there exists a
+prelie product on ginducing its bracket if, and only if, there exists a very good pointedg-module.
+Proof. As the category of finite-dimensional modules over gis semi-simple, for any good
+pointedg-module (M,m), there exists a Vsuch that ((M,m),V)∈ GM(g). Using the bijections
+of the preceding section, PL(g)is non-empty if, and only if, LI(g)is non-empty, if, and only
+if, the set GM(g)is non-empty, if, and only if, the set GM′(g)is non-empty, if, and only if,
+VGM(g)is non-empty. ✷
+Letgbe a finite-dimensional complex simple Lie algebra. In order to prove that gis not
+prelie, we have to prove that ghas no very good pointed modules.
+Lemma 18 Let(M,m)be a very good g-module and let M′be a submodule of M. There
+existsm′∈M, such that the following application is epic:
+Υm′:/braceleftbiggM′−→M′
+x−→x.m′.
+Proof. Asgis semi-simple, let M′′be a complement of M′inMand letm=m′+m′′the
+decomposition of minM=M′⊕M′′. Letu∈M′. AsΥmis bijective, there exists a (unique)
+x∈g, such that Υm(x) =u. Sou+0 =x.m′+xm′′in the direct sum M=M′⊕M′′. Hence,
+Υm′(x) =uandΥm′is epic. ✷
+Corollary 19 Let(M,m)be a very good g-module. Then Mis not isomorphic to (g,ad)
+and its trivial component is (0).
+Proof. First, note that for all x∈g,((g,ad),x)is not very good: it is obvious if x= 0and
+ifx/ne}ationslash= 0,Υx(x) = [x,x] = 0, soΥxis not monic.
+If(M,m)is very good, let us consider its trivial component M′. By the preceding lemma,
+there exists m′∈M′, such that Υm′:M′−→M′is epic. As M′is trivial, Υm′= 0, and finally
+M′= (0). ✷
+2.2 Representations of small dimension
+We first give the representations of gwith dimension smaller than the dimension of g. We shall
+say that a g-module Missmall if:
+•Mis simple.
+•The dimension of Mis smaller than the dimension of g.
+•Mis not trivial and not isomorphic to (g,ad).
+Remark. By corollary 19, any very good module is the direct sum of small modules.
+Letnbe the rank of gand letΓ(a1,...,an)be the simple g-module of highest weight (a1,...,a n).
+The dimension of Γ(a1,...,an)is given in [4] in chapter 15 for sln+1, (formula 15.17), in chapter 24
+12forsp2n,so2nandso2n+1(exercises 24.20, 24.30 and 24.42). It turns out from these f ormulas
+that ifb1≤a1,...bn≤an, thendim(Γ(b1,...,bn))≤dim(Γ(a1,...,an)), with equality if, and only
+if,(b1,...,bn) = (a1,...,a n). Direct computations of dim(Γ(0,...,0,3,0,...,0))then shows that it is
+enough to compute the dimensions of Γ(a1,...,an)with alla1≤2. With the help of a computer,
+we obtain:
+g dim(g) small modules V dim(V)
+sln, n≥9 n2−1 Γ(1,0,...,0),Γ(0,...,0,1) n
+Γ(0,1,0,...,0),Γ(0,...,0,1,0) n(n−1)/2
+Γ(2,0,...,0),Γ(0,...,0,2) (n+1)/2
+sln,3≤n≤8n2−1Γ(0,...,0,1,...,0),the1in position i/parenleftbign
+i/parenrightbig
+Γ(2,0,...,0),Γ(0,...,0,2) (n+1)/2
+sl2 3 Γ1 2
+sp2n,n= 2orn≥4n2n+1 Γ(1,0,...,0) 2n
+Γ(0,1,0,...,0) 2n+1(n−1)
+sp6 21 Γ(1,0,0) 6
+Γ(0,1,0),Γ(0,0,1) 14
+so2n, n≥8n(2n−1) Γ(1,0,...,0) 2n
+so2n,4≤n≤7n(2n−1) Γ(1,0,...,0) 2n
+Γ(0,0,1,0,...,0),Γ(0,...,0,1) 2n−1
+so6 15 Γ(1,0,0) 6
+Γ(0,1,0),Γ(0,0,1) 4
+Γ(0,2,0),Γ(0,0,2) 10
+so2n+1, n≥7n2n+1 Γ(1,0,...,0) 2n+1
+so2n+1,2≤n≤6n2n+1 Γ(1,0,...,0) 2n+1
+Γ(0,...,0,1) 2n
+e6 78 Γ(1,0,0,0,0,0) 27
+e7 133 Γ(1,0,0,0,0,0,0) 56
+e8 78 × ×
+f4 52 Γ(1,0,0,0) 26
+g2 14 Γ(1,0) 7
+2.3 Proof in the generic cases
+Proposition 20 Letgbesln, withn/ne}ationslash= 6, orso2n, withn≥2, orsp2n, withn≥2, oren,
+with6≤n≤8. Thengis not prelie.
+Proof. Letgbe a simple finite-dimensional Lie algebra, with a prelie pro duct inducing its
+Lie bracket. From the preceding results, ghas a very good pointed g-module (M,m), and the
+submodules appearing in the decomposition of Minto simples modules are all small. Let us
+show that it is not possible in these different cases.
+First case. Ifg=sln, withn≥9, then the decomposition of a very good pointed g-module
+into simples would have asubmodules of dimension n,bsubmodules of dimensions n(n−1)/2
+andcsubmodules of dimension n(n+1)/2. So:
+n2−1 =an+bn(n−1)
+2+cn(n+1)
+2.
+Let us assume that nhas a prime factor p/ne}ationslash= 2. Thenp|n,p|n(n±1)and2|n(n±1), so
+p|n(n±1)
+2. As a consequence, p|n2−1, sop|1: this is absurd. So n= 2kfor a certain k≥4,
+asn≥9. Then:
+a2k+b2k−1(2k−1)+c2k−1(2k+1) = 22k−1,
+13so2k−1|22k−1andk= 1: contradiction.
+Second case. Similarly, if g=sp2nwithn/ne}ationslash= 3, we would have a,b∈Nsuch that a2n+
+b2n+1(n−1) =n2n+1. Ifb≥2, thenn2n+1≥22n+1(n−1), so2n+1(n−2)≤0and
+n≤2: contradiction. So b= 0or1. Ifb= 0, thena=n+1
+2: absurd. If b= 1, thena= 1+1
+2n:
+absurd.
+Third case. Ifg=so2n, withn≥8, then we would have 2n|n(2n−1), son−1
+2∈N: absurd.
+Other cases. Let us sum up the possible dimensions of the small modules in a n array:
+gdimensions of the small modules dim(g)
+sl2 2 3
+sl3 3,6 8
+sl4 4,6,10 15
+sl5 5,10,15 24
+sl7 7,21,28,35 48
+sl8 8,28,36,56 63
+so6 4,6,10 15
+so8 8 28
+so10 10,16 45
+so12 12,32 66
+so14 14,64 91
+sp6 6,14 21
+e6 27 78
+e7 56 133
+e8 × 248
+In all cases except so6, (there is a prime integer which divides all the dimensions o f the small
+submodules but not the dimension of g), or (there is a unique dimension of small modules and it
+does not divide the dimension of g), or (there is no small modules). So gis not prelie. For so12,
+it comes from the fact that 4divides12and32but divides not 66.
+So none of these Lie algebras is prelie. ✷
+2.4 Case of so2n+1
+We assume that n≥2. Let us recall that:
+so2n+1=
+
+
+A B −tF
+C−tA−tE
+E F 0
+|B,Cskew symmetric
+
+,
+whereA,B,C aren×nmatrices, E,Fare1×nmatrices.
+Lemma 21 Let(M,m)be a very good pointed module over g=so2n+1. ThenMis isomor-
+phic toM2n+1,n(C)as ag-module, with the action of ggiven by the (left) matricial product.
+Proof. Let us first assume that n≥7. Thenghas a unique small module of dimension 2n+1,
+which is the standard representation C2n+1. So ifMis a direct sum of copies of this module;
+comparing the dimension, there are necessarily ncopies of this module, so M≈M2n+1,n(C).
+14Ifn≤6, let us sum up the possible dimensions of the small modules in an array:
+gdimensions of the small modules dim(g)
+so5 5,4 10
+so7 7,8 21
+so9 9,16 36
+so11 11,32 55
+so13 13,64 78
+In all these cases, the only possible decomposition of Misncopies of the standard representation
+of dimension 2n+1, so the conclusion also holds. ✷
+The elements of M2n+1,n(C)will be written as/parenleftigX
+Y
+z/parenrightig
+, whereX,Y∈Mn(C)andz∈M1,n(C).
+Proposition 22 For any m=/parenleftigX
+Y
+z/parenrightig
+∈M2n+1,n(C),(M2n+1,n(C),m)is not very good.
+Proof. Let us denote:
+M={A∈M2n+1,n((C)|(M2n+1,n(C),A)is very good }.
+Let us recall that:
+SO2n+1=
+
+B∈M2n+1,2n+1(C)|tB
+0I0
+I0 0
+0 0I
+B=
+0I0
+I0 0
+0 0I
+
+
+.
+It is well-known that if B∈SO2n+1andA∈so2n+1, thenBAB−1∈so2n+1.
+First step. Let us prove that m∈ M if, and only if, Bm∈ M for allB∈SO2n+1. Let us
+assume that m∈ M and letB∈SO2n+1. Let us take A∈Ker(ΥBm). ThenABm= 0, so
+B−1ABm= 0andB−1AB∈Ker(Υm). Asm∈ M,Υmis bijective, so A= 0:ΥBmis monic.
+AsMandso2n+1have the same dimension, ΥBmis bijective.
+Second step. Let us prove that m∈ M if, and only if, mP∈ M for allP∈GL(n). Indeed,
+m−→mPis an isomorphism of g-modules for all P∈GL(n).
+Third step . Let us prove that/parenleftigX
+Y
+z/parenrightig
+∈ Mif, and only if,/parenleftbigg
+QXP
+tQ−1YP
+zP/parenrightbigg
+∈ Mfor allP,Q∈GL(n).
+Indeed, if/parenleftigX
+Y
+z/parenrightig
+∈ Mand ifP,Q∈GL(n), by the first and second steps:
+
+Q0 0
+0tQ−10
+0 0 1
+
+X
+Y
+z
+P∈ M,
+as the left-multiplying matrix is an element of SO2n+1.
+Fourth step . Let/parenleftigX
+Y
+z/parenrightig
+∈ M, let us prove that Xis invertible. If Xis not invertible, for a
+good choice of PandQ,QXP has its first row and first column equal to 0. By the third step,
+m′=/parenleftbigg
+QXP
+tQ−1YP
+zP/parenrightbigg
+∈ M, butΥm′(En+1,2−En+2,1) = 0, whereEi,jis the elementary matrix with
+only a1in position (i,j): this is a contradiction. So Xis invertible.
+Last step. Let us assume that Mis not empty and let m=/parenleftigX
+Y
+z/parenrightig
+∈ M. ThenXis invertible.
+For a good choice of PandQ, we obtain an element m′=/parenleftigIn
+Y′
+z′/parenrightig
+∈ M. Let us then choose a
+15non-zero skew-symmetric matrix B∈Mn(C)(this exists as n≥2), then the following element
+is inso2n+1:
+A=
+−BY B 0
+tYBY−tYB0
+0 0 0
+∈so2n+1.
+An easy computation shows that A.m′= 0, soA∈Ker(Υm′): contradiction, m′/∈ M. SoMis
+empty. ✷
+Corollary 23 For alln≥2,so2n+1is not prelie.
+2.5 Proof for sl6andg2
+The Lie algebra sl6has dimension 35, and the possible dimensions of its small modules are 6,15,
+20. So if(M,m)is a very good pointed module, Mis the direct sum of a simple of dimension 15
+and a simple of dimension 20. As there is only one simple of dimension 20, that is to say Λ3(V)
+whereVis the standard representation, Mcontains Λ3(V). By lemma 18, in order to prove that
+sl6is not prelie, it is enough to prove that for any m∈Λ3(V),Υmis not epic. This essentially
+consists to show that the rank of a certain 20×35matrix is not 20and this can be done directly
+using MuPAD pro 4, see section 4.2.
+The proof for g2is similar: if (M,m)is a very good module, then M≈V⊕V, whereVis
+the only small module of g, that is to say its standard representation. So M≈M7,2(C)as a
+g-module. It remains to show that for any m,(M7,2(C),m)is not very good. This essentially
+consists to show that a certain 14×14matrix is not invertible and this can be done directly
+using MuPAD pro 4, see section 4.3.
+The proof for f4would be similar: if (M,m)is a very good module, then M≈V⊕V, where
+Vis the only small module of g, that is to say its standard representation. So M≈M26,2(C)
+as af4-module. It would remain to show that for any m,(M26,2(C),m)is not very good. This
+would consist to show that a certain 52×52matrix is not invertible.
+We finally prove:
+Theorem 24 Letgbe a simple, finite-dimensional complex Lie algebra. If it is not isomor-
+phic tof4, it is not prelie.
+We conjecture:
+Theorem 25 Letgbe a simple, finite-dimensional complex Lie algebra. Then it is not prelie.
+Remark. As a corollary, we obtain that gis not associative. This result is also proved
+in a different way in [7], with the help of compatible products . Indeed, we have the following
+equivalences:
+the prelie product ⋆is compatible
+⇐⇒ ∀x,y,z∈g,[x,y ⋆z] = [x,y]⋆z+y ⋆[x,z]
+⇐⇒ ∀x,y,z∈g,x⋆(y ⋆z)−(y ⋆z)⋆x−(x⋆y)⋆z+(y ⋆x)⋆z−y ⋆(x⋆z)+y ⋆(z ⋆x)
+⇐⇒ ∀x,y,z∈g,−(y ⋆z)⋆x−+y⋆(z ⋆x) = 0
+⇐⇒⋆is associative.
+As from [7], the only admissible associative product on gis0,gis not associative.
+163 Dendriform products on cofree coalgebra
+3.1 Preliminaries and results on tensor coalgebras
+LetVbe a vector space. The tensor algebra T(V)has a coassociative coproduct, given for all
+v1,...,vn∈Vby:
+∆(v1...vn) =n/summationdisplay
+i=0v1...vi⊗vi+1...vn.
+Let us recall the following facts:
+1. Let us fix a basis (vi)i∈IofV. We define TIas the set of words win letters the elements
+ofI. For an element w=i1...ik∈ SΛ, we put l(w) =k, and:
+vw=/productdisplay
+i∈Ivi1...vik.
+Then(vw)w∈TIis a basis of T(V), and the coproduct is given by:
+∆(vw) =/summationdisplay
+w1w2=wvw1⊗vw2.
+2. LetT+(V)be the augmentation ideal of T(V). It is given a coassociative, non counitary
+coproduct ˜∆defined by ˜∆(x) = ∆(x)−x⊗1−1⊗xfor allx∈T+(V). In other terms,
+puttingT+I=TI−{∅} ,(vw)w∈T+Iis a basis of T+(V)and:
+˜∆(vw) =/summationdisplay
+w1w2=w
+w1,w2∈T+Ivw1⊗vw2.
+3. Let˜∆(n):T+(V)−→T+(V)⊗(n+1)be then-th iterated coproduct of T+(V). Then:
+Ker/parenleftig
+˜∆(n)/parenrightig
+=n/circleplusdisplay
+k=1Tk(V).
+In particular, Prim(T(V)) =V.
+4.(Tn(V))n∈Nis a gradation of the coalgebra T(V).
+Lemma 26 InT+(V)⊗T+(V),Ker(˜∆⊗Id−Id⊗˜∆) =Im(˜∆).
+Proof.⊇comes from the coassociativity of ˜∆.
+⊆: let us consider X=/summationdisplay
+w1,w2∈T+Ixw1,w2vw1⊗vw2∈Ker(˜∆⊗Id−Id⊗˜∆) =Im(˜∆). Then:
+(˜∆⊗Id)◦˜∆(X) =/summationdisplay
+w1,w2,w3∈T+Ixw1w2,w3vw1⊗vw2⊗vw3
+= (Id⊗˜∆)◦˜∆(X) =/summationdisplay
+w1,w2,w3∈T+Ixw1,w2w3vw1⊗vw2⊗vw3.
+So, for all w1,w3∈ T+I,w2∈ TI,xw1w2,w3=xw1,w2w3. In particular, xi1...ik,j1...jk=xi1,i2...ikj1...jl
+for alli1,...,ik,j1,...,jl∈I. We denote by xi1...ikj1...jlthis common value. Then:
+∆(X) =/summationdisplay
+w∈T+Ixw/summationdisplay
+w1w2=w
+w1,w2∈SΛ+vw1⊗vw2=/summationdisplay
+w∈T+Ixw˜∆(xw).
+SoX∈Im(˜∆). ✷
+17Lemma 27 LetWbe a subspace of T+(V), such that T(V) = (1)⊕V⊕W. There exists a
+coalgebra endomorphism φofT(V), such that φ|V=IdVandφ
+/circleplusdisplay
+n≥2Tn(V)
+=W. Moreover,
+φis an automorphism.
+Proof. Similar to the proof of lemma 3. ✷
+Corollary 28 LetCbe a cofree coalgebra and let W⊂C+, such that C= (1)⊕Prim(C)⊕W.
+There exists a unique coalgebra isomorphism φ:T(Prim(C))−→Csuch that φ|Prim(C)=
+IdPrim(C)andφ(T≥2(Prim(C)) =W.
+3.2 Left ideal associated to a dendriform Hopf algebra
+LetAbe a dendriform Hopf algebra [8, 10], that is to say the produc t (denoted by ∗) ofAcan
+be split on A+as∗=≺+≻, with:
+1. For all x,y,z∈A+:
+(x≺y)≺z=x≺(y∗z), (1)
+(x≻y)≺z=x≻(y≺z), (2)
+(x∗y)≻z= (x≻y)≻z. (3)
+2. For all a,b∈A+:
+˜∆(a≺b) =a′∗b′⊗a′′≺b′′+a′∗b⊗a′′+b′⊗a≺b′′+a′⊗a′′≺b+b⊗a,(4)
+˜∆(a≻b) =a′∗b′⊗a′′≻b′′+a∗b′⊗b′′+b′⊗a≻b′′+a′⊗a′′≻b+a⊗b.(5)
+We used the following notations: A+is the augmentation ideal of Aand˜∆ :A+−→
+A+⊗A+is the coassociative coproduct defined by ˜∆(a) = ∆(a)−a⊗1−1⊗afor all
+a∈A+.
+Proposition 29 LetAbe a dendriform Hopf algebra. Then the following applicatio n is a
+monomorphism of coalgebras:
+ΘA:/braceleftbiggT(Prim(A))−→A
+v1...vn−→vn≺(vn−1≺(...≺(v2≺v1)...)
+Moreover, if Ais connected as a coalgebra, then ΘAis an isomorphism, so Ais a cofree coalgebra.
+Proof. For allv1,...,vn∈Prim(A), we put:
+ω(v1,...,vn) =vn≺(vn−1≺(...≺(v2≺v1)...).
+An easy induction using (4) proves that:
+∆(ω(v1,...,vn)) =n/summationdisplay
+i=0ω(v1,...,vi)⊗ω(vi+1,...,vn).
+SoΘAis a morphism of coalgebras. Let us prove that this morphism i s injective. If not, its kernel
+would contain primitive elements of T(Prim(A)), that is to say elements of Prim(A): absurd.
+IfAis connected, this morphism is surjective: let us take x∈A, let us prove that x∈Im(Θ).
+AsAis connected, for all x∈A, there exists n≥1such that ˜∆(n)(x) = 0. Let us proceed by
+induction on n. Ifn= 1, thenx∈g⊆Im(Θ). Ifn≥2, then˜∆(n−1)(x)∈g⊗n. Let us put:
+˜∆(n−1)(x) =/summationdisplay
+i1,...,in∈Iai1...invi1⊗...⊗vin.
+18Then:
+˜∆(n−1)(x) =˜∆(n−1)
+/summationdisplay
+i1,...,in∈Iai1...invi1...in
+.
+By the induction hypothesis, x−/summationdisplay
+i1,...,in∈Iai1...invi1...in∈Im(Θ), sox∈Im(Θ). As a conclusion,
+Θis an isomorphism, so (vw)w∈TIis a basis of A. ✷
+Proposition 30 LetAbe a dendriform Hopf algebra, connected as a coalgebra. Let u s put
+A≺2=Prim(A)≺A+. Then:
+1.A+=Prim(A)⊕A≺2.
+2.A≺2=A+≺A+.
+3.A≺2is a left ideal of A.
+Proof.
+1. Using ΘA:
+ΘA(Prim(A)) =Prim(A),ΘA
+/circleplusdisplay
+n≥2Prim(A)⊗n
+=A≺2,ΘA
+/circleplusdisplay
+n≥1Prim(A)⊗n
+=A+.
+AsΘAis an isomorphism, we obtain the result.
+2.⊆. AsPrim(A)⊆A+,A≺2⊆A+≺A+.
+⊇. Note that A+=Vect(ω(v1,...,vn)|n≥1,v1,...,vn∈Prim(A)). Letx=ω(v1,...,vm)
+andy=ω(w1,...,w n)∈A+. We put x′=ω(v1,...,vm−1). Then, by (1):
+x≺y= (vm≺x′)≺y=vm≺(x′∗y).
+Asx′∗y∈A+,x≺y∈A≺2.
+3. It is enough to prove that A+∗A≺2⊆A≺2. Letx,y,z∈A+.
+x∗(y≺z) =x≺(y≺z)+(x≻y)≺z∈A+≺A+=A≺2.
+SoA≺2is a left ideal of A.
+✷
+3.3 Dendriform products on a tensorial coalgebra
+Let us now consider an associative product ∗on the coalgebra T(V), such that:
+1.(T(V),∗,∆)is a Hopf algebra.
+2.T(V)≥2=/circleplusdisplay
+n≥2Tn(V)is a left ideal of (T(V),∗).
+Proposition 31 We define a product ≺onT+(V)in the following way: for all v,v1,...,vn∈
+V,w∈T+(V),/braceleftbiggv≺w=wv,
+(v1...vn)≺w= ((v1...vn−1)∗w)vn.
+We also put ≻=∗− ≺. Then(T(V),≺,≻,˜∆)is a dendriform Hopf algebra.
+19Proof. Let us first prove (1). Let u1,...,u k,v1,...,vl,w1,...,w m∈V.
+(u1...uk≺v1...vl)≺w1...wm= ((u1...uk−1∗v1...vl)uk)≺w1...wm
+= (u1...uk−1∗v1...vl∗w1...wm)uk
+=u1...uk≺(v1...vl∗w1...wm).
+Let us now prove (4). We take a=u1...uk,b=v1...vl∈T+(V). We denote ˜a=u1...uk−1.
+˜∆(a≺b) =˜∆((˜a∗b)uk)
+= ˜a∗b⊗uk+˜∆(˜a∗b)(1⊗uk)
+= ˜a∗b⊗uk+˜a⊗buk+b⊗˜auk+˜a′∗b⊗˜a′′uk
++˜a′⊗(˜a′′∗b)uk+˜a∗b′⊗b′′uk+b′⊗(˜a∗b′′)uk+˜a′∗b′⊗(˜a′′∗b′′)uk.
+Moreover, using (1):
+a′∗b′⊗a′′≺b′′= ˜a∗b′⊗b′′uk+˜a′∗b′⊗(˜a′′∗b′′)uk,
+a′∗b⊗a′′= ˜a∗b⊗uk+˜a′∗b⊗˜a′′uk,
+b′⊗a≺b′′=b′⊗(˜a∗b′′)uk,
+a′⊗a′′≺b= ˜a⊗buk+˜a′⊗(˜a′′∗b)uk,
+b⊗a=b⊗˜auk.
+So (4) is satisfied. As T(V)is a Hopf algebra, (4)+(5) is satisfied, so (5) also is.
+Let us prove (2). For all x,y,z∈T+(V), we put φ(x,y,z) = (x≻y)≺z−x≻(y≺z). A
+direct computation using (4) and (5) shows that:
+˜∆(φ(x,y,z)) =x′y′z′⊗Φ(x′′,y′′,z′′)+y′z′⊗Φ(x,y′′,z′′)+x′z′⊗Φ(x′′,y,z′′)
++x′y′⊗Φ(x′′,y′′,z)+z′⊗Φ(x,y,z′′)+y′⊗Φ(x,y′′,z)+x′⊗Φ(x′′,y,z).
+Moreover:
+φ(x,y,z) = (x≻y)≺z−x∗(y≺z)+x≺(y≺z).
+By definition of ≺,(x≻y)≺zandx≺(y≺z)∈T≥2(V). In the same way, y≺z∈T≥2(V),
+left ideal of T(V), sox∗(y≺z)∈T≥2(V). finally, φ(x,y,z)∈T≥2(V).
+Let us now prove that φ(x,y,z) = 0 by induction on n=l(x) +l(y) +l(z). Ifn= 3,
+thenx,y,z∈V, so are primitive. So ˜∆(φ(x,y,z)) = 0 , andφ(x,y,z)∈V. By the preced-
+ing point, φ(x,y,z)∈V∩T≥2(V) = (0) , soφ(x,y,z) = 0 . Let us assume the result at all
+rank< n. By the induction hypothesis applied to x′,y′,z′and others, ˜∆(φ(x,y,z)) = 0 , so
+φ(x,y,z)∈V∩T≥2(V) = (0) . Hence, (2) is satisfied. As ∗is associative, (1)+(2)+(3) is satis-
+fied, so (3) also is. ✷
+Remark. In the dendriform Hopf algebra T(V),T(V)≺2=T(V)≥2.
+By the dendriform Cartier-Quillen-Milnor-Moore theorem, T(V)is now the dendriform en-
+veloping algebra of the brace algebra V=Prim(T(V)). By [12], the brace structure on V
+induced by the dendriform structure of Ais given, for all a1,...,a n∈V, by:
+/an}bracketle{ta1,...,a n/an}bracketri}ht
+=n−1/summationdisplay
+i=0(−1)n−1−i(a1≺(a2≺(...≺ai)...)≻an≺(...(ai+1≻ai+2)≻...)≻an−1)
+=n−1/summationdisplay
+i=0(−1)n−1−i(a1...ai)≻an≺(...(ai+1≻ai+2)≻...)≻an−1)
+=πV/parenleftiggn−1/summationdisplay
+i=0(−1)n−1−i(a1...ai)≻an≺(...(ai+1≻ai+2)≻...)≻an−1)/parenrightigg
+,
+20where we denote by πVthe canonical projection on VinT(V). We obtain,as T(V)≺2=Ker(πV):
+/an}bracketle{ta1,...,an/an}bracketri}ht=πV((a1...an−1)≻an)
+=πV((a1...an−1)∗an)−πV((a1...an−1)≺an)
+=πV((a1...an−1)∗an).
+In other terms, identifying VandT+(V)/T(V)≥2,Vbecomes a left (T(V),∗)-module, and the
+brace structure of Vis given by this module structure.
+3.4 Dendriform structures on a cofree coalgebra
+Theorem 32 LetAbe a cofree coalgebra. We define:
+1.DD(A) ={(≺,≻)|(A,≺,≻,∆)is a dendriform Hopf algebra }.
+2.LI(A) =/braceleftbigg
+(∗,I)|(A,∗,∆)is a Hopf algebra and
+Iis a left ideal of Asuch that A+=Prim(A)⊕I/bracerightbigg
+.
+There is a bijection between these two sets, given by:
+ΦA:/braceleftbiggDD(A)−→ LI (A)
+(≺,≻)−→(≺+≻,A+≺A+).
+Proof. By proposition 30, ΦAis well-defined. We now define the inverse bijection ΨA. Let
+(∗,I)∈ LI(A). In order to lighten the notation, we put V=Prim(A). By corollary 28, there
+exists a isomorphism of coalgebras φI:T(V)−→A, such that φI(v) =vfor allv∈Vand
+φI(T(V)≥2) =I. Let˜∗be the product on T(V), making φIan isomorphism of Hopf algebras.
+ThenT(V)≥2is a left ideal of T(V). By proposition 31, there exists a dendriform structure
+(T(V),˜≺,˜≻)onT(V). Let(A,≺,≻)be the dendriform Hopf algebra structure on A, making
+φIan isomorphism of dendriform Hopf algebras. We then put ΨA(∗,I) = (≺,≻). Note that
+≺+≻=∗, as˜≺+˜≻=˜∗.
+Let us show that ΦA◦ΨA=IdLI(A). Let(∗,I)∈ LI(A). We put ΦA◦ΨA(∗,I) = (∗′,I′).
+Then, as ΦI: (T(V),˜≺,˜≻)−→(T(V),≺,≻)is an isomorphism of dendriform algebras:
+I= ΦI(T(V)≥2) =φI(T(V)˜≺2) =A≺2.
+By definition of ΦA,I′=A≺2=I. Moreover, ∗′=≺+≻=∗.
+Let us show that ΨA◦ΦA=IdDD(A). Let(≺,≻)∈ DD(A). We put ΨA◦ΦA(≺,≻) = (≺′,≻′),
+andΦA(≺,≻) = (∗,I). AsφI: (T(V),˜≺,˜≻)−→(A,≺′,≻′)is an isomorphism of dendriform
+algebras, we have to prove that φI: (T(V),˜≺,˜≻)−→(A,≺,≻)is an isomorphism of dendriform
+algebras. Let a=u1...uk,b=v1...vl∈T+(V). First:
+φI(a˜≺b)−φI(a)≺φI(b)∈φI(T≥2(V))+A+≺A+=I+I=I.
+Let us prove that φI(a˜≺b) =φI(a)≺φI(b)by induction on k. Fork= 1, let us proceed by
+induction on l. Forl= 1,φI(a˜≺b) =φI(v1u1). Moreover, in A:
+˜∆(φI(u1)≺φI(v1)) =˜∆(u1≺v1)
+=v1⊗u1,
+˜∆(φI(u1˜≺v1)) = ( φI⊗φI)◦˜∆(u1≺v1)
+=φI(v1)⊗φI(u1)
+=v1⊗u1.
+21SoφI(u1˜≺v1)−φI(u1)≺φI(v1)∈Prim(A)∩I= (0). Let us suppose the result for all l′< l.
+Then, by the induction hypothesis applied to b′′:
+˜∆(φI(u1˜≺b)) =˜∆(φI(bu1))
+= (φI⊗φI)(b⊗u1+b′⊗u1˜≺b′′)
+=φI(b)⊗φI(u1)+φI(b)′⊗φI(u1)˜φI(b′′)
+=˜∆(φI(u1)≺φI(b)).
+SoφI(u1˜≺b))−φI(u1)≺φI(b)∈Prim(A)∩I= (0). This prove the result for k= 1. Let us
+assume the result at all rank k′< k. We put u1...uk−1= ˜a. Then, using the first step:
+φI(a˜≺b) =φI(uk˜≺(˜a˜∗b))
+=φI(uk)≺(φI(˜a)∗φI(˜b))
+= (φI(uk)≺φI(˜a))≺φI(˜b)
+= (φI(uk≺˜a))≺φI(˜b)
+=φI(a)≺φI(˜b).
+SoΨA◦ΦA=IdDD(A). ✷
+4 MuPAD computations
+We here give the different MuPAD procedures we used in this tex t.
+4.1 Dimensions of simple modules
+The following procedures compute the dimension of the simpl e module Γ(a1,...,an)forsln,sp2n,
+so2nandso2n+1.
+dimsl:=proc(a)
+local n,res;
+begin
+n:=nops(a)+1;
+res:=product(product((sum(a[k],k=i..j-1)+j-i)/(j-i) ,j=i+1..n),i=1..n-1);
+return(res);
+end_proc;
+dimsp:=proc(a)
+local n,k,s,l,res;
+begin
+n:=nops(a);l:=[];
+for k from 1 to n do l:=l.[sum(a[x],x=k..n)+n-k]; end_for;
+res:=product(product((l[i]-l[j])*(l[i]+l[j]+2),j=i+ 1..n),i=1..n-1);
+res:=res*product(l[j]+1,j=1..n)/product((2*n-2*j-1) !,j=0..n-1);
+return(res);
+end_proc;
+dimsoodd:=proc(a)
+local n,k,l,res;
+begin
+n:=nops(a); l:=[];
+for k from 1 to n do l:=l.[sum(a[x],x=k..n)-a[n]/2+n-k]; en d_for;
+22res:=product(product((l[i]-l[j])*(l[i]+l[j]+1),j=i+ 1..n),i=1..n-1);
+res:=res*product(2*l[j]+1,j=1..n)/product((2*n-2*j- 1)!,j=0..n-1);
+return(res);
+end_proc;
+dimsoeven:=proc(a)
+local n,k,l,res;
+begin
+n:=nops(a); l:=[];
+for k from 1 to n-2 do l:=l.[sum(a[x],x=k..n-2)+a[n-1]/2+a [n]/2+n-k];end_for;
+l:=l.[a[n-1]/2+a[n]/2+1,-a[n-1]/2+a[n]/2];
+res:=product(product((l[i]-l[j])*(l[i]+l[j]),j=i+1. .n),i=1..n-1);
+res:=res/product((2*n-2*j)!,j=1..n-1)*2^(n-1);
+return(res);
+end_proc;
+The following procedures give the representations of dimen sion smaller than dim(g)forg=
+sln,sp2n,so2nandso2n+1.
+smallsl:=proc(n)
+local ens,prod,i,d,dg;
+begin
+dg:=n^2-1;
+print(Unquoted,"dimension of g: ".expr2text(dg));
+ens:=[];
+for i from 1 to n-1 do ens:=ens.[{0,1,2}]; end_for;
+prod:=combinat::cartesianProduct::list(ens[x]$x=1.. n-1);
+for i from 1 to 3^(n-1) do
+d:=dimsl(prod[i]);
+if d<=dg then print(Unquoted,"heighest weight ".expr2tex t(prod[i]).
+" of dimension ".expr2text(d));
+end_if;
+end_for;
+end_proc;
+smallsp:=proc(n)
+local ens,prod,i,d,dg;
+begin
+dg:=n*(2*n+1);
+print(Unquoted,"dimension of g: ".expr2text(dg));
+ens:=[];
+for i from 1 to n do ens:=ens.[{0,1,2}]; end_for;
+prod:=combinat::cartesianProduct::list(ens[x]$x=1.. n);
+for i from 1 to 3^n do
+d:=dimsp(prod[i]);
+if d<=dg then print(Unquoted,"heighest weight ".expr2tex t(prod[i]).
+" of dimension ".expr2text(d));
+end_if;
+end_for;
+end_proc;
+smallsoodd:=proc(n)
+local ens,prod,i,d,dg;
+23begin
+dg:=n*(2*n+1);
+print(Unquoted,"dimension of g: ".expr2text(dg));
+ens:=[];
+for i from 1 to n do ens:=ens.[{0,1,2}]; end_for;
+prod:=combinat::cartesianProduct::list(ens[x]$x=1.. n);
+for i from 1 to 3^n do
+d:=dimsoodd(prod[i]);
+if d<=dg then print(Unquoted,"heighest weight ".expr2tex t(prod[i]).
+" of dimension ".expr2text(d));
+end_if;
+end_for;
+end_proc;
+smallsoeven:=proc(n)
+local ens,prod,i,d,dg;
+begin
+dg:=n*(2*n-1);
+print(Unquoted,"dimension of g: ".expr2text(dg));
+ens:=[];
+for i from 1 to n do ens:=ens.[{0,1,2}]; end_for;
+prod:=combinat::cartesianProduct::list(ens[x]$x=1.. n);
+for i from 1 to 3^n do
+d:=dimsoeven(prod[i]);
+if d<=dg then print(Unquoted,"heighest weight ".expr2tex t(prod[i]).
+" of dimension ".expr2text(d));
+end_if;
+end_for;
+end_proc;
+4.2 Computations for sl6
+The procedure testsl6 produces the matrix used in section 2.4, corresponding to th e action of
+sl6overΛ3(V), whereVis the standard representation of sl6.
+basis:=[[1,2,3],[1,2,4],[1,2,5],[1,2,6],[1,3,4],[1, 3,5],[1,3,6],[1,4,5],[1,4,6],
+[1,5,6],[2,3,4],[2,3,5],[2,3,6],[2,4,5],[2,4,6],[2, 5,6],
+[3,4,5],[3,4,6],[3,5,6],[4,5,6]]:
+indexbasis:=proc(B)
+local i;
+begin
+for i from 1 to 20 do if B=basis[i] then return(i); end_if; end _for;
+end_proc:
+actionbasis:=proc(A,B)
+local vec,vec1,liste,liste2,coef,numero,i,j,k,n;
+begin
+liste:=[]; i:=B[1]; j:=B[2]; k:=B[3];
+vec:=matrix(6,1); vec[i]:=1; vec:=A*vec;
+for n from 1 to 6 do liste:=liste.[[vec[n],[n,j,k]]]; end_f or;
+vec:=matrix(6,1); vec[j]:=1; vec:=A*vec;
+for n from 1 to 6 do liste:=liste.[[vec[n],[i,n,k]]]; end_f or;
+24vec:=matrix(6,1); vec[k]:=1; vec:=A*vec;
+for n from 1 to 6 do liste:=liste.[[vec[n],[i,j,n]]]; end_f or;
+liste2:=[];
+for n from 1 to nops(liste) do
+coef:=(liste[n])[1];
+i:=((liste[n])[2])[1]; j:=((liste[n])[2])[2]; k:=((li ste[n])[2])[3];
+if (coef<>0) then
+if (i<j) and (j<k) then liste2:=liste2.[[coef,[i,j,k]]]; end_if;
+if (i<k) and (k<j) then liste2:=liste2.[[-coef,[i,k,j]]] ; end_if;
+if (j<i) and (i<k) then liste2:=liste2.[[-coef,[j,i,k]]] ; end_if;
+if (j<k) and (k<i) then liste2:=liste2.[[coef,[j,k,i]]]; end_if;
+if (k<i) and (i<j) then liste2:=liste2.[[coef,[k,i,j]]]; end_if;
+if (k<j) and (j<i) then liste2:=liste2.[[-coef,[k,i,j]]] ; end_if;
+end_if;
+end_for;
+vec:=matrix(20,1);
+for n from 1 to nops(liste2) do
+coef:=(liste2[n])[1];
+numero:=indexbasis((liste2[n])[2]);
+vec1:=matrix(20,1); vec1[numero]:=coef; vec:=vec+vec1 ;
+end_for;
+return(vec);
+end_proc:
+actionvector:=proc(A,vec)
+local i,res;
+begin
+res:=matrix(20,1);
+for i from 1 to 20 do res:=res+vec[i]*actionbasis(A,basis[ i]); end_for;
+return(res);
+end_proc:
+testsl6:=proc()
+local vec,res,i,j,mat;
+begin
+res:=[]; vec:=[];
+for i from 1 to 20 do vec:=vec.[a[i]]; end_for;
+for i from 1 to 5 do
+mat:=matrix(6,6); mat[i,i]:=1; mat[i+1,i+1]:=-1;
+res:=res.actionvector(mat,vec);
+end_for;
+for i from 1 to 5 do
+for j from i+1 to 6 do
+mat:=matrix(6,6); mat[i,j]:=1;
+res:=res.actionvector(mat,vec);
+mat:=matrix(6,6); mat[j,i]:=1;
+res:=res.actionvector(mat,vec);
+end_for;
+end_for;
+return(res);
+end_proc;
+254.3 Computations for g2
+The procedure testg2 produces the matrix used in section 2.4, corresponding to th e action of
+g2onM7,2(C).
+H1:=matrix([[1,0,0,0,0,0,0],[0,-1,0,0,0,0,0],[0,0,2 ,0,0,0,0],[0,0,0,0,0,0,0],
+[0,0,0,0,-2,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,-1]]) :
+H2:=matrix([[0,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,-1 ,0,0,0,0],[0,0,0,0,0,0,0],
+[0,0,0,0,1,0,0],[0,0,0,0,0,-1,0],[0,0,0,0,0,0,0]]):
+Y1:=matrix([[0,0,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0, 0,0,0,0],[0,0,1,0,0,0,0],
+[0,0,0,2,0,0,0],[0,0,0,0,0,0,0],[0,0,0,0,0,-1,0]]):
+Y2:=matrix([[0,0,0,0,0,0,0],[0,0,0,0,0,0,0],[0,-1,0 ,0,0,0,0],[0,0,0,0,0,0,0],
+[0,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,0]]):
+Y3:=-Y1*Y2+Y2*Y1: Y4:=-1/2*(Y1*Y3-Y3*Y1):
+Y5:=1/3*(Y1*Y4-Y4*Y1): Y6:=Y2*Y5-Y5*Y2:
+X1:=matrix([[0,1,0,0,0,0,0],[0,0,0,0,0,0,0],[0,0,0, 2,0,0,0],[0,0,0,0,1,0,0],
+[0,0,0,0,0,0,0],[0,0,0,0,0,0,-1],[0,0,0,0,0,0,0]]):
+X2:=matrix([[0,0,0,0,0,0,0],[0,0,-1,0,0,0,0],[0,0,0 ,0,0,0,0],[0,0,0,0,0,0,0],
+[0,0,0,0,0,1,0],[0,0,0,0,0,0,0],[0,0,0,0,0,0,0]]):
+X3:=X1*X2-X2*X1: X4:=1/2*(X1*X3-X3*X1):
+X5:=-1/3*(X1*X4-X4*X1): X6:=-(X2*X5-X5*X2):
+g2:=[H1,H2,X1,X2,X3,X4,X5,X6,Y1,Y2,Y3,Y4,Y5,Y6]:
+testg2:=proc()
+local i,j,res,A,M;
+begin
+A:=matrix(7,2);
+for i from 1 to 7 do for j from 1 to 2 do A[i,j]:=a[i,j]; end_for; end_for;
+res:=[];
+for i from 1 to 14 do
+M:=g2[i]*A;
+res:=res.linalg::stackMatrix(M[1..7,1],M[1..7,2]);
+end_for;
+return(res);
+end_proc;
+References
+[1] Frédéric Chapoton, Algèbres pré-lie et algèbres de Hopf liées à la renormalisat ion, C. R.
+Acad. Sci. Paris Sér. I Math. 332(2001), no. 8, 681–684.
+[2] Frédéric Chapoton and Muriel Livernet, Pre-Lie algebras and the rooted trees operad , Inter-
+nat. Math. Res. Notices 8(2001), 395–408, arXiv:math/0002069.
+[3] Alain Connes and Dirk Kreimer, Hopf algebras, Renormalization and Noncommutative ge-
+ometry , Comm. Math. Phys 199(1998), no. 1, 203–242, arXiv:hep-th/9808042.
+[4] William Fulton and Joe Harris, Representation theory , Graduate Texts in Mathematics, vol.
+129, Springer-Verlag, New York, 1991, A first course, Readin gs in Mathematics.
+[5] Wee Liang Gan and Travis Schedler, The necklace Lie coalgebra and renormalization alge-
+bras, J. Noncommut. Geom. 2(2008), no. 2, 195–214.
+[6] Dirk Kreimer, Combinatorics of (pertubative) Quantum Field Theory , Phys. Rep. 4–6
+(2002), 387–424, arXiv:hep-th/0010059.
+26[7] F. Kubo, Compatible algebra structures of Lie algebras , Ring theory 2007, World Sci. Publ.,
+Hackensack, NJ, 2009, pp. 235–239.
+[8] Jean-Louis Loday, Dialgebras , Lecture Notes in Math., vol. 1763, Springer, Berlin, 2001.
+[9] Jean-Louis Loday and Maria Ronco, Combinatorial hopf algebras , arXiv:0810.0435, 2009.
+[10] Jean-Louis Loday and Maria O. Ronco, Hopf algebra of the planar binary trees , Adv. Math.
+139(1998), no. 2, 293–309.
+[11] Jean-Michel Oudom and Daniel Guin, Sur l’algèbre enveloppante d’une algèbre pré-Lie , C.
+R. Math. Acad. Sci. Paris 340(2005), no. 5, 331–336, arXiv:math/0404457.
+[12] Maria Ronco, A Milnor-Moore theorem for dendriform Hopf algebras , C. R. Acad. Sci. Paris
+Sér. I Math. 332(2001), no. 2, 109–114.
+[13] Pepijn van der Laan and Ieke Moerdijk, Families of Hopf algebras of trees and pre-Lie
+algebras , Homology, Homotopy Appl. 8(2006), no. 1, 243–256, arXiv:math/0402022.
+27
\ No newline at end of file
diff --git a/1001.0852.txt b/1001.0852.txt
new file mode 100644
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+++ b/1001.0852.txt
@@ -0,0 +1,472 @@
+arXiv:1001.0852v1  [astro-ph.SR]  6 Jan 2010Mon. Not. R. Astron. Soc. 000, 1–??(2009) Printed 30 May 2018 (MN L ATEX style file v2.2)
+UBVRI observations of the flickering of RS Ophiuchi at
+Quiescence⋆
+R. K. Zamanov1†, S. Boeva1, R. Bachev1, M. F. Bode2, D. Dimitrov1,
+K. A. Stoyanov1, A. Gomboc3, S. V. Tsvetkova1, L. Slavcheva-Mihova1,
+B. Spassov1, K. Koleva1, B. Mihov1
+1Institute of Astronomy, Bulgarian Academy of Sciences, 72 T sarighradsko Shousse Blvd., 1784 Sofia, Bulgaria
+2Astrophysics Research Institute, Liverpool John Moores Un iversity, Twelve Quays House, Birkenhead, CH41 1LD, UK
+3Faculty of Mathematics and Physics, University of Ljubljan a, Jadranska 19, 1000 Ljubljana, Slovenia
+Accepted .. .. .. ... Received .. .. 2009
+ABSTRACT
+We report observations of the flickering variability of the recurren t nova RS Oph at
+quiescence on the basis of simultaneous observations in 5 bands ( UBVRI). RS Oph
+has flickering source with ( U−B)0=−0.62±0.07,(B−V)0= 0.15±0.10,(V−R)0=
+0.25±0.05.
+We find for the flickeringsourcea temperature Tfl≈9500±500K, and luminosity
+Lfl∼50−150L⊙(using a distance of d= 1.6 kpc).
+We also find that on a ( U−B) vs (B−V) diagram the flickering of the symbiotic
+starsdiffers fromthat ofthe cataclysmicvariables.The possible so urceofthe flickering
+is discussed.
+The data are available upon request from the authors and on the we b
+www.astro.bas.bg/ ∼rz/RSOph.UBVRI.2010.MNRAS.tar.gz.
+Key words: stars: individual: RS Oph – binaries: symbiotic – binaries: novae, cata -
+clysmic variables
+1 INTRODUCTION
+Inthesymbiotic recurrentnovaRSOphiuchi(HD162214), a
+near-Chandrasekhar-mass white dwarf (WD) accretes mate-
+rial from a red giant companion (e.g., Hachisu & Kato 2001;
+Sokoloski et al. 2006). It experiences nova eruptions appro x-
+imately every 20 yr (Evans et al. 2008), with the most recent
+eruption having occurred on 2006 February 12 (Narumi et
+al. 2006). Fekel et al. (2000) found that RS Oph has an or-
+bital period of 455 days and give red giant and white dwarf
+(WD)masses of2 .3M⊙andclose to1 .4M⊙respectivelywith
+a separation between the components of a= 2.68×1013cm.
+For the range of spectral types suggested (Worters et al.
+2007) for the red giant in the RS Oph system, its radius
+is smaller than its Roche lobe, and accretion onto the WD
+may occur only from the red giant wind.
+The flickering (stochastic light variations on timescales
+⋆based on data from Bulgarian observatories Rozhen and Belo-
+gradchik
+†e-mail: rkz@astro.bas.bg; sboeva@astro.bas.bg;
+bachevr@astro.bas.bgof a few minutes with amplitude of a few ×0.1 magnitudes)
+is a variability observed in the three main types of binaries
+thatcontainwhitedwarfs accreting material from acompan-
+ion mass-donor star: cataclysmic variables (CVs), superso ft
+X-ray binaries, and symbiotic stars (Sokoloski 2003). The
+flickering of RS Oph has been detected by Walker (1977).
+The systematic searches for flickering variability in sym-
+biotic stars and related objects (Dobrzycka et al. 1996a;
+Sokoloski, Bildsten & Ho 2001; Gromadzki et al. 2006)
+have shown that among ∼200 symbiotic stars known, only
+8 present flickering – RS Oph, T CrB, MWC 560, Z And,
+V2116 Oph, CH Cyg, RT Cru, o Cet and V407 Cyg.
+Here we investigate the flickering variability of RS Oph
+in theUBVRI bands and discuss its possible origin.
+2 OBSERVATIONS
+On the night of 2008 July 6, we observed RS Oph simulta-
+neously with four telescopes equipped with CCD cameras.
+The 2m RCC telescope of the National Astronomical Obser-
+vatory Rozhen equipped with a dual channel focal reducer
+c/circlecop†rt2009 RAS2Zamanov, Boeva, Bachev, et al.
+Table 1. CCD observations of RS Oph. In the table are given as follows: the telescope, band, UT-start and UT-end of the run, exposur e
+time, number of CCD images obtained, average magnitude in th e corresponding band, minimum – maximum magnitudes in each b and,
+standard deviation of the mean, typical observational erro r.
+telescope band UT Exp-time N pts average min-max stdev err
+start-end [sec] [mag] [mag]-[mag] [mag] [mag]
+2008 July 6 JD2454654
+2.0 m Rozhen U 19:27 - 21:26 300 20 12.546 12.293 – 12.721 0.103 0.011
+50/70 cm Schmidt B 19:46 - 21:29 100,60,20 50 12.471 12.380 – 12.615 0.051 0.005
+2.0 m Rozhen V 18:54 - 21:29 30 196 11.215 11.079 – 11.333 0.052 0.007
+60 cm Rozhen R 19:36 - 21:33 15,20 301 10.277 10.172 – 10.362 0.036 0.002
+60 cm Belogr I 19:42 - 21:22 60 74 8.990 8.855 – 9.098 0.057 0.100
+2009 July 21 JD2455034
+50/70 cm Schmidt U 20:58 - 21:52 120 21 12.020 11.863 – 12.166 0.092 0.013
+60 cm Rozhen B 20:17 - 21:38 40 70 12.030 11.876 – 12.277 0.098 0.007
+60 cm Belogr V 20:12 - 21:31 30 81 11.009 10.891 – 11.236 0.092 0.004
+60 cm Belogr R 20:12 - 21:31 10 80 10.219 10.121 – 10.406 0.074 0.003
+60 cm Rozhen I 20:17 - 21:38 10 70 9.172 9.070 – 9.325 0.066 0.003
+2009 July 23 JD2455036
+2.0 m Rozhen U 19:23 - 21:14 120 41 11.659 11.482 – 11.845 0.081 0.028
+50/70 cm Schmidt B 20:53 - 21:21 30 46 11.978 11.857 – 12.099 0.062 0.007
+2.0 m Rozhen V 19:24 - 21:15 10 467 10.932 10.766 – 11.093 0.070 0.008
+60 cm Rozhen I 19:40 - 21:17 5 766 9.096 8.970 – 9.216 0.050 0.005
+60 cm Belogr I 19:55 - 21:24 10 404 9.118 8.998 – 9.206 0.048 0.003
+Figure 1. Variability of RS Oph in the UBVRI bands on 2008
+July 6.Figure 2. Variability of RS Oph in the UBVRI bands on 2009
+July 21.
+c/circlecop†rt2009 RAS, MNRAS 000, 1–??Optical flickering of RS Oph 3
+Figure 3. Variability of RS Oph in the UBVI bands on 2009
+July 23.
+observed in UandVbands. In the Uband a Photomet-
+rics CCD (1024x1024 px, field of view 4.9’x4.9’) has been
+used, and in the Vband a VersArray 1330B (1340x1300,
+6.3’x6.3’). The 60 cm Rozhen telescope observed in the R
+band (equipped with a FLI PL09000 CCD with 3056 x
+3056 pixels and 5.7’x5.7’); the 50/70 cm Schmidt telescope
+of NAO Rozhen in the Bband (SBIG STL11000M CCD,
+4008x2672 px, 16’x24’), and the 60 cm telescope of the Bel-
+ogradchick Astronomical Observatory in the Iband (SBIG
+ST8 CCD, 1530x1020px, 6.4’x4.2’).
+On the night of 2009 July 21, we observed RS Oph si-
+multaneously with 3 telescopes. The 50/70 cm Schmidt tele-
+scope observed in the Uband, the 60 cm Rozhen telescope
+– repeating BandIbands, and the 60 cm Belogradchick
+telescope – repeating VandRbands.
+On the night of 2009 July 23, we observed RS Oph
+simultaneously with 3 telescopes. The 2m RCC telescope
+observed simultaneously in UandVbands, the 50/70 cm
+Schmidt telescope – in B, and the 60 cm Rozhen and Belo-
+gradchick telescopes – in Iband.
+All the CCD images have been bias subtracted, flat
+fielded, and standard aperture photometry has been per-
+formed. The data reduction and aperture photometry are
+done with IRAF and have been checked with alternative
+software packages. The comparison stars of Henden and Mu-
+nari (2006) have been used.
+The results of our observations are summarized in Ta-Figure 4. (U−R), (B−R), and (V−R) colours of RS Oph
+versusRband magnitude for 2008 July 6.
+ble 1 and plotted in Figs.1, 2and 3. For each run we measure
+the minimum, maximum, and average brightness in the cor-
+responding band, plus the standard deviation of the run.
+3 COLOUR VARIATIONS
+As can be seen in Figs. 1, 2, 3 and Table 1, during our
+observations RS Oph exhibited variability on a time scale of
+1-30 minutes with amplitude 0.3 mag in V. The amplitude
+increases to shorter wavelengths and decreases to longer.
+3.1 Magnitude - Colour relation
+In Fig.4 we plot the colours of RS Oph versus the Rband
+magnitude for 2008 July 6. The Rband is chosen because we
+have the shortest exposure time. In order to relate magni-
+tudes in different bands, taken at non-matching start times
+and different exposure times, we linearly interpolated the
+light curves. The colours have been calculated over 3 grids
+with time resolutions 70, 120 and 360 seconds, to match the
+poorest filter sampling in the corresponding colour relatio n:
+70 s. for ( V−R) vs.R, 120 s for ( B−R) vs.R, and 360 s
+for (U−R) vs.R. Linear fits (of type y=a+bx) to the
+data points in Fig.4 give:
+(U−R) =−9.73(±1.15) +1.17(±0.11)R (1)
+(B−R) =−5.91(±0.71) +0.79(±0.07)R (2)
+(V−R) =−0.86(±0.29) +0.17(±0.03)R (3)
+c/circlecop†rt2009 RAS, MNRAS 000, 1–??4Zamanov, Boeva, Bachev, et al.
+Figure 5. Dereddened fluxes of the flickering light source of RS Oph. The solid line represents a black body fit.
+a)2008 July 6 (circles), Tbb= 9492 K, radius R= 2.6R⊙, located at distance d= 1.6 kpc.
+b)2009 July 21 (diamonds) and 2009 July 23 (pluses) plotted tog ether. The fit is Tbb= 9370 K, radius R= 4.7R⊙.
+c)All data normalized to V band and plotted together. The fit is Tbb= 9236 K. Symbols are identical to those in a) and b).
+The errors of the coefficients are given in brackets. These
+relations are obtained on the basis of the observations from
+2008July6. Theyarevalidovertherange10 .20≤R≤10.35
+mag. The Spearman’s (rho) rank correlation gives ρ= 0.85
+(significance 8 .10−6) for Eq.1, ρ= 0.66 (1.10−7) for Eq.2,
+ρ= 0.57 (2.10−10) for Eq.3. The significance in Eq.1-Eq.3
+is always <<0.001 indicating that all these correlations are
+highly significant and changes in the colours of RS Oph are
+correlated with brightness variations.
+3.2 Colours of the flickering source
+Bruch (1992) proposed that the light curve of CVs can
+be separated into two parts – constant light and variable
+(flickering) source. We assume that all the variability is
+due to flickering. In these suppositions the flickering light
+source is considered 100% modulated. Following these sug-gestions, we calculate the fluxof the flickeringlight source as
+Ffl=Fav−Fmin, whereFavis the average flux during the
+run andFminis the minimum flux during the run (corrected
+for the typical error of the observations). Fflhas been cal-
+culated for each band, using the values given in Table 1 and
+Bessel (1979) calibration for the fluxes of a zero magnitude
+star.
+Adopting these results, we find that the flickering light
+source contributes about 6% of the light in the Rband, 7%
+inV, 10% in B, and 13% in U.
+Following Snijders (1987) we assume interstellar extinc-
+tionEB−V= 0.73 and an extinction law as given in Cardelli
+et al.(1989). Using the colour relations (Eq.1–Eq.3, deriv ed
+in Sect.3.1) in the interval R= 10.28−10.34 mag (in other
+words average R= 10.28 and we calculate average values for
+other bands, minimal brightness in R= 10.34, and we de-
+rive minimal fluxes in other bands), we obtain colours of the
+c/circlecop†rt2009 RAS, MNRAS 000, 1–??Optical flickering of RS Oph 5
+flickering light source in RS Oph as ( U−B)0=−0.67±0.06,
+(B−V)0= 0.08±0.06, (V−R)0= 0.28±0.08. On 2009
+July 21 we obtain ( B−V)0= 0.22 and ( V−R)0= 0.22.
+Using the entire runs in Uband from 2009 July 23 and B
+andVfrom 2009 July 21 (as plotted in Fig.5) we calculate
+(B−V)0= 0.22 and ( U−B)0=−0.57. The values are
+similar to the colours of the flickering source in 1983 July-
+August ( U−B)0=−0.71 and (B−V)0= 0.07−0.11 (Bruch
+1992).
+4 FLICKERING
+4.1 Temperature and size of the flickering source
+The derived ( U−B)0colour corresponds to a B4V star
+(Teff≈17000 K) and a black body with T≈9000 K. The
+(B−V)0colour corresponds toan A5V( Teff≈9000 K) star
+and a black body with T≈10000 K. ( V−R)0corresponds
+to a F8V star ( Teff= 7000 K). These estimates give an
+approximate temperature of the flickering light source Tfl=
+9000−12000 K.
+For the flickering light source we obtain magnitudes
+(corrected for interstellar extinction):
+2008 July 6 : U= 11.40,B= 12.06,V= 11.92,R=
+11.55 mag;
+2009 July 21 : B= 10.73,V= 10.52,R= 10.29,I=
+9.94 mag;
+2009 July 23 : U= 10.00,V= 10.76,I= 10.11−10.45
+mag.
+The errors are in U±0.04, inB±0.05, inV±0.08, in
+R±0.10 inI±0.12.
+We adopt d= 1.6±0.3 kpc as given in Bode (1987). In
+Fig.5 we plot these magnitudes transformed to fluxes. Using
+a black body fit ( nfit1droutine of IRAF), we calculate for
+the flickering light source: Tfl= 9492±300 K;Rfl= 2.6±
+0.3R⊙;Lfl∼50L⊙(2008 July 6) and Tfl= 9370±300 K;
+Rfl= 4.7±0.3R⊙;Lfl∼150L⊙(2009 July 21 and 23). In
+Fig.5c we plot all points normalized to the corresponding V
+band flux. The fit gives Tfl= 9236±200 K;
+OurI-band data from 2008 July 6 have considerably
+larger observational errors due to a technical problem whic h
+gave as a consequence a variable dark current pattern of the
+CCD.Ufrom 2009 July 21 and Bfrom 2009 July 23 are
+short. They are therefore not used in the black body fits nor
+plotted in Fig.5.
+There are several different estimates of the mass accre-
+tion rate in RS Oph. An estimate of ˙Macc≈2.10−7M⊙yr−1
+arises from the required mass accreted on to the WD to
+give rise to the outburst plus the latest inter-outburst tim e
+scale of 21 years (Osborne et al. 2010). Assuming RWD=
+0.003R⊙(Starrfield et al. 1991), this results in a total ac-
+cretion luminosity Lacc≈GMwd˙M/Rwd≈2900L⊙. Alter-
+natively, Walder et al. (2008) explore the hydrodynamics of
+windaccretion inthesystemandfind ˙Macc≈10−8M⊙yr−1,
+i.e.Lacc≈146L⊙(which compares to Lacc≈190−740L⊙
+from IUE observations at quiescence - Snijders 1987). The
+spectral energy distribution of the accretion disk around t he
+WDissimilar tomid-Bspectral type(Dobrzyckaetal. 1996)
+with luminosity 100 −600L⊙. In practice therefore, the dif-
+ferent estimates give Lacc∼140−2800L⊙, which means
+that 5%-35% of the total accretion luminosity is emitted by
+the flickering source (in other words Lfl/Lacc∼0.05−0.35).Figure 6. Position of the flickering light source on a ( U−B) vs
+(B−V) diagram. The dashed line is the main sequence. The solid
+line is a black body, (blue) triangles – CVs (from Bruch 1992) ,
+(black) squares – symbiotic stars (MWC 560, V407 Cyg), RS Oph
+is marked with (red) plus symbols, T CrB – (red) squares.
+4.2 Time delay
+We searched for time lags between the light curves in differ-
+ent bands, using the interpolation cross-correlation meth od
+(Gaskell & Sparke, 1986), butfound no delays larger than10
+sec, which is shorter than our time resolution (the exposure
+times).
+4.3 Difference between symbiotics and CVs?
+Fig. 6representa2-colour diagram, ( U−B)0versus(B−V)0
+for the flickering light source in a number of CVs and sym-
+biotics. The data for the CVs and T CrB are from Bruch
+(1992). We also plot 2 points for MWC 560 and 2 for
+V407 Cyg, which are calculated from the data of Gromadzki
+et al. (2006)
+The crosses represent RS Oph. The three crosses refer
+to the three epochs of observations – 1983, 2008, 2009. They
+lie well outside the area of the CVs. To address the possible
+errors of our measurement, we separated our light curves of
+RS Oph (plotted in Fig. 1) into two parts (from UT 19:46
+to UT 20:26 and UT 20:27 – 21:22). We obtained for the
+flickering source ( U−B)0=−0.61 (B−V)0= 0.11 for the
+first time interval, and ( U−B)0=−0.74 (B−V)0= 0.15
+for the second. This indicates that for RS Oph the errors
+are approximately equal to the size of the symbols.
+To search for differences between the flickering of
+symbiotic stars and CVs we performed a two-dimensional
+Kolmogorov-Smirnov test (Peacock 1983; Fasano & Frances-
+chini 1987), using the data plotted on Fig.6. We compare
+colours of the flickering of the recurrent novae RS Oph and
+T CrB with those of the CVs. The test gives a probability
+of 2×10−3that both distributions are extracted from the
+same parent population.
+We also performed the same test but comparing CVs
+with all symbiotic stars. The test gives a probability of
+4×10−4that both distributions are extracted from the same
+c/circlecop†rt2009 RAS, MNRAS 000, 1–??6Zamanov, Boeva, Bachev, et al.
+parentpopulation. Thenumberofpointsisnothigh,butsuf-
+ficient, as shown in Fasano & Franceschini (1987), and the
+difference is significant. This result indicates that in this di-
+agram there are clues to the cause of the differences between
+the flickering of these two classes of accreting white dwarfs .
+In the symbiotics of course, the mass donor is a red
+giant and the orbital periods are >100 d. In the CVs the
+mass donors are late-type dwarfs and the orbital periods
+are∼1000 times shorter. To the best of our knowledge
+this is the first evidence that flickering differs in these type s
+of accreting WDs and the physical cause of this difference
+warrants further investigation (see below).
+5 DISCUSSION
+In quiescence RS Oph varies irregulary between V= 10.0−
+12.2 (Collazzi et al. 2009, AAVSO light curves). This vari-
+ability is observed before as well as after the 2006 outburst .
+Darnley et al.(2008) detected an increase of the brightness
+inBfromB= 13.5 toB= 11.9 mag and from V= 12.0
+toV= 10.5 for a year after the 2006 outburst. Worters
+et al. (2007) detected an increase in VfromV= 11.3 to
+V= 11.9 for 2 weeks. The brightness of RS Oph at the
+time of our flickering observations is well inside these limi ts
+(unfortunately, there does not seem to be any published U
+light curve). The derived parameters of the flickering can be
+considered as typical for quiescence.
+After the 2006 outburst the flickering of RS Oph dis-
+appeared (Zamanov et al. 2006) probably as a result of (1)
+destruction of the accretion disk from the nova explosion or
+(2) a change in the inner disk associated with jet produc-
+tion (Sokoloski et al. 2008). The flickering resumed by day
+241 of the outburst (Worters et al. 2007). For the symbiotic
+star CH Cyg, Sokoloski & Kenyon (2003) observed changes
+in the flickering in association with the mass ejection event .
+Observations of CH Cyg and RS Oph therefore indicate that
+a connection does exist between the flickering behaviour and
+the ejection of matter from the WD.
+Different sites for the origin of the flickering have been
+discussed. They are all related to the accretion process:
+(i)the brightspot (the region of impact of thestream of
+transferred matterfrom themass donor staron theaccretion
+disk);
+(ii)the boundary layer (between the innermost accre-
+tion disk and the white dwarf surface);
+(iii)inside the accretion disk itself.
+The findings of Bruch (2000) for HT Cas, V2051 Oph,
+IP Peg and UX UMa demonstrated that the flickering in
+these CVs can originate in both regions (i)and(ii).
+(i)The temperature and the size of the bright spot
+are derived for a number of CVs. A few examples of tem-
+peratures are: for OY Car Wood et al (1989) calculated
+black body T= 13800 ±1300 K, and color temperature
+T= 9000 K; Marsh (1988) for IP Peg - T= 11200 K;
+Zhang & Robinson (1987) for U Gem - T= 11600 ±500 K;
+Robinson et al. (1978) give T= 16000 K for the bright
+spot in WZ Sge. The temperature of the optical flickering
+source of RS Oph Tfl≈10000 K (see Sect.4.1) is similar to
+the temperature of the bright spot for the CVs. The mass
+transfer in RS Oph may not be from Roche lobe overflow(as in CVs), but via stellar wind accretion. As a result of
+thesupersonic motion of theWDthrough thered giant wind
+there should be a shock cone and an accretion wake. Around
+the WD, it is likely that an accretion disk and/or a cocoon
+is formed. In the place where the matter flowing through
+the accretion wake encounters the accretion disk/cocoon a
+bright spot could be formed (analogous to the bright spot
+formed in the case of CVs, where the mass flow from the
+inner Lagrangian point encounters the accretion disk).
+(ii)Another possible site for the origin of the flickering
+in RS Oph is the boundary layer between the white dwarf
+andaccretiondisk.Bruch&Duschl(1993)estimatedthesize
+of the boundary layer ( ǫ) in RS Oph as ǫ= 2.20. In their
+modelLd/Lblis connected with the size of the boundary
+layer (Ldis the the luminosity of the accretion disk, Lblis
+the luminosity of the boundary layer; see Fig 1 in Bruch
+& Duschl 1993). If we assume Ld+Lbl≈Lacc≈110−
+2000L⊙, andLbl≈Lfl≈5−150L⊙, then we calculate
+Ld/Lbl= 1.2−12, and the lower value agrees with the size of
+the boundary layer as estimated by Bruch & Duschl (1993).
+However, Tflof RS Oph as derived in Sect. 4.1 is too low
+for a boundary layer.
+(iii)For UU Aqr, Baptista & Bortoletto (2008) found
+no evidence of flickering generated in regions (i)and(ii).
+They suggested that the flickering in UU Aqr is generated
+in the accretion disk itself and a possible reason can be tur-
+bulence generated after the collision of disk gas with the
+density-enhanced spiral wave in the accretion disk.
+The radial temperature profile of a steady-state accre-
+tion disk is:
+T4
+eff=3G˙MaccMwd
+8πσR3/bracketleftbigg
+1−/parenleftBigRwd
+R/parenrightBig1/2/bracketrightbigg
+, (4)
+whereσis the Stefan-Boltzmann constant, Ris the radial
+distance from the WD. Using the parameters for RS Oph, a
+temperature Tfl∼9500 K (the temperature of the flickering
+light source as derived in Sect. 4.1) should be achieved at a
+distance R≈0.5−1R⊙from the WD. If (iii)is the place
+for the origin of the flickering of RS Oph, then it comes from
+R<∼1R⊙from the WD.
+To understand more fully the nature of the flickering
+variability of RS Oph we need to acquire a set of spectra
+simultaneously with photometry and with time resolution
+∼30 seconds (see also Sokoloski 2003). Such spectra poten-
+tially can directly give the spectrum of the flickering light
+source.
+6 CONCLUSIONS
+We report our CCD observations of the flickering variability
+of the recurrent nova RS Oph, simultaneously with 4 tele-
+scopes in the UBVRI bands. RS Oph has a flickering source
+with (U−B)0=−0.62±0.07, (B−V)0= 0.15±0.10,
+(V−R)0= 0.25±0.05.
+For theflickeringlight source in RSOph(1) we estimate
+Teff≈9500±500 K, which is similar to the temperature of
+the bright spot in CVs; (2) using a distance of d= 1.6 kpc,
+we find size Rfl≈3.5±0.5R⊙, and luminosity Lfl∼
+50−150L⊙.
+We find that on the ( U−B) vs (B−V) diagram the
+c/circlecop†rt2009 RAS, MNRAS 000, 1–??Optical flickering of RS Oph 7
+flickering of the symbiotic stars differs from the flickering o f
+the cataclysmic variables.
+The possible sites for the origin of the flickering are
+briefly discussed and it is suggested that time-resolved spe c-
+troscopy should be carried out to explore this in more detail .
+ACKNOWLEDGMENTS
+RKZ,AG,andKASacknowledgethepartialsupportbyBul-
+garian NSF (HTC01-152) and Slovenian Research Agency
+(BI-BG/09-10-006). We are grateful to the referee, Dr.
+P. Woudt, for very helpful comments on the initial version
+of this paper.
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+arXiv:1001.0853v1  [math.DG]  6 Jan 2010RIEMANNIAN SUBMERSIONS WITH DISCRETE SPECTRUM
+G. PACELLIBESSA, J. FABIOMONTENEGRO,ANDPAOLO PICCIONE
+ABSTRACT . WeprovesomeestimatesonthespectrumoftheLaplacian
+of the total space of a Riemannian submersionin terms of the s pectrum
+of the Laplacian of the base and the geometry of the fibers. Whe n the
+fibers of the submersions are compact and minimal, we prove th at the
+total space is discrete if and onlyif the base is discrete. Wh en the fibers
+are not minimal, we prove a discreteness criterion for the to tal space
+in terms of the relative growth of the mean curvature of the fib ers and
+the mean curvature of the geodesic spheres in the base. We dis cuss in
+particularthecase ofwarpedproducts.
+1. INTRODUCTION
+LetMbe a complete Riemannian manifold and △=div◦gradbe the
+Laplace-Beltrami operator acting on the space of smooth fun ctions onM
+withcompactsupport. Theoperator △isessentiallyself-adjoint,thusithas
+a unique self-adjoint extension, to an unbounded operator, denoted by △,
+whose domain is the set of functions f∈L2(M)so that△f∈L2(M).
+Recall that the spectrum of a self-adjoint operator A, denoted by σ(A),
+is formed by all λ∈Rfor whichA−λIis not injective or the inverse
+operator(A−λI)−1is unbounded,[7]. In thispaperweare goingtostudy
+thespectrumof −△,(theoperator △isnegative),andwereferto σ(−△)as
+thespectrumof Mandinthiscaseonly,wedenoteby σ(M). Itisimportant
+(in our study) to distinguish the various types of elements o f the spectrum
+ofMin order to have a better understanding of the relations betw eenM
+andσ(M). This way, it is said that the set of all eigenvalues of σ(M)is
+thepoint spectrum σp(M), while the discrete spectrum σd(M)is the set
+of all isolated1eigenvalues of finite multiplicity. The essential spectrum
+σess(M) =σ(M)\σd(M)isthecomplementofthediscretespectrum.
+There is a vast literature studying the spectrum of complete Riemannian
+manifolds, among that, we point out geometric restrictions implying that
+the spectrum is purely continuous ( σp(M) =∅), see [9], [10], [12], [16],
+[21], [23] or implying that the spectrum if discrete ( σess(M)=∅), see [1],
+[4], [11], [15], [17], [18].
+Date:January5th,2010.
+The authors were partially supported by CNPq-CAPES (Brazil ) and MEC project
+PCI2006-A7-0532(Spain).
+1Isolatedin thesensethat forsome ε >0onehasthat (λ−ε,λ+ε)∩σ(△M) =λ.
+12 G. PACELLIBESSA,J. FABIOMONTENEGRO,ANDPAOLOPICCIONE
+In[1],Baiderstudiedtheessentialspectrumofwarpedprod uctmanifolds
+W=X×γY= (X×Y,dX2+γ2(x)dY2), whereγ:X→Ris a
+positivesmooth function. The Laplace-Beltrami operator △Wrestricted to
+C∞
+0(X)⊗C∞
+0(Y)hasthisform △W=A0⊗1Y+γ−2⊗(△Y),whereA0is
+an elliptic operator, symmetric relative to the density γndXwith the same
+symbolas △X. Baider showedthatif σess(Y) =∅then
+(1.1) σess(W) =∅ ⇐⇒ σess(A1) =∅,
+whereA1=A0+λ∗(Y)γ−2andλ∗(Y) = infσ(△Y). Whenγ≡1then
+W=X×YandA1=△X+λ∗(Y). In general, one can not substitute
+σess(A1) =∅byσess(X) =∅. There are examples of warped manifolds
+Rn×γS1,with discrete spectrum, therefore σess(A1) =∅, see [1], but the
+spectrumσess(Rn) = [0,∞). Riemannian manifolds whose Laplacian has
+emptyessentialspectrumare sometimescalled discretein theliterature.
+In this paper we consider Riemannian submersions π:M→Nand we
+provesomespectralestimatesrelatingthe(essential)spe ctrumofMandN.
+Riemannian submersions were introduced in the sixties by B. O’Neill and
+A. Gray (see [13, 19, 20]) as a tool to study the geometry of a Ri emannian
+manifold with an additional structure in terms of certain co mponents, that
+is, the fibers and the base space. When M(and thus also N) is compact,
+estimatesontheeigenvaluesoftheLaplacianof Mhavebeenstudiedin[5],
+under the assumption that the mean curvature vector of the fib ers isbasic,
+i.e.,π-related to some vector field on the basis. We will consider he re the
+noncompact case, assuminginitiallythat thefibers are mini mal.
+An important class of examples are Riemannian homogeneous s paces
+G/K, whereGis a Lie group endowed with a bi-invariant Riemannian
+metricandKisaclosedsubgroupof G,see[19]fordetails. Theprojection
+G→G/Kis a Riemannian submersions with totally geodesic fibers, an d
+withfibers diffeomorphicto K.
+Another important class of examples of manifolds that can be described
+as the total space of Riemannian submersions with minimal fib ers are the
+homogeneous 3-dimensional Riemannian manifolds with isometry group
+of dimension four, see [22]. This class includes the special linear group
+SL(2,R)endowedwithafamilyofleft-invariantmetrics,whichisth etotal
+spaceofRiemanniansubmersionswithbasegivenbythehyper bolicspaces,
+and fibers diffeomorphicto S1.
+GivenaRiemanniansubmersion π:M→Nwithcompactminimalfibers,
+weprovethat
+σess(M) =∅ ⇐⇒ σess(N) =∅,
+seeTheorem1. ThisresultcoincideswithBaider’sresultwh enM=X×Y
+is a product manifold, Yis compact, N=Xandπ:X×Y→Xis the
+projectiononthefirst factor.
+Theorem 1. Letπ:M→Nbe a Riemannian submersion with compact
+minimalfibers. ThenRIEMANNIAN SUBMERSIONS WITH DISCRETE SPECTRUM 3
+i.σess(N)⊂σess(M),σp(N)⊂σp(M), thusσ(N)⊂σ(M).
+ii.infσess(N) = infσess(M). Therefore, Mis discrete if and only if
+Nis discrete.
+A few remarks on this result are in order. First, we observe th at for
+the inequality infσess(M)≤infσess(N), Lemma 3.7, we need only the
+compactness of the fibers with uniformly bounded volume, mea ning that
+0< c2≤vol(Fp)≤C2for allp∈N. Second, the example of [1] shows
+thattheassumptionofminimalityofthefibersisnecessaryi nTheorem1. In
+fact, one has examples of Riemannian submersions having com pact fibers
+with discrete base and non discrete total space, or with disc rete total space
+butnotdiscretebase, seeExample4.2.
+In the second part of the paper we study the essential spectru m of the
+total spacewhen theminimalityassumptionon thefiber is dro pped. In this
+case, we prove that a sufficient condition for the discretene ss of the total
+space is that the growth of the mean curvature of the fibers at i nfinity is
+controlled by the growth of the mean curvature of the geodesi c spheres in
+thebasemanifold. Inordertostateourresult,letusintrod ucethefollowing
+terminology. Thecutlocus cut(p)ofapointpinaRiemannian n-manifold
+issaidtobe thin,ifits(n−1)-Hausdorffmeasurezero, Hn−1(cut(p)) = 0.
+Theorem 2. Letπ:M→Nbe a Riemannian submersion with compact
+fibers,andassumethat Nhasa pointx0withthincutlocus. If thefunction
+h:M→Rdefined by
+h(q) = (△Nρp0)π(q)+gN/parenleftbig
+(gradNρp0)π(q),dπq(Hq)/parenrightbig
+isproperthenσess(M) =∅. Hereρp0isthedistancefunctionin Ntop0.
+The Theorem 2 can be interpreted geometrically in terms of th e mean
+curvatureofthegeodesicspheresin thebaseand themean cur vatureofthe
+fibers. Namely, the Laplacian of the distance function ρp0(p)is exactly the
+valueofthemeancurvatureofthegeodesicsphere Sp=ρ−1
+p0/parenleftbig
+ρp0(p)/parenrightbig
+atthe
+pointp. Thus, assumption says that the sum of the mean curvature of t he
+geodesic balls in Nand the mean curvature of the fibers must diverge at
+infinity.
+Theorem 1 is proved in Section 3 and Theorem 2 in Section 4. An a lter-
+nativestatementofTheorem2canbegivenintermsofradialc urvature,see
+Corollary 4.2. Thereare twobasicingredientsfortheproof ofourresults.
+•The Decomposition Principle, that relates the fundamental tone of
+the complement of compact sets with the infimum of the essenti al
+spectrum,seeProposition3.2;
+•Two estimates of the fundamental tones of open sets in terms o f
+the divergence of vector fields, proved recently in [2] and [3 ], see
+Propositions3.4 and3.5.
+2. RIEMANNIAN SUBMERSIONS4 G. PACELLIBESSA,J. FABIOMONTENEGRO,ANDPAOLOPICCIONE
+2.1.Preliminaries. Given manifolds MandN, a smooth surjective map
+π:M→Nis a submersion if the differential dπ(q)has maximal rank for
+everyq∈M. Ifπ:M→Nis a submersion, then for all p∈Nthe
+inverseimage Fp=π−1(p)is a smoothembedded submanifold of M, that
+will be called the fiberatp. IfMandNare Riemannian manifolds, then a
+submersion π:M→Niscalleda Riemanniansubmersion ifforallp∈N
+and allq∈ Fp, therestrictionof dπ(q)to theorthogonalsubspace TqF⊥
+pis
+an isometryonto TpM.
+Givenp∈Nandq∈ Fp,atangentvector ξ∈TqMissaidtobe vertical
+ifitistangentto Fp,anditis horizontal ifitbelongstotheorthogonalspace
+(TqFp)⊥. LetD= (TF)⊥⊂TMdenote the smooth rank kdistribution
+onMconsisting of horizontal vectors. The orthogonal distribu tionD⊥is
+clearly integrable, the fibers of the submersion being its ma ximal integral
+leaves. Given ξ∈TM, its horizontal and vertical components are denoted
+respectively by ξhandξv. The second fundamental form of the fibers is a
+symmetrictensor SF:D⊥×D⊥→ D, defined by
+SF(v,w) = (∇M
+vW)h,
+whereWisaverticalextensionof wand∇MistheLevi–Civitaconnection
+ofM.
+For any given vector field X∈X(N), there exists a unique horizontal
+/tildewideX∈X(M)whichisπ-relatedtoX,thisis,forany p∈Nandq∈ Fp,then
+dπq(/tildewideXq) =Xp, calledhorizontal lifting ofX. A horizontal vector field
+/tildewideX∈X(M)iscalledbasicifitisπ-relatedtosomevectorfield X∈X(N).
+If/tildewideXand/tildewideYarebasicvectorfields,thentheseobservationsfollowseas ily.
+(a)gM(/tildewideX,/tildewideY) =gN(X,Y)◦π.
+(b)[/tildewideX,/tildewideY]hisbasicandit is π-related to [X,Y].
+(c)(∇M
+/tildewideX/tildewideY)hisbasicand itis π-related to ∇N
+XY,
+where∇Nis theLevi-Civitaconnectionof gN.
+Letusnowconsiderthegeometryofthefibers. First,weobser vethatthe
+fibers are totally geodesic submanifolds of Mexactly when SF= 0. The
+mean curvature vector of the fiber is the horizontal vector field Hdefined
+by
+(2.1) H(q) =k/summationdisplay
+i=1SF(q)(ei,ei) =k/summationdisplay
+i=1(∇M
+eiei)h
+where(ei)k
+i=1is a local orthonormalframe for thefiber through q. Observe
+thatHis not basic in general. For instance, when n= 1, i.e., when the
+fibers are hypersurfaces of M, thenHis basic if and only if all the fibers
+have constant mean curvature. The fibers are minimalsubmanifolds of M
+whenH≡0.RIEMANNIAN SUBMERSIONS WITH DISCRETE SPECTRUM 5
+2.2.Differentialoperators. Letπ:M→NbeaRiemanniansubmersion.
+Besides the natural operations of lifting a vector or vector fields inNto
+horizontalvectorsandbasicvectorfieldsonehasthatfunct ionsonNcanbe
+lifted to functionson Mthat are constant along thefibers. Such operations
+preserves the regularity of the lifted objects. One can also (locally) lift
+curves in the base γ: [a,b]→Nto horizontal curves /tildewideγ: [a,c)→Mwith
+the same regularity as γwith arbitrary initial condition on the fiber Fγ(a).
+Wewillneedformulasrelatingthederivativesof π-relatedobjectsin Mand
+N. Let usstart withdivergenceofvectorfields.
+Lemma2.1. Let/tildewideX∈X(M)beabasicvectorfield, π-relatedtoX∈X(N).
+The following relation holds between the divergence of /tildewideXandXatp∈N
+andq∈Fp.
+(2.2)divM(/tildewideX)q= divN(X)p+gM(/tildewideXq,Hq)
+= divN(X)p+gN/parenleftbig
+dπq(/tildewideXq),dπq(Hq)/parenrightbig
+.
+In particular,ifthefibersareminimal,then divM(/tildewideX) = divN(X).
+Proof.Formula (2.2) is obtained by a direct computation of the left -hand
+side, using a local orthonormal frame e1,...,ek,ek+1,...,ek+nofTM,
+wheree1,...,ekare basic fields. The equality follows using equalities (a)
+and (c)in Subsection2.1,and formula(2.1)forthemeancurv ature. /square
+Givena smoothfunction f:N→R, denoteby ˜f=f◦π:M→Rits
+liftingtoM. Itiseasytoseethatthegradient gradM˜fof˜fisthehorizontal
+lifting of the gradient gradNf. If we denote with a tilde /tildewideXthe horizontal
+lifting of a vector field X∈X(N), then the previous statement can be
+writtenas
+(2.3) gradM˜f=/tildewidergradNf.
+Now, given a function f:M→R, one can define a function fav:N→R
+byaveraging fon each fiber
+fav(p) =/integraldisplay
+FpfdFp,
+wheredFpis the volume element of the fiber Fprelative to the induced
+metric. Weareassumingthatthisintegralisfinite. Astothe gradientofthe
+averaged function fav,we havethefollowinglemma.
+Lemma 2.2. Letp∈Nandv∈TpNand denoteby Vthesmooth normal
+vector field along Fpdefined by the property dπq(Vq) =vfor allq∈ Fp.
+Then, foranysmoothfunction f:M→R
+(2.4)gN/parenleftbig
+gradNfav(p),v/parenrightbig
+=/integraldisplay
+Fp/bracketleftbig
+gM/parenleftbig
+gradMf,V/parenrightbig
++f·gM(H,V)/bracketrightbig
+dFq.6 G. PACELLIBESSA,J. FABIOMONTENEGRO,ANDPAOLOPICCIONE
+Proof.Astandardcalculationasinthefirstvariationformulafort hevolume
+functional of the fibers. Notice that when f≡1, thenfavis the volume
+function of the fibers, and (2.4) reproduces the first variati on formula for
+thevolume. /square
+Observe that, in (2.4), the gradient gradMfneed not be basic or even
+horizontal2. An averaging procedure is available also to produce vector
+fieldsXavon the basis out of vector fields Xdefined in the total space. If
+X∈X(M), letXav∈X(N)bedefined by
+(Xav)p=/integraldisplay
+Fpdπq/parenleftbig
+Xq/parenrightbig
+dFp(q).
+Observe that the integrand above is a function on Fptaking values in the
+fixed vector space TpN. IfX∈X(M)is a basic vector field, π-related
+to the vector field X∗∈X(N), then(Xav)p= vol(Fp)·(X∗)p, wherevol
+denotes the volume. Using the notion of averaged field, equal ity (2.4) can
+berewrittenas
+gradN(fav) =/parenleftbig
+gradMf+f·H/parenrightbig
+av.
+Remark2.3.From the above formula it follows easily that the averaged
+mean curvature vector field Havvanishes at the point p∈Nif and only if
+pis a critical point of the function z/mapsto→vol(Fz)inN. This happens, in
+particular,whentheleaf Fpisminimal. Whenallthefibersareminimal,or
+moregenerallywhentheaveragedmeancurvaturevectorfield Havvanishes
+identically,thenthevolumeofthefibers isconstant.
+Corollary2.4. Letπ:M→NbeaRiemanniansubmersionwithcompact
+minimalfibers F. Leth∈L2(N). Iff∈C∞
+0(M)suchthatfav= 0for all
+q∈Nthen
+(2.5)/integraldisplay
+M/tildewideh△MfdM= 0.
+Proof.Supposefirstthat hissmooth. BytheDivergenceTheorem,Fubini’s
+Theoremfor Riemanniansubmersionsand 2.4wehave
+/integraldisplay
+M/tildewideh△MfdM=−/integraldisplay
+MgM(gradM/tildewideh,gradMf)dM
+=−/integraldisplay
+N/integraldisplay
+FqgM(gradM/tildewideh,gradMf)dFqdN
+=−/integraldisplay
+NgN(gradM/tildewideh,gradNfav)dN
+= 0
+2Infact,agradientisbasic ifandonlyif it ishorizontal.RIEMANNIAN SUBMERSIONS WITH DISCRETE SPECTRUM 7
+Ifh∈L2(N)there exists a sequence of smooth functions hk∈C∞(N)
+convergingto hwithrespect to the L2-norm. On theotherhand
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+M/tildewideh△MfdM/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+M(/tildewidehk−/tildewideh)△MfdM/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤/integraldisplay
+M/vextendsingle/vextendsingle/vextendsingle/tildewidehk−/tildewideh/vextendsingle/vextendsingle/vextendsingle|△Mf|dM
+≤/parenleftbigg/integraldisplay
+M/vextendsingle/vextendsingle/tildewidehk−/tildewideh/vextendsingle/vextendsingle2dM/parenrightbigg1/2
+·/parenleftbigg/integraldisplay
+M|△Mf|2dM/parenrightbigg1/2
+=/ba∇dbl△Mf/ba∇dblL2(M)·/parenleftBigg/integraldisplay
+N/integraldisplay
+Fq/vextendsingle/vextendsinglehk−h/vextendsingle/vextendsingle2dFqdM/parenrightBigg1/2
+= vol(Fq)1/2·/ba∇dbl△Mf/ba∇dblL2(M)·/ba∇dblhk−h/ba∇dblL2(N)
+Sincehk→hinL2(N)then 2.5 holds. Observe that we used that the
+volumeoftheminimalfibers isconstant,seeRemark 2.3. /square
+3. SPECTRAL ESTIMATES IN RIEMANNIAN SUBMERSIONS
+3.1.Generalities on the Laplace-Beltrami operator. LetΩ⊂Mbe an
+open set in complete Riemannian manifold. The fundamental t one ofΩis
+defined by
+λ∗(Ω) = inf/integraltext
+Ω|gradf|2
+/integraltext
+Ωf2,
+wheretheinfimumistakenoverallsmoothnonzerofunctions f∈C∞
+0(Ω).
+Thefundamentaltonehasthefollowingmonotonicityproper ty: ifA⊂B
+are open subsets then λ∗(A)≥λ∗(B). IfΩis an open subset of Mwith
+compactclosure,orif λ∗(Ω)/\e}atio\slash∈σess(M),thenλ∗(Ω)coincideswiththefirst
+eigenvalueλ1(Ω)oftheDirichletproblem
+/braceleftbigg
+△Mu+λu=0inΩ,
+u=0on∂Ω.
+WhenΩ =Mthenλ∗(M) = infσ(M).
+The Laplace-Beltrami operator in Nof a smooth function f:N→R
+and the Laplace-Beltrami operator in Mof its extension ˜f=f◦πare
+related bythefollowingformula.
+Lemma3.1. Letf:N→Rbeasmoothfunctionandset ˜f=f◦π. Then,
+forallp∈Nand allq∈ Fp:
+(3.1)(△M˜f)q= (△Nf)p+gM/parenleftbig
+(gradM˜f)q,Hq/parenrightbig
+= (△Nf)p+gN/parenleftbig
+(gradNf)p,dπq(Hq)/parenrightbig
+.
+Theprooffollowseasilyfrom(2.2)appliedtothevectorfiel dsX= gradM˜f
+andX∗= gradNf, using(2.3).8 G. PACELLIBESSA,J. FABIOMONTENEGRO,ANDPAOLOPICCIONE
+3.2.Decomposition Principle. LetK⊂Mbe a compact set of the same
+dimension of M. The Laplace-Beltrami operator △ofMacting on the
+spaceC∞
+0(M\K)ofsmoothcompactlysupportedfunctionsof M\Khas
+a self-adjointextension,denoted by △′. TheDecompositionPrinciple [11]
+saysthatσess(M) =σess(M\K).Ontheotherhand,
+0≤λ∗(M\K) = infσ(M\K)≤infσess(M\K) =σess(M),
+thusµ= sup{λ∗(M\K),K⊂Mcompact} ≤σess(M). We will show
+thatinfσess(M)≤µ. Tothat wewillsupposethat µ<∞,otherwisethere
+is nothing to prove. Let K1⊂K2⊂ ···be a sequence of compact sets
+withM=∪∞
+i=1Ki. Wehavethat
+λ∗(M)≤λ∗(M\K1)≤λ∗(M\K2)≤ ··· →µ.
+Givenε >0, there exists f1∈C∞
+0(M\K1)with/ba∇dblf1/ba∇dblL2= 1and/integraltext
+M|gradf1|2≤λ∗(M\K1)+ε<µ+ε. Thisis/a\}b∇acketle{t(−△−µ−ε)f1,f1/a\}b∇acket∇i}htL2<0.
+Wecansupposethat supp(f1)⊂(K2\K1). Thereexists f2∈C∞
+0(M\K2)
+with/ba∇dblf2/ba∇dblL2= 1and/integraltext
+M|gradf2|2≤λ∗(M\K2) +ε < µ+ε. This is
+equivalent to /a\}b∇acketle{t(−△ −µ−ε)f2,f2/a\}b∇acket∇i}htL2<0.Moreover,/integraltext
+Mf1f2= 0since
+supp(f1)∩supp(f2) =∅. This way, we obtain an orthonormal sequence
+{fk} ⊂C∞
+0(M)suchthat /a\}b∇acketle{t(−△−µ−ε)fk,fk/a\}b∇acket∇i}htL2<0. By3.3wehavethat
+(−∞,µ]∩σess(M)/\e}atio\slash=∅andinfσess(M)≤µ. This proves the following
+proposition.
+Proposition3.2. Theinfimumoftheessentialspectrumischaracterizedby
+(3.2) infσess(M) = sup/braceleftbig
+λ∗(M\K) :Kcompactsubsetof M/bracerightbig
+.
+In particular, σess(M)isemptyif andonlyifgiven anycompactexhaustion
+K1⊂K2⊂ ··· ⊂Kn⊂...ofM, thelimit lim
+n→∞λ∗(M\Kn)isinfinite.
+LetHbe a Hilbert space and A:D ⊂H→Hbe a densely defined
+self-adjointoperator. Given λ∈R,wewriteA≥λif/a\}b∇acketle{tAx,x/a\}b∇acket∇i}ht ≥λ/ba∇dblx/ba∇dbl2for
+allx∈ D. BytheSpectralTheoremfor(unbounded)self-adjointoper ators,
+we have that A≥λiffσ(A)⊂[λ,+∞). Let us write A >−∞if there
+existsλ∗∈Rsuch thatA≥λ∗.
+Lemma3.3. LetA:D ⊂H→Hbeaself-adjointoperatorwith A>−∞,
+and letλ∈Rbe fixed. Assume that for all ε >0there exists an infinite
+dimensional subspace Gε⊂ Dsuch that /a\}b∇acketle{tAx,x/a\}b∇acket∇i}ht<(λ+ε)/ba∇dblx/ba∇dbl2for all
+x∈Gε. Then,
+σess(A)∩(−∞,λ]/\e}atio\slash=∅.
+This lemma is well known, see [8] but for sake of completeness we
+present hereitsproof.
+Proof.First we will show that σ(A)∩(−∞,λ] =σ(A)∩[λ∗,λ]/\e}atio\slash=∅.
+Takeεk= 1/k,k≥1. By our hypothesis there exists xk/\e}atio\slash= 0such that
+/a\}b∇acketle{tAxk,xk/a\}b∇acket∇i}ht<(λ+ 1/k)/ba∇dblxk/ba∇dbl2, and thusσ(A)∩[λ∗,λ+ 1/k]/\e}atio\slash=∅for all
+k≥1. Sinceσ(A)is closed, it follows σ(A)∩(−∞,λ]/\e}atio\slash=∅. We mayRIEMANNIAN SUBMERSIONS WITH DISCRETE SPECTRUM 9
+supposethat σ(A)∩(−∞,λ]/\e}atio\slash⊂σess(A),otherwisethereisnothingtoprove.
+Thus/parenleftbig
+σ(A)\σess(A)/parenrightbig
+∩(−∞,λ] ={λ1,...,λn}
+is a finite set of eigenvalues of Aof finite multiplicity. Denote by Hi⊂ D
+theλi-eigenspace of A,i= 1,...,n, and setX=/circleplustext
+iHi⊂ D. This
+is clearly an invariant subspace of A. SinceXhas finite dimension, then
+D=X⊕X1whereX1=X⊥∩ Dis also invariant by A. Denote by
+A1the restriction of Ato the Hilbert space X1which is still self-adjoint.
+Clearly,σ(A1) =σ(A)\{λ1,...,λn}andσess(A1) =σess(A). Inparticular,
+we haveσ(A1)∩(−∞,λ]⊂σess(A1). Using the infinite dimensionality
+of the space Gε, it is now easy to see that the assumptions of our lemma
+hold for the operator A1, and the first part of the proof applies to obtain
+σess(A)∩(−∞,λ] =σess(A1)∩(−∞,λ]/\e}atio\slash=∅. /square
+Letusrecallfrom[2]and[3]thefollowingestimatesforthe fundamental
+toneofopen setsofRiemannianmanifolds.
+Proposition 3.4. LetΩ⊂Mbe an open set of a Riemannian manifold.
+Then
+(3.3) λ∗(Ω)≥1
+4sup
+X
+inf
+Ωdiv(X)
+sup
+Ω/ba∇dblX/ba∇dbl
+2
+,
+wherethesupremumintakenoverallsmoothvectorfields XinΩsatisfying
+inf
+Ωdiv(X)>0,andsup
+Ω/ba∇dblX/ba∇dbl<+∞.
+Proposition 3.5. LetΩ⊂Mbe an open set of a Riemannian manifold.
+Given anysmoothvector field X∈X(Ω)then
+(3.4) λ∗(Ω)≥inf
+Ω/bracketleftBig
+div(X)−/vextendsingle/vextendsingleX/vextendsingle/vextendsingle2/bracketrightBig
+.
+Equality in (3.4)holds when λ∗(Ω)∈σ(Ω)\σess(Ω), by considering the
+fieldX=−∇(logf), wherefis the positive eigenfunction associated to
+λ∗(Ω).
+Remark3.6.Propositions (3.4) and (3.5) hold for vector fields Xof class
+C1(Ω\F)∩L∞(Ω)suchthat div(X)∈L1(Ω), whereF⊂Mis aclosed
+subset with (n−1)-Hausdorffmeasure Hn−1(F∩Ω) = 0, see [3, Lemma
+3.1].
+3.3.Fundamental tones estimates of Riemannian submersions. LetM
+andNbeconnectedRiemannianmanifoldsand π:M→NbeaRiemannian
+submersion. Denoteby △Mand△NtheLaplacianoperatoronfunctionsof
+(M,gM)andof(N,gN)respectively. Wewanttocomparethefundamental
+tones of open subsets Ω⊂Nwith the fundamental tones of its lifting
+/tildewideΩ =π−1(Ω).10 G. PACELLIBESSA,J. FABIOMONTENEGRO,ANDPAOLOPICCIONE
+Lemma 3.7. Assume that the fibers of π:M→Nare compact. Let Ωbe
+an open subset of N, and denote by /tildewideΩthe open subset of Mgiven by the
+inverseimage π−1(Ω). Then
+(3.5)/bracketleftbigg
+inf
+p∈Ωvol(Fp)/bracketrightbigg
+·λ∗(/tildewideΩ)≤/bracketleftbigg
+sup
+p∈Ωvol(Fp)/bracketrightbigg
+·λ∗(Ω).
+Inparticular,if thefibersare minimal,then
+(3.6) λ∗(/tildewideΩ)≤λ∗(Ω).
+Moreover, if inf
+p∈Ωvol(Fp)>0andsup
+p∈Ωvol(Fp)<∞then
+(3.7) inf
+p∈Ωvol(Fp)·infσess(M)≤sup
+p∈Ωvol(Fp)·infσess(N).
+Proof.Letε>0andchoosefε∈C∞
+0(Ω)such that
+(3.8)/integraldisplay
+Ω/vextendsingle/vextendsinglegradNfε/vextendsingle/vextendsingle2</parenleftbig
+λ∗(Ω)+ε/parenrightbig/integraldisplay
+Ωf2
+ε.
+Let us consider the function ˜fε=fε◦π. By the assumption that the fibers
+ofπare compact, ˜fεhas compact support in M. Using Fubini’s Theorem
+forsubmersionswehave
+/integraldisplay
+/tildewideΩ/vextendsingle/vextendsingle˜fε/vextendsingle/vextendsingle2dM=/integraldisplay
+Ω/parenleftBigg/integraldisplay
+Fp/vextendsingle/vextendsingle˜fε/vextendsingle/vextendsingle2dFp/parenrightBigg
+dN=/integraldisplay
+Ωvol(Fp)· |fε/vextendsingle/vextendsingle2dN.
+Thus
+(3.9)/integraldisplay
+/tildewideΩ/vextendsingle/vextendsingle˜fε/vextendsingle/vextendsingle2dM≥inf
+p∈Ωvol(Fp)·/integraldisplay
+Ω/vextendsingle/vextendsinglefε/vextendsingle/vextendsingle2dN.
+Similarly,using(2.3), wehave
+/integraldisplay
+/tildewideΩ/vextendsingle/vextendsinglegradM˜fε/vextendsingle/vextendsingle2=/integraldisplay
+/tildewideΩ/vextendsingle/vextendsingle/tildewidergradNfε/vextendsingle/vextendsingle2
+=/integraldisplay
+Ω/parenleftBig/integraldisplay
+Fp/vextendsingle/vextendsingle/tildewidergradNfε/vextendsingle/vextendsingle2dFp/parenrightBig
+dN (3.10)
+=/integraldisplay
+Ωvol(Fp)·/vextendsingle/vextendsinglegradNfε/vextendsingle/vextendsingle2,
+thus
+(3.11)/integraldisplay
+/tildewideΩ/vextendsingle/vextendsinglegradM˜fε/vextendsingle/vextendsingle2≤sup
+p∈Ωvol(Fp)·/integraldisplay
+Ω/vextendsingle/vextendsinglegradNfε/vextendsingle/vextendsingle2.RIEMANNIAN SUBMERSIONS WITH DISCRETE SPECTRUM 11
+Using(3.8), (3.9)and (3.11), wethenobtain
+inf
+p∈Ωvol(Fp)·λ∗(/tildewideΩ)≤inf
+p∈Ωvol(Fp)·/integraltext
+/tildewideΩ/vextendsingle/vextendsinglegradM˜fε/vextendsingle/vextendsingle2
+/integraltext
+/tildewideΩ/vextendsingle/vextendsingle˜fε/vextendsingle/vextendsingle2
+≤sup
+p∈Ωvol(Fp)·/integraltext
+Ω/vextendsingle/vextendsinglegradNfε/vextendsingle/vextendsingle2
+/integraltext
+Ω|fε|2(3.12)
+<sup
+p∈Ωvol(Fp)·[λ∗(Ω)+ε].
+This proves (3.5). If all the fibers are minimal (or more gener ally if the
+averaged mean curvature vector field Havvanishes identically on N, see
+Remark 2.3), then the volume of the fibers is constant, and ine quality (3.6)
+follows from (3.5). To prove the inequality (3.7) we pick a co mpact subset
+K⊂Mand setK0=π(K)and let/tildewideK=π−1(K0). The set/tildewideKis compact
+by the assumption that the fibers of πare compact. Let Ω =N\K0and
+/tildewideΩ =π−1(Ω) =M\/tildewideK. Clearly,/tildewideΩ⊂M\Kandthusλ∗(/tildewideΩ)≥λ∗(M\K).
+Hence, using(3.5)weget
+λ∗(M\K)≤λ∗(/tildewideΩ)≤sup
+p∈Ωvol(Fp)
+inf
+p∈Ωvol(Fp)λ∗(Ω)≤sup
+p∈Ωvol(Fp)
+inf
+p∈Ωvol(Fp)infσess(N).
+Takingthesupremumoverallcompactsubset K⊂Mintheleft-handside,
+weobtainthedesiredinequality. /square
+Now consider the case that the fibers of the submersion π:M→Nare
+compactand minimal.
+Lemma 3.8. Letπ:M→Nbe a Riemannian submersion with compact
+and minimal fibers F. Then for every open subset Ω⊂N, denoting by/tildewideΩ
+theinverseimage π−1(Ω), onehasthat
+(3.13) λ∗(/tildewideΩ) =λ∗(Ω).
+Proof.In view of (3.6), it suffices to show the inequality λ∗(/tildewideΩ)≥λ∗(Ω).
+To this aim, we will use the estimate in (3.4). We observe init ially that it
+suffices to prove the inequality when Ωis bounded. Namely, the general
+case follows from λ∗(Ω) = lim n→∞λ∗(Ωn), by considering an exhaustion
+ofΩby a sequence ofbounded open subsets Ωn. Note that Ωis bounded if
+andonlyif/tildewideΩisbounded,bythecompactnessofthefibers. Let fbethefirst
+eigenfunctionoftheproblem △Nu+λu= 0inΩwithDirichletboundary
+conditions,thatcan beassumedto bepositivein Ω.
+SetX=−gradN/parenleftbig
+logf/parenrightbig
+, so thatdivN(X)−|X|2=λ1(Ω)is constant
+inΩ. If/tildewideXis the horizontal lifting of X, then clearly |/tildewideXq|=|Xπ(q)|for all
+q∈/tildewideΩ. Moreover, by Lemma2.1, since H= 0,divM(/tildewideX)q= divN(X)π(q).12 G. PACELLIBESSA,J. FABIOMONTENEGRO,ANDPAOLOPICCIONE
+Using(3.4), wethenobtain:
+λ∗(/tildewideΩ)≥inf
+/tildewideΩ/bracketleftbig
+divM(/tildewideX)−|/tildewideX|2/bracketrightbig
+= inf
+Ω/bracketleftbig
+divN(X)−|X|2/bracketrightbig
+=λ∗(Ω).
+ThisprovesLemma(3.8). /square
+Corollary3.9. Assumethatthefibersof πarecompactandminimal. Then,
+σess(M) =∅ifand onlyif σess(N) =∅.
+The above result applies in particular to Riemannian coveri ngs, yielding
+thefollowing
+Corollary 3.10. IfMis a finite covering of N, thenσess(M)/\e}atio\slash=∅if and
+onlyifσess(N)/\e}atio\slash=∅.
+3.4.Proof of Theorem 1. The item ii. of Theorem 1 follows from this
+Lemma 3.8. For if we take a sequence of compact sets K1⊂K2⊂ ···
+withN=∪∞
+i=1Ki. Likewisewe have M=∪∞
+i=1/tildewideKi, where/tildewideKi=π−1(Ki).
+By the proof of (3.2) we have that infσess(N) = limi→∞λ∗(N\Ki)and
+infσess(M) = limi→∞λ∗(M\/tildewideKi). However,λ∗(N\Ki) =λ∗(M\/tildewideKi),
+bytheLemma(3.8). Beforeweproveitemi. weneedthefollowi nglemma.
+Lemma 3.11. Letπ:M→Nbe a Riemannian submersion with compact
+minimal fibers F. Iff∈L2(N)and△Nf∈L2(N)then/tildewidef∈L2(M)
+and△M/tildewidef=/tildewide△Nf∈L2(M). In other words, if f∈Dom(△N)then
+/tildewidef∈Dom(△M).
+Proof.Let/tildewidef=f◦πbetheliftingof f. By Fubini’sTheoremwehave
+/integraldisplay
+M/tildewidef2dM=/integraldisplay
+N/parenleftbig/integraldisplay
+Fpf2dFp/parenrightbig
+dN= vol(Fp)/integraldisplay
+Nf2dN <∞.
+This shows that /tildewidef∈L2(M). To show that △M/tildewidef∈L2(M)we will
+show that △M/tildewidef=/tildewide△Nf. Everyϕ∈C∞
+0(M)can be decomposed as
+ϕ=ϕ1+ϕ2whereϕ1is constant along the fibers Fand(ϕ2)av= 0,
+see [5]. Moreover, ϕ1andϕ2has compact support. Observe that we can
+defineψ:N→Rbyψ(π(p)) =ϕ1(p)so thatϕ1=/tildewideψ. By theLemma3.1
+we have that △Mϕ1(p) =△Nψ(π(p))for everyp∈M. By Corollary 2.4RIEMANNIAN SUBMERSIONS WITH DISCRETE SPECTRUM 13
+/integraltext
+M/tildewidef△Mϕ2dM= 0, therefore
+/integraldisplay
+M/tildewidef△MϕdM=/integraldisplay
+M/tildewidef△Mϕ1dM
+=/integraldisplay
+N/parenleftbig/integraldisplay
+Fpf△Mϕ1dFp/parenrightbig
+dN
+=/integraldisplay
+N/parenleftbig
+f△Nψ/integraldisplay
+FpdFp/parenrightbig
+dN
+= vol(Fp)/integraldisplay
+Nf△NψdN
+= vol(Fp)/integraldisplay
+Nψ△NfdN
+=/integraldisplay
+N/parenleftbig/integraldisplay
+Fpψ△NfdFp/parenrightbig
+dN
+=/integraldisplay
+M/tildewiderψ△NfdM
+=/integraldisplay
+Mϕ1/tildewide△NfdM
+=/integraldisplay
+Mϕ/tildewide△NfdM
+/square
+To show that σp(N)⊂σp(M)we takeλ∈σp(N)andf∈L2(N)with
+−△Nf=λfin distributional sense. This implies that /tildewider−△Nf=λ/tildewidef.
+By Lemma 3.11, −△M/tildewidef=λ/tildewidefshowing that λ∈σp(M). To show that
+σess(N)⊂σess(M)we takeµ∈σess(N). There exists an orthonormal
+sequenceoffunctions fk∈Dom(△)suchthat /ba∇dbl−△Nfk−µfk/ba∇dblL2(N)→0
+ask→ ∞. By Lemma3.11,wehavethat /tildewidefk∈Dom(△M). Now
+/ba∇dbl−△M/tildewidefk−µ/tildewidefk/ba∇dbl2
+L2(M)=/integraldisplay
+M|−△M/tildewidefk−µ/tildewidefk|2dM
+=/integraldisplay
+N/integraldisplay
+Fq|−△Nfk−µfk|2dFqdN
+= vol(Fq)/integraldisplay
+N|−△Nfk−µfk|2dN
+= vol(Fq)/ba∇dbl−△Nfk−µfk/ba∇dbl2
+L2(N)→0
+Thisshowsthat µ∈σess(M), theproofofTheorem1 isconcluded.
+Corollary 3.12. LetGbe a Lie group endowed with a bi-invariant metric.
+Then,σess(G)is empty if and only if for some (hence for any) compact
+subgroupK⊂G, the Riemannian homogeneous space G/Khas empty
+essentialspectrum.14 G. PACELLIBESSA,J. FABIOMONTENEGRO,ANDPAOLOPICCIONE
+Proof.ApplyTheorem1totheRiemanniansubmersion G/mapsto→G/K,which
+has minimalandcompact fibers. /square
+Other interesting examples of applications of Theorem 1 ari se from non
+compactLiegroups.
+Example. Consider the 2×2special linear group SL(2,R). There exists
+a2-parameter family of left-invariant Riemannian metrics gκ,τ, withκ<0
+andτ/\e}atio\slash= 0, for which/parenleftbig
+SL(2,R),gκ,τ/parenrightbig
+→H2(κ)is a Riemannian sub-
+mersion with geodesic fibers diffeomorphic to the circle S1. An explicit
+description of these metrics can be found, for instance, in [ 24]. Endowed
+with these metrics, SL(2,R)is one of the eight homogeneous Riemannian
+3-geometries,as classifiedin [22], anditsisometrygroupha s dimension 4.
+Proposition 3.13. Forallκ<0andτ/\e}atio\slash= 0,
+σ/parenleftbig
+SL(2,R),gκ,τ/parenrightbig
+=σess/parenleftbig
+SL(2,R),gκ,τ/parenrightbig
+=/bracketleftbig
+−κ
+4,+∞/parenrightbig
+.
+Proof.Itisknownthatthespectrum σ/parenleftbig
+H(κ)/parenrightbig
+=σess/parenleftbig
+H(κ)/parenrightbig
+=/bracketleftbig
+−κ
+4,+∞/parenrightbig
+,
+see[8]. By Lemma3.8
+λ∗/parenleftbig
+SL(2,R),gκ,τ/parenrightbig
+=λ∗/parenleftbig
+H(κ)/parenrightbig
+=−κ
+4,
+henceσ/parenleftbig
+SL(2,R),gκ,τ/parenrightbig
+⊂/bracketleftbig
+−κ
+4,+∞/parenrightbig
+. Ontheotherhand,by Theorem1
+/bracketleftbig
+−κ
+4,+∞/parenrightbig
+=σess/parenleftbig
+H(κ)/parenrightbig
+⊂σess/parenleftbig
+SL(2,R),gκ,τ/parenrightbig
+.
+Thisprovestheproposition. /square
+4. MEAN CURVATURE OF GEODESIC SPHERES VERSUS MEAN
+CURVATURE OF THE FIBERS . PROOF OF THEOREM 2.
+Wewillnowdroptheminimalityandthecompactnessassumpti ononthe
+fibers, however, we will make some assumptions on the curvatu re of the
+base and the fibers of the submersion. Assume that (N,gN)has apolep0
+or more generally has a point p0withthin cut locus , see the Introduction.
+Forp∈N\ {p0}, letγp: [0,1]→Nbe the unique affinely parameter-
+ized geodesic in (N,gN)such thatγp(0) =p0andγp(1) =p. Theradial
+curvature function of (N,gN,p0), denoted by κp0:N→R, is defined by
+κp0(p) = max
+σsec(σ), wheresecis the section curvatureand the maximum
+is taken overall 2-planesσ⊂TpNcontaining the direction γ′
+p(1). Finally,
+let us denote by ρp0:N→[0,+∞)the distance function in Ngiven by
+ρp0(p) = distN(p,p0).
+Wearenowready for
+Proofof Theorem 2. Assume first that π:M→Nis a Riemannian sub-
+mersionsatisfyingthefollowingassumptions:
+(a)(N,gN)has apolep0.
+(b) the function h(q) = (△Nρp0)π(q)+gN/parenleftbig
+(gradNρp0)π(q),dπq(Hq)is
+proper.RIEMANNIAN SUBMERSIONS WITH DISCRETE SPECTRUM 15
+Consider the lifting /tildewideρp0:M→Rdefined by/tildewideρp0=ρp0◦π. Then by
+Lemma3.1 wehavethat h=△M/tildewideρp0. Moreover,by(2.3),
+|gradM/tildewideρp0|=|gradNρp0| ≡1.
+IfK1⊂K2⊂ ··· ⊂Kn⊂...isacompactexhaustionof M,thenby(3.3)
+appliedtoX= gradM/tildewideρp0
+(4.1) λ∗(M\Kn)≥1
+4/bracketleftbig
+inf
+M\Knh/bracketrightbig2.
+Sincehis proper, the right-hand side in the above inequality tends to+∞
+asn→ ∞,thus,by Proposition3.2, σess(M) =∅.
+Ifp0hasthincutlocus,thesameproofaboveholds,since X= gradM/tildewideρp0
+satisfiestheProposition3.4 and therefore4.1,seeRemark 3 .6. /square
+Corollary 4.1. If the fibers of π:M→Nare compact and if the function
+l(p) =△Nρp0(p)−max
+q∈Fp/ba∇dblHq/ba∇dblisproperthen σess(M) =∅.
+Adifferent statementcan beobtainedintermsofradial curv ature.
+Corollary 4.2. Assume that G: [0,+∞[→Ris a smooth function such
+that:
+(4.2) κp0(p)≤ −G/parenleftbig
+ρp0(p)/parenrightbig
+for allp∈N. Denote by ψ: [0,+∞[→Rthe solution of the Cauchy
+problem:
+ψ′′(t) =G(t)ψ(t), ψ(0) = 0, ψ′(0) = 1,
+andsetℓ(t) = (n−1)ψ′(t)/ψ(t),t>0. If
+(4.3) lim
+q→∞/bracketleftBig
+ℓ/parenleftbig
+ρp0/parenleftbig
+π(q)/parenrightbig
++gN/parenleftbig
+(∇Nρp0)π(q),dπq(Hq)/parenrightbig/bracketrightBig
+= +∞
+thenσess(M) =∅.
+Proof.Using the Hessian Comparison Theorem [14, Chapter 2], under the
+assumption(4.2)onehas:
+(4.4) Hess(ρN)≥ψ′
+ψ·/parenleftbig
+gN−dρp0⊗dρp0/parenrightbig
+.
+Consideringanorthogonalbasisof TpNoftheform {∇Nρp0,e1,...,en−1},
+where{e1,...,en−1}is an orthonormal basis of TpSp, and taking the trace
+ofthesymmetricbilinearforms inthetwosidesof(4.4), weg et
+(4.5) △Nρp0≥(n−1)ψ′
+ψ.
+It is clear that (4.3) implies that h(q) =△Nρp0+gN/parenleftbig
+gradNρp0,dπq(Hq))
+isproper. /square16 G. PACELLIBESSA,J. FABIOMONTENEGRO,ANDPAOLOPICCIONE
+4.1.Warpedproducts. Let(N,gN)and(F,gF)beRiemannianmanifolds
+and letψ:N→R+be a smooth function. The warped product manifold
+M=N×ψFistheproductmanifold N×FendowedwiththeRiemannian
+metricgN+ψ2gF. It is immediate to see that the projection π:M→N
+ontothefirst factor isa Riemanniansubmersion,withfiber Fp={p}×F.
+AmongRiemanniansubmersions,warpedproductsarecharact erizedbythe
+followingproperties:
+•the horizontal distribution is integrable, and its leaves a re totally
+geodesic;
+•thefibers are totallyumbilical.
+Forwarpedproducts,theresultsofthepapercanbestatedin amoreexplicit
+form in terms of the warping function f. The mean curvature of the fibers
+aregivenby
+(4.6) H=−dim(F)gradM/tildewideψ
+/tildewideψ,
+where/tildewideψistheliftingof ψ.
+Proposition4.3. LetM=N×ψFbea warpedproduct,with Fcompact.
+(a)Ifσess(N)/\e}atio\slash=∅, and0<infNψ≤supNψ<+∞.
+Thenσess(M)/\e}atio\slash=∅.
+(b)If(N,gN)hasa polep0andthefunction
+△Nρp0−1
+ψgN/parenleftbig
+∇Nρp0,∇Nψ/parenrightbig
+isproper,then σess(M)/\e}atio\slash=∅.
+Proof.Part (a) follows from Proposition 3.7, observing that the vo lume of
+thefiberFp={p}×Fequalsψ(p)dim(F)vol(F).
+Part(b) followsfromTheorem 2 andformula(4.6). /square
+4.2.Example. LetRn= [0,∞)×Sn−1/∼endowed with the smooth
+metricds2=dt2+f2(t)dθ2, wheref(0) = 0,f′(0) = 1. The equivalence
+relation∼isthefollowing
+(t,θ)∼(s,α)⇔t=s= 0 ort=s>0 andθ=α.
+The radial sectional curvatures Kradalong the geodesic issuing the origin
+0 ={0} ×Sn−1/∼is given by Krad(t,θ) =−f′′(t)
+f(t). Let us consider
+W=Rn×S1withmetricdw2=ds2+g2(ρ(x))d2
+S1,whereρisthedistance
+function to the origin in (Rn,ds2)andg: [0,∞)→(0,∞)is a smooth
+function. Choosing f(t) =tet2we haveKrad(t,θ) =−4t3−6t, thus
+limt→∞Krad(t,θ) =−∞. By Donnelly-Li’s Theorem [11], the spectrum
+of(Rn,ds2)isdiscrete. Choosing g(t) =et−t2. Aneasycomputationyields
+that
+i. Thevolume vol(W,dw2) =∞.RIEMANNIAN SUBMERSIONS WITH DISCRETE SPECTRUM 17
+ii. Thelimit µ= limsupr→∞log(vol(BW(r)))
+r<∞,whereBW(r)is
+thegeodesicball centered at apoint p= (0,ξ)∈Wand radiusr.
+The items i.and ii. imply by Brooks’ Theorem [6] that σess(W)/\e}atio\slash=∅. This
+gives an example of a Riemannian submersion π: (W,dw2)→(Rn,ds2)
+where the base space is discrete but the total space is not, wh ile thefiber is
+compactbut notminimal.
+AnexampleofaRiemanniansubmersion π: (Rn×S1,dw2)→(Rn,ds2)
+where the total space is discrete but the base space is not, wh ile thefiber is
+compactbut notminimalispresented in[1, Proposition4.3] .
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+G. PACELLIBESSA
+DEPARTAMENTO DE MATEM´ATICA,
+UNIVERSIDADE FEDERAL DO CEARA
+BLOCO914 – C AMPUS DO PICI
+60455-760 – F ORTALEZA - CE – B RAZIL
+BESSA@MAT.UFC.BR
+J. FABIOMONTENEGRO
+DEPARTAMENTO DE MATEM´ATICA,
+UNIVERSIDADE FEDERAL DO CEARA
+BLOCO914 – C AMPUS DO PICI
+60455-760 – F ORTALEZA - CE – B RAZIL
+FABIO@MAT.UFC.BR
+PAOLOPICCIONE
+DEPARTAMENTO DE MATEM´ATICA-IME,
+UNIVERSIDADE DE S˜AOPAULO
+RUA DOMAT˜AO1010
+05508-090 S ˜AOPAULO-SP – B RAZIL
+PICCIONE .P@GMAIL.COM
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+arXiv:1001.0854v1  [astro-ph.GA]  6 Jan 2010VARIABLE STARS, THE GALACTIC HALO
+AND GALAXY FORMATION
+C. Sterken, N. Samus and L. Szabados (Eds.) 2010
+SS433: Evolution of Radio Structure in LandC
+Frequency Ranges
+A. A. Chuprikov & I. A. Guirin
+Astro Space Center of P.N. Lebedev Physical Institute of Russi an
+Academy of Sciences, Moscow, Russia
+Abstract. We analyze results of data processing of observations of well-
+known object titled SS433 with the and VLBA during more than 10 yea rs. Data
+have been processed with the software project titled ’Astro Space Locator’ (ASL
+for Windows) . The Multi Frequency Synthesis (MFS) method has been used
+for reconstruction of radio maps at 18 and 6 centimeter wavelengt h ranges (L
+and C ranges). High quality images of SS433 for several epochs are presented.
+Evolution of its radio structure is demonstrated. Astrophysical p arameters of
+object and their changes in time are discussed. Any polarization phe nomena
+are not taken into account. We present results of processing of d ata of RR
+polarization for all the observational sessions
+1. Introduction
+The object titled SS433 (J1911+0458) is known more than 30 years. It asso-
+ciated with the Super Nova Remnant titled W50which is approximately 10000
+years old. SS433 is very weak in optic (the visible magnitude value is about
++14). On the other hand, object is a powerful source of radio a nd X-ray emis-
+sion. This close binary system concists of the A-class star a nd some dark com-
+ponent. Distance to SS433 is approximately 5.5 kpc(see Brun dell(2004)). Thus,
+1 milliarcsecond in map corresponds to the linear size of app roximately 1 As-
+tronomical Unit (1 AU = 1 .5·108km). There are the following reasons of our
+interest to this object :
+•SS433 is included into source list of the scientific program o f the ’Radioas-
+tron’ ground-space VLBI mission (see Hirabayashi(2000)). Thus, it is necessary
+to have detailed information about the radio structure of th is object and its
+variability
+•object is sufficienly weak in L and C frequency ranges. It is rel evant for
+advancing of calibration procedures implemented into our s oftware project
+•SS433 has been observed with VLBA many times for different epoc hs
+In this paper we analyze results of processing of data of the f ollowing obser-
+vational sessions : GP025, GP050, BC174, BM111 . All the data were tranferred
+from archive of the NRAO (National Radio Astronomy Observatory, USA) and
+processed with with the software project titled ’Astro Spac e Locator’ (ASL for
+Windows) (see Chuprikov(2002)).
+6364 Chuprikov & Guirin
+2. Method and Results of Data Processing
+The data processing consists of the following stages :
+•Amplitude calibration of all the data using values from GC an d TY tables
+and some additional information
+•Single band fringe fitting (the primary phase calibration) o f all the data.
+Estimation of the optimum value of solution interval
+•Averaging of all the data over each frequency band
+•Multi band fringe fitting of the atmosphere and ionosphere ca librator
+data. Estimation of gain values for every frequency and ever y time interval
+•Application of gains, compensation of atmosphere and ionos phere delay
+for all the data
+•Self-calibration. Final averaging, editing of the data, an dImaging
+Amplitude calibration of the data was made with usage of two s tandard
+calibration tables ( ’Gain Curve’ (GC) and’System Temperature’ (TY) ) that
+were established with the VLBA correlator during the primar y data process-
+ing. The standard procedure of primary phase calibration al lows to reconstruct
+the visibility function phase. Then, we can obtain the dirty map of our source
+and estimate the signal-to-noise ratio (SNR). For weak sour ces, such as SS433,
+SNR depends conciderably on the phase calibration solution interval value. It
+is necessary, to find the optimal solution interval for every particular data set.
+In our case, this value changes between 10 seconds and 8 minut es. After phase
+of visibility function is reconstructed, all the data could be averaged in time
+and frequency. SNR will be increased due to this averaging. T he main difficult
+stage of the VLBI data processing is usage of phase calibrato rs for compensation
+of interferometer model errors as well as for compensation o f atmosphere and
+ionosphere delays. For this goal, it is necessary to include into any VLBI session
+schedule some additional sources (phase calibrators). The re are two criteria for
+selection of such sources :
+•They are close to the main source (at the distance no more than some
+degrees)
+•They are point-like and bright enough
+In our case, the sources titled J1913+0508, J1929+0507, J1907+0127 and
+J1950+0807 were used as such calibrators. As mentioned above, they are u sed
+to estimate the gain values for compensation of delays cause d by atmosphere,
+ionosphere and any uncertainties of the interferometer mod el.
+It is well known (see Doolin (2009)) that SS433 has at least 4 p eriodical
+modes :
+•Orbital period is approximately 13.08 days
+•Jet traces a cone with angle of approximately 40 degrees with period of
+162.375 days
+•There is a nodding superimposed in the precession of the jet a xis with a
+period of 6.06 daysSS433: EvolutionofRadioStructurein LandCFrequencyRanges 65
+•There is a prcession of ruff independently on the jet precessi on. This
+period is equal to 552.5 days
+Thus, a radio structure of object is very changeble in time. F igure 1 demon-
+strates changes in radio structure of SS433 in C range (6 cm). The ruff is seen
+clearly in the left picture. Its size is about 10 mas (or 10 AU) . The length of
+visible part of jet is approximately 30 mas (30 AU). The right picture shows the
+same object at other epoch. Here, the size of ruff seems to be co nciderably less
+(about 5 - 10 AU). Moreover, it is clear that position angles o f jet are strongly
+different for these two epochs (difference is about 20 degrees) a nd length of jet
+in the right picture is about 1.5 times less than in the left pi cture.
+Figure 1. The 6 cm radio images reconstructed of SS433. Session BM111,
+18/11/1998 (LEFT). Session BC174, 8/5/2008 (RIGHT)
+Figure 2 demonstrates changes in radio structure of SS433 in other wave-
+length range (18 cm, L range). It is clear, that object is much more changeble
+in this range. The length of visible part of jet is here about 2 00 mas (200 AU).
+Ruff is not seen so clearly in this frequency range. It seems to be visible in the
+right picture. Size of ruff seems to be about 20 - 40 AU. Again, i t is clear that
+position angles of jet are strongly different for these two epo chs (difference is
+about 40 degrees). In the right picture, length of jet is abou t 4 times less than
+in the left picture.
+3. Conclusions
+We made processing of data of some VLBA observational sessio ns had been
+made during 1998 - 2008 GP025, GP050, BC174, BM111 . The main goal of
+this investigation is to reveal the radio structure of SS433 and its changing in
+time. Results of these data processing are :
+•Theradioimages reconstucted inbothL (18 cm) andC(6cm) wav elength
+ranges confirms the existence of precessing ruff in radio stru cture of the object.66 Chuprikov & Guirin
+Figure 2. The 18 cm radio images reconstructed of SS433. Session GP050,
+30/01/1999 (LEFT). Session GP025, 20/02/2000 (RIGHT)
+The size of the ruff is approximately 40 AU for the L range. It mo re compact
+for the C range (less than 20 AU for the C range)
+•This investigation demonstrates that calibration procedu res of ’Astro
+Space Locator’ are relevant for processing of interferomet rical data for weak
+radio sources
+We’ll continue our investigations to reveal more detail inf ormation of vari-
+ability of radio structure of SS433 during one orbital perio d as well as the mod-
+ulation of this variability.
+4. Acknowlegements
+The authors would like to thank Dr. Vitaly Goranskij (Sternb erg Astronomical
+Institute, Moscow University, Russia) for very usefuldisc ussionabout thenature
+of the SS433.
+References
+Chuprikov, A. 2002, Proc. of the European VLBI Network Sympos ium held in Bonn,
+Germany, on June 25th-28th 2002, Max-Plank-Institut fuer Rad ioastronomie, 27
+Doolin, S., & Bludell, k. M. 2009, ApJ, 698, L23
+Blundel, K. M., & Bowler, M. G. 2004, ApJ, 616, L159
+Hirabayashi, H. 2000, Proc. of the VSOP Symposium, 19-21 Januar y 2000, Institute
+of Space and Astronautical Science Yoshinodai 3-1-1, Sagamihar a, Kanagawa
+229-8510, Japan, p.3
\ No newline at end of file
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+arXiv:1001.0855v1  [cond-mat.str-el]  6 Jan 2010Electronic structure of Fe 1.04(Te0.66Se0.34)
+Fei Chen1, Bo Zhou1, Yan Zhang1, Jia Wei1, Hong-Wei Ou1, Jia-Feng Zhao1,
+Cheng He1, Qing-Qin Ge1, Masashi Arita2, Kenya Shimada2, Hirofumi Namatame2,
+Masaki Taniguchi2, Zhong-Yi Lu3, Jiangping Hu4, Xiao-Yu Cui5, D. L. Feng1∗
+1Department of Physics, Surface Physics Laboratory (Nation al Key Laboratory),
+and Advanced Materials Laboratory, Fudan University, Shan ghai 200433, P. R. China
+2Hiroshima Synchrotron Radiation Center and Graduate Schoo l of Science,
+Hiroshima University, Hiroshima 739-8526, Japan.
+3Department of Physics, Renmin University of China, Beijing 100872, P. R. China
+4Department of Physics, Purdue University, West Lafayette, IN 47907, USA and
+5Swiss Light Source, Paul-Scherrer Institut, 5232 Villigen , Switzerland
+(Dated: October 31, 2018)
+We report the electronic structure of the iron-chalcogenid e superconductor, Fe 1.04(Te0.66Se0.34),
+obtained with high resolution angle-resolved photoemissi on spectroscopy and density functional cal-
+culations. In photoemission measurements, various photon energies and polarizations are exploited
+to study the Fermi surface topology and symmetry properties of the bands. The measured band
+structure and their symmetry characters qualitatively agr ee with our density function theory calcu-
+lations of Fe(Te 0.66Se0.34), although the band structure is renormalized by about a fac tor of three.
+We find that the electronic structures of this iron-chalcoge nides and the iron-pnictides have many
+aspects in common, however, significant differences exist ne ar theΓ-point. For Fe 1.04(Te0.66Se0.34),
+there are clearly separated three bands with distinct even o r odd symmetry that cross the Fermi en-
+ergy (EF) near the zone center, which contribute to three hole-like F ermi surfaces. Especially, both
+experiments and calculations show a hole-like elliptical F ermi surface at the zone center. Moreover,
+no sign of spin density wave was observed in the electronic st ructure and susceptibility measurements
+of this compound.
+I. INTRODUCTION
+The discovery of superconductivity with the super-
+conducting transition temperature ( Tc) up to 55 K in
+iron-pnictides [1–3] has generated great interests. The
+FeAs layer is considered as the key structure for super-
+conductivity in systems ranging from SmO 1−xFxFeAs,
+Ba1−xKxFe2As2[4, 5], to LiFeAs [6, 7]. Recently, certain
+iron-chalcogenides, eg.Fe1+xSe, Fe 1+yTe1−xSex[8, 9],
+have been found to be superconducting as well. Fe 1+xSe
+shows superconductivity at 8 K under ambient pres-
+sure [8] and 37 K under a 7 GPa hydrostatic pressure
+[10], which is comparable to Ba 1−xKxFe2As2(Tc=38 K)
+[5]. Because iron-chalcogenides do not involve arsenic, it
+would be particularly important for applications. Fur-
+thermore, although Fe 3dorbitals play a vital role in the
+iron-based high temperature superconductors, the anions
+seem also play an important role on various aspects, not-
+ing LaOFeP possesses a T cof merely 5 K. Besides the
+size effect, the polarizability of the anions has even been
+suggested to be crucial for the superconductivity [11].
+Therefore, iron-chalcogenides provide an opportunity to
+study the role of anions in iron-based superconductors.
+The iron-pnictides and iron-chalcogenides have many
+things in common. The FeSe(Te) layer in Fe 1+yTe1−xSex
+is isostructural to the FeAs or FeP layer in iron-pnictides.
+Moreover, the phase diagram of Fe 1+yTe1−xSexresem-
+∗Electronic address: dlfeng@fudan.edu.cn
+Figure 1: (color online) (a) Real and imaginary parts of the
+magnetic susceptibility of Fe 1.04(Te0.66Se0.34) single crystal at
+10 Gauss. The data were taken with the zero field cool (ZFC)
+procedure. (b) The low temperature magnetic susceptibilit y
+data at 10 Gauss and 1 T magnetic field.
+bles that of the iron-pnictides, where the competition
+between magnetism and superconductivity has been ob-
+served in both cases. The undoped Fe 1+yTe exhibits a
+spin density wave (SDW) ground state. With sufficient
+selenium doping, the SDW is suppressed, and the super-
+conductivity occurs at a T cas high as 15 K [9].2
+Figure 2: (color online) (a) Photoemission intensity distr ibution integrated over [EF−5meV,E F+ 5meV]window for
+Fe1.04(Te0.66Se0.34). (b) The Fermi surfaces are constructed based on the measur ed Fermi crossings, which are labeled by
+squares, circles, crosses and triangles for the β,γ,δandδ′bands respectively. (c) The photoemission intensity along the cut
+#1 in the Γ−M direction, and (d) its second derivative with respect to en ergy. (e) The data in panel c is repotted after dividing
+the angle integrated energy distribution curve. (f) The EDC ’s for data in panel c. (g) The MDC’s in the box area of panel
+c. (h) The MDC’s in the box area of panel e. The crosses mark the feature positions based on a double Lorenzian fit.(i) The
+photoemission intensity along the cut #2 in the Γ−X direction, and (j) its second derivative. Data were taken w ith circularly
+polarized 22 eV photons at HSRC.
+There are also critical differences between the iron-
+pnictides and iron-chalcogenides, in particular, between
+their structures of magnetic ordering. A common
+collinear commensurate antiferromagnetic (AFM) order-
+ing has been identified in all iron-pnictides. However,
+the magnetic state of the Fe 1+yTe1−xSexfamily has a
+bi-collinear commensurate or incommensurate antiferro-
+magnetic ordering depending on the concentration of the
+interstitial iron [12, 13]. It is still in a heated debate
+about the origin of magnetic ordering in iron-based su-
+perconductors. While models based on local moments
+have been suggested to understand both magnetic order-
+ings [14–19], the collinear AFM in the iron-pnictides in
+principle can originate from nesting mechanism between
+the hole pockets at Γand the electron pockets at M
+[20], but the bi-collinear magnetic structure is inconsis-
+tent with this picture since there is no Fermi surface at
+X. Is there a connection between the electronic struc-
+ture and magnetic ordering in the iron-chalcogenides? If
+there is, what is the connection? The answers of these
+fundamental questions require a deep understanding of
+the electronic structures of iron-chalcogenides. However ,
+there is just few data reported on the electronic structure
+of iron-chalcogenides [21].
+In this Article, we investigate the electronic structure
+of Fe1.04Te0.66Se0.34with high resolution angle-resolved
+photoemission spectroscopy (ARPES) and band calcula-
+tion. The measured Fermi surfaces and the band struc-
+ture are identified and found to qualitatively agree withthe density function theory (DFT) calculations. The
+orbital characters of individual bands are studied by
+polarization-dependence studies and found to agree with
+the calculation as well. No obvious effect of the fluctu-
+ating SDW is observed on the electronic structure. Fur-
+thermore, we found that although most aspects of the
+electronic structure of this iron-chalcogenide are simila r
+to the iron-pnictides, there are clearly three separated
+bands at the zone center for the iron-chalcogenides, while
+there appear just two separated features for the normal
+state of iron-pnictides. Moreover, the symmetry prop-
+erties of the iron-chalcogenide bands near the zone cen-
+ter are different from those of the iron-pnictides. The
+difference and similarity between the iron-pnictides and
+iron-chalcogenides in their electronic structure may shed
+light on our understanding of the role of anions and the
+superconductivity in iron-based superconductors.
+II. MATERIAL AND EXPERIMENTAL SETUP
+Fe1.04(Te0.66Se0.34) single crystal was synthesized with
+the NaCl/KCl-flux method. Fe powder, Te powder
+and Se powder were weighed according to the ratio of
+Fe:Te:Se=1:0.7:0.3 (mole), and pressed into thin plates.
+Then FeTe(Se) polycrystal was acquired by reacting
+the plate in an evacuated quartz tube at 1173 K for
+24 hours. FeTe(Se) polycrystal and the NaCl/KCl-
+flux were weighed according to the ratio of FeTe(Se):3
+Figure 3: (color online) (a, b, c) Photoemission data taken
+with 22 eV circularly polarized photons at HSRC along three
+momentum cuts as indicated in the Brillouin zone sketch. (d,
+e, f) Second derivative with respect to energy for data in
+panels a, b, c respectively. (g, h, i) Momentum distribution
+curves near EFfor data in panels a, b, c respectively.
+NaCl/KCl=1:10 (mass). They were thoroughly grounded
+into a mixture, and loaded into an evacuated quartz tube.
+The tube was kept at 1223 K for 24 hours and then slowly
+cooled to 873 K in 100 hours. Finally the quartz tube
+was cooled in the furnace after shutting off the power.
+Fe1.04(Te0.66Se0.34) single crystal was obtained after dis-
+solving the flux in deionized water. The element com-
+positions of this single crystal were determined through
+energy-dispersive x-ray (EDX) analysis with dense sam-
+pling spots across a 0.4 ×0.4 mm2surface area. The
+EDX result shows that the sample is homogeneous, and
+the maximal deviation of its compsitions is within 1.8%.
+The temperature dependence of the magnetic suscepti-
+bility (Fig. 1) does not show any signs of SDW or struc-
+tural transition. The resistivity data indicate that cryst al
+reaches the zero resistance at about 9 K. However, the
+susceptibility measurements show that although regions
+of the sample become superconducting at 9 K, it reaches
+a bulk superconducting state at 2 K, with a transition
+width less than 3 K (10 %-90%). This indicates that the
+bulk of the single crystal is quite homogeneous. With
+1 T magnetic field, superconductivity is suppressed, and
+there is no sign of field-induced meta-magnetic transition.
+The photoemission data have been taken with Scienta
+R4000 electron analyzers at Beamline 9 of Hiroshima syn-
+chrotron radiation center (HSRC) and the Surface and
+Interface Spectroscopy (SIS) Beamline of Swiss Light
+Source (SLS). The typical angular resolution is 0.3 de-
+Figure 4: (color online) (a, b, c) Photoemission data taken
+with 22 eV light at HSRC, and 50 eV, and 100 eV circularly
+polarized light at SLS respectively. All three momentum cut s
+cross the Γ-Z line ( kx=0,ky=0,kz) in the reciprocal space.
+(d, e, f) Second derivative with respect to energy for data in
+panels a, b, c respectively. (g, h, i) Momentum distribution
+curves near EFfor data in panels a, b, c respectively. (j, k, l)
+Energy distribution curves for data in panels a, b, c respec-
+tively after divided by the Fermi-Dirac distribution funct ion.
+gree, and the typical energy resolution is 15 meV. The
+sample was cleaved in situ , and measured under ultra-
+high-vacuum better than 5×10−11torr. The sample ag-
+ing effects are carefully monitored to ensure they do not
+cause artifacts in our analyses and conclusions. The SIS
+beamline is equipped with an elliptically polarized undu-
+lator (EPU), which could switch the photon polarization
+between horizontal, vertical, or circular mode. This facil -
+itates the polarization dependence studies, which is use-
+ful in determining the orbital characters of the bands [22].
+III. BAND STRUCTURE AND FERMI
+SURFACE
+The photoemission intensities distribution of
+Fe1.04(Te0.66Se0.34) at the Fermi energy is shown
+in Fig. 2(a). The data were taken at 15 K with 22 eV
+photons. Similar to the iron-pnictides, the spectral4
+Figure 5: (color online) The calculated electronic band str uc-
+ture of FeTe 0.66Se0.34along high-symmetry lines in the irre-
+ducible Brillouin zone.
+weight is mostly located around Γand M. In order to
+resolve the details of the Fermi crossings, Fig. 2(c) shows
+the photoemission intensity along the cut #1 in the
+Γ−M direction, several bands could be resolved. For a
+better visualization of the bands, Fig. 2(d) shows the
+second derivative with respect to energy for the data in
+Fig. 2(c). Three bands, α,β, andγ, could be clearly
+identified. The top of the αband is very close to the
+Fermi energy, but it is hard to judge whether it crosses
+EFbased on the energy distribution curves (EDC’s) in
+Fig. 2(f). By judging from the momentum distribution
+curves (MDC’s) in Fig. 2(g) near Γ, one finds that it
+crosses the Fermi level with a very small-sized Fermi
+surface. In order to check whether the spectral weight
+around M represents band crossings, the data in Fig. 2(c)
+is renormalized by its angular integrated spectrum and
+shown in Fig. 2(e). In this way, another band, δ, is
+resolved. The MDC’s in the boxed region are shown
+in Fig. 2(h), where one observes an electron-pocket
+type of dispersion. This is similar to the BaFe 2As2
+[23], the δband is quite weak in such an experimental
+geometry due to the strong orbital-dependence of the
+matrix element [22]. Similarly, Figs. 2(i) and (j) show
+the photoemission intensity and its second derivative
+plot along the Γ-X direction, where α,β, andγbands
+are observed. Therefore, there are totally three bands
+near the Γ-point, and they all cross the Fermi surface
+and form three hole pockets. Based on the identified
+band dispersions, the Fermi crossings are determined
+and shown in Fig. 2(b). We note because of symmetry
+constraints, only crossings for one elliptical Fermi surfa ce
+could be observed around M or M’ [22]. The experimen-
+tal Fermi surfaces are determined by fitting the Fermi
+crossings with symmetry in consideration. Assuming
+theβ,γ, andδFermi surfaces to be cylindrical, one
+could estimate the electron concentration based on the
+Figure 6: (color online) The calculated Fermi surface of
+FeTe0.66Se0.34.
+Luttinger theorem. We obtained 0.08 holes per unit
+cell. This is not inconsistent with the chemical formula,
+considering variations of the Fermi surface volume
+caused by kzdispersion of the band structure.
+To further illustrate the behavior of the αband, Fig. 3
+shows three nearby cuts taken with 22 eV photons. When
+approaching (kx= 0,ky= 0), theαband disperses
+rather rapidly with its top in each cut moving towards
+EF. Based on the peak positions in the MDC’s [Fig. 3(j)],
+one could observe a Fermi crossing of the αband very
+close to(kx= 0,ky= 0), giving a small hole-like Fermi
+surface. However, this Fermi crossing is not observed at
+several other photon energies such as 50 eV and 100 eV
+(Fig. 4). Since these momentum cuts sample through
+(kx,ky) = (0,0)at different kz, theαFermi surface is
+thus a closed pocket. Moreover, in Figs. 4(j-l), the EDC’s
+have been divided by the temperature-broadened Fermi-
+Dirac distribution, where both the αandβbands appear
+to be degenerate within the experimental resolution at
+22 eV. Based on the calculations below, it suggests that
+this data cut should be very close to the zone center, Γ.
+IV. ELECTRONIC BAND STRUCTURE
+CALCULATION
+To understand the data, we have calculated the
+electronic band structure for FeTe 0.66Se0.34. In the
+calculations the plane wave basis method was used
+[24]. We adopted the generalized gradient approxima-
+tion of Perdew-Burke-Ernzerhof [25] for the exchange-
+correlation potentials. The ultrasoft pseudopotentials5
+Figure 7: (color online) Comparison of the band structure
+of the ARPES data and the DFT calculation results along
+theΓ-M cut. Note: the energy scale of the calculated band
+structure at the right side is 3.125 times of the energy scale
+of the experimental data at the left side.
+[26] were used to model the electron-ion interactions. Af-
+ter the full convergence test, the kinetic energy cut-off
+and the charge density cut-off of the plane wave basis
+were chosen to be 600 eV and 4800 eV, respectively. The
+Gaussian broadening technique was used and a mesh of
+16×16×8k-points were sampled for the irreducible
+Brillouin-zone integration. The internal atomic coordi-
+nates within a cell were determined by the energy min-
+imization. The doping effect upon electronic structures
+was studied by using virtual crystal calculations.
+Fig. 5 displays the calculated band structure, there are
+indeed three bands near Γ, and two bands near M that
+cross the Fermi energy. In particular, the inner-most
+band near Γdoes show significant dispersion along the
+Γ−Zdirection. As a result, our calculations give five
+Fermi surfaces as shown in Fig. 6. The calculated band
+structure to a large extent resembles those of the iron-
+arsenide superconductors.
+Qualitatively, the calculated Fermi surfaces agree well
+with our experiments. However, there are some impor-
+tant quantitative discrepancies. Fig. 7 illustrates the
+measured band structure along Γ-M as reproduced from
+Fig. 2(c), together with the calculated bands. One finds
+that the size of the calculated γFermi surface is much
+smaller than the measured one. However, the calculated
+α,β, andγbands match the data after scaled by 3.125
+and shifted down by 45 meV, except the Fermi crossings
+of the measured and calculated γband are different. The
+scaling factor illustrates the correlation effects in this m a-
+terial. The experimental Fermi velocity of α,β, andγ
+bands are 0.62 eVÅ, 0.4eVÅ, and 0.137 eVÅ respec-
+tively. On the other hand, the measured Fermi surface
+around M is much smaller than the calculated ones. Simi-
+lar to the iron-pnictides [27], the renormalization factor s
+of the bands vary in different regions of the Brillouin
+zone.
+We note that in order to obtain accurate band renor-
+Figure 8: (color online) (a) Cartoon of the polarization de-
+pendence experiment, and the symmetry of the orbitals with
+respect to the mirror plane defined by sample normal and Γ-
+X. (b) The photoemission intensity along the Γ-X direction
+measured at the σgeometry, and (d) its second derivative
+with respect to energy. (c) The photoemission intensity alo ng
+theΓ-X direction measured at the πgeometry, and (e) its
+second derivative. Data were taken with 100 eV photons at
+SLS, and the temperature was 10 K.
+malization factors, it is crucial to compare the data with
+the calculation conducted for the same Se doping . We
+have calculated the band structures of FeTe 1−xSexwith
+a series of doping, and found that the band structure
+aroundΓevolves rapidly with increasing Se concentra-
+tion. Similar conclusion can be drawn from the published
+FeSe and FeTe band structures by Subedi and coworkers
+[28].
+V. POLARIZATION DEPENDENCE
+For a multi-band and multi-orbital superconductor, it
+is crucial to understand the orbital characters of the band
+structure near EF. In photoemission, such information
+can be obtained to a large extent in the polarization de-
+pendence. Fig. 8(a) illustrates two types of experimental
+setup with linearly polarized light. The incident beam
+and the sample surface normal define a mirror plane. For
+theπ(orσ) experimental geometry, the electric field di-
+rection ( ˆε) of the incident photons is in (or out of) the
+mirror plane. The matrix element of the photoemission6
+process can be described by
+|Mk
+f,i| ∝ |∝angbracketleftφk
+f|ˆε·r|φk
+i∝angbracketright|2
+, whereφk
+iandφk
+fare the initial and final state wave-
+functions respectively [29]. In our experimental setup,
+the momentum of the final-state photoelectron is in the
+mirror plane, and φk
+fcan be approximated by a plane
+wave. Therefore, φk
+fis always even with respect to the
+mirror plane. In the πgeometry, (ˆε·r)is even, to give a
+finite photoemission matrix element, |φk
+i∝angbracketrightmust be even
+with respect to the mirror plane. Thus only even state
+is probed in the πexperimental geometry. On the other
+hand, one could similarly deduce that only odd state is
+observed in the σgeometry.
+In contrast to the data measured with circularly polar-
+ized light in Figs. 2(i) and (j), only the βband is observed
+in theσgeometry [Figs. 8(b) and (d)], while just the α
+andγbands are observed in the πgeometry [Figs. 8(c)
+and (e)]. Based on the symmetry of different orbitals il-
+lustrated in Fig. 8(a), the βband is odd with respect to
+the mirror plane, while the αandγbands are even along
+theΓ-X direction. Therefore, the βband has to be made
+ofdxyand/ordyzorbitals, while the αandγbands may
+be consisted of dx2−y2,dz2, and/or dxz.
+These experimental findings of the symmetry proper-
+ties of the band structure are well captured by the band
+structure calculation. In Fig. 9, the orbital characters of
+the bands are shown by the false color plot. Near the
+Fermi energy, the αband is mainly consisted of dxz/dyz
+orbitals, which should be purely dxzalong the Γ−X
+direction. The βband is consisted of mainly dxzand
+dyzorbitals, and some dxyorbitals; while the γband is
+consisted of dx2−y2orbital. Along the Γ−Zdirection,
+the band structure near EFhas some contributions from
+thepzorbital and small contributions from the dz2or-
+bital. They all have even symmetry and thus can be
+observed in the πexperimental geometry. The small el-
+lipsoidal Fermi surface near zone center is mainly con-
+tributed by the dz2orbital for FeAs-based compounds,
+while for Fe(Te 0.66Se0.34), Te/Se pzorbital plays an im-
+portant role. To have a more quantitative picture, we
+have listed the contributions of various orbitals to the
+states at EFin Table. I, which are the coefficients of the
+calculated corresponding Bloch wavefunctions projected
+into the orbitals.
+VI. DISCUSSION
+Although the chalcogen ions contribute little spectral
+to the density-of-states (DOS) near the Fermi energy, the
+Fe3drelated band structure in iron-chalcogenides does
+show significant difference compared with that of iron-
+pnictides. Recently, it is even proposed that the polar-
+ization of the As porbitals might be the cause of the
+unconventional superconductivity in FeAs-based super-
+conductors [11]. Therefore in this regard, the electronic
+Figure 9: (color online) Contributions of various Fe 3d
+and Te/Se porbitals to the calculated band structure of
+Fe(Te 0.66Se0.34).
+consequences related to the chalcogen or pnicogen anions
+in the iron-based superconductors are particularly inter-
+esting to explore.
+In general, Fe 1.04(Te0.66Se0.34) has similar Fermi sur-
+face and band structure as the iron-pnictides [22, 30].
+However, there are some important differences. For ex-
+ample, the three bands near Γare well separated in this
+iron-chalcogenides, each with distinct symmetry. On the
+other hand in recent polarization dependence studies of
+the BaFe 1.82Co0.18As2, only two features around Γwere
+observed [22]. Moreover, the inner feature is a mixture of
+orbitals of both even and odd symmetries, while the outer
+feature is even in symmetry. Moreover, our calculations
+show that the Te 5porbitals contribute to the density
+of states near the Fermi energy, while Fe 3dz2orbital
+contributes very little in this iron-chalcogenide. Fur-
+thermore, there is a small ellipsoidal Fermi surface near
+the zone center of Fe 1.04(Te0.66Se0.34), while for iron-
+pnictides, such a small Fermi pocket has not been un-
+ambiguously observed in the paramagnetic normal state.
+Compared with the electronic structure of Fe 1+yTe
+obtained earlier [31], the Fe 1.04(Te0.66Se0.34) electronic
+structure behaves differently in the following two aspects.
+First, three bands α,β, andγ, are clearly observed
+aroundΓfor Fe1.04(Te0.66Se0.34), whereas only two bands
+were distinguished in Fe 1+yTe. Secondly, a weak Fermi
+surface was observed around X-point in Fe 1+yTe, which
+was argued to be a folded Fermi surface by the spin den-
+sity wave. For Fe 1.04(Te0.66Se0.34), neutron scattering ex-7
+Table I: The contributions of Fe 3dand Te/Se porbitals to the
+bands in Fe(Te 0.66Se0.34) nearEFalong the Γ−Xdirection.
+dz2dxzdyzdx2−y2dxypzpx+py
+α0.0391 0.6702 0 0.0182 0 0.0791 0.0767
+β 0 0 0.6579 0 0.2359 0 0.0463
+γ0.0033 0.0434 0 0.9391 0 -0.0001 0.0069
+periments have found incommensurate short-range mag-
+netic order below 50 K [12], however, our measurements
+with two different photon polarizations confirm the ab-
+sence of states near EFaround X. Furthermore, no band
+splitting that is associated with the SDW in BaFe 2As2
+and SrFe 2As2is observed here. This might suggest that
+such a short range magnetic order should be very weak
+in Fe1.04(Te0.66Se0.34).
+VII. CONCLUSION
+To summarize, we have studied the electronic struc-
+ture of Fe 1.04(Te0.66Se0.34). Both the ARPES and DFT
+calculations reveal one inner closed Fermi pocket and twoouter cylindrical Fermi surfaces near Γ, and two electron-
+like Fermi surfaces near the M-point. There are no states
+near the Fermi energy around the X-point. Polarization
+dependence measurements further elucidate the symme-
+try of the band structure. The ARPES results qualita-
+tively agree with the DFT calculations. Compared with
+the iron-pnictides, although many aspects of the band
+structures are similar, there are also significant differ-
+ences, particularly in their electronic structures near Γ
+at the paramagnetic normal state. Our results provide
+a comprehensive picture on the electronic structure of
+Fe1.04(Te0.66Se0.34), and shed new light on the role of
+anions in iron-based superconductors.
+Acknowledgments
+Part of this work was performed at the Surface and In-
+terface Spectroscopy beamline, Swiss Light Source, Paul
+Scherrer Institute, Villigen, Switzerland. We thank C.
+Hess and F. Dubi for technical support. This work was
+supported by the NSFC, MOE, MOST (National Basic
+Research Program No.2006CB921300), and STCSM of
+China.
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+arXiv:1001.0856v1  [math.PR]  6 Jan 2010Stochastic integrals for spde’s: a comparison
+Robert C. Dalang∗
+Ecole Polytechnique F´ ed´ erale de Lausanne
+Llu´ ıs Quer-Sardanyons†
+Universitat Aut` onoma de Barcelona
+Abstract
+We present the Walsh theory of stochastic integrals with res pect to martingale mea-
+sures, alongsideof theDa PratoandZabczyk theory of stocha stic integrals withrespect
+to Hilbert-space-valued Wiener processes and some other ap proaches to stochastic in-
+tegration, and we explore the links between these theories. We then show how each
+theory can be used to study stochastic partial differential eq uations, with an emphasis
+on the stochastic heat and wave equations driven by spatiall y homogeneous Gaus-
+sian noise that is white in time. We compare the solutions pro duced by the different
+theories.
+∗Partly supported by the Swiss National Foundation for Scientific Re search.
+†Partly supported by the grant MEC-FEDER Ref. MTM20006-06427 from the Direcci´ on General de
+Investigaci´ on, Ministerio de Educaci´ on y Ciencia, Spain.
+1Contents
+1 Introduction 3
+2 Stochastic integrals with respect to a Gaussian spatially homogeneous
+noise 5
+2.1 Stochastic integration with respect to a cylindrical Wi ener process . . . 5
+2.2 Spatially homogeneous noise as a cylindrical Wiener pro cess . . . . . . . 6
+2.3 The real-valued stochastic integral for spatially homo geneous noise . . . 9
+2.4 Examples of integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
+2.5 The Dalang-Mueller extension of the stochastic integra l . . . . . . . . . 13
+3 Infinite-dimensional integration theory 17
+3.1 Nuclear and Hilbert-Schmidt operators . . . . . . . . . . . . . . . . . . . 17
+3.2 Hilbert-space-valued Wiener processes . . . . . . . . . . . . . . . . . . . 19
+3.3H-valued stochastic integrals . . . . . . . . . . . . . . . . . . . . . . . . 2 0
+3.4 The case where H=R. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
+3.5 The case Tr Q= +∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
+4 Spde’s driven by a spatially homogeneous noise 28
+4.1 The random field approach . . . . . . . . . . . . . . . . . . . . . . . . . 29
+4.2 Examples: stochastic heat and wave equations . . . . . . . . . . . . . . 30
+4.3 Random field solutions with arbitrary initial condition s . . . . . . . . . 31
+4.4 Spatially homogeneous spde’s in the infinite-dimension al setting . . . . . 35
+4.5 Relation with the random field approach . . . . . . . . . . . . . . . . . . 37
+4.6 Relation with the Dalang-Mueller formulation . . . . . . . . . . . . . . . 44
+21 Introduction
+The theory of stochastic partial differential equations (spd e’s) developed on the one
+hand, from the work of J.B. Walsh [28], and on the other hand, t hrough work on
+stochastic evolution equations in Hilbert spaces, such as i n [10]. An important mile-
+stone in the latter approach is the book of Da Prato and Zabczy k [11].
+These two approaches led to the development of two distinct s chools of study for
+spde’s, based on different theories of stochastic integratio n: the Walsh theory, which
+emphasizes integration with respect to worthy martingale m easures, and a theory of
+integration with respect to Hilbert-space-valued process es, as expounded in [11]. A
+consequence of the presence of these separate theories is th at the literature published
+by each of the two schools is difficult to access when one has bee n trained in the
+other school. This is unfortunate since both approaches hav e advantages and in some
+problems, using both approaches can be useful (one example o f this is [7]).
+The objective of this paper is to help create links between th ese two schools of
+study. It is addressed to researchers who have some familiar ity with at least one of
+the two approaches. We develop both theories, and explore th e links between the two.
+Then we show how each theory is used to study spde’s. The Walsh theory emphasizes
+solutions that are random fields, while [11] centers around s olutions in Hilbert spaces
+of functions. Each theory is presented rather succinctly, t he main focus being on
+relationships between the theories. We show that these theo ries often (but not always)
+lead to the same solutions to various spde’s.
+It should be mentioned that the general theory of integratio n with respect to
+Hilbert-space-valued processes and its generalizations— such as the stochastic integral
+with respect to cylindrical processes—was well-developed several years before [28] and
+more than a decade before reference [11] appeared: see, for i nstance, the book of
+M´ etivier and Pellaumail [18]. This reference, and several others, are cited in [11] and
+[28]. However, J.B.Walshpreferredtodevelophisowninteg ral, eventhoughherealized
+that the two were related (see the Notes at the end of [28]).
+Here, we present in Section 2.1 a modern formulation of the th eory of stochastic
+integrals withrespecttocylindrical Wienerprocesses, as developedin[18], asaunifying
+integral behind most of those that were introduced later on. This integral is briefly
+recalled in Section 2.1. In Section 2.2, we show how spatiall y homogeneous noise that
+is white in time can be viewed as a cylindrical Wiener process on a particular Hilbert
+space. Emphasizing this type of noise is natural, since in re cent years, following in
+particular the papers of Mueller [20], Dalang and Frangos [4 ], Dalang [3] and Peszat
+and Zabczyk [23, 24], this type of noise has been used by sever al researchers. This is
+dueinparttothefactthat it leads toatheoryof non-linears pde’sinspatial dimensions
+greater than 1, while non-linear spde’s driven by space-tim e white noise generally only
+have a solution in spatial dimension 1. In Section 2.3, we sho w (Proposition 2.6) that
+the Walsh stochastic integral and the extension presented b y Dalang [3] and Nualart
+and Quer-Sardanyons [21] can be viewed as integrals as define d in Section 2.1. Section
+2.4 gives a wide class of integrable processes. In Section 2. 5, we discuss the relationship
+between this integral and the function-valued stochastic i ntegral introduced by Dalang
+3and Mueller in [6].
+In Section 3, we sketch the construction of the infinite dimen sional stochastic in-
+tegral in the setup of Da Prato and Zabczyk [11]. We also make u se of the more
+recent presentation of Pr´ evˆ ot and R¨ ockner [25]. In Secti on 3.1, we recall some basic
+properties of Hilbert-Schmidt operators. Section 3.2 give s the relationship between a
+Hilbert-space-valued Wiener process and a cylindrical Bro wnian motion, in the case
+wherethecovariance operatorhasfinitetrace. Hilbert-spa ce-valued stochastic integrals
+are defined in Section 3.3. In particular, we show in Proposit ion 3.4 how this infinite-
+dimensional stochastic integral can be written as a series o f Itˆ o stochastic integrals.
+This is used in Section 3.4 to show how the integrals of Sectio n 2 can be interpreted in
+the infinite-dimensional context. The case of covariance op erators with infinite-trace
+is discussed in Section 3.5.
+It is well-known that in certain cases, the Hilbert-space-v alued integral is equivalent
+to a martingale-measure stochastic integral. For instance , it is pointed out in [11,
+Section 4.3] that whentherandomperturbationis space-tim e whitenoise, thenWalsh’s
+stochastic integral in [28] is equivalent to an infinite-dim ensional stochastic integral as
+in [11] (see also [15]). Of course, space-time white noise is only a special case of
+spatially homogeneous noise, and we are interested in compa ring solutions to spde’s
+driven by this more general noise. The function-valued appr oach of [6] gives solutions
+to spde’s for which it is not known if a random field solution ex ists, and the Hilbert-
+space approach is even more general. However, for a wide clas s of spde’s that have
+solutions in two or more of these formulations, such as the st ochastic heat equation
+(d≥1) and wave equation ( d∈ {1,2,3}) driven by spatially homogeneous noise, we
+will show that the solutions turn out to be equivalent. One do es not expect this to be
+the case in all situations. Indeed, there are a few cases in wh ich a solution exists with
+one approach and is known not to exist in one of the others. For instance, for noise
+concentrated on a hyperplane, as considered in [5], the auth ors establish existence of
+function-valued solutions and show that there is no random fi eld solution.
+In Section 4, we first discuss the random field approach to the s tudy of spde’s, with
+an emphasis on the stochastic heat and wave equations. We use the stochastic integral
+of Section 2.3 to extend the result of [3] to arbitrary initia l conditions (Theorem 4.3).
+We then discuss the Hilbert-space-valued approach to the st udy of the same equations,
+taking the stochastic wave equation and the approach of [24] as a primary example.
+In particular, we show that the mild random field solution of T heorem 4.3, when
+interpreted as a Hilbert-space-valued process, yields the solution given in [24]. This
+is achieved by identifying the multiplicative non-lineari ty with an appropriate Hilbert-
+Schmidt operator, and using the relationships between stoc hastic integrals exposed
+in Section 3. Since the two solutions are defined using differen t Hilbert spaces, the
+embedding from one Hilbert space to the other has to be writte n explicitly. Finally,
+in Section 4.6, we compare the random field solution of the sto chastic wave equation
+with the function-valued solution constructed in [6]. Agai n, in cases where both types
+of solutions are defined, that is, in spatial dimensions d∈ {1,2,3}, we show that
+the random field solution yields the function-valued soluti on (Theorem 4.12). Overall,
+Section4unifiestheexistingliteratureonthestochastich eatandwaveequationsdriven
+4by spatially homogeneous noise, and clarifies the relations hips between the various
+approaches.
+2 Stochastic integrals with respect to a Gaus-
+sian spatially homogeneous noise
+In this section, we recall the notion of cylindrical Wiener p rocess and the stochastic
+integral with respect to such processes. Then we introduce a spatially homogeneous
+Gaussian noise that is white in time, and show how to interpre t this noise as a cylindri-
+cal Wiener process. Buildingonmaterial presentedin[21], wethenrelate thestochastic
+integral with respect to this particular cylindrical Wiene r process with Walsh’s mar-
+tingale measure stochastic integral and the extension give n by Dalang in [3]. We also
+discuss the extension given in Dalang and Mueller [6].
+2.1 Stochastic integration with respect to a cylindrical
+Wiener process
+Fix a Hilbert space Vwith inner product /a\}b∇acketle{t·,·/a\}b∇acket∇i}htV. Following [14, 18], we define the
+general notion of cylindrical Wiener process in V.
+Definition 2.1. LetQbe a symmetric (self-adjoint) and non-negative definite bou nded
+linear operator on V. A family of random variables B={Bt(h), t≥0, h∈V}is a
+cylindrical Wiener process onVif the following two conditions are fulfilled:
+1. for any h∈V,{Bt(h), t≥0}defines a Brownian motion with variance
+t/a\}b∇acketle{tQh,h/a\}b∇acket∇i}htV;
+2. for alls,t∈R+andh,g∈V,
+E(Bs(h)Bt(g)) = (s∧t)/a\}b∇acketle{tQh,g/a\}b∇acket∇i}htV,
+wheres∧t:= min(s,t). IfQ= IdVis the identity operator in V, thenBwill be called
+astandard cylindrical Wiener process . We will refer to Qas thecovariance ofB.
+LetFtbe theσ-field generated by the random variables {Bs(h), h∈V,0≤s≤t}
+and theP-null sets. We define the predictable σ-field as the σ-field in [0,T]×Ω
+generated by the sets {(s,t]×A, A∈ Fs,0≤s<t≤T}.
+We denote by VQthe Hilbert space Vendowed with the inner-product
+/a\}b∇acketle{th,g/a\}b∇acket∇i}htVQ:=/a\}b∇acketle{tQh,g/a\}b∇acket∇i}htV, h,g ∈V.
+We can now define the stochastic integral of any predictable s quare-integrable process
+with values in VQ, as follows. Let ( vj)jbe a complete orthonormal basis of the Hilbert
+spaceVQ. For any predictable process g∈L2(Ω×[0,T];VQ), it turns out that the
+following series is convergent in L2(Ω,F,P) and the sum does not depend on the
+chosen orthonormal system:
+g·B:=∞/summationdisplay
+j=1/integraldisplayT
+0/a\}b∇acketle{tgs,vj/a\}b∇acket∇i}htVQdBs(vj). (2.1)
+5We notice that each summand in the above series is a classical Itˆ o integral with respect
+to a standard Brownian motion, and the resulting stochastic integral is a real-valued
+random variable. The stochastic integral g·Bis also denoted by/integraltextT
+0gsdBs. The
+independence of the terms in the series (2.1) leads to the iso metry property
+E/parenleftBig
+(g·B)2/parenrightBig
+=E/parenleftBigg/parenleftbigg/integraldisplayT
+0gsdBs/parenrightbigg2/parenrightBigg
+=E/parenleftbigg/integraldisplayT
+0/ba∇dblgs/ba∇dbl2
+VQds/parenrightbigg
+.
+We note that there is an alternative way of defining this integ ral: one can start
+by defining the stochastic integral in (2.1) for a class of simplepredictable VQ-valued
+processes, and then use the isometry property to extend the i ntegral to elements of
+L2(Ω×[0,T];VQ) by checking that these simple processes are dense in this se t.
+2.2 Spatially homogeneous noise as a cylindrical Wiener
+process
+We now define the Gaussian random noise that will play a centra l role in this paper.
+On a complete probability space (Ω ,F,P), we consider a family of mean zero Gaussian
+random variables W={W(ϕ), ϕ∈ C∞
+0(Rd+1)}, whereC∞
+0(Rd+1) denotes the space of
+infinitely differentiable functions with compact support, wi th covariance
+E(W(ϕ)W(ψ)) =/integraldisplay∞
+0dt/integraldisplay
+Rddx/integraldisplay
+Rddyϕ(t,x)f(x−y)ψ(t,y) (2.2)
+=/integraldisplay∞
+0dt/integraldisplay
+Rddxf(x)(ϕ(t)∗˜ψ(t))(x),
+where “∗” denotes convolution in the spatial variable and ˜ ϕ(x) :=ϕ(−x).
+In the above, fis a non-negative and non-negative definite continuous func tion
+onRd\ {0}which is integrable in a neighborhood of 0 and is the Fourier t ransform
+of a non-negative tempered measure µonRd. That is, by definition of the Fourier
+transform on the space S′(Rd) of tempered distributions, for all ϕbelonging to the
+spaceS(Rd) of rapidly decreasing C∞functions,
+/integraldisplay
+Rdf(x)ϕ(x)dx=/integraldisplay
+RdFϕ(ξ)µ(dξ),
+and there is an integer m≥1 such that
+/integraldisplay
+Rd(1+|ξ|2)−mµ(dξ)<∞. (2.3)
+We have denoted by Fϕthe Fourier transform of ϕ∈ S(Rd):
+Fϕ(ξ) =/integraldisplay
+Rdϕ(x)e−2πiξ·xdx.
+6Themeasure µis called the spectral measure ofWandis necessarily symmetric(see [27,
+Chap. VII, Th´ eor` eme XVII]), and fis necessarily an even function. The covariance
+(2.2) can also be written, using elementary properties of th e Fourier transform, as
+E(W(ϕ)W(ψ)) =/integraldisplay∞
+0dt/integraldisplay
+Rdµ(dξ)Fϕ(t)(ξ)Fψ(t)(ξ).
+Remark 2.2. We observe that, in (2.2), it is possible to take a slightly mo re general
+spatial correlation: the function fcould be replaced by a non-negative and non-negative
+definite tempered measure (for instance, see [6, Section 2]) . Formula (2.2), which we
+use for the sake of clarity in the exposition, corresponds to the case where this measure
+is absolutely continuous with respect to Lebesgue measure o nRd, with density f.
+It is natural to associate a Hilbert space with W: letUthe completion of the
+Schwartz space S(Rd) endowed with the semi-inner product
+/a\}b∇acketle{tϕ,ψ/a\}b∇acket∇i}htU=/integraldisplay
+Rddx/integraldisplay
+Rddyϕ(x)f(x−y)ψ(y) =/integraldisplay
+Rdµ(dξ)Fϕ(ξ)Fψ(ξ),(2.4)
+ϕ,ψ∈ S(Rd), and associated semi-norm /ba∇dbl · /ba∇dblU. ThenUis a Hilbert space that may
+contain Schwartz distributions (see [3, Example 6]).
+Remark 2.3. Let˜L2(Rd,dµ)be the subspace of L2(Rd,dµ)consisting of functions
+φsuch that ˜φ=φ. It is not difficult to check that one can identify Uwith the set
+{Ψ∈ S′(Rd) : Ψ =F−1φ,whereφ∈˜L2(Rd,dµ)}, with inner product
+/a\}b∇acketle{tF−1φ,F−1ϕ/a\}b∇acket∇i}htU=/a\}b∇acketle{tφ,ϕ/a\}b∇acket∇i}htL2(Rd,dµ), φ,ϕ ∈˜L2(Rd,dµ).
+We fix a time interval [0 ,T] and we set UT:=L2([0,T];U). This set is equipped
+with the norm given by
+/ba∇dblg/ba∇dbl2
+UT=/integraldisplayT
+0/ba∇dblg(s)/ba∇dbl2
+Uds.
+We nowassociate acylindrical Wienerprocessto W, as follows. Adirect calculation
+using (2.2) shows that the generalized Gaussian random field {W(ϕ), ϕ∈ C∞
+0([0,T]×
+Rd)}is a random linear functional, in the sense that W(aϕ+bψ) =aW(ϕ)+bW(ψ),
+andϕ/mapsto→W(ϕ) is an isometry from ( C∞
+0([0,T]×Rd),/ba∇dbl · /ba∇dblUT) intoL2(Ω,F,P). The
+following lemma identifies the completion of C∞
+0([0,T]×Rd) with respect to /ba∇dbl·/ba∇dblUT.
+Lemma 2.4. The space C∞
+0([0,T]×Rd)is dense in UT=L2([0,T];U)for/ba∇dbl·/ba∇dblUT.
+Proof.Following [21], we will use the notation ϕ1(·) to indicate that ϕ1is a function
+t/mapsto→ϕ1(t) of the time-variable, and ϕ2(⋆) to indicate that ϕ2is a function x/mapsto→ϕ2(x)
+of the spatial variable.
+LetCdenote theclosure of C∞
+0([0,T]×Rd) inUTfor/ba∇dbl·/ba∇dblUT. Clearly,Cis a subspace
+ofUT. The proof can be split into three parts.
+7Step 1.We show that elements of UTof the form ϕ1(·)ϕ2(⋆), whereϕ1∈ C∞
+0(R+;R)
+with support included in [0 ,T] andϕ2∈ S(Rd), belong to C. Using the fact that
+/integraldisplayd
+Rdxf(x)(|ϕ2|∗|˜ϕ2|)(x)<∞
+becausefis a tempered function by (2.3) and |ϕ2| ∗ |˜ϕ2|decreases rapidly, together
+with dominated convergence, one checks that there is a seque nce (ϕn
+2)n⊂ C∞
+0(Rd) such
+that limn→∞/ba∇dblϕ2−ϕn
+2/ba∇dblU= 0. Then, by the very definition of the norm in UT, one
+easily proves that lim n→∞/ba∇dblϕ1ϕ2−ϕ1ϕn
+2/ba∇dblUT= 0. Therefore, ϕ1(·)ϕ2(⋆)∈UT.
+Step 2.Suppose that we are given ϕ1∈L2([0,T];R) andϕ2∈ S(Rd). We show that
+ϕ1(·)ϕ2(⋆)∈C. Indeed, let ( ϕn
+1)n∈ C∞
+0(R+) be such that, for all n, the support of
+ϕn
+1is contained in [0 ,T] andϕn
+1→ϕ1inL2([0,T];R). Thenϕn
+1ϕ2∈Cby Step 1,
+and one checks that ϕn
+1ϕ2converges, as ntends to infinity, to ϕ1ϕ2inUT. Therefore,
+ϕ1(·)ϕ2(⋆)∈C.
+Step 3.Suppose that ϕ∈UT. We show that ϕ∈C. Indeed, let ( ej)jbe a complete
+orthonormal basis of Uwithej∈ S(Rd), for allj. Then, since ϕ(s)∈Ufor any
+s∈[0,T],
+/ba∇dblϕ/ba∇dbl2
+UT=/integraldisplayT
+0/ba∇dblϕ(s)/ba∇dbl2
+Uds=∞/summationdisplay
+j=1/integraldisplayT
+0/a\}b∇acketle{tϕ(s),ej/a\}b∇acket∇i}ht2
+Uds.
+In particular, for any j≥1, the function s/mapsto→ /a\}b∇acketle{tϕ(s),ej/a\}b∇acket∇i}htUbelongs to L2([0,T];R).
+Thus, it follows from Step 2 that
+ϕn(·) :=n/summationdisplay
+j=1/a\}b∇acketle{tϕ(·),ej/a\}b∇acket∇i}htUej
+belongs toC. Moreover, it is straightforward to verify that /ba∇dblϕ−ϕn/ba∇dbl2
+UT→0 asn→ ∞.
+This shows that ϕ∈C.
+Therefore, takingintoaccount theabovelemma, W(ϕ)canbedefinedforall ϕ∈UT
+followingthestandardmethodforextendinganisometry. Th isestablishes thefollowing
+property.
+Proposition 2.5. Fort≥0andϕ∈U, setWt(ϕ) =W(1[0,t](·)ϕ(⋆)). Then the
+processW={Wt(ϕ), t≥0, ϕ∈U}is a cylindrical Wiener process as defined in
+Section 2.1, with Vthere replaced by UandQ=IdU. In particular, for any ϕ∈U,
+{Wt(ϕ), t≥0}is a Brownian motion with variance t/ba∇dblϕ/ba∇dblUand for all s,t≥0and
+ϕ,ψ∈U,E(Wt(ϕ)Ws(ψ)) = (s∧t)/a\}b∇acketle{tϕ,ψ/a\}b∇acket∇i}htU.
+With this proposition, it becomes possible to use the stocha stic integral defined in
+Section 2.1. This defines the stochastic integral g·Wfor allg∈L2(Ω×[0,T];U)≡
+L2(Ω;UT). By definition of U, the complete orthonormal basis ( ej)jin the definition
+ofg·Wcan be chosen such that ( ej)j⊂ S(Rd).
+8Before discussing this further, we first relate the statemen t of Proposition 2.5 to
+Walsh’stheoryofstochasticintegrals withrespecttomart ingalemeasures. Letusrecall
+that Walsh’s theory of stochastic integration is based on th e concept of martingale
+measure, which is a stochastic process of the form {Mt(A),Ft, t∈[0,T], A∈ Bb(Rd)},
+whereBb(Rd) denotes the set of bounded Borel sets of Rd, and (Ft)tis a filtration
+satisfying the usual conditions . For the precise definition of a martingale measure, we
+refer to [28, Chapter 2]. Hence, in order to use Walsh’s const ruction, one has first
+to extend the generalized random field {W(ϕ), ϕ∈ C∞
+0(R+×Rd)}to a martingale
+measure. More precisely, using an approximation procedure similar to the one used in
+Lemma 2.4, one extends the definition of Wto indicator functions of bounded Borel
+sets inR+×Rd(for details see [4] or [26, p.13]). Then one sets
+Mt(A) =W(1[0,t](·)1A(⋆)), t∈[0,T], A∈ Bb(Rd). (2.5)
+Moreover, if we let ( Ft)tbe the filtration generated by {Mt(A), A∈ Bb(Rd)}
+(completed and made right-continuous), then the process {Mt(A),Ft, t∈[0,T], A∈
+Bb(Rd)}defines a worthy martingale measure in the sense of Walsh [28] . Itscovariation
+measure is determined by
+/a\}b∇acketle{tM(A),M(B)/a\}b∇acket∇i}htt=t/integraldisplay
+Rd/integraldisplay
+Rd1A(x)f(x−y)1B(y)dxdy,
+t∈[0,T],A,B∈ Bb(Rd), and its dominating measure coincides with the covariance
+measure (see [4]).
+One easily checks that, for ϕ∈ S(Rd),
+Wt(ϕ) =/integraldisplay
+R+/integraldisplay
+Rd1[0,t](s)ϕ(x)M(ds,dx),
+where the integral on the right-hand side is Walsh’s stochas tic integral.
+2.3 The real-valued stochastic integral for spatially ho-
+mogeneous noise
+The aim of this section is to exhibit the relationship betwee n the stochastic integral
+constructed in Section 2.1 and the random field approach of Wa lsh [28] and Dalang
+[3]. Recall that the stochastic integral with respect to Mdefined in [28] only allows
+function-valued integrands, and this theory was extended i n [3] in order to cover more
+general integrands, such as certain processes with values i n the space of (Schwartz)
+distributions. We are going to show that these two integrals can be interpreted in the
+context of Section 2.1.
+Recall that Walsh’s stochastic integral g·Mis defined when g∈ P+, whereP+is
+the set of predictable processes ( ω,t,x)/mapsto→g(t,x;ω) such that
+/ba∇dblg/ba∇dbl2
++:=E/parenleftbigg/integraldisplayT
+0dt/integraldisplay
+Rddx/integraldisplay
+Rddy|g(t,x)|f(x−y)|g(t,y)|/parenrightbigg
+<∞.
+9Forg∈ P+, we can consider that g∈L2(Ω;UT) and set
+/ba∇dblg/ba∇dbl2
+0:=E(/ba∇dblg/ba∇dbl2
+UT) =E/parenleftbigg/integraldisplayT
+0dt/integraldisplay
+Rddx/integraldisplay
+Rddyg(t,x)f(x−y)g(t,y)/parenrightbigg
+.(2.6)
+In [3], Dalang considered the set P0, which is the completion with respect to /ba∇dbl·/ba∇dbl0
+of the subset E0ofP+that consists of functions g(s,x;ω) such that x/mapsto→g(s,x;ω)∈
+S(Rd), for allsandω, and he defined the stochastic integral g·Mfor allg∈ P0.
+Finally, in order to use the stochastic integral of Section 2 .1, let (ej)j⊂ S(Rd)
+be a complete orthonormal basis of U, and consider the cylindrical Wiener process
+{Wt(ϕ)}defined in Proposition 2.5. For any predictable process g∈L2(Ω×[0,T];U),
+the stochastic integral of gwith respect Wis
+g·W=/integraldisplayT
+0gsdWs:=∞/summationdisplay
+j=1/integraldisplayT
+0/a\}b∇acketle{tgs,ej/a\}b∇acket∇i}htUdWs(ej), (2.7)
+and the isometry property is given by
+E/parenleftBig
+(g·W)2/parenrightBig
+=E/parenleftBigg/parenleftbigg/integraldisplayT
+0gsdWs/parenrightbigg2/parenrightBigg
+=E/parenleftbigg/integraldisplayT
+0/ba∇dblgs/ba∇dbl2
+Uds/parenrightbigg
+. (2.8)
+We note that the right-hand side of (2.7) is essentially the d efinition of W(ϕ) in [19].
+We also use the notation/integraldisplayT
+0/integraldisplay
+Rdg(s,y)W(ds,dy)
+instead of/integraltextT
+0gsdWs.
+Proposition 2.6. (a) Ifg∈ P+, theng∈L2(Ω×[0,T];U)andg·M=g·W, where
+the left-hand side is a Walsh integral and the right-hand sid e is defined as in (2.7).
+(b) Ifg∈ P0, theng∈L2(Ω×[0,T];U)andg·M=g·W, where the left-hand side
+is a Dalang integral and the right-hand side is defined as in (2.7).
+Proof.Let us prove part (a) in the statement. We first observe that if g∈ P+, then
+/ba∇dblg/ba∇dbl2
+L2(Ω×[0,T];U)=E/parenleftbigg/integraldisplayT
+0dt/integraldisplay
+Rddx/integraldisplay
+Rddyg(t,x)f(x−y)g(t,y)/parenrightbigg
+≤ /ba∇dblg/ba∇dbl2
++<+∞. (2.9)
+This implies that g∈L2(Ω×[0,T];U).
+Secondly, in order to check the equality of the integrals, we use the fact that the set
+of elementary processes is dense in ( P+,/ba∇dbl· /ba∇dbl+) (see [28, Proposition 2.3]). Hence, by
+inequality (2.9), it suffices to show that both integrals coin cide whengis an elementary
+process of the form
+g(t,x;ω) = 1(a,b](s)1A(x)X(ω), (2.10)
+where 0 ≤a < b≤T,A∈ Bb(Rd) andXis a bounded and Fa-measurable random
+variable.
+10On one hand, when ghas the particular form (2.10), according to [28] and (2.5),
+/integraldisplayT
+0/integraldisplay
+Rdg(t,x)M(dt,dx) = [Mb(A)−Ma(A)]X
+=/bracketleftbig
+W(1(0,b](·)1A(⋆))−W(1(0,a](·)1A(⋆))/bracketrightbig
+X
+=W(1(a,b](·)1A(⋆))X.
+On the other hand, by the very definition of the integral (2.7) ,
+/integraldisplayT
+0gtdWt=∞/summationdisplay
+j=1/integraldisplayb
+aX/a\}b∇acketle{t1A,ej/a\}b∇acket∇i}htUdWt(ej)
+=X∞/summationdisplay
+j=1/a\}b∇acketle{t1A,ej/a\}b∇acket∇i}htU[Wb(ej)−Wa(ej)]
+=X∞/summationdisplay
+j=1/a\}b∇acketle{t1A,ej/a\}b∇acket∇i}htUW(1(a,b](·)ej)
+=XW(1(a,b](·)1A(⋆)),
+which implies that/integraldisplayT
+0/integraldisplay
+Rdg(t,x)M(dt,dx) =/integraldisplayT
+0gtdWt,
+for allgof the form (2.10). This concludes the first part of the proof.
+Concerning part (b), let us point out that P0is the completion of E0with respect
+to/ba∇dbl· /ba∇dbl0(see (2.6)), where the latter coincides with the norm in L2(Ω×[0,T];U) for
+smooth elements. Hence, since E0⊂ P+⊂L2(Ω×[0,T];U), any/ba∇dbl · /ba∇dbl0-limitgof a
+sequence (gn)n⊂ E0will determine a well-defined element in L2(Ω×[0,T];U).
+Moreover, as a consequence of this, we will only need to check the equality of
+the integrals for integrands ginE0. Since such elements are contained in P+, Dalang’s
+integral ofgwithrespecttothemartingalemeasure MturnsouttobeaWalsh integral,
+so that we can conclude by using the first part of the proof.
+Remark 2.7. According to Proposition 2.6, when one integrates an elemen t ofP+, it
+is possible to use either the Walsh integral or the integral w ith respect to a cylindrical
+Wiener process. However, the Walsh integral enjoys additio nal properties, in part
+because it is possible to make use of the dominating measure, which can be very useful
+in certain estimates. For example, establishing H¨ older co ntinuity of the solution to the
+1-dimensional stochastic wave equation, in which a Walsh int egral appears, is an easy
+exercise [28, Exercise 3.7], while for the 3-dimensional stochastic wave equation, this
+is quite involved [9].
+2.4 Examples of integrands
+Inthis section, weaim to provideusefulexamples of randomd istributionswhich belong
+toL2(Ω×[0,T];U), that is, for which we can define the stochastic integral (2. 7) with
+respect toW.
+11Recall that an element Θ ∈ S′(Rd) is anon-negative distribution with rapid decrease
+if Θ is a non-negative measure and if
+/integraldisplay
+Rd(1+|x|2)k/2Θ(dx)<+∞,
+for allk>0 (see [27]).
+Recall that µis the spectral measure of W. We consider the following hypothesis.
+Hypothesis 2.8. LetΓbe a function defined on R+with values in S′(Rd)such that,
+for allt>0,Γ(t)is a non-negative distribution with rapid decrease, and
+/integraldisplayT
+0dt/integraldisplay
+Rdµ(dξ)|FΓ(t)(ξ)|2<∞. (2.11)
+In addition, Γis a non-negative measure of the form Γ(t,dx)dtsuch that, for all T >0,
+sup
+0≤t≤TΓ(t,Rd)<∞.
+The main examples of integrands are provided by the followin g proposition (see [21,
+Proposition 3.3 and Remark 3.4]). In comparison with the ana logous result by Dalang
+[3, Theorem 2], Proposition 2.9 does not require that the sto chastic process Zhave a
+spatially homogeneous covariance (see Hypothesis A in [3]) .
+Proposition 2.9. Assume that Γsatisfies Hypothesis 2.8. Let Z={Z(t,x),(t,x)∈
+[0,T]×Rd}be a predictable process such that
+sup
+(t,x)∈[0,T]×RdE(|Z(t,x)|p)<∞, (2.12)
+for somep≥2. Then, the random measure G={G(t,dx) =Z(t,x)Γ(t,dx), t∈[0,T]}
+is a predictable process with values in Lp(Ω×[0,T];U). Moreover,
+E/parenleftbig
+/ba∇dblG/ba∇dbl2
+UT/parenrightbig
+=E/bracketleftbigg/integraldisplayT
+0dt/integraldisplay
+Rdµ(dξ)|F(Γ(t)Z(t))(ξ)|2/bracketrightbigg
+and
+E/parenleftBig
+/ba∇dblG/ba∇dblp
+UT/parenrightBig
+≤C/integraldisplayT
+0dt/parenleftBigg
+sup
+x∈RdE(|Z(t,x)|p)/parenrightBigg/integraldisplay
+Rdµ(dξ)|FΓ(t)(ξ)|2.
+The integral of G={G(t,dx) =Z(t,x)Γ(t,dx), t∈[0,T]}with respect to Wwill
+be also denoted by
+G·W=/integraldisplayT
+0/integraldisplay
+RdΓ(s,y)Z(s,y)W(ds,dy). (2.13)
+It is worth pointing out two key steps in the proof of this prop osition (see [21]): the
+first is to check that under Hypothesis 2.8, Γ belongs to UT=L2([0,T];U); the second
+is to notice that if Γ and Zsatisfy, respectively, Hypothesis 2.8 and condition (2.12 ),
+thenG(t) =Z(t,⋆)Γ(t,⋆) defines a distribution with rapid decrease, almost surely.
+12Remark 2.10. We note that [2] presents a further extension of Walsh’s stoc hastic
+integral, with which it becomes possible to integrate certa in random elements of the
+formZ(t,z)Γ(t,⋆), whereΓis a tempered distribution which is not necessarily non-
+negative. This extension is useful for studying the stochast ic wave equation in high
+spatial dimensions.
+2.5 The Dalang-Mueller extension of the stochastic inte-
+gral
+We briefly summarize here the function-valued stochastic in tegral constructed in [6].
+This is an extension of Walsh’s stochastic integral, where o ne integrates processes that
+take values in L2(Rd) (or a weighted L2-space) and the value of the integral is in the
+sameL2-space.
+Suppose that s/mapsto→Γ(s)∈ S′(Rd) satisfies:
+1) For alls≥0,FΓ(s) is a function and
+/integraldisplayT
+0dssup
+ξ∈Rd/integraldisplay
+Rdµ(dη)|FΓ(s)(ξ−η)|2<+∞.
+2) For allφ∈ C∞
+0(Rd), sup0≤s≤TΓ(s)∗φis a bounded function on Rd.
+Suppose that s/mapsto→Z(s)∈L2(Rd) satisfies:
+3) For 0 ≤s≤T,Z(s)∈L2(Rd) a.s.,Z(s) isFs-measurable, and s/mapsto→Z(s) is
+mean-square continuous from [0 ,T] intoL2(Rd).
+For such Γ and Z, one sets
+IΓ,Z:=/integraldisplayT
+0ds/integraldisplay
+RddξE/parenleftbig
+|FZ(s)(ξ)|2/parenrightbig/integraldisplay
+Rdµ(dη)|FΓ(s)(ξ−η)|2<+∞.(2.14)
+Then the stochastic integral
+vΓ,Z=/integraldisplayT
+0/integraldisplay
+RdΓ(s,⋆−y)Z(s,y)M(ds,dy) (2.15)
+is defined as an element of L2(Ω×Rd,dP×dx), such that
+E/parenleftBig
+/ba∇dblvΓ,Z/ba∇dbl2
+L2(Rd)/parenrightBig
+=IΓ,Z. (2.16)
+This definition is obtained in three steps.
+a) If, in addition to 1), Γ( s)∈ C∞(Rd), for 0≤s≤T, and in addition to 3), Z(s)∈
+C∞
+0(Rd) and there is a compact K⊂Rdsuch that supp Z(s)⊂K, for 0≤s≤T, then
+vΓ,Z(x) =/integraldisplayT
+0/integraldisplay
+RdΓ(s,x−y)Z(s,y)M(ds,dy),
+13where the right-hand side is a Walsh stochastic integral. Eq uality (2.16) is checked by
+direct calculation (see [6, Lemma 1]).
+b) If Γ is as in a) and Zsatisfies 3), then one checks that
+lim
+m→∞lim
+n→∞IΓ,Z−(Z1[−m,m])∗ψn= 0,
+where (ψn)⊂C∞
+0(Rd) is a sequence that converges to the Dirac distribution, and one
+sets
+vΓ,Z= lim
+m→∞lim
+n→∞vΓ,(Z1[−m,m])∗ψn,
+where the limits are in L2(Ω×Rd,dP×dx).
+c) If Γ satisfies 1) and 2), and Zsatisfies 3), then one checks that
+lim
+n→∞IΓ−Γ∗ψn,Z= 0
+and one sets
+vΓ,Z= lim
+n→∞vΓ∗ψn,Z,
+where the limit is in L2(Ω×Rd,dP×dx): see [6, Theorem 6].
+In comparison with the stochastic integral of Section 2.3, w e remark that the pro-
+cessZverifies sups∈[0,T]E(/ba∇dblZ(s)/ba∇dbl2
+L2(Rd))<+∞, rather than (2.12), and the resulting
+integralvΓ,Z, as a random function of x, belongs to L2(Ω×Rd).
+We now relate this stochastic integral to the one defined in Se ction 2.3.
+Proposition 2.11. Assume that ΓandZsatisfy conditions 1), 2) and 3) above. Then:
+(i) For almost all x∈Rd, the element Γ(·,x−⋆)Z(·,⋆)belongs toL2(Ω×[0,T];U).
+Hence, as in (2.7), we can define the (real-valued) stochastic integral
+IΓ,Z(T,x) :=/integraldisplayT
+0/integraldisplay
+RdΓ(s,x−y)Z(s,y)W(ds,dy),for a.a.x∈Rd.
+(ii)IΓ,Z(T,⋆)∈L2(Ω×Rd)and/ba∇dblIΓ,Z(T,⋆)/ba∇dbl2
+L2(Ω×Rd)=IΓ,Z.
+(iii)IΓ,Z(T,⋆) =vΓ,ZinL2(Ω×Rd).
+Proof.We will split the proof in three steps, which essentially cor respond to the
+construction of the Dalang-Mueller integral vΓ,Z.
+Step 1.Let us assume first that Γ and Zsatisfies the hypotheses in a) above. Then,
+as we pointed out there, for all x∈Rd, the stochastic integral vΓ,Z(x) can be defined
+as a Walsh stochastic integral. Hence, by Proposition 2.6(a ), the integrand ( s,y)/mapsto→
+Γ(s,x−y)Z(s,y) defines an element in L2(Ω×[0,T];U) and, for all x∈Rd,vΓ,Z(x) =
+IΓ,Z(T,x). Condition (ii) in the statement can be deduced from this la tter equality
+and (2.16).
+Step 2.Assume now that Γ is as in Step 1 and Zsatisfies condition 3). Then, as in b)
+above, there exists a sequence of processes ( Zn)nsuch that, for all n≥1,Znsatisfies
+14the hypotheses in a) and IΓ,Zn−Zconverges to zero as ntends to infinity. For this
+sequence,
+vΓ,Z:= lim
+n→∞vΓ,Zn= lim
+n→∞IΓ,Zn(T,⋆) (2.17)
+by Step 1, where the limit is in L2(Ω×Rd).
+We now check property (i) in the statement of the proposition . Observe that, by
+Proposition 2.9,
+/integraldisplay
+Rddx/ba∇dblΓ(·,x−⋆)[Zn(·,⋆)−Z(·,⋆)]/ba∇dbl2
+L2(Ω×[0,T];U)(2.18)
+=/integraldisplay
+RddxE/parenleftbigg/integraldisplayT
+0ds/integraldisplay
+Rdµ(dη)|F/parenleftbig
+Γ(s,x−⋆)[Zn(s,⋆)−Z(s,⋆)]/parenrightbig
+(η)|2/parenrightbigg
+.
+Use the very last lines in the proof of [6, Lemma 1] to see that t his is equal to IΓ,Zn−Z.
+Since this quantity converges to zero as n→ ∞, we deduce that there exists a subse-
+quence (nj)jsuch that, for almost all x∈Rd,
+lim
+j→∞/vextenddouble/vextenddoubleΓ(·,x−⋆)Znj(·,⋆)−Γ(·,x−⋆)Z(·,⋆)/vextenddouble/vextenddouble
+L2(Ω×[0,T];U)= 0.
+This implies that, for almost all x∈Rd, the element ( s,y)/mapsto→Γ(s,x−y)Z(s,y) belongs
+toL2(Ω×[0,T];U), and we can define the (real-valued) stochastic integral
+IΓ,Z(T,x) :=/integraldisplayT
+0/integraldisplay
+RdΓ(s,x−y)Z(s,y)W(ds,dy), (2.19)
+and
+IΓ,Z(T,x) = lim
+j→∞IΓ,Znj(T,x) inL2(Ω).
+Notice that
+/ba∇dblIΓ,Zn(T,⋆)−IΓ,Z(T,⋆)/ba∇dbl2
+L2(Ω×Rd)=/ba∇dblIΓ,Zn−Z(T,⋆)/ba∇dbl2
+L2(Ω×Rd).(2.20)
+By the isometry property (2.8), this is equal to (2.18), and t herefore to IΓ,Zn−Z, which
+tends to 0 as n→ ∞. Therefore, using Step 1, we see that
+/ba∇dblIΓ,Z(T,⋆)/ba∇dbl2
+L2(Ω×Rd)= lim
+n→∞/ba∇dblIΓ,Zn(T,⋆)/ba∇dbl2
+L2(Ω×Rd)= lim
+n→∞IΓ,Zn=IΓ,Z,
+which proves (ii). The arguments following (2.20) and (2.18 ) prove (iii).
+Step 3. In this final part, we assume that Γ and Zsatisfy conditions 1), 2) and 3).
+Then, it is a consequence of step c) above that there exists (Γ n)nsuch that, for all
+n≥1, Γnverifies the assumptions of the previous step and
+lim
+n→∞IΓn−Γ,Z= 0.
+In order to prove parts (i), (ii) and (iii) for this case, one c an follow exactly the same
+lines as we have done in Step 2. We omit the details.
+15As we will explain in Section 4.6, for the particular case of t he stochastic wave
+equation, it is useful to consider stochastic integrals of t he formvΓ,Zwhich take values
+in some weighted L2-space. We now describe this situation.
+Fixk > dand letθ:Rd→Rbe a smooth function for which there are constants
+0<c<C such that
+c(1∧|x|−k)≤θ(x)≤C(1∧|x|−k).
+The weighted L2-spaceL2
+θis the set of measurable g:Rd→Rsuch that /ba∇dblg/ba∇dblθ<+∞,
+where
+/ba∇dblg/ba∇dbl2
+θ=/integraldisplay
+Rd|g(x)|2θ(x)dx.
+Consider a function s/mapsto→Γ(s)∈ S′(Rd) that satisfies 1), 2) above, and, in addition,
+4) There is R>0 such that for s∈[0,T], supp Γ(s)⊂B(0,R).
+For a stochastic process Z, we consider the following hypothesis:
+5) For 0 ≤s≤T,Z(s)∈L2
+θa.s.,Z(s) isFs-measurable, and s/mapsto→Z(s) is mean-
+square continuous from [0 ,T] intoL2
+θ.
+Then the stochastic integral
+vθ
+Γ,Z=/integraldisplayT
+0/integraldisplay
+RdΓ(s,⋆−y)Z(s,y)M(ds,dy) (2.21)
+is defined as an element of L2(Ω×Rd,dP×θ(x)dx), such that
+E(/ba∇dblvθ
+Γ,Z/ba∇dbl2
+L2
+θ)≤Iθ
+Γ,Z,
+where
+Iθ
+Γ,Z:=/integraldisplayT
+0dsE(/ba∇dblZ(s,⋆)/ba∇dbl2
+L2
+θ) sup
+ξ∈Rd/integraldisplay
+Rdµ(dη)|FΓ(s)(ξ−η)|2.
+This definition is obtained by showing that Zn(s,⋆) :=Z(s,⋆)1[−n,n](⋆) also satisfies 5)
+as well as 3). Therefore, vθ
+Γ,Zn=vΓ,Znis defined as an element of L2(Ω×Rd,dP×dx),
+and one checks that this element also belongs to L2(Ω×Rd,dP×θ(x)dx), and
+lim
+n→∞Iθ
+Γ,Z−Zn= 0,
+provided that Γ satisfies 1), 2) and 4). Then one sets
+vθ
+Γ,Z= lim
+n→∞vΓ,Zn,
+where the limit is in L2(Ω×Rd,dP×θ(x)dx): see [6, Theorem 12].
+163 Infinite-dimensional integration theory
+In this section, we first sketch the construction of the infini te dimensional stochastic
+integral in the setup of Da Prato and Zabczyk in [11]. For this , we will define the
+general concept of Hilbert-space-valued Q-Wiener process and study its relationship
+with the cylindrical Wiener process considered in Section 2 .1. Then we will show that
+thestochastic integral constructedinSection 2.1 canbein sertedintothis moreabstract
+setting. In particular, we will treat specifically the case o f the standard cylindrical
+Wiener process given by the spatially homogeneous noise des cribed in Section 2.2.
+We begin by recalling some facts concerning nuclear and Hilb ert-Schmidt operators
+on Hilbert spaces.
+3.1 Nuclear and Hilbert-Schmidt operators
+LetE,Gbe Banach spaces and let L(E,G) be the vector space of all linear bounded
+operators from EintoG. We denote by E∗andG∗the dual spaces of EandG,
+respectively.
+An element T∈L(E,G) is said to bea nuclearoperator if thereexist two sequences
+(aj)j⊂Gand (ϕj)j⊂E∗such that
+T(x) =∞/summationdisplay
+j=1ajϕj(x),for allx∈E,
+and∞/summationdisplay
+j=1/ba∇dblaj/ba∇dblG/ba∇dblϕj/ba∇dblE∗<+∞.
+The space of all nuclear operators from EintoGis denoted by L1(E,G). When
+endowed with the norm
+/ba∇dblT/ba∇dbl1= inf
+
+∞/summationdisplay
+j=1/ba∇dblaj/ba∇dblG/ba∇dblϕj/ba∇dblE∗:T(x) =∞/summationdisplay
+j=1ajϕ(x), x∈E
+
+,
+it is a Banach space.
+LetHbe a separable Hilbert space and let ( ej)jbe a complete orthonormal basis
+inH. ForT∈L1(H,H), thetraceofTis
+TrT=∞/summationdisplay
+j=1/a\}b∇acketle{tT(ej),ej/a\}b∇acket∇i}htH. (3.1)
+Oneproves that if T∈L1(H) :=L1(H,H), thenTrTis awell-defined real numberand
+its value does not depend on the choice of the orthonormal bas is (see, for instance, [11,
+Proposition C.1]). Further, according to [11, Proposition C.3], a non-negative definite
+operatorT∈L(H) is nuclear if and only if, for an orthonormal basis ( ej)jonH,
+∞/summationdisplay
+j=1/a\}b∇acketle{tT(ej),ej/a\}b∇acket∇i}htH<+∞.
+17Moreover, in this case, Tr T=/ba∇dblT/ba∇dbl1.
+LetVandHbe two separable Hilbert spaces and ( ek)ka complete orthonormal
+basis ofV. A bounded linear operator T:V→His said to be Hilbert-Schmidt if
+∞/summationdisplay
+k=1/ba∇dblT(ek)/ba∇dbl2
+H<+∞.
+It turns out that the above property is independent of the cho ice of the basis in V. The
+set of Hilbert-Schmidt operators from VintoHis denoted by L2(V,H). The norm in
+this space is defined by
+/ba∇dblT/ba∇dbl2=/parenleftBigg∞/summationdisplay
+k=1/ba∇dblT(ek)/ba∇dbl2
+H/parenrightBigg1/2
+, (3.2)
+and defines a Hilbert space with inner product
+/a\}b∇acketle{tS,T/a\}b∇acket∇i}ht2=∞/summationdisplay
+k=1/a\}b∇acketle{tS(ek),T(ek)/a\}b∇acket∇i}htH. (3.3)
+Finally, let us point out that (3.1) and (3.2) imply that if T∈L2(V,H), thenTT∗∈
+L1(H), whereT∗is the adjoint operator of T, and
+/ba∇dblT/ba∇dbl2
+2= Tr (TT∗). (3.4)
+We conclude this section by recalling the definition and some properties of the
+pseudo-inverse of bounded linear operators (see, for insta nce, [25, Appendix C]).
+LetT∈L(V,H) and KerT:={x∈V:T(x) = 0}. Thepseudo-inverse of the
+operatorTis defined by
+T−1:=/parenleftBig
+T|(KerT)⊥/parenrightBig−1
+:T(V)→(KerT)⊥.
+Notice that Tis one-to-one on (Ker T)⊥(the orthogonal complement of Ker T) and
+T−1is linear and bijective.
+IfT∈L(V) is a boundedlinear operator defined on VandT−1denotes the pseudo-
+inverse ofT, then (see [25, Proposition C.0.3]):
+1. (T(V),/a\}b∇acketle{t·,·/a\}b∇acket∇i}htT(V)) defines a Hilbert space, where
+/a\}b∇acketle{tx,y/a\}b∇acket∇i}htT(V):=/a\}b∇acketle{tT−1(x),T−1(y)/a\}b∇acket∇i}htV, x,y ∈T(V).
+2. Let (ek)kbe an orthonormal basis of (Ker T)⊥. Then (T(ek))kis an orthonormal
+basis of (T(V),/a\}b∇acketle{t·,·/a\}b∇acket∇i}htT(V)).
+Finally, accordingto[25, CorollaryC.0.6], if T∈L(V,H)andweset Q:=TT∗∈L(H),
+then we have Im Q1/2= ImTand
+/vextenddouble/vextenddouble/vextenddoubleQ−1/2(x)/vextenddouble/vextenddouble/vextenddouble
+H=/ba∇dblT−1(x)/ba∇dblV, x∈ImT,
+whereQ−1/2is the pseudo-inverse of Q1/2.
+183.2 Hilbert-space-valued Wiener processes
+The stochastic integral presented in Da Prato and Zabczyk [1 1] is defined with respect
+to a class of Hilbert-space-valued processes, namely Q-Wiener processes, which we now
+introduce.
+We consider a separable Hilbert space Vand a linear, symmetric (self-adjoint)
+non-negative definite and bounded operator QonVsuch that Tr Q<+∞.
+Definition 3.1. AV-valued stochastic process {Wt, t≥0}is called a Q-Wiener
+process if (1) W0= 0, (2)Whas continuous trajectories, (3) Whas independent
+increments, and (4) the law of Wt− Wsis Gaussian with mean zero and covariance
+operator (t−s)Q, for all0≤s≤t.
+We recall that according to [11, Section 2.3.2], condition ( 4) above means that for
+anyh∈Vand 0≤s≤t, the real-valued random variable /a\}b∇acketle{tWt−Ws,h/a\}b∇acket∇i}htVis Gaussian,
+with mean zero and variance ( t−s)/a\}b∇acketle{tQh,h/a\}b∇acket∇i}htV. In particular, using (3.1), we see that
+E(/ba∇dblWt/ba∇dbl2
+V) =tTrQ, which is one reason why the assumption Tr Q<∞is essential.
+Let (ej)jbe an orthonormal basis of Vthat consists of eigenvectors of Qwith
+corresponding eigenvalues λj,j∈N∗. Let (βj)jbe a sequence of independent real-
+valuedstandardBrownianmotionsonaprobabilityspace(Ω ,F,P). Thenthe V-valued
+process
+Wt=∞/summationdisplay
+j=1/radicalbig
+λjβj(t)ej (3.5)
+(where the series converges in L2(Ω;C([0,T];V))), defines a Q-Wiener process on V
+(see (2.1.2) in [25]). We note that/radicalbig
+λjej=Q1/2(ej). In the special case where Vis
+finite-dimensional, say dim V=n, thenQcan be identified with an n×n-matrix which
+is the variance-covariance matrix of {Wt}, and{Wt}has the same law as {Q1/2W0
+t},
+where{W0
+t}is a standard Brownian motion with values in Rn.
+If{Wt, t≥0}is aQ-Wiener process on V, there is a natural way to associate
+to it a cylindrical Wiener process in the sense of Definition 2 .1. Namely, for any
+h∈Vandt≥0, we setWt(h) :=/a\}b∇acketle{tWt,h/a\}b∇acket∇i}htV. Using polarization, one checks that
+{Wt(h), t≥0, h∈V}is a cylindrical Wiener process on Vwith covariance operator
+Q. Note that in this case, Wt(ej) =/radicalbig
+λjβj(t), so the Brownian motions βjin (3.5)
+are given by βj(t) =Wt(vj), where
+vj=λ−1/2
+jej=Q−1/2(ej),forj≥1 withλj/\e}atio\slash= 0. (3.6)
+In particular, ( vj)jis a complete orthonormal basis of the space VQof Section 2.1.
+However, it is not true in general that any cylindrical Wiene r process is associated
+to aQ-Wiener process on a Hilbert space. Indeed, we have the follo wing result (see
+[18, p.177]).
+Theorem 3.2. LetVbe a separable Hilbert space and Wa cylindrical Wiener process
+onVwith covariance Q. Then, the following three conditions are equivalent:
+1.Wis associated to a V-valuedQ-Wiener process W, in the sense that /a\}b∇acketle{tWt,h/a\}b∇acket∇i}htV=
+Wt(h), for allh∈V.
+192. For any t≥0,h/mapsto→Wt(h)defines a Hilbert-Schmidt operator from Vinto
+L2(Ω,F,P).
+3.TrQ<+∞.
+If any one of the above conditions holds, then the norm of the Hi lbert-Schmidt operator
+h/mapsto→Wt(h), as an element of L2/parenleftbig
+V,L2(Ω,F,P)/parenrightbig
+, is given by
+/ba∇dblWt/ba∇dbl2=E(/ba∇dblWt/ba∇dbl2
+V) =tTrQ.
+As a consequence of the above result, if dim V= +∞and ifWis a standard
+cylindrical Wiener process on V, that isQ= IdV, then there is no Q-Wiener process
+Wassociated to W. However, as we will explain in Section 3.5, it will be possib le
+to find a Hilbert-space-valued Wiener process with values in a larger Hilbert space V1
+which will correspond to Win a certain sense.
+3.3H-valued stochastic integrals
+We now sketch the construction of the infinite-dimensional s tochastic integral of [11].
+LetVandHbe two separable Hilbert spaces and let {Wt, t≥0}be aQ-Wiener
+process defined on V. We note by ( Ft)tthe (completed) filtration generated by W. In
+[11], the objective is to construct the H-valued stochastic integral
+/integraldisplayt
+0ΦsdWs, t∈[0,T],
+where Φ is a process with values in the space of linear but not n ecessarily bounded
+operators from VintoH.
+Consider the subspace V0:=Q1/2(V) ofVwhich, endowed with the inner product
+/a\}b∇acketle{th,g/a\}b∇acket∇i}ht0:=/a\}b∇acketle{tQ−1/2h,Q−1/2g/a\}b∇acket∇i}htV,
+is a Hilbert space. Here Q−1/2denotes the pseudo-inverse of the operator Q1/2(see
+Section 3.1). Let us also set
+L0
+2:=L2(V0,H),
+which is the Hilbert space of all Hilbert-Schmidt operators fromV0intoH, equipped,
+as in (3.3), with the inner product
+/a\}b∇acketle{tΦ,Ψ/a\}b∇acket∇i}htL0
+2=∞/summationdisplay
+j=1/a\}b∇acketle{tΦ˜ej,Ψ˜ej/a\}b∇acket∇i}htH,Φ,Ψ∈L0
+2, (3.7)
+where (˜ej)jis any complete orthonormal basis of V0. In particular, using the fact that
+we can take
+˜ej=/radicalbig
+λjej=Q1/2(ej), j≥1, λj>0, (3.8)
+where the ( ej)jare as in (3.5) (see condition 2. in the final part of Section 3. 1) and
+applying (3.4), the norm of Ψ ∈L0
+2can be expressed as
+/ba∇dblΨ/ba∇dbl2
+L0
+2=/ba∇dblΨ◦Q1/2/ba∇dbl2
+L2(V,H)= Tr (ΨQΨ∗).
+20We note that in the case where dim V=n<+∞and dimH=m<+∞, then it is
+natural to identify Ψ ∈L0
+2with anm×n-matrix and Qwith ann×n-matrix. The
+norm of Ψ corresponds to a classical matrix norm of Ψ Q1/2(whose square is the sum
+of squares of entries of Ψ Q1/2).
+Let Φ = {Φt, t∈[0,T]}be a measurable L0
+2-valued process. We define the norm
+of Φ by
+/ba∇dblΦ/ba∇dblT:=/bracketleftbigg
+E/parenleftbigg/integraldisplayT
+0/ba∇dblΦs/ba∇dbl2
+L0
+2ds/parenrightbigg/bracketrightbigg1/2
+.
+The aim of [11, Chapter 4], is to define the stochastic integra l with respect to Wof any
+L0
+2-valued predictable process Φ such that /ba∇dblΦ/ba∇dblT<∞. More precisely, Da Prato and
+Zabczyk first consider simpleprocesses, which are of the form Φ t= Φ01(a,b](t), where
+Φ0is anyFa-measurable L(V,H)-valued random variable and 0 ≤a<b≤T. For such
+processes, the stochastic integral takes values in Hand is defined by the formula
+/integraldisplayt
+0ΦsdWs:= Φ0(Wb∧t−Wa∧t), t∈[0,T]. (3.9)
+The map Φ /mapsto→/integraltext·
+0ΦsdWsis an isometry between the set of simple processes and the
+spaceMHof square-integrable H-valued ( Ft)-martingales X={Xt, t∈[0,T]}en-
+dowed with the norm /ba∇dblX/ba∇dbl= [E(/ba∇dblXT/ba∇dbl2
+H)]1/2. Indeed, as it is proved in [11] (see also
+[25, Proposition 2.3.5]), the isometry property for simple processes reads
+E/parenleftBigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayT
+0ΦtdWt/vextenddouble/vextenddouble/vextenddouble/vextenddouble2
+H/parenrightBigg
+=/ba∇dblΦ/ba∇dbl2
+T. (3.10)
+Remark 3.3. The appearance of /ba∇dbl·/ba∇dblTcan be understood by considering the case where
+Φ(t) = Φ01(a,b](t), whereΦ0∈L(V,H)is deterministic and 0≤a<b≤T. Indeed, in
+this case, using (3.9)and the representation (3.5),
+E/parenleftBigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayT
+0ΦtdWt/vextenddouble/vextenddouble/vextenddouble/vextenddouble2
+H/parenrightBigg
+=E
+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay
+j/radicalbig
+λj(βj(b)−βj(a))Φ0(ej)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2
+H
+,
+and the right-hand side is equal to
+/summationdisplay
+jλj(b−a)/ba∇dblΦ0(ej)/ba∇dbl2
+H= (b−a)/summationdisplay
+j/vextenddouble/vextenddouble/vextenddoubleΦ0(Q1/2ej)/vextenddouble/vextenddouble/vextenddouble2
+H= (b−a)/vextenddouble/vextenddouble/vextenddoubleΦ0◦Q1/2/vextenddouble/vextenddouble/vextenddouble2
+L2(V,H)
+=E/parenleftbigg/integraldisplayT
+0/ba∇dblΦs/ba∇dbl2
+L0
+2ds/parenrightbigg
+.
+Once the isometry property (3.10) is established, a complet ion argument is used to
+extend the above definition to all L0
+2-valued predictable processes Φ satisfying /ba∇dblΦ/ba∇dblT<
+∞. The integral of Φ is denoted by
+Φ·W=/integraldisplayT
+0ΦtdWt
+21and the isometry property (3.10) is preserved for such proce sses:
+E(/ba∇dblΦ·W/ba∇dbl2
+H) =/ba∇dblΦ/ba∇dbl2
+T.
+The details of this construction can be found in [11, Chapter 4].
+Let us conclude this section by providing a representation o f the stochastic integral
+Φ·Win terms of ordinary Itˆ o integrals of real-valued processe s. Indeed, observe first
+that the expansion (3.5) can be rewritten in the form
+Wt=∞/summationdisplay
+j=1βj(t)˜ej, (3.11)
+where (˜ej)jis defined in (3.8).
+Proposition 3.4. Let(fk)kbe a complete orthonormal system in the Hilbert space
+H. Assume that Φ ={Φt, t∈[0,T]}is anyL0
+2-valued predictable process such that
+/ba∇dblΦ/ba∇dblT<∞. Then
+/integraldisplayT
+0ΦtdWt=∞/summationdisplay
+k=1
+∞/summationdisplay
+j=1/integraldisplayT
+0/a\}b∇acketle{tΦt(˜ej),fk/a\}b∇acket∇i}htHdβj(t)
+fk. (3.12)
+Proof.First of all, we will prove that, under the standing hypothes es, the right-hand
+side of (3.12) is a well-defined element in L2(Ω;H). For this, we will check that
+E
+∞/summationdisplay
+k=1
+∞/summationdisplay
+j=1/integraldisplayT
+0/a\}b∇acketle{tΦt(˜ej),fk/a\}b∇acket∇i}htHdβj(t)
+2
+=/ba∇dblΦ/ba∇dbl2
+T,
+where the right-hand side is finite, by assumption.
+Since (βj)jis a family of independent standard Brownian motions,
+E
+∞/summationdisplay
+k=1
+∞/summationdisplay
+j=1/integraldisplayT
+0/a\}b∇acketle{tΦt(˜ej),fk/a\}b∇acket∇i}htHdβj(t)
+2
+=∞/summationdisplay
+k,j=1E/bracketleftBigg/parenleftbigg/integraldisplayT
+0/a\}b∇acketle{tΦt(˜ej),fk/a\}b∇acket∇i}htHdβj(t)/parenrightbigg2/bracketrightBigg
+,
+and the right-hand side is equal to
+∞/summationdisplay
+k,j=1/integraldisplayT
+0E[/a\}b∇acketle{tΦt(˜ej),fk/a\}b∇acket∇i}ht2
+H]dt=E
+/integraldisplayT
+0∞/summationdisplay
+j=1/ba∇dblΦt(˜ej)/ba∇dbl2
+Hdt
+=E/bracketleftbigg/integraldisplayT
+0/ba∇dblΦt/ba∇dbl2
+L0
+2dt/bracketrightbigg
+,
+and the last term is equal to /ba∇dblΦ/ba∇dbl2
+T. Hence, the series on the right-hand side of (3.12)
+defines an element in L2(Ω;H) and its norm is given by /ba∇dblΦ/ba∇dblT. Therefore, by the
+isometry property of the stochastic integral (see (3.10)), in order to prove equality
+(3.12), we only need to check this equality for simple proces ses. Namely, assume that
+22Φ is of the form Φ t= Φ01(a,b](t), where Φ 0is aFa-measurable L(V,H)-valued random
+variable and 0 ≤a<b≤T. Then, by (3.9),
+/integraldisplayT
+0ΦtdWt= Φ0(Wb−Wa).
+On the other hand,
+∞/summationdisplay
+k=1
+∞/summationdisplay
+j=1/integraldisplayT
+0/a\}b∇acketle{tΦt(˜ej),fk/a\}b∇acket∇i}htHdβj(t)
+fk=∞/summationdisplay
+k,j=1/a\}b∇acketle{tΦ0(˜ej),fk/a\}b∇acket∇i}htH(βj(b)−βj(a))fk,
+and the right-hand side is equal to
+∞/summationdisplay
+j=1(βj(b)−βj(a))Φ0(˜ej) = Φ0
+∞/summationdisplay
+j=1(βj(b)−βj(a))˜ej
+= Φ0(Wb−Wa),
+where the last equality follows from (3.11). The proof is com plete.
+3.4 The case where H=R
+We consider a cylindrical Wiener process Won some separable Hilbert space Vwith
+covarianceQ, such that Tr Q<+∞. By Theorem 3.2, Wis associated to a V-valued
+Q-Wiener process W. We shall check that the stochastic integral with respect to W,
+constructed in Section 2.1, is equal to an integral with resp ect toW, constructed in
+[11] and sketched in Section 3.3, when the Hilbert space Hin which the integral takes
+its values is H=R.
+In Section 2.1, we defined the Hilbert space VQand the stochastic integral
+g·W=/integraldisplayT
+0gsdWs,
+forany predictablestochastic process g∈L2(Ω×[0,T];VQ), withtheisometry property
+E/parenleftbig
+(g·W)2/parenrightbig
+=E/parenleftbigg/integraldisplayT
+0/ba∇dblgs/ba∇dbl2
+VQds/parenrightbigg2
+.
+For anys∈[0,T] andg∈L2(Ω×[0,T];VQ), we define an operator Φg
+s:V→Rby
+Φg
+s(η) :=/a\}b∇acketle{tgs,η/a\}b∇acket∇i}htV, η∈V. (3.13)
+We denote by L0
+2the setL2(V0,H), withV0=Q1/2(V) andH=R.
+Proposition 3.5. Under the above assumptions, Φg={Φg
+s, s∈[0,T]}defines a
+predictable process with values in L0
+2=L2(V0,R), such that
+E/parenleftbigg/integraldisplayT
+0/ba∇dblΦg
+s/ba∇dbl2
+L0
+2ds/parenrightbigg
+=E/parenleftbigg/integraldisplayT
+0/ba∇dblgs/ba∇dbl2
+VQds/parenrightbigg
+. (3.14)
+23Therefore, the stochastic integral of Φgwith respect to Wcan be defined as in Section
+3.3 and in fact,/integraldisplayT
+0Φg
+sdWs=/integraldisplayT
+0gsdWs. (3.15)
+Proof.We first check (3.14). Let ejbe as in (3.5), ˜ ejbe as in (3.8) and vjbe as in
+(3.6), so that ˜ ej=Q(vj). By (3.7) with H=R, and by (3.13),
+/ba∇dblΦg
+s/ba∇dbl2
+2=∞/summationdisplay
+j=1/a\}b∇acketle{tgs,˜ej/a\}b∇acket∇i}ht2
+V=∞/summationdisplay
+j=1/a\}b∇acketle{tgs,Qvj/a\}b∇acket∇i}ht2
+V=∞/summationdisplay
+j=1/a\}b∇acketle{tgs,vj/a\}b∇acket∇i}ht2
+VQ=/ba∇dblgs/ba∇dbl2
+VQ.
+We conclude that (3.14) holds. We note for later reference th at this equality /ba∇dblΦg
+s/ba∇dbl2=
+/ba∇dblgs/ba∇dblVQremains valid even if Tr Q= +∞.
+Since, by hypothesis, the right hand-side of (3.14) is finite , we deduce that Φgis
+a square integrable process with values in L0
+2and the stochastic integral/integraltextT
+0Φg
+sdWsis
+well-defined.
+It remains to prove (3.15). For this, we apply Proposition 3. 4 in the following
+situation:H=R, with one basis vector fk= 1, Φ is defined in (3.13), and the
+sequence of independent standard Brownian motions in (3.12 ) is given by βj(t) =
+Wt(vj). Therefore,/integraldisplayT
+0Φg
+tdWt=∞/summationdisplay
+j=1/integraldisplayT
+0Φg
+t(˜ej)dβj(t),
+and the right-hand side is equal to
+∞/summationdisplay
+j=1/integraldisplayT
+0/a\}b∇acketle{tgt,˜ej/a\}b∇acket∇i}htVdWt(vj) =∞/summationdisplay
+j=1/integraldisplayT
+0/a\}b∇acketle{tgt,vj/a\}b∇acket∇i}htVQdWt(vj) =/integraldisplayT
+0gtdWt.
+This completes the proof.
+3.5 The case Tr Q= +∞
+In Proposition 2.5, we showed that the covariance operator o f the standard cylindrical
+Wiener process {Wt(g), t≥0, g∈U}associated with the spatially homogeneous
+noise that we considered in Section 2.2 is Q= IdU, which implies that Tr Q= +∞.
+Therefore, we cannot make use of Proposition 3.5 since, in th is case, there is no Q-
+Wiener process associated to W. However, there is the related notion of cylindrical
+Q-Wiener process , which we now define.
+Let (V,/ba∇dbl·/ba∇dblV) be a Hilbert space. Let Qbe a symmetric non-negative definite and
+bounded operator on V, possibly such that Tr Q= +∞. Let (ej)jbe an orthonormal
+basis ofVthat consists of eigenvectors of Qwith corresponding eigenvalues λj,j∈N∗.
+DefineV0=Q1/2(V) as in Section 3.3.
+It is always possible to find a Hilbert space V1and a bounded linear injective
+operatorJ: (V,/ba∇dbl·/ba∇dblV)→(V1,/ba∇dbl·/ba∇dblV1) such that the restriction J0=J|V0: (V0,/ba∇dbl·/ba∇dblV0)→
+(V1,/ba∇dbl · /ba∇dblV1) is Hilbert-Schmidt. Indeed, as explained in [25, Remark 2. 5.1], we may
+24chooseV1=V,/a\}b∇acketle{t·,·/a\}b∇acket∇i}htV1=/a\}b∇acketle{t·,·/a\}b∇acket∇i}htV,αk∈(0,∞) for allk≥1 such that/summationtext∞
+k=1α2
+k<+∞,
+and defineJ:V→Vby
+J(h) :=∞/summationdisplay
+k=1αk/a\}b∇acketle{th,ek/a\}b∇acket∇i}htVek, h∈V, (3.16)
+where (ek)kis an orthonormal basis of V. Then, for g∈V0,g=/summationtext∞
+k=1/a\}b∇acketle{tg,˜ek/a\}b∇acket∇i}htV0˜ek,
+where ˜ek=Q1/2(ek),k≥1, we have
+J0(g) =∞/summationdisplay
+k=1αk/a\}b∇acketle{tg,˜ek/a\}b∇acket∇i}htV0/radicalbig
+λkek=∞/summationdisplay
+k=1αk/a\}b∇acketle{tg,˜ek/a\}b∇acket∇i}htV0˜ek,
+and soJ0: (V0,/ba∇dbl·/ba∇dblV0)→(V,/ba∇dbl·/ba∇dblV) is clearly Hilbert-Schmidt.
+As an operator between Hilbert spaces, from V0toV1,J0has an adjoint J∗
+0:V1→
+V0. However, if we consider V0andV1as Banach spaces, it is more common to consider
+the adjoint ˜J∗
+0:V∗
+1→V∗
+0.
+Proposition 3.6 ([11, Proposition 4.11] and [25, Proposition 2.5.2]) .
+1. DefineQ1=J0J∗
+0:V1= ImJ0→V1.Q1is symmetric (self-adjoint), non-
+negative definite and TrQ1<+∞.
+2. Let˜ej=Q1/2(ej), where(ej)jis a complete orthonormal basis in V, and let
+(βj)jbe a family of independent real-valued standard Brownian mo tions. Then
+Wt:=∞/summationdisplay
+j=1βj(t)J0(˜ej), t≥0, (3.17)
+is aQ1-Wiener process in V1.
+3. LetI:V0→V∗
+0be the one-to-one mapping which identifies V0with its dual V∗
+0,
+and consider the following diagram:
+V∗
+1˜J∗
+0−→V∗
+0I−1
+−→V0J0−→V1.
+Then, for all s,t≥0andh1,h2∈V∗
+1,
+E(/a\}b∇acketle{th1,Ws/a\}b∇acket∇i}ht1/a\}b∇acketle{th2,Wt/a\}b∇acket∇i}ht1) = (t∧s)/angbracketleftBig
+(I−1◦˜J∗
+0)(h1),(I−1◦˜J∗
+0)(h2)/angbracketrightBig
+V0,(3.18)
+where/a\}b∇acketle{t·,·/a\}b∇acket∇i}ht1denotes the dual form on V∗
+1×V1.
+4. ImQ1/2
+1=ImJ0and
+/ba∇dblh/ba∇dbl0=/ba∇dblQ−1/2
+1J0(h)/ba∇dblV1=/ba∇dblJ0(h)/ba∇dblQ1/2
+1(V1), h∈V0,
+whereQ−1/2
+1denotes the pseudo-inverse of Q1/2
+1. Thus,J0:V0→Q1/2
+1(V1)is an
+isometry.
+25Remark 3.7. (a) Part 3 in the Proposition’s statement is commonly abbrev iated in
+the following formal form (see, for instance, [23, Proposit ion 1.1]): for all s,t≥0and
+h1,h2∈V∗
+1,
+E(/a\}b∇acketle{th1,Ws/a\}b∇acket∇i}ht1/a\}b∇acketle{th2,Wt/a\}b∇acket∇i}ht1) = (t∧s)/a\}b∇acketle{th1,h2/a\}b∇acket∇i}htV0.
+(b) TheQ1-Wiener process {Wt, t≥0}obtained in Proposition 3.6 is usually also
+called acylindricalQ-Wiener process. As it is pointed out in [11, p.98], if TrQ<+∞,
+then we can take αk= 1in (3.16), so V1=VandJ= IdV, and we get the classical
+concept ofQ-Wiener process. In this case, one can take V∗
+0=VQ,I−1=Q|VQand the
+equality(3.18)reduces to
+E(/a\}b∇acketle{th1,Ws/a\}b∇acket∇i}ht1/a\}b∇acketle{th2,Wt/a\}b∇acket∇i}ht1) = (t∧s)/a\}b∇acketle{tQh1,h2/a\}b∇acket∇i}htV.
+Proof of Proposition 3.6. Statement 1. follows from (3.4) and the fact that J0is
+Hilbert-Schmidt. Concerning 2., we observe that for h∈V1,
+E(/a\}b∇acketle{tWt,h/a\}b∇acket∇i}ht2
+V1) =E
+
+∞/summationdisplay
+j=1βj(t)/a\}b∇acketle{tJ0(˜ej),h/a\}b∇acket∇i}htV1
+2
+,
+and the right-hand side is equal to
+t∞/summationdisplay
+j=1/a\}b∇acketle{tJ0(˜ej),h/a\}b∇acket∇i}ht2
+V1=t∞/summationdisplay
+j=1/a\}b∇acketle{t˜ej,J∗
+0(h)/a\}b∇acket∇i}ht2
+V0=t/ba∇dblJ∗
+0(h)/ba∇dbl2
+V0
+=t/a\}b∇acketle{tJ∗
+0(h),J∗
+0(h)/a\}b∇acket∇i}htV0=t/a\}b∇acketle{tJ0J∗
+0(h),h/a\}b∇acket∇i}htV1.
+Let us prove now part 3. For the sake of clarity, we will prove t he statement for
+s=tandh1=h2. Hence, let t≥0 andh∈V∗
+1. We denote by /a\}b∇acketle{t·,·/a\}b∇acket∇i}ht0the dual form on
+V∗
+0×V0. Then, by (3.17), the relation between J0andJ∗
+0, and the properties of Iand
+the family ( βj)j, we obtain
+E(/a\}b∇acketle{th,Wt/a\}b∇acket∇i}ht2
+1) =E
+/angbracketleftBigg
+h,∞/summationdisplay
+j=1βj(t)J0(˜ej)/angbracketrightBigg2
+1
+,
+and the right-hand side is equal to
+t∞/summationdisplay
+j=1/a\}b∇acketle{th,J0(˜ej)/a\}b∇acket∇i}ht2
+1=t∞/summationdisplay
+j=1/a\}b∇acketle{t˜J∗
+0(h),˜ej/a\}b∇acket∇i}ht2
+0=t∞/summationdisplay
+j=1/a\}b∇acketle{t(I−1◦˜J∗
+0)(h),˜ej/a\}b∇acket∇i}ht2
+V0
+=t/ba∇dbl(I−1◦˜J∗
+0)(h)/ba∇dbl2
+V0.
+For 4., we refer the reader to [25, Proposition 2.5.2].
+Let{Wt, t≥0}be as in (3.17). A predictable stochastic process {Φt, t∈[0,T]}
+will be integrable with respect to Wif it takes values in L2(Q1/2
+1(V1),H) and
+E/parenleftbigg/integraldisplayT
+0/ba∇dblΦt/ba∇dbl2
+L2(Q1/2
+1(V1),H)dt/parenrightbigg
+<+∞.
+26By part 4 of Proposition 3.6, we have
+Φ∈L0
+2=L2(V0,H)⇐⇒Φ◦J−1
+0∈L2(Q1/2
+1(V1),H).
+Definition 3.8. For any square integrable predictable process Φwith values in L0
+2such
+that
+E/parenleftbigg/integraldisplayT
+0/ba∇dblΦt/ba∇dbl2
+L0
+2dt/parenrightbigg
+<+∞,
+theH-valued stochastic integral Φ·Wis defined by
+/integraldisplayT
+0ΦsdWs:=/integraldisplayT
+0Φs◦J−1
+0dWs.
+We note that the class of integrable processes with respect t oWdoes not depend
+on the choice of V1.
+We now relate this notion of stochastic integral with the sto chastic integral with
+respect to the cylindrical Wiener process of Section 2.1. Le t{Wt, t∈[0,T]}be
+a cylindrical Wiener process with covariance Qon the Hilbert space V, and letg∈
+L2(Ω×[0,T];VQ) bea predictable process, so that g·Wis well definedas in Section 2.1.
+By Proposition 3.6, we can consider the cylindrical Q-Wiener process {Wt, t∈[0,T]}
+defined by
+Wt=∞/summationdisplay
+j=1βj(t)J0(˜ej) (3.19)
+as in formula (3.17) with βj(t) =Wt(vj), wherevj=Q−1/2(ej), ˜ej=Q1/2(ej) and
+(ej)jdenotes a complete orthonormal basis in Vconsisting of eigenvalues of Q, so that
+(vj)jis a complete orthonormal basis in VQ. This process takes values in some Hilbert
+spaceV1.
+Forg∈L2(Ω×[0,T];VQ), we define, as in (3.13), the operator
+Φg
+s(η) =/a\}b∇acketle{tgs,η/a\}b∇acket∇i}htV, η∈V,
+which takes values in H=R. Recall that V0=Q1/2(V) andVQ=Q−1/2(V).
+Proposition 3.9. The process {Φg
+s, s∈[0,T]}defines a predictable process with values
+inL2(V0,R), such that
+E/parenleftbigg/integraldisplayT
+0/ba∇dblΦg
+s/ba∇dbl2
+2ds/parenrightbigg
+=E/parenleftbigg/integraldisplayT
+0/ba∇dblgs/ba∇dbl2
+VQds/parenrightbigg
+,
+and /integraldisplayT
+0Φg
+sdWs=/integraldisplayT
+0gsdWs.
+Proof.First, we will prove that Φg
+s∈L2(V0,R), fors∈[0,T]. As in the first part of
+the proof of Proposition 3.5, /ba∇dblΦg
+s/ba∇dbl2=/ba∇dblgs/ba∇dblVQ. This gives the equality of expectations
+in the statement of the proposition, and the right-hand side is finite by assumption.
+27Concerning the equality of integrals, we note that by definit ion,
+/integraldisplayT
+0ΦsdWs:=/integraldisplayT
+0Φs◦J−1
+0dWs,
+where the right-hand side is defined using the finite-trace ap proach of Section 3.3.
+We note that by Proposition 3.6, part 4, ( J0(˜ej))jis a complete orthonormal basis of
+Q1/2
+1(V1).
+According to Proposition 3.4 with H=R, a single basis element fk= 1 ofH,
+βj(s) =Ws(vj), and ˜ejthere replaced by J0(˜ej), formula (3.12) becomes
+/integraldisplayT
+0Φg
+s◦J−1
+0dWs=∞/summationdisplay
+j=1/integraldisplayT
+0Φg
+s◦J−1
+0(J0(˜ej))dβj(s),
+and the right-hand side is equal to
+∞/summationdisplay
+j=1/integraldisplayT
+0Φg
+s(˜ej)dWs(vj) =∞/summationdisplay
+j=1/integraldisplayT
+0/a\}b∇acketle{tgs,˜ej/a\}b∇acket∇i}htVdWs(vj)
+=∞/summationdisplay
+j=1/integraldisplayT
+0/a\}b∇acketle{tgs,vj/a\}b∇acket∇i}htVQdWs(vj).
+The last expression is equal to g·W.
+Remark 3.10. Proposition 3.9 allows us in particular to associate the spa tially homo-
+geneous noise of Section 2.2, viewed as a cylindrical Wiener process with covariance
+IdUin Proposition 2.5, with a cylindrical Q-Wiener process as in Proposition 3.6, with
+Q= IdU, on the Hilbert space Uof Section 2.2, and to relate the associated stochastic
+integrals.
+4 Spde’s driven by a spatially homogeneous
+noise
+This section is devoted to presenting a class of spde’s in Rddriven by a spatially homo-
+geneous noise. In Section 4.1, we present the real-valued ap proach using the notion of a
+mildrandomfieldsolutionof theequation. Section4.2 gives two examples: thestochas-
+tic heat equation in any spatial dimension and the stochasti c wave dimension in spatial
+dimensions d= 1,2,3. In Section 4.3, we establish an existence and uniqueness r esult
+which extends a theorem of [3]. In Section 4.4, we present the infinite-dimensional for-
+mulation of these spde’s. In Section 4.5, we examine the rela tionship between these two
+formulations, and conclude that they are equivalent (see Pr oposition 4.9). In Section
+4.6, we examine the relationship with the approach of [6].
+We are interested in the following class of non-linear spde’ s:
+Lu(t,x) =σ(u(t,x))˙W(t,x)+b(u(t,x)), (4.1)
+28t≥0,x∈Rd, whereLdenotes a general second order partial differential operator
+with constant coefficients, with appropriate initial condit ions. The coefficients σandb
+are real-valued functions and ˙W(t,x) is the formal notation for the Gaussian random
+perturbation described at the beginning of Section 2.2.
+IfLis first order in time, such as the heat operator L=∂
+∂t−∆, where ∆ denotes
+the Laplacian operator on Rd, then we impose initial conditions of the form
+u(0,x) =u0(x)x∈Rd, (4.2)
+for some Borel function u0:Rd→R. IfLis second order in time, such as the wave
+operatorL=∂2
+∂t2−∆, then we have to impose two initial conditions:
+u(0,x) =u0(x),∂u
+∂t(0,x) =v0(x), x∈Rd, (4.3)
+for some Borel functions u0,v0:Rd→R.
+4.1 The random field approach
+We now describethe notion of mild random field solution to equation (4.1). Recall that
+we are given a filtered probability space (Ω ,F,(Ft),P), where ( Ft)tis the filtration
+generated by the standard cylindrical Wiener process Wof Proposition 2.5, and we
+fix a time horizon T >0. A real-valued adapted stochastic process {u(t,x),(t,x)∈
+[0,T]×Rd}is amild random field solution of (4.1) if the following stochastic integral
+equation is satisfied:
+u(t,x) =I0(t,x)+/integraldisplayt
+0/integraldisplay
+RdΓ(t−s,x−y)σ(u(s,y))W(ds,dy)
++/integraldisplayt
+0ds/integraldisplay
+RdΓ(s,dy)b(u(t−s,x−y)), a.s., (4.4)
+for all (t,x)∈[0,T]×Rd. In (4.4), Γ denotes the fundamental solution associated
+toLandI0(t,x) is the contribution of the initial conditions, which we defi ne below.
+The stochastic integral on the right hand-side of (4.4) is as defined in Section 2.3. In
+particular, we need to assumethat for any ( t,x), the fundamental solution Γ( t−·,x−⋆)
+satisfies Hypothesis 2.8, and to require that s/mapsto→Γ(t−s,x−⋆)σ(u(s,⋆)),s∈[0,t],
+defines a predictable process taking values in the space Uof Section 2.2 such that
+E/parenleftbigg/integraldisplayt
+0/ba∇dblΓ(t−s,x−⋆)σ(u(s,⋆))/ba∇dbl2
+Uds/parenrightbigg
+<+∞
+(see Sections 2.2 and 2.4). As we will make explicit in Sectio n 4.3, these assumptions
+will be satisfied under certain regularity assumptions on th e coefficients bandσ(see
+Theorem 4.3).
+The last integral on the right-hand side of (4.4) is consider ed in the pathwise sense,
+and we use the notation “Γ( s,dy)” because we will assume that Γ( s) is a measure on
+29Rd. Concerning the term I0(t,x), ifLis a parabolic-type operator and we consider the
+initial condition (4.2), then
+I0(t,x) = (Γ(t)∗u0)(x) =/integraldisplay
+Rdu0(x−y)Γ(t,dy). (4.5)
+On the other hand, in the case where Lis second order in time with initial values (4.3),
+I0(t,x) = (Γ(t)∗v0)(x)+∂
+∂t(Γ(t)∗u0)(x)
+=/integraldisplay
+Rdv0(x−y)Γ(t,dy)+∂
+∂t/parenleftbigg/integraldisplay
+Rdu0(x−y)Γ(t,dy)/parenrightbigg
+.(4.6)
+4.2 Examples: stochastic heat and wave equations
+In the case of the stochastic heat equation in any space dimen siond≥1 and the
+stochastic wave equation in dimensions d= 1,2,3, following [3, Section 3] (see also [21,
+Examples 4.2 and 4.3]), the fundamental solutions are well- known and the conditions
+in Hypothesis 2.8 can be made explicit.
+Indeed, let Γ be the fundamental solution of the heat equatio n inRd,d≥1, so that
+Γ(t,x) = (4πt)−d/2exp/parenleftbigg
+−|x|2
+4t/parenrightbigg
+.
+In particular, we have FΓ(t)(ξ) = exp(−4π2t|ξ|2),ξ∈Rd, and, because
+/integraldisplayT
+0exp(−4π2t|ξ|2)dt=1
+4π2|ξ|2(1−exp(−4π2T|ξ|2)),
+we conclude that condition (2.11) in Hypothesis 2.8 holds if and only if
+/integraldisplay
+Rdµ(dξ)
+1+|ξ|2<+∞. (4.7)
+Now let Γ dbe the fundamental solution of the wave equation in Rd, withd= 1,2,3.
+This restriction on the space dimension is due to the fact tha t the fundamental solution
+inRdwithd>3 is no longer a non-negative distribution (for results on th e stochastic
+wave equation in spatial dimension d>3, we refer the reader to [2]: see Remark 2.10).
+It is well known (see [13, Chapter 5]) that
+Γ1(t,x) =1
+21{|x|<t},Γ2(t,x) =1
+2π(t2−|x|2)−1/2
++,Γ3(t)(dx) =1
+4πtσt(dx),
+whereσtdenotestheuniformsurfacemeasureonthethree-dimension al sphereof radius
+t, with total mass 4 πt2. This implies that, for each t, Γd(t) has compact support.
+Furthermore, for all dimensions d≥1, the Fourier transform of Γ d(t) is
+FΓd(t)(ξ) =sin(2πt|ξ|)
+2π|ξ|.
+30Elementary estimates show that there are positive constant sc1andc2depending on
+T >0 such that
+c1
+1+|ξ|2≤/integraldisplayT
+0sin2(2πtξ)
+4π2|ξ|2dt≤c2
+1+|ξ|2.
+Therefore, Γ dsatisfies condition (2.11) if and only if (4.7) holds.
+Ford= 1,I0(t,x) is given by the so-called d’Alembert’s formula (see, for in stance,
+[12, p.68]):
+I1
+0(t,x) =1
+2[u0(x+t)+u0(x−t)]+1
+2/integraldisplayx+t
+x−tv0(y)dy, x ∈R.(4.8)
+Ford= 2 (see [12, p.74]),
+I2
+0(t,x) =1
+2πt/integraldisplay
+|x−y|<tu0(y+tv0)+∇u0(y)·(x−y)
+(t2−|x−y|2)1/2dy, x ∈R2.
+Finally, for d= 3 (see [12, p.77]), for x∈R3,
+I3
+0(t,x) =1
+4πt2/integraldisplay
+R3(tv0(x−y)+u0(x−y)+∇u0(x−y)·y)σt(dy).(4.9)
+It is important to remark that in the above formulas, we have i mplicitly assumed that
+all integrals that appear are well defined. Indeed, in Lemma 4 .2 below, we will exhibit
+sufficient conditions on u0andv0under which such integrals exist and are uniformly
+bounded with respect to tandx.
+4.3 Random field solutions with arbitrary initial condi-
+tions
+The aim of this section is to prove the existence and uniquene ss of a mild random field
+solution to the stochastic integral equation (4.4).
+We are interested in solutions that are Lp-bounded, as in (4.10) below, and L2-
+continuous. This is only possible under certain assumption s on the initial conditions.
+In particular, the initial conditions will have to be such th at the following hypothesis
+is satisfied.
+Hypothesis 4.1. (t,x)/mapsto→I0(t,x)is continuous and sup(t,x)∈[0,T]×Rd|I0(t,x)|<+∞.
+For the particular case of the heat equation in any spatial di mension and the wave
+equation with d∈ {1,2,3}, sufficient conditions for Hypothesis 4.1 to hold are given in
+the next lemma.
+Lemma 4.2. Consider the following two sets of hypotheses:
+(i)Heat equation. u0:Rd→Ris measurable and bounded.
+(ii)Wave equation. Whend= 1,u0is bounded and continuous, and v0is bounded
+and measurable. When d= 2,u0∈C1(R2)and there is q0∈]2,∞]such that
+u0,∇u0,v0all belong to Lq0(R2). Whend= 3,u0∈C1(R3),u0and∇u0are
+bounded, and v0is bounded and continuous.
+31Then under condition (i) or (ii), Hypothesis 4.1 is satisfied.
+Proof.Assume first that Lis the heat operator on Rd,d≥1, with initial condition u0
+satisfying (i). Then, by (4.5),
+sup
+(t,x)∈[0,T]×Rd|I0(t,x)| ≤ /ba∇dblu0/ba∇dbl∞sup
+t∈]0,T]/integraldisplay
+Rd(2πt)−d/2exp/parenleftbigg
+−|y|2
+2t/parenrightbigg
+dy
+=/ba∇dblu0/ba∇dbl∞<+∞.
+Secondly, assume that Lis the wave operator on Rd,d= 1,2,3, and that condition
+(ii) is satisfied. We make explicit the dependence on the spac e dimension by denoting
+Id
+0(t,x),d= 1,2,3, the term I0(t,x).
+By (4.8), if d= 1 it is clear that
+sup
+(t,x)∈[0,T]×R|I1
+0(t,x)| ≤C(/ba∇dblu0/ba∇dbl∞+/ba∇dblv0/ba∇dbl∞).
+To deal with the case d= 2, we refer to [19, p.808–809]. In this reference, the expli cit
+formula Γ 2(t,x) =1
+2π(t2−|x|2)−1/2
++was used to show that
+sup
+(t,x)∈[0,T]×R|I2
+0(t,x)| ≤C(/ba∇dblu0/ba∇dbl∞+/ba∇dbl∇u0/ba∇dbl∞+/ba∇dblv0/ba∇dbl∞).
+Finally, for the case d= 3 we have, by (4.9):
+|I3
+0(t,x)| ≤C(/ba∇dblv0/ba∇dbl∞+/ba∇dblu0/ba∇dbl∞+/ba∇dbl∇u0/ba∇dbl∞) sup
+s∈]0,T]σs(R3)
+s2,
+whereσsdenotestheuniformsurfacemeasureonthethree-dimension al sphereofradius
+s. In particular, the total mass of σsis proportional to s2and, therefore, I3
+0(t,x) is
+uniformly bounded with respect to t∈[0,T] andx∈R3.
+Finally, the continuity property of ( t,x)/mapsto→I0(t,x) follows from the hypotheses and
+the explicit formulas for I0(t,x) given in Section 4.1. This concludes the proof.
+The next theorem discusses existence and uniqueness of mild random field solutions
+to equation (4.4). Since this theorem covers rather general initial conditions, it is an
+extension of Theorem 13 in [3]. Indeed, in this reference, on ly vanishing initial data
+could be considered, because of the spatially homogeneous c ovariance required for the
+processZin theconstruction of the stochastic integral used therefo r the wave equation
+whend= 3 (see [3, p.10 and Theorem 2]). Of course, in the case of the s tochastic
+wave equation in spatial dimensions d= 1,2, there are many results on existence and
+uniqueness of mild random field solutions with non-vanishin g initial conditions: see for
+instance [1, 4, 19, 20].
+Theorem 4.3. Assume that Hypotheses 2.8 and 4.1 are satisfied and that σand
+bare Lipschitz functions. Then there exists a unique mild rando m field solution
+{u(t,x),(t,x)∈[0,T]×Rd}of equation (4.4). Moreover, the process uisL2-continuous
+and for all p≥1,
+sup
+(t,x)∈[0,T]×RdE(|u(t,x)|p)<+∞. (4.10)
+32Proof.The proof is similar to those of [19, Theorem 1.2] and [3, Theo rem 13]. We
+define the Picard iteration scheme
+u0(t,x) =I0(t,x),
+un+1(t,x) =u0(t,x)+/integraldisplayt
+0/integraldisplay
+RdΓ(t−s,x−y)σ(un(s,y))W(ds,dy)
++/integraldisplayt
+0/integraldisplay
+Rdb(un(t−s,x−y))Γ(s,dy)ds, (4.11)
+forn≥0. We prove by induction on nthat the process {un(t,x),(t,x)∈[0,T]×Rd}
+is well defined and, for p≥1,
+sup
+(t,x)∈[0,T]×RdE(|un(t,x)|p)<+∞, (4.12)
+for everyn≥0.
+Notice that by Hypothesis 4.1, the process u0is locally bounded, and the Lipschitz
+property on σyields
+sup
+(t,x)∈[0,T]×Rd|σ(u0(t,x))|p<+∞.
+By Proposition 2.9, this implies that the stochastic integr al
+I0(t,x) =/integraldisplayt
+0/integraldisplay
+RdΓ(t−s,x−y)σ(u0(s,y))W(ds,dy)
+is well-defined and
+E(|I0(t,x)|p)≤C/integraldisplayt
+0dssup
+z∈Rd/parenleftbig
+1+|u0(s,z)|p/parenrightbig/integraldisplay
+Rdµ(dξ)|FΓ(t−s)(ξ)|2
+≤Csup
+(s,z)∈[0,T]×Rd(1+|u0(s,z)|p)/integraldisplayT
+0dsJ(s), (4.13)
+where
+J(s) =/integraldisplay
+Rdµ(dξ)|FΓ(s)(ξ)|2.
+In order to deal with the pathwise integral
+J0(t,x) =/integraldisplayt
+0ds/integraldisplay
+RdΓ(s,dy)b(u0(t−s,x−y)),
+we apply H¨ older’s inequality with respect to the finite meas ure Γ(s,dy)dson [0,T]×Rd
+and use the Lipschitz property of b:
+|J0(t,x)|p≤C/integraldisplayt
+0ds/integraldisplay
+RdΓ(s,dy)/parenleftbig
+1+|u0(t−s,x−y)|p/parenrightbig
+. (4.14)
+33The latter term is uniformly bounded with respect to tandx. Together with (4.13),
+this implies that {u1(t,x),(t,x)∈[0,T]×Rd}is a well-defined measurable process.
+Further, by (4.13), (4.14) and Hypothesis 2.8,
+sup
+(t,x)∈[0,T]×RdE(|u1(t,x)|p)<+∞.
+Consider now n >1 and assume that {un(t,x),(t,x)∈[0,T]×Rd}is a well-
+defined measurable process satisfying (4.12). Using the sam e arguments as above, one
+proves that the integrals In+1(t,x) andJn+1(t,x) exist, so that the process un+1is
+well-defined and is uniformly bounded in Lp(Ω). This proves (4.12).
+The next step consists in showing that the bound (4.12) is uni form with respect to
+n, that is
+sup
+n≥0sup
+(t,x)∈[0,T]×RdE(|un(t,x)|p)<+∞. (4.15)
+Indeed, the same kind of estimates as in the first part of the pr oof show that for n≥1,
+E(|un+1(t,x)|p)≤C/parenleftBigg
+1+/integraldisplayt
+0ds/parenleftBigg
+1+ sup
+z∈RdE(|un(s,z)|p)/parenrightBigg
+(J(t−s)+1)/parenrightBigg
+,
+We conclude that (4.15) holds by the version of Gronwall’s Le mma presented in [3,
+Lemma 15].
+Now we show that the sequence ( un(t,x))n≥1converges in Lp(Ω). Following the
+same lines as in the proof of [3, Theorem 13], let
+Mn(t) := sup
+(s,x)∈[0,t]×RdE(|un+1(s,x)−un(s,x)|p).
+Using the Lipschitz property of bandσ, and applying the same arguments as above,
+we obtain the estimate
+Mn(t)≤C/integraldisplayt
+0dsMn−1(s)(J(t−s)+1).
+Hence, weapplyagain[3, Lemma15]toconcludethat( un(t,x))n≥1converges uniformly
+inLp(Ω) to a limit u(t,x). The process {u(t,x),(t,x)∈[0,T]×Rd}has a measurable
+version that satisfies equation (4.4). Indeed, let us sketch the calculations concerning
+the stochastic integral term In(t,x) of (4.11): we will prove that
+lim
+n→∞sup
+(t,x)∈[0,T]×RdE(|In(t,x)−I(t,x)|p) = 0.
+By the Lipschitz property of σ, Proposition 2.9 and Hypothesis 2.8,
+E(|In(t,x)−I(t,x)|p)
+≤E/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/integraldisplayT
+0/integraldisplay
+RdΓ(t−s,x−y)[σ(un−1(s,y))−σ(u(s,y))]W(ds,dy)/vextendsingle/vextendsingle/vextendsinglep/parenrightbigg
+≤C/integraldisplayT
+0dssup
+z∈RdE(|un−1(s,z)−u(s,z)|p)/integraldisplay
+Rdµ(dξ)|FΓ(t−s)(ξ)|2
+≤Csup
+(s,z)∈[0,T]×RdE(|un−1(s,z)−u(s,z)|p)
+34and this last term converges to zero, as ntends to infinity. The pathwise integral term
+can be studied in a similar manner. Therefore, the process usolves (4.4). Finally,
+uniqueness of the solution can be checked by standard argume nts.
+4.4 Spatially homogeneous spde’s in the infinite-
+dimensional setting
+Stochastic partial differential equations of the form (4.1) o nRdand driven by a spa-
+tially homogeneous Wiener process have been studied, in the context of Da Prato and
+Zabczyk [11], in a series of works: [16, 17, 22, 23, 24]. The ai m of this section is to
+sketch the formulation used in those papers, focusing mostl y on the one used by Peszat
+and Zabczyk in [24]. Then, in Section 4.5, we will compare the ir solution with the mild
+random field solution of Section 4.1.
+In [24], the stochastic wave equation with d= 1,2,3 and the stochastic heat equa-
+tion in any space dimension are considered. This meshes well with the case considered
+in Section 4.1, in which the fundamental solution associate d to the underlying differ-
+ential operator is a non-negative distribution. However, w e note that the stochastic
+wave equation in higher dimensions ( d >3) can also be formulated and solved in the
+infinite-dimensional setting, but using a slightly different formulation (see [22]).
+To begin with, wenotice that in [24]—and, indeed, in theabov e mentioned compan-
+ion papers—the slightly more general spatially correlated noise described in Remark
+2.2 is used. More precisely, one considers a spatially homog eneous Wiener process
+{W∗
+t, t≥0}with values in the space S′(Rd) of tempered distributions. If we denote
+by/a\}b∇acketle{t·,·/a\}b∇acket∇i}htthe usual duality action of S′(Rd) onS(Rd), this means that for all ϕ∈ S(Rd),
+{/a\}b∇acketle{tW∗
+t,ϕ/a\}b∇acket∇i}ht, t∈R+}is a centered Gaussian process and there exists Λ ∈ S′(Rd) such
+that for all ϕ,ψ∈ S(Rd) ands,t∈R+,
+E(/a\}b∇acketle{tW∗
+s,ϕ/a\}b∇acket∇i}ht/a\}b∇acketle{tW∗
+t,ψ/a\}b∇acket∇i}ht) = (s∧t)/a\}b∇acketle{tΛ,ϕ∗˜ψ/a\}b∇acket∇i}ht,
+where˜ψ(x) =ψ(−x). The Schwartz distribution Λ must be the Fourier transform of a
+symmetric and non-negative tempered measure µonRd.
+Remark 4.4. In the particular case where Λhas a density fsatisfying the conditions
+of Section 2.2, we recover the covariance operator of the cyl indrical Wiener process W
+on the Hilbert space Udefined in Proposition 2.5 (see (2.2)):
+E(/a\}b∇acketle{tW∗
+s,ϕ/a\}b∇acket∇i}ht/a\}b∇acketle{tW∗
+t,ψ/a\}b∇acket∇i}ht) = (s∧t)/integraldisplay
+Rddx/integraldisplay
+Rddyϕ(x)f(x−y)ψ(y)
+=E(Ws(ϕ)Wt(ψ)). (4.16)
+For the sake of clarity in the exposition, we assume that the s patial correlation of
+the noise is given by Λ = f, as just described in Remark 4.4.
+LetUbe the Hilbert space defined in Section 2.2, and let U∗be the dual of U. The
+following characterization of U∗is given in [23, Proposition 1.2]. Recall that ˜L2(Rd,µ)
+stands for the subspace of L2(Rd,µ) consisting of all functions φsuch that ˜φ=φ.
+35Lemma 4.5. A distribution g∈ S′(Rd)belongs toU∗if and only if there is φ∈
+˜L2(Rd,µ)such thatg=F(φµ). Moreover, if g1=F(φ1µ)andg2=F(φ2µ), with
+φ1,φ2∈˜L2(Rd,µ), then
+/a\}b∇acketle{tg1,g2/a\}b∇acket∇i}htU∗=/a\}b∇acketle{tφ1,φ2/a\}b∇acket∇i}ht˜L2(Rd,µ).
+Remark 4.6. The previous lemma allows us to determine the explicit form of the
+isometryI:U→U∗. More precisely, as stated in Remark 2.3, any element g∈Ucan
+be written in the form g=F−1φ, withφ∈˜L2(Rd;dµ). Then, for such g,I(g)∈U∗is
+defined by
+I(g) =F(φµ).
+Moreover, we have the following lemma whose proof is straigh tforward. In this
+lemma,˜S(Rd) denotes the family of functions ϕ∈ S(Rd) such that ˜ ϕ=ϕ.
+Lemma 4.7. Letϕ∈Ube such that ϕ∈˜S(Rd). ThenI(ϕ) =ϕ∗f.
+As it has been explained in [23, p.191] (see, in particular, P roposition 1.1 therein),
+W∗may be regarded as a U∗-valued cylindrical Q-Wiener process with Q= IdU∗.
+More precisely, let U∗
+1bea Hilbert space such that there exists a denseHilbert-Sch midt
+embedding J∗:U∗→U∗
+1(see Proposition 3.6). Then
+W∗
+t=∞/summationdisplay
+j=1βj(t)J∗(e∗
+j), (4.17)
+where (e∗
+j)jis a complete orthonormal basis in U∗, and theβj(t) are independent
+standard Brownian motions (note that Q1/2=Q−1/2= IdU∗). Therefore, we will
+be able to define Hilbert-space-valued stochastic integral s with respect to W∗, as has
+been described in Section 3.5. Note that U∗is sometimes called the reproducing kernel
+Hilbert space associated to W∗(see, for instance, [11, Section 2.2.2] or [23, p.191]).
+In [24], mild solutions to the formal equation (4.1) are cons idered in a Hilbert space
+Hof the form L2
+ϑ=L2(Rd,ϑ(x)dx), whereϑis a strictly positive even function such
+thatϑ(x) =e−|x|, for|x| ≥1. Let us also denote by H1
+ϑthe weighted Sobolev space
+which is the completion of S(Rd) with respect to the norm
+/ba∇dblψ/ba∇dblH1
+ϑ=/parenleftbigg/integraldisplay
+Rd/bracketleftbig
+|ψ(x)|2+|∇ψ(x)|2/bracketrightbig
+ϑ(x)dx/parenrightbigg1/2
+.
+For simplicity, we will restrict ourself to equation (4.1) w henL=∂2
+∂t2−∆ with
+spatial dimensions d= 1,2,3. Let Γ be the fundamental solution associated to L, let
+u0∈H1
+ϑ,v0∈L2
+ϑ, and fix a time horizon T >0. By definition, a mildL2
+ϑ-valued
+solution of (4.1) with L=∂2
+∂t2−∆, is an Ft-adapted process {u(t), t∈[0,T]}with
+values inL2
+ϑsatisfying
+u(t) =∂
+∂t(Γ(t)∗u0)+Γ(t)∗v0+/integraldisplayt
+0Γ(t−s)∗b(u(s))ds
++/integraldisplayt
+0Γ(t−s)∗σ(u(s))dW∗
+s. (4.18)
+36The stochastic integral on the right-hand side of (4.18) has to be defined. This requires
+interpreting the integrand Γ( t−s)∗σ(u(s)) in the framework of Section 3.5.
+Recall that, as in Section 3.5 and since Q= IdU∗and soU∗= (U∗)0, we will be
+able to define the stochastic integral with respect to W∗of any predictable process Φ
+taking values in the space L2(U∗,H), whereH=L2
+ϑ. Therefore, it is necessary to
+interpret Γ( t−s)∗σ(u(s)) as an element of L2(U∗,H).
+LetU∗,0be the dense subspace of U∗consisting of all g=F(φµ) withφ∈˜S(Rd).
+According to [24, p.427], it holds that U∗,0⊂ Cb(Rd), the space of bounded and con-
+tinuous functions on Rd. Foru∈L2
+ϑandt>0, define the following operator:
+K(t,u)(η) = Γ(t)∗(uη), η∈U∗,0. (4.19)
+Then it is shown in Lemma 3.3 of [23] that, for all t >0 andu∈L2
+ϑ,K(t,u) has
+a unique extension to a Hilbert-Schmidt operator from U∗intoL2
+ϑ. Thus extended,
+K(t,·) becomes a bounded linear operator from L2
+ϑintoL2(U∗,L2
+ϑ). Therefore, if uis
+anL2
+ϑ-valued adapted process, we can define the stochastic integr al as follows:
+/integraldisplayt
+0(Γ(t−s)∗σ(u(s)))dW∗
+s:=/integraldisplayt
+0K(t−s,σ(u(s)))dW∗
+s. (4.20)
+In the formulation above, σ(u(s)) denotes the function σ(u(s))(x) :=σ(u(s,x)),x∈
+Rd, which belongs to L2
+ϑ.
+This definition of the stochastic integral (4.20) is the one t hat is used in the mild
+formulation (4.18). The main result in [24] on existence and uniqueness of a solution
+to equation (4.18) is the following (see [24, Theorem 0.1]).
+Theorem 4.8. Assume that d∈ {1,2,3}and that the coefficients bandσare Lipschitz
+functions. Suppose that there is κ >0such that Λ +κdxis a nonnegative measure
+(wheredxdenotes Lebesgue measure), and the spectral measure µsatisfies
+/integraldisplay
+Rdµ(dξ)
+1+|ξ|2<+∞. (4.21)
+Then, for arbitrary u0∈H1
+ϑandv0∈L2
+ϑ, there exists a unique L2
+ϑ-valued solution to
+equation (4.18).
+As we mentioned at the beginning of this section, this result in [24] was extended
+in [22] to higher spatial dimensions.
+4.5 Relation with the random field approach
+We now examine the relationship between the random field solu tion to equation (4.4)
+and theL2
+ϑ-valued solution to equation (4.18). For this, we assume tha t the cylindrical
+Wiener process Wconsidered in the beginning of Section 4.1 and the cylindric alQ-
+Wiener process W∗(withQ= IdU∗) that appears in (4.18) are related as follows.
+37Let (ej)jbe a complete orthonormal basis of the Hilbert space Usuch thatej∈
+˜S(Rd), for allj≥1. Assume that the e∗
+jand theβj(t) that appear in (4.17) are given
+by
+e∗
+j=I(ej) and βj(t) =Wt(ej), (4.22)
+whereIis the isometry described in Remark 4.6. Recall that J∗:U∗→U∗
+1denotes
+a Hilbert-Schmidt embedding between U∗and a possibly larger Hilbert space U∗
+1;
+moreover, by Proposition 3.6, ( J∗(e∗
+j))jdefines a basis in U∗
+1.
+Let us consider {u(t,x),(t,x)∈[0,T]×Rd}, the mild random field solution of (4.4)
+as given in Theorem 4.3, in the case where Lis the wave operator in spatial dimension
+d= 1,2 or 3 (so as to have a specific form for I0(t,x)). Then for all ( t,x)∈[0,T]×Rd,
+u(t,x) =/integraldisplay
+Rdv0(x−y)Γ(t,dy)+∂
+∂t/parenleftbigg/integraldisplay
+Rdu0(x−y)Γ(t,dy)/parenrightbigg
++/integraldisplayt
+0/integraldisplay
+RdΓ(t−s,x−y)σ(u(s,y))W(ds,dy)
++/integraldisplayt
+0/integraldisplay
+Rdb(u(t−s,x−y))Γ(s,dy)ds, a.s., (4.23)
+The initial conditions u0andv0satisfy the hypotheses specified in Lemma 4.2. The
+coefficients σandbare Lipschitz functions. Recall that ( t,x)/mapsto→u(t,x) is mean-square
+continuous and satisfies
+sup
+(t,x)∈[0,T]×RdE(|u(t,x)|2)<+∞. (4.24)
+This section is devoted to proving the following result.
+Proposition 4.9. Let{u(t,x),(t,x)∈[0,T]×Rd}be the mild random field solution
+of(4.23). Letu(t) =u(t,⋆). Then{u(t), t∈[0,T]}is the mild L2
+ϑ-valued solution of
+(4.18).
+Proof.In view of the integral equations (4.23) and (4.18), it is cle ar that the most
+delicate part in the proof corresponds to the analysis of the stochastic integral terms.
+Hence, we will start by assuming that both the initial condit ions and the drift term b
+vanish. In this case, {u(t,x),(t,x)∈[0,T]×Rd}solves the integral equation
+u(t,x) =/integraldisplayt
+0/integraldisplay
+RdΓ(t−s,x−y)σ(u(s,y))W(ds,dy), a.s.
+for all (t,x)∈[0,T]×Rd. Let us use the following notation for the above stochastic
+integral:
+I(t,x) :=/integraldisplayt
+0/integraldisplay
+RdΓ(t−s,x−y)σ(u(s,y))W(ds,dy).
+For any (t,x)∈[0,T]×Rd, the above integral is a real-valued random variable and it
+is well-defined because the integrand satisfies the hypothes es described in Section 2.4,
+38that is, Γ(t−·,x−⋆) verifies Hypothesis 2.8 and {σ(u(s,y)),(s,y)∈[0,t]×Rd}is a
+predictable process such that
+sup
+(s,y)∈[0,T]×RdE(|σ(u(s,y))|2)≤C/parenleftBigg
+1+ sup
+(s,y)∈[0,t]×RdE(|u(s,y)|2)/parenrightBigg
+<+∞.(4.25)
+Letu(t) =u(t,⋆),t∈[0,T]. We aim to prove that {u(t), t∈[0,T]}defines
+a square-integrable stochastic process with values in the w eighted space L2
+ϑwhich
+satisfies
+u(t) =/integraldisplayt
+0Γ(t−s)∗σ(u(s))dW∗
+s, t∈[0,T].
+Hence, our objective is to prove that {I(t,⋆), t∈[0,T]}defines an element in L2(Ω×
+[0,T];L2
+ϑ) and
+I(t,⋆) =/integraldisplayt
+0Γ(t−s)∗σ(u(s))dW∗
+s.
+In order to simplify the notation, we will write Z(s,y) :=σ(u(s,y)) and letZ(s)
+denote the function Z(s)(y) =Z(s,y),y∈Rd.
+We will split the proof into several steps.
+Step 1. We shall check that {I(t,⋆), t∈[0,T]}belongs to L2(Ω×[0,T];L2
+ϑ) and
+that, for any fixed ( t,x)∈[0,T]×Rd, the real-valued stochastic integral I(t,x) can be
+written as a stochastic integral with respect to a Hilbert-s pace-valued Wiener process.
+Notice that the norm of I(·,⋆) inL2(Ω×[0,T];L2
+ϑ) coincides with the norm of u(·)
+in the same space, and the latter is given by
+E/parenleftbigg/integraldisplayT
+0dt/integraldisplay
+Rddxϑ(x)|u(t,x)|2/parenrightbigg
+.
+By (4.24) and the fact that ϑis integrable over Rd, this quantity is finite. In particular,
+we also deduce that Zbelongs toL2(Ω×[0,T];L2
+ϑ).
+On the other hand, let us recall that I(t,x) is a stochastic integral with respect
+to the cylindrical Wiener process {Ws(h), s∈[0,T], h∈U}(see Section 2.2) with
+covariance operator Q= IdUands/mapsto→Γ(t−s,x−⋆)Z(s) is a predictable process in
+L2(Ω×[0,T],U) by Proposition 2.9. Hence, by Proposition 3.9, the stochas tic integral
+I(t,x) may be written as
+I(t,x) =/integraldisplayt
+0Φt,x
+sdWs, (4.26)
+where similar to (3.19),
+Wt=∞/summationdisplay
+j=1Wt(ej)J(ej),
+J:U→U1is a Hilbert-Schmidt embedding from Uinto a possibly larger space U1
+(note thatU1need not be the dual of U∗
+1mentioned after (4.22)), and {Φt,x
+s,s∈[0,t]}
+39is the predictable and square integrable process with value s in the space L2(U,R) of
+Hilbert-Schmidt operators from UintoR, given by
+Φt,x
+s(h) =/a\}b∇acketle{tΓ(t−s,x−⋆)Z(s),h/a\}b∇acket∇i}htU, h∈U.
+Moreover,
+E/parenleftbigg/integraldisplayt
+0/ba∇dblΦt,x
+s/ba∇dbl2
+L2(U,R)ds/parenrightbigg
+=E/parenleftbigg/integraldisplayt
+0/ba∇dblΓ(t−s,x−⋆)Z(s)/ba∇dbl2
+Uds/parenrightbigg
+.
+Step 2.Recall that we aim to prove that
+I(t,⋆) =/integraldisplayt
+0(Γ(t−s)∗Z(s))dW∗, t∈[0,T], (4.27)
+where this equality must be understood in L2(Ω×[0,T];L2
+ϑ).
+Lett∈[0,T] and (fk)kbe a complete orthonormal basis in L2
+ϑ. We will find a
+suitableexpansionof I(t,⋆) intermsof( fk)k. Indeed, by(4.26) andsince I(t,⋆) defines
+a square integrable L2
+ϑ-valued random variable, we have the following representat ion:
+I(t,⋆) =∞/summationdisplay
+k=1/bracketleftbigg/integraldisplay
+Rddxϑ(x)/parenleftbigg/integraldisplayt
+0Φt,x
+sdWs/parenrightbigg
+·fk(x)/bracketrightbigg
+fk. (4.28)
+Then,bydefinitionofthestochasticintegralwithrespectt oWandusingrepresentation
+(3.12) in Proposition 3.4 (for H=R), for allx∈Rd,
+/integraldisplayt
+0Φt,x
+sdWs=/integraldisplayt
+0Φt,x
+s◦J−1dWs
+=∞/summationdisplay
+j=1/integraldisplayt
+0Φt,x
+s◦J−1(J(ej))dWs(ej),
+=∞/summationdisplay
+j=1/integraldisplayt
+0Φt,x
+s(ej)dβj(s), (4.29)
+where we have made use of (4.22). Hence, plugging (4.29) into (4.28), we see that
+I(t,⋆) =∞/summationdisplay
+k=1
+/integraldisplay
+Rddxϑ(x)
+∞/summationdisplay
+j=1/integraldisplayt
+0Φt,x
+s(ej)dβj(s)
+·fk(x)
+fk. (4.30)
+Step 3. We now give an analogous representation for the stochastic i ntegral on the
+right-hand side of (4.27). For this, we will again apply Prop osition 3.4 directly to the
+right-hand side of (4.27); notice that here, H=L2
+ϑ, and theJ∗in (4.17) cancels with
+the (J∗)−1in the definition of the stochastic integral. Therefore, tak ing (4.22) into
+account, we see that
+/integraldisplayt
+0Γ(t−s)∗Z(s)dW∗
+=∞/summationdisplay
+k=1
+∞/summationdisplay
+j=1/integraldisplayt
+0/a\}b∇acketle{tΓ(t−s)∗(Z(s)I(ej)),fk/a\}b∇acket∇i}htL2
+ϑdβj(s)
+fk,(4.31)
+40where (fk)kandβjare as in Step 2. Recall that, on the left-hand side of (4.31),
+Γ(t−s)∗Z(s)istheformal notation for theHilbert-Schmidtoperator de finedonU∗and
+taking values in L2
+ϑsuch that, for any η∈U0,∗, (Γ(t−s)∗Z(s))(η) =K(s,Z(s))(η) =
+Γ(t−s)∗(Z(s)η).
+By Lemma 4.7, I(ej) =ej∗f(becauseej∈˜S(Rd)), so equality (4.31) can be
+written in the form
+/integraldisplayt
+0Γ(t−s)∗Z(s)dW∗
+=∞/summationdisplay
+k=1
+∞/summationdisplay
+j=1/integraldisplayt
+0/parenleftbigg/integraldisplay
+Rddxϑ(x) [Γ(t−s)∗(Z(s)(ej∗f))](x)·fk(x)/parenrightbigg
+dβj(s)
+fk.
+ApplyingFubini’sTheoremandcomparingthelatter express ionwith(4.30), weobserve
+that, in order to prove (4.27), it suffices to check that, for al most allx∈Rdand any
+ϕ∈˜S(Rd),
+Φt,x
+s(ϕ) = [Γ(t−s)∗(Z(s)(ϕ∗f))](x), s∈[0,t].
+By definition of the operator Φt,x
+sand expanding the convolutions on the right-hand
+side above, this equality is equivalent to
+/a\}b∇acketle{tΓ(t−s,x−⋆)Z(s),ϕ/a\}b∇acket∇i}htU=/integraldisplay
+RdΓ(t−s,dz)Z(s,x−z)/integraldisplay
+Rddyf(x−z−y)ϕ(y).
+Notice that this is precisely the statement of Lemma 4.10 bel ow. Therefore, we can
+conclude that (4.27) holds.
+Step 4.Let us finally sketch the extension of what we have proved so fa r to the case of
+equations (4.23) and (4.18). That is, we consider a general L ipschitz continuous drift b
+and initial conditions u0,v0satisfying the hypotheses specified at the beginning of the
+section. Hence, {u(t,x),(t,x)∈[0,T]×Rd}satisfies (4.23).
+One proves that the process {u(t), t∈[0,T]}belongs to L2(Ω×[0,T];L2
+ϑ) as we
+have done in Step 1. Indeed, an immediate consequence of the p roof of Theorem 4.3
+is that each term in equation (4.23) is bounded in L2(Ω), uniformly with respect to
+(t,x)∈[0,T]×Rd. This clearly implies that each term in (4.23) defines an elem ent in
+L2(Ω×[0,T];L2
+ϑ).
+It follows that the stochastic integral/integraltextt
+0Γ(t−s)∗σ(u(s))dW∗is well-defined and,
+by Steps 2 and 3 above, we have
+/integraldisplayt
+0Γ(t−s)∗σ(u(s))dW∗=/integraldisplayt
+0/integraldisplay
+RdΓ(t−s,⋆−y)σ(u(s,y))W(ds,dy),
+where the⋆symbol on the right-hand side stands for the variable in L2
+ϑ.
+Concerning the pathwise integral in (4.18), we have
+/integraldisplayt
+0Γ(t−s)∗b(u(s))ds=/integraldisplayt
+0ds/integraldisplay
+RdΓ(t−s,dy)b(u(s,⋆−y))
+=/integraldisplayt
+0ds/integraldisplay
+RdΓ(s,dy)b(u(t−s,⋆−y)).
+41It is also clear that the contributions of the initial condit ions in equations (4.23) and
+(4.18) coincide as elements in L2([0,T];L2
+ϑ). We have therefore proved that {u(t), t∈
+[0,T]}is the mild solution of (4.18), which concludes the proof of P roposition 4.9.
+We now state and prove the following technical lemma, which w as used in the proof
+of Proposition 4.9.
+Lemma 4.10. Fixt∈[0,T]. Then, for all ϕ∈˜S(Rd)andx∈Rd, the stochastic
+process{Φt,x
+s(ϕ), s∈[0,t]}given by
+Φt,x
+s(ϕ) =/a\}b∇acketle{tΓ(t−s,x−⋆)Z(s),ϕ/a\}b∇acket∇i}htU
+coincides, as an element in L2(Ω×[0,t]), with{Kt,x
+s(ϕ), s∈[0,t]}, where
+Kt,x
+s(ϕ) =/integraldisplay
+RdΓ(t−s,dz)Z(s,x−z)/integraldisplay
+Rddyf(x−z−y)ϕ(y).
+Proof.In order to prove the statement, we will first approximate {Φt,x
+s(ϕ), s∈[0,t]}
+by a sequence of smoothprocesses.
+More precisely, as it has been explained in [21, Proposition 3.3], for any ( s,x)∈
+[0,t]×Rd, we can regularize the element Γ( t−s,x−⋆)Z(s)ofUby means of an approx-
+imation of the identity ( ψn)n⊂C∞
+0(Rd), and we can assume that ψnis symmetric, for
+alln, and|Fψn| ≤1. Then, for any s∈[0,t], setJt,x
+n(s) :=ψn∗(Γ(t−s,x−⋆)Z(s)).
+Again by [21, Proposition 3.3], Jt,x
+n(s) belongs to S(Rd) and, asn→ ∞,Jt,x
+nconverges
+to Γ(t−·,x−⋆)ZinL2([0,t]×Ω;U). Define
+Φt,x
+n,s(h) :=/a\}b∇acketle{tJt,x
+n(s),h/a\}b∇acket∇i}htU, h∈U.
+Thisoperator is well-defined because Jt,x
+n(s) is asmooth function and, in fact, it defines
+an element in L2([0,t]×Ω;L2(U,R)).
+Moreover, Φt,x
+n→Φt,xinL2([0,t]×Ω;L2(U,R)), asn→ ∞. Indeed, this is an
+immediate consequence of the fact that the norm of Φt,x
+n−Φt,xis given by
+E/parenleftbigg/integraldisplayt
+0/ba∇dblΦt,x
+n−Φt,x/ba∇dbl2
+L2(U,R)ds/parenrightbigg
+=E
+/integraldisplayt
+0∞/summationdisplay
+j=1|/a\}b∇acketle{tJt,x
+n(s)−Γ(t−s,x−⋆)Z(s),ej/a\}b∇acket∇i}htU|2ds
+
+=E/parenleftbigg/integraldisplayt
+0/ba∇dblJt,x
+n(s)−Γ(t−s,x−⋆)Z(s)/ba∇dbl2
+Uds/parenrightbigg
+,
+where (ej)jis a complete orthonormal basis in U. The last term above tends to zero
+because, as mentioned before, Jt,x
+n→Γ(t−·,x−⋆)ZinL2([0,t]×Ω;U).
+Therefore, for any ϕ∈˜S(Rd) (in fact, for any ϕ∈U), the sequence of real-
+valued processes (Φt,x
+n(ϕ))nconverges to Φt,x(ϕ) inL2(Ω×[0,t]). In particular, Φt,x
+n(ϕ)
+converges weakly to Φt,x(ϕ), that is, for any 0 ≤a<b≤tandA∈ F,
+E/parenleftbigg
+1A/integraldisplayb
+adsΦt,x
+n,s(ϕ)/parenrightbigg
+−→E/parenleftbigg
+1A/integraldisplayb
+adsΦt,x
+s(ϕ)/parenrightbigg
+. (4.32)
+42We will conclude the proof by checking that the left-hand sid e of (4.32) also con-
+verges to
+E/parenleftbigg
+1A/integraldisplayb
+adsKt,x
+s(ϕ)/parenrightbigg
+. (4.33)
+For this, note that, by definition of Φt,x
+n,s, the left-hand side of (4.32) can be written as
+E/parenleftbigg
+1A/integraldisplayb
+ads/a\}b∇acketle{tJt,x
+n(s),ϕ/a\}b∇acket∇i}htU/parenrightbigg
+.
+BecauseJt,x
+n(s) andϕare smooth functions, we can explicitly compute the inner pr od-
+uct in the above expression:
+/a\}b∇acketle{tJt,x
+n(s),ϕ/a\}b∇acket∇i}htU=/integraldisplay
+Rddy/integraldisplay
+Rddz Jt,x
+n(s,y)f(y−z)ϕ(z)
+=/integraldisplay
+RddyJt,x
+n(s,y)(f∗ϕ)(y)
+=/integraldisplay
+Rddy/parenleftbigg/integraldisplay
+RdΓ(t−s,dz)ψn(y−x+z)Z(s,x−z)/parenrightbigg
+(f∗ϕ)(y)
+=/integraldisplay
+RdΓ(t−s,dz)Z(s,x−z)/parenleftbigg/integraldisplay
+Rddyψn(y−x+z)(f∗ϕ)(y)/parenrightbigg
+,
+and so the term on the left-hand side of (4.32) equals
+E/parenleftbigg
+1A/integraldisplayb
+ads/integraldisplay
+RdΓ(t−s,dz)Z(s,x−z)/integraldisplay
+Rddyψn(x−z−y)(f∗ϕ)(y)/parenrightbigg
+.(4.34)
+Sinceϕ∈˜S(Rd), the function y/mapsto→(f∗ϕ)(y) is continuous in Rdand lim |y|→∞(f∗
+ϕ)(y) = 0. This implies that, for any x,z∈Rd,
+lim
+n→∞/integraldisplay
+Rddyψn(x−z−y)(f∗ϕ)(y) = (f∗ϕ)(x−z).
+Moreover, because ψnandϕbelongto ˜S(Rd), we can apply the definition of the Fourier
+transform of tempered distributions:
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Rddyψn(x−z−y)(f∗ϕ)(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+Rdµ(dξ)Fψn(x−z−·)(ξ)Fϕ(ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤/integraldisplay
+Rdµ(dξ)|Fϕ(ξ)|<+∞.
+Thus, in order to apply the Dominated Convergence Theorem in (4.34), it remains to
+prove that
+E/parenleftbigg
+1A/integraldisplayb
+ads/integraldisplay
+RdΓ(t−s,dz)|Z(s,x−z)|/parenrightbigg
+<+∞,
+43and this follows from the hypothesis on Γ and the process Z. So we have proved that
+the limit of (4.34), as ngoes to infinity, is
+E/parenleftbigg
+1A/integraldisplayb
+ads/integraldisplay
+RdΓ(t−s,dz)Z(s,x−z)/integraldisplay
+Rddyf(x−z−y)ϕ(y)/parenrightbigg
+.
+This shows that the left-hand side of (4.32) converges to (4. 33), which concludes the
+proof.
+4.6 Relation with the Dalang-Mueller formulation
+In this section, we examine the relationship between the mil d random field solution
+to equation (4.23) and the solution introduced by Dalang and Mueller in [6], which
+is based on the L2-valued stochastic integration framework that was summari zed in
+Section 2.5. Let L2
+θbe the space defined in Section 2.5.
+In [6], the authors consider solutions to the following stoc hastic wave equation in
+Rd, for anyd≥1:
+∂2u
+∂t2(t,x)−∆u(t,x) =σ(u(t,x))˙W(t,x), (4.35)
+with initial conditions
+u(0,x) =u0(x),∂u
+∂t(0,x) =v0(x), x∈Rd,
+whereu0,v0:Rd→Rare appropriate Borel functions. The noise ˙W(t,x) corresponds
+essentially to the spatially homogeneous Gaussian noise de scribed in Section 2.2 (in [6]
+a spatial correlation as described in Remark 2.2 was conside red). For simplicity, we
+will restrict to the case where the noise is as defined in Secti on 2.2, with covariance f
+as in (2.2), and associated spectral measure µ.
+We denote by H−1(Rd) the Sobolev space of distributions such that
+/ba∇dblv/ba∇dbl2
+H−1(Rd):=/integraldisplay
+Rddξ1
+1+|ξ|2|Fv(ξ)|2<+∞.
+According to [6, Section 5], an adapted L2
+θ-valued process {u(t,⋆), t∈[0,T]}is amild
+L2
+θ-valued solution to (4.35) if t/mapsto→u(t,⋆) is mean-square continuous from [0 ,T] into
+L2
+θand the following L2
+θ-valued stochastic integral equation is satisfied:
+u(t,⋆) = Γ(t)∗v0+∂
+∂t(Γ(t)∗u0)+/integraldisplayt
+0/integraldisplay
+RdΓ(t−s,⋆−y)σ(u(s,y))M(ds,dy),(4.36)
+where Γ denotes the fundamental solution of the wave equatio n inRd. The stochastic
+integral in (4.36) takes values in L2
+θand is defined in the final part of Section 2.5. The
+main result on existence and uniqueness of solutions to equa tion (4.36) is the following
+(see [6, Theorem 13]).
+Theorem 4.11. Assume that the spectral measure µsatisfies (4.7),u0∈L2(Rd),
+v0∈H−1(Rd)andσis a Lipschitz function. Then equation (4.36) has a unique mild
+L2
+θ-valued solution.
+44In order to be able to compare the solution of the above equati on with the mild
+random field solution to (4.23), we consider space dimension sd∈ {1,2,3}and we set
+b= 0. The main result of this section is the following.
+Theorem 4.12. Letd∈ {1,2,3}, and let {u(t,x),(t,x)∈[0,T]×Rd}be the mild
+random field solution of (4.23)(withb= 0). Letu(t) =u(t,⋆). Then{u(t), t∈[0,T]}
+is theL2
+θ-valued solution of (4.36).
+Proof.For simplicity, we assume that the initial conditions vanis h (the extension to
+the general case is straightforward). Recall that d∈ {1,2,3}and{u(t,x),(t,x)∈
+[0,T]×Rd}satisfies the integral equation
+u(t,x) =IΓ,Z(t,x), a.s. (4.37)
+for all (t,x)∈[0,T]×Rd, where
+IΓ,Z(t,x) :=/integraldisplayt
+0/integraldisplay
+RdΓ(t−s,x−y)Z(s,y)W(ds,dy)
+andZ(s,y) :=σ(u(s,y)). In order to prove that {u(t,⋆), t∈[0,T]}is the solution of
+(4.36), we observe that u(t,⋆)∈L2
+θa.s., since
+E(/ba∇dblu(t,⋆)/ba∇dbl2
+L2
+θ) =/integraldisplay
+RdE(u(t,x)2)θ(x)dx<∞
+by (4.24). Next, we note that t/mapsto→u(t,⋆) from [0,T] intoL2
+θis mean-square continuous,
+since
+E(/ba∇dblu(t,⋆)−u(s,⋆)/ba∇dbl2
+L2
+θ) =/integraldisplay
+RdE((u(t,x)−u(s,x))2)θ(x)dx,
+and we observe that as s→t, by (4.24) and since ( t,x)/mapsto→u(t,x) isL2(Ω)-continuous,
+the right-hand side converges to 0 by the Dominated Converge nce Theorem.
+Define
+vθ
+Γ,Z(t,⋆) =/integraldisplayt
+0/integraldisplay
+RdΓ(t−s,⋆−y)σ(u(s,y))M(ds,dy),
+where the stochastic integral is defined as in (2.21). It rema ins to show that
+IΓ,Z(t,⋆) =vθ
+Γ,Z(t,⋆) inL2(Ω×Rd,dP×θ(x)dx). (4.38)
+For this, set Zn(s,y) =Z(s,y)1[−n,n]d(y), so that, by definition,
+vθ
+Γ,Z(t,⋆) = lim
+n→∞vθ
+Γ,Zn(t,⋆) inL2(Ω×Rd,dP×θ(x)dx),
+wherevθ
+Γ,Zn(t,⋆) is defined as in (2.15). By Proposition 2.11, vθ
+Γ,Zn(t,⋆) =IΓ,Zn(t,⋆)
+inL2(Ω×Rd,dP×dx), therefore also in L2(Ω×Rd,dP×θ(x)dx). In order to establish
+(4.38), it suffices to show that
+E(/ba∇dblIΓ,Z(t,⋆)−IΓ,Zn(t,⋆)/ba∇dbl2
+L2
+θ)−→0 asn→ ∞. (4.39)
+45Set Γk:=ψk∗Γ, whereψkis as in b) of Section 2.5. The expectation in (4.39) is
+equal to
+/integraldisplay
+Rddxθ(x)E/parenleftbig
+|IΓ,Zn(t,x)−IΓ,Z(t,x)|2/parenrightbig
+=/integraldisplay
+Rddxθ(x)E/parenleftbigg/integraldisplayt
+0ds/ba∇dblΓ(t−s,x−⋆)[Zn(s,⋆)−Z(s,⋆)]/ba∇dbl2
+U/parenrightbigg
+≤C(A1+A2), (4.40)
+whereCis a positive constant and
+A1=/integraldisplay
+Rddxθ(x)E/parenleftbigg/integraldisplayt
+0ds/ba∇dblΓk(t−s,x−⋆)[Zn(s,⋆)−Z(s,⋆)]/ba∇dbl2
+U/parenrightbigg
+,
+A2=/integraldisplay
+Rddxθ(x)
+×E/parenleftbigg/integraldisplayt
+0ds/ba∇dbl[Γ(t−s,x−⋆)−Γk(t−s,x−⋆)][Zn(s,⋆)−Z(s,⋆)]/ba∇dbl2
+U/parenrightbigg
+.
+By Proposition 2.9 and the definition of Zn,
+A2≤Csup
+(r,y)∈[0,T]×RdE(|Z(r,y)|2)/integraldisplayT
+0ds/integraldisplay
+Rdµ(dξ)|FΓ(s)(ξ)|2|1−Fψk(ξ)|2,
+and this last expression tends to 0 when k→ ∞.
+Concerning the term A1, since Γk,ZnandZare functions of the space variable, we
+smoothZnandZby convolving with ψℓ, so that we can write the U-norm explicitly,
+then we use Fatou’s Lemma and the fact that Γ k≥0 to see that
+A1≤/integraldisplay
+Rddxθ(x)E/parenleftbigg/integraldisplayt
+0ds/integraldisplay
+Rddy/integraldisplay
+RddzΓk(t−s,x−y)|Zn(s,y)−Z(s,y)|
+×f(y−z)Γk(t−s,x−z)|Zn(s,z)−Z(s,z)|/parenrightBig
+. (4.41)
+Let us prove that, for any fixed k≥1, the right-hand side of (4.41) converges to 0 as
+n→ ∞. Indeed, in order to apply the Dominated Convergence Theore m, observe that
+the integrand in (4.41) converges to 0 pointwise and its abso lute value is bounded, up
+to some positive constant, by
+Γk(t−s,x−y)|Z(s,y)|f(y−z)Γk(t−s,x−z)|Z(s,z)|,
+which is such that
+/integraldisplay
+Rddxθ(x)
+×E/parenleftbigg/integraldisplayt
+0ds/integraldisplay
+Rddy/integraldisplay
+RddzΓk(t−s,x−y)|Z(s,y)]f(y−z)Γk(t−s,x−z)|Z(s,z)|/parenrightbigg
+≤Csup
+(r,y)∈[0,T]×RdE(|Z(r,y)|2)/integraldisplayT
+0ds/integraldisplay
+Rdµ(dξ)|FΓ(s)(ξ)|2<+∞.
+46Therefore, taking into account that A2→0 ask→ ∞, uniformly with respect to n,
+we deduce that the left-hand side of (4.40) tends to 0 wheneve rn→ ∞. This proves
+(4.39), and concludes the proof.
+References
+[1] R. Carmona and D. Nualart. Random nonlinear wave equatio ns: Smoothness of
+the solutions. Probab. Theory Relat. Fields 79, No.4, 469-508 (1988).
+[2] D. Conus and R. C. Dalang. The non-linear stochastic wave equation in high
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+[3] R. C. Dalang. Extending the martingale measure stochast ic integral with applica-
+tions to spatially homogeneous s.p.d.e.’s. Electr. J. Prob ab.4, Paper No.6, 29 p.
+(1999).
+[4] R. C. Dalang and N. E. Frangos. The stochastic wave equati on in two spatial
+dimensions. Ann. Probab. 26, no. 1, 187-212 (1998).
+[5] R. C. Dalang and O. L´ evˆ eque. Second-order hyperbolic s .p.d.e.’s driven by ho-
+mogeneous Gaussian noise on a hyperplane. Trans. Amer. Math . Soc.358, no. 5,
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+[7] R.C.Dalang, C.MuellerandL.Zambotti.Hittingpropert iesofparabolics.p.d.e.’s
+with reflection. Ann. Probab. 34(2006), 1423-1450.
+[8] R. C. Dalang and M. Sanz-Sol´ e. Regularity of the sample p aths of a class of
+second-order spde’s. J. Funct. Anal. 227(2005), no. 2, 304-337.
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+Probab. 25(1997), 133-151.
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+Theory Related Fields 116(2000), no. 3, 421-443.
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+[26] L. Quer-Sardanyons. The stochastic wave equation: stu dy of the law and approx-
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+48Robert C. Dalang Llu´ ıs Quer-Sardanyons
+Institut de Math´ ematiques Departament de Matem` atiques
+Ecole Polytechnique F´ ed´ erale Universitat Aut` onoma de B arcelona
+Station 8 08193 Bellaterra (Barcelona)
+CH-1015 Lausanne Spain
+Switzerland
+robert.dalang@epfl.ch quer@mat.uab.cat
+49
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+1 NUCLEAR EMULSION FILM DETECTORS  FOR PROTON RADIOGRAPHY:  DESIGN AND TEST OF THE FIRST PROTOTYPE S. BRACCINI†, A. EREDITATO, I. KRESLO, U. MOSER, C. PISTILLO, S. STUDER Albert Einstein Center for Fundamental Physics,  Laboratory for High Energy Physics (LHEP), University of Bern,   Sidlerstrasse 5, CH-3012 Bern, Switzerland P. SCAMPOLI Dipartimento di Scienze Fisiche, Università di Napoli Federico II,   Complesso Universitario di Monte S. Angelo, I-80126, Napoli, Italy  and  Department of Radiation Oncology, Inselspital, University of Bern,   Murtenstrasse 35, CH-3010 Bern, Switzerland Proton therapy is nowadays becoming a wide spread clinical practice in cancer therapy and sophisticated treatment planning systems are routinely used to exploit at best the ballistic properties of charged particles. The information on the quality of the beams and the range of the protons is a key issue for the optimization of the treatment. For this purpose, proton radiography can be used in proton therapy to obtain direct information on the range of the protons, on the average density of the tissues for treatment planning optimization and to perform imaging with negligible dose to the patient. We propose an innovative method based on nuclear emulsion film detectors for proton radiography, a technique in which images are obtained by measuring the position and the residual range of protons passing through the patient’s body. Nuclear emulsion films interleaved with tissue equivalent absorbers can be fruitfully used to reconstruct proton tracks with very high precision. The first prototype of a nuclear emulsion based detector has been conceived, constructed and tested with a therapeutic proton beam at PSI. The scanning of the emulsions has been performed at LHEP in Bern, where a fully automated microscopic scanning technology has been developed for the OPERA experiment on neutrino oscillations. After track reconstruction, the first promising experimental results have been obtained by imaging a simple phantom made of PMMA with a step of 1 cm. A second phantom with five 5 x 5 mm2 section aluminum rods located at different distances and embedded in a PMMA structure has been also imaged. Further investigations are in progress to improve the resolution and to image more sophisticated phantoms.                                                            † Corresponding author - E-mail: Saverio.Braccini@cern.ch.  2 1.   Introduction Proton therapy is nowadays becoming a wide spread clinical practice in cancer therapy and many hospital based centers are under construction or planned1. The high precision of this innovative technique in radiation therapy allows a very high local control of the pathology with minimal secondary effects. This is particularly advantageous in the treatment of tumors located near organs at risk and in pediatric radiation oncology where the irradiation of healthy tissues may lead to severe permanent consequences on the quality of life of the patients and to the induction of secondary cancers. To exploit at best the ballistic properties of charged particles, sophisticated treatment planning systems are routinely used to calculate the dose given to the target and to the nearby organs to be spared. To accomplish this task, imaging techniques represent more and more a crucial issue and the precise knowledge of the behavior of the beam inside the patient is of paramount importance for the optimization of the treatment in proton therapy.  2.   Proton radiography with nuclear emulsions The possibility to use high-energy protons to obtain medical images of the patient’s body – the so-called proton radiography – is an interesting research issue since many years2.  This imaging methodology is based on the use of protons having a fixed energy, larger than the one used for therapy, thus penetrating the patient’s body. By measuring the residual range, proton radiography allows obtaining images directly proportional to the average density of the traversed material, thus reducing range uncertainties in proton therapy treatment planning, usually based on computed tomography images to which correction factors are applied. Moreover, the dose given to the patients is order of magnitudes less with respect to ordinary X-ray radiography since every single proton passing through the patient’s body brings valuable information. Important results in proton radiography have been obtained at PSI using scintillator based detectors3 and several research groups in Europe and US are exploring this imaging modality using sophisticated and expensive semiconductor based detectors4. We propose5 an innovative method based on nuclear emulsion film detectors. This technique relies on high precision charged particle tracking inside the emulsions, as routinely performed in neutrino physics experiments. As presented in Fig. 1, nuclear emulsion films interleaved with tissue equivalent absorbers can be fruitfully used to reconstruct proton tracks with very high  3 precision to obtain images through the measurement of the position and the residual range of protons passing through the patient’s body. 
+ Figure 1 - Schematic principle of a detector for proton radiography based on nuclear emulsion films.  The proposed technique has good potentialities since a detector of this kind is very cheap and easy to install and remove. These characteristics are surely valuable in a clinical environment where routine daily treatment has absolute priority. Another possible application is the characterization of proton beams for new proton therapy clinical facilities. 3.   Construction and test of the first prototype To study the feasibility of this new application of nuclear emulsions, simulations based GEANT3 and SRIM has been performed in order to image two different phantoms. The first ‘step’ phantom is made of PMMA with thicknesses of 3 and 4 cm, thus presenting a step of 1 cm. The second ‘rod’ phantom is made of PMMA with a total thickness of 4.5 cm in which five 5 x 5 mm2 section aluminum rods are embedded in different positions.  Several configurations of the stack – dubbed ‘brick’ - made of nuclear emulsions and polystyrene layers have been considered in order to optimize the resolution on the measurement of the range of the protons and to minimize the number of emulsions to be employed. A 138 MeV proton beam, uniformly spread on a 10 x 10 cm2 surface, has been considered. The emulsion films used for this work consist of 2 layers of 44 µm thick sensitive emulsion coated on both sides of a 205 µm thick triacetate base.  Following the results of the simulations, two identical bricks have been constructed. Each brick is composed of two parts. The first part is made of 30 emulsions, interleaved with 1.5 mm polystyrene absorber plates and allows the tracking of passing through protons using a minimal number of emulsions. The second part is composed by 40 emulsions interleaved with 0.5 mm polystyrene and allows a precise measurement of the range. The Bragg peak is located in the second part, where the spatial resolution is maximal.  4 The clinical beam of the Gantry 1 at PSI has been used for the first beam tests, as presented in Fig. 2 (left). In order to be able to reconstruct the proton tracks without saturating the emulsions, a maximum of 106 protons on a 10 x 10 cm2 surface are needed. This corresponds to an average dose to the patient smaller than 10-5 Gy. For this purpose a special low intensity beam was set-up by the PSI team and an accurate measurement of the intensity was performed using a coincidence scintillating telescope before performing the irradiation. The irradiation was performed with 138 MeV protons uniformly spread on the surface by means of spot scanning.  
+ 
+ Figure 2 - First beam tests of proton radiography with nuclear emulsions at the Gantry1 at PSI. The brick with the ‘step’ phantom on the top is highlighted in the inset (left). The density of the mini-tracks is plotted for different emulsion sheets (right).  After the development, the scanning of the emulsions has been performed at LHEP in Bern, where a fully automated microscopic scanning technology has been developed for the OPERA experiment on neutrino oscillations.  Before full track reconstruction, mini-tracks contained only in one emulsion sheet have been searched for. As presented in Fig. 2 (right), in the case of the ‘step’ phantom, a clear decrease of the mini-track density has been observed in correspondence of the Bragg peaks produced by protons crossing 3 and 4 cm of PMMA, respectively. Emulsion 1, which is the nearest to the phantom, shows a proton density of about 106 protons per 10 x 10 cm2, as expected. The Bragg peak corresponding to protons passing through 4 cm of PMMA is located in correspondence of Emulsion 50, according to the simulations. Emulsion 55 is traversed only by protons passing through 3 cm of PMMA.  After full track reconstruction, the residual range is measured for all the proton tracks. For the ‘step’ phantom’, a clear image of the 1 cm PMMA step is obtained, as shown in Fig. 3 (left). A preliminary estimation of the resolution of about 2.5 mm is obtained. For the ‘rod’ phantom, the measurement of the range as a function of the position shows the image of the five rods, as presented in Fig. 3 (right). The first peak has a better resolution and corresponds to the rod  5 nearest to the brick. It is interesting to note that by measuring the average scattering angle only with Emulsion 1, the effect due to the five rods can clearly be put in evidence. 
+  
+ Figure 3 – Proton radiography image of the ‘step’ phantom (left). Projected proton radiography of the ‘rod’ phantom (right).  Following these first positive results, further investigations are in progress at LHEP to test new kind of nuclear emulsion films, improve the resolution and image more sophisticated phantoms. Acknowledgments The authors would like to acknowledge the Proton Therapy centre of the Paul Scherrer Institute (PSI) in Villigen (Switzerland) for setting-up and providing the proton beam. References 1.  S. Braccini, Scientific and technological development of hadrontherapy, these proceedings. 2.  A.M. Koehler, Proton Radiography, Science, 160 (1968) 303. 3.  U.Schneider and E. Pedroni, Proton Radiography as a tool for quality control in proton therapy, Med. Phys. 22 (1995) 4. 4.   M. Petterson et al., IEEE NSS Conference Record 2006, vol. 1, 2276 – 2280; D. Menichelli, et al., Development of a proton computed radiography apparatus, Nuclear Science Symposium and Medical Imaging Conference of IEEE, Dresden, October 19-25, 2008; F. Sauli, AQUA: Advance Quality Assurance for hadron therapy, presented at IEEE Nuclear Science Symposium and Medical Imaging Conference, Dresden, 19-25 October, 2008.  5.  S. Braccini et al., First results on proton radiography with nuclear emulsion detectors, to be submitted to JINST.   
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+arXiv:1001.0858v2  [hep-th]  16 Mar 2011Desy 10-001, ESI 2203, ZMP-HH/09-33 - January 2010
+Local causal structures, Hadamard statesand theprin-
+ciple of local covariance in quantum field theory.
+Claudio Dappiaggi1,a,Nicola Pinamonti2,b,Martin Porrmann3,c
+1Erwin Schr¨ odinger Institut f¨ ur Mathematische Physik, Boltzman ngasse 9, A-1090 Wien, Austria.
+2II. Institut f¨ ur Theoretische Physik, Universit¨ at Hamburg, Lu ruper Chaussee 149, D-22761 Hamburg,
+Germany.
+3Quantum Research Group, School of Physics, University of KwaZu lu-Natal and National Institute for
+Theoretical Physics, Private Bag X54001, Durban, 4001, South A frica
+aclaudio.dappiaggi@esi.ac.at,bnicola.pinamonti@desy.de,cmartin.porrmann@desy.de
+Version of November 6, 2018
+Abstract . In the framework of the algebraic formulation, we discuss and ana lyse some new features of
+the local structure of a real scalar quantum field theory in a stron gly causal spacetime. In particular we
+use the properties of the exponential map to set up a local version of a bulk-to-boundary correspondence.
+The bulk is a suitable subset of a geodesic neighbourhood of any but fi xed pointpof the underlying
+background, while the boundary is a part of the future light cone ha vingpas its own tip. In this regime,
+we providea novelnotion for the extended ∗-algebraofWick polynomialson the saidcone and, on the one
+hand, we prove that it contains the information of the bulk counter part via an injective ∗-homomorphism
+while, on the other hand, we associate to it a distinguished state who se pull-back in the bulk is of
+Hadamard form. The main advantage of this point of view arises if one uses the universal properties of
+the exponentialmap and ofthe light cone in orderto showthat, for any twogiven backgrounds MandM′
+and for any two subsets of geodesic neighbourhoods of two arbitr ary points, it is possible to engineer the
+above procedure such that the boundary extended algebras are related via a restriction homomorphism.
+This allows for the pull-back of boundary states in both spacetimes a nd, thus, to set up a machinery
+which permits the comparison of expectation values of local field obs ervables in MandM′.
+MSC: 81T20, 81T05
+1Contents
+1 Introduction 2
+2 Frames and Cones 5
+2.1 Frames and the Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
+2.2 Double Cones and Their Past Boundary . . . . . . . . . . . . . . . . . . . . . . . 8
+3 Algebras of Observables on the Bulk and on the Boundary 11
+3.1 Quantum Algebras on D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
+3.2 Quantum Field Theory on the Boundary . . . . . . . . . . . . . . . . . . . . . . . 17
+3.3 Natural Boundary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
+3.4 Extended Algebra on the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . 22
+3.5 Interplay Between the Algebras and the States on Dand on C+
+p. . . . . . . . . 26
+4 Interplay with General Covariance and Comparison between Spacetimes 33
+4.1 Comparison of Expectation Values in Different Spacetimes . . . . . . . . . . . . . 34
+4.2 An Application: Extracting the Curvature . . . . . . . . . . . . . . . . . . . . . . 36
+5 Summary and Outlook 39
+A Hadamard States 40
+1 Introduction
+In the framework of quantum field theory over curved backgrou nds, we witnessed a considerable
+series of leaps forward due to a novel use of advanced mathema tical techniques combined with
+new physical insights leading to an improved understanding of the underlying foundations of
+the theory. It is far from our intention to give a recollectio n of all of them, but we would
+like to draw attention at least to some of them. On the one hand , in [9], the principle of
+general local covariance was formulated leading to the real isation of a quantum field theory as
+a covariant functor between the category of globally hyperb olic (four-dimensional) Lorentzian
+manifolds with isometric embeddings as morphisms and the ca tegory ofC∗-algebras with unit-
+preserving monomorphisms as morphisms and also to the new in terpretation of local fields as
+natural transformations from compactly supported smooth f unction to suitable operators. On
+the other hand, the presence of a nontrivial background come s with the grievous problem of the
+a priori absence of a sufficiently large symmetry group to identify a na tural ground state as in
+Minkowski spacetime where Poincar´ e invariance enables th is.
+Nonetheless, it isstill possibletoidentify aclass of phys ically relevant states asthosefulfilling
+the so-called Hadamard condition. This guarantees that the ultraviolet behaviour of the chosen
+state mimics that of the Minkowski vacuum at short distances as well as that the quantum
+fluctuations of observables such as the smeared components o f the stress-energy tensor are
+2bounded. From a practical point of view, the original charac terisation of the Hadamard form
+was realised by means of the local structureof the integral k ernel of the two-point function of the
+selected quasi-freestate inasuitablysmallneighbourhoo dofabackgroundpoint. Unfortunately,
+such a criterion is rather difficult to check in a concrete exam ple and a real step forward has
+been achieved in [38, 39] in which the connection between the Hadamard condition and the
+microlocal properties of the two-point function is proven a nd fully characterised.
+This result has prompted a series of interesting developmen ts in the analysis of physically
+relevant states in a curved background, but we focus mainly o n a few recent advances ( cf.
+[13, 14, 15]) where it has been shown that, either in asymptot ically flat or in cosmological
+spacetimes, it ispossibletoexploit theconformal structu reof themanifold toidentify apreferred
+null submanifold of codimension one, the conformal boundar y. On the latter it is possible to
+coherently encode the information of the bulk algebra of obs ervables and to identify a state
+fulfilling suitable uniqueness properties whose pull-back in the bulk satisfies the Hadamard
+condition, being at the same time invariant under all spacet ime isometries.
+The main problem in the above construction is the need to find a rigid and global geometric
+structure which acts as an auxiliary background out of which the bulk state is constructed.
+Hence the local applicability of a similar scheme seems rath er limited; yet one of the main goals
+of the present paper is to show that such a procedure can indee d be set up at a local level and
+for all spacetimes of physical interest. In particular, thi s statement is established on the basis
+of a careful use of some rather well-known geometrical objec ts.
+To be more precise, the point of view taken is the following: i f one considers an arbitrary but
+fixed point pin a strongly causal four-dimensional spacetime, it is alwa ys possible to single out
+a geodesic neighbourhood where the exponential map is a loca l diffeomorphism. Within this set
+we can also always select asecond point qsuch that the doublecone D≡D(p,q).=I+(p)∩I−(q)
+is a globally hyperbolic spacetime. This line of reasoning h as a twofold advantage: on the one
+hand, one can single out a local natural null submanifold of c odimension one, C+
+p, as the portion
+ofJ+(p) contained in the closure of D, while, on the other hand, we are free to repeat the very
+same construction for a second point p′with associated double cone D′in another spacetime
+M′. Since the exponential map is invertible and the tangent spa cesTp(M) andTp′(M′) are
+isomorphic, it turns out that it is possible to engineer all t he geometric data in such a way that
+the two boundaries C+
+pandC+
+p′can be related by a suitable restriction map, the only freedo m
+being the choice of a frame at pand atp′.
+These two advantages can be used to draw some important concl usions on the structure of
+local quantum field theories. More precisely, we shall focus on a real scalar field theory in D,
+withgeneric mass mand withgeneric curvaturecoupling ξ. Theassociated quantumobservables
+are described by the Borchers-Uhlmann algebra or rather by t he extended algebra of fields. In
+particular, we shall show that it is possible to construct a s calar field theory also on Cpand,
+as a novel result, that also, in the boundary, there exists a n atural notion of extended algebra
+which is made precise here. Apart from the check of mathemati cal consistency of our definition,
+we reinforce our proposal by showing that there exists an inj ective∗-homomorphism Π between
+the bulk and the boundary counterparts. The relevance of thi s result is emphasised by the
+identification of a natural state on the boundary, whose pull -back in Dvia Π turns out still
+3to be invariant under a change of the frame (hence, physicall y speaking, it is the same for all
+inertial observers at p) and to be of Hadamard form. This result provides a potential candidate
+for a local vacuum in the large class of backgrounds to be cons idered here.
+Yet westillhavenotmadeprofitableuseofthesecondadvanta geoutlinedbefore. Asamatter
+of fact, we can now consider two arbitrary strongly causal sp acetimesMandM′as well as two
+points therein so that the relevant portions of the two bound aries, CpandCp′, say, associated
+with the double cones, can be related by a suitable restricti on map. The construction of the
+boundary field theory shows that such a map becomes an injecti ve homomorphism between the
+boundary extended algebras, hence allowing for the constru ction of a local Hadamard state in
+two different backgrounds starting from the same building blo ck on the boundary.
+The results presented in the present work have some antecede nts in the concept of relative
+Cauchyevolutiondevelopedin[9]. Knowingatheory(andits correspondingHadamardfunction)
+in the neighbourhood of a Cauchy surface, such a method permi ts one to reconstruct the theory
+in the neighbourhood of any other Cauchy surface of the same s pacetime. The deformation
+arguments, see [18], play a crucial role in obtaining the Had amard property for the deformed
+state in particular. Another related key result is also the o ne presented in [45] about the local
+quasi-equivalence of quasi-free Hadamard states. In the pr esent paper, using the null cones as
+hypersurfaces on which to encode the quantum information, w e succeed in giving an extended
+algebra of observables without knowing the state in a neighb ourhood of such a surface. Hence,
+based on the new method presented here, it is possible to dete rmine quantum states out of their
+form on null surfaces alone and thus in a spacetime-independ ent way.
+We are now in a position to have a reference state with respect to which we can compare the
+expectation values of the same field observables in two differe nt spacetimes. In particular, if one
+of these is (a portion of) Minkowski spacetime, it is obvious that the result of the comparison
+will be related to the geometric data of the second backgroun d which can now be assessed with a
+crystal clear procedure. Furthermore, we shall show that th is method admits an interpretation
+within the language of category theory, so that it becomes ma nifest that our proposal is not in
+contrast with the principle of general local covariance, bu t can actually be seen as a generalisa-
+tion. As a matter of fact, it reduces to the latter whenever is ometric embeddings are involved, in
+which case the fields recover their interpretation as natura l transformations as in [9], i.e., they
+transform in a covariant manner under local isometries.
+To reinforce the above procedure we also provide an explicit example of this “comparison”
+strategy considering a massless real scalar field minimally coupled to scalar curvature both
+in Minkowski and in a Friedman-Robertson-Walker spacetime with flat spatial sections. We
+demonstrate how the difference of the expectation values of th e regularised squared scalar fields
+in these two spacetimes can be expanded into a power series of a suitable local coordinate system
+(null-advanced) yielding, at firstorder, acontribution de pendenton thestructureof theso-called
+scale factor of the curved background.
+Since we have already extensively discussed the plan of acti on, we only briefly sketch the
+synopsisof the paper. InSection 2we shall analyse all thege ometric structureneeded. Although
+most of the material, devoted to the construction of frames a nd of the exponential map, is
+rather well-known in the literature, we try nonetheless to r ecollect it here to provide guidance
+4through the construction of the main geometric objects requ ired, the boundary in particular.
+In Section 3 we shall tackle the problem of constructing a qua ntum scalar field theory on a null
+cone; in particular, in Subsections 3.1 and 3.2 we discuss th e structure of the bulk and boundary
+algebras therein while, in Subsection 3.3, we identify the d istinguished boundary state. The
+novel construction of the extended algebra on the boundary i s presented in Subsection 3.4 and
+all theseresultsareconnectedtothebulkcounterpartinSu bsection3.5. Eventually, inSection 4,
+we discuss, by means of the language of categories, the schem e which leads to the possibility
+to compare field theories on different spacetimes. The concret e example mentioned above is in
+Subsection 4.2. Section 5 summarises the paper and sets out a few conclusions as well as possible
+future investigations.
+2 Frames and Cones
+As outlined in the introduction, the keyword of this paper is “comparison,” i.e.our ultimate
+goal will be to correlate quantum field theories in different ba ckgrounds both at the level of
+algebras and of states and, moreover, to try in the process al so to extract information on the
+local geometry. To this avail one needs a crystal clear contr ol both of the underlying background
+and of its properties. Therefore, we cannot consider arbitr ary manifolds, but need to focus only
+on those which are of physical relevance insofar as they can c arry a full-fledged quantum field
+theory.
+If we keep in mind this perspective, we shall henceforth call spacetime a four-dimensional,
+Hausdorff, connected smooth manifold Mendowed with a Lorentzian metric whose signature
+is (−,+,+,+). Then, consequently, Mis also second countable and paracompact [19, 20].
+Customarily one also requires that Mbe globally hyperbolic (see for example [1] or [9]) in
+order to have a well-defined Cauchy problem for the equations of motion ruling the dynamics of
+standard free field theories.
+The next natural step is the identification of further local g eometric structures which could
+serve as a useful tool in the comparison of two different field th eories on two different spacetimes,
+MandM′. It is known that, for a real scalar field theory, it suffices to r equire that the two
+spacetimes are either isometrically embedded into each oth er or conformally related [9, 34]. The
+drawback of this approach is that only few pairs MandM′fulfil such criteria and potentially
+interesting cases, such as when Mcoincides with Minkowski spacetime and M′with de Sitter,
+are excluded.
+A natural alternative would be to consider pairs of spacetim esMandM′related by a global
+diffeomorphism, but, unfortunately, these maps do no preserv e the geometric structures at the
+heart of the quantum or even of the classical field theory. A ty pical example of such a problem
+arises in connection with the equations of motion of a dynami cal system whenever these are
+constructed out of the spacetime metric. The action of a gene ric diffeomorphism preserves their
+form only in special cases, viz.when they are related to isometries. Hence we would return to
+the original scenario.
+Apart from these remarks we should also keep in mind the idea, briefly sketched in the
+5introduction, to exploit a bulk-to-boundary reconstructi on procedure along the lines of [13, 14].
+At a global level, this requires the existence of a conformal boundary structure, a feature shared
+only by a certain class of manifolds. Since we want to conside r a scenario as general as possible,
+a viable alternative is to focus only on the localstructures of the underlying spacetimes. In the
+remainder of this section, we show how to substantiate this h euristic idea if one carefully uses
+certain properties of the exponential map.
+2.1 Frames and the Exponential Map
+The aim of this subsection is to introduce the basic geometri c tools to be used. Most of the
+concepts are certainly well-known in the literature and the reader might refer either to [26] for a
+full-fledged analysis of those related to bundles and their p roperties or to [28, 33] for a discussion
+focused on the differential geometric aspects. Nonetheless, it is worthwhile to recapitulate part
+of them since they will play a pivotal role in this paper and we can, at the same time, fix the
+notation.
+Consider an arbitrary four-dimensional differentiable mani foldM. To any point p∈M, we
+can associate
+•alinear frame Fpof the tangent space, i.e., a non-singular linear mapping e:R4→Tp(M),
+or, equivalently, an assignment of an ordered basis e1, ...,e4ofTp(M).
+It is straightforward to infer that the set of all such linear framesFMat an arbitrary but fixed
+p∈Mnaturally comes with a right and free action of the group GL(4,R) which is tantamount
+to the possible changes of basis in R4,i.e., (A,e)/mapsto→eAwhereeAdenotes the ordered basis Ai
+jei
+for allA∈GL(4,R). ThusFMcan be endowed with the following additional structure:
+•Given a four-dimensional differentiable manifold M, aframe bundle is the principal bundle
+/tildewidestFM=F[GL(4,R),π′,M] built from the disjoint union/unionsqtext
+p/tildewideFpM, where/tildewideFpMis identified
+with the typical fibre GL(4,R) andπ′:/tildewidestFM→Mis the projection map. Furthermore, the
+tangent bundle TMcan be constructed as the associated bundle TM=/tildewidestFM×GL(4,R)R4.
+We emphasise the well-known fact that the structure introdu ced last guarantees that the
+typical fibre of the tangent bundle at any point pisR4regardless of the chosen manifold, a fact
+we shall use in the forthcoming discussion. Following [28, 3 3], recall that
+•for anyp∈M, ifDpis the set of all vectors vinTp(M) such that the geodesic γv: [0,1]→
+Madmitsvas tangent vector in 0, then the exponential map at pis expp:Dp→Mwith
+expp(v) =γv(1);
+•for any point p∈Mthere always exists a neighbourhood /tildewideOof the 0-vector in Tp(M) such
+that the exponential map is a diffeomorphism onto an open subse tO⊂M. Furthermore,
+whenever/tildewideOisstar-shaped, Oiscalled a normal neighbourhood , andtheinversemaptherein
+will be denoted exp−1
+p:O→/tildewideO.
+6Although the existence of open sets where the exponential ma p is a diffeomorphism suggests
+a way to compare local quantum field theories on different manif olds, we also need to single out a
+preferred structure of codimension 1, since we wish to imple ment a bulk-to-boundary procedure.
+To this avail all the manifolds are henceforth endowed with a smooth Lorentzian metric, which
+entails the following additional features:
+•Since a linear frame at a point p∈Mcan be seen as the assignment of an ordered basis
+ofR4, one can endow this latter vector space with the standard Min kowski metric η,
+which, by construction, is invariant under the Lorentz grou pSO(3,1). In this case the
+frame bundle becomes FM=F[SO(3,1),π′,M] which is also referred to as the bundle of
+orthonormal frames over M. Furthermore, if the spacetime is oriented and time-orient ed,
+we can further reduce the group to SO0(3,1), the component of SO(3,1) connected to the
+identity.
+•Every point in a Lorentzian manifold admits a normal neighbo urhood (see Proposition 7
+and also Definition 5 in Chapter 5 of [33]).
+•There is always a choice of coordinates, called normal coordinates , such that, in these
+coordinates, the pull-back of the metric gunder the inverse of the exponential map equals
+η(the Minkowski metric in standard coordinates) on the inver se image of the point p.
+•Since we shall ultimately need to single out a sort of preferr ed codimension 1 structure,
+it is rather important that, in a Lorentzian manifold, the so -called Gauss lemma holds
+true (Lemma 1 in Chapter 5 of [33]). In particular, this entai ls that, given any p∈M,
+if we consider the null cone /tildewideC⊂Tp(M) havingpas its own tip, then the subset /tildewideC∩/tildewideO
+is mapped into a local null cone in O⊂Mwhich consists of initial segments of all null
+geodesics starting at p.
+We are now in a position to outline the building blocks of our g eometric construction. Let
+us consider two spacetimes ( M,g) and (M′,g′) and two generic points p∈Mandp′∈M′,
+together with their normal neighbourhoods OpandOp′. If we equip each tangent space with
+an orthonormal basis via a frame, e:R4→Tp(M) ande′:R4→Tp′(M′), we are also free to
+introduce a map ie,e′:Tp(M)→Tp′(M′) which is constructed simply by identifying the elements
+of the two ordered bases.
+The strategy is now to exploit the fact that the exponential m ap is a diffeomorphism (hence
+invertible) in a geodesic neighbourhood to introduce a map ıe,e′:Op→Op′such that
+ıe,e′.= expp′◦ie,e′◦exp−1
+p. (1)
+It is important to stress a few further aspects of this last de finition:
+•The mapıe,e′is well defined only when exp−1
+p′can be inverted on the image of ie,e′◦exp−1
+p,
+that is when ie,e′◦exp−1
+p(Op)⊂/tildewideOp′. Therefore, for the sake of notational simplicity, when
+we writeıe,e′it is always assumed that such a requirement is satisfied. Fur thermore, for
+every point pwe can always consider a sufficiently smaller subset of Op, retaining all its
+properties, where the above inclusion holds true.
+7•The mapıe,e′, which maps a sufficiently small OtoO′, is not unique, in the sense that it
+depends on the chosen orthonormal frames eande′. We have always the freedom to act
+with an element of the structure group of the fibre (be it SO(3,1) orSO0(3,1) depending
+on the scenario considered) which maps an orthonormal basis into a second one, and this
+either onTp(M) orTp′(M′). Such arbitrariness cannot be lifted and, for this reason, we
+have explicitly indicated the two frames in the mapping ıe,e′.
+•Despite the freedom mentioned above, the map ıe,e′is invariant under the action of a
+single element of the structure group of the fibre on both eande′,i.e., there exists an
+equivalence relation: We say that
+ıe,e′∼ı˜e,˜e′ (2)
+if and only if there exists an element Λ ∈SO0(3,1) such that ˜ e= Λeand ˜e′= Λe′. This
+equivalence relation shall actually play a relevant role in the discussion of Section 4.
+As a related point, notice that, if the spacetime Mis isometrically embedded into M′,
+a scenario close to the hypotheses in [9], each isometry φ:M→M′induces an isomorphism
+between theorthonormal framebundles FMandFM′sincethemetric structureispreserved. In
+this case every local character of themanifold Mis preserved under φ(see for exampleChapter 3
+of [33]) and, hence, one can consider a sufficiently small subs et of the normal neighbourhood of
+anyp∈Mas well as of φ(p)∈M′so that our construction yields the following commutative
+diagram,
+Opexp−1
+p−−−−→Tp(M)
+φ/arrowbt/arrowbtie,(φ∗◦e)
+Oφ(p)←−−−−
+expφ(p)Tφ(p)(M′).
+Notice that the presence of φ∗◦ein place of a generic e′can be justified as follows: If we
+call (,)pthe inner product between vectors in Tp(M), then for any v,w∈Tp(M), one has
+(v,w)p= (φ∗(v),φ∗(w))φ(p), which, upon introduction of a local frame e:R4→Tp(M), yields
+(v,w)p= (e(vi),e(wi))p= (φ∗◦e(vi),φ∗◦e(wi))φ(p)wherevi,wi∈R4. Moreover, if a generic
+e′is used in place of φ∗◦ethere is no guarantee that the previous diagram commutes. A
+counterexample can actually be constructed considering tw o isometrically related spacetimes,
+which are not rotationally invariant and taking for e′,φ∗◦erotated by some generic angle.
+2.2 Double Cones and Their Past Boundary
+Theanalysis oftheprevioussubsectionisafirststeptoward s thesetupofafull-fledgedprocedure
+which allows for the local comparison of quantum field theori es on different spacetimes. We shall
+now single out a preferred submanifold of codimension 1 on wh ich to apply a bulk-to-boundary
+reconstruction.
+To this avail we have to ensure in the first place that one can co nsistently assign to the
+background Ma well-defined quantum field theory. Since we are only interes ted in local quan-
+tities, the usual hypothesis of global hyperbolicity of the spacetime can be moderately relaxed
+8and, henceforth, we shall assume Mto bestrongly causal [2],i.e., for every point p∈M,
+there exists an arbitrarily small convex, causally convex n eighbourhood O′
+p, which means that
+no non-spacelike curve intersects O′
+pin a disconnected set. In other words, O′
+pitself is globally
+hyperbolic.
+From a physical point of view, this simply forces us to requir e that, ultimately, the theory
+coincides with the usual quantisation procedure on each of t hese subsets, while, from a geomet-
+rical perspective, the discussion of the preceding section still holds true since we are entitled to
+selectO′
+p⊆Op, the normal neighbourhood of p, in such a way that the exponential map is a
+local diffeomorphism also on O′
+p. Furthermore, a rather useful class of sets is constructed o ut of
+the so-called double cones ,
+D(p′,q) =I+(p′)∩I−(q)⊂M,
+whereI±stand, respectively, for the chronological future and past whileq∈O′
+p′. Notice that
+bothp′andqcan be arbitrary but, for our construction, we shall always s uppose that at least
+one of them coincides with p, henceforth p′≡p. It is also interesting that D(p,q) is an open and
+still globally hyperbolic subset of O′
+p. In the forthcoming discussion the boundary of this region
+will also be relevant and we point out that the closure D(p,q) is a compact set (see for example
+Chapter 8 in [46]) which coincides with J+(p)∩J−(q). Furthermore, it is also important to
+recall both that the set of (the closures of) double cones can be used as a base of the topology
+ofO′
+pand that, under the previous assumptions, we can also freely consider the image of D(p,q)
+under the inverse exponential map exp−1
+p, denoted by U(p,q). The reader should bear in mind
+thatU(p,q) is not necessarily the closure of a doublecone in Tp(M)∼R4with respect to the flat
+metric since only (portions of) cones in Tp(M), havingpas their tip, are mapped in (portions
+of) those in Opand vice versa.
+Nonetheless, this construction allows for the identificati on of the main geometrical structure
+needed, since the very existence of D(p,q) and the properties of this set as well as of J+(p)
+under the exponential map suggest to consider C+
+p.=∂J+(p)∩D(p,q) as the natural boundary
+on which to encode data from a field theory in the bulk. The bulk here means D(p,q) which
+is a genuine globally hyperbolic submanifold of Mon which a full-fledged quantum field theory
+can indeed be defined.
+From a geometrical point of view, a few interesting intrinsi c properties of C+
+pcan readily be
+inferred, namely, to start with, C+
+pis generated by future directed null geodesics in particula r
+originating from p. Notice that the latter are not complete since the set we are i nterested in is
+constrained to D(p,q)⊂O′
+pand, therefore, its image under exp−1
+pinTp(M) identifies a portion
+of a futuredirected null cone C+constructed with respect to theflat metric η, wherethis portion
+is topologically equivalent to I×S2,I⊆R. Yet all these properties are universal, thus they
+do not depend on the choice of a specific frame eatp. This is not the case for the form of
+the image of C+
+pinC+under exp−1
+por the pull-back of the metric in normal coordinates under
+exp∗
+p. These clearly depend upon the coordinate system considere d (individuated by e) and,
+hence, the possible choices of eand of coordinates on C+
+pdeserve a more detailed discussion.
+If one starts from the observation that the double cones of in terest all lie in a normal neigh-
+bourhood, a first natural guess is to select the standard norm al coordinates constructed out of
+9the framee. In this setting the metric can be expanded as
+gµν(q) =ηµν−1
+3Rµανβ(p)σα(q,p)σβ(q,p)+O(3),
+whereσ(q,p) is the so-called Synge’s world function, i.e.half of the square of the geodesic dis-
+tance between pandq(see Section 2.1 of [35]). Here σαandσβdenote the covariant derivatives
+ofσperformed at p, whereasO(3) is a shortcut to stress that the metric is approximated up to
+cubic quantities in the normal coordinates.
+Unfortunately, both the coordinate system and the expansio n are not well suited to be used
+in the analysis of the geometry of the null cone C+
+p, since one would like to have a local chart
+where it is manifest that C+
+pis a null hypersurface. Furthermore, for later purposes, we also
+need to discuss some properties of the metric in the vicinity ofC+
+pas a whole and not only
+in a neighbourhood of p. To this end, it is useful to employ the so called retarded coordinates
+as introduced in [36, 37]. We also refer to the review [35], wh ich has the advantage to clearly
+discuss the explicit relation between these new coordinate s and the normal ones (or, also, the
+Fermi-Walker ones).
+Let us briefly recall the construction of these retarded coor dinates. Consider a timelike
+geodesic ˜γthroughpwith unit tangent vector u. In this setting one can define a coordinate r
+as the field
+r(q).=−σα(q,p′)uα(p′),
+whereq∈Oandp′∈˜γare connected by a light-like geodesic originating from p′and pointing
+towards thefuture. With σα(q,p′) wemean thecovariant derivative at pof thegeodesic distance.
+The net advantage of ris that, on C+
+p, it can be read as an affine parameter of the null geodesics
+emanating from p. In other words, once an orthonormal frame eis chosen in such a way that
+e0(p′) =u, the scalar field ronC+
+p′is unambiguously fixed.
+We can now define the full retarded coordinates as ( u,r,xA), whereulabels the family of
+forward null cones with tips lying on ˜ γ(see equation (154) and the preceding discussion in [35]),
+and
+C+
+p=/braceleftbig
+p′∈D(p,q)|u(p′) = 0/bracerightbig
+,
+whilexAare local coordinates on S2. Notice that one could alternatively switch to the more
+common local chart ( θ,ϕ) ofS2atp′and we shall do so whenever needed.
+Moreover, in this coordinate system, the most generic form o f the metric reads [12]
+ds2=−αdu2+2υAdudxA−2e2βdudr+g′
+ABdxAdxB, (3)
+whereα,υA,βandg′
+ABare smooth functions depending on the coordinates. Notice t hat here
+r∈(0,∞) whileuranges over an open set I⊆Rwhich contains 0. Moreover, the x-coordinates
+on the sphere give rise to a volume element with respect to (3) of the form
+/radicalBig/vextendsingle/vextendsingleg′
+AB/vextendsingle/vextendsingledxA∧dxB=/radicalbig
+|gAB||sinθ|dθ∧dϕ, (4)
+10where the symbol | · |under the square root is kept to recall that we are actually re ferring to
+the determinant of the matrices involved. Notice also that, depending on the chosen coordinates
+xAonS2, the switch to ( θ,ϕ) yields a harmless additional contribution to the metric co efficients;
+this justifies the two symbols g′
+ABandgAB, although, henceforth, we shall mostly stick to the
+last one.
+It is also remarkable that, whenever Rµν˙γµ˙γν= 0,γa generator of C+
+p, one can prove that,
+onC+
+p, (3) simplifies (see formula (2.36) in [12]) to
+ds2=−αdu2−2dudr+r2(dθ2+sin2θdϕ2), (5)
+where the standard coordinates ( θ,ϕ) on the 2-sphere are used in place of xA. Apart from being
+much simpler, this form is more closely related to the standa rd Bondi one which is canonically
+used in the implementation of bulk-to-boundary techniques as devised in [13, 14] for a large class
+of asymptotically flat and of cosmological spacetimes. Unfo rtunately, contrary to these papers,
+here the scenario is much more complicated and, furthermore , the cone does not seem to display
+any particular symmetry group to be exploited, such as for ex ample the BMS in [13]. Yet, the
+situation is not as desperate as one might think, since, ulti mately, for our purposes it will only
+be relevant that the metric at pbecomes the Minkowski one, our coordinates being construct ed
+out of an orthonormal frame at p. In particular, this means that, at p,/radicalbig
+|gAB|will become
+proportional to r, which, in this special scenario, can be seen both as the affine null parameter
+introduced above, or, equivalently, as the standard radial coordinate in Minkowski spacetime
+constructed out of the orthonormal frame in Tp(M)∼R4.
+Before concluding this section, we briefly compare (5) with t he corresponding expression in
+Minkowski spacetime, where the flat metric can be written as
+ds2=−dU2+2dUdr+r2(dθ2+sin2θdϕ2), (6)
+U.=t+rdenoting the light coordinate constructed out of the time an d spherical coordinates.
+The cone with tip at 0 is characterised by U= 0 and, also in this case, ris an affine parameter
+along the null geodesics emanating from 0. It is important to stress that the pull-back of (3)
+under exp∗
+ptends to (6) when approaching the point expp(0) =p.
+Finally, we comment on the behaviour of D(p,q) under (1). As mentioned before, exp−1
+p
+does not map a double cone in Minto one in Tp(M), but, nonetheless, we can still adapt the
+choice ofqin such a way that ıe,e′(D(p,q)) is properly contained in a sufficiently large double
+coneD(p,q′)⊂O′
+p.
+3 Algebras of Observables on the Bulk and on the Boundary
+In the previous section, we focused on the introduction and a nalysis of the main geometric tools
+needed. Inparticular, werecall oncemorethat themaingeom etrical objects arethedoublecones
+D(p,q) which are globally hyperbolic spacetimes in their own righ t. Since, in the forthcoming
+discussion, neither pnorqwill play a distinguished role, we shall omit them, hence usi ngD
+11in place of D(p,q). More importantly, we are now entitled to introduce a well- defined classical
+field theory and, for the sake of simplicity, we shall hencefo rth only deal with a free real scalar
+field with generic mass mand generic curvature coupling ξ.
+Let us recollect some standard properties of such a physical system along the lines, e.g., of
+[47]. Consider ϕ:D→Rwhich fulfils the following equation of motion,
+Pϕ.=/parenleftbig
+✷g+ξR+m2/parenrightbig
+ϕ= 0,m2>0 andξ∈R, (7)
+where✷g=−∇µ∇µis the d’Alembert wave operator constructed out of the metri cgwhile
+Ris the scalar curvature. Since this is a second-order hyperb olic partial differential equation,
+each solution with smooth and compactly supported initial d ata on a Cauchy surface can be
+constructed as the image of the following map,
+∆ :C∞
+0(D)→C∞(D), (8)
+where ∆ is the causal propagator defined as the difference of the advanced and the retarded
+fundamental solutions. Furthermore, each ϕf.= ∆(f) satisfies the following support property,
+supp(ϕf)⊆J+(supp(f))∪J−(supp(f)),
+and, ifS(D) denotes the set of solutions of (8) with smooth compactly su pported initial data
+on any Cauchy surface Σ of D, then this turns out to be a symplectic space when endowed wit h
+the weakly non-degenerate symplectic form,
+σ(ϕf,ϕh) =/integraldisplay
+Σdµ(Σ)(ϕf∇nϕh−ϕh∇nϕf) =/integraldisplay
+Ddµ(D)(f∆h),∀f,h∈C∞
+0(D). (9)
+Here the integral is independent of the choice of Cauchy surf ace Σ, as can be noticed from the
+last equation, while dµ(Σ),dµ(D) andnare, respectively, the metric-induced measures and the
+vector normal to Σ.
+As a last ingredient, these properties can be exploited in co mbination with the fact that, by
+construction, Dis contained in a larger globally hyperbolic open set ( O′in the notation of the
+previous section), in order to conclude that ϕfcan be unambiguously extended to a solution of
+the very same equation throughout O′. This can be proved by recalling that both DandO′are
+globally hyperbolic and by invoking the uniqueness of the ca usal propagator. As a consequence
+we are entitled to consider the restriction of ϕfonC+
+pwhich yields
+ϕf/vextendsingle/vextendsingle
+C+
+p∈C∞(C+
+p). (10)
+3.1 Quantum Algebras on D
+After the setup of a classical field theory, we consider a suit able quantisation scheme to be de-
+scribedas a two-fold process: in a firststep we shall select a suitable algebra of fieldswhich fulfils
+the necessary commutation relations and, second, we choose a quantum state as a functional on
+this algebra in order to compute the expectation values of th e relevant observables.
+12Thus let us proceed in logical sequence starting from D, the bulk spacetime, where we can
+introduce Fb(D) as the subset of sequences with a finite number of elements ly ing in
+/circleplusdisplay
+n≥0⊗n
+sC∞
+0(D),
+wheren= 0 yields Cby definition while ⊗n
+sdenotes the n-fold symmetric tensor product.
+According to this definition it is customary to denote a gener icF∈Fb(D) as a finite sequence
+{Fn}nwhere each Fn∈⊗n
+sC∞
+0(D). We can now promote Fb(D) to a topological∗-algebra
+equipping it with
+•a tensor product ·Ssuch that
+(F·SG)n=/summationdisplay
+p+q=nS(Fp⊗Gq),
+whereSis the operator which realises total symmetrisation;
+•a∗-operation via complex conjugation, i.e.,{Fn}∗
+n={Fn}nfor allF∈Fb(D);
+•the topology induced by the natural one of ⊗n
+sC∞
+0(D).
+The above more traditional realisation of Fb(D) can be replaced by a novel point of view,
+thoroughly developed in [7, 5]. To be specific, consider Fb(D) as a suitable subset of the func-
+tionals over C∞(D), the smooth field configurations. Explicitly, F∈Fb(D) yields a functional
+F:C∞(D)→Rout of the standard pairing between ⊗nC∞(D) and⊗nC∞
+0(D), denoted by
+/a\}b∇acketle{t,/a\}b∇acket∇i}ht, via
+F(ϕ).=∞/summationdisplay
+n=01
+n!/a\}b∇acketle{tFn,ϕn/a\}b∇acket∇i}ht. (11)
+Inordertograsptheconnection between thetwo perspective s, it is usefulto introduceaGˆ ateaux
+derivative,
+F(n)(ϕ)h⊗n=dn
+dλnF(ϕ+λh)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+λ=0,∀h∈C∞(D),
+so thatFn≡F(n)(0). We shall use alternatively both pictures in the forthco ming analysis.
+The key point in the subsequent quantisation scheme consist s in the modification of the
+algebraic product ·Sto yield a new one, ⋆, which is constructed out of the causal propagator ∆,
+unambiguously defined according to (8),
+(F ⋆G)(ϕ) =∞/summationdisplay
+n=0in
+2nn!/angbracketleftbig
+F(n)(ϕ),∆⊗nG(n)(ϕ)/angbracketrightbig
+,∀F,G∈Fb(D). (12)
+By direct inspection one realises that F ⋆Gstill lies in Fb(D) and, more importantly, that
+(Fb(D),⋆) gets the structure of a∗-algebra under the operation of complex conjugation.
+13It is important to notice that, up to now, we have not used the e xistence of the equation of
+motion (7) and, therefore, we can refer to Fb(D) as anoff-shell algebra. On the other hand,
+if one wants to encompass also the dynamics of the classical s ystem, one needs only to divide
+Fb(D) by the ideal Iwhich is the set of elements in Fb(D) generated by those of the form
+PjFn(x1,...,x n), wherePjis the operator in (7) applied to the j-th variable in Fn∈⊗n
+sC∞
+0(D).
+The outcome is the on-shell algebra Fbo(D).=Fb(D)
+Iwhich inherits the∗-operation from Fb(D)
+and is nothing but the more commonly used field algebra.
+At this stage, it is important to remark that neither Fb(D) nor its on-shell version Fbo(D)
+contain all the elements needed to fully analyse the underly ing quantum field theory. As a
+matter of fact, objects such as the components of the stress- energy tensor involve products of
+fields evaluated at the same spacetime point, an operation wh ich isa priorinot well-defined due
+to the distributional nature of the fields themselves. To cir cumvent this obstruction, a standard
+procedure calls for the regularisation of these ill-defined objects by means of a suitable scheme
+which goes under the name of Hadamard regularisation. We sha ll not dwell here on the technical
+details but only highlight some aspects in the appendix. The interested reader is referred to
+[23, 24] for a full account.
+In the functional language used before, the problem mention ed above translates into the
+impossibility to include in Fb(D) objects of the form
+F(ϕ) =/integraldisplay
+Ddµ(g)f(x)ϕ2(x),
+wheredµ(g) is the metric-induced volume form, while fis a test function in C∞
+0(D) andϕ∈
+C∞(D). Actually, the star product (12) applied to a couple of such fields involves the ill-defined
+pointwise product of ∆ with itself.
+To solve this problem, we shall follow the line of reasoning o f [7]. Namely, we introduce a
+new class of functionals, Fe(D), which have a finite number of non-vanishing derivatives, t he
+n-th of which has to be a symmetric element of the space of compa ctly supported distributions
+E′(Dn), whose wave front sets, moreover, satisfy the following re striction
+WF(Fn)∩/braceleftbig/parenleftbig
+D×V+/parenrightbign∪/parenleftbig
+D×V−/parenrightbign/bracerightbig
+=∅, (13)
+whereV±corresponds to the forward and to the backward causal cone in the tangent space,
+respectively. The closure symbol indicates that we also inc lude the tip of the cone in the set of
+future or past directed causal vectors.
+We can make Fe(D) a∗-algebra if we extend naturally the∗-operation of Fb(D) and endow
+it with a new product, ⋆H, whose well-posedness was first proved in [8, 6, 23, 24]. The e xplicit
+form is realised as
+(F ⋆HG)(ϕ) =∞/summationdisplay
+n=01
+n!/angbracketleftbig
+F(n)(ϕ),H⊗nG(n)(ϕ)/angbracketrightbig
+, (14)
+whereH∈D′(D2) is the so-called Hadamard bi-distribution. We shall briefl y introduce and
+discuss it in the appendix, but, for our purposes, it is impor tant to recall that, on the one
+14hand, it satisfies the microlocal spectrum condition, hence yielding a natural substitute for the
+notion of positivity of energy out of its wave front set, whil e on the other hand it suffers from
+an ambiguity. At the level of the integral kernel, only the an tisymmetric part of the Hadamard
+bi-distribution is fixed to i/2 times the causal propagator ∆. Also the singular structure is
+unambiguously determined by the choice of the background. O therwise there always exists the
+freedomtoaddasmoothsymmetricfunctionwhich, in ourscen ario, means that, if H,H′aretwo
+Hadamard distributions, then the integral kernel of H−H′is a symmetric element of C∞(D2).
+Yet, as far as the algebra is concerned, this freedom boils do wn to an algebraic isomorphism
+iH′,H: (Fe(D),⋆H)→(Fe(D),⋆H′), namely ( cf.[23, 5])
+iH′,H=αH′◦α−1
+H,
+αH(F).=∞/summationdisplay
+n=01
+n!/angbracketleftbig
+H⊗n,F(2n)/angbracketrightbig
+.(15)
+As for the algebra generated by compactly supported smooth f unctions, also the extended
+one,Fe, hasitson-shellcounterpart, Feo, constructedfromthequotientwiththeidealgenerated
+by the equation of motion applied to the elements of Fe.
+One of the advantages of the formalism employed is the possib ility to easily transcribe the
+overall construction in terms of categories, hence yieldin g a crystal-clear mathematical picture of
+the relevant structures and their relations. This was first a dvocated and utilised in the seminal
+paper [9], where the principle of general local covariance w as formulated in this language to
+which we shall also stick. In particular, we shall now recast the above discussion in this different
+perspective, while the actual relation with [9] will only la ter be outlined in Section 4. Hence,
+we shall use the following categories:
+DoCo: The objects are defined as follows: For every spacetime M, as in the previous section,
+we consider the oriented and time-oriented double cones D(p,q) with the property that
+there exists a normal neighbourhood Op⊂Mcentred inpthat contains D(p,q). Hence,
+an object is a triple D≡[D(p,q),Op,e]. Recall that since both Opand the double cones
+are globally hyperbolic, the choice of time and space orient ation is always possible. The
+morphisms are the maps ıe,e′:Op→Op′introduced in (1) such that ıe,e′/vextendsingle/vextendsingle
+D/parenleftbig
+D(p,q)/parenrightbig
+⊂
+D′(p′,q′). Although the subscript which refers to eande′is a small abuse of notation, we
+feel that its presence might help the reader to focus on the in gredients here at play. The
+composition rule for the morphisms is defined in terms of the c omposition of the maps ıe,e′
+and hence the associativity derives from the associativity of the composition. Notice that,
+as per (1), an arrow between two objects exists only if the sou rce double cone is sufficiently
+small so that its image under exp−1
+pis contained in the domain of the definition of expp′.
+This caveat does not spoil the associativity property of the composition of arrows.
+DoCoiso: This is the subcategory of DoCoobtained by keeping the same objects but restricting the
+possible morphisms of DoCoto those which are isometric embeddings.
+Algi: The objects are unital, topological,∗-algebras Fi(D) withi=b,bo, constructed as above,
+while fori=e,eo, we further restrict the objects to the equivalence classes generated by
+15identifying extended algebras which are isomorphic under t he map (15). The morphisms
+are equivalence classes of injective, unit-preserving,∗-homomorphisms.
+Sincethekey ingredients toconstructboth Fb(D) and Fbo(D) arejustthecausal propagator
+∆ from (8) and the operator (7) realising the equations of mot ion, their uniqueness in any D
+suggests that the association of a suitable algebra,
+Fi:D→Fi(D),i=b,bo, (16)
+with each double cone Dcan be promoted to a functor between DoCoisoandAlgi. To this
+end, in order to define the action of Fion morphisms of DoCoiso, notice that Fi(D) is an
+algebra generated by smooth and compactly supported functi ons on D, hence Fi(ıe,e′) is just
+the injective∗-homomorphism which associates with every element in Fi(D) its image under
+ıe,e′. Furthermore, Fidefined in that way enjoys the covariance property and maps id D, the
+identity of DinDoCoiso, to id Fi(D), the identity of Fi(D) inAlgi,i.e.,
+Fi(ıe,e′)◦Fi(˜ıe′,˜e′) =Fi(ıe,e′◦˜ıe′,˜e′) and Fi(idD) = id Fi(D).
+Notice that, in the case of the on-shell algebra, the equatio n of motion is left unchanged by any
+morphism in DoCoiso.
+It is also important to note that singling out extended algeb ras which are related via (15)
+ensures that the ambiguity in the choice of the Hadamard bi-d istribution does not spoil the
+well-posedness of (16) when i=e. All these assertions can be proved noticing that, due to the
+discussion presented after (1), the category DoCoisois just a subcategory of the category of local
+manifolds introduced in [9] where similar results are discu ssed.
+It would be desirable to extend the functor (16) to the catego ryDoCothat has a larger group
+of morphisms. Unfortunately, this is not straightforward a nd, actually, not even possible. If we
+consider two generic globally hyperbolic regions DandD′inDoCo, related by ıe,e′as in (1), we
+can draw the following diagram,
+DF−−−−→ Fi(D)
+ıe,e′/arrowbt
+D′−−−−→
+FFi(D′).(17)
+To have a functor between DoCoand some Algirequires existence of a well-defined morphism
+F(ıe,e′) between Fi(D) and Fi(D′). But this is not possible since, in general, ıe,e′is not an
+isometry and, thus, it does neither map solutions of (7) in Dinto those of the same equation
+(but out of a different metric) in D′, nor does it preserve the causal propagator. Hence it spoils
+the⋆-operation and does not preserve the singular structure of t he Hadamard bi-distribution
+which only depends upon the underlying geometry.
+As an aside, a positive answer to the present question will on ly be possible considering
+the off-shell classical∗-algebras ( Fb,·S), but, as soon as quantum algebras are employed, the
+16situation looks grim. Yet it is possible to circumvent this o bstruction making profitable use of
+the geometric properties of the portion of the future direct ed light cone in any Din order both
+to set up a bulk-to-boundary correspondence and to later com pare the outcome on the boundary
+of different spacetimes.
+3.2 Quantum Field Theory on the Boundary
+In order to fulfil our goals, it is mandatory as a first step to un derstand how to construct a full-
+fledged quantum field theory on the light cone and this will be t he main aim of this subsection.
+Our procedure derives from the experience gathered in simil ar scenarios where a field theory on
+a null surface was constructed such as, e.g., in [13, 14, 15, 31] (see also [21, 43, 11] for further
+analyses in similar contexts).
+Therefore, following the same philosophy as in these papers , we shall first show that it is
+possible to assign to the boundary a natural field algebra and that there is a “natural” choice
+for the relevant quantum state. To this avail, in this subsec tion, we shall consider the cone as an
+abstractmanifoldonits own, notregardingit asaparticula rportionoftheboundaryofaspecific
+globally hyperbolic double cone D, since the connection with the bulk theory will be presented
+only later. Nevertheless, we have to keep in mind that the alg ebra to be constructed has to
+be large enough in order to contain the images of some suitabl e projections of all the elements
+of the algebra in the bulk D. This will be the most delicate point of the whole constructi on
+because it is not sufficient to consider an algebra generated b y compactly supported data on the
+cone; we have to extend this set to more generic elements.
+Let us start with the introduction of three distinct sets in R×S2⊂R4relevant for the
+following construction, namely, employing the standard co ordinates, we define
+C+
+p=/braceleftbig
+(V,θ,ϕ)∈R×S2|V∈I⊂R, (θ,ϕ)∈S2/bracerightbig
+, (18)
+whereIis the open interval/parenleftbig
+0,V0(θ,ϕ)/parenrightbig
+with a positive, bounded smooth function V0(θ,ϕ) on
+the sphere. The other two regions will be denoted CpandC, respectively, where the coordinate
+Vis allowed to extend over (0 ,∞) or the full real line, so that C+
+p⊂Cp⊂C. We stress that,
+with a slight abuse of notation, we employ the symbol C+
+pas in the previous sections although
+we are not referring to an actual cone since, ultimately, (18 ) will indeed coincide with J+
+p∩∂D,
+if we employ the same conventions and nomenclatures as in the preceding analysis. Furthermore
+notice that Cis not completely independent of pwhich still is required to be a point in this set.
+Yet, to avoid worsening an already cumbersome notation, we l eave such a dependence implicit.
+As a natural next step, we need to identify a suitable space of functions on the boundary
+and, to this avail, viewing Cpimmersed in R4, we define
+S(Cp).=/braceleftBig
+ψ∈C∞(Cp)|ψ=hf/vextendsingle/vextendsingle
+Cp,f∈C∞
+0(R4) andh∈C∞(Cp)/bracerightBig
+, (19)
+wherehvanishes uniformly on S2, asV→0, while each derivative along Vtends to a constant
+uniformly on S2. As far as this subsection is concerned we can safely choose hto be equal to
+17V. Furthermore, S(Cp) turns out to be a symplectic space when endowed with the foll owing
+strongly non-degenerate symplectic form,
+σC(ψ,ψ′).=/integraldisplay
+Cp/bracketleftBig
+ψdψ′
+dV−dψ
+dVψ′/bracketrightBig
+dV∧dS2,∀ψ,ψ′∈S(Cp), (20)
+wheredS2is the standard measure on the unit 2-sphere.
+The reason for such an apparently strange choice of S(Cp) is the later need to relate the
+theory on these sets to the one in the bulk of a double cone. Hen ce the most natural choice
+of compactly supported smooth functions on Cpwould not fit into the overall picture since a
+general solution of the Klein-Gordon equation (7) with smoo th compactly supported initial data
+on some Dwould propagate on the light cone Cpto a function which is also supported on the
+tip. This point corresponds to V= 0 in the above picture and, thus, it does not lie in Cp. On
+the contrary, we shall show in the next subsection that (19) i s indeed the natural counterpart
+on the boundary which arises from the set of solutions of (7).
+In order to introduce the relevant algebra of observables, w e follow the same philosophy as in
+Subsection 3.1, introducing Ab(Cp), whose generic element F′is a sequence{F′
+n}nwith a finite
+number of elements in /circleplusdisplay
+n≥0⊗n
+sS(Cp), (21)
+where⊗n
+sagain denotes the symmetrised n-fold tensor product and the first term in the sum
+isC. Notice that the′-superscript is introduced in this subsection in order to av oid a potential
+confusion with the similar symbols used for the counterpart in the bulk. In order to promote
+(21) to a full topological∗-algebra, we have to endow it with
+•a∗-operation which is the complex conjugation, i.e.,{F′
+n}∗
+n={F′n}nfor allF′∈Ab(Cp);
+•multiplication of elements such that for any F′,G′∈Ab(Cp),
+(F′·SG′)n=/summationdisplay
+p+q=nS(F′
+p⊗G′
+q);
+•the topology induced by the natural topology of S(Cp), namely the topology of smooth
+functions on Cp.
+Although well-defined, this algebra is not suited to be put in relation to data in the bulk and,
+thus, the above product has to be deformed once more. To this a vail, we employ the functional
+point of view as in (11), i.e., ifX.=/braceleftbig
+Φ∈C∞(Cp)|V−1Φ∈C∞(C)/bracerightbig
+, thenF′∈Ab(Cp) yields
+a functional F′:X→Rout of/a\}b∇acketle{t,/a\}b∇acket∇i}ht, the pairing between ⊗nXand⊗nS(Cp),
+F′(Φ) =∞/summationdisplay
+n=01
+n!/a\}b∇acketle{tF′
+n,Φn/a\}b∇acket∇i}ht,∀Φ∈X.
+18By direct inspection of the definition of S(Cp) in (19), one notices that /a\}b∇acketle{t,/a\}b∇acket∇i}htis nothing but
+the standard inner product on (0 ,∞)×S2between compactly supported functions and smooth
+ones, taken with respect to the measure dV∧dS2.
+Although the theory on Cphas no equation of motion built in and, hence, no causal propa -
+gator such as (8), we can nonetheless introduce a new ⋆B-product on Ab(Cp), namely
+(F′⋆BG′)(Φ).=∞/summationdisplay
+n=0in
+2nn!/angbracketleftbig
+F′
+n(Φ),∆n
+σCG′
+n(Φ)/angbracketrightbig
+,∀Φ∈X, (22)
+where ∆ σCis the integral kernel of (20), i.e.,
+∆σC/parenleftbig
+(V,θ,ϕ),(V′,θ′,ϕ′)/parenrightbig
+=−∂2
+∂V∂V′sign(V−V′)δ(θ,θ′), (23)
+whereδ(θ,θ′) stands for δ(θ−θ′)δ(ϕ−ϕ′) and the derivatives have to be taken in the weak
+sense. Notice that ∆ σCis defined as a distribution on C∞
+0(C2
+p) which, by direct inspection, can
+be extended also to S2(C2
+p). Finally,⋆Bis well-defined because only a finite number of elements
+appear in the sum on the right-hand side of (23) and, thus, con vergence is not an issue here. We
+can hence conclude this subsection with a proposition whose proof follows from the preceding
+discussion.
+Proposition 3.1. The pair/parenleftbig
+Ab(Cp),⋆B/parenrightbig
+equipped with the∗-operation introduced above is a
+well defined∗-algebra.
+In the next section we shall discuss the form of a certain clas s of quantum states on this
+algebra to eventually use them to extend the boundary algebr a of observables analogous to the
+procedure on the bulk.
+3.3 Natural Boundary States
+The next step in our construction is the introduction of an ex tended algebra on the boundary,
+but, in the present scenario, there is no standard definition of an Hadamard state or of a bi-
+distribution; and this lack of a class of a priori physically relevant states hinders an imitative
+repetition of the use of the function Has in (14). Therefore, a natural bi-distribution on C+
+pis
+needed and, for this purpose, our choice is the following wea k limit,
+ω/parenleftbig
+(V,θ,ϕ),(V′,θ′,ϕ′)/parenrightbig.=−1
+πlim
+ǫ→0+1
+(V−V′−iǫ)2δ(θ,θ′), (24)
+which has the advantage of being at the same time a well-define d element of D′(C2), where
+C∼R×S2andD′denotes the space of distributions over the test functions i nC∞
+0(C).
+Such an expression already appeared in different, albeit rela ted scenarios where a scalar
+quantum field theory was studied [13, 15, 16, 31, 27, 43, 44]. I t is important to remark that in
+thefirsttwoof thesepapers, (24)was actually usedasthebui ldingblock toconstructaquasi-free
+19pure state for a scalar field theory built on a three-dimensio nal null cone which represented the
+conformal boundary of a suitable class of spacetimes. In all these cases, the particular geometric
+structure as well as the presence of a particular symmetry gr oup entails that (24) fulfils suitable
+uniqueness properties and, furthermore, gives rise to a ful l-fledged Hadamard state in the bulk.
+Therefore we shall employ the above expression as the natura l candidate bi-distribution on the
+boundary, proving in the next sections that, when we realise C+
+pas part of the boundary of a
+double cone, we can also construct a physically meaningful s tate on the bulk Dout of (24).
+The above bi-distribution can be read as a functional ω:S(Cp)×S(Cp)→R, but it
+is actually much more convenient to recall that Cp⊂C. Within this perspective, since Cis
+topologically R×S2and each element in S(Cp), together with the V-derivative, also lies in
+L2(C,dV∧dS2), the following expression for the 2-point function is mean ingful,
+ω(ψ,ψ′) =−1
+πlim
+ǫ→0+/integraldisplay
+R2×S2dVdV′dS2(θ,ϕ)ψ(V,θ,ϕ)ψ′(V′,θ,ϕ)
+(V−V′−iǫ)2, (25)
+wherethe delta function over the angular coordinates is alr eady integrated out. Thedistribution
+(25) satisfies a suitable continuity condition, as for examp le shown in [30],
+|ω(ψ,ψ′)|/lessorequalslantC/parenleftbig
+/ba∇dblψ/ba∇dblL2/ba∇dbl∂Vψ/ba∇dblL2/parenrightbig/parenleftbig
+/ba∇dblψ′/ba∇dblL2+/ba∇dbl∂Vψ′/ba∇dblL2/parenrightbig
+<∞, (26)
+which allows for the extension of ωto the space of square-integrable functions whose derivati ve
+along theV-coordinate is also an element of L2. Furthermore, this also entails the possibility
+to perform a Fourier-Plancherel transform along V(cf.Appendix C in [30]) resulting in a much
+more manageable form for (25),
+ω(ψ,ψ′) =/integraldisplay
+R×S2dkdS2(θ,ϕ) 2kΘ(k)/hatwideψ(k,θ,ϕ)/hatwideψ′(k,θ,ϕ), (27)
+where Θ(k) is the step function equal to 1 if k/greaterorequalslant0 and 0 otherwise. It should be stressed that
+the presence of Θ( k) corresponds to the physical intuition of taking only posit ive frequencies,
+also because, on the cone, the only causal directions are the lines at constant angular variables.
+Under special circumstances this idea has a clear connectio n with the geometrical bulk data as
+well as with the Hadamard property of a bulk state constructe d out of (24) ( cf.for example
+[29, 30, 15]).
+As a last step in this subsection, we underline that the above analysis entails two relevant
+remarks. The first one concerns the wave front set of the bi-di stributionωonC2∼(R×S2)2.
+This was already studied in Lemma 4.4. of [30] yielding
+WF(ω)⊆A∪B, (28)
+where
+A=/braceleftbig/parenleftbig
+(V,θ,ϕ,ζ V,ζθ,ζϕ),(V′,θ′,ϕ′,ζV′,ζθ′,ζϕ′)/parenrightbig
+∈(T∗C)2\{0}|
+V=V′,θ=θ′,ϕ=ϕ′, 0<ζV=−ζV′,ζθ=−ζθ′,ζϕ=−ζϕ′/bracerightbig
+(29)
+20and
+B=/braceleftbig/parenleftbig
+(V,θ,ϕ,ζ V,ζθ,ζϕ),(V′,θ′,ϕ′,ζV′,ζθ′,ζϕ′)/parenrightbig
+∈(T∗C)2\{0}|
+θ=θ′,ϕ=ϕ′,ζV=ζV′= 0,ζθ=−ζθ′,ζϕ=−ζϕ′/bracerightbig
+. (30)
+Although, at this stage, this result is only an aside, it will play a pivotal role in the discussion
+of Subsections 3.4 and 3.5. In particular, if we recall that Cp⊂C, it turns out that the wave
+front set of ωonC∞
+0(C2
+p) can only be smaller or equal to A∪B, and actually it corresponds to
+A∪Brestricted to ( T∗Cp)2.
+As a second remark, note that it is possible to construct a new algebra, say Ab(C) on the full
+Cstarting from (19) and considering the set L2(C,dV∧dS2) in place of S(Cp), while keeping
+the same∗-operation and composition rule. On the one hand, it is strai ghtforward, that Ab(Cp)
+is a∗-subalgebra of Ab(C) while, on the other hand, we can see that the two-point funct ion
+ωas in (24) can be used as a building block of a quasi-free state forAb(C). Hence the same
+conclusion can be drawn for Ab(Cp) since, by construction, the antisymmetric part of ωis equal
+toi
+2∆σ. The only possible issue is positivity, but this is solved by direct inspection of (27) whose
+right-hand side is manifestly greater than 0 once ψ=ψ′. It is important to point out, for the
+sake of completeness, that an almost identical analysis app ears in [13, 15], though performed at
+the level of Weyl algebras. In summary, we get
+Proposition 3.2. The Gaussian (quasi-free) state constructed out of the distr ibutionωenjoys
+the following properties:
+1. It is a well-defined algebraic state on Ab(Cp)and on Ab(C).
+2. It is a vacuum with respect to ∂V, in the sense given in [40].
+3. It is invariant under the change of the local frame, hence in variant under the action of
+SO0(1,3).
+Proof.The first point can be analysed by checking linearity, positi vity and normalisability of
+the state on Ab(Cp). Since the state is quasi-free, it is enough to examine thes e properties for
+the functional ωonS(Cp)×S(Cp) where they follow from the previous discussion.
+The proof of the second point derives from the observation th atω, as a state on Ab(C),
+is invariant under translations and from the fact that the Fo urier-Plancherel transform of the
+integral kernel of ωalong theV-direction only contains positive frequencies, as is clear from
+(27).
+Thethirdpointcan beproved recalling aresultin[13], name ly,ωonAb(C) isinvariant under
+the action of an infinite-dimensional group, the so-called B ondi-Metzner-Sachs group (BMS). In
+short, if one switches from the coordinates ( V,θ,ϕ) to (V,z,¯z) obtained out of a stereographic
+projection, the BMS maps
+/braceleftBigg
+z/mapsto→z′= Λ(z).=az+b
+cz+d,ad−bc= 1,a,b,c,d∈C,
+V/mapsto→V′=KΛ(z,¯z)(V+α(z,¯z)),
+21where ¯ztransforms as the complex conjugate of z,α(z,¯z)∈C∞(S2) and
+KΛ(z,¯z).=1+|z|2
+|az+d|2−|bz+c|2.
+Hence, by direct inspection of the above formulae, the BMS gr oup is seen to be the regular
+semidirect product SL(2,C)⋊C∞(S2). Most notably one observes that there exists a proper
+subgroup which is homomorphism to SO(3,1) and thus the state ωturns out to be invariant
+under the group sought-for.
+3.4 Extended Algebra on the Boundary
+In the previous subsection, we introduced the boundary alge bra together with a suitable notion
+of⋆-product, but this is still not sufficient to intertwine the bo undary data with those on the
+bulk because we lack a counterpart for the extended algebra o f observables on Cp. Yet, due to
+the results of the last subsection, we have all the ingredien ts to construct it.
+As a starting point, we define the building block of the extend ed algebra as follows.
+Definition 3.1. We callAnthe set of elements F′
+n∈D′(Cn
+p)that fulfil the following properties:
+1.Compactness : TheF′
+nare compact towards the future, i.e., the support of F′
+nis contained
+in a compact subset of Cn∼(R×S2)n.
+2.Causal non-monotonic singular directions : The wave front set of F′
+ncontains only
+causal non-monotonic directions which means that
+WF(F′
+n)⊆Wn.=/braceleftbig
+(x,ζ)∈(T∗Cp)n\{0}|(x,ζ)/\e}atio\slash∈V+
+n∪V−
+n,(x,ζ)/\e}atio\slash∈Sn/bracerightbig
+,(31)
+where(x,ζ)≡(x1,...,x n,ζ1,...,ζn)∈V+
+nif, employing the standard coordinates on Cp,
+for alli= 1, ...,n,(ζi)V>0orζivanishes. The subscript Vhere refers to the component
+along theV-direction on Cp. Analogously, we say (x,ζ)∈V−
+nif every (ζi)V<0orζi
+vanishes. Furthermore, (x,ζ)∈Snif there exists an index isuch that, simultaneously,
+ζi/\e}atio\slash= 0and(ζi)V= 0.
+3.Smoothness Condition : The distribution F′
+ncan be factorised into the tensor product
+of a smooth function and an element of An−1when localised in a neighbourhood of V= 0,
+i.e., there exists a compact set O⊂Cpsuch that, if Θ∈C∞
+0(Cp)so that it is equal to 1on
+OandΘ′.= 1−Θ, then for every multi-index Pin{1,...,n}and for every i/lessorequalslantn,
+f.=˜F′
+n(uxPi+1,...,xPn)Θ′
+xP1···Θ′
+xPi∈C∞(Ci
+P), (32)
+where˜F′
+n:C∞
+0(Cn−i−1
+p)→D′(Ci
+p)is the unique map from C∞
+0(Cn−i−1
+p)toD′(Ci
+p)deter-
+mined byF′
+nusing the Schwartz kernel theorem. Furthermore, uxPi+1,...,xPn∈C∞
+0(Cn−i
+p),
+and we have specified the integrated variables xPi+1, ...,xPn. For every j/lessorequalslanti,∂V1···∂Vjf
+lies inC∞(Ci
+p)∩L2(Ci
+p,dVP1∧dS2
+P1···dVPi∧dS2
+Pi)∩L∞(Ci
+p), while the limit of fasVj
+tends uniformly to 0vanishes uniformly in the other coordinates.
+22Remark:
+a) In property2 of Definition 3.1 we required that WF( F′
+n)∩Sn=∅,viz., nospatial directions
+are present in the wave front set of F′
+n. Even if such an extra condition is not stipulated in
+the definition of the elements on Fe, here we have to add it because, later on, we want to
+multiply elements of Awithωand in the wave front set of ωthere are spatial directions,
+to be specific, the intersection WF( ω)∩S2is not empty.
+b) Thanks to the smoothness condition, the last requirement in Definition 3.1, the distribu-
+tions inAncan be extended over ⊗nS(Cp) and such an extension is unique.
+Notice that S(Cp) is strictly contained in Aand that the candidate to play the role of the
+extended algebra on Cpthus is
+Ae(Cp) =/circleplusdisplay
+n/greaterorequalslant0An
+s,
+whereAn
+sis the subset of totally symmetric elements in Andefined in Definition 3.1. Moreover,
+thefirstspaceinthepreviousdirectsumis Candonly sequenceswithafinitenumberofelements
+are considered. We can now endow this set with the structure o f∗-algebra by introducing the
+∗-operation{F′
+n}∗.={F′n}for allF′∈Ae. The composition law arises from a modification
+of⋆Bby means of the state constructed in the sequel of (24). It is a priori clear that such
+a procedure intrinsically depends on the particular ωconsidered. Nonetheless, our choice will
+later be justified both through its connection with the bulk d ata and by well-posedness of the
+new structure. If we stick to the functional representation , we can thus introduce
+⋆ω:Ae(Cp)×Ae(Cp)→Ae(Cp),
+(F′⋆ωG′)(Φ) =∞/summationdisplay
+n=01
+n!/angbracketleftbig
+F′(n)(Φ),ωnG′(n)(Φ)/angbracketrightbig
+,(33)
+for allF′,G′∈Ae(Cp) and for all Φ∈C∞(Cp).
+Proposition 3.3. The operation (33)is a well-defined product in Ae.
+Proof.As a starting point, notice that (33) is bilinear by construc tion and that, by definition of
+Ae(Cp), there are only a finite number of non-vanishing elements F′(n)andG′(n). Accordingly,
+(33) consists of finite linear combinations of terms that for mally look like
+S/integraldisplay
+C2kpF′
+j(x1,...,x j)ω(x1,y1)···ω(xk,yk)G′
+l(y1,...,yl)k/productdisplay
+i=1dµ(xi)dµ(yi) (34)
+withk/lessorequalslantjandk/lessorequalslantl, whileSrealises symmetrisation in the non-integrated variables a nddµis
+the measure dV∧dS2(θ,ϕ) onCp, written here in the usual coordinates. Therefore, the proo f
+amounts to showing that it is possible to give a rigorous mean ing to expressions like (34) and
+that the result of such an operation is an element of Aj+l−2k.
+23First, wefollow theproofof[23], and, tothisavail, examin eif (34)canbeseenasthetarget of
+1∈C∞(C2k
+p) under the linear map determined, with the help of the Schwar tz kernel theorem,
+by the distribution resulting from multiplication of the tw o distributions ω⊗k⊗I⊗(j+l−2k)∈
+D′(Cj+l) andF′
+j⊗G′
+l∈Aj+l, whereIdenotes the identity operator on A1. Let us start by
+discussing the well-posedness of the multiplication of dis tributions presented above. This can
+be checked by examining the structure of the wave front set of the single objects to verify that
+their composition never contains the zero section. The key i ngredients for this can be readily
+inferred using Theorem 8.2.9 in [25],
+WF(F′
+j⊗G′
+l)⊂/parenleftbig
+Wj∪{0}/parenrightbig
+×/parenleftbig
+Wl∪{0}/parenrightbig
+\{0}, (35)
+and
+WF(ω⊗k⊗Ij+l−2k)⊂/parenleftbig
+A∪B∪{0}/parenrightbigk×{0}\{0}, (36)
+where, as usual, we have not specified the dimension of the zer o section in the cotangent space.
+Furthermore, A,BandWjare defined in (29), (30) and (31), respectively. It is now pos sible to
+apply Theorem 8.2.10 in [25] since the above wave front sets n ever sum up to the zero section.
+This is tantamount to realising that for every nandm, sinceAn⊂V+
+n×V−
+nandV±
+n∩Wn=∅,
+Bn×{R3}m∩W2n+m=∅andAn∩(Wn×Wn) =∅. The outcome is that, in (34), the pointwise
+product of F′
+j⊗G′
+l∈D′(Cj+l
+p) withω⊗k⊗Ij+l−2k∈D′(Cj+l
+p) is still a well-defined element of
+D′(Cj+l
+p), whose wave front set satisfies the inclusion
+WF/parenleftbig
+F′
+j⊗G′
+l·ω⊗k⊗Ij+l−2k/parenrightbig
+⊂WF(Fj⊗Gl)∪{0}+WF/parenleftbig
+ω⊗k⊗Ij+l−2k/parenrightbig
+∪{0}, (37)
+where, asusual, thesumoftwo wavefrontsetsisdefinedasthe sumonthefibresofthecotangent
+spaces. Unfortunately, this does not suffice to show the well- posedness of (34). Since F′
+jandG′
+l
+are not compactly supported on Cj
+pandCl
+p, respectively, their product does not lie in E′/parenleftbig
+Cj+l
+p/parenrightbig
+.
+Hence we cannot directly test the linear map stemming from ( F′
+j⊗G′
+l)·/parenleftbig
+ω⊗k⊗Ij+l−2k/parenrightbig
+on the
+unit constant function on C2k
+pin order to infer that (34) is an element in Aj+l−2k.
+Let us hence proceed by showing that in the case k= 1 we can test the linear map arising
+from (F′
+j⊗G′
+l)·/parenleftbig
+ω⊗Ij+l−2/parenrightbig
+on 1 and that the result of this operation is an element of Aj+l−2.
+The case for a generic karises from of a recursive application of the very same proce dure
+and, eventually, the application of an operator realising t he total symmetrisation. Thus we are
+interested in /integraldisplay
+C2p(F′
+j⊗G′
+l)·/parenleftbig
+ω⊗Ij+l−2/parenrightbig
+dµ(x1)dµ(y1),
+whereF′
+j∈AjandG′
+l∈Al. We exploit property 3 of Definition 3.1 and notice that, if th e
+smoothness condition holds for a compact set O, it also holds for every larger compact set O1
+containing O∈Cp. We can thus find a common set O1for which the smoothness condition
+property is true at the same time for F′
+j= (Θ +Θ′)F′
+jandG′
+l= (Θ + Θ′)G′
+lwith respect to
+a common compactly supported function Θ equal to 1 on O1. Effectively, the above integral is
+divided into the sum of four different ones, which we now analys e separately.
+24Part I)The first term is
+/integraldisplay
+C2p(Θ(x1)F′
+j⊗Θ(y1)G′
+l)·/parenleftbig
+ω⊗Ij+l−2/parenrightbig
+dµ(x1)dµ(y1). (38)
+In this case the integral can be considered as the smearing of a distribution in D′/parenleftbig
+Cj+l
+p/parenrightbig
+with
+a test function in C∞
+0(C2
+p). Hence, Theorem 8.2.12 of [25] ensures that, using the nota tion
+introducedthere, theresultof (38) isadistributionwhose wave frontset is contained in Wj+l−2∪
+(Wj×Wl)◦(A×{0})⊂Wj+l−2as given in (31). Notice that, in the proof of the last inclusi on,
+we have used (35) and (36). Hence property 2 in Definition 3.1 h olds. That said, property 1 is
+automatically satisfied since, by hypothesis, F′
+j∈AjandG′
+l∈Al, while property 3 holds true
+for the resulting distribution as (32) is valid a priori in all variables and, thus, left untouched
+for those which have not been integrated out in (38).
+Part II) The second term is
+/integraldisplay
+C2p(Θ′(x1)F′
+j⊗Θ′(y1)G′
+l)·/parenleftbig
+ω⊗Ij+l−2/parenrightbig
+dµ(x1)dµ(y1). (39)
+The analysis is rather simple if we make profitable use of (32) in the integrated variables and
+interpret the previous integral in the weak sense. Namely, l etf.= Θ′(x1)F′
+j(u) for some u∈
+C∞
+0/parenleftbig
+Cj−1
+p/parenrightbig
+andf′.= Θ′(xj+1)G′
+l(u′) for someu′∈C∞
+0/parenleftbig
+Cl−1
+p/parenrightbig
+, then we have that the operation
+ω(f,f′) is well defined due to the continuity property (26) satisfied byω. Hence, properties 1,
+2 and 3 in Definition 3.1 hold true because they are satisfied by Θ′(x1)F′
+j⊗Θ′(y1)G′
+l.
+Parts III & IV) The remaining two terms are substantially identical and we t reat only one
+of them. Hence consider
+/integraldisplay
+C2p(ΘF′
+j⊗Θ′G′
+l)·/parenleftbig
+ω⊗Ij+l−2/parenrightbig
+dµ(x1)dµ(y1). (40)
+In order to cope with this integral we introduce a new larger f actorisation η+η′= 1 with
+η∈C∞
+0(Cp) such that η= 1 on a large compact set properly containing the closure of s upp(Θ)
+so that both supp( η′Θ′)∩supp(Θ) =∅andη′Θ′=η′. If nowG′
+lis substituted with ( η+η′)G′
+l
+we obtain another splitting. On the one hand, since Θ′η∈C∞
+0(Cp), the analysis of Θ F′
+j⊗Θ′ηG′
+l
+boils down to that of case I, while, on the other hand,Θ(V)[Θ′η′](V′)
+(V−V′)2turns out to be smooth on
+Cp, since by construction ( V−V′)2>0 forVon the support of Θ and V′on that ofη′. Hence,
+if we write the smoothness condition (32) by means of ηasη′(xj+1)G′
+l(tl−1) =f(xj+1), where
+tl−1∈C∞
+0/parenleftbig
+Cl−1
+p/parenrightbig
+, we obtain that u.= Θω(f) is a compactly supported smooth function on C,
+thus yielding that F′
+j(u) is a well-posed operation as uis compactly supported. Furthermore,
+in order to conclude the analysis of the present case, due to T heorem 8.2.12 in [25], we notice
+that the wave front set of (40) is contained in Wj+l−2given in (31) and that property 3 in
+Definition 3.1 holds true just by applying (32) before smeari ng it.
+The result of this subsection is that/parenleftbig
+Ae(Cp),⋆ω/parenrightbig
+is a full-fledged topological∗-algebra. Fur-
+thermore, duetothecompactness propertystated inDefiniti on 3.1, thesubalgebra( Ae/parenleftbig
+C+
+p),⋆ω/parenrightbig
+25defined by restriction of the domain of the test distribution s toC+
+pis a well-defined topolog-
+ical∗-algebra and, thus, we are ready to discuss the intertwining relations between bulk and
+boundary data.
+3.5 Interplay Between the Algebras and the States on Dand on C+
+p
+We are now in the position to discuss a connection between the field theories described above,
+hence setting up a bulk-to-boundary correspondence and ide ntifying an Hadamard state in
+the bulk. The whole subsection is devoted to this issue, but, as a starting point, we need to
+recapitulate the geometric structure in order to clearly re late Subsections 2.2 and 3.2.
+Recall that we consider the globally hyperbolic subset O′contained in a geodesic neighbour-
+hood of an arbitrary but fixed point pin a strongly causal spacetime M. InO′we single out a
+double cone D≡D(p,q), which plays the role of the bulk spacetime, while the set ∂J+(p)∩D
+is our selected boundary. Up to the choice of an orthonormal f rame inp, the latter can be
+seen as the locus u= 0 in the natural coordinate system/parenleftbig
+u,r,xA/parenrightbig
+introduced in Subsection 2.2
+which is furthermore endowed with the metric (3). In terms of the structure of Subsection 3.2,
+we can identify the boundary as C+
+pwith a small caveat with respect to the coordinates used.
+While it is always possible to switch from xA(A= 1, 2) to the standard ( θ,ϕ), the role of V
+as a coordinate is played by r, the affine parameter on the null geodesics of the cone. As a las t
+point, the role of the function hin (19) is taken in general by4/radicalbig
+|gAB|, wheregABare the metric
+components appearing in (3) evaluated at u= 0 and|·|is kept to indicate the determinant. It
+is interesting to notice, that, whenever the conditions for the reduction of (3) to (5) are fulfilled,
+hcan be set to V≡r, (see also the relation between the volume elements of the sp here in
+different coordinates (4)). Furthermore, in the retarded coo rdinates used, the exponential map
+becomes an identity. Hence, if not strictly necessary, we sh all not indicate it anymore.
+Now we proceed in two steps. The first one consists in proving t he possibility of introducing
+a well-defined map from the extended algebra in Dto the one on C+
+p⊂Cp, while, in the second,
+we prove that this map is also well-behaved with respect to th e algebra structures.
+Theorem 3.1. LetDbe a double cone and regard the portion C+
+pof the boundary as part of a
+coneCp. Let us introduce the linear map Π :Fe(D)→Ae(C+
+p)by setting
+Πn(Fn).=4/radicalbig
+|gAB|1···4/radicalbig
+|gAB|n∆⊗n(Fn)/vextendsingle/vextendsingle
+(C+
+p)n, (41)
+where∆is the causal propagator (8),|C+
+pdenotes the restriction on C+
+pand the subscripts 1,
+...,nentail dependence of the root on the coordinates of the i-th cone. Then, the following
+properties hold true:
+1)ˆΠn, the integral kernel of Πn, is equal to⊗nˆΠ1and is an element of D′/parenleftbig
+(C+
+p×D)n/parenrightbig
+. The
+wave front set of ˆΠnsatisfies
+WF(ˆΠn)⊂(WF(ˆΠ1)∪{0})n\{0}. (42)
+Furthermore, if (x,ζx;y,ζy)∈WF(ˆΠ1), then:
+26(a) neither ζxnorζyvanish;
+(b)(ζx)r/\e}atio\slash= 0
+(c)(ζx)r/greaterorequalslant0if and only if−ζyis future directed.
+2) The image of Fe(D)underΠlies in Ae(Cp).
+Proof.We prove the above properties in two separate steps.
+I) Construction of/parenleftbig
+4/radicalbig
+|gAB|∆C/parenrightbig⊗nand the wave front set of its integral kernel
+In the normal neighborhood Op⊂Mwhich contains Dwe can select a subset O′⊂(J−(D)\
+J−(p))⊂Opwhich is a globally hyperbolic open set that extends DoverC+
+p, but neither over
+pnor over the future of D. The existence of a similar set results from the global hyper bolicity
+ofMwhich contains Opand thus also D. Let us indicate by ˆ∆∈D′(O′×D) the integral kernel
+of ∆ :C∞
+0(D)→C∞(O′) defined by restricting the map in (8). It holds true that
+WF(ˆ∆) =/braceleftbig
+(x1,ζ1;x2,ζ2)∈T∗O′×T∗D\{0}/vextendsingle/vextendsingle(x1,ζ1)∼(x2,−ζ2)/bracerightbig
+, (43)
+where the equivalence relation ( x1,ζ1)∼(x2,ζ2) means that there exists a null geodesic γwith
+respect to the metric ginDwhich contains both xandy. Furthermore, gµν(ζ1)νandgµν(ζ2)ν
+are the tangent vectors of the affinely parametrised geodesic γinxandy, respectively [38, 39].
+We proceed by restricting one entry of the causal propagator1onC+
+p, while leaving the
+other localised in D. To this end, let us define the embedding χofC+
+p×DtoO′×D, whose
+action, in retarded coordinates on O′, is defined as χ(r,θ,ϕ;x2) = (0,r,θ,ϕ;x2). According to
+Theorem 8.2.4 of [25], the restriction of the first entry of ˆ∆ on C+
+pby means of the pullback
+underχis well defined provided that Mχ∩WF(ˆ∆) =∅, whereMχis the set of normals of χ. In
+order to verify this statement about the empty intersection , we notice that, using the definition
+employed in such a theorem,
+Mχ⊂Nχ=/braceleftbig
+(x1,ζ1;x2,ζ2)∈T∗O′×T∗D/vextendsingle/vextendsinglex1∈C+
+p⊂O′,ζ1= (ζ1)udu, (ζ1)u∈R/bracerightbig
+.
+We shall now prove that Nχ∩WF(ˆ∆) =∅, consider ( x1,ζ1;x2,ζ2)∈Nχand the null geodesic
+γ′originating from x1whose tangent vector in x1is equal to g−1(ζ1). Notice that, in the
+retarded coordinates, the only non-vanishing component of g−1(ζ1) is (g−1(ζ1))rwhich implies
+thatγ′is contained in C+
+pand, in particular, does not enter D. For this reason the intersection
+ofNχwith WF( ˆ∆) is the empty set and hence the hypotheses of Theorem 8.2.4 o f [25] are
+fulfilled. Thus ˆ∆C, the pullback of ˆ∆ underχ, can be defined in one and only one way and
+WF(ˆ∆C)⊂χ∗WF(ˆ∆). In particular, this entails that, if ( x,ζx;y,ζy)∈WF(ˆ∆C) withx∈C+
+p
+andy∈D, it enjoys properties (a), (b) and (c) stated above.
+In order to verify (b), suppose this were not true and conside r (x,ζx;y,ζy)∈WF(ˆ∆C),
+where (ζx)r= 0. Thus there should exist an element ( x,ζ′
+x;y,ζy) such that χ∗(x,ζ′
+x;y,ζy) =
+(x,ζx;y,ζy), whereζ′
+xis a null covector whose components are ( ζ′
+x)r= 0, (ζ′
+x)θ= (ζx)θand
+(ζ′
+x)ϕ= (ζx)ϕwhile (ζ′
+x)uis a fixed number in R. Sinceg−1(ζ′
+x) has to be null and since
+1The same procedure was employed in the proof of Proposition 4 .3 in [30] or in the work [22].
+27(g−1(ζ′
+x))u= 0, the only possibility is ( ζ′
+x)θ= (ζ′
+x)ϕ= 0. Hence the only non-vanishing compo-
+nent ofg−1(ζ′
+x) is ther-component, which implies that ζ′
+x= (ζ′
+x)udu, thus (x,ζ′
+x;y,ζy)∈Nχ. At
+this point we have reached a contradiction because WF( ˆ∆)∩Nχ=∅so that (ζx)rhas to be dif-
+ferent from zero. Notice that (a) and (c) result from the cons traint imposed by the equivalence
+relation∼in the wave front set of ˆ∆ in (43) and from the observation that the projection of
+ζ′
+xonC+
+punderχ∗does not change the causal direction (past/future). In orde r to accomplish
+this part of the proof, we only need to multiply every ˆ∆Cwith4/radicalbig
+|gAB|⊗1 which is smooth
+because it is the 4-th order root of a smooth positive functio n. Hence the wave front set of the
+resulting distribution is left unchanged. Thus we define ˆΠnas the tensor productof distributions
+ˆΠn.= (hˆ∆C)⊗n∈D′((C+
+p×D)n), whereh=4/radicalbig
+|gAB|⊗1, and, due to Theorem 8.2.9 in [25],
+WF(ˆΠn) enjoys the inclusion (42). By the Schwartz kernel theorem w e obtain the linear map
+Πn= (4/radicalbig
+|gAB|∆C)⊗nwhose integral kernel is ˆΠn.
+II) On the image of Π
+Notice that, since every F∈Fe(D) is composed of a finite number of Fn, it is sufficient to
+prove that the generic Fnis mapped to an element of An
+sby Π or, rather, by Π n. Moreover, the
+pointwise product of ˆΠn, the integral kernel of Π n, andIn⊗Fn,Ithe unit constant function
+inD′(Cp), is well-defined because their wave front sets do not sum up t o the zero section, as
+one can infer from (13), from Theorem 8.2.9 in [25] and from (4 2) along with the discussion
+presented above. More precisely, we have that WF( In⊗Fn)⊂{0}×WF(Fn) where{0}is
+the zero section in T∗C+
+pnwhile every element in WF( ˆΠn) has non-vanishing components on
+that cotangent space, thus WF( In⊗Fn)+WF( ˆΠn) cannot contain the zero section. Hence the
+H¨ ormander criterion for the multiplication of distributi ons, Theorem 8.2.10 in [25], is fulfilled
+and, moreover, the resulting distribution ( ˆΠn)·(In⊗Fn) can be tested on any compactly
+supported smooth characteristic function ηon the support of Fnyielding the Π n(Fn) sought-
+after. Theorem 8.2.12 in [25] guarantees that Π n(Fn)∈D′/parenleftbig
+C+
+pn/parenrightbig
+and that WF(Π n(Fn))⊂/braceleftbig
+(x1,ζx1;...;xn,ζxn;y1,0;...;yn,0)∈WF/parenleftbig
+(ˆΠn)·(In⊗Fn)/parenrightbig/bracerightbig
+⊂Wnas in (31). In order to
+verify the last inclusion, we notice that WF( ˆΠ1)∩(S1×T∗D) =∅and that, making once more
+use of Theorem 8.2.10 in [25], WF(ˆΠn)·(In⊗Fn)∩(V±
+n×{0}) =∅, where{0}is the zero
+section inT∗Dn. Thus the wave front set of Π n(Fn) is contained in Wn, and this is tantamount
+to the second condition in Definition 3.1.
+As for the first one, this can be shown to hold true since, by con struction, supp( Fn)⊂Kn⊂
+Dn,Ka compact set, and hence, dueto the supportproperty of ∆, the re exists another compact
+setK′constructed as the closure of J−(K)∩C+
+pinCsuch thatK′ncontains the support of
+Πn(Fn).
+The third and last requirement can also be established by rec alling that the singular support
+of the causal propagator (8) is contained in the set of the nul l geodesics. Furthermore, those
+emanating from the support of any Fn(recall that supp( Fn)⊂K) intersect Cpon a compact set
+that is disjoint from pin particular. Hence the causal propagator is a smooth funct ion whenever
+one entry is smoothly localised2on the support of Fnand the other one on a neighbourhood of
+2The localisation is realised by pointwise multiplication w ith smooth functions of suitable support.
+28p. Furthermore, even after multiplication both by the functi on Θ′as in Definition 3.1 and by
+4/radicalbig
+|gAB|, such smooth function is square-integrable and bounded, to gether with its V-derivative,
+in a suitable open set of Cp∼R+×S2such thatV∈(0,V0), because it is a restriction to C+
+pof
+a smooth function defined in a neighbourhood of pmultiplied by4/radicalbig
+|gAB|. For the same reason,
+the limitV→0 of4/radicalbig
+|gAB|ˆ∆Ctends to zero whenever one entry of the causal propagator is
+localised on some compact set in D. Finally, notice that Π n(Fn) is totally symmetric whenever
+Fnhas this property, and this completes the proof that Π n(Fn)∈An
+s.
+As an intermediate step, we proceed by discussing the effect of the map Π on the symplectic
+form.
+Proposition 3.4. The projection Π :Fb(D)→Ae(C+
+p)is a symplectomorphism, i.e., for every
+f,h∈C∞
+0(D),
+σ(ϕf,ϕh) =σC(Π1f,Π1h), (44)
+withσtaken as in (9).
+Proof.Letϕf= ∆fandϕh= ∆h, where ∆ is as in (8), and consider both a Cauchy surface Σ
+ofDand the portion of O1.=D∩I−(Σ) whose boundary is formed by the null surface C+
+pand
+by Σ. Then the current
+Jµ.=ϕf∂µϕh−ϕh∂µϕf
+satisfies/integraltext
+Σdµ(Σ)nµJµ=σ(ϕf,ϕh) withnµthe unit future directed vector normal to Σ. Hence
+we can apply the divergence theorem to JµinO1considered as a subregion of a larger globally
+hyperbolicspacetime, O′, that contains D. Theresultis that, since ∇µJµ= 0inO1in particular,
+the following identity holds,
+σ(ϕf,ϕh) =/integraldisplay
+C+
+pdµ(C+
+p)nµJµ.
+Furthermore, the right-hand side of the preceding equation can be rewritten in terms of the
+retarded coordinates on C+
+pand, if one uses the relation between the volume elements on t he
+sphere (4) in spherical and local coordinates, it becomes
+/integraldisplay
+R+×S2/radicalbig
+|gAB|/bracketleftBig
+ϕf∂
+∂rϕh−ϕh∂
+∂rϕf/bracketrightBig
+dr∧dS2, (45)
+where both ϕfandϕhare evaluated on C+
+p, a legitimate operation as explained at the beginning
+of this section, and, furthermore, they vanish on the comple ment of C+
+pinCp. Finally, due to
+the antisymmetry of the preceding expression, we can consid er4/radicalbig
+|gAB|ϕf= Π1fas well as
+4/radicalbig
+|gAB|ϕh= Π1h, and a direct inspection shows that (45) equals σC(Π1f,Π1g) as given in (20)
+settingr=V.
+Remark: Notice that the ideal Igenerated by the equations of motion (7) is mapped by Π
+to 0∈Ae(C), because ∆ is a weak solution of (7). Hence the image of both Fb(D) and Fbo(D)
+under Π lie in Ae(C); actually they coincide.
+On the basis of this remark, we stress another important prop erty of Π.
+29Proposition 3.5. The map Πis injective when acting on the on-shell extended algebra Feo(D).
+Proof.Let us recall that the action of Π on F={Fn}n∈Fe(D) is determined by the actions
+of Πn= Π⊗n
+1on its components Fn. We shall hence analyse the kernel of Π 1seen as a map from
+T′(D) to some functionals on S(C+
+p), where the elements of T′(D) are the compactly supported
+symmetric distributions whose wave front sets enjoy (13). W e shall prove that ker(Π 1) =K.=/braceleftbig
+Pu/vextendsingle/vextendsingleu∈T′(D)/bracerightbig
+. Givenu∈E′(D) there exists a sequence of uj∈C∞
+0(D) whose support is
+contained in K, a proper compact subset of D, such that uj→uweakly forj→∞. Consider
+then the following chain of equalities
+−/integraldisplay
+M∆(f)uj=/integraldisplay
+M∆(f)P∆A(uj) =σ(∆(f),∆(uj)) =σC(Π1(f),Π1(uj)),
+wherePis the operator realising the equation of motion and ∆ Ais its advanced fundamental
+solution. Furthermore, Σ does intersect neither KnorJ+(K), and hence, on Σ, ∆ A(uj) is
+equal to ∆ R(uj). Notice that in order to obtain the second equality, we have to choose two
+elements of a family of Cauchy hypersurfaces, Σ and Σ′, which do not intersect Kand such that
+J+(K)∩Σ =∅andJ−(K)∩Σ′=∅, and integrate by parts twice. The resulting integral on Σ′
+vanishes due to the support property of ∆ A, while the integral on Mis zero since P(∆(f)) = 0.
+Thus the remaining integral is precisely the symplectic for m computed on Σ. Furthermore, the
+last equality derives from Proposition 3.4. We proceed by wr iting
+/integraldisplay
+M∆(f)uj=−2/integraldisplay
+Cp4/radicalbig
+|gAB|∆(uj)∂
+∂V/parenleftBig
+4/radicalbig
+|gAB|∆(f)/parenrightBig
+dVdS2.
+Passing now to the weak limit yields Π 1(u)(−2∂VΠ1f) =u(∆(f)). From the previous discussion
+we obtain that, letting S.=−2∂VΠ1(C∞
+0(D)), the condition Π 1(u)(S) = 0 is equivalent to
+u(∆(D(D))) = 0, and the latter implies u=Pu′for someu′∈E′(D). Since, as a functional,
+Π1(u) are defined on a set larger than Swe have that ker(Π 1)⊂K. In order to obtain the
+opposite inclusion, let us define by Rthe operator that realises the restriction on C+
+pand the
+subsequent multiplication by4/radicalbig
+|gAB|. Notice that Π 1is defined as the composition R◦∆, where
+now ∆ is the map from T′(D) toD′(O′), where O′is the normal neighbourhood containing D.
+Hence ker(Π 1)⊃ker(∆) = K. Note that Kis contained in the ideal Idivided out of Fe(D)
+in order to obtain Feo(D). The proof can then be concluded by applying a similar proce dure
+to Πnto verify that also ker(Π n) is contained in the ideal I.
+Before continuing the analysis of the map Π acting on the exte nded algebra Fe(D), we show
+that the pull-backs both of the symplectic form σCand of the boundary state ωhave a nice
+interplay with the symplectic form in the bulk and with the Ha damard states in general. The
+next proposition deals with the singular structure of the st ateωwhen pulled back in the bulk.
+Proposition 3.6. Under the assumptions of Theorem 3.1
+Hω.= Π∗ω (46)
+is an Hadamard bi-distribution constructed as the pull-bac k ofωas in(24)underΠas in(41).
+30Proof.The proof of this proposition can be performed by restrictin g attention to the compactly
+supported smooth functions on D. Let us start by showing that Hωis a good distribution on
+C∞
+0(D2), hence continuous in the topology of compactly supported s mooth functions. To this
+end, notice that Hω(f,g) =ω(Π1(f),Π1(g)), moreover, ω(Π1(f),Π1(g)) enjoys the continuity
+stated in (26). Furthermore, the L2norms present in (26) are controlled by the supremum
+norms of Π 1(f), Π1(g) and their r-derivatives on some compact set in C(here taken as the
+entire cone). The proof of the continuity of Hωcan be concluded employing the continuity of
+∆ :C∞
+0(M)→C∞(M), whichisspoiltneitherbytherestrictionon C+
+pnorbythemultiplication
+by4/radicalbig
+|gAB|.
+Notice that the antisymmetric part of Hωequals the symplectic form (9) which is preserved
+by the action of Π (Proposition 3.4), and Hωsatisfies the equation weakly because so does ∆
+which is used in the definition of Π. Furthermore, Hω(f,f) is positive for every f, becauseω
+is a state for the boundary algebra. We thus conclude that Hωis the two-point function of
+a quasi-free state for Fb(D). Hence, in order to prove the Hadamard property, due the the
+work of Radzikowksi [38], it is only necessary to check that t he wave front set of Hωsatisfies
+the microlocal spectrum condition. This can be verified foll owing the procedure envisaged in
+[30, 22]. For completeness, we shall summarise here the main steps of such a proof.
+1.)It suffices to show that the microlocal spectrum condition hol ds locally in D, namely, when
+Hωis restricted on a generic compact set K2withK⊂D. We hence have to show that
+WF(Hω) =/braceleftbig
+(x1,ζ1;x2,−ζ2)∈T∗K2\{0}/vextendsingle/vextendsingle(x1,ζ1)∼(x2,ζ2),ζ1⊲0/bracerightbig
+, (47)
+where∼istheequivalencerelationof (43), while ζ1⊲0indicatesthat ζ1isafuturedirectedvector.
+According to the preceding discussion, to show the inclusio n⊃, we make use of Theorem 5.8 of
+[41] which can be applied once ⊂holds and yields the other relation as thesis.
+2.)In order to show that ⊂holds in (47), notice that the past directed null geodesics o riginating
+fromKinOpintersect C+
+pon a region contained in a compact set N⊂C+
+p. We stress that
+p/\e}atio\slash∈C+
+p, and hence p/\e}atio\slash∈N. Thus, if we smoothly localise ˆΠ1(the integral kernel of Π 1) on
+N′×K, whereN′is the complement of NinC+
+p, the resulting object is described by a smooth
+function which is square-integrable on C+
+pand so is its V-derivative, also when an entry of ˆΠ1
+is kept fixed in K.
+3.)We shall henceintroducea partition of unity on C+
+p, ΘN+Θ′
+N= 1, such that Θ N∈C∞
+0(C+
+p)
+is equal to 1 on the compact set N. Hence it vanishes on the intersection of a sufficiently small
+neighbourhood of pwithC+
+p. Inserting two such partitions of unity in Π∗ωand employing
+multilinearity, Hωbecomes the sum of four terms, ω/parenleftbig
+(ΘN+Θ′
+N)ˆΠ1⊗(ΘN+Θ′
+N)ˆΠ1/parenrightbig
+.
+4.)The only one which contributes to WF( Hω) isω(ΘNˆΠ1⊗ΘNˆΠ1). In this case, we notice that,
+due to the form of WF( ω) given in (28) and to the constraint enjoyed by WF( ˆΠ1) as discussed
+in 1) of Theorem 3.1, we can apply Theorem 8.2.13 of [25] in ord er to obtain that the inclusion
+sought holds for the wave front set of this term.
+5.)All the other three terms have vanishing wave front sets. Let us briefly discuss them
+separately. In particular, due to the regularity shown by Θ′
+NˆΠ1when restricted to K, the
+31composition of (Θ′
+NˆΠ1⊗Θ′
+NˆΠ1) withωonC+
+p2can be computed and yields a smooth function
+onK2. The remaining two terms can be addressed similarly. To this end, let us concentrate on
+ω(Θ′
+NˆΠ1⊗ΘNˆΠ1). At this point, notice that the supports of Θ′
+Nand of Θ Nhave non-vanishing
+intersection; however, such an intersection is contained i n a further compact set R. We can
+thus insert another partition of unity, Θ R+ Θ′
+R, in order to divide such a term in two parts
+ω(Θ′
+N(ΘR+ Θ′
+R)ˆΠ1⊗ΘNˆΠ1/parenrightbig
+. Hence,ω/parenleftbig
+(Θ′
+NΘ′
+R)ˆΠ1⊗ΘNˆΠ1) has a vanishing wave front set
+because the supports of Θ′
+NΘ′
+Rand ΘNare disjoint and ωis represented by a smooth function
+on their Cartesian product. Furthermore, since both Θ′
+NΘRand Θ Nare inC∞
+0(C+
+p), we can
+estimate the wave front set of ω/parenleftbig
+Θ′
+NΘRˆΠ1⊗ΘNˆΠ1/parenrightbig
+employing once more Theorem 8.2.13 of
+[25] to obtain that it is equal to the empty set.
+We can now prove asecond theorem which focuseson the effect of t he mapΠ onthe algebraic
+structures and on the boundary state. In particular, we shal l individuate an Hadamard state in
+the bulk.
+Theorem 3.2. Under the assumptions of Theorem 3.1, one has:
+1)Πinduces a unit-preserving∗-homomorphism between the algebras/parenleftbig
+Fe(D),⋆Hω/parenrightbig
+and/parenleftbig
+Ae(C+
+p),⋆ω/parenrightbig
+.
+2)Πis an injective∗-homomorphism when acting “on shell” on/parenleftbig
+Feo(D),⋆Hω/parenrightbig
+.
+Proof.We only prove the first statement. The second arises by direct inspection from this one
+and from Proposition 3.5. First notice that Π automatically preserves the∗-operation because
+Πn= Πn, hence
+Πn(Fn)∗= Πn(F∗
+n).
+Thus we only need to verify the statement on the ⋆-products. In particular, we look for ⋆Hω:
+Fe(D)×Fe(D)→Fe(D) such that,
+Π(F ⋆HωG) = (ΠF)⋆ω(ΠG),∀F,G∈Fe(D), (48)
+and, at the same time/parenleftbig
+Fe,⋆Hω/parenrightbig
+is isomorphic to/parenleftbig
+Fe,⋆H/parenrightbig
+.
+The natural candidate arises from the analysis performed in Proposition 3.6 and, in partic-
+ular, from the distribution Hωintroduced in (46) to be plugged in (14) in place of H. This is a
+well-defined procedure since Hωis of Hadamard form as proved in Proposition 3.6, and, hence,
+(Fe,⋆H) turns out to be isomorphic to ( Fe,⋆Hω), the isomorphism being realised as in (15).
+We are thus left with the task to verify (48) for every FandGinFe(D). If we exploit both the
+definitions and the bilinearity of all the ⋆-products involved, this reduces to the requirement to
+show that
+Πl+m−2k/parenleftbig
+(Fl⊗Gm)(Hω⊗k)/parenrightbig
+= (ΠlFl⊗ΠmGm)(ω⊗k)
+forl+m−2k/greaterorequalslant0. The last relation directly results from bilinearity, (46 ) and from the fact
+that Π k= Πk1⊗Πk2for allk1,k2>0 such that k1+k2=k.
+32Remark: Notice that it is possible to turn the injective homomorphis m into a bijection if
+we restrict attention to the local von Neumann algebras defin ed as the double commutant in
+the GNS representation of a quasifree Hadamard state of the C∗-algebra generated by the local
+Weyl operators constructed out of the symplectic forms (9) a nd (20) in Dand on the boundary
+C+
+p, respectively. This last claim is based on the invertibilit y of Π 2on the weak solutions of
+the Klein-Gordon equation, where we recall that the Goursat problem with compact initial data
+onCp, in general, yields a solution of (7) whose restriction on an y Cauchy surface of Dis not
+compact.
+Alas, the von Neumann algebra mentioned does not contain rel evant physical observables
+such as the components of the regularised stress-energy ten sor or the regularised squared fields,
+objects we would like to use in order to extract information a bout the local geometric data such
+as the scalar curvature.
+4 Interplay with General Covariance and Comparison between
+Spacetimes
+We are now in the position to collect all our results in a compr ehensive framework which will
+exhibit both a nice interplay with the principle of local cov ariance, as devised in [9], and the
+possibility to compare quantum field theories on different bac kgrounds both at the level of
+algebras and of states.
+To this avail, the construction in Subsection 3.5 will play a pivotal role, and the natural
+language we adopt is that of categories, which was already in troduced in Subsection 2.1. In
+particular, notice that the construction of an extended alg ebra of observables on a double cone
+Dcan be realised as a suitable functor between the categories DoCoisoandAlg, although it is
+not possible to extend such a functor to DoCo.
+Notwithstandingthis obstruction, theadditional structu rewhicharises fromboththebound-
+ary and the field theory defined thereon allows us to circumven t the above problem in a way
+that also puts us in a position to compare field theories in diffe rent spacetimes. This requires
+the introduction of a further category, namely,
+BAlg: The objects of this category are the extended boundary (top ological∗-)algebras presented
+in Subsection 3.4, constructed on all possible C+
+p, while the morphisms are the unit pre-
+serving∗-homomorphisms among them.
+The key point consists in making a profitable use of the∗-homomorphisms Π introduced in
+Theorem 3.1, in order to establish that Π ◦Findeed defines a functor between the two categories
+DoCoandBAlg, whichadmits thefollowing pictorial, butinspiringdiagr ammatic representation,
+DF−−−−→ Fe(D)Π−−−−→ Ae(C+
+p)
+ıe,e′/arrowbt/arrowbtαıe,e′
+D′−−−−→
+FFe(D′)−−−−→
+Π′Ae(C+
+p′).(49)
+33The arrow αıe,e′traces back to the analysis in Subsection 3.2, where it was sh own that the
+boundary theory can be constructed and analysed independen tly of the specific bulk. Hence,
+αıe,e′is the∗-homomorphism αıe,e′:Ae(C+
+p)→Ae(C+
+p′) whose action on the F′∈Ae(C+
+p) is
+defined as follows: By means of the push-forward, it is
+αıe,e′(F′
+n) =ıe,e′∗F′
+n,
+on/parenleftbig
+ıe,e′(C+
+p)/parenrightbign⊂(C+
+p′)n, whileαıe,e′(F′
+n) = 0 on the complement of/parenleftbig
+ıe,e′(C+
+p)/parenrightbignin (C+
+p′)n.
+Such an operation is well-defined because F′
+nhas compact support towards the future and, when
+extended on the closure of C+
+p,ıe,e′mapsptop′. We hence have the following
+Proposition 4.1. Consider Ae:DoCo→BAlg, whose action on the objects and morphisms is
+seen as follows,
+•the action of Aeon the objects of DoCois such that Ae(D) = Π◦Fe(D) =Ae(C+
+p);
+•the action of Aeon the morphisms ıe,e′is such that Ae(ıe,e′) =αıe,e′.
+Then Aeis a functor between the two categories.
+Proof.In order to show that Ae:DoCo→BAlgis a functor, notice that the covariance property
+holds and the identity is preserved, i.e.,
+Ae(ıe,e′)◦Ae(˜ıe′,˜e′) =Ae(ıe,e′◦˜ıe′,˜e′), Ae(idD) =idAe(C+
+p),
+as can be seen by direct inspection of the definition of Ae(ıe,e′).
+As for the connection with the principle of general local cov ariance, we recall that, in the
+mostgeneral case, it isnotpossibletofindadirect relation between F(D) andF(D′), unlessthe
+embedding Π : F(D′)→F(C′) can be inverted on the image of Π composed with αıe,e′. This
+is indeed what happens whenever, e.g,ıe,e′is an isometry (or, at worst, a conformal isometry
+[34]) which preserves the base point p. Hence we are working in DoCoiso.
+Under this assumption, the discussion about causality, usu ally an integral part of the rea-
+soning as in [9, 7], does not have to be performed directly, si nce its essence is already encoded
+in the analysis of the properties of the map Π. A similar state ment holds true also for the
+time-slice axiom (see [10], in particular). Especially, si nce the theory on the boundary is, to a
+certain extent, non-dynamical, there is no such axiom in our boundary framework and the one
+in the bulk is automatically assured by Π and its properties.
+4.1 Comparison of Expectation Values in Different Spacetime s
+The aim of this section is to clarify in which sense one can com pare two field theories on two
+different backgrounds. We shall first explain the procedure ab stractly and then give a concrete
+example.
+34Bearing in mind (49), it is straightforward to realise that, whenever we assign a state ωon
+Ae(C′
+p), wecanpullitbackeitheron Fe(D′)viaΠ′oronFe(D)viaαıe,e′◦Π, andtheinformation
+about the bulk geometry is indeed restored by Π and Π′. This is a rather general feature which
+holds true regardless of the global structure of the spaceti mes in which DandD′are embedded.
+Yet, on practical grounds, it is natural to choose one of the t wo double cones embedded in the
+four-dimensionalMinkowski spacetime, whereourcapabili ty ofperformingexplicitcomputations
+of physical quantities is enhanced due to the large symmetry of the background.
+To be more specific on this point, consider a double cone Das a subset of ( R4,η) while D′
+lies in a generic strongly causal spacetime M′, chosen in such a way that there exist two frames
+eande′inTp(R4) andTp′(M′), respectively, yielding a well-defined ıe,e′:D→D′. At this
+point we can apply the construction discussed for the local fi elds and algebras, related on the
+boundary via the map αıe,e′.
+As a next step, following also the general philosophy of [9], we consider observables con-
+structed out of the same local fields either on DorD′. Here we suppose that there exists
+ıe,e′:D→D′and hence, from the same field, we form Φ( f)∈Fe(D) and Φ(ıe,e′∗f)∈Fe(D′),
+where Φ is a local field in the sense of [9]. We compute their exp ectation values on the pull-back
+of a suitable boundary state yielding an Hadamard counterpa rt in the bulk. In general, the
+difference between the results depends on the geometric data o f both DandD′, and thus we
+are ultimately comparing quantum field theories on different b ackgrounds.
+Nonetheless, from a computational point of view, this proce dure is still too involved, since
+one has to cope with the singular structure of the chosen stat e(s). Even if we restrict attention
+to those fulfilling the Hadamard condition, one would still n eed to take care of the regularisation
+procedure of the observables, a hassle which can be avoided. The idea is to consider on each
+double cone two bulk Hadamard states constructed out of the p ull-back of different boundary
+counterparts andtoworkwiththeirdifference. Inthiscaseth eintegral kernelof suchadifference
+isknowntobesmooth, anadvantagewhichstronglyreducesth ecomputational efforts. Although
+the price to pay is the introduction of two states on the light cone, a natural candidate as one
+of them is the distinguished reference state ωwhich arises from (46) in Proposition 3.6. Hence
+we are left with the need to assign only one extra datum.
+Before we discuss an explicit example of this procedure, we s tress the most important prop-
+erties of both ωand of its pull-back in the bulk, say Π∗ω. These can be inferred from both
+Proposition 3.2 and Theorem 3.2:
+1.Local Lorentz invariance : According to the third item of Proposition 3.2, ωturns out to be
+invariant under a set of geometric transformations on the fu ll cone Cwhich contains the
+boundary. In particular, ωis always invariant under the natural action of the subgroup of
+the Lorentz group which corresponds to isometries of the nei ghbourhood where the state
+is defined. Since the map Π is constructed substantially out o f the causal propagator (8) in
+a geodesic neighbourhood, its action on ωvia pull-back does not spoil the above property,
+i.e., Π∗ωis invariant under the above assumptions.
+2.Microlocal structure : The wave front set of ωis contained in the union of (29) and (30)
+and, most notably, it does not contain directions to the past . In particular, this allows for
+35the proof of Proposition 3.6 according to which Π∗ωsatisfies the Hadamard property in
+the bulk double cone.
+3.Behaviour as a “vacuum” : The boundary state ωturns out to be invariant under rigid
+translations of the V-coordinate, or, in other words, it is a vacuum with respect t o the
+transformation generated by the vector ∂V. This statement can be proved exactly as
+in [29, 15] for the counterpart on the conformal boundary of a n asymptotically flat or
+of a cosmological spacetime, viz., from the explicit form of the two-point function (27).
+This also entails that the energy computed on the cone with re spect to∂Vis minimised.
+Unfortunately, this property has not a strong counterpart i n the bulk, but, if the bulk can
+be realised as an open set in ( R4,η), then Π∗ωis seen to coincide with the Minkowski
+vacuum for massless fields.
+4.2 An Application: Extracting the Curvature
+In this subsection we shall present an explicit application which follows the guidelines given
+above. For simplicity consider on the one hand a double cone Drealised as an open subset of
+Minkowski spacetime ( R4,η), where the metric ηhas the standard diagonal form with respect
+to the Cartesian coordinates ( t,x,y,z) induced by the standard orthonormal frame eofR4. As
+D′we consider a double cone which can be embedded in a homogeneo us and isotropic solution
+of Einstein’s equation with flat spatial section. This is a Fr iedman-Robertson-Walker spacetime
+(M′,g), whereg=a2(t)ηanda(t)∈C∞(I,R+) withI⊆Randa(0) = 1. Here trefers to
+the so-called conformal time and thus we still consider the c oordinates ( t,x,y,z) induced by
+the standard frame of R4, indicated as e′to distinguish it from the previous one. Furthermore,
+notice that, in view of the special form of g,e′=e
+a(t).
+Since the underlying spacetimes are conformally related, t heir causal structures and, in
+particular, the double cones coincide. Consider two points p= (0,0,0,0) andq= (t′,0,0,0) and
+the corresponding double cones D(p,q)⊂R4andD′(p,q)⊂M′. In this framework the map
+ıe,e′:D(x0,x1)→D′(x0,x1) turns out to be trivial. Next, we choose a minimally coupled real
+scalar field theory, viz.,
+φ:D→R,✷φ= 0,
+where✷isthed’Alembertwaveoperatorconstructedoutofthemetri cinD. Westressoncemore
+that we consider the very same equation also in D′. If we follow the guidelines of Subsection 3.1,
+we can construct Fe(D) and Fe(D′) and their counterparts on the boundaries CpandC′
+p. As
+outlined in the previous subsection, we now consider two alg ebraic states on Ae(Cp), one,ω, is
+thereferencestate, whiletheothercanbearbitrary, provi dedthatthepull-backtothebulkviaΠ
+still fulfils the Hadamard condition. This requirement is no t too restrictive since, e.g., any state
+which differs from ωby a smooth function on the boundary and vanishes in a neighbo urhood of
+the tip is an admissible choice.
+In particular, consider another Gaussian state ω′:Ae(Cp)→C, whose two-point function
+36has the following form,
+ω′(ψ1,ψ2) =ω(ψ1,ψ2)+1
+4π/integraldisplay
+R+×R+×S2drdr′dS2(θ,ϕ)ψ1(r,θ,ϕ)ψ2(r′,θ,ϕ). (50)
+Notice that the integral on the right-hand side entails that the integral kernel of ω′−ωis not
+smooth because it contains a δ-like singularity in the angular coordinates. Despite this factω′
+can be pulled back to every spacetime still yielding an Hadam ard bi-distribution.
+Thelast statement can beproved operatingin thesameway as i ntheproofof Proposition3.6
+withωreplaced by ω′−ω. The key new feature, yielding WF/parenleftbig
+Hω′−ω/parenrightbig
+=∅, is the fact that, while
+(ζx)ris never equal to zero for every ( x,ζx;y,ζy)∈WF(Π 1), every (x,ζx;y,ζy)∈WF(ω′−ω) is
+such that (ζx)r= (ζy)r= 0.
+We are now ready to consider the expectation values of suitab le observables from the point
+of view of both DandD′. The most natural one is the expectation value of φ2(f), where
+f∈C∞
+0(D) (and hence also in C∞
+0(D′)), with respect to the bulk states constructed out of
+ω′andω. Notice that φ2(f) is a shortcut for saying that we are actually considering uf=
+f(x)δ(x−y)∈A2
+s(D)⊂Fe(D). One of the advantages of this construction is that, in this
+case, we are allowed to keep a more general stance, namely, we can substitute f(x) by a Dirac
+function peaked at the point xt= (t,0,0,0) witht∈(0,t′). This is tantamount to consider
+:φ2: (xt), whereu=δ(x−xt)δ(x−y)∈A2
+s(D).
+Now (41) can be used to evaluate Π 2uin both ( R4,η) and (M′,g) by means of the explicit
+form of the causal propagator. In Minkowski spacetime it loo ks like (see [17] or [35])
+∆(x,x′).=−δ(t−t′−|x−x′|)
+4π|x−x′|+δ(t−t′+|x−x′|)
+4π|x−x′|, (51)
+wheretis the time coordinate and xthe three-dimensional spatial vector in Euclidean coordi-
+nates. The counterpart of ∆ in D′can be directly evaluated exploiting the conformal trans-
+formation between ( M′,g) and (R4,η). The d’Alembert wave equation in the first spacetime
+(M′,g) corresponds in the flat one to
+✷φ−a′′
+aφ= 0, (52)
+where′stands for time derivation and ✷=−∇µ∇µ. Furthermore, the causal propagator /tildewide∆ of
+this partial differential equation is related to the one in M′via
+∆M′(x,y) =1
+a(tx)a(ty)/tildewide∆(x,y). (53)
+If we follow the procedure discussed in Chapter 4 of [35], we g et
+/tildewide∆(x,x′) = ∆(x,x′)+V(x,x′)
+4π/parenleftbig
+Θ(t−t′−/vextendsingle/vextendsinglex−x′/vextendsingle/vextendsingle)−Θ(−t+t′−/vextendsingle/vextendsinglex−x′/vextendsingle/vextendsingle)/parenrightbig
+,
+37whereV(x,x′) is a smooth function whose explicit form is derived from the Hadamard recursive
+relations for (52). In particular, it also holds that V(x,x) =−a′′(x)
+2a(x).
+We are now ready to compare the expectation values in ( ω′−ω). The Minkowski side of this
+operation yields, by direct computation,
+(ω′−ω)(ΠR4
+2u) =1
+4,
+while in the cosmological setting
+(ω′−ω)(ΠM′
+2u) =/bracketleftBigg
+(4π)/integraldisplay∞
+0/tildewide∆(xt,r∗)
+a(t)a(r∗)r∗a(r∗)a(r∗)2dr∗/bracketrightBigg2
+,
+where we have rewritten the integral in the r-variable in terms of r∗, the affine parameter of the
+null cone in Minkowski spacetime. The defining relation betw een the two variables is
+dr=a2(r∗)dr∗.
+The above integral can be rewritten by means of (53) as
+(ω′−ω)(ΠM′
+2u) =/bracketleftBigg
+4π/integraldisplay∞
+0∆(xt,r∗)a(r∗)2
+a(t)r∗dr∗+/integraldisplayt/2
+0V(xt,r∗)a(r∗)2
+a(t)r∗dr∗/bracketrightBigg2
+,
+in which the first integral yields via (51)
+(ω′−ω)(ΠM′
+2u) =/bracketleftBigg/integraldisplay∞
+0δ(t−2r∗)a(r∗)2
+a(t)dr∗+/integraldisplayt/2
+0V(xt,r∗)a(r∗)2
+a(t)r∗dr∗/bracketrightBigg2
+.
+Let us now expand a(t) in a power series around the point x0,
+(ω′−ω)(ΠM′
+2u) =/bracketleftbigg1
+2a(t/2)2
+a(t)+a′′(x0)t2
+4+O(t3)/bracketrightbigg2
+,
+where, in the derivative, we have exploited that, at first ord er int,
+V(xt,r∗) =a′′(x0)
+a(x0)+O(t),
+also due to the rotational symmetry of M′. If we now expand both a(t) anda(t/2) in a Taylor
+series , we obtain
+(ω′−ω)(ΠM′
+2u) =/bracketleftBigg
+a
+2/bracketleftBigg
+1+/parenleftBigg
+−1
+2a′′
+a+3
+2/parenleftbigga′
+a/parenrightbigg2/parenrightBigg
+t2/bracketrightBigg
++a′′t2
+4+O(t3)/bracketrightBigg2
+=
+/bracketleftBigg
+a
+2+3a
+4/parenleftbigga′
+a/parenrightbigg2
+t2+O(t3)/bracketrightBigg2
+,
+38where all functions atogether with their derivatives are evaluated at x0. We can summarize the
+discussion, finally calculating the difference between ( ω′−ω)(ΠR4
+2u) and (ω′−ω)(ΠM′
+2u),
+(ω′−ω)(ΠR4
+2u)−(ω′−ω)(ΠM′
+2u) =3
+4(a′(x0))2t2+O(t3),
+witha(x0) = 1. The interpretation of this result is that, exactly as ex pected, the above com-
+parison yields a result which, at first order, allows us to ext ract via a measurement precise
+information on the a priori unknown geometric data, in this c ase, the derivative of the scale
+factor at the point x0in a Friedman-Robertson-Walker universe.
+5 Summary and Outlook
+In this paper we have achieved a twofold goal: on the one hand w e propose a novel way to look
+at the properties of a local quantum field theory in a suitable curved background, while, on the
+other hand, the very same construction yields a mechanism wh ich allows for the comparison of
+expectation values of field observables in different spacetim es.
+More specifically, starting from a careful analysis of the un derlying geometry, we realise that
+only moderate assumptions are needed to reach our goals, viz., our general setting consists of
+an arbitrary strongly causal manifold Min which we identify an arbitrary but fixed double
+coneD≡D(p,q) =I+(p)∩I−(q) strictly contained in a normal neighbourhood of p. Since D
+is globally hyperbolic, we can consider therein a real scala r field theory along the lines of (7)
+and therefore follow the general quantisation scheme which particularly calls for the association
+of a Borchers-Uhlmann algebra of observables with the chose n system. This algebra can be
+extended, both enlarging the set of its elements and the defin ing product, in order to encompass
+alsoa priorimore singular objects, such as the Wick polynomials, which c onstitute the so-called
+extended algebra. The very deep reason for choosing D⊂Mlies in its boundary and, more
+properly, on the portion of J+(p) which it contains. This is a differentiable submanifold of
+codimension 1 on which it is possible to construct a genuine f ree scalar field theory, following
+exactly the same procedure successfully employed for the ca usal boundary of an asymptotically
+flat or cosmological spacetime in [13, 14]. The main novel res ult in this framework arises from
+the construction of an extended algebra also for the boundar y theory— Ae(Cp) in the main
+body—whose well-posedness is justified both by its mathemat ical properties and by its relation
+to the bulk counterpart. Hence, the latter is embedded in Ae(Cp) by means of Π, an injective
+∗-homomorphism.
+The advantage of this picture is the possibility to make use o f a long tradition, originating
+from [27], which allows us to exploit the geometrical proper ties of the boundary to identify
+for the algebra thereon a natural state which can be pulled-b ack to the bulk via Π, yielding a
+counterpart which satisfies the microlocal spectrum condit ion, hence is of Hadamard form. This
+guarantees that we can identify a local state in Dwhich is physically well-behaved. In physical
+terms it means that this state is the same for all inertial obs ervers atp. In other words, the
+bulk state as well as the one on the boundary are invariant und er a natural action of SO0(3,1).
+39Thus it can be identified as a sort of local vacuum on a curved sp acetime independent of the
+frame.
+The second goal of comparing expectation values on different b ackgrounds is based on the
+above construction. More precisely, we consider not just on e but actually two regions as above
+in twoa priori different spacetimes MandM′. The construction of the field theories proceeds
+as usual, but now we make use of the invertibility of the expon ential map in geodesic neigh-
+bourhoods in order to engineer the double cones D⊂MandD′⊂M′so that we can map
+the boundary in Dto that in D′via a local diffeomorphism. This procedure can be brought
+to the level of boundary extended algebras which thus can be r elated by means of a suitable
+restriction homomorphism. The advantage is the previously unknown possibility to carefully use
+the distinguished state identified on each Cpin order to compare the bulk expectation values
+of field observables constructed for the theories in DandD′. The important point is that this
+new perspective is completely compatible with the standard principle of general local covariance
+when applicable as devised in [9], and, actually, it complem ents it corroborating its significance.
+Furthermore, sincenothingpreventsusfromchoosingoneof thespacetimes astheMinkowski
+one, one can concretely check how the proposed machinery all ows for the comparison of the
+expectation values of the field observables, making manifes t the role and the magnitude of the
+geometric quantities. We stress this point by means of a simp le example involving a massless
+minimally coupled field in the flat and in a cosmological space time. It seems safe to claim that
+there are several possibilities to apply our procedure to ma ny other cases of physical interest.
+These are certainly not the only roads left open, and actuall y even the identified bulk Hadamard
+state should be studied in more detail. As a matter of fact, it is interesting to understand
+whether it is connected in any way with the states of minimum e nergy that appear in Friedman-
+Robertson-Walker spacetimes [32]. We leave this as well as t he myriad of other questions for
+future investigations.
+Acknowledgements.
+The work of C.D. is supported by a Junior Research Fellowship from the Erwin Schr¨ odinger
+Institute and he gratefully acknowledges it, while that of N .P. is supported by the German DFG
+Research Program SFB 676. We would like to thank Bruno Bertot ti, Romeo Brunetti, Klaus
+Fredenhagen, Thomas-Paul Hack and Valter Moretti for profit able discussions on the topics of
+the present paper. C.D. also gratefully acknowledges the su pport of the National Institute for
+Theoretical Physics of South Africa for his stay at the Unive rsity of KwaZulu-Natal.
+A Hadamard States
+This appendix briefly recollects some properties of Hadamar d states which are used throughout
+the main text. Since most of the material has already been pro ved in several different alterna-
+tive ways in the literature, we limit ourselves to giving the main statements and the necessary
+references. Let us stress that, from a physical perspective , Hadamard states are the natural can-
+40didates for physical ground states of a quantum field theory o n a curved background, since their
+ultraviolet behaviour mimics that of the Minkowski vacuum a t short distances and, furthermore,
+they guarantee that the quantum fluctuations of the expectat ion values of observables, such as
+the smeared components of the stress-energy tensor, are fini te.
+Inthesubsequentdiscussionwealways assumethatwearedea lingwithaquasi-freestateona
+suitablefieldalgebraconstructedonaglobally hyperbolic spacetime( M,g) fromafieldsatisfying
+an equation of motion such as (7). We stick to this assumption because it is consistent with
+the main body of the paper, but the reader should keep in mind t hat such an hypothesis could
+be relaxed (see for example [42]). As a starting point we stat e a global criterion characterising
+Hadamard states [38, 39].
+Definition A.1. A stateωsatisfies the Hadamard condition and is thus called an Hadamard
+stateif and only if
+WF(ω) =/braceleftbig
+(x,kx;y,−ky)∈T∗M2\{0}/vextendsingle/vextendsingle(x,kx)∼(y,ky),kx⊲0/bracerightbig
+,
+where, in this expression, ωactually stands for the integral kernel of the two-point fun ction
+associated with ω. The relation (x,kx)∼(y,ky)indicates that there exists a null geodesic γ
+connecting xtoysuch thatkxis coparallel and cotangent to γatxandkyis the parallel
+transport of kxfromxtoyalongγ. The requirement kx⊲0means that the covector kxis future
+directed.
+The above condition on the wave front set is rather useful and often employed on practical
+grounds to check whether a given state really is Hadamard or n ot. Nonetheless, it is possi-
+ble to provide another definition via the so-called Hadamard form , which has been rigorously
+introduced in [27].
+Definition A.2. A stateωis said to be of the (local) Hadamard form if and only if in
+any convex normal neighbourhood the integral kernel of the a ssociated two-point function can be
+written as
+ω(x,y) =H(x,y)+W(x,y),
+where
+H(x,y) = lim
+ǫ→0+U(x,y)
+σǫ(x,y)+V(x,y)lnσǫ(x,y)
+λ2, (54)
+and the limit is to be understood in the weak sense. Here, U,V, as well as Ware smooth
+functions, while λis a reference length; furthermore,
+σǫ(x,y).=σ(x,y)±2iǫ/parenleftbig
+T(x)−T(y)/parenrightbig
++ǫ2
+withǫ >0. In the above formula, Tis a time function, such that ∇Tis timelike and future
+directed on the full spacetime (M,g). In addition, if we apply (7)either to the x- or to the
+y-variable, the result has to be a smooth function.
+41The existence of a time function Tis guaranteed on any globally hyperbolic manifold [3, 4]
+as these can be decomposed as Σ ×R, where Σ is a smooth Cauchy surface and Ris the range
+of the time function T.
+A completely satisfactory definition of the Hadamard form re quires some more work to rule
+out spacelike singularities, to circumvent convergence pr oblems of the series V, which is only
+asymptotic, and, finally, to assure that the definition depen ds neither on a special choice of the
+temporal function Tnor on the convex normal neighbourhood employed.
+Instrict terms, wehaveonly definedthelocal Hadamardform h ere. A stronger andmoresat-
+isfactorydefinition, theso-calledglobalHadamardform,h asbeenintroducedin[27]. Itreinforces
+the local form extending it from the convex normal neighbour hoodsto certain “causally-shaped”
+neighbourhoods of a Cauchy surface, thereby ruling out spac elike singularities. However, in [39],
+it has been shown that the local Hadamard form already implie s the global Hadamard form.
+Another important fact is that the singular structure (54) i s completely determined by the
+geometry of the background and the equation of motion. This o f course does not hold for W
+which encodes the full state dependence.
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+arXiv:1001.0859v2  [math.GR]  5 Jan 2011ON THE RANK OF COMPACT p-ADIC LIE GROUPS
+BENJAMIN KLOPSCH
+Abstract. The rank of a profinite group Gis the basic invariant
+rk(G) := sup {d(H)|H≤G}, where Hranges over all closed
+subgroups of Gandd(H) denotes the minimal cardinality of a
+topological generating set for H. A compact topological group G
+admits the structure of a p-adic Lie group if and only if it contains
+an open pro- psubgroup of finite rank.
+For every compact p-adic Lie group Gone has rk( G)≥dim(G),
+where dim( G) denotes the dimension of Gas ap-adic manifold. In
+this paper we consider the converse problem, bounding rk( G) in
+terms of dim( G).
+Every profinite group Gof finite rank admits a maximal finite
+normalsubgroup,itsperiodicradical π(G). Oneofourmainresults
+is the following. Let Gbe a compact p-adic Lie group such that
+π(G) = 1, and suppose that pis odd. If {g∈G|gp−1= 1}is
+equal to {1}, then rk( G) = dim( G).
+1.Introduction
+In the 1960s Lazard developed a sophisticated theory of p-adic Lie
+groups. One of the high points of his work was the algebraic charac-
+terisation of such groups, providing a solution to the p-adic analogue
+of Hilbert’s 5th problem. A more recent approach to the theory of
+compact p-adic Lie groups, inspired by Lazard’s work and focusing on
+group-theoretic aspects, is given in [1].
+Therankof a profinite group Gis the basic invariant rk( G) :=
+sup{d(H)|H≤G}, whereHranges over all closed subgroups of G
+andd(H) denotes the minimal cardinality of a topological generating
+set forH. According to [1, Corollary 8.34], a topological group G
+admits the structure of a compact p-adic Lie group if and only if it is a
+profinite group of finite rank which contains an open pro- psubgroup.
+Moreover, theanalyticstructurewhichmakes Gintoap-adicLiegroup,
+whenever it exists, is unique. See [1, Interlude A] for a wide range o f
+alternative characterisations of pro- pgroups of finite rank.
+Another key invariant of a compact p-adic Lie group Gis itsdi-
+mension, dim(G), as ap-adic manifold. Algebraically, dim( G) can be
+2000Mathematics Subject Classification. 20E18, 22E20.
+Key words and phrases. compact p-adic Lie group, rank, dimension, number of
+generators.
+12 BENJAMIN KLOPSCH
+described as d(U) whereUis any uniformly powerful open pro- psub-
+group of G. Consequently, for every compact p-adic Lie group Gone
+has rk(G)≥dim(G). Uniformly powerful pro- pgroups are rather spe-
+cialp-adic analytic groups which are torsion-free and in many ways
+behave like lattices. For instance, one has rk( U) = dim( U) for every
+uniformly powerful pro- pgroupU.
+Straightforward examples show that, in general, to bound the ran k
+rk(G) of a compact p-adic Lie group Gin terms of dim( G), one needs
+to restrain the torsion of G. In the first instance, it is thus natural to
+consider groups which are torsion-free.
+Question 1.1.Is it true that for every torsion-free compact p-adic Lie
+groupGone has rk( G) = dim(G)?
+In Section 2 we show that a result of Laffey [8], which bounds the
+number of generators of finite p-groups, readily implies
+Proposition 1.2. LetGbe a torsion-free compact p-adic Lie group,
+and suppose that pis odd. Then rk(G) = dim(G).
+It is remarkable that a p-group argument yields such a result, but
+unfortunately the method of proof does not appear to lead furth er.
+Proposition 1.2 is restricted to groups without any torsion and falls
+short of dealing with 2-adic Lie groups. An analogue of Laffey’s the-
+orem for p= 2 is given in [3, Corollary 3.10]; it has, however, rather
+weak implications in the present context.
+Inthe present paper we take a new approach which leads to the mor e
+general Theorems 1.3 and 1.4 below and which, we think, will be more
+suitable for dealing with 2-adic Lie groups. Indeed, our results are
+formulated for compact p-adic Lie groups which possess no non-trivial
+finite normal subgroups, a condition which also arises in the study of
+lattices. Every profinite group Gof finite rank admits a maximal finite
+normalsubgroup π(G),theperiodicradical ofG. IfGisp-adicanalytic,
+so isG/π(G) and dim( G) = dim( G/π(G)). These facts provide a
+natural reduction to groups Gsatisfying π(G) = 1. For every prime ℓ,
+we define the ℓ-rankof a profinite group Gto be rk ℓ(G) := rk(S) where
+Sis any Sylow pro- ℓsubgroup of G. Our main result is
+Theorem 1.3. LetGbe a compact p-adic Lie group with π(G) = 1,
+and suppose that pis odd. Then one has
+max{rkℓ(G)|ℓ >2prime}= rkp(G) = dim(G)
+and
+rk2(G)≤/braceleftBigg
+⌊3dim(G)/2⌋ifp≡41,
+dim(G) ifp≡43.
+A theorem of Guralnick and Lucchini, generalising work of Kov´ acs,
+shows that every profinite group Gsatisfies rk( G)≤sup{rkℓ(G)|
+ℓprime}+1; see [4, 10]. This and related results allow us to proveON THE RANK OF COMPACT p-ADIC LIE GROUPS 3
+Theorem 1.4. LetGbe a compact p-adic Lie group with π(G) = 1,
+and suppose that pis odd. Then one has
+(1) rk(G)≤max{dim(G),rk2(G)}+1;
+(2) rk(G)≤max{dim(G)+1,rk2(G)}ifGis prosoluble;
+(3) rk(G)≤dim(G)+1ifp≡43;
+(4) rk(G) = dim(G)ifGhas trivial (p−1)-torsion, i.e. if {g∈G|
+gp−1= 1}is equal to {1}.
+Remark 1.5.Relatively simple examples show that Theorems 1.3 and
+1.4 are to some extent best possible. Indeed, for p≡41 the matrix
+group
+S:=/a\}⌊ra⌋ketle{t/parenleftbigg
+0 1
+1 0/parenrightbigg
+,/parenleftbigg√−1 0
+0√−1/parenrightbigg
+,/parenleftbigg
+−1 0
+0 1/parenrightbigg
+/a\}⌊ra⌋ketri}ht ≤GL2(Zp)
+has order 16, acts irreducibly on the abelian lattice Z2
+pand requires a
+minimumof3generators. Thusforany r∈Nandanyprime p≡41,the
+2r-dimensional compact p-adic Lie group G= (S⋉Z2
+p)rhasπ(G) = 1
+and rk 2(G) = 3r= 3dim(G)/2.
+For any prime p, any divisor m/\e}ati≀\slash= 1 ofp−1 and any d∈N, the
+groupG=T⋉ Zd
+p, whereT∼=Cmconsists of the scalar matrices in
+GLd(Zp) corresponding to mth roots of unity in Zp, hasπ(G) = 1 and
+rk(G) =d+1 = dim( G)+1.
+While the focus of the present paper is on p-adic Lie groups for
+odd primes p, it remains a challenging open problem to describe the
+precise relation between rank and dimension for compact 2-adic Lie
+groups. Basic examples indicate that things are somewhat different
+for the prime 2. For instance, for every r∈Nther-fold direct power
+of the pro-2 dihedral group D=C2⋉ Z2satisfiesπ(Dr) = 1 and
+rk(Dr) = 2r= 2dim(Dr).
+We recall that a pro- pgroupGis said to be d-maximal ifd(H)<
+d(G) for every proper open subgroup HofG. It is known that, for
+oddp, everyd-maximal finite p-group is nilpotent of class at most 2;
+cf. [3] and references therein. Our proof of Theorem 1.3 indicates that
+Question 1.1 for p= 2 is linked to the long standing problem of under-
+standing the structure of d-maximal finite 2-groups and pro-2 groups.
+Of particular relevance would be a positive answer to
+Question 1.6.Is it true that every d-maximal pro-2 group is soluble?
+Besides using the concept of d-maximal groups, we employ in our
+proof of Theorem 1.3 standard techniques from the theory of p-adic
+analytic pro- pgroups. Furthermore we take advantage of the descrip-
+tion of maximal ℓ-subgroups of general linear groups over Qp, given
+in [9], and the classification of indecomposable ZpC-modules for cyclic
+groupsCof order p, provided in [5]. Of independent interest is our4 BENJAMIN KLOPSCH
+elementary proof of the following auxiliary result, which one may com-
+pare with more general, asymptotic estimates for the minimal numbe r
+of generators of finite linear groups; for instance, see [6, 2, 7, 1 2].
+Theorem 1.7. Suppose that pis odd. Let d∈N, and let ℓbe prime
+withℓ/\e}ati≀\slash=p. Letd0∈ {0,1}such that d≡2d0, and set m(p,ℓ) :=
+min{n∈N|ℓdividespn−1}. Then
+rkℓ(GLd(Zp)) = rk ℓ(GLd(Fp)) =
+
+⌊d/m(p,ℓ)⌋ifp/\e}ati≀\slash≡ℓ1,
+(3d−d0)/2ifℓ= 2andp≡41,
+d in all other cases.
+Notation. Throughout, pandℓdenote primes. The field of p-adic
+numbers and the ring of p-adic integers are denoted by QpandZp. The
+commutator subgroup of a group Gis denoted by [ G,G]. The centre of
+Gis denoted by Z( G), the centraliser in Gof a subset S⊆Gby CG(S).
+The minimal cardinality of a generating set for a group Gis denoted by
+d(G). Likewise the minimal cardinality of a generating set for a module
+Mover a ring Ris denoted by dR(M). Forn∈N0it is customary to
+denote by Gnthen-fold direct power of G; in a few places, we use
+the same notation, Gnforna power of p, to denote the characteristic
+subgroupof Ggenerated byall nthpowers ofelements of G. All wreath
+products considered are permutational wreath products.
+WhenGis a topological group, subgroups HofGare tacitly taken
+to be closed, d(G) tacitly refers to the minimal number of topological
+generators, etc. A key concept in the theory of p-adic Lie groups is
+that of a powerful pro- pgroup. We refer to [1, Part I] for standard
+properties of uniformly powerful pro- pgroups. A profinite group Gis
+said to be just infinite if it is infinite and if every proper quotient of G
+is finite.
+2.A short proof using Laffey’s bound
+Clearly, Proposition 1.2 is a consequence of Theorem 1.3. But it also
+admits a short independent proof, based on a result of Laffey [8].
+Proof of Proposition 1.2. For every prime ℓwithℓ/\e}ati≀\slash=p, the Sylow pro- ℓ
+subgroup of Gis finite. Since Gis torsion-free, we conclude that G
+is a pro- pgroup. Let Ube a uniformly powerful open normal sub-
+group of G. Thend(U) = dim( G) and descending to an open sub-
+group, if necessary, it suffices to show that d(G)≤dim(G). The open
+subgroups Upn,n∈N, provide a base for the neighbourhoods of 1
+inG. Consequently, since Gis torsion-free, we find m∈Nsuch that
+{x∈G|xp∈Upm} ⊆U. (In fact, a short argument shows that
+alreadym= 1 works.) As Uis uniformly powerful, this implies that
+for alln∈Nwithn≥m,
+Ω1(G/Upn) =/a\}⌊ra⌋ketle{txUpn∈G/Upn|xp∈Upn/a\}⌊ra⌋ketri}ht=Upn−1/UpnON THE RANK OF COMPACT p-ADIC LIE GROUPS 5
+has sizepdim(U). Sincepis odd, we deduce from [8, Corollary 2] that
+d(G) = limsup
+n∈Nd(G/Upn)≤limsup
+n∈Nlogp|Ω1(G/Upn)|= dim(G).
+/square
+3.Maximal ℓ-subgroups of GLd(Qp)
+Forr∈N0we denote by Wr(ℓ) ther-fold iterated permutational
+wreath product Cℓ≀ ··· ≀Cℓof cyclic groups of order ℓ. This finite
+permutation group of degree ℓrhas order |Wr(ℓ)|=ℓ(ℓr−1)/(ℓ−1)and re-
+quiresd(Wr(ℓ)) =rgenerators. Recall that the Sylow ℓ-subgroups of a
+symmetric group Sym( n) of finite degree ncan be described in terms of
+iterated wreath products: each of them is isomorphic to/producttextt
+i=1Wi(ℓ)ni,
+wheren=/summationtextt
+i=0niℓiis theℓ-adic expansion of nwith coefficients
+0≤ni< ℓ.
+The Sylow ℓ-subgroups of finite symmetric groups feature in the
+classification of maximal ℓ-subgroups of general linear groups; see [9].
+We define the following numerical invariants:
+m(p,ℓ) :=/braceleftBigg
+min{n∈N|ℓdividespn−1}ifp/\e}ati≀\slash=ℓ,
+p−1 if p=ℓ,
+a(p,ℓ) :=/braceleftBigg
+max{b∈N0|ℓbdividespm(p,ℓ)−1}ifp/\e}ati≀\slash=ℓ,
+1 if p=ℓ.
+In addition, we define for p≡43 (andℓ= 2),
+c(p,2) := max {b∈N|2bdividesp2−1}.
+Furthermore, we recall that for c∈Nwithc≥3 the semidihedral
+group of order 2c+1is given by the presentation
+Dsemi
+2c+1=/a\}⌊ra⌋ketle{tx,y|x2=y2c= 1, yx=y−(1+2c−1)/a\}⌊ra⌋ketri}ht.
+We require the following information about maximal ℓ-subgroups of
+general linear groups over the p-adic field Qp.
+Proposition 3.1. Letd∈Nand suppose that (p,ℓ)/\e}ati≀\slash= (2,2). Then the
+groupGLd(Qp)has exactly one conjugacy class of maximal ℓ-subgroups.
+LetLbe one of the maximal ℓ-subgroups of GLd(Qp).
+(1)Ifℓis odd, then L∼=Cℓa≀S, wherea=a(p,ℓ)andSis a Sylow
+ℓ-subgroup of Sym(⌊d/m(p,ℓ)⌋).
+(2)Suppose that ℓ= 2and that pis odd. If p≡41, thenL∼=C2a≀S,
+wherea=a(p,2)andSis a Sylow 2-subgroup of Sym(d). Ifp≡43
+andd≡20, thenL∼=Dsemi
+2c+1≀S, wherec=c(p,2)andSis a Sylow 2-
+subgroup of Sym(d/2). Ifp≡43andd≡21, thenL∼=C2×(Dsemi
+2c+1≀S),
+wherec=c(p,2)andSis a Sylow 2-subgroup of Sym((d−1)/2).6 BENJAMIN KLOPSCH
+Proof.The proposition is a special instance of Theorems II.4, IV.2 and
+IV.3 in [9]. Indeed, m(p,ℓ) is equal to the degree of the ℓth cyclotomic
+fieldQp(ζℓ) overQp, whereζℓdenotes a primitive ℓth root of unity,
+anda(p,ℓ) = max{b∈N|Qp(ζℓ) contains a primitive ℓbth root of 1 }.
+Moreover, if ℓ= 2 and p≡43, thenc=c(p,2) takes the same val-
+ues as the invariant γ(Qp) used in [9, Section IV]. We observe that
+Theorem IV.4 in op. cit. is not used, as we do not discuss the case
+p=ℓ= 2. /square
+Our aim is to prove Theorem 1.7. In view of Proposition 3.1, we
+consider for r∈N0the groups
+Xa,r(ℓ) :=Cℓa≀Wr(ℓ), a ∈N,
+Yc,r:=Yc,r(2) :=Dsemi
+2c+1≀Wr(2), c ∈Nwithc≥3.
+We require the following lemma.
+Lemma 3.2. Letn∈N, and suppose that σ∈Sym(n)is a permu-
+tation of ℓ-power order acting without fixed points. Let h∈GLn(Zℓ)
+be a monomial matrix with permutation part corresponding to σ. Let
+Mbe aZℓ/a\}⌊ra⌋ketle{th/a\}⌊ra⌋ketri}ht-submodule of the natural Zℓ/a\}⌊ra⌋ketle{th/a\}⌊ra⌋ketri}ht-moduleV=Zn
+ℓ. Then
+dZℓ/angbracketlefth/angbracketright(M)≤2n/ℓ.
+Proof.We observe that nis a multiple of ℓ, and we argue by induction
+onn. First suppose that n=ℓso thatσis anℓ-cycle. We are to show
+thatdZℓ/angbracketlefth/angbracketright(M)≤2. Clearly, this is true if ℓ= 2. Now suppose that
+ℓ >2. Then we write det( h) =λℓ(1 +µ), where λ∈Z∗
+ℓand either
+µ= 0 orµ∈ℓZℓ\ℓ2Zℓ. After a change of basis, we may assume that
+the action of hon the standard Zℓ-basise1,...,e ℓofVis given by
+eh
+i=λei+1fori∈ {1,...,ℓ−1}andeh
+ℓ=λ(1+µ)e1.
+For the purpose of bounding dZℓ/angbracketlefth/angbracketright(M) we may as well assume that
+λ= 1, and we treat the two possibilities for µseparately.
+First consider the case µ= 0. Then his the permutation matrix as-
+sociated to the ℓ-cycleσandVthe corresponding integral permutation
+module. The Zℓ/a\}⌊ra⌋ketle{th/a\}⌊ra⌋ketri}ht-moduleMisfreeasa Zℓ-moduleanddecomposesas
+a direct sum of indecomposable Zℓ/a\}⌊ra⌋ketle{th/a\}⌊ra⌋ketri}ht-modules. Each indecomposable
+summand IofMsatisfiesdZℓ/angbracketlefth/angbracketright(I) = 1 and dim Zℓ(I)∈ {1,ℓ−1,ℓ};
+see [5, Theorem 2.6]. Since dim Zℓ(M)≤n=ℓand since /a\}⌊ra⌋ketle{th/a\}⌊ra⌋ketri}htdoes
+not act trivially on M, it follows that Mis the sum of at most two
+indecomposable Zℓ/a\}⌊ra⌋ketle{th/a\}⌊ra⌋ketri}ht-modules, and hence dZℓ/angbracketlefth/angbracketright(M)≤2, as wanted.
+Nextconsider thecase µ∈ℓZℓ\ℓ2Zℓ. Thentheminimumpolynomial
+ofh−1 overQℓis (X+ 1)ℓ−(1 +µ), an Eisenstein polynomial.
+Thus the ring Zℓ/a\}⌊ra⌋ketle{th/a\}⌊ra⌋ketri}htis isomorphic to the ring of integers Oof a totally
+ramified extension KofQℓof degree ℓ. Furthermore, the Zℓ/a\}⌊ra⌋ketle{th/a\}⌊ra⌋ketri}ht-module
+Vcorresponds to the O-module O, andMto an ideal of the principal
+ideal domain O. ThusdZℓ/angbracketlefth/angbracketright(M)≤1, finishing the argument for n=ℓ.ON THE RANK OF COMPACT p-ADIC LIE GROUPS 7
+Now suppose that n > ℓand that the permutation group /a\}⌊ra⌋ketle{tσ/a\}⌊ra⌋ketri}htis
+intransitive: /a\}⌊ra⌋ketle{tσ/a\}⌊ra⌋ketri}ht ≤Sym(n1)×Sym(n2), wheren=n1+n2withpositive
+summands. Write σ1andσ2for the images of σunder projection
+into Sym( n1) and Sym( n2) respectively, and note that each of these
+permutationsactswithoutfixedpoints. Themonomialmatrix hadmits
+acorresponding blockdecomposition h=h1⊕h2, withthepermutation
+part of the monomial matrix hicorresponding to σifori∈ {1,2}.
+Clearly, the module Vadmits a Zℓ/a\}⌊ra⌋ketle{th/a\}⌊ra⌋ketri}ht-submodule Wsuch that V1:=
+V/W∼=Zn1
+ℓandV2:=W∼=Zn2
+ℓare naturally modules for Zℓ/a\}⌊ra⌋ketle{th1/a\}⌊ra⌋ketri}htand
+Zℓ/a\}⌊ra⌋ketle{th2/a\}⌊ra⌋ketri}ht. Setting M1:= (M+W)/WandM2:=M∩W, we deduce by
+induction that
+dZℓ/angbracketlefth/angbracketright(M)≤dZℓ/angbracketlefth1/angbracketright(M1)+dZℓ/angbracketlefth2/angbracketright(M2)≤2(n1+n2)/ℓ= 2n/ℓ.
+Finally, suppose that n > ℓand that the permutation group /a\}⌊ra⌋ketle{tσ/a\}⌊ra⌋ketri}htis
+transitive. Then nis equal to the order of σ, hence an ℓ-power:n=ℓk
+withk≥2. Clearly, σℓ∈Sym(n) acts with ℓorbits of size ℓk−1and
+has no fixed points. Applying the argument given before, we deduce
+thatdZℓ/angbracketlefth/angbracketright(M)≤dZℓ/angbracketlefthℓ/angbracketright(M)≤2n/ℓ. /square
+Proposition 3.3. Leta∈Nandr∈N0. Then
+rk(Xa,r(ℓ)) =/braceleftBigg
+3·2r−1ifℓ= 2,a≥2andr≥1,
+ℓrin all other cases.
+Proof.We write X:=Xa,r(ℓ) and argue by induction on r. Ifr= 0,
+the group X∼=Cℓais cyclic of order ℓaand the assertion holds. Now
+suppose that r≥1.
+First we derive a lower bound for rk( X). LetE:=Ea,r(ℓ) denote
+the homocyclic group of exponent ℓawhich arises as the base subgroup
+ofX=Cℓa≀Wr(ℓ). Clearly, rk( X)≥rk(E) =ℓr. Next we derive for
+ℓ= 2,a≥2 andr≥1 the stronger lower bound rk( X)≥3·2r−1.
+Indeed, the 2-group Sdisplayed in Remark 1.5 has d(S) = 3 and is
+easily embedded into C4≀C2. Sincea≥2, the group C4≀C2embeds
+intoC2a≀C2, and the decomposition X∼=(C2a≀C2)≀Wr−1(2) shows
+that the 2r−1-fold direct power of Sembeds into X. We conclude that
+rk(X)≥d(S2r−1) = 3·2r−1.
+It remains to give an upper bound for rk( X). Consider an arbitrary
+subgroup H≤X, and decompose X∼=Xa,r−1(ℓ)≀CℓasX=B⋊C,
+whereB=B1×...×BℓwithBi∼=Xa,r−1(ℓ) fori∈ {1,...,ℓ}and
+C∼=Cℓ. In addition, we decompose the homocyclic base group EofX
+intoE=E1×...×Eℓ, whereEi=E∩Bi∼=(Cℓa)ℓr−1is the homocyclic
+base group of the wreath product Bi∼=Cℓa≀Wr−1(ℓ) fori∈ {1,...,ℓ}.8 BENJAMIN KLOPSCH
+First consider the case H≤B. Induction on ryields
+d(H)≤ℓ/summationdisplay
+i=1rk(Bi) =ℓ·rk(Xa,r−1)
+=/braceleftBigg
+3·2r−1ifℓ= 2,a≥2 andr≥2,
+ℓrin all other cases.(3.1)
+Indeed, by projecting Hinto the first factor B1ofB, then projecting
+the kernel N1of the first projection into the second factor B2ofB, and
+so on, we find a descending subnormal series H=N0≥N1≥...≥
+Nℓ= 1 such that, for i∈ {1,...,ℓ}, the quotient Ni−1/Niembeds
+intoBi. Clearly, d(H)≤/summationtextℓ
+i=1d(Ni−1/Ni)≤/summationtextℓ
+i=1rk(Bi) and (3.1) is
+justified.
+It remains to consider the case where His not contained in B, and
+it will be enough to prove that under this hypothesis d(H)≤3ℓr−1.
+Decompose X∼=Cℓa≀Wr(ℓ) asX=E⋊W, whereE=E1×...×Eℓ
+withEi=E∩Bifori∈ {1,...,ℓ}as before and W∼=Wr(ℓ) =
+X1,r−1(ℓ). By induction on r, we have d(HE/E)≤rk(W) =ℓr−1.
+Furthermore, if ℓ= 2, we have d(H∩E)≤rk(E) =ℓr= 2·ℓr−1, thus
+d(H)≤d(HE/E)+d(H∩E)≤3ℓr−1, as wanted.
+Now suppose that ℓ≥3. Choose h∈H\Band regard H∩Eas a
+Zℓ/a\}⌊ra⌋ketle{th/a\}⌊ra⌋ketri}ht-module. In view of the inequality d(H)≤d(HE/E)+dZℓ/angbracketlefth/angbracketright(H∩
+E),itsufficestoprovethat dZℓ/angbracketlefth/angbracketright(H∩E)≤2ℓr−1. Notethat hpermutes
+cyclically the ℓfactorsEiofE. Working with a pre-image MofH∩E
+in the base group of the ℓ-adic group Zℓ≀W, we apply Lemma 3.2 to
+deduce that dZℓ/angbracketlefth/angbracketright(H∩E)≤2ℓr/ℓ= 2ℓr−1. /square
+We remark that, for ℓ≥5, one can easily extend the argument
+given in the proof of Proposition 3.3 to show that every subgroup Hof
+X:=Xa,r(ℓ) =Cℓa≀Wr(ℓ) withd(H) =ℓr= rk(X) is contained in the
+homocyclic base group E∼=Cℓr
+ℓa. It remains a curious open problem to
+classify subgroups witnessing the rank of Xa,r(ℓ) for the small primes
+ℓ∈ {2,3}.
+Proposition 3.4. Letc∈Nwithc≥3, and let r∈N0. Then
+rk(Yc,r) = 2r+1.
+Proof.We reason similarly as in the proof of Proposition 3.3. Write
+Y:=Yc,rand argue by induction on r. Ifr= 0, the group Y∼=Dsemi
+2c+1
+is metacyclic but not cyclic, and the assertion follows. From now on
+suppose that r≥1.
+By considering the base subgroup of Y=Dsemi
+2c+1≀Wr(2), we obtain
+rk(Y)≥rk((Dsemi
+2c+1)2r) = 2rrk(Dsemi
+2c+1) = 2r+1,
+and it remains to show that rk( Y)≤2r+1. Consider an arbitrary
+subgroup H≤Y, and decompose Y∼=Yc,r−1≀C2asY=B⋊C, whereON THE RANK OF COMPACT p-ADIC LIE GROUPS 9
+B=B1×B2withBi∼=Yc,r−1,i∈ {1,2}, andC∼=C2. IfH≤B, then
+induction on ryields
+d(H)≤rk(B) = 2rk(Yc,r−1) = 2·2r= 2r+1.
+Thus it remains to consider the case where His not contained in B.
+Decompose Y∼=Dsemi
+2c+1≀Wr(2) asY=E⋊W, whereE∼=(Dsemi
+2c+1)2r
+andW∼=Wr(2). Since Dsemi
+2c+1is metacyclic, Eis metabelian and, in
+fact, decomposes via a characteristic subgroup Finto homocyclic parts
+E/F∼=(C2)2randF∼=(C2c)2r.
+Chooseh∈H\Band regard ( H∩E)F/FandH∩FasZ2/a\}⌊ra⌋ketle{th/a\}⌊ra⌋ketri}ht-
+modules. In view of the inequality
+d(H)≤d(HE/E)+dZ2/angbracketlefth/angbracketright((H∩E)F/F)+dZ2/angbracketlefth/angbracketright(H∩F),
+it suffices to establish: (i) d(HE/E)≤2r−1, (ii)dZ2/angbracketlefth/angbracketright((H∩E)F/F)≤
+2r−1and (iii) dZ2/angbracketlefth/angbracketright(H∩F)≤2r.
+SinceHE/Eembeds into W∼=X1,r−1(2), Proposition 3.3 yields (i).
+Considering(ii),wewrite E=E1×E2,whereEi=E∩Bi∼=(Dsemi
+2c+1)2r−1
+fori∈ {1,2}. It is convenient to set K:=H∩Eand to use ·to denote
+reduction modulo F. Note that htransposes the two factors in the
+decomposition E=E1×E2of the elementary 2-group E. Thush
+conjugates K∩E2injectively into E1∼=E/E2, and we conclude that
+dZ2/angbracketlefth/angbracketright(K)≤/parenleftbig
+d(K/E2)−d(K∩E2)/parenrightbig
++d(K∩E2) =d(K/E2).
+AsE/E2∼=E1is an elementary 2-group of rank 2r−1, this proves (ii).
+Finally,halso transposes the two factors Fi:=F∩Ei,i∈ {1,2},
+ofF=F1×F2. Working with the pre-image MofH∩Fin the
+base group of the 2-adic group ( Z2⋉Z∗
+2)≀W, we apply Lemma 3.2 to
+deduce (iii). /square
+Again, it would be interesting to give a complete description of all
+subgroups witnessing the rank of Yc,r. We are ready to prove Theo-
+rem 1.7 and the following complementary result.
+Proposition 3.5. Suppose that pis odd, and let d∈N. LetGbe a
+maximal finite p-subgroup of GLd(Qp). Thenrk(G) =⌊d/(p−1)⌋.
+Proof of Theorem 1.7 and Proposition 3.5. The claims are direct con-
+sequences of Propositions 3.1, 3.3 and 3.4, combined with the descrip -
+tion of Sylow subgroups of finite symmetric groups in terms of wreat h
+products. /square
+4.Bounding the ℓ-rank
+Proposition 4.1. Suppose that pis odd. Let Gbe a finite p-group,
+and letMbe a faithful ZpG-module which is free and of finite rank as
+aZp-module. Then
+d(G)+dZpG(M)≤dimZp(M).10 BENJAMIN KLOPSCH
+Proof.Clearly, we may suppose that Gis not trivial. Let C=/a\}⌊ra⌋ketle{tx/a\}⌊ra⌋ketri}htbe a
+central subgroup of Gof orderp. According to [5, Theorem 2.6], there
+are three types of indecomposable ZpC-modules: they are the trivial
+moduleZpof dimension 1, the free module ZpCof dimension p, and
+the (p−1)-dimensional module ZpC/Φp(x)ZpCwhere Φ p= 1 +X+
+...+Xp−1. As aZpC-module, Mis the direct sum of indecomposable
+modules of these three types.
+SetN:= CM(C) ={m∈M|mx=m}. SinceCis central in G
+and since Cacts faithfully on M, the set Nconstitutes a proper ZpG-
+submodule of M. We distinguish two cases: N={0}andN/\e}ati≀\slash={0}.
+First suppose that N/\e}ati≀\slash={0}. In this case, Gacts naturally on the
+quotient module M1:=M/Nwith image G1≤GL(M/N) and kernel
+G2, say. The finite group G2acts faithfully on M2:=N; indeed, the
+QpG-module Qp⊗ZpMis completely reducible and Qp⊗ZpM2admits
+a complement isomorphic to Qp⊗ZpM1. Clearly, d(G)≤d(G1)+d(G2)
+anddZpG(M)≤dZpG1(M1)+dZpG2(M2). Therefore induction gives
+d(G)+dZpG(M)≤d(G1)+d(G2)+dZpG1(M1)+dZpG2(M2)
+≤dimZp(M1)+dim Zp(M2)
+= dim Zp(M).
+Now suppose that N={0}. Then, as a ZpC-module, Mnecessarily
+decomposes into a direct sum of indecomposable ZpC-modules each of
+which is isomorphic to ZpC/Φp(x)ZpC. Clearly,
+dZpG(M)≤dZpC(M) = (p−1)−1dimZp(M).
+Ontheotherhand,Proposition3.5showsthat d(G)≤(p−1)−1dimZp(M)
+and thus
+d(G)+dZpG(M)≤2(p−1)−1dimZp(M)≤dimZp(M),
+as wanted. /square
+Lemma 4.2. LetGbe an insoluble pro- pgroup, and suppose that pis
+odd. Then for every open normal subgroup NofGthere exists an open
+normal subgroup KofGsuch that K/subset⋊r⋉⋊tdbleqlNandd(K)≥d(N).
+Proof.Clearly, the assertion holds if Gis not finitely generated. Now
+suppose that d(G)<∞, and let Nbe an open normal subgroup of G.
+Let Φ(N) =Np[N,N] denote the Frattini subgroup of N. SinceGis
+insoluble, we have M:= [Φ(N),N]/\e}ati≀\slash= 1. Observe that Mis normal
+inG, but not necessarily open. Nevertheless, working modulo an open
+normal subgroup UofGsuch that U/subset⋊r⋉⋊tdbleqlMU⊆Φ(N), we find an open
+normal subgroup LofGsuch that L⊆MU⊆Φ(N) andMU/L∼=
+Cp. Now an argument similar to the one given in the proof of [3,
+Theorem 3.3] shows that
+K:={g∈N|[Φ(N),g]⊆L}
+satisfiesd(K)≥d(N). One checks easily that K/triangleleftequalGandK/subset⋊r⋉⋊tdbleqlN./squareON THE RANK OF COMPACT p-ADIC LIE GROUPS 11
+Recall that π(G) denotes the periodic radical, viz. the unique maxi-
+mal finite normal subgroup, of a profinite group Gof finite rank.
+Lemma 4.3. LetGbe a profinite group of finite rank and Ha subgroup
+ofG. IfHis normal or open in G, thenπ(H)≤π(G).
+Proof.In both cases, the union of all conjugates of π(H) inGis a finite
+normal subset of Gconsisting of elements of finite order. By Dicman’s
+Lemma, this set generates a finite normal subgroup of G. /square
+For the next two results, we denote by dℓ(G) the minimal cardinality
+of a generating set for a Sylow pro- ℓsubgroup of a profinite group G.
+Proposition 4.4. LetGbe a compact p-adic Lie group such that
+π(G) = 1, and suppose that pis odd. Then dp(G)≤dim(G).
+Proof.By Lemma 4.3, we may assume without loss of generality that
+Gis a pro-pgroup. If dim( G) = 0, then G=π(G) = 1 and there is
+nothing further to prove. Hence suppose that dim( G)≥1. Choose
+a normal subgroup NofGsuch that G/Nis just infinite. Note that
+π(G/N) = 1 and π(N) = 1, by Lemma 4.3. Thus, if N/\e}ati≀\slash= 1 we apply
+induction to deduce that
+d(G)≤d(G/N)+d(N)≤dim(G/N)+dim(N) = dim(G).
+It remains to consider the case that N= 1, i.e. that Gis just infinite.
+First suppose that Gis soluble. Then Gis virtually abelian; we find
+an abelian open normal subgroup AofG. PutC:= CG(A). Then
+|C: Z(C)| ≤ |C:A|<∞, and hence [ C,C] is finite. Since Gis
+just infinite, we conclude that Cis abelian and self-centralising. Thus
+Proposition 4.1 shows that d(G)≤dim(C) = dim(G).
+Next suppose that Gis insoluble. For a contradiction assume that
+d(G)>dim(G). By Lemma 4.2, we find a strictly descending chain
+G=N1⊃N2⊃N3⊃...
+of open normal subgroups NiofGsuch that d(Ni)≥d(G) for all
+i∈N. Fix a uniformly powerful open normal subgroup UofG. Since
+rk(U) = dim( U)< d(G), we have Ni/\e}ati≀\slash⊆Ufor each i∈N, witnessed
+bygi∈Ni\U, say. Since Gis compact, there is a subsequence of gi,
+i∈N, which converges to an element g∈G\U. Clearly, we also have
+g∈/intersectiontext{Ni|i∈N}=:M. SinceMis a normal subgroup of infinite
+index inGand since Gis just infinite, we conclude g∈M={1} ⊆U,
+a contradiction. /square
+Proposition 4.5. LetGbe a compact p-adic Lie group such that
+π(G) = 1, and let d:= dim(G). Then for every prime ℓwithℓ/\e}ati≀\slash=p,
+dℓ(G)≤rkℓ(GLd(Zp)) = rk ℓ(GLd(Fp)).12 BENJAMIN KLOPSCH
+Proof.Arguing as at the beginning of the proof of Proposition 4.4, we
+may suppose that Gis just infinite. Let Sbe a Sylow pro- ℓsubgroup
+ofG, and choose a uniformly powerful open normal pro- psubgroup U
+ofG. ThenSis a finite ℓ-group, and we claim that Sacts faithfully
+onUby conjugation. Indeed, since Gis just infinite, C G(U) is either
+trivialoropenin G. Intheformercasethereisnothingfurther toshow.
+In the latter case, writing C:= CG(U), we note that |C: Z(C)| ≤ |C:
+C∩U|<∞so that [C,C] is a finite normal subgroup of G. SinceGis
+just infinite, we conclude that Cis abelian andpro- p. HenceS∩C= 1,
+and again Sacts faithfully on U.
+SinceUisuniformlypowerful, theset Uadmitsthestructureofa Zp-
+Lie lattice with additive group isomorphic to Zd
+p, whered:= dim(U) =
+dim(G), and the faithful action of SonUtranslates into an embedding
+ofSintoGL d(Zp). Dividingoutthefirstprincipalcongruencesubgroup
+GL1
+d(Zp), we obtain an embedding of Sinto GL d(Fp). /square
+Proof of Theorem 1.3. LetGbe a compact p-adic Lie group such that
+π(G) = 1, and suppose that pis odd. The assertion about rk 2(G) fol-
+lows directly from Proposition 4.5 and Theorem 1.7. For the assertion s
+about rk ℓ(G),ℓ >2, it suffices to show that
+(i) rkp(G)≥dim(G),
+(ii) rk ℓ(G)≤dim(G) for every odd prime ℓ.
+For every uniformly powerful open pro- psubgroup UofGone has
+d(U) = dim(U) = dim(G), and this implies (i). Let ℓbe an odd prime.
+It is known that
+rkℓ(G) = sup{dℓ(H)|H≤Gan open subgroup };
+cf. [1, Proposition 3.11]. Thus (ii) follows from Lemma 4.3 and Propo-
+sitions 4.4 and 4.5, in conjunction with Theorem 1.7. /square
+5.Bounding the rank
+We will make use of the following theorem which was proved for sol-
+uble groups by Kov´ acs, and for all finite groups (using the classific ation
+of finite simple groups) by Lucchini [10] and Guralnick [4].
+Theorem 5.1. If every Sylow subgroup of a finite group Gcan be
+generated by delements, then d(G)≤d+1.
+Corollary 5.2. For every profinite group Gone has
+(5.1) rk( G)≤sup{rkℓ(G)|ℓprime}+1.
+Proof of Theorem 1.4. Claims (1) and (3) are direct consequences of
+Corollary5.2andTheorem1.3. Todeduceclaims(2)and(4)weusethe
+additional analysis given in [11], which distinguishes between groups of
+zero and non-zero presentation rank.
+First suppose that Gis prosoluble. Then all its finite quotients have
+zero presentation rank. Hence [11, Theorem 1] shows that equalit y inON THE RANK OF COMPACT p-ADIC LIE GROUPS 13
+(5.1) implies rk( G)≤rkℓ(G)+1 for some odd prime ℓ. Thus claim (2)
+follows from Corollary 5.2 and Theorem 1.3.
+Finally suppose that Ghas trivial ( p−1)-torsion and without loss
+of generality assume that dim( G)≥1. Then rk p(G) = dim( G) and
+rkℓ(G) = 0≤dim(G)−1 for primes ℓwithp≡ℓ1. Furthermore,
+Proposition 4.5 and Theorem 1.7 show that rk ℓ(G)≤ ⌊dim(G)/2⌋ ≤
+dim(G)−1 for all remaining primes ℓ. Corollary 5.2 gives rk( G)∈
+{dim(G),dim(G)+1}, and [11, Theorems 1 and 2] rule out the possi-
+bility rk( G) = dim(G)+1. This proves claim (4). /square
+Acknowledgements. The author thanks Frank Calegari for posing a
+question which led to the present article. He is also grateful to Gabr iele
+Nebe for pointing out a gapin the arguments given inan earlier version
+of this paper.
+References
+[1] J.D. Dixon, M.P.F. du Sautoy, A. Mann and D. Segal, Analytic pro- pgroups,
+Second edition, Cambridge Studies in Advanced Mathematics 61, Cambridge
+University Press, Cambridge, 1999.
+[2] J.D. Dixon and L.G. Kov´ acs, Generating finite nilpotent irreducible linear
+groups, Quart. J. Math. Oxford 44(1993), 1–15.
+[3] J. Gonz´ alez-S´ anchez and B. Klopsch, Onw-maximal groups , J. Algebra 328
+(2011), 155–166.
+[4] R. Guralnick, On the number of generators of a finite group , Arch. Math.
+(Basel)53(1989), 521–523.
+[5] A. Heller and I. Reiner, Representations of cyclic groups in rings of integers I ,
+Ann. of Math. 76(1962), 73–92.
+[6] I.M. Isaacs, The number of generators of a linear p-group, Can. J. Math. 24
+(1972), 851–858.
+[7] L.G. Kov´ acs and G.R. Robinson, Generating finite completely reducible linear
+groups, Proc. Amer. Math. Soc. 112(1991), 357–364.
+[8] T.J. Laffey, The minimum number of generators of a finite p-group, Bull. Lon-
+don Math. Soc. 5(1973), 288–290.
+[9] C.R. Leedham-Green and W. Plesken, Some remarks on Sylow subgroups of
+general linear groups , Math. Z. 191(1986), 529–535.
+[10] A. Lucchini, A bound on the number of generators of a finite group , Arch.
+Math. (Basel) 53(1989), 313–317.
+[11] A. Lucchini, Some questions on the number of generators of a finite group ,
+Rend. Sem. Mat. Univ. Padova 83(1990), 202–222.
+[12] A. Lucchini, F. Menegazzo and M. Morigi, On the number of generators and
+composition length of finite linear groups , J. Algebra 243(2001), 427–447.
+Department of Mathematics, Royal Holloway, University of L on-
+don, Egham TW20 0EX, United Kingdom
+E-mail address :Benjamin.Klopsch@rhul.ac.uk
\ No newline at end of file
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+1 SCIENTIFIC AND TECHN OLOGICAL DEVELOPMENT   
+OF HADRONTHERAPY  
+SAVERIO BRACCINI* 
+Albert Einstein Centre for Fundamental Physics,  
+Laboratory for High Energy Physics  (LHEP), University of Bern,  
+ Sidlerstrasse 5, CH -3012 Bern, Switzerland  
+Hadrontherapy is a no vel technique of cancer radiation therapy which employs beams of 
+charged hadrons, protons and carbon ions in particular. Due to their physical and 
+radiobiological properties, they allow one to obtain a more conformal treatment with 
+respect to photons used in conventional radiation therapy, sparing better the healthy 
+tissues located in proximity of the tumour and allowing a higher control of the disease. 
+Hadrontherapy is the direct application of research in high energy physics, making use of 
+specifically co nceived particle accelerators and detectors. Protons can be considered 
+today a very important tool in clinical practice due to the several hospital -based centres in 
+operation and to the continuously increasing number of facilities proposed worldwide. 
+Very promising results have been obtained with carbon ion beams, especially in the 
+treatment of specific radio resistant tumours. To optimize the use of charged hadron 
+beams in cancer therapy, a continuous technological challenge is leading to the 
+conception an d to the development of innovative methods and instruments. The present 
+status of hadrontherapy is reviewed together with the future scientific and technological 
+perspectives of this discipline.  
+1.   Introduction  
+Hadrontherapy – often also denominated ‘particle  therapy’ - is a collective word 
+used to indicate the treatment of tumours through external irradiation by means 
+of accelerated hadronic particles. Several kind of particles have been and are the 
+subject of intensive clinical and radiobiological studies: n eutrons, protons, pions, 
+antiprotons, helium, lithium, boron, carbon and oxygen ions. Among all these 
+possibilities, only two of them – protons and carbon ions – are nowadays used in 
+clinical practice and represent the focus of an ongoing remarkable scient ific and 
+technological development. For this reason, only proton and carbon ion therapy 
+are discussed in this paper.  
+Protons and carbon ions are more advantageous in cancer radiation therapy 
+with respect to X -rays mainly because of three reasons. The relea se of energy 
+along their path inside the patient’s body is characterized by a large deposit 
+                                                             
+* E-mail: Saverio.Braccini@cern.ch .  2 
+localized in the last few millimetres at the end of their range, in the so called 
+Bragg peak region, where they produce severe damage to the cells while sparing 
+both traversed and deeper located healthy tissues. Moreover, they penetrate the 
+patient with minimal diffusion and, using their electric charge, few millimetre 
+FWHM ‘pencil beams’ of variable penetration depth can be precisely guided 
+towards any part of the t umour. The third reason pertains to carbon ions - and 
+light ions in general - and is based on radiation biology. Since, for the same 
+range, carbon ions deposit about a factor 24 more energy in the Bragg peak 
+region with respect to protons, the produced ion ization column is so dense to be 
+able to induce direct multiple strand brakes in the DNA, thus leading to non -
+repairable dama ge. This effect is quantified by  an enhancement of the Radio 
+Biological Effectiveness (RBE) and opens the way to the treatment of t umours, 
+which are resistant to X -rays and protons at the doses prescribed by standard 
+medical protocols.  
+In order to treat deep -seated tumours, depths of the order of 25 cm in soft 
+tissues have to be reached. This directly translates into the maximum ener gies of 
+proton and carbon ion beams which must be 200 MeV and 4 500 MeV (i.e. 375 
+MeV/u), respectively. As far as the beam currents are concerned, the limit is set 
+by the amount of dose to be delivered to the tissues, typically 2 Gy per litre per 
+minute. T his translates into beam currents on target of 1 nA and 0.1 nA for 
+protons and carbon ions, respectively.  
+ 
+Figure 1. General layout of a proton therapy centre featuring one accelerator, three treatment rooms 
+equipped with rotating gantries and one room e quipped with a fixed horizontal beam. The example 
+reported here is based on a system commercialized by the company IBA (Belgium).   3 
+The maximum energy and current determine the main characteristics of the 
+accelerator: 4 -5 metre diameter cyclotrons, both at r oom temperature and 
+superconducting, and 6 -8 metre diameter synchrotrons are today in use for 
+proton therapy while for carbon ion therapy only 20 -25 metre diameter 
+synchrotrons are available.  
+As presented in Fig. 1, a modern hadrontherapy centre is based o n a 
+complex facility in which the accelerator is connected to several treatment 
+rooms by means of beam transport lines. The treatment rooms are equipped with 
+fix beams – usually horizontal but vertical and 45˚ are also in use – or rotating 
+gantries, which,  for proton therapy, are about 10 m high, 100 tons structures 
+supporting a set of magnets, able to irradiate the patient from any direction, 
+exactly like in conventional X -ray radiation therapy.  
+The first idea of using accelerated protons and ions in cance r radiation 
+therapy dates back to 1946 when Bob Wilson wrote a very illuminating seminal 
+paper1 in which all the basic principles and potentialities of this discipline are 
+stated. I personally find this work very remarkable and still incredibly actual, 
+especially if one considers the fact that precise imaging techniques and enough 
+powerful accelerators were almost a dream at that time.  
+2.   Present status of proton and carbon ion therapy  
+The first proton therapy treatment took place in Berkeley2 in 1954, followe d by 
+Uppsala in 1957. This pioneering work opened the way to the intensive activity 
+performed at the Harvard cyclotron where physicists and radiation oncologists 
+worked together for many decades on three clinical studies: neurosurgery for 
+intracranial lesi ons (3 687 patients), eye tumours (2 979 patients) and head -neck 
+tumours (2 449 patients). The results obtained by the Harvard group represented 
+the basis for the successive clinical and technical developments of this 
+discipline.  
+A fundamental milestone w as accomplished in 1990 when the first patient 
+was treated at the Loma Linda University Medical Center in California, the first 
+hospital based proton therapy centre. This facility featured the first rotating 
+gantries designed for routine treatment. It has to be remarked that, up to this 
+moment, all the hadrontherapy facilities were based on existing particle 
+accelerators designed for fundamental research, often sharing human resources 
+and beam time with other activities. Moreover, some of these centres made  use 
+of low energy – about 70 MeV – cyclotrons, in which only the treatment of 
+ocular pathologies was possible.   4 
+In the las t twenty years, a  progressive development of proton therapy took 
+place. From being practiced only in specialized research nuclear and particle 
+physics laboratories, proton therapy is becoming a widely recognized clinical 
+modality in oncology. As reported in Fig. 2, many hospital based centres are 
+nowadays active in the world.  
+ 
+Figure 2. Hospital based proton therapy facilities3 in oper ation at the end of 2008.   5 
+More that 50 000 patients have been treated by proton therapy worldwide, 
+very often affected by pathologies located near critical structures, the so called 
+organs at risk (OAR). It is important to remark the importance of proton t herapy 
+for paediatric patients, for whom  the irradiation of critical organs may lead to 
+severe permanent consequences on the quality of life and to possible induction 
+of secondary cancers.  
+Many proton therapy centres are nowadays under construction or in a  phase 
+of advanced project and more that ten new hospital based facilities are expected 
+to be operational in the next five years, mostly located in US, Europe and Japan. 
+It is important to acknowledge the important contribution to this dvelopment 
+given by the main commercial companies active in this field: Optivus (USA), 
+IBA (Belgium), Varian/Accel (USA/Germany), Hitachi (Japan) and Mitsubishi 
+(Japan).  
+Carbon ion therapy is still on its way to become a commonly accepted 
+clinical option. Due to its specifici ty and to the complex radiobiological models 
+essential for treatment planning, this irradiation modality is nowadays facing a 
+phase of intense clinical and technical research.  
+Since the construction of the Heavy Ion Medical Accelerator in Chiba 
+(HIMAC), wh ich treated the first patient in 1994, Japan is the most advanced 
+country in carbon ion therapy. By the end of 2009, about 5000 patients have 
+been treated and many radio resistant tumours have been shown to be 
+controllable.  
+In Europe, treatments started at  the ‘pilot project’ for carbon ion therapy at 
+GSI in Germany in 1997. Since then, about 400 patents have been treated and 
+this pr oject lead to the construction of the first European hospital based centre in 
+Heidelberg (HIT). This new centre features a gig antic 600 tons, 25 metre long 
+carbon ion gantry – the only one constructed so far - and treated the first patient 
+in November 2009. In Italy, the Centro Nazionale di Adroterapia Oncologica 
+(CNAO) is in a phase of very advanced construction and will be the second 
+hospital based centre in Europe. Two more centres are under construction in 
+Germany by the company Siemens and two projects, Etoile in France and 
+MedAustron in Austria have been approved.  
+In a few years from now, several hospital based centres for carbon ion 
+therapy will be operational and will produce the necessary knowledge needed to 
+exploit at best the clinical potential of this highly ionizing radiation.   6 
+3.   Future perspectives  
+Since a few years, hadrontherapy is facing a deep change which is bringi ng a 
+relatively small scientific community into a much larger multi disciplinary 
+contest in which physics, biology, engineering, medicine, law, management and 
+finance came together and play all an important role. This is mainly due to the 
+large size of thi s kind of projects which, being of the order of 100 million Euro, 
+have a big impact not only on the scientific and technological side but also on 
+the financial, logistic and management aspects. Needles to say that in a complex 
+project, such as the construc tion of a hadrontherapy centre, many details have to 
+be carefully evaluated and a solid multi -competence project team represents the 
+key to face all the unavoidable problems and compromises to be assessed from 
+the conception to the running of the facility.  
+In my opinion, two main forces – not always pointing towards the same 
+direction - are driving the developments of this discipline. On one hand, to be 
+competitive with the continuously increasing performances of conventional 
+radiation therapy, innovative t ools and techniques are needed to exploit at best 
+the higher potential of hadron beams. On the other hand, to be able to offer this 
+treatment modality to a larger number of patients, the cost, the size and the 
+complexity of the equipment have to be reduced .  
+To face these challenges, many scientific and technological developments 
+are ongoing and many new ideas are appearing at the horizon. A non -exhaustive 
+summary of some selected topics is reported here.  
+3.1.   Dose delivery systems  
+The dose delivery system repre sents a key issue in the full t reatment process 
+since allows the  transform ation of  the beam coming out from the accelerator into 
+a clinical three -dimensional dose distribution which has to comply with the 
+medical prescription.  
+Up to present and in almost a ll the centres, the ‘passive spreading’ method - 
+presented in Fig. 3 – is used. Although relatively simple, this method requires a 
+large amount of mechanical work and does not allow exploiting at best the 
+ballistic properties of charged hadron beams. In pa rticular, extra doses are 
+produced at the edges of the tumour.  
+In the last fifteen years, two innovative techniques have been developed in 
+two research laboratories in Europe to ‘paint’ the tumour by means of a small 
+‘pencil beam’ driven by magnetic force s and without the use of any passive 
+device, as presented in Fig. 4. The ‘spot scanning’4 and the ‘raster scanning’5 
+allow obtaining very conformal dose distributions and, by the superposition of  7 
+non-uniform dose distributions coming from many directions, they allow very 
+complex treatments by Intensity Modulated Particle Therapy (IMPT). Scanning 
+techniques are nowadays becoming mature and commercial companies are 
+offering this modality, often in parallel with the standard passive spreading.  
+ 
+Figure 3. In ‘ passive spreading’, the beam is widened and flattened by means of scatterers and 
+adapted to the volume to be irradiated by means of personalized collimators and compensators. 
+Different penetration depths are superimposed by means of a range shifter to obta in the Spread Out 
+Bragg Peak (SOBP) covering the tumour.  
+To fully exploit the potentialities of scanning and IMPT, a new dedicated 
+gantry is now under commissioning at PSI. The ‘Gantry 2’6 allows scanning 
+with parallel beams in a two -dimensional surface an d rotates only of 180˚ to 
+allow an easier access to the patient with a more compact structure. This 
+innovative device futures also a robotic couch for patient positioning and 
+advanced imaging modalities. The possibility to use X -rays parallel to the proton  
+beam and passing through a special aperture in the 90˚ magnet is foreseen to 
+optimize the treatment of moving organs.  
+  
+  
+Figure 4. Developed at PSI, the ‘spot scanning’ technique consists on chopping the beam from a 
+cyclotron to obtain discrete dose sp ots of given energy and direction which are used to form a 
+uniform dose distribution (left). Developed at GSI, the ‘raster scanning’ consi sts on the use of a  
+pencil beam to paint a slice of the tumour; the irradiation of successive slices , corresponding to  
+different beam energies,  gives a three -dimensional uniform dose distribution (right).  
+It has to be remarked that most of the tumours treated so far by particle 
+therapy – like for example head and neck cancers - are not subject to motion due  8 
+to respiration . The treatment of moving organs, like in the case of lung cancer, 
+represents an important challenge and a very actual theme of research. In Japan, 
+a synchronization ‘gating’ between the beam and the patient’s respiration is 
+already used and the irradiatio n takes place only when the tumour is in the 
+appropriate position. More sophisticated techniques of active compensation are 
+under study at GSI where the realisation of an on -line feedback system will 
+allow the  change of position and energy of the beam to c orrect organ motion 
+effects.  
+Due to the much higher magnetic rigidity, a conventional magnet ion gantry 
+is necessarily very big, like the one constructed at HIT. To make carbon ion 
+gantries more compact, the use of superconductivity is under study together  with 
+more ‘exotic’ beam transport techniques such as the Fixed Field Alternating 
+Gradient (FFA G). Non -isocentric gantries are also  proposed. In this case the 
+movement of the beam is limited – and therefore the number of magnets – which 
+implies moving the patient to perform irradiations from any direction.  
+3.2.   Imaging and quality assurance  
+In modern radiation therapy imaging is essential and computed tomography 
+(CT) is the basis of treatment planning. In the near future, the role of MRI and 
+PET functional imagi ng will surely be more important. In particular, innovative 
+molecules7 for PET imaging are nowadays in a phase of ad vanced study and will  
+provide fundamental information for treatment planning on hypoxia and on the 
+localization  of tumours which are almost  invisible to CT.  
+ 
+  
+Figure 5. SAMBA (Strip Accurate Monitor for Beam Applications) is composed by two 2 mm strip 
+ionization chambers (left). This detector is capable to monitor on -line the beam intensity and the 
+transverse position with sub -millimetric precision during spot scanning (right).  
+On-line monitoring of hadron beams during treatment is of paramount 
+importance for quality assurance. Many interesting developments in the field of 
+particle detectors have been performed in the last years as the inno vative strip  9 
+ionization chamber8 developed by the TERA Foundation and INFN -Torino for 
+the Gantry 2 (Fig. 5).  
+In radiation therapy, the control of the dose distribution relies on off -line 
+dosimetry and not on direct on -line measurements. At GSI, an innovati ve 
+technique9 – the in -beam PET – has been developed exploiting the fact that 
+carbon ions form 11C nuclei by interacting with the patient’s body (Fig. 6). Since 
+11C is a positron emitter, its distribution can be measured by means of a PET 
+camera installed in the treatment room and compared with the treatment 
+planning. Studies are underway also to exploit this technique for proton -
+therapy10. 
+  
+  
+Figure 6. The treatment room at GSI equipped with the on -beam PET detector (left). Measured and 
+simulated data de monstrate that the dose released by carbon ions is effectively conform with respect 
+to the treatment planning.  
+The precise determination of the range is essential in proton therapy and, 
+being based on CT data, correction factors have to be applied in the t reatment 
+planning to calculate the density of the traversed material.  Proton radiography 
+can be used to obtain direct information on the range for treatment planning 
+optimization and to perform imaging with negligible dose to the patient. Proton 
+radiograp hy is an interesting theme of research and detectors based on silicon 
+trackers and nuclear film emulsions are  proposed11. 
+3.3.   Particle accelerators  
+Contrary with respect to conventional radiation therapy, where each treatment 
+room is equipped with a linac, onl y one radiation source is available in a 
+hadrontherapy facility. For this reason, the choice of the particle accelerator is of 
+paramount importance and its up -time represents a very critical issue.   10 
+At present , four commercial companies offer proton accele rators - IBA 
+and Varian/Accel offer cyclotrons, Hitachi and Mitsubishi synchrotrons - while 
+for carbon ion therapy the only commercial solution is the synchrotron proposed 
+by Siemens.  
+Cyclotrons are characterized by a beam of fixed energy which has to be 
+degraded with passive absorbers in the first part of the beam line – in the so 
+called Energy Selection System (ESS) – and, with respect to the human 
+breathing cycle, present a continuous beam, advantageous for the treatment of 
+moving organs.  
+On the other hand, the beam of a synchrotron presents a cycle – the so 
+called ‘spill’ – which lasts about 2 seconds and in which the beam is present for 
+about 0.5 seconds. This time structure is similar to the human respiratory cycle, 
+posing problems for the treatment of moving organs. The energy can be varied 
+from spill to spill and no passive absorbers are needed.  
+To reduce size, cost and complexity, single room facilities look to be a very 
+promising solution, which may lead to a deep change of the entire discipline i n 
+the next future . The company Still River System is proposing a 10 Tesla 
+superconducting synchrocyclotron to be directly installed in a 180˚ rotating 
+gantry (Fig. 7 – left). The construction of the first prototype is in a very 
+advanced phase and it surely represents a ver y interesting development to 
+follow. With a longer research and development timescale, the company 
+TomoTherapy, together with the Lawrence Livermore National Laboratories, is 
+proposing a single room system based on an innovative accelerating technique, 
+the Dielectric Wall Accelerator (DWA). Studies are underway to prove that 
+beams suitable for proton therapy can be obtained by means of this new 
+technology.  
+ 
+  
+Figure 7. The single room proton therapy facility proposed by Still River Systems (left). With a l inac 
+the beam energy can be continuously changed by switching off the last modules and by varying the 
+power in the last active module (right).   11 
+Radio frequency linacs12 represent a very interesting solution since they 
+naturally allow very fast changes of th e beam energy in a time scale of 1 ms, 
+consistent with the repetition rate of the accelerator. As presented in Fig. 7 
+(right), thanks to the rapid  ‘bunch -to-bunch’ energy variation, linacs are ideal 
+radiation sources for spot scanning.  
+Initiated by the TE RA Foundation, a system based on a high current 
+cyclotron followed by a linac post accelerator is proposed by the company 
+ADAM. This solution allows proton therapy and radioisotope production with a 
+single accelerator complex.  
+For carbon ion therapy, the c ompany IBA is studying an innovative multi -
+particle superconducting cyclotron and the first prototype of this machine is 
+under construction in Caen (France). A similar 300 MeV proton and 300 MeV/u 
+superconducting cyclotron named SCENT is proposed by INFN -LNS and IBA 
+and will allow full treatments in proton therapy and carbon ion therapy up to a 
+maximum penetration of 16 cm in water. A linear post accelerator named 
+CABOTO is proposed by the TERA Foundation with the scope of boosting 
+carbon ions to the energy  of 400 MeV/u to treat any kind of tumour.  
+Looking into the very far future, the acceleration of protons and ions by 
+means of laser plasma devi ces represents a fascinating fi eld of research13. It has 
+to be remarked that, at p resent, only the maximum energy  of 67.5 MeV for 
+protons has been recently achieved. Moreover, protons are produced with a very 
+wide energy spread peaked at low energies , making of the dose distribution 
+system a very difficult challenge . 
+4.   Conclusions  
+Since its  beginning s, particle physics  has always offered medicine and biology 
+tools and techniques to study, detect and cure cancer. Hadrontherapy represents 
+today one of the major contributions  of particle physics to the m edical filed and  
+is now facing a very exciting phase at the forefront of science and technology.  
+Acknowledgements  
+I would sincerely acknowledge Ugo Amaldi who first introduced and than 
+guided me into this fascinating field of application of particle physics to 
+medicine. I would like to thank Vesna Sossi for having given me t he possibility 
+to present hadrontherapy at the 11th ICATPP Conference in Como.  
+  12 
+ 
+References  
+1.   R. R. Wilson, Radiological use of fast protons, Radiology  47, 487 (1946).  
+2.   C. A. Tobias et al., Pituitary Irradiation with High -Energy Proton Beams A 
+Preliminary Report, Cancer Research  18, 121 (1958).  
+3.  Particle Therapy CoOperative Group (PTCOG), www.ptcog.com and 
+ptcog.web.psi.ch.  
+4.  E. Pedroni et al., The 200 -MeV proton therapy project at the Paul Scherrer 
+Institute: conceptual design and practical realization, Med. Phys.  22, 37 
+(1995).  
+5.  T. Haberer et al., Magnetic scanning system for heavy ion therapy, Nucl. 
+Instr.  Methods A330, 296 (1993).  
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+Pharmacology's new PET ?, Clin. Pharmacol. Ther.  81, 792 (2007).  
+8.  S. Braccini, An Innovative Strip Ionization Chamber for on -line Monitoring 
+of Hadron Beams, presented at PTCOG 44, Zu rich, Switzerland, 2006.  
+9.  W. Engh ardt et al, Charged hadron tumour therapy monitoring by means of 
+PET, Nucl. Instr. Methods  A525 , 284 (2004).  
+10.  K. Parodi et al, The feasibility of in -beam PET for accurate monitoring of 
+proton therapy: results of a comprehensive experimental study, IEEE Tran s. 
+Nucl. Sci.  52, 778 (2005).  
+11.  S. Braccini et al., Nuclear emulsion film detectors for proton radiography: 
+design and tests of the first prototype, these proceedings.  
+12.  U. Amaldi, S. Braccini and P. Puggioni, High frequency linacs for 
+hadrontherapy, Reviews o f Accelerator Science and Technology (RAST) , 
+Vol. II, World Scientific, 2009.  
+13.  U. Amaldi and S. Braccini, Present and Future of Hadrontherapy, in 
+Superstrong Fields in Plasmas , AIP, 248 (2006).  
\ No newline at end of file
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+arXiv:1001.0861v2  [gr-qc]  13 Jan 2010Quantum Black Hole in the Generalized Uncertainty
+Principle Framework
+A. Bina1∗, S. Jalalzadeh2,3†,
+A. Moslehi1
+1Department of Physics, Faculty of Science, Arak University , Arak 879, Iran
+2Department of Physics, Shahid Beheshti University G.C., Ev in, Tehran 19839, Iran
+3Research Institute for Astronomy and Astrophysics of Marag ha (RIAAM)- Maragha, Iran,P. O. Box: 55134-441
+October 30, 2018
+Abstract
+In this paper we study the effects of the Generalized Uncertai nty Principle (GUP) on canonical quantum
+gravity of black holes. Through the use of modified partition function that involves the effects of the GUP,
+we obtain the thermodynamical properties of the Schwarzsch ild black hole. We also calculate the Hawk-
+ingtemperature andentropyfor the modification of theSchwa rzschild blackhole in thepresence oftheGUP.
+PACS: 98.80.Qc, 02.40.Gh, 04.70.-s
+1 Introduction
+The discovery of temperature and entropy of black holes is one of t he most important achievements in
+gravitational physics. Since entropy is a statistical concern, it ha s been in a great interest [1]-[3]. In the early
+seventies Bekenstein proposed the quantization of black holes [1]. H e showed that the gravitational surface of
+a black hole is proportional to its temperature and so the event hor izon area is proportional to the entropy.
+He concluded that the event horizon of non-extremal black holes b ehaves like an adiabatic invariant, thus the
+event horizon should have discrete spectrum in the general relativ ity framework [2].
+Similarities between these rules and thermodynamics was first invest igated by Hawking [3]. Later on he
+discoveredthe evaporation of black holes by a series of semi-classic al calculations. It meant that the black holes
+like black bodies emit thermal radiation proportional to their gravita tional surface. In addition the entropy
+is equivalent to one-forth of the event horizon area. Hawking’s calc ulations introduced the relation between
+classicalmechanicsofablackholeandits thermodynamics. These re sultsledto deepercorrespondencebetween
+classical gravity, quantum mechanics and statistical mechanics.
+Duringthese yearsthe entropyofblackholes hasbeen studied by d ifferent approaches,forexampleby string
+theory, loop quantum gravity and canonical gravity [4]. Although t heses methods, involves curved geometry,
+we can perform Hawking radiation calculations in a flat space. Also, up to now these calculations have been
+done for extremal and near extremal black holes [5].
+The existence of a minimal length is one of the most interesting predic tions of the theories related to
+quantum gravity [6, 7]. From the perturbative string theory this le ngth is due to the fact that the strings
+cannot influence through distances smaller than their size. One inte resting property of the existence of the
+minimal length is the modification of the standard commutation relatio n between position and momentum in
+usual quantum mechanics [8] which is called the generalized uncertain ty principle (GUP). Noncommutativity
+between space-time coordinates was first studied by Snyder [9] an d [10]-[13].
+The noncommutativity theory is of great interest because of its int eresting predictions in particle physics,
+for example the mixture of IR/UV and non-locality [18], Lorenz violatio n [19], canonical noncommutative with
+deformed rather than broken symmetries [14, 15], Lie-algebraic no ncommutativity [16, 17] and modern physics
+∗email: amir.bina82@gmail.com(a-bina@arshad.araku.ac. ir)
+†email: s-jalalzadeh@sbu.ac.ir
+1at small scales [19, 20]. Also in the past much attention years has bee n paid to these fields (for applications of
+the GUP and non-commutativity in minisuperspace dynamics see [21]-[2 4]).
+While studying the thermodynamics ofa blackhole, the analysisofits t emperature leadsto many questions.
+Whenthe initialstateofablackholeisapurequantumoneandevolves toamixedfinalstate[25], firsttheblack
+hole attracts all the information behind its event horizon and then d isappears through thermal distribution.
+This causes the violation of the unitarity principle and is associated wit h the information loss assumption. The
+uncertainty principle is one of the ways to escape this information los s [26],[27]. The Schwarzchild radius of a
+black hole with the Plank mass is of the order of the Plank length. Since this length is the wavelength of a
+particle with the Plank mass, if the mass of the black hole becomes lowe r than this mass, then we have a mass
+inside the volume smaller than that allowed by the uncertainty principle . Zeldovich proposed that black holes
+with masses smaller than the Plank mass are related to stable element ary particles [28].
+There are many questions about minimal length during the study of b lack holes [29, 30, 31, 32]. By
+investigation of the space-time for string scattering with an increa sing higher orders in perturbation theory,
+the size of the string reduces relative to the Schwarzchild radius of the collision region, thus the production of
+black holes becomes impossible in such a way but the length approach w ill make it possible but complicated
+[33]. Recentlystringandloopquantumgravitytheorieshavesuccee dedtoaccountforthe entropy-eventhorizon
+area [29].
+Inthispaperinsection2, wereviewamodeltosolvetheWheeler-Dew ittequationoftheSchwarzschildblack
+hole and derive its thermodynamics. In section 3, we examine Hilbert s pace representation in the generalized
+uncertainty principle framework. In section 4, we study the Schwa rzschild black hole with the generalized
+uncertainty relation and finally derive the thermodynamical proper ties of the quantum black hole.
+2 The Model
+The Wheeler-Dewitt equation for a Schwarzschild black hole where th e Hamiltonian involves only coordi-
+nates and momenta ( a,pa) i.e.H=p2
+a
+2a+1
+2a, can be obtained as [34, 35]
+/planckover2pi12G2
+c6a−s−1d
+da/parenleftbigg
+asd
+daψ(a)/parenrightbigg
+= (a−2GM
+c2)ψ(a), (1)
+whereaandP2
+a=−/planckover2pi12G2
+c6a−sd
+da/parenleftbig
+asd
+daψ(a)/parenrightbig
+are phase coordinates deduced from the phase space coordinate s
+mandPm, by means of an appropriate canonical transformation and also m(t) =M(t,r) andPm(t) =/integraltext∞
+−∞drPM(t,r). The variable mcan be defined as mass Mof the hole when Einstein’s equations are satisfied
+[34] andsis a factor ordering parameter. In particular, if we choose s= 2 and identifying Rs=2GM
+c2, we have
+/planckover2pi12G2
+c61
+a/parenleftbiggd2
+da2+2
+ad
+da/parenrightbigg
+ψ(a) = (a−Rs)ψ(a). (2)
+Let us consider the following transformations
+
+
+ψ(a) =1
+aU(a)
+ξ=a−Rs,(3)
+where the variable ξindicates the gravitational degrees of freedom of the Schwarzsc hild black hole, and define
+the appropriate constants and consider the fact that the energ y of excitations associated with variable ais
+not positive [35]. The physical reason is simply that the total energy of the black hole is included and the
+ADM energy is equal to zero. Then, the quantum equation (2) turn s into
+/parenleftBigg
+−ℓ2
+pEp
+2d2
+dξ2+1
+2Ep
+ℓ2pξ2/parenrightBigg
+U(ξ) =Rs
+4ℓpEsU(ξ), (4)
+whereEs=Mc2is the black hole ADM energy and ℓp=/radicalBig
+G/planckover2pi1
+c3andEp=/radicalBig
+c5/planckover2pi1
+G. It can easily be shown that
+Eq.(4) agrees with the Beckenstein’s proposal [1] and represents a quantum linear oscillator with energy levels
+of
+Rs(n)
+4ℓpEs(n) =/parenleftbigg
+n+1
+2/parenrightbigg
+Ep. (5)
+2Also according to above equation one can get the mass of the black h ole as
+M2(n) =2/planckover2pi1c
+G/parenleftbigg
+n+1
+2/parenrightbigg
+. (6)
+2.1 Black Hole Entropy
+Details of the thermodynamical properties of a black hole can be obt ained from its partition function
+[35]-[37]. For a quantum mechanical system the Feynman’s path integ ral approach is a useful method for
+determination of the free energy and partition function. In this wa y, to include the quantum effects [56, 57] a
+corrected potential in considered. In the case of the black hole wit h respect to Eq.(4) the quantum equation
+is similar to the equation of a one-dimensional harmonic oscillator with f requency of /planckover2pi1ω=/radicalBig
+3
+2πEp, and the
+corrected potential
+V(ξ) =3Ep
+4πℓ2p/parenleftBigg
+ξ2+βℓ2
+pEp
+12/parenrightBigg
+, (7)
+which leads to the following partition function [35, 36]
+ZQ=/radicalbigg
+2π
+3exp(−β2E2
+p
+16π)
+βEp. (8)
+From the fact that the internal energy of a black hole is equal to its gravitational energy, i.e.
+E=−∂ln(ZQ)
+∂β=Mc2, (9)
+we get
+E2
+p
+8πβ2−Mc2β+1 = 0. (10)
+The positive solution for this equation when Ep≪Mc2is
+β=βH/bracketleftbigg
+1−1
+βHMc2/bracketrightbigg
+, (11)
+whereβH=8πMc2
+E2p=1
+kTH, is the Hawking’s temperature. The entropy of the black hole using t he partition
+function and internal energy is defined as
+S
+k= lnZQ+β¯E . (12)
+Putting the Hawking’s temperature in this equation we obtain
+S
+k=As
+4ℓ2p/bracketleftBigg
+1−1
+8πE2
+p
+(Mc2)2/bracketrightBigg2
+−1
+2ln
+As
+4ℓ2p/bracketleftBigg
+1−1
+8πE2
+p
+(Mc2)2/bracketrightBigg2
+−1
+2ln(24)+1, (13)
+whereAs= 4πR2
+sis the area horizon.
+In terms of the Bekenstein-Howking relation SBH/k=As/4ℓ2
+pand ignoring terms of higher order, one
+can find the logarithmic correction to the entropy as is acquired usin g different procedures in [37, 38]
+S
+k=SBH
+k−1
+2ln/parenleftbiggSBH
+k/parenrightbigg
++O/parenleftbig
+S−1
+BH/parenrightbig
+. (14)
+This result has the interesting feature that the coefficient of the fi rst correction, the logarithmic one, agrees
+with the one obtained in loop quantum gravity [39], as well as in string th eory [40]. The form of this correction
+was already obtained from other papers by other considerations [3 8, 41, 42].
+33 Hilbert Space Representation in the GUP Framework
+In this section, we briefly study the modified Heisenberg algebra. In one dimension, deformation of the
+Heisenberg algebra generated by X,Pis given by
+[X,P] =i/planckover2pi1(1+σP2), σ> 0 (15)
+whereσis the deformation parameter and this commutation relation can be s een in perturbative string theory
+[6]. The above equation by Kemf, Mangano and collaborators leads to the following relation [44, 45]
+∆x∆p≥/planckover2pi1
+2(1+σ(∆p)2+σ/an}bracketle{tP/an}bracketri}ht2), (16)
+so that the canonical Heisenberg algebra is satisfied in the limit σ→0. By paying attention to the above
+equation the uncertainty in momentum will be
+∆p=∆x
+/planckover2pi1σ±/radicalBigg/parenleftbigg∆x
+/planckover2pi1σ/parenrightbigg2
+−1
+σ−/an}bracketle{tP/an}bracketri}ht2. (17)
+So the minimal uncertainty in position is /an}bracketle{tP/an}bracketri}htdependent, that is
+∆xmin(/an}bracketle{tP/an}bracketri}ht) =/planckover2pi1√σ/radicalbig
+1+σ/an}bracketle{tP/an}bracketri}ht2. (18)
+It is clear that the smallest uncertainty in position occurs when /an}bracketle{tP/an}bracketri}ht= 0 and is equal to ∆ xmin=/planckover2pi1√σ. This
+relation says that it is impossible to consider any physical state as th e eigenstate of the position [44, 45],
+therefor in the presence of ∆ xmin, the definition of a state |ψn/an}bracketri}ht ∈ D(D ⊂ Hfrom a Hilbert space) such that
+lim
+n→∞(∆xmin)|ψn/angbracketright= lim
+n→∞/an}bracketle{tψ|(X−/an}bracketle{tψ|X|ψ/an}bracketri}ht)2|ψ/an}bracketri}ht= 0, (19)
+is impossible. It means that the eigenstates of the position are no lon ger physical states and should be assumed
+as formal states. Consequently in the GUP approach we cannot wo rk in the space configuration and have to
+use “quasi-position” states.
+Eq.(16) involvesboth the lowand highenergyregionswhicharerelate dtoquantum mechanicsandquantum
+gravity limits respectively. These limits as a sample of applications of th e GUP have been derived through
+string theory [50, 51]. The quantum mechanical limit is given by
+σ(∆p)2+σ/an}bracketle{tp/an}bracketri}ht2≪1/ma√sto→(∆p)2+/an}bracketle{tp/an}bracketri}ht2≪1
+σ, (20)
+Also the quantum gravity limit is of the form
+σ(∆p)2+σ/an}bracketle{tp/an}bracketri}ht2∼1/ma√sto→(∆p)2+/an}bracketle{tp/an}bracketri}ht2∼1
+σ. (21)
+3.1 Representation in Momentum Space
+With respect to the lack of non vanishing minimal uncertainty in momen tum the Heisenbergh algebra can
+be represented in momentum space wave function ψ(p) =/an}bracketle{tp|ψ/an}bracketri}ht. On a dense domain in the Hilbert space X
+andPplay as operators such that [43, 46, 48, 49]
+Pψ(p) =pψ(p),
+Xψ(p) =i/planckover2pi1/bracketleftbigg
+(1+σp2)∂
+∂p+σp/bracketrightbigg
+ψ(p). (22)
+As can be seen XandPare symmetric and this representation is easily seen to respect the commutation
+relation (15). The scalar product of two arbitrary wave functions on the mentioned dense domain is given by
+/an}bracketle{tΦ|Ψ/an}bracketri}ht=/integraldisplay+∞
+−∞dpΦ∗(p)Ψ(p). (23)
+4It should be noted that in the presence of minimal uncertainty in pos ition the momentum operator is still
+self-adjoint and also the functional analysis of the position operat or changes. Note that the definition of inner
+product and representation of XandPoperators in this paper are different from corresponding definition on
+Kemf, Mangano and collaborators [44, 45]. In fact they used the f ollowing representation for Hilbert Space of
+states
+P.ψ(p) =pψ(p)
+X.ψ(p) =i/planckover2pi1(1+σp2)∂pψ(p) (24)
+And also
+/an}bracketle{tΨ|Φ/an}bracketri}ht=/integraldisplay+∞
+−∞dp
+1+σp2Ψ∗(p)Φ(p). (25)
+3.2 Maximal Localization States
+As mentioned before when there is uncertainty in position eigenstat es the eigenstates of position are not
+physical states therefor in order to gain information about positio n we have to write the matrix elements of
+position operator in another basis e.g. momentum basis. This leads to the study of states which are called
+“maximal localization states”. As a result we consider the state |ψml
+χ/an}bracketri}ht, maximally localized around position χ
+with the following properties
+/an}bracketle{tψml
+χ|X|ψml
+χ/an}bracketri}ht=χ, (26)
+and
+(∆x)|ψml
+χ/angbracketright= ∆xmin. (27)
+Paying attention to the smallest uncertainty in position and consider ing the standard deviation in uncertainty
+relation, for any state in the Heisenbergh algebra we have
+/an}bracketle{tψ|(X−/an}bracketle{tX/an}bracketri}ht)2−/parenleftbigg|/an}bracketle{t[X,P]/an}bracketri}ht|
+2(∆p)2/parenrightbigg2
+(P−/an}bracketle{tP/an}bracketri}ht)2|ψ/an}bracketri}ht ≥0, (28)
+which immediately implies
+∆x∆p≥|/an}bracketle{t[X,P]/an}bracketri}ht|
+2. (29)
+It is clear that if the state |ψ/an}bracketri}htobeys ∆x∆p=|/angbracketleft[X,P]/angbracketright|
+2then it will obey
+/parenleftbigg
+X−/an}bracketle{tX/an}bracketri}ht+/an}bracketle{t[X,P]/an}bracketri}ht
+2(∆p)2(P−/an}bracketle{tP/an}bracketri}ht)/parenrightbigg
+|ψ/an}bracketri}ht= 0. (30)
+By solving the above equation we determine the differential equation governing the maximal localization states
+/bracketleftbigg
+i/planckover2pi1(1+σp2)∂p−χ+i/planckover2pi1σp+i/planckover2pi11+σ(∆p)2
+2(∆p)2p/bracketrightbigg
+ψ(p) = 0. (31)
+Thus the maximal localization states are given by
+ψml
+χ(p) =/radicalbigg
+2√σ
+π(1+σp2)−1exp(−iχ
+/planckover2pi1√σtan−1(√σp)). (32)
+In the non-deformedcasethe planewavesinmomentum spaceorDir acδ-functionin positionspacearemaximal
+localized states, but here having deformation ψml
+χ(p) can be considered as a generalization of plane waves.
+53.3 Transformation to Quasi-Position Wave Functions
+In order to investigate the probability of a state being in a maximally loc alized state around position χ,
+we consider the scalar product of arbitrary states |φ/an}bracketri}hton the states |ψml
+χ/an}bracketri}htso thatφ(χ) =/an}bracketle{tψml
+χ|φ/an}bracketri}htis called
+the quasi-position wave function. Any wave function in the momentu m representation can be transformed
+into its quasi-position counterpart by the following generalized Four ier transformation (this result is similar to
+Ref.[23, 44])
+φ(χ) =/radicalbigg
+2√σ
+π/integraldisplay+∞
+−∞dp
+1+σp2exp(iχ
+/planckover2pi1√σtan−1(√σp))φ(p). (33)
+Note that in the limit σ→0 the usual wave function φ(χ) =/an}bracketle{tχ|φ/an}bracketri}htis determined and the above equation
+reduces to a plane wave in the momentum space.
+4 Quantum Black Hole with the Generalized Uncertainty Relat ion
+The commutation relation (15) leads us to the generalized uncertain ty relation (GUR) (16) [43]-[49]. Ac-
+cording to Eq.(22) the position and momentum operators are then r epresented in momentum space by
+
+
+X=i/planckover2pi1/bracketleftbig
+(1+σp2)∂p+σp/bracketrightbig
+,
+P=p.(34)
+The Wheeler-DeWitt equation for the Schwarzschild black hole (4) wit h the GUR then becomes
+/braceleftBigg
+−/planckover2pi12
+ℓ2p/bracketleftBigg
+/parenleftbig
+(1+σp2)∂p/parenrightbig2+2σp/parenleftbig
+(1+σp2)∂p/parenrightbig
++2σ2p2+σ/bracketrightBigg
++ℓ2
+p
+/planckover2pi12p2/bracerightBigg
+ψ(p) =Rs
+2ℓpEpEsψ(p),(35)
+wherepis canonical momenta conjugate to ξ. The exact solution of Eq.(35) is given in [43, 46]
+Rs(n)
+4ℓpEs(n) =Ep
+/parenleftbigg
+n+1
+2/parenrightbigg/radicalBigg
+1+/parenleftbiggσ/planckover2pi12
+2ℓ2p/parenrightbigg2
++/parenleftbigg
+n2+n+1
+2/parenrightbiggσ/planckover2pi12
+2ℓ2p
+. (36)
+and also for the mass of black hole we have
+M2(n) =2/planckover2pi1c
+G
+/parenleftbigg
+n+1
+2/parenrightbigg/radicalBigg
+1+/parenleftbiggσ/planckover2pi12
+2ℓ2p/parenrightbigg2
++/parenleftbigg
+n2+n+1
+2/parenrightbiggσ/planckover2pi12
+2ℓ2p
+. (37)
+Now, we define
+c=1/radicalbig
+1+σp2, s=√σp/radicalbig
+1+σp2,2ν= 1+/radicalbigg
+1+4
+σ2, (38)
+the normalized energy eigenfunctions are
+ψn(p) = 2νΓ(ν)/radicalBigg
+n!(n+ν)√σ
+2πΓ(n+2ν)cν+1Cν
+n(s), (39)
+wheren= 0,1,2,...andCν
+n(s) is theGegenbauer polynomial
+Cν
+n(s) =(−1)n
+2nn!Γ(2ν+n)Γ(2ν+1
+2)
+Γ(2ν)Γ(2ν+1
+2+n)(1−s2)1/2−νdn
+dsn(1−s2)ν+n−1/2. (40)
+Also one can write (39) as [55]
+ψn(p) = 2νΓ(ν)/radicalBigg
+n! (n+1)√σ
+2πΓ(n+2ν)cν+1Γ(n+2ν)F(−n,n+2ν; 1/2+ν;−1/2s+1/2)
+Γ(n+1)Γ(2ν). (41)
+6Note that using Eq.(33), one can acquire ψn(χ) also according to the equations (20), (21) and (37) we conclude
+that in quantum gravity regime the mass of a black hole is proportiona l with the quantum number nin
+contrast, according to Eq.(6) with ordinary scales of energy in sta ndard quantization of black holes, that mass
+is proportional to√nthat it agrees with Beckenstein’s proposal [1].
+4.1 Black Hole Entropy with the GUR
+When there is not any deformation in the system, the coordinates a nd momenta variables xiandpjare
+canonically conjugate i.e. {xi,pj}=δij,{xi,xj}={pi,pj}= 0, thus the thermodynamics of the system is
+calculated by using the following partition function
+Z=1
+2π/planckover2pi1/integraldisplay
+e−βH(x,p)dxdp. (42)
+In the more general deformed case with the following generalized co mmutation relations
+[Xi,Pj] =i/planckover2pi1fij(X,P),
+[Pi,Pj] =i/planckover2pi1hij(X,P),
+[Xi,Xj] =i/planckover2pi1gij(X,P). (43)
+where operators XiandPjare new coordinates and momentum variables respectively and fij,gijandhijare
+the deformation functions that obey properties like bilineary, Libniz rules and Jacobi identity. In the classical
+limit/planckover2pi1→0 the above relations reduce to the deformed Poisson brackets
+{Xi,Pj}=fij(X,P),{Pi,Pj}=hij(X,P),{Xi,Xj}=gij(X,P). (44)
+These relations are anti-symmetric, bilinear and obey the Libniz rules and Jacobi identity [49, 52]. Then the
+partition function for deformed case can be interpreted in terms o fXandP[53] as
+Zdeformed=1
+2π/planckover2pi1/integraldisplay
+e−βH(X,P)dXdP
+J. (45)
+According to Eq.(15), g(X,P) =h(X,P) = 0 and {X,P}=f(X,P) = 1 +σP2, therefore the partition
+function of the quantum black hole becomes
+ZGUP=1
+2π/planckover2pi1/integraldisplay
+dXexp[−βV(X)]/integraldisplay
+dPexp/parenleftBig
+−βcℓp
+2/planckover2pi1P2/parenrightBig
+1+σP2. (46)
+Now by inserting the modified potential (7) , the corrected partitio n function will be
+ZGUP
+Q=ℓp
+/planckover2pi1/radicalbiggπ
+3βEpσexp/bracketleftbigg
+−/parenleftbiggβ2Ep2
+16π+cℓpβ
+2/planckover2pi1σ/parenrightbigg/bracketrightbigg
+Γ/parenleftbigg1
+2,cℓpβ
+2/planckover2pi1σ/parenrightbigg
+. (47)
+For the case ofcℓpβ
+2/planckover2pi1σ≫1 the above equation leads to
+ZGUP
+Q=/radicalbigg
+2π
+3exp/parenleftBig
+−β2Ep2
+16π−/planckover2pi1σ
+cℓpβ/parenrightBig
+βEp. (48)
+Now similar to the non-deformed case we put, E=−∂ln(ZGUP
+Q)
+∂β=Ep2
+8πβ+1
+β−/planckover2pi1
+cℓpβ2σ=Mc2. Hence, the
+temperature of the quantum black hole in the GUP framework in term of the Hawking temperature becomes
+β=βH/bracketleftbigg
+1−1
+βHMc2+MEp
+(βHMc2−1)(βHMc2−2)σ/bracketrightbigg
+. (49)
+The entropy is accounted for as before and by using the obtained t emperature, it is given as follows
+SGUP
+k=As
+4ℓ2p/bracketleftbigg
+1−1
+βHMc2/bracketrightbigg2
++As
+4ℓ2p/bracketleftbigg
+1−1
+βHMc2/bracketrightbiggMEp
+(βHMc2−2)(βHMc2−1)σ
+−1
+2ln/parenleftBigg
+As
+4ℓ2p/bracketleftbigg
+1−1
+βHMc2/bracketrightbigg2
++As
+4ℓ2p/bracketleftbigg
+1−1
+βHMc2/bracketrightbiggMEp
+(βHMc2−2)(βHMc2−1)σ/parenrightBigg
+−2Epσ
+c2/bracketleftbigg
+βH/parenleftbigg
+1−1
+βHMc2/parenrightbigg/bracketrightbigg−1
+−1
+2ln(24)+1. (50)
+7Finally the definition of GUP Hawking-Bekenstein entropy
+SGUP
+BH
+k=SBH
+k/parenleftBigg
+1+E3
+p
+8πM2c6σ/parenrightBigg
+, (51)
+leads to
+SGUP
+k=SGUP
+BH
+k−1
+2ln/parenleftbiggSGUP
+BH
+k/parenrightbigg
+−2Mc2/parenleftbiggSGUP
+BH
+SBH−1/parenrightbigg
++O/parenleftbig
+SGUP−1
+BH/parenrightbig
+. (52)
+As the log-type correction is similar to the existing results that are d erived from other methods [29, 31, 32] . It
+is shown that this result has the same form as the non-deformed ca se, again the logarithmic correction to the
+entropy appears with a −1/2 factor and it is clear that we get the non-deformed entropy in the limitσ→0. As
+we can see, these thermodynamical quantities are modified due to t he presence of the deformation parameter
+σ. In particular the GUP lessens the value of the entropy, which can b e understood from the fact that GUP
+reduces the accessible physical states. Similar results can be reac hed by surveying noncommutativity [36, 54].
+5 Conclusion
+In summary, we first introduced a model for quantum black holes an d showed that its Wheeler-Dewitt equa-
+tion issimilarto the equationofaone-dimensionalharmonicoscillator. We then reviewedthe thermodynamical
+properties of quantum black holes for which the logarithmic correct ion of the entropy with a −1/2 factor ap-
+peared. Next we presented the Hilbert space in the generalized unc ertainty principle framework and obtained
+the relevant generalized Fourier transformation which gives the qu asi-position wave function from momentum
+space. In the next step we studied the quantum black hole in this fra mework and obtained its wave functions
+and energy eigenvalues and argued that in a quantum gravity regime the mass of a black hole is proportional
+to the integer n, in contrast to ordinary scales of energy in standard quantization of black hole where the mass
+is proportional to√n. Finally we determined the thermodynamical properties in this scena rio by introducing
+the relevant Bekenstein-Hawking entropy in the GUP framework an d concluded that again the logarithmic
+correction of the entropy appears with the factor −1/2 and also the value of the entropy diminishes which
+can be comprehended from the fact that the GUP reduces the ava ilable physical states .
+Acknowledgmentsan
+We would like to thank H. R. Sepangi for a careful reading of the man uscript and A.B. is grateful to M.
+Poorakbar for helpful discussions. This work has been supported by the Research Institute for Astronomy and
+Astrophysics of Maragha, Iran. We also would like to thank the anon ymous referee for valuable comments to
+improve the paper.
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+arXiv:1001.0862v1  [math.AC]  6 Jan 2010ABSTRACT LOCAL COHOMOLOGY FUNCTORS
+YUJI YOSHINO, TAKESHI YOSHIZAWA
+Abstract. We propose to define the notion of abstract local cohomology func tors.
+The ordinary local cohomologyfunctor RΓIwith support in the closed subset defined
+by an ideal Iand the generalized local cohomology functor RΓI,Jdefined in [16] are
+characterized as elements of the set of all the abstract local coh omology functors.
+Introduction
+LetRbe a commutative noetherian ring and Ibe an ideal of R. We denote the
+category of all R-modules by R-Mod and also denote the derived category consisting of
+all left bounded complexes of R-modules byD+(R-Mod). Then the section functor ΓI:
+R-Mod→R-Mod and its derived functor RΓI:D+(R-Mod)→D+(R-Mod) (called
+the local cohomology functor) are basic tools not only for the theo ry of commutative
+algebras but also for algebraic geometry. They are actually extens ively studied by
+many authors. See, for example, [3], [4], [7] and [8].
+To give a way of generalizing such classical local cohomology functor s, the authors
+have introduced, together with Ryo Takahashi inthe paper [16], t he generalized section
+functorΓI,J:R-Mod→R-Mod and the generalized local cohomology functor RΓI,J:
+D+(R-Mod)→D+(R-Mod) associated with a given pair of ideals I,J. The aim of
+this paper is to characterize these functors among the set of fun ctors, and show how
+naturally the functors ΓI,JandRΓI,Jappear in the context of functors.
+Our strategy is the following.
+As for the section functors ΓIandΓI,J, we consider the set S(R) of all the left exact
+radical functors on R-Mod. Actually, ΓIandΓI,Jare elements of S(R). A radical
+functor, or more generally a preradical functor, has its own long h istory in the theory
+of categories and functors. See [6] or [11] for the case of module c ategory. One of
+the most useful and important facts is that there is a bijective cor respondence between
+S(R) andtheset ofhereditary torsiontheories for R-Mod([15, Chapter VI, Proposition
+3.1]). In this paper, after giving some characterizations of element s ofS(R), we shall
+show that S(R) is a complete lattice, and we can define a product and a quotient for a
+couple of elements of S(R). As a consequence, we shall prove that a left exact radical
+functorγisoftheform ΓIforanideal IofRifandonlyif γsatisfies akindofascending
+chain condition inside the set S(R) (Theorem 5.3). Moreover we also prove that ΓI,J
+is nothing but a quotient of ΓIbyΓJ(Theorem 5.6).
+Asforthederivedfunctors RΓIandRΓI,J, weconsiderthesetofisomorphismclasses
+ofabstractlocalcohomologyfunctors, whichweshalldefineinDefi nition1.10. Wesaya
+trianglefunctor δ:D+(R-Mod)→D+(R-Mod)isanabstract localcohomologyfunctor
+12 YUJI YOSHINO, TAKESHI YOSHIZAWA
+if it defines a stable t-structure on D+(R-Mod) which divides indecomposable injective
+R-modules. (See Definition 1.10 for the precise meaning.) Actually RΓIandRΓI,J
+are abstract local cohomology functors. We note here that the n otion of t-structure
+was introduced and studied first in the paper [1], but what we need in this paper is the
+notion of stable t-structure introduced by Miyachi [13]. We denote byA(R) the set of
+all the isomorphism classes of abstract local cohomology functors onD+(R-Mod). We
+shall show that A(R) bijectively corresponds to the set of specialization-closed subse ts
+of Spec(R). In fact, we prove in Theorem 2.12 that each abstract local coho mology
+functor is of the form RΓWwithWbeing a specialization-closed subset of Spec( R).
+After these observation, we define a product and a quotient for a couple of elements
+ofA(R), in section 3. Finally we shall prove that the functor RΓIis characterized as
+an element of A(R) which satisfies a kind of ascending chain condition (Theorem 5.4).
+Moreover, RΓI,Jis a quotient of RΓIbyRΓJinA(R) (Theorem 5.7).
+The organization of the paper is the following.
+In section 1, we recall some basic concepts and properties fromth e theory of functors
+and the torsion theory, and we give the definition of abstract local cohomology functors
+(Definition 1.10). Since Miyachi’s results [13] concerning stable t-st ructure is essential
+for this definition, we include the precise statement and a rough pro of of Miyachi’s
+Theorem in section 1 (Theorem 1.8).
+In section 2, we observe some necessary and sufficient conditions f or a functor to be
+left exact radical functor (Theorem 2.7) and prove that an abstr act local cohomology
+functor is always a derived functor of a section functor with suppo rt in a specialization-
+closed subset (Theorem 2.12).
+In section 3, we define the closure operation for preradical funct or in the set of left
+exact radical functors (Definition 4.1), and define the quotient in S(R) andA(R) as
+mentioned above.
+In section 4, we give characterization of the section functors ΓIandΓI,Jas elements
+ofS(R), respectively inTheorem5.3andTheorem 5.6. Wealsocharacterize thederived
+functorsRΓIandRΓI,Jaselements of A(R), respectively inTheorem 5.4andTheorem
+5.7.
+1.Preliminaries on functors and the definition of abstract lo cal
+cohomology functors
+Throughoutthepaper, Ralwaysdenotesacommutative noetherianring, and R-Mod
+denotes the category consisting of all R-modules and R-module homomorphisms.
+In the first half of this section, we are interested in covariant func tors from R-Mod
+to itself. Let γ1andγ2be such functors. Recall that γ1is said to be a subfunctor of
+γ2, denoted by γ1⊆γ2, ifγ1(M) is a submodule of γ2(M) for all M∈R-Mod and
+ifγ1(f) is a restriction of γ2(f) toγ1(M) for allf∈HomR(M,N). Let1denote the
+identity functor on R-Mod. Note from the definition that if γ⊆1, thenγ(M) is a
+submodule of Mfor allM∈R-Mod and γ(f) is a restriction of fontoγ(M) for all
+f∈HomR(M,N). First of all we shall make several remarks about subfunctors o f1.ABSTRACT LOCAL COHOMOLOGY FUNCTORS 3
+Remark 1.1. (1) Ifγis a subfunctor of 1, thenγis an additive R-linear functor from
+R-Mod to R-Mod.
+In fact, the mapping Hom R(M,N)→HomR(γ(M),γ(N)), which is induced by γ,
+mapsfto its restriction f|γ(M)as explained above. It is obvious that the restriction
+mapping is additive and R-linear.
+(2) Ifγ1andγ2are subfunctors of 1, then their composition functor γ1·γ2is also a
+subfunctor of 1.
+In fact, for an R-moduleM, sinceγ2(M)⊆M, we have γ1·γ2(M)⊆γ2(M)⊆M. If
+f∈HomR(M,N), then it is easily seen that γ1·γ2(f) =f|γ1·γ2(M). It therefore follows
+γ1·γ2⊆1.
+The following observations will be used later in this paper.
+Lemma 1.2. Letγ,γ1andγ2be subfunctors of 1and assume that they are left exact
+functors on R-Mod.
+(1)IfNis anR-submodule of M, then the equality γ(N) =N∩γ(M)holds.
+(2)For allR-moduleM, we have γ1·γ2(M) =γ1(M)∩γ2(M) =γ2·γ1(M). In
+particular, the equality γ1·γ2=γ2·γ1holds.
+(3)The idempotent property holds for γ, i.e.γ2=γ.
+(4)Ifγ1is isomorphic to γ2as functors on R-Mod, thenγ1is identical with γ2as
+subfunctors of 1, i.e.γ1∼=γ2impliesγ1=γ2.
+Proof.(1)Theequality γ(N) =N∩γ(M)easilyfollowsfromthefollowingcommutative
+diagram with exact rows.
+0−−−→N−−−→M−−−→M/N−−−→0/arrowtp/arrowtp/arrowtp
+0−−−→γ(N)−−−→γ(M)−−−→γ(M/N)
+(2) Applying the functor γ1to a submodule γ2(M)⊆Mand using (1), we have
+γ1·γ2(M) =γ1(M)∩γ2(M). Similarly γ2·γ1(M) =γ2(M)∩γ1(M). Therefore
+γ1·γ2(M) =γ2·γ1(M) holds for all M∈R-Mod, hence we have γ1·γ2=γ2·γ1.
+(3)Apply theresult of(2)andweseethat γ2(M) =γ·γ(M) =γ(M)∩γ(M) =γ(M)
+for allM∈R-Mod. Hence γ2=γ.
+(4) Suppose φ:γ1→γ2is an isomorphism of functors. Then, φ(M) :γ1(M)→
+γ2(M) is an isomorphism of R-modules for any R-module M. Applying the functor γ1
+to thisR-module homomorphism, we have the following commutative diagram.
+γ1(M)φ(M)−−−→ γ2(M)
+/uniontext|/arrowtp/uniontext|/arrowtp
+γ2
+1(M)γ1(φ(M))−−−−−→ γ1·γ2(M),
+where the left vertical arrow is an equality by (3). Thus it follows tha tγ2(M) =
+γ1·γ2(M) for allM∈R-Mod, hence γ2=γ1·γ2as functors. Considering φ−1, we can4 YUJI YOSHINO, TAKESHI YOSHIZAWA
+show that γ1=γ2·γ1as well. Since γ1·γ2=γ2·γ1as we have shown in (2), we have
+γ1=γ2as desired. /square
+Let us recall some definitions for functors from the theory of cat egories.
+Definition 1.3. Letγbe a functor R-Mod→R-Mod.
+(1) A functor γis called a preradical functor if γis a subfunctor of 1.
+(2) A preradical functor γis called a radical functor if γ(M/γ(M)) = 0 for every
+R-module M.
+(3) A functor γis said to preserve injectivity if γ(I) is an injective R-module
+whenever Iis an injective R-module.
+We should remark that a left exact radical functor is sometimes calle d a torsion
+radical or an idempotent kernel functor, which depends on the au thors. (E.g. O.
+Goldman [6], J. Lambek [10]).
+Example 1.4. LetWbe a subset of Spec( R). Recall that Wis said to be closed under
+specialization (or specialization-closed) if p∈Wandp⊆q∈Spec(R) implyq∈W.
+WhenWis closed under specialization, we can define the section functor ΓWwith
+support in Was
+ΓW(M) ={x∈M|Supp(Rx)⊆W},
+for allM∈R-Mod. Then it is easy to see that ΓWis a left exact radical functor that
+preserves injectivity.
+For the later use we need the notion of torsion theory. See [14] or [1 5] for the detail
+of the torsion theory.
+Definition 1.5. A torsion theory for R-Mod is a pair (T,F) of classes of R-modules
+satisfying the following conditions:
+(1) Hom R(T,F) = 0.
+(2) If Hom R(M,F) = 0, then M∈T.
+(3) If Hom R(T,M) = 0, then M∈F.
+Atorsiontheory ( T,F)forR-Modis called hereditary if Tisclosed under submodules.
+Remark 1.6. It is easily observed that the following hold for a torsion theory ( T,F)
+forR-Mod. (Cf. [14] or [15].)
+(1)Tis closed under quotient modules, direct sums and extensions.
+(2)Fis closed under submodules, direct products and extensions.
+(3) Forevery R-moduleM, thereisauniqueexactsequence0 →T→M→F→0
+withT∈TandF∈F.
+It is well known that there is a one-to-one correspondence betwe en the set of left
+exact radical functors and the set of hereditary torsion theorie s. In fact, if γis a left
+exact radical functor, then one obtains a hereditary torsion the ory (Tγ,Fγ) by setting
+(∗)/braceleftBigg
+Tγ={T∈R-Mod|γ(T) =T},
+Fγ={F∈R-Mod|γ(F) = 0}.ABSTRACT LOCAL COHOMOLOGY FUNCTORS 5
+Conversely, given a hereditary torsion theory ( T,F) forR-Mod, one can define a left
+exact radical functor γin such a way that the submodule γ(M) of anR-module Mis
+the sum of all submodules of Mwhich belong to the class T.
+We denote byD+(R-Mod) the derived category of R-Mod consisting of all left-
+bounded complexes of R-modules. It is known that D+(R-Mod) has structure of tri-
+angulated category. We always regard an R-module Mas a complex···→0→M→
+0→···inD+(R-Mod) concentrated in degree zero. In this way, R-Mod is a full
+subcategory ofD+(R-Mod).
+We recall some definitions and notation from the theory of triangula ted categories.
+LetTandT′be general triangulated categories. An additive functor δ:T →T′
+is called a triangle functor provided that δ(X[1])∼=δ(X)[1] for any X∈T, and the
+diagram δ(X)→δ(Y)→δ(Z)→δ(X)[1] is a triangle in T′whenever X→Y→Z→
+X[1] is a triangle in T. For any functor δ:T →T′, we denote
+Im(δ) ={X′∈T′|X′∼=δ(X) for some X∈T},
+Ker(δ) ={X∈T |δ(X)∼=0},
+which we regard as full subcategories of TandT′respectively. For a full subcategory
+U⊆T, the perpendicular full subcategories are defined as
+U⊥={X∈T |HomT(U,X) = 0 for all U∈U},
+⊥U={X∈T |HomT(X,U) = 0 for all U∈U}.
+The notion of stable t-structure is introduced by Miyachi [13]. Rec all that a full
+subcategory of a triangulated category is called a triangulated sub category if it is
+closed under the shift functor [1] and making triangles.
+Definition 1.7. A pair (U,V) of full triangulated subcategories of a triangulated cat-
+egoryTis called a stable t-structure on Tif it satisfies the following conditions:
+(i) Hom T(U,V) = 0.
+(ii) For any X∈T, there is a triangle U→X→V→U[1] with U∈Uand
+V∈V.
+The following theorem proved by Miyachi is a key to our argument. We shall refer
+to this theorem as Miyachi’s Theorem.
+Theorem 1.8. [13, Proposition 2.6] LetTbe a triangulated category and Ua full
+triangulated subcategory of T. Then the following conditions are equivalent for U.
+(1)There is a full subcategory VofTsuch that (U,V)is a stable t-structure on T.
+(2)The natural embedding functor i:U→Thas a right adjoint ρ:T →U.
+If it is the case, setting δ=i◦ρ:T →T, we have the equalities
+U= Im(δ)andV=U⊥= Ker(δ).
+Proof.Although a proof of the theorem is given in [13, Proposition 2.6], we nee d in the
+later part of the present paper how the adjoint functor corresp onds to the subcategory.
+For this reason we briefly recall the proof of the theorem.6 YUJI YOSHINO, TAKESHI YOSHIZAWA
+Assume that (U,V) is a stable t-structure on T. Then, for any X∈T, there is a
+triangleU→X→V→U[1] withU∈UandV∈V. We first note that Uis uniquely
+determined by Xup to isomorphisms. In fact, this can be easily proved only by using
+the conditions (i) and (ii) in the definition of a stable t-structure. Sim ilarly, given a
+morphism f:X1→X2inT, we can easily see that it induces a morphism of triangles
+U1−−−→X1−−−→V1−−−→U1[1]
+g/arrowbt f/arrowbt/arrowbt g[1]/arrowbt
+U2−−−→X2−−−→V2−−−→U2[1],
+whereU1,U2∈UandV1,V2∈V, and the morphism gis uniquely determined, so that
+it depends only on f. In such a way we can define a functor ρ:T →Uby setting
+ρ(X) =Uandρ(f) =gunder the notation above. By this construction, every X∈T
+is embedded in a triangle of the form i◦ρ(X)→X→V→i◦ρ(X)[1],whereV∈V.
+Then, for any U∈U, since Hom T(i(U),V) = 0, we have
+HomU(U,ρ(X)) = Hom T(i(U),i◦ρ(X))∼=HomT(i(U),X).
+Therefore ρis a right adjoint of i. In this case, if X∈Ker(δ) whereδ=i◦ρ, then the
+above triangle shows that X∼=V∈V. Hence we have Ker( δ)⊆V.
+Conversely assume that ihas a right adjoint ρ:T →U. Then there is an adjunction
+morphism φ:i◦ρ→1, where1is the identity functor on T. Therefore every X∈T
+can be embedded in a triangle of the form
+(1) i◦ρ(X)φ(X)−−−→X−−−→VX−−−→i◦ρ(X)[1].
+It follows from the property of adjunction morphisms that for any objectU∈ U,
+HomT(i(U),φ(X)) is an isomorphism, and hence Hom T(i(U),VX) = 0. This implies
+thatVX∈U⊥. Thus one can see that ( U,U⊥) is a stable t-structure on T.
+Let (U,V) be a stable t-structure on T, and letρbe a right adjoint of i:U→T. Set
+δ=i◦ρas above. Then we have shown that Ker( δ)⊆V, and the inclusion V⊆U⊥
+holds obviously from the definition. Now assume X∈U⊥. Thenφ(X) = 0 in the
+triangle (1), as it is an element of Hom T(i◦ρ(X),X) andρ(X)∈U. Therefore the
+trianglesplitsoffandwehave VX∼=X⊕i◦ρ(X)[1]. However Hom T(i◦ρ(X)[1],VX) = 0,
+sincei◦ρ(X)[1]∈UandVX∈U⊥. This implies that i◦ρ(X) = 0, hence X∈Ker(δ).
+Thus it follows that U⊥⊆Ker(δ). Hence we have shown Ker( δ) =V=U⊥.
+To prove thatU⊆Im(δ), letX∈U. Then the morphism X→VXin the triangle
+(1) is zero, since VX∈U⊥. Therefore we have i◦ρ(X)∼=X⊕VX[−1]. Since Hom T(i◦
+ρ(X),VX[−1]) = 0, this implies VX= 0 and φ(X) is an isomorphism. Thus X∈
+Im(δ). /square
+Remark 1.9. Let (U,V) be a stable t-structure on T, and let ρbe a right adjoint
+functor of i:U→T. Setδ=i◦ρas in the theorem.
+(1) It is known and is easy to see that the functors ρandδare triangle functors.
+(2) The functor ρ, henceδas well, is unique up to isomorphisms, by the uniqueness
+of right adjoint functors.ABSTRACT LOCAL COHOMOLOGY FUNCTORS 7
+(3) As we have shown in the last paragraph of the proof, an object X∈Tbelongs
+toU= Im(δ) if and only if the morphism φ(X) :δ(X)→Xis an isomorphism.
+Now we can define an abstract local cohomology functor which is a ma in theme of
+this paper.
+Definition 1.10. We denoteT=D+(R-Mod) in this definition. Let δ:T →Tbe a
+triangle functor. We call that δis an abstract local cohomology functor if the following
+conditions are satisfied:
+(1) Thenaturalembedding functor i: Im(δ)→Thasarightadjoint ρ:T →Im(δ)
+andδ∼=i◦ρ. (Hence, by Miyachi’s Theorem, (Im( δ),Ker(δ)) is a stable t-
+structure onT.)
+(2) Thet-structure (Im( δ),Ker(δ)) divides indecomposable injective R-modules, by
+which we mean that each indecomposable injective R-module belongs to either
+Im(δ) or Ker( δ).
+Example 1.11. We denote by ER(R/p) the injective hull of an R-module R/pfor
+a prime ideal p∈Spec(R). Note that any indecomposable injective R-module is
+isomorphic to ER(R/p) for some p∈Spec(R), sinceRis assumed to be noetherian.
+LetWbeaspecialization-closedsubsetofSpec( R). AswehaveexplainedinExample
+1.4, the section functor ΓW:R-Mod→R-Mod is a left exact radical functor. Hence
+we can define the right derived functor RΓW:D+(R-Mod)→D+(R-Mod). We claim
+thatRΓWis an abstract local cohomology functor.
+In fact, it is known that D+(R-Mod) is triangle-equivalent to the triangulated cate-
+goryK+(Inj(R)),whichisthehomotopycategoryconsistingofallleft-boundedin jective
+complexes over R. Throughthisequivalence, foranyinjective complex I∈K+(Inj(R)),
+RΓW(I) =ΓW(I) is the subcomplex of Iconsisting of injective modules supported in
+W. Hence every object of Im( RΓW) (resp. Ker( RΓW)) is an injective complex whose
+componentsaredirect sumsof ER(R/p)withp∈W(resp.p∈Spec(R)\W). Inpartic-
+ular, ifp∈W(resp.p∈Spec(R)\W), thenER(R/p)∈Im(RΓW) (resp.ER(R/p)∈
+Ker(RΓW)). Since Hom R(ER(R/p),ER(R/q)) = 0 for p∈Wandq∈Spec(R)\W,
+we can see that Hom K+(Inj(R))(I,J) = Hom K+(Inj(R))(I,ΓW(J)) for any I∈Im(RΓW)
+andJ∈K+(Inj(R)). Hence it follows from the above equivalence that RΓWis a right
+adjoint of the natural embedding i: Im(RΓW)→D+(R-Mod).
+Remark 1.12. EvenifRisanon-commutativering, Definition1.10isvalidfordefining
+an abstract local cohomology functor over R. We can give such an example over non-
+commutative rings in a similar way to Example 1.11.
+For this, let Rbe a non-commutative ring. We define Spec(R) to be the set of iso-
+morphism classes of all indecomposable injective left R-modules. Assume that a subset
+WofSpec(R) satisfies that Hom R(E,E′) = 0 for all E∈WandE′∈Spec(R)\W.
+For an injective left R-module I, ifIdecomposes as I=/circleplustext
+iEiwith every Eibeing
+indecomposable, then we define ΓW(I) to be the submodule/circleplustext
+Ei∈WEi. For aninjective
+complex I∈K+(Inj(R)) we also define ΓW(I) just by applying the functor ΓWon each
+component of I. Then, through the equivalence D+(R-Mod)∼=K+(Inj(R)), it defines8 YUJI YOSHINO, TAKESHI YOSHIZAWA
+the triangle functor RΓWonD+(R-Mod). Then it is quite similarly proved that RΓW
+satisfies the conditions in Definition 1.10.
+2.Characterization of abstract local cohomology functors
+LetWbe a specialization-closed subset of Spec( R) andΓWbe a section functor with
+support in W. We have pointed out in Example 1.11 that the right derived functor
+RΓWis an abstract local cohomology functor. In this section we shall pr ove that every
+abstract local cohomology functor is of this form. We will do this aft er a sequence of
+lemmas and propositions.
+First of all we shall show every left exact radical functor preserv es injectivity.
+Lemma 2.1. Letγbe a left exact radical functor and p∈Spec(R). Thenγ(ER(R/p))
+is identical to either ER(R/p)or0.
+Proof.Let (Tγ,Fγ) be a hereditary torsion theory for R-Mod corresponding to γ,
+which is defined in ( ∗) after Remark 1.6. Then there is an exact sequence 0 →T→
+ER(R/p)→F→0 withT∈TγandF∈Fγ. IfT= 0, then ER(R/p)∼=F∈Fγ,
+therefore γ(ER(R/p)) = 0. If T/\e}atio\slash= 0, then there is an element x∈T∩R/psuch that
+Rx∼=R/p. SinceTγis closed under taking submodules, we have R/p∈Tγ. Letybe
+an arbitrary element of ER(R/p). Then there is a filtration of the R-module Ry;
+0 =Mn/subsetnoteqlMn−1/subsetnoteql···/subsetnoteqlM0=Ry,
+such that Mi/Mi+1∼=R/qifor some qi∈Spec(R) (0≤i < n). See [12, Theorem 6.4].
+It is known that pny= 0 forn≫1, hence all the qi(0≤i < n) contain p. SinceTγ
+is closed under quotients and extensions, it results that Ry∈Tγ. Therefore it follows
+thaty∈Ry=γ(Ry)⊆γ(ER(R/p)) =T. Since this holds for every y∈ER(R/p), we
+haveER(R/p) =T∈Tγ, henceγ(ER(R/p)) =ER(R/p). /square
+As a result of this lemma we have the following.
+Proposition 2.2. Ifγis a left exact radical functor on R-Mod, thenγpreserves
+injectivity.
+Proof.As in the proof of the lemma, let ( Tγ,Fγ) be a hereditary torsion theory for
+R-Mod corresponding to γ. For an injective R-module E, it is known that it has a
+decomposition into indecomposable injective R-modules, say E=/circleplustext
+λ∈ΛER(R/pλ).
+We setE1=/circleplustext
+λ∈Λ1ER(R/pλ) andE2=/circleplustext
+λ∈Λ2ER(R/pλ), where
+Λ1={λ∈Λ|γ(ER(R/pλ)) =ER(R/pλ)},Λ2={λ∈Λ|γ(ER(R/pλ)) = 0}.
+It follows from Lemma 2.1 that E=E1⊕E2. SinceTγis closed under taking direct
+sums andFγis closed under taking direct products and submodules, we have E1∈Tγ
+andE2∈Fγ. Therefore we have an equality γ(E) =γ(E1)⊕γ(E2) =E1, which is an
+injective R-module. /square
+Nextweshallshowthatevery leftexactpreradical functorwhich preserves injectivity
+is of the form ΓWfor a specialization-closed subset Wof Spec(R). We begin with the
+following lemma.ABSTRACT LOCAL COHOMOLOGY FUNCTORS 9
+Lemma 2.3. Letγbe a left exact preradical functor which preserves injectiv ity. Then
+the following hold for a prime ideal pofR.
+(1)γ(ER(R/p))is either ER(R/p)or0.
+(2)γ(R/p)is either R/por0.
+Proof.(1) Since γ(ER(R/p)) is an injective submodule of an indecomposable injective
+moduleER(R/p), it is a direct summand of ER(R/p). Thus the indecomposability of
+ER(R/p) forcesγ(ER(R/p)) is either ER(R/p) or 0.
+(2) It follows from Lemma 1.2(1) that γ(R/p) =R/p∩γ(ER(R/p)). Therefore γ(R/p)
+is either R/por 0 by (1). /square
+Definition 2.4. For a left exact preradical functor γwhich preserves injectivity, we
+define a subset Wγof Spec(R) as follows:
+Wγ={p∈Spec(R)|γ(R/p) =R/p}.
+Note from the proof of Lemma 2.3 that Wγis the same as the set {p∈Spec(R)|
+γ(ER(R/p)) =ER(R/p)}.
+Lemma 2.5. Letγbe a left exact preradical functor which preserves injectiv ity. Then
+Wγis closed under specialization.
+Proof.Letp∈Wγ. For any prime ideal q⊇p, there is a commutative diagram
+R/pp−−−→ R/q/vextenddouble/vextenddouble/vextenddouble/arrowtp/uniontext|
+γ(R/p)−−−→γ(R/q),
+wherepis a natural projection map. Thus it follows from this diagram that R/q=
+γ(R/q), henceq∈Wγ. /square
+Now we are able to prove the following proposition.
+Proposition 2.6. Letγbe a left exact preradical functor which preserves injectiv ity.
+Then the equality γ=ΓWγholds as subfunctors of 1, whereWγis a specialization-closed
+subset of Spec(R)defined in Definition 2.4.
+Proof.We prove the equality γ(M) =ΓWγ(M) for any R-module M, which is enough
+for the proof, since the both functors are subfunctors of 1.
+First of all, we consider the case that Mis a finite direct sum of indecomposable
+injective R-modules/circleplustextn
+i=1ER(R/pi). Thenγ(M) =/circleplustext
+pi∈WγER(R/pi) =ΓWγ(M) from
+Lemma 2.3 and Remark 1.1(1). (Note that, since γis an additive functor, γcommutes
+with finite direct sums. This is used in the first equality above. )
+Next, we consider the case that Mis a finitely generated R-module. Since the
+injective hull ER(M) ofMis a finite direct sum of indecomposable injective modules,
+we have already shown that γ(ER(M)) =ΓWγ(ER(M)). Thus, using Lemma 1.2(1),
+we haveγ(M) =M∩γ(ER(M)) =M∩ΓWγ(ER(M)) =ΓWγ(M).10 YUJI YOSHINO, TAKESHI YOSHIZAWA
+Finally, we show the claimed equality for an R-module Mwithout any assumption.
+We should notice that an element x∈Mbelongs to γ(M) if and only if the equality
+γ(Rx) =Rxholds. In fact, this equivalence is easily observed from the equality
+γ(Rx) =Rx∩γ(M) that we showed in Lemma 1.2(1). This equivalence is true for the
+section functor ΓWγas well. So x∈Mbelongs to ΓWγ(M) if and only if ΓWγ(Rx) =
+Rx. Since the claim is true for finitely generated R-module Rx, we have γ(Rx) =
+ΓWγ(Rx). Therefore, we see that x∈γ(M) if and only if x∈ΓWγ(M), and the proof
+is completed. /square
+Recall that, for a left exact functor γ:R-Mod→R-Mod, we can define the right
+derived functor Rγ:D+(R-Mod)→D+(R-Mod) which is of course a triangle functor.
+Theorem 2.7. The following conditions are equivalent for a left exact pre radical func-
+torγonR-Mod.
+(1)γis a radical functor.
+(2)γpreserves injectivity.
+(3)γis a section functor with support in a specialization-close d subset of Spec(R).
+(4)Rγis an abstract local cohomology functor.
+Proof.The implications (1) ⇒(2)⇒(3)⇒(1) and (3)⇒(4) are already proved
+respectively in Propositions 2.2 and 2.6, Examples 1.4 and 1.11. We have only to prove
+(4)⇒(1).
+Assume that Rγis an abstract local cohomology functor. We have to show that
+γ(M/γ(M)) = 0 for any R-module M. It is enough to show that γ(E/γ(E)) = 0
+for any injective R-module E. In fact, for any R-module M, taking the injective hull
+E(M) ofM, we have γ(M/γ(M))⊆γ(E(M)/γ(E(M))) by Lemma 1.2 (1).
+Note that the natural inclusion γ⊂1of functors on R-Mod induces a natural
+morphism φ:Rγ→1of functors onD+(R-Mod). Since (Im( Rγ),Ker(Rγ)) is a
+stable t-structure on D+(R-Mod), it follows from the proof of Miyachi’s Theorem 1.8
+that every injective R-module Eis embedded in a triangle
+Rγ(E)φ(E)−−−→E−−−→V−−−→Rγ(E)[1],
+withRγ(E)∈Im(Rγ) andV∈Ker(Rγ). SinceEis an injective R-module and since
+Rγis the right derived functor of a left-exact functor, Rγ(E) =γ(E) is a submodule
+ofEvia the morphism φ(E). Therefore we have V∼=E/γ(E) inD+(R-Mod). In
+particular, H0(Rγ(E/γ(E)))∼=H0(Rγ(V)) = 0. Since γis left exact functor, it
+is concluded that γ(E/γ(E)) = 0 as desired. This completes the proof of Theorem
+2.7. /square
+Remark 2.8. (1) The equivalences among the conditions (1), (2) and (3) in Theor em
+2.7 already appear in several literatures, but they arenot explicitly written. A new and
+significantfeatureofTheorem2.7isthattheyareequivalent aswell tothecondition(4).
+(2) It is well known that there is a one-to-one correspondence be tween the set of left
+exactradicalfunctorsandthesetofGabrieltopologies([15, Cha pterVI.Theorem5.1]).
+Therefore, adding to Theorem 2.7, giving a left exact preradical fu nctor on R-ModABSTRACT LOCAL COHOMOLOGY FUNCTORS 11
+satisfying one of the conditions (1)-(4) is equivalent to giving a Gabr iel topology on
+the ringR.
+More generally than Theorem 2.7, we are able to prove that every ab stract lo-
+cal cohomology functor is the derived functor of a section functo r with support in
+specialization-closed subset. Before proceeding to this theorem, we prepare lemmas
+that will be necessary for its proof.
+Lemma 2.9. LetX∈ D+(R-Mod)and letWbe a specialization-closed subset of
+Spec(R).
+(1)X∼=0⇐⇒RHomR(R/p,X)p= 0for allp∈Spec(R).
+(2)X∈Im(RΓW)⇐⇒RHomR(R/q,X)q= 0for allq∈Spec(R)\W.
+(3)X∈Ker(RΓW)⇐⇒RHomR(R/p,X)p= 0for allp∈W.
+Proof.(1) Suppose X/\e}atio\slash∼=0. Since Xis a left bounded complex, there is an integer i0
+such that Hi(X) = 0 for i < i0andHi0(X)/\e}atio\slash= 0. Now take p∈AssR(Hi0(X)). Since
+Hi0(X) is the initial cohomology of X, we have isomorphisms of R-modules
+Hi0(RHomR(R/p,X)p)∼=HomD+(R-Mod)(R/p,X[i0])p∼=HomR(R/p,Hi0(X))p,
+the last term of which is non-trivial. Therefore RHomR(R/p,X)p/\e}atio\slash= 0.
+(2) Recall from Example 1.11 that Xbelongs to Im( RΓW) if and only if Xis quasi-
+isomorphic to an injective complex whose components are direct sum s ofER(R/p) with
+p∈W. Note that
+HomR(R/q,ER(R/p))q= 0 ifp/\e}atio\slash=q. (∗)
+Hence if X∈Im(RΓW), then it is easy to see that RHomR(R/q,X)q= 0 for any
+q∈Spec(R)\W.
+Conversely assume that RHomR(R/q,X)q= 0 for any q∈Spec(R)\W. Since
+(Im(RΓW), Ker(RΓW)) is a stable t-structure on D+(R-Mod), there is a triangle
+RΓW(X)φ(X)−−−→X−−−→V−−−→RΓW(X)[1],
+as in the proof of Miyachi’s Theorem 1.8. Replacing Xwith its injective resolution
+I, the morphism φ(X) is isomorphic to the natural inclusion ΓW(I)⊂I. Hence V
+is isomorphic inD+(R-Mod) to the quotient complex I/ΓW(I), which is an injective
+complex whose components aredirect sums of ER(R/q) withq∈Spec(R)\W. Suppose
+V/\e}atio\slash∼=0. Then, as in the proof of (1), we can take an associated prime idea lQof
+the initial cohomology Hi0(V) ofVand soRHomR(R/Q,V)Q/\e}atio\slash= 0. Since Hi0(V)
+is a submodule of a direct sum of injective modules ER(R/q) withq∈Spec(R)\W,
+the associated prime Qequals one of those q∈Spec(R)\W. SinceRΓW(X) is in
+Im(RΓW), it follows from what we have proved in the first half of this proof an d the
+assumption on XthatRHomR(R/Q,X)Q=RHomR(R/Q,RΓW(X))Q= 0, but this
+forcesRHomR(R/Q,V)Q= 0. This is a contradiction, hence we conclude V∼=0 and
+X∼=RΓW(X)∈Im(RΓW).
+(3) Suppose RΓW(X)∼=0. Taking an injective resolution IofX, we have ΓW(I) is a
+nullcomplexand Xisquasi-isomorphicto I/ΓW(I). Replacing IwithI/ΓW(I)ifneces-
+sary, we may assume that Iconsists of injective modules E(R/q) withq∈Spec(R)\W.12 YUJI YOSHINO, TAKESHI YOSHIZAWA
+Therefore it follows from ( ∗) above that RHomR(R/p,X)p∼=HomR(R/p,I)p= 0 for
+allp∈W.
+Conversely assume that RHomR(R/p,X)p= 0 for all p∈Wand take a triangle
+RΓW(X)−−−→X−−−→V−−−→RΓW(X)[1],
+as in the proof of (2). Then, since RΓW(V)∼=0, it follows from the first part of this
+proof that RHomR(R/p,V)p= 0 for all p∈W. Hence we can deduce from the triangle
+thatRHomR(R/p,RΓW(X))p= 0 for all p∈Was well. On the other hand we know
+from (2) that RHomR(R/p,RΓW(X))p= 0 even for p∈Spec(R)\W. Thus (1) forces
+thatRΓW(X) = 0, hence X∈Ker(RΓW). /square
+We have the following corollary as a result of this lemma, in which RΓmdenotes
+the right derived functor of the section functor with support in th e closed (hence
+specialization-closed) subset V(m) ={m}.
+Corollary 2.10. Let(R,m,k)be a noetherian local ring and let X/\e}atio\slash∼=0∈D+(R-Mod).
+IfX∈Im(RΓm), thenRHomR(ER(k),X)/\e}atio\slash∼=0.
+Proof.Suppose RHomR(ER(k),X) = 0. Then we have
+(2)RHomR(ER(k),RHomR(k,X))∼=RHomR(k,RHomR(ER(k),X)) = 0.
+SinceX(/\e}atio\slash∼=0) belongs to Im( RΓm), we note from Lemma 2.9(1)(2) that RHomR(k,X)
+/\e}atio\slash= 0, which is a complex of k-vector spaces, and hence it is isomorphic to a direct sum of
+k[n] (n∈Z) inD+(R-Mod). Thus the equality (2) forces that RHomR(ER(k),k) = 0.
+Therefore we have only to prove that RHomR(ER(k),k)/\e}atio\slash= 0 for a noetherian local ring
+(R,m,k).
+By an obvious isomorphism RHomR(k,ER(k))∼=k, we have
+RHomR(ER(k),k)∼=RHomR(ER(k),RHomR(k,ER(k)))
+∼=RHomR(k,RHomR(ER(k),ER(k)))
+∼=RHomR(k,/hatwideR).
+Now letFbe a minimal free resolution of kwhich belongs to D−(R-Mod). Then the
+last complex in the above isomorphism is isomorphic to the complex Hom R(F,/hatwideR)∼=
+Hom/hatwideR(F⊗R/hatwideR,/hatwideR). Since F⊗R/hatwideRis a free resolution of kover/hatwideR, we obtain an
+isomorphism RHomR(ER(k),k)∼=RHom/hatwideR(k,/hatwideR), which is a nontrivial complex, as it
+iswell-known thatits nthcohomologymoduleExtn
+/hatwideR(k,/hatwideR) isnontrivial if n= depth( /hatwideR).
+/square
+Lemma 2.11. As in the previous lemma, let X∈ D+(R-Mod)and letWbe a
+specialization-closed subset of Spec(R).
+(1)IfX∈Ker(RΓW)andRHomR(X,ER(R/q)) = 0for allq∈Spec(R)\W, then
+X∼=0.
+(2)IfX∈Im(RΓW)andRHomR(ER(R/p),X) = 0for allp∈W, thenX∼=0.ABSTRACT LOCAL COHOMOLOGY FUNCTORS 13
+Proof.(1) Assume that X∈Ker(RΓW) andRHomR(X,ER(R/q)) = 0 for all q∈
+Spec(R)\W. Then, as in the proof of Lemma 2.9 (3), Xis isomorphic inD+(R-Mod)
+to an injective complex whose components are direct sums of ER(R/q) withq∈
+Spec(R)\W. Suppose that X/\e}atio\slash∼=0. Then the initial nontrivial cohomology Hi0(X) has
+anassociatedprimeideal qwhichbelongstoSpec( R)\W,andHom R(Hi0(X),ER(R/q))
+/\e}atio\slash= 0 for such a q. SinceER(R/q) is an injective module, note that
+H−i0(RHomR(X,ER(R/q)))∼=HomR(Hi0(X),ER(R/q)),
+hence this is a nontrivial module. This contradicts to that RHomR(X,ER(R/q)) = 0.
+(2) Assume X∈Im(RΓW) andRHomR(ER(R/p),X) = 0 for all p∈W. Suppose
+X/\e}atio\slash∼=0 and we shall show a contradiction. It follows from Lemma 2.9(1)(2) that there
+is a prime ideal P∈Wsuch that RHomR(R/P,X)P/\e}atio\slash= 0. Take such a Pas maximal
+among these prime ideals and set Y=RHomR(R/P,X). LetQ∈Spec(R). IfP/\e}atio\slash⊂Q,
+then (R/P)Q= 0, hence
+YQ∼=RHomR(R/P,X)Q∼=RHomRQ((R/P)Q,XQ) = 0.
+(Weshouldnoticethat RHomR(R/P,−)commutes withtakinglocalization, since R/P
+is a finitely generated R-module. ) Thus
+RHomR(R/Q,Y)Q=RHomRQ((R/Q)Q,YQ) = 0
+for allQ∈Spec(R)\V(P), hence we have Y∈Im(RΓV(P)) by Lemma 2.9(2). Thus, as
+in the proof of Lemma 2.9(2), Yis isomorphic to a complex which consists of injective
+modules of the form ER(R/p) withp∈V(P). On the other hand, if P/subsetornotdbleqlQ, then we
+have
+RHomR(R/Q,Y)Q∼=RHomR(R/Q,RHomR(R/P,X))Q∼=RHomR(R/P,RHomR(R/Q,X))Q,
+∼=RHomRQ((R/P)Q,RHomR(R/Q,X)Q),
+where we notice that RHomR(R/Q,X)Q= 0 by the maximality of P. Therefore we
+haveRHomR(R/Q,Y)Q= 0 for all Q∈V(P)\{P}. Setting W′=V(P)\{P}, we see
+thatW′is a specialization-closed subset of Spec( R). It follows from Lemma 2.9(3)
+thatY∈Ker(RΓW′). As a result, as in the proof of Lemma 2.9(3), we have that Yis
+isomorphic to an injective complex consisting of direct sums of copies ofER(R/P).
+Now we note that RHomR(ER(R/P),Y) = 0. In fact, this is isomorphic to
+RHomR(ER(R/P),RHomR(R/P,X))∼=RHomR(R/P,RHomR(ER(R/P),X)),
+which vanishes by the assumption. Note also that ER(R/P) has a structure of RP-
+module. As we have shown above, Yis isomorphic to a complex Iconsisting of direct
+sums ofER(R/P). In general, the equality Hom R(M,N) = Hom RP(M,N) holds for
+RP-modules MandN. This equality extends to complexes and we can see that Ihas
+a structure of complex over RP. Therefore we have isomorphisms
+RHomR(ER(R/P),Y)∼=HomR(ER(R/P),I)
+= Hom RP(ER(R/P),I)
+∼=RHomRP(ER(R/P),Y).14 YUJI YOSHINO, TAKESHI YOSHIZAWA
+Tosumupwehavesuchasituationthat Y(/\e}atio\slash∼=0)∈D+(RP-Mod)belongstoIm( RΓPRP)
+andRHomRP(ER(R/P),Y) = 0. But this contradicts Corollary 2.10. /square
+Nowweareabletoprovethefollowingtheorem, which isamainresult of thissection.
+Theorem 2.12. Given an abstract local cohomology functor δonD+(R-Mod), there
+exists a specialization-closed subset W⊆Spec(R)such that δis isomorphic to the right
+derived functor RΓWof the section functor ΓW.
+Proof.In this proof we denote T=D+(R-Mod). Suppose that δ:T →T is an
+abstract local cohomology functor. It then follows that it gives a s table t-structure
+(Im(δ),Ker(δ)) onT. We divides the proof into several steps.
+(1st step) : Consider the subset W={p∈Spec(R)|ER(R/p)∈Im(δ)}of Spec(R).
+ThenWis a specialization-closed subset.
+To see this, we have only to show that ER(R/p)∈Im(δ) implies ER(R/q)∈Im(δ)
+for prime ideals p⊆q. Assume contrarily that there are prime ideals p⊆qso
+thatER(R/p)∈Im(δ) butER(R/q)/\e}atio\slash∈Im(δ). Since the t-structure (Im( δ),Ker(δ))
+divides indecomposable injective modules, we must have ER(R/q)∈Ker(δ). Then,
+from the definition of t-structures, we have Hom T(ER(R/p),ER(R/q)) = 0, which says
+that there are no nontrivial R-module homomorphisms from ER(R/p) toER(R/q).
+However, a natural nontrivial map R/p→R/q֒→ER(R/q) extends to a non-zero
+mapER(R/p)→ER(R/q). This is a contradiction, hence it is proved that Wis
+specialization-closed. /squaresolid
+Our final goal is, of course, to show the isomorphism δ∼=RΓW. Notice that, since
+the both functors δandRΓWare abstract local cohomology functors, we have two
+stable t-structures (Im( δ),Ker(δ)) and (Im( RΓW),Ker(RΓW)) onT.
+(2nd step) : Note that if p∈W, thenER(R/p)∈Im(δ)∩Im(RΓW). On the other
+hand, ifq∈Spec(R)\W, thenER(R/q)∈Ker(δ)∩Ker(RΓW).
+This is clear from the definition of W./squaresolid
+(3rd step) : To prove the theorem, it is enough to show that Im( δ) = Im(RΓW).
+In fact, by Miyachi’s Theorem 1.8, an abstract local cohomology fun ctorδ(resp.
+RΓW) is uniquely determined by the full subcategory Im( δ) (resp. Im( RΓW)). See
+also Remark 1.9(2). /squaresolid
+(4th step) : Now we prove the inclusion Im( δ)⊆Im(RΓW).
+To do this, assume X∈Im(δ). Then there is a triangle in T;RΓW(X)→
+X→V→RΓW(X)[1], where V∈Ker(RΓW). Letqbe an arbitrary element of
+Spec(R)\W. Since (Im( δ),Ker(δ)) and (Im( RΓW),Ker(RΓW)) are stable t-structures
+and since ER(R/q) belongs to Ker( δ)∩Ker(RΓW), it follows that
+HomT(X,ER(R/q)[n]) = Hom T(RΓW(X),ER(R/q)[n]) = 0
+for any integer n. Then by the above triangle we have Hom T(V,ER(R/q)[n]) = 0
+for anyn. This is equivalent to that RHomR(V,ER(R/q))∼=0. In fact, the n-th
+cohomology module of RHomR(V,ER(R/q)) is just Hom T(V,ER(R/q)[n]) = 0. SinceABSTRACT LOCAL COHOMOLOGY FUNCTORS 15
+V∈Ker(RΓW), Lemma 2.11(1) forces V∼=0, therefore X∼=RΓW(X). Hence we have
+X∈Im(RΓW) as desired. /squaresolid
+(5th step) : For the final step of the proof, we show the inclusion I m(δ)⊇Im(RΓW).
+LetX∈Im(RΓW). Then there are triangles δ(X)→X→Y→δ(X)[1] with
+Y∈Ker(δ), andRΓW(Y)→Y→V→RΓW(Y)[1] with V∈Ker(RΓW). Letpbe
+an arbitrary prime ideal belonging to W. Similarly to the proof of the 4th step, since
+ER(R/p)∈Im(δ)∩Im(RΓW), we see that
+HomT(ER(R/p)[n],Y) = Hom T(ER(R/p)[n],V) = 0
+for any integer n, hence we have Hom T(ER(R/p)[n],RΓW(Y)) = 0 for any n. This
+showsRHomR(ER(R/p),RΓW(Y)) = 0, thenbyLemma2.11(2)wehave RΓW(Y) = 0.
+ThusY∈Ker(RΓW). Then, since (Im( RΓW),Ker(RΓW)) is a stable t-structure, the
+morphism X→Yin the triangle δ(X)→X→Y→δ(X)[1] is zero. It then follows
+thatδ(X)∼=X⊕Y[−1]. Since there is no nontrivial morphisms δ(X)→Y[−1] in
+T, it is concluded that δ(X)∼=X, henceX∈Im(δ) as desired, and the proof is
+completed. /square
+3.Lattice structure of the set of abstract local cohomology
+functors
+For a given commutative noetherian ring Rwe are considering the following sets.
+Definition 3.1. (1) We denote by S(R) the set of all left exact radical functors on
+R-Mod.
+(2) We denote by A(R) the set of the isomorphism classes [ δ] whereδranges over
+all abstract local cohomology functors D+(R-Mod)→D+(R-Mod) .
+(3) We denote by sp( R) the set of all specialization-closed subsets of Spec( R).
+All these sets are bijectively corresponding to one another. Actu ally we can define
+mappings among these sets. First of all, by using Definition 2.4, we are able to give a
+mapping
+S(R)−→sp(R) ;γ/mapsto→Wγ,
+which has the inverse mapping
+sp(R)−→S(R) ;W/mapsto→ΓW.
+See Proposition 2.6 and Theorem 2.7. We also have a mapping
+S(R)−→A(R) ;γ/mapsto→[Rγ],
+which is surjective by Theorem 2.12. It is injective as well. In fact, sin ceγ(M) =
+H0(Rγ(M)) forγ∈S(R) andM∈R-Mod,γis uniquely determined by Rγ.
+To sum up we have the following result as a corollary of Theorems 2.7 an d 2.12.
+Corollary 3.2. The mapping W/mapsto→ΓW(resp.γ/mapsto→[Rγ]) gives a bijection sp(R)→
+S(R)(resp.S(R)→A(R)).
+Note that ΓSpec(R)=1andΓ∅=0(the zero functor).16 YUJI YOSHINO, TAKESHI YOSHIZAWA
+Remark 3.3. (1) Recall that a subcategory of a triangulated category is said to be
+thick if it is a triangulated subcategory and is closed under taking dire ct summands.
+M. J. Hopkins gave the following theorem in [9]. Let P(R) denote the thick sub-
+category ofD(R-Mod) consisting of all the complexes which are quasi-isomorphic to
+boundedcomplexes offinitelygeneratedprojective R-modules. Thentherearebijective
+mappings/braceleftbigg
+thick subcategories
+ofP(R)/bracerightbigg
+−→
+←−/braceleftbigg
+specialization-closed
+subsets of Spec( R)/bracerightbigg
+.
+Therefore, taking Corollary 3.2 into account, the set set S(R) bijectively corresponds
+to the set of thick subcategories of P(R).
+(2) There are bijective maps among the following three sets: S(R), the set of hereditary
+torsion theories on R-Mod and the set of specialization-closed subsets of Spec( R).
+These bijections have already appeared in the papers of M. H. Bijan -Zadeh [2] and P.
+Cahen [5]. (We should note that a torsion theory in their papers mean s a hereditary
+one in our sense.)
+Letγ1,γ2∈S(R). It is easy to see that γ1⊆γ2as functors if and only if Wγ1⊆Wγ2
+as subsets of Spec( R). Hence the one-to-one correspondence in Corollary 3.2 preserv es
+the inclusion relation.
+Recall that a partially ordered set is called a lattice if every couple of e lements have
+a least upper bound and a greatest lower bound, and a lattice is called complete if
+every subset has a least upper bound and a greatest lower bound.
+If{Wλ|λ∈Λ}is a set of specialization-closed subsets of Spec( R), then/intersectiontext
+λWλand/uniontext
+λWλare also closed under specialization. By this reason sp( R) is a complete lattice.
+In view of Corollary 3.2 we can define/intersectiontextand/uniontextfor any subsets of S(R). Actually,
+if{γλ|λ∈Λ}is a set of elements in S(R), thenγ:=/intersectiontext
+λγλ(resp.δ:=/uniontext
+λγλ) is
+well-defined as an element of S(R) so that Wγ=/intersectiontext
+λWγλ(resp.Wδ=/uniontext
+λWγλ). In
+this way we have shown that S(R) has a structure of complete lattice and the bijective
+mapping sp( R)→S(R) in Corollary 3.2 gives an isomorphism as lattices.
+We can define a lattice structure as well on the set A(R) so that the bijection
+A(R)∼=S(R) is an isomorphism as complete lattices. More precisely, we define the
+order on A(R) by
+[Rγ1]⊆[Rγ2]⇐⇒γ1⊆γ2
+forγ1,γ2∈S(R). Notice that/intersectiontext
+λ[Rγλ] = [R(/intersectiontext
+λγλ)], and/uniontext
+λ[Rγλ] = [R(/uniontext
+λγλ)].
+Summing all up we have the following result.
+Theorem 3.4. The mapping S(R)→A(R)whichmaps γto[Rγ](resp.sp(R)→A(R)
+which sends Wto[RΓW]) gives an isomorphism of complete lattices.
+4.Closure operation and quotients
+Definition 4.1. Letγbe a preradical functor on R-Mod, which is not necessarily a
+left exact radical functor. We can define the closure (or the cove r) ¯γofγinS(R) asABSTRACT LOCAL COHOMOLOGY FUNCTORS 17
+the smallest left exact radical functor containing γ. By virtue of Remark 3.3, ¯ γis the
+intersection of all the left exact radical functors which contain γ.
+¯γ=/intersectiondisplay
+γ⊆γ′∈S(R)γ′.
+For a preradical functor γ, we define a subset of Spec( R) by the following:
+Wγ:={p∈Spec(R)|∃q⊆ps.t.γ(R/q)/\e}atio\slash= 0},
+which is clearly closed under specialization. Note that this generalizes the definition of
+Wγfor a left exact radical functor γin Definition 2.4. In fact, if γ∈S(R), then this
+definition of Wγagrees with Definition 2.4.
+Proposition 4.2. Letγbe a preradical functor.
+(1)ThenΓWγ⊆¯γ.
+(2)If, in addition, γis left exact, then ¯γ=ΓWγ.
+Proof.(1) By virtue of Corollary 3.2, it is sufficient to prove that Wγ⊆W¯γ.
+Suppose p∈Wγandγ⊆γ′∈S(R). Then there is a prime ideal q⊆psuch that
+γ(R/q)/\e}atio\slash= 0. Since γ(R/q)⊆γ′(R/q), we have γ′(R/q)/\e}atio\slash= 0, hence q∈Wγ′. SinceWγ′
+is specialization closed, we have p∈Wγ′. This shows that Wγ⊆Wγ′for anyγ′∈S(R)
+which contains γ. ThusWγ⊆/intersectiontext
+γ⊆γ′∈S(R)Wγ′=W¯γ.
+(2) We shall prove γ⊆ΓWγ. This is enough to show (2). In fact, if ΓWγis a left exact
+radical functor containing γ, then by (1) it is the minimum among such functors, hence
+ΓWγ= ¯γ. Now we prove that
+(3) γ(M)⊆ΓWγ(M),
+for allM∈R-Mod.
+First of all, we note that γ(ER(R/p)) = 0 unless p∈Wγ. In fact, if γ(R/p) = 0,
+then applying Lemma 1.2(1) to R/p⊆ER(R/p) we have R/p∩γ(ER(R/p)) = 0. Since
+R/p⊆ER(R/p) is an essential extension, it follows that γ(ER(R/p)) = 0.
+Secondly, we prove the equation (3) in the case that Mis a finite direct sum of
+indecomposable injective R-modules/circleplustextn
+i=1ER(R/pi). In this case, by what we re-
+marked above, we have γ(M) =/circleplustext
+pi∈Wγγ(ER(R/pi)) and this is a submodule of/circleplustext
+pi∈WγER(R/pi) =ΓWγ(M). Thus the claim is true in this case.
+Thirdly, we consider the case that Mis a finitely generated R-module. Since the
+injective hull ER(M) ofMis a finite direct sum of indecomposable injective modules,
+we have already shown that γ(ER(M))⊆ΓWγ(ER(M)). Thus, using Lemma 1.2(1),
+we haveγ(M) =M∩γ(ER(M))⊆M∩ΓWγ(ER(M)) =ΓWγ(M).
+Finally, we show the claim (3) for an R-module Mwithout any assumption. We
+should notice that an element x∈Mbelongs to γ(M) if and only if the equality
+γ(Rx) =Rxholds. (See Lemma 1.2(1). Also see the proof of Proposition 2.6.) This
+equivalence is true for the left exact radical functor ΓWγas well. So x∈Mbelongs to
+ΓWγ(M) if and only if ΓWγ(Rx) =Rx. Since the claim (3) is true for finitely generated
+R-module Rx, we have γ(Rx)⊆ΓWγ(Rx). Therefore, we conclude that if x∈γ(M),
+thenx∈ΓWγ(M). Hence γ(M)⊆ΓWγ(M). /square18 YUJI YOSHINO, TAKESHI YOSHIZAWA
+Example 4.3. (1) Let ( R,m) be a local artinian ring with m/\e}atio\slash= 0. Then, since
+Spec(R) ={m}, there are only two subsets of Spec( R) which are closed under special-
+ization, namely∅and Spec( R). Therefore, by Corollary 3.2, we have S(R) ={1,0},
+where0denotes thezero functor. Wedefine afunctor γ:R-Mod→R-Modby γ(M) =
+mMfor allM∈R-Mod and γ(f) =f|mM:mM→mNfor allf∈HomR(M,N). It is
+clear that γis a non-zero functor and γ⊆1. Therefore it follows from the definition
+that ¯γ=1. However, since γ(R/m) = 0, we have Wγ=∅and hence ΓWγ=0. Thus
+¯γ/\e}atio\slash=ΓWγin this case. Note that γis not a left exact functor.
+(2) LetIbe an ideal of R. Then Hom R(R/I,−) is a left exact preradical functor.
+It follows that WHomR(R/I,−)is the set of prime ideals containing I, which is a closed
+subset of Spec( R) denoted by V(I). We denote ΓI=ΓV(I). Thus we obtain from
+Proposition 4.2 the equality
+HomR(R/I,−) =ΓI.
+We can show from Lemma 1.2(2) that the set S(R) admits multiplication.
+Lemma 4.4. Ifγ1,γ2∈S(R), thenγ1·γ2=γ2·γ1∈S(R).
+Proof.It is easy to see that if γ1andγ2are left exact preradical functor, then so is
+γ1·γ2. Ifγ1,γ2∈S(R), and ifIis an injective R-module, then, since γ2(I) is injective
+as well, we see that γ1·γ2(I) is also injective. Thus γ1·γ2∈S(R). The commutativity
+of multiplication follows from Lemma 1.2(2). /square
+We can also define the ‘quotient’ in S(R).
+Lemma 4.5. Letγ1,γ2∈S(R). Suppose that γ1⊆γ2. Then the set Sγ1,γ2={γ∈
+S(R)|γ·γ2=γ1}has a unique maximal element with respect to inclusion relat ion.
+Proof.FirstofallweshouldnoticefromLemma1.2(2)andfromtheassumpt ionγ1⊆γ2
+thatSγ1,γ2contains γ1, henceSγ1,γ2is non-empty. By virtue of Corollary 3.2, an
+elementγ∈S(R) belongs to Sγ1,γ2if and only if Wγ∩Wγ2=Wγ1. Therefore, setting
+W=/uniontext
+γ∈Sγ1,γ2Wγ, we can see that it satisfies W∩Wγ2=Wγ1. It is clear that Wis
+a unique maximal subset of Spec( R) which is closed under specialization and satisfies
+W∩Wγ2=Wγ1. ThusΓWis a unique maximal element in Sγ1,γ2. /square
+Definition 4.6. Forγ1,γ2∈S(R) withγ1⊆γ2, we denote by γ1/γ2the unique
+maximal element of Sγ1,γ2in Lemma 4.5 and call it the quotient of γ1byγ2.
+It is easy to verify that γ/1=γfor allγ∈S(R), and0/γ=0ifγ/\e}atio\slash=0∈S(R).
+(Note from the definition that 0/0=1.)
+By virtue of Theorem 3.4 we can also define the quotients for abstra ct local coho-
+mology functors in A(R).
+Definition 4.7. Letδ1,δ2be abstract local cohomology functors on D+(R-Mod) and
+assume that [ δ1]⊆[δ2] in the lattice structure of A(R). Then, by Theorem 3.4, there
+areγ1,γ2∈S(R) such that δi∼=Rγi(i= 1,2) andγ1⊆γ2inS(R). Under these
+circumstances we define the abstract local cohomology functor δ1/δ2to be the the
+right derived functor R(γ1/γ2) ofγ1/γ2∈S(R). We call δ1/δ2the quotient of δ1byδ2.ABSTRACT LOCAL COHOMOLOGY FUNCTORS 19
+5.Characterization of ΓIandΓI,J
+Weareconcerned withthe following two types of subsets inSpec( R) which areclosed
+under specialization, and their corresponding left exact radical fu nctors.
+Definition 5.1. (1) LetIbe an ideal of Rand setV(I) ={p∈Spec(R)|p⊇I}. It
+is known that V(I) is a closed subset of Spec( R) and conversely every closed subset
+is of this form. We set ΓI:=ΓV(I)the corresponding left exact radical functor, which
+we refer to as the section functor with the closed support defined I. We denote the
+right derived functor of ΓIbyRΓI, which we call the local cohomology functor with
+the closed support defined by I. See [3].
+(2) LetI,Jbe a pair of ideals of R. We set
+W(I,J) ={p∈Spec(R)|In⊆p+Jfor some n >0},
+which is closed under specialization. The corresponding left exact ra dical functor
+ΓW(I,J)is denoted by ΓI,J, which is called the section functor defined by the pair
+of ideals I,J. We also denote the right derived functor of ΓI,JbyRΓI,J, which we call
+the (generalized) local cohomology functor defined by the pair I,Jof ideals. See [16].
+Note that, since ΓI,ΓI,J∈S(R), the derived functors RΓIandRΓI,Jare abstract
+local cohomology functors.
+The aim of this section is to characterize ΓIandΓI,Jas elements of S(R), by which
+we will be able to characterize RΓIandRΓI,Jas elements of A(R).
+We start with the following observation.
+Lemma 5.2. LetW⊆Spec(R)be closed under specialization. We set Min(W)to be
+the set of prime ideals which are minimal among the primes in W, i.e.
+Min(W) ={p∈W|ifq⊆pforq∈W, thenq=p}.
+ThenW=/uniontext
+p∈Min(W)V(p). Furthermore Wis a closed subset of Spec(R)if and only
+ifMin(W)is a finite set.
+Proof.SinceWis closed under specialization, a prime ideal qbelongs to Wif and only
+ifqcontains a prime pin Min(W). This proves the equality W=/uniontext
+p∈Min(W)V(p).
+IfW=V(I) for an ideal IofR, then Min( W) is just a set of minimal prime ideals of
+I, which is known to be a finite set. Conversely, if Min( W) is a finite set{p1,...,pn},
+then we have W=V(p1)∪···∪V(pn) =V(p1∩···∩pn), which is a closed subset of
+Spec(R). /square
+Now we characterize ΓIas elements of S(R).
+Theorem 5.3. The following conditions are equivalent for γ∈S(R).
+(1)γ=ΓIfor an ideal IofR.
+(2)γsatisfies the ascending chain condition in the following sen se: If there is an
+ascending chain of left exact radical functors
+γ1⊆γ2⊆···⊆γn⊆···⊆γ
+with/uniontext
+nγn=γ, then there is an integer N >0such that γN=γN+1=···=γ.20 YUJI YOSHINO, TAKESHI YOSHIZAWA
+(3)If there is an ascending chain of preradical functor
+γ′
+1⊆γ′
+2⊆···⊆γ′
+n⊆···⊆γ
+with/uniontext
+nγ′n=γ, then there is an integer N >0such that γ′
+N=γ′
+N+1=···=γ.
+Proof.(1)⇒(2). Suppose that Γ=ΓI. Note from Corollary 3.2 that we have
+Wγ1⊆Wγ2⊆···⊆Wγn⊆···⊆V(I)
+such that/uniontext
+nWγn=V(I). Since Min( V(I)) is a finite set, we can take an enough
+large integer N >0 so that WγNcontains all such prime ideals in Min( V(I)). Then
+WγN=WγN+1=···=V(I), henceγN=γN+1=···=γ.
+(2)⇒(1). ByLemma5.2itissufficient toshowthatMin( Wγ)isafiniteset. Contrarily,
+assume that Min( Wγ) is an infinite set. Then, we can choose infinitely many distinct
+prime ideals p1,p2,...,pn,...in Min(Wγ). SetW′=/uniontext
+p∈Min(W)\{pi|i∈N}V(p),Wn=
+W′∪V(p1)∪V(p2)∪···∪V(pn) andγn=ΓWnfor eachn∈N. Note that Wγ=/uniontext
+nWn
+andWn/subsetnoteqlWn+1for eachn >0. Then there is an ascending chain of left exact radical
+functors
+γ1⊆γ2⊆···⊆γn⊆···⊆γ
+with/uniontext
+nγn=γ. From the condition (2), there is an integer N >0 such that γN=
+γN+1=···=γ. Thus we have WN=WN+1=···=Wγ. But this is a contradiction.
+Therefore Min( Wγ) is a finite set.
+(2)⇒(3). Note that if γ′
+1andγ′
+2are preradical functors and if γ′
+1⊆γ′
+2, then¯γ′
+1⊆¯γ′
+2.
+Suppose that there is an ascending chain of preradical functor;
+γ′
+1⊆γ′
+2⊆···⊆γ′
+n⊆···⊆γ
+with/uniontext
+nγ′
+n=γ. Then we have an ascending chain of left exact radical functors
+γ′
+1⊆γ′
+2⊆···⊆γ′
+n⊆···⊆γ=γ.
+Sinceγ′
+n⊆γ′nfor eachn, we have/uniontext
+nγ′
+n⊆/uniontext
+nγ′n. Henceγ=/uniontext
+nγ′n⊆/uniontext
+nγ′n⊆γ=γ.
+Here we should notice that, since/uniontext
+nγ′
+n∈S(R), we have γ=/uniontext
+nγ′
+n=/uniontext
+nγ′
+n. Then,
+from the condition (2), there is an integer N >0 such that γ′
+N=γ′
+N+1=···=γ.
+The implication (3) ⇒(2) is clear. /square
+By virtue of Theorem 3.4 we can state the same theorem in terms of a bstract local
+cohomology functors.
+Theorem 5.4. The following conditions are equivalent for an abstract loc al cohomology
+functorδonD+(R-Mod).
+(1)δ∼=RΓIfor an ideal IofR.
+(2)δsatisfies the ascending chain condition in the following sen se: If there is an
+ascending chain in A(R)
+[δ1]⊆[δ2]⊆···⊆[δn]⊆···⊆[δ]
+with/uniontext
+n[δn] = [δ], then there is an integer N >0such that [δN] = [δN+1] =
+···= [δ].ABSTRACT LOCAL COHOMOLOGY FUNCTORS 21
+To characterize the functor ΓI,Jfor pairs of ideals IandJ, we prepare the following
+lemma.
+Lemma 5.5. LetIandJbe ideals of R. Then, W(I,J)is the largest specialization
+closed subset WofSpec(R)which satisfies W∩V(J) =V(I+J).
+Proof.Setting
+W′=/uniondisplay
+{W⊆Spec(R)|Wis specialization closed and W∩V(J) =V(I+J)},
+we can see that W′is also closed under specialization, and clearly it is the largest
+among such subsets. To prove the lemma we show W′=W(I,J).
+Ifp∈W′, then there is a specialization closed subset Wcontaining psuch that
+W∩V(J) =V(I+J). Then, since V(p)⊆W, we see V(p+J) =V(p)∩V(J)⊆
+V(I+J), hence ( I+J)n⊆p+Jfor a large integer n. In particular, In⊆p+Jand
+thusp∈W(I,J).
+Conversely, if p∈W(I,J), then it can be seen that V(p)∩V(J)⊆V(I+J).
+Therefore a closed subset W=V(p)∪V(I) satisfies W∩V(J) =V(I+J), hence
+p∈W⊆W′. /square
+Theorem 5.6. The following conditions are equivalent for γ∈S(R).
+(1)γ=ΓI,Jfor a pair of ideals I,JofR.
+(2)γ=γ1/γ2for left exact radical functors γ1⊆γ2, the both of which satisfy the
+ascending chain condition in Theorem 5.3.
+Proof.(1)⇒(2) : Assume that γ=ΓI,J, and set γ1=ΓI+J,γ2=ΓJ. SinceW(I,J)∩
+V(J) =V(I+J), it follows from Corollary 3.2 that γ·γ2=γ1, hence that γ∈Sγ1,γ2,
+where we use the notation in Lemma 4.5. Then Lemma 5.5 forces that γ=ΓI,Jis the
+maximal element of Sγ1,γ2, thus we have γ=γ1/γ2.
+(2)⇒(1) : Suppose γ=γ1/γ2whereγ1⊆γ2satisfy the ascending chain condition.
+By virtue of Theorem 5.3, we may write γ1=ΓIandγ2=ΓJfor some ideals Iand
+J. Note that, since γ1⊆γ2, we must have V(I)⊆V(J). Thus W(I,J)∩V(J) =
+V(I+J) =V(I) holds, and hence ΓI,Jis an element of Sγ1,γ2. It then follows from
+the definition of quotients that ΓI,J⊆γ. On the other hand, since γ∈Sγ1,γ2, we have
+γ·γ2=γ1, henceWγ∩V(J) =V(I). Then we see from Lemma 5.5 that Wγ⊆W(I,J).
+Thusγ⊆ΓI,J. /square
+We can state the same theorem in terms of A(R).
+Theorem 5.7. The following conditions are equivalent for an abstract loc al cohomology
+functorδ.
+(1)δ∼=RΓI,Jfor a pair of ideals I,JofR.
+(2)δ∼=δ1/δ2for abstract local cohomology functors δ1⊆δ2, the both of which
+satisfy the ascending chain condition in Theorem 5.4.
+References
+[1]A. A. Beilinson ,J. Bernstein andP. Deligne , Faisceaux Pervers, Ast´ erisque 100, 1982.22 YUJI YOSHINO, TAKESHI YOSHIZAWA
+[2]M. H. Bijan-Zadeh , Torsion theories and local cohomology over commutative and Noet herian
+rings, J. London Math. Soc. 19, 1979, no. 3, 402–410.
+[3]M. P. Brodmann andR. Y. Sharp ,Local cohomology: an algebraic introduction with geometri c
+applications , Cambridge University Press, 1998.
+[4]W. Bruns andJ. Herzog ,Cohen-Macaulay rings, revised version , Cambridge University Press,
+1998.
+[5]P. Cahen , Torsion theory and associated primes, Proc. Amer. Math. Soc. 38, 1973, 471–476.
+[6]O. Goldman , Rings and modules of quotients. J. Algebra 13, 1969, 10–47.
+[7]A. Grothendieck ,Local cohomology , Lecture Notes in Mathematics 41, Springer, 1967.
+[8]R. Hartshorne ,Residues and duality , Lecture Notes in Mathematics 20, Springer, 1966.
+[9]M. J. Hopkins ,Global methods in homotopy theory, In Homotopy theory (Durh am, 1985) ,
+London Math. Soc. Lecture Note Ser. 117, Cambridge University Press, 1987, 73–96.
+[10]L. Lambek ,Torsion theories, additive semantics, and rings of quotien ts, Lecture Notes in Math-
+ematics177, Springer, 1971.
+[11]J. M. Maranda Injective structures, Trans. Amer. Math. Soc. 110, 1964, 98–135.
+[12]H. Matsumura ,Commutative ring theory , Cambridge University Press, 1986.
+[13]J. Miyachi , Localization of Triangulated Categories and Derived Categories, J . Algebra, 141,
+1991, 463–483.
+[14]B. Stenstr ¨om,Rings and Modules of Quotients , Lecture Notes in Mathematics 237, Springer,
+1971.
+[15] —,Rings of Quotients , Springer, 1975.
+[16]R. Takahashi, Y.Yoshino andT.Yoshizawa , Local cohomologybased on anonclosed support
+defined by a pair of ideals, J. Pure Appl. Algebra 213, 2009, no. 4, 582–600.
+Yuji Yoshino
+Graduate School of Natural Science and Technology,
+Okayama University, Okayama 700-8530, Japan
+E-mail address :yoshino@math.okayama-u.ac.jp
+Takeshi Yoshizawa
+Graduate School of Natural Science and Technology,
+Okayama University, Okayama 700-8530, Japan
+E-mail address :tyoshiza@math.okayama-u.ac.jp
\ No newline at end of file
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+arXiv:1001.0863v1  [math.ST]  6 Jan 2010Correction to: “Blind maximum likelihood
+separation of a linear-quadratic mixture”
+Shahram Hosseini, Yannick Deville
+Laboratoire d’Astrophysique de Toulouse-Tarbes, Univers it´ e de Toulouse, CNRS ,
+14 avenue Edouard Belin - 31400 Toulouse - France.
+shosseini@ast.obs-mip.fr , ydeville@ast.obs-mip.fr
+Abstract
+An error occurred in the computation of a gradient in [1]. The equa-
+tions (20) in Appendix and (17) in the text were not correct. T he current
+paper presents the correct version of these equations.
+I Summary of [1]
+In [1] (see Appendix for an authors’ version of this article), we pro posed a
+maximum likelihood approach for blindly separating a linear-quadratic m ixture
+defined by (Eq. (2) in [1]):
+x1=s1−l1s2−q1s1s2 x2=s2−l2s1−q2s1s2(I.1)
+wheres1ands2are two independent sources. The log-likelihood for Nsamples
+of the mixed signals x1andx2reads (Eq. (12) in [1]):
+L=Et[logfS1(s1(t))]+Et[logfS2(s2(t))]−Et[log|J(s1(t),s2(t))|] (I.2)
+whereEt[.] represents the time average operator on the Nsamples,fs1(.) and
+fs2(.) are the probability density functions (pdf) of the sources s1ands2and
+Jis the Jacobian of the mixture which reads (Eq. (4) in [1])
+J= 1−l1l2−(q2+l2q1)s1−(q1+l1q2)s2. (I.3)
+Maximizing the log-likelihood requires that its gradient with respect to the pa-
+rametervector w= [l1,l2,q1,q2],i.e.∂L
+∂w, vanishes. Defining the score functions
+of the two sources as (Eq. (13) in [1])
+ψi(u) =−∂logfSi(u)
+∂ui= 1,2
+we can write (Eq. (14) in [1])
+∂L
+∂w=−Et[ψ1(s1)∂s1
+∂w]−Et[ψ2(s2)∂s2
+∂w]−Et[1
+J∂J
+∂w] (I.4)
+1Rewriting (I.1) in the vector form x=f(s,w) and considering was the inde-
+pendent variable and sas the dependent variable, we can write, using implicit
+differentiation (Eq. (15) in [1])
+0=∂f
+∂s∂s
+∂w+∂f
+∂w(I.5)
+which yields (Eq. (16) in [1])
+∂s
+∂w=−(∂f
+∂s)−1∂f
+∂w(I.6)
+Note that∂f
+∂sis the Jacobian matrix of the mixing model. Considering (I.1), we
+can write (Appendix in [1])
+∂f
+∂s=/parenleftbigg1−q1s2−l1−q1s1
+−l2−q2s21−q2s1/parenrightbigg
+and∂f
+∂w=/parenleftbigg−s20−s1s20
+0−s10−s1s2/parenrightbigg
+,
+which implies, from (I.6)
+∂s
+∂w=−1
+J/parenleftbigg1−q2s1l1+q1s1
+l2+q2s21−q1s2/parenrightbigg
+./parenleftbigg−s20−s1s20
+0−s10−s1s2/parenrightbigg
+(I.7)
+and yields (Eq. (19) in [1])
+∂s1
+∂w=1
+J[(1−q2s1)s2,(l1+q1s1)s1,(1−q2s1)s1s2,(l1+q1s1)s1s2]
+∂s2
+∂w=1
+J[(l2+q2s2)s2,(1−q1s2)s1,(l2+q2s2)s1s2,(1−q1s2)s1s2](I.8)
+Using (I.8), we obtain the first two terms of the gradient (I.4). To o btain the
+third term, we need to compute∂J
+∂w. This partial derivative was computed
+inaccurately in [1] so that Equations (20), and thus (17), in [1] are erroneous.
+II Correct versions of Equations (20) and (17)
+in [1]
+In [1] we did not consider the implicit dependence of s1ands2onwand
+computed the derivative of Jwith respect to wignoring this dependence. Con-
+sideringJ=g(w,s(w)), the correct equation for∂J
+∂wreads
+∂J
+∂w=∂J
+∂w|scte+∂J
+∂s∂s
+∂w(II.1)
+Equation (20) in [1] only corresponded to the first term on the right side of the
+above relation which reads, following (I.3), as:
+∂J
+∂w|scte=−[l2+q2s2,l1+q1s1,l2s1+s2,s1+l1s2] (II.2)
+2We now compute the gradient (II.1) entirely. Considering (I.3), we c an write
+∂J
+∂s=−[q2+l2q1,q1+l1q2] (II.3)
+Using(II.1), (II.2), (II.3)and(I.7)wefinallyobtainthefollowingequ ationwhich
+must replace the equation (20) in [1]
+∂J
+∂w= [−(l2+q2s2)−(q2+l2q1)(1−q2s1)s2/J−(q1+l1q2)(l2+q2s2)s2/J,
+−(l1+q1s1)−(q1+l1q2)(1−q1s2)s1/J−(q2+l2q1)(l1+q1s1)s1/J,
+−(l2s1+s2)−(q2+l2q1)(1−q2s1)s1s2/J−(q1+l1q2)(l2+q2s2)s1s2/J,
+−(l1s2+s1)−(q1+l1q2)(1−q1s2)s1s2/J−(q2+l2q1)(l1+q1s1)s1s2/J]
+(II.4)
+Inserting (I.8) and (II.4) in (I.4), we obtain the following expression for the
+gradient which must replace Equation (17) in [1]
+∂L
+∂w=−Et[(ψ1(s1)(1−q2s1)s2+ψ2(s2)(l2+q2s2)s2
+−(l2+q2s2)−(q2+l2q1)(1−q2s1)s2/J−(q1+l1q2)(l2+q2s2)s2/J)/J,
+(ψ1(s1)(l1+q1s1)s1+ψ2(s2)(1−q1s2)s1
+−(l1+q1s1)−(q1+l1q2)(1−q1s2)s1/J−(q2+l2q1)(l1+q1s1)s1/J)/J,
+(ψ1(s1)(1−q2s1)s1s2+ψ2(s2)(l2+q2s2)s1s2
+−(l2s1+s2)−(q2+l2q1)(1−q2s1)s1s2/J−(q1+l1q2)(l2+q2s2)s1s2/J)/J,
+(ψ1(s1)(l1+q1s1)s1s2+ψ2(s2)(1−q1s2)s1s2
+−(l1s2+s1)−(q1+l1q2)(1−q1s2)s1s2/J−(q2+l2q1)(l1+q1s1)s1s2/J))/J]
+(II.5)
+References
+[1] S. Hosseini, Y. Deville, Blind maximum likelihood separation of a linear-
+quadratic mixture, Proceedings of the Fifth International Confe rence on In-
+dependentComponentAnalysisandBlindSignalSeparation(ICA200 4),pp.
+694-701, ISSN 0302-9743, ISBN 3-540-23056-4, Springer-Ver lag, vol. LNCS
+3195, Granada, Spain, Sept. 22-24, 2004.
+3APPENDIX: Authors’ version of [1]
+1 Introduction
+It is well known that the independence hypothesis is not sufficient fo r separat-
+ing general nonlinear mixtures because of the very large indetermin acies which
+make the nonlinear BSS problem ill-posed. A natural idea for reducing the in-
+determinacies is to constrain the structure of mixing and separatin g models to
+belong to a certain set of transformations. This supplementary co nstraint can
+be viewed as a regularization of the initially ill-posed problem.
+In this paper, we study a linear-quadratic mixture model which may b e
+considered as the simplest (nonlinear) version of a general polynom ial model.
+Our main aim is to develop an approach which can be easily extended to h igher-
+order polynomial models. Hence, we propose a recurrent separat ing structure
+whose realization does not require the knowledge of the explicit form of the
+inverse of the mixing model. We develop a rigorous method to identify t he
+parameters of the separating structure in a maximum likelihood fram ework.
+The algorithm is developed so that the inverse of the mixing structur e is not
+required to be known. Thus, it can be extended to more general po lynomial
+mixtures.
+2 mixing and separating models
+Supposeu1andu2are two independent random signals. Given the following
+nonlinear instantaneous mixture model
+xi=ai1u1+ai2u2+biu1u2i= 1,2 (1)
+we would like to estimate u1andu2up to a permutation and a scaling factor
+(and possibly an additive constant). For simplicity, let’s denote s1=a11u1and
+s2=a22u2.s1ands2will be referred to as the sourcesin the following. (1) can
+be rewritten as
+x1=s1−l1s2−q1s1s2
+x2=s2−l2s1−q2s1s2 (2)
+in whichl1=−a12/a22andl2=−a21/a11represent the linear contributions of
+the sources in the mixture, and q1=−b1/(a11a22) andq2=−b2/(a11a22) repre-
+sent the quadratic contributions. The negative signs are chosen f or simplifying
+the notations of the separating structure.
+Solving the model (2) for s1ands2leads to the following two pairs of solu-
+tions, which may be considered as two direct separating structure s:
+(∫1,∫2)1= ((−b1+/radicalbig
+∆1)/2a1,(−b2+/radicalbig
+∆2)/2a2)
+(∫1,∫2)2= ((−b1−/radicalbig
+∆1)/2a1,(−b2−/radicalbig
+∆2)/2a2) (3)
+4−0.5 0 0.5−0.500.5(a)
+s1s2
+−1 −0.5 0 0.5 1−0.6−0.4−0.200.20.40.60.8(b)
+x1x2
+−1.6−1.4−1.2 −1−0.8−0.60.511.522.5(c)
+y1y2
+−1 −0.5 0 0.5 1−0.8−0.6−0.4−0.200.20.40.6(d)
+y1y2
+Figure 1: Case when J >0 for all the source values. Distribution of (a) sources,
+(b) mixtures, (c) output of the first direct separating structur e, (d) output of
+the second direct separating structure.
+where ∆ i=b2
+i−4aici,a1=q2+l2q1,a2=q1+l1q2,b1=q1x2−q2x1+l1l2−1,
+b2=q2x1−q1x2+l1l2−1,c1=x1+l1x2andc2=x2+l2x1. It can be easily
+verified that ∆ 1= ∆2=J2, whereJis the Jacobian of the mixing model (2)
+and reads
+J= 1−l1l2−(q2+l2q1)s1−(q1+l1q2)s2 (4)
+According to the variation domain of the two sources, three differe nt cases may
+be considered:
+1)J <0 for all the values of s1ands2. In this case (3) becomes:
+(∫1,∫2)1= (s1,s2) (5)
+(∫1,∫2)2= (−q1+l1q2
+q2+l2q1s2−l1l2−1
+q2+l2q1,−q2+l2q1
+q1+l1q2s1−l1l2−1
+q1+l1q2) (6)
+Thus, the first direct separating structure in (3) leads to the act ual sources and
+the second direct separating structure leads to another solution , equivalent to
+the first one up to a permutation, a scaling factor, and an additive c onstant.
+2)J >0 for all the values of s1ands2. In this case, the first structure
+leads to the permuting solution, defined by (6), and the second str ucture to the
+actual sources ( s1,s2). An example is shown in Fig. 1 for the numerical values
+l1=−0.2,l2= 0.2,q1=−0.8,q2= 0.8 andsi∈[−0.5,0.5].
+3)J >0 for some values of the sources and J <0 for the other values. In
+this case, each structure leads to the non-permuted sources (5 ) for some values
+of the observations and to the permuted sources (6) for the oth er values. An
+example is shown in Fig. 2 (with the same coefficients as in the second ca se,
+but forsi∈[−2,2]). The permutation effect is clearly visible in the figure. One
+may also remark that the straight line J= 0 in the source plane is mapped to
+a conic section in the observation plane (shown by asterisks).
+Thus, it is clear that the direct structures may be used for separa ting the
+sourcesifthe Jacobianofthe mixingmodel isalwaysnegativeoralway spositive,
+5−2 −1 0 1 2−3−2−1012(a)
+s1s2
+−5 0 5 10−6−4−20246(b)
+x1x2
+−3 −2 −1 0 1−2−1012345(c)
+y1y2
+−4 −2 0 2 4−3−2−1012(d)
+y1y2
+Figure 2: Case when J >0 for some values of the sources and J <0 for the
+other values. Distribution of (a) sources, (b) mixtures, (c) outp ut of the first
+direct separatingstructure, (d) output ofthe second direct se parating structure.
+i.e.forallthesourcevalues. Otherwise, althoughthesourcesarese paratedsam-
+ple by sample , each retrieved signal contains samples of the two sources. This
+problem arises because the mixing model (2) is not bijective. This the oretically
+insoluble problem should not discourage us. In fact, our final objec tive is to ex-
+tend the idea developed in the current study to more general polyn omial models
+which will be used to approximate the nonlinear mixtures encountere d in the
+real world. If these real-world nonlinear models are bijective, we ca n logically
+suppose that the coefficients of their polynomial approximations ta ke values
+which make them bijective on the variation domains of the sources. T hus, in
+the following, we suppose that the sources and the mixture coefficie nts have nu-
+merical values ensuring that the Jacobian Jof the mixing model has a constant
+sign.
+The natural idea to separate the sources is to form a direct separ ating struc-
+ture using any of the equations in (3), and to identify the paramete rsl1,l2,
+q1andq2by optimizing an independence measuring criterion. Although this
+approach may be used for our special mixing model (2), as soon as a more
+complicated polynomial model is considered, the solutions ( ∫1,∫2) can no longer
+be determined so that the generalization of the method to arbitrar y polyno-
+mial models seems impossible. To avoid this limitation, we propose a recu rrent
+structure shown in Fig. 3. Note that, for q1=q2= 0, this structure is reduced
+to the basic H´ erault-Jutten network. It may be checked easily th at, for fixed
+observations defined by (2), y1=s1andy2=s2corresponds to a steady state
+for the structure in Figure 3.
+The use ofthis recurrent structure is more promising because it ca n be easily
+generalized to arbitrary polynomial models. However, the main prob lem with
+this structure is its stability. In fact, even if the mixing model coeffic ients are
+exactlyknown, the computation ofthe structureoutputs requir esthe realization
+6/G71/G32/G6C/G31
+/G6C/G32
+/G78/G32/G79/G32/G71/G31/G79/G31/G78/G31
+/G2B
+/G2B/G58
+Figure 3: Recurrent separating structure.
+of the following recurrent iterative model
+y1(n+1) =x1+l1y2(n)+q1y1(n)y2(n)
+y2(n+1) =x2+l2y1(n)+q2y1(n)y2(n) (7)
+where a loop on nis performed for each couple of observations ( x1,x2) until
+convergence is achieved.
+It can be shown that this model is locally stable at the separating poin t
+(y1,y2) = (s1,s2), if and only if the absolute values of the two eigenvalues of
+the Jacobian matrix of (7) are smaller than one. In the following, we s uppose
+that this condition is satisfied.
+3 Maximum likelihood estimation of the model
+parameters
+LetfS1,S2(s1,s2) be the joint pdf of the sources, and assume that the mixing
+model is bijective so that the Jacobian of the mixing model has a cons tant sign
+on the variation domain of the sources. The joint pdf of the observ ations can
+be written as
+fX1,X2(x1,x2) =fS1,S2(s1,s2)
+|J(s1,s2)|(8)
+Taking the logarithm of (8), and considering the independence of th e sources,
+we can write:
+logfX1,X2(x1,x2) = logfS1(s1)+logfS2(s2)−log|J(s1,s2)|(9)
+Given N samples of the mixtures X1andX2, we want to find the maximum
+likelihood estimator for the mixture parameters w= [l1,l2,q1,q2]. This estima-
+tor is obtained by maximizing the joint pdf of all the observations (su pposing
+that the parameters in ware constant), which is equal to
+E=fX1,X2(x1(1),x2(1),···,x1(N),x2(N)) (10)
+Ifs1(t) ands2(t) are two i.i.d. sequences, x1(t) andx2(t) are also i.i.d. so that
+E=/producttextN
+i=1fX1,X2(x1(i),x2(i)) and logE=/summationtextN
+i=1logfX1,X2(x1(i),x2(i)). The
+7cost function to be maximized can be defined as L=1
+NlogE, which will be
+denoted using the temporal averaging operator Et[.] as
+L=Et[logfX1,X2(x1(t),x2(t))] (11)
+Using (9):
+L=Et[logfS1(s1(t))]+Et[logfS2(s2(t))]−Et[log|J(s1(t),s2(t))|] (12)
+Maximizing this cost function requires that its gradient with respect to the
+parameter vector w,i.e.∂L
+∂w, vanishes. Defining the score functions of the two
+sources as
+ψi(u) =−∂logfSi(u)
+∂ui= 1,2 (13)
+and considering that∂log|J|
+∂w=1
+J∂J
+∂w, we can write
+∂L
+∂w=−Et[ψ1(s1)∂s1
+∂w]−Et[ψ2(s2)∂s2
+∂w]−Et[1
+J∂J
+∂w] (14)
+Rewriting (2) in the vector form x=f(s,w) and considering was the inde-
+pendent variable and sas the dependent variable, we can write, using implicit
+differentiation
+0=∂f
+∂s∂s
+∂w+∂f
+∂w(15)
+which yields
+∂s
+∂w=−(∂f
+∂s)−1∂f
+∂w(16)
+Note that∂f
+∂sis the Jacobian matrix of the mixing model. Using (14) and (16),
+the gradient of the cost function Lwith respect to the parameter vector wis
+equal to (see the appendix for the computation details)
+∂L
+∂w=−Et[(ψ1(s1)(1−q2s1)s2+ψ2(s2)(l2+q2s2)s2−(l2+q2s2))/J,
+(ψ1(s1)(l1+q1s1)s1+ψ2(s2)(1−q1s2)s1−(l1+q1s1))/J,
+(ψ1(s1)(1−q2s1)s1s2+ψ2(s2)(l2+q2s2)s1s2−(l2s1+s2))/J,
+(ψ1(s1)(l1+q1s1)s1s2+ψ2(s2)(1−q1s2)s1s2−(s1+l1s2))/J](17)
+In practice, the actual sources and their density functions are u nknown and will
+be replaced by the reconstructed sources, i.e.by the outputs of the separating
+structure of Fig 3, yi, in an iterative algorithm. The score functions of the
+reconstructedsourcescanbe estimated byanyofthe existingpa rametricornon-
+parametric methods. In our work, we used a kernel estimator bas ed on third-
+order cardinal splines. Using (17), the cost function (12) can be m aximized by a
+gradient ascent algorithm which updates the parameters by the ru lew(n+1) =
+w(n)+µ∂L
+∂w. The learning rate parameter µmust be chosen carefully to avoid
+the divergence of the algorithm. Note that the algorithm does not r equire
+the knowledge of the explicit inverse of the mixing model (direct sepa rating
+structures (3)). Hence, it can be easily extended to more genera l polynomial
+mixing models.
+8Appendix: details of gradient computation
+Considering (2), we can write
+∂f
+∂s=/parenleftbigg
+1−q1s2−l1−q1s1
+−l2−q2s21−q2s1/parenrightbigg
+and∂f
+∂w=/parenleftbigg
+−s20−s1s20
+0−s10−s1s2/parenrightbigg
+,
+which implies, from (16)
+∂s
+∂w=−1
+J/parenleftbigg1−q2s1l1+q1s1
+l2+q2s21−q1s2/parenrightbigg
+./parenleftbigg−s20−s1s20
+0−s10−s1s2/parenrightbigg
+which yields
+∂s1
+∂w=1
+J[(1−q2s1)s2,(l1+q1s1)s1,(1−q2s1)s1s2,(l1+q1s1)s1s2]
+∂s2
+∂w=1
+J[(l2+q2s2)s2,(1−q1s2)s1,(l2+q2s2)s1s2,(1−q1s2)s1s2](18)
+Considering (4)
+∂J
+∂w=−[l2+q2s2,l1+q1s1,l2s1+s2,s1+l1s2] (19)
+(17) follows directly from (14), (18) and (19).
+9
\ No newline at end of file
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+++ b/1001.0864.txt
@@ -0,0 +1,2226 @@
+arXiv:1001.0864v2  [gr-qc]  7 Jul 2010Desy Preprint 10-002 - ZMP-HH/10-1 - January 2010
+On the initial conditions and solutions of the semi-
+classical Einstein equations in a cosmological scenario.
+Nicola Pinamonti
+II. Institut f¨ ur Theoretische Physik, Universit¨ at Hamburg, Lu ruper Chaussee 149, D-22761 Hamburg,
+Germany.
+Dipartimento di Matematica, Universit` a di Roma “Tor Vergata”, V ia della Ricerca Scientifica, I-00133
+Roma, Italy.
+nicola.pinamonti@desy.de
+Abstract . In this paper we shall discuss the backreaction of a massive quant um scalar field on the
+curvature, the latter treated as a classical field. Furthermore, we shall deal with this problem in the
+realm of cosmological spacetimes by analyzing the Einstein equations in a semiclassical fashion. More
+precisely, we shall show that, at least on small intervals of time, solu tions for this interacting system exist.
+This result will be achieved providing an iteration scheme and showing t hat the series, obtained starting
+from the massless solution, converges in the appropriate Banach s pace. The quantum states with good
+ultraviolet behavior (Hadamard property), used in order to obtain the backreaction, will be completely
+determined by their form on the initial surface if chosen to be lightlike . Furthermore, on small intervals
+of time, they do not influence the behavior of the exact solution. On large intervals of time the situation
+is more complicated but, if the spacetime is expanding, we shall show t hat the end-point of the evolution
+does not depend strongly on the quantum state, because, in this lim it, the expectation values of the
+matter fields responsible for the backreaction do not depend on th e particular homogeneous Hadamard
+state at all. Finally, we shall comment on the interpretation of the se miclassical Einstein equations for
+this kind of problems. Although the fluctuations of the expectation values of pointlike fields diverge, if
+the spacetime and the quantum state have a large spatial symmetr y and if we consider the smeared fields
+on regions of large spatial volume, they tend to vanish. Assuming th is point of view the semiclassical
+Einstein equations become more reliable.
+Contents
+1 Introduction 2
+2 Geometry of the spacetime, classical and quantum matter 5
+2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
+2.2 Scalar fields, quantization and the stress tensor . . . . . . . . . . . . . . . . . . . 7
+13 Homogeneous states and Hadamard property in flat FRW 9
+3.1 Characterization of pure, homogeneous states in terms o f solutions of the wave
+equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
+4 Semiclassical treatment of backreaction and existence of solutions 14
+4.1 Semiclassical approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
+4.2 Initial conditions at the beginning of the universe . . . . . . . . . . . . . . . . . . 15
+4.3 Expectation value of φ2on the states ω1,0, and further renormalization freedom. 18
+4.4 Recursiveconstructionsandboundssatisfiedby χkandbytheauxiliaryfunctionalΦ 20
+4.5 Existence of exact solutions of the semiclassical Einst ein equations out of initial
+data given at the beginning of the universe . . . . . . . . . . . . . . . . . . . . . 23
+5 Further considerations on the semiclassical solutions 27
+5.1 Expectation value of /a\}b∇acketle{tφ2/a\}b∇acket∇i}htat late times . . . . . . . . . . . . . . . . . . . . . . . . 27
+5.2 Variance of /a\}b∇acketle{tT/a\}b∇acket∇i}htwhen smeared on large volume regions . . . . . . . . . . . . . . . 29
+6 Summary of the results and final comments 33
+A Appendix 33
+A.1 Deformation quantization and the extended algebra of fie lds . . . . . . . . . . . . 33
+A.2 Functional derivatives of aandτ. . . . . . . . . . . . . . . . . . . . . . . . . . . 35
+A.3 Bounds satisfied by χkand its functional derivatives . . . . . . . . . . . . . . . . 36
+A.4 Proof of Proposition 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
+1 Introduction
+Despite the presence of many attempts, no widely accepted de scription of the quantization
+procedure necessary to treat quantum gravity is, up to now, a vailable; however, nowadays we
+know how such a theory should look like, at least in some regim es. For example, treating fields
+as quantum object over a fixed curved classical background, m any interesting effects have been
+discovered, like for example Hawking radiation in the case o f black hole physics [Ha75, FH90]
+or particle creation in the contest of cosmological spaceti me [Pa68, Pa69]. Another interesting
+regime, where some progress could bemade towards an eventua l quantum gravity, is the analysis
+of the backreaction of matter quantum fields on curvature. In the present paper we would like
+to analyze some aspects of this problem in the context of cosm ological spactimes.
+In this respect the recent developments in the theory of quan tum fields on curved spacetimes
+provided by the algebraic approach to quantization [Ha92], have been really fruitful. Here we
+think for exampleat the work [BFV03] whereit has beenshown, that thequantization is nothing
+but a functor that associates to every spacetime its corresp onding algebra of fields/observables.
+It is thus possible to quantize simultaneously on every spac etime and, as a byproduct, the
+renormalization ambiguities, present in the definition of p ointwise product of fields, acquire a
+clearer interpretation [HW01]. In order to use similar idea s, a good control on the form of
+2the quantum states has to be available and, in this respect, a n important contribution is the
+one of Radzikowksi [Ra96] (see also [BFK95]). In that paper h e has shown that, considering
+a free quantum theory and employing techniques proper of the H¨ ormander microlocal analysis
+[H¨ o89], it is sufficient to prescribe the form of the wave fron t set of the two-point function of
+a state in order to fix the form of its singularity (Hadamard fo rm). Using similar ideas and
+similar methods it is also possible to show that many of the fe atures of quantum fields on flat
+spacetime can be translated in a conceptual clear way to the c urved case. More precisely, the
+Wick polynomial, their time ordered products and the stress tensor have been addressed in
+[HW01, HW02, HW05, Mo03], whereas the study of the perturbat ive construction of interacting
+field theories can be found in [BF00, HW03, BF09, BDF09].
+Coming back to the analysis of the backreaction of quantum ma tter on classical curvature,
+a milestone is provided by the seminal paper of Wald [Wa77]. I n that work, he has given a set
+of five axioms that a renormalization scheme should satisfy i n order to produce a stress tensor
+whose expectation values can be safely used in the semiclass ical Einstein equations
+Gµν= 8π/a\}b∇acketle{tTµν/a\}b∇acket∇i}ht (1)
+in such a way that a physically reasonable backreaction emer ges. Nowadays, the first four
+requirements on theexpectation values of the stress tensor (consistency, covariance, conservation
+of/a\}b∇acketle{tTµν/a\}b∇acket∇i}htand standard flat spacetime limit) are automatically obtain ed in every locally covariant
+theory constructed as in [BFV03] (see also the nice lectures [BF09]). However, the fifth one,
+which requires that the expectation values of the stress ten sor depend on the derivatives of the
+coefficients of the metric up to the second order is more proble matic. Nevertheless, even if it
+cannot be satisfied in general, it is possible to fix part of the renormalization freedom in order
+to have that at least the expectation values of the trace /a\}b∇acketle{tT/a\}b∇acket∇i}ht:=/a\}b∇acketle{tTµµ/a\}b∇acket∇i}htmeet this requiremeant
+[Wa78b].
+Making use of these methods, in the case of cosmological spac etimes, the backreaction of a
+massless conformally coupled scalar field has been studied b y Wald in [Wa78a] and in this case
+an exact solution can be found. Later on, another important c ontribution in this direction is the
+oneduetoStarobinsky[St80](seealso [Vi85]), whereanaly zingthetrace ofthestresstensor, and
+its anomalous term [Wa78b], the backreaction on curvature w as computed. Following similar
+ideas the analysis of the backreaction of massive scalar fiel ds has been analyzed in [SS02] and
+more recently in [DFP08]. Unfortunately while in the case of massless conformally coupled
+fields the backreaction can be analytically computed, essen tially because the quantum state
+does not influence the dynamics of the spactime, when the field s have a mass, the situation
+is more complicated and the details of the quantum state have to be taken into account. A
+nice study of the semiclassical problem, also in terms of num erical integration, can be found
+in the series of works of Anderson [An85, An86]. The analysis of the problem of definition of
+the expectation values of the stress tensor in relation with the Casimir effect is presented in
+[GNW09]. The semiclassical Einstein equations have also be en employed by Flanagan and Wald
+in [FW96] in order to study the validity of the averaged null e nergy condition on semiclassical
+solutions. Other interesting works at the interface of clas sical gravity and quantum matter
+are [GM86, RV99, PS93, PR99, AE00, Sh08]. Recently the dynam ical system associated with
+3the semiclassical backreaction has been studied in [EG10], where similar equations, as the one
+studied in the present paper, have been derived also for more general kinds of matter fields (with
+different curvature coupling).
+In this paper we shall follow the discussion presented in [DF P08] where some approximated
+solutions of the semiclassical Einstein equations have bee n displayed. In the latter work the
+matter has been described by a free scalar field whose couplin g constant to the curvature is
+equal to 1/6 and with a non vanishing mass m. Notice that, despite the simplicity of the treated
+model, the solutions presented in that paper have very nice p hysical properties. On the one
+hand they show, at late times, a phase of exponential expansi on (de Sitter), whose acceleration
+corresponds to a renormalization freedom of such a model, wh ile on the other hand, at the
+beginning of the model a phase typical of the power law inflati onary scenario emerges. In a
+certain sense that model provides an effective cosmological c onstant that varies in time and that
+could be used in order to model the present acceleration of ou r universe.
+Here, we shall show that exact solutions of that model exist a nd that many of the previously
+discussed physical properties shown by the approximated so lutions persist. To this end, since we
+are here interested in the cosmological scenario we shall co nsider the category1Fof spacetimes
+with a large spatial symmetry namely the flat Friedmann Rober tson Walker (FRW). Thanks to
+the result stated in [BFV03] and [HW01], the fields realizing the components of the stress tensor
+can be unambiguously found on every spacetime, hence also on every element of F. However, in
+order to have a well defined system of equations governing the backreaction, we must be able to
+unambiguously select a class of states (one for every elemen t ofF) with good ultraviolet behavior
+in a spacetime independent way. Eventually, the validity of the semiclassical Einstein equations
+will be employed as a constraint on the class F. The difficult part in this program is to give, for
+every element of F, a state which is not based on the particular spacetime. Here we would like
+to give a prescription for obtaining those natural states. F urthermore, in order to give meaning
+to observables like the stress tensor, the states must have g ood ultraviolet behavior, namely the
+singular part of their two-point function must be as close as possible to the singularity shown
+by the vacuum in Minkowksi spacetime. Unfortunately this is not always possible and in many
+cases, a state, determined by some initial values on a spacel ike Cauchy surface, does not satisfy
+the necessary regularity conditions. From the physical poi nt of view, a natural candidate for a
+quantum state would be the choice of a vacuum, but unfortunat ely such concept is not available
+in curved spacetime. In any case we could ask for the validity of some physical properties in
+order to select a class of states that behaves like the vacuum in Minkowski spacetime, or at least
+that are close to it. To this end, Parker [Pa69] introduced th e adiabatic states, namely states
+that are close to the vacuum. Afterwards, L¨ uders and Robert s [LR90] made their definition
+precise employing the construction of Parker to select cert ain initial conditions at finite time
+in a FRW spacetime. Unfortunately, in general, the states ob tained in that way do not satisfy
+the Hadamard condition, hence they do not have the same ultra violet singularity as the vacuum
+in Minkowski spacetime, as shown by Junker and Schrohe [JS02 ]; nevertheless they satisfy a
+1The elements of Fare the oriented and time oriented flat FRW spacetimes and the morphisms can be chosen
+to be the orientation preserving conformal transformation s
+4condition weaker than the microlocal spectral condition gi ven in [BFK95]. Another possibility
+would be to consider states that minimize the smeared energy density. Olbermann [Ol07] has
+constructed such states and has shown that they are of Hadama rd form. Unfortunately, since
+this procedure depends upon the smearing over a finite interv al of time, it is not possible to get
+those states only out of the initial conditions.
+In the present work we shall follow another procedure close t o the one presented in [DMP06,
+Mo06, Mo08] in the context of asymptotically flat spacetimes and already employed in cosmo-
+logical scenarios in [DMP09a, DMP09b]. We shall see that, if the initial surface is chosen to be
+lightlike (coinciding with the null past infinity in the conf ormally related spacetime), we can give
+a class of states (that do not rely on the particular spacetim e) that can be used in order to solve
+the coupled semiclassical system. Eventually, we shall com ment on the late time behavior of the
+solutions showing that, if a de Sitter phase of exponentiall y accelerated expansion has emerged,
+the value of that acceleration does not depend on the particu lar homogeneous Hadamard state.
+Finally we shall give a new interpretation of the semiclassi cal Einstein equations when consid-
+ered as a non local equation. We shall actually show that, whe n smeared on large spatial volume
+regions, while the expectation value of the trace, /a\}b∇acketle{tT/a\}b∇acket∇i}ht, does not differ from the pointlike one, its
+fluctuations tend to vanish.
+In the subsequent section we shall introduce the geometric s etup and the basic facts about
+the quantization we shall use in the next, this section is use d to fix the notation and to recollect
+some results used throughout the subsequent part of the work . In the third section a discussion
+about the regularity and the Hadamard property shown by a cla ss of states will be discussed,
+the states constructed in a spacetime independent way are pr esented in this section. The fourth
+section contains the main theorems about the existence of so lutions of the semiclassical Einstein
+equations in a cosmological scenario. In the fifth section we shall discuss the behavior of the
+semiclassical solutions at late times and we shall see that t he variance of the trace vanish when
+smeared on large volume regions. Finally, some comments and remarks are collected in the
+sixth section. The appendix contains a discussion about def ormation quantization, the technical
+estimates and a technical proof of a proposition used in the m ain text.
+2 Geometry of the spacetime, classical and quantum matter
+2.1 Geometry
+Since we are dealing with problems in cosmology we shall assu me homogeneity and isotropy.
+Hence, every element ( M,g) in the class of spacetimes under investigation is such that the
+smooth manifold Mis of the form M=I×Σ whereI⊂Ris an interval representing the
+cosmological time and Σ is a three dimensional hypersurface that can be closed o pen or flat.
+The metric g, takes the well known Friedmann-Robertson-Walker form:
+g=−dt2+a2(t)/bracketleftbiggdr2
+1−κr2+r2dS2(θ,ϕ)/bracketrightbigg
+, (2)
+5wheredS2(θ,ϕ) is the standard metric of the unit sphere in spherical coord inates and κ∈
+{+1,0,−1}distinguishes between the topologies of Σ, in particular be tween open, flat or closed
+spacelike hypersurfaces respectively. Furthermore, a(t) is thescaling factor representing the
+“story” of the universe and it is the single dynamical degree of freedom present in the system.
+In the next, also supported by recent observations that seem to confirm this hypothesis, we
+shall restrict our attention to the flat case, κ= 0. With this assumption the metric (2) becomes
+conformally flat. This last fact, when a(t) is given, can be manifestly seen passing from the
+cosmological time tto theconformal time τ. The latter is determined by the diffeomorphism
+τ(t) :=τ0−/integraldisplayt0
+t1
+a(t′)dt′,
+whereτ0andt0are two fixed constants. Since this transformation is a diffeom orphism and ais
+a field on the manifold M, it is clear that acan be seen both as a function of tor, as a function
+ofτ. For this reason, in the next, we shall indicate a◦τ−1(τ) bya(τ) as it is custom for scalar
+fields over manifolds.
+We would like to stress that choosing a(t) as the single degree of freedom, we are at the same
+time selecting a fixed (free) background on which a(t) evolves. The matter evolution could be
+analyzed on that background too, however, since we restrict our attention to conformally flat
+spacetimes, another natural choice for the background woul d be Minkowksi spacetime ( M,gM)
+taken with the standard Minkowksi metric
+gM:=−dτ2+dr2+r2dS2(θ,ϕ),
+where (r,θ,ϕ) represents a point in R3in the standard spherical coordinates. In order to fix the
+notation, in the next, we shall indicate the generic element xofMby (τ,x); notice that this
+defines a coordinate system also on M. Later it will also be useful, to consider as a background,
+the static Einstein universe ( Me,ge). The advantage of this last selection is in the compactness
+of its spatial sections. All these spacetimes are connected by a chain of conformal embeddings
+[Pi09]ηandη′such that
+(M,g)֒→
+η(M,gM)֒→
+η′(Me,ge) (3)
+where the conformal factor of ηis Ω2:=a−2, while the one of η′is
+Ω′2:=/parenleftbigg(1+(τ+r)2)(1+(τ−r)2)
+4/parenrightbigg−1
+.
+We would like to stress that in every spacetime under investi gation, namely for every a(t), there
+is a conformal Killing vector, ∂τgenerating conformal translations τ→τ+λ. We shall use
+these conformal symmetry in order to define a suitable quantu m state used for the computation
+of backreaction.
+62.2 Scalar fields, quantization and the stress tensor
+As a model for matter we shall consider a real scalar field, nam ely a field satisfying
+Pϕ= 0, P=−✷g+ξRg+m2. (4)
+For simplicity, we shall choose the coupling to the metric to beξ= 1/6, corresponding to the
+so-called conformal coupling, whereas we shall allow the ma ss to be non vanishing. In this
+respectmshall play the role of the “coupling constant” for the matter gravity interaction, or
+the interaction between the field ϕand the scaling factor a.
+We shall employ the quantization prescription proper of the algebraic approach ([BFV03,
+HW01]). First of all we select the ∗−algebra of local observables / fields A(M) and afterwards,
+choosingasuitablestate ω(anormalized, positivelinearfunctionalover A(M)), theexpectation
+values for the elements of A(M) are obtained. In order to keep the paper self-consistent, a n
+explicit construction of A(M) will begiven in the appendix A.1. For our purposeit is impor tant
+to stress that, being Ma globally hyperbolic spacetime, A(M) is uniquely determined by
+the equation of motion and the unique causal propagator2E. The whole procedure is hence
+functorial as discussed in [BFV03]. In order to compute the b ackreaction of matter on gravity
+by means of the semiclassical Einstein equations (1), the ex pectation values of the component of
+the stress tensor need to be computed. Hence, since the point wise product of fields is singular,
+the quantum state has to be selected in a physically reasonab le way. In other words a certain
+regularization procedure has to be implemented and it must b e meaningful on that state.
+We proceed now to discuss this aspect in detail. On the field al gebra A(M) (constructed as
+discussed on the appendix A.1), the functional ωrepresenting the state can be thought as being
+completely described by a suitable set of distributions ωn∈D′(Mn), the so called n−point
+functions . In this paper we shall restrict our attention to states that are quasi-free, namely
+states whose n−point functions vanish for odd nwhile for even nthey can be given in terms of
+the two-point function ω:=ω2only by means of the formula
+ωn(x1,...,x n) =/summationdisplay
+π∈Pnω(xπ(1),xπ(2))...ω(xπ(n−1),xπ(n)) (5)
+where the sum is taken over the set Pnof permutations of the first nnatural numbers with the
+constraints
+πn(2i−1)<πn(2i),1≤i≤n/2 and πn(2i−1)<πn(2i+1),1≤i<n/2.
+Let us proceed to discuss the regularization prescription w e shall use to give meaning to observ-
+ables like the component of Tµνor the field square. In order to implement it in practice, we
+must have control on the ultraviolet behavior of our states. For this reason, we shall impose a
+constraint on the states, asking that their two-point funct ions have singularities that look, as
+much as possible, similar to the ultraviolet singularity di splayed by the Minkowski vacuum. This
+is achieved requiring that the two-point function ωsatisfies the so called microlocal spectrum
+condition [Ra96, BFK95].
+2The construction, and the proof of existence and uniqueness can be found for example in [BGP96].
+7Definition 2.1. A stateωonA(M)satisfies the microlocal spectrum condition ( µSC) if its
+two-point function ωis a distribution on C∞
+0(M2)whose wave front set has the form
+WF(ω) ={(x,y,kx,ky)∈T∗M2\{0},(x,kx)∼(y,−ky), kx⊲0}.
+The relation ( x,kx)∼(y,−ky) is satisfied in Mif the points xandyare connected by a null
+geodesic whose cotangent vector in xiskxand inyis−ky. Furthermore kx⊲0 ifkxis future
+directed. Notice that, thanks to the work of Sanders [Sa09], it suffices to give a condition only
+on the two-point function. In the above mentioned paper it is actually shown that every state of
+a free scalar field algebra, whose two-point function satisfi es the microlocal spectrum condition,
+hasn−point functions that enjoy the microlocal spectrum conditi on too (for the definition in
+that case we refer to the work [BFK95]).
+Notice that, the two-point function of a quasi-free state fo r fields satisfying the equation of
+motion (4) and the microlocal spectrum condition, has an ult raviolet singularity that is similar
+to the one of the Minkowski vacuum. In other words, its integr al kernel can be expanded in the
+following way
+ω(x,y) := lim
+ǫ→0+/parenleftbiggU(x,y)
+σǫ(x,y)+V(x,y)logσǫ(x,y)
+λ2/parenrightbigg
++W(x,y) (6)
+whereU,VandWare smooth functions on M2,λis a length scale used in order to have the
+logarithm with a dimensionless argument and σǫ(x,y) =σ(x,y)+iǫ(T(x)−T(y))+ǫ2where
+Tis a generic time function and σis the Synge world function, namely half of the squared
+geodesic distance taken with sign [DB60, KW91]. Furthermor e,ǫis a regularization parameter
+that eventually tends to 0+(weak limit). A two-point function whose singular part is of the
+form (6) is said to be of Hadamard form . It is important to notice that the coefficient U
+andVare completely determined out of the geometry and of the equa tion of motion, while the
+coefficientWreally characterizes the state. The distribution construc ted out of the first two
+terms in (6) is called Hadamard singularity and it is indicated by H, then its integral kernel
+is
+H(x,y) := lim
+ǫ→0+/parenleftbiggU(x,y)
+σǫ(x,y)+V(x,y)logσǫ(x,y)
+λ2/parenrightbigg
+. (7)
+Hence, in order to regularize the states satisfying the micr olocal spectrum condition we can
+simply subtract the singular part Hfromω, what remains is a smooth function whose coinciding
+point limit can be safely computed. As a side remark, we notic e that, since we are interested in
+local fields, for our purpose it is enough to have an explicit f orm ofHon small domains. The
+question about global existence of His hence not an important one in our approach. Coming
+backtothemainpoint, itishencepossibletoobtainawellpo sedstresstensoronstates satisfying
+theµSC. On practical ground this is achieved by operating on the r egularized two-point function
+ω−Hwith a certain differential operator beforecomputing the wel l posed coinciding point limit.
+This procedureis called point splitting regularization . Furthermore, we notice that on those
+states the fluctuations of the observables computed on simil ar states are always finite.
+8Let us discuss this point in some details. On the class of stat es that satisfy the µSC, we could
+extend the algebra of fields A(M) in order to encompass also pointwise product of fields. Here
+we shall indicate by F(M), the extended algebra of fields, namely the ∗−algebra that contains
+also the Wick powers as constructed in [HW01], or more recent ly by means of deformation
+quantization methods in [BDF09, BF09]. For completeness, t he main steps of that construction
+can be found in the appendix A.1. We are now ready to discuss th e form of the stress tensor as
+an element of F(M). To this end, following the discussion introduced in [Mo03 ], we consider
+the following operation on smooth field configurations ϕ
+Tab(ϕ) :=∂aϕ∂bϕ−1
+6gab/parenleftbig
+∂cϕ∂cϕ+m2ϕ2/parenrightbig
+−ξ∇a∂bϕ2+ξ/parenleftbigg
+Rab−R
+6gab/parenrightbigg
+ϕ2+/parenleftbigg
+ξ−1
+6/parenrightbigg
+gab✷ϕ2.
+Notice that for every compactly supported smooth function f,
+Tab(ϕ)(f) :=1
+2/a\}b∇acketle{tf,Tab(ϕ)/a\}b∇acket∇i}ht
+is an element of F(M), and we shall use it in order to compute the backreaction in ( 1). Another
+element of F(M) that will be useful is φ2(f). This is nothing but/integraltext
+fϕ2/2. We shall indicate
+by/a\}b∇acketle{tφ2/a\}b∇acket∇i}htωand/a\}b∇acketle{tTab/a\}b∇acket∇i}htωthe expectation values of the two observables introduced ab ove computed
+on the state ωregularized according to the previous discussion. Hence /a\}b∇acketle{tφ2/a\}b∇acket∇i}htωand/a\}b∇acketle{tTab/a\}b∇acket∇i}htωare
+obtained replacing the smooth field configuration ϕ(x)ϕ(y) with the smooth function ω−H
+inTab(ϕ)(f) andφ2(f) before performing the coinciding point limit (contractin g with the delta
+function). Further details on this regularization, can bef ound in appendixA.1. In the procedure
+presented above, there is an ambiguity in the definition of H, whose form is describedin [HW01];
+we call it renormalization freedom, later on we shall give mo re details on this crucial aspect.
+3 Homogeneous states and Hadamard property in flat FRW
+3.1 Characterization of pure, homogeneous states in terms o f solutions of
+the wave equation.
+In this subsection we shall proceed to discuss the form of the states we are interested in, we
+shall make use of some previous works in order to get the form o f the two-point function of
+the pure homogeneous states on FRW spacetimes in particular . We start with the definition
+of the class C(M) of pure and homogeneous quasi-free states, we shall employ in the present
+paper. Although the set C(M) does not contain all the possible pure and homogeneous stat es
+forA(M), that definition is motivated by the works of L¨ uders and Rob erts [LR90] (Theorem
+2.3 of [LR90] in particular) and the one of Olbermann [Ol07] s ee also the nice work of Degner
+and Verch [DV09] for a review. Furthermore, we shall restric t our attention to that class.
+Definition 3.1. We shall indicate by C(M)the set of pure,homogeneous and quasi-free
+statesωforA(M)whose two-point function ω∈D′(M×M), using the standard Minkowksian
+9coordinates, takes the following form, in the distribution al sense3,
+ω((τ,x);(τ′,x′)) :=1
+(2π)3/integraldisplay
+R3Sk(τ)
+a(τ)Sk(τ′)
+a(τ′)eik·(x−y)dk, (8)
+where(τ,x)is a generic element of (I×R3,g),k=|k|. For every k≥0,Skis a smooth function
+satisfying
+d2
+dτ2Sk(τ)+(m2a(τ)2+k2)Sk(τ) = 0, (9)
+and
+Sk(τ)d
+dτSk(τ)−d
+dτSk(τ)Sk(τ) =i. (10)
+Furthermore, for every τ, the map k/ma√sto→Sk(τ), and its first time derivative, are measurable
+functions that are polynomially bounded at large k. When restricted on a generic interval [0,k0]⊂
+R+they are in L2([0,k0],k2dk).
+It is not guaranteed that the state, constructed out of a gene ric solution Skof the equation
+(9) (also with the constraint on the Wronskian), is of Hadama rd type. It is also usually not
+easy to impose some ad hoc initial conditions for Sk(τ) on a spacelike hypersurface in order
+to have an Hadamard state. This problem, in connection with t he concept of adiabatic states,
+has been discussed in the paper of Junker and Schrohe [JS02] w ho showed that it is possible to
+get Hadamard states only if the recursive construction of ad iabatic vacua does not break down.
+However, as shown by L¨ uders and Roberts [LR90], one cannot e xclude such possibility. Another
+possible explicit construction of Hadamard states is the on e presented by Olbermann [Ol07],
+but this construction requires the smearing on a finite time i nterval. It is hence very difficult to
+use similar ideas in treating backreaction problems. Here w e would like to give an alternative
+construction employing ideas similar to the one presented i n [Mo08, DMP09a, DMP09b] and
+adjusting them in order to be suitable for the purposeof stud ying the gravity-matter interacting
+system. First of all, we start selecting a particular soluti onχk(τ) of (9), when the spacetime M
+admits the limit τ→ −∞, namely the one that satisfies the initial conditions
+lim
+τ→−∞eikτχk(τ) =1√
+2k,lim
+τ→−∞eikτd
+dτχk(τ) =−i/radicalbigg
+k
+2. (11)
+Wecan nowintroducethefollowingpropositionthat summari zessomeresultsobtainedin[LR90,
+Ol07, DV09]
+Proposition 3.1. If the limits in (21) are well posed, every solution Skof (9) which determines
+a stateω∈C(M)according to the definition 3.1 can be realized as a linear com bination of the
+χk(the solution of the equation (9) satisfying the asymptotic conditions (11)):
+Sk(τ) :=A(k)χk(τ)+B(k)χk(τ), (12)
+3Here we mean that ωhas to be constructed as the weak limit ǫ→0+of the expression on the right hand side
+of (8) regularized multiplying the integrand by an extra fac tore−ǫ|k|.
+10and the coefficient A(k)andB(k)are such that
+|A(k)|2−|B(k)|2= 1.
+Furthermore, for every τ, the functions, on R+,k/ma√sto→A(k)χk(τ)andk/ma√sto→B(k)χk(τ), together
+withk/ma√sto→A(k)d
+dτχk(τ)andk/ma√sto→B(k)d
+dτχk(τ), are at most of polynomial growth and locally
+square integrable.
+Notice that, even if we restrict our attention to the class of pure states4, we have still some
+freedom in the definition of the state, as expressed in the pre ceding proposition by the choice of
+A(k) andB(k) in theSkof (12) used in finding the two-point function (8). Motivated by this
+expression, we shall indicate by ωA,Bthe quasi-free state whose two-point function is
+ωA,B((τ,x);(τ′,x′)) :=1
+(2π)3/integraldisplay
+R3dkeik(x−x′)Sk(τ)
+a(τ)Sk(τ′)
+a(τ′)(13)
+where we have highlighted the dependance upon A(k) andB(k) in the state.
+We are now going to find sufficient conditions that have to be ful filled byA(k) andB(k)
+in order to ensure that ωA,B∈C(M) satisfies the Hadamard condition. The strategy we shall
+pursue is the following one. We prove that ω1,0is of Hadamard form provided some regularity
+is shown by a(t) in (2) and then that ωA,Bsatisfies the µSC if and only if B(k) is rapidly
+decreasing for large k. Notice that, in order to prove that a certain distribution s atisfies the
+microlocal spectrum condition, it is not enough to test it on some Cauchy surface because no
+remnant of causality survives the projection of any vectors on a spacelike hypersurface. This
+is not true for projection on lightlike hypersurfaces, and s uch is the reason why the arguments
+used in [Mo08], [Ho00] and [DMP09b] works. We would like to gi ve sufficient conditions on
+the form of the spacetime, and on its past asymptotic form in p articular, in order to guarantee
+thatω1,0is of Hadamard form. In the case of asymptotic de Sitter space time, it was already
+shown in [DMP09b] that the state ω1,0satisfies the microlocal spectrum condition, and that
+if the spacetime is precisely the de Sitter one, ω1,0becomes the well known Bunch Davies
+state [BD78]. Furthermore, in the case of asymptotic flat spa cetimes, Moretti has proven the
+Hadamardpropertyforananaloguestate [Mo08]; asimilar pr oofcan alsobefoundin[Ho00]. We
+arenowgoingtogeneralizetheseresultsinordertoencompa ssthecaseconsideredhere. Theidea
+we would like to use, is to consider ω1,0as the pullback ( η◦η′)∗(˜ω) associated with the conformal
+embedding η◦η′introduced in (3) where ˜ ωis the two-point function of a certain state in a
+hyperbolicsubsetof theEinstein universethat contains th eimage ofMunderη′◦ηtogether with
+its past boundary ℑ−∪i−. Notice that on the latter, the field theory looks like a massl ess Klein
+Gordonfieldperturbedwithanexternalpotential oftheform m2a(τ)2(1+(τ+r)2)(1+(τ−r)2)/4,
+and hence ˜ωis a state for this particular theory. It happens that, if tha t potential becomes
+regular on the extended spacetime in a neighborhood of ℑ−∪i−then the state ω1,0, defined in
+such a way, turns out to be of Hadamard form.
+4More details in this respect can be found in [LR90, Ol07]
+11Theorem 3.1. Suppose that M=I×R3whereI= (0,t0)and thata(t)is such that the region
+t→0corresponds to the past null infinity of (M,gM)in(Me,ge)and furthermore such that the
+smooth function
+(1+(τ+r)2)(1+(τ−r)2)m2a(τ)2(14)
+defined onMseen as embedded in the Einstein universe (Me,ge)can be smoothly extended in a
+neighborhood O⊂Mecontaining the past boundary of Mhere indicated as ℑ−∪i−. Under these
+hypotheses the state ω1,0, constructed out of the solutions χkthat satisfy the initial condition on
+ℑ−satisfies the µSC.
+Proof.The proof can be found following similar ideas as those prese nted in [Mo08, Ho00] or
+alternatively in [DMP09b], we shall summarize here the main strategy and we shall discuss only
+the key point in detail. We would like to show that the two-poi nt function ω1,0satisfies the
+microlocal spectrum condition, hence we have to analyze its wave front set and to show that
+WF(ω1,0) =/braceleftbig
+(x,y,kx,ky)∈T∗M2\{0},(x,kx)∼(y,−ky), kx⊲0/bracerightbig
+. (15)
+Notice that, in showing the preceding equality, the difficult part is to prove the inclusion ⊂. If
+it holds true, the other inclusion descends from the fact tha tω1,0is a state [SV01, SVW02] and
+out of an application of the H¨ ormander’s theorem of propaga tion of singularity [DH72]. In order
+to show the validity of inclusion ⊂we can rewrite ω1,0(f,g) asw(ΩEf↾ℑ−,ΩEg↾ℑ−) wherew
+is a certain distribution on ℑ−×ℑ−defined employing the Fourier Plancherel transform5along
+theτ−coordinate on ℑ−:
+w(ψ1,ψ2) :=/integraldisplay
+S2/integraldisplay∞
+02k/hatwiderψ1(k,θ)/hatwiderψ2(k,θ)dkdS2,
+wheredS2is the measure on the two dimensional unit sphere S2andθis a shortcut for the
+standard spherical coordinates. A discussion on such a dist ribution can be found in theorem
+3.1 of [Mo08]. Furthermore, Ω Ef↾ℑ−is the restriction to ℑ−of the wave function E(f)
+constructed by means of the causal propagator E, multiplied by a certain factor Ω. That
+restriction can be safely computed in an Einstein universe a nd the multiplication by the factor
+Ω corresponds to the conformal transformations that maps so lutions ofPinMto solutions of Pe
+inMe. We have indicated by Pethe operator −✷e+1
+6Re+m2a2Ω2where✷eis the d’Alembert
+operator associated with the Einstein metric. Hence, the tw o-point function ω1,0can be seen as
+a composition of distributions
+ω1,0:=w◦(ΩE↾ℑ−⊗ΩE↾ℑ−).
+Because of the non compact nature of the restriction of Eonℑ−, which persists also if the other
+entry is localized on a compact set of M, the previous composition of distributions cannot be
+straightforwardly justified by an application of theorem 8. 2.13 of [H¨ o89]. On the other hand, in
+5This operation is introduced and discussed in the appendix o f [Mo08].
+12proposition 4.3 in [Mo08], it is proven that a similar compos ition is well defined and that the
+resulthas thedesiredwave front set providedthefollowing two facts hold: Firstof all considering
+a compact set OinsideM, there exists a neighborhood Oi−ofi−inMesuch that there are no
+lightlike geodesics connecting any point in Owith those in Oi−, and, second, if one entry of
+the causal propagator is restricted on Oand the other on ℑ−, the causal propagator itself must
+fall off sufficiently fast towards i−inM. Both requirements hold under the hypotheses of the
+present theorem, the first descends from the fact that Mis conformally flat and the latter is a
+consequence of the fact that m2a(τ)2(1+(τ+r)2)(1+(τ−r)2) can be smoothly extended on a
+neighborhood of i−inMe, hence the same result as the one stated in lemma 4.2 of [Mo08] holds
+in the present case too.
+Notice that the constraints given above are only sufficient bu t not necessary, in fact in [DMP09b]
+it has been proven that ω1,0is a well defined Hadamard state also when the spacetime is
+asymptotically de Sitter in the past, and in such a case the po tential that appear in the equation
+of motion on the Einstein universe is singular on i−. Hence the hypothesis of smooth extension
+of the function (14) over ℑ−∪i−inMecan be relaxed. Finally, notice that employing similar
+techniques to the one presented in [DMP09a] and in [DMP09b] b ased on a careful analysis of
+the modes χk(τ), it is actually possible to obtain the Hadamard property fo r the state ω1,0
+also when (14) can be extended over ℑ−∪i−only with the regularity of C1(Me). We would
+like to conclude this section by introducing a theorem that g ives some necessary and sufficient
+condition for the state ωA,Bto be of Hadamard form.
+Theorem 3.2. Suppose that ω1,0, constructed as in (13), is of Hadamard form. Then the
+quasi-free state ωA,B∈C(M), satisfies the microlocal spectrum condition if and only if |B(k)|
+decreases rapidly for large k, in other words, there are constants cnandksuch that
+|k|n|B(k)| ≤cn,∀ |k|>k.
+Proof.The proof of this fact can be easily done using the formulatio n of the state on a given
+Cauchy surface, see [LR90] for a discussion. Notice that, wh en restricted on a Cauchy surface,
+the difference between the two-point functions ωA,B−ω1,0becomes smooth if and only if B(k)
+decreases rapidly (and this holds for the time derivative of one or both of its entries too). This
+proves that the rapid decrease of |B(k)|is a necessary condition for the validity of µSC. Finally,
+in order to prove that it is also sufficient, we can make use of a f ormula like the one given in
+equation 3.60 of [KW91] and reconstruct ωA,B−ω1,0out of its form on a Cauchy surface. We can
+extend the result about the smoothness to the whole M2by noticing that every of the four terms
+appearing in the formula 3.60, above cited, can be obtained c omposingωA,B−ω1,0(sometime
+with a time derivative applied to one or both of its entries) w ith the tensor product E⊗E
+(sometime taken with a time derivative applied to one or both of its entries). The proof can
+be easily finished noticing that Emaps compactly supported smooth functions on the Cauchy
+surface to smooth functions in the spacetime.
+134 Semiclassical treatment of backreaction and existence of so-
+lutions
+4.1 Semiclassical approximation
+We shall now proceed to treat the backreaction of quantum mat ter on gravity, and, as discussed
+in the introduction, we shall do it for flat FRW spacetimes. Wi th such a large symmetry, the
+Einstein equations take the well known Friedmann Robertson Walker form, namely, being the
+stress tensor of matter diagonal and equal to Tµν=diag(−ρ,P,P,P ), if we indicate by ˙ athe
+derivative of awith respect to the cosmological time tandH:=a−1˙athey become
+3H= 8πρ−3κ
+a2,3/parenleftBig
+˙H+H2/parenrightBig
+=−4π(ρ+3P).
+Because of the conservation of the stress tensor we can inste ad solve the equation for the trace
+ofGµνand ofTµν
+−R= 8πT
+and then employ the first FRW equation as a constraint, that ne eds to be imposed only at a
+fixed time, in order to fix only an initial condition. We shall a ssume this point of view and, in the
+semiclassical scenario, we have simply to substitute the cl assicalTwith the expectation value
+of the corresponding observable /a\}b∇acketle{tT/a\}b∇acket∇i}htcomputed in the chosen quantum state. More precisely,
+as discussed in the introduction, we have to select a class of states (one for every a(t)) and
+then the semiclassical equation −R= 8π/a\}b∇acketle{tT/a\}b∇acket∇i}htacts as a constraint on this class. Notice that, by
+employing the initial values on the null boundary, we have al ready disentangled the problem of
+the definition of the class of states from the geometric prope rty of the spacetime.
+Hence the backreaction of such system can be treated simply e mploying the equation for the
+trace. But in the trace, taking the conformal coupling and a n on-vanishing mass, the explicit
+form of the state become important. More precisely, the expe ctation value of the trace of the
+stress tensor in a state ωis
+/a\}b∇acketle{tT/a\}b∇acket∇i}htω:=2[V1]
+8π2+/parenleftbigg
+−3/parenleftbigg1
+6−ξ/parenrightbigg
+✷−m2/parenrightbigg
+/a\}b∇acketle{tφ2/a\}b∇acket∇i}htω.
+Above [V1] is the coinciding point limit of the second Hadamard coeffici ent (see [SV01, Mo03,
+DFP08] for more details). This contribution to the trace is t he anomalous one, it coincides
+with the well known trace anomaly displayed by the stress ten sor [Wa78b] in the massless case.
+Coming back to the dynamical equations, in the case of a flat FR W metric, the equation of
+motion for the scaling factor is
+−6/parenleftBig
+˙H+2H2/parenrightBig
+=−8πm2/a\}b∇acketle{tφ2/a\}b∇acket∇i}htω−1
+30π/parenleftBig
+˙HH2+H4/parenrightBig
+. (16)
+In writing the preceding equation we have performed some cho ices of the renormalization pa-
+rameters, in particular we have chosen some of them in such a w ay that the higher derivatives
+of the coefficients of the metric disappear. There is an extra r enormalization freedom that we
+14have not considered here, but it can always be reabsorbed in t he freedom present in /a\}b∇acketle{tφ2/a\}b∇acket∇i}htω. It
+seems important to notice that, apart from these renormaliz ation freedom, the trace anomaly
+is fixed and it is really a c−number. For the discussion of these and other points we recal l the
+work [DFP08].
+Hence, in the case of vanishing m, the state does not enter anymore in (16) and an exact
+solution can be easily found6. This solution can be written in an implicit form as
+e4tH+=e2H+
+H/vextendsingle/vextendsingle/vextendsingle/vextendsingleH+H+
+H−H+/vextendsingle/vextendsingle/vextendsingle/vextendsingle(17)
+where the time of the big bang is fixed to be t= 0 andH2
++= 360πin natural units, that is
+G=/planckover2pi1=c= 1.
+The massless solution (17) presents three branches for H >0 (the first 0 <H2<H2
++/2, the
+secondH2
++/2<H2<H2
++and the last H2>H2
++), we concentrate ourself on the upper one and
+we discuss some of its properties. Notice that, it correspon ds to a universe that shows, at the
+beginning, a phase typical of the power law inflationary scen ario, while it ends up in a phase
+which corresponds to a de Sitter universe. It is a remarkable fact that the form of the initial
+singularity is lightlike; hence, for example, the horizon p roblem is not present in this simple
+model. The Hubble constant Hof the de Sitter phase shown in the asymptotic future is equal
+toH+, and it is many order of magnitude too big in order to explain t he present acceleration
+of the universe. Despite of this fact, in [DFP08], it has been shown that, in the case of massive
+fields, this parameter is not fixed and it needs to be interpret ed as an extra renormalization
+freedom. In the next we shall make use of the lightlike nature of the initial singularity to see if
+it is possible to obtain an exact solution also in the case of m assive fields, and we shall look for
+solutions that are similar to the massless one at the beginni ng of the universe. In this respect,
+we shall treat H+as a renormalization constant. Furthermore, since, in the m assless case, only
+the anomaly appears in the trace and it is a c−number it has a vanishing variance, hence, such
+equation should hold also for large H(as is the case close to the initial singularity t= 0 in the
+(17)). Of course, the situation changes when the mass is differ ent from zero, and in this case the
+result provided by the semiclassical Einstein equations ne eds to be interpreted more carefully.
+We shall comment on this points in the next section.
+4.2 Initial conditions at the beginning of the universe
+In this subsection we would like to discuss the existence of s olutions of the semiclassical Einstein
+equations (16) that look as much as possible close to (17), at least on small intervals of proper
+time just after the initial singularity. Hence, we shall ass ume the form of the singularity to be
+lightlike just as in the massless case (17). As previously di scussed this task can be accomplished
+only after acarefulanalysis of theclass of states wehaveto useinordertoobtain theexpectation
+values ofφ2in the different spacetimes we are considering, in any case the requirement of having
+a lightlike initial singularity is essential in order to app ly the results of the preceding sections
+and to have Hadamard states.
+6see [Wa78a, DFP08] for more details
+15The states we would like to chose are the one constructed in th e preceding part of this paper
+and denoted by ωA,BwithBof rapid decrease. We shall start considering only the state ω1,0,
+and we notice that these states are defined by the requirement to be vacuum states with respect
+to the conformal time on the initial singularity (this proce dure works only for lightlike initial
+singularities). Before proceedingwith thediscussion of e quation (16) we have to discuss theform
+of the expectation value of φ2on these states and to fix once and forever the renormalizatio n
+freedom, according to the picture presented in [BFV03].
+Since we would like to impose initial conditions on ℑ−or better at τ→ −∞, where the
+Hubble constant Husually diverges, it is useful the rewrite the dynamical equ ation for its
+inverse that we shall call X:=H−1. With this in mind, we can rewrite the equation (16) as
+follows
+dX
+dt= 1−H2
+cX2
+1−H2cX2+C2
+mX4
+1−H2cX2/a\}b∇acketle{tφ2/a\}b∇acket∇i}htω1,0, (18)
+whereH2
+c:=H2
++/2 = 180πandC2
+m:= 240π2m2are two constants. Notice that, since /a\}b∇acketle{tφ2/a\}b∇acket∇i}htω1,0
+is a functional of a(t) and ofτ, the proof of the existence of solutions of (16) needs some effo rt,
+and the analysis of the functional dependance of /a\}b∇acketle{tφ2/a\}b∇acket∇i}htω1,0uponX=H−1needs to be carefully
+performed in particular.
+We intend to prove the existence of solutions of (16) (or bett er of (18)) in a certain compact
+setBcof a Banach space B. To this end let us start introducing these spaces of functio ns.
+Definition 4.1. Consider the set Bas the subset of elements of C1([0,t0])that vanish in 0and
+whose first derivative is bounded on [0,t0].Bis a Banach space when equipped with the norm
+/ba∇dblf/ba∇dblB:= sup
+[0,t0]/vextendsingle/vextendsingle/vextendsingle˙f(t)/vextendsingle/vextendsingle/vextendsingle.
+For every2
+3<c<1we define Bcas the closed ball in Bof radius 1/2−c/2centered in
+f0(t) :=1+c
+2t(t∈[0,t0]). (19)
+The strategy of the proof of existence of solutions of (18) is the following: We shall build a
+sequenceXninBcby a recursive use of a certain map T:Bc→Bdefined as
+T(X)(t) :=/integraldisplayt
+0/bracketleftbigg1−2H2
+cX2
+1−H2cX2+C2
+mX4
+1−H2cX2/a\}b∇acketle{tφ2/a\}b∇acket∇i}htω1,0[X]/bracketrightbigg
+dt′(20)
+and we shall show that, since Tit is contractive on Bc,Xnconverges to a solution. Hence the
+existence and uniqueness descends from an easy application of the Banach fixed point theorem.
+Notice that, if we succeed in obtaining that result, since ce rtain physical properties (like the
+presence of a phase of power law inflation) are enjoyed by ever y element of Bcthose properties
+have to be shown by the solution too.
+We shall now discuss how it is possible to associate a spaceti me to anX∈Bcin such a way
+that the initial conditions are implemented.
+16Proposition 4.1. Fix the three constant t0,a0andτ0. Then every element XofBcdetermines
+a spacetime (M,g[X])whereM= [0,t0]×R3and whereg[X]is the metric (2)constructed out
+of the following scaling factor
+a[X](t) :=a0exp/parenleftbigg
+−/integraldisplayt0
+tX(t′)−1dt′/parenrightbigg
+.
+The Hubble parameter shown by (M,g[X])is then
+H(t) =X(t)−1,
+and the corresponding conformal time
+τ[X](t) :=τ0−/integraldisplayt0
+ta[X](t′)−1dt′,
+is a functional of Xtoo. Finally, the spacetime enjoys the following initial co nditions
+X(0) = 0, a(t0) =a0, τ(t0) =τ0 (21)
+per construction.
+For everyXinBcwe have immediately, from the definition, that the initial co nditions (21)
+given above are satisfied, and the functional a[H] andτ[H] are, moreover, well defined on Bc.
+Notice that, on Bc
+lim
+t→0/integraldisplayt0
+tX−1(t′)dt′=∞
+out of this, we have that a(0) = 0 sufficiently fast in order to ensure that τ[X]→ −∞fort→0.
+This last requirement is necessary in order for the resultin g spacetime to have a lightlike past
+boundary or, in other words, that the initial surface t= 0 is of null type. As already discussed,
+this last requirement is a necessary prerequisite for the co nstruction of Hadamard states on
+(M,g). We have in fact, the following two propositions
+Proposition 4.2. For every spacetime characterized by a metric of FRW type, wh ose scaling
+factora(t)is such that X(t) =a(t)
+˙a(t)is inBc∩C∞([0,t0]), we can construct ω1,0as in(13)and it
+determines a quasi-free state ω1,0which, moreover, satisfies the microlocal spectrum conditi on.
+Proof.Iftheanalogue of function(14) can besmoothly extended ove rℑ−∪i−theproofdescends
+from theorem 3.1, otherwise, also according to the discussi on given after that theorem we can
+adapt the proof of theorem 3.1 given in [DMP09b] to the presen t case.
+Notice that, if we drop the requirement of being smooth, and w e replace it only with X∈
+Bc, the construction of the state is still valid, and, even if th e resulting two-point function is
+not of Hadamard form, we can still apply the procedure to comp ute expectation values of φ2.
+Unfortunately, for fields that contain derivatives, it is no t anymore guaranteed that the point
+17splitting regularization works for these states (they are a diabatic states of suitable order, in the
+sense given in [JS02]).
+We shall conclude this subsection by noticing that, since a[X] is positive, τ[X](t) can always
+be inverted, furthermore, on Bcsome useful inequalities hold. We shall summarize them in th e
+next proposition:
+Proposition 4.3. The following inequalities hold for every XinBcwith2/3<c<1and for
+t∈[0,t0]
+c≤X(t)
+t≤1,1≤t
+X(t)≤1
+c,/parenleftbiggt
+t0/parenrightbigg1
+c
+≤a[X](t)
+a0≤t
+t0.
+The proof descends from a straightforward application of th e definition of aandτout ofH
+and from the bounds satisfied by the elements of Bc.
+4.3 Expectation value of φ2on the states ω1,0, and further renormalization
+freedom.
+We are now in place to discuss the form /a\}b∇acketle{tφ2/a\}b∇acket∇i}htω1,0constructed out of point splitting regularization
+on the quasi-free state ω1,0introduced above in (13), namely we can construct it by
+/a\}b∇acketle{tφ2(x)/a\}b∇acket∇i}htω1,0= lim
+y→x[ω1,0(x,y)−H(x,y)]+αR(x)+βm2(22)
+whereHis the Hadamard parametrix (7) constructed in the spacetime (M,g), andαandβ
+are renormalization constants. At this point it is importan t to stress that αandβare really
+constants, in the sense that they do not depend on the form of t he spacetime. In other words, if
+they can be fixed in a specific spacetime, they are fixed on all of them. This is a consequence of
+the general covariance presented in [BFV03] which uses the h ypothesis that the fields transform
+covariantly under isometries used in combination with spac etime deformation arguments.
+According to the discussion presented in [DFP08] we would li ke to have Minkowksi spacetime
+as asolution of thesemiclassical equation (16), whentheMi nkowksi vacuumis employed. Thisis
+tantamount to require that the expectation value of φ2on the Minkowksi vacuum in Minkowksi
+spacetime vanishes and this fixes the renormalization const antβto a certain value. We notice
+also that the other renormalization constant αcan be used in order to adjust the parameter
+Hcpresent in the equation (18), hence from now on we shall consi der it as a renormalization
+constant [DFP08].
+Once we have a prescription to fix the renormalization freedo m we can start to discuss
+the Hadamard regularization; to this end we need to know the H adamard parametrix Hin
+some details. For our purposes, since the modes χare easily treated when analyzed on the
+conformal Minkowksi background, it would be desirable to pe rform the subtraction ω1,0−H
+directly on Minkowksi spacetime. To this avail we have to ana lyze howω1,0−Htransforms
+under the push forward η∗associated to the conformal transformation η: (M,g)→(M,gM)
+introduced in (3). Notice that η∗mapsω1,0toa(t1)a(t2)ω1,0seen as a distribution on Minkowski
+spacetime, and, moreover, since the conformal transformat ion preserves the Hadamard property
+18[Pi09], also a(t1)a(t2)ω1,0satisfies the µSC inM. We can hence regularize it by means of the
+Hadamard parametrix HMconstructed in Minkowksi spacetime for the field theory PMφ= 0
+withPM:=−✷M+a2m2. Notice that, under the conformal transformation η, the mass term
+onMbecomes a position dependent potential a(t)2m2.
+In this respect, we already know that the microlocal spectru m condition, together with the
+causal propagator, are preserved by a conformal transforma tion [Pi09], and this holds also if the
+conformally coupled quantum field has a mass. Of course, unde r a conformal transformation,
+the symmetric part of the Hadamard function is not preserved , hence the coinciding point limit
+of the difference does not vanish in general, actually, taking the same scale λin the logarithmic
+divergence, we obtain that
+lim
+x1→x2/bracketleftbigg
+H(x1,x2)−1
+a(τ1)a(τ2)HM(x1,x2)/bracketrightbigg
+=m2
+8π2loga(τ2)+cR(x2),
+wherecis a fixed constant. Notice that an unexpected logarithmic de pendance in the scaling
+factor log(a(τ)) appears and we have to carefully useit. On the other end, th e term proportional
+toR, which appears when the two different regularization schemes are used, falls into the class
+of the standard regularization freedom [HW01] and, thanks t o the previous discussion we can
+incorporate it in the definition of Hcand forget it.
+Furthermore, since we are only interested in computing the e xpectation value of φ2, it is
+possible to regularize by subtracting an approximated vers ion of the Hadamard parametrix
+constructed just out of the first Hadamard coefficient (see [SV 01, Mo03] for more details)
+H0
+M:= lim
+ǫ→0/bracketleftbigg1
+8π21
+σǫ+V0logσǫ
+λ2/bracketrightbigg
+.
+According to the analysis performed in chapter 4 of [Fr75], t heV0(x,x′) for spacelike separated
+xandx′can be found as the result of the following integral
+V0(x,x′) :=1
+2U(x,x′)/integraldisplay1
+0/bracketleftbiggPMU
+U/bracketrightbigg
+(x,γ(s))ds
+where the integration is done in the affine parameter sof the geodesic γconnecting xand
+x′withγ(0) =xandγ(1) =x′. The previous integral can be easily performed on Minkwoksi
+spacetime where U= 1/(8π2),PM=−✷M+m2a2, andthegeodesics aresimplythestraight lines
+connecting xandx′. Furthermore, if we choose xandx′as (τ,x) and (τ,x′) in the Minkowksian
+coordinates, we obtain
+V0(x,x′) =1
+(4π)2m2a(τ)2.
+We can now write the Hadamard parametrix at first order, for xandx′lying on the same
+Cauchy surface Σ τ, as
+H0
+M(x,x′) :=1
+(4π)2/parenleftbigg2
+σǫ+m2a(τ)2log/parenleftBigσǫ
+λ2/parenrightBig/parenrightbigg
+.
+19Notice that the first two contributions are the same as the one of a massive Klein Gordon field
+propagating in Minkowski spacetime with the mass equal to m2a2(τ) (where here τis fixed by
+the choice of Σ τ). Furthermore, we know that in this case, the coinciding poi nt limit of the
+following expression yields
+lim
+x′→x/bracketleftBigg
+H0
+M(x,x′)−1
+(2π)3/integraldisplay1
+2/radicalbig
+k2+m2a(τ)2eik(x−x′)dk/bracketrightBigg
+=m2a2
+(4π)2/parenleftbigg
+−log(ma)2
+λ2+c/parenrightbigg
+which has to be understood in the sense of distributions, whi lecis a constant. Hence, as a side
+remark, it is clear that the Hadamard regularization and the first order adiabatic regularization
+[Pa69] for the computation of the expectation values of φ2coincides up to the choice of a
+renormalization constant. Furthermore, with a judicious c hoice ofλ(proportional to m), a
+direct computation shows that for every xandx′lying on the same Cauchy surface Σ τ
+lim
+x′→x/bracketleftbigg
+H0
+M(x,x′)−1
+(2π)3/integraldisplay
+Θ(|k|−ma(τ))/parenleftbigg1
+2|k|−m2a(τ)2
+4|k|3/parenrightbigg
+eik(x−x′)dk/bracketrightbigg
+=m2a2
+8π2(−loga+c′),
+and hence we could use the right hand side to regularize the st ates in order to obtain the
+expectation value of φ2. Once again we incorporate the constant c′in the renormalization
+constantβ, that we shall fix a posteriori according to the discussion gi ven above.
+Eventually, also in the computation of the expectation valu e ofφ2, we are free to restrict
+bothxandx′to the same Cauchy surface, determined by τ1=τ2=τand, after the subtraction,
+we can perform the coinciding point limit
+/a\}b∇acketle{tφ2(x′)/a\}b∇acket∇i}htω1,0:=
+lim
+x→x′1
+(2π)3a(τ)2/integraldisplay
+dkeik(x−x′)/bracketleftbigg
+χk(τ)χk(τ)−Θ(|k|−ma(τ))/parenleftbigg1
+2|k|−m2a(τ)2
+4|k|3/parenrightbigg/bracketrightbigg
++βm2,
+(23)
+whereβcan now be fixed to the value −1/(8π2) by the requirement that φ2vanishes in the limit
+offlat spacetime, furthermore, theintegrals presentabove aretobeintendedinthedistributional
+sense. Notice that the log( a) terms are not manifestly present anymore in the expectatio n value
+ofφ2. We conclude this section with a remark. Since the map t→τis a diffeomorphism,
+the previous expression can be easily given in cosmological coordinates, just employing a(t)
+andχk(t) at the place of a(τ) andχk(τ), where, as discussed above, a(τ) = (a◦τ−1)(τ) and
+χk(t) := (χk◦τ)(t).
+4.4 Recursive constructions and bounds satisfied by χkand by the auxiliary
+functional Φ
+The aim of the present section is to find suitable bounds satis fied byω1,0(φ2), for small values
+ofaand close to t= 0. In order to accomplish this task we need some better contr ol on the
+explicit form of the χkappearing in (23). Hence, here we shall give a recursive cons truction
+20forχk, used in the definition of ω1,0usingmas a perturbation parameter. While the analysis
+of the state, and of the modes χkin particular, is easier if performed in the conformal time τ,
+the form of the differential equation (16) suggests the use of t he cosmological time t. Here, as
+previously said since the map τfrom the cosmological time to the conformal is a diffeomorphis m,
+without ambiguity we shall write χk(t) for (χk◦τ)(t) whereχk(τ) is a solution of (9) with the
+initial conditions (11). Following similar ideas to the one presented in [FH90] though in another
+context, we can write χk(τ) by means of perturbation theory, hence in term of a Dyson ser ies
+over the massless solution χ0
+kand treating m2a(τ)2as the perturbation potential (a similar
+construction can be found in the proof of Theorem 4.5 in [DMP0 9a]). Finally, since we would
+like to use the cosmological time instead of the conformal on e, we can simply compose χkwith
+τ(writingχk(τ(t))) and then change the coordinates in every integral appear ing in that series
+from the conformal time to the cosmological one. Following t his procedure we obtain
+χk[a,τ](t) =∞/summationdisplay
+n=0χn
+k[a,τ](t), (24)
+where,
+χn
+k[a,τ](t) =−/integraldisplayt
+0sin(k(τ(t)−τ(t′)))
+ka(t′)m2χn−1
+k[a,τ](t′)dt′, (25)
+and
+χ0
+k[a,τ](t) =e−ikτ(t)
+√
+2k.
+The advantage of usingthe cosmological time is in the fact th at it becomes clear how χkdepends
+onτ. Notice that τappears only in sin( k(τ(t)−τ(t′))) of (25) and in the form of χ0
+k. This
+decrease the number of functional derivatives that needs to be performed in the computation of
+the functional derivatives of χkwith respect to the small variations δXofX. The first thing
+we would like to check is that the perturbative construction is well defined, namely let us show
+that the sum in (24) converges uniformly. We shall state the r esult about convergence in the
+following proposition and, although the proof is almost str aightforward we would like to give it
+since it contains some useful remarks that we shall use also l ater.
+Proposition 4.4. Let(M,g)be the flat FRW spacetime determined by X∈Bcaccording to
+the statement of proposition 4.1, where ais its scaling factor and τthe corresponding conformal
+time. Then the series (24), constructed over (M,g), converges absolutely to χkon[0,t0], where
+χkis a solution of equation (9), when read in conformal time, and, moreover, it enjoys the
+initial condition (11). Furthermore χn
+k, given in (24), andχksatisfy the following bounds
+|χn
+k(t)| ≤1√
+2k(mt)2n
+n!,|χk(t)| ≤1√
+2kexp/parenleftbiggm2a(t)t
+k/parenrightbigg
+,|χk(t)| ≤1√
+2kexp/parenleftbig
+m2t2/parenrightbig
+.
+Introducing the following notation
+χ>n
+k[a,τ](t) =∞/summationdisplay
+i=n+1χi
+k[a,τ](t),
+21we have also that/vextendsingle/vextendsingleχ>n
+k(t)/vextendsingle/vextendsingle≤Cn(t)1
+kn+1
+2.
+Proof.By direct inspection, under the hypotheses stated in the pro position, if the series con-
+verges, it has to tend to a solution of the equation of motion ( 9) which satisfies the wanted initial
+conditions (11). More precisely, notice that
+/parenleftbigg∂2
+∂τ2+k2/parenrightbigg
+χn
+k(τ)+m2a2χn−1
+k(τ) = 0 and/parenleftbigg∂2
+∂τ2+k2/parenrightbigg
+χ0
+k(τ) = 0.
+Hence, let us apply the operator realizing the equation of mo tion (9) to the truncated series
+(χk−χ>n
+k). Using the inequalities given in the proposition, we obtai n that the absolute value of
+thereminderofthelatteroperationissmallerthen a2
+0m2(mt0)2n/(n!√
+2k), whichtendsuniformly
+to zero over [0 ,t0] for largen. At the same time, for every n>0, limτ→−∞eikτχn
+k(τ) = 0 and
+the same holds for the first τ−derivative of χn
+k.
+What remains to be shown is the convergence and the validity o f the estimates given in the
+statement of the proposition. We proceed analyzing the n−th element of the sum. Looking at
+the recursion relations (25), it is a straightforward task t o obtain
+|χn
+k(t)| ≤/integraldisplayt
+0a(t′)
+km2|χn−1
+k(t′)|dt′,|χn
+k(t)| ≤/integraldisplayt
+0(τ(t)−τ(t′))a(t′)m2|χn−1
+k(t′)|dt′.
+Theexponentialboundspresentedinthepropositioncanbeo btainednoticingthat, W(t1,...,tn)
+being any totally symmetric integrable function,
+/integraldisplay
+0≤t1≤t2···≤tn≤tW(t1,...,tn)dt1...dtn=1
+n!/integraldisplay
+[0,t]nW(t1,...,tn)dt1...dtn.(26)
+Hence, out of (26), for any V(t),
+1+/integraldisplayt
+0dt1V(t1)+/integraldisplayt
+0dt1V(t1)/integraldisplayt1
+0dt2V(t2)+···= exp/integraldisplayt
+0dt1V(t1).
+Together with |χ0
+k|= (2k)−1
+2, the last observation entails both the absolute convergenc e of the
+series and the following bounds
+|χn
+k(t)| ≤1√
+2kn!/parenleftbigg/integraldisplayt
+0m2(τ(t)−τ(t′))a(t′)dt′/parenrightbiggn
+,
+|χk(t)| ≤1√
+2kexp/integraldisplayt
+0m2
+ka(t′)dt′,|χk(t)| ≤1√
+2kexp/integraldisplayt
+0m2(τ(t)−τ(t′))a(t′)dt′.
+Finally, since a(t) is growing monotonically and since τ(t)−τ(t′) =/integraltextt
+t′a(t′′)−1dt′′, the estimates
+presented in the proposition descend straightforwardly.
+22Thepreviouspropositionyields suitableestimates for χk[a,τ](t). We shallpresent thederiva-
+tion of further useful estimates also for its functional der ivativeDχk[a,τ,δX](t) with respect to
+the small variation δXofX, in appendix A.3. We shall conclude this subsection, by disc ussing
+some properties of the auxiliary functional Φ on Bcdefined as
+Φ[X] := 2π2a2/a\}b∇acketle{tφ2/a\}b∇acket∇i}htω1,0 (27)
+which corresponds to the expression (23) multiplied by 2 π2a2and evaluated in the proper time
+tinstead of the conformal time τ. For that expression we have the following proposition whos e
+long and technical proof can be found in appendix A.4.
+Proposition 4.5. The following inequality holds on Bc
+|Φ[X](t)| ≤C1(a0,c,t0)t.
+Furthermore, the functional derivative7ofΦ, with respect to the small perturbation δXofX,
+satisfies the following inequality
+|DΦ[X,δX](t)| ≤C2(a0,c,t0)t0/ba∇dblδX/ba∇dblB,
+whereC1(ℓt0,c,t0)andC2(ℓt0,c,t0)are uniformly bounded for t0→0whenℓandcare fixed.
+Above/ba∇dbl·/ba∇dblBis the norm of the Banach space Bintroduced in 4.1.
+4.5 Existence of exact solutions of the semiclassical Einst ein equations out
+of initial data given at the beginning of the universe
+Inthissubsection, weshallpresentthemainresultofthisp aper,namelytheexistenceofsolutions
+of the semiclassical Einstein equations uniquelydetermin ed by some initial conditions andby the
+form of the state ω1,0in particular. Later on we shall generalize the result to oth er Hadamard
+states in C(M). The proof we shall give is very similar to the Picard-Linde l¨ of proof of existence
+of solutions of first order differential equations together wi th the complication that the potential
+is a functional of the solution and not simply a function. Non etheless, we shall show that it will
+be possible to use similar methods thanks to the estimates pr ovided by proposition 4.5. Hence,
+let us start stating the following proposition which we shal l use in order to prove that there exist
+solutions of (16),
+Proposition 4.6. Fixℓand leta0=ℓt0, then, ift0is sufficiently small, the image of Bcunder
+the map Tdefined in (20), is contained in Bc. Furthermore, on Bc,Tis a contraction.
+Proof.First of all we have to show that TmapsBctoBc. In this respect, proposition 4.2 and
+the subsequent remark ensure that we can compute Φ for every e lementXofBc; hence we have
+to prove that
+/ba∇dblT(X)−f0/ba∇dblB≤1−c
+2,
+7A nice introduction of these concepts can be found in [Ha82].
+23where the norm /ba∇dbl·/ba∇dblBis the one introduced in definition 4.1 of Bcandf0is the center of the
+ballBcintroduced in (19). Fixing the two constants C1= 120m2H−2
+candC2= (1−c)H−2
+c, the
+previous inequality is equivalent to
+−C2+(2−c)X2(t)≤C1X4(t)
+a2(t)Φ(t)≤X2(t)∀t∈[0,t0].
+SinceX2≤t2≤t2
+0, ift0is sufficiently small, this chain of inequalities is satisfied if
+|Φ(t)| ≤1
+C1/vextendsingle/vextendsingle/vextendsingle/vextendsinglea2(t)
+X2(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∀t∈[0,t0]
+which holds true for a certain t0thanks both to the bounds, satisfied by the elements of Bc,
+given in proposition 4.3 and to the estimates for |Φ[X](t)|found in proposition 4.5.
+The next step consist to show that Tis a contraction on Bcwith respect to the norm of B,
+that is, for every X1andX2inBc
+/ba∇dblT(X1)−T(X2)/ba∇dblB<C/ba∇dblX1−X2/ba∇dblB,
+with 0< C <1. This property descends once more from the results stated i n proposition 4.5,
+noticing that for every 0 ≤λ≤1 we can consider
+Xλ:=λX1+(1−λ)X2=X2+λ[X1−X2],
+whereXλis inBc. Furthermore, indicating by δXthe difference X1−X2, we have
+/vextendsingle/vextendsingle/vextendsingle/vextendsingled
+dtT(X1)(t)−d
+dtT(X2)(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1
+0d
+dλd
+dtT(Xλ)(t)dλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sup
+λ∈[0,1]/vextendsingle/vextendsingle/vextendsingle/vextendsingleDd
+dtT(Xλ,δX)(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
+In order to conclude the proof we have to analyze the function al derivative ofdT
+dtinBc. In
+this latter step, the contribution which requires some care is the one arising from the functional
+derivative of X4a−2Φ, which, using both the estimates (33) derived in the the app endix and the
+result of proposition 4.5, can be shown to satisfy the inequa lity
+sup
+t∈[0,t0]/vextendsingle/vextendsingle/vextendsingle/vextendsingleD/parenleftbiggX4Φ
+a2/parenrightbigg
+[X,δX](t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C3t0/ba∇dblδX/ba∇dblB,
+for everyXinBc, whereC3is some constant. It is not difficult to show that similar estim ates
+hold also for the other contributions to the functional deri vative ofdT
+dt. Hence, there is certainly
+a sufficiently small t0for whichC <1 and hence the map Tacts as a contraction therein.
+Theorem 4.1. Fixℓand leta0=ℓt0. Ift0is chosen sufficiently small, it exists a unique
+solutionX=H−1of(16)inBcwhich can be constructed recursively starting from the mass less
+solutionX0:=H−1
+0implicitly given in (17). In other words the sequence
+Xn:=T(Xn−1),
+converges in Bcto a solution of the semiclassical Einstein equations.
+24Proof.On account of proposition 4.6, the proof descends out of a str aightforward application of
+the Banach fixed point theorem to the contraction TonBc.
+Before concluding this subsection we would like to briefly co mment on the result obtained
+above. First of all notice that the method to obtain the solut ion is constructive, in the sense
+that for every n,Xnis closer to the solution then Xn−1. Furthermore, notice that if we start
+fromH0, which is a smooth function in Bc, everyHnis also smooth, because Tmaps smooth
+functions to smooth ones. Unfortunately out of the previous theorem, we cannot conclude that
+the foundsolution is smooth. In order to address this proble m we should discussthe convergence
+of the series Xnin the Fr´ echet spaces of smooth functions C∞[0,t0], but this is out of the scope
+of the present paper. For our purpose, on account of the prece ding discussion, it is enough to
+notice that we can find a smooth spacetime Xnas close as we want, in the norm of B, to the
+the exact solution Xprovided by the previous theorem. Furthermore, the regular ity displayed
+by the generic element of Bcis enough in order for equation (16) to be meaningful.
+Asdiscussedatthebeginningofthissectionwehaveusedonl ypartoftheinformationpresent
+in the semiclassical Einstein equations, namely only the tr ace. What is left is completely set
+once the parameter a0is fixed in agreement with the expectation value of T00. Let us start
+noticing that in equation (16) nothing depends explicitly o na0and actually, also in /a\}b∇acketle{tφ2/a\}b∇acket∇i}htω1,0
+as given in equation (23), a0does not really appear. The latter statement becomes eviden t,
+re-scaling everything with respect to a0, that is, using k:=a−1
+0k,τ:=a0τ,a:=a−1
+0aand
+χr
+k(τ) =√a0χk(τ) which now satisfy the equation
+∂2
+∂τ2χr
+k(τ)+/parenleftBig
+k2+m2a/parenrightBig
+χr
+k(τ) = 0.
+Hence, once a solution of (16) is obtained, we can always find o ther solutions by choosing a
+differenta0. In this sense, a0is really a free parameter that can be fixed using the informat ion
+stored in the expectation value of T00. Actually, it can be shown that ω1,0(T00) diverges when
+H=X−1→ ∞; hence, the choice the state ω1,0really provides the exact solution presented
+here, in the sense that the latter condition permits really t o start the iterative construction of
+H=X−1out of the upper brunch H0of the massless solution (17).
+We conclude the present section noticing that the same resul t about the uniqueness and the
+existence of the solution of the semiclassical Einstein equ ations holds also for other choices of
+Hadamard states in C(M) (see definition 3.1) provided suitable conditions are sati sfied. We
+have in fact the following theorem
+Theorem 4.2. LetωA,B∈C(M)be constructed out of AandBinC2(R+), and such that,
+with respect to the auxiliary function f(k) :=A(k)B(k)k2, the following conditions are satisfied
+a)Bis rapidly decreasing,
+b)A(k)B(k)and|B(k)|2are contained in L1(R+,kdk),
+c)f(0) =f′(0) = 0,
+25d)f′′,f′∈L1(R+,dk).
+Leta0=ℓt0; then, ift0is sufficiently small, a unique solution of the semiclassical Einstein
+equations exists.
+Proof.The proof of the present theorem can be performed following t he one of theorem 4.1. For
+this reason, here, we shall discuss only the differences. Let u s start noticing that what changes
+in this case is the value of /a\}b∇acketle{tφ2/a\}b∇acket∇i}ht. More precisely, instead of dealing with the map T, we have to
+deal with T′which is defined as follows
+T′(X) :=T(X)+X4
+1−H2cX2ΦA,B
+a2
+where Φ A,B:= 2π2a2/parenleftbig
+/a\}b∇acketle{tφ2/a\}b∇acket∇i}htωA,B−/a\}b∇acketle{tφ2/a\}b∇acket∇i}htω1,0/parenrightbig
+which can be rewritten as
+ΦA,B(t) =/integraldisplay∞
+0/bracketleftBig
+2|B(k)|2χkχk(t) +A(k)B(k)χkχk(t)+A(k)B(k)χkχk(t)/bracketrightBig
+k2dk.
+Since we already have the estimates for χk−χ0
+kgiven in the appendix A.3, it is useful to divide
+ΦA,Bin two parts, namely
+ΦA,B= Φ0
+A,B+Φ+
+A,B
+where Φ0
+A,Bis constructed as Φ A,Bwithχ0
+kin the place of χk. Let us introduce the closed set
+B′
+dformed by the elements of Bsuch that
+B′
+d:=/braceleftbigg
+X∈B,1−td≤d
+dtX(t)≤1+td/bracerightbigg
+.
+We have that Φ+
+A,Bsatisfies on B′
+d, similar inequalities as the ones stated in proposition 4.5 for
+Φ. In order to verify the last statement, we could make profita ble use of the fact that B(k) is
+rapidly decreasing and that B(k)χkandA(k)χkare locally square integrable with respect to
+the measure k2dk. Hence the first bound
+|Φ+
+A,B(t)| ≤Ct, ∀t∈[0,t0]
+descends straightforwardly from the inequalities in propo sition 4.4, while the second one, about
+the variation of Φ+under the small perturbation δXofX,
+|DΦ+
+A,B(t)| ≤Ct0/ba∇dblδX/ba∇dblB,∀t∈[0,t0]
+can be proved using the results stated in the appendix A.3. Φ0
+A,Bdoes not satisfy similar
+inequalities, and it is for such a reason that we have introdu cedB′
+d, a closed set different then
+Bc. For Φ0
+A,B, there exists a constant δ1such that
+|Φ0
+A,B(t)| ≤δ1,∀t∈[0,t0]
+26and this inequality descends straightforwardly form the pr operties a) and b) stated in the hy-
+potheses and from the fact that |χ0
+k|kis constant. The variation of Φ0
+A,Bunder the small
+perturbation δXofXinvolves only the variation of e2ikτ, present in the second and third term
+of Φ0
+A,B. From properties c) and d), it descends the following unifor m decay
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
+A(k)B(k)k2e2ikτ(t)dk/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C
+τ2(t),∀t∈[0,t0].
+Furthermore, operating in a similar way as in appendix A.2, s ince|δX(t)| ≤t/ba∇dblδX/ba∇dblB, we obtain
+the uniform estimate
+sup
+t∈[0,t0]/vextendsingle/vextendsingle/vextendsingle/vextendsingleDτ(t)
+τ2(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Csup
+t′∈[0,t0]/vextendsingle/vextendsingle/vextendsingle/vextendsinglet′2
+X2(t′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/ba∇dblδX/ba∇dblB
+valid for sufficiently small t0inB′
+dand for some constant C. Combining these arguments we
+easily obtain
+|DΦ0
+A,B(t)| ≤C/ba∇dblδX/ba∇dblB.∀t∈[0,t0]
+Finally, let us notice that the results given in proposition 4.5 hold also on B′
+d.
+Although the behavior of Φ0
+A,Bis worse then that of Φ+
+A,Bor of Φ A,B, the inequalities given
+above are sufficient in order to obtain the existence and uniqu eness of the solution in B′
+d. This
+is possible because, on B′
+d,|Xa−1(t)| ≤Cwhich is not valid for the generic element of Bc.
+Actually, for a sufficiently large d(depending on δ1in particular) and a sufficiently small t0, the
+mapT′introduced above, is contractive in B′
+das it can be seen adapting the proof of proposition
+4.6 to encompass the case under the present investigation. O ut of this observation the proof
+of the present proposition can then be easily concluded, by a pplying the Banach fixed point
+theorem on B′
+d.
+5 Further considerations on the semiclassical solutions
+5.1 Expectation value of /a\}b∇acketle{tφ2/a\}b∇acket∇i}htat late times
+Out of direct inspection we notice that, at late times, the ma ssless solution (17) tends to the de
+Sitter spacetime. We would like to see if such behavior persi sts also for massive fields8. To this
+end notice that in [DFP08] it is argued that, in the case of a ma ssive conformally coupled scalar
+field, if the expectation value of φ2has the form βm2+αRthen, the same behavior at late times
+can be inferred. Such an estimate was obtained analyzing the adiabatic approximation for the
+state. It would be desirable to see if the assumption made in [ DFP08] for the expectation value
+ofφ2is reliable for the exact solution of the semiclassical Eins tein equations.
+Westartnoticingthat, inthecaseofanexactdeSitterspace time, theexpectationvaluesof φ2
+in the Bunch Davies [BD78] state has exactly that behavior, a nd hence, there is only one choice
+of the cosmological constant which solves the semiclassica l Einstein equations. Unfortunately, it
+is very difficult to compute the evolution of the expectation v alue ofφ2in the exact solution we
+8See also the work [PRV08].
+27have found before. Despite the difficulties present in the ana lysis of the semiclassical solution
+at finite times, we would like to stress that if a de Sitter phas e exists at late times, it has to be
+exactly the one found in [DFP08]. Such a remark originates fr om the fact that the expectation
+value ofφ2, in an expanding flat universe ( H >0), is state independent (at least on the class of
+homogeneous pure Hadamard states) as asserted in the follow ing theorem.
+Theorem 5.1. In an expanding H >0flat FRW with smooth scale factor awhich diverges in
+the limitt→ ∞, choosing a quasi-free state ωA,Bwhose two-point function is as in (13)and of
+Hadamard form, the expectation value of φ2inωA,Bequals the one computed on ω1,0.
+Proof.The proof of this theorem can be obtained considering the for mχk. Since both ωA,B
+andω1,0are Hadamard states, it holds for their two-point functions
+ωA,B(φ2(x))−ω1,0(φ2(x)) =
+1
+(2π)31
+a(t)2/integraldisplay/bracketleftBig
+2|B(k)|2χkχk(t) +A(k)B(k)χkχk(t)+A(k)B(k)χkχk(t)/bracketrightBig
+dk(28)
+wherex= (t,x) in cosmological coordinates and k=|k|. Because of theorem 3.2, B(k) is
+rapidly decreasing and A(k) of polynomial growth, hence we can pass the coinciding poin t limit
+under the sign of k-integration. We aim to prove that the preceding expression vanishes in the
+limita→ ∞. To this end we shall show that the previous kintegration is bounded in time. To
+prove the last statement we have to analyze the behavior of χkfor large times. To this avail, let
+us consider the functions χkin cosmological time ( χk(t) := (χk◦τ)(t)), and let us notice that
+they have to satisfy the following equation
+¨χk+H˙χk+/parenleftbiggk2
+a2+m2/parenrightbigg
+χk= 0
+where ˙χkand ¨χkindicate respectively the first and second derivative of χkwith respect to the
+cosmological time t. Hence the previous equation descends straightforwardly o ut of a change of
+variables in (9). Let us introduce the positive quantity
+Q(t,k) :=˙χk˙χk(t)+/parenleftbiggk2
+a2+m2/parenrightbigg
+χkχk(t).
+We would like to show that, at fixed k,Q(t,k) decreases in time. To this end notice that, since
+H≥0
+˙Q(t,k) =−2H|˙χk(t)|2−2k2
+a(t)2H(t)|χk(t)|2≤0,
+where we have indicated by ˙Q(t,k) the∂
+∂tQ(t,k). Hence, for every t>t0, we have that
+Q(t,k) =Q(t0,k)+/integraldisplayt
+t0˙Q(t′,k)dt′≤Q(t0,k).
+28Out of the form of Q(t,k) we obtain that, for every t>t0,
+|χk(t)|2≤Q(t0,k)
+m2.
+It is now possible to use the preceding inequality to estimat e thek-integral present in (28), and
+since both Q(t0,k)|B(k)|2andQ(t0,k)|A(k)B(k)|are integrable in k, we have that it exists a
+positive constant C(t0), such that, for every xin the future of the surface Σ t0,
+/vextendsingle/vextendsingleωA,B(φ2(x))−ω1,0(φ2(x))/vextendsingle/vextendsingle≤C(t0)
+a(t)2.
+Out of the preceding remark, we can straightforwardly evalu ate the limit t→ ∞and, sincea(t)
+diverges per hypothesis therein, the difference vanishes.
+5.2 Variance of /a\}b∇acketle{tT/a\}b∇acket∇i}htwhen smeared on large volume regions
+Asdiscussedintheintroduction, thesemiclassical Einste inequationsarevalidwhenthequantum
+fluctuations of the expectation value of the stress tensor ar e negligible.
+In this paper, in order to obtain the dynamics of the spacetim e, due to the large symmetry
+present in the model under investigation, we have used only t he trace of the stress tensor. Notice
+that the anomalous terms in the trace are c−numbers; hence, they have vanishing fluctuations,
+and this already means that, in the case of massless fields, th e found semiclassical equation and
+its solution are meaningful at every times (not only when the curvature is small).
+Furthermore, in the case of a massive field conformally coupl ed with the curvature, the
+source for the fluctuations in the trace of the stress tensor c an only be the expectation value of
+φ2(x). Unfortunately we have to notice that point-like fields lik eφ2(x), although represented by
+smooth functions, could have divergent variance. This is in deed the case even if the Hadamard
+regularization is employed, as can be seen noticing that in
+φ2(x)⋆Hφ2(y) =φ2(x)φ2(y)+4H(x,y)φ(x)φ(y) +2H(x,y)H(x,y) (29)
+the product H(x,y)H(x,y) diverges in the coinciding point limit. Above His the Hadamard
+singularity and ⋆His the product in the extended algebra of observables F(M), as discussed in
+[BF09, BDF09]. Some further comments on such a product are gi ven in appendix A.1. We shall
+use that expression in the computation of the variance of φ2(f), namely on a stat ωit looks like
+∆ω(φ2(f)) :=ω(φ2(f)⋆Hφ2(f))−ω(φ2(f))ω(φ2(f)). (30)
+Furthermore, ifthestateisquasi-free, indicatingby Wthesmoothpartofthetwo-point function,
+it becomes
+/integraldisplay
+[2W(x,y)W(x,y)+4H(x,y)W(x,y)+2H(x,y)H(x,y)]f(x)f(y)dµ(x)dµ(y)
+29which, on the state ω1,0we have chosen, is equal to
+2/integraldisplay
+ω2
+1,0(x,y)f(x)f(y)dµ(x)dµ(y).
+whereω2
+1,0is the product of distribution ω1,0·ω1,0. Although ω2
+1,0is a well defined distribution,
+due to the form of its wave front set, its coinciding point lim it is not well defined and actually it
+diverges. Hence, the expectation value of φ2(x) has divergent fluctuations, but, luckily enough,
+we can overcome this difficulty by a suitable smearing procedu re we are going to introduce.
+To this end, we could use once again the spatial symmetry and c onsider a smearing function
+fn1,n2such that, for large n1andn2it tends to have support on the whole Cauchy surface Σ τ,
+and it is normalized in such a way that its integral on Mis equal to one. More precisely, we
+consider a compactly supported smooth function f∈C∞
+0(M) centered in xτ:= (τ,0), and such
+that
+f(xτ) = 1,/integraldisplay
+Mfdµ(g) = 1,0≤f(x)≤1.
+Notice that, in writing the preceding expression, we are usi ng the conformal coordinates on M.
+We can then generate the desired fn1,n2as
+fn1,n2(τ′,x) =n1
+n3
+2f/parenleftbigg
+n1(τ′−τ)+τ,x
+n2/parenrightbigg/radicalbig
+|g(n1(τ′−τ)+τ,x/n2)|/radicalbig
+|g(τ′,x)|(31)
+and, as anticipated before, for large n1,fn1,n2tends to have support on Σ τwhile, for large n2,
+even if its spatial support becomes larger and larger,/integraltext
+Mfn1,n2dµ(g) = 1.
+We can then smear the fields in (16) and subsequently in (30) wi thfn1,n2and analyze the
+behavior of the smeared quantity for large n1andn2. In other words, notice that, since the
+trace of the Einstein tensor Gis smooth and equal to −R, in the limit ( n1,n2)→ ∞
+lim
+(n1,n2)→∞/a\}b∇acketle{tgµνGµν,fτ
+n1,n2/a\}b∇acket∇i}ht=−R(τ),
+and, thanks to the continuity properties of R, the result does not depend on the order in which
+the limits are taken. The analysis of the trace of /a\}b∇acketle{tT/a\}b∇acket∇i}htfollows similarly. In fact, the anomalous
+part of the trace depends continuously on the coefficients of t he metric, and its derivatives, up
+to the second order, while the expectation value of /a\}b∇acketle{tφ2(x)/a\}b∇acket∇i}hthas vanishing wave front set and,
+hence, it is continuous too. Thus, the equation −R= 8π/a\}b∇acketle{tT/a\}b∇acket∇i}htω1,0, together with its solutions,
+holds exactly in the same way for the smeared quantity.
+We proceed to analyze the variance of these quantities, and w e notice immediately that the
+variances of the geometric entities present in (16) vanish f or largen1andn2, no matter in which
+order the limits are performed. We have to be more careful whe n analyzing the variance of the
+expectation values of /a\}b∇acketle{tφ2/a\}b∇acket∇i}htbut, luckily enough, we have the following theorem which sho ws that
+the fluctuations of φ2vanish if the limits are taken in a suitable order.
+30Theorem 5.2. Letfn1,n2be some smearing functions constructed as in (31)out of some f
+with the properties stated above. Fix n1, then the variance of φ2(fn1,n2), computed according to
+formula (30)and evaluated on the state ω1,0, vanishes for large n2in the weak sense, that is
+lim
+n2→∞∆ω1,0(φ2(fn1,n2)) = 0.
+Proof.First of all notice that, ω1,0is of Hadamard form and hence its wave front set satisfies
+the microlocal spectrum condition. This implies that the po intwise product of ω1,0with itself,
+that we shall indicate by ω2
+1,0, is a well defined distribution because the H¨ ormander crite rion for
+multiplication of distributions is fulfilled (the sum of the wave front set of the distribution ω1,0
+with itself does not contain the zero section). Furthermore , theorem 8.2.10 of [H¨ o89] yields that
+thewave front set of theproduct ω2
+1,0hasto becontained in ( WF(ω1,0)∪{0})+(WF(ω1,0)∪{0}),
+where the sum is meant in the cotangent space. This implies, t hat the singular support of ω2
+1,0
+can only contain points connected by a lightlike geodesic. H ence, we can divide in two parts the
+distribution
+ω2
+1,0(fn1,n2⊗fn1,n2) :=ω2
+1,0u(fn1,n2⊗fn1,n2)+ω2
+1,0u′(fn1,n2⊗fn1,n2).
+Above we have split the distribution ω2
+1,0multiplying the test functions by a suitable partition
+of unitu+u′= 1 onM×Mand using the linearity of ω2
+1,0. We shall now discuss how we have
+to construct uin order to have that the intersection of the singular suppor t ofω2
+1,0u′with the
+support offn1,n2⊗fn1,n2is empty for every n2and for fixed n1. Consider a compactly supported
+smooth function ρonR3equal to one on a neighborhood of the origin. Then we shall ind icate
+byuthe smooth function on M×M,
+u: ((τ,x);(τ′,x′))/ma√sto→ρ(x−x′),
+where we have used standard coordinates introduced above. L et us start analyzing ω2
+1,0u, and
+suppose to test it on a test function h∈C∞
+0(M×M) of the following form
+h((τ1,x);(τ1,y)) :=ft(τ1,τ2)f+(x−y)f−(x+y),
+whereft∈C∞
+0(I×I) andf+,f−∈C∞
+0(R3). Furthermore, these functions are chosen in such
+a way that his positive and, for a fixed n1,h≥fn1,1⊗fn1,1. Hence, due to the translation
+invariance satisfied by ω2
+1,0and the multiplication by the cutoff u, the following continuity
+condition holds true
+|ω2
+1,0u(h)| ≤C
+/summationdisplay
+|a|≤q/ba∇dblDaft/ba∇dbl∞
+/ba∇dblf+/ba∇dbl1
+/summationdisplay
+|b|≤q/ba∇dblDbf−/ba∇dbl∞
+
+for suitable values of Candq. Notice that, because of the localization introduced by u,Cdoes
+not depend on the support of f−. Out of this observation we have that, for hn∈C∞
+0(M×M)
+defined ashn((τ1,x);(τ1,y)) :=h((τ1,x/n);(τ1,y/n))/n6,
+lim
+n→∞ω2
+1,0u(hn) = 0.
+31Thelatter limit implies alsothat lim n2→∞ω2
+1,0u(fn1,n2⊗fn1,n2) = 0, because, for fixed n1, thanks
+to the special choice of ft,f+andf−and to the positivity of ω2
+1,0u,
+ω2
+1,0u(fn1,n2⊗fn1,n2)≤ω2
+1,0u(hn2).
+We have now to analyze the second contribution to ω2
+1,0, namelyω2
+1,0u′, and we notice that,
+per construction, the singular support of ω2
+1,0u′has vanishing intersection with the support of
+fn1,n2⊗fn1,n2. Hence, when tested on fn1,n2⊗fn1,n2,ω2
+1,0u′has a smooth integral kernel, and,
+for everyn2and for sufficiently large n1,
+ω2
+1,0u′(fn1,n2⊗fn1,n2)
+is represented by an ordinary integral. We can change the var iable of spatial integration in such
+a way that the preceding operation can be rewritten as
+lim
+n2→∞/integraldisplay
+(ω2
+1,0u′)(τ1,τ2,n2(x−y))n2
+1f(n1τ,x)f(n1τ′,y)dµ(τ1,τ2)dxdy.
+Sincefhascompact supportand( ω2
+1,0u′)isbounded,wecanpassthelimitunderthesignofinte-
+gration. Weobtainthatthelimitforlarge n2vanishesprovidedthatlim n2→∞(ω2
+1,0u′)(τ1,τ2,n2(x−
+y)) = 0.
+Due to the spatial translational and rotational invariance , it is enough to check this last
+requirement for ω1,0(without the square), for a single direction and when one of t he points is in
+the spatial origin of the employed coordinate system. Hence , using the conformal coordinates,
+let us consider x:= (τ1,0) andy:= (τ2,r,0,0), withr >>0. We can rewrite the two-point
+functionω1,0as
+ω1,0(x,y) =2
+(2π)21
+a(τ1)a(τ2)/integraldisplay∞
+0sin(kr)
+rχk(τ1)χk(τ2)kdk,
+where, we have madeuseof the factthat ris chosen to bestrictly positive andthefact that ( x,y)
+is not contained in the singular support of ω1,0. We can now split the k−integral on (0 ,∞) in
+two parts, as similarly done in the proof of proposition 4.5, namely on the interval (0 ,1) and on
+(1,∞). Sinceχk(τ1)χk(τ2) is locally integrable in the measure kdk, due to the estimates given
+in proposition 4.4 and, making use of the Riemann Lebesgue th eorem, we can conclude that the
+contribution on (0 ,1) vanishes when rdiverges. Let us proceed to analyze the k−integral on
+the remaining interval (1 ,∞). Hence, we can now make use of the perturbative constructio n in
+m2for theχkgiven in (24) and analyze the different contributions separat ely. First of all we
+notice that the sum of the contributions to χk(τ1)χk(τ2) of order equal or larger to m4lie in
+L1(R,kdk), and, once more, due to results stated in proposition 4.4, b y the Riemann Lebesgue
+lemma, asrdiverges this contribution tends to zero. We are left with th e analysis of the order
+1 andm2, The former consists of a well known distribution that vanis hes for large r, whereas
+the latter (order m2) can be estimated, using an argument similar to the one used t o treat (42).
+Therefore, for large rand for some constant C, this contribution can be shown to be bounded
+byCr−1(1+log(r)) which tends to zero when rdiverges.
+32This discussion provides a new and stronger physical interp retation of the semiclassical Ein-
+stein equations on spacetime with large spatial symmetry as the cosmological one.
+6 Summary of the results and final comments
+In this paper we have rigorously constructed some solutions of the semiclassical Einstein equa-
+tions in cosmological spacetimes. All these solutions disp lay a phase of power law inflation after
+theinitial singularity. Wehave alsoseenthat, ifastabled eSitter phaseisasymptotically present
+in the future, then the acceleration of the latter does not de pend on the particular homogenous
+pure state in which the quantum theory is evaluated, and henc e it can be considered as a uni-
+versal character. Finally we have discussed the interpreta tion of these equations, showing that,
+when smeared on large spatial regions the equation −R= 8π/a\}b∇acketle{tT/a\}b∇acket∇i}htcontinues to be valid, and, in
+this case, since the fluctuations of /a\}b∇acketle{tT/a\}b∇acket∇i}httend to vanish, it acquires an even stronger interpretation .
+We have been able to derive those results because we have sele cted a class of spacetimes and
+we have used an universal quantization scheme for the matter fields on that class, employing
+ideas typical of local covariance [BFV03]. Furthermore, he re we have also been able to give
+unambiguously a quantum state on every element of the class o f considered spacetimes, using
+and even generalizing, the results in [DMP09a, DMP09b]. Fur thermore, the employed methods
+permit to characterize the quantum states in an intrinsic wa y. In other words, using a lightlike
+initial surface and giving initial data thereon, we provide d a prescription which can be used to
+construct states with a good ultraviolet behavior.
+There are of course many questions that are still open. In pri nciple, similar ideas could be
+used to treat also more complicated problems like, for examp le, the backreaction of collapsing
+matter forming a black hole. Another interesting aspect, th at should be addressed in the future,
+is the relation between the results obtained here on the vani shing of the fluctuations by smearing
+the fields on infinite volume regions and the approach typical of the so called stochastic gravity
+[HV08].
+Acknowledgments.
+I would like to thank Romeo Brunetti, Claudio Dappiaggi, Kla us Fredenhagen, Thomas-Paul
+Hack and Valter Moretti for useful discussions, suggestion s and comments on the subject of this
+paper. Supported in part by the German DFG Research Program S FB 676 and by the ERC
+Advanced Grant 227458 OACFT “Operator Algebras and Conform al Field Theory”.
+A Appendix
+A.1 Deformation quantization and the extended algebra of fie lds
+The standard procedure for quantizing the system (4) will be to choose the symplectic space of
+the theory ( S(M),σ) and then to construct the C∗-algebra generated by the Weyl operator, see
+33for example [Di80]. Here we shall follow another path, that i s we introduce the quantization as
+it is custom in the so called deformation quantization [BF09 , BDF09]. Let us define the off shell
+Borchers-Uhlmann algebra as
+A(M) :=∞/circleplusdisplay
+nC∞
+0(M)⊗Sn
+where only sequences with a finite number of elements are take n into account, ⊗Sstands for
+the symmetric tensor product and the first space ( n= 0) in the previous direct sum is C. The
+elements of A(M) can be alternatively seen as the functionals on smooth field configurations
+ϕ. To wit, to every F:={Fn} ∈A(M), we can associate
+F(ϕ) :=/summationdisplay
+n1
+n!/a\}b∇acketle{tFn(x1,...,x n),ϕ(x1)·····ϕ(xn)/a\}b∇acket∇i}ht,
+whereϕis inC∞(M) and/a\}b∇acketle{t·,·/a\}b∇acket∇i}htis the standard pairing. Out of this transformation the set A(M)
+is interpreted as the set of functionals with smooth functio nal derivatives F(n)and with only a
+finite number of non vanishing F(n). If we indicate by F(n)(0) the functional derivatives of F
+we notice that F(n)(0) :=Fn.
+The set A(M), equipped with the pointwise product becomes a topologica l∗−algebra with
+the complex conjugation taken as the ∗−operation and whose topology is induced by the one of
+the compactly supported smooth functions. Such algebra is c onsidered to be “off shell” because
+no information about the equations of motion has been employ ed in its construction. At the
+end, the equation of motion needs to be implemented singling out the ideal Igenerated by
+elementsPxif(x1,...,x n), wherePxiis the operator realizing the equation of motion applied on
+the variable xi, we shall call this last case “on shell”.
+Thequantizationof A(M)isrealizedintroducingaparticularstarproducton A(M), namely
+letFandGbe two elements of A(M)
+F ⋆G:=/summationdisplay
+nin
+2nn!/a\}b∇acketle{tF(n),E⊗nG(n)/a\}b∇acket∇i}ht (32)
+whereEis the causal propagator (the advanced minus the retarded fu ndamental solution).
+Furthermore, also ( A(M),⋆) is a topological ∗-algebra with the complex conjugation as ∗-
+operation.
+In order to analyze the backreaction we need to deal with the s tress tensor and similar
+fields. Hence, we need to introduce pointwise products of fiel ds (like the one involved in the
+stress tensor) in the algebra of observables. Unfortunatel y this is usually not possible in A(M),
+becauselocal fields, like /a\}b∇acketle{tf(x)δ(x,y),ϕ(x)ϕ(y)/a\}b∇acket∇i}ht, cannotbemultipliedwithrespecttotheproduct
+rule given in (32). Yet, since this kind of fields are needed in our discussion, we shall proceed as
+follows, first of all we isomorphically deform the algebra an d then we enlarge the deformed one
+by including these and other more involved fields. The first st ep consists in the deformation of
+the⋆product introduced in (32) to the so-called ⋆H−product. The latter is realized substituting
+in (32)E(the causal propagator) with −2iH. The new algebra ( A(M),⋆H) turns out to be
+34isomorphic to the old one; further details about this isomor phism can be found in [BF09]. After
+thedeformation, wecansafely enlargethealgebra( A(M),⋆H)to(F(M),⋆H)insuchaway that
+more general local fields like the components of the stress te nsor are included in ( F(M),⋆H). To
+be more precise, F(M) is defined as the set of functionals over smooth field configur ations, i.e.,
+they are infinitely often differentiable andevery F∈F(M) has symmetric functional derivatives
+F(n)(0)∈E′(Mn) (the set of distributions with compact support). Their wav e front set satisfies
+WF(F(n)(0))∩/parenleftBig
+V+∪V−/parenrightBig
+=∅,
+whereV±are the subsets of the cotangent space T∗(Mn) formed by elements whose fiber
+components are future, respectively past, pointing causal or null vectors.
+We proceed now to discuss the form of the states and their inte rplay with the previously
+discussed deformation. Following the discussion presente d in [BDF09, BF09], when we consider
+the enlarged algebra, we have to restrict the class of admiss ible states to the one satisfying
+the Hadamard condition. Furthermore, the deformation must not be visible on the expectation
+values of the original algebra. Thus, in order to preserve th e expectation values of the elements
+ofA(M) under the deformation, the states on ( A(M),⋆H) are constructed out of those on
+(A(M),⋆) by means of the push-forward induced by the isomorphism bet ween the two algebras.
+Let us give a closer look to the effects of this push-forward on t hen−point functions for the
+deformed algebra. To this end, let us fix a state ωon (A(M),⋆), then we shall indicate by : ωn:
+then−point function used in evaluating the elements of ( A(M),⋆H) and we shall call them
+regularized n−point functions . Notice that all : ωn: are obtained from the non regularized
+ones by the generating formula [BF00, HW01]
+:ωn: (x1,...,x n) :=1
+inδn
+δf(x1)...δf(xn)ω/parenleftbigg
+exp/parenleftbigg1
+2H(f⊗f)−iϕ(f)/parenrightbigg/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+f=0
+where eventually the ω(ϕ(x1)·····ϕ(xn)), needs to be substituted by ωn. Notice that, if ω
+satisfies the microlocal spectrum condition, then the all th e :ωn: are smooth functions, hence
+the state ωcan be safely extended to ( F(M),⋆H). Furthermore, the regularization employed in
+this paper coincides with the push forward of the state under the deformation described above,
+in fact it holds that : ω2:=ω−H. Finally, we stress once more that, the stress tensor Tµνand
+φ2constructed in the main text of this paper are elements of F(M).
+A.2 Functional derivatives of aandτ
+In this subsection we shall derive some estimates, for the fu nctionalsaandτ, needed to establish
+the proofs of the main theorems. Let us start by recalling the form of the scaling factor aas a
+functional of X. Fixing theinitial condition a0at thecosmological time t0and its firstfunctional
+derivative we can write
+a[X](t) :=a0exp/braceleftbigg
+−/integraldisplayt0
+tX−1(t′)dt′/bracerightbigg
+, Da [X,δX](t) :=a[X](t)/integraldisplayt0
+tX−2(t′)δX(t′)dt′
+35and, per direct inspection, we have the following estimate v alid for every XinBcandt∈[0,t0],
+|Da[X,δX](t)| ≤/vextendsingle/vextendsingle/vextendsingle/vextendsinglea[X](t)/parenleftbigg1
+t−1
+t0/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglesup
+t′∈[0,t0]/vextendsingle/vextendsingleX−2(t′)t′2/vextendsingle/vextendsingle/ba∇dblδX/ba∇dbl∞≤a0
+t01
+c2/ba∇dblδX/ba∇dbl∞.(33)
+Furthermore, out of both the preceding inequalities and the definition of functional derivative
+as a directional derivative, it holds that
+|a1(t)−a2(t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1
+0Da[Xλ,δX](t)dλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤a0
+t01
+c2/ba∇dblδX/ba∇dbl∞,∀t∈[0,t0]
+whereδX=X2−X1andXλ=X1+λδXand the last inequality descends from (33) together
+with the proposition 4.3.
+Let us continue analyzing the form of the conformal time as a f unctional of aand the first
+functional derivative of τ◦a[X]
+τ[a](t) :=τ0−/integraldisplayt0
+t1
+adt′, D(τ◦a)[X,δX] =Dτ[a,Da[X,δX]]
+and it satisfies the following inequality
+|D(τ◦a)[X,δX](t)| ≤2/integraldisplayt0
+t1
+a/parenleftbigg1
+t′−1
+t0/parenrightbigg
+dt′sup
+t′∈[0,t0]/vextendsingle/vextendsingleX−2(t′)t′2/vextendsingle/vextendsingle/ba∇dblδX/ba∇dbl∞.
+Hence, multiplying by a sufficiently large power of tm we obtain that the inequalities,
+/vextendsingle/vextendsingle/vextendsinglet1/cD(τ◦a)[X,δX](t)/vextendsingle/vextendsingle/vextendsingle≤C(a0,t0,c) sup
+t′∈[0,t0]/vextendsingle/vextendsingleX−2(t′)t′2/vextendsingle/vextendsingle/ba∇dblδX/ba∇dbl∞≤2t1
+c
+0
+a0c/ba∇dblδX/ba∇dbl∞,(34)
+hold in the ball Bcconstructed out of cas introduced in the definition 4.1.
+A.3 Bounds satisfied by χkand its functional derivatives
+In the following we shall derive some useful estimates satis fied by the functions χkconstructed
+perturbatively in (24), this appendix is meant as a completi on of the estimates presented in
+proposition 4.4, where the convergence of (24) was proven. A ll the forthcoming formulas are
+meanttobevalidforeveryspacetimethatarisesfromanelem entXofBcintroducedindefinition
+4.1, along the lines of proposition 4.1 (Notice that they hol ds also on B′
+dintroduced in the proof
+of theorem 4.2). For the same reason t∈[0,t0] everywhere. For our later purpose, in order to
+shorten some formulas, it is useful to introduce the followi ng two functionals
+F1[a](t) :=/integraldisplayt
+0a(t′)m2
+kdt′, F 2[a](t) :=/integraldisplayt
+0(τ(t)−τ(t′))a(t′)m2dt′(35)
+36that satisfy |F1[a](t)| ≤a0
+t0kt2m2and|F2[a](t)| ≤m2t2. Following a procedure similar to the
+one presented in the proof of proposition 4.4, we can derive t he following estimates
+/vextendsingle/vextendsingleχ>n
+k[a](t)/vextendsingle/vextendsingle≤1√
+2k(F1/2[a](t))n+1expF1/2[a2](t). (36)
+In the remaining part of this appendix we shall analyze the fu nctional derivatives of χk−χ0
+k
+with respect to X∈Bc; more precisely we shall compute
+D(χk−χ0
+k)[X,δX] =Da(χk−χ0
+k)[a,τ,Da[X,δX]]+Dτ(χk−χ0
+k)[a,τ,Dτ[X,δX]].
+Let us start from Da(χk−χ0
+k)[a,τ,Da[X,δX]], which, exploiting the properties of the recursive
+series (24), the form of Da[X,δX] and the identity (26), turns out to satisfy the following
+inequality
+/vextendsingle/vextendsingleDa(χk−χ0
+k)[a,τ,Da(X,δX)](t)/vextendsingle/vextendsingle≤eF1/2(t)
+k√
+2km2t/ba∇dblDa/ba∇dbl∞
+where/ba∇dbl·/ba∇dbl∞is the uniform norm on C[0,t0], whileDaχ0
+k= 0. The second contribution is more
+laborious; let us consider the recursive sum and let us conce ntrate on the contribution to the
+n-th order which looks like
+/integraldisplay
+0≤tn≤tn−1···≤teikτ(tn)
+√
+2kn/productdisplay
+i=1sin(k(τ(ti−1)−τ(ti)))
+km2a(ti)dt1...dtn (37)
+where, only in the previous formula, t0corresponds to the time t. The action of the τ-functional
+derivatives on (37) is twofold: On the one hand, we have to der ive every factor of the form
+sin(2k(τ−τ′)) and, on the other hand, we have to derive τinχ0
+k. Let us first of all consider
+the functional derivatives of sin(2 k(τ−τ′)) and, using (26), we obtain that the n−th order is
+bounded by
+1
+(n−1)!1√
+2kFn−1
+1/2(t)·/integraldisplayt
+0m24
+ca0/parenleftbiggt0
+t′/parenrightbigg1
+c
+a(t′)dt′·/ba∇dblδX/ba∇dbl∞
+where/ba∇dbl· /ba∇dbl∞is the uniform norm on the elements of Band where we have used the following
+estimate descending from (34)
+|Dτ[X,δX](t1)−Dτ[X,δX](t2)| ≤4
+ca0/parenleftbiggt0
+t2/parenrightbigg1
+c
+/ba∇dblδX/ba∇dbl∞, t 1>t2.
+The absolute value of the functional derivative of eikτin (37) is also bounded by a similar
+expression
+1
+(n−1)!1√
+2kFn−1
+1/2(t)·/integraldisplayt
+0m22
+ca0/parenleftbiggt0
+t′/parenrightbigg1
+c
+a(t′)dt′·/ba∇dblδX/ba∇dbl∞.
+Both these last two estimates can be summed in order to give an exponential. Hence
+/vextendsingle/vextendsingleDτ(χk−χ0
+k)[X,δX](t)/vextendsingle/vextendsingle≤expF1/2(t)√
+2k6m2
+2c−1t0/ba∇dblδX/ba∇dbl∞
+37and, together with the a-functional derivative, we obtain
+/vextendsingle/vextendsingleD(χk−χ0
+k)[X,δX](t)/vextendsingle/vextendsingle≤/bracketleftbigg1
+ka0
+t01
+c2+6m2
+2c−1/bracketrightbiggexpF1/2(t)√
+2km2t0/ba∇dblδX/ba∇dbl∞. (38)
+A further inequality we would like to present in this subsect ion is the following
+/vextendsingle/vextendsingle/vextendsingle(χk−χ0
+k)[X](t)·Dχ0
+k[X,δX](t)/vextendsingle/vextendsingle/vextendsingle≤expF1/2(t)
+2k2m2
+2c−1t0/ba∇dblδX/ba∇dbl∞ (39)
+which can be shown to hold in a similar way as the bound found ab ove for the functional
+derivative of eikτ. We conclude this subsection with two inequalities, valid f orn≥0, that can
+be shown to hold in a similar way as before:
+/vextendsingle/vextendsingleDχ>n
+k[X,δX](t)/vextendsingle/vextendsingle≤Fn
+1/2(t)expF1/2(t)√
+2k/bracketleftbigg1
+ka0
+t01
+c2+6m2
+2c−1/bracketrightbigg
+m2t0/ba∇dblδX/ba∇dbl∞ (40)
+and
+/vextendsingle/vextendsingle/vextendsingleχ>n
+k[X](t)·Dχ0
+k[X,δX](t)/vextendsingle/vextendsingle/vextendsingle≤Fn
+1/2(t)expF1/2(t)
+2k2m2
+2c−1t0/ba∇dblδX/ba∇dbl∞. (41)
+A.4 Proof of Proposition 4.5
+Proof.The proof is done dividing the expression for Φ in a finite numb er of parts and showing
+thateverycontributionseparatelysatisfiesthedesiredin equalities withrespecttosomeconstants
+that have the properties stated in proposition 4.5. In the ne xt, since most of the computations
+are rather long but straightforward, we shall concentrate o nly on the key points, and we shall
+make extensive use of all the estimates derived in the previo us appendix. Throughout this proof
+we shall make use of proposition 4.1 in order to associate a sp acetime to an element XofBc.
+According to the same proposition, we shall indicate X−1asH, and thetwhich appears in
+some estimates has to be thought as being contained in the int erval [0,t0]. Let us start dividing
+the integral over kin two parts
+Φ[X] =/integraldisplay1
+0dkk2/bracketleftBig
+χkχk−χ0
+kχ0
+k/bracketrightBig
++/integraldisplay∞
+1dkk2/bracketleftbigg
+χkχk−χ0
+kχ0
+k+a2m2
+4k3/bracketrightbigg
+−m2a2log(am),
+where the regularization is needed only for large k, whereasχ,aandτare functionals of X
+as discussed previously. Notice that, the zeroth order in m2, present in /a\}b∇acketle{tφ2/a\}b∇acket∇i}htthroughχ0
+kχ0
+k,
+vanish due both to the discussion about the regularization f reedom and to the requirement that
+Minkowksi spacetime, with respect to the Minkwoksi vacuum, is a solution of the problem. For
+this reason we have subtracted it in (27) and we shall not cons ider it anymore. Let us start
+considering the first part of the integral, to which we shall r efer to as the infrared part and we
+shall indicate it by ΦI. In this respect we shall write
+/bracketleftBig
+χkχk−χ0
+kχ0
+k/bracketrightBig
+=/parenleftBig
+χk−χ0
+k/parenrightBig/parenleftbig
+χk−χ0
+k/parenrightbig
++/parenleftBig
+χk−χ0
+k/parenrightBig
+χ0
+k+χ0
+k/parenleftbig
+χk−χ0
+k/parenrightbig
+38and, hence, using the estimates (36) for χkand the one for the functional derivatives (38) and
+(39), withF2as in (35), we obtain
+/vextendsingle/vextendsingleΦI(t)/vextendsingle/vextendsingle≤1
+4/bracketleftbig
+F2(t)2exp(2·F2(t))+2F2(t)expF2(t)/bracketrightbig
+≤1
+4/parenleftbig
+m4t4+2m2t2/parenrightbig
+e2m2t2
+whereF2(t)≤m2t2, as it can be seen by direct inspection of the definition of bot hF2(t) andτ,
+and
+/vextendsingle/vextendsingleDΦI[X,δX](t)/vextendsingle/vextendsingle≤/braceleftbigg
+[F2(t)expF2(t)+1]/bracketleftbigga0
+t01
+c2+3m2
+2c−1/bracketrightbigg
++1m2
+2c−1/bracerightbigg
+expF2(t)m2t0/ba∇dblδX/ba∇dbl∞.
+This conclude the analysis of the contribution of the lower e nergies to the value of Φ, we can
+now proceed to analyze the higher frequencies which we shall indicate as ΦU. We shall split
+it in powers of m2and discuss the first three powers separately. Afterwards, b efore summing
+everything together, we shall analyze the remainder.
+I) order. Let us start considering the first order in m2and its contribution to ΦUin particular,
+ΦU
+1[a,τ](t) :=/integraldisplay∞
+1k2/bracketleftbigg
+χ1
+kχ0
+k[a,τ](t)+χ0
+kχ1
+k[a,τ](t)+a2(t)m2
+4k3/bracketrightbigg
+dk−m2a2(t)log(a(t)m)
+which can be expanded as
+ΦU
+1[a,τ](t) :=/integraldisplay∞
+1dkm2
+4k/integraldisplayt
+0cos(2k(τ−τ′))∂a2
+∂t(t′)dt′−m2a(t)2log(ma(t))
+where both aandτhave to be thought as functionals of Xand where we have written the
+perturbative series (24) using t, instead of τ, as the time variable. Furthermore, τhas to be
+intended as τ[X](t) whileτ′asτ[X](t′). Let us rewrite it in the following form
+ΦU
+1[a,τ](t) :=/integraldisplay∞
+1dkm2
+8k2/integraldisplayt
+0sin(2k(τ−τ′))∂
+∂t/parenleftbigg
+a∂a2
+∂t/parenrightbigg
+(t′)dt′−m2a(t)2log(ma(t))
+and, using the fact that the tderivative of Xis positive, we can obtain the following bound
+|ΦU
+1(t)| ≤m2a3(t)X−1(t)+m2a(t)t+m2a2(t)log(ma(t)).
+Let us proceed to analyze the functional derivative
+DΦU
+1[a,τ,δX] =DaΦU
+0[a,τ,Da[X,δX]] +DτΦU
+0[a,τ,Dτ[X,δX]],
+and let us analyze both term separately, first of all, setting F:=Da[X,δX],
+DaΦU
+1[a,τ,F](t) :=/integraldisplay∞
+1m2
+8k2/integraldisplayt
+0sin(2k(τ−τ′))/bracketleftBig
+F(8a2H2+4a2˙H)+8˙Fa2H+2a2¨F/bracketrightBig
+(t′)dt′dk−
+−2m2a(t)Flog(ma(t))−m2a(t)F ,
+39where as usual H=X−1and the dots indicates the derivatives with respect to the co smological
+timet. Using the estimates present in proposition 4.3 and (33), it is now an easy task to obtain
+the desired bounds. Notice in particular that the second ord er time derivative of Fmultiplied by
+a2can be estimated by /ba∇dblδX/ba∇dblB(the norm of B), as it can be verified directly from the definition
+ofa[X]. Let us proceed with the second functional derivative, and now setting G:=Dτ[X,δX]
+we obtain
+DτΦU
+1[a,τ,G](t) :=/integraldisplay∞
+1m2
+4k/integraldisplayt
+0cos(2k(τ−τ′))/bracketleftbig
+G(t)−G(t′)/bracketrightbig/bracketleftBig
+6a3H2+2a3˙H/bracketrightBig
+(t′)dt′dk.
+We could in principle integrate by parts in order to obtain an extra 1/(2k) factor. Unfortunately
+this would not help much, since, due to the extra t-derivative the left hand side would depend
+on¨Hand we do not have control on the second t-derivative of HinBc. At the same time we
+have to remember thatcoskx
+kis integrable in kon [1,∞) for everyx>0 and that the k-integral
+present above needs to be understood in the distributional s ense, hence taken with the following
+ǫ-prescription
+DτΦU
+1[a,τ,G](t′) := lim
+ǫ=0+/integraldisplay∞
+1e−ǫkm2
+4k/integraldisplayt
+0cos(2k(τ−τ′))/bracketleftbig
+G(t)−G(t′)/bracketrightbig
+(6a3H2+2a3˙H)(t′)dt′dk.
+Since [G(t)−G(t′)] (6a3H2+2a3˙H) isL1(0,t), at fixedǫ, the absolute value of the integrand
+indk∧dtis integrable. We can thus switch the order of integration to obtain
+DτΦU
+1[a,τ,G](t) :=m2/integraldisplayt
+0/bracketleftbig
+G(t)−G(t′)/bracketrightbig
+(6a3H2+2a3˙H)(t′) lim
+ǫ=0+/integraldisplay∞
+1e−ǫk
+4kcos(2k(τ−τ′))dkdt′.
+(42)
+We are now ready to perform the integral in kand to notice that the result is a smooth function
+everywhere in τ−τ′but in 0 where a logarithmic divergence in τ−τ′appears when the limit
+ǫ= 0+is computed. Let us call L(τ−τ′) the result of the k-integration in the limit ǫ= 0.
+Notice that L(τ(t)−τ′(t′))/a(t′) is absolutely integrable for t′in [t−δ,t] and, moreover, we can
+adjustδin such a way both that this integral is equal to 1 and that |L(τ(t)−τ′(t′))| ≤1 fort′
+in (0,t−δ). Hence, since a(t′)[G(t)−G(t′)] (6a3H2+2a3˙H)(t′) is uniformly bounded for t′in
+[0,t], we can split the t′-integral in (42) in two parts, namely over [0 ,t−δ] and over [ t−δ,t].
+Both can be uniformly estimated and finally the following abs olute bound for/vextendsingle/vextendsingleDτΦU
+1/vextendsingle/vextendsinglecan be
+established
+/vextendsingle/vextendsingleDτΦU
+1(t)/vextendsingle/vextendsingle≤sup
+t′∈[0,t]/vextendsingle/vextendsinglea(t′)/bracketleftbig
+G(t)−G(t′)/bracketrightbig/vextendsingle/vextendsingle/ba∇dbl6a3H2+2a3˙H/ba∇dbl∞+/integraldisplayt
+0/vextendsingle/vextendsingle/vextendsingle/bracketleftbig
+G(t)−G(t′)/bracketrightbig
+(6a3H2+2a3˙H)/vextendsingle/vextendsingle/vextendsingledt′.
+Combining all the estimates, we obtain the desired behavior , namely that the absolute value of
+the functional derivatives is controlled by Ct0/ba∇dblδX/ba∇dbl.
+II) order. Thecomputation at thesecond orderis moreinvolved dueto ap airof integrals which
+needs to be addressed, but at the same time there are no subtle ties with the extra derivatives
+40which could necessitate control on the derivatives of the va riation higher than the first one. We
+start reminding the form of this term
+ΦU
+2[a,τ] :=/integraldisplay∞
+1k2/bracketleftBig
+χ2
+kχ0
+k[a,τ]+χ1
+kχ1
+k[a,τ]+χ0
+kχ2
+k[a,τ]/bracketrightBig
+dk,
+which can be more explicitly rewritten as
+ΦU
+2[a,τ](t) :=/integraldisplay∞
+1dkm4
+k/bracketleftBigg/bracketleftbigg1
+2k/integraldisplayt
+0/parenleftbig
+sin(k(τ−τ′))/parenrightbig2∂a2
+∂t(t′)dt′/bracketrightbigg2
++1
+8k/bracketleftbigg/integraldisplayt
+0sin(2k(τ−τ′))∂a2
+∂t(t′)dt′/bracketrightbigg2/bracketrightBigg
+−
+−/integraldisplay∞
+1dkm4
+4k2/integraldisplayt
+0sin(2k(τ−τ′))∂a2
+∂t(t′)/integraldisplayt
+t′a(t′′)dt′′dt′+/integraldisplay∞
+1dkm4
+4k2/integraldisplayt
+0sin(2k(τ−τ′))a3(t′)dt′.
+(43)
+Out of such expression, an estimate satisfied by |ΦU
+2|can be easily derived and it looks like
+/vextendsingle/vextendsingleΦU
+2[a,τ](t)/vextendsingle/vextendsingle≤m4/bracketleftbigg5
+4(a2Ht)2+1
+2a3Ht2+1
+4a3t/bracketrightbigg
+.
+Let us proceed to the analysis of the functional derivatives of Φ and let us notice that the
+analysis of DaΦ2can also be easily done yielding
+/vextendsingle/vextendsingleDaΦU
+2[a,τ,Da](t)/vextendsingle/vextendsingle≤m4
+8/bracketleftbig
+6a3H2t2+3a2Ht2+ta3/bracketrightbig
+/ba∇dblDa/ba∇dbl∞+m4
+8/bracketleftbig
+2a2Ht2+at2/bracketrightbig/vextenddouble/vextenddouble/vextenddoublea(˙Da)/vextenddouble/vextenddouble/vextenddouble
+∞.
+A little bit more laborious is the handling of the τ-directional derivative, because, operating as
+above, an extra factor 2 kappears in the formula. Furthermore, it could make the k-integral
+divergent when the trigonometric functions are replaced wi th one in the maximization procedure
+of the second, third and fourth factor in (43). In order to avo id this problem, we can integrate
+by parts after performing the functional derivative, along the following lines:
+Dτ/integraldisplayt
+0sin(2k(τ−τ′))∂a2
+∂tdt′=/integraldisplayt
+0sin(2k(τ−τ′))∂
+∂t/bracketleftbigg
+a/parenleftbigg∂a2
+∂t/parenrightbigg
+D(τ−τ′)/bracketrightbigg
+dt′.
+The price to be paid is in the extra time derivative, we have to deal with. In fact, proceeding
+in that way we obtain the following estimate
+/vextendsingle/vextendsingleDτΦU
+2[a,τ,Dτ](t)/vextendsingle/vextendsingle≤C1/integraldisplayt
+0a|D(τ−τ′)|dt′+C2/integraldisplayt
+0a2/vextendsingle/vextendsingle/vextendsingle/vextendsingled
+dt′D(τ−τ′)/vextendsingle/vextendsingle/vextendsingle/vextendsingledt′.
+It is now a straightforward task to obtain the desired inequa lities.
+Rest2)The contribution of the order larger than 2 in m2is
+ΦU
+>2[a,τ] :=/integraldisplay∞
+1dkk2/bracketleftBig
+χ1
+kχ2
+k+χ1
+kχ2
+k+/parenleftBig
+χ0
+k+χ1
+k/parenrightBig
+χ>2
+k+χ>2
+k/parenleftbig
+χ0
+k+χ1
+k/parenrightbig
++χ>1
+kχ>1
+k/bracketrightBig
+.
+41Hence, employing the inequality (36), and the definition of F1in (35), we obtain the bounds
+/vextendsingle/vextendsingleΦU
+>2[a,τ](t)/vextendsingle/vextendsingle≤/integraldisplay∞
+1dk1
+k2/parenleftbigg
+m2a0
+t0t2/parenrightbigg3/bracketleftbigg
+1+/parenleftbigg
+1+m2
+ka0
+t0t2/parenrightbigg
+exp(F1[a])+m2
+ka0
+t0t2exp(2F1[a])/bracketrightbigg
+whereF1(t)≤m2
+ka0
+t0t2, more explicitly we can rewrite it as
+/vextendsingle/vextendsingleΦU
+>2[a,τ](t)/vextendsingle/vextendsingle≤3/parenleftbigg
+m2a0
+t0t2/parenrightbigg3/parenleftbig
+1+m2a0t0/parenrightbig
+exp/parenleftbig
+2m2a0t0/parenrightbig
+.
+We do not analyze the functional derivative of this contribu tion, because, due to the extra k
+appearing in the functional derivative of the trigonometri c functions with respect to τ, we have
+to further split it.
+III) order The contribution at third order in m2is
+ΦU
+3[a,τ] :=/integraldisplay∞
+1dkk2/bracketleftBig
+χ3
+kχ0
+k[a,τ]+χ2
+kχ1
+k[a,τ]+χ1
+kχ2
+k[a,τ]+χ0
+kχ3
+k[a,τ]/bracketrightBig
+and its explicit form is
+ΦU
+3[a,τ](t) :=/integraldisplay∞
+1dkm6
+k2·
+/bracketleftbigg/integraldisplay
+0<t3<t2<t1<tsin(k(τ−τ1))sin(k(τ1−τ2))sin(k(τ2−τ3))cos(k(τ−τ3))a(t1)a(t2)a(t3)dt1dt2dt3+
++/integraldisplayt
+0/integraldisplay
+0<t3<t2<tsin(k(τ−τ1))sin(k(τ−τ2))sin(k(τ2−τ3))cos(k(τ1−τ3))a(t1)a(t2)a(t3)dt1dt2dt3/bracketrightbigg
+(44)
+An estimate of the form/vextendsingle/vextendsingleDΦU
+3(t)/vextendsingle/vextendsingle≤C(t0,a0,c)/ba∇dblδX/ba∇dbl∞
+holds for the first functional derivative. Notice that, in th e derivation of the preceding estimate
+starting from (44), a problem could in principlearise when t heτfunctional directional derivative
+is computed because in such a case an extra kfactor appears and the kintegral is logarithmic
+divergent. Hence, in order to obtain a finite k-integration, before performing the functional
+derivative, we have to rewrite the product of the trigonomet ric functions as a sum of products
+of trigonometric functions depending only on τ−τ1,τ1−τ2,τ2−τ3for the first line in (44) and
+onτ−τ1,τ−τ2,τ2−τ3for the second one. Subsequently we can safely perform a coup le of
+integration by parts. The outcome is that the result of the t-integrations is of order 1 /(2k) ink
+and this extra kpermits to obtain finite bounds for the k-integration also for the τ-functional
+derivative. The price to be paid is that, afterwards, in the tiintegrations the derivatives of
+a(t) might appear. This also holds for the adirectional functional derivative which has to be
+carefully computed. More precisely we obtain
+/vextendsingle/vextendsingleDτΦU
+3(t)/vextendsingle/vextendsingle≤/bracketleftbigg/integraldisplay∞
+1dkm6
+2k2/bracketrightbigg/bracketleftbig
+112t2a4H+6ta3/bracketrightbig/integraldisplayt
+0a|Dτ|
+42and
+/vextendsingle/vextendsingleDaΦU
+3(t)/vextendsingle/vextendsingle≤/bracketleftbigg/integraldisplay∞
+1dkm6
+2k3/bracketrightbigg/bracketleftbig
+84Ha3t3+6a3t2/bracketrightbig
+/ba∇dblDa/ba∇dbl∞+/bracketleftbigg/integraldisplay∞
+1dkm6
+2k3/bracketrightbigg/bracketleftbig
+28a2t3/bracketrightbig/vextenddouble/vextenddouble/vextenddoublea(˙Da)/vextenddouble/vextenddouble/vextenddouble
+∞,
+out of which it is straightforward to obtain the wanted inequ ality.
+Rest3)The contribution of the order larger than 3 in m2is
+ΦU
+>3[a,τ] :=/integraldisplay∞
+1dkk2/bracketleftBig
+χ0
+kχ>3
+k+χ>3
+kχ0
+k+χ1
+kχ>2
+k+χ>2
+kχ1
+k+χ>1
+kχ>1
+k/bracketrightBig
+and, using (41) and (40) together with (36), we can find a const antC′(a0,t0,c), with the desired
+properties, i.e.,/vextendsingle/vextendsingleDΦU
+>3(t)/vextendsingle/vextendsingle≤C′(a0,t0,c)/ba∇dblδX/ba∇dbl∞,∀t∈[0,t0].
+After a rather long but straightforward computation, we are ready to collect all the estimates
+presented above hence obtaining that
+|Φ(t)| ≤C(a0,c,t0)t,|DΦ[X,δX](t)| ≤C(a0,c,t0)/ba∇dblδX/ba∇dbl∞,∀t∈[0,t0],
+whereC(t0ℓ,c,t0) remains bounded for small t0. The thesis of the proposition can be obtained
+noticing that /ba∇dblδX/ba∇dbl∞≤t0/ba∇dblδX/ba∇dblBin particular.
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+47
\ No newline at end of file
diff --git a/1001.0865.txt b/1001.0865.txt
new file mode 100644
index 0000000000000000000000000000000000000000..093bb709998fe223fc2e31a12d5dcab004cdadda
--- /dev/null
+++ b/1001.0865.txt
@@ -0,0 +1,431 @@
+Observation of vortex nucleation in a rotating two-dimensional lattice of
+Bose-Einstein condensates.
+R. A. Williams, S. Al-Assam, and C. J. Foot
+Clarendon Laboratory, Department of Physics, University of Oxford,
+Parks Road, Oxford, OX1 3PU, United Kingdom
+(Dated: November 19, 2018)
+We report the observation of vortex nucleation in a rotating optical lattice. A87Rb Bose-Einstein
+condensate was loaded into a static two-dimensional lattice and the rotation frequency of the lattice
+was then increased from zero. We studied how vortex nucleation depended on optical lattice depth
+and rotation frequency. For deep lattices above the chemical potential of the condensate we observed
+a linear dependence of the number of vortices created with the rotation frequency, even below the
+thermodynamic critical frequency required for vortex nucleation. At these lattice depths the system
+formed an array of Josephson-coupled condensates. The eective magnetic eld produced by rotation
+introduced characteristic relative phases between neighbouring condensates, such that vortices were
+observed upon ramping down the lattice depth and recombining the condensates.
+PACS numbers: 03.75.Lm, 67.85.Hj, 74.50.+r, 74.81.Fa
+Ultracold quantum gases in rotating optical lattices
+nd themselves at the intersection of two elds which
+have generated an impressive body of experimental and
+theoretical work. The versatile and clean potentials of-
+fered by optical lattices have proved to be an incredi-
+bly adept system for exploring a wide range of funda-
+mental problems in condensed matter physics such as
+the Mott-insulator transition [1], Anderson localization
+[2, 3] and Tonks-Girardeau gases [4]. Similarly rotating
+quantum gases have provided an array of striking results,
+from the nucleation of vortex lattices in Bose gases [5, 6]
+to rotating Fermi gases at the BCS-BEC crossover [7].
+The close analogy between the physics of rapidly rotating
+neutral atoms and electrons under a magnetic eld has
+led to considerable interest in the possibility of achiev-
+ing strongly-correlated quantum Hall states in a rapidly
+rotating atomic gas [8, 9].
+Rotating optical lattices for cold atoms have gener-
+ated a large amount of interest in their own right, which
+can broadly be divided into three main areas: (i) Weak
+rotating lattices have been used to pin vortices [10] and
+theoretical work has shown that rich vortex lattices struc-
+tures, including lattices of doubly quantized vortices, are
+predicted to emerge [11, 12]; (ii) Stronger optical lat-
+tice potentials with a large number of atoms per site
+(100-1000) realize the physics of Josephson junction ar-
+rays (JJA) under magnetic elds [13, 14]; (iii) Arguably
+the most interesting regime is for a dilute gas in the
+tight binding regime in a rotating lattice where fractional
+quantum Hall physics is predicted to occur [15{17]. In
+all three regimes rich structures emerge that depend on
+the relative density of vortices and lattice sites.
+We report on the rst experiments with a rotating op-
+tical lattice in the `deep' lattice regime where lattices
+depths were reached such that a 2D array of weakly linked
+condensates was created, forming a Josephson junction
+array. Under rotation this system realizes the uniformlyfrustrated JJA [13, 18]. We have investigated the nucle-
+ation of vortices in a rotating lattice, starting from a non-
+rotating condensate loaded into a static lattice, and then
+increasing the rotation rate of the lattice. There has been
+one previous experiment employing a rotating optical lat-
+tice by Tung et al. [10] at JILA, where it was reported
+that heating due to mechanical instabilities and aber-
+rations limited the rotating lattice to depths less than
+30% of the condensate's chemical potential. We were
+not limited to this weak lattice regime and our system is
+markedly dierent to the JILA experiment where a vor-
+tex lattice was created by an elegant evaporative spin-up
+technique [6] before a weak co-rotating optical lattice was
+imposed.
+The rotating lattice used in the experiments described
+in this paper was generated by dual-axis acousto-optic
+de
ectors and a novel optical system, described in detail
+in [19]. The two-dimensional optical lattice was formed in
+the focal plane of a custom-made multi-element lens with
+a numerical aperture of 0.27. Generating the rotation
+of the lattice acousto-optically gave intrinsically smooth
+rotation and allowed the rotation rate to be varied during
+an experimental run.
+The Coriolis force on neutral atoms in the rotating
+frame has the same form as the Lorentz force experi-
+enced by charged particles in a magnetic eld. Rotation
+at angular frequency 
+ gives rise to the eective mag-
+netic vector potential A=
+r, and the Peierls phase
+gained by an atom tunnelling from site ito sitejis
+Aij= (m=h)Rrj
+riA(r0)
dr0, wheremis the mass of the
+atom. The sum of the link phases around a lattice pla-
+quette of side dobeys the propertyP
+plaquetteAij= 2f,
+wheref= 2m
+d2=his the mean number of vortices per
+lattice plaquette.
+The experiments started with a87Rb condensate con-
+taining 2105atoms with no visible thermal compo-
+nent, held in a trap formed by an axial symmetric Ioe-arXiv:1001.0865v1  [cond-mat.quant-gas]  6 Jan 20102
+Pritchard trap and a single red-detuned light sheet in
+the radial plane increasing the axial trapping frequency
+to givef!r;!zg= 2f20:1;53:0gHz. The residual ra-
+dial anisotropy of the combined magnetic and optical
+trap was!y=!x= 1:0080:003. The dipole sheet trap
+was creating using broadband light with a spectral width
+
3 nm centred on 865 nm in order that the light
+had a short coherence length. This ensured there were
+no interference eects, e.g. from multiple re
ections at
+the vacuum cell wall, adding noise to the trapping po-
+tential. With narrow-band light it was observed such
+perturbations acted to spin down the cloud and caused
+irregularities in the lling of the lattice.
+The 2D optical lattice was formed in the radial plane
+of the trapped condensate by four circularly polarized
+beams at= 830 nm intersecting in the focal plane of a
+high N.A. objective lens, achieving a lattice constant of
+2m. One pair of beams was detuned by 10 MHz to the
+other pair ensuring interference between the orthogonal
+optical lattices did not aect the atoms. The optical
+lattice was initially static as it was ramped to its nal
+depthV0in the range 100 Hz V04000 Hz [20]. The
+radial Thomas-Fermi radius of the condensate was 16 m,
+resulting in the lling of 16 lattice sites across the
+diameter of the condensate and around 200 sites in all.
+The lattice enhanced interaction energy at the central
+sites rose above the chemical potential of the unperturbed
+condensate ( 500 Hz) meaning an array of individual
+condensates was not formed until V01000 Hz. The
+central wells contained around 1500 atoms.
+After the condensate had been loaded into the static
+optical lattice 
+ was increased linearly from zero to its
+nal value in 320 ms, followed by a further 100 ms of
+rotation at the nal value. While the lattice was still ro-
+tating the lattice depth was then ramped down to zero
+in 12 ms, allowing the condensates to merge and con-
+verting phase dierences around plaquettes to vortices.
+Previously Scherer et al [21] have investigated vortex for-
+mation by merging three independent condensates, and
+Schweikhard et al [22] have observed the thermal activa-
+tion of vortices on a two-dimensional lattice of Josephson-
+coupled BECs, however our experiment takes place in
+the rotating frame. Rotation gives rise to an additional
+phase 2faround a plaquette of the optical lattice, sim-
+ulating the eect of a magnetic eld on charged particles.
+After the optical lattice was ramped down the cloud was
+then immediately released from the magnetic and optical
+traps and destructively imaged after 20 ms time-of-
ight
+expansion. We also repeated the sequence but holding
+the cloud after the lattice was ramped down for an ad-
+ditional 500 ms, to allow any vortex-antivortex pairs to
+annihilate (such pairs could be created in a static lat-
+tice [22]). This generally improved visibility of vortices
+in the harmonic trap and we observed the same number
+of vortices in the system within experimental error. The
+uncertainty in the number of vortices observed for a given
+Ωmin/2π   (Hz)
+V0  (Hz)(b)
+500 1000 1500012345
+2000 2500 3000 3500 4000 0670 2 4 6 8 10024681012141618
+Ω/2π  (Hz)N
+  
+V0 = 190Hz
+V0 = 1200Hz(a)
+vFIG. 1: (a) Number of vortices nucleated as a function of
+optical lattice rotation frequency 
+ for two lattice depths,
+V0= 190 Hz and V0= 1200 Hz. The error bars denote
+the standard deviation in vortex number for three experi-
+mental images. Inset: an example of a vortex lattice created
+by the rotating optical lattice of depth V0= 1200 Hz at 
+ =
+2= 10 Hz. (b) The minimum rotation frequency needed
+to nucleate a single vortex, 
+ min, as a function of the lattice
+depthV0(1 Hz was the minimum frequency used).
+
+ arose from diculties in determining the presence of
+vortices near the edge of a condensate.
+Figure 1(a) shows the number of vortices, Nv, observed
+as a function of the optical lattice rotation frequency, 
+,
+for lattice depths of V0= 190 Hz and V0= 1200 Hz. The
+behaviour of Nv(
+) has a characteristic shape which de-
+pends on the exact vortex nucleation process. The rst
+BEC stirring experiments used stirring beams of a similar
+size to the condensate and observed that no vortices were
+nucleated below 
+ 0:7!r, where there is excitation of
+thel= 2 surface mode [23]. It was later shown that suf-
+ciently small stirring beams nucleate vortices without
+discrete resonances of surface modes [24], and the mini-
+mum frequency for vortex nucleation, 
+ min, matched the
+frequency at which a single vortex at the center of a con-
+densate is energetically stable, 
+ c[25]. For our experi-
+mental conditions 
+ c=24 Hz. ForV0= 190 Hz, far
+below the1000 Hz needed to split the condensate into
+weakly linked islands, Fig. 1(a) shows a similar trend
+to that observed in [24], i.e. Nvdid not exhibit sharp
+resonances and 
+ min
+c. We checked this behaviour
+was an equilibrium property by repeating the experiment
+with rotation times up to 1 s, and observed the same re-
+sponse. We infer that a weak ( V0) rotating lattice
+stirs up a condensate in a non-resonant fashion. For a3
+0 0.2 0.4 0.6 0.8 1 1.20102030405060
+Ω/ωr  
+V0 = 600Hz
+V0 = 1100HzNv
+FIG. 2: Number of vortices nucleated as a function of 
+ =
+!r. The solid line is the number of vortices expected for a
+cloud of xed radius, R0= 16m, while the dashed line in
+the number of vortices expected for a centrifugally distorted
+cloud of radius R=R0[1(
+=!r)2]3=10. We observed the
+optical lattice prevented the cloud from reaching Rfor a given
+
+, allowing rotation above the critical frequency 
+ > !rto
+be achieved.
+rotating lattice of depth V0= 1200 Hz, however, Nv
+displayed a distinctly dierent trend, with the number
+of vortices observed following a linear dependence on 
+even below 
+ c. This change in behaviour is indicative
+of entering the regime where the system forms an array
+of Josephson-coupled condensates, rather than simply a
+condensate perturbed by a weak rotating lattice as for
+V0= 190 Hz. Figure 1(b) highlights the transition be-
+tween the two regimes, showing the dependence of 
+ min
+onV0. 
+minfalls below 
+ casV0rises above = 500 Hz,
+and falls to 1 Hz (the minimum rotation frequency used)
+forV01000 Hz, corresponding to the lattice depth
+at which condensates are expected to be well localized
+on sites and can only communicate by tunneling. The
+rotation induced phase dierences between neighbouring
+condensates are converted to vortices when the lattice
+is ramped down in the rotating frame. This nucleation
+mechanism creates vortices near the centre of the con-
+densate even when 
+ <
+c. Vortices do not have to spin
+in from the edge of the cloud, which is what we observed
+for stirring with a weak lattice potential.
+The measurements of the number of vortices Nvwas
+extended to higher rotation frequencies 
+ as shown in
+Figure 2. The ratio 
+ =!ris used where !ris the ra-
+dial trapping frequency taking into account the slight en-
+hancement by the Gaussian envelope of the lattice beams:
+forV0= 600 Hz,!r=2= 21:5 Hz and for V0= 1100 Hz,
+!r=2= 22:6 Hz. The density of vortices for a given 
+,
+whether a lattice is present or not, is nv=m
+=h. The
+total number of vortices is then Nv=m
+R2=h. For a ro-
+tating BEC in equilibrium at 
+ the centrifugal distortion
+of the Thomas-Fermi radius is R=R0[1(
+=!r)2]3=10
+[26]. The predicted number of vortices for a cloud dis-
+playing this behaviour is denoted by the dashed line in
+(a) (b)
+(c) (d)FIG. 3: Images after release from the rotating optical lattice
+showing the pattern of vortices: (a) Condensate before rota-
+tion; (b) 
+=2= 4 Hz; (c) 
+ =2= 7 Hz; (d) 
+ =2= 17 Hz.
+Fig. 2. The solid line presumes the radius stays constant
+atR0= 16m. ForV0= 1100 Hz, Nvrises linearly with
+
+, whereas for the weaker V0= 600 Hz lattice Nvshows
+a faster, nonlinear dependence on 
+. We attribute this
+behaviour to the deeper lattice suppressing the eect of
+centrifugal distortion. For the deeper lattice atoms may
+only redistribute themselves by tunneling between sites.
+While the tunneling parameter (the Josephson-coupling
+energy [13]) is around 2 kHz at central lattice sites this
+drops to<50 Hz at peripheral lattice sites due to the
+drop in atom number, giving tunneling times at the edge
+of the lattice comparable to the experimental duration.
+Bloch oscillations may also play a role in inhibiting the
+atoms from spreading out in the lattice [27].
+As a result of the optical lattice inhibiting the spread-
+ing out of the cloud we were able to rotate the cloud
+above the critical frequency 
+ > !r. As seen in Fig. 2
+the number of vortices dropped for rotation above !r, due
+to increased heating of the condensate. However vortices
+could be observed up to 
+ = 1 :15!rforV0= 600 Hz
+before heating and reduction of vortex visibility meant it
+was no longer possible to observe vortices. The heating
+at 
+!rwas worse for the deeper V0= 1100 Hz lattice
+and vortices could not be observed above 
+ = 1 :02!r. A
+cloud in a harmonic trap without the presence of an op-
+tical lattice should be expelled for 
+ >!rthough Bretin
+et al. were able to achieve 
+ = 1 :06!rusing additional
+quartic connement [28].
+The structure of the vortex patterns formed in a
+bosonic JJA under rotation are heavily in
uenced by the
+presence of the lattice. For values of the frustration pa-
+rameter (the mean number of vortices per lattice plaque-
+tte)f= 2m
+d2=h=p=q(where p and q are integers) the
+ground state vortex patterns have been calculated [13],
+and have a unit cell of size qqsites. Between these
+rational fraction values of fintermediate patterns are
+formed. Figure 3 shows examples of the vortex patterns4
+0 500 1000 1500 2000 2500 3000 35000
+V0 (Hz)Condensate fraction N0/N
+0.10.30.50.60.7
+0.4
+0.2
+FIG. 4: Condensate fraction N0=Nas a function of lattice
+depthV0after ramping up the rotation rate to 15 Hz in 40 ms
+followed by 35 ms of rotation at 15 Hz.
+we observe. We do not reproducibly observe the pre-
+dictedqqunit cell structure as we scan 
+ and hence
+f. There are several possible reasons for this: (i) for
+d= 2m and!r=2= 20:1 Hz the maximum value of
+fwe can reach is f1=5. The nite size of the system
+then becomes an issue, as it will be dicult to realize
+the periodic structure for q= 5 when a maximum of 200
+sites are lled by the condensate; (ii) the energy land-
+scape of cold atoms in a rotating lattice is known to be
+very glassy, that is closely spaced minima in energy can
+be separated by large barriers, making it hard to reach
+the true ground state [29]. Lack of equilibration on ex-
+perimental timescales might then account for the absence
+ofqqvortex structures; (iii) any intrinsic heating from
+the rotating optical lattice will make it dicult to reach
+the ground state.
+The issue of nite size can be addressed by increasing
+!zor using a larger condensate, enabling more lattice
+sites to be lled. The values of fachievable can be in-
+creased by implementing tighter radial trapping to allow
+higher 
+ to be reached or increasing the lattice constant
+d. Thef= 1=2 case is particularly interesting due to
+competition between an Ising-type and BKT type phase
+transition [14]. Vortex patterns in a bosonic JJA under
+rotation for 0f1=2 will be explored in future work.
+An important question regarding the viability of fu-
+ture rotating optical lattice experiments is the issue of
+heating. Figure 4 displays the fraction of atoms in the
+condensate, N0=N, after rotation at 
+ =2= 15 Hz (
+ =
+2was increased from 0 to 15 Hz in 40 ms followed by
+35 ms of rotation at 15 Hz) for dierent lattice depths,
+showing an approximately linear dependence. We found
+at 
+=2= 15 Hz,V0= 1100 Hz we could rotate for
+around 300 ms before heating signicantly reduced the
+condensate fraction, with longer rotation times achiev-
+able for smaller 
+ or V0or both. The main sources of
+heating were 
uctuations in the lattice depth during ro-
+tation (at the1 % level) and residual imperfections in
+the lattice upon rotation. All the data in this paper were
+taken without using any additional evaporation during or
+after the rotation process (unlike previous stirring experi-
+ments [5, 23]). The implementation of an additional cool-ing scheme in the optical lattice such as that described
+by Griessner et al. [30] could enable suciently low tem-
+peratures to be achieved to realise quantum Hall physics
+in future experiments with the rotating lattice [15{17].
+In conclusion, the use of acousto-optic de
ection to
+generate smooth rotation of an optical lattice at a well-
+dened frequency has allowed us to explore a new method
+of vortex nucleation. For lattice depths V0<we found
+the weak lattice acted as a stirring mechanism. For
+deeper lattices the system was split into well-localized
+condensates at each site and a linear dependence of vor-
+tex number on 
+ was observed for rotation frequencies
+below the thermodynamic critical frequency for vortex
+nucleation. This implied vortices were created locally at
+a lattice plaquette.
+This research is supported by EPSRC, QIP IRC
+(GR/S82176/01) and ESF.
+[1] M. Greiner et al., Nature 415, 39 (2002).
+[2] J. Billy et al., Nature 453, 891 (2008).
+[3] G. Roati et al., Nature 453, 895 (2008).
+[4] T. Kinoshita et al., Science 305, 1125 (2004).
+[5] Abo-Shaeer et al., Science 292, 476 (2001).
+[6] V. Schweikhard et al., Phys. Rev. Lett. 92, 040404 (2004)
+[7] M. W. Zwierlein et al., Nature 435, 1047 (2005).
+[8] N. K. Wilkin and J. M. F. Gunn, Phys. Rev. Lett. 84, 6
+(2000).
+[9] N. R. Cooper et al., Phys. Rev. Lett. 87, 120405 (2001).
+[10] S. Tung et al., Phys. Rev. Lett. 97, 240402 (2006).
+[11] K. Kasamatsu and M. Tsubota, Phys. Rev. Lett. 97,
+240404 (2006).
+[12] H. Pu et al., Phys. Rev. Lett. 94, 190401 (2005).
+[13] K. Kasamatsu, Phys. Rev. A 79, 021604 (2009).
+[14] M. Polini et al., Phys. Rev. Lett. 95, 010401 (2005).
+[15] R. Bhat et al., Phys. Rev. A 76, 043601 (2007).
+[16] R. N. Palmer and D. Jaksch, Phys. Rev. Lett. 96, 180407
+(2006).
+[17] A. S. Srensen et al., Phys. Rev. Lett. 94, 086803 (2005).
+[18] N. R. Cooper, Advances in Physics 57, 539 (2008).
+[19] R. A. Williams et al., Opt. Express 16, 16977 (2008).
+[20]V0was calibrated by Kapitza-Dirac diraction with an
+uncertainty of 10%. The measured V0agreed with that
+calculated from measured parameters within this error.
+[21] D. R. Scherer et al., Phys. Rev. Lett. 98, 110402 (2007).
+[22] V. Schweikhard et al., Phys. Rev. Lett. 99, 030401 (2007)
+[23] K. W. Madison et al., Phys. Rev. Lett. 84, 806 (2000).
+[24] C. Raman et al., Phys. Rev. Lett. 87, 210402 (2001).
+[25] A. L. Fetter and A. A. Svidzinsky, J. Phys. C 13, R135
+(2001).
+[26] A. L. Fetter, Rev. Mod. Phys. 81, 647 (2009).
+[27] T. Wang and S. Yelin, arXiv:quant-ph/0610114v3.
+[28] V. Bretin et al., Phys. Rev. Lett. 92, 050403 (2004).
+[29] D. S. Goldbaum and E. J. Mueller, Phys. Rev. A 79,
+063625 (2009).
+[30] A. Griessner et al., Phys. Rev. Lett. 97, 220403 (2006).
\ No newline at end of file
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@@ -0,0 +1,106 @@
+arXiv:1001.0866v1  [quant-ph]  6 Jan 2010Comment on ‘Effective polar potential in the central
+force Schr¨ odinger equation’
+Francisco M. Fern´ andez
+INIFTA (UNLP, CCT La Plata-CONICET), Divisi´ on Qu´ ımica Te´ orica, Blvd. 113
+S/N, Sucursal 4, Casilla de Correo 16, 1900 La Plata, Argentina
+E-mail:fernande@quimica.unlp.edu.arEffective polar potential 2
+Abstract. We analyze a recent pedagogical proposal for an alternative trea tment of
+the angular part of the Schr¨ odinger equation with a central pote ntial. We show that
+the authors’ arguments are unclear, unconvincing and misleading.
+Inarecent paperShikakhwa andMustafa[1]put forwardanaltern ativepedagogical
+discussion oftheangularpartofthesolutiontotheSchr¨ odinger e quationforaquantum–
+mechanical model with a central force:
+−¯h2
+2m∇2ψ+V(r)ψ=Eψ (1)
+They devoted part of the paper to show that this equation is separ able in spherical
+coordinates: ψ(r,θ,φ) =R(r)Θ(θ)eimφ, wherem= 0,±1,..., a discussion that appears
+in almost every introductory textbook on quantum mechanics or qu antum chemistry
+[2,3].
+In particular, the authors concentrated on the polar equation
+1
+sinθd
+dθsinθdΘ
+dθ−m2
+sin2θ=−l(l+1)Θ (2)
+wherel= 0,1,...is the angular–momentum quantum number. They proposed to
+convert this Sturm–Liouville equation into the Schr¨ odinger–like one
+−1
+2d2y(θ)
+dθ2+m2−1
+4
+2sin2θy(θ) =Wy(θ) (3)
+wherey(θ) = sin1
+2θΘ(θ) andW=Wl= (1/2)[l(l+ 1) + 1/4]. We want to call the
+reader’s attention to the misleading notation used by the authors w ho calledEto the
+eigenvalue of this equation as if it where the energy of the central–fi eld model (1) (see
+their equations (12) and (14)). To avoid such misunderstanding we choose the symbol
+Wfor the eigenvalue of the polar equation. We think that for a pedago gical discussion
+it would have been more reasonable that the authors had chosen a r igid rotator in which
+case the eigenvalue of the polar equation is proportional to the ene rgy of the system.
+It is worth adding that the transformation of a Sturm–Liouville prob lem like (2) into a
+Schr¨ odinger equation like (3) is well–known since long ago.Effective polar potential 3
+If we rewrite |m|= 0,1,...,lasl=|m|+n,n= 0,1,...then we derive the correct
+form of the eigenvalue of the polar equation in terms of mandn
+Wl=1
+2/parenleftbigg
+l+1
+2/parenrightbigg2
+=1
+2/parenleftbigg
+n+|m|+1
+2/parenrightbigg2
+(4)
+which shows that Wldoes not depend on the sign of m, as expected from the fact
+that the effective polar potential in Eq. (3) depends on m2. On the other hand, the
+authors’ polar energy Em
+n(see their equation (14)) depends on mand does not clearly
+reflect the degeneracy just mentioned. In order to derive their e xpression the authors
+resorted to the following unconvincing and rather misleading argume nt “These solutions
+are for non–negative m; those for negative mare–as is well known–directly proportional
+to these solutions.” The proportionality factor is in fact e±i|m|φand reflects part of
+the degeneracy of the central–field models; for that reason one s hould not neglect it so
+lightly.
+The authors state that “The solutions |Pm
+l(cosθ)|2represent the probability of
+finding the particle at a certain angle θ”. They seemed to have forgotten the
+normalization factor Nm
+land that the polar part of the volume element is sin θdθ
+because the actual probability for finding the particle between θandθ+dθis well
+known to be |Nm
+lPm
+l(cosθ)|2sinθ. Besides, |Pm
+l(cosθ)|2is not asolution to the polar
+equation.
+The eigenvalue Wlincreases with |m|as shown by Eq. (4). In order to explain this
+behaviour Shikakhwa and Mustafa [1] plotted the polar potential fo r increasing values
+of|m|and showed that the minimum increases. We agree that this graphica l procedure
+is illustrative, but the well–known Hellmann–Feynman theorem [2] is mo re elegant and
+rigorous, and should be added to the discussion. If we consider the eigenvalue equation
+ˆAy=Wy(y(0) =y(π) = 0) for the operator
+ˆA=−1
+2d2
+dθ2+λ
+sin2θ(5)
+whereλis real, then that theorem states that
+dW
+dλ=/angbracketleftBig
+sin−2θ/angbracketrightBig
+>0 (6)Effective polar potential 4
+Clearly, as |m|increasesλincreases and Wlincreases.
+Summarizing, we think that the paper by Shikakhwa and Mustafa [1] is not suitable
+for pedagogical purposes for the following reasons: first, they a pparently mistook the
+eigenvalue of the polar equation for the total energy of the centr al field model, second,
+the treatment of the polar equation (3) is unclear and misleading. In particular, the
+polar eigenvalue in their equation (14) does not clearly reveal the de generacy coming
+from the sign of m, and the argument for the restriction to m≥0 in their expressions
+for the polar eigenvalue and eigenfunction is unconvincing and unnec essary. In fact, the
+correct result follows straightforwardly from the form of Wlas we have already shown
+above. Besides, thesame simple argument clearly shows that P|m|
+l(cosθ) =P|m|
+|m|+n(cosθ)
+which is consistent with the textbook treatment of the problem [2, 3]. We should also
+add the sloppy discussion of the probability of finding the particle in a g iven region of
+space.
+[1] Shikakhwa M S and Mustafa M 2010 Eur. J. Phys. 31151.
+[2] Cohen-Tannoudji C, Diu B, and Lalo¨ eF 1977 Quantum Mechanics (John Wiley &Sons, New York).
+[3] Eyring H, Walter J, and Kimball G E 1944 Quantum Chemistry (John Wiley & Sons, New York).
\ No newline at end of file
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+arXiv:1001.0867v2  [cond-mat.stat-mech]  18 Mar 2010Critical behavior of the pure and random-bond two-dimensio nal triangular Ising
+ferromagnet
+Nikolaos G. Fytas and Anastasios Malakis
+Department of Physics, Section of Solid State Physics,
+University of Athens, Panepistimiopolis, GR 15784 Zografos , Athens, Greece
+(Dated: October 28, 2018)
+We investigate the effects of quenched bond randomness on the critical properties of the two-
+dimensional ferromagnetic Ising model embedded in a triang ular lattice. The system is studied in
+both the pure and disordered versions by the same efficient two -stage Wang-Landau method. In
+the first part of our study we present the finite-size scaling b ehavior of the pure model, for which
+we calculate the critical amplitude of the specific heat’s lo garithmic expansion. For the disordered
+system, the numerical data and the relevant detailed finite- size scaling analysis along the lines of
+the two well-known scenarios - logarithmic corrections ver sus weak universality - strongly support
+the field-theoretically predicted scenario of logarithmic corrections. A particular interest is paid to
+the sample-to-sample fluctuations of the random model and th eir scaling behavior that are used as
+a successful alternative approach to criticality.
+PACS numbers: 75.10.Nr, 05.50.+q, 64.60.Cn, 75.10.Hk
+I. INTRODUCTION
+Understanding the role played by impurities on the
+nature of phase transitions is of great importance, both
+from experimental and theoretical perspectives. First-
+order phase transitions are known to be dramatically
+softened under the presence of quenched randomness [1–
+10], while continuous transitions may have their expo-
+nents altered under random fields or random bonds [11,
+12]. There are some very useful phenomenological ar-
+guments and some, perturbative in nature, theoreti-
+cal results, pertaining to the occurrence and nature of
+phase transitions under the presence of quenched ran-
+domness [2, 5, 13, 14]. Historically, the most celebrated
+criterion is that suggested by Harris [11]. This criterion
+relates directly the persistence, under random bonds, of
+the non random behavior to the specific heat exponent
+αpof the pure system. According to this criterion, if
+αp>0, then disorder will be relevant, i.e., under the ef-
+fect of the disorder, the system will reach a new critical
+behavior. Otherwise, if αp<0, disorder is irrelevant and
+the critical behavior will not change.
+Pure systems with a zero specific heat exponent ( αp=
+0) are marginal cases of the Harris criterion and their
+study, upon the introduction of disorder, has been of
+particular interest [15]. The paradigmatic model of the
+marginal case is, of course, the general random 2d Ising
+model (random-site, random-bond, and bond-diluted)
+and this model has been extensively investigated and de-
+bated [see Ref. [16] and references therein]. Several re-
+cent studies, both analytical (renormalization group and
+conformal field theories) and numerical (mainly Monte
+Carlo (MC) simulations) devoted to this model, have
+provided very strong evidence in favor of the so-called
+logarithmic corrections’s scenario [17–21]. According to
+this, the effect of infinitesimal disorder gives rise to a
+marginal irrelevance of randomness and besides logarith-
+mic corrections, the critical exponents maintain their 2dIsing values. In particular, the specific heat is expected
+to slowly diverge with a double-logarithmic dependence
+of the form C∝ln(lnt), where t=|T−Tc|/Tcis the
+reduced critical temperature [17–21]. Here, we should
+mention that there is not full agreement in the literature
+and a different scenario, the so-called weak universality
+scenario [22–25], predicts that critical quantities, such
+as the magnetization, zero-field susceptibility, and cor-
+relation length display power-law singularities, with the
+corresponding exponents β,γ, andνchanging continu-
+ously with the disorder strength; however this variation
+is such that the ratios β/νandγ/νremain constant at
+the pure system’s value. The specific heat of the disor-
+dered system is, in this case, expected to saturate, with a
+correspondingcorrelationlength’sexponent ν≥2/d[12].
+In general, a unitary and rigorous physical description
+of critical phenomena in disordered systems still lacks
+and certainly, lacking such a description, the study of
+further models for which there is a general agreement
+in the behavior of the corresponding pure cases is very
+important. In this sense, the triangular Ising ferromag-
+net is a further suitable candidate for testing the above
+predictions that has not been previously investigated in
+the literature. Thus, our investigation will be related to
+the extensive relevant literature concerning the critical
+properties of the disordered 2d square Ising model [15–
+23, 26–59]. In particular, our discussion will focus on the
+main point of dispute over the last two decades, concern-
+ing the two well-known conflicting scenarios mentioned
+above and we will provide additional new evidence in fa-
+vor of the well-established scenario of strong universality.
+We should note here that, the theoreticallypredicted sce-
+narioofstronguniversalityhasbeen confirmed byseveral
+MC studies on the square lattice starting from the early
+90’s to nowadays [see Ref. [16] for a detailed historical
+review].
+As mentioned above, it is alwaysimportant to consider
+further models for which the critical properties of the2
+corresponding pure versions are exactly known. This is
+also the case for the present model under consideration,
+namely the triangularIsing model, called hereafteras the
+TrIM, defined as usual by the Hamiltonian
+H=−J/summationdisplay
+<ij>sisj, (1)
+where the spin variables sitake on the values −1,+1,
+< ij > indicates summation over all nearest-neighbor
+pairs of sites, and J >0 is the ferromagnetic exchange
+interaction. The TrIM belongs to the same universality
+class with the corresponding square Ising model (SqIM),
+sharing the same values of critical exponents and a log-
+arithmic behavior of the specific heat [60, 61]. Addition-
+ally, the critical temperature of the model and also the
+critical amplitude A0of Ferdinand and Fisher’s [62] spe-
+cific heat’s logarithmic expansion [see also the discussion
+in Sec. III and Eq. (7)] are exactly known from the early
+work of Houtappel [63] to be Tc= 4/ln3 = 3.6409···
+andA0= 0.499069···, respectively. Nevertheless, it ap-
+pears that for the TrIM a verification of the finite-size
+scaling (FSS) properties of the model using high qual-
+ity data from MC simulation is still lacking. Thus, the
+first part of this work is devoted to the investigation of
+the FSS behavior of the model, especially the estimation
+of the amplitudes and other relevant coefficients in the
+specific heat’s logarithmic expansion and also to the es-
+timation of the critical exponents. In this sense, the aim
+of this first part is twofold: First, to provide the first
+detailed FSS analysis of the pure model and, second, to
+present a concrete reliability test of the proposed numer-
+ical scheme.
+Our main focus, on the other hand, is the case with
+bond disorder given by the bimodal distribution
+P(Jij) =1
+2[δ(Jij−J1)+δ(Jij−J2)] ; (2)
+J1+J2
+2= 1 ;J1> J2>0 ;r=J2
+J1,
+so thatrreflects the strength of the bond randomness
+andwefix2 kB/(J1+J2) = 1tosetthetemperaturescale.
+The value ofthe disorderstrength considered throughout
+this paper is r= 1/3. The resulting quenched disor-
+dered (random-bond) version of the Hamiltonian defined
+in Eq. (1) reads now as
+H=−/summationdisplay
+<ij>Jijsisj (3)
+and will be referred in the sequel as the random-bond
+triangular Ising model (RBTrIM). The corresponding
+random-bondSqIMwillbedenotedhereafterrespectively
+as RBSqIM. The model on the square lattice has the ad-
+vantage that the critical temperature is exactly known
+as a function of the disorder strength rby duality re-
+lations [64]. For the RBTrIM there exist only several
+approximations for the critical frontier of the site- andbond-diluted cases, obtained via renormalization-group
+techniques [65] and, to our knowledge, a study of the
+critical behavior of the model is lacking.
+The rest of the paper is laid out as follows: In Sec. II
+we present the necessarysimulation details of our numer-
+ical scheme. In Sec. III we discuss the FSS behavior of
+the pure model, testing with our high accuracy numeri-
+cal data the exact expansion of the critical specific heat.
+Then, in Sec. IV we present a detailed FSS analysis for
+the random version of the model, including - apart from
+the classical FSS techniques - concepts from the scaling
+theory of disordered systems. Our results and the rele-
+vant discussion clearly favors the scenario of strong uni-
+versality in marginal disordered systems. Finally, Sec. V
+summarizes our conclusions.
+II. SIMULATION DETAILS
+Resorting to large scale MC simulations is often nec-
+essary [66], especially for the study of the critical be-
+havior of disordered systems. It is also well known [67]
+that for such complex systems traditional methods be-
+come very inefficient and that in the last few years sev-
+eral sophisticated algorithms, some of them are based
+on entropic iterative schemes, have been proven to be
+very effective. The present numerical study of the RB-
+TrIM will be carried out by applying our recent and ef-
+ficient entropic scheme [59, 68, 69]. In this approach we
+follow a two-stage strategy of a restricted entropic sam-
+pling, which is described in our study of random-bond
+Ising models (RBIM) in 2d [59] and is very similar to
+the one applied also in our numerical approach to the
+3d random-field Ising model (RFIM) [69]. In these pa-
+pers, we have presented in detail the various sophisti-
+cated routes used for the restriction of the energy sub-
+spaceandthe implementationofthe Wang-Landau(WL)
+algorithm [70]. Further details and an up to date imple-
+mentation of this approach, especially for the study of
+disordered systems, is provided in our recent paper on
+the universality aspects of the pure and random-bond 2d
+Blume-Capel model [71].
+We do not wish to reproduce here the details of our
+two-stage implementation and the practice followed in
+our scheme for improving accuracy by repeating the sim-
+ulations. However, we should like to include a brief dis-
+cussion on the approximate nature of the WL method.
+The usual WL recursion proceeds by modifying the den-
+sity of states G(E) according to the rule G(E)→fG(E)
+andinitially one chooses G(E) = 1 and f=f0=e. Once
+the accumulative energyhistogramis sufficiently flat, the
+modification factor fis redefined as: fj+1=/radicalbig
+fj, with
+j= 0,1,...andtheenergyhistogramresettozerountil f
+is very close to unity (i.e. f=e10−8≈1.000 000 01). As
+has been reported by many authors in the study of sev-
+eral models, once fis close enough to unity, systematic
+deviations become negligible. However, the WL recur-
+sion violates the detailed balance from the early stages of3
+/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48
+/s76/s32/s61/s32/s50/s48/s48
+/s76/s32/s61/s32/s49/s50/s48
+/s32/s32/s42
+/s113
+/s113
+FIG. 1: Disorder distribution of the susceptibility maxima of
+two lattices with linear sizes L= 120 and L= 200 of the
+RBTrIM. The running averages over the samples is shown by
+the thick solid lines.
+the process and care is necessary in setting up a proper
+protocol of the recursion. In spite of the fact that the
+WL method has produced very accurate results in sev-
+eral models, it is fair to saythat there is not a safe way to
+access possible systematic deviations in the general case.
+This has been pointed out and critiqued in a recent re-
+view by Janke [72]. For the 2d Ising model (where exact
+results are available for judgement) the WL method has
+been shown to converge very rapidly. Furthermore, from
+our experience and especially from our recent study of
+the 2d RBSqIM [59], for which the exact phase diagram
+is known by duality relations, our high-level WL imple-
+mentation has produced excellent results, enabling us to
+discriminate between competing theoretical predictions
+on that model. Since the RBTrIM is expected to have a
+similar “entropy structure” to the corresponding square
+model, we anticipate and it will be verified in the se-
+quel, that, our WL scheme produces sufficiently accurate
+estimates enabling us, also in this case, to distinguish
+between competing theoretical predictions, as we have
+already done in the corresponding model on the square
+lattice.
+Using this scheme we performed extensive simulations
+for several lattice sizes in the range L= 20−200, over
+large ensembles {1,···,q,···,Q}of random realizations
+(Q= 500). Let us note here that for the pure model
+we simulated for each lattice size, 200 independent runs
+(WL random walks). It is well known that, extensive
+disorder averaging is necessary when studying random
+systems, where usually broad distributions are expected
+leading to a strong violation of self-averaging [73, 74]. A
+measure from the scaling theory of disordered systems,
+whose limiting behavior is directly related to the issue
+of self-averaging [73, 74] may be defined with the help of
+the relativevarianceofthe sample-to-samplefluctuations
+of any relevant singular extensive thermodynamic prop-
+ertyZas follows: RZ= ([Z2]av−[Z]2
+av)/[Z]2
+av. Figure 1presents evidence that the above number of random re-
+alizations is sufficient in order to obtain the true average
+behavior and not a typical one. In particular, we plot
+in this figure (for lattice sizes L= 120 and L= 200)
+the disorder distribution of the susceptibility maxima χ∗
+q
+and the corresponding running average, i.e. a series of
+averages of different subsets of the full data set - each
+of which is the average of the corresponding subset of a
+larger set of data points, over the samples for the simu-
+lated ensemble of Q= 500 disorder realizations. A first
+striking observation from this figure is the existence of
+very large variance of the values of χ∗
+q, indicating the ex-
+pected violation of self-averaging for this quantity. This
+figure illustrates that the simulated number of random
+realizations is sufficient in order to probe correctly the
+averagebehaviorofthe system, sincealreadyfor Q≈300
+the average value of χ∗
+qappears quite stable.
+Closely related to the above issue of self-averaging in
+disordered systems is the manner of averaging over the
+disorder. This non-trivial process may be performed in
+two distinct ways when identifying the finite-size anoma-
+lies, such as the peaks ofthe magnetic susceptibility. The
+first way corresponds to the average over disorder real-
+izations ([ ...]av) and then taking the maxima ([ ...]∗
+av),
+or taking the maxima in each individual realization first,
+and then taking the average ([ ...∗]av). In the present
+paper we have undertaken our FSS analysis using both
+ways of averaging and have found comparable results for
+the values of the critical exponents, as will be discussed
+in more detail below. Closing this brief outline, let us
+comment on the statistical errors of our numerical data.
+The statistical errors of our WL scheme on the observed
+average behavior, were found to be of relatively small
+magnitude (of the order of the symbol sizes) when com-
+pared to the relevant disorder-sampling errors (due to
+the finite number of simulated realizations). Thus, the
+error bars in most of our figures below concerning the
+average [ ...]∗
+avand used also in the corresponding fitting
+attempts, reflect the disorder-sampling errors and have
+been estimated using groups of 50 realizations via the
+jackknife method [67]. On the other hand for the case
+[...∗]avthe errorbars shownreflect the sample-to-sample
+fluctuations.
+III. PURE MODEL
+In this Section we proceed to investigate the critical
+behavior of the pure TrIM defined in Eq. (1). Our aim
+is to observe the exact critical behavior of the model and
+also to estimate the whole set of critical exponents, pay-
+ing particular attention to the FSS behavior of the crit-
+ical specific heat. As mentioned above, the numerical
+data shown below in Figs. 2 - 4 have been estimated as
+averages over 200 independent runs together with the
+corresponding error bars.
+Figure 2(a) gives the shift behavior of the pseudocrit-
+ical temperatures corresponding to the peaks of the fol-4
+/s49/s48 /s49/s48/s48/s49/s48/s49/s48/s48/s49/s48/s48/s48/s48 /s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48 /s50/s48/s48/s51/s46/s54/s51/s46/s55/s51/s46/s56/s51/s46/s57/s52/s46/s48/s52/s46/s49/s52/s46/s50
+/s40/s98/s41
+/s32/s40 /s32/s108/s110/s32/s60/s77/s110
+/s62/s32/s47/s32 /s32/s75/s41
+/s75/s61/s75
+/s99
+/s76/s32/s110/s32/s61/s32/s49/s58/s32 /s32/s61/s32/s48/s46/s57/s57/s56/s40/s52/s41
+/s32/s110/s32/s61/s32/s50/s58/s32 /s32/s61/s32/s48/s46/s57/s57/s57/s40/s53/s41
+/s32/s110/s32/s61/s32/s52/s58/s32 /s32/s61/s32/s48/s46/s57/s57/s57/s40/s51/s41
+/s32/s32/s84/s42
+/s76/s59/s32/s90
+/s76/s32/s90/s32/s61/s32/s67
+/s32/s90/s32/s61/s32
+/s32/s90/s32/s61/s32 /s32/s60/s124/s77/s124/s62/s32/s47 /s32/s75
+/s32/s90/s32/s61/s32 /s32/s108/s110/s32/s60/s77/s62/s32/s47/s32 /s32/s75
+/s32/s90/s32/s61/s32 /s32/s108/s110/s32/s60/s77/s50
+/s62/s32/s47/s32 /s32/s75
+/s32/s90/s32/s61/s32 /s32/s108/s110/s32/s60/s77/s52
+/s62/s32/s47/s32 /s32/s75/s84
+/s99/s32/s61/s32/s51/s46/s54/s52/s49/s50/s40/s53/s41/s32 /s84
+/s99/s40/s101/s120/s97/s99/s116/s41/s32/s61/s32/s51/s46/s54/s52/s48/s57
+/s61/s32/s49/s46/s48/s48/s52/s40/s53/s41
+/s40/s97/s41
+FIG. 2: (a) Simultaneous fitting of the form (6) of 6 pseudo-
+critical temperatures defined in the text. (b) FSS of several
+powers of the logarithmic derivatives [Eq. (5)] of the order
+parameter at the critical temperature in a log-log scale. Th e
+solid lines are linear fittings giving the value ν= 1 for the
+correlation length’s exponent.
+lowing six quantities: specific heat C, magnetic suscepti-
+bilityχ, derivative of the absolute order parameter with
+respect to inverse temperature K= 1/T[75]
+∂∝an}bracketle{t|M|∝an}bracketri}ht
+∂K=∝an}bracketle{t|M|H∝an}bracketri}ht−∝an}bracketle{t|M|∝an}bracketri}ht∝an}bracketle{tH∝an}bracketri}ht, (4)
+and logarithmic derivatives of the first ( n= 1), second
+(n= 2), and fourth ( n= 4) powers of the order parame-
+ter with respect to inverse temperature [75]
+∂ln∝an}bracketle{tMn∝an}bracketri}ht
+∂K=∝an}bracketle{tMnH∝an}bracketri}ht
+∝an}bracketle{tMn∝an}bracketri}ht−∝an}bracketle{tH∝an}bracketri}ht. (5)
+Fitting our data for the whole lattice range to the ex-
+pected power-law behavior
+T∗
+L;Z=Tc+bL−1/ν, (6)
+whereZstands for the different thermodynamic quan-
+tities mentioned above, we find the critical temperature
+to beTc= 3.6412(5) which is in excellent agreement
+with the exact value 3 .6409···. Additionally, our esti-
+mate of the critical exponent νof the correlation length
+isν= 1.004(5),alsoinexcellentagreementwiththevalue
+ν= 1. Additional estimates for the critical exponent ν
+may be obtained from the scaling behavior of the loga-
+rithmic derivatives (5), which scale as ∼L1/νwith the/s48 /s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48 /s50/s48/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48
+/s65
+/s48/s32/s61/s32/s48/s46/s52/s57/s57/s48/s40/s57/s41
+/s66/s32/s61/s32/s48/s46/s49/s51/s56/s40/s53/s41/s67
+/s99/s32/s61/s32/s65
+/s48/s32/s108/s110/s32/s76/s32/s43/s32/s66
+/s32/s32/s67
+/s99
+/s76
+FIG. 3: FSS of the specific heat data at the exact critical
+temperature.
+lattice size [75]. The FSS of these logarithmic derivatives
+at the critical temperature is shown in Fig. 2(b) and in
+all three cases a value ν≈1 is obtained consistent with
+the estimate from panel (a) of Fig. 2 and with the exact
+valueν= 1.
+Weknowturntothemostinterestingissueinthestudy
+of the pure model, which is the specific heat’s logarith-
+mic expansion, as mentioned in the introduction. For
+the square lattice, it was shown in 1969 in the pioneer-
+ing work of Ferdinand and Fisher [62] that close to the
+critical point the specific heat obeys the following FSS
+expansion
+CL(T) =A0lnL+B(T)+B1(T)lnL
+L+B2(T)1
+L+···,
+(7)
+where the value of the critical amplitude A0is
+0.494358···. As pointed out in Ref. [62], this is the same
+with the amplitude A0in the temperature expansion of
+the specific heat close to the critical point and this was
+already known from the original paper of Onsager [76].
+The first Bcoefficients are given in Ref. [62] and further
+details have been presented in Refs. [68, 77–79]. In par-
+ticular, at the critical temperature the constant term B
+is 0.138149···and as it is also well-known [62, 78] the
+coefficient B1is zero.
+The universality of the above expansion has been al-
+ready pointed out and discussed by Fisher [60]. For the
+three most common 2d lattices, i.e. the square, plane
+triangular, and honeycomb, a unified approach has been
+presented by Wu et al.[79], from which one can find also
+forthe plane triangularlattice the first B-coefficientsand
+the amplitude A0at the critical temperature. As shown
+by Wuet al.[79] for all the three lattices, the coeffi-
+cientB1is zero at the critical temperature and as in the
+square lattice, also for the triangular lattice, the crit-
+ical amplitude of the expansion is A0= 0.499069···,
+equal to the amplitude of the temperature expansion
+close to the critical point obtained by Houtappel [63].
+From the work of Wu et al.[79] we find, using their
+closed form expressions, for the plane triangular lat-
+tice (with periodic boundary conditions and aspect ratio5
+/s49/s48 /s49/s48/s48/s48/s46/s53/s48/s46/s53/s53/s48/s46/s54/s48/s46/s54/s53/s48/s46/s55/s48/s46/s55/s53
+/s49/s48 /s49/s48/s48/s49/s49/s48/s49/s48/s48/s49/s48/s48/s48
+/s32/s32/s77
+/s99
+/s76/s32/s47/s32 /s32/s61/s32/s48/s46/s49/s50/s53/s50/s40/s52/s41
+/s40/s97/s41
+/s32/s47/s32 /s32/s61/s32/s49/s46/s55/s53/s52/s40/s53/s41
+/s40/s98/s41
+/s32/s32/s99
+/s76
+FIG. 4: Magnetic exponent ratios of the pure TrIM: FSS in a
+log-log scale of (a) the order parameter and (b) the magnetic
+susceptibility at the critical temperature. In both cases l inear
+fittings are applied.
+R= 1) the following values B=B(Tc) = 0.14185···
+andB2=B2(Tc) =−0.15003···.
+It is of interest at this point to examine the compat-
+ibility of our numerical data with the above expansion
+for the case of the critical specific heat data, i.e. the
+data of the specific heat at the exact critical tempera-
+ture of the TrIM. In Fig. 3 we consider only the first two
+terms in the above expansion (i.e. we set also B2= 0)
+and pay attention in estimating the critical amplitude
+A0and the constant B-contribution. From the results
+of the fitting in the total lattice range L= 20−200, as
+also shown in the figure, we observe that the estimated
+value for the critical amplitude A0= 0.4990(9) is very
+close to the exact value A0= 0.499069···[63]. For the
+firstB-coefficient we find the value B= 0.138(5) which
+is in good agreement with the exact result 0 .14185···.
+Let us note here that if we try to fit the data of Fig. 3
+including also the coefficient B2of the expansion (7) we
+get the estimates 0 .4952(59), 0 .153(16), and −0.20(12)
+for the critical amplitude A0and the coefficients Band
+B2, respectively. However, if we fix the values of A0and
+Bto their exact ones, the estimate for B2we get from
+the fitting is −0.158(11), in excellent agreement with the
+exact value, whereas if we fix the value of BandB2to
+their exact ones we get the estimate 0 .4990(3) for thecritical amplitude A0.
+Finally, Figs. 4(a) and (b) present our estimations for
+the magnetic exponent ratios β/νandγ/ν. For the esti-
+mation of β/νwe use the values of the order parameter
+at the exact critical temperature. As shown in panel
+(a), in a log-log scale, the linear fitting provides the es-
+timateβ/ν= 0.1252(4). In panel (b) we show the FSS
+of the critical susceptibility, also in a log-log scale. The
+straight line is a linear fitting for L≥20 giving the esti-
+mateγ/ν= 1.754(5). Thus, our results presented in this
+Sectionforthepure2dTrIMmodelareinexcellentagree-
+ment with the exact results and also with the expected
+logarithmic expansion of the specific heat [62, 79]. This
+consists a very strong accuracy test of the proposed two-
+stage WL entropic sampling in restricted energy spaces.
+IV. RANDOM MODEL
+We now present our numerical results for the random-
+bond version of the triangular Ising model for disorder
+strength r= 1/3. From simple universality-type theoret-
+ical arguments, this system is also expected to undergo
+a second-order transition between the ferromagnetic and
+paramagnetic phases and in particular it should be also
+expected that this transition will be in the same univer-
+sality class as the RBSqIM.
+We start our analysis of the RBTrIM by presenting
+the general shift behavior of various pseudocritical tem-
+peratures of the model. Figure 5(a) illustrates the shift
+behavior of 7 pseudocritical temperatures defined as the
+temperature where the corresponding average thermo-
+dynamic property attains its maximum. The first 6
+are as those defined in Sec. II for the corresponding
+pure model. The last pseudocritical temperature is a
+newly introduced temperature, defined as the tempera-
+ture where the ratio R[χ∗]avdefined in Sec. II, becomes
+maximum. The solid lines show an excellent simultane-
+ous power-law fitting attempt of the form (6) giving the
+valueTc= 3.4642(52) for the critical temperature of the
+random model and a value ν= 0.997(6) for the critical
+exponent νof the correlation length. The fitting shown
+in Fig. 5(a) has been performed for all lattice sizes and
+it was very stable when shifting the L-range to larger
+values. Note also that a simple fitting attempt using
+only the newly defined pseudocritical temperature de-
+fined with the help of the sample-to-sample fluctuations
+gives a value Tc= 3.4677(47) for the critical tempera-
+ture and a value ν= 0.994(8) for the correlation length’s
+exponent. These overall estimates for the exponent ν
+consist a strong indication that the RBTrIM shares the
+same value of νas the pure version, thus reinforcing the
+scenario of logarithmic corrections (strong universality).
+Figure 5(b) illustrates again the shift behavior of the 6
+pseudocritical temperatures of panel (a), estimated now
+via the second way of averaging discussed in Sec. I, i.e.
+by taking the average over the individual pseudocriti-6
+/s48 /s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48 /s50/s48/s48/s51/s46/s50/s51/s46/s52/s51/s46/s54/s51/s46/s56/s52/s46/s48
+/s48 /s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48 /s50/s48/s48/s51/s46/s52/s51/s46/s53/s51/s46/s54/s51/s46/s55/s51/s46/s56/s51/s46/s57
+/s48 /s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48 /s50/s48/s48/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52/s48/s46/s48/s53
+/s32/s32
+/s84
+/s99/s32/s61/s32/s51/s46/s52/s54/s52/s50/s40/s53/s50/s41
+/s32/s61/s32/s48/s46/s57/s57/s55/s40/s54/s41/s84
+/s91/s90/s93/s42
+/s97/s118
+/s76/s32/s90/s32/s61/s32/s67
+/s32/s90/s32/s61/s32
+/s32/s90/s32/s61/s32 /s32/s60/s124 /s77/s124 /s62/s32/s47/s32 /s32/s75
+/s32/s90/s32/s61/s32 /s32/s108/s110/s32/s60/s77/s62/s32/s47/s32 /s32/s75
+/s32/s90/s32/s61/s32 /s32/s108/s110/s32/s60/s77/s50
+/s62/s32/s47/s32 /s32/s75
+/s32/s90/s32/s61/s32 /s32/s108/s110/s32/s60/s77/s52
+/s62/s32/s47/s32 /s32/s75/s40/s97/s41
+/s32/s90/s32/s61/s32/s82
+/s91 /s93
+/s97/s118 
+/s40/s98/s41
+/s84
+/s99/s32/s61/s32/s51/s46/s52/s54/s56/s50/s40/s53/s57/s41
+/s32/s61/s32/s48/s46/s57/s57/s54/s40/s56/s41
+/s32/s91/s84/s42
+/s90/s93
+/s97/s118
+/s76/s32/s90/s32/s61/s32/s67
+/s32/s90/s32/s61/s32
+/s32/s90/s32/s61/s32 /s32/s60/s124 /s77/s124 /s62/s32/s47/s32 /s32/s75
+/s32/s90/s32/s61/s32 /s32/s108/s110/s32/s60/s77/s62/s32/s47/s32 /s32/s75
+/s32/s90/s32/s61/s32 /s32/s108/s110/s32/s60/s77/s50
+/s62/s32/s47/s32 /s32/s75
+/s32/s90/s32/s61/s32 /s32/s108/s110/s32/s60/s77/s52
+/s62/s32/s47/s32 /s32/s75
+/s40/s99/s41
+/s32/s40/s91/s84/s42
+/s93/s41
+/s97/s118
+/s76/s91/s84/s42
+/s93
+/s97/s118/s41/s32/s126/s32/s76/s45/s32/s110
+/s110/s32/s61/s32/s49/s46/s48/s48/s49/s40/s51/s41/s32/s61/s32/s100/s32/s47/s32/s50
+FIG. 5: (a) - (b) Shift behavior of several pseudocritical
+temperatures defined in the text. The error bars in panel
+(b) reflect the sample-to-sample fluctuations. (c) FSS of the
+sample-to-sample fluctuations of the pseudocritical tempe ra-
+ture of the magnetic susceptibility shown in panel (b).
+cal temperatures. The error bars shown in this panel
+reflect the sample-to-sample fluctuations of the pseudo-
+critical temperatures. Again, the results obtained from
+the simultaneous fitting attempt Tc= 3.4682(59) and
+ν= 0.996(8), as also shown in the figure, agree excel-
+lently with the estimates of panel (a), providing further
+evidence in favorofthe accuracyofour numericalscheme
+and the stronguniversalityhypothesis. Noteworthythat,
+if we fix the exponent νto the exact value ν= 1 we get
+the most accurate estimates for the critical temperature
+to be 3.4663(16) and 3 .4669(19) from the corresponding
+fittings of panels (a) and (b) of Fig. 5.
+Using now the above sample-to-sample fluctuations of/s49/s48 /s49/s48/s48/s49/s48/s49/s48/s48/s49/s48/s48/s48
+/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50 /s48/s46/s48/s51 /s48/s46/s48/s52 /s48/s46/s48/s53 /s48/s46/s48/s54/s48/s46/s57/s50/s48/s46/s57/s54/s49/s46/s48/s48/s49/s46/s48/s52/s49/s46/s48/s56/s76
+/s109/s105/s110/s32/s61/s32/s54/s48/s40/s97/s41
+/s32/s32/s91/s40 /s32/s108/s110/s32/s60/s77/s110
+/s62/s32/s47/s32 /s32/s75/s41/s42
+/s93
+/s97/s118
+/s76/s32/s110/s32/s61/s32/s49/s58/s32 /s32/s61/s32/s49/s46/s48/s48/s53/s40/s55/s41
+/s32/s110/s32/s61/s32/s50/s58/s32 /s32/s61/s32/s49/s46/s48/s48/s50/s40/s53/s41
+/s32/s110/s32/s61/s32/s52/s58/s32 /s32/s61/s32/s49/s46/s48/s48/s49/s40/s54/s41
+/s40/s98/s41
+/s32/s32/s101/s102/s102
+/s49/s32/s47/s32/s76
+/s109/s105/s110/s32/s110/s32/s61/s32/s49
+/s32/s110/s32/s61/s32/s50
+/s32/s110/s32/s61/s32/s52
+FIG. 6: (a) FSS of the logarithmic derivatives of the order
+parameter in a log-log scale. The solid lines are simple line ar
+fittings for the larger lattice sizes ( Lmin= 60). (b) Values
+of effective exponents νeffobtained from the data of panel
+(a) from several fitting attempts in the range ( Lmin−Lmax).
+The solid line marks the proposed estimate ν= 1.
+the pseudocritical temperatures and the theory of FSS in
+disordered systems as introduced by Aharony and Har-
+ris [73] and Wiseman and Domany [74], one may fur-
+ther examine the nature of the fixed point that controls
+the critical behavior of the disordered system. According
+to the theoretical predictions [73, 74], the pseudocritical
+temperatures T∗
+Zofthe disorderedsystem are distributed
+with a width δ[T∗
+Z]av, that scales with the system size as
+δ([T∗
+Z]av)∼L−n, (8)
+wheren=d/2 orn= 1/νr, depending on whether the
+disordered system is controlled by the pure or the ran-
+dom fixed point, respectively. In panel (c) of Fig. 5 we
+plot these sample-to-sample fluctuations of the pseudo-
+critical temperature of the magnetic susceptibility. The
+solid line shows a very good power-law fitting giving the
+value 1.001(3) for the exponent nof the theory, which
+is in agreement with the case n=d/2, indicating that
+the random model is controlled by the pure fixed point.
+We should note here that analogous results to those dis-
+cussed here in panel (c) for the case of the site-diluted
+Ising model on the square lattice have been presented by7
+/s48 /s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48 /s50/s48/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48/s50/s46/s50
+/s49/s46/s52 /s49/s46/s53 /s49/s46/s54 /s49/s46/s55/s49/s46/s54/s49/s46/s56/s50/s46/s48/s50/s46/s50/s50/s46/s52
+/s76
+/s109/s105/s110/s32/s61/s32/s54/s48
+/s32/s32/s91/s67/s42
+/s93
+/s97/s118
+/s108/s110/s32/s40/s108/s110/s32/s76/s41/s76
+/s109/s105/s110/s32/s61/s32/s54/s48
+/s32/s32/s91/s67/s93/s42
+/s97/s118/s32/s59/s32/s91/s67/s42
+/s93
+/s97/s118
+/s76/s32/s91/s67/s93/s42
+/s97/s118
+/s32/s91/s67/s42
+/s93
+/s97/s118/s32
+FIG. 7: FSS of the specific heat maxima obtained via the
+two distinct ways of disorder averaging. The solid and dotte d
+lines are double-logarithmic fittings of the form (9) for lat tice
+sizes in the range L= 60−200. The inset shows the data
+for the case [ C∗]avas a function of the double logarithm of
+L. The solid line is an excellent linear fitting.
+Tomita and Okabe, using the probability-changing clus-
+ter algorithm [48].
+As in the pure case, the second alternative estimation
+ofνis carried out by analyzing the divergency of the log-
+arithmic derivatives of the order parameter. In Fig. 6(a)
+we illustrate in a double-logarithmic scale the size depen-
+dence of the first- (filled squares), second- (filled circles),
+and fourth-order (filled triangles) logarithmic derivatives
+(averaged over the individual maxima). The solid lines
+show linear fittings for the sizes L≥60. In all cases
+a valueν= 1 is obtained for the critical exponent ν,
+providing further evidence to the strong universality sce-
+nario emerged from Fig. 5. Figure 6(b) illustrates our
+method to evaluate and discuss the stability of the esti-
+mation for the exponent νfrom the scaling behavior of
+the logarithmic derivatives of panel (a). It shows val-
+ues of effective exponents ( νeff) determined by impos-
+ing a lower cutoff ( Lmin) and applying simultaneous fit-
+tings in windows ( Lmin−Lmax), where as for the pure
+case,Lmax= 200 and Lmin= 20,40,60,80, and 100
+as a function of 1 /Lmin. The effective estimates show
+a finite-size effect for small values of the lower cutoff,
+whereas and for L≥60 a clear trend towards the value
+ν= 1 of the Ising universality class is obtained. Let us
+note here that the same picture emerged from the FSS
+of the disorder-averaged logarithmic derivatives of the
+form [∂ln∝an}bracketle{tMn∝an}bracketri}ht/∂K]∗
+avthat corresponds to the first way
+of averaging, but is omitted here for brevity. We should
+note here that a similar cross-over behavior in the esti-
+mates of the critical exponent νhas been observed in the
+case of the 2d site-diluted SqIM by Ballesteros et al.[39]
+and has been explained as logarithmic corrections. Thus,
+summarizingourestimatesforthe criticalexponent ν, we
+feel that it is clear that it maintains the value ν= 1 of
+the pure case, indicating again the validity of the strong
+universality scenario.We continue the presentation of our results by showing
+in Fig. 7 the FSS of the specific heat maxima averaged
+overdisorder: [ C]∗
+av(upfilledtriangles)and[ C∗]av(down
+open triangles). Using these data for the larger sizes
+L≥60, we tried to observe the quality of the fittings,
+assuming a double-logarithmic divergence of the form
+[C]∗
+av;[C∗]av∼C1+C2ln(lnL), (9)
+or a simple power law
+[C]∗
+av;[C∗]av∼C∞+C3Lα/ν. (10)
+Although it is rather difficult to numerically distinguish
+between the above scenarios, our detailed fitting at-
+tempts indicated that the double-logarithmic scenario
+[Eq. (9)] applies better to the numerical data and this
+is generally true for both [ C]∗
+avand [C∗]avdata.
+In fact, the double-logarithmic fitting is shown in
+the main panel, whereas in the corresponding inset of
+Fig. 7 the data of [ C∗]avare plotted as a function of
+ln(lnL). The solid line shown is an excellent linear fit
+forL≥60. Let us now give some details on the qual-
+ity of the applied fittings. We used the following sets
+of data points ( Lmin−Lmax), withLmax= 200 and
+Lmin= 20,40,60,80,and 100. The quality of the fittings
+indicated a very good trend for the values of χ2/DoF for
+the double logarithmic fittings (9) in the range: 0 .2−0.7
+and for both sets of data points. However, a strong relia-
+bility test in favor of the logarithmic corrections scenario
+is provided by the stability of the coefficient C2, for both
+[C]∗
+av(C2≈1.43(5)) and [ C∗]av(C2≈1.48(4)) data. On
+the other hand, the estimated values of the exponent α/ν
+of the power law (10), for both [ C]∗
+avand [C∗]av, fluctu-
+ate in the range [ −0.12(9),−0.05(6)] (as Lminincreases)
+with the fitting procedure becoming rather unstable as
+we move to larger values of Lmin. The conclusion is that
+our numerical data are more properly described by the
+double logarithmic form (9), in agreement with the MC
+findings of Selke et al.[43] and Ballesteros et al.[39]
+for the site-diluted SqIM and also with those of Wang
+et al.[28] for the strong disorder regime ( r= 1/4 and
+r= 1/10) of the RBSqIM.
+In Fig. 8 we provide estimates for the magnetic expo-
+nent ratios β/νandγ/νof the RBTrIM. In panel (a) we
+plot the average magnetization at the estimated critical
+temperature, as a function of the lattice size Lin a log-
+log scale. The solid line is a linear fitting for L≥20
+giving within error bars the value of the pure model, i.e.
+β/ν= 0.1253(5)≈0.125. Additional estimate for the
+ratioβ/νcan be obtained from the FSS of the derivative
+of the absolute order parameter with respect to inverse
+temperature defined in Eq. (4) which is expected to scale
+asL(1−β)/νwith the system size [75]. Thus, in panel (b)
+of Fig. 8 we plot the data for ∂∝an}bracketle{t|M|∝an}bracketri}ht/∂Kaveraged over
+disorder as a function of L, also in a double-logarithmic
+scale. The solid line is a linear fitting for the larger lat-
+tice sizes L≥60, which combined with the value ν= 1,
+gives an estimate of 0 .1247(4) for the ratio β/ν. Finally,8
+in panel (c) we present the FSS of the maxima of the
+average magnetic susceptibility [ χ]∗
+av(filled squares) and
+also the average of the individual maxima [ χ∗]av(open
+triangles). The solid and dotted lines present linear fit-
+tings using the total lattice range spectrum, giving the
+estimates 1 .751(5) and 1 .756(9) for the ratio γ/νin very
+good agreement with the expected value 1 .75 of the pure
+system. For the average [ χ]∗
+avthe error bars indicate the
+statistical errors due to the finite number of the realiza-
+tions, as discussed in Sec. I. For the average [ χ∗]avthe
+errors bars shown reflect now the relatively large sample-
+to-sample fluctuations.
+Finally, using the latter sample-to-samplefluctuations,
+we construct the ratio R[χ∗]avand plot it as a function of
+the inverse linear size, as shown in the inset of Fig. 8(c).
+Clearly,forthepresentmodelthelimitingvalueof R[χ∗]av
+is non-zero, indicating, as expected also for marginal dis-
+ordered systems [74], a strong violation of self-averaging.
+V. CONCLUSIONS
+The effects induced by the presence of quenched bond
+randomness on the critical behavior of the 2d Ising spin
+model embedded in the triangular lattice have been in-
+vestigated by an efficient entropic scheme based on the
+Wang-Landau algorithm. In the first part of our study
+we presented the finite-size scaling behavior of the pure
+model, for which we calculated with high accuracy the
+critical exponents and the coefficients of the specific
+heat’s logarithmic expansion at the critical point. Our
+results are in full agreement with the exact expansionpresented by Wu et al.[79].
+In the second part of our study we investigated the
+critical properties of the disordered system. The pre-
+sented detailed finite-size scaling analysis along the lines
+of the two existing scenarios - strong versus weak univer-
+sality-stronglysupportsthe scenarioofstronguniversal-
+ity. Thus, our results are in agreement with the behavior
+predicted originally on theoretical basis many years ago
+by Dotsenko and Dotsenko [17], Jug [18], Shalaev [19],
+Shankar [20], and Ludwig [21] and verified by simula-
+tions in recent years for the square random Ising model
+by several authors [39–41, 43, 44, 48, 58, 59]. Particular
+interest was paid to the sample-to-sample fluctuations of
+the random model and their scaling behavior that were
+used as a successful alternative approach to estimate the
+critical temperature and the correlation length’s expo-
+nent. Closing, we would like to note that another inter-
+esting candidate, that has not been studied before in the
+triangularlattice, is the 3-statePottsmodel, which, in its
+pure version, has a positive specific-heat exponent. Dis-
+order would be relevant in this case and could provide a
+further complementary study of the present work, analo-
+gous to the early transfer-matrix calculations of Derrida
+et al.[80] of the random-bond model on the square lat-
+tice.
+Acknowledgments
+The authors would like to thank Ralph Kenna and
+Walter Selke for their useful comments and a critical
+reading of the manuscript.
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+/s49/s48 /s49/s48/s48/s48/s46/s53/s48/s46/s53/s53/s48/s46/s54/s48/s46/s54/s53/s48/s46/s55/s48/s46/s55/s53/s48/s46/s56
+/s49/s48 /s49/s48/s48/s49/s49/s48/s49/s48/s48/s49/s48/s48/s48/s49/s48 /s49/s48/s48/s49/s49/s48/s49/s48/s48/s47/s32 /s61/s32/s48/s46/s49/s50/s53/s51/s40/s53/s41/s40/s97/s41
+/s32/s32/s91/s77/s93
+/s97/s118/s40/s84/s61/s84
+/s99/s41
+/s76
+/s48/s46/s48/s48 /s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s50
+/s32/s32
+/s82
+/s91/s42
+/s93
+/s97/s118
+/s49/s47/s76
+/s32/s32/s91 /s93/s42
+/s97/s118/s32/s59/s32/s91/s42
+/s93
+/s97/s118
+/s76/s32/s91 /s93/s42
+/s97/s118/s58/s32 /s47/s32 /s61/s32/s49/s46/s55/s53/s49/s40/s53/s41
+/s32/s91/s42
+/s93
+/s97/s118/s58/s32 /s47/s32 /s61/s32/s49/s46/s55/s53/s54/s40/s57/s41/s40/s99/s41/s40/s98/s41
+/s76
+/s109/s105/s110/s32/s61/s32/s54/s48/s40/s49/s45 /s41/s32/s47/s32 /s32/s61/s32/s48/s46/s56/s55/s53/s51/s40/s52/s41/s32
+/s32/s47/s32 /s32/s61/s32/s48/s46/s49/s50/s52/s55/s40/s52/s41
+/s32/s32/s91/s40 /s32/s60/s124/s77/s124/s62/s32/s47/s32 /s32/s75/s41/s42
+/s93
+/s97/s118
+/s76
+FIG. 8: (a) FSS of the average order parameter at the crit-
+ical temperature. (b) FSS of the disorder-averaged inverse -
+temperature derivative of the absolute order parameter. (c )
+FSS of the disorder-averaged magnetic susceptibility. The in-
+set shows the limiting behavior of the ratio R[χ∗]av. In all
+main panels (a) - (c) a double logarithmic scale is considere d.
+Additionally, all fitting attempts are performed in the com-
+plete lattice range L= 20−200 since no deviation in the
+estimated values of the critical exponents was observed whe n
+shifting the value of the lower cutoff Lmin.
\ No newline at end of file
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@@ -0,0 +1,428 @@
+arXiv 1001.0869 
+ 
+Systematic investigation of influence of n-type doping on electron 
+spin dephasing in CdTe 
+ 
+D. Sprinzl, P. Horodyská, N. Tesa řová, E. Rozkotová, E. Belas, R. Grill,  P. Malý, 
+and P. Němec* 
+Faculty of Mathematics and Physics, Char les University in Prague, Ke Karlovu 3, 
+121 16 Prague 2, Czech Republic 
+ 
+ABSTRACT 
+We used time-resolved Kerr rotation technique to study the electron spin 
+coherence in a comprehensive set of  bulk CdTe samples with various 
+concentrations of electrons that were supplied by n-type doping. The electron spin 
+coherence time of 40 ps was observed at temperature of 7 K in p-type CdTe and 
+in n-type CdTe with a low concentration of electrons. The increase of the 
+concentration of electrons leads to a substantial prolongation of the spin 
+coherence time, which can be as long as 2.5 ns at 7 K in optimally doped samples, and to a modification of the g factor of electrons. The influence of the 
+concentration of electrons is the most pronounced at low temperatures but it has a 
+sizable effect also at room temperature. The optimal concentration of electrons to 
+achieve the longest spin cohe rence time is 17-times higher in CdTe than in GaAs 
+and the maximal low-temperature value of the spin coherence time in CdTe is 70-times shorter than the corresponding value in GaAs. Our data can help in cross-checking the predictions  of various theoretical m odels that were suggested 
+in literature as an explanati on of the observed non-monotonous doping 
+dependence of the electron spin  coherence time in GaAs.  
+ 
+PACS numbers: 78.47.J-, 72.25.Dc, 61.72.uj 
+ 
+ 
+I. INTRODUCTION  
+ 
+The utilization of the electron spin, in addition to its charge, is a heart of the emerging 
+new branch of electronics – spintronics. The dephasing time of a spin coherence, which 
+directly determines the survival of information encoded using the spin of  carriers, is one of 
+the most important material parameters for sp intronics and quantum co mputation. Therefore, 
+the observed suppression of the spin dephasing in n-type GaAs1 motivated a rather intensive 
+research in this field. And even though the influence of n-type doping on the spin relaxation 
+was experimentally observed also in se veral other bulk materials (namely, ZnSe2, GaN3, 4, 
+InSb5, 6, InAs7, and ZnO8) the topic remains controversial. Up to now, a systematic  study of 
+the dependence of the electron spin dephasing time on the concentration of electrons ( n) was 
+reported only in GaAs9 and, very recently, in InSb6 (in other materials, typically, only two or 
+three different doping levels were studied). And the obtained results are quite different in these two materials. In GaAs the increase of n leads to a prolongation of the spin dephasing 
+time for n up to 3 × 10
+15 cm-3 that is followed by its  decrease for higher n (Ref. 9). On the 
+other hand, in InSb the spin re laxation time was reported to decrease  with n (Ref. 6), which, 
+in fact, contradicts the earlier results5 reported for this material. Also from the theoretical 
+point of view this issue is not fully understood. Namely, two different spin relaxation mechanisms (SRM) were stated to be dominant in n-type semiconductors at low temperatures 
+– Elliot-Yafet (EY)
+10, 11 and D’yakonov-Perel (DP)12, 13. Alternatively, the hyperfine 
+interaction and/or a change  of dominant SRM with n were also considered.r4, 9, 14 In addition, a 
+ 1arXiv 1001.0869 
+ 
+considerably different mechanisms were used to explain th e observed non-monotonous 
+dependence of the spin dephasing time on n in GaAs.9, 12, 13 In this paper, we would like to 
+contribute to this topic by providing another model material where the dependence of the 
+electron spin dephasing time on n was measured systematically. We selected CdTe for this 
+research because it has the same crystal struct ure and nearly the same band gap as GaAs but 
+rather different material parame ters (e.g., effective masses, dielec tric constants and spin-orbit 
+interaction). Moreover, the comparis on of the recently published data in n-type doped CdTe 
+quantum wells15 with our data measured in the corre sponding bulk material can help in the 
+understanding of the role of the quantum confinement in the spin dynamics. Prior to this work 
+only room temperature data in intent ionally undoped bulk CdTe were reported16-18. 
+ 
+ 
+II. EXPERIMENTAL 
+ 
+CdTe bulk single crystals were prepared by  a vertical gradient freeze method. All 
+undoped as-grown samples were p-type (with a hole concentration p ≈ 1014 – 1016 cm-3) due 
+to an unintentional doping by foreign acceptors. N-type CdTe single crys tals were prepared by 
+the intentional donor doping (using i ndium). The electron concentration n can be tuned in the 
+interval 107 – 1018 cm-3 using a post-growth annealing of In -doped crystals in the various Cd 
+overpressures.19 The highest electron concentration, which is similar to the donor doping 
+level, was obtained by the annealing at the Cd-saturated overpressure. Crystals with a lower 
+electron concentration can be prepared by th e annealing in a lower Cd overpressure. We 
+studied eight n-type samples with  n spanning from 1.5 × 1013 cm-3 to 3.2 × 1017 cm-3 and also, 
+as a reference, the as-grown p-type sample with p ≈ 1016 cm-3 (the carrier concentration was 
+determined at room temperature by the Hall effect measurement). 
+The dephasing of electron spin coherence was studied by the time-resolved Kerr 
+rotation (KR) technique using a femtosecond Ti -sapphire laser (Tsunami, Spectra Physics). 
+Spin-polarized electrons were optically injected  by laser pulses with a duration of 80 fs and a 
+repetition rate of 82 MHz, which were spectrally tuned to match the band gap energy of CdTe 
+(i.e., during the measurements of the temperat ure dependences the wave length of laser pulses 
+was tuned to follow the shrinkage of a band gap of CdTe with the sample temperature20). The 
+pump laser polarization was modul ated by a photoelastic modulator from left circular to right 
+circular at 50 kHz that eliminated the buildup  of a nuclear spin po larization via hyperfine 
+interaction. The energy fluence of the pump pulses was about 1 μJ.cm-2, which corresponds to 
+a concentration of photoexcit ed carriers of about  5 × 1016 cm-3, and the probe pulses were 
+always at least 10 times weaker. The sample was mounted in a closed-cycle He cryostat 
+placed between the poles of an electromagnet, wh ich created a transverse magnetic field up to 
+≈ 0.7 T. 
+ 
+ 
+III. RESULTS AND DISCUSSION 
+ 
+In Fig. 1 (a) we show the KR signals meas ured in CdTe crystals with different 
+concentration of electrons as  a function of the time delay between pump and probe pulses 
+(Δt). The signals can be fitted by a function 
+ 
+() ( ) (∑
+=Δ Δ − = Δ2
+1cos / exp
+iL i i t t t A t KRω),       ( 1 )  
+ 
+ 2arXiv 1001.0869 
+ 
+0 1000 2000 3000024681012
+0.0 0.2 0.4 0.60204020002500
+102103
+101310151017-1.75-1.70-1.65-1.60
+S7
+S8S6S5S4S3S2
+  KR (arb. units)
+Time delay (ps)S1(a)
+  T2* (ps)
+Magnetic field (T)S6
+p - type
+   T2* (ps) 
+(b)  g factor 
+n (cm-3 )(c)
+ 
+ 
+Fig. 1. (a) Time-resolved KR signal measured in n -CdTe with different concentr ations of electrons (in cm-3) - 
+sample S1: 1.5 × 1013, S2: 3.7 × 1014, S3: 1.3 × 1015, S4: 3.9 × 1015, S5: 1.4 × 1016, S6: 4.9 × 1016, S7: 6.6 × 1016, 
+S8: 3.2 × 1017; the data are normalized and offset for clarity. The measurement was done in a magnetic field of 
+0.685 T at a temperature of 7 K. Inset: Dependence of the spin coherence time  on the transverse magnetic 
+field for the sample S6 (with an optimal co ncentration of electrons) and the reference p-CdTe (points). The upper 
+line depicts the inhomogeneous dephasing *
+2T
+() h2 / 0 / 1 / 1*
+2*
+2 B g T TBμΔ + ≈  (Ref. 22) in S6 (with a Gaussian 
+distribution of g factors Δ g = 0.013) and the horizontal lower line illustrates the independence of  on the 
+magnetic field for the p-type sample. (b) and (c) Dependence of the spin coherence time and the g factor, 
+respectively, on the con centration of electrons n for a magnetic field of 0 T and a temperature of 7 K. The 
+corresponding values in the reference p-CdTe sample are shown as horizontal dashed lines. *
+2T
+*
+2T
+ 
+ 
+where Ai and t i describe the amplitude and the decay time  of the signal envelope, respectively, 
+and ωL is  the Larmor frequency, which is a direct measure of the electron g 
+factor,() ( B gB L)μω/h=  (μB is the Bohr magneton and B is the magnetic field). In the 
+investigated samples the signal envelope c ould not be described well by a simple mono-
+exponential decay. Instead, a double-exponential decay  seems to be more appropriate. Similar 
+double-exponential (or multi-exponential) decay  was observed also in other materials  
+and we attribute the shorter and the longer tim e constants to the lifetime of photo-injected 
+carriers and to the electron tr ansverse spin coherence time , respectively.  In Fig. 1 (b) we 
+show a dependence of the zero field values of  on the concentration of electrons  n. For 
+small n, the obtained values of are the same as that measured in the reference p-type 
+sample, which is probably connected with th e influence of photoexcited carriers with a 
+concentration of about  5 × 10  cm . When n exceeds the value of ≈ 5 × 10  cm  there is a 
+rather substantial prolongati on of the spin coherence time  (up to ≈  2.5 ns for the samples 
+with the optimal value of th e concentration of electrons  n ≈ 5 × 10  cm ) that is followed 
+by a decrease of  for n ≥ 10  cm . This prolongation, which is up to 60-times in samples 
+with n ≈ n with respect to  value observed in the reference p-type sample [the measured 
+value ≈ 40 ps is schematically shown as a hor izontal dashed line in Fig. 1 (b)], is 
+accompanied also by a change of the magnetic field dependence of . In p-type sample,  
+does not depend on the magnetic field but in the sample with n  the values of  decrease B
+3, 4, 15, 21
+*
+2T4
+*
+2T
+*
+2T
+16 -3 15 -3
+*
+2T
+opt 16 -3
+*
+2T17 -3
+opt *
+2T
+*
+2T
+*
+2T*
+2T
+opt *
+2T
+ 3arXiv 1001.0869 
+ 
+with a magnetic field – see In set in Fig. 1 (a). Similar dopi ng-induced change of magnetic 
+field dependence of  was observed also in other semiconductors.  This field 
+dependence of  (i.e., the time constant that charac terizes the decay of the oscillation 
+envelope) in the sample with n  can be a consequence of a “d ephasing” of precession angles 
+within the excited spin population that arises from a spread in the electron g factor.  
+Alternatively, the decrease of  with a magnetic field can be  explained by an additional 
+magnetic field-dependent SRM.   *
+2T1, 3, 8, 15, 22
+*
+2T
+opt
+22
+*
+2T
+3
+Our data further reveal that in CdTe ther e is also a systematic dependence of the g 
+factors on the concentration of electrons. In fact, from the KR  experiment we can determine 
+only the magnitude of the g factor but not its sign. Neverthe less, we can obtain the sign of the 
+g factor from a comparison with the previously reported23 low-temperature value for bulk 
+CdTe g = -1.65 ±  0.03. The measured dependence of the electron g factor on n is shown in 
+Fig. 1 (c). The obtained doping dependence of the g factors is quite similar to that of  - for 
+n < 5 × 10*
+2T
+15 cm-3 the values of the g factors in n-type samples are the same as that in the 
+p-type sample and above this concentr ation there is a reduction of the g factor magnitude. The 
+similar values of the g factors in p-type and n-type samples clearly show s that we measure the 
+spin relaxation of electrons also in the p-type sample (because the electron and the hole g 
+factors are known to be quite different ). Up to now, the doping dependence of g factors was 
+not studied systematically in  n-type semiconductors. We will come  back to the discussion of 
+this effect later on. 
+ The measured non-monotonous dependence of the spin coherence time on the 
+concentration of electrons [Fig. 1 (b)] is quite different from that measured recently in InSb6 
+but it is rather similar to that observed in GaAs (see Fig. 3 in Ref. 9). The major similarity between GaAs and CdTe is that in both materi als there exist an optimal concentration of 
+electrons n
+opt where  is the largest. Nevertheless, there are significant differences also 
+between GaAs and CdTe: in GaAs the largest values of (≈ 180 ns) are obtained for n*
+2T
+*
+2Topt ≈ 
+0.3 × 1016 cm-3, in CdTe the largest values of (≈ 2.5 ns) are obtained for n*
+2Topt ≈ 5 × 1016 
+cm-3. We note that in samples with a low concentration of electrons the value of  is rather 
+similar in both materials – 65 ps in GaAs*
+2T
+24 and 40 ps in CdTe. The exact physical origin of 
+the peak in the doping dependence of in GaAs is still an opened question – this density was 
+even called a ‘magic’ electron density.*
+2T
+22 In Ref. 9 this maximum, which was observed for n 
+close to the metal-to-insulator transition (MIT) in GaAs, was assigned to the crossover between relaxation mechanisms originating fr om the hyperfine inter action of localized 
+electrons with lattice nuclei and from the spin-o rbit interaction of free electrons in the metallic 
+regime. This explanation was adopted also in Ref. 14 where the bias-dependent electron spin 
+lifetimes were measured by the Henle effect using the cw laser. On the other hand, it was argued in Ref. 13 that this peak can be e xplained solely by a breakdown of the motional 
+narrowing in DP SRM – i.e., that MIT does not ha ve to be considered. Similarly, in Ref. 12 
+this non-monotonous dependence of on n was explained by DP SRM only - increases 
+with n in the nondegenerate regime  (i.e., for low values of  n) due to a decrease of the 
+momentum scattering time but it decreases in th e degenerate regime (i.e., for high values of  n) 
+due to an enhancement of the inhomogeneous broadening. Our re sults clearly revealed that a 
+considerably higher value of n*
+2T*
+2T
+opt is required in CdTe than in  GaAs for a maximal suppression 
+of the electron spin dephasing. One possibility is that it is connect ed with the MIT. In general, 
+MIT occurs above the critical Mott concentration nc that is given by nc1/3aB = 0.25, where aB BB is 
+the exciton Bohr radius.25 In GaAs nc ≈ 2 ×  1016 cm-3 (Ref. 9), so, nopt ≈  0.15 n c. For 
+uncompensated CdTe MIT occurs for nc ≈ 15 × 1016 cm-3 but, for some degree of 
+ 4arXiv 1001.0869 
+ 
+compensation in the samples, nc could be as high as 90 × 1016 cm-3 (Ref. 26). Taking this 
+uncertainty in the determination of nc into account, we obtained for CdTe the relation nopt ≈  
+(0.05 - 0.3) nc, which is quite in line with the results observed in GaAs. These results seems to 
+indicate that there might be indeed a connect ion between the MIT and the peak in the doping 
+dependence of , as suggested in the earlier reports.*
+2T9, 14 However, because we cannot 
+exclude the possibility that this apparent correlation between nopt and n c is just a coincidence, 
+for its verification it would be necessary to  measure the systematic doping dependence of  
+yet in another material. *
+2T
+0 100 200 300-1.7-1.6-1.5
+0 100 200 300101102103
+  g factor
+Temperature (K)(b)p - typeS6(b)  T2* (ps)
+Temperature (K)(a)p - typeS6(a)
+ 
+Fig. 2. Temperature dependence of the spin coherence time (a) and the g factor (b) measured in the n-CdTe 
+sample S6 and the reference p-CdTe sample for a magnetic field of 0 T. The lines in (b) are fits by the function 
+(Ref. 23), where a = -1.65, b  = 2.5 × 10*
+2T
+()2cT bT a T g+ + =-4, c = 1.3 × 10-6 for the sample S6, and a = 
+-1.73, b = 3.4 × 10-4, c = 1.9 × 10-6 for the p-type sample. 
+ 
+ 
+The dominant SRM for samples with nopt is still an opened que stion even for GaAs, 
+which is the most thoroughly investigated semiconductor.9-13 For example, it was suggested in 
+Ref. 10 that EY SRM is dominant in strongly n-doped semiconductors. On the contrary, in 
+Ref. 12, where the electron spin relaxation was investigated from a fully microscopic kinetic 
+spin Bloch equation approach, it was concluded that EY SRM is less important than DP SRM. 
+On the other hand, in Ref. 11 it was argued th at at low temperatures DP SRM is not 
+applicable below MIT, where th e electrons are strongly localized. In any case it should be 
+stressed that in Ref. 10 rather simple approxi mate formulas were used while in Refs. 12 and 
+13 much more complex and accurate theoretical  description was provided. Our major aim in 
+this paper is to provide an experimental data for a second model material where the 
+microscopic calculations can be cross-checked with  the experimental results that could help in 
+solving the ongoing controversy in the field. In Fig. 2 the temperature dependence of the spin 
+dephasing is shown for the p-type doped and the optimally n-type doped CdTe samples. There 
+is a clear difference between these two samples – in the n-type sample  decreases 
+monotonously with increasing temperature while in the p-type sample  increases for 
+temperatures up to 100 K and then  starts to decrease. Even though the precise determination 
+of the dominant SRM in CdTe at various temper atures and concentrations of electrons is out 
+of the scope of this paper, we can discuss the role of various SR Ms in CdTe qualitatively. 
+First, we address a possible role of the hyperfin e interaction for a spin relaxation of the donor-
+bound electrons. The averaging of the nuclear field from the atoms, which are within the 
+localization volume of the impurity bound electr on, leads to the de phasing time that is *
+2T
+*
+2T
+ 5arXiv 1001.0869 
+ 
+proportional to the number of the atoms and in versely proportional to the strength of the 
+hyperfine interaction. 9, 27, 28 In GaAs 100% of nuclei have spin  3/2  (see Table 1 in Ref. 28) 
+and, therefore, the hyperfine constant, which characterizes the strength of the hyperfine 
+interaction and thus the precession frequency of th e electron spin in the nuclear field, is rather 
+large: AGaAs = 90 μ eV (Ref. 29). In CdTe only a fract ion of the nuclei have a magnetic 
+moment (8% of Te and 25% of Cd ions) and these have a spin 1/ 2 (see Table 1 in Ref. 28). As 
+a result, the spin interaction with the nuclei is  rather weak in CdTe. If we suppose that the 
+hyperfine interaction is the dominant SRM in GaAs for nopt where ≈ 180 ns (Ref. 9) and if 
+we take, for simplicity, the value A*
+2T
+Cd = 12 μ eV as the hyperfine consta nt of CdTe (Ref. 27) 
+and if we assume that the loca lization volume of the bound electron in CdTe is similar to that 
+in GaAs, the electron dephasing time due to hyper fine interaction should be approximately 4-
+times longer in CdTe than in GaAs. Even though this is only a rough estimate, as the actual 
+localization volume of the elect rons in not known for CdTe, it suggests that the hyperfine 
+interaction is rather unlikel y to be the dominant SRM in n-type CdTe as the measured 
+maximal value of  in CdTe is 70-times shorter than the corresponding value in GaAs. The 
+Bir-Aronov-Pikus (BAP) SRM, in which electrons exchange their spins with holes, is also ineffective due to the lack of holes  in n-type semiconductors. Cons equently, the dominant 
+SRM is probably the DP or EY SRM (or their co mbination). At this time we are not able to 
+evaluate their relative importance - for this it is necessary to perform a fully microscopic 
+calculation of EY SRM.*
+2T
+12 We just want to mention that the smaller value of  measured in 
+optimally doped CdTe compared to that obser ved in GaAs might be connected with the 
+stronger spin-orbit interaction in CdTe compared  to that in GaAs (the spin-orbit splitting Δ*
+2T
+0 = 
+0.80 eV in CdTe and 0.341 eV in GaAs)30.  
+In Fig. 2 (b) we show the corres ponding temperature dependence of g factors. While at 
+low temperatures the g factors in the n-type and p-type samples are rather different [cf. Fig. 1 
+(c)], at 300 K they are nearly identical. Overall, the temperature dependence of the g factors is 
+quite similar to those measured in GaAs23, 31, 32 and CdTe23. The g factors are affected by the 
+lattice temperature by two competing processes:  As the temperature increases the band gap 
+energy decreases, which is making the g factors more negative.31, 32 On the other hand, with 
+the increasing temperature the electrons po pulate higher Landau levels and the band’s 
+nonparabolicity makes the g factors more positive.31 The second effect could be also 
+responsible for the observe d doping dependence of the g factors [see Fig. 1 (c)] – as the 
+concentration of electrons increases they popul ate higher Landau levels that in turn makes 
+their average g factor less negative.  
+ 
+ 
+IV. CONCLUSIONS 
+ 
+In conclusion, we performed a detailed meas urement of the electron spin dephasing in a 
+comprehensive set of n-CdTe bulk crystals with various c oncentrations of electrons (from 1.5 
+× 1013 cm-3 to 3.2 ×  1017 cm-3) and also, as a reference, in the p-CdTe sample with a 
+concentration of holes ≈ 1016 cm-3. The major goal of this Brief Report was to provide the 
+experimental data for a second bulk model material where the suppression of the spin 
+dephasing time of electrons by the n-type doping was studied system atically. In particular, we 
+showed that in the samples with a low concentration of electrons  is comparable in CdTe 
+and GaAs. For the optimal concentration of electrons, which is 17-times higher in CdTe than 
+in GaAs,  is significantly prolonged in both ma terials but the maximal value of  is 70-
+times shorter in CdTe than in GaAs. We belie ve that our CdTe data can help in cross-*
+2T
+*
+2T*
+2T
+ 6arXiv 1001.0869 
+ 
+checking the predictions of various  theoretical models that were  suggested as an explanation 
+of the observed non-monotonous doping dependence of  in GaAs.  *
+2T
+ 
+ 
+ACKNOWLEDGEMENTS 
+ 
+This work was supported by Ministry of Education of the Czech Republic (research 
+centre LC510 and the research plan MSM0021620834) , Grant Agency of the Czech Republic 
+(grant no. 202/09/H041) and by grant no. SVV- 2010-261306 of the Charles University in 
+Prague. 
+ 
+ 
+REFERENCES 
+ 
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+B 53, 7911 (1996). 
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+the electron spin relaxa tion time of 130 ps mentioned in the paper is, by convention, obtained 
+as a double of the actually measured decay time c onstant of 65 ps, which is investigated here. 
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+Oxford University Press, New York, 1989, p. 1. 
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+ 
+ 8
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+arXiv:1001.0870v1  [astro-ph.HE]  6 Jan 2010Mon. Not. R. Astron. Soc. 000, 1–7 (2009) Printed 20 November 2018 (MN L ATEX style file v2.2)
+In-depth studies of the NGC 253 ULXs with
+XMM-Newton: remarkable variability in ULX1, and
+evidence for extended coronae
+R. Barnard1
+1Department of Physics and Astronomy, The Open University, W alton Hall, Milton Keynes, MK7 6AA, UK
+ABSTRACT
+We examined the variabilityofthree ultra-luminousX-raysources(U LXs) in the 2003,
+110 ks XMM-Newton observation of NGC253. Remarkably, we discov ered ULX1 to
+be three times more variable than ULX2 in the 0.3–10 keV band, even t hough ULX2
+is brighter. Indeed, ULX1 exhibits a power density spectrum that is consistent with
+the canonical high state or very high/steep power law states, but not the canonical
+low state. The 0.3–10 keV emission of ULX1 is predominantly non-ther mal, and may
+be related to the very high state. We also fitted the ULX spectra wit h disc blackbody,
+slim disc and convolution Comptonization ( simpl⊗diskbb) models. The brightest
+ULX spectra are usually described by a two emission components (dis c blackbody
++ Comptonized component); however, the simplmodel results in a single emission
+component, and may help determine whether the well known soft ex cess is a feature
+of ULX spectra or an artifact of the two-component model. The simplmodels were
+rejected for ULX3 (and also for the black hole + Wolf-Rayet binary I C10 X-1); hence,
+we infer that the observed soft-excesses are genuine features of ULX emission spectra.
+We use an extended corona scenario to explain the soft excess see n in all the highest
+quality ULX spectra, and provide a mechanism for stellar mass black h oles to exhibit
+super-Eddington luminosities while remaining locally sub-Eddington.
+Key words: X-rays: general – X-rays: binaries – Galaxies: individual: NGC 253 –
+black hole physics
+1 INTRODUCTION
+In Barnard et al. (2008a) we identified a new spectral state
+in the confirmed black hole + Wolf-Rayet binary IC10 X-1,
+as well as the BH+WR candidate NGC 300 X-1 and the
+well known black hole binary LMC X-1. Both IC10 X-1
+and NGC300 X-1 exhibited ∼0.0001–0.1 Hz power den-
+sity spectrum (PDS) where the power is inversely propor-
+tional to the frequency, like the canonical high state. How-
+ever, the 0.3–10 keV for these sources emission is predomi-
+nantly non-thermal, while the canonical emission spectrum
+for high state black hole X-ray binaries is dominated by a
+multi-temperature disk blackbody that contributes ∼90% of
+the 1–10 keV flux (McClintock & Remillard 2006).
+In Barnard et al. (2008a) we proposed that the non-
+thermal high state was due to a persistently high accretion
+rate maintaining a stable corona; most black hole X-ray bi-
+naries are transient, and we suggested that the canonical
+soft high state was caused by the ejection of the corona dur-
+ing outburst. After finding out from Soria & Kuncic (2008)
+that∼90% of X-ray sources with luminosities >1039erg s−1
+have non-thermal spectra, we suggested that the same phys-ical processes may apply in these sources as in IC10 X-1 and
+NGC 300 X-1. The ultra-luminous X-ray sources (ULXs) are
+an intriguing class of extragalactic X-ray sources that are not
+associated with the galaxy nucleus, yet exhibit X-ray lumi-
+nosities >2×1039erg s−1, the Eddington limit for a ∼15
+M⊙black hole (see e.g. Roberts 2007, for a review).
+Our survey of X-ray sources in the 2003 XMM-
+Newton observation of the nearby spiral galaxy NGC 253
+(Barnard et al. 2008b) yielded three X-ray sources with
+0.3–10 keV luminosities >2×1039erg s−1. We shall call
+these ULX1–ULX3, and their known properties are pre-
+sented in Table 1; these properties are taken from our survey
+(Barnard et al. 2008b). In this paper, we examine the vari-
+ability of these sources. We also perform additional spectr al
+fitting; we model their spectra with slim disc models, and
+“physical” Comptonization models where the inner disc pro-
+vides the seed photons. In Section 2 we discuss the observa-
+tion and data analysis, followed by the results in Section 3.
+Our discussion and conclusion is presented in Section 4.
+c/circlecopyrt2009 RAS2R. Barnard
+1.1 ULX variability
+Few ultra-luminous X-ray sources (ULXs) have sufficiently
+high quality data for timing analysis. The PDS of M82 X-
+1 has been characterised by a quasi-periodic oscillations
+(QPOs) at 54 mHz (Strohmayer & Mushotzky 2003) and
+114 Hz (Dewangan et al. 2006b), on top of a broken power
+law continuum (Dewangan et al. 2006b). QPOs have also
+been found in Holmberg IX X-1 (along with a power law con-
+tinuum Dewangan et al. 2006a) and NGC 5408 X-1 (with a
+broken power law continuum Strohmayer et al. 2007).
+Heil et al. (2009) conducted a timing study of ULXs
+observed with XMM-Newton. They analysed 19 observa-
+tions of 16 ULXs and found 6 ULXs with PDS best de-
+scribed by power laws or broken power laws: M82 X-1, NGC
+1313X-1, NGC 1313 X-2, NGC 55 ULX, Ho IX X-1 and
+NGC 5408 X-1; the others showed no significant variability.
+Heil et al. (2009) obtained upper limits to the variability o f
+these ULXs, and concluded that the intrinsic variability wa s
+lower.
+1.2 ULX emission spectra
+Two emission components are often used to describe the X-
+ray spectra of ULXs: a thermal component, and a compo-
+nent, e.g. power law, to represent inverse Compton scatter-
+ing of cool photons on hot electrons. (see e.g. Stobbart et al .
+2006; Gladstone et al. 2009). These fits often result in a
+“soft excess”, where the power law component diverges from
+the thermal component Such spectra have been deemed
+by some as non-standard and unphysical, assuming that
+the seed photons are provided by the inner disc (see e.g.
+Roberts et al. 2005; Gon¸ calves & Soria 2006).
+Gladstone et al. (2009) conducted a survey of XMM-
+Newton observations of ULXs with >10000 source counts
+in their EPIC spectra; their preferred model consists of a
+disc blackbody and a Comptonized component with the seed
+photons tied to the inner disc temperature. They found two
+universal features: a soft excess and roll-over in the spect rum
+above∼3 keV. They found that disc + Comptonized corona
+emission models fitted the data well; however, the resulting
+coronae were cool (k T≃1–3 keV) and optically thick ( τ∼
+5–30). The combination of a cool disc and broken hard com-
+ponent led Gladstone et al. (2009) to define a new, ultralu-
+minous spectral state. Such coronae are distinct from coro-
+nae of Galactic binaries ( τ∼1), and Gladstone et al. (2009)
+use this difference to reject the scenario of sub-Eddington
+accretion onto intermediate mass black holes.
+2 OBSERVATION AND ANALYSIS
+In Barnard et al. (2008b) we conducted a spectral survey of
+X-ray sources in the 2003 XMM-Newton observation of NGC
+253; full details of our data reduction are given in that pape r.
+The present work concerns new variability studies of three
+ULXs. In Barnard et al. (2007) we showed that the combi-
+nation of lightcurves from the different cameras on-board
+XMM-Newton can lead to artefacts in the PDS. To ensure
+fidelity of the PDS, we performed our variability analysis
+on the unbinned pn events files for each source. The total
+exposure time was ∼110 ks; however, several background
+Figure 1. Detail of a three colour pn + MOS image of NGC253
+showing ULX1, ULX2 and ULX3. Red represents 0.3–2.0 keV,
+green represents 2.0–3.0 keV, and blue represents 4.0–10 ke V.
+The image is log-scaled, and the source extraction regions f or the
+ULXs are labeled.
+flares occurred during the observation; the flares were re-
+moved (see Barnard et al. 2008b, for details), leaving ∼50
+ks of good data.
+3 RESULTS
+Figure 1 shows a three colour detail of the combined EPIC
+(pn + MOS1 + MOS2) image of NGC 253, overlaid with the
+source extraction regions; the image is log-scaled, north i s
+up and east is left.
+3.1 Variability
+We present the 0.3–10 keV pn lightcurves of ULX1, ULX2
+and ULX3 in Figure 2, binned to 100 s from the unbinned
+events. Intervals with strong background flares have been
+filtered out. The background contribution is negligible in
+each case. The x- and y-axes are plotted on the same scale
+for each source. For each lightcurve we provide the fraction al
+r.m.s. variability calculated by the program lcurve from
+the FTOOLs analysis suite1. We note that ULX3 is rather
+fainter than ULX1 and ULX2 and there are two reasons
+for this. Firstly, ULX3 is located at the corner of a chip in
+the pn image, and the collecting area is reduced when the
+1http://heasarc.nasa.gov/lheasoft/ftools/ftools menu.html
+c/circlecopyrt2009 RAS, MNRAS 000, 1–7NGC253 ULXs: variability and extended coronae 3
+Table 1. Known properties of the three ULXs, taken from our survey of X -ray sources in NGC253 (Barnard et al. 2008b). For each
+source we give the source coordinates as returned by the sour ce detection routine; these have uncertainties of ∼2′′. We then give the
+number of net source counts in pn and MOS detectors. Next we gi ve the details of the best fit spectral model: absorption ( NH), blackbody
+temperature (k T) and photon index (Γ), along with the corresponding χ2/dof and 0.3–10 keV luminosity / 1039erg s−1. Numbers in
+parentheses indicate 90% confidence limits on the final digit .
+Source Position pn MOS NH/1021atom cm−2kT/keV Γ χ2/dof L/1039
+ULX1 00 47 22.56 −25 20 51 10537 10390 2.00(2) 0.73(5) 2.14(4) 347/323 2.90(12 )
+ULX2 00 47 32.98 −25 17 50 11614 12014 2.9(2) 0.98(6) 1.94(5) 374/374 4.10(19)
+ULX3 00 47 42.41 −25 14 59 773 4156 6.0(11) 0.94(8) 3.4(5) 84/76 2.4(4)
+Figure 2. Lightcurves of the three ULXs, binned to 100 s from
+the unbinned 0.3–10 keV pn events. X- and y- axes are set to
+the same scales. We give the fractional r.m.s. variability o f each
+lightcurve.
+(FLAG==0) filter is applied; lastly, ULX3 is more heavily
+absorbed than the other two. ULX1 is particularly variable,
+ranging in intensity over ∼0.0–0.4 count s−1; analysis of the
+ULX1 lightcurves in the 0.3–2.5 keV and 2.5–10 keV bands
+showed no evidence for energy dependence in this variabilit y.
+For each source, we constructed 0.3–10 keV PDS that
+were averaged over several intervals consisting of 64 bins
+with a resolution of 100 s; intervals of background flaring
+were excluded. Figure 3 shows the resulting PDS for ULX1
+(black) and ULX2 (grey) , which have comparable inten-
+sities. The PDS are normalized to give the (r.m.s./mean)2
+power and the expected noise is subtracted; the PDS are
+binned by to give 56 frequencies per bin. The PDS of ULX1
+(black) is well described by a power law; if the power at fre-
+quencyνisP(ν), thenP(ν)∝ν−γ. We show the best fit
+power law to the ULX1 PDS in Fig. 3; γ= 1.0±0.5, with
+χ2/ = 3 for 6 degrees of freedom (dof). Fitting the ULX1
+PDS with zero power yields χ2/dof = 26/8, an unaccept-
+able fit; hence the observed power in ULX1 is significant.
+The PDS for ULX2 shows no strong evidence for variability,
+even though ULX2 is slightly brighter than ULX1; fitting
+zero power to the ULX2 PDS yields χ2/dof = 5/6. Hence
+the observed power from ULX1 is intrinsic, rather than an
+artefact of the observation. Indeed, the PDS of the combined
+pn+MOS1+MOS2 lightcurves of ULX2 and ULX3 were also
+featureless. These results show that 10,000 source counts a reFigure 3. Power density spectra constructed from 0.3–10 keV
+pn events for ULX1 (black) and ULX2 (grey); the expected nois e
+is subtracted. The PDS of ULX1 features strong variability a t
+low frequencies; we provide the best fit power law model with γ
+= 1.0±0.5 (χ2/dof = 3/6).
+sufficient for detecting γ∼1 PDS in sources where the r.m.s.
+variability is ∼30%, but not for sources with 10% variability.
+The frequency range of our PDS is ∼2×10−4–5×10−3
+Hz. Churazov et al. (2001) showed that Cygnus X-1 exhibits
+similar power law noise with γ∼1 in this frequency range
+during its low state and high state; however, the r.m.s.2
+power of Cygnus X-1 at ∼10−4Hz is∼20 times higher in
+the high state than in the low state. This is unsurprising, be -
+cause the high state PDS is characterisd by a γ∼1 power law
+over the 0.0001–10 Hz range, while its low state is similar,
+except for the interval ∼2×10−3–2×10−1Hz where γ∼0.
+The variability of ULX1 is consistent with that of Cygnus
+X-1 in the high state, but not its low state.
+We compared the 10−2–10 Hz PDS of Cygnus X-1 in the
+high and low states (Churazov et al. 2001), with the 10−2–
+10 Hz PDS of other black hole X-ray binaries in the high, low
+and steep power law states (McClintock & Remillard 2006).
+The high state PDS of Cygnus X-1 has rather more power
+at 10−2Hz than the high states of other black hole binaries;
+this may be a consequence of its high mass donor. The low
+state PDS of Cygnus X-1 shows variability at 10−2Hz that
+is as high as, or higher than the PDS of other black holes in
+the low state. It is difficult to distinguish between the high
+and very high states variability alone; we therefore infer t hat
+ULX1 is in a high accretion rate state that may be related
+to the high or very high/ state.
+c/circlecopyrt2009 RAS, MNRAS 000, 1–74R. Barnard
+Figure 4. Unfolded 0.3–10 keV pn and combined MOS spectra
+of ULX1, modelled with disc blackbody and power law emission
+components, and suffering photo-electric absorption by mat erial
+in the line of sight. A constant of normalization accounts fo r the
+differences in pn and MOS responses.
+3.2 Spectral analysis
+The emission spectra of the brightest ULXs are generally
+well described by disc emission plus a Comptonization com-
+ponent (see e.g. Stobbart et al. 2006). Figure 4 shows the
+unfolded 0.3–10 keV pn and spectrum of ULX1, simulta-
+neously modeled with a disc blackbody and a power law
+component to represent Comptonization; line-of-sight ab-
+sorption is included, and a constant of normalization ac-
+counts for differences in the pn and MOS responses. The
+Comptonized component contributes most of the 0.3–10 keV
+emission; however, the thermal emission dominates the ∼1–5
+keV range. The power law component dominates the spec-
+trum below ∼1 keV; this ”soft excess” is sometimes deemed
+unphysical, as the seed photons for the Comptonization are
+assumed to come from the inner disc (see e.g. Roberts et al.
+2005; Gon¸ calves & Soria 2006). In this section, we model the
+spectra of ULX1–ULX3 with several alternative models.
+3.2.1 Disc blackbody models
+The emission spectra of distant ULXs are often successfully
+described by disc blackbody models (see e.g. Isobe et al.
+2009). However, for these models to be meaningful, they
+must have inner disc temperatures and luminosities that are
+consistent with the black hole mass, MBH; the temperature
+is governed by M−1/4
+BH, while the luminosity should not ex-
+ceed∼0.5LEdd(Mitsuda et al. 1984).
+Successful fits to the NGC253 ULX spectra with mean-
+ingful disc blackbody models would indicate that these sys-
+tems were in the canonical high state; disc temperatures
+range over ∼0.7–1.5 keV for most known black hole binaries
+in the high state (McClintock & Remillard 2006). Therefore
+we simultaneously fitted absorbed disc blackbody models to
+the pn and combined MOS spectra for each ULX; a constant
+of normalisation is included to account for differences in th e
+pn and MOS responses. We reject the absorbed disc black-
+body model for ULX1 and ULX3. The disc blackbody fit for
+ULX2 is acceptable: NH= 1.56±0.08×1021atom cm−2, kTin= 1.68±0.04 keV, and χ2/dof = 397/376 (good fit probabil-
+ity, g.f.p. = 0.22). However, the temperature for ULX2 is
+rather higher than known black hole binaries, and is incon-
+sistent with the modelled 0.3–10 keV luminosity of 3 ×1039
+erg s−1. F-testing shows the blackbody + power law model
+to be a better fit; the probability that this improvement is
+due to chance is 1.4 ×10−5.
+3.2.2 Slim disc models
+Slim disc models have also been applied to some ULXs (see
+e.g. Okajima et al. 2006). Kawaguchi (2003) provides a suite
+of tabular models for XSPEC1, including a standard disc,
+thermal and Comptonized slim discs, with and without rel-
+ativistic effects. Each model is described by four parame-
+ters; the mass of the accretor, M, is given in solar masses;
+the accretion rate, ˙M, is normalised to LEdd/c2, so that
+Eddington accretion corresponds to ˙M≃10 (Kawaguchi
+2003); the viscosity parameter, α, is limited to the range
+0.01–1; the normalisation is defined as (10 kpc/ d)2, whered
+is the distance. The distance was fixed to 4 Mpc, following
+Grimm et al. (2003).
+We applied these slim disc models to our ULX targets.
+For ULX1, no thermal model resulted in good fits, meaning
+that Comptonization was required. The best results were
+gleaned from the Comptonized slim disc model with rela-
+tivistic correction: NH= 1.53±0.06×1021atom cm−2,M
+= 17.5±1.5 M ⊙,˙M= 12.8±0.8LEdd/c2,α= 0.11±0.03,
+andχ2/dof = 351/324 (g.f.p. = 0.14). For ULX2, all slim
+disc models failed, with g.f.p. <2×10−6. ULX3 was best
+described by a modified blackbody with relativistic correc-
+tion:NH= 3.7±0.2×1021atom cm−2,M= 8±7 M⊙,˙M=
+10.7±0.7LEdd/c2,α= 0.36±0.10,χ2/dof = 94/77 (g.f.p. =
+0.09). F-testing showed that our two-component model has
+a 99.6% probability of providing a significant improvement.
+3.2.3 Self-consistent Comptonization of a disc blackbody
+Steiner et al. (2008) have produced a convolution model for
+XSPEC v12, simpl , which describes the Comptonization of
+any input spectrum; the modeling can include up-scattering
+and down-scattering, or upscattering only. It uses only two
+free parameters: the photon index of the resulting power law ,
+and the fraction of scattered photons. Fitting our ULXs with
+simpl⊗diskbb XSPEC emission model will help determine
+whether the observed soft excess is a feature of the ULX
+spectrum or the model.
+Best fitsimpl models are provided in Table 2; these in-
+clude up-scattering and down-scattering. We reject all simpl
+fits to ULX3. We found acceptable fits for ULX1 and ULX2;
+however, the emission parameters could not be constrained.
+The best fit to ULX1 is presented in Fig. 5. F-testing tells us
+that we must prefer the two-component models, since they
+provided a lower χ2for the same degrees of freedom in each
+case, and this is always a significant improvement. We there-
+fore infer that the soft excess is a genuine feature of ULX
+spectra; such a feature could be produced by an extended
+corona that has access to low energy photons from the outer
+disc.
+1http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/mode ls/slimdisk.html
+c/circlecopyrt2009 RAS, MNRAS 000, 1–7NGC253 ULXs: variability and extended coronae 5
+1 10 0.5 2 510−510−4Photons cm−2 s−1 keV−1
+Energy (keV)
+Figure 5. Unfolded 0.3–10 keV pnand combined MOSspectra of
+ULX1 with the best-fit simplmodel. A constant of normalization
+accounts for the differences in pn and MOS responses.
+One might expect the simpl and two component models
+to give equally acceptable results, as they are both describ -
+ing the Comptonization of a disc blackbody, with no spec-
+ified corona geometry. However, the simpl model does not
+include photons outside the energy range of the spectrum;
+hence, the softer photons in the cooler regions of the disc
+are excluded. We further note that the simpl model assumes
+that all photons have equal probability of being scattered,
+and that photons of all energies are scattered by the same
+amount (Steiner et al. 2008). The latter may approximate a
+compact corona, but is unlikely to be true for an extended
+corona, as the photon energies will vary by several orders of
+magnitude.
+We also fitted simpl models to our XMM-Newton spec-
+tra for the known black hole + Wolf-Rayet binary IC10 X-1
+(see Barnard et al. 2008a, and references within). The best
+fit model yielded χ2/dof = 763/650 (g.f.p. = 0.0014) and is
+rejected; the two component model yields χ2/dof = 699/650
+(g.f.p = 0.08). Hence we infer the soft excess to be a genuine
+feature of the IC10 X-1 emission spectrum also.
+4 DISCUSSION AND CONCLUSIONS
+ULX1 is a factor of ∼3 times more variable than ULX2,
+despite the fact that ULX2 is brighter. Furthermore, the
+∼0.0002–0.005 Hz PDS of ULX1 is well described by P( ν)
+∝ν−1.0±0.5; the power at 0.0002 Hz is a factor ∼10
+higher than expected for the canonical low state for known
+black hole binaries (McClintock & Remillard 2006). How-
+ever, the 0.0002 Hz power of ULX1 is consistent with the
+high state of Cygnus X-1 (Churazov et al. 2001). It is diffi-
+cult to differentiate between the high state and very high
+state (a.k.a. steep power law state) using only the PDS
+(McClintock & Remillard 2006); however, the emission of
+ULX1 is more like the very high / steep power law state
+than the high state.
+The best studied ULX spectra exhibit are well described
+by thermal + Comptonization emission models. They often
+exhibit soft excesses which have been labeled unphysical, a s-
+suming that the seed photons are provided by the inner disc
+(Roberts et al. 2005; Gon¸ calves & Soria 2006, e.g.). We havetested this assumption using simpl models. We have rejected
+these models for ULX3 and IC10 X-1, and find that our two-
+component models are preferred for ULX1 and ULX2. We
+therefore infer that these soft excesses are genuine featur es
+of ULX spectra.
+The disc blackbody contributes 1.3 ±0.2×1039erg s−1
+to the unabsorbed 0.3–10 keV flux from our best two-
+component fit to ULX1; a distance of 4 Mpc is assumed,
+but the distance is not well known. This is plausible for a
+black hole with mass ∼20 M⊙; the most massive known stel-
+lar mass black hole, in IC10 X-1, has a mass of ∼20–35 M ⊙
+(Silverman & Filippenko 2008).
+We propose an explanation for the soft excess that cru-
+cially involves an extended corona. In Barnard et al. (2008a )
+we proposed that the canonical high soft state observed in
+the transient black hole binary systems is caused by the ejec -
+tion of the disc corona as the X-ray luminosity increases by
+several orders of magnitude during outburst. We also pro-
+posed that a steady high accretion rate could maintain a
+stable corona, resulting in a non-thermal high state. This
+theory is supported by observations of the high mass black
+hole binary LMC X-1, which appears to be in a stable, non-
+thermal high state (Barnard et al. 2008a, and references
+within).
+The ”non-standard” fits to the ULXs resemble the emis-
+sion generally seen in the Galactic X-ray binaries with neu-
+tron star primaries; White et al. (1988) found neutron star
+LMXB spectra to be dominated by a Comptonized emis-
+sion component, with an additional blackbody component
+at high luminosities, while Church & Ba/suppress luci´ nska-Church
+(2001) found that the blackbody component was present at
+low luminosities also. The thermal and non-thermal emis-
+sion components are spatially distinct in the neutron star
+LMXBs, as demonstrated by the high inclination LMXBs
+known as the dipping sources where the X-ray source is
+experiences increased absorption by material in the outer
+accretion disc on the orbital period; the thermal com-
+ponent is completely and abruptly removed by dipping,
+while the Comptonized component experiences more grad-
+ual changes in absorption (see e.g. White & Swank 1982;
+Church & Ba/suppress luci´ nska-Church 1995; Barnard et al. 2001).
+Corona sizes can be estimated for high inclination
+X-ray binaries, from the ingress times of the inten-
+sity dips caused by photo-electric absorption of X-rays
+by cold material in the outer accretion disc (see e.g.
+Church 2001, for details). Measured corona diameters for
+the dipping sources range from 20,000 km to 700,000
+km with radius increasing with luminosity (Church 2001;
+Church & Ba/suppress luci´ nska-Church 2004); the measured coronae
+extend over 10–50% of the accretion disc radius, with this
+fraction increasing with luminosity. Such extended corona e
+would have access to an enormous reservoir of photons ≪1
+keV that could account for the soft excess.
+We note that further, independent evidence for ex-
+tended coronae has been discovered by Schulz et al. (2009).
+They found broadened emission lines in Chandra observa-
+tions of Cygnus X-2, requiring Doppler velocities of ∼1100–
+2700 km s−1. Schulz et al. (2009) infer from these results
+that the corona of Cygnus X-2 is hot, dense and extended
+up to∼1010cm.
+If the coronae in ULXs are similarly extended, as our
+results suggest, then the Eddington limit may be relaxed
+c/circlecopyrt2009 RAS, MNRAS 000, 1–76R. Barnard
+Table 2. Best fit self-consistent Comptonization models to the ULXs. The spectral model used was const*wabs*simpl(diskbb) with
+constaccounting for the differences in pn and MOS response. The mod els include up-scattering and down-scattering. We show the
+absorption ( NH/1020atom cm−2), photon index (Γ), scattering fraction ( fscat), and inner disc temperature (k Tin). We then give the
+χ2/dof and good fit probability. The emission parameters are un constrained for all models.
+Model NH/1020atom cm−2ΓfscatkTinχ2/dof [g.f.p]
+ULX1 6 1.1 0.3 1.0 359/323 [0.08]
+ULX2 16 3.2 0.6 1.3 386/374 [0.32]
+ULX3 24 1.1 0.4 0.8 110/76 [7E-3]
+and many observed ULXs may be explained by stellar mass
+black holes accreting at a consistently high rate, but radi-
+ating at a locally sub-Eddington level. We propose that the
+non-thermal component of the observed ULX spectrum is
+provided by an extended corona that Comptonizes photons
+from the inner regions of the disc: the hot photons from the
+inner disc and also cool photons from further out. The ob-
+served thermal component would then be the portion of the
+disc emission that is unscattered.
+This state may well be related to the canonical very
+high / steep power law state. Indeed, if the high soft state is
+caused by ejection of the corona during transient outburst,
+then the very high state could indicate that the corona was
+regenerating; an extended corona could make the very high
+state appear brighter in X-rays than the high soft state if
+sufficient cool photons were inverse-Compton scattered into
+the observed energy range. It is unfortunate that the highes t
+quality X-ray observations of Galactic black hole binaries ,
+from RXTE, are insensitive to energies below 2 keV; sensi-
+tive spectra could reveal a soft excess, and extended corona ,
+in the very high state also.
+The extended corona scenario means that many ULXs
+could contain stellar mass black holes that exhibit a super-
+Eddington luminosity while being locally sub-Eddington.
+Furthermore, it may be directly tested, as IC10 X-1 is a
+high inclination system that eclipses on a ∼35 hour period
+(Prestwich et al. 2007; Silverman & Filippenko 2008); if our
+scenario is correct, then we would observe photo-electric a b-
+sorption dips on the orbital period, and could determine the
+size of its corona from the ingress time.
+ACKNOWLEGMENTS
+We thank the anonymous referee for their constructive and
+informative comments. This work is based on observations
+with XMM-Newton, an ESA science mission with instru-
+ments and contributions directly funded by ESA member
+states and the US (NASA). Astronomy research at the Open
+University is funded by an STFC Rolling Grant.
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+arXiv:1001.0871v2  [astro-ph.CO]  5 Nov 2010Astronomy& Astrophysics manuscriptno.aa13999 c∝circlecopyrtESO 2018
+October31,2018
+The galaxycluster YSZ-LXandYSZ-Mrelations
+from the WMAP5-yr data
+J.-B. Melin1, J.G. Bartlett2,J. Delabrouille3,M.Arnaud4, R.Piffaretti4,and G.W.Pratt4
+1DSM/Irfu/SPP,CEA/Saclay,F-91191 Gif-sur-YvetteCedex
+e-mail:jean-baptiste.melin@cea.fr
+2APC–Universit´ e ParisDiderot, 10rue Alice Domon etL´ eoni e Duquet, 75205 ParisCedex 13, France
+e-mail:bartlett@apc.univ-paris7.fr
+3APC–CNRS,10rue Alice Domon etL´ eonie Duquet, 75205 ParisC edex 13, France
+e-mail:delabrouille@apc.univ-paris7.fr
+4DSM/Irfu/SAp, CEA/Saclay, F-91191 Gif-sur-YvetteCedex
+e-mail:monique.arnaud@cea.fr, rocco.piffaretti@cea.fr, gabri el.pratt@cea.fr
+Received ;accepted
+ABSTRACT
+We use multifrequency matched filters to estimate, in the WMA P 5-year data, the Sunyaev–Zel’dovich (SZ) fluxes of 893 ROSA T
+NORAS/REFLEXclustersspanningtheluminosityrange LX,[0.1−2.4] keV=2×1041−3.5×1045ergs−1.Thefiltersarespatiallyoptimised
+by using the universal pressure profile recently obtained fr om combining XMM-Newton observations of the REXCESS sample and
+numerical simulations. Although the clusters are individu ally only marginally detected, we are able to firmly measure t he SZ signal
+(>10σ) when averaging the data in luminosity /mass bins. The comparison between the bin-averaged SZ signa l versus luminosity
+andX-raymodelpredictionsshowsexcellentagreement,imp lyingthatthereisnodeficitinSZsignalstrengthrelativet oexpectations
+from the X-ray properties of clusters. Using the individual cluster SZ flux measurements, we directly constrain the Y500−LXand
+Y500−M500relations,where Y500is the Compton y–parameter integratedover asphere of radiu sr500. TheY500–M500relation,derived
+for the first time in such a wide mass range, has a normalisatio nY∗
+500=[1.60±0.19]×10−3arcmin2atM500=3×1014h−1M⊙, in
+excellent agreement with the X-ray prediction of 1 .54×10−3arcmin2, and a mass exponent of α=1.79±0.17, consistent with the
+self-similarexpectationof5 /3.Constraintsontheredshiftexponent areweakduetotheli mitedredshiftrangeofthesample,although
+theyare compatible withself-similarevolution.
+Key words. Cosmology: observations, Galaxies: cluster: general, Gal axies: clusters: intracluster medium, Cosmic background r adi-
+ation, X-rays:galaxies: clusters
+1. Introduction
+Capability to observe the Sunyaev-Zel’dovich (SZ) e ffect has
+improved immensely in recent years. Dedicated instruments
+now produce high resolution images of single objects (e.g.
+Kitayamaet al. 2004; Halversonet al. 2009; Nordet al. 2009)
+andmoderatelylargesamplesofhigh–qualitySZmeasuremen ts
+of previously–known clusters (e.g., Mroczkowskiet al. 200 9;
+Plaggeetal. 2009). In addition, large-scale surveys for cl usters
+using the SZ effect are underway, both from space with the
+Planck mission (Valenzianoetal. 2007; Lamarreet al. 2003)
+and from the ground with several dedicated telescopes, such
+as the South Pole Telescope (Carlstrometal. 2009) leading t o
+the first discoveries of clusters solely through their SZ sig nal
+(Staniszewskietal. 2009).Theseresultsopenthewayforab et-
+terunderstandingofthe SZ−Massrelationand,ultimately,for
+cosmologicalstudieswithlargeSZ clustercatalogues.
+The SZ effect probes the hot gas in the intracluster medium
+(ICM). Inverse Compton scattering of cosmic microwave back -
+ground (CMB) photons by free electrons in the ICM creates a
+unique spectral distortion (Sunyaev& Zeldovich 1970, 1972 )
+seen as a frequency–dependent change in the CMB surface
+brightness in the direction of galaxy clusters that can be wr it-
+ten as∆iν(ˆn)=y(ˆn)jν(x), where jνis a universal function of
+the dimensionless frequency x=hν/kTcmb. The Compton y–parameteris givenby the integral of the electronpressure a long
+theline–of–sightinthe direction ˆ n,
+y=/integraldisplay
+ˆnkTe
+mec2neσTdl, (1)
+whereσTistheThomsoncrosssection.
+Most notably, the integrated SZ flux from a cluster directly
+measuresthetotalthermalenergyofthegas.Expressingthi sflux
+intermsoftheintegratedCompton y–parameter YSZ–definedby/integraltext
+dΩ∆iν(ˆn)=YSZjν(x) – we see that YSZ∝/integraltext
+dΩdlneTe∝/integraltext
+neTedV. For this reason, we expect YSZto closely correlate
+with total cluster mass, M, and to provide a low–scatter mass
+proxy.
+This expectation, borne out by both numerical simulations
+(e.g.,daSilvaet al.2004;Motl etal.2005;Kravtsovetal.2 006)
+and indirectly from X–ray observations using YX, the prod-
+uct of the gas mass and mean temperature (Nagaietal. 2007;
+Arnaudetal.2007;Vikhlininet al.2009),stronglymotivat esthe
+use of SZ cluster surveys as cosmological probes. Theory pre -
+dicts the cluster abundance and its evolution – the mass func -
+tion – in terms of Mand the cosmological parameters. With
+a good mass proxy, we can measure the mass function and its
+evolution and hence constrain the cosmological model, incl ud-
+ingthepropertiesofdarkenergy.Inthiscontexttherelati onship2 J.-B.Melinet al.:The galaxycluster YSZ-LXandYSZ-Mrelations fromthe WMAP 5-yr data
+between the integrated SZ flux and total mass, YSZ−M, is fun-
+damental as the required link between theory and observatio n.
+Unfortunately,despite its importance, we are only beginni ng to
+observationally constrain the relation (Bonamenteet al. 2 008;
+Marroneetal. 2009).
+Several authors have extracted the cluster SZ signal from
+WMAP data (Bennettet al. 2003; Hinshawetal. 2007, 2009).
+However, the latter are not ideal for SZ observations: the
+instrument having been designed to measure primary CMB
+anisotropies on scales larger than galaxy clusters, the spa -
+tial resolution and sensitivity of the sky maps render clus-
+ter detection difficult. Nevertheless, these authors extracted
+the cluster SZ signal by either cross–correlating with the
+general galaxy distribution (Fosalbaet al. 2003; Myerset a l.
+2004; Hern´ andez-Monteagudoetal. 2004, 2006) or ‘stack-
+ing’ existing cluster catalogues in the optical or X-ray
+(Lieuetal. 2006; Afshordiet al. 2007; Atrio-Barandelaet a l.
+2008; Bielby&Shanks 2007; Diego&Partridge 2009). These
+analyses indicate that an isothermal β-model is not a good de-
+scription of the SZ profile, and some suggest that the SZ signa l
+strengthis lower than expectedfrom the X-raypropertiesof the
+clusters(Lieuet al. 2006; Bielby&Shanks2007).
+Recent in–depth X-ray studies of the ICM pressure profile
+demonstrate regularity in shape and simple scaling with clu s-
+ter mass. Combining these observations with numerical simu -
+lations leads to a universal pressure profile (Nagaiet al. 20 07;
+Arnaudetal. 2009) that is best fit by a modified NFW profile.
+Theisothermalβ–model,ontheotherhand,doesnotprovidean
+adequate fit. From this newly determined X–ray pressure pro-
+file, we can infer the expectedSZ profile, y(r), and the YSZ−M
+relationat lowredshift(Arnaudet al.2009).
+It is in light of this recent progress from X–ray obser-
+vations that we present a new analysis of the SZ e ffect in
+WMAP with the aim of constraining the SZ scaling laws.
+We build a multifrequency matched filter (Herranzetal. 2002 ;
+Melinet al.2006)basedontheknownspectralshapeofthethe r-
+mal SZ effect and the shape of the universal pressure profile of
+Arnaudetal. (2009). This profile was derived from REXCESS
+(B¨ ohringeret al.2007),asampleexpresslydesignedtomea sure
+the structural and scaling properties of the local X–ray clu ster
+populationby meansof an unbiased,representativesamplin g in
+luminosity. Using the multifrequency matched filter, we sea rch
+fortheSZ effect inWMAP froma catalogueof893clustersde-
+tected by ROSAT, maximising the signal–to–noise by adaptin g
+the filter scale to the expected characteristic size of each c lus-
+ter. The size is estimated through the luminosity–mass rela tion
+derivedfromthe REXCESS samplebyPratt etal. (2009).
+We then use our SZ measurementsto directly determine the
+YSZ−LXandYSZ−Mrelationsandcomparetoexpectationsbased
+ontheuniversalX–raypressureprofile.Ascomparedtothepr e-
+vious analyses of Bonamenteet al. (2008) and Marroneet al.
+(2009),thelargenumberofsystemsinourWMAP /ROSATsam-
+pleallowsustoconstrainboththenormalisationandslopeo fthe
+YSZ−LXandYSZ−Mrelations overa wider mass range and in
+thelargerapertureof r500. Inaddition,the analysisisbasedona
+morerealisticpressureprofilethanintheseanalyses,whic hwere
+basedonanisothermal β–model.Besidesprovidingadirectcon-
+straintontheserelations,thegoodagreementwithX–raypr edic-
+tions impliesthat there is in fact no deficit in SZ signal stre ngth
+relative to expectationsfrom the X-ray properties of these clus-
+ters.
+The discussion proceeds as follows. We first present the
+WMAP 5–year data and the ROSAT cluster sample used, a
+combination of the REFLEX and NORAS catalogues. We thenpresent the SZ model based on the X–ray–measured pressure
+profile (Sec. 3). In Sec. 4, we discuss our SZ measurements,af -
+ter first describing how we extract the signal using the match ed
+filter. Section 5 details the error budget. We compare our mea -
+sured scaling laws to the X–raypredictionsin Sect. 6 and 7 an d
+thenconcludeinSec.8.Finally,wecollectusefulSZdefinit ions
+andunitconversionsin theAppendices.
+Throughoutthispaper,weusetheWMAP5–onlycosmolog-
+ical parameters set as our ‘fiducial cosmology’, i.e. h=0.719,
+ΩM=0.26,ΩΛ=0.74,where his the Hubbleparameterat red-
+shiftz=0 in units of 100 km /s/Mpc. We note h70=h/0.7
+andE(z) is the Hubble parameter at redshift znormalised to
+its present value. M500is defined as the mass within the ra-
+diusr500at which the mean mass density is 500 times the
+critical density,ρcrit(z), of the universe at the cluster redshift:
+M500=4
+3πρcrit(z)500r3
+500.
+2. TheWMAP-5yrdataandtheNORAS/REFLEX
+clustersample
+2.1. The WMAP-5yrdata
+We work with the WMAP full resolution coadded five year
+sky temperature maps at each frequency channel (downloaded
+from the LAMBDA archive1). There are five full sky maps cor-
+responding to frequencies 23, 33, 41, 61, 94 GHz (bands K,
+Ka, Q, V, W respectively).The correspondingbeam full width s
+at half maximum are approximately 52.8, 39.6, 30.6 21.0 and
+13.2 arcmins. The maps are originally at HEALPix2resolution
+nside=512(pixel=6.87arcmin).Althoughthisisreasonablyad-
+equate to sample WMAP data, it is not adapted to the multi-
+frequencymatched filter algorithm we use to extract the clus ter
+fluxes. We oversample the original data, to obtain nside =2048
+maps, by zero-paddingin harmonic space. In detail, this is p er-
+formed by computing the harmonic transform of the original
+maps, and then performing the back transform towards a map
+withnside=2048,withamaximumvalueof ℓofℓmax=750,850,
+1100,1500,2000fortheK,Ka,Q,V,Wbandsrespectively.The
+upgraded maps are smooth and do not show pixel edges as we
+would have obtained using the HEALPix upgrading software,
+basedonthetreestructureoftheHEALPixpixelisationsche me.
+This smooth upgrading scheme is important as the high spatia l
+frequencycontent inducedby pixel edges would have been am-
+plified through the multifrequencymatched filters implemen ted
+inharmonicspace.
+In practice, the multifrequency matched filters are imple-
+mented locally on small, flat patches (gnomonic projection o n
+tangentialmaps),whichpermitsadaptationofthefiltertot helo-
+calconditionsofnoiseandforegroundcontamination.Wedi vide
+the sphere into 504 square tangential overlapping patches ( 100
+deg2each, pixel=1.72 arcmin). All of the following analysis is
+doneontheseskypatches.
+The implementation of the matched filter requires knowl-
+edge of the WMAP beams. In this work, we assume symmet-
+ric beams, for which the transfer function bℓis computed, in
+eachfrequencychannel,fromthenoise-weightedaverageof the
+transferfunctionsofindividualdi fferentialassemblies(asimilar
+approachwasusedinDelabrouilleet al.2009).
+1http://lambda.gsfc.nasa.gov /
+2http://healpix.jpl.nasa.govJ.-B.Melinet al.:The galaxycluster YSZ-LXandYSZ-Mrelations fromthe WMAP 5-yr data 3
+Fig.1.Inferredmassesforthe893NORAS /REFLEXclustersas
+a function of redshift. The cluster sample is flux limited. Th e
+right vertical axis gives the corresponding X-ray luminosi ties
+scaled by E(z)−7/3. The dashed blue lines delineate the mass
+range over which the L500-M500relation from Pratt et al. (2009)
+wasderived.
+2.2. The NORAS/REFLEXclustersampleand derived X–ray
+properties
+We construct our cluster sample from the largest published X -
+ray catalogues: NORAS (B¨ ohringeret al. 2000) and REFLEX
+(B¨ ohringeret al. 2004), both constructed from the ROSAT Al l-
+SkySurvey.WemergetheclusterlistsgiveninTables1,6and 8
+from B¨ ohringeret al. (2000) and Table 6 from B¨ ohringeret a l.
+(2004) and since the luminosities of the NORAS clusters are
+given in a standard cold dark matter (SCDM) cosmology ( h=
+0.5,ΩM=1), we converted them to the WMAP5 cosmology.
+We also convert the luminosities of REFLEX clusters from the
+basicΛCDM cosmology ( h=0.7,ΩM=0.3,ΩΛ=0.7) to the
+morepreciseWMAP5cosmology.Removingclustersappearing
+in both catalogues leaves 921 objects, of which 893 have mea-
+suredredshifts.Weusethese893clustersintheanalysisde tailed
+inthenextSection.
+TheNORAS/REFLEXluminosities LX,measuredinthesoft
+[0.1–2.4]keV energy band, are given within various apertures
+depending on the cluster. We convert the luminosities LXto
+L500, the luminosities within r500, using an iterative scheme.
+This scheme is based on the mean electron density profile of
+theREXCESS cluster sample (Crostonet al. 2008), which al-
+lows conversion of the luminosity between various aperture s,
+and the REXCESS L500–M500relation (Pratt etal. 2009), which
+implicitly relates r500andL500. The procedure thus simultane-
+ouslyyieldsanestimateoftheclustermass, M500,andthecorre-
+spondingangularextent θ500=r500/Dang(z),whereDang(z)isthe
+angular distance at redshift z. In the following we consider val-
+ues derived from relations both corrected and uncorrected f or
+Malmquist bias. The relations are described by the followin g
+powerlawmodels3:
+E(z)−7/3L500=CMM500
+3×1014h−1
+70M⊙αM
+(2)
+3Sinceweconsiderastandardself-similarmodel,weusedthe power
+law relations given in Appendix B of Arnaud et al. (2009). The y are
+derived as in Prattet al. (2009) with the same luminosity dat a but for
+masses derived from a standard slope M500–YXrelation.Table1.Valuesfortheparametersofthe LX−Mrelationderived
+fromREXCESS data(Prattet al.2009; Arnaudet al.2009)
+Correctedfor MB log/parenleftbigg
+CM
+1044h−2
+70[ergs−1]/parenrightbigg
+αMσlogL−logM
+no 0.295 1.50 0.183
+yes 0.215 1.61 0.199
+wherethenormalisation CM,theexponentαMandthedispersion
+(nearlyconstantwithmass)aregiveninTable1.The L500–M500
+relation was derived in the mass range [1014–1015]M⊙. These
+limits are shown in Fig. 1. Note that we assume the relation is
+validforlowermasses.
+The final catalogue of 893 objects contains the position of
+theclusters(longitudeandlatitude),themeasuredredshi ftz,the
+derived X-ray luminosity L500, the mass M500and the angular
+extentθ500. The clusters uniformly cover the celestial sphere at
+Galactic latitudes above |b|>20deg. Their luminosities L500
+range from 0.002 to 35 .01044erg/s, and their redshifts from
+0.003to 0.460.Figure1showsthe masses M500asa functionof
+redshiftzfor the cluster sample (red crosses). The correspond-
+ingcorrectedluminosities L500canbereadontherightaxis.The
+typicalluminositycorrectionfrommeasured LXtoL500is about
+10%. The progressive mass cut-o ffwith redshift (only the most
+massive clusters are present at high z) reflects the flux limited
+natureofthesample.
+3. Thecluster SZmodel
+InthisSectionwedescribetheclusterSZmodel,basedonX-r ay
+observations of the REXCESS sample combined with numer-
+ical simulations, as presented in Arnaudet al. (2009). We us e
+the standard self-similar model presented in their Appendi x B.
+Givena cluster mass M500and redshift z, the modelpredictsthe
+electronic pressure profile. This gives both the SZ profile sh ape
+andY500, theSZfluxintegratedina sphereofradius r500.
+3.1. Clustershape
+The dimensionless universal pressure profile is taken from
+Eq. B1 andEq.B2ofArnaudet al.(2009):
+P(r)
+P500=P0
+xγ(1+xα)(β−γ)/α(3)
+wherex=r/rswithrs=r500/c500andc500=1.156,α=1.0620,
+β=5.4807,γ=0.3292andwith P500definedinEq.4below.
+This profile shape is used to optimise the SZ signal detec-
+tion.AsdescribedbelowinSect.4,weextractthe YSZfluxfrom
+WMAP data for each ROSAT system fixing c500,α,β,γto the
+abovevalues,butleavingthenormalisationfree.
+3.2. Normalisation
+Themodelallowsustocomputethephysicalpressureprofilea s
+a functionof mass and z, thus the YSZ-M500relation by integra-
+tion ofP(r) tor500. For the shape parameters given above, the
+normalisation parameter P0=8.130h3/2
+70=7.810 and the self-
+similardefinitionof P500(Arnaudetal.2009,Eq.5andEq.B2),
+P500=1.65×10−3E(z)8/3M500
+3×1014h−1
+70M⊙2/3
+h2
+70keVcm−3,(4)4 J.-B.Melinet al.:The galaxycluster YSZ-LXandYSZ-Mrelations fromthe WMAP 5-yr data
+Fig.2.Left:EstimatedSZfluxfromacylinderofapertureradius5 ×r500(Ycyl
+5r500)asafunctionoftheX-rayluminosityinanaperture
+ofr500(L500), for the 893 NORAS /REFLEX clusters. The individualclusters are barely detect ed. The bars give the total 1 σerror.
+Right:Red diamonds are the weighted average signal in 4 logarithmi cally–spaced luminosity bins. The two high luminosity bins
+exhibit significant SZ cluster flux. Note that we have divided the vertical scale by 30 between Fig. left and right. The thic k and
+thin bars give the 1 σstatistical and total errors, respectively. Green triangl es (shifted up by 20% with respect to diamonds for
+clarity)showtheresultofthesameanalysiswhenthefluxeso ftheclustersareestimatedatrandompositionsinsteadoft ruecluster
+positions.
+oneobtains:
+Y500[arcmin2]=Y∗
+500/parenleftiggM500
+3×1014h−1M⊙/parenrightigg5/3
+E(z)2/3/parenleftiggDang(z)
+500Mpc/parenrightigg−2
+,(5)
+whereY∗
+500=1.54×10−3/parenleftigh
+0.719/parenrightig−5/2arcmin2. Equivalently,one
+canwrite:
+Y500[Mpc2]=Y∗
+500/parenleftiggM500
+3×1014h−1M⊙/parenrightigg5/3
+E(z)2/3(6)
+whereY∗
+500=3.27×10−5/parenleftigh
+0.719/parenrightig−5/2Mpc2. Details of unit con-
+versionsaregivenin AppendixB. Themassdependence( M5/3
+500)
+and the redshift dependence ( E(z)2/3) of the relation are self-
+similar by construction. This model is used to predict the Y500
+value for each cluster. These predictions are compared to th e
+WMAP-measuredvaluesinFigs.3,4, 5and6.
+4. ExtractionoftheSZflux
+4.1. MultifrequencyMatchedFilters
+We usemultifrequencymatchedfilters toestimate clusterflu xes
+fromtheWMAPfrequencymaps.Byincorporatingpriorknowl-
+edge of the cluster signal, i.e., its spatial and spectral ch aracter-
+istics, the method maximally enhances the signal–to–noise of a
+SZ cluster source by optimally filtering the data. The univer sal
+profile shape described in Sec. 3 is assumed, and we evaluate
+the effects of uncertainty in this profile as outlined in Sec. 5
+where we discuss our overall error budget. We fix the position
+andthe characteristicradius θsofeachclusterandestimate only
+its flux. The position is taken from the NORAS /REFLEX cat-
+alogue andθs=θ500/c500withθ500derived from X-ray data
+as described in Sec. 2.2. Below, we recall the main features o f
+the multifrequency matched filters. More details can be foun d
+inHerranzet al. (2002)orMelinet al. (2006).
+Consideraclusterwithknownradius θsandunknowncentral
+y–valueyopositionedataknownpoint xoonthesky.Theregionis coveredby the fiveWMAP maps Mi(x) at frequenciesνi=23,
+33, 41, 61, 94 GHz ( i=1,...,5). We arrange the survey maps
+into a column vector M(x) whoseithcomponent is the map at
+frequencyνi.Themapscontainthe clusterSZsignalplusnoise:
+M(x)=yojνTθs(x−xo)+N(x) (7)
+whereNis the noise vector(whose componentsare noise maps
+at the different observation frequencies) and jνis a vector with
+components given by the SZ spectral function jνevaluated at
+eachfrequency.Noisein thiscontextrefersto bothinstrum ental
+noise as well as all signals other than the cluster thermal SZ ef-
+fect;it thusalsocomprisesastrophysicalforegrounds,fo rexam-
+ple,the primaryCMB anisotropy,di ffuseGalactic emission and
+extragalactic point sources. Tθs(x−xo) is the SZ template, tak-
+ingintoaccounttheWMAPbeam,atprojecteddistance( x−xo)
+from the cluster centre, normalised to a central value of uni ty
+beforeconvolution.Itiscomputedbyintegratingalongthe line–
+of–sight and normalising the universal pressure profile (Eq . 3).
+The profileis truncatedat 5 ×r500(i.e. beyondthe virial radius)
+so thatwhat isactuallymeasuredis thefluxwithin acylinder of
+apertureradius5×r500.
+X-rayobservationsaretypicallywell-constrainedoutto r500.
+Our decision to integrate out to 5 ×r500is motivatedby the fact
+that for the majority of clusters the radius r500is of order the
+Healpix pixel size (nside =512, pixel=6.87 arcmin). Integrating
+only out to r500would have required taking into account that
+only a fraction of the flux of some pixels contributesto the tr ue
+SZ flux in a cylinder of aperture radius r500. We thus obtain the
+total SZ flux of each cluster by integrating out to 5 ×r500, and
+thenconvertthisto thevalueina sphereofradius r500fordirect
+comparisonwiththe X-rayprediction.
+Themultifrequencymatchedfilters Ψθs(x)returnaminimum
+varianceunbiasedestimate, ˆ yo, ofyowhencenteredonthe clus-
+ter:
+ˆyo=/integraldisplay
+d2xΨθst(x−xo)·M(x) (8)
+where superscript tindicates a transpose (with complex conju-
+gation when necessary). This is just a linear combinationof theJ.-B.Melinet al.:The galaxycluster YSZ-LXandYSZ-Mrelations fromthe WMAP 5-yr data 5
+maps, each convolved with its frequency–specific filter ( Ψθs)i.
+TheresultexpressedinFourierspaceis:
+Ψθs(k)=σ2
+θsP−1(k)·Fθs(k) (9)
+where
+Fθs(k)≡jνTθs(k) (10)
+σθs≡/bracketleftigg/integraldisplay
+d2kFθst(k)·P−1·Fθs(k)/bracketrightigg−1/2
+(11)
+withP(k) being the noise power spectrum, a matrix in fre-
+quencyspacewithcomponents Pijdefinedby∝angbracketleftNi(k)N∗
+j(k′)∝angbracketrightN=
+Pij(k)δ(k−k′). The quantityσθsgives the total noise variance
+through the filter, corresponding to the statistical errors quoted
+in thispaper.Theotheruncertaintiesareestimated separa telyas
+describedinSec.5.1.Thenoisepowerspectrum P(k)isdirectly
+estimated from the maps: since the SZ signal is subdominant
+at each frequency, we assume N(x)≈M(x) to do this calcula-
+tion. We undertake the Fourier transform of the maps and aver -
+agetheircross-spectrainannuliwithwidth ∆l=180.
+4.2. Measurementsofthe SZflux
+ThederivedtotalWMAP fluxfromacylinderofapertureradius
+5×r500(Ycyl
+5r500)forthe893individualNORAS /REFLEXclusters
+isshownasafunctionofthemeasuredX–rayluminosity L500in
+the left-hand panel of Fig. 2. The clusters are barely detect ed
+individually.Theaveragesignal–to–noiseratio(S /N)ofthetotal
+populationis 0.26andonly29 clustersare detected at S/N>2,
+the highestdetectionbeingat 4.2.However,onecan disting uish
+the deviation towards positive flux at the very high luminosi ty
+end.
+In the right-hand panel of Fig. 2, we average the 893 mea-
+surements in four logarithmically–spaced luminosity bins (red
+diamonds plotted at bin center). The number of clusters are 7 ,
+150,657,79fromthelowesttothehighestluminositybin.He re
+and in the following, the bin average is defined as the weighte d
+mean of the SZ flux in the bin (weight of 1 /σ2
+θs). The thick er-
+rorbarscorrespondtothestatisticaluncertaintiesonthe WMAP
+data only, while the thin bar gives the total errors as discus sed
+in Sec. 5.1. The SZ signal is clearly detected in the two highe st
+luminositybins(at6.0and5.4 σ,respectively).Asademonstra-
+tivecheck,wehaveundertakentheanalysisasecondtimeusi ng
+randomclusterpositions.Theresultisshownbythegreentr ian-
+glesin Fig.2andisconsistentwithnoSZ signal,asexpected .
+In the following Sections, we study both the relation be-
+tween the SZ signal and the X-ray luminosity and that with the
+massM500. We consider Y500, the SZ flux from a sphere of ra-
+diusr500, converting the measured Ycyl
+5r500intoY500as described
+in Appendix A. This allows a more direct comparison with the
+modelderivedfromX–rayobservations(Sec.3).Beforepres ent-
+ingtheresults,we first discussthe overallerrorbudget.
+5. Overallerrorbudget
+5.1. Errordue todispersion inX-rayproperties
+The errorσθsonY500given by the multifrequency matched fil-
+teronlyincludesthestatisticalSZmeasurementerror,due tothe
+instrument (beam, noise) and to the astrophysical contamin ants
+(primary CMB, Galaxy, point sources). However, we must also
+takeintoaccount:1)uncertaintiesontheclustermassesti mationfrom the X-ray luminosities via the L500−M500relation, 2) un-
+certaintiesontheclusterprofileparameters.Thesearesou rcesof
+erroronindividual Y500estimates(actualparametersforeachin-
+dividualcluster may deviate somewhatfrom the averageclus ter
+model).These deviationsfromthe mean,however,induce add i-
+tionalrandomuncertaintiesonstatisticalquantitiesderivedfrom
+Y500,i.e.binaveraged Y500valuesandthe Y500–L500scalingrela-
+tion parameters. Their impact on the Y500–M500relation, which
+dependsdirectlyonthe M500estimates,isalsoanadditionalran-
+domuncertainty.
+Theuncertaintyon M500isdominatedbytheintrinsicdisper-
+sioninthe L500–M500relation.ItseffectisestimatedbyaMonte
+Carlo (MC) analysis of 100 realisations. We use the dispersi on
+atz=0 asestimatedby Prattet al. (2009),givenin Table 1. For
+eachrealisation,wedrawarandommasslog M500foreachclus-
+ter from a Gaussian distribution with mean given by the L500–
+M500relation and standard deviation σlogL−logM/αM. We then
+redo the full analysis (up to the fitting of the YSZscaling rela-
+tions)withthenewvaluesof M500(thusθs).
+The second uncertainty is due to the observed dispersion
+in the cluster profile shape, which depends on radius as shown
+in Arnaudet al. (2009, σlogP∼0.10 beyond the core). Using
+new 100 MC realisations, we estimate this error by drawing a
+cluster profile in the log–log plane from a Gaussian distribu -
+tion with mean given by Eq. 3 and standard deviation depend-
+ing on the cluster radius as shown in the lower panel of Fig. 2
+inArnaudet al.(2009).
+The total error on Y500and on the scaling law parameters
+is calculated from the quadratic sum of the standard deviati on
+of both the above MC realisations and the error due to the SZ
+measurementuncertainty.
+5.2. The Malmquistbias
+TheNORAS/REFLEXsampleisfluxlimitedandisthussubject
+to the Malmquist bias (MB). This is a source of systematic er-
+ror.Ideallywe shoulduse a L500–M500relationwhichtakesinto
+account the specific bias of the sample, i.e. computed from th e
+trueL500–M500relation, with dispersion and bias according to
+each surveyselection function.We have an estimate of the tr ue,
+ie MB corrected, L500–M500relation, from the published analy-
+sis ofREXCESS data (Table 1). However, while the REFLEX
+selection function is known and available, this is not the ca se
+for the NORAS sample. This means that we cannot perform a
+fully consistent analysis. In order to estimate the impact o f the
+MalmquistBiaswethuspresent,inthefollowing,resultsfo rtwo
+cases.
+In the first case, we use the published L500–M500relation
+derived directly from the REXCESS data, i.e. not corrected for
+theREXCESS MB(hereafterthe REXCESS L500–M500relation).
+Notethatthe REXCESS isasub-sampleofREFLEX.Usingthis
+relation should result in correct masses if the Malmquist bi as
+for the NORAS/REFLEX sample is the same as that for the
+REXCESS . TheY500–M500relation derived in this case would
+also be correct and could be consistently comparedwith the X –
+ray predicted relation. We recall that this relation was der ived
+from pressure and mass measurements that are not sensitive t o
+theMalmquistbias.However L500wouldremainuncorrectedso
+that theY500–L500relationderivedin this case shouldbe viewed
+as a relation uncorrected for the Malmquist bias. In the seco nd
+case, we usethe MBcorrected L500–M500relation(hereafterthe
+intrinsicL500–M500relation). This reduces to assuming that the
+Malmquist bias is negligible for the NORAS /REFLEX sample.
+The comparisonof the two analyses providesan estimate of th e6 J.-B.Melinet al.:The galaxycluster YSZ-LXandYSZ-Mrelations fromthe WMAP 5-yr data
+direction and amplitude of the e ffect of the Malmquist bias on
+our results. The REXCESS L500–M500relation is expectedto be
+closer to the L500–M500relation for the NORAS /REFLEX sam-
+ple than the intrinsic relation. The discussions and figures cor-
+respond to the results obtained when using the former, unles s
+explicitlyspecified.
+The choice of the L500–M500relation has an effect both on
+the estimated L500,M500andY500values and on the expectation
+fortheSZsignalfromtheNORAS /REFLEXclusters.However,
+foraclusterofgivenluminositymeasuredagivenaperture, L500
+dependsweakly on the exact value of r500due to the steep drop
+of X–ray emission with radius. As a result, and although L500
+andM500(or equivalently r500) are determined jointly in the it-
+erative procedure described in Sec. 2.2, changing the under ly-
+ingL500–M500relation mostly impacts the M500estimate: L500
+is essentially unchanged(median di fference of∼0.8%) and the
+differencein M500simply reflectsthe di fferencebetweenthe re-
+lations at fixed luminosity. This has an impact on the measure d
+Y500via the value of r500(the profile shape being fixed) but the
+effectisalsosmall(<1%).Thisisduetotherapidlyconverging
+natureofthe YSZflux(seeFig.11ofArnaudet al.2009).Onthe
+other hand, all results that depend directly on M500, namely the
+derivedY500–M500relation or the model value for each cluster,
+that variesas M5/3
+500(Eq.5), dependsensitively on the L500–M500
+relation.M500derivedfromtheintrinsicrelationishigher,anef-
+fect increasing with decreasing cluster luminosity (see Fi g. B2
+ofPrattet al. 2009).
+5.3. Otherpossible sources of uncertainty
+The analysis presented in this paper has been performed on
+the entire NORAS/REFLEX cluster sample without removal of
+clusters hosting radio point sources. To investigate the im pact
+of the point sources on our result, we have cross-correlated the
+NVSS (Condonet al. 1998) and SUMMS (Mauchet al. 2003)
+catalogues with our cluster catalogue. We conservatively r e-
+moved from the analysis all the clusters hosting a total radi o
+fluxgreaterthan1Jywithin5 ×r500. Thisleaves328clustersin
+thecatalogue,removingthemeasurementswithlargeuncert ain-
+ties visible in Figure 2 left. We then performedthe full anal ysis
+on these 328 objects up to the fitting of the scaling laws, find-
+ingthattheimpactonthefittedvaluesismarginal.Forexamp le,
+fortheREXCESS case,thenormalisationofthe Y500–M500rela-
+tiondecreasesfrom1.60to 1.37(1.6statistical σ) andthe slope
+changesfrom1.79to 1.64(1statistical σ). The statistical errors
+ontheseparametersdecreaserespectivelyfrom0.14to 0.30 and
+from 0.15 to 0.40 due to the smaller number of remaining clus-
+tersinthesample.
+The detection method does not take into account superposi-
+tioneffectsalongthelineofsight,adrawbackthatisinherentto
+any SZ observation. Thus we cannot fully rule out that our flux
+estimates are not partially contaminated by low mass system s
+surrounding the clusters of our sample. Numerical simulati ons
+give a possible estimate of the contamination: Hallmanet al .
+(2007) suggest that low-mass systems and unbound gas may
+contribute up to 16 .3%+7%
+−6.4%of the SZ signal. This would lower
+ourestimatedclusterfluxesby ∼1.5σ.
+6. TheYSZ-L500relation
+6.1. WMAPSZmeasurementsvs. X–raymodel
+We first consider bin averaged data, focusing on the luminosi ty
+rangeL500∼>1043ergs/swhere the SZ signal is significantlyde-Fig.4.Estimated SZ flux Y500(in a sphere of radius r500) as a
+function of the mass M500averaged in 4 mass bins. Red dia-
+monds are the WMAP data. Blue stars correspond to the X-ray
+basedmodelpredictionsandareshiftedtohighermassesby2 0%
+forclarity.Themodelisinverygoodagreementwiththedata .
+tected (Fig. 2 right). The left panel of Fig. 3 shows Y500from a
+sphere of radius r500as a function of L500, averagingthe data in
+six equally–spaced logarithmic bins in X–ray luminosity. B oth
+quantitiesare scaled accordingto theirexpectedredshift depen-
+dence. The results are presented for the analyses based on th e
+REXCESS (reddiamonds)andintrinsic(redcrosses) L500–M500
+relations. Forthe reasonsdiscussed in Sec. 5.2, the derive ddata
+pointsdonotdiffersignificantlybetweenthetwoanalyses(Fig.3
+left),confirmingthat themeasured Y500–L500relationis insensi-
+tivetothefinerdetailsoftheunderlying L500–M500relation.
+We also apply the same averaging procedure to the model
+Y500valuesderivedfor each cluster in Sec. 3. The expectedval-
+ues for the same luminosity bins are plotted as stars in the le ft-
+handhandpanelofFig.3.The Y500–L500relationexpectedfrom
+thecombinationofthe Y500–M500(Eq.5) and L500–M500(Eq.2)
+relationsissuperimposedtoguidetheeye.Theright-handp anel
+of Fig. 3 shows the ratio between the measured data points and
+those expected from the model. As discussed in Sec. 5.2, the
+model values depend on the assumed L500–M500relation. The
+difference is maximum in the lowest luminosity bin where the
+intrinsic relation yields ∼40% higher value than the REXCESS
+relation (Fig. 3 left panel). The SZ model prediction and the
+data are in good agreement, but the agreement is better when
+theREXCESS L500–M500is used in the analysis (Fig. 3 right
+panel). This is expected if indeed the agreement is real and t he
+effectiveMalmquistbiasfortheNORAS /REFLEXsampleisnot
+negligibleandissimilar tothatofthe REXCESS .
+6.2.Y500-L500relationfit
+Working now with the individual flux measurements, Y500, and
+L500values,wefit an Y500–L500relationoftheform:
+Y500=Y∗L
+500/parenleftiggE(z)−7/3L500
+1044h−2erg/s/parenrightiggαL
+Y
+E(z)βL
+Y/parenleftiggDang(z)
+500Mpc/parenrightigg−2
+(12)
+using the statistical error on Y500given by the multifrequency
+matched filter. The total error is estimated by Monte Carlo (s ee
+Sec. 5.1) but is dominated by the statistical error. The resu lts
+are presentedin Table 2. As alreadystated in Sec. 6.1, the fit ted
+valuesdependonlyweaklyonthechoiceof L500–M500relation.J.-B.Melinet al.:The galaxycluster YSZ-LXandYSZ-Mrelations fromthe WMAP 5-yr data 7
+Fig.3.Left:Bin averaged SZ flux from a sphere of radius r500(Y500) as a functionof X-ray luminosity in a aperture of r500(L500).
+The WMAP data (red diamonds and crosses), the SZ cluster sign al expected from the X–ray based model (blue stars) and the
+combinationofthe Y500–M500andL500–M500relations(dashanddotteddashedlines)aregivenfortwoan alyses,usingrespectively
+the intrinsic L500–M500and the REXCESS L500–M500relations. As expected, the data points do not change signifi cantly from one
+case to the other showing that the Y500–L500relation is rather insensitive to the finer details of the und erlyingL500–M500relation.
+Right:Ratio of data points to model for the two analysis. The points for the analysis undertaken with the intrinsic L500–M500are
+shiftedtolowerluminositiesby20%forclarity.
+Table 2. Fitted parameters for the observed YSZ-L500relation. The X-ray based model gives Y∗L
+500=0.89|1.07×
+10−3(h/0.719)−5/2arcmin2,
+αL
+Y=1.11|1.04andβL
+Y=2/3forthe REXCESS andintrinsic L500–M500relation,respectively.
+L500−M500Y∗L
+500[10−3(h/0.719)−2arcmin2]αL
+Y βL
+Y
+REXCESS 0.92±0.08stat [±0.10tot] 1.11 (fixed) 2 /3(fixed)
+0.88±0.10stat [±0.12tot] 1.19±0.10stat [±0.10tot] 2 /3(fixed)
+0.90±0.13stat [±0.16tot] 1.11 (fixed) 1 .05±2.18stat [±2.25tot]
+Intrinsic 0.95±0.09stat [±0.11tot] 1.04 (fixed) 2 /3(fixed)
+0.89±0.10stat [±0.12tot] 1.19±0.10stat [±0.10tot] 2 /3(fixed)
+0.89±0.13stat [±0.16tot] 1.04 (fixed) 2 .06±2.14stat [±2.21tot]
+7. TheYSZ–M500relationand itsevolution
+In this Section, we study the mass and redshift dependence
+of the SZ signal and check it against the X–ray based model.
+Furthermore, we fit the Y500–M500relation and compare it with
+theX–raypredictions.
+7.1. Massdependence andredshift evolution
+Figure4showsthebinaveragedSZfluxmeasurementasafunc-
+tionof masscomparedto the X–raybasedmodelprediction.As
+expected,theSZclusterfluxincreasesasafunctionofmassa nd
+iscompatiblewith the model.Inordertostudythe behaviour of
+theSZfluxwithredshift,wesubdivideeachofthefourmassbi ns
+into three redshift bins corresponding to the following ran ges:
+z<0.08, 0.08<z<0.18,z>0.18. The result is shown
+in the left panel of Fig. 5. In a given mass bin the SZ flux de-
+creases with redshift, tracing the Dang(z)−2dependence of the
+flux.Inparticular,inthehighestmassbin(1015M⊙),the SZflux
+decreases from 0.007 to 0 .001arcmin2while the redshift varies
+fromz<0.08toz>0.18.Themassandtheredshiftdependence
+areingoodagreementwiththemodel(stars)describedinSec .3.
+Since the Dang(z)−2dependenceisthe dominante ffectin the
+redshift evolution, we multiply Y500byDang(z)2and divide itby the self-similar mass dependence M5/3
+500. The expected self-
+similar behaviour of the new quantity Y500Dang(z)2/M5/3
+500as a
+function of redshift is E(z)2/3(see Eq. 5). The right panel of
+Fig. 5 shows Y500Dang(z)2/M5/3
+500as a function of redshift for the
+three redshift bins z<0.08, 0.08<z<0.18,z>0.18. The
+pointshavebeencenteredattheaveragevalueofthecluster red-
+shifts in each bin. The model is displayed as blue stars. Sinc e
+the modelhas a self-similar redshift dependenceand E(z)2/3in-
+creases only by a factor of 5% over the studied redshift range ,
+the model stays nearly constant. The blue dotted line is plot ted
+through the model and varies as E(z)2/3. The data points are in
+good agreement with the model, but clearly, the redshift lev er-
+age of the sample is insu fficient to put strong constraints on the
+evolutionofthescalinglaws.
+We now focus on the mass dependence of the relation. We
+scaletheSZfluxwiththeexpectedredshiftdependenceandpl ot
+itasafunctionofmass.TheresultisshowninFig.6forthehi gh
+mass end.Thefigure showsa verygoodagreementbetweenthe
+datapointsandthemodel,whichisconfirmedbyfittingtherel a-
+tiontotheindividualSZ fluxmeasurements(see nextSection ).8 J.-B.Melinet al.:The galaxycluster YSZ-LXandYSZ-Mrelations fromthe WMAP 5-yr data
+Fig.5.Evolution of the Y500-M500relation.Left:The WMAP data from Fig. 4 are divided into three redshift bins : z<0.08 (blue
+diamonds),0.08<z<0.18(greencrosses),z >0.18(red triangles).We observethe expectedtrend: at fixed mass,Y500decreaseswith
+redshift. This redshift dependence is mainly due to the angu lar distance ( Y500∝Dang(z)−2). The stars give the prediction of the
+model.Right:We divide Y500byM5/3
+500Dang(z)−2and plot it as a function of zto search for evidence of evolution in the Y500-M500
+relation.Thethickbarsgivethe1 σstatistical errorsfromWMAP data.Thethinbarsgivethetot al1 sigmaerrors.
+Fig.6.Left:Zoomonthe>5×1013M⊙massrangeofthe Y500−M500relationshowninFig.4.Thedatapointsandmodelstarsare
+nowscaledwiththeexpectedredshiftdependenceandarepla cedatthemeanmassoftheclustersineachbin. Right:Ratiobetween
+dataandmodel.
+Table 3. Fitted parameters for the observed YSZ-M500relation. The X-ray based model gives Y∗
+500=1.54×
+10−3(h/0.719)−5/2arcmin2,αY=5/3andβY=2/3.
+L500–M500relation Y∗
+500[10−3(h/0.719)−2arcmin2]αY βY
+REXCESS 1.60±0.14stat [±0.19tot] 5 /3 (fixed) 2 /3(fixed)
+1.60±0.15stat [±0.19tot] 1.79±0.15stat [±0.17tot] 2 /3(fixed)
+1.57±0.23stat [±0.29tot] 5 /3 (fixed) 1 .05±2.18stat [±2.52tot]
+intrinsic 1 .37±0.12stat [±0.17tot] 5 /3 (fixed) 2 /3(fixed)
+1.36±0.13stat [±0.17tot] 1.91±0.16stat [±0.18tot] 2 /3(fixed)
+1.28±0.19stat [±0.24tot] 5 /3 (fixed) 2 .06±2.14stat [±2.48tot]
+7.2.Y500–M500relationfit
+Using the individual Y500measurements and M500estimated
+fromtheX–rayluminosity,wefit a relationofthe form:
+Y500=Y∗
+500/parenleftiggM500
+3×1014h−1M⊙/parenrightiggαY
+E(z)βY/parenleftiggDang(z)
+500Mpc/parenrightigg−2
+(13)The results are presented in Table 3 for the analysis underta ken
+using the REXCESS and that using the intrinsic L500–M500re-
+lation. The pivot mass 3 ×1014h−1M⊙, close to that used by
+Arnaudetal. (2009), is slightly larger than the average mas s of
+thesample(2.8|2.5×1014M⊙fortheREXCESS —intrinsic L500–
+M500relation, respectively). We use a non-linear least-square sJ.-B.Melinet al.:The galaxycluster YSZ-LXandYSZ-Mrelations fromthe WMAP 5-yr data 9
+fit built on a gradient-expansion algorithm (IDL curvefit fun c-
+tion).Inthefittingprocedure,onlythestatistical errors givenby
+the matched multifilter ( σY500) are taken into account. The total
+errors on the final fitted parameters, taking into account unc er-
+tainties in X–ray properties, are estimated by Monte Carlo a s
+describedinSec. 5.
+We first discuss the results obtained using the REXCESS
+L500–M500relation, which is expected to be close to the opti-
+mal case (see discussion in Sec. 5.2). First, we keep the mass
+and redshift dependence fixed to the self-similar expectati on
+(αY=5/3,βY=2/3)andwe fit onlythe normalisation.We ob-
+tainY∗
+500=1.60×10−3(h/0.719)−2arcmin2, in agreementwith
+the X-ray prediction Y∗
+500=1.54×10−3(h/0.719)−5/2arcmin2
+(at0.4σ).Then,werelaxtheconstrainton αYandfitthenormal-
+isationandmassdependenceatthesametime.Weobtainavalu e
+forαY=1.79, slightly greater than the self-similar expectation
+(5/3)by0.8σ.Tostudytheredshiftdependenceofthee ffect,we
+fix the mass dependence to αY=5/3 and fitY∗
+500andβYat the
+same time. We obtain a somewhatstronger evolution βY=1.05
+than the self-similar expectation(2 /3).The difference,however,
+is not significant (0 .2σ). As already mentioned above (see also
+Fig. 5 right), the redshift leverage is too small to get inter esting
+constraintsonβY.
+As cluster mass estimates depend on the assumption of the
+underlying L500–M500relation, so does the derived Y500–M500
+relation as well. However, the e ffect is small. The normalisa-
+tion is shifted from (1.60±0.14stat [±0.19tot])10−3arcmin2
+to(1.37±0.12stat [±0.17tot])10−3arcmin2whenusingthein-
+trinsicL500–M500relation.Thedifferenceisless thantwo statis-
+tical sigmas,andforthemassexponent,it islessthanone.
+8. Discussionand conclusions
+In this paper we have investigated the SZ e ffect and its scal-
+ing with mass and X-ray luminosity using WMAP 5-year data
+of the largest published X-ray-selected cluster catalogue to
+date, derived from the combined NORAS and REFLEX sam-
+ples. Cluster SZ flux estimates were made using an optimised
+multifrequency matched filter. Filter optimisation was ach ieved
+through priors on the pressure distribution (i.e., cluster shape)
+andthe integrationaperture(i.e., clustersize). The pres suredis-
+tribution is assumed to follow the universal pressure profil e of
+Arnaudetal.(2009),derivedfromX-rayobservationsofthe rep-
+resentative local REXCESS sample. This profile is the most re-
+alistic available for the general population at this time, a nd has
+beenshowntobeingoodagreementwithrecenthigh-qualityS Z
+observations from SPT (Plaggeetal. 2009). Furthermore, ou r
+analysis takes into account the dispersion in the pressure d istri-
+bution.Theintegrationapertureisestimatedfromthe L500−M500
+relationofthesame REXCESS sample.Weemphasisethatthese
+two priors determine only the input spatial distribution of the
+SZ fluxforuse bythemultifrequencymatchedfilters; the prio rs
+give no information on the amplitude of the measurement. As
+the analysis uses minimal X-ray data input, the measured and
+X-raypredictedSZ fluxesare essentiallyindependent.
+We studied the YSZ−LXrelation using both bin averaged
+analyses and individual flux measurements. The fits using ind i-
+vidual flux measurements give quantitative results for cali brat-
+ing the scaling laws. The bin averaged analyses allow a direc t
+quantitativecheckofSZfluxmeasurementsversusX-raymode l
+predictions based on the universal pressure profile derived by
+Arnaudetal.(2009)from REXCESS .Anexcellentagreementis
+found.Using WMAP 3-year data, both Lieuetal. (2006) and
+Bielby&Shanks (2007) found that the SZ signal strength is
+lower than predicted given expectations from the X-ray prop er-
+ties of their clusters, concluding that that there is some mi ssing
+hotgasintheintra-clustermedium.Theexcellentagreemen tbe-
+tween the SZ and X-ray propertiesof the clusters in our sampl e
+showsthatthereisinfactnodeficitinSZsignalstrengthrel ative
+to expectations from X-ray observations. Due to the large si ze
+and homogeneous nature of our sample, and the internal con-
+sistency of our baseline cluster model, we believe our resul ts
+to be robust in this respect. We note that there is some con-
+fusion in the literature regarding the phrase ‘missing bary ons’.
+The ‘missing baryons’ mentioned by Afshordiet al. (2007) in
+the WMAP 3-year data are missing with respect to the univer-
+salbaryonfraction,butnotwithrespecttotheexpectation sfrom
+X-raymeasurements.Afshordiet al.(2007)actuallyfoundg ood
+agreement between the strength of the SZ signal and the X-ray
+properties of their cluster sample, a conclusion that agree s with
+our results. This goodconvergencebetween SZ direct measur e-
+ments and X–ray data is an encouraging step forward for the
+predictionandinterpretationofSZ surveys.
+UsingL500asamassproxy,wealsocalibratedthe Y500–M500
+relation,finding a normalisationin excellentagreementwi th X-
+ray predictions based on the universal pressure profile, and a
+slope consistent with self-similar expectations. However , there
+is some indication that the slope may be steeper, as also indi -
+cated from the REXCESS analysis when using the best fitting
+empirical M500–YXrelation (Arnaudet al. 2009). M500depends
+on the assumed L500−M500relation, making the derived Y500–
+M500relationsensitivetoMalmquistbiaswhichwecannotfully
+account for in our analysis. However, we have shown that the
+effect of Malmquist bias on the present results is less than 2 σ
+(statistical).
+Regardingevolution,wehaveshownobservationallythatth e
+SZ flux is indeed sensitive to the angular size of the cluster
+through the diameter distance e ffect. For a given mass, a low
+redshift cluster has a bigger integrated SZ flux than a simila r
+system at high redshift, and the redshift dependence of the i n-
+tegrated SZ flux is dominated by the angular diameter distanc e
+(∝D2
+ang(z) ). However, the redshift leverage of the present clus-
+ter sample is too small to put strong contraints on the evolut ion
+of theY500–L500andY500–M500relations. We have nevertheless
+checked that the observed evolution is indeed compatible wi th
+theself-similarprediction.
+Inthisanalysis,wehavecompensatedforthepoorsensitivi ty
+andresolutionoftheWMAPexperiment(regardingSZscience )
+withthelargenumberofknownROSATclusters,leadingtosel f-
+consistent and robust results. We expect further progress u sing
+upcomingPlanckall-skydata.WhilePlanckwillo fferthepossi-
+bilityofdetectingtheclustersusedinthisanalysistohig herpre-
+cision, thus significantly reducing the uncertainty on indi vidual
+measurements, the question of evolution will not be answere d
+with the present RASS sample due to its limited redshift rang e.
+Acomplementaryapproachwill thusbetoobtainnewhighsen-
+sitivity SZ observations of a smaller sample. The sample mus t
+berepresentative,coverawidemassrange,andextendtohig her
+z (e.g.,XMM-Newtonfollow-upofsamplesdrawnfromPlanck
+andgroundbasedSZ surveys).Thiswoulddelivere fficientcon-
+straintsnot onlyon the normalisationand slope of the YSZ−LX
+andYSZ−Mrelations,butalsotheirevolution,openingtheway
+fortheuseofSZ surveysforprecisioncosmology.
+Acknowledgements. The authors wish to thank the anonymous referee for
+useful comments. J.-B. Melin thanks R. Battye for suggestin g introduction
+of thehdependance into the presentation of the results. The author s also10 J.-B.Melinet al.:The galaxycluster YSZ-LXandYSZ-Mrelations fromthe WMAP 5-yr data
+acknowledge the use of the HEALPix package (G´ orski et al. (2 005)) and of
+the Legacy Archive for Microwave Background Data Analysis ( LAMBDA).
+Support for LAMBDA is provided by the NASA O ffice of Space Science. We
+also acknowledge use of the Planck Sky Model, developed by th e Component
+Separation Working Group (WG2) of the Planck Collaboration , for the
+estimation of the radio source flux in the clusters and for the development of
+the matched multifilter, although the model was not directly used in the present
+work.
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+Vikhlinin, A.,Burenin, R.,Ebeling, H.etal. 2009, ApJ,692 , 1033AppendixA: SZ fluxdefinitions
+In this Appendix, we give the definitions of SZ fluxes we used.
+TableA.1givestheequivalencebetweenthem.Inthispaper, we
+mainly use Y500as the definition of the SZ flux. This flux is the
+integratedSZ flux from a sphere of radius r500. It can be related
+toYnr500, the fluxfroma sphereofradius n×r500byintegrating
+overtheclusterprofile:
+Ynr500=Y500/integraltextnr500
+0drP(r)4πr2
+/integraltextr500
+0drP(r)4πr2(A.1)
+whereP(r) is given by Eq. 3. The ratio Ynr500/Y500is given in
+TableA.1for n=1,2,3,5,10.
+Inpractice,anexperimentdoesnotdirectlymeasure Y500but
+the SZ signal of a cluster integrated along the line of sight a nd
+withinanangularaperture.ThiscorrespondstotheCompton pa-
+rameter integratedovera cylindricalvolume.In Sec. 4, we e sti-
+mateYcyl
+5r500, the flux from a cylinder of aperture radius 5 ×r500
+using the matched multifilter. Given the cluster profile, we c an
+deriveYnr500fromYcyl
+nr500:
+Ycyl
+nr500=Ynr500/integraltext∞
+0dr/integraltext
+rsinθ<nr500dθP(r)2πr2
+/integraltextnr500
+0drP(r)4πr2(A.2)
+The ratio Ynr500/Ycyl
+nr500is given in Table A.1 for n=
+1,2,3,5,10. In the paper, we calculate Y500fromY500=
+0.986/1.814×Ycyl
+5r500.
+AppendixB: SZ unitsconversion
+In this Appendix, we provide the numerical factor needed for
+the SZ flux unitsconversionand derivethe relation betweent he
+recentlyintroduced YXparameterandtheSZflux YSZ.Thelatter
+will allow readers to easily convert between SZ fluxes given i n
+thispaperandthosereportedinotherpublications.
+GiventhedefinitionofSZ flux:
+Ycyl
+nr500=/integraldisplay
+Ωnr500dΩy (B.1)
+whereΩnr500is the solid angle coveredby n×r500, and the fact
+thattheComptonparameter yisunitless,theobservationalunits
+for the SZ flux are those of a solid angle and usually given in
+arcmin2.
+The SZ flux can be also computed in units of Mpc2and the
+conversionis givenby
+YSZ[Mpc2]=60−2/parenleftbiggπ
+180/parenrightbigg2
+YSZ[arcmin2]/parenleftiggDang(z)
+1Mpc/parenrightigg2
+=8.46×10−8YSZ[arcmin2]/parenleftiggDang(z)
+1Mpc/parenrightigg2
+(B.2)
+Table A.1. EquivalenceofSZfluxdefinitions
+n 123510
+Ynr500/Y50011.5051.6901.8141.873
+Ynr500/Ycyl
+nr5000.8270.9300.9630.9860.997J.-B.Melinet al.:The galaxycluster YSZ-LXandYSZ-Mrelations fromthe WMAP 5-yr data 11
+whereDang(z)isthe angulardistancetothe cluster.
+TheX-rayanalogueoftheintegratedSZComptonisationpa-
+rameter is YX=Mgas,500TXwhose natural units are M ⊙keV,
+whereMgas,500is the gas mass in r500andTXis thespec-
+troscopic temperature excluding the central 0 .15r500region
+(Kravtsovetal. 2006). To convertbetween YSZandYX, we first
+have
+YSZ[Mpc2]=/integraldisplayr500
+0drσTTe(r)
+mec2ne(r)4πr2(B.3)
+whereσTistheThomsoncrosssection(inMpc2),mec2theelec-
+tronmass(inkeV), Te(r)theelectronictemperature(inkeV)and
+ne(r) the electronic density. By assuming that the gas tempera-
+tureTg(r) is equal to the electronic temperature Te(r) and writ-
+ingthegasdensityas ρg(r)=µempne(r),wherempistheproton
+massandµe=1.14themeanmolecularweightperfreeelectron,
+oneobtains:
+YSZ[Mpc2]=σT
+mec21
+µemp/integraldisplayr500
+0drρg(r)Tg(r)4πr2
+=CXSZMgas,500TMW=ACXSZYX (B.4)
+where,asin Arnaudet al.(2009),we defined
+CXSZ=σT
+mec21
+µemp=1.416×10−19Mpc2
+M⊙keV(B.5)
+Themassweighted temperatureis definedas:
+TMW=/integraltextr500
+0drρg(r)Tg(r)4πr2
+/integraltextr500
+0drρg(r)4πr2(B.6)
+andthefactor A=TMW/TXtakesintoaccountforthedi fference
+betweenmassweightedandspectroscopicaveragetemperatu res.
+Arnaudetal. (2009)find A∼0.924.
\ No newline at end of file
diff --git a/1001.0872.txt b/1001.0872.txt
new file mode 100644
index 0000000000000000000000000000000000000000..44c9d3bee0f8ae9df3e8c8fecdd6fa4cdc3ffc13
--- /dev/null
+++ b/1001.0872.txt
@@ -0,0 +1,737 @@
+Symmetry, Conserved Charges, and Lax 
+Representations of Nonlinear Field Equations: 
+A Unified Approach 
+ 
+C. J. Papachristou 
+ 
+Department of Physical Sciences, Nava l Academy of Greece, Piraeus 18539, Greece 
+E-mail:  papachristou@snd.edu.gr  
+ 
+Abstract:   A certain non-Noetherian connection between symmetry and integrability 
+properties of nonlinear field equations in conservation-law form is studi ed. It is shown that 
+the symmetry condition alone may lead, in a rather  straightforward way, to the construction of 
+a Lax pair, a doubly infinite set of (generally  nonlocal) conservation laws, and a recursion 
+operator for symmetries. Applications include  the chiral field equation and the self-dual 
+Yang-Mills equation.   
+ 
+Keywords: Nonlinear equations; Symmetries;  Conservation laws; Lax pair.  
+ 
+ 
+1.  Introduction   
+ 
+In a recent paper [1] an analytical method was described for constructing a Lax pair 
+for the Ernst equation of General Relativity. The starting point was the symmetry 
+condition (or linearized form) of the fiel d equation. The latter equation is in 
+conservation-law form, and thus so is its associated symmetry condition. A doubly 
+infinite hierarchy of conser vation laws was then construc ted by a recursive process, 
+and the conserved “charges” were used as Laurent coefficients in a series 
+representation (in powers of the sp ectral parameter) of a function Ψ which was seen to 
+satisfy the sought-for Lax pair.  
+    It is natural to inquire whether this technique can also be a pplied to other nonlinear 
+partial differential equations (PDEs) of Math ematical Physics. This article describes a 
+general, non-Noetherian framework for conne cting integrability ch aracteristics of a 
+given nonlinear PDE to the symmetry properties of this PDE. It is remarkable that, by 
+starting with the symmetry condition, one may discover a number of important things 
+such as the existence of a recursion opera tor [2,3] for symmetries, a doubly-infinite 
+set of (typically nonlocal) conservation laws , and a Lax pair which “linearizes” the 
+nonlinear field equation.  
+    To illustrate the use of the method, application is made to two familiar nonlinear PDEs: the chiral field equation and the self-dual Yang-Mills equation. In these 
+examples, the corresponding Lax pairs and infinite sequences of c onservation laws are 
+constructed explicitly. Moreover, the recursion operators for symmetries are derived. 
+In the case of the real Erns t equation, treated previously in [1], although a recursion 
+operator doesn’t seem to exist for that par ticular form of the equation (due to the 
+coordinate “pathology” which results in th e explicit appearance of an independent 
+variable in the PDE), one still gets an in teresting “hidden” symmetry transformation 
+which leads to new approximate solutions for stationary gravitational fields with axial symmetry [4].  
+ 2  C. J. Papachristou  
+2.  Th e General Idea   
+ 
+Let F [u]=0 be a nonlinear PDE in the dependent variable u and the independent  
+variables x, y, ... . The bracket notation [ u] indicates that the function F may depend 
+explic itly o n the va riables u, x, y, ... , as well as on partial derivatives, of various 
+orders, of  u with respec t to the inde pendent variables, denoted ux , uy , uxx , uyy , uxy , 
+etc. W e adopt the defin ition acco rding to  whic h the  PDE is in tegrab le if it ha s an 
+associated  Lax-pair representation , i.e., if  it can be exp ressed  as an integ rability 
+condition for solution of a linear system  of PDEs for an auxiliary field Ψ :  
+ 
+                       (; ;)0 1,2iLu iλΨ==                                             (1) 
+ 
+where the d ifferentia l expression s Li are lin ear in Ψ, and  where λ is a (generally 
+complex) “spectral” param eter.  
+    It has bee n obs erved that integrable PDEs often have an infinite num ber of 
+symmetries which m ay be produced, for exam ple, with the aid of one or m ore 
+recurs ion operato rs (see, e.g., [2,3] and the r eferences th erein ). This  connection  
+between symmetry and integrabi lity m ay be attributed to a variety of factors. For 
+exam ple, an integ rable PDE m ay have an underlying  Ham iltonian struc ture in whic h 
+the Lagrangian density possesses an infinite number of variational symmetries. In this  
+case, the Noether th eorem  provides the connection  between symmetry and 
+integ rability , the latter  manifesting itse lf in the presen ce of  an inf inite se t of 
+conservation  laws. As is often the cas e, the ex istence of these laws is asso ciated with a 
+Lax structure for the nonlinear problem.  
+    Non-Noetherian connections between symmetry and integrability, however, m ay 
+also exist. L et us recall that the non linear PDE F [u]=0 is a consistency condition for 
+solution of the system  (1). On the othe r hand, the (generally com plex) function Ψ will 
+satisfy som e PDE of its own, also derived from  system (1). This PDE will be linear in 
+Ψ and will contain  u as a “param etric” function:  
+ 
+                    (; )0 GuΨ=                                                       (2) 
+ 
+where the expression  G is linea r in Ψ, and where u is a so lution of  F [u]=0  . We may 
+say that the system  (1) is a Bäcklund transform ation relati ng the nonlinear PDE  
+F [u]=0  to the linear PDE (2).  Now, we already k now an equation of th e form (2): it is  
+the symmetry condition  (linearized for m) of F [u]=0  . Let [] uu Quα′=+  be an 
+infinitesimal symm etry transf ormation f or the latter PDE, where α is an inf inites imal 
+param eter (we note that any symm etry of a PDE can be expressed as a transform ation 
+of the dependent variable alone [2,3], i.e., is equivalent to a “vertical” sy mmetry). The 
+symmetry ch aracter istic Q  [u] then satisf ies a lin ear PDE of  the f orm  
+ 
+                    (; )0mod[] SQ u Fu =                                                 (3) 
+ 
+where “m od F [u]” signif ies that the PDE on the lef t is sa tisfied when u is a solution 
+of the nonlinear PDE F [u]=0  . Now, if it happens that Eqs.(2) and  (3) become 
+identical when Ψ≡Q (i.e., if the functions G and S are the same), then th e solu tion Ψ 
+of the Lax pair (1) will also be a sym metry characteristic of  F [u]=0  :  
+ 
+                                                                   (4) (; )0mod[] Su Fu Ψ=
+  Symmetry, Conserved Charges, and Lax Representati ons 3 
+    Of course, as th e examples of the self-dual Yang-Mills equation [5] and the E rnst 
+equation [1,4] have taught us, it is possi ble that a given nonlinear PDE adm it more 
+than one Lax represen tation. W hat we are s eeking here is a Lax pair which functions 
+as a Bäcklund transform ation connecting the nonlinear PDE F [u]=0 to its (line ar) 
+symmetry condition  (3). The symmetry conditi on its elf is  thus “bu ilt” into the L ax 
+pair, and a very fundam ental connection b etween symm etry and integ rability  is 
+estab lished.  
+    W ith regard to the complex param eter λ of the Lax pair (1), w e rem ark th e 
+following: Since the role of such a param eter is generally nontriv ial, it will be 
+required that λ be nonzero (as well as, of course, fini te in m agnitude). W e then expect  
+that the solu tion Ψ of system  (1), for  a given u satisf ying F [u]=0  , will be an analytic 
+function of  λ for λ≠0 . This solution m ay thus be represented as a L aurent series 
+expansion in powers of  λ, with  u-dependent coefficients:  
+ 
+                                                               (5) ()(; ) []nn
+nu Q λλ+∞
+=−∞Ψ=∑ u
+± 
+where the functions Q(n)[u] may be local or nonlocal in u. Now, we recall that Ψ is 
+assum ed to be a symm etry characteristic of F [u]=0  , and t his must be  true for all 
+values of  λ in the L ax pair. Subs tituting Eq.(5 ) into Eq.(4) ( which is lin ear in  Ψ), and 
+equating  coefficients of  all λn to zero, we find a doubly infinite set of linear PDEs for  
+Q(n), of the form   
+ 
+                                   (6) ()();0 mod[], 0,1,2,nSQ u Fu n = =±"
+ 
+All Lauren t coefficients  Q(n)[u] are thus seen to  be symm etry characteristics for th e 
+nonlinear PDE F [u]=0 , and the presence o f this infinite set of s ymmetries is  
+intim ately r elated to the Lax pair.  
+    Substitu ting the expansion (5) into the Lax pair (1), and e quating coef ficients of  all 
+powers of  λ to zero,  we obtain a pair  of linear P DEs containing Q(n) and (say) Q(n+1). 
+In essence, this is a Bäcklund trans formation for the sym metry condition (3 ). This 
+differential recursion relation constitutes a recursion operator  [2,3] for t he PDE   
+F [u]=0  , in the spirit of a new perception of th is concept originally proposed by this 
+author [5,6] and, independently, by Marv an [7]. W e thus have a m ethod for the 
+explic it con struction of  such an ope rator. S tarting with any symm etry Q(0), we can, in 
+principle, use this operator to deri ve a double infinity of symm etries Q(n) (although not  
+all of them  will necessarily b e nontrivial).  
+    Finally, suppose that F [u] is a divergence, so  that the PDE F [u]=0 h as the for m of 
+a conservation law. Then, its symmetry conditi on (3) also is in such form. Given that 
+an infinite n umber of symmetry characteristics Q(n)[u] are availab le, we imm ediately 
+obtain a doubly infinite collec tion of conserv ation laws f or F [u]=0 from Eq.( 6) 
+(where now  the function S is a divergen ce). Typically,  the recurs ion operato r 
+connecting the Q(n) to each other is an integro- differential operator; thus, th e 
+conserved “currents” are generally  expected to be nonlocal in  u .  
+ 
+ 
+ 
+ 4  C. J. Papachristou  
+3.  Analytical Description of the Method   
+ 
+Our objective is the following: Given a nonlinear PDE F [u]=0 in conservation-law 
+form, we se ek a Lax pair whose solution is a symmetry characteristic for this PDE, 
+and, in the process, we expect to derive a recursion operato r for symm etries as well a s 
+an infinite set of (nonlocal) c onservation laws. Although the solution u of the PDE 
+may depend on m ore than two independent vari ables, we r estrict ours elves to th e case 
+where F [u] is a divergence in only two of the m:  
+ 
+                                                             (7) [] [] [] 0xy Fu DAuDBu ≡+ =
+ 
+where Dx and Dy denote tota l deriv atives (see Ap pendix), which will also be indica ted 
+by using subscripts: x x DAA≡ , etc. W e will assum e, in general, th at u is square-
+matrix-valued, and so are the functions A, B, F.  
+    Let [] u Quδα=  be an infinitesim al symm etry of Eq.(7) (where α is a n 
+infinitesimal param eter and Q is the m atrix-v alued sym metry chara cteristic).  We 
+write, in f inite f orm,  
+ 
+                       [] uQ u∆=                                                             (8) 
+ 
+where, in general, ∆ denotes the Fréchet derivative of any function f [u], with respe ct 
+to the ch aracteristic Q (see Appendix). The symmetry cond ition for the PDE (7) is  
+ 
+                                                                       (9) [] 0mod[] Fu Fu ∆=
+ 
+where  
+ 
+                                       [] [] []xy Fu D Au D Bu ∆= ∆+∆
+  
+(since F réchet der ivative s and total d erivatives commute). Putting  
+ 
+              [] (;), [] (;) AuG Qu Bu HQu ∆≡ ∆≡                                      (10) 
+ 
+(where the functions G and H are linear in  Q), we rewrite  Eq.(9) in  the form  of a  
+linea r PDE for Q :  
+ 
+                                 (11) (; ) (;) (; )0mod[]xy SQu DGQ u DHQ u Fu ≡+ =
+ 
+We note that S(Q ; u) is a diverg ence, so th at the sym metry condition (11 ) is a 
+conservation law for the corre sponding nonlinear PDE (7).  
+    Equation (11) suggests that we introduce a “potential” function K , such that G=K y 
+and H=−Kx (subsc ripts denote  total differentia tions).  We assum e that K is linear ly 
+dependent on som e new function Q′, and we write:  
+ 
+             (;) (;), (;) (;)y GQ uDKQu HQu DKQux′ ′ = =−                            (12) 
+ 
+  Symmetry, Conserved Charges, and Lax Representati ons 5 
+Clearly, this  system  is integrab le for Q′ (mod F [u]) if Q satisf ies th e symmetry 
+condition (1 1). The integrability req uirement for Q , on the other h and, will yield 
+some linear PDE f or . It is possible that, by an a ppropriate choice of the function 
+, this PDE will b e just the symm etry condition (1 1) for QQ′
+(; ) KQ u′ ′:  
+ 
+                                              (; )0mod[ SQ u Fu] ′=  .    
+ 
+That is,  will also b e a symmetry charac teristic. The system  (12) the n constitu tes a 
+Bäcklund transform ation (BT) for the symm etry condition (11). This BT m ay be 
+viewed as an inver tible recursion op erator for symmetries of  the nonlinear PDE (7). 
+Such an operator will, in principle, produ ce a doubly infinite sequence of symm etry 
+characteristics QQ′
+(n) (n= ±1, ±2, ...) from any given characteristic Q(0).  
+    To better display the recursiv e character of the BT (12), we rewrite this system  as 
+follows:  
+ 
+     () ()
+() (() (1)
+() (1);;
+;;nn
+y
+nn
+xGQ u DKQ u
+HQ u DKQ u+
++=
+=− )                                           (13) 
+ 
+(n= 0, ±1, ±2, ...), where G and H are line ar in Q(n), while K is linea r in Q(n+1). Now, 
+since all the Q(n) satisfy the PDE (11), the BT (13)  also yields a double infinity of 
+conservation laws for the field equation (7):  
+ 
+            () ()() ();; 0monn
+xyDGQu DHQu Fu + = d[]                                 (14) 
+ 
+Starting with a known sy mmetry characteristic Q , we can evaluate the conserved 
+“charges” Q(n) (n= 0, ±1, ±2, ...) as fo llows: ( a) We take the BT (13) with n=0 and s et 
+Q(0)=Q on the left-hand  side.  Then,  Q(1) is found by integration. To f ind Q(2) we 
+similarly integrate th e BT (13) with  n=1, etc. We thus obtai n all positively-indexed 
+charges Q(n). (b) We take the BT (13) with n= −1 and set Q(0)=Q on t he right-hand 
+side. W e then solve for Q(−1) . Working sim ilarly for n= −2, −3, ... , we obtain all 
+negatively-indexed charges Q(n).  
+    We now introduce a com plex param eter λ, which we require to  be nonzero and of 
+finite m agnitude. Mu ltiplying bo th sides of  Eq.( 13) by λn, summ ing over all integral 
+values of  n, and taking into account that the functions G, H and K are linear in their 
+respective Q’s, we find the following pair of PDEs:  
+ 
+         () ()
+() ()1;;
+1;;nn nn
+y
+n n
+nn nn
+x
+nnGQ u DK Q u
+HQ u DK Qλλλ
+λλλ+∞ +∞
+=−∞ =−∞
++∞ +∞
+=−∞ =−∞⎛⎞ ⎛⎞
+= ⎜⎟ ⎜⎟⎜⎟ ⎜⎟⎝⎠ ⎝
+⎛⎞ ⎛⎞
+=− ⎜⎟ ⎜⎟⎜⎟ ⎜⎟⎝⎠ ⎝⎠∑∑
+∑∑ u⎠                            (15) 
+ 
+ 
+ 
+ 
+ 6  C. J. Papachristou  
+We set   
+ 
+                                                                        (16) ()(; ) []nn
+nu Q λλ+∞
+=−∞Ψ=∑ u
+ 
+Equation (16) has the form  of a Laur ent expansion of a com plex function Ψ in powers 
+of λ, for a given solution u of the field equation (7).  We note that Ψ is a linear 
+combination of symmetry characteristics of Eq.(7), hence Ψ itself is a symmetry 
+charac teristic of  that PDE. Substituting Eq.(16) into Eq.(15),  we rewrite the la tter in 
+the form  of a system  of linear PDEs for Ψ:  
+ 
+           (;) (;), (;) (;)y x DKu G u DK u H u λ λ Ψ=Ψ Ψ=−Ψ                        (17) 
+ 
+The consistency of this system  requires that Ψ satisfy the linear PDE (11),   
+ 
+                       .  (; ) (;) (; )0(mod[])xy Su DG u DH u Fu Ψ≡ Ψ+ Ψ=
+ 
+This ve rifies that Ψ is a symmetry c haracteristic. More over, the system  (17) is linear 
+in Ψ, and its solvability dem ands that u satisfy the nonlinear PDE (7) [this was 
+required from the start in order that the BT (13), by which the charges Q(n) appearing  
+in the Laurent expansion (16) are defined, m ay be integrable for Q(n) and Q(n+1)] . We  
+thus conclude that the linear system  (17) constitutes a Lax pair for the field equation 
+(7), and that, m oreover, the solution Ψ of this system  is a symmetry characteristic of 
+that equ ation.  
+    A final comm ent before closing this section: The whole idea was based on the 
+assum ption that an a uto-Bäcklund  transformation of  the form (12) e xists for the 
+symmetry c ondition  (11). I t is po ssible,  how ever, that n o choic e for the function 
+ in Eq.(12) e xists su ch th at Q (; ) KQ u′ ′ be a symm etry characteristic when Q is such a 
+characteristic. In this case, the m ethod desc ribed above m ay still furnish an infinite 
+number of c onservation laws as well as a Lax pair, albeit not a recursion operator for  
+producing infinite sets of symm etries. Moreover, the solution Ψ of the Lax pair w ill 
+no longer represent a symm etry of the fi eld equation (although, of course, it will 
+somehow be related to a symm etry, since the symm etry condition w as the starting 
+point for constructing the Lax pair). The exam ple of the Er nst equa tion, exam ined in 
+detail in [ 1], made this  point c lear. In this ca se, the ab sence of  an inf inite se t of 
+symmetries is not a pro perty of  the gravit ation al field equations  them selves (whic h, 
+when properly form ulated, do exhibit such an infinite set [8]) but is a consequence of 
+the chosen real form  of the Ernst eq uation,  in w hich a spatial coordinate m akes an 
+explicit app earance.  
+ 
+ 
+4.  Chiral Field Equation   
+ 
+The chiral f ield equation (a two-dim ensional r eduction  of the se lf-dual Yang-M ills 
+equation, to be discussed later) is of  the f orm  
+ 
+                                                     (18) 1 1[] ( )( )0tt xx Fg gg gg− −≡+ =
+ 
+  Symmetry, Conserved Charges, and Lax Representati ons 7 
+where g is a GL(N,C)-valued function of  t and  x (as usual, subscripts denote total 
+differentia tions with res pect to the se variab les). Let [] g Qg δα=  be an inf initesimal 
+symmetry of  Eq.(18), with symm etry charac teristic Q [g]. We have that , 
+where ∆ denotes th e Fréchet deri vative with re spect to  Q (see Appendix). Moreover , 
+by the commutativity o f the Fréchet deriva tive with total de rivatives,  [] gQ g∆=
+ 
+                                 11
+11[] ( ) ( )
+ˆˆ() ()tt x
+tt xxx FgD gg D gg
+DAgQ DAgQ−−
+−−∆= ∆+∆
+=+       
+ 
+where we have introduced the “covariant derivative” operators  
+ 
+                              11 ˆˆ [, ], [ ,tt t x x x AD gg A D gg−−=+ =+ ]
+t− 
+(the square brackets denote commutators). It can be sh own that th ese opera tors 
+commute, as expected from the fact that the “c onnection s” g−1gt and g−1gx are pure 
+gauges. The symmetry condition (9) reads:  
+ 
+                                        (19) 1 ˆˆ (; )( )( )0mod[]tt xx SQ g DADA gQ Fg−≡+ =
+ 
+and it is obviously in conservation-law for m.  
+    We now seek an auto-Bäcklund transform ation (BT) of the form  (12) for the linear  
+PDE (19). This m ust be of the for m  
+ 
+                                         11 ˆ ˆ () , ( )tx x Ag Q K AgQ K−−==
+ 
+for some fun ction . Let u s try (; ) KQ g′1(; ) KQ g gQ−′ ′= :  
+ 
+                                        (20) 11 1 1 ˆ ˆ () ( ), ( ) (tx x Ag Q gQ AgQ gQ−− − −′′= = )t−
+ 
+Integrabi lity for Q′ clearly requires that Q satisfy Eq.(19). The integrability condition 
+for Q can be written (by taking into account that covariant derivatives commute),  
+ 
+                                               1 ˆˆ[, ]( )0txAA gQ−= .      
+ 
+After a somewhat leng thy calcu lation, and by using the operator identity  
+ 
+                              []ˆˆ ˆ ˆ [],
+ˆˆ mod []tt xx tt xx
+tt xxAD AD DA DA Fg
+DA DA Fg+= +−
+=+          
+ 
+we find that the above in tegrab ility condition y ields the PDE  
+ 
+                                     1 ˆˆ() ( )0modtt xx DA DA gQ Fg−′ + = []
+ 8  C. J. Papachristou  
+which is  just the symm etry cond ition (19) for Q′. We conclud e that Eq.(2 0) is indeed 
+an auto-BT for the aforem entioned symmetry condition. Th is BT is eq uivalen t to a 
+recurs ion op erato r for symmetries of the field eq uation (18 ). It can be re written in th e 
+form (13), as follows:  
+ 
+        () ()
+() (1( ) 1(1)
+1( ) 1(1)ˆ
+ˆnn
+t x
+n
+xtAg Q DgQ
+Ag Q DgQ−−
+− −=
+=− )n+
++                                        (21) 
+ 
+(n= 0, ±1, ±2, ...). The conservation laws of the for m (14) (which form  a doubly 
+infinite set ) are wri tten, i n this c ase,  
+ 
+     ()1( ) ˆˆ() 0modn
+tt xx DA DA gQ Fg−+ = []
+1)−                                (22) 
+ 
+(where all conserved “charges” Q(n) are symm etry characteristics ), while  the Lax p air 
+(17) reads,  
+ 
+              11 1 ˆˆ () () , ( ) (xt t x Dg Ag Dg Ag λλ−− −Ψ=Ψ Ψ=− Ψ                  (23) 
+ 
+    The proof of the Lax-pair property of th e line ar sys tem (23) is sketch ed as follows : 
+By the integ rability cond ition 1 1() ( )xt g g− −Ψ= Ψtx , we get:  
+ 
+                                             (24) 1 ˆˆ (; )( )( )0tt xx SgD ADA g−Ψ≡ + Ψ=
+ 
+On the other hand, the integrability condition 1 ˆˆ[, ]( )0txAA g λ−Ψ=, yields:  
+ 
+                                                1(; ) [], 0 Sg Fgg−⎡⎤Ψ− Ψ=⎣⎦
+ 
+which, in view of Eq.(24), becom es 1[], 0 Fg g−⎡ ⎤Ψ=⎣ ⎦
+t. This is va lid ind epend ently  
+of  Ψ if  F [g]=0, i.e., if  g is a solutio n of the field equa tion (18). W e conclude that the 
+linea r syst em (23) is in deed a Lax pair for the nonlinea r PDE (18), the  solution Ψ of 
+which pair i s a symm etry charac teristic [as foll ows from  Eq.(24)]. W e note tha t this 
+Lax pair is different from that found se veral years ago by Zakharov and Mikhailov 
+[9].  
+    We conclude this section by giving an  example of using the BT (21) to find 
+conserved charges Q(n). Let us co nsider the s ymmetry charac teristic , 
+where M is an arbit rary constan t matrix. The BT (21) with n=0, integrated for Q(0)Q gM=
+(1), 
+yields  
+                                                      ,     (1)[, ] Qg XM=
+ 
+where X  is the potential of Eq.(18), defi ned by the system  of equations  
+ 
+                  1 1,tx x gg X gg X− −= =−                                            (25) 
+ 
+  Symmetry, Conserved Charges, and Lax Representati ons 9 
+We note th at Q(1) is th e characteristic  of a potential symmetry  [3,6]. Higher-o rder 
+charges Q(n) with  n>1 (which a lso are h igher-ord er potential sy mmetries) a re 
+similarly found by recursive integr ation of the BT (21) with  n= 1, 2, etc.  
+    To find negatively-indexed charges and corresponding symmetries, w e begin with 
+the BT (21 ) with n=−1, which we integrat e for Q(−1). The result is a rathe r 
+uninteresting local sym metry: (1)Q−g=Λ, where Λ is any  constan t matrix. Iterati ng 
+for n=−2, however, we find a new characte ristic Q(−2), given by the system  of 
+equations  
+ 
+                      11 1 1() , (tt x x x QQ gg gg g Q Qgg gg g−− − −−= Λ − =−Λ)t
+ 
+(where we have put Q(−2)= Q , for brevity ). High er-ord er, negatively-indexed charges  
+are obtained by further iteration.  
+    Unfortunately, in cont rast to th e “interna l” sy mmetries co nsidered  above, the local 
+coordinate symm etries [such as (0)
+t Qg=, (0)
+x Q g=, etc.] do not yield any new 
+results by applying the BT (21). These latte r symmetries, however, play an equally 
+important role as internal ones in problem s in m ore than two dim ensions, as the 
+exam ple discussed in th e next se ction will show.   
+ 
+ 
+5.  Self-Dual Yang-Mills Equation   
+ 
+The self-dua l Yang-Mi lls (SDYM) equation is writ ten in the for m  
+ 
+                                 1 1[] ( ) ( )0yy zz FJ JJ JJ− −≡+ =                                        (26) 
+ 
+where J is assum ed SL(N,C)-valued (i.e., det  J=1). The four independent variables 
+(appearing as subscripts) are constructed from  the coordinates of an underlying 
+complexified Euclide an space in such a way that y and z becom e the com plex 
+conjugates of  y and  z, respectively, when the above space is  real. As usual, subscripts 
+denote to tal deriva tives with respe ct to thes e vari ables.  
+    Let [] J QJ δα=  be an  infinites imal symm etry of Eq.(26), where the c haracteristic 
+Q is subjec t to the cond ition that 1( ) trJQ−0=, required for producing new SL(N,C) 
+solutions from old ones. The symmetry c ondition is, in analogy with Eq.(19),  
+ 
+                1 ˆˆ (; )( )( )0mod[]yy zz SQ J DA DAJQ FJ−≡+ =                          (27) 
+ 
+where we have introduced the covariant derivatives  
+ 
+                                      11 ˆ ˆ [, ], [ ,yy y z z z AD JJ A D JJ−−=+ =+ ]
+0 
+(note ag ain that these operato rs commute). An auto-BT for the lin ear PDE (27)  
+[analogous to that of Eq.(20)], which is consistent with the physical requirement 
+, is the fol lowing:  1( ) trJQ−=
+ 10  C. J. Papachristou  
+            11 1 1 ˆ ˆ () ( ), ( ) ( )yz z AJ Q JQ AJQ JQ−− − −
+y′ ′ = =−                         (28) 
+ 
+Integrabi lity for Q′ requi res tha t Q satisfy Eq.(27). Integrabi lity for Q , expressed by 
+the condition , and upon using the operator identity  1 ˆˆ[, ]( )0yzAA JQ−=
+ 
+                              []ˆˆ ˆ ˆ [],
+ˆˆ mod []yy z z yy zz
+yy zzAD AD DA DA FJ
+DA DA FJ+= +−
+=+      
+ 
+leads us ag ain to Eq.(2 7), this time for Q′. The BT (28) may be regarded as an 
+invertible recursion operato r for the SDYM equation. It  can b e re-express ed as  
+ 
+              () ()
+() ()1( ) 1(1)
+1( ) 1(1)ˆ
+ˆnn
+y z
+nn
+zyAJ Q DJQ
+AJ Q DJQ−−
+−−=
+=−+
++                                       (29) 
+ 
+(n= 0, ±1, ±2, ...). From  this we get a double infinity of conservation laws of the for m  
+ 
+             ()1( ) ˆˆ () 0modn
+yy zz DA DA JQ FJ−+ = []                                 (30) 
+ 
+    Finally, the Lax pair f or SDYM [a nalogous to those of Eqs.(17) and (23)] is  
+ 
+       11 1 ˆˆ () () , ( ) (zy y z DJ AJ DJ AJ λλ−− − 1)−Ψ=Ψ Ψ=− Ψ                   (31) 
+ 
+The proof of the Lax-pair p ropert y is ske tched as follo ws: By the  integ rability 
+condition 1 1() ()zy yz JJ− −Ψ− Ψ=0 , we get:  
+ 
+          1 ˆˆ (; )( )( )0yy zz SJ DADAJ−Ψ≡ + Ψ=                                   (32) 
+ 
+On the other hand, the integrability condition 1 ˆˆ[, ]( )0zyAA J λ−Ψ=, yields:  
+ 
+                                                1(; ) [], 0 SJ FJJ−⎡⎤Ψ− Ψ=⎣⎦
+ 
+which, in view of Eq.(32), becom es 1[], 0 FJ J−⎡ ⎤Ψ=⎣ ⎦. This is va lid ind epend ently  
+of  Ψ if  F [J ]=0, i.e., if  J is an SDYM solution. W e conclude tha t the linea r syst em 
+(31) is indeed a Lax pair for th e SDYM equation (26), the solution Ψ of which pair is  
+a symmetry charac terist ic [as follows from  Eq.(32)]. This Lax pair can b e shown to be 
+equivalent to that reported previously by this author [5], although the system atic 
+method for e xplicitly con structing this system  is presented here for the firs t time.  
+    We now give exam ples of using the BT (29) to find conserved charges Q(n). Let us 
+consider th e symm etry charac teristic , where M is a con stant trace less 
+matrix. The BT (29) wit h n=0, integ rated for Q(0)Q JM=
+(1), yields  
+ 
+  Symmetry, Conserved Charges, and Lax Representati ons 11 
+                                                      ,     (1)[, ] QJ XM=
+ 
+where X  is the potential of Eq.(26), defi ned by the system  of equations  
+ 
+                  1 1,yz z JJ X JJ X− −=y=−
+J                                           (33) 
+ 
+We note th at Q(1) is th e characte ristic of a po tential symm etry [ 3,6]. Higher-o rder 
+charges Q(n) with  n>1 (which a lso are h igher-ord er potential sy mmetries) a re 
+similarly found by recursive integr ation of the BT (29) with  n= 1, 2, etc.  
+    To find negatively-indexed charges and corresponding symmetries, w e begin with 
+the BT (29) with n=−1, which we integrate for Q(−1). The resul t is a familiar lo cal 
+symmetry: , where Λ is any constant  trace less matrix. Ite rating for n=−2, 
+we find a new characteristic Q(1)Q−=Λ
+(−2), given by the system  of equations  
+ 
+                 11 1 1() , (yy z z z QQ JJ JJ J Q QJJ JJ J−− − −−= Λ − =−Λ)y     
+ 
+(where we have put Q(−2)= Q , for brevity ). High er-ord er, negatively-indexed charges  
+are obtained by further iteration.  
+    In the preceding exa mple, the in itial symm etry characte ristic Q(0) represented a n 
+“inte rnal” symm etry (a symm etry in the fiber space). Loc al coordin ate symm etries 
+(symm etries in the ba se space), how ever, a lso le ad to the d iscovery of in finite se ts of 
+potential sy mmetries an d assoc iated conser vation laws for SDYM. As an exam ple, 
+consider the obvious symm etry of y-trans lation, represen ted by the charac teristic 
+. The BT (29) with n=0, integ rated f or Q(0)
+y Q J=(1), gives  
+ 
+                                                            (1)
+y Q JX=
+ 
+[wher e X is the SDYM poten tial defined in  Eq.(33)], which is another potential 
+symmetry. Higher-ord er poten tial symmetries, whose chara cteristics  Q(n) (n>0) app ear 
+as conserved charges in conservation laws of the for m (30), can be found by repeated 
+applic ation  of the recu rsion operato r (29). Th e infinite  sets of potentia l symmetries 
+generated by coordinate transformations have been shown to possess a rich Lie-
+algebraic s tructure [10,1 1].  
+    To conclude our example, let us fi nd som e negatively-indexed symmetries. The BT 
+(29) with n=−1 and , integra ted for Q(0)
+y Q J=(−1), give s: (1)
+z Q−J= , which is the 
+characteristic for the obvious z-translationa l sym metry. The first non trivial resul t is 
+found for n=−2, yield ing a chara cteristic Q(−2) which is defined by the set of equations  
+ 
+               11 1 1() , (yy zz z z Q QJJ JJJ Q QJJ JJJ−− − −−= − =− )zy      
+ 
+(where we have put Q(−2)= Q , for brevity).  
+ 
+ 
+ 
+ 12  C. J. Papachristou  
+6.  Summary  
+ 
+Motivated by the results of [1] for the Erns t equation, we have proposed a general, 
+non-Noetherian schem e for connecting sy mmetry and integrabil ity properties of 
+nonlinear PDEs in conservation-law for m. We have shown that, by starting with the  
+symmetry c ondition  (which i s itself a loc al conservat ion law for th e assoc iated 
+nonlinea r PDE), one may derive s ignificant m athem atical o bjects such as a recursion 
+operator for symmetries,  a Lax pair, and an infinite collection of  (generally nonlocal) 
+conserva tion laws. Such objects are usually so ught by trial-and-erro r processes, thu s 
+any system atic technique for their discovery is useful.  
+    The method was illus trated by using tw o physically sign ificant exam ples, nam ely, 
+the chiral field equation and the self-dua l Yang-Mills (SDYM) equation. The latter 
+PDE has been shown to constitute a prot otype equation from  which several other 
+integrable P DEs are derived by reduction [ 12,13]. Thus, the results regarding SDYM  
+may also prove useful for the st udy of other nonlin ear problem s.  
+ 
+ 
+7.  Appendix: Total  Derivat ives and Fréchet Deri vatives   
+ 
+To m ake this article as self-contained as po ssible, we define two key concepts that are 
+being used,  nam ely, the total derivative a nd the Fréche t deriva tive. The reade r is 
+referred to the extensive review article [14] by this author for m ore details. (It should 
+be noted, however, that our present definiti on of  the Fréchet derivative corresponds to 
+the definitio n of the Lie deriva tive in that a rticle. Since these two de rivatives a re 
+local ly indistinguishable, this discrepanc y in term inology should not cause any 
+concern m athem atically. )  
+    We consi der the s et of all PDEs  of the form  F [u]=0, where, for s implicity, th e 
+solution s u (which m ay be m atrix-valued) are assum ed to be functions of only tw o 
+variables x and t : u=u(x,t). In  genera l, [] (,,,,,,,,)x txx ttxt Fu Fxtuuuuuu ≡ ". 
+Geom etrically, we say that the function F is defined in a jet spa ce [2,15] with 
+coordinates x, t, u, and as m any partial derivatives of  u as needed  for the  give n 
+problem . A solution of the PDE F [u]=0 is then a  surface in this je t spac e.  
+    Let F [u] be a given function in the jet space.  When differentiating su ch a function 
+with respe ct to  x or  t, both im plicit (through u) and explicit dependence of F on these 
+variab les m ust be  taken  into  accou nt. If u is a scal ar qu antity, we define th e total 
+deriva tive o perators  Dx  and Dt  as follows:  
+ 
+                                 xx xx xt
+x t
+tt xt tt
+xtDu u uxu u u
+Du u utu u u∂∂ ∂ ∂=+ + + +∂∂ ∂ ∂
+∂∂ ∂∂=+ + ++∂∂ ∂ ∂"
+"           
+ 
+(note that the operators ∂/∂x and ∂/∂t concern on ly the e xplicit dependence of F on x 
+and t). If, however, u is m atrix-valued, the above representation has only sym bolic 
+significance and cannot be used for actual cal culations. W e must therefore define the 
+total derivat ives Dx  and Dt  in more genera l terms.  
+ 
+  Symmetry, Conserved Charges, and Lax Representati ons 13 
+    We define a linea r operato r Dx , acting on functions F [u] in the jet space and having 
+the followin g propert ies:  
+    1. On fun ctions  f (x, t) in th e base space,  
+ 
+                                             (,) /xxDfxt fx f =∂∂≡∂ .  
+ 
+    2. On fun ctions F [u]= u  or  ux , ut , etc., in th e “fiber” spa ce,  
+ 
+                              ,, ,x xx x xxx t tx xt Duu Du u Duu u == ==  etc.   
+ 
+    3. The operato r Dx is a derivation on the algebra of all functions F [u] in the jet 
+space (i.e., the Leibniz rule is s atisfied):  
+ 
+                           () () [] [] [] [] [] []xxDFuGu DFu Gu Fu DGu = +x  .    
+ 
+    We similarly define the operator Dt . Extension to higher-order total derivatives is 
+obvious (although these latter derivatives are no longer derivations, i.e. , they do not 
+satisfy th e Leibniz rule ). The following notat ion has been use d in this arti cle:  
+ 
+                                      [] [], [] []xx t t DFu Fu DFuFu ≡≡  .     
+ 
+Finally, it can be shown that, for any m atrix-valued functions A and B in the jet spac e, 
+we have  
+ 
+                                   11 1 1 1() ,( )xx t t AA AA A AA−− − − −=− =−1A−
+and  
+ 
+                  [, ][,][, ], [, ][,][, ]x xx t t t DAB AB AB DAB AB AB =+ =+     
+ 
+where square bracke ts denote commutato rs.  
+    Let now [] u Quδα  be an  infinite simal symm etry tran sformation (with  
+charac teristic Q [u] ) for the  PDE F [u]=0. We define the Fréchet de rivative  with  
+respec t to th e charac teristic Q as a lin ear opera tor ∆ acting on functions F [u] in the jet 
+space and ha ving the following prope rties:  
+    1. On fun ctions  f (x, t) in th e base space,   
+ 
+                                                        (,)0 fxt∆=  
+ 
+(this is a consequence of our liberty to c hoose all our symmetries to be in “vertical” 
+form [2,3]).  
+    2. On  F [u]= u ,  
+                                                         [] uQ u∆=  .    
+ 
+    3. The operator ∆ commutes with tota l deriv ative operators of any order.  
+ 
+ 14  C. J. Papachristou  
+    4. The Leibniz rule is satisfi ed:  
+ 
+                             () () [] [] [] [] [] [] FuG u FuGu FuGu ∆= ∆ +∆  .     
+ 
+The following properties can be proven:  
+ 
+                            () [], () [xx x t t t uu Qu u u Q ]u ∆=∆= ∆=∆=     
+                   11() ()1A AA A−−∆=−∆−  ;   [, ][,][, ] ABA B AB ∆=∆+∆       
+ 
+where A and B are any matrix-va lued functions in the je t spac e.  
+    If the solution  u of the PDE is a sc alar function  (thus so is the chara cteristic Q ), the 
+Fréchet de rivative with respec t to Q adm its a differential-operator representation of 
+the form   
+ 
+             xt xx tt xt
+xt xx tt xtQQ Q Q Q Quu u u u u∂∂ ∂ ∂ ∂ ∂∆=+++ + + +∂∂ ∂ ∂ ∂ ∂"    
+ 
+Such repres entations, h owever, are not valid for PDEs in matrix for m. In these case s 
+we must resort to the general definiti on of the Fréchet derivative given above.  
+    Finally,  by using the Fréchet  derivative,  the symmetry co ndition  for a  PDE F [u]=0 
+can be expressed as follows [2,3]:  
+ 
+                                                    .      [] 0mod[] Fu Fu ∆=
+ 
+This condition yields a linear PD E for the symmetry characteristic Q , of the form   
+ 
+                                                 (; )0mod[] SQ u Fu =  .       
+ 
+ 
+ 
+ 
+ 
+ 
+Acknow ledgment   
+ 
+I thank Ka thleen O’Sh ea-Arapog lou for caref ully reviewing the m anuscrip t and 
+making a number of useful suggestions.  
+ 
+ 
+  Symmetry, Conserved Charges, and Lax Representati ons 15 
+References   
+ 
+1. C. J. Papachristou and B. K. Harrison, Elect. J. Theor. Phys. (EJTP) 6, No. 22 
+(2009) 29.  
+2. P. J. Olver, Applications of Lie Groups to Differential Equations , 2nd ed. 
+(Springer-Verlag, 1993).  
+3. G. W. Bluman and S. Kumei, Symmetries and Differential Equations  
+(Springer-Verlag, 1989).  
+4. C. J. Papachristou and B. K. Harrison, Phys. Lett. A 190 (1994) 407.  
+5. C. J. Papachristou, J. Phys. A: Math. Gen. 24 (1991) L  1051.  
+6. C. J. Papachristou, Phys. Lett. A 145 (1990) 250.  
+7. M. Marvan, Another Look on Recursion Operators , in Differential Geometry 
+and Applications  (Proc. Conf., Brno, 1995) 393.  
+8. Y. Nakamura, J. Math. Phys. 24 (1983) 606.  
+9. V. E. Zakharov and A. V. Mikhailov, Sov. Phys. JETP 47 (1978) 1017.  
+10. C. J. Papachristou, Phys. Lett. A 154 (1991) 29.  
+11. A. D. Popov and C. R. Preitsc hopf, Phys. Lett. B 374 (1996) 71.  
+12. R. S. Ward, Phil. Trans. R. Soc. A 315 (1985) 451.  
+13. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations 
+and Inverse Scattering (Cambridge University Press, 1991).  
+14. C. J. Papachristou, Symmetry and Integrability of Classical Field Equations , 
+arXiv:  0803.3688.  
+15. C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their 
+Applications  (Academic Press, 1982).  
+ 
+ 
\ No newline at end of file
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new file mode 100644
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--- /dev/null
+++ b/1001.0873.txt
@@ -0,0 +1,939 @@
+Rheologica Acta manuscript No.
+(will be inserted by the editor)
+Shear localisation with 2D Viscous Froth and its relation to the
+Continuum Model
+Joseph D. Barry, Denis Weaire, Stefan Hutzler
+School of Physics, Trinity College Dublin, Ireland
+Received: date / Revised version: date
+Abstract Simulations of monodisperse and polydis-
+perse (2(A) = 0:130:002) 2D foam samples undergo-
+ing simple shear are performed using the 2D Viscous
+Froth (VF) Model. These simulations clearly demon-
+strate shear localisation. The dependence of localisation
+length on the product V(shearing velocity Vtimes
+external wall friction coecient ) is examined and is
+shown to agree qualitatively with other published ex-
+perimental data. A wide range of localisation lengths is
+found at low V, an eect which is attributed to the
+existence of distinct yield and limit stresses. The gen-
+eral Continuum Model is extended to incorporate such
+an eect and its parameters are subsequently related to
+those of the VF Model. A Herschel-Bulkley exponent of
+a= 0:3 is shown to accurately describe the observed
+behaviour. The localisation length is found to be inde-
+pendent of Vfor monodisperse foam samples.
+Correspondence to : barryjo@tcd.iePaper presented at 5th Annual European Rheology
+Conference (AERC), April 15-17, 2009, Cardi, United
+Kingdom.
+1 Introduction
+Foam is dened as a two-phase system in which a dis-
+persed phase of gas is enclosed by a continuous phase
+of liquid [29]. In aqueous foams, the dispersed phase is
+typically air, and the liquid phase water with an added
+surfactant. To simplify the task of understanding the
+rheology of these systems, it has become popular to con-
+centrate on two-dimensional (2D) foams, which consist
+of a single planar layer of bubbles. It is the rheology
+of these systems which is under scrutiny in this paper,
+using the 2D VF Model.
+We endeavour to understand the mechanisms which
+cause localisation of shear at a moving boundary, as re-
+ported in a number of recent experiments (see below).arXiv:1001.0873v2  [cond-mat.soft]  3 Mar 20102 Joseph D. Barry et al.
+This behaviour is observed in the VF simulations that
+we will discuss. In the literature, the extent of this shear
+localisation eect is measured by a localisation length. In
+this paper, two dierent denitions of localisation length
+will be employed.
+Simulation results for polydisperse froths subjected
+to simple shear will be presented, where the localisa-
+tion length is found to vary with V(viscous drag times
+driving velocity; why this is the important parameter
+to consider is explained in Sec. 4). For low V, a wide
+range of localisation lengths is found; see Sec. 4. This
+eect is attributed to the existence of distinct yield and
+limit stresses. We give qualitative evidence of this in Sec.
+5, where we extend the general Continuum Model to in-
+corporate such an eect. In Sec. 6 we proceed to relate
+parameters of the VF Model to the Continuum Model
+which is shown to accurately predict the observed be-
+haviour for a Herschel-Bulkley exponent of a= 0:3. The
+localisation length is found to be independent of Vin
+the monodisperse case.
+In experiments with 2D foams, often a single layer of
+bubbles is conned between two narrowly spaced glass
+plates (a Hele-Shaw cell). There are also other types of
+quasi-2D systems, such as the Bragg raft [3], where a sin-
+gle layer of bubbles 
oats on a liquid pool, and the con-
+ned bubble raft, where a Bragg raft is trapped under-
+neath a glass plate. The most important distinction be-
+tween these experimental realisations is concerned with
+the presence of viscous drag. When a foam is in contactwith one or two conning plates, there is a drag force
+associated with any movement of the foam relative to
+the plate(s). As we shall see, the drag force in the VF
+Model (see eq. 2) plays a role in the localisation of 
ow
+in our foam samples.
+Fig. 1 A T1 neighbour-swapping event triggered by apply-
+ing shear. a) Initial conguration, b) A and B lose their com-
+mon edge, creating an unstable four-fold vertex point, c) a
+new edge is created between C and D and d) nal congura-
+tion. Data taken from a VF simulation.
+When a 2D foam is subjected to an applied shear
+stress, after an initial transient, it yields and begins to
+
ow. The foam yields locally when the yield stress is
+reached. A more detailed description of the stress-strain
+relation will be required when interpreting the simula-
+tion results presented in this paper; see Sec. 5. At the
+local level, yielding is due to plastic events, i.e. T1 topo-Shear localisation with 2D Viscous Froth and its relation to the Continuum Model 3
+logical changes of the foam structure (see Fig. 1). Two
+neighbouring bubbles (A and B) lose a common edge
+which is subsequently gained by two proximate bubbles
+(C and D), which become neighbours. We describe the
+incorporation of these topological changes into the VF
+Model in Sec. 2.
+When 
ow is concentrated in one region and not in
+another, the 
ow is said to have localised. Debreg eas et
+al.[8] were the rst to report denitive evidence of shear
+localisation in 2D aqueous foams. Their experiments ex-
+hibit shear localisation next to a moving boundary in a
+Couette geometry, with an exponential decay in the mea-
+sured foam velocity proles. Similar results have been re-
+ported by Wang et al. [25] and Krishan and Dennin [23]
+for straight and circular geometries, respectively. These
+results have been interpreted within the framework of
+the Continuum Model [6, 14, 15] where shear localisation
+is attributed to the presence of a drag force. This notion
+is further supported by the work of [12] which studies the
+eects of drag forces at high shear rates using the VF
+Model. However there are also quasi-static simulations
+showing localisation (discussed below) in which there is
+no such wall drag. This suggests that there is more than
+one mechanism that may lead to shear localisation. We
+will return to this point in Sec. 5.
+Experimental work by Katgert et al. [19, 20] on the
+shearing of bidisperse foams in a Hele-Shaw cell (straight
+geometry, that is, simple shear) shows Herschel-Bulkley
+behaviour (discussed below) and supplies further evi-dence of shear localisation in 2D foams. In this case,
+however, the velocity proles are not exponential. Such
+non-exponential velocity proles, together with their ve-
+locity dependence can be obtained from an extension or
+generalisation of the original Continuum Model [28, 30].
+Furthermore, these experiments show that the locali-
+sation length decreases as the velocity of the moving
+boundary increases. In this paper, we show velocity pro-
+les from our VF simulations which exhibit qualitatively
+similar behaviour.
+Of particular current interest in these types of exper-
+iments is the dependence of (local) shear stress on shear
+rate. This eect is captured by the Herschel-Bulkley con-
+stitutive relation,
+=y+cv_a(1)
+whereis stress,yis the yield stress, the coecient
+cvis the so-called consistency, _ is strain rate and ais
+the HB exponent. Katgert and co-workers report a=
+0:36. They also note that in the monodisperse case (i.e.
+bubbles of equal size), the localisation length is found to
+be independent of shear rate. We too nd this to be the
+case in our simulations; see Sec. 4.
+Shear localisation has been studied computationally
+using other microscopic (bubble scale) models. Quasi-
+static models, as explored by [2, 13, 31, 32] might shed
+light on behaviour at very low strain rates. Results re-
+ported by Kabla [16, 18] show localisation next to the
+boundaries in quasi-static shearing simulations ( 2(A)4 Joseph D. Barry et al.
+0:06 using the denition eq. 3), consistent with experi-
+mental observation. In these simulations of simple shear,
+where there is no wall drag, either wall may be regarded
+as the one that moves. Recent results by Wyn [33] sug-
+gest that as the second moment of the bubble area dis-
+tribution2(A) is increased in such simulations to val-
+ues approaching 0.2 or higher, shear-banding can occur
+in regions away from the moving boundary. The width
+of these shear bands has a square-root dependence on
+2(A). This type of behaviour has yet to be observed in
+experiments. In this paper, we only examine the rheol-
+ogy of foam samples of disorder 2(A) = 0:13 but will
+probe higher values of disorder in future work in order
+to search for similar eects.
+With the aid of the Surface Evolver software [4],
+quasi-static simulations are increasingly easy to imple-
+ment and, with modern computing, are certainly fast.
+But are they suitable for rheology? In quasi-statics, the
+foam is relaxed to equilibrium at each step. There is
+therefore no relevant time scale present and so no con-
+cept of shear-rate. It makes no sense to consider Herschel-
+Bulkley type relations or to discuss the dependence of
+localisation on boundary velocity.
+What then, are the alternatives to quasi-static sim-
+ulations? Bubble models [9], where a foam is modeled
+as a collection of interacting disks appear to represent
+at some level the dynamics of 2D foams. Langlois et
+al.[24] report a Herschel-Bulkley exponent of a= 0:54
+(2(A)0:03). In addition, shear localisation is ob-served when wall drag is present. For dry foams though,
+where low liquid fraction causes bubbles to become more
+polygonal in shape, this approach is no longer accurate
+[12].
+In this paper, we adopt the 2D Viscous Froth (VF)
+Model [11, 21] as a more realistic model for dry 2D
+foam dynamics. We have performed an extensive study
+of shear localisation with the VF Model in a straight ge-
+ometry which shows realistic dynamics and a rich variety
+of behaviour, particularly at low V.
+For a summary of the experimental and theoretical
+work presented in this section, see [27].
+2 The 2D Viscous Froth Model and its
+implementation
+The model describes the motion of a soap lm in the
+2D systems described above, with wall drag [21]. In the
+present case, bubble areas are kept constant. The foam is
+considered to be suciently dry (liquid fraction less than
+0.01) so that a soap lm may be accurately described by
+a curved line and the junctions are represented by points.
+In the present simulations, a soap lm is approximated
+as a system of connected straight line segments. The
+motion of a point sjoining these segments is given via
+the equation
+v?(s) =
P
K(s) (2)Shear localisation with 2D Viscous Froth and its relation to the Continuum Model 5
+Fig. 2 A diagram illustrating the various forces involved in
+lm motion with the two-dimensional VF Model. BandB0
+indicate the two bubbles the central soap lm is separating.
+Note that this lm is in contact with a surface (in the plane
+of the page) which results in a drag force when the lm is
+moving.
+whereis the wall drag coecient, v?(s) is the velocity
+of a pointsin the direction of the normal vector N(s)
+to the soap lm, 
Prepresents relative pressure dier-
+ences between neighbouring cells, 
is a constant surface
+tension force (in 2D), and K(s) is local lm curvature
+calculated from the relative positions of adjacent (dis-
+crete) lm segment points. See Fig. 2 for an illustration
+of the forces involved. Setting = 0 in eq. 2 recovers the
+Young-Laplace law, corresponding to soap lms that are
+arcs of circles.
+Throughout the implementation of the model, lm
+segments adjacent to the three-fold vertex points are
+held an angle of2
+3radians relative to each other, in
+accordance with Plateau's rules for a soap froth. De-tails on the numerics of this calculation are best found
+in [11]. It should be noted that for high rates of strain,
+one would expect surface tensions in the soap lms to
+vary to the point that this equilibrium condition would
+no longer apply (for example, because of the Marangoni
+eect). At least for lower rates of strain, the2
+3rule is
+reasonable.
+The VF model may be conveniently incorporated into
+a Surface Evolver [4] script (as pioneered by Cox [7]),
+thus allowing for the use of various SE features. The
+procedure for performing (T1) topological changes in the
+Surface Evolver is as follows, and is illustrated in Fig. 1.
+The distance along the lm between neighbouring three-
+fold vertex points is calculated at each timestep. When
+this lm length becomes smaller than a predened criti-
+cal cut-o length, lcthen it is deleted using the Evolver's
+`edgeweed' command. A four-fold vertex is temporarily
+formed to maintain the topology of neighbouring cells
+(see Fig. 1(b)). The Evolver's `pop' command is then
+employed, which scans the foam for vertices which do
+not have a legal topology and replaces the four-fold ver-
+tex with a proper local topology. This results in a new
+lm of eectively negligible length oriented in the per-
+pendicular direction to the old lm (see Fig. 1(c)).
+Further details on the implementation of the VF Model
+can be found in the papers by Kern et al. [21] and Green
+et al. [11].6 Joseph D. Barry et al.
+3 Sample Creation
+A semi-periodic monodisperse sample is created using
+the standard method outlined in the Surface Evolver
+documentation (which is supplied with the software pack-
+age). Disordered semi-periodic samples are created by
+the following process (illustrated in Fig. 3).
+Points are placed at random in the unit cell using
+a uniform distribution to determine the x and y posi-
+tions; see Fig. 3(a). New points are added to the box if
+they are more than a predened minimum distance rmin
+from any other point. The process is continued until the
+desired number of points have been successfully placed.
+A lower value of rminresults in more polydisperse sam-
+ples. These points are translated to boxes to the left
+and right, and re
ected (see Fig. 3(a)) through the lines
+y= 0 andy= 1 to boxes above and below, based on
+the method of De Fabritiis and Coveney [10] (as indi-
+cated by the background triangles in Fig. 3(a)). With
+the software package Qhull [1], the Voronoi Diagram (a
+particular way of tessellating the plane into regions of
+convex polygons) of these points is calculated; see Fig.
+3(b). This is then passed into the Surface Evolver. The
+box in the centre is isolated (see Fig. 3(c)) and keeping
+the areas of each of the cells xed, the Surface Evolver
+performs line minimization on the structure. The result-
+ing structure is our nal two-dimensional half-periodic
+(i.e. periodic in the x-direction only) foam data le; see
+Fig. 3(d). Of interest here (as a result of the re
ection)is that the straight line boundaries at y= 0 andy= 1
+naturally occur as a result of this process.
+4 Simulation Details and Results
+Using the above methods for foam sample creation, we
+create one monodisperse foam sample and ve foam sam-
+ples of polydispersity 2(A) = 0:130:002, where the
+measure of polydispersity is dened as the second mo-
+ment of the area distribution,
+2(A) =
+1A
+A2
+: (3)
+HereAdenotes the area of a bubble, and Athe mean
+bubble area.
+The foam samples consist of Nb= 100 bubbles in
+a square unit cell of area 1; it is too computationally
+expensive to run larger samples in a VF simulation. In
+our dimensionless simulation units, our system size L=
+1 and mean bubble area A= 0:01. We dene a new
+length scale A1=2, the square root of the mean bubble
+area. In these new units, L= 10 A1=2. The width Wlof
+one layer of bubbles in our square sample is given by
+Wl=L=p
+Nb=A1=2(4)
+We proceed to move the top boundary in the posi-
+tive x-direction with velocity Vby incrementally moving
+vertices at y= 1 a distance Vdt per timestep dt. The
+VF algorithm, as outlined in Sec. 2 is used to determine
+the dynamics of the foam during each timestep. Typical
+values for the displacement of the shearing boundary perShear localisation with 2D Viscous Froth and its relation to the Continuum Model 7
+Fig. 3 The creation of a semi-periodic polydisperse two-dimensional foam sample as required for our simple shear simulations.
+a) Points are placed in the central unit cell and translated/re
ected into adjacent boxes (as indicated by the background
+triangles). b) The Voronoi Diagram of these points is calculated. c) The central area is isolated and made into a half-periodic
+(in x-direction) data-le. d) Keeping cell areas constant, the Surface Evolver performs line minimisation on the structure. We
+refer to this shown structure as Sample 1 later in the text.
+timestep are in the range 106A1=2Vdt103A1=2
+(depending on what values of Vandare used). No-slip
+boundary conditions are maintained by xing vertices ly-
+ing on the boundaries while the VF algorithm is being
+implemented. Our boundary conditions are thus8
+>><
+>>:v(L) =V
+v(0) = 0(5)
+Multiple simulations are run for dierent values of
+V(wall drag coecient times boundary velocity) with
+a xed value of surface tension 
. To see why this is
+the appropriate parameter to look at, consider again the
+equation of motion for the VF Model, as given by eq. 2.8 Joseph D. Barry et al.
+By settingv?=V^v?, whereVis the boundary velocity
+and ^v?is our rescaled dimensionless velocity, we can
+rewrite our equation of motion as
+(V)^v?(s) =
P
K(s) (6)
+It is clear that, given any initial state conguration, its
+development in time is determined by V. Furthermore,
+we see evidence of this Vdependence if we rewrite the
+Herschel-Bulkley relation (see eq. 1) in terms of our VF
+parameters. As stress in 2D has dimensions of force per
+length, on dimensional grounds, we see that
+=y+ ^cv
1aAa1=2La(V)a(7)
+where the 2D surface tension 
has dimensions of force,
+Vhas dimensions of force per length and ^ cvis a dimen-
+sionless parameter of order unity which may be related
+to2(A). In this derivation, we dene the strain rate
+term of eq. 1 as the nominal shear rate of the system,
+_=V=L.
+To calculate 
ow proles, bubble centre positions are
+determined. We subsequently divide our foam into bins
+of widthWland calculate the average velocity of bubbles
+centres in each bin over time. A sketch of our simulation
+setup is illustrated in Fig. 4 (polydisperse sample).
+Fig. 5 shows examples of averaged steady state ve-
+locity proles. We say that a simulation has reached
+a steady state once there is no longer any appreciable
+change in our velocity prole in time. Typically, we av-
+Fig. 4 (a) A polydisperse foam consisting of 100 bubbles
+(Sample 5) in equilibrium. Ldenotes our system size. (b)
+The same foam being sheared at a velocity V.
+erage our steady state velocity proles over the range
+110, where the imposed strain is dened as
+=
x=L and
xis equal to the total displacement of
+the moving boundary. Note that there is a clear change in
+the 
ow proles as we vary V. We nd that localisation
+occurs close to the moving boundary in all but two of our
+simulations. (In one of these cases, for V= 0:01
A1=2,
+localisation switches to the stationary boundary, while in
+the second case, for V= 0:005
A1=2, a shear band oc-
+curs in the centre of the sample away from either bound-
+ary (data not shown).) Localisation of 
ow can also be
+made visible by plotting the positions at which T1 topo-
+logical changes occur in our samples, as done by [33]. An
+example is shown in Fig. 6.
+At this stage, it is unclear what the form of the ve-
+locity proles is. We have attempted to use exponential
+ts and ts from the general Continuum Model [28] in
+the data tting process but this approach does not yield
+consistently good ts to our velocity proles which areShear localisation with 2D Viscous Froth and its relation to the Continuum Model 9
+0 2 4 6 8 10
+Distance from moving boundary [A1/2]00.20.40.60.81Average bubble velocityλV = 0.01 γ / Α1/2
+λV = 0.025 γ / Α1/2
+λV = 0.05 γ / Α1/2
+λV = 0.1 γ / Α1/2
+Fig. 5 Examples of velocity proles for dierent values of
+V. Proles shown are for Sample 1. The corresponding lo-
+calisation lengths for these proles are denoted by lled cir-
+cles in Fig. 7.
+0 2 4 6 8 10
+Strain0246810y location of T1 [A1/2]
+Fig. 6 The location of T1 topological changes as a function
+of strain for a polydisperse sample. After an initial transient,
+where T1s happen everywhere in the foam, the 
ow localises
+and T1s are found to occur mostly next to the moving bound-
+ary (aty= 10 A1=2). Shown data is for Sample 1 where
+V= 0:05
=A1=2
+.clearly quite noisy, presumably due to the small system
+size. To obtain a measure of the width of the 
owing
+region from these noisy proles, we use the following
+denition of localisation length, denoted by lint[27]
+lint=1
+VZL
+0v(y)dy : (8)
+0 0.05 0.1 0.15 0.2 0.25 0.3
+λV [γ / A1/2]0246810Localisation Length lint[A1/2] Sample 1
+Sample 2
+Sample 3
+Sample 4
+Sample 5
+Fig. 7 Localisation lengths for a range of V. Each symbol
+represents one simulation run. For low V, there is a wide
+range of lengths, while for high V, only the rst layer of
+bubbles 
ows. Filled symbols indicate simulations where V
+is xed and is varied. Open symbols indicate simulations
+whereis xed and Vis varied.
+This integral, which has the required dimensions of
+length, is calculated numerically for each of our velocity
+proles using the Trapezoidal Rule. Fig. 7 shows a varia-
+tion of localisation length with V. Note that for low V
+we nd large scatter in the localisation lengths, however,
+this scatter decreases as Vis increased. For high V,
+the length converges towards the minimum localisation
+length,lmin=A1=2, the width of one bubble layer (see10 Joseph D. Barry et al.
+eq. 4). This is because the rst layer of bubbles always
+
ows.
+The ratio of the intrinsic timescale of the VF Model
+to the external timescale (as imposed by the nominal
+shear rate _=V=L), otherwise known as the Deborah
+numberDe, is given by
+De=(V)A
+
L; (9)
+as dened in [21]. A small Deborah number ( De1)
+indicates that the foam has enough time available to re-
+equilibrate, even as the applied shear attempts to bring
+the foam out of equilibrium (and vice versa for large De).
+In our simulations, 0 :001De0:03. We therefore
+conclude that we are close to the quasi-static regime in
+all of the discussed simulations.
+While not shown here, similar simulation runs have
+been performed for a monodisperse foam ( 2(A) = 0).
+The localisation length is found to be independent of V
+and is determined to be l=A1=2(the same as lminin
+our polydisperse simulations).
+Our simulation results are broadly consistent with
+the ndings of Katgert et al. [19, 20], where the localisa-
+tion length is found to decrease with increasing V, and
+rate independence of localisation length is found in the
+monodisperse case.
+We wish to gain an understanding of these VF sim-
+ulation results by attempting to capture the observed
+behaviour by a continuum model. Such a model must
+include a constitutive relation which relates the local(wall) drag force of the VF model to an averaged drag
+force in the continuum description. However, this would
+not be enough to explain the observed simulation results,
+as (according to the general Continuum Model [28]) it
+would result in a zero localisation length in all cases.
+Therefore, results suggest that internal dissipation (rep-
+resented by the shear rate term in the HB relation; see
+eq. 1) should also be included, although it is not clear
+how this dissipation arises in the VF simulations. We
+will also appeal to the idea of the existence of a stress
+overshoot in order to explain the variation of localisation
+length at low V.
+5 The Continuum Model
+Up until now, we have discussed microscopic (bubble
+scale) models of 2D foam rheology. An alternative way
+of describing a foam is to treat it as a continuum. The
+generalised Continuum Model [28, 30] (for steady shear)
+combines the Herschel-Bulkley constitutive relation (see
+eq. 1) with the following expression for the variation of
+(wall) drag force Fdper unit area as a function of local
+velocityv
+Fd=cdvb; (10)
+wherecdis the drag coecient and bis the drag exponent
+(the Bretherton law gives b=2
+3[5]). These two expres-
+sions can be related by a force balance, which leads to
+the following dierential equation [14, 30]Shear localisation with 2D Viscous Froth and its relation to the Continuum Model 11
+d
+dydv(y)
+dya
+=cd
+cvv(y)b; (11)
+which can be solved using the boundary conditions v(0) =
+Vandv(L) = 0. These are equivalent to the boundary
+conditions imposed in our VF simulations; see eq. 5 (al-
+beit that here the distance yis measured downward from
+the shearing boundary).
+Upon inspection, it is clear that that a velocity prole
+of the following form satises eq. 11, and exhibits 
ow
+localisation:
+v(y) =V(1y=y 0)n(12)
+wherey0andnmay be obtained by inserting eq. 12 into
+eq. 11 and equating prefactors and exponents [27]. This
+gives
+y0=1 +a
+aba(1 +b)cvVab
+(1 +a)cd 1
+1+a
+(13)
+and
+n=1 +a
+ab: (14)
+Eq. 12 clearly satises the rst boundary condition,
+v(0) =Vof eq. 11. The second boundary condition is
+satised if we take our sample size L!1 [27]. This
+approach is valid so long as the size of the sample is
+much greater than the localisation length ( Ll) which
+may be dened as
+l=V
+dv(0)
+dy; (15)an alternative denition to that of the previous section;
+see eq. 8. Inserting eq. 12 into eq. 15 leads to the follow-
+ing expression for localisation length as a function of the
+shearing velocity V,
+l=a(1 +b)cv
+(1 +a)cd 1
+1+a
+Vab
+1+a: (16)
+Its relation to the denition of localisation length lint
+(see eq. 8) is
+lint=l= (1 +a)=(1 + 2ab); (17)
+as given by [27]. For a < b (as is the case for our VF
+simulations; see Sec. 6), localisation length therefore de-
+creases as the shearing velocity Visincreased .
+The possibility of having a range of localisation lengths
+at lowV(as in Fig. 7) can be accounted for by extending
+the Continuum Model to incorporate what we will refer
+to as a stress overshoot. This we will now proceed to do.
+In a recent paper [28], Weaire et al. introduced the
+idea of distinct yield yand limitlstresses as a possi-
+ble mechanism for localisation in the absence of viscous
+drag. An illustration of the typical stress vs strain pic-
+ture is shown in Fig. 8. The constitutive stress relation
+thus becomes
+=l+cv_a(18)
+whereldenotes the limit stress. When the magnitude of
+the stress overshoot, 
=ylis set to zero we recover
+the original Herschel-Bulkley relation (see eq. 1).12 Joseph D. Barry et al.
+Fig. 8 An illustration of a stress vs strain relation incorpo-
+rating the idea of distinct yield and limit stresses, denoted
+byyandlrespectively. The lled circles indicate that the
+foam can co-exist at the same stress at the boundary between
+
owing and non-
owing regions.
+If shear localisation is present in a foam, there exists
+at least one point yBwhich lies on the boundary between
+
owing and stationary regions. In our VF simulations,
+this corresponds to the point at which the velocity prole
+intercepts the v= 0 axis (see, for example Fig. 5). At
+this point, the system can co-exist at the same value of
+stress in both static and 
owing regions, as indicated by
+the lled dots in Fig. 8. The stress at yBcan take on any
+value between landyas the foam is sheared. From
+eq. 18 we see that this leads to the inequality
+0cv_(yB)a
 : (19)
+AsVis increased, on average we expect the viscous
+stresscv_(y)abetween 0 and yBto cause the stress in
+the 
owing region to lie closer to yso that the stress
+overshoot is less evident. However, at low V, the eect
+is obvious (see Fig. 10) and may have important eects.The dierential equation given by eq. 11 may be
+solved numerically, yielding velocity prole solutions of
+the kind we envisage, which are valid if they satisfy the
+inequality given by eq. 19. As the local strain rate is
+dened as
+_(y) =dv(y)
+dy; (20)
+in this case [27], the quantity _ (yB) can be directly mea-
+sured from the calculated velocity proles, provided they
+intersect the v= 0 axis at some nite value.
+The results of these calculations can be seen in Fig.
+9, where we have solved the model numerically for the
+valuesa= 0:5,b=cd=cv=
= 1. The upper bound
+l+(V) corresponds to where the shear stress (yB) =l,
+where the viscous stress cv_(yB)a= 0 and the ana-
+lytic solution for localisation length given by eq. 16. The
+lower bound l(V) corresponds to where the shear stress
+(yB) =yand where the viscous stress cv_(yB)a=
.
+Thus, for a given V,l(V)l(V)l+(V) gives the
+range of allowed solutions, indicated by the shaded re-
+gion in Fig. 9.
+We note that Fig. 9 is qualitatively similar to Fig.
+7, with a large range of possible localisation lengths at
+lowVand convergent behaviour at high V. Remarkably,
+the model predicts that for low V, the localisation length
+can take any value 0 <l<1. This prediction, of course,
+holds only in the presence of viscous drag.
+An important question to be answered is how does
+one dene the critical velocity Vcbelow which the foam
+can take on a wide range of localisation lengths? If oneShear localisation with 2D Viscous Froth and its relation to the Continuum Model 13
+0 1 2 3 4 5
+V00.511.52Localisation Lengtha=0.5 , b=1 , cd=1, cv=1 , ∆=1
+Vcl+(V)
+l-(V)
+Fig. 9 Results from a numerical solution of the Continuum
+Model incorporating the inequality given by eq. 19. For low
+V, there is a large range of possibilities for localisation length,
+while for high V, the range of allowed lengths converges to a
+narrow band. Vcis our critical velocity below which a large
+range of localisation lengths is possible; see eq. 22. l+(V)
+andl(V) denote the upper and lower bounds to the range
+of allowable solutions, respectively.
+assumes that as V!0, the velocity prole becomes
+approximately linear, then _ =V
+l, wherelis the local-
+isation length. If we are on the lower bound, from eq.
+19 we see that 
=cvV
+l
a, or expressing it in a more
+convenient form,
+l=cv
+
1
+aV : (21)
+We are interested in the point where this line inter-
+sects the upper bound l+(V), which is given by eq. 16.
+We solve this pair of simultaneous equations (eq. 16, 21)
+in terms of Vand choose to dene the point of intersec-
+tion as our critical velocity Vc(see Fig. 9). This yieldsVc=
a+1
+a(1+b)a(1 +b)
+1 +a 1
+1+b1
+cd(cv)1
+a 1
+1+b
+:(22)
+To make a more quantitative comparison between
+continuum theory and the VF results presented in Sec.
+4, a more detailed study of the relationship between the
+parameters of both models is required. We present such
+a study in the next section.
+In one of the earliest publications on this subject,
+Kabla & Debregeas [17] attribute shear localisation in
+quasi-statics to what they call `self amplication'. This
+idea is qualitatively the same as the ideas presented in
+this section. This approach may have the capacity to
+explain other results in the literature, particularly [26]
+where shearing experiments are performed for an ordi-
+nary Bragg raft (where there are no conning plates)
+in a straight geometry. In these experiments (as in our
+VF simulations) variations in the averaged velocity pro-
+les are observed between experiments but averages over
+several experiments converge much better.
+6 Relating the Continuum Model to the Viscous
+Froth Model
+We now proceed to relate the parameters of the (mi-
+croscopic) VF Model and the (macroscopic) Continuum
+Model. This is done using a combination of numerical
+and analytic approximations.
+To demonstrate preliminary evidence of existence of
+the stress overshoot in simulation, we have performed14 Joseph D. Barry et al.
+quasi-static calculations using the Surface Evolver (of
+the type mentioned in in Sec. 1) for 29 foam samples of
+disorder2(A) = 0:130:03 withNb= 100 bubbles.
+This eectively sets the viscous stress cv_(y)ato zero,
+thereby allowing us to obtain an accurate estimate of the
+magnitude of the stress overshoot, 
. The foam samples
+are created using the process outlined in Sec. 3. The
+shear stress xy(dened in [22]) is recorded for each
+simulation and subsequently averaged; see Fig. 10.
+As there is localisation in these simulations (at either
+the moving or stationary boundary) which aects our
+stress measurements, the limit stress lreported here
+must be treated as an approximate measurement. The
+value of the yield stress y, however, is exact, as up to
+a strain of unity, the foam is in the elastic regime and
+the bubble motion has not yet localised. We measure the
+magnitude of the stress overshoot to be 
= 0:1
=A1=2,
+which corresponds to a 17% overshoot. In the calculation
+shown in Fig. 12, a 20% overshoot is used.
+The drag force per unit area for the Continuum Model
+is given by eq. 10 and acts in the direction of shear. We
+wish to relate this to the the drag force of the VF Model,
+v?, which is a force per length and acts in the normal
+direction to a soap lm (see Fig. 2). Trivially, the drag
+exponent,b= 1. The numerical prefactor cdmay be
+calculated analytically for a 2D hexagonal honeycomb
+structure, which serves as a reasonable approximation.
+We also take into account the direction in which the drag
+0 2 4 6 8 10
+Strain00.20.40.6Stress [γ / A1/2]µ2(Α)=0.13
+∆ = 0.1σy
+σlFig. 10 Averaged shear stress data for 29 foam samples of
+disorder2(A) = 0:130:03. The stress overshoot is clearly
+evident. The yield stress yis taken to be the maximum stress
+value, which occurs at a strain of unity. The limit stress l
+is the stress average from a strain of 2 to 10. Calculation
+performed using quasi-static simulations.
+force is dened and the orientation of soap lms in the
+foam.
+The drag force per unit area must be proportional to
+the total length of the soap lms in that area. For the
+honeycomb, this yields
+cd/s
+2p
+3
+A: (23)
+In the VF simulations, it is observed that bubbles
+move on average only in the direction of shear. The mag-
+nitude of this `apparent' velocity is denoted by vappin
+Fig. 11. However, the drag force for the VF model by
+denition points in the direction of the normal to a soap
+lm, and so we project vappin this direction (see Fig.
+11(i)). This results in jv?j=jvappjcos, whereisShear localisation with 2D Viscous Froth and its relation to the Continuum Model 15
+the relative angle between the normal vector to the soap
+lm and the shear direction. To relate the normal drag
+force to the actual drag force of the Continuum Model,
+we need to project v?in the shear direction (see Fig.
+11(ii)), resulting in jvj=jv?jcos=jvappj(cos)2.
+θ|vapp||v
+Т|=|vapp|cosθ
+|v|=|v
+Т
+|cos θsoap /f_ilm
+(i)
+(ii)
+Direction of shear 
+Fig. 11 Two projections are necessary to relate the veloc-
+ityvappof a soap lm segment in the VF model to the local
+velocity vof the Continuum Model: (i) projection of the av-
+erage velocity of a soap lm segment vappin the direction of
+the normal to that segment, and (ii) projection of the normal
+velocity of the soap lm segment v?back in the direction of
+shear.is the angle between the normal vector to the soap
+lm segment and the shear direction. It is the magnitude of
+the vectors that is displayed in the gure.
+Finally, we consider how the orientation of the soap
+lms in our foam might aect the drag force. We assume
+that the foam is isotropic and proceed to average over
+all possible values of . As<(cos)2>= 1=2, our nal
+expression for the continuum drag force coecient cdis
+cd= ^cd=1
+2s
+2p
+3
+A ; (24)giving the (continuum) drag force the required dimen-
+sions of force per area.
+In Sec. 4, we showed how the viscous stress has a
+Vdependence using dimensional arguments (see eq. 7).
+Using these arguments, but rather dening the strain
+rate as a locally changing quantity (see eq. 20), we see
+that
+cv= ^cv
1aAa1=2a(25)
+where the Herschel-Bulkley exponent aand the dimen-
+sionless quantity ^ cvare free parameters.
+Using all of the approximations calculated in this sec-
+tion, we proceed to solve eq. 11 numerically, accepting
+solutions only if they obey the inequality given by eq.
+19, as done in Sec. 5. The key dierence here is that
+localisation length is a function of the product V.
+The upper bound for the Continuum Model predic-
+tion is formulated in terms of Vby inserting eq. 24 and
+eq. 25 into eq. 16, resulting in
+l+(V) =2a^cv1aAa1=2
+(1 +a) ^cd1
+1+a
+(V)a1
+1+a:(26)
+The corresponding lower bound must be found numeri-
+cally. To simplify this calculation, we x and allowV
+to vary.
+A comparison of the VF and Continuum Model re-
+sults can be seen in Fig. 12, where values of a= 0:3
+and ^cv= 0:26 are chosen as they give a reasonable
+prediction for both upper and lower bounds (although16 Joseph D. Barry et al.
+0:2<a< 0:4 gives a reasonable t to the upper bound).
+The shaded region between these bounds indicates the
+range of all allowable localisation lengths, as predicted
+by the Continuum Model. Filled and open symbols rep-
+resent the VF simulation results, which are the same as
+in Fig. 7, only with the minimum localisation length of
+lmin=A1=2subtracted to coincide with the Continuum
+Model predictions which give l= 0 forV!1 .
+0 0.05 0.1 0.15 0.2 0.25 0.3
+λV [γ / A1/2]02468Localisation Length (lint-lmin) [A1/2]
+upper, lower bounds
+Sample 1
+Sample 2
+Sample 3 
+Sample 4 
+Sample 5
+(λV)c = 0.056HB exponent a = 0.3
+Fig. 12 A comparison of the VF simulation results (open
+and lled symbols) and the prediction for the range of al-
+lowed localisation lengths as given by the Continuum Model
+(shaded region). A Herschel-Bulkley exponent of a= 0:3 and
+a stress overshoot of 20 % is found to give a good t to the
+data. The critical cross-over point, ( V)c, as given by eq.
+27 indicates the point below which the system yields a wide
+range of localisation lengths.
+The denition for the critical cross-over point, as
+given by eq. 22 may also be formulated in terms of V.
+This is achieved by inserting eq. 25 into eq. 21 and nd-
+ing the point at which this line intersects eq. 26. Alter-natively, one may insert eq. 24 and eq. 25 into eq. 22.
+This gives
+(V)c=s
+2a
1+a
+a( ^cv1aAa1=2)1=a
+(1 +a) ^cd; (27)
+which is illustrated by the dotted lines in Fig. 12. The
+calculated critical cross-over point ( V)c= 0:056
=A1=2
+fullls its promise of oering a reasonable estimate of
+the point below which the system yields a wide range of
+localisation lengths.
+While the comparison between the VF Model and the
+Continuum Model presented in this section gives a fasci-
+nating theoretical explanation for the simulation results
+discussed, its details are far from precise. The location
+of the upper bound in Fig. 12 is simply an estimate,
+and further simulations may be needed to determine its
+exact location. In addition, many approximations were
+taken in relating the parameters of the two models. How-
+ever, it is remarkable that despite these approximations,
+a robust prediction can still me made.
+7 Outlook
+The apparent agreement of the simulation results in this
+paper with published experimental work suggests that
+the 2D VF Model may have further potential for describ-
+ing realistic foam dynamics. For more detailed studies to
+be conducted, however, the VF algorithm will need to be
+improved to decrease the required computation time for
+these types of simulations. Issues that we will address in-
+clude the eect of 2(A) on localisation with the 2D VFShear localisation with 2D Viscous Froth and its relation to the Continuum Model 17
+Model and on the value of the HB exponent. In addition,
+the dependence of the magnitude of the stress overshoot
+
on2(A) will be investigated as it is critical to our
+understanding of its role as a mechanism for shear local-
+isation. It will be of interest to observe what happens to
+the location of the shear-band for samples with higher
+2(A), in light of the results published in [33]. We also
+intend to compare our VF simulations with simulations
+using the Soft Disk Model [24].
+8 Acknowledgements
+The author would like to acknowledge IRCSET Embark
+for funding this project. IITAC, the HEA, the National
+Development Plan and the Trinity Centre for High Per-
+formance Computing are acknowledged for the use of the
+computing facilities at TCD. S.J. Cox, Aberystwyth is
+thanked for his useful input and correspondence in rela-
+tion to this work. This publication has emerged from re-
+search conducted with the nancial support of the Euro-
+pean Space Agency (MAP AO-99-108:C14914/02/NL/SH
+and AO-99-075:C14308/00/NL/SH). This material is based
+upon works supported by the Science Foundation Ire-
+land under Grant No. (08/RFP/MTR1083 and STTF
+08). We would also like to thank the anonymous referees
+whose input helped greatly in improving the quality of
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+arXiv:1001.0875v1  [math.MG]  6 Jan 2010Approximately gaussian marginals andthe
+hyperplane conjecture
+R. Eldan and B.Klartag∗
+Abstract
+We discuss connections between certain well-known open pro blems re-
+lated to the uniform measure on ahigh-dimensional convex bo dy. In partic-
+ular, we show that the “thin shell conjecture” implies the “h yperplane con-
+jecture”. This extends a result by K. Ball, according to whic h the stronger
+“spectral gap conjecture” implies the “hyperplane conject ure”.
+1 Introduction
+Little is currently known about the uniform measure on a gene ral high-
+dimensional convex body. Many aspects of the Euclidean ball or the unit
+cube are easy to analyze, yet it is difficult to answer even som e of the sim-
+plest questions regarding arbitrary convex bodies, lackin g symmetries and
+structure. Forexample,
+Question 1.1 Is there a universal constant c >0such that for any dimen-
+sionnand a convex body K⊂RnwithVoln(K) = 1, there exists a
+hyperplane H⊂Rnfor which Voln−1(K∩H)> c?
+Here, of course, Volkstands for k-dimensional volume. A convex body
+is a bounded, open convex set. Question 1.1 is referred to as t he “slicing
+problem” or the“hyperplane conjecture”, and wasraised byB ourgain [5,6]
+inrelation tothemaximal function inhighdimensions. Itwa sdemonstrated
+by Ball [2] that Question 1.1 and similar questions are most n aturally for-
+mulated in the broader class of logarithmically concave den sities.
+∗The authors were supported in part by the Israel Science Foun dation and by a Marie Curie
+ReintegrationGrantfromtheCommissionoftheEuropeanCom munities.
+1A probability density ρ:Rn→[0,∞)is calledlog-concave if it takes
+the form ρ= exp(−H)for a convex function H:Rn→R∪ {∞}. A
+probability measure is log-concave if it has a log-concave d ensity. The uni-
+form probability measure on a convex body is an example of a lo g-concave
+probability measure, as well as the standard gaussian measu re onRn. A
+log-concave probability density decays exponentially at i nfinity (e.g., [17,
+Lemma2.1]),andthushasmomentsofallorders. Foraprobabi lity measure
+µonRnwith finite second moments, we consider its barycenter b(µ)∈Rn
+and covariance matrix Cov(µ)defined by
+b(µ) =/integraldisplay
+Rnxdµ(x), Cov (µ) =/integraldisplay
+Rn(x−b(µ))⊗(x−b(µ))dµ(x)
+where for x∈Rnwe write x⊗xfor then×nmatrix(xixj)i,j=1,...,n. A
+log-concave probability measure µonRnisisotropicifitsbarycenter liesat
+the origin and its covariance matrix is the identity matrix. For an isotropic,
+log-concave probability measure µonRnwedenote
+Lµ=Lf=f(0)1/n
+wherefis the log-concave density of µ. It is well-known (see, e.g., [17,
+Lemma3.1]) that Lf> c,for some universal constant c >0. Define
+Ln= sup
+µLµ
+where the supremum runs over all isotropic, log-concave pro bability mea-
+suresµonRn. AsfollowsfromtheworksofBall[2],Bourgain[5],Fradeli zi
+[11], Hensley [12] and Milman and Pajor [20], Question 1.1 is directly
+equivalent tothe following:
+Question 1.2 Is it true that supnLn<∞?
+Seealso MilmanandPajor [20] andthesecond author’s paper [ 16]for a
+surveyofresultsrevolvingaroundthisquestion. Foraconv exbodyK⊂Rn
+we write µKfor the uniform probability measure on K. A convex body
+K⊂Rniscentrally-symmetric if K=−K. It isknown that
+Ln≤Csup
+K⊂RnLµK (1)
+where the supremum runs over all centrally-symmetric conve x bodiesK⊂
+Rnfor which µKis isotropic, and C >0is a universal constant. In-
+deed, the reduction from log-concave distributions to conv ex bodies was
+2provenbyBall[2](see[16]forthestraightforwardgeneral izationtothenon-
+symmetriccase), andthereduction fromgeneral convex bodi estocentrally-
+symmetric ones was outlined, e.g., in the last paragraph of [ 15]. The best
+estimate known to date is Ln< Cn1/4for a universal constant C >0(see
+[16]), which slightly sharpens an earlier estimate by Bourg ain [7,8,9].
+Our goal in this note is to establish a connection between the slicing
+problem and another open problem in high-dimensional conve x geometry.
+Write| · |for the standard Euclidean norm in Rn, and denote by x·ythe
+scalar product of x,y∈Rn. We say that a random vector XinRnis
+isotropic and log-concave if it is distributed according to an isotropic, log-
+concave probability measure. Let σn≥0satisfy
+σ2
+n= sup
+XE(|X|−√n)2(2)
+wherethe supremum runs over all isotropic, log-concave ran dom vectors X
+inRn. The parameter σnmeasures the width of the “thin spherical shell”
+of radius√nin which most of the mass of Xis located. See (5) below
+for another definition of σn, equivalent up to a universal constant, which is
+perhapsmorecommonintheliterature. Itisknownthat σn≤Cn0.41where
+C >0is a universal constant (see [19]), and it is suggested in the works
+of Anttila, Ball and Perissinaki [1] and of Bobkov and Koldob sky [4] that
+perhaps
+σn≤C (3)
+for a universal constant C >0. Again, up to a universal constant, one may
+restrict attention in (2) to random vectors that are distrib uted uniformly in
+centrally-symmetric convex bodies. This essentially foll ows from the same
+technique asin the case of the parameter Lnmentioned above.
+The importance of the parameter σnstems from the central limit the-
+orem for convex bodies [18]. This theorem asserts that most o f the one-
+dimensional marginals of an isotropic, log-concave random vector are ap-
+proximately gaussian. The Kolmogorov distance to the stand ard gaussian
+distribution of a typical marginal has roughly the order of m agnitude of
+σn/√n. Therefore, the conjectured bound (3) actually concerns th e qual-
+ityof the gaussian approximation tothe marginals of high-d imensional log-
+concave measures. Our main result reads as follows:
+Inequality1.1 Foranyn≥1,
+Ln≤Cσn (4)
+whereC >0isa universal constant.
+3Inequality 1.1 states, in particular, that an affirmative an swer to the slic-
+ing problem follows from the thin shell conjecture (3). This sharpens a
+result announced by Ball [3], according to which a positive a nswer to the
+slicing problem is implied by the stronger conjecture sugge sted by Kan-
+nan, Lov´ asz and Simonovits [13]. The quick argument leadin g from the
+latter conjecture to (3) is explained in Bobkov and Koldobsk y [4]. Write
+Sn−1={x∈Rn;|x|= 1}for theunit sphere, and denote
+σn=1√nsup
+X/vextendsingle/vextendsingleEX|X|2/vextendsingle/vextendsingle=1√nsup
+Xsup
+θ∈Sn−1E(X·θ)|X|2,
+wherethe supremum runs over all isotropic, log-concave ran dom vectors X
+inRn.
+Lemma1.3 Foranyn≥1,
+σ2
+n≤1
+nsup
+XE(|X|2−n)2≤Cσ2
+n, (5)
+wherethesupremum runsoverall isotropic, log-concave ran dom vectors X
+inRn. Furthermore,
+1≤σn≤Cσn≤˜Cn0.41.
+Here,C,˜C >0are universal constants.
+Inequality 1.1 maybe sharpened, inview of Lemma1.3,to theb ound
+Ln≤Cσn,
+for a universal constant C >0. This is explained in the proof of Inequality
+1.1 in Section 3. Our argument involves a certain Riemannian structure,
+which ispresented inSection 2.
+Asthereaderhasprobablyalreadyguessed,weusetheletter sc,˜c,c′,C,˜C,C′
+to denote positive universal constants, whose value is not n ecessarily the
+sameindifferentappearances. Furthernotationandfactst obeusedthrough-
+out the text: The support Supp(µ)of a Borel measure µonRnis the min-
+imal closed set of full measure. When µis log-concave, its support is a
+convexset. ForaBorelmeasure µonRnandaBorelmap T:Rn→Rkwe
+define the push-forward of µunderTto be the measure ν=T∗(µ)onRk
+with
+ν(A) =µ(T−1(A))for any Borel set A⊂Rk.
+Note that for any log-concave probability measure µonRn, there exists an
+invertible affine map T:Rn→Rnsuch that T∗(µ)is isotropic. When
+4Tis a linear function and k < n, we say that T∗(µ)is a marginal of µ.
+ThePr´ ekopa-Leindler inequality impliesthat anymargina l ofalog-concave
+probability measure is itself a log-concave probability me asure. The Eu-
+clidean unit ball is denoted by Bn
+2={x∈Rn;|x| ≤1}, and its volume
+satisfiesc√n≤Voln(Bn
+2)1/n≤C√n.
+We write ∇ϕfor the gradient of the function ϕ, and∇2ϕfor the hessian
+matrix. For θ∈Sn−1we write ∂θfor differentiation in direction θ, and
+∂θθ(ϕ) =∂θ(∂θϕ).
+Acknowledgements. Wewould like to thank Daniel Dadush, Vitali Mil-
+man, Leonid Polterovich, Misha Sodin and Boris Tsirelson fo r interesting
+discussions related to this work, and to Shahar Mendelson fo r pointing out
+that there is adifference between extremal points and expos ed points.
+2 ARiemannianmetricassociatedwithacon-
+vexbody
+The main mathematical idea presented in this note is a certai n Riemannian
+metric associated witha convex body K⊂Rn. Our construction isaffinely
+invariant: WeactuallyassociateaRiemannianmetricwitha nyaffineequiva-
+lenceclassofconvexbodies(twoconvexbodiesin Rnareaffinelyequivalent
+if there exists an invertible affine transformation that map s one to the other.
+Thus, all ellipsoids are affinely equivalent).
+Beginbyrecallingthetechniquefrom[16]. Supposethat µisacompactly-
+supported Borel probability measure on Rnwhose support is not contained
+in a hyperplane. Denote by K⊂Rnthe interior of the convex hull of
+Supp(µ), soKis a convex body. The logarithmic Laplace transform ofµ
+is
+Λ(ξ) = Λµ(ξ) = log/integraldisplay
+Rnexp(ξ·x)dµ(x) (ξ∈Rn).(6)
+The function Λis strictly convex and C∞-smooth on Rn. Forξ∈Rn
+letµξbe the probability measure on Rnfor which the density dµξ/dµis
+proportional to x/ma√sto→exp(ξ·x). Differentiating under the integral sign, we
+seethat
+∇Λ(ξ) =b(µξ),∇2Λ(ξ) =Cov(µξ) (ξ∈Rn),
+5whereb(µξ)is the barycenter of the probability measure µξandCov(µξ)
+is the covariance matrix. We learned the following lemma fro m Gromov’s
+work[10]. Aproof isprovided for the reader’s convenience.
+Lemma2.1 Inthe above notation,
+/integraldisplay
+Rndet∇2Λ(ξ)dξ=Voln(K).
+Proof:It is well-known that the open set ∇Λ(Rn) ={∇Λ(ξ);ξ∈Rn}
+isconvex, andthatthemap ξ/ma√sto→ ∇Λ(ξ)isone-to-one (see, e.g.,Rockafellar
+[22,Theorem 26.5]). Denote by Kthe closure of K. Then,
+∇Λ(Rn)⊆K (7)
+since for any ξ∈Rn, the point ∇Λ(ξ)∈Rnis the barycenter of a cer-
+tain probability measure supported on the compact, convex s etK. Next
+we show that ∇Λ(Rn)contains all of the exposed points of Supp(µ). Let
+x0∈Supp(µ)bean exposed point, i.e.,there exists ξ∈Rnsuch that
+ξ·x0> ξ·xfor allx0/ne}ationslash=x∈Supp(µ). (8)
+Weclaim that
+lim
+r→∞∇Λ(rξ) =x0. (9)
+Indeed, (9)followsfrom(8)andfrom thefact that x0belongs tothesupport
+ofµ: Themeasure µrξconvergesweaklytothedeltameasure δx0asr→ ∞,
+hence the barycenter of µrξtends tox0. Therefore x0∈∇Λ(Rn). Any
+exposed point of Kis an exposed point of Supp(µ), and we conclude that
+alloftheexposedpointsof Karecontainedin ∇Λ(Rn). FromStraszewicz’s
+theorem (see, e.g., Schneider [23, Theorem 1.4.7]) and from (7) we deduce
+that
+K=∇Λ(Rn).
+The set∇Λ(Rn)is open and convex, hence necessarily ∇Λ(Rn) =K.
+SinceΛis strictly-convex, its hessian is positive-definite every where, and
+according tothe change of variables formula,
+Voln(K) =Voln(∇Λ(Rn)) =/integraldisplay
+Rndet∇2Λ(ξ)dξ.
+/square
+6Recall that µis any compactly-supported probability measure on Rn
+whosesupportisnotcontainedinahyperplane. Foreach ξ∈Rnthehessian
+matrix∇2Λ(ξ) =Cov(µξ)ispositive definite. For ξ∈Rnset
+g(ξ)(u,v) =gµ(ξ)(u,v) =Cov(µξ)u·v(u,v∈Rn).(10)
+Thengµ(ξ)is a positive-definite bilinear form for any ξ∈Rn, and thus gµ
+isaRiemannian metric on Rn. Wealso set
+Ψµ(ξ) = logdet∇2Λ(ξ)
+det∇2Λ(0)= logdetCov(µξ)
+detCov(µ)(ξ∈Rn).(11)
+We say that Xµ= (Rn,gµ,Ψµ,0)is the “Riemannian package associated
+withthe measure µ”.
+Definition2.2 A “Riemannian package of dimension n” is a quadruple
+X= (U,g,Ψ,x0)whereU⊂Rnis an open set, gis a Riemannian metric
+onU,x0∈UandΨ :U→Ris afunction with Ψ(x0) = 0.
+SupposeX= (U,g,Ψ,x0)andY= (V,h,Φ,y0)areRiemannianpack-
+ages. A map ϕ:U→Vis an isomorphism of XandYif the following
+conditions hold:
+1.ϕisaRiemannianisometrybetweentheRiemannianmanifolds (U,g)
+and(V,h).
+2.ϕ(x0) =y0.
+3.Φ(ϕ(x)) = Ψ(x)for anyx∈U.
+Whensuchanisomorphism existswesaythat XandYareisomorphic, and
+wewriteX∼=Y.
+Letusdescribeanadditional construction ofthesameRiema nnianpack-
+age associated with µ, a construction which is dual to the one mentioned
+above. Consider the Legendre transform
+Λ∗(x) = sup
+ξ∈Rn[ξ·x−Λ(ξ)] (x∈K).
+ThenΛ∗:K→Risastrictly-convex C∞-function, and ∇Λ∗:K→Rnis
+theinversemapof ∇Λ :Rn→K(seeRockafellar [22,ChapterV]).Define
+Φµ(x) = logdet∇2Λ∗(b(µ))
+det∇2Λ∗(x)(x∈K),
+and forx∈Kset
+h(x)(u,v) =hµ(x)(u,v) =/bracketleftbig
+∇2Λ∗/bracketrightbig
+(x)u·v(u,v∈Rn).
+7ThenhµisaRiemannian metric on K. Notethe identity
+/bracketleftbig
+∇2Λ(ξ)/bracketrightbig−1=/bracketleftbig
+∇2Λ∗/bracketrightbig
+(∇Λ(ξ)) ( ξ∈Rn).
+Using this identity, it is a simple exercise to verify that th e Riemannian
+package ˜Xµ= (K,hµ,Φµ,b(µ))is isomorphic to the Riemannian pack-
+ageXµ= (Rn,gµ,Ψµ,0)described earlier, with x=∇Λ(ξ)being the
+isomorphism.
+The constructions Xµand˜Xµare equivalent, and each has advantages
+over the other. It seems that Xµis preferable when carrying out computa-
+tions, as the notation is usually less heavy in this case. On t he other hand,
+the definition ˜Xµis perhaps easier to visualize: Suppose that µis the uni-
+form probability measure on K. In this case ˜Xµequips the convex body
+Kitself with a Riemannian structure. One is thus tempted to im agine, for
+instance, how geodesics look on K, and what is a Brownian motion in the
+bodyKwith respect to this metric. The following lemma shows that t his
+Riemannian structure on Kisinvariant under linear transformations.
+Lemma2.3 Supposeµandνare compactly-supported probability mea-
+sures onRnwhose support is not contained in a hyperplane. Assume that
+there exists alinear map T:Rn→Rnsuch that
+ν=T∗(µ).
+ThenXµ∼=Xν.
+Proof:It is straightforward to check that the linear map Tt(the trans-
+posed matrix) is the required isometry between the Riemanni an manifolds
+(Rn,gν)and(Rn,gµ). However,perhapsabetterwaytounderstandthisiso-
+morphism,istonotethattheconstructionof Xµmaybecarriedoutinamore
+abstract fashion: Suppose that Vis ann-dimensional linear space, denote
+byV∗the dual space, and let µbe a compactly-supported Borel probability
+measure on Vwhosesupport isnot contained inaproper affinesubspace of
+V. The logarithmic Laplace transform Λ :V∗→Ris well-defined, as is
+the family of probability measures µξ(ξ∈V∗)on the space V. Fora point
+ξ∈V∗and twotangent vectors η,ζ∈TξV∗≡V∗, set
+gξ(η,ζ) =/integraldisplay
+Vη(x)ζ(x)dµξ(x)−/parenleftbigg/integraldisplay
+Vη(x)dµξ(x)/parenrightbigg/parenleftbigg/integraldisplay
+Vζ(x)dµξ(x)/parenrightbigg
+.
+(12)
+Amomentofreflectionrevealsthatthedefinition(12)ofthep ositive-definite
+bilinear form gξis equivalent to the definition (10) given above. Addition-
+ally, there exists a linear operator Aξ:V∗→V∗, which is self-adjoint and
+8positive-definite withrespect to thebilinear form g0, that satisfies
+gξ(η,ζ) =g0(Aξη,ζ)for allη,ζ∈V∗.
+Hence we may define Ψ(ξ) = logdet Aξ, which coincides with the defi-
+nition (11) of Ψµabove. Therefore, Xµ= (V∗,g,Ψ,0)is the Riemannian
+packageassociatedwith µ. Backtothelemma,weseethat Xµisconstructed
+from exactly the samedata as Xν,hence they must be isomorphic. /square
+Corollary 2.4 Supposeµandνare compactly-supported probability mea-
+sures onRnwhose support is not contained in a hyperplane. Assume that
+there exists anaffine map T:Rn→Rnsuch that
+ν=T∗(µ).
+ThenXµ∼=Xν.
+Proof:TheonlydifferencefromLemma2.3isthatthemap Tisassumed
+to be affine, and not linear. It is clearly enough to deal with t he case where
+Tis atranslation, i.e.,
+T(x) =x+x0(x∈Rn)
+for acertain vector x0∈Rn. From the definition (6) wesee that
+Λν(ξ) =ξ·x0+Λµ(ξ) (ξ∈Rn).
+Addingalinearfunctionaldoesnotinfluencesecondderivat ives,hence gµ=
+gνand alsoΨµ= Ψν. Therefore Xµ= (Rn,gµ,Ψµ,0)is trivially isomor-
+phic toXν= (Rn,gν,Ψν,0). /square
+Ann-dimensional Riemannian package is of “log-concave type” i f it
+is isomorphic to the Riemannian package Xµassociated with a compactly-
+supported, log-concave probability measure µonRn. Note that accord-
+ing to our terminology, a log-concave probability measure i s absolutely-
+continuous with respect to the Lebesgue measure on Rn, hence its support
+isnever contained in ahyperplane.
+Lemma2.5 SupposeX= (U,g,Ψ,ξ0)is ann-dimensional Riemannian
+package of log-concave type. Let ξ1∈U. Denote
+˜Ψ(ξ) = Ψ(ξ)−Ψ(ξ1) (ξ∈U). (13)
+Then also Y= (U,g,˜Ψ,ξ1)is ann-dimensional Riemannian package of
+log-concave type.
+9Proof:Letµbe a compactly-supported log-concave probability mea-
+sure onRnwhose associated Riemannian package Xµ= (Rn,gµ,Ψµ,0)is
+isomorphic to X. Thanks to the isomorphism, we may identify ξ1with a
+certain point in Rn, which will still be denoted by ξ1(with a slight abuse of
+notation). Wenow interpret the definition (13) as
+˜Ψ(ξ) = Ψ(ξ)−Ψ(ξ1) (ξ∈Rn).
+Inorder toprove the lemma, weneed todemonstrate that
+Y= (Rn,gµ,˜Ψ,ξ1) (14)
+is of log-concave type. Recall that µξ1is the compactly-supported prob-
+ability measure on Rnwhose density with respect to µis proportional to
+x/ma√sto→exp(ξ1·x). A crucial observation is that µξ1is log-concave. Set
+ν=µξ1, and note the relation
+Λν(ξ) = Λµ(ξ+ξ1)−Λµ(ξ1) ( ξ∈Rn).(15)
+It suffices to show that the Riemannian package Yin (14) is isomorphic to
+Xν= (Rn,gν,Ψν,0). Weclaim that an isomorphism ϕbetweenXνandY
+issimply the translation
+ϕ(ξ) =ξ+ξ1(ξ∈Rn).
+Inorder tosee that ϕis indeed anisomorphism, note that (15) yields
+∇2Λν(ξ) =∇2Λµ(ξ+ξ1) ( ξ∈Rn), (16)
+henceϕis a Riemannian isometry between (Rn,gν)and(Rn,gµ), with
+ϕ(0) =ξ1. The relation (16) implies that ˜Ψ(ϕ(ξ)) = Ψ ν(ξ)for allξ∈Rn.
+Henceϕis an isomorphism between Riemannain packages, and the lemm a
+isproven. /square
+Remark. Whenµis a product measure on Rn(such as the uniform
+probability measure on the cube, or the gaussian measure), s traightforward
+computations of curvature show that the manifold (Rn,gµ)is flat (i.e., all
+sectional curvatures vanish). We were not able to extract me aningful infor-
+mation from the local structure of the Riemannian manifold (Rn,gµ)in the
+general case.
+3 Inequalities
+Proof of Lemma 1.3: First, note that for any random vector XinRnwith
+finitefourth moments,
+E(|X|−√n)2≤1
+nE(|X|−√n)2(|X|+√n)2=1
+nE(|X|2−n)2.
+10Thisproves theinequality onthe left in(5). Regarding thei nequality on the
+right, weusethe bound
+E|X|41|X|>C√n≤Cexp/parenleftbig
+−√n/parenrightbig
+(17)
+which follows from Paouris theorem [21]. Here 1|X|>C√nis the random
+variable that equals one when |X|> C√nand vanishes otherwise. Apply
+again the identity |X|2−n= (|X|−√n)(|X|+√n)to conclude that
+E(|X|2−n)2=E(|X|2−n)21|X|≤C√n+E(|X|2−n)21|X|>C√n
+≤(C+1)2nE(|X|−√n)2+E|X|41|X|>C√n, (18)
+whereC≥1is the universal constant from (17). A simple computation
+showsthat σn≥√
+2,asiswitnessedbythestandardgaussianrandomvector
+inRn, or by the example in the next paragraph. Thus the inequality on the
+right in (5) follows from (17) and (18). Our proof of (5) utili zed the deep
+Paouris theorem. Another possibility could be to use [19, Th eorem 4.4] or
+thedeviation inequalities for polynomials proved first byB ourgain [7].
+In order to prove the second assertion in the lemma, observe t hat since
+EX= 0,
+E(X·θ)|X|2=E(X·θ)(|X|2−n)≤/radicalbig
+E(X·θ)2E(|X|2−n)2≤C√nσn,
+where weused the Cauchy-Schwartz inequality, thefact that E(X·θ)2= 1
+and(5). Itremainstoprovethat σn≥2. Tothisend,considerthecasewhere
+Y1,...,Y nare independent random variables, all distributed accordi ng to
+the density t/ma√sto→e−I(t+1)on the real line, where I(a) =afora≥0
+andI(a) = +∞fora <0. ThenY= (Y1,...,Y n)is a random vector
+distributed according to an isotropic, log-concave probab ility measure on
+Rn, and
+E/summationtextn
+j=1Yj√n|Y|2= 2√n.
+Thiscompletes the proof. /square
+Whenϕis a smooth real-valued function on a Riemannian manifold
+(M,g), wedenote itsgradient at thepoint x0∈Mby∇gϕ(x0)∈Tx0(M).
+HereTx0(M)stands for the tangent space to Mat the point x0. The sub-
+scriptgin∇gϕ(x0)means that the gradient is computed with respect to
+the Riemannian metric g. The usual gradient of a function ϕ:Rn→R
+at a point x0∈Rnis denoted by ∇ϕ(x0)∈Rn, without any subscript.
+The length of a tangent vector v∈Tx0(M)with respect to the metric gis
+|v|g=/radicalbig
+gx0(v,v).
+11Lemma3.1 SupposeX= (U,g,Ψ,ξ0)is ann-dimensional Riemannian
+package of log-concave type. Then, for any ξ∈U,
+|∇gΨ(ξ)|g≤√nσn.
+Proof:Suppose first that ξ=ξ0. Weneed toestablish the bound
+|∇gΨ(ξ0)|g≤√nσn (19)
+for any log-concave package X= (U,g,Ψ,ξ0)of dimension n. Any such
+packageXis isomorphic to Xµ= (Rn,gµ,Ψµ,0)for a certain compactly-
+supported log-concave probability measure µonRn. Furthermore, accord-
+ing to Corollary 2.4, we may apply an appropriate affine map an d assume
+thatµis isotropic. Thusour goal is toprove that
+|∇gµΨµ(0)|gµ≤√nσn. (20)
+Sinceµis isotropic, ∇2Λµ(0) =Cov(µ) =Id, whereIdis the identity
+matrix. Consequently, the desired bound (20) isequivalent to
+|∇Ψµ(0)| ≤√nσn.
+Equivalently, weneed to show that
+∂θlogdet∇2Λµ(ξ)
+det∇2Λµ(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ξ=0≤√nσn for allθ∈Sn−1.
+A straightforward computation shows that ∂θlogdet∇2Λµ(ξ)equals the
+trace of the matrix/parenleftbig
+∇2Λµ(ξ)/parenrightbig−1∇2∂θΛµ(ξ). Sinceµisisotropic,
+∂θlogdet∇2Λµ(ξ)
+det∇2Λµ(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+ξ=0=△∂θΛµ(0) =/integraldisplay
+Rn(x·θ)|x|2dµ(x)≤√nσn,
+according to the definition of σn, where△stands for the usual Laplacian
+inRn. This completes the proof of (19). The lemma in thus proven in the
+special case where ξ=ξ0.
+The general case follows from Lemma2.5: When ξ/ne}ationslash=ξ0, we may con-
+sider the log-concave Riemannian package Y= (U,g,˜Ψ,ξ), where˜Ψdif-
+fers from Ψby an additive constant. Applying (19) with the log-concave
+packageY, wesee that
+|∇gΨ(ξ)|g=|∇g˜Ψ(ξ)|g≤√nσn.
+12/square
+The next lemma is a crude upper bound for the Riemannian dista nce,
+valid for any Hessian metric (that is, a Riemannian metric on U⊂Rnin-
+duced by the hessian of aconvex function).
+Lemma3.2 Letµbe a compactly-supported probability measure on Rn
+whose support is not contained in a hyperplane. Denote by Λits logarith-
+mic Laplace transform, and let Xµ= (Rn,gµ,Ψµ,0)be the associated
+Riemannian package. Thenfor any ξ,η∈Rn,
+d(ξ,η)≤/radicalbig
+Λ(2ξ−η)−Λ(η)−2∇Λ(η)·(ξ−η),(21)
+whered(ξ,η)is the Riemannian distance between ξandη, with respect to
+theRiemannianmetric gµ. Inparticular, whenthebarycenterof µliesatthe
+origin,
+d(ξ,0)≤/radicalbig
+Λ(2ξ). (22)
+Proof:Thebound (21) isobvious when ξ=η. Whenξ/ne}ationslash=η,weneed to
+exhibitapathfrom ηtoξwhoseRiemannianlengthisatmosttheexpression
+onthe right in (21). Set θ= (ξ−η)/|ξ−η|andR=|ξ−η|. Consider the
+interval
+γ(t) =η+tθ (0≤t≤R).
+Thispath connects ηandξ, and itsRiemannian length is
+/integraldisplayR
+0/radicalBig
+gµ(γ(t))(θ,θ)dt=/integraldisplayR
+0/radicalbig
+[∂θθΛ](η+tθ)dt
+=/integraldisplayR
+0/radicalbigg
+d2Λ(η+tθ)
+dt2dt≤/radicalBigg/integraldisplay2R
+0(2R−t)d2Λ(η+tθ)
+dt2dt/integraldisplayR
+0dt
+2R−t,
+according to the Cauchy-Schwartz inequality. Clearly,/integraltextR
+0dt/(2R−t) =
+log2≤1. Regardingtheotherintegral, recallTaylor’sformulawit hintegral
+remainder:
+/integraldisplay2R
+0(2R−t)d2Λ(η+tθ)
+dt2dt= Λ(η+2Rθ)−[Λ(η)+2Rθ·∇Λ(η)].
+The inequality (21) is thus proven. Furthermore, Λ(0) = 0 , and when the
+barycenter of µlies at the origin, also ∇Λ(0) = 0 . Thus (22) follows from
+(21). /square
+13Thevolume-radius of aconvex body K⊂Rnis
+v.rad.(K) = (Voln(K)/Voln(Bn
+2))1/n.
+This is the radius of the Euclidean ball that has exactly the s ame volume
+asK. WhenE⊆Rnis an affine subspace of dimension ℓandK⊂E
+is a convex body, we interpret v.rad.(K)as(Vol(K)/Vol(Bℓ
+2))1/ℓ. For a
+subspace E⊂Rn, denote by ProjE:Rn→Ethe orthogonal projection
+operator onto EinRn. A Borel measure µonRnisevenorcentrally-
+symmetric ifµ(A) =µ(−A)for any measurable A⊂Rn.
+Lemma3.3 Letµbe an even, isotropic, log-concave probability measure
+onRn. Let1≤t≤√nanddenoteby Bt⊂Rnthecollection ofall ξ∈Rn
+withd(0,ξ)≤t,whered(0,ξ)is asin Lemma3.2. Then,
+Voln(Bt)1/n≥ct√n, (23)
+wherec >0is a universal constant. Here, as elsewhere, Volnstands for
+theLebesgue measure on Rn(and not the Riemannian volume).
+Proof:It suffices to prove the lemma under the additional assumptio n
+thattis aninteger. According toLemma3.2,
+Kt:={ξ∈Rn;Λ(2ξ)≤t2} ⊆Bt.
+LetE⊂Rnbe anyt2-dimensional subspace, and denote by fE:Rn→
+[0,∞)the density of the isotropic probability measure (ProjE)∗µ. Then
+fEis a log-concave function, according to the Pr´ ekopa-Leind ler inequality,
+andfEis also an evenfunction. According tothe definition above,
+fE(0)1/t2=LfE≥c.
+Note that the restriction of Λto the subspace Eis the logarithmic Laplace
+transform of (ProjE)∗µ. It isproven in[17, Lemma2.8] that
+v.rad.(Kt∩E)≥ctfE(0)1/t2≥c′t. (24)
+Thebound (24)holds for anysubspace E⊂Rnof dimension t2. From[14,
+Corollary 3.1]wededuce that
+v.rad.(Kt)≥˜ct.
+SinceKt⊆Bt,the bound (23) follows. /square
+14Lemma3.4 Letµbe a compactly-supported, even, isotropic, log-concave
+probability measure on Rn. Denote by Kthe interior of the support of µ, a
+convex body in Rn. Then,
+Voln(K)1/n≥c/σn,
+wherec >0isauniversal constant.
+Proof:Sett= max{√n/σn,1}. Then1≤t≤√nandσn≤C√n,
+according to Lemma 1.3. Recall the definition of the set Bt⊂Rnfrom
+Lemma 3.3. Consider the Riemannian package Xµ= (Rn,gµ,Ψµ,0)that
+isassociated with the measure µ. According to Lemma3.1,for any ξ∈Bt,
+Ψµ(0)−Ψµ(ξ)≤√nσnd(0,ξ)≤t√nσn≤Cn.
+SinceΨµ(ξ) = logdet ∇2Λµ(ξ)andΨµ(0) = 0,then
+det∇2Λµ(ξ)≥e−Cnfor anyξ∈Bt.
+FromLemma2.1,
+Voln(K) =/integraldisplay
+Rndet∇2Λµ(ξ)dξ≥/integraldisplay
+Btdet∇2Λµ(ξ)dξ≥e−CnVoln(Bt)
+asΛµis convex and hence det∇2Λµ(ξ)≥0for allξ. Lemma 3.3 yields
+that
+Voln(K)1/n≥e−C/parenleftbigg
+ct√n/parenrightbigg
+≥c′
+σn.
+Thelemmaisproven. /square
+Proof of Inequality 1.1 : LetK⊂Rnbe a centrally-symmetric convex
+body such that theuniform probability measure µKis isotropic. Then,
+LµK=1
+Voln(K)1/n≤Cσn
+thanks to Lemma 3.4. In view of (1), the bound Ln≤Cσnis proven. The
+desired inequality (4) now follows from Lemma1.3. /square
+The following proposition is not applied in this article. It is neverthe-
+less included as it may help understand the nature of the elus ive quantity/vextendsingle/vextendsingleEX|X|2/vextendsingle/vextendsinglefor an isotropic, log-concave random vector XinRn.
+Proposition 3.5 SupposeXis an isotropic random vector in Rnwith finite
+third moments. Then,
+/vextendsingle/vextendsingleEX|X|2/vextendsingle/vextendsingle2≤Cn3/integraldisplay
+Sn−1/parenleftbig
+E(X·θ)3/parenrightbig2dσn−1(θ)
+whereσn−1is the uniform Lebesgue probability measure on the sphere
+Sn−1,andC >0is auniversal constant.
+15Proof:DenoteF(θ) =E(X·θ)3forθ∈Rn. ThenF(θ)is a homoge-
+nous polynomial of degree three, and its Laplacian is
+△F(θ) = 6E(X·θ)|X|2.
+Denotev=EX|X|2∈Rn. Thefunction
+θ/ma√sto→F(θ)−6
+2n+4|θ|2(θ·v) (θ∈Rn)
+is a homogenous, harmonic polynomial of degree three. In oth er words, the
+restriction F|Sn−1decomposes into spherical harmonics as
+F(θ) =6
+2n+4(θ·v)+/parenleftbigg
+F(θ)−6
+2n+1(θ·v)/parenrightbigg
+(θ∈Sn−1).
+Sincespherical harmonics of different degrees areorthogo nal toeach other,
+/integraldisplay
+Sn−1F2(θ)dσn−1(θ)≥36
+(2n+4)2/integraldisplay
+Sn−1(θ·v)2dσn−1(θ) =36
+n(2n+4)2|v|2.
+/square
+Remark.AccordingtoProposition3.5,ifwecouldshowthat/vextendsingle/vextendsingleE(X·θ)3/vextendsingle/vextendsingle≤
+C/nfor a typical unit vector θ∈Sn−1, we would obtain a positive answer
+toQuestion 1.1. It isinteresting tonote that thefunction
+F(θ) =E|X·θ| (θ∈Sn−1)
+admits tight concentration bounds. For instance,
+/integraldisplay
+Sn−1(F(θ)/E−1)2dσn−1(θ)≤C/n2
+whereE=/integraltext
+Sn−1F(θ)dσn−1(θ), whenever Xis distributed according to
+a suitably normalized log-concave probability measure on Rn. The nor-
+malization we currently prefer here is slightly different f rom the isotropic
+normalization. The details will be explained elsewhere, as well as some
+relations to the problem of stability inthe Brunn-Minkowsk i inequality.
+References
+[1] Anttila, M.,Ball,K.,Perissinaki, I., Thecentral limit problem forcon-
+vex bodies . Trans. Amer.Math. Soc.,355, no. 12, (2003), 4723–4735.
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+inRn.Studia Math., 88, no. 1, (1988), 69–84.
+[3] Ball, K.,lecture at the Institut Henri Poincar´ e, Paris ,June 2006.
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+bodies.Geometric aspects of functional analysis, Lecture Notes in
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+rael
+e-mail address: [roneneld,klartagb]@tau.ac.il
+18
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+arXiv:1001.0876v1  [astro-ph.CO]  6 Jan 2010Studying the kinematics of faint stellar
+populations with thePlanetary Nebula
+Spectrograph
+MichaelR.Merrifield∗andThe PN.SConsortium†
+∗Schoolof Physics&Astronomy,University ofNottingham,UK
+†http://www.astro.rug.nl/~pns/
+Abstract. Galaxiesarefaintenoughwhenoneobservesjusttheirlight distributions,butinstudying
+their full dynamical structure the stars are spread over the six dimensions of phase space rather
+than just the three spatial dimensions, making their densit ies very low indeed. This low signal is
+unfortunate,as stellar dynamicshold important clues to th ese systems’ life histories, and the issue
+is compoundedbythe factthat themost interestinginformat ioncomesfromthefaintest outerparts
+ofgalaxies,wheredynamicaltimescales(andhencememorie sofpasthistory)arelongest.
+Toextractthisinformation,wehaveconstructedaspecial- purposeinstrument,thePlanetaryNeb-
+ula Spectrograph, which observes planetary nebulae as kine matic tracers of the stellar population,
+and allows one to study the stellar dynamics of galaxies down to extremely low surface bright-
+nesses. Here, we present results from this instrument that i llustrate how it can uncover the nature
+of low surface-brightnessfeatures such as thick disks by st udying their kinematics, and trace faint
+kinematicpopulationsthatarephotometricallyundetecta ble.
+Keywords: Galaxydynamics,Planetarynebulae
+PACS:98.10.+z,98.62.Dm
+INTRODUCTION
+Because stars are well-spaced, even the inner parts of galax ies can appear faint, com-
+parable in surface brightness to the darkest of night skies. Not surprisingly, then, when
+we start exploring the full six-dimensional phase space of v elocities and positions that
+starsoccupyingalaxies,thesignalperphasespaceelement becomesalmostvanishingly
+small. In the case of the Milky Way, for example, the fascinat ing stellar streams in the
+halo[1]havewidthsofonly ∼20pc, andvelocitydispersionsofmaybe ∼10kms−1. If
+we were to divide the Galaxy, with a linear dimension of ∼20kpc and velocities span-
+ning∼600kms−1, into resolution elements this small, we would end up with ∼1014
+of them, many more than there are stars in the Milky Way, so the average occupation
+numberperelementwouldbevery smallindeed.
+Thus,ifwearetostudythedetailedkinematicsofgalaxies, weneedtodotwothings.
+First, we must be able to detect stellar populations at extre mely low phase densities,
+which essentially means detecting individual stars. Secon d, we need to come up with
+ways to combine the information from these individual detec tions in order to bring
+together enough signal to say anything about the properties of the galaxy, and hence
+reconstructitsdetaileddynamicalstructureinorderto le arn aboutitsevolution.
+Planetary nebulae (PNe) offer an ideal tracer of such faint s tellar populations, asthey are readily individually detectable from their emissi on lines, even in quite distant
+galaxies,and the Dopplershiftsin these linescan be used to measure theirline-of-sight
+velocities.Tomakesuchmeasurementsefficientlyinasingl estep,wehavedesignedand
+built a customized instrument, the Planetary Nebula Spectr ograph or PN.S [2]. In this
+paper,wepresentexamplesfromacoupleoftheprojectsthat wehavebeenundertaking
+withthisinstrument,toillustratestheprinciplesofinve stigatinggalaxiesintheultimate
+low-densityregimeofphasespace.
+THETHICK DISKIN NGC 891
+When one traces edge-on stellar disks away from the plane out to very low surface
+brightnesses, an excess of light above the extrapolation of the normal disk population
+is frequently found [3]. The nature of such “thick disks” rem ains highly controversial:
+theycouldjustbetheextremetailofthenormaldiskpopulat ion,ortheycouldrepresent
+some more dramatic event, such as the debris from a merger. On e exciting new clue
+to help distinguish between these possibilities was uncove red when it was found that
+the thick disk in the edge-on system FGC 227 appears to be rota ting in the opposite
+directiontothenormalthin-diskpopulation,indicatingt hatitwouldhavetobeadistinct
+component such as might be formed in a merger [4]. However, th e surface brightness
+of this component is so faint that, even with ten hours of inte gration on an 8-metre
+telescope, conventional spectroscopy had such large error bars that the data were also
+consistentwithco-rotatingthinand thickdisks.
+As Figure 1 illustrates, we can do a great deal better with a ra ther shorter integration
+on a 4-metre telescope using PN.S. The upper panel shows the n umber counts of PNe
+detectedintheedge-onspiralNGC891asafunctionofdistan cefromitsplane.Thelines
+show a possible decomposition of the disk into thin and thick exponential components
+[5]. The data drop below the model close to the plane, as PNe ar e difficult to detect
+against the high surface brightness of the galaxy in this reg ion, and some PNe will also
+belostinNGC 891’sstrongdustlane. However,intheregionw eareinterestedin away
+from theplane, thePN numbercountstrace thelightdistribu tionvery well, underlining
+theirnatureas generic tracers ofthestellarpopulation.F urther, weare clearly detecting
+them in significant numbers out to beyond the transition wher e the thick disk becomes
+thedominantcomponent.
+The lower panel in this figure shows the observed mean rotatio n speed for these PNe
+asafunctionofdistancefromtheplane.Thereisplentyofsi gnaltodistinguishbetween
+co- and counter-rotating thin and thick disk components, an d in this system the two are
+clearly rotating in the same direction. Indeed, we can go fur ther and fit a simplesingle-
+componentkinematicmodeltothedata. Thisheuristicmodel isoftheform
+/angbracketleftv/angbracketright=α(vc−βσz(z)2/vc), (1)
+whereαprovidesthefactorthatallowsfortheline-of-sightproje ctioneffectsthatreduce
+the line-of-sight velocity below the mean rotational veloc ity [6], while βparameterizes
+allthetermsrelatedtotheshapeofthevelocityellipsoidi ntheasymmetricdriftequation
+[7]; here it is assumed to be a constant. The circular speed of the galaxy, vc, can be
+inferred from gas kinematics, while the vertical velocity d ispersion, σz(z), is estimatedFIGURE1. Propertiesofthe PNe in thedisk ofthe edge-ongalaxyNGC891 asa functionofdistance
+from the galactic plane. The upper panel shows the number cou nts of PNe as detected with PN.S, with
+lines showing a model decomposition of conventional photom etry into thin disk and thick disk [5]. The
+lower panel shows the mean rotation velocity of these PNe, wi th a line showing the simplest possible
+single-populationmodelfortheirkinematics.
+by solving the Jeans equation for a self-gravitating sheet w hose density distribution is
+given by the double exponential model in the upper panel of Fi gure 1. Clearly, the data
+areessentiallyconsistentwiththissimplesingle-compon entmodel:thereisnokinematic
+evidencethatthethickdiskin NGC891isin anyway adistinct entity.
+A FAINTKINEMATIC POPULATION IN M31
+Asafurtherillustrationofthewayinwhichfaintkinematic componentscanbedetected,
+Figure2showstheresultsofananalysisofthe2615PNethatw erefoundusingPN.Sin
+a survey of M31 [9]. In an inclined disk system like this, the p hase space is essentially
+reducedtofourdimensions,twospatialandtwovelocity.We canmeasurethreeofthese
+components,twospatialandtheline-of-sightvelocity,so eachPNhasonlyoneunknown
+coordinate. Thus, while one cannot measure the energy and an gular momentum of any
+single PN, there is only a one-dimensional family of possibi lities, as illustrated by the
+linesectionsinthefigure. IfgroupofPNeshareacommonorbi t(and henceenergy and
+angular momentum), then their lines will all intersect at a s ingle point, creating a local
+excess in this plane. Such an excess of crossing lines is, ind eed, found in this figure, at
+the point indicated by the box. The location of the orbit to wh ich it corresponds does
+not appear in any way special in an optical imageof M31, but, a s theinset imagein theFIGURE2. Plotofthepossiblevaluesofangularmomentumandenergy(w iththelocalcircularenergy
+subtracted, so a circular orbit lies at the bottom of the plot ) for each of the PNe in the M31 survey data.
+Thewhite box showsthe pointof the largest local excessof PN e. The inset showsthe orbit to whichthis
+pointcorresponds,superimposedona mid-infraredimageof thegalaxy[8].
+figureshows,mid-infrareddatarevealabrightringofemiss ionatthisradius,indicating
+that it lies at one of the major orbital resonances. Thus, eit her we have detected the
+population of stars born in this ring of star formation, or, m ore likely since PNe come
+from old stars, we are picking out the relatively small popul ation of objects that have
+become dynamically trapped at this resonance. Whatever the ir origin, these data again
+illustrate the power of PN.S to identify kinematic componen ts that are at far too low a
+densityin phasespace tobedetectablebymoreconventional means.
+REFERENCES
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+6. E.Neistein,D. Maoz,H. Rix,andJ. L.Tonry, AJ117,2666–2675(1999).
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+ 1 DNA heats up: Energetics of genome ejection from phage revealed by isothermal titration calorimetry.  Meerim Jeembaeva1,2, Bengt Jönsson2, 3,§, Martin Castelnovo4,§, and Alex Evilevitch1,2*  1Department of Physics, Carnegie Mellon University, 5000 Forbes Ave, 15213 PA, USA  2Department of Biochemistry and 3Department of Biophysical Chemistry, Center for Molecular Protein Science, Lund University, P O Box 124, SE-221 00, Lund, Sweden 4Université de Lyon, Laboratoire de Physique, École Normale Supérieure de Lyon, 46 Allée d'Italie, 69364 Lyon Cedex 07, France  § These authors contributed equally  * Corresponding author: alexe@andrew.cmu.edu  ABSTRACT  Most bacteriophages are known to inject their double-stranded DNA into bacteria upon receptor binding in an essentially spontaneous way. This downhill thermodynamic process from the intact virion toward the empty viral capsid plus released DNA is made possible by the energy stored during active packaging of the genome into the capsid. Only indirect measurements of this energy have been available until now using either single-molecule or osmotic suppression techniques. In this paper, we describe for the first time the use of isothermal titration calorimetry to directly measure the heat released (or equivalently the enthalpy) during DNA ejection from phage !, triggered in solution by a solubilized receptor. Quantitative analyses of the results lead to the identification of thermodynamic determinants associated with DNA ejection. The values obtained were found to be consistent with those previously predicted by analytical models and numerical simulations. Moreover, the results confirm the role of DNA hydration in the energetics of genome confinement in viral capsids.    2 Keywords: Bacteriophage, Isothermal Titration Calorimetry, Enthalpy, Genome Ejection, Capsid  Introduction  Packaging of double-stranded DNA into viral protein capsids is a highly energy-consuming and complex process in bacteriophages and eukaryotic motor-packaged double-stranded DNA viruses such as herpes simplex1. The first experimental evidence of the internal mechanical force associated with ATP driven DNA packaging was provided by Smith et al.2 who found using optical tweezers that a force of ~50 pN was required for complete DNA packaging by the portal complex in phage "29. In the reverse process, during infection, the highly stressed DNA inside the capsid is released into the host cells leading to reproduction of the virion. The presence of an internal capsid pressure has been verified by in vitro measurements on phage !, in which DNA ejection was found to be suppressed by an external osmotic pressure of tens of atmospheres in the host solution3. These two experimental techniques can be used to obtain estimates of the internal forces and pressures associated with genome packaging. From a theoretical perspective, the stored internal energy resulting from genome packaging is considered mainly as a sum of interaxial DNA–DNA repulsion and bending energies as was originally suggested as early as 1978 by Riemer et al.4. This is motivated by the high DNA volume fraction inside the capsid5 and small capsid size compared to DNA persistence length. Recent improvements of these analytical models were done6-8 by using a phenomenological description of DNA-DNA interactions based on osmotic stress measurements on dense hexagonal phases of DNA9. This has the advantage of implicitly including hydration forces, which are known to be dominant in DNA–DNA interactions at short length scale9-10. This ansatz allowed quantitative comparisons of internal forces using both single-molecule and osmotic suppression techniques 2, 3, 6-8, 11-21. Similarly, the total free energy of packaging, which is the sum of the internal energy and the entropic penalty mainly associated with DNA confinement, has been computed as a function of DNA packaging density using molecular dynamics (MD) simulations for the phages "29 and #1521-22.  In order to experimentally dissect the thermodynamic determinants of DNA  3 packaging and release energetics, we present for the first time the use of isothermal titration calorimetry (ITC) to directly measure the heat released by DNA ejection from a phage. In contrast to single-molecule or osmotic suppression techniques, this approach allows one to address experimentally the relative weight and origin of enthalpic and entropic contributions to the free energy of DNA packing inside bacteriophages. Calorimetric techniques have been successfully used in recent decades to measure energetic contributions in various studies involving proteins and nucleic acids23-41. In particular, differential scanning calorimetry (DSC) has been used in studies on virus stability, assembly and maturation transitions of viral capsids36-38, 42-44. This paper describes direct measurements of the change in enthalpy associated with DNA ejection from phage !.  In ITC, the enthalpy change ($H) associated with DNA ejection is measured at constant temperature by titrating phage !  into a cell containing the !-receptor protein LamB extracted and purified from E. coli, as shown schematically in Figure 1. The LamB protein trimer binds to the end of the phage tail inducing a conformational change in the tail-portal complex that triggers DNA release45. Approximately 90% of the phage ! particles release their DNA under these experimental conditions46.  We were able to measure DNA ejection enthalpies for phage ! with different genome lengths of 37.7 and 48.5 kbp (corresponding to 78 and 100% of wild type !-DNA length), and at different temperatures between 22 and 32°C. The enthalpy values obtained at given DNA packaging densities were analyzed and compared with theoretical values22, and good quantitative agreement was found. Furthermore, by studying the variation of the enthalpy with temperature, we were able to deduce the entropy and free energy associated with DNA ejection. In particular, we were able to demonstrate that the entropy of the ejection process is mainly dominated by the local hydration properties of the DNA rather than by its global conformation.  Results and Discussion  Isothermal titration calorimetry measurements of DNA ejection from phage  The enthalpy of DNA ejection from phage ! was studied using ITC. Phage  4 particles were titrated into LamB solution and the heat change associated with the reaction in the sample cell was measured versus time. This heat change, measured at constant pressure, gives the change in enthalpy of the reaction. The temperature in the reference cell is continuously equilibrated to that in the sample cell after each titration. The differential power between reference cell and sample cell is recorded in µcal/s, as shown schematically in Figure 1.   Since DNA ejection is an exothermic process, the heat change in the cell is negative and occurs immediately after 20 µL of the phage solution is titrated into the sample cell. The increase in sample cell temperature causes the feedback system to reduce the heating of sample cell, leading to the negative peak of differential power as shown in Figure 2, where the experimental data recorded at 27°C for the wild type (WT) phage ! (DNA length 48.5 kbp) (in TM buffer with 1% oPOE detergent (octyl-polyoxyethylene)) when titrated into LamB solution (also in TM and 1% oPOE) (blue curve) is shown. Several minutes are required for the temperature change in the sample cell to return to the baseline value once the reaction is complete. This time is dependent on the duration of DNA ejection and on the experimental conditions, e.g. the volume of phage solution added, sample mixing rate, and reference cell heating rate. The observed period of heat change was within the time frame for complete DNA ejection from ! (within a few minutes at 25°C), observed in our previous light scattering measurements47. After the temperature change in the sample cell had returned to the baseline value, the phage solution was again titrated into the LamB solution to check the reproducibility of the measurements. The two heat release peaks shown in Figure 2 are almost identical. The phage solution was titrated into the cell at least 3 times in each experiment, allowing the experimental error to be determined. Furthermore, several sets of data were collected per sample during separate measurements as a further check of the reproducibility of the data.   Enthalpy of DNA ejection Integration of the area under the heat change peak over time, provides the reaction enthalpy, which includes in particular DNA ejection from the phage; mixing of phage in LamB solution; dilution of LamB; and the pressure-volume work associated with titration  5 of one volume into another. The contributions to the enthalpy not arising from the enthalpy of DNA ejection can also be experimentally estimated in a standard way by titrating the phage solution into the buffer, the buffer into the LamB solution,  and buffer into buffer .The results are given in Table 1. (The concentrations of phage and LamB were the same within each set of measurements.) The area under the red curve  in Figure 2 corresponds to the titration of phage into buffer, which is the second largest contribution to the enthalpy after the DNA ejection enthalpy, as can be seen in Table 1. The enthalpy resulting from phage titration into LamB (blue peak) is significantly larger than the enthalpy resulting from phage titration into buffer without LamB (red peak); both signals are exothermic, and are clearly distinguishable from the baseline noise in the calorimetric titration curve. The enthalpy associated with DNA ejection from the phage is calculated in a standard way as: !Hej = !H(phage in LamB) - !H(phage in buffer) - !H(buffer in LamB) + !H(buffer in buffer).           (1) The term !H(buffer in buffer) must be added since it is subtracted twice in Eq. (1) above. Table 1 gives all the measured enthalpy contributions and calculated !Hej for DNA ejection from 37.7 and 48.5 kbp phage ! at 27°C for one set of data. Several sets of data were collected for each phage strain with different phage batches, and the average values of the ejection enthalpy <!Hej> obtained for these two mutants are shown in Table 1.  The energy change associated with the ejection of DNA from the phages, defined as !Hej = !HDNA outside virion – !HDNA inside virion, is measured here as an enthalpy change, !H = !U + p!V, where !U is the change in internal energy, p is the atmospheric pressure acting on the system and !V is the change in volume of the solution surrounding the phages during DNA ejection. Since this volume change is very small, the change in internal energy is approximately equal to the measured change in enthalpy. Other contributions to the enthalpy during the ejection process, such as phage-receptor binding and conformational changes in the portal-tail are negligible, see Methods. If the internal energy of DNA ejection is assumed to be equal to the internal energy required for the reverse process of genome packaging (i.e., if one neglects dissipation due to DNA friction in the portal and tail), then the internal energy of ejection measured here using ITC can be directly compared with existing theoretical models of DNA packaging  6 energy. For the sake of clarity, the experimentally measured thermodynamic quantities are labeled “ejection” and theoretical values “packaging” in the discussion below. The enthalpies measured at 27°C, !Hej = –(1.9 ± 0.3) x 10-16 J/virion for WT 48.5 kbp phage !,  and !Hej = –(5.3 ± 0.8) x 10-17 J/virion for the 37.7 kbp phage are given in Table 1. The observed decrease in measured enthalpy as the DNA length is decreased provides the evidence that ITC measurements of DNA ejection from phages are sensitive to genome size. Moreover, the values obtained are in quantitative agreement with the numerical results of DNA packaging Molecular Dynamics (MD) simulations by Petrov et al.22. MD method allows the thermodynamic analysis of DNA packaging by providing packaging force as a function of packaged DNA length at equilibrium. The area under the force curve provides the free energy of packaging, !A, while the molecular mechanics energy provides the internal energy, !U. The difference between the free energy and the internal energy is proportional to the change in entropy of the process, –T!S. Note that in these simulations, the change in entropy resulting from DNA packaging is always negative since it is associated with the loss of conformational degree of freedom upon DNA confinement. Among the phages analyzed thermodynamically using the MD method, bacteriophage #15 is an appropriate candidate for direct comparison with !, bearing in mind that the structural identity and DNA packaging density is central parameter in the energetics of DNA release15. Like phage !, bacteriophage #15 also has a spherically shaped T=7 icosahedral capsid with a diameter of ~ 675 Å (compared to ~ 630 Å for phage !) and has a slightly shorter DNA length of 39.7 kbp (compared to 48.5 kbp in !). Due to the cylindrical protein core in #15, absent in !, the DNA packaging density is similar in both phages: 0.52 for fully packed #15 (Anton Petrov, personal communication) and 0.48 for WT !48. The relative packaging density, %, is defined as the ratio of the volume of packaged DNA to the capsid volume48. The thermodynamic analysis described in ref.22 for #15 was performed at 27°C, as was the case in our ITC measurements. Interpolation of the internal energy values from ref.22 to the packaging density % = 0.48 (corresponding to WT phage !), provides an internal energy value of !Upack = +1.7 x 10-16 J/virion, which is very close to our experimental value of !Hej= –(1.9 ± 0.3) x 10-16 J/virion. The packaging density of 0.38, corresponding to 37.7 kbp  7 DNA, yielded a value of !Upack = 5.2 x 10-17 J/virion, which again agrees well with our measured value of !Hej = –(5.3 ± 0.8) x 10-17 J/virion. These simulations thus provide evidence of the validity of the ITC technique for the direct measurement of DNA packaging energetics.   Free energy and entropy of DNA ejection Further insights into the thermodynamics of DNA packaging, and in particular the associated entropy, can be obtained by comparing our results to the analytical inverse spool model described by Tzlil et al.6-7 and Purohit et al.8, 48, which has been shown to quantitatively describe osmotic suppression data. This model provides the free energy of DNA packaging based on the assumption that the genome packed inside a spherical viral capsid has an idealized inverse spool geometry. In this model, the free energy is written as the sum of bending and short-range repulsion energies. The mean field free energy of the inverse spool model is written as: ! E(R,ds)=L3F0(c2+cds)exp("dsc)"4#$kBT3%ds2Rout2"R2+RoutlogRout"Rout2"R2R& ' ( ( ) * + + & ' ( ( ) * + +  (2) where Rout is the internal radius of the capsid (29.5 nm), R is the radius of the internal region of the inverse spool devoid of any DNA due to overwhelming bending stress, ds is the uniform interaxial DNA–DNA spacing within the capsid, L is the length of the DNA within the capsid, kB is Boltzmann’s constant, and T is temperature. F0 and c are experimentally determined constants describing the interaction between neighboring DNA double helices, and lp is the persistence length of DNA. F0, c and lp are dependent on the salt concentration in the host solution (in our experimental system we used 50 mM Tris and 10 mM MgSO4). Using this salt condition, Grayson et al.49-50 showed that osmotic suppression data were effectively well described by the model using values of lp  & 50 nm, F0 = 12 000 pN/nm, and c = 0.30 nm at T=310K (37°C). The two parameters Fo and c are only slightly dependent on temperature (Don Rau, private communication). Inserting the additional constraint of the conservation of DNA volume within the capsid (relating L, ds, Rout, and R), and the optimal interaxial distance ds, found by free energy minimization, into this free energy expression, gives the equilibrium free energy that has been used to compute ejection forces in recent studies6-7, 15, 48, 51-52. The changes in  8 enthalpy (!H) and entropy (!S) of the ejection process can then be calculated using the thermodynamic relationships: ! "H="G+T"S="G#Td"G()dT and ! T"S="H#"G. The values calculated in this way for the packaging enthalpy in WT phage (48.5 kbp) and in the phage with a DNA length of 37.7 kbp, –1.9 x 10-16 J/virion and –6.1 x 10-17 J/virion, respectively, roughly matched the experimental values. (Here we adopted the qualitative trends of the variation of Fo and c with temperature to calculate the values at 27°C (Don Rau, Private communication)). It should be noted that our use of both the inverse spool model and the set of parameters used in Grayson et al.6-7, 15, 48, 51-52 is mainly motivated by the quantitative agreement of the model with the osmotic suppression data.  Interestingly, the values of computed free energy, !G, obtained for packaged DNA lengths of 37.7 and 48.5 kbp at 27°C, are one order of magnitude smaller than the values of the enthalpy measured with ITC; the calculated free energy of ejection (!Gej) was on the order of –5 x 10-17 J/virion while the measured enthalpy (!Hej) was –1.9 x 10-16 J/virion for the WT phage ! at 27°C. Using the difference between the theoretical free energy and the experimentally determined enthalpy, we estimate that the entropy associated with DNA packaging is positive, reflecting an increase in the disorder of the system. This result may appear surprising at first glance, mainly because when DNA is packaged one would intuitively expect it to become more ordered, leading to a reduction in conformational entropy. This behavior is indeed observed in the simulations by Petrov et al.22. However, as the DNA helices come closer together in the capsid, the hydration properties of DNA change, a fact that is not explicitly included in the simulations. More precisely, as the DNA becomes hexagonally packed, the ordered water molecules directly surrounding the DNA are released, increasing the net disorder of the system, thus increasing its entropy. The positive change in entropy associated with the dehydration process upon DNA packing has been observed by Leikin et al. in dense hexagonal phases of DNA probed by the osmotic stress method53. In their study, they were able to convert osmotic stress data into changes in entropy and enthalpy per base pair. The changes in both enthalpy and entropy associated with DNA packing induced by osmotic pressure were positive and decreasing functions of temperature. The enthalpy associated with DNA density, measured by Leikin et al.53, ranged from 4 x 10-18 J at low density to 1.9 x  9 10-16 J at the highest density possible in their experimental setup (corresponding roughly to the packaging of 48.5 kbp DNA in phage !). Since DNA is not strongly bent into hexagonal phases in this type of experiment, we conclude from the similarity between the enthalpy in this case and that measured in phage !, that short-range DNA–DNA interactions, and thus the hydration properties of DNA, constitute the main contribution to the energetics of DNA packaging. This comparison can only be made qualitatively because of the different ionic conditions used by us and Leikin et al.53.  In order to further investigate the thermodynamic parameters associated with DNA ejection, we also performed enthalpy measurements for DNA ejection from WT phage ! at different temperatures. The results are reported in Table 2 and shown in Figure 3a. We found that the change in enthalpy associated with DNA ejection is a strongly decreasing function of temperature. The reason for this behavior is probably that as the temperature is increased, the enthalpy of DNA outside the capsid increases faster than the enthalpy of DNA inside the capsid, due to enhanced fluctuations within the DNA coil. Therefore, the absolute value of the difference in enthalpy, !HDNA outside virion – !HDNA inside virion, which is what we measure and which is negative, will decrease (i.e., become less negative) as the temperature is increased. Assuming that !Hej(T) is almost linear in the narrow temperature range studied here (22-32°C), it is possible to extract an approximate constant specific heat capacity for DNA ejection per virion, !Cp, by fitting a line to our data, using: ! "Hej(T)#"Hej(293K)+"Cp$(T%293K)=%3.78$10%16+2.17$10%17$(T%293K) (3) where we used T=293K as a reference temperature and thus !Hej(293K) is the reference enthalpy value. The value for ! "Cp=2.17#10$17JK$1/virion can now be used to calculate how !Sej and !Gej vary with temperature using: ! "Sej(T)#"Sej(293K)+"Cp$lnT293% & ' ( ) *     (4) where !Sej(293K) is the reference entropy at 20°C.  There is no direct way to obtain the reference value of the entropy, !Sej(293K), from experimental data. However, since the total free energy of DNA ejection, !Gej, is negative (since ejection is spontaneous), this sets a lower bound on the entropy such that !Sej(293K) > –1.25x10-18 J/K. A likely upper bound is !Sej(293K) " 0, which would lead  10 to !Gej " !Hej within our experimental temperature range (22-32°C), implying that the entropy is positive and dominates the DNA ejection process above 27°C. However, we chose the reference value of the change in entropy !Sej(293K) to be – 1.1 x 10-18 J/K to obtain the value of !Gej at 27°C given by the inverse spool model, as discussed above, mainly because the associated values of !Hej from the model are quantitatively consistent with our experimental values for two different DNA lengths, and the model provides a good description of the osmotic suppression data49. Furthermore, this choice of reference entropy is also consistent with data presented by Leikin et al. for osmotic stress experiments of bulk DNA53, where !Gej << !Hej. The absolute values of !Hej, !Gej and !Sej calculated using experimental values of !Hej and Cp (for the reference entropy value !Sej(293K) = –1.1 x 10-18 J/K) are shown in Figure 3b. According to this thermodynamic calculation based only on the temperature dependence of enthalpy and a particular value of the reference entropy, –T!Sej is positive (implying that the DNA ejection entropy is negative) within our experimental temperature range (22-32°C), and decreases as the temperature increases, as was observed by Leikin et al. for dense phases of DNA53. This observation therefore strengthens the suggestion that the entropy of DNA ejection is strongly related to hydration, as discussed above. This implies that at lower temperatures (20-35°C) the DNA ejection process is dominated by the enthalpy, which is negative and promotes DNA release, while positive entropy strives to keep the DNA inside the capsid, see Figure 3b. However, as can be observed from linear extrapolation of data in this figure, at higher temperatures, the entropy changes sign and starts to dominate the DNA ejection process.  Conclusions This is the first report on measurements of the heat released in association with receptor-triggered DNA ejection from phage ! using isothermal titration calorimetry. By using DNA with two different lengths, we have shown that the heat released, or equivalently the enthalpy, increases with DNA length, and agrees with numerical simulations by Petrov et al.22 for a different phage with a similar internal DNA density, and with the analytical inverse spool model of DNA packaging6-8. This supports the use of ITC for direct measurements on DNA ejection energetics. We also found the ejection  11 enthalpy to have a strong dependence on temperature; the absolute value decreasing with increasing temperature. This in turn implies a strong temperature variation of –T!Sej, which changes sign as the temperature is increased. It was also found that the free energy for ejection is one order of magnitude smaller than the enthalpy, which indicates a negative change in entropy associated with DNA ejection within the experimental temperature range used here (22-32°C). The sign of this ejection entropy is easily interpreted in terms of dehydration of the DNA backbone upon helix packing. The hydration entropy dominates over the loss of conformational entropy of the DNA coil upon confinement in the capsid, and has the opposite sign. Similar thermodynamic behavior has been observed in osmotic stress measurements on bulk DNA53. Our measurements show that at lower temperatures the DNA ejection process is enthalpy dominated (where entropy prevents DNA release), while at higher temperatures the entropic term dominates ejection.  The release of phage DNA into cells in vivo leads to the condensation of ejected DNA due to molecular crowding in the cytoplasm54. It is, however, likely that the energetics of the in vivo phage infection process are similar to that in vitro at lower temperatures (where enthalpy dominates the free energy), since the confinement of DNA in the capsid provides a major energy contribution. However, at higher temperatures, where the entropic contribution dominates the energy, the in vivo energetics may be different, for example, reflecting the condensation of the ejected DNA.   Materials and Methods Purification of the phage and LamB receptor Wild type bacteriophage ! with a genome length of 48.5 kbp was produced by thermal induction of the lysogenic E. coli strain AE1. The shorter-genome phage !b221, with 37.7 kbp DNA was extracted from plaques by a confluent lysis technique. The details of phage purification have been described elsewhere3. The phages were purified by CsCl equilibrium centrifugation. The sample was then dialyzed against TM buffer (10 mM MgSO4 and 50 mM Tris-HCl/pH 7.4). The final titer was 1013 virions/mL, determined by plaque assay55.  The phage ! receptor, the LamB protein, was purified from pop 154, a strain of E.  12 coli K12 in which the LamB gene has been transduced from Shigella sonnei 3070. This protein has been shown to cause complete in vitro ejection of DNA from phage !, without having to add the solvents required for DNA ejection from the WT E. coli receptor56. Purified LamB was solubilized from the outer membrane with 1 vol. % of the detergent (oPOE) in TM buffer47, 57. The purity of the LamB protein was checked by SDS-PAGE and it was confirmed that no contaminants were present.   Isothermal titration calorimetry  Calorimetric measurements were performed using the isothermal titration calorimeter (VP-ITC) manufactured by MicroCal, Inc. (Northampton, MA, USA) with an active cell volume of 1.4 mL. Prior to each measurement, the VP-ITC calorimeter was tested and equilibrated with Milli-Q water at the relevant temperature. Milli-Q water was used in the reference cell during all measurements. The optimal mixing speed in the cell was found to be 260 rpm. After degassing, the LamB solution in the TM buffer (pH 7.4 with 1% oPOE) was loaded into the sample cell using a 2.5-mL Hamilton syringe. The ratio between LamB receptor trimers and phage particles was maintained at approximately 1000:1 to ensure maximum “opening” efficiency of the phage without undue time delay. We have previously verified, by dynamic light scattering, that phage–receptor binding is not a rate-limiting step in the DNA ejection from phage ! when LamB is present at such a high excess47. Twenty µL of phage ! solution with a viral concentration on the order of 1013 virions/mL was titrated into LamB solution with a concentration of ~0.4 mg/mL. The TM buffer, which contained 1 vol. % oPOE was added to both the phage and LamB solutions to avoid enthalpy contributions from the solvent and surfactant mixing. Measurements were performed within the temperature range 22-32°C. Raw data were analyzed using Origin 7.0 software (baseline correction and peak integration).  Other enthalpy contributions Several energetic contributions could be involved in addition to the genome ejection enthalpy determined here during ITC. These could be associated with binding of the receptor to the phage and the enthalpy required for the activation of DNA release  13 (involving conformational changes to the tail and portal after receptor binding). In an in vitro study, the binding enthalpy between the receptor and the phage T5 was found to be on the order of ~10-19 J/virion58. In two different studies using light scattering to study rate of DNA ejection, the enthalpy required for the activation of DNA ejection from phages T5 and ! could be determined to be ~10-19 J/virion, through the temperature dependence of ejection rates59-60. Thus, both the receptor binding energy and the activation energy for genome release are on the order of 10-19 J/virion, which is 3 orders of magnitude lower than the genome ejection enthalpy measured here (~10-16 J/virion, see Results and Discussion). Hence, we can neglect these enthalpy contributions in the thermodynamic analysis of the genome release process.   ITC is sensitive to differences in concentration of the oPOE surfactant between the syringe solution containing the phages and the sample cell containing LamB. Therefore, it is essential to maintain the same concentration of oPOE in the LamB and the phage solution, in both the cell and the syringe.   Acknowledgments: We dedicate this work to the memory of Mark Evilevitch (AE’s father), who gave us the idea and inspiration for this study. We acknowledge the help of Thom Leiding in setting up the experiments. We are also grateful to C.S. Raman, Philip Serwer, William Gelbart, Charles Knobler, Anton Petrov, Gerd Olofsson, and Cecilia Leal for advice and valuable discussions. This work was supported by grants from the Swedish Research Council and Royal Physiographic Society (to AE). MJ was supported by Lilly and Sven Lawski’s Foundation.   14 REFERENCES 1. Nurmemmedov, E., Castelnovo, M., Catalano, C. E. & Evilevitch, A. (2007). Biophysics of viral infectivity: matching genome length with capsid size. Q Rev Biophys 40, 327-56. 2. Smith, D. E., Tans, S. J., Smith, S. B., Grimes, S., Anderson, D. L. & Bustamante, C. (2001). The bacteriophage straight phi29 portal motor can package DNA against a large internal force. Nature 413, 748-52. 3. Evilevitch, A., Lavelle, L., Knobler, C. M., Raspaud, E. & Gelbart, W. M. (2003). Osmotic pressure inhibition of DNA ejection from phage. Proc Natl Acad Sci U S A 100, 9292-5. 4. Riemer, S. C. & Bloomfield, V. A. (1978). 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(2007). Real-time observations of single bacteriophage lambda DNA ejections in vitro. Proc Natl Acad Sci U S A 104, 14652-7. 51. Evilevitch, A., Fang, L. T., Yoffe, A. M., Castelnovo, M., Rau, D. C., Parsegian, V. A., Gelbart, W. M. & Knobler, C. M. (2008). Effects of salt concentrations and bending energy on the extent of ejection of phage genomes. Biophys J 94, 1110-20. 52. Evilevitch, A., Castelnovo, M., Knobler, C. M. & Gelbart, W. M. (2004). Measuring the force ejecting DNA from phage. Journal of Physical Chemistry B 108, 6838-6843. 53. Leikin, S., Rau, D. C. & Parsegian, V. A. (1991). Measured entropy and enthalpy of hydration as a function of distance between DNA double helices. Phys Rev A 44, 5272-5278. 54. Jeembaeva, M., Castelnovo, M., Larsson, F. & Evilevitch, A. (2008). Osmotic pressure: resisting or promoting DNA ejection from phage? J Mol Biol 381, 310-23. 55. Silhavy, T. J. (1984). Experiments with Gene Fusions. (Silhavy, T. J., Berman, M. L. & Enquist, L. 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Biophys J 93, 3999-4005.     18 Figure Legends  Figure 1: Schematic illustration of the experimental ITC setup showing the titration of phage ! solution (from the syringe) into the LamB receptor solution in the sample cell. The change in temperature in the sample cell due to the heat developed by DNA ejection from the phages is equilibrated by heating the reference cell containing water as reference solution. The power required to equilibrate the temperature to that in the sample cell is expressed in µcal/s versus time. Integration of the area under the titration peak then gives the change in enthalpy during the ejection process once the pV-work associated with titration has been subtracted.  Figure 2: The differential power (in µcal/s) versus time (s) obtained when titrating 20 µL WT phage ! (48.5 kbp DNA) into the LamB receptor solution in the sample cell at 27°C (blue curve). Both the phage and LamB solutions contained 10 mM MgSO4, 50 mM Tris, pH 7.4 (TM buffer) and 1 vol. % oPOE. The red curve shows the enthalpy change for the titration of phage ! solution into TM buffer and 1% oPOE without LamB. The figure shows the negative heat change “peaks” during two titrations between which the temperature was allowed to return to the baseline value. At least 3 consecutive titrations were made for each sample in order to ensure reproducibility. The area under the blue peak gives the sum of the reaction enthalpy associated with DNA ejection from the phages and the pV-work of titration. The enthalpy of phage in buffer titration (that includes the pV-work), the area under the red peak, is subtracted from the area under the blue peak and divided by the number of phage particles added to give the enthalpy of DNA ejection in J/virion.   Figure 3: a) Enthalpy of DNA ejection for the 48.5 kbp WT phage ! at 22, 24, 27 and 32°C. Data are shown for several batches of phages at each temperature. The dashed line shows a linear fit to the average enthalpy value at a given temperature. b) Predicted values of free energy, !Gej, and entropy, -T!Sej, caused by DNA ejection as a function of temperature, based on the measured enthalpy, !Hej, and a reference entropy value of !Sej(293 K) = –1.1 x 10-18 J/K.  19   1 Tables Table 1: Enthalpy data from ITC measurements on ! phages with DNA of two different lengths (37.7 and 48.5 kbp). The number of phages in a 20 µL injection volume used to calculate !Hej was 1.3x1011 for the 48.5 kbp DNA and 8.6x1010 for the 37.7 kbp DNA. The average values of the enthalpy for DNA ejection given in parentheses in the last column were obtained from 6 different ITC runs for each phage mutant at 27°C.   DNA length (kbp)  !H (phage in LamB) (µcal) !H (phage in buffer)  (µcal) !H (buffer in LamB) (µcal) !H (buffer in buffer) (µcal)  !Hej  (µcal)   !Hej, J/virion 37.7 –3.169 –1.656  –1.433 –0.687  –0.767  –3.7 x 10-17  (<!Hej>= – (5.3±0.5) x 10-17 ) 48.5  –8.099 –2.469 –1.471 –0.910 –5.069 –1.7 x 10-16 (<!Hej>= – (1.9±0.3) x 10-16 )    Table 2: Average values of the DNA ejection enthalpy, <!Hej >, for the 48.5 kbp WT phage ! at 4 different temperatures. The results were obtained from several measurements at each temperature.   Temperature (°C) <!Hej> (J/virion) 22 – (3.7 ±0.4) x 10-16 24 – (2.7 ±0.3) x 10-16 27 – (1.9 ±0.3) x 10-16 32 – (1.4 ±0.2) x 10-16  
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+arXiv:1001.0878v1  [cond-mat.mtrl-sci]  6 Jan 2010Enhancement of semiconducting single-wall carbon
+nanotubes photoluminescence
+Etienne Gaufr` es1, Nicolas Izard2+, Laurent Vivien1∗, Sa¨ ıd Kazaoui2, Delphine
+Marris-Morini1and Eric Cassan1
+1: Institut d’Electronique Fondamentale, CNRS-UMR 8622, U niversit´ e Paris-Sud 11,
+91405 Orsay, France
+2: National Institute of Advanced Industrial Science and Te chnology (AIST), 1-1-1 Higashi,
+Tsukuba, Ibaraki 305-8565, Japan
++: Currently at Institut d’Electronique Fondamentale
+laurent.vivien@u-psud.fr
+Photoluminescence propertiesofsemiconducting singlewall carbon nanotubes
+(s-SWNT) thin films with different metallic single wall carbon nanotubes
+(m-SWNT) concentrations are reported. s-SWNT purified samples are ob-
+tained by polymer assisted selective extraction. We show that a few m-SWNT
+in the sample generates a drastic quenching of the emission. Theref ore,
+highly purified s-SWNT films are a strongly luminescent material and a g ood
+candidate for future applications in photonics, such as near infrar ed emitters,
+modulators and detectors. c/circlecopyrt2018 Optical Society of America
+OCIS codes: 160.2540, 160.4236, 160.4760, 260.2510, 300.6280
+Single wall carbon nanotube (SWNT) is a unique monodimensional mate rial, exhibiting
+unusual electronic and optical properties, making it an ideal candid ate for future opto-
+electronic [1] and all-optical devices [2]. The nanotube structure, defined by the so-called
+(n,m) chiral index, fixes all the nanotubes properties, such as met allic or semiconducting
+behavior. Semiconducting nanotubes (s-SWNT) are of particular in terest in photonics for
+their direct band-gap, allowing near-IR luminescence (PL) from elec tron-hole recombination.
+In the last few years, numerous impressive results have been obta ined on the photolumines-
+cence properties of SWNT, where luminescence arise from well-isolat ed s-SWNT, either in
+solution by wrapping with surfactant [3–5], or dispersion in gelatin mat rix [6,7]. Recently,
+photoluminescent composite gels have also been demonstrated [8]. I ndeed, s-SWNT are ex-
+tremely sensitive to their environment. An incomplete isolation, resu lting in small bundles
+1or metallic nanotubes (m-SWNT) in contact with s-SWNT, will lead to th e creation of
+alternative non-radiative de-excitation paths and ultimately an effe ctive quenching of the
+photoluminescence. It has been estimated that only around 10 % of the s-SWNT embedded
+in gelatin matrix effectively participate in the fluorescence, the rema inder being quenched
+by left-over m-SWNT, bundles or impurities [7]. The direct influence of m-SWNT removal
+on the nanotubes optical properties was never really studied. Sev eral approaches have been
+explored in the past few years for optimizing the diameter and chiralit y selection of SWNTs,
+such as chemical functionalization, DNA, polymer wrapping and dens ity gradient centrifuga-
+tion techniques [9–14]. The possibility to achieve chiral specific SWNT e xtraction is indeed
+very promising for future photonic applications. Recently, we demo nstrated that the use of
+poly-9,9-di-n-octyl-fluorenyl-2,7-diyl (PFO) in toluene as an extra cting agent followed by
+ultracentrifugation allows to obtain quasi metallic-free SWNT [15]. U sing sample with a
+constant concentration in s-SWNT but differents concentrations in m-SWNT, we show in
+this paper that the presence of even a few metallic nanotubes quen che the strong intrinsic
+photoluminescence properties of s-SWNT.
+Samples were prepared as previously described [15]. First, SWNT pow der (as-prepared
+HiPco, Carbon Nanotechnologies Inc.), poly-9,9-di-n-octyl-fluore nyl-2,7-diyl (PFO) (Sigma-
+Aldrich) and toluene were mixed in the ratio of SWNT (5mg):PFO (20mg) :toluene (30ml)
+and homogenized by sonication (for 1 h using a water-bath sonicato r and 15 min using
+a tip sonicator). Then this mixture was centrifugated for 5-60 min u sing either a desktop
+centrifuge (anglerotortype, 10.000g)oranultracentrifuge (sw ing rotortype, 150.000g),after
+which the upper 80 % of the supernatant solution was collected. We s hall focus on two types
+of SWNT solution: solution centrifugated at 10.000g for 15 min (labele d L) and solution
+centrifugated at 150.000 g for 2 hours (labeled S). We also consider the non centrifugated
+solution (labeled R) for comparison purpose. At this stage, the con centration of solutions R
+and L were adjusted by dilution to match the most intense absorptio n peak around 1200 nm
+(1.03 eV) of sample S to assure the same amount of s-SWNTs in all off t he solutions (cf.
+Figure 1a). Solutions were then spin coated several times onto glas s substrate at 500 RPM
+for 60 s to achieve a layer thickness around 200 nm [16]. Lastly, sam ples were annealed at
+150oC for 15 min.
+Absorption spectra of R, L and S samples in toluene are presented in Figure 1a. Sharp
+peaks in the range of 1050-1400 nm (1.18-0.89 eV, labeled E 11) and 600-900 nm (2.07-
+1.38 eV, E 22) are the optical absorption bands corresponding respectively to the first and
+second transitions between Van Hove singularities in the s-SWNT den sity of states. Peaks
+in the range of 500-600 nm (2.48-2.07 eV, M 11) correspond to the absorption bands of m-
+SWNTs [17]. Peak intensities gradually diminished and eventually vanishe d when all the m-
+SWNTs were removed by ultracentrifugation in sample S. The absorp tion background also
+2drastically reduced after the first low gravity (G) centrifugation. Indeed, the centrifugation
+step permits the effective removal of amorphous carbon and cata lyst impurities initially
+present in the HiPCO SWNT powder. Raman spectra for samples L and S were recorded at
+514.5nm (2.41eV) and theradial breathing mode (RBM) andtangent ial mode (TM) regions
+are reported in Figure 1 b and c, respectively. Raman spectroscop y is a resonant process for
+SWNT, and the RBM peak at 190 cm−1is specific to s-SWNT while the RBM peak at
+270 cm−1is specific to m-SWNT [18]. We observed that m-SWNT peaks present in sample
+L completely vanished in sample S. This is coherent with the TM regions, where the broad
+asymmetric line shape between 1400 and 1600 cm−1of sample L, characteristic of a metallic
+behavior, completely disappears in sample S. Further electric measu rements presented in
+reference [15] confirm the absence of m-SWNT in sample S. Accordin g to absorption and
+Raman measurements, the first low G centrifugation step (sample L ) effectively remove the
+initially present amorphous carbon and catalyst impurities, but also r emove some m-SWNT.
+The high G centrigugation step (sample S) remove all the remaining m- SWNT. Thus, the
+composition of samples L and S differ only by the presence of m-SWNT in sample L.
+The PL excitation-emission maps of each samples were obtained using a Titane Sapphire
+(Ti:Sa) laser pumped by a cw Ar laser. Ti:Sa laser delivers continuous ligh t at a wavelength
+rangefrom 700nm to 840nm (1.77 to 1.48 eV). Photoluminescence sig nal is then recorded at
+room temperature using a monochromator (JobinYvon 550) couple d with a cooled InGaAs
+detector with a 3 nm step. Emission intensity as a function of both ex citation (from 700
+to 840 nm / 1.77 to 1.48 eV) and emission wavelength (from 1000 to 160 0 nm / 1.24 to
+0.78 eV) was recorded. The experimental values were corrected b y taking into account the
+system response. Results are presented in Figure 2a, b and c, eac h map being normalized by
+themaximal intensity. Figure2dshowsthechirality mapforthephot oluminescent nanotubes
+from raw nanotubes in sample R and PFO purified s-SWNT in sample S. We discovered, as
+previously observed by Chen et al. [13], a chiral selectivity of PFO poly mer. Indeed, chirality
+in the starting material were evenly distributed inside the measuring range, with chiral
+angle ranging from 3.96◦in (12,1) tube to 27.8◦in (8,7) tubes. In contrast, PFO-extracted
+nanotubes consist of only two tubes in the measuring range, 73 % of (8,6) and 27 % of (8,7)
+nanotubes [19]. Those two nanotubes possess near armchair stru cture with high chiral angles
+above 25◦. The exact mechanism of PFO’s high chiral selectivity is still unknown, but the
+aromatic structure of the polymer may play a major role in the wrapp ing interaction with
+nanotubes [14].
+In order to determine the influence of the purification process, SW NT photoluminescence
+intensity for an excitation energy of 1.7 eV is displayed in Figure 3. Par ticular care was given
+to obtain the three samples with roughly the same concentration of (8,6) s-SWNT as shown
+on Figure 1a, which permit a comparison of the photoluminescence int ensity level. Further-
+3more, photoluminescence intensity of S and L samples were correct ed by the absorption peak
+intensity at 1.7 eV after background substraction. It was not pos sible to properly remove
+all background contribution in R sample; uncorrected photolumines cence intensities of all
+three samples are presented in the inset of Figure 3. The most strik ing feature is the impres-
+sive increase of (8,6) nanotube photoluminescence intensity with th e degree of purification.
+Indeed, while sample R presents a rather low signal, samples L and S ex hibit PL signal
+enhancement. An increase by a factor of 3 is observed in the PL inte nsity of (8,6) nanotube
+for sample S as compared to L. PL intensities of L and S samples increa se by a factor of 2
+and 6 respectively compared to sample R (unpurified). An analog incr ease was also observed
+for the intensity of the (8,7) nanotube, but as the relative concen tration in (8,7) nanotube
+change between samples, the comparison is not as straightforwar d. It is remarkable that even
+if the concentration of (8,7) tube is higher in the starting material, t he photoluminescence
+intensity is still 3 times higher in the PFO selectively extracted materia l. As the number
+of quenching centers constituted by m-SWNT decreases, most of the exciton recombination
+occurring in the (8,6) nanotubes likely contribute to the luminescenc e phenomenon, leading
+to an intensity increase of S sample by factor of 3 and 6 as compared to L and R samples,
+respectively.
+In conclusion, we have studied the photoluminescence of pure semic onducting SWNT thin
+films, extracted by the PFO centrifugation technique. We found th at the removal of m-
+SWNTs leads to an enhancement of the photoluminescence propert ies. Indeed, after removal
+of m-SWNT, the same quantity of (8,6) s-SWNT displays a 6-fold incre ase in photolumines-
+cence intensity. The use of PFO as a matrix leads to the formation of well isolated nanotube
+thin films. These results are of major importance for the future us e of s-SWNT thin film
+matrix in photonic applications.
+N. Izard thanks the Japan Society for the Promotion of Science fo r financial support.
+References
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+Polymer Structure and Solvent Effects on the Selective Dispe rsion of Single-Walled Car-
+bon Nanotubes , J. Am. Chem. Soc. 130, 3543 (2008)
+15. N. Izard, S. Kazaoui, K. Hata, T. Okazaki, T. Saito, S. Iijima an d N. Minami,
+Semiconductor-enriched single wall carbon nanotube netwo rks applied to field effect tran-
+sistors, Appl. Phys. Lett. 92, 243112 (2008)
+16. Layer thickness was estimated from ellipsometry spectroscop y and direct profilometer
+measurement.
+17. I.W. Chiang, B.E. Brinson, A.Y. Huang, P.A. Willis, M.J. Bronikowski, J .L. Margrave,
+R.E. Smalley and R.H. Hauge, Purification and Characterization of Single-Wall Car-
+bon Nanotubes (SWNTs) Obtained from the Gas-Phase Decompos ition of CO (HiPco
+Process), J. Phys. Chem. B 105, 8297 (2001)
+518. J.-L. Sauvajol, E. Anglaret, S. Rols and L. Alvarez, Phonons in single wall carbon nan-
+otube bundles , Carbon 40, 1697 (2002)
+19. Determined using method described in Ref. [13]
+6Fig. 1. (color online) (a) The optical absorption spectra in toluene, (b) and (c) the RBM and
+TM Raman spectra at 2.41 eV of R, L and S samples. Arrows indicates r espectively position
+of the Raman laser probe (left) and where the concentration adju stment was performed
+(right).
+7Fig.2.(coloronline) Photoluminescence map(emissionwavelength vs excitationwavelength)
+of PFO- embedded SWNT thin film spin-coated onto glass substrate. The layer thickness
+is around 200 nm. (a) Sample R, (b) sample L and (c) sample S. (d) Chir al map showing
+the wrapping preference of PFO (in blue / dark gray) as compared t o raw HiPCO SWNT
+sample (in yellow / light gray).
+80.90 0.95 1.00 1.05 1.10 1.15 1.20
+Transition energy (eV)024681012Photoluminescence (a.u.)(8,6)
+(8,7)S
+L
+0.9 1.0 1.1 1.204812
+S
+L
+R
+Fig. 3. (color online) Photoluminescence intensities of samples L and S which differ in m-
+SWNT concentration. Intensity was normalized taking into account the net intensity of light
+absorbed in s-SWNT at the pumping enegy (1.7eV). Inset: Photolum inescence intensities of
+L and S samples compared to R sample, without normalization.
+9
\ No newline at end of file
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+++ b/1001.0879.txt
@@ -0,0 +1,950 @@
+arXiv:1001.0879v1  [cs.LG]  6 Jan 2010Linear Probability Forecasting
+Fedor Zhdanov and Yuri Kalnishkan
+Computer Learning Research Centre,
+Department of Computer Science,
+Royal Holloway University of London,
+Egham, Surrey, TW20 0EX, UK
+{fedor,yura }@cs.rhul.ac.uk
+Abstract
+Multi-class classification is one of the most important task s in ma-
+chine learning. In this paper we consider two online multi-c lass classi-
+fication problems: classification by a linear model and by a ke rnelized
+model. The quality of predictions is measured by the Brier lo ss function.
+We suggest two computationally efficient algorithms to work w ith these
+problems and prove theoretical guarantees on their losses. We kernelize
+one of the algorithms and prove theoretical guarantees on it s loss. We
+perform experiments and compare our algorithms with logist ic regression.
+1 Introduction
+Onlinepredictionisawideareaofmachinelearning(seeCesa-Bianchi a nd Lugosi,
+2006). Its algorithms can be applied to different data mining problems (see for
+example Freund and Schapire, 1997). Online prediction provides effic ient algo-
+rithms which adapt to a predicted process “on fly”. In online regres sion frame-
+work we assume the existence of some input at each step and try to predict
+an outcome on this input. This process is repeated step by step. We consider
+multi-dimensional Briergame where outcomesand predictions come f rom a sim-
+plex and can be thought of as probability distributions on the vertice s of the
+simplex. If the outcomes are identified with vertices of the simplex th is problem
+can be thought of as the multi-class classification problem of the give n input.
+In the simple case the dependence between the input and its outcom e is
+assumed to be linear; linear regression minimising the expected loss is s tudied
+in statistics. As opposite to the traditional statistical setting, th e learner in
+online prediction does not make any statistical assumptions about t he data
+generating process. Its goal is to predict as well as the best linear function on
+input. Instead of looking for the best linear function, our learner c onsiders all
+linear functions and makes his prediction by mixing them in a certain way at
+each prediction step. We prove theoretical bounds on the cumulat ive loss of the
+learner in comparison with the cumulative loss of the best linear funct ion (wesay the learner competes with these functions). We consider the s quare loss:
+meansquareerroris oneofthe benchmarkmeasuresforclassifica tionalgorithms
+(see Brier, 1950).
+We use Vovk’sAggregatingAlgorithm (a generalizationof the Bayesia nmix-
+ture)to mixfunctions (asin AggregatingAlgorithmRegression,AAR : seeVovk,
+2001). This method has previously been applied to the case when pos sible out-
+comeslieinasegmentoftherealline, andsothepredictionwasone-d imensional.
+We develop two algorithms to solve the problem of multi-dimensional pr edic-
+tion. The first algorithm applies a variant of AAR to predict each coor dinate of
+the outcome separately, and then combines these predictions in a c ertain way to
+get probability prediction. The other algorithm is designed to give pro bability
+predictions directly; these are first computationally efficient online r egression
+algorithm designed to solve linear and non-linear multi-class classificat ion prob-
+lems. We derive theoretical bounds on the losses of both algorithms . We come
+to an unexpected conclusion that the component-wise algorithm is b etter than
+the second one asymptotically, but worse in the beginning of the pre diction
+process. Their performance on benchmark data sets is very similar .
+One component of the prediction of the second algorithm has the me aning
+of a remainder. In practice this situation is quite common. For examp le, in a
+football match either one team wins or the other, and the remainde r is a draw
+(see Vovk and Zhdanov (2008) for online prediction experiments in f ootball).
+When we analyse a precious metal alloy we may look for a description of the
+following kind: the alloy has 40% of gold, 35% of silver, and some addition
+(e.g., copper and palladium). It is common for financial applications to predict
+the direction of the price: the price can go up, down, or stay close t o the
+current value. We perform classification experiments with linear algo rithms
+and compare them with logistic regression.
+A description of the framework can be found in Section 2, descriptio n of the
+algorithms can be found in Section 3, and derivation of the theoretic al bounds
+can be found in Section 4.
+We look for a way to extend the class of experts using the kernel tr ick. We
+kernelize the second algorithm and prove a theoretical bound on its loss. The
+cumulative loss ofthe kernelized algorithm is compared with the cumula tive loss
+of any finite set of functions from the RKHS given by a kernel param eter it uses.
+Kernelization process is described in Section 5. Our experiments are shown in
+Section 6. Section 7 makes the conclusions and shows some possibilitie s for
+prospective work.
+2 Framework
+A game of prediction contains three components: a space Ω of outc omes, a
+decision space Γ, and a loss function λ: Ω×Γ→R. We are interested in the
+generalisation of the Brier game from Brier (1950) where the space of outcomes
+Ω =P(Σ) is the set of all probability measures on a finite set Σ with delements,
+Γ :={(γ1,...,γ d) :/summationtextd
+i=1γi= 1,γi∈R}is a hyperplane in d-dimensional space
+2containing all the outcomes, and for any y∈Ω we define the loss
+λ(y,γ) =/summationdisplay
+σ∈Σ(γ{σ}−y{σ})2.
+For example, if Ω = {1,2,3},ω= 1,γ{1}= 1/2,γ{2}= 1/4, andγ{3}= 1/4,
+λ(ω,γ) = (1/2−1)2+(1/4−0)2+(1/4−0)2= 3/8. Brier loss is one of the most
+important loss functions used to assess the quality of classification algorithms.
+The game of prediction is being played repeatedly by a learner receivin g some
+input vectors xt∈X⊆Rn, and follows prediction protocol 1.
+Protocol 1 Protocol of forecasting game
+L0:= 0.
+fort= 1,2,...do
+Reality announces a signal xt∈X⊆Rn.
+Learner announces γt∈Γ⊆Rd.
+Reality announces yt∈Ω⊆Rd.
+Lt:=Lt−1+λ(yt,γt).
+end for
+We find an algorithm which is capable of competing with all linear function s
+(we call them experts) ξt= (ξ1
+t,...,ξd
+t)′onx:
+ξ1
+t= 1/d+α′
+1xt
+...
+ξd−1
+t= 1/d+α′
+d−1xt (1)
+ξd
+t= 1−ξ1−···−ξd−1= 1/d−/parenleftiggd−1/summationdisplay
+i=1αi/parenrightigg′
+xt,
+whereαi= (α1
+i,...,αn
+i)′, i= 1,...,d−1. In the model (1) the prediction for
+the last component of an outcome is calculated from the predictions for other
+components. Denote α= (α′
+1,...,α′
+d−1)′∈Θ =Rn(d−1). Then any expert
+can be presented as ξt=ξt(α). Let also LT(α) =/summationtextT
+t=1λ(yt,ξt(α)) be the
+cumulative loss of an expert αoverTtrials.
+3 Derivation of the algorithms
+In this section we describe how we apply the Aggregating Algorithm (A A) pro-
+posed in Vovk(1990) to mix experts and make predictions. The algor ithm keeps
+weightsPt−1(dα) for the experts at each prediction step t, and updates them
+by the exponential weighting scheme after the actual outcomes is announced:
+Pt(dα) =βλ(yt,ξt(α))Pt−1(dα), β∈(0,1). (2)
+3Hereβ=e−η, where η∈(0,∞) is a learning rate parameter. This weight
+update ensures that the experts which predict badly at the step treceive less
+weight. The weights are then normalized P∗
+t(dα) =Pt(dα)
+Pt(Θ).
+The prediction of the algorithm is a combination of the experts’ pred ictions.
+It is suggested in Kivinen and Warmuth (1999) that the prediction is s imply
+the weighted average of the experts’ predictions with weights Pt(dα). The Ag-
+gregating Algorithm uses more sophisticated prediction scheme, an d sometimes
+achieves better theoretical performance. It first defines a generalised prediction
+at any step tas a function gt: Ω→Rsuch that
+gt(y) = logβ/integraldisplay
+Θβλ(y,ξt(α))P∗
+t−1(dα) (3)
+for ally∈Ω. It is a weighted average (in a general sense) of the experts’ los ses
+for each possible outcome. It then predicts any γtsuch that
+λ(y,γt)≤gt(y) (4)
+for all possible y∈Ω. If such prediction can be found for any weights distribu-
+tion on experts the game is called perfectly mixable . Perfectly mixable games
+and other types of games are analyzed in Vovk (1998). It is also sho wn there
+that for countable (and thus finite) number of experts the AA ach ieves the best
+possible theoretical guarantees.
+3.1 Proof of mixability
+In this section we prove that our game is perfectly mixable and show a function
+that can be used to give predictions satisfying (4).
+It isshownin Theorem1Vovk and Zhdanov(2008) that the Briergam ewith
+finite number of outcomes is perfectly mixable iff η∈(0,1]. The two authors of
+that paper consider the outcome space of dprobability measures concentrated
+in points of Σ. We denote this space by R(Σ). They consider experts giving
+predictions from all probability measures P(Σ). We need to prove that the
+inequality (4) holds for our experts (1) (who can give predictions ou tside of the
+probability simplex) and our outcome space Ω (the whole probability sim plex,
+not only its vertices). Lemma 2 describes the first part, but first w e need to
+state an additional statement. The following lemma shows that any v ector from
+Rdcan be projected into simplex without increasing the Brier loss.
+Lemma 1. For any ξ= (ξ1,...,ξ d)∈Rdthere exists θ= (θ1,...,θ d)∈ P(Σ)
+such that for any y∈Ωwe have λ(y,θ)≤λ(y,ξ).
+Proof.The Brier loss of a prediction γis a square Euclidean distance between
+γand the actual outcome yin ad-dimensional space. The proof follows from
+the fact that Ω is a convex and closed set in Rd.
+4Lemma 2. LetP(dα)be any probability distribution on Θ. Then for any η∈
+(0,1]there exists γ∈Γsuch that for any y∈ R(Σ)we have
+λ(y,γ)≤logβ/integraldisplay
+Θβλ(y,ξ(α))P(dα).
+Proof.By Lemma 1 for any ξ(α) we can find θ(α)∈ P(Σ) such that the loss of
+experts decreases: λ(y,θ(α))≤λ(y,ξ(α)) for any y∈ R(Σ). Thus we have
+logβ/integraldisplay
+Θβλ(y,θ(α))P(dα)≤logβ/integraldisplay
+Θβλ(y,ξ(α))P(dα)
+for anyy∈ R(Σ). We can take the same prediction γ∈Γ that satisfies the
+necessary inequality with θinstead of ξ. By Theorem 1 in Vovk and Zhdanov
+(2008) such prediction exists for any η∈(0,1] (β∈[e−1,1)).
+A way to convert the generalised prediction into the prediction of AA is
+called a substitution function. We prove that we can use the same su bstitution
+function and the same learning rate parameter ηas for the case of finite number
+of possible outcomes. Such a function is proposed in Vovk and Zhdan ov (2008).
+This is an extension of Lemma 4.1 from Haussler et al. (1998).
+Lemma 3. LetP(dα)be a probability distribution on Θand put
+f(y) = logβ/integraldisplay
+Θβλ(y,ξ(α))P(dα)
+for every y∈Ω. Then if γis such a prediction that λ(z,γ)≤f(z)for any
+z∈ R(Σ)thenλ(y,γ)≤f(y)for anyy∈Ω.
+Proof.For the typographical reasons we will write ξinstead of ξ(α). It is easy
+to ensure that λ(y,γ)−λ(y,ξ) =/summationtext
+σ∈Σy{σ}[λ(zσ,γ)−λ(zσ,ξ)] forzσ{ρ}= 0
+ifσ/ne}ationslash=ρandzσ{ρ}= 1 ifσ=ρ. We also have that λ(y,γ)−f(y)≤0 is
+equivalent to/integraltext
+Θβλ(y,ξ)−λ(y,γ)P(dα)≤1. Thus due to the convexity of the
+exponent function/integraltext
+Γβ/summationtext
+σ∈Σy{σ}[λ(zσ,ξ)−λ(zσ,γ)]P(dα)≤/summationtext
+σ∈Σy{σ}= 1.
+Let us denote the i-th possible outcome from R(Σ) byy{i},i= 1,...,d. We
+use the substitution function defined by the following proposition:
+Proposition 1. Letri=g(y{i}), andx+= max(x,0). Define s∈Rby the
+requirement
+d/summationdisplay
+i=1(s−ri)+= 2.
+If the prediction of the Aggregating Algorithm is given by
+γi=(s−ri)+
+2, i= 1,...,d
+then(4)holds.
+5This function allows us to avoid weights normalization in calculating the
+generalized prediction at each step (avoid∗in the weights distribution), which
+would be computationally inefficient. Suppose we can get only r=gt(y) +C
+instead of gt(y), where Cis the same for all y. Then predictions γtdefined by
+the substitution function from Proposition 1 will be the same as if we c alculated
+the generalized prediction with weights normalization.
+3.2 Algorithm for multidimensional outcomes
+We set the prior weights distribution P0over the set Θ = Rn(d−1)of experts α
+to have the Gaussian density with a parameter a >0:
+(aη/π)n(d−1)/2e−aη/bardblα/bardbl2dα.
+Instead of taking the integral in (3) we get a shifted generalised pr edictionrby
+calculating ri=gT(y{i})−gT(y{d}) (we omit the index Tinrfor brevity).
+Eachcomponent of r= (r1,...,r d) correspondsto one ofthe possible outcomes,
+sord= 0. Other components, i= 1,...,d−1:
+ri= logββgT(y{i})+/summationtextT−1
+t=1gt(yt)
+βgT(y{d})+/summationtextT−1
+t=1gt(yt)= logβ/integraltext
+Θe−ηQ(α,y{i})dα/integraltext
+Θe−ηQ(α,y{d})dα
+where by Q(α,y) we denote the quadratic form:
+Q(α,y) =T/summationdisplay
+t=1d/summationdisplay
+i=1((yi
+t−ξi(xt))2.
+Hereyt= (y1
+t,...,yd
+t) are the outcomes on the steps before TandyT=
+(y1
+T,...,yd
+T) is apossibleoutcome on the step T.
+LetC=/summationtextT
+t=1xtx′
+tben×nmatrix. The quadratic form Qcan be divided
+into a quadratic part, a linear part, and a remainder: Q=Q1+Q2+Q3. Here
+Q1(α,y) =α′Aα
+is a quadratic part of Q(α,y). HereAis a square matrix with n(d−1) rows
+(see the expression for Ain the algorithm below). The linear part is equal to
+Q2(α,y) =h′α−2d−1/summationdisplay
+i=1(yi
+T−yd
+T)α′
+ixT,
+wherehi=−2/summationtextT−1
+t=1(yi
+t−yd
+t)xt,i= 1,...,d−1 make up a big vector h=
+(h′
+1,...,h′
+d−1)′. The remainder is equal to
+Q3(α,y) =T−1/summationdisplay
+t=1d/summationdisplay
+i=1(yi
+t−1/d)2+d/summationdisplay
+i=1(yi
+T−1/d)2.
+Ratio for rican be calculated using the following lemmas. The integral
+evaluates as follows:
+6Lemma 4. LetQ(α) =α′Aα+b′α+c, whereα,b∈Rn,cis a scalar and Ais
+a symmetric positive definite n×nmatrix. Then
+/integraldisplay
+Rne−Q(α)dα=e−Q0πn/2
+√
+detA,
+whereQ0= min α∈RnQ(α).
+The proof of this lemma can be found in Harville (1997, Theorem 15.12.1 ).
+Following this lemma, we can rewrite riasri=F(A,bi,zi), i= 1,...,d−1,
+where
+F(A,bi,zi) = min
+α∈ΘQ(α,yi)−min
+α∈ΘQ(α,yd).
+Variables bi,ziand the precise formula for Fare defined by the following lemma
+Lemma 5. Let
+F(A,b,z) = min
+α∈Rn(α′Aα+b′α+z′α)−min
+α∈Rn(α′Aα+b′α−z′α),
+whereb,z∈RnandAis a symmetric positive definite n×nmatrix. Then
+F(A,b,z) =−b′A−1z.
+Proof.This lemma is proven by taking the derivative of the quadratic forms in
+Fbyαand calculating the minimum: min α∈Rn(α′Aα+c′α) =−(A−1c)′
+4cfor
+anyc∈Rn(see Harville, 1997, Theorem 19.1.1).
+We can see that bi=h+(x′
+T,...,x′
+T,0,x′
+T,...,x′
+T)′∈Rn(d−1), where0is a
+zero-vector from Rn. We also have zi= (−x′
+T,...,−x′
+T,−2x′
+T,−x′
+T,...,−x′
+T)′.
+Thus we can calculate d−1 differences ri, assign rd= 0, and then apply
+the substitution function from proposition 1 to get predictions. Th e resulting
+algorithm is Algorithm 1. We will further call it mAAR (multi-dimensional
+Aggregating Algorithm for Regression).
+3.3 Component-wise algorithm
+In this section we derive the component-wise algorithm. It gives pre dictions for
+each component of the outcome separately, and then combines th em in a special
+way.
+First we explain why we should not directly use the algorithm and the th e-
+oretical bound proposed in Vovk (2001). Vovk’s experts do not allo w us to take
+advantage of the fact that only one outcome is possible to happen a t each mo-
+ment. They are more suitable for the case when each input vector xcan belong
+to many classes simultaneously in case of classification. In other wor ds, they are
+centered around the center 1 /2 of the prediction interval [0 ,1]:ξi= 1/2+αix.
+Assume that the number of outcomes is very large and the distribut ion on ex-
+perts is normal N(0,σ2) with small σ. Then the average experts’ prediction is
+(1/2,...,1/2,1−(d−1)/2)),andtheaveragelossoftheexpertsontrialswiththe
+same outcome y=y{i}(we can take y= (1,0,...,0)) is (d−1)/22+(d−1)2/22.
+7Algorithm 1 mAAR for the Brier game
+Fixn,a >0.C= 0,h= 0.
+fort= 1,2,...do
+Read new xt∈X.
+C=C+xtx′
+t,A=aI+
+2C···C
+.........
+C···2C
+
+Setbi=h+(x′
+t,...,x′
+t,0,x′
+t,...,x′
+t)′, where 0 is a zero-vector from Rnis
+placed at i-th position, i= 1,...,d−1.
+Setzi= (−x′
+t,...,−x′
+t,−2x′
+t,−x′
+t,...,−x′
+t)′, where−2x′
+tis placed at i-th
+position, i= 1,...,d−1.
+Calculate ri:=−b′
+iA−1zi,rd:= 0, i= 1,...,d−1.
+Solve/summationtextd
+i=1(s−ri)+= 2 ins∈R.
+Setγi
+t:= (s−ri)+/2,ω∈Ω, i= 1,...,d.
+Output prediction γt∈ P(Ω).
+Read observation yt.
+hi=hi−2(yi
+t−yd
+t)xt,h= (h′
+1,...,h′
+d−1)′.
+end for
+Components of experts (1) concentrate around the point 1 /d, and so experts
+have the average loss ( d−1)/d2+(1−1/d)2. This loss is smaller than the loss
+of Vovk’s experts for large values of d.
+Our component-wise experts are expressed by
+ξi
+t= 1/d+α′
+ixt, i= 1,...,d. (5)
+Thederivationofthecomponent-wisealgorithm(furthercAARsta ndsforcomponent-
+wise Aggregating Algorithm Regression) is similar to the derivation of A lgo-
+rithm 1 for two outcomes. The initial distribution on each component of ex-
+perts (5) is given by
+(a˜η/π)n/2e−a˜η/bardblαi/bardbl2dαi.
+Note that the value for ˜ ηhere will be different from 1 since the loss function by
+each component is half of the Brier loss λ(y,γ) = (y−γ)2+(1−y−(1−γ))2.
+We will further see that ˜ η= 2. The loss of expert ξ(αi) over the first Ttrials is
+T/summationdisplay
+t=1(yi
+t−1/d−α′
+ixt)2=α′
+i/parenleftiggT/summationdisplay
+t=1xtx′
+t/parenrightigg
+αi−2α′
+i/parenleftiggT/summationdisplay
+t=1(yi
+t−1/d)xt/parenrightigg
++T/summationdisplay
+t=1(yi
+t−1/d)2.
+Instead of the substitution function from Proposition 1 we use the substitution
+function suggested in Vovk (2001) for the one-dimensional game:
+γi
+T=1
+2+gT(0)−gT(1)
+2
+8Therefore, the substitution function can be represented as
+γi
+T=1
+2+1
+2log˜β˜βgT(0)
+˜βgT(1)
+=1
+2+1
+2log˜β/integraltext
+Rne−˜ηα′
+iBαi+2˜ηα′
+i(E+(0−1/d)xT)−˜η(W+1/d2)dαi/integraltext
+Rne−˜ηα′
+iBαi+2˜ηα′
+i(E+(1−1/d)xT)−˜η(W+(1−1/d)2)dαi
+=1
+d+1
+2F/parenleftbigg
+B,−2E−d−2
+dxT,xT/parenrightbigg
+=1
+d+/parenleftiggT−1/summationdisplay
+t=1(yi
+t−1/d)x′
+t+d−2
+2dx′
+T/parenrightigg/parenleftigg
+aI+T/summationdisplay
+t=1xtx′
+t/parenrightigg−1
+xT(6)
+fori= 1,...,d. HereB=aI+/summationtextT
+t=1xtx′
+t,E=/summationtextT−1
+t=1(yi
+t−1/d)xt,W=/summationtextT−1
+t=1(yi
+t−1/d)2,˜β=e−˜η. The transitions are justified using Lemma 4 and
+Lemma 5.
+Then this method projects its prediction onto the prediction simplex such
+that the loss does not increase. We use the projection algorithm su ggested
+in Michelot (1986).
+Algorithm 2 Projection of a point from Rnonto probability simplex.
+Initialize I=∅,x=1∈Rd.
+LetγTbe the prediction vector and |I|is the dimension of the set I.
+while1do
+γT=γT−/summationtextd
+i=1γi
+T−1
+d−|I|;
+γi
+T= 0,∀i∈I;
+Ifγi
+T≥0 for alli= 1,...,dthen break;
+I=I/uniontext{i:γi
+T<0};
+Ifγi
+T<0 for some ithenγi
+T= 0;
+end while
+4 Theoretical bound
+We derive the theoretical bounds for the losses of Algorithm 1 and o f a naive
+component-wise algorithm predicting in the same framework.
+4.1 Component-wise algorithm
+We prove here the theoretical bound for the loss of cAAR. The follo wing lemma
+is the main tool helping us to prove our theorems. It is easy to prove the
+following statement (Lemma 1 from Vovk (2001)):
+Lemma 6. If the learner follows the Aggregating Algorithm in a perfec tly mix-
+able game, then for every positive integer T, every sequence of outcomes of the
+9lengthT, and any initial weights distribution on experts P0(dα)it suffers loss
+satisfying for any α∈Θ
+LT(AA(η,P0))≤logβ/integraldisplay
+ΘβLT(α)P0(dα). (7)
+Proof.We proceed by induction in T: forT= 0 the inequality is obvious, and
+forT >0 we have:
+LT(AA(η,P0))≤LT−1(AA(η,P0))+gT(ωT)
+= logβ/integraldisplay
+ΘβLθ
+T−1P0(dθ)+logβ/integraldisplay
+Θβλ(ωT,ξθ
+t)βLθ
+T−1
+/integraltext
+ΘβLθ
+T−1P0(dθ)P0(dθ)
+= logβ/integraldisplay
+ΘβLθ
+TP0(dθ).
+Here the second equality follows from the inductive assumption, the defini-
+tion (3) of gT, and (2).
+The loss of the component-wise algorithm by one component is bound ed as
+in the following theorem.
+Theorem 1. Let the outcome space in the prediction game be [A,B],A,B∈R.
+Assume experts’ predictions at each step are ξt=C+α′xt, whereα∈Rn,
+C∈Ris the same for all the experts α, and/bardblxt/bardbl∞≤X,∀t. There exists a
+prediction algorithm producing γi∈R,i= 1,...,dsuch that for any a >0,
+every positive integer T, every sequence of input vectors and outcomes of the
+lengthTand any α∈Rnwe have
+T/summationdisplay
+t=1(γt−yt)2≤T/summationdisplay
+t=1(ξt−yt)2+a/bardblα/bardbl2
+2+n(B−A)2
+4ln/parenleftbiggTX2
+a+1/parenrightbigg
+.(8)
+Proof.We need to prove that the game is perfectly mixable (see (4)) and fin d
+the optimal parameter ηfor the algorithm. Implications similar to the ones in
+the proof of Lemma 2 from Vovk (2001) lead to the inequality η≤2
+(B−A)2.
+Clearly, Lemma 6 holds for our case, so we need only to calculate the d ifference
+between the right-hand side of (7)
+logβ/integraldisplay
+Rndα(aη/π)n/2exp/bracketleftigg
+−ηα′/parenleftigg
+aI+T/summationdisplay
+t=1xtx′
+t/parenrightigg
+α
++η2α′/parenleftiggT/summationdisplay
+t=1(yt−C)xt/parenrightigg
+−ηT/summationdisplay
+t=1(yt−C)2/bracketrightigg
+.
+andthelossofthebestexpert α′
+0/parenleftig
+aI+/summationtextT
+t=1xtx′
+t/parenrightig
+α0−2α′
+0/parenleftig/summationtextT
+t=1(yt−C)xt/parenrightig
++
+/summationtextT
+t=1(yt−C)2.Hereα0is the point where the minimum of the quadratic form
+10is attained. Then due to Lemma 4 this difference will be equal to
+1
+2ηlndet/parenleftigg
+I+1
+aT/summationdisplay
+t=1xtx′
+t/parenrightigg
+≤n(B−A)2
+4ln/parenleftbiggTX2
+a+1/parenrightbigg
+.
+We bound the determinant of a symmetric positive definite matrix by t he prod-
+uct of its diagonal elements (see Beckenbach and Bellman (1961), C hapter 2,
+Theorem 7) and use η=2
+(B−A)2.
+Interestingly,thetheoreticalboundfortheregressionalgorith mdependsonly
+on the size of the prediction interval but not on the location of it. It also does
+not depend on the concentration point of experts. We use the com ponent-wise
+algorithm to predict each component separately.
+Theorem 2. If/bardblxt/bardbl∞≤X,∀t,then for any a >0, every positive integer T,
+every sequence of outcomes of the length T, and any α∈Rn(d−1)the lossLTof
+the component-wise algorithm satisfies
+LT≤LT(α)+da/bardblα/bardbl2
+2+nd
+4ln/parenleftbiggTX2
+a+1/parenrightbigg
+. (9)
+Proof.We extend the class of experts in (1) in (5). The algorithm predicts
+each component of the outcome separately. Summing theoretical bounds (8) for
+dcomponents of the outcome, taking αd=−/summationtextd−1
+i=1α′
+i, and using the Cauchy
+inequality /bardbl/summationtextd−1
+i=1αi/bardbl2
+2≤(d−1)/summationtextd−1
+i=1/bardblαi/bardbl2
+2we get the bound. To give proba-
+bility forecasts we can project prediction points on the prediction s implex using
+Algorithm 2. The bound will then hold by Lemma 1.
+4.2 Linear forecasting
+The theoretical bound for the loss of the Algorithm 1 is
+Theorem 3. If/bardblxt/bardbl∞≤X,∀t,then for any a >0, every positive integer T,
+every sequence of outcomes of the length T, and any α∈Rn(d−1)mAAR(2a)
+satisfies
+LT(mAAR(2 a))≤LT(α)+2a/bardblα/bardbl2
+2+n(d−1)
+2ln/parenleftbiggTX2
+a+1/parenrightbigg
+.(10)
+Proof.WeapplymAARwiththeparameter b= 2a. Recallthat C=/summationtextT
+t=1xtx′
+t.
+Following the line of the proof of Theorem 1 with η= 1 we get the theoretical
+bound.
+We can derive a slightly better theoretical bound: in the determinan t ofA
+one should subtract the second block raw from the first one and th en add the
+first block column to the second one, then repeat this d−2 times.
+11Proposition 2. In the conditions of Theorem 3 mAAR( a) satisfies
+LT(mAAR( a))≤LT(α)+a/bardblα/bardbl2
+2
++n(d−2)
+2ln/parenleftbiggTX2
+a+1/parenrightbigg
++n
+2ln/parenleftbiggTX2d
+a+1/parenrightbigg
+.(11)
+The theoretical bound (10) is worse asymptotically by dthan the bound (9)
+of the component-wise algorithm, but it is better in the beginning, es pecially
+when the norm of the best expert /bardblα/bardblis large. This can happen in the impor-
+tant case when the dimension of the input vector is larger than the s ize of the
+prediction set: n >> T.
+5 Kernelization
+In some cases the linear model can be considered not rich enough to describe
+data well, and a more complicated model is needed. We use a popular in c om-
+puter learning kernel trick, firstly applied to the AAR in Gammerman e t al.
+(2004). We derive an algorithm competing with all sets of functions f rom an
+RKHS with d−1 elements.
+5.1 Derivation of the algorithm
+Definition 1. Let us take x1,...,x n∈X. Akernel function is a nonnegative
+function K:Rn×Rn→Rsatisfying/summationtextn
+i,j=1K(xi,xj)ξiξj≥0 for all positive
+integersn, allx1,...,x n∈X, andξ1,...,ξ n∈R.
+An RKHS contains all linear regressors /an}bracketle{tΦ(·),h/an}bracketri}htHdefined by means of a
+feature map (for all the definitions see Sch¨ olkopf and Smola, 2002 ). It can also
+be defined in a different equivalent way as a functional Hilbert space w ith con-
+tinuous evaluation functional ϕ:f∈ F /ma√sto→f(x) for each x∈X. We will use
+the notation cF(x) for the norm of this functional: cF(x) := supf:/bardblf/bardblF≤1|f(x)|
+and for the embedding constant cF:= supx∈XcF(x) and assume cF<∞.
+Our algorithm competes with the following experts:
+ξ1
+t= 1/d+f1(xt)
+...
+ξd−1
+t= 1/d+fd−1(xt) (12)
+ξd
+t= 1−ξ1−···−ξd−1.
+Heref1,...,f d−1∈ Fare any functions from some RKHS F. We start by
+rewriting mAAR in the dual form. Denote
+/tildewideYi=−2(yi
+1−yd
+1,...,yi
+T−1−yd
+T−1,−1/2),
+Yi=−2(yi
+1−yd
+1,...,yi
+T−1−yd
+T−1,0)
+/tildewidek(xT) = (x′
+1xT,...,x′
+TxT)′,
+/tildewideK= (x′
+s,xt)s,tis the matrix of scalar products
+12fori= 1,...,d−1,s,t= 1,...,T. We show that the predictions of mAAR can
+be represented in terms of variables defined above. We will need the following
+matrix property.
+Proposition 3. LetB,Cbe matrices such that the number of rows in Bequals
+to the number of columns in C, and identity matrices I. IfaI+CBandaI+BC
+are nonsingular then
+B(aI+CB)−1= (aI+BC)−1B. (13)
+Proof.This is equivalent to ( aI+BC)B=B(aI+CB). That is true because
+of distributivity of matrix multiplication.
+Let us set A=
+aI+
+2/tildewideK···/tildewideK
+.........
+/tildewideK···2/tildewideK
+
+.
+Lemma 7. On trialTvaluesrifori= 1,...,d−1in mAAR can be represented
+as
+ri=/parenleftig
+/tildewideY1···Yi···/tildewideYd−1/parenrightig
+·A−1/parenleftig
+/tildewidek(xT)′···2/tildewidek(xT)′···/tildewidek(xT)′/parenrightig′
+.(14)
+Proof.ByM= (x1,...,x T) denote a matrix n×Tof column input vectors.
+Let us set
+B=
+2M···M
+.........
+M···2M
+,C=
+M′···0
+.........
+0···M′
+.
+Thenhifrom the algorithm mAAR equals hi=MYi∈Rn. Decompose b′
+i=/parenleftig
+/tildewideY1···Yi···/tildewideYd−1/parenrightig
+C,where only the i-th block uses Yi. The matrix
+Ais equalA=aI+BC.Using proposition 3
+ri=−b′
+iA−1zi=−/parenleftig
+/tildewideY1···Yi···/tildewideYd−1/parenrightig
+·(aI+CB)−1C/parenleftbig
+−x′
+T··· −2x′
+T···−x′
+T/parenrightbig′.
+Note that/tildewideK=M′Mand/tildewidek(xT) =M′xT, thus (14) holds.
+If instead of dot product in /tildewideK,/tildewidek(xT) we can choose a different kernel (clas-
+sical examples of kernels are Gaussian (RBF): K(xi,xj) =e−/bardblxi−xj/bardbl2
+2σ2, Vapnik’s
+polynomial K(xi,xj) = (xi·xj+1)d, etc.). To get predictions one can use the
+same substitution function from Proposition 1. We call this algorithm mKAAR
+(K for Kernelized).
+135.2 Theoretical bound for the kernelized algorithm
+To derive a theoretical bound for the loss of mKAAR we will use the fo llowing
+matrix determinant identity lemma.
+Lemma 8 (Matrix determinant identity) .LetB,Care as in Proposition 3,
+andais a real number. Then det(aI+BC) = det(aI+CB).
+Proof.The proof is by considering a block matrix identity.
+The main theorem follows from the property of RKHS called Represen ter
+theorem (see Sch¨ olkopf and Smola, 2002, Theorem 4.2).
+Theorem 4 (Representer theorem) .Denote by g: [0,∞)→Ra strictly mono-
+tonic increasing function. Assume Xis an arbitrary set, and Fis a Reproduc-
+ing Kernel Hilbert Space of functions on Xwith the given kernel K:X2→
+R. Assume we also have a positive integer Tand an arbitrary loss function
+c: (X×R2)T→R/uniontext{∞}. Then each minimizer f∈ Fof
+c((x1,y1,f(x1)),...,(xT,yT,f(xT)))+g(/bardblf/bardblF)
+admits a representation of the form f(x) =/summationtextT
+i=1αiK(xi,x)for anyx∈Xand
+realsαi,i= 1,...,T.
+The theoretical bound for the loss of mKAAR is proven in the following
+theorem.
+Theorem 5. Assume Xis an arbitrary set of inputs and Fis a Reproducing
+Kernel Hilbert Space of functions on Xwith the given kernel K:X2→R. Then
+for anya >0, anyf1,...,f d−1∈ F, any positive integer T, and any sequence
+of inputs and outputs (x1,y1),...,(xT,yT)
+LT(mKAAR) ≤LT(f)+ad−1/summationdisplay
+i=1/bardblfi/bardbl2
+F+1
+2lndetA (15)
+Here the matrix /tildewideKis a matrix of kernel values K(xi,xj),i,j= 1,...,T.
+Proof.The bound follows from Theorem 3 for mAAR and the Representer
+theorem. Let us first consider the case with scalar product kerne l. Denote
+C=/summationtextT
+t=1xtx′
+t. By Lemma 8 and calculations similar to ones in the proof of
+Lemma 7 we have the equality of determinants. So we can use any oth er kernel
+instead of scalar product to get the term with the determinant. Th e Represen-
+ter theorem assures that the minimum of the expression LT(f)+a/summationtextd−1
+i=1/bardblfi/bardbl2
+F
+byf-s is reached on a linear regressor.
+We can represent the bound (15) in another form which is more familia r
+from the on-line prediction literature:
+14Corollary 1. Under assumptions of Theorem 5 and if we know the number of
+stepsTin advance and are given F >0, the mKAAR reaches the performance
+LT(mKAAR) ≤LT(f)+2cFF/radicalbig
+(d−1)T, (16)
+for anyf1,...,f d−1∈ F:/summationtextd−1
+i=1/bardblfi/bardbl2
+F≤F.
+Proof.Bounding the logarithm of the determinant we have lndet A≤(d−
+1)Tln/parenleftig
+1+2c2
+F
+a/parenrightig
+.We canchoosethevalue for awherethe minimum isachieved:
+a=cF√
+(d−1)T
+F.
+6 Experiments
+We run our algorithms on six real world time-series data sets. In the time series
+we consider there are no signals attached to the outcomes. Howev er we can take
+vectors consisting of previous observations (we shall take ten of those) and use
+them as signals. Data set DEC-PKT1contains an hours worth of all wide-area
+traffic between Digital Equipment Corporation and the rest of the w orld. Data
+set LBL-PKT-41consists of observations of another hour of traffic between the
+Lawrence Berkeley Laboratory and the rest of the world. We tran sformed both
+the data sets in such a way that each observationis the number of p acketsin the
+correspondingnetworkduringafixedtimeintervalofonesecond. Theotherfour
+datasets2(C4,C9,E5,E8) relate to transportation data. Two of them (C9,C11)
+contain low-frequency monthly traffic measures. Two of them (E5,E 8) contain
+high-frequency day traffic measures. On each of these data sets the following
+operations were performed: subtraction of the mean value and div ision by the
+maximum absolute value. The resulting time series are shown in Figure 1 .
+We used ten previous observations as an input vector for tested a lgorithms
+at each prediction step. We are solving the 3-class classification pro blem: we
+predict whether the next value in a time series will be more than the pr evious
+value plus a precision parameter ǫ, less than that value, or lies in the 2 ǫtube
+around the previous value. The precision ǫis chosen to be the median of all the
+changes in a data set. In order to assess the quality of predictions , we calculate
+the cumulative square loss at the last two thirds of each time series ( test set)
+and divide it by the number of examples (MSE). Since we are considerin g the
+online setting, we could calculate the cumulative loss from the beginnin g of each
+time series. However our approach is not sensitive to starting effec ts, it allows
+us to choose the ridge parameter afairly on the training set, and it allows us to
+compare the performance of our algorithms with batch algorithms, which would
+be normally used to solve this problem.
+The square loss on the test set takes into account the quality of an algorithm
+only at the very end of the prediction process, and does not consid er the quality
+during the process. We introduce another quality measure: at eac h step in the
+1Data sets can be found http://ita.ee.lbl.gov/html/traces .html.
+2Data sets can be found http://www.neural-forecasting-com petition.com/index.htm.
+1505001000150020002500300035004000−0.4−0.200.20.40.60.81
+(a) DEC-PKT series05001000150020002500300035004000−0.200.20.40.60.811.2
+(b) LBL-PKT series020406080100120140160180−1−0.8−0.6−0.4−0.200.20.40.60.81
+(c) C4 series0 50 100 150 200 250−1−0.8−0.6−0.4−0.200.20.40.60.8
+(d) C9 series
+0100200300400500600700800−0.4−0.200.20.40.60.81
+(e) E5 series0100200300400500600700800−1−0.8−0.6−0.4−0.200.20.40.6
+(f) E8 series
+Figure 1: Time series from 6 data sets.
+test set we calculate MSE of an algorithm until this step. After all th e steps we
+average these MSEs (AMSE). Clearly, if one algorithm is better than another
+on the whole test set (its total MSE is smaller) but was often worse o n many
+parts of the test set (total MSEs of many parts of the set is large r), this measure
+takes it into account.
+We comparethe performance of our algorithmswith the multinomial lo gistic
+regression (mLog), because it is a standard classification algorithm which gives
+probability predictions:
+γi
+mLog=eθix
+/summationtextd
+i=1eθix
+for all the components of the outcome i= 1,...,d. In our case d= 3. Here pa-
+rameters θ1,...,θ dare estimated from the training set. We apply this algorithm
+in two regimes: batch regime, where the algorithm learns only on the t raining
+set and is tested on the test set (and thus θis not updated on the test set);
+and in the online regime, where at each step new parameters θare found, and
+only one next outcome is predicted. The second regime is more fair to compare
+with online algorithms, but the first one is standard and faster. In b oth regimes
+logistic regression does not have theoretical guarantees on the s quare loss.
+We also compare our algorithms with the simple predictor predicting th e
+average of the ten previous outcomes (and thus it always gives pro bability pre-
+dictions).
+We are not aware of other efficient algorithms for online probability pr e-
+diction, and thus logistic regression and simple predictor as the only b aselines.
+Component-wisealgorithmswhichcouldbe usedforonlineprediction( e.g., Gra-
+dientDescent, Kivinen and Warmuth1997,RidgeRegression,Hoerl and Kennard
+2000), have to use normalization by Algorithm 2. Thus they have to b e applied
+in a different way than they are described in the corresponding pape rs, and can
+not be fairly compared with our algorithms.
+16The ridge for our algorithms is chosen to achieve the best MSE on the
+training set: the first third of each series. The results are shown in Table 1.
+We highlight the most precise algorithms for different data sets. We a lso show
+time needed to make predictions on the whole data set. The algorithm s were
+implemented in Matlab R2007b and run on the laptop with 2Gb RAM and
+processor Intel Core 2, T7200, 2.00GHz.
+As we can see from the table, online methods perform better than t he batch
+method. Online logistic regression performs well, but is very slow. Our al-
+gorithms perform similar to each other and comparable to the online lo gistic
+regression, but are much faster.
+7 Discussion
+Weconsideranimportantgeneralizationoftheonlineclassificationpr oblem. We
+presented new algorithms which give probability predictions in the Brie r game.
+Both algorithms do not involve any numerical integration, and can be easily
+computed. Both algorithms have theoretical guarantees on their cumulative
+losses. One of the algorithms is kernelized and a theoretical bound is proven for
+the kernelized algorithm. We performed experiments with linear algor ithmsand
+showed that they perform relatively well. We compared them with the logistic
+regression: the benchmark algorithm giving probability predictions.
+Competing with linear experts in the case where possible outcomes lie in a
+morethan2-dimensionalsimplexwasnotwidelyconsideredbyotherr esearchers,
+so the comparison of theoretical bounds can not be performed. K ivinen and
+Warmuth’s work Kivinen and Warmuth (2001) includes the case when t he pos-
+sible outcomes lie in a more than 2-dimensional simplex and their algorith m
+competes with all logistic regression functions. They use the relativ e entropy
+loss function Land get a regret term of the order O(/radicalbig
+LT(α)) which is upper
+unbounded in the worst case. Their prediction algorithm is not compu tationally
+efficient and it is not clear how to extend their results for the case wh en the
+predictors lie in an RKHS.
+We can prove lower bounds for the regret term of the order O(d−1
+dlnT) for
+the case of the linear model (1) using methods similar to ones describ ed in Vovk
+(2001), and lower bounds for the regret term of the order O(√
+T) for the case of
+RKHS. Thus we can say that the order of our bounds by time step is o ptimal.
+Multiplicative constants may possibly be improved though.
+Acknowledgments
+Authors are grateful for useful comments and discussions to Ale xey Chernov,
+Vladimir Vovk, and Alex Gammerman. This work has been supported by EP-
+SRC grant EP/F002998/1 and ASPIDA grant from the Cyprus Rese arch Pro-
+motion Foundation.
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+C Michelot. A finite algorithm for finding the projection of a point onto the
+canonical simplex of rn. J. Optim. Theory Appl. , 50:195–200, 1986.
+Bernhard Sch¨ olkopf and Alexander J. Smola. Learning with kernels: Support
+Vector Machines, regularization, optimization, and beyon d. MIT Press, Cam-
+bridge, MA, USA, 2002.
+Vladimir Vovk. Aggregating strategies. In Proceedings of the Third Annual
+Workshop on Computational Learning Theory , pages 371–383, San Mateo,
+CA, 1990. Morgan Kaufmann.
+18Vladimir Vovk. A game of prediction with expert advice. J. Comput. System
+Sci., 56:153–173, 1998.
+Vladimir Vovk. Competitive on-line statistics. International Statistical Review ,
+69:213–248, 2001.
+Vladimir Vovk and Fedor Zhdanov. Prediction with expert advice for t he Brier
+game. In ICML ’08: Proceedings of the 25th International Conference on
+Machine Learning , pages 1104–1111, New York, NY, USA, 2008. ACM.
+19Set/Algorithm MSE AMSE Time
+DEC-PKT
+cAAR 0.45906 0.45822 0.578
+mAAR 0.45906 0.45822 1.25
+mLog 0.46107 0.46265 0.375
+mLog Online 0.45751 0.45762 2040.141
+Simple 0.58089 0.57883 0
+LBL-PKT
+cAAR 0.48147 0.479 0.579
+mAAR 0.48147 0.479 1.266
+mLog 0.47749 0.47482 0.391
+mLog Online 0.47598 0.47398 2403.562
+Simple 0.57087 0.5657 0.016
+C4
+cAAR 0.64834 0.65447 0.015
+mAAR 0.64538 0.65312 0.062
+mLog 0.76849 0.77797 0.016
+mLog Online 0.68164 0.7351 4.328
+Simple 0.69037 0.69813 0.016
+C9
+cAAR 0.63238 0.64082 0.015
+mAAR 0.63338 0.64055 0.063
+mLog 0.97718 0.91654 0.031
+mLog Online 0.71178 0.75558 10.625
+Simple 0.6509 0.65348 0
+E5
+cAAR 0.34452 0.34252 0.078
+mAAR 0.34453 0.34252 0.219
+mLog 0.31038 0.30737 1.109
+mLog Online 0.30646 0.30575 446.578
+Simple 0.58212 0.58225 0
+E8
+cAAR 0.29395 0.29276 0.078
+mAAR 0.29374 0.29223 0.25
+mLog 0.31316 0.30382 0.109
+mLog Online 0.27982 0.27068 83.125
+Simple 0.69691 0.70527 0.016
+Table 1: The square losses and prediction time (sec) of different algo rithms
+applied for time series prediction. cAAR and mAAR state for the deriv ed
+algorithms, mLog states for the logistic regression, mLogOnline sta tes for online
+logistic regression, and Simple stands for the simple average predict or.
+20
\ No newline at end of file
diff --git a/1001.0880.txt b/1001.0880.txt
new file mode 100644
index 0000000000000000000000000000000000000000..93d3ecedd7c23e63794239be00458b5c0e2810ad
--- /dev/null
+++ b/1001.0880.txt
@@ -0,0 +1,973 @@
+ 
+- 1 - 证券成交量/价的行为是否像是一种几率波
+  
+ 
+ 
+石磊磊 a,b,c 
+2 0 0 5年1 0月1 7日
+  
+ 
+a
+北京师范大学管理学院系统科学系
+ 北京100875
+  
+b
+中国科学技术大学近代物理系
+ 合肥230026
+  
+c
+中意人寿保险有限公司北京分公司代理人
+ 北京100738
+  
+E-mail: Shileilei8@yahoo.com.cn  或 leilei.shi@hotmail.com   
+ 
+ 
+摘要 : 考虑到在股票市场中成交金额对成交量和价格波动存在某种约束关系
+ 本文
+作者通过成交金额来研究成交量与价格波动之间的相互关系
+ 我们发现
+ 随着交易
+时间的延长
+ 累计交易量在交易价格区间逐渐地在成交价格平均值附近呈现峰化的
+分布特征
+ 这一特征与体系在此间交易价格涨落的路径
+ 时间序列或总成交量的大
+小无关
+ 为了解释这种量价行为
+ 我们运用物理学的方法
+ 提出了成交能量关系假
+说
+推导出一个不显含时间变量的证券成交量 /价几率波方程
+ 并且得到两组解析的
+成交量随价格变化的分布函数
+ 通过实证检验
+ 我们证明了在股票市场中存在着相
+干特性
+ 初步验证了该模型的有效性
+ 成交量价的行为像是一种几率波
+ 这样
+我们
+试图提出一个适用于描述金融交易市场的
+ 微观和动态的成交价格波动几率的统一理论
+  
+ 
+JEL分类
+ G12; D30; D40  
+关键词
+ 价格波动
+ 成交量峰化
+ 成交量 /价行为
+ 相干
+几率波  
+  
+- 2 -  
+Does Security Transaction Volume-Price Behavior Resemble A 
+Probability Wave? 
+Leilei Shi a,b,c 
+(October 17, 2005) 
+ 
+a Department of Systems Science, Sc hool of Management, Beijing Normal  University, Beijing 100875, China 
+b Department of Modern Physics, University of Sc ience and Technology of China, Hefei 230026, China 
+c Agents, Generali-China Life Insurance Co. Ltd. (Beijing Branch), Beijing 100738, China 
+E-mail addresses: Shileilei8@yahoo.com.cn, or, leilei.shi@hotmail.com  
+ 
+ 
+Abstract 
+ 
+Motivated by how transaction amount constrain trading volume and price volatility in stock 
+market, we, in this paper, study the relation between volume and price if amount of transaction is 
+given. We find that accumulative trading volume gr adually emerges a kurtosis near the price mean 
+value over a trading price range when it takes a longer trading time, regardless of actual price 
+fluctuation path, time series, or total transaction volume in the time interval. To explain the 
+volume-price behavior, we, in terms of physics, pr opose a transaction energy hypothesis, derive a 
+time-independent transaction volume-price probability wave equation, and get two sets of analytical volume distribution eigenfunctions over a trading price range. By empiric test, we show the existence of coherence in stock market and demonstrate the model validation at this early stage. The volume-price behaves like a probability wave. Thus, we attempt to offer a unified, micro, and 
+dynamic wave theory on price volatility probability in financial market. 
+ JEL Classifications : G12; D30; D40  
+Keywords:  Price volatility; V olume kurtosis; V olume-price behavior; Coherence; Probability wave 
+ 
+   
+- 3 - 1. Introduction 
+ 
+Although there are many trading price models in financial market, none of them has the explicit 
+price formation mechanism that is expressed by an analytical expression. Fama [1] and Ross [2] 
+launched efficient market hypothesis and arbitrag e pricing theory, respectively, based on rational 
+trading assumption. Black and Scholes [3], together with Merton [4], derived a Black-Scholes-Merton model in terms of Samuels on’s log-normal process or economic Brownian 
+motion [5] that could be traced to Bachelier’s dissertation regarding an option pricing problem [6]. 
+In addition, Engle [7] formulated ARCH model, which was later developed to GARCH model by Bollerslev [8], to estimate price volatility e rror or nonlinear item. In recent years, some 
+econophysicists begin using formulation in physics to develop asset pricing models in financial 
+market. For example, McCauley and Gunaratne [9 ] showed how the Fokker-Plank formulation of 
+fluctuations can be used with a local volatility to generate an exponential distribution for asset return. 
+In the past 20 years, there was a growing body of research studying price and volume. Gallant et 
+al. [10] undertook a comprehensive investigatio n of price and volume co-movement using daily 
+New York Stock Exchange data from 1928 to 1987. Gervais et al. [11] claimed the existence of 
+high-volume return premium in stock market. Moreover, Zhang [12], an econophysicist, presented an argument for a square-root relationship between price changes and demand. Over the last few years, spin models are used in studying pri ce and volume as the most popular models in 
+econophysics [13]. Plerou et al. [14] applied a spin model and empirically addressed how stock 
+prices respond to changes in demand. They found that large price fluctuations occur when demand 
+is very small. 
+Ausloos and Ivanova [15] studied price and volume by introducing the notion of a generalized 
+kinetic energy. A generalized momentum is also borrowed from classical mechanics. It is defined 
+as the product of normalized transaction volume ti mes the average rate of price change during a 
+price moving average period. They emphasized at the close that these concepts might also serve in 
+a dynamic equation framework. Wang and Pandey [16] followed the same terminology with 
+somewhat different definitions. They defined trading momentum as the product of relative price 
+velocity and a time-dependent “mass”, a normalized trading volume in a time interval, i.e. the volume liquidity. However, traditional literature on price and volume mainly focuses on the correlation between return and total volume (over a trading price range) in a given time interval. 
+Some scholars attempted to explain the behavior of price and volume. Admati and Pfleiderer 
+[17] developed a theory in which concentrated-tra ding patterns arise endogenously as a result of 
+the strategic behavior of liquidity traders and informed traders. Wang [18] used ICAPM to establish a theoretical links between prices and volume. Econophysicists, for example, Gabaix et al. [19] proposed a theory to provide a unified way to understand the power-law tailed 
+distributions of return and volume, the non-normal distributions that have been caught much 
+attention by econophysicists since Mandelbrot’s finding [20]. But current theories credited the correlation between price and volume to a variet y of factors, for examples, (optimal) trading 
+motive and information quality etc. “What is surprising is how little we really know about trading 
+volume” [21]. 
+Soros [22] guessed: In natural sciences, the phenomenon most similar to that in financial 
+economics probably exists in quantum physics, in which scientific observation generates 
+Heisenberg’s uncertainty principle
+ Unfortunately, it is impossible for economics to become 
+science
+  
+Among econophysicists, however, Schaden [23] prudently discussed the possible generic 
+aspects of quantum finance to model secondary financial markets and the challenges we probably 
+had to face in this potential interdisciplinary field. Piotrowski and Sladkowski [24] published a 
+series of papers on quantum finance and currently proposed the price model that uses complex amplitudes whose squared modules describe price movement probabilities, inspired by quantum mechanical evolution of physical particles. Kleinert [25] applied path integrals to the price 
+fluctuation of assets, considering the prices as a function of time. Baaquie et al. [26] developed a 
+derivative pricing model based on a Hamiltonian formulation, and wrote that it was too difficulty to solve Merton-Garman Hamiltonian analytically in most pricing problems [27]. Jimenez and Moya [28], on the other hand, showed that it is possible to obtain quantum mechanics principles using information and game theory etc. These re searches are based on existed formulation and  
+- 4 - principle in quantum mechanics. 
+There is a celebrated dictum in Walls Street : Cash is king. Inspir ed by Soros’s guess and 
+motivated by how transaction amount constrain trading volume and price volatility in stock market, 
+we study the relation between volume and price th rough amount of transaction in terms of physics. 
+Stock market appears a complex system because of  a variety of interacted and coupled trading 
+agents in this open and fully competitive market. Thus, it is probably a key for us to model it successfully how we observe the system and find a simplified methodology.  
+Price is volatile upward and downward to its mean  value in intraday transactions on individual 
+stock. We observe that accumulative trading volu me gradually emerges a kurtosis near the price 
+mean value over a trading price range when it takes a longer trading time, regardless of actual price fluctuation path, time series, or total transaction volume in the time interval [29,30]. 
+Moreover, the volumes are not distributed normally. Whereas some of the distributions appear to 
+be normal, others show to be wave, and the others exhibit to be exponent. These phenomena can not be explained by a current economic and finance mainstream theory—both a rational trading assumption and a price volatility random walk hypothesis.  
+Is the volume-price behavior driven by a kind of restoring force? Unlike previous literature, this 
+paper investigates how total trading volume distributes over a trading price range in a given time 
+interval and why it does in market dynamic pe rspective. Solving the volume-price probability 
+wave equation that is derived from a transaction energy hypothesis rather than an existed formulation in finance or physics, we find two sets of analytical volume distribution 
+eigenfunctions over a trading price range in a given time interval (volume liquidity distribution 
+eigenfunctions). By empiric test, we demonstrat e our model validation at this early stage.  
+The rest of the paper is organized as follows. In section 2, we use the absolute of zero-order 
+Bessel eigenfunction model to fit and test the volume distributions over a trading price range and 
+draw a major conclusion that our observation holds true; In order to explain our observation and 
+volume-price behavior, we propose a transaction en ergy hypothesis and use a potential function to 
+describe price volatility in Section 3; We, then, assume that there is the restoring force that drives 
+trading toward an equilibrium price, and identify the linear potential (energy) that can represent 
+for effective supply-demand quantity restoring force in stock market in Section 4; Section 5 
+constructs a Hamiltonian that is equal to the sum of transaction dynamic energy and potential 
+energy, derives a time-independent security transaction volume-price probability wave equation, and gets two sets of analytical solutions; In Section 6 are analyses, discussions, and possible 
+applications (practical uses); And final are summaries and conclusions. 
+ 
+2. Empirical results 
+ 
+In order to demonstrate that a stock total tran saction shows a volume kurtosis near the price 
+mean value over a trading price range in an intraday time interval, we test a group of individual stock daily trading data using econometric methodology. We collect the first 30 individual stock data samples in Shanghai 180 Index in June, 2003. Each sample is total transaction volume of a stock, which distributes over a trading price range, in an intraday trading time interval. There are 
+total 630 (
+2130× ) samples. However, 11 samples are halt samples and one sample data is lost. 
+Total tested samples are 618. 
+ 
+-20 -10 10 200.20.40.60.81
+ 
+Fig. 1. The absolute zero-order Bessel Eigenfunctions 
+ 
+We use the absolute of zero-order Bessel eigenf unction as a test model (see fig. 1). It is  
+- 5 - i i m m i i p p JC pP ε ω +− = )] ([ )(0 0 ,      ( N iL3,2,1= ) (1) 
+where )] ([0 0 pp Ji m−ω  is a zero-order Bessel eigenfunction, and its absolute normalized is a 
+theoretical volume distribution probability at price ip; mC is a normalized constant; )(i ipP  
+is an observed volume probability that is equal to  the ratio of accumulative trading volume at price 
+ip to total volume over a trading price range; 0p is an equilibrium price if it exists; mω is an 
+eigenvalue constant; And iε is a set of random error [31].  
+Origin 6.0 Professional is friendly used in our te st. Fig. 2 illustrates the fitting results of three 
+typical samples. One appears to be normal, another shows to be wave, and the other exhibits to be exponent.   
+5.64 5.66 5.68 5.70 5.72 5.74 5.76 5.78 5.80 5.82 5.840.000.020.040.060.080.100.120.14Data: Sheet1_D
+Model: Probawave   Chi^2 =  0.00062
+R^2 =  0.73917
+  P1 0.1065 ±0.01008P2 31.67703 ±2.21972P3 5.74198 ±0.0047(Volume) Probability
+Price (yuan)600002
+(2003/06/13)
+ 
+Fig. 2 (a). Fitting goodness in sample 1 
+(Its price mean value is RMB5.74 yuan) 
+ 
+5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.40.000.010.020.030.040.050.060.070.08Data: Sheet1_D
+Model: Probawave   Chi^2 =  0.00012R^2 =  0.596
+  
+P1 0.0457 ±0.00319P2 22.55 ±0.60702P3 6.128 ±0.00486(Volume) Probability
+Price (yuan)000682
+(2002/01/08)
+ 
+Fig. 2 (b). Fitting goodness in sample 2 
+(Its price mean value is RMB6.08 yuan) 
+ 
+10.78 10.80 10.82 10.84 10.86 10.88 10.90 10.92 10.94 10.96 10.98 11.000.000.050.100.150.200.250.30
+Data: Sheet1_D
+Model: Probawave 
+  
+Chi^2 =  0.00074R^2 =  0.82767
+  
+P1 0.2555 ±0.02446P2 211.41957 ±2.65737
+P3 10.85254 ±0.00516(Volume) Probability
+Price (yuan)600008
+(2003/06/09)
+ 
+Fig. 2 (c). Fitting goodness in sample 3 
+(Its price mean value is RMB10.87 yuan) 
+ 
+The empiric test results that 94.34% of total samples shows significance at 95% level. The 
+volume distribution has a main kurtosis over a trading price range in great majority situations. Thus, we conclude that our observation holds true.  
+- 6 - Is there any a kind of restoring force that driv es trading toward an equilibrium price to form 
+those tested patterns? Does the volume-price behavior resemble a probability wave? What is the coherence in this economic system if it does? All of these questions are leading us to find a unified 
+theory to reply and explain them. 
+ 
+3. Transaction energy hypothesis 
+ 
+In this section, we are going to establish a transaction energy hypothesis so that we can study 
+the volume-price behavior in details.  
+The object of our study is accumulative trading volume (i.e. effective supply-demand quantity 
+or transaction volume) distribution over a trading price range in intraday transaction on individual 
+stocks. The reference or the origin of independent  generalized coordinate is the price zero point. 
+And, the benchmark is amount of transaction in a given time interval.  
+A transaction system is defined as a set of accumulative interacting trading in a given time 
+interval, for example, a set of intraday transa ctions on a stock. In order to study collective 
+interaction behavior in stock market, we define transaction volume 
+v. It is accumulative trading 
+volume or effective supply-demand quantity at a corresponding price p, if not specified. Its 
+dimension is [share]. Total transaction volume V is the sum of transaction volume over a trading 
+price range. Its dimension is [share], too. 
+If transaction volume is fixed, but trading time interval is adjusted, then, there is a different 
+effective supply-demand or trading activity in th e market. The shorter it takes time, the more 
+active the transaction is in the market. The balance between supply and demand is more difficult to be changed. The market is more stable. Therefore, we introduce transaction volume liquidity, which is defined as 
+tvvt=,                ( 2 )  
+where t is a trading time interval for the transaction volume, and its dimension is [time]; tv is 
+transaction volume liquidity, and its dimension is [share][time]-1. 
+Similarly, if transaction amount is fixed, but tr ading time interval is adjusted, then, there is 
+different transaction amount liquidity in the market. The shorter it takes time, the better the 
+transaction amount liquidity is in the market. The market is more stable. We introduce transaction amount in a given time interval or transacti on amount liquidity to measure effective cash 
+supply/demand liquidity in the market. It is  
+t t pvtpv
+tmm === ,             ( 3 )  
+where p is price, and its dimension is [currency unit][share]-1; m is transaction amount or the 
+product of transaction volume and price, and its dimension is [currency unit]; And tm is 
+transaction amount liquidity, its dimension is [currency unit][time]-1, and it is benchmark in this 
+study. 
+In order to keep dimension consistence in derivation later, we introduce the notion of 
+transaction energy. It is transaction amount liquid ity rate or transaction amount acceleration at a 
+price. It is defined as 
+ttt tpvtpv
+tmE === ,             ( 4 )  
+where  
+2tv
+tvvt
+tt==              ( 5 )  
+is transaction volume liquidity rate or transaction volume acceleration, and its dimension is 
+[share][time]-2; E is transaction energy (cash liquidity energy) and its dimension is [currency 
+unit][time]-2. 
+The transaction volume probability is 
+VvP= .                ( 6 )   
+- 7 - Let  
+),( ),( ) 1(t t vpW vpTEP PEE + ≡−+≡ .        ( 7 )  
+where we define ),(tvpT  and ),(tvpW  are dynamic energy (distribution energy) and 
+potential energy, respectively. Their dimension is [currency unit][time]-2, too. Potential energy 
+represents the interaction between the volume v and the volume vV− in a given time interval, 
+namely, the interaction between the volume liquidity tv and the volume liquidity t tvV−. 
+When we focus on a stationary trading state, the potential is time independent. Thus, we write 
+equation (7) as  
+0)(2
+=++− pWpVvEt.            ( 8 )  
+Equation (8) is a transaction energy hypothesis in a differential expression. It is that transaction 
+energy is equal to the sum of its dynamic energy  and potential energy over a trading price range, 
+which could be analogous to a conserved energy system in physics. 
+If 1 /=Vv , i.e. there is no price volatility in a given trading time interval, then, 0)(=pW ; 
+If 0)(>pW , then, 1 /<Vv , i.e. there is price volatility in the time interval. The potential 
+energy can describe price volatility.  
+Equation (8) is a holonomic constraint (a generalized velocity independent constraint, e.g. 
+0),(=Φtvp ) at price coordinate. According to classic dynamics [32,33], the number of degrees 
+of freedom is equal to the number of generalized coordinates minus the number of independent 
+equations of constraint. In our study, generalized coordinates are price and transaction volume in a given time interval (transaction volume liquidity ). Thus, the number of independent generalized 
+coordinates is one if we choose price as an independent generalized coordinate. When we do so, it 
+is a one-dimensional problem describing the volume-price behavior as a whole.  
+4. Linear potential (energy) 
+ 
+In Section 3, we propose a transaction energy hypothesis and know that potential energy can be 
+used in describing price volatility. Now, we assume that there is the restoring force that drives 
+trading price toward an equilibrium price in a tr ansaction system. Let us study how the force is 
+produced in a financial economics perspective, and find what can represent for it in terms of a potential. 
+According to economic theory, buying or selling behavior is constrained. Demand quantity is 
+constrained by budget or amount of the cash hold by  buyers, while supply quantity is constrained 
+by the volume of a product or a stock which is ho ld by sellers. Accumulative trading or transaction 
+volume is constrained by both the price and amount of transactions. On the other hand, trading 
+price and its volatility are constrained by amount of transactions and the volume. 
+How does the volume-price behave in a set of intraday transactions on an individual stock? 
+Assume that a stock initial price is its equilibr ium price, and total amount of cash used in buying 
+stock (demand quantity) and total stock shares to be sold (supply quantity) are given. 
+Relative to present price, first, we assume that the price volatility is positive if demand quantity 
+is greater than supply quantity, whereas it is negative if demand quantity is less than supply quantity. 
+Now, let us consider a restoring force in the system. Relative to initial price, if the balance 
+between supply quantity and demand quantity is constantly maintained on intraday transactions, 
+then, the trading price is its initial or equilibrium pr ice. There is no price vo latility at all (Here, the 
+transaction potential energy is equal to zero in the time interval). Any a variable is exclusively determined by the other two among the volume, price, and amount of transaction. 
+Relative to an equilibrium price, supply quantity is frequently not equal to demand quantity 
+because of separate individual buying and selling decisions in stock market. If demand quantity is 
+greater than supply quantity in a short term interv al, then, the trading price is volatile positively 
+and deviates from its equilibrium price. The more th e price deviates from it, the higher the price is. 
+The same amount of remained cash could buy less volume of stock shares. The demand quantity is 
+reduced. When the demand quantity is less than the supply quantity, the price will be volatile 
+negatively and show reversal. It adjusts toward its equilibrium price. When reaching equilibrium  
+- 8 - price, the price volatility will continue in the same direction and deviate from equilibrium again 
+because of price motion inertia. The more the price deviates from equilibrium, the lower the price is. The same amount of remained cash could buy more volume of stock shares. The demand 
+quantity is increased. When the demand quantity is greater than the supply quantity once again, 
+the price will be volatile in a positive direction and adjust toward its equilibrium point again until it returns to its equilibrium price once a more time . Hereafter, the price will repeatedly deviate 
+from its equilibrium price and adjust toward it.  
+In a transaction system, trading price deviates from its equilibrium price because of quantity 
+imbalance between supply and demand in a short term. The deviation changes the imbalance and causes the price reversal. When the price returns to its equilibrium price, it will continue its volatility in the same direction because of price motion inertia until it appears reversal again and 
+then returns to its equilibrium price. It demonstr ates that there exists an effective supply-demand 
+quantity (i.e. transaction volume) restoring force that  drives trading toward its equilibrium price in 
+addition to generalized transactio n force in a transaction system.  
+Because transaction potential energy is proportional to price, it is a linear potential. Moreover, 
+the price moves upward and downward to its equilibr ium price in a set of intraday transactions on 
+individual stock. And, the volume-price behavior is driven by effective supply-demand quantity 
+restoring force. Therefore, the linear potential that can represent for the restoring force is 
+()ppA ppA pW −≈−= ) ( )(0 ,          ( 9 )  
+where 0p is an equilibrium price, p is its price mean value; A is a transaction coefficient, 
+and its dimension is [share][time]-2. 
+Differentiating equation (8) by using equation (4) and (9),  we define transaction restoring force 
+quantitatively in this holonomic constraint system as 
+ttt
+tt R vVv
+Vvv ApWF 
+
+−−=
+
+
+−−=−=∂∂−= 12
+,       ( 1 0 )  
+where RF is restoring force. Thus, it  is clear that coefficient A is the magnitude of effective 
+supply-demand quantity restoring force in a tr ansaction system. The minus sign means that the 
+force is always toward to its equilibrium price. 
+From the definition, equation (10), we can perceive that the restoring force is equal to zero if 
+supply quantity and demand quantity is constantly balanced, i.e. 1 /=Vv  is satisfied. The linear 
+potential can represent for an effective supply-demand restoring force in stock market.  
+ 5. Probability wave equation and its solutions 
+ 
+We have proposed a transaction energy hypothesis in Section 3 and identified the linear 
+potential that can represent for effective supply-demand quantity restoring force in Section 4. 
+Based on both, we will derive a differential equation describing the volume-price behavior and find its solution. 
+Suppose that a transaction volume-price probability wave function 
+)(pψ  is 
+ BiSeR p/)(⋅=ψ ,              ( 1 1 )  
+where R is the wave amplitude, S its action or Hamiltonian principal function describing the 
+volume-price behavior, and B a constant to make its phase dimensionless [34,35]. 
+  From equation (11), we have 
+piB
+pS
+∂∂−=∂∂ ψ
+ψ.              ( 1 2 )  
+Assume that the volume-price behavior satisfies a Hamilton-Jacobi equation in this interacting 
+energy system (see paragraph 5 in discussion 6.2.3. in Section 6) as 
+0=),(+pSpHtS
+∂∂
+∂∂,             ( 1 3 )  
+where we construct the Hamiltonian that is equa l to the sum of transaction dynamic energy and 
+potential energy. It is written as   
+- 9 - )( , ),( pWpSpTpSpH +
+
+
+
+∂∂=∂∂.          ( 1 4 )  
+According to transaction energy hypothesis, equation (7) or (8), the transaction energy is always 
+equal to the sum of transaction dynamic energy and potential energy over a trading price range no matter whether it is a constant. In comparison with transaction energy or potential energy, 
+moreover, dynamic energy (distribution energy) is an error item when total transaction volume is 
+large enough according to probability and statistics, and the error change could be ignored. On the 
+other hand, when we study the error change, we co uld regard both transaction energy and potential 
+energy as “infinite” constants. In addition, our major objective at initial derivation is to identify 
+transaction momentum at price coordinate so that we can study its non-linear or second order item 
+(dynamic or distribution energy) by building its second order differential equation in addition to studying price volatility linear item (potential energy)
+*. The momentum is supposed to be defined 
+in this economic system no matter whether the system is conservative or dissipative. Thus, we 
+could assume that both transaction energy and potential energy are constants at first in our 
+derivation†. After we establish the (momentum second order differential) equation, we, then, study 
+in details in this volume-price probability wave equation whether it is conservative or dissipative (please see its solutions later). 
+Now, we can find a special solution for equation (13) as follow: 
+EtpStpS −= )( ),(1 ,             ( 1 5 )  
+or 
+EtS−=∂∂,               ( 1 6 )  
+where we suppose that the action in wave functi on (11) describe the volume-price behavior in a 
+given time interval i.e. transaction amount liquidity behavior; E is any time and price explicit 
+independent energy (not necessary to be a constant over a trading price range), and its dimension 
+is [currency unit][time]-2.  
+Because the dimension of ),(tpS  or )(1pS  is the same as that of Et or tpv, we can 
+write the action as 
+() β α +−= Et pv tpSt ),( ,            ( 1 7 )  
+where α is any a dimensionless constant and β is any a constant with dimension [currency 
+unit][time]-1; In convenience, let 1=α  and 0=β , then, we define 
+Et pvSt−≡ .              ( 1 8 )  
+By equation (18), we critically define the generalized momentum or transaction momentum at 
+price coordinate as 
+tvpSQ=∂∂≡ .              ( 1 9 )  
+According to action fo rce definition in classical dynamics, we have transaction force TF as 
+ttt
+T vdtdv
+dtdQF ==≡ .             ( 2 0 )   
+Equation (20) shows that the force is transaction volume liquidity rate or transaction volume 
+acceleration. Its demission is [share][time]-2. 
+Substituting the momentum, equation (19), into equation (8), we write the transaction energy 
+hypothesis or the Hamilton-Jacobi equation (13) as 
+                                                           
+* In fact, Engle [7] estimates price volatility error or non-linear  item by ARCH model. But econometrics does not 
+have its analytical expression. 
+† Actually, we also assume that potential energy is a constant at first in the derivation of Sch ǒrdinger’s 
+equation although it is not in great majority cases in physics. We mainly focus on investigating a 
+momentum second order item, kinetic energy, by building a second order differential equation as well [34,35].  
+- 10 - 0=)(+ +2
+pWpS
+VpE
+
+
+
+∂∂− .           ( 2 1 )  
+We, then, construct a transaction energy functional ()ψ,pG  using equation (12) as 
+() ( )
+
+
+
+∂∂
+
+
+
+
+∂∂+ −=p ppVBE W pGψψψψ ψ** ,2
+.       ( 2 2 )  
+According to security trading regu lation, transaction priority is given by price first and time first 
+regardless of information quality and participant’s rationality. If current trading price is cp, then, 
+the next trading priority is offered to minimize the price volatility with respect to cp. Therefore, 
+we have its mathematic expression as 
+0 ),(=∫dpp Gψδ .             ( 2 3 )  
+It is that actual trading price path is chosen  by its energy functional to minimize its wave 
+function ψ with respect to price variations.  
+We, therefore, derive and write a time-independent transaction wave equation as 
+[] 0 ) (0 22 2
+=−−+
+
+
++ ψψψppAEdpd
+dpdpVB,       ( 2 4 )  
+and its natural boundary conditions are 
+0 (0)=ψ , ∞<)(0pψ , and 0)(→+∞ψ . 
+(Please see Appendix A for the derivation of a time-dependent equation) 
+To solve equation (24), we first choose 0p as an origin in ''po− coordinate and a natural 
+unit 12=
+BV. Obviously, the equation is supposed to keep its validation regardless of an origin 
+selection. After solving the equation in a new coordinate, we let 0 ' ppp−=  and, then, get final 
+solution for this problem. 
+We get two sets of analytical solutions from equation (24). If ttvpE '= , then, we have a set of 
+analytical solutions as  
+ )] ([ )(0 0 pp JC pm m m − =ω ψ ,       ( L2,1,0=m ) (25) 
+where )] ([0 0 pp Jm−ω  are zero-order Bessel eigenfunctions, mC are normalized constants; 
+and mω are eigenvalues ( 0>mω ) and satisfy 
+ .2const vVvF FA vtt R T tt m ==+=−=ω    ( L2,1,0=m ) (26) 
+The absolute of function (25) is 
+)] ([ )(0 0 pp JC pm m m − =ω ψ ,      ( L2,1,0=m ) (27) 
+where )(pmψ  is transaction volume density  or probability at price p in a given time interval, 
+i.e. the volume liquidity density or probability (see fig. 1). It has already been tested in Section 2 
+(see discussion 6.2.2. in Section 6). 
+If transaction energy is a consta nt over a trading price range (a conserved system in a physics 
+perspective), then, we have the other set of analytical solutions from equation (24) as follows (see 
+Appendix B): 
+()0 2,1, =)(0ppA mF eC pmppA
+m mm− −⋅−−ψ ,  ( L2,1,0=m ) (28) 
+with  
+ 0.2+1= >=constmEAm
+m ,       ( L2,1,0=m ) (29) 
+where ) ,1,(0ppA mFm− −  is a confluent hypergeometric function or the first Kummer’s  
+- 11 - function; mA is the magnitude of restoring force or eigenvalue constant; mC are normalized 
+constants; m is the order of the eigenfunctions. The absolute of function (28) is  
+()0 2,1, )(0ppA mF eC pmppA
+m mm− −⋅ =−−ψ , ( L2,1,0=m ) (30) 
+where )(pmψ  is transaction volume probability at price pin a given time interval. 
+If 0=m , then, ( )1 2,1,00 0≡−ppA F . The volume distribution )(0pψ  is exponent 
+over a trading price range, which is sufficiently covered by function (27) (Compare Fig. 1 with Fig. 
+3 (a)); The eigenfunctions (30) are plotted in fig. 3, in which 10 2,1,0and m= . 
+ 
+ 
+(a) )(0pψ                                ( b )  )(1pψ  
+ 
+(c) )(2pψ                             ( d )  )(10pψ  
+Fig. 3. The eigenfuntions if the magnitude of re storing force is a consta nt over a trading price 
+range 
+ 
+6. Analyses, Discussions, and Possible Applications 
+ 
+  In this section, we will have some analyses, discussions, and possible applications regarding our model.    
+6.1. Analyses on abnormal volume distribution patterns 
+ 
+In our test, 35 samples among 618 lack significance at 95% level. They are characterized by at 
+least two of trading volume kurtosis over a trading price range (see fig. 4). Let us explain it. 
+Relative to an equilibrium price, the quantity imbalance between supply and demand has been 
+increased too much so as to cause their equilibriu m price to have a step or jump change in those 
+trading samples. Price is volatile around from one eq uilibrium price to another. In such a situation, 
+the volume distribution over a trading price range can be represented by the superposition of two eigenfunctions as follows: 
+() )] ([ )] ([ )(02 0 01 0 pp J pp JC pm m m m − +− = ω ω ψ . ( L2,1,0=m ) (31) 
+Fig. 4 shows the test results of two typical samples fitted by eigenfunctions (31). Among 35 
+samples, 34 samples have significance at 95% level. 
+This test characters an equilibrium price step or jump change from time to time. A transaction 
+state still remains uncertainty if an eigenvalue is known (see Fig. 4 (a)). 
+The rest sample lacking significance is approximately a uniform distribution (see fig. 5). Fig. 5 
+and fig. 6 are the sample test results fitted by  function (31) and the first-order function (30), 
+respectively. If it is fitted by the first order eigenfunctions (30), then, the test shows significance at -15 -10 -5 5 10 150.20.40.60.81
+-40 -20 20 400.20.40.60.81
+-600 -400 -200 200 400 6000.10.20.30.40.5-4 -2 2 40.20.40.60.81 
+- 12 - 95% level (see fig. 6).  
+ 
+5.50 5.52 5.54 5.56 5.58 5.60 5.620.000.020.040.060.080.100.120.140.16Data: Sheet1_D
+Model: Probawave200   
+Chi^2 =  0.00058
+R^2 =  0.72535
+  
+P1 0.13061 ±0.01811P2 166.59 ±8.9586
+P3 5.522 ±0.00443
+P4 5.589 ±0.00533(Volume) Probability
+Price (yuan)600001
+(2003/06/06)
+ 
+Fig. 4 (a). Fitting goodness by the superposition of two functions with an eigenvalue 
+(31 among 34 samples) 
+ 
+11.95 12.00 12.05 12.10 12.15 12.200.000.020.040.060.080.100.120.14Data: Sheet1_D
+Model: Probawave3   Chi^2 =  0.00069R^2 =  0.49625  P1 0.07498 ±0.01252P2 301.20149 ±8.48523P3 11.99797 ±0.01934P4 34.08417 ±3.87234P5 12.11409 ±0.01709(Volume) Probability
+Price (yuan)600018
+(2003/06/24)
+ 
+Fig. 4 (b). Fitting goodness by the superposition of two functions with two eigenvalues 
+ (3 among 34 samples) 
+ 
+6.00 6.05 6.10 6. 15 6.20 6.25 6.300.000.020.040.060.080.10Data: Sheet1_D
+Model: Probawave3   Chi^2 =  0.00035
+R^2 =  0.26511
+  P1 0.09532 ±0.02989P2 309.60762 ±27.27173P3 6.07494 ±0.01338P4 324.66671 ±12.10562
+P5 6.1504 ±0.00784(Volume) Probability
+Price (yuan)600005
+(2003/06/06)
+ 
+Fig. 5. The sample tested by superposition function (31) at 95% level 
+( 29.0 27.02 2=<=critR R ) 
+ 
+6.00 6.05 6.10 6. 15 6.20 6.25 6.300.000.020.040.060.080.10Data: Sheet1_D
+Model: ProbawaveA1   Chi^2 =  0.00028
+R^2 =  0.3523
+  P1 0.11529 ±0.01003
+P2 29.69449 ±2.46174
+P3 6.15089 ±0.00448(Volume) Probability
+Price (yuan)600005
+(20030606)
+ 
+Fig. 6. The sample tested by the first order eigenfunctions at 95% level 
+( 16.0 35.02 2=>=critR R )  
+- 13 -  
+Combining test results in this section and those in Section 2, we conclude that whereas trading 
+price waves around an equilibrium price constantly, the equilibrium price itself characterizes a 
+step or jump change from time to time. An eige nvalue, together with an equilibrium price, 
+determines a stationary state. In addition, price random characteristic is an extremely conditional 
+case in our model. It is the case that its transacti on energy or the magnitude of restoring force is a 
+constant over a trading price range—a conserved ener gy system in a physics perspective. It is rare 
+( %16.06181=  in this paper). 
+ 
+6.2. Discussions 
+ 
+  In this subsection, we mainly discuss transaction momentum, two sets of analytical solutions, and transaction energy hypothesis (together with methodology).  
+6.2.1. Transaction momentum (the volume liquidity) and transaction force 
+ 
+According to equation (19), transaction momentum does be transaction volume liquidity at a 
+price. In stock market, there is no price change at all if there is no actual trading (volume). The 
+volume plays an exclusive role in determining price change and its liquidity controls the rise or 
+drop of an equilibrium price by its weight. This can explain why large price fluctuations occur when demand is very small [14]. When demand is very small, a small trading volume can produce big impact on its equilibrium price, and even caus e its step or jump change. The market is much 
+more unstable.  
+In addition, transaction momentum rate or tran saction volume acceleration is transaction force. 
+The force is defined by equation (20). It describes how we make trading volume, namely, trading behavior in stock market. 
+A skeptical reader may argue why transactio n volume acceleration is not acceleration. Of 
+course, one may define the volume acceleration to be acceleration rather than the force. Let us see 
+what happen if we do so. 1) The transaction volume acceleration is acceleration if an d only if we choose the volume as an 
+independent generalized coordinate. In this way, equation (8) is not a holonomic constrain in 
+the study of volume and price because it includes generalized velocity. Not all generalized 
+force can be derivable by differentiating (to the volume) function (s) in this non-holonomic constraint system [32]. 
+2) If the volume is chosen as an independent generalized coordinate, we would study price as a 
+function of the volume. It would be another research topic. Osborne [36], a pioneer in 
+econophysics, observed that price as a function of volume does not exist empirically, and then explained why volume is the function of price, and this is not invertible. McCauley [37] showed that price as a fu nction of volume does not exist mathematically. 
+3) If the volume is chosen as an independent generalized coordinate, we probably have to study 
+the correlation between the volume and price. In this approach, no analytical solution can be obtained. 
+4) If the volume is chosen as an independent generalized coordinate, it would be inconvenient 
+for us to observe the volume-price behavior because we choose price as an independent 
+coordinate in financial market. 
+Thus, it is reasonable for us to choose price rath er than volume as an independent variable. The 
+transaction volume acceleration is transaction for ce rather than acceleration in equation (4). 
+ 
+6.2.2. Equation solutions 
+ 
+In Section 5, we get two sets of analytical solutions from transaction volume-price probability 
+wave equation (24). One is a set of zero-order Be ssel eigenfunctions (25) if equation (26) is 
+satisfied. In this scenario, transaction energy is not a constant over a trading price range. It is a 
+dissipative energy system in a physics perspective. However, there is coherence between transaction force and restoring force. It is the coherence that the sum of both is equal to an eigenvalue constant over a trading price range. From here, we find that our observed volume  
+- 14 - distribution pattern is the consequence of a kind of coherence. In Section 2, our empiric test has 
+already demonstrated that equation (26) is satisfi ed in great majority situations. The volume-price 
+behaves like a probability wave. 
+The other is a set of eigenfunctions (28) if tr ansaction energy or the magnitude of restoring 
+force is a constant over a trading price rang e. Under such a condition, restoring force is 
+independent on transaction force. There is no coherence. The function can describe volume 
+uniform distribution or price random walk characteristic. 
+In physics, the square of a wave function modulus has the meaning of a probability (density). 
+However, we show that it is not true in this model. The volume distribution kurtosis does not behave as steep as the square of a wave functi on modulus does. A tentative explanation is that 
+function (27) is the consequence of coherence already. 
+ 
+6.2.3. Methodology and transaction energy hypothesis 
+ 
+Unlike traditional approaches, we take a transaction system as a “black box” in this paper. What 
+we need to know is input (total amount of transaction) and output (the volume distribution over a 
+trading price range) regardless of what really happens inside the box. We study accumulative or 
+collective trading behavior rather than we plunge ourselves into dark to study individual trading behavior. Actually, we are unable to know exact price path by a one-dimensional problem in a trading system because of separate individual buy ing and selling decisions in a fully competitive 
+stock market.  
+Equation (7) is a pure mathematical definition on which a universal theory can be based. When 
+we recast it to transaction energy  hypothesis, equation (8), it has some economic and financial 
+meanings. First, when transaction energy is given, its dynamic energy is known, and then, 
+potential energy can be determined by the hypothesis. On the other hand, dynamic energy 
+(distribution energy) can be determined if transaction energy and potential energy are given. We actually propose what dynamic energy is supposed to be over a trading price range in a given potential field when transaction energy is known. By this hypothesis, we study the volume 
+liquidity nonlinear distribution after we choose a linear potential to represent for a restoring force 
+in stock market. 
+Second, the hypothesis is a holonomic constraint at price coordinate. When we choose price as 
+an independent variable, we simplify an apparently two-dimensional problem (volume and price) to one-dimensional. Otherwise, if we had considered price as a function of time, we would have 
+had to study the volume and price separately rather than whole. For example, Ausloos and Ivanova 
+[15], and Wand and Pandey [16] actually searched for the correlation between the volume and price when a generalized velocity is used. In addition, all the generalized forces are derivable by differentiating function(s) in a ho lonomic constraint system [32]. 
+Third, one may confuse that price is static in equations (4), (7), and (8) because it is not a 
+function of time. It is not true. Price fluctuatio n or dynamic behavior does not depend on time but 
+depend on actual trading, i.e. effective supply or demand. There is no price ch ange at all if there is 
+no actual trading in stock market no matter how long it takes. Equations (4), (7), and (8) express 
+transaction energy states at a pric e. Specifically, equation (8) is a differential expression at a price. 
+Price in the equation is dynamic and not static although it is not a function of time. Does a dynamic variable have to be a function of time? 
+Forth, with regard to a question on supposition, equation (13), it is clear that the equation is 
+equivalent to transaction energy hypothesis, equation  (8), if we assume that the action in equation 
+(11) describes the volume-price behavior or equation (16) is satisfied. Can we suppose that the 
+volume function be expressed by a wave function, equation (11)? Actually, we can also derive the same result as that, equation (24), if we assume that the volume function is an exponent function 
+instead of a wave function. Sch ǒrdinger [34] got particle wave equation assuming that particle 
+behavior satisfies an exponent function rather than  a wave function. Therefore, our supposition is 
+equation (8), together with equation (11) in which the action describes volume-price behavior in a 
+given time interval or transaction amount liquidity behavior. There is no need for any an extra supposition at all.   Finally, we examine our hypothesis by its analytical solutions, equations (1 or 27) and (30). By 
+empirical test, we demonstrate that it hold true at this early stage. In my opinion, most interesting 
+and incredible is probably the hypothesis in this paper. For example, can we extend to derive  
+- 15 - Galaxy density distribution in astrophysics [38] or electron density distribution in double-slit 
+electron interference experiment in quantum physics from a similar hypothesis?    
+6.3. Possible applications 
+ 
+There are many possible applications using this effective supply-demand volume-price wave 
+model. For example, we understand that transaction momentum or volume liquidity plays an 
+exclusive role in determining the rise or drop of an equilibrium price through its weight in a 
+transaction system. We can estimate both risk an d opportunity on a market by the momentum. In 
+addition, it is convenient for us to model an entire price volatility distribution very well in which fat tail and cluster exist, and forecast the volatility probability. Trading institutions, moreover, can 
+use the ratio of cash (demand quantity) to the market value of stocks (supply quantity) they hold to 
+measure the risk or opportunity they have in financial market. Whereas it is difficult to distinguish between current behavioral theory and traditional rational theory because of mathematical and predictive similarities [39], furthermore, this model provides a new methodology to study behavioral finance quantitatively because of measurable coherence between transaction force and 
+restoring force. With this model, we are able to simulate actual trading in stock market. For 
+example, how much amount of money are we going to use to trade a stock price from current $5 dollar per share to expected $10 dollar per share in a speculative strategy, or to boost Shanghai Stock Index from present 1000 points to prospect 1500 points in the market strategy that 
+accelerates transaction liquidity, avoids cr isis, and makes the market profitable? 
+ 
+7. Summaries a nd conclusions 
+ 
+We test our observation that there is a stationary transaction volume distribution over a trading 
+price range on individual stocks in an intraday trading time interval. It results that the volume has a main kurtosis near the price mean value in gr eat majority situations, and concludes that our 
+observation holds true. In order to explain the volume-price behavior, we propose a transaction 
+energy hypothesis and find the linear potential that can repres ent for effective supply-demand 
+restoring force. Based on both, we derive a time-independent security transaction volume-price probability wave equation and get two sets of analytical transaction volume distribution eigenfunctions over a trading pri ce range. One is a set of zero-order Bessel eigenfunctions (25) if 
+there is coherence. From this solution, we find that our observed volume distribution pattern is the 
+consequence of a kind of coherence. The other is a set of eigenfunctions (28) if transaction energy 
+or the magnitude of restoring force is a constant over a trading price range. It can describe the volume uniform distribution over a trading price range or price random walk characteristic. By empiric test, we demonstrate the model validation at this early stage. The volume-price behaves 
+like a probability wave. 
+ 
+Acknowledgement 
+ 
+I especially appreciate Suhua Liao, my mother , for understanding and supporting my career 
+constantly.  
+- 16 - Appendix A: Derivation of a time-dependent volume-price probability wave equation 
+ 
+Suppose that a transaction volume-price probability wave function )(pψ  is 
+BiSeR p/)(⋅=ψ ,              ( A - 1 )  
+where R is the wave amplitude, S its action or Hamiltonian principal function, and B a 
+constant to make its phase dimensionless [34,35]. 
+  From equation (A-1), we have 
+piB
+pS
+∂∂−=∂∂ ψ
+ψ             ( A - 2 )  
+and 
+tiB
+tS
+∂∂−=∂∂ ψ
+ψ.              ( A - 3 )  
+Assume that the volume-price behavior satisfies a Hamilton-Jacobi equation in this interacting 
+energy system as 
+0=),(+pSpHtS
+∂∂
+∂∂,             ( A - 4 )  
+where we construct a Hamiltonian that is equal to the sum of transaction dynamic energy and 
+potential energy. It is written as  
+)( , ),( pWpSpTpSpH +
+
+
+
+∂∂=∂∂.          ( A - 5 )  
+In equation (A-5), dynamic energy and potential energy are constrained by transaction energy 
+hypothesis, equation (8). 
+Let us consider two scenarios in a transaction system. If we assume 
+EtS−=∂∂              ( A - 6 )  
+or 
+ψψEtiB=∂∂,              ( A - 7 )  
+then, we get a time-independent equation (24). In equation (A-6), we suppose that the action in 
+function (A-1) describe volume-price behavior or transaction amount in a given time interval.  
+Otherwise, we use equation (A-3) a nd (19), and write equation (A-4) as 
+0)(2
+=+
+
+
+
+∂∂+∂∂− pWpS
+Vp
+tBiψ
+ψ.          ( A - 8 )  
+Making due allowance for the complex nature of ψ, we recast equation (A-8) as 
+0)(*
+=+
+
+
+
+∂∂
+
+
+
+
+∂∂+∂∂− pWpS
+pS
+Vp
+tBiψ
+ψ.         ( A - 9 )  
+We construct a transaction energy functional ),(ψpG  using equation (A-2) as follow: 
+
+
+
+
+∂∂
+
+
+
+
+∂∂+
++∂∂−=p ppVBpWtBi pGψψψψψ
+ψψ** )( ),(2
+.   (A-10) 
+Using equation (23), we thus write a time-dependent volume-price wave equation as 
+ψψψψ)(2 22
+pWBV
+p pptBVi −∂∂+∂∂=∂∂− .         ( A - 1 1 )  
+Obviously, if equation (A-7) is satisfied, then, equation (A-11) is a time-independent equation 
+(24).   
+- 17 - Appendix B 
+ 
+Given a differential equation 
+0='+''1+
+'22
+ψψψ
+
+
+
+−ApE
+dpd
+p dpd         ( B - 1 )  
+and natural boundary conditions 
+∞<)0(ψ  and 0)(→±∞ψ ,           ( B - 2 )  
+where E is a constant and 0 ' ppp−= , find its solution. 
+In equation (B-1), there is a regular singular point at 0='p  and non-regular singular points at 
+∞±='p . 
+For 0'≥p : 
+If 0'→p , then, let  
+ρψ ' )'(0 p p∝ .              ( B - 3 )  
+Substituting (B-3) into (B-1), we have identical equation as 
+0=+)1(ρρρ−  
+or 0=ρ .                ( B - 4 )  
+So, we write  
+ . )'(0 const p∝ψ              ( B - 5 )  
+If ∞→+'p , then, the equation (B-1) is approximately as 
+0=)'(
+')'(
+22
+p A
+dpp d
+∞+∞+−ψψ.           ( B - 6 )  
+Because of eigenvalue 0>A , the asymptotic equation (B-6) has a solution as 
+' ±~)'(pAe p∞+ψ .             ( B - 7 )  
+According to natural boundary condition 0)(+→∞ψ , only '~)'(pAe p−
+∞+ψ  is 
+acceptable.  
+Assume that a general solu tion for equation (B-1) is 
+)'( )'()'( )'( =)'('
+0 pu e pup p ppA⋅∝⋅⋅−
+∞+ψψψ .     (B-8) 
+Substituting the trial solution (B-8) back into equation (B-1), we yield 
+0=)'('+')'(2'1+
+')'(
+22
+pupA E
+dppduAp dppud
+
+
+
+−
+
+
+
+− .    (B-9) 
+Introducing a new variable ' 2= pAξ  into the equation above, we produce 
+() 0=
+221 )() 1(+)(
+22
+ξξξξξξξ u
+AE
+ddu
+dud
+
+−− − ,       ( B - 1 0 )  
+Comparing equation (B-10) with a confluent hypergeometric equation 
+()()()() 0= + ξ22
+ξαξξξγξξuddu
+dud− − ,         ( B - 1 1 )  
+we have 
+1=γ  and 
+
+−
+AE121=α .           ( B - 1 2 )  
+The analytical solution near the origin 0
+ξ  for equation (B-11) is the confluent 
+hypergeometric function or the first Kummer’s function ),,(ξγαF , where 
+()()
+(),!,,
+0k
+k kk
+kF ξγαξγα∑∞
+==  ( 0≠γ  and not a negative integer)   (B-13)  
+- 18 - () ,)() (
+αααΓ+Γ=k
+k             ( B - 1 4 )  
+xxx)1()(+Γ=Γ .              ( B - 1 5 )  
+The function ),,(ξγαF  has the important property that, if )3,2,1,0 ( L=−= mmα , it 
+reduces to a polynomial of order m so as to satisfy natural boundary condition. Hence, the 
+eigenvalues mA from (B-12) are 
+mEAm2+1= , )0>(E            ( B - 1 6 )  
+or 
+22
+)2+1(=
+mEAm .              ( B - 1 7 )  
+Thus, the solution for equation (B-9) is 
+[] ]' 2,1,[ ,1, )'( pA mF mF pum m m −=−= ξ .        ( B - 1 8 )  
+Substituting the solutions (B-18) into (B-8), we have the eigenfunctions )'(pmψ  for equation 
+(B-1) as, 
+() ' 2,1, =)'('pA mF eC pmpA
+m mm−⋅−ψ ,        ( B - 1 9 )  
+where mC is a normalized constant, L,3,2,1,0=m  
+ 
+If 0'≤p , Let ' =' p p− , then, the equation (B-1) is 
+0='+' '1+
+'22
+ψψ ψ
+
+
+
+−− ApE
+pdd
+p pdd.         ( B - 2 0 )  
+Using the same method, we can derive the solution as 
+mE
+mEAm2+1=21=+−*,           ( B - 2 1 )  
+or 
+22
+)2+1(=
+mEAn , ( 0< =E E− ) .          ( B - 2 2 )  
+Their eigenfunctions fo r equation (B-20) are 
+() ' ,1, =)'('pA mF e pmpA
+mm−⋅−ψ         ( B - 2 3 )  
+Combining (B-19) and (B-23), we have a general solution )'(pψ  for equation (B-1) over the 
+range ∞ ∞− +<'<p  as 
+() ' 2,1, =)'('pA mF e pmpA
+mm−⋅−ψ ,        ( B - 2 4 )  
+with  22
+)2+1(=mEAm .             ( B - 2 5 )  
+where L,3,2,1,0=m ; 0>E  if 0'≥p , and 0< =E E−  if 0'≤p . 
+                                                           
+* Since ttvp E '= , 0< =E E−  holds if 0'≤p  
+  
+- 19 - References 
+ 
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+ 
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+arXiv:1001.0882v2  [hep-th]  27 Jan 2010Communications in Mathematical Physics manuscript No.
+(will be inserted by the editor)
+Exotic smooth R4, noncommutative algebras and
+quantization
+Torsten Asselmeyer-Maluga1, Jerzy Kr´ ol2
+1GermanAerospace center,Rutherfordstr.2,12489 Berlin,t orsten.asselmeyer-maluga@dlr.de
+2University of Silesia, ul. Uniwesytecka 4, 40-007 Katowice , iriking@wp.pl
+Received: date / Accepted: date
+Abstract The paper shows deep connections between exotic smoothings of
+smallR4, noncommutativealgebrasoffoliationsandquantization.At first, based
+on the close relation of foliations and noncommutative C⋆-algebraswe show that
+cyclic cohomology invariants characterize some small exotic R4. Certain exotic
+smoothR4’s define a generalized embedding into a space which is K-theoretic
+equivalent to a noncommutative Banach algebra.
+Furthermore, we show that a factor IIIvon Neumann algebra is naturally re-
+lated with nonstandard smoothing of a small R4and conjecture that this factor
+is the unique hyperfinite factor III1. We also show how an exotic smoothing of
+a smallR4is related to the Drinfeld-Turaev (deformation) quantization of th e
+Poisson algebra ( X(S,SL(2,C),{,}) of complex functions on the space of flat
+connections X(S,SL(2,C) over a surface S, and that the result of this quantiza-
+tionistheskeinalgebra( Kt(S),[,])forthedeformationparameter t= exp(h/4).
+This skein algebra is retrieved as a II1factor of horocycle flows which is Morita
+equivalent to the II∞factor von Neumann algebra which in turn determines
+the unique factor III1as crossed product II∞⋊θR∗
++. Moreover, the structure of
+Casson handles determine the factor II1algebra too. Thus, the quantization of
+the Poisson algebra of closed circles in a leaf of the codimension 1 foliat ion of
+S3gives rise to the factor III1associated with exotic smoothness of R4.
+Finally, the approachto quantization via exotic 4-smoothnessis con sidered as
+a fundamental question in dimension 4 and compared with the topos a pproach
+to quantum theories.
+Contents
+1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
+2. Exotic R4and codimension-1 foliation . . . . . . . . . . . . . . . . . . 4
+3. Exotic R4and operator algebras . . . . . . . . . . . . . . . . . . . . . 52 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+4. The connection between exotic smoothness and quantization . . . . . 12
+5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
+A. Non-cobordant foliations of S3detected by the Godbillon-Vey class . . 28
+1. Introduction
+Even though non-standard smooth R4’s exist as a 4-dimensional smooth mani-
+folds, one still misses a direct coordinate-likepresentation which wo uld allow the
+usage of global,exotic smooth functions. Such functions aresmoo th in the exotic
+smoothness structure but fail to be differentiable in a standard wa y determined
+by the topological product of axes. On the way toward uncovering peculiarities
+of exotic R4’s we obtained some unexpected connections of these with quantum
+theories and string theory formalism, see [2,3]. It seems however t hat any com-
+plete knowledge of the connection especially to quantum theories is s till missing.
+The whole complex of problems is strongly connected to a future the ory of quan-
+tum gravity where one has to include exotic smoothness structure s in the formal
+functional integration over the space of metrics or connections. The presented
+paper aims to fill this gap and shed light on the above-mentioned conn ection
+with quantum theories. The relation with string theory will be addres sed in our
+forthcoming paper. At first we will show how exotic smooth R4’s are related
+with certain noncommutative Banach and C⋆-algebras. Then we will discuss the
+main result of the paper: the explanation how the factor IIIvon Neumann al-
+gebra corresponds to small exotic R4and the explicit quantization procedure of
+the Poisson algebra of loops on surfaces driven by a small exotic R4. Moreover,
+the geometric content of the quantization and the structure of t his factor IIIis
+carefully worked out. We observe and discuss the possible relation o f this kind of
+quantization driven by 4-exotics, with the topos approach to (qua ntum) theories
+of physics.
+We have observed in our previous paper [2], that the exoticness of some
+small exotic R4’s is localized on some compact submanifolds and depend on
+the embedding of the submanifolds. Or, a small exotic structures o f theR4is
+determined by the so-called Akbulut cork (a 4-dimensional compact contractible
+submanifold of R4with a boundary), and its embedding given by an attached
+Casson handle. The boundary of the cork is a homology 3-sphere co ntaining a 3-
+sphereS3such that the codimension-1foliationsare determined by the foliatio ns
+ofS3.Wehaveexplainedthisimportantpointindetailsinourpreviouspaper s[2,
+3]. Following this topological situation we want to localize exotica on a co mpact
+submanifold of R4such that we are able to relate the topological data, like
+foliations and Casson handles, with some other structures on the s ubmanifold.
+Thechangesofthestructuresweregraspedalreadybytechniqu esfromconformal
+field theory and WZW models and gerbes on groupoids and can be relat ed with
+string theory and correspond to the changes of exotic smoothne ss onR4(see [2,
+3]). We follow this philosophy further in this paper. Especially we will find :
+–a compact submanifold of R4such that its K-theory is deformed towards
+theK-theory of some noncommutative algebra when changing the stand ard
+smoothness on R4toward exotic one. This compact submanifold is again the
+S3part of the boundary of the Akbulut cork of exotic R4, i.e.S3⊂R4. We
+describethenoncommutativeBanachalgebrawhose K-theoryisthedeformed
+K-theory of the commutative algebra C(S3),S3⊂R4when the smoothnessExotic smooth R4, noncommutative algebras and quantization 3
+is changed from standard to exotic. We say that such an exotic R4contains
+an embedded S3or that exotic R4deforms the K-theoryof the 3-sphere. This
+is the content of section 3.2 and theorem 2.
+–Next, given a codimension-1 foliation ( S3,F) ofS3whereFis an inte-
+grable subbundle F⊂TS3, one can associate a C⋆- algebra C(S3,F) to
+the leaf space S3/Fof the foliation. Hurder and Katok [28] showed that ev-
+eryC∗algebra of a foliation with non-trivial Godbillon-Vey invariant contains
+a factor IIIsubalgebra. Based on the relation of codimension-1 foliations of
+S3and small exotic smooth structures on R4, as established in [2], one has a
+factorIIIalgebra corresponding to the exotic R4’s. In subsection 3.3 we con-
+struct the C⋆- algebra C(M,F) of the foliation with a nontrivial Godbillon-
+Vey invariant and relate the exotic R4to the factor IIIvon Neumann algebra.
+Furthermore we conjecture that the factor IIIis in fact the unique hyperfinite
+factorIII1. In subsection 3.4 we relate the smoothing of R4with the cyclic
+cohomology invariants of the C⋆- algebra C(M,F). This is done via the KK
+theoryofKasparov, allowingthedefinition ofthe Ktheoryofthelea fspaceof
+the foliation, K(V/F), and the analytic assembly map µofConnes and Baum
+using finally the Connes pairing between K theory and cyclic cohomolog y.
+In section 4 we turn to the quantization procedure as related to no nstandard
+smoothings of R4. Based on the dictionary between operator algebra and fo-
+liations one has the corresponding relation of small exotic R4’s and operator
+algebras. This is a noncommutative C⋆algebra which can be seen as the algebra
+of quantum observables of some theory.
+–First, in subsection 3.3.2 we recognized the algebra as the hyperfinit e factor
+III1von Neumann algebra. From Tomita-Takesaki theory it follows that any
+factorIIIalgebraMdecomposes as a crossed product into M=N⋊θR∗
++
+whereNis a factor II∞. Via Connes procedure one can relate the factor III
+foliation to a factor IIfoliation. Then we obtain a foliation of the horocycle
+flow on the unit tangent bundle over some genus gsurface which determines
+the factor II∞. This foliation is in fact determined by the horocycles which
+are closed circles.
+–Next we are looking for a classical algebraic structure which would giv e the
+above mentioned noncommutative algebra of observables as a resu lt of quan-
+tization. The classical structure is recovered by the idempotent o f theC⋆
+algebra and has the structure of a Poisson algebra. The idempoten ts were
+already constructed in subsection 3.3.1 as closed curves in the leaf o f the fo-
+liation of S3. As noted by Turaev [42], closed curves in a surface induces a
+Poisson algebra: Given a surface SletX(S,G) be the space of flat connec-
+tions on G=SL(2,C) bundles on S; this space carries a Poisson structure as
+is shown in subsection 4.2. The complex functions on X(S,SL(2,C)) can be
+considered as the algebra of classical observables forming the Pois son algebra
+(X(S,SL(2,C)),{,}).
+–Next in the subsection 4.3 we find a quantization procedure of the ab ove
+Poisson algebra which is the Drinfeld-Turaev deformation quantizat ion. It is
+shown that the result of this quantization is the skein algebra ( Kt(S),[,])
+for the deformation parameter t= exp(h/4) (t=−1 corresponds to the
+commutative Poisson structure on ( X(S,SL(2,C)),{,})).
+–This skein algebra is directly related to the factor III1von Neumann algebra
+derived from the foliation of S3. In fact the skein algebra is constructed in4 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+subsection 4.4 as the factor II1algebra Morita equivalent to the factor II∞
+which in turn determines the factor III1of the foliation.
+–Next, in subsection 4.5weshowthat atthe levelof4-exoticsmooth structures
+main building blocks of these, i.e. Casson handles, determine the fact orII1
+algebras. This was already shown in [1,4].
+–Finally, in subsection 4.6 we argue that the appearance of the facto rIII1
+and quantization in context of small exotic R4’s can be lifted to a general
+and fundamental tool applicable for local relativistic algebraic quan tum field
+theories in dimension 4. This is quite similar to the topos approach to (q uan-
+tum) theories as has been recently actively developed (see e.g. [18, 33,27]).
+We show that under some natural suppositions regarding the role o f the fac-
+torIII1and the quantization of the Poisson algebra ( X(S,SL(2,C)),{,}) of
+complex functions on X(S,SL(2,C), in field theories, our 4-exotics approach
+can be seen as a way how to reformulate quantum theories and make them
+classical. The price to pay is the change of smoothing of R4and this works
+in dimension 4. In the topos approach one is changing logic and set the ory
+into intuitionistic ones valid in topoi and this is quite universal procedu re.
+However, we think that the specific choice of dimension 4 by our 4-ex otics
+formalismcanbe particularlywellsuited to thetaskofquantizationo fgravity
+in dimension 4 (cf. [32,31,4]).
+Then we have a closed circle: the exotic R4determines a codimension-1 foliation
+of a 3-sphere which produces a factor III1used in the algebraic quantum field
+theory (see [12]) as vacuum. In [2] we constructed a relation betwe en the leaf
+space of the foliation and the disks in the Casson handle. Thus the co mplicated
+structure of the leaf space is directly related to the complexity of t he Casson
+handle making them to a noncommutative space.
+2. Exotic R4and codimension-1 foliation
+Here we present the main line of argumentation in our previous paper [2]:
+1. In Bizacas exotic R4one starts with the neighborhood N(A) of the Akbulut
+corkAin the K3 surface M. The exotic R4is the interior of N(A).
+2. This neighborhood N(A) decomposes into Aand a Casson handle represent-
+ing the non-trivial involution of the cork.
+3. From the Casson handle we construct a grope containing Alexand ers horned
+sphere.
+4. Akbuluts construction gives a non-trivial involution, i.e. the doub le of that
+construction is the identity map.
+5. From the grope we get a polygon in the hyperbolic space H2.
+6. This polygon defines a codimension-1 foliation of the 3-sphere insid e of the
+exoticR4with an wildly embedded 2-sphere, Alexanders horned sphere. This
+foliation agreeswith the correspondingfoliationofthe homology3-s phere∂A.
+This codimension-1 foliations of ∂Ais partly classified by the Godbillon-Vey
+class lying in H3(∂A,R) which is isomorphic to H3(S3,R).
+7. Finally we get a relation between codimension-1foliations ofthe 3-s phereand
+exoticR4.
+This relation is very strict, i.e. if we change the Casson handle then we must
+change the polygon. But that changes the foliation and vice verse. Finally weExotic smooth R4, noncommutative algebras and quantization 5
+obtained the result:
+The exotic R4(of Bizaca) is determined by the codimension-1 foliations w ith
+non-vanishing Godbillon-Vey class in H3(S3,R3)of a 3-sphere seen as subman-
+ifoldS3⊂R4.
+3. Exotic R4and operator algebras
+In our previous paper [2] we uncover a relation between an exotic (s mall)R4
+and non-cobordant codimension-1 foliations of the S3classified by Godbillon-
+Vey classas element of the cohomologygroup H3(S3,R). By using the S1-gerbes
+it was possible to interpret the integer elements H3(S3,Z) as characteristic class
+of aS1-gerbe over S3. We also discuss a possible deformation to include the full
+real case as well. Here we will use the idea to relate an operator algeb ra to the
+foliation. Then the invariants of the operator algebra will reflect th e invariants
+of the foliation.
+3.1. Twisted K-theory and algebraic K-theory . It is known that ordinary K-
+theory of a (compact) manifold Mis the algebraic K-theory of the (commuta-
+tive) algebra of continuous, complex valued functions on M,C(M) ([17], Chap.
+2, Sec.1). Then twisted K-theory of M(w.r.t. the twisting [ H]∈H3(M,Z)) can
+be similarly determined as an algebraic K-theoryof some generalizedalgebra. In
+fact this algebra must be a noncommutative (non-unital) Banach alg ebra which
+we are going to describe now [6].
+To describe twisted K-theory on a manifold Mone can follow the idea of
+ordinarynon-twisted K-theoryto construct a (representation)space Asuch that
+K0(M) = [M,A] where [ M,A] is the space of homotopy classes of continuous
+maps. As shown by Atiyah [5], the representationspace for the K-theory of Mis
+the topological space Fred(H) of Fredholm operators in an infinite dimensional
+Hilbert space H, with the norm topology . Thus
+K0(M) = [M,Fred(H)].
+This is based on the fact that a family of deformations of Fredholm op erators
+parametrizedby Mwhosekerneland co-kernelhavelocallyconstantdimensions,
+are given by vector bundles on M. The formal difference of these vector bundles
+determines the element of K0(M).
+The twisting of this K-theorycan be performed by the elements of H3(M,Z).
+One ofthe interpretationsofthese integral3-rdcohomologyclas sesis by the pro-
+jective, infinite dimensional bundles on M, namely given a class [ H]∈H3(M,Z)
+we can represent it by a projective PU(H) bundle Ywhose class is [ H]. Here
+U(H) is the group of unitary operators on H. Let us see briefly how it is possible
+[6,3].
+The classifying space of the third cohomology group of Mis the Eilenberg-
+MacLanespace K(Z,3).Theprojectiveunitarygroupon H,PU(H) =U(H)/U(1),
+can now be determined. A model for K(Z,3) is the classifying space of PU(H),
+i.e.,K(Z,3) =BPU(H).Thismeansthat H3(M,Z) = [M,K(Z,3)] = [M,BPU (H)]
+and the realization of H3(M,Z) is as follows:
+Isomorphism classes of principal PU(H)bundles over Mcorrespond one to
+one to the classes from H3(M,Z).6 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+Now let us associate to a class [ H] (torsion or not) a PU(H) bundle Yrep-
+resenting the class. Let Fred(H) acts on PU(H) by conjugations .We can form
+an associated bundle
+Y(Fred) =Y⊗PU(H)Fred(H)
+Let [M,Y(Fred)] denote the space of all homotopy classes of sections of the
+bundleY(Fred). Then one can define the twisted K-theory:
+Definition 1. The twisted by [H]∈H3(M,Z)K-theory of M, i.e.K(M,[H])
+is given by the homotopy classes [M,Y(Fred)]of the sections of Y(Fred), i.e.
+K(M,[H]) = [M,Y(Fred)] (1)
+Now we are searching for bundles of algebras on Mwhose sections would
+be an algebra such that its K-theory is the K(M,[H]). One natural candidate
+is the bundle End(P) of endomorphisms of the projective PU(H)-bundle P
+representing the class [ H]∈H3(M,Z). However, this choice gives rise to the
+trivialK-theory [6]. Instead one should take the algebra Kof compact operators
+inHwith the norm topology. This is a noncommutative Banach algebra with out
+unity. Therefore, we attach to the projective PU(H) bundle P, representing
+[H]∈H3(M,Z), the bundle KPof non-unital algebras. The fiber of the bundle
+KPatx∈Mis the algebra of compact operators acting on Px. The algebra
+of sections of the bundle KPis an noncommutative Banach non-unital algebra
+ΓKP. Now one obtains the result ([6], Definition 3.4 and Theorem 3.2 of [37]) :
+Theorem 1. LetMbe a compact manifold and [H]∈H3(M,Z), the group
+K(M,[H])is canonically isomorphic to the algebraic K-theory of the noncom-
+mutative non-unital Banach algebra ΓKP.
+3.2. Exotic R4andnoncommutative Banach algebras. Theinterpretationoftwisted
+K-theory for a manifold Mas the algebraic K-theory of some noncommutative
+Banach algebra can be seen as deformation of a space towards the noncommu-
+tative space using the twisting of K-theory. We are interested in the following
+special smooth submanifolds of M:
+Definition 2. We say that a compact submanifold L⊂Mof a manifold Mis
+embedded in Mas a noncommutative subspace orembedded in a generalized
+smooth sense when the following two conditions hold
+1.Lis embedded in Min ordinary sense,
+2. when a smooth structure on Mis changed, the K-theory of Lis deformed
+toward the algebraic K- theory of some noncommutative Banach algebra.
+From [3], Theorem 2, we know that small exotic R4’s (corresponding to the
+integral 3-rd cohomologies of S3) deform K-theory of S3toward twisted K-
+theory. Twisted K-theory of the compact manifold S3by [H]∈H3(S3,Z), is in
+turn the K-theory of the noncommutative Banach algebra ΓKPas we explained
+in the previous subsection.
+In our case of S3we have the twisted K-theory with twist τ=k[]∈
+H3(S3,Z):Exotic smooth R4, noncommutative algebras and quantization 7
+Kτ+n(S3) =/braceleftBigg
+0,n= 0
+Z/k,n= 1(2)
+which can be interpreted as the algebraic K-theory of the noncommutative al-
+gebra. Finally we can formulate:
+Theorem 2. Small exotic R4, corresponding to the integral 3-rd cohomologies
+ofS3, deform the K-theory of S3toward the K-theory of some noncommutative
+algebra. Thus some small non-standard smooth structuresof R4deform embedded
+S3towards a noncommutative space.
+The deformation is performed by using the commutative algebra of c omplex-
+valued continuous functions on S3and changing it to the noncommutative non-
+unital Banach algebra of sections of the bundle KPonS3:
+δP:C(S3)→ΓKP.
+HerePis the projective bundle on S3whose Dixmier-Douady class is [ H]∈
+H3(S3,Z) and the deformed exotic smooth R4is determined by the same class
+(see Subsec. 3.1). We see that the exotic smoothness of R4islocalized onS3
+making it a noncommutative space, while the standard smooth R4corresponds
+rather to the ordinary, non-twisted space S3⊂R4and the commutative algebra
+C(S3).
+The above case of small exotic R4and embedded S3in it, is the example
+of the generalized embedding in a sense of our definition 2 and this is th e only
+known example to us of this phenomenon. In fact, any other Rn,n/\e}a⊔io\slash= 4 excludes
+any generalized smooth embedding. There is a possibility to be more ex plicit in
+the description of this generalized embeddings, namely by generalize d Hitchin’s
+structures on S3and their relation to exotics [2]. However we do not address
+this issue here.
+3.3. Leaf space and factor III C∗-algebras. Givenafoliation( M,F)ofamanifold
+M, i.e. an integrable subbundle F⊂TMof the tangent bundle TM. The leaves
+Lof the foliation ( M,F) are the maximal connected submanifolds L⊂Mwith
+TxL=Fx∀x∈L. We denote with M/Fthe set of leaves or the leaf space.
+Now one can associate to the leaf space M/FaC∗algebraC(M,F) by using
+the smooth holonomy groupoid Gof the foliation. For a codimension-1 foliation
+there is the Godbillon-Vey invariant [23] as element of H3(M,R). Hurder and
+Katok [28] showed that the C∗algebra of a foliation with non-trivial Godbillon-
+Vey invariant contains a factor IIIsubalgebra. In the following we will construct
+thisC∗algebra and discuss the factor IIIcase.
+3.3.1. The smooth holonomy groupoid and its C∗algebra. Let (M,F) be a foli-
+ated manifold. Now we shall construct a von Neumann algebra W(M,F) canoni-
+cally associated to ( M,F) and depending only on the Lebesgue measure class on
+thespace X=M/Fofleavesofthefoliation.Theclassicalpointofview, L∞(X),
+will onlygive the center Z(W) ofW. Accordingto Connes [17], we assign to each
+leafℓ∈Xthe canonical Hilbert space of square-integrable half-densities L2(ℓ).8 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+This assignment, i.e. a measurable map, is called a random operator fo rming a
+von Neumann W(M,F). The explicit construction of this algebra can be found
+in [16]. Here we remark that W(M,F) is also a noncommutative Banach algebra
+which is used above. Alternatively we can construct W(M,F) as the compact
+endomorphismsofmodules overthe C∗algebraC∗(M,F) ofthe foliation( M,F)
+also known as holonomy algebra. From the point of view of K theory, b oth al-
+gebrasW(M,F) andC∗(M,F) are Morita-equivalent to each other leading to
+the same Kgroups. In the following we will construct the algebra C∗(M,F) by
+using the holonomy groupoid of the foliation.
+Given a leaf ℓof (M,F) and two points x,y∈ℓof this leaf, any simple path
+γfromxtoyon the leaf ℓuniquely determines a germ h(γ) of a diffeomorphism
+from a transverse neighborhood of xto a transverse neighborhood of y. The
+germ of diffeomorphism h(γ) thus obtained only depends upon the homotopy
+class ofγin the fundamental groupoidofthe leaf ℓ, and is called the holonomyof
+the path γ. The holonomy groupoid of a leaf ℓis the quotient of its fundamental
+groupoid by the equivalence relation which identifies two paths γandγ′from
+xtoy(both in ℓ) iffh(γ) =h(γ′). The holonomy covering ˜ℓof a leaf is the
+covering of ℓassociated to the normal subgroup of its fundamental group π1(ℓ)
+given by paths with trivial holonomy. The holonomy groupoid of the fo liation is
+the union Gof the holonomy groupoids of its leaves.
+Recall a groupoid Gis a category where every morphism is invertible. Let G0
+be a set of objects and G1the set of morphisms of G, then the structure maps
+ofGreads as:
+G1t×sG1m→G1i→G1s
+⇒
+tG0e→G1 (3)
+wheremis the composition of the composable two morphisms (target of the fi rst
+is the source of the second), iis the inversion of an arrow, s, tthe source and
+target maps respectively, eassigns the identity to every object. We assume that
+G0,1are smooth manifolds and all structure maps are smooth too. We re quire
+that the s, tmaps are submersions, thus G1t×sG1is a manifold as well. These
+groupoids are called smoothgroupoids.
+Given an element γofG, we denote by x=s(γ) the origin of the path γ
+and its endpoint y=t(γ) with the range and source maps t,s. An element γ
+ofGis thus given by two points x=s(γ) andy=r(γ) ofMtogether with an
+equivalence class of smooth paths: the γ(t),t∈[0,1]withγ(0) =xandγ(1) =y,
+tangent to the bundle F(i.e. withd
+dtγ(t)∈Fγ(t),∀t∈R) identifying γ1andγ2
+as equivalent iff the holonomy of the path γ2◦γ−1
+1at the point xis the identity.
+The graph Ghas an obvious composition law. For γ,γ′∈G, the composition
+γ◦γ′makes sense if s(γ) =t(γ). The groupoid Gis by construction a (not
+necessarily Hausdorff) manifold of dimension dim G= dimV+dimF. We state
+thatGis a smooth groupoid, the smooth holonomy groupoid .
+Then the C∗algebraC∗
+r(M,F) of the foliation ( M,F) is theC∗algebraC∗
+r(G)
+of the smooth holonomy groupoid G. For completeness we will present the ex-
+plicit construction (see [17] sec. II.8). The basic elements of C∗
+r(M,F)) are
+smooth half-densities with compact supports on G,f∈C∞
+c(G,Ω1/2), where
+Ω1/2
+γforγ∈Gis the one-dimensional complex vector space Ω1/2
+x⊗Ω1/2
+y, where
+s(γ) =x,t(γ) =y, andΩ1/2
+xis the one-dimensional complex vector space ofExotic smooth R4, noncommutative algebras and quantization 9
+maps from the exterior power ΛkFx,k= dimF, toCsuch that
+ρ(λν) =|λ|1/2ρ(ν)∀ν∈ΛkFx,λ∈R.
+Forf,g∈C∞
+c(G,Ω1/2), the convolution product f∗gis given by the equality
+(f∗g)(γ) =/integraldisplay
+γ1◦γ2=γf(γ1)g(γ2)
+Then we define via f∗(γ) =f(γ−1) a∗operation making C∞
+c(G,Ω1/2) into a
+∗algebra.Foreachleaf Lof(M,F)onehasanaturalrepresentationof C∞
+c(G,Ω1/2)
+on theL2space of the holonomy covering ˜LofL. Fixing a base point x∈L,
+one identifies ˜LwithGx={γ∈G, s(γ) =x}and defines the representation
+(πx(f)ξ)(γ) =/integraldisplay
+γ1◦γ2=γf(γ1)ξ(γ2)∀ξ∈L2(Gx).
+The completion of C∞
+c(G,Ω1/2) with respect to the norm
+||f||= sup
+x∈M||πx(f)||
+makes it into a C∗algebraC∗
+r(M,F). Among all elements of the C∗algebra,
+there are distinguished elements, idempotent operators or proje ctors having a
+geometric interpretation in the foliation. For later use, we will const ruct them
+explicitly (we follow [17] sec. II.8.βclosely). Let N⊂Mbe a compact subman-
+ifold which is everywhere transverse to the foliation (thus dim( N) = codim( F)).
+A small tubular neighborhood N′ofNinMdefines an induced foliation F′of
+N′overNwith fibers Rk, k= dimF. The corresponding C∗algebraC∗
+r(N′,F′)
+is isomorphic to C(N)⊗ KwithKtheC∗algebra of compact operators. In
+particular it contains an idempotent e=e2=e∗,e= 1N⊗f∈C(N)⊗ K,
+wherefis a minimal projection in K. The inclusion C∗
+r(N′,F′)⊂C∗
+r(M,F) in-
+duces an idempotent in C∗
+r(M,F). Now we consider the range map tof the
+smooth holonomy groupoid Gdefining via t−1(N)⊂Ga submanifold. Let
+ξ∈C∞
+c(t−1(N),s∗(Ω1/2)) be a section (with compact support) of the bundle of
+half-density s∗(Ω1/2) overt−1(N) so that the support of ξis in the diagonal in
+Gand/integraldisplay
+t(γ)=y|ξ(γ)|2= 1∀y∈N.
+Then the equality
+e(γ) =/summationdisplay
+s(γ)=s(γ′)
+t(γ′)∈N¯ξ(γ′◦γ−1)ξ(γ′)
+defines an idempotent e∈C∞
+c(G,Ω1/2)⊂C∗
+r(M,F). Thus, such an idempotent
+is given by a closed curve in Mtransversal to the foliation.10 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+3.3.2. Some information about the factor IIIcase.In our case of codimension-1
+foliations of the 3-sphere with nontrivial Godbillon-Vey invariant we h ave the
+result of Hurder and Katok [28]. Then the corresponding von Neuma nn algebra
+W(S3,F) contains a factor IIIalgebra. At first we will give an overview about
+the factor III.
+Remember a von Neumann algebra is an involutive subalgebra Mof the
+algebra of operators on a Hilbert space Hthat has the property of being the
+commutant of its commutant: ( M′)′=M. This property is equivalent to saying
+thatMis an involutive algebra of operators that is closed under weak limits.
+A von Neumann algebra Mis said to be hyperfinite if it is generated by an
+increasing sequence of finite-dimensional subalgebras. Furtherm ore we call Ma
+factor if its center is equal to C. It is a deep result of Murray and von Neumann
+that every factor Mcan be decomposed into 3 types of factors M=MI⊕
+MII⊕MIII. The factor Icase divides into the two classes InandI∞with the
+hyperfinite factors In=Mn(C) the complex square matrices and I∞=L(H)
+the algebra of all operators on an infinite-dimensional Hilbert space H. The
+hyperfinite IIfactors are given by II1=CliffC(E), the Clifford algebra of
+an infinite-dimensional Euclidean space E, andII∞=II1⊗I∞. The case III
+remained mysterious for a long time. Now we know that there are thr ee cases
+parametrized by a real number λ∈[0,1]:III0=RWthe Krieger factor induced
+by an ergodic flow W,IIIλ=Rλthe Powers factor for λ∈(0,1) andIII1=
+R∞=Rλ1⊗Rλ2the Araki-Woods factor for all λ1,λ2withλ1/λ2/∈Q. We
+remark that all factor IIIcases are induced by infinite tensor products of the
+other factors. One example of such an infinite tensor space is the F ock space in
+quantum field theory.
+But now we are interested in an explicit construction of a factor IIIvon
+Neumann algebra of a foliation. The interesting example of this situat ion is
+given by the Anosov foliation Fof the unit sphere bundle V=T1Sof a compact
+Riemann surface Sof genus g >1 endowed with its Riemannian metric of
+constant curvature −1. In general the manifold Vis the quotient V=G/Tof
+the semi-simple Lie group G=PSL(2,R), the isometry group of the hyperbolic
+planeH2, by the discrete cocompact subgroup T=π1(S), and the foliation F
+ofVis given by the orbits of the action by left multiplication on V=G/Tof
+the subgroup of upper triangular matrices of the form
+/parenleftbigg
+1t
+0 1/parenrightbigg
+t∈R
+The von Neumann algebra M=W(V,F) of this foliation is the (unique) hyper-
+finite factor of type III1=R∞. In the appendix A we describe the construction
+of the codimension-1 foliation on the 3-sphere S3. The main ingredient of this
+construction is the convex polygon Pin the hyperbolic plane H2having curva-
+ture−1. The whole construction don’t depend on the number of vertices o fP
+but on the volume vol(P) only. Thus without loss of generality, we can choose
+the even number 4 gforg∈Nof vertices for P. As model of the hyperbolic
+plane we choose the usual upper half-plane model where the group SL(2,R) (the
+real M¨ obius transformations) and the hyperbolic group PSL(2,R) (the group
+of all orientation-preserving isometries of H2) act via fractional linear transfor-
+mations. Then the polygon Pis a fundamental polygon representing a Riemann
+surfaceSof genus g. Via the procedure above, we can construct a foliation onExotic smooth R4, noncommutative algebras and quantization 11
+T1S=PSL(2,R)/TwithT=π1(S). This foliation is also induced from the
+foliation of T1H(as well as the foliation of the S3) via the left action above. The
+difference between the foliation on T1Sand onS3is given by the different usage
+of the polygon P. Thus the von Neumann algebra W(S3,F) of the codimension-
+1 foliation of the 3-sphere contains a factor IIIalgebra in agreement with the
+results in [28]. Currently we don’t know whether it is the hyperfinite III1factor
+R∞. The Reeb components of this foliation of S3(see appendix A) are repre-
+sented by a factor I∞algebra and thus don’t contribute to the Godbillon-Vey
+class. Putting all things together we will get
+Theorem 3. A small exotic R4as represented by a codimension-1 foliation of
+the 3-sphere S3with non-trivial Godbillon-Vey invariant is also associat ed to a
+von Neumann algebra W(S3,F)induced by the foliation which contains a factor
+IIIalgebra.
+We conjecture that this factor IIIalgebra is the hyperfinite III1factorR∞.
+Now one may ask, what is the physical meaning of the factor III? Because
+of the Tomita-Takesaki-theory, factor IIIalgebras are deeply connected to the
+characterization of equilibrium temperature states of quantum st ates in statis-
+tical mechanics and field theory also known as Kubo-Martin-Schwing er (KMS)
+condition. Furthermore in the quantum field theory with local obser vables (see
+Borchers[12] for an overview)one obtains close connections to To mita-Takesaki-
+theory. For instance one was able to show that on the vacuum Hilber t space with
+one vacuum vector the algebra of local observables is a factor III1algebra. As
+shown by Thiemann et. al. [34] on a class of diffeomorphism invariant t heories
+there exists an unique vacuum vector. Thus the observables algeb ra must be of
+this type.
+3.4. Cyclic cohomology and K-theoretic invariants of leaf s paces.As Connes [16]
+showed, interesting K-theoretic invariants of foliations can be des cribed by Kas-
+parov’s KK theory. For completeness we will give a short introductio n. Then we
+will apply the theory to our case.
+Lets begin with KK theory. Let AandBbe graded C∗-algebras. E(A,B) is
+the set of all triples ( E,φ,F), where Eis a countably generated graded Hilbert
+module over B,φis a graded ∗-homomorphismfrom Ato the bounded operators
+B(E) overE, andFis an operator in B(E) of degree 1, such that [ F,φ(a)],
+(F2−1)φ(a), and(F−F∗)φ(a) areallcompactoperators(i.e.in K(E)) forall a∈
+A(i.e.Fis a Fredholm operator). The elements of E(A,B) are called Kasparov
+modules for ( A,B).D(A,B) is the set of triples in E(A,B) for which[ F,φ(a)],
+(F2−1)φ(a), and (F−F∗)φ(a) are 0 for all a. The elements of D(A,B) are
+called degenerate Kasparov modules. Then one defines a homotopy ∼hbetween
+two triples ( Ei,φi,Fi) fori= 0,1 by a triple in E(A,[0,1]×B) relating the two
+triples in an obvious way. The KK groups are given by
+KK(A,B) =E(A,B)/∼h
+the equivalence classes. Especially we get for A=Cthe usual K homology
+KK(C,B) =K0(B) and for B=Cthe K theory KK(A,C) =K0(A) =
+K(A). A delicate part of KK theory is the existence of a intersection (or cup)12 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+product⊠so that elements in KK(A,C) andKK(C,B) combine to an element
+inKK(A,B). See chapter VIII in the nice book [11].
+LetEbe theC∗module on C∗(V,F) of a foliation Ffor a manifold V(as-
+sociated to the von Neumann algebra W(V,F)). LetNi
+→Vbe a transversal
+of the foliation constructed above (i.e. a submanifold with dim N=codim(F)
+transversal to the foliation). It defines a tranverse bundle which is associated to
+a bundle τover the classifying space BGof the holonomy groupoid Gof the
+foliation ( V,F). Using KK theory we can define the K theory K(V/F) of the
+leaf space V/Fvia theC∗algebraC∗(V,F). By geometric methods Baum and
+Connes were able to construct a purely geometric group K∗,τ(BG) together with
+an isomorphism µ:K∗,τ(BG)→K∗(V/F) (analytic assembly map, see [16]).
+OnVthe foliation determines a Haefliger structure Γvia the holonomy, i.e. we
+have a continuous map (unique up to homotopy) BG→BΓmapping elements
+ofK∗,τ(BG) to elements in K∗,τ(BΓ). The last K group is related to the usual
+cohomology via Chern-Weil theory.
+Now we specialize to the codimension-1 foliation over S3. Then we have es-
+pecially the relation between K3,τ(BΓ1) andH3(BΓ1,R) and by the analytic
+assembly map µabove, we have
+K3,τ(BΓ1)→K3,τ(BG)µ→K3(S3/F) =K1(S3/F)
+where the last equality is the Bott isomorphism. Via Connes pairing bet ween K
+theory and cyclic cohomology we have a map
+K3,τ(BΓ1)→HC1(C∗(S3,F))
+to the cyclic cohomology of the foliation. Thus we have shown
+Theorem 4. Let a small exotic R4be determined by a codimension-1 foliation
+on the 3-sphere S3. Every such foliation defines an element in the K theory
+K3,τ(BΓ1)determining an element in the cyclic cohomology HC1(C∗(S3,F))of
+the foliation (the transverse fundamental class).
+It is interesting to note that we already had found an interpretatio n of an
+element in the cyclic cohomology. In [4] we studied the change of the connection
+by changing the smoothness structure on a compact manifold to ge t a singular
+connection representing the change. This singular connection had a close con-
+nection to a cyclic cohomology class used to proof that the set of sin gular forms
+builds an algebra, the Temperley-Lieb algebra or the factor II1algebra. In sub-
+section 4.4 we will show that there is a subfactor in the von Neumann a lgebra
+W(S3,F) of the foliation which contains a factor II1algebra.
+4. The connection between exotic smoothness and quantizati on
+Inthissectionwedescribeadeeprelationbetweenquantizationand thecodimension-
+1 foliation of the S3determining the smoothness structure on a small exotic R4.Exotic smooth R4, noncommutative algebras and quantization 13
+Foliation Operator algebra
+leaf operator
+closed curve transversal to foliation projector (idempotent operator)
+holonomy linear functional (state)
+local chart center of algebra
+Table 1. relation between foliation and operator algebra
+4.1. From exotic smoothness to operator algebras. In subsection 3.3.1 we con-
+structed (following Connes [17]) the smooth holonomy groupoid of a f oliation
+Fand its operator algebra C∗
+r(M,F). The correspondence between a foliation
+and the operator algebra (as well as the von Neumann algebra) is vis ualized by
+table 1. As extract of our previous paper [2], we obtained a relation b etween
+exoticR4’s and codimension-1 foliations of the 3-sphere S3. Furthermore we
+showed in subsection 3.3.2 that the codimension-1 foliation of S3consists partly
+of Anosov-like foliations and Reeb foliations. Using Tomita-Takesaki- theory,one
+has a continuous decomposition (as crossed product) of any fact orIIIalgebra
+Minto a factor II∞algebraNtogether with a one-parameter group1(θλ)λ∈R∗
++
+of automorphisms θλ∈Aut(N) ofN, i.e. one obtains
+M=N⋊θR∗
++.
+Butthatmeans,thereisafoliationinducedfromthefoliationofthe S3producing
+thisII∞factor.Aswesawinsubsection3.3.2onehasacodimension-1foliation F
+as part of the foliation of the S3whose von Neumann algebra is the hyperfinite
+factorIII1. Connes [17] (in section I.4 page 57ff) constructed the foliation F′
+canonically associated to Fhaving the factor II∞as von Neumann algebra. In
+our case it is the horocycle flow: Let Pthe polygon on the hyperbolic space
+H2determining the foliation of the S3(see appendix A). Pis equipped with
+the hyperbolic metric 2 |dz|/(1− |z|2) together with the collection T1Pof unit
+tangent vectors to P. A horocycle in Pis a circle contained in Pwhich touches
+∂Pat one point. Then the horocycle flow T1P→T1Pis the flow moving an unit
+tangent vector along a horocycle (in positive direction at unit speed ). As above
+the polygon Pdetermines a surface Sof genus g >1 with abelian torsion-less
+fundamental group π1(S) so that the homomorphism π1(S)→Rdetermines an
+unique (ergodicinvariant)Radonmeasure.Finallythe horocycleflow determines
+a factorII∞foliation associated to the factor III1foliation. We remark for later
+usage that this foliation is determined by a set of closed curves (the horocycles).
+Using results of our previous papers, we have the following picture:
+1. Every small exotic R4is determined by a codimension-1 foliation (unique up
+to cobordisms) of some homology 3-sphere Σ(as boundary ∂A=Σof a
+contractable submanifold A⊂R4, the Akbulut cork).
+2. Thiscodimension-1foliationon Σdeterminesviasurgeryalongalinkuniquely
+a codimension-1 foliation on the 3-sphere and vice verse.
+3. This codimension-1 foliation ( S3,F) onS3has a leaf space which is deter-
+mined by the von Neumann algebra W(S3,F) associated to the foliation.
+1The group R∗
++is the group of positive real numbers with multiplication as group operation
+also known as Pontrjagin dual.14 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+4. The vonNeumann algebra W(S3,F) containsahyperfinite factor III1algebra
+as well as a factor I∞algebra coming from the Reeb foliations.
+Thus bythis procedurewe get anoncommutativealgebrafrom anex oticR4. The
+relation to the quantum theory will be discussed now. We remark tha t we have
+already a quantum theory represented by the von Neumann algebr aW(S3,F).
+Thus we are in the strange situation to construct a (classical) Poiss on algebra
+together with a quantization to get an algebra which we already have .
+4.2. The observable algebra and Poisson structure. In this section we will de-
+scribe the formal structure of a classical theory coming from the algebra of
+observables using the concept of a Poisson algebra. In quantum th eory, an ob-
+servable is represented by a hermitean operator having the spect ral decomposi-
+tion via projectors or idempotent operators. The coefficient of th e projector is
+the eigenvalue ofthe observable or one possible result ofa measure ment. At least
+one of these projectors represent (via the GNS representation ) a quasi-classical
+state. Thus to construct the substitute of a classical observab le algebra with
+Poisson algebra structure we have to concentrate on the idempot ents in the C∗
+algebra.
+In subsection 3.3.1, an idempotent was constructed in the C∗algebra of the
+foliation and geometrically interpreted as closed curve transversa l to the folia-
+tion. Such a curve meets every leaf in a finite number of points. Furt hermore
+the 3-sphere S3is embedded in the some 4-space with the tubular neighborhood
+S3×[0,1]. Then we have a thickened curve S1×[0,1] or a closed curve on a
+surface. Thus we have to consider closed curves in surfaces. Now we will see that
+the set of closed curves on a surface has the structure of a Poiss on algebra.
+Let us start with the definition of a Poisson algebra. Let Pbe a commutative
+algebra with unit over RorC. APoisson bracket onPis a bilinearform {,}:
+P⊗P→Pfulfilling the following 3 conditions:
+–anti-symmetry {a,b}=−{b,a}
+–Jacobi identity {a,{b,c}}+{c,{a,b}}+{b,{c,a}}= 0
+–derivation {ab,c}=a{b,c}+b{a,c}.
+Then aPoisson algebra is the algebra ( P,{,}). Now we consider a surface S
+together with a closed curve γ. Additionally we have a Lie group Ggiven by
+the isometry group. The closed curve is one element of the fundame ntal group
+π1(S). From the theory of surfaces we know that π1(S) is a free abelian group.
+Denote by Zthe freeK-module ( Ka ring with unit) with the basis π1(S), i.e.Z
+is a freely generated K-modul. Recall Goldman’s definition of the Lie bracket in
+Z(see [24]). For a loop γ:S1→Swe denote its class in π1(S) by/a\}b∇acke⊔le{⊔γ/a\}b∇acke⊔∇i}h⊔. Letα,β
+be two loops on Slying in general position. Denote the (finite) set α(S1)∩β(S1)
+byα#β. Forq∈α#βdenote by ǫ(q;α,β) =±1 the intersection index of αand
+βinq. Denote by αqβqthe product of the loops α,βbased in q. Up to homotopy
+the loop ( αqβq)(S1) is obtained from α(S1)∪β(S1) by the orientation preserving
+smoothing of the crossing in the point q. Set
+[/a\}b∇acke⊔le{⊔α/a\}b∇acke⊔∇i}h⊔,/a\}b∇acke⊔le{⊔β/a\}b∇acke⊔∇i}h⊔] =/summationdisplay
+q∈α#βǫ(q;α,β)(αqβq). (4)Exotic smooth R4, noncommutative algebras and quantization 15
+According to Goldman [24], Theorem 5.2, the bilinear pairing [ ,] :Z×Z→Z
+given by (4) on the generators is well defined and makes Zto a Lie algebra.
+The algebra Sym(Z) of symmetric tensors is then a Poisson algebra (see Turaev
+[42]).
+The whole approach seems natural for the construction of the Lie algebra
+Zbut the introduction of the Poisson structure is an artificial act. F rom the
+physical point of view, the Poisson structure is not the essential p art of classical
+mechanics. More important is the algebra of observables, i.e. funct ions over the
+configurationspaceformingthePoissonalgebra.Thuswewilllookfo rthealgebra
+of observables in our case. For that purpose, we will look at geomet ries over
+the surface. By the uniformization theorem of surfaces, there is three types of
+geometrical models: spherical S2, Euclidean E2and hyperbolic H2. LetMbe
+one of these models having the isometry group Isom(M). Consider a subgroup
+H⊂Isom(M) of the isometry group acting freely on the model Mforming the
+factor space M/H. Then one obtains the usual (closed) surfaces S2,RP2,T2
+and its connected sums like the surface of genus g(g >1). For the following
+construction we need a group Gcontaining the isometry groups of the three
+models. Furthermore the surface Sis part of a 3-manifold and for later use we
+havetodemand that Ghastobe alsoaisometrygroupof3-manifolds.According
+to Thurston [40] there are 8 geometric models in dimension 3 and the la rgest
+isometry group is the hyperbolic group PSL(2,C) isomorphic to the Lorentz
+groupSO(3,1).It is known that every representation of PSL(2,C) can be lifted
+to the spin group SL(2,C). Thus the group Gfulfilling all conditions is identified
+withSL(2,C). This choice fits very well with the 4-dimensional picture.
+Now we introduce a principal Gbundle on S, representing a geometry on the
+surface. This bundle is induced from a Gbundle over S×[0,1] having always
+a flat connection. Alternatively one can consider a homomorphism π1(S)→G
+represented as holonomy functional
+hol(ω,γ) =Pexp
+/integraldisplay
+γω
+∈G
+with the path ordering operator Pandωas flat connection (i.e. inducing a flat
+curvature Ω=dω+ω∧ω= 0). This functional is unique up to conjugation
+induced by a gauge transformation of the connection. Thus we hav e to consider
+the conjugation classes of maps
+hol:π1(S)→G
+forming the space X(S,G) of gauge-invariant flat connections of principal G
+bundles over S. Now (see [38]) we can start with the construction of the Poisson
+structure on X(S,G).The construction based on the Cartan form as the unique
+bilinearform of a Lie algebra. As discussed above we will use the Lie gro up
+G=SL(2,C) but the whole procedure works for every other group too. Now w e
+consider the standard basis
+X=/parenleftbigg
+0 1
+0 0/parenrightbigg
+, H=/parenleftbigg
+1 0
+0−1/parenrightbigg
+, Y=/parenleftbigg
+0 0
+1 0/parenrightbigg16 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+of the Lie algebra sl(2,C) with [X,Y] =H,[H,X] = 2X,[H,Y] =−2Y. Fur-
+thermore there is the bilinearform B:sl2⊗sl2→Cwritten in the standard
+basis as
+0 0−1
+0−2 0
+−1 0 0
+
+Now we consider the holomorphic function f:SL(2,C)→Cand define the
+gradient δf(A) alongfat the point Aasδf(A) =ZwithB(Z,W) =dfA(W)
+and
+dfA(W) =d
+dtf(A·exp(tW))/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+t=0.
+The calculation of the gradient δtrfor the trace tralong a matrix
+A=/parenleftbigg
+a11a12
+a21a22/parenrightbigg
+is given by
+δtr(A) =−a21Y−a12X−1
+2(a11−a22)H .
+Given a representation ρ∈X(S,SL(2,C)) of the fundamental group and an
+invariant function f:SL(2,C)→Rextendable to X(S,SL(2,C)). Then we
+consider two conjugacy classes γ,η∈π1(S) represented by two transversal
+intersecting loops P,Qand define the function fγ:X(S,SL(2,C)→Cby
+fγ(ρ) =f(ρ(γ)). Letx∈P∩Qbe the intersection point of the loops P,Qand
+cxa path between the point xand the fixed base point in π1(S). The we define
+γx=cxγc−1
+xandηx=cxηc−1
+x. Finally we get the Poisson bracket
+/braceleftbig
+fγ,f′
+η/bracerightbig
+=/summationdisplay
+x∈P∩Qsign(x)B(δf(ρ(γx)),δf′(ρ(ηx))),
+wheresign(x)isthesignoftheintersectionpoint x.Thusthespace X(S,SL(2,C))
+has a natural Poisson structure (induced by the bilinear form on th e group) and
+the Poisson algebra ( X(S,SL(2,C),{,}) of complex functions over them is the
+algebra of observables.
+4.3. Drinfeld-Turaev Quantization. Now we introduce the ring C[[h]] of formal
+polynomials in hwith values in C. This ring has a topological structure, i.e. for
+a given power series a∈C[[h]] the set a+hnC[[h]] forms a neighborhood. Now
+we define a Quantization of a Poisson algebra Pas aC[[h]] algebra Phtogether
+with the C-algebra isomorphism Θ:Ph/hP→Pso that
+–the modul Phis isomorphic to V[[h]] for aCvector space V
+–leta,b∈Panda′,b′∈PhbeΘ(a) =a′,Θ(b) =b′then
+Θ/parenleftbigga′b′−b′a′
+h/parenrightbigg
+={a,b}Exotic smooth R4, noncommutative algebras and quantization 17
+Figure 1. crossings L∞,Lo,Loo
+One speaks of a deformation of the Poisson algebra by using a defor mation
+parameter hto get a relation between the Poisson bracket and the commutator .
+Now we have the problem to find the deformation of the Poisson algeb ra
+(X(S,SL(2,C)),{,}). The solution to this problem can be found via two steps:
+at first find another description of the Poisson algebra by a struct ure with one
+parameterataspecialvalueandsecondlyvarythisparametertog etthedeforma-
+tion. Fortunately both problems were already solved (see [41,42]) . The solution
+of the first problem is expressed in the theorem: The Skein modul K−1(S×[0,1])
+(i.e.t=−1) has the structure of an algebra isomorphic to the Poisson al gebra
+(X(S,SL(2,C),{,}).(seealso[14,13])Thenwehavealsothe solutionofthesec-
+ond problem: The skein algebra Kt(S×[0,1])is the quantization of the Poisson
+algebra(X(S,SL(2,C),{,})with the deformation parameter t= exp(h/4).(see
+also [14]) To understand these solutions we have to introduce the sk ein module
+Kt(M) of a 3-manifold M(see [36]).
+For that purpose we consider the set of links L(M) inMup to isotopy and
+construct the vector space CL(M) with basis L(M). Then one can define CL[[t]]
+as ring of formal polynomials having coefficients in CL(M). Now we consider the
+linkdiagramofalink,i.e.theprojectionofthelinktothe R2havingthecrossings
+in mind. Choosing a disk in R2so that one crossing is inside this disk. Differ
+three links by the three crossings Loo,Lo,L∞(see figure 1) inside of the disk
+then these links are skein related. Then in CL[[t]] one writes the skein relation2
+L∞−tLo−t−1Loo. Furthermore let L⊔Obe the disjoint union of the link with
+a circle then one writes the framing relation L⊔O+(t2+t−2)L. LetS(M) be
+the smallest submodul of CL[[t]] containing both relations, then we define the
+Kauffman bracket skein modul by Kt(M) =CL[[t]]/S(M). We list the following
+general results about this modul:
+–The modul K−1(M) fort=−1 is a commutative algebra.
+–LetSbe a surface then Kt(S×[0,1]) caries the structure of an algebra.
+The algebra structure of Kt(S×[0,1]) can be simple seen by using the diffeo-
+morphism between the sum S×[0,1]∪SS×[0,1] alongSandS×[0,1]. Then the
+productaboftwoelements a,b∈Kt(S×[0,1])isalinkin S×[0,1]∪SS×[0,1]cor-
+respondingtoalinkin S×[0,1]viathediffeomorphism.Thealgebra Kt(S×[0,1])
+is in general non-commutative for t/\e}a⊔io\slash=−1.
+2The relation depends on the group SL(2,C).18 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+For the following we will omit the interval [0 ,1] and denote the skein algebra
+byKt(S). Then we have:
+Theorem 5. The skein algebra (Kt(S),[,])is the quantization of the Poisson
+algebra(X(S,SL(2,C),{,})with deformation parameter t= exp(h/4).
+Ad hocthe skein algebra is not directly related to the foliation. We used only
+the fact that there is an idempotent in the C∗algebra represented by a closed
+curve. It is more satisfying to obtain a direct relation between both construction.
+Then the von Neumann algebra of the foliation is the result of a quant ization in
+the physical sense. This construction is left for the next subsect ion.
+4.4. Temperley-Lieb algebra and the operator algebra of the foliation. In this
+subsection we will describe a direct relation between the skein algebr a and the
+factorIII1constructed above. At first we will summarize some of the results
+above.
+1. The foliation of the 3-sphere S3has non-trivial Godbillon-Vey class. The
+corresponding von Neumann algebra must contain a factor IIIalgebra.
+2. We conjectured that the von Neumann algebra is the hyperfinite factorIII1
+determined by a factor II∞algebra via Tomita-Takesaki theory.
+3. In the von Neumann algebra there are idempotent operators giv en by closed
+curves in the foliation.
+4. The set of closed curves carries the structure of the Poisson a lgebra whose
+quantization is the skein algebra determined by knots and links. Thus the
+skein algebra can be seen as a quantization of the fundamental gro up.
+Thus our main goal in this subsection should be a direct relation betwe en a
+suitable skein algebra and the von Neumann algebra of the foliation. A s a first
+step we remark that a factor II∞algebra is the tensor product II∞=II1⊗I∞.
+Thus the main factor is given by the II1factor, i.e. a von Neumann algebra with
+finite trace. From the point of view of invariants, both factors II∞andII1are
+Morita-equivalent leading to the same K-theoretic invariants.
+Now we are faced with the question: Is there any skein algebra isomo rphic to
+the factor II1algebra? Usually the skein algebra is finite or finitely generated
+(as module over the first homology group). Thus we have to constr uct a finite
+algebra reconstructing the factor II1in the limit. Following the theory of Jones
+[29], one uses a tower of Temperley-Lieb algebras as generated by p rojection
+(or idempotent) operators. Thus, if we are able to show that a ske in algebra
+constructed from the foliation is isomorphic to the Temperley-Lieb a lgebra then
+we have constructed the factor II1algebra.
+For the construction we go back to factor II∞foliation discussed above and
+identified as the horocycle foliation. Let Pthe polygon with hyperbolic metric
+used in subsection 3.3.2 and in appendix A. Remember a horocycle is a cir cle in
+the interior of Ptouching the boundary at one point. Now we consider the flow
+inT1Palongahorocyclewithunit speedwhichinduces acodimension-1foliatio n
+inT1P. Before we will describe the details of the approach, we will present the
+main idea. The horocycle foliation is parametrized by the set of horoc ycles onP.
+Every horocycle meets the boundary of Pat one point, which we mark. Using
+the horocycles we obtain a flow for every pair of marked points. The set of unitExotic smooth R4, noncommutative algebras and quantization 19
+Figure 2. fig. a) an example for a horocycle honPand fig. b) marked points on the boundary
+of the polygon P
+Figure 3. product structure
+tangent vectors labels the leaves of the foliation and can be describ ed by curves
+between the marked points. Then we group the marked points and a ssume that
+we have the same number of marked points on the left and on the righ t side of
+P. It is obvious that two polygons with the same number of marked poin ts on
+one side can be put together (product operation, see figure 3). S pecial attention
+should be given to flows between two marked points of the same side. Such flows
+appear if one resolves the singularities of the flow (see figure 5). Th en we obtain
+the relations of the Temperley-Lieb algebra which is the skein algebra of a disk
+with marked points on the boundary (see [36] 26.9.). As Jones [29] sh owed: the
+limit case is the factor II1.
+Theorem 6. The leaf space of the horocycle foliation of a surface S(of genus
+g >1) represented by a horocycle foliation of a polygon PinH2can be given
+the structure of an algebra by connected sum of the polygons. This algebra is
+isomorphic to the hyperfinite factor II1algebra given as tower of Temperley-
+Lieb algebras (and equal to the skein algebra of a marked disk ).
+After the general overview we will now present the details. Given a p olygon
+Pas covering space of a surface S(of genus g >1) with nonpositive curvature.
+Denote by γvthe geodesic with initial tangent vector vand bydist(γv(t),γw(t))
+the distance between two points on two curves. We call the two tan gent vectors
+v,wof the cover Pasymptotic if the distance dist(γv(t),γw(t)) is bounded as
+t→ ∞. For a unit tangent vector v∈T1Pdefine the Busemann function bv:
+P→Rby
+bv(q) = lim
+t→∞(dist(γv(t),q)−t)
+This function is differentiable and the gradient −∇qbvis the unique vector at q
+asymptotic to v. We define alternatively the horocycle h(v) (determined by v)20 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+Figure 4. a)flow linebetween two marked points defined viatwo horocycl es, b) simplepicture
+as substitute
+Figure 5. resolution of the flow singularities
+as the level set b−1
+v(0). Clearly h(v) is the limit as R→ ∞of the geodesic circles
+of radius Rcentered at γv(R). LetW(v) be the set of vectors wasymptotic to
+vwith footpoints on h(v), i.e.
+W(v) ={−∇qbv|q∈h(v)}.
+The curves W(v), v∈T1Pare the leaves of the horocycle foliation WofT1P
+which can be lifted to a horocycle foliation WonT1S. Thus the set of unit tan-
+gent vectors labels the leaves of the foliation or the leaf space is par ametrized
+by unit tangent vectors. Furthermore we remark that every hor ocycle is also
+determined by a unit tangent vector. By definition, the set of unit t angent vec-
+tors is completely determined by curves in P. Therefore consider a finite set of
+horocycles and mark the boundary points of these cycles, say m1,...,m n. Then
+by uniqueness of the flow, there is a curve from the boundary point m1in the
+interior of Pmeeting a point ofollowed by a curve from this point oto another
+boundary point m2(see figure 4). Thus in general we obtain curves in Pgoing
+from one marked boundary point mkto another marked boundary point ml.
+Now we will construct a set of generators for these curves. Form ally we can
+group the marked points into two classes (see figure 3), lying on the left or
+right (we assume w.l.o.g. an even number). We remark that one can int roduce
+a product like in figure 3. If we connect all marked points with each ot her then
+we obtain flows intersecting each other, i.e. we obtain singular flows. But the
+singularities or intersection points can be solved to get non-singular flows. The
+figure 5 shows the method3. It is easy to see that we obtain the generators
+(see figure 6) of the skein algebra of a disk with marked points. Lets denote
+3The method was used in the theory of finite knot invariants (Va ssiliev invariants) and is
+known as STU relation.Exotic smooth R4, noncommutative algebras and quantization 21
+Figure 6. example for generators e1ande3of the Temperley-Lieb algebra
+these generators by ei. By simple graphical manipulation (see [36]) we obtain
+the relations:
+e2
+i=ei, eiej=ejei:|i−j|>1,
+eiei+1ei=ei, ei+1eiei+1=ei+1, e∗
+i=ei (5)
+where the ∗operation is a simple change of the orientation of the curves. Now we
+need a delicate argument to get the desired relation to factor II1. Up to now we
+have only countable infinite curves between marked points on the dis k. To repre-
+sent the wholeset ofunit tangentvectors,weneed allpossible cur veson the disk.
+We can approximate between two curves represented by two gene ratorsek,en
+using the linear combination aek+benand introducing the sum operation. The
+coefficients a,bare complex numbers representing a 2-dimensional deformation
+of the curve. The set of countable infinite generators e1,e2,...together with its
+linear span L(e1,e2,...) is dense in the set of all possible curves, i.e. a Cauchy
+sequences of curves in L(e1,e2,...) approximate every arbitrary curve. Thus we
+have constructed the factor II1algebra as skein algebra.
+We will finish this subsection with one remark. In the factor Temperle y-Lieb
+algebra there is an unique idempotent operator, the Jones-Wenzl idempotent,
+which is related to Connes idempotent operator in the operator alge bra of the
+foliation by our construction.
+4.5. Casson handle and operator algebra. For amusement we will also discuss a
+direct relation between the Casson handle and the factor II1algebra represented
+by the Temperley-Lieb algebra. This will support the result of our pr evious
+subsection. The following discussion is adapted from the book [1].
+According to Freedman ([22] p.393), a Casson handle is represente d by a
+labeled finitely-branching tree Qwith base point ⋆, having all edge paths in-
+finitely extendable away from ⋆Each edge should be given a label + or −and
+each vertex corresponds to a kinky handle where the self-plumbing number of
+that kinky handle equals the number of branches leaving the vertex . The sign
+on each branch corresponds to the sign of the associated self plum bing.
+Now we are ready to describe the relation to the algebra of the prev ious
+subsection. Lets take start with the tree Q. Every path in this tree represents
+one leave in the foliation of the S3. Two different paths in the tree represent two
+different leaves in the foliation. Then we have to consider two paths in the tree
+Q, the reference path for the given leaf and a path for the another leaf of the
+foliation. Thus, a pair of two paths corresponds to one element of t he algebra.
+To recover all relations of such pairs of paths in a tree we have to co nsider the22 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+so-called string algebra according to Ocneanu [35]. For that purpo se we define
+a non-negative function µ:Edges→Ctogether with the adjacency matrix △
+acting on µby
+△µ(x) =/summationdisplay
+v∈Edges
+s(v)=x
+r(v)=yµ(y)
+wheres(v) andr(v) denote the source and the range of an edge v. A path in
+the tree is a succession of edges ξ= (v1,v2,...,v n) wherer(vi) =s(vi+1) and
+we write ˜ vfor the edge vwith the reversed orientation. Then, a string on the
+tree is a pair of paths ρ= (ρ+,ρ−), withs(ρ+) =s(ρ−),r(ρ+)∼r(ρ−) which
+means that r(ρ+) andr(ρ−) ending on the same level in the tree and ρ+,ρ−
+have equal lengths i.e. |ρ+|=|ρ−|expressing the previous described property
+r(ρ+)∼r(ρ−) too. Now we define an algebra String(n)with the linear basis of
+then-strings, i.e. strings with length nand the additional operations:
+(ρ+,ρ−)·(η+,η−) =δρ−,η+(ρ+,η−)
+(ρ+,ρ−)∗= (ρ−,ρ+)
+where·can be seen as the concatenation of paths. We normalize the funct ionµ
+byµ(root) = 1. Now we choose a function µin such a manner that
+△µ=βµ (6)
+for a complex number β. Then we can construct elements enin the algebra
+String(n+1)by
+en=/summationdisplay
+|α|=n−1
+|v|=|w|=1/radicalbig
+µ(r(v))µ(r(w))
+µ(r(α))(α·v·˜v,α·w·˜w) (7)
+fulfilling the algebraic relations
+e2
+i=ei, eiej=ejei:|i−j|>1,
+eiei+1ei=ei, ei+1eiei+1=τ·ei+1, e∗
+i=ei (8)
+whereτ=β−2. The trace of the string algebra given by
+tr(ρ) =δρ+,ρ−β−|ρ|µ(r(ρ))
+defines on A∞= (/uniontext
+nString(n),tr) an inner product by /a\}b∇acke⊔le{⊔x,y/a\}b∇acke⊔∇i}h⊔=tr(xy∗) which is
+after completion the Hilbert space L2(A∞,tr).
+Now we will determine the parameter τ. Originally, Ocneanu introduce its
+string algebra to classify the splittings of modules over an operator algebra.
+Thus, to determine this parameter we look for the simplest generat ing structure
+in the tree.The simplest structurein ourtreeisone edgewhichis con nectedwith
+two other edges. This graph is represented by the following adjace nce matrix
+
+0 1 1
+1 0 0
+1 0 0
+Exotic smooth R4, noncommutative algebras and quantization 23
+having eigenvalues 0 ,√
+2,−√
+2. According to our definition above, βis given by
+the greatest eigenvalue of this adjacence matrix, i.e. β=√
+2 and thus τ=β−2=
+1
+2. Then, without proof, westate that this algebrais givenby the Cliffo rd algebra
+onR∞, i.e. by the hyperfinite factor II1algebra.
+4.6. The approach via topoi. In this section we propose to consider exotic R4
+allowing for a fundamental approach to theories of physics at least in dimen-
+sion 4. This is in some analogy with the approach via topos theory, rec ently
+under active development. The topos techniques are universally va lid in any di-
+mension, but non-classical, intuitionistic logic is an essential part. No w we will
+make some additional suppositions to understand foliation-based q uantization
+presented above in the context of 4-dimensional quantum field the ories.
+Firstly we assume (see subsection 3.3.2): (A) the appearance of the factor
+III1in the von Neumann algebra of the codimension-1 foliations o fS3⊂R4.
+Then this factor III1indicates that exotic 4-smoothness of R4can be connected
+with the regime of relativistic local QFT or with the statistical limit of qu antum
+systems [25].
+Let us suppose that indeed in the case of a small exotic R4: (B)the factor
+III1, obtained from the codimension-1 foliation of S3which corresponds to the
+exoticR4, is the factor III1algebra of quantum observables of some relativis-
+tic local QFT (RQFT) defined on standard R4(with the Minkowskian metric ).
+Equivalently one could perform a construction of a RQFT on R4with suitable
+scaling asymptotic conditions on states of the theory and the algeb ra of local
+observables of the theory must be the factor III1. Then this factor is isomorphic
+to the von Neumann algebra of the codimension-1 foliation of S3⊂R4. Note
+that in the case of standard smooth R4the Godbillon-Vey invariant vanishes,
+so we do not have the corresponding foliations of S3, hence the III1does not
+appear.
+Moreover, we obtained the factor III1, corresponding to some nonstandard
+smoothingof R4,asdeformationquantizationofthe classicalPoissonalgebraun-
+derlying the observables of some classical theory. We see that sta ndard smooth-
+ing ofR4does not give rise to this kind of quantization, since the factor III1
+algebra does not appear. So, standard smooth R4should correspond to some
+classical theory. Let us suppose that: ( C)there exists a classical theory, defined
+on Euclidean or Minkowskian R4, which determines the theory whose algebra of
+observables is the Poisson algebra (X(S,SL(2,C),{,})of complex functions on
+X(S,SL(2,C)).
+We do not discuss the details of the determination nor the specific th eories
+appearing in the assumptions ( B) and (C). We just assume the existence of
+such theories and are interested in their algebraic structures. No w we state the
+result which is already implicitly contained in the constructions of the p revious
+sections
+under(A), (B), (C) the quantization corresponds to the change of a smoothing
+ofR4(from an exotic to the standard one).
+This is what we call the 4-exotics approach to (quantum) theories o f physics. It
+bearssomecommonfeatureswiththetoposapproachto theories of physics (TA) .
+Namely, in TA one tries to reformulate quantum theory internally in so me topos
+such that the results resembles rather classical, though logically int uitionistic,
+theory. So one can say that in TA24 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+quantization corresponds to the change of a category (from a topos to Set) and
+the corresponding change of logic.
+Certainly, as we already mentioned, our 4-exotics approach up to n ow, is limited
+to dimension 4 and to specific theories. The appearance of the fact orIII1is a
+common feature for allsmall exotic R4’s (as being classified by the codimension-
+1 foliations of S3), so we do not vary between these exotics at this stage.
+Let us present some more details of the approaches. The main tech nical in-
+gredients of the 4-exotics approach of the quantization (related to a small exotic
+R4) was already developed in the previous sections. Let us focus now o n TA.
+The topos approach to theories of physics as reformulated after D¨ oring and
+Isham [18,19,20,21] by Caspers, Heunen, Landsman and Spitter s [27,15,26] is
+based on the algebraic formulation of QM i.e. given by a C⋆algebra, A, whose
+self-adjoint elements represent the algebra of observables of a q uantum theory.
+Given such an algebra A(noncommutative) one tries to build the spectrum
+ΣAof it by the analogy with the Gelfand-Naimark spectrum Xexisting for
+a commutative C⋆algebra. This means that one tries to model the algebra A
+by the ’isomorphic’ to it object of ’functions’ f:ΣA→RwhereRis the
+suitable counterpart to the set of real numbers. The above goal can not be
+achieved for the ’true’ (i.e. in the mathematical sense based on clas sical logic)
+real numbers, true topological compact space ΣAand true continuous functions
+onΣA. The big achievement of the topos approach to quantum theories is to
+solve this problem for some categories in mathematics, namely topoi. ThusΣA,
+Rare objects in a topos τAuniquely determined by the algebra Aandf’s are
+morphisms in τA. The ’isomorphisms’is now understood as internalin the topos.
+The topos τAwhich corresponds to the noncommutative algebra Aand makes
+it a commutative internal one is the topos of covariant functors on C(A) where
+C(A)isthecategoryofcommutativesubalgebrasof Aandtheirhomomorphisms.
+This whole procedure requires however the internal logic and set th eory of topoi
+whichisnotclassicalbutrather intuitionistic .Ingeneral,theusualmathematical
+constructions can be formulated in topoi provided they make no us e of the set
+theoretical axiom of choice or the logical rule of the excluded middle.
+The existence of the internal spectrum ΣAfor a noncommutative C⋆algebra
+Awas established in a series of papers by Banashewski and Mulvey [7,8 ,9].ΣAis
+a (completely regular) locale in τAwhich means that it generalizes ordinary
+topological spaces towards the pointfree topologies. Such a locale ΣAplays a
+role of quantum phase space of a system under consideration and d etermines the
+space of its subobjects. Similarly as closed linear subspaces of a Hilbe rt space
+Hdefine a quantum logic (according to the original proposition by Birkh off and
+von Neumann [10]) or propositions about the system described by H(i.e. the
+open subsets of ΣA) define the logic of our internal system as well about the
+algebraA. The space ofopen subsets of ΣAis a Heyting algebra.The elementary
+fact of the theory of Heyting algebras is that these are distributiv e in opposition
+to the nondistributive lattice of projections on closed linear subspa ces ofH.
+Thus the logic of previously nondistributive quantum lattice of proje ctions was
+weakened to the distributive Heyting algebras, i.e. to the intuitionist ic logic
+of topoi. The noncommutativity of the original C⋆algebra was inverted to the
+internalcommutative C⋆algebrain τA.Thepricetopayis,however,theessential
+use of intuitionistic logic of topoi. Next people try to develop all ingred ients ofExotic smooth R4, noncommutative algebras and quantization 25
+this internal intuitionistic theory as if it were a classical theory and w e will
+present these below following [27].
+The topos techniques are also applicable in our case of the factor II I1algebra
+and the Banach algebra of sections, ΓKPas discussed in Secs. 3.1, 3.2, when
+extended to a C⋆algebra with unity. Given the factor III1, sayN, we can follow
+the topos procedure:
+1. We build the category C(N) whose objects are unital commutative subalge-
+bras,C⊂N, (this set of subalgebras is partially ordered by the inclusion),
+and morphisms between 2 objects, C,D⊂N, i.e.HomC(N)(C,D), contains a
+singlearrow,when C⊆D,and∅(noarrow)otherwise. C(N)isusuallyconsid-
+ered as the category of classical windows of the noncommutative ( quantum)
+algebra of observables describing a quantum system.
+2. Itisknownthatforanycategory KandthecategorySetofsetsandfunctions,
+the category of functors K→Set and natural transformations between these
+functors, i.e. SetK, is a topos. In our case we take the topos
+τN= SetC(N)
+3. The tautological functor N:C→C,C∈ C(N) fromC(N) to Set defines
+an internal in τNcommutative unitalC⋆algebraN. This algebra is called
+by Landsman et.all,[26] the Bohrification of NsinceNis noncommutative
+whereas Nis a commutative algebra, and the Bohr doctrine of quantum
+physics relays,roughly,on seeing its alsoexperimental results, in a final stage,
+via the classical (commutative) way.
+4. Now given a commutative internal C⋆algebra in τNwe can make use of
+the results of Banashewski and Mulvey regarding Gelfand spectra [9] of the
+internal in topoi commutative C⋆algebras. Namely, there exists an internal
+inτNspectrum ofthe algebra Nwhich is a completely regularlocale ΣN. The
+object of all locale maps from ΣNto the object of complex numbers in τN,
+C,i.e.τN(ΣN,C) is a commutative internal C⋆algebra which is isomorphic
+(internally) to N:
+τN(ΣN,C)≃τNN
+Now we can collect physical counterparts of the above construct ion:
+1. The quantum phase space of fields system given by the algebra of local ob-
+servables N(the factor III1) is the above locale ΣN(internal in τN).
+2. The internalopen subsets of ΣNcorrespondto the arrowsin τN, 1→ O(ΣN),
+i.e. a frame corresponding to the locale ΣN, where 1 is the terminal object
+inτNandO(ΣN) is the lattice of (open) subobjects of ΣN. This lattice
+of open subsets represents the quantum projections or propos itions about
+the quantum system described by the algebra N. However, this lattice is a
+distributive Heyting algebra rather than the external non-distrib utive von
+Neumann lattice of projections.
+3. Each observable o∈Ndetermines a map between locales δ(o) :ΣN→ ℜ
+whereℜis an interval-domain object in τNwhich, for the task of the correct
+interpreting the quantum theory observables via internal ’functio ns’, plays
+the role of internal real numbers. It is worth mentioning that ℜis not an
+object of real numbers in the topos τN. It is rather suitably smeared version
+of the internal reals.26 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+4. Given an open subset-interval of the external real numbers, ∆⊆R, we have
+the corresponding internal open subset of ℜhence the arrow in τN,∆:
+1→ O(ℜ). Besides, the map δ(o) :ΣN→ ℜas above gives rise to the
+map of the frames: δ−1(o) :O(ℜ)→ O(ΣN). The composition of both, i.e.
+1∆→ O(ℜ)δ−1
+→ O(ΣN), gives the proposition [ o∈∆] : 1→ O(ΣN).
+5. In the Birkhof and von Neumann lattice of projections L(H) on the closed
+linear subspaces [ o∈∆]’s these are images of the spectral projections E(∆)H
+ofowhereois an observable.The pairingof a state ρand a proposition o∈∆
+is given by the Born rule (for pure states) < ρ,o∈∆ >which is a number in
+[0,1]. In the topos interpretation of QM we have the corresponding pa iring
+given by
+< o∈∆,ρ >= 1[o∈∆]→ O(ΣN)χρ→ΩN
+whereχρisthecharacteristicfunctionofthestate ρasansubobjectof O(ΣN)
+andΩNis the subobject classifier of τN.
+The above topos procedure applies also to the factor III1algebra and via the
+uniqueness of this factor to an algebraic relativistic QFT (RAQFT) de fined
+on Minkowskian space Mn. To deal with the algebraic and local structure of
+this QFT one notices that an AQFT can be defined as a functor AQFT :
+O(Mn)→CSTAR [27] from the category of open subobjects in the Minkowski
+space and their inclusions to the category of C⋆algebras and their morphisms.
+From this more general perspective one can take as a topos for th is AQFT
+τAQFT= SetO(Mn)and a single internal commutative C⋆algebra. As a result of
+the topos approach to AQFT on the 4-Minkowski space we obtain an intuition-
+istic classical-like internal theory representing the local algebraic Q FT.
+Let us observe that the 4-exotic approach to the factor III1and RAQFT can
+give a somewhat similar result. The factor III1of a relativistic algebraic QFT
+defined on M4can be obtained by the deformation of the Poisson algebra of
+some classical theory as was presented in Secs. 4.1, 4.2. However, this whole
+procedure (via our conjectures at the beginning of this section) c orresponds to
+the change of exotic smooth structure on R4, to the standard smooth R4(ew.
+with Minkowskian metric hence M4). The point is that one can in principle
+formulate a classical field theory on an exotic R4and this should correspond to
+some quantized AQFT on M4. Thus the task of quantization of some AQFT on
+M4and the reverse task to obtain a classical theory from a quantum r elativistic
+AQFT, can be formulated in the 4-exotic paradigm as:
+Let CFT be a classical field theory on R4orM4which by assumption (C)
+determines a classical theory with the algebra of observabl es equal to the Pois-
+son algebra (X(S,SL(2,C),{,})of complex functions on X(S,SL(2,C)). The
+quantization of the Poisson algebra can be obtained as a CFT o n an exotic R4.
+Then changing the smooth structure to the standard R4gives the factor III1al-
+gebra which is the algebra of quantum observables of the quan tized theory. This
+quantum theory is equivalent by assumption (B) to some RAQFT onM4. The
+equivalence can be understood as a duality of theories. Conv ersely, a quantized
+RAQFT on the standard smooth R4with Minkowski metric, is derived from a
+classical field theory on certain exotic smooth R4.
+Hence the classical theory on the exotic R4is dual in the above sense to some
+quantized AQFT. A relativistic AQFT on standard smooth R4with Minkowski
+metric,becomesaclassicalfieldtheorywiththePoissonalgebrawhe nformulatedExotic smooth R4, noncommutative algebras and quantization 27
+on certain small exotic R4. In this case the quantization is performed via the
+change of the smooth structure of the exotic R4to the standard one. This is
+analogous to the change of the logic from the intuitionistic logic of top oi to the
+classical logic of Set in the topos approach, which corresponds to q uantization.
+We do not have explicit descriptions of a RAQFT on an exotic R4or even
+classical field theory on it since we do not have an exotic metric nor th e global
+exotic smooth structures glued from local coordinate patches. H owever the local
+descriptionsarecompletelyequivalentwiththestandardcase.Mor eover,wehave
+the relation between algebrasof observablesand exotic smoothing s ofR4worked
+out in previous subsections. From that algebraic point of view all we n eed is the
+knowledge about the existence of theories which have the quantum algebra of
+observables spanned on the factor III1and the classical algebra spanned on a
+Poisson algebra.
+A natural question one can ask is the relation of the (intuitionistic) c lassical
+theories obtained from the topos approach and via 4-exotics, pro vided one starts
+with the same quantum RAQFT4. Probably this requires more thorough un-
+derstanding of both approaches with a well recognized relation bet ween internal
+intuitionistic and external theories and their physical contents. L et us just make
+a few comments at this point.
+–Toposes serve as a generalization of topological spaces. This gives rise to
+the intuitionistic driven translation of the noncommutative C⋆algebras to
+commutative ones, though internal in toposes.
+–In the 4-exotics approach to quantum physics we rather deal with generalized
+geometries as given by gerbes on manifolds and so called B-fields as the
+extension of (pseudo-) Riemannian geometry of manifolds [2,3].
+–The 4-exotics approach is essentially 4-dimensional. The factor III1von Neu-
+mann algebra is unique. When one wants to vary different exotic R4’s in this
+approach, the net of algebras suitably embedded into each other s hould be
+probably considered. However even in dimension 2 the description of such
+nets appears as quite nontrivial task [30].
+The topos approach is universally valid for any unital C⋆algebra and intuition-
+ism is an essential technical and conceptual ingredient of this. How ever,one does
+not have direct relation of classical (internal in toposes) and exte rnal quantum
+observables. From the other side, the 4-exotics approach is cert ainly up to now
+bounded to the limiting case of theories, namely those which have the factorIII1
+as the subalgebra of the observables algebra, and works exclusive ly in dimension
+4. However, dimension 4 is of exceptional importance for physics an d the factor
+III1is also quite essential for relativistic local field theories. All constru ctions
+areperformedinclassicalmathematicswithout necessitytorefer tointuitionism.
+We hope to present soon the more detailed analysis of the 4-exotics approach
+with some valid for physics applications.
+5. Conclusions
+In this paper we presented a variety of relations between codimens ion-1 folia-
+tions of the 3-sphere S3and noncommutative algebras. By using the results of
+4The connection of topos theory and intuitionism with exotic R4’s, though from the model-
+theoretic perspective, was already proposed in [32,31].28 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
+our previous paper [2], we obtain a relation between (small) exotic smo othness
+of theR4and noncommutativity via the noncommutative leaf space of the foli-
+ation and the Casson handle. Thus we get our main result of this pape r:
+The Casson handle carries the structure of a noncommutative space determined
+by a factor II1algebra which is related to the skein algebra of the disk with marked
+points and to the leaf space of the horocycle foliation.
+Thus we have obtained a direct link between noncommutative spaces and ex-
+otic 4-manifolds which can be used to get a direct relation to quantum field
+theory. One of the central elements in the algebraic quantum field t heory is the
+Tomita-Takesaki theory leading to the III1factor as vacuum sector [12]. As a
+possiblecandidateonehasloopquantumgravitywithanuniquediffeom orphism-
+invariant vacuum state [34].
+The attentive reader came across with the fact that there are on e hyperfinite
+III1factor but many Casson handles. We will discuss this possible contra diction
+now.In the subsection3.3.2we presentedthe relation III1=R∞=Rλ1⊗Rλ2for
+theAraki-Woodsfactorforall λ1,λ2withλ1/λ2/∈QtothePowersfactors IIIλ=
+Rλ.Thuswehaveacontinuumof III1factorsparametrizedbyirrationalnumbers
+λ1/λ2/∈Q. It is known by Freedmans work [22] that all Casson handles can be
+parametrized by a dual tree or by the Cantor continuum. Thus eve ry particular
+Casson handle is given by a real (irrational) number. All the corresp onding
+factors are equivalent to III1and we need finer invariants to distinguish them
+from each other. One possible invariant is the flow of weights introdu ced by
+Connes [17] and related to the Godbillon-Vey invariant. That closes t he circle
+to the elements of H3(S3,R) and we obtain the description of the whole group
+H3(S3,R) in contrast to the integer case in [3].
+Intheretrospective,wehavemanycloseandunexpectedrelation sbetweenex-
+otic smooth R4and string theory especially WZW models and D-brane charges.
+With this paper we have also included quantum aspects in our model. Bu t
+string or M theory is usually formulated in 10 or 11 dimensions whereas our
+approach is based on 4 dimensions. Usually one compactifies partly th e 10- or
+11-dimensional space to get the desired 4-dimensional spacetime. In our forth-
+coming paper we will show that the structure of the compactified sp ace as 6-
+dimensional Calabi-Yau manifold or as 7-dimensional G2manifold can be also
+induced by the smoothnes structure of the spacetime.
+A. Non-cobordant foliations of S3detected by the Godbillon-Vey
+class
+In [39], Thurston constructed a foliation of the 3-sphere S3depending on a poly-
+gonPin the hyperbolic plane H2so that two foliations are non-cobordant if the
+corresponding polygons have different areas. We will present this c onstruction
+now.
+Consider the hyperbolic plane H2and its unit tangent bundle T1H2, i.e the
+tangent bundle TH2where everyvector in the fiber has norm 1. Thus the bundle
+T1H2is aS1-bundle over H2. There is a foliation FofT1H2invariant under the
+isometries of H2which is induced by bundle structure and by a family of parallel
+geodesics on H2. The foliation Fis transverse to the fibers of T1H2. LetPbe
+any convex polygon in H2. We will construct a foliation FPof the three-sphere
+S3depending on P. Let the sides of Pbe labeled s1,...,s kand let the anglesExotic smooth R4, noncommutative algebras and quantization 29
+have magnitudes α1,...,α k. LetQbe the closed region bounded by P∪P′,
+whereP′is the reflection of Pthrough s1. LetQǫ, beQminus an open ǫ-disk
+about each vertex. If π:T1H2→H2is the projection of the bundle T1H2, then
+π−1(Q) is a solid torus Q×S1(with edges) with foliation F1induced from F.
+For each i, there is an unique orientation-preserving isometry of H2, denoted Ii,
+which matches sipoint-for-pointwith its reflected image s′
+i. We glue the cylinder
+π−1(si∩Qǫ) to the cylinder π−1(s′
+i∩Qǫ) by the differential dIifor each i >1,
+to obtain a manifold M= (S2\{k punctures })×S1, and a (glued) foliation F2,
+induced from F1. To get a complete S3, we have to glue-in ksolid tori for the
+k S1×punctures .Now we choose a linear foliation of the solid torus with slope
+αk/π(Reeb foliation). Finally we obtain a smooth codimension-1 foliation FP
+of the 3-sphere S3depending on the polygon P.
+Nowweconsidertwocodimension-1foliations F1,F2depending ontheconvex
+polygons P1andP2inH2. As mentioned above, these foliations F1,F2are
+defined by two one-forms ω1andω2withdωa∧ωa= 0 and a= 0,1. Now we
+define the one-forms θaas the solution of the equation
+dωa=−θa∧ωa
+and consider the closed 3-form
+ΓFa=θa∧dθa (9)
+associated to the foliation Fa. As discovered by Godbillon and Vey [23], ΓF
+depends only on the foliation Fand not on the realization via ω,θ. ThusΓF,
+theGodbillon-Vey class , is an invariant of the foliation. Let F1andF2be two
+cobordantfoliations then ΓF1=ΓF2. In case ofthe polygon-dependent foliations
+F1,F2, Thurston [39] obtains
+ΓFa=vol(π−1(Q)) = 4π·Area(Pa)
+and thus
+–F1is cobordant to F2=⇒Area(P1) =Area(P2)
+–F1andF2are non-cobordant ⇐⇒Area(P1)/\e}a⊔io\slash=Area(P2)
+Wenotethat Area(P) = (k−2)π−/summationtext
+kαk.TheGodbillon-Veyclassisanelement
+of the deRham cohomology H3(S3,R). Furthermore we remark that the classi-
+fication is not complete. Thurston constructed only a surjective h omomorphism
+from the group of cobordism classes of foliation of S3into the real numbers R.
+Acknowledgment
+T.A. wants to thank C.H. Brans and H. Ros´ e for numerous discussio ns over the
+years about the relation of exotic smoothness to physics. J.K. ben efited much
+from the explanations given to him by Robert Gompf regarding 4-smo othness
+several years ago, and discussions with Jan Sladkowski.30 Torsten Asselmeyer-Maluga, Jerzy Kr´ ol
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+Communicated by name
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+arXiv:1001.0883v2  [math.GT]  30 Nov 2010SPHERICAL COMPLEXES ATTACHED TO SYMPLECTIC LATTICES
+WILBERD VAN DER KALLEN AND EDUARD LOOIJENGA
+ABSTRACT . To the integral symplectic group Sp(2g,Z)we associate two
+posets of which we prove that they have the Cohen-Macaulay pr operty.
+As an application we show that the locus of marked decomposab le prin-
+cipally polarized abelian varieties in the Siegel space of g enusghas the
+homotopy type of a bouquet of (g−2)-spheres. This, in turn, implies that
+the rational homology of moduli space of (unmarked) princip al polarized
+abelian varieties of genus gmodulo the decomposable ones vanishes in
+degree≤g−2. Another application is an improved stability range for the
+homology of the symplectic groups over Euclidean rings. But the original
+motivation comes from envisaged applications to the homolo gy of groups
+of Torelli type.
+The proof of our main result rests on a refined nerve theorem fo r posets
+that may have an interest in its own right.
+INTRODUCTION
+This paper is about quasi-unimodular symplectic lattices, so let us begin
+with explaining that notion: a symplectic lattice is a free abelian group Lof
+finite rank endowed with a symplectic form (a,b)∈L×L/ma√s⊔o→/ang⌊ra⌋k⌉⊔l⌉f⊔a,b/ang⌊ra⌋k⌉⊔righ⊔. It is
+said to be unimodular if the associated map a∈L/ma√s⊔o→/ang⌊ra⌋k⌉⊔l⌉f⊔a,/ang⌊ra⌋k⌉⊔righ⊔ ∈Hom(L,Z)is
+an isomorphism; the rank of Lis then even and half that rank is called the
+genus ofL. Let us say that Lisquasi-unimodular of genus gifLbecomes
+unimodular of genus gonce we divide out by its radical. For example, the
+intersection pairing on an oriented surface with finite first Bet ti number is
+quasi-unimodular with the genus equal to that of the surface. Th e unimodu-
+lar sublattices of Lmake up a poset (with respect to inclusion) and our main
+result (Theorem 1.1) says that this poset is Cohen-Macaulay of dime nsion
+equal to the genus.
+However, for the applications we have in mind, the poset of unimodu-
+lar decompositions of L(where we now assume that Lis unimodular) is
+more relevant. We derive from our main result that this poset is al so Cohen-
+Macaulay (Theorem 3.1). This has the following interesting cons equence:
+consider in the Siegel space of genus gthe locus of decomposables, i.e., the
+locus that parametrizes marked principally polarized abelian v arieties that
+are decomposable as polarized varieties. This is a locally finite u nion of
+symmetric submanifolds (each being isomorphic to a product of two Sie gel
+2010 Mathematics Subject Classification. 05E18, 11E57, 19B14.
+Key words and phrases. integral symplectic group, Cohen-Macaulay poset.
+12 WILBERD VAN DER KALLEN AND EDUARD LOOIJENGA
+spaces) and a nonempty intersection of such submanifolds is also i somor-
+phic to a product of Siegel spaces (with perhaps more factors). A S iegel
+space is contractible and so the locus of decomposables comes with a Le ray
+covering by closed subsets. According to a classical result of Weil , the homo-
+topy type of this locus is then that of the nerve of this covering. This nerve
+is just a poset of unimodular decompositions and so we find (Corollary 3 .3)
+that the locus of decomposables has the homotopy type of a bouquet of
+(g−2)-spheres. (It would be desirable—and very interesting—to have a
+reasonable presentation of the homology of this complex in degree g−2
+as a module over the symplectic group.) This implies that the rati onal ho-
+mology of the pair (Ag,Ag,dec), whereAgis the moduli space of principally
+polarized abelian varieties and Ag,decparametrizes the decomposables, van-
+ishes in degree ≤g−2.
+As an aside we observe that our results and arguments remain vali d if we
+work over an arbitrary Euclidean ring Rrather than Z(so that Lis now a
+R-module). This makes it possible to improve Charney’s stability rang e for
+the homology groups of the groups {Sp(2g,R)}g≥0by a unit or two (Theorem
+4.1). We may paraphrase this by saying that from her perspective , Euclidean
+rings behave as if they were fields.
+Finally we show (Theorem 5.1) that for a closed orientable surface of
+genusg, the separating curve complex modulo the Torelli group has the
+same homotopy type as the locus of decomposables in Siegel space of genus
+g, so has the homotopy type of a bouquet of (g−2)-spheres. (One of us [6]
+recently proved that this also holds for separating curve comple x itself.)
+We shall not review the individual sections, but we wish to point ou t the
+central role played by our Nerve Theorem 2.3: while the first half of t he
+nerve theorem may be familiar from Mirzaii-Van der Kallen ([8, T hm. 4.3]);
+it is the second half that we believe is new and makes the theorem do the job
+that is needed here. We take the occasion to observe that its proof ill ustrates
+our belief that it is best when a proof of a statement in any given cate gory—
+here a homotopy category—does not leave that category. This means tha t
+we have expunged from the main argument all homology and the menag erie
+that usually accompanies it, such as the use of spectral sequen ces, local
+systems and the Hurewicz theorem (and that remains so when we us e it
+to derive the results that motivated this paper). The shortnes s of the proof
+of the Mirzaii-Van der Kallen part may be regarded as a testimony to the
+efficiency of this approach.
+1. T HE COMPLEX OF UNIMODULAR SUBLATTICES
+LetLbe a free abelian group of finite rank endowed with a symplectic
+form(a,b)∈L×L/ma√s⊔o→/ang⌊ra⌋k⌉⊔l⌉f⊔a,b/ang⌊ra⌋k⌉⊔righ⊔. We say that Lisunimodular if the associated
+mapa∈L/ma√s⊔o→/ang⌊ra⌋k⌉⊔l⌉f⊔a,/ang⌊ra⌋k⌉⊔righ⊔ ∈Hom(L,Z)is an isomorphism. Then Lhas even rank
+and half that rank is called the genus ofg. We say Lisquasi-unimodular if theSPHERICAL COMPLEXES ATTACHED TO SYMPLECTIC LATTICES 3
+associated map a∈L/ma√s⊔o→¯a:=/ang⌊ra⌋k⌉⊔l⌉f⊔a,/ang⌊ra⌋k⌉⊔righ⊔ ∈Hom(L,Z)has a torsion free cokernel.
+So ifLo⊂Ldenotes the kernel of this map (often referred to as the radical
+ofL), then the induced symplectic form on ¯L:=L/Lois unimodular; the
+genus ofLis then by definition that of ¯L.
+We now fix a quasi-unimodular symplectic lattice Lof genus g. The col-
+lection of unimodular sublattices of L(the trivial lattice included) make
+up a poset that we shall denote by U(L). Its standard height function is
+then given by the genus ( =half the rank). A maximal chain is of the form
+0=u0/subset⋉⋊teqlu1/subset⋉⋊teql···/subset⋉⋊teqlugand so dim U(L) =g.
+The main result of this article is
+Theorem 1.1. The poset U(L)is Cohen-Macaulay of dimension g.
+We shall prove Theorem 1.1 with induction on g. Forg=0,U(L)is a
+singleton and there is nothing to show. We assume the theorem veri fied
+for genera < g. Let us first isolate the two essential cases needed for the
+induction step.
+SinceU(L)has a minimal element (namely 0), it is contractible and hence
+spherical of dimension g. Ifu∈ U(L), thenU(L)<u=U(u)<uis spherical of
+dimension g(u)−1for a similar reason. On the other hand, U(L)>ucan be
+identified with U(u⊥)>0and hence if g(u)> 0, this is spherical of dimension
+g−g(u)−1by our induction hypothesis. If Lis unimodular, then L∈ U(L)
+is a maximal element and then U(L)>uis even contractible. So the only case
+to deal with here is when Lisnotunimodular and u={0}: we must show
+thatU(L)>0is then(g−2)-connected.
+Similarly, if u/subset⋉⋊teqlu′is an ordered pair in U(L), thenu′′:=u⊥∩u′has
+genusg(u′′) =g(u′)−g(u)and so if g(u′)−g(u)< g, then by our induction
+hypothesis, U(L)(u,u′)∼=U(u′′)(0,u′′)is spherical of dimension g(u′′)−2. So
+we may restrict to the case when u′=0andg(u) =L. In other words,
+we must show that when Lis unimodular, then U(L)(0,L)is(g−3)-connected.
+LetLbe as above, i.e., quasi-unimodular of genus g. We denote the poset
+of isotropic sequences in Lthat project to a partial basis of ¯LbyI(L). For the
+induction step we need the following proposition, which is a specia l case of
+a result due to Barbara van den Berg ([1, Prop. 1.6.1]). It is ins pired by a
+similar result in the Utrecht thesis (1979) of Maazen [7], whi ch says that the
+poset of partial bases of a free finitely generated module over a Eucl idean
+ring is Cohen-Macaulay. (See also the Appendix.) We here deriv e the result
+in question from Maazen’s theorem.
+Proposition 1.2. LetLbe a quasi-unimodular lattice of genus g. Then the
+posetI(L)of isotropic sequences that map to a partial basis of ¯Lis Cohen-
+Macaulay of dimension g−1.
+Proof. We first show that I(L)is spherical of dimension g−1. Consider the
+posetI(¯L)of nonzero primitive isotropic sublattices of ¯L. According to a4 WILBERD VAN DER KALLEN AND EDUARD LOOIJENGA
+theorem of Solomon-Tits [2, IV 5, Thm 2], I(¯L)is Cohen-Macaulay of di-
+mension g−1. So ifa∈I(¯L)is of rank k+1, thenI(¯L)>ais spherical of
+dimension g−2−k. Letf:I(L)→I(¯L)be the poset map that assigns to an
+isotropic sequence in Lthe span of its image in ¯L. Let ˜abe the preimage of
+aunder the projection L→¯L. Thenf/ais the set of sequences (v0,...,vs)
+in ˜athat map to a partial basis of a. Or equivalently, if we first fix a basis
+(w1,...,w r)ofLo, that(v0,...,vs,w1,...,w r)is a partial basis of ˜ a. Ac-
+cording to Maazen ([7, Thm. III 4.2] or Theorem 6.1 below), this subp oset
+is spherical of dimension k. So condition ( Cop) of Corollary 2.2 is fulfilled
+forn=g−1and we conclude that I(L)is spherical of dimension g−1.
+We next verify the other properties needed for Cohen-Macaulayness . Let
+v∈ I(L)have length k+1. ThenI(L)<vis the poset of all proper subse-
+quences of v. This is essentially the boundary of a k-simplex and hence a
+(k−1)-sphere. For a similar reason, if v′is a subsequence of vof length
+k′+1, thenI(L)>v′∩I(L)<vis spherical of dimension k−k′−2.
+Finally, if L′denotes the orthogonal complement of the span IofvinL,
+thenL′is quasi-unimodular of genus g−k−1with radical Lo⊕IandI(L)>v
+consists of the sequences obtained by shuffling vwith elements of I(L′). By
+the above discussion combined with [3, Cor. 1.7] for the shuffling, I(L)>vis
+therefore spherical of dimension g−k−2. /square
+We useI(¯L)to index full subcomplexes of U(L)>0as follows. For a
+primitive vector v∈¯Ldenote by Lvthe set of u∈Lwith/ang⌊ra⌋k⌉⊔l⌉f⊔¯u,v/ang⌊ra⌋k⌉⊔righ⊔=0and put
+Xv:=U(Lv). For an isotropic sequence v= (v0,...,vk), we put Lv:=∩iLvi
+andXv:=∩iXvi=X(Lv). The collection {U(Lv)>0}v∈I(¯L)covers the poset
+ofu∈ U(L)with0 < g(u)< g (which we shall denote by U(L)(0,g)): if
+u∈ U(L)(0,g), then there is a primitive isotropic v∈¯Lperpendicular to ¯ u
+and for any such vwe have u∈ U(Lv)>0. The fact that U(Lv)>0=∅ifvhas
+lengthgwill not bother us. Observe that if v′is a subsequence of v, then
+U(Lv)>0⊂ U(Lv′)>0.
+Proof of Theorem 1.1 in case Lis unimodular. We must show that U(L)(0,g)is
+(g−3)-connected. We do this by verifying the hypotheses (i) and (ii) of
+Theorem 2.3 for X=U(L)(0,g), forA:=I(¯L)indexing the collection of
+full subposets {Xv:=U(Lv)>0}v∈I(¯L)ofXand with n=g−2. We take for
+height functions the standard ones on these posets: ht (u) =g(u) −1and
+ht(v) =|v|−1(here|v|stands for the length of v).
+By our induction hypothesis, the subset Xv=U(Lv)>0is(g−2−ht(v))-
+connected. This verifies 2.3-(i).
+Givenu∈X, thenX<u=U(L)(0,u)is(g(u) −3)-connected by induction.
+The poset Auofv∈ I(¯L)>0with ¯u⊥vis justI(¯u⊥)and hence is (g−g(u)−
+2)-connected by Proposition 1.2. Since g−g(u)−2= (g−2)−ht(u) −1,
+2.3-(ii) is also satisfied.SPHERICAL COMPLEXES ATTACHED TO SYMPLECTIC LATTICES 5
+Hence Theorem 2.3 applies. Since Ais(g−3)-connected (it is even
+(g−2)-connected), it follows that Xis(g−3)-connected. /square
+For the case when Lis not unimodular, we intend to invoke both halves
+of Theorem 2.3. We shall use the following elementary observation:
+Lemma 1.3. Letv= (v0,...,vk)∈ I(¯L). Ifu∈ U(L)is of genus k+1and
+such that vis contained in ¯u, then for every u′∈ U(Lv),u′+uis unimodular
+of genusg(u′)+k+1.
+Proof. Choose a lift ei∈uofviand extend (e0,...,ek)to a symplectic basis
+ofu:(e0,...,ek;f0,...,fk). Foru′as in the lemma, we have ei⊥u′. Since
+u′is unimodular, there exists a f′
+i∈u′such that fi−f′
+iis perpendicular to
+u′. Then(e0,...,ek;f0−f′
+0,...,fk−f′
+k)spans a unimodular lattice ˜ uand
+u+u′is the perpendicular direct sum of ˜ uandu′. Sou′+uis unimodular
+of genus g(u′)+k+1. /square
+Proof of Theorem 1.1 in case Lis not unimodular. We wish to apply 2.3 to
+the case when X=U(L)>0, but with the same collection of subposets
+{Xv=U(Lv)>0}v∈I(¯L)and value of n(namely g−2) as in the unimodu-
+lar case. The conditions (i) and (ii) are still satisfied for this value of n. We
+also know that A=I(¯L)is(g−2)-connected.
+Letv= (v0,...,vk)∈ I(¯L)k. Fori=0,...,k we choose ui∈Xof genus 1
+such that vi∈¯uiandu0,...,u kare pairwise perpendicular. For v′<v, we
+letuv′∈Xbe the span of the uiwithvi∈v−v′. Thenuv′∈Xis of genus
+|v|−|v′|and is contained in Xv′. According to Lemma 1.3, the sublattice
+u+uv′is unimodular for any u∈Xv−v′. This is a fortiori so when u∈Xv,
+but since uv′∈Xv′, we then in fact have u+uv′∈Xv′.
+So condition 2.3-(iii) is verified for sv(v′) :=uv′andev(v′,u) :=u+uv′.
+It remains to see that the resulting poset map
+^sv: (A<v)op⋊ ⋉Xv→X
+is null-homotopic. The image Vvof^svis the union of Xvand the images of
+sv,ev. Letu∅denote the span of u0,...,u k. Note that sv(v′) =uv′≤u∅
+and that if u∈Xv, thenu+u∅∈Xby Lemma 1.3. So ^svis homotopic to
+the constant map with value u∅via the relations (A<v)op⋊ ⋉Xv→Vv∋u≤
+u+u∅≥u∅. /square
+2. G ENERALITIES ON POSETS
+We here collect some general results regarding posets that were used in
+the proofs of the previous section. A poset Xdefines a simplicial complex
+with vertex set Xfor which its k-simplices are chains x0< x1<···< xk.
+So the poset structure makes that every simplex has a natural ord er of its
+vertices, in particular, has a natural orientation. The geometric realization
+|X|ofXis by definition that of the associated complex. So an element of
+|X|is given by a function φ:X→R≥0whose support is a chain x0< x1<
+···< xkand/summationtext
+iφ(xi) =1. We often allow ourselves the common abuse of6 WILBERD VAN DER KALLEN AND EDUARD LOOIJENGA
+terminology when we say that Xenjoys a given (topological) property (such
+as connectedness or dimension), when we actually mean this to hol d for|X|.
+AZ-valued function fdefined on the poset Xis called a height function if it
+is strictly increasing: x < y impliesf(x)< f(y). If for every x∈X, dim(X≤x)
+(the supremum of the set of nfor which there exists a chain x0< x1<···<
+xn=xinX) is finite, then a standard choice for fisf(x) :=dim(X≤x). We
+call a height function bounded if its image is contained in a finite interval.
+Given a height function fonX, we take −fas height function on Xop.
+We use the convention that the dimension of the empty set is −1.
+Recall that a poset Xis said to be n-spherical if its geometric realization
+is of dimension nandXis(n−1)-connected (we agree that the empty
+set is(−1)-spherical). It is said to be Cohen-Macaulay of dimension nif in
+addition
+(i) for every x∈X,X<xresp.X>xis spherical of dimension dim X≤x−1
+resp.n−1−dimX≤xand
+(ii) for every ordered pair x < y inX,X>x∩X<yis spherical of dimension
+dimX≤y−dimX≤x−2.
+IfX,Yare posets we define its join X∗Yas in [9, 1.8] to be the disjoint
+union of XandYequipped with the ordering which agrees with the given
+orderings on XandYand which is such that any element of Xis less than
+any element of Y. Note that this is asymmetric in XandY. It is clear that
+with this definition an iterated join X1∗X2∗··· ∗Xnhas as underlying set
+the disjoint union of X1,...,X nwith the given partial order on any piece Xi
+and with any element of Xidominating X1∪ ··· ∪Xi−1. In the case of two
+posetsX,Ywe will also make use of the thick join X⋊ ⋉Ywhich is symmetric
+inXandY. It is defined as follows. Letting X×Ydenote the product of X
+andYin the category of posets, then X⋊ ⋉Yis the disjoint union of X,Y,
+X×Yequipped with the ordering which is obtained by adding to the give n
+orderings on X,Y,X×Ythe relations x <(x,y)> yforx∈X,y∈Y. The
+poset map h(X,Y) :X⋊ ⋉Y→X∗Ywhich maps xtox,(x,y)toy,yto
+y, is a homotopy equivalence [7, Prop. II 1.2]. If Xisd-connected and Yis
+d′-connected, then X∗Yisd+d′+2-connected (and hence X⋊ ⋉Yis too).
+Letf:X→Ybe a map of posets. Recall that f/y={x∈X|fx≤y}and
+y\f={x∈X|fx≥y}. We define the mapping cylinder off,M(f), to be
+the disjoint union of XandYequipped with the ordering which agrees with
+the given orderings on XandYand which is such that for x∈X,y∈Y, one
+hasx > y if and only if fx≥y. Observe that Yis a deformation retract of
+M(f). Dually we define Mop(f) :=M(fop)op, wherefop:Xop→Yop. (As
+a set map fopequalsf.) Again Yis a deformation retract, but in Mop(f)
+any element of x∈Xis less than fx∈Y. IfYis a poset equipped with
+a height function ht we let M(f,≤k)denote the disjoint union of Xand
+Y≤k:={y∈Y|ht(y)≤k}with the ordering induced from M(f).
+Let us also define the mapping cone off, although this notion will play a
+less prominent role in this paper: it is the disjoint union C(f)of a singletonSPHERICAL COMPLEXES ATTACHED TO SYMPLECTIC LATTICES 7
+{vf}andM(f)whose ordering extends the one on M(f)and for which vfis
+smaller than every point of Xand is incomparable with the points of Y.
+Proposition 2.1. Letf:X→Ybe a map of posets and let htbe a bounded
+height function on Y. ThenM(f,≤k) =Xfork≪0,M(f,≤k) =M(f)∼Y
+fork≫0. Furthermore, for every y∈Yof height k, the link of yinM(f,≤
+k−1)can be identified with Y<y∗(y\f), and so M(f,≤k)is obtained from
+M(f,≤k−1)by putting for every ywith ht(y) =ka cone over Y<y∗(y\f). If
+in addition, Y<y∗(y\f)is(n−1)-connected for all y∈Y, thenfisn-connected
+(which means that the pair (M(f),X)isn-connected).
+Proof. The first assertion is clear and so is the second. If Y<y∗(y\f)is(n−1)-
+connected for all y∈Y, then|M(f)|is gotten, up to homotopy, from |X|by
+successive attachment of cells of dimension ≥n+1and so the pair (M(f),X)
+isn-connected. /square
+We get the following slight variation on Theorem 9.1. of Quillen [9] .
+Compare also [8, Thm 3.8].
+Corollary 2.2. Letf:X→Ybe a map of posets. Assume Yis endowed with a
+bounded height function htwith the property that for some integer nand some
+set mapt:Y→Zone of the following is true:
+(C)for every y∈Y,Y<yis(t(y)−2)-connected and y\fis(n−t(y)−1)-
+connected or dually,
+(Cop)for every y∈Y,Y>yis(n−t(y)−2)-connected and f/yis(t(y)−1)-
+connected.
+Thenfisn-connected.
+Proof. In case ( C), first observe that Y<y∗(y\f)is(t(y) −2)+(n−t(y) −
+1) +2-connected, hence n−1-connected, so that the result follows from
+Proposition 2.1. In case ( Cop), pass to the opposite. /square
+In many cases of interest, t=ht does the job. The same is true for the
+following nerve theorem, which is the technical result that make s the proof
+of Main Theorem 1.1 possible. The first half of the nerve theorem is fa miliar
+from Mirzaii-Van der Kallen ([8, Thm. 4.3]). We reprove it here to illustrate
+the efficiency of the present setup. The second half is our key adva nce.
+Theorem 2.3 (Nerve Theorem for Posets) .LetXandAbe posets both en-
+dowed with bounded height functions. Assume that Aoplabels fullsubposets of
+Xin the sense for every a∈Awe are given a subposet Xawith the property
+that ifx∈Xa, then also any y < x is inXaanda < b impliesXa⊃Xb. Letn
+be an integer such that for some set maps tX:X→ZandtA:A→Z
+(i)for every a∈A,A<ais(tA(a)−2)-connected, Xais(n−tA(a)−1)-
+connected and
+(ii)for every x∈X,X<xis(tX(x)−2)-connected and the full subcomplex
+AxofAspanned by the a∈Awithx∈Xais(n−tX(x)−1)-connected.8 WILBERD VAN DER KALLEN AND EDUARD LOOIJENGA
+ThenAis(n−1)-connected if and only if Xis.
+If in addition to (i) and (ii) there exist for every a∈A, poset maps sa:
+(A<a)op→Xandea: (A<a)op×Xa→Xsuch that
+(iii) ifb < a , thensa(b)∈Xband for all x∈Xaone hasea(b,x)∈Xb
+andsa(b)≤ea(b,x)≥x, and
+(iv)the resulting poset map ^sa: (A<a)op⋊ ⋉Xa→Xis null-homotopic,
+then there even exists an n-connected map |A|→|X|(so thatXisn-connected
+whenAis).
+Proof. Denote by Z⊂Aop×Xthe subset of pairs (a,x)∈A×Xwithx∈Xa
+with the induced partial order. We have poset projections
+f:Zop→Aandg:Z→X.
+We now divide the proof in a number of steps. Since |Zop|=|Z|, the first
+assertion of the theorem follows from:
+Step 1: Both fandgaren-connected. For every a∈A, we have that
+a\f={(b,y)∈Z|b≥a,y∈Xb}contains {a}×Xaas deformation retract
+with retraction given by (b,y)/ma√s⊔o→(a,y). So by (i), a\fis(n−ht(a) −1)-
+connected. By that same assumption, A<ais(ht(a) −2)-connected.
+According to Corollary 2.2, fis then-connected. A similar argument
+applied to g:Z→X, withg/xinstead of x\g, yields that gisn-connected.
+From now on we assume that the conditions (iii) and (iv) are satisfie d.
+Step 2: Construction of a diagram, commutative up to homotop y.Leta∈A
+and putk=ht(a). We shall need the following diagram of poset maps
+((A<a)op⋊ ⋉Xa)ophop
+− −−−→
+∼A<a∗(Xa)op1∗jop
+− −−−→
+∼A<a∗(a\f)op
+˜sop
+a/arrowbt/arrowbtattaching map
+Zop− −−−→M(f,≤k−1) M(f,≤k−1).
+Here the top horizontal maps are homotopy equivalences: the first ma p is
+the opposite of the natural map h:Xa⋊ ⋉(A<a)op→Xa∗(A<a)opencoun-
+tered before (it sends b∈A<atob,x∈Xatoxand(b,x)tob) and the
+second map is the join of the identity map of A<aand the opposite of the
+homotopy equivalence j:x∈(Xa)/ma√s⊔o→(a,x)∈(a\f). The first lower hori-
+zontal map is the inclusion. The vertical map on the left is the opp osite of
+the poset map
+˜sa: (A<a)op⋊ ⋉Xa→Z,
+˜sa/parenleftbig
+b <(b,x)> x/parenrightbig
+=/parenleftbig
+(b,sa(b))≤(b,ea(b,x))≥(a,x)/parenrightbig
+,
+whereb∈A<aandx∈Xa. Notice that g˜sa= ^sa. The vertical map on the
+right is the embedding of the link of a∈M(f,≤k)inM(f,≤k−1)that
+appears in Proposition 2.1 (it is given by A<a⊂Aand(a\f)op⊂Zop).SPHERICAL COMPLEXES ATTACHED TO SYMPLECTIC LATTICES 9
+We check that the diagram is homotopy commutative. The two maps fr om
+((A<a)op⋊ ⋉Xa)optoM(f,≤k−1)(whose underlying set is A≤k−1⊔Zop)
+are given by
+/parenleftbig
+b >(b,x)< x/parenrightbig
+/ma√s⊔o→/braceleftBigg/parenleftbig
+(b,sa(b))≥(b,ea(b,x))<(a,x)/parenrightbig
+,/parenleftbig
+b=b <(a,x)/parenrightbig
+.
+The first two elements in the second line lie in A<a; all other elements
+lie inZop. The definition of the partial order on M(f,≤k−1)is such
+that we see that the maps define together one from the product poset
+(0 < 1)×((A<a)op⋊ ⋉Xa)optoM(f,≤k−1)and so are homotopic.
+Step 3: Conclusion. Let˜Z⊃Zbe the union of the mapping cones of the
+poset maps ˜ sa: (A<a)op⋊ ⋉Xa→Z(these mapping cones have Zin com-
+mon). Since M(f,≤k)is obtained from M(f,≤k−1)by putting a cone over
+eachA<a∗(Xa)with ht(a) =k, repeated use of the above diagram yields
+a homotopy equivalence |˜Z|˜−→|M(f)|. Since |A|is a deformation retract
+of|M(f)|, we find a homotopy equivalence |˜Z|˜−→|A|. Choose a homotopy
+inverseH:|A|˜−→|˜Z|.
+Each composite ^sa=g˜sais null-homotopic by assumption (iv). This
+implies that the map |g|extends to a continuous map G:|˜Z|→|X|. As|˜Z|is
+obtained from |Z|by attaching cells of dimension ≥n+1, this map is still
+n-connected. So GHis as desired. /square
+3. T HE COMPLEX OF UNIMODULAR DECOMPOSITIONS
+We suppose Lunimodular of genus g > 0 . We call a subset u⊂ U(L)>0
+aunimodular decomposition ofLif the natural map ⊕u∈uu→Lis an iso-
+morphism. If uandu′are unimodular decompositions, then we say that
+u′refines u(and we write u′≥u) if every member of u′is contained in
+one of u. This makes the collection of such decompositions a poset that we
+shall denote by D(L). We chose this convention for the partial order (rather
+than its opposite) as to have the standard height function on D(L)assign
+touthe value |u|−1(the value 0being taken by the unique minimal ele-
+ment(L)). Notice that D(L)is a subcategory of the category whose objects
+are finite subsets of U(L)>0and whose morphisms are surjections between
+these subsets. We sometimes write D+(L)forD(L)>(L), the subposet of strict
+decompositions.
+Recall that the barycentric subdivision of a poset Xis the poset X′whose
+elements are chains in Xand for which <is the relation ‘subchain of’. Its
+geometric realization is homeomorphic to that of X.
+Now observe that there is a natural poset map
+f: (U(L)>0)′→D(L)
+which assigns to a chain 0/n⌉ga⊔ionslash=u0/subset⋉⋊teqlu1···/subset⋉⋊teqluk⊂Lthe decomposition whose
+members are u0,u1∩u⊥
+0,...,u k∩u⊥
+k−1and (in the case when uk/n⌉ga⊔ionslash=L)u⊥
+k.10 WILBERD VAN DER KALLEN AND EDUARD LOOIJENGA
+Theorem 3.1. The poset D(L)of unimodular decompositions of Lis Cohen-
+Macaulay of dimension g−1.
+We first prove a ‘simplicial counterpart’ of this theorem.
+Lemma 3.2. LetXbe a finite set with at least two elements. Denote by D+(X)
+the poset of strict decompositions of X(this excludes the trivial decomposition
+{X}). ThenD+(X)is spherical of dimension |X|−2.
+Proof. We prove this with induction on d:=|X|−2. The case d=0is trivial,
+so suppose d > 0 . The poset F(X)of proper nonempty subsets of Xis the
+the barycentric subdivision of the boundary of the simplex spanned by X
+and is hence a combinatorial d-sphere. The poset F(X)′of nested proper
+nonempty subsets of Xcan be identified with the barycentric subdivision
+ofF(X)and so is still a combinatorial d-sphere. Now consider the map
+g+:F(X)′→D+(X)which assigns to ∅/subset⋉⋊teqlX0/subset⋉⋊teqlX1/subset⋉⋊teql···/subset⋉⋊teqlXh/subset⋉⋊teqlXthe
+partition {X0,X1−X0,...,X h−Xh−1,X−Xh}. This is a map of posets to
+which we want to apply Corollary 2.2. Let Y={Y0,...,Y h+1}be a partition
+ofX. Then
+D+(X)>Y∼=D+(Y0)∗D+(Y1)∗···∗D +(Yh+1)
+and so by our induction hypothesis, D+(X)>Yis spherical of dimension −1+/summationtexth+1
+i=0/parenleftbig
+|Yi|−1/parenrightbig
+=|X|−3−h=d−1−h. On the other hand, g+/ucan
+be identified with F({0,...,h+1})′, hence is a sphere of dimension h. Now
+apply version ( Cop) of Corollary 2.2. /square
+Proof of Theorem 3.1. We proceed with induction on g≥1. Forg=1,
+D(L) ={L}and so there is nothing to show. Assume g > 1 and the theorem
+proved for genera < g. We apply version ( Cop) of Corollary 2.2 to f. Let
+u={u0,...,u h}∈ D(L), so that ht (u) =h. Then we readily observe that
+D(L)>u∼=D+(u0)∗···∗D +(uh)
+and soD(L)>uis spherical of dimension −1+/summationtexth
+i=0/parenleftbig
+g(ui)−1/parenrightbig
+=g−2−h. On
+the other hand, f+/ucan be identified with the poset of nested nonempty
+subsets of {0,...,h}. This is contractible as it has the entire set as its maximal
+element. So the conditions ( Cop) of 2.2 are satisfied. Combining this with
+Theorem 1.1 yields that D(L)is spherical of dimension g−1. In passing we
+showed that D(L)>uis spherical. Notice that D(L)<ucan be identified with
+the poset D+({0,...,h})of all strict decompositions of the set {0,...,h}. This
+is spherical of dimension h−1by Lemma 3.2 above. Similarly on finds that
+ifu<u′, thenD(L)(u,u′)is spherical of dimension |u′|−|u|−2. /square
+The poset D(L)appears in the moduli space of principally polarized
+abelian varieties. Consider the complexification of L,LC:=C⊗ZLand
+extend/ang⌊ra⌋k⌉⊔l⌉f⊔,/ang⌊ra⌋k⌉⊔righ⊔bilinearly to LC. OnLCwe also have the Hermitian form Hde-
+fined by H(v,w) :=√
+−1/ang⌊ra⌋k⌉⊔l⌉f⊔v,w/ang⌊ra⌋k⌉⊔righ⊔, whose signature is (g,g). The space S(L)
+of complex subspaces F⊂LCof dimension gon which /ang⌊ra⌋k⌉⊔l⌉f⊔,/ang⌊ra⌋k⌉⊔righ⊔is zero and
+His positive definite can be understood as the moduli space of complexSPHERICAL COMPLEXES ATTACHED TO SYMPLECTIC LATTICES 11
+structures on the real torus (R/Z)⊗ZLthat are polarized by H. It is a sym-
+metric space for the symplectic group of LRand a choice of symplectic basis
+yields an isomorphism with the Siegel upper half space of genus g. Notice
+that a unimodular decomposition uofLdetermines a proper embedding of/producttext
+u∈uS(u)inS(L)with image a totally geodesic analytic submanifold. We
+shall denote that submanifold by S(u).
+Corollary 3.3. The locus S(L)decin the genus gSiegel space S(L)which
+parametrizes the principally polarized abelian varieties that ar e decomposable
+as polarized varieties is a closed analytic subvariety of S(L)which has the ho-
+motopy type of a bouquet of (g−2)-spheres. In fact, the (closed) covering of
+S(L)decby its irreducible components is a Leray covering (which mean s that ev-
+ery nonempty finite intersection is contractible) whose nerv e is identified with
+D+(L)opvia the correspondence u/ma√s⊔o→S(u).
+Proof. It is clear that S(L)decis the union of the S({u,u⊥}), whereuruns
+over the unimodular sublattices of Lthat are neither 0norL. It is known
+that this union is locally finite. So S(L)decis a closed analytic subvariety
+ofS(L)and as there are no inclusion relations among the S({u,u⊥}), these
+yield the distinct irreducible components of S(L)dec. In view of Weil’s nerve
+theorem it now suffices to prove the last statement.
+It is a classical result (see for instance [5, §6.9]) that any principally po-
+larized abelian variety Aof positive dimension has a unique decomposition
+into indecomposables. Precisely, if {Ai}iis the collection of abelian subvari-
+eties of positive dimension of Athat receive from Aa principal polarization
+and are minimal for this property, then the natural map/producttext
+iAi→Ais an
+isomorphism. It follows that u/ma√s⊔o→S(u)defines an bijection from D+(L)to
+the collection of nonempty intersections of the irreducible componen ts of
+S(L)dec. If we give the latter the poset structure defined by inclusion, t hen
+this is in fact a poset isomorphism from D+(L)op. EachS(u)∼=/producttext
+u∈uS(u)
+is contractible and so the covering has the asserted properties. /square
+Corollary 3.4. The locus Ag,dec⊂ Agof decomposables in the moduli space of
+principal polarized abelian varieties of genus gis a quasiprojective subvariety.
+If˜Ag→Agis a finite covering defined by a torsion free subgroup of Sp(2g,Z)
+of finite index, and ˜Ag,decdenotes its preimage in ˜Ag, then the pair (˜Ag,˜Ag,dec)
+is(g−2)-connected. In particular, Hk(Ag,Ag,dec;Q) =0fork≤g−2.
+Sketch of Proof. The first statement is in fact well-known, so let us just out-
+line its proof. It follows from Corollary 3.3 that Ag,dec⊂ Agis a closed
+analytic subvariety. The Baily-Borel theory shows that its closure ( relative
+to the Hausdorff topology) in the Baily-Borel compactification A∗
+gofAgis
+projective, as it is the image of an analytic morphism from the union of
+A∗
+k×A∗
+g−k,k=1,...,⌊1
+2g⌋toA∗
+g. SinceA∗
+gis projective, and has a projec-
+tive boundary, it follows that Ag,decis quasiprojective.
+As to the remaining statements, let Sgbe the Siegel space attached to
+the standard symplectic lattice Z2g. According to Corollary 3.3, Sg,dechas12 WILBERD VAN DER KALLEN AND EDUARD LOOIJENGA
+the homotopy type of a bouquet of (g−2)-spheres. So we can construct
+a relative CW complex (Z,Sg,dec)obtained from Sg,decby attaching cells of
+dimension ≥g−1in a Sp(2g,Z)-equivariant manner as to ensure that Z
+is contractible and no nontrivial element of Sp (2g,Z)fixes a cell. If ˜Agis
+defined by the subgroup Γ⊂Sp(2g,Z), thenΓacts freely on Z(as it does
+on the contractible Sg) and so there is a Γ-equivariant homotopy equiva-
+lenceZ→Sgrelative to Sg,dec. It follows that there is also a homotopy
+equivalence Γ\Z→˜Agrelative to ˜Ag,dec. Hence (˜Ag,˜Ag,dec)is(g−2)-
+connected. /square
+Remark 3.5.This corollary has a counterpart for the moduli space of stable
+genusgcurves with compact jacobian (see [6, Cor. 1.2]).
+4. I MPROVED HOMOLOGICAL STABILITY FOR THE SYMPLECTIC GROUPS
+As an aside we show in this section that our main result yields a sli ght
+improvement of a result of Charney. We denote the basis elements of Z2g
+by(e1,e−1,....eg,e−g)and endow Z2gwith the symplectic form that this
+notation suggests: /ang⌊ra⌋k⌉⊔l⌉f⊔ei,e−i/ang⌊ra⌋k⌉⊔righ⊔=sign(i)and/ang⌊ra⌋k⌉⊔l⌉f⊔ei,ej/ang⌊ra⌋k⌉⊔righ⊔=0whenj/n⌉ga⊔ionslash=±i. Let us
+writeGgfor the algebraic group attached to Sp (2g,Z)so that for any ring
+R,Gg(R) =Sp(2g,R). The obvious embedding of Gg(R)inGg+1(R)(which
+identifies the Gg(R)as theGg+1(R)-stabilizer of the last two basis vectors)
+induces a map on homology that is known to be an isomorphism in low de-
+gree for many choices of R: Charney [4], following up on earlier work of
+Vogtmann, proved such a result for noetherian rings Rof finite noetherian
+dimension. Her stability range is phrased in terms of the connect ivity prop-
+erties of a poset HUg(R)whose elements are what she calls split unimodular
+sequences inR2g: these are sequences of pairs/parenleftbig
+(v1,v−1),...,(vk,v−k)/parenrightbig
+inR2g
+such that /ang⌊ra⌋k⌉⊔l⌉f⊔vi,v−i/ang⌊ra⌋k⌉⊔righ⊔=sign(i)and/ang⌊ra⌋k⌉⊔l⌉f⊔vi,vj/ang⌊ra⌋k⌉⊔righ⊔=0whenj/n⌉ga⊔ionslash=±i(so this is equiva-
+lent to giving a kand a symplectic embedding of Z2kinZ2g). The result may
+the be stated as follows: if a≥dim(R) +2is an integer such that HUg(R)
+is⌊1
+2(g−a−1)⌋-connected for all g, thenHi(Gg(R))→Hi(Gg+1(R))is an
+isomorphism for g≥2i+a+1and surjective for g=2i+a. This is supple-
+mented by the theorem that the connectivity assumption is fulfill ed for the
+caseRis Dedekind domain resp. a principal ideal domain for a=5resp.
+a=4. The point we wish to make is that we can do slightly better when R
+is a Euclidean ring:
+Theorem 4.1. IfRis a Euclidean ring, then we may take a=2:HUg(R)is
+⌊1
+2(g−3)⌋-connected and hence Hi(Gg(R))→Hi(Gg+1(R))is an isomorphism
+forg≥2i+3and surjective for g=2i+2.
+There is corresponding statement with twisted coefficients (se e Theorem
+4.3 of op. cit. ) that we will not bother to explicate. The proof of Theorem
+4.1 rests on the observation that if we view the contents of the Theore ms
+1.1 and 3.1 as properties regarding Gg(R)forR=Z, then both statements
+and proofs remain valid if we let Rbe any a Euclidean ring (that propertySPHERICAL COMPLEXES ATTACHED TO SYMPLECTIC LATTICES 13
+enters in the proof of Theorem 6.1, a result that is due to Maazen). And
+we may replace dim (R)with zero because Charney uses dim (R)only to deal
+with projective modules. In our case they are free.
+For the proof of 4.1 we need
+Lemma 4.2. LetAbe a set endowed with a partition Pinto a finite number
+of (nonempty) subsets. Then the poset of sequences in Awhich hit every part
+ofPat most once is spherical of dimension |P|−1.
+Proof. Denote this poset by Xand letYbe the allied poset of nonempty finite
+subsets of Athat meet every part at most once. So we have an evident poset
+mapf:X→Y. Since|Y|can be identified with the iterated join of the
+distinct parts of A(which are |P|in number), it is spherical of dimension
+|P|−1. Observe that for any y∈Y,f/yis the poset of nonempty sequences
+whose terms are distinct and lie in y. This is well-known to be spherical of
+dimension |y|−1([7, Thm. 2.1], or [10, Lemma 2.13(ii)]). On the other
+hand,|Y>y|can be identified with the iterated join of the distinct parts of
+Anot hit by yand as there are |P|−|y|such parts, |Y>y|is spherical of
+dimension |P|−|y|−1. Now apply Corollary 2.2. /square
+Proof of Theorem 4.1. Consider the poset map f:HUg(R)→D+(R2g)
+which assigns to/parenleftbig
+(v1,v−1),...,(vk,v−k)/parenrightbig
+the collection of genus 1 sum-
+mandsUi:=Rvi+Rv−i,i=1,...,k and the perp of their sum in R2g.
+Given u∈ D+(R2g), denote by t(u)the number of genus 1 summands in
+uand bys(u)the number of summands of higher genus. Then clearly
+2s(u) +t(u)≤g, or equivalently, s(u)≤1
+2(g−t(u)). Observe that f/u
+is a poset of the type that appears in Lemma 4.2 above: a member of A
+is an oriented basis of any genus 1 summand in uandPis the obvious
+partition defined by those summands. As there are t(u)parts, the lemma
+tells us that f/uis(t(u) −2)-connected. On the other hand, D+(R2g)>u
+is a join of all the D+(u),u∈uand hence is connected of dimension
+−2+/summationtext
+u∈u(g(u)−1) =g−2−|u|. In view of the inequality
+t(u)+(g−|u|)−2=g−s(u)−2≥ ⌈1
+2(g+t(u))⌉−2≥ ⌊1
+2(g−3)⌋,
+it follows from the ( Cop)-variant of Corollary 2.2 that HUg(R)is⌊1
+2(g−3)⌋-
+connected. /square
+5. R ELATION TO THE SEPARATED CURVE COMPLEX
+A variation of the complex D+(L)considered above appears in the study
+of the Torelli groups. Let Sbe a closed connected orientable surface of genus
+g. ThenH1(S)has the structure of a unimodular lattice of genus gso that we
+have defined D(H1(S)). An isotopy class of embedded (unoriented) circles
+inSis called separating if the complement of a representative decomposes
+Sinto two connected components of positive genus. Notice that then t he
+homology of the components determines a nontrivial unimodular split ting
+ofH1(S). The separated curve complex Csep(S)ofSis the simplicial complex14 WILBERD VAN DER KALLEN AND EDUARD LOOIJENGA
+whose vertex set are the isotopy classes of separating curves; a fi nite set
+of these spans a simplex precisely when its elements can be repr esented by
+embedded circles that are pairwise disjoint. The mapping clas s groupΓ(S)
+(=the group of connected components of the orientation preserving diff eo-
+morphisms of S) acts on both H1(S)andCsep(S). The action on H1(S)has
+the full integral symplectic group Sp (H1(S))as its image; the kernel of this
+action is called the Torelli group ofS, denoted here by T(S). The map that
+assigns to a separating curve its associated unimodular split ting ofH1(S)
+extends to a Sp (H1(S))-equivariant poset map T(S)\Csep(S)→D+(H1(S)).
+This is however not an isomorphism: think of the case when gembedded
+circles split off ggenus 1 surfaces with a hole, so that what remains is a
+sphere with gholes. Nevertheless:
+Theorem 5.1. The poset map T(S)\Csep(S)→D+(H1(S))is a homotopy
+equivalence, in particular, T(S)\Csep(S)is(g−3)-connected.
+This may be regarded as a natural companion of a result of one of us [6]
+that states Csep(S)is(g−3)-connected as well.
+Before getting into the proof, we give a construction of T(S)\Csep(S)
+entirely in terms of D+(H1(S)).
+In what follows a treeis a finite simplicial complex of dimension 1that is
+contractible. This amounts to giving a pair T= (T0,T1), whereT0is a finite
+set (the vertex set) and T1is anonempty collection of two-element subsets of
+T0(the set of edges) such that |T0|=|T1|+1and its geometric realization is
+connected. The degree of a vertex v∈T0, degT(v), is the number of elements
+ofT1that contain v.
+We denote by T00⊂T0the collection of vertices of degree ≤2. The
+identity|T0|=|T1|+1is equivalent to/summationtext
+v∈T0(degT(v) −2) = −2. Ifdi(T)
+denotes the number of vertices of degree i, then this amounts to/summationtext
+i≥3(i−
+2)di=d1−2, which shows that if we fix a bound on |T00|=d1+d2, then
+we have only finitely many isomorphism classes.
+Notice that if Eis a nonempty set of edges of T1, and if we contract every
+connected component of T−E, then we have formed a quotient tree πE:
+T→TEwith the property that πEmapsEmaps bijectively onto the edge set
+ofTE.
+Lemma 5.2. LetEandE′be nonempty sets of edges of a tree Tsuch that there
+exists an isomorphism h:TE∼=TE′with the property that πE′andhπEhave
+the same restriction to T00. ThenE=E′.
+Proof. We prove this with induction on |T1|. For|T1|=1there is nothing to
+show and so suppose |T1|> 1. We prove that there is a terminal edge eofT
+on which πEandπE′coincide, i.e., for which either e /∈E∪E′ore∈E∩E′.
+This suffices: in the first case we can apply the induction hypothe sis to the
+treeT/e, obtained by contracting e, and the images of EandE′inT/e. In
+the second case, πE(e)is a terminal edge with terminal vertex πE(v)andSPHERICAL COMPLEXES ATTACHED TO SYMPLECTIC LATTICES 15
+similarly for πE′(e)andπE′(v). Since the isomorphism htakesπE(v)to
+πE′(v), it will take πE(e)toπE′(e). So here too the induction hypothesis can
+be invoked to the tree T/eand the images of EandE′inT/e.
+Choose a terminal vertex v∈T00and lete={v,v′}∈T1be the corre-
+sponding terminal edge. If deg (v′) =2(so thatv′∈T00), thenπEandπE′
+coincide on e.
+So it remains to consider the case when deg (v′)≥3. This implies that
+T−eis disconnected. Assume that e∈E. Thenesubsists as an end edge
+inTEandπE(v)is terminal in TE. But then πE′(v)is also terminal in TE′. If
+e /∈E′(so thatvis contracted by πE′), then this can only happen if all but
+one of the components of T−eget contracted by πE′. So ifv′′∈T00is a
+terminal vertex of such a component, then πE′mapsvandv′′to the same
+vertex. Then πEwill have the same property. But elies on the geodesic
+string in Tfromvtov′′and soπE(v)/n⌉ga⊔ionslash=πE(v′), contradiction. So elies inE′
+also. /square
+Letube a finite set with at least two elements. We define a u-tree as a tree
+Tendowed with a map i:u→T0from to uto the vertex set T0ofTwith the
+property that its image contains T00. In view of the preceding observation,
+there are only finitely many isomorphism classes of u-trees. In fact, the
+number of vertices not in usuch a tree can have is at most |u|−2and so its
+number of edges will be at most 2|u|−3. This bound is attained when iis
+injective, all vertices in the image of ihave degree 1 and all other vertices
+have degree 3. Since a tree automorphism is determined by its res triction to
+the set of terminal vertices, an u-tree has no automorphism other than the
+identity. We denote the set of isomorphism classes of u-trees by T(u). We
+say that a u-tree is strict if the map i:u→T0is injective and we denote the
+corresponding subset of T(u)by˙T(u).
+We turnT(u)into a partially ordered set by stipulating that (T′,i′)≤(T,i)
+ifT′is obtained from Tby means of a contraction map whose composite
+withiyieldsi′. Lemma 5.2 shows that this contraction is already deter-
+mined by the associated injection T′
+1⊂T1. SoT(u)≤(T,i)can be identified
+with the poset of nonempty subsets of T1and hence |T(u)≤(T,i)|with the
+simplexσ(T). By the above computation, the dimension of T(u)is at most
+2|u|−4.
+Lemma 5.3. The poset T(u)is contractible.
+Proof. Ifuhas just two elements, then T(u)is a singleton (it has a single
+edge) and that is certainly contractible. Suppose therefore tha t|u|≥3. Then
+the cone C(u)onu(which has one vertex ∗not in uand edges {∗,u},u∈u)
+is clearly a u-tree. Notice that T(u)≤C(u)is the simplex on uand hence is
+contractible. We also have a deformation retraction of T(u)→T(u)≤C(u)
+which assigns to every u-tree inT(u)theu-tree inT(u)≤C(u)obtained by
+contracting all the internal edges, that is, the edges that do not have a vertex
+of degree one. /square16 WILBERD VAN DER KALLEN AND EDUARD LOOIJENGA
+We return to the poset D+(L)of strict unimodular decompositions of L.
+Observe that any u∈ D+(L)is a finite subset of the set of all unimodular
+latticesuwith0/n⌉ga⊔ionslash=u/n⌉ga⊔ionslash=Land that the relation u′≤udetermines (and
+is given by) a surjection u→u′. So the poset D+(L)is a subcategory of
+the category of sets. We define a poset TD(L)oftree decompositions ofL:
+its underlying set is the disjoint union of the isomorphism classes of strict
+u-trees, where uruns over D+(L):
+T D(L) :=/unionsqdisplay
+u∈D(L)˙T(L)
+and we stipulate that (T′,i′:u′֒→T′
+0)≤(T,i:u֒→T0)ifurefines u′and
+there exists an edge contraction T→T′extending the surjection u→u′
+associated to the refinement property. It follows from Lemma 5.2 th at this
+edge contraction is unique, so that we have defined a partial orde r indeed.
+Notice that there is an obvious forgetful map of posets
+p:T D(L)→D+(L).
+Proposition 5.4. The poset map p:T D(L)→D+(L)is a homotopy equiva-
+lence.
+Proof. For every u∈ D+(L),p/umay be identified with T(u), hence is
+contractible. Now apply Proposition (1.6) of [9]. /square
+Proof of Theorem 5.1. A simplex of Csep(S)is given by a closed one-
+dimensional submanifold A⊂Ssuch that every connected component of
+S−Ahas negative Euler characteristic and the associated graph T(S,A)
+(having the set of connected components of S−Aresp.Aas its set of
+vertices resp. edges) is a tree. Since a connected component of S−Aof
+zero genus has at least ≥3boundary components, all vertices of degree
+≤2correspond with connected components of S−Awith nonzero genus,
+hence positive genus. Moreover, the first homology groups of the connec ted
+components of positive genus define a decomposition of H1(S). This gives
+T(S,A)the structure of a tree decomposition of H1(S). Another choice A′
+forAyields the same tree decomposition of H1(S)if and only if there ex-
+ists an orientation preserving self-homeomorphism of Sthat takes AtoA′
+and induces the identity map in H1(S). It is now easy to identify the poset
+mappin Proposition 5.4 with the poset map T(S)\Csep(S)→D+(H1(S))
+introduced at the beginning of this section. The theorem follows. /square
+6. A PPENDIX
+In this appendix we recall the proof of the theorem of Maazen on the
+Cohen-Macaulayness of the poset O(n)of partial bases in a free module Rn
+over a Euclidean ring.
+Given a partial basis w= (w1,...,w r), we write O(n)wfor the poset of
+partial bases (v0,...,vs)so that(v0,...,vs,w1,...,w r)is a partial basis of
+Rn. Observe that GL (n)acts transitively on partial bases of a given length.SPHERICAL COMPLEXES ATTACHED TO SYMPLECTIC LATTICES 17
+We write pn:Rn→Rfor taking the last coordinate. For a,b∈Rwith
+||b||> 0, letq(a,b)∈Rbe the scalar given by the Euclidean structure, so
+that||a−q(a,b)b||<||b||. Fork≥0the subposet O(n,k)ofO(n)consists
+of the(v0,...,vs)with||pn(vi)||≤kfor alli. SoO(n,0)∼=O(n−1).
+Ifw= (w1,...,w r)∈ O(n)and||pn(wi)||=kfor some i≤r, we define
+ρw,i:Rn→Rnbyρw,i(v) =v−q(pn(v),pn(wi))wi. This induces a retract
+fromO(n)wtoO(n,k−1)w:=O(n,k−1)∩O(n)w. The key observation is
+that ifO(n)wisd-connected, then so is O(n,k−1)w. The other tools that
+Maazen uses have been published in [3] and [10], to which we refe r freely.
+Theorem 6.1 (Maazen) .LetRbe a Euclidean ring and d∈Z.
+(i)O(n)isd-connected if n≥d+2,
+(ii)O(n)wisd-connected for w∈ O(n)ifn≥d+|w|+2,
+(iii)O(n,k)wisd-connected for w∈ O(n), if there is an iwith
+||pn(wi)||=k+1≥1andn≥d+|w|+2,
+(iv)O(n,k)isd-connected if n≥d+2andk≥1.
+Proof. By induction on d. Ford <−1there is nothing to prove. Let n≥d+2.
+Choosew∈ O(n,1)\O(n,0)with|w|=1. Thenpn(w1)is a unit, so that the
+link ofwinO(n,0)is all ofO(n,0). Apply [10, Lemma 2.13(ii)] to conclude
+thatO(n,1)isd-connected. By induction on kit follows from [10, Lemma
+2.13(i)] that O(n,k)isd-connected for k≥1. Taking the limit k→ ∞ we
+see that O(n)isd-connected. For case (ii) we may assume that wconsists
+of the last |w|elements of the standard basis. By [3, Cor. 1.7], with Sequal
+to the span of wandF=O(n−|w|), case (ii) follows. Then case (iii) follows
+by means of the retract. /square
+From [3, Cor. 1.7] with n=|w|we see that O(m)>wis(m−|w|−2)-
+connected for w∈ O(m). One concludes as in 1.2 that
+Corollary 6.2. O(n)is Cohen-Macaulay.
+REFERENCES
+[1] B. van den Berg: On the abelianization of the Torelli group , Thesis Utrecht (2003),
+available at
+igitur-archive.library.uu.nl/dissertations/2003-101 3-162512/UUindex.html
+[2] K. S. Brown, Buildings, vii+215 pp., Springer-Verlag, New York, 1989.
+[3] R. Charney: On the problem of Homology Stability for Congruence subgrou ps, Comm. in
+Algebra, 12(17), 2081–2123 (1984).
+[4] R. Charney: A generalization of a theorem of Vogtmann , J. of Pure and Appl. Alg. 44,
+107–125 (1987).
+[5] O. Debarre: Complex tori and abelian varieties, SMF/AMS Texts and Monographs, 11,
+x+109 pp., Paris, 2005.
+[6] E. Looijenga: Connectivity of complexes of separating curves, arXiv:1001.0823
+[7] H. Maazen: Homology stability for the general linear group , Thesis Utrecht (1979),
+available at
+http://www.staff.science.uu.nl/ ∼kalle101/maazen1979.pdf
+[8] B. Mirzaii, W. van der Kallen: Homology stability for unitary groups, Doc. Math. 7,
+143–166 (2002).18 WILBERD VAN DER KALLEN AND EDUARD LOOIJENGA
+[9] D. Quillen: Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv.
+in Math. 28, 101–128 (1978).
+[10] W. van der Kallen: Homology stability for Linear Groups, Inventiones Math. 60, 269–
+295 (1980).
+E-mail address :W.vanderKallen@uu.nl, E.J.N.Looijenga@uu.nl
+MATHEMATISCH INSTITUUT , UNIVERSITEIT UTRECHT , P.O. B OX80.010, NL-3508 TA
+UTRECHT (NEDERLAND )
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+arXiv:1001.0884v1  [physics.gen-ph]  6 Jan 2010CONTENTS 1
+Hubbard-Stratonovich transformations to self-energies w ith coset decomposition to
+anomalous pair condensates for the standard model of electr oweak interactions
+(Classical field theory with self-energy matrices of the irr educible propagator parts)
+Bernhard Mieck1
+(article finished at 25 November 2009)
+Abstract
+The standard model of the strong and electroweak interactions is transformed from the ordinary path
+integral with the Lagrangians of quarks and leptons and with the Ab elian and non-Abelian gauge fields to
+correspondingself-energies. We apply the precise formulationin te rms ofmasslessMajoranaFermi fields with
+’Nambu’doublingwhichnaturallyleadstotheappropriateHST’s(Hubb ard-Stratonovich-transformations)of
+the self-energies and to the subsequent coset decomposition for the SSB (spontaneous symmetry breaking).
+The total coset decomposition of the Fermi fields is given by the dime nsionN0= 90 for the symmetry
+breaking SO( N0,N0)/U(N0)⊗U(N0) where the densities of fermions, related to the invariant subgrou p
+U(N0), are contained in a background functional for the remaining SO( N0,N0)/U(N0) coset field degrees
+of freedom of total Fermi fields which are composed of quark and le pton pairs. We find that the Higgs
+field partially enters into a sum with the self-energies for the gauge fi eld strength tensors so that it may be
+very difficult to observe pure, solely Higgs fields without a considerab le contribution from the Abelian and
+non-Abelian gauge field strength self-energies. The self-energy m atrices of the standard model, representing
+the irreducible terms of propagation, allow rigorous derivations for the so-called ”effective”, classical field
+theories as the Skyrme-like models from gradient expansions of the determinant remaining from the bilinear,
+anomalous doubled, fermionic field integrations.
+Keywords : ’Nambu’ doubling of fields, Hubbard-Stratonovich transformatio ns to self-energies, coset de-
+composition for pair condensates, standard model of strong and electroweak forces.
+PACS : 12.10.Dm , 12.39.Fe , 12.39.Dc , 11.15.Tk
+Contents
+1 Introduction 2
+1.1 The SO(90 ,90)/U(90)⊗U(90) self-energy matrix for the total standard model . . . . . . . . . 2
+2 Anomalous doubling of Fermi fields within the standard model 3
+2.1 Symmetry breaking source actions and anomalous pair con densates . . . . . . . . . . . . . . . . 3
+2.2 The Lagrangian of the standard model in terms of currents and auxiliary Higgs fields . . . . . . 8
+3 Self-energies of gauge field strength tensors and quartic Fermi fields 12
+3.1 Gaussian transformations with self-energy matrices of the gauge fields . . . . . . . . . . . . . . 12
+4 HST and coset decomposition of anomalous doubled Fermi fields 15
+4.1 Transformation of current-current terms or quartic, bi linear Fermi fields to self-energies . . . . 15
+5 Summary and conclusion 18
+5.1 The combination of the Higgs field with the gauge self-ene rgies . . . . . . . . . . . . . . . . . . 18
+1e-mail: ”bjmeppstein@arcor.de”; freelance activity 2009 ; current location : Zum Kohlwaldfeld 16, D-65817 Eppstein, Germany.2 1 INTRODUCTION
+1 Introduction
+1.1 The SO (90,90)/U(90)⊗U(90)self-energy matrix for the total standard model
+Although the standard model with the SU c(3)×SUL(2)×UY(1) gauge interactions contains numerous un-
+determined parameters, it gives profound insight into the s pontaneous symmetry breaking (SSB) of gauge
+symmetries with the Higgs mechanism [1, 2, 3]. The latter phe nomenon causes the various masses of quarks
+and leptons which initially enter into the Lagrangian only a s massless Majorana Fermi fields. The presented
+treatment starts out from the path integral for the standard model with axial gauge constraints and transforms
+the anti-commuting Fermi fields of leptons and quarks and als o the gauge fields with their gauge field strength
+tensorstocorrespondingself-energies after suitableHST s(Hubbard-Stratonovich-transformations). Apartfrom
+the Lagrangians of the standard model, we introduce symmetr y breaking condensate ’seeds’ for fermionic co-
+herent wavefunctions and for the even-numbered pairs of ’Na mbu’ or ’anomalous’ doubled Fermi fields [4, 5].
+The self-energies of the gauge field strength tensors only co ntribute as background fields in common with the
+density parts of the total self-energy of the Fermi fields whe reas the parts of the total self-energy, replacing the
+anomalous doubled Fermi field constituents, are the remaini ng field degrees of freedom in a coset decomposi-
+tion. The various, prevailing transformations in this pape r follow in analogy to the coset decomposition for the
+strong interaction case [6]. However, one has to extend the c oset decomposition SO( N0,N0)/U(N0)⊗U(N0)
+from the dimension N0= 24 for the QCD case with up-down (isospin) quark fields to N0= 90 under inclusion
+of the electroweak force. The dimension N0= 90 is attained from counting the independent components of
+quark and lepton Fermi fields with consideration of missing r ight-handed neutrinos. We finally achieve the
+effective generating function (1.1) of coset matrices ˆT(Ψ)(xp) = exp{−ˆY(xp)}for anomalous doubled Fermi
+fields Ψ(xp) ={ψ(xp);ψ∗(xp)}of leptons and quarks with the ’Nambu’ metric tensor ˆS for the process of
+doubling fields2
+Z/bracketleftbigˆT(Ψ)(xp);ˆJ;Jψ;ˆJψψ/bracketrightbig
+=/integraldisplay
+d[ˆT−1
+(Ψ)(xp)dˆT(Ψ)(xp)]/tildewide∆/parenleftBig
+ˆT−1;AB
+(Ψ);Mψ;Nψ(xp),ˆTAB
+(Ψ);Mψ;Nψ(xp);ˆJAB
+ψψ;Mψ;Nψ(xp)/parenrightBig
+(1.1)
+×DET/bracketleftBig
+ˆMAB
+Mψ;Nψ(xp,yq)/bracketrightBig1/2
+exp/braceleftbigg
+−ı
+2/integraldisplay
+Cd4xpd4yqJT,A
+ψ;Mψ(xp)ˆSAB′ˆTB′B′′
+(Ψ);Mψ;M′
+ψ(xp)/integraldisplay
+Cd4y′
+q′×
+×/bracketleftBig
+ˆM−1;B′′A′′
+M′
+ψ;N′′
+ψ(xp,y′
+q′)−ˆ1B′′A′′
+M′
+ψ;N′′
+ψδpq′δ(4)(xp−y′
+q′)/bracketrightBig /parenleftBig
+/an}bracketle{tˆHˆS/an}bracketri}ht−1/parenrightBigA′′A′
+N′′
+ψ;N′
+ψ(y′
+q′,yq)ˆTA′B
+(Ψ);N′
+ψ;Nψ(yq)JB
+ψ;Nψ(yq)/bracerightbigg
+;
+ˆMAB
+Mψ;Nψ(xp,yq) =ˆ1AB
+Mψ;Nψδpqδ(4)(xp−yq)+ (1.2)
++/bracketleftBig/parenleftbig
+/an}bracketle{tˆHˆS/an}bracketri}ht−1/parenrightbig
+δˆHˆS/parenleftbigˆT−1
+(Ψ),ˆT(Ψ)/parenrightbig
++/parenleftbig
+/an}bracketle{tˆHˆS/an}bracketri}ht−1/parenrightbigˆT−1
+(Ψ)ˆSˆJˆT(Ψ)/bracketrightBigAB
+Mψ;Nψ(xp,yq) ;
+/an}bracketle{tˆHˆS/an}bracketri}ht=ˆS/an}bracketle{tˆH/an}bracketri}ht; (1.3)
+δˆHˆS/parenleftbigˆT−1
+(Ψ),ˆT(Ψ)/parenrightbig
+=/parenleftBig
+ˆT−1
+(Ψ)ˆS/an}bracketle{tˆH/an}bracketri}htˆT(Ψ)/parenrightBig
+−/parenleftBig
+ˆS/an}bracketle{tˆH/an}bracketri}ht/parenrightBig
+=/parenleftBig
+exp/braceleftbig− −−−−−− →
+[ˆY,...]−/bracerightbigˆS/an}bracketle{tˆH/an}bracketri}ht/parenrightBig
+−/parenleftBig
+ˆS/an}bracketle{tˆH/an}bracketri}ht/parenrightBig
+. (1.4)
+The path integral (1.1) on the non-equilibrium time contour with the two branches ’ p,q=±’ consists of
+the effective kinetic part /an}bracketle{tˆHˆS/an}bracketri}htwith effective gauge terms from a saddle point computation of a background
+averaging functional (denoted by /an}bracketle{t.../an}bracketri}ht) with the self-energy ”densities” of Fermi fields as the inva riant subgroup
+in the coset decomposition for the SSB. In correspondence to appendix D of [6], one can perform a gradient
+2Since the precise derivation of (1.1) involves several type s of gauge-field-’dressed’ coset matrices, we mark the final, remaining
+coset matrix ˆT(Ψ)(xp) = exp{−ˆY(xp)}by the subscript ’ (Ψ)’ for the total, anomalous doubled Fermi fields. The part /tildewide∆(...) (first
+line in (1.1)) denotes a condensate ’seed’ functional with a condensate ’seed’ matrix ˆJAB
+ψψ;Mψ;Nψ(xp) for ’Nambu’ pairs of fermions.3
+expansion with the gradient term (1.4) up to fourth order for an effective Lagrangian (Derrick’s theorem [7]);
+hence, one has obtained a reliable, convenient, non-pertur bative, classical method for the dynamics within the
+total standard model from the spacetime evolution of the cos et matrices ˆT(Ψ)(xp) (compare e. g. the ’Skyrme
+model’ [8, 9]). It is also possible to calculate the energy mo mentum tensor of (1.1-1.4) or of a gradient-
+expanded version from the infinitesimal spacetime variatio ns (with inclusion of the variations of the coset
+integration measure d[ˆT−1
+(Ψ)(xp)dˆT(Ψ)(xp)], SO(90,90)/U(90)) in order to couple to gravity; the corresponding
+energy-momentum tensor of (1.1-1.4) has then to act as a sour ce tensor in the classical Einstein-field equations
+[10, 11]. In this manner one can encompass all known interact ions and their non-perturbative dynamics into
+the spacetime evolution of the coset matrices ˆT(Ψ)(xp) where the locally Euclidean spacetime coordinates dxµ
+p
+in the generating function (1.1) are related by the inverse s quare root ˆ g−1/2
+µν(xp)dxµ
+p=dzp,νof the coordinate
+metric tensor ˆ gµν(xp) to the curved spacetime dzp,µwhich are determined from the Einstein field equations.
+According to chap. 5 (”Derivation of a nontrivial topology a nd the chiral anomaly”) in [6], one can also
+conclude for Hopf invariants Π 3(S2) =Zfrom the quaternion eigenvalues and matrix elements of the g enerator
+ˆY(xp)withinthecoset matrix ˆT(Ψ)(xp); however, thereoccursananomalycancellation withinthe total standard
+model so that the total sum of Hopf invariants of the combined lepton and quark sectors should add to zero.
+This vanishing sumof Hopf invariants Π 3(S2) =Zshouldtherefore constrain scattering and decays of combin ed
+quark and lepton fields, as e. g. in neutron decay n→p+e−+ν. The nontrivial topological configuration
+of fields can be rather involved for the case of the Hopf invari ants; as one considers the pre-image of apoint
+from theS2sphere within the internal space of fields to the compactified three dimensional coordinate space,
+one can attain closed, one dimensional loops or toroidal con figurations [12]. In the case of the original Skyrme
+model, one assigns nontrivial configurations or ’solitons’ of homotopy Π 3(S3) =Zto Baryons whereas our
+precise derivation [6] for the solely strong interaction le ads to nonzero Hopf invariants Π 3(S2) =Zfrom the
+non-vanishing axial anomaly of QCD; in consequence, the tra nsformed path integral in [6] with coset matrices
+is more closely related to the Skyrme-Faddeev model [13]. It is therefore also of interest for the total standard
+model to what extent nontrivial topological configurations (as the restrictive, vanishing sum of Hopf invariants)
+can determine prevailing field combinations as baryons or me sons with the leptonic sector of electrons and
+neutrions.
+2 Anomalous doubling of Fermi fields within the standard mode l
+2.1 Symmetry breaking source actions and anomalous pair con densates
+The applied path integral for the standard model is given by c ontour time integrals (2.1) for forward ηp=+= +1
+and backward ηp=−=−1 propagation which is considered by the contour time metric (2.2), the contour time
+coordinates xµ
+p=±(2.3) and contour time derivatives ˆ∂p=±,µwithin the actions
+/integraldisplay
+Cd4xp...=/integraldisplay
+L3d3/vector x/parenleftbigg/integraldisplay+∞
+−∞dx0
++...+/integraldisplay−∞
++∞dx0
+−.../parenrightbigg
+=/integraldisplay
+L3d3/vector x/parenleftbigg/integraldisplay+∞
+−∞dx0
++...−/integraldisplay+∞
+−∞dx0
+−.../parenrightbigg
+=/integraldisplay
+L3d3/vector x/parenleftbigg/summationdisplay
+p=±/integraldisplay+∞
+−∞dx0
+pηp.../parenrightbigg
+; (2.1)
+ηp=/braceleftbig
++1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+p=+;−1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+p=−/bracerightbig
+;ηq=/braceleftbig
++1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+q=+;−1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+q=−/bracerightbig
+; ”p”,”q” =±; (2.2)4 2 ANOMALOUS DOUBLING OF FERMI FIELDS WITHIN THE STANDARD MOD EL
+xµ
+p=/parenleftbig
+x0
+p,/vector x/parenrightbig
+;xµ
++=/parenleftbig
+x0
++,/vector x/parenrightbig
+;xµ
+−=/parenleftbig
+x0
+−, /vector x/parenrightbig
+;
+ˆ∂p,µ=/parenleftbigg
+∂
+∂x0p,∂
+∂/vector x/parenrightbigg
+;ˆ∂+,µ=/parenleftbigg
+∂
+∂x0
++,∂
+∂/vector x/parenrightbigg
+;ˆ∂−,µ=/parenleftbigg
+∂
+∂x0
+−,∂
+∂/vector x/parenrightbigg
+.(2.3)
+The appropriate description of the standard model starts ou t from massless Majorana fields of fermions which
+obtain their corresponding masses from the spontaneous sym metry breaking with the Higgs phenomenon.
+We combine the strongly interacting quark fields qm(xp) and lepton fields /tildewidelm(xp) of the three basic families
+m= 1,2,3 into one single fermionic spinor field ψm(xp) with the distinguishing labels ψ= ”/tildewidel”,”q” where the
+tilde ’/tildewide’ over/tildewidelm(xp) indicates the missing field degree of freedom for the right- handed neutrinos νm,R(xp)≡0.
+The derivation of the nonlinear sigma model SO(90 ,90)/U(90)⊗U(90) within this paper follows according to
+the formulation of Ref. [14] (C.P. Burgess and G.D. Moore, ”T he Standard Model : A Primer”); however, we
+emphasize with our derivation one additional, important po int which concerns the anomalous doubling (or also
+’Nambu’ doubling [4, 5]) within the standard model given in t erms of left-handed and right-handed Majorana
+fields. Therefore, we double the total, two spin-component fi eldψm(xp) (2.4) of leptons /tildewidelm(xp) and quarks
+qm(xp) by their complex conjugates /tildewidel∗
+m(xp),q∗
+m(xp) or in total by ψ∗
+m(xp) which is taken into account by the
+first uppercase letters A,B,C= 1,2 (2.8) of the Latin alphabet. Hence, one has anomalous doubl ed Fermi
+fields/tildewideLA
+m(xp) (2.6),QA
+m(xp) (2.7) or in total ΨA
+m(xp) (2.5) with the corresponding capital letters′/tildewideL′,′Q′or′Ψ′
+ψm(xp) =/parenleftBig
+/tildewidelm(xp);qm(xp)/parenrightBigT
+;ψ= ”/tildewidel”,”q” ; (2.4)
+ΨA(=1,2)
+m(xp) =/parenleftBigψ=”/tildewidel”/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
+/tildewidelm(xp)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+A=1,/tildewidel∗
+m(xp)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+A=2;ψ=”q”/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
+qm(xp)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+A=1, q∗
+m(xp)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+A=2/parenrightBigT
+; (2.5)
+/tildewideLA(=1,2)
+m(xp) =/parenleftBig
+/tildewidelm(xp)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+A=1,/tildewidel∗
+m(xp)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+A=2/parenrightBigT
+; (2.6)
+QA(=1,2)
+m(xp) =/parenleftBig
+qm(xp)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+A=1, q∗
+m(xp)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+A=2/parenrightBigT
+; (2.7)
+A,B,C, ... = 1,2. (2.8)
+According to the formulation of Ref. [14], we point out the ad ditional fact of the standard model that it is
+entirely specified in terms of ’Nambu’ or anomalous doubled F ermi fields ΨA
+m(xp),/tildewideLA
+m(xp),QA
+m(xp) [4, 5]. In
+consequence this ’Nambu’ doubling naturally leads to a cose t decomposition for anomalous doubled pairs of
+Fermi fields with a removal of the self-energy densities as ba ckground fields.
+We defineinrelation (2.9) thetotal gauge groupSU c(3)⊗SUL(2)⊗UY(1) (correspondingto Ref. [14]) with
+the first Greek letters α,β,γ= 1,...,8 for the eight gluon fields Gα
+µ(xp), with the first lowercase Latin letters
+a,b,c= 1,2,3 for the SU L(2) gauge boson fields Wa
+µ(xp) and with the Greek letters κ,λ,µ,ν,ρ of spacetime
+indices for the weak hypercharge gauge boson field Bµ(xp). Furthermore, the indices r,s= 1,2,3 andf,g= 1,2
+denote the matrix elements ( ˆλα)rsof (gluon) Gell-Mann matrices and the Pauli-iso-spin matri ces (ˆτa)fgfor the
+electroweak doublets. However, we depart from the formulat ion of Ref. [14] concerning the 4 ×4 Dirac gamma
+matrices (ˆγµ)IJ, (I,J,K= 1,...,4) and separate these into ’Nambu’ doubled parts A,B= 1,2 consisting of
+2×2 Pauli spin matrices ( ˆ/vector σ)iAjB(iA,jB=↑,↓) for the massless Majorana Fermi fields. Consequently, the
+2×2 Pauli spin matrices within the 4 ×4 Dirac gamma matrices transform the upper ’ ↑’ and lower ’ ↓’ spin
+components of the Majorana fields. The particular list of defi nitions is described in (2.10) to (2.16) where we
+especially hint to the split of the 4 ×4 Dirac gamma matrices (ˆ γµ)IJ, (I,J,K= 1,...,4) into the 2 ×2 Pauli2.1 Symmetry breaking source actions and anomalous pair con densates 5
+spin matrices for the anomalous doubling of Fermi fields with in the off-diagonal blocks ’ A/ne}ationslash=B’ of (ˆγµ)AB
+iAjB
+SUc(3) ×SUL(2) × UY(1)
+↓ ↓ ↓
+8Gα
+µ(xp) 3 Wa
+µ(xp) Bµ(xp)
+α, β, γ= 1,...,8a, b, c= 1,2,3κ,λ,µ,ν,ρ = 0,1,2,3
+↓ ↓ ↓
+(ˆλα)rs (ˆτa)fg (ˆγµ)IJ= (ˆγµ)AB
+iAjB
+r, s= 1,2,3 f,g= 1,2I, J, K= 1,2,3,4 ;
+A, B, C = 1,2
+iA, jB, kC=↑,↓.(2.9)
+ˆλα: (gluon) Gell-Mann matrices ; (2.10)
+ˆτa: Pauli (iso)-spin matrices of the weak interaction ; (2.11)
+ˆηµν:= diag/parenleftbig
+−1,+1,+1,+1/parenrightbig
+;ε0123= +1; (mostly ’+’ convention, cf. [14]); (2.12)
+/parenleftbig
+ˆγµ/parenrightbig
+IJ: ˆγ0=/parenleftbigg0−ˆı
+−ˆı0/parenrightbigg
+IJ;ˆβ=ıˆγ0=/parenleftbigg0ˆ1
+ˆ1 0/parenrightbigg
+IJ;ˆ/vector γ=/parenleftBigg
+0−ıˆ/vector σ
+ıˆ/vector σ0/parenrightBigg
+IJ;(I,J=1,...,4);(2.13)
+ˆ/vector σ:=/parenleftbig
+ˆσ1,ˆσ2,ˆσ3/parenrightbig
+= Pauli spin-matrices ;
+/parenleftbig
+ˆγµ/parenrightbigAB
+iAjB: ˆγ0=/parenleftbigg0−(ˆı)iAjB
+−(ˆı)iAjB0/parenrightbiggAB
+;ˆβ=ı/parenleftbig
+ˆγ0/parenrightbigAB
+iAjB;ˆ/vector γ=/parenleftBigg
+0−ı(ˆ/vector σ)iAjB
+ı(ˆ/vector σ)iAjB0/parenrightBiggAB
+;(2.14)
+(ˆ/vector σ)iAjB:=/parenleftbig
+(ˆσ1)iAjB,(ˆσ2)iAjB,(ˆσ3)iAjB/parenrightbig
+= Pauli spin-matrices ; (iA,jB=↑,↓);
+ψ(xp) =ψ†(xp)ˆβ; (2.15)
+ˆγ5:=−ıˆγ0ˆγ1ˆγ2ˆγ3=/parenleftbiggˆ1 0
+0−ˆ1/parenrightbigg
+; (2.16)
+ˆPL=/parenleftbigˆ1+ ˆγ5/parenrightbig
+2=/parenleftbiggˆ1 0
+0 0/parenrightbigg
+;ˆPR=/parenleftbigˆ1−ˆγ5/parenrightbig
+2=/parenleftbigg0 0
+0ˆ1/parenrightbigg
+.
+Inthefollowing the detailed labeling withthevarious indi ces is definedfor thelepton andquarksectors of Fermi
+fields; this also allows to conclude for the dimension N0= 90 of the coset decomposition SO( N0,N0)/U(N0)⊗
+U(N0) for the total self-energy SO( N0,N0) with coset matrices ˆT(Ψ)(xp) SO(N0,N0)/U(N0) for anomalous
+pairs of fields and the unitary subgroup U( N0) for self-energy densities as the vacuum or background stat es.
+The left-handed′H=L′, two-component spin′iA=↑,↓′lepton fields (2.17-2.20) consist of the three families
+m,n= 1,2,3 with the left-handed neutrino /tildewidelL= ”νl” and left-handed electron /tildewidelL= ”eL”; the latter distinction
+is also redundantly contained within the Pauli-iso-spin in dicesf,g= 1,2 for the neutrino f,g= 1 and electron
+f,g= 2 where both notations will conveniently be applied in para llel. We therefore attain the dimension ( m=
+1,2,3)×(f= 1,2)×(H=L)×(iA=↑,↓) = 3·2·1·2 = 12withintheleft-handedleptonsector. Theright-hande d
+sector of leptons (2.19,2.20) lacks the right-handed neutr ino field degree of freedom /tildewidelf=1,H=R= ”νH=R”≡0
+with the remaining right-handed electron part (2.20). As we count the contributing dimension of the right-
+handed lepton sector, we are left with a total of ( m= 1,2,3)×(f= 2)×(H=R)×(iA=↑,↓) = 3·1·1·2 = 6
+components
+/tildewidelm,f=1,H=L,iA=↑,↓(xp) =νm,H=L,iA=↑,↓(xp) ; (or/tildewidelL= ”νL”) ; (2.17)
+/tildewidelm,f=2,H=L,iA=↑,↓(xp) =em,H=L,iA=↑,↓(xp) ; (or/tildewidelL= ”eL”) ; (2.18)6 2 ANOMALOUS DOUBLING OF FERMI FIELDS WITHIN THE STANDARD MOD EL
+/tildewidelm,f=1,H=R,iA=↑,↓(xp) =νm,H=R,iA=↑,↓(xp)≡0 ; (or/tildewidelf=1,H=R= ”νH=R”≡0) ; (2.19)
+/tildewidelm,f=2,H=R,iA=↑,↓(xp) =em,H=R,iA=↑,↓(xp) ; (or/tildewidelR= ”eR”). (2.20)
+The electroweak doublet structure of quarks q= ”u”,q= ”d” (2.21,2.22) has equal numbers of right- and
+left-handed parts within the three families m= 1,2,3 where the notation of the 2 ×2 iso-spin matrices with
+f,g= 1,2 equivalently refers to the u(p)- andd(own)-quark components. In comparison to the lepton sector,
+one has also to include the indices r,s= 1,2,3 of the Gell-Mann matrices for the gluons so that one acquire s
+a total of (m= 1,2,3)×(f= 1,2)×(r= 1,2,3)×(H=L,R)×(iA=↑,↓) = 3·2·3·2·2 = 72 components
+within the quark sector
+qm,f=1,r=1,2,3,H=L,R,iA=↑,↓(xp) =um,r=1,2,3,H=L,R,iA=↑,↓(xp) ; (orq= ”u”) ; (2.21)
+qm,f=2,r=1,2,3,H=L,R,iA=↑,↓(xp) =dm,r=1,2,3,H=L,R,iA=↑,↓(xp) ; (orq= ”d”). (2.22)
+As we add the 18 components of the lepton sector to the 72 compo nents of the quark sector to N0= 90 and as
+we consider the ’Nambu’ metric tensor ˆSAB(2.23) of the anomalous doubling, one finally achieves the to tal self-
+energy matrix SO( N0,N0) for the standard model to be decomposed into the coset part S O(N0,N0)/U(N0) of
+anomalous pairs and the unitary sub-group part U( N0) of self-energy densities. Note that the ’Nambu’ metric
+tensorˆSAB(2.23) has off-diagonal entries compared to previous investi gations [15, 16, 6]. This is caused by the
+change oftheanomalous doubleddensity offieldsfrom ψ∗(xp)·ψ(xp) =1
+2(Ψ†,A(xp)ˆSA=BΨB(xp)) withˆSA=B=
+diag{ˆ1;−ˆ1}of Refs. [15, 16, 6] to the considered case with ψ∗(xp)·ψ(xp) =1
+2(ΨT,A(xp)ˆSA/negationslash=BΨB(xp)) which
+contains the off-diagonal metric ˆSA/negationslash=B(2.23) and transposition ΨT,A(xp) instead of the hermitian conjugation
+Ψ†,A(xp) as in previous investigations [15, 16, 6]
+ˆSAB=/parenleftbiggˆ0−ˆ1
+ˆ1ˆ0/parenrightbiggAB
+. (2.23)
+Apart from the Lagrangian for the dynamics of Fermi and gauge boson fields, we introduce a symmetry break-
+ing source action AS[ˆJ,Jψ,ˆJψψ] (2.24) of the fermionic part with three source fields ˆJAB
+Mψ;Nψ(yq,xp),JA
+ψ;Mψ(xp),
+ˆJAB
+ψψ;Mψ;Nψ(xp) where the bilinear parts have even, complex, commuting sou rcesˆJAB
+Mψ;Nψ(yq,xp),ˆJAB
+ψψ;Mψ;Nψ(xp)
+for anomalous doubled Fermi fields and where the linear symme try breaking part is caused by anti-commuting,
+doubled source fields JA
+ψ;Mψ(xp) for coherent macroscopic wavefunctions3. The source matrix ˆJAB
+Mψ;Nψ(yq,xp) is
+usedfor generating bilinear observables of Fermi fields by d ifferentiation whereas thematrix field ˆJAB
+ψψ;Mψ;Nψ(xp)
+with anti-symmetric sub-matrices ˆjψψ;Mψ;Nψ(xp),ˆj†
+ψψ;Mψ;Nψ(xp) acts as a condensate seed for anomalous paired
+fermionic fields after setting the matrix source field to equi valent values ˆJAB
+ψψ;Mψ;Nψ(x+) =ˆJAB
+ψψ;Mψ;Nψ(x−) on
+the two branches p=±of the time contour. Similar considerations hold for the ant i-commuting source field
+JA
+ψ;Mψ(xp) which can be chosen to generate an odd number of Fermi fields f or observables and for condensate
+seeds of macroscopic wavefunctions from equivalent values JA
+ψ;Mψ(x+) =JA
+ψ;Mψ(x−) on the two time contour
+branches. In relations (2.24-2.33), we can therefore list t he source action AS[ˆJ,Jψ,ˆJψψ] (2.24) with the par-
+ticular property of missing right-handed neutrinos which i s separately specified for the source fields JA
+ψ;Mψ(xp)
+andˆJAB
+ψψ;Mψ;Nψ(xp) in eqs. (2.26-2.33) and is also denoted by the tilde ’ /tildewide’ over the lepton sector /tildewidelM/tildewidel(xp)
+AS[ˆJ,Jψ,ˆJψψ] =1
+2/integraldisplay
+Cd4xp/parenleftbigg
+JT,A
+ψ;Mψ(xp)ˆSABΨB
+Mψ(xp)+ΨT,A
+Mψ(xp)ˆSABJB
+ψ;Mψ(xp)/parenrightbigg
++ (2.24)
+3We summarize the various indices for lepton (2.17-2.20) and quark fields (2.21,2.22) by collective indices ’ Mψ’, ’Nψ’ for brevity,
+cf. e.g. relations (2.34-2.36).2.1 Symmetry breaking source actions and anomalous pair con densates 7
++1
+2/integraldisplay
+Cd4xpΨT,A
+Mψ(xp)/parenleftbiggˆj†
+ψψ;Mψ;Nψ(xp) 0
+0 ˆjψψ;Mψ;Nψ(xp)/parenrightbiggAB
+/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright
+ˆJAB
+ψψ;Mψ;Nψ(xp)ΨB
+Nψ(xp)+
++1
+2/integraldisplay
+Cd4xpd4yqΨT,A
+Mψ(yq)ˆJAB
+Mψ;Nψ(yq,xp) ΨB
+Nψ(xp) ;
+ΨA
+Mψ(xp) =/parenleftbigg
+/tildewidelM/tildewidel(xp),/tildewidel∗
+M/tildewidel(xp);qMq(xp), q∗
+Mq(xp)/parenrightbiggT
+; (2.25)
+JA
+ψ;Mψ(xp) =/parenleftbigg
+j/tildewidel;M/tildewidel(xp), j∗
+/tildewidel;M/tildewidel(xp);jq;Mq(xp), j∗
+q;Mq(xp)/parenrightbiggT
+; (2.26)
+0≡/tildewidelm,f=1,H=R,iA(xp) =νm,H=R,iA(xp) ; (2.27)
+0≡j/tildewidel;m,f=1,H=R,iA=jν;m,H=R,iA(xp) ; (2.28)
+j/tildewidel/tildewidel;m,f,H 1,iA;n,g,H2,iB(xp) =−jT
+/tildewidel/tildewidel;m,f,H 1,iA;n,g,H2,iB(xp) ; (2.29)
+jqq;m,f,r,H 1,iA;n,g,s,H 2,iB(xp) =−jT
+qq;m,f,r,H 1,iA;n,g,s,H 2,iB(xp) ; (2.30)
+j/tildewidelq;m,f,H 1,iA;n,g,s,H 2,iB(xp) =−jT
+/tildewidelq;m,f,H 1,iA;n,g,s,H 2,iB(xp) ; (2.31)
+j/tildewidel/tildewidel;m,f=1,H1=R,iA;n,g,H2,iB(xp) =jν/tildewidel;m,H1=R,iA;n,g,H2,iB(xp)≡0 ; (2.32)
+j/tildewidelq;m,f=1,H1=R,iA;n,g,s,H 2,iB(xp) =jνq;m,H1=R,iA;n,g,s,H 2,iB(xp)≡0. (2.33)
+Since one has to assign many indices for the lepton and quark s ectors with its various subspaces, we have
+defined collective indices Mψ,NψorM/tildewidel,N/tildewidelandMq,Nq(2.34-2.36) with the extension that a bar over an
+additionally listed index, as e.g. in Mq(r,iA) (2.36), denotes the omittance of these over-barred indice s in the
+prevailing total collection of these
+Mψ:=/braceleftbiggMψ=/tildewidel:={/tildewidel,m,f,H,i A}without right-handed neutrinos
+Mψ=q:={q,m,f,r,H,i A}(2.34)
+Nψ:=/braceleftbiggNψ=/tildewidel′:={/tildewidel′,n,g,H′,jB}without right-handed neutrinos
+Nψ=q′:={q′,n,g,s,H′,jB}(2.35)
+Mq(r,iA) :={q,m,f, r/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+canceled,H, iA/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+canceled}={q,m,f,H, }; etc. further examples .(2.36)
+Similartothesourceterm AS[ˆJ,Jψ,ˆJψψ](2.24)ofFermifields, wealsoincludeasourceaction Asg[ˆj(ˆG)
+α,ˆj(ˆW)
+a,ˆj(ˆB)]
+(2.37) for thegauge fieldstrength tensors withanti-symmet ric, even- andreal-valued source matrices ˆj(ˆB)µν(xp),
+ˆj(ˆW)µν
+a(xp),ˆj(ˆG)µν
+α(xp) (2.39); however, we omit an anomalous kind of doubling as in the case of the Fermi fields
+whereas weallow for therather general extension of thespac etime metrictensor ˆ ηµλˆηνρ(2.38) by corresponding
+θ-terms for possible, nontrivial θvacua (see the description for possible, relevant changes b y theseθ-terms in
+Refs. [17])
+Asg[ˆj(ˆG)
+α,ˆj(ˆW)
+a,ˆj(ˆB)] =/integraldisplay
+Cd4xp/parenleftbigg
+ˆj(ˆG)µν
+α(xp) ˆη(g3,θ3)
+µν,λρˆGλρ
+α(xp)+ (2.37)
++ˆj(ˆW)µν
+a(xp) ˆη(g2,θ2)
+µν,λρˆWλρ
+a(xp)+ˆj(ˆB)µν(xp) ˆη(g1,θ1)
+µν,λρˆBλρ(xp)/parenrightbigg
+;8 2 ANOMALOUS DOUBLING OF FERMI FIELDS WITHIN THE STANDARD MOD EL
+ˆη(g1,θ1)
+µν,λρ= ˆηµλˆηνρ+g2
+1θ1
+16π2ˆεµνλρ; ˆη(g2,θ2)
+µν,λρ= ˆηµλˆηνρ+g2
+2θ2
+16π2ˆεµνλρ; (2.38)
+ˆη(g3,θ3)
+µν,λρ= ˆηµλˆηνρ+g2
+3θ3
+16π2ˆεµνλρ;
+ˆj(ˆB)µν(xp) =−ˆj(ˆB)νµ(xp) ; ˆj(ˆW)µν
+a(xp) =−ˆj(ˆW)νµ
+a(xp) ; (2.39)
+ˆj(ˆG)µν
+α(xp) =−ˆj(ˆG)νµ
+α(xp).
+Finally, we can state the general structure of the generatin g functionZ[ˆJ,Jψ,ˆJψψ;ˆj(ˆG),ˆj(ˆW),ˆj(ˆB)] (2.43) for the
+standard model with the source actions AS[ˆJ,Jψ,ˆJψψ] (2.24), Asg[ˆj(ˆG)
+α,ˆj(ˆW)
+a,ˆj(ˆB)] (2.37), with the integration
+over the anti-commuting fields of leptons and quarks (2.6,2. 7) and the integration over the gauge fields Gµ
+α(xp),
+Wµ
+a(xp),Bµ(xp) whose chosen axial gauge conditions (2.40-2.42) are speci fied by the auxiliary, real fields
+s(G)
+α(xp),s(W)
+a(xp),s(B)(xp) within the standard Fourier integral representation of co rresponding delta functions
+containing constant vectors n(G)µ,n(W)µandn(B)µ
+δ/parenleftbig
+n(B)µBµ(xp)/parenrightbig
+=/integraldisplay
+d[s(B)(xp)] exp/braceleftBig
+ı/integraldisplay
+Cd4xps(B)(xp)n(B)µBµ(xp)/bracerightBig
+; (2.40)
+3/productdisplay
+a=1δ/parenleftbig
+n(W)µWµ
+a(xp)/parenrightbig
+=/integraldisplay
+d[s(W)
+a(xp)] exp/braceleftBig
+ı/integraldisplay
+Cd4xps(W)
+a(xp)nµ
+(W)Wa
+µ(xp)/bracerightBig
+; (2.41)
+8/productdisplay
+α=1δ/parenleftbig
+n(G)µGµ
+α(xp)/parenrightbig
+=/integraldisplay
+d[s(G)
+α(xp)] exp/braceleftBig
+ı/integraldisplay
+Cd4xps(G)
+α(xp)nµ
+(G)Gα
+µ(xp)/bracerightBig
+; (2.42)
+Z[ˆJ,Jψ,ˆJψψ;ˆj(ˆG),ˆj(ˆW),ˆj(ˆB)] =/integraldisplay
+d[/tildewidelM/tildewidel(xp),/tildewidel∗
+M/tildewidel(xp)]d[qMq(xp),q∗
+Mq(xp)]× (2.43)
+×/integraldisplay
+d[φ†(xp),φ(xp)] exp/braceleftbigg
+ı/integraldisplay
+Cd4xp/parenleftBig
+Lff(xp)+LH(xp)/parenrightBig
+−ıAS[ˆJ,Jψ,ˆJψψ]/bracerightbigg
+×/integraldisplay
+d[Gµ
+α(xp),s(G)
+α(xp)]/integraldisplay
+d[Wµ
+a(xp),s(W)
+a(xp)]/integraldisplay
+d[Bµ(xp),s(B)(xp)]
+×exp/braceleftbigg
+ı/integraldisplay
+Cd4xp/parenleftBig
+Lgj(xp)+Lgg(xp)/parenrightBig
+−ıAsg[ˆj(ˆG)
+α,ˆj(ˆW)
+a,ˆj(ˆB)]/bracerightbigg
+.
+2.2 The Lagrangian of the standard model in terms of currents and auxiliary Higgs fields
+It remains to describe the detailed form of the kinetic, ferm ionic part Lff(xp) and Higgs part LH(xp) within
+the total Lagrangian Ltot(xp) where the gauge boson - current coupling Lgj(xp) determines the interaction of
+fermions and where one also has the additional, non-Abelian self-interaction in Lgg(xp) among the gauge fields
+Ltot(xp) =Lff(xp)+LH(xp)+Lgj(xp)+Lgg(xp). (2.44)
+Although we follow the detailed description and specificati on of Lagrangian parts according to [14], we again
+outline the various parts of the standard model Lagrangian Ltot(xp), in order to emphasize the natural ap-
+pearance of an anomalous doubling of the Fermi fields, due to t he proper formulation with massless Majorana
+Fermi fields and symmetry breaking terms [4, 5]. Therefore, o ne is guided by the formulation, corresponding
+to Ref. [14], to the coset decomposition into ’Nambu’ double d pairs within the self-energy and remaining sub-
+group parts of the self-energy densities as background or va cuum state. We define from [14] left-, right-handed,
+vectorial and axial gamma matrices (ˆ γµ
+H=L,R)AB
+iAjB(2.45,2.46), (ˆ γµ
+5,H=L,R)AB
+iAjB(2.47,2.48) which separate into a2.2 The Lagrangian of the standard model in terms of currents and auxiliary Higgs fields 9
+symmetric (anti-symmetric) block structure of Pauli spin m atrices in the off-diagonal blocks A/ne}ationslash=Bof vectorial
+(ˆγµ
+H=L,R)AB
+iAjB(axial (ˆγµ
+5,H=L,R)AB
+iAjB) gamma matrices, respectively. Thevectorial gammamatric es (ˆγµ
+H=L,R)AB
+iAjB
+enter into the kinetic part Lff(xp) (2.49) of fermions with an additional ’ −ıˆSABεp’ term (2.50) for further
+analytic properties and convergence of non-equilibrium Gr een functions
+/parenleftbig
+ˆγµ
+H=L/parenrightbigAB
+iAjB=/parenleftBigg
+0/parenleftbig
+−ˆı, ı/vector σ/parenrightbigT
+iAjB /parenleftbig
+−ˆı, ı/vector σ/parenrightbig
+iAjB0/parenrightBiggAB
+; (2.45)
+/parenleftbig
+ˆγµ
+H=R/parenrightbigAB
+iAjB=/parenleftBigg
+0/parenleftbig
+−ˆı,−ı/vector σ/parenrightbigT
+iAjB /parenleftbig
+−ˆı,−ı/vector σ/parenrightbig
+iAjB0/parenrightBiggAB
+; (2.46)
+/parenleftbig
+ˆγµ
+5,H=L/parenrightbigAB
+iAjB=/parenleftBigg
+0 −/parenleftbig
+−ˆı, ı/vector σ/parenrightbigT
+iAjB /parenleftbig
+−ˆı, ı/vector σ/parenrightbig
+iAjB0/parenrightBiggAB
+; (2.47)
+/parenleftbig
+ˆγµ
+5,H=R/parenrightbigAB
+iAjB=/parenleftBigg
+0/parenleftbig
+−ˆı,−ı/vector σ/parenrightbigT
+iAjB
+−/parenleftbig
+−ˆı,−ı/vector σ/parenrightbig
+iAjB0/parenrightBiggAB
+; (2.48)
+Lff(xp) =−1
+2/parenleftBigg/tildewidelM/tildewidel(xp)
+/tildewidel∗
+M/tildewidel(xp)/parenrightBiggT,A/parenleftBig/parenleftbig
+ˆγµ
+H/parenrightbigAB
+iAjBˆ∂p,µ−ıˆSABεpδiAjB/parenrightBig
+δM/tildewidel(iA);N/tildewidel(jB)/parenleftBigg/tildewidelN/tildewidel(xp)
+/tildewidel∗
+N/tildewidel(xp)/parenrightBiggB
++(2.49)
+−1
+2/parenleftbiggqMq(xp)
+q∗
+Mq(xp)/parenrightbiggT,A/parenleftBig/parenleftbig
+ˆγµ
+H/parenrightbigAB
+iAjBˆ∂p,µ−ıˆSABεpδiAjB/parenrightBig
+δMq(iA);Nq(jB)/parenleftbiggqNq(xp)
+q∗
+Mq(xp)/parenrightbiggB
+=−1
+2ΨT,A
+Mψ(xp)/parenleftBig/parenleftbig
+ˆγµ
+H/parenrightbigAB
+iAjBˆ∂p,µ−ıˆSABεpδiAjB/parenrightBig
+δMψ(iA);Nψ(jB)ΨB
+Nψ(xp) ;
+εp=ηpε+;ε+>0 ;ε+→0+. (2.50)
+The Higgs part LH(xp) (2.51), whose non-zero vacuum value of φa(xp) = (φ1(xp), φ2(xp))Tcauses mass terms
+of fermions and couplings among quarks according to the Koba yashi-Maskawa matrix [1], is also given in
+correspondence to [14]; however, we reformulate this part b y introducing an anti-symmetric matrix ˆFAB
+ψψ;Mψ;Nψ
+(2.52) for the ’Nambu’ doubled Fermi fields ΨT,A
+Mψ(xp)ˆFAB
+ψψ;Mψ;Nψ(xp) ΨB
+Nψ(xp), similar to the kinetic part which
+is also in total anti-symmetric, due to the anti-symmetric d erivative operator ˆ∂p,µand symmetric, vectorial
+gamma matrices (2.45,2.46)
+LH(xp) =−/parenleftbigˆ∂p,µφ(xp)/parenrightbig†/parenleftbigˆ∂µ
+pφ(xp)/parenrightbig
++µ2/parenleftbig
+φ†(xp)φ(xp)/parenrightbig
+−λ/parenleftBig
+φ†(xp)φ(xp)/parenrightBig2
+−λ/parenleftBigµ2
+2λ/parenrightBig2
++ (2.51)
+−φ1(xp)/parenleftBig
+ν†
+m,L(xp)ˆfmnen,R(xp)+u†
+m,L(xp)ˆhmndn,R(xp)−u†
+m,R(xp) ˆgmndn,L(xp)/parenrightBig
++
+−φ∗
+1(xp)/parenleftBig
+e†
+m,R(xp)ˆfmnνn,L(xp)+d†
+m,R(xp)ˆhmnun,L(xp)−d†
+m,L(xp) ˆgmnun,R(xp)/parenrightBig
++
+−φ2(xp)/parenleftBig
+e†
+m,L(xp)ˆfmnen,R(xp)+d†
+m,L(xp)ˆhmndn,R(xp)+u†
+m,R(xp) ˆgmnun,L(xp)/parenrightBig
++
+−φ∗
+2(xp)/parenleftBig
+e†
+m,R(xp)ˆfmnen,L(xp)+d†
+m,R(xp)ˆhmndn,L(xp)+u†
+m,L(xp) ˆgmnun,R(xp)/parenrightBig
+;
+LH(xp) =−/parenleftbigˆ∂p,µφ(xp)/parenrightbig†·/parenleftbigˆ∂µ
+pφ(xp)/parenrightbig
++µ2/parenleftbig
+φ†(xp)·φ(xp)/parenrightbig
+−λ/parenleftBig
+φ†(xp)·φ(xp)/parenrightBig2
+−λ/parenleftBigµ2
+2λ/parenrightBig2
++ (2.52)
+−/parenleftBigg/tildewidelM/tildewidel(xp)
+/tildewidel∗
+M/tildewidel(xp)/parenrightBiggT,A
+ˆFAB
+M/tildewidel;N/tildewidel/parenleftBig
+φ1(xp),φ2(xp);ˆfmn/parenrightBig/parenleftBigg/tildewidelN/tildewidel(xp)
+/tildewidel∗
+N/tildewidel(xp)/parenrightBiggB
++10 2 ANOMALOUS DOUBLING OF FERMI FIELDS WITHIN THE STANDARD MOD EL
+−/parenleftbiggqMq(xp)
+q∗
+Mq(xp)/parenrightbiggT,A/parenleftbigg
+ˆHAB
+Mq;Nq/parenleftBig
+φ1(xp),φ2(xp);ˆhmn/parenrightBig
++ˆGAB
+Mq;Nq/parenleftBig
+φ1(xp),φ2(xp);ˆgmn/parenrightBig/parenrightbigg/parenleftbiggqNq(xp)
+q∗
+Nq(xp)/parenrightbiggB
+=−/parenleftbigˆ∂p,µφ(xp)/parenrightbig†·/parenleftbigˆ∂µ
+pφ(xp)/parenrightbig
++µ2/parenleftbig
+φ†(xp)·φ(xp)/parenrightbig
+−λ/parenleftBig
+φ†(xp)·φ(xp)/parenrightBig2
+−λ/parenleftBigµ2
+2λ/parenrightBig2
++
+−1
+2ΨT,A
+Mψ(xp)ˆFAB
+ψψ;Mψ;Nψ/parenleftBig
+φ1(xp),φ2(xp)/parenrightBig
+ΨB
+Nψ(xp).
+In analogy, the Lagrangian Lgj(xp) (2.53) with anomalous doubled Fermi fields within the vario us currents
+(apart fromthepureHiggs currents j(φ)µ
+B(xp) andj(φ)µ
+W;a(xp)) follows theprinciple”(transposed’Nambu’ doubled
+FermifieldΨT,A
+Mψ(xp))×(anti-symmetricmatrixoroperator)AB
+Mψ;Nψ×(anomalousdoubledFermifieldΨB
+Nψ(xp))”
+as inLff(xp) (2.49) or LH(xp) (2.52)
+Lgj(xp) =−g1
+2Bµ(xp)/parenleftBig
+j(ax.)µ
+B(xp)+j(φ)µ
+B(xp)/parenrightBig
+−g2
+2Wa
+µ(xp)/parenleftBig
+j(vec.)µ
+W;a(xp)+j(ax.)µ
+W;a(xp)+j(φ)µ
+W;a(xp)/parenrightBig
++
+−g3
+2Gα
+µ(xp)/parenleftBig
+j(vec.)µ
+G;α(xp)+j(ax.)µ
+G;α(xp)/parenrightBig
+. (2.53)
+We therefore include anti-symmetric and symmetric Pauli-i sospin-matrices ˆ τ(−)
+a, ˆτ(+)
+a(2.54) and the eight
+Gell-Mann (gluon) matrices ˆλ(−)
+α,ˆλ(−)
+α(2.55)
+/parenleftbig
+ˆτ(−)
+a/parenrightbig
+fg=/parenleftBigˆτa−ˆτT
+a
+2/parenrightBig
+fg;/parenleftbig
+ˆτ(+)
+a/parenrightbig
+fg=/parenleftBigˆτa+ ˆτT
+a
+2/parenrightBig
+fg; (2.54)
+/parenleftbigˆλ(−)
+α/parenrightbig
+rs=/parenleftBigˆλα−ˆλT
+α
+2/parenrightBig
+rs;/parenleftbigˆλ(+)
+α/parenrightbig
+rs=/parenleftBigˆλα+ˆλT
+α
+2/parenrightBig
+rs, (2.55)
+which combine with the vectorial and axial gamma matrices (ˆ γµ
+H) (2.45,2.46), (ˆ γµ
+5,H) (2.47,2.48) to completely
+anti-symmetric matrix couplings among the bilinear anomal ous doubled Fermi fields. Relations (2.56-2.65)
+encompass all the various currents of the standard model (ac cording to Ref. [14]) which, however, are entirely
+reformulated in terms of bilinear, anomalous doubled Fermi fields coupled to anti-symmetric matrices. Further-
+more, one has to incorporate the weak hyper-charges e(Y)
+/tildewidel;H,e(Y)
+q;H(2.58) and left- and right-handed charge values
+e(G)
+q;H(2.65) of strongly interacting quarks, in order to attain th e proper couplings for the standard model. Note
+that one has only left-handed current components within j(vec.)µ
+W;a(xp) (2.60),j(ax.)µ
+W;a(xp) (2.61) according to the
+SUL(2) gauge group of weak interactions
+j(vec.)µ
+B(xp)≡0 ; (2.56)
+j(ax.)µ
+B(xp) =−1
+2/parenleftBigg/tildewidelM/tildewidel(xp)
+/tildewidel∗
+M/tildewidel(xp)/parenrightBiggT,A
+ı/parenleftbig
+ˆγµ
+5,H/parenrightbigAB
+iAjBe(Y)
+˜l,HδM/tildewidel(iA);N/tildewidel(jB)/parenleftBigg/tildewidelN/tildewidel(xp)
+/tildewidel∗
+N/tildewidel(xp)/parenrightBiggB
++ (2.57)
+−1
+2/parenleftbiggqMq(xp)
+q∗
+Mq(xp)/parenrightbiggT,A
+ı/parenleftbig
+ˆγµ
+5,H/parenrightbigAB
+iAjBe(Y)
+q,HδMq(iA);Nq(jB)/parenleftbiggqNq(xp)
+q∗
+Nq(xp)/parenrightbigg
+=−1
+2ΨT,A
+Mψ(xp)ı/parenleftbig
+ˆγµ
+5,H/parenrightbigAB
+iAjBe(Y)
+ψ,HδMψ(iA);Nψ(jB)ΨB
+Nψ(xp) ;
+/parenleftbig
+e(Y)
+˜l,H=L=−1 ;e(Y)
+˜l,H=R= +2/parenrightbig
+;/parenleftbig
+e(Y)
+q,H=L= +1/3 ;e(Y)
+q=u,H=R=−4/3 ;e(Y)
+q=d,H=R= +2/3/parenrightbig
+; (2.58)
+j(φ)µ
+B(xp) =ı/bracketleftBig
+φ†(xp)·/parenleftbigˆ∂µ
+pφ(xp)/parenrightbig
+−/parenleftbigˆ∂µ
+pφ(xp)/parenrightbig†·φ(xp)/bracketrightBig
+; (2.59)2.2 The Lagrangian of the standard model in terms of currents and auxiliary Higgs fields 11
+j(vec.)µ
+W;a(xp) =−1
+2/parenleftBigg/tildewidelM/tildewidel(xp)
+/tildewidel∗
+M/tildewidel(xp)/parenrightBiggT,A
+ı/parenleftbig
+ˆτ(−)
+a/parenrightbig
+fg/parenleftbig
+ˆγµ
+L/parenrightbigAB
+iAjBδM/tildewidel(f,iA);N/tildewidel(g,jB)/parenleftBigg/tildewidelN/tildewidel(xp)
+/tildewidel∗
+N/tildewidel(xp)/parenrightBiggB
++ (2.60)
+−1
+2/parenleftbiggqMq(xp)
+q∗
+Mq(xp)/parenrightbiggT,A
+ı/parenleftbig
+ˆτ(−)
+a/parenrightbig
+fg/parenleftbig
+ˆγµ
+L/parenrightbigAB
+iAjBδMq(f,iA);Nq(g,jB)/parenleftbiggqNq(xp)
+q∗
+Nq(xp)/parenrightbiggB
+=
+=−1
+2ΨT,A
+Mψ(xp)ı/parenleftbig
+ˆτ(−)
+a/parenrightbig
+fg/parenleftbig
+ˆγµ
+L/parenrightbigAB
+iAjBδMψ(f,iA);Nψ(g,jB)ΨB
+Nψ(xp) ;
+j(ax.)µ
+W;a(xp) =−1
+2/parenleftBigg/tildewidelM/tildewidel(xp)
+/tildewidel∗
+M/tildewidel(xp)/parenrightBiggT,A
+ı/parenleftbig
+ˆτ(+)
+a/parenrightbig
+fg/parenleftbig
+ˆγµ
+5,L/parenrightbigAB
+iAjBδM/tildewidel(f,iA);N/tildewidel(g,jB)/parenleftBigg/tildewidelN/tildewidel(xp)
+/tildewidel∗
+N/tildewidel(xp)/parenrightBiggB
++ (2.61)
+−1
+2/parenleftbiggqMq(xp)
+q∗
+Mq(xp)/parenrightbiggT,A
+ı/parenleftbig
+ˆτ(+)
+a/parenrightbig
+fg/parenleftbig
+ˆγµ
+5,L/parenrightbigAB
+iAjBδMq(f,iA);Nq(g,jB)/parenleftbiggqNq(xp)
+q∗
+Nq(xp)/parenrightbiggB
+=
+=−1
+2ΨT,A
+Mψ(xp)ı/parenleftbig
+ˆτ(+)
+a/parenrightbig
+fg/parenleftbig
+ˆγµ
+5,L/parenrightbigAB
+iAjBδMψ(f,iA);Nψ(g,jB)ΨB
+Nψ(xp) ;
+j(φ)µ
+W;a(xp) =ı/bracketleftBig
+φ†(xp)ˆτa/parenleftbigˆ∂µ
+pφ(xp)/parenrightbig
+−/parenleftbigˆ∂µ
+pφ(xp)/parenrightbig†ˆτaφ(xp)/bracketrightBig
+; (2.62)
+j(vec.)µ
+G;α(xp) =−1
+2/parenleftbiggqMq(xp)
+q∗
+Mq(xp)/parenrightbiggT,A
+ı/parenleftbigˆλ(−)
+α/parenrightbig
+rs/parenleftbig
+ˆγµ
+H/parenrightbigAB
+iAjBδMq(r,iA);Nq(s,jB)/parenleftbiggqNq(xp)
+q∗
+Nq(xp)/parenrightbiggB
+(2.63)
+=−1
+2QT,A
+Mq(xp)ı/parenleftbigˆλ(−)
+α/parenrightbig
+rs/parenleftbig
+ˆγµ
+H/parenrightbigAB
+iAjBδMq(r,iA);Nq(s,jB)QB
+Nq(xp) ;
+j(ax.)µ
+G;α(xp) =−1
+2/parenleftbiggqMq(xp)
+q∗
+Mq(xp)/parenrightbiggT,A
+ı/parenleftbigˆλ(+)
+α/parenrightbig
+rs/parenleftbig
+ˆγµ
+5,H/parenrightbigAB
+iAjBδMq(r,iA);Nq(s,jB)/parenleftbiggqNq(xp)
+q∗
+Nq(xp)/parenrightbiggB
+(2.64)
+=−1
+2QT,A
+Mq(xp)ı/parenleftbigˆλ(+)
+α/parenrightbig
+rs/parenleftbig
+ˆγµ
+5,H/parenrightbigAB
+iAjBδMq(r,iA);Nq(s,jB)QB
+Nq(xp) ;
+/parenleftbig
+e(G)
+q;H=L= +1, e(G)
+q;H=R=−1/parenrightbig
+. (2.65)
+The Lagrangian Lgg(xp) with quadratic terms of the gauge field strength tensors ˆGα
+µν(xp) (2.67), ˆWa
+µν(xp)
+(2.68),ˆBµν(xp) (2.69) is given in relation (2.66) with the generalized met ric tensors ˆ η(g1,θ1)
+µν,λρ, ˆη(g2,θ2)
+µν,λρ, ˆη(g3,θ3)
+µν,λρ
+(2.38) ofθ-terms, the gauge couplings to the Higgs field densities ( φ†(xp)·φ(xp)), (φ†(xp)ˆτaφ(xp)) and the
+auxiliary, real fields s(G)
+α(xp),s(W)
+a(xp),s(B)(xp) and constant vectors nµ
+(G),nµ
+(W),nµ
+(B)for axial gauge fixing,
+respectively. Furthermore, we hint to the various coupling parameters g3,g2,g1of the strong, weak and hyper
+UY(1) interactions which have already occurred in the couplin gLgj(xp) (2.53) of the gauge boson fields Gα
+µ(xp),
+Wa
+µ(xp),Bµ(xp) to their corresponding currents
+Lgg(xp) =−1
+4ˆGαµν(xp) ˆη(g1,θ1)
+µν,λρˆGαλρ(xp)+ (2.66)
+−1
+4ˆWaµν(xp) ˆη(g2,θ2)
+µν,λρˆWaλρ(xp)−g2
+2
+4Wa
+µ(xp)Wµ
+a(xp)/parenleftbig
+φ†(xp)φ(xp)/parenrightbig
++
+−1
+4ˆBµν(xp) ˆη(g3,θ3)
+µν,λρˆBλρ(xp)−g2
+1
+4Bµ(xp)Bµ(xp)/parenleftbig
+φ†(xp)φ(xp)/parenrightbig
++
+−g1g2
+2Bµ(xp)Wa
+µ(xp)/parenleftbig
+φ†(xp) ˆτaφ(xp)/parenrightbig
++
++Gα
+µ(xp)s(G)
+α(xp)nµ
+(G)+Wa
+µ(xp)s(W)
+a(xp)nµ
+(W)+Bµ(xp)s(B)(xp)nµ
+(B);
+ˆGα
+µν(xp) =ˆ∂p,µGα
+ν(xp)−ˆ∂p,νGα
+µ(xp)+g3fα
+βγGβ
+µ(xp)Gγ
+ν(xp) ; (2.67)123 SELF-ENERGIES OF GAUGE FIELD STRENGTH TENSORS AND QUARTIC FERMI FIELDS
+ˆWa
+µν(xp) =ˆ∂p,µWa
+ν(xp)−ˆ∂p,νWa
+µ(xp)+g2εa
+bcWb
+µ(xp)Wc
+ν(xp) ; (2.68)
+ˆBµν(xp) =ˆ∂p,µBν(xp)−ˆ∂p,νBµ(xp). (2.69)
+3 Self-energies of gauge field strength tensors and quartic F ermi fields
+3.1 Gaussian transformations with self-energy matrices of the gauge fields
+One might infer that proper HSTs with field strength tensors ( 2.67-2.69) of gauge fields cannot simplify the
+gauge field interactions, due to the self-interaction of thr ee and four vertex contributions of the non-Abelian
+gauge fields Gα
+µ(xp),Wa
+µ(xp) [18]; however, we have already demonstrated in Ref. [6] for the strong SU c(3)
+interaction of gluon fields that one has never to use more than the reproducing property of Gaussian Fourier
+transformations (the ’HST’) in order to disentangle the glu on interactions. One starts out from the Gaus-
+sian identities (3.1-3.3) for each gauge field strength tens orˆGµν
+α(xp) (2.67), ˆWµν
+a(xp) (2.68), ˆBµν(xp) (2.69)
+with analogous source fields ˆj(ˆG)µν
+α(xp),ˆj(ˆW)µν
+a(xp),ˆj(ˆB)µν(xp) and introduces the complementary self-energies
+ˆS(ˆG)
+αµν(xp),ˆS(ˆW)
+aµν(xp),ˆS(ˆB)
+µν(xp), as Gaussian integration variables with anti-symmetric s pacetime indices. In
+consequence the quadratic parts of the field strength tensor s with linear coupling to source fields (first two lines
+in (3.4)) transform to the action (3.5) with the quadratic se lf-energies and remaining linear couplings between
+gauge field strength tensors and corresponding self-energi es (last line in (3.4))
+1≡/integraldisplay
+d[ˆS(ˆG)
+αµν(xp)] exp/braceleftbiggı
+4/integraldisplay
+Cd4xp/parenleftBig
+ˆS(ˆG)µν
+α(xp)−ˆGµν
+α(xp)−2ˆj(ˆG)µν
+α(xp)/parenrightBig
+× (3.1)
+׈η(g3,θ3)
+µν,λρ/parenleftBig
+ˆS(ˆG)λρ
+α(xp)−ˆGλρ
+α(xp)−2ˆj(ˆG)λρ
+α(xp)/parenrightBig/bracerightbigg
+;
+1≡/integraldisplay
+d[ˆS(ˆW)
+aµν(xp)] exp/braceleftbiggı
+4/integraldisplay
+Cd4xp/parenleftBig
+ˆS(ˆW)µν
+a(xp)−ˆWµν
+a(xp)−2ˆj(ˆW)µν
+a(xp)/parenrightBig
+× (3.2)
+׈η(g2,θ2)
+µν,λρ/parenleftBig
+ˆS(ˆW)λρ
+a(xp)−ˆWλρ
+a(xp)−2ˆj(ˆW)λρ
+a(xp)/parenrightBig/bracerightbigg
+;
+1≡/integraldisplay
+d[ˆS(ˆB)
+µν(xp)] exp/braceleftbiggı
+4/integraldisplay
+Cd4xp/parenleftBig
+ˆS(ˆB)µν(xp)−ˆBµν(xp)−2ˆj(ˆB)µν(xp)/parenrightBig
+× (3.3)
+׈η(g1,θ1)
+µν,λρ/parenleftBig
+ˆS(ˆB)λρ(xp)−ˆBλρ(xp)−2ˆj(ˆB)λρ(xp)/parenrightBig/bracerightbigg
+;
+exp/braceleftbigg
+−ı/integraldisplay
+Cd4xp/parenleftbigg
+ˆGµν
+α(xp)ˆη(g3,θ3)
+µν,λρ
+4ˆGλρ
+α(xp)+ˆj(ˆG)µν
+α(xp) ˆη(g3,θ3)
+µν,λρˆGλρ
+α(xp)+ˆWµν
+a(xp)ˆη(g2,θ2)
+µν,λρ
+4ˆWλρ
+a(xp)+ (3.4)
++ˆj(ˆW)µν
+a(xp) ˆη(g2,θ2)
+µν,λρˆWλρ
+a(xp)+ˆBµν(xp)ˆη(g1,θ1)
+µν,λρ
+4ˆBλρ(xp)+ˆj(ˆB)µν(xp) ˆη(g1,θ1)
+µν,λρˆBλρ(xp)/parenrightbigg/bracerightbigg
+=
+=/integraldisplay
+d[ˆS(ˆG)
+αµν(xp)]/integraldisplay
+d[ˆS(ˆW)
+aµν(xp)]/integraldisplay
+d[ˆS(ˆB)
+µν(xp)]×exp/braceleftbigg
+ıA/parenleftBig
+ˆS(ˆG)
+α,ˆj(ˆG)
+α;ˆS(ˆW)
+a,ˆj(ˆW)
+a;ˆS(ˆB),ˆj(ˆB)/parenrightBig
++
+−ı
+2/integraldisplay
+Cd4xp/parenleftbigg
+ˆGµν
+α(xp) ˆη(g3,θ3)
+µν,λρˆS(ˆG)λρ
+α(xp)+ˆWµν
+a(xp) ˆη(g2,θ2)
+µν,λρˆS(ˆW)λρ
+a(xp)+ˆBµν(xp) ˆη(g1,θ1)
+µν,λρˆS(ˆB)λρ(xp)/parenrightbigg/bracerightbigg
+;
+A/parenleftBig
+ˆS(ˆG)
+α,ˆj(ˆG)
+α;ˆS(ˆW)
+a,ˆj(ˆW)
+a;ˆS(ˆB),ˆj(ˆB)/parenrightBig
+=/integraldisplay
+Cd4xp/parenleftbigg
+ˆS(ˆG)µν
+α(xp)ˆη(g3,θ3)
+µν,λρ
+4ˆS(ˆG)λρ
+α(xp)+ (3.5)3.1 Gaussian transformations with self-energy matrices of the gauge fields 13
++ˆS(ˆW)µν
+a(xp)ˆη(g2,θ2)
+µν,λρ
+4ˆS(ˆW)λρ
+a(xp)+ˆS(ˆB)µν(xp)ˆη(g1,θ1)
+µν,λρ
+4ˆS(ˆB)λρ(xp)+
+−ˆj(ˆG)µν
+α(xp) ˆη(g3,θ3)
+µν,λρˆS(ˆG)λρ
+α(xp)−ˆj(ˆW)µν
+a(xp) ˆη(g2,θ2)
+µν,λρˆS(ˆW)λρ
+a(xp)−ˆj(ˆB)µν(xp) ˆη(g1,θ1)
+µν,λρˆS(ˆB)λρ(xp)+
++ˆj(ˆG)µν
+α(xp) ˆη(g3,θ3)
+µν,λρˆj(ˆG)λρ
+α(xp)+ˆj(ˆW)µν
+a(xp) ˆη(g2,θ2)
+µν,λρˆj(ˆW)λρ
+a(xp)+ˆj(ˆB)µν(xp) ˆη(g1,θ1)
+µν,λρˆj(ˆB)λρ(xp)/parenrightbigg
+.
+It is the linear coupling ˆGµν
+α(xp) ˆη(g3,θ3)
+µν,λρˆS(ˆG)λρ
+α(xp),ˆWµν
+a(xp) ˆη(g2,θ2)
+µν,λρˆS(ˆW)λρ
+a(xp),ˆBµν(xp) ˆη(g1,θ1)
+µν,λρˆS(ˆB)λρ(xp)
+(last line in (3.4)) with linear and quadratic gauge fields Gµ
+α(xp),Gν
+α(xp) in the field strength tensors ˆGµν
+α(xp)
+(2.67) and the linear coupling between gauge fields and curre nts that allows to eliminate all gauge fields
+Gµ
+α(xp),Wµ
+a(xp),Bµ(xp) by Gaussian integration at the expense of current-current interactions (quartic in the
+anomalous doubled Fermi fields) and the occurrence of the sel f-energies ˆS(ˆG)
+αµν(xp),ˆS(ˆW)
+aµν(xp),ˆS(ˆB)
+µν(xp). We
+exemplify latter description to Gaussian integrations for the SUc(3) case with the gluon gauge fields Gβµ(xp),
+Gγν(xp) and add an anti-hermitian term −ıˆe(ˆG)
+p(3.7) for convergent properties
+exp/braceleftbigg
+−ı
+2/integraldisplay
+Cd4xpˆGµν
+α(xp) ˆη(g3,θ3)
+µν,λρˆS(ˆG)λρ
+α(xp)/bracerightbigg
+= (3.6)
+= exp/braceleftbigg
+−ı
+2/integraldisplay
+Cd4xp/parenleftBig
+ˆ∂µ
+pGαν(xp)−ˆ∂ν
+pGαµ(xp)+g3fα
+βγGβµ(xp)Gγν(xp)/parenrightBig
+ˆη(g3,θ3)
+µν,λρˆS(ˆG)λρ
+α(xp)/bracerightbigg
+=
+= exp/braceleftbigg
+−ı
+2/integraldisplay
+Cd4xpGβµ(xp)/bracketleftBig
+−ıˆe(ˆG)
+p+g3fα
+βγˆη(g3,θ3)
+µν,λρˆS(ˆG)λρ
+α(xp)/bracketrightBig
+Gγν(xp)/bracerightbigg
+×
+×exp/braceleftbigg
+−ı
+2/integraldisplay
+Cd4xp/parenleftBig
+−Gαν(xp) ˆη(g3,θ3)
+µν,λρ/parenleftbigˆ∂µ
+pˆS(ˆG)λρ
+α(xp)/parenrightbig
++Gαµ(xp) ˆη(g3,θ3)
+µν,λρ/parenleftbigˆ∂ν
+pˆS(ˆG)λρ
+α(xp)/parenrightbig/parenrightBig/bracerightbigg
+=
+= exp/braceleftbigg
+−ı
+2/integraldisplay
+Cd4xpGβµ(xp)/bracketleftBig
+−ıˆe(ˆG)
+p+g3fα
+βγˆη(g3,θ3)
+µν,λρˆS(ˆG)λρ
+α(xp)/bracketrightBig
+Gγν(xp)/bracerightbigg
+×
+×exp/braceleftbigg
+−ı/integraldisplay
+Cd4xpGαµ(xp) ˆη(g3,θ3)
+µν,λρ/parenleftbigˆ∂ν
+pˆS(ˆG)λρ
+α(xp)/parenrightbig/bracerightbigg
+;
+/parenleftBig
+ˆe(ˆG)
+p/parenrightBig
+βµ,γν=ηpe(ˆG)
++δβγδµν;e(ˆG)
++>0. (3.7)
+Similar transformations for the SU L(2) gauge fields Wbµ(xp),Wcν(xp) and for U Y(1) withBµ(xp),Bν(xp) with
+their gauge field strength tensors result into the HST relati ons (3.8) with the self-energies ˆS(ˆG)
+αµν(xp),ˆS(ˆW)
+aµν(xp),
+ˆS(ˆB)
+µν(xp), being anti-symmetric in spacetime indices, and into the a uxiliary, real, axial gauge determining
+fieldss(G)
+α(xp),s(W)
+a(xp),s(B)(xp) with constant vectors n(G)µ,n(W)µ,n(B)µ. Therefore, one achieves follow-
+ing relation (3.8) for the gauge-gauge and gauge-fermion cu rrent Lagrangians Lgg(xp) (2.66), Lgj(xp) (2.53)
+with the source action Asg[ˆj(ˆG)
+α,ˆj(ˆW)
+a,ˆj(ˆB)] after integrating Gaussian terms of gauge fields Gβµ(xp)... Gγν(xp),
+Wbµ(xp)... Wcν(xp),Bµ(xp)... Bν(xp) with linear couplings to the fermion currents and to the der ivatives
+of the gauge field strength self-energies
+/integraldisplay
+d[Gµ
+α(xp),s(G)
+α(xp)]/integraldisplay
+d[Wµ
+a(xp),s(W)
+a(xp)]/integraldisplay
+d[Bµ(xp),s(B)(xp)]× (3.8)
+×exp/braceleftbigg
+ı/integraldisplay
+Cd4xp/parenleftBig
+Lgg(xp)+Lgj(xp)/parenrightBig
+−ıAsg[ˆj(ˆG)
+α,ˆj(ˆW)
+a,ˆj(ˆB)]/bracerightbigg
+=143 SELF-ENERGIES OF GAUGE FIELD STRENGTH TENSORS AND QUARTIC FERMI FIELDS
+=/integraldisplay
+d[ˆS(ˆG)
+αµν(xp),s(G)
+α(xp)]/integraldisplay
+d[ˆS(ˆW)
+aµν(xp),s(W)
+a(xp)]/integraldisplay
+d[ˆS(ˆB)
+µν(xp),s(B)(xp)]×
+×exp/braceleftbigg
+ıA/parenleftBig
+ˆS(ˆG)
+α,ˆj(ˆG)
+α;ˆS(ˆW)
+a,ˆj(ˆW)
+a;ˆS(ˆB),ˆj(ˆB)/parenrightBig/bracerightbigg
+×det/bracketleftBig
+ˆM(ˆG)
+βµ,γν/parenleftbigˆS(ˆG)λρ
+α(xp)/parenrightbig/bracketrightBig−1/2
+×
+×det/bracketleftBig
+ˆM(ˆW)
+bµ,cν/parenleftbigˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)/parenrightbig/bracketrightBig−1/2
+×det/bracketleftBig
+ˆM(ˆB)
+µ,ν/parenleftbigˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)/parenrightbig/bracketrightBig−1/2
+×
+×exp/braceleftbiggı
+2/integraldisplay
+Cd4xp/parenleftbigg
+J(ˆG)βµ(xp)ˆM(ˆG)−1
+βµ,γν/parenleftbigˆS(ˆG)λρ
+α(xp)/parenrightbig
+J(ˆG)γν(xp)+
++J(ˆW)bµ(xp)ˆM(ˆW)−1
+bµ,cν/parenleftbigˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)/parenrightbig
+J(ˆW)cν(xp)+
++J(ˆB)µ(xp)ˆM(ˆB)−1
+µ,ν/parenleftbigˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)/parenrightbig
+J(ˆB)ν(xp)/parenrightbigg/bracerightbigg
+.
+Apart from the already given action A(ˆS(ˆG)
+α,ˆj(ˆG)
+α;ˆS(ˆW)
+a,ˆj(ˆW)
+a;ˆS(ˆB),ˆj(ˆB)) in (3.5), it remains to outline the
+various abbreviations of terms as the matrices ˆM(ˆG)
+βµ,γν(3.9),ˆM(ˆW)
+bµ,cν(3.11),ˆM(ˆB)
+µ,ν(3.14), containing their
+corresponding gauge field strength self-energy ˆS(ˆG)λρ
+α(xp),ˆS(ˆW)λρ
+a(xp) and Higgs field combinations, in inverse
+squarerootsofdeterminants det[ ...]−1/2andasinverted matrices ˆM(ˆG)−1
+βµ,γν,ˆM(ˆW)−1
+bµ,cν,ˆM(ˆB)−1
+µ,νwithingeneralized
+current-current interactions J(ˆG)βµ(xp)...J(ˆG)γν(xp),J(ˆW)bµ(xp)...J(ˆW)cν(xp),J(ˆB)µ(xp)...J(ˆB)ν(xp). We
+therefore briefly list the definitions of the various abbrevi ations in eqs. (3.9-3.18) where one has also to include
+convergence generating epsilon terms (3.16-3.18) from the Gaussian integrations of the HST transformations
+ˆM(ˆG)
+βµ,γν/parenleftbigˆS(ˆG)λρ
+α(xp)/parenrightbig
+=/bracketleftBig
+−ıˆe(ˆG)
+p+g3fα
+βγˆη(g3,θ3)
+µν,λρˆS(ˆG)λρ
+α(xp)/bracketrightBig
+; (3.9)
+J(ˆG)
+αµ(xp) = ˆη(g3,θ3)
+µν,λρ/parenleftbigˆ∂ν
+pˆS(ˆG)λρ
+α(xp)/parenrightbig
+−s(G)
+α(xp)n(G)µ+g3
+2/parenleftBig
+j(vec.)
+G;αµ(xp)+j(ax.)
+G;αµ(xp)/parenrightBig
+; (3.10)
+ˆM(ˆW)
+bµ,cν/parenleftbigˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)/parenrightbig
+=/bracketleftBig
+−ıˆe(ˆW)
+p+g2εa
+bcˆη(g2,θ2)
+µν,λρˆS(ˆW)λρ
+a(xp)+ (3.11)
++g2
+2
+2ηµνδbc/parenleftbig
+φ†(xp)φ(xp)/parenrightbig/bracketrightBig
+;
+J(ˆW)
+aµ(xp, Bµ(xp)) =J(ˆW)
+aµ(xp)+g1g2
+2Bµ(xp)/parenleftbig
+φ†(xp)ˆτaφ(xp)/parenrightbig
+; (3.12)
+J(ˆW)
+aµ(xp) =/bracketleftbigg
+ˆη(g2,θ2)
+µν,λρ/parenleftbigˆ∂ν
+pˆS(ˆW)λρ
+a(xp)/parenrightbig
+−s(W)
+a(xp)n(W)µ+ (3.13)
++g2
+2/parenleftBig
+j(vec.)
+W;aµ(xp)+j(ax.)
+W;aµ(xp)+j(φ)
+W;aµ(xp)/parenrightBig/bracketrightbigg
+;
+ˆM(ˆB)
+µν/parenleftbigˆS(ˆW)λρ
+a(xp),φ†(xp),φ(xp)/parenrightbig
+=−ıe(ˆB)
+pδµν+g2
+1
+2ˆηµν/parenleftbig
+φ†(xp)φ(xp)/parenrightbig
++ (3.14)
+−g2
+1g2
+2
+4/parenleftbig
+φ†(xp)ˆτbφ(xp)/parenrightbigˆM(ˆW)−1
+bµ,cν/parenleftbigˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)/parenrightbig /parenleftbig
+φ†(xp)ˆτcφ(xp)/parenrightbig
+;
+J(ˆB)
+µ(xp) = ˆη(g1,θ1)
+µν,λρ/parenleftbigˆ∂ν
+pˆS(ˆB)λρ(xp)/parenrightbig
+−s(B)(xp)n(B);µ+g1
+2/parenleftBig
+j(ax.)
+B;µ(xp)+j(φ)
+B;µ(xp)/parenrightBig
++ (3.15)
+−g1g2
+2/parenleftbig
+φ†(xp)ˆτbφ(xp)/parenrightbigˆM(ˆW)−1
+bµ,cν/parenleftbigˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)/parenrightbig
+J(ˆW)cν(xp) ;15
+/parenleftBig
+ˆe(ˆG)
+p/parenrightBig
+βµ,γν=ηpe(ˆG)
++δβγδµν;e(ˆG)
++>0 ; (3.16)
+/parenleftBig
+ˆe(ˆW)
+p/parenrightBig
+bµ,cν=ηpe(ˆW)
++δbcδµν;e(ˆW)
++>0 ; (3.17)
+/parenleftBig
+ˆe(ˆB)
+p/parenrightBig
+µ,ν=ηpe(ˆB)
++δµν;e(ˆB)
++>0. (3.18)
+Since the various transformations of this section appear to be involved, we give further details in appendices of
+subsequent articles for the various steps which lead to (3.8 ) with the particular defined parts of composed cur-
+rentsJ(ˆG)
+αµ(xp)(3.10),J(ˆW)
+aµ(xp, Bµ(xp))(3.12), J(ˆW)
+aµ(xp)(3.13),J(ˆB)
+µ(xp)(3.15)andwiththecorrespondingprop-
+agators ˆM(ˆG)
+βµ,γν(ˆS(ˆG)λρ
+α(xp)) (3.9), ˆM(ˆW)
+bµ,cν(ˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)) (3.11), ˆM(ˆB)
+µν(ˆS(ˆW)λρ
+a(xp),φ†(xp),φ(xp))
+(3.14) for the strong, electroweak and hypercharge cases, r espectively.
+4 HST and coset decomposition of anomalous doubled Fermi fiel ds
+4.1 Transformation of current-current terms or quartic, bi linear Fermi fields to self-
+energies
+As we combine eq. (3.8) for the HST to the self-energies of the gauge field strength tensors with the fermion
+partsLff(xp) (2.49), LH(xp) (2.52), one acquires the path integral (4.1) with anti-com muting integration
+variablesd[/tildewideLA
+M/tildewidel(xp)],d[QA
+Mq(xp)]
+Z[ˆJ,Jψ,ˆJψψ;ˆj(ˆG),ˆj(ˆW),ˆj(ˆB)] =/integraldisplay
+d[ˆS(ˆG)
+αµν(xp),s(G)
+α(xp)]/integraldisplay
+d[ˆS(ˆW)
+aµν(xp),s(W)
+a(xp)]/integraldisplay
+d[ˆS(ˆB)
+µν(xp),s(B)(xp)]×(4.1)
+×/integraldisplay
+d[φ†(xp),φ(xp)] exp/braceleftbigg
+ıA/parenleftBig
+ˆS(ˆG)
+α,ˆj(ˆG)
+α;ˆS(ˆW)
+a,ˆj(ˆW)
+a;ˆS(ˆB),ˆj(ˆB)/parenrightBig/bracerightbigg
+×det/bracketleftBig
+ˆM(ˆG)
+βµ,γν/parenleftbigˆS(ˆG)λρ
+α(xp)/parenrightbig/bracketrightBig−1/2
+×
+×det/bracketleftBig
+ˆM(ˆW)
+bµ,cν/parenleftbigˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)/parenrightbig/bracketrightBig−1/2
+×det/bracketleftBig
+ˆM(ˆB)
+µ,ν/parenleftbigˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)/parenrightbig/bracketrightBig−1/2
+×
+×/integraldisplay
+d[/tildewideLA
+M/tildewidel(xp)]d[QA
+Mq(xp)] exp/braceleftBig
+ı/integraldisplay
+Cd4xp/parenleftBig
+Lff(xp)+LH(xp)/parenrightBig
+−ıAS[ˆJ,Jψ,ˆJψψ]/bracerightBig
+×
+×exp/braceleftbiggı
+2/integraldisplay
+Cd4xp/parenleftbigg
+J(ˆG)βµ(xp)ˆM(ˆG)−1
+βµ,γν/parenleftbigˆS(ˆG)λρ
+α(xp)/parenrightbig
+J(ˆG)γν(xp)+
++J(ˆW)bµ(xp)ˆM(ˆW)−1
+bµ,cν/parenleftbigˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)/parenrightbig
+J(ˆW)cν(xp)+
++J(ˆB)µ(xp)ˆM(ˆB)−1
+µ,ν/parenleftbigˆS(ˆW)λρ
+a(xp), φ†(xp), φ(xp)/parenrightbig
+J(ˆB)ν(xp)/parenrightbigg/bracerightbigg
+.
+In the following we reorder the various terms according to th eir number of Fermi fields occurring in the various
+actions within (4.1). At first we note the zero order terms of f ermionic field degrees of freedom in the first
+three lines of (4.1) which entirely consist of gauge field str ength self-energies, Higgs field combinations and the
+auxiliary, real gauge fixing integration variables. Howeve r, there are even more zero order terms of Fermi fields
+hidden in the generalized currents J(ˆG)βµ(xp) (3.10), J(ˆW)bµ(xp) (3.13) and J(ˆB)µ(xp) (3.15) which we separate
+in relations (4.2-4.8) by introducing a tilde ’ /tildewide’ over the parts of currents without bilinear fermionic field content
+J(ˆG)
+αµ(xp) =g3
+2/parenleftBig
+j(vec.)
+G;αµ(xp)+j(ax.)
+G;αµ(xp)/parenrightBig
++/tildewideJ(ˆG)
+αµ/parenleftbigˆS(ˆG)λρ
+α(xp),s(G)
+α(xp)/parenrightbig
+; (4.2)16 4 HST AND COSET DECOMPOSITION OF ANOMALOUS DOUBLED FERMI FIE LDS
+/tildewideJ(ˆG)
+αµ/parenleftbigˆS(ˆG)λρ
+α(xp),s(G)
+α(xp)/parenrightbig
+= ˆη(g3,θ3)
+µν,λρ/parenleftbigˆ∂ν
+pˆS(ˆG)λρ
+α(xp)/parenrightbig
+−s(G)
+α(xp)n(G);µ; (4.3)
+J(ˆW)
+aµ(xp) =g2
+2/parenleftBig
+j(vec.)
+W;aµ(xp)+j(ax.)
+W;aµ(xp)/parenrightBig
++/tildewideJ(ˆW)
+aµ/parenleftbigˆS(ˆW)λρ
+a(xp),s(W)
+a(xp),φ†(xp),φ(xp)/parenrightbig
+;(4.4)
+/tildewideJ(ˆW)
+aµ(xp) =/tildewideJ(ˆW)
+aµ/parenleftbigˆS(ˆW)λρ
+a(xp),s(W)
+a(xp),φ†(xp),φ(xp)/parenrightbig
+(4.5)
+=g2
+2j(φ)
+W;aµ(xp)+ ˆη(g2,θ2)
+µν,λρ/parenleftbigˆ∂ν
+pˆS(ˆW)λρ
+a(xp)/parenrightbig
+−s(W)
+a(xp)n(W);µ;
+J(ˆB)
+µ(xp) =g1
+2j(ax.)
+B;µ(xp)+g1
+2ˆN(ˆW)
+µ,cν/parenleftbigˆS(ˆW)λρ
+a(xp),φ†(xp),φ(xp)/parenrightbig
+J(ˆW)cν(xp)+ (4.6)
++/tildewideJ(ˆB)
+µ/parenleftbigˆS(ˆB)λρ(xp),s(B)(xp),φ†(xp),φ(xp)/parenrightbig
+;
+/tildewideJ(ˆB)
+µ/parenleftbig
+xp/parenrightbig
+=/tildewideJ(ˆB)
+µ/parenleftbigˆS(ˆB)λρ(xp),s(B)(xp),φ†(xp),φ(xp)/parenrightbig
+(4.7)
+=g1
+2j(φ)
+B;µ(xp)+ ˆη(g1,θ1)
+µν,λρ/parenleftbigˆ∂ν
+pˆS(ˆB)λρ(xp)/parenrightbig
+−s(B)(xp)n(B);µ;
+ˆN(ˆW)
+µ,cν/parenleftbig
+xp/parenrightbig
+=ˆN(ˆW)
+µ,cν/parenleftbigˆS(ˆW)λρ
+a(xp),φ†(xp),φ(xp)/parenrightbig
+(4.8)
+=−g2/parenleftbig
+φ†(xp) ˆτbφ(xp)/parenrightbigˆM(ˆW)−1
+bµ,cν/parenleftbigˆS(ˆW)λρ
+a(xp),φ†(xp),φ(xp)/parenrightbig
+.
+These zero order terms of Fermi fields, as the first three lines of (4.1) and the current-current interaction with
+only ’tilded’ currents (4.3,4.5,4.7) are only kept as backg round averaging actions for the anomalous doubled
+parts of self-energies of Fermi fields. Aside from these zero order terms of Fermi fields, there also appears
+a linear term of Fermi fields which is coupled to the symmetry b reaking, ’Nambu’ doubled, anti-commuting
+source field JA
+ψ;Mψ(xp) (2.26). Moreover, one has several bilinear parts of ’Nambu ’ doubled Fermi fields which
+comprise the Lagrangians Lff(xp) (2.49), LH(xp) (2.52), the couplings to the generating matrix ˆJAB
+Mψ;Nψ(xp,yq)
+for observables and the condensate ’seed’ matrix ˆJAB
+ψψ;Mψ;Nψ(xp) and which also result from the generalized
+current-current interactions (last three lines of (4.1)). The latter bilinear Fermi fields follow from the product
+of ’tilded’ currents (4.3,4.5,4.7) without any Fermi fields with the already defined bilinear, fermionic currents
+j(ax.)µ
+B(xp) (2.57),j(vec.)µ
+W;a(xp) (2.60),j(ax.)µ
+W;a(xp) (2.61),j(vec.)µ
+G;α(xp) (2.63),j(ax.)µ
+G;α(xp) (2.64). These bilinear
+parts of Fermi fields could be directly removed by integratio n according to the relation of anti-commuting
+variablesξiand an anti-symmetric matrix ˆMij(the symmetric part of ˆM= +ˆMTcancels in (4.9)!)
+/integraldisplay
+d[ξ] exp/braceleftBig
+−ξiˆMijξj/bracerightBig
+=/parenleftBig
+det/bracketleftbigˆMij/bracketrightbig/parenrightBig1/2
+;ˆM=−ˆMT; (i,j=1,...,even-numbered dimension ),(4.9)
+if the path integral (4.1) did not contain the current-curre nt interactions of purely bilinear, fermionic currents
+or quartic Fermi fields. In order to eliminate these quartic, fermionic field combinations by integration, one
+therefore has to perform additional HST transformations of quartic Fermi fields, similar to that in Ref. [6] for
+the strong interaction, but in the present case with three di fferent gauge field interactions and corresponding
+self-energies. We skip the details of these transformation s to purely bilinear Fermi fields and refer to [6] as an
+example with the derivation in the QCD case (details will be l isted in appendices of further articles in order
+to outline the various steps of the HST to anomalous doubled s elf-energies of Fermi fields with ’hinge’ fields as
+density terms). The coset decomposition to the self-energy matrixˆT(Ψ)(xp) (4.10-4.12) (SO( N0,N0)/U(N0);
+N0= 90), replacing the anomalous doubled Fermi field pairs, is o btained in analogy to the purely strong QCD
+interaction case of Ref. [6], after removal of several gauge -field-’dressed’ coset matrices
+ˆTAB
+(Ψ)Mψ;Nψ(xp) =/parenleftBig
+exp/braceleftBig
+−ˆYA′B′
+(Ψ)M′
+ψ;N′
+ψ(xp)/bracerightBig/parenrightBigAB
+Mψ;Nψ; (4.10)4.1 Transformation of current-current terms or quartic, bi linear Fermi fields to self-energies 17
+ˆYA′/negationslash=B′
+(Ψ)M′
+ψ;N′
+ψ(xp) =/parenleftbigg0 ˆX(Ψ)M′
+ψ;N′
+ψ(xp)
+ˆX†
+(Ψ)M′
+ψ;N′
+ψ(xp) 0/parenrightbiggA′B′
+;(4.11)
+ˆX(Ψ)M′
+ψ;N′
+ψ(xp) =−/parenleftBig
+ˆX(Ψ)M′
+ψ;N′
+ψ(xp)/parenrightBigT
+∈C. (4.12)
+One eventually succeeds into following path integral (4.13 ) of anomalous doubled parts from self-energy ma-
+trices for Fermi fields within a coset decomposition under in clusion of the invariant integration measure
+d[ˆT−1
+(Ψ)(xp)dˆT(Ψ)(xp)] of SO(N0,N0)/U(N0), (N0= 90); furthermore, the path integral consists of several
+background fields as the gauge field strength self-energies, Higgs fields and unitary subgroup parts with self-
+energy ”densities” which only substitute the ”density” ter ms composed of fermions. This background averaging
+is marked by the brackets /an}bracketle{td[ˆT−1
+(Ψ)(xp)dˆT(Ψ)(xp)]....../an}bracketri}ht
+Z/bracketleftbigˆT(Ψ)(xp);ˆJ;Jψ;ˆJψψ/bracketrightbig
+=/angbracketleftbigg/integraldisplay
+d[ˆT−1
+(Ψ)(xp)dˆT(Ψ)(xp)]/tildewide∆/parenleftBig
+ˆT−1;AB
+(Ψ);Mψ;Nψ(xp),ˆTAB
+(Ψ);Mψ;Nψ(xp);ˆJAB
+ψψ;Mψ;Nψ(xp)/parenrightBig
+×
+×DET/bracketleftBig
+ˆMAB
+Mψ;Nψ(xp,yq)/bracketrightBig1/2
+exp/braceleftbigg
+−ı
+2/integraldisplay
+Cd4xpd4yqJT,A
+ψ;Mψ(xp)ˆSAB′ˆTB′B′′
+(Ψ);Mψ;M′
+ψ(xp)/integraldisplay
+Cd4y′
+q′×(4.13)
+×/bracketleftBig
+ˆM−1;B′′A′′
+M′
+ψ;N′′
+ψ(xp,y′
+q′)−ˆ1B′′A′′
+M′
+ψ;N′′
+ψδpq′δ(4)(xp−y′
+q′)/bracketrightBig/parenleftBig
+ˆH−1
+ˆS/parenrightBigA′′A′
+N′′
+ψ;N′
+ψ(y′
+q′,yq)ˆTA′B
+(Ψ);N′
+ψ;Nψ(yq)JB
+ψ;Nψ(yq)/bracerightbigg/angbracketrightbigg
+;
+ˆMAB
+Mψ;Nψ(xp,yq) =ˆ1AB
+Mψ;Nψδpqδ(4)(xp−yq)+ (4.14)
++/bracketleftBig/parenleftbigˆH−1
+ˆS/parenrightbig
+δ(ˆHˆS(ˆT−1
+(Ψ),ˆT(Ψ))+/parenleftbigˆH−1
+ˆS/parenrightbigˆT−1
+(Ψ)ˆSˆJˆT(Ψ)/bracketrightBigAB
+Mψ;Nψ(xp,yq) ;
+ˆHˆS=ˆSˆH; (4.15)
+δˆHˆS(ˆT−1
+(Ψ),ˆT(Ψ)) =/parenleftBig
+ˆT−1
+(Ψ)ˆSˆHˆT(Ψ)/parenrightBig
+−/parenleftBig
+ˆSˆH/parenrightBig
+=/parenleftBig
+exp/braceleftbig− −−−−−− →
+[ˆY,...]−/bracerightbigˆSˆH/parenrightBig
+−/parenleftBig
+ˆSˆH/parenrightBig
+; (4.16)
+ˆHAB
+Mψ;Nψ(xp,yq) =/bracketleftbigg/parenleftBig/parenleftbig
+ˆγν
+H/parenrightbigAB
+iAjB∂
+∂xνp−ıˆSABεpδiAjB/parenrightBig
+δMψ(iA);Nψ(jB)+ˆFAB
+ψψ;Mψ;Nψ(xp)+ (4.17)
++/parenleftbig
+/ˆG(xp)/parenrightbig
+Mq;Nq+/parenleftbig
+/ˆW(xp)+/ˆB(xp)/parenrightbig
+Mψ;Nψ+1
+16ˆSAB′/parenleftbigg(κ)=0,...,3/summationdisplay
+ˆs(Ψ,ˆB)(κ)B′A′
+Mψ;Nψ(xp)+
++(κ)=0,...,3/summationdisplay
+(a)=1,2,3ˆs(Ψ,ˆW)(a;κ)B′A′
+Mψ;Nψ(xp)+(κ)=0,...,3/summationdisplay
+(α)=1,...,8ˆs(Q,ˆG)(α;κ)B′A′
+Mq;Nq(xp)/parenrightbigg
+ˆSA′B/bracketrightbigg
+ηpδpqδ(4)(xp−yq) ;
+/ˆG(xp) =δψ=”q”Gγ
+ν(xp)ı/bracketleftBig/parenleftbigˆλ(−)
+γ/parenrightbig
+rs/parenleftbig
+ˆγν
+H/parenrightbigAB
+iAjB+/parenleftbigˆλ(+)
+γ/parenrightbig
+rs/parenleftbig
+ˆγν
+5,H/parenrightbigAB
+iAjBe(G)
+q;H/bracketrightBig
+δMq(r,iA);Nq(s,jB); (4.18)
+/ˆW(xp) =Wc
+ν(xp)ı/bracketleftBig/parenleftbig
+ˆτ(−)
+c/parenrightbig
+fg/parenleftbig
+ˆγν
+L/parenrightbigAB
+iAjB+/parenleftbig
+ˆτ(+)
+c/parenrightbig
+fg/parenleftbig
+ˆγν
+5,L/parenrightbigAB
+iAjB/bracketrightBig
+δMψ(f,iA);Nψ(g,jB); (4.19)
+/ˆB(xp) =Bν(xp)ı/parenleftbig
+ˆγν
+5,H/parenrightbigAB
+iAjBe(Y)
+ψ;HδMψ(iA);Nψ(jB); (4.20)
+Gγ
+ν(xp) =g3
+2/tildewideJ(ˆG)µ
+β/parenleftbigˆS(ˆG)λρ
+β(xp),s(G)
+β(xp)/parenrightbigˆM(ˆG)−1;β,γ
+µ,ν/parenleftbigˆS(ˆG)α
+λρ(xp)/parenrightbig
+; (4.21)
+Wc
+ν(xp) =g2
+2/tildewideJ(ˆW)µ
+b(xp)ˆM(ˆW)−1;b,c
+µ,ν(xp)+g1g2
+4/tildewideJ(ˆB)
+µ(xp)ˆM(ˆW)−1;µ,κ(xp)ˆN(ˆW)c
+κ;ν(xp)+ (4.22)
++g2
+1g2
+8/tildewideJ(ˆW)
+bµ(xp)ˆN(ˆW)(bµ)
+κ(xp)ˆM(ˆB)−1;κλ(xp)ˆN(ˆW)c
+λ;ν(xp) ;
+Bν(xp) =g1
+2/tildewideJ(ˆB)µ(xp)ˆM(ˆW)−1
+µ,ν(xp)+g2
+1
+4/tildewideJ(ˆW)bκ(xp)ˆN(ˆW)µ
+(bκ)(xp)ˆM(ˆB)−1
+µ,ν(xp). (4.23)18 5 SUMMARY AND CONCLUSION
+Apart from a condensate ’seed’ functional /tildewide∆(ˆT−1;AB
+(Ψ);Mψ;Nψ(xp),ˆTAB
+(Ψ);Mψ;Nψ(xp);ˆJAB
+ψψ;Mψ;Nψ(xp)), one achieves
+the square root of the determinant and the bilinear source fie ld partJT,A
+ψ;Mψ(xp)... JB
+ψ;Nψ(yq) with matrix
+ˆMAB
+Mψ;Nψ(xp,yq) (4.14). This matrix ˆMAB
+Mψ;Nψ(xp,yq) is composed of the ’Nambu’ doubled, kinetic Hamiltonian
+part (4.15,4.17) with self-energy densities ˆs(Ψ,ˆB)(κ)B′A′
+Mψ;Nψ(xp),ˆs(Ψ,ˆW)(a;κ)B′A′
+Mψ;Nψ(xp),ˆs(Q,ˆG)(α;κ)B′A′
+Mq;Nq(xp) from the
+background averaging functional, the generating source ma trixˆJAB
+Mψ;Nψ(xp;yq) for observables and the gradient
+termδˆHˆS(ˆT−1
+(Ψ),ˆT(Ψ)) (4.16). Furthermore, one achieves the effective gauge fields /ˆG(xp) (4.18), Gγ
+ν(xp) (4.21),
+/ˆW(xp) (4.19), Wc
+ν(xp) (4.22), /ˆB(xp) (4.20), Bν(xp) (4.23) consisting of the matrices ˆM(ˆG)
+βµ,γν(3.9),ˆM(ˆW)
+bµ,cν(3.11),
+ˆM(ˆB)
+µ,ν(3.14) with self-energies for gauge field strength tensors a nd tilded currents /tildewideJ(ˆG)
+αµ(xp) (4.3),/tildewideJ(ˆW)
+aµ(xp)
+(4.5),/tildewideJ(ˆB)
+µ(xp) (4.7) also composed of self-energy matrices in place of the gauge field strength tensors without
+any Fermi fields. A convenient approximation for the backgro und averaging in (4.13) follows from a saddle
+point computation so that the effective gauge field variables ( 4.18,4.21), (4.19,4.22), (4.20,4.23) and self-energy
+densities ˆs(Ψ,ˆB)(κ)B′A′
+Mψ;Nψ(xp),ˆs(Ψ,ˆW)(a;κ)B′A′
+Mψ;Nψ(xp),ˆs(Q,ˆG)(α;κ)B′A′
+Mq;Nq(xp) are replaced by definite, fixed spacetime func-
+tions/an}bracketle{t/ˆG(xp)/an}bracketri}ht,/an}bracketle{t/ˆW(xp)/an}bracketri}ht,/an}bracketle{t/ˆB(xp)/an}bracketri}htand/an}bracketle{tˆs(Ψ,ˆB)(κ)B′A′
+Mψ;Nψ(xp)/an}bracketri}ht,/an}bracketle{tˆs(Ψ,ˆW)(a;κ)B′A′
+Mψ;Nψ(xp)/an}bracketri}ht,/an}bracketle{tˆs(Q,ˆG)(α;κ)B′A′
+Mq;Nq(xp)/an}bracketri}ht. These fixed
+spacetime functions are determined from first order variati ons of the background averaging functional ’ /an}bracketle{t.../an}bracketri}ht’
+where the anti-hermitian ingredient of latter background f unctional has to comply with the already introduced,
+various anti-hermitian epsilon terms for a stable, proper c onvergence of Green functions.
+5 Summary and conclusion
+5.1 The combination of the Higgs field with the gauge self-ene rgies
+Apart from the case of the strong interaction (3.9), the Higg s fields add to various parts, as to the self-energies
+of the electroweak interaction in (3.11,3.14) and also to th eir generalized currents (3.12,3.15). Therefore, it may
+be difficult to observe pure Higgs field contributions in exper iments, in particular because the Higgs field adds
+to self-energies for the gauge field parts in various, rather involved combinations. However, it turns out that
+one conclude for coset matrices ˆT(Ψ)(xp) = exp{−ˆY(xp)}in an effective generating functional with effective,
+composed gauge fields, determined from a saddle point comput ation of a background functional. The total
+dimensionN0= 90 for the SSB with SO( N0,N0)/U(N0)⊗U(N0) is the exact, relevant number of the total
+standard model which is, however, very different from the larg e SUc(Nc) expansions with the number of colour
+degrees of freedom from N c= 3 to N c→ ∞[19, 20, 21]. In addition to this aspect of a large dimension
+N0= 90, it is also of interest to what extent the vanishing axial anomaly and corresponding vanishing sum of
+Hopf invariants can constrain the topology of field pairs of f ermions or anomalous self-energy parts within the
+standard model [22]. Hence, it remains to determine and to in vestigate detailed, nontrivial field combinations of
+the coset matrix ˆT(Ψ)(xp) with vanishing sum of Hopf invariants for the anti-symmetr ic, quaternion eigenvalues
+and total matrix elements within the generator ˆY(xp), similar to the Skyrme-Faddeev field theory for the strong
+interaction case [13, 12, 6].
+In this paper we have demonstrated how to obtain a classical fi eld theory with self-energy matrices in a
+coset decomposition for simplifying the total path integra l of the quantized standard model of electroweak
+interactions. Since one concentrates on the self-energy ma trices, representing the irreducible propagator parts
+of a quantum many-body theory, our derived classical field th eory also has a semiclassical notion. The given
+approach of this article and of Ref. [6] generalizes to super -symmetric QCD and to the super-symmetric caseREFERENCES 19
+of electroweak forces which will be considered separately. The various steps of sections 3 and 4 can be directly
+transferred to super-symmetric cases where one can perform a similar anomalous doubling of Fermi fields, but
+with inclusion of similar ’Nambu’ doubling for the bosonic p artners in the total chiral super-fields.
+References
+[1] M. Srednicki, ” Quantum Field Theory ”, (Cambridge University Press, Cambridge, New York, 2007)
+[2] A. Das, ” Field Theory (A Path Integral Approach) ”, (2nd edition, ”World Scientific Lecture Notes in
+Physics Vol. 75”, New York, London, Singapore, 2006)
+[3] A. Das, ” Lectures on Quantum Field Theory ”, (World Scientific, 2008)
+[4] Y. Nambu, Phys. Rev. Lett. 4(1960), 380
+[5] J. Goldstone, Nuovo Cimento 19(1961), 154
+[6] B. Mieck, ” BCS-like action and Lagrangian from the gradient expansion o f the determinant of Fermi fields
+in QCD-type, non-Abelian gauge theories with chiral anomali es (Derivation for an effective action of BCS
+quark pairs with the Hopf invariant Π3(S2) =Zas a nontrivial toplogy.) ”, arXiv:0906.2095v1
+[7] R. Rajamaran, ” Solitons and instantons (An introduction to solitons and in stantons in quantum field
+theory)”, (North-Holland, Amsterdam, 1984)
+[8] I. Zahed and G.E. Brown, ” The Skyrme Model , Phys. Rep. 142(No. 1 & 2) (1986), 1-102
+[9] G.E.Brown(editor), ” Selected Papers, with Commentary, of Tony Hilton Royle Skyrm e”, (WorldScientific
+series in 20th Century Physics - Vol. 3, World Scientific, 199 4)
+[10] M.P. Hobson, G. Efstathiou and A.N. Lasenby, ” General Relativity (An Introduction for Physicists) ”,
+(Cambridge University Press, Cambridge, 2006)
+[11] M. Burgess, ” Classical Covariant Fields ”, (Cambridge Monographs on Mathematical Physics, Cambrid ge
+University Press, Cambridge, 2004)
+[12] N. Manton, P. Sutcliffe, ” Topological Solitons ”, (Cambridge Monographs on Mathematical Physics, Cam-
+bridge University Press, Cambridge, New York, 2004)
+[13] L.D. Faddeev, ” Quantization of solitons ”, ’Princeton preprint IAS-75-QS70’ (1975)
+[14] C.P. Burgess and G.D. Moore, ” The Standard Model : A Primer ”, (Cambridge University Press, Cam-
+bridge, 2006)
+[15] B. Mieck, ” Coherent state path integral and super-symmetry for condens ates composed of bosonic and
+fermionic atoms ”, Fortschr. Phys. (”Progress of Physics”) 55(No. 9-10) (2007), 989-1120
+[16] B. Mieck, ” Ensemble averaged coherent state path integral for disorde red bosons with a repulsive interaction
+(Derivation of mean field equations) ”, Fortschr. Phys. (”Progress of Physics”) 55(No. 9-10) (2007), 951-
+98820 REFERENCES
+[17] S. Weinberg, ” The Quantum Theory of Fields (Vol. 2, Modern Applications) ”, (Cambridge University
+Press, 1996)
+[18] T. Muta, ” Foundations of Quantum Chromodynamics (An Introduction to Pe rturbative Methods in Gauge
+Theories) ”, (World Scientific Lecture Notes in Physics - Vol. 5, World S cientific, 1987)
+[19] E. Witten, ” Baryons in the 1/N Expansion ”, Nucl. Phys. B160(1979), 57-115
+[20] E. Witten, ” Global Aspects of Current Algebra ”, Nucl. Phys. B223(1983), 422-432
+[21] E. Witten, ” Current Algebra, Baryons, and Quark Confinement ”, Nucl. Phys. B223(1983), 433-444
+[22] K. Fujikawa and H. Suzuki, ” Path Integrals and Quantum Anomalies ”, (Clarendon Press, Oxford, 2004)
\ No newline at end of file
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+arXiv:1001.0885v1  [cond-mat.soft]  6 Jan 2010Polyelectrolyte Electrophoresis in Nanochannels: A Dissi pative Particle Dynamics
+Simulation
+Jens Smiatek1,∗and Friederike Schmid2,†
+1Institut f¨ ur Physikalische Chemie, Universit¨ at M¨ unste r, D-48149 M¨ unster, Germany
+2Institut f ¨ur Physik, Johannes Gutenberg-Universit ¨at, Staudinger Weg 7, D-55099 Mainz, Germany
+(Dated: August 1, 2018)
+We present mesoscopic DPD-simulations of polyelectrolyte electrophoresis in confined nanogeome-
+tries, for varying salt concentration and surface slip cond itions. Special attention is given to the
+influence of electroosmotic flow (EOF) on the migration of the polyelectrolyte. The effective poly-
+electrolyte mobility is found to depend strongly on the boun dary properties, i.e., the slip length and
+the width of the electric double layer. Analytic expression s for the electroosmotic mobility and the
+total mobility are derived which are in good agreement with t he numerical results. The relevant
+quantity characterizing the effect of slippage is found to be the dimensionless quantity κδB, where
+δBis the slip length, and κ−1an effective electrostatic screening length at the channel b oundaries.
+PACS numbers: 82.35.Rs 47.57.jd 47.61.-k 82.45+z 83.50.Lh
+I. INTRODUCTION
+In recent years there is growing interest in techniques
+for manipulating single nanoparticles or macromolecules
+in micro- and nanochannel systems. The flow profiles in
+these channelsand the motion ofthe macromoleculescan
+be controlled on the nanoscale by pressure gradients and
+electric fields, and by exploiting smart channel geome-
+tries. This explains the great potential and the broad
+applicability of nanochannel devices, e.g., for analyzing
+tiny DNA or protein samples by electrophoresis1–11.
+Suchsystemsrepresentachallengefortheoryandcom-
+putersimulationduetotheirhighcomplexity. Inbiotech-
+nological applications, the molecules of interest are often
+charged and dissolved in buffer solutions with high salt
+concentrations. Thus electrostatic and hydrodynamic ef-
+fects compete with each other, resulting, among other, in
+a remarkable ’electrohydrodynamic screening’ effect: In
+free solution electrophoresis ( i.e., in a constant electric
+field), the counterion layer surrounding a charged par-
+ticle not only screens the electrostatic interactions, but
+also the dominant contribution to the hydrodynamic in-
+teractions, i.e., those that are generated by the applied
+field12,13. Therefore, simulations of electrophoresis that
+neglect the electrostatic and hydrodynamic interactions
+altogether often give results that are in good semiquan-
+titative agreement with experiments1,14,15. More sophis-
+ticated approaches that still allow to avoid the explicit
+representation of charges have been devised as well16,17.
+Nevertheless, it is clear that such simplified treatments
+disregard important physics, especially in confined ge-
+ometries where electrostatic interactions compete with
+the regular steric interactions with the confining walls18.
+Simulations that take full account of electrostatic and
+hydrodynamic interactions are clearly desirable. Such
+simulations have recently been carried out for polyelec-
+trolyte electrophoresis in free solutions19,20, but, to the
+best knowledge of the present authors, not yet for mi-
+crochannels.In many microchannels, an additional effect comes
+into play, which significantly modifies the effective elec-
+trophoretic response of particles to electric fields: The
+same electric field that drives the polyelectrolyte may
+also induce a total net fluid flow in the microchannel, the
+’electroosmotic flow’ (EOF). Many materials commonly
+used in microtechnology like PDMS (Polydimethylsilox-
+ane) acquire charges if brought in contact with water,
+either by the ionization or dissociation of surface groups
+or the adsorption of ions from solution. To screen the
+charges on the channel walls, a diffuse layer of oppositely
+charged ions forms in front of the walls. In the presence
+of an external electric field E(ext), these ions are pulled
+along, dragging the surrounding fluid with them. One
+gets a characteristic ’plug’ flow profile which saturates at
+a fluid velocity
+vEOF=µEOFE(ext)(1)
+outside of the diffuse layer, with the so-called electroos-
+motic mobility µEOF. The effective migration speed vP
+of particles in microchannels in response to the exter-
+nal fields then results from two contributions15: The
+bare electrophoretic mobility µeof the polyelectrolyte
+in a fluid at rest, and the background EOF velocity,
+vP=vEOF+µeE(ext). Experimentally, it has been found
+that the former may even dominate over the latter, such
+that the polyelectrolyte effectively migrates in the direc-
+tionopposite to the applied field4.
+Now if the diffuse layer is thin compared to the chan-
+nel dimensions, Eq. (1) can be regarded as an effective
+boundary condition for a steady-state EOF velocity field
+vEOF(r) inside the channel. (We note that in a steady-
+state situation, the external field in the vicinity to a wall
+is necessary parallelto the wall.) Cummings et al.21have
+shown that for laminar incompressible flow, this bound-
+ary condition effectively defines the flow inside the chan-
+nel. Provided Eq. (1) also holds at the inlet and outlet
+boundaries of the channel, it becomes valid everywhere
+in the channel22. Due to this remarkable similitude be-
+tween the steady-state velocity field of an EOF and the2
+externally applied electric field, the net electrophoretic
+velocity of nanoparticles or polyelectrolytes can be writ-
+ten approximately as
+vP= (µEOF+µe)E(ext)=:µtE(ext)(2)
+with the effective total mobility µt=µEOF+µe. We note
+that this expression relies on two assumptions which are
+both not obvious: The nanoparticles or polyelectrolytes
+stay well outside the diffuse layer covering the wall, and
+they themselves do not influence the EOF.
+The EOF amplitude at planar walls with no-slip
+boundary conditions has been calculated a long time ago
+by Smoluchowski23. On the nanoscale, however, the no-
+slip boundary condition does not necessarily apply. Ex-
+periments have indicated24–26, that the velocity profile
+is not strictly continuous at walls, i.e., fluids exhibit a
+certain amoung of slippage. This effect can be enhanced
+significantly by using superhydrophobic walls which are
+covered by a thin gas layer. It can also be tuned to some
+extent by designing nanopatterned surfaces with alter-
+nating hydrophobic and hydrophilic sections27. In all of
+these cases, the appropriate mesoscopic boundary condi-
+tion is the ’partial slip’ boundary condition,
+vx(±zB) =∓δB∂
+∂zvx(z)|z=±zB, (3)
+wherezBis the position of the hydrodynamic boundary
+(which is usually close to the physical boundary, but not
+necessarily identical), and the ’slip length’ δBcharacter-
+izes the amount of slippage. No-slip boundaries corre-
+spond toδB= 0, full-slip to δB→ ∞. From Eq. (3),
+one would expect that the EOF amplitude is enhanced
+in the presence of slippage27, and this is indeed found in
+simulations28,29. The effect of slippage on EOF has been
+calculated within the linearized Poisson-Boltzmann the-
+ory, the Debye-H¨ uckel approximation, by Joly et al.28.
+Below, we will derive a general expression which is also
+valid beyond the Poisson-Boltzmann theory.
+In this paper we present Dissipative Particle Dynam-
+ics (DPD)-simulations of polyelectrolyte electrophoresis
+in microchannels with varying slip lengths, at varying
+salt concentrations. We treat the solvent and all ions ex-
+plicitly, and all charges interact viaunscreened Coulomb
+interactions. This allows to investigate the interplay of
+solvent and polyelectrolyte, of electrostatic and hydro-
+dynamic interactions, of electrophoresis and EOF in full
+detail, with almost no approximation. (Our only approx-
+imation is to neglect image charge effects, i.e., the di-
+electric constant is taken to constant everywhere.) Our
+results indicate that the hydrodynamic boundary condi-
+tions stronglyinfluence the total mobility of the polyelec-
+trolyte, and the total mobility can be tuned from positive
+to negative by varying the slip length. The simulation
+data are compared with a simple analytical expression,
+which is derived based on the assumption that the flow
+profile in the channel follows the Stokes equation. The
+numerical results are in very good agreement with the
+theory.The paper is organized as follows. The theory is pre-
+sented in section 2. Section 3 focuses on the simulation
+method and the parametersused in our simulations. The
+numericalresultswill be shownin section4. We conclude
+with a brief summary in section 5.
+II. THEORETICAL CONSIDERATIONS: EOF
+IN THE PRESENCE OF SLIPPAGE
+We consider for simplicity a planar slit channel with
+identical walls at z=±L/2, exposed to an external elec-
+tric fieldExin thexdirection. The electrostatic poten-
+tial Φ(x,y,z) then takes the general form Φ( x,y,z) =
+ψ(z)+Exx+const. where we can set ψ(0) = 0 for sim-
+plicity. The electrolyte in the channel is taken to contain
+ndifferent ion species iwith local number density ρi(z)
+and valency Zi, which results in a net charge density
+ρ(z) =/summationtextn
+i=1(Zie)ρi(z). The electric field then generates
+a force density fx(z) =ρ(z)Exin the fluid. Comparing
+the Poisson equation for the electrostatic potential ψ,
+∂2ψ(z)
+∂z2=−ρ(z)
+ǫr(4)
+(whereǫris the dielectric constant), with the Stokes
+equation30
+ηs∂2vx(z)
+∂z2=−fx(z) =−ρ(z)Ex (5)
+(with the shear viscosity ηs), one finds immediately
+∂zzvx(z) =∂zzψ(z) (ǫrEx/ηs). For symmetry reasons,
+the profiles vxandψmust satisfy the boundary condi-
+tion∂zvx|z=0=∂zψ|z=0= 0 at the center of the channel.
+This gives the relation
+vx(z) =ǫrEx
+ηsψ(z)+vEOF, (6)
+where we have used ψ(0) = 0 and identified the fluid
+velocity at the center of the channel with the EOF veloc-
+ity,vx(0) =vEOF. We further define ψB:=ψ(±zB) (for
+no-slip boundaries, ψBis the so-called Zeta-Potential23).
+Insertingthe partial-slipboundarycondition forthe flow,
+Eq. (3), we finally obtain the following simple expression
+for the electroosmotic mobility,
+µEOF=vEOF/Ex=µ0
+EOF(1+κδB),(7)
+where we have defined the inverse ’surface screening
+length’
+κ:=∓∂zψ
+ψ|z=±zB, (8)
+andµ0
+EOFis the well-knownSmoluchowskiresult23forthe
+electroosmotic mobility at sticky walls,
+µ0
+EOF=−ǫrψB/ηs. (9)3
+The remaining task is to determine the screening pa-
+rameterκ. If the surface charges are very small and the
+ions in the liquid are uncorrelated, it can be calculated
+analytically within the linearized Debye-H¨ uckel theory31.
+The Debye-H¨ uckel equation for the evolution of the po-
+tentialψin an electrolyte solution reads ∂zzψ=κ2
+Dψ
+with the inverse Debye-H¨ uckel screening length
+κD=/radicalBigg/summationtextn
+i=1(Zie)2ρi,0
+ǫrkBT, (10)
+whereρi,0is the density of ions ifar from the surface. It
+is solved by an exponentially decaying function,
+ψ(z)∝(eκDz+e−κDz−2). (11)
+Inserting that in Eq. (8), one finds κ=κD,i.e., the sur-
+facescreeninglengthisidenticalwiththeDebyescreening
+length. Eq. (9) with κ=κDbasically corresponds to the
+result of Joly et al.28.
+Unfortunately, the range of validity of the Debye-
+H¨ uckel theory is limited, it breaks down already for mod-
+erate surface potentials ψBand/or for highly concen-
+trated ion solutions. Nevertheless, the exponential be-
+havior often persists even in systems where the Debye-
+H¨ uckel approximation is not valid. For high ion con-
+centrations, detailed studies based on integral equa-
+tions have lead to the conclusion that the Debye-H¨ uckel
+approximation can still be used in a wide parameter
+range, ifκDis replaced by a modified effective screening
+length32–34. For high surface charges, analytical solu-
+tions are again available in the so-called ’strong coupling
+limit’, where the profiles are predicted to decay expo-
+nentially with the Guy-Chapman length35. This limit
+is very special and rarely encountered. At intermediate
+coupling regimes, the decay length must be obtained em-
+pirically, e.g., byfitting the chargedistribution ρ(z)to an
+exponential behavior, which is characterized by the same
+exponential behavior than ψ(z) by virtue of the Poisson
+equation,
+n/summationdisplay
+i=q(Zie)ρi(z)∝∂2ψ(z)
+∂z2∝(eκz+e−κz).(12)
+Puttingeverythingtogether, the totalnet electrophoretic
+mobilityµt(Eq. 2) ofa polyelectrolytein the channel can
+be expressedin termsofthe electroosmoticmobility µEOF
+as
+µt
+µEOF= 1+µe
+µ0
+EOF(1+κδB), (13)
+wheretheratio µe/µ0
+EOFdependsonlyweaklyontheionic
+strength of the electrolyte and the slip length of the sur-
+face. The main effect of slippage is incorporated in the
+factor (1+ κδB)−1.III. SIMULATION METHOD
+A. Dissipative Particle Dynamics
+Newtonian fluids are effectively modeled by the Dis-
+sipative Particle Dynamics (DPD) method36,37. DPD is
+a coarse-grained,momentum-conservingsimulation tech-
+nique which creates a well-defined canonical ensemble.
+The basic DPD equations are defined in terms of the
+forces on one particle, which involve two-particle interac-
+tions
+/vectorFDPD
+i=/summationdisplay
+i/negationslash=j/vectorFC
+ij+/vectorFD
+ij+/vectorFR
+ij (14)
+with a conservative force /vectorFC
+ij
+/vectorFC
+ij=−/vector∇ijUij(rij). (15)
+(whererijdenotes the distance between the centers of
+particlesiandj), a dissipative force /vectorFD
+ij
+/vectorFD
+ij=−γDPDωD(rij)(ˆrij·/vector vij)ˆrij (16)
+with the friction coefficient γDPD, and a random force /vectorFR
+ij
+/vectorFR
+ij=/radicalbig
+2γDPDkBT ωR(rij)ˇζijˆrij.(17)
+Hereˇζij=ˇζjiis a symmetric random number with zero
+mean and unit variance, and the weighting functions of
+the dissipative and the stochastic force are related by a
+fluctuation-dissipation theorem,
+ωD(rij) = [ωR(rij)]2≡ωDPD(rij),(18)
+which ensures that an equilibrium simulation samples a
+canonical ensemble36,37. Otherwise, the weight function
+ωDPDis arbitrary and will be chosen linear here as often
+in the literature,
+ωDPD(rij) =/braceleftbigg1−rij
+rc:rij<rc
+0 :rij≥rc(19)
+wherercdenotes the cut-off radius.
+B. Tunable slip boundaries
+The hydrodynamic boundary condition at the walls is
+realized with a recently developed method38that allows
+to implement arbitrary partial-slip boundary conditions:
+Weintroduceanadditionalcoordinate-dependentviscous
+force that mimicks the wall/fluid friction
+FL
+i=FD
+i+FR
+i (20)
+with a dissipative contribution
+FD
+i=−γLωL(z) (vi−vwall) (21)4
+couplingtotherelativevelocity( vi−vwall)oftheparticle
+with respect to the wall, and a stochastic force
+FR
+i,α=/radicalbig
+2γLkBT ωL(z)χi,α (22)
+which satisfies the fluctuation-dissipation relation and
+thus ensures that the local equilibrium distribution is
+again a Boltzmann distribution. Here αisα=x,y,z
+andχi,αis a Gaussian distributed random variable with
+mean zero and variance one: ∝angb∇acketleftχi,α∝angb∇acket∇ight= 0,∝angb∇acketleftχi,αχj,β∝angb∇acket∇ight=
+δijδαβ. The viscous coupling between fluid and wall is
+achieved by the locally varying viscosity γLωL(z) with
+ωL(z) = 1−z/zcup to a cut-off distance zc. The pref-
+actorγLcan be used to tune the strength of the friction
+force and hence the value of the slip length. Within this
+approach it is possible to tune the slip length δBsystem-
+atically from full-slip to no-slip, and to derive an analytic
+expression for the slip length as a function of the model
+parameters38.
+C. Simulation details
+We have studied the electrophoresis of charged poly-
+mers of length N= 20 in electrolyte solutions, confined
+by a planar slit channel with charged walls. All particles,
+polymer, solventandions, aremodeledexplicitly. We use
+a simulation box of size (12 σ×12σ×10σ) which is peri-
+odic inx- andy-direction and confined by impermeable
+walls in the z-direction. The walls repel the particles
+viaa soft repulsive WCA potential39of rangeσand am-
+plitudeǫ. (Hence the accessible channel width for the
+particles is actually Lz= 8σ). Ions and monomers repel
+each other with the same WCA potential. In addition,
+chain monomers are connected by harmonic springs
+Uharmonic =1
+2k(rij−r0)2(23)
+with the spring constant k= 25ǫ/σ2andr0= 1.0σ. Neu-
+tral solvent particles have no conservative interactions
+except with the walls.
+Thewallcontainsimmobilized, negativelychargedpar-
+ticlesatrandompositions. Everysecondmonomeronthe
+polyelectrolyte carries a negative charge. The solvent
+contains the positive counterions for the walls and the
+polyelectrolyte, and additional (positive and negative)
+saltions. Allchargesaremonovalent,andthesystemasa
+wholeiselectroneutral. Inadditiontotheirotherinterac-
+tions, charged particles interact viaa Coulomb potential
+with the Bjerrum length λB=e2/4πǫrkBT= 1.0σ, and
+they are exposed to an external field Ex=−1.0ǫ/eσ.
+Specifically, we have studied systems with a surface
+charge density of σA=−0.208eσ−2. The total counte-
+rion density was ρ= 0.06σ−3and the salt density varied
+betweenρs=0.05625, 0.0375, 0.03, 0.025, and 0 .015σ−3.
+We use DPD simulations with a friction coefficient
+γDPD= 5.0σ−1(mǫ)1/2. The density of the solvent par-
+ticles wasρ= 3.75σ−3, and the temperature of the sys-
+tem wasT= 1.0ǫ/kB. For these parameters, the shearTable I: Slip lengths δBfor different layer friction coefficients
+γL, compared with theoretical value δT
+Baccording to Ref. 38.
+γL[σ−1(mǫ)1/2]δB[σ]±δB[σ]δT
+B[σ]
+0.1 14 .977 1.879 14 .000
+0.25 5 .664 0.783 5.458
+0.5 2 .626 0.521 2.613
+0.75 1 .765 0.409 1.664
+1.0 1 .292 0.423 1.190
+6.1 0 .000 0.197 0.000
+viscosity of the DPD fluid – as determined by fitting
+the amplitude of Plane Poiseuille flows38– is given by
+ηs= (1.334±0.003)σ−2(mǫ)1/2. The DPD timestep was
+δt= 0.01σ(m/ǫ)1/2.
+Tunable-slip boundary conditions were used with fric-
+tion coefficients γL=0.1, 0.25, 0.5, 0.75, 1.0, and
+6.1σ−1(mǫ)1/2. The range of the viscous layer was
+zc= 2.0σ. Only the solvent particles interact with
+the tunable-slip boundaries. By performing Plane
+Poiseuille and Plane Couette flow simulations with the
+above given parameters, the slip length δBand the hy-
+drodynamic boundary positions zBcan be determined
+independently38. The hydrodynamic boundary position
+is found at |zB|= (3.866±0.266)σin all simulations.
+The corresponding slip lengths are presented in Table I
+together with the theoretical values predicted by the an-
+alytic expression in Ref. 38. The comparison shows that
+the simulated results are in good agreementwith the the-
+ory.
+The electrostatics were calculated by P3M40and
+the ELC (electrostatic layer correction)-algorithm41for
+2D+hslabwise geometries. All simulations have been
+carried out with the freely available software package
+ESPResSo42,43
+IV. NUMERICAL RESULTS
+Fig. 1 shows the average ionic distributions of an-
+ionsρaand cations ρcfor the salt concentration ρs=
+0.05625σ−3. Here the ’cations’ include the positively
+charged salt ions and the counterions of the wall and
+the polyelectrolyte, and the ’anions’ only the negatively
+charged salt ions. Due to the presence of the polyelec-
+trolyte in the middle of the channel, the average cation
+density there is slightly increased. To determine the in-
+verse effective screening length κ, we have thus fitted the
+following function
+∆ρ= ∆ρ0(e−κz+eκz)+c (24)
+to the ionic difference ∆ ρ=ρc−ρa. The exponential
+fit describes the data verywell (black solid line in Fig. 1).
+The fit parameters for κare listed in Table II, along with5
+ 0  0.1  0.2  0.3  0.4 
+-4-3-2-1 0 1 2 3 4∆ρ [σ-3]
+z [σ] 0  0.1  0.2  0.3  0.4 
+-4-3-2-1 0 1 2 3 4ρ [σ-3]
+z [σ]
+κ  ∼ 2.305 σ−1Exponential fit
+Figure 1: Distribution of the ionic difference ∆ ρ=ρc−ρa
+between cations and anions in the solution (not counting
+the polyelectrolyte) for an exemplary salt concentration o f
+ρs= 0.05625σ−3and the surface charge density σA=
+−0.208eσ−2. The black line corresponds to an exponential
+fit (Eq. (24)) with an effective inverse screening length of
+κ= 2.305±0.025σ−1.Inset:Distribution of cations (cir-
+cles) and anions (triangles) for the same system.
+the values for the Debye-H¨ uckel screening parameter κD
+(Eq. (10)). The decay lengths are overall very different
+from those predicted by the Debye-H¨ uckel theory. We
+conclude that the system is outside the validity region of
+the linearized Poisson-Boltzmann approximation. This
+is perhaps not surprising, given that the individual ion
+profiles(inset ofFig.1) atthe wallsdeviatestronglyfrom
+their bulk value, i.e., these deviations can hardly be con-
+sidered as small perturbations. The surface charge is too
+high. On the other hand, the electrostatic coupling con-
+stant Ξ = 2 πZ3λ3
+BσA∼0.2 (Z= 1 is the valency of
+the cations) is still much smaller than unity, hence we
+are still in a ’weak coupling’ regime. This is also evi-
+dent from the fact that the effective screening param-
+eterκdiffers strongly from the Guy Chapman length,
+µ−1= 2πλBZσA= 1.31σ.
+The EOFprofilesforthe samesalt concentration( ρs=
+0.05625σ−3) are shown in Fig. 2. The different curves
+correspond to different hydrodynamic boundary condi-
+tions (slip lengths). As expected, the flow velocity in-
+creases drastically for larger slip lengths. All curves are
+Table II: Fitted inverse screening lengths κand Debye-H¨ uckel
+screening parameter κDfor different salt concentrations ρs
+and the fixed counterion density of ρ= 0.06σ−3.
+ρs[σ−3]κ[σ−1]±κ[σ−1]κD[σ−1]
+0.015 1.996 0.041 0.98
+0.0225 2.011 0.049 1.02
+0.03 1.983 0.041 1.07
+0.0375 2.182 0.047 1.11
+0.05625 2.305 0.025 1.21-2.5-2-1.5-1-0.5 0 0.5
+-4-3-2-1 0 1 2 3 4vx(z) [σ(m/ε)-1/2]
+z [σ]
+Figure 2: Exemplary flow profiles for a salt concentration
+ρs= 0.05625σ−3for varyingslip lengths (from bottom to top:
+δB= (14.98,5.66,2.63,1.77,1.29,0.00)σ.) The black lines are
+the theoretical predictions obtained by integrating the St okes
+equation (Eq. (5)) with a fitted inverse screening length of
+κ= 2.305σ−1.
+ 0   10  20  30  40  50 
+ 0   10  20  30  40  50µEOF / µEOF
+ρs [σ-3]0
+κ δB1 + κ δ   B-2-1 0
+ 0  0.02  0.04  0.06 ψB [kBT/e]
+Figure 3: Ratio µEOF/µ0
+EOFplotted against δBκfor the differ-
+ent salt concentrations and screening lengths given in Tabl e I
+and II. The black line is the theoretical prediction of Eq. (7 )
+with slope 1+ κδB.Inset:Surface potential as obtained from
+µ0
+EOFusing Eq. (9) (circles) and indepently by a test charge
+method (triangles) as a function of the salt concentration ρs.
+ingoodagreementwiththetheoreticalpredictions,which
+were obtained by integrating the Stokes equation (24)
+numerically with the correct partial-slip boundary con-
+ditions. In agreement with our earlier studies at zero salt
+concentration29, we thus find that a description based on
+the Stokes equation – a continuum equation – remains
+valid even for very narrow channels.
+The flow velocity in the middle of the channel gives
+the EOF mobility. Fig. 3 compares our numerical re-
+sults for all salt concentrations and slip lengths with the
+theoretical prediction of Eq. (7), where µ0
+EOFhas been
+determined by a linear regression for each salt concen-
+tration independently. We find good agreement between
+simulation data and theory. This confirms the validity of
+our theoretical result, Eq. (7). It also demonstrates that
+the polyelectrolyte, which was present in all simulations,
+does not perturb the EOF even in very narrow channels.6
+ 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
+-4-3-2-1 0 1 2 3 4Monomer Probability
+z [σ]σ
+σ = 1.51373 +/− 0.031532~ exp(−0.5 (z /   )  )
+Figure 4: Normalized monomer distribution inside the chan-
+nel for the salt concentration ρs= 0.05625σ−3.
+The values of the EOF mobility for zero slip length,
+µ0
+EOF, can be used to determine the surface potential ψB
+for the different salt concentrations viaEq. (9). As a
+consistency check, we have also determined ψBindepen-
+dently by inserting a test charge into the ion layer at
+z=zB. Theresultsareshownin theinsetofFig.3. Both
+methods give identical results. The surface potential is
+found to be largelyindependent ofthe salt concentration.
+After investigating the EOF of the solvent, we dis-
+cuss the properties of the polyelectrolyte. The proba-
+bility distribution for finding a monomer at a given po-
+sitionzis shown in Fig. 4 for the salt concentration
+ρs= 0.05625σ−3. It is approximately Gaussian with
+a peak in the middle of the channel and a variance
+Var∼2.28σ. Thus the polyelectrolyte mainly ’senses’
+the EOF in the middle of the channel, and the details
+of the flow profiles close to the channel walls have very
+little influence on its net mobility: The assumption that
+the total mobility is governed by a single EOF velocity
+vEOF(Eq. (2)) is probably legitimate.
+The influence of the ion profiles on the chain structure
+of the polyelectrolyte can be investigated by consider-
+ing static properties like the radius of gyration R2
+g=
+(1/2N2)/summationtextN
+i,j=1<(/vectorRi−/vectorRj)2>and the end-to-end ra-
+diusR2
+e=<(/vectorRN−/vectorR1)2>44. The results for these
+parameters are shown in Table III. Both characteristic
+Table III: Radius of gyration Rgand end to end radius Re
+for a polyelectrolyte with N= 20 monomers for different salt
+concentrations ρs.
+ρs[σ−3]Rg[σ] Re[σ]
+0.015 3 .2218±0.047 10.6480±0.0314
+0.0225 3 .1661±0.0041 10.2736±0.0266
+0.03 3 .1486±0.0451 10.1777±0.0292
+0.0375 3 .1279±0.0045 10.0819±0.0287
+0.05625 3 .0825±0.0045 9.8331±0.0280-1 0 1 2 3 4 5 6
+-5 0 5 10 15 20 25 30 35 40µt/|µEOF|
+κ δB-1500-1000-500 0
+ 0 200 400 600 800xcm [σ]
+t [σ(m/ε)1/2]
+Figure 5: Ratio µt/|µEOF|plotted against δBκfor all slip
+lengths (Table I) and salt concentrations (Table II). The
+black line is the theoretical prediction of Eqn. (13) with ab -
+solut values of |µEOF|. In the limit δBκ→ ∞, the to-
+tal mobility of the polyelectrolyte is equal to the electroo s-
+motic mobility µEOF. The ratio µe/µ0
+EOFhas been fitted
+to−3.778±0.128. Negative values of µt/|µEOF|indicate
+negative total mobilities of the polyelectrolyte. Inset:To-
+tal displacement of the polyelectrolytes center of mass for
+different boundary conditions and a salt concentration of
+ρs= 0.05625σ−3. The total mobility becomes negative for
+|µe| ≪ |µEOF|. The lines correspond from top to bottom to
+the slip lengths δB≈(0.00,1.292,1.765,2.626,5.664,14.98)σ.
+Thus larger slip lengths indirectly enhance the total mobil ity
+of the polyelectrolyte.
+lengthsdecreasewithincreasingsaltconcentrationdueto
+a more effective screening of electrostatic interactions, in
+accordance with standard theories45. The ratio between
+the end-to-end radii and the gyration radii is unusually
+large, which is most likely a squeezing effect due to the
+presence of the channel walls46,47. Specific flow-induced
+effects such as shear-induced elongation48are probably
+less important, since the flow profile is basically constant
+inside the channel. It should be noted that the pure
+electrophoretic mobility µeof the chain is presumably
+modified by the confinement as an indirect effect of the
+elongation. This effect has not been investigated in the
+present study.
+The total mobility of the polyelectrolyte for varying
+boundary conditions and salt concentrations is finally
+presentedinFig.5. ThetheoreticalpredictionofEq.(13)
+agrees well with the numerical results with the single fit
+parameterµe/µ0
+EOF=−3.778±0.128. It is remarkable
+thatthis parametercanbe settoaconstant, i.e., it seems
+to be largely independent of the salt concentration ρs.
+Sinceµ0
+EOFdoes not depend on ρs(see Fig. 3, inset), this
+means that µeis also independent of ρsfor the range of
+salt concentrations considered here49.
+For no-slip boundary conditions with δB≈0, we
+find ordinary behaviour where the polyelectrolyte fol-
+lows the applied electric field. In the presence of wall
+slip, however, the EOF becomes stronger and eventu-
+ally dominates. Then the total mobility may become
+negative, i.e., the polyelectrolyte migrates in a direc-7
+tion which is opposite to the applied force. The inset of
+Fig. 5 illustrates this by showing the total displacement
+of the chain’s center of mass for the salt concentration
+ρs= 0.05625σ−3and various slip lengths. In nearly all
+cases except δB≈0, the total mobility of the polyelec-
+trolyte is negative.
+To summarize this section, both the assumptions and
+the predictions of section II are supported by our numer-
+ical results. The total mobility of the polyelectrolyte can
+therefore be adequately described by Eqs. (7) and (13).
+V. CONCLUSIONS
+We have presented mesoscopic DPD simulations of
+polyelectrolyte electrophoresis in narrow microchannels,
+taking full account of hydrodynamic and electrostatic in-
+teractions. A particular focus was put on studying the
+effects of the hydrodynamic boundary conditions at the
+channel walls on the electroosmotic flow and on the net
+electrophoretic mobility of the polyelectrolyte. We have
+shown that they can be incorporated into a single dimen-
+sionless parameter (1+ κδB), whereδBis the slip length
+andκthe (local) inverse screening length of the charge
+distribution at the wall. This was derived analytically
+and supported by our numerical data. It remained valid
+even for very narrow channels, where the chain confor-
+mations were affected by the confinement.
+We have shown that wall slip massively enhances the
+EOF and hence influences the total mobility of the poly-
+electrolyte. If the EOF mobility µEOFand the free drain-ing mobility µeoppose each other, i.e., if the effective
+charges on the polyelectrolyte and the walls have the
+same sign, the mobility may even become negative. As
+mentioned in the introduction, this effect has also been
+observed experimentally4. In the other case, where the
+signofthe chargeson the polyelectrolyteand the wallare
+opposite, the main effect of slip is to enhance the total
+mobility of the polyelectrolyte.
+In summary, the total mobility of polyelectrolytes in
+microchannels results from an interplay of electroos-
+motic, electrophoretic, electrostatic and slippage effects.
+The latter have a particularly strong influence and can
+be used to design channelswith improvedproperties. For
+example, the characteristics of the channel walls could
+be designed to tune effective slip lengths27and hence
+flow velocities, which offers the possibility to optimize
+the time which is needed for polymer migration or sep-
+aration techniques. This could be an important aspect
+for future applications in microchannels or micropumps
+to accelerate measuring times.
+Acknowledgments
+We thank Christian Holm, Burkhard D¨ unweg, Ulf D.
+Schiller, MarcelloSegaandKaiGrassforniceandfruitful
+discussions and the Arminius PC2Cluster at Paderborn
+University, HLRS Stuttgart and NIC J¨ ulich for comput-
+ing time. J. S. especially thanks Stefanie G¨ urtler and
+Theodor A. Smiatek. Financial funding from the Volk-
+swagen Stiftung is gratefully acknowledged.
+∗Electronic address: jens.smiatek@uni-muenster.de
+†Electronic address: Friederike.Schmid@Uni-Mainz.DE
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+arXiv:1001.0887v1  [cs.CE]  6 Jan 2010StableFeatureSelectionforBiomarkerDiscovery
+ZengyouHe, Weichuan Yu
+Laboratoryfor Bioinformaticsand ComputationalBiology,
+DepartmentofElectronicand ComputerEngineering,
+TheHong KongUniversityofScience andTechnology,Kowloon ,Hong Kong,China.
+May 30,2018
+Abstract
+Feature selection techniques have been used as the workhors e in biomarker discovery applications for
+a long time. Surprisingly, the stability of feature selecti on with respect to sampling variations has long
+been under-considered. It is only until recently that this i ssue has received more and more attention. In
+this article, we review existing stable feature selection m ethods for biomarker discovery using a generic
+hierarchal framework. We have two objectives: (1) providin g an overview on this new yet fast growing
+topic for a convenient reference; (2) categorizing existin g methods under an expandable framework for
+futureresearchanddevelopment.
+Keywords: Featureselection;biomarkerdiscovery;stability; machi nelearning
+1 Introduction
+Recent advances in genomics and proteomics enable the disco very of biomarkers for diagnosis and treat-
+ment of complex diseases at the molecular level [1]. A biomar ker may be defined as “a characteristic that is
+objectively measured and evaluated as an indicator of norma l biological processes, pathogenic processes, or
+pharmacologic responses toatherapeutic intervention” [2 ].
+Thediscovery ofbiomarkers fromhigh-throughput “omics” d ataistypically modeledasselecting themost
+discriminating features(orvariables) forclassification (e.g. discriminating healthyversusdiseased, ordifferen t
+tumor stages) [3, 4]. In the language of statistics and machi ne learning, this is often referred to as feature
+selection. Featureselection hasattracted strongresearc hinterest inthepastseveral decades. Forrecent reviews
+of feature selection techniques used inbioinformatics, th e reader isreferred to [5, 6, 3, 7].
+While many feature selection algorithms have been proposed , they do not necessarily identify the same
+candidate feature subsets if we repeat the biomarker discov ery procedure [8]. Even for the same data, one
+may find many different subsets of features (either from the s ame feature selection method or from different
+feature selection methods) that can achieve the same or simi lar predictive accuracy [9, 10, 11]. In practice,
+high reproducibility of feature selection is equally impor tant as high classification accuracy [12]. It is widely
+1believed that a study that cannot be repeated has little valu e [13]. Consequently, the instability of feature
+selection results will reduce our confidence in discovered m arkers.
+Thestability issue in feature selection has received much a ttention recently. In this article, weshall review
+existing methods for stable feature selection in biomarker discovery applications, summarize them with an
+unified framework and provide aconvenient reference for fut ure research and development.
+Thisarticle differs from existing review papers onfeature selection in the following aspects:
+•Compared to current feature selection reviews [5, 6, 3, 7], t his review focuses only on those feature
+selection approaches that incorporate “stability” into th e algorithmic design.
+•Thisarticle mainlyfocuses on“methods” forfindingreliabl emarkersrather than“metrics” ofmeasuring
+the stability of selected feature subsets [14], although we also list these metrics for completeness.
+The remainder of the paper is organized as follows. In sectio n 2, we discuss several sources that cause the
+instability of feature selection. In section 3, we summariz e available stable feature selection algorithms and
+describe different classesofmethodsindetail. Insection 4,weprovidealistofstabilitymeasuresandillustrate
+their definitions. Wegive adiscussion insection 5. Finally , weconclude this paper in section 6.
+2 Causesof Instability
+Thereare mainly three sources of instability inbiomarker d iscovery:
+1. Algorithm design without considering stability: Classi c feature selection methods aim at selecting a
+minimumsubset offeaturestoconstruct aclassifierofthebe stpredictiveaccuracy[8]. Theyoftenignore
+“stability” in thealgorithm design.
+2. Theexistence ofmultiplesetsoftruemarkers: Itispossi ble that thereexist multiplesetsofpotential true
+markers in real data. On the one hand, when there are many high ly correlated features, different ones
+maybe selected under different settings [8]. Ontheother ha nd, eventhere areno redundant features, the
+existence of multiple non-correlated sets of real markers i salso possible [15].
+3. Small number of samples in high dimensional data: In the an alysis of gene expression data and pro-
+teomics data, there are typically only hundreds of samples b ut thousands of features. It has been experi-
+mentally verifiedthat therelatively small number of sample s inhigh dimensional data isoneof themain
+sources oftheinstability problem infeature selection [16 ,17]. Tounderstand thenatureoftheinstability
+of selected feature subset, Ein-Dor et al[18]developed ane wmathematical model andconcluded that at
+least thousands of samples areneeded to achieve stable feat ure selection.
+Here we list three sources that can cause the instability of f eature selection in biomarker discovery. We
+believe that there may be still other sources that can affect the stability of feature selection. The identification
+2Stable Feature Selection
+Ensemble
+Feature
+Selection
+Data
+PerturbationFunction
+PerturbationGroup
+Feature
+Selection
+Knowledge-Driven
+Group Formation
+PathwayData-Driven
+Group Formation
+Clustering Density
+EstimationPrior Feature 
+RelevanceSample
+Injection
+Transductive
+Learning
+Artificial
+Training
+Sample s
+Figure 1: Ahierarchical framework for stable feature selec tion methods.
+of these sources is of primary importance for future researc h and development. On the one hand, knowing
+the reason enables us to better understand the problem. On th e other hand, such knowledge will facilitate the
+design of new methods for stable biomarker discovery.
+3 ExistingMethods
+Todate,therearemanymethodsavailableforstablefeature selection. Wewishtocoverallexistingmethods
+inasystematic and expandable manner. Fig.1illustrates ou r approach tosummarizing different methods based
+on the way they treat different sources of instabilities. Br iefly, the ensemble feature selection method and the
+method using prior feature relevance incorporate stabilit y consideration into the algorithm design stage. To
+handle data with highly correlated features, the group feat ure selection approach treats feature cluster as the
+basic unit in the selection process to increase robustness. The sample injection method tries to increase the
+sample size to address the small-sample-size vs. large-fea ture-size issue. In the following sections, we will
+discuss each category in detail.
+3.1 Ensemble FeatureSelection
+In statistics and machine learning, ensemble learning meth ods combine multiple learned models under the
+assumption that“two(ormore)headsarebetterthanone”. Ty picalensemblelearningmethodssuchasbagging
+[19] and boosting [20] have been widely used in classificatio n and regression. Ensemble feature selection
+techniques use atwo-step procedure that is similar toensem ble classification and regression:
+31. Creating aset of different feature selectors.
+2. Aggregating the results of component selectors togenera te the ensemble output.
+The second step is typically modeled as a rank aggregation pr oblem. Rank aggregation combines multiple
+rankings intoaconsensus ranking, whichhasbeenwidelystu died inthecontext ofwebsearchengines [21]. In
+most cases, the strategies rely on thefollowing informatio n:
+•Theordinal rank associated witheach feature.
+•Thescore assigned to each feature.
+To date, many rank aggregation approaches have been propose d and the reader is referred to [14] for a
+survey of popular aggregation methods used inbioinformati cs.
+Both theoretical and experimental results have suggested t hat the generation of a set of diversecomponent
+learners is one of the keys to the success of ensemble learnin g [22]. To construct diverse local learners, two
+strategies are widely used: data perturbation and function perturbation.
+3.1.1 Data Perturbation
+Dataperturbation tries torun component learners withdiff erent sample subsets (e.g., Bagging [19], Boost-
+ing[20])orindistinct featuresubspaces (e.g.,RandomSub space [23]). Inensemblefeatureselection withdata
+perturbation, different samplings of the original data are generated to construct different feature selectors, as
+described in Fig.2. Several recent methods [24, 25, 26, 27] f all into this category. These methods can be fur-
+ther distinguished according to the sampling method, the co mponent feature selection algorithm and the rank
+aggregation method (see Table 1).
+Thecombination ofdatasamplingandensemblelearningforf eatureselectionisprobablythemostintuitive
+ideatohandle selection instability withrespect tosampli ngvariation. Thesuperiority ofsuchstrategy hasbeen
+verified both experimentally [24,27] and theoretically [25 ,26].
+Interestingly, all methods listed in Table 1 are based on the same aggregation scheme, i.e., linear combina-
+tion. Note that it is also feasible to combine data perturbat ion with other complicated aggregation procedures
+such asthose ones used infunction perturbation (see next su bsection).
+3.1.2 FunctionPerturbation
+Hereweusefunctionperturbation torefertothoseensemble featureselectionmethodsinwhichthecompo-
+nentlearnersaredifferentfromeachother. Thebasicideai stocapitalizeonthestrengthsofdifferentalgorithms
+to obtain robust feature subsets.
+Function perturbation isdifferent from data perturbation in twoperspectives:
+•It uses different feature selection algorithms rather than the same feature selection method.
+4Original DataSample 1
+Sampling
+Sampling
+Sample k
+Sampling…Feature
+SelectionF1
+Feature
+SelectionF2
+Feature
+SelectionFkAggregatingF~Sample 2
+Figure 2: Ensemble feature selection using data perturbati on. We first use different sub-samplings of training
+data toselect features and then build aconsensus output wit harank aggregation method.
+•It typically conducts local feature selection on the origin al data (without sampling).
+Existingensemblefeatureselectionmethodsinthiscatego ry[30,31,32,33]differmainlyintheaggregation
+procedure:
+•Thedistance synthesis method isused in[30].
+•TheMarkov chain based rank aggregation method [21] is utili zed in [31].
+•Thelinear combination method is used in[32].
+•Theconcept of stacking [34] isapplied to the aggregation of feature selection results in [33].
+Compared to data perturbation, function perturbation is le ss flexible since the ensemble scale is limited by
+the number of available feature selection algorithms in the system. As a result, no more than four component
+feature selectors are used in the ensemble learning process [30,31, 32,33].
+3.2 FeatureSelection withPriorFeatureRelevance
+In most biomarker discovery applications, wetypically ass ume that all features are equally relevant before
+theselection procedure. Inpractice, someprior knowledge maybeavailable tobiastheselection towardssome
+features assumed to be more relevant [35, 36]. It has been sho wn that the use of prior knowledge on relevant
+features induces alarge gain in stability with improved cla ssification performance [35].
+5Table1: Classification ofdataperturbation basedensemble featureselection methods. Herelinear combination
+methods aggregate rankings using the (weighted) min,max, orsumoperation. The filter method is a general
+feature selection strategy that attempts torank features s olely according totheir relevance totarget class.
+Reference Sampling Method Feature Selector Aggregation Me thod
+Davis et al [24] Random Subset Filter Method Linear Combinat ion
+Bach [25] Boostrap Lasso[28] Linear Combination
+Meinshausen and Buhlmann [26] Random Subset Randomized Las so Linear Combination
+Abeel et al [27] Random Subset SVM-REF[29] Linear Combinati on
+Fig.3showsthatthereareseveralmethodsforobtainingsuc hkindofpriorknowledge. Onefeasiblemethod
+is to seek advices from domain experts or relevant publicati ons. For instance, in gene expression data classifi-
+cation, one biologist mayknow or guess that some genes are li kely to bemore relevant [35].
+Another more interesting method is to obtain such prior know ledge from relevant data sets using transfer
+learning [36]. Transferlearningfocuses onextracting kno wledgefromsourcetaskandapplying ittoadifferent
+but related task [37]. In [36], those features that have been identified as markers from other data sets are
+considered tobe more relevant inthe new feature selection t ask.
+Thoughthepriorknowledgeishelpfulinimprovingthestabi lityoffeatureselection,usingsuchinformation
+deserves certain limitations since biomarker discovery ai ms at finding new features rather than knownones.
+3.3 GroupFeature Selection
+One motivation for group feature selection is that groups of correlated features commonly exist in high-
+dimensional data, and such groups areresistant tothe varia tions of training samples [16]. If each feature group
+is considered as acoherent single entity, potentially wema yimprove the selection stability.
+Existing group feature selection algorithms follow the pro cedure described in Fig. 4. There are two key
+steps: group formation and group transformation.
+Groupformation istheprocess ofidentifying groups ofasso ciated features. Therearetypically twoclasses
+ofmethodsforthispurpose: knowledge-drivenmethodsandd ata-drivenmethods. Theknowledge-drivengroup
+formation methodutilizes domain knowledge tofacilitate t hegeneration offeature groups. Forexample, genes
+normally function in co-regulated groups, making it feasib le to search genes in the same pathway for group
+identification. Incontrast, thedata-drivengroupformati onmethodfindsfeatureclustersusingonlyinformation
+contained inthe input data.
+Group transformation generates a single coherent represen tation for each feature group. The transforma-
+tion method can range from simple approaches like feature va lue mean [38] to complicated methods such as
+principal component analysis [39].
+6Relevant Data Set
+Prior Feature 
+RelevanceTransfer Learning
+f1f2f3f4f5f6f7f8f9f10 f11f12f13 f14 f15f9,f11
+f3Domain Knowledge
+Figure 3: Stable feature selection using prior knowledge on features. Theprior knowledge onrelevant features
+are either obtained from domain experts or extracted from re levant data sets via transfer learning.
+Feature Group
+TransformationFeature
+SelectionDomain
+Knowledge
+Input 
+DataFeature Group
+GenerationResult
+Figure4: Ageneric groupfeature selection framework. Inth efirststep, weidentify feature groups usingeither
+knowledge-driven methodsordata-driven approaches. Inth esecondstep,wetransform eachfeaturegroupinto
+a single entity. Finally, weconduct feature selection inth e transformed space.
+7Protein-Protein Interaction network
+Features
+f1f2f3f4f5f6
+Case Controls1
+s2
+s3
+s4
+s5
+s6Samplesg1g2g3g4
+s1
+s2
+s3
+s4
+s5
+s6Transformed FeaturesIn this example, we assume that the new feature 
+g1is obtained from the output of a transformation 
+function with a group of features f4, f5and f6as
+input.
+In the most simple case, we can calculate g1as:
+g1= (f4+ f5+ f6) / 3.
+Figure 5: Anillustration of knowledge-driven group format ion and feature transformation. Thepathway infor-
+mation is used to guide the search of correlated features (ge nes or proteins). Each identified feature group is
+transformed into anew feature for further analysis.
+In the following subsections, we will discuss existing grou p feature selection methods according to their
+group formation strategies.
+3.3.1 Knowledge-Driven Group Formation
+Recentadvances intheconstruction oflarge proteinnetwor ks makeitfeasible tofindgenesorproteins that
+have coherent expression patterns inthesamepathway1. Usingavailable protein-protein interaction (PPI)net-
+works, anumber of approaches have been proposed toincorpor ate thepathway information into thebiomarker
+discovery procedure. As shown in Fig.5, the basic idea is to fi nd a group of associated genes or proteins from
+thesamepathway, andthentransform thisgroup intoanewent ity for subsequent feature selection andclassifi-
+cation. Ithasbeenshownthatsuchknowledge-based methodi scapableofachieving morereproducible feature
+selection and higher accuracy [40].
+Wecanfurtherdistinguishavailablemethodsinthiscatego ryaccordingtotheirtargetdata: geneexpression
+data and proteomics data.
+Before summarizing existing approaches on biomarker disco very using gene pathways, we would like to
+discuss a closely related problem: gene set significance tes ting. Testing the statistical significance of gene
+pathways or clusters has been extensively investigated. We ll-known examples include the gene set enrichment
+1For simplicity, this paper uses the terms “gene ontology”, “ pathway”, and “gene set” interchangeably, although they ma y not be
+strictlyequivalent.
+8analysis [41] and the maxmean approach [42]. The reader is re ferred to [43, 44] for comprehensive reviews
+of existing approaches on gene set analysis. Here we highlig ht the fact that gene set significance testing is
+different from pathway-guided biomarker discovery since t hey havedifferent objectives. Theobjective of gene
+set significance testing is to find whether a given gene set sat isfies the hypothesis, while biomarker discovery
+aims at searching for asmall subset of genes that can disting uish cases from controls as accurately aspossible.
+Their intrinsic connection is that we can utilize pathway si gnificance assessment method as a filter method (a
+special typeoffeatureselectiontechniquethatconsiders eachentityindividually) forrankingpathwaymarkers.
+Some methods have been proposed to identify markers not as in dividual genes but gene sets [38, 39, 40,
+45, 46, 47, 48, 49, 50]. In Table 2, we provide a brief summary o f existing biomarker identification methods
+that use gene pathway information as prior knowledge. These methods exploit different strategies for group
+generation and transformation. In group generation, we can use all genes in the pathway for a clear biological
+interpretation. Alternatively, we can search for a subset o f genes so as to obtain one more discriminating
+group. To effectively represent each group, various summar y statistics have been applied, ranging from mean
+to principal component analysis.
+Table 2: Summary of gene pathway biomarker discovery method s. In the generation of gene groups, weeither
+accept all genes in agiven pathway or use heuristic search me thods (such as greedy algorithm) tofind asubset
+ofdiscriminatinggenes. Here“Notransformation”meansth atweuseallthegenesinthegrouptorepresentthis
+gene set. GXNA(Gene eXpression Network Analysis) is asoftw are package developed in [51] for identifying
+a subset of differentially expressed genes from agiven path way.
+Reference Group Generation Group Transformation
+Guoet al [38] Useall genes Meanand median
+Rapaport et al [39] Useall genes Principal component analys is
+Chuang et al [40] Greedy search Sum of z-scores
+Tai and Pan[45] Useall genes Notransformation
+Leeet al [46] Greedy search Sum of z-scores
+Hwangand Park[47] Greedy search Mean
+Yousef et al [48] GXNA[51] Notransformation
+Chenand Wang [49] Useall genes Principal component analysi s
+Suet al [50] Useall genes Sumof log-likelihood ratio
+Recently, such knowledge-driven approach has also been app lied to proteomics data for biomarker discov-
+ery at the level of protein group [52, 53]. Compared to gene ex pression data, more research efforts toward this
+direction are desired in future research.
+The pathway-guided group formation method has the advantag e that new transformed features are bio-
+logically interpretable since the underlying disease proc ess may be dependent on perturbations of different
+9pathways. Thus, prediction models based on pathways may app roximate the true disease process more closely
+than gene-based models[49]. Itsmaindisadvantage isthat w emaygroup unrelated genes or proteins since the
+reliability of the predicted interactions inPPInetwork is still questionable [54].
+3.3.2 Data-Driven Group Formation
+Insteadofrelyingonprior knowledgeofbiology, thedata-d riven groupformation methodidentifiesfeature
+clusters using either cluster analysis [55, 56, 57, 58, 59, 6 0, 61, 62, 63] or density estimation [8, 16]. As sum-
+marizedinTable3,clustering-based methodsutilizepopul arpartitionalgorithmssuchashierarchical clustering
+ork-means to generate feature groups. It should be noted that mo st existing clustering-based methods do not
+explicitly consider the stability of feature group. Altern atively, kernel density estimation is utilized in [8, 16]
+based on the observation that dense core regions are stable r espect tosamplings of dimensions.
+Table 3: Summary of clustering-based group feature selecti on methods according to clustering algorithms and
+group transformation methods.
+Reference Clustering Methods Group Transformation
+Hastie et al [55] Hierarchical clustering Mean
+Jornsten and Yu[56] Integrated clustering and group select ion Mean
+Auet al [57] K-modes Onemost discriminating feature
+Maet al [58] K-means Asubset of most discriminating features
+Maand Haung [59] Hierarchical clustering/ K-means Asubset of most discriminating features
+Yousel et al [60] K-means Notransformation
+Parket al [61] Hierarchical clustering Mean
+Shin et al [62] Hierarchical clustering Onemost discrimina ting feature
+Tanget al [63] Fuzzy k-means Notransformation
+There is another class of related methods assigning compara ble coefficients to correlated, important vari-
+ables. The “elastic net” [64] is a typical example in this cat egory. We omit these methods in this survey since
+they didn’t explicitly identify feature groups.
+The data-driven group feature selection method fully explo its the characteristics of target data so that it is
+widely applicable. One main drawback is that it is not easy to interpret and validate the selected feature group
+biologically. One possible remedy is to use a hybrid strateg y that combines the data-driven method with the
+knowledge-driven method, asrecently discussed in [65,66] .
+103.4 FeatureSelection withSampleInjection
+Inbiomarker discovery applications, thenumber of feature s istypically larger than thesample size. Thisis
+one of the main sources of instability in feature selection. To increase the reproducibility of feature selection,
+one natural idea is to generate more samples. However, the ge neration of real sample data from patients and
+healthy people is usually expensive and time-consuming. Wi th this practical limitation in mind, people begin
+to seek other alternative methods for the samepurpose.
+Fromthe viewpoint of data analysis, there aretwo data augme ntation strategies:
+•Utilizing test data to increase the sample size in feature se lection process, which can be modeled as a
+transductive learning problem [67].
+•Generating some artificial training samples according to th edistribution of available training data.
+Inthe following sections, wewill introduce each method ind etail.
+3.4.1 Method UsingTransductive Learning
+Different frominductive learning algorithms, thetransdu ctive learning algorithm isnotrequired toproduce
+a general hypothesis that can predict the label of any unobse rved data [67]. As illustrated in Fig. 6, it is only
+required to predict the labels of a given test set of samples. In other words, we can use both training data and
+testing data inthe learning procedure.
+Transductive learning has been used to increase sample size in some recent papers [68, 69]. The main idea
+istotakeadvantageoftheinformationembeddedinthetestd atasothattheroleoftestsamplesischangedfrom
+passivetoactive. Thatis,theunlabeledtestsamplesareincorporatedintot hefeatureselectionandclassification
+process.
+3.4.2 Method UsingArtificial Training Samples
+Theideaistogenerateanumberofartificialtrainingsample saccordingtothedistribution ofgivensamples.
+Then, feature subsets can be assessed using both the generat ed data and the original data. In Fig.7, we provide
+an example to illustrate the effect of injected artificial sa mples on model selection.
+There are many methods for generating artificial training sa mples. For instance, we can first pick one
+trainingsample xirandomlyandthengenerateapoint zfromstandardnormaldistribution. Finally,wegenerate
+the new artificial point as: y=xi+hz, wherehisaconstant.
+There are twoways in which the artificial points participate in feature selection. Onemethod is to treat the
+injected points astheoriginal samplesinthetraining proc ess [70]. Another methodistousetheinjected points
+only inthe evaluation stage [71].
+11Inductive Learning
+Distribution
+of ExamplesTraining Data 
+Hypothesis LabelsUnlabeledExamples
+Transductive Learning
+Training Data
+Labels of
+Test SamplesLearning
+AlgorithmLearning
+Algorithm
+Unlabeled Test Data
+Figure 6: An illustration of inductive learning and transdu ctive learning. Intuitively, wecan consider inductive
+learning as “an education system for all-round development ” while transductive learning is an “examination-
+oriented education system”.
+(a) Before Point InjectionCase ControlArtificial Samples
+(b) After Point Injection
+Figure7: Theeffect ofartificial training samplesonmodel s election. Theseparating hyperplane obtained from
+the data set with injected samples isdifferent from that of t heoriginal samples.
+124 Stability Measure
+In stable feature selection, one important issue is how to me asure the “stability” of feature selection algo-
+rithms, i.e., how to qualify the selection sensitivity to va riations in the training set. The stability measure can
+be used in different contexts. Onthe one hand, it is indispen sable for evaluating different algorithms in perfor-
+mance comparison. Ontheother hand, it canbeused for intern al validation infeature selection algorithms that
+take into account stability.
+Noticing that there is already a nice review paper [14] on the stability of ranked gene lists, here we would
+like toprovide amore comprehensive list that includes eval uation methods from different domains.
+Measuring stability requires a similarity measure for feat ure selection results. This depends on the repre-
+sentations used by different algorithms. Formally, let tra ining examples be described by a vector of features
+F= (f1,f2,...,fm), then there arethree types of representation methods [72]:
+•Asubset of features: S={s1,s2,..,sk},si∈ {f1,f2,...,fm}.
+•Aranking vector: R= (r1,r2,..,rm),1≤ri≤m.
+•Aweighting-score vector: W= (w1,w2,..,wm),wi∈R+.
+Ingeneral,weareinterestedinstabilitymeasuresthattak emorethantwosubsets(orrankings)intoaccount.
+In this review, we use measures defined on two subsets (or rank ings) for the sake of notation simplification.
+As pointed out in [14], there are essentially two approaches for generalizing the definition. One approach is
+to summarize pairwise stability measures through averagin g. Another approach is to consider all subsets (or
+rankings) simultaneously inthe specification of stability measure.
+In the following, wewill summarize available stability mea sures according to the representation of feature
+selection results.
+4.1 FeatureSubset
+There is a wide variety of similarity measures available for the comparison of sets. Table 4 summarizes
+availablestabilitymeasures. Onemayfindthatmostofthese measuresaredefinedusingthephysical properties
+of two sets, e.g., the ratio of the intersection to the union. One exception is the “percentage of overlapping
+featuresrelated”[13],whichincorporatesadditional fea turecorrelationinformationintothemeasuredefinition.
+This is definitely plausible for biomarker discovery applic ations since there are always some highly correlated
+features inthe “omics” data.
+MS1: The relative Hamming distance between the masks correspon ding to two subsets is used to measure
+the stability [73].
+MS2: The Tanimoto distance metric measures the amount of overla p between two sets of arbitrary cardi-
+nality. It takes values in [0,1], with 0 meaning no overlap be tween the two sets and 1 meaning two sets are
+identical. In fact, this measure is equivalent tothe Jaccar d’s index:|S∩S′|
+|S∪S′|.
+13Table 4: A list of stability measures when the feature select ion algorithm produces a subset of features as
+output. Here SandS′are twosubsets of features.
+Index Description Formula
+MS1[73] Relative Hammingdistance 1−|S\S′|+|S′\S|
+m
+MS2[72, 74,75] Tanimoto distance/Jaccard’s index 1−|S|+|S′|−2|S∩S′|
+|S|+|S′|−|S∩S′|
+MS3[8,11,16] Dice-Sorensen’s index2|S∩S′|
+|S|+|S′|
+MS4[11] Ochiai’s index2|S∩S′|√
+|S||S′|
+MS5[76] Percentage of overlapping features|S∩S′|
+|S|,|S∩S′|
+|S′|
+MS6[13] Percentage of overlapping features related|S∩S′|+c12
+|S|,|S∩S′|+c21
+|S′|
+MS7[77] Kuncheva’s stability measure|S∩S′|m−c2
+c(m−c)
+MS8[78] Consistency1
+|S∪S′|/summationtext
+f∈S∪S′freq(f)−1
+m−1
+MS9[78] Weighted consistency/summationtext
+f∈S∪S′(freq(f)
+|S|+|S′|·freq(f)−1
+m−1)
+MS10[24] Length adjusted stability max{0,/summationtext
+f∈S∪S′(freq(f)
+2|S∪S′|−α|S|+|S′|
+2m)}
+MS3: TheDice-Sorensen’s index isthe harmonic mean of|S∩S′|
+|S|and|S∩S′|
+|S′|.
+MS4: The Ochiai’s index is the geometric mean of|S∩S′|
+|S|and|S∩S′|
+|S′|. It has been shown that the perfor-
+mance of the Ochiai’s index issimilar with that of Jaccard’s index and Dice-Sorensen’s index [11].
+MS5: This measure is originally named as: “Percentage of Overla pping Genes (POG)” in the context of
+gene expression data analysis.
+MS6: It is an extension of POG, which incorporates highly correl ated features between two sets into the
+stability evaluation. In the formula, c12(orc21) denotes the number of features in S(orS′) that are not shared
+but are significantly positively correlated with at least on e feature in S′(orS). The normalized form of this
+measure isalso presented in[13].
+MS7: Thisstability measure assumes that SandS′have the same size (cardinality), i.e., |S|=|S′|=c.
+MS8 and MS9 : In both definitions, freq(f)denotes the number of occurrences (frequency) of feature f
+inS∪S′. It has been proved that both measures take values in [0,1]. T he (weighted) consistency value is 1 if
+two sets areidentical and 0if they aredisjoint.
+MS10: In the formula, αis one user-specified parameter and is set to 10 [24]. Note tha t(|S|+|S′|)/2
+corresponds to the median required in[24] since there are on ly two sets inour formulation.
+4.2 Ranking List
+The problem of comparing ranking lists is widely studied in d ifferent contexts such as voting theory and
+web document retrieval. Table 5 shows some distance measure s for two ranking lists. One typical example is
+14MR2, in which the Spearman’s correlation is adapted to place moreweights on those top ranked features since
+these features are more important than irrelevant features in thestability evaluation.
+Table 5: A list of stability measures when the feature select ion algorithm generates a ranking list as output.
+HereRandR′are twodifferent ranking lists.
+Index Description Formula
+MR1[72] Spearman’s rank correlation coefficient 1−6m/summationtext
+i=1(ri−r′
+i)2
+m(m2−1)
+MR2[12] Canberra distancem/summationtext
+i=1|min{ri,k+1}−min{r′
+i,k+1}|
+min{ri,k+1}+min{r′
+i,k+1}
+MR3[79] Overlap scorem/summationtext
+i=1e−αi|{rj|j < i}∩{r′
+j|j < i}|
+MR1: The Spearman’s rank correlation coefficient takes values i n [-1,1], with 1 meaning that the two
+ranking lists are identical and avalue of -1meaning that the y have exactly inverse orders.
+MR2: The Canberra distance is a weighted version of Spearmans fo otrule distance [12], i.e.,m/summationtext
+i=1|ri−r′
+i|
+ri+r′
+i.
+Since themost important features areusually located at the top ofthe ranking list [12],thedistance calculation
+in the table only considers top kranked features.
+MR3: The overlap score is originally proposed in [79] and here we follow [14] to reformulate it with the
+assumption that only top ranked features are important. In t he formula, αis a user-specified parameter to
+control the decreasing rate.
+4.3 Weighting-ScoreVector
+The computational issue of combinatorial search for featur e subset can to some extent be alleviated by
+usingafeature weightingstrategy[80]. Allowingfeaturew eightstotakereal-valued numbersinsteadofbinary
+ones enables us to use well-established optimization techn iques in algorithmic development. For instance, the
+RELIEFalgorithm[81]isonerepresentative ofsuchkindofm ethods,whichgeneratesaweightingscorevector
+as output.
+Theweighting score vector isseldom usedindefining stabili ty measure. Table6listsonestability measure
+MW1 [72]. The Pearson’s correlation coefficient ranges from -1 to 1. A value of 1 indicates a perfect positive
+correlation,avalueof0meansthatthereisnocorrelation, whileavalueof-1meansthattheyareanti-correlated.
+In the formula, uWanduW′are the means of weight scores of WandW′,respectively.
+5 Discussions
+Wesummarizethreesourcesofinstability forfeatureselec tioninsection2. Amongthesesources, probably
+thesmallnumber ofsamplesinhighdimensional featurespac eisthemostdifficultoneinbiomarker discovery.
+15Table 6: The Pearson’s correlation coefficient measure. Her eWandW′are two different weighting score
+vectors.
+Index Description Formula
+MW1[72] Pearson’s correlation coefficient/summationtext
+i(wi−uW)(w′
+i−uW′)/radicalBig/summationtext
+i(wi−uW)2/summationtext
+i(w′
+i−uW′)2
+Besidesfeatureselection, otherdataanalysistasksalsof acethesamechallenges. Researchprogressesinrelated
+fields will facilitate the development of effective stable f eature selection methods aswell.
+Group feature selection is the most extensively studied met hod among existing stable feature selection ap-
+proaches. Thisis because there are manycorrelated feature s inhigh dimensional space. However, such feature
+grouping strategy can only partially alleviate selection i nstability since westill need toface the reproducibility
+issueinthetransformedspace. Inthisregard,ensemblefea tureselectionisprobablymorepromisingtoprovide
+ageneral-purpose solution. Oneimmediatehybridstrategy istocombinegroupfeatureselectionwithensemble
+feature selection, i.e., firstperform feature grouping and thenuse ensemble feature selection inthenew feature
+space.
+The group feature selection strategy is only helpful when mu ltiple sets of true markers are generated due
+to the existence of redundant features. However, it is also p ossible that multiple sets of true markers share no
+correlated features. The feature selection problem in this case is much harder than finding a minimal optimal
+feature set for classification [82]. Toour knowledge, there is still no available method and measure that aim at
+handling stability issues inthis context. Thegeneral prob lem isopen and needs more research efforts.
+With respect to stability index, most available measures ar e defined over feature subsets since the feature
+subset can be obtained from rankings or scores (but not vice- versa). The major problem is that there is still
+no consensus on the best stability measure. Therefore, acom prehensive comparison study on existing stability
+measures should beconducted infuture research.
+Infact, the biomarker discovery process involves manyproc edures. Hereweonly discuss feature selection
+techniques for stable biomarker identification. The develo pment of biomarker classifier is also very important.
+The readers arereferred toarecent review [83] for research progress towards this direction.
+Finally,wewouldliketoraisethefollowingquestions inth epursuit ofstablebiomarker discoverymethods
+for future research:
+•Howtodirectly measure the stability of feature(s) without sampling training data?
+•Can we propose new methods that are capable of explicitly con trolling the stability of reported feature
+subset?
+•Arethere other special requirements for biomarker discove ry rather than stability?
+166 Conclusions
+To discover reproducible markers from “omics” data, the sta bility issue of feature selection has received
+much attention recently. This review summarizes existing s table feature selection methods and stability mea-
+sures. Stable feature selection is a very important researc h problem, from both theoretical perspective and
+practical aspect. More research efforts should be devoted t o this challenging topic.
+Acknowledgments
+This work was supported with the general research fund 62170 7 from the Hong Kong Research Grant
+Council, aresearch proposal competition awardRPC07/08.E G25andapostdoctoral fellowship from theHong
+Kong University of Science and Technology.
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+arXiv:1001.0888v1  [astro-ph.SR]  6 Jan 2010Mem.S.A.It.Vol., 1
+c/circlecopyrtSAIt 2008 Memorie della
+Numericalsimulationsofthequietchromosphere
+Jorrit Leenaarts1
+Astronomical Institute - Utrecht University - PO Box 80000 N L–3508 TA Utrecht,
+The Netherlands e-mail: j.leenaarts@uu.nl
+Abstract. Numerical simulations of the solar chromosphere have becom e increasingly
+realistic over the past 5 years. However, many observed chro mospheric structures and be-
+havior are not reproduced. Current models do not show fibrils in CaII 8542 Å, and neither
+reproduce the CaII 8542 Å bisector. The emergent H αline core intensity computed from
+the models show granulation instead of chromospheric shock s or fibrils. I discuss these
+deficiencies and speculate about what physics should be incl uded to alleviate these short-
+comings.
+Key words. Sun: chromosphere, Methods: numerical
+1. Introduction
+The solar chromosphere is in many ways the
+most complicated and most interesting region
+ofthesolar atmosphere.
+The chromosphere is the interface layer
+between the convection zone /photosphere and
+the corona where the value of the plasma
+β-parameter changes from larger to smaller
+than unity. Consequently, its overall charac-
+ter changes from a convective overshoot layer
+into a regime where the three-dimensional
+(3D) magnetic topology determines the struc-
+ture. The chromosphereis convectivelystable,
+so the dominant input of mechanical energy
+comes in the form of waves excited by gran-
+ules and the buffeting of magnetic concentra-
+tions in the photosphere. It also absorbs ra-
+diation from both the photosphere ( e.g.,H−,
+Balmer-continuum) and the corona (He con-
+tinua). Thermal conduction in the corona has
+a strong influence on the shape of the transi-
+Send offprint requests to :J. Leenaartstion region, and hence of the upper chromo-
+sphere. The dominant chromospheric energy
+loss is through radiation in strong lines. Thus,
+a comprehensive model of the chromosphere
+cannot be made without modeling the under-
+lying photosphere and upper convection zone
+andtheoverlyinglowercorona.
+The wide range of pertinent physical pro-
+cesses require that a comprehensive descrip-
+tion of the chromosphereneeds to incorporate
+acarefulcombinationofatomicandmolecular
+physics, radiative transfer, MHD and plasma
+physics.Thiscomplicatedphysicsrequiresnu-
+mericalmodeling.
+From the above discussion I conclude that
+a “minimallysatisfyingself-consistentmodel”
+ofthechromosphereshould
+–betime-dependent;
+–bethree-dimensional;
+–modelradiativeheatingandcooling;
+–have a lower boundary in the convection
+zone,with a boundaryconditionthat feeds
+enoughenergyintothesimulationtomain-2 Leenaarts: Simulations ofthe quiet chromosphere
+tain a the radiative energy loss at the solar
+effectivetemperature;
+–have an upper boundary in the corona. If
+the model cannot self-consistently main-
+tain a corona, it must provide enough en-
+ergyfluxtokeepplasmaneartheboundary
+at coronaltemperatures.
+Within these constraints one can add features
+and physics to the model to increase its real-
+ism, while at the same time trying to keep the
+numberoffreeparametersto aminimum.
+The above list does not imply that mod-
+elswithoutthosepropertiesaresomehow“bad
+models”. To the contrary, simpler models are
+often instrumental in identifying and charac-
+terizing relevant physical processes. However,
+athoroughunderstandingofthechromosphere
+ultimatelyrequiresaself-consistentmodelthat
+reproducesall observedchromosphericbehav-
+ior.
+2. Currentstatus
+The current state-of-the-art in self-consistent
+chromospheric modeling is the modeling
+performed with the radiation-magneto-
+hydrodynamics (RMHD) Oslo Stagger
+Code (OSC Hansteen 2004) and the MPI-
+parallelized Bifrost code currently under
+development(Gudiksenet al.2010).
+Thesecodesarebasedonthemethodspio-
+neered by Nordlund (1982) in 3D convection
+simulations. They use the single-fluid MHD
+approach to model the solar gas and mag-
+netic field. Radiative transfer in the photo-
+sphere and lower chromosphere is treated us-
+ing multi-group opacity binning with scatter-
+ing (Skartlien 2000) with typically 4 radi-
+ation bins. This multi-group scheme is not
+well suited to model the few strong chromo-
+spheric lines that cause the dominant radia-
+tive energy losses in the upper chromosphere.
+Therefore both codes employ parameterized
+radiative cooling in those lines in the up-
+per chromospherebased on the detailed radia-
+tive transfer in a 1D dynamicalchromospheric
+model (computed with the RADYN code, see
+e.g.,Carlsson& Stein 2002). Radiative losses
+in the corona are computed using an opti-callythinradiativelossfunction.Heatconduc-
+tion parallel to the magnetic field is included.
+Two-dimensional simulations cannot maintain
+a corona by themselves and require addi-
+tional heating at the upper boundary. Three-
+dimensional models however self-consistently
+maintaina coronabyJoule heating.
+Similar codes exist, such as Muram
+(V¨ ogleret al.2005),CO5BOLD(Freytaget al.
+2002; Wedemeyeret al. 2004) and RADMHD
+(Abbett 2008). However these codes so far do
+not support a corona (CO5BOLD, Muram) or
+lack detailed treatment of the radiation field
+(RADMHD).
+OSC has been used to study dynamic
+fibrils (DFs) by Hansteenet al. (2006) and
+De Pontieuet al. (2007a). They conclude that
+DFs are driven by magneto-acoustic shocks
+based on the similarity in dynamic behavior
+of wave-guidedshocksin a 2D simulationand
+observations of DFs in the core of H α. Their
+work has been extended to 3D in simulations
+byMart´ ınez-Sykoraet al. (2009b).
+3D simulations of flux emergence from
+the convection zone through the chromo-
+sphere into the corona were performed by
+Mart´ ınez-Sykoraetal. (2008, 2009a) A simi-
+larsimulationshowedthepresenceofAlfv´ enic
+waves with properties very much like what
+has been observed with the Hinode spacecraft
+(DePontieuetal. 2007b).
+For a good overview of 3D solar atmo-
+sphere modeling and post-processing in the
+form of 3D radiative transfer I refer the
+readertoCarlsson(2008,2009)andreferences
+therein.
+Despite the impressive progress that has
+been madein the last 5 years,many,oftensur-
+prisinglybasic propertiesof the chromosphere
+are not yet reproduced in the models. In the
+next section I discuss some of those observa-
+tionalfactsandspeculateaboutwhatismissing
+inthe models.Leenaarts: Simulations ofthe quiet chromosphere 3
+3. Open questionsandchallenges
+3.1. Lack ofextended magnetic
+structurein CaII8542Å
+ThefirstthingonenoticesinCaII8542Åline-
+core images are the elongated structures de-
+lineating magnetic field lines around network
+regions. The NLTE CaII 8542 Å computation
+fromasnapshotofa3Dsimulationofthechro-
+mosphere with OSC of Leenaartset al. (2009)
+does, however, not show any such structures,
+despitethepresenceofarelativelystrongmag-
+neticfield.Insteaditshowsthesignatureofos-
+cillations and shocks, both in relatively field-
+free regions as in regions with stronger mag-
+netic fields. Similar structures are observed in
+CaII 8542 Å quiet regionsaway from the net-
+work.Thelack offibrilsinthe simulationisin
+my opinion the most pressing challenge mod-
+elsofthechromospherecurrentlyface.
+There can be many reasons for this lack.
+It might be that such fibrils require large-scale
+magnetic field configurationsthat do not fit in
+thetypicalsizeofthecomputationaldomainof
+16×16×16Mm3.Itmightbethatthesimula-
+tion cannot form fibrils due to a lack of grid
+resolution. Alternatively, fibril-like structures
+mightexist in the models,but the lackof reso-
+lution inhibits mass loading of the fibrils, pos-
+sibly due to too large numerical viscosity (see
+also Sec. 3.2). This lack of mass then trans-
+lates into optically thin fibrils. Another possi-
+bility is that CaII 8542Å opacityin the fibrils
+is caused by physical processes currently not
+modeled in the simulations, for example non-
+equilibrium ionization or plasma e ffects that
+cannot be modeled within the MHD assump-
+tion.
+3.2. Linebroadeningandbisector
+The spatially averaged CaII 8542 Å line pro-
+filecomputedin3DNLTEfromthesameOSC
+snapshot shows a too narrow and too deep
+linecorecomparedtoobservations(seeFigs.2
+and 3 of Leenaartsetal. 2009). These calcu-
+lations did not employ any non-physical mi-
+croturbulence, and had a rather large grid cell
+size of64×64×32km3. Thetoo narrowcorewas attributed to this lack of spatial resolu-
+tion. Indeed, a newer simulation by Carlsson
+(private communication) with a cell size of
+32×32×16km3showsashallowerandwider
+averageCaII 8542Å core, but not yet as wide
+and shallow as observed. The increase in grid
+resolution causes an increase in amplitude of
+the velocity variations in the mid and upper
+chromosphere,andhenceawideningoftheav-
+erageprofile.
+This result suggests that with even higher
+resolution the width of the line core might at
+least be qualitatively explained. However, the
+observed profile is at least partially formed in
+fibrils and as long as the models do not repro-
+duce these the core width and depth remains
+notcompletelyunderstood.
+The low resolution simulation also did not
+reproducethe observed inverse C-shape of the
+CaII8542Ålinebisector.Similarresultswere
+obtained earlier by Uitenbroek (2006) who
+computed the NLTE line profiles from both
+a 3D simulation that did not extend into the
+corona and a dynamic 1D model. It would be
+interesting to see whether the bisector shape
+can be reproduced by high-resolution simula-
+tions, or that it is determined by phenomena
+currentlynotpresentinthe simulations.
+3.3. Lackof Hαopacity
+The identification of DFs as magneto-acoustic
+shocks guided along magnetic field lines has
+been established based on the excellent agree-
+ment of dynamical properties observed in H α
+with 2D simulations (Hansteenet al. 2006;
+De Pontieuet al.2007a).
+In a simulation with identical reso-
+lution and magnetic field configuration,
+Leenaartsetal. (2007) investigated the e ffect
+of non-equilibrium hydrogen ionization on
+the chromosphere. They found that the hy-
+drogenn=2 level populations and column
+densities are much higher in the simulated
+DFs than elsewhere in the chromosphere.
+This is consistent with the observations where
+DFs appear dark on top of a brighter, deeper
+formed background. However, a subsequent
+NLTE column-by-column radiative transfer
+computation employing non-equilibrium4 Leenaarts: Simulations ofthe quiet chromosphere
+Fig.1.Vertically emergent H αline core intensity from a 2D radiation-MHD simulation with
+non-equilibrium hydrogen ionization taken into account. S tartup effects are visible the first 3
+minutes. There is a weak imprint of upward propagating shock s, superimposed on a slower-
+evolving pattern caused by granulation and reversed granul ation. No parabolic outlines of DFs
+are presentasintheobservations(seeFig. 7ofDePontieuet al. 2007a).
+Fig.2.Results of a NLTE radiative transfer computation of the H αline from a 3D radiation-
+MHDsimulationwithanLTEequationofstate.Theblacklinei npanelsaandbindicatesthecut
+shown in panel c and d. Panel a: vertically emergent intensit y in the far blue wing; b: vertically
+emergentintensityatlinecenter;c:verticallyemergenti ntensityalongthecutindicatedinpanels
+aandbforthebluewing(grey,left-handscale)andlinecent er(black,righthandscale);d: τ=1
+heightforthebluewing(grey)andlinecenter(black).
+temperatures and electron densities, but as-
+sumingstatisticalequilibriumforthehydrogen
+level-populations,showedthat the upperchro-
+mosphere is transparent in the H αline core in
+thissimulation.
+Fig. 1 showsthe time-evolutionof the ver-
+ticallyemergentHαlinecenterintensity.There
+are magneticfield concentrationsat x=4 Mm
+andx=12 Mm along which DFs run up inthechromosphere.Theirtopsshouldappearas
+paraboliccurvesinthis xt-slice,butinsteadthe
+sliceshowsaslowlyevolvingpatternofgranu-
+lation and reverse granulation. Superimposed
+on this pattern is a weak imprint of upward
+propagating shocks (the diagonal curves ema-
+nating from the magnetic field concentrations,
+forexampletheonestartingat( x,t)=(5.5,10)
+Analysis of the contribution functions showsLeenaarts: Simulations ofthe quiet chromosphere 5
+that this imprint forms below 1 Mm, much
+lower than the typical height of DFs (see Fig.
+16ofDe Pontieuet al.2007a).
+Fig.2showstheresultsofaNLTEcolumn-
+by-column radiative transfer computation of a
+snapshotofa3Dsimulationofflux-emergence
+byMart´ ınez-Sykoraet al.(2008).Thissimula-
+tion used an LTE equation of state unlike the
+2D simulation that used a more realistic non-
+equilibrium equation of state. Panels a and b
+show that the Hαline core shows granulation,
+even though itsτ=1 height lies between 0.3
+and 1.7 Mm (panel d). The line source func-
+tion is strongly scattering and set in the upper
+photosphere, where it is sensitive to the tem-
+peraturepatternofthe granulation.
+Both the 2D non-equilibrium simulation
+and the 3D LTE simulation show an H αline
+center intensity pattern that is set in the upper
+photosphere, in stark contrast to the observa-
+tionswhichshowfibrilarstructureinallbutthe
+quietest areas of the atmosphere.Even in such
+quiet areas the observationsdo not show gran-
+ulation, but signatures of shockwaves instead
+(Ruttenet al.2008).
+Some of the reasons that might cause the
+absenceoffibrilsinthesimulatedCaII8542Å
+line core also apply here: too small compu-
+tational domain, too low spatial resolution or
+the limitations of the MHD assumption. The
+2D simulation necessarily had a rather simple
+magnetic field topology, which might inhibit
+the formation of fibrils and the mass loading
+oftheupperchromosphere.The3Dsimulation
+did not suffer from this, but it had lower reso-
+lution and did not model non-equilibriumion-
+ization. High resolution 3D models with non-
+equilibrium hydrogen ionization might have a
+chromospherethat isopticallythickinH α.
+Hydrogen line formation is however noto-
+riouslydifficulttomodel.Thenon-equilibrium
+ionizationofhydrogendoesnotonlya ffectthe
+electron density and temperature in the chro-
+mosphere, it also has a strong e ffect on hydro-
+gen line formation. The standard NLTE line
+formationassumptionofstatisticalequilibrium
+is notvalidbecauseoftheslow collisionaland
+radiativetransitionratesascomparedtothehy-
+drodynamicaltimescale.Inadditionatleastthe
+Lyman-αndβlines need to be modeled withpartialfrequencyredistribution(PRD)because
+of their strong influence on the H αline. Both
+time-dependentradiative transfer and PRD ef-
+fects were ignored in the results of Figs. 1
+and 2. Inclusion of these e ffects might signif-
+icantlyincreasetheH αopacity.
+Thus, proper 3D modeling of the H αline
+requires full time-dependent radiative transfer
+with PRD in tandemwith the MHD evolution,
+a Herculean task that has not even been done
+in 1D hydrodynamic simulations. So far all
+1D simulations perform radiative transfer
+in complete redistribution (Carlsson& Stein
+2002; Rammacher&Ulmschneider 2003;
+Kaˇ sparov´ aetal. 2009).
+An easier way out is to perform the 3D
+radiation-MHD simulations with approximate
+non-equilibrium hydrogen ionization follow-
+ing the method developed by Sollum (1999)
+as was done in 2D by Leenaartset al. (2007)
+in order to get reasonably accurate values for
+the electron density and temperature. One can
+then perform time-dependent radiative radia-
+tivetransferwithPRD onaseriesofsnapshots
+from the simulation to get the H αline pro-
+files.Thisseparatesthedetailedradiativetrans-
+fer from the MHD computation, which signif-
+icantly reduces the di fficulty of the problem.
+However, even with this simplification it re-
+mains a daunting task to accurately model H α
+lineformationinthedynamicchromosphere.
+4. Summary& conclusions
+Multidimensional radiation-MHD simulations
+of the solar chromosphere are becoming in-
+creasingly realistic. They have, amongst other
+things,beenusedtostudydynamicfibrils,flux
+emergence and the propagation of Alfv´ enic
+wavesintothecorona.
+The modelsyet fail to reproducesome ob-
+servationalfacts:
+–simulatedCaII8542Å coreimagesdonot
+showfibrils;
+–the averagesimulated CaII 8542Å core is
+deeper and narrower than observed and its
+bisector not show the observed inverse C-
+shape;
+–the simulated Hαline core shows granu-
+lation, both in 2D simulations with non-6 Leenaarts: Simulations ofthe quiet chromosphere
+equilibriumeffectstakenintoaccountsand
+in3Dwith anLTEequationofstate.
+In order to reproduce these the models
+probablyrequirehigherresolution,largercom-
+putational domains, an improved treatment of
+radiation and non-equilibrium hydrogen ion-
+ization and inclusion of plasma e ffects. In or-
+dertomodelhydrogentransitions,inparticular
+Hα, 3D NLTE time-dependentradiative trans-
+fercodesincludingPRDneedtobedeveloped.
+Acknowledgements. J.L. acknowledges financial
+support by the European Commission through
+the SOLAIRE Network (MTRN-CT-2006-035484)
+and the Netherlands Organisation for Scientific
+Research (NWO).
+References
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+arXiv:1001.0889v1  [cs.CG]  6 Jan 2010Asynchronous deterministic rendezvous in bounded
+terrains
+Jurek Czyzowicz∗†David Ilcinkas‡Arnaud Labourel∗§Andrzej Pelc∗¶
+Abstract
+Two mobile agents (robots) have to meet in an a priori unknown boun ded terrain modeled
+as a polygon, possibly with polygonal obstacles. Agents are modeled as points, and each of
+them is equipped with a compass. Compasses of agents may be incohe rent. Agents construct
+their routes, but the actual walk of each agent is decided by the ad versary: the movement of the
+agent can be at arbitrary speed, the agent may sometimes stop or go back and forth, as long as
+the walk of the agent in each segment of its route is continuous, doe s not leave it and covers all
+of it. We consider several scenarios, depending on three factors : (1) obstacles in the terrain are
+present, or not, (2) compasses of both agents agree, or not, ( 3) agents have or do not have a map
+of the terrain with their positions marked. The cost of a rendezvou s algorithm is the worst-case
+sum of lengths of the agents’ trajectories until their meeting. Fo r each scenario we design a
+deterministic rendezvous algorithm and analyze its cost. We also pro ve lower bounds on the
+cost of any deterministic rendezvous algorithm in each case. For all scenarios these bounds are
+tight.
+keywords: mobile agent, rendezvous, deterministic, polygon, obstacle
+∗D´ epartement d’informatique, Universit´ e du Qu´ ebec en Ou taouais, Gatineau, Qu´ ebec J8X 3X7, Canada. E-mails:
+jurek@uqo.ca ,labourel.arnaud@gmail.com ,pelc@uqo.ca
+†Partially supported by NSERC discovery grant.
+‡LaBRI, Universit´ e Bordeaux I, 33405 Talence, France. E-ma il:david.ilcinkas@labri.fr
+§This work was done during this author’s stay at the Universit ´ e du Qu´ ebec en Outaouais as a postdoctoral fellow.
+¶Partially supported by NSERC discovery grant and by the Rese arch Chair in Distributed Computing at the
+Universit´ e du Qu´ ebec en Outaouais.1 Introduction
+The problem and the model. Twomobileagents (robots) modeled aspoints startingat diffe rent
+locations of an a priori unknown bounded terrain have to meet . The terrain is represented as a
+polygon possibly with a finite number of polygonal obstacles . We assume that the boundary of the
+terrain is included in it. Thus, formally, a terrain is a set P0\(P1∪···∪P k), where P0is a closed
+polygon and P1,...,Pkare disjoint open polygons included in P0. We assume that an agent knows
+if it is at an interior or at a boundary point, and in the latter case it is capable of walking along the
+boundary in both directions (i.e., it knows the slope(s) of t he boundary at this point). However,
+an agent cannot sense the terrain or the other agent at any vic inity of its current location. Meeting
+(rendezvous) is defined as the equality of points representi ng agents at some moment of time.
+We assume that each agent has a unit of length and a compass. Co mpasses of agents may be
+incoherent, however we assume that agents have the same (clo ckwise) orientation of their system
+of coordinates. An additional tool, which may or may not be av ailable to the agents, is a map
+of the terrain. The map available to an agent is scaled (i.e., it accurately shows the distances),
+distinguishes the starting positions of this agent and the o ther one, and is oriented according to
+the compass of the agent. (Hence maps of different agents may ha ve different North.)
+All our considerations concern deterministic algorithms. The crucial notion is the routeof the agent
+which is a finite polygonal path in the terrain. The adversary initially places an agent at some
+point in the terrain. The agent constructs its route in steps in the following way. In every step
+the agent starts at some point v; in the first step, vis the starting point chosen by the adversary.
+The agent chooses a direction α, according to its compass, and a distance x. If the segment of
+lengthxin direction αstarting in vdoes not intersect the boundary of the terrain, the step ends
+when the agent reaches point uat distance xfromvin direction α. Otherwise, the step ends at
+the closest point of the boundary in direction α. If the starting point vin a step is in a segment
+of the boundary of the terrain, the agent has also an option (i n this step) to follow this segment of
+the boundary in any of the two directions till its end or for so me given distance along it. Steps are
+repeated until rendezvous, or until the route of the agent is completed.
+We consider the asynchronous version of the rendezvous problem. The asynchrony of the age nts’
+movements is captured by the assumption that the actual walk of each agent is decided by the
+adversary: the movement of the agent can be at arbitrary spee d, the agent may sometimes stop or
+go back and forth, as long as the walk of the agent in each segme nt of its route is continuous, does
+not leave it and covers all of it. More formally, the route in a terrain is a sequence ( S1,S2,...,S k)
+of segments, where Si= [ai,ai+1] is the segment corresponding to step i. In our algorithms the
+route is always finite. This means that the agent stops at some point, regardless of the moves of
+the other agent. We now describe the walk fof an agent on its route. Let R= (S1,S2,...,S k) be
+the route of an agent. Let ( t1,t2,...,tk+1), where t1= 0, be an increasing sequence of reals, chosen
+by the adversary, that represent points in time. Let fi: [ti,ti+1]→[ai,ai+1] be any continuous
+function, chosen by the adversary, such that fi(ti) =aiandfi(ti+1) =ai+1. For any t∈[ti,ti+1],
+we define f(t) =fi(t). The interpretation of the walk fis as follows: at time tthe agent is at the
+pointf(t) of its route and after time tk+1the agent remains inert. This general definition of the
+2walk and the fact that it is constructed by the adversary capt ure the asynchronous characteristics
+of the process. Throughout the paper, rendezvous means deterministic asynchronous rendezvous.
+Agents with routes R1andR2and with walks f1andf2meet at time t, if points f1(t) andf2(t)
+are equal. A rendezvous is guaranteed for routes R1andR2, if the agents using these routes meet
+at some time t, regardless of the walks chosen by the adversary. The trajec tory of an agent is the
+sequence of segments on its route until rendezvous. (The las t segment of the trajectory of an agent
+may be either the last segment of its route or any of its segmen ts or a portion of it, if the other
+agent is met there.) The cost of a rendezvous algorithm is the worst case sum of lengths of segments
+of trajectories of both agents, where the worst case is taken over all terrains with the considered
+values of parameters, and all adversarial decisions.
+We consider several scenarios, depending on three factors: (1) obstacles in the terrain are present,
+or not, (2) compasses of both agents agree, or not, (3) agents have or do not have a map of the
+terrain. Combinations of the presence or absence of these fa ctors give rise to eight scenarios. For
+each scenario we design a deterministic rendezvous algorit hm and analyze its cost. We also prove
+lower bounds on the cost of any deterministic rendezvous alg orithm in each case. For all scenarios
+these bounds are tight.
+One final precision has to be made. For all scenarios except th ose with incoherent compasses
+and the presence of obstacles (regardless of the availabili ty of a map), agents may be anonymous,
+i.e., they execute identical algorithms. By contrast, with the presence of obstacles and incoherent
+compasses, anonymity would preclude feasibility of rendez vous in some situations. Consider a
+square with one square obstacle positioned at its center. Co nsider two agents starting at opposite
+(diagonal) corners of the larger square, with compasses poi nting to opposite North directions.
+If they execute identical algorithms and walk at the same spe ed, then at each time they are in
+symmetric positions in the terrain and hence rendezvous is i mpossible. The only way to break
+symmetry for a deterministic rendezvous in this case is to eq uip the agents with distinct labels
+(which are positive integers). Hence, this is the assumptio n we make for the scenarios with the
+presence of obstacles and incoherent compasses (both with a nd without a map). For any label µ,
+we denote by |µ|the length of the binary representation of the label, i.e., |µ|=⌊logµ⌋+1.
+Our results. The cost of our algorithms depends on some of the following pa rameters (different
+parameters for different scenarios, see the discussion in Sec tion 4): Dis the distance between
+starting positions of agents in the terrain (i.e., the lengt h of the shortest path between them
+included in the terrain), Pis the perimeter of the terrain, (i.e., the sum of perimeters of all
+polygons P0,P1,...,Pk),xis the largest perimeter of an obstacle, and landLare the smaller and
+larger labels of agents, respectively, for the two scenario s that require different labels, as remarked
+above., i.e., for the scenarios with the presence of obstacl es and incoherent compasses.
+Our rendezvous algorithms rely on two different ideas: either meeting in a uniquely defined point
+of the terrain, or meeting on a uniquely defined cycle. It turn s out that a uniquely defined point
+can be found in all scenarios except those with the presence o f obstacles and incoherent compasses.
+In this case even anonymous agents can meet. On the other hand , with the presence of obstacles
+and incoherent compasses, such a uniquely defined point may n ot exist, as witnessed by the above
+3quoted example of a square with one square obstacle position ed at its center. For these scenarios
+we resort to the technique of meeting at a common cycle, break ing symmetry by different labels of
+agents.
+We first summarize our results concerning rendezvous when ea ch of the agents is equipped with a
+map showing its own position and that of the other agent. If co mpasses of the agents are coherent,
+then we show a rendezvous algorithm at cost D, which is clearly optimal. Otherwise, and if the
+terrain does not contain obstacles, then we show an algorith m whose cost is again D, and hence
+optimal. Finally, with incoherent compasses in the presenc e of obstacles, we show a rendezvous
+algorithm at cost O(D|l|); in the latter scenario we show that cost Ω( D|l|) is necessary for some
+terrains.
+Our results concerning rendezvous without a map are as follo ws. If compasses of the agents are
+coherent, then we show a rendezvous algorithm at cost O(P). We also show a matching lower
+bound Ω( P) in this case. If compasses of the agents are incoherent, but the terrain does not
+contain obstacles, then we show a rendezvous algorithm at co stO(P) and again a matching lower
+bound Ω( P). Finally, in the hardest of all scenarios (presence of obst acles, incoherent compasses
+and no map) we have a rendezvous algorithm at cost O(P+x|L|) and a matching lower bound
+Ω(P+x|L|). Table 1 summarizes our results.
+Rendezvous with a map Rendezvous without a map❵❵❵❵❵❵❵❵❵❵❵❵obstaclescompassescoherent incoherent❵❵❵❵❵❵❵❵❵❵❵❵obstaclescompassescoherent incoherent
+noDD noΘ(P)Θ(P)
+yes Θ(D|l|) yes Θ(P+x|L|)
+Table 1: Summary of results
+Related work. The rendezvous problem was first described in [30]. A detaile d discussion of the
+large literature on rendezvous can be found in the excellent book [4]. Most of the results in this
+domain can be divided into two classes: those considering th e geometric scenario (rendezvous in
+the line, see, e.g., [10, 11, 20, 31], or in the plane, see, e.g ., [7, 8]), and those discussing rendezvous
+in graphs, e.g., [2, 5]. Some of the authors, e.g., [2, 3, 6, 10 , 22] consider the probabilistic scenario
+where inputs and/or rendezvous strategies are random. Rand omized rendezvous strategies use
+random walks in graphs, which were thoroughly investigated and applied also to other problems,
+such as graph traversing [1], on-line algorithms [15] and es timating volumes of convex bodies [18].
+A generalization of the rendezvous problem is that of gather ing [19, 22, 23, 24, 27, 32], when more
+than two agents have to meet in one location.
+If graphs are unlabeled, deterministic rendezvous require s breaking symmetry, which can be ac-
+complished either by allowing marking nodes or by labeling t he agents. Deterministic rendezvous
+with anonymous agents working in unlabeled graphs but equip ped with tokens used to mark nodes
+was considered e.g., in [26]. In [33] the authors studied the task of gathering many agents with
+uniquelabels. In [17, 25, 34] deterministic rendezvous in g raphs with labeled agents was considered.
+However, in all the above papers, the synchronous setting wa s assumed. Asynchronous gathering
+undergeometric scenarios has been studied, e.g., in [13, 19 , 28] in different models than ours: agents
+4could not remember past events, but they were assumed to have at least partial visibility of the
+scene. The first paper to consider deterministic asynchrono us rendezvous in graphs was [14]. The
+authors concentrated on complexity of rendezvous in simple graphs, such as the ring and the infi-
+nite line. They also showed feasibility of deterministic as ynchronous rendezvous in arbitrary finite
+connected graphs with knownupper bound on the size. Further improvements of the above re sults
+for the infinite line were proposed in [31]. Gathering many ro bots in a graph, under a different
+asynchronous model and assuming that the whole graph is seen by each robot, has been studied in
+[23, 24].
+2 Rendezvous with a map
+We start by describing the following procedure that finds a un ique shortest path from the starting
+position of one agent to the other. The procedure works in all scenarios in which agents have a
+map of the terrain with their positions indicated.
+Procedure pathUniquePath (pointv, pointw)
+1 point u:=v; pathp:={v};
+2S={ps|psis a shortest path between vandw};
+3while(u/\e}atio\slash=w)do
+4U:=all paths psofSsuch that the first segment of the subpath of psleading from u
+towis the first in clockwise order around ustarting from the direction vw;
+5p′:=/intersectiontext
+ps∈Ups;
+6 extend pwith the connected part of p′containing u;
+7u:= new end of path p;
+8returnp;
+Lemma 2.1 Procedure UniquePath computes a unique shortest path from vtow, independent of
+the agent computing it.
+Proof:All shortest paths between two points inside a terrain can be computed as in [21]. The path
+computed by the call of UniquePath (v,w) is a shortest path, since it is composed, by construction,
+of parts of shortest paths between vandw. The path is computed in a deterministic way without
+using the compass direction of the agent or the unit of length of the agent. Hence, it is unique. /square
+2.1 Coherent compasses
+If agents have a map and coherent compasses, then they can eas ily agree on one of their two
+starting positions and meet at this point at cost D, which is optimal. This is done by the following
+Algorithm RVCM (rendezvous with a map and coherent compasse s).
+5Algorithm RVCM
+Letvbe the northernmost of the two starting positions of the agen ts. If both agents have the
+same latitude, let vbe the easternmost of them. Let wbe the other starting position. The agent
+starting at vremains inert. The agent starting at wcomputes the path p=UniquePath (w,v)
+and moves along puntilv.
+Theorem 2.1 Algorithm RVCM guarantees rendezvous at cost D, for any two agents with a map
+and coherent compasses, in any terrain.
+Proof:Theposition vcomputedbythetwoagents isthesame, sincetheyhave cohere nt compasses.
+The agents will eventually meet in v. The cost of rendezvous is D, sincepis of length D./square
+2.2 Incoherent compasses
+2.2.1 Terrains without obstacles
+In an empty polygon there is a unique shortest path between st arting positions of the agents [9],
+and agents with a map can meet in the middle of this path at cost D, which is optimal. This is
+done by Algorithm RVM (rendezvous with a map, without obstac les).
+Algorithm RVM
+The agent computes the (unique) shortest path between the st arting positions of the two agents.
+Then, it moves along this shortest path until the middle of it .
+Theorem 2.2 Algorithm RVMguarantees rendezvous at cost Dfor any two agents with a map,
+in any terrain without obstacles.
+Proof:In a polygon without obstacles, the shortest path between tw o points is unique and can be
+computed as in [21]. The two agents will eventually meet in th e middle of this shortest path. The
+cost of rendezvous is D, since the path is of length D. /square
+2.2.2 Terrains with obstacles
+This is the first of the two scenarios where agents cannot alwa ys predetermine a meeting point.
+Therefore they compute a common embedding of a ring on which t hey are initially situated, and
+theneach agent executes therendezvous algorithm from[14] forthis ring. Rendezvousis guaranteed
+to occur on the ring, but the meeting point depends on the walk s of the agents determined by the
+adversary. This is done by Algorithm RVMO (rendezvous with a map, with obstacles).
+6Algorithm RVMO
+Phase 1: computation of the embedding∗Rof a ring of size 4.
+Letvbe the starting position of the agent and let wbe the starting position of the other agent.
+The agent computes the embedding Rof a ring, composed of four nodes v,a,wandb, wherea
+is the midpoint of UniquePath (v,w),bis the midpoint of UniquePath (w,v), and the four edges
+are the respective halves of these paths.
+Phase 2: rendezvous on R.
+This phase consists in applying the rendezvous algorithm fr om [14] for ring R, whose size (four)
+is known to the agents.
+aThis embedding is not necessarily homeomorphic with a circl e, it may be degenerated.
+Theorem 2.3 Algorithm RVMO guarantees rendezvous at cost O(D|l|)for arbitrary two agents
+with a map, in any terrain.
+Proof: Leta1anda2be the two agents that have to meet. The embedding Rof the ring is
+the same for the two agents by Lemma 2.1. The algorithm from [1 4] guarantees rendezvous and
+has complexity expressed in terms of the total number of edge traversals by the agents before
+rendezvous occurs, equal to O(n|l|), where nis the number of nodes of the ring and lis the smaller
+of the two labels of the agents. Since the ring has size four an d each of its edges has length D/2,
+the total cost of rendezvous is O(D|l|). /square
+The following lower bound shows that the cost of Algorithm RV MO cannot be improved for some
+terrains. Indeed, it implies that for all D >0, there exists a polygon with a single obstacle, for
+which the cost of any rendezvous algorithm for two agents, st arting at distance D, is Ω(D|l|).
+Theorem 2.4 For any rendezvous algorithm A, for any D >0, and for any integers k2≥k1>0,
+there exist two labels l1andl2of lengths at most k1and at most k2, respectively, and a polygon
+with a single obstacle of perimeter 2D, such that algorithm Aexecuted by agents with labels l1and
+l2starting at distance D, requires cost Ω(Dk1). This holds even if the two agents have a map.
+Proof:The idea of the proof is based on an argument from [17]. For y >0, we consider a terrain
+Tthat is a hexagon of side y+2 with one hexagonal obstacle of side ywith the same center. The
+two agents start at positions uandvinT, as depicted in Figure 1. The compasses of agents point
+in opposite directions. Observe that D= 3y. We call slicesthe six trapezoids bounded by two
+corresponding parallel sides of the two hexagons and by the s egments linking the corresponding
+vertices of the hexagons. To avoid ambiguity, we say that an a gent in the segment shared by two
+slices is in the first of them in clockwise order. Note that age nts start in two different slices with
+two slices in between.
+Fix a rendezvous algorithm A. We assume that both agents always move at the same constant
+speed. We divide the execution of algorithm Ainto periods during which each agent traverses a
+distance y. During any period, an agent can only visit the slice where it starts the period and one
+7y+2
+yu
+vslices
+Figure 1: Terrain T
+of the two adjacent slices. The behavior of an agent with labe ll, running algorithm A, yields the
+following sequence of integers from the set {−1,0,1}, called the behavior code. The i-th term of the
+behavior code of an agent is −1 if the agent ends period iin the slice preceding (in clockwise order)
+the slice in which it began the period, 1 if it ends period iin the slice following it (in clockwise
+order), and 0 if it begins and ends period iin the same slice. Due to the symmetry of the figure
+and to opposite compasses an agent with a given label has the s ame behavior code if it starts at
+pointuor at point v. Note that two agents with the same prefix of length k1of their behavior
+codes cannot accomplish rendezvous during the first k1periods, since they start separated by at
+least two slices, and they cannot be in the same slice during a ny period.
+There are less than 3k1/2<2k1behavior codes of length at most k1/2. Hence it is possible to pick
+two distinct labels l1andl2of lengths at most k1, respectively, such that the prefix of length k1/2
+of their behavior codes is the same. For these labels, algori thmAdoes not accomplish rendezvous
+before both agents have travelled a distance yk1/2 = Ω(Dk1). /square
+3 Rendezvous without a map
+3.1 Coherent compasses
+Itturnsout that agents can recognize theouter boundaryof t heterrain even withoutamap. Hence,
+if their compasses are coherent, they can identify a uniquel y defined point on this boundary and
+meet in this point. This is done by Algorithm RVC (rendezvous with coherent compasses) at cost
+O(P).
+8Algorithm RVC
+From its starting position v, the agent follows the half-line αpointing to the North, as far as
+possible. When it hits the boundary of a polygon P(i.e., either the external boundary of the
+terrain or the boundary of an obstacle), it traverses the ent ire boundary of P. Then, it computes
+the point uwhich is the farthest point from vinP ∩α. It goes around Puntil reaching uagain
+and progresses on α, if possible. If this is impossible, the agent recognizes th at it went around
+the boundary of P0. It then computes the northernmost points in P0. Finally, it traverses the
+boundary of P0until reaching the easternmost of these points.
+Theorem 3.1 Algorithm RVCguarantees rendezvous at cost O(P)for any two agents with co-
+herent compasses, in any terrain.
+Proof:The first phase of the algorithm that consists in reaching P0and making the tour of the
+boundary of P0costs at most 3 P, since the boundary of each polygon of the terrain is travers ed
+at most twice and the total length of parts of αinside the terrain is at most P. Reaching the
+rendezvous point costs at most P. The agents will eventually meet in the easternmost of the
+northernmost points of P0, since they have coherent compasses and this point is unique ./square
+The following lower bound shows that the cost of Algorithm RV C is asymptotically optimal, for
+some polygons even without obstacles. This lower boundΩ( P) holdseven if the distance Dbetween
+starting positions of agents is bounded and if their compass es are coherent.
+Theorem 3.2 There exists a polygon of an arbitrarily large perimeter P, for which the cost of any
+rendezvous algorithm for two agents with coherent compasse s starting at any distance D >0, is
+Ω(P).
+Proof: Consider the polygon P′obtained by attaching to each side of a regular k-gon, whose
+center is at distance D/8 from its boundary, a rectangle of length 3 D/8 and of height equal to the
+side length of the k-gon. The polygon Pis the polygon obtained by gluing two copies of P′by the
+small side of one of the rectangles, as depicted in Fig. 2. Let Pbe the perimeter of the polygon P.
+We choose k= Θ(P/D). There are two types of rectangles in P, twopassingones (they share one
+side) and the 2 k−2normalones.
+Consider all rotations of the polygon Paround its center of symmetry by angles 2 πi/k, fori=
+0,...,k−1. We will prove that any deterministic rendezvous algorith m requires cost Ω( P) in at
+least one of the rotated polygons. Each agent starts in the ce nter of a different k-gon. We say that
+an agent has penetrated a rectangle if it has moved at distance D/8 inside the rectangle. In order
+to accomplish rendezvous, at least one agent has to penetrat e a passing rectangle. Each time one
+agent penetrates a rectangle, the adversary chooses a rotat ion, so that all previously penetrated
+rectangles, including the current one, are normal rectangl es. This choice is coherent with the
+knowledge previously acquired by the agents, since normal r ectangles are undistinguishable from
+each other and an agent needs to penetrate a rectangle in orde r to distinguish its type. Hence,
+9the two agents have to penetrate a total of k−1 rectangles before the adversary cannot rotate the
+figure to prevent the penetration of a passing rectangle. It f ollows that at least one of the agents
+has to traverse a total distance of Ω( kD) = Ω(P) before meeting.
+ypassingrectanglesrectanglesnormal
+3D
+8D
+8x
+Figure 2: Polygon P
+/square
+3.2 Incoherent compasses
+3.2.1 Terrains without obstacles
+In this section, we use the notion of medial axis, proposed by Blum [12], to define a unique point
+of rendezvous inside the terrain. Observe that we cannot use the centroid for the rendezvous point
+since, as we also consider non-convex terrains, the centroi d is not necessarily inside the terrain.
+Themedial axis M(P) of a polygon Pis defined as the set of points inside Pwhich have more
+than one closest point on the boundary of P. Actually, M(P) is a planar tree contained in P,
+in which nodes are linked by either straight-line segment or arcs of parabolas [29]. We define the
+medial point of a polygon Pas either the central node of M(P) or the middle of the central edge
+ofM(P), depending on whether M(P) has a central node or a central edge. Remark that the
+medial point of Pis unique and is inside P. The medial axis of a polygon Pcan be computed as
+in [16]. Algorithm RV (rendezvous without obstacles, witho ut a map and with possibly incoherent
+compasses) determines the unknown (empty) polygon and guar antees meeting in its medial point.
+Algorithm RV
+At its starting position, the agent chooses an arbitrary hal f-lineαwhich it follows until it hits
+the boundary of the polygon P0. It traverses the entire boundary of P0and computes the medial
+pointvofP0. Then, it moves to vby a shortest path and stops.
+Theorem 3.3 Algorithm RVguarantees rendezvous at cost O(P)for any two agents, in any ter-
+rain without obstacles.
+10Proof:The cost of reaching the boundary of Pand completing a tour of it is at most 2 P. The
+agent can compute the medial point of the polygon and reach it at cost at most P. The two agents
+will eventually meet at the medial point, since it is unique. /square
+The lower bound from Theorem 3.2 shows that the cost of Algori thm RV cannot be improved for
+some polygons.
+3.2.2 Terrains with obstacles
+Our last rendezvous algorithm, Algorithm RVO, works for the hardest of all scenarios: rendezvous
+with obstacles, no map, and possibly incoherent compasses. Here again it may be impossible to
+predetermine a meeting point. Thus agents identify a common cycle and meet on this cycle. The
+difference between the present setting and that of Algorithm RVMO, where a map was available,
+is that now agents may start outside of the common cycle and ha ve to reach it before attempting
+rendezvous on it. Also the common cycle is different: rather th an being composed of two shortest
+paths between initial positions of the agents (a map seems to be needed to find such paths), it is
+the boundary of a (possible) obstacle Oin which the medial point of the outer polygon is hidden.
+These changes have consequences for the cost of the algorith m. The fact that the medial point
+of the outer polygon has to be found and the obstacle Ohas to be reached is responsible for the
+summand Pin the cost. The only bound on the perimeter of this obstacle i sx. Finally, the fact
+that the adversary may delay the agent with the smaller label and force the other agent to make its
+tours of obstacle Obefore the agent with the smaller label even reaches the obst acle, is responsible
+for the summand x|L|, rather than x|l|, in the cost.
+Acycleis a polygonal path whose both extremities are the same point . Atourof a cycle Cis
+any sequence of all the segments of Cin either clockwise or counterclockwise order starting fro m a
+vertex of C. By extension, a partial tour ofCis a path which is a subsequence of a tour of Cwith
+the first or the last segment of the subsequence possibly repl aced by a subsegment of it.
+11Algorithm RVO
+Phase 1: Computation of the medial point of P0
+At its starting position z, the agent chooses an arbitrary half-line αwhich it follows as far as
+possible. When it hits the boundary of a polygon P, it traverses the entire boundary of P. Then,
+it computes the point wwhich is the farthest point from zinP ∩α. It goes around Puntil
+reaching wagain and progresses on α, if possible. If this is impossible, the agent recognizes th at
+it went around the boundary of P0. The agent computes the medial point vofP0.
+Phase 2: Moving to the medial point of P0
+Letube the current position of the agent. The agent follows the se gmentuvas far as possible.
+Similarly as in the first phase of the algorithm, if the agent h its a polygon P, it traverses the
+entire boundary of P. Then, it computes the point wwhich is the farthest point from uinP∩uv.
+It goes around Puntil reaching wagain and progresses on α, if possible. If this is impossible and
+if the point vhas not been reached, the agent recognizes that the point vis inside an obstacle
+O, and executes phase 3. If the agent reaches v, it does not enter phase 3 of the algorithm and
+stops.
+Phase 3: Rendezvous around the medial obstacle of the terrain
+Theagent goes aroundtheobstacle Ountil it reaches avertex s. Theagent producesthemodified
+labelµ∗consisting of the binary representation of the label µof the agent followed by a 1 and
+then followed by |µ|zeros. This phase consists of |µ∗|stages. In stage i, the agent completes
+two tours of the boundary of O, starting and ending in s, clockwise if the i-th bit of µ∗is 1 and
+counterclockwise otherwise.
+Letu1u2andu2u3be consecutive segments in clockwise order (resp. counterc lockwise order) of a
+cycle. For a given walk fof an agent a, we say that the agent traverses in a clockwise way (resp.
+in a counterclockwise way ) a vertex u2of a cycle at time tiff(t) =u2and there exist positive reals
+ǫ1andǫ2and points yandzsuch that y=f(t−ǫ1) is an internal point of u1u2,z=f(t+ǫ2) is
+an internal point of u2u3and the agent walks in u1u2∪u2u3during the time period [ t−ǫ1,t+ǫ2].
+Before establishing the correctness and cost of Algorithm RVO, we need to show the following two
+lemmas.
+Lemma 3.1 Consider two agents on cycle C. Suppose that one agent executes a tour of Cin some
+sense of rotation, starting and ending in v. If during the same period of time, the other agent either
+traverses vfor the first time in the other sense of rotation or does not tra verse it at all, then the
+two agents meet.
+Proof:Letf1andf2be the walks of agents a1anda2, respectively. Let t′be the moment when
+agenta1starts its tour of Cat some vertex v. Lett′′be the moment when agent a1ends its tour,
+if agenta2does not traverse vin the same period of time, or, otherwise, the first moment aft er
+t′when agent a2traverses v. We cut cycle Cat vertex vobtaining the path pwith extremities v′
+andv′′that are copies of v. The walks f1andf2, during the time period [ t′,t′′], can be transposed
+inp, since neither of the two agents traverses vduring the period ( t′,t′′). For any t∈[t′,t′′], let
+di(t) be the distance of agent aifromv′at timet, counted on p. The two functions d1andd2
+12are continuous, since the walks of both agents on pare continuous. Notice that, since the first
+traversal of vby agent a2may be only in the sense of rotation opposite to that of agent a1, we have
+f1(t′) =v′and either f2(t′′) =v′orf1(t′′) =v′′. Letδ(t) =d1(t)−d2(t). We have δ(t′) =d′≤0
+andδ(t′′) =d′′≥0, since d1(t′) = 0 and d1(t′′)≥d2(t′′). The function δis thus a continuous
+function from the interval [ t′,t′′] onto some interval [ c′,c′′], where c′≤d′andc′′≥d′′. Since 0
+belongs to the interval [ c′,c′′], there must exist a moment tin the interval [ t′,t′′], for which δ(t) = 0.
+For this moment, f1(t) =f2(t) and the rendezvous occurs. /square
+Lemma 3.2 Consider two agents on a cycle Cand letk≥0be an integer. If an agent executes
+either a partial tour of Cfollowed by at most ktours of C, or at most ktours of Cfollowed by a
+partial tour of C, while the second agent executes k+2tours ofC, then the two agents meet.
+Proof:Assume for contradiction that the two agents never meet. Dur ing each tour of Cby the
+second agent, the first agent has to traverse the starting pos itionvof the second agent, in view of
+Lemma 3.1. Hence, the first agent has traversed k+2 times vertex v. Notice that an agent cannot
+traverse vwithout executing a tour of Casvis an extremity of a segment of its route. Hence the
+first agent has completed at least k+1 complete tours of Cstarting and ending at v. Finally, the
+first agent has started executing its tours at point v, a contradiction. /square
+Theorem 3.4 Algorithm RVOguarantees rendezvous at cost O(P+x|L|)for any two agents in
+any terrain for which xis the largest perimeter of an obstacle.
+Proof:Leta1anda2be the two agents that have to meet. The first phase of the algor ithm that
+consists in reaching P0and making the tour of the boundary of P0costs at most 3 P, since the
+boundary of each polygon of the terrain is traversed at most t wice and the total length of parts of
+αinside the terrain is at most P. For the same reason as in phase 1, the total cost of phase 2 is a t
+most 3P.
+If the medial point of P0is inside the terrain, then the agents meet at the end of phase 2 at total
+cost of at most 12 P. Otherwise, both agents eventually enter phase 3 of the algo rithm and they
+are on the boundary of the obstacle Ocontaining the medial point of P0. The cost follows from
+the fact that each agent travels a distance O(x|L|) in phase 3. Indeed, each agent executes at most
+2|L|+1 stages and each stage costs at most 2 x. Hence it remains to show that rendezvous occurs
+in this case as well.
+Assume for contradiction that the two agents never meet. Not ice that the modified label l∗cannot
+be the suffix of the modified label L∗. Indeed, if |l∗|=|L∗|then the two labels are different since
+l/\e}atio\slash=L, and otherwise the second part of l∗, consisting of 1 followed by |l|zeros, cannot be the
+suffix of L∗. Hence, there exists an index isuch that the ( |l∗| −i)-th bit of l∗differs from the
+(|L∗|−i)-th bit of L∗. We call important stages the ( |l∗|−i)-th stage of the agent with label land
+the (|L∗|−i)-th stage of the agent with label L.
+Forj= 1,2, lettjbe the moment when agent ajenters its important stage and let t′be the first
+moment when both agents have finished the execution of the alg orithm. Suppose by symmetry
+13thatt1≤t2, i.e., agent a1was the first to enter its important stage. Then a2must have entered
+its important stage during the first tour of the important sta ge ofa1. Otherwise, agent a2would
+have completed 2 i+2 tours between t2andt′, while agent a1would have completed at most 2 i+1
+tours. Hence, the two agents would have met in view of Lemma 3. 2. Hence, from the time t2, agent
+a2completes one tour in some sense of rotation, starting and en ding at a vertex v, while agent
+a1either traverses vfor the first time in the other sense of rotation or does not tra verse it at all.
+Hence by Lemma 3.1, the two agents meet. /square
+The following result gives a lower bound matching the cost of Algorithm RVO.
+Theorem 3.5 There exist terrains for which the cost of any rendezvous algo rithm is Ω(P+x|L|).
+This holds for arbitrarily small D >0.
+Proof:Since our lower bound is expressed as a sum, in order to prove i t, we show two examples,
+oneinwhich thefirstsummandisas small as possibleandthebo undis equal to theother summand,
+and vice-versa. The first example uses the polygon from Theor em 2.4:Pmust be at least xand
+in this example we have P= Θ(x) and the lower bound is Ω( x|L|). Indeed, consider two integers
+m2≤m1. By Theorem 2.4, applied for k1=k2=m1, and for any rendezvous algorithm A, there
+exists a label Lof length m1, such that the sum of lengths of segments of the route produce d by
+the execution of Aby an agent a1with label Lis Ω(xm1). The adversary chooses as the initial
+position of the second agent a2any point outside a path pof length Θ( xm1), which is a prefix of
+the route of agent a1. This point can be chosen arbitrarily close to the initial po sition of the first
+agent. The label of agent a2is of length m2. Suppose that the start of agent a2is delayed by the
+adversary and occurs when pis entirely traversed by agent a1. The two agents do not meet during
+this traversal of pby the first agent and so the cost of rendezvous is Ω( xm1) = Ω(x|L|). The second
+example is given by the proof of Theorem 3.2. Indeed, in this e xample there are no obstacles and
+hencex=x|L|= 0, while the lower bound is Ω( P). /square
+4 Discussion of parameters
+We presented rendezvous algorithms, analyzed their cost an d proved matching lower bounds in all
+considered scenarios. However, it is important to note that the formulas describing the cost depend
+on the chosen parameters in each case. All our results have th e following form. For a given scenario
+we choose some parameters (among D,P,x,l,L), show an algorithm whose cost in any terrain
+isO(f), where fis some simple function of the chosen parameters, and then pr ove that for some
+class of terrains any rendezvous algorithm requires cost Ω( f), which shows that the complexity of
+our algorithm cannot be improved in general, for the chosen p arameters.
+This yields the question which parameters should be chosen. In the case of complexities Dand
+Θ(P), this choice does not seem controversial, as here DandPare very natural parameters, and
+the only ones in these simple cases. However, for the two scen arios with incoherent compasses and
+with the presence of obstacles, there are several other poss ible parameters, and their choice may
+14raise a doubt. As mentioned in the introduction, in these two scenarios, distinct labels of agents
+are necessary to break symmetry, since rendezvous is imposs ible for anonymous agents. Hence any
+rendezvous algorithm has to use labels landLas inputs, and thus the choice of these labels as
+parameters seems natural. By contrast, the choice of parame terxmay seem more controversial.
+Why do we want to express the cost of a rendezvous algorithm in terms of the largest perimeter of
+an obstacle? Are there other natural choices of parameter se ts? What are their implications?
+Let us start by pondering the second question. It is not hard t o give examples of other natural
+choices of parameters for the two scenarios with incoherent compasses and with the presence of
+obstacles. For example, in the hardest scenario (without a m ap), we could drop parameter xand
+try to express the cost of the same Algorithm RVO only in terms ofD,P,l, andL. Sincex≤P, we
+would get O(P|L|) instead of O(P+x|L|). Incidentally, as in our lower bound example of terrains
+we have x= Θ(P), this new complexity O(P|L|) is optimal for the same reason as the former one.
+Another possibility would be adding, instead of dropping a p arameter. We could, for example, add
+the parameter Pewhich is the length of the external perimeter of the terrain, i.e., the perimeter of
+polygonP0. Thenit becomes natural to modify Algorithm RVO as follows. Thefirsttwo phases are
+the same. In the third phase, the agent goes around obstacle Oand compares its perimeter to Pe.
+If the perimeter of Ois smaller (or equal), then the algorithm proceeds as before , and if it is larger,
+then the agent goes back to the boundary of P0and executes Phase 3 on this boundary instead
+of the boundary of O. The new algorithm has complexity O(P+min(x,Pe)|L|). Its complexity is
+again optimal because in our lower bound example we can choos e the parameter y= min(x,Pe)
+and enlarge the largest of the two boundaries by lengthy but t hin zigzags. Thus we can preserve
+the lower bound Ω( P+min(x,Pe)|L|), even when xandPediffer significantly.
+The reason why we chose parameters D,P,l,L, andxinstead of just D,P,landL, is that
+complexity O(P+x|L|) shows a certain continuity of the complexity of Algorithm R VO with
+respect to the sizes of obstacles: when the largest obstacle decreases, this complexity approaches
+O(P) and it becomes O(P) if there are no obstacles. In this case our algorithm coinci des with
+Algorithm RV. This is not the case with complexity O(P|L|). On the other hand, this choice
+coincides with O(P+ min(x,Pe)|L|) in many important cases, for example for convex obstacles
+(as then we have x < Pe).
+It is then natural to ask what happens if we add parameter xin the scenario with incoherent
+compasses and with the presence of obstacles but with the map . Obviously we could still use
+Algorithm RVO and get complexity O(P+x|L|). However, our lower bound argument in this
+scenario gives in fact only Ω( D+min(x,D)|l|). In our example we had D= Θ(x) but we only get
+Ω(D+x|l|) even if Dis much larger than x. On the other hand, if Dis much smaller than xwe
+can only get the lower bound Ω( D|l|) because it matches the complexity of RVMO in this case.
+Hence it is natural to ask if there exists a rendezvous algori thm with cost O(D+min(x,D)|l|) for
+arbitrary terrains in this scenario. We leave this as an open question.
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+18
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+arXiv:1001.0890v1  [cs.DS]  6 Jan 2010How to meet asynchronously (almost) everywhere
+Jurek Czyzowicz∗†Arnaud Labourel∗‡ ¶Andrzej Pelc∗§
+Abstract
+Two mobile agents (robots) with distinct labels have tomeet
+in an arbitrary, possibly infinite, unknown connected graph
+or in an unknown connected terrain in the plane. Agents
+are modeled as points, and the route of each of them only
+depends on its label and on the unknown environment. The
+actual walk of each agent also depends on an asynchronous
+adversary that may arbitrarily vary the speed of the agent,
+stop it, or even move it back and forth, as long as the
+walk of the agent in each segment of its route is continuous,
+does not leave it and covers all of it. Meeting in a graph
+means that both agents must be at the same time in some
+node or in some point inside an edge of the graph, while
+meeting in a terrain means that both agents must be at the
+same time in some point of the terrain. Does there exist a
+deterministic algorithm that allows any two agents to meet
+in any unknown environment in spite of this very powerfull
+adversary? We give deterministic rendezvous algorithms
+for agents starting at arbitrary nodes of any anonymous
+connected graph (finite or infinite) and for agents starting a t
+any interior points with rational coordinates in any closed
+region of the plane with path-connected interior. While
+our algorithms work in a very general setting – agents can,
+indeed, meet almost everywhere – we show that none of the
+above few limitations imposed on the environment can be
+removed. On the other hand, our algorithm also guarantees
+the following approximate rendezvous for agents starting at
+arbitrary interior points of a terrain as above: agents will
+eventually get at an arbitrarily small positive distance fr om
+each other.
+1 Introduction
+The problem and the model. Two mobile agents
+(robots)modeledaspointsstartingatdifferentlocations
+∗D´ epartement d’informatique, Universit´ e du Qu´ ebec en
+Outaouais, Gatineau, Qu´ ebec J8X 3X7, Canada. E-mails:
+jurek@uqo.ca ,labourel.arnaud@gmail.com ,pelc@uqo.ca
+†Partially supported by NSERC discovery grant.
+¶Supported by the ANR-project“ALADDIN”, and the´ equipe-
+projet INRIA “C ´EPAGE”.
+‡Thiswork wasdone during thisauthor’s stay atthe Universit ´ e
+du Qu´ ebec en Outaouais as a postdoctoral fellow.
+§Partially supported by NSERC discovery grant and by the
+Research Chair in Distributed Computing at the Universit´ e du
+Qu´ ebec en Outaouais.of an unknown environment have to meet. This task is
+known in the literature as the rendezvous problem, and
+hasbeenstudiedundertwoalternativescenarios. Either
+the agentsmoveina network,modeled byanundirected
+connected graph (the graph scenario ), or they move in
+(some subset of) the plane (the geometric scenario ).
+In this paper we study the asynchronous version of
+the rendezvous problem, under both above scenarios:
+each agent designs its route and an adversary controls
+the speed of each agent, can vary this speed, stop
+the agent, or even move it back and forth, as long as
+the walk of the agent in each segment of its route is
+continuous, does not leave it and covers all of it. In the
+asynchronous version of the graph scenario, meeting at
+a node may be impossible even in the two-node graph,
+asthe adversarycan desynchronizethe agentsandmake
+them visit nodes at different times. Thus it is necessary
+to relaxthe requirement and allowagents to meet either
+in a node or inside an edge. Such a definition of meeting
+is natural, e.g., when agents are robots traveling in a
+labyrinth. We consider an embedding of the underlying
+graph in the three-dimensional Euclidean space, with
+nodes of the graph being points of the space and
+edges being pairwise disjoint line segments joining them
+(hence there areno edge crossings). Agentsaremodeled
+as points moving inside this embedding.
+If nodes of the graph are labeled and the labeling
+is known, then agents can decide to meet at a pre-
+determined node and the rendezvous problem reduces
+to graph exploration. However, in many applications,
+when rendezvous is needed in a network of unknown
+topology, such unique labeling of nodes may be unavail-
+able, or agents may be unable to perceive such labels,
+e.g., due to security reasons. Hence it is important to
+design rendezvous algorithms for agents operating in
+anonymous graphs, i.e., graphs without unique labeling
+of nodes. It is important to note that the agents have to
+be able to locallydistinguish ports at a node: otherwise,
+an agent may even be unable to visit all neighbors of a
+node of degree 3 (after visiting the second neighbor, the
+agentcannotdistinguishtheportleadingtothe firstvis-
+ited neighbor from that leading to the unvisited one).
+Consequently, agents initially located at two nodes of
+degree 3, might never be able to meet. This justifies a
+common assumption made in the literature: all ports ata node are locally labeled by distinct positive integers.
+Degrees of nodes can be either finite or infinite. No co-
+herencebetween those locallabelings is assumed. When
+an agent leaves a node, it is aware of the port number
+by which it leaves and when it enters a node, it is aware
+of the entry port number. It can also verify, at each
+node, whether a given positive integer is a port num-
+ber at this node. Agents know neither the graph, nor
+the initial distance between them. They cannot mark
+the nodes or the edges in any way. Rendezvous has to
+be accomplished regardless of local labelings of ports.
+Each agent terminates its walk at the time of meeting
+the other agent.
+In the geometric scenario, we assume that the
+terrain in which agents operate is a closed subset of the
+Euclidean plane, i.e., it contains limits of all converging
+sequences of points in it. The boundary of the terrain
+is defined as the set of points having arbitrarily close
+points both in the terrain and outside of it. Since the
+terrain is closed, the boundary is included in it. All
+other points of the terrain are its interiorpoints. Each
+agent can only distinguish if it is currently in an interior
+point of the terrain or in its boundary. Agents do not
+know the terrain in which they operate, they cannot
+“see”anyvicinityofthe currentlyvisitedpoint andthey
+cannot leave any marks. Agents are equipped with a
+compassandwiththesameunitoflength. Thussystems
+of coordinates of agents are aligned and the origin for
+each agent is at its starting point. Again, an agent
+terminates its walk at the time of meeting the other
+agent.
+If agents are identical, i.e., they do not have dis-
+tinct identifiers, and execute the same algorithm, then
+deterministic rendezvous is impossible, e.g., in the ring
+(graph scenario) or in the plane (geometric scenario):
+the adversary will make the agents move always in the
+same direction at the same speed, keeping them at the
+same distance at all times, thus they will never meet.
+Hence we assume that agents have distinct identifiers,
+called labels, which are two different positive integers.
+This is their only way to break symmetry. We assume
+that each agent knows its own label but not the la-
+bel of the other agent. This excludes, e.g., rendezvous
+strategies of the type “waiting for mommy”, in which
+the agent with smaller label remains idle and the other
+agent explores the graph or the terrain. We do not im-
+pose any restriction on the memory of the agents: from
+the computational perspective they are viewed as Tur-
+ing machines.
+Two important notions used to describe movements
+of agents are the routeof each agent and its walk.
+Roughly speaking, each agent chooses the route where
+it moves and the adversary describes the walk on thisroute, deciding howthe agent moves. More precisely,
+these notions are defined as follows. The adversary
+initially places an agent with label ℓat some node of
+the graph or at some point in the terrain. Given this
+label and this starting point, the route is chosen by
+the agent and is defined as follows. In the case of the
+graph, the agent chooses one of the available ports at
+the current node. After getting to the other end of
+the corresponding edge, the agent chooses one of the
+available ports at this node, and so on, indefinitely
+(until rendezvous). The resulting route of the agent
+is the corresponding sequence of edges, which is a (not
+necessarily simple) path in the graph. The route in a
+terrain is a sequence ( S1,S2,...) of segments, where
+Si= [ai,ai+1], defined in stages as follows, given the
+agent’slabel and the starting point. In stage ithe agent
+starts at point ai, anda1is the starting point chosen
+by the adversary. The agent chooses a direction αand
+distance x. If the segment of length xin direction α
+starting in vintersects the boundary of the terrain at
+some distance y≤xfromv, the agent becomes aware
+of it in the intersection point wclosest to v. In this
+case, the stage ends at wand in the next stage the
+agent chooses the reverse direction αand the distance
+y(that will cause it to return to v). If the segment of
+lengthxin direction αstarting in vdoes not intersect
+the boundary of the terrain, the stage ends when the
+agent reaches point uat distance xfromvin direction
+α. Stages are repeated indefinitely (until rendezvous).
+We now describe the walk fof an agent on its
+route. Let R= (S1,S2,...) be the route of an
+agent. In the graph scenario this is a (not necessarily
+simple) infinite path in the (spatial embedding of) the
+graph, and in the geometric scenario it is an infinite
+polygonal line in the plane. Let ( t1,t2,...), where
+t1= 0, be an increasing sequence of reals, chosen
+by the adversary, that represent points in time. Let
+fi: [ti,ti+1]→[ai,ai+1] be any continuous function,
+chosen by the adversary, such that fi(ti) =aiand
+fi(ti+1) =ai+1. For any t∈[ti,ti+1], we define
+f(t) =fi(t). The interpretation of the walk fis as
+follows: at time tthe agent is at the point f(t) of
+its route. This general definition of the walk and the
+fact that it is constructed by the adversary capture
+the asynchronous characteristics of the process. The
+movement of the agent can be at arbitrary speed, the
+agent may sometimes stop or go back and forth, as long
+as the walk in each segment of the route is continuous
+and covers all of it.
+Notice that the power of the asynchronous adver-
+sary to produce any continuous walk on the routes de-
+termined by the agents implies the following signifi-
+cantdifferencewithrespecttothesynchronousscenario.Whileinthelatterscenariotherelativemovementofthe
+agents depends only on their routes, in our setting, this
+movement is also controlled by the adversary.
+Agents with routes R1andR2and with walks f1
+andf2meet at time t, if points f1(t) andf2(t) are
+identical. A rendezvous is guaranteed for routes R1and
+R2, if the agents using these routes meet at some time
+t, regardless of the walks chosen by the adversary. A
+rendezvous algorithm executed by agents in a graph or
+in a terrain produces routes of agents, given the label of
+each agent and its starting point.
+It should be stressed that, while routes of agents
+are formally defined as infinite sequences of segments,
+our results imply that in any instance of the rendezvous
+problem, meeting will occur at some finite time, and
+thus each agent will compute only finitely many seg-
+mentsofitsroute. As mentionedabove,agentscompute
+their routes in stages, and given any walk chosen by the
+adversary, each stage is completed in finite time. There
+is no stopping issue in our solution: rendezvous always
+occurs at some stage for each of the agents and then
+both agents stop. Another feature of our rendezvous
+algorithms is that in the choice of consecutive segments
+of its route an agent does not use the knowledge of the
+walk to date. Thus the route depends only on the label
+of the agent, on the environment (graph or terrain), and
+on the starting point chosen by the adversary, but not
+on its other actions.
+Our results. We give two deterministic algo-
+rithms, the first for rendezvous in the graph scenario
+and the second in the geometric scenario. For the graph
+scenario, our algorithm accomplishes rendezvous in any
+connected countable∗(finite or infinite) graph, for arbi-
+trary starting nodes. A consequence of this very general
+result is the positive answer to the following question
+from[12]: Isdeterministicasynchronousrendezvousfea-
+sible in any finite connected graph without knowing any
+upperboundonitssize? (In[12]theauthorspresenteda
+deterministic asynchronous rendezvous algorithm in ar-
+bitraryfiniteconnectedgraphswith knownupperbound
+on the size.)
+For the geometric scenario, our algorithm accom-
+plishes rendezvous for agents starting at any interior
+points with rational coordinates in any closed region of
+the plane with path-connected interior. (Recall that a
+subsetTofthe planeis path-connected, ifforanypoints
+u,v∈T, there is a continuous function h: [0,1]−→P,
+such that P⊆Tandh(0) =u,h(1) =v.) On the other
+hand, our algorithm guarantees the following approxi-
+mate rendezvous for agents starting at arbitrary interior
+∗A graph is countable if the set of its nodes is countable, i.e. ,
+if there exists a one-to-one function from this set into the s et of
+natural numbers.points of a terrain as above: agents will eventually get
+atanarbitrarilysmallpositivedistancefromeachother.
+This implies the perhaps surprising result that if agents
+have arbitrarily small positive visibility ranges (rather
+than 0 visibility range as we assume) and they start in
+arbitrarypoints of the (empty) plane, then they will see
+each other in finite time, regardless of the actions of the
+adversary.
+Discussion of limitations. While our algorithms
+work in a very general setting – agents can, indeed,
+meet almost everywhere – it turns out that none of
+the few limitations imposed on the environment can be
+removed. For the graph scenario, the only limitation is
+connectivity of the graph. It is clear that rendezvous
+in disconnected graphs is impossible, if the agents start
+in different connected components. For the geometric
+scenario, let us review the limitations one by one. First,
+we assume that the terrain is closed. This assumption
+cannot be entirely removed for the following technical
+reason. Consider the construction of a route in an open
+disc. An agent starting at any point, that chooses in
+the first stage an arbitrary direction and a sufficiently
+large distance, at some point would have to leave the
+disc. Since it does not see anything in its vicinity, it
+cannot know where the boundary is before hitting it,
+and it cannot hit it, as it is not allowed to leave the
+terrain. It follows that the agent could not construct
+further segments of its route. The second assumption
+is that agents start at interior points of the terrain.
+This assumption cannot be removed either. Indeed,
+suppose that the terrain is a closed disc with a semi-
+circle attached to it. This is a closed subset of the
+plane with nonempty path-connected interior. Suppose
+that one agent starts in the disc and the other at the
+end of the semi-circle. Since agents need to move along
+polygonal lines, the second agent could not move at all
+and the first one cannot reach it. Our next assumption
+is that the interior of the terrain is path-connected. To
+show that this assumption cannot be removed, consider
+two disjoint closed discs joined by an arc of a circle.
+This terrain is closed and path-connected, but if each
+agent starts inside a different disc, again they cannot
+meet, because agents need to move along polygonal
+lines, and hence cannot traverse the joining arc. The
+final assumption is that the starting points of the agents
+have rational coordinates. In Section 4 we prove that
+if the agents start in arbitrary points, then rendezvous
+cannot be guaranteed even in the plane. We show,
+however, that for arbitrary starting points approximate
+rendezvous is guaranteed.
+Related work. The rendezvous problem was first
+described in [25]. A detailed discussion of the large
+literature on rendezvous can be found in the excellentbook [4]. Most of the results in this domain can be
+dividedintotwoclasses: thoseconsideringthegeometric
+scenario (rendezvous in the line, see, e.g., [9, 10, 17],
+or in the plane, see, e.g., [7, 8]), and those discussing
+rendezvous in graphs, e.g., [2, 5]. Some of the authors,
+e.g., [2, 3, 6, 9, 18] consider the probabilistic scenario
+where inputs and/or rendezvous strategies are random.
+Randomized rendezvous strategies use random walks in
+graphs, which were thoroughly investigated and applied
+also to other problems, such as graph traversing [1], on-
+line algorithms [13] and estimating volumes of convex
+bodies [15]. A generalization of the rendezvous problem
+is that of gathering [16, 18, 19, 20, 23, 27], when more
+than 2 agents have to meet in one location.
+If graphs are unlabeled, deterministic rendezvous
+requiresbreakingsymmetry, which can be accomplished
+either by allowing marking nodes or by labeling the
+agents. Deterministic rendezvous with anonymous
+agents working in unlabeled graphs but equipped with
+tokens used to mark nodes was considered e.g., in [22].
+In [28] the authors studied gathering many agents with
+unique labels. In [14, 21, 29] deterministic rendezvous
+in graphs with labeled agents was considered. How-
+ever, in all the above papers, the synchronous setting
+was assumed. Asynchronous gathering under geomet-
+ric scenarios has been studied, e.g., in [11, 16, 24] in
+different models than ours: agents could not remem-
+ber past events, but they were assumed to have at least
+partial visibility of the scene. The first paper to con-
+sider deterministic asynchronous rendezvous in graphs
+was [12]. The authors concentrated on complexity of
+rendezvousinsimplegraphs,suchasthe ringandthein-
+finite line. They also showed feasibility of deterministic
+asynchronous rendezvous in arbitrary finite connected
+graphswith knownupperboundonthesize. Furtherim-
+provementsof the above results for the infinite line were
+proposedin [26]. Gatheringmanyrobotsin agraph, un-
+der a different asynchronous model and assuming that
+the whole graph is seen by each robot, has been studied
+in [19, 20].
+2 Preliminary notions and results
+A fundamental notion on which our algorithms are
+based is that of a tunnel. Consider any graph Gand
+two routes R1andR2starting at nodes vandw,
+respectively. We say that these routes form a tunnel,
+if there exists a prefix [ e1,e2,...,e n] of route R1and a
+prefix [en,en−1,...,e 1] of route R2, for some edges ei
+in the graph, such that ei={vi,vi+1}, wherev1=v
+andvn+1=w. Intuitively, the route R1has a prefix
+Pending at wand the route R2has a prefix which
+is the reverse of P, ending at v. By a slight abuse of
+terminology we will also say that prefixes [ e1,e2,...,e n]and [en,en−1,...,e 1] form a tunnel.
+Proposition 2.1. If routes R1andR2form a tunnel,
+then they guarantee rendezvous.
+Proof.Consider an embedding of the graph Gin the
+three-dimensional Euclidean space, with nodes of the
+graphbeingpointsofthespaceandedgesbeingpairwise
+disjoint line segments joining them. Consider routes R1
+andR2starting at nodes vandw, respectively. Let
+agentaiexecute route Ri. LetPbe the polygonal line
+joiningvwithw, corresponding to the prefixes of the
+routes, given by the tunnel. Let Dbe its length defined
+as the sum of lengths of edges in the corresponding
+prefixes of the routes. (For non-simple paths in the
+graph, the same edge is counted many times.) Consider
+any walks f1onR1andf2onR2. Lett′be the first
+moment when an agent leaves its starting point and let
+t′′be the moment when an agent gets to the end of P
+other than its starting point. For any t∈[t′,t′′], let
+d1(t) be the distance of agent a1from its starting point
+vat timet, counted on the route R1, and let d2(t) be
+the distance of agent a2from its target point vat time
+t, counted on the route R2. Letδ(t) =d2(t)−d1(t). We
+haveδ(t′) =Dandδ(t′′) =d≤0. Thefunction δisthus
+acontinuousfunction from the interval [ t′,t′′] ontosome
+interval[ d′,D], whered′≤d, in viewofthecontinuityof
+walksf1andf2. Since 0 belongs to the interval [ d′,D],
+there must exist a moment tin the interval [ t′,t′′], for
+whichδ(t) = 0. For this point we have f1(t) =f2(t),
+and the rendezvous occurs.
+We now recallsome basic facts from set theory, that
+will be used in further considerations.
+Proposition 2.2. The set of rational numbers and the
+set of positive rational numbers are countable. The
+cartesian product of two countable sets is countable. The
+set of all finite sequences with terms in a countable set
+is countable.
+3 Rendezvous in the graph scenario
+LetG= (V,E) be the connected graph in which the
+rendezvous must be performed. Let Snbe the set of
+sequences of npositive integers. Let P={(i,j,s′,s′′)|
+i,j∈N,i < jand∃ns.t.s′,s′′∈ Sn}. There exists
+a bijection from the set of positive integers onto P,
+in view of Proposition 2.2. Let ( ϕ1,ϕ2,...) be a fixed
+enumeration of P. All agents have to agree on the same
+enumeration. It is easy to produce a formula computing
+φkfor anyk. This formula is included in the rendezvous
+algorithm. For a finite path rinG, we denote by rthe
+path with the same edges as in r, but in the reverse
+order. Remark that randrform a tunnel.We first give a high-level idea of the algorithm
+referring to lines of the pseudo code given below. We
+“force” the routes of any two agents to form a tunnel
+for every possible combination of starting nodes and
+labelsofthetwoagents. ByProposition2.1, thissuffices
+to guarantee rendezvous. Any starting configuration of
+robotiplaced at node vand robot jplaced at node w
+by the adversarycorresponds to a quadruple ( i,j,s′,s′′)
+wheres′is a sequence of ports inducing a path from v
+towands′′is a sequence of ports inducing the reverse
+path from wtov.
+Each agent constructs its route in phases. At the
+beginning and at the end of each phase the agent is in
+its starting node. At phase kthe previouslyconstructed
+initial part of the route rhistis extended while the
+agent processes quadruple ϕk(some of the extensions
+are null). This extension guarantees that the routes
+of agents of the corresponding starting configuration
+will form a tunnel. When agent with label lprocesses
+quadruple ϕk= (i,j,s′,s′′) nothing happens if l/ne}ationslash=i
+andl/ne}ationslash=j(line 4). If l=i, agentitries to extend its
+route to guarantee rendezvous with agent junder the
+hypothesis that a path from vtowcorresponds to the
+sequence s′of ports and the reversepath correspondsto
+the sequence s′′. For this to happen, the agent first tries
+to follow the path r(s′) induced by the sequence s′of
+ports (lines 8-11). This attempt is considered successful
+if the following conditions are satisfied:
+•at consecutive nodes of the traversed path, ports
+with numbers from the sequence s′are available,
+•the reverse path corresponds to the sequence s′′
+of ports.
+When the attempt is successful (the condition of
+line 13 is satisfied) the agent is at node wand it
+simulates the first k−1 phases of the execution of
+the algorithm by agent with label jstarting from w.
+The effect of this simulation is the path rsim. Upon
+completion of this part, agent with label ireturns to
+w. Now the agent is able to further extend its path
+to form a tunnel with the route of agent j(line 16).
+Finally, whether the attempt to follow the path r(s′) is
+successful or not, the agent with label ibacktracks to v
+(line 17). If l=j, the above actions are performed with
+the roles of iandjreversed and the role of s′ands′′
+reversed.
+Algorithm GraphRV calls the recursive function
+GraphRVREC . This function is called in two different
+modes controlled by the boolean mode. In the “main”
+mode (mode=true) the function is executed indef-
+initely, until rendezvous. In the “simulation” mode
+(mode=false), the function is executed for all values
+up to a given p, or until rendezvous, whichever comes
+first. The symbol⌢denotes the concatenation of se-quences.
+Algorithm GraphRV
+INPUT: A starting node v∈Vand a label lof the
+agent.
+OUTPUT: A rendezvous route r.
+GraphRVREC (v,l,0,true);
+function
+GraphRVREC (nodev,labell,integerp,boolean mode)
+1k:= 1;r:=λ;
+2whilenot rendezvous and ( k≤pormode)do
+3 letϕk= (i,j,s′,s′′);rhist:=r;
+4ifl=iorl=jthen
+5ifl=ithens1:=s′;s2:=s′′;l′:=j;
+6elses1:=s′′;s2:=s′;l′:=i;
+7 let s1= (p1,...,p n);m:= 1;r(s1) :=λ;
+8whilem≤nandpmis a port do
+9 r(s1):=r(s1)⌢(em)
+whereemcorresponds to port pm;
+10 let ambe the port corresponding
+toemat its other endpoint;
+11 m:=m+1;
+12r:=r⌢r(s1);
+13ifs2= (an,...,a 1)then
+14 let wbe the current node;
+15 rsim:=GraphRVREC (w,l′,k−1,false);
+16 r:=r⌢rsim⌢r(s1)⌢
+rhist⌢r(s1)⌢rsim;
+17r:=r⌢r(s1);
+18k=k+1;
+19returnr
+vrv
+rwqw
+rv⌢q⌢rw⌢q⌢rv⌢q⌢rw⌢q· · ·
+route ofagentj:
+rv⌢q⌢rw⌢q⌢rv⌢q⌢rwroute ofagenti:
+tunnel:rw⌢q⌢rv⌢q⌢rw⌢q⌢rv⌢q. . .
+Figure 1: Tunnel between the routes of two agents
+Theorem 3.1. Algorithm GraphRV guarantees asyn-
+chronous rendezvous for arbitrary two agents starting
+from any nodes of an arbitrary connected graph.Proof.Letvandi(resp.wandj) be the starting
+node and the label of the first agent (resp. the second
+agent). There exists a path qlinkingvtow, since the
+graphGis connected. Let s′(resp.s′′) be the finite
+sequence of ports corresponding to the path q(resp.
+the path q). In view of Proposition 2.1, it suffices to
+prove that the routes of the two agents form a tunnel.
+We show that, after the phase corresponding to the
+quadruple ϕk= (i,j,s′,s′′) during the execution of
+Algorithm GraphRV for agents iandj, the routes of
+the two agents form a tunnel. Observe that this phase
+eventually occurs during any execution of GraphRVREC ,
+since all recursive calls of any phase k′< kare done
+with a parameter pstrictly smaller than k′. Thus all
+these phases are completed in finite time.
+First, we show by induction that, at the beginning
+of phase kof any execution of Algorithm GraphRV, each
+agent is at its starting node. This is clearly true for
+k= 1. Assume that the property holds for k−1. It
+follows that during the execution of the phase k−1,
+the paths rhistandrsimare cycles. Hence, after the
+execution of line 16, the agent ends in node w. After
+theexecutionofline17, theagentreturnstothestarting
+nodevof the phase k−1. So, the agent starts phase k
+in the same node, and the property is true for all k.
+Letrv(resp.rw) be the output of the execution of
+the first k−1 phases of Algorithm GraphRV for agent i
+(resp.j) starting in node v(resp.w). At the beginning
+ofphasek, theportionofthe routeconstructedbyagent
+iisrv. After the execution of line 12, the portion
+of the route constructed by agent iisrv⌢q, since
+the agent has started the phase in node v. The path
+rsimcomputed by the recursive call of GraphRVREC
+is equal to rw. It follows that at the end of phase
+k, the portion of the route constructed by agent iis
+ρ=rv⌢q⌢rw⌢q⌢rv⌢q⌢rw⌢q. Similarly, at
+the end of phase k, the portion of the route constructed
+by agent jisρ′=rw⌢q⌢rv⌢q⌢rw⌢q⌢rv⌢q.
+By construction, the part rv⌢q⌢rw⌢q⌢rv⌢q⌢rw
+ofρand the part rw⌢q⌢rv⌢q⌢rw⌢q⌢rvofρ′
+form a tunnel (see Fig 1).
+4 Rendezvous in the geometric scenario
+In this section we consider the problem of rendezvous in
+aterrainincludedintheEuclideanplane. Asannounced
+in the introduction, we restrict attention to closed
+subsets of the plane whose interior is path-connected.
+We observed that these restrictions cannot be removed.
+Fix a system of coordinates Σ with the y-axis
+pointing to North (shown by compasses of the agents)
+and with the unit of length equal to that of the
+agents. Points with rational coordinates in Σ will be
+calledrational. A polygonal line all of whose vertices,including extremities, are rational will be called a
+rational line . For any point u, let Σ ube the shift of
+the system Σ with origin at point u.
+Lemma 4.1. In any path-connected, open subset Sof
+the plane and for any rational points u,v∈S, there
+exists a rational polygonal line included in S, with
+extremities uandv, all of whose vertices are rational.
+Proof.By path-connectivity of S, there exists a path
+pincluded in Swith extremities uandv, which is a
+continuous image of the interval [0 ,1]. Let dbe the
+distance from ptocS- the complement of S. Since
+p∩cS=∅and both pandcSare closed sets, we have
+d >0. Partition the plane into squares of side length
+at most d/2 with rational vertices. Let Qbe the set
+of squares intersecting p. Since pis a bounded set,
+Qis finite. Consider the graph Gpwith node set Q,
+such that x,y∈Qare adjacent if pcontains a point
+belongingtoacommonboundaryof xandy. SinceGpis
+connected, there exists a path ( x1,...,x k) inGplinking
+squaresx1containing uandxkcontaining v. Letp∗
+be the polygonal path ( uw1,w1w2,...,wk−1wk,wkv),
+wherewiis the center of square xi, for alli= 1,...,k.
+The path p∗is rational and contained in the union of
+squares from Q. Since each point of p∗is at distance at
+mostd√
+2/2 from some point of p, we have p∗∩cS=∅,
+hencep∗is included in S.
+We define the following graph GT= (V,E), for a
+given terrain T. The set of nodes Vis the union of
+two disjoint subsets V1,V2. The set V1is the set of all
+interior, rational points of Tand the set V2is defined
+below.
+For each pair of points p1,p2, such that p1∈V1and
+p2is any rational point of the plane, we consider the
+segment s=p1p2. Ifsdoes not intersect the boundary
+ofT, thenp2must belong to V1and we add the edge
+{p1,p2}toE. Ifsintersects the boundary of T, we add
+a new node vtoV2and we add the edge {p1,v}toE.
+Note that such a node vis always added to V2when
+pointp2is on the boundary of Tor outside of T(and it
+mayormaynotbe addedto V2whenp2isin the interior
+ofT). Since each node in V2corresponds to a pair of
+rational points p1,p2, there is a countable number of
+nodes in V2, each of them having degree 1. The unique
+port at any node in V2has number 1 and ports at any
+node in V1are defined as follows. Let ( z1,z2,...) be
+any fixed enumeration of all pairs of rational numbers.
+Letzi= (q1,q2). Letube the point in the plane with
+coordinates ( q1,q2) in the system Σ p. The port at p
+corresponding to edge {p,u}has number i.
+Algorithm GeometricRV
+The algorithm is a direct application of Algorithm
+GraphRV to the graph GT. The agent operating in anunknown terrain Tdesigns a route in the corresponding
+unknown graph GTas follows. When the agent chooses
+a port at a node p∈V1, this edge corresponds to
+some rational point qiin the plane that the agent
+tries to reach from p. Two cases may occur. The
+agent either walks in the interior of Tuntil reaching qi,
+which corresponds to the traversal of an edge between
+two nodes of V1, or it hits the boundary of T, which
+corresponds to a visit of a node p′∈V2. At a node
+p′∈V2there is no choice of port, since its degree is
+1. The agent takes the unique port which leads to the
+(already visited) node p∈V1. The resulting route is
+a sequence of segments joining rational interior points
+of the terrain Tand of pairs of consecutive segments
+(vb,bv), where vis a rational interior point of Tandb
+is a point on the boundary of T.
+Since by Lemma 4.1 graph GTis connected, ren-
+dezvous is guaranteed in the graph GT, which implies
+rendezvous in T.
+Theorem 4.1. Algorithm GeometricRV guarantees
+asynchronous rendezvous for arbitrary two agents
+starting from arbitrary rational interior points of any
+closed terrain Twith path-connected interior.
+Theorem 4.1 should be contrasted with the follow-
+ing negative result showing that the restriction on the
+starting points of the agents cannot be removed, even
+for rendezvous in the (empty) plane.
+Proposition 4.1. There is no algorithm that guaran-
+tees asynchronous rendezvous of arbitrary agents start-
+ing from arbitrary points in the plane.
+Proof.Consider the agent with label ℓoperating in the
+empty plane. Since the terrain is fixed, the route of
+this agent depends only on the starting point. Let
+R= (e1,e2,...) be the route of the agent with label
+1 starting at a fixed point v. Consider the route R2(w)
+ofthe agentwith label 2startingat point w. Since there
+are no boundary points in the terrain, for any starting
+pointsw′andw′′, routeR2(w′′) is a parallel shift of
+routeR2(w′) by the vector ( w′,w′′). Both of them are
+polygonal lines.
+We will say that two routes are almost disjoint if
+all vertices of each of them are outside the other route.
+Observe that if, for some starting point w, routeR2(w)
+of agent 2 is almost disjoint from route Rof agent 1,
+then the adversarycan avoidrendezvousofagents1 and
+2 following these routes, by first moving agent 1 to the
+end of the first segment of its route, then moving agent
+2 to the end of the first segment of its route and so on,
+alternating traversalsof agents on consecutive segments
+of their routes. Hence, in order to prove our result, it isenough to show the existence of a point w∗, such that
+routeR2(w∗) is almost disjoint from route R.
+Let (f1,f2,...) be the sequence of vectors corre-
+sponding to consecutive segments of the route R2(w),
+for any starting point w. For any fixed point w, de-
+note byfj(w) the segment corresponding to vector fj
+on route R2(w). Let (p1,p2,...) be any sequence that
+orders all couples ( ei,fj), for all positive integers i,j.
+Foranyk, letpk= (eik,fjk) We will constructby induc-
+tion a descending sequence of closed discs ( D1,D2,...)
+of positive radii, satisfying the following invariant. For
+allpoints w∈Dk, both endpoints ofthe segment eikare
+outside of the segment fjk(w) and both endpoints of the
+segment fjk(w) are outside of the segment eik. Suppose
+that the invariant is satisfied for k−1 (fork= 1 we may
+take asD0any disc with radius 1). The set of points w
+inside disc Dk−1that may possibly violate the invariant
+forkiscontainedintheunionoffoursegments: twoseg-
+ments parallel to fjkand two segments parallel to eik.
+There exists a closed disc of positive radius contained
+inDk−1which is disjoint from those four segments. Let
+Dkbe such a disc. Thus the invariant is satisfied for k.
+This completes the construction by induction.
+The intersection of all discs Dkis non-empty. Let
+w∗be a point in this intersection. Since ( p1,p2,...)
+enumerated all couples ( ei,fj), it follows that route
+R2(w∗) is almost disjoint from route R, and hence
+agents 1 and 2 starting at points vandw∗, respectively,
+do not meet for some walks chosen by the adversary.
+While Proposition 4.1 shows that rendezvous of
+agents starting from arbitrary points is impossible, it
+turns out that a slightlyeasiertaskcan be accomplished
+in this setting. For any ǫ >0, we say that routes R1
+andR2of agents guarantee ǫ-approximate rendezvous ,
+if at some point tof time, the agents get at distance at
+mostǫfrom each other, regardless of the walks chosen
+by the adversary.
+Theorem 4.2. Algorithm GeometricRV guarantees ǫ-
+approximate rendezvous for any ǫ >0, for arbitrary
+agents starting from arbitrary interior points of any
+closed terrain Twith path-connected interior.
+Proof.Fix anǫ >0. Consider any two agents starting
+at interior points vandwof the terrain T. Letρ >0
+be the distance from wto the boundary of T. Choose a
+pointw′with rational coordinates in Σ v, in the interior
+of the disc Dof radius ρ/2 centered at w. By Lemma
+4.1 (applied to the system Σ vinstead of Σ), there exists
+a rational polygonal line Pin the system Σ v, included
+in the interior of Tand joining vandw′. Letd >0
+be the distance between Pand the boundary of T. Let
+r= min(ρ/2,d/2,ǫ). Choose a point w′′with rationalcoordinates in the system Σ v, at distance less than r
+fromw. LetPvbe the polygonal line Pextended by
+the segment w′w′′. Note that Pvis at distance at least
+rfrom the boundary of T. Indeed, if x∈P, then the
+distance from xto the boundary is at least d > r, and if
+x∈w′w′′, then the distance from xto the boundary is
+at leastρ/2≥r, as the entire segment w′w′′is included
+in discD.
+Let Φ be the translation by the vector ( w′′w) and
+letPwbe the image of Pvwith respect to Φ. The point
+wis one of the extremities of Pw. Since the distance
+fromw′′towis less than r, the entire polygonal line
+Pwis in the interior of T. The polygonal line Pwis
+rational in the system Σ w.
+Letfbe any walk of the first agent on any route
+with the initial part Pvstarting at v, and let gbe
+any walk of the second agent on any route with the
+initial part Pwstarting at w. Consider the composition
+g∗= Φ−1◦g. Henceg∗is a walk on a route with initial
+partPv, starting at w′′. By Theorem 4.1, Algorithm
+GeometricRV guarantees rendezvous of agents at some
+point in time t, for the walk fstarting at vand the walk
+g∗starting at w′′.
+Consider the positions at time tof both agents
+starting at vandw, in walks fandg. Since f(t) =
+g∗(t) = Φ−1(g(t)), and Φ is a translation by a vector
+of length less than r(hence less than ǫ), it follows that
+the distance between f(t) andg(t) is less than ǫ. Since
+walksfandgwere arbitrarily chosen by the adversary,
+this guarantees ǫ-approximate rendezvous.
+A consequence of Theorem 4.2 is that if agents have
+arbitrarily small positive visibility ranges (rather than
+0 visibility range as we assumed) and they start in
+arbitrary points of the (empty) plane, then Algorithm
+GeometricRV guarantees that they will see each other
+in finite time, regardless of the actions of the adversary.
+5 Conclusion
+We provided deterministic asynchronous rendezvous
+algorithms for graphs and for terrains in the plane. We
+studied onlythe feasibilityofrendezvousand ourresults
+are very general: for the graph scenario, we showedthat
+rendezvousispossibleinanyconnectedcountable(finite
+or infinite) graph, starting from any nodes, without any
+information on the graph. The only thing an agent
+needs to know is its own label. In particular, this result
+implies a positive solution of a problem from [12].
+Our algorithms rely on an arbitrary fixed enumera-
+tionofquadruples( i,j,s′,s′′), whereiandjarepositive
+integers and s′ands′′are finite sequences of positive in-
+tegers. The complexity of the algorithm (measured by
+the worst-case length of paths that the agents have totraverseuntil rendezvous)depends on this enumeration,
+but we do not think any enumeration can make the al-
+gorithm efficient. Thus a natural interesting question
+left for further investigations is the following:
+Does there exist a deterministic asyn-
+chronous rendezvous algorithm, working for
+all connected countable unknown graphs, with
+complexity polynomial in the labels of the
+agents and in the initial distance between
+them?
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\ No newline at end of file
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+arXiv:1001.0891v3  [hep-ph]  9 Jan 2010Some Characteristic Parameters of Proton from the Bag Model
+Z. G. Tan,1,∗L. Y. Huang,1and C. B. Yang2
+1Department of Electronics and Communication Engineering,
+Changsha University,Changsha, 410003,P.R.China
+2Institute of Particle Physics, Hua-Zhong Normal University , Wuhan, 430079, China
+We treat the mass of a proton as the total static energy which c an be separated into two parts
+that come from the contribution of quarks and gluons respect ively. We adopt the essential of the bag
+model ofhadrontodiscuss thestructure ofaprotonand findth atthe calculated temperature, proton
+radius, the bag constant are compare well with QCD results if a proton is a thermal equilibrium
+system of quarks and gluons.
+PACS numbers: 12.39.Ba, 14.20.Dh
+I. INTRODUCTION
+Exploring proton structure [1–3] is still one of most
+important subjects for more profound enhancement of
+human knowledge on strong interactions. It is also very
+helpful for people to search for a new matter state–quark
+gluon plasma (QGP) which is the deconfined state of
+strongly interacting matter. Most theoretical investi-
+gations focus on Quantum Chromodynamics (QCD) [4].
+However,there are a few phenomenological models about
+nucleon structure and interactions. The classical string
+model describesmesons asstringsegmentsexecuting lon-
+gitudinal expansion and contraction [5]. The bag model
+describes quarks being confined inside a hadron [6]. In
+high energy collisions, the string model represents the
+process of particle production in the fragmentation of
+a stretching string by creating pairs of quark and anti-
+quark. It worked well [7] for elementary collisions where
+strings can be formed among the few initial partons and
+breakup toformthe softfinalstatehadrons. However,in
+relativistic heavy ion collisions (RHIC), there are thou-
+sands initial partons. It is intractable to pair partons
+and have a string for each pair. Even if the strings are
+formed, they must be modified by the presence of many
+other color charges. The independent fragmentation ap-
+proach [7], though valid for high Q2partonic processes
+in the vacuum, cannot explain experimentally observed
+largep/πratio in central Au+Aucollisions at RHIC [8].
+Another possible hadronization mechanism, the decon-
+fined quark (fled out from the bag) recombinationmodel,
+has been able to reproduce spectra for almost all stable
+particles for different colliding systems. It provides a
+natural explanation for the baryon/meson ratio and the
+nuclear suppression factor observed at RHIC[9].
+A natural mechanism for quark confinement is given
+by the bag model [6]. While the bag model has a few
+different versions, we shall in this paper keep the essen-
+tial characteristics of the phenomenology of quark con-
+finement. The gluons are mediate bosons which transfer
+∗tanzg@iopp.ccnu.edu.cnthe interactions between quarks . Their effect can be re-
+placed by the bag pressure which confines the quarks in
+a hadron. On the other hand, if all interactions among
+partonsin the bagare neglected, we assume that the par-
+tons might be treated as a thermally equilibrated system
+with a given volume. Then properties of a hadron can be
+investigated and some characteristic parameters for the
+hadron can be obtained in the bag model.
+The organization of this paper is as follows. In Sec.II,
+we will discuss some features about a thermally equili-
+brated QGP system. Then in Sec.III, we get an estimate
+of the maximum kinetic energy of a confined quark in a
+spherical cavity of radius R. Combining the discussions
+in section II and III by assuming its origin from the con-
+tribution of gluons, the magnitude of the hadronic bag
+pressure is discussed in Sec.IV. The last section is for
+conclusions and discussions.
+II. FREE EQUILIBRATED QGP
+Let us first consider a thermally equilibrated quark-
+gluon plasma (QGP) system at first. When its temper-
+atureTand volume Vare given, the total energy and
+particle number can be calculated:
+E=Nf/summationdisplay
+i=−Nfgi
+(2π¯h)3/integraldisplay
+fi(T)p0dΓ
+=Nf/summationdisplay
+i=−NfgiV
+2π2¯h3/integraldisplay
+fi(T)p0|p|2dp, (1)
+N=Nf/summationdisplay
+i=−Nfgi
+(2π¯h)3/integraldisplay
+fi(T)dΓ
+=Nf/summationdisplay
+i=−NfgiV
+2π2¯h3/integraldisplay
+fi(T)|p|2dp, (2)
+whereNfis the number of flavors and gi=NcNsis
+the degeneracy number for a parton and equals to the
+product of quantum numbers of quark’s color and spin.
+fiis the distribution function which is of Fermi-Dirac for2
+0.06 0.08 0.1 0.12 0.14 0.16 0.180.511.52R (fm)
+T (GeV)R = 0.0524 T −4/3
+FIG. 1: The relation between the radius of a proton and its
+temperature when take its mass as the total energy. The
+numerical simulative formula is given on the legend as well.
+quarks and Bose-Einstein for gluons
+fi=1
+1+e(p0∓µq)/T’-’ for quark, (3)
+’+’for anti-quark
+fi=1
+ep0/T−1for gluon. (4)
+Hereµqis the quark’s chemical potential. For the case
+when the number density of the quarks is the same as
+that of the anti-quarks, µq= 0.
+For a massless quark gas with zero chemical potential
+µq= 0, Eqs.(1,2) can be solved analytically. For example
+for the case with only two flavors, we get the densities
+for energy and quark number as (¯ h= 1)
+ǫ=E
+V=7
+4(gq+g¯q)π2
+30T4+ggπ2
+30T4
+=37
+30π2T4, (5)
+n=N
+V=3ζ(3)
+2π2(gq+g¯q)T3+ggΓ(3)ζ(3)
+2π2T3
+=34×1.202
+π2T3. (6)
+Now if we treat a proton as a free equilibrated QGP
+system asuse the aboveresults, we canget the relationof
+the proton radius and temperature. The result is shown
+in Fig. (1).
+From Fig.(1), we can see, the radius of a proton is
+decreased with its internal temperature. The numerical
+simulative formula is
+RT4/3= 0.0524 (7)
+If the temperature is 100 MeV, the corresponding radius
+ofprotonis about 1fm. Ifa proton is compressedto have
+a radius of 0.6 fm, the inner temperature will be about
+170 MeV, which is close to the critical temperature, sothat a proton may break and quarks may flee out from
+the bag.
+III. QUARKS CONFINED IN A HADRON BAG
+Let’s give some discussions about quark wave-function
+from theory. We assume the quarks are confined in a
+spherical cavity of radius R, they are free fermions in-
+side it but cannot fly out because all contributions from
+gluons are attributed to the bag constant. Then the sur-
+face of the sphere bag becomes the maximum range that
+quarks can arrive. On the bag boundary where the cur-
+rent of fermion must be zero. A hadron’s wave-function
+is then product of those for quarks.
+The Dirac equation for a free massless fermion in the
+bag is
+(iγµ∂µ−m)ψ= 0 with m= 0, (8)
+where∂µ= (p0,p). We will in this paper use the Dirac
+representation
+γ0=/parenleftbigg
+I0
+0−I/parenrightbigg
+,
+and
+γi=/parenleftbigg
+0σi
+−σi0/parenrightbigg
+,
+whereIis a 2×2 unit matrix and σiare the Pauli matri-
+ces. We write the four-component wave function for the
+massless fermion ψas
+ψ=/parenleftbiggψ+
+ψ−/parenrightbigg
+where both ψ+andψ−are two dimensional spinors.
+Eq.(8) becomes
+/parenleftbigg
+p0−σ·p
+σ·p−p0/parenrightbigg/parenleftbiggψ+
+ψ−/parenrightbigg
+= 0 (9)
+The lowest energy solution for the above equation is the
+s1/2state given by[10]
+ψ+(r,t) =Ne−ip0tj0(p0r)χ+
+ψ−(r,t) =Ne−ip0tσ·ˆ rj1(p0r)χ−,
+wherejlis the spherical Bessel function which can be
+expressed by an elementary function
+jl(x) = (−1)lxl/parenleftbigg1
+xd
+dx/parenrightbigglsinx
+x, (10)
+χ±aretwo-dimensionalspinors,and Nisanormalization
+constant. The confinement of the quarks is equivalent
+to the requirement that the normal component of the
+vector current Jµ=¯ψγµψvanishes at the surface. This3
+condition is the same as the requirement that the scalar
+density¯ψψof the quark vanishes at the bag surface r=
+R. This leads to
+¯ψψ/vextendsingle/vextendsingle
+r=R= [j0(p0R)]2−σ·ˆ rσ·ˆ r[j1(p0R)]2= 0
+or
+[j0(p0R)]2−[j1(p0R)]2= 0 (11)
+From Eq.(10), solution of the above equation is given by
+p0
+mR= 2.04,orp0
+m=2.04
+R. (12)
+This result means that in order to keep the bag from
+being broken, the kinetic energy of any quark can not
+larger than p0
+mdetermined by the radius of the bag.
+IV. BAG PRESSURE
+Takep0
+masthe upper limit, we can separatethe energy
+of quarks from the total of a proton
+Eq=(gq+g¯q)V
+2π2¯h3/integraldisplayp0
+m
+0p3dp
+1+ep/T(R).(13)
+For simplicity, we have neglected the chemical poten-
+tial, and treat the quarks as massless. Then gq=g¯q=
+NcNsNf= 3×2×2 = 12.
+The energy from gluon contribution provides the pres-
+sure effect directed from the region outside the bag
+B=M−Eq
+V. (14)
+HereMis the mass of the hadron.
+From Eq.(14), we can easily learn that the the bag
+pressure will change with radius as shown in Fig.(2). We
+also give the numerical formula
+B1/4= 0.17R−0.65. (15)
+The average kinetic energy of each quark can be calcu-
+lated as
+¯Eq=Eq
+Nq=/integraltextp0
+m
+0p3dp
+1+ep/T(R)
+/integraltextp0m
+0p2dp
+1+ep/T(R). (16)The energy carried by the valence quarks in a proton
+is then 3 ׯEq. So that contributions from gluons and
+sea quarks to the energy is M−3¯Eq. The bag pressure
+decreases with the radium. This may be used to explain
+why the resonance particle (which has large radius thus
+smallbagpressure)is usuallyunstablebecause theirbags
+are more fragile.
+0.8 11.2 1.4 1.6 1.8 2100110120130140150160170180190200
+B1/4 = 0.17 R−0.65
+R (fm)B1/4 (MeV)
+FIG. 2: The bag pressure change with the radium of proton.
+The open circle is calculated with Eq.(14), and the line is
+its numerical simulation results, while the star is with the
+scenario that three quarks are surrounded by gluons.
+V. CONCLUSION
+In this work we have discussed some feature of a pro-
+ton from the bag model. Especially, we got the relations
+between temperature and the radius of the proton when
+a proton is treated as a non-interacting thermal equilib-
+rium QGP system. The bag pressure comes from the
+contribution of gluons.
+Acknowledgments
+We are grateful to the financial support from China
+ChangSha University under grant No.SF080101. We also
+thank Prof. A. Bonasera for stimulating discussions and
+comments.
+[1] Marc Vanderhaeghen, Nucl. Phys.A 805:210-220,2008.
+[2] J. Sowinski, Nucl.Phys.A 790:485-488,2007,
+[3] A. Bhattacharya, S. N. Banerjee, B.Chakraborti, S.
+Banerjee, Nucl. Phys. Proc. Suppl. 142 (2005) 13-15
+[4] F. Wilczek, Ann. Rev. Nucl. and Part. Sci. 32, 177(1982)
+[5] X. Artru and G. Mennessier, Nucl. Phys. B70,
+93(1974),B.Andersson, G. Gustafson and B. S¨ oderberg,
+Z. Phys. C20,317(1983)[6] C. D. Detar and J. F. Donoghue, Ann. Rev. Nucl. Part.
+Sci. 33,235(1983)
+[7] Torbj¨ orn Sj¨ ostrand, Leif L¨ onnblad, Stephen Mrenna, Pe-
+ter Skrands, JHEP 0605(2006) 026
+[8] R. C. Hwa and C. B. Yang, J. Phys. G30(2004) S1117-
+S1120
+[9] C. B. YANG,Int. J. Mod. Phys. E 16, No. 10, 3148(2007)
+[10] C. Y. Wong, Introduction to High-Energy Heavy ion Col-4
+lisions, World Scientific Co., Singapore,1994.
+[11] Z. G. Tan, S. Terranova and A. Bonasera, Int. J. Mod.
+Phys. E17 No 8(2008)1577-1589;[12] Z. G. Tan, A. Bonasera, C. B. Yang, D. M. Zhou and
+S. Terranova. Int. J. Mod. Phys. E16, Nos.7&8 (2007)
+2269-2275
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+arXiv:1001.0893v2  [cond-mat.dis-nn]  11 Mar 2010Sample-to-sample fluctuations and bond chaos in the m-component spin glass
+T. Aspelmeier
+Max Planck Institute for Dynamics and Self Organization, Bu nsenstr. 10, 37073 G¨ ottingen, Germany and
+Scivis GmbH, Bertha-von-Suttner-Str. 5, 37085 G¨ ottingen , Germany
+A. Braun
+Institut f¨ ur Theoretische Physik, Universit¨ at G¨ otting en,
+Friedrich-Hund-Platz 1, 37077 G¨ ottingen, Germany
+We calculate the finite size scaling of the sample-to-sample fluctuations of the free energy ∆ Fof
+themcomponent vector spin glass in the large- mlimit. This is accomplished using a variant of the
+interpolating Hamiltonian technique which is used to estab lish a connection between the free energy
+fluctuations and bond chaos. The calculation of bond chaos th en shows that the scaling of the free
+energy fluctuations with system size Nis ∆F∼Nµwith1
+5≤µ <3
+10, and very likely µ=1
+5exactly.
+I. INTRODUCTION
+Spin glass physics1continue to be a very exciting and difficult topic. One of the important ongoing issues are
+the finite size corrections to thermodynamic quantities2–11. Such finite size corrections are usually impossible to
+calculate within replica theory (which is otherwise extremely success ful for the spin glass and other problems), due to
+the massless modes which are often encountered and which preven t going beyond zero-loop order in a perturbation
+expansion. Nevertheless, many results are by now established for the Sherrington-Kirkpatrick spin glass12, either
+numerically or analytically (see references cited above).
+An observable which is particularly interesting is the finite size scaling o f the sample-to-sample fluctuations of
+the free energy. This quantity provides a link between two apparen tly distant fields, namely spin glass physics
+and extreme value statistics13. In extreme value statistics, the question is the probability distribu tion of extremal
+events such as the maximum (or minimum) of a set of random numbers . The classic results of extreme value theory
+state that such extremal events follow one of three possible limiting distributions (the Weibull, Gumbel or Fr´ echet
+distribution). Recently, a fourth universality class was found, the Tracy-Widom distribution for the smallest (or
+largest) eigenvalue of a Gaussian random matrix14. Similarly, one could ask what the distribution of the ground
+state energy of a (Sherrington-Kirkpatrick or other) spin glass is , which is a question of extreme value statistics. In a
+statistical mechanics setting, however, it can be generalized to th e question of what the distribution of the freeenergy
+is at finite temperature. This is a very difficult question indeed (both f or finite and zero temperature). To keep it
+simpler, we merely ask what the widthof the distribution is (i.e. the sample-to-sample fluctuations), and t his width
+∆Fwill scale in some way with the system size N, i.e. ∆F∼Nµwith an exponent µ. These free energy fluctuations
+have been considered for the Sherrington-Kirkpatrick model by n umerical investigations, see e.g. Refs. 6,7,10. There
+also exist heuristic arguments for µ=1
+46andµ=1
+69,11,15–17, and the limit µ≤1
+4has been shown18,19. All these
+results show that the Sherrington-Kirkpatrick model does not fa ll in any of the four established universality classes
+of extreme value statistics. For a different famous replica symmetr ic spin glass model, the spherical spin glass20,
+the situation is different. Since its groundstate energy is the smalles t eigenvalue of a Gaussian random matrix, this
+model falls into the Tracy-Widom universality class21. This implies that the fluctuations of the free energy scale as
+∆F∼N1/3, which has been confirmed recently by a replica calculation11.
+In this work we consider the fully connected mcomponent vector spin glass in the limit of large m. This model is
+known to be replica symmetric22, which suggests that its free energy fluctuations might fall into th e same universality
+class as the spherical model. Furthermore it might be expected tha t due to replica symmetry the fluctuations in
+this model are simpler to calculate than in the Ising spin glass. Unfort unately, this hope is not entirely justified, as
+we will see. As mentioned above, it is usually impossible to calculate sube xtensive quantities within replica theory.
+We circumvent this problem by using a connection between the sample -to-sample fluctuations of the free energy and
+bond chaos, which was derived for the Ising spin glass19, and which we generalize here to vector spin glasses. This
+connection allows us to calculate the sample-to-sample fluctuations by calculating bond chaos instead, and this is
+possible within a large deviation approximation, combining techniques f rom Ref. 23 and 24. As it will turn out,
+the large deviation approximation is not good enough to obtain the fin al answer, and we must resort to additional
+resources, such as our previous knowledge of the scaling propert ies of the large- mmodel25–28and some additional
+scaling assumptions. To forestall our main result, we obtain µ≤1
+5≤3
+10and most likely µ=1
+5exactly. This shows
+that the large- mspin glass does not belong to any of the four universality classes and probably to a different class
+than the Ising spin glass (unless it so happens that e.g. µ=1
+4for both the Ising and the large- mspin glass, which is
+not entirely ruled out by our results but which we deem unlikely). The r esultµ=1
+5was first suggested in Ref. 25.2
+However, the derivation required the existence of a gap in the eigen value spectrum of the inverse susceptibility matrix
+which was later shown not to exist26. Here, we provide a different explanation for this value of µ.
+This paperis organizedas follows. In sectionII we will showhow toder ivethe exactconnection between the sample-
+to-sample fluctuations of the free energy, ∆ FN, and bond chaos for the m-component spin glass using interpolating
+Hamiltonians. We will see that the fluctuations only depend on the finit e size scaling of averages of powers of the
+link overlap qLbetween two copies of a spin glass with different (but correlated) dis order. The averages of the link
+overlaps will be calculated in section III by calculating the probability d istribution of the link overlap, Pǫ(qL), where
+the parameter ǫmeasures the degree of correlation. We will use these results in sec tion IV to derive the exponent µ
+of the free energy fluctuations.
+II. CONNECTION BETWEEN THE SAMPLE-TO-SAMPLE FLUCTUATIONS AND BOND CHAOS
+We use the technique of interpolating Hamiltonians29in order to derive an exact connection between the sample-
+to-sample-fluctuations of the free energy, ∆ FN, and bond chaos in the m-component spin glass. To do so, we adapt
+the method from Ref. 19, which was developed for the Ising spin glas s, to the m-component vector spin glass. The
+Hamiltonian of the m-component spin glass is
+H=/radicalbigg
+1
+N/summationdisplay
+i<jJij/vector si/vector sj, (1)
+where/vector siarem-component spins and Jijare independent Gaussian random variables with unit variance. The s pins
+are assumed to be normalized in such a way that /vector s2
+i=m.
+The main idea for calculating the sample-to-sample fluctuations of th e free energy is to use the following two
+Hamiltonians,
+Ht=−/radicalbigg
+1−t
+N/summationdisplay
+i<jJij/vector si/vector sj−/radicalbigg
+t
+N/summationdisplay
+i<jJ′
+ij/vector si/vector sjand
+H′
+τ=−/radicalbigg
+1−τ
+N/summationdisplay
+i<jJij/vector si/vector sj−/radicalbiggτ
+N/summationdisplay
+i<jJ′′
+ij/vector si/vector sj, (2)
+with 0≤t,τ≤1 andJ′
+ijandJ′′
+ijadditional independent Gaussian random variables with unit variance , and express
+the free energy fluctuations in terms of them. These Hamiltonians in terpolate between a spin glass with a given
+set of coupling constants {Jij}fort= 0 (orτ= 0) and an identical spin glass with a different, independent set of
+coupling constants {J′
+ij}or{J′′
+ij}fort= 1 (resp. τ= 1). For all 0 < t,τ <1 we have the same type of spin glass but
+with coupling constants√1−tJij+√
+tJ′
+ij(or√1−τJij+√τJ′′
+ij), which are still Gaussian random variables with
+unit variance. We denote the disorder average with respect to all c oupling constants {Jij},{J′
+ij}and{J′′
+ij}asE···.
+Thermal averages will be denoted as ∝an}bracketle{t···∝an}bracketri}ht.
+The free energy fluctuations can be written as β2∆F2
+N=E(logZ1−logZ0)(logZ′
+1−logZ′
+0), withZtandZ′
+τbeing
+the partition functions with respect to the Hamiltonians HtandH′
+τ. This can be seen by writing log Z=−βFand
+using the independence of the different sets of bonds for EF1F′
+0=EF0F′
+1=EF1F′
+1=F2andEF0F′
+0=F2(the
+overbardenotes a disorder averagedquantity of the original sys tem of interest, i.e. Eq. (1)). We use the idea in Ref. 29
+to represent this expression by integrals in the form
+β2∆F2
+N=/integraldisplay1
+0dt/integraldisplay1
+0dτE∂
+∂tlogZt∂
+∂τlogZ′
+τ. (3)
+To calculate the right hand side, we follow Ref. 19 and obtain
+E∂
+∂tlogZt∂
+∂τlogZ′
+τ=β2
+16N2E/summationdisplay
+ijkl/parenleftbig
+∝an}bracketle{t(/vector si/vector sj)(/vector sk/vector sl)∝an}bracketri}htt−∝an}bracketle{t/vector si/vector sj∝an}bracketri}htt∝an}bracketle{t/vector sk/vector sl∝an}bracketri}htt/parenrightbig/parenleftbig
+∝an}bracketle{t(/vector si/vector sj)(/vector sk/vector sl)∝an}bracketri}htτ−∝an}bracketle{t/vector si/vector sj∝an}bracketri}htτ∝an}bracketle{t/vector sk/vector sl∝an}bracketri}htτ/parenrightbig
++β2
+8N√1−t√1−τ/parenleftBig
+E/summationdisplay
+ij/parenleftbig
+∝an}bracketle{t/vector si/vector sj∝an}bracketri}htt∝an}bracketle{t/vector si/vector sj∝an}bracketri}htτ/parenrightbig
+−m2N/parenrightBig
+. (4)
+The subscript torτon the thermalaveragesindicates whether the averageis to be ta kenin a system with Hamiltonian
+HtorH′
+τ.3
+We introduce the link overlap between two replicas with potentially diffe rent coupling constants as q13
+L=
+2
+N(N−1)/summationtext
+i<j∝an}bracketle{t/vector si/vector sj∝an}bracketri}htt∝an}bracketle{t/vector si/vector sj∝an}bracketri}htτ=1
+N(N−1)/parenleftBig/summationtext
+µν∝an}bracketle{t/parenleftbig/summationtext
+is(1)
+iµs(3)
+iν/parenrightbig2∝an}bracketri}ht −m2N/parenrightBig
+. As we will shortly need not only 2 but up
+to 4 replicas, we label the replicas with upper indices (1) to (4), wher e replicas 1 and 2 have Hamiltonian Htand
+replicas 3 and 4 have Hamiltonian H′
+τ. Analogously to q13
+L, we define the other possible link overlaps q14
+L,q23
+Landq24
+L.
+Then equation (4) can be expressed in the following way
+E∂
+∂tlogZt∂
+∂τlogZ′
+τ=(N−1)2β4
+16E∝an}bracketle{t(q13
+L−q23
+L)(q13
+L−q14
+L)∝an}bracketri}ht+(N−1)β2
+8√1−t√1−τE∝an}bracketle{tq13
+L∝an}bracketri}ht
+=(N−1)2β4
+16/parenleftBig
+[(q13
+L)2]−[q13
+L]2/parenrightBig
++(N−1)β2
+8√1−t√1−τ[q13
+L]. (5)
+The notation [ ...] in the last line stands for the average taken with the bond average d probability distribution Pǫ(qL)
+of finding a given link overlap qL. The parameter ǫindicates the statistical “distance” between the sets of bonds in
+the two replicas involved and will be defined in detail below. In principle, one would have to consider simultaneous
+multi-replica overlaps (such as, for instance, the term E∝an}bracketle{tq13
+Lq23
+L∝an}bracketri}ht). Fortunately, it follows from replica symmetry,
+which holds for the m-component spin glass in the large- mlimit, that the corresponding joint probability distribution
+P123
+ǫ(q13
+L,q23
+L) factorizesinto P123
+ǫ(q13
+L,q23
+L) =Pǫ(q13
+L)Pǫ(q23
+L). Asimilarstatement holdsforthe four-replicaprobability
+distribution P1234
+ǫ(q13
+L,q24
+L).
+Up to now the overlaps qab
+Ldepend on the two parameters tandτ. However, the only important quantity is how
+much the sets of bonds in the two replicas aandbdiffer. For example, when t=τ= 0, the bonds in replicas 1
+and 3 (say) are identical. On the other hand, when t= 0 and τ= 1 (or vice versa), the bonds in replicas 1 and 3
+are completely uncorrelated. The degree of correlation between t he two sets of bonds can be measured by the one
+parameter ǫdefined by1√
+1+ǫ2=√1−t√1−τ, as shown in Ref. 19. When ǫ= 0, the bonds are identical; when
+ǫ=∞, the bonds are completely uncorrelated.
+The parameter ǫcan be used to eliminate the integration variable τfrom Eq. (3) by a variable transformation. The
+integralover tcanthenbe evaluatedexactly, andasaresult, theconnectionbet weenthesample-to-sample-fluctuations
+of the free energy and bond chaos in the m-component spin glass is found to be (see Ref. 19 for details)
+β2∆F2
+N=(N−1)2β4
+16/integraldisplay∞
+0dǫf2(ǫ)/parenleftbig
+[(q13
+L)2]−[q13
+L]2/parenrightbig
++(N−1)β2
+4/integraldisplay∞
+0dǫg2(ǫ)[q13
+L],
+=:(N−1)2β4
+16I21+(N−1)β2
+4I22 (6)
+with the nonnegative functions f2(ǫ) =2ǫlog(1+ǫ2)
+(1+ǫ2)2andg2(ǫ) =ǫlog(1+ǫ2)
+(1+ǫ2)3/2. Note that bond chaos enters Eq. (6)
+through the measure of distance ǫof the bonds of the two replicas between which the link overlap q13
+Lis calculated.
+The integrals I21andI22will be calculated below.
+The analog of Eq. (6) was called “second route to chaos” in Ref. 19, hence the first index is 2 on I21andI22. In
+addition to this result, however, there is a another exact relation b etween the fluctuations and bond chaos (the “first
+route to chaos”). It stems from using only the first interpolating H amiltonian of equation (2), Ht, and the relation
+2β2∆F2
+N=E(logZ1−logZ0)2, which is easy to derive. Proceeding similarly to above, one can prove the following
+equality:
+β2∆F2
+N=−(N−1)2β4
+16/integraldisplay∞
+0dǫf1(ǫ)/parenleftbig
+[(q13
+L)2]−[q13
+L]2/parenrightbig
++(N−1)β2
+4/integraldisplay∞
+0dǫg1(ǫ)[q13
+L]
+=:−(N−1)2β4
+16I11+(N−1)β2
+4I12. (7)
+The only difference between this equation and Eq. (6) is the minus sign in front of the first term and the weight
+functions f1(ǫ) =4ǫ2
+(1+ǫ2)2arcsin1√
+1+ǫ2andg1(ǫ) =2
+(1+ǫ2)3/2arcsin1√
+1+ǫ2in the integrals I11andI12instead of f2(ǫ)
+andg2(ǫ) as inI21andI22. The minus sign of the first term implies that the second term is an upp er bound of
+β2∆F2
+N.
+III. CALCULATING Pǫ(qL)
+Eq. (6) involves moments of the link overlap taken with the probability densityPǫ(qL). This function can in
+principle be calculated by taking two replicas with bond realizations dra wn with parameter ǫand constraining the4
+replicas to have link overlap qL. This constrained system has free energy Fǫ,J(qL), and the (non disorder averaged)
+probability density Pǫ,J(qL) then follows to be
+Pǫ,J(qL) =exp(−βFǫ,J(qL))/integraltext∞
+0dq′
+Lexp(−βFǫ,J(q′
+L)). (8)
+Finally,Pǫ,J(qL) must be averaged over the disorder.
+Unfortunately, this task is too difficult in general. Instead, we will ca lculate only the disorder averaged extensive
+part of the free energy, denoted by Nmfǫ(qL), and the probability density defined by
+P0
+ǫ(qL) =exp(−βNmf ǫ(qL))/integraltext∞
+0dq′
+Lexp(−βNmf ǫ(q′
+L)). (9)
+This is the large deviation approximation to Pǫ(qL). Averages taken with respect to P0
+ǫ(qL) will be denoted by [ ...]0.
+The tail of this distribution will be the same as that of Pǫ(qL), but in general there will be deviations. In fact, this
+is the point where this paper differs most from Ref. 19 since in that pu blication, the difference between the true and
+the approximative distribution was only quantitative, whereas here it is substantial. We know that for ǫ= 0 the link
+overlap distribution consists of a δpeak at the Edwards-Anderson value and 0 elsewhere since the larg e-mspinglass
+is replica symmetric. We will see below, however, that we do not obser ve this peak in P0
+ǫ(qL) at all. It must therefore
+be generated from the finite size corrections to the extensive par t of the free energy.
+Hence the finite size corrections are very important in this calculatio n, but we have no direct way of calculating
+them. In order to overcome this problem, we will show that P0
+ǫ(qL) will be valid for ǫlarger than some crossover
+value, and we will use additional arguments and simulation results to fi ll the gap for smaller ǫ.
+A. Replica calculation
+The first task is to calculate the extensive part of the disorder ave raged free energy Nmfǫ(qL) of two replicas
+constrained to have link overlap qL. To this end we calculate the partition function Zǫ,J(qL) of this two replica
+system. This system has vector spins /vector sx
+iwith (/vector sx
+i)2=mfor every spin i= 1,...,Nand the two real replicas
+x= (10),(2ǫ) (this notation denotes the replica number in its first entry and the value ofǫin its second entry). These
+two replicas differ in their coupling constants Kij(ξ) =1√
+1+ξ2K0
+ij+ξ√
+1+ξ2K′
+ij, whereK0
+ijandK′
+ijare independent
+Gaussian random numbers with unit variance, by choosing ξ= 0 for the first replica and ξ=ǫfor the second replica.
+Furthermore, the two replicas have link overlap qL, which is enforced by a δfunction in the partition function as
+follows:
+Zǫ,J(qL) = Tr /vector s/parenleftBig
+δ/parenleftbig
+qL−1
+N(N−1)/summationdisplay
+µν/parenleftbig/summationdisplay
+is(10)
+iµs(2ǫ)
+iν/parenrightbig2−m2
+N−1/parenrightbig/parenrightBig
+×exp/parenleftBigβ√
+N/summationdisplay
+i<jKij(0)/vector s(10)
+i/vector s(10)
+j+β√
+N/summationdisplay
+i<jKij(ǫ)/vector s(2ǫ)
+i/vector s(2ǫ)
+j/parenrightBig
+(10)
+We follow the replica calculation of Viana23for thentimes replicated partition function, write the δ-function in an
+integral representation with parameters zα, and introduce the traceless tensor Tµν
+α23to separate the diagonal part of
+Qµν
+αβ(with/summationtext
+α,β
+µ,νQµν
+αβ= 2/summationtext
+α<β
+µ,νQµν
+αβ+2/summationtext
+α,µ<νTµν
+α+/summationtext
+α,µ/parenleftbig
+Qαα+Tµµ
+α/parenrightbig
+). After evaluating the trace and regarding5
+only terms up to third order in the tensors Q,TandRwe get
+EZn
+ǫ,J(qL)∝/integraldisplay/parenleftbig/productdisplay
+αdzα/parenrightbig
+e−N/summationtext
+αqLzα−m2/summationtext
+αzα/integraldisplay
+dΛµν
+αβexp/parenleftBig
+N/bracketleftBig
+τ/summationdisplay
+α<β
+µ,ν/parenleftbig
+Qµν
+αβ/parenrightbig2+/parenleftbig
+τ+r
+m+2/parenrightbig/summationdisplay
+α,µ<ν/parenleftbig
+Tµν
+α/parenrightbig2
++/parenleftbig
+τ+r′
+m+2/parenrightbig/summationdisplay
+α,µ/parenleftbig
+Tµµ
+α/parenrightbig2+τ′/summationdisplay
+α/negationslash=β
+µ,ν/parenleftbig
+Rµν
+αβ/parenrightbig2+/summationdisplay
+α,µ,ν/parenleftbig2zα
+β2+1√
+1+ǫ2/parenrightbig(−1)/parenleftbig
+Rµν
+αα/parenrightbig2
++ω
+3!/parenleftBig/summationdisplay
+α<β<γ
+µ,ν,ρQµν
+αβQνρ
+βγQρµ
+γα+m2
+(m+2)(m+4)/summationdisplay
+α
+µ<ν<ρTµν
+αTνρ
+αTρµ
+α+15m2
+(m+2)(m+4)/summationdisplay
+α,µ/parenleftbig
+Tµµ
+α/parenrightbig3
++3m
+m+2/summationdisplay
+α<β
+ν,µ<ρQµν
+αβQνρ
+βαTρµ
+α+3/summationdisplay
+(α<β)/negationslash=γ
+µ,ν,ρQµν
+αβRνρ
+βγRρµ
+γα+3/summationdisplay
+α<β
+µ,ν,ηQµν
+αβRνη
+βαRηµ
+αα
++3m
+m+2/summationdisplay
+α/negationslash=β
+(µ<ν),ρTµν
+αRνρ
+αβRρµ
+βα+3/summationdisplay
+α
+µ<ν,ηTµν
+αRνη
+ααRηµ
+αα/parenrightBig/bracketrightBig/parenrightBig
+, (11)
+withτ=1
+2(1−1
+β2),τ′=1
+2(1−√
+1+ǫ2
+β2),r=−1,r′= 2m−1,ω= 1 and the matrix Λµν
+αβis given by
+Λ =/parenleftbigg
+Q(10)R
+R Q(2ǫ)/parenrightbigg
+, (12)
+with the above separation for Qµν
+αβand only Tµµ
+αon the diagonal of the matrices Q. We split the tensor Rinto its
+diagonal matrices pµν
+d=Rµν
+ααand the rest. This equation is solved by a saddle point integration for the tensors Qµν
+αβ,
+Tµν
+αandRµν
+αβand the parameters zα.
+B. Solving the saddle point equations
+Solving the saddle point equations is the remaining task to derive the c haracteristic form of the overlap distribution
+Pǫ(qL). As the m-component spin glass was shown to be replica symmetric in the limit m→ ∞22, we calculate the
+saddle points in the replica symmetric case. The ansatz for a replica s ymmetric scenario is as follows23,24:
+Qµν
+αβ=Qδµν, Tµν
+α= 0
+Rµν
+αβ=Pδµν, pµν
+d=pdδµνandzα=z (13)
+To briefly justify these equations, one has the usual interpretat ionQµν(10)
+αβ=∝an}bracketle{ts(10)
+αµs(10)
+βν∝an}bracketri}htfrom the replica calculation.
+Due to the isotropy of the model, averaged quantities like this reduc e toxδµνwith some mean value x, depending on
+the quantity at hand. Thus EZn
+ǫ,J(q) is given by
+EZn
+ǫ,J(qL)∝/integraldisplay/parenleftbigN−1
+2π/parenrightbigndze−2NnqLz−m2nz/integraldisplay
+dΛµν
+αβexp/parenleftBig
+N/bracketleftBig
+τmn(n−1)Q2+τ′nm(n−1)P2+nm
+2p2
+d
+−mn1
+2β2/parenleftbig2z
+β2+1√
+1+ǫ2/parenrightbig(−1)p2
+d+2ωn(n−1)(n−2)
+3!mQ3
++ωn(n−1)(n−2)mQP2+2ωn(n−1)mQPp d/bracketrightBig/parenrightBig
+(14)
+This leads to four different saddle point equations (we use Nmas the large parameter)
+0 =τQ−Q2−P2+Ppd (15)
+0 =τ′P−2QP+Qpd (16)
+0 =pd−1
+β2pd/parenleftbig2z
+β2+1√
+1+ǫ2/parenrightbig(−1)−2QP (17)
+qL
+m=/parenleftBigpd
+β2/parenleftbig2z
+β2+1√
+1+ǫ2/parenrightbig/parenrightBig2
+, (18)6
+where line 3 and line 4 can be combined to
+pd−2QP=/radicalbiggqL
+m. (19)
+In the following, we substitute/radicalbigqL
+m→q.
+From equation (14) the free energy is (after taking the usual rep lica limit n→0)
+βfǫ(q) = const .+pdq−p2
+d
+2−q2
+2(1−2τ′)+τQ2+τ′P2−2
+3Q3−2QP2+2QPpd (20)
+1. Above and at the critical temperature
+Above (τ <0) and at the critical temperature ( τ= 0), the Ising spin glass is replica symmetric, just as the m-
+component spin glass. Therefore the solutions of the saddle point e quations are the same as in Ref. 24. Inserting
+them into the free energy (equation (20)) gives
+above T c:βfǫ(q) =q2
+2/parenleftBig
+1−β2
+√
+1+ǫ2/parenrightBig
++O(q4) (21)
+at Tc:βfǫ(q) =/braceleftBig1
+6q3+3ǫ2
+16q2+O(ǫ4,q4)ǫ2≪q≪1
+q2
+2/parenleftBig
+1−1√
+1+ǫ2/parenrightBig
++O(q4)q≪ǫ2≪1(22)
+AtTcthe probability distribution Pǫ(q)∝e−Nmβf ǫ(q)consists of two parts with different dominating exponents
+depending on ǫbeing smaller or larger than N−1/6:
+Pǫ(q)∝/braceleftBigg
+e−Nmq3
+6ǫ≪N−1/6
+e−Nmq2ǫ2
+4N−1/6≪ǫ(23)
+2. Below the critical temperature
+The equations (15), (16) and (19) can not be solved for general qandǫ. Due to this, we calculate corrections to
+the two following solvable cases, q= 0 and ǫ= 0, perturbatively in various limits. The exact result for q= 0 is
+P= 0
+Q=τ (24)
+pd= 0
+To find the solution for ǫ= 0, we rewrite equations (15) and (16) in terms of the new variables a=Q+Pand
+b=Q−P
+(τ+pd)a−τ−τ′
+2(a−b)−a2= 0 (25)
+(τ−pd)b−τ−τ′
+2(a−b)−b2= 0 (26)
+with the solution ( ǫ= 0, soτ=τ′)
+Q=τ (27)
+P=pd=q
+1−2τ, (28)
+fora∝ne}ationslash= 0 and b∝ne}ationslash= 0. With equation (20) we get βf0(q) =βf0=1
+3τ3. We will use this free energy as the reference free
+energy as it is the energy of the unconstrained system. For q > τ−2τ2=qEAthe solution with b= 0 maximizes the7
+free energy (the solution with a= 0 is an unphysical one). In terms of ∆ q=q−(τ−2τ2) it is (to lowest order in ∆ q)
+Q=P=τ+pd
+2
+pd=τ+∆q
+1−2τ, (29)
+and the difference to βf0is
+∆f0(q) =c0∆q3, (30)
+withc0=1
+6(1−2τ)3. Now we will calculate corrections to the first solution, f0, in various different limits of ǫandq.
+3.ǫ→ ∞
+In the limit ǫ→ ∞, i.e.τ′→ −∞Eq. (16) yields the solution (to leading order)
+P=−Qpd
+τ′+O(1
+(τ′)2) (31)
+For equation (15) this leads to (again in leading order)
+0 = (τ+p2
+d
+τ′)Q−Q2(32)
+Forτ′=−∞,pdis equal to qand the free energy difference to f0is
+βf∞(q)−βf0=qpd−p2
+d
+2+O(p4
+d) =q2
+2+O(q4) (33)
+4. Perturbative solution for ǫ2≪q≪1
+We use the saddle point equations for a(25) and b(26), insert a=a0+a1,b=b0+b1andpd=pd0+pd1, where
+the index 0 denotes the undisturbed solution ( ǫ= 0) and the index 1 the first order corrections to it. The results ar e
+(∆τ=τ−τ′>0)
+a1=−∆τpd0−pd1a0
+τ+pd0
+b1=∆τpd0−pd1b0
+τ−pd0⇒Q1=ǫ2q2
+4τ2(1−2τ)
+P1=−ǫ2q
+4τ(34)
+and the correction to pdispd1=−ǫ2q
+2. For the free energy these results yield:
+βfǫ(q)−βf0=2τ+1
+16τǫ4q2+O(ǫ4q4) (35)
+5. Perturbative solution for q≪ǫ2≪1
+In this case the suitable reference solution is the one with q= 0, equations (24). We again introduce corrections
+(Q→Q+∆Q) to this solution and combine equations (16) and (19) in lowest order to
+τ′∆P−2Q∆P+Q(∆q+2Q∆P) = 0, (36)
+where ∆qis understood to be the correction to q= 0. The solution is
+∆P=−τ∆q
+τ′−2τ+2τ2ǫ2≪1→∆q
+1−2τ−ǫ2∆q
+4τ(1−2τ), (37)8
+∆Q= 0 and
+∆pd= ∆qτ′−2τ
+τ′−2τ+2τ2ǫ2≪1→∆q
+1−2τ−ǫ2∆q
+2(1−2τ). (38)
+The result for the free energy then is
+βfǫ(q)−βf0=1
+16τ(1−2τ)ǫ4q2+O(q3). (39)
+The restriction ǫ2≪1 is basically unnecessary and we could calculate the free energy in th e limitq≪min(1,ǫ2).
+However, since the regime of large ǫwill not contribute to the sample-to-sample-fluctuations we neglec t it already at
+this point, and write βfǫ(q)−βf0(q) =f(ǫ)q2withf(ǫ) =ǫ41
+16τ(1−2τ)for small ǫ.
+C.P0
+ǫ(q)
+Although we are interested in the probability density of the link overla pqL, the more practical quantity for the
+calculation turned out to be q=/radicalbig
+qL/m. We therefore formulate our results in terms of qinstead of qLfor the time
+being. This is not a serious restriction since we have the simple relation
+[(q13
+L)n]0=mn/integraldisplay
+dqq2nP0
+ǫ(q) =:mn[q2n]0. (40)
+The probability distribution P0
+ǫ(q)∼e−Nmβ(fǫ(q)−f0(q))divides into 4 parts, depending on the range of ǫ. For
+ǫ≪N−1/4thecontributiontotheprobabilitydistributionof qofbothequations(35)and(39)(bothwith fǫ(q)∼ǫ4q2)
+is negligible, therefore it is approximately a constant in that range fo r allq∈[0,qEA]. Eq. (30) implies that P0
+ǫ(q) has
+an exponentially decaying tail for q > qEAwithe−Nmc0(q−qEA)3. We define a function Θ( q−qEA) which combines
+both properties. Instead of this plateau in P0
+ǫ(q) there should be a δpeak at q=qEAwhich we do not see in
+our calculation. This is due to the fact that we have neglegted finite s ize corrections to the free energy which are
+dominating in this regime and which we will implement in the next subsectio n.
+The two solutions we found perturbatively in Eqs. (35) and (39) bot h produce a probability distribution of the
+forme−Nmβc xǫ4q2with different constants cx, but hold in different ranges of ǫ, depending on the relation of ǫ2and
+q. The order of ǫdetermining the crossover from one regime to the other is where ǫ2is of the same order as qand
+Nǫ4q2(in the range N−1/4≪ǫ≪ǫ0) is of order one such as to be the dominating part of the free energ y. This leads
+toǫ∼N−1/8as the crossover value. Thus we get the final result
+Pǫ(q)∝
+
+Θ(q−qEA)ǫ≪N−1/4
+e−Nmc1ǫ4q2N−1/4≪ǫ≪N−1/8
+e−Nmc2ǫ4q2N−1/8≪ǫ≪ǫ0
+e−Nmf(ǫ)q2ǫ0< ǫ,(41)
+withc1=2τ+1
+16τandc2=1
+16τ(1−2τ).
+D. The overlap distribution at ǫ= 0
+In order to account for the missing δpeak inP0
+ǫ(q) for small ǫ, we have to consider the finite size corrections to
+the free energy, which are impossible to calculate. However, not all is lost since at least we know that at ǫ= 0 they
+grow asN1−ywithy= 2/528, and we expect that this scaling is independent of the value of ǫ. We can therefore trust
+our result derived above when Nmβ(fǫ(qEA)−f0)≫N3/5. SinceqEA=O(1), it follows that this is the case when
+N3/5≪Nǫ4orN−1/10≪ǫ. We then have
+Pǫ(q)∝
+
+δFS(q−qEA)ǫ≪N−1/10
+e−Nc2ǫ4q2N−1/10≪ǫ≪ǫ0
+e−Nf(ǫ)q2ǫ0< ǫ.(42)9
+ 0.01 0.1 1 10 100
+ 0.01  0.1  1  10  100m(∆ qL13)2
+Nm-1/µ`m=12
+m=10
+m=8
+m=7
+m=6
+m=5
+m=4
+m=3
+α=1.4
+FIG. 1: Finite size scaling plot of the variance of the link ov erlap at temperature T= 0.6. See text for explanation.
+The function δFS(q−qEA) stands for a function which goes to a δpeak in the thermodynamic limit. Note that the
+new regime completely replaces the first two regimes we calculated in E q. (41).
+What we need in the actual calculation of the sample-to-sample fluct uations of the free energy are the expressions
+[(q13
+L)2]−[q13
+L]2and [q13
+L]. In the regimes where we have an explicit expression for P0
+ǫ(q), we can calculate this directly
+(see below). In the regime just found, however, we do not have th is information available. We must therefore resort
+to other methods and use a finite size scaling ansatz of the form
+1
+m2([(q13
+L)2]−[q13
+L]2) =N−αFm(Nβǫ) (43)
+forǫ≪N−1/10and with a scaling function Fm(x) whose properties will be discussed below. Similarly, we assume
+that in this regime [ q13
+L] can be written as
+1
+m[q13
+L] =Gm(Nρǫ) (44)
+with an exponent ρand a scaling function Gm(x). These scaling functions and exponents will enter the calculation o f
+the fluctuations below.
+We assume m→ ∞such that the system is replica symmetric. Therefore, the left han d side of Eq. (43), which is
+the variance of q13
+L/m, goes to 0 for N→ ∞since it is just the squared width of the peak in P0
+ǫ(qL). The exponents
+αandβare unknown, but we can obtain αfrom a simulation at ǫ= 0 by measuring the width of the peak in the
+distribution of qL. This is shown in Fig. 1. There is, however, a complication involved. In a simulation, mmust be
+finite, and for finite m, the thermodynamic limit is a replica symmetry broken phase and the v ariance of q13
+Lwillnot
+tend to 0, thus apparently making α= 0. In order to overcome this problem, we recall that in Refs. 25,26 it was
+shown that at T= 0 there is a critical number of spin components n0∼Nµ′(with an exponent µ′= 2/5) above which10
+the system does not depend on the number of components any mor e and is thus identical to the replica symmetric
+m=∞limit. We conjecture that something similar happens at finite tempera ture, i.e. when m≫Nµ′, the system
+is in a replica symmetric phase, whereas for m≪Nµ′it is in a different phase. For this reason we have plotted the
+variance (∆ q13
+L)2ofq13
+Lagainstx=Nm−1/µ′, such that we can expect replica symmetric behavior for small xand a
+crossover to a constant variance for large x, and this is precisely what can be observed (although the crossove r is so
+slow that we do not see the expected plateau yet). For the determ ination of αonly the data for small xare relevant.
+For the simulation, we have implemented a parallel tempering Monte Ca rlo algorithm using system sizes from
+N= 64 to N= 216 with up to 29 different temperatures in the range of [0 .6 : 1.6]. In that way we produced for
+two different replicas (with the same set of coupling constants since we are considering ǫ= 0) at least 64 statistically
+independent sets of spin configuration for every temperature. F rom these, we calculated the variance of the link
+overlap (∆ q13
+L)2= [(q13
+L)2]−[q13
+L]2. According to the scaling ansatz above, m−2(∆q13
+L)2∼(Nm−1/µ′)−αm−α/µ′at
+ǫ= 0, such that mα/µ′−2(∆q13
+L)2∼(Nm−1/µ′)−α. The data in Fig. 1 shows on the one hand that α/µ′−2≈1 and
+on the other hand that α≈1.4. The observation α/µ′−2≈1 implies α≈6
+5. Together with the second, more direct,
+observation of α, the data shows that α≥6
+5, and as we will see below, this is all we need to know.
+IV. CALCULATING [qn]0AND THE SAMPLE-TO-SAMPLE FLUCTUATIONS OF THE FREE ENERGY
+The remaining task is to calculate [ qn]0and insert it into equation (6). To do so, we use steepest descent m ethods
+to write for the regime ǫwithN−1/10≪ǫ≪ǫ0(withP0
+ǫ(q) =1
+Nqe−βNmf ǫ(q)and the normalization constant Nq)
+[qn]0=1
+Nq/integraldisplay∞
+0dqqne−Nmcxǫ4q2+O(q4), (45)
+neglect the term of order q4in the exponent (note that we set the upper bound from 1 to ∞, which introduces only
+exponentially small errors), and get (with Nq=Γ(1
+2)
+2√
+Nmcxǫ4)
+[qn]0=1
+Γ(1
+2)/integraldisplay∞
+0dxxn−1
+2(Nmcxǫ4)−n/2e−x
+=(Nmcxǫ4)−n/2
+Γ(1
+2)Γ(n+1
+2). (46)
+We then have
+[q2]0=1
+2Nmcxǫ4(47)
+[q4]0=3
+(2Nmcxǫ4)2. (48)
+With this, we calculate the sample-to-sample-fluctuationsthrough Eqs. (6) and (7) by taking the leading order in Nof
+everyintegralinto account. The first integrals, I21, (withf2(ǫ) = 2ǫ3+O(ǫ5)) separatesinto three integrationintervals
+corresponding to our three regimes. We neglect the range of ǫ > ǫ0, because it gives a contribution of order 1 which
+we are not interested in. For the part ǫ≪N−1/10we have the scaling ansatz for [( q13
+L)2]−[q13
+L]2=m2N−αFm(Nβǫ).
+This ansatz must match m2([q4]0−[q2]2
+0) from the neighboring regime at the crossover point ǫ=N−1/10, which there
+goes asN−2ǫ−8∼N−6/5(see Eqs. (47) and (48)). This implies that the scaling function Fm(x) decays as x−γ/m2
+forx→ ∞with an exponent γobeyingγ(β−1
+10) =6
+5−α.
+This leads to
+I21=/integraldisplayN−1/10
+0dǫ2ǫ3N−αFm(Nβǫ)+/integraldisplayǫ0
+N−1/10dǫ2ǫ31
+2(Nc2ǫ4)2
+∼2N−α−4β/integraldisplayNβ−1/10
+0dxFm(x)x3+N−2/integraldisplayǫ0
+N−1/10dǫǫ−5
+c2
+2+...
+∼const.1N−α−4
+γ(6
+5−α)−2
+5+const.2N−8/5, (49)
+as long as γ >4. Ifγ <4, the first part of the integral is of order O(N−8/5).11
+The second integral of Eq. (6), I22, can be estimated in a similar way with the scaling ansatz1
+m[q13
+L] =Gm(Nρǫ).
+The scaling function Gm(x) should have the properties Gm(x)x→∞→x−ηandGm(x)x→0→const.and should scale as
+N−1ǫ−4forǫ=N−1
+10in order to match the neighboring regime, which yields the scaling relat ionρ=3
+5η+1
+10. This
+gives rise to a contribution
+I22=/integraldisplayN−1/10
+0dǫǫ3Gm(Nρǫ)+/integraldisplayǫ0
+N−1/10dǫǫ3
+2Nc2ǫ4+...
+= const. 3N−12
+5η−2
+5+const. 4logN
+N, (50)
+provided that η >4. Ifη <4 the first part of the integral is O(N−1) instead. We therefore find the scaling exponent
+of the sample-to-sample-fluctuations with the system size, ∆ FN∼Nµ, to be
+µ= max/bracketleftbigg1
+5,4
+5−α
+2−2
+max(γ,4)/parenleftbigg6
+5−α/parenrightbigg
+,3
+10−6
+5max(η,4)/bracketrightbigg
+. (51)
+As decribed in section 3, we haveanother, slightly different route to chaosand will use it now to check for consitency
+with the above result. From Eq. (7) we take the first integral I11=m2/integraltextǫ0
+0dǫf1(ǫ)([q4]−[q2]2) and use the same
+scaling ansatz as above for equation (49) and obtain contributions of the form
+I11=m2/integraldisplayǫ0
+0dǫf1(ǫ)([q4]−[q2]2)∼const.5N−α−3
+γ(6
+5−α)−3
+10+const.6N−3/2+... (52)
+This integral is positive and has a negative prefactor in Eq. (7). The fluctuations can not be negative, however,
+therefore the leading order of this term mustbe compensated by the second integral I12. We use the scaling function
+Gm(Nρǫ) again, which yields
+I12=m/integraldisplayǫ0
+0dǫg1(ǫ)[q2]∼const.7N−3
+5η−1
+10+const.8N−7/10. (53)
+Thissecondintegral,togetherwithits leadingprefactor NfromEq.(7), must atleastcancelthe termoforder O(N1/2)
+which is contained in N2I11. This is only possible if η≥3
+2. Hence we obtain a limit on ηand no contradiction to the
+result derived above.
+V. CONCLUSION
+The purpose of this work was to calculate the finite size scaling of the sample-to-sample fluctuations of the free
+energy in the m-component vector spin glass in the limit of large m. The result is Eq. (51). Although this equation
+looks unpromising at first sight, it is in fact very informative. First, w e have the solid result µ≥1
+5. Second, from our
+numerical work we know that α≥6
+5. But under this condition, the second term in the max function in Eq. (51) is≤1
+5
+and can thus simply be omitted. Third, the last term in the max functio n is≤3
+10, such that we obtain1
+5≤µ≤3
+10.
+Even better, the third term is greater than1
+5only for η >12, which seems an unlikely large value. We therefore
+conjecture that µ=1
+5is in fact the exact answer. But be that as it may, the exponent ηcould easily be measured in
+simulations, and work along these lines is in progress.
+Since our result is not an exact mathematical proof, we will summariz e here the main assumptions on which it
+rests because they may have been obscured by the technicalities. The first ingredient is the connection between the
+fluctuations and bond chaos of the link overlap, Eqs. (6) and (7). T hey are mathematically exact equalities and pose
+no problem. The second ingredient is the calculation of bond chaos. T his is done using large deviation statistics and
+replica theory. We believe that replica theory in principle gives the cor rect results. It became apparent, however, that
+for themcomponent spin glass the “small” deviations play a crucial role. The sm all deviations statistics are caused
+by finite size corrections of the free energy, which can not be calcu lated within replica theory. However, by reference
+to earlier results28we know at least the finite size scaling of the free energy correction s and can thus estimate the
+point where our large deviation calculation becomes valid. The region o f the small deviations is then covered by a
+scaling ansatz, which is assumed to cross over smoothly to the regio n of the large deviations. The introduction of the
+scaling ansatz also introduced a number of unknown exponents. Ho wever, only three of these exponents are actually
+relevant for our results, and one of them, α, has been measured experimentally. Moreover, the precise values of the
+exponents are largely irrelevant: if α≥6
+5(as we have checked numerically) and η≤12, then µ=1
+5is exact.12
+The result µ=1
+5(or even the range1
+5≤µ≤3
+10) is very interesting because it demonstrates that the large- mmodel
+is fundamentally different from the spherical model20, even though their free energies are identical22. While the width
+of the distribution of ground state energies of the spherical spin g lass scales as N1/3, this behavior is definitely ruled
+out by our results.
+It would be interesting to see whether a result similar to ours could be obtained using the methods of Ref. 11. We
+believe a corresponding replica calculation for the large- mmodel ought to be feasible.
+Acknowledgments
+This work was supported by the German Science Foundation (DFG) t hrough grant no. AS 136/2-1.
+1M. M´ ezard, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
+2G. Parisi, F. Ritort, and F. Slanina, J. Phys. A 26, 247 (1993).
+3G. Parisi, F. Ritort, and F. Slanina, J. Phys. A 26, 3775 (1993).
+4M. Palassini and S. Caracciolo, Phys. Rev. Lett. 82, 5128 (1999).
+5B. Drossel, H. Bokil, M. A. Moore, and A. J. Bray, Eur. Phys. J. B13, 369 (2000).
+6J.-P. Bouchaud, F. Krzakala, and O. C. Martin, Phys. Rev. B 68, 224404 (2003).
+7S. Boettcher, Eur. Phys. J. B 38, 83 (2004).
+8A. Billoire, Phys. Rev. B 73, 132201 (2006).
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+10G. Parisi and T. Rizzo, Phys. Rev. B 79, 134205 (2009).
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+14C. A. Tracy and H. Widom, in Calogero-Moser-Sutherland models , edited by J. F. van Diejen and L. Vinet (Springer Verlag,
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\ No newline at end of file
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+arXiv:1001.0894v1  [astro-ph.GA]  6 Jan 2010Astronomy& Astrophysics manuscriptno.12484 c/circlecopyrtESO 2018
+November7,2018
+H2CO andCH 3OHmaps ofthe OrionBar photodissociation region⋆
+S.Leurini1,2,B.Parise2,P.Schilke2,3,J. Pety4, R.Rolffs2
+1ESO,Karl-Scharzschild-Strasse 2,D-85748, Garching-bei -M¨ unchen, Germany
+e-mail:sleurini@mpifr.de
+2Max-Planck-Institut f¨ ur Radioastronomie, Aufdem H¨ ugel 69, 53121 Bonn, Germany
+3Physikalisches Institut,Universit¨ at zuK¨ oln, Z¨ ulpich er Str.77, 50937 K¨ oln, Germany
+4Institut de Radioastronomie Millim´ etrique, 300Rue de laP iscine, 38406 SaintMartind’H` eres, France
+November 7,2018
+ABSTRACT
+Context. A previous analysis of methanol and formaldehyde towards th e Orion Bar concluded that the two molecular species may
+trace different physical components, methanol the clumpy material, a ndformaldehyde the interclumpmedium.
+Aims.To verify this hypothesis, we performed multi-line mapping observations of the two molecules to study their spatial dis tribu-
+tions.
+Methods. Theobservations wereperformed withtheIRAM-30m telescop e at218and241GHz,withanangular resolutionof ∼11′′.
+Additional data for H 2CO from the Plateaude Bure array arealso discussed. The data were analysed using anLVGapproach.
+Results.Both molecules are detected in our single-dish data. Our dat a show that CH 3OH peaks towards the clumps of the Bar, but
+itsintensitydecreases belowthe detectionthresholdinth e interclumpmaterial.Whenaveragingover alargeregionof the interclump
+medium, the strongest CH 3OH line is detected with a peak intensity of ∼0.06 K. Formaldehyde also peaks on the clumps, but it is
+alsodetected inthe interclump gas.
+Conclusions. We verified that the weak intensity of CH 3OH in the interclump medium is not caused by the di fferent excitation con-
+ditions of the interclump material, but reflects a decrease i n the column density of methanol. The abundance of CH 3OH relative to
+H2COdecreases by atleast one order of magnitude from the dense clumps tothe interclumpmedium.
+Key words. ISM: individual objects (OrionBar)- ISM:abundances -ISM: molecules - ISM:structure
+1. Introduction
+Given its proximity and nearly edge-on orientation, the so-
+called Orion Bar is one of the clearest examples of a photon-
+dominated region (PDR). For this reason, the Orion Bar has
+been extensively observed in past decades to test theoretic al
+models of PDR structure, chemistry, and energetics. Its nea rly
+edge-on orientation make direct observations of the gas str at-
+ification possible, from ionised gas to neutral atomic gas to
+molecular gas as a function of the increasing distance from t he
+ionisation source (e.g., Tielensetal. 1993; vanderWerf et al.
+1996; vanderWiel et al. 2009). These studies show that the
+molecular gas consists of clumpy molecular cores embedded
+in an interclump gas. While the clumps have densities of sev-
+eral 106cm−3(YoungOwl etal. 2000; Lis&Schilke 2003),
+the interclump material has lower densities (104–105cm−3,
+Hogerheijdeet al. 1995; YoungOwl et al. 2000). Such observa -
+tional results are successfully reproduced by theoretical works
+(e.g.,Gorti&Hollenbach2002).
+Observations of methanol (CH 3OH) and formaldehyde
+(H2CO) in the Orion Bar have been reported in the past
+(Hogerheijdeetal.1995;Jansen etal.1995;Leuriniet al.2 006).
+Leuriniet al. (2006) studied the excitation of both molecul ar
+species in the line of sight of one molecular clump through
+a multi-line analysis at 290 GHz. Their findings (high densit y
+Send offprint requests to : S.Leurini
+⋆Based on observations carried out with the IRAM-30m tele-
+scope and the Plateau de Bure interferometer. IRAM is suppor ted by
+INSU/CNRS(France),MPG (Germany) andIGN (Spain).and relatively low temperature for CH 3OH, high temperature,
+and relativelylow densityfor H 2CO) suggest that methanoland
+formaldehyde do not trace the same material, but the first is
+foundinthedensegasassociatedwiththeclumps,thesecond in
+the warmer andless dense gasof the interclumpmaterial. Sin ce
+both molecular species can e fficiently form on grain surfaces
+(Hidakaet al. 2004), the authors suggested that photodisso cia-
+tion of methanol to form formaldehyde (LeTeu ffet al. 2000)
+takes place in the interclump medium, while the high density
+shields CH 3OH in the clumps and prevents its photodissocia-
+tion.Whereasgrainsurfacereactionsaretheonlyviablero uteto
+methanol, formaldehyde can also form in the gas phase via the
+reaction CH 3+O (e.g., LeTeuffet al. 2000), which could con-
+tributetotheinterclumpabundanceofH 2CO.
+However, the study of Leuriniet al. (2006) on CH 3OH and
+H2CO is based on low angular resolution ( ∼20′′), single-
+pointingdata.Toconfirmthatthetwospeciesindeedtracedi ffer-
+entmedia,wemappedalargeareaoftheOrionBarinformalde-
+hyde and methanol at 218 and 241 GHz, respectively, with an
+angular resolutionof ∼11′′. We also analysed existing interfer-
+ometricdataforthe218GHzformaldehydelines,tocomparet he
+observed fluxes from single-dish and interferometer and der ive
+constraints on the size of the emitting region. This work rep -
+resents a first observational attempt to study the distribut ion of
+CH3OH andH 2CO in the OrionBar andtoverifytheirdi fferent
+origin.2 S.Leurini etal.:H 2CO andCH 3OH maps of the OrionBar
+(a)
+Fig.1.In grey scale, the integrated intensity of the H 2CO (30,3−20,2) line (left) and of the CH 3OH-A(50−40) transition (right)
+observed with the IRAM-30m telescope. Black contours(4, 8, and 12 times 0.032 Jy beam−1km s−1) show the distribution of the
+H13CN (1-0) emission observed with the Plateau de Bure interfer ometer (Lis& Schilke 2003). For H 2CO the contours start from
+1.8K kms−1instepsof1.8(equalto3 σ),forCH 3OH from0.3K kms−1instepsof0.3(equalto3 σ).Intheleft panel,thedashed
+lines indicate the strips used for Fig. 6; the solid lines out line the regions used to derive the average H 2CO and CH 3OH emission
+in the interclump material (Figs. 3(b) and 4(c)); the dotted lines mark the region shown in the right panel. In the right pa nel, the
+numberslabelthepositionsoftheclumpsdetectedinH13CN;thedottedcirclesoutlinethemosaicofsevenfieldsobse rvedwiththe
+PdBI(see§2.2).
+2. Observations
+2.1. IRAM-30mtelescope
+The observations were performed in February and March 2007
+using the HERA multi-beam receiver (Schusteret al. 2004) at
+theIRAM-30mtelescope.TheHERA1andHERA2pixelswere
+tuned to 241.850 GHz in lower side band (LSB), to detect the
+CH3OH (5k−4k) band.On February23and 24,theHERA2 re-
+ceivers were tuned to 218.349 GHz (LSB) to detect the H 2CO
+(3Ka,Kc−2Ka,Kc−1) band. The backend used was VESPA with
+a bandwidth of 160 MHz and a resolution of 0.3125 MHz for
+the CH 3OH setup (corresponding to a velocity resolution of ∼
+0.4kms−1),160MHzand1.25MHzfortheH 2COsetup(corre-
+spondingto∼1.7kms−1).Inthisway,theCH 3OH(5k−4k)band
+is fully covered with the only exception being the k=−1,0-E
+lines.TheH 2COsetupcoversthe3 0,3−20,2and32,2−22,1lines,
+whilethe3 2,1−22,0transitionliesoutsidetheobservedfrequency
+range.
+We used the on-the-fly mode with a rotation of the multi-
+beam system of 9.7◦, to ensure a Nyquist sampling between
+the rows. The reference position used as the centre of the map
+wasα2000=05h35m25s.3,δ2000=−05◦24′34′′.0, correspond-
+ing to the “Orion Bar (HCN)” position of Schilkeet al. (2001) ,
+themostmassiveclumpseeninH13CN(Lis&Schilke2003),as
+well as the target of the spectral survey of Leuriniet al. (20 06).
+The OFF position was chosen to be (500′′,0′′) from the centre
+ofthemap.
+Thepointingwascheckedonseveralcontinuumsourcesand
+found to be accurate within 2′′. The observations were done
+under varying weather conditions, and the T sysspan the range
+540–1200KforHERA1and700–1600KforHERA2.Thehalf-
+power beam width of the IRAM-30m telescope at 218 GHz is
+∼11′′and∼10′′at 242 GHz. We used a beam e fficiency of0.55 at 218 GHz, and of 0.50 at 242 GHz to convert antenna
+temperaturesT⋆
+aintomain-beamtemperaturesT mb1.
+Theregioncoveredbyourobservationsisshownin Fig.1.
+2.2. PlateaudeBure interferometer
+We observed the H 2CO 30,3−20,2line at 218.2 GHz with the
+Plateau de Bure interferometer in March, April, and Decembe r
+2004. We observed a 7-field mosaic centred on α2000=
+05h35m25.280s,δ2000=−05◦24′39′′.300. The field positions
+followed a compact hexagonal pattern to ensure Nyquist sam-
+pling in all directions.The imaged field-of-viewis almost a cir-
+clewitharadiusof22 .5′′.OnMarch28and29,thesixantennas
+were used in the 6 Cp configurationwith 3 and 1.5 mm precip-
+itable water vapour, which translated into system temperat ures
+of 450 K and 250 K. On April 22, the six antennas were used
+in the more compact 6Dp configurationwith 6 to 10 mm PWV,
+leadingtoa systemtemperatureofabout1000K.OnDecember
+12, the six antennas were used in the 6Cp configuration, with
+3.5 mm PWV, leading to a system temperature of about 700 K.
+Taking the time for calibration and data filtering into accou nt,
+this translates into an on–source integration time of useful data
+of5.3hoursforafull6-antennaarray.Thetypical1mmresol u-
+tion is 2.4′′. We used the 30m data described above to produce
+themissingshortspacings.
+The data processing was done with the GILDAS2software
+suite (Pety 2005). Standard calibration methods implement ed
+in the GILDAS/CLIC program were applied using close bright
+quasars(0528+134and 0607−157)as phase and amplitudecal-
+ibrators. The absolute flux was calibrated using simultaneo us
+1http://www.iram.es/IRAMES/telescope.html
+2Seehttp://www.iram.fr/IRAMFR/GILDAS for more informa-
+tionabout the GILDASsoftwares.S.Leurini etal.:H 2CO andCH 3OH maps of the OrionBar 3
+Fig.2.Top: uvcoverage for one field of the hybrid 30m +PdBI
+mosaic.Bottom:Associated dirty beam. Please note that the
+dirty beam has many secondary side-lobes as high as 40% of
+themainlobe.
+measurementsofthe PdBIprimaryfluxcalibratorMWC349. uv
+tables were then producedat a relatively coarse velocity re solu-
+tionof1.7km/s.
+All other processing was performed with the
+GILDAS/MAPPING software. The single-dish map from
+the IRAM-30m was used to create the short-spacing pseudo-
+visibilities unsampled by the Plateau de Bure interferome-
+ter (Rodr´ ıguez-Fern´ andezet al. 2008). These were then me rged
+with the interferometric observations. Each mosaic field wa s
+imaged independently, and a dirty mosaic was built through a
+linear combination of these dirty images (Guethet al. 1995) .
+The dirty mosaic was then deconvolved using the standard
+H¨ ogbom CLEAN algorithm. While all these procedures are
+standard and usually easy to use in the GILDAS /MAPPING
+software, we had to take special precautions with this data
+set because the short usable integration time at PdBI implie d
+1) a low signal-to-noise ratio of the interferometric data a nd
+2) dirty beams for each field with large secondary side-lobes
+(see Fig. 2). As the 30m data show that the extended signal
+fills most of the field-of-view observed with the interferom-
+eter, we subtracted the spectrum averaged over the observed
+field-of-view from the 30m data before any processing and we
+added this averaged spectrum again after the deconvolution
+of the 30m+PdBI hybrid data set to recover the correct flux
+scale. This simplifies the deconvolution considerably by th e
+usualCLEANalgorithms( i.e.muchfewerCLEANcomponents
+are needed) because it avoids the deconvolution of extended
+uniform intensities. Second, we tapered the visibility wei ghtsTable 1.Detectedmethanolandformaldehydetransitionsin the
+IRAM-30mtelescopedata
+line νEupclump1 interclump
+[MHz] [K]
+H2CO30,3−20,2218222 21.0 YaY
+H2CO32,2−21,1218475 68.0 YaY
+CH3OH-A50−40241791 34.8 YaY
+CH3OH-A5±3−4±3241832 84.6 YbN
+CH3OH-A52−42241842c72.5 YbN
+CH3OH-E53−43241843 82.5 YbN
+CH3OH-E51−41241879 55.9 YaN
+CH3OH-E5±2−4±2241904∼60dYbN
+adetected at the original resolution of the IRAM-30m telesco pe
+data
+bdetected at aresolution of ∼20′′
+cblended withthe 5 3−43-E
+dEup=60.7 K for the 5 −2−4−2line, 57.1 K for the 5 +2−4+2tran-
+sition.
+withanaxis-symmetricalGaussianof80m FWHMtogetamore
+symmetricalbeam(goingfrom4 .1′′×1.2′′fornaturalweighting
+to3.3′′×1.8′′aftertapering)andtoimprovethesignal-to-noise
+ratio for the extended structures. The final noise rms measur ed
+at the centred of the mosaic is about 0.3 K in channels of 1.7
+km/swidth.Thisleadstoamaximumsignal-to-noiseratioof18
+for the hybrid data cube. However, the produced data cube has
+its brightness dynamical range limited by the poor dirty bea m.
+The cleaned data cube was finally scaled from Jy /beam to Tmbtemperaturescale usingthesynthesisedbeamsize.
+3. Observationalresults
+3.1. single-dishdata
+Figures 3(a) and 4(a) show the spectra at 218.4 and 241.8 GHz
+towardsthecentralpositionofthemap.Bothformaldehydet ran-
+sitionsaredetected.Onlythe 5 0−40-Aand51−41-Emethanol
+transitions are detected at the original resolution of ∼10′′.
+However, when increasing the signal-to-noise ratio by smoo th-
+ing to a lower resolution of ∼20′′, CH3OH transitionswith up-
+per level energieslower than 85 K are also detected (Fig. 4(b )).
+The detected lines of both molecular species are reported in
+Table1.
+The region mapped with the IRAM-30m telescope is pre-
+sented in Fig. 1, where in the left panel we show the integrate d
+intensity of the strongest detected H 2CO line, while the right
+panel shows the emission from the strongest CH 3OH transition
+alongtheOrionBaralone.Forcomparison,theintegratedin ten-
+sity of the H13CN (1-0) from Lis& Schilke (2003) is overlaid
+onourdata.Bothmoleculesclearlytracethemolecularmedi um
+associated with the Orion Bar, and peak to the southwest of it .
+Thispeakprobablybelongstothelow-intensitybridgethat con-
+nects Orion South to the Orion Bar. This structure was detect ed
+in previous observations at velocities between 7 and 9 km s−1
+(e.g., Tauberet al. 1995). The southwest peak of CH 3OH and
+H2CO detected in our data has a peak velocity of 8 km s−1. In
+addition,formaldehydeisdetectedtowardsthenorthofthe map,
+atapositionthatspatiallycoincideswithcontinuumemiss ionat
+350µm (Liset al. 1998) and with a velocity between 9 and 11
+kms−1.
+Along the Bar, both molecules trace the elongated region
+where molecular clumps are located, with an absolute emis-4 S.Leurini etal.:H 2CO andCH 3OH maps of the OrionBar
+(a)
+(b)
+Fig.3.Spectrum of the H 2CO (3Ka,Kc−2Ka,Kc−1) band towards clump 1 (top) and averaged over the interclump medium (bottom)
+fromtheIRAM-30mtelescope.
+sion peak on clump 3 of Lis&Schilke (2003). Moreover, both
+species are detected on a secondary peak, northeast of clump 1
+(α2000=05h35m29.6s,δ2000=−05◦23′59′′.9), not covered by
+the observationsof Lis&Schilke (2003), but close to a peak o f
+CO(6-5) (Fig. 5), which could represent an additional clump of
+dense gas along the Orion Bar. To better visualise the overal l
+morphologyofthe OrionBar,in Fig. 5we showthe 20cm con-
+tinuumemissionfromYusef-Zadeh(1990),whichtracesthei on-
+isation front; the CO(6-5) integrated emission (Liset al. 1 998),
+which shows the temperature distribution of the molecular g as;
+andtheH13CN (1-0)emission,whichtracesthedenseclumps.
+InFig.3and4wecomparetheH 2COandCH 3OHspectrum
+towardsclump1 to the correspondingspectra averagedovert he
+region of the interclump gas outlined in Fig. 1 (left). The in te-
+gratedintensityoftheCH 3OH(50−40)-Atransitionintheinter-
+clump medium is a factor ∼9 lower than in the clump, the one
+of H2CO 32,2−22,1line a factor of 7 (see Table 2). For a better
+visualisation of the formaldehyde and methanol morphologi es,
+we show in Fig. 6 the variation in the integrated intensities of
+the CH 3OH-A(50−40) and H 2CO (32,2−22,1) lines for three
+strips in the map; the exact location of the strips is shown in
+Fig. 1. The strip across clump1 (Fig. 6(a)) suggestsa larger ex-
+tension of the H 2CO emission with respect to CH 3OH, whilethe variation in the integratedintensity of the two species along
+the Bar is verysimilar (Fig. 6(c)). Althoughwe present data for
+the weakest of the two H 2CO lines, the signal-to-noise ratio in
+the methanol data is still lower than for H 2CO and could bias
+our results. From these strips and fromthe comparisonbetwe en
+the H2CO emission and the CO(6-5) map, it also emerges that
+formaldehyde extends to a greater distance from the clumps i n
+the molecular cloud than towards the ionisation front. We al so
+smoothed the data to lower resolutions, to verify whether th e
+CH3OH and H 2CO emission are similarly a ffected by beam di-
+lution.At aresolutionof50′′,theH 2COemissionisstill ∼37%
+oftheintensityofthe originaldata,whileat the sameresol ution
+the CH 3OH peak intensity dropsdown to ∼18% of the original
+value.Thesetestssuggestthatformaldehydeemissionisas soci-
+atedwiththeclumpsbutalsowiththeinterclumpmedium,whi le
+methanolemissionisonlyfoundinthedensemolecularclump s.
+In Sect. 4, we investigatewhether the di fferent morphologiesof
+the two molecularspeciesstem froman excitationalor obser va-
+tionalbias,orwhethertheyimplydi fferentabundancesofH 2CO
+andCH 3OH inthetwo media.
+Finally, the spectral resolution used for the methanol ob-
+servations allows us to study the velocity field along the Bar .
+The channel maps of the (5 0−40)-Aline (Fig. 7, smoothedS.Leurini etal.:H 2CO andCH 3OH maps of the OrionBar 5
+(a)
+(b)
+(c)
+Fig.4.Spectrumof the CH 3OH (5k−4k) bandtowardsclump1 at a resolutionof ∼10′′(top),smoothedto a resolutionof ∼20′′
+(middle),andaveragedovertheinterclumpmedium(bottom) .Dataare fromtheIRAM-30mtelescope.
+to 0.77 km s−1) show that clump 1 peaks around 9.5 km s−1
+and clump 3 around 10.5, as for H13CN (Lis& Schilke 2003).
+The secondary peak detected to the northeast of clump 1 has a
+peak velocity around 11 km s−1, while the molecular cloud to
+the south of the Bar is red-shifted with respect to the clumps
+(vlsr∼8 km s−1). This trend in velocity (blue-shifted velocitiesin the northeast, red-shifted values in the southwest) was a lso
+foundbyYoungOwl etal. (2000) intheirHCN maps.
+3.2. Hybrid interferometric+single-dishdata
+Figure 8 compares the 30m (top panels) and hybrid 30m +PdBI
+(bottompanels)datacubes.Theleftcolumndisplaystheima ges6 S.Leurini etal.:H 2CO andCH 3OH maps of the OrionBar
+Table 2.Lineparameters
+TMB/integraltext
+TMBdv∆vaTMB/integraltext
+TMBdv∆va
+[K] [km s−1] [km s−1] [K] [Kkm s−1] [km s−1]
+H2CO (30,3−20,2) CH 3OH(5 0−40-A)
+clump1 3.2 8.9 2.0 1.1 1.8 1.4
+interclump 0.3 1.2 3.1 0.06 0.2 4.0
+acorrected for the spectral resolution, ∆v=/radicalbig
+(∆vobs)2−(∆vres)2
+(a)
+Fig.5.Distribution of the CO(6-5) peak brightness temperature
+(colourimage).Theblackcontours(from40%ofthepeakinte n-
+sity,instepsof10%)showthe20cmcontinuumemission,whic h
+tracestheionisationfront.Thedottedcontours(from15%o fthe
+peakintensity,instepsof10%)aretheintegratedintensit yofthe
+H2CO 30,3−20,2line; the blue contours (as in Fig. 1) represent
+thedenseclumps;thesquaremarksthepositionofthesecond ary
+peakidentifiedinH 2CO andCH 3OH.
+integrated between 8.3 and 11.7 km /s. The middle column dis-
+plays the spectra averaged over the whole interferometric fi eld-
+of-view.Thehybridspectraareidenticaltothe30moneswit hin
+thenoiselevel,confirmingthatalltheextendedemissionfil tered
+outbythePdBIiscorrectlyrecoveredfromthesingle-dishd ata.
+TherightcolumnofFig.8displaysthespectraaveragedover the
+centralclumpmarkedintheleftcolumn.Thehybridspectrum is
+brighter than the 30m spectrum implying beam dilution in the
+single-dishdata.
+From the PdBI data and the hybrid 30m +PdBI data, we can
+thereforeconcludethat the H 2CO emission comesfrombothan
+extended component and a compact one, still unresolved at th e
+resolutionoftheIRAM-30mtelescope.
+4. Discussion
+4.1. Formaldehyde
+Lineratiosofformaldehydelinescanbeusedtoinferthephy sics
+of the gas in dense molecular regions (Mangum&Wootten
+1993). Because ofthe coarsespectral resolutionof our obse rva-
+tions(∆v∼1.7kms−1),thelineprofilesoftheH 2COtransitions
+are poorly resolved. As a result, we do not have a correct mea-
+sureofthepeakintensityofthelines,dilutedoverthebeam ,but
+our values are a lower limit to the true main-beam line intens i-
+ties.However,theareasareconservedquantitiesandcanbe used
+tostudytheexcitationofH 2CO in theOrionBar.Figure 9 shows the ratio of the integrated intensities of the
+30,3−20,2line to the 3 2,2−22,1line. The analysis is limited to
+the region where the signal-to-noise ratio of the 3 2,2−22,1inte-
+grated intensity is at least equal to 3. The ratio increases f rom
+thenortheast,wheretheclumpsare,to thesouthwest.
+We ran LVG calculations for column densities of H 2CO be-
+tween 1012and1015cm−2, temperaturesin the range20–200K,
+and densities of 104–108cm−3, and calculated the behaviour of
+theχ2valueoftheobserved3 0,3−20,2and32,2−22,1lineinten-
+sities as function of these quantities. Two possible regime s are
+found (see Fig. 10), independent of temperature. At low dens i-
+ties (n<106cm−3), theχ2value dependsonly on the productof
+the H2density and H 2CO column density (low-densitybranch).
+For higher densities, the χ2value varies only slowly with H 2
+densityandismostlyafunctionofH 2CO columndensity(high-
+densitybranch).
+For the values in the interclump gas, assuming a beam fill-
+ing factor of unity, we find a nominal minimum for the in-
+terclump medium at T=180 K, n(H 2)=6.6×105cm−3,
+NH2CO=1.4×1013cm−2. However, as can be seen in Fig. 10,
+these parameters are not well constrained. Using the 1 σconfi-
+dence level, we derived a lower limit to the kinetic temperat ure
+of the interclumpequal to 76 K, and the product nH2×NH2COis
+equalto∼3×1018cm−5fordensitiessmallerthan106cm−3.By
+comparison with PDR models, YoungOwl et al. (2000) found
+densities of hydrogennuclei of 104-105cm−3in the interclump;
+similarly,Hogerheijdeetal.(1995)foundaninterclumpH 2den-
+sity of 3×104cm−3by modelling mm line observations. Thus,
+by using a molecular hydrogen density between 5 ×103and
+5×104cm−3we can constrain the column density of H 2CO to
+values in the range 6 ×1013−6×1014cm−2. The high-density
+branchdoesnotappeartoberelevantfortheinterclumpmedi um.
+InSect. 3.2 we analysedthe combinedPdBI +30mdata and
+concluded that the H 2CO emission is composed of two compo-
+nents, one extended and one unresolved at the resolution of t he
+30m telescope at this frequency. However from these data, it is
+difficult to infer the exact size of clump 1 and the contribution
+of the interclump gas to the total emission of H 2CO. In Fig. 8,
+we marked clump 1 with an ellipse of 15′′×5′′(P.A.=47◦) and
+showedthattheIRAM-30mdataarea ffectedbybeamdilutionat
+this scale, implying a smaller size for the clump. Therefore , for
+clump1,wecorrectedthemain-beamlineintensitiesforthe con-
+tribution from the interclump and for the beam dilution assu m-
+inga typicalsize forthe clumpof 7′′(see Lis&Schilke 2003).
+As expected giventhe similar value of the line ratio on clump 1
+and in the interclump, the values derived are similar to thos e of
+the interclump: T=170 K, n(H 2)=6.6×105cm−3,NH2CO=
+1.6×1014cm−2.The1σcontourgivesavalueof3 .8×1019cm−5
+for the product nH2×NH2COforNH2CO>5×1013cm−2and
+n(H2)<5×105cm−3. For the low-density branch, we derive
+a lower limit of 50 K for the kinetic temperature. For the high -
+densitybranch,moreappropriatefortheclump,thecolumnd en-S.Leurini etal.:H 2CO andCH 3OH maps of the OrionBar 7
+Fig.8.Topline:IRAM-30mdata. Bottomline: hybrid30m+PdBIdeconvolveddatacube. Leftcolumn: Imagesoftheintegratedline
+intensity of the H 2CO (30,3−20,2) transition between 8.3 and 11.7 km /s. The values of the contour levels are shown on the colour
+scale.Middle column: Spectra averagedoverthe whole interferometricfield-of-v iewmarkedas the red circle on the images. Right
+column:Spectraaveragedoverthecentralclumpmarkedwiththeblue ellipse onthe images.
+sityisconstrainedtovalues2 .8×1014−5.4×1014cm−2.Alower
+limit of 80 K can be derived for the kinetic temperature.It ma y
+appear surprising that the temperature is so little constra ined,
+although the ratio of these lines has been used as a tempera-
+turetracer(Mangum&Wootten1993;Hogerheijdeet al.1995) .
+However, for the values of the line ratio we observe, the LVG
+predictions show a more complex behaviour with temperature
+(seealsoFig.13ofMangum&Wootten1993).Takingthemea-
+surement errors into account by calculating the χ2value, as we
+do,and usingthe line intensitiesandnot just the line ratio sthen
+constrainsonlyalowerlimit tothetemperature.
+The results of the LVG analysis of H 2CO are presented in
+Table3.
+4.2. Methanol
+Leuriniet al. (2004) have studied the excitation of the CH 3OH
+(5k−4k) band as function of the temperature, density, and
+methanolcolumndensity ofthe gasfora rangeofphysicalcon -
+ditionsapplicabletothe OrionBar. Theyconcludedthat lin era-
+tios within this band are not sensitive to the temperature of the
+gas for temperatures higher than 30 K, but only depend on the
+density.
+The methanol emission from clumps 1 and 3 is studied in
+a separate paper (Pariseet al. 2009): clump 1 is found to be
+warmer than clump 3, although the errors are large (3 σranges:
+T1∼45+47
+−17K,T3∼35+17
+−15K), but with similar columndensities
+anddensities( NCH3OH∼3×1014cm−2,n≥5×106cm−3).These
+valueswere derivedby modellingthe 5 k−4kand 6k−5kbands
+observedwiththeIRAM-30m(observationspresentedinthis pa-
+per) and the APEX telescopes, respectively.For the models, the
+IRAM data were smoothed to the resolution of the APEX data
+(FWHMbeam∼20′′).Asourcesizeof10′′wasused,sincesev-eral clumps fall in the beam of the observations. This is equi v-
+alent to assuming that all clumps have similar physical cond i-
+tions.SincetheCH 3OHemissionwasfoundtobeopticallythin,
+wecancorrecttheresultsforasourcesizeof7′′(asassumedfor
+the model of H 2CO) and derive the equivalent column density.
+This corresponds to a total column density of 5 .4×1014cm−2,
+orto2.7×1014cm−2forCH 3OH-A,assumingthatthetwosym-
+metric states have the same column density. The 1 σconfidence
+levelis2.3×1014−3.1×1014cm−2.
+For the interclump medium, we analysed the CH 3OH spec-
+trum averaged over the area of the interclump medium out-
+lined in Fig. 5. We ran models for the excitation of methanol
+in the range of densities 104−108cm−3, column densities
+1012−1018cm−2, and temperatures 20 −200 K. The observed
+main-beambrightnesstemperatureof the (5 0−40)-Aline in the
+interclumpmediumis0 .06±0.01K.Assumingthattheemission
+comesfromanextendedsource(e.g.,beamfillingfactor ηc=1),
+wefindasituationsimilartotheonedescribedforH 2COandwe
+can identifytwo di fferentregimes.In the first ( n<106cm−2), a
+fit tothe data isfoundfor NCH3OH−A<1.3×1014cm−2: thecol-
+umndensitydecreaseswithincreasingdensity,buttheirpr oduct
+is almost constant (between 2 ×1017cm−5, 50 K≤T<200 K,
+and 2×1018cm−5, 20 K<T<50 K). For higher densities,
+the 50−40line thermalises. In this regime, the column density
+increases slowly ( NCH3OH−A=1.3×1012−2×1013cm−2). For
+both regimes, the dependence on the kinetic temperature is n ot
+strongandtheresultspresentedarevalidfortemperatures inthe
+range20–200K.However,increasingthetemperatureoftheg as
+implies a decrease in the column density of methanol for the
+low-density case ( T≥50 K,NCH3OH−A<3×1013cm−2, see
+Fig.11).
+Asalreadydiscussedintheprevioussection,fromtheliter a-
+ture we know that the density of the interclump gas is less tha n8 S.Leurini etal.:H 2CO andCH 3OH maps of the OrionBar
+Table 3.Summaryofthemodelresults
+NH2CO−pNCH3OH−A[CH3OH−A]/[H2CO−p]
+[1013cm−2] [1013cm−2]
+clump1 (28−54)a23−31 0.4−1.1
+interclump (6−60)b(0.04−4)b7×10−4−0.7
+(0.04−0.4)c(7×10−4−7×10−2)c
+aassuming n=106cm−3
+bassuming n=(5×103−5×104)cm−3
+cassuming T>50K
+106cm−3, and therefore that the low-density regime applies to
+themodellingofthe interclump.Thebestfit tothe dataisfou nd
+forNCH3OH−A∼3×1012cm−2,T∼80K,n∼9×104cm−3.For
+densitiesintherange5 ×103−5×104cm−3,andforthevalues
+oftheproduct N×ngivenabove,we derivecolumndensitiesin
+the range 4×1011−4×1013cm−2. However, if we allow only
+temperaturesabove 50 K, as found in our data and by other au-
+thors,the inferredmethanolcolumndensity in the interclu mpis
+4×1011≤N≤4×1012cm−2(see Table3).
+5. Abundancesof methanolandformaldehydein
+theOrion Bar
+From the analysis performed in the previous section, we can
+computethe abundanceofmethanolrelativeto formaldehyde in
+the interclumpmediumandin the denseclumps,andverifytha t
+the dropof intensityof the methanolemission in the intercl ump
+medium is the result of the di fferent physics of the interclump
+relativetothedenseclumps,orwhetheritreflectsarealdec rease
+intheCH 3OH abundance.
+GivenanestimateofH 2columndensity,wecouldalso infer
+theabundanceofbothmoleculesrelativetomolecularhydro gen.
+An estimate of the H 2column density can be derived from the
+continuumemissionat(sub)millimetrewavelengths,assum inga
+given temperature for the dust and optically thin emission. The
+continuumemission at 350 µm of the Orion Bar was studied by
+Lisetal. (1998). These authors defined an average temperatu re
+for the dust in the OMC-1 region of 55 K on the basis of its
+FIR colours (Werneretal. 1976) and excluded colder tempera -
+turesfortheOrionBar giventhe goodagreementofthe 350 µm
+continuumemissionandtheCO(6-5)emission,whichorigina ted
+in the outer PDR layers. However, Werneret al. (1976) found
+an average FIR colour temperature of 75 K in the Orion Bar,
+which could imply that the Orion Bar is significantly warmer
+than the rest of OMC-1. For a dust temperature of 55 K in the
+interclumpmedium,andforthemeanvalueofthe350 µmemis-
+sion over the region used to derive the average spectra of the
+interclump medium (Fig. 1), we derive a column density of H 2
+of 5.3×1021cm−2. This translates into to 3 .4×1021cm−2for
+Td=75K.Avalueof0.101gcm−2wasusedforthedustopacity
+(Ossenkopf&Henning1994).Forclump1,foratemperatureof
+45K(asderivedfrommethanol),andundertheassumptiontha t
+thedust andthegasare thermallycoupledat the highdensity of
+theclumps(e.g.,Kr¨ ugel& Walmsley1984),theH 2columnden-
+sity is 2×1023cm−2. Thisis notthe averagecolumndensityon
+thebeam,butitwascorrectedforasourcesizeof7′′andforthe
+contribution of the interclump medium at the position of clu mp
+1. These values translate into abundancesof H 2CO in the range
+(11−110)×10−9for the interclump,(1 .4−2.7)×10−9for the
+clumps.Formethanol,we obtainanabundancerelativeto H 2of(0.08−8)×10−9for the interclump, (1 .2−1.6)×10−9for the
+clumps.
+However,theseestimatesarea ffectedbylargeuncertainties:
+a difference of 20 K in the dust temperature implies an uncer-
+tainty of a factor of ∼2 in the estimate of the H 2column den-
+sity in the interclump, uncertainty that could be even great er if
+the average temperature in the region of the interclump used
+in our analysis is higher than 75 K. Similarly, the estimate o f
+the H2column density on the clumps is a ffected by large un-
+certainties because of the assumption that the dust is therm ally
+coupled to the gas at high densities and because of the large e r-
+rors on the temperature of the gas from the methanol spectrum
+(28<T1<92K,20<T3<52K).
+On the other hand, the uncertainties in the determination
+of the abundance of CH 3OH relative to H 2CO are related only
+to the errors in the estimate of the column density of the two
+molecules, which already include the uncertainties in the t em-
+perature. Therefore, we believe [CH 3OH−A]/[H2CO−p] to
+be more solidly estimated than [CH 3OH−A]/[H2] or [H 2CO−
+p]/[H2].Usingthecolumndensitiesderivedintheprevioussec-
+tion, we infer an abundance of CH 3OH relative to H 2CO of the
+order 0.4–1.1 in the dense clumps, and 7 ×10−4– 7×10−1in the
+interclump.If we assume that the temperatureof the intercl ump
+mediumis higherthan 50 K, then abundanceof methanoldrops
+down to 7×10−4– 7×10−2. Since it is a reasonable assumption
+that the temperature of the interclump medium is 50 K or even
+higher, we conclude that the abundance of methanol relative to
+formaldehyde decreases by at least one order of magnitude in
+theinterclumpmediumincomparisontothe denseclumps.
+A possible explanation for the decrease in abundance of
+methanolrelativeto formaldehydefrom the clumpsto the int er-
+clumpmediumisphotodissociationinalow-densityenvirom ent.
+Although the photodissociation rates of CH 3OH and H 2CO are
+of the same order of magnitude (LeTeu ffet al. 2000), one of
+the products of the photodissociation of methanol is formal de-
+hyde. Therefore, both molecules can be destroyed by the ra-
+diation field in the interclump medium, but H 2CO may be re-
+plenished through the photodissociation of CH 3OH. However,
+large uncertainties a ffect the measurement of the photodisso-
+ciation rates, and so more accurate models would be required
+to test this hypothesis. Alternatively, as already discuss ed by
+Leuriniet al. (2006), H 2CO could be formed in the interclump
+mediumthroughgasphase reactions, which are not e fficient for
+theformationofCH 3OH(Lucaet al.2002).
+6. Conclusion
+In a previous analysis of methanol and formaldehyde towards
+the Orion Bar, we suggested that the two moleculesmight trac e
+differentenvironmentsandthat,whileCH 3OHisassociatedwith
+theclumpymolecularcoresoftheBar,H 2COisfoundinthein-S.Leurini etal.:H 2CO andCH 3OH maps of the OrionBar 9
+(a)
+(b)
+(c)
+Fig.6.VariationintheintegratedintensityoftheCH 3OH-A(50−
+40) and H 2CO 32,2−22,1lines observed with the IRAM-30m
+telescope across clumps 1 ( a), the interclump medium between
+clump1and2( b)andalongtheBar( c).InFig.1weoutlinethe
+stripsusedtoextractthe plots.
+terclumpmaterial.Totest thishypothesis,wemappedtheOr ion
+Bar in both molecular species with the IRAM-30m telescope,
+withaspatialresolutionslightlyhigherthantheexpected sizeof
+the clumps. Additional data were taken with the IRAM Plateau
+deBureInterferometerin H 2CO.
+Both molecules are detected in the Orion Bar in our single-
+dishdata.OurdatashowthatCH 3OHpeakstowardstheclumps
+oftheBar,butitsintensitydecreasesbelowthedetectiont hresh-
+Fig.7.Channel maps of the CH 3OH-A(50−40) emission be-
+tween 11.3 and 7.4 km s−1observed with the IRAM-30m tele-
+scope.
+Fig.9.Ratio of the integrated intensity of the H 2CO 30,3−20,2
+transition to the 3 2,2−22,1. The solid black contours show
+the integrated intensity of the 3 0,3−20,2line (levels are from
+1.8Kkms−1instepsof1.8).TypicalerrorsalongtheBarareof
+theorderof0.3.
+old in the interclump at individual positions. By averaging over
+a large region of the interclump medium, the strongest of the
+CH3OH lines in our setup (5 0−40-A) is detected with a peak
+intensityof∼0.06K.Ontheotherhand,contrarytothehypoth-
+esis formulated in our previousstudy, formaldehydeis dete cted
+towardstheclumpsandtheinterclumpgas.
+Using an LVG program, we studied the excitation of H 2CO
+and CH 3OH in the Orion Bar. We suggest that formaldehydeis10 S.Leurini etal.:H 2CO andCH 3OH maps of the OrionBar
+Fig.10.χ2square distribution of the 3 0,3−20,2and 32,2−22,1
+line intensitiesin the [ n(H2),N(H2CO)] planefora temperature
+of 100 K for the two sets of observationsclump and interclump
+showninFig.3.Theplotshows,foreachspatialregion,thel ow-
+densitybranchandthehigh-densitybranchdiscussedinthe text.
+Note that to derive the column densities given in §4.1, one has
+tomultiplywith thelinewidths.
+Fig.11.Results of the statistical equilibrium calculations for
+CH3OH. The solid and dashed lines show the line intensity of
+the 50−40-Aline as measured towards the interclump medium
+(0.06K) for temperaturesof 20 and 80 K, respectively,as fun c-
+tionsofdensityandcolumndensityofCH 3OH-A.
+present in both components of the Bar (clumps and interclump
+material), with column densities up to 5 .4×1014cm−2in the
+clumpsandbetween6 ×1013−6×1014cm−2intheinterclump.
+These values only refer to para-formaldehyde.From the anal y-
+sis of methanol, we concluded that the reason for the drop of
+intensity of CH 3OH in the interclump medium is not the dif-
+ferent physical conditions with respect to the clumps, but a real
+drop in its column density compared to the clumps (down to
+4×1011cm−2for temperaturesabove 50 K). The abundanceof
+methanol relative to formaldehydedecreases by at least one or-
+der of magnitudein the interclumpmediumcomparedto that in
+thedenseclumps.Despite the large errors on our estimates, our observations
+revealthatthecolumndensityofmethanolandformaldehyde in
+the clumps are of the same order of magnitude, while the col-
+umn density of methanol in the interclump is lower than that
+of formaldehyde. This may be a result of photodissociation o f
+CH3OH in the unshielded interclump gas, or it may reflect that
+H2CO can be produced in the gas phase more e fficiently than
+CH3OH. Definitive observational conclusions would require a
+much deeper integration towards the interclump gas and a bet -
+tercharacterisationofitsdensityandtemperature.Mored etailed
+chemicalmodelsofPDRs,includinggrainsurfacereactions ,are
+clearlyneededtoproperlyinterprettheseobservationalr esults.
+Acknowledgements. We are grateful to Helmut Wiesemeyer for his support be-
+fore and during the observations, and to the observers who ob tained part of the
+data presented in this work during pooled observations at th e IRAM-30m tele-
+scope.
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\ No newline at end of file
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+arXiv:1001.0895v2  [math.PR]  13 Mar 2013The Annals of Applied Probability
+2013, Vol. 23, No. 3, 957–986
+DOI:10.1214/12-AAP861
+c/circlecopyrtInstitute of Mathematical Statistics , 2013
+AVERAGING OVER FAST VARIABLES IN THE FLUID LIMIT
+FOR MARKOV CHAINS: APPLICATION TO THE SUPERMARKET
+MODEL WITH MEMORY
+By M. J. Luczak1and J. R. Norris2
+London School of Economics and University of Cambridge
+We set out a general procedure which allows the approximatio n
+of certain Markov chains by the solutions of differential equ ations.
+The chains considered have some components which oscillate rapidly
+and randomly, while others are close to deterministic. The l imiting
+dynamics are obtained by averaging the drift of the latter wi th re-
+spect to a local equilibrium distribution of the former. Som e general
+estimates are proved under a uniform mixing condition on the fast
+variable which give explicit error probabilities for the flu id approx-
+imation. Mitzenmacher, Prabhakar and Shah [In Proc. 43rd Ann.
+Symp. Found. Comp. Sci. (2002) 799–808, IEEE] introduced a vari-
+ant with memory of the “join the shortest queue” or “supermar ket”
+model, and obtained a limit picture for the case of a stable sy stem
+in which the number of queues and the total arrival rate are la rge.
+In this limit, the empirical distribution of queue sizes sat isfies a dif-
+ferential equation, while the memory of the system oscillat es rapidly
+and randomly. We illustrate our general fluid limit estimate by giving
+a proof of this limit picture.
+1. A general fluid limit estimate. We describe a general framework to
+allow the incorporation of averaging over fast variables in to fluid limit es-
+timates for Markov chains, building on the approach used in [ 2]. The main
+results of this section, Theorems 1.5and1.6, establish explicit error proba-
+bilities for the fluid approximation under assumptions whic h can be verified
+Received March 2010; revised August 2011.
+1Supported in part by a STICERD grant at the LSE, and by EPSRC Le adership
+Fellowship EP/J004022/2. Part of this work was accomplishe d while visiting the Mittag-
+Leffler Institute.
+2Supported by EPSRC Grant EP/E01772X/1.
+AMS 2000 subject classifications. Primary 60J28; secondary 60K25.
+Key words and phrases. Join the shortest queue, supermarket model, supermarket
+model with memory, law of large numbers, exponential martin gale inequalities, fast vari-
+ables, correctors.
+This is an electronic reprint of the original article published by the
+Institute of Mathematical Statistics inThe Annals of Applied Probability ,
+2013, Vol. 23, No. 3, 957–986 . This reprint differs from the original in pagination
+and typographic detail.
+12 M. J. LUCZAK AND J. R. NORRIS
+from knowledge of the transition rates of the Markov chain. A lso see [1] for
+related results.
+1.1.Outline of the method. LetX=(Xt)t≥0beacontinuous-timeMarkov
+chain with countable state-space Sand with generator matrix Q=(q(ξ,ξ′):
+ξ,ξ′∈S). Assume that the total jump rate q(ξ) is finite for all states ξ,
+and that Xis nonexplosive. Then the law of Xis determined uniquely by
+Qand the law of X0. Make a choice of fluid coordinates xi:S→R, for
+i=1,...,d, and write x=(x1,...,xd):S→Rd. Consider the Rd-valued pro-
+cessX=(Xt)t≥0given by Xt=(X1
+t,...,Xd
+t)=x(Xt). CallXtheslowor
+fluid variable . Define for each ξ∈Sthedrift vector
+β(ξ)=Qx(ξ)=/summationdisplay
+ξ′/ne}ationslash=ξ(x(ξ′)−x(ξ))q(ξ,ξ′).
+Also, make a choice of an auxiliary coordinate y:S→I, for some countable
+setI, and set Yt=y(Xt). Call the process Y=(Yt)t≥0thefast variable . For
+ξ∈Sandy′∈Iwithy′/ne}ationslash=y(ξ), writeγ(ξ,y′) for the total rate at which Y
+jumps to y′whenXis atξ. Thus
+γ(ξ,y′)=/summationdisplay
+ξ′:y(ξ′)=y′q(ξ,ξ′).
+Choose a subset UofRdand a function b:U×I→Rd. Choose also, for
+eachx∈U, a generator matrix Gx=(g(x,y,y′):y,y′∈I) having a unique
+invariant distribution πx= (π(x,y):y∈I). These choices are to be made
+so thatβ(ξ) is close to b(x(ξ),y(ξ)) andγ(ξ,y′) is close to g(x(ξ),y(ξ),y′)
+whenever x(ξ)∈Uandy′∈I. Define for x∈U
+¯b(x)=/summationdisplay
+y∈Ib(x,y)π(x,y).
+Then, under regularity assumptions to be specified later, th ere exists a func-
+tionχ:U×I→Rdsuch that
+Gχ(x,y)=/summationdisplay
+y′∈Ig(x,y,y′)χ(x,y′)=b(x,y)−¯b(x). (1)
+Make a choice of such a function χ. Callχthecorrector for b.
+Fixx0∈U. We will assume that ¯bis Lipschitz on U. Then the differential
+equation ˙ xt=¯b(xt) has auniquemaximal solution ( xt)t<ζinUstarting from
+x0. Fixt0∈[0,ζ). Then for t≤t0,
+xt=x0+/integraldisplayt
+0¯b(xs)ds. (2)AVERAGING OVER FAST VARIABLES 3
+Define for ξ∈Swithx(ξ)∈U
+¯x(ξ)=x(ξ)−χ(x(ξ),y(ξ)).
+LetTbe a stopping time such that Xt∈Ufor allt≤T. Then, under
+regularity assumptions to be specified later, for t≤T,
+¯x(Xt)=¯x(X0)+Mt+/integraldisplayt
+0¯β(Xs)ds, (3)
+whereM=M¯xis a martingale and where
+¯β=Q¯x=β−Q(χ(x,y)). (4)
+On subtracting equations ( 2) and (3) we obtain for t≤T∧t0
+Xt−xt=X0−x0+χ(Xt,Yt)−χ(X0,Y0)+Mt+/integraldisplayt
+0∆(Xs)ds
+(5)
++/integraldisplayt
+0(β(Xs)−b(Xs,Ys))ds+/integraldisplayt
+0(¯b(Xs)−¯b(xs))ds,
+where ∆= Gχ(x,y)−Q(χ(x,y)).
+The discussion in the present paragraph is intended for orie ntation only,
+and will play no essential role in the derivation of our resul ts. FixU0⊆U
+such that for all ξ,ξ′∈Swithx(ξ)∈U0andq(ξ,ξ′)>0 we have x(ξ′)∈U.
+Assume that Tis chosen so that Xt∈U0for allt≤T. Define for ξ∈Swith
+x(ξ)∈U0thediffusivity tensor α(ξ)∈Rd⊗Rdby
+αij(ξ)=/summationdisplay
+ξ′/ne}ationslash=ξ(¯xi(ξ′)−¯xi(ξ))(¯xj(ξ′)−¯xj(ξ))q(ξ,ξ′) (6)
+and define for t≤T
+Nt=Mt⊗Mt−/integraldisplayt
+0α(Xs)ds.
+Then, under regularity assumptions, Nis a martingale in Rd⊗Rd. Choose
+a function a:U0×I→Rd⊗Rdand set
+¯a(x)=/summationdisplay
+y∈Ia(x,y)π(x,y).
+This choice is to be made so that α(ξ) is close to a(x(ξ),y(ξ)) whenever
+x(ξ)∈U0. Suppose we can also find a corrector for a, that is, a function
+˜χ:U0×I→Rd⊗Rdsuch that
+G˜χ(x,y)=a(x,y)−¯a(x). (7)4 M. J. LUCZAK AND J. R. NORRIS
+Then, for t≤T,
+/integraldisplayt
+0α(Xs)ds= ˜χ(Xt,Yt)−˜χ(X0,Y0)−˜Mt+/integraldisplayt
+0˜∆(Xs)ds
+(8)
++/integraldisplayt
+0(α(Xs)−a(Xs,Ys))ds+/integraldisplayt
+0¯a(Xs)ds,
+where˜∆=G˜χ(x,y)−Q(˜χ(x,y)) and, under suitable regularity conditions,
+˜M=M˜χis a martingale up to T.
+The martingale terms Mand˜Min (5) and (8) can be shown to be small,
+under suitable conditions, using the following standard ty pe of exponential
+martingaleinequality. Intheformgivenhereitmaybededuc ed,forexample,
+from [2], Proposition 8.8, by setting f=θφ,A=θ2eθJε/2 andB=θδ.
+Proposition 1.1. Letφbe a function on S. Define
+Mt=Mφ
+t=φ(Xt)−φ(X0)−/integraldisplayt
+0Qφ(Xs)ds.
+WriteJ=J(φ)for the maximum possible jump in φ(X), thus
+J= sup
+ξ,ξ′∈S,q(ξ,ξ′)>0|φ(ξ′)−φ(ξ)|.
+Define a function α=αφonSby
+α(ξ)=/summationdisplay
+ξ′/ne}ationslash=ξ{φ(ξ′)−φ(ξ)}2q(ξ,ξ′).
+Then, for all δ,ε∈(0,∞)and all stopping times T, we have
+P/parenleftbigg
+sup
+t≤TMt≥δand/integraldisplayT
+0α(Xt)dt≤ε/parenrightbigg
+≤exp{−δ2/(2εeθJ)},
+whereθ∈(0,∞)is determined by θeθJ=δ/ε.
+Now, ifβ,γ,αare well approximated by b,g,aand if we can show that
+the corrector terms in ( 5) and (8) are insignificant, then we may hope to
+use these equations to show that the path ( xt:t≤t0) provides a good (first
+order) approximation to ( Xt:t≤t0) and, moreover, that the fluctuation
+process ( Xt−xt:t≤t0) is approximated (to second order) by a Gaussian
+process ( Ft:t≤t0) given by
+Ft=F0+Bt+/integraldisplayt
+0∇¯b(xs)Fsds,
+where (Bt:t≤t0) is a zero-mean Gaussian process in Rdwith covariance
+E(Bs⊗Bt)=/integraldisplays∧t
+0¯a(xr)dr.AVERAGING OVER FAST VARIABLES 5
+Our aim in the rest of this section is to give an explicit form o f the first
+order approximation with optimal error scale, that is, of th e same order as
+the scale of deviation predicted by the second order approxi mation. The
+next subsection contains some preparatory material on corr ectors. A reader
+who wishes to understand only the statement of the fluid limit estimate can
+skip directly to Section 1.3.
+1.2.Correctors. In order to implement the method just outlined, it is
+necessary either to come up with explicit correctors or to ap peal to a general
+result which guarantees the existence, subject to verifiabl e conditions, of
+correctors with good properties. In this subsection we obta in such a general
+result. In fact, we shall find conditions which guarantee the existence, for
+each bounded measurable function fonU×I, of a good corrector for f,
+that is to say, a function χ=χfonU×Isuch that
+Gχ(x,y)=f(x,y)−¯f(x),
+where
+¯f(x)=/summationdisplay
+y∈If(x,y)π(x,y).
+Moreover, we shall see that χfdepends linearly on fand we shall obtain a
+uniform bound and a continuity estimate for χf.
+Assume that there is a constant ν∈(0,∞) such that, for all x∈Uand all
+y∈I, the total rate of jumping from yunderGxdoes not exceed ν. Then
+we can choose an auxiliary measurable space E, with aσ-fieldE, a family of
+probability measures µ=(µx:x∈U) on (E,E) and a measurable function
+F:I×E→Isuch that, for all x∈Uand ally,y′∈Idistinct,
+g(x,y,y′)=νµx({v∈E:F(y,v)=y′}). (9)
+LetN= (N(t):t≥0) be a Poisson process of rate ν. Fixx∈Uand let
+V=(Vn:n∈N) be a sequence of independent random variables in E, all
+with law µx. Thus
+g(x,y,y′)=νP(F(y,Vn)=y′)
+for all pairs of distinct states y,y′and alln. Fix a reference state ¯ y∈I.
+Giveny∈I, setZ0=yand¯Z0= ¯yand define recursively for n≥0,
+Zn+1=F(Zn,Vn+1),¯Zn+1=F(¯Zn,Vn+1).
+SetYt=ZN(t)and¯Yt=¯ZN(t). ThenY=(Yt)t≥0and¯Y=(¯Yt)t≥0are both
+Markov chains in Iwith generator matrix Gx, starting from yand ¯y, respec-
+tively,3and are realized on the same probability space. We call the tr iple
+(ν,µ,F) acoupling mechanism . Define the coupling time
+Tc=inf{t≥0:Yt=¯Yt}.
+3The process Yintroduced here is not the fast variable, also denoted Yin the rest of
+the paper: the current Yis to be considered as a local approximation of the fast varia ble.6 M. J. LUCZAK AND J. R. NORRIS
+Assume that, for some positive constant τ, for allx∈Uand ally,¯y∈I,
+m(x,y,¯y)=E(x,y,¯y)(Tc)≤τ. (10)
+Fix a bounded measurable function fonU×Iand set
+χ(x,y)=E(x,y)/integraldisplayTc
+0(f(x,Yt)−f(x,¯Yt))dt. (11)
+Thenχis well defined and, for all x∈Uand ally∈I,
+|χ(x,y)|≤2τ/ba∇dblf/ba∇dbl∞. (12)
+Proposition 1.2. The function χis a corrector for f.
+Proof. In the proof we suppress the variable x. Note first that, if in-
+stead of taking ¯Z0= ¯y, we start ¯Zrandomly with the invariant distribution
+π, then we change the value of χby a constant independent of y. Hence, it
+will suffice to establish the corrector equation Gχ=f−¯fin this case. Fix
+λ>0 and define
+φλ(y)=E/integraldisplayTλ
+0f(Yt)dt,¯φλ=E/integraldisplayTλ
+0f(¯Yt)dt,
+whereTλ=T1/λ, withT1an independent exponential random variable of
+parameter 1. Then, since Yand¯Ycoincide after Tc,
+¯φλ−φλ(y)=E/integraldisplayTλ∧Tc
+0(f(¯Yt)−f(Yt))dt→χ(y)
+asλ→0. By elementary conditioning arguments, ( G−λ)φλ+f= 0 and
+λ¯φλ=¯f, so
+(G−λ)(¯φλ−φλ)=f−¯f.
+On passing to the limit λ→0 in this equation, using bounded convergence
+we find that Gχ=f−¯f, as required. /square
+We remark that the corrector χ(x,·) in fact depends only on f,Gxand
+the choice of ¯ y, as the preceding proof makes clear. The further choice of a
+coupling mechanism is a way to obtain estimates on χ.
+The following estimate will be used in dealing with the ∆ term in (5). We
+write/ba∇dblµx−µx′/ba∇dblfor the total variation distance between µxandµx′.
+Proposition 1.3. For allx,x′∈Uand ally∈I,
+|χ(x,y)−χ(x′,y)|≤2τsup
+z∈I|f(x,z)−f(x′,z)|
+(13)
++2ντ2/ba∇dblf/ba∇dbl∞/ba∇dblµx−µx′/ba∇dbl.AVERAGING OVER FAST VARIABLES 7
+Proof. By a standard construction (maximal coupling) there exists a
+sequence of independent random variables (( Vn,V′
+n):n∈N) inE×Esuch
+thatVnhas distribution µx,V′
+nhas distribution µx′andP(Vn/ne}ationslash=V′
+n) =
+1
+2/ba∇dblµx−µx′/ba∇dbl=supA∈E|µx(A)−µx′(A)|, for alln. Write (Ft)t≥0for the filtra-
+tion of the marked Poisson process obtained by marking Nwith the random
+variables ( Vn,V′
+n). Construct ( Y,¯Y) fromNand (Vn:n∈N) as above. Sim-
+ilarly construct ( Y′,¯Y′) fromNand (V′
+n:n∈N). Recall that Tc=inf{t≥
+0:Yt=¯Yt}and setT′
+c=inf{t≥0:Y′
+t=¯Y′
+t}. Setλ=1
+2ν/ba∇dblµx−µx′/ba∇dbland set
+D=inf{t≥0:(Yt,¯Yt)/ne}ationslash=(Y′
+t,¯Y′
+t)}.
+Then the process t/ma√sto→1{D≤t}−λtis an (Ft)t≥0-supermartingale and Tcis
+an (Ft)t≥0-stopping time. So, by optional stopping, we have P(D≤Tc)≤
+λE(Tc)≤λτ. Moreover, by the strong Markov property, on {D≤Tc}, we
+haveE(Tc−D|FD)=m(x,YD,¯YD)≤τso, for any function g:I→Rd, with
+|g|≤/ba∇dblf/ba∇dbl∞,
+E/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayTc
+D∧Tcg(Yt)dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤τ/ba∇dblf/ba∇dbl∞P(D≤Tc)≤λτ2/ba∇dblf/ba∇dbl∞.
+On the other hand,
+/integraldisplayD∧Tc
+0g(Yt)dt=/integraldisplayD∧T′
+c
+0g(Y′
+t)dt
+so
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleE/integraldisplayTc
+0g(Yt)dt−E/integraldisplayT′
+c
+0g(Y′
+t)dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2λτ2/ba∇dblf/ba∇dbl∞=ντ2/ba∇dblf/ba∇dbl∞/ba∇dblµx−µx′/ba∇dbl.
+We apply this estimate with g=f(x,·) to obtain
+|χ(x,y)−χ(x′,y)|
+=/vextendsingle/vextendsingle/vextendsingle/vextendsingleE/integraldisplayTc
+0(f(x,¯Yt)−f(x,Yt))dt−E/integraldisplayT′
+c
+0(f(x′,¯Y′
+t)−f(x′,Y′
+t))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤2τsup
+z∈I|f(x,z)−f(x′,z)|+/vextendsingle/vextendsingle/vextendsingle/vextendsingleE/integraldisplayTc
+0f(x,¯Yt)dt−E/integraldisplayT′
+c
+0f(x,¯Y′
+t)dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle
++/vextendsingle/vextendsingle/vextendsingle/vextendsingleE/integraldisplayTc
+0f(x,Yt)dt−E/integraldisplayT′
+c
+0f(x,Y′
+t)dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+≤2τsup
+z∈I|f(x,z)−f(x′,z)|+2ντ2/ba∇dblf/ba∇dbl∞/ba∇dblµx−µx′/ba∇dbl
+as required. /square
+To summarize, we have shown the following proposition.8 M. J. LUCZAK AND J. R. NORRIS
+Proposition 1.4. Assume conditions ( 9) and (10). Then, for any bound-
+ed measurable function fonU×I, there exists a corrector χfforfsatisfying
+the estimates ( 12) and (13).
+1.3.Statement of the estimates. Recall the context of Section 1.1. We
+consider a continuous-time Markov chain Xwith countable state-space S
+and generator matrix Q. We choose fluid coordinates x:S→Rdand an
+auxiliary coordinate y:S→I. We choose also a subset U⊆Rd, which pro-
+vides a means of localization, together with a map b:U×I→Rd, and a
+familyG=(Gx:x∈U) of generator matrices on I, each having a unique
+invariant distribution πx. Also choose, as in the preceding subsection, a
+coupling mechanism for G. This comprises a constant ν >0, an auxiliary
+spaceE, a function F:I×E→Iand a family of probability distributions
+µ=(µx:x∈U) onEsuch that
+g(x,y,y′)=νµx({v∈E:F(y,v)=y′}), x∈U,y,y′∈Idistinct.
+Define for x∈U
+¯b(x)=/summationdisplay
+y∈Ib(x,y)π(x,y).
+WriteXt=x(Xt) andassumethat ( xt)0≤t≤t0is asolution in Uto ˙xt=¯b(xt).
+We use a scaled supremum norm on Rd: fix positive constants σ1,...,σdand
+define for x∈Rd
+/ba∇dblx/ba∇dbl= max
+1≤i≤d|xi|/σi.
+We now introduce some constants Λ ,B,τ,J,J 1(b),J(µ),Kwhich charac-
+terize certain regularity properties of Q,bandG. Assume that, for all ξ∈S,
+allx∈Uand ally,y′∈I,
+q(ξ)≤Λ,/ba∇dblb(x,y)/ba∇dbl≤B, m (x,y,y′)≤τ. (14)
+Herem(x,y,y′) is the mean coupling time for Gxstarting from yandy′,
+defined in the preceding subsection, which dependson the cho ice of coupling
+mechanism. Write Jfor the set of pairs of points in Ubetween which X
+can jump, thus
+J={(x,x′)∈U×U:x=x(ξ),x′=x(ξ′) for some ξ,ξ′∈Swithq(ξ,ξ′)>0}.
+Set
+J= sup
+(x,x′)∈J/ba∇dblx−x′/ba∇dbl,
+J1(b)= sup
+(x,x′)∈J,y∈I/ba∇dblb(x,y)−b(x′,y)/ba∇dbl,
+J(µ)= sup
+(x,x′)∈J/ba∇dblµx−µx′/ba∇dbl.AVERAGING OVER FAST VARIABLES 9
+WriteKfor the Lipschitz constant of ¯bonU; thus, for all x,x′∈U,
+/ba∇dbl¯b(x)−¯b(x′)/ba∇dbl≤K/ba∇dblx−x′/ba∇dbl. (15)
+Recall from Section 1.1the definitions of the drift vector βforxand the
+jump rate γfory. Define
+T=inf{t≥0:Xt/∈U}.
+Fix constants δ(β,b),δ(γ,g)∈(0,∞) and consider the events
+Ω(β,b)=/braceleftbigg/integraldisplayT∧t0
+0/ba∇dblβ(Xt)−b(x(Xt),y(Xt))/ba∇dbldt≤δ(β,b)/bracerightbigg
+(16)
+and
+Ω(γ,g)=/braceleftbigg/integraldisplayT∧t0
+0/summationdisplay
+y′/ne}ationslash=y(Xt)|γ(Xt,y′)−g(x(Xt),y(Xt),y′)|dt≤δ(γ,g)/bracerightbigg
+. (17)
+Theorem 1.5. Letε>0be given and set δ=εe−Kt0/7. Assume that
+J≤εand
+max{/ba∇dblX0−x0/ba∇dbl,δ(β,b),2τBδ(γ,g),2τB,2Λt0(τJ1(b)+ντ2BJ(µ))}≤δ.
+Set¯J=J+ 4τBand assume that δ≤Λ¯Jt0/4. Further assume that the
+following tube condition holds:
+forξ∈Sandt≤t0/ba∇dblx(ξ)−xt/ba∇dbl≤2ε=⇒x(ξ)∈U.
+Then
+P/parenleftBig
+sup
+t≤t0/ba∇dblXt−xt/ba∇dbl>ε/parenrightBig
+≤2de−δ2/(4Λ¯J2t0)+P(Ω(β,b)c∪Ω(γ,g)c).
+The proof of this result follows the initial stages of the pro of of the more
+elaborate Theorem 1.6below. We will not write it out separately but give
+further indications immediately before the statement of Th eorem1.6. The
+reader will understand clearly the role of the inequalities which appear as
+hypotheses by following the proof. Here is an informal guide to their mean-
+ings. The tube condition, together with J≤ε, allows us to localize the other
+hypotheses to Uby trappingtheprocess insidea tubearoundthe limit path;
+these conditions can be satisfied by choosing Usufficiently large. The con-
+ditions/ba∇dblX0−x0/ba∇dbl≤δandδ(β,b)≤δenforce that the initial conditions and
+drift fields match closely. This requires, in particular, th at the fluid and aux-
+iliary coordinates provide sufficient information to nearly determine β. The
+condition on δ(γ,g) forces a close match between the local behavior of the
+fast variable and the idealized fast process used to compute the corrector.10 M. J. LUCZAK AND J. R. NORRIS
+The condition 2 τB≤δallows us to control the size of the corrector, balanc-
+ing the mean recurrence time of the fast variable τagainst the range of the
+drift field b. The condition on 2Λ t0(τJ1(b)+ντ2BJ(µ)) is needed for local
+regularity of the corrector, allowing us to pass back from th e idealized fast
+process at one point xto the actual fast variable when the fluid variable is
+nearx. Finally, the condition δ≤Λ¯Jt0/4 ensures we are in the “Gaussian
+regime” of the exponential martingale inequality, where ba d events cannot
+occur by a small number of large jumps. For a nontrivial limit ing dynamics,
+ΛJshould be of order 1, while for a useful estimate Λ J2should be small;
+thus, as expected, we can attempt to use the result when the Ma rkov chain
+takes small jumps at a high rate.
+It is sometimes possible to improve on the constant Λ ¯J2appearing in the
+preceding estimate, thereby obtaining useful probability bounds for smaller
+choices of ε. However, to do this we have to make hypotheses expressed in
+terms of a corrector. Fix ¯ y∈Iand denote by χthe corrector for bgiven by
+(11). Define for ξ∈Swithx(ξ)∈U
+¯x(ξ)=x(ξ)−χ(x(ξ),y(ξ)).
+Define, for ξ∈Ssuch that x(ξ)∈Uandx(ξ′)∈Uwhenever q(ξ,ξ′)>0,
+αi(ξ)=/summationdisplay
+ξ′/ne}ationslash=ξ{¯xi(ξ′)−¯xi(ξ)}2q(ξ,ξ′), i=1,...,d.
+Note that, since we shall be interested only in upper bounds, we deal here
+only with the diagonal terms of the diffusivity tensor defined a t (6). Choose
+functions ai:I→[0,∞) such that, for all ξ∈Uwhereαi(ξ) is defined,
+αi(ξ)≤ai(y(ξ)), i=1,...,d. (18)
+For simplicity, we do not allow ato depend on the fluid variable x(ξ). Since
+we can localize our hypotheses near the (compact) limit path , we do not
+expect to lose much precision by this simplification. On the o ther hand, by
+permitting a dependence on the fast variable we can sometime s do signifi-
+cantly better than Theorem 1.5, as we shall see in Section 2. Set
+¯a(x)=/summationdisplay
+y∈Ia(y)π(x,y), x∈U.
+We introduce two further constants Aand¯A, with¯A≤A≤Λ¯J2. Assume
+that, for all x∈Uand ally∈I,
+ai(y)≤Aσ2
+i,¯ai(x)≤¯Aσ2
+i, i=1,...,d. (19)
+Note that the corrector bound ( 12) gives/ba∇dblχ(x(ξ),y(ξ))/ba∇dbl≤2τB, soαi(ξ)≤
+Λ¯J2σ2
+iand so (19) holds with A=¯A=Λ¯J2andai(y) =Aσ2
+iand ¯ai(x)=AVERAGING OVER FAST VARIABLES 11
+Aσ2
+i. Thus Theorem 1.5follows directly from ( 20) below. The new inequal-
+ities required on the left-hand side of ( 21) can be understood roughly as
+imposing that the ratio of the averaged diffusivity to a unifor m bound on
+the diffusivity is not too small compared to the mean recurrenc e time of the
+fast variable; so an effective averaging takes place.
+Theorem 1.6. Assume that the hypotheses of Theorem 1.5hold and
+thatδ¯J≤At0/4. Then
+P/parenleftBig
+sup
+t≤t0/ba∇dblXt−xt/ba∇dbl>ε/parenrightBig
+≤2de−δ2/(4At0)+P(Ω(β,b)c∪Ω(γ,g)c). (20)
+Moreover, under the further conditions δ¯J≤¯At0/4and
+1
+t0max{τ,τδ(γ,g),Λt0ντ2J(µ)}≤¯A/(20A)≤Λτ, (21)
+we have
+P/parenleftBig
+sup
+t≤t0/ba∇dblXt−xt/ba∇dbl>ε/parenrightBig
+≤2de−δ2/(4¯At0)+2de−(¯A/A)2t0/(6400Λτ2)
+(22)
++P(Ω(β,b)c∪Ω(γ,g)c).
+Proof. Consider the stopping time
+T0=inf{t≥0:/ba∇dblXt−xt/ba∇dbl>ε}.
+By the tube condition, we have T0≤T. Moreover, for any t<T0and any
+ξ′∈Ssuch that q(Xt,ξ′)>0, we have
+/ba∇dblx(ξ′)−xt/ba∇dbl≤J+/ba∇dblXt−xt/ba∇dbl≤2ε
+so by the tube condition x(ξ′)∈U.
+Recall that χis the corrector for bgiven by ( 11). For the proof of ( 22),
+we shall use ( 11) to also construct a corrector ˜ χfora. Set˜δ=¯At0/10. Note
+from (12) the bounds
+/ba∇dblχ(x,y)/ba∇dbl≤2τB≤δ,|˜χi(x,y)|≤2τAσ2
+i≤˜δσ2
+i.
+Theinequality involving ˜δandfurthersuch inequalities below,which depend
+on the first inequality in assumption ( 21), will not be used in the proof
+of (20). Write ∆ = Gχ(x,y)−Q(χ(x,y)) = ∆ 1+∆2and˜∆ =G˜χ(x,y)−
+Q(˜χ(x,y))=˜∆1+˜∆2, where
+∆1(ξ)=/summationdisplay
+y′/ne}ationslash=y(ξ){g(x(ξ),y(ξ),y′)−γ(ξ,y′)}χ(x(ξ),y′) (23)
+and
+∆2(ξ)=/summationdisplay
+ξ′/ne}ationslash=ξq(ξ,ξ′){χ(x(ξ),y(ξ′))−χ(x(ξ′),y(ξ′))} (24)12 M. J. LUCZAK AND J. R. NORRIS
+and where ˜∆1and˜∆2are defined analogously. Then, on Ω( γ,g), fort≤
+T∧t0,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt
+0∆1(Xs)ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤2τBδ(γ,g)≤δ
+and, using Proposition 1.3,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt
+0∆2(Xs)ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤2Λt0(τJ1(b)+ντ2BJ(µ))≤δ.
+Similarly, for t≤T∧t0,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt
+0˜∆i
+1(Xs)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2τAδ(γ,g)σ2
+i≤˜δσ2
+i
+and/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt
+0˜∆i
+2(Xs)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2Λt0ντ2AJ(µ)σ2
+i≤˜δσ2
+i.
+TakeM=M¯xas in equations ( 3) and (5) and consider the event
+Ω(M)=/braceleftBig
+sup
+t≤T0∧t0/ba∇dblMt/ba∇dbl≤δ/bracerightBig
+.
+Then,on Ω( β,b)∩Ω(γ,g)∩Ω(M), wecan estimate thetermsin ( 5) to obtain
+fort≤T0∧t0,
+/ba∇dblXt−xt/ba∇dbl≤7δ+K/integraldisplayt
+0/ba∇dblXs−xs/ba∇dblds,
+so that/ba∇dblXt−xt/ba∇dbl≤εby Gronwall’s lemma. Note that this forces T0≥t0
+and hence, supt≤t0/ba∇dblXt−xt/ba∇dbl≤ε. Setρ=3¯A/2 and consider the event
+Ω(a)=/braceleftbigg/integraldisplayT0∧t0
+0ai(Ys)ds≤ρt0σ2
+ifori=1,...,d/bracerightbigg
+.
+By condition ( 18), on Ω(a) we have
+/integraldisplayT0∧t0
+0αi(Xs)ds≤ρt0σ2
+i.
+Set
+Ji=J(¯xi)= sup
+ξ,ξ′∈S,x(ξ),x(ξ′)∈U,q(ξ,ξ′)>0|¯xi(ξ)−¯xi(ξ′)|, i=1,...,d,
+anduse( 12)toseethat Ji≤¯Jσi.Determine θi∈(0,∞)byθieθiJi=δ/(ρt0σi);
+thenθi≤δ/(ρt0σi), soθiJi≤2δ¯J/(3¯At0)≤1/4, since we assumed that
+δ¯J≤¯At0/4. Since e1/4≤4/3, we have ρeθiJi≤2¯A. We now apply the ex-
+ponential martingale inequality, Proposition 1.1, substituting ±¯xiforφfor
+i=1,...,dand substituting δσiforδandρt0σ2
+iforε. We thus obtain
+P(Ω(M)c∩Ω(a))≤2de−δ2/(4¯At0).AVERAGING OVER FAST VARIABLES 13
+If we take ¯A=A, then, using ( 18) and (19), we have Ω( a)=Ω, so the proof
+of (20) is now complete.
+Setη=16Λτ2A2. We shall complete the proof of ( 22) by showing that
+P(Ω(a)c∩Ω(γ,g))≤2de−˜δ2/(4ηt0).
+Take˜Mas in (8), withaas in (18). Then, for t≤T,
+/integraldisplayt
+0a(Ys)ds= ˜χ(Xt,Yt)−˜χ(X0,Y0)−˜Mt
+(25)
++/integraldisplayt
+0˜∆(Xs)ds+/integraldisplayt
+0¯a(Xs)ds,
+where˜∆=G˜χ(x,y)−Q(˜χ(x,y)). Consider the event
+Ω(˜M)=/braceleftBig
+sup
+t≤T0∧t0|˜Mi
+t|≤˜δσ2
+ifori=1,...,d/bracerightBig
+.
+Then, on Ω( γ,g)∩Ω(˜M), we can estimate the terms in ( 25) to obtain
+/integraldisplayT0∧t0
+0ai(Ys)ds≤(5˜δ+¯At0)σ2
+i≤ρt0σ2
+i.
+Hence, it will suffice to show that
+P(Ω(˜M)c)≤2de−˜δ2/(4ηt0).
+For this, we again use the exponential martingale inequalit y. Takeφ(ξ) =
+±˜χi(x(ξ),y(ξ)) in Proposition 1.1and note that αφ(ξ)≤16Λτ2A2σ4
+i, so
+/integraldisplayT0∧t0
+0αφ(Xs)ds≤16Λτ2A2σ4
+it0=ηt0σ4
+i.
+Set
+˜Ji=J(φ)= sup
+ξ,ξ′∈S,x(ξ),x(ξ′)∈U,q(ξ,ξ′)>0|φ(ξ)−φ(ξ′)|, i=1,...,d,
+then˜Ji≤4τAσ2
+i. Determine ˜θi∈(0,∞) by˜θie˜θi˜Ji=˜δ/(ηt0σ2
+i). Then ˜θi≤
+˜δ/(ηt0σ2
+i) so˜θi˜Ji≤¯A/(40ΛτA)≤1/2 and so e˜θi˜Ji≤2. Hence,
+P(Ω(˜M)c)≤2dexp{−˜δ2/(2ηt0e˜θi˜Ji)}≤2de−˜δ2/(4ηt0)
+as required. /square14 M. J. LUCZAK AND J. R. NORRIS
+2. The supermarket model with memory. The supermarket model with
+memory is a variant, introduced in [ 8], of the “join the shortest queue”
+model, which has been widely studied [ 3–7,9]. We shall rigorously verify the
+asymptotic picture for large numbers of queues derived in [ 8]. This will serve
+as an example to illustrate the general theory of the precedi ng sections. The
+explicit form of the error probabilities in Theorem 1.6is used to advantage
+in dealing with the infinite-dimensional character of the li mit model.
+Fixλ∈(0,1) andan integer n≥1. Weshall consider thelimiting behavior
+asN→∞of the following queueing system. Customers arrive as a Pois son
+process of rate Nλat a system of Nsingle-server queues. At any given time,
+thelength ofoneofthequeuesiskept underobservation. Thi squeueiscalled
+thememory queue . On each arrival, an independent random sample of size
+nis chosen from the set of all Nqueues. For simplicity, we sample with
+replacement, allowing repeats and allowing the choice of th e memory queue.
+The customer joins whichever of the memory queue or the sampl ed queues
+is shortest, choosing randomly in the event of a tie. Immedia tely after the
+customer has joined a queue, we switch the memory queue, if ne cessary, so
+that it is the currently shortest queue among the queues just sampled and
+the previous memory queue. The service requirements of all c ustomers are
+assumed independent and exponentially distributed of mean 1.
+WriteZk
+t=ZN,k
+tfor the proportion of queues having at least kcustomers
+at timet, and write Ytfor length of the memory queue at time t. SetZt=
+(Zk
+t:k∈N) andXt=(Zt,Yt). ThenX=(Xt)t≥0is a Markov chain, taking
+values in S=S0×Z+, whereS0is the set of nonincreasing sequences in
+N−1{0,1,...,N}with finitely many nonzero terms. We shall treat Yas a
+fast variable and prove a fluid limit for ZasN→∞.
+2.1.Statement of results. LetDbe the set of nonincreasing sequences4
+z=(zk:k∈N) in the interval [0 ,1] such that
+m(z):=/summationdisplay
+kzk<∞.
+Define for z∈Dandk∈N
+µ(z,k)=k/productdisplay
+j=1zn
+j
+1−pj−1(z), (26)
+where
+pk−1(z)=n(zk−1−zk)zn−1
+k
+4To lighten the notation, we shall sometimes move the coordin ate index from a super-
+script to a subscript, allowing the nth power of the kth coordinate to be written zn
+k. We
+shall also write the time variable sometimes as a subscript, sometimes as an argument.AVERAGING OVER FAST VARIABLES 15
+and wherewe take z0=1. Setµ(z,0)=1 for all z. An elementary calculation
+(maximizing over zkwhile keeping zk−1fixed) shows that in the case n≥2,
+pk−1(z)≤zn
+k−1(1−1/n)n−1≤(1−1/n)n/2≤e−1/2<1. (27)
+In the case n=1 we have pk−1(z)=zk−1−zk≤1 and it is possible that 0 /0
+appears in the product ( 26). For definiteness we agree to set 0 /0=1 in this
+case. Note that µ(z,k)≥µ(z,k+1) for all k≥0. Define for z∈D
+vk(z)=λzn
+k−1µ(z,k−1)−λzn
+kµ(z,k)−(zk−zk+1)
+and consider the differential equation
+˙z(t)=v(z(t)), t≥0. (28)
+By asolution to (28) inDwe mean a family of differentiable functions
+zk:[0,∞)→[0,1] such that for all t≥0 andk∈Nwe have ( zk(t):k∈N)∈D
+and
+˙zk(t)=vk(z(t)).
+Theorem 2.1. For allz(0)∈D, the differential equation ˙z(t)=v(z(t))
+has a unique solution in Dstarting from z(0). Moreover, if (w(t):t∈D)is
+another solution in Dwithzk(0)≤wk(0)for allk, thenzk(t)≤wk(t)for
+allkand allt≥0.
+There is a fixed point of these dynamics a∈Dgiven by setting a0= 1
+and defining
+ak+1=λan
+kµ(a,k), k≥0. (29)
+The components of adecay super-geometrically. Set
+α=n+1
+2+/radicalBig
+n2+1
+4. (30)
+Thenα∈(2n,2n+1).
+Theorem 2.2. We have
+lim
+k→∞1
+kloglog/parenleftbigg1
+ak/parenrightbigg
+=α.
+Assume for simplicity that we start the queueing system from the state
+where all queues, except the memory queue, are empty and wher e the mem-
+ory queue has exactly one customer. Write ( z(t):t≥0) for the solution to
+(28) starting from 0. Then zk(t)≤akfor allkandt. Our main result shows
+that (z(t):t≥0) is a good approximation to the process of empirical dis-
+tributions of queue lengths ( ZN(t):t≥0) for large N. The sense of this
+approximation is reasonably sharp. In particular, as a stra ightforward corol-
+lary, we obtain that, on a given time interval [0 ,t0], for any r >α−1, with
+high probability as N→∞, no queue length exceeds rloglogN.16 M. J. LUCZAK AND J. R. NORRIS
+Theorem 2.3. Setκ=(2α)−1and define
+d=d(N)=sup{k∈N:Nak>Nκ}.
+Fix a function φonNsuch that φ(N)/Nκ→0andlogφ(N)/loglogN→∞
+asN→∞. Setρ=4/(1−λ)whenn=1and setρ=2n/(1−e−1/2)when
+n≥2. Set˜ad+1=N−1an
+d+ρdad+1. Then
+lim
+N→∞d(N)/loglogN=1/α. (31)
+Moreover, for all t0≥0, we have
+lim
+N→∞P/parenleftBigg
+sup
+t≤t0sup
+k≤d|ZN,k
+t−zk
+t|√ak≥/radicalbigg
+φ(N)
+N/parenrightBigg
+=0 (32)
+and
+lim
+R→∞limsup
+N→∞P(ZN,d+1
+t≥R˜ad+1for some t≤t0)=0 (33)
+and
+lim
+N→∞P(ZN,d+2
+t=0for allt≤t0)=1. (34)
+The argument used to prove this result would apply without mo dification
+starting from any initial condition z(0) for the limit dynamics ( 28) such that
+zk(0)≤akfor allk, with suitable conditions on the convergence of ZN(0)
+toz(0). It may be harder to move beyond initial conditions which do not
+lie below the fixed point. We do note here, however, a family of long-time
+upper bounds for the limit dynamics which might prove useful for such an
+extension. Fix j∈Nand define a(j)
+k=a(k−j)+for eachk∈Z+; thena(j)is a
+fixed point of the modified equation
+˙wk(t)=vk(w(t))+(wj(t)−wj+1(t))1{k=j}.
+Since the added term is always nonnegative, a similar argume nt to that used
+to prove Theorem 2.1in the next subsection also shows that, if z(0)≤a(j)
+and (z(t):t≥0) is a solution of the original equation, then z(t)≤a(j)for
+allt.
+2.2.Existence and monotonicity of the limit dynamics. The differential
+equation ( 28) characterizes the limit dynamics for the fluid variables in our
+queueing model. Our analysis of its space of solutions will r est on the ex-
+ploitation of certain nonnegativity properties which have a natural proba-
+bilistic interpretation. We shall use the following standa rd property of dif-
+ferential equations: if b=(b1,...,bd) is a Lipschitz vector field on Rdsuch
+thatb1(x)≥0 whenever x=(x1,...,xd) withx1=0, and if ( x(t):t≤t0) is
+a solution to ˙ x(t)=b(x(t)) withx1(0)≥0, thenx1(t)≥0 for all t≤t0.AVERAGING OVER FAST VARIABLES 17
+We consider first a truncated, finite-dimensional system. Fi xd∈Nand
+define a vector field u=u(d)onDby setting uk(z)=vk(z) fork≤d−1 and
+ud(z)=λzn
+d−1µ(z,d−1)−λzn
+dµ(z,d)−zd (35)
+anduk(z)=0 for k≥d+1. Set D(d)={(x1,...,xd,0,0,...):0≤xd≤···≤
+x1≤1}.
+Proposition 2.4. For allx(0)∈D(d), the differential equation ˙x(t)=
+u(x(t))has a unique solution (x(t):t≥0)inD(d)starting from x(0).
+Proof. In the proof, we consider D(d) as a subset of Rd. The func-
+tionuis continuous on D(d) and is differentiable in the interior of D(d)
+with bounded partial derivatives. [In the case n= 1, the singularity in
+(∂/∂xj)µ(x,k) forj≤kasxj−1−xj→1 is canceled by the factor xkby
+whichitismultiplied,since xk≤xjonD(d).]Forx∈D(d)wehave u1(x)≤0
+whenx1=1, and ud(x)≥0 whenxd=0. Moreover, for k=1,...,d−1, if
+xk=xk+1thenpk(x)=0 so
+uk+1(x)=xn
+k(µ(x,k)−µ(x,k+1))≤µ(x,k)−µ(x,k+1)
+≤xn
+k−1µ(x,k−1)−xn
+kµ(x,k)≤uk(x).
+The conclusion now follows by standard arguments. /square
+Proof of Theorem 2.1.Suppose that ( w(t):t≥0) is a solution to
+˙w(t) =v(w(t)) inDstarting from w(0), with z(0)≤w(0), that is to say
+zk(0)≤wk(0) for all k. Fixdand write x(t) =z(d)(t) for the solution to
+˙x(t)=u(d)(x(t)) inD(d) starting from ( z1(0),...,zd(0),0,0,...). Sety(t)=
+(w1(t),...,w d(t),0,0,...) and note that x(0)≤y(0) andy(t)∈D(d) for all
+t. We shall show that x(t)≤y(t) for allt. Now consider D(d) as a subset of
+Rd. We have
+˙y(t)=u(y(t))+wd+1(t)ed,
+whereed=(0,...,0,1). Note that wd+1(t)≥0 for all t. Nowuis Lipschitz
+onD(d) and for k=1,...,dwe can show that5
+x,y∈D(d), x≤y, x k=yk=⇒uk(x)≤uk(y).
+Hence, by a standard argument z(d)(t) =x(t)≤y(t)≤w(t) for all t. The
+same argument shows that z(d)(t)≤z(d+1)(t) for all t, so the limit zk(t)=
+limd→∞z(d)
+k(t) exists for all kandt, andz(t)≤w(t) for allt.
+5An elementary calculation shows that µ(x,j)≤µ(y,j) for all jwhenever x≤y. This
+will also be shown by a soft probabilistic argument in Sectio n2.6. The further condition
+xk=ykgives the inequality
+yn
+k−1−xn
+k−1≥n(yk−1−xk−1)xn−1
+k=pk−1(y)−pk−1(x)≥y2n
+k
+1−pk−1(y)−x2n
+k
+1−pk−1(x).18 M. J. LUCZAK AND J. R. NORRIS
+Fixkand take d≥k+1. Then the following equation holds for all t:
+z(d)
+k(t)+/integraldisplayt
+0λz(d)
+k(s)nµ(z(d)(s),k)ds+/integraldisplayt
+0z(d)
+k(s)ds
+=zk(0)+/integraldisplayt
+0λz(d)
+k−1(s)nµ(z(d)(s),k−1)ds+/integraldisplayt
+0z(d)
+k+1(s)ds.
+On letting d→∞, we see by monotone convergence that
+zk(t)+/integraldisplayt
+0λzk(s)nµ(z(s),k)ds+/integraldisplayt
+0zk(s)ds
+=zk(0)+/integraldisplayt
+0λzk−1(s)nµ(z(s),k−1)ds+/integraldisplayt
+0zk+1(s)ds.
+Sincez(t)∈Dfor allt, all integrands in this equation are bounded by 1. It
+is now straightforward to see that ( z(t):t≥0) is a solution.
+Now
+wk(t)+/integraldisplayt
+0λwk(s)nµ(w(s),k)ds+/integraldisplayt
+0wk(s)ds
+=wk(0)+/integraldisplayt
+0λwk−1(s)nµ(w(s),k−1)ds+/integraldisplayt
+0wk+1(s)ds.
+By summing these equations over k∈{1,...,d}we see that the map t/ma√sto→/summationtextd
+k=1wk(t)−λtis nonincreasing for all d. Hence,m(w(t))≤m(w(0))+λt<
+∞. The equations can then be summed over all kand rearranged to obtain
+m(w(t))=m(w(0))+λt−/integraldisplayt
+0w1(s)ds.
+On the other hand,
+m(z(d)(t))=m(z(d)(0))+λt−/integraldisplayt
+0z(d)
+1(s)ds−λ/integraldisplayt
+0z(d)
+d(s)nµ(z(d)(s),d)ds
+so
+m(w(t))−m(z(d)(t))≤m(w(0))−m(z(d)(0))
+(36)
++λ/integraldisplayt
+0zd(s)nµ(z(s),d)ds.
+Ifw(0)=z(0) then the right-hand side tends to 0 as d→∞so we must have
+z(t)=w(t) for allt./square
+2.3.Properties of the fixed point. Recall the definition ( 29) of the fixed
+pointa. Sinceµ(a,k)≤1 for all k, we have
+ak≤λ1+n+···+nk−1AVERAGING OVER FAST VARIABLES 19
+soak→0 ask→∞. Theorem 2.2is then a straightforward corollary of the
+following estimate.
+Proposition 2.5. There is a constant C(λ,n)<∞such that, for all
+k≥0,
+C−1aα
+k≤ak+1≤Caα
+k, (37)
+whereαis given by ( 30).
+Proof. Note that, since a1=λ, we have pk−1(a) =ak−1−ak≤λ∨
+(1−λ)<1 for all kwhenn= 1. On the other hand, equation ( 27) gives
+pk−1(z)≤e−1/2forallkwhenn≥2.Thenfrom/summationtext∞
+k=1pk−1(a)≤1,weobtain
+a constant c<∞, which may depend on λwhenn=1, such that
+∞/productdisplay
+k=11
+1−pk−1(a)≤c.
+Then for k≥0
+λa2n
+kk−1/productdisplay
+j=1an
+j≤ak+1≤cλa2n
+kk−1/productdisplay
+j=1an
+j,
+so fork≥1,
+c−1a2n+1
+ka−n
+k−1≤ak+1≤ca2n+1
+ka−n
+k−1. (38)
+Note that λaα
+0=λ=a1≤λ−1aα
+0. FixA≥1/λand suppose inductively that
+A−1aα
+k−1≤ak≤Aaα
+k−1.
+On using these inequalities to estimate ak−1in (38), we obtain
+(cAn/α)−1aα
+k≤ak+1≤cAn/αaα
+k,
+where we have used the fact that 1 −n/α=α−2n. Hence, the induction
+proceeds provided we take A≥cα/(α−n)./square
+2.4.Choice of fluid coordinates and fast variable. In the remaining sub-
+sections we apply Theorem 1.6to deduce Theorem 2.3. Definedas in Theo-
+rem2.3and take as auxiliary space I=Nwhenn= 1 and I=Z+when
+n≥2. Make the following choice of fluid and auxiliary coordinat es: for
+ξ=(z,y)∈Swithz=(zk:k∈N), set
+xk(ξ)=zk, k=1,...,d, y (ξ)=y.
+Thus our fluid variable is Xt=x(Xt)=(Z1
+t,...,Zd
+t) and our fast variable is
+Yt=y(Xt). Note that when n=1, ifY0≥1 thenYt≥1 for allt, soYtakes
+values in I=N.20 M. J. LUCZAK AND J. R. NORRIS
+Let us compute the drift vector β(ξ) forXwhenXis in state ξ=(z,y)∈
+S. Note that Xkmakes a jump of size 1 /Nwhen a customer arrives at a
+queue of length k−1, and makes a jump of size −1/Nwhen a customer
+departs from a queue of length k, otherwise Xkis constant. The length
+of the queue which an arriving customer joins depends on the l ength of
+the memory queue yand on the lengths of the sampled queues. Denote the
+vector ofsampledqueuelengthsby V=V(z)=(V1,...,Vn) andwrite V(1)≤
+V(2)≤···≤V(n)for the ordered queue lengths. Define min( v)=v1∧···∧vn
+and setM=min(V). ThenM=V(1)and
+P(M≥k)=zn
+k.
+A new customer will go to a queue of length at least kif and only if M≥k
+andy≥k. So the rate for an arrival to a queue of length exactly k−1 is
+NλP(M≥k−1)1{y≥k−1}−NλP(M≥k)1{y≥k}.
+The rate for a departure from a queue of length kisN(zk−zk+1). Hence,
+settingz0=1, we have
+βk(ξ)=λzn
+k−11{y≥k−1}−λzn
+k1{y≥k}−(zk−zk+1).
+We now compute (an approximation to) the jump rates γ(ξ,y′) forY
+whenXis in state ξ=(z,y)∈S. The rate of departures from the memory
+queue is at most 1. Arrivals to the system occur at rate Nλ. Occasionally,
+the memory queue falls in the sample, an event of probability no greater
+thann/Nand hence, of rate no greater than λn. Assuming that the the
+memory queue does not fall in the sample, the length of the mem ory queue
+after an arrival is given by
+F(y,V)=(y+1)1{y≤M−1}+y1{M≤y≤P}+P1{y≥P+1}, (39)
+whereP=p(V) is given by P=M+1 when n=1 and P=(M+1)∧V(2)
+otherwise. Hence, we have
+/summationdisplay
+y′/ne}ationslash=y(ξ)|γ(ξ,y′)−NλP(F(y,V(z))=y′)|≤1+λn. (40)
+2.5.Choice of limit characteristics and coupling mechanism. Define
+U={x∈Rd:0≤xd≤···≤x1≤1 andx1≤(λ+1)/2 andxk≤2akfor allk}.
+The condition x1≤(λ+1)/2 ensures that 1 −x1is uniformly positive on U.
+Defineb:U×Z+→Rdby
+bk(x,y)=λxn
+k−11{y≥k−1}−λxn
+k1{y≥k}−(xk−xk+1), (41)
+where we set x0=1 and xd+1=0. Then, for ξ∈Swithx(ξ)∈U, we have
+β(ξ)=b(x(ξ),y(ξ))+(0,...,0,zd+1). (42)AVERAGING OVER FAST VARIABLES 21
+It is convenient to specify our choice of the generator matri ces (Gx:x∈U)
+andourchoice ofcouplingmechanismat thesametime. Set ν=Nλandtake
+as auxiliary space E= (Z+)n. Define a family of probability distributions
+µ= (µx:x∈U) onE, taking µxto be the law of a random sample V=
+V(x)=(V1,...,Vn) with
+P(V1≥k)=···=P(Vn≥k)=xk
+fork=0,1,...,d+1. Note that
+/ba∇dblµx−µx′/ba∇dbl≤2nd/summationdisplay
+k=1|xk−x′
+k|. (43)
+Then define for distinct y,y′∈Z+,
+g(x,y,y′)=NλP(F(y,V(x))=y′),
+whereFis given by ( 39). We take as coupling mechanism thetriple ( ν,µ,F).
+Note that F(y,v) =F(¯y,v) for all y,¯y∈Iwhenever p(v) = minI. For
+x∈Uwe have
+P(p(V(x))=1)≥1−x1>1−λ
+2,
+whenn=1, whereas for n≥2 we have
+P(p(V(x))=0)≥(1−x1)2>/parenleftbigg1−λ
+2/parenrightbigg2
+.
+Hence, we obtain, in all cases, m(x,y,¯y)≤τ, where we set
+τ=4
+Nλ(1−λ)2.
+Forξ∈Swithx=x(ξ)∈Uwe can realize a sample V(z) (from the
+distribution of queue lengths) and the sample V(x) on the same probability
+space by setting Vi(x) =Vi(z)∧d. WriteM(x) = min( V(x)) andP(x) =
+p(V(x)). Then M(x)=M(z)∧dandP(x)=P(z)∧(d+1) when n=1 and
+P(x)=P(z)∧dwhenn≥2. The difference between the two cases is that
+there is no second shortest queue in the sample when n=1. We have, for
+n=1,
+P(P(z)/ne}ationslash=P(x))≤P(M(z)≥d+1)=zd+1
+and, for n≥2,
+P(P(z)/ne}ationslash=P(x))≤P(P(z)≥d+1)≤nzdzn−1
+d+1.
+NowP(x)=P(z) implies M(x)=M(z) and hence, F(y,V(x))=F(y,V(z))
+for ally. Hence,
+P(F(y,V(z))/ne}ationslash=F(y,V(x)))≤P(P(x)/ne}ationslash=P(z)).22 M. J. LUCZAK AND J. R. NORRIS
+On combining this with ( 40) we obtain
+/summationdisplay
+y′/ne}ationslash=y(ξ)|γ(ξ,y′)−g(x(ξ),y(ξ),y′)|≤1+λn+Nλnzd+1. (44)
+2.6.Local equilibrium distribution. TheMarkov chain determinedby the
+generator Gxhas a unique closed communicating class, which is contained
+in{0,1,...,d}. Hence, Gxhas a unique equilibrium distribution πxwhich is
+supported on {0,1,...,d}. Consider a continuous-time Markov chain6Y=
+(Yt)t≥0with generator Gx, and initial distribution πx. Setµ(x,k)=P(Y0≥
+k). ThenYjumpsinto {0,1,...,k}fromjat rateα=Nλ(1−xn
+k+1−pk(x)),
+for allj≥k+1. On the other hand, Yjumps out of {0,1,...,k}only from
+k, and that at rate β=Nλxn
+k+1. Since the long run rates of such jumpsmust
+agree, we deduce that αµ(x,k+1)=βπx(k). Hence, we obtain
+µ(x,k+1)(1−pk(x))=xn
+k+1µ(x,k)
+and so
+µ(x,k)=k/productdisplay
+j=1xn
+j
+1−pj−1(x), k=1,...,d. (45)
+Hence, our present notation is consistent with the definitio n (26). Note also
+that, for z∈D,µ(z,k) depends only on z1,...,zk; in particular, if x=
+(z1,...,zd) thenµ(z,k)=µ(x,k) for allk≤d. Note that ¯bis given by
+¯bk(x)=λxn
+k−1µ(x,k−1)−λxn
+kµ(x,k)−(xk−xk+1), k=1,...,d, (46)
+wherex0=1 and xd+1=0. Hence, ¯b=u(d)as defined in Section 2.2.
+A comparison of ( 35) and (41) now shows that ¯b=u(d).
+Recall that ρ=4/(1−λ) whenn=1 and that ρ=2n/(1−e−1/2) when
+n≥2. Then for x∈Uandk≥1 we have
+µ(x,k)=xn
+kµ(x,k−1)/(1−pk−1(x))≤ρan
+kµ(x,k−1)
+≤ρan
+kµ(x,k−1)/(1−pk−1(a))
+so, for all x∈Uand inductively for all k≥1, we obtain
+µ(x,k)≤ρkµ(a,k). (47)
+The following argument shows that µ(z,k)≤µ(z′,k) for all kwhenever
+z≤z′. Fixd≥kand setx′=(z′
+1,...,z′
+d). Assume that x≤x′. By a standard
+construction we can realize samples V=V(x) andV′=V(x′) on a common
+probability space such that Vi≤V′
+ifor alli. Then we can construct Markov
+6See footnote 3.AVERAGING OVER FAST VARIABLES 23
+chainsYandY′, having generators GxandGx′, respectively, on the canon-
+ical space of a marked Poisson process of rate Nλ, where the marks are
+independent copies of ( V,V′), as follows. Set Y0=Y′
+0=1 and define recur-
+sively at each jump time Tof the Poisson process YT=F(YT−,VT) and
+Y′
+T=F(Y′
+T−,V′
+T), where ( VT,V′
+T) is the mark at time T. Then, since Fis
+nondecreasing in both arguments, we see by induction that Yt≤Y′
+tfor all
+t. Hence, by convergence to equilibrium,
+µ(z,k)=µ(x,k)= lim
+t→∞P(Yt≥k)≤lim
+t→∞P(Y′
+t≥k)=µ(x′,k)=µ(z′,k).
+2.7.Corrector upper bound. We take as our reference state ¯ y= minI
+and note that, under the coupling mechanism, we have ¯Yt≤Ytfor allt.
+Then the kth component of the corrector for bis given by
+χk(x,y)=λEy/integraldisplayTc
+0(xn
+k−11{¯Ys<k−1≤Ys}−xn
+k1{¯Ys<k≤Ys})ds,
+so forx∈Uand ally∈Iwe have
+|χk(x,y)|≤τxn
+k−1≤Can
+k−1/N.
+Now fixy≤k−2 and consider the stopping time T=inf{t≥0:Yt=k−1}.
+Note that Ycan enter state k−1 only from k−2 and does so at rate
+Nλxn
+k−1, whereas Tcoccurs in state k−2 at rate at least Nλ(1−λ)2/4.
+Hence,Py(T≤Tc)≤Can
+k−1and so, by the Markov property,
+Ey/integraldisplayTc
+01{¯Ys<k−1≤Ys}ds≤Ey(1{T≤Tc}m(x,k−1,¯YT))≤Can
+k−1τ≤Can
+k−1/N.
+Hence, we obtain, for x∈Uand ally∈I,
+|χk(x,y)|≤C(an
+k−11{y≥k−1}+a2n
+k−1)/N. (48)
+2.8.Quadratic variation upper bound. Thegrowthrateat ξofthequadratic
+variation of the corrected kth coordinate is given by
+αk(ξ)=/summationdisplay
+ξ′/ne}ationslash=ξ{¯xk(ξ′)−¯xk(ξ)}2q(ξ,ξ′).
+Recall that ¯ xk=xk−χk(x,y). We estimate separately, writing x=x(ξ) and
+y=y(ξ),
+/summationdisplay
+ξ′/ne}ationslash=ξ{xk(ξ′)−xk}2q(ξ,ξ′)≤N−2(Nxk+Nλxn
+k−11{y≥k−1})
+and
+/summationdisplay
+ξ′/ne}ationslash=ξ{χk(x(ξ′),y(ξ′))−χk(x,y)}2q(ξ,ξ′)≤CN−1x2n
+k−1.24 M. J. LUCZAK AND J. R. NORRIS
+Wheny≤k−2, we can improve the last estimate by splitting the sum in
+two and using
+/summationdisplay
+ξ′/ne}ationslash=ξ,y(ξ′)≤k−2{χk(x(ξ′),y(ξ′))−χk(x,y)}2q(ξ,ξ′)≤CN−1x4n
+k−1
+and
+/summationdisplay
+ξ′/ne}ationslash=ξ,y(ξ′)≥k−1{χk(x(ξ′),y(ξ′))−χk(x,y)}2q(ξ,ξ′)≤CN−1x3n
+k−1.
+We used ( 48) for the first inequality and for the second used
+/summationdisplay
+ξ′/ne}ationslash=ξ,y(ξ′)≥k−1q(ξ,ξ′)≤CNxn
+k−1.
+On combining these estimates we obtain
+αk(ξ)≤C(xk+xn
+k−11{y≥k−1}+x3n
+k−1)≤ak(y(ξ))/N,
+where
+ak(y(ξ))=C(ak+an
+k−11{y≥k−1})/N.
+Then, using the estimate ( 47) and the limit ( 31), we have
+¯ak(x)=C(ak+an
+k−1µ(x,k−1))≤Cρd−1ak/N≤C(logN)Cak/N,
+so we have for all x∈Uandy∈I
+ak(y)≤Aak,¯ak(x)≤¯Aak
+withA=C/(Na1−n/α
+d) and¯A=C(logN)C/N. Itisstraightforward tocheck
+that, for Nsufficiently large, we have ¯A≤A≤Λ¯J2.
+2.9.Truncation estimates. A specific feature of the problem we consider
+is that the limit dynamics is infinite dimensional, while the general fluid
+limit estimate applies in a finite-dimensional context. In t his subsection we
+establish some truncation estimates which will allow us to r educe to finitely
+many dimensions.
+Let (z(t):t≥0) be the solution in Dto ˙z(t)=v(z(t)) starting from 0, as
+in Theorem 2.3. Let (x(t):t≥0) be the solution to ˙ x(t)=¯b(x(t)) starting
+from 0.
+Lemma 2.6. We have
+d/summationdisplay
+k=1|zk(t)−xk(t)|≤tad+1.AVERAGING OVER FAST VARIABLES 25
+Proof. Since¯b=u(d), we have x(t)=z(d)(t) for allt, and so, from ( 36),
+we obtain
+d/summationdisplay
+k=1|zk(t)−xk(t)|≤d/summationdisplay
+k=1(zk(t)−z(d)
+k(t))≤λ/integraldisplayt
+0zd(s)nµ(z(t),d)ds
+≤tλan
+dµ(a,d)=tad+1. /square
+Denote by Ak(t) the number of arrivals to queues of length at least kby
+timet. Note that NZk+1
+t≤Ak(t) for allk≥1 and all t. Recall that
+˜ad+1=N−1an
+d+ρdad+1.
+Lemma 2.7. There is a constant C(λ,n)<∞such that, for all t≥0
+and allN, we have
+E(Ad(T∧t))≤CeCtN˜ad+1.
+Proof. Consider the function fonU×Igiven by f(x,y)=1{y≥d}and
+note that ¯f(x)=µ(x,d). Letχbe the corrector for fgiven by ( 11). Then,
+for allx∈Uand ally∈I,
+|χ(x,y)|≤2τ/ba∇dblf/ba∇dbl∞=2τ=CN−1
+and, whenever x=x(ξ) andx′=x(ξ′) withq(ξ,ξ′)>0, by the estimates
+(13) and (43),
+|χ(x,y)−χ(x′,y)|≤2ντ2/ba∇dblf/ba∇dbl∞/ba∇dblµx−µx′/ba∇dbl≤CN−2. (49)
+By optional stopping,
+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT∧t
+0Q(χ(x,y))(Xs)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle=|E(χ(XT∧t,YT∧t)−χ(X0,Y0))|≤CN−1.
+Now
+Q(χ(x,y))(ξ)=1{y≥d}−µ(x(ξ),d)−∆1(ξ)−∆2(ξ),
+where ∆ 1,∆2are given by ( 23), (24). We use ( 44) to obtain the estimate
+|∆1(ξ)|≤2τ(1+λn+Nλnzd+1)=C(zd+1+N−1)
+and from ( 49) deduce that
+|∆2(ξ)|≤N(1+λ)CN−2=CN−1.
+So
+E/integraldisplayT∧t
+01{Ys≥d}ds≤CN−1+E/integraldisplayT∧t
+0(µ(Xs,d)+CZd+1
+s+CN−1)ds.26 M. J. LUCZAK AND J. R. NORRIS
+Setg(t)=E(Ad(T∧t)), then
+g(t)=NλE/integraldisplayT∧t
+0(Xd
+s)n1{Ys≥d}ds≤Nλ2nan
+dE/integraldisplayT∧t
+01{Ys≥d}ds
+≤Can
+d/parenleftbigg
+1+/integraldisplayt
+0(Nρdµ(a,d)+1+g(s))ds/parenrightbigg
+≤CN˜ad+1(1+t)+C/integraldisplayt
+0g(s)ds.
+Here we have used the estimate µ(x,d)≤ρdµ(a,d) forx∈U. The claimed
+estimate now follows by Gronwall’s lemma. /square
+FixR∈(0,∞) and define
+˜T=inf{t≥0:Ad(t)≥RN˜ad+1}∧T.
+Lemma 2.8. There is a constant C(λ,n)<∞such that, for all t≥0
+and allN, we have
+E(Ad+1(˜T∧t))≤C(1+t)Rn˜an
+d+1+CtRn+1N˜an+1
+d+1.
+Proof. We argue as in theprecedingproof,except now taking f(x,y)=
+1{y≥d+1}, for which ¯f(x)=0. We obtain
+E/integraldisplay˜T∧t
+01{Ys≥d+1}ds≤CN−1+CE/integraldisplay˜T∧t
+0(Zd+1
+s+CN−1)ds
+and hence,
+E(Ad+1(˜T∧t))=NλE/integraldisplay˜T∧t
+0(Zd+1
+s)n1{Ys≥d+1}ds
+≤Nλ(R˜ad+1)nE/integraldisplay˜T∧t
+01{Ys≥d+1}ds
+≤C(1+t)Rn˜an
+d+1+CtRn+1N˜an+1
+d+1. /square
+2.10.Proof of Theorem 2.3.Recall that αis defined by ( 30) and that
+α∈(2n,2n+1). Note that α2−(2n+1)α+n=0. Recall that κ=(2α)−1
+and
+d=d(N)=sup{k∈N:Nak>Nκ}.
+The asymptotic growth rate ( 31) follows from Theorem 2.2. We shall use
+without further comment below the inequalities
+C−1a1/α
+k≤ak+1≤Caα
+k, k≥0,AVERAGING OVER FAST VARIABLES 27
+proved in Proposition 2.5and the inequalities
+ad+1≤N−(1−κ)≤ad, d≤loglogN,
+the last being valid for all sufficiently large N.
+By the truncation estimate, Lemma 2.6, we have
+sup
+t≤t0sup
+k≤d|zk(t)−xk(t)|√ak≤t0ad+1√ad≤Ct0a1−1/(2α)
+d+1≤Ct0N−1/2.
+Sinceφ(N)→ ∞asN→ ∞it will therefore suffice to show ( 32) with
+(z(t):t≥0) replaced by ( x(t):t≥0).
+We apply the general procedure of Section 1.3. Take as norm scales σk=√akso that
+/ba∇dblx/ba∇dbl=max
+k|xk|/√ak, x∈Rd.
+We now identify suitable regularity constants Λ ,B,τ,J,J 1(b),J(µ),K. We
+writeCfor a finite positive constant which may depend on λandnand
+whose value may vary from line to line. We shall see that, as N→ ∞,
+the inequalities between these regularity constants requi red in Theorem 1.6
+become valid. The maximum jump rate is bounded above by
+Λ=N(1+λ)=CN.
+We refer to the form of b(x,y) given at ( 41) and note that, for x∈Uand
+y∈I,
+/ba∇dblb(x,y)/ba∇dbl≤B=2na−1/2+n/α
+d=Ca−1/2+n/α
+d.
+We showed in Section 2.5the following upper bound on the mean coupling
+time of our coupling mechanism:
+m(x,y,¯y)≤τ=4
+Nλ(1−λ)2=CN−1.
+We refer to Section 1.3for the definitions of the jump bounds J,J1(b),J(µ)
+and leave the reader to check the validity of the following in equalities:
+J≤N−1a−1/2
+d, J 1(b)≤CN−1a−1/2+(n−1)/α
+d, J(µ)≤2nN−1.
+Recall from ( 46) the form of ¯b. In estimating the Lipschitz constant Kfor¯b
+onU, first note that, for x∈Uand forj=1,...,k−1,/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂
+∂xjxn
+k−1µ(x,k−1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cxn
+k−1µ(x,k−1)(x−1
+j+1).
+Here we have used the explicit form ( 26) ofµ(x,k−1) and the fact that
+(1−pj−1(x))−1≤ConU. Also note the inequalities
+x2n−1
+k−1/radicalbiggak−1
+ak≤22n−1a2n−1/2−α/2
+k−1≤C,∞/summationdisplay
+j=1√aj≤C.28 M. J. LUCZAK AND J. R. NORRIS
+We find, after some further straightforward estimation, tha t we can take
+K=C.
+Recall the choice of function φin the statement of Theorem 2.3. Set
+ε=/radicalbigg
+φ(N)
+N, δ=εe−Kt0/7, δ(β,b)=δ, δ(γ,g)=δ/(2τB).
+Recall that X0=(1/N,0,...,0) andx0=0 and that the driving rate νfor
+the coupling mechanism is equal to Nλ. It is now straightforward to check
+that all the inequalities required in the statement of Theor em1.5are valid,
+for all sufficiently large N.
+Now we check the tube condition of Theorem 1.5. The inequalities 0 ≤
+xd(ξ)≤··· ≤x1(ξ)≤1 hold for all ξ∈S. By a monotonicity property es-
+tablished in the proof of Theorem 2.1, we have xk(t)≤akfor allt≥0
+and fork= 1,...,d. Hence, for Nsufficiently large, if /ba∇dblx(ξ)−x(t)/ba∇dbl ≤2ε
+for some t≥0, thenxk(ξ)≤ak+2ε√ak≤2akandx1(ξ)≤a1+2ε√a1≤
+λ+(1−λ)/2≤(1+λ)/2, sox(ξ)∈Uand the tube condition is satisfied.
+Now we turn to the extra conditions needed to apply Theorem 1.6. We
+noted in Section 2.8the quadratic variation bounds
+ak(y)≤Aσ2
+k,¯ak(x)≤¯Aσ2
+k,
+valid for all x∈Uandy∈I, where
+A=C/(Na1−n/α
+d),¯A=C(logN)C/N
+and where ¯A≤A≤Λ¯J2for sufficiently large N. It is now straightforward to
+check, also for Nsufficiently large, that the remaining inequalities require d
+in the statement of Theorem 1.6hold. Theorem 1.6therefore applies to give
+P/parenleftBig
+sup
+t≤t0/ba∇dblXt−xt/ba∇dbl>ε/parenrightBig
+≤2de−δ2/(4¯At0)+2de−(¯A/A)2t0/(6400Λτ2)
+(50)
++P(Ω(β,b)c∪Ω(γ,g)c).
+Now, for Nsufficiently large, we have d≤loglogNand, by our choice of φ
+andκ,
+δ2/(4¯At0)≥φ(N)/((logN)Ct0)≥logN
+and
+(¯A/A)2t0/(6400Λτ2)≥logN.
+Hence, the first and second terms on the right-hand side of ( 50) tend to 0
+asN→∞.
+Recall from ( 16) and (17) the form of the events Ω( β,b) and Ω( γ,g).
+In the present example, the complementary exceptional even ts arise either
+as a result of truncation or because of finite Neffects in the fast variableAVERAGING OVER FAST VARIABLES 29
+dynamics, as shown by ( 42) and estimate ( 44). Recall that δ(β,b) =δand
+δ(γ,g)=δ/(2τB). Then
+Ω(β,b)c⊆/braceleftbigg/integraldisplayT∧t0
+0Zd+1
+t√addt≥δ(β,b)/bracerightbigg
+⊆/braceleftbigg
+Ad(T∧t0)≥Nδ√ad
+t0/bracerightbigg
+. (51)
+It is straightforward to check that, for all sufficiently larg eN,δ(γ,g)≥
+2t0(1+λn), which implies that
+Ω(γ,g)c⊆/braceleftbigg/integraldisplayT∧t0
+0(1+λn+NλnZd+1
+t)dt≥δ(γ,g)/bracerightbigg
+(52)
+⊆/braceleftbigg
+Ad(T∧t0)≥δ
+4λnt0τB/bracerightbigg
+.
+To see that P(Ω(β,b)c∪Ω(γ,g)c)→0 asN→ ∞, we use the bound on
+E(Ad(T∧t0)) proved in Lemma 2.7and Markov’s inequality. It then suffices
+to show that in the limit N→∞,
+C(an
+d+ρdNad+1)eCt0≪/radicalbig
+φ(N)Nad
+4nt0.
+For the term involving an
+dthis is easy. For the other term, involving Nad+1,
+we can check that, in fact,
+Nad+1≪/radicalbig
+Nad, ρd≤(logN)C≪/radicalbig
+φ(N).
+This completes the proof of ( 32). Limit ( 33) follows immediately from Lem-
+ma2.7using Markov’s inequality. Finally, note that, as N→∞,
+˜ad+1≤CN−1an
+d+(logN)Cad+1≤C(N−1+(logN)CN−1+κ)→0
+and
+N˜an+1
+d+1≤C(N−(1−κ)(n+1)/α+(logN)CN1−(1−κ)(n+1))→0.
+Then the limit ( 34) follows from ( 33) and Lemma 2.8using Markov’s in-
+equality.
+2.11.Monotonicity of the queueingmodel. Hereweproveanaturalmono-
+tonicity property of the supermarket model with memory whic h is a micro-
+scopic counterpart of the monotonicity of solutions to the d ifferential equa-
+tion (28) shown in Theorem 2.1. We do not rely on this result in the rest of
+the paper.
+First we construct, on a single probability space, for all ξ= (z,y)∈S,
+a version X=X(ξ) of the supermarket model with memory starting from
+ξ. Sety1=y1(ξ)=yand determine yi=yi(ξ)∈Z+fori=2,...,Nby the
+conditions
+y2≤···≤yN, z k=|{i∈{1,...,N}:yi≥k}|/N, k ∈N.30 M. J. LUCZAK AND J. R. NORRIS
+Weworkonthecanonical spaceofamarkedPoissonprocessofr ateN(1+λ),
+where the marks are either, with probability 1 /(1+λ), independent copies
+of a uniform random variable Jin{1,...,N}or, with probability λ/(1+λ),
+independentcopiesofauniformrandomsample( J1,...,Jn) from{1,...,N}.
+Fixξ=(z,y)∈Sanddefineaprocess X=X(ξ)=(Xt:t≥0)inSasfollows.
+SetXt=ξforallt<T,whereTisthefirstjumptimeofthePoissonprocess.
+If the first mark is a random variable, Jsay, take the sequence y1,...,yN
+and replace yJby (yJ−1)+to obtain a sequence u1,...,uNsay; set ˜y1=u1
+and write u2,...,uNin nondecreasing order to obtain ˜ y2≤···≤˜yN. If the
+first mark is a random sample, ( J1,...,Jn) say, select components ( yi:i∈
+{1,J1,...,Jn}) and write these in nondecreasing order, w1≤···≤wmsay;
+replacew1byw1+1andwritetheresultingsequence,again innondecreasing
+order,v1≤ ··· ≤vmsay; set ˜ y1=v1and write v2,...,vmcombined with
+the unselected components ( yi:i /∈{1,J1,...,Jn}) in nondecreasing order to
+obtain ˜y2≤···≤˜yN. SetXT=((Zk
+T:k∈N),YT), where
+Zk
+T=|{i∈{1,...,N}:˜yi≥k}|/N, k ∈N, Y T= ˜y1,
+and repeat the construction from XTin the usual way.
+Forξ,ξ′∈Swriteξ≤ξ′ifyi(ξ)≤yi(ξ′) fori=1,...,N.
+Theorem 2.9. Letξ,ξ′∈Swithξ≤ξ′. ThenXt(ξ)≤Xt(ξ′)for all
+t≥0.
+Proof. It will suffice to check that the desired inequality holds at th e
+first jump time T, that is to say, with obvious notation, that ˜ yi≤˜y′
+ifor all
+i. Note that if ai≤bifor allifor two sequences ( a1,...,an) and (b1,...,bn),
+then the same is true for their nondecreasing rearrangement s. In the case
+where the first mark is a random variable J, sinceyi≤y′
+ifor alli, we have
+ui≤u′
+ifor alliand so ˜yi≤˜y′
+ifor alli. On the other hand, when the first
+mark is a random sample ( J1,...,Jn), we have wj≤w′
+jfor allj, sovj≤v′
+j
+for allj, and so ˜ yi≤˜y′
+ifor alli./square
+REFERENCES
+[1]Ball, K. ,Kurtz, T. G. ,Popovic, L. andRempala, G. (2006). Asymptoticanalysis
+of multiscale approximations to reaction networks. Ann. Appl. Probab. 161925–
+1961.MR2288709
+[2]Darling, R. W. R. andNorris, J. R. (2008). Differential equation approximations
+for Markov chains. Probab. Surv. 537–79.MR2395153
+[3]Graham, C. (2000). Chaoticity on path space for a queueing network with selection
+of the shortest queue among several. J. Appl. Probab. 37198–211. MR1761670
+[4]Graham, C. (2005). Functional central limit theorems for a large netwo rk in which
+customers join the shortest of several queues. Probab. Theory Related Fields 131
+97–120.MR2105045AVERAGING OVER FAST VARIABLES 31
+[5]Luczak, M. J. andMcDiarmid, C. (2006). On the maximum queue length in the
+supermarket model. Ann. Probab. 34493–527. MR2223949
+[6]Luczak, M. J. andMcDiarmid, C. (2007). Asymptotic distributions and chaos for
+the supermarket model. Electron. J. Probab. 1275–99.MR2280259
+[7]Luczak, M. J. andNorris, J. (2005). Strong approximation for the supermarket
+model.Ann. Appl. Probab. 152038–2061. MR2152252
+[8]Mitzenmacher, M. ,Prabhakar, B. andShah, D. (2002). Load balancing with
+memory. In Proc. 43rd Ann. Symp. Found. Comp. Sci. 799–808. IEEE, Los
+Alamitos, CA.
+[9]Vvedenskaya, N. ,Dobrushin, R. L. andKarpelevich, F. I. (1996). A queueing
+system with a choice of the shorter of two queues – an asymptot ic approach.
+(Russian) Problemy Peredachi Informatsii 3220–34; translation in Probl. Inf.
+Transm. 3215–27.MR1384927
+School of Mathematical Sciences
+Queen Mary, University of London
+Mile End Road
+London E1 4NS
+United Kingdom
+E-mail: m.luczak@qmul.ac.ukStatistical Laboratory
+Centre for Mathematical Sciences
+University of Cambridge
+Wilberforce Road
+Cambridge, CB3 0WB
+United Kingdom
+E-mail: j.r.norris@statslab.cam.ac.uk
\ No newline at end of file
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+arXiv:1001.0896v3  [astro-ph.CO]  17 Feb 2010Astronomy& Astrophysics manuscriptno.13279 c∝circleco√yrtESO 2018
+October23,2018
+Spitzerdeepandwide Legacymid- andfar-infrared number counts
+andlower limits of cosmicinfrared background
+M. B´ ethermin1,H.Dole1, A.Beelen1,and H.Aussel2
+1Institutd’Astrophysique Spatiale (IAS),bat121, F-91405 Orsay, France; Universit´ e Paris-Sud11and CNRS(UMR8617)
+2LaboratoireAIM,CEA /DSM-CNRS-Universit´ eParisDiderot,IRFU /Serviced’Astrophysique,Bt.709,CEA-Saclay,91191Gif-s ur-
+Yvette Cedex, France
+Received 11September 2009 /Accepted 05 January 2009
+Abstract
+Aims.We aim to place stronger lower limits on the cosmic infrared b ackground (CIB) brightness at 24 µm, 70µm and 160µm and
+measure the extragalacticnumber counts at these wavelengt hs ina homogeneous way from various surveys.
+Methods. UsingSpitzerlegacy data over 53.6 deg2of various depths, we build catalogs with the same extractio n method at each
+wavelength. Completeness and photometric accuracy are est imated with Monte-Carlo simulations. Number count uncerta inties are
+estimated with a counts-in-cells moment method to take gala xy clustering into account. Furthermore, we use a stacking a nalysis to
+estimatenumber counts ofsources notdetectedat70 µmand160µm.Thismethodisvalidatedbysimulations. Theintegration ofthe
+number counts gives new CIB lower limits.
+Results. Numbercountsreach35 µJy,3.5mJyand40mJyat24 µm,70µm,and160µm,respectively.Wereachdeeperfluxdensities
+of0.38mJyat70,and3.1at160 µmwithastackinganalysis.Weconfirmthenumbercountturnov erat24µmand70µm,andobserve
+itforthe firsttimeat160 µmatabout 20mJy, together withapower-law behavior below 10 mJy.These mid-andfar-infraredcounts:
+1)arehomogeneously builtbycombiningfieldsofdi fferentdepths andsizes,providingalegacyoveraboutthreeo rdersofmagnitude
+in flux density; 2) are the deepest to date at 70 µm and 160µm; 3) agree with previously published results in the common m easured
+fluxdensity range; 4) globally agree withthe Lagache et al. ( 2004) model, except at 160 µm, where the model slightly overestimates
+the counts around 20and 200 mJy.
+Conclusions. These counts are integrated to estimate new CIB firm lower lim its of 2.29+0.09
+−0.09nW.m−2.sr−1, 5.4+0.4
+−0.4nW.m−2.sr−1, and
+8.9+1.1
+−1.1nW.m−2.sr−1at 24µm, 70µm, and 160µm, respectively, and extrapolated to give new estimates of t he CIB due to galaxies
+of 2.86+0.19
+−0.16nW.m−2.sr−1, 6.6+0.7
+−0.6nW.m−2.sr−1, and 14.6+7.1
+−2.9nW.m−2.sr−1, respectively. Products (point spread function, counts, C IB
+contributions, software) are publicly available for downl oad at http://www.ias.u-psud.fr/irgalaxies/.
+Key words. Cosmology: observations - Cosmology: di ffuse radiation- Galaxies: statistics- Galaxies: evolution - Galaxies: photom-
+etry-Infrared: galaxies
+1. Introduction
+The extragalactic background light (EBL) is the relic emiss ion
+of all processes of structure formation in the Universe. Abo ut
+half of this emission, called the Cosmic Infrared Backgroun d
+(CIB) is emitted in the 8-1000 µm range, and peaks around
+150µm. It is essentially due to the star formation (Pugetet al.
+1996; Fixsenetal. 1998; Hauseret al. 1998; Lagacheet al.
+1999; Gispertetal. 2000; Hauser& Dwek 2001; Kashlinsky
+2005; Lagacheet al.2005).
+The CIB spectral energy distribution (SED) is an im-
+portant constraint for the infrared galaxies evolution
+models (e.g. Lagacheet al. (2004); Franceschiniet al.
+(2009); Le Borgneet al. (2009); Pearson&Khan (2009);
+Rowan-Robinson (2009); Valianteetal. (2009)). It gives th e
+budget of infrared emission since the first star. The distrib ution
+of the flux of sources responsible for this background is also a
+critical constraint. We propose to measure the level of the C IB
+andthefluxdistributionofthesourcesat3wavelengths(24 µm,
+70µmand160µm).
+In the 1980’s, the infrared astronomical satellite (IRAS)
+and COBE/DIRBE performed the first mid-infrared (MIR) and
+far-infrared (FIR) full-sky surveys. Nevertheless, the de tectedsources were responsible for a very small part of the CIB.
+Between 1995 and 1998, the ISO (infrared space observatory)
+performed deeper observationsof infrared galaxies. Elbaz et al.
+(2002) resolved into the source more than half of the CIB
+at 15µm. At larger wavelengths, the sensitivity and angular
+resolution was not su fficient to resolve the CIB (Doleet al.
+2001).
+TheSpitzerspace telescope (Werneretal. 2004), launched
+in 2003, has performed deep infrared observations on wide
+fields. The multiband imaging photometers for Spitzer(MIPS)
+(Riekeet al.2004)mappedtheskyat24 µm,70µmand160µm.
+About 60% of the CIB was resolved at 24 µm (Papovichet al.
+2004) and at 70µm (Frayeret al. 2006). Because of confusion
+(Doleetal. 2003), only about 7% were resolved at 160 µm
+(Doleetal. 2004). Doleet al. (2006) managed to resolve most
+ofthe70µm and160µmbystacking24µm sources.
+The cold mission of Spitzer is over, and lots of data are
+now public. We present extragalactic number counts built
+homogeneously by combining deep and wide fields. The large
+sky surface used significantly reduces uncertainties on num ber
+counts. In order to obtain very deep FIR number counts, we
+used a stackinganalysis andestimate the level of the CIB in t he2 B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits
+Fieldname Surface area 80% completeness flux Scalingfactor
+24µm70µm160µm24µm70µm160µm24µm70µm160µm
+deg2µJy mJy
+FIDELeCDFS 0.23 0.19 - 60. 4.6 - 1.0157 1 -
+FIDELEGS 0.41 - 0.38 76. - 45. 1.0157 - 0.93
+COSMOS 2.73 2.41 2.58 96. 7.9 46. 1 0.92 0.96
+SWIRELH 10.04 11.88 11.10 282. 25.4 92. 1.0509 1.10 0.93
+SWIREEN1 9.98 9.98 9.30 261. 24.7 94. 1.0509 1.10 0.93
+SWIREEN2 5.36 5.34 4.98 267. 26.0 90. 1.0509 1.10 0.98
+SWIREES1 7.45 7.43 6.71 411. 36.4 130. 1.0509 1.10 0.98
+SWIRECDFS 8.42 8.28 7.87 281. 24.7 88. 1.0509 1.10 0.98
+SWIREXMM 8.93 - - 351. - - 1.0509 - -
+Total 53.55 45.51 42.91
+Table 1. Size, 80% completeness flux density and calibration scaling factor (see Sect. 2.1) of the used fields. Some fields are not
+usedat allwavelengths.
+threeMIPSbandswiththem.
+2. Data,sourceextractionandphotometry
+2.1. Data
+We took the public Spitzermosaics1from different observation
+programs: the GOODS /FIDEL (PI: M. Dickinson), COSMOS
+(PI: D. Sanders) and SWIRE (PI: C. Lonsdale). We used only
+thecentralpartofeachfield,whichwasdefinedbyacutof50%
+ofthemediancoverageforSWIREfieldsand80%fortheother.
+The total area covers 53.6 deg2, 45.5 deg2, 42.9 deg2at 24µm,
+70µm and 160µm respectively. The surface of the deep fields
+(FIDEL, COSMOS) is about 3.5 deg2. Some fields were not
+used at all wavelengthsfor di fferent reasons: There is no public
+release of FIDEL CDFS data at 160 µm; the pixels of the EGS
+70µm are not square; XMM is not observedat 70 and 160 µm.
+Table1summarisesthefield names,sizes andcompletenesses .
+In 2006, new calibration factors were adopted for MIPS
+(Engelbrachtet al. 2007; Gordonet al. 2007; Stansberryet a l.
+2007).TheconversionfactorfrominstrumentalunittoMJy /sris
+0.0454(resp.702and41.7)at24 µm(resp.70µmand160µm).
+The COSMOS GO3 and SWIRE (released 22 Dec. 2006) mo-
+saics were generated with the new calibration. The FIDEL mo-
+saics were obtained with other factors at 24 µm and 160µm
+(resp. 0.0447and 44.7).The 70 µm and 160µm COSMOS mo-
+saics were color corrected (see Sect. 2.3). Consequently we ap-
+plied a scaling factor (see Table 1) before the source extrac tion
+toeachmosaictoworkonahomogeneoussampleofmaps(new
+calibrationandnocolorcorrection).
+2.2. Source extraction andphotometry
+The goal is to build homogeneous number counts with well-
+controlled systematics and high statistics. However, the fi elds
+present various sizes and depths. We thus employed a single
+extractionmethodat agivenwavelength,allowingtheheter oge-
+neousdatasetstocombineina coherentway.
+1from the Spitzer Science Center website:
+http://data.spitzer.caltech.edu /popular/2.2.1. Mid-IR/far-IRdifferences
+The MIR (24µm) and FIR (70µm and 160µm) maps have
+different properties: in the MIR, we observe lots of faint
+blendedsources;intheFIR,duetoconfusion(Doleet al.200 4),
+all these faint blended sources are only seen as background
+fluctuations. Consequently, we used di fferent extraction and
+photometry methods for each wavelength. In the MIR, the
+priority is the deblending: accordingly we took the SExtrac tor
+(Bertin&Arnouts 1996) and PSF fitting. In the FIR, we used
+efficient methods with strong background fluctuations: wavelet
+filtering,thresholddetectionandaperturephotometry.
+2.2.2. Pointspread function(PSF)
+The 24µm empirical PSF of each field is generated with the
+IRAF (image reduction and analysis facility2) DAOPHOT
+package (Stetson 1987) on the 30 brightest sources of each
+map. It is normalized in a 12 arcsec radius aperture. Apertur e
+correction(1.19)iscomputedwiththeSTinyTim3(Krist2006)
+theoretical PSF for a constant νSνspectrum. The difference of
+correction between a Sν=ν−2and aν2spectrum is less than
+2%. So, the hypothesis on the input spectrum is not critical f or
+thePSF normalization.
+At 70µm and 160µm, we built a single empirical PSF
+from the SWIRE fields. We used the Starfinder PSF extrac-
+tion routine (Diolaitiet al. 2000), which median-stacks th e
+brightest non-saturated sources (100 mJy <S70<10 Jy and
+300 mJy<S160<1 Jy). Previously,fainter neighboringsources
+were subtracted with a first estimation of the PSF. At 70 µm
+(resp. 160µm), the normalization is done in a 35 arcsec (resp.
+80 arcsec) aperture, with a sky annulus between 75 arcsec
+and 125 arcsec (resp. 150 arcsec and 250 arcsec); the apertur e
+correction was 1.21 (resp. 1.20). The theoretical signal in the
+sky annulus and the aperture correction were computed with
+the S Tiny Tim Spitzer PSF for a constant νSνspectrum. These
+parameters do not vary more than 5 % with the spectrum of
+sources. Pixels that were a ffected by the temporal median
+filtering artifact, which was sometimes present around brig ht
+sources,were maskedpriorto theseoperations.
+2http://iraf.noao.edu/
+3http://ssc.spitzer.caltech.edu /archanaly/contributed/stinytim/B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits 3
+2.2.3. Source extractionand photometry
+At 24µm, we detected sources with SExtractor. We chose a
+Gaussian filter (gauss 5.09x9.conv) and a background filter of
+thesize of64×64pixels.Thedetectionandanalysisthresholds
+weretunedforeachfield.We performedPSF fitting photometry
+withtheDAOPHOTallstarroutine.Thisroutineisverye fficient
+forblendedsourcesfluxmeasurement.
+At 70µm and 160µm, we applied the a-trou wavelet
+filtering (Starcketal. 1999) on the maps to remove the large
+scale fluctuations(10pixels)onwhichwe performedthe sour ce
+detection with a threshold algorithm (Doleet al. 2001, 2004 ).
+The threshold was tuned for each field. Photometry was done
+by aperture photometry on a non filtered map at the positions
+foundonthe wavelet filtered map.At 70 µm, we used 10arcsec
+aperture radius and a 18 arcsec to 39 arcsec sky annulus. At
+160µm, we used an aperture of 20 arcsec and a 40 arcsec to 75
+arcsec annulus. Aperture corrections were computed with th e
+normalized empirical PSF: 3.22 at 70 µm and 3.60 at 160 µm.
+In order to estimate the uncertainty on this correction, ape rture
+correctionswere computedusingfive PSF built on fivedi fferent
+SWIRE fields. The uncertainty is 1.5% at 70 µm and 4.5% at
+160µm.
+2.3. Colorcorrection
+The MIPS calibration factors were calculated for a 10000 K
+blackbody (MIPS Data Handbook 20074). However, the galax-
+iesSED(spectralenergydistribution)aredi fferentandtheMIPS
+photometricbandsarelarge( λ/∆λ≈3).Thus,colorcorrections
+wereneeded.We used(likeShupeet al.(2008)andFrayeret al .
+(2009)) a constant νSνspectrum at 24µm, 70µm and 160µm.
+Consequently,allfluxesweredividedby0.961,0.918and0.9 59
+at 24µm, 70µm and 160µm due to this color correction.
+Another possible convention is νSν∝ν−1. This convention is
+more relevant for the local sources at 160 µm, whose spectrum
+decreasesquicklywith wavelength.Nevertheless,the reds hifted
+sources studied by stacking are seen at their peak of the cold
+dustemission,andtheirSEDagreesbetterwiththeconstant νSν
+convention. The di fference of color correction between these
+two conventions is less than 2 %, and this choice is thus not
+critical. We consequently chose the constant νSνconvention to
+more easily compare our results with Shupeet al. (2008) and
+Frayeret al. (2009).
+3. Catalogproperties
+3.1. Spurioussources
+Our statistical analysis may su ffer from spurious sources. We
+havetoestimate howmanyfalsedetectionsarepresentinama p
+and what their flux distribution is. To do so, we built a catalo g
+with the flipped map. To build this flipped map, we multiplied
+the values of the pixels of the original map by a factor of -1.
+Detection and photometry parameters were exactly the same a s
+for normal catalogs. At 24 µm, there are few spurious sources
+(<10%) in bins brighter than the 80% completeness limit flux
+density. At 70µm and 160µm, fluctuations of the background
+due to unresolved faint sources are responsible for spuriou s
+detections. Nevertheless, the ratio between detected sour ce
+4http://ssc.spitzer.caltech.edu /mips/dh/[h]
+Figure1. Flux distribution of sources extracted from normal
+(solid line) and flipped (dash line) maps, at 70µm in FIDEL
+eCDFS. The vertical dashed line represents the 80% complete -
+nessfluxdensity.
+numbersandspurioussourcenumbersstayedreasonable(bel ow
+0.2) down to the 80% completeness limit (see the example of
+FIDELCDFS at 70 µm inFig.1).
+3.2. Completeness
+The completeness is the probability to extract a source of a
+given flux. To estimate it, we added artificial sources (based on
+empirical PSF) on the initial map and looked for a detection i n
+a 2 arcsec radius at 24 µm around the initial position (8 arcsec
+at 70µm and16arcsec at160 µm).Thisoperationwasdonefor
+different fluxes with a Monte-Carlo simulation. We chose the
+numberofartificialsourcesineachrealizationinawaythat they
+have less than 1% probabilityto fall at a distance shorter th an 2
+PSF FWHM (fullwidthat halfmaximum).Thecompletenessis
+plotted in Fig. 2, and the 80% completeness level is reported in
+Table1.
+3.3. Photometricaccuracy
+The photometric accuracy was checked with the same Monte-
+Carlo simulation.For di fferentinputfluxes,we built histograms
+of measured fluxes and computed the median and scatter of
+these distributions. At lower fluxes, fluxes are overestimat ed
+and errors are larger. These informations were used to estim ate
+the Eddington bias (see next section). The photometric accu -
+racyat70µminFIDELCDFSisplottedasanexampleinFig.3.
+We also compared our catalogs with published catalogs.
+At 24µm, we compared it with the GOODS CDFS catalog of
+Charyetal.(2004),andtheCOSMOScatalogofLeFloc’het al.
+(2009). Theirfluxeswere multipliedby a correctivefactor t o be
+compatible with the νSν=constant convention. Sources were
+considered to be the same if they are separated by less than 2
+arcsec. We computed the standard deviation of the distribut ion
+of the ratio between our and their catalogs. In a 80-120 µJybin4 B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits
+Figure2. Completenessat24 µm(left),70µm(center),and160µm(right)asafunctionofthesourcefluxforallfields.Thedashed
+linerepresents80%completeness.
+[h]
+Figure3. Ratiobetweenmeasuredfluxandinputfluxcomputed
+fromMonteCarlosimulationsat70 µminFIDELeCDFS.Error
+barsrepresent1σdispersion.Theverticaldashedlinerepresents
+the80%completenessfluxdensity.
+in the CDFS, we found a dispersion of 19%. In a 150-250 µJy
+bin in COSMOS, we founda scatter of 13%. The o ffset is+3%
+with COSMOS catalog and -1% with GOODS catalog. At 70
+and 160µm, we comparedour catalogs with the COSMOS and
+SWIRE teamones.Inallcases, thescatter islessthan15%,an d
+the offset is less than 3%. At all wavelengths and for all fields,
+theoffsetis lessthanthecalibrationuncertainty.3.4. Eddingtonbias
+Figure4. Eddingtonbias: ratio between the numberof detected
+sources and the number of input sources at 70 µm in FIDEL
+eCDFS. The vertical dashed line represents the 80% complete -
+nessfluxdensity.
+When sources become fainter, photometric errors increase.
+In addition, fainter sources are more numerous than brighte r
+ones (in general dN/dS∼S−r). Consequently, the number of
+sources in faint bins are overestimated. This is the classic al
+Eddingtonbias(Eddington1913,1940).TheexampleofFIDEL
+CDFS at 70µm isplottedinFig. 4.B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits 5
+Tocorrectforthise ffectat70µmand160µm,weestimated
+a correction factor for each flux bin. We generated an input
+flux catalog with a power-law distribution (r =1.6 at 70µm,
+r=3 at 160µm). We took into account completeness and
+photometric errors (coming from Monte-Carlo simulations) to
+generate a mock catalog. We then computed the ratio between
+the number of mock sources found in a bin and the number of
+input sources. This task was done for all fields. This correct ion
+is more important for large r (at 160 µm). At 24µm, thanks
+to the PSF fitting, the photometric error is more reduced and
+symmetrical. Less faint sources are thus placed in brighter flux
+bins. Because of this property and the low r (about 1.45), thi s
+correctioncanbeignoredfor24 µm counts.
+4. Numbercounts
+4.1. Removingstars fromthe catalogs
+To computeextragalacticnumbercountsat 24 µm, we removed
+the stars with the K−[24]<2 colorcriterionand identification
+procedure following Shupeetal. (2008). The K band mag-
+nitudes were taken from the 2MASS catalog (Skrutskieet al.
+2006). We ignored the star contribution at 70 µm and 160µm,
+whichisnegligible( <1%inallusedfluxdensitybins)according
+totheDIRBE Faint Countmodel(Arendtetal. 1998).
+4.2. 24µm numbercounts
+We counted the number of extragalactic sources for each field
+and in each flux bin. We subtracted the number of spurious
+detections (performed on the flipped map). We divided by the
+completeness. As a next step, the counts of all fields were
+combinedtogetherwithameanweightedbyfieldsize.Actuall y,
+a weighting by the number of sources in each field overweighs
+the denser fields and biases the counts. Counts from a field
+were combined only if the lower end of the flux bin was larger
+then orequal to the 80% completeness.We thus reached71 µJy
+(71µJy to 90µJy bin) in the counts. However, to probe fainter
+flux densities, we used the data from the deepest field (FIDEL
+eCDFS) between a 50 and 80 % completeness, allowing us to
+reach35µJy.
+Our number counts are plotted in Fig. 5 and are writ-
+ten in Table 2. We also plot data from Papovichetal. (2004),
+Shupeet al. (2008) and LeFloc’het al. (2009), and model pre-
+dictionsfromLagacheetal.(2004)andLeBorgneet al.(2009 ).
+ThePapovichet al.(2004)fluxesaremultipliedbyafactor1. 052
+to take into account the update in the calibration, the color cor-
+rection and the PSF. This correction of flux also implies a cor -
+rectiononnumbercounts,accordingto:
+/parenleftBigdN
+dSfS2.5
+f/parenrightBig
+Sf=/parenleftBig
+c1.5dN
+dSiS2.5
+i/parenrightBig
+cSi, (1)
+whereSiis the initial flux, Sfis the corrected flux and c the
+corrective factor ( Sf=cSi). A correction of the flux thus
+corresponds to a shift in the abscissa (factor c) and in the
+ordinate (factor c1.5). Papovichet al. (2004) do not subtract
+stars and thus overestimate counts above 10 mJy. We have a
+very good agreement with their work below 10 mJy. We also
+have a very good agreement with Shupeet al. (2008). The
+LeFloc’het al. (2009) fluxes are multiplied by 1.05 to take in toaccount a difference of the reference SED: 10 000K versus
+constantνSν, and byanothercorrectionof 3% correspondingto
+the offset observedin Sect. 3.3. Thereis an excellentagreement
+withtheirwork.
+The Lagacheet al. (2004)5and LeBorgneetal. (2009)6
+generally agree well with the data, in particular on the fain t
+end below 100µJy, and on the position of the peak around
+300µJy. However, the Lagacheetal. (2004) model slightly
+underestimates (about 10%) the counts above 200 µJy. The
+LeBorgneet al.(2009)modelisflatterthanthedata,andagre es
+reasonablywell above600 µJy.
+4.3. 70µm numbercounts
+Counts in the flux density bins brighter than the 80% com-
+pleteness limit were obtained in the same way as at 24 µm
+(Fig. 6 and Table 3). In addition, they were corrected from th e
+Eddington bias (c.f. Sect. 3.4). We reached about 4.9 mJy at
+80% completeness (4.9 to 6.8 bin). We used CDFS below 80%
+completeness limit to probe fainter flux density level. We cu t
+these counts at 3.5 mJy. At this flux density, the spurious rat e
+reached 50%. We used a stacking analysis to probe fainter flux
+densitylevels(c.f.section5).
+We can see breaksin the countsaround10 mJy and 20 mJy.
+These breaks appear between points built with a di fferent set
+of fields. Our counts agree with earlier works of Doleet al.
+(2004), Frayeret al. (2006) and Frayeret al. (2009). Howeve r,
+these works suppose only a Poissonian uncertainty, which
+underestimatestheerrorbars(seeSect.4.5).Ourdataalso agree
+well with these works. The Lagacheetal. (2004) model agrees
+well with our data. The Le Borgneet al. (2009) model gives a
+reasonable fit, despite an excess of about 30% between 3 mJy
+and10mJy.
+4.4. 160µm numbercounts
+The 160µm number counts were obtained exactly in the same
+way as at 70µm. We used COSMOS and EGS to probe counts
+below the 80% completeness limit. We reached 51 mJy at
+80% completeness (51 mJy to 66 mJy bin) and 40 mJy for
+the 50% spurious rate cut (figure 7 and table 4). We used a
+stackinganalysistoprobefainterfluxdensitylevels(c.f. Sect.5).
+OurcountsagreewiththeearlierworksofDoleet al.(2004)
+and Frayeretal. (2009) . We find like Frayeret al. (2009) that
+the Lagacheet al. (2004) model overestimates the counts by
+about 30% above 50 mJy (see the discussion in Sect. 7.2). On
+the contrary, the LeBorgneet al. (2009) model underpredict s
+thecountsbyabout20%between50mJyand150mJy.
+4.5. Uncertaintieson numbercountsincludingclustering
+Shupeet al. (2008) showed that the SWIRE field-to-field vari-
+anceissignificantlyhigherthanthePoissonnoise(byafact orof
+5Lagache et al. (2004) model used a ΛCDM cosmology with
+ΩΛ=0.73,ΩM=0.27andh=0.71
+6LeBorgne et al. (2009) model used a ΛCDM cosmology with
+ΩΛ=0.7,ΩM=0.3and h=0.76 B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits
+<S>Smin SmaxdN/dS.S2.5σpoissonσclusteringσclus.+calib.Ωused
+(inmJy) (ingal.Jy1.5.sr−1) deg2
+0.040 0.035 0.044 17.5 1.0 1.1 1.3 0.2
+0.050 0.044 0.056 21.4 1.0 1.1 1.4 0.2
+0.064 0.056 0.071 28.2 1.2 1.5 1.8 0.2
+0.081 0.071 0.090 36.2 1.5 1.9 2.4 0.2
+0.102 0.090 0.114 52.6 1.3 1.9 2.9 0.6
+0.130 0.114 0.145 64.1 1.0 1.7 3.1 3.4
+0.164 0.145 0.184 78.7 1.1 2.2 3.8 3.4
+0.208 0.184 0.233 89.8 1.3 2.8 4.5 3.4
+0.264 0.233 0.295 96.5 1.5 3.3 5.1 3.4
+0.335 0.295 0.374 112.0 0.8 1.8 4.8 37.2
+0.424 0.374 0.474 103.7 0.6 1.7 4.5 46.1
+0.538 0.474 0.601 91.9 0.6 1.5 4.0 53.6
+0.681 0.601 0.762 81.2 0.6 1.5 3.6 53.6
+0.863 0.762 0.965 72.8 0.7 1.6 3.3 53.6
+1.094 0.965 1.223 65.3 0.8 1.6 3.1 53.6
+1.387 1.223 1.550 60.8 0.9 1.7 3.0 53.6
+1.758 1.550 1.965 56.7 1.0 1.8 2.9 53.6
+2.228 1.965 2.490 55.4 1.2 2.1 3.0 53.6
+2.823 2.490 3.156 54.0 1.5 2.3 3.2 53.6
+3.578 3.156 4.000 55.9 1.8 2.7 3.5 53.6
+5.807 4.000 7.615 54.8 1.5 2.9 3.6 53.6
+11.055 7.615 14.496 46.9 2.3 3.6 4.1 53.6
+21.045 14.496 27.595 36.4 3.3 4.4 4.6 53.6
+40.063 27.595 52.531 43.4 5.9 7.7 7.9 53.6
+76.265 52.531 100.000 47.7 9.9 12.0 12.2 53.6
+Table 2. Differentialnumbercountsat 24 µm.σclusteringisthe uncertaintytakinginto accountclustering(see Sect . 4.5).σclus.+calib.
+takesintoaccountbothclusteringandcalibration(Engelb rachtet al.2007).
+<S>Smin SmaxdN/dS.S2.5σpoissonσclusteringσclus.+calib.Ωused
+(inmJy) (ingal.Jy1.5.sr−1) deg2
+4.197 3.500 4.894 2073. 264. 309. 342. 0.2
+5.868 4.894 6.843 2015. 249. 298. 330. 0.2
+8.206 6.843 9.569 1690. 289. 332. 353. 0.2
+11.474 9.569 13.380 2105. 123. 202. 250. 2.6
+16.044 13.380 18.708 2351. 148. 228. 281. 2.6
+22.434 18.708 26.159 1706. 153. 208. 240. 2.6
+31.369 26.159 36.578 2557. 69. 124. 218. 38.1
+43.862 36.578 51.146 2446. 73. 123. 211. 45.5
+61.331 51.146 71.517 2359. 90. 141. 217. 45.5
+85.758 71.517 100.000 2257. 112. 164. 228. 45.5
+157.720 100.000 215.440 2354. 121. 198. 257. 45.5
+339.800 215.440 464.160 2048. 200. 276. 311. 45.5
+732.080 464.160 1000.000 2349. 381. 500. 526. 45.5
+Table 3. Differentialnumbercountsat 70 µm.σclusteringisthe uncertaintytakinginto accountclustering(see Sect . 4.5).σclus.+calib.
+takesintoaccountbothclusteringandcalibration(Gordon et al. 2007).
+<S> Smin SmaxdN/dS.S2.5σpoissonσclusteringσclus.+calib.Ωused
+(inmJy) (ingal.Jy1.5.sr−1) deg2
+45.747 40.000 51.493 16855. 1312. 2879. 3519. 3.0
+58.891 51.493 66.289 14926. 1243. 2704. 3243. 3.0
+75.813 66.289 85.336 13498. 1319. 2648. 3104. 3.0
+97.596 85.336 109.860 12000. 1407. 2442. 2835. 3.0
+125.640 109.860 141.420 10687. 457. 991. 1621. 36.2
+161.740 141.420 182.060 7769. 425. 773. 1211. 42.9
+208.210 182.060 234.370 7197. 472. 810. 1184. 42.9
+268.040 234.370 301.710 5406. 487. 734. 979. 42.9
+345.050 301.710 388.400 5397. 585. 843. 1063. 42.9
+444.200 388.400 500.000 4759. 662. 891. 1059. 42.9
+750.000 500.000 1000.000 6258. 685. 1158. 1380. 42.9
+1500.000 1000.000 2000.000 4632. 989. 1379. 1487. 42.9
+Table4. Differentialnumbercountsat160 µm.σclusteringistheuncertaintytakingintoaccountclustering(seeSect .4.5).σclus.+calib.
+takesintoaccountbothclusteringandcalibration(Stansb erryet al.2007).B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits 7
+[h]
+Figure5. Differential number counts at 24 µm.Filled circle : points obtained with ≥80% completeness; filled diamond : points ob-
+tained with a 50% to 80% completeness; open triangle : Papovichet al. (2004) GTO numbercountsobtainedwith PSF fi tting pho-
+tometry;opensquare :Shupeet al.(2008)SWIREnumbercountsobtainedwithapert urephotometry; opendiamond :LeFloc’het al.
+(2009) COSMOS number counts obtained with PSF fitting photom etry;continuousline : Lagacheet al. (2004) model; dashed line
+and grey region : LeBorgneet al. (2009) model and 90% confidence region. Erro r bars take into accounts clustering (see section
+4.5)andcalibrationuncertainties(Engelbrachtet al.200 7).
+three in some flux bins). They estimated their uncertainties on
+number counts with a field bootstrap method. We used a more
+formalmethodtodealwith thisproblem.
+The uncertainties on the number counts are Poissonian
+only if sources are distributed uniformly. But, actually, t he
+infrared galaxies are clustered. The uncertainties must th us be
+computed taking into account clustering. We first measured
+the source clustering as a function of the flux density with
+the counts-in-cells moments (c-in-c) method (Peebles 1980 ;
+Szapudi 1998; Blake &Wall 2002). We then computed the
+uncertaintiesknowingtheseclusteringpropertiesofthes ources,
+the source density in the flux density bins and the field shapes .ThedetailsareexplainedintheappendixA.
+Thisstatistical uncertaintycanbecombinedwiththe Spitzer
+calibration uncertainty (Engelbrachtetal. 2007; Gordone t al.
+2007; Stansberryet al. 2007) to compute the total uncertain ty
+ondifferentialnumbercounts.8 B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits
+Figure6. Differential number counts at 70 µm.Filled circle : points obtained with ≥80% completeness; filled diamond : points
+obtainedwithlessthan50%spurioussourcesandlessthan80 %completeness; filledsquare :stackingnumbercounts(clear:FIDEL
+eCDFS,dark:COSMOS); opensquare :Doleetal.(2004)numbercountsinCDFS,BootesandMarano; opentriangle :Frayeret al.
+(2006) in GOODS and Frayeretal. (2009) in COSMOS; cross: Frayeret al. (2006) deduced from background fluctuations; con-
+tinuousline : Lagacheet al. (2004) model; dashedline andgrey region : LeBorgneet al. (2009) model and 90% confidenceregion.
+Errorbarstakeintoaccountclustering(see Sect.4.5)andc alibrationuncertainties(Gordonet al.2007).
+5. DeeperFIR numbercountsusing astacking
+analysis
+5.1. Method
+ThenumbercountsderivedinSect.4showthatdowntothe80%
+completeness limit, the source surface density is 24100 deg−2,
+1200 deg−2, and 220 deg−2at 24, 70, and 160 µm, respectively,
+i.e. 20 times (resp. 110 times) higher at 24 µm than at 70µm
+(resp. 160µm). These differences can be explained by the
+angular resolution decreasing with increasing wavelength , thus
+increasing confusion, and the noise properties of the detec tors.
+There are thus many 24 µm sources without detected FIR
+counterparts. If we want to probe deeper into the FIR number
+counts, we can take advantage of the information provided bythe 24µm data, namely the existence of infrared galaxies not
+necessarilydetectedinthe FIR,andtheirpositions.
+We used a stacking analysis (Doleet al. 2006) to determine
+the FIR/MIR color as a function of the MIR flux. With this
+information, we can convert MIR counts into FIR counts. The
+stacking technique consists in piling up very faint far-inf rared
+galaxies which are not detected individually,but are detec ted at
+24µm. For this purpose, it makes use of the 24 µm data prior
+to tracking their undetected counterpart at 70 µm and 160µm,
+where most of the bolometric luminosity arises. This method
+wasusedbyDoleet al.(2006),whomanagedtoresolvetheFIR
+CIB using 24µm sources positions, as well many other authors
+(e.gSerjeantetal.(2004);Dyeet al.(2006);Wanget al.(20 06);B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits 9
+Figure7. Differential number counts at 160 µm.Filled circle : points obtained with ≥80% completeness; filled diamond : points
+obtained with less than 50% spurious sources and less than 80 % completeness; filled square : stacking number counts (clear:
+FIDEL/GTOCDFS, middle:COSMOS, dark:SWIRE EN1); opensquare :Dole etal.(2004)numbercountsinCDFS andMarano;
+open triangle : Frayeret al. (2009) in COSMOS; continuous line : Lagacheet al. (2004) model; dashed line and grey region :
+LeBorgneet al. (2009) model and 90% confidence region. Error bars take into account clustering (see Sect. 4.5) and calibr ation
+uncertainties(Stansberryet al. 2007).
+Devlinetal. (2009); Dyeet al. (2009); Marsdenet al. (2009) ;
+Pascaleet al. (2009)).
+To derive the 70µm or 160µm versus 24µm color, we
+stackedtheFIRmaps(cleanedofbrightsources)attheposit ions
+ofthe24µmsourcessortedbyflux,andperformedaperturepho-
+tometry(sameparametersasinSect. 2).We thusget
+SFIR=f(S24), (2)
+whereSFIRis the average flux density in the FIR of the
+populationselected at 24 µm,S24the averageflux density at 24
+microns,andf is the functionlinkingbothquantities.We de rive
+f empirically using the S FIRversus S 24relation obtained fromstacking.
+Wecheckedthat fisasmoothmonotonicfunction,inagree-
+ment with the expectation that the color varies smoothly wit h
+the redshift and the galaxy emission properties. Assuming t hat
+the individualsourcesfollowthisrelationexactly,the FI Rnum-
+bercountscouldbededucedfrom
+dN
+dSFIR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleSFIR=f(S24)=dN
+dS24/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleS24/slashBigdSFIR
+dS24/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleS24. (3)10 B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits
+In practice, the two first terms are discrete. In addition, th e last
+termiscomputednumericallyinthesame S24bin,usingthetwo
+neighboringfluxdensitybins(k-1andk +1).We finallyget
+dN
+dSFIR(SFIRk)=/parenleftBiggdN
+dS24/parenrightBigg
+k/slashBigSFIR,k+1−SFIR,k−1
+S24,k+1−S24,k−1, (4)
+where<SFIR>is measured by stacking. In reality, sources do
+not follow Eq. 2 exactly, but exhibit a scatter around this me an
+relation
+SFIR=f(S24)+σ. (5)
+Our method is still valid under the conditionσ
+f≪1, and we
+verifyitsvaliditywith simulations(see nextsection).
+To obtain a better signal to noise ratio, we cleaned the
+resolved bright sources from the FIR maps prior to stacking.
+We used 8 S24bins per decade. We stacked a source only if
+the coverage was more than half of the median coverage of the
+map. Uncertainties on the FIR mean flux were estimated with
+a bootstrap method. Furthermore, knowing the uncertaintie s on
+the 24µm numbercounts and the mean S24fluxes, we deduced
+theuncertaintiesontheFIRnumbercountsaccordingtoEq.4 .
+At 70µm, we used the FIDEL eCDFS (cleaned at
+S70>10 mJy) and the COSMOS (cleaned at S70>50 mJy)
+fields. At 160µm, we used 160µm the GTO CDFS (cleaned at
+S160>60mJy),the COSMOS (cleanedat S160>100mJy)and
+the SWIRE EN1 (no clean to probethe S160>20 mJy sources)
+fields.
+5.2. Validationonsimulations
+We used the Fernandez-Condeetal. (2008) simulations7to
+validate our method. These simulations are based on the
+Lagacheet al. (2004) model, and include galaxy clustering. We
+employed 20 simulated mock catalogs of a 2.9 deg2field each,
+containing about 870 000 mock sources each. The simulated
+maps have the same pixel size as the actual Spitzer mosaics
+(1.2”, 4”, and 8” at 24 µm, 70µm, and 160µm, resp.) and
+are convolved with our empirical PSF. A constant standard
+deviation Gaussian noise was added. We applied the same
+method as for the real data to produce stacking number counts .
+At 160µm, for bins below 15 mJy, we cleaned the sources
+brighter than 50 mJy. Figure 8 shows 160 µm number counts
+from the mock catalogs down to S160=1 mJy (diamond) and
+thenumbercountsdeducedfromthestackinganalysisdescri bed
+in the previous section (triangle). The error bars on the figu re
+arethestandarddeviationsofthe20realization.Thereisa good
+agreement between the stacking counts and the classical cou nts
+(better than 15%). Nevertheless,we observeda systematic b ias,
+intrinsic to the method, of about 10% in some flux density
+bins. We thus combined this 10% error with the statistical
+uncertainties to compute our error bars. We also validated t he
+estimation of the statistical uncertainty in the stacking c ounts:
+we checkthatthe dispersionofthe countsobtainedbystacki ng,
+coming from different realizations, was compatible with our
+estimation of statistical uncertainties. The results are t he same
+at 70µm.
+7Publiclyavailable at http: //www.ias.u-psud.fr/irgalaxies/Figure8. Simulated number countsat 160 µm, computedfrom
+stackingcounts (triangle) andfromtheinputmockcatalog (solid
+line).Thegoodagreement(betterthan15%)validatesthestack-
+ingcountsmethod(seeSect.5.2).Twentyrealizationsofth e2.9
+deg2field maps with about 870 000 mock sources each were
+used.
+5.3. Results
+At 70µm, the stacking number counts reach 0.38 mJy (see
+Fig. 6 and Table 5). The last stacking point is compatible wit h
+the Frayeret al. (2006) P(D) constraint. The stacking point s
+also agree very well with the Lagacheetal. (2004) model.
+The LeBorgneet al. (2009) model predicts slightly too many
+sources in 0.3 to 3 mJy range. The turnover around 3 mJy
+and the power law behavior of the faint counts ( S70<2mJy),
+observed by Frayeret al. (2006) are confirmed with a better
+accuracy.
+At 160µm, the stacking counts reach 3.1 mJy (see Fig. 7
+and Table 6). We observed for the first time a turnover at about
+20 mJy, and a power-lawdecreaseat smaller flux densities. Th e
+stacking counts are lower than the Lagacheetal. (2004) mode l
+around20mJy (about30%).Below 15mJy,the stackingcounts
+agree with this model.The LeBorgneet al. (2009) model agree
+quite well with our points below 20 mJy. The results at 160 µm
+will bediscussedinSect. 7.2.
+6. NewlowerlimitsandestimatesoftheCIB at
+24µm,70µm and160µm
+6.1. 24µm CIB:lowerlimitandestimate
+By integrating the measured 24 µm number counts between
+35µJy and 0.1 Jy, we can estimate a lower value of the CIB
+at this wavelength. The counts were integrated with a trapez e
+method. We estimated the uncertaintyon the integral by addi ng
+(on 10000 realisations) a random Gaussian error to each data
+point with theσgiven by the count uncertainties taking into
+account clustering. We then added the 4% calibration error o f
+the instrument (Engelbrachtet al. 2007). We found 2 .26+0.09
+−0.09
+nW.m−2.sr−1. The very bright source counts ( S24>0.1 Jy) are
+supposed to be euclidian ( dN/dS=CeuclS2.5). We used the
+three brightest points to estimate Ceucl. We founda contributionB´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits 11
+<S> dN/dS.S2.5σclus.σclus.+calib. Field
+(inmJy) (ingal.Jy1.5.sr−1)
+0.38±0.05 246. 72. 76. FIDELCDFS
+0.64±0.07 517.109. 122. FIDELCDFS
+0.94±0.03 646. 80. 105. COSMOS
+1.48±0.04 1218. 151. 198. COSMOS
+2.14±0.05 1456. 183. 239. COSMOS
+3.27±0.07 1657. 211. 273. COSMOS
+Table 5. Stacking extragalactic number counts at 70 µm.σclus.
+is the uncertainty taking into account clustering (see Sect . 4.5).
+σclus.+calib.takes into account both clustering and calibration
+(Gordonet al.2007).
+<S> dN/dS.S2.5σclus.σclus.+calib. Field
+(inmJy) (ingal.Jy1.5.sr−1)
+3.11±0.46 6795.2163. 2485. GTOCDFS
+4.71±0.16 9458.1236. 2104. COSMOS
+6.74±0.22 13203. 1627. 2880. COSMOS
+9.65±0.26 18057. 2307. 3986. COSMOS
+12.95±0.37 19075. 2388. 4182. COSMOS
+19.82±0.48 22366. 2944. 4987. SWIREEN1
+25.71±0.81 20798. 2811. 4682. SWIREEN1
+33.74±0.98 16567. 2671. 4004. SWIREEN1
+45.18±2.08 20089. 4849. 6049. SWIREEN1
+Table6. Stackingextragalacticnumbercountsat160 µm.σclus.
+is the uncertainty taking into account clustering (see Sect . 4.5).
+σclus.+calib.takes into account both clustering and calibration
+(Stansberryet al. 2007).
+Figure9. Cumulative contribution to the surface brightness of
+the 24µm CIB as a function of S24µm. The colored area rep-
+resents the 68% and 95% confidence level. The shaded area
+represents the S24<35µJy power-law extrapolation zone (see
+Sect. 6.1). The 4% calibration uncertainty is not represent ed.
+The table corresponding to this figure is available online at
+http://www.ias.u-psud.fr/irgalaxies/.
+to the CIB of 0.032+0.003
+−0.003nW.m−2.sr−1. Consequently, very
+brightsourcesextrapolationisnotcriticalfortheCIBest imation
+(1% of CIB). The contributionof S24>35µJy is thus 2.29+0.09
+−0.09
+nW.m−2.sr−1(c.f.Table7).24µm70µm160µm
+Scut,resolved mJy 0.035 3.50 40.0
+Scut,stacking -0.38 3.1
+νBν,resolved nW.m−2.sr−12.29+0.09
+−0.093.1+0.2
+−0.21.0+0.1
+−0.1
+νBν,resolved+stacking -5.4+0.4
+−0.48.9+1.1
+−1.1
+νBν,tot 2.86+0.19
+−0.166.6+0.7
+−0.614.6+7.1
+−2.9
+Table 7. SummaryofCIB resultsfoundinthisarticle.
+We might have wanted to estimate the CIB value at 24
+µm. To do so, we needed to extrapolate the number counts
+on the faint end. Below 100 µJy, the number counts exhibit a
+power-law behavior (Fig. 5). We assumed that this behavior ( of
+the form dN/dS=CfaintSr) still holds below 35 µJy. r and
+Cfaintare determined using the four faintest bins. We found
+r=1.45±0.10 (compatible with 1 .5±0.1 of Papovichet al.
+(2004)). Our new estimate of the CIB at 24 µm due to infrared
+galaxiesisthus2.86+0.19
+−0.16nW.m−2.sr−1.Theresultsareplottedin
+Fig. 9. We concludethat resolvedsources downto S24=35µJy
+accountfor80%ofthe CIBatthiswavelength.
+6.2. 70µm and160µm CIB:lowerlimitandestimate
+At 70µm and 160µm, the integration of the number counts
+was done in the same way as at 24 µm, except for the stacking
+counts, which are correlated. To compute the uncertainties
+on the integral, we added (on 10000 realizations) a Gaussian
+error simultaneously to the three quantities and completel y
+recomputedthe associated stacking counts: 1- the mean dens ity
+flux given by the stacking; 2- the 24 µm number counts;
+3- the mean 24µm flux density. At 70 µm, and 160µm,
+the calibration uncertainty is 7 % (Gordonet al. 2007) and
+12% (Stansberryet al. 2007), respectively. We estimated
+the CIB surface brightness contribution of resolved source s
+(S70>3.5 mJy and S160>40 mJy) of 3.1+0.2
+−0.2nW.m−2.sr−1and
+1.0+0.1
+−0.1nW.m−2.sr−1. The contribution of S70>0.38 mJy and
+S160>3.1 mJy is 5.4+0.4
+−0.4nW.m−2.sr−1and 8.9+1.1
+−1.1nW.m−2.sr−1,
+respectively.
+Below 2 mJy at 70 µm, and 10 mJy at 160 µm, the stacking
+counts are compatible with a power-law. Like at 24 µm, we
+assumed that this behavior can be extrapolated and determin ed
+the lawwith the fivefaintest binsat 70 µm, andthefourfaintest
+at 160µm. We found a slope r=1.50±0.14 at 70µm, and
+1.61±0.21at160µm.Theslopeofthenumbercountsat70 µm
+is compatible with the Frayeretal. (2006) value (1 .63±0.34).
+The slope at 160µm is measured for the first time. Our new
+estimate of the CIB at 70 µm and 160µm due to infrared
+galaxies is thus 6.6+0.7
+−0.6nW.m−2.sr−1, and 14.6+7.1
+−2.9nW.m−2.sr−1,
+respectively. We conclude that resolved and stacking-stud ied
+populationsaccount for 82% and 62% of the CIB at 70 µm and
+160µm, respectively. These results are summarized in Table 7,
+andFig.10and11.
+7. Discussion
+7.1. New lowerlimitsof theCIB
+The estimations of CIB based on number counts ignore a
+potential diffuse infrared emission like dust in galaxy clusters
+(Montier&Giard 2005). The extrapolation of the faint sourc e12 B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits
+Figure10. Cumulative contribution to the surface brightness
+of the 70µm CIB as a function of S70µm. The colored area
+represents the 68% and 95% confidence level. The shaded ar-
+eas represent the 0 .38<S70<3.3 mJy stacking counts zone
+and theS70<0.38 mJy power-law extrapolation zone (see
+Sect. 6.2). The 7% calibration uncertainty is not represent ed.
+The table corresponding to this figure is available online at
+http://www.ias.u-psud.fr/irgalaxies/.
+Figure11. Cumulative contribution to the surface brightness
+of the 160µm CIB as a function of S160µm. The colored area
+represents the 68% and 95% confidence level. The shaded ar-
+eas represent the 3 .1<S160<45 mJy stacking counts zone
+and theS160<3.1 mJy power-law extrapolation zone (see
+Sect. 6.2). The 12% calibration uncertainty is not represen ted.
+The table corresponding to this figure is available online at
+http://www.ias.u-psud.fr/irgalaxies/.
+counts supposes no low luminosity population, like populat ion
+III stars or faint unseen galaxies. Accordingly, this type o f
+measurementcanprovidein principleonlya lowerlimit.
+At 24µm, Papovichet al. (2004) found 2 .7−0.7
+1.1nW.m−2.sr−1
+usingthecountsandtheextrapolationofthefaintsourceco unts.Weagreewiththisworkandsignificantlyreducedtheuncerta in-
+tiesonthisestimation.Dole etal.(2006)foundacontribut ionof
+1.93±0.23nW.m−2.sr−1forS24>60µJysources(afterdividing
+theirresultsby1.12tocorrectanapertureerrorintheirph otom-
+etryat24µm).Ouranalysisgives2 .10±0.08nW.m−2.sr−1fora
+cut at 60µJy, which agrees very well. Rodighieroet al. (2006)
+gave a total value of 2 .6 nW.m−2.sr−1, without any error bar.
+Charyetal. (2004) found 2 .0±0.2 nW.m−2.sr−1, by integrating
+sources between 20 and 1000 µJy (we find 2.02±0.10 for the
+sameinterval).
+At 70µm, using the number counts in the ultra deep
+GOODS-N and a P(D) analysis, Frayeretal. (2006) found
+aS70>0.3 mJy source contribution to the 70 µm CIB of
+5.5±1.1 nW.m−2.sr−1. Using the stacking counts, we found
+5.5±0.4 nW.m−2.sr−1for the same cut, in excellent agreement
+and with improved uncertainties. In Doleet al. (2006), the c on-
+tribution at 70µm of the S24<60µJy sources was computed
+withanextrapolationofthe24 µmnumbercountsandthe70 /24
+color.Theyfound7 .1±1.0nW.m−2.sr−1, but the uncertaintyon
+the extrapolation took only into account the uncertainty on the
+70/24 color and not the uncertainty on the extrapolated 24 µm
+contribution, and was thus slightly underestimated. This i s in
+agreeswithourestimation.
+At 160µm they found with the same method,
+17.4±2.1 nW.m−2.sr−1(a corrective factor of 1.3 was ap-
+plied due to an error on the map pixel size). This estimation i s
+a little bit higherthan ourestimation,and can be explained by a
+smallcontribution(oftheorderof15%)ofthesourcecluste ring
+(Bavouzet2008).
+Our results can also be compared with direct measurements
+made by absolute photometers. These methods are biased
+by the foreground modeling, but do not ignore the extended
+emission. Fixsenetal. (1998) found a CIB brightness of
+13.7±3.0at 160µm, in excellentagreementwith ourestimation
+(14.6+7.1
+−2.9nW.m−2). From the discussion in Doleet al. (2006)
+(Sect. 4.1), the Lagacheetal. (2000) DIRBE WHAM (FIRAS
+calibration)estimationat140 µmand240µmof12nW.m−2.sr−1
+and 12.2 nW.m−2.sr−1can be also compared with our value at
+160µm. A more recent work of Odegardet al. (2007) found
+25.0±6.9 and 13.6±2.5 nW.m−2.sr−1at 140µm and 240µm re-
+spectively(resp.15 ±5.9nW.m−2.sr−1and12.7±1.6nW.m−2.sr−1
+with the FIRAS scale). Using ISOPHOT data, Juvelaet al.
+(2009)giveanestimationoftheCIBsurfacebrightnessbetw een
+150µmand180µm of20.25±6.0±5.6nW.m−2.sr−1.
+The total brightnessdue to infrared galaxies at 160 µm cor-
+respondsto the total CIB level at this wavelength.We thusha ve
+probably resolved the CIB at this wavelength. Nevertheless ,
+the uncertainties are relatively large, and other minor CIB
+contributorscannotbe excluded.
+In addition, upper limits can be deduced indirectly from
+blazar high energy spectrum. Stecker&deJager (1997) gave
+an upper limit of 4 nW.m−2.sr−1at 20µm using Mkn 421.
+Renaultet al. (2001) found an upper limit of 4.7 nW.m−2.sr−1
+between5 and15µm with Mkn501.Thisisconsistentwith our
+lowerlimitat 24µm.
+An update of the synthetic EBL SED of Doleet al. (2006)
+with the new BLAST (balloon-borne large-aperture submil-
+limeter telescope) lower limits from Devlinetal. (2009) an dB´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits 13
+our values is plotted in Fig. 12. The BLAST lower limits are
+obtained by stacking of the Spitzer 24 µm sources at 250µm,
+350µm and500µm (Devlinet al.2009; Marsdenet al.2009).
+7.2. 160µm numbercounts
+Atmost,weobserveda30%overestimationoftheLagacheet al .
+(2004)modelcomparedtothe160 µmnumbercounts(Sect.4.4
+and 5.3 and Fig. 7), despite goodfits at other wavelengths. Th is
+model uses mean SEDs of galaxies sorted into two populations
+(starburst and cold), whose luminosity functions evolve se pa-
+rately with the redshift. A possible explanation of the mode l
+excess is a slightly too high density of local cold galaxies. By
+decreasingthedensityofthislocalpopulationalittle,th emodel
+might be able to better fit the 160 µm number counts without
+significantly affecting other wavelengths, especially at 70 µm
+(more sensitive to warm dust rather than cold dust), and in th e
+submillimetre range (more sensitive to redshifted cold dus t at
+faintfluxdensitiesforwavelengthslargerthan500 µm).
+The LeBorgneetal. (2009) model slightly overpredicts
+faint 160µm sources, probably because of the presence of too
+many galaxies at high redshift; this trend is also seen at 70
+µm. With our number counts as new constraints, their inversio n
+shouldgivemoreaccurateparameters.
+Herschel was successfully launched on May 14th, 2009
+(togetherwith Planck).Itwillobserveinfraredgalaxiesbetween
+70µmand500µmwithanimprovedsensitivity.Itwillbepossi-
+ble to directly observethe cold dust spectrumof high-z ULIR G
+(ultraluminousinfraredgalaxy)andmedium-zLIRG(lumino us
+infrared galaxy). PACS (Photodetectors Array Camera and
+Spectrometer) will make photometric surveys in three bands
+centred on 70µm, 100µm and 160µm.Herschel will allow
+us to resolve a significant fraction of the background at thes e
+wavelengths (Lagacheetal. 2003; Le Borgneet al. 2009).
+SPIRE(spectralandphotometricimagingreceiver)willobs erve
+around 250µm, 350µm and 500µm, and will be quickly
+confusionlimited.Inbothcases,thestackinganalysiswil lallow
+us to probe fainter flux density levels, as it is complementar yto
+SpitzerandBLAST.
+8. Conclusion
+With a large sample of public Spitzerextragalactic maps, we
+built new deep, homogeneous, high-statistics number count s in
+threeMIPSbandsat24 µm, 70µm and160µm.
+At 24µm, the results agree with previous works. These
+counts are derived from the widest surface ever used at this
+wavelength(53.6deg2).Usingthesecounts,wegiveanaccurate
+estimation of the galaxy contribution to the CIB at this wave -
+length(2.86+0.19
+−0.16nW.m−2.sr−1).
+At 70µm, we used the stacking method to determine the
+counts below the detection limit of individual sources, by
+reaching 0.38 mJy, allowing us to probe the faint flux density
+slope of differential number counts. With this information, we
+deduced the total contribution of galaxies to the CIB at this
+wavelength(6.6+0.7
+−0.6nW.m−2.sr−1).At 160µm, our counts reached 3 mJy with a stacking
+analysis. We exhibited for the first time the maximum in
+differential number counts around 20 mJy and the power-law
+behavior below 10 mJy. We deduced the total contribution of
+galaxies to the CIB at this wavelength (14 .6+7.1
+−2.9nW.m−2.sr−1).
+Herschel will likely probe flux densities down to about 10 mJy
+at thiswavelength(confusionlimit,Le Borgneet al. (2009) ).
+The uncertainties on the number counts used in this work
+take carefully into account the galaxy clustering, which is
+measuredwiththe”counts-in-cells”method.
+We presented a method to build very deep number counts
+with the information provided by shorter wavelength data
+(MIPS 24µm) and a stacking analysis. This tool could be used
+onHerschel SPIRE data with a PACS priorto probefainterflux
+densitiesin thesubmillimetrerange.
+We publicly release on the website
+http://www.ias.u-psud.fr/irgal/, the following products: PSF,
+number counts and CIB contributions. We also release a stack -
+inglibrarysoftwarewritteninIDL.
+Acknowledgements. We wish to acknowledge G. Lagache, who has generated
+simulations used in this work. We acknowledge J.L. Puget, G. Lagache, D.
+Marcillac, B. Bertincourt, A. Penin and all members of the co smology group
+of IAS for their comments and suggestions. We wish to thank th e members of
+ANR D-SIGALE for their valuable comments, in particular D. L e Borgne for
+providing us an electronic version of his model. We also than k E. Le Floc’h
+for quickly providing us the table of his counts. This work is based in part on
+archival data obtained with the Spitzer Space Telescope, wh ich is operated by
+the Jet Propulsion Laboratory, California Institute of Tec hnology under a con-
+tract with NASA. Support for this work was provided by an awar d issued by
+JPL/Caltech. This publication makes use of data products from th e Two Micron
+All Sky Survey, which is a joint project of the University of M assachusetts and
+the Infrared Processing and Analysis Center /California Institute of Technology,
+funded by the National Aeronautics and Space Administratio n and the National
+Science Foundation.
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+AppendixA: Uncertaintieson numbercounts
+includingclustering
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+Weconsideraclusteredpopulationwithasurfacedensity ρ.The
+expected number of objects in a field of the size ΩisN=ρΩ.
+In the Poissonian case, the standard deviation around this v alue
+is√
+N. Fora clustereddistribution,thestandarddeviation σNis
+givenby(Wall &Jenkins2003)
+σN=/radicalBig
+y.¯N2+¯N. (A.1)B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits 15
+FigureA.1. Amplitude of the auto correlation as a function of
+thefluxdensityofthesourcesat24 µm, andbest power-lawfit.
+Theexpectedvalueof yisgivenby(Peebles1980)
+y=/integraltext
+field/integraltext
+fieldw(θ)dΩ1dΩ2
+Ω2, (A.2)
+wherew(θ)istheangulartwopointsautocorrelationfunctionof
+thesources.
+A.2. Measuringsource clusteringasafunctionoffluxdensit y
+We assume the classical power law description w(θ)=
+A(S,λ)θ1−γwith an indexγ=1.8. So,ydepends only on A
+andonthe shapeofthefield:
+y=A(S,λ)/integraltext
+field/integraltext
+fieldθ1−γdΩ1dΩ2
+Ω2. (A.3)
+Theuncertaintyon yis givenby(Szapudi1998):
+σy=/radicalBigg
+2
+Ncell¯N2. (A.4)
+To measure A(S,λ), we cut our fields in 30′×30′square
+boxes,inwhichwecountthenumberofsourcesandcomputethe
+variance in five, three and three flux density bins at 24 µm, 70
+µm and 160µm. We calculate the associate A(S,λ) combining
+Eq.A.1andA.3
+A(S,λ)=σ2
+N−¯N
+¯N2×Ω2
+/integraltext
+field/integraltext
+fieldθ1−γdΩ1dΩ2. (A.5)
+The fit of A(S24,24µm) versusS24(see Fig. A.1) gives ( χ2=
+2.67forfivepointsandtwofitted parameters)
+A(S,24µm)=(2.86±0.29).10−3/parenleftBiggS
+1mJy/parenrightBigg0.90±0.15
+. (A.6)
+The measured exponent in A.6 of 0 .90±0.15 corresponds to
+γ/2, which is the expected value in the case of a flux-limited
+survey in an Euclidean universe filled with single luminos-
+ity sources. We fix this exponent to fit A(S70,70µm) and
+A(S160,160µm). We find A(1mJy,70µm)=(0.25±0.08).10−3
+andA(1mJy,160µm)=(0.3±0.03).10−3.FigureA.2. Relativeerroronthenumbercountasafunctionof
+the field size. We have chosen A=0.019 andρ=38.5 deg−2
+(values for a 80-120mJy flux density bin at 160 µm). The field
+isa square.
+A.3. Computeuncertaintiesdue toclustering
+With this model of A(S,λ) and the field shape, we compute y
+(Eq. A.3). Assuming ¯N=N(N to be the number of detected
+sources in a given field and flux density bin), we deduce σ(N)
+from Eq. A.1, and consequently the error bar on the number
+countsfora singlefield.
+Tocomputethefinaluncertaintyonthecombinedcounts,we
+usethe followingrelation
+σcomb,dN
+dS=/radicalBig/summationtext
+iΩ2
+iσ2
+i,dN
+dS/summationtext
+iΩi, (A.7)
+whereσcomb,dN
+dSis the uncertainty on the combined number
+counts,Ωithe solid angle of the i-th field, and σi,dN
+dSthe uncer-
+tainty on the number counts in the i-th field (given by Var(N),
+Eq.A.1).
+A.4. Discussion aboutclustering andnumbercount
+uncertainties
+For a clustered distribution of sources, the uncertainties on the
+number counts are driven by two quadratically combined term s
+(Eq. A.1): a Poissonian term√
+Nand a clustering term√y.N
+(see Fig. A.2). We have N∝Ωandy∝Ω(γ−1)/2(Blake& Wall
+2002). When the uncertainty is dominated by the Poissonian
+term (small field), the relative uncertainty is thus proport ional
+to√
+Ω−1/2. Whentheuncertaintyisdominatedbytheclustering
+term (large field), the relative error is proportional to Ω(1−γ)/4
+(Ω−0.2forγ=1.8).
+Consequently uncertainties decrease very slowly in the the
+clustering regime. Averaging many small independent fields
+gives more accurate counts than a big field covering the same
+surface. For example, in the clustering regime, if a field of
+10deg2hasarelativeuncertaintyof0.2,therelativeuncertainty
+is 0.2/√
+10=0.063 for the mean of ten fields of this size, and
+0.2×10−0.2=0.126forasinglefieldof100deg2.Consequently,
+if one studies the counts only, many small fields give better16 B´ etherminetal.: Spitzerdeepwide legacyMIR /FIR counts andCIBlower limits
+results than one very large field. But, this is not optimal if o ne
+studies the spatial properties of the galaxies, which requi res
+largefields.
\ No newline at end of file
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@@ -0,0 +1,560 @@
+arXiv:1001.0898v1  [astro-ph.HE]  6 Jan 2010Astronomy& Astrophysics manuscriptno.arxiv c/circlecopyrtESO 2018
+November7,2018
+Constraints on the kinematicsof the44Ti ejecta
+ofCassiopeia Afrom INTEGRAL/SPI
+Pierrick Martin1,2,J¨ urgen Kn¨ odlseder1,2, Jacco Vink3,Anne Decourchelle4,and Matthieu Renaud5
+1Centre d’Etude Spatiale des Rayonnements (CESR),9, avenue colonel Roche, BP44346, 31028 Toulouse cedex 4, France
+2Universit´ e de Toulouse (UPS),Centre National de laRecher che Scientifique (CNRS),UMR 5187
+3Astronomical Institute,UniversityUtrecht, POBox80000, 3508 TA Utrecht,The Netherlands
+4Laboratoire AIM,CEA /DSM-CNRS-Universit´ eParisDiderot,DAPNIA /Sap, CEA-Saclay,91191 Gif-sur-Yvette,France
+5Laboratoire APC,CNRS-UMR7164-Universit´ e ParisDiderot ,10, rue A. Domonet L.Duquet, 75205 ParisCedex13, France
+Received 7March2008 /Accepted3April 2009
+ABSTRACT
+Context. The medium-lived44Ti isotope is synthesised by explosive Si-burning in core-c ollapse supernovae. It is extremely sensi-
+tive to the dynamics of the explosion and therefore can be use d to indirectly probe the explosion scenario. The young supe rnova
+remnant Cassiopeia A is to date the only source of gamma-ray l ines from44Ti decay. The emission flux has been measured by
+CGRO/COMPTEL,BeppoSAX /PDS andINTEGRAL /IBIS.
+Aims.The high-resolution spectrometer SPI on-board the INTEGRA L satellite can provide spectrometric information about th e
+emission.The lineprofilesreflectthekinematics ofthe44TiinCassiopeia Aandcanthusplace constraints onitsnucle osynthesis and
+potentiallyon the associatedexplosion process.
+Methods. Using 4 years of data from INTEGRAL /SPI, we have searched for the gamma-ray signatures from the d ecay of the44Ti
+isotope. The overwhelming instrumental background noise r equired an accurate modelling and a solid assessment of the s ystematic
+errors inthe analysis.
+Results.Duetothestrongvariabilityoftheinstrumentalbackgroun dnoise,ithasnotbeenpossibletoextractthetwolinesat67 .9and
+78.4keV. Regarding the high-energy line at 1157.0keV, no si gnificant signal is seen in the 1140-1170keV band, thereby su ggesting
+thatthelinesignalfromCassiopeiaAisbroadened bytheDop pler effect.Fromourspectrum,wederivea ∼500kms−1lowerlimitat
+2σonthe expansion velocityof the44Tiejecta.
+Conclusions. Our result does not allow us to constrain the location of44Ti since the velocities involved throughout the remnant,
+derived from optical andX-raystudies, are allfar above our lower limit.
+Key words. Gamma rays: observations – ISM: supernova remnants – ISM: in dividual (Cassiopeia A) – Nuclear reactions, nucle-
+osynthesis, abundances
+1. Introduction
+Until very recently, Cassiopeia A (hereafter Cas A) occupie d a
+specific place among galactic supernova remnants as being th e
+result of the most recent galactic supernova known to date (t he
+youngest one might now be G1.9 +0.3, with an estimated age
+of about a century, see Reynoldset al. 2008). Based on an opti -
+cal study of outer high-velocityknots, Thorstensenet al. ( 2001)
+derived an explosion date of 1671 or slightly later for Cas A.
+Combined with the 3.4kpc distance estimated by Reedet al.
+(1995),theexplosionthateventuallyleadtoCasAmightwel lbe
+the transient 6th magnitude star noticed by Flamsteed in 168 0.
+From the light-echo scattered by interstellar dust, the eve nt has
+now been identified as an SNIIb, which indicates that the pro-
+genitor was a red supergiant that had lost most of its hydroge n
+envelopeat themomentofcollapse(Krauseet al.2008).
+CasAretainsinterestingfeaturesthatcouldbevaluablecl ues
+inthelong-standinge fforttounderstandthesupernovaexplosion
+mechanism.Indeed,growingevidencehasarisenthattheexp lo-
+sionand/ortheexpansionofCasAwereasymmetricandhighly
+turbulent. X-ray images taken with Chandra have revealed a j et
+(andto a lesser extenta counter-jet)protrudingin thenort h-east
+direction far beyond the nearly spherical forward shock (Vi nk
+2004;Hwangetal.2004).Anotherindicationforanasymmetr icprocess is the substantial mixing suggested by the Chandra X -
+rayimagesinwhichFe-richmaterialappears,inatleast two ex-
+tendedregions,atagreaterprojectedradiusthanSi-richm aterial
+(Hugheset al.2000; Hwanget al.2000).Ona smallerscale,th e
+widevarietyintheso-calledopticalFast-MovingKnots(FM Ks)
+spectraregardingtherelativestrengthsofOandSlines,to gether
+with the absence of a correlationbetween these spectral pro per-
+ties and the velocity of the FMKs, also imply some turbulence
+orunevenburning(Chevalier&Kirshner1979).
+Inadditiontothesepeculiarities,CasAisknownastheonly
+source of gamma-rayline emission from44Ti decay. The unsta-
+ble44Ti isotope is a by-product of explosive Si-burning result-
+ing from alpha-rich freeze-out (Woosleyet al. 1973; Theet a l.
+1998).The44Ti isexpectedto liedeepintheejecta anditsyield
+isverysensitivetotheearlydynamicsoftheexplosionandt othe
+positionofthemasscut(the boundarythat separatestheeje cted
+material from the material that falls back onto the compact o b-
+ject). As such,44Ti provides indirect evidence about the flow
+pattern that mediated the explosionenergyand can therefor ebe
+usedtoprobetheexplosionscenario.
+After a mean lifetime of 85 yrs (Ahmadet al. 2006),44Ti
+decays by electron capture to44Sc and then to44Ca, thereby
+emitting three de-excitation photons at 67.9, 78.4 and 1157 .0
+keV. Given its youth and relative proximity, Cas A constitut es2 PierrickMartinet al.:Constraints onthe kinematics of th e44Ti ejecta of Cassiopeia A from INTEGRAL /SPI
+an ideal target to search for the characteristic decay emiss ion
+of the medium-lived44Ti. The first detection of the44Ti emis-
+sion from Cas A was reported by Iyudinet al. (1994) from
+CGRO/COMPTEL observations at 1157.0keV, with a down-
+ward revision of the flux by Duprazet al. (1997). Subsequent
+detectionsofthelow-energylinesbyBeppoSax /PDS(Vinket al.
+2001) and recently by INTEGRAL /IBIS/ISGRI (Renaudet al.
+2006) have allowed investigators to tighten the flux value fo r
+the44Ti emission from Cas A to (2.5 ±0.3)×10−5phcm−2s−1
+(Renaudet al. 2006). With the age and distance mentioned
+above for Cas A and assuming that the44Ti is not strongly
+ionized (which could delay its decay by electron capture, se e
+Mochizukiet al. 1999), this flux translates into an initial44Ti
+massofabout1.6×10−4M⊙.
+While the total44Ti flux, and hence the total44Ti mass, is
+ratherwell constrainedtoday,little isknownaboutthe kin emat-
+icsofthe44Tiejectaasnoneoftheabove-mentionedinstruments
+had the capabilities required for fine spectrometry. The hig h-
+resolution spectrometer SPI on-board the INTEGRAL satelli te
+opened the way for a determinationof line profiles and thus fo r
+an estimate of the44Ti velocity. In particular, the high-energy
+line at 1157.0keV is expected to reflect the motion of the44Ti
+ejecta. Such a measurement is essential to achieve a complet e
+understandingofthenucleosynthesisof44TiinCasAandisrel-
+evanttofundamentalquestionssuchashowcore-collapsesu per-
+novaeexplodeandwhyCasAistheonlysourceof44Tiemission
+inthesky(Theet al.2006).
+InthispaperwepresenttheINTEGRAL /SPIobservationsof
+the44Tidecaylineat1157.0keVandtheirimplicationsinterms
+of velocity of the44Ti ejecta. From this constraint, the site of
+thenucleosynthesisof44Tiisthendiscussedinthecontextofall
+otherobservationaldataavailableonCasA.
+2. Dataanalysis
+2.1. Instrument characteristics
+The SPI instrument onboard the INTEGRAL gamma-ray space
+observatory is a high-resolution spectrometer with imagin g ca-
+pabilities (Vedrenneet al. 2003; Roqueset al. 2003). The sp ec-
+trometryofgammaradiationwith energiesbetween20keVand
+8 MeV is performed by an array of 19 high-purity germanium
+detectors,with a spectral resolutionof about2 keV FWHM at 1
+MeV. All gamma-ray sources, di ffuse or point-like, emitting in
+thisenergyrangecanalsobeimagedindirectlythankstoaco ded
+masksystemwithanangularresolutionof2.8◦andafully-coded
+field of view of 16◦x16◦. For a given pointing, a coded-mask
+achieves a spatial modulation of the celestial signal and th e re-
+construction of the sky intensity distribution exploits th e corre-
+lations between the recorded image and the mask pattern or a
+decoding array derived from it (see Fenimore&Cannon 1978).
+The instrument response of SPI, however, is quite complex an d
+inversionbycorrelationmethodsisnot possible.Moreinfo rma-
+tionshouldbeaccumulatedandtheobservationofanastroph ys-
+ical object or region with SPI therefore consists of a series of
+pointingsaround the target, following a so-called ditheri ng pat-
+tern(thusaddingatemporalmodulationoftheastrophysica lsig-
+nal).Theskysignalisthenreconstructedthroughamodel-fi tting
+approach.
+ThephotoninteractionsrecordedbySPIaredividedintotwo
+classes: single events, be they from astrophysical or instr umen-
+tal origin,depositingall of their energyin a single detect or,and
+multiple events, in which case the interactions involve sev eral
+adjacent detectors (which therefore form a so-called pseud o-detector). In the work presented here, only single events (h ere-
+after SE) and double events (hereafter ME2) were considered .
+Detector 2 and 17 failed during orbit 142 and 210 respectivel y.
+Apart from a reduction in e ffective area, this led to an increase
+of the background in the SE recorded by the detectors adjacen t
+tothe failedones.
+2.2. Data set
+An observation group (hereafter OG) has been built from all
+pointingswithin20◦aroundCas Aachievedbetweenrevolution
+7andrevolution539,plusallpointingsoutsidethegalacti cplane
+(|b|≥25◦) duringthe same period. The formerare mainly dedi-
+catedobservationsoftheCasAremnantcomplementedbysome
+pointings from the periodic Galactic Plane Scan, while the l at-
+ter are empty fields (at the energy considered) that will help to
+constrainthebackgroundlevel.AllSEandME2datawithener -
+giesbetween1130and1200keVwereconsidered(ME2account
+for about a quarter of all counts at these energies). This ini tial
+selection of pointings has been screened for abnormally hig h
+counting rates such as those occurring as the satellite cros ses
+the radiation belts or experiences solar flares. The eventua l OG
+is a three-dimensional space with the following characteri stics:
+8328 pointings (time axis), 19 single detectors and 42 doubl e
+detectors (space axis) and 70 one-keV bins ranging from 1130
+to1200keV(energyaxis).Thisamountsto6.65Msofe ffective
+observation time towards Cas A and 12.5 Ms of e ffective ob-
+servation time of empty regions, spread over a 4.2 year inter val
+betweenDecember2002andMarch2007.
+The corresponding raw spectrum and count rate are pre-
+sented in Fig. 1. Most of these reflect the instrumental back-
+ground noise, the origin and handling of which are detailed i n
+thefollowingsection.
+2.3. Instrumentalbackground noise
+The constant bombardmentof the satellite by solar and cosmi c-
+ray particles generates a very strong background noise thro ugh
+the constant activation or excitation of the on-board mater ial.
+This background noise manifests in the form of fluorescence
+and nuclear lines on top of a continuum (Jeanet al. 2003). It
+overwhelms the astrophysical signal by a factor of 100 and ex -
+hibitstime,spectralandevenspatialvariationswithinth edetec-
+tor plane. The strong time variability of the background noi se
+is illustrated in the middle plot of Fig. 1, where the increas ing
+trend is due to the evolution of the solar activity that modifi es
+the cosmic-ray flux at the satellite position, while the disc onti-
+nuities are the result of the selected pointings of the OG bei ng
+non-contiguous in time. In addition, some of the instrument al
+backgroundlines correspondexactly to the transitions sea rched
+forbythenucleargamma-rayastronomers.Inparticular,ab ack-
+ground line exists at 1157.0 keV (see upper plot of Fig. 1),
+and corresponds to the production and subsequent nuclear de -
+excitation of44Ca in the instrument. Care should therefore be
+taken when analysing the data to ensure complete background
+linesubtraction.
+Standardmethodsforanalysingthedataofacoded-maskin-
+strument make use of detector patterns , which give the relative
+background count rates in each detector at a given energy and
+are calibrated fromempty field observations.For a given obs er-
+vation, the detector pattern is scaled to the data and the excess
+counts are then accounted for by fitting a source model or ex-
+ploited by some image reconstruction algorithm. In the spec ificPierrickMartinet al.:Constraints onthe kinematics of the44Ti ejecta of Cassiopeia A from INTEGRAL /SPI 3
+Fig.1.RawSPIspectrumwitharrowindicatingthenominalenergyof thehigh-energy44Tiline(top).Meancountrateinthe1140-
+1170 keV band as a function of pointing: black crosses are SPI data and the red line is the sum of backgroundand source model s
+(middle). Residual count rate as a function of pointing afte r subtraction of the fitted background and source models to th e data,
+pointingshavebeenrebinnedingroupsof20forclarity(bot tom).4 PierrickMartinet al.:Constraints onthe kinematics of th e44Ti ejecta of Cassiopeia A from INTEGRAL /SPI
+case of astrophysical gamma-raylines, however,the signal s are
+so weak that this approach becomes prohibitive. The variati ons
+of the background noise level from pointing to pointing are o f
+the order of the searched signals and the relative count rate s
+in the detectors are subject to some long-term evolution. Th e
+search for a weak source using a detector pattern would there-
+fore require a frequent scaling of the pattern (for each poin ting
+in fact) and a few readjustments of it to cope with long-term
+variations.With such a largenumberof degreesof freedom,t he
+resultingsensitivityisfartoolowformostastrophysical gamma-
+raylines.Bringingthe signalsto theforethusnecessitate salter-
+nativemethodsinvolvingfewer parametersso as to improvet he
+sensitivity of the instrument. The true challenge here is th e re-
+productionofthetimeevolutionofthebackgroundnoise,at each
+energyandforeachdetector.
+In the approachwe have used here, the backgroundis mod-
+elled as a linear combination of various templates , which are
+time-series intended to account for the di fferent sources and
+temporal trends of the background noise. It should be em-
+phasized, however, that despite all e fforts (see for example
+Weidenspointneret al. 2003), the true nature of the backgro und
+remains mostly unknown, and as a consequence our approach
+is empirical. The central element of this stage is the so-cal led
+GEDSAT activity tracer, which represents the count rates of all
+eventsthatsaturatethedetectors,thatisalleventswithi ncoming
+energies above 8 MeV. It gives an estimate of the flux of high-
+energycosmic-ray(and solar) particles, and thereforetra ces the
+instantactivationorexcitationrateoftheon-boardmater ial.The
+useoftheGEDSATinourmodellingisanhypothesisonthetime
+variabilityoftherealbackgroundand,assuch,itcouldint roduce
+some bias in the results. Yet, many analyses of SPI gamma-ray
+line observations have been performed under this prescript ion
+(seeKn¨ odlsederet al.2005;Wang etal.2007;Diehlet al.20 06,
+for the annihilation,60Fe and26Al lines respectively) and pro-
+duced successful results, so that adopting a similar strate gy for
+thestudyofthe44Tilinesseemsjustified;inadditionwewillen-
+surethattheresultsarenota ffectedbysystematice ffectsthrough
+a thoroughanalysisoftheresiduals.
+For the low-energy lines at 67.9 and 78.4keV, the back-
+groundnoiseturnedoutto beso highandso variablethat it wa s
+impossible to build a satisfactory model. For the analysis o f the
+1157.0keV line, on the contrary, the background proved to be
+quitewellrepresentedbyasimplethree-componentbackgro und
+modelofthefollowingform:
+BGMp,d,e=Aj(p,d,e)×LIVEp,d
++Bk(p,d,e)×(Tp−T0)×LIVEp,d
++Cl(p,d,e)×GEDSAT p,d×LIVEp,d(1)
+In the above formula, BGMis the resulting background model
+andtheindices p,dandeindicatethepointing,thedetectorand
+theenergybinorbandconsidered. A,BandCarethescalingco-
+efficientsof each component,and the functions j,kandldefine
+theparametersetforeachcomponent. TpandT0arerespectively
+the meandate ofpointing pandthe date ofthe beginningof the
+mission,and GEDSAT p,dandLIVEp,daretherateofsaturating
+eventsandthelivetimeofdetector dduringpointing p.Thefirst
+component (controlled by factors Aj) is a constant component,
+supposedly accountingfor the part of the backgroundthat or ig-
+inates from the long-lived natural radioactivity of the ele ments
+making up the satellite. The second component (controlled b y
+factorsBk) is a linear growth from the day the satellite was put
+into space, and accountsfor any build-upof a specific medium -
+lived radioisotope due to a continuousactivation by cosmic -ray
+Fig.2.Images of the sky residuals from our analysis (top) and
+froma typicalsimulation(bottom).
+bombardment. The third component (controlled by factors Cl)
+models the prompt componentof the cosmic-ray induced back-
+ground,thatis,allbackgroundradiationarisingfromshor t-lived
+nuclei or excitation states (short compared to the typical d ura-
+tionofapointing).The Aj,BkandClfactorscannotbeestimated
+a prioriwith sufficient accuracy and should therefore be deter-
+minedthroughfitting tothedata.
+2.4. Modelfitting
+Inthiswork,wewanttoobtainspectralinformationonafiduc ial
+source of44Ti emission whose position is well established. The
+44Ti signal is therefore extracted through model fitting (as op -
+posed to image reconstruction). Cas A is considered as a poin tPierrickMartinet al.:Constraints onthe kinematics of the44Ti ejecta of Cassiopeia A from INTEGRAL /SPI 5
+Fig.3.Distribution of the sky residuals obtained from our anal-
+ysis (black crosses, only non-zero points are shown) compar ed
+totheexpectedstatistical distributionobtainedfromsim ulations
+(redcurve).
+source at (l,b)=(111.73◦,-2.13◦), since the size of the remnant
+(roughly 5’ on the Chandra images from Hwanget al. 2004)
+is far below the SPI angular resolution. This source model is
+mapped into the data-space through a convolution with the in -
+strument response function and then forms the sky model. All
+background and sky components are then fitted simultaneousl y
+to the data using a Maximum Likelihood criterion for Poisson
+statistics. The source model is fitted independentlyfor eac h en-
+ergy bin to extract the source spectra, while each component of
+the backgroundmodel is fitted independently for each detect or,
+energy bin and for 3 time periods delimited by the failures of
+detectors2 and 17 (because the failureshave changedthe bac k-
+groundnoiseinthedetectorssurroundingthedeadonesqual ita-
+tivelyandquantitatively).
+2.5. Assessment ofsystematic effects
+Before proceeding with the interpretation of the result, it is im-
+portantto check that our analysis of the SPI data is satisfac tory.
+We want to be sure that our modelling of the background has
+not introduced systematic errors that might have prevented the
+detectionofa signal.
+The residuals left from the subtraction of all fitted models
+(background and sky) to the data are projected along the time
+and energy dimensions of the data-space. It is checked using
+chi-squareteststhattheresidualsarestatisticallydist ributedand
+donotsuspiciouslygroupovercertainperiodsorenergyran ges.
+The temporalresidualsfor the presentanalysisare shown in the
+lower plot of Fig. 1, where they have been rebinned for clar-
+ity.Thereducedχ2whenresidualsaredeterminedona pointing
+basis is 1.0. Yet, when computing the residuals over increas ing
+time-scales, the reduced χ2increases up to values of about 2.0,
+indicating that long-term systematic e ffects are present. Some
+long-termtrendsmaynotbeproperlyaccountedforbyourbac k-
+groundmodel,whichcanbe explainedbythe fact thatourmain
+ingredient,theGEDSATrate,tracesinstantactivityonlya ndnot
+build-up effects. However, the propagation of these systematics
+from the data-space to the final results is not straightforwa rd.
+Firstly, the instrument response is quite complex and image re-
+constructionisbasedonspecificcorrelationsbetweendete ctors.
+Secondly, when working with large OG including several thou -sand pointings and spanning a long time range, it may well be
+thatsystematic errorsaverageout.
+In order to assess the impact of the systematic errors on the
+result, we have studied the distribution of sky residuals. S ky
+residuals correspond to a back-projection of the residuals from
+the detector plane to the sky, through the coded mask. It is th e
+sky space, rather than the data space, that we are interested in,
+so that this analysis tells us more than the temporal residua ls
+discussedpreviously.Formally,skyresidualsresultfrom thefol-
+lowingcalculation:
+Iα,δ=(Nα,δ−Uα,δ)/radicalbigUα,δ(2)
+withNα,δ=Nnorm∗/summationdisplay
+p,dRα,δ,p,d×np,d
+andUα,δ=Unorm∗/summationdisplay
+p,dRα,δ,p,d×µp,d
+whereNnorm=/summationtext
+p,dnp,d/summationtext
+α,δ/summationtext
+p,dRα,δ,p,d×np,d
+andUnorm=/summationtext
+p,dµp,d/summationtext
+α,δ/summationtext
+p,dRα,δ,p,d×µp,d
+whereIα,δis the image of sky residuals, expressed in num-
+ber ofσ.Nα,δandUα,δare respectively the data counts and
+model(backgroundandsky)countsback-projectedonthepla ne
+of the sky from the np,dandµp,d, which are respectively the
+data and model counts recorded by detector dduring pointing
+p, for a given energy range. Rα,δ,p,dis the instrument response
+function (IRF) that gives for each direction in the sky the ex -
+pected distribution of events over the detector plane (deri ved
+from Sturneretal. (2003)). A normalisation is necessary to en-
+sure that the numberof back-projectedcountsis identicalt o the
+numberofcountsinthedataspace.
+To allow qualitative and quantitative assessment of our sky
+residuals, simulations were performed for the same OG, usin g
+as the true background function our fitted background model
+and as the true source function a point-source at the positio n
+of Cas A with a flux of 2.5 ×10−5phcm−2s−1. The sky residu-
+als obtained in the 1150-1165keV band are presented in Fig. 2 ,
+together with a typical simulation. The images are qualitat ively
+similar, suggesting that our residuals are close to what is e x-
+pectedfromstatistical fluctuationsonly.Thisisbetteril lustrated
+in Fig. 3, where the distribution of our residuals is compare d to
+the purely statistical distribution obtained from 10 simul ations.
+Both distributions match, thereby indicating that the syst ematic
+errorsrevealedbythetemporalresidualsdonotconvertint oany
+bias on the sky properties. We can therefore consider that ou r
+∼500kms−1lower limit on the expansion velocity of the44Ti
+ejectaisrobust.
+3. Resultsanddiscussion
+3.1. The spectrum of CassiopeiaA
+The model-fitting procedure described in 2.4 yields the sour ce
+spectrumpresentedinFig.4,whereSEandME2havebeencom-
+bined. The spectrum has been fitted by a Gaussian shape with a
+flat continuum underneath and the best-fit profile correspond s
+to a line flux of (1.6 ±1.2)×10−5phcm−2s−1at a position of
+1155.8±1.5keV.Thisisdefinitelynotsignificant(althoughcon-
+sistent with theexpectedflux andpositionwithin theerrorb ars)
+and it suggests that the 1157.0keV line signal from Cas A may6 PierrickMartinet al.:Constraints onthe kinematics of th e44Ti ejecta of Cassiopeia A from INTEGRAL /SPI
+Fig.4.Cassiopeia A spectrumat 1157.0keV combiningSE and
+ME2;theredsolidcurveisthefit ofa Gaussianshape.
+Fig.5.χ2-curve for the total line width (including intrinsic and
+instrumental broadening), assuming a line flux of 2.5 ×10−5
+phcm−2s−1andnopositionshift.
+be broadened by the Doppler e ffect, with its flux spread over
+a too wide energy range and consequently lost in the statisti cal
+fluctuations. From this nearly flat spectrum we can derivea st a-
+tistical lowerlimitontheline width.
+Assuming a total line flux of 2.5 ×10−5phcm−2s−1and no
+position shift, the 2 σlower limit on the measured line width is
+3.8keV FWHM (see the χ2-curve shown in Fig. 5). The spec-
+tral shape eventually appearing in the source spectrum resu lts
+from the convolution of the intrinsic astrophysical 1157.0 keV
+line profile with the spectral response of the instrument. Un der
+the assumption that both profiles can be described by Gaussia n
+functions, the measured line width is a quadratic sum of the i n-
+strumental and astrophysical line widths. Since the spectr al res-
+olution of SPI at 1MeV is about 2keV FWHM, our lower limit
+on the measured line width translates into a lower limit on th e
+astrophysicalwidthof about3.2keV FWHM, whichin turnim-
+pliesan∼500kms−1Dopplervelocityat1157.0keV.Thisvalue
+is an order of magnitude lower limit on the expansion velocit y
+ofthe44Tiejecta,sinceourspectrumclearlydoesnotallowusto
+discriminatebetweendi fferentsophisticatedvelocitypatterns.3.2. The ironand velocitydistributioninCassiopeiaA
+From SPI and IBIS we have no detailed positional information
+about the44Ti emission but the constraint on the expansion ve-
+locity derived above, together with some nucleosynthesis c on-
+siderations,canbeinterpretedinthecontextofallobserv ational
+data currently available. Since44Ti is synthesized in explosive
+Si-burning,a strong spatial correlationbetween56Fe and44Ti is
+expected. So a good starting point for an interpretation of o ur
+44Tiresultisnodoubtaninventoryofalltheironrecordedsof ar
+inCas A.
+From XMM-Newton data, Willingaleet al. (2003) per-
+formed an extensive spectral analysis of Cas A and determine d
+a shocked ejecta mass of 2.2M ⊙and a swept-up CSM mass of
+7.9M⊙. The abundances they obtained from spectral modelling
+indicate a total X-ray emitting iron mass of 0.058M ⊙, which is
+a sizeable fraction of the canonical 0.1M ⊙of iron supposedly
+ejectedincore-collapseevents.Accountingforpre-exist ingiron
+in the progenitorreducesthe value to about 0.04M ⊙for the ini-
+tialsolarmetallicity,whichstillconstitutesasubstant ialamount.
+Ifweconsiderthatthereverse-shockedejectamaycontains ome
+fraction of cold material (as suggested by Hwang&Laming
+2003), presently undetectable in X-rays because of an earli er
+reverse-shock crossing and subsequent cooling or because o f a
+lowerdensity,theworkofWillingaleet al.(2003)seemstoi ndi-
+cate that a fair amount of the iron produced by Cas A currently
+liesbetweentheforwardandreverseshocks,withrmsradial ve-
+locitiesofabout1700-1800kms−1.
+The total iron yield of the Cas A event remains however
+uncertain. The low brightness of the event recorded (or not,
+see Stephenson&Green 2005) by Flamsteed in the 1680s sug-
+gests a low56Ni yield, but the visual extinction towards Cas A
+might range from 4 to 8 magnitudes. Taking this into account,
+Youngetal. (2006) estimated that the maximum56Ni mass that
+could have been produced in the event is 0.2M ⊙. The recent
+identification of Cas A as the remnantof a SNIIb explosionhas
+allowed us to tighten the constraints on the total iron yield ; it
+very likely lies between 0.07 and 0.15M ⊙, as indicated by the
+canonical SNIIb SN1993J (see Krauseet al. 2008, and refer-
+ences therein). So in the most optimistic scenario, Cas A har -
+boursabout0.09-0.011M ⊙in additionto the 0.04-0.06M ⊙esti-
+matedbyWillingaleet al.(2003).
+From nucleosynthesis considerations, a substantial fract ion
+of these 0.09-0.11M ⊙of iron can be expected to lie inside the
+reverse shock. Indeed, in a spherical supernova explosion56Fe
+is thought to form the deepest ejecta. According to the analy tic
+hydrodynamic model used by Laming& Hwang (2003) and
+Hwang&Laming(2003),0.2 M ⊙of materialare still freely ex-
+panding,sothemostofironcouldstillbethere,possiblyac com-
+paniedby44Ti.Thefreeexpansionvelocityat thereverse-shock
+position was estimated to around 5000 kms−1by Morseet al.
+(2004) from nearly undecelerated optical clumps. Therefor e,
+the expansion velocities of the unshocked56Fe and44Ti ejecta
+shouldrangebetween0and5000kms−1.
+An alternative has been put forward in the particular con-
+text of44Ti production.Maeda& Nomoto(2003) showed that a
+bipolar supernova explosion caused by a pair of opposite hy-
+drodynamic jets fed by accretion brings the material on the
+jet axis to higher temperatures than that reached in spheric al
+models, and hence causes a more rapid expansion and hence
+a stronger alpha-rich freeze-out for that material. This re sults
+primarily in an augmented44Ti yield, but the hydrodynamics
+associated with that scenario also leads to an inversion of t he
+velocities along the jet where44Ti and56Ni have the highestPierrickMartinet al.:Constraints onthe kinematics of the44Ti ejecta of Cassiopeia A from INTEGRAL /SPI 7
+velocities around 20000 kms−1, followed by28Si and16O. The
+simulation of Maeda& Nomoto(2003) has been invoked to ex-
+plain the high44Ti mass and44Ti/56Ni ratio inferred for Cas A
+(Prantzos2004)andalsoprovidesanexplanationfortheout ward
+increasing iron abundance suggested by Laminget al. (2006)
+and the outward increasing sulfur-to-oxygen ratio reporte d by
+vandenBergh(1971)andFesen(2001)fortheopticalFMKsof
+thejet.Moreover,Laming&Hwang(2003)showedfromacom-
+parison of Chandra spectra of individual knots with self-si milar
+hydrodynamical simulations that more energy was injected i n
+the direction of the north-east jet. In the specific case of Ca s A,
+the jet lies within±20◦of the plane of the sky (Fesen 2001), so
+the20000kms−1ofthe44Tiejectainthisjet-inducednucleosyn-
+thesisscenariotranslatesintomaximumvelocitiesalongt heline
+ofsightofabout±6400kms−1.
+The44Ti ejecta can therefore be present in three distinct lo-
+cations: the freely-expandingejecta, the reverse-shocke d ejecta
+and the jet. All three sites are associated with expansion ve loc-
+ities that are far higher than our ∼500kms−1lower limit and
+so the only conclusion we can draw from our INTEGRAL /SPI
+observations is that the44Ti does not lie strictly at the mass cut
+(theasymptoticvelocityofwhichistheoreticallyzero),c ontrary
+towhatis obtainedin sphericallysymmetricmodels.
+4. Conclusion
+We have searched for the three44Ti decay lines in Cassiopeia
+A with the high-resolution spectrometer SPI on board
+INTEGRAL. Due to the very high instrumental background
+noise with strong time variability, it was not possible to ex tract
+the two lines at 67.9 and 78.4keV. For the high-energy line at
+1157.0keV, no significant signal is seen in the 1140-1170keV
+band,suggestingthatthe1157.0keVlinesignalfromCassio peia
+AisbroadenedbytheDopplere ffect.Fromourspectrum,wede-
+rive a∼500kms−1lower limit at 2σon the expansion velocity
+ofthe44Ti ejecta.
+Wheninterpretedinthecontextofallobservationaldatacu r-
+rentlyavailable,especiallyinopticalandX-rays,thisre sultdoes
+not allow us to constrain the nucleosynthesis site of44Ti or the
+explosion process since the velocities involved throughou t the
+remnant are all far above our lower limit. Further progress o n
+this issue may follow from renewed e fforts to understand and
+model the background noise of the SPI instrument at low en-
+ergies. In the medium term, interesting results could also a rise
+from the search for the44Sc Kαfluorescence emission that fol-
+lowstheelectroncaptureof44Ti.InvestigationswithChandraas
+reported by Theiling&Leising (2006) have been fruitless, b ut
+line detection for specific regions may be achieved through e x-
+tendedobservations.
+Acknowledgements. The SPI project has been completed under the responsibil-
+ity and leadership of CNES. We are grateful to ASI, CEA, CNES, DLR, ESA,
+INTA,NASAand OSTC for their support.
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+arXiv:1001.0899v1  [cond-mat.stat-mech]  6 Jan 2010Dissipative electromagnetism from a nonequilibrium
+thermodynamics perspective
+Asja Jeli´ c, Markus H¨ utter, and Hans Christian ¨Ottinger
+Department of Materials, Institute of Polymers,
+ETH Zurich, 8093 Zurich, Switzerland.
+(Dated: November 13, 2018)
+Abstract
+Dissipative effects in electromagnetism on macroscopic scal es are examined by coarse graining
+the microscopic Maxwell equations with respect to time. We i llustrate a procedure to derive
+the dissipative effects on the macroscopic scale by using a Gre en-Kubo type expression in terms
+of the microscopic fluctuations and the correlations betwee n them. The resulting macroscopic
+Maxwell equations areformulatedwithintheGeneralEquati onfortheNon-EquilibriumReversible-
+Irreversible Coupling (GENERIC) framework, accounting al so for inhomogeneous temperature.
+PACS numbers: 05.70.Ln, 03.50.De, 05.40.-a
+1I. INTRODUCTION
+The dynamics of matter can be described on different levels of detail. For example,
+microscopic descriptions resolve short length scales and fast proc esses. In principle, one
+can follow the microscopic dynamics over very long times to arrive at t he behavior on the
+macroscopically relevant time scales. However, doing so is often not desirable, and one
+can learn significantly more by using a coarse-grained description. F or example, the flow
+of water in a complex geometry is usually studied not in terms of the dy namics of water
+molecules, but rather in terms of the continuum equations of hydro dynamics. This is so
+because key features of interest emerge only on the length and tim e scales much larger than
+the molecular ones. The intrinsic time scale for the relaxation of inhom ogeneities in the
+macroscopic velocity field is closely related to the shear viscosity η. This material specific
+property can either be determined experimentally on purely macros copic grounds. Another,
+conceptually very interesting, routegoesvia theso-calledfluctua tion-dissipation theorem[1–
+4], which establishes a relation between micro- and macroscopic prop erties. In particular,
+the Green-Kubo formula relates the shear viscosity to the correla tions between fluctuations
+of the microscopic shear stress, σxy, by way of
+η=V
+kBT∞/integraldisplay
+0/an}b∇acketle{tσxy(t)σxy(0)/an}b∇acket∇i}hteqdt, (1)
+withVthe sample volume, Tthe absolute temperature, and /an}b∇acketle{t.../an}b∇acket∇i}hteqthe equilibrium aver-
+age. The integration range [0 ,∞] stands symbolically for an integration range much longer
+than any microscopic relaxation time. However, it is implicitly assumed t hat this range is
+shorter than macroscopic time scales, which implies that micro- and m acroscopic timescales
+are clearly separated. Relations of this type are frequently used in molecular dynamics sim-
+ulations to determine macroscopic transport properties [3], wher e transport coefficients are
+expressed in terms of two-time correlations of current densities.
+The motivation for the work presented here related to the Maxwell equations rests on
+the following idea. The Green-Kubo relation (1), and generalizations thereof, not only
+tells ushowto relate microscopic fluctuations to macroscopic transport coeffi cients. More
+fundamentally, it indicates thatmicroscopic fluctuations give rise to dissipative processes
+in more coarse-grained descriptions. In view of electromagnetism, this serves as a strong
+2motivation to believe that, e.g., fluctuations in the polarization and ma gnetization by way
+of particle vibrations and fast spin dynamics must be observable on m acroscopic scales as
+dissipativeprocesses. AlthoughtheMaxwellequationsbelongtoth emoststudieddifferential
+equations in physics, they do not account for the dissipative pheno mena beyond Ohmic
+resistance andthefrequency dependent imaginarypartsofthep ermittivity andpermeability
+in the linear response regime. In view of the above discussion, we ant icipate that the electric
+(Ohmic) resistance is related to the fluctuations of the electric cur rent of the unbound
+charges. However, in addition also the bound particles fluctuate, g iving rise to fluctuations
+in the polarization and the magnetization. How those fluctuations giv e rise to dissipative
+effects onthe macroscopic scale is precisely what we want to elabora te onin this manuscript.
+Thermodynamics comes into play in the Green-Kubo relation (1) by wa y of the absolute
+temperature T. In addition, the occurrence of a “thermal” variable is also of funda mental
+importance in a purely macroscopic model with dissipative effects, sin ce the latter lead
+to an entropy increase, and in turn to a change in the thermal stat e. In contrast, the
+Maxwell equations are usually taken as a set of isolated equations, d ecoupled from the other
+macroscopic variables, e.g., temperature and density. This means t hat the evolution of
+the electromagnetic field is generally obtained in the approximation th at these macroscopic
+variables have set values or, at best, are given as functions of time . In order to obtain a
+rigorous and reliable theory, one needs to include these macroscop ic quantities in the list
+of variables and to consider their dynamics in conjunction with the tim e evolution of the
+electromagnetic field. Only then one can capture the interplay betw een the thermodynamic
+behavior and the electromagnetic field, and dissipative effects in elec tromagnetism can be
+addressed in a consistent manner.
+To perform a thorough theoretical analysis of the interplay betwe en the thermodynamic
+properties and the electromagnetic field is a demanding task, espec ially if one wants to
+account for dissipation effects beyond a linear response theory. L iu and coworkers have
+incorporated dissipative effects into the Maxwell equations [5–11]. T he resulting theory
+describes the dynamics of macroscopic systems that are exposed to electromagnetic fields
+and contain electric charges and currents. In particular, the driv ing forces for the dissipative
+effects on the macroscopic scale have been established. In the dyn amic equations, the new
+dissipative terms occur as anaddition to the commonly used macrosc opic Maxwell equations
+when the system is out of equilibrium. The validity of this theory was at first confined to
+3the low-frequency regime, but was later generalized to higher freq uencies [11].
+In this paper the goal is to use the reversible Maxwell equations to d erive macroscopic
+Maxwell equations for dissipative electromagnetism in a medium, valid a lso outside of the
+linear response regime. As the guideline to complete this task, the Ge neral Equation for the
+Non-EquilibriumReversible-Irreversible Coupling(GENERIC)formalis mofnon-equilibrium
+thermodynamics [4, 12, 13] is used for two reasons. First, thermo dynamic consistency of the
+macroscopic model is ensured. And second, the formalism is equippe d with a scheme for
+temporal coarse-graining, the most important part of which is con cerned with the emergence
+of dissipative effects upon coarse-graining, i.e., a generalization of t he Green-Kubo relation
+(1). This scheme will be illustrated below specifically on the Maxwell equ ations in order to
+obtain the macroscopic equations that possess the GENERIC stru cture. These equations
+are then compared to the dissipative Maxwell equations suggested by Liu [5].
+The paper is organized as follows. In Sec. II, the microscopic and co mmon macroscopic
+Maxwell equations are briefly presented. The usual coarse-grain ing procedure and its short-
+comings are recapitulated. In Sec. III, a GENERIC formulation of t he Maxwell equations
+together with the equation for the energy density is presented. W e illustrate the procedure
+to obtain, by temporal coarse-graining using Green-Kubo-type e xpressions, the dissipative
+effects in the time evolution equations for the macroscopic variables .
+II. MICRO- AND MACROSCOPIC MAXWELL EQUATIONS
+The classical derivation of the macroscopic Maxwell equations for t he electromagnetic
+fields in a medium from the microscopic ones,
+˙b=−∇×e, ∇·b= 0, (2a)
+˙e=1
+ǫ0µ0∇×b−1
+ǫ0j, ∇·e=1
+ǫ0ρ , (2b)
+is often performed through spatial averaging [14]. Throughout th e manuscript we denote the
+partial time derivative by ˙A≡∂
+∂tA. The quantities ǫ0andµ0stand for the permittivity and
+the permeability, respectively, in vacuum. Starting from the micros copic Maxwell equations
+containing only fields eandb, one separates the free from the bound parts of the charge ρ
+andof thecurrent density j. Uponspatial averaging, the expressions forthe boundpartsof ρ
+andjaresimplified byintroducingthepolarization Pandthemagnetization M. WithEand
+4Bthe spatially averaged electric field and magnetic induction, respect ively, the macroscopic
+Maxwell equations can then be formulated conveniently as
+˙B=−∇×E, ∇·B= 0, (3a)
+˙D=∇×H−Je, ∇·D=ρe, (3b)
+where one has introduced the electric displacement D=ǫ0E+P, assuming that quadrupol
+moments are negligible, and the magnetic field H=B/µ0−M. Doing so, the source terms
+of the macroscopic Maxwell equations are given by the averages of the free charge density
+and free current density, ρeandJe, respectively. In addition, the evolution equation for the
+energy density of the electromagnetic field can be written in the for m
+˙εem=−∇·(E×H)−Je·E, (4)
+whereE×His the Poynting vector.
+The structure of the equations (3) imposes the interpretation th at the field variables are
+DandB, the temporal equations are their time evolution equations, while th e other two
+represent the constraints that must be satisfied at all times. The fieldsEandHthen need
+to be expressed as some functions of DandBin order to render equations (3) closed. Such
+closing relations are known as constitutive equations, which contain the information about
+the medium, and are usually considered as given in macroscopic electr odynamics. The usual
+choice is to use linear relations between ( E,B) and (D,H), valid if the electromagnetic field
+is sufficiently weak, in order to obtain the properties of the permittiv ityǫand permeability
+µthat describe the response of a medium on the electromagnetic field . Then the real
+parts of the functions ǫ(ω) andµ(ω) express the oscillatory motion of the bound charges
+and the imaginary parts describe dissipation. However, in many case s, like the passage of
+the laser beam through a substance, or for a medium in a strong mag netic field, the linear
+response theory is not valid anymore and one needs to work in the re gime of nonlinear
+electrodynamics, with the consequence of loosing all the simple relat ions.
+A common constitutive relation concerns the current density. It is known that Ohmic
+conductors satisfy the condition Eeq=0at equilibrium, and that in non-equilibrium sit-
+uations the force to relax Dto equilibrium is proportional to E. Such behavior is often
+expressed by Je=σEwith conductivity σ. This interrelation suggests to draw the following
+analogy. Instationary, equilibrium situations, withvanishing electric currentJe, one obtains
+5from (3) the conditions
+∇×Heq=0,∇×Eeq=0. (5)
+We therefore assume that in non-equilibrium situations on the macro scopic scale there are
+forces to drive BandDtowards equilibrium with those forces being closely related to ∇×H
+and∇×Ewith associated transport coefficients. Conditions (5) have been o btained also by
+Liu by minimizing the total energy of the system under the constrain ts of the non-temporal
+Maxwell equations ∇·D=ρeand∇·B= 0 [5].
+The commonly used macroscopic Maxwell equations (3) can be obtain ed not only by spa-
+tial averaging (see e.g. [14]), which is also called “truncation” in [15], bu t also by ensemble
+averaging [15, 16]. In both cases, only single time properties are con sidered, while tem-
+poral correlations and two-time ensemble averages are neglected . In the remainder of this
+manuscript, we will thus refer to these two techniques as “spatial” averaging, in contrast to
+temporal coarse-graining which will be discussed in detail below. Neg lecting coarse-graining
+with respect to time to arrive at the spatially averaged Maxwell equa tions (3) may be a valu-
+able ansatz under many circumstances [14]. However, it is also known that coarse-graining
+with respect to space andtime can lead to a better understanding, and some aspects of a
+model emerge clearly only on longer time scales [4, 17]. Such an examp le is hydrodynamics,
+where the shear viscosity given by (1) and other transport coeffic ients in the Navier-Stokes
+equations arise at the continuum level of description, but are abse nt in a molecular model.
+Therefore, we will examine specifically the effect of coarse-graining in time by using Green-
+Kubo-type expressions to learn more about dissipative effects in ele ctromagnetic systems
+with nonlinear constitutive relations. In order to obtain a closed set of the macroscopic
+time evolution equations for electromagnetism in a medium, a scheme f or temporal coarse-
+graining of the “spatially averaged” Maxwell equations is presented following the GENERIC
+formalism [4, 12, 13]. The full coarse-graining procedure together with the expressions ob-
+tained for the additional dissipative terms is presented in the followin g sections.
+III. GENERIC FORMULATION
+A. Formalism
+The main points of the GENERIC framework of nonequilibrium thermod ynamics
+6[4, 12, 13] can be summarized briefly in the following way. In analogy to equilibrium thermo-
+dynamics, major importance comes to, first, choosing a complete s et of variables x, which
+describes the situation of interest to the desired detail. The rever sible contributions to the
+time evolution for this set of variables, ˙x|rev, is formulated in close reference to classical
+Hamiltonian mechanics. In particular, it is related to the energy grad ient by way of a Pois-
+son operator L, i.e.,˙x|rev=L·(δE/δx). The Poisson bracket associated to L, given by
+{A,B}= (δA/δx,L·δB/δx) with appropriate scalar product ( ...), must be antisymmetric
+and satisfy the Jacobi identity, which are both abstract feature s that capture the nature of
+reversibility. In order to formulate the irreversible part of the dyn amics one is motivated
+by the reversible part. For the reversible dynamics, the energy pla ys a distinct role. First,
+it is a conserved quantity for closed systems and, second, it drives the reversible dynamics.
+In parallel, for irreversible dynamics entropy is a fundamentally impor tant quantity, which
+must not decrease for closed systems. In analogy to the reversib le dynamics, it is assumed
+in the GENERIC framework that the irreversible contributions to th e time evolution of x,
+˙x|irrev, is driven by the entropy gradient, i.e., that it is of the form ˙x|irrev=M·(δS/δx), with
+Ma generalized friction matrix. The latter is required to be (Onsager- Casimir) symmetric,
+and contains transport coefficients and relaxation times associate d to the corresponding dis-
+sipative effects. The condition that the friction matrix is positive sem i-definite ensures that
+˙S≥0 is fulfilled.
+In summary, the time evolution of xcan be expressed in terms of four building blocks E,
+S,LandMas
+˙x=L(x)·δE(x)
+δx+M(x)·δS(x)
+δx. (6)
+The two different contributions to the time evolution, reversible and irreversible, are not
+independent. Rather, they are interrelated by the two degenera cy requirements
+L(x)·δS(x)
+δx=0,M(x)·δE(x)
+δx=0. (7)
+The first condition expresses the reversible nature of the Lcontribution to the dynamics,
+demonstratingthefactthatthereversible dynamicscapturedin Ldoesnotaffecttheentropy
+functional. The second one expresses the conservation of the to tal energy of an isolated
+system by the irreversible contribution to the system dynamics cap tured in M.
+A particular feature of the GENERIC is that it is applicable on different levels of descrip-
+tion, including reversible Hamiltonian mechanics and dissipative macros copic field theories.
+7The corresponding four building blocks obviously differ between the d ifferent levels. How-
+ever, there are abstract procedures to relate them. Most impor tantly, the friction matrix
+on a coarse-grained level can be expressed in terms of the more mic roscopic dynamics [4].
+This procedure helps to understand how dissipative effects can aris e in coarse-grained de-
+scriptions. It is particularly this issue that we want to address in rela tion to the Maxwell
+equations of electrodynamics.
+B. Choice of variables, energy and entropy functionals
+First, we identify a natural set of variables for a dynamic descriptio n of dissipative elec-
+tromagnetism. In order to describe a body in an electromagnetic fie ld, we make the local
+equilibrium assumption, so that temperature, internal energy and entropy densities are de-
+fined at each point of the nonequilibrium system. Although one is free to choose any of
+these three variables for describing the “thermal” state of the bo dy, it will become clear
+below that it is particularly useful to choose the energy density ε, since it is the density of
+a variable which is conserved for a closed system. The energy densit yεstands for the total
+energy density inside the system volume, consisting of both the inte rnal energy of matter
+and the energy of the electromagnetic field. As far as the variables describing the electro-
+magnetic state are concerned, we keep in mind that our primary inte rest is in the effect
+of coarse-graining the Maxwell equations with respect to time. Hen ce, we use the already
+spatially averaged fields DandBas used in the temporal Maxwell equations in (3). Since
+our focus here is on the temporal coarse-graining and on constru cting the dissipative effects
+in a macroscopic formulation of electromagnetism, we consider only b odies at rest to avoid
+additional complexity, i.e., we do not include the velocity field in the set o f variables. This
+restriction, in turn, also means that the mass density is a constant , as one can easily infer
+from the mass balance equation. In other words, we restrict our a ttention to materials for
+which the volume changes are negligible compared to the other effect s of interest, namely
+materials with vanishingly small isothermal compressibility and therma l expansion coeffi-
+cient. The full set of variables is thus given by x= (ε,D,B). The total energy and entropy
+8of the system, expressed in terms of these variables, are given by the functionals
+E=/integraldisplay
+ε d3r, (8a)
+S=/integraldisplay
+s(ε,D,B)d3r . (8b)
+The functional derivatives of EandSwill be needed further below for the calculation of
+time evolution of variables xby using the general evolution equation (6). In order to connect
+the expressions for these functional derivatives to known quant ities, one refers to the Gibbs
+thermodynamic relation. The latter gives the change in the energy d ensityεof a thermally
+isolated medium at rest in the form of the total differential
+dε=Tds+E·dD+H·dB, (9)
+under the assumption of local equilibrium, where the fields EandH, as well as the tem-
+perature T, are given as the partial derivatives of the energy density εwith respect to the
+appropriate variables. The first term on the right side in (9) is identic al to the ordinary
+thermodynamic relation for the internal energy density in the abse nce of the electromagnetic
+field and for a constant mass density. The last two terms represen t the change of the energy
+density due to changes in the electromagnetic fields. The form of th ese two terms is well
+known from the standard textbooks on electrodynamics [14, 18], w here the total electro-
+magnetic energy increment is given as a volume integral over the incr ementsδDandδB
+multiplied by EandH, respectively. Relation (9) is also in agreement with the total energ y
+density used in [19] for a system at rest. We point out that the prec ise form of the energy
+densityεof the system is not required below for the derivation of the macros copic Maxwell
+equations in the GENERIC form.
+The functional derivatives δE/δxandδS/δxare now obtained by reordering Eq.(9) in
+terms of the entropy total differential, and calculating the approp riate partial derivatives
+therefrom. One finds the following expressions needed for the GEN ERIC time evolution
+9equation (6):
+δE
+δx=
+1
+0
+0
+, (10a)
+δS
+δx=1
+T
+1
+−E
+−H
+. (10b)
+C. Reversible dynamics
+The reversible contribution to the time evolution of the variables x= (ε,D,B) is related
+to the energy gradient δE/δxby way of the Poisson operator L. At this stage, we point out
+that the effects arising through coarse-graining in time by using the Green-Kubo formula
+are completely comprised in the friction matrix M, i.e., in the irreversible part of the time
+evolution. Correspondingly, temporal coarse-graining does not a ffect the reversible part of
+the time evolution of the variables x. Therefore, the Poisson operator
+L=
+LεεLεDLεB
+LDεLDDLDB
+LBεLBDLBB
+(11)
+can be build up from the already established Maxwell equations (3) an d the adequate evo-
+lution equation for the total energy density, given only by the first term in Eq.(4) for a
+medium at rest.
+We approach the construction of Lin the following manner, having in mind that the
+reversible contributions are of the form L·(δE/δx). Since only the first element of the
+vectorδE/δxis non-zero, Eq.(10a), terms LDεandLBεmust reproduce the time evolution
+equations (3). The antisymmetry of Lis then used to specify the elements LεDandLεB.
+With this, the ε-component of the degeneracy requirement L·(δS/δx) =0can be satisfied
+by choosing Lεεaccordingly. Notice that this term is then also in consistence with the time
+evolution for ε, and it is antisymmetric as well. The other two components of the deg eneracy
+condition can be fulfilled by choosing LDB=∇×,LBD=−∇×, andLDD=LBB=0. In
+10summary, one obtains the Poisson operator
+L=
+E·∇×H−H·∇×E−H·∇×E·∇×
+∇×H 0 ∇×
+−∇×E −∇× 0
+, (12)
+with all derivative operators acting on everything to the right, i.e., a lso on functions mul-
+tiplied to the right side of the operator L. At this point we mention that, in general, the
+symbol “ ·” in (6) implies not only summation over discrete indices. If field variable s are
+involved the operators LandMare written in terms of two space arguments ( r,r′), and an
+integration over r′must be performed when multiplied with a function of r′from the right.
+However, in the case of the field equations being local, one can expre ssLandMin terms of
+a single variable ronly [4], and no integration is implied when these operators are multiplied
+from the right. Such single variable notation is used for the Poisson o perator (12). However,
+in case of the friction matrix M, discussed in the next section, it will be beneficial to keep
+the general form in terms of randr′.
+The reversible time evolution of the variables xas obtained from L·δE/xtakes the form
+˙εrev=−∇·(E×H), (13a)
+˙Drev=∇×H, (13b)
+˙Brev=−∇×E, (13c)
+in accord with (3).
+We note here that the lower-right 2 ×2 block matrix in L, Eq.(12), describing the
+electromagnetic part of the system, has been used previously by M arsden and Ratiu in
+[20]. Therein, it was given in terms of a Poisson bracket at which we can arrive by using
+{F,G}=/integraltext
+(δF/δx)·L·(δG/δx)d3r. In our work, their Poisson bracket is embedded in a
+thermodynamic formulation with a thermal variable ε.
+The degeneracy requirement and the antisymmetry of the Poisson operator are satisfied
+by construction. For the Jacobi identity, it is useful to observe t hat in our system the
+time evolution of the entropy density s(ε,D,B) has no reversible contributions. Since the
+transformation of variables does not affect the Jacobi identity, w e choose to prove the Jacobi
+identity in the set of variables x′= (s,D,B), in which the Poisson operator L′takes a much
+11simpler form. Namely, only two elements, L′DBandL′BD, are non-zero, and L′coincides
+with the Poisson bracket in [20], for which the Jacobi identity is fulfilled . With this, the
+Poisson operator Lis completely constructed.
+D. Friction matrix obtained from microscopic fluctuations
+Coarsegrainingwithrespect totimefromonelevel ofdescription(L 1)toanotherone(L2)
+in principle leads to two kinds of irreversible effects on the coarser lev el. First, dissipative
+effects on level L1 reappear on level L2. Second, certain effects t hat are slower than the
+time resolution on level L1 and faster than the time resolution of L2 w ill be expressed as
+rapid fluctuations on level L2, and hence emerge as irreversible effe cts on that coarser level
+of description. It is this latter kind of irreversible effects after coa rse-graining which we
+concentrate on. The coarse-graining procedure is captured in th e following five steps and
+illustrated for the case of dissipative electromagnetism.
+First, we express the friction matrix M, which captures the dissipative contributions
+to the GENERIC evolution equation (6), in terms of a Green-Kubo-t ype expression. The
+matrixMcan be calculated from the microscopic description by a generalizatio n of the
+Green-Kubo relation for the viscosity (1). If ˙xfdenotes the fluctuations on all components
+ofx, we write [4],
+M(r,r′) =1
+kB/integraldisplayτ
+0dt/an}b∇acketle{t˙xf(r,t)˙xf,T(r′,0)/an}b∇acket∇i}ht, (14)
+which can be expressed alternatively, in component notation, as
+[1+ε(xi)ε(xk)]Mik(r,r′) =1
+kBτ/an}b∇acketle{t△xf
+i(r)△xf
+k(r′)/an}b∇acket∇i}ht, (15)
+withε(xj) =±1dependingontheparityof xjundertime-reversal. Here, τisanintermediate
+time scale separating the slow degrees of freedom (on the coarser level L2) from the fast ones
+(on the finer level L1). The quantity △xf
+j(r) is given by △xf
+j(r) =/integraltextτ
+0˙xf
+j(r,t)dt, and the
+brackets /an}b∇acketle{t.../an}b∇acket∇i}htindicate the average over an ensemble of microscopic trajectories consistent
+with the slow macrostate xover the whole time interval, see pages 357–358 in [4] for more
+details. The superscript Tin (14) denotes the usual matrix transpose.
+Second, for formulating the form of the fluctuations ˙xfthe type of the variables xis
+important. Specifically, choosing densities of (conserved) extens ive variables is particularly
+12useful. Doing so is in close analogy to the common procedure in the the ory of fluctuations
+in classical thermodynamics [1, 21]. Furthermore, the Green-Kubo relation (1), and similar
+relations for the thermal conductivity and diffusion coefficient, indic ate that current-current
+correlations are quantities of fundamental importance. For exam ple, the stress tensor is the
+current of momentum. Current densities play a major role inthe for mulationofconservation
+laws in the form of field equations for density variables of extensive q uantities, e.g., internal
+energy density, momentum density. Hence, from this perspective it is tempting to use
+density variables in the set x. How this can be exploited further is discussed in the third
+step below. In our example, the energy density εis hence a good variable, in contrast to
+the temperature T, or the (not conserved) entropy density s. Further dissipative processes
+we are considering here originate from the fluctuations in the partic le positions and spins.
+Thus, fluctuations in DandBarise due to fluctuations in densities of extensive variables,
+namely in the polarization Pand the magnetization M.
+Third, we make an ansatzfor the structural form of the fluctuations. We assume that
+the evolution equations for the fluctuations are similar in structure to their macroscopic
+counterparts, in the spirit of Onsager’s regression hypothesis. T his hypothesis states that
+the fluctuations about the equilibrium state decay, on the average , according to the same
+laws that govern the decay of macroscopic deviations from equilibriu m [2–4]. In view of
+the first component of ˙xfwe note that (4) represents the change in electromagnetic energ y
+only, while the corresponding equation for the thermal energy, εth, of the body will be of
+the form ˙ εth=Je·E−∇·Jqwith heat flux Jq, such that the total energy, i.e. the integral
+ofε=εem+εth, is conserved. Approximating the fluctuating contributions by whit e noise
+leads to expressions for ∆ xfof the form
+∆xf=
+−∇·/parenleftbig
+W(q)+E×W(H)+W(E)×H/parenrightbig
+∇×W(H)−W(j)
+−∇×W(E)
+, (16)
+where all Ware (small) increments of Wiener processes, which are integrals of w hite noise
+over a time interval τ. The superscripts to the Wiener processes indicate their physical
+origin.
+Fourth, by specifying the second moments of the Wiener processe s, and all correlations
+between them, the construction of the Mmatrix is completed. We assume that all Ware
+13decorrelated because of the difference in the underlying microscop ic mechanisms, but that
+Wqhas a non-zero correlation only with the electric current fluctuatio nsWj, which will
+give rise to the thermoelectric effects. Hence we write the following r elations
+/an}b∇acketle{tW(α)(r)W(β)(r′)/an}b∇acket∇i}ht= 2kBτδ(r−r′)C(αβ)(r), (17)
+withα,β∈ {q,j,E,H}, where the non-zero C(αβ), i.e.,C(qq),C(jj),C(EE),C(HH)and
+C(qj)≡C(jq),T, represent the correlation functions, with implicit r-dependence, e.g., through
+temperature. Note that we have assumed only local correlations b etween the Wiener pro-
+cesses, i.e., between the fluctuations. Hence, we are dealing with a lo cal field formulation.
+However, non-local effects can be included if so desired.
+In the fifth and final step, the firction matrix Mis calculated. Using the fluctuations
+(16) in conjunction with the correlations (17) we calculate the Gree n-Kubo relations for the
+elements of the matrix M(r,r′) by way of (15). The final result for the friction matrix is
+M(r,r′) =
+MεεMεDMεB
+MDεMDD0
+MBε0MBB
+, (18a)
+with
+Mεε=∇·C(qq)(r)·(∇′δ(r−r′))+∇·E(r)×C(HH)(r)·(∇′×E(r′)δ(r−r′))
++∇·H(r)×C(EE)(r)·(∇′×H(r′)δ(r−r′)), (18b)
+MDD=−∇×C(HH)(r)·(∇′δ(r−r′))×+δ(r−r′)C(jj)(r), (18c)
+MBB=−∇×C(EE)(r)·(∇′δ(r−r′))×, (18d)
+MεD=∇·C(qj)(r)δ(r−r′)+∇·E(r)×C(HH)(r)·(∇′δ(r−r′))×, (18e)
+MDε=C(qj),T(r)·(∇′δ(r−r′))−∇×C(HH)(r)·(∇′×E(r′)δ(r−r′)), (18f)
+MεB=∇·H(r)×C(EE)(r)·(∇′δ(r−r′))×, (18g)
+MBε=−∇×C(EE)(r)·(∇′×H(r′)δ(r−r′)), (18h)
+where it is understood that contractions ·and cross products ×are executed in the order of
+occurrence from the right to the left.
+Given the expression (18) for M, it is straightforward to show that the degeneracy con-
+ditionM·(δE/δx) =0is indeed satisfied. The origin of this result goes back to the ansatz
+14for the fluctuations ∆ xfin (16). Their specific form ensures that the fluctuations occur on
+a submanifold of constant energy, as can be shown readily by
+/integraldisplay
+∆xf,T·δE
+δxd3r= 0, (19)
+as required by the GENERIC. Equation (19) is fulfilled already by writin g the first compo-
+nent of˙xf, namely the fluctuations in the total energy density, as in (16), wh ich requires it
+to be the divergence of a vector field. It is then straightforward t o show that (19) is satisfied
+if boundary terms can be neglected.
+If dissipative matrix M(r,r′) is applied to the entropy gradient δS/δx(r′), including an
+integration over r′, one gets the irreversible contributions to the time evolution equat ions.
+Using the definitions
+E∗=1
+TC(EE)·(∇×H), (20a)
+H∗=−1
+TC(HH)·(∇×E), (20b)
+J∗
+e=C(jj)·E
+T+C(qj),T·∇1
+T, (20c)
+one obtains for the irreversible part of the time evolutions of DandB
+˙Dirr=∇×H∗−J∗
+e, (21a)
+˙Birr=−∇×E∗, (21b)
+where all functions are evaluated at r. We recognize in ˙Dirrthe Ohmic conduction C(jj),
+whileC(qj)represents the thermoelectric coupling [1].
+E. Final set of evolution equations
+The final electromagnetic time evolution equations can be convenien tly written in the
+following way. Combining the reversible (13) and irreversible (21) con tributions to the time
+evolution equations of the fields DandB, the resulting temporally coarse-grained Maxwell
+equations differ from (3) in that the fields E,HandJein (3) have to be replaced by E+E∗,
+H+H∗andJe+J∗
+e, respectively. The additional fields E∗,H∗andJ∗
+egive rise to the
+irreversible contributions to ˙Dand˙B. They are proportional to the quantities ∇ ×H,
+∇ ×E,Eand∇T, as expected from conditions (5) and the discussion in Sec. II, and
+15represent the nonequilibrium forces that tend to restore equilibriu m. We mention that the
+full macroscopic Maxwell equations obtained here can be used to de rive expressions for the
+complex permittivity ǫ(ω) and permeability µ(ω) when examined in the linear response
+regime [6, 8].
+The frequently considered balance equation for the energy densit y can be obtained from
+(13a) and the result of the application of M(r,r′) to the entropy gradient δS/δx(r′), namely,
+˙ε=−∇·(E×H)−∇·/parenleftbigg
+E×H∗+E∗×H+C(qq)·∇1
+T+C(qj)·E
+T/parenrightbigg
+.(22)
+The first term on the right side, which is the divergence of the well kn own Poynting vec-
+tor, represents the reversible part of the time evolution equation , while the second term is
+the irreversible part, describing the heat conduction (through C(qq)) and the Peltier effect
+(through C(qj)), as well as a modification of the electromagnetic energy flux, i.e., of the
+Poynting vector, due to the additional fields E∗andH∗. This modification of the energy
+flux was found also by Liu [7, 9].
+For the time evolution of the entropy density sone can write, using the chain rule and
+rearranging the expression,
+˙s=−∇·/parenleftbigg1
+TC(qq)·∇1
+T+1
+TC(qj)·E
+T/parenrightbigg
++
+∇1
+T
+E
+T
+1
+T(∇×E)
+1
+T(∇×H)
+·
+C(qq)C(qj)0 0
+C(qj),TC(jj)0 0
+0 0 C(HH)0
+0 0 0 C(EE)
+·
+∇1
+T
+E
+T
+1
+T(∇×E)
+1
+T(∇×H)
+.(23)
+The central matrix of the last term is positive semidefinite as can be s een after inserting
+the microscopic expressions for the correlation functions (17), a nd therefore the entropy
+production rate is indeed non-negative. The obtained entropy pro duction rate corresponds
+to the one proposed by Liu in [5, 6].
+IV. DISCUSSION AND CONCLUSIONS
+Coarse-grained dissipative Maxwell equations of electromagnetism have been obtained
+and examined from the perspective of nonequilibrium thermodynamic s by using the
+GENERIC formalism. We have illustrated a scheme to relate the dissipa tive effects on
+16the macroscopic scale to the fast microscopic fluctuations in the po larization and magneti-
+zation in the spirit of a generalized fluctuation-dissipation theorem. In particular, the study
+is applicable also to materials having nonlinear constitutive relations be tween (E,B) and
+(D,H). It emerged clearly that the difference between spatial and temp oral coarse-graining
+is of fundamental importance. On the one hand, the usual proced ure of averaging the mi-
+croscopic Maxwell equations only spatially (or by single-time ensemble s averages) leads to
+terms equivalent to the reversible contributions in the GENERIC for mulation, L·(δE/δx).
+On the other hand, coarse-graining the spatially averaged Maxwell equations further with
+respect to time leads to new irreversible effects captured in terms o fM·(δS/δx), as shown
+here. The friction matrix Mhas been related to two-time correlations of fast processes
+on the finer level of description. Using the Green-Kubo expression (14), the correlations
+of microscopic fluctuations give rise to dissipative processes such a s Ohmic currents, the
+thermoelectric effect, and to other irreversible contributions to t he electric and magnetic
+fields.
+In the above investigations, the following points have to be highlighte d. First, we pre-
+sented an illustration of the coarse-graining procedure in order to obtain the structure of
+fluctuations and, in turn, the structure of the dissipative effects in electromagnetism. Sec-
+ond, the identification of current densities occurring in ˙xfwas of fundamental importance,
+since correlations between them lead to the new dissipative process es on the macroscopic
+scale. To aid the relation between the Green-Kubo-type expressio n forM(14) and the
+correlations between current densities, it has proven useful to c hoose density variables of
+extensive quantities in the set x. Using Onsager’s regression hypothesis, the evolution equa-
+tions of the fluctuations takes the form of conservation laws, inclu ding fluctuating current
+densities. We point out that the correlations (17) are similar in spirit t o the Green-Kubo
+relation (1), in the sense that they define transport coefficients. In contrast, the expression
+forM(14) in conjunction with the entropy gradient leads to the full form of the dissipa-
+tive processes. The third point concerns the details behind the cor relations (17). In order
+to specify the dissipative processes beyond their structure, one needs to discuss in more
+detail the physics behind the correlation matrices Cαβ, withα,β∈ {q,j,E,H}. That is,
+the transport coefficients appearing in the final expressions thro ughCαβcan be obtained,
+e.g., via molecular dynamics simulations, or be derived based on explicit e xpressions for the
+microscopic fluxes and the correlations between them [22].
+17The scheme of temporal coarse-graining for obtaining the macros copic Maxwell equations
+differs from Liu’s [5, 6], who started by making an ansatzfor the entropy production rate.
+However, both procedures lead to identical modifications of the Ma xwell equations, and also
+the energy equation (22) is in agreement with the corresponding re lations in [5, 6]. The
+obtained Maxwell relations and extensions of them are successfully applied to the problems
+of nematic liquid crystals [5, 6], colloidal magnetic and electric fluid, and ferrofluids [7–
+9]. In nematic liquid crystals, the dissipative couplings between the ne w thermodynamic
+forces∇×Eand∇×Hwith the additional thermodynamic forces specific for this system
+are permitted and are equipped with transport coefficients, thus b eing valid also in case of
+the oscillatory instabilities, when usually employed static Maxwell equa tions no longer hold
+[5, 6]. Liu’s equations for ferrofluids also give rise to the dissipative fo rces that account for
+the special spin-up behavior of ferrofluids under a rotating exter nal field [7–9].
+A cross relationship, similar to the one suggested by Liu [5], occurre d in the dissipative
+terms (21) obtained by us: C(HH)in˙Dirraccounts for the correlations of the fluctuations
+in magnetization, i.e. the magnetization relaxation time, and C(EE)in˙Birraccounts for the
+correlations of the fluctuations in polarization, i.e. the polarization r elaxation time. This is
+the consequence of the nature of Maxwell equations and of the mu tual dependence of the
+electric and magnetic fields. It is naturally obtained by the form (16) for the increments in
+fluxes ∆xf.
+Aspects, different from the ones discussed here, of dissipative eff ects in electromagnetism
+andoflinking electromagnetism withhydrodynamics canbefoundin[1, 19, 23]. Decomposi-
+tion of the electromagnetic fields, in particular polarization and magn etization, into the slow
+and fast parts has been considered by Felderhof and Kroh [19]. In this way they extended
+the irreversible thermodynamics approach used by de Groot and Ma zur [1] for the hydrody-
+namics of magnetic and dielectric fluids in interaction with the electrom agnetic field. In a
+subsequent paper [23], Felderhof studied semirelativistic hydrodyn amic evolution equations
+for a medium with polarization and magnetization.
+[1] S.R. de Groot and P. Mazur, Non-equilibrium Thermodynamics (North-Holland, Amsterdam,
+1962)
+18[2] R. Kubo, M. Toda and N. Hashitsume, Nonequilibrium Statistical Mechanics , vol. II of Sta-
+tistical Physics (Springer, Berlin, 1991) 2nd edition
+[3] D.J. Evans and G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids (Academic
+Press, 1990)
+[4] H.C. ¨Ottinger, Beyond Equilibrium Thermodynamics (Wiley, Hoboken, N.J., 2005)
+[5] M. Liu, Phys. Rev. Lett. 70, 3580 (1993)
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+[11] Y. Jiang and M. Liu, Phys. Rev. Lett. 77, 1043 (1996)
+[12] M. Grmela and H.C. ¨Ottinger, Phys. Rev. E 56, 6620 (1997)
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+[14] J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1975)
+[15] F.N.H. Robinson, Macroscopic Electromagnetism (Pergamon, Oxford, 1973)
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+19
\ No newline at end of file
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+arXiv:1001.0900v2  [astro-ph.HE]  11 May 2010Astronomy& Astrophysics manuscriptno.13952 c∝circlecopyrtESO 2018
+October30,2018
+Evidenceof heavy-elementashesinthermonuclear X-ray bur sts
+withphotospheric superexpansion
+J. J. M.in ’t Zand1& N.N.Weinberg2
+1SRONNetherlands Institute forSpace Research, Sorbonnela an 2, 3584 CA Utrecht,the Netherlands
+2AstronomyDepartment andTheoreticalAstrophysicsCenter ,UniversityofCalifornia,Berkeley,601CampbellHall,Be rkeley, CA
+94720, U.S.A.
+Acceptedfor publication inA &AonApril 28, 2010
+ABSTRACT
+Asmallsubsetofthermonuclear X-rayburstsonneutronstar sexhibitsuchastrongphotospheric expansionthatforafew seconds the
+photosphereislocatedataradius rph>∼103km.Such‘superexpansions’implyalargeandrapidenergyre lease,afeaturecharacteristic
+of pure He burst models. Previous calculations have shown th at during a pure He burst, the freshly synthesized heavy-ele ment ashes
+of burning can be ejected in a strong radiative wind and produ ce significant spectral absorption features. We search the b urst data
+catalogs and literature and find 32 superexpansion bursts, 2 4 of which were detected with BeppoSAX and three with RXTE at h igh
+time resolution. We findthat these bursts exhibit the follow ing interesting features: (1) At least 31 are from (candidat e) ultracompact
+X-ray binaries in which the neutron star accretes hydrogen- deficient fuel, suggesting that these bursts indeed ignite i n a helium-rich
+layer.(2) Intwoof theRXTEburstswe detectstrong absorpti onedges duringthe expansion phase. The edge energies and de pths are
+consistent with the H-like or He-like edge of iron-peak elem ents with abundances >∼100 times solar, suggesting that we are seeing
+the exposed ashes ofnuclear burning. (3) Thesuperexpansio n phase isalways followedbya moderate expansion phase duri ng which
+rph∼30 km and the luminosity is near the Eddington limit. (4) The d ecay time of the bursts, τdecay, ranges from short ( ≈10s)
+to intermediate (>∼103s). However, despite the large range of τdecay, the duration of the superexpansion is always a few seconds,
+independent ofτdecay. By contrast, the duration of the moderate expansion is alwa ys of orderτdecay. (5) The photospheric radii rph
+during the moderate expansion phase are much smaller than st eady state wind models predict. We show that this may be furth er
+indicationthat the windcontains highly non-solar abundan ces of heavy elements.
+Keywords. X-rays: binaries –X-rays: bursts–Nuclear reactions, nucl eosynthesis, abundances –stars:neutron–radiative trans fer–
+X-rays: individual (4U1722-30, 4U 0614 +09, 4U 1820-30)
+1. Introduction
+X-raybursts are thermonuclearshell flashes that result fro mthe
+unstableburningoffuelaccretedonthesurfaceofaneutron star
+(NS) from a Roche-lobe filling companion star (Grindlayet al .
+1976; Woosley&Taam 1976; Maraschi&Cavaliere 1977;
+Lamb& Lamb 1978; for reviews, see Lewinet al. 1995;
+Strohmayer&Bildsten 2006). Approximately20% of all burst s
+exhibit some degree of photospheric radius expansion (PRE;
+see, e.g., Gallowayetal. 2008a) due to a flux exceeding the
+Eddington limit, resulting in a radiatively driven expansi on
+of the photosphere (Lewinet al. 1984; Ebisuzakiet al. 1983;
+Tawaraet al. 1984a,b). During the expansion,the emission a rea
+increases while the bolometric luminosity remains nearly c on-
+stant (near the Eddington limit). As a result, the photosphe re
+coolsandthedetectedX-rayfluxdecreases.Eventuallytheb olo-
+metric luminosity decreases, the photosphere returns to th e NS
+surface,andtheX-rayfluxrisesagain,signifyingthestart ofthe
+mainburst.
+Inasmallfractionofbursts(weestimatenomorethanafew
+tenths of a percent), the PRE is so extreme that for a few sec-
+ondsthedetectedX-rayfluxdecreasestolessthan1%ofitspe ak
+value.SuchextremePREscreatetheillusionofaprecursort othe
+mainburst,asfirstreportedbyHo ffmanetal.(1978).Giventhat
+the peaktemperatureof a burstis typically2 .5keV, close to the
+softendofatypicalX-raydevice’sbandpass,suchadipimpl ies
+Send offprint requests to : J.J.M.in’tZand, email jeanz@sron.nlthattheeffectivetemperaturedecreasestolessthan0 .2keV(as-
+suming pure black body radiation at a constant bolometric flu x
+and a hydrogen column density of 1022H-atomscm−2, typical
+for an X-ray burst source). This corresponds to a radius expa n-
+sionfactorof100ormore,implyingthatthephotospheremov es
+outtoradii rph>∼103km.WecallsuchanextremePREa super-
+expansion to distinguish it from the much more common mod-
+eratePRE withan expansionfactorofa few.
+In X-ray burst models, bursts that ignite in helium-rich lay -
+ershavehighpeakluminositiesthat oftenexceedtheEdding ton
+limit and drive strong radiative winds (Hanawa&Sugimoto
+1982; Paczy´ nski 1983; Fujimotoetal. 1987; Woosleyet al.
+2004). Superexpansion bursts are the most luminous of all X-
+ray bursts and may be the result of nearly pure helium ignitio n;
+assuchtheyserveasausefultestbedforheliumburstmodels .In
+addition,calculationsshow that heavy-elementashes of nu clear
+burning can be ejected in the winds of PRE bursts and produce
+strongspectral absorptionfeatures(Weinbergetal. 2006b ). The
+brightness and powerful, long-lasting winds of superexpan sion
+bursts makes them ideal candidates for detecting spectral e vi-
+denceofheavy-elementash ejection.
+Helium bursts can be of either short or long duration de-
+pendingonthepropertiesoftheaccretion(Fujimotoetal.1 981;
+Bildsten 1998). If the NS accretes a H /He mixture, the bursts
+tend to be short (seconds). If, however, the system is an ul-
+tracompact X-ray binary (UCXB) and the NS accretes hydro-
+gen deficient material, the bursts tend to be much longer (sev -2 in’tZandand Weinberg:Superexpansion X-raybursts
+eral tens of minutes, i.e., the ‘intermediate-duration bur sts’;
+in’tZandetal. 2005; Cumminget al. 2006). This is because in
+UCXBs,unlikeinmixedH /Heaccretors,thereisnostableCNO
+burning to keep the freshly accreted envelope hot. As a resul t,
+theheliumfuelignitesatagreaterdepth,wherethethermal time
+is longer. As we will show, the superexpansion bursts that we
+findareall associatedwith(candidate)UCXBs.
+The volume of data on superexpansion has increased sub-
+stantially since the last detailed publication on extreme P REs
+(vanParadijset al. 1990). At that time only 4 superexpansio n
+bursts were known. It therefore makes sense to revisit the su b-
+ject.In§2wereportonanadditional28superexpansionbursts,
+most of them found in archival BeppoSAX and RXTE data. In
+§3 we examine the spectroscopic features of the best-covered
+superexpansion bursts. We find evidence for a variable absor p-
+tion edge that may be due to exposed heavy-element ashes of
+nuclear burning. In §4 we describe the spectroscopic features
+of the other superexpansionbursts. We find that in all 32 burs ts
+the superexpansion phase is followed by a moderate expansio n
+phase. In§5 we show that the duration of the two phases are
+uncorrelated.We place these results in the context of model s of
+helium-rich X-ray bursts and radiation-driven NS winds in §6
+andconcludein§7.
+2. Searchforsuperexpansionbursts
+We searched for bursts with precursors in the literature
+(Gallowayet al. 2008a; Kuulkerset al. 2003 and other releva nt
+literature; see Table 1) and data of the Proportional Counte r
+Array (PCA; operational since January 1996) on RXTE up
+to and including 2008 and in all data of the two Wide Field
+Cameras (WFCs; operational between June 1996 and April
+2002) on BeppoSAX. Both instrument packagesconsist of pro-
+portional counters with similar e fficiency curves and spectral
+resolution (20% full-width at half maximum), and bandpasse s
+that start at 2 keV. The PCA (Jahodaet al. 2006) consists of
+5 proportional counter units (PCUs) that each have a propane -
+filled and a xenon-filled gas layer. The xenon layer is the main
+detector and is read out at high time and energy resolution
+between 2 and 60 keV. The propane layer is used for anti-
+coincidence. It has only 0.125 s time resolution and no energ y
+resolution. However, since it has additional sensitivity b etween
+1.5 and 2 keV, it is useful for studying the soft phases of X-ra y
+bursts. The net collecting area of the xenon layers combined is
+about8000cm2,thefieldofview2◦(full-widthtozeroresponse)
+andthereisnoangularresolution.Thetimeresolutioncanb eset
+to1µsatbest.ThetwoWFCs (Jageret al.1997)eachhada net
+collectingareaof140cm2,afieldofviewof40◦,anangularres-
+olution of 5′in a bandpass of 2 to 28 keV. The time resolution
+was0.5ms.TheobservationprogramofthePCAisinlargepart
+dedicated to bursting NSs, yielding large exposuretimes (o f or-
+der megaseconds;Gallowayet al. 2008a) despite the small fie ld
+ofview.Thanksto thewide field ofview ofthe WFCs, longex-
+posuretimes were acquiredonthe majorityof burstingNSs, r e-
+sultinginanunprecedentedabilitytodetectrareandunexp ected
+eventssuchassuperexpansionbursts.
+Oursearchthrough ≈3400burstsandtheliteratureturnedup
+32 superexpansionbursts from 8 di fferent systems, see Table 1.
+All the bursts that have been unambiguously identified with a n
+object(7outof8systems)arefrom(candidate)UCXBs aslisted
+by in’tZandet al. (2007). The UCXB candidacy of the only
+transient, XB 1715-321, has been put forward in Jonkeret al.
+(2007). The one burst not unambiguouslyidentifiedwith an ob -
+ject is from 4U 1708-23. Of the 32 bursts listed, 22 are fromFig.1.Time profiles of the 2-30 keV photon flux of 24 X-ray
+bursts detected from 4U 1722-30 with BeppoSAX WFC (this
+includes21burstswithprecursor,seeTable1,andthreewit hout
+atMJD50521.804,50527.181and50532.411).Thephotonflux
+is scaled between -1 and 4 phot s−1cm−2. The horizontal dotted
+linesindicateafluxlevelof0phots−1cm−2. TheMJD indicated
+in eachpanelrefersto t=0 s. The profileshavebeenalignedto
+thestart timeoftheprecursor.
+4U1722-30(intheglobularclusterTerzan2),21ofwhichwer e
+detected with BeppoSAX-WFC (see Fig. 1 and Kuulkerset al.in’tZandand Weinberg:Superexpansion X-raybursts 3
+Table 1.Measurementsof32burstswithsuperexpansion.
+Duration (s)⋄
+Instr./MJD SuperModerate Ref.‡
+Precursor expansion expansionτ¶
+decay
+phase phasetsephasetme
+4U 0614+09
+RXTE/51944.903 0.05 1.2 100 40(2) 1
+A 1246-588
+WFC/50286.290 3.0 1.5 5438(5) 2
+WFC/51539.874 1.5 6.0 2519(1) 2
+4U 1708-23 (probably)
+SAS-C/43181.834 4.2 6.0 304300(50) 3,4
+XB1715-321
+SAS-C/42957.620 2.4 1.5 36 30∆3
+Hakucho/45170.231 3 4.0 105 85(5) 5
+4U 1722-30
+WFC/50318.279 4.0 2.0 1618(3) 6
+WFC/50330.196 2.0 5.0 2016(6) 6
+WFC/50348.938 3.0 3.0 1516(5) 6
+WFC/50368.307 2.0 5.0 1419(3) 6
+WFC/50526.311 1.0 4.0 1521(6) 6
+WFC/50536.895 2.5 3.5 11 9(3) 6
+WFC/50538.439 3.0 1.0 1519(6) 6
+WFC/50553.130 2.5 1.5 1712(2) 6
+WFC/50892.706 3.0 2.5 1815(2) 6
+WFC/50904.813 2.5 4.0 2119(3) 6
+WFC/51057.579 1.5 3.0 2219(4) 6
+WFC/51231.379 2.0 5.0 1726(11) 6
+WFC/51270.560 2.0 5.5 2514(2) 6
+WFC/51278.690 2.5 2.5 1513(1) 6
+WFC/51422.838 2.0 6.5 2228(5) 6
+WFC/51431.282 4.0 5.0 2726(5) 6
+WFC/51453.377 1.5 5.5 2024(6) 6
+WFC/51461.331 4.0 3.5 1923(4) 6
+WFC/51610.000 3.0 3.5 2017(3) 6
+WFC/51639.966 1.5 5.0 3223(5) 6
+WFC/51956.091 2.0 3.5 4129(10) 6
+RXTE/50395.292 3.6 1.6 2330.2(0.1) 7
+SLX1735-269
+I’GRAL/52897.733 2.0 7.0 482600(100) 8
+4U 1820-30
+RXTE/51430.074×15.0 2.3 1400 2500 9
+M15 X-2
+Ginga/47454.730 1.5 5.5 88 60 10
+WFC/51871.593 1.5 7.5 169155(11)
+⋄See Fig. 4 for a plot of tseandtmevs.τdecay.¶Numbers in paren-
+theses represent 1 σuncertainties;‡1 - Kuulkers et al. (2009), 2 -
+in’tZandet al. (2008), 3 - Ho ffman et al. (1978), 4 - Lewinet al.
+(1984), 5 - Tawara etal. (1984b), 6 - Kuulkers et al. (2003), 7 -
+Molkov et al.(2000),8-Molkov et al.(2005),9-Strohmayer & Brown
+(2002),10-van Paradijset al.(1990);∆Thisnumber isratheruncertain
+due toincomplete coverage of the burst;×superburst
+2003). Almost all bursts from 4U 1722-30have a precursor;fo r
+a burster with precursors, it has relatively frequent burst s (once
+every few days). The superexpansionburst from 4U 1820-30 is
+a superburst(Strohmayer& Brown2002, hereafterSB02).
+3. Spectralanalysisof the3PCAbursts
+The data for the three PCA superexpansionbursts are of highe r
+quality than the WFC burst data due to the PCA’s large pho-
+ton collecting area. The quality is especially good because at
+least three PCUs were employedduringthese observations(t his
+is generally true of only ≈5% of the PCA’s observation time).Nonetheless, the spectroscopic resolution of the PCA is mod est
+(17% FWHM at 6 keV) and only relatively broad spectral fea-
+turescanberesolved.
+High-time-resolution spectrally-resolved data are avail able
+for the first 30 s of the burst from 4U 0614 +091. Such data are
+available for much of the 4U 1722-30 burst, except between 3
+and 8 s after burst onset. The superburst from 4U 1820-30 had
+goodspectralcoveragethroughoutbutonlyatamodesttimer es-
+olution of 16 s (the ‘Standard-2’data collecting mode). We d e-
+scribe the spectral data and analysis method in more detail i n
+AppendixA.
+Typically burst spectra are well fit by an absorbed
+black body model (Swanket al. 1977; Ho ffmanet al. 1977;
+Strohmayer&Bildsten 2006). Deviationsfrom this model hav e
+been reported for the superexpansion bursts from M15 X-2
+(vanParadijset al. 1990) and the PCA superexpansion super-
+burstfrom4U1820-30(SB02).Inthelattercase,SB02detect ed
+abroademissionlinebetween5.8and6.4keVandanabsorptio n
+edge at 8−9 keV at times>100s after burst onset (at earlier
+times the blackbody temperature was evolving too quickly fo r
+the Standard-2 accumulations to resolve). To model these fe a-
+tures, SB02 included a Gaussian emission line and an absorp-
+tion edge in their spectral model. With these components, th ey
+found acceptable fits with reduced χ2∼1; both the line and
+edgewerestronglyrequiredtoadequatelymodelthedata.SB 02
+further noted that features of the Gaussian line—its finite w idth
+and an energy centroid that is often significantly less than 6 .4
+keV—suggestedthat the line was producedby reflection from a
+relativistic accretiondisk. Indeed,excellent fits to the d ata were
+obtainedbymodelingthelineasanFefluorescencefeaturefr om
+an accretiondisk and accountingfor smearingof the absorpt ion
+edge due to disk motions (Ballantyne&Strohmayer 2004 and
+Ballantyne2004).
+3.1. Absorptionedges
+As shown in Appendix B (see top row of Fig. B.1), the bursts
+from4U 0614+091and 4U 1722-30also show significant devi-
+ationfromanabsorbedblackbody.Thedeviationsaresimila rto
+those seen in the 4U 1820-30 superburst: a broad emission lin e
+at 6to7 keVandan edgeat ≈10keV thatvarieswithtime.
+Motivated by the success of Ballantyne&Strohmayer
+(2004) at fitting the deviationsin 4U 1820-30with a model tha t
+includes a reflection component, we attempted to fit the spec-
+tra of the 4U 0614 +091 and 4U 1722-30 bursts using the same
+black body plus reflection model as Ballantyne & Strohmayer
+(provided to us by D. Ballantyne, see Appendix B). Although,
+as shown in Fig. B.1, the model fits some of the spectra, it fail s
+to fit all the spectra, particularlyspectra just after the en d of the
+superexpansion phase. Examples of when the black body plus
+reflection model fail are shown in the top row of Fig. 2. These
+includetwo spectra of 4U 0614 +091within a few secondsafter
+the superexpansionand one spectrumof 4U 1722-30duringthe
+first 7safterthesuperexpansion(asindicatedbythegreyba rin
+Fig.3).Inthesethreeexamples,theedgesaredeeperoratlo wer
+energiesthanthereflectionmodelcanaccommodate.
+It is worth noting that during the superexpansion, the flux
+of both bursts drops below the pre-burst level (see top panel
+of Fig. 3). This may be because the expanding shell and super-
+Eddington flux disrupt the disk and temporarily stop the accr e-
+tion. Alternatively, the disk is not disrupted and we just do not
+seetheaccretionfluxbecausetheexpandingshellsurrounds and
+obscures the inner disk. Whatever the cause, it is perhaps no t
+surprising that the disk reflection model fails at times imme di-4 in’tZandand Weinberg:Superexpansion X-raybursts
+1001000104[phot s−1 keV−1]4U 0614+09 at 1.7 s / Black body + reflection fit
+10 5 200.511.52Data/Model10010001044U 0614+09 at 2.5 s / Black body + reflection fit
+10 5 200.511.521010010004U 1722−30 / Black body + reflection fit
+10 5 200.811.21.4
+10 5 200.5 1 1.5 2Data/Model
+Energy [keV]4U 0614+09 at 1.7 s / Fit with edge included
+10 5 200.5 1 1.5 2
+Energy [keV]4U 0614+09 at 2.5 s / Fit with edge included
+10 5 200.8 1 1.21.4
+Energy [keV]4U 1722−30 / Fit with edge included
+Fig.2.Spectral fits to the PCA burstsfrom 4U 0614 +09and 4U 1722-30duringtime intervalswhen the absorbedbla ck bodyplus
+reflection model fails to fit the data (see Fig. B.1 for example time intervals when the black body plus reflection model prov idesa
+good fit to these bursts). Left plots: The 4U 0614+09 spectrum between 1.65 and 1.78 s after burst onset (this is the second data
+point after the superexpansionphase in Fig. 3). Middle plots: The 4U 0614+09spectrumbetween 2.53and 2.65s after burst onset
+(ninth data point). Right plots: The 4U 1722-30 spectrum for the 7s long interval indicated by the grey bar in Fig. 3. The upper
+panelsshowtheobservedphotonflux(crosses)andthefittedb lackbodyplusreflectionmodel(histogram).Themiddlepane lsshow
+the fractionaldeviationof thedata pointsfromthe blackbo dyplusreflectionmodel.Thelower panelsshowthe deviation safteran
+absorptionedgeisincludedinthemodel.
+atelyfollowingthesuperexpansion.Furthermore,evenwhe nthe
+reflection model does fit the data, it often yields unreasonab le
+results;forinstance,itrequiresthatthemajorityofthefl uxbein
+the reflectioncomponentrather thanthe directblack bodyco m-
+ponent. This results in unreasonable values for the black bo dy
+luminosityandradius.
+The failure of the reflection model to fit some of the spec-
+tral data from 4U 0614 +091 and 4U 1722-30 prompted us
+to add another component to the spectral model: an absorp-
+tion edge stemming from the primary (non-reflected) spheri-
+cally symmetric burst emission. Weinberget al. (2006b) hav e
+suggested that such a signature should appear in the spectra
+of radius expansion bursts due to the presence of heavy ashes
+of nuclear burning in the photosphere (see §6.2). To model
+such an edge, we multiply the spectral energy distribution b y
+M(E)=exp[−τopt(E/Eedge)−3] for photon energies E>Eedge
+andM(E)=1 otherwise1. The fit parameters are the optical
+depthofthelineτoptandtheedgeenergy Eedge.Ourfullspectral
+model thusaccountsfor three features: an absorbedblack bo dy,
+the reflection of the burst flux from an inner accretion disk, a nd
+anabsorptionedgeformedin thephotosphere.
+We find that including such an edge significantly improves
+thefits2,asillustratedinthelowerrowofplotsinFig.2.Theevo-
+lutionofthe absorptionedgeparameters τoptandEedgeis shown
+1This is multiplicativemodel edgeinXSPEC.
+2For some spectra the edge can be satisfactorily fit by allowin g for
+unrealistic values of NH, the black body temperature and the accretion
+flux. However, for the 4U 0614 +091 burst, the edge immediately after
+the superexpansion has such a low energy (see below) that a go od fitinFig.3forthethreePCAbursts.Theabsorptionedgefrom4U
+0614+091exhibitsapeculiarincreasein Eedgefrom4.63±0.05
+keV just after the superexpansionto 8 .5±0.1 keV less than 2 s
+later (see also left and middle panels of Fig. 2). For the next 2
+s,Eedgeremains nearly constant at 8.5 keV, after which no sig-
+nificant edge is detected. While Eedgeis increasing,τoptis large
+(between2and3),afterwhichitbecomessubstantiallyless than
+1.
+The fit to the first spectral data point from 4U 1722-30 (8 s
+afterburstonset)yieldsanabsorptionedgewith Eedge≃11keV
+andτopt≃1. The parametersremain near these values until the
+temperature peaks at 23 s after onset, after which the parame -
+ters quickly change to Eedge≃6−7keV andτopt≃0.5. Since
+there is no spectral data immediatelyfollowing the superex pan-
+sion (from 3 to 8 s), we do not know if the increase in Eedge
+seen in the spectra from 4U 0614 +091occursin the case of 4U
+1722-30.
+The absorption edge from 4U 1820-30 is shallow, but still
+detected with significance thanks to the high quality signal pro-
+videdbythelongdurationofthe burst.Thisisparticularly clear
+when co-adding all the data accumulated over a long time span
+(e.g.,all databetween5000and10000s).
+Although we obtain reasonable fits to the data once we in-
+clude an edge in the model, the limited spectral resolution o f
+the PCA precludes an unambiguous identification of these fea -
+tures as absorption edges. Higher spectral resolution obse rva-
+cannot be obtained even if one allows for unrealistic values of these
+standard spectral parameters.in’tZandand Weinberg:Superexpansion X-raybursts 5
+Fig.3.Time profiles of various spectral parameters for 3 superexpa nsion bursts detected with RXTE-PCA from 4U 0614 +09
+(left; see also Kuulkerset al. 2009), 4U 1722-30 (center; se e also Molkovet al. 2000) and 4U 1820-30 (right; see also
+Strohmayer&Brown 2002; Ballantyne&Strohmayer 2004). Tim e is in seconds since burst onset (see Table A.1). Top panels:
+total PCA countrate for the xenon layers (solid line, in unit s of 104s−1) and propane layer (dot-dashed grey line, in units of
+1,250s−1), after subtractingthe pre-burst levels. The horizontald ashed lines mark a count rate of zero. Second panels: bolometric
+flux of the fitted black bodyif no reflectioncomponentis inclu ded,correctedfordetector dead time and for absorption.Th e errors
+were not determined,but they becomelarge when k T<1 keV.Third panels: black bodytemperaturein keV. Fourth panels: black
+body radius assuming isotropic emission. Assumed distance s: 3.2 kpc (4U 0614 +09; Kuulkersetal. 2009), 9.5 kpc (4U 1722-30;
+Kuulkerset al. 2003) and 7.6 kpc (4U 1820-30;Kuulkerset al. 2003);Fifth and sixth panels: absorption edge energy Eedgein keV
+andopticaldepthτopt.Bottompanels: reducedχ2valueforeachfittotheblackbody +reflection+edgemodel.Weleftoutthedata
+for4U1820-30between2000and2500s,becausethesearenotu nderstood(seeBallantyne&Strohmayer2004).Thegreyvert ical
+barsfor4U 0614+09and4U 1722-30delimit the time intervalsfor the spectra s hownin Fig. 2. NHwas fixed for4U 0614 +09and
+4U1820-30andfreefor4U1722-30becauseafixedvaluedidnot yieldgoodfits.
+tionsofsuperexpansionbursts, forinstancewithXMM-Newt on
+orChandra,wouldbeextremelyuseful.
+3.2. Otherspectral featuresof the3 PCAbursts
+Theevolutionofsomeoftheotherspectralparametersissho wn
+in Fig. 3. We do not show the results for the reflection parame-
+ters as these are not of immediate interest. The top panel sho ws
+the light curve for both the xenon and propane detector layer s.
+The most conspicuous feature is the significant dip in the lig ht
+curveduringthesuperexpansionphase;thefluxdropsbelowt he
+pre-burst level signifying that the large burst flux tempora rily
+disrupts or obscures the accretion on to the NS. The propane
+layer is most sensitive at 1.5 to 2 keV and the signal from thislayer is strongest when the black bodytemperature kT≈1keV
+immediatelyfollowingthesuperexpansion.Thelightcurve from
+4U1722-30hasa2slongspikeneartheendofthesuperexpan-
+sion phase that is especially pronouncedin the propanelaye r. It
+maybeduetoabriefcontractionandre-expansionofthephot o-
+sphere, although interpreting the spike is di fficult because there
+is no spectral data available during this time interval. The re are
+indications of similar spikes in the 4U 1722-30 bursts detec ted
+with the WFC (Fig. 1). A similar feature has been reported in a
+burstfromM15X-2(vanParadijset al.1990).
+The3PCA burstsshowa phaseofmoderateexpansionafter
+the superexpansion. During the moderate expansion phase, t he
+black body temperature increases and the radius decreases. We
+define the end of the moderate expansion phase to be the time6 in’tZandand Weinberg:Superexpansion X-raybursts
+at which the temperature peaks, which is attributed to the ph o-
+tosphere ‘touching-down’ on the NS surface (e.g., Lewinet a l.
+1993; Kuulkerset al. 2003; Gallowayet al. 2008a,b). This is
+nicely illustrated in Fig. 3 in the bursts from 4U 1722-30 and
+4U1820-30,wheretouchdownoccurs23and1400safterburst
+onset, respectively. The moment of touch down in 4U 0614 +09
+was not coveredby the PCA, but was observedwith FREGATE
+onHETE-IItobeat 90s (Kuulkerset al.2009).
+In the bursts from 4U 0614 +09 and 4U 1722-30, we find
+that at the start of the superexpansion phase, when the X-ray
+fluxbeginstodrop,thephotosphereis apparentlymovingout at
+velocitiesvph≈3×104kms−1andvph≈103kms−1, respec-
+tively (left and middle panels of Fig. 3; this estimate is sim ply
+thechangeinapparentradiusdividedbythetimeoverwhicht hat
+changeoccurred).Itisdi fficulttodeterminewhetherthesemea-
+surementsreflectthetruevelocitiesofthephotospheredue toun-
+certainties,e.g.,intheblackbodycolorcorrection(Lond onet al.
+1986; Pavlovet al. 1991). Calculations by Madejetal. (2004 )
+show that the observed color temperature of a black body fit
+may differ by a factor of 1.3 to 1.4 from the e ffective temper-
+ature for a 1 keV black body and about 1.6 for a 3 keV black
+body. Consequently, inferred radii would need to be correct ed
+bya factorof1.7to 2.6.Inaddition,the PCA calibrateddata do
+notcoverphotonenergiesbelow3keVand,thus,thepeakofth e
+thermal spectrum when the expansion is large. We nonetheles s
+considertheexpansionvelocitiestobereliabletowithina factor
+ofafew.
+The radii during the moderate expansion phase of the 4U
+1820-30 superburst are unreasonably small (i.e., <10km).
+It cannot be attributed to an underestimated value of the dis -
+tance as this is well measured both optically (Heasleyet al.
+2000) and from comparingthe burst peak flux to the Eddington
+limit of a hydrogen-poor photosphere (e.g., Zdziarskiet al .
+2007; Gallowayet al. 2008a). A color correction would, how-
+ever, increase the measured radii to acceptable values (see also
+Strohmayer&Brown2002).Thesuperburstalsobehavessome-
+what unusually between 2000 and 2500 s after burst onset. The
+behavior is unexplained and the data in this time frame is ex-
+cludedinbothouranalysisandthatofBallantyne&Strohmay er
+(2004).
+4. Spectroscopicfeaturesof theotherbursts
+When there is sufficient sensitivity, other superexpansionbursts
+show some or all of the features seen in the 3 PCA bursts. The
+edge is difficult to detect in bursts measured with instruments
+otherthanthePCA, althoughtheburstfromM15X-2measured
+withGingashowedindicationsofedgesinthespectrum(seeF ig.
+6invanParadijset al.1990).Theonefeaturethatis alwaysseen
+is a phaseof moderateexpansionfollowingthesuperexpansi on.
+Thespikeinlow-energyfluxafewsecondsaftersuperexpansi on
+isnotalwaysseen(e.g.,itisnotseeninthePCA burstsfrom4 U
+0614+09and4U1820-30).
+The burst from 4U 0614 +09 has a very fast rise and the
+precursor is merely 0.05 s long (Fig. 3). For such short dura-
+tions, smaller detectorsmight fail to catch the precursor a nd in-
+stead first see the burst during the end of the superexpansion
+phase, when the photospheric radius is decreasing. An examp le
+of this might be the long burst detected from 2S 0918-549with
+theBeppoSAXWFCs(in’tZandetal.2005).Thisburstlacksa
+precursor and yet spectroscopic analysis of the rising phas e re-
+vealsablackbodyradiusthatisrapidlydecreasing,asifth eburst
+was preceded by a superexpansion phase. The very long burst
+fromSLX1737-282( τdecay=659±32s,seealsoin ’tZandet al.Fig.4.Duration of radius expansion phases versus decay time
+τdecayfor each superexpansion burst. For data, see Table 1.
+Crossesmarkthedurationofaburst’ssuperexpansionphase (tse)
+andasterisksmarkthedurationofitsmoderateexpansionph ase
+(tme).Thedashedlinemarkstherelationships tse,tme=τdecay.
+2002; Falangaet al. 2008) also lacks a precursordespite hav ing
+a radius that decreases during the rise, although the decrea se is
+notasextremeasin2S0918-549.
+Thereare a few burststhat also show two expansionphases,
+but whose expansion factor in the first phase is too small (be-
+tween5and10)toqualifyasasuperexpansion.TheirX-raysi g-
+nalisnotcompletelylost,buttheirlightcurvestillshows adeep
+dip. This is true of all ordinary X-ray bursts detected from 4 U
+1820-30 with the PCA (Gallowayetal. 2008a), a second PCA
+burst from 4U 1722-30 detected at MJD 54526.679 (Galloway,
+priv. comm.), a burst from M15 X-2 (Smale 2001), and a few
+bursts from the non-UCXBs 4U 1705-44, KS 1731-260, SAX
+J1747.0-2853,andSAX J1808.4-3658.
+5. Measurementof expansiondurations
+We define the duration of the superexpansion phase, tse, as the
+time between the end of the precursor and the start of the main
+burst. In practice (allowing for statistical uncertaintie s), this is
+the time over which the flux is less than about 10% of its peak
+value. We define the duration of the moderate expansion phase ,
+tme, as the time between the end of the superexpansion and the
+time of peak temperature or spectral hardness, which is usua lly
+considered to be the moment when the expanded photosphere
+‘touchesdown’ontheNS(see §3).Thedecaytime τdecayisde-
+termined by fitting an exponential function to the full-band pass
+light curve after touch down and measuring the e-foldingdec ay
+time. Since all the detectors are of a similar type (i.e., a xe non-
+filled proportionalcounter), the systematic errorsintrod ucedby
+usingobserved photonratesinsteadof derivedcalibratedenergy
+fluxesarenegligiblecomparedtostatistical errors.
+Table1givesthedurationmeasurementsforthe32superex-
+pansionbursts.Thedurations tseandtmeareplottedasafunctionin’tZandand Weinberg:Superexpansion X-raybursts 7
+ofτdecayin Fig. 4. The diagram reveals two important features:
+first,tmeisproportionaltoτdecay(theyareapproximatelyequal).
+Thisisnotsurprising;thelongertheburst,thelongerthefl uxre-
+mains near its peak value. Since the peaks are super-Eddingt on
+inthesebursts,thedurationofthesuper-Eddingtonphases hould
+be proportionalto the decay time (see AppendixC). Second, tse
+is independent ofτdecay; in all bursts the superexpansionlasts a
+few seconds. We discuss possible explanations for the lack o f
+correlationbetween tseandτdecayin§6.3.
+6. Discussion
+6.1. AssociationbetweensuperexpansionburstsandUCXBs
+All the superexpansion bursts that we find, including those i n
+the literature, are from (candidate) UCXBs (see in ’tZandet al.
+2007). A characteristic of UCXBs is that they are hydrogende -
+ficient (e.g., Nelemans 2008). As a result, X-ray bursts from
+UCXBs are most likely fueled entirely by helium and /or heav-
+ierelements3.Furthermore,sincethereisnostableCNOburning
+in-betweenbursts,thefreshlyaccretedNSenvelopeinanUC XB
+is colder than that of a hydrogen-rich system accreting at th e
+samerate.BurstsfromUCXBsthereforetendtooccuratgreat er
+depth, where the thermal time is longer. This gives rise to so -
+called ‘intermediate-duration bursts’ with durations of s everal
+tens of minutes (in’tZandet al. 2005; Cummingetal. 2006).
+Because of the large and rapid energy release, the peak lumi-
+nosityofhelium-richbursts(ofboththeshort-andinterme diate-
+duration variety) often exceeds the Eddington limit, resul ting
+in phases of photospheric radius expansion (Paczy´ nski 198 3;
+Weinberget al.2006b;Cumminget al. 2006).
+6.2. Evidenceforhea vy-element ashesinphotosphere
+The convective region that forms during an X-ray burst is wel l-
+mixed with the freshly synthesized ashes of nuclear burning .
+Weinberget al.(2006b)solvefortheevolutionofthevertic alex-
+tentof theconvectiveregioninPRE burstswith strongradia tive
+winds. Theyfind that the convectiveregionextendsoutwards to
+sufficiently shallow depths that the base of the wind lies well
+within the ashes of burning. As a result, some ashes are eject ed
+in the wind and exposed at the photosphere. The column den-
+sity ofejected ashes, which consist primarilyof heavyelem ents
+(near the Fe peak; see below),is expectedto be especially la rge
+fortheenergetic,helium-richburststhatexhibitsuperex pansion.
+The spectral signature of the ashes is therefore expected to be
+detectable at even modest spectral resolution (Weinberget al.
+2006b; see also vanParadijs1982andFosteretal. 1987).
+We now discuss two features in the data that suggest the
+presence of heavy-element ashes in the photosphere: absorp -
+tion edges at the end of the superexpansion phase ( §6.2.1)
+and smaller-than-expected radii during the moderate expan sion
+phase(§6.2.2).
+6.2.1. Absorptionedges
+Which elements might be responsible for the observed absorp -
+tion edges? We first consider the early-time edge in the spec-
+trum of 4U 0614+09, which increases from an initial energy of
+3It is unclear whether free-falling αparticles, in the absence of pro-
+tons, will spall heavier accreted nuclei already present in the NS atmo-
+spheretosuchadegreethatsignificantamountsofprotonsar eproduced
+(cf.,Bildstenetal. 1992,2003).Eedge=4.6keV to 8.5 keV within ≃2seconds. Since the pho-
+tosphere is at large radii ( rph>∼30km) during this time, the
+gravitational redshift should be negligible ( z<∼0.1). Although
+rphis decreasing with time, suggesting that material is moving
+away from the observer and back on to the NS, the rate of de-
+crease appears much too slow ( ∼100kms−1) to produce any
+appreciable doppler redshift.4This suggests that the variations
+inEedgearenotduetovariationsinredshiftbutratherareintrin-
+sic to the source. One possibility is that the shift in Eedgeis due
+to variations in which elements and ionization states domin ate
+the absorption as the photosphere contracts and Teffincreases.
+If correct, bound-bound transitions should also be a source of
+absorption.The limited spectral resolution of the PCA make s it
+difficult to detect narrow X-ray absorption lines, however (such
+lines have never been confirmed in PCA spectra as far as we
+know).
+Wenowconsiderthelate-time8.5keVedgein4U0614 +09.
+The measured photosphericradii at this time ( rph≈10km) im-
+ply that the photosphere is close to the NS surface and, thus,
+thattheedgeisgravitationallyredshifted.FortypicalNS param-
+eters (e.g., M=1.4M⊙,R=10km)z≈0.3 at the surface and
+the intrinsic energy of the 8 .5keV edge would be 11keV. This
+is close to both the hydrogen-like and helium-like edges of N i
+(10.8keVand10.3keV,respectively).Thattheedgemaybedue
+to Ni is supported by calculations which show that at the larg e
+ignitiondepth5impliedbytheintermediatedurationoftheburst
+from 4U 0614+09 (ybase∼1010g cm−2; Kuulkerset al. 2009),
+Ni is the dominantproduct of constant pressure helium burni ng
+(Hashimotoetal. 1983).
+The 4U 1722-30moderate expansion phase edge at Eedge≈
+11keV is also close to the unshifted hydrogen-like Ni edge,
+consistent with the large measured photospheric radius dur ing
+this phase. After the moderate expansion phase, when the pho -
+tosphereisat theNSsurface, Eedge≈7±1 keV,consistentwith
+a gravitationallyredshiftedhydrogen-likeNi edge.
+We can also use the measured optical depth of the edge,
+τopt, to demonstrate that not only must the absorption be due
+to a heavy element such as Ni, but that its abundance must be
+much higher than the solar value. We do so by assuming that
+τopt=τbf(Ee)≡Nσbf(Ee), where Nis the hydrogenic col-
+umn density of the element above the photosphere, σbf(E)≃
+6.3×10−18(Eedge/E)3Z−2cm2is the bound-free cross-section,
+and we take Eedge=13.6Z2eV (see, e.g., Bildsten et al. 2003).
+We approximatethe hydrogeniccolumn density as N(T,ρ,Z)=
+fζHNe, wherefis the fraction of atoms belonging to that ele-
+ment(the‘numberfraction’), ζHisthefractionin thehydrogen-
+like state at a given temperature and density from Saha equil ib-
+rium(weassumeanelementiseitherfullyionizedorhydroge n-
+like),Ne≈σ−1
+This the electron numbercolumn density, and σTh
+is the Thomson cross-section. Such an approximation should
+give a reasonable estimate of N(see, e.g., Weinberget al.
+2006b).
+Inthe top panelof Fig. 5 we show ζHas a functionof effec-
+tive temperature kTeffandrph=(LEdd/4πσT4
+eff)1/2for44Ti and
+56Ni at a density6ofρ=0.01gcm−3. In the bottom panel we
+4If instead it was a radial free-fall, the redshift could be si gnificant
+even at these large radii. The ratio of observed to emitted fr equency
+wouldbeνobs/νem=1−βwhereβ=v/c=(2GM/c2r)1/2isthefreefall
+velocity. Thus, for r≃30km haveνobs≃0.6νem.
+5We choose to express depth in terms of the mass contained in a
+column of unit cross-sectional area.
+6In the wind models, dlnρ/dlnr>∼−3 andρ≈1gcm−3atr=R
+(see Paczy´ nski &Pr´ oszy´ nski 1986); since the observed ex pansion fac-8 in’tZandand Weinberg:Superexpansion X-raybursts
+Fig.5.Fraction of44Ti and56Ni in a hydrogenic state ζH(top
+panel) and the optical depth of the edge τbf(Ee)≡Nσbf(Ee)
+(bottom panel ) as a function of e ffective temperature ( bottom
+axis) and blackbody radius rph≡(LEdd/4πσT4
+eff)1/2(top axis)
+assuming a density of ρ=0.01g cm−3. Results are shown for
+massfractionsof X=0.1and1.
+show the implied optical depth of the edge τbf(Ee)≡Nσbf(Ee)
+formassfractionsof X=0.1and1.Asthefigureshows,amass
+fractionX>∼0.1isrequiredinordertoobtainanopticaldepthas
+large as the measured values τopt>∼1 at the observed tempera-
+tures.Furthermore,elements Z<∼22cannotproducesu fficiently
+large optical depths even for X=1. A Ni (Fe) mass fraction of
+X∼0.1is∼1000(100)timeslargerthanthe solarabundance.
+Rotational broadening is expected to be small compared to
+thePCAspectralresolutionandthewidthofanabsorptioned ge.
+The NS in 4U 0614 +09 has a spin frequency of νspin=415
+Hz (Strohmayeret al. 2007), resulting in a maximum relative
+Doppler shift ofv/c=2πνspinRNS/c≃0.1 and a FWHM of the
+broadeningprofileapproximatelyafactor2lower(n.b.,vie wing
+angle effects and the smaller rotation velocity of an expanded
+photosphere tend to further reduce the broadening). The oth er
+two bursters have no measured spin frequencies, but the high -
+estconfirmedspinfrequencyevermeasuredforabursterison ly
+50%larger(620Hz;Munoet al. 2002).
+6.2.2. Smallradiiduringmoderate expansionphase
+Steady state models of radiativewinds from NSs predict phot o-
+spheric radii of 100 −1000km for mass outflow rates of ˙M≈
+1017−1019g s−1(Paczy´ nski& Pr´ oszy´ nski 1986; Joss& Melia
+1987; Nobiliet al. 1994). The measured photosphericradii d ur-
+tors are≈3−5 during the moderate expansion phase, we expect the
+photospheric density to be ≈10−2gcm−3. ForζH<∼0.1, the optical
+depthτbfscales almost linearlywith ρ.ing the moderate expansion are much smaller than this ( rph≈
+30−50km).Themodelswerenotcalculatedforlower ˙Mfornu-
+mericalreasonsandperhapstheobservedradiiaresmallbec ause
+the outflow rates are ˙M<1017gs−1. Thisseems unlikely,how-
+ever.Fromenergyconservation,Paczy´ nski&Pr´ oszy´ nski (1986)
+showedthattheradiatively-drivenmassoutflowratefroman eu-
+tronstar isgivenby ˙M≃α˙M0,whereα≡(Lbase−LEdd)/LEddis
+the excessluminosityat the base of the ignitionlayer relat iveto
+Eddingtonand ˙M0=1.7×1018g s−1is a constant that depends
+onlyontheneutronstarmassandradius(takentobe M=1.4M⊙
+andR=10km). Such a low ˙M(i.e.,α<∼0.05)would therefore
+seem to require fine-tuning as it would imply that all the burs ts
+aresuper-Eddingtonbyonlytheslightest amount.
+Analternativepossibilityisthattheoutflowstructureist run-
+cated at a radius where absorption becomes su fficiently more
+important than electron scattering; material above this ra dius
+has a lower effective Eddington limit and is blown away. The
+wind models either ignored absorption (Ebisuzakiet al. 198 3;
+Paczy´ nski&Pr´ oszy´ nski 1986; Nobiliet al. 1994) or assum ed
+thewindconsistsofsolar abundances(Joss&Melia 1987).Th e
+presence of heavy element ashes in the wind can, however,sig -
+nificantlymodifythestructureofthe outflow.
+To see why, note that at the temperatures observed during
+the moderate expansion phase ( kTeff≃1−2 keV), the fraction
+of Ni in the hydrogen-like state is 0 .001−0.4 from Saha equi-
+librium at a density ρ=0.01g cm−3(this assumes the Ni is
+either fully ionized or hydrogen-like; Ni is the dominant pr od-
+uct of burning at these depths, see §6.2.1). The radiative force
+on hydrogenic Ni in the ground state is, in the optically thin
+limit (i.e.,ignoringlineblanketing), Frad=πhe2f1→2F(E)/mec2
+(Bildstenet al.2003;Michaud1970),wherethefluxperunite n-
+ergyattheLyαtransition F(E)isnearlyconstantacrosstheline
+(E=8.0keV) and f1→2=0.42 is the oscillator strength for
+hydrogenicNi (Fuhret al. 1981). At the observedtemperatur es,
+theradiativeaccelerationonNiisthen arad≈1015−1016cms−2
+assuming a blackbody spectrum7, a factor of>∼100 larger than
+the gravitational acceleration at the observedphotospher icradii
+rph≈30−50km. The force acting on the bound electrons is
+therefore>∼100 times larger than that acting on the free elec-
+tronsand line drivingwill becomedynamicallysignificant o nce
+the abundanceratio of boundto free electronsexceeds ∼1/100
+(see,e.g.,Gayley1995).Thissuggeststhatoncematerialr eaches
+≈50km and a significant fraction of ions become hydrogenic,
+line-driving will accelerate the flow to velocities much hig her
+thanthev≈103kms−1foundintheaforementionedwindmod-
+els.
+Sinceρ∝v−1in a steady state wind, the structure of the
+outflow above≈50km may be much more tenuous than the
+wind models suggest. If line-driving accelerates the mater ial to
+∼104−105kms−1, the column density of overlying material
+will be a factor of ∼10−100 smaller than that found in the
+wind models. The radius of the electron scattering photosph ere
+maythereforebesmallerbythesamefactor,inbetteragreem ent
+with the observed rph. One caveat is that it is not clear whether
+linesor electronscatteringwill bethe dominantsourceof o pac-
+ity; a wind calculation that accounts for the absorption opa city
+from heavy elements is thereforeneededin orderto reliably es-
+timate the location of the photosphere. While the electron s cat-
+7The spectra of type I bursts are somewhat harder than a blackb ody
+attheobserved Teff(London et al.1986;Pavlovetal.1991;Madej et al.
+2004; Majczyna et al. 2005). The radiative force acting on hy drogenic
+ions maytherefore be largerthanour estimates,althoughth e ionization
+fractionwillalsobe smaller.in’tZandand Weinberg:Superexpansion X-raybursts 9
+tering opacity dominates the free-free and bound-free opac ity
+for the measured temperatures and densities of the photosph ere
+(even if the composition is primarily heavy elements; see, e .g.,
+Cox&Giuli1968),thebound-boundopacitymaybeimportant.
+6.3. Superexpansionduration
+If superexpansion is due to a super-Eddington flux driving th e
+photosphere to large radii, as the time profiles suggest, why is
+thesuperexpansionduration tsealwaysshort(afewseconds)and
+independent of the decay time τdecay∼10−1000s (see§5)?
+Sincethelongeraburst,thelongerthefluxremainsnearitsp eak
+value, one might have expected tseto increase with increasing
+τdecay,asis thecasewith themoderateexpansionduration tme.
+We now describe a possible explanationforthis lack of cor-
+relationbetween tseandτdecay. InAppendixCwe showwhythe
+lackofcorrelationisunlikelytheresultofeitherabrief( second-
+long) phase of highly super-Eddington flux or hydrodynamic
+burning.
+X-ray burst wind models have all assumed a steady state
+outflow (Ebisuzakiet al. 1983; Paczy´ nski&Pr´ oszy´ nski 19 86;
+Joss&Melia 1987; Nobilietal. 1994) and do not therefore de-
+scribethetime-dependentresponseoftheoverlyinglayers inthe
+very first moments of radiative driving. These models do, how -
+ever,show that it takes ≈1s fortime-independentconditionsto
+be established (see tcharin table 1 of Joss&Melia 1987). Given
+thattse≈1s, this suggests that the superexpansion may be the
+resultofa transientstage inthedevelopmentofthewind.
+Analyzing this possibility requires a time-dependent wind
+calculation and is beyond the scope of this paper. It is nonet he-
+less worth noting that the superexpansionis consistent wit h the
+ejectionofageometricallythinshellofmaterialofinitia lcolumn
+depthyej≫yph,whereyph≃1gcm−2isthedepthofthephoto-
+sphere on the NS surface. This shell, which may correspond to
+the layer of material above where the flux first becomes super-
+Eddington8, will become transparent when it reaches an expan-
+sion factor r/R∼(yej/yph)1/2. Once it becomes transparent—at
+a timetseaccording to this picture—the observer suddenly sees
+theunderlyingneutronstaragain.Sincethewindhashadeno ugh
+time to reach a steady state by this time, the photosphere is l o-
+catedat themoderateexpansionvalue rph≈30−50km.
+The nuclear energy release from He burning
+(≈1MeVnucleon−1) is approximately 1% of the gravita-
+tional binding energy at the NS surface, and therefore at
+most 1% of the accreted mass can be ejected. This implies
+yej<∼0.01ybase. If we assume the velocity of the ejected
+material is roughly constant with radius, the initial depth of
+the ejected column is yej=yph(vphtse/R)2and we have the
+constraintybase>∼108(vphtse/104km)2gcm−2. Since even short
+duration X-ray bursts have ignition depths ybase>108gcm−2,
+this constraint is easily satisfied in the case of 4U 1722-30
+(vph≈103kms−1andtse=5s; see§3.2 and Table 1). In the
+case of 4U 0614+09, the only other superexpansion burst for
+which there is an estimate of vph, we get the strong constraint
+ybase>∼109g cm−2forvph≈3×104kms−1andtse=1.2s.
+This burst is an intermediate-duration burst and the igniti on
+8During the burst rise, the flux first exceeds the Eddington lim it at
+some depth deep below the photosphere. This is because the el ectron
+scatteringopacityincreases towardsthesurface due tothe temperature-
+dependent Klein-Nishina corrections (Paczy´ nski 1983). A given flux
+can therefore be sub-Eddington in the deeper layers and at a c ertain
+point become super-Eddington as it di ffuses upward intolower density,
+cooler material.depth inferred from the cooling time is indeed consistent wi th
+ybase∼1010gcm−2(Kuulkerset al. 2009).
+7. Summaryand conclusions
+Thevolumeandqualityofdataonsuperexpansionburstshasi n-
+creasedsubstantiallysincethefirstdetailedstudynearly 20years
+ago(vanParadijsetal.1990).Wefound28newsuperexpansio n
+bursts in the burst catalogsandliterature, severalof whic h were
+detectedwithmuchhighertemporalandspectralresolution than
+the4originalsuperexpansionbursts.Instudyingthislarg ersam-
+ple,wefoundthatsuperexpansionburstshavethefollowing fea-
+tures:
+1. At least 31 of the 32 superexpansionburstsare from(candi -
+date)ultracompactX-raybinaries.
+2. Three PCA bursts show significant spectral deviations fro m
+the absorbed black body model that typically describe X-
+ray burst spectra. The spectral deviations can be explained
+asabsorptionedges,withedgeenergiesanddepthsthatvary
+withtime.
+3. Superexpansion is seen in short duration bursts,
+intermediate-duration bursts, and superbursts and the
+superexpansion phase is always followed by a moderate
+expansionphase.
+4. Thedurationofthesuperexpansionisalwaysafewseconds ,
+independentofburstduration,whilethedurationofthemod -
+erateexpansionphaseisproportionaltoburstduration.
+5. The photospheric radius during the moderate expansion
+phase is significantly smaller than that predicted by models
+ofradiation-drivenX-rayburstwinds.
+Weshowedthattheabsorptionedgesandsmallphotospheric
+radii of the moderate expansion phases may indicate the pres -
+ence of heavy-elementashes in the wind. In particular,the e dge
+energies and optical depths appear consistent with a redshi fted
+hydrogen-likeorhelium-likeNiedgeandhighNimassfracti ons
+(X>∼0.1);Ni isexpectedto bethe dominantproductof burning
+at the large ignition depths inferred from the duration of th ese
+bursts. While we do not have a good explanation for why the
+edgeenergiesseemtoshiftoverthecourseofafewseconds,w e
+speculatethatit maybe dueto variationsin whichelementsa nd
+ionization states dominate as rphdecreases and the temperature
+increases.
+Thebrevityofthesuperexpansionanditslackofcorrelatio n
+with burst duration may be the result of a transient stage in t he
+wind’sdevelopment,perhapsduringwhichashell ofmateria lis
+ejectedtolargeradiibythesuddenonsetofsuper-Eddingto nflux
+deepbelowthephotosphere.Atime-dependentwindcalculat ion
+using realistic compositions and opacities is needed in ord er to
+assess theviabilityofthisproposedexplanation.
+The spectral deviations seen in the PCA bursts suggest that
+the spectra of superexpansion bursts have the potential to p ro-
+vide great insight into burst and NS physics. High spectral r es-
+olution observations with sub-2 keV energy coverage, as can
+be achievedwith Chandra orXMM-Newton ,areclearly needed.
+The shortest burst recurrence times for X-ray sources that e x-
+hibitalmostexclusivelysuperexpansionbursts(e.g.,4U1 722-30
+and4U 1812-12)areof order5 to 10days(e.g.,in ’tZandet al.
+2007),implyingrequiredexposuretimesof ∼500ksifonewere
+toplanadedicatedobservation.Thereisoftensomepredict abil-
+ity to when an X-ray burst will occur (e.g., Ubertiniet al. 19 99;
+Gallowayetal. 2008a) and with careful planning less demand -
+ingexposuretimes(saybyafactoroftwo)maybe feasible.10 in’tZandand Weinberg:Superexpansion X-raybursts
+Acknowledgements. We are grateful to David Ballantyne for providing us with
+his reflection models, Craig Markwardt for guidance in the PC A background
+calculation, Duncan Galloway for pointing out a second RXTE burst from 4U
+1722-30, Erik Kuulkers for discovering the rich RXTE burstf rom 4U 0614+09,
+Nikolai Shaposhnikov forcalling ourattention tobackgrou nd issueswiththe4U
+1722-30data,andthankLarsBildsten,PhilChang,DuncanGa lloway,Alexander
+Heger, Jelle Kaastra, Laurens Keek, Frits Paerels, Tod Stro hmayer and Eliot
+Quataert for helpful discussions.
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+AppendixA: PCA spectraldatapreparationand
+analysismethod
+Details about the PCA data collecting modes, analysis setti ngs,
+and time and spectral resolution are provided in Table A.1. F or
+4U 0614+09 and 4U 1722-30 data were obtained in the burst
+catcher mode. For 4U 1820-30 standard-2 data were obtained.
+ThePCAspectrawereaccumulatedfromallactivePCUs.While
+the spectra for 4U 1820-30 were drawn from only the upper
+xenon layer (‘LR1’), those of the other two bursts were from
+all layers(the ‘burst catcher’modedoesnot allow separati onof
+PCUs orlayers).
+Avaryingtimeresolutionwasemployedinthetime-resolved
+spectroscopy, based on the resolution available, the stati stical
+qualityof the data andthe changesin the spectrum.The spect ra
+were rebinned so as to have at least 15 photons per bin in or-
+dertoensureapplicabilityofthe χ2statistic. Contributionsfrom
+particle and cosmic backgrounds, as predicted through vers ion
+3.6ofthe tool pcabackest ,were subtracted.
+Response matrices were generated with version 11.7 of the
+toolpcarmf.Asystematicerrorof1.5%wasassumedperphoton
+energy channel. The instrument team recently suggested9that
+the systematic error should be set to 0.5%. While our fitted pa -
+rameter valuesdo not changeif we adopt their prescription, do-
+ingsoincreasesthe χ2valuessomewhat.Perhapsthelargersys-
+tematicuncertaintyisduetoaspectrum(afew-keVblackbod y)
+that is different from that of the Crab source (a power law with
+aphotonindexof2.1)whichwassolelyusedforthecalibrati on;
+thisshouldbeinvestigatedfurther.
+Burst spectra were corrected for detector dead time follow-
+ingtheproceduredescribedintheRXTECookBook10.Thetime
+resolutionforsuch a correctionis limited to 0.125s, so int erpo-
+lationswerenecessaryfortheprecursorin4U0614 +09.
+The burst spectra were compared with models between 3
+keV and 30 keV using XSPECversion 12.5.0 (Arnaud 1996).
+The 3 keV threshold signifies the photon energy below which
+theinstrumentresponseisnotwellcalibrated.Theupperth resh-
+oldsignifiesthephotonenergyabovewhichtheburstemissio nis
+9http://www.universe.nasa.gov /xrays/programs/rxte/pca/
+doc/rmf/pcarmf-11.7/
+10http://heasarc.gsfc.nasa.gov /docs/xte/recipes/pcadeadtime.htmlin’tZandand Weinberg:Superexpansion X-raybursts 11
+Table A.1. Detailsofthedata(analysis)forthe3PCA superexpansionb ursts.See alsoFigs. 2, 3, andB.1.
+4U 0614+09 4U 1722-30 4U 1820-301
+Starttimeinspacecraft clockseconds2223941163.45 90054034.88 179459214.38
+UTC 4-Feb-01 8-Nov-96 9-Sep-99
+21:52:34 7:00:31 1:47:58
+Active PCUs 0,1,2,3 0,1,2,3,4 0,2,3
+Time interval of superexpansion30.04-1.2 3.63-5.26 15-17
+Endtime of moderate expansion389 23 1400
+Time interval withavailable burstcatcher mode data30.93/35.93 -0.25 /3.25 n/a
+7.75/97.50
+Time/channel resolutionburst catcher mode data 8ms /64 2ms /64 n /a
+Time/channel resolutionscience event mode data 125 µs/64 16µs/64 n/a
+Time/channel resolutionresolution standard-2 data 16s /129 16s /129 16s /129
+Time interval of initialspectral study (Fig.B.1)37.0-23.0 – 325.1-1861.1
+Time interval of ‘pre-burst’ spectrum3-740/-100+5760/+92204-3140/-2320
+14U1820-30dataisaccumulatedonlyfortheupperxenonlayer s;2Inspacecraftclockseconds (i.e.,seconds sinceJan.1,199 4,0:00:00.00UTC);
+3Timeinseconds sincestarttime;4For4U1722-30 dataweretaken aftertheburstbecause pre-burstdataarecontaminatedbyarecen temergence
+of the spacecraft from the SouthAtlantic Geomagnetic Anoma ly.
+expectedto be negligible.At those energies,the spectrumi s ex-
+pectedtobedominatedbytheaccretionflux.Thus,somehandl e
+isobtainedontheaccretionemission.
+Accountingforthechangingaccretionemissionisnottrivi al.
+In super-Eddingtonbursts, one cannot simply subtract the s pec-
+trum accumulatedbeforethe burst becausethe super-Edding ton
+flux and expanding photosphere can a ffect the accretion disk
+(e.g., Ballantyne&Strohmayer 2004). Indeed, as shown in th e
+main text, the observed flux drops below the pre-burst (i.e., ac-
+cretion) flux during the superexpansion phase. The flux durin g
+the moderate expansion phase may also a ffect the accretion. It
+isdifficulttodifferentiateaccretionfluxfromburstflux,because
+the burst emission dominatesoverthe most sensitive part of the
+bandpass. Only at the highest energies does the accretion em is-
+sion dominate, but the statistical quality is insu fficient to infer
+a spectral shape. We assume that the spectral shape of the ac-
+cretion emission remains, throughoutthe burst, identical to that
+of the pre-burst emission and fit its (‘accretion’) normaliz ation.
+When possible we determined an empirical 3-30 keV spectral
+shapefromdataina ∼1000stimeintervalpriortotheburst(this
+isnotalwaysexactly1000ssincethereisnotalwaysdataava il-
+able for a continuous 1000 s interval, see Table A.1). The dat a
+from4U 0614+09are consistent with a power law, and the data
+from 4U 1722-30 and 4U 1820-30 are consistent with a comp-
+tonizedspectrum,allwithlow-energyabsorption.Thisacc retion
+modelwasincorporatedintoourfullspectralmodeloftheda ta.
+Quoted errors on spectral parameters are single-parameter
+1σvalues.
+AppendixB: PCAspectralmodel
+To understand the X-ray burst emission process, we first accu -
+mulated an optimum-quality spectrum for each burst by iden-
+tifying long data stretches when the spectral shape, as mea-
+sured by the flux ratio between two photon energy ranges, re-
+mains approximately constant. The times over which we accu-
+mulateddataforthisinitialspectralstudyaregiveninTab leA.1.
+We then tested whether the resulting spectra are consistent with
+the canonical burst model, consisting of black body radiati on
+(leaving free the temperature and normalization), low-ene rgy
+absorption by cold gas of cosmic abundances (following the
+model for atomic cross sections and cosmic abundances of
+Morrison&McCammon 1983; leaving free the equivalent H-atom column density NH) and accretion emission (leaving free
+the accretion normalization; see above). The fractional de via-
+tions of the spectra with respect to this model are shown in th e
+upperrowofFig.B.1.Notshownaretheresultsbetween20and
+30keVbecauseofthelargeerrorbarsinthatbandpass.Noneo f
+the spectra fit the model. The shapes of the deviations are gen -
+erally similar with a maximum at 6 to 7 keV and a minimum
+between 10 and 13 keV. The amplitude of the deviations of the
+different bursts are similar and far larger than instrumental un -
+certainties(0.5%to 1.5%perchannel).
+Oneapproachtomodelingthedeviationsistoincludebotha
+broadGaussianemissionlinecenteredbetween6and7keVand
+an absorption edge close to 10 keV. A more physically moti-
+vatedapproachistoaccountforscattering(‘reflection’)a ndrel-
+ativistic blurringof the burst’sblackbodyradiationbya h ot ac-
+cretion disk (Ballantyne& Strohmayer2004; Ballantyne 200 4).
+Ballantyne& Strohmayer(2004)usedsuchareflectionmodelt o
+fit the deviationsin the 4U 1820-30superburst. Their reflect ion
+model accounts for reprocessing of thermal photons from the
+neutronstarbytheaccretiondisk(includingcomptonizati on,flu-
+orescence,recombination,andphoto-electricabsorption ),which
+is allowed to come into thermal and ionization balance. The
+modelincludesanumberofionizationstagesofC,N,O,Mg,Si
+andFe,andtheparametersincludethebolometricfluxofther e-
+flection,thetemperatureoftheblackbody,metalabundance and
+the ionization degree ξ=4πFX/nHof the scattering medium,
+withFXthe bolometric flux of the direct black body radiation
+andnHthe hydrogennumberdensity.See Ballantyne (2004) for
+furtherdetailsofthismodel.
+We used the Ballantyne (2004) reflection model. In our im-
+plementation, we assume solar abundances and fit for two of
+the reflection parameters: ξand the bolometric flux. Assuming
+solar abundances is only relevant to Fe since it is the only el -
+ement in the model with spectral features in the PCA band-
+pass. The temperature is given by that of the black body com-
+ponent.Notethatthismodelisquitedi fferentfromreflectionby
+a power law as commonly applied in AGN and Galactic black
+hole spectra. In bursts, the radiant energy per frequency de cade
+(the ‘νFν’ spectrum) peaks for black bodies at photon energies
+where photoelectricabsorption by metals is veryimportant , un-
+like for power law emission in AGN and Galactic black holes
+(Ballantyne2004).12 in’tZandand Weinberg:Superexpansion X-raybursts
+10 5 200.8 0.9 1 1.1 1.2Data/Model4U 0614+09 / Black body fit
+10 5 200.8 1 1.2 1.44U 1722−30 / Black body fit
+10 5 200.9 0.95 1 1.05 1.14U 1820−30 / Black body fit
+10 5 200.8 0.9 1 1.1 1.2Data/Model
+Energy [keV]4U 0614+09 / Black body + reflection fit
+10 5 200.9 0.95 1 1.05 1.1
+Energy [keV]4U 1820−30 / Black body + reflection fit
+Fig.B.1. Spectral fits to the PCA bursts from 4U 0614 +09 (left plots), 4U 1722-30 ( middle plot ) and 4U 1820-30 ( right plots )
+duringtimeintervalswhentheabsorbedblackbodyplusrefle ctionmodelprovidesagoodfittothedata(seeFig.2forexamp letime
+intervalswhen the black bodyplus reflection model fails to p rovidea goodfit to these bursts). The upper row shows the frac tional
+deviationsforamodelthatincludestheabsorbedblackbody componentbutnotthereflectionorabsorptionedgecomponen ts.The
+lowerrow showsthe deviationsfora modelthat includesthe a bsorbedblackbodyandreflection componentbut not the absor ption
+edgecomponent.4U1722-30isnotshownbecauseat thattime d oesa reflectioncomponentexplainthedeviations.Thespect raare
+takenduringthemoderateexpansion.Theerrorbarsinclude the1.5%systematicerror.
+Thereflectionmodelalsoincludesaconvolutionfunctionfo r
+relativistic blurringby the accretiondisk followingFabi anet al.
+(1989). It is parametrized by the power law index for the radi al
+dependence of the emissivity (set to -3), the inner disk radi us
+(set to 10G M/c2withMthe neutron star mass), the outer disk
+radius (set to 200G M/c2) and the inclination angle (set to 30o).
+Wheneverthese parameterswere left free duringfitting, the im-
+provement in the fit was marginal. For practical purposes the y
+were,therefore,keptfixed.
+The complete applied model can be written in XSPEC
+as:constant*(wabs 1*(bbodyrad+rdblur*atable{reflection})+
+wabs2*comptt/po)withconstant the detector dead time factor,
+wabs1andwabs2low-energy absorption (di fferent for the two
+instancesin thisformula), bbodyrad the Planckfunction, rdblur
+the relativistic blurringfunction, reflection the model for the re-
+flectionofthePlanckfunctionagainsttheaccretiondiskfo llow-
+ingBallantyne(2004), comptta comptonizationcomponentfol-
+lowingTitarchuk(1994)and poapowerlawfunction.Thelatter
+two components are representative of the non-burst emissio n.
+The model has six free parameters: NH, kTbb, Rbb, the flux and
+ξofthereflectioncomponentandtheaccretionnormalization of
+compttorpoexpressedrelativetothepre-burstvalue.
+Including reflection in the model is highly successful in re-
+ducing the deviations in the spectra of 4U 0614 +09 and 4U
+1820-30 for the particular time interval specified in Table A .1
+(the“timeintervalofinitialspectralstudy”).Wefind χ2
+ν=2.3and
+0.75, respectively (the value for 4U 1820-30 is consistent w ithBallantyne& Strohmayer 2004). The remaining deviations ar e
+shown in the lower row of Fig. B.1. The black bodyplusreflec-
+tion model does not, however,fit all the spectra of 4U 0614 +09
+and 4U 1820-30 (see §3.1). Furthermore, it almost never pro-
+vides a good fit to the spectra of 4U 1722-30 (the middle-right
+panelinFig.2isrepresentativeofthetypicalspectraldev iations
+foundat all times in black bodyplus reflection model fits to th e
+4U1722-30spectra).
+Even when the black body plus reflection model provides a
+goodfit,thefluxofthereflectioncomponent,afreeparameter ,is
+often higher than that of the direct black body component.Th is
+is due to the fact that the fitting procedure primarily tries t o fit
+thenarrowfeatures–thebroademissionlineandtheabsorpt ion
+edge. Since the continua of the reflection and direct black bo dy
+componentsare very similar in the PCA bandpass, their contr i-
+butions are difficult to disentangle. High reflection fluxes were
+also found by Ballantyne&Strohmayer (2004) in the late-tim e
+burst spectra of 4U 1820-30 (see their Fig. 1). As a result of a
+highreflectionflux,thefluxofthedirectblackbodycomponen t
+becomes so low that the black body radius becomes substan-
+tially lower than 10 km. This is physically unrealistic. Wit hout
+a propermodel for the geometryof the accretion disk and inde -
+pendentmeasurementsoftheironabundanceinthedonorstar ,it
+is impossible to meaningfullyconstrain the reflection flux i n an
+independentmanner.
+We calculated bolometric luminosities and emission region
+radii by fitting the spectra without the reflection (or edge) c om-in’tZandand Weinberg:Superexpansion X-raybursts 13
+ponent. After fitting, the absorption and the accretion norm al-
+ization was set to zero,the deadtime correctionset to 1, and the
+responsematrixin XSPECsettoadummyvalue.Thefluxcould
+thenbeestimatedinabroaderenergyrangethanthePCA band-
+pass. The unabsorbed flux was calculated between 0.05 and 30
+keV, essentially a bolometric bandpass for a black body with a
+temperature between 0.5 and 3 keV. Since the fits are often for -
+mallyunacceptable,noerrorswerecalculated.
+We tried several other functionsto model the spectra. These
+include combinations of two di fferent black bodies, combina-
+tions of one black body with a power law (with a free spec-
+tral index), gauss function or relativistic iron line (foll owing
+Laor 1991, model laorinXSPEC), and single reflection mod-
+els of a power law against a neutral or ionized disk (followin g
+Magdziarz&Zdziarski 1995, models pexrav/pexrivinXSPEC).
+While some of these models fit some of the spectra reasonably
+well, they only do so over limited time intervals in any given
+burst. A broken power law fits the data with the same goodness
+(χ2
+ν)at anytime,but thereis nophysicalbasisforsuchamodel.
+AppendixC: Ruling outalternativeexplanationsfor
+thelackof correlationbetween tseandτdecay
+Herewe first considerthe possibilitythat the transitionbe tween
+the superexpansion phase and the moderate expansion phase i s
+due not to a shell ejection, as suggested in §6.3, but rather to
+a transition in the luminosity at the NS surface. In particul ar, if
+theluminosityexceedsnotonlythe localEddingtonluminosity,
+LEdd≡4πGMc/κ0,butalsotheEddingtonluminosityatinfinity,
+LEdd,∞≃LEdd(1+z)2, a strong radiation-drivenwind will form
+(κ0=0.2cm2g−1is the hydrogen-freeelectronscattering opac-
+ity). Such a wind can potentially drive the photosphere to ve ry
+large radii (>∼100km). If, however, LEdd<L<LEdd,∞, a com-
+parativelymild quasi-static expansion is su fficient to reduce the
+luminosityat thephotospherebelow LEdd(Hanawa&Sugimoto
+1982; Paczy´ nski 1983; Paczy´ nski&Pr´ oszy´ nski 1986). In this
+case, the PRE will be less extreme. This suggests that perhap s
+the large radius expansion factors reached during the super ex-
+pansion occur because initially L>LEdd,∞. Then, after a time
+tse≈seconds, the luminosity decreases to LEdd<L<LEdd,∞
+and the photosphere retreats closer to the NS surface, marki ng
+the start of the moderateexpansionphase. This phase contin ues
+foradurationoforderthecoolingtimeafterwhichthelumin os-
+ity decreases to L<LEddand the photosphere settles on to the
+NS surface.
+Totestthisidea,wecarryouttime-dependentcoolingcalcu -
+lationsforarangeofignitiondepths.Tosolvetheheattran sport
+equationweusethetechniquedescribedinWeinberg& Bildst en
+(2007); Cumming&Macbeth (2004) carry out similar calcula-
+tions for superbursts. We assume that the atmosphere is radi a-
+tive and initially at its peak temperature, in accord with mo dels
+of the rise of X-ray bursts(see e.g.,Cumming& Bildsten 2000 ;
+Weinberget al. 2006b). We assume a peak temperature at the
+base of the ignition layer ybaseofTbase,initial=fTmax, wherefis
+factorlessthanbutoforderunityand Tmax=(3gybase/a)1/4isthe
+criticaltemperatureatwhichradiationpressurecomplete lydom-
+inates(g=(1+z)GM/R2is the gravitationalaccelerationand a
+istheradiationconstant).Wechosethevalueof fbysolvingfor
+Tbase,initialsuchthatthefluence(thecoolingcurveintegratedover
+the durationof the burst) equalsthe total nuclearenergyre lease
+E=4πR2ybaseQnuc/(1+z)assuming Qnuc=1.6MeV nucleon−1,
+correspondingtoheliumburningcompletelytonickel.Alth oughFig.C.1. Model calculations of the radiative cooling following
+the burst rise. Thesolid lines showthe luminosity Lforignition
+depthsybase=3×108,109,and1010g cm−2withf=0.88,0.85,
+and 0.78, respectively. The values for fwere chosen such that
+thefluenceequalsthenuclearenergyreleasedinburningthe en-
+tireHecolumntoNi.TheEddingtonlimitattheNS surfaceand
+at infinity (assuming z=0.31 andM=1.4 M⊙) are drawn as
+dottedlines.
+ourconclusionsare notsensitiveto the exactvalueof f, forref-
+erence,we find f≈0.8−0.9.
+InFig.C.1 we show timeprofilesofthecalculatedluminos-
+ityLfor three ignition depths ybase. The dotted lines mark LEdd
+andLEdd,∞for a 1.4 M⊙NS and an opacity κ0=0.2cm2g−1.
+An observer would see an Eddington-limitedlight curve wher e
+L>LEdd(theexcessluminosityis usedto drivetheoutwardex-
+pansion of the overlying layers) and a light curve that follo ws
+theplottedluminositywhere L<LEdd.
+As the figure shows, the profilesall have the same shape re-
+gardless ofybaseandf: an early flat phase followed by a break
+anda steep decay.The factor fprimarilydeterminesthe overall
+verticalscalewhile ybasedeterminesthethermaltimeoftheigni-
+tionlayerandthereforethe locationof thebreak.Thekeyre sult
+is that the early-time light curve is very flat, barely decrea sing
+until after the break, at which point it rapidly falls below LEdd.
+Suchalightcurvecannotexplainthefeaturesofthesuperex pan-
+sionasthereisnosecond-long,depth-independentphasedu ring
+whichL>LEdd,∞.
+The light curve is flat because we assumed that the initial
+profilewastheradiativeprofile( T∝ynwithn≈1/4)andthere-
+foreL∝T4/yis constant until the thermal wave reaches ybase
+and the ignition layer begins to cool. If we had assumed a shal -
+lowerinitialprofile,theearly-timefluxwoulddecreasewit htime
+(see, e.g., the superburst calculations of Cumming&Macbet h
+2004 who assume an initial profile with n=1/8 corresponding
+toanisobaricdeflagration).However,absentanystrongmot iva-
+tionforassuminganon-radiativeinitialprofile,weconclu dethat
+a variation in the early time Lis unlikely to explain the lack of
+correlationbetween tseandτdecay.14 in’tZandand Weinberg:Superexpansion X-raybursts
+Since the superexpansion phase always occurs within sec-
+onds of burst onset, perhaps it is associated not with post-p eak
+cooling, as assumed above, but rather with the physics of the
+burningduringthe burst rise. Due to their largerignition d epths
+and helium mass fractions, the peak energy generation rate i n
+superexpansion bursts can be considerably higher than in or di-
+nary short-duration bursts. If the ignition is deep enough, the
+heating time due to burning theat=(dlnTb/dt)−1can become
+shorter than the local dynamical time tdyn≈10−6s. Such hy-
+drodynamic burning could drive strong shocks through the up -
+perlayersoftheatmosphere(Zingaleet al.2001;Weinberge t al.
+2006a). However,when we solve the rise using the prescripti on
+described in Weinbergetal. (2006b) assuming an initially p ure
+He layer, we find that theat>tdynunlessybase>∼1011gcm−2.
+This suggests that strong shocks do not form during the rise o f
+superexpansionburstsastheyigniteat ybase<∼1010g cm−2.
\ No newline at end of file
diff --git a/1001.0901.txt b/1001.0901.txt
new file mode 100644
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+++ b/1001.0901.txt
@@ -0,0 +1,960 @@
+arXiv:1001.0901v1  [hep-ph]  6 Jan 2010QCDMAPT: program package for
+Analytic approach to QCD
+A.V. Nesterenkoa,1and C. Simolob
+aBLTPh, Joint Institute for Nuclear Research, Dubna, 141980, R ussian Federation
+bISAC–CNR, I–40129, Bologna, Italy
+Abstract
+A program package, which facilitates computations in the framewor k of Analytic
+approachtoQCD,isdevelopedanddescribedindetails. Thepackage includesthe
+explicit expressions for relevant spectral functions calculated up to the four–loop
+level and the subroutines for necessary integrals.
+PACS: 11.15.Tk; 11.55.Fv; 12.38.Lg
+Key words: Nonperturbative QCD; Dispersion relations
+PROGRAM SUMMARY
+Manuscript Title: QCDMAPT: program package for Analytic approach to QCD
+Authors: A.V. Nesterenko and C. Simolo
+Program Title: QCDMAPT
+Journal Reference:
+Catalogue identifier:
+Licensing provisions: none
+Programming language: Maple 9 and higher
+Computer: Any which supports Maple 9
+Operating system: Any which supports Maple 9
+Keywords: Nonperturbative QCD; Dispersion relations
+PACS:11.15.Tk, 11.55.Fv, 12.38.Lg
+Classification: 11.1, 11.5, 11.6
+Nature of problem: SubroutineshelpingcomputationswithinAnalyticapproac htoQCD
+Solution method: A program package for Maple is provided. It includes the expl icit
+expressions for relevant spectral functions and the subrou tines for basic integrals used
+in the framework of Analytic approach to QCD.
+Running time: Template program running time is about a minute (depends on C PU)
+1E-mail: nesterav@theor.jinr.ru1 Introduction
+The stronginteractions display two fundamental features, name ly, theasymptotic
+freedom (at high energies) and color confinement (at low energies) . The first
+feature allows one to study the strong interaction processes in th e ultraviolet
+domain by making use of perturbation theory. However, theoretic al description
+of hadron dynamics at low energies requires nonperturbative meth ods.
+In general, there is a variety of nonperturbative approaches to h andle the
+strong interaction processes at low energies. In this work we will fo cus on
+the so–called “dispersive” (or “analytic”) approach to Quantum Ch romodynam-
+ics (QCD). The basic idea of this approach is to merge the perturbat ive results
+with the nonperturbative constraints arising from relevant disper sion relations.
+In turn, this eliminates some intrinsic difficulties of perturbation theo ry and ex-
+tends its range of applicability towards the infrared domain. One of t he pos-
+sible implementations of this approach within QCD is the so–called “mass less”
+Analytic Perturbation Theory [1–3]. The latter has been successf ully employed
+in the studies of various strong interaction processes (see, e.g., p apers [4–14],
+reviews [3,15,16] and references therein). The incorporation of effects due to
+the nonvanishing mass of lightest hadron state has been implemente d into ap-
+proach in hand within the so–called “massive” Analytic Perturbation T heory, see
+Refs. [17,18] for the details.
+A central object of the current approach is the so–called spectr al function,
+which can be calculated by making use of the strong running coupling αs(Q2)
+(see Sect. 3). At the one–loop level the perturbative running cou pling and rele-
+vant spectral function have a quite simple form (Eqs. (32) and (9) , respectively).
+However, at the higher loop levels the strong running coupling has a r ather cum-
+bersome structure, and the calculation of corresponding spectr al functions1rep-
+resents a rather complicated task.
+The primary objective of this paper is to calculate (by hands) the ex plicit
+expressions fortheaforementioned spectral functions attheh igher looplevels and
+to incorporate them (together with proper subroutines for nece ssary integrals)
+into a single program package. In turn, the latter will facilitate comp utations
+within the approach in hand.
+The layout of the paper is as follows. Section 2 constitutes a brief de scription
+of the Analytic approach to QCD. The calculation of spectral funct ions and the
+numerical evaluation of basic integrals is considered in Sects. 3 and 4 , respec-
+tively. Section 5 describes the template program and includes the list of package
+1At the higherlooplevelsthe spectral functions canalsobe calculate dnumerically. However,
+it requires a lot of computation resources and essentially slows down the overall computation
+process.
+2commands. The basic results are summarized in the Conclusions (Sec t. 6). Ap-
+pendix A contains explicit expressions for the perturbative strong running cou-
+pling up to the four–loop level. Appendix B contains explicit expression s for the
+spectral functions calculated up to the four–loop level.
+2 Analytic approach to QCD
+In general, in the framework of perturbation theory the high ener gy behavior of
+the strong correction d(Q2) to a physical observable D(Q2) can be approximated
+by the power series in the strong running coupling αs(Q2). Namely, at the ℓ–loop
+level
+d(ℓ)
+pert(Q2) =ℓ/summationdisplay
+j=1dj/bracketleftBig
+α(ℓ)
+s(Q2)/bracketrightBigj=ℓ/summationdisplay
+j=1dj/parenleftbigg4π
+β0/parenrightbiggj/bracketleftBig
+a(ℓ)
+s(Q2)/bracketrightBigj, Q2→ ∞,(1)
+whereQ2=−q2>0 is the spacelike kinematic variable, djstands for the rele-
+vant perturbative expansion coefficient, α(ℓ)
+s(Q2) is theℓ–loopperturbative strong
+running coupling (see App. A), β0= 11−2nf/3 denotes the one–loop perturba-
+tiveβfunction expansion coefficient, nfis the number of active quarks, and
+a(ℓ)
+s(Q2)≡α(ℓ)
+s(Q2)β0/(4π) is the so–called “couplant”. However, perturbative
+expansion (1) is valid in the ultraviolet domain only. In particular, at an y loop
+leveld(ℓ)
+pert(Q2) (1) possesses unphysical singularities in the infrared domain. This
+fact contradicts the general principles of local Quantum Field Theo ry and essen-
+tially complicates the analysis of low energy experimental data. Besid es, pertur-
+bative expansion (1) can not be directly employed in the theoretical description
+of physical observables depending on the timelike2kinematic variable s=q2>0.
+AsithasbeennotedintheIntroduction, onecanovercometheafo rementioned
+difficulties of perturbative approach by invoking relevant dispersion relations.
+Thus, in the framework of “massless” Analytic Perturbation Theor y (APT) [1–3]
+the theoretical expressions for the strong corrections d(Q2) andr(s) to physical
+observables D(Q2)andR(s), depending onspacelike ( Q2=−q2>0)andtimelike
+(s=q2>0) kinematic variables, take the form (see papers [2,3] and refere nces
+therein for the details)
+d(ℓ)
+APT(Q2) =ℓ/summationdisplay
+j=1dj/parenleftbigg4π
+β0/parenrightbiggj¯A(ℓ)
+SL,j(z),¯A(ℓ)
+SL,j(z) =∞/integraldisplay
+0̺(ℓ)
+j(σ)
+σ+zdσ, z=Q2
+Λ2,(2)
+r(ℓ)
+APT(s) =ℓ/summationdisplay
+j=1dj/parenleftbigg4π
+β0/parenrightbiggj¯A(ℓ)
+TL,j(w),¯A(ℓ)
+TL,j(w) =∞/integraldisplay
+w̺(ℓ)
+j(σ)dσ
+σ, w=s
+Λ2.(3)
+2For example, the experimental data on the so–called R(s)–ratio of electron–positron an-
+nihilation into hadrons can only be examined by making use of both pert urbation theory and
+relevant dispersion relation, see, e.g., Ref. [19].
+3Here and further Λ denotes the QCD scale parameter and ̺(ℓ)
+j(σ) stands for the
+spectral function corresponding to j-thpower of the ℓ–loopcouplant (see Sect. 3).
+Ingeneral, theeffects duetothemassofthelightest hadronstat ecanbesafely
+neglected at intermediate and high energies only. In particular, suc h effects play
+an essential role in theoretical description of the strong interact ion processes at
+low energies. As it has been noted in the Introduction, the effects d ue to the
+nonvanishing mass of the lightest hadron state have been account ed for within
+so–called “massive” Analytic Perturbation Theory (MAPT) [17,18]. T hus, in the
+framework of MAPT the theoretical expressions for the above–m entioned strong
+corrections read
+d(ℓ)
+MAPT(Q2,m2) =ℓ/summationdisplay
+j=1dj/parenleftbigg4π
+β0/parenrightbiggj
+A(ℓ)
+SL,j(z,χ), z=Q2
+Λ2, χ=m2
+Λ2,(4)
+A(ℓ)
+SL,j(z,χ) =z
+z+χ∞/integraldisplay
+χ̺(ℓ)
+j(σ)σ−χ
+σ+zdσ
+σ, (5)
+r(ℓ)
+MAPT(s,m2) =ℓ/summationdisplay
+j=1dj/parenleftbigg4π
+β0/parenrightbiggj
+A(ℓ)
+TL,j(w,χ), w=s
+Λ2, (6)
+A(ℓ)
+TL,j(w,χ) =θ(w−χ)∞/integraldisplay
+w̺(ℓ)
+j(σ)dσ
+σ, (7)
+whereθ(x) is the unit step function ( θ(x) = 1 ifx≥0 andθ(x) = 0 otherwise).
+It is worth noting that in the limit m→0 Eqs. (4) and (6) coincide with Eqs. (2)
+and (3), respectively: d(ℓ)
+MAPT(Q2,0) =d(ℓ)
+APT(Q2) andr(ℓ)
+MAPT(s,0) =r(ℓ)
+APT(s). Be-
+sides,d(ℓ)
+MAPT(Q2,m2)→d(ℓ)
+APT(Q2) forQ2≫m2andr(ℓ)
+MAPT(s,m2) =r(ℓ)
+APT(s) for
+s > m2(see Refs. [17,18] for the details). It is worthwhile to emphasize als o that
+expressions (2), (3) and (4)–(7) represent the nonpower func tional expansions3of
+the strong corrections d(Q2) andr(s).
+3 Calculation of the spectral functions
+In general, there is no unique way to restore the aforementioned s pectral func-
+tion̺(ℓ)
+j(σ) by making use of the perturbative expression for the strong run ning
+coupling α(ℓ)
+s(Q2) (discussion of this issue can be found in Refs. [7,16,20]). In
+what follows we adopt the definition proposed in Refs. [1–3]:
+̺(ℓ)
+j(σ) =1
+2πilim
+ε→0+/parenleftBigg/braceleftbigg
+a(ℓ)
+s/bracketleftBig
+−Λ2(σ+iε)/bracketrightBig/bracerightbiggj
+−/braceleftbigg
+a(ℓ)
+s/bracketleftBig
+−Λ2(σ−iε)/bracketrightBig/bracerightbiggj/parenrightBigg
+(8)
+3In other words, for j≥2¯A(ℓ)
+SL,j(z)/negationslash=/bracketleftBig
+¯A(ℓ)
+SL,1(z)/bracketrightBigj
+and¯A(ℓ)
+TL,j(w)/negationslash=/bracketleftBig
+¯A(ℓ)
+TL,1(w)/bracketrightBigj
+. Nonethe-
+less, in the ultraviolet asymptotic ( z→ ∞)¯A(ℓ)
+SL,j(z)→/bracketleftbig
+a(ℓ)
+s(Q2)/bracketrightbigj, i.e., the expansions (2)
+and (4) reproduce the perturbative power series (1).
+4Figure 1: The one–loop perturbative couplant (Eq. (32), dot–das hed curve) and
+the one–loop first–order expansion functions: APT (Eq. (10), da shed curves) and
+MAPT (Eqs. (11), (12), solid curves). The values of parameters: Λ = 350MeV,
+m= 270MeV.
+(σisadimensionlessvariable). Attheone–looplevel( ℓ= 1)thefirst–order( j= 1)
+spectral function (8) can easily be calculated (see Eq. (32)):
+̺(1)
+1(σ) =1
+2πilim
+ε→0+/parenleftbigg
+a(1)
+s/bracketleftBig
+−Λ2(σ+iε)/bracketrightBig
+−a(1)
+s/bracketleftBig
+−Λ2(σ−iε)/bracketrightBig/parenrightbigg
+=1
+2πilim
+ε→0+/bracketleftBigg1
+ln(−σ−iε)−1
+ln(−σ+iε)/bracketrightBigg
+=1
+y2+π2,(9)
+wherey= lnσ. Eventually, this leads to the following expressions4for the one–
+loop (ℓ= 1) first–order ( j= 1) expansion functions (2), (3), (5), and (7):
+¯A(1)
+SL,1(z) =1
+lnz+1
+1−z,¯A(1)
+TL,1(w) =1
+2−1
+πarctan/parenleftBigglnw
+π/parenrightBigg
+,(10)
+A(1)
+SL,1(z,χ) =1
+lnz+z
+1−z1+χ
+z+χ−z
+z+χχ/integraldisplay
+0̺(1)
+1(σ)
+σ+z/parenleftbigg
+1−χ
+σ/parenrightbigg
+dσ,(11)
+A(1)
+TL,1(w,χ) =θ(w−χ)/bracketleftBigg1
+2−1
+πarctan/parenleftBigglnw
+π/parenrightBigg/bracketrightBigg
+, (12)
+see papers [3,18] and references therein for the details. The fun ctions (10)–(12)
+are shown in Fig. 1.
+4It is assumed that arctan xis a continuously increasing function of its argument: −π/2≤
+arctanx≤π/2 for−∞< x <∞.
+5Figure 2: The four–loop spectral functions ̺(4)
+j(σ) (8):̺(4)
+1(σ) (solid curve),
+̺(4)
+2(σ) (dotted curve), 10 ·̺(4)
+3(σ) (dashed curve), and 102·̺(4)
+4(σ) (dot–dashed
+curve). The values of parameters: nf= 3,y= lnσ.
+At the higher loop levels ( ℓ≥2) perturbative couplants a(ℓ)
+s(Q2) have a cum-
+bersome structure (see App. A). Hence, the calculation of the ℓ–loop spectral
+function of j–th order ̺(ℓ)
+j(σ) (8) (1≤j≤ℓ) represents a rather complicated
+task. The numerical calculation of the spectral functions (8) req uires a lot of
+computational resources and essentially slows down the running of the program.
+Nonetheless, this problem can be resolved5in the following way.
+Firstofall, itisconvenient toexpressthespectralfunctions ̺(ℓ)
+j(σ)(8)interms
+of the real and imaginary parts of the ℓ–loop couplant a(ℓ)
+s(Q2) on a physical cut:
+lim
+ε→0+a(ℓ)
+s/bracketleftBig
+−Λ2(σ∓iε)/bracketrightBig
+≡A(ℓ)
+Re(σ)∓iπA(ℓ)
+Im(σ). (13)
+In this case Eq. (8) can be represented in a concise form:
+̺(ℓ)
+j(σ) =Kj/summationdisplay
+k=0/parenleftBiggj
+2k+1/parenrightBigg
+(−1)kπ2k/bracketleftBig
+A(ℓ)
+Im(σ)/bracketrightBig2k+1/bracketleftBig
+A(ℓ)
+Re(σ)/bracketrightBigj−2k−1,(14)
+whereKj=j/2+(jmod 2)/2−1, and
+/parenleftBiggn
+m/parenrightBigg
+=n!
+m!(n−m)!(15)
+5An alternative way to overcome this problem is to construct a set of explicit expressions
+which approximate the nonpower expansion functions (2), (3), (5 ), and (7) accurately enough,
+see, e.g., Ref. [21].
+6is the binomial coefficient. In particular, at any loop level
+̺(ℓ)
+1(σ) =A(ℓ)
+Im(σ), (16)
+̺(ℓ)
+2(σ) = 2A(ℓ)
+Im(σ)A(ℓ)
+Re(σ), (17)
+̺(ℓ)
+3(σ) =A(ℓ)
+Im(σ)/braceleftBigg
+3/bracketleftBig
+A(ℓ)
+Re(σ)/bracketrightBig2−π2/bracketleftBig
+A(ℓ)
+Im(σ)/bracketrightBig2/bracerightBigg
+, (18)
+̺(ℓ)
+4(σ) = 4A(ℓ)
+Im(σ)A(ℓ)
+Re(σ)/braceleftBigg/bracketleftBig
+A(ℓ)
+Re(σ)/bracketrightBig2−π2/bracketleftBig
+A(ℓ)
+Im(σ)/bracketrightBig2/bracerightBigg
+. (19)
+Then, one has to calculate (by hands) the real and imaginary parts (13) of
+the couplants a(ℓ)
+s(Q2) (32)–(35) on a physical cut. In turn, this will enable
+one to construct the spectral functions ̺(ℓ)
+j(σ) (14) corresponding to any integer
+power (j≥1) of the couplant a(ℓ)
+s(Q2) up to the four–loop level. The explicit
+expressions for the calculated functions A(ℓ)
+Im(σ) andA(ℓ)
+Re(σ) (1≤ℓ≤4) are given
+in App. B (see also App. C of Ref. [6] and App. C of Ref. [16]). The four –loop
+(ℓ= 4) spectral functions ̺(4)
+j(σ) (1≤j≤4) are shown in Fig. 2.
+4 Basic integrals
+The explicit expressions for the spectral functions ̺(ℓ)
+j(σ) (14) enable one to
+compute the corresponding nonpower expansion functions in spac elike (Eqs. (2)
+and (5)) and timelike (Eqs. (3) and (7)) domains. Specifically, in the f ramework
+of massless APT the spacelike expansion functions (2) read
+¯ASL(z) =∞/integraldisplay
+0̺(σ)
+σ+zdσ. (20)
+For the numerical evaluation of ¯ASL(z) it is convenient to split Eq. (20) into three
+terms6, namely
+¯ASL(z) =0/integraldisplay
+−1r(x)e1/x
+e1/x+zdx+1/integraldisplay
+−1̺y(y)
+1+ze−ydy+1/integraldisplay
+0r(x)
+1+ze−1/xdx, (21)
+where
+̺y(y) =̺(σ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+σ=ey, ̺ x(x) =̺y(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+y=1/x, r(x) =̺x(x)
+x2.(22)
+6This was first proposed by Dr. I.L.Solovtsov.
+7Figure 3: The four–loop “massless” APT expansion functions ¯A(4)
+SL,j(z) (2) and
+¯A(4)
+TL,j(w) (3):j= 1 (solid curves), j= 2 (dotted curves, scaled ×10),j= 3
+(dashed curves, scaled ×102), andj= 4 (dot–dashed curves, scaled ×103). The
+values of parameters: nf= 3, Λ = 350MeV, z=Q2/Λ2,w=s/Λ2.
+The numerical integration of Eq. (21) is implemented within QCDMAPT lib rary
+by the subroutine APTSL(see Sect. 5). In turn, the timelike massless APT ex-
+pansion functions (3) take the form
+¯ATL(w) =∞/integraldisplay
+w̺(σ)dσ
+σ. (23)
+Similarly to the previous case, it is worth representing this equation in the fol-
+lowing way:
+¯ATL(w) =
+
+J1/parenleftbigg
+−1,1
+lnw/parenrightbigg
++J2(−1,1)+J1(0,1),lnw <−1
+J2(lnw,1)+J1(0,1),−1≤lnw <1
+J1/parenleftbigg
+0,1
+lnw/parenrightbigg
+,1≤lnw(24)
+where
+J1(x1,x2) =x2/integraldisplay
+x1r(x)dx, J 2(y1,y2) =y2/integraldisplay
+y1̺y(y)dy. (25)
+The numerical integration of Eq. (24) is implemented within QCDMAPT lib rary
+by the subroutine APTTL(see Sect. 5). The four–loop massless APT expansion
+functions ¯A(4)
+SL,j(z) and¯A(4)
+TL,j(w) (1≤j≤4) are shown in Fig. 3.
+8Figure 4: The four–loop MAPT expansion functions A(4)
+SL,j(z,χ) (5) and
+A(4)
+TL,j(w,χ) (7):j= 1 (solid curves), j= 2 (dotted curves, scaled ×10),
+j= 3 (dashed curves, scaled ×102), andj= 4 (dot–dashed curves, scaled ×103).
+The values of parameters: nf= 3, Λ = 350MeV, m= 270MeV, z=Q2/Λ2,
+w=s/Λ2.
+In the framework of massive APT the spacelike expansion functions (5) read
+ASL(z,χ) =z
+z+χ∞/integraldisplay
+χ̺(σ)σ−χ
+σ+zdσ
+σ. (26)
+For the numerical evaluation of ASL(z,χ) it is convenient to represent Eq. (26) in
+the following way:
+ASL(z,χ) =
+
+z
+z+χ/bracketleftbigg
+I1/parenleftbigg
+−1,1
+lnχ/parenrightbigg
++I2(−1,1)+I3(0,1)/bracketrightbigg
+,lnχ <−1
+z
+z+χ/bracketleftBig
+I2(lnχ,1)+I3(0,1)/bracketrightBig
+,−1≤lnχ <1
+z
+z+χI3/parenleftbigg
+0,1
+lnχ/parenrightbigg
+,1≤lnχ(27)
+where
+I1(x1,x2) =x2/integraldisplay
+x1r(x)e1/x−χ
+e1/x+zdx, (28)
+I2(y1,y2) =y2/integraldisplay
+y1̺y(y)ey−χ
+ey+zdy, (29)
+9I3(x1,x2) =x2/integraldisplay
+x1r(x)1−χe−1/x
+1+ze−1/xdx. (30)
+The numerical integration of Eq. (27) is implemented within QCDMAPT lib rary
+by the subroutine MAPTSL(see Sect. 5). In turn, the timelike MAPT expansion
+functions (7) take the form
+ATL(w,χ) =θ(w−χ)∞/integraldisplay
+w̺(σ)dσ
+σ=θ(w−χ)¯ATL(w), (31)
+where¯ATL(w) is defined in Eq. (24). The numerical integration of Eq. (31) is
+implemented within QCDMAPT library by the subroutine MAPTTL(see Sect. 5).
+Thefour–loopMAPTexpansion functions A(4)
+SL,j(z,χ)andA(4)
+TL,j(w,χ)(1≤j≤4)
+are shown in Fig. 4.
+5 Description of the template program
+The provided template program ( template.mw ) is self–explaining. Besides, both
+template program and QCDMAPT library ( QCDMAPT.lib ) contain detailed com-
+ments.
+The template program is organized as follows (see template.pdf ). On the
+first page the QCDMAPT library is loaded and a sample set of input para meters
+is specified: nf= 3 active quarks, Q= 700MeV, s= (700MeV)2,m= 270MeV,
+and Λ = 350MeV. The evaluation of expansion functions for the given set of pa-
+rameters is presented on page 2. In particular, the first computa tion deals with
+the first–order ( j= 1)ℓ–loop(ℓ= 1,2,3,4) expansion functions, namely, a(ℓ)
+s(Q2)
+(Eqs. (32)–(35)), ¯A(ℓ)
+SL,1(z) (Eq. (2)), ¯A(ℓ)
+TL,1(w) (Eq. (3)), A(ℓ)
+SL,1(z,χ) (Eq. (5)), and
+A(ℓ)
+TL,1(w,χ)(Eq.(7)). Thesecondcomputationevaluatesfour–loop( ℓ= 4)expan-
+sion functions of j–thorder ( j= 1,2,3,4), namely, [ a(4)
+s(Q2)]j(Eq. (35)), ¯A(4)
+SL,j(z)
+(Eq. (2)), ¯A(4)
+TL,j(w)(Eq. (3)), A(4)
+SL,j(z,χ) (Eq. (5)), and A(4)
+TL,j(w,χ)(Eq. (7)). The
+data arrays of the first–order one–loop expansion functions are computed and rel-
+evant plots are presented on page 3 (see Fig. 1 and its caption for t he details).
+The data arrays of the massless APT four–loop ( ℓ= 4) expansion functions of
+j–th order ( j= 1,2,3,4) are computed and corresponding plots are presented on
+page 4 (see Fig. 3 and its caption for the details). The data arrays o f the MAPT
+four–loop ( ℓ= 4) expansion functions of j–th order ( j= 1,2,3,4) are computed
+and relevant plots are presented on page 5 (see Fig. 4 and its captio n for the
+details).
+The summary of QCDMAPT package commands is given below.
+10•PERTURBATIVE QCD
+AlphaPert ℓL(z)=α(ℓ)
+s(Q2) (ℓ= 1,2,3,4;z=Q2/Λ2):ℓ–loop perturbative
+strong running coupling, see App. A.
+APertℓL(z)=a(ℓ)
+s(Q2) =α(ℓ)
+s(Q2)β0/(4π) (ℓ= 1,2,3,4;z=Q2/Λ2):ℓ–loop
+perturbative QCD couplant, see Eqs. (32)–(35).
+betaj=βj(j= 0,1,2,3): perturbative ( j+1)–loop expansion coefficient of the
+QCDβfunction, see Eqs. (36)–(39).
+Bj=Bj=βj/βj+1
+0(j= 1,2,3): combination of QCD βfunction perturbative
+expansion coefficients, see App. A.
+•SPECTRAL FUNCTIONS
+AReℓLY(y)=A(ℓ)
+Re(σ)/vextendsingle/vextendsingle/vextendsingle
+σ=ey(ℓ= 1,2,3,4): real part of the ℓ–loop perturbative
+QCD couplant on a physical cut, see Eqs. (13), (22), and App. B.
+AImℓLY(y)=A(ℓ)
+Im(σ)/vextendsingle/vextendsingle/vextendsingle
+σ=ey(ℓ= 1,2,3,4): imaginary part of the ℓ–loop pertur-
+bative QCD couplant on a physical cut, see Eqs. (13), (22), and Ap p. B.
+RhoℓLjpY(y)=̺(ℓ)
+j(σ)/vextendsingle/vextendsingle/vextendsingle
+σ=ey(ℓ= 1,2,3,4;j= 1,...,ℓ): spectral function (in
+terms of y= lnσvariable) corresponding to j–th power of the ℓ–loop per-
+turbative QCD couplant, see Eqs. (8), (14), (22), and App. B.
+RℓLjpX(x)=r(ℓ)
+j(x) =x−2̺(ℓ)
+j(σ)/vextendsingle/vextendsingle/vextendsingle
+σ=e1/x(ℓ= 1,2,3,4;j= 1,...,ℓ): spectral
+function (in terms of x= 1/lnσvariable) corresponding to j–th power of
+theℓ–loopperturbative QCD couplant, see Eqs. (8), (14), (22), and A pp. B.
+•BASIC INTEGRALS
+APTSL(RhoY,RX,z) =¯ASL(z): numerical integrationforthemassless APTspace-
+like expansion functions, see Eqs. (20)–(22) with ̺y(y) =RhoY(y) and
+r(x) =RX(x).
+APTTL(RhoY,RX,w) =¯ATL(w): numerical integrationforthemassless APTtime-
+like expansion functions, see Eqs. (23)–(25), (22) with ̺y(y) =RhoY(y) and
+r(x) =RX(x).
+MAPTSL(RhoY,RX,z,chi) =ASL(z,χ): numericalintegrationfortheMAPTspace-
+like expansion functions, see Eqs. (26)–(30), (22) with ̺y(y) =RhoY(y) and
+r(x) =RX(x).
+11MAPTTL(RhoY,RX,w,chi) =ATL(w,χ): numerical integration for the MAPT
+timelike expansion functions, see Eqs. (31), (23)–(25), (22) with ̺y(y) =
+RhoY(y) andr(x) =RX(x).
+•APT EXPANSION FUNCTIONS
+ASLℓLjp(z)=¯A(ℓ)
+SL,j(z) (ℓ= 1,2,3,4;j= 1,...,ℓ): APT spacelike expansion
+function ( ℓ–loop level, j–th order), see Eq. (2).
+ATLℓLjp(w)=¯A(ℓ)
+TL,j(w) (ℓ= 1,2,3,4;j= 1,...,ℓ): APT timelike expansion
+function ( ℓ–loop level, j–th order), see Eq. (3).
+AlphaSL ℓL(z)=¯A(ℓ)
+SL,1(z)4π/β0(ℓ= 1,2,3,4): APT spacelike ℓ–loop effective
+coupling, see Eq. (2).
+AlphaTL ℓL(w)=¯A(ℓ)
+TL,1(w)4π/β0(ℓ= 1,2,3,4): APT timelike ℓ–loop effective
+coupling, see Eq. (3).
+•MAPT EXPANSION FUNCTIONS
+ASLmℓLjp(z,chi) =A(ℓ)
+SL,j(z,χ) (ℓ= 1,2,3,4;j= 1,...,ℓ): MAPT spacelike
+expansion function ( ℓ–loop level, j–th order), see Eq. (5).
+ATLmℓLjp(w,chi) =A(ℓ)
+TL,j(w,χ) (ℓ= 1,2,3,4;j= 1,...,ℓ): MAPT timelike
+expansion function ( ℓ–loop level, j–th order), see Eq. (7).
+AlphaSLm ℓL(z,chi) =A(ℓ)
+SL,1(z,χ)4π/β0(ℓ= 1,2,3,4): MAPTspacelike ℓ–loop
+effective coupling, see Eq. (5).
+AlphaTLm ℓL(w,chi) =A(ℓ)
+TL,1(w,χ)4π/β0(ℓ= 1,2,3,4): MAPTtimelike ℓ–loop
+effective coupling, see Eq. (7).
+6 Conclusions
+A program package, which facilitates computations in the framewor k of the An-
+alytic approach to QCD, is developed. It includes the explicit express ions for
+relevant spectral functions calculated up to the four–loop level a nd the subrou-
+tines for necessary integrals.
+This work was partially supported by the grants RFBR-08-01-0068 6, BRFBR-
+JINR-F08D-001, NS-1027.2008.2, and JINR grant.
+12A Perturbative QCD running coupling
+This Section contains explicit expressions for the perturbative str ong running
+coupling α(ℓ)
+s(Q2)≡4πa(ℓ)
+s(Q2)/β0up to the four–loop level. Specifically,
+a(1)
+s(Q2) =1
+lnz, z=Q2
+Λ2, (32)
+a(2)
+s(Q2) =1
+lnz−B1ln(lnz)
+ln2z, (33)
+a(3)
+s(Q2) =1
+lnz−B1ln(lnz)
+ln2z
++1
+ln3z/braceleftbigg
+B2
+1/bracketleftBig
+ln2(lnz)−ln(lnz)−1/bracketrightBig
++B2/bracerightbigg
+, (34)
+a(4)
+s(Q2) =1
+lnz−B1ln(lnz)
+ln2z
++1
+ln3z/braceleftbigg
+B2
+1/bracketleftBig
+ln2(lnz)−ln(lnz)−1/bracketrightBig
++B2/bracerightbigg
++1
+ln4z/braceleftBigg
+B3
+1/bracketleftbigg
+−ln3(lnz)+5
+2ln2(lnz)
++2ln(lnz)−1
+2/bracketrightbigg
+−3B1B2ln(lnz)+1
+2B3/bracerightBigg
+, (35)
+whereBj=βj/βj+1
+0isthecombinationofQCD βfunctionperturbativeexpansion
+coefficients:
+β0= 11−2
+3nf, (36)
+β1= 102−38
+3nf, (37)
+β2=2857
+2−5033
+18nf+325
+54n2
+f, (38)
+β3=149753
+6+3564ζ(3)−/bracketleftbigg1078361
+162+6508
+27ζ(3)/bracketrightbigg
+nf
++/bracketleftbigg50065
+162+6472
+81ζ(3)/bracketrightbigg
+n2
+f+1093
+729n3
+f. (39)
+In these equations nfstands for the number of active quarks and ζ(x) denotes the
+Riemann ζfunction, ζ(3)≃1.202. Theone–andtwo–loopcoefficients ( β0andβ1)
+are scheme–independent, whereas the expressions given for β2andβ3are calcu-
+lated in the MS subtraction scheme (see papers [22–25] and references ther ein for
+13the details). Notethatexpressions (33)–(35)correspond toap proximate solutions
+of the perturbative renormalization group (RG) equation for the Q CD invariant
+charge, see, e.g., Ref. [26], Sect. 2 of review [15], and App. A of revie w [16].
+Nonetheless, the difference between the approximate running cou plingα(ℓ)
+s(Q2)
+(33)–(35) and the exact solution of perturbative RG equation α(ℓ)
+ex(Q2) is not
+controllable at every considered loop level.
+B Spectral functions
+As it has been mentioned in Sect. 3, the spectral functions ̺(ℓ)
+j(σ) (8) can be
+represented in the following form:
+̺(ℓ)
+j(σ) =Kj/summationdisplay
+k=0/parenleftBiggj
+2k+1/parenrightBigg
+(−1)kπ2k/bracketleftBig
+A(ℓ)
+Im(σ)/bracketrightBig2k+1/bracketleftBig
+A(ℓ)
+Re(σ)/bracketrightBigj−2k−1,(40)
+where
+lim
+ε→0+a(ℓ)
+s/bracketleftBig
+−Λ2(σ∓iε)/bracketrightBig
+≡A(ℓ)
+Re(σ)∓iπA(ℓ)
+Im(σ), (41)
+ℓ–loop perturbative couplants a(ℓ)
+s(Q2) are given by Eqs. (32)–(35), and Kj=
+j/2+ (jmod 2)/2−1. The functions A(ℓ)
+Im(σ) andA(ℓ)
+Re(σ), calculated up to the
+four–loop level (1 ≤ℓ≤4), are listed below.
+One–loop level:
+A(1)
+Im(σ) =1
+y2+π2, A(1)
+Re(σ) =y
+y2+π2, y = lnσ.(42)
+Two–loop level:
+A(2)
+Im(σ) =1
+(y2+π2)2/bracketleftBig
+2yS(2)
+1(y)+(π2−y2)S(2)
+2(y)/bracketrightBig
+, (43)
+A(2)
+Re(σ) =1
+(y2+π2)2/bracketleftBig
+(y2−π2)S(2)
+1(y)+2π2yS(2)
+2(y)/bracketrightBig
+, (44)
+where
+S(2)
+1(y) =y−B1
+2ln(y2+π2), (45)
+S(2)
+2(y) = 1−B1/bracketleftbigg1
+2−1
+πarctan/parenleftbiggy
+π/parenrightbigg/bracketrightbigg
+, (46)
+andBj=βj/βj+1
+0is the combination of QCD βfunction perturbative expansion
+coefficients (see App. A).
+14Three–loop level:
+A(3)
+Im(σ) =1
+(y2+π2)3/bracketleftBig
+(3y2−π2)S(3)
+3(y)−y(y2−3π2)S(3)
+4(y)/bracketrightBig
+, (47)
+A(3)
+Re(σ) =1
+(y2+π2)3/bracketleftBig
+y(y2−3π2)S(3)
+3(y)+π2(3y2−π2)S(3)
+4(y)/bracketrightBig
+,(48)
+where
+S(3)
+1(y) =1
+2ln(y2+π2), (49)
+S(3)
+2(y) =1
+2−1
+πarctan/parenleftbiggy
+π/parenrightbigg
+, (50)
+S(3)
+3(y) =y2−π2−B1/bracketleftBig
+yS(3)
+1(y)−π2S(3)
+2(y)/bracketrightBig
++B2
+1/braceleftbigg
+S(3)
+1(y)/bracketleftBig
+S(3)
+1(y)−1/bracketrightBig
+−π2/bracketleftBig
+S(3)
+2(y)/bracketrightBig2−1/bracerightbigg
++B2, (51)
+S(3)
+4(y) = 2y−B1/bracketleftBig
+S(3)
+1(y)+yS(3)
+2(y)/bracketrightBig
++B2
+1S(3)
+2(y)/bracketleftBig
+2S(3)
+1(y)−1/bracketrightBig
+.(52)
+Four–loop level:
+A(4)
+Im(σ) =1
+(y2+π2)4/braceleftbigg
+4y(y2−π2)/bracketleftBig
+S(4)
+3(y)+S(4)
+5(y)+S(4)
+7(y)/bracketrightBig
++/bracketleftBig
+4π2y2−(y2−π2)2/bracketrightBig/bracketleftBig
+S(4)
+4(y)+S(4)
+6(y)+S(4)
+8(y)/bracketrightBig/bracerightbigg
+, (53)
+A(4)
+Re(σ) =1
+(y2+π2)4/braceleftbigg/bracketleftBig
+(y2−π2)2−4π2y2/bracketrightBig/bracketleftBig
+S(4)
+3(y)+S(4)
+5(y)+S(4)
+7(y)/bracketrightBig
++4π2y(y2−π2)/bracketleftBig
+S(4)
+4(y)+S(4)
+6(y)+S(4)
+8(y)/bracketrightBig/bracerightbigg
+, (54)
+where
+S(4)
+1(y) =1
+2ln(y2+π2), (55)
+S(4)
+2(y) =1
+2−1
+πarctan/parenleftbiggy
+π/parenrightbigg
+, (56)
+S(4)
+3(y) =B2
+1y/braceleftbigg
+S(4)
+1(y)/bracketleftBig
+S(4)
+1(y)−1/bracketrightBig
+−π2/bracketleftBig
+S(4)
+2(y)/bracketrightBig2+B2
+B2
+1−1/bracerightbigg
+−B2
+1π2S(4)
+2(y)/bracketleftBig
+2S(4)
+1(y)−1/bracketrightBig
+, (57)
+S(4)
+4(y) =B2
+1/braceleftbigg
+S(4)
+1(y)/bracketleftBig
+S(4)
+1(y)−1/bracketrightBig
+−π2/bracketleftBig
+S(4)
+2(y)/bracketrightBig2
++yS(4)
+2(y)/bracketleftBig
+2S(4)
+1(y)−1/bracketrightBig
++B2
+B2
+1−1/bracerightbigg
+, (58)
+15S(4)
+5(y) =B3
+1/braceleftBigg
+S(4)
+1(y)/bracketleftbigg
+3π2/parenleftBig
+S(4)
+2(y)/parenrightBig2−/parenleftBig
+S(4)
+1(y)/parenrightBig2/bracketrightbigg
++5
+2/bracketleftbigg/parenleftBig
+S(4)
+1(y)/parenrightBig2−π2/parenleftBig
+S(4)
+2(y)/parenrightBig2/bracketrightbigg
++S(4)
+1(y)/parenleftBigg
+2−3B2
+B2
+1/parenrightBigg
++1
+2/parenleftBiggB3
+B3
+1−1/parenrightBigg/bracerightBigg
+, (59)
+S(4)
+6(y) =B3
+1S(4)
+2(y)/braceleftbigg
+π2/bracketleftBig
+S(4)
+2(y)/bracketrightBig2−3/bracketleftBig
+S(4)
+1(y)/bracketrightBig2
++5S(4)
+1(y)−3B2
+B2
+1+2/bracerightbigg
+, (60)
+S(4)
+7(y) =y(y2−3π2)+B1/bracketleftBig
+2π2yS(4)
+2(y)−(y2−π2)S(4)
+1(y)/bracketrightBig
+, (61)
+S(4)
+8(y) = 3y2−π2−B1/bracketleftBig
+(y2−π2)S(4)
+2(y)+2yS(4)
+1(y)/bracketrightBig
+. (62)
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+18
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+arXiv:1001.0902v1  [math.AT]  6 Jan 2010HOMOLOGICAL DIMENSIONS OF RING SPECTRA
+MARK HOVEY AND KEIR LOCKRIDGE
+Abstract. We define homological dimensions for S-algebras, the generalized
+rings that arise in algebraic topology. We compute the homol ogical dimensions
+of a number of examples, and establish some basic properties . The most
+difficult computation is the global dimension of real K-theory KOand its
+connective version koat the prime 2. We show that the global dimension of
+KOis 1, 2, or 3, and the global dimension of kois 4 or 5.
+Introduction
+The authors have been engaged in trying to develop homological dime nsions for
+the ring objects that arise in algebraic topology. These are cohomo logy theories
+with some kind of cup product that is associative up to infinitely coher ent homo-
+topy. These are commonly called S-algebras, among other names, and a standard
+reference is [EKMM97].
+SuchS-algebras are analogous to rings, but they have no elements, and t hey do
+not have abelian module categories. What they do have is a triangulat ed derived
+category that generalizes the derived category of an ordinary as sociative ring R.
+Indeed, in this case, there is an Eilenberg-MacLane spectrum S-algebra HR, and
+D(HR) is naturally equivalent to D(R) as a triangulated category.
+Thus, to develop the ring theory of S-algebras E, we need to work with their
+derived categories D(E). In previous papers [HL09c, HL09b], the authors have
+developed definitions of right global dimension r .gl.dim.Eand ghost dimension
+gh.dim.EforS-algebras Eand proved that these generalize the usual notions of
+right global dimension and weak dimension for rings. We have also stud iedS-
+algebras for which these dimensions are 0.
+The object of this paper is to study the homologicaldimensions ofth eS-algebras
+that arise in nature. After an initial section in which we recall the bas ics of our
+theory of global dimension and ghost dimension of S-algebras, we discuss examples
+inSection2. Thesphere, MU,andBPallhaveinfiniteglobalandghostdimensions,
+whereasEnhas ghostand globaldimension n. For acommutative S-algebraEsuch
+thatE∗is Noetherian and has finite global dimension, such as EnorK, we have
+gl.dim.E= gh.dim.E= gl.dim.E∗.
+Thisfailswhengl .dim.E∗=∞,however. Weshowthat,attheprime2,gl .dim.KO
+is either 1, 2, or 3, and gl .dim.kois either 4 or 5, whereas both rings KO∗and
+ko∗have infinite global dimension. We also show that gl .dim.tmfis finite at the
+prime 3.
+Date: June 7, 2018.
+1991Mathematics Subject Classification. 55P43, 16E10, 18E30.
+12 MARK HOVEY AND KEIR LOCKRIDGE
+In Section 3 we prove some basic properties of global and ghost dime nsion. The
+most important point is that these dimensions are not Morita invarian t. In fact,
+it is already true that the global and weak dimensions of ordinary ring s are not
+invariant under derived equivalences, and we give an example commun icated to us
+by Lidia Angeleri H¨ ugel. We have been unable to generalize many of th e basic
+properties of the global dimension of Noetherian rings, however. T his might be
+because we have no intrinsic definition of Ebeing a Noetherian S-algebra; we
+just assume E∗is right Noetherian. So, for example, we do not know whether
+r.gl.dim.E= gh.dim.EwhenE∗is right Noetherian. We end the paper with a
+brief section on S-algebras Ewith global dimension 1. We had originally thought
+this would mean E∗would have to be 1-Gorenstein, in analogy with the fact that
+gl.dim.E= 0 implies E∗is quasi-Frobenius, and in fact this is true, but only with
+additional assumptions on E∗.
+The authors would like to thank Lidia Angeleri H¨ ugel for the example mentioned
+above, and Ben Wieland for pushing us to determine the global dimens ion ofKO.
+1.Homological dimensions
+The object of this section is to define our various homological dimens ions of ring
+spectra, and to prove basic relations between them. Recall that Eis anS-algebra,
+andD(E) denotes the derived category of right E-modules. This is a compactly
+generated triangulated category, and when Eis the Eilenberg-MacLane spectrum
+HRof an ordinary ring R, thenD(E) is equivalent to the usual unbounded derived
+category of R.
+We begin by defining projective, injective, and flat objects of D(E).
+Definition 1.1. An object X∈ D(E) is said to be projective (resp.injective ,
+resp.flat) ifX∗is a projective (resp. injective, resp. flat) E∗-module. A map
+f:X− →YinD(E) is said to be ghostiff∗= 0, and fis said to be phantom if
+D(E)(A,f) = 0 for all compact A∈ D(E).
+The basic properties of these objects and maps are summed up in th e following
+proposition.
+Proposition 1.2. Suppose Eis anS-algebra.
+(1)IfMis a projective or injective E∗-module, then there is an X∈ D(E)
+(necessarily projective or injective )withX∗∼=M.
+(2)IfMis a flat E∗-module, and D(E)is a Brown category (see[HPS97,
+Section 4.1]) , for example if E∗is countable, then there is an X∈ D(E)
+(necessarily flat )such that X∗∼=M.
+(3)Xis a projective object of D(E)if and only if the natural map
+D(E)(X,Y)− →HomE∗(X∗,Y∗)
+is an isomorphism for all Y∈ D(E). This is true if and only if every ghost
+with domain Xis null.
+(4)Xis an injective object of D(E)if and only if the natural map
+D(E)(Y,X)− →HomE∗(Y∗,X∗)
+is an isomorphism for all Y∈ D(E). This is true if and only if every ghost
+with codomain Xis null.HOMOLOGICAL DIMENSIONS OF RING SPECTRA 3
+(5)Xis a flat object of D(E)if and only the natural map
+X∗⊗E∗Y∗− →π∗(X∧EY)
+is an isomorphism for all left E-modules Y. This is true if and only if every
+ghost with domain Xis phantom.
+Proof.Thefirstpartiswell-known. Infact, it isprovedin[BKS04, Propositio nA.4]
+that every E∗-module of projective or injective dimension ≤2 is realizable. We
+prove part (3) next. If Xis projective, then the universal coefficient spectral se-
+quence of [EKMM97, Theorem IV.4.1] implies that
+D(E)(X,Y)∼=HomE∗(X∗,Y∗).
+This in turn implies that there are no nontrivial ghosts with domain X. Now, if
+there are no nontrivial ghosts with domain X, construct a projective module P∗
+and an epimorphism P∗− →X∗. This is then realizable, as we have just seen, by
+a mapP− →X, whose cofiber X− →Yis a ghost. This map is thus null, so Xis
+a retract of P, and hence projective. The proof of part (4) is similar. Part (5) is
+proved in [HL09b].
+Turning to part (2), suppose Mis a flat E∗-module. Then M∗is a directed
+colimit of finitely generated projective E∗-modules Qi. We can then realize this
+system by compact projective E-modules Pi, withπ∗Pi∼=Qi, by part (3). If
+D(E) is a Brown category, we can then take the minimal weak colimit [HPS9 7,
+Section 4.2] of the diagram of the Pito obtain an XwithX∗∼=colimQi∼=M./square
+Once we have projective, injective, and flat objects, we should th en be able
+to define the projective dimension of an arbitrary object. This is a lit tle more
+complicated than in the abelian setting, but has in fact already been w orked out
+by Christensen in [Chr98].
+Definition 1.3. LetEbe anS-algebra, and Xan object of D(E). We de-
+fine the projective dimension (resp.constructible flat dimension ) ofX,
+written proj .dim.X(resp. con .flat dim.X), inductively as follows. We have
+proj.dim.X= 0 (resp. con .flat dim.X= 0) if and only if Xis projective (resp.
+flat). Then we define proj .dim.X≤n+1 (resp. con .flat dim.X≤n+1) if and
+only if there is an exact triangle
+Y− →P− →/tildewideX− →ΣY
+wherePis projective (resp. flat), proj .dim.Y≤n(resp. con .flat dim.X≤n),
+andXis a retract of /tildewideX. We define the flat dimension ofX, written flat dim .X,
+to be the smallest integer nfor which any composite of n+1 ghosts with domain X
+is phantom, or ∞if there is no such n. The reader may well wonder why we have
+twonotionsofflat dimension. The answeris that we havebeen unable to provethey
+are equivalent, and the constructible flat dimension is the more obvio us one, but
+the flat dimension is the more useful one. See Proposition 1.4 below an d [HL09b]
+for details.
+We can define injective dimension similarly to how we defined projective dimen-
+sion, but it would be more usual to define the injective dimension ofX, written
+inj.dim.X, inductively as follows. We define inj .dim.X= 0 if and only if Xis
+injective, and inj .dim.X≤n+1 if and only if there is an exact triangle
+Σ−1Y− →/tildewideX− →I− →Y4 MARK HOVEY AND KEIR LOCKRIDGE
+whereIis injective, inj .dim.Y≤n, andXis a retract of /tildewideX.
+The major difference between this definition and the definition of the analogous
+dimensions in abelian categories is the fact that Xitself need not appear in an
+exact triangle with things of smaller dimension, but must only be a retr act of such
+a thing. Without this condition, Proposition 1.4 below would be false.
+Proposition 1.4. LetEbe anS-algebra, and let Xbe an object of D(E).
+(1) proj.dim.X≤nif and only if every composite of n+1ghostsfn+1◦fn◦
+···◦f1is null, where the domain of f1isX. This is true if and only if in
+the universal coefficient spectral sequence
+Es,t
+2= Exts,t
+E∗(X∗,Y∗)⇒ D(E)(X,Y)t−s
+we have Es,∗
+∞= 0for alls > nand all objects YofD(E).
+(2) inj.dim.X≤nif and only if every composite of n+1ghostsfn+1◦fn◦···◦f1
+is null, where the codomain of fn+1isX. This is true if and only if in the
+universal coefficient spectral sequence
+Es,t
+2= Exts,t
+E∗(Y∗,X∗)⇒ D(E)(Y,X)t−s
+we have Es,∗
+∞= 0for alls > nand all objects YofD(E).
+(3) flat dim .X≤con.flat dim.X, andflat dim.X≤nif and only if the
+following equivalent properties hold :
+(a)In the universal coefficient spectral sequence
+E2
+s,t= TorE∗
+s,t(X∗,Y∗)⇒πt−s(X∧EY)
+we have E∞
+s,∗= 0for alls > nand all objects YofD(Eop).
+(b)There is an exact triangle
+A− →Xg− →W− →ΣA
+in which proj.dim.A≤nandgis phantom.
+(c)Every map F− →Xfrom a compact E-module Ffactors through a
+compact Bwithproj.dim.B≤n.
+Proof.For part (1), the first part is contained in [Chr98, Theorem 3.5]. The second
+partfollowsfromtheconstructionandnaturalityoftheuniversal coefficientspectral
+sequence [EKMM97, Section IV.5]. Indeed, suppose Es,∗
+∞= 0 for all s > nand all
+Y, and suppose
+Xf1− →X1f2− → ···fn+1− −− →Xn+1
+is a composite of n+ 1 ghosts. Each of the maps fi:Xi− →Xi+1has positive
+filtration. Therefore, their composition will have filtration ≥n+ 1 by Proposi-
+tion IV.4.4 of [EKMM97], and therefore will be null. Conversely, suppos eXhas
+proj.dim.X≤n. In the construction of the universal coefficient spectral seque nce,
+we construct exact triangles
+Σ−1X1− →P0g0− →Xh1− →X1
+Σ−1X2− →P1g1− →X1h2− →X2
+···
+Σ−1Xj+1− →Pjgj− →Xjhj+1− −− →Xj+1HOMOLOGICAL DIMENSIONS OF RING SPECTRA 5
+wherePjis projective and π∗gjis onto for all j, and sohj+1is a ghost for all j. A
+map has filtration n+1 if and only if it factors through the composite
+Xh1− →X1h2− → ···hn+1− −− →Xn+1.
+Thuseverymapoffiltration n+1isacompositeof n+1ghosts. Thus,proj .dim.X≤
+nimplies every map of filtration n+1 is null.
+Part (2) is then completely dual. Part (3) is proved in [HL09b]. /square
+The following fact is also useful.
+Proposition 1.5. LetEbe anS-algebra, and let Xbe an object of D(E). Then
+proj.dim.X≤proj.dim.E∗X∗,inj.dim.X≤inj.dim.E∗X∗,
+and
+flat dim.X≤con.flat dim.X≤flat dim.E∗X∗.
+The first two statements of this proposition are obvious given Prop osition 1.4,
+and the last statement is proved in [HL09b].
+We can now define our homological dimensions.
+Definition 1.6. Suppose EisanS-algebra. Define the (right) global dimension
+ofE, r.gl.dim.E, by
+r.gl.dim.E= sup
+Xproj.dim.X= sup
+Xinj.dim.X.
+These two numbers are equal because they are both equal to the longest nontrivial
+composition of ghosts in D(E) (or∞if there are arbitrarily long nontrivial com-
+positions of ghosts). In case Eis a commutative S-algebra, we just refer to the
+global dimension , gl.dim.E. Define the ghost dimension ofE, gh.dim.E, by
+gh.dim.E= sup
+Xcompactproj.dim.X= sup
+Xflat dim.X.
+It is proven in [HL09b] that the two definitions of ghost dimension give n above
+coincide.
+We then have the following theorem that sums up the basic propertie s of these
+definitions.
+Theorem 1.7. Suppose Eis anS-algebra.
+(1) gh.dim.E≤r.gl.dim.E.
+(2) r.gl.dim.E≤r.gl.dim.E∗, with equality if E=HRfor an ordinary ring
+R.
+(3) gh.dim.E≤w.dim.E∗, with equality if E=HRfor an ordinary ring R.
+(4) gh.dim.E= gh.dim.Eop, whereEopisEwith the opposite multiplication.
+The first part of this theorem is obvious. Part (2) is proved in [HL09a ], though it
+is originally in the second author’s thesis. Parts (3) and (4) are prov ed in [HL09b].
+S-algebras of global dimension 0 are called semisimple . They are studied
+in [HL09c]. In particular, there are semisimple ring spectra with r .gl.dim.E∗=∞.
+Similarly, S-algebras of ghost dimension 0 are called von Neumann regular , and
+are also studied in [HL09c].6 MARK HOVEY AND KEIR LOCKRIDGE
+2.Examples
+We would, of course, like to compute gh .dim.Eand r.gl.dim.Efor various S-
+algebras E. Some of this was done in [HL09c], where the authors classified the
+semisimple S-algebras Eif either E∗is commutative or local. Those spectra are
+rather unusual, however. We address some of the more common S-algebras Ein
+this section, concentrating on the case when E∗is Noetherian.
+We begin with the sphere spectrum, and the following unsurprising re sult.
+Proposition 2.1. LetSbe the sphere S-algebra. Then gh.dim.S= gl.dim.S=
+∞.
+Proof.This is due to Christensen [Chr98], who provides bounds on proj .dim.RPn
+(which he calls the length). In particular, a lower bound for proj .dim.RPkis
+given by the longest nonzero chain of Steenrod operations in its hom ology, since
+Steenrod operations are obviously ghosts. And this longest chain is easily seen to
+grow without bound as kgrows. /square
+Corollary 2.2. LetS(p)be thep-local sphere S-algebra, where pis an integer prime.
+Thengh.dim.S(p)= gl.dim.S(p)=∞.
+Proof.Use reduced power operations in the cohomology of skeleta of BZ/pto
+replace Steenrod operations in RPk. We do not need to know the exact length of
+a nontrivial composition of these operations, we just need to know that this length
+grows without bound as we take larger skeleta. /square
+Recall that the length of the longest regular sequence in a ring is oft en called
+thedepth.
+Theorem 2.3. IfEis a commutative S-algebra, then
+depthE∗≤gh.dim.E≤min{w.dim.E∗,r.gl.dim.E} ≤r.gl.dim.E∗.
+Proof.In view of Theorem 1.7, it suffices to prove the first inequality. Let
+x1,x2,...,x n
+be a regular sequence in E∗. The main algebraic input we need is the computation
+that
+Exti
+E∗(E∗/(x1,...,x n),E∗) = 0 ifi/ne}ationslash=n
+and is nonzero if i=n. One can prove this by induction on n(or by the Koszul
+resolution), using the exact sequences
+0− →E∗/(x1,...,x k−1)xk−→E∗/(x1,...,x k−1)− →E∗/(x1,...,x k)− →0,
+where we have ignored suspensions for simplicity.
+We also need the fact that there is a E-module E/(x1,...,x n) realizing the E∗-
+moduleE∗/(x1,...,x n). One can alsoconstruct these by induction, using the exact
+triangles
+E/(x1,...,x i−1)xi− →E/(x1,...,x i−1)− →E/(x1,...,x i)− →E/(x1,...,x i−1),
+ignoring suspensions again.
+The universal coefficient spectral sequence
+Exts,t
+E∗(E∗/(x1,...,x n),E∗)⇒ D(E)(E/(x1,...,x n),E)
+then has only one non-vanishing line, where s=n. It therefore collapses, and so
+there is an element in E∞of filtration n. Thus gh .dim.E≥n, as required. /squareHOMOLOGICAL DIMENSIONS OF RING SPECTRA 7
+It is tempting to believe that Theorem 2.3 works in the noncommutativ e case
+as well, as long as we take regular sequences in the center Z(E∗). The algebraic
+calculation works fine, but we do not seem to be able to construct th e necessary
+mapsx:M− →Mfor anE-module Mand anx∈Z(E∗). We can construct such
+maps for E-bimodule maps x:E− →E, but an element in the center of E∗need
+not give such a map, so far as we know.
+Corollary 2.4. We have
+gh.dim.MU= gh.dim.BP=∞,
+whilegh.dim.En= gl.dim.En=nandgh.dim.K= gl.dim.K= 1.
+HereEndenotes Morava E-theory, with
+En∗∼=WFpn[[u1,...,u n−1]][u,u−1].
+This is known to be a commutative S-algebra by [GH04]. The coefficient ring has
+global dimension n, andp,u1,...,u n−1is a regular sequence, so Theorem 2.3 gives
+the desired result.
+This corollary suggests the following more general theorem.
+Theorem 2.5. Suppose Eis a commutative S-algebra such that E∗is Noetherian
+withgl.dim.π∗E <∞. Thengh.dim.E= gl.dim.E= gl.dim.E∗.
+Proof.Thepoint is that depth R= gl.dim.RwhenRisNoetherian commutativeof
+finite global dimension, so the result follows from Theorem 2.3. This alg ebraic fact
+is a corollary of Serre’s characterization of regular local rings, but we cite [BH84,
+Section 2] because it contains an interesting non-commutative gen eralization as
+well. /square
+We suspect that Theorem 2.5 may be true even if Eis not a commutative S-
+algebra. MuchlessisknownaboutnoncommutativeNoetherianrings offiniteglobal
+dimension, however.
+We now address the global dimension of real K-theory. Unfortunately, we have
+not been able to determine the exact value of gl .dim.KO, but we do at least bound
+it.
+Theorem 2.6. LetKOdenote2-local periodic real K-theory and kodenote2-local
+connective real K-theory, both of which are commutative S-algebras. Then
+1≤gl.dim.KO≤3
+and
+4≤gl.dim.ko≤5.
+Proof.ThisdependsontheresultsofBousfield[Bou90]andWolbert[Wol98 ]. Bous-
+field wrote his paper using naive KO-module spectra, as opposed to objects in
+D(KO). However, Wolbert explains why Bousfield’s results hold in D(KO) as well,
+and proves analogous results for D(ko). We note that there is an error in Wolbert’s
+paper [Sag08], but this error is about the realizability of ko∗andKO∗-modules and
+does not affect this theorem.
+Bousfield constructs an abelian category of CRT-modules, which ar e modules
+over the 3-object additive category consisting of {KO∗,K∗,KSC ∗}and the various
+standard maps between them. Here KSC=KO∧C(η2) is self-conjugate K-
+theory. There is a functor from KO-module spectra to this category that takes X8 MARK HOVEY AND KEIR LOCKRIDGE
+toπCRT
+∗(X), which is the set {π∗X,π∗(C(η)∧X),π∗(C(η2)∧X)}together with
+the maps between them. He then proves that πCRT
+∗(X) has projective dimension
+≤1 for every KO-module spectrum X. TheKO-modules Psuch that πCRT
+∗is
+projective as a CRT-module are coproducts of suspensions of KO,K, andKSC,
+and every projective CRT-module arises this way. Furthermore, m aps between
+projective CRT-modules are realizable as maps of the correspondin gKO-modules.
+This means that for every KO-module spectrum X, there is a cofiber sequence
+Q− →P− →X− →ΣQ
+inD(KO) in which QandPare coproducts of suspensions of copies of KO,K,
+andKSC. SinceK=KO∧C(η) andKSC=KO∧C(η2), we see that any 2-fold
+composite of ghosts out of PorQis trivial. A simple argument then shows that
+any 4-fold composite of ghosts out of Xis trivial, so gl .dim.KO≤3.
+The argument for kois a little more complicated. Wolbert [Wol98] describes the
+connectiveversionofcrt-modules, and showsthat if Xisako-module, then πcrt
+∗(X)
+has projective dimension at most 2 in the category of crt-modules. Wolbert then
+gives a version of the universal coefficient spectral sequence
+Es,t
+2= Exts,t
+crt(πcrt
+∗(X),πcrt
+∗(Y))⇒ D(ko)(X,Y)t−s.
+As before, any 2-fold composite of ghosts has filtration 1 in this spe ctral sequence
+(sincekandkscare 2-cell complexes in D(ko)). Therefore, any 6-fold composite of
+ghosts will have filtration 3, but the spectral sequence is trivial ab ove filtration 2.
+Thus every 6-fold composite of ghosts is trivial, so gl .dim.ko≤5.
+To see that gh .dim.ko≥4, we use the fact that ko∧A(1) =HF2, whereA(1)
+is the usual 8-cell complex with whose cohomology is A< x > / (Sq2nx|n≥2).
+Using this, we can compute D(ko)(HF2,HF2). It is the subring of the Steenrod
+algebra generated by Sq1and Sq2. Since every element of the Steenrod algebra is a
+ghost, the nontrivial element Sq2Sq1Sq2Sq1is a nontrivial composite of 4 ghosts,
+and is a self-map of the compact object HF2inD(ko). Thus gh .dim.ko≥4./square
+The proof of this theorem is of course dependent on knowing an awf ul lot about
+KOandko. We have much less information about higher analogues of KO, such as
+the spectrum tmfof topological modular forms [Hop02]. However, it is often the
+case with such spectra Ethat there is a finite type 0 spectrum Xsuch that E∧Xis
+a Noetherian S-algebra of finite global dimension. For example, KO∧C(η) =KU,
+and at the prime 3, there is a 3-cell complex Tsuch that tmf∧Tis a wedge of two
+copies of BP <2>[Beh06, Lemma 2, after Corollary 2.4.6]. Presumably a larger
+such finite complex also exists at the prime 2 for tmf, though such a result has not
+been proven as yet.
+In general, given spectra XandY, we can define the term “ Ycan be built from
+Xinℓsteps” in the same way that we defined the projective dimension. Th at is,
+we say that Ycan be built from Xin 0 steps if Yis a retract of a coproduct of
+suspensions of X. We then say that Ycan be built from Xinℓsteps if there is an
+exact triangle
+Z− →W− →/tildewideY− →ΣZ
+whereWcan be built from Xin 0 steps, Zcan be built from Xinℓ−1 steps, and
+Yis a retract of /tildewideY.
+Theorem 2.7. Suppose Eis anS-algebra and Xis a spectrum with the following
+properties.HOMOLOGICAL DIMENSIONS OF RING SPECTRA 9
+(1)E∧Xis anE-algebra, so also an S-algebra, with r.gl.dim.(E∧X) =m <
+∞.
+(2)As an object of D(S),proj.dim.X=k.
+(3)Scan be built from Xinℓsteps.
+Thengl.dim.E≤(k+1)(ℓ+1)(m+1)−1.
+Note that if Yis in the thick subcategory generated by X, thenYcan be built
+fromXin a finite number of steps. In particular, if Xis a type 0 finite spectrum,
+thenScan be built from Xin a finite number of steps. Hence we get the following
+corollary.
+Corollary 2.8. Suppose Eis anS-algebra and Xis a type 0finite spectrum such
+thatE∧Xis anE-algebra with finite right global dimension as an S-algebra. Then
+Ehas finite right global dimension.
+Here is the proof of Theorem 2.7.
+Proof.Note that if Mis anE-module, then M∧X∼=M∧E(E∧X) is anE∧X-
+module. Similarly, if fis a map of E-modules, then f∧1Xis a map of E∧X-
+modules.
+Now, let MandNbeE-modules. By induction on t, one can see that if
+proj.dim.Z=t, then every t+ 1-fold ghost f:M− →Ninduces a ghost f∧
+1Z:M∧Z− →N∧ZofE-modules. Taking Z=X, we see that if fis ak+1-fold
+ghost, then f∧1Xis a ghost, necessarily as a map of E∧X-modules. Hence, if
+f:M− →Nis a (k+1)(m+1)-fold ghost, then f∧1Xis an (m+1)-fold ghost, and
+hence is null as a map of E∧X-modules, and in particular as a map of E-modules.
+But then we can proceed by induction on ℓto see that if Ycan be built from X
+inℓsteps, then any ( k+1)(m+1)(ℓ+1)-fold ghost fhasf∧1Ynull as a map of
+E-modules. Taking Y=Scompletes the proof. /square
+Corollary 2.9. At the prime 3, the spectrum tmfof topological modular forms has
+finite global dimension.
+Proof.As mentioned above, there is a 3-cell complex Tsuch that
+tmf∧T≃BP <2>∧Σ8BP <2>
+by [Beh06]. The complex Thas cells in dimensions 0, 4, and 8, so is obviously type
+0. The spectrum tmf∧Tis also called tmf0(2), and is a commutative tmf-algebra
+(it is the connective cover of TMF0(2), which is the spectrum of sections over a
+certain stack of a sheaf of commutative S-algebras; see [Beh06]). The homotopy
+ring oftmf0(2) is polynomial over Z3(if we use the completed version) on two
+generators [Hil07, Proposition 2.3], and therefore tmf0(2) has global dimension 3
+by 2.5. /square
+3.Properties
+In this section, we examine some general properties of global and g host dimen-
+sion. Most of these properties concern the relationship between t he global dimen-
+sion or ghost dimension of an S-algebra Eand some other S-algebra Frelated to it.
+We discuss the cases when Fis a smashing Bousfield localization of E, whenFis
+a freeE-module, and when Fis Morita equivalent to E. We end with a discussion
+of the relationship between ghost and global dimension. We find this p articularly10 MARK HOVEY AND KEIR LOCKRIDGE
+unsatisfactory, however, because we are unable to prove anyth ing along the lines
+of the well-known algebraic fact that if Ris Noetherian or right perfect (which is
+equivalent to flats being projective), then r .gl.dim.R= w.dim.R.
+Proposition 3.1. Suppose Lis a smashing Bousfield localization functor, and E
+is anS-algebra. Then
+r.gl.dim.E≥r.gl.dim.LE.
+We do not know if this theorem is true if the localization functor is not s mashing.
+Proof.Notefirstthat LEisagainan S-algebra[EKMM97, ChapterVIII].Themain
+point is that because Lis smashing, the category LD(E) ofL-localE-modules
+is equivalent to the category D(LE) [EKMM97, Proposition VIII.3.2]. Thus a
+nontrivial composite of ghosts in D(LE) is the same thing as a nontrivial composite
+of ghosts between L-local objects in D(E). Hence r .gl.dim.E≥r.gl.dim.LE./square
+The same thing is true for ghost dimension, though we need to assum eD(E) is
+a Brown category. Recall from Section 4.2 of [HPS97] that this mea ns homology
+theories, and morphisms between them, are representable.
+Proposition 3.2. Suppose Lis a smashing Bousfield localization functor, and E
+is anS-algebra such that D(E)is a Brown category. Then
+gh.dim.E≥gh.dim.LE.
+Proof.Because D(E) is a Brown category, every object of D(E) is the minimal
+weak colimit of the compact objects mapping to it by [HPS97, Theore m 4.2.4]. But
+then we can follow the proof of [HS99, Theorem 6.2 (b,c)] to show th at the compact
+objectsof LD(E) arethe retractsofobjectsofthe form LF, forFcompact in D(E).
+Thus, if we have a nontrivial composite of ghosts
+X0− →X1− → ··· − → Xn
+inLD(E), where X0is compact in LD(E), we can write X0as a retract of LFfor
+some compact object F∈ D(E). Then the composite
+F− →LF− →X0− →X1
+must be nontrivial, and gives us a nontrivial composite of ghosts out of a compact
+object in D(E). /square
+In general, there is not much relationship between the global dimens ion of an
+S-algebra Eand a general E-algebra F. At one extreme we have the smashing
+localizations discussed above. The other extreme is E-algebras which are free over
+E.
+Proposition 3.3. Suppose E− →Fis a map of S-algebras such that F∗is free over
+E∗. Thenr.gl.dim.E≤r.gl.dim.Fandgh.dim.E≤gh.dim.F.
+Note that the inequalities in this proposition can certainly be strict, a s we can
+see by looking at the inclusion of ordinary rings from ZtoZ[x], for example. We
+think this proposition should hold even if F∗is only assumed to be faithfully flat
+overE∗, but have not been able to prove it.HOMOLOGICAL DIMENSIONS OF RING SPECTRA 11
+Proof.Consider the extension functor F∧E(−):D(E)− → D(F) and its right ad-
+joint, the restriction functor. Because F∗is flat over E∗, the natural map
+F∗⊗E∗X∗− →π∗(F∧EX)
+is an isomorphism. Indeed, both sides are homology functors on D(E), and the
+given map is an isomorphism for X=E, so it is an isomorphism for all X∈ D(E).
+This means that F∧E(−) preserves ghosts.
+In particular, if gis a composition of nghosts, and F∗is flat over E∗, thenF∧Eg
+is a composition of nghosts. If the domain of gis a compact object of D(E), then
+the domain of F∧Egis a compact object of D(F). However, F∧Egmay be zero,
+even ifgis nonzero. This cannot happen if F∗is free over E∗, however, for then
+F∧EXis a coproduct of copies of Xas anE-module. Thus gis a retract of the
+restriction of F∧Eg, so ifgis nontrivial, so is F∧Eg. /square
+We now discuss Morita equivalence. Recall from the work of Scwede a nd Ship-
+ley [Sch04] that two S-algebras EandFare called Morita equivalent if there is a
+chain of Quillen equivalences from the model category of E-modules to the model
+category of F-modules. Schwede and Shipley prove that, if EandFare Morita
+equivalent and cofibrant in the model structure on S-algebras (we can always as-
+sume this, since weak equivalences of S-algebras induce Quillen equivalences of the
+module categories), then there is an E−F-bimodule Mand anF−E-bimodule
+N, both of which are compact both as EandF-modules, so that the functors
+Φ(X) =X∧EMand Ψ(Y) =Y∧FN
+are inverse equivalences, with Φ: D(E)− → D(F) and Ψ going back the other way.
+Furthermore, Mgenerates D(F) andNgenerates D(E), in the sense that the
+smallest localizing subcategory containing M(resp.N) isD(E) (resp.D(F)).
+This is analogous to the usual Morita situation with ordinary rings, wit h one
+important difference. The generators MandNdo not have to be projective. This
+means that neither Φ nor Ψ need preserve projective objects, so that we do NOT
+expect global dimension or ghost dimension to be Morita invariant.
+Indeed, we thank Lidia Angeleri H¨ ugel for the following example com ing from
+the theory of tilting modules, a reference for which is [ASS06]. Recall that, for an
+ordinary ring A, a tilting module is a module Twhose projective resolution gives
+an equivalence of categories from the derived category of Ato the derived category
+of End A(T) by using the derived tensor product. In particular, a tilting module
+defines a Morita equivalence in the above sense between the Eilenber g-MacLane
+S-algebras HAandHB. Now, let Abe the path algebra of the quiver 1 − →2− →3,
+so thatAis isomorphic to the ring of lower triangular 3 ×3 matrices. There are
+then three projective indecomposable A-modules P1,P2, andP3, corresponding to
+the quiver representations
+k=− →k=− →k,0− →k=− →k,and 0− →0− →k.
+There are also three dual injective indecomposables I1,I2,I3=P1corresponding
+to the representations
+k− →0− →0, k=− →k− →0,andk=− →k=− →k.
+We claim that T=P3⊕I1⊕P1is a tilting module. Indeed, like any path algebra,
+Ahas global dimension ≤1 (and in fact r .gl.dim.A= 1), so proj .dim.T≤1.
+We also need Ext1(T,T) = 0, and this boils down to Ext1(I1,P3) = 0, which is12 MARK HOVEY AND KEIR LOCKRIDGE
+straightforward. The last condition for Tto be a tilting module is for there to be
+an exact sequence
+0− →A− →T1− →T2− →0
+whereT1andT2are finite direct sums of direct summands of T, so finite direct
+sums ofP3,I1, andP1. ButAcorresponds to the representation
+k− →k⊕k− →k⊕k⊕k
+where the maps are inclusions of the obvious summands. Thus we can just take
+T2=I1andT1=P3⊕P1⊕P1with the obvious surjection from T1toT2.
+So we have a Morita equivalence between HAandHB, where
+B= End A(T).
+One can compute Bdirectly. It is a 5-dimensional algebra generated by the or-
+thogonalidempotents e1,e2,e3and twoother elements α,β, where the only nonzero
+products involving αandβare
+e3α=αe1=αande2β=βe3=β.
+LetMbe the right B-module which has dimension 1 over kwheree2acts by
+the identity and the other generators of Bact trivially. Then one can check that
+proj.dim.M= 2, so
+r.gl.dim.B≥2>r.gl.dim.A= 1.
+In fact, r .gl.dim.B= 2. Note that both AandBare Noetherian, so their weak
+dimensions are also different. Thus HAandHBhave different ghost dimensions
+as well.
+We should mention that the property of having global dimension 0 IS M orita
+invariant, at least if one of the two S-algebras has commutative homotopy [HL09c,
+Proposition 2.11]. We now prove that the finiteness of global dimensio n is at least
+Morita invariant.
+Proposition 3.4. Suppose EandFare Morita equivalent S-algebras. Then
+r.gl.dim.E <∞if and only if r.gl.dim.F <∞. Similarly, gh.dim.E <∞if
+and only if gh.dim.F <∞.
+For example, this means that the endomorphism S-algebra of any finite type 0
+spectrum has infinite global dimension. In general, we expect r .gl.dim.Eto be
+always infinite if Eis a finite spectrum.
+Proof.As mentioned above, we can assume that EandFare cofibrant S-algebras,
+andwehavecompactgenerators MofD(F) andNofD(E) thatarealsobimodules,
+sothat smashingwith MandNgiveinverseequivalences. Nowsuppose fis aghost
+map inD(E). Thenf∧EMhas the property that D(F)(M,f∧EM)∗= 0. Since
+Mis a compact generator of D(F),Fis in the thick subcategory generated by
+M. Thus,Fcan be built from Min finitely many steps (see the discussion before
+Theorem 2.7). In particular, there is an integer ksuch that, if fis a composite of
+kghosts in D(E), thenf∧EMis a ghost in D(F). In particular, if Fhas finite
+global dimension, say n, then if fis a composite of k(n+1) ghosts in D(E), then
+f∧EMis null. Since smashing with Mis an equivalence of categories, this means
+thatfis null, so Ehas finite global dimension.
+Reversing the roles of EandFgives the reverse implication. For the ghost
+dimension, we repeat the same argument using a compact object as the source ofHOMOLOGICAL DIMENSIONS OF RING SPECTRA 13
+our first ghost map. This causes no problems since the equivalences of categories
+preserve compact objects. /square
+We now look for a relationship between the global dimension and the gh ost
+dimension. In algebra, we have the well-known inequality
+r.gl.dim.R≤w.dim.R+pure gl .dim.R,
+where pure gl .dim.Ris the pure global dimension of R. This can actually be
+replaced by the maximum projective dimension of a flat module. Indee d, in a
+projective resolution of an arbitrary R-module M, the syzygies Mkare flat when-
+everk≥w.dim.R. But this means our projective resolution is also a projective
+resolution of the flat module Mw.dim.R, giving us the desired inequality.
+One might expect the analogue of the pure global dimension to be the phantom
+dimension , defined below.
+Definition 3.5. Suppose Eis anS-algebra. Define the phantom dimension of
+E, phan.dim.E, to be the smallest integer nsuch that every composite of n+ 1
+phantom maps in D(E) is zero, or ∞if there exist arbitrarily long such nonzero
+composites.
+Recall that a phantom map is a map ffor which [ C,f]∗= 0 for every compact
+E-module C. So, if we think of a map whose cofiber is a ghost as the homotopy-
+theoreticanalogueofan epimorphism, then a map whosecofiber is a p hantom is the
+homotopy-theoretic analogue of a pure epimorphism, and the phan tom dimension
+should be analogous to the pure global dimension. Furthermore, th e phantom
+dimension is obviously invariant under Morita equivalences, since it only mentions
+compact objects, which are preserved by any equivalence of cate gories.
+Notethatif E∗iscountable, or, moregenerally,if D(E) isaBrowncategory,then
+phan.dim.E≤1 [Nee97, Chr98]. Very little else about the phantom dimension is
+known, except that it can be greater than one (as is shown in the ab ove papers).
+The natural conjecture is that phan .dim.HRshould be the pure global dimension
+ofR, but this is false for R=Z/4, whose pure global dimension is 0 whereas
+phan.dim.Z/4 = 1. The problem here is the difference between finitely presented
+R-modules and compact objects of D(R).
+In any case, we have the following proposition.
+Proposition 3.6. Suppose Eis anS-algebra. Then
+r.gl.dim.E <(gh.dim.E+1)(phan .dim.E+1).
+In particular, if D(E)is a Brown category, then
+r.gl.dim.E≤2gh.dim.E+1.
+This proposition is quite a bit weaker than the inequality for ordinary r ings,
+mentioned above. This is because we cannot construct a resolution of anE-module
+inthesamewayaswecaninalgebra. Butpossiblythispropositioncanb eimproved.
+Proof.Suppose gh .dim.E=nand phan .dim.E=k. There is nothing to prove if
+either of these is infinite, so assume they are finite. Then any n+1-fold composite
+of ghosts is phantom, so any ( n+1)(k+1)-fold composite of ghosts is a composite
+ofk+1 phantom maps, and so is null. /square14 MARK HOVEY AND KEIR LOCKRIDGE
+4.Gorenstein rings
+Recall that one of the themes of [HL09c] was that if Eis anS-algebra and
+r.gl.dim.E= 0, then there are very severe restrictions on E∗. In particular, we
+show that E∗must be quasi-Frobenius, though much more is true. Since quasi-
+Frobenius rings are the same as 0-Gorenstein rings, a natural con jecture might be
+that if r.gl.dim.E=n, thenE∗isn-Gorenstein. Unfortunately, this is easily seen
+to be false, since KO∗is notn-Gorenstein for any n, as we show in this section.
+However, we also show that if r .gl.dim.E= 1 andE∗is a Noetherian domain, then
+E∗is 1-Gorenstein. We do not know if this statement is true for larger n, though
+it seems unlikely.
+Recall that a (possibly noncommutative) ring Ris called Gorenstein if it is
+left and right Noetherian and Rhas finite injective dimension as a left or right
+R-module. This generalizes the usual definition of Gorenstein in the co mmutative
+case, which is much used in algebraic geometry. If Ris Gorenstein, the right and
+left injective dimensions of Rmust coincide, and if they are at most n,Ris called
+n-Gorenstein. These generalizations of quasi-Frobenius rings, whic h are just 0-
+Gorenstein rings, have been the object of much recent study. Ch apter 9 of [EJ00]
+is a good place to start.
+We begin by showing that KO∗is not Gorenstein.
+Proposition 4.1. The ring KO∗is notn-Gorenstein for any n, though it is Noe-
+therian and gl.dim.KO <∞.
+Proof.IfKO∗were Gorenstein, it would be Cohen-Macaulay, which would mean
+that its depth would be equal to its Krull dimension. But
+KO∗=Z(2)[η,w,v,v−1]/(η3,2η,wη,w2−4v).
+The only prime ideas in KO∗are (η) and the maximal ideal ( η,2,w). So the Krull
+dimension is 1. But there are no non-zero divisors in the maximal ideal, so the
+depth is 0, and so KO∗is not Cohen-Macaulay. /square
+We now consider the case when gl .dim.E= 1.
+Proposition 4.2. Suppose Eis anS-algebra with
+r.gl.dim.E= r.gl.dim.Eop= 1,
+and suppose that there are no nonzero maps from an injective (left or right )E∗-
+module to a projective (left or right )E∗-module. Then inj.dim.E∗≤1as either a
+left or right E∗-module. Consequently, if E∗is also left and right Noetherian, then
+E∗is a1-Gorenstein ring.
+Proof.LetR=E∗. It suffices to show that inj .dim.R≤1. We just do this on one
+side, since the hypotheses are left-right symmetric. For this, we e mbedRinto an
+injective module I0, giving us the short exact sequence
+0− →R− →I0− →R1− →0.
+We can realize this uniquely by an exact triangle in D(E)
+E− →J0− →M1δ0− →ΣEHOMOLOGICAL DIMENSIONS OF RING SPECTRA 15
+in which δ0is a ghost. We can then embed π∗M1into an injective module I1, and
+get an analogous exact triangle
+M1− →J1− →M2δ1− →ΣM1
+in which δ1is a ghost. Now the composite
+M2δ1− →ΣM1δ0− →Σ2E
+is necessarily trivial, because r .gl.dim.E= 1. It is represented in the universal
+coefficient spectral sequence
+Es,t
+2= Exts,t
+E∗(π∗M2,E∗)⇒ D(E)(M2,E)t−s
+on the 2-line by the extension
+0− →R− →I0− →I1− →M2− →0
+which is trivial if and only if inj .dim.R≤1. This element of E2is a permanent
+cycle, since it represents a map, but it must not survive the spectr al sequence.
+Therefore, there must be a differential, necessarily a d2, that hits it. The source of
+such a differential is a map π∗M2− →E∗. But there are no nonzero maps like this,
+becauseM2is the quotient of an injective module. Thus, the extension above mu st
+be 0, so inj .dim.R≤1. /square
+Now consider the case where Ris commutative Noetherian. Then every injective
+module is a direct sum of copies of the injective hulls E(R/p) of prime ideals p
+(see [Lam99, Section 3I]). Every element of E(R/p) is killed by pnfor some n.
+Thus, if Ris a domain, the only possible injective module that can map to a
+projective module is E(R) itself. But E(R) is divisible, so every element can be
+divided by every element of R. Thus as long as Ris a Noetherian domain that is
+not a field, there are no nonzero maps from an injective to a projec tive module.
+We have therefore proved the following corollary.
+Corollary 4.3. Suppose Eis anS-algebra with r.gl.dim.E= 1, andE∗is a
+commutative Noetherian domain that is not a field. Then E∗is1-Gorenstein.
+References
+[ASS06] Ibrahim Assem, Daniel Simson, and Andrzej Skowro´ n ski,Elements of the represen-
+tation theory of associative algebras. Vol. 1 , London Mathematical Society Student
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+ogy45(2006), no. 2, 343–402. MR MR2193339 (2006i:55016)
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+Math. Soc. 281(1984), no. 1, 197–208. MR MR719665 (85e:16046)
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+MR MR2055748 (2005b:20102)
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+Adv. Math. 136(1998), no. 2, 284–339. MR MR1626856 (99g:18007)
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+Gruyter Expositions in Mathematics, vol. 30, Walter de Gruy ter & Co., Berlin, 2000.
+MR 2001h:1601316 MARK HOVEY AND KEIR LOCKRIDGE
+[EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May ,Rings, modules, and algebras
+in stable homotopy theory , American Mathematical Society, Providence, RI, 1997,
+With an appendix by M. Cole. MR 97h:55006
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+(2007), no. 12, 4075–4086 (electronic). MR MR2341960 (2008 k:55008)
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+preprint, 2009.
+[HL09c] Mark Hovey and Keir Lockridge, Semisimple ring spectra , New York J. Math. 15
+(2009), 219–243. MR MR2511136
+[Hop02] M.J.Hopkins, Algebraic topology and modular forms , Proceedingsofthe International
+Congress of Mathematicians, Vol. I (Beijing, 2002) (Beijin g), Higher Ed. Press, 2002,
+pp. 291–317. MR MR1989190 (2004g:11032)
+[HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland ,Axiomatic stable homo-
+topy theory , Mem. Amer. Math. Soc. 128(1997), no. 610, x+114. MR MR1388895
+(98a:55017)
+[HS99] Mark Hovey and Neil P. Strickland, Morava K-theories and localisation , Mem. Amer.
+Math. Soc. 139(1999), no. 666, viii+100. MR MR1601906 (99b:55017)
+[Lam99] T. Y. Lam, Lectures on modules and rings , Graduate Texts in Mathematics, vol. 189,
+Springer-Verlag, New York, 1999. MR MR1653294 (99i:16001)
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+Department of Mathematics, Wesleyan University, Middletow n, CT 06459
+E-mail address :hovey@member.ams.org
+Department of Mathematics, Wake Forest University, Winston -Salem, NC 27109
+E-mail address :lockrikh@wfu.edu
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+arXiv:1001.0903v3  [math.CO]  5 Oct 2011KALEIDOSCOPICAL CONFIGURATIONS
+ING-SPACES
+T. BANAKH, O. PETRENKO, I.V. PROTASOV, S. SLOBODIANIUK
+Abstract. LetGbe a group and Xbe aG-space with the action G×X→X, (g,x)/mapsto→gx. A subset F
+ofXis called a kaleidoscopical configuration if there exists a c oloringχ:X→Csuch that the restriction
+ofχon each subset gF,g∈G, is a bijection. We present a construction (called the split ting construction)
+of kaleidoscopical configurations in an arbitrary G-space, reduce the problem of characterization of kaleido-
+scopical configurations in a finite Abelian group Gto a factorization of Ginto two subsets, and describe all
+kaleidoscopical configurations in isometrically homogene ous ultrametric spaces with finite distance scale. Also
+we construct 2cmany (unsplittable) kaleidoscopic configurations of cardi nalitycin the Euclidean space Rn.
+Introduction
+LetXbe a set and Fbe a family of subsets of X(the pair (X,F) is called a hypergraph ). Following [4], we
+say that a coloring χ:X→κofX(i.e. a surjective mapping of Xonto a cardinal κ) is
+•F-surjective if the restriction χ|Fis surjective for all F∈F;
+•F-injective ifχ|Fis injective for all F∈F;
+•F-bijective orF-kaleidoscopical ifχ|Fis bijective for all F∈F.
+A hypergraph ( X,F) is called kaleidoskopical if there exists an F-kaleidoscopical coloring χ:X→κ. The
+adjactive ”kaleidoscopical” appeared in definition [5] of s-regular graph Γ( V,E) (each vertex v∈Vhas degree
+s) admitting a vertex ( s+ 1)-colloring such that each unit ball B(v,1) ={u∈V:d(u,v) = 1}has the vertices
+of all colors ( dis a path matric on V). These graphs can be considered as a graph version of Hamming co des
+[6].
+We shall consider hypergraphs related to G-space. Let Gbe a group. A G-space is a setXendowed with
+an actionG×X→X, (g,x)/mapsto→gx. AllG-spaces are suppose to be transitive (for any x,y∈Xthere exists
+g∈Gsuch thatgx=y). For a subset A⊆X, we putG[A] ={gA:g∈G}.
+A subsetA⊆Xis called a kaleidoscopical configuration if the hypergraph ( X,G [A]) is kaleidoscopical (in
+words, if there exists a coloring χ:X→ |A|such thatχ|gAis bijective for every g∈G).
+In Section 1 we show that kaleidoscopical configurations are tightly connected with classical combinatorial
+themeTransversality and, in the case X=Gand (left) regular action of GonG, with factorization problem,
+well known in Factorization Theory of groups , see [11], [12].
+In Section 2 we introduce and describe the kaleidoscopical configur ations (called splittable) which arise from
+the chains of G-invariant equivalences (imprimitivities) on X. If aG-spaceXis primitive (the only G-invariant
+equivalences on XareX×Xand ∆ X) then the only splittable configurations in XareXand the singletons.
+In Section 3 we prove that every kaleidoscopical configuration in iso metrically homogeneous metric space
+with finite distance scale is splittable. For n≥2, we construct a plenty of kaleidoscopical configurations of
+cardinality cinRn. These configurations are non-splittable because Rnis isometrically primitive. We don’t
+know whether there exists a finite non-singleton or countable kaleid oscopical configurations in Rn,n≥2.
+In Section 4 we study the problem of splittability of kaleidoscopic confi gurations in finite Abelian groups
+and reformulate this problem in terms of the semi-Haj´ os property , see [11], [12].
+1.Transversality and factorization
+Let (X,F) be a hypergraph. A subset T⊆Xis called an F-transversal if |F/intersectiontextT|= 1 for each F∈F.
+1991Mathematics Subject Classification. 05B45; 05C15, 05E15; 05E18; 20K01.
+Key words and phrases. Kaleidoscopical configuration, G-space, transversality, splitting, factorization of a gro up, Haj´ os
+property.
+12 T. BANAKH, O. PETRENKO, I.V. PROTASOV, S. SLOBODIANIUK
+Proposition 1.1. A hypergraph (X,F)is kaleidoscopical if and only if Xcan be partitioned into F-transversals.
+Proof. For a kaleidoscopic hypergraph ( X,F), letχ:X→κbe a kaleidoscopical coloring. Then/unionsqtext
+α<κχ−1(α)
+is a partition of XintoF-transversal.
+On the other hand, if/unionsqtext
+α<κTαis a partition of XintoF-transversal then the coloring χ:X→κdefined as
+χ(x) =α⇔x∈Tαis kaleidoscopical. /square
+LetXbe aG-spaceAbe a kaleidoscopical configuration in X. IfTisG[A]-transversal then AisG[T]-
+transversal and gTisG[A] transversal for each g∈G.
+We say that a kaleidoscopical configuration AinXishomogeneous if there exist a G[A]-transversal Tand
+a subsetH⊆Xsuch thatX=/unionsqtext
+h∈HhT.
+A subsetAof a group Gis defined to be complemented inGif there exists a subset B⊆Gsuch that
+the multiplication mapping µ:A×B→G, (a,b)/mapsto→ab, is bijective. Following [12], we call the set Ba
+complementer factor toA, and say that G=ABis a factorization of G. In this case, we have
+G=/unionsqdisplay
+a∈AaB=/unionsqdisplay
+b∈BAb.
+A subsetA⊆Gis called doubly complemented if there are factorization G=AB=BCfor some subsets B,C
+ofG.
+Proposition 1.2. For two subsets A,Bof a groupGthe following conditions are equivalent:
+(1)BisG[A]-transversal;
+(2)G=AB−1is a factorization of G.
+Proof. (1)⇒(2) For each g∈G,g−1A∩B/\e}atio\slash=∅, sog∈AB−1. Ifg=a1b−1
+1=a2b−1
+2for somea1,a2∈A,
+b1,b2∈B, theng−1a1=b1andg−1a−1
+2=b2and by (1), b1=b2anda1=a2, witnessing that G=AB−1is a
+factorization of G.
+(2)⇒(1) Fix any g∈G. The inclusion g−1∈AB−1impliesgA∩B/\e}atio\slash=∅. Ifga1=b1andga2=b2for some
+a1,a2∈A,b1,b2∈B, theng−1=a1b−1
+1=a2b−1
+2and by (2), b1=b2, witnessing that |gA∩B|= 1. /square
+Corollary 1.3. Each kaleidoscopic configuration in group Gis complemented.
+Proof. Given a kaleidoscopical configuration A⊂G, fix aA-kaleidoscopical coloring χ:G→C. We choose a
+colorc∈C, consider the monochrome class B=χ−1(b)⊂Gand observe that for every g∈G|gA∩B|= 1
+by the definition of A-kaleidoscopical coloring. By Proposition 1.2, G=AB−1is a factorization, so Ais
+complemented in G. /square
+Proposition 1.4. A subsetAof a groupGis doubly complemented if and only if Ais a homogeneous kalei-
+doscopical configuration.
+Proof. LetG=AB=BCbe a factorization of G. By proposition 1.2, B−1is aG[A]-transversal. Since
+G=/unionsqtext
+c∈Cc−1B, we conclude that Ais a homogeneous kaleidoscopical configuration.
+LetAbe a homogeneous kaleidoscopical configuration. We choose a G[A]-transversal Tand a subset H⊆G
+such thatG=/unionsqtext
+h∈HhT. By proposition 1.2, G=AT−1. SinceG=/unionsqtext
+h∈HhT,G=T−1H−1is a factorization.
+Hence,Ais doubly complemented. /square
+Corollary 1.5. For a subset Aof an Abelian group G, the following statements are equivalent:
+(1)Ais complemented;
+(2)Ais a kaleidoscopical configuration;
+(3)Ais a homogeneous kaleidoscopical configuration.
+Question 1.6. Is each complemented subset of a (finite) group kaleidoscopi cal?
+Proposition 1.7. LetXbe aG-space,x∈X,Gx={g∈G:gx=x},γx:G→X,γx(g) =gx,s:X→G
+be a section of γx. LetAbe a subset of X,TbeG[A]-transversal. Then
+(1)s(T)is aG[γ−1
+x(A)]-transversal;
+(2)|G|=|Gx||A||T|.
+Proof. The statement (1) is evident. The statement (2) follows from (1) a nd proposition 1.2. /squareKALEIDOSCOPICAL CONFIGURATIONS 3
+Corollary 1.8. LetAbe a kaleidoscopical configuration in a finite G-spaceXwith a kaleidoscopical coloring
+χ:G→k. Thenχ−1(0) =···=χ−1(k−1)and|X|=|A||χ−1(0)|.
+Proof. We may suppose that Gis a subgroup of the group of all permutations of XsoGis finite. Since
+|G|=|X||Gx|, we can apply proposition 1.7(2). /square
+Proposition 1.9. Letκbe an infinite cardinal, (X,F)be a hypergraph such that |F|=κand|F|=κfor each
+F∈F. If|F∩F′|< cfκfor all distinct F,F′∈Fthen there are a disjoint family TofF-transversals such
+that|T|=κ,|T|=κfor eachT∈T
+Proof. We enumerate F={Fα:α < κ}and choose inductively the subsets {Vα⊂Fα:α < κ}such that
+the family {Fα\Vα:α < κ}is disjoint and |Fα\Vα|=κfor eachα < κ . LetFα\Vα={tαβ:β < κ},
+Tβ={tαβ:α<κ}. ThenT={Tβ:β <κ}is the desired family. /square
+For a hypergraph ( X,F),x∈XandA⊆X, we put
+St(x,(F)) =/uniondisplay
+{F∈F:x∈F},
+St(A,F) =/uniondisplay
+{St(a,F) :a∈A}.
+Proposition 1.10. A hypergraph (X,F)is kaleidoscopical provided that, for some infinite cardina lκ, the
+following two conditions are satisfied:
+(1)F≤κand|F|=κfor eachF∈F;
+(2)for any subfamily A⊂Fof cardinality |A|< κand any subset B⊂X\(/uniontextA)of cardinality |B|< κ
+the intersection St(B,F)∩(/uniontextA)has cardinality less then κ.
+Proof. Letλ=|F|andF={Fα:α<λ}be an injective enumeration of F. By induction we shall construct a
+transfinite sequence ( χα:Fα→κ)α<λof bijective colorings such that for any ordinals α<β <λ
+(1αβ) the colorings χαandχβcoincide on Fα∩Fβ;
+(2αβ) no distinct points a∈Fαandb∈Fβwithχα(a) =χβ(b) lie in some hyperedge F∈F.
+Assume that for some ordinal γ < λ we have constructed a sequence of colorings ( χα)α<γsatisfying the
+conditions (1 αβ) and (2 αβ) for allα<β <γ .
+Let us define a bijective coloring χγ:Fγ→κ. First we show that the union
+F′
+γ=/uniondisplay
+α<γFγ∩Fα
+has cardinality |F′
+γ|<κ. Observe that for each α<γ we getFα/\e}atio\slash⊂Fγ. Assuming conversely that Fα/subsetnoteqlFγand
+taking any point v∈Fγ\Fαwe conclude that the intersection Fα∩St(v,F)⊃Fα∩Fγ=Fαhas cardinality
+≥κ, which contradicts the condition (2) of the theorem.
+Therefore, for each α<γ we can choose a point vα∈Fα\Fγ. Then for the set B={vα:α<γ}the set
+F′
+γ⊂Fγ∩St(B,F) has cardinality |F′
+γ| ≤ |Fγ∩St(A,F)|<κaccording to (2).
+For every point x∈Fγ\F′
+γand every ordinal α<γ consider the sets St(x,F)∩FαandCα(x) =χα(St(x,F)∩
+Fα)⊂κ. The condition (2) implies that the set C(x) =/uniontext
+α<γCα(x) has cardinality |C(x)|<κ.
+Let≺be any well-order on the set Fγsuch thatF′
+γcoincides the initial segment {x∈Fγ:x<y}for some
+pointy∈Fγ. Consider the coloring χγ:Fγ→κdefined byχγ(x) =χα(x) ifx∈Fγ∩Fαfor someα<γ and
+χγ(x) = minκ\(C(x)∪{χ(y) :y≺x})
+ifx∈Fγ\F′
+γ.
+Let us show that the coloring χγ:Fγ→κis bijective. The injectivity of χγfollows from the definition of
+χγand the conditions (2 αβ),α<β <γ .
+The surjectivity of χγwill follow as soon as we check that for each color c∈κ\χγ(F′
+γ) the setFγ(c) =
+{x∈Fγ\F′
+γ:c∈C(x)}has cardinality <κ. Observe that c∈C(x) if and only if there is α<γ and a point
+a∈Fα\Fγsuch thatχα(a) =candx∈ St(a,F). The set Ac=/uniontext
+α<γχ−1
+α(c)\Fγhas size |Ac| ≤γ < κ
+and by the condition (2), the set Fγ(c)⊂Fγ∩St(Ac,F) has cardinality <κ. This completes the proof of the
+bijectivity of the coloring χγ.4 T. BANAKH, O. PETRENKO, I.V. PROTASOV, S. SLOBODIANIUK
+The conditions (1 αγ) and (2 α,γ) for allα<γ follow from the definition of the coloring χγ. This completes
+the inductive step of the construction of the sequence ( χα)α<λ.
+After completing the inductive construction, let χ:V→κbe any coloring such that χ|Fα=χαfor all
+α<λ . The conditions (1 αβ) guarantee that the coloring χis well-defined. The bijectivity of the colorings χα,
+α<λ , ensures the kaleidoscopicity of the coloring χ. /square
+We conclude this section with short discussion of possibilities of trans fering above notions and results of
+quasigroups.
+We recall that a quasigroup is a setXendowed with a binary operation ∗:X×X→Xsuch that, for every
+a,b∈X, the system of equations a∗x=b,y∗a=bhas a unique solution x=a\b,y=b/ainX.
+In an obvious way the notion of a kaleidoscopical configuration gene ralizes to quasigroup.
+A subsetAof a quasigroup Xis called
+•kaleidoscopical if there is a coloring χ:X→Csuch thatχ|x∗A:x∗A→Cis bijective for all x∈X;
+•complemented if there is a subset B⊂Xsuch that the right division δ:B×A→X,δ(b,a) =b/ais
+bijective;
+•doubly complemented if there exists a complemented subset B⊂Xsuch that the multiplication µ:
+A×B→X,µ(a,b) =a∗b, is bijective;
+•self-complemented if the maps µ:A×A→X,µ(x,y) =x∗y, andδ:A×A→X,δ(x,y) =x/y, are
+bijective.
+It follows from the proof of proposition 1.2 that each kaleidoscopica l subset in a semigroup is complemented.
+In contrast, Proposition 1.4 does not generalize to quasigroup.
+Example 1.11. There exists a quasigroup Xof order|X|= 9that contains a self-complemented subset A⊂X,
+which is not kaleidoscopical.
+Proof. It is well-known that finite quasigroups can be identified with Latin squ ares, i.e.,n×nmatrices whose
+rows and columns are permutations of the set {1,...,n}. Forr,s≤nan (r×s)-matrix (xij) is called a partial
+Latin (r×s)-rectangle ifxij∈ {1,2,...,n}andxlj/\e}atio\slash=xij/\e}atio\slash=xikfor any 1 ≤i/\e}atio\slash=l≤rand 1≤j/\e}atio\slash=k≤s. By a
+result of Ryser [7] (see also Lemma 1 in [1, p.214]) each partial latin ( r×s)-rectangle can be completed to a
+Latin (n×n)-square if and only if each number i∈ {1,...,n}appears in the rectangle not less than r+s−n
+times. This extension result allows us to find a quasigroups operation onX={1,..., 9}whose multiplication
+table has the following first three columns:
+∗1 2 3
+11 4 5
+26 2 7
+38 9 3
+44 1 6
+55 6 1
+62 7 8
+77 8 2
+83 5 9
+99 3 4
+Looking at this table we can see that the set A={1,2,3}is self-complemented as A∗A=X=A/A.
+Assuming that Ais kaleidoscopic, find a coloring χ:X→Asuch thatχ|x∗Ais bijective for each x∈X.
+Since 1∗A={1,4,5}and 4∗A={4,1,6}, the elements 5 and 6 have the same color, which is not possible as
+5∗A={5,6,1}andχ|5∗Ais bijective. /square
+Corollary 1.8 implies that the size |K|of any kaleidoscopic subset Kin a finite group Gdivides the cardinality
+|G|ofG. The same is true for any finite transitive G-spaceX.
+2.Splitting
+In this section we present a simple construction of kaleidoscopic con figurations in arbitrary G-space, called
+the splitting construction. Kaleidoscopic subsets constructed in t his way will be called splittable.KALEIDOSCOPICAL CONFIGURATIONS 5
+First we recall some definitions. A map ϕ:X→YbetweenG-spaces is called equivariant ifϕ(gx) =gϕ(x)
+for allg∈Gandx∈X. It is easy to see that each equivariant map between transitive G-spaces is surjective
+and homogeneous.
+A function ϕ:X→Yis defined to be homogeneous if it isκ-to-1 for some non-zero cardinal κ. The latter
+means that |ϕ−1(y)|=κfor ally∈Y.
+Proposition 2.1. Letκbe a non-zero cardinal, π:X→Ybe anκ-to-1 equivariant map between two G-spaces
+ands:Y→Xbe a section of ϕ. LetK⊂Ybe a kaleidoscopic subset and χ:Y→Cbe anK-kaleidoscopic
+coloring. Then:
+(1)the preimage ¯K=π−1(K)is a kaleidoscopic configuration in Xwith respect to any coloring ¯χ:X→
+C×κsuch that for each y∈Ythe restriction ¯χ|ϕ−1(y) :π−1(y)→ {χ(y)}×κis bijective;
+(2)the image ˜K=s(K)is a kaleidoscopic configuration in Xwith respect to the ˜K-kaleidoscopic coloring
+˜χ=χ◦π:X→C.
+Proof. 1. Given any element g∈G, we need to check that the restriction ¯ χ|g¯K:g¯K→C×κis bijective.
+To see that it is surjective, take any color ( c,α)∈C×κand using the surjectivity of χ|gK:gK→C, find a
+pointy∈gKwithχ(y) =c. Since the restriction ¯ χ|π−1(y) :π−1(y)→ {c}×κis bijective, there is a point
+x∈π−1(y)⊂π−1(gK) =g¯Kwith ¯χ(x) = (c,α), so ¯χ|g¯Kis surjective.
+To see that it is injective, take any two distinct points x,x′∈g¯K. Ifπ(x) =π(x′), then for the point
+y=π(x) =π(x′)∈gπ(K) =π(gK) the injectivity of the restriction ¯ χ|π−1(y) implies that ¯ χ(x)/\e}atio\slash= ¯χ(x′).
+Ifπ(x)/\e}atio\slash=π(x′), then the injectivity of χ|gKguarantees that χ(π(x))/\e}atio\slash=χ(π(x′)) and then ¯ χ(x)/\e}atio\slash= ¯χ(x′) as
+¯χ(x)∈ {χ(π(x))}×κand ¯χ(x)∈ {χ(π(x′))}×κ.
+2. Given any element g∈G, we need to check that the restriction ˜ χ|g˜K:g¯K→Cis bijective. To see that
+˜χ|g˜Kis surjective, observe that
+˜χ(g˜K) =χ◦π(g˜K) =χ(gπ(˜K)) =χ(gK) =C
+by the surjectivity of χ|gK:gK→C.
+To see that ˜ χ|g˜Kis injective, take any two distinct points x,x′∈˜K=s(K) and observe π(x)/\e}atio\slash=π(x′).
+Sinceπis equivariant, π(gx) =gπ(x)/\e}atio\slash=gπ(x′) =π(gx′). Sinceπ(gx),π(gx′)∈gKandχ|gKis injective,
+˜χ(x) =χ(π(gx))/\e}atio\slash=χ(π(gx′)) = ˜χ(π(gx′)) are we are done. /square
+Iterating the constructions from Proposition 2.1, we get the so-c alledsplitting construction of kaleidoscopical
+configurations.
+Proposition 2.2. LetX0→X1→ ··· →Xmbe a sequence of G-spaces linked by homogeneous G-equivariant
+mapsπi:Xi→Xi+1,i<m. LetKi⊂Xi,i≤m, be subsets such that for every i<meither the restriction
+πi|Ki:Ki→Ki+1is bijective or else Ki=π−1
+i(Ki+1). If the set Kmis kaleidoscopic in the G-spaceXm,
+then for every i≤mthe setKiis kaleidoscopic in the G-spaceXi.
+Proof. This proposition can be derived from Proposition 2.1 by the reverse in duction on i∈ {m,m−1,..., 0}.
+/square
+Proposition 2.2 can be alternatively written in terms of invariant equiv alence relations.
+Given an equivalence relation E⊂X×Xon a setXletX/E ={[x]E:x∈X}be the quotient space
+consisting of the equivalence classes [ x]E={y∈X: (x,y)∈E},x∈X. Denote by qE:X→X/E ,qE:x/mapsto→
+[x]E, the quotient map. For a subset K⊂XletK/E ={[x]E:x∈K} ⊂X/E and [K]E=/uniontext
+x∈K[x]E⊂X.
+LetEbe an equivalence relation on a set X. A subsetK⊂Xis defined to be
+•E-parallel ifK∩[x]E= [x]Efor allx∈K;
+•E-orthogonal ifK∩[x]E={x}for allx∈K.
+Given two equivalence relations E⊂FonXwe can generalize these two notions defining K⊂Xto be
+•F/E-parallel if [K]E∩[x]F= [x]Ffor allx∈K;
+•F/E-orthogonal if [K]E∩[x]F= [x]Efor allx∈K.
+Observe that a set K⊂XisE-parallel (E-orthogonal) if and only if it is E/∆X-parallel (E/∆X-orthogonal).
+Here ∆ X={(x,x) :x∈X}stands for the smallest equivalence relation on X.6 T. BANAKH, O. PETRENKO, I.V. PROTASOV, S. SLOBODIANIUK
+An equivalence relation Eon aG-spaceXis calledG-invariant if for each ( x,y)∈Eand anyg∈Gwe get
+(gx,gy )∈E. For aG-invariant equivalence relation EonXthe quotient space X/E is aG-space under the
+induced action
+G×X/E→X/E, (g,[x]E)/mapsto→[gx]E
+of the group G. In this case the quotient projection q:X→X/E is equivariant. G-Invariant equivalence
+relations on G-spaces are also called imprimitivities .
+Proposition 2.3. Let∆X=E0⊂E1⊂ ··· ⊂Embe a sequence of G-invariant equivalence relations on a
+transitiveG-spaceX. A subsetK⊂Xis kaleidoscopic provided
+(1)the projection K/Emis kaleidoscopic in the G-spaceX/Em;
+(2)for everyi<mthe setKisEi+1/Ei-parallel or Ei+1/Ei-orthogonal.
+Proof. For everyi≤mconsider the G-spaceXi=X/Eiand the subset Ki=K/EiinXi. SinceE0= ∆X,
+the spaceX0coincides with X. Next, for every i < m , consider the equivariant map πi:Xi→Xi+1,
+πi: [x]Ei/mapsto→[x]Ei+1. This map is homogeneous because of the transitivity of the G-spaceXi.
+We claim that the maps πisatisfy the requirements of Proposition 2.2. Indeed, if KisEi+1/Ei-parallel,
+thenKi=π−1
+i(Ki+1). IfKisEi+1/Ei-orthogonal, then the restriction πi|Ki:Ki→Ki+1is bijective.
+Now Proposition 2.2 implies that the set K=K0is kaleidoscopic in X=X0. /square
+Proposition 2.3 suggests the following notion that will be cenral in our subsequent discussion.
+Definition 2.4. A (kaleidoscopic) subset Kin aG-spaceXis calledsplittable if there is an increasing sequence
+ofG-invariant equivalence relations
+∆X=E0⊂E1⊂ ··· ⊂Em=X×X
+such that for every i<m the setKis eitherEi+1/Ei-parallel or Ei+1/Ei-orthogonal.
+Proposition 2.3 implies that each splittable subset in a transitive G-space is kaleidoscopic. What about the
+inverse implication?
+Problem 2.5. For whichG-spacesXevery kaleidoscopic configuration K⊂Xis splittable?
+3.Kaleidoscopical configurations in matric spaces
+Here we consider each metric space ( X,d) as aG-space endowed with the natural action of its isometry
+groupG= Iso(X). If this action is transitive, then the metric space Xis called isometrically homogeneous .
+Let us recall that a metric space ( X,d) isultrametric if the metric dsatisfies the strong triangle inequality
+d(x,z)≤max{d(x,y),d(y,z)}
+for allx,y,z∈X. It follows that for every ε≥0 the relation
+Eε={(x,y)∈X2:d(x,y)≤ε} ⊂X×X
+is an invariant equivalence relation on X.
+Theorem 3.1. Let(X,d)be an isometrically homogeneous ultrametric space with the finite distance scale
+d(X×X) ={ε0,ε1,...,ε n}where 0 =ε0<ε1<...<ε n. Then every kaleidoscopical configuration KinX
+is(Eε0,Eε1,...,E εn)-splittable.
+Proof. Assume conversely that Kis not (Eε0,Eε1,...,E εn)-splittable. Then for some k < n the setKis
+neitherEεk+1/Eεk-parallel nor Eεk+1/Eεk-orthogonal. We can assume that kis the smallest number with that
+property. By [ x]εiwe shall denote the closed εi-ball [x]Eεicentered at a point x∈X.
+SinceKis notEεk+1/Eεk-orthogonal, there are two points u,v∈Ksuch thatεk<d(u,v) =εk+1. SinceK
+is notEεk+1/Eεk-parallel, there are points w∈Kandz∈Xsuch thatεk<infx∈Kd(z,x) =d(z,w) =εk+1.
+SinceXis isometrically homogeneous, we can find an isometry ϕ:X→Xsuch thatϕ(w) =z. Then
+ϕ([w]εk) = [z]εkand we can define an isometry φ:X→Xletting
+φ(x) =
+
+ϕ(x) ifx∈[w]εk,
+ϕ−1(x) ifx∈[z]εk,
+x otherwise.KALEIDOSCOPICAL CONFIGURATIONS 7
+The isometry φswaps the balls [ w]εkand [z]εkbut does not move points outside the union [ w]εk∪[z]εk. Since
+Kisχ-kaleidoscopic, the restrictions χ|φ(K) andχ|Kare bijections onto C. Consequently, χ(w) =χ(z′) for
+some point z′∈[z]εk. Taking into account that d(w,z′) =d(w,z) =εk+1=d(u,v) andXis an isometrically
+homogeneous ultrametric space, we can construct an isometry ψ:X→Xsuch thatψ(u) =wandψ(v) =z′.
+For this isometry, w,z′∈ψ(K) and hence χ|ψ(K) is not injective, contradicting the choice of the coloring
+χ. /square
+Problem 3.2. Let{0,1}ωbe the Cantor space endowed with the standard ultrametric ge nerating the product
+topology. Describe all kaleidoscopical configurations in {0,1}ω.
+Remark 3.3. All closed kaleidoscopical configurations in {0,1}ωcan be characterized with usage of Theorem
+3.1. Among them there are plenty of non-splittable configurations.
+AG-spaceXis called primitive if eachG-invariant equivalence relation on Xis equal to ∆ Xor toX×X.
+It follows that each splitting configuration Kin a primitive G-spaceXis trivial, i.e. either K=XorKis
+singleton. It is natural to ask if every kaleidoscopical configuratio n in a primitive G-space trivial.
+The answer to this question is affirmative if Xis2-transitive in the sense that for any pairs ( x,y),(x′,y′)∈
+X2\∆Xthere isg∈Xsuch that (x′,y′) = (gx,gy ).
+An example of a primitive G-space, which is not 2-transitive is the Euclidean space Rnof dimension n≥2
+endowed with the action of its isometry group Iso( Rn). It turns out that Rncontain 2cmany unsplittable
+kaleidoscopic configurations of cardinality c.
+To construct a kaleidoscopic subset in Rnuse proposition1.10 and the following auxiliary definition.
+Let (X,d) be a metric space. By S(x,r) ={y∈X:d(x,y) =r}we shall denote the sphere of radius r
+centered as a point x∈X.
+Definition 3.4. A subsetKof a metric space ( X,d) is called rigid if for any pairwise distinct points x,y,z∈K
+and numbers rx,ry,rz∈d(K×K) the spheres S(x,rx),S(y,ry),S(z,rz) have no common point in X\K.
+Theorem 3.5. LetXbe metric space and G⊂Iso(X)be a group of isometries of X. Each infinite rigit subset
+K⊂Xof cardinality |K| ≥ |G|is kaleidoscopical.
+Proof. The kaleidoscopicity of the set Kwill follow from proposition 1.10 as soon as we check that the
+hypergraph ( V,F) = (X,{gK:g∈G}) satisfies the conditions (1)–(2) for the cardinal κ=|K|. Since
+|G| ≤κ=|K|=|gK|for allg∈G, the condition (1) is satisfied.
+To show that (2) holds, take any subset A⊂Gof cardinality |A|< κ and any subset B∈X\AK of
+cardinality |B|< κ. We need to show that |St(B,F)∩AK|< κ. This will follow from max {|A|,|B|}< κas
+soon as we check that |St(b,F)∩aK| ≤2 for every b∈Banda∈A. Assuming conversely that St(b,F)∩aK
+contains three pairwise distinct points x,y,z we shall obtain a contradiction with the metric independence of
+Kasd(b,x),d(b,y),d(b,z)∈d(K×K) andbis the common point of the spheres S(x,d(b,x)),S(y,d(b,y)),
+S(z,d(b,z)). /square
+In light of Theorem 3.5 it is important to construct a rigit subsets in me tric spaces.
+Lemma 3.6. Any algebraic independent subset Lof affine line in the Euclidean space Rnof dimension n≥1
+is rigit.
+Proof. Identify algebraic independent Lwith a subset of Rand letYbe any subset of /suppress L with cardinality less
+thenc. It’s enough to show that there are no a/\e}atio\slash=b/\e}atio\slash=c∈Y,ra,rb,rc∈d(Y×Y)\{0}andx∈Rn\(Y∪{p})
+withd(x,a) =ra,d(x,b) =rb,d(x,c) =rc. It follows from the theorem of cosines applying to cos( ∠abx) =
+−cos(∠cbx) and the observation that there are no such values that (( a−b)(b−c)+r2
+b)(a−c)−(a−b)r2
+c−(b−c)r2
+a=
+0}. Now the proof of the last statement. Let ra=s1−s2,rb=z1−z2,rc=t1−t2wheres1,s2,t1,t2,z1,z2∈A.
+It is a polinom with variable atakingra,rb,rcas linear functions of aif some of the numbers s1,s2,t1,t2,z1,z2
+are equal to aor constants. If z1orz2is equal toathent1ort2is. Then the coefficient of a2in the equation
+isb−c−2z2+b+ 2t2orb−c−2z2+b+ 2t2+b−cand in any case is nonzero that is impossible. The same if
+one of{z1,z2}is equal toc. Ifz1orz2equals toband none of them equals to aorcthen . Ifz1/\e}atio\slash=a,b,c and
+z2/\e}atio\slash=a,b,c then as a polinom with variables z1,z2the coefficient of z1z2is 0 only if |z1−z2|=|s1−s2|=|t1−t2|
+but then (a−b)(b−c)(a−c) = 0. /square
+Now we are able to prove the promised:8 T. BANAKH, O. PETRENKO, I.V. PROTASOV, S. SLOBODIANIUK
+Theorem 3.7. LetLis continual algebraic independent subset of a line in Rn. Each subset K⊂Lof
+cardinality cis kaleidoscopic in Rn. Consequently, the Euclidean space Rncontains 2cmany kaleidoscopic
+subsets.
+Proof. It follows from Theorem 3.5 and Lemma 3.6 /square
+Problem 3.8. The Euclidean space Rnof dimension n≥2, does it contain a non-trivial finite or countable
+kaleidoscopical subset K⊂Rn?If such a set Kexists, then its cardinality |K|is not less that the chromatic
+number of Rn.
+Let us recall that the chromatic number χ(X) of a metric space Xis equal to the smallest number κof
+colors for which there is a coloring of Xwithout monochrome points on the distance 1. It is known that
+4≤χ(R2)≤7 but the exact value of χ(R2) is not known. There is a conjecture that χ(Rn) = 2n+1−1, see
+[10,§47].
+Problem 3.9. Is every finite kaleidoscopical configuration in a (finite) pr imitiveG-space trivial?
+Some examples of infinite G-spaces with only trivial finite kaleidoscopical configurations can be found in [4,
+chapter 8]
+A space Rncan also be considered as a G-space with respect to the group G=Aff(Rn) ={λx+a:λ∈
+R\{0},a∈Rn}of all affine transformations. The only kaleidoscopical configuratio nsKof cardinality |K|<c
+in this space are singletons as any line that contains more then one po int of kaleidoscopical configuration has
+no distinct points of the same color. On the other hand, every affine subspace of Rnis kaleidoscopical.
+Question 3.10. Is there any non-splitting kaleidoscopical configuration i nRnwith action of Aff(Rn)
+Restricting ourself with only translations of Rn, we get a kaleidoscopical configuration of any size κ, 1≤
+κ≤c. It follows from well-known decomposition of Rninthe direct sum of rationals and the observation that
+Zhas a kaleidoscopical configuration of any finite size.
+4.Haj´os properties in groups and G-spaces
+In this section we reveal the relation of splittability of kaleidoscopic c onfigurations in finite Abelian groups
+to the Haj´ os property introduced in [2] and studied in [8], [11], [12].
+We recall that an Abelian group Ghas theHaj´ os property if for each factorization G=ABeitherAorB
+is periodic. A subset Aof a group Gis called periodic ifA=gAfor some non-zero element g∈G. Finite
+Abelian groups with Haj´ os property were classified in [8]:
+Theorem 4.1 (Haj´ os-Sands) .A finite Abelian group Ghas the Haj´ os property if and only if Gis isomorphic
+to a subgroup of a group that has one of the following types:
+(pn,q),(p2,q2),(p2,q,r),(p,q,r,s ),(p,p),(p,3,3),(32,3),
+(p3,2,2),(p2,2,2,2),(p,22,2),(p,2,2,2,2),(p,q,2,2),(2n,2),(22,22),
+wherep<q<r<s are distinct primes and n∈N.
+A groupGis of type (n1,...,n k) ifGis isomorphic to the direct sum of cyclic groups Cn1⊕···⊕Cnk.
+Now let us define two weakenings of the Haj´ os property.
+Definition 4.2. An Abelian group Gis defined to have
+•thesemi-Haj´ os property if each complemented subset A/subsetnoteqlGeither is periodic or has a periodic
+complementer factor in G;
+•thedemi-Haj´ os property if for each factorization G=ABone of the factors A,B either is periodic or
+has a periodic complementer factor.
+It is clear that for each Abelian group G
+Haj´ os⇒semi-Haj´ os ⇒demi-Haj´ os .
+Problem 4.3. Is the semi-Haj´ os property of finite Abelian groups equival ent to the demi-Haj´ os property?KALEIDOSCOPICAL CONFIGURATIONS 9
+The demi-Haj´ os property was (implicitly) defined in [9] and follows fro m the quasi-periodicity of any fac-
+torization of the group. In contrast to the Haj´ os property, at the moment we have no classification of finite
+Abelian groups possessing the demi-Haj´ os property. It is even no t known if each finite cyclic group has the
+demi-Haj´ os property, see Problem 5.4 in [12]. The best known positiv e result on the semi-Haj´ os property is
+the following version of Theorem 5.13 [12]:
+Theorem 4.4 (Bruijn-Szab´ o-Sands) .Each finite Abelian group Gof square-free order |G|has the semi-Haj´ os
+property.
+We say that a number nissquare-free ifnis not divisible by the square p2of any prime number p.
+Surprisingly, the following problem of Fuchs and Sands [3, p.364], [9], [12 , p.120] posed in 60-ies still is open:
+Problem 4.5. Has each finite Abelian group the demi-Haj´ os property?
+The “semi” version of this problem also is open:
+Problem 4.6. Has each finite Abelian group the semi-Haj´ os property?
+The semi-Haj´ os property is tightly connected with the splittability o f kaleidoscopical configurations. In
+order to state the precise result, let us generalize the definition of the semi-Haj´ os property to G-spaces.
+Definition 4.7. AG-spaceXhas thesemi-Haj´ os property if for each kaleidoscopic subset K/subsetnoteqlXthere is a
+G-invariant equivalence relation E/\e}atio\slash= ∆XonXsuch thatKisE-parallel or E-orthogonal and the set K/E is
+kaleidoscopic in the G-spaceX/E .
+For finite Abelian groups this definition of the semi-Haj´ os property agrees with that given in Definition 4.2.
+Proposition 4.8. A finite Abelian group Ghas the semi-Haj´ os property if and only if it has that proper ty as
+aG-space.
+Proof. Assume that the group Ghas the semi-Haj´ os property. To show that the G-spaceGhas the semi-
+Haj´ os property, take any kaleidoscopic subset A⊂G. By Corollary ??,Ais complementable and hence
+has a complementer factor B. SinceGhas the semi-Haj´ os property, either Ais periodic or else Ahas a
+periodic complementer factor. In the latter case we can assume th at the complementer factor Bis periodic.
+Consequently there is a non-trivial cyclic subgroup H⊂Gsuch that either A+H=AorB+H=B.
+Consider the quotient group G/H and the quotient homomorphism q:G→G/H . By Lemma 2.6 of [12], the
+imagesA/H =q(A) andB/H =q(B) form a factorization G/H =A/H·B/H of the quotient group G/H .
+Consequently, the set A/H is complemented in G/H and by Corollary ??, it is kaleidoscopic in G/H .
+The subgroup Hinduces aG-invariant equivalence relation E={(x,y)∈G:x−y∈H}whose quotient
+spaceG/E coincides with the quotient group G/H . We claim that the set Ais eitherE-parallel or E-orthogonal.
+By the choice of the group H, we getA=A+HorB=B+H. In the first case the set AisE-parallel. In
+the second case AisE-orthogonal as ( A−A)∩H⊂(A−A)∩(B−B) ={0}.
+Now assuming that the G-spaceGhas the semi-Haj´ os property, we shall prove that the group Ghas the
+semi-Haj´ os property. Given any complemented subset A⊂Gwe need to show that either Ais periodic
+or elseAhas a periodic complementer factor. By Corollary ??, the setAis kaleidoscopic in the G-space
+G. The semi-Haj´ os property of the G-spaceGguarantees the existence of an invariant equivalence relation
+E/\e}atio\slash= ∆GonGsuch thatAisE-parallel or E-orthogonal and A/E is kaleidoscopic in G/E . It follows that the
+equivalence class H= [0]Eof zero is a subgroup of the group G. Taking into account that EisG-invariant,
+we conclude that ( x,y)∈Eiffx−y∈[0]E. So,G/E coincides with the quotient group G/H . The setA/H ,
+being kaleidoscopical, is complemented in G/H according to Corollary ??. Consequently, there is a subset
+BH⊂G/H such thatG/H =A/H·BH. Letq:G→G/H be the quotient map and s:G/H→Gbe any
+section ofq.
+Now consider two cases. If AisE-parallel, then A=A+His periodic and complemented as B=s(BH) is
+a complementer factor to AinG. IfAisE-orthogonal, then the complete preimage B=q−1(BH) is a periodic
+complementer factor to AinG. /square
+Now we reveal the relation between the semi-Haj´ os property and the splittability of kaleidoscopic sets.
+Proposition 4.9. If each kaleidoscopic subset of a transitive G-spaceXis splittable, then Xhas the semi-Haj´ os
+property.10 T. BANAKH, O. PETRENKO, I.V. PROTASOV, S. SLOBODIANIUK
+Proof. To show that Xhas the semi-Haj´ os property, fix any kaleidoscopic subset K⊂X. By our assumption,
+Kis (E0,...,E m)-splittable by some increasing chain of invariant equivalence relation s ∆X=E0⊂ ··· ⊂
+Em=X×X. For every i≤mconsider the quotient G-spaceXi=X/Eiand letqi:X→Xibe the quotient
+projection. Also let Ki=qi(K)⊂Xi. By Proposition 2.2, Kiis kaleidoscopic in the G-spaceXi. In particular,
+K1is kaleidoscopic in X1=X/E1. By Definition 2.4, K=K0is eitherE1-parallel or E1-orthogonal. This
+means that Xhas the semi-Haj´ os property. /square
+Theorems 3.1 and Proposition 4.9 imply
+Corollary 4.10. Each isometrically homogeneous ultrametric space with fini te distance scale has the semi-
+Haj´ os property.
+AG-spaceYis defined to be a quotient of aG-spaceXifYis the image of Xunder aG-equivariant map
+f:X→Y.
+Proposition 4.11. Each kaleidoscopical subset of a G-spaceXis splittable provided that:
+(1)each quotient G-space ofXhas the semi-Haj´ os property and
+(2)Xadmits no strictly increasing infinite sequence (En)n∈ωofG-invariant equivalence relations.
+Proof. Assume that some kaleidoscopic subset K⊂Xis not splittable. Let K0=K,E0= ∆X, and
+X0=X/E0=X. SinceXhas the semi-Haj´ os property, there is a G-invariant equivalence relation E1/\e}atio\slash= ∆X
+onX0such that the set K1=K0/E0is kaleidoscopic in the G-spaceX1=X0/E1andK0is eitherE1-parallel
+orE1-orthogonal.
+By our assumption, Kis not splittable, so X1is not a singleton. The G-spaceX1=X/E1, being a
+quotient of X, has the semi-Haj´ os property. Consequently, for the kaleidosc opic setK1⊂X1there is a
+G-invariant equivalence relation ˜E2/\e}atio\slash= ∆X1onX1such that the set K1is˜E2-parallel or ˜E2-orthogonal and
+the quotient set K2=K1/˜E1is kaleidoscopical in the G-spaceX2=X1/˜E1. Letq1
+2:X1→X2be the
+quotient projection. The composition q1
+2◦q1:X→X2determines the G-invariant equivalence relation
+E2={(x,x′)∈X2:q1
+2◦q1(x) =q1
+2◦q1(x′)}onXsuch thatX/E2=X2andK2=K/E2andK1is either
+E2/E1-parallel or E2/E1-orthogonal.
+Continuing by induction, we shall produce an infinite increasing seque nce (En)n∈ωofG-invariant equivalence
+relations on Xsuch that for every n∈Nthe setKn=K/Enis kaleidoscopic in the G-spaceX/EnandKis
+eitherEn/En−1-parallel or En/En−1-orthogonal. But the existence of an infinite strictly increasing seq uence
+ofG-invariant equivalence relations on Xcontradicts our assumption. /square
+Since each quotient group of a finite Abelian group Gis isomorphic to a subgroup of G, Proposition 4.11
+implies:
+Corollary 4.12. If each subgroup of a finite Abelian group Ghas the semi-Haj´ os property, then each kaleido-
+scopic subset K⊂Gis splittable.
+Question 4.13. Assume that a finite Abelian group Ghas the semi-Haj´ os property. Has each subgroup of G
+that property?
+The classification of finite Abelian groups with Haj´ os property given in Theorem 4.1 implies that this
+property is inherited by subgroups. Because of that, Corollary 4.1 2 implies:
+Corollary 4.14. For a finite Abelian group Gwith the Haj´ os property, each kaleidoscopical subset K⊂Gis
+splittable.
+Also Proposition 4.11 and Theorem 4.4 imply:
+Corollary 4.15. For a finite Abelian group Gof square-free order |G|each kaleidoscopical subset K⊂Gis
+splittable.
+Remark 4.16. It follows from Proposition 4.9 and Corollary 4.12 that Problems ??and 4.6 are equivalent
+(and both are open and apparently difficult).
+According to an old result of Haj´ os [2], if in a factorization Z=A+Bof the infinite cyclic group Zthe
+factorAis finite, then the factor Bis periodic. We do not know if the same is true for the groups Znwith
+n≥2.KALEIDOSCOPICAL CONFIGURATIONS 11
+Problem 4.17. Assume that Zn=A+Bis a factorization with finite factor A. Is the factor Bperiodic?
+HasAa periodic complementer factor?
+References
+[1] M. Aigner, G. Ziegler, Proofs from The Book , Springer-Verlag, Berlin, 2010.
+[2] G. Haj´ os, Sur la factorisation des groupes ab` eliens ,ˇCasopis Pˇ est. Mat. Fys. 74(1949), 157–162.
+[3] J.M. Irwin, E.A. Walker, Topics in Abelian Groups , Glenview, Illinois, 1963.
+[4] I. Protasov, T. Banakh, Ball Structures and Colorings of Graphs and Groups , Math. Stud. Monogr. Ser, Vol. 11, VNTL
+Publisher, Lviv, 2003.
+[5] I. Protasov, K. Protasova, Kaleidoscopical Graphs and Hamming codes , Voronoi’s Impact on Modern Science, Book 4, Volume
+1, Proc. 4th Intern. Conf. on Analytic Number Theory and Spat ial Tesselations, Institute of Mathematics, NAS of Ukraine ,
+Kyiv, 2008, 240–245.
+[6] K.D. Protasova, Kaleidoscopical Graphs , Math. Stud., 18(2002), 3–9.
+[7] H.J. Ryser, A combinatorial theorem with an application to latin rectan gles, Proc. Amer. Math. Soc. 2(1951), 550–552.
+[8] A.D. Sands, On the factorisation of finite Abelian groups II , Acta Math. Acad. Sci. Hungar. 13(1962), 153–169.
+[9] A.D. Sands, On a conjecture of G. Haj´ os , Glasgow Math. J. 15(1974), 88–89.
+[10] A. Soifer, The mathematical coloring book , Springer, 2009.
+[11] S. Szab´ o, Topics in Factorization of Abelian Groups , Birkh¨ auser Basel, 2005.
+[12] S. Szab´ o, A. Sands, Factoring groups into subsets , CRC Press, 2009.
+Faculty of Mechanics and Mathematics, Lviv University, Unive rsytetska 1, Lviv 79000, Ukraine
+E-mail address :t.o.banakh@gmail.com
+Faculty of Cybernetics, Kyiv University, Volodymyrska 64, K yiv 01033, Ukraine
+E-mail address :opetrenko72@gmail.com; i.v.protasov@gmail.com;
+Faculty of Mechanics and Mathematics, Kyiv University, Volo dymyrska 64, Kyiv 01033, Ukraine
+E-mail address :slobodianiuk@yandex.ru
\ No newline at end of file
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+arXiv:1001.0904v2  [gr-qc]  1 Feb 2010A new approach for doing theoretical and numeric work with
+Lemaˆ ıtre–Tolman–Bondi dust models.
+Roberto A. Sussman‡
+‡Instituto de Ciencias Nucleares, Universidad Nacional Aut ´ onoma de
+M´ exico (ICN-UNAM), A. P. 70–543, 04510 M´ exico D. F., M´ exi co.
+(Dated: November 9, 2018)
+We introducequasi–local integral scalar variables for the studyof spherically symmetric Lemaˆ ıtre–
+Tolman–Bondi (LTB) dust models. Besides providing a covari ant, and theoretically appealing, in-
+terpretation for the parameters of these models, these vari ables allow us to study their dynamics
+(in their full generality) by means of fluid flow evolution equ ations that can be handled with simple
+numerical techniques and has a significant potential for ast rophysical and cosmological applications.
+These evolution equations can also be understood in the fram ework of a gauge invariant and covari-
+ant formalism of spherical non–linear perturbations on a FL RW background. The covariant time
+splitting associated the new variables leads, in a natural w ay, to rephrase the known analytic solu-
+tions within an initial value framework in which covariant s calars are given by simple scaling laws.
+By using this re–parametrization of the analytic solutions , we re–examine and provide an alterna-
+tive outlook to various theoretical issues already treated in the literature: regularity conditions, an
+Omega parameter, as well as the fitting of a given LTB model to r adial profiles of density or velocity
+at different cosmic times. Other theoretical issues and nume ric applications will be examined in
+separate articles.
+PACS numbers: 98.80.-k, 04.20.-q, 95.36.+x, 95.35.+d
+I. INTRODUCTION.
+Spherically symmetric LTB dust models are among
+the best known exact solutions of Einstein’s equations
+[1]. Since they provide access to non–linear effects as-
+sociated with inhomogeneous sources by means of ana-
+lytic and/or tractable numeric solutions, they have re-
+ceived widespread attention in the literature (see [2, 3]
+for comprehensive reviews). These exact solutions are
+used mostly to study cosmological inhomogeneities, as
+for example the series of articles in the references [7–
+10], but have also been employed to examine a variety of
+issues, for example as useful toy models to describe grav-
+itational collapse under a classical [4, 5] and quantum
+[6] approach. LTB models have also been widely used
+[11–18] in the context of the ongoing widespread theo-
+retical and empiric effort to explore the possibility that
+cosmic observations could be explained without resorting
+to dark energy, but by taking into consideration the fully
+non–linear effects of inhomogeneities in the context of
+General Relativity (see [19] for a review). These models
+are also helpful as theoretical tools to probe various aver-
+aging techniques applied to cosmological inhomogeneities
+[14, 15, 20–22], in particular, the scalar averaging formal-
+ism developed by Buchert and coworkers [23]. One finds
+in the extensive literature on these models a preferred
+set of free functions and analytic solutions (implicit and
+parametric) that has become a sort of standard [2, 3, 24]
+(even in numeric work on these models [14–18]).
+Given the success of these models and the fact that
+their standard parametrization is adequate and works in
+practice, then it is legitimate to ask for the justification
+of an article proposing and discussing alternative vari-
+ables. The answer to this question is simple: alternativevariables may provide or motivate either new theoretical
+and empiric developments or allow us to grasp known
+results under a different perspective, thus illuminating
+possible unsuspected connections with other models or
+theoretical issues. In this context, we propose an alter-
+native description of LTB dust models that is based on
+a set of covariant scalar variables defined by applying
+to arbitrary scalars the same type of integral construct
+that yields the Misner–Sharp quasi–local mass function
+[26–28]. Hence, we find it natural to call these variables
+quasi–local scalars.
+The quasi–local scalars mentioned above have already
+been used for a dynamical system study of dust LTB
+models [29] and in the application of Buchert’s averaging
+formalism to LTB models [21, 22], as well as in a numeric
+study of inhomogeneous dark energy sources associated
+with an LTB metric, but with an energy–momentum ten-
+sor that is more general than dust [30, 31]. The pure dust
+case merits the separate study that we provide here, not
+only because dust is an adequate source to model cold
+dark matter in cosmological scales (and a much less ar-
+tificial source for an LTB metric than the fluid tensor in
+[30, 31]), but because dust sources allow for a qualitative
+analytic framework that complements numeric work and
+yields very useful theoretical and practical information.
+The potential of the quasi–local variables for ana-
+lytic and qualitative work on LTB models follows read-
+ily from parametrizing their known analytic expressions
+in terms of these variables and their fluctuations. The
+new variables lead in a natural way to an initial value
+parametrization of LTB models, which in turn leads to
+the introduction of a FLRW–like metric and scaling laws
+for density, Hubble expansion, spatial curvature, shear
+and all other covariant quantities of physical or geomet-2
+ric relevance. Various elements of this parametrization
+have been already considered in the literature [13–17],
+but only as a useful coordinate ansatzes that are jus-
+tified because the involved expressions resemble FLRW
+parameters. In this article we provide an appealing the-
+oretical and covariant context for these expressions that
+could allow for new empiric or numeric results and/or a
+better understanding of existing work.
+The quasi–local scalar representation applied the LTB
+dust case leads to a complete description of the dynamics
+in the context of “fluid flow” (or “1+3”) evolution equa-
+tions that are fully general, but can be handled by simple
+numerical methods, and thus have a significant potential
+for empiric modeling of cosmological dust sources. This
+scalar representation also leads to a description of LTB
+dust models in the framework of spherical, non–linear,
+gauge invariant and covariant (GIC) perturbations over
+a FLRW background (in the context of the fluid flow
+perturbation formalism derived by Ellis and Bruni [32–
+34]). The resulting perturbation framework is also con-
+sistent with the traditional perturbation approach [37],
+as the evolution equations of quasi–local scalars and their
+fluctuations generalize the evolution equations of linear
+perturbations of spherical dust sources in the comoving
+gauge [38].
+The plan of the article is described below. We sum-
+marize in section II the metric, field equations, analytic
+solutions and basic regularity of LTB models in the con-
+ventional variables. In section III we introduce the rep-
+resentation of local covariant scalars and their evolution
+equations in the context of the covariant time slicing asso-
+ciated with a fluid flow approach to the dynamics of the
+models. The quasi–local scalars and their fluctuations
+are introduced and discussed in section IV, while section
+V deals with some theoretical and practical implications
+of these scalars. In particular, the quasi–local scalars
+lead naturally to an initial value approach in which ini-
+tial value functions are given by the new variables eval-
+uated at a fiducial (initial) hypersurface defined by the
+1+3 time slicing. This approach leads to a FLRW–like
+form for the LTB metric, which has been suggested previ-
+ously (see [13, 14]) but as an ansatz without a theoretical
+context. The initial value approach also leads to simple
+scaling laws for all scalars and an intuitive gauge for the
+radial coordinate. Hence, in section VI we rewrite the
+analytic solutions of the models in terms of the above
+mentioned initial value parametrization, which leads in
+section VII to an elegant characterization of the coor-
+dinate locus of the curvature singularities that is sim-
+pler and more intuitive than that using conventional vari-
+ables. In section VIII we rewrite the Hellaby–Lake con-
+ditions [24, 39] to avoid shell–crossings in terms of the
+initial value functions, while in section IX we rewrite the
+analytic solutions in the same initial value approach, but
+using a suitably defined Omega parameter and the quasi–
+local expansion factor as initial value functions (which
+can be used also to parametrize the Hellaby–Lake condi-
+tions).In section X we address (as an application) an inter-
+esting theoretical issue raised by Hellaby and Krasinski
+(see [7, 10, 18]), namely: the possibility of mapping by
+an LTB model of a given density or velocity radial pro-
+file at one cosmic time to another density or velocity
+profile at a latter time. We show how the initial value
+parametrization of the analytic solutions greatly simpli-
+fies the understanding and handling of this problem. In
+section XI we derive three equivalent systems of fluid
+flow evolution equations for the quasi–local scalars and
+their fluctuations. Each of these systems provides a full
+description of the dynamics of the models, and all are
+effectively equivalent to systems of ordinary differential
+equations subjected to constraints, hence they are tech-
+nically more accessible than (and as fully general as) the
+system that arises for the local scalars in the 1+3 fluid
+flow framework. We explain how initial conditions can be
+set up and argue that these equations have a significant
+potential for numeric computation of quantities of ob-
+servational interest (null geodesics, red shift factor, area
+and luminosity distances, etc). In section XII we show
+how the quasi–local scalars, their fluctuations and their
+evolution equations can be understood in the framework
+of a gauge invariant and covariant formalism of spherical
+non–linear perturbations.
+In section XIII we derive a fourth system of fluid flow
+evolution equations, but now for the Omega and Hub-
+ble parameters introduced in section IX. These evolution
+equations lead to a dynamical system (like the one in
+[29]). We comment on the relation between the perturba-
+tion formalism of section XII and these parameters (and
+others proposed in the literature [16, 17]), thus provid-
+ing a theoretical context for quantities given as coordi-
+nate ansatzes on the basis of their resemblance to FLRW
+Omega and Hubble parameters. We discuss in section
+XIV various aspects of special LTB configurations, such
+as: closed models, models with simultaneous big–bang or
+collapsing singularities, simultaneous maximal expansion
+and a mixed elliptic/hyperbolic configuration. Our con-
+clusions and suggestions for future work are summarized
+in section XV. In Appendix A we discuss various regular-
+ity issues of the quasi–local scalars and their fluctuations,
+while Appendix B discusses the relation between quasi–
+local spatial curvature (which determines the kinematic
+class: parabolic, hyperbolic, elliptic) and local spatial
+curvature.
+II. LTB DUST MODELS IN THE
+CONVENTIONAL VARIABLES.
+Lemaitre–Tolman–Bondi (LTB) dust models [1, 2] are
+the spherically symmetric solutions of Einstein’s equa-
+tions characterized by the LTB line element and the3
+energy–momentum tensor
+ds2=−c2dt2+R′2
+1 +Edr2+R2(dθ2+ sin2θdϕ2),(1)
+Tab=ρc2uaub, ua=δa
+0,(2)
+whereR=R(t,r),R′=∂R/∂r ,E=E(r) and
+ρ=ρ(t,r) is the rest–mass density. The field equations
+reduce to
+˙R2=2M
+R+E, (3)
+2M′=κρR2R′, (4)
+whereκ= 8πG/c2,M=M(r) and ˙R=ua∇aR=
+∂R/∂ (ct).
+A. Kinematic equivalence class and analytic
+solutions.
+It is common usage in the literature (see [2, 3, 24])
+to classify the solutions of (3) in “kinematic equivalence
+classes” given by the sign of E, which determines the
+existence of a zero of ˙R2. SinceE=E(r), the sign of
+this function can be, either the same in the full range of
+r, in which case we have LTB models of a given kine-
+matic class, or it can change sign in specific ranges of r,
+defining LTB models with regions of different kinematic
+class (see [24]). The solutions of the Friedman–like field
+equation (3) for each kinematic class take the following
+well known parametric form:
+Parabolic models or regions: E= 0.
+c(t−tbb) =2
+3η3, R = (2M)1/3η2, (5)
+Hyperbolic models or regions: E≥0.
+R=M
+E(coshη−1), c (t−tbb) =M
+E3/2(sinhη−η),
+(6)
+Elliptic models or regions: E≤0.
+R=M
+|E|(1−cosη), c (t−tbb) =M
+|E|3/2(η−sinη),
+(7)
+wheretbb=tbb(r) is called “big bang time”, as it marks
+the coordinate locus of the central expanding curvature
+singularity: R(t,r) = 0 for r≥0. It emerges as an “in-
+tegration constant” in the integration of (3). Notice that
+the locus of central curvature singularity is distinct from
+that of the center of symmetry R(t,0) = 0 (see Appendix
+A 1). Besides the kinematic class, LTB models that ad-
+mit (at least) one symmetry center can be classified as
+“open” or “closed”, respectively corresponding to the hy-
+persurfaces of constant tbeing topologically equivalent to
+R3orS3(see Appendix A 3).In order to deal with a given specific model by means
+of (5)–(7), we need to prescribe the three “conventional”
+free functions
+M(r), E (r), t bb(r). (8)
+However, the metric (1) is invariant under rescalings of
+the radial coordinate r=r(¯r), hence it is always possible
+to eliminate one of the free functions in (8) by a suitable
+choice of the radial coordinate, thus leaving only two ba-
+sic irreducible free functions.
+Given a choice of free functions (8), the solutions (5)–
+(7) can be used to find the remaining relevant quantities
+that may be required for a specific problem, a procedure
+that has been used abundantly in the literature [2, 3].
+B. Regular LTB models
+We will assume henceforth the existence of (at least)
+one symmetry center marked by r= 0 (see Appendix
+A 1), and will denote by “regular LTB model” any LTB
+configuration for which the condition
+sign(R′) = sign( M′) = sign(√
+1 +E), (9)
+holds for all tand allrnot marking a symmetry center
+(sinceM′(0) = 0, E(0) = 0 and R′(t,0) = 1 [24, 25]).
+Conditions (9) imply that if R′, M′,√
+1 +Ehave a zero
+under regular conditions, it must be a common zero of
+the same order at some worldline r=rtv>0. Violation
+of (9),i.e.R′= 0 forM′/ne}ationslash= 0,√
+1 +E/ne}ationslash= 0, results in a
+shell crossing singularity for which ρ→ ∞ occurs with
+R > 0. Condition (9) together with (4) also implies
+thatρ≥0 holds everywhere and that it is bounded
+everywhere except at the coordinate locus of a central
+singularity ( i.e. points for which R(t,r) = 0 holds
+that are not the worldline of a symmetry center). The
+restrictions on the free parameters (8) that guarantee
+the fulfillment of (9) are the well known Hellaby–Lake
+(necessary and sufficient) conditions [24, 25, 39] given by
+Parabolic and hyperbolic models or regions:
+R′>0⇔ {M′≥0, E′≥0, t′
+bb≤0},(10)
+Elliptic models or regions:
+±R′>0⇔ ±M′≥0,±t′
+bb≤0,
+±/bracketleftbiggM′
+M−3
+2E′
+E+ct′
+bb|E|3/2
+2πM/bracketrightbigg
+≥0,
+(11)
+where only expanding configurations are considered in
+(10) and the ±sign in (11) accounts for the fact that R′<
+0 occurs in elliptic models whose that admit a second
+symmetry center (see Appendix A 3). The equal sign
+holds only at symmetry centers and at values of rwhere
+R′= 0.4
+III. COVARIANT OBJECTS AND THEIR “1+3”
+EVOLUTION EQUATIONS.
+The normal geodesic 4–velocity in (1)–(2) defines a
+natural time slicing in which the space slices are the
+hypersurfaces3T[t], orthogonal to uaand marked by
+arbitrary constant values of t[46]. The metric of the
+3T[t] is simply hab=gab+uaub=gijδi
+aδj
+b(or (1) with
+dt= 0). The proper volume element and 3–dimensional
+Ricci scalar associated with these hypersurfaces are
+dVp=/radicalbig
+det(hab) d3x=R2R′sin2θdrdθdϕ√
+1 +E,(12)
+3R=−2(ER)′
+R2R′=−2E′
+R′R−2E
+R2, (13)
+Given this time splitting, the covariant objects in dust
+LTM models are ρand3Rin (4) and (13), together with
+the shear and electric Weyl tensors, σab=˜∇(aub)−
+(Θ/3)habandEabucudCabcd, where ˜∇a=hb
+a∇b, and
+Cabcdis the Weyl tensor, and the expansion scalar, Θ,
+is given by
+Θ = ˜∇aua=2˙R
+R+˙R′
+R′. (14)
+Since LTB models (as all spherically symmetric space-
+times) are locally rotationally symmetric (LRS), all their
+covariant objects are either scalars or entirely charac-
+terized by covariant scalars [36]. Therefore, the spatial
+trace–less shear and electric Weyl tensors can be given
+in terms of single scalar functions, Σ ,E, in a covariant
+manner:
+σab= Σ Ξab⇒ Σ =1
+6σabΞab=−1
+3/parenleftBigg˙R′
+R′−˙R
+R/parenrightBigg
+,
+(15)
+Eab=EΞab⇒ E =1
+6EabΞab=−κ
+6ρ+M
+R3,(16)
+where Ξab≡hab−3χaχbandχa=√
+hrrδa
+ris the unit
+vector orthogonal to uaand to the 2–spheres orbits of
+SO(3).
+A description of the dynamics of LTB models, which
+is an equivalent alternative to the field equations and
+their analytic solutions (5)–(7), follows by considering
+the following set of covariant scalars
+A=/braceleftbig
+ρ,Θ,Σ,E,3R/bracerightbig
+, (17)
+which completely characterize these models. Hence, their
+dynamics is completely determined by suitable evolution
+equations and constraints for these scalars. Following
+the “fluid flow” or covariant “1+3” framework of Ehlers,
+Ellis, Bruni, Dunsbury and van Ellst [32–34, 36], and
+considering (15) and (16), the 1+3 evolution equationsfor the scalars (17) are
+˙Θ =−Θ2
+3−κ
+2ρ−6Σ2, (18a)
+˙ρ=−ρΘ, (18b)
+˙Σ =−2Θ
+3Σ + Σ2−E, (18c)
+˙E=−κ
+2ρΣ−3E/parenleftbiggΘ
+3+ Σ/parenrightbigg
+, (18d)
+while the spacelike and Hamiltonian constraints are
+/parenleftbigg
+Σ +Θ
+3/parenrightbigg′
++ 3 ΣR′
+R= 0,κ
+6ρ′+E′+ 3ER′
+R= 0,
+(19)
+/parenleftbiggΘ
+3/parenrightbigg2
+=κ
+3ρ−3R
+6+ Σ2, (20)
+Solving this system of partial differential equations de-
+termines the metric functions and remaining quantities
+though the definitions (3), (4) and (12)–(16) that relate
+these functions to the scalars in (17). However, it can
+be a difficult system to use in numerical work because
+time and radial derivatives cannot be (in general) de-
+coupled. In the following section we will introduce an
+alternative set of covariant scalars whose corresponding
+evolution equations (see section XI) are wholly equivalent
+to (18)–(20) but are easier to handle.
+IV. QUASI–LOCAL SCALAR FUNCTIONS AND
+FLUCTUATIONS.
+The function Mthat appears in (3)–(4) is for LTB
+models the Misner–Sharp quasi–local mass–energy func-
+tion, which is an important invariant in spherically sym-
+metric spacetimes [26–28]. It is basically the integral
+along a spherical comoving domain of the field equation
+(4), which will be well defined if we assume the existence
+of a symmetry center (at r= 0) and can be given as
+2M=2G
+c2/integraldisplay
+D(r)ρFdVp=κ/integraldisplayr
+0ρR2R′dx, (21)
+F ≡√
+1 +E=/bracketleftbigg
+˙R2+ 1−2M
+R/bracketrightbigg1/2
+. (22)
+where the spherical comoving domain D(r) is defined fur-
+ther below and we have used the notation/integraltextr
+0...dx=/integraltextx=r
+x=0...dx. It is evident from (21) and (22) that Mis a
+proper volume integral “weighed” by the invariant scalar
+Fwhich generalizes the “ γ” factor of Special Relativity
+[27, 28]. The integral definition (21) motivates the in-
+troduction of similar “weighed” proper volume integral
+functions which turn out to be very useful in the study
+of LTB models. In order to define such integrals, we re-
+mark that every regular3T[t] contains comoving regions5
+(containing a symmetry center) that are diffeomorphic to
+the product manifold
+D(r) =ϑ(r)×S2(θ,φ), ϑ (r)≡ {x|0≤x≤r},(23)
+wherex= 0 marks a symmetry center. Because of spher-
+ical symmetry every scalar function on a domain D(r) is
+equivalent to a real valued function on the real interval
+ϑ(r).
+A. Quasi–local scalar variables.
+Consider now the following
+Definition 1: Quasi–local scalar functions Aq. Letϑ(r)
+be a radial domain in a regular3T[t]. For every scalar
+function A[t0] :ϑ(r)→R, there is a “quasi–local dual
+function” Aq[t0] :ϑ(r)→Rsuch that
+Aq[t0](r) =/integraltextr
+0AFdVp/integraltextr0
+0FdVp=/integraltextr
+0A(t0,x)R2(t0,x)R′(t0,x) dx/integraltextr
+0R2(t0,x)R′(t0,x)dx,
+(24)
+whereFis given by (22). Since it is clear that tis an
+arbitrary but fixed parameter in these integrals, we will
+omit (unless it is necessary) to express explicitly the
+time dependence of scalars AandAq.
+Comment 1 : Definition (24) is invariant under arbitrary
+rescalings of the radial coordinate r=r(¯r) that do not
+violate (9) and (A7). In particular, the proper length
+ℓ=/integraltext√grrdrcan be used as the integration variable at
+each3T[t] (though it is not useful as a global integral
+variable because ℓ=ℓ(t,r)).
+Comment 2 : In general, not all hypersurfaces3T[t] in
+LTB models are fully regular, since the central singular-
+ities (expanding or collapsing) are not simultaneous and
+will intersect some of the3T[t]. For these3T[t] the do-
+main (23) must be suitably restricted and the integrals in
+(24) must be treated as improper integrals. We discuss
+this issue in A 5.
+Comment 3. The quasi–local functions Aqin (24) are
+not “averages”, as they do not comply with the prop-
+erties of average distributions of a continuous random
+variable. They can be recast as proper volume averages
+with “weight factor” Fif we define them as functionals.
+See [22] for discussion and clarification of this issue.
+B. Properties.
+The definition (24) leads to the definition of a “quasi–
+local volume” given by
+Vq(r) =/integraldisplay
+D(r)FdVp= 4π/integraldisplayr
+0R2R′dx=4π
+3R3(r),(25)so that
+˙Vq
+Vq=3˙R
+R= Θq,V′
+q
+Vq=3R′
+R, (26)
+where we expressed (14) as Θ = [ln( R2R′)]˙ and the com-
+mutation of ∂/∂t with the integrals in (24). The quasi–
+local functions comply with the following properties:
+A′
+q= (Aq)′=V′
+q
+Vq[A−Aq], (27a)
+A(r)−Aq(r) =1
+Vq(r)/integraldisplayr
+0A′Vqdx, (27b)
+˙Aq= (Aq)˙ = ( ˙A)q+ (ΘA)q−ΘqAq, (27c)
+C. A representation of covariant quasi–local
+scalars.
+The introduced quasi–local variables provide a com-
+plete and very useful representation of covariant scalars
+that is alternative to (17). Applying the definition (24)
+to (3), (13) and (21) we obtain the quasi–local duals of
+ρ,3Rand Θ
+2mq=2M
+R3, kq=−E
+R2,H2
+q=˙R2
+R2= 2mq−kq.(28)
+where, to simplify notation, we have defined
+2m≡κ
+3ρ, k ≡3R
+6,H ≡Θ
+3, (29)
+Notice that M, R, ˙R=ua∇aRandF=√
+1 +Eare
+invariants in spherical symmetry [26–28], hence mq, kq
+andHqare covariant quantities. As a consequence of
+(28), the scalars Σ and Ein (15) and (16), associated with
+the shear and electric Weyl tensors, become expressible
+as deviations or fluctuations of the local scalars H, m(or
+Θ,ρ) with respect to their quasi–local duals:
+Σ =−(H−H q),E=−(m−mq). (30)
+Since the shear and electric Weyl tensors are covariant
+objects, then these fluctuations are also covariant.
+D. Fluctuations of quasi–local duals.
+A convenient way to relate the local covariant scalars
+Ain (17) and their quasi–local duals Aqfollows by intro-
+ducing the relative deviations (fluctuations)
+δ(A)≡A−Aq
+Aq=A′
+q/Aq
+3R′/R=1
+Aq(r)/integraldisplayr
+0A′Vqdx,(31)
+where we used (27a) and (27b). Considering (30) and
+(31), each scalar Ain (17) can be uniquely expressed in6
+terms of its associated Aqandδ(A):
+m=mq/bracketleftBig
+1 +δ(m)/bracketrightBig
+,H=Hq/bracketleftBig
+1 +δ(H)/bracketrightBig
+,
+k=kq/bracketleftBig
+1 +δ(k)/bracketrightBig
+,Σ =−Hqδ(H),E=−mqδ(m).
+(32)
+By applying the definition (31) to the Friedman–like
+equation for H2
+qin (28) we obtain the following important
+constraint
+2H2
+qδ(H)= 2mqδ(m)−kqδ(k), (33)
+while the relation between δ(m)andδ(k)and radial gra-
+dients of the conventional variables MandE
+M′
+M=3R′
+R/bracketleftBig
+1 +δ(m)/bracketrightBig
+,E′
+E=3R′
+R/bracketleftbigg2
+3+δ(k)/bracketrightbigg
+,(34)
+follows readily from (28) and (31).
+Sincem, k,Handmq, kq,Hqare all covariant scalars,
+theirδ(A)are also covariant. In fact, these fluctuations
+can be understood in the context of a covariant gauge in-
+variant and non–linear perturbation formalism (see sec-
+tion XII and references [30, 31]). Hence, the set
+{Aq,δ(A)}={mq, kq,Hq, δ(m), δ(k), δ(H)}, (35)
+constitutes a complete covariant scalar representation
+that is equivalent and alternative to (17).
+There are important regularity issues associated with
+variables (35), which we discuss in detail in Appendix A.
+For example, any one of the δ(A)might diverge if Aq→0
+under regular conditions (no violation of (9) and finite
+curvature scalars). However, as we show in Appendix
+A 4 this does not lead to a curvature singularity, as the
+Riemann tensor frame components do not diverge (see
+Appendix A 2). Also, the regular radial range of some
+of the hypersurfaces3T[t] is necessarily restricted by a
+central singularity, which does not invalidate the regu-
+larity of these scalars away from the singular coordinate
+surface (see Appendix A 5). While these issues must be
+taken into consideration, they do not affect the usage of
+the variables (35).
+V. SOME THEORETICAL AND PRACTICAL
+IMPLICATIONS OF QUASI–LOCAL SCALARS.
+The quasi–local variables and their relative fluctua-
+tions introduced in the previous section lead in a natural
+way to an initial value parametrization of LTB models,
+which can be very helpful for numeric and qualitative
+work. Various elements of this parametrization has been
+already considered in the literature [14–17], but only as
+a useful set of coordinate ansatzes whose justification is
+merely their resemblance with FLRW parameters. How-
+ever, these ansatzes can acquire a clear covariant meaning
+by its relation with quasi–local scalars.A. A FLRW–like metric and scaling laws for
+covariant scalars.
+Consider the dimensionless scale factor
+L≡R
+Ri, (36)
+whereRi≡R(ti,r) andt=timarks a fiducial initial
+hypersurface3Ti=3T[ti]. We shall denote henceforth all
+quantities evaluated at t=tiby the subindex i. Bearing
+in mind (36), we can re–write (28) as
+mq=mqi
+L3, kq=kqi
+L2, (37)
+H2
+q=˙L2
+L2= 2mq−kq=2mqi−kqiL
+L3, (38)
+which, if we identify Lwith a position dependent FLRW
+scale factor, are formally identical to the scaling laws for
+density, spatial curvature and Hubble factor in FRLR
+dust universes. Applying this parametrization to the
+LTB metric (1) yields the FLRW–like line element
+ds2=−c2dt2+L2/bracketleftBigg
+Γ2R′
+i2dr2
+1−kqiR2
+i+R2
+i/parenleftbig
+dθ2+ sin2θdφ2/parenrightbig/bracketrightBigg
+,
+(39)
+where the new dimensionless metric function Γ is
+Γ≡R′/R
+R′
+i/Ri= 1 +L′/L
+R′
+i/Ri, Γi= 1. (40)
+The local density ( m=κρ/3) and spatial curvature ( k=
+3R/6) satisfy the following scaling laws in terms of L,Γ
+and initial value functions
+m=mqi
+L3[1 +δ(m)] =mi
+L3Γ, (41a)
+k+kqi
+L2[1 +δ(k)] =ki
+L2Γ/bracketleftBigg
+1 +Γ−1
+3 (1 +δ(k)
+i)/bracketrightBigg
+.
+(41b)
+where we used (4), (13) and (32). Comparing (41) with
+(37), and bearing in mind (29) and (31), we obtain the
+following scaling laws for δ(m)andδ(k)
+1 +δ(m)=1 +δ(m)
+i
+Γ, (42a)
+2
+3+δ(k)=2/3 +δ(k)
+i
+Γ, (42b)
+δ(m)−3
+2δ(k)=δ(m)
+i−(3/2)δ(k)
+i
+Γ. (42c)
+Applying (31) to the Hamiltonian constraint (38) yields
+the scaling law for δ(H)
+2δ(H)=2mqδ(m)−kqδ(k)
+2mq−kq=2mqiδ(m)−kqiLδ(k)
+2mqi−kqiL.
+(43)7
+Inserting (43) into (30) allows us to find scaling laws sim-
+ilar to the ones above for H= 3Θ and the scalars Σ ,E,
+respectively associated by (15) and (16) to the shear and
+electric Weyl tensor. Using (31), we can express the scal-
+ing laws (42) as relations between the gradients H′
+q, m′
+q
+andk′
+qandL′. Notice that it is always possible to use
+(38) and (43) to eliminate any one of the three pairs
+Hq, δ(H)ormq, δ(m)orkqandδ(k)in terms of the other
+two.
+The scaling laws (41), (42), and (43) depend on both
+Land Γ. While the effects of their dependence on Lis
+easy to grasp because the qualitative time dependence
+ofL(for a fixed r) can be appreciated directly from the
+Friedman–like equation (38), there is no simple way to
+guess the qualitative behavior of Γ (either in the torr
+directions). As a consequence, there is not much we can
+do with these scaling laws as long as we lack an expression
+for Γ in terms of Land the initial value functions (we
+obtain this expression in section VII).
+B. A radial coordinate gauge.
+The metric of LTB models in the form (39) is also in-
+variant under an arbitrary rescaling r=r(¯r). Since L
+and Γ are time dependent, it is natural when using (39)
+to identify this radial coordinate gauge freedom with the
+freedom to choose the initial value function Ri(r). How-
+ever, because of (A6) and (A7), the choice for a function
+Riis not completely arbitrary: it depends on the topol-
+ogy of the space slices3T[t] (see [25] and Appendix A 3).
+For open models (or LTB regions in which (9) allows for
+R′>0 to hold everywhere) Rican be any monotonously
+increasing function complying with Ri(0) = 0 and R′
+i>0
+for allr. Evidently, the simplest choice in these cases is
+Ri=R0r, (44)
+whereR0is an arbitrary constant characteristic length
+scale. The gauge (44) is a popular choice in the literature
+[14–17], not only due to its simplicity, but because setting
+r=Ri/R0allows us to regard radial dependence as a
+dependence on an fiducial value of an invariant quantity
+(R) that has a clear physical and geometric meaning.
+Also, the choice of R0provides a physical length scale
+for the radial coordinate.
+However, other radial coordinate gauges are often used
+(for example setting Mas radial coordinate, as in [7–
+10]). And, as pointed before, (44) cannot be used in cases
+where a regular zero of R′occurs. For closed models, Ri
+must be selected so that it vanishes at both centers of
+symmetry and R′
+ihas a common same order zero with
+F=√
+1 +E(to avoid the surface layer singularity asso-
+ciated with (123), see [40]). Moreover, even if we use the
+parametrization associated with (39), we are not forced
+to use the radial coordinate gauge for fixing Ri, it is still
+possible to keep this initial value function unspecified and
+use the gauge freedom to fix any one of the other initial
+value functions.VI. THE ANALYTIC SOLUTIONS IN TERMS
+OF AN INITIAL VALUE APPROACH.
+We rephrase in this section the analytic solutions (5)–
+(7) in terms of {mqi, kqi}, which are related to the free
+parameters MandEin (5)–(7) by
+M=mqiR3
+i, E =−kqiR2
+i. (45)
+A third free function (the “bang time”, tbb) necessarily
+appears as an “integration constant” in both, the solu-
+tions of ˙Lin (38) and those of ˙Rin (3). However, as
+we show below, if we rewrite (5)–(7) in the context of an
+initial value problem associated with mqi, kqiandL(con-
+sidering that Li= 1), the bang time tbbcan be obtained
+as a function of mqi, kqi.
+Of course, other combinations of initial value functions,
+such as{mqi,Hqi}or{kqi,Hqi}, can also be considered,
+as (38) relates the three available functions. Since the
+fluctuations follow from the gradients of these functions
+from (31), any two of {mqi, kqi,Hqi}forms an irreducible
+set of initial value functions. As we show in section XI,
+these initial value functions provide also the basic initial
+conditions for the evolution equations under a numeric
+approach.
+A. Parabolic models or regions: kqi= 0
+We express MandRin terms of mqiandLfrom (36)
+and (45) in (5), after re–arranging terms we get the fol-
+lowing closed analytic expression for L
+L=/bracketleftbigg
+1 +3
+2/radicalbig
+2mqic(t−ti)/bracketrightbigg2/3
+, (46)
+where we are only considering expanding configurations
+(Lincreases for t > ti). The bang time follows by setting
+L= 0 and t=tbbin (46)
+ctbb=cti−2
+3/radicalbig2mqi=cti−2
+3Hqi. (47)
+B. Hyperbolic models or regions: kqi<0
+We obtain the following implicit solution of the form
+t=t(R) by eliminating the parameter ηfrom the equa-
+tion forRin (6) and substituting in the equation for t:
+E3/2
+Mc(t−tbb) =Zh(¯R), (48)
+where ¯R= (E/M )RandZhis the function
+u/ma√sto→Zh(u) =u1/2(2 +u)1/2−arccosh(1 + u).(49)
+We express then M, E andRin (48) in terms of mqi, kqi
+andLfrom (36) and (45). The result is
+yic(t−ti) =Zh(xiL)−Zh(xi), (50)8
+where
+xi=|kqi|
+mqi, y i=|kqi|3/2
+mqi. (51)
+Settingt=tbbandL= 0 in (50) and using (49) yields
+the bang time as a function of mqiand|kqi|:
+ctbb=cti−Zh(xi)
+yi. (52)
+C. Elliptic models or regions: kqi>0
+In this case, the implicit solution t=t(R) follows from
+eliminating ηfrom the first equation in (7) and substi-
+tuting in the second one. The resulting implicit solution
+has two branches, an “expanding” one (0 < η < π with
+˙R > 0) and a “collapsing” one ( π < η < 2πwith ˙R < 0):
+|E|3/2
+Mc(t−tbb) =/braceleftbigg
+Ze(¯R) expanding phase
+2π−Ze(¯R) collapsing phase,
+(53)
+where ¯R= (|E|/M)RandZeis given by
+u/ma√sto→Ze(u) = arccos(1 −u)−u1/2(2−u)1/2.(54)
+Proceeding as in the hyperbolic case, we transform (53)
+into
+yic(t−ti)+Ze(xi) =
+
+Ze(xiL) expanding phase
+2π−Ze(xiL) collapsing phase
+(55)
+where
+xi=kqi
+mqi, y i=k3/2
+qi
+mqi, (56)
+It follows from (55)–(54) that Lis restricted by 0 < L≤
+Lmax, where the maximal expansion is Lmax= 2/xi=
+2mqi/kqi, characterized by ˙L= 0 and Hq= 0.
+Settingt=tbbandL= 0 in the expanding phase
+of (55) yields a bang time function, while t=tcolland
+L= 0 in the collapsing phase yields the “crunch” time
+associated with the collapsing singularity. The maximal
+expansion time follows by substituting t=tmaxandL=
+Lmaxin either branch of (55). These times are given by
+ctbb=cti−Ze(xi)
+yi, (57a)
+ctmax=ctbb+π
+yi=cti+π−Ze(xi)
+yi, (57b)
+ctcoll=ctbb+2π
+yi=cti+2π−Ze(xi)
+yi.(57c)
+Notice that (in general) tmax=tmax(r) andtcoll=tcoll(r),
+so neither one coincides with a3T[t] hypersurface (liketbb(r)). For every comoving observer r= const., the time
+evolution is contained in the range tbb(r)< t < t coll(r).
+Rewriting the analytic solutions (5)–(7) as in (46), and
+as the implicit forms (50) and (55) is very useful, as it al-
+lows us to relate (by implicit radial derivation) the gradi-
+ents ofL(and thus of R) with gradients of the intial value
+functions miq, kiq(see section 11). Notice that the time
+splitting associated with uaand the scaling of RwithRi
+inLimposes on LTB models a constraint between tbb
+andmqi, kqi, since (47) and (52) imply that any choice
+of these functions uniquely determines the “age” ti−tbb
+of the initial slice t=tifor any comoving observer. Also,
+the initial slice t=tiin elliptic models always occurs in
+the expanding phase [25].
+VII. CURVATURE SINGULARITIES.
+The new variables allow us to express the coordinate
+locus of curvature singularities exclusively in terms of L
+and Γ by means of (41), (42) and (43). Notice that for
+reasonable initial value functions (bounded and contin-
+uous), all scalars Aq=mq, kq,Hqdiverge as L→0,
+whereas local scalars A=m, k,Hcan also diverge if
+Γ→0 (even if L > 0). Following the standard criteria
+that define a curvature singularity and considering (A4),
+we can identify two known possible singular surfaces
+L(t,r) = 0, central singularity (58a)
+Γ(t,r) = 0 shell crossing singularity .(58b)
+Notice that if Γ >0, then all scalars AandAqonly
+diverge at the central singularity L= 0, but if Γ →
+0, then all the relative fluctuations δ(A)diverge (with
+Aq/ne}ationslash= 0), so that local scalars Adiverge while their quasi–
+local duals Aqremain bounded. This is an obviously
+unphysical effect of shell crossings that must be avoided.
+It is important to remark that characterizing the cur-
+vature singularities as in (58a) and (58b) represents an
+improvement in clarity over the conventional variables.
+In these latter variables “ R= 0” can be either a regular
+symmetry center, or a curvature singularity. Likewise,
+“R′= 0” can be either a shell crossing singularity or a
+regular “turning value” r=rtvassociated with a closed
+model. There is no such ambiguity in our parametriza-
+tion, since L >0 holds at a symmetry center and Γ >0
+holds at the turning value.
+In order to test the avoidance of (58b) we need to
+compute Γ. For this purpose, we derive both sides of
+the solutions (46), (50) and (55), then we use (31) and
+L′/L= (1−Γ)R′
+i/Rito eliminate m′
+qi,k′
+qiandL′in terms
+ofδ(m)
+i, δ(k)
+iand Γ. The result is
+•Parabolic models or regions.
+Γ = 1 +δ(m)
+i−δ(m)
+i
+L3/2. (59)9
+•Hyperbolic and elliptic models or regions.
+Γ = 1 + 3( δ(m)
+i−δ(k)
+i)/parenleftbigg
+1−Hq
+Hqi/parenrightbigg
+−3Hqc(t−ti)/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg
+, (60)
+whereHqandHqifollow from (38), while c(t−ti)
+is given by (50) and (55).
+Inserting (59) or (60) (depending on the kinematic class)
+into (41), (42) and (43) leads to closed analytic expres-
+sions for all these scaling laws in terms of Land ini-
+tial value functions in the context of the initial value
+approach described in previous sections. The functional
+forms of Γ and Hqin (38) and (59) allow us to obtain
+analytic expressions for all the scaling laws of covariant
+scalars (as functions of Land initial value functions).
+From (9), (31), (40) and (42a) applied to A=m, con-
+dition (9) for standard regularity that avoids (58b) now
+reads
+Γ>0∀(ct,r) such that L >0 (61)
+The fulfillment of this condition is examined in the fol-
+lowing section.
+VIII. THE HELLABY–LAKE CONDITIONS.
+We obtain in this section the Hellaby–Lake conditions
+(10)–(11) that fulfill the regularity condition (61) [24, 25,
+39], but now as restrictions on the initial value functions.
+These conditions guarantee absence of shell crossings for
+allt. Notice from (9) and (40) that avoidance of this
+singularity already excludes a surface layer singularity
+associated with (123).
+In order to examine the Hellaby–Lake conditions in the
+initial value parametrization, we need to relate the gra-
+dientsM′andE′appearing in (10)–(11) with the initial
+fluctuations by specializing (34) to t=ti
+M′
+M=3R′
+i
+Ri/bracketleftBig
+1 +δ(m)
+i/bracketrightBig
+,E′
+E=3R′
+i
+Ri/bracketleftbigg2
+3+δ(k)
+i/bracketrightbigg
+.(62)
+We will also need an expression for the radial gradient t′
+bb
+in terms of our initial value functions. This expression
+follows by differentiating both sides of (47) and (52) with
+respect to r:
+ct′
+bb
+3R′
+i/Ri=c(ti−tbb)δ(m)
+i=2δ(m)
+i
+3/radicalbig2mqi, parabolic
+(63a)
+ct′
+bb
+3R′
+i/Ri=δ(m)
+i−δ(k)
+i
+Hqi−c(ti−tbb)/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg
+,
+hyperbolic and elliptic (63b)
+wheretbbin (63b) is given by (52) with Zh(xi) orZe(xi),
+respectively, for hyperbolic and elliptic models. As men-
+tioned before, Rican be always be prescribed as a radial
+coordinate gauge.A. Parabolic models/regions.
+From (59), if L→0 then Γ → −δ(m)
+i/L3/2, while Γ →
+1 +δ(m)
+iforL→ ∞ . Considering (42a), the necessary
+and sufficient condition for (61) to hold is simply
+−1≤δ(m)
+i≤0, (64)
+which from (31) implies m′
+qi≤0. It is evident from (62)
+and (63a) that (64) implies t′
+bb/R′
+i≤0 andM′≥0, and
+so it is equivalent to the Hellaby–Lake conditions (10).
+B. Hyperbolic models/regions.
+For a fixed rall initial value functions are finite, con-
+sider then ti> t1> tbbbutt1≈tbb, so that L≈0 and
+Hq→ ∞ , and thus (60) becomes
+Γ≈ −3Hq
+Hqi/bracketleftbigg
+δ(m)
+i−δ(k)
+i−Hqic(ti−t1)/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg/bracketrightbigg
+=−3Hq/bracketleftbiggct′
+bb
+3R′
+i/Ri+c(t1−tbb)/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg/bracketrightbigg
+,
+(65)
+where we used (63b). Now consider t→ ∞ so thatL→
+∞for allr. From (38) and (50) we have in this limit
+Hq→0 and
+Hqc(t−ti)≈1+Zh(xi) + ln(2xiL)−2
+xiL+O(L−2),(66)
+hence Γ in (60) becomes
+Γ≈1 +3
+2δ(k)
+i+O(L−2). (67)
+Bearing in mind, from (9) and (A6), that R′
+i>0 for
+hyperbolic models/regions and that (61) must hold as
+t1→tbbin (65), and since the scaling law (42a) and
+(65)–(67) must hold for all tandr, the conditions for
+(61) are [47]
+ct′
+bb≤0, δ(k)
+i≥ −2
+3, δ(m)
+i≥ −1. (68)
+wheret′
+bbis given in terms of our initial value functions
+by (63b). Notice, from (42b), that all regular hyperbolic
+models comply with δ(k)≥ −2/3 for all their time evolu-
+tion. From (34) and (62), it is evident that (68) are com-
+pletely equivalent to the Hellaby–Lake conditions (10).
+Since these conditions are necessary and sufficient, (68)
+are also necessary and sufficient [24, 25, 39].
+C. Elliptic models/regions.
+We look again at Γ in (60), but now considering that
+Hqchanges sign (from positive to negative) at some t=10
+tmaxwhereL=Lmax= 2mqi/kqi= 2/xi, so that 0 < L≤
+Lmaxand we have the presence of a second (collapsing)
+singularity as L→0 whent→tcoll, withtmaxandtcoll
+given by (57). Near the maximal expansion L→Lmax
+and soHq→0, while c(t−ti)→c(tmax−ti). Thus, we
+have in this limit the following necessary (not sufficient)
+condition for (61):
+Γ = 1 + 3( δ(m)
+i−δ(k)
+i)>0. (69)
+Near the initial singularity (bang) we have Hq→ ∞ as
+t→tbb, while near the collapsing singularity we have
+Hq→ −∞ ast→tcollwithtcollgiven by (57). We
+consider t=t1such that ti> t1> tbbbutt1≈tbb, as
+well ast=t2such that tcoll> t2> tiwitht2≈tcoll. For
+these times Γ takes the form
+Γ≈ −3Hq
+Hqi/bracketleftbigg
+δ(m)
+i−δ(k)
+i−Hqic(ti−t1)/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg/bracketrightbigg
+=−3Hq/bracketleftbiggct′
+bb
+3R′
+i/Ri+c(t1−tbb)/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg/bracketrightbigg
+,
+(70)
+Γ≈3|Hq|
+Hqi/bracketleftbigg
+Hqic(t2−ti)/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg
++δ(m)
+i−δ(k)
+i/bracketrightbigg
+= 3|Hq|/bracketleftbiggct′
+bb
+3R′
+i/Ri+c(t2−tbb)/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg/bracketrightbigg
+,
+(71)
+Since (70) must hold as t1→tbband (71) must hold as
+t2→tcoll, each expression furnishes a second and third
+necessary condition for (61). Since both conditions that
+follow from (70) and (71) must hold for all tin the range
+tbb(r)< t < t coll(r) for all r, these two conditions taken
+together are then the necessary and sufficient conditions
+for the fulfillment of (61), which can be written as
+ct′
+bb
+3R′
+i/Ri≤0,ct′
+coll
+3R′
+i/Ri≥0, δ(m)
+i≥ −1,(72)
+with
+ct′
+coll
+3R′
+i/Ri=/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg
+c(tcoll−tbb) +ct′
+bb
+3R′
+i/Ri,(73)
+where we used c(tcoll−ti) =c(tcoll−tbb)−c(ti−tbb), with
+tbbandtcollgiven by (57), and the equal sign holds only
+at a symmetry center. The fact that one of the three con-
+ditions in (72) to avoid shell crossings in elliptic models
+is basically a sign condition on the gradient t′
+collhas not
+been, apparently, noticed in the extensive literature that
+uses the conventional variables. The reader is requested
+to compare the simplicity and elegance of (72) with the
+cumbersome form (11).
+Conditions (72) imply the following necessary (but not
+sufficient) condition for (61)
+δ(m)
+i−3
+2δ(k)
+i≥0, (74)which, because of (42c), implies that δ(m)−(3/2)δ(k)≥0
+necessarily holds for all times if (61) holds. Since condi-
+tion (74) implies (69), then the latter (also necessary but
+not sufficient) follows from (72) too.
+It is straightforward to show from (31), (57), (62),
+(63b) and (73) that conditions (72) are equivalent to
+the Hellaby–Lake conditions (11). We have expressed
+these conditions in terms of mqi, kqi, δ(m)
+i, δ(k)
+i, but it
+is straightforward to rewrite them in terms of other sets
+initial value functions, like mqi,Hqi, δ(m)
+i, δ(H)
+i. Because
+of (A6) and (40), we do not need to express (72) in terms
+of the±signs for R′, as it is understood that the signs
+oft′
+bbandt′
+collmust be, respectively, the same and the
+opposite of the sign of R′
+i(which because of (A6) will be
+the same as the sign of R′).
+IX. OMEGA AND HUBBLE EXPANSION
+PARAMETERS.
+The fact that Hqbehaves in (38) as a Hubble scalar of a
+FLRW dust universe, has motivated some authors [15, 16]
+dealing with cosmological and astrophysical applications
+to define an “Omega” parameter analogous to that of a
+FLRW dust cosmology
+ˆΩ≡κρq
+3H2q=2mq
+H2q=2mq
+2mq−kq=2mqi
+2mqi−kqiL,(75)
+so that
+ˆΩ−1 =kq
+2mq−kq=kqiL
+2mqi−kqiL, (76)
+where we use the symbol ˆΩ, and not Ω q, because this
+quantity is not the quasi–local dual of the scalar 2 m/H2
+under the definition (24) (see section XIII). The following
+FLRW–like scaling laws hold for ˆΩ and ˆΩ−1:
+ˆΩ =ˆΩi
+ˆΩi−(ˆΩi−1)L, ˆΩ−1 =(ˆΩi−1)L
+ˆΩi−(ˆΩi−1)L,
+(77)
+while the Hamiltonian constraint (38) can be rewritten
+as an analogue of the FLRW Hubble expansion
+H2
+q=H2
+qi/bracketleftBiggˆΩi
+L3+1−ˆΩi
+L2/bracketrightBigg
+. (78)
+The relative fluctuation (43) of the quasi–local Hubble
+factor,δ(H), takes the appealing form
+2δ(H)=ˆΩδ(m)+ (ˆΩ−1)δ(k), (79)
+which can be given in terms of metric functions L,Γ
+and initial value functions ˆΩi, δ(m)
+iandδ(H)
+iby means of
+(42a), (42b), (43) and (77).
+We can identify ˆΩiin (77), (78) and (79) as analogous
+to anr−dependent FLRW Omega factor at a fiducial11
+cosmic time ti
+ˆΩi=2mqi
+H2
+qi=2mqi
+2mqi−kqi, ˆΩi−1 =kqi
+H2
+qi=kqi
+2mqi−kqi,
+(80)
+Notice that the sign of ˆΩi−1 in any region of rdeter-
+mines the sign of ˆΩ−1 for all times, which is the same
+as withkqiandkq. Hence, ˆΩ−1 behaves as the FLRW
+Omega factor for spatial curvature, so that ˆΩi−1 deter-
+mines the kinematic class: parabolic if ˆΩi= 1, elliptic if
+ˆΩi>1 and hyperbolic if 0 <ˆΩi<1. Also, irrespective
+of the kinematic class we have for every LTB model or
+region: ˆΩ→1 asL→0 (near the central curvature
+singularity). For all hyperbolic models or regions ˆΩ is
+bounded between 0 and 1 for all choices of ˆΩi,Hqi(or
+mqi, kqi), hence: ˆΩ→0 asL→ ∞ , whereas for elliptic
+models or regions ˆΩ→ ∞ asL→LmaxbecauseHq→0.
+A. Analytic solutions in terms of ˆΩi
+It is straightforward to parametrize the analytic solu-
+tions in terms of ˆΩi,Hqi. For the parabolic case we have
+ˆΩi= 1, and thus we just obtain (46) with/radicalbig2mqi=Hqi.
+For the hyperbolic and elliptic models or regions we sub-stitute
+mqi=H2
+qiˆΩi, k qi=H2
+qi(ˆΩi−1), (81)
+in (50) and (55). Since ˆΩi,Hqiare roughly equivalent
+to inhomogeneous generalization of fiducial Omega and
+Hubble factors, parametrizing the analytic solutions with
+these initial value functions could be more intuitive than
+doing it with the density and spatial curvature profiles
+mqi, kqi. The solutions (50) and (55) become
+•Hyperbolic models or regions: ˆΩi−1≤0.
+c(t−ti) =W−Wi
+Hqi. (82)
+•Elliptic models or regions: ˆΩi−1≥0.
+Hqic(t−ti) =
+
+W−Wi expanding phase
+πˆΩi(ˆΩi−1)−3/2−W−Wi
+collapsing phase(83)
+where the functions WandWiare
+Hyperbolic models or regions
+W=/bracketleftBig
+ˆΩi+ (1−ˆΩi)L/bracketrightBig1/2
+L1/2
+1−ˆΩi−ˆΩi
+2(1−ˆΩi)3/2arccosh/parenleftbigg2L
+ˆΩi+ 1−2L/parenrightbigg
+, (84a)
+Wi=1
+1−ˆΩi−ˆΩi
+2(1−ˆΩi)3/2arccosh/parenleftbigg2
+ˆΩi−1/parenrightbigg
+. (84b)
+Elliptic models or regions
+W=ˆΩi
+2(ˆΩi−1)3/2arccos/parenleftbigg2L
+ˆΩi+ 1−2L/parenrightbigg
+−/bracketleftBig
+ˆΩi−(ˆΩi−1)L/bracketrightBig1/2
+L1/2
+ˆΩi−1, (85a)
+Wi=ˆΩi
+2(ˆΩi−1)3/2arccos/parenleftbigg2
+ˆΩi−1/parenrightbigg
+−1
+ˆΩi−1. (85b)
+Settingt=tbbandL= 0 in (82) and in the expanding
+phase of (83) yields the bang time for hyperbolic and
+elliptic configurations
+ctbb=cti−Wi
+Hqi, (86)while the maximal expansion and collapse times, t=tmax
+andt=tcoll, given by (57) are
+tmax=ctbb+πˆΩi
+2Hqi[ˆΩi−1]3/2, (87a)
+tcoll=ctbb+πˆΩi
+Hqi[ˆΩi−1]3/2. (87b)
+We can also parametrize Γ, as well as the Hellaby–Lake
+conditions, in terms of ˆΩi,Hqiand their gradients. For12
+the parabolic case, we substitute into (59) and (64)
+δ(m)
+i=2H′
+qi/Hqi
+3R′
+i/Ri, (88)
+while for hyperbolic and elliptic models or regions we
+substitute into (60), (68) and (72) equation (81) plus
+δ(m)
+i=Ri
+3R′
+i/bracketleftBiggˆΩ′
+i
+ˆΩi+2H′
+qi
+Hqi/bracketrightBigg
+, (89a)
+δ(k)
+i=Ri
+3R′
+i/bracketleftBiggˆΩ′
+i
+ˆΩi−1+2H′
+qi
+Hqi/bracketrightBigg
+, (89b)
+whereRican be specified as a choice of radial coordinate.
+The initial value functions ˆΩi,Hqican also be used
+in the analytic solutions given in the forms (5)–(7), in
+terms of the parameter η. For this purpose, we eliminate
+MandEby means of
+ˆΩi=2M
+2M+ERi,H2
+qi=2M+ERi
+R3
+i, (90)
+whereRican be prescribed as a radial coordinate gauge.
+The bang time is then given by (86).
+X. MAPPING OF DENSITY TO DENSITY OR
+VELOCITY TO VELOCITY PROFILES.
+Krasi´ nski and Hellaby [7, 10] have examined an inter-
+esting proposal: the possibility that arbitrary radial pro-
+files of density (or velocity or density and velocity) given
+at two different cosmic times for a finite radial range can
+be fit (or “mapped”) by a unique LTB model. The initial
+value parametrization of the analytic solutions is ideally
+suited to examine this issue, which can also be under-
+stood in terms of the equivalence of boundary and initial
+conditions to solve the evolution equations of sections
+XI and XIII. Our aim here is to illustrate how the ini-
+tial value parametrization can be helpful, hence we only
+examine in detail the mapping between the quasi–local
+density profiles by a hyperbolic model, since all other
+similar mappings can be handled in the same manner.
+Consider two radial profiles of mq. Since t=tiis
+arbitrary, then mqican be one of the profiles, while we
+takemqj=mq(ctj,r) for some tj> tias the second
+profile. If mqiandmqjare known, then
+Lj(r) =L(ctj,r) =/parenleftbiggmqi
+mqj/parenrightbigg1/3
+(91)
+follows from (37) (remember that Li= 1). For a
+parabolic model, we have from (46)
+c(tj−ti) =2
+3/radicalbig2mqi/bracketleftBigg/parenleftbiggmqi
+mqj/parenrightbigg1/2
+−1/bracketrightBigg
+, (92)a constraint that will not be satisfied unless mqiandmqj
+have very restricted forms. For a hyperbolic model we
+rewrite (50) as
+F(|kqi|)≡Zh(α|kqi|)−Zh(β|kqi|)
+β|kqi|3/2=c(tj−ti),(93)
+whereZhis given by (49) and
+α=1
+m2/3
+qim1/3
+qj, β =1
+mqi. (94)
+Since a unique LTB model is determined by prescribing
+mqiand|kqi|, and we assume mqiandmqjknown, then
+(93) becomes a constraint to find the missing initial value
+function |kqi|that determines the LTB model. Solving
+this constraint for known mqi, mqjandc(tj−ti) can only
+be done numerically, but by looking at the properties of
+the function F(|kqi|) we can find the conditions for the
+existence of this solution, as it was done in [7, 10].
+Considering that in an expanding model we have mqi>
+mqjfortj> ti, we have α > β in general, and since Zhis
+monotonously increasing, then the left hand side of (93)
+is positive (like the right hand side). We also have
+F(|kqi|)≈√
+2
+3β/parenleftBig
+α3/2−β3/2/parenrightBig
++O(|kqi|),|kqi| ≈0,
+(95a)
+F(|kqi|)≈α−β
+β|kqi|+O(|kqi|−3/2),|kqi| → ∞,
+(95b)
+which, together with similar expansions of dF/d|kqi|,
+show that F(|kqi|) is monotonously decreasing and
+bounded by
+0< F(|kqi|)≤√
+2
+3β/parenleftBig
+α3/2−β3/2/parenrightBig
+=√
+2
+3/bracketleftBig
+m−1/2
+qj−m−1/2
+qi/bracketrightBig
+(96)
+Hence, considering (91), a unique solution of (93) must
+exist if
+1 +3
+2/radicalbig
+2mqic(tj−ti)</parenleftbiggmqi
+mqj/parenrightbigg1/2
+=L3/2
+j, (97)
+which is the same result found in [7]. Comparison with
+(46) shows that the density-density mapping requires the
+LTB model that maps mqitomqjto expand faster than
+a parabolic model. For the elliptic case we follow the
+same method: using the solution (55) (in the expanding
+and collapsing phases) as a constraint to find the missing
+initial value function kqi. If the local density profiles mi
+andmjare known (instead of mqiandmqj), then we
+have to use the scaling law (41a) with Γ given by (40)
+3mi
+mjR2
+iR′
+i= 3LjΓjR2
+iR′
+i= (R3
+iL3
+j)′, (98)13
+which yields Ljby integration (as miandmjare known
+andRican be fixed as a coordinate gauge). It is straight-
+forward to show that the analytic solutions for the hy-
+perbolic class yield a unique LTB model mapping these
+profiles if (97) holds.
+Krasi´ nski and Hellaby also examined the mapping of
+profiles of velocities defined as v≡˙R/Ri=LHq. To deal
+with this case we eliminate kqiaskqi= 2mqi− H2
+qi, so
+that the initial value functions are mqiandHqi. Then,
+from the Hamiltonian constraint (38) we obtain
+Lj=2mqiR2
+i
+v2
+j−v2
+i+ 2mqiR2
+i, (99)
+which inserted in the solutions (50) and (55) yields the
+constraints needed to obtain the missing initial value
+function mqi. We can also obtain the LTB model that
+results from assuming that the profiles ˆΩiand ˆΩjare
+known. In this case we obtain Ljfrom (77)
+Lj=ˆΩi
+ˆΩj1−ˆΩj
+1−ˆΩi, (100)
+which substituted in the solutions (82) and (83) yields
+the missing initial value function Hqiby means of the
+constraint
+Hqi=W(ˆΩi,ˆΩj)−Wi(ˆΩi)
+c(tj−ti), (101)
+whereWandWitake the forms (84a)–(84b) or (85a)–
+(85b) with Lgiven by (100) for hyperbolic or elliptic
+models or regions. Evidently, obtaining the missing ini-
+tial value function by means of constraints like (93) and
+(101) does not guarantee the fulfillment of regularity con-
+ditions (like absence of shell crossings). This has to be
+verified independently by means of the Hellaby–Lake con-
+ditions once the LTB model mapping the profiles has
+been found.
+The issue of mapping two profiles can also be under-
+stood in terms of the systems of evolution equations that
+we derive in the following section. These systems are
+partial differential equations, hence initial and boundary
+conditions are given as functions of r(or radial profiles).
+The selection of two initial value functions is sufficient
+to determine a unique solution. These functions can be,
+eithermqi,Hqiormqi,kqiorˆΩi,Hqi. Hence if we know
+only one of these functions (evaluated at t=ti) and the
+same function at t=tj, then we may use one of the avail-
+able constraints to find the second initial value function,
+and thus guarantee (if the constraint has a solution) that
+the evolution system yields a unique LTB model. We
+have proceeded in this section (following [7, 10]) by us-
+ing the analytic solutions as the constraints that link the
+boundary and initial conditions through L. The results
+are the same as those of [7, 10], but the initial value
+parametrization of the solutions makes the whole proce-
+dure a lot more natural and straightforward.XI. EVOLUTION EQUATIONS FOR A
+NUMERICAL TREATMENT.
+The analytic solutions, either in terms of the con-
+ventional variables M, E, ct bb, Ror the initial value
+parametrization mqi, kqi, Ri, Lthat we have presented,
+are mostly useful for qualitative work (as illustrated in
+the previous section). We remark that these solutions
+are either parametric or implicit, and thus a pure ana-
+lytic framework based on them is necessary limited. For
+concrete applications in models of less idealized cosmo-
+logical inhomogeneities and observations there is no other
+way but to consider a numeric framework.
+The 1+3 fluid flow system (18)–(20) discussed in sec-
+tion IV provides a fully covariant description and can be
+used to determine the dynamics of LTB models by means
+of numeric techniques. However, the associated spacelike
+constraints (19) are partial differential equation on rthat
+are not easy to solve, making it difficult (in general) to
+decouple the time and radial derivatives. As we show in
+this section, the quasi–local scalars and their fluctuations
+(in the initial value parametrization) provide more con-
+venient variables for a numeric treatment of LTB models.
+A. Quasi–local evolution equations.
+An alternative system to the evolution equations (18)–
+(20) follows by considering a 1+3 fluid flow evolu-
+tion equations for the quasi–local scalars (35), involv-
+ingmq,Hqand their corresponding relative fluctuations
+δ(m), δ(H). This system can be obtained by eliminating
+ρ,Θ,Σ andEin (18) in terms of the new variables by
+means of (32) and using (31) to deal with the constraints
+(19). Considering the notation (29): H= Θ/3, m =
+κρ/3 andk=3R/6, this framework leads to the follow-
+ing evolution equations
+˙mq=−3mqHq, (102a)
+˙Hq=−H2
+q−mq, (102b)
+˙δ(m)=−3(1 +δ(m))Hqδ(H), (102c)
+˙δ(H)=−(1 +δ(H))Hqδ(H)+mq
+Hq(δ(H)−δ(m)),
+(102d)
+while the spacelike constrains (19) become the equations
+defining δ(m)andδ(H)through (27a) and (31), and the
+Hamiltonian constraint (20) reduces to (38). This is the
+particular case of pure dust in [30, 31].
+The system (102) is equivalent to (18), but it is much
+more practical and easier to use in a numerical treat-
+ment, as it is not necessary to solve any radial differen-
+tial equation as a precondition to solve it. In practice,
+we can handle it as a system of non–linear autonomous
+ordinary differential equations, where renters as a pa-
+rameter (see Appendix B of [29]). Notice that once we
+have solved (102) all local scalars in (17) can be deter-
+mined by means of (32), (38) and (43).14
+Sinceδ(H)diverges as Hq→0, as the maximal expan-
+sionct=ctmaxis reached in elliptic models or regions
+(see A 4), it is better to use in these cases the alterna-
+tive variable Σ = −Hqδ(H)(see (32)). This leads to the
+alternative system
+˙mq=−3mqHq, (103a)
+˙Hq=−H2
+q−mq, (103b)
+˙δ(m)=−3(1 +δ(m)) Σ, (103c)
+˙Σ =−2 ΣHq+ Σ2+mqδ(m), (103d)
+which is entirely equivalent to (102), and has the advan-
+tage (besides its use for recollapsing configurations) that
+Σ is the scalar associated with the shear tensor by (15).
+We may supplement either (102) or (103) with the fol-
+lowing extra pair of differential equations
+˙L=LHq, (104a)
+˙Γ = 3 Γ Hqδ(H)=−3 Γ Σ, (104b)
+which follow directly from (32), (38) and (40). By includ-
+ing these extra equations we can use the systems above
+to determine numerically the metric functions in (39).
+In fact, if we substitute the scaling laws (37)–(38), (41),
+(42) and (43) into (104) we can work only with these two
+evolution equations given as
+˙L=[2mqi−kqiL]1/2
+L1/2, (105a)
+˙Γ =−3
+22mqi[δ(m)
+i+ 1−Γ]−kqiL[δ(k)
+i+2
+3(1−Γ)]
+[2mqi−kqiL]1/2,
+(105b)
+whose solution ( Land Γ) allows us to compute all scalars
+by means of these scaling laws. However, from a numeric
+point of view it might be more convenient and practical
+to work with (102) or (103) supplemented by (104) than
+with (105).
+Solving any one of the systems of evolution equations
+described above is needed to find radial null geodesics
+and their corresponding red shift factor, z, which are in
+turn needed for computing observable quantities (lumi-
+nosity distance, etc). Given a numerical solution of either
+system, radial geodesics ct=ctN(r) in the past null cone
+andzfollow from solving [3]
+cdtN(r)
+dr=−R′
+N
+(1 +E)1/2=−LNΓNR′
+i
+[1−kqiR2
+i]1/2,
+(106a)
+d
+drln(1 +z) =˙R′
+N
+(1 +E)1/2
+=LNΓNHqN(1 + 3δ(H)
+N)R′
+i
+[1−kqiR2
+i]1/2,(106b)
+where the subindex Nmeans evaluation at the null curve
+ct=ctN(r),i.e.AN=A(ctN(r),r). Since the systems(102) or (103) or (105) can be handled in practice as
+constrained ordinary differential equations, solving nu-
+merically (106) can be achieved with simple numerical
+techniques.
+B. Initial conditions.
+An obvious choice of initial value functions for the sys-
+tems (102)–(104) or (103)–(104) is
+mqi,Hqi, δ(m)
+i=m′
+qi/mqi
+3R′
+i/Ri,
+δ(H)
+i=H′
+qi/Hqi
+3R′
+i/Rior Σi=−H′
+qi
+3R′
+i/Ri(107)
+where we used (31) and Rican be prescribed as a radial
+coordinate gauge.
+Sincekqidetermines the solutions of (38) (or (3) for
+R=RiL), this initial value function is closely related to
+the kinematic evolution of the models (or specific regions
+of them). Hence, it is more practical and intuitive to
+choose instead of (107) the initial value functions
+mqi, k qi, δ(m)
+i=m′
+qi/mqi
+3R′
+i/Ri, δ(k)
+i=k′
+qi/kqi
+3R′
+i/Ri.
+(108)
+These initial conditions are sufficient to solve (105), but
+to solve (102)–(104) or (104)–(103) we also need Hqiand
+δ(H)
+ior Σi, which readily follow from (38) and (43)
+Hqi= [2mqi−kqi]1/2,
+δ(H)
+i=2mqiδ(m)
+i−kqiδ(k)
+i
+2[2mqi−kqi]
+or Σ i=−2mqiδ(m)
+i−kqiδ(k)
+i
+2[2mqi−kqi]1/2,
+(109)
+The main advantage of using (108) as initial conditions
+is the fact that they are exactly the same initial value
+functions that we employed in the parametrization of the
+analytic solutions and in the forms of Γ in (59)–(60), and
+the Hellaby–Lake conditions. Hence, given a choice of
+these functions, a time evolution free from shell crossings
+can be immediately tested by means of (64), (68) and
+(72).
+The evolution equations (103) and have already been
+used for a dynamical systems approach to LTB mod-
+els [29], while a suitable generalization of (102) was
+employed for a numeric study of cosmological models
+endowed with the LTB metric but with an energy–
+momentum tensor of an anisotropic fluid [31].
+A comprehensive numeric study of LTB models by
+means of the systems (102), (103) or (105) is beyond
+the scope of this article, and thus is presently under ex-
+amination in a separate work. Our purpose has been
+to show how these systems, constructed with quasi–local
+variables, are potentially promising for this purpose.15
+XII. GAUGE INVARIANT AND COVARIANT
+SPHERICAL PERTURBATIONS ON A FLRW
+BACKGROUND.
+The connection between the system (102) and a per-
+turbation formalism on a FLRW background is clearly
+suggested by the fact that the evolution equations (102a)
+and (102b) for the quasi–local scalars mq,Hqare for-
+mally identical to the energy balance and Raychaudhuri
+equations of a dust FLRW cosmology. Also, from (28),
+(29), (37) and (38), we have ˙kq=−2kqHq, which is the
+same evolution equation for3Rin a FLRW spacetime,
+while the δ(A)defined by (31) and its relation with the
+scalars in (32) clearly highlights a sort of perturbation
+definition. We provide in this section a rigorous charac-
+terization of this intuitive resemblance.
+A perturbation formalism between an idealized space-
+time, ¯S(a dust FLRW cosmology), and an “lumpy” uni-
+verse model, S(a dust LTB model), follows by defining
+a “background model” in S, constructed by objects (in
+S) that are the images of a suitable map Φ between ob-
+jects in ¯Sto objects in S[32]. Though, perturbations
+in general are not uniquely defined by such an abstract
+map Φ, because of the ambiguity of the gauge freedom
+associated with choices of coordinates and hypersurfaces.
+This fact requires carefully defining perturbed variables
+that are “gauge–invariant” [37]. However, in the specific
+case that concerns us we face a simpler task, as both the
+idealized FLRW and the perturbed LTB spacetimes are
+LRS (locally rotationally symmetric) and can be com-
+pletely described by scalar functions [36]. Hence, Φ can
+be a map of scalars to scalars, while the fact that both
+spacetimes are spherically symmetric and are given in the
+same normal geodesic coordinate (or frame) representa-
+tion, greatly suppresses most of the gauge freedom that
+we would find for a general S.[48]
+The map Φ can be defined rigorously as follows. Let
+¯XandXbe, respectively, the sets of smooth integrable
+scalar functions in ¯SandS, then for all covariant FLRW
+scalars ¯A∈¯X(we denote FLRW objects with an over–
+bar) the map
+Φ :¯X→X, ¯A/ma√sto→Φ(¯A) =Aq∈X, (110)
+defines a “background model” (associated to a FLRW
+cosmology) in LTB models through the quasi–local
+scalarsAq(which are LTB objects satisfying FLRW dy-
+namics). Given their common normal geodesic represen-
+tation and considering the perturbations
+δ(A)=A−Φ(¯A)
+Φ(¯A)(111)
+associated with (110), LTB models characterized by the
+metric (1) and source (2) can be considered then as spher-
+ical non–linear “perturbed” dust FLRW cosmologies in
+the “comoving” gauge.
+The perturbation scheme that we described fits nat-
+urally to the gauge invariant and covariant (GIC) ap-
+proach of Dunsbury, Ellis and Bruni [32, 33]. Followingthese authors, a perturbation scheme on FLRW cosmolo-
+gies is covariant if the “lumpy” model Sis described
+by variables defined in the framework of the 1+3 fluid
+flow variables associated with the system (18). Although
+our description of LTB spacetimes is not based on these
+scalars (the representation (17)) but on the quasi–local
+representation (35), it is still a covariant description be-
+causemqandHq, are covariant scalars by virtue of their
+connection with the invariants M, R and their deriva-
+tives in (28). Hence, the formalism based on mqandHq
+would also be covariant.
+As commented by Ellis and Bruni [32], by virtue of the
+Stewart–Walker gauge invariance lemma [35], all covari-
+ant objects in Sthat would vanish in the background ¯S(a
+FLRW cosmology in this case) are gauge invariant (GI),
+to all orders, and also in the usual sense (as in [37]). The
+tensorial quantities in LTB models that vanish for a dust
+FLRW cosmology in the 1+3 formalism are the shear and
+electric Weyl tensors σabandEab, given by (15) and (16)
+in terms of the scalar functions Σ and E, which from (30),
+can be written in terms of the fluctuations m−mqand
+H−H q. From (31), it is evident that these fluctuations
+andδ(A), as well as the radial gradients m′
+qandH′
+qare
+all “first order” quantities (in the perturbation scheme)
+that are GI to all orders, though mq,Hqare not (which
+is expected because these are “zero order” background
+variables). As a consequence, the fluid flow dynamics of
+LTB models in the quasi–local scalar representation (35)
+also provides a rigorous characterization of LTB models
+as spherical, non–linear GIC perturbations on a FLRW
+background.
+We note that spatial curvature k=3R/6 and its quasi–
+local dual kq=3Rq/6 do not appear in the dynamic
+equations (102a)–(102d), though they can be used when
+specifying initial conditions as in (108) (see also the ap-
+pendices of [31]). Spatial curvature is GI only if the
+FLRW cosmology ¯Sis spatially flat, but its associated
+variables δ(k)andk′
+qare GI. In fact, from (33), (37) and
+(38) we can always eliminate either one of mq,Hqor
+δ(m), δ(H)in terms of kqandδ(k), and construct a sys-
+tem of evolution equations equivalent to (102a)–(102d),
+but describing the dynamics in terms of spatial curvature
+kqand its perturbation δ(k).
+Given the correspondence between (102) and the dy-
+namics of non–linear perturbations on a FLRW back-
+ground, it is important to examine the connection with
+linear perturbation theory of dust sources. For this pur-
+pose we derive a second order equation for the density
+perturbation δ(m)by differentiating both sides of (102a)
+and use the remaining equations (102b)–(102d) to elimi-
+nate all derivatives except ¨δ(µ)and ˙δ(m). We obtain
+¨δ(m)−[˙δ(m)]2
+1 +δ(m)+ 2Hq˙δ(m)−κ
+2ρqδ(m)/parenleftBig
+1 +δ(m)/parenrightBig
+= 0,
+(112)
+which is an exact non–linear equation for δ(m)(a sim-
+ilar equation was obtained in [41]). For near homoge-16
+neous conditions, as assumed in linear perturbations with
+“small” perturbations |δ(m)| ≪1, (112) reduces to
+¨δ(m)+ 2Hq˙δ(m)−κ
+2ρqδ(m)= 0, (113)
+an equation that is formally identical to the evolution
+equation for linear density perturbations of a dust source
+around a FLRW background (characterized by ρq,Hq) in
+the comoving gauge [38], which for dust is a synchronous
+gauge as well.
+XIII. A THEORETICAL CONTEXT FOR THE
+OMEGA PARAMETER.
+The parameter ˆΩ that we discussed in section IX has
+been introduced in the literature [15, 16] (together with
+Hq) as useful ansatzes justified by their resemblance to
+the corresponding FLRW Hubble and Omega parame-
+ters. Another generalization of the FLRW Hubble factor,
+suggested by Moffat and Tartarsky [42] and used in var-
+ious articles [17], follows by defining two expansion fac-
+tors, a radial and an azimuthal one, which can be given
+in terms of Hqandδ(H)as:
+H⊥≡˙R
+R=˙L
+L=Hq, (114a)
+H/bardbl≡˙R′
+R′=˙L
+L+˙Γ
+Γ=Hq−3Σ =Hq(1 + 3δ(H)),
+(114b)
+Evidently, H⊥andH/bardblcorrespond, respectively, to ex-
+pansion factors of proper lengths in the direction orthog-
+onal and parallel to radial rays orthogonal to the orbits
+of SO(3). Moffat and Tartarsky define from these expan-
+sion factors an “effective” Hubble parameter as
+H2
+eff=H2
+⊥+ 2H⊥H/bardbl=HqH=H2
+q(1 +δ(H)),(115)
+so that an Omega parameter follows as Ω eff=κρ/(3H2
+eff).
+However, both pairs Hq,ˆΩ andHeff,Ωeffare basically
+useful quantities in the qualitative or numeric application
+of LTB models, since the proper local covariant general-
+ization of the FLRW Hubble parameter to LTB models
+is neither Hq, norHeff, but the expression given by equa-
+tion (41) of [43]. Nevertheless, it is still interesting to
+discuss the theoretical assumptions underlying these ex-
+pressions.
+While both HqandHeffare covariant quantities, the
+theoretical context for Hqmay be easier to justify, as this
+scalar is the quasi–local dual of the fluid flow expansion
+scalarH= Θ/3, and thus it is a GIC background variable
+in the perturbation formalism discussed in section XII (it
+is the image of the FLRW Hubble parameter under the
+map (110)). However, H2
+qis not the quasi–local dual of
+H2, since the definition (24) implies that ( H2)q/ne}ationslash= (Hq)2,
+henceH2
+q(and thus) ˆΩ are not images under the per-
+turbation map (110) of the Omega and squared Hubbleparameters of a FRLW dust spacetime. As a consequence
+ˆΩ is not a background variable in the perturbation for-
+malism, though its gradient defined as
+∆≡ˆΩ′/ˆΩ
+3R′/R=δ(m)−2δ(H)= (1−ˆΩ) (δ(m)−δ(k)),(116)
+is a GIC perturbation in this formalism (since δ(m)and
+δ(H)are), though we have ∆ /ne}ationslash= (Ω−ˆΩ)/ˆΩ in general
+(this relation only holds in the linear limit). On the other
+hand, it is hard to find a theoretical context for Heff,Ωeff
+besides their resemblance to FLRW quantities and their
+utility in computations.
+A. Evolution equations in terms of ˆΩand∆.
+Since ˆΩ andHqin (75) and (78) are constructed from
+mqandkq, both quantities can easily be computed from
+a numeric solution of the systems (102) or (103). The
+initial value functions mqiandkqiin (107), (108) and
+(109) can be given in terms of ˆΩiandHqifrom (81),
+whileδ(m)
+iandδ(k)
+ifollow from (89).
+However, given the practical utility of ˆΩ and the con-
+nection of its gradient ∆ to the perturbation formalism
+of section XII, it is still useful to construct a system of
+evolution equations that include evolution laws for these
+quantities. Rewriting the system (102) in terms of ˆΩ and
+∆ yields the following dimensionless system
+∂H
+∂τ=−H2/parenleftbigg
+1 +1
+2ˆΩ/parenrightbigg
+, (117a)
+∂ˆΩ
+∂τ= H ˆΩ/parenleftBig
+ˆΩ−1/parenrightBig
+, (117b)
+∂∆
+∂τ= H/bracketleftBig/parenleftBig
+δ(H)+ ∆/parenrightBig
+ˆΩ−/parenleftBig
+1 + 3∆ + 4 δ(H)/parenrightBig
+δ(H)/bracketrightBig
+,
+(117c)
+∂δ(H)
+∂τ=−H/bracketleftbigg/parenleftBig
+1 +δ(H)/parenrightBig
+δ(H)+1
+2/parenleftBig
+δ(H)+ ∆/parenrightBig
+ˆΩ/bracketrightbigg
+,
+(117d)
+where we have introduced the dimensionless variables
+H≡Hq
+H0, τ≡H0c(t−ti), (118)
+withH0an inverse length scale (cm−1), which can be
+identified as the cosmological Hubble scale at a specific
+fiducial cosmic time (we can also identify R0=H−1
+0when
+specifying the radial coordinate gauge through Ri). The
+initial conditions for this system are basically Hqi,ˆΩi
+and their radial gradients, which makes their specifica-
+tion very intuitive and practical, as these initial value
+functions could tend at a given asymptotic limit to the
+Hubble and Omega factors in a FLRW background at a
+fiducial cosmic time. The system (117) can also be sup-
+plemented by the differential equations (104) for Land
+Γ.17
+B. A dynamical system
+The system (117) clearly suggests a dynamical systems
+approach, as discussed in [44] for FLRW and Bianchi
+models. This approach is based on the fluid flow equa-
+tions [32–34] and an Omega parameter constructed as
+the ratio of density to the the squared expansion scalar
+H2= Θ2/9. As shown in [44], if the evolution parameter
+is defined by ∂/∂ξ = (1/H)∂/∂τ the Raychaudhuri equa-
+tion decouples from the remaining evolution equations,
+leading to a reduced system that can be analyzed qualita-
+tively: critical points, invariant subspaces, etc. For LTB
+models associated with evolution equations like (117), it
+is more convenient (see [29]) to define ξin terms of H in
+(118) by means of the coordinate transformation [29]
+τ=τ(ξ,¯r), r = ¯r, ξ = ln H (119)
+so that for every scalar A(τ,r) =A(τ(ξ,r),r) =A(ξ,r)
+and derivatives associated with the 4–velocity flow ∂/∂τ
+(withrconstant) become
+/bracketleftbigg∂A
+∂τ/bracketrightbigg
+r=∂A
+∂ξ/bracketleftbigg∂ξ
+∂τ/bracketrightbigg
+r=∂A
+∂ξH
+⇒∂
+∂ξ=1
+H∂
+∂τ=1
+Hq∂
+c∂t. (120)
+Hence, the three equations (117b)–(117d) become inde-
+pendent of H and effectively decouple from (117a), lead-
+ing to the reduced system
+∂ˆΩ
+∂ξ=ˆΩ/parenleftBig
+ˆΩ−1/parenrightBig
+, (121a)
+∂δ(H)
+∂ξ=−/parenleftBig
+1 +δ(H)/parenrightBig
+δ(H)−1
+2/parenleftBig
+δ(H)+ ∆/parenrightBig
+ˆΩ,(121b)
+∂∆
+∂ξ=/parenleftBig
+δ(H)+ ∆/parenrightBig
+ˆΩ−/parenleftBig
+1 + 3∆ + 4 δ(H)/parenrightBig
+δ(H),
+(121c)
+which is very similar to that analyzed in [29] (where
+δ(m)and a dimensionless Σ were used instead of ∆ and
+δ(H)). Notice that this dynamical system is character-
+ized by a 3–dimensional phase space parametrized by
+{ˆΩ,∆, δ(H)}, which contains the FLRW dust case as an
+invariant set given by the line [ ˆΩ = ˆΩ(ξ),∆ = 0, δ(H)=
+0]. Therefore, this dynamical systems approach also pro-
+vides a nice and appealing theoretical context to justify
+the role of ˆΩ as an inhomogeneous generalization of the
+FLRW dust Omega parameter.
+XIV. REGULARITY OF SPECIAL LTB
+CONFIGURATIONS.
+A. Closed elliptic models and regular zeroes of
+R′= 0.
+In closed elliptic models the3T[t] are homeomorphic to
+S3(see Appendix A 3). There are two symmetry centers,atr= 0 and r=rc. SinceR(t,0) =R(t,rc) = 0 for
+allt, thenR′(rtv) = 0 where 0 < rtv< rcmust hold
+for allt. The regularity conditions (A6) and (A7) imply
+that for all rthe sign of R′is the same for all3T[t].
+Hence, (9) requires the function Ri(r) to be selected so
+thatRi(0) =Ri(rc) = 0 and R′
+i(rtv) = 0, with
+R′
+i>0 and 0 <F ≤ 1 for 0 < r < r tv,
+R′
+i<0 and −1≤ F<0 forrtv< r < r c,
+(122)
+whereF=±√1 +E, withF(0) = 1,F(rc) =−1 and
+F′(0) =F′(rc) =F(rtv) = 0. Since R, R′andFare
+bounded, if (9) holds then the maximal coordinate range
+is 0≤r≤rcin all regular3T[t].
+Equations (122) provide the coordinate ranges where
+the “+” or “ −” signs hold in specifying the Hellaby–Lake
+conditions (11) in the conventional variables. However,
+the±signs are no longer needed with the initial value
+parametrization, as it is clear that fulfillment of (72) (the
+equivalent of (11)) implies by its construction that t′
+bband
+t′
+collmust have, respectively, the opposite and same sign
+(and a common zero) as R′
+i/Ri, and from (A6), this sign
+will be the same for R′/Rat all3T[t].
+If only the first sign equality in (9) holds, then there
+would exist a range of rfor which
+sign(R′
+i) = sign( M′)/ne}ationslash= sign(F), (123)
+so thatM′andR′
+ihave a common zero, but either F
+has a zero that is not common to that zero, or Fhas
+no zeroes. In these situations, ρin (4) remains bounded,
+and so there is no curvature singularity, but we have a
+surface layer discontinuity at r=rtv[24, 25, 40]. This
+is the reason why when R′vanishes regularly at some
+r=rtv, this worldline must lie, either in an elliptic model
+or in an elliptic region. Besides the surface layer, (123)
+also implies an ill–defined metric coefficient grrin (1) and
+(39) (which leads in turn to an ill–defined proper radial
+length between different comoving dust layers becomes).
+Once these issues are taken under consideration, initial
+conditions complying with the Hellaby–Lake conditions
+(72) can be provided for this class of models, whether for
+analytic/qualitive or numeric work.
+B. Simultaneous big–bang.
+In general, the initial curvature singularity given by
+(58a) is not simultaneous, since it is marked by the curve
+ct=ctbb(r) in the ( ct,r) coordinate plane. For parabolic
+models, the condition ct′
+bb= 0 in (63a) implies δ(m)
+i= 0,
+and thus m′
+qi= 0, which corresponds to the FLRW limit.
+However, non–trivial hyperbolic and elliptic LTB models
+follow by setting the initial value functions so that ctbbis
+a constant (see [10]). In order to examine these cases, we
+use (52), (57) and (63b) to rewrite Γ in (60) in terms of
+ctbbandct′
+bb, which after setting t′
+bb= 0 and tbb=t(0)
+bb=18
+constant, leads to
+Γ = 1 + 3( δ(m)
+i−δ(k)
+i)−3Hqc(t−t(0)
+bb)/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg
+,
+(124)
+whereHqfollows from (38) and
+c(t−t(0)
+bb) =/braceleftbigg
+Zh(xiL)/yihyperbolic
+Ze(xiL)/yielliptic, (125)
+withxi, yi, ZhandZegiven by (51), (49), (56) and (54).
+Sincexiandyidepend only on mqiandkqi, a simulta-
+neous big–bang t′
+bb= 0 implies the following constraint
+on these initial value functions
+c(ti−t(0)
+bb) =/braceleftbigg
+Zh(xi)/yihyperbolic
+Ze(xi)/yielliptic, (126)
+which, in turn, implies (by differentiation) the following
+constraint between the δ(m)
+iandδ(k)
+i(hence, between the
+gradients m′
+qiandk′
+qi)
+δ(m)
+i/bracketleftBig
+1−Hqic(ti−t(0)
+bb)/bracketrightBig
+=δ(k)
+i/bracketleftbigg
+1−3
+2Hqic(ti−t(0)
+bb)/bracketrightbigg
+.
+(127)
+As a consequence of (126), a simultaneous big–bang al-
+lows us to prescribe only one of the two functions mqi
+andkqi, the other one must be found by solving the al-
+gebraic constraint (126) or solving (127) as a differential
+equation for d kqi/dmqi(the function Ri(r) remains free
+and can be fixed as a radial coordinate gauge). Since
+we need both functions mqiandkqito solve (102) or
+(103), finding previously a numeric solution of the alge-
+braic constraint (126) is not a problem (it is part of the
+work needed to provide appropriate initial conditions).
+However, assuming that mqiis selected and bearing in
+mind that
+0< Zh(xi) for xi>0, (128a)
+0< Ze(xi)< π for 0< xi<2, (128b)
+it is evident that the hyperbolic branch of (126) does not
+appear to be restrictive at all to find |kqi|. For the elliptic
+branch, a necessary condition for a solution of (126) that
+yieldskqiis
+0< c(ti−t(0)
+bb)k3/2
+qi< πmqi, (129)
+which, again, does not seem to be restrictive (see [10] for
+comparison). Still, since we cannot solve (126) analyti-
+cally, nor invert (125) to find L, the case of a simultane-
+ous big–bang may be easier to handle analytically with
+the conventional variables M, E andRby means of (48)
+or (53), since we only need to fix tbb=t(0)
+bband then pre-
+scribe the functions MandEtaking care to fulfill the
+Hellaby–Lake conditions (10) and (11). Nevertheless, we
+can still use mqiandkqito examine qualitatively these
+regularity conditions in these models through (124).•Hyperbolic models
+We examine (124) in the limits L≈0 andL→ ∞
+along constant but arbitrary r(so that the time
+dependence is concentrated on L). Consider the
+following series expansions of Hqc(t−t(0)
+bb):
+Hqc(t−t(0)
+bb)≈2
+3+x1/2
+i
+15L+O(L3/2)
+forL≪1, (130a)
+Hqc(t−t(0)
+bb)≈1 +2−ln(2xiL)
+xiL+O(L−2)
+forL≫1. (130b)
+where we used (49), (38) and (125). Taking the
+leading terms in these expansions, Γ in (124) be-
+comes
+Γ≈1 +δ(m)
+i (L≪1),
+Γ≈1 +3
+2δ(k)
+i (L≫1),
+leading to the following conditions for the fulfill-
+ment of (61)
+δ(k)
+i≥ −2
+3, δ(m)
+i≥ −1. (131)
+which are the same as (68) with t′
+bb= 0.
+•Elliptic models
+The behavior of Γ in (124) around L≈0 fort≈t(0)
+bb
+follows from an expansion similar to (130b), but
+usingZeinstead of Zh. This yields also Hqc(t−
+t(0)
+bb)≈2/3 +O(L), and lead to Γ ≈1 +δ(m)
+i.
+Considering now L≈0 but in the collapsing phase,
+so thatt≈tcoll, we have Hqc(t−t(0)
+bb)→ −∞ .
+Hence, Γ in this limit takes the form
+Γ≈3|Hq|c(t−t(0)
+bb)/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg
+, (132)
+while Γ for t≈tmaxtakes the form (69). Therefore,
+the conditions for the fulfillment of (61) are simply
+δ(m)
+i≥ −1, δ(m)
+i−3
+2δ(k)
+i≥0. (133)
+Notice, by comparing with (73), that the second
+condition above is equivalent to ct′
+coll≥0, hence
+(133) are simply (72) with t′
+bb= 0. It is important
+to notice that a simultaneous big–bang t′
+bb= 0 does
+not imply a simultaneous collapsing singularity or
+maximal expansion ( t′
+collandt′
+maxare not zero).
+C. Simultaneous maximal expansion and
+simultaneous collapsing time.
+These elliptic LTB configurations follow by setting the
+maximal expansion and collapse times, tmax, tcoll, in (57)19
+to constants. They have been examined in [10], hence
+the reader can compare the treatment of these cases in
+this reference with our treatment in this subsection.
+The condition for t′
+max= 0 in elliptic models follows
+directly from (57) as
+ctmax=cti+γ0, π−Ze(xi) =γ0yi, (134)
+whereγ0is a positive constant and xi, yiandZeare
+given by (54) and (56). As in the case t′
+bb= 0, we can only
+prescribe one of mqiandkqi, with the other one obtained
+by solving numerically the constraint (134). If mqiis
+prescribed, then a necessary condition for the existence of
+a solution for kqifollows directly from (128b) and (134):
+0< γ0k3/2
+qi< πmqi, (135)
+which looks like (129) and also does not appear to be
+restrictive at all. Still, it is necessary to work (134) nu-
+merically or to make further assumptions on mqiorkqito
+get more information. The locus of the big–bang and col-
+lapse singularities are now given by inserting (134) into
+(57)
+ctbb=cti+γ0−π
+yi, ct coll=cti+γ0+π
+yi,(136)
+which indicates a time symmetric location of tbbandtcoll
+with respect to tmax. The gradients of tbbandtcollare
+ct′
+bb
+3R′
+i/Ri=π
+yi/parenleftbigg3
+2δ(k)
+i−δ(m)
+i/parenrightbigg
+,
+ct′
+coll
+3R′
+i/Ri=π
+yi/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg
+. (137)
+where we used (56) and (31).
+Ift′
+coll= 0, then (57) implies the following constraint
+similar to (134)
+ctcoll=cti+ǫ0, 2π−Ze(xi) =ǫ0yi, (138)
+whereǫ0is a positive constant. Again, we can only pre-
+scribe one of mqiandkqi, and obtain the other by solving
+(138). From (128b) we have π <2π−Ze(xi)<2π, hence,
+if we prescribe mqia necessary condition for finding kqi
+as a solution of (138) is
+π < ǫ0k3/2
+qi
+mqi<2π, (139)
+which, again, does not seem to be restrictive, though fur-
+ther information requires either numerical work on (138)
+or making assumptions on mqiorkqi. The big–bang and
+maximal expansion times and their gradients are
+ctbb=cti+ǫ0−2π
+yi,
+ctmax=cti+ǫ0−π
+yi, (140a)
+ct′
+bb
+3R′
+i/Ri=2π
+yi/parenleftbigg3
+2δ(k)
+i−δ(m)
+i/parenrightbigg
+,
+ct′
+max
+3R′
+i/Ri=π
+yi/parenleftbigg
+δ(m)
+i−3
+2δ(k)
+i/parenrightbigg
+, (140b)By comparing (137) and (140b) with (72), then suffi-
+cient conditions to fulfill (61) in either case ct′
+max= 0 and
+ct′
+coll= 0 are simply (133). As with the case ct′
+bb= 0,
+it is not problematic to work out these cases in solving
+(102) or (103) numerically. As in the case t′
+bb= 0, these
+cases are easier to handle analytically with the conven-
+tional variables, since for a given choice of MandE, we
+haveyi=|E|3/2/M, and thus ctbbfollows directly from
+(136) or (140a). Of course, the three free parameters
+must comply with the Hellaby–Lake conditions (11).
+D. Mixed hyperbolic/elliptic configurations.
+LTB models admit more than one kinematic class in
+their full radial domain. This type of “mixed” configu-
+rations can be constructed either by smoothly matching
+LTB regions with one kinematic class to regions of an-
+other (as in [24]), or simply by choosing the initial value
+function kqi(which determines the kinematic class) so
+that it changes sign in its radial domain.
+A particularly interesting mixed configuration is an el-
+liptic region surrounded by a hyperbolic exterior given
+by the choice
+kqi
+
+>0 for 0 ≤r < rb,elliptic region
+= 0 for r=rb, interface
+<0 forr > rb, hyperbolic region.(141)
+The fact that δ(k)
+i→ −∞ ifkqi→0 (from its definition,
+see Appendix A 4), and thus δ(k)→ −∞ for allt, signals
+a potential problem in using the quasi–local variables to
+study an elliptic/hyperbolic model. However, we notice
+thatδ(k)does not appear in (102) nor in (103). Also, δ(k)
+i
+only appears in the initial conditions (108) in the form
+kqiδ(k)
+i, which does not diverge as kqi→0. Hence, this
+behavior of δ(k)
+ihas no consequences for the numerical
+integration of these evolution equations for these config-
+urations.
+For analytic or qualitative work, the limit kqi→0 must
+be handled carefully [24]. While regularity conditions for
+these mixed models follow also from (60), even if δ(k)
+i→
+−∞ askqi→0, we cannot simply set kqi= 0 in (60).
+Instead, we examine Γ at constant values of rasr→rb
+(orkqi→0). The following series expansions around
+kqi= 0 hold in this limit
+Hqc(t−ti)≈2
+3/parenleftbigg
+1−1
+L3/2/parenrightbigg
++O(kqi),(142a)
+1−Hq
+Hqi≈1−1
+L3/2+O(kqi), (142b)
+so that Γ in (60) takes the form
+Γ≈1 +/parenleftbigg
+1−1
+L3/2/parenrightbigg
+δ(m)
+i+O(kqi), (143)20
+which (up to leading order) is the form of Γ in (59).
+Hence, Γ passes smoothly from its elliptic to its hyper-
+bolic form, approaching the form for parabolic models
+wherever kqi≈0 forr≈rb.
+Since the limit kqi→0 implies xi→0 in either the
+elliptic (r < rb) or hyperbolic side ( r < rb), we should
+also obtain for Lthe approximated parabolic form (46)
+forr≈rbfrom the analytic solutions (50) and (55) in
+the limit xi→0. Expanding Zh(u) andZe(u) around
+u= 0 yields
+Zh(u)≈√
+2
+3u3/2+√
+2
+6u5/2, (144a)
+Ze(u)≈√
+2
+3u3/2−√
+2
+6u5/2, (144b)
+Hence, taking only the leading term in ZhandZeand
+substituting into (50) and the expanding phase of (55)
+yields for both elliptic and hyperbolic sides
+L3/2≈1 +3
+2√mqic(t−ti) +O(xi), (145)
+which up to the leading term coincides with (46). We
+did not consider the collapsing phase in (55) because,
+from (57), we have ctmax→ ∞ askqi→0, hence the
+worldline marking the interface r=rbis contained in
+the expanding phase for all t.
+The conditions for the fulfillment of (61) can be ob-
+tained jointly for the elliptic and hyperbolic regions,
+bearing in mind that for r≈rbwe have δ(k)
+i→ −∞ but
+also Γ and Ltake the forms (143) and (145). For both
+the elliptic and hyperbolic regions we have as r→rb
+ct′
+bb
+3R′
+i/Ri≈δ(m)
+i
+3√mqi, (146)
+henceδ(m)
+i≤0 (withδ(m)
+i= 0 only at r= 0) is a suf-
+ficient condition for ct′
+bb<0 in (68) and (72) to hold in
+the full radial range (note, from (31), that this condition
+is equivalent to m′
+qi≤0). Since ctcoll(r)→ ∞ asr→rb,
+the collapsing singularity is contained entirely in the el-
+liptic region, hence the existence of the hyperbolic region
+forr > rbhas no effect on the condition ct′
+coll≥0 in (72).
+The only remaining condition is δ(k)
+i≥ −2/3 in the
+hyperbolic region. Evidently, this condition does not hold
+in the limit r→rbbecauseδ(k)
+i→ −∞ , but in this limit
+Γ has the form (143), and so (61) is not violated. As
+long asδ(k)
+iremains negative we have k′
+qi>0 (from (31)
+withkqi<0), and thus E′>0 holds, hence condition
+(10) (equivalent to (61)) also holds. In the elliptic side,
+we have kqi→0 withk′
+qi<0, which implies from (B3)
+thatE′>0 andki=3Ri/6<0 nearrb. In fact local
+spatial curvature k=3R/6 is negative for all times near
+r=rb, even if kq>0 all the way up to r=rb. This
+follows directly by applying the integral property (27b)
+withr=rbas integration limit and considering thatk′(t,rb)≤0 andkq(t,rb) = 0 hold for all t:
+k(t,rb) =1
+R3(t,rb)/integraldisplayrb
+0k′(t,x)R3(t,x)dx <0.(147)
+This provides an example of how elliptic dynamics does
+not (necessarily) imply positive local curvature (see Ap-
+pendix B).
+Notice that if instead of a hyperbolic region in (141),
+we have a parabolic region ( kq= 0) for r > rb, then the
+results above still apply to the elliptic region in 0 ≤r <
+rb. Hence, we would still have k <0 nearr=rbin this
+region, and thus k <0 would also hold in the parabolic
+region. This is an example of how local spatial curvature
+can be negative in a parabolic region not containing a
+symmetry.
+XV. CONCLUSION AND FURTHER WORK.
+We have provided a comprehensive examination of
+LTB dust models in terms of quasi–local scalars and their
+fluctuations, which are covariant objects that can be re-
+lated to covariant scalars in the “fluid flow” or “1+3”
+formalism. The motivation, and contents of the article,
+together with a summary of previous work, concepts and
+ideas, have been given in detail in the introduction. As
+we have shown throughout the article, the initial value
+parametrization that emerges from these scalars is use-
+ful in analytic, qualitative and numerical studies of the
+models. The “fluid flow” evolution equations for these
+scalars are an appealing and practical alternative to the
+analytic solutions conventionally used in the literature,
+as they are fully general and at the same time techni-
+cally simple: they can be effectively handled as ordinary
+differential equations, and thus can be very handy for a
+numeric treatment of the models. These evolution equa-
+tions can also be understood in terms of a gauge invariant
+and covariant (GIC) perturbation formalism, consistent
+with the covariant fluid flow approach of Ellis et al [32–
+34], as well as the traditional gauge invariant approach
+[37]. Under this formalism, the dynamics of LTB mod-
+els can be cast in terms of the dynamics of spherical,
+non–linear GIC perturbations on a FLRW background.
+Regarding analytic and qualitative work, the quasi–
+local scalar representation and its associated initial value
+description provide an appealing theoretical context to
+understand the conventional parameters of the models,
+leading to an understanding and appreciation of previ-
+ous work under a new perspective. This is specially evi-
+dent in the fact that all scalars can be expressed by sim-
+ple scaling laws that can be analyzed qualitatively, as
+well as in the formulation of regularity conditions (the
+Hellaby–Lake conditions [24, 25, 39]) as restrictions on
+initial value functions. We have also looked at the prob-
+lem, proposed by Krasi´ nski and Hellaby [7, 9], of finding
+the existence of a unique LTB model that is consistent
+with the “mapping” of arbitrary radial profiles of den-
+sity or velocity at different cosmic times. Also, we have21
+provided a theoretical context for an Omega parameter
+for LTB models, which has been defined in the literature
+as an ansatz [15, 16]. Given the widespread use of LTB
+dust solutions for dealing with a wide variety of prob-
+lems and models in General Relativity and Cosmology,
+the new approach to these solutions presented here has
+a significant potential for applications.
+In this article we have mostly introduced, discussed
+and justified the use of a new scalar representation for
+studying LTB models. However, the formalism that we
+have presented is potentially useful in future work using
+these models, hence we are looking at its direct applica-
+tion in separate articles, on the make, dealing with new
+results on important theoretical and practical issues, such
+as: (i) the asymptotic properties and boundary condi-
+tions of scalars in the radial direction, (ii) the conditions
+for constructing LTB models with scalars having radial
+profiles of “clumps” or “voids”, (iii) the conditions for
+the existence of a positive “back–reaction” in LTB mod-
+els, which would allow us to mimmic the effect of an
+accelerated cosmic expansion in the context of Buchert’s
+scalar averaging [21, 22]. Together with these articles
+under elaboration, we are also conducting a comprehen-
+sive numeric study of the models, based on the evolu-
+tion equations derived here. Finally, as shown by [45],
+the extension of the formalism of quasi–local scalars can
+be suitably modified in order to apply it to the quasi–
+spherical Szekeres spacetimes. The main justification
+of the present article is then to motivate and induce
+researchers to consider the introduced variables in the
+study and application of LTB models, as well as to serve
+as the theoretical reference for the use of these variables.
+Appendix A: Regularity issues of the new variables.
+The regularity of LTB models in the conventional
+variables has been extensively discussed in the litera-
+ture [2, 3, 24, 25, 39]. We examine this issue in terms of
+the quasi–local variables and their fluctuations.
+1. Symmetry centers.
+We have only considered in this article LTB models
+having (at least) one symmetry center, which is a regular
+timelike worldline corresponding to a fixed point of the
+rotation group SO(3). This is a sufficient condition for
+integrals in (24) to be finite in a domain ϑ(r) defined by
+(23). The symmetry center can be marked as r= 0 (and
+r=rcif there is a second one). The following conditions
+hold:R(t,0) = ˙R(t,0) = 0. Considering (36) and (40)
+andR′→1 asr→0, we can mark the symmetry center
+by a zero of Ri(r) (distinct from the locus of a curvature
+singularity: L= 0) [25]. Hence
+L(ct,0)>0, Γ(ct,0) = 1. (A1)SinceE(0) =M(0) =M′(0) =E′(0) = 0, then mqi(0) =
+mi(0) andkqi(0) =ki(0) must hold, and thus Hqi(0) =
+Hi(0). For any scalar Aand its dual function Aqwe have
+A(t,0) =Aq(t,0), A′(t,0) =A′
+q(t,0) = 0,(A2)
+The same conditions hold at a second symmetry cen-
+terr=rcin closed elliptic models (see Appendix A 3).
+Notice that a central singularity is now associated with
+L(ct,r) = 0.
+2. The Riemann tensor.
+Regardless of which parameters or variables we may
+use, the regularity at each spacetime point or surface
+in LTB models can be characterized by the continu-
+ity and finiteness of the Riemann tensor R(e)(f)(g)(h) =
+Rabcdea
+(e)eb
+(f)ec
+(g)ed
+(h), in an orthonormal tetrad basis ea
+(e)
+[3, 24]. Considering the natural tetrad associated with
+the LTB metric in its form (39)
+ea
+(0)=ua,ea
+(1)=/radicalbig
+1−kqiR2
+i
+/suppress L ΓR′
+iδa
+r,
+ea
+(2)=1
+LRiδa
+θ,ea
+(3)=1
+LRisinθδa
+φ, (A3)
+the nonzero Riemann tensor basis components are given
+readily in terms of the new variables mqandδ(m)by
+R(0)(1)(0)(1) = 3m−2mq=mq(1 + 3δ(m)),
+R(1)(2)(1)(2) = 3m−mq=mq(2 + 3δ(m)),
+R(0)(2)(0)(2) =mq,R(2)(3)(2)(3) = 2mq.
+(A4)
+These basis components also involve other scalars like
+H,Hq, k, kqandδ(H), δ(k), which are related to m, mq
+andδ(m)by the constraints (37), (38), (33) and the
+Hamiltonian constraint (20) for the local covariant
+scalars
+H2= 2m−k+ Σ2= 2m−k+ (Hqδ(H))2. (A5)
+Clearly, it is sufficient for the regularity of LTB models
+based on (A4) that all the involved scalars (not only m
+andmq) are bounded and continuous, but it is not a nec-
+essary condition, since HandHqcould diverge because
+kandkqdiverge, with mandmqremaining bounded.
+Also, we note that the relative fluctuations δ(A)might
+diverge in some cases when AandAqare bounded.
+3. Topology of the space slices.
+Given the existence of (at least) one symmetry center,
+the admissible topologies (homeomorphic classes) of the
+3T[t] are22
+•“Open models” :3T[t] homeomorphic to R3.
+There is only one symmetry center, at r= 0.
+As a consequence of the regularity condition (9),
+R′>0 andE >−1 must hold for all r, and so this
+topology is compatible with regions or models of all
+kinematic classes (hyperbolic, parabolic or elliptic
+with−1< E < 0). We discuss the asymptotic
+radial regime associated with this topology in a
+separate article.
+•“Closed models” :3T[t] homeomorphic to S3.
+There are two symmetry centers, at r= 0 and
+r=rc. Since R(t,0) =R(t,rc) = 0 for all t,
+then there must exist a turning value of Rso that
+R′(rtv) = 0 where 0 < rtv< rc.
+Since (9) is valid in each3T[t] andF=√1 +Edoes not
+depend on t, a zero of F(characteristic of S3topology)
+or the fulfillment of F>0 (characteristic of R3topology)
+will be common to all3T(t). Hence, all3T[t] belong to
+the same topological class and thus
+sign[R′(ti,r)] = sign[ R′(tj,r)], (A6)
+must hold for every rand for any arbitrary pair of distinct
+hypersurfaces3T[ti],3T[tj]. As consequence, (9) can be
+given as an initial condition
+sign (R′
+i) = signM′= signF, (A7)
+specified on an arbitrary fiducial or “initial” hypersurface
+3T[ti].
+4. Possible blowing up of the δ(A).
+From their definition (31), it is evident that relative
+fluctuations δ(A)will diverge under certain regular con-
+ditions ( i.econtinuous and finite AandAq) if in the
+integration domain ϑ(r) of (24) Aqhas a zero that is not
+a common same order zero of A. This situation occurs
+in various situations that do not violate regularity, such
+as the zero of Hqatt=tmaxas elliptic models or regions
+pass from expansion ( Hq>0) to collapse ( Hq<0), or in
+mixed elliptic/hyperbolic configurations where kqpasses
+from positive to negative at some comoving radius r=rb
+(see section XIV D). However, the quantities δ(m), δ(k)
+andδ(H)in all basis components in (A4) only appear in
+terms such as mqδ(m), kqδ(k)andHqδ(H), which do not
+diverge at a zero of mq,kqorHq. Hence, the blowing up
+ofδ(H)andδ(k)because of zeroes of kqorHqdoes not
+affect the regularity of the solutions, as conveyed by the
+continuity and finiteness of (A4). Nevertheless, in these
+cases it will be preferable to use Aqδ(A), instead of δ(A),
+for studying the time evolution of scalars (which justifies
+the use of (103) over (102) for numeric work involving
+expanding/collapsing regions or models).5. Restrictions of the radial range due to a
+curvature singularity.
+When we introduced the quasi–local scalars in (24), we
+assumed that the integration range ϑ(r) defined by (23)
+was fully regular. However, in general, the coordinate
+locus (58a) of the central singularity (expanding or col-
+lapsing) is not simultaneous ( i.e.not marked by t=tbb=
+const., so that t′
+bb/ne}ationslash= 0). Consider the case of an expand-
+ing non–simultaneous singularity (big bang), marked by
+the curve [ tbb(r),r], witht′
+bb≤0, in the ( ct,r) coordinate
+plane, so that L(tbb(r),r) = 0 (the collapsing singularity
+is analogous). The hypersurfaces3T[t] fort≤tbbare
+only regular for the semi open subset
+¯ϑ(r)≡ {x|rbb< x≤r} ⊂ϑ(r), (A8)
+whererbbis the intersection of tbb(r) and the constant t
+value associated with the3T[t]. As a consequence, the
+integration range in the definition (24) for these3T[t]
+must be ¯ϑ(r), notϑ(r). For a collapsing singularity in
+elliptic models we have exactly the same situation, but
+the involved hypersurfaces are those with t≥tcoll(r),
+with the lower radial bound given by r=rcollmarking
+the intersection of tcoll(r) and the3T[t].
+However, this range restriction has no consequence in
+the definition and usage of the quasi–local scalars Aq,
+as the involved integrals can be treated simply as stan-
+dard improper integrals. We define at each3T[t] with
+t≤tbb(r) the incumbent integrals with their lower in-
+tegration limit as y=rbb+ǫ, for an arbitrarily small
+ǫ >0, and then obtain the limit as ǫ→0. Off course,
+since scalars like m, k,Hdiverge in this limit, mq, kqand
+Hqmight diverge as well, but the functions are well de-
+fined and behaved in the range ¯ϑ(r). This restriction only
+prevents the Aqfrom taking values r < rbbfort≤tbb(r),
+and so all results that involve these scalars can be trivially
+extended to include hypersurfaces3T[t] fort≤tbb(r) and
+t≥tcoll(r).
+Appendix B: Local vs. quasi–local spatial curvature.
+The kinematic class of LTB models is determined by
+the sign of the initial quasi–local spatial curvature kqi(or
+E=−kqiR2
+i). From (37), the sign of kqidetermines the
+sign ofkq. It is evident that parabolic models (contain-
+ing a symmetry center) are spatially flat, as kqi=ki= 0
+trivially implies kq=k= 0. However, it is not obvi-
+ous if a given sign of kqin regular hyperbolic or elliptic
+LTB models also determines the sign of the local spatial
+curvature k=3R/6. In order to examine this point, we
+rewrite (41b) as
+k=kq/bracketleftBig
+1 +δ(k)/bracketrightBig
+=kqi
+3ΓL2/bracketleftbigg
+Γ + 3/parenleftbigg
+δ(k)
+i+2
+3/parenrightbigg/bracketrightbigg
+.(B1)
+We examine the relation between kqandkand their
+initial values for hyperbolic and elliptic models.23
+Negative spatial curvature: hyperbolic models
+In hyperbolic models or regions containing a center,
+Γ>0 andδ(k)
+i≥ −2/3 hold everywhere (conditions (61)
+and (68)), therefore, as a consequence of (37) and (B1),
+we have:
+kq<0⇒k <0, (B2)
+Hence, since all regular hyperbolic models or regions
+comply with kq<0, then all these models or regions have
+also negative local spatial curvature. As explain further
+below, the converse is not true, as local spatial curvature
+can be negative in certain elliptic regions in which kq≥0.
+Elliptic models and positive spatial curvature.
+The relation between kqandkis more complicated
+in elliptic models or regions, since standard regularity
+(conditions (61) and (72)) do not place a lower bound
+onδ(k)
+i. Hence, the possibility that δ(k)
+i<−2/3 occurs
+cannot be ruled out, and so k <0 can happen in regions
+wherekq>0. It is straightforward to show from (62)
+thatδ(k)
+i≤ − 2/3 can only occur in a regular elliptic
+model or region (Γ >0) ifE′= 0 for some 0 < rtv< rin
+a domain ϑ(r). As a consequence, we have δ(k)
+i>−2/3
+for all regular elliptic models in which E′≤0 holds for all
+r(withE′= 0 only at the symmetry center), and so (B1)implies in this case that kq>0⇒k >0 everywhere.
+If there is a zero of E′at somer=r∗, thenδ(k)
+i<−2/3
+will hold in some regions without violating regularity con-
+ditions. Since −1≤E≤0 andE(0) = 0, then for r≈0
+we must have E′<0. Thus, the only possible configu-
+ration is: E′≤0 andδ(k)
+i>−2/3 for 0≤r < r∗, with
+E′≥0 andδ(k)
+i<−2/3 forr > r∗. Further insight into
+this situation comes from rewriting (13) as
+E′=RR′(kq−3k) =−4RR′/bracketleftbigg
+kq+k′
+q/kq
+2R′/R/bracketrightbigg
+.(B3)
+If the regularity condition (A7) holds, then R′/√
+1 +E >
+0, and so for E′(r∗) = 0 to occur the local curvature
+kmust decrease sufficiently to reach k(r∗) =kq(r∗)/3,
+henceE′>0 (orδ(k)
+i<−2/3) leads to further decreas-
+ing ofk, so that k(r∗)< kq(r∗)/3 holds for r > r∗. A
+situation in which k <0 occurs with kq>0 can easily be
+conceived: since k′andk′
+qare both monotonously nega-
+tive, we have a curvature clump and so kq> k, thus, if
+kqdecays to zero sufficiently fast kmight become neg-
+ative. This happens in the elliptic side of the mixed el-
+liptic/hyperbolic configuration examined in section XIV
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+[46] We use the notation [ t] to emphasize that tis a fixed
+arbitrary parameter.
+[47] This equation corrects an error in [25], where −2/3≤
+δ(k)
+i≤0 was incorrectly assumed.
+[48] The coordinate resemblance between FLRW dust cos-
+mologies and LTB models is further highlighted by using
+metric form (39) for the latter.
\ No newline at end of file
diff --git a/1001.0905.txt b/1001.0905.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6c9cbd0b63955ed74d0a429def782d558150cd81
--- /dev/null
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@@ -0,0 +1,1168 @@
+arXiv:1001.0905v2  [math.CT]  12 Jan 2010The cohomological comparison arising from the
+associated abelian object
+Dominique Bourn
+Abstract
+We make explicit some conditions on a semi-abelian category Dsuch
+thatthecohomology grouphomomorphisms jn
+A:Hn
+M(D/Y)(A)→Hn
+D/Y(A),
+induced by the inclusion j:AbD֌Dof the abelian objects of D, are ac-
+tually isomorphisms. These conditions hold when Dis the category Gpof
+groups, and this allows us togive a newinsight on the Eilenbe rg-Mac Lane
+cohomology of groups. They hold also when Dis the category D=K-Lie
+of Lie-algebras.
+Introduction
+In a non-abelian context, there are several ways to realize the co homology
+groupsHn
+E(A) of a category E(provided, let us say, it is finitely complete and
+exact) with coefficients in an internal abelian group AofE: by means of sim-
+plicial objects in E, as introduced by Duskin [15] and Glenn [16], but also by
+means of internal n-groupoids in E, as in [5]. When EisAbthe category of
+abelian groups, the equivalence between the category of internal n-groupoids
+inAband the category of chain complexes of length nmake that these Hn
+coincide with the Yoneda’s Extn. When EisGpthe category of groups, the
+equivalencebetween the categoryofinternal n-groupoidsin Gpand the category
+of crossedn-fold complexes make that these Hncoincide (again see [5]) with the
+Opextnof the Eilenberg-Mac Lane cohomologyof groups, via the interpret ation
+(independently) given by Holt [18] and Huebschmann [19].
+Now, any left exact functor: U:E→E′which preserves the regular epi-
+morphisms provides natural comparison group homomorphisms Un
+A:Hn
+E(A)→
+Hn
+E′(U(A)). We shall be especially interested by the following situation. Let
+Dbe a semi-abelian category [21], as are the categories Gpof groups and K-
+LieofK-Lie algebras. Now, let Ybe an object in D, andD/Ythe associated
+slice category. We determine conditions on D, such the group homomorphisms
+jn
+A:Hn
+M(D/Y)(A)→Hn
+D/Y(A), induced by the inclusion jY:M(D/Y)֌D/Y
+of the commutative objects of D/Y(induced itself by the inclusion j:AbD֌D
+of the abelian objects of D) are actually group isomorphisms. These conditions
+are both global ( Dperi-abelian, see Definition 4.1 below) and local (the object
+Yhas projective dimension 1).
+These conditions hold, for any object Y, in the category Gpof groups and
+also in the category K-LieofK-Lie algebras. In the case of the category Gp,
+1this leads to the rather unexpected observation that, in a way, th e categories
+AbandGpare so close to each other (through the ”peri-abelian” connection )
+that the Eilenberg-Mac Lane cohomology is unable to discriminate the m. We
+thank G. Janelidze for our helpful discussions.
+The first section of the article is devoted to some recalls about n-groupoids.
+The second one deals with the properties, in the regular Mal’cev cont ext, of the
+inclusionj:MD֌Dfrom the commutative objects in D, and of its extensions
+to then-groupoids. The third one gives some recalls about the realization of
+the cohomology groups by means of n-groupoids; and the forth one introduces
+the notion of peri-abelian category and provides the final result.
+1 Internal groupoids and n-groupoids
+We shall suppose all our categories finitely complete. Given the follow ing right
+hand side commutative square, we denote the kernel equivalence r elation off
+byR[f] and the induced map between the kernel equivalences by R(x):
+R[f]
+R(x)
+/d15/d15p0/d47/d47
+p1/d47/d47X
+x
+/d15/d15f/d47/d47s0/d111/d111Y
+y
+/d15/d15
+R[f′]/d47/d47
+/d47/d47X′
+f′/d47/d47 /d111/d111Y′.
+An internal groupoid X1inEwill be presented (see [4]) as a reflexive graph
+(d0,d1) :X1⇒X0endowed with an operation d2:
+R[d0]2R(d2)
+/d19/d19p2/d47/d47p1/d47/d47
+p0/d47/d47R[d0]d2
+/d18/d18p1/d47/d47
+p0/d47/d47X1d1/d47/d47
+d0/d47/d47X0s0/d111/d111
+making the previous diagram satisfy all the simplicial identities, includin g the
+ones concerning the degeneracies. In the set theoretical conte xt, this operation
+d2associates the composite ψ.φ−1with any pair ( φ,ψ) of arrows of X1with
+same domain. Any equivalence relation Ron an object XinEprovides an
+internal groupoid:
+R[p0]2p3
+/d19/d19p2/d47/d47p1/d47/d47
+p0/d47/d47R[p0]p2
+/d17/d17p1/d47/d47
+p0/d47/d47Rp1/d47/d47
+p0/d47/d47Xs0/d111/d111
+which, in some formal circumstances, will be denoted by R1.
+LetGrdEdenote the category of internal groupoids and internal functor s
+inE, and () 0:GrdE→Ethe forgetful functor associating with the groupoid
+2X1its ”object of objects” X0. This functor is a left exact fibration. Any fibre
+(above a given object X) has a terminal object ∇1(X) which is the undiscrete
+relation on the object X:
+X×Xp1/d47/d47
+p0/d47/d47Xs0/d111/d111
+and an initial object ∆1(X) which is the discrete equivalence relation on X:
+X1X/d47/d47
+1X/d47/d47X1X/d111/d111
+They produce respectively a right adjoint and a left adjoint to the f orgetful
+functor () 0.
+An internal functor f1:X1→Y1is ()0-cartesian if and only if the following
+square is a pullback in E, in other words if and only if it is internally fully
+faithful:
+X1f1/d47/d47
+(d0,d1)/d15/d15Y1
+(d0,d1)/d15/d15
+X0×X0f0×f0/d47/d47Y0×Y0
+We shall need also the following classical definition:
+Definition 1.1. The internal functor f1is said to be a discrete fibration when
+any of the following squares is a pullback:
+X1d1/d47/d47
+d0/d47/d47
+f1
+/d15/d15X0
+f0
+/d15/d15/d111/d111
+Y1d1/d47/d47
+d0/d47/d47Y0/d111/d111
+1.1 Internal n-groupoids
+They are defined by induction on the integer n∈N([5]). The category 2- GrdE
+of internal 2-groupoids is defined as the category of internal gro upoids inside
+the fibres of the fibration () 0. We get a forgetful functor () 1: 2-GrdE→GrdE
+associatingwithany2-groupoid X2itsunderlyinggroupoid X1of1-cells. Again,
+it is a left exact fibration. Now, suppose defined the left exact fibra tion: ()n−2:
+(n-1)-GrdE→(n-2)-GrdE, then-groupoidsaretheinternalgroupoidsinsidethe
+fibresof() n−2. In this way, aninternal n-groupoidXndetermines an underlying
+diagram of reflexive graphs in Ewe shall denote by:
+Xn:Xnd1/d47/d47
+d0/d47/d47Xn−1s0/d111/d111d1/d47/d47
+d0/d47/d47Xn−2···X1s0/d111/d111d1/d47/d47
+d0/d47/d47X0s0/d111/d111
+3By∇n(respt. ∆n), we denotethe right(resp. left) adjointtothe fibration() n−1
+which gives the terminal (resp. initial) objects in the fibres of () n−1. To get
+that, given a n-groupoidXn, define by induction the object X≏
+nof its ”parallel”
+n-cells, from the following string of kernel equivalence relations ; begin with:
+X≏
+1p1/d47/d47
+p0/d47/d47X1s0/d111/d111(d0,d1)/d47/d47X0×X0
+whose left hand side part determines the upper level of the 2-grou poid∇2X1,
+then construct by induction:
+X≏
+np1/d47/d47
+p0/d47/d47Xns0/d111/d111(d0,d1)/d47/d47X≏
+n−1
+whose left hand side part determines the upper level of the ( n+ 1)-groupoid
+∇n+1Xn. The (n+ 1)-groupoid ∆n+1Xnis given by the discrete equivalence
+relation on Xn.
+Definition 1.2. We shall say that the n-groupoidXnis pointed when the ter-
+minal map Xn→∇n(Xn−1)in the fibre of ()n−1is split.
+Finally, as an internal groupoid in the fibres of () n−1, then-groupoidXn
+will be denoted in the following way inside the category ( n-1)-GrdE:
+/contintegraltext
+n−1Xnd1
+n−1/d47/d47
+d0
+n−1/d47/d47Xn−1, /d111/d111
+where/contintegraltext
+n−1Xnis the integral ( n-1)-groupoid of the n-cells of the n-groupoid
+Xn:
+/contintegraltext
+n−1Xn:Xnd1·d1/d47/d47
+d0·d0/d47/d47Xn−2s0.s0/d111/d111d1/d47/d47
+d0/d47/d47Xn−3···X1s0/d111/d111d1/d47/d47
+d0/d47/d47X0.s0/d111/d111
+When the category Eis regular [1], we call aspherical a groupoid X1which
+isconnected , namely such that its core(d0,d1) :X1→X0×X0is a regular
+epimorphism,andwhichhasitsobjectofobjects X0withglobalsupport,namely
+such that terminal map X0→1 is a regular epimorphism. A n-groupoidXnis
+aspherical when it is connected , namely such that its core(d0,d1) :Xn→X≏
+n−1
+is a regular epimorphism, and which has its underlying ( n-1)-groupoid Xn−1
+aspherical.
+Definition 1.3. An-functorfn:Xn→Ynis said to be a discrete fibration
+4when any of the following squares (at the highest level) is a p ullback:
+Xnd1/d47/d47
+d0/d47/d47
+fn
+/d15/d15Xn−1
+fn−1
+/d15/d15/d111/d111
+Ynd1/d47/d47
+d0/d47/d47Yn−1/d111/d111
+Proposition 1.1. Supposen≥1and the(n+1)-functorfn+1:Xn+1→Yn+1
+such that the underlying n-functorfnis()n−1-cartesian. Then fn+1is a dis-
+crete fibration if and only if it is ()n-cartesian. Suppose moreover the cate-
+goryEregular, the (n+ 1)-groupoidXn+1asherical and the (n+ 1)-groupoid
+Yn+1connected, then there is a kind of converse: when fn+1is at the same
+time()n-cartesian and a discrete fibration, then the underlying n-functorfnis
+()n−1-cartesian.
+Proof.Consider the following diagram:
+Xn+1fn+1/d47/d47
+(d0,d1)/d15/d15
+d0
+/d41/d41Yn+1
+(d0,d1)/d15/d15
+d0
+/d116/d116X≏
+n
+p0/d15/d15p1/d15/d15f≏
+n/d47/d47Y≏
+n
+p0/d15/d15p1/d15/d15
+Xnfn/d47/d47
+(d0,d1)/d15/d15Yn
+(d0,d1)/d15/d15
+X≏
+n−1f≏
+n−1/d47/d47Y≏
+n−1
+The fact that the n-functorfnis ()n−1-cartesian means that the lower square
+is a pullback. Then the middle square is a pullback. Accordingly the bow
+quadrangleisapullback(whichpreciselymeansthat fn+1isadiscretefibration)
+if and only if the the upper square is a pullback (which means that fn+1is ()n-
+cartesian).
+Suppose now that Eis regular and that the bow quadrangle and the upper
+square are pullbacks. If moreover the two upper vertical arrows are regular
+epimorphisms (which means that the ( n+ 1)-groupoids Xn+1andYn+1are
+connected), then the middle square is a pullback. When moreover th e left hand
+side lower vertical arrow is a regular epimorphism (which follows from t he fact
+thatXn+1is aspherical), then the lower square is a pullback (by the Barr-Kock
+theorem), and the underlying n-functorfnis ()n−1-cartesian.
+1.2 The regular Mal’cev context
+A category Dis a Mal’cev category when any reflexive relation is an equivalence
+relation, see [13] and [14]. Equivalently Dis a Mal’cev category when any
+5reflexive sub-graph of an internal groupoid is a groupoid. In this co ntext, any
+internal category is a groupoid. The categories Gpof groups,Rgof non unitary
+commutative rings and K-LieofK-Lie algebras are Mal’cev categories. When
+Dis a Mal’cev category, then any slice category D/Yis still a Mal’cev category.
+It appears that the context of Mal’cev categories particularly fits with the
+notion of commutator of equivalence relations [9], [23]. It is then poss ible to
+define an object XinDas being commutative when we have [∇X,∇X] = 0, i.e.
+when the commutator [ ∇X,∇X] is trivial, or, equivalently, when the object X
+is equipped with a (unique possible) Mal’cev operation p:X×X×X→X. We
+shall denote by MDthe subcategory of the commutative objectsXinDand by
+j:MD֌Dthe inclusion functor. A morphism f:X→Yis central when we
+have [R[f],∇X] = 0. It has an abelian kernel relation (or is commutative as an
+object in the slice category D/Y) when we have [ R[f],R[f]] = 0.
+InGp, a commutative object is an abelian group. Given any group Y, the
+slice category Gp/Yis still a Mal’cev category, and an object of Gp/Y, namely
+a group homomorphism X→Y, is commutative in Gp/Yif and only if its
+kernel is abelian. Given any finitely complete category E, the fibres of the
+fibration () 0:GrdE→Eare Mal’cev categories, so that there is a natural
+notion of commutative (actually we say abelian) groupoid. Recall that, when D
+is a Mal’cev category, then any internal groupoid in Disabelian.
+Recall a category Dis regular [1] when the regular epimorphisms are stable
+under pullback and any effective equivalence relation admits a quotien t. It
+is exact when, moreover, any equivalence relation is effective. Any v ariety of
+Universal Algebra is exact, and in particular the category Gp. We shall need
+the intermediate notion of efficiently regular category [8]:
+Definition 1.4. A regular category Dis said to be efficiently regular when any
+equivalence relation Ton an object Xwhich is a subobject j:T֌R[f] of
+an effective equivalence relation R[f] onXby an effective monomorphism in C
+(which means that jis the equalizer of some pair of maps in C) is itself effective.
+Any exact categoryisefficiently regular. The category GpTop(resp.AbTop)
+of topological (resp. topological abelian) groups is efficiently regula r, but not
+exact. More generally any category TopTof topological protomodular algebras
+(whereTis a protomodular theory) is efficiently regular: it is a regular categor y
+according to [3], and clearly an equivalence relation TonXis effective if and
+only if the object Tis endowed with the topology induced by the topological
+product, which is the case when j:T֌R[f] is an effective monomorphism.
+WhenEis efficiently regular, such is any slice category E/Y, and any fibre of
+the fibration () 0:GrdE→E. The main fact in an efficiently regular Mal’cev
+category Dis that any commutative object Xhas adirection which is given by
+6the following diagram [8]:
+X×X×Xp2/d47/d47
+(p0,p)/d47/d47
+p0
+/d15/d15X×X
+p0
+/d15/d15νX/d47/d47/d47/d47/d95 /d95 /d95A
+/d15/d15
+X×Xp1/d47/d47
+p0/d47/d47/d79/d79
+X/d47/d47/d79/d79
+1/d79/d79
+where the map νXis the quotient of the upper horizontal equivalence relation
+(where the maps piare the product projections, and the map pis the Mal’cev
+operation which makes Xcommutative). Any commutative square in this dia-
+gram is a pullback. When moreover Xhas a global support, the section s0of
+p0can be extented to he quotient (by the dotted arrow), Abecomes an internal
+abelian group in Dand the upward right hand side square becomes a pushout.
+For the same general reasons, any aspherical groupoid X1, being abelian in the
+Mal’cev context, admits a direction , namely an internal abelian group AinD
+such that the following downward square is a pullback, while, at the sa me time,
+the upward square is a pushout:
+X≏
+1/d47/d47/d47/d47
+p1
+/d15/d15p0
+/d15/d15A
+/d15/d15
+X1/d47/d47/d47/d47/d79/d79
+(d0,d1)/d15/d15/d15/d151/d79/d79
+X0×X0
+More generally, any aspherical n-groupoidXnadmits a directionAgiven by
+the same kind of diagram (see [10]):
+X≏
+n/d47/d47/d47/d47
+p1
+/d15/d15p0
+/d15/d15A
+/d15/d15
+Xn/d47/d47/d47/d47/d79/d79
+(d0,d1)/d15/d15/d15/d151/d79/d79
+X≏
+n−1
+Anyn-functorfn:Xn→Ynbetween two aspherical n-groupoids produces a
+group homomorphism between their respective directions.
+Remark 1.1. This group homomorphism is an isomorphism if and only if the
+n-functorfnis()n−1-cartesian, see [5].
+Finally when Dis moreover finitely cocomplete, the inclusion functor j:
+MD֌Dadmits a left adjoint (see [7]) which will be denoted by M.
+72 The reg-epi reflections
+Now, more generally, we shall consider a regular Mal’cev category Dandj:
+C֌Da full replete inclusion which admits a reg-epi reflection I:D→C.
+Recall:
+Definition 2.1. Letj:C֌Dbe a full replete inclusion and Da regular
+category. We shall say that the reflection I:D→Cis a reg-epi reflection when
+any projection ηX:X։IXis a regular epimorphism.
+In the regular context, being a reg-epi reflection is equivalent to s aying that
+Cisstableundersubobjects. So, when DisafinitelycocompleteregularMal’cev
+category, the reflection M:D→MDto the subcategory MDof the commuta-
+tive objects in Dis a reg-epi reflection. On the other hand, when Iis a reg-epi
+reflection, a map h:C→C′inCis a regular epimorphism in Cif and only
+if it is a regular epimorphism in D, so that the inclusion jpreserves regular
+epimorphisms.
+Any resultofthelastpartofthis sectionwillbe borrowedfrom[11]. Thereg-
+epi reflections have strong left exact properties, among which th e preservation
+of internal groupoids:
+Proposition 2.1. WhenDis a regular Mal’cev category, any reg-epi reflec-
+tionIpreserves the pullbacks of split epimorphims along split ep imorphisms.
+Accordingly it preserves the kernel equivalence relations of split epimorphisms,
+and the image I(X1)of any internal groupoid X1is an internal groupoid.
+However the kernel equivalence relation of any map is not preserve d byI
+in general, and we shall need to identify a certain class of maps having this
+property. Following [20], we have the following:
+Definition 2.2. A mapf:X→YinDis said to be I-trivial when the following
+square is a pullback:
+XηX/d47/d47/d47/d47
+f
+/d15/d15IX
+If
+/d15/d15
+YηY/d47/d47/d47/d47IY
+Any map in CisI-trivial. The isomorphismsare I-trivial, the I-trivial maps
+are stable under composition and such that, when g.fandgareI-trivial, then
+fisI-trivial. Also I-trivial maps are stable under those pullbacks which are
+preserved by the reflection I. On the other hand, a I-trivial map fis certainly
+I-cartesian, namely universal among the maps above I(f).
+Proposition 2.2. Suppose Dis a regular Mal’cev category, Iis a reg-epi re-
+flection and fisI-trivial. Then we have I(R[f])≃R[I(f)], and the maps
+pi:R[f]→Xands0:X֌R[f]are stillI-trivial. In other words, the
+reflectionIpreserves the kernel equivalence relations of the I-trivial maps.
+8Proposition 2.3. Suppose Dis an efficiently regular Mal’cev category and I
+a reg-epi reflection. Then a regular epimorphism is I-trivial if and only if it
+is the pullback of some map of the subcategory C. TheI-trivial regular epi-
+morphisms coincide with the I-cartesian regular epimorphisms, they are stable
+under pullbacks along any map, and these pullbacks are prese rved byI.
+Again following [20], we shall need also the following extension of the clas s
+ofI-trivial maps:
+Definition 2.3. Suppose Dis a regular Mal’cev category and Ia reg-epi reflec-
+tion . We call a map f:X→Y I-normal when the projection p0:R[f]→X
+isI-trivial.
+Then, by Proposition 2.2, any I-trivial map is I-normal. Considering any
+finitely cocomplete efficiently regular Mal’cev category Dand the reg-epi reflec-
+tionM:D→MD, a regular epimorphism f:X։YisM-normal if and only
+if it is central is the classical sense (i.e. [ R[f],∇X] = 0) [20], [11]. Finally we
+get:
+Proposition 2.4. Suppose Dis a regular Mal’cev category and Ia reg-epi
+reflection. If the regular epimorphism f:X։YisI-normal, then it is I-trivial
+if and only if its kernel equivalence relation is preserved b yI(i.e.R[I(f)]≃
+I(R[f])). Consequently a split epimorphism fisI-trivial if and only if it is
+I-normal.
+We shall need also the following extension of the last assertion:
+Proposition 2.5. Suppose Dis a regular Mal’cev category and Ia reg-epi
+reflection. Any split epimorphism tbetween two I-normal maps fandf′in the
+slice category D/Y:
+Xt/d47/d47
+f/d25/d25/d52/d52/d52/d52/d52X′s/d111/d111
+f′/d4/d4/d9/d9/d9/d9/d9
+Y
+isI-trivial.
+2.1 The reg-epi reflections I:n-GrdD→n-GrdC
+Suppose Dis a regular Mal’cev category and I:D→Ca reg-epi reflection. We
+noticed that Ipreserves the internal groupoids. Consequently, for any n∈N,
+the functor Iextends naturally to a functor: n-GrdD→n-GrdC(for sake of
+simplicity still denoted by I) which, again, is a reg-epi reflection. We are now
+going to investigate it more precisely.
+Fromnowon, weshallsuppose DisanefficientlyregularMal’cevcategory(in
+order to be able to define the direction of aspherical groupoids) an dI:D→C
+is a reg-epi reflection. We shall assume moreover that the reflect ionI:D→C:
+1) induces an isomorphism AbC≃AbD, whereAbDdenotes the category of
+9abelian groups in D, or equivalently the category of pointed commutative ob-
+jects inD
+2) preserves the groupoids ∇1(X) (i.e. is such that I(X×X)≃IX×IX).
+The main example we have in mind satisfying those conditions is the reg- epi
+reflectionM:D→MDtowards the commutative objects in a finitely cocom-
+plete efficiently regular Mal’cev category D. It obviously satisfies Condition 1;
+it is shown in [11] that, under the assumption that Dis efficiently regular, it
+satisfies also Condition 2.
+What is important for us, with the condition AbC≃AbD, is that any as-
+pherical groupoid X1has its core ( d0,d1) :X1։X0×X0I-normal:
+Lemma 2.1. Under the assumptions precised at the beginning of this sect ion,
+any aspherical groupoid X1has its core X1։X0×X0I-normal.
+Proof.Indeed, let Abe its direction. As an abelian group in D, it lies also in C.
+According to the Proposition 2.3 the following upper pullbacks show th atp0is
+I-trivial and, consequently, the core ( d0,d1) :X1։X0×X0isI-normal:
+X≏
+1/d47/d47/d47/d47
+p1
+/d15/d15p0
+/d15/d15A
+/d15/d15
+X1/d47/d47/d79/d79
+(d0,d1)/d15/d15/d15/d151/d79/d79
+X0×X0
+The functor Ipreserves the aspherical groupoids, since Ipreserves the regular
+epimorphism and is such that we have I(X×X) =I(X)×I(X). Here are our
+first problematical observations:
+There is no reasonwhy, in general, the functor Iwould preserve the direction
+of the aspherical groupoids (whichwouldbe equivalenttosayingthatthefunctor
+X1։IX1is ()0-cartesian), since the objects X≏
+1are not preserved by I.
+And no reason why the image of any aspherical n-groupoid, 2≤n, would be
+aspherical. However we can get the following precisions:
+Proposition 2.6. LetX2be an aspherical 2-groupoid, then d0:X2→X1is
+I-trivial and η2:X2→IX2is a discrete fibration.
+Proof.Since the 1-groupoids/contintegraltext
+1X2andX1are both aspherical, the cores
+(d0.d0,d1.d1) :X2→X0×X0and (d0,d1) :X1→X0×X0areI-normal.
+Thus the split epimorphism d0:X2→X1, which makes commute these cores,
+isI-trivial by Proposition 2.5, and η2:X2→IX2is a discrete fibration.
+Proposition 2.7. LetX2be an aspherical 2-groupoid, then its core (d0,d1) :
+X2→X≏
+1isI-trivial.
+10Proof.We haved0=p0.(d0,d1) :X2→X≏
+1→X1. The map d0:X2→X1is
+I-trivial. This is equally the case for p0
+X≏
+1p1/d47/d47
+p0/d47/d47X1s0/d111/d111(d0,d1)/d47/d47X0×X0
+since (d0,d1) :X1→X0×X0isI-normal. Accordingly ( d0,d1) :X2→X≏
+1is
+I-trivial.
+Proposition 2.8. LetXnbe an aspherical n-groupoid,n≥2, then the n-
+functorηn:Xn→IXnis a discrete fibration and its core (d0,d1) :Xn→X≏
+n−1
+isI-trivial.
+Proof.By induction. We saw it is true when n= 2. Suppose it is true as far
+as the level n-1. The (n-1)-groupoids/contintegraltext
+n−1XnandXn−1being aspherical, the
+maps (d0,d1) :Xn→X≏
+n−2and (d0,d1) :Xn−1→X≏
+n−2areI-trivial by the
+inductive assumption. Accordingly, the map d0:Xn→Xn−1isI-trivial and
+ηn:Xn→IXnis a discrete fibration. Moreover we have d0=p0.(d0,d1) :
+Xn→X≏
+n−1→Xn−1. The map d0:Xn→Xn−1isI-trivial. This is equally
+the case for the map p0:
+X≏
+n−1p1/d47/d47
+p0/d47/d47Xn−1s0/d111/d111(d0,d1)/d47/d47X≏
+n−2
+since (d0,d1) :Xn−1→X≏
+n−2isI-trivial by the inductive assumption. Accord-
+ingly (d0,d1) :Xn→X≏
+n−1isI-trivial.
+So the onlylevelwherethe coreofanaspherical n-groupoidis not necessarily
+I-trivial is the level 1.
+Definition 2.4. A groupoid X1will be said to be I-specific when it is aspherical
+and such that its core (d0,d1) :X1→X0×X0isI-trivial. More generally, a
+n-groupoidXnwill be said to be I-specific when it is aspherical and such that
+Xn−1isI-specific.
+We are now going to show that this special class of aspherical n-groupoids
+has a ”better” behaviour with respect to I:
+Example 2.1. Since any aspherical groupoid X1has its core X1։X0×X0
+I-normal, then any pointed groupoid is I-specific, following Proposition 2.4.
+Actually this last point is equivalent to the condition AbC≃AbD.
+In presence of the assumption 2) about the functor I, the groupoid X1isI-
+specific if and only if the functor η1:X1→IX1is ()0-cartesian or equivalently,
+(following Proposition 2.4), if and only if we have ∇2(I(X1)) =I(∇2(X1)).
+Proposition 2.9. LetXnbe an aspherical n-groupoid, then it is I-specific if
+and only if, for all 1≤k≤n, we have∇k+1(I(Xk)) =I(∇k+1(Xk)).
+11Proof.By induction. The equivalence holds at level 1. Suppose it holds as far
+as the level n-1. Then consider the following diagram where all the left hand
+side horizontal arrows are the maps η:
+X≏
+n/d47/d47/d47/d47
+p1
+/d15/d15p0
+/d15/d15(ηXn)≏
+/d47/d47
+I(X≏
+n)ǫn/d47/d47
+I(p1)
+/d15/d15I(p0)
+/d15/d15I(Xn)≏
+p1
+/d15/d15p0
+/d15/d15
+XnηXn/d47/d47/d47/d47/d79/d79
+(d0,d1)/d15/d15I(Xn)1I(Xn)/d47/d47/d79/d79
+I((d0,d1))/d15/d15I(Xn)/d79/d79
+(d0,d1)
+/d15/d15
+X≏
+n−1/d47/d47/d47/d47
+(ηXn−1)≏/d47/d47I(X≏
+n−1)ǫn−1/d47/d47I(Xn−1)≏
+SupposeXnI-specific. Then Xn−1isI-specific, and the inductive assumption
+(∇k+1(I(Xk)) =I(∇k+1(Xk)), 1≤k≤n−1) implies that ǫn−1is an iso-
+morphism. Since the middle vertical diagram is a kernel equivalence re lation
+((d0,d1) :Xn→X≏
+n−1beingI-trivial by Proposition 2.8), the factorization ǫn
+is an isomorphism, and we get ∇n+1(I(Xn)) =I(∇n+1(Xn))
+Conversely, suppose we have ∇k+1I(Xk)) =I(∇k+1(Xk)) for all 1≤k≤n.
+The inductive assumption is satisfied as far as level n-1. Accordingly Xn−1, and
+thusXn, isI-specific
+Corollary 2.1. Given any I-specificn-groupoidXn, the groupoid I(Xn)is
+aspherical.
+Proposition 2.10. LetXnbe an aspherical n-groupoid, then it is I-specific if
+and only if ηn:Xn→IXnis()n−1-cartesian. Actually it is I-specific if and
+only ifη1:X1→IX1is()0-cartesian.
+Proof.Recall that when Xnis aspherical, the map ηn:Xn→IXnis a discrete
+fibration. The direct proof will be given by induction. This is true for n= 1,
+since (d0,d1) :X1→X0×X0I-trivial means precisely η1:X1→IX1()0-
+cartesian. Suppose the result holds as far as the level n-1. IfXnisI-specific,
+thenX(n−1)isI-specific. By the inductive assomption ηn−1:Xn−1→IXn−1
+is ()n−2-cartesian, and by Proposition 1.1 ηn:Xn→IXnis ()n−1-cartesian.
+Again the converse is true for n= 1. Suppose it holds as far as the level n-1.
+Ifηn:Xn→IXnis ()n−1-cartesian, then, since ηn:Xn→IXnis a discrete
+fibration and the n-groupoids XnandI(Xn) are aspherical, the ( n−1)-functor
+ηn−1:Xn−1→IXn−1is ()n−2-cartesian by Proposition 1.1; and by the in-
+ductive assumption the ( n-1)-groupoid Xn−1, and thus the n-groupoidXn, are
+I-specific. The last point comes from the fact that any ηk:Xk→IXkis a
+discrete fibration by Proposition 1.1.
+As a consequence, by Remark 1.1, we then get what we were aiming to :
+Theorem 2.1. Given any I-specificn-groupoidXn, the groupoids I(Xn)and
+Xnhave same direction.
+123 The cohomology groups with coefficients in A
+LetEbe an efficiently regular category, and Abe an internal abelian group in
+E. Then, as usual, define H0
+E(A) as the abelian group HomE(1,A) of global sec-
+tions ofA, andH1
+E(A) as the abelian group of A-torsors, namely objects Xwith
+global support endowed with a simply transitive action of the group A. There
+areseveralequivalentwaystorealizethe groups Hn
+E(A),n≥2: bymeansofsim-
+plicial objects, as introduced by Duskin [15] and Glenn [16], but also b y means
+ofn-groupoids, as in [5]. In this last case, when E=A/C, whereAis an abelian
+category, these Hn
+A/C(A) coincide with the Yoneda Extn(C,A) via the equiva-
+lence between the extensions of length ninAand the aspherical n-groupoids
+inA. When E=Gp/Y, theseHn
+Gp/Y(A) coincide with the Opextn(Y,A) of
+the Eilenberg-Mac Lane cohomology of groups, via the interpretat ion given by
+Holt [18] and Huebschmann [19], and the equivalence between the cro ssedn-fold
+extensions and the aspherical n-groupoids in Gp, see [12] and [5]. When E=D
+is supposed to be a Mal’cev category, the abelian group Hn+1
+D(A) appears to
+be made of the component classes of the aspherical n-groupoids with direction
+A, see [10], while the group H1
+D(A) is made of the component classes of the
+commutative objects with direction A. The addition is given by a tensor prod-
+uct analogous to the tensor product of torsors. The inverse of t he class of Xn
+inHn+1
+D(A) is given by the class of its dual Xop
+nwhich is defined by inverting
+the role of the domain and the codomain maps at the last level, the one of the
+n-cells.
+3.1 The cohomological comparison
+Now, suppose we are under the assumptions of section 2.1, and let Abe in
+AbC=AbD. Then clearly the groups H0
+C(A) andH0
+D(A), whose elements are
+the sections (in CandDrespectively) of the terminal map A→1, are the
+same. Moreover, since the inclusion functor j:C֌Dis left exact and, having
+a reg-epi reflection I, preserves the regular epimorphisms, it induces, for any
+k∈N, a group homomorphism jk
+A:Hk
+C(A)→Hk
+D(A).
+Proposition 3.1. The homomorphism j1
+A:H1
+C(A)→H1
+D(A)is an isomor-
+phism.
+Proof.This homomorphism j1
+Ais injective: let Xbe a commutative object in C
+with global support and direction A; the image of its class by j1
+Ais 0 when the
+objectXis pointed in D, which is the case if and only if it is pointed in C. Let
+us showj1
+Ais surjective: let Ybe a commutativeobject in Dwith globalsupport
+and direction A. Then consider the following pullback of split epimorphisms:
+Y×Yν/d47/d47/d47/d47
+p1
+/d15/d15p0
+/d15/d15A
+/d15/d15
+Y/d47/d47/d47/d471eA/d79/d79
+13SinceAis inC, then, by Proposition 2.3, the map p0:Y×Y→YisI-trivial
+and these pullbacks are preserved by I. They produces the following diagram
+where any of the upper squares is a pullback:
+Y×Yν/d47/d47/d47/d47
+/d47/d47
+p1
+/d15/d15p0
+/d15/d15I(Y×Y)Iν/d47/d47
+p1
+/d15/d15p0
+/d15/d15A
+/d15/d15
+Y/d47/d47/d47/d47ηY/d47/d47/d47/d47
+/d15/d15/d15/d15IY/d47/d47/d47/d47
+/d15/d15/d15/d151eA/d79/d79
+1 1
+We haveI(Y×Y) =IY×IY. Consequently the two left hand side vertical
+diagramsarekernel equivalence relations. Since the upper left han d side squares
+arepullbacks,theBarr-Kocktheoremimpliesthatthelowersquare isapullback,
+and that consequently the map ηYis an isomorphism, and Yis inC.
+Proposition 3.2. The homomorphism j2
+A:H2
+C(A)→H2
+D(A)is injective.
+Proof.LetX1be an aspherical groupoid in Cwith direction A. Saying that its
+image byj2
+Ais 0 is saying that there is a X1-torsorYinD, namely a discrete
+fibration:
+Y×Yφ1/d47/d47/d47/d47
+p1
+/d15/d15p0
+/d15/d15X1
+d1/d15/d15d0/d15/d15
+Yφ0/d47/d47/d47/d47/d79/d79
+/d15/d15/d15/d15X0/d79/d79
+1
+Again (Proposition 2.3) these pullbacks are preserved by Iand produces the
+following diagram:
+Y×Yφ1/d47/d47/d47/d47
+/d47/d47
+p1
+/d15/d15p0
+/d15/d15IY×IYIφ1/d47/d47
+p1
+/d15/d15p0
+/d15/d15X1
+d1/d15/d15d0/d15/d15
+Y
+φ0/d47/d47/d47/d47ηY/d47/d47/d79/d79
+IYIφ0/d47/d47/d79/d79
+X0/d79/d79
+Accordingly the left hand side squares are pullbacks, and since YandIYhave
+global support, the map ηYis an isomorphism for the same reasons as in the
+previous proposition. Consequently Yis inC, and the groupoid X1is in the
+class of 0 in H2
+C(A).
+Saying that the homomorphism j2
+Ais surjective is saying that any aspher-
+ical groupoid Y1inDwith direction Ais in the same component class as an
+14aspherical groupoid T1inCwith direction A, namely that there is a pair of
+()0-cartesian maps with Z1aspherical (see [5]):
+Y1θ1←−Z1ψ1−→T1
+SinceZ1։Z0×Z0is a regular epimorphism, ψ1()0-cartesian and T1inC,
+then, according to Proposition 2.3, the map Z1։Z0×Z0isI-trivial, and the
+groupoidZ1isI-specific. So, saying that j2
+Ais surjective is saying that, for
+any aspherical groupoid Y1inDwith direction A, there is aI-specific groupoid
+Z1inDtogether with a () 0-cartesian map θ1:Z1→Y1(which implies that
+the direction of Z1isA). Accordingly, saying that j2
+Ais surjective for any
+abelian group object Ais saying that, for any aspherical groupoid Y1, there is
+aI-specific groupoid Z1inDtogether with a () 0-cartesian map θ1:Z1→Y1.
+Definition 3.1. We shall say that the category DisI-specific, when, for any
+aspherical groupoid Y1, there is a I-specific groupoid Z1inDtogether with a
+()0-cartesian map θ1:Z1→Y1.
+With this definition, our previous discussion about the surjectivity o fj2
+A
+becomes:
+Proposition 3.3. Suppose Dis an efficiently regular Mal’cev category and I
+is a reg-epi reflection satisfying Conditions 1) and 2) of Sec tion 2.1. Then, the
+group homomorphism j2
+Ais an isomorphism for any abelian group Aif and only
+if the category DisI-specific.
+Now we get:
+Proposition 3.4. Suppose DisI-specific, then for any aspherical n-groupoid
+YninD, there is aI-specificn-groupoidZninDtogether with a ()n−1-cartesian
+mapθn:Zn→Yn.
+Proof.By induction. We suppose the result holds as far the level n-1. Let
+Ynbe any aspherical n-groupoid and θn−1:Zn−1→Yn−1make explicit the
+result at the level n-1, withZn−1I-specific and θn−1()n−2-cartesian. Then
+takeθn:Zn→Ynthe ()n−1-cartesian map above θn−1. By definition, the
+(n-1)-groupoid Zn−1beingI-specific, so is the n-groupoidZn.
+Theorem 3.1. Suppose DisI-specific and the assumptions of Proposition 3.3
+hold. For any abelian group AinAbC=AbD, the group homomorphisms
+jn
+A:Hn
+C(A)→Hn
+D(A)are isomorphisms.
+Proof.Let us show that jn
+Ais injective. Let Xnbe any aspherical n-groupoid
+inCwith direction Asuch that its image by jn
+Ais 0. This means that there is
+15aXn-torsorYn−1inD, namely a discrete fibration:
+Y≏
+n−1φn/d47/d47
+p1
+/d15/d15p0
+/d15/d15Xn
+d1/d15/d15d0/d15/d15
+Yn−1φn−1/d47/d47/d79/d79
+(d0,d1)/d15/d15/d15/d15Xn−1/d79/d79
+Y≏
+n−2
+SinceDisI-specific, we are able to complete the previous diagram with a I-
+specific(n−1)-groupoid Zn−1andadiagramwhereallthesquaresarepullbacks:
+Z≏
+n−1θ≏
+n−1/d47/d47
+p1
+/d15/d15p0
+/d15/d15Y≏
+n−1φn/d47/d47
+p1
+/d15/d15p0
+/d15/d15Xn
+d1/d15/d15d0/d15/d15
+Zn−1θn−1/d47/d47/d79/d79
+(d0,d1)/d15/d15/d15/d15Yn−1φn−1/d47/d47/d79/d79
+(d0,d1)/d15/d15/d15/d15Xn−1/d79/d79
+Z≏
+n−2θ≏
+n−2/d47/d47Y≏
+n−2
+SinceXnis inCwe have the following factorizations where all the squares are
+pullbacks (again Proposition 2.3):
+Z≏
+n−1/d47/d47/d47/d47
+p1
+/d15/d15p0
+/d15/d15I(Z≏
+n−1)I(φn.θ≏
+n−1)/d47/d47
+p1
+/d15/d15p0
+/d15/d15Xn
+d1
+/d15/d15d0
+/d15/d15
+Zn−1ηn−1/d47/d47/d47/d47/d79/d79
+(d0,d1)/d15/d15/d15/d15I(Zn−1)
+I(φn−1.θn−1)/d47/d47/d79/d79
+(d0,d1)/d15/d15/d15/d15Xn−1/d79/d79
+Z≏
+n−2/d47/d47/d47/d47I(Z≏
+n−2)
+Now, since Zn−1isI-specific, we have certainly I(Z≏
+n−1) =I(Zn−1)≏and
+I(Z≏
+n−2) =I(Zn−2)≏, and thus a Xn-torsor in C:
+I(Zn−1)≏I(φn.θ≏
+n−1)/d47/d47
+p1
+/d15/d15p0
+/d15/d15Xn
+d1/d15/d15d0/d15/d15
+I(Zn−1)
+I(φn−1.θn−1)/d47/d47/d79/d79
+(d0,d1)/d15/d15/d15/d15Xn−1/d79/d79
+I(Zn−2)≏
+16Let us show now that jn
+Ais surjective. Let Ynbe any aspherical n-groupoidin D
+with direction A. According to Proposition 3.4, there is a I-specificn-groupoid
+ZninDtogether with a () n−1-cartesian map θn:Zn→Yn. Then the following
+pair of maps shows that the image by jn
+AofIZn(which is in C) is in the same
+component class as Yn:
+IZnηn←−Znθn−→Yn
+sinceηnis ()n−1-cartesian by Proposition 2.10.
+4 Examples of I-specific categories
+4.1 The protomodular context
+Given any finitely complete category E, we denote by PtEthe category whose
+objects are the split epimorphisms in Eand whose morphisms are the commu-
+tative squares between them:
+X′x/d47/d47
+f′
+/d15/d15X
+f
+/d15/d15
+Y′
+y/d47/d47s′/d79/d79
+Ys/d79/d79
+The codomain functor: PtE→Eis a fibration which is called the fibration
+of points and whose cartesian maps are those previous squares which are pu ll-
+backs. The fibre PtYEaboveYhas the split epimorphisms above Yas objects
+and the commutative triangles between them as morphisms. Recall t hat the
+category Eisprotomodular [6] when the change of base functors with respect
+to this fibration are conservative (i.e. reflect the isomorphisms) an d that any
+protomodular category is necessarily a Mal’cev category [2]. The ca tegoryGp
+of group, the category Rgof non unitary commutative rings, the category K-Lie
+ofK-Lie algebras and any additive category are pointed protomodular. It is the
+case also of the category GpTopof topological groups, and more generally of
+any category TopTof topological protomodular algebras [3].
+Given any finitely complete category E, any fibre of the fibration () 0:
+GrdE→Eis protomodular. When moreover Eis protomodular, any slice
+category E/Yis protomodular, while any fibre PtYEis pointed protomodular.
+It appears that the context of pointed protomodular categories particularly fits
+with the treatment of exact sequences [6]; in particular the (split) s hort five
+lemma holds in it [2].
+4.2 Peri-abelian categories
+Consider first a finitely cocomplete regular pointed protomodular ca tegoryD
+andC=AbDthe subcategory of the abelian (group) objects (actually, Dbeing
+pointed, we have AbD=MDsince any commutative object, being pointed,
+is abelian). Then the inclusion AbD֌Dhas a reg-epi reflection which will
+17be denoted by A. This reflection obviously satisfies Condition 1) of Section
+2.1. It satisfies Condition 2) since, the category Dbeing pointed, it preserves
+any product by Proposition 2.1. More generally, for any object YinD, any
+inclusionAb(PtYD)֌PtYDproduces a reg-epi reflection denoted by AY,
+since the category of PtYDof split epimorphisms in DaboveYis still finitely
+cocomplete, regular, and pointed protomodular, and this reflectio nAYsatisfies
+Conditions 1) and 2). In the case D=Gpthe category of groups, the functor
+AYis classically described by the following diagram where ¯X=X/[kerf,kerf],
+since [kerf,kerf] is a normal subgroup of X:
+X
+f
+/d32/d32/d64/d64/d64/d64/d64ηf/d47/d47/d47/d47¯X
+¯f/d126/d126/d126/d126/d126/d126/d126
+Ys/d96/d96/d64/d64/d64/d64/d64/d62/d62/d126/d126/d126/d126/d126
+It is clear that ker¯f= kerf/[kerf,kerf] =A(kerf), so that the change of
+base with respect to the fibration of points along the initial map αY: 1→Y
+preserves the associated abelian object. Consequently this is the case for the
+change of base along any map Y′→Y. This property is far from being satisfied
+by any finitely cocomplete regular pointed protomodular or even sem i-abelian
+category (i.e. non only regular, but also exact, see [21]) as shown by the next
+lemma below.
+Definition 4.1. We shall say that a finitely cocomplete, regular, pointed pro -
+tomodular category Dis peri-abelian when the change of base functor along any
+mapY′→Ywith respect to the fibration of points preserves the associa ted
+abelian object.
+It is clear that this definition still holds in the finitely cocomplete, regu lar,
+non-pointed Mal’cev context, but we shall only focus here on the co ntext pre-
+cised in the definition. So, here, ”peri-abelian” will imply ”finitely cocom plete,
+regular, pointed protomodular”. If ( AbPt)Ddenotes the subcategory of the
+abelian objects in the fibres of the fibration of points, it is equivalent to saying
+that the reg-epi reflection A()is cartesian, i.e. it preserves the cartesian maps:
+(AbPt)D
+/d31/d31/d64/d64/d64/d64/d64/d64/d64/d64/d47/d47 /d47/d47PtD
+/d4/d4/d8/d8/d8/d8/d8/d8/d8A()
+/d9/d9
+D
+For the same reasons as for the category Gp, it is clear that the categories
+Rgof non unitary commutative rings and K-LieofK-Lie algebras are peri-
+abelian. When Xis a topological group, endowing the normal subgroups ker f
+and [kerf,kerf] with the induced topology allows us to use the same argument
+as above to show that the category GpTopof topological groups is peri-abelian.
+18Lemma 4.1. Suppose Dis peri-abelian. Then the change of base functors with
+respect to the fibration of points along any map do reflectthe abelian objects.
+Proof.It is sufficient to prove it for the initial map αY: 1֌Y. So, suppose
+kerf=Ais abelian, where : X→Yis a split epimorphism. Since Dis peri-
+abelian, we have also ker ¯f=A; more precisely, the factorization determined
+byηfis an isomorphism. So, by the short five lemma, the map ηfis itself an
+isomorphism, and fis in (AbPt)D.
+Accordingly the semi-abelian category DiGpof digroups (a digroup is a set
+endowed with two group structures whose only coherence conditio n is to have
+same unit element) is not peri-abelian since its change of base functo rs with
+respect to the fibration of points do not reflect the abelian object s.
+We are going now to extend the preservation of abelian objects at t he level
+of the fibration of points to the preservation of commutative obje cts at the level
+of the slice categories.
+Suppose Dis not only regular, but efficiently regular. Consider the slice
+category D/Y. Then the construction of the associated commutative object in
+D/Ycan be derived from the construction of the associated abelian obj ect in
+the fibresPtUD. For that, given any map f:X→Y, consider the following
+diagram,where the twolowerverticalmaps ¯ p0arethe associatedabelian objects
+of the pointed objects p0:R[f]2→R[f] andp0:R[f]→Xrespectively :
+R2[f]p1/d47/d47
+p2/d47/d47
+/d15/d15/d15/d15
+p0
+/d15/d15R[f]d1/d47/d47/d47/d47
+/d15/d15/d15/d15X
+η/d15/d15/d15/d15
+f
+/d116/d116R[f]2p1/d47/d47
+p2/d47/d47
+¯p0
+/d15/d15R[f]¯q/d47/d47/d47/d47
+¯p0
+/d15/d15¯X
+¯f
+/d15/d15
+R[f]p0/d47/d47
+p1/d47/d47Xf/d47/d47Y
+Since the left hand side vertical rectangles are pullbacks of split epim orphisms,
+and the two upper vertical arrows are regular epimorphisms, then , as shown in
+[11], in the regular Mal’cev category D, the two left hand side levels of squares
+are pullbacks, and produce two discrete fibrations. In particular, since the lower
+one is a discrete fibration above an equivalence relation, the middle ho rizontal
+diagram is an equivalence relation. Since Dis efficiently regular, it is an ef-
+fective equivalence relation. Let ¯ qbe its coequalizer and ¯f,ηbe the induced
+factorizations. Now since all the squares on the left hand side are p ullbacks,
+and the maps d1and ¯qare regular epimorphisms, the two squares on the right
+hand side are pullbacks. Moreover, the map ¯fhas an abelian kernel equivalence
+relation as a quotient of the map ¯ p0which has an abelian kernel relation. It
+is straighforward to show that ¯fis the associated abelian object of ffrom the
+fact that ¯p0is the associated abelian object to p0:R[f]→X.
+19Proposition 4.1. WhenDis efficiently regular and peri-abelian, then, given
+any maph:Y′→YinD, the change of base functor h∗:D/Y→D/Y′
+preserves the associated commutative object.
+Proof.Consider the following diagram were the lower (and thus any) square is
+a pullback:
+R[f′]R(g)/d47/d47
+p1
+/d15/d15p0
+/d15/d15R[f]
+p1
+/d15/d15p0
+/d15/d15
+X′g/d47/d47
+f′
+/d15/d15X
+f
+/d15/d15
+Y′
+h/d47/d47Y
+Then, with the notation of the previous construction, the two follo wing vertical
+rectangles are the same:
+R[f′]R(g)/d47/d47
+p0
+/d15/d15R[f]
+p0
+/d15/d15R[f′]R(g)/d47/d47
+qX′/d15/d15/d15/d15R[f]
+qX/d15/d15/d15/d15
+X′g/d47/d47
+f′
+/d15/d15X
+f
+/d15/d15¯X′¯g/d47/d47
+¯f′
+/d15/d15¯X
+¯f
+/d15/d15
+Y′
+h/d47/d47Y Y′
+h/d47/d47Y
+The lower left hand side square is a pullback by assumption, while the up per
+left hand side square is a pullback since Dis peri-abelian. Accordingly the right
+hand side rectangle is a pullback. If the upper right hand side square is shown
+to be a pullback, the lower right hand side square will be a pullback since the
+two upper vertical arrows are regular epimorphism. Now consider t he following
+horizontal rectangle made of two pullbacks:
+R[f′]¯p0/d47/d47
+qX′/d15/d15/d15/d15X′g/d47/d47
+f′
+/d15/d15X
+f
+/d15/d15¯X′
+¯f′/d47/d47Y′
+h/d47/d47Y
+it is the same as the following one:
+R[f′]R(g)/d47/d47
+qX′/d15/d15/d15/d15R[f]
+qX/d15/d15/d15/d15¯p0/d47/d47X
+f
+/d15/d15¯X′
+¯g/d47/d47¯X¯f/d47/d47Y
+Now, the whole rectangle and the right hand side square are pullback s, accord-
+ingly the left hand side square is a pullback.
+204.3 The projective assumption
+We shall suppose, unless otherwisestated, that the pointed cate goryDis finitely
+cocomplete, efficiently regular, protomodular and peri-abelian. This holds in
+particular when Dis semi-abelian and peri-abelian, as are the categories Gp,Rg
+andK-Lie. This holdsalsowhen Dis the category GpTopoftopologicalgroups.
+Under our conditions, it is clear that the reg-epi reflections A:D→AbD,
+AY:PtYD→AbPtYDandMY:D/Y→M(D/Y) satisfy Condition 1) of
+Section 2.1. We already noticed that the two first ones satisfy Cond ition 2). It
+is shown in [11] that, in presence of the efficiently regular assumption , the third
+one satisfies also Condition 2).
+As usual, we have the following:
+Definition 4.2. Given any homological (i.e. pointed regular and protomodul ar)
+category E, we shall say that the object Yhas projective dimension 1when there
+is an exact sequence with HandKprojective objects in E(with respect to the
+regular epimorphisms):
+1/d47/d47K/d47/d47k/d47/d47Hh/d47/d47/d47/d47Y/d47/d471
+We are now going to prove that, when the object Yin our category Dhas
+projective dimension 1, the slice category D/YisMY-specific, whereMYis
+the reg-epi reflection associated with the inclusion of the commuta tive objects
+M(D/Y)֌D/Y. For that, we shall need some recall about the construction
+of the normalization of a groupoid.
+4.4 The normalization functor
+The category Dbeing pointed, we can associate with any internal groupoid U1
+its Moore normalization, namely the 1-chain complex (= map) given by t he left
+hand side vertical arrow in the following pullback:
+K[d0]κ/d47/d47
+µU1/d15/d15U1
+(d0,d1)
+/d15/d15
+U0(0,1U0)/d47/d47U0×U0
+Clearly this construction extends to a functor: µ:GrdD→Ch1Dwhich maps
+()0-cartesian functors onto pullback squares. For us, the main fact will be that,
+sinceDis efficiently regular and protomodular, the restriction of this funct or
+µto the subcategory of aspherical (=connected since Dis pointed) groupoids
+determines an equivalence of categories onto the category of cen tral extensions
+inD(see Proposition 17 in [6]). So, let us consider any aspherical groupo id in
+D/Y:
+H1d1/d47/d47
+d0/d47/d47H0s0/d111/d111h/d47/d47/d47/d47Y
+21Lemma 4.2. Suppose Dis a pointed regular and protomodular category. If the
+kernelKof the regular epimorphism his projective in D, then the aspherical
+groupoidα∗
+Y(H1)inDis pointed. If Dis finitely cocomplete, the aspherical
+groupoidα∗
+Y(H1)isA-specific. If moreover Dis efficiently regular and peri-
+abelian, any aspherical groupoid H1inD/Y(aboveh) isMY-specific.
+Proof.Consider the following diagram where any square is a pullback:
+NK1κ/d47/d47
+/d47/d47
+¯κ/d47/d47
+µK1/d15/d15/d15/d15K1/d47/d47k1/d47/d47
+(d0,d1)/d15/d15/d15/d15H1
+(d0,d1)/d15/d15/d15/d15
+K
+/d15/d15/d47/d47(0,1)/d47/d47K×K
+p1
+/d15/d15p0
+/d15/d15/d47/d47R(k)/d47/d47R[h]
+p1
+/d15/d15p0
+/d15/d15
+1/d47/d47K
+/d15/d15/d47/d47k /d47/d47H0
+h/d15/d15/d15/d15
+1αY/d47/d47Y
+The groupoid α∗
+Y(H1) is given by the middle vertical diagram, and denoted
+byK1. Its normalization µK1is a regular epimorphism, as a pullback of a
+regular epimorphism. The object Kbeing projective, this regular epimorphism
+µK1is split. Since the groupoid K1is aspherical and the restriction of the
+normalization functor µto the aspherical groupoids produces an equivalence
+of categories with the central extensions in D, the splitting of µK1determines
+a splitting of the core ( d0,d1) :K1։K×Kwhich makes the groupoid K1
+a pointed groupoid. According to Example 2.1, the fact that K1is pointed
+makes itA-specific where A:D→AbDgive the associated abelian object. This
+implies that η1:K1→A(K1) is ()0-cartesian in D.
+When, moreover, Dis efficiently regular and peri-abelian, the functor α∗
+Y:
+D/Y→Dpreserves the associated commutative objects, and we have η1=
+α∗
+Y(ηY
+1), withηY
+1:H1→MY(H1) inD/Y. It remains to show this last map
+is ()0-cartesian in D/Y. Since pulling back along αY: 1֌Ydetermines
+a functorα∗
+Y:D/Y→Dwhich reflects isomorphisms, when it is restricted to
+regularepimorphismsof D/Y(thisisthe shortfivelemma), thepullbackmaking
+the functor η1a ()0-cartesian morphism in Dis reflected in D/Y. Accordingly
+the functor ηY
+1is ()0-cartesian in D/Yand the groupoid H1isMY-specific.
+Whence the following:
+Theorem 4.1. Under the conditions on Dassumed at the beginning of Section
+4.3, when the object Yhas projective dimension 1, the category D/YisMY-
+specific.
+22Proof.Keeping the notations of the previous definition, let Yhave projective
+dimension 1. Let us consider any internal aspherical groupoid in D/Y:
+Y1d1/d47/d47
+d0/d47/d47Y0s0/d111/d111q/d47/d47/d47/d47Y
+Since the domain Hofh:H։Yis projective, there is a factorization θ:H→
+Y0such thatq.θ=h. Letθ1:H1→Y1be the () 0-cartesian map above θ.
+The maph:H։Ybeing a regular epimorphism, it is clear that the groupoid
+H1is aspherical in the category D/Y. Since, moreoever, the kernel Kofhis
+projective, according to the previous lemma, the groupoid H1produces, inside
+the category D/Y, theMY-specific groupoid associated with Y1we are looking
+for.
+As a straightforward consequence, we get:
+Theorem 4.2. Under the conditions on Dassumed at the beginning of Section
+4.3, when the object Yhas projective dimension 1, then, for any abelian group A
+in the slice category D/Yand any integer n, the cohomology groups Hn
+M(D/Y)(A)
+andHn
+D/Y(A)are isomorphic.
+It is well known that any subgroup of a free group is free. Accordin gly,
+given any group Y, the following exact sequence makes explicit the projective
+dimension1ofthegroup Y, whereFU(Y) isthefreegroupabovethe underlying
+setU(Y):
+1/d47/d47K/d47/d47k/d47/d47FU(Y) /d47/d47/d47/d47Y/d47/d471
+So that, for any group Y, the previous theorem holds. Our groups Hn
+Gp/Y(A)
+are nothing but the ones given by Eilenberg-Mac Lane cohomology, s ince the
+category of crossed n-fold exensions with kernel Aby means of which these
+cohomology groups are realized in [18] and [19], coincides with the cate gory
+of aspherical n-groupoids with direction Ain the slice category Gp/Y(see [5]).
+The translation of our theorem means that any class of crossed n-fold exensions,
+for a given Y-module structure on the abelian group A:
+1→A→Yn... Y1→Y0q→Y→1
+has an equivalent representation where Z1is abelian:
+1→A→Zn... Z1→Z0¯q→Y→1
+This was first made explicit in Proposition 2.7 in the Holt’s article about th e
+Eilenberg Mac-Lane cohomology groups [18], while the Huebschmann’s method
+was quite different [19]. In a way, our isomorphisms show that, via th e ”peri-
+abelian” connection and the projective dimension, the categories GpandAbare
+so close to each other that the Eilenberg-Mac Lane cohomology is un able to
+discriminate them.
+23On the other hand the so-called Shirshov-Witt theorem asserts th at any Lie
+subalgebra of a free Lie algebra is free. Accordingly the previous th eorem also
+holds in the category K-Lie, for any K-Lie algebra Y.
+Since the category GpTopof topological groups satisfies all the conditions
+given at the beginning of Section 4.3, a natural question would be wet her it is
+possible to adapt, or not, the last step of our method (dealing with t he projec-
+tive dimension) to obtain similar cohomology isomorphisms in this topolog ical
+setting.
+References
+[1] M. Barr, Exact categories , Springer L.N. in Math., vol. 236, 1971, 1-120.
+[2] F. Borceux and D. Bourn, Mal’cev, protomodular, homological an d semi-
+abelian categories,Kluwer,Mathematics andits application, vol.566 , 2004,
+479 pp.
+[3] F. Borceux and M.M. Clementino, Topological protomodular algebr as,
+Topology and its applications,, 153, 2006, 3085-3100.
+[4] D Bourn, The shift functor and the comprehensive factorizatio n for the
+internal groupoids, Cahiers de Top. et G´ eom. Diff. Cat´ egoriques ,28, 1987,
+197-226.
+[5] D. Bourn, The tower of n-groupoids and the long cohomology seq uence, J.
+Pure Appl. Algebra, 62, 1989, 137-183.
+[6] D. Bourn, 3×3 lemma and protomodularity, J. of Algebra, 236, 2001,
+778-795.
+[7] D.Bourn, Commutator theory in regular Mal’cev categories , FieldsInstitute
+Communications, vol.43, 2004, 61-75.
+[8] D. Bourn, Baer sums in homological categories, Journal of Algeb ra,308,
+2007, 414-443.
+[9] D. Bourn and M. Gran, Centrality and connectors in Maltsev cate gories,
+Algebra Universalis, 48, 2002, 309-331.
+[10] D. Bourn and D. Rodelo, Cohomology without projective, Cahier s de Top.
+et G´ eom. Diff. Cat´ egoriques, 48, 2007, 104-153.
+[11] D. Bourn and D. Rodelo, Comprehensive factorization and unive rsalI-
+normal extension in the Mal’cev context, Preprint, 2009.
+[12] R. Brown and P.J. Higgins, The equivalence between the ∞-groupoids and
+crossed complexes, Cahiers de Top. et G´ eom. Diff. Cat´ egoriques ,22, 1981,
+371-386.
+24[13] A. Carboni, J. Lambek and M.C. Pedicchio, Diagram chasing in Mal’ce v
+categories, J. Pure Appl. Algebra, 69, 1991, 271-284.
+[14] A. Carboni, M.C. Pedicchio and N. Pirovano, Internal graphs and internal
+groupoids in Mal’cev categories , CMS Conference Proceedings, 13, 1992,
+97-109.
+[15] J.W. Duskin, Simplicial methods and the interpretation of triple cohomo l-
+ogy, Memoirs Amer. Math .Soc. 163, 1975.
+[16] P.G. Glenn, Realization of cohomology classes in arbitary exact ca tegories,
+J. Pure Appl. Algebra, 25, 1982, 33-105.
+[17] M. Gran, Applications of categorical Galois Theory in Universal Alg ebra,
+Fields Institute Communications, vol.43, 2004, 243-280.
+[18] D. Holt, An interpretation of the cohomology groups H(G,M), J. Algebra,
+60, 1979, 307-318.
+[19] J. Huebschmann, Crossed n-fold extensions of groups and co homology,
+Comment. Math. Helv., 55, 1980, 302-314.
+[20] G. Janelidze and G.M. Kelly, Galois theory and a general notion of c entral
+extension, J. Pure Appl. Algebra, 97, 1994, 135-161.
+[21] G. Janelidze, L. Marki and W. Tholen, Semi-abelian categories, J . Pure
+Appl. Algebra, 168, 2002, 367-386.
+[22] S. Mac Lane, Homology, Springer, 1963.
+[23] M.C. Pedicchio, A categorical approach to commutator theory , Journal of
+Algebra, 177, 1995, 647-657.
+Universit´ e du Littoral, Calais, France
+bourn@lmpa.univ-littoral.fr
+25
\ No newline at end of file
diff --git a/1001.0906.txt b/1001.0906.txt
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+arXiv:1001.0906v1  [hep-th]  6 Jan 2010KIAS-P10001
+N= 2SCFTs: AnM5-braneperspective
+Bin Chen1, Eoin´O Colg´ ain2, Jun-Bao Wu2and HosseinYavartanoo2
+1Department of Physics,
+and State KeyLaboratory of Nuclear Physics and Technology,
+Peking University,
+Beijing 100871, P.R.China
+2Korea Institute for Advanced Study,
+Seoul, Korea
+Abstract
+Inspiredbytherecentlydiscoveredholographicdualitybe tweenN= 2SCFTs
+and half-BPS M-theorybackgrounds,westudyprobeM5-brane s. Thoughour
+main focus is supersymmetric M5-branes whose worldvolume h as anAdSn
+factor, we also consider some other configurations. Of speci al mention is the
+identification of AdS5andAdS3probes preserving supersymmetry,with only
+thelattersupportingaself-dualfield strength.1 Introduction
+Four-dimensional N= 2supersymmetric theories are truely remarkable. Compared t o
+N= 4supersymmetric theories, which are finite, they are much ric her in physics, but
+yet still solvable. Especially, they provide a window to stu dy the nonperturbative aspects
+of quantum field theories and have been widely studied for mor e than fifteen years since
+Seiberg and Witten’s monumental works [1, 2]. However, the s urprises they present to us
+havenot come to the end. Recent developmentson four-dimens ionalN= 2superconfor-
+maltheoriesstartingfrom[3]havedrawnlotsofattention. Thisclassofgeneralizedquiver
+theories could be constructed geometrically by wrapping M5 branes on Riemann surfaces
+with genus and punctures. The electric-magnetic duality an d the Argyres-Seiberg duality
+[4] have since been generalized to these theories. It turns o ut that the gauge couplings of
+the theory are encoded in the complex structure moduli of the Riemann surface, includ-
+ing the position of the punctures. More interestingly, it wa s conjectured in [5] that the
+Nekrasovpartitionfunctionofthesetheorieswith SU(2)gaugegroupscould berelated to
+theconformalblocksandcorrelationfunctionsoftheLiouv illetheory. Non-localoperators
+inthesefour-dimensionaltheorieshavebeen studiedin [6, 7, 8, 9,10](See also[11]).
+The holographic dual of these theories in the large N limit wa s neatly studied in [12].
+The theory on the gravity side of this AdS/CFT correspondenc e is M-theory on back-
+groundswhicharetheproductsof AdS5spacetimeandsix-dimensionalinternalmanifolds
+withSU(2)×U(1)isometry. Thesegravitybackgroundsbelongtothegeneralg eometries
+foundin [13]. Of particularinterest amongthesebackgroun dsis theso-called Maldacena-
+N´ u˜ nez(MN) geometry, which was first discovered in [14] by c onsidering the IR limit of
+M5-branes wrapped on a Riemann surface. In thiscase, thesix -dimensionalinternal man-
+ifold is simply an S4fibered over the Riemann surface. On the other side, the field t heory
+corresponding to the MN geometry comes from M5-branes wrapp ing the same Riemann
+surface. It is remarkable, that in this case, there are no pun ctures on the Riemann surface
+and that the building block of the quiver gauge theory is a str ongly coupled superconfor-
+maltheoryTNwiththreeSU(N)globalsymmetries,yetwithoutacouplingconstant. One
+motivationofthiswork istounderstand thisintriguing TNtheoryfrom itsgravitydual.
+Probebranesplayanimportantroleinthegravitysideofhol ographiccorrespondences.
+These branes are intrinsically stringy but accessible. Amo ng other things, they could be
+dualto local operators [15, 16, 17], loop operators [18, 19, 20], surface operators [21, 22],
+ordomainwalls[23]inthefieldtheorysideofvariousAdS/CF Tcorrespondences. Adding
+suitablebranes can alsoadd flavortothefield theory [24].
+In this paper, we plan to start the search for interesting pro be M5-branes in the LLM
+geometries studied in [12]. We mainly focus on the simplest M N background which we
+1haverederivedin theappendixto includethefluxes. Theform ofthesolutionis
+ds2
+11= ˜κ2/3/parenleftbigg1
+2W1/3ds2
+AdS5+W−2/3
+4/bracketleftbigg
+W(dx2+dy2)
+y2
++Wdθ2+cos2θ(dφ2
+1+sin2φ1dφ2
+2)+2sin2θ/parenleftbigg
+dχ+dx
+y/parenrightbigg2/bracketrightbigg/parenrightbigg
+, (1)
+H4= ˜κ/parenleftbigg
+−1
+4W2[3+cos2θ]sinθcos2θdθ(dχ+dx
+y)+1
+4cos3θ
+Wdxdy
+y2/parenrightbigg
+d2Ω,
+whereφ1,φ2parameterize a two-sphere, x,ydenote a hyperbolic Riemann surface Σ2of
+constantnegativecurvature,and Wis
+W= 1+cos2θ. (2)
+The constant ˜κdenotes an additional scale factor that will be accounted fo r by ensuring
+thatthefluxis correctly quantised.1
+This geometry may also be obtained from the most general solu tionsof eleven dimen-
+sionalsupergravitypreserving N= 2superconformalsymmetry[13]asdescribedin[12].
+Althoughwe focus on the above background, we believe our res ults can be generalized to
+moregeneralLLMgeometries. Intheliterature,someBPSpro bebraneshavebeenstudied
+in[12], and half-BPS M2-branedual toloopoperatorhaveapp eared in [6].
+Starting with Killing spinors of MN geometry and kappa symme try for M5, one can
+search for BPS M5-branes. As a first step, one needs to determi ne the Killing spinors
+preserved by the MN solution. Luckily, this has already been done for the most general
+solution [13] and the Killing spinors corresponding to the a nalytically continued solution
+correspondingtoMNhaveappearedin[6]. Thelatterappears withoutderivation,sointhe
+appendix we validate their claim by following a similar deco mposition to that appearing
+in [13] (see also [25])2. The result of that exercise is that the eleven-dimensional Killing
+spinors of the MN solution can be expressed in terms of AdS5(ψ) andS2(χ+) Killing
+spinorsas
+ǫ=eλ/2ψ⊗(1+iσ3⊗γ(4))χ+⊗e−i
+2φ0γ10eiχ/2ǫ0,
+ǫc=eλ/2ψc⊗(1−iσ3⊗γ(4))χ+⊗e−i
+2φ0γ10e−iχ/2γ7ǫ0, (3)
+wherethesuperscript cdenotes theconjugateand
+sinφ0=√
+2cosθ√
+W,cosφ0=−sinθ√
+W, e2λ=˜κ2/3W1/3
+8. (4)
+1Itis relatedto the κin [12] by ˜κ= 24Nκ.
+2We also curesome typosin [6].
+2Theconstantspinor ǫ0satisfiesthefollowingprojectionconditions
+γ9ǫ0=ǫ0, iγ78ǫ0=ǫ0. (5)
+Withpossibledualnon-localobjectsinfieldtheoryinourmi nd,wepayprincipalatten-
+tion to M5-branes whose worldvolumes have AdSm(2≤m≤5)factors. The brane with
+worldvolume AdS5×S1has been studied previously in [12]. However, we find that al-
+thoughturningonself-dualthree-formfieldstrengthonthe worldvolumeinasuitableway
+doesnotbreak supersymmetry,theequationsofmotionofM5- braneswillnotbesatisfied.
+This comes as some surprise as the Kaluza-Klein reduction of the two-form potential on
+theS1(χ) gives rise to a U(1)gauge field in AdS5corresponding to a global symmetry
+rotatingthephaseofadualbifundamentalfield [12].
+Moreover, we find BPS M5-branes with an AdS3factor. This brane should be dual
+to some two-dimensional object in the field theory side. Howe ver it is not dual to the
+supersymmetric surface operator studied in [7, 9], since th is brane wraps the Riemann
+surface in the six-dimensional internal space, instead of i ntersecting with this Riemann
+surface at a point. In this case, we find that we can turn on a sui table self-dual three-
+form field strength on the worldvolume such that the BPS condi tion and the equations of
+motionarebothsatisfied. WealsofindBPSM5-branesnotembed dedalongthe AdS5radial
+direction that satisfy the equations of motion hinting that there should be non-BPS AdS
+branestherealso. Inaddition,inspiredbysomeprobeM5-br anesinAdS7×S4[26,27],we
+turntosearchingforM5-branesinMNbackgroundwithmoreco mplicatedworldvolumes.
+As a result, we find M5-branes with an AdS3×S1andAdS2×S2factors which are
+completely embedded in the AdS5part of the background. We explicitly illustrate that
+genericallythesebranes arenon-supersymmetric.
+In the next section, we move to review the M5-brane equations of motion and BPS
+condition. With these tools at hand, we study various probe M 5-branes in section 3. In
+section4,weexaminemoreexoticembeddingsin AdS5beforeconcluding. Sometechnical
+detailsarelocated intheappendices.
+2 M5-branereview
+Inthissection,wereviewtheM5-branecovariantequations ofmotionsincurvedspacetime
+and discuss the condition for the M5-brane probe to preserve supersymmetry. For earlier
+work on various aspects of the M5-brane, see [29, 30, 31, 32, 3 3, 34] (For a review of
+M-theorybranes,see[35]). Thissectionechoesthebriefre viewoftheM5-branepresented
+in[27]and werefer thereader thereforafurtheraccount oft heM5-braneaction.
+3Focusing solely on the bosoniccomponents, we simplyhave tw o equations of motion:
+ascalarand atensorequation. Thescalarequationtakes the form
+Gmn∇mEc
+n=Q√−gǫµ1···µ6/parenleftbig1
+6!Ha
+7µ1···µ6+1
+(3!)2Ha
+4µ1µ2µ3Hµ4µ5µ6/parenrightbig
+Pc
+a(6)
+andthetensorequationis oftheform
+Gmn∇mHnpq=Q−1(4Y−2(mY+Ym)+mYm)pq. (7)
+Here our notation is as follows: indices from the beginning( middle) of the alphabet refer
+to frame(coordinate) indices, and the underlined indices r efer to target space ones. More
+detailsofourconventionsmay befoundintheappendices.
+Appearing in the equations of motion, we have the following q uantities which are de-
+fined intermsoftheself-dual3-form field strength hon theM5-braneworldvolume
+kn
+m=hmpqhnpq, (8)
+Q= 1−2
+3Trk2, (9)
+mq
+p=δq
+p−2kq
+p, (10)
+Hmnp= 4Q−1(1+2k)q
+mhqnp (11)
+Notethathmnpisself-dualwithrespecttoworldvolumemetricbutnot Hmnp. Theinduced
+metricissimply
+gmn=Ea
+mEb
+nηab (12)
+where
+Ea
+m=∂mzmEa
+m. (13)
+Herezmisatargetspacetimecoordinate,whichbecomesafunctiono fworldvolumecoor-
+dinateξthrough the embedding, and Ea
+mis the component of target space vielbein. From
+theinduced metric,wecan define anothertensor
+Gmn= (1+2
+3k2)gmn−4kmn. (14)
+Wealsohave
+Pc
+a=δc
+a−Em
+aEc
+m. (15)
+Note that in the scalar equation of motion, the covariant der ivative∇mEc
+ninvolvesnot
+onlytheLevi-CivitaconnectionoftheM5-braneworldvolum ebutalsothespinconnection
+ofthetarget spacetimegeometry. Moreprecisely,onehas
+∇mEc
+n=∂mEc
+n−Γp
+mnEc
+p+Ea
+mEb
+nωc
+ab (16)
+4whereΓp
+mnis the Christoffel symbol with respect to the induced worldv olume metric and
+ωc
+abisthespinconnectionofthebackgroundspacetimepulledba ck to theworldvolume.
+Moreover, there is a background 4-form field strength H4a1···a4and its Hodge dual
+7-formH7a1···a7:
+H4=dC3
+H7=dC6+1
+2C3∧H4 (17)
+Theframeindiceson H4andH7intheaboveequations(6)and(7)havebeenconvertedto
+worldvolumeindiceswithfactors of Ec
+m. From thebackgroundfluxes, wecan define
+Ymn= [4⋆H−2(m⋆H+⋆Hm)+m⋆Hm]mn, (18)
+where
+⋆Hmn=1
+4!√−gǫmnpqrsHpqrs (19)
+ThefieldHmnpis defined by
+H=dA2−C3, (20)
+whereA2is a 2-form gauge potential and C3is the pull-back of the bulk gauge potential.
+Fromitsdefinition, HsatisfiestheBianchi identity
+dH=−H4 (21)
+whereH4isthepull-backofthetarget space 4-formflux.
+In general, the supersymmetric embeddings of a probe brane i n a background may be
+determinedfromthekappa-symmetrycondition
+Γκǫ=±ǫ. (22)
+Here,Γκdenotesthegammamatrixassociatedtotheprobe, ǫdenotestheKillingspinorof
+thebackgroundand thesignaccounts for thechoice between b rane and anti-braneprobes.
+Theamountofunbroken supersymmetrymaybe determinedby ke eping track oftheaddi-
+tionalprojectionconditionsthat arisefromtheaboveequa tion.
+SpecializingtotheMNbackgroundwithM5-braneprobes,the kappasymmetrymatrix
+ΓM5maybewrittenfollowing[28]
+ΓM5=1
+6!√−gǫj1···j6[Γ<j1···j6>+40Γ <j1j2j3>hj4j5j6]. (23)
+Heregis the determinant of the induced worldvolume metric compon ent,hj4j5j6is the
+self-dual3-form ontheM5-braneand Γ<j1···jn>isdefined as
+Γ<j1···jn>=Ea1
+j1···Ean
+jnΓa1···an, (24)
+5whereΓa1···anistheproductoftheGammamatricesin orthonormalframe.
+We pause here to make a brief comment. Denoting the worldvolu me of the M5 by
+ξa,a= 0,···5, in the case of a simple probe configuration, we may rewrite th e above
+projector(22)as
+[αΓ012345+β(Γ012−Γ345)]ǫ=±ǫ, (25)
+whereα,βdenote arbitrary factors. Demanding it to be a projector, it is essential the left
+hand side squares to unity. In that event, βdrops out completely and α2= 1, meaning
+thatα=±1. Theimplicationofthisobservation,atleastforthesimpl eprobesconsidered
+in this paper, is that if the M5-probe is not supersymmetric, then supersymmetry cannot
+be restored by introducing h. So the task in the rest of the paper is pretty straightforwar d:
+identifysupersymmetricprobesandthenturnon htoseeifitpreservessupersymmetry. At
+each stage, itisalso imperitivetoensurethat theequation sofmotionare satisfied.
+3 Supersymmetricprobes
+In this section, we focus on the kappa-symmetry condition (2 2) and isolate probes that
+will preserve some supersymmetry. We descend in dimension o f the part in AdS5from
+d=5 to d=2 and in each case, we enumerate the possibilities. T hroughout we differentiate
+between probes that are AdSi.e. those incorporating the radial direction rofAdS5and
+thoselocatedat afixed r. We beginby examiningthe AdSprobes.
+3.1 Supersymmetric AdSprobes
+Inthissubsection,wedescendfrom AdS5toAdS2andidentifythesupersymmetricprobes
+(ifany),beforeexaminingtheadditionalconstraintscomi ngfromtheequationsofmotion.
+As a warm-up, we begin with the AdS5M5-brane probe which received someattentionin
+[12].
+AdS5probes
+In general, one can consider studying the probe brane with wo rldvolumeAdS5×Cin the
+MNbackground,where Cdenotesacurveinthesix-dimensionalspacetransverseto AdS5.
+We consider the Cto be parameterised by σi.ezm(σ). UsingAdS5Poincar´ e coordinates,
+anaturalchoicefortheM5embeddingis
+ξ0=x0, ξi=xi, ξ4=r, ξ5=σ, (26)
+wherei= 1,2,3.
+6Adopting the gauge choice σ=χ, while permitting embeddings of the form x≡
+x(χ),y≡y(χ),thekappasymmetrymatrix ΓM5simplifiesto
+ΓM5=1√gχχΓ01234Γ<χ>, (27)
+wheretheinducedmetriccomponentis
+gχχ=/parenleftbigg˜κ
+W/parenrightbigg2/3/parenleftBigg
+W
+4/parenleftbigg(∂χx)2+(∂χy)2
+y2/parenrightbigg
++sin2θ
+2/parenleftbigg
+1+(∂χx)
+y/parenrightbigg2/parenrightBigg
+,(28)
+andtheinducedgammamatrixis
+Γ<χ>= 14⊗12⊗/parenleftbigg˜κ
+W/parenrightbigg1/3/bracketleftbiggsinθ√
+2/parenleftbigg
+1+(∂χx)
+y/parenrightbigg
+γ9+W1/2
+2/parenleftbigg(∂χx)
+yγ7+(∂χy)
+yγ8/parenrightbigg/bracketrightbigg
+.
+(29)
+Utilisingρ01234=i, therequirement forsupersymmetry ΓM5ǫ=±ǫthenreduces to
+˜κ1/3W−1/3
+√gχχ/bracketleftbiggsinθ√
+2/parenleftbigg
+1+(∂χx)
+y/parenrightbigg
+γ9+W1/2
+2(∂χx)
+yγ7+W1/2
+2(∂χy)
+yγ8/bracketrightbigg
+ǫ0=±ǫ0,(30)
+providedφ0=π. This means that θ=π
+2andW= 1. In addition, we rquire the probe to
+belocatedat apointontheRiemann surface:
+∂χx=∂χy= 0, (31)
+so that the terms proportional to γ7andγ8disappear. The projection condition γ9ǫ0=ǫ0
+alsosinglesoutthepositivesignaboveindicatingthatthe probeisanM5-braneasopposed
+toan anti-M5-brane.
+Therefore,thecurve Cisexclusivelyalongthe χ-direction. Asnotedin[12],the θ=π
+2
+conditioncorrespondstothe S2shrinking,sothesuperconformalsymmetry SU(2)×U(1)
+symmetryofthebackgroundispreservedbythisprobe. Also, nosupersymmetryisbroken
+bythisprobe.
+Now that we have a supersymmetric probe, we may inquire wheth er it is possible to
+turnon self-dual h. As explainedearlier, thisproblemreduces toensuring
+a(Γ012−Γ349)ǫ= 0, (32)
+wherewehavedefined
+h=a
+2(E012+E349). (33)
+Again using the decomposition (125) and the relationship ρ01234=i⇒ρ34=iρ012, then
+itispossibletoshowthatthisconditionis satisfied.
+7Having verified the kappa-symmetry condition is satisfied, i t remains to show that the
+equationsofmotionare satisfied. Theinduced metricmaybew ritten
+ds2
+ind=˜κ2/3
+2/bracketleftbiggdxµdxµ+dr2
+r2+dχ2/bracketrightbigg
+, (34)
+wherewehaveused Poincar´ ecoordinates.
+TheRHSofthetensorequation(7)iszeroasthebackground4- formfluxdoesnotpull
+back totheM5worldvolume. Thetensorequationis thensimpl y
+Gmn∇mHnpq= 0. (35)
+As for scalar equation, the RHS vanishestrivially when c/ne}ationslash= 10. Forthe case with c= 10,
+the RHS is non-vanishing for general θdue to thevolAdS5∧dθ∧dχpart ofH7. How-
+ever, the coefficient is proportional to cosθ, so it vanishes when it gets pulled back to the
+worldvolumeat θ=π
+2. So, neglectingthiscase, thescalarequationissimply
+Gmn∇mEc
+n= 0. (36)
+Thisequationisquicklyconfirmedtobesatisfiedasitonlyha sonenon-trivialcomponent:
+∇rE4
+r=∂r˜κ1/3
+√
+2r+˜κ1/3
+√
+2r2= 0. (37)
+Theansatzweconsiderfor his
+h=a
+2/parenleftbig
+E012+E349/parenrightbig
+=a˜κ
+4√
+2/parenleftbigg1
+r3dtdx1dx2+1
+r2dx3drdχ/parenrightbigg
+, (38)
+whereaisafunctionofr. Followingthetreatmentin [27], Hmaybeexpressed as
+H=a˜κ
+(1+a2)√
+21
+r3dtdx1dx2+a˜κ
+(1−a2)√
+21
+r2dx3drdχ. (39)
+Asthebackground4-formfluxdoesn’tpullback,theBianchii dentity(21)issimply dH=
+0. Thismeans thata
+(1+a2)r3=constant. (40)
+Switchingthelocationof drin theflux ansatzabovewouldmakethis
+a
+(1−a2)r2=constant. (41)
+OncetheBianchi issatisfied, onemay return tothetensorequ ation. HereGmnis diagonal
+and the only term of interest is ∇rHrx3χwhich is not zero unless ais a constant. So, we
+8reachacontradictionandtheconclusionisthatthereisnos upersymmetric AdS5M5-brane
+probewithself-dual 3-form h.
+AdS4probes
+ForAdS4probes, a quick look at appendix B reveals that we must mix the spinorψwith
+its conjugate ψc. This is because ρ0124η+andη−both have the same eigenvalue under ρ4.
+Thus,weconsider
+ρ0124ψ=cψc, (42)
+wherecis a constant. The overall effect of this mixing is that the MN Killing spinor gets
+related toitsconjugatethroughthekappa-symmetrycondit ion.
+AdoptingtheM5embedding
+ξ0=x0, ξi=xi(i= 1,2), ξ3=r,{ξ4,ξ5} ⊂M6, (43)
+whereM6denotes the space transverse to AdS5, the kappa-symmetry condition may be
+re-writtenas
+cγ7Γ(2)√g2(1+iσ3⊗γ(4))χ+e−i
+2φ0γ10eiχǫ0= (1+iσ3⊗γ(4))χ+e+i
+2φ0γ10ǫ0,(44)
+where we have multiplied across by eiχ/2γ7and used Γ(2)≡Γ<ξ4ξ5>. One may quickly
+recognizethatanecessary conditionsforsupersymmetryar e
+/bracketleftbig
+γ7Γ(2),σ3⊗γ(4)/bracketrightbig
+= 0, (45)
+/braceleftbig
+γ7Γ(2),γ10/bracerightbig
+= 0. (46)
+Thelatterconditionmaybeignoredif φ0=π. Thedirectionstransversetothe AdS5space
+are the product of a two-sphere with a four-dimensional spac eM6=S2×M4. Thus, in
+general,Γ(2)canbealinearcombinationoftwoanti-symmeterisedgammam atrices,either
+alongS2, alongS1⊂S2with a direction in M4, or alongM4. The three possibilities for
+γ7Γ(2)are, respectively
+iσ3⊗γ7, σi⊗γ7γ(4)γµ,12⊗γ7γµν, (47)
+wherei= 1,2andµ,ν= 7,8,9,10. All three choices fail to satisfy (45), so there is no
+supersymmetricprobewiththisembedding.
+AdS3probes
+ForAdS3probes there is no mixing required between the conjugate MN K illing spinors,
+soforsimplicity,wesimplyuse ǫ=ψ⊗ξandignoretheconjugate. Referringto(128)and
+9(129), this allows us to identify η+as Poincar´ e and η−as superconformal Killing spinors,
+respectively.
+Aftersometrial anderror, oneidentifiesonlyonepromising candidateembedding
+ξ0=x0, ξ1=x1, ξ2=r,{ξ3,ξ4} ⊂Σ2, ξ5=χ. (48)
+UsingtheAdSprojection3
+ρ014ψ=−ψ, (49)
+thekappa-symmetryconditionbecomes
+Γ<xyχ>√g3ǫ0=γ789ǫ0=−iǫ0, (50)
+where the probe is required to be at θ=π
+2to preserve supersymmetry. The AdS projector
+ρ014ψ=−ψmeans thatρ01η+=−η+andρ01η−=η−provided the probe is located at
+x2=x3= 0inAdS5. So wecan preserve 4 Poincar´ e and 4 superconformal supersy mme-
+tries. Asinthecaseof AdS5,theSU(2)×U(1)superconformalsymmetryispreservedby
+thisprobe.
+For this supersymmetric probe, the equations of motion can b e shown to be satisfied.
+Here, theinducedmetricis now
+ds2
+ind=˜κ2/3
+2/bracketleftBigg
+−dx2
+0+dx2
+1+dr2
+r2+1
+2(dx2+dy2)
+y2+/parenleftbigg
+dχ+dx
+y/parenrightbigg2/bracketrightBigg
+.(51)
+Thescalar equation(6)reduces totwo non-trivialcomponen ts
+Grr∇rE4
+r=Gyy∇yE8
+y= 0, (52)
+whichmaybeeasily verifiedto hold. Theansatzfor h
+h=a˜κ
+4√
+2/parenleftbiggdx0∧dx1∧dr
+r3+dx∧dy∧dχ
+2y2/parenrightbigg
+, (53)
+ensuresthat boththetensorequation(7)and Bianchi(21)ar e satisfiedwhen aisconstant.
+AdS2probes
+As in the case of AdS4treated earlier, here we also need to mix MN Killingspinorco nju-
+gates. Again,wechoose
+ρ04ψ=cψc, (54)
+withcconstant. We alsoadopttheM5embedding
+ξ0=x0, ξ1=r,{ξ2,...,ξ5} ⊂M6. (55)
+3ThesignchoicehereidentifiestheprobeasanM5. Ananti-M5m aybeconsideredbychangingsign.
+10In similarfashion tostepstaken before, thekappa-symmetr yconditionnowreads
+cγ7Γ(4)√g4(1+iσ3⊗γ(4))χ+e−i
+2φ0γ10eiχǫ0= (1+iσ3⊗γ(4))χ+e+i
+2φ0γ10ǫ0,(56)
+wherenowγ7Γ(4)≡γ7Γ<ξ2...ξ5>isalinearcombinationofthebuildingblocks
+iσ3⊗γ7γµν, σi⊗γ7γ(4)γµνρ,12⊗γ7γ(4). (57)
+These correspond to the probe wrapping S2, wrappingS1⊂S2, and the probe not wrap-
+pingS2, respectively. Before we can even consider talking about pr ojection conditions, a
+necessary condition for supersymmetry is that γ7Γ(4)commutes with σ3⊗γ(4). In much
+the same way as for AdS2, this condition is not satisfied, thus ruling out the possibi lity of
+asimplesupersymmetric AdS2.
+3.2 Other supersymmetricprobes
+In this section, we repeat the steps of the last section for pr obes at a fixed value of r. We
+catalogue the possibilities below. In general, one may cons ider change the embedding
+along a spatial direction of AdS5into the radial direction r, getting new M5-brane config-
+urations. Weanticipatethat thosenon-supersymmetriccon figurationsare also solutionsto
+the equations of motion. For M4, these branes should correspond to domain walls on the
+fieldtheoryside. Itwouldbeinterestingtoeventuallystud yhowthegaugetheorychanges
+whenonecrosses thedomainwall.
+M4probes
+HereagainwehavenomixingbetweentheMNKillingspinorand itsconjugate,soweopt
+tojustworkwith ǫ=ψ⊗ξ. Introducingtheprojector
+ρ0123ψ=±iψ, (58)
+weadoptthefollowingembeddingfortheM5-brane
+ξ0=x0, ξi=xi(i= 1,2,3),{ξ4,ξ5} ⊂M6. (59)
+After some preliminary trial and error using the background projectors (3), one finds that
+therearetwo promisingcandidatesforembeddings:
+{ξ4,ξ5} ⊂Σ2and{ξ4,ξ5} ⊂S2. (60)
+For the first embedding, we take χto be a function of x,yi.e.χ≡χ(x,y). We also
+neglectθas there is no γ10projector acting on the MN Killing spinors. Making use of
+11ρ0123ψ=iψ, thekappasymmetryconditionbecomes
+iΓ<xy>√g2ǫ0=ǫ0, (61)
+whereg2denotestheinducedmetric. Theinduced gammamatrices are
+Γ<x>= ˜κ1/3/bracketleftbiggW1/6
+2yγ7+sinθW−1/3
+√
+2yγ9+∂xχsinθW−1/3
+√
+2γ9/bracketrightbigg
+,
+Γ<y>= ˜κ1/3/bracketleftbiggW1/6
+2yγ8+∂xχsinθW−1/3
+√
+2γ9/bracketrightbigg
+. (62)
+Settingθ= 0,wefindthatthisconfigurationissupersymmetricwith4Poin car´ esupersym-
+metriespreserved. Onemay alsoswitchon h-field oftheform
+h=a
+2(E012+E378). (63)
+In thesecond case, thekappasymmetryconditionbecomes
+σ3χ+=±χ+. (64)
+On top of the projector (58), this breaks supersymmetry furt her. So we are left with 2
+Poincar´ esupersymmetries. Thereisnoconditionon θandprovidedweavoid θ=π
+2where
+theS2shrinks. It is easy to show that the aboveprojector also supp orts a self-dual h-field
+like
+h=a
+2(E012+E378). (65)
+Theequationsofmotiongivemoreconstraints. Oneofthesca larequationsrequire θshould
+bezero, forbothvanishing hand thenon-zero h-field givenabove.
+M3probes
+We find there is no simplesupersymmetricprobe obtained by mi xingψwith its conjugate
+ψc. Thedifficultiespresentedintreatingtherelevantfactor eiχwillalsobefoundinan M1
+probe,which wechoosenotto pursue.
+M2probes
+As in the case of M4, we confine our attention to ǫ=ψ⊗ξ. Introducing the projection
+condition
+ρ01ψ=±ψ, (66)
+weconsideran embeddingoftheform
+ξ0=x0, ξ1=x1{ξ2,ξ3} ⊂S2,{ξ4,ξ5} ⊂Σ2. (67)
+12Thisembeddingpreserves supersymmetryprovidedweallowf or theadditionalprojector
+σ3χ+=±χ+. (68)
+Asiγ78commutes with γ10, supersymmetry does not pick out a specific angle for θpro-
+vided we avoid the S2shrinking. In total 2 Poincar´ e supersymmetriesare preser ved. This
+ansatz satisfies the M5 equations of motion when θ= 0. We can get non-BPS brane with
+anAdS2factor from the brane with an M2factor. This non-BPS brane should be dual to
+someone-dimensionalobjectin thefield theory side.
+4 More non-supersymmetricprobes
+InspiredbysomeM5-branesin AdS7×S4[26,27],inthissectionweconsiderfibrationsof
+AdS5thatmaygiverisetoloopoperators( AdS2×S1⊂AdS5, AdS 2×S2⊂AdS5)and
+surface operators ( AdS3×S1⊂AdS5). We begin by analysing the equations of motion
+fortheAdS3M5-braneprobes whichturnouttobeinstructivein makingco mmentsabout
+theAdS2case.
+WebeginwithanM5whoseworldvolumeis AdS3×S1×Σ2withAdS3×S1inAdS5.
+Wewriteds2
+AdS5as:
+ds2
+AdS5= cosh2ρ(−cosh2ζdτ2+dζ2+sinh2ζdϕ2)+dρ2+sinh2ρdα2.(69)
+Theveilbeinofthebackground geometryis:
+E0=˜κ1/3W1/6
+√
+2coshρcoshζdτ, E1=˜κ1/3W1/6
+√
+2coshρdζ, (70)
+E2=˜κ1/3W1/6
+√
+2coshρsinhζdϕ, E3=˜κ1/3W1/6
+√
+2dρ, (71)
+E4=˜κ1/3W1/6
+√
+2sinhρdα, E5=˜κ1/3W−1/3cosθ
+2dφ1, (72)
+E6=˜κ1/3W−1/3cosθsinφ1
+2dφ2, E7=˜κ1/3W1/6
+2dx
+y, (73)
+E8=˜κ1/3W1/6
+2dy
+y, E9=˜κ1/3W−1/3sinθ√
+2(dχ+dx
+y),(74)
+E10=˜κ1/3W1/6
+2dθ. (75)
+TheansatzoftheM5-braneis:
+ξ0=τ,ξ1=ζ,ξ2=ϕ, (76)
+13ξ3=α,ξ4=x,ξ5=y, (77)
+ρ=ρ0,θ=θ0. (78)
+Theinduced metricis:
+d˜s2= ˜κ2/31
+2W1/3
+0(−cosh2ρ0cosh2ζdτ2+cosh2ρ0dζ2+cosh2ρ0sinh2ζdϕ2)
++ ˜κ2/3(1
+2W1/3
+0sinh2ρ0dα2+W1/3
+0(dx2+dy2)
+4y2+1
+2W2/3
+0sin2θ0dx2
+y2),(79)
+where
+W0= 1+cos2θ0. (80)
+Theansatzfor his:
+h=a
+2κ(W1/2
+0
+2√
+2cosh3ρ0sinhζcoshζδτdζdϕ +W1/2
+0
+4√
+2sinhρ0
+y2dαdxdy).(81)
+From thiswecan get:
+kn
+m=/parenleftBigg
+−a2
+2I30
+0a2
+2I3/parenrightBigg
+, (82)
+Trk2=3
+2a4,Q= 1−a4, (83)
+H= 2aκ−1(W1/2
+0
+2√
+2(1+a2)cosh3ρ0sinhζcoshζdτωdζωdϕ
+(84)
++W1/2
+0
+4√
+2(1−a2)sinhρ0
+y2dαdxdy) (85)
+FromH4= 0,we getthat dH=H4issatisfied if aisconstant.
+Theresultsfor Gmnis:
+Gττ= (1+a2)2gττ,Gζζ= (1+a2)2gζζ,Gϕϕ= (1+a2)2gϕϕ(86)
+Gαα= (1−a2)2gαα,Gxx= (1−a2)2gxx,Gyy= (1−a2)2gyy. (87)
+Thetensorequationsnowbecome:
+Gmn∇mHnpq= 0. (88)
+Theaboveansatzsatisfytheseequationswhen θ0= 0.
+Thescalarequationsbecome:
+Gmn∇mEc
+n= 0, (89)
+14forc/ne}ationslash= 3. Whenθ0= 0, The equations for c/ne}ationslash= 3are satisfied automatically. For the
+equation with c= 3, the RHS is non zero due to H7. Among three terms in H4, only the
+followingcontributes:
+−2√
+2(3+cos2θ)
+˜κ1/3W7/6e789(10), (90)
+whoseHodgedualis:
+−2√
+2(3+cos2θ)
+˜κ1/3W7/6e0123456, (91)
+Thescalar equationwith c= 3is:
+√
+2
+˜κ1/3W1/6
+0(3(1+a2)2sinhρ
+coshρ+(1−a2)2coshρ
+sinhρ) =2√
+2(3+cos2θ0)(1−a4)
+˜κ1/3W7/6
+0(92)
+By usingθ0= 0,weget
+3(1+a2)2sinhρ
+coshρ+(1−a2)2coshρ
+sinhρ= 4(1−a4). (93)
+By themean valueinequality,theabsolutevalueoftheLHS is notless than
+2√
+3|1−a4|, (94)
+sothisequationhas real rootwhen |a| ≤1.
+If we consider M5 with AdS2×S2×Σ2withAdS2×S2insideAdS5, it seems that
+theonlychangeis thescalar equationwith c= 3,weget
+2(1+a2)2sinhρ
+coshρ+2(1−a2)2coshρ
+sinhρ= 4(1−a4). (95)
+Wehavesimilarresultsasabove. Wehavealsoverifiedthatth erearenoM5-branesolutions
+withfactorAdS2×S1⊂AdS5.
+4.1 Supersymmetry
+In this subsection we identify the conditions for supersymm etry to be preserved by the
+aboveprobes. Webeginwith AdS3bysolvingtheKillingspinorequationforan AdS3×S1
+fibration of AdS5. The result of the analysis stipulates that the probe has to b e located at
+ρ=∞for supersymmetry to be preserved. Therefore, we validate o ur claim that the
+above probe located at finite ρis non-supersymmetric. We also sketch the calculation for
+AdS2×S2.
+In each case weemploythesamefibrationwithvielbein
+Ea= coshρ¯Ea
+AdSn, en=dρ, eα= sinhρ¯Eα
+S4−n, (96)
+15wherea= 0,..,n−1andα=n+1,..,4. .
+ForAdS3×S1,weintroducethefollowingdecompositionforthe AdS5gammamatri-
+ces
+ρa=τa⊗σ3, ρ3= 1⊗σ1, ρ4= 1⊗σ2, (97)
+where
+{τa,τb}= 2ηab, (98)
+andσidenote the Pauli matrices. By further writing the AdS5spinor,ψas the product
+ψ≡χ⊗ξ, theKillingspinorequationon AdS5become
+1
+coshρξ+sinhρ
+coshρiσ2ξ=σ3ξ, (99)
+∂ρξ=σ1
+2ξ, (100)
+∂αξ+iσ3
+2coshρξ=σ2
+2sinhρξ. (101)
+Here, we have used the Killing spinor equation on AdS3:∇aχ=τa
+2χ. Writing the two-
+dimensionalcomplexspinoras
+ξ=/parenleftBigg
+ξ1
+ξ2/parenrightBigg
+, (102)
+(99)tellsus thatthetwocomponentsarenotindependent:
+sinh/parenleftBigρ
+2/parenrightBig
+ξ1= cosh/parenleftBigρ
+2/parenrightBig
+ξ2. (103)
+Then solving (100) and (101), we find that the solution is of th e final form for the AdS5
+Killingspinoris
+ψ≡χ⊗e−iα/2/parenleftBigg
+cosh/parenleftbigρ
+2/parenrightbig
+sinh/parenleftbigρ
+2/parenrightbig/parenrightBigg
+, (104)
+whereαdenotes oneoftheanglesofthe S2.
+We can now determine the supersymmetry condition on a flat pro be in theAdS3×S1
+directions. Theprojectioncondition ρ0124ψ=iψimplies
+sinhρ= coshρ. (105)
+So, supersymmetrycan bepreserved at ρ=∞.
+ForAdS2×S2the calculation runs as follows. We introduce the gamma matr ix de-
+composition
+ρa=τa⊗1, ρ2=τ3⊗σ3, ρα=τ3⊗σα, (106)
+andthefollowingdecompositionfor thespinor
+ψ=fABχA⊗ξB, (107)
+16wheretheAdS2spinorχAandtheS2spinorξBsatisfytheKillingspinorequations
+∇aχ±=±1
+2τ3τaχ±,
+∇αξ±=±1
+2σaξ±. (108)
+Thecomponentsoftherespectivespinorsarealso related vi a
+τ3χ+=χ−, σ3ξ+=ξ−. (109)
+With this set-up, placing ψdirectly into the AdS5Killing spinor equation, one arrives at
+thefollowingform oftheKillingspinor
+ψ= cosh(ρ
+2)χ+⊗ξ++sinh(ρ
+2)χ−⊗ξ−, (110)
+withf+−=f−+= 0. One can then readily verify that the probe projector ρ0134ψ=iψ
+can onlybesolvedwhen ρ=∞.
+5 Conclusion
+In this paper, motivated by the recent advances in our unders tanding of N= 2SCFTs,
+we have attempted to identify simple probe M5-branes in the M N background preserving
+supersymmetry. In addition to the known AdS5×S1probe, our analysis identified an
+AdS3×Σ2×S1counterpart embedding that breaks supersymmetry further. As the M5
+also wraps the Riemann surface in this case, its interpretat ion as a two-dimensionalobject
+in the dual field theory is still unclear. In addition, one unu sual aspect of our study is the
+realization that the AdS5×S1M5-brane probe does not support a self-dual h-field. It is
+ruledoutby theequationsofmotion.
+WehavealsoidentifiedotherBPSandnon-BPSprobesthatshou ldcorrespondtosome
+non-local objects in the dual theory. We hope to study these o bjects more in future work
+and provide some better illumination of their properties. A s a future direction, one can
+immediatelyimaginegeneralizingtheprobeswehaveidenti fiedinMNtothemoregeneral
+classofLLMgeometries. Anotheropenavenueconcernsbackr eactingtheprobebranesin
+theliteraturealongthelinesofworkpioneered by Luninfor AdS4×S7andAdS7×S4.
+Acknowledgements
+We are grateful to Nadav Drukker, Bo Feng, Nakwoo Kim, Sungja y Lee, Tianjun Li,
+Takuya Okuda, Soo-Jong Rey, Takao Suyama, Satoshi Yamaguch i and Hyun Seok Yang
+forusefuldiscussionsandcomments. JWandHYwishtothankt heInstituteofTheoretical
+17Physics(ITP),theChineseAcademyofSciencesforhospital ityduringtheprocessofcom-
+piling this paper. E ´OC is also grateful to the Center of Mathematical Sciences, Z hejiang
+University for a warm stay and, finally, JW thanks the School o f Physics, Peking Uni-
+versity for gracious hospitality. The work of BC was partial ly supported by NSFC Grant
+No.10775002,10975005and NKBRPC (No. 2006CB805905).
+A Maldacena-N ´u˜nezbackground
+Theansatzforthed=7 metricappearing in[14]is
+ds2
+7=e2f(r)(−dt2+dx2
+i+dr2)+e2g(r)
+y2(dx2+dy2). (111)
+From the supersymmetry variations in [14], the general solu tion wherex,ydefine a Hy-
+perbolicspacemay bewritten
+e5λ=e2ρ+1
+2+C1e−2ρ
+e2ρ+1
+4,
+e2g=eλ(e2ρ+1
+4),
+e2f=C2e2ρeλ,
+e2f/parenleftbiggdr
+dρ/parenrightbigg2
+=e−4λ. (112)
+Hereρ→ ∞corresponds to the boundary of AdS7,C2is a trivial integration constant
+that may be absorbed by a volume rescaling, and when C1= 0, the solution interpolates
+betweenAdS7andAdS5×Σ. Thesolutionin theIR has thefixed pointvalues
+e5λ= 2, e2g−λ=1
+4, ef+2λ=1
+r, (113)
+makingthed=7 metric
+ds2
+7=eλ/bracketleftbigg1
+2ds2
+AdS5+1
+4(dx2+dy2)
+y2/bracketrightbigg
+. (114)
+Uplifting this solution to d=11 makes use of section 4 from [3 6], and in particular, the
+followingformula
+ds2
+11=˜∆1/3ds2
+7+g−2˜∆−2/3/parenleftBigg
+X−1
+0dµ2
+0+2/summationdisplay
+i=1X−1
+i(dµ2
+i+µ2
+i(dφi+gAi)2)/parenrightBigg
+,
+∗11H4= 2g2/summationdisplay
+α=0/parenleftBig
+X2
+αµ2
+α−˜∆Xα/parenrightBig
+vol7+g˜∆X0vol7+1
+2g2/summationdisplay
+α=0X−1
+α∗7dXα∧d(µ2
+α)
++1
+2g22/summationdisplay
+i=1X−2
+id(µ2
+i)∧(dφi+gAi)∧∗7Fi, (115)
+18where∗7andvol7are the Hodge dual and the volume form with respect to the metr ic
+ds2
+7and∗11is the Hodge dual of the uplifted metric. In addition, we have the following
+relationships:
+X0≡(X1X2)−2,˜∆ =2/summationdisplay
+α=0Xαµ2
+α,2/summationdisplay
+α=0µ2
+α= 1. (116)
+Makingcontactbetweenthetwoactions,i.e. (89)of[14]and (4.6)of[36],meansadopting
+thefollowingidentifications
+g= 2, X0=X2=e2λ, X1=e−3λ,2Ai
+MN=Ai
+µ0= cosθcosψ, µ 1= sinθ, µ 2= cosθsinψ. (117)
+Withtheseidentifications
+˜∆ =e−3λ(1+cos2θ) =e−3λW. (118)
+Wealsoobtainthemetric
+ds2
+11=1
+2W1/3ds2
+AdS5+W−2/3
+4/bracketleftbigg
+W(dx2+dy2)
+y2
++Wdθ2+cos2θ(dψ2+sin2ψdφ2
+2)+2sin2θ/parenleftbigg
+dφ1+dx
+y/parenrightbigg2/bracketrightbigg
+,(119)
+where we have used A1
+MN=y/4dx. One may also determine the fluxes from the formula
+above. Theterm
+2g2/summationdisplay
+α=0/parenleftBig
+X2
+αµ2
+α−˜∆Xα/parenrightBig
+vol7+g˜∆X0vol7 (120)
+givesthefollowingcontributionto H4:
+−1
+4W2[3+cos2θ]sinθcos2θdθ(dφ1+dx
+y)sinψdψdφ 2. (121)
+Thenextterm iszero andthelastterm
+1
+2g22/summationdisplay
+i=1X−2
+id(µ2
+i)∧(dφi+gAi)∧∗7Fi, (122)
+becomes
+1
+4cos3θ
+Wdxdy
+y2sinψdψdφ 2. (123)
+Up to relabeling of coordinates, this is the form of the solut ion appearing in the introduc-
+tion.
+19B MN Killingspinors
+Parallel to the treatment in [13], we introduce a decomposit ion for the D=11 gamma ma-
+tricessatisfying
+{ΓM,ΓN}= 2ηMN, M,N= 0,...,10. (124)
+Thesemay bere-expressed intermsoflower-dimensionalgam mamatrices as
+Γa=ρa⊗σ3⊗γ(4),
+Γi= 14⊗σi⊗γ(4),
+Γµ= 14⊗12⊗γµ, (125)
+wherea= 0,...,4denoteAdS5directions,i= 1,2denote directions along the S2and
+µ= 7,...,10label gamma matrices along the remaining (x,y,χ,θ)directions, respec-
+tively.γ(4)is simply the product γ78910and we adopt the sign choice Γ012345678910 =−1.
+Consequently,thisimpliesthat ρ01234=i.
+Throughoutthispaper, wewillmakeuseoftheexplicitconst ructionofKillingspinors
+onAdSspacetimesappearing in[37]. Writingthe AdS5metricas
+ds2
+AdS5=1
+r2/parenleftbig
+dxµdxµ+dr2/parenrightbig
+, (126)
+thesolutionstotheKillingspinorequation
+Daψ=1
+2ρaψ, (127)
+maybeexpressedas
+ψ+=r−1/2η+, (128)
+ψ−=r1/2η−+r−1/2xαραη−, (129)
+wherea= 0,···,4labels theAdS5coordinates including r, andα= 0,···,3omitsr.
+The constant spinors η+,η−correspond to Poincar´ e and superconformal Killing spinor s
+andare subjecttotheadditionalprojectioncondition[37]
+ρrη±=±η±. (130)
+Notethatreplacing ψwithitsconjugate ψcresultsinasignchangeintheKillingspinor
+equation (127). This knock-on effect of this change is that η+andη−get interchanged in
+thesolution.
+Fromhereon,wewritedownageneralexpressionforaKilling spinorpreservedbythe
+MNbackgroundas
+ǫ=ψ⊗ξ+ψc⊗ξc, (131)
+20wherewejustfocuson thePoincar´ eKillingspinors
+ψ=r−1/2η+, ψc=r−1/2η−. (132)
+The rest of this section concerns the identification of ξandξcfrom the Killing spinor
+equationfortheMN background.
+Webeginbyexaminingtheeleven-dimensionalKillingspino requation
+∇mη+1
+288[Γnpqr
+m−8δn
+mΓpqr]H4npqrη= 0. (133)
+AsinLLM[13],theanalyticallycontinuedsolutionmaybere ducedonAdS5,thenS2,be-
+forethedifferentialconstraintsontheremainingfour-di mensionalspacemaybeextracted.
+WithintheframeworkofLLMwecanincorporatethetwosignch oicesintheAdS5Killing
+spinorequationas:
+DaΨ = (ib)i
+2ρaΨ, (134)
+whereb=−1forΨ =ψandb= 1forΨ =ψc.
+Introducing the gamma matrix decomposition introduced ear lier (125) and following
+thestepsas outlinedinappendixF ofLLM, onearrivesat thef ollowingequations
+/bracketleftBig
+γµ∂µλ+a
+12e−3λ−2Aγ(4)γµνF(2)
+µν+iabm/bracketrightBig
+ǫ= 0, (135)
+/bracketleftBig
+ie−Aγ(4)+γµ∂µA−a
+4e−3λ−2Aγ(4)γµνF(2)
+µν−iabm/bracketrightBig
+ǫ= 0, (136)
+/bracketleftbigg
+∇µ−iabm
+2γµ−a
+4e−3λ−2AF(2)
+µνγνγ(4)/bracketrightbigg
+ǫ= 0, (137)
+witha=±1andF(2)defined in termsofthefour-form flux by
+H4=F(2)∧d2Ω. (138)
+AsinLLMcombining(135)and(136)toremovethe F(2)terms,wefindthecondition
+∂µ(A+3λ)γµǫ+ie−Aγ(4)ǫ+2iabmǫ= 0. (139)
+Takingm=1
+2, thesolutiontothisequationis
+ǫ=e−iabγ10φ0/2ǫ0, (140)
+where
+sinφ0=√
+2cosθ√
+W,cosφ0=−sinθ√
+W, (141)
+andǫ0satisfiestheprojectioncondition
+(iγ10γ(4)+1)ǫ0= 0. (142)
+21Returning then to (135), we may determine the second project ion condition from the
+requirement that it is satisfied by the MN solution. This amou nts to the following being
+satsified
+/bracketleftbigg
+−sinθcosθ
+3√
+2Wγ10−acosθ
+3√
+2Wγ910−a(3+cos2θ)
+6Wγ78+iabm/bracketrightbigg
+ǫ= 0.(143)
+Using (140), m=1
+2,γ9ǫ0=αǫ04whereα=±1, we may expand this expression to get
+twoparts,oneproportionaltotheidentityand theotherpro portionalto γ10:
+γ10/bracketleftbigg
+−sc
+3√
+2W(1−s√
+W)+c√
+2W+αac
+3√
+2W(1−s√
+W)+αb(3+c2)c
+3√
+2W√
+W/bracketrightbigg
+ǫ0= 0
+14/bracketleftbigg
+absc2
+3W√
+W+1
+2ab(1−s√
+W)+αbc2
+3W+αa(3+c2)
+6W(1−s√
+W)/bracketrightbigg
+ǫ0= 0.(144)
+Here wehave employedtheshorthand c≡cosθ,s≡sinθto compress theseexpressions.
+Theseare satisfiedprovided
+a=b=−α,witha2= 1. (145)
+The final part of the Killing spinor may be determined by solvi ng (137) directly to
+determine the functional dependence of ǫ0. From the projectors iγ78ǫ0=αǫ0andγ9ǫ0=
+αǫ0, we can track the dependence on the sign α. After examining (137) one determines
+that
+∂θǫ0=∂xǫ0=∂yǫ0= 0. (146)
+Thedependenceon χmayeasilybedeterminedfromtheKillingspinorequationin thexor
+χdirections. For easeof illustrationwefocus on the xdirection. After asmall calculation
+thisbecomes
+/bracketleftbiggy√
+2/parenleftbigg
+∂x−∂χ
+y/parenrightbigg
++1
+2/parenleftbigg
+−1√
+2γ78−sinθ
+2√
+Wγ98/parenrightbigg
+−iab
+4γ7+a√
+2cosθ
+4√
+Wγ789/bracketrightbigg
+e−i
+2abγ10φ0ǫ0(χ) = 0.(147)
+Inthisexpression,afterusingtheprojectors,therearete rmsproportionalto 14,γ8andγ810.
+The latter two cancel independently of their own accord, but the term proportional to the
+identitybecomes/bracketleftbigg
+∂χ+1
+2γ78/bracketrightbigg
+ǫ0(χ) = 0. (148)
+Usingiγ78ǫ0=αǫ0,ǫ0(χ)may bewrittensimplyas
+ǫ0(χ) =ei
+2αχ˜ǫ0, (149)
+4Implying iγ78ǫ0=αǫ0from(142).
+22where˜ǫ0is aconstant spinor. We can therefore confirm theform ofthe K illingspinors for
+MNappearinginthetext(3). Theextra γ7hasbeenintroducedtoensurethattheprojection
+conditionsarecorrect.
+C Someconnections
+In this appendix, we list the connections appear in section 4 . The nontrivial independent
+componentsoftheLevi-Civitaconnectionfortheinducedme triceq. (79)are:
+Γζ
+ττ=−Γζ
+ϕϕ= coshζsinhζ, (150)
+Γτ
+τζ=sinhζ
+coshζ, (151)
+Γϕ
+ϕζ=coshζ
+sinhζ, (152)
+Γx
+yx= Γy
+yy=−1
+y, (153)
+Γy
+xx=1
+y(1+2sin2θ0
+W0). (154)
+Some of the nonzero independent components of the spin conne ction with respective
+tothevielbeinsin thatsectionare:
+ω1
+00=√
+2sinhζ
+˜κ1/3W1/6coshρcoshζ, (155)
+ω1
+22=−√
+2coshζ
+˜κ1/3W1/6coshρsinhζ, (156)
+ω3
+00=−ω3
+11=−ω3
+22=√
+2sinhρ
+˜κ1/3W1/6coshρ, (157)
+ω3
+44=−√
+2coshρ
+˜κ1/3W1/6sinhρ, (158)
+ω8
+77=2
+˜κ1/3W1/6, (159)
+ω10
+00=−ω10
+ii=−2sinθcosθ
+3˜κ1/3W1/6(1+cos2θ),i= 1,···,4,7,8,(160)
+ω10
+55=ω10
+66=2
+˜κ1/3W1/6/parenleftbiggsinθ
+cosθ−2sinθcosθ
+3(1+cos2θ)/parenrightbigg
+, (161)
+ω10
+99=2
+˜κ1/3W1/6/parenleftbigg
+−cosθ
+sinθ−2sinθcosθ
+3(1+cos2θ)/parenrightbigg
+. (162)
+Theremainingnon-vanishingcomponentsare:
+ω5
+66,ω7
+98,ω7
+89,ω8
+79, (163)
+buttheyarenot needed inthecomputationsin thispaper.
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+arXiv:1001.0908v1  [math.GN]  6 Jan 2010Sequential order under CH
+Chiara Baldovino,
+Department of Mathematics, University of Trieste
+Via Alfonso Valerio 12/1, 34127 Trieste – Italy,
+chiara.baldovino@polito.it
+June 28, 2018
+1 INTRODUCTION
+LetXbe a topological space and M⊆X; thesequential closure ofMis
+seqcl(M) ={x∈X:∃(xn)n∈ω⊆M1,limn∈ωxn=x}.For every ordinal
+α≤ω1, theα-sequential closure ofMis inductively defined as follows:
+-seqcl0(M) =Mandseqcl1(M) =seqcl(M);
+-seqclα+1(M) =seqcl(seqclα(M));
+-seqclα(M) =/uniontext
+β<αseqclβ(M) ifαis a limit ordinal.
+A topological space Xis said to be sequential if
+seqclω1(M) =M,∀M⊆X;
+this definition is equivalent to that according to which a space Xis said to
+besequential if every sequentially closed subset of Xis closed.
+Thesequential order of a sequential space Xis an ordinal invariant of
+the space defined as
+σ(X) = min{α≤ω1:∀M⊆X,seqcl α(M) =M}.
+1By the notation ( xn)n∈ω⊂Mwe mean that ( xn)n∈ωis a sequence and that xn∈M
+for every n∈ω.
+1While the problem naturally posed in the sixties concerning the possibilit y
+to produce examples of sequential spaces of any sequential orde r up to and
+including ω1inZFCwascompletely solved intheaffirmative byArhangel’ski˘ ı
+and Franklin (cf. [1]), it turns out difficult to construct compact seq uential
+spaces without additional assumptions of the Theory of the Sets, even of se-
+quential order 3; indeed, up to now, 2 is the maximum order of seque ntiality
+of a compact space in ZFC. In this context the work due to Baˇ skiro v and
+concisely presented in a Doklady article (see [3]) gathers a certain pr omi-
+nence: in this paper the author suggests a scheme of constructio n to produce
+compact sequential spaces of any order as quotient spaces of βωunder the
+assumption of the Continuum Hypothesis. Since Baˇ skirov’s work is v ery
+concise and devoid of any proof and check, we write down the const ruction
+with some essential alterations with respect to the original work in o rder to
+complete it in all details and explain where the Continuum Hypothesis is
+essentially used.
+In order to make comprehensive our survey concerning the possib ility to ob-
+tain compact sequential spaces of the greatest order, we have t o mention the
+construction under CH due to Kannan (cf. [9]) and the more recent con-
+structions under MA due to Dow (see [4] and [5]). Under the Continuu m
+Hypothesis, Kannan ensures that it is possible to construct compa ct sequen-
+tial spaces of any order while, under MA, Dow manages to give an exa mple
+of a compact sequential space of order 4, the best upper bounds under this
+axiom up to now.
+While Baˇ skirov suggests a construction from top to down, Kannan and Dow
+present a construction from down to top. Indeed Baˇ skirov work s inβωand
+by assuming to have constructed all the spaces of sequential ord er a successor
+ordinal less than a fixed successor ordinal α+1 he gives a starting decompo-
+sition on βω; the Continuum Hypothesis guarantees him that in ω1steps he
+can purify the starting decomposition in such a way that in the space asso-
+ciated to the last decomposition there is a new point fit to produce a s pace
+of sequential order α+1. Instead Kannan and Dow start from the natural
+numbers with the discrete topology. If we want to summarize the ide a of
+Kannan, we can say that he generalizes the construction of the on e-point
+compactification of the Mr´ owka-Isbell space. On the other hand , Dow con-
+structs by transfinite induction on cthree suitable families of subsets of ωin
+such a way that the Stone space associated to the Boolean algebra generated
+by the elements of these subsets admits a point of sequential orde r 4.
+There is a remarkable reason to determine the maximum possible sequ ential
+2order in the presence of the PFA which implies Martin’s axiom and c=ω2;
+indeed in 1989 Balogh solved the Moore-Mr´ owka problem proving tha t each
+compact space of countable tightness is sequential under PFA (se e [2]). If
+there is some finite bound on the sequential order of compact sequ ential
+spaces in models of PFA, it would mean that compact spaces of count able
+tightness are a few steps away from being Fr´ echet-Urysohn. In [5, Propo-
+sition 3.1], Dow points out that there are obstructions to extend his type
+of construction to produce compact sequential spaces of order greater than
+4. However the problem if there exists a bound on the sequential or der of
+compact sequential spaces in models of PFA is still open.
+2 PRELIMINARY FACTS
+We are interested in the construction of compact sequential spac es of sequen-
+tial order 1 and 2 in ZFC as quotient spaces of βω; we choose to present the
+following constructions to enter into the scheme of the main constr uction we
+will present.
+Let us consider the space
+K1=βω/ω∗=βω/≈1
+where
+x≈1y⇔(x=y∨(x∈ω∗∧y∈ω∗))
+and let us denote by j1the natural quotient mapping from βωtoK1. Triv-
+ially the one-point elements of the quotient under the relation of equ ivalence
+≈1are images of the points of ω, while the natural quotient mapping col-
+lapses all the free ultrafilters to a single point P. The topology of the space
+K1=βω/ω∗=ω∪{P}withP /∈ωis a topology τsuch that the points of
+ωare isolated while Phas a fundamental system of (open) neighborhoods
+formed by {UP,F}={{P}∪j1(ω\F) :F∈[ω]<ω}as we prove in the follow-
+ing lemma.
+Lemma 2.1 LetP=j1(ω∗)∈K1; a fundamental system of neighborhoods
+ofPis given by the collection {j1(βω\F) :F∈[ω]<ω}.
+PROOF. If A=βω\FwithF∈[ω]<ω, thenAis a saturated open subset of
+βωandA⊇ω∗whencej1(A) is an open neighborhood of P.
+Suppose now that Vis an arbitrary neighborhood of PinK1: we want to
+3prove that there exists F∈[ω]<ωsuch that j1(βω\F)⊆V, i.e. such that
+βω\F⊆j−1
+1(V). Towards a contradiction suppose that |βω\j−1
+1(V)|=ω;
+then we can set L=βω\j−1
+1(V) and consider a free ultrafilter U ∈ω∗⊆
+j−1
+1(V) such that L∈ U. Sincej−1
+1(V) is open, there exists an infinite set H
+ofωwithH∈ Usuchthat H∗∪H⊆j−1
+1(V)(inparticular H⊆j−1
+1(V)). Now
+H,L∈ Uand then it follows that H∩L/\e}a⊔io\slash=∅(and still better |H∩L|=ω).
+Hence we can fix a point m∈H∩L: on the one hand, it holds that m∈
+H⊆j−1
+1(V), while, on the other hand, it results that m∈L⊆βω\j−1
+1(V).
+A contradiction. /square
+Notice that the fundamental neighborhoods {UP,F}ofPare clopen sub-
+sets; we refer to these neighborhoods as elementary .
+It is easy to see that the space K1has the same topology as a convergent
+sequence and hence it is trivially a Hausdorff compact space.
+Now we want to find a suitable relation of equivalence in βωin such a
+way that the corresponding quotient space is a Hausdorff compact space of
+sequential order 2. Let Mbe an infinite MAD family on ω; to every element
+M∈ Mwe can associate the unique element M∗⊆ω∗in the following way:
+M/maps⊔o→M∗={U ∈ω∗:M∈ U}.
+It turns out that if M1,M2∈ MwithM1/\e}a⊔io\slash=M2thenM∗
+1∩M∗
+2=∅: indeed
+if the intersection was not empty, then there would exist a free ultr afilter
+V ∈ω∗such that M1∈ VandM2∈ V; thusM1∩M2∈ Vbut|M1∩M2|< ω
+and the ultrafilter would be fixed against the hypothesis.
+We also remark that the subset ω∗\/uniontext
+M∈MM∗is not empty: if it was empty,
+then{M∗∪ω:M∈ M}would be an infinite and open cover of βωfrom
+which it would be impossible to extract a finite subcover; this fact clas hes
+with the compactness of βω.
+Let us take into account the space K2=βω/≈2where two free ultra-
+filter ofβωare equivalent under the relation ≈2if they belong to the same
+M∗withM∈ M; moreover all the free ultrafilters belonging to the subset
+ω∗\/uniontext
+M∈MM∗are equivalent, while the relation does not identify any point
+inω. Then, if we denote by j2the natural quotient mapping from βωtoK2,
+it holds that j2leaves the points of ωunaltered, while it collapses every M∗
+withM∈ Mto a single point and the non-empty subset ω∗\/uniontext
+M∈MM∗to
+another single point too.
+Letusset L0=j2(ω),L1={j2(M∗) :M∈ M}andx∞=j2(ω∗\/uniontext
+M∈MM∗).
+4We say that the points in the set L0have level 0 while the points of L1have
+level 1 and the point x∞has level 2. We will prove that the levels of the
+points coincide with their sequential order with respect to the set L0.
+We already know that the points in L0are isolated in K2; now we prove the
+following lemma about the neighborhoods of the points of level 1.
+Lemma 2.2 Lety∈L1withy=j2(M∗)andM∈ M; then the collec-
+tion{Uy,F}={{y} ∪j2(M\F) :F∈[M]<ω}is a fundamental system of
+neighborhoods for y.
+PROOF. On the one hand, since M∗∪Mis a saturated open subset of
+βωand since every fixed ultrafilter in βωis isolated, it is evident that
+M∗∪(M\F) is a saturated open subset of βωtoo for every F∈[M]<ω.
+Therefore j2(M∗∪(M\F)) ={y}∪j2(M\F) is an open neighborhood of y
+for every F∈[M]<ω.
+On the other hand, let us suppose that Vis an arbitrary neighborhood of y;
+we want to prove that there exists F∈[M]<ωsuch that j2(M\F)⊆V,
+i.e. such that M\F⊆j−1
+2(V). Let us suppose by contradiction that
+|M\j−1
+2(V)|=ω; then let us set L=M\j−1
+2(V) and consider a free ul-
+trafilterUwithL∈ U: it holds that M∈ U(sinceL⊆M) and then
+thatU ∈M∗⊆j−1
+2(V). Asj−1
+2(V) is open, there exists H∈ Usuch
+thatH∗∪H⊆j−1
+2(V) (in particular H⊆j−1
+2(V)). In this way it turns
+out that H,L∈ U; thenH∩L∈ UwhenceH∩L/\e}a⊔io\slash=∅(even better
+|H∩L|=ωsinceUis free). Let us fix a point m∈H∩L: on the one
+hand, it holds that m∈H⊆j−1
+2(V), while, on the other hand, it results
+thatm∈L⊆M\j−1
+2(V). A contradiction. /square
+We want to remark that the fundamental neighborhoods Uy,Fof a general
+pointyof level 1 are clopen in K2: indeed M∗∪(M\F) is a saturated closed
+subset of βωfor eachF∈[M]<ω; let us call elementary these neighborhoods.
+We state in advance the following remark to the lemma about the neigh bor-
+hoods of the point x∞.
+Remark 2.3 For every D∈[ω]ωthe subfamily MD={M∈ M:|M∩D|=
+ω}is such that/uniontextMD⊇∗D; suppose by contradiction that/uniontextMD/notsuperseteql∗D, i.e.
+suppose that there exists a subset E⊂Dsuch that |E|=ωand(/uniontextMD)∩
+E=∗∅. Since the family Mis maximal, there exists a subset A∈ M\M D
+such that |A∩E|=ωbutA∩E⊂Dand hence |A∩D|=ω; therefore Ais
+an element of MDbut this contradicts what we have just supposed.
+5Let us fix an arbitrary D∈[ω]ω; if for every finite union/uniontext
+M∈FMwith
+F ∈[M]<ωit turns out that/uniontext
+M∈FM/notsuperseteql∗D, then|MD| ≥ω. Indeed if
+|MD|< ω, by taking F=MD, it holds that/uniontextF ⊇∗Dbecause of what we
+have just remarked above.
+Lemma 2.4 The collection of the clopen subsets K2\/uniontext
+x∈GUx(whereGis
+a finite set and for every x∈Gthe clopen subset Uxis an elementary neigh-
+borhood of the point xinK2that can have level 0or1) is a base at the point
+x∞.
+PROOF. In an obvious way, K2\/uniontext
+x∈GUx(whereGis a finite set and for
+everyx∈Gthe clopen subset Uxis an elementary neighborhood of the point
+xof level 0 or 1) is open and closed in K2since its complementary subset is
+a finite union of clopen subsets.
+Now let Abe an open subset containing x∞and letC=K2\Abe the
+complementary closed subset. For every x∈C, letUxbe an elementary
+clopen neighborhood of x; trivially, by taking all the clopen subsets Uxwith
+x∈C, we cover C. Let us consider j−1
+2(C): it is a closed subset in βωand
+then it is compact. The subsets j−1
+2(Ux) (withx∈C) form an open cover of
+j−1
+2(C); then there exists a finite subcover/uniontext
+x∈Gj−1
+2(Ux)⊇j−1
+2(C). It turns
+out that
+j2(/uniondisplay
+x∈Gj−1
+2(Ux)) =/uniondisplay
+x∈Gj2(j−1
+2(Ux)) =/uniondisplay
+x∈GUx⊇j2(j−1
+2(C)) =C;
+by returning to the complementary subsets, we are able to conclud e that
+K2\/uniontext
+x∈GUx⊆K2\C=A. /square
+Let us call elementary these clopen neighborhoods of the point x∞. Now
+we are finally able to prove the following lemma.
+Lemma 2.5 K2is a compact sequential Hausdorff space of sequential order
+2.
+PROOF. It is trivial to part any point j2(n)∈L0from any other point
+Q∈K2: indeed we can take respectively the open disjoint neighborhoods
+j2({n}) andj2(βω\{n}). We have to analyse the other following two cases
+in order to conclude that K2is a Hausdorff space
+61. If we have to separate two points P1,P2∈K2of level 1, i.e. such that
+j−1
+2(P1) =M∗
+1,j−1
+2(P2) =M∗
+2withM1,M2∈ M, then we notice that
+M∗
+1∪M1andM∗
+2∪M2are open subsets of βωand that F=M1∩M2
+is finite and hence closed in βω; thus it turns out that M∗
+1∪(M1\F)
+andM∗
+2∪(M2\F) are disjoint saturated open subset of βωand then
+their images j2(M∗
+1∪(M1\F)) andj2(M∗
+2∪(M2\F)) are disjoint open
+neighborhoods of P1andP2respectively.
+2. If we have to part the point x∞=j2(ω∗\/uniontext
+M∈MM∗) from any point
+Pof level 1, i.e. such that j−1
+2(P) =M∗
+1withM1∈ M, then we
+notice that M∗
+1∪M1is a saturated clopen subset of βωand that also
+its complement is saturated and clopen; therefore j2(M∗
+1∪M1) and
+j2(βω\(M∗
+1∪M1)) are disjoint open neighborhoods of Pandx∞re-
+spectively (the latter one is a neighborhood of x∞since
+j−1
+2(x∞) =ω∗\/uniondisplay
+M∈MM∗⊆βω\(M∗
+1∪M1).
+We can also conclude that K2is compact, since j2is a continuous function
+from the compact space βωto the Hausdorff space K2.
+Finally we can prove that the space K2has sequential order 2; more precisely
+we show that its sequential order, σ(K2), is less than or equal to 2 (and, in
+particular, that K2is a sequential space) and then that it is exactly 2.
+LetS⊆K2be an arbitrary subset such that S/\e}a⊔io\slash=S: we want to show
+thatS=seqcl2(S); clearly it is enough to prove that S\S⊆seqcl2(S).
+Let us consider a point x∈S\Sand let us set P=j−1
+2(x). Trivially
+P /∈ {{n}:n∈ω}: indeed the images of points of ωinK2can not belong
+toS\Ssince they are isolated. Therefore there are two cases to study:
+1st)P=M∗withM∈ M;
+2nd)P=ω∗\(/uniontext
+M∈MM∗).
+Let us take them into account:
+1st) First ofall if x∈S\S, by Lemma 2.2we canassert that x∈S∩j2(M).
+This fact allows us to conclude that S∩j2(M) is infinite, since K2
+satisfies the T1separation axiom, and then we can write S∩j2(M) as
+{j2(mn) :n∈ω}wheren/maps⊔o→mnis injective and mn∈Mfor every
+n∈ω. Weprovethat, forevery neighborhood VofxinK2,j2(M)\Vis
+7finite - this will imply that ( j2(mn))n∈ω→xinK2and hence that x∈
+seqcl1(S)⊆seqcl2(S). Towards a contradiction, suppose that j2(M)\V
+is infinite; then there exists M′∈[M]ωsuch that j2(M′) =j2(M)\V.
+Let us consider a free ultrafilter UwithM′∈ U. SinceVis an open
+neighborhood of xinK2, it turns out that j−1
+2(V) is a saturated open
+subset of βωsuch that j−1
+2(x) =M∗⊆j−1
+2(V). Then it holds that
+U ∈(M′)∗⊆M∗⊆j−1
+2(V) and hence there exists an infinite subset
+T∈ Usuch that T∗∪T⊆j−1
+2(V) and, in particular, it turns out that
+j2(T)⊆j2(T∪T∗)⊆V; sinceT∩M′/\e}a⊔io\slash=∅(they are both elements
+ofU), we can fix h∈T∩M′and on the one hand we obtain that
+j2(h)∈j2(T)⊆V, while on the other hand we have j2(h)∈j2(M′) =
+j2(M)\Vand we reach a contradiction.
+2nd) Sincex∈Sby Lemma 2.4 it holds that that either in Sthere are at
+least a countable infinity of points yn∈L1or, if|S∩L1|<∞, inS
+there are infinite points of level 0 (let us call Dthe set consisting of
+thesepoints) such thatitisnotpossibletocover Dwithafinitenumber
+of elementary neighborhoods of points of level 1. In the former ca se by
+Lemma 2.4 any sequence extracted from S∩L1converges to x∞and
+thenx∈seqcl1(S). In the latter case by Remark 2.4, since it is not
+possible to cover Dwith a finite number of elementary neighborhoods
+of points of level 1, i.e. since for every finite union of elements M∈
+Mit turns out that/uniontextM/notsuperseteql∗D, it follows that |MD|=∞: then
+there are infinite points of level 1, {yα}α∈A⊆L1, such that {yα}α∈A⊆
+seqcl1(j2(D))⊆seqcl1(S) by the former case; moreover a sequence
+extracted from this set has to converge to x∞as we have remarked
+above and hence it turns out that x∞∈seqcl2(S).
+Now we have to prove that the sequential order of K2is exactly 2: we will
+show that there is a subset D⊆K2with the property that x∞∈Dand
+that no sequence extracted from Dconverges to x∞. Let us consider the
+subsetj2(ω); it is clear that x∞∈j2(ω). Now we prove that no sequence
+extracted from j2(ω) converges to x∞; towards a contradiction, suppose that
+there exists a sequence ( j2(mn))n∈ω→x∞. In an obvious way, the set
+H={mn:n∈ω}is infinite since K2satisfies the T1separation axiom;
+hence there exists an infinite set ˜M∈ Msuch that |H∩˜M|=ω. If we
+call ˜ythe unique point in L1such that j−1
+2(˜y) =˜M∗, by Lemma 2.2 it turns
+out that ˜ y∈j2(H). Therefore we are in the first situation we have studied
+above and so we can assert that there is a subsequence ( mni)i∈ωof (mn)n∈ω
+8such that ( j2(mni))i∈ω→˜y; on the other hand by hypothesis we know that
+(j2(mn))n∈ω→x∞and hence that ( j2(mni))i∈ω→x∞. This is a contradic-
+tion since K2is a Hausdorff space and x∞/\e}a⊔io\slash= ˜y. /square
+We want to remark that the space K2we have just constructed is trivially
+homeomorphic to the one-point compactification of the Mr´ owka-I sbell space
+Ψ(M).
+By referring to the type of construction of the space K2, we could think that
+a good idea to construct a space with a larger order of sequentiality could be
+to associate a new infinite MAD family HMto every M∈ M; we will prove
+that in this way we do not construct a space of higher sequential or der.
+Then let Mbe an infinite MAD family on ωand let us suppose to associate a
+new infinite MAD family HMto everyM∈ M; let us consider the partition
+P={{n}:n∈ω} ∪ {H∗:H∈/uniontext
+M∈MHM} ∪ {M∗\/uniontext
+H∈HMH∗:M∈
+M} ∪ {ω∗\/uniontext
+M∈MM∗}. Let us set KIII=βω/≈where≈is the relation
+of equivalence associated to Pand letjIIIbe the natural quotient mapping
+fromβωtoKIII. The space KIIIconsists of the following elements.
+1st) The isolated points of the form jIII(n) for every n∈ω.
+2nd) The points of the form xH=jIII(H∗) for every H∈ HMand every
+M∈ M; a fundamental system of neighborhoods for xHis given by
+{{xH}∪jIII(H\F)}F∈[H]<ω.
+We refer to these neighborhoods with the symbols UxH,F.
+3rd) The points of the form yM=jIII(M∗\/uniontext
+H∈HMH∗) for every M∈ M;
+a fundamental system of neighborhoods for yMis given by
+{({yM}∪jIII(M))\/uniondisplay
+xH∈GUxH,F}G
+whereGisafiniteset; werefertotheseneighborhoodswiththesymbols
+WyM.
+4th) The point p∞=jIII(ω∗\/uniontext
+M∈MM∗) which has a fundamental system
+of neighborhoods given by
+{jIII(βω)\/uniondisplay
+yM∈KWyM}K
+whereKis a finite set.
+9We could think that the points of the sets
+{jIII(H∗) :H∈/uniondisplay
+M∈MHM},{jIII(M∗\/uniondisplay
+H∈HMH∗) :M∈ M},
+{jIII(ω∗\/uniondisplay
+M∈MM∗)}
+have sequential orders respectively 1, 2 and 3 with respect to the setjIII(ω).
+We want to show that the point jIII(ω∗\/uniontext
+M∈MM∗) does not have sequential
+order 3 with respect to the set jIII(ω). In an obvious way we point out that
+x∞∈jIII(ω): indeed there are always infinitely many points of ωout of
+the union of any finite number of neighborhoods of points of the thir d type;
+otherwise there would not be space enough for the other elements of the
+MAD family M. Now let us fix a countably infinite set {Mn:n∈ω} ⊆ M
+and, for every n∈ω, let us fix an infinite subset Hn∈ HMn; moreover, for
+everyn∈ω, let us call znthe unique point in KIIIsuch that j−1
+III(zn) =
+H∗
+n. We assert that for every n∈ωit is possible to extract a subsequence
+{mni}i∈ωfromjIII(ω) with{mni}i∈ω→zn: indeed for every n∈ωit is
+enough to put into the subsequence the image under jIIIof a countably
+infinite number of points belonging to Hn. Therefore for every n∈ωit
+turns out that zn∈seqcl1(jIII(ω)). We claim that ( zn)n∈ω→p∞: consider
+an open subset Ω ⊆βωsuch that ω∗\/uniontext
+M∈MM∗⊆Ω; we want to prove
+that the set N={n∈ω:H∗
+n/notsubseteqlΩ}is finite. Towards a contradiction,
+suppose that Nis infinite; then, in particular, it turns out that the set
+M′={M∈ M:∃H∈ HM,H∗/notsubseteqlΩ}is infinite and hence that the set
+M′′={M∈ M:M∗/notsubseteqlΩ}is infinite. Now consider the infinite open cover
+A={Ω}∪{M∗∪ω:M∈ M′′}ofβω; from this open cover it is not possible
+to extract a finite subcover since the set M′′is infinite. A contradiction.
+We can conclude that p∞∈seqcl2(jIII(ω)) and hence that it does not have
+sequential order 3 with respect to the set jIII(ω); it follows that KIIIdoes not
+have sequential order 3.
+3 BAˇSKIROV’S IDEA
+In this section we want to explain the general scheme of the constr uction
+suggested by Baˇ skirov’s in [3]; the idea is to work in a completely differe nt
+way to produce compact sequential spaces of order a successor ordinal and
+10compact sequential spaces of order a limit ordinal. The constructio n of the
+compact spaces of order a successor ordinal will be carried out by transfinite
+induction on the order of sequentiality. When we will have construct ed com-
+pact spaces Kα+1of sequential order α+1< ω1for every successor ordinal
+less than ω1, then it will be easy to get compact spaces of sequential order
+a limit ordinal βfor every β≤ω1. Indeed, if we want to construct a com-
+pact sequential space of order a limit ordinal β= supα+1<β{α+1}, we can
+consider the disjoint sum of the spaces Kα+1i.e.
+Zβ=/circleplusdisplay
+α+1<βKα+1.
+Trivially the sequential order of this space is β. It is easy to prove that also
+its one-point compactification Kβ=Z∗
+βhas also sequential order β: indeed
+the point ∞we have added to make compact the space has sequential order
+1 with respect to each subset A⊆Kβsuch that ∞ ∈A.
+Therefore the problem reduces to construct compact spaces wh ose sequential
+order is a successor ordinal number. We will construct a compact s paceKα+1
+of sequential order α+1 for every successor ordinal number α+1< ω1; each
+Kα+1will be a quotient space of βω, i.e.Kα+1=βω/≈α+1where the
+relation≈α+1is such that only natural numbers are one-point elements of
+thequotient. Forevery α+1< ω1wewill denoteby jα+1thenaturalquotient
+mapping jα+1:βω→Kα+1. We will prove by transfinite induction that for
+eachα+1< ω1the space Kα+1satisfies the following conditions.
+S.1 The space Kα+1can be uniquely represented in the form of
+Kα+1=L0/unionsqdisplay/parenleftbig/unionsqdisplay
+γ≤αLγ+1/parenrightbig
+.
+The points of level γ+ 1 with γ∈[0,α], i.e. the points belonging to
+the setLγ+1, have sequential order equal to γ+1 with respect to L0,
+the subset consisting of the images of the points of ωunderjα+1.
+S.2 The set Lα+1consists of only one point.
+S.3 Every point in Kα+1of nonzero level has a basis formed by clopen
+subsetscalledelementary; moreover if Uisanelementary neighborhood
+ofapointoflevel γ+1,thentherelation ≈α+1restrictedto /tildewideU=j−1
+α+1(U)
+produces a compact space homeomorphic to Kγ+1.
+11S.4 For every γ≤α, if a nonconstant sequence ( xn)n∈ωof points xn∈
+Lγn+1, with nondecreasing levels, converges to a point x∈Lγ+1, then
+for the sequence ( γn+1)n∈ωof ordinal numbers it holds that sup {γn+
+1}=γ.
+S.5 For every γ≤α, from every injective sequence ( xn)n∈ωof points xn∈
+Lγn+1with nondecreasing levels such that supn∈ω{γn+ 1}=γ, it is
+possible to extract a subsequence converging to a point of level γ+1.
+S.6 If{Ni}i∈ωis a countable family of pairwise disjoint infinite subsets Ni
+ofωand if it holds that for every i∈ωa relation of type βi+ 1 is
+given on Niin such a way that the sequence of ordinals ( βi+1)i∈ωis
+not decreasing and sup {βi+1}=α, then it is possible to extend the
+relation obtained on/uniontext∞
+i=1Nito a relation of βωof typeα+1.
+From the first three conditions we trivially deduce other two proper ties.
+S.7 IfUis an elementary neighborhood of a point xof levelγ+1 inKα+1,
+then its level in U=/tildewideU/(≈α+1|/tildewideU) is equal to γ+1.
+S.8 IfUis an elementary neighborhood of a point xof levelγ+1 inKα+1,
+thenU\{x} ⊆/uniontext
+γ′<γLγ′+1.
+The compact sequential spaces K1andK2will be taken as bases of the
+recursion. Let us begin to check that properties S.1 to S.6 hold for t hese
+spaces.
+Check of the properties of K1
+S.1 The space K1=ω∪ {P}can be uniquely represented in the form of
+K1=L0/unionsqtextL1; we denote by L0the one-point elements of the quotient
+that are images of the points of ωunderj1, whileL1consists of only
+one point that is the image of ω∗under the natural quotient mapping.
+The unique point of L1has sequential order 1 with respect to L0.
+S.2 The set L1consists of the unique limit point of the sequence.
+S.3 The unique point of L1has a basis formed by the clopen elementary
+neighborhoods UP,F: the space obtained by restricting the relation ≈1
+to/tildewideUP,F=j−1
+1(UP,F) is homeomorphic to K1.
+12S.4 ForK1we do not have anything to prove.
+S.5 Obvious.
+S.6 It is necessary to prove S.6 starting from the space K2.
+Check of the properties of K2
+S.1 The space K2can be uniquely represented in the form of
+K2=L0/unionsqdisplay
+L1/unionsqdisplay
+L2
+where we denote by L0the one-point elements of the quotient that are
+images of the points of ωunderj2; the quotient mapping j2collapses
+everyM∗withM∈ Mto a single point (and L1consists of these
+points), while it collapses ω∗\/uniontext
+M∈MM∗to another single point that
+givesL2.
+The points of L1have sequential order 1 with respect to L0, while the
+unique point of L2has sequential order 2 with respect to L0.
+S.2 The set L2consists of the unique point
+x∞=j2(ω∗\/uniondisplay
+M∈MM∗).
+S.3 Every point y∈L1has a basis formed by the clopen elementary neigh-
+borhoods Uy,Fand the space obtained by restricting the relation ≈2to
+/tildewideUy,F=j−1
+2(Uy,F) is homeomorphic to the compact sequential space K1.
+The point x∞∈L2has a basis given by the clopen elementary neigh-
+borhoods K2\/uniontext
+x∈GUx; the space obtained by restricting the relation
+≈2toj−1
+2(K2\/uniontext
+x∈GUx) is a compact space homeomorphic to K2.
+S.4 ForK2Property S.4 is obvious.
+S.5 From every noncostant sequence ( xn)n∈ωof points with nondecreasing
+levels such that sup {l(xn)}= 0 it is possible to extract a subsequence
+converging toapointoflevel 1: indeed, sincethefamily MisMAD,the
+sequence ( xn)n∈ωhas infinite intersection with at least one element M1
+of the family Mand hence we can extract a subsequence converging to
+13the point j2(M∗
+1) of level 1. From every noncostant sequence ( xn)n∈ωof
+points with nondecreasing levels such that sup {l(xn)}= 1 it is possible
+to extract a subsequence converging to x∞: indeed the points of the
+sequence are eventually in every neighborhood of x∞.
+S.6 If{Ni}i∈ωis a countable family of pairwise disjoint infinite subsets
+Ni⊂ωand if it holds that for every i∈ωa relation of type βi+ 1
+is given on Niin such a way that the sequence of ordinals βi+ 1 is
+nondecreasing and that sup {βi+1}= 1 (and hence in such a way
+thatβi+ 1 = 1 for every i∈ω), then we can extend the so obtained
+relation on/uniontext∞
+i=1Nito a relation on βωof type 2: indeed it is enough
+to complete the almost disjoint family {Ni}i∈ωto a MAD family and
+then put into relation the elements of βωin the way we have already
+seen when we constructed K2.
+Now we are sure we can take the compact sequential spaces K1andK2as
+bases of the induction. Moreover we will assume that for all β+1< α+1 the
+compact sequential spaces Kβ+1(in which properties S.1 to S.6 hold) have
+been constructed; then we will able to construct the compact spa ceKα+1
+with sequential order α+1 satisfying conditions S.1 to S.6.
+4 SOME PROPAEDEUTIC LEMMAS
+Before showing the construction in all details, let us prove some use ful very
+technical lemmas.
+Lemma 4.1 The intersection of any countable family of open subsets of ω∗
+is either empty or contains a non-empty open subset.
+PROOF. Let {Ai}i∈ωbe a countable family of open subsets of ω∗whose
+intersection contains a point U. For every i∈ωthere exists a subset Ni⊂ω
+such that
+U ∈N∗
+i⊂Ai.
+The intersection of any finite collection of the sets N∗
+iis not empty and
+open and hence the intersection of any finite collection of the sets Niis
+infinite. Then there exists an increasing sequence of integers nisuch that
+ni∈N1∩N2∩...∩Ni. Let us set N={ni:i∈ω}; sinceN\Niis finite for
+eachi∈ω, it holds that N∗⊂N∗
+ifor eachi∈ωand then we can conclude
+14thatN∗⊂/intersectiontext
+iAiwhereN∗/\e}a⊔io\slash=∅as|N|=ω. /square
+Lemma 4.2 Let{N∗
+i}i∈ωbe a countably infinite family of clopen subsets of
+ω∗; let us suppose that ω∗\/uniontext
+i≤¯ıN∗
+i/\e}a⊔io\slash=∅for every ¯ı∈ω. Then there exists
+∆⊂ωsuch that |∆|=ωand∆∗∩/uniontext
+i∈ωN∗
+i=∅.
+PROOF. Let us set ∆ 1=N∗
+1,∆2=N∗
+1∪N∗
+2,...,∆¯ı=N∗
+1∪N∗
+2∪...∪N∗
+¯ı
+and so on; it is clear that for every ¯ ı∈ω, ∆¯ıis a clopen subset of ω∗. Notice
+thatω∗\∆1⊇ω∗\∆2⊇...⊇ω∗\∆¯ı⊇ω∗\∆¯ı+1⊇...; let us set Ci=ω∗\∆i
+for every i∈ωandE={Ci:i∈ω}. The family Esatisfies the finite
+intersection property i.e. the intersection of a finite number of elem ents ofE
+isnotempty: indeedthesetoftheindicesofthisfinitenumberofelem entshas
+a maximum nand their intersection is equal to ω∗\/uniontext
+j≤nN∗
+j⊇ω∗\/uniontext
+j≤nN∗
+j
+which is a non-empty subset by hypothesis. Since ω∗is compact, it follows
+that/intersectiontextE=/intersectiontext
+i∈ωCi/\e}a⊔io\slash=∅. Therefore it turns out that
+∅ /\e}a⊔io\slash=/intersectiondisplay
+i∈ωCi=/intersectiondisplay
+i∈ω∆C
+i= [/uniondisplay
+i∈ω(∪j≤iN∗
+j)]C= (/uniondisplay
+i∈ωN∗
+i)C=ω∗\/uniondisplay
+i∈ωN∗
+i.
+Wecanconclude thatthefamily {∆C
+i}i∈ωconsisting ofopensubsets of ω∗has
+non-empty intersection and hence, by Lemma 4.1, this intersection contains
+an open subset A⊆ω∗\/uniontext
+i∈ωN∗
+i. Then there exists ∆ ⊂ωwith|∆|=ω
+such that ∆∗⊆ω∗\/uniontext
+i∈ωN∗
+iand hence such that ∆∗∩/uniontext
+i∈ωN∗
+i=∅./square
+Lemma 4.3 LetP=Q∪Rbe a family of infinite subsets of ωsuch that
+-Qis an almost disjoint family;
+-|Q| ≤ωand|R| ≤ω;
+- for every element Qi∈ Qand every element Rn∈ Rit turns out that
+|Qi∩Rn|< ω.
+Then there exists L∈[ω]ωsuch that L∗⊇/uniontext
+Qi∈QQ∗
+iandL∗∩R∗
+n=∅for
+everyRn∈ R.
+15PROOF. Let us set Q={Qi:i∈ω}with|Qi∩Qj|< ωfori/\e}a⊔io\slash=j.
+Obviously we can suppose R /\e}a⊔io\slash=∅and then we can write R={Rn:n∈ω}
+withn/maps⊔o→Rnnot necessarily injective. If |Q|< ω, then we set L=/uniontextQi.
+If instead |Q|=ω, we setL=/uniontext
+n∈ω(Qn\ ∪n′<nRn′). For every n∈ω,Rn
+intersects Lonly in those points in which Rn, in case, intersects the subsets
+Qnwithn= 0,...,n; these points are in a finite number. Furthermore for
+everyn∈ω,Qn\Lconsists of a finite number of points and exactly of those
+in which Qnintersects Rnwithn <n(and these again are certainly in a
+finite number); therefore Q∗
+n⊆L∗for every n∈ωand hence/uniontext
+nQ∗
+n⊆L∗.
+/square
+In the following lemma we will take into account a countable family of
+infinite pairwise disjoint subsets of ω,{˜Ni}i∈ωand a relation ≈onU=/unionsqtext˜N∗
+i⊂ω∗. Let us set H=U/≈and letjbe the quotient mapping
+j:U→H. We assume that the subsets ˜N∗
+iaredistinguished relative to ≈,
+i.e. that j−1(j(˜N∗
+i)) =˜N∗
+ifor every i∈ω. Now let us prove the lemma.
+Lemma 4.4 Let{˜Ni}i∈ωbe a countable family of infinite pairwise disjoint
+subsets˜Ni⊂ωand let≈be a relation on U=/unionsqtext˜N∗
+i⊂ω∗where the subsets
+˜N∗
+iare distinguished relative to ≈and the spaces ˜N∗
+i/≈are zero-dimensional
+compact spaces. Let us suppose that the set B={xn:n∈ω}is devoid of
+any accumulation point in Hand that for every xnthere exists an index in
+and a clopen neighborhood U(xn)such that U(xn)⊆˜N∗
+in/≈. Moreover let
+us suppose that/uniontext
+n∈ωU(xn)/\e}a⊔io\slash=H. Then
+i) there exist pairwise disjoint clopen subsets Unwithn∈ωsuch that
+xn∈Unfor every n∈ω;
+ii) there exists ˜N′⊂ωsuch that
+(˜N′)∗∩/uniondisplay
+i∈ω˜N∗
+i=j−1(/unionsqdisplay
+n∈ωUn) =/unionsqdisplay
+n∈ωE∗
+n.
+PROOF. Since the subsets {˜N∗
+i}iare disjoint and distinguished relative to
+≈it holds that
+(˜N∗
+i/≈)∩(˜N∗
+j/≈) =∅
+for every i,j∈ωwithi/\e}a⊔io\slash=j. Moreover ˜N∗
+i/≈is open and closed in H
+for every i∈ω, sincej−1(˜N∗
+i/≈) =˜N∗
+iis open and closed in U. We need
+16to remark that, for every i∈ω,˜N∗
+i/≈intersects only a finite number of
+the neighborhoods {U(xn)}nbecause of the hypothesis that the subset B is
+devoid of any accumulation point in H.
+By transfinite induction we are going to construct the subsets Unwithn∈ω
+such that xn∈Unfor every n∈ω.
+Let us consider the point x1and the clopen subset U(x1)⊆˜N∗
+i1/≈; since
+Bis devoid of any accumulation point in H, there exists an open neigh-
+borhood A1⊆Hofx1such that A1∩B={x1}. Nowx1∈[(A1∩˜N∗
+i1/≈
+)∩U(x1)] =D1: thissubset isopenin ˜N∗
+i1/≈andhence, since ˜N∗
+i1/≈iszero-
+dimensional, there exists a clopen subset U1⊆D1of˜N∗
+i1/≈withx1∈U1;
+trivially U1is clopen also in H. NowH\U1is an open subset of Hand it
+contains B\{x1}which is devoid of any accumulation point in H\U1; thus
+there exists an open neighborhood A2⊆Hofx2such that A2∩B={x2}.
+Hencex2∈[(H\U1)∩A2∩(˜N∗
+i2/≈)∩U(x2)] =D2: this is an open subset
+of˜N∗
+i2/≈and then, since ˜N∗
+i2/≈is zero-dimensional, there exists a clopen
+subsetU2⊆D2of˜N∗
+i2/≈withx2∈U2; trivially U2is clopen also in Hand
+moreover it results that U1∩U2=∅. Notice that H\(U1⊔U2) is a non-empty
+clopen subset of Hand that B\{x1,x2} ⊆H\(U1⊔U2).
+Let us suppose that for every n≤nthere exists a clopen subset Un⊆H
+withxn∈UnandUn⊆U(xn) and that Un∩Un′=∅for every n′,n≤n;
+moreover suppose that for every n≤nit holds that Bn=B\{x1,...,x n} ⊆
+H\/unionsqtext
+j≤nUj. Let us prove that these properties hold also for n+ 1. By
+inductive hypothesis xn+1∈(H\/unionsqtext
+j≤nUj) whereH\/unionsqtext
+j≤nUjis open, since
+the finite union/unionsqtext
+j≤nUjis clopen; moreover there exists an open neighbor-
+hoodAn+1⊆Hofxn+1such that An+1∩B={xn+1}. It turns out that
+xn+1∈[(H\/unionsqtext
+j≤nUj)∩An+1∩˜N∗
+in+1/≈]∩U(xn+1) =Dn+1and that Dn+1
+is open in ˜N∗
+in+1/≈. Now, since ˜N∗
+in+1/≈is zero-dimensional, there exists
+a clopen subset Un+1⊆Dn+1of˜N∗
+in+1/≈withxn+1∈Un+1: we trivially
+remark that Un+1is also clopen in H, thatUn+1∩Un′=∅for every n′<n+1
+and that H\/unionsqtext
+j≤n+1Uj⊇B\{x1,...,x n+1}.
+Therefore Un⊆˜N∗
+in/≈is a clopen neighborhood of xninHfor every n∈ω
+andj−1(Un) is a clopen subset of ˜N∗
+in; then for every n∈ωit holds that
+j−1(Un) =E∗
+nwhereEnis an infinite subset of ω. Moreover we can assert
+that/unionsqtext
+n∈ωUn/\e}a⊔io\slash=Hsince/unionsqtext
+n∈ωUn⊆/uniontext
+n∈ωU(xn). Now we want to prove
+that/unionsqtext
+n∈ωUnis clopen in U/≈; trivially/unionsqtext
+n∈ωUnis open and now we show
+that it is also closed. If we take a point z∈H\/unionsqtext
+n∈ωUnthere exists an
+17indexizsuch that z∈(˜N∗
+iz/≈)\/unionsqtext
+n∈ωUn. Since ˜N∗
+iz/≈intersects only
+a finite number of the clopen subsets we have just constructed (w e denote
+these clopen subsets by Uj1,...,U jn), then/unionsqtextn
+i=1Uji∩(˜N∗
+iz/≈) is closed in
+˜N∗
+iz/≈since/unionsqtextn
+i=1Ujiis closed in H; the subset ˜N∗
+iz/≈ \(/unionsqtextn
+i=1Uji) is an
+open subset to which zbelongs and hence there exists an open neighborhood
+ofzin˜N∗
+iz/≈(and then in H) disjoint from/unionsqtextUn. Finally we can conclude
+thatj−1(/unionsqtextUn) =/unionsqtextj−1(Un) is clopen in Uand then there exists a clopen
+subset (˜N′)∗ofω∗with˜N′⊆ωand|˜N′|=ωsuch that
+(˜N′)∗∩/uniondisplay˜N∗
+i=/unionsqdisplay
+j−1(Un) =/unionsqdisplay
+E∗
+n.
+/square
+Remark 4.5 We remark that Lemma 4.4 still holds when we consider a
+family/braceleftBig
+˜Nγ:γ∈ω1/bracerightBig
+of infinite subsets ˜Nγ⊂ωkeeping all the other hypotheses.
+5 Construction of a Baˇ skirov’s space of order
+an arbitrary successor ordinal
+Finally we will show how to construct the space Kα+1by assuming that all
+compact sequential spaces Kβ+1(of sequential order β+1) with β+1< α+1
+have been constructed and that properties S.1 to S.6 hold in each of these
+space; moreover we will check that properties S.1 to S.6 hold in Kα+1too.
+We will carry out the construction when αis a successor ordinal, but we will
+remark from time to time what it is necessary to change if we have to w ork
+in the case in which αis a limit ordinal).
+It will be very important to take the set Γ into account: it is the set o f all
+the families Cξwhose elements are countable pairwise disjoint clopen subsets
+ofω∗; under the Continuum Hypothesis, we can write Γ as
+Γ ={Cξ:ω≤ξ < ω1}. (5.1)
+Roughly speaking, our type of construction ensures that, by a nu mber of
+steps of cardinality equal to the cardinality of Γ, we are able to exha ust the
+18whole Γ; moreover at each stage α < ω 1of the inductive construction, it will
+be essential the fact that αis a countable ordinal in order to guarantee that
+we can continue the process and hence it is crucial that we can enum erate Γ
+as in (5.1).
+Let us begin the construction. Let {Ni}i∈ωbe a family of pairwise disjoint
+infinite subsets Ni⊂ω. For every i∈ωthe closures of Niinβω, namely Ni,
+isaclopensubset of βωwhichishomeomorphictoit; forevery i∈ωletusset
+a decomposition of type βi+1 onNitaking care that the sequence of ordinals
+S= (βi+1)i∈ωis nondecreasing and such that sup {βi+1}=α. Notice that
+itispossible toextract aninjective subsequence S′= (βin+1)n∈ω⊆Sinsuch
+a way that the sequence S′converges upwards to α; sinceαis a successor
+ordinal, this mean that there are infinite n∈ωsuch that the decomposition
+set onNinis a relation of type α.2
+For every i∈ω, letjβi+1:Ni→Kβi+1be the quotient mapping. Let us
+check that the following properties hold for every ¯ ı∈ω.
+T.1N∗
+¯ı\/uniontext
+i′<¯ıN∗
+i′/\e}a⊔io\slash=∅: indeed the subsets Niare pairwise disjoint and,
+in particular, the subsets Niwithi= 1,...,¯ıare pairwise disjoint;
+then the corresponding N∗
+iare pairwise disjoint. Hence it holds that
+N∗
+¯ı\(/uniontext
+i′<¯ıN∗
+i′)/\e}a⊔io\slash=∅and obviously N∗
+¯ı\/uniontext
+i′<¯ıN∗
+i′/\e}a⊔io\slash=∅.
+T.2/uniontext
+i′≤¯ıN∗
+i′/\e}a⊔io\slash=ω∗: indeed we have already pointed out that the subsets
+N∗
+iare pairwise disjoint; then N∗
+¯ı+1is a non-empty subset disjoint from
+all the subsets N∗
+iwithi≤¯ı. Moreover for every i≤¯ıthe subset N∗
+iis
+closed and the finite union/uniontext
+i≤¯ıN∗
+iis closed too; thus it follows that
+ω∗\/uniontext
+i≤¯ıN∗
+i=ω∗\/uniontext
+≤¯ıN∗
+i⊃N∗
+¯ı+1.
+T.3 For every i′<¯ıit holds that Ni′∩N¯ı=∅.
+T.4 Property T.4 takes into account the families Cξwithξ≤¯ıbut the fam-
+iliesCξhave indices from ωtoω1not included and so, at the moment,
+we do not have to consider this property.
+In view of T.3 and the relations set on each Niwithi∈ω, we have
+defined a relation QωonUω=/uniontext
+i∈ωN∗
+i. Letjω
+α+1be the quotient mapping
+2Ifαis a limit ordinal we will have to set decompositions of type βi+1 onNiin such
+a way that the sequence ( βi+1)i∈ωis nondecreasing and sup {βi+1}=α. Also in this
+case it is possible to extract an injective subsequence S′= (βin+1)n∈ω⊆Sin such a way
+that the sequence S′converges upwards to α.
+19jω
+α+1:Uω→Uω/Qω.
+We say that Cξ∈Γ is anω−family if Cξconsists of elements that can be de-
+composed into two subfamilies L0andL1satisfying the following conditions.
+U.1/uniontextL0∩Uω=∅.
+U.2 For every c∈ L1there exists i < ω, a point xc∈Ni/≈βi+1of level
+γc+1 and an elementary neighborhood Ucofxcsuch that c=/tildewiderUc∩ω∗
+where/tildewiderUc=j−1
+βi+1(Uc).
+U.3 The set {xc:c∈ L1}has no accumulation points in Uω/Qω.
+U.4 It holds that sup {γc+1 :c∈ L1}< α.
+Let us rewrite these properties in order to make clear the new notio n.
+i)L0consists of elements C∗
+nwhere the subsets Cn⊂ωare transversal to
+the subsets Ni, i.e. every Cn⊂ωintersects every Niin a finite number
+of points (in this way we are respecting U.1);
+ii)L1consists of elements C∗
+mwhere for every mthere exist i∈ω, a point
+xm∈Ni/≈βi+1of levell(xm)< αand an elementary neighborhood
+U(xm) such that C∗
+m=j−1
+βi+1U(xm)∩ω∗. A further necessary require-
+ment isthat theset {xm}is devoid ofany accumulation point in Uω/Qω
+and that sup {l(xm)}< α. We want to point out that by l(xj) we mean
+a successor ordinal. (In this way we are respecting U.2-U.3-U.4).
+Notice that it is possible to find an ω-family: for example, we can use Lemma
+4.2 since the subsets N∗
+icomply with the hypotheses; in this way we find an
+infinite subset ∆ ω⊂ωsuch that ∆ ωintersects every Niin a finite number of
+points. We can decompose this infinite set in an infinite number of infinit e
+subsetsTn⊂ωthat again intersect every Niin a finite number of points; we
+setL0={T∗
+n:n∈ω}. It is clear that L=L0is anω-family.
+Among all the ω-families let us take the one with the minimum index ω; we
+write it as Cω=L0⊔L1withL0={N∗
+ω,n:n∈J0},L1={N∗
+ω,n:n∈J1}
+andJ0∩J1=∅. Of course, by construction, the ω-familyCωcomplies with
+the following properties.
+U.1/uniontextL0∩Uω=∅.
+20U.2 For every N∗
+ω,nwithn∈J1there exist in∈ω, a point xn∈Nin/≈βin+1
+of levell(xn)< αand an elementary neighborhood U(xn) such that
+N∗
+ω,n=j−1
+βin+1(U(xn))∩ω∗.
+U.3 The set {xn:n∈J1}has no accumulation point in Uω/Qω.
+U.4 It holds that sup {l(xn) :n∈J1}=βω< αwithβωthat can take up
+valuefrom1to αnotincluded. Withoutlossofgeneralitywecanalways
+assume that the levels of the points are ordered in a nondecreasing way.
+Forevery n∈J1itturnsoutthat ˆU(xn) =U(xn)\ωisaclopenneighborhood
+ofxninN∗
+in/≈βin+1. We can apply Lemma 4.4 since the family {Ni:i∈ω},
+the points xnwithn∈J1and the relation Qωdefined on Uω=/unionsqtextN∗
+isatisfy
+the hypotheses. We remark that/uniontextˆU(xn)/\e}a⊔io\slash=Uω/Qωsince inUω/Qωthere
+are points of level αwhich/uniontextˆU(xn) does not cover.3Therefore it is possible
+to find pairwise elementary neighborhoods Unwithxn∈Unand a subset
+N′
+ω⊂ωsuch that
+(N′
+ω)∗∩Uω=/unionsqdisplay
+n∈J1(jω
+α+1)−1(Un) =/unionsqdisplay
+n∈J1E∗
+n.
+Let us define C′=L0∪{(jω
+α+1)−1(Un) :n∈J1}.
+Now if we set Q={Nω,n:n∈J0}andR={Ni:i∈ω}, thenP=Q∪R
+is a family of subsets with the following properties:
+-Qis an almost disjoint family;
+-|Q| ≤ωand|R| ≤ω;
+- for every Nω,n∈ Qand every Ni∈ Rit holds that |Nω,n∩Ni|< ω.
+Therefore, by Lemma 4.3, there exists a subset N′′
+ω∈[ω]ωsuch that
+/uniondisplay
+n∈J0N∗
+ω,n⊆(N′′
+ω)∗and (N′′
+ω)∗∩N∗
+i=∅,∀Ni∈ R.
+Trivially it follows that/uniontext
+n∈J0N∗
+ω,n⊆N′′ωandN′′ω∩Uω=∅. Let us recapitu-
+late:
+3Notice that/uniontextˆU(xn)/\e}a⊔io\slash=Uω/Qωalso in the case in which αis a limit ordinal: indeed
+at the beginning of the construction we put decompositions of type βi+1 on the subsets
+Niin such a way that sup {βi+1}=α; hence in Uω/Qωthere certainly exists a point of
+levelβω+1< αthat/uniontextˆU(xn) does not cover.
+211)N′′
+ω⊇∗Nω,nfor every n∈J0;
+2)|N′′
+ω∩Ni|< ωfor every i∈ω.
+Notice that ( N′′′
+ω)∗= (N′
+ω)∗∪(N′′
+ω)∗is a clopen subset of ω∗. Now it turns
+out that
+(N′′′
+ω)∗∩Uω= [(N′
+ω)∗∪(N′′
+ω)∗]∩Uω= (jω
+α+1)−1(/unionsqdisplay
+n∈J1Un)∪∅=/unionsqdisplay
+n∈J1E∗
+n
+and then we can conclude that N′′′
+ω⊇∗Enfor every n∈J1andN′′′
+ω⊇∗Nω,n
+for every n∈J0. Let us set
+Mn=/braceleftbiggN′′′
+ω∩Enifn∈J1
+N′′′
+ω∩Nω,nifn∈J0.
+Certainly M∗
+n=E∗
+nfor every n∈J1andM∗
+n=N∗
+ω,nfor every n∈J0.
+For every n∈ωlet us fix a point ln∈Mn\(/uniontextn−1
+j=0Mj∪{l0,...,ln−1}) - it is
+possible since the family {Mn}is almost disjoint - and let us set L={li:
+i∈ω}.
+Let us define
+Nω=/unionsqdisplay
+n∈ω(Mn\n−1/uniondisplay
+j=0Mj)\{li:i∈ω}=/unionsqdisplay
+n∈ωHn
+whereHn= (Mn\/uniontextn−1
+j=0Mj)\{li:i∈ω}. Notice that N∗
+ω⊇/uniontextC′(in-
+deed from every Mnwe removed only a finite number of points) and that
+(N′′′
+ω)∗\N∗
+ω/\e}a⊔io\slash=∅(sinceN′′′
+ω\Nω={li:i∈ω}) whence |ω\Nω|=ω.
+Now let us take into account Nω=/unionsqtext
+n∈ωHnwhereM∗
+n=H∗
+nfor every n∈ω;
+we want to remark that on each Hnwithn∈J1we have already a decom-
+position of type l(xn) by construction.
+Now if|J1|=ω, on every Hnwithn∈J0let us put a decomposition of type
+1; then let us order the subsets Hnin such a way that the types of decom-
+position that we have put on them form a nondecreasing sequence w ith the
+supremum equal to βω< α.
+If|J1|< ω, it turns out that sup {l(xn) :n∈J1}=βωis a successor ordinal;
+let us put a decomposition of type βωon every Hnwithn∈J04and then
+4In this case |J0|=ω.
+22let us order the subsets Hnin such a way that the types of decomposition
+that we have put on them form a nondecreasing sequence whose su premum
+is equal to βω< α.
+IfJ1=∅, we choose to put a decomposition of type a successor ordinal
+βω< αon every Hnwithn∈J0and then we proceed as in the latter case.
+This time let us apply property S.6 to the subsets Hnand toNω: it turns
+out that {Hn}n∈ωis a countably infinite family of infinite pairwise disjoint
+subsets of Nωand on every Hnis given a relation of some type in such a way
+that the supremum of the nondecreasing sequence consisting of t he types of
+decomposition has supremum βωwithβωthat can take up value from 1 to
+αnot included. Then the relation on/uniontext∞
+n=1Hnobtained in this way can be
+extended to a relation ≈βω+1onNωof typeβω+ 1 where βω+ 1 can have
+value a successor ordinal from 2 up to α.5
+Remark 5.1 We want to remark that for the points constructed by the de-
+compositions on the Hnwithn∈J0it is always possible to find a fundamen-
+tal system of elementary neighborhoods contained in Nω/≈βω+1and such
+that their inverse images through jω
+α+1have empty intersection with Uωsince
+H∗
+n∩Uω=∅; from now on, we consider only these neighborhoods as elemen -
+tary neighborhoods of those points.
+Let us check the following properties:
+T.1N∗ω\/uniontext
+i∈ωN∗
+i/\e}a⊔io\slash=∅: indeed it turns out that ( Nω)∗⊇/uniontextL0while/uniontextL0∩
+(/uniontext
+i∈ωN∗
+i) =∅; hence, if L0/\e}a⊔io\slash=∅, it follows that N∗
+ω\/uniontextN∗
+i⊇/uniontextL0/\e}a⊔io\slash=∅.
+On the other hand if L0=∅, then the family {Hn:n∈J1}of pairwise
+disjoint subsets of ωis infinite; thus we can construct an infinite subset
+T⊂Nωin this way: we choose a point tm∈Hmfor every m∈J1and
+we setT={tm:m∈J1}. The non-empty subset T∗is such that T∗⊆
+N∗
+ω, whileT∗∩N∗
+i=∅for every i∈ω. We want to check it: certainly
+T∗∩H∗
+n=∅for every n∈J1and hence T∗∩(/unionsqtextH∗
+n) =∅; if there is an
+indexi∈ωsuch that |T∩Ni|=ω, then (T∩Ni)∗⊂(N∗
+ω∩/uniontext
+i∈ωN∗
+i),
+while we know that ( N∗
+ω∩/uniontext
+i∈ωN∗
+i) =/unionsqtextH∗
+n.
+T.2/uniontext
+i≤ωN∗
+i/\e}a⊔io\slash=ω∗: notice that the set Lis such that L∗∩N∗
+ω=∅and
+thatL∗∩N∗
+i=∅for every i < ω(indeed for every i < ωit holds that
+5In the case in which αis a limit ordinal, βω+1 can take up value on the successor
+ordinals from 2 up to an ordinal strictly less than α.
+23|Ni∩L|< ω; if there exists an index i∈ωsuch that |L∩Ni|=ω, then
+(L∩Ni)∗⊂((N′′′
+ω)∗∩/uniontext
+i∈ωN∗
+i), whileweknowthat(( N′′′
+ω)∗∩/uniontext
+i∈ωN∗
+i) =/unionsqtextH∗
+nandL∗∩(/unionsqtextH∗
+n) =∅). Then we obtain that/uniontext
+i≤ωN∗
+i∩L∗=∅
+whereL∗is open in ω∗whenceω∗\L∗is a closed subset that contains/uniontext
+i≤ωN∗
+i; so it contains its closure and it follows that/uniontext
+i≤ωN∗
+i∩L∗=∅.
+At the end, we can conclude that/uniontext
+i≤ωN∗
+i/\e}a⊔io\slash=ω∗.
+T.3 For every i∈ω, the relations ≈βi+1and≈βω+1coincide on Ni∩Nω:
+indeedN∗
+ω∩N∗
+i⊆/unionsqtextH∗
+n(withn∈J1), the relation on N∗
+ωextends
+the relations placed on the subsets H∗
+n(withn∈J1) and these last
+relations coincide with the relations we put on the subsets N∗
+i.
+Then a relation Qω+1is defined on Uω+1=/uniontextω
+i=1N∗
+i.
+T.4 A family Cξ∈Γ with index ξ≤ωis not an ( ω+ 1)−family. Notice
+that the families Cξwithξ < ωdo not exist and hence we have only to
+prove that the family Cωis not an ( ω+1)−family.
+We say that Cξ∈Γis an(ω+1)−family if Cξcan be decomposed into
+two subfamilies Lω+1
+0andLω+1
+1satisfying the following conditions.
+U.1/uniontextLω+1
+0∩Uω+1=∅.
+U.2 For every c∈ L1ω+1there exists i≤ω, a point xc∈Ni/≈βi+1
+of levelγc+1and an elementary neighborhood Ucofxcsuch that
+c=j−1
+βi+1(Uc)∩ω∗.
+U.3 The set/braceleftbig
+xc:c∈ Lω+1
+1/bracerightbig
+is devoid of any accumulation point in
+Uω+1/Qω+1.
+U.4 It holds that sup/braceleftbig
+γc+1 :c∈ Lω+1
+1/bracerightbig
+< α.
+Remember that Cωis theω-family with minimum index we have just
+used in order to construct Nω/≈βω+1; moreover notice that if ω > ω
+the family Cωis not an ω-familyand then it neither is an ( ω+1)-family.
+Towards a contradiction, suppose that it is an ( ω+1)-families: in Lω
+0
+we put the elements that lie in Lω+1
+0and all those elements c∈ Lω+1
+1
+such that ωis the only value of the index ifor which U.2 is satisfied;
+thesecare such that c∩Uω=∅by Remark 5.1. Instead in Lω
+1we put
+all the other c∈ Lω+1
+1which are left: they obviously satisfy U.4 since
+we are estimating the supremum on a lesser number of elements; mor e-
+over they satisfy U.3 since if the points xchad accumulation points in
+Uω/Qωthen they would have accumulation points in Uω+1/Qω+1due
+24to the fact that the new relation respects the old ones.
+Ifinstead ω=ω,thenCω=L0∪L1isnotan( ω+1)-familysincetheele-
+mentsof Cωwouldhavealltostayin Lω+1
+1butthecorrespondinginfinite
+points{xc:c∈ Lω+1
+1}, which are all in the compact space N∗
+ω/≈βω+1,
+must have an accumulation point in Uω+1/Qω+1⊇N∗
+ω/≈βω+1.
+Suppose to have constructed infinite subsets Nγ⊂ω(withγ < δ < ω 1)
+and relations ≈βγ+1of typeβγ+ 1< α+ 1 onNγsatisfying the following
+conditions.
+T.1N∗γ\/uniontext
+γ′<γN∗
+γ′/\e}a⊔io\slash=∅.
+T.2/uniontext
+γ′≤γN∗
+γ′/\e}a⊔io\slash=ω∗.
+T.3 For every γ′< γ, the relations ≈βγ′+1and≈βγ+1coincide on Nγ′∩Nγ.
+T.4 A family Cξwith index ξ≤γis not a ( γ+1)−family.
+By T.3 and the decompositions set on the Nγfor every γ < δ, a decompo-
+sitionQδis defined on Uδ=/uniontext
+γ<δN∗
+γ. Letjδ
+α+1be the quotient mapping
+jδ
+α+1:Uδ→Uδ/Qδ.
+We say that an element Cξ∈Γ is aδ−family if Cξcan be decomposed into
+two subfamilies L0andL1satisfying the following conditions.
+U.1/uniontextL0∩Uδ=∅.
+U.2 Forevery c∈ L1there exists γ < δ, a point xc∈Nγ/≈βγ+1oflevelγc+
+1andanelementaryneighborhood Ucofxcsuchthat c=j−1
+βγ+1(Uc)∩ω∗.
+U.3 The set {xc:c∈ L1}has no accumulation point in Uδ/Qδ.
+U.4 It holds that sup {γc+1 :c∈ L1}< α.
+We can rewrite these properties in the following way:
+i)L0has to consist of elements C∗
+nwhere the subsets Cnare transversal
+to the subsets Nγ, i.e. every Cn⊂ωintersects every Nγin a finite
+number of points (in this way we are respecting U.1);
+25ii)L1has to consist of elements C∗
+mwhere for every mthere exists γ∈
+δ, a point xm∈Nγ/≈βγ+1of levell(xm)< αand an elementary
+neighborhood U(xm) such that C∗
+m=j−1
+βγ+1(U(xm))∩ω∗. A further
+necessary request is that the set {xm}is devoid of any accumulation
+point in Uδ/Qδand that sup {l(xm)}< α. We want to remark that
+byl(xm) we mean a successor ordinal. (In this way we are respecting
+U.2-U.3-U.4)
+Let us show that it is possible to find a δ-familyCξ: for example, we can
+use again Lemma 4.2, since the subsets N∗
+γwithγ < δ < ω 1comply with
+the hypotheses. We want to remark that here the fact that δis a countable
+ordinal is essential in order to apply Lemma 4.2. In this way we are able to
+find a subset ∆ δ⊂ωwith|∆δ|=ωand such that ∆ δintersects every Nγin
+a finite number of points. We can decompose this infinite set in an infinit e
+number of infinite subsets Tn⊂ωwhich again intersect every Nγin a finite
+number of points; we set L0={T∗
+n:n∈ω}. It is clear that L=L0is a
+δ-family.
+Among all the δ-families in Γ, let us take the one with the minimum index
+δ: we can write it as Cδ=L0⊔L1whereL0={N∗
+δ,n:n∈J0},L1={N∗
+δ,n:
+n∈J1}andJ0∩J1=∅. Of course, by construction, the δ-familyCδwill
+comply with the following properties.
+U.1/uniontextL0∩Uδ=∅;
+U.2 For every N∗
+δ,nwithn∈J1there exist an index γn∈δ, a point xn∈
+Nγn/≈βγn+1of levell(xn)< αand an elementary neighborhood U(xn)
+such that
+N∗
+δ,n=j−1
+βγn+1(U(xn))∩ω∗.
+U.3 The set {xn:n∈J1}has no accumulation point in Uδ/Qδ.
+U.4 It holds that sup {l(xn) :n∈J1}=βδ< αwithβδthat can take up
+value from 1 to αnot included. Without loss of generality, we can al-
+ways assume thatthelevels ofthepointsareorderedinanondecre asing
+way.
+For every n∈J1it holds that ˆU(xn) =U(xn)\ωis a clopen neighbor-
+hood of xninN∗
+γn/≈βγn+1. In order to apply Lemma 4.4, we need to
+26rewrite/uniontext
+γ∈δN∗
+γas a disjoint union; notice that, since δis a countable ordi-
+nal, we can enumerate {N∗
+γ}γ∈δas{N∗
+γi}i∈ω. Let us set ˜Nγ1=Nγ1,˜Nγ2=
+Nγ2\Nγ1,...,˜Nγk=Nγk\/uniontextk−1
+i=1Nγi. It holds that/unionsqtext
+k∈ω˜Nγk=/uniontext
+k∈ωNγk,
+whence/unionsqtext
+k∈ω˜N∗
+γk=/uniontext
+k∈ωN∗
+γk; we have only to prove the non-trivial inclu-
+sion/unionsqtext
+k∈ω˜N∗
+γk⊇/uniontext
+k∈ωN∗
+γk: ifx∈/uniontext
+k∈ωN∗
+γk, then there is ¯ ı∈ωsuch that
+x∈N∗
+γ¯ı=/bracketleftbig/unionsqdisplay
+k≤¯ı˜Nγk/bracketrightbig∗=/unionsqdisplay
+k≤¯ı˜N∗
+γk⊆/unionsqdisplay
+k∈ω˜N∗
+γk.
+Notice that, for every k∈ω,˜N∗
+γkis distinguished relative to Qδ. Finally
+we can apply Lemma 4.4 since the countable family {˜Nγ}γ<δ, the points
+{xn}n∈J1and the relation Qδdefined on Uδ=/unionsqtext
+γ<δ˜N∗
+γ=/uniontext
+γ<δN∗
+γsatisfy
+the hypotheses.6We remark that/uniontextˆU(xn)/\e}a⊔io\slash=Uδ/Qδsince inUδ/Qδthere are
+points of level αthat/uniontextˆU(xn) does not cover.7
+Therefore it is possible to find pairwise disjoint elementary neighborh oods
+Unwithxn∈Unand a subset N′
+δ⊂ωsuch that
+(N′
+δ)∗∩Uδ=/unionsqdisplay
+(jδ
+α+1)−1(Un) =/unionsqdisplay
+E∗
+n.
+Let us define C′=L0∪{(jδ
+α+1)−1(Un) :n∈J1}.
+Let us set Q={Nδ,n:n∈J0}andR={Nγ:γ∈δ}. ThenP=Q∪Ris a
+family with the following properties:
+-Qis an almost disjoint family;
+-|Q| ≤ωand|R| ≤ω;
+- for every Nδ,n∈ Qand every Nγ∈ Rit holds that |Nδ,n∩Nγ|< ω.
+Therefore by Lemma 4.3 there exists a subset N′′
+δ∈[ω]ωsuch that
+/uniondisplay
+n∈J0N∗
+δ,n⊆(N′′
+δ)∗and (N′′
+δ)∗∩N∗
+γ=∅,∀Nγ∈ R;
+obviously it holds that/uniontext
+n∈J0N∗
+δ,n⊆N′′
+δandN′′
+δ∩Uδ=∅. HenceN′′
+δis such
+that:
+6At most we have to restrict the neighborhoods of the points {xn}in such a way that
+each of them belongs to some ˜N∗
+γ/Qδfor some γ < δ.
+7Notice that/uniontextˆU(xn)/\e}a⊔io\slash=Uδ/Qδalso in the case in which αis a limit ordinal: indeed at
+the beginning of the construction we put decompositions of type βi+1 on the subsets Ni
+in such a way that sup {βi+1}=α; hence in Uδ/Qδthere certainly exists a point of level
+βδ+1< αthat/uniontextˆU(xn) does not cover.
+271)N′′
+δ⊇∗Nδ,nfor every n∈J0;
+2)|N′′
+δ∩Nγ|< ωfor every γ∈δ.
+Let us remark that ( N′′′
+δ)∗= (N′
+δ)∗∪(N′′
+δ)∗is a clopen subset of ω∗. Now
+it turns out that
+(N′′′
+δ)∗∩Uδ= [(N′
+δ)∗∪(N′′
+δ)∗]∩Uδ= [(jδ
+α+1)−1(/unionsqdisplay
+n∈J1Un)]∪∅=/unionsqdisplay
+n∈J1E∗
+n
+and hence N′′′
+δ⊇∗Enfor every n∈J1andN′′′
+δ⊇∗Nδ,nfor every n∈J0. Let
+us set
+Mn=/braceleftbiggN′′′
+δ∩Enifn∈J1
+N′′′
+δ∩Nδ,nifn∈J0.
+Certainly M∗
+n=E∗
+nfor every n∈J1andM∗
+n=N∗
+δ,nfor every n∈J0.
+For every n∈ωlet us fix a point ln∈Mn\(/uniontextn−1
+j=0Mj∪{l0,...,ln−1}) and let
+us setL={li:i∈ω}. Let us define
+Nδ=/unionsqdisplay
+n∈ω(Mn\n−1/uniondisplay
+j=0Mj)\{li:i∈ω}=/unionsqdisplay
+n∈ωHn.
+Notice that Nδis such that N∗
+δ⊇/uniontextC′(indeed from every Mnwe removed
+only a finite number of points) and at the same time that ( N′′′
+δ)∗\N∗
+δ/\e}a⊔io\slash=∅
+(sinceN′′′
+δ\Nδ={li:i∈ω}) whence |ω\Nδ|=ω.
+Therefore it holds that Nδ=/unionsqtext
+n∈ωHn, withM∗
+n=H∗
+nfor every n∈ω.
+Remember that on each Hnwithn∈J1we have already a decomposition of
+typel(xn).
+If|J1|=ω, on every Hnwithn∈J0let us set a decomposition of type 1 and
+let us order the subsets Hnin such a way that the types of decomposition
+that we have put on them form a nondecreasing sequence with supr emum
+equal to βδ< α.
+If|J1|< ω, it turns out that sup {l(xn) :n∈J1}=βδis a successor ordinal;
+let us set decompositions of type βδon every Hnwithn∈J08and let us
+order the subsets Hnin such a way that the types of decomposition that we
+have put on them form a nondecreasing sequence with supremum eq ual to
+βδ< α.
+8In this case |J0|=ω
+28IfJ1=∅, we choose to put a decomposition of type a successor ordinal
+βδ< αon every Hnwithn∈J0and then we proceed as in the latter case.
+Let us apply property S.6 to the subsets Hnand toNδ: indeed {Hn}n∈ωis
+a countably infinite family of infinite pairwise disjoint subsets of Nδand on
+everyHnis given a relation of some type in such a way that the supremum
+of the nondecreasing sequence consisting of the types of decomp osition has
+supremum βδwithβδthat can take up value from 1 to αnot included. Then
+the relation on/uniontext∞
+n=1Hnobtained in this way can be extended to a relation
+≈βδ+1onNδof typeβδ+1 with 2 ≤βδ+1≤α.9
+Remark 5.2 We want to remark that for the points constructed by the de-
+compositions on the Hnwithn∈J0it is always possible to find a fun-
+damental system of elementary neighborhoods contained in Nδ/≈βδ+1and
+such that their inverse images through jδ
+α+1have empty intersection with Uδ
+sinceH∗
+n∩Uδ=∅; from now on, we consider only these neighborhoods as
+elementary neighborhoods of those points.
+Let us check that properties T.1 to T.4 hold.
+T.1N∗
+δ\(/uniontext
+γ∈δN∗
+γ) =N∗
+δ\(/uniontext
+γ∈δ˜N∗
+γ)/\e}a⊔io\slash=∅: the check is the same as in the
+case ofNω(see page 23).
+T.2/uniontext
+γ≤δN∗γ/\e}a⊔io\slash=ω∗: the check of this property is similar to that we have
+just done in the case of Nω(see page 23).
+T.3 For every γ∈δ, the relations ≈βγ+1and≈βδ+1coincide on Nγ∩Nδ:
+indeedN∗
+γ∩N∗
+δ⊆Uδ∩N∗
+δ⊆/uniontextH∗
+n(withn∈J1), the relation on
+N∗
+δextends the relations set on the subsets H∗
+nand these last relations
+coincide with those we put on the subsets N∗
+γby construction. Then a
+relationQδ+1is defined on Uδ+1=/uniontextδ
+γ=1N∗
+γ.
+T.4 A family Cξwith index ξ≤δis not a ( δ+1)−family.
+We say that Cξ∈Γis a(δ+ 1)−family if Cξcan be decomposed into
+two subfamilies Lδ+1
+0andLδ+1
+1satisfying the following conditions.
+U.1/uniontextLδ+1
+0∩Uδ+1=∅;
+9In the case in which αis a limit ordinal it turns out that 2 ≤βδ+1< α
+29U.2 For every c∈ L1δ+1there exist γ≤δ,a pointxc∈Nγ/≈βγ+1
+of levelγc+1and an elementary neighborhood Ucofxcsuch that
+c=j−1
+βγ+1(Uc)∩ω∗.
+U.3 The set/braceleftbig
+xc:c∈ Lδ+1
+1/bracerightbig
+is devoid of any accumulation point in
+Uδ+1/Qδ+1.
+U.4 It holds that sup/braceleftbig
+γc+1 :c∈ Lδ+1
+1/bracerightbig
+< α.
+Remember that Cδis theδ-family with minimum index we have just
+used in the construction of Nδ/≈βδ+1. Ifδ > δthe families Cξwith
+ξ≤δare notδ-families and then they neither are ( δ+ 1)-families.
+Towards a contradiction, suppose that they are ( δ+1)-families; then in
+Lδ
+0we put the elements that lie in Lδ+1
+0and all those elements c∈ Lδ+1
+1
+such that δis the only value of the index γfor which U.2 is satisfied; by
+Remark 5.2 these care such that c∩Uδ=∅. Instead in Lδ
+1we put all
+theother c∈ Lδ+1
+1that areleft: they obviously satisfy U.4, since we are
+estimatingthesupremumonalessernumberofelements; moreover they
+satisfy U.3, since if the points xchad accumulation points in Uδ/Qδ,
+then they would have accumulation points in Uδ+1/Qδ+1, due to the
+fact that the new relation respects the old ones.
+On the other hand, if δ=δ, then the families Cξwithξ < δare not
+δ-families and by what we have just remarked they neither are ( δ+1)-
+families; on the other hand Cδ=Lδ
+0∪Lδ
+1is not a ( δ+1)-family, since
+the elements of Cδwould have all to stay in Lδ+1
+1but the corresponding
+infinite points xc, which are all in the compact space N∗
+δ/≈βδ+1, must
+have an accumulation point in Uδ+1/Qδ+1⊇N∗
+δ/≈βδ+1.
+Therefore, by transfinite induction, we have defined a relation Qω1on/uniontext
+γ<ω1Nγwhich coincides with ≈βγ+1on eachNγ.
+Let us prove the following lemma.
+Lemma 5.3 If a family Cξ∈Γis not a ϑ-family then it is not a δ-family
+for every δ > ϑ.
+PROOF. We prove that if Cξ∈Γ is aδ-family then it is also a ϑ-family.
+Let us suppose that Cξis aδ-family; then it can be decomposed into two
+subfamilies Lδ
+0andLδ
+1satisfying the following conditions.
+U.1/uniontextLδ
+0∩Uδ=∅.
+30U.2 Forevery c∈ Lδ
+1thereexist γ < δ, apointxc∈Nγ/≈βγ+1oflevelγc+1
+and an elementary neighborhood Ucofxcsuch that c=j−1
+βγ+1(Uc)∩ω∗.
+U.3 The set/braceleftbig
+xc:c∈ Lδ
+1/bracerightbig
+has no accumulation point in Uδ/Qδ.
+U.4 It holds that sup/braceleftbig
+γc+1 :c∈ Lδ
+1/bracerightbig
+< α.
+We want to show that Cξis also a ϑ-family. In Lϑ
+0we put the elements
+that lie in Lδ
+0and all those elements c∈ Lδ
+1for which the only ordinals that
+fit for U.2 are larger than or equal to ϑ; thesecare the inverse images of
+elementary neighborhoods of points constructed by starting fro m someFn
+whereF∗
+n∈ Lζ
+0withζ≥ϑ. We know that the elementary neighborhoods of
+these points are contained in Fn/≈βζ+1and then by Remarks 5.1 and 5.2
+it follows that for each of these cit holds that c∩Uϑ=∅. On the other
+hand inLϑ
+1we put all the other c∈ Lδ
+1that are left. They obviously satisfy
+U.4, since we are estimating the supremum on a lesser number of eleme nts;
+moreover they satisfy U.3, since if the points xchad accumulation points in
+Uϑ/Qϑ, then they would have accumulation points in Uδ/Qδdue to the fact
+that the new relations respect the old ones. /square
+Now we can state the following fundamental remark.
+Remark 5.4 For every ϑ < ω 1,Cϑ∈Γis not an ω1-family. Towards a
+contradiction suppose that there exists an index ϑ < ω 1such that Cϑis an
+ω1-family. By transfinite induction we proved that, for every ϑ < ω 1, a family
+Cξwithξ≤ϑis not a(ϑ+1)-family and hence Cϑis not a(ϑ+1)-family.
+On the other hand we supposed that Cϑis anω1-family and then by Lemma
+5.3 it is a (ϑ+ 1)-family. A contradiction. Let us point out that there can
+not exist ω1-families in Γ, since the elements of the set Γhave indeces that
+go fromωincluded to ω1not included.
+Finally we define the relation ≈α+1onβωin this way:
+- it coincides with Qω1on/uniontext
+γ<ω1Nγ;
+- two free ultrafilter belonging to ω∗\/uniontext
+γ<ω1N∗
+γare equivalent under the
+relation≈α+1.
+Let us call Kα+1the space obtained by the quotient of βωwith this relation
+andjα+1the natural quotient mapping. Let us remark that, by property
+31U.2,ω∗\/uniontext
+γ<ω1N∗
+γis not empty: indeed for every γ∈ω1it holds that Bγ=
+ω∗\/uniontext
+γ′≤γN∗
+γ′is a closed subset of ω∗and the subsets Bγ(withγ∈ω1) are
+such that Bγ1⊇Bγ2for every γ1< γ2; moreover the family of closed subsets
+{Bγ}γ∈ω1has the finite intersection property by T.2 proved for every step
+γ∈ω1. Thus, due to the compactness of ω∗, it follows that
+/intersectiondisplay
+γ∈ω1Bγ=/intersectiondisplay
+γ∈ω1(ω∗\/uniondisplay
+γ′≤γN∗
+γ′) =ω∗\/uniondisplay
+γ∈ω1N∗
+γ/\e}a⊔io\slash=∅.
+Thenjα+1collapses ω∗\/uniontext
+γ<ω1N∗
+γto a single point which we call x∞.
+If an element of Kα+1is a point of the decompositions ≈βδ+1and≈βγ+1,
+then the point lies in the same level in Nδ/≈βδ+1andNγ/≈βγ+1: indeed
+whenever we reconsider a point that was in some previous decompos itions we
+takecarethatthereexistsanelementaryneighborhoodofitthat accompanies
+the point in the new decomposition; in this way the level of the point is
+preserved and the definition of Lβ+1as the set of the points that lie in the
+levelβ+ 1 in some Nγ/≈βγ+1is correct. If a point of the space Kα+1is a
+point of the decompositions ≈βδ+1and≈βγ+1withδ > γ, then the problem
+reduces to examine what happens in Nδ/≈βδ+1as regards its elementary
+neighborhoods. We have to remark that in the construction of the space
+Kα+1we paid attention to the fact that for every level 0 < β+1≤α10every
+point of level β+1 had a basis of clopen subsets homeomorphic to the space
+Kβ+1which is compact and sequential by inductive hypothesis.
+Now we have to understand which are the elementary neighborhood s of the
+uniquepointoflevel α+1inKα+1,i.e. ofthepoint x∞=jα+1(ω∗\/uniontext
+γ<ω1N∗
+γ).
+On this subject let us prove the following lemma.
+Lemma 5.5 The collection of the clopen subsets Kα+1\/uniontext
+x∈GUx(whereG
+is a finite set and for every x∈Gthe clopen subset Uxis an elementary
+neighborhood in Kα+1of the point xthat can have level equal to a successor
+ordinal smaller than or equal to α) is a basis at the point x∞.
+PROOF. In an obvious way Kα+1\/uniontext
+x∈GUxis a clopen subset of Kα+1
+containing x∞. LetAbe an open subset of Kα+1containing x∞and let
+C=Kα+1\Abe the complementary closed subset. For every x∈C, letUx
+be an elementary clopen neighborhood of x; trivially, by taking all the clopen
+neighborhoods Ux, withx∈C, we cover C. Let us consider j−1
+α+1(C): it is a
+10strictly smaller than αin the case in which αis a limit ordinal.
+32closed subset of βωand then it is compact. If we take all the open subsets
+j−1
+α+1(Ux) (withx∈C) they form an open cover of j−1
+α+1(C); then there exists
+a finite subcover/uniontext
+x∈Gj−1
+α+1(Ux)⊇j−1
+α+1(C). Hence it turns out that
+jα+1(/uniondisplay
+x∈Gj−1
+α+1(Ux)) =/uniondisplay
+x∈Gjα+1(j−1
+α+1(Ux)) =
+/uniondisplay
+x∈GUx⊇jα+1(j−1
+α+1(C)) =C
+and,bypassingtothecomplementarysubsets, wecanconcludeth atKα+1\/uniontext
+x∈GUx⊆
+Kα+1\C=A. /square
+We callelementary each of these neighborhoods of the point x∞.
+6 Check of the properties of Kα+1
+Now we want to check that the space Kα+1satisfies all the requested prop-
+erties.
+Lemma 6.1 Kα+1is a Hausdorff space and it is compact.
+PROOF. Trivially the points of L0can be separated from every other point
+sincetheyareisolated. Moreover, ifwewanttoseparate x∞=jα+1(ω∗\/uniontext
+γ<ω1N∗
+γ)
+from any other point x, it is enough to take respectively the open disjoint
+elementary neighborhoods Kα+1\UxandUx.
+Suppose now to have to part two points x1andx2of level smaller than α+1;
+it is possible to face up with two different situations.
+1) Thereexists ϑ∈ω1suchthat x1,x2∈jα+1(Nϑ); noticethat jα+1(Nϑ)≃
+Kβ+1(withβ+ 1< α+ 1) which is a Hausdorff space by inductive
+hypothesis. Then in jα+1(Nϑ) there are two open neighborhoods Vx1
+andVx2with empty intersection; they are also open in Kα+1and hence
+Vx1andVx2are open neighborhoods of x1andx2respectively with
+empty intersection.
+2) There is no ϑ∈ω1such that x1,x2∈jα+1(Nϑ); therefore there are
+ϑ1,ϑ2∈ω1such that x1∈jα+1(Nϑ1)≃Kβ+1withβ+ 1< α+
+1 andx2∈jα+1(Nϑ2)≃Kγ+1withγ+ 1< α+ 1. Now I=
+jα+1(Nϑ1)∩jα+1(Nϑ2)isaclopen subset of Kα+1andhence jα+1(Nϑ1)\I
+andjα+1(Nϑ2)\Iare disjoint open neighborhoods of x1andx2respec-
+tively.
+33Therefore it turns out immediately that Kα+1is compact since jα+1is a con-
+tinuous function from the compact space βωto the Hausdorff space Kα+1./square
+Beforeprovingthesequentialityofthespace Kα+1weneedtodemonstrate
+that properties S.4 and S.5 hold.
+Remark 6.2 InKα+1, if a nonconstant sequence (xn)n∈ωof points xn∈
+Lγn+1with nondecreasing levels converges to a point x∈Lγ+1, then for
+the sequence (γn+ 1)of ordinal numbers it holds that sup{γn+ 1}=γ.
+(Properties S.4 )
+PROOF. For every γ+ 1< α+ 1 we apply the inductive hypothesis, since
+we have supposed that property S.4 holds in Kγ+1for every γ+1< α+1.
+Now we have to prove that for a non-constant sequence of points xn∈Lγn+1
+(where the sequence ( γn+1) is not decreasing) that converges to the point
+x∞∈Lα+1it holds that sup {γn+ 1}=α. Towards a contradiction, let us
+suppose that sup {γn+1}< α. In principle there are two different cases we
+have to analyse:
+1) from the sequence ( xn)n∈ωwe can extract an injective subsequence
+(xni)i∈ω;
+2) fromthesequence ( xn)n∈ωwecannot extract anyinjective subsequence
+(xni)i∈ω.
+We can avoid considering the latter case: indeed, since ( xn)n∈ωis a non-
+constant sequence, there are at least two points that appear infi nite times
+and then the sequence is not convergent to any point against the h ypothesis.
+Intheformercasethesequence( xni)i∈ωhastoconvergeto x∞too. If{xni}i∈ω
+was devoid of any accumulation point in Uω1/Qω1, then by Remark 4.5 it
+would be possible to find a countable infinity of pairwise disjoint clopen s ub-
+setsofω∗; moreover theseclopensubsets wouldsatisfythepropertiestob ean
+ω1-family (notice that sup {γn+1}< α) and this would be inconsistent with
+Remark 5.4. Then the subset S={xni:i∈ω}has at least an accumulation
+point in Uω1/Qω1; thus there exists a point y∈Uω1/Qω1(where certainly
+l(y) =δ+ 1< α+ 1) such that y∈{xni:i∈ω}. Then let us consider
+an elementary neighborhood of y,Uy, which has to be homeomorphic to the
+spaceKδ+1; we can assert that infinite points of Ssuch that the supremum
+of their levels is equal to an ordinal number η < αare inUy. We denote this
+34set of points by S′⊆S; we know that property S.5 holds in Kδ+1and then
+from the injective sequence S′it is possible to extract a sequence converging
+to a point of level η+ 1< α+ 1. Therefore the sequence ( xn)n∈ωadmits a
+subsequence which converges to a point of level strictly smaller tha nα+ 1
+against the hypothesis. /square
+Remark 6.3 InKα+1, from every injective sequence S= (xn)n∈ωof points
+with nondecreasing levels such that sup{l(xn)}=η≤αit is possible to
+extract a subsequence converging to a point of level η+1. (Property S.5 )
+PROOF. If η= 0 then the sequence ( xn)n∈ωis formed by points of ω;
+therefore there is an index γ∈ω1such that |{xn}n∈ω∩Nγ|=ω: other-
+wise, if it turns out that |{xn}n∈ω∩Nγ|< ωfor every γ∈ω1, then from
+Nω1={xn}n∈ωwe are able to construct an ω1-family and this is a contradic-
+tion. Then in Nγ/≈α+1there are infinite points of the above sequence but
+Nγ/≈α+1≃Kβ+1withβ+ 1< α+ 1 and hence, since property S.5 holds
+inKβ+1by inductive hypothesis, it is possible to extract a subsequence con -
+verging to a point of level 1 from the starting sequence.
+Suppose now that 0 < η < α ; let us choose an injective subsequence S′=
+(xni)i∈ω⊆Sin such a way that the sequence of the levels of the points con-
+verges upwards to η; ifS′was devoid of any accumulation point in Uω1/Qω1,
+then by Remark 4.5 it would be possible to find a countable infinity of pair -
+wise disjoint clopen subsets of ω∗; moreover these clopen subsets would sat-
+isfy the properties to be an ω1-family (notice that sup {l(xni)}< α) and this
+would be inconsistent with Remark 5.4. Thus S′must have at least an ac-
+cumulation point in Uω1/Qω1and hence there exists a point y∈Uω1/Qω1
+withl(y) =δ+1< α+1 such that y∈{xni:i∈ω}. Then let us consider
+an elementary neighborhood of y,Uy, which has to be homeomorphic to the
+spaceKδ+1; we can assert that infinite points of the set {xni:i∈ω}such
+that the limit and hence the supremum of their levels is equal to η < αare
+inUy. We denote this set of points by S′′⊆S′; we know that property S.5
+holds inKδ+1andthen fromthe injective sequence S′′it is possible to extract
+a sequence converging to a point of level η+1< α+1.
+Ifη=αthen let us choose again an injective subsequence S′= (xni)i∈ω⊆S
+in such a way that the levels of the points xniconverges upwards to α; the
+sequence S′converges to x∞, since its points fall eventually in every neigh-
+borhood of x∞. /square
+35Now we are able to prove the sequentiality of Kα+1.
+Lemma 6.4 Kα+1is sequential.
+PROOF. Let us begin by proving that Bα+1=Kα+1\{x∞}is sequential,
+i.e. by showing that if Fis a sequentially closed subset of Bα+1then it is
+closed. Let us suppose that Fis sequentially closed and let us show that for
+everyx∈Bα+1\Fthere exists an elementary neighborhood ˆUxofxsuch that
+ˆUx⊆Bα+1\F. Ifx∈Bα+1\F, then there exists an open neighborhood of x,
+Ux⊆Bα+1with the peculiarity that Ux≃Kβ+1(withβ+1< α+1) which
+is a compact sequential space. Notice that x /∈F∩Ux; ifF∩Ux=∅, then
+Uxis an elementary neighborhood containing xsuch that Ux⊆Bα+1\F. If
+insteadF∩Ux/\e}a⊔io\slash=∅, sinceFis sequentially closed in Bα+1, thenF∩Uxis
+sequentially closed in Ux(otherwise, if F∩Uxis not sequentially closed in
+Ux, hence we have a sequence in F∩Uxwith its limit point in Ux\F; we
+can see this sequence as a sequence in Fwith its limit point out of Fand
+thenFis not sequentially closed against the hypothesis). It follows that
+F∩Uxis closed in UxsinceUxis sequential and hence it is compact; let us
+consider the open cover of F∩Uxformed by elementary neighborhoods of
+points in F∩Uxnot containing x. From this open cover it is possible to
+extract a finite subcover/uniontextn
+i=1Uyi⊇F∩Ux. ThenˆUx=Ux\/uniontextn
+i=1Uyiis an
+open neighborhood of xwhich has empty intersection with F. Thus we can
+conclude that Bα+1is sequential.
+Now we have still to demonstrate that, if Fis sequentially closed in Kα+1
+andx∞/∈F, thenx∞/∈F. Towards a contradiction, suppose that x∞/∈F
+and, at the same time, x∞∈F. Sincex∞/∈Fthen either Fis finite (and
+in this case the point x∞/∈Fagainst the hypothesis) or Fis infinite and in
+this second case from Fit is not possible to extract any injective sequence of
+pointswithnondecreasing levels such thatthesupremum oftheleve ls isequal
+toα; indeed if such a sequence existed, by Remark 6.3 from this sequenc e it
+would possible toextract a subsequence converging to x∞andthen x∞would
+stay inF(sinceFis sequentially closed) against the hypothesis. Now if α
+is a successor ordinal, there exists at most a finite number of points of level
+α=γ0inFthat we call z1,z2,...,z m; let us consider an elementary neigh-
+borhood Uzifor each of these points and let us set G1=F\/uniontextm
+i=1Uzi⊆F.
+We assert that either G1is finite (and in this case it turns out that x∞/∈F
+against the hypothesis) or G1is infinite and in this second case from G1it is
+not possible to extract any injective sequence of points with nonde creasing
+36levels such that the supremum of the levels is equal to α−1 =γ1; indeed if
+such a sequence existed, by Remark 6.3 from this sequence it would p ossi-
+ble to extract a subsequence converging to a point of level αdifferent from
+z1,z2,...,z mand then also this point would stay in Fagainst our assump-
+tion. If instead αis a limit ordinal, it is not true that for every γ∈αthere
+existsx∈Fsuch that l(x)> γ+1 (otherwise x∞∈Fwhich is sequentially
+closed) and hence there exists an index γ∈αsuch that for every x∈Fit
+turns out that l(x)≤γ+1< α. Therefore we can assert that in Fthere are
+at most a finite number of elements of level γ+1 that we call y1,y2,...,y k:
+indeed if we had an infinite number of these points, it would possible to e x-
+tract a subsequence converging to a point of level ( γ+1)+1 and this point
+would stay again in Fbut this is against what we have just remarked. Let us
+consider an elementary neighborhood Uyifor each of these points and let us
+callG1=F\/uniontextk
+i=1Uyi⊆F. We can say that either G1is finite (and in this
+case the point x∞/∈F) orG1is infinite and in this second case from G1it is
+not possible to extract any injective sequence of points with nonde creasing
+levels such that the supremum of the levels is equal to γ1=γ < α; indeed if
+such a sequence existed, by Remark 6.3 from this sequence it would b e pos-
+sible to extract a subsequence converging to a point of level γ+ 1 different
+fromy1,y2,...,y kand then also this point would stay in Fagainst what we
+have assumed.
+In each case we have constructed a sequentially closed subset G1from which
+it is not possible to extract any injective sequence of points with non decreas-
+ing levels such that the supremum of the levels is equal to γ1< α; moreover
+G1is the complement of a finite number of elementary neighborhoods in F.
+Then it is possible to repeat the procedure and to find step by step a decreas-
+ing sequence of ordinals γ0> γ1> γ2> ... > γ n> ...and corresponding
+subsetsG1⊇G2⊇...⊇Gn⊇.... This sequence has to be finite and then
+we find a finite set Gnafter a finite number of steps. Trivially we can cover
+Gnby a finite number of elementary neighborhoods; moreover Gnhas been
+constructed as complement of a finite number of elementary neighb orhoods
+inF. Then it turns out that it is possible to cover Fwith finitely many
+elementary neighborhoods of points of level smaller than α+1 and hence it
+follows that x∞/∈F. A contradiction. /square
+Now we want to show that every point in Kα+1belongs to the closure of
+L0, i.e that the set L0is dense in Kα+1.
+37Remark 6.5 For every x∈Kα+1it holds that x∈L0, i.e.L0=Kα+1.
+PROOF. Let xbe a point in Kα+1withl(x) =β+ 1< α+ 1 and let V
+be a non-empty neighborhood of x; then there exists an open elementary
+neighborhood Ux≃Kβ+1⊆Vand it turns out that j−1
+α+1(Ux) is a non-empty
+open subset in βω. Therefore there exists a free or a fixed ultrafilter Usuch
+thatU ∈j−1
+α+1(Ux). IfUis fixed we trivially finish; if Uis a free ultrafilter,
+sincej−1
+α+1(Ux) is an open subset, there is an infinite subset U′ofωwith
+U′∈ Usuch that ( U′)∗∪U′⊆j−1
+α+1(Ux); thenU′⊆ω(with|U′|=ω) is such
+thatU′⊆j−1
+α+1(Ux) and hence W=jα+1(U′)⊆Ux; we can conclude that
+Ux∩L0⊇W∩L0/\e}a⊔io\slash=∅.
+Now let us consider x∞∈Kα+1and letUbe an open neighborhood of x∞.
+ByLemma 5.5 thereexists anopensubset Ax∞=Kα+1\/uniontextUx⊆U; therefore
+j−1
+α+1(Ax∞) is a non-empty open subset of βωand hence we can proceed as
+above. /square
+Since we have proved that the space Kα+1is sequential and that L0=
+Kα+1, Remark 6.2 allows us to conclude that the level of each point is larger
+or equal to its order of sequentiality with respect to L0. We have to prove a
+last remark before concluding that the level of each point is exactly equal to
+its sequential order with respect to the set L0.
+Remark 6.6 LetAbe a closed subset in/uniontext
+γ+1≤ηLγ+1withη≤α. Then it
+follows that A∩/uniontext
+γ+1≤η+1Lγ+1=seqcl(A).
+PROOF. Since Ais closed in/uniontext
+γ+1≤ηLγ+1, thenAis sequentially closed in/uniontext
+γ+1≤ηLγ+1and hence there is no sequence in Aconverging to some point
+of/uniontext
+γ+1≤ηLγ+1\A. Notice that by Remark 6.2 it turns out that seqcl(A)⊆
+A∩/uniontext
+γ+1≤η+1Lγ+1. Wewanttoprovethat A∩/uniontext
+γ+1≤η+1Lγ+1⊆seqcl(A). Let
+xbe a point in ( A\A)∩(/uniontext
+γ+1≤η+1Lγ+1); trivially it holds that l(x) =η+1.
+SinceKα+1is sequential and x∈A, it turns out that there exists an index
+β∈ω1such that x∈seqclβ(A); we state that β= 1. Towards a contradic-
+tion, let us suppose that x /∈seqcl1(A), i.e. let us suppose that no sequence
+inAconverges to x. Thenxis the limit point of a sequence whose elements
+are in some sequential closure of Aand not in A, i.e.xis the limit point
+of a sequence ( yβi+1)i∈ωwith sup {βi+ 1} ≥η+ 1 but this is absurd since
+l(x) =η+1: indeed if it was correct, in Kα+1there would exist a sequence
+(yβi+1)i∈ωwith sup{βi+1} /\e}a⊔io\slash=ηconverging to a point of level η+1 and this
+38is inconsistent with Remark 6.2. /square
+Finally we can prove the following crucial lemma.
+Lemma 6.7 InKα+1the order of sequentiality of a point of level β+1with
+respect to L0isβ+1andKα+1is a space with sequential order α+1.
+PROOF. Notice that the points of level 0 and 1 have sequential orde r respec-
+tively 0 and 1 with respect to the set L0. Now consider the set/uniontext
+γ+1≤ηLγ+1
+withη≤α; it complies with the hypotheses of Remark 6.6 since it is closed
+in/uniontext
+γ+1≤ηLγ+1and hence it holds that
+/uniondisplay
+γ+1≤ηLγ+1/intersectiondisplay /uniondisplay
+γ+1≤η+1Lγ+1=
+{y∈/uniondisplay
+γ+1≤η+1Lγ+1:∀Uy,(Uy∩/uniondisplay
+γ+1≤ηLγ+1)/\e}a⊔io\slash=∅}=
+/uniondisplay
+γ+1≤η+1Lγ+1=seqcl(/uniondisplay
+γ+1≤ηLγ+1).
+This result together with Remark 6.5 allows us to conclude that the lev el of
+each point is smaller or equal to its order of sequentiality with respec t to the
+setL0. But we have already remarked that the level of each point is larger
+or equal to its order of sequentiality with respect to the set L0and hence
+we can conclude that the level of each point is exactly equal to its or der of
+sequentiality with respect to the set L0.
+Then the space Kα+1has sequential order equal to α+1, since x∞∈L0and
+x∞has sequential order equal to α+1 with respect to L0. /square
+Let us finally check that properties S.1 toS.6 hold in the space Kα+1.
+S.1 The space Kα+1can be uniquely represented in the form of
+Kα+1=L0/unionsqdisplay
+(/unionsqdisplay
+γ≤αLγ+1).
+The points of level γ+ 1 with γ∈[0,α], i.e. the points belonging to
+the setLγ+1, have sequential order equal to γ+1 with respect to L0:
+see Lemma 6.7.
+39S.2 The set Lα+1consists of the unique point x∞.
+S.3 Every point in Kα+1of nonzero level has a basis formed by clopen
+subsets called elementary; if Uis an elementary neighborhood of a
+point of level γ+1, then the relation ≈α+1restricted to /tildewideU=j−1
+α+1(U)
+produces a compact space homeomorphic to Kγ+1: for the points of
+level smaller than α+ 1, S.3 is true by inductive hypothesis while for
+x∞the property is correct since each of its elementary neighborhood is
+homeomorphic to the whole space Kα+1(see Lemma 5.5).
+S.4 For every γ≤α, if a nonconstant sequence ( xn)n∈ωof points xn∈
+Lγn+1, with nondecreasing levels, converges to a point x∈Lγ+1, then
+for the sequence ( γn+1)n∈ωof ordinal numbers it holds that sup {γn+
+1}=γ: see Remark 6.2.
+S.5 For every γ≤α, from every injective sequence ( xn)n∈ωof points xn∈
+Lγn+1with nondecreasing levels such that supn∈ω{γn+ 1}=γ, it is
+possible to extract a subsequence converging to a point of level γ+1:
+see Remark 6.3.
+S.6 If{Ni}i∈ωis a countable family of pairwise disjoint infinite subsets Ni
+ofωand if it holds that for every i∈ωa relation of type βi+ 1 is
+given on Niin such a way that the sequence of ordinals ( βi+1)i∈ωis
+not decreasing and sup {βi+1}=α, then it is possible to extend the
+relation obtained on/uniontext∞
+i=1Nito a relation of βωof typeα+1: we have
+just constructed it.
+Remark 6.8 Notice that every Baˇ skirov’s space of sequential order a su c-
+cessor ordinal is a scattered space such that the sequential order of each point
+is equal to its scattering level.
+Finally we can state the following theorem.
+Theorem 6.9 (CH) Let αbe any ordinal less than or equal to ω1. There
+exists a compact sequential Hausdorff quotient space of βωwith sequential
+orderα.
+Acknowledgments. I wish to thank Gino Tironi, Professor at the University
+of Trieste, for giving me the opportunity to work on this challenging t opic,
+raising my interest in topology and for patiently answering all my ques tions.
+40I am grateful to Camillo Costantini, Assistant Professor at the Univ ersity of
+Torino, for helping me to get more familiarity with some techniques whic h
+are of use to investigate the notions related to my research field.
+References
+[1] A. V. Arhangel’skii andS. P. Franklin, Ordinal invariants for topological
+spaces, Michigan Math. J. 15(1968), 313–320.
+[2] Zolt´ an T. Balogh, On compact Hausdorff spaces of countable tightness ,
+Proc. Amer. MAth. Soc., 105(3)(1989), 755–764.
+[3] A. I. Baˇ skirov, The classification of quotient maps and sequential bicom-
+pacta, Soviet Math. Dokl. 15(1974), 1104–1109.
+[4] Alan Dow, On MAD families and sequential order ,
+http://www.math.uncc.edu/ ˜adow/Others.htmal under the title “On
+the sequential order of Compact spaces”.
+[5] Alan Dow, Sequential order under MA , TopologyAppl. 146-147 (2005),
+501–510.
+[6] Ryszard Engelking, General Topology , Translated fromthe Polish by the
+author.2nded. SigmaSeriesinPureMathematics, 6.Berlin: Helderm an
+Verlag, 1989.
+[7] S.P. Franklin, Spaces in which sequences suffice , Fund. Math. 57(1965),
+107–115.
+[8] S.P. Franklin, Spaces in which sequences suffice. II , Fund. Math. 61
+(1967), 51–56.
+[9] V. Kannan, Ordinal invariants in topology II. Sequential order of com-
+pactifications , Compositio Math. 39(2)(1979), 247–262.
+[10] V. Kannan, Ordinal invariants in topology III. Simplest compact space s
+for sequential order , Math. Today 1(1983), 65-80.
+41[11] JanvanMill, An introduction to βω, InHandbookofset-theoretic topol-
+ogy, North- Holland, Amsterdam, (1984), 503-567.
+[12] Walter Rudin, Homogeneity problems in the thoery of ˇCech compactifi-
+cations, Duke Math. J 23(1956), 409-419.
+42
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+arXiv:1001.0910v2  [math.AP]  6 Jun 2011Partial Differential Equations
+Barenblatt profiles for a nonlocal porous medium equation
+Piotr Bilera, Cyril Imbertb, Grzegorz Karcha
+aInstytut Matematyczny, Uniwersytet Wroc/suppress lawski, pl. Grun waldzki 2/4, 50–384 Wroc/suppress law, Poland
+bUniversit´ e Paris-Dauphine, CEREMADE (UMR CNRS 7534), Pla ce de Lattre de Tassigny, 75775 Paris Cedex 16, France
+Received ... ; accepted after revision ... .
+Presented by ...
+Abstract
+We study a generalization of the porous medium equation invo lving nonlocal terms. More precisely, explicit self-
+similar solutions with compact support generalizing the Ba renblatt solutions are constructed. We also present a
+formal argument to get the Lpdecay of weak solutions of the corresponding Cauchy problem .
+R´ esum´ e
+Solutions auto-similaires pour une ´ equation des milieux p oreux non locale
+Cettenoteestconsacr´ ee` al’´ etuded’uneg´ en´ eralisati on nonlocale del’´ equationdesmilieuxporeux.Pluspr´ eci s´ ement,
+on obtient des formules explicites de solutions auto-simil aires ` a support compact qui ressemblent fortement aux solu -
+tions de type Barenblatt. On donne aussi un argument formel q ui permet d’obtenir des estimations Lpdes solutions
+faibles du probl` eme de Cauchy.
+Version fran¸ caise abr´ eg´ ee
+Nous consid´ erons le probl` eme de Cauchy pour l’´ equation non loc ale suivante
+∂tu−∇·/parenleftbig
+u∇α−1/parenleftbig
+|u|m−1/parenrightbig/parenrightbig
+= 0, (1)
+avecm >1,x∈Rd,α∈(0,2),t >0, ` a laquelle on ajoute une condition initiale u(0,x) =u0(x). Ici∇βest
+un op´ erateur int´ egral singulier g´ en´ eralisant le gradient usuel (β= 1) et li´ e au laplacien fractionnaire.
+Le r´ esultatprincipal de cette note sont des formules explicites de solutions auto-similairesqui se propagent
+` a une vitesse finie.
+Th´ eor` eme 1 (Solutions auto-similaires)
+La fonction
+u(t,x) =Ct−d
+d(m−1)+α/parenleftbigg/parenleftBig
+R2−|x|2t−2
+d(m−1)+α/parenrightBigα
+2
++/parenrightbigg1
+m−1
+(ainsi que ses translations en x) est une solution auto-similaire de l’´ equation (1) pour R >0quelconque et
+une constante C >0convenable.
+Email addresses: biler@math.uni.wroc.pl (Piotr Biler), imbert@ceremade.dauphine.fr (Cyril Imbert),
+karch@math.uni.wroc.pl (Grzegorz Karch).
+Preprint submitted to the Acad´ emie des sciences 26 octobre 2018Ensuite, nous pr´ esentons un calcul formel qui permet d’obtenir des estimations en norme Lp(Rd) des
+solutions faibles du probl` eme de Cauchy, en particulier celles const ruites par Caffarelli et V´ azquez [3] dans
+le casm= 2 et telle que |u0(x)| ≤Ce−c|x|pour deux constantes Cetc.
+Calcul formel 1 (Asymptotique du probl` eme de Cauchy)
+´Etant donn´ ee une fonction 0≤u0∈L1(Rd)∩L∞(Rd), les normes Lp(Rd),1≤p <∞, des solutions faibles
+utendent vers 0quandt→ ∞avec le taux alg´ ebrique suivant
+/ba∇dblu(t)/ba∇dblp≤C(d,α,m,p )/ba∇dblu0/ba∇dblα+d(m−1)/p
+α+d(m−1)
+1 t−d
+d(m−1)+α(1−1/p)for allt >0.
+1. Introduction
+We study a nonlocal generalization of the porous medium equation
+∂tu−∇·/parenleftbig
+u∇α−1/parenleftbig
+|u|m−1/parenrightbig/parenrightbig
+= 0, (1)
+wherem >1,α∈(0,2),x∈Rd,t >0, supplemented with an initial condition
+u(0,x) =u0(x). (2)
+The pseudodifferential (vector-valued) operator ∇βin (1) is defined via the Fourier transform as ∇βu=
+F−1(iξ|ξ|β−1Fu). This definition is consistent with the usual gradient: ∇1=∇; the components of ∇0
+are the Riesz transforms; moreover we have ∇ · ∇α−1=∇α
+2· ∇α
+2=−(−∆)α
+2, where ( −∆)α
+2denotes the
+fractional Laplace operator: ( −∆)α
+2u=F−1(|ξ|αFu). It can also be defined by real analysis tools as follows
+∇α−1u=∇I2−αu, whereIβforβ∈(0,d) is an integral smoothing operator, called the Riesz potential (see
+[12, Ch. V])
+Iβ(u)(x) =−Cβ/integraldisplayu(x+z)
+|z|d−βdz
+with some Cβ>0. Note that then ∇α−1u(x) =∇I2−α(u)(x) = (d+α−2)C2−α/integraltext
+(u(x+z)−u(x))z
+|z|d+αdz,
+α∈(0,2).
+Eq. (1) can be interpreted as a transport equation of the type ∂tu=∇·(uv) for some velocity vectorfield
+vwhich is a potential; more precisely, v=∇pwherep=I2−α(|u|m−1). It can be interpreted as a nonlocal
+pressure in the case of nonnegative initial data. Then, of course ( −∆)2−α
+2p=|u|m−1, see [3], [4] for that
+notation.
+Notice that for α= 2 and nonnegative initial data we recover the Boussinesq equation (m= 2), and the
+usual porous media equation ( m >1):∂tu=∇·/parenleftbig
+u∇/parenleftbig
+um−1/parenrightbig/parenrightbig
+,t >0,x∈Rd.
+Recently, L. Caffarelli and J. L. V´ azquez ([3], [4]) studied equation ( 1) in the case m= 2. They proved
+the existence of weak solutions for nonnegative bounded integrab le initial data with exponential decay at
+infinity. They also treat the case of bounded and compactly suppor ted initial data, which propagate with
+finite speed. It is shown in [3] that self-similar solutions can be constr ucted by considering an obstacle
+problem for the fractional Laplace operator. In this note, we con tribute to those results constructing explicit
+compactly supported self-similar solutions . Moreover, we show a kind of hypercontractivity estimates, i.e.
+the optimal decay in Lpof general solutions with u0∈L1(Rd).
+Results on equation (1) in this note are multidimensional generalizatio ns of those obtained in [1] for
+a model of the dynamics of dislocations in crystals (for the integral ofuwhend= 1 and m= 2). The
+structure of (1) suggests that it should enjoy the conservation of mass and some comparison properties as
+was shown in [1]. For an analysis of a related nonlocal equation, see [7].
+Complete proofs of all results announced in this note will be published in [2].
+2. Self-similar solutions
+The equation (1) has the following scaling property:
+2ifu(t,x) is a solution ,so isℓdλu(ℓt,ℓλx)
+for each ℓ >0 andλ=1
+d(m−1)+α. We look for nonnegative solutions that are invariant under that sc aling,
+i.e.of the following form
+u(t,x) =1
+tdλΦα,m/parenleftBigx
+tλ/parenrightBig
+,whereλ=1
+d(m−1)+α, (3)
+for a function Φ α,m:Rd→R+satisfying the following nonlinear and nonlocal equation in Rd
+−λ∇·(yΦα,m) =∇·/parenleftbig
+Φα,m∇α−1Φm−1
+α,m/parenrightbig
+. (4)
+Before stating our main result, we recall the definition of weak solut ions for (1) introduced in [3] in the case
+m= 2.
+Definition 2.1 (Weak solutions) A function u: (0,T)×Rd→Ris aweak solution of (1) in QT=
+(0,T)×Rdsubmittedto the initial condition u(0,x) =u0(x)ifu∈L1(QT),I2−α(|u|m−1)∈L1(0,T;W1,1
+loc(Rd)),
+u∇I2−α(|u|m−1)∈L1(QT)and
+/integraldisplay /integraldisplay
+u(ϕt−∇I2−α(|u|m−1)·∇ϕ)dxdt+/integraldisplay
+u0(x)ϕ(0,x)dx= 0 (5)
+for each test function ϕ∈C1(QT)such that ϕhas compact support in the space variable x, and vanishes
+neart=T.
+Theorem 2.2 (Self-similar solutions) For each α∈(0,2],m >1andR >0, the function
+Φα,m(y) =/parenleftBig
+k(R2−|y|2)α
+2
++/parenrightBig1
+m−1withk=/parenleftbiggd
+d(m−1)+α/parenrightbigg/parenleftBigg
+Γ/parenleftbigd
+2/parenrightbig
+2αΓ/parenleftbig
+1+α
+2/parenrightbig
+Γ/parenleftbigd+α
+2/parenrightbig/parenrightBigg
+(6)
+is a solution of (4). Consequently, the function
+u(t,x) =t−d
+d(m−1)+α/parenleftbigg
+k/parenleftBig
+R2−|x|2t−2
+d(m−1)+α/parenrightBigα
+2
++/parenrightbigg1
+m−1
+(7)
+is a weak solution of (1), satisfies the equation in the pointw ise sense for |x| /negationslash=Rt1
+d(m−1)+α, and is
+min/braceleftBig
+α
+2(m−1),1/bracerightBig
+-H¨ older continuous at the interface |x|=Rt1
+d(m−1)+α.
+Whenα= 2, we recover the classical Kompaneets–Zel’dovich– Barenblatt –Pattleformulas, see, e.g., [13].
+Note also that given M >0 there exists a unique R >0 such that/integraltext
+Φα,m(y)dy=M.
+Remark 2.3 As mentioned above, self-similar solutions of (1 ) have been proved to exist in [3] by studying
+the following obstacle problem for the fractional Laplacia n
+P≥Φ, V= (−∆)α
+2P≥0,eitherP= Φ or V= 0,
+withα∈(0,2)andΦ(y) =C−a|y|2. The novelty of our approach is that we exhibit the explicit s olution of
+this obstacle problem: P(y) =Iα(Φα,2)(y), whereIαis the Riesz potential, and Φα,2is defined in (6) with
+m= 2and a suitable R >0.
+Those explicit self-similar solutions express one of the most importan t features of the porous medium
+equation: the property of finite propagation speed. In the case o f the classical porous medium equation
+(α= 2), this property is established using comparison with suitably large self-similar solutions, cf.[13].
+For the generalized porous medium equation (1) with m= 2, special supersolutions have been used for
+comparison, cf.[4].
+The proof of Theorem 2.2 is based on an application of the following fun damental technical fact.
+3Lemma 2.4 For allβ∈(0,2)andγ >0, we have
+Iβ/parenleftBig
+(1−|y|2)γ
+2
++/parenrightBig
+=
+
+Cγ,β,d×2F1/parenleftbiggd−β
+2,−γ+β
+2;d
+2;|y|2/parenrightbigg
+for|y| ≤1,
+˜Cγ,β,d|y|β−d×2F1/parenleftbiggd−β
+2,2−β
+2;d+γ
+2;1
+|y|2/parenrightbigg
+for|y| ≥1,(8)
+withCγ,β,d= 2−βΓ(γ
+2+1)Γ(d−β
+2)
+Γ(d
+2)Γ(β+γ
+2+1)and˜Cγ,β,d= 2−βΓ(γ
+2+1)Γ(d−β
+2)
+Γ(d
+4)Γ(d+γ
+2+1), where 2F1is the hypergeometric function.
+The verification of (8) consists in passing to Fourier transforms an d calculating certain integrals (the so
+called (Sonine–)Weber–Schafheitlin integrals) involving Bessel and h ypergeometric functions 2F1,cf.[11].
+Proof of Theorem 2.2. Letφα(y) =/parenleftbig
+1−|y|2/parenrightbigα
+2
++. Observethat φα∈L1(Rd). We next show that I2−α(φα)∈
+W1,1
+loc(Rd). By Lemma 2.4 with γ=α∈(0,2),β= 2−α, and2F1(a,−1;c;z) = 1−a
+cz, we get
+I2−α(φα)(y) =
+
+Cα,2−α,d/parenleftbigg
+1−d+α−2
+d|y|2/parenrightbigg
+if|y| ≤1,
+˜Cα,2−α,d|y|2−(d+α)2F1/parenleftbiggd+α
+2−1,α
+2;d+α
+2;1
+|y|2/parenrightbigg
+if|y| ≥1,
+which is a locally integrable function. Recalling that ∇α−1=∇I2−α, we then deduce by the chain rule that
+fory∈B1,∇α−1(φα)(y) =−(dKα,d)−1ywhereKα,dis defined in Corollary 2.5 below. For |y| ≥1, one
+uses∂
+∂z2F1(a,b;c;z) =ab
+c2F1(a+1,b+1;c+1;z), hence ∇α−1(φα) is locally integrable. To conclude, we
+remark that
+φ1
+m−1α(y)∇α−1φα(y) =−φ1
+m−1α(y)(Kα,dd)−1y,for ally∈Rd,
+becauseφα(y) = 0 for |y| ≥1.
+Hence, scaling the variables, we immediately obtain that the function Φα,mdefined in (6) satisfies
+∇α−1(Φm−1
+α,m) =−λyfor|y|< R. Now, for all y∈Rdthe identity
+−λyΦα,m= Φα,m∇α−1Φm−1
+α,m
+follows because Φ α,m(y) = 0 for all |y| ≥R, so (4) holds with λ=1
+d(m−1)+αandk=dλKα,ddefined in (6).
+It is straightforward to verify using (4) that ugiven by formula (7) is a weak solution of (1) in each strip
+(t0,T)×Rd, 0< t0< T <∞. Moreover, the H¨ older continuity is easy to check. /square
+The following known result (with an important probabilistic interpreta tion) proved by Getoor [6, Th.
+5.2], see also [9, App.] for a related calculation, is an immediate consequ ence of Lemma 2.4. For its proof, it
+suffices to use the relation ( −∆)α
+2= ∆I2−α.
+Corollary 2.5 For each α∈(0,2], the identity Kα,d(−∆)α
+2/parenleftBig/parenleftbig
+1−|y|2/parenrightbigα
+2
++/parenrightBig
+=−1 inB1holds with the
+explicit constant Kα,d=Γ(d
+2)
+2αΓ(1+α
+2)Γ(d+α
+2).
+3. The Cauchy problem and asymptotics
+We now briefly discuss the questions of the existence of weak solutio ns and their uniqueness in the case
+m= 2. In [1], viscosity solutions have been considered which permitted t he authors to prove regularity and
+uniqueness of solutions to (1) in one space dimension. In higher dimen sions, a construction of mild solutions
+is achieved in [3] through a parabolic regularization of (1) with a suitab le cutoff of the singular kernel of
+I2−α, and then a passage to the limit. The uniqueness of weak solutions an d,a fortiori , the validity of
+the full comparison principle seem to be difficult questions, cf.a discussion in [3]. Another construction of
+weak solutions to (1) with m >1 as limits of mild solutions of parabolically perturbed equation (1) will be
+published in [2].
+4Formal computation 3.1 (Decay of solutions for the Cauchy pr oblem) Suppose that for 0≤u0∈
+L1(Rd)∩L∞(Rd). ThenLp(Rd)norms (1≤p <∞) of any sufficiently regular and global in time nonnegative
+weak solution uof (1)–(2) such that/integraltext
+u(t,x)dx=/integraltext
+u0(x)dxdecay algebraically
+/ba∇dblu(t)/ba∇dblp≤C(d,α,m,p )/ba∇dblu0/ba∇dblα+d(m−1)/p
+α+d(m−1)
+1 t−d
+d(m−1)+α(1−1/p)for allt >0. (9)
+Our computation below can be applied, for example, to weak solutions constructed by Caffarelli and
+V´ azquez [3], who studied problem (1)–(2) with m= 2 and with initial conditions satisfying 0 ≤u0(x)≤
+Ce−c|x|for some constants c,Cand allx∈Rd.
+Formal proof of (9). In order to prove the announced Lpestimates of solutions (similar to those for degen-
+erated partial differential equations like the porous medium equatio n ine.g.[14], [5, Ch. 2]), we recall the
+Stroock–Varopoulos inequality for q≥1
+/integraldisplay
+|w|q−2w(−∆)α
+2wdx≥4(q−1)
+q2/integraldisplay/vextendsingle/vextendsingle/vextendsingle∇α
+2|w|q
+2/vextendsingle/vextendsingle/vextendsingle2
+dx (10)
+valid for each w∈Lq(Rd) such that ( −∆)α
+2w∈Lq(Rd). Note that the constant in (10) is the same as for
+the usual Laplacian operator −∆ (i.e.α= 2). The proof is given, e.g., in [10, Prop. 1.6] and [10, Th. 2.1
+combined with (1.7)].
+We will also need the following Nash inequality
+/ba∇dblv/ba∇dbl2(1+α
+d)
+2≤CN/ba∇dbl∇α
+2v/ba∇dbl2
+2/ba∇dblv/ba∇dbl2α
+d
+1 (11)
+valid for all functions vwithv∈L1(Rd),∇α
+2v∈L2(Rd) with a constant CN=C(d,α). The proof of (11)
+ford= 1 can be found in, e.g., [8, Lemma 2.2], and this extends easily to the general case d≥1.
+Moreover, we will need the Gagliardo–Nirenberg type inequality: for p >1 andp≥m−1,
+/ba∇dblu/ba∇dbla
+p≤CN/vextenddouble/vextenddouble∇α
+2|u|r
+2/vextenddouble/vextenddouble2
+2/ba∇dblu/ba∇dblb
+1 (12)
+witha=p
+p−1d(r−1)+α
+dandb=pα+d(m−1)
+d(p−1)andr=p+m−1. This inequality is a consequence of the Nash
+inequality (11) written for v=|u|r
+2,i.e./ba∇dblu/ba∇dblr(1+α
+d)
+r≤CN/vextenddouble/vextenddouble∇α
+2|u|r
+2/vextenddouble/vextenddouble2
+2/ba∇dblu/ba∇dblrα
+dr
+2, and two H¨ older inequalities for
+theLqnorms:/ba∇dblu/ba∇dblp≤ /ba∇dblu/ba∇dblγ
+r/ba∇dblu/ba∇dbl1−γ
+1and/ba∇dblu/ba∇dblr
+2≤ /ba∇dblu/ba∇dblδ
+p/ba∇dblu/ba∇dbl1−δ
+1,which hold with γ=p−1
+r−1r
+pandδ=r−2
+p−1p
+r.
+Combining the above three inequalities, we get (12).
+The computation, which lead to (9), consists in getting a differential inequality of the formd
+dt/integraltext
+|u|pdx≤
+−K/ba∇dblu/ba∇dbla
+p/ba∇dblu/ba∇dbl−b
+1for some positive constant Kand where aandbappear in (12).
+Multiply (1) by up−1withp >1, integrate by parts, and use the relation ∇·∇α−1=−(−∆)α
+2to get
+1
+pd
+dt/integraldisplay
+updx=−(p−1)/integraldisplay
+up−1∇u·∇α−1(um−1)dx
+=−p−1
+p/integraldisplay
+up(−∆)α
+2um−1dx
+≤−4(p−1)(m−1)
+(p+m−1)2/vextenddouble/vextenddouble/vextenddouble∇α
+2/parenleftBig
+up+m−1
+2/parenrightBig/vextenddouble/vextenddouble/vextenddouble2
+2
+after applying the Stroock–Varopoulos inequality (10) with w=um−1andq=p
+m−1+1. To estimate the
+right hand side of the above inequality, we use (12). We finally have:
+d
+dt/integraldisplay
+|u|pdx≤ −K/ba∇dblu/ba∇dbla
+p/ba∇dblu/ba∇dbl−b
+1
+5withK=1
+CN4p(p−1)(m−1)
+(p+m−1)2. The above inequality leads to the differential inequalityd
+dtf(t)≤ −KM−bf(t)a
+p
+for the function f(t) =/ba∇dblu(t)/ba∇dblp
+p,M=/ba∇dblu0/ba∇dbl1, anda/p >1, which immediately gives the algebraic decay of
+theLpnorms for p≥m−1:f(t)≤/parenleftBig
+K/parenleftBig
+a
+p−1/parenrightBig
+M−bt/parenrightBig−1a
+p−1. Finally, we obtain the desired estimate with
+C(d,α,m,p ) =/parenleftbigg
+K/parenleftbigga
+p−1/parenrightbigg/parenrightbigg−1
+a−p
+=/bracketleftbigg4(m−1)(d(m−1)+α)
+CNd×p
+(p+m−1)2/bracketrightbigg−d
+d(m−1)+α(1−1
+p)
+.
+Ofcourse,this issufficienttogetthe conclusionofTheorem3.1fore ach1≤p <∞since1
+a
+p−1=d(p−1)
+d(m−1)+α,
+and interpolating between L1(Rd) andLp(Rd) withpsufficiently large. /square
+Remark 3.2 In our recent work [2], we present a more subtle ite rative argument, which allows us to show
+that also
+/ba∇dblu(t)/ba∇dbl∞≤C/ba∇dblu0/ba∇dblα
+d(m−1)+α
+1t−d
+d(m−1)+αfor allt >0.
+Acknowledgments. The preparation of this paper was supported by the Polish Ministry o f Science
+(MNSzW) grant N201 418839, and a PHC POLONIUM project 0185, 2 009–2010, and the Foundation
+for Polish Science operated within the Innovative Economy Operatio nal Programme 2007–2013 funded by
+European Regional Development Fund (Ph.D. Programme: Mathema tical Methods in Natural Sciences).
+The authors wish to thank Juan Luis V´ azquez, R´ egis Monneau and Jean Dolbeault for fruitful discussions
+they had together.
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+[14] J. L. V´ azquez, Smoothing and Decay Estimates for Nonli near Diffusion Equations. Equations of Porous Medium Type,
+Oxford Lecture Series in Mathematics and its Applications 3 3, Oxford University Press, Oxford, (2006).
+6
\ No newline at end of file
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+arXiv:1001.0911v1  [math.GT]  6 Jan 2010Conjugations on 6-manifolds with free integral cohomology
+Martin Olbermann
+November 4, 2018
+Abstract
+In this article, we show the existence of conjugations on many simply -connected spin
+6-manifolds with free integral cohomology. In a certain class the on ly condition on X6
+to admit a conjugation with fixed point set M3is the obvious one: the existence of a
+degree-halving ring isomorphism between the Z2-cohomologies of XandM.
+1 Introduction
+A conjugation space is a space together with an involution sa tisfying a certain cohomological
+pattern. The definition of a conjugation space [ HHP05] has its origin in the observation that
+there are many examples of involutions with this cohomologi cal pattern, e.g. the complex
+conjugation on the projective space CPnand on complex Grassmannians, natural involutions
+on smooth toric manifolds [ DJ91] and on polygon spaces [ HK98]. There is always a degree-
+halving isomorphism κbetween the Z2-cohomologies of the space and its fixed point set.
+Hausmann, Holm and Puppe [ HHP05] also discovered that in all these examples there is an
+even richer structure in the equivariant cohomology of the s pace itself and the space of fixed
+points of the involution. This extra structure is also part o f the definition of a conjugation
+space. Their main examples of conjugation spaces are conjug ation complexes: every cell
+complex with the property that each cell is a unit disk in Cnwith complex conjugation,
+and with equivariant attaching maps, is a conjugation space . They also prove that coadjoint
+orbits of semi-simple Lie groups with the Chevalley involut ion are conjugation spaces. One
+can form connected sums of manifolds with conjugations to ob tain new conjugation spaces.
+Moreover, there are several bundle constructions yielding conjugations on the total space,
+provided conjugations on base and fiber.
+As a general assumption, we are interested in smooth conjuga tions on connected, closed,
+smooth manifolds. These have to be of even dimension. A point with the trivial involution
+is a conjugation space. In dimension 2, only the sphere admit s conjugations. Dimension 4
+is special since the quotient space of the involution can be g iven a smooth structure [ HH09].
+Hambleton and Hausmann prove that equivariant diffeomorphis m classes of oriented con-
+nected conjugation 4-manifolds correspond to diffeomorphis m classes of pairs ( X,Σ), where
+Xis an oriented 4-dimensional Z2-homology sphere and Σ is a closed connected subsurface
+ofX. Hence the study of conjugations is “reduced” to a difficult no n-equivariant problem:
+a generalization of four-dimensional knot theory. Another result by Hambleton and Haus-
+mann is that a simply-connected spin 4-manifold admitting a conjugation is homeomorphic
+1a connected sum of S2×S2’s. Thus there are for example no smooth conjugations on the
+Kummer surface, although its Z2-cohomology ring is the same as the one of a connected sum
+ofS2×S2’s.
+The problem of realizing 3-manifolds as fixed point sets of co njugations was considered in
+[Olb07]. The present paper focuses on the identification of the resu lting conjugation space. It
+shows that for a certain class of 6-manifolds, the obvious co ndition on X6to admit a conju-
+gation with fixed point set a given M3(namely the existence of a degree-halving isomorphism
+between the Z2-cohomologies) is the only condition.
+We focus on a class of 6-manifolds which has been considered i n connection with the
+question of the existence of simply-connected asymmetric m anifolds. In the smooth category,
+amanifold Xis called asymmetricif every (smooth) action of afinite(or e quivalently compact
+Lie) groupon Xis trivial. Oneof theproblemsin findingsimply-connected c losed asymmetric
+manifolds is that every explicit construction of a manifold seems to produce a non-trivial
+symmetry group. Thus we should be interested in more implici t descriptions of the manifolds
+under consideration. For example we could use abstract clas sification results as the following,
+proved by Wall.
+Theorem 1.1 ([ Wal66])Diffeomorphism classes of six-dimensional manifolds which are
+simply-connected, spin, have free integer cohomology, and zero odd cohomology, correspond
+bijectively to isomorphism classes of the following invari ants:
+1. the rank mof the second cohomology with integer coefficients H2,
+2. a trilinear form µonH2corresponding to the evaluation of triple cup products on th e
+fundamental class,
+3. and a linear form P1onH2corresponding to the Poincar´ e dual of the first Pontrjagin
+class,
+subject to the conditions:
+µ(x,x,y)≡µ(x,y,y)(mod 2) for all x,y∈H2,
+P1(x)≡4µ(x,x,x)(mod 24) for all x∈H2.
+V. Puppeconsidered the class of such manifolds andproved ce rtain asymmetry statements
+for manifolds in this class [ Pup07]. In particular Puppe proved the following theorem which
+gives a relation between conjugations and asymmetric manif olds.
+Theorem 1.2 ([ Pup07])Some simply-connected spin 6-manifolds do not admit a non-t rivial
+orientation-preserving Zp-action. Moreover, they are either asymmetric, or conjugat ion
+spaces.
+In our construction [ Olb07] of 6-dimensional manifolds with conjugations we (necessa rily)
+obtain manifolds whose Z2-cohomology is concentrated in even degrees. This implies t hat for
+integral cohomology, no torsion of even order is possible. O dd torsion might occur, however:
+2in the last step of the construction (surgery in the middle di mension), we only control the
+Z2-cohomology of the result, since we find a hyperbolic summand only inZ2-cohomology. In
+the first part of this article, a more detailed analysis of the intersection form in the middle
+dimension leads to a better choice for the surgery in the midd le dimension. (In particular we
+use information given by universal coefficient spectral sequ ences.) From this we deduce our
+first result:
+Theorem 1.3 Every closed orientable 3-dimensional manifold such that H1(M;Z)contains
+no 2-primary torsion can be realized as fixed point space of a s mooth conjugation on a closed
+simply-connected spin 6-manifold with free integral and ze ro odd cohomology.
+(Under the conjugation space isomorphism κofZ2-cohomology algebras dividing the degree
+by two, the second Wu class of the 6-manifold will be mapped to the first Wu class of its
+fixed point set, since we have κ(Sq2k(x)) =Sqk(κ(x)), see [FP05]. This implies that only
+orientable 3-manifolds may occur as fixed point sets of conju gations on such 6-manifolds.)
+In the second part, we analyze which 6-manifolds are produce d by our construction. The
+main result of the article is
+Theorem 1.4 LetXbe a closed simply connected spin 6-manifold with free and on ly even
+integral cohomology. Let Mbe a closed oriented 3-manifold such that H1(M;Z)contains no
+2-primary torsion, and such that there exists a ring isomorp hismH∗(X;Z2)→H∗(M;Z2)
+dividing all degrees by 2. Then there exists a smooth conjugat ion onXwith fixed point set
+M.
+From this we deduce:
+Corollary 1.5 Every closed simply connected spin 6-manifold with free and only even integral
+cohomology, such that x2is divisible by 2 for all x∈H2(M;Z)admits a conjugation. In
+particular the examples from [ Pup07] are not asymmetric, but do admit conjugations.
+Remark 1.6 This corrects the wrong statement in [ Kre09a] that Puppe’s examples are asym-
+metric. In order to modify his argument such that it gives asym metric manifolds, Kreck moved
+to the non-smooth category [ Kre09b].
+AnotherimmediatecorollaryisapartialconversetoP.A.Sm ith’stheoremaboutgroupactions
+on spheres.
+Corollary 1.7 EveryZ2-homology 3-sphere is the fixed point set of an involution (a c onju-
+gation) on S6.
+In a forthcoming paper [ Olb10], we show that our surgery approach to conjugations, and
+the computation of the transfer map, can be used not only for e xistence questions, but are
+also suited to give classification results. In particular we classify all involutions on S6with
+three-dimensional fixed point set.
+The obvious open question is whether in the main theorem 1.4the condition that x2
+is divisible by 2 for all x∈H2(M;Z) is necessary. An example where the condition is
+3violated is that every homotopy projective CP3admits a conjugation with fixed point set
+RP3. (Dovermann, Masuda and Schultz show the existence of such i nvolutions [ DMS86], and
+all these involutions must be conjugations.)
+Acknowledgements. The author was partially supported by a research fellowship of the
+DFG. We would like to thank Diarmuid Crowley, Ian Hambleton, Jean-Claude Hausmann,
+Matthias Kreck and Arturo Prat-Waldron for many helpful dis cussions and remarks.
+2 Construction of conjugation 6-manifolds with free integr al
+cohomology
+2.1 Conjugations on manifolds
+A conjugation on a topological space Xis an involution τ:X→X, which we consider as an
+action of the group Z2∼=C={id,τ}onX, and which satisfies the following cohomological
+pattern: We denote the Borel equivariant cohomology of XbyH∗
+C(X;Z2). It is a module
+overH∗
+C(pt;Z2) =Z2[u]. The restriction maps in equivariant cohomology are denot ed by
+ρ:H∗
+C(X;Z2)→H∗(X;Z2) andr:H∗
+C(X;Z2)→H∗
+C(Xτ;Z2)∼=H∗(Xτ;Z2)[u].
+Definition 2.1 Xis a conjugation space if
+•Hodd(X;Z2) = 0,
+•there exists a (ring) isomorphism κ:H2∗(X;Z2)→H∗(Xτ;Z2)
+•and a (multiplicative) section σ:H∗(X;Z2)→H∗
+C(X;Z2)ofρ
+•such that the so-called conjugation equation holds:
+rσ(x) =κ(x)uk+terms of lower degree in u.
+One does not need to require that κandσbe ring homomorphisms, it is a consequence of
+the definition. Moreover, the “structure maps” κandσare unique, and natural with respect
+to equivariant maps between conjugation spaces.
+A conjugation manifold is a conjugation space consisting of a smooth manifold Xwith a
+smooth involution τ. As a consequence, Xmust be even-dimensional, say of dimension 2 n,
+andMis of dimension n.
+In [Olb07] we proved that it is possible to give a definition of conjugat ion spaces without
+the non-geometric maps κandσ, which is moreover well-adapted to the case of conjugation
+manifolds,wherethefixedpointsethasanequivarianttubul arneighbourhood. Anapplication
+of this characterization is theorem 2.2below.
+2.2 Review and outline
+In this section we review the construction from [ Olb07], and we explain how it is modified in
+the next section.
+4Theorem 2.2 ([ Olb07])LetXbe a closed 6-manifold with a smooth involution τsuch that
+the fixed point set Mis a 3-manifold with trivial normal bundle. Using the equiva riant tubular
+neighbourhood theorem, we write X= (M×D3)∪V, where the involution restricts to a free
+involution τonVsuch that W:=V/τis a6-manifold with boundary ∂W=M×RP2. Then
+Xis a conjugation space if and only if restriction to the bound ary
+H∗(W;Z2)→H∗(M×RP2;Z2)
+induces an isomorphism:
+H∗(W;Z2)→H∗(M×RP2;Z2)//circleplusdisplay
+i>jHi(M;Z2)uj.
+Translated to homology this is equivalent to the condition t hat inclusion of the boundary
+H∗(M×RP2;Z2)→H∗(W;Z2)
+induces an isomorphism:
+/circleplusdisplay
+i≤jHi(M;Z2)⊗Hj(RP2;Z2)→H∗(W;Z2).
+Thus, in order to construct a conjugation on some 6-manifold with fixed point set M, it
+suffices to find the “right” nullbordism WofM×RP2. (The trivial normal bundle condition
+is always satisfied if Mis oriented and Xis simply-connected.)
+We want the manifolds Xto be among those classified by Wall, which means simply-
+connected spin and with H3(X) = 0. This gives conditions on W, and on its normal 2-type:
+Definition 2.3 The normal 2-type of a compact manifold Nis a fibration B2(N)→BO
+which is obtained as a Postnikov factorization of the stable normal bundle map N→BO:
+There is a 3-connected map N→B2(N), the fibration B2(N)→BOis 3-coconnected (i.e.
+πi(BO,B 2(N)) = 0fori >3), and the composition is the stable normal bundle map. This
+determines B2(N)→BOup to fiber homotopy equivalence.
+Given a closed oriented 3-manifold M, we set m=rk(H1(M;Z2)) + 1. We construct
+a fibration B→BOwhich is 3-coconnected, and such that Bis connected, π1(B) =Z2,
+π2(B)∼=Zm, on which π1(B) acts by multiplication with −1, andπ3(B) = 0. More precisely,
+we use the fiber bundle S2→CP∞→HP∞induced from the identification C∞∼=H∞. The
+antipodal map on each fiber induces a free involution τonCP∞. NowB=BSpin×Qm,
+where
+Qm= (CP∞×···×CP∞×S∞)/(τ,...,τ,−1),
+where we have mfactorsCP∞. We have the projection on the last factor p:Qm→RP∞=
+BO(1). Since the involution τis free, we can identify up to homotopy Qm∼=(CP∞)m/τm.
+The map B→BOis the composition B=BSpin×QmBπ×p→BO×BO(1)⊕→BO.
+Lemma 2.4 ([ Olb07])IfXis simply-connected spin, with free and only even cohomolog y,
+τis a conjugation on Xwith fixed point set Mwhich acts by −1onH2(X), andWis defined
+as before, then the normal 2-type of WisB→BO.
+5Now start with a closed oriented 3-manifold M. By a bordism calculation, we show that
+there exists a manifold Wwith boundary M×RP2and normal 2-type B→BO:
+Theorem 2.5 ([ Olb07])The bordism group of 5-manifolds with normal B-structures is triv-
+ial:ΩB
+5= 0. (A normal B-structure is a lift of the normal bundle map to B. For a more
+precise definition, see [ Sto68].) Using surgery below the middle dimension, we may assume
+thatB→BOis the normal 2-type for W.
+Thus the map W→Bis 3-connected, and so in particular π1(W) =Z2. We would like
+to obtain a manifold W0with the same boundary, and such that we get an isomorphism
+/circleplusdisplay
+i≤jHi(M;Z2)⊗Hj(RP2;Z2)∼=H∗(W0;Z2).
+ForWthis map is an isomorphism except in dimension 3, where the ma p is injective, but
+has a possibly non-trivial cokernel. It is this cokernel we w ant to kill using surgery. The
+cokernel is isomorphic to H3(W;Z2)/rad, whereradis the radical of the intersection form on
+H3(W;Z2). We find that the intersection form on H3(W;Z2)/radis hyperbolic, and we find
+disjointly embedded 3-spheres in Wwith trivial normal bundle mapping to generators for a
+Lagrangian. We use the following lemma.
+Lemma 2.6 ([ Olb07])The Hurewicz map π3(W)→H3(W;Λ)and the map H3(W;Λ)→
+H3(W;Z2)/radare both surjective.
+By surgery on these 3-spheres we kill the cokernel in the midd le dimension and obtain a
+manifold W0with the desired properties. The double cover V0ofW0has boundary M×S2.
+We glue V0along its boundary to M×D3and obtain a 6-manifold X=M×D3∪V0. The
+involution on Xwhich is ( id,−id) onM×D3and which equals the unique non-trivial deck
+transformation on V0is a conjugation with fixed point set M.
+Theorem 2.7 ([ Olb07])Every closed orientable 3-manifold can be realized as fixed p oint set
+of a smooth conjugation on a closed simply-connected spin 6- manifold.
+The drawback of the surgery procedure is that we have not cont rolled the Λ = Z[Z2]-
+valued quadratic form on H3(W;Λ). (We only extracted partial information which made sure
+that we found disjointly embedded 3-spheres in Wwith trivial normal bundle we could do
+surgery on.) So we cannot say precisely which 6-manifolds wi th conjugations we obtain.
+In the following we will show that H3(W;Λ)/radis a free Λ-module and that the induced
+quadratic form is non-degenerate and thus stably hyperboli c. (Morally, the quadratic form
+is given by the map H3(W;Λ)→H3(W,∂W;Λ), and the absence of (odd) Z-torsion in
+H2(∂W;Λ) implies that it is possible to do the necessary surgeries without creating odd
+torsion.) The intersection form on H3(W;Λ) maps to the Z2-valued intersection form on
+H3(W;Z2) using the map ǫ: Λ→Z2defined by a+bT/mapsto→a+b.
+We conclude that (after possibly stabilizing Wby connected sum with copies of S3×S3)
+we can do surgery on generators of a Lagrangian for H3(W;Λ)/rad. This kills H3(W;Z2)/rad
+as before, but we also know that the effect on Λ-homology is just to killH3(W;Λ)/rad. In
+6particular we see that the normal 2-type of W0is equal to the normal 2-type of W. By the
+Mayer-Vietoris sequence for X=M×D3∪V0, we see that H2(X) is a free Z-module on
+which the conjugation acts by multiplication with −1 (see [Olb07]). Since H1(X) andH2(X)
+are free over Z, Poincar´ e duality implies that all homology of Xis free over Z. But since we
+have a degree-halving Z2-homology isomorphism from XtoM, we see that the free homology
+ofXis concentrated in even degrees.
+2.3 Proof of theorem 1.3
+Recall from [ Olb07] that the normal 2-type for WisB=BSpin×Qm→BO, whereQmis
+a fibration over RP∞with fiber K(Zm,2), andπ1(B) acts on π2(B) by multiplication with
+−1. Since π3(B) = 0 we have a ( −1)-quadratic form on π3(W) = Ker(π3(W)→π3(B)). Let
+Λ =Z[Z2]∼=Z[T]//an}bracketle{tT2−1/an}bracketri}ht, with the involution a+bT/mapsto→a−bT. Then the quadratic form
+consists of a ( −1)-hermitian map λ:π3(W)×π3(W)→Λ and a quadratic refinement ˜ µ:
+π3(W)→Λ/Z·1. (This is a slight modification of the usual quadratic form [ Wal99] counting
+(self-)intersections. Since we consider the self-interse ction on elements of the homotopy group
+and not on regular homotopy classes of immersions, the value s of ˜µare defined only modulo
+Z·1. See [Kre99] for details.)
+Themap λfitsin acommutative diagram withtheHurewicz map π3(W)→H3(W;Λ) and
+the Λ-valued intersection pairing on H3(W;Λ). It also maps to H3(W;Z2) and the Z2-valued
+intersection pairing, using the map ǫ: Λ→Z2defined by a+bT/mapsto→a+b.
+Wehaveseenthatthemap π3(W)→H3(W;Λ)issurjective, andthatthemap H3(W;Λ)→
+H3(W;Z2) induces a surjection onto H3(W;Z2)/rad, and that the latter carries a hyperbolic
+bilinear form.
+It turns out that a detailed analysis of the long exact sequen ces of the pair ( W,∂W) with
+Λ- andZ2-coefficients leads to a good understanding of the quadratic f orm. So let us analyze
+the commutative diagram
+H4(W,∂W;Λ) /d47/d47
+/d15/d15H3(∂W;Λ)
+/d15/d15/d47/d47H3(W;Λ) /d47/d47
+/d15/d15H3(W,∂W;Λ) /d47/d47
+/d15/d15H2(∂W;Λ)
+/d15/d15/d47/d47H2(W;Λ)
+/d15/d15
+H4(W,∂W;Z2)/d47/d47H3(∂W;Z2)/d47/d47H3(W;Z2)/d47/d47H3(W,∂W;Z2)/d47/d47H2(∂W;Z2)/d47/d47H2(W;Z2)
+In the following we want to use (“universal”) Poincar´ e dual ity with group ring coefficients.
+For this we need oriented covers (see [ Ran02], Def. 4.56) of our manifolds. The homology
+groups with group ring coefficients are just the integral homo logy groups of the oriented
+covering space, with left action of the fundamental group gi ven by deck transformations.
+The definition of cohomology groups with group ring coefficien ts is more difficult. For finite
+fundamental groups, they are defined in the following way: Th e cohomology groups of the
+covering space are naturally right modules over the group ri ng, if one uses the action of the
+fundamental group given by deck transformations. In order t o make them left modules, we
+need an involution on the group ring. We use the “involution t wisted by the first Stiefel-
+Whitney class”. In our case this is a+bT/mapsto→a−bT. The result is that the Λ-cohomology
+7groups of our manifolds are the integral cohomology groups o f their double covers, but Tacts
+by (−1) times the deck transformation.
+With this definition we have universal Poincar´ e duality, i. e. isomorphisms of Λ-modules
+Hk(W;Λ)∼=H6−k(W,∂W;Λ),
+Hk(W;Λ)∼=H6−k(W,∂W;Λ),
+Hk(∂W;Λ)∼=H5−k(∂W;Λ).
+More precisely, universal Poincar´ e duality is described i n [Ran02].
+UsingPoincar´ e duality, H4(W,∂W;Λ)→H3(∂W;Λ) identifies with themap H2(W;Λ)→
+H2(∂W;Λ). We have H2(W;Λ)∼=Zm
++andH2(∂W;Λ)∼=H2(M;Z)−⊕H2(S2)+(we indicate
+theT-actions by subscripts ±).
+So we have to look at the double cover of the map M×RP2→Qm. Going through
+our construction of this map [ Olb07], we see that the map M×S2→(CP∞)mmaps the
+2-skeleton of Mto a point, and S2=CP1diagonally into ( CP∞)m. Hence the map
+Zm
++→H2(M;Z)−⊕H2(S2)+
+maps every Z-summandisomorphically to H2(S2)+, and has cokernel H2(M;Z)∼=H1(M)+⊗
+H2(S2)−. AndH2(∂W;Λ)→H2(W;Λ) is the map H2(M)+⊕H2(S2)−→Zm
+−sending
+H2(M)+to 0 and the generator of H2(S2)−to (1,1,...,1). We get an exact sequence:
+0→H1(M)+⊗H2(S2)−→H3(W;Λ)→H3(W,∂W;Λ)→H2(M)+→0.
+Note that H2(M)+is free over Z.
+From the condition that ensures that we obtain a conjugation space (which is fulfilled for
+Wexcept in the middle dimension) we know that the map on H2withZ2-coefficients restricts
+to an isomorphism
+H0(M;Z2)⊗H2(RP2;Z2)⊕H1(M;Z2)⊗H1(RP2;Z2)→H2(W;Z2).
+Thus the map H2(∂W;Z2)→H2(W;Z2) has kernel isomorphic to H2(M;Z2)∼=Zm−1
+2. By
+Poincar´ e duality, the map H4(W,∂W;Z2)→H3(∂W;Z2) has cokernel Zm−1
+2. So we get the
+commutative diagram with exact rows:
+0 /d47/d47H1(M)+⊗H2(S2)−/d47/d47H3(W;Λ) /d47/d47
+/d15/d15H3(W,∂W;Λ) /d47/d47
+/d15/d15H2(M)+/d47/d470
+0 /d47/d47Zm−1
+2/d47/d47H3(W;Z2) /d47/d47H3(W,∂W;Z2) /d47/d47Zm−1
+2/d47/d470
+By Poincar´ e duality we see that H3(W,∂W;Λ)∼=H3(W;Λ) isZ-free since its Z-torsion
+is theZ-torsion in H2(W;Λ), which is 0.
+Now let us apply the universal coefficient spectral sequence
+Extp
+Λ(Hq(W;Λ),Λ) =⇒Hp+q(W;Λ).
+8Wehave H1(W;Λ) = 0, Extp
+Λ(H2(W;Λ),Λ) = Extp
+Λ(Zm
+−,Λ) = 0for p >0, andExtp
+Λ(H0(W;Λ),Λ) =
+Extp
+Λ(Z+,Λ) = 0 for p >0, which implies that
+H3(W;Λ)∼=HomΛ(H3(W;Λ),Λ).
+Again, the right hand side is naturally a right Λ-module, and we must use the involution
+on Λ to turn it into a left Λ-module, i.e. we multiply the T-action with ( −1). By standard
+theory, the map H3(W;Λ)→H3(W,∂W;Λ)∼=HomΛ(H3(W;Λ),Λ) in the middle dimension
+defines the intersection product, i.e. the map sends x/mapsto→(y/mapsto→λ(y,x)). (Observe that with
+our conventions, the intersection product λis conjugate-linear in the first and linear in the
+second variable.)
+Similarly, the map H3(W;Z2)→H3(W,∂W;Z2)∼=HomZ2(H3(W;Z2),Z2) defines the Z2-
+valued intersection product, i.e. the map sends x/mapsto→(y/mapsto→λZ2(y,x)).
+We also consider the homology universal coefficient spectral sequence:
+TorΛ
+p(Hq(W;Λ),Z2)⇒Hp+q(W;Z2).
+This can also be interpreted as the Leray-Serre spectral seq uence for the fibration V→W→
+RP∞.
+The following diagram shows the E2-term with possible differentials in low degrees.
+012345q
+0 1 2 3 4 5 pZ2Zm
+2H3(W;Λ)⊗ΛZ2H4(W;Λ)⊗ΛZ2
+Z2Zm
+2TorΛ
+1(H3(W;Λ),Z2)
+Z2Zm
+2
+Z2Z2Z2···
+···
+···
+···
+···/d108/d108 /d106/d106 /d103/d103/d109/d109
+The map H3(W;Λ)→H3(W;Z2) can be factorized through H3(W;Λ)⊗ΛZ2. The map
+H3(W;Λ)→H3(W;Λ)⊗ΛZ2is a surjection, and H3(W;Λ)⊗ΛZ2→H3(W;Z2) can be
+observed at the edge in the universal coefficient spectral seq uence. We can compare this to
+the corresponding sequence for Q1. Recall that we have maps W→Qm→Q1inducing
+isomorphisms on π1. We will use the naturality of the universal coefficient spect ral sequence.
+Note that TorΛ
+p(Z+,Z2)∼=TorΛ
+p(Z−,Z2)∼=Z2for allp≥0. Thus for all p≥0, we
+have TorΛ
+p(Hq(Q1;Λ),Z2) = 0 if qis odd, and TorΛ
+p(Hq(Q1;Λ),Z2) =Z2ifqis even. (The
+universal cover of Q1isCP∞.) We know that H3(Q1;Z2) = 0 and that the non-zero class
+inH4(Q1;Z2) comes from H4(Q1;Λ). So no non-zero differential in the spectral sequence for
+Q1ends up in E0,4, and the d3-differentials E3,0
+3→E2,0
+3,E4,0
+3→E2,1
+3andE5,0
+3→E2,2
+3are
+9all non-zero. By naturality, the same must be true in the spec tral sequence for W. It follows
+thatd2: TorΛ
+2(Zm
+−,Z2) =Zm
+2→H3(W;Λ)⊗ΛZ2can have rank at most m−1, and that it is
+the only possibly non-zero differential ending up in E0,3.
+This implies that the map H3(W;Λ)⊗ΛZ2→H3(W;Z2) has a cokernel of rank exactly
+m−1 (which is the sum of the ranks of the spaces E2,1
+∞,E1,2
+∞, andE3,0
+∞in the spectral
+sequence), and a kernel of rank at most m−1.
+But we saw that the image of H3(∂W;Z2) inH3(W;Z2) is the radical of the intersection
+form and has rank m−1. And the map H3(W;Λ)→H3(W;Z2) induces a surjection onto
+H3(W;Z2)/rad. So the images of H3(W;Λ) and H3(∂W;Z2) inH3(W;Z2) must be comple-
+ments. This implies that the image of H3(W;Λ) inH3(W;Z2) injects into H3(W,∂W;Z2).
+But this also implies that all Λ-torsion in H3(W;Λ) maps to 0 in H3(W,∂W;Z2), re-
+spectively in H3(W;Z2): Since the Z2-intersection form is 0 on a complement of the image
+ofH3(W;Λ), it suffices to show that if x∈H3(W;Λ) is torsion, then ǫ(λ(x,y)) = 0 for all
+y∈H3(W;Λ). Wehavethreecases: If xisZ-torsion, then λ(x,y) = 0forall y. If(1+T)x= 0,
+thenTλ(x,y) =λ(−Tx,y) =λ(x,y), soλ(x,y) is a multiple of (1+ T) andǫ(λ(x,y)) = 0 for
+ally. And similarly if (1 −T)x= 0, then Tλ(x,y) =λ(−Tx,y) =λ(−x,y) =−λ(x,y), so
+λ(x,y) is a multiple of (1 −T) andǫ(λ(x,y)) = 0 for all y.
+LetH1(M)∼=S⊕Zm−1, whereSis oddtorsion. Then H2(M)∼=Zm−1. Now thestructure
+theorem forfinitely generated Z-freeΛ-modules[ CR62] says that every such moduleisadirect
+sum of summands isomorphic to Λ, Z+orZ−. We want to apply this to the exact sequence
+0→S−⊕Zm−1
+−→H3(W;Λ)→H3(W,∂W;Λ)→Zm−1
++→0.
+Recall that H3(W,∂W;Λ) isfreeover Z, whichimpliesthatover Z, weget twosplitshortexact
+sequences 0 →S⊕Zm−1→H3(W;Λ)→K→0 and 0→K→H3(W,∂W;Λ)→Zm−1
++→0,
+whereKis the image of the map H3(W;Λ)→H3(W,∂W;Λ).
+By applying the structure theorem, we see that the modules H3(W;Λ)/{Z−torsion}and
+H3(W,∂W;Λ) have the form H3(W;Λ)/{Z−torsion}∼=Za
+−⊕Zb
++⊕Λc, andH3(W,∂W;Λ)∼=
+Za
++⊕Zb
+−⊕Λc. We have also used that H3(W,∂W;Λ)∼=HomΛ(H3(W;Λ)/{Z−torsion},Λ)
+withT-action multiplied by −1. Note that only the numbers a,b,care uniquely determined;
+the decomposition itself into cyclic summands is NOT unique .
+Choosesuchadecomposition for H3(W;Λ)/{Z−torsion}, andusethedualdecomposition
+for Hom Λ(H3(W;Λ),Λ). By tensoring the exact sequence over Λ with Q[Z2]∼=Q−⊕Q+, we
+get
+0→Qm−1
+−→Qa+c
+−⊕Qb+c
++→Qb+c
+−⊕Qa+c
++→Qm−1
++→0.
+It follows that a=b+(m−1). SoH3(W;Λ)/{Z−torsion}∼=Zb+(m−1)
+−⊕Zb
++⊕Λc.
+Now we really use the assumption that Sconsists only of odd torsion elements: Then
+Ext1
+Λ(Z±,S−) = 0, and we obtain H3(W;Λ)∼=S−⊕Zb+(m−1)
+−⊕Zb
++⊕Λc. Then
+rk(H3(W;Λ)⊗ΛZ2) =rk(S−⊕Zb+(m−1)
+−⊕Zb
++⊕Λc)⊗ΛZ2
+= 0+b+(m−1)+b+c= 2b+c+(m−1).
+We have seen that themap H3(W;Λ)→H3(W;Z2) is 0on thesubmodule S−⊕Zb+(m−1)
+−⊕Zb
++
+ofH3(W;Λ) and that the map H3(W;Λ)⊗ΛZ2→H3(W;Z2) has kernel of rank at most
+10m−1. It follows that b= 0, and since the image has even rank, cmust be even, say c= 2k.
+So our exact sequence becomes
+0→S−⊕Zm−1
+−→S−⊕Zm−1
+−⊕Λ2k→Zm−1
++⊕Λ2k→Zm−1
++→0.
+TheZ-torsionSismappedisomorphicallytoitselfunderthefirstmap, andth eintersection
+form is 0 if one of the arguments is in S. So let us again look at the sequence modulo the
+Z-torsionS:
+0→Zm−1
+−(i1,i2)−→Zm−1
+−⊕Λ2k→Zm−1
++⊕Λ2k→Zm−1
++→0.
+On the submodule Λ2k⊆H3(W;Λ), we know that if we tensor the quadratic form with Z2,
+it becomes hyperbolic. So we can choose basis vectors ei,fifor Λ2ksuch that ǫ(λ(ei,ej)) =
+ǫ(λ(fi,fj)) = 0 and ǫ(λ(ei,fj)) =δij.
+What can we say about the matrix Aof the intersection form λwith respect to the ei,fj?
+•Ais a skew-hermitian matrix.
+•Ais the matrix for the component Λ2k→Λ2kof the middle map in the exact sequence.
+•ǫ(A) =/parenleftbigg0I
+I0/parenrightbigg
+. This means in particular that Ahas a determinant which is a
+nonzerodivisor, which implies that the map Λ2k→Λ2kis injective.
+So we see that the middle map in the spectral sequence is injec tive on Λ2k. Let us denote its
+image by L⊆Zm−1
++⊕Λ2k. We have L∼=Λ2kas Λ-modules. We also get that i1is injective,
+thus it has a finite cokernel, which we denote by G−.
+Our next claim is that G−does not have non-trivial 2-torsion. Let v∈Zm−1
+−be a
+representative for a 2-torsion element in the cokernel, and suppose that x= (2v,−w)∈
+Zm−1
+−⊕Λ2kis in the image of ( i1,i2). This is equivalent to λ(2v,y) =λ(w,y) for all y∈
+H3(W;Λ). It suffices to show that wis divisible by 2, because then ( v,−w/2) is in the image
+of (i1,i2), andvrepresents 0 ∈G−.
+Note that λ(2v,y) is divisible by 2(1+ T) for allyand that Tw=−w, since it is in the
+image of a map from Z−. Thus we can write
+w=/summationdisplay
+αi(1−T)ei+βi(1−T)fi,
+whereαi,βi∈Z. Sety=fj. Thenλ(2v,y) =λ(w,y) implies that
+/summationdisplay
+αi(1+T)λ(ei,fj)+βi(1+T)λ(fi,fj)
+is divisible by 2(1 + T). Now for i/ne}ationslash=j, we have ǫ(λ(ei,fj)) = 0 , and this implies that
+(1+T)λ(ei,fj) is divisible by 2(1+ T). Similarly, (1+ T)λ(fi,fj) is divisible by 2(1+ T) for
+alli. But (1 + T)λ(ej,fj) is not divisible by 2, and so αjis divisible by 2. Repeating the
+argument with y=ejshows that βjis divisible by 2. Thus we have shown that wis divisible
+by 2.
+By a diagram chase, we see that the cokernel G−ofi1is isomorphic to K/L. (Recall that
+Kis the image of the middle map, Lis the image of the middle map restricted to the free
+11submodule.) It follows that Kis an extension of the finite Λ-module G−by the free Λ-module
+L∼=Λ2k. We must have K∼=Z4kasZ-module, since Kis a submodule of H3(W,∂W;Λ).
+ThusKis a direct sum of summands isomorphic to Λ, Z+orZ−as Λ-module.
+If we tensor the exact sequence
+0→Λ2k→K→G−→0
+over Λ with Z2, we get the (right) exact sequence
+Z2k
+2→K⊗ΛZ2→0→0.
+(We have G−⊗ΛZ2= 0 since G−has trivial 2-torsion.) But since K⊗ΛZ2has rank ≤2k,
+Kmust be a free Λ-module, i.e. K∼=Λ2k.
+But since Kis free, this means that the exact sequence
+0→Zm−1
+−(i1,i2)−→Zm−1
+−⊕Λ2k→K→0
+splits. Thus there is a new decomposition H3(W;Λ)/S∼=Zm−1
+−⊕Λ2ksuch that the first
+summand Zm−1
+−is exactly the kernel for the map to H3(W,∂W;Λ). Choose basis vectors aj
+for this submodule. We may choose basis vectors for the secon d summand with the same
+properties as the ei,fi. By abuse of notation, and since we do not need the old generat ors
+any more, we denote these new basis vectors by ei,fiagain.
+Thenλ(aj,y) = 0 for all y∈H3(W;Λ). The generators aj,ei,fifor
+H3(W;Λ)/S∼=Zm−1
+−⊕Λ2k
+can beused to give generators for H3(W,∂W;Λ)∼=HomΛ(H3(W;Λ),Λ). For each jwe have a
+uniquesuchmapsending ajto(1−T)andallothergeneratorsto0. Foreach iwehaveaunique
+such map sending eito 1 and all other generators to 0 and a unique such map sending fito 1
+and all other generators to 0. These maps form generators for H3(W,∂W;Λ)∼=Zm−1
++⊕Λ2k,
+i.e. our distinguished decomposition of H3(W;Λ) induces a distinguished decomposition of
+H3(W,∂W;Λ).
+Now we see that Λ2k⊆H3(W;Λ) has to map into Λ2k⊆H3(W,∂W;Λ) since λ(aj,y) = 0
+for allyalso implies λ(y,aj) = 0 for all y. The map Λ2k→Λ2kmust have zero cokernel, so
+it is bijective.
+SoH3(W;Λ)/radis a free Λ-module generated by ei,fiand the induced quadratic form
+is non-degenerate. But this implies we finally can use the sta ndard theory of non-degenerate
+quadratic forms. The only difference is that our quadratic refi nement ˜µhas values in Λ /Z·1,
+whereas Wall’s quadratic refinement takes values in Λ //an}bracketle{tx+¯x|x∈Λ/an}bracketri}ht= Λ/Z·2. Wall proved
+thatL6(Λ)∼=Z2, generated by the Arf invariant [ Wal99]. This implies that ˜L6(Λ) is the
+trivial group: there is a lift of ˜ µfrom Λ/Z·1 to Λ/Z·2 with zero Arf invariant.
+Thus, afterstabilizing Wbyconnected sumwithcopies of S3×S3, wemay assumethat the
+restriction of λto the free submodule generated by ei,fiis hyperbolic. We may also assume
+that the eiare generators for a Lagrangian. Since the Hurewicz map π3(W)→H3(W;Λ)
+12is surjective, we obtain disjointly embedded spheres in Wwith trivial normal bundle, more
+precisely with a unique trivialization such that the normal B-structure extends over the
+surgery cobordism.
+Now the diagram on p. 73 of Ranicki’s book [ Ran02] shows that the effect of doing surgery
+on theeion the homology of Wwith coefficients in Λ is exactly killing the free submodule
+ofH3(W;Λ) generated by the ei,fi. In particular, the resulting manifold W0still has 2-type
+B. Thus the Mayer-Vietoris sequence for X=M×D3∪V0contains
+H2(M×S2)→H2(M×D3)⊕H2(V0)→H2(X)→0.
+We deduce that H2(X)∼=Coker(H2(S2)→H2(V0)). But in [ Olb07] we have seen that if W0
+has normal 2-type B, then this cokernel is isomorphic to Zm−1
+−.
+So we have constructed a conjugation on a simply connected sp in manifold Xwith fixed
+point set M. But since H1(X) andH2(X) are free over Z, Poincar´ e duality implies that all
+homology of Xis free over Z. And since we have a degree-halving Z2-homology isomorphism
+fromXtoM, we see that the free homology of Xis concentrated in even degrees.
+3 Analysis of the result of the construction
+3.1 The Z2-cohomology and homology of Qm
+Theintegral cohomology of K(Zm,2) is apolynomial ringonthestandardclasses v1,...,vm∈
+H2(K(Zm,2)) which correspond to the dual basis of the standard basis o fZm. We get dual
+generators ei=v∗
+ifor the second integral homology corresponding to the stand ard basis of
+Zm. More generally we get a basis ei1...ik= (vi1...vik)∗for the integral homology in degree
+2k. Let us abuse notation and denote by vialso the corresponding class in Z2-cohomology,
+and byei1...ikalso the corresponding class in Z2-homology.
+In [Olb07] it was proved that
+H∗(Qm;Z2)∼=Z2[q,t,x1,...xm−1]/t3withdeg(q) = 4,deg(t) = 1,deg(xi) = 2.
+Here we have to be more precise about the isomorphism. We deno te the maps
+Qm= (CP∞×···×CP∞)/τm→Qn= (CP∞×···×CP∞)/τn
+[a1,...,am]/mapsto→[ai1,...,ain]
+bypri1,...in. There is a fibration RP2→Q1→HP∞whose Serre spectral sequence collapses
+at theE2-term. We get generators t∈H1(Q1;Z2) andq∈H4(Q1;Z2). Using the fibration
+(CP∞)m−1→Qmprm→Q1we transport the classes tandqtoH∗(Q1;Z2). Fori= 1,...m−1
+we denote by xi∈H2(Qm) the class which maps to vi∈H2(K(Zm−1,2)), and to 0 under the
+diagonal map ∆ = pr1,...,1:Q1→Qm.
+To compute the maps π:K(Zm,2)→Qmandpri:Qm→Q1inZ2-cohomology for
+i < m, we first look at the case m= 2. We consider the commutative diagram whose rows
+13are fibrations:
+S2×S2
+/d15/d15/d47/d47CP∞×CP∞
+/d15/d15/d47/d47HP∞×HP∞
+id
+/d15/d15
+(S2×S2)/(−1,−1) /d47/d47Q2
+pr1,pr2
+/d15/d15/d47/d47HP∞×HP∞
+Q1×Q1/d54/d54/d109/d109/d109/d109/d109/d109/d109/d109/d109/d109/d109/d109/d109/d109
+Theelements q1,q2∈H4(HP∞×HP∞;Z2) (coming fromthetwo factors) pullback to v2
+1,v2
+2∈
+H4(CP∞×CP∞;Z2), and to 0 ∈H4((S2×S2)/(−1,−1);Z2). Thus under π:K(Z2,2)→Q2,
+the element qmaps to v2
+2, the element tmaps to 0, and the element xmaps to v1+v2. And
+underpr1:Q2→Q1, the element tmaps to t, andqmaps to q+x2
+1(there is no term t2x1
+since this would map nontrivially to ( S2×S2)/(−1,−1)).
+Now for the general case we consider the diagram (where i < m):
+K(Zm,2)
+/d15/d15/d47/d47K(Z2,2)
+/d15/d15
+Q1∆/d47/d47Qmpri,m/d47/d47
+(pri,prm)/d38/d38/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78/d78Q2
+/d15/d15
+Q1×Q1
+Hereq1,q2∈H4(Q1×Q1;Z2)mapto q+x2
+1,q∈H4(Q2;Z2)andtov2
+i,v2
+m∈H4(K(Zm,2);Z2).
+Hencex1∈H2(Q2;Z2) maps to xi∈H2(Qm;Z2) underpri,m. Under pri:Qm→Q1, we
+havet/mapsto→t,q/mapsto→q+x2
+i, and under π:K(Zm,2)→Qm, we get t/mapsto→0,xi/mapsto→vi+vm,q/mapsto→v2
+m.
+Similarly, we compute the effect of pri,j:Qm→Q2fori < j < m . We obtain q/mapsto→q+x2
+j,
+x1/mapsto→xi+xj,t/mapsto→t. And for prijk:Qm→Q3withi < j < k we getq/mapsto→q+x2
+k,x1/mapsto→xi+xk,
+x2/mapsto→xj+xk,t/mapsto→tin the case k < mandq/mapsto→q,x1/mapsto→xi,x2/mapsto→xj,t/mapsto→tin the case k=m.
+We will use monomials in the generators as standard bases for theZ2-cohomology groups
+and use the dual bases for the corresponding Z2-homology groups. For example ( qx1)∗will
+denote the element of the sixth Z2-homology group which pairs to 1 with qx1and to 0 with
+all other monomials in the generators.
+The transfer map in Z2-homology tr:H∗(Qm;Z2)→H∗(K(Zm,2);Z2) is induced by
+the change of coefficients (modules over the group ring of the f undamental group of Qm)
+given by Z2→Z2[Z2], 1/mapsto→1 +T. We use the short exact sequence Z2→Z2[Z2]→Z2
+which induces a long exact sequence in homology. The fact tha tZ2[Z2]→Z2inducesπ∗:
+H∗(K(Zm,2);Z2)→H∗(Qm;Z2) shows that Im(tr) =Ker(π∗). From the fact that the
+involution τmonK(Zm,2) is trivial on Z2-homology it follows that Im(π∗)⊆Ker(tr). (The
+composition Z2[Z2]→Z2→Z2[Z2] is multiplication by 1 + T, which induces id+ (τm)∗.)
+But since the ranks of H∗(Qm;Z2) equal those of H∗(K(Zm,2);Z2) in even degrees, we get
+equality Im(π∗) =Ker(tr) in these degrees.
+14We first look at tr:H2(Qm;Z2)→H2(K(Zm,2);Z2). From the above computation of π∗
+it follows that tr((xi)∗) = 0 for all i= 1,...,m−1 andtr((t2)∗) =/summationtextm
+j=1ej.
+Thekernelof π∗:H4(K(Zm,2);Z2)→H4(Qm;Z2)isgeneratedby/summationtext
+j,j/negationslash=ieij,i= 1,...,m−
+1 (just dualize π∗), and the image of π∗has generators q∗,(xixj)∗fori≤j. To compute the
+transfer map in degree 4, we first deduce that for m= 2, the remaining generator ( t2x1)∗
+must be mapped to e12. Then we use naturality with respect to the projections prj,kto see
+that for general m, the transfer map is ( t2xi)∗/mapsto→/summationtext
+j,j/negationslash=ieij.
+In degree 6, the transfer has image generated by
+/summationdisplay
+i≤j≤keijk,m/summationdisplay
+j=1eiijfori= 1,...,m−1,m/summationdisplay
+k/negationslash=i,j
+k=1eijkfori < j < m,
+and kernel generated by
+(xiq)∗,(xixjxk)∗fori≤j≤k,
+which follows again from our earlier computation of π∗. Again we first compute the cases for
+smallm≤3, and then use the naturality of the transfer with respect to projections to Q1,
+Q2andQ3to see that the transfer is
+(t2q)∗/mapsto→/summationdisplay
+i≤j≤keijk,(t2x2
+i)∗/mapsto→m/summationdisplay
+j=1eiijfori= 1,...,m−1,
+(t2xixj)∗/mapsto→m/summationdisplay
+k/negationslash=i,j
+k=1eijkfori < j < m.
+3.2 The transfer map ΩB
+6→Ω˜B
+6
+The projection map π∗: Ω˜B
+6→ΩB
+6is given by [ M→˜B]/mapsto→[M→˜B→B]. The composition
+tr◦π∗is equal to 1+ τ∗, whereτdenotes the nontrivial deck transformation ˜B→˜B. This is
+multiplication with −1 onH2(˜B), but it also changes the orientation of M, because the ori-
+entation is obtained by composition with ˜B→BSO, and the nontrivial deck transformation
+of the double cover BSO→BOreverses orientation of a bundle. By the computation of Ω˜B
+6
+we see that τ∗is the identity. So tr◦π∗is multiplication by 2. In particular it follows that
+the cokernel of tris a finite group consisting of elements of order (dividing) 2 .
+We can compute the transfer map using the Atiyah-Hirzebruch spectral sequence.
+E2
+p,q∼=Hp(Qm;ΩSpin
+q)
+/d15/d15/d47/d47ΩB
+6
+/d15/d15
+˜E2
+p,q∼=Hp(K(Zm,2);ΩSpin
+q) /d47/d47Ω˜B
+6
+For the facts needed and not proved here see Zubr [ Zub75] (for the computation of Ω˜B
+6)
+and [Olb07] (for the computation of ΩB
+6).
+15For the terms on the second pages of the spectral sequences we use the transfer maps
+in homology induced by the short exact coefficient sequences Z−→Z[Z2]→ZandZ2→
+Z2[Z2]→Z2. We get:
+E2
+0,6=˜E2
+0,6=E2
+1,5=˜E2
+1,5= 0.
+The map
+E2
+2,4∼=Zm→˜E2
+2,4∼=Zm
+is injective with cokernel H2(Qm)∼=Zm−1
+2. More precisely its image is the kernel of the map
+H2(K(Zm,2))∼=Zm→H2(Qm)∼=Zm−1
+2, which is generated by 2 ei,i= 1,...,m−1 and/summationtextm
+j=1ej. (Compare with the Z2-homology transfer.)
+We have
+E2
+3,3∼=0→˜E2
+3,3∼=0.
+We do already know the Z2-homology transfer map
+E2
+4,2∼=Z(m+1
+2)
+2→˜E2
+4,2∼=Z(m+1
+2)
+2.
+We have
+E2
+5,1∼=Z(m
+2)+1
+2→˜E2
+5,1∼=0.
+Finally we see that
+E2
+6,0∼=Z(m+2
+3)→˜E2
+6,0∼=Z(m+2
+3)
+is injective with cokernel isomorphic to H6(Qm)∼=Z(m+1
+3)+m−1
+2 . More precisely the image in
+E2
+6,0∼=H6(K(Zm,2)) is the kernel of the composition H6(K(Zm,2))→H6(K(Zm,2);Z2)→
+H6(Qm;Z2). We obtain generators
+m/summationdisplay
+j=1eiijfori= 1,...,m−1,m/summationdisplay
+k/negationslash=i,j
+k=1eijkfori < j < m,
+/summationdisplay
+i≤j≤keijk,2eijkfori≤j≤k < m, 2eimmfori < m.
+For the third pages of the spectral sequences we get the same a s before in the (2 ,4)-terms.
+We get the zero map E3
+4,2∼=Zm
+2→˜E3
+4,2∼=Zm
+2, since the image of the map on the second page
+is also in the image of the ˜d2-differential. We get a zero map E3
+5,1∼=Z2→˜E3
+5,1∼=0.
+Finally for the (6 ,0)-terms, we have to restrict both domain and codomain of the second
+page to subgroups with quotient Z(m
+2)
+2:
+The˜d2-differential is the composition of reduction modulo 2 and the dual ofSq2. It sends
+eijjfori/ne}ationslash=jtoeijand all other generators to 0. Thus we can consider ˜E3
+6,0as the subset of
+˜E2
+6,0with generators eijkfori < j < k ,eiiifor alli,eiij+eijjand 2eiijfor alli < j. We saw
+thatE2
+6,0can also be considered as a subset of ˜E2
+6,0. We have to compute the image of the
+generators under ˜d2.
+Thed2-differential is the composition of reduction modulo 2 and the dual ofSq2+tSq1.
+Since reduction modulo 2 is surjective, its image is the imag e of the dual of Sq2+tSq1. Its
+generators are ( xixj)∗fori < jand (t2xi)∗+(x2
+i)∗for alli.
+16E3
+6,0∼=Z(m+2
+3)
+/d15/d15/d47/d47E2
+6,0∼=Z(m+2
+3) /d47/d47
+/d15/d15/an}bracketle{t(xixj)∗,(t2xi)∗+(x2
+i)∗/an}bracketri}ht∼=Z(m
+2)
+2
+/d15/d15
+˜E3
+6,0∼=Z(m+2
+3) /d47/d47˜E2
+6,0∼=/an}bracketle{teijk|i≤j≤k/an}bracketri}ht∼=Z(m+2
+3) /d47/d47/an}bracketle{teij/an}bracketri}ht∼=Z(m
+2)
+2
+We begin with the case m= 1. Here E2
+6,0is generated by e111, and this generator
+maps to 0. In the case m= 2 we use the diagonal map and the case m= 1 to compute
+d2(e111+e112+e122+e222) = 0, weusethemod2transfercomputationstoseethat e111+e112/mapsto→
+(t2x1)∗+(x2
+1)∗. For the remaining generators we use the coefficient changes Z[Z2]→Z−and
+Z−→Z[Z2] which induce projection and transfer maps respectively. T heir composition
+Z[Z2]→Z[Z2] is multiplication with 1 −T. OnH6(K(Zm,2)) this is multiplication with 2.
+The other generators are in the image of the projection map H6(K(Zm,2))→H6(Qm;Z−).
+This allows to compute that d2(2e111) =d2(2e122) = 0.
+Form= 3, we compute e111+e112+e113+e122+e123+e133+e222+e223+e233+e333/mapsto→0
+using the diagonal map, d2(2e111) =d2(2e222) = 0 and d2(2e112) =d2(2e122) =d2(2e133) =
+d2(2e233) = (x1x2)∗using the projection map. Finally we use the maps pr1,1,2:Q2→Q3,
+pr1,2,2:Q2→Q3andpr2,1,2:Q2→Q3and the case m= 2 to compute e111+e112+e113/mapsto→
+(t2x1)∗+(x2
+1)∗,e221+e222+e223/mapsto→(t2x2)∗+(x2
+2)∗+µ(x1x2)∗, ande123/mapsto→(x1x2)∗.
+For the general case, we use the Z2-homology transfer computation to conclude that/summationtextm
+j=1eiij/mapsto→(t2xi)∗+ (x2
+i)∗+/summationtext
+j<kµijk(xjxk)∗for certain µijk∈Z2, and that all other
+generators mapto alinear combination of generators of thef orm (xixj)∗. Usingtheprojection
+map, we compute that 2 eijk/mapsto→0 in the cases i < j < k < m andi=j=k < m,
+2eijk/mapsto→(xixj)∗fori < j=k < m and 2eijk/mapsto→(xjxk)∗fori=j < k < m . Also
+we obtain 2 eimm/mapsto→/summationtext
+j/negationslash=i(xixj)∗. Finally we compare with the case m= 3 to see that/summationtextm
+k/negationslash=i,j
+k=1eijk/mapsto→(xixj)∗and/summationtext
+i≤j≤keijk/mapsto→0.
+This finally allows us to identify generators for the kernel E3
+6,0:
+/summationdisplay
+i≤j≤keijk,2eiiifori < m,m/summationdisplay
+j=12eiijfori < m, 2eimm+/summationdisplay
+k/negationslash=i,k/negationslash=meimkfori < m,
+2eijj+/summationdisplay
+k/negationslash=i,k/negationslash=jeijkfori/ne}ationslash=jandi,j < m, 2eijkfori < j < k.
+This means that the classes F∈H6(K(Zm,2)) belonging to E3
+6,0are exactly those which
+satisfy:
+/an}bracketle{tvivivj,F/an}bracketri}ht=/an}bracketle{tvkvkvl,F/an}bracketri}ht(mod2) for all i,j,k,l∈ {1,...m}
+/an}bracketle{tvivjvk,F/an}bracketri}ht=1
+2(/an}bracketle{tvivivj,F/an}bracketri}ht+/an}bracketle{tvivjvj,F/an}bracketri}ht+/an}bracketle{tvivivk,F/an}bracketri}ht
++/an}bracketle{tvivkvk,F/an}bracketri}ht+/an}bracketle{tvjvjvk,F/an}bracketri}ht+/an}bracketle{tvjvkvk,F/an}bracketri}ht) (mod2) for all i < j < k.
+Therearealmost nomorehigherdifferentialsinvolved. Theon lytermwhichcouldpossibly
+be different on the ∞-page isE3
+4,2since there is room for a d3fromE3
+7,0. Equipped with this
+17information about the ∞-page, we compute the extensions. We denote the filtrations o f Ω˜B
+6
+and ΩB
+6by˜FjandFjso that˜Fj/˜Fj−1∼=˜E∞
+j,6−jandFj/Fj−1∼=E∞
+j,6−j.
+Actually we know from [ Olb07] that
+ΩB
+6∼=Z(m+2
+3)+m⊕Z4
+and from Zubr that
+Ω˜B
+6∼=Z(m+2
+3)+m.
+We have a commutative diagram with short exact sequences:
+F5
+/d15/d15/d47/d47ΩB
+6/d47/d47
+/d15/d15E3
+6,0
+/d15/d15
+˜F5/d47/d47Ω˜B
+6/d47/d47˜E3
+6,0
+We see that F5∼=Zm⊕Z4and˜F5∼=Zm. We apply the snake lemma, using that the middle
+vertical map has cokernel a Z2-vector space, and the right vertical map is injective. We ge t
+exact sequences 0 →Z4→F5→˜F5, 0→Z4→ΩB
+6→Ω˜B
+6, and for each of these sequences
+the last map has cokernel a finite-dimensional Z2-vector space. In particular the dimension
+of this vector space for the first exact sequence is at most m.
+Now we consider the commutative diagram with short exact seq uences:
+F2∼=Zm
+/d15/d15/d47/d47F5∼=Zm⊕Z4/d47/d47
+/d15/d15F5/F2
+/d15/d15
+˜F2∼=Zm /d47/d47˜F5∼=Zm /d47/d47˜F5/˜F2∼=Zm
+2
+We also have the short exact sequence E∞
+4,2→F5/F2→Z2. By applying the snake lemma
+again, we see that F5/F2contains an element of order 4, soit is isomorphicto Z4⊕Zk
+2for some
+k≤m−1. We may also assume that the generator of the Z4-summand is in the kernel of the
+vertical map. But we also know that the left vertical map is in jective with cokernel Zm−1
+2,
+and that the middle vertical map has kernel Z4and cokernel a Z2-vector space of dimension
+at mostm. Considering all this, we can deduce: F5/F2∼=Z4⊕Zm−1
+2, andE3
+4,2=E∞
+4,2. And
+F5→˜F5has cokernel Zm
+2, i.e. the image of F5in˜F5is exactly ˜F2which is also the same as
+all elements divisible by 2.
+Since˜E3
+6,0is free over Z, the short exact sequence for ΩSpin
+6(K(Zm,2)) splits. Let
+ˆA(M,x) =/an}bracketle{texˆA(M),[M]/an}bracketri}ht=/an}bracketle{t4x3−p1(M)x
+24,[M]/an}bracketri}ht. Zubr [ Zub75] proves that there is an in-
+jective map
+ΩSpin
+6(K(Zm,2))→Zm⊕Z(m+2
+3)
+[f:M→B]/mapsto→/parenleftBig
+(ˆA(M,f∗(vi))),(/an}bracketle{tf∗(vivjvk),[M]/an}bracketri}ht)/parenrightBig
+with image given by the restrictions
+/an}bracketle{tf∗(vivivj),[M]/an}bracketri}ht=/an}bracketle{tf∗(vivjvj),[M]/an}bracketri}ht(mod2)
+18for alli,j. These conditions corresponds exactly to the inclusion of ˜E3
+6,0into˜E2
+6,0, and the
+map toZmis exactly the splitting of the short exact sequence to ˜F5.
+Theorem 3.1 The transfer map ΩB
+6→Ω˜B
+6has kernel the torsion subgroup Z4. Under Zubr’s
+inclusion map ΩSpin
+6(K(Zm,2))→Zm⊕Z(m+2
+3), the image of the transfer map corresponds
+to those classes [f:M→B]such that
+ˆA(M,f∗(vi)) = 0 ( mod2)for alli,
+/an}bracketle{tf∗(vivivj),[M]/an}bracketri}ht=/an}bracketle{tf∗(vkvkvl),[M]/an}bracketri}ht(mod2)for alli,j,k,l∈ {1,...m},
+/an}bracketle{tf∗(vivjvk),[M]/an}bracketri}ht=1
+2/an}bracketle{tf∗(v2
+ivj+viv2
+j+v2
+ivk+viv2
+k+v2
+jvk+vjv2
+k),[M]/an}bracketri}ht(mod2)
+for alli < j < k.
+Proof: It remains to show that for every [ f:M→B]∈ΩB
+6, we have ˆA(˜M,f∗(vi)) =
+0 (mod2) for all i. This would indeed imply that the splitting for the short exa ct sequence
+forΩSpin
+6(K(Zm,2)) alsoinducesasplittingfortheshortexactsequencefor ΩB
+6. Byprojecting
+onto the ith factor, we may assume m= 1. Recall the fibre bundle S2→CP∞→HP∞,
+which restricts to S2→CP3→H1=S4. We use that ΩSpin
+6(K(Z,2))∼=ΩSpin
+6(CP3)→
+ΩSpin
+6(S4)∼=Z2is given by
+[f:M→K(Z,2)]/mapsto→ˆA(M,f∗v) (mod2).
+This follows for example from a comparison of the Atiyah-Hir zebruch spectral sequences. The
+mapCP∞×S∞→S4×S∞is equivariant with respect to ( τ,−1) onCP∞×S∞and (id,−1)
+onS4×S∞. It induces a map between the quotient spaces: Q1→S4×RP∞. By naturality
+of the transfer map it suffices to show that the transfer ΩSpin
+6(S4×RP∞;L)→ΩSpin
+6(S4) is
+zero. But ΩSpin
+6(S4×RP∞;L)∼=ΩSpin
+7(S4×RP∞,S4), and via this isomorphism, the transfer
+map becomes the boundary map in the long exact sequence of the pair (S4×RP∞,S4), see
+the lemma below. But S4certainly is a retract of S4×RP∞, and so this boundary map is
+zero.q.e.d.
+Lemma 3.2 Given a line bundle L→X, we have the double cover S(L)→X, where
+S(L)is the corresponding sphere bundle and D(L)the disk bundle. Under the identification
+ΩSpin
+n(X)∼=ΩSpin
+n(D(L))and the Thom isomorphism ΩSpin
+n(X;L)→ΩSpin
+n+1(D(L),S(L)), the
+transferΩSpin
+n(X;L)→ΩSpin
+n(S(L))corresponds to the boundary map ΩSpin
+n+1(D(L),S(L))→
+ΩSpin
+n(S(L))in the long exact sequence of the pair (D(L),S(L)).
+Proof: Given [f: (Mn+1,∂M)→(D(L),S(L))]∈ΩSpin
+n+1(D(L),S(L)), we may assume f
+is transverse to the zero section Xof the line bundle. Then the corresponding element in
+ΩSpin
+n(X;L) is given by [ f−1(X)→X], and its image under the transfer map is represented
+byf−1(X)×XS(L)→S(L). But this is spin bordant to ∂M→S(L): Letp: [0,1]×S(L)→
+D(L) be given by ( t,v)/mapsto→tv. ThenM×D(L)([0,1]×S(L))→S(L) defines the spin bordism
+we need. The geometric interpretation is that we cut Malongf−1(X).q.e.d.
+193.3 Identifying the result of the construction
+We want to know which are the conjugation manifolds Xwe obtain by our construction.
+So we should (by Wall’s result) compute the first Pontrjagin c lass ofXand the triple cup
+products of degree 2 cohomology classes.
+By Zubr’s result it follows that if we equip our conjugation s paceXwith a map to
+K(Zm−1,2) which is an isomorphism on π2, then it suffices to know the spin bordism class of
+X.
+The process of starting with a manifold Wwith normal B-structure and boundary M×
+RP2and taking the union of the double cover VwithM×D3along the common boundary
+M×S2as described earlier makes sense for any such W, not only those Wwhich produce
+conjugation spaces. It can be interpreted as a map of sets of b ordism classes
+ΩB
+6(∂=M×RP2)→ΩSpin
+6(K(Zm,2),∂=M×S2)→ΩSpin
+6(K(Zm−1,2)).
+We still have to make precise how to obtain the map X→K(Zm−1,2) fromV→K(Zm,2):
+Letp:K(Zm,2)→K(Zm−1,2)bethemapinducedbythegrouphomomorphism Zm→Zm−1
+defined by ei/mapsto→eifori < mandem/mapsto→ −/summationtextei. The use of this map pcorresponds to the
+choice of the basis given by e1,...em−1for
+/an}bracketle{tei/an}bracketri}ht//an}bracketle{t/summationdisplay
+ei/an}bracketri}ht∼=H2(K(Zm,2))/ImH2(S2)
+which is isomorphic to H2(V)/Ker(H2(V)→H2(X))∼=H2(X) in the case where we produce
+a conjugation space X. We see that in integral cohomology p∗sendsvi/mapsto→vi−vm. We
+obtain a map V→K(Zm−1,2) as the composition of V→K(Zm,2) withp. The restriction
+ofV→K(Zm−1,2) to the boundary M×S2is nullhomotopic, which implies that the cor-
+responding element in H2(V;Zm−1) comes from H2(V,M×S2;Zm−1)∼=H2(X,M;Zm−1).
+SinceH2(X)→H2(V) is injective, this means that the element comes from a uniqu ely de-
+termined H2(X,Zm−1), or in other words that the map V→K(Zm−1,2) extends uniquely
+up to homotopy to X→K(Zm−1,2).
+In the above we use bordism classes of manifolds with a fixed bo undary. These bordism
+sets of manifolds with boundary are torsors ΩB
+6(∂=M×RP2) for the bordism group ΩB
+6,
+and Ω˜B
+6(∂=M×S2) for the bordism group Ω˜B
+6= ΩSpin
+6(K(Zm,2)).
+Moreover, the map
+ΩB
+6(∂=M×RP2)→ΩSpin
+6(K(Zm−1,2))
+is equivariant with respect to the group homomorphism given as the composition
+ΩB
+6tr→ΩSpin
+6(K(Zm,2))p∗→ΩSpin
+6(K(Zm−1,2)).
+To compute the group homomorphism, we use the injections ΩSpin
+6(K(Zr,2))→Zr⊕
+H6(K(Zr,2)) given by the ˆA-genera and triple cup products. Let us denote the generator s of
+Zrbydiand the generators of H6(K(Zr,2)) byeijkas before.
+20The image of [ Mf→K(Zm,2)] under p∗is [Mp◦f→K(Zm−1,2)]. In order to write it as a
+linear combination/summationtext
+iλidi+/summationtext
+i≤j≤kµijkeijkwe have to compute the coefficients
+λi=ˆA(M,(p◦f)∗(vi))
+=ˆA(M,f∗(vi−vm))
+=ˆA(M,f∗vi)−ˆA(M,f∗vm)+1
+2/an}bracketle{tf∗(−v2
+ivm+viv2
+m),[M]/an}bracketri}htand
+µijk=/an}bracketle{t(p◦f)∗(vivjvk),[M]/an}bracketri}ht
+=/an}bracketle{tf∗((vi−vm)(vj−vm)(vk−vm)),[M]/an}bracketri}ht
+=/an}bracketle{tf∗(vivjvk−vivjvm−vivkvm−vjvkvm+viv2
+m+vjv2
+m+vkv2
+m−v3
+m),[M]/an}bracketri}ht.
+Thus we obtain:
+di/mapsto→difori < m,
+dm/mapsto→ −/summationdisplay
+idi,
+eijk/mapsto→eijkfori≤j≤k < m,
+eijm/mapsto→ −/summationdisplay
+k/negationslash=i,jeijk−2eiij−2eijjfori < j < m,
+2eiim/mapsto→ −di−2/summationdisplay
+j/negationslash=ieiij−6eiiifori < m,
+eiim+eimm/mapsto→/summationdisplay
+j≤k
+j,k/negationslash=ieijk+/summationdisplay
+j/negationslash=ieiijfori < m,
+emmm/mapsto→ −/summationdisplay
+i≤j≤keijk.
+This implies that under p∗: ΩSpin
+6(K(Zm,2))→ΩSpin
+6(K(Zm−1,2)), the generators of the
+image of the transfer map are mapped in the following way:
+2di/mapsto→2difori < m,
+2dm/mapsto→ −/summationdisplay
+i2di,
+/summationdisplay
+i≤j≤keijk/mapsto→0,
+2eiii/mapsto→2eiiifori < m,
+m/summationdisplay
+j=12eiij/mapsto→ −di−6eiiifori < m,
+2eimm+/summationdisplay
+k/negationslash=i,k/negationslash=meimk/mapsto→di+6eiii+2/summationdisplay
+j/negationslash=ieiijfori < m,
+2eijj+/summationdisplay
+k/negationslash=i,k/negationslash=jeijk/mapsto→ −2eiijfori/ne}ationslash=jandi,j < m,
+2eijk/mapsto→2eijkfori < j < k.
+21Thus the image of ΩB
+6→ΩSpin
+6(K(Zm−1,2)) has generators difor 1≤i≤m−1, and
+2eijkfor 1≤i≤j≤k≤m−1.
+We know that the construction produces manifolds Xsuch that the Z2-cohomology ring
+ofXis isomorphic to the Z2-cohomology ring of Mwith all degrees multiplied by 2. But our
+calculation shows that by choosing different Win different bordismclasses producingdifferent
+conjugation 6-manifolds Xwith the same fixed point set M, we can modify the bordism class
+ofXbydiand 2eijk, which means that we can produce conjugations with fixed poin t set
+Mon any simply-connected spin 6-manifold with free cohomolo gy, which has Z2cohomology
+ring isomorphic to the Z2cohomology ring of Musing an isomorphism dividing all degrees
+by 2. This proves theorem 1.4.
+Finally, we prove corollary 1.5. Postnikov proved that those trilinear forms ton finite-
+dimensional Z2-vector spaces Vwith values in Z2which occur as the triple cup products of
+one-dimensional Z2-cohomology classes ofaclosedoriented 3-manifold areexa ctly thosewhich
+satisfyt(v,v,w) =t(v,w,w) for all v,w∈V[Pos48]. IfH1(M;Z) contains no summand
+Z2, then we have the additional condition t(v,v,w) = 0 for all v,w∈V(sinceH2(M;Z)
+contains no summand Z2, and so v2=Sq1(v) = 0 for all v∈H1(M;Z2)). Sullivan [ Sul75]
+constructed closed oriented 3-manifolds with free integra l cohomology realizing all trilinear
+formstsatisfying this additional condition.
+(Matthias Kreck explained to me how to prove the realizabili ty in a different way: The
+additional condition means exactly that the trilinear form comes from a Z-valued trilinear
+form on a free Z-module. Trilinear forms on Zmare in bijection with H3(K(Zm,1)). Now
+˜ΩSpin
+3(K(Zm,1))→H3(K(Zm,1)) is surjective, as one can see from the Atiyah-Hirzebruch
+spectral sequence. By surgery below the middle dimension, e very bordism class contains 2-
+connected maps M→K(Zm,1), so that H1(M;Z)∼=Zm, with the desired trilinear form
+given by triple cup products.)
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+23
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+arXiv:1001.0912v1  [hep-th]  6 Jan 2010FIELDS IN NONAFFINE BUNDLES. IV.
+Harmonious non-Abelian currents in string defects.
+Brandon Carter
+LUTh (CNRS), Observatoire Paris - Meudon.
+3 January, 2010.
+.
+Abstract. This article continues the study of the category of harmonious field
+models that was recently introduced as a kinetically non-linear gener alisation of the
+well known harmonic category of multiscalar fields over a supporting brane wordsheet
+in a target space with a curved Riemannian metric. Like the perfectly harmonious case
+of which a familiar example is provided by ordinary barotropic perfect fluids, another
+important subcategory is the simply harmonious case, for which it is s hown that as well
+as “wiggle” modes of the underlying brane world sheet, and sound ty pe longitudinal
+modes, there will also be transverse shake modes that propagate at the speed of light.
+Models of this type are shown to arise from a non-Abelian generalisat ion of the Witten
+mechanism for conducting string formation by ordinary scalar fields with a suitable
+quartic self coupling term in the action.
+1 Introduction
+This article continues a series in which a systematic covariant differen tiation procedure
+was developed [1, 2] and applied [3] to multiscalar field models for which t he relevant
+target space lacks the usual linear (vectorial or affine) structur e, but is curved, as
+in the prototypical example [4] of a harmonic map, which is governed b y dynamical
+equations involving non linear dependence on the fields but only linear d ependence
+on their space-time gradients. As well as the possibility of confineme nt to string or
+brane worldsheets, and the gauging of internal symmetries of the target space, two
+different kinds of generalisation of the ordinary harmonic category were considered.
+The first [2, 3] was that of forced-harmonic models in which a harmon ic type kinetic
+term is supplemented in the Lagrangian by a self coupling term ˆVhaving the form of a
+predetermined scalar field on the target space. The second kind wa s that of harmonious
+models, whose definition will be recapitulated in the next section, and and which differ
+from ordinary harmonic and forced-harmonic models in that their dy namical equations
+involve nonlinear dependence not just on the fields but also on their g radients.
+Nonlinear gradient dependence of the kind in question has long been f amiliar in the
+context of irrotational fluid models such as are relevent for super fluidity [5, 6] (and
+1perhaps also for cosmology [7, 8]). However the target space in th ese examples is of
+the usual flat vectorial kind, as also is that of their cosmic string su pported analogues
+of both the singly [9, 10] and multiply [11] conducting kinds that have b een studied in
+recent years.
+Interest in such string supported fields began when Witten [12] dre w attention to
+the fact that they would arise naturally by condensation in the core s of string (or other)
+topological defects in spacetime field models of the commonly conside red – kinemati-
+cally linear – kind in which the only nonlinearity is that of a scalar self coup ling term
+responsible for spontaneous symmetry breaking of the vacuum. I t was subsequently
+recognised that suitable macroscopic models [13, 14] for the descr iption of such fields
+in the thin string limit would need to involve nonlinear field gradient depen dence of
+the type qualifiable as harmonious, though only of the rather trivial kind in which the
+target space is flat and the relevant internal group Abelian
+The main purpose of the present article is to show that a straightfo rward – still
+kinematically linear – extension of the class of models proposed by Witt en will give rise
+to nonlinearly harmonious string models of a more interesting kind, in w hich the target
+space is curved and the relevant symmetry group non Abelian.
+Before proceeding it is important to warn that the subject treate d under the ti-
+tle “non-Abelian string conductivity” by Kibble and his collaborators [1 5] needs to be
+distinguished – e.g. by the insertion of another hyphen, so as to obt ain “non-Abelian-
+string conductivity” – from the subject of of the present study, which might appropri-
+ately be entitled “non-Abelian string-conductivity”. The point of th is nuance is that
+Kibble and his collaborators were concerned with a generalisation of W itten’s model
+wherein, instead of being attributable to the spontaneous breakd own of an Abelian
+U{1}symmetry, the string formation was attributable to the spontane ous breakdown
+of a non-Abelian SU {2}symmetry, but the currents considered by these authors were
+nevertheless merely Abelian in the sense of being generated only by a distinct U {1}
+symmetry subalgebra that had survived the breakdown. In contr ast, the present work
+will be concerned with a different kind of generalisation of Witten’s mod el, wherein –
+although the string itself will just be Abelian, in the sense of being att ributable to the
+spontaneous breakdown only of an Abelian U {1}symmetry subgroup – the currents
+therein will actually be non-Abelian in the sense of being generated, n ot just by a U {1}
+action, but by theactionof a surviving non-AbelianSU {2}orhigher symmetry algebra.
+Although the generalisation of the category of underlying kinetically linear space-
+time field models is straightforward, the Witten mechanism itself cann ot be directly
+employed for the construction of a curved target space, as it dep ends on a weak cylin-
+drical symmetry ansatz that will not be self consistently applicable in the nonAbelian
+case. It will be shown below in Section 8 how Witten’s weak symmetry an satz can be
+replaced for this purpose by a more general local geodicity ansatz , that works as a good
+approximation, and can do the job, so long as the currents involved are not too strong
+compared with a scale that will be estimated in Section 9.
+2It is to be commented that the evident incompatibility of cylindrical sy mmetry with
+a non degenerate mapping into a curved (spherical or more genera l) target space means
+that for a string loop retaining several non-commuting currents it will be impossible
+(as in the simple Goto-Nambu case) to attain a stationary circular vo rton type equilib-
+rium state: such a loop will be condemned to go on oscillating until comp letion of the
+dissipation of all currents except those of an Abelian subgroup.
+2 Harmonious field models
+According to the definition of the preceeding article [3], a multiscalar fi eldΦwith local
+components χAinaq-dimensionaltargetspaceover abraneworldsheet Sofdimension d
+(ford≤n, wherenis the dimension of the background spacetime)will beof harmonious
+type if it is governed by a scalar Lagrangian Lthat is specified by some equation of
+state as a diffeomorphically invariant function just of the relevant t arget space metric
+ˆgAB– which is supposed to have been prescribed in advance – and of the c orresponding
+horizontal projection ˆwABof the inverse gijof the underlying spacetime metric gij
+on the worldsheet. It is to be understood that the horizontality of the projection is
+specified with respect to a some gauge form Aiwith vectorial components AiAon the
+target space. We thereby obtain a prescription of the form
+ˆwAB=gijΦA
+| | |iΦB
+| | |j, (1)
+in which the relevant projector will be specified in terms of the field gr adient compo-
+nentsχA
+,iby
+ΦA
+| | |i=χA
+,i+AA
+i. (2)
+In terms of the symmetric target space tensor κABgiven by the definition
+κAB=−2∂L
+∂ˆwAB(3)
+and therefor such that
+κA
+CˆwBC=κB
+CˆwAC=−2∂L
+∂ˆgAB, (4)
+it was shown in the preceeding article [3] that, for a system of this ha rmonious type,
+the intrinsic dynamical equations will be expressible in terms of a set o f surface currents
+with components given by
+JAi=gijκABΦB
+| | |j. (5)
+as the pseudo-conservation laws
+DiJAi= 0, (6)
+3inwhichDiisabitensoriallycovariant differentiationoperatorofthenotso simp lekind
+introduced in earlier work [1, 2], andwhich will give rise to genuine curre nt conservation
+laws [3] only when suitable internal symmetry conditions are satisfied .
+One of the questions that arises in the study of any system of differ ential equations
+of motion is the orientation of the characteristic surfaces, with no rmal direction λi
+say, along which infinitesimal discontinuities can be propagated. For the transverse
+“wiggle”perturbationmodesoftheextrinsicevolutionofthesuppo rtingworldsheet, itis
+easily shown [10] that – regardless of the internal dynamics – the re levant characteristic
+equation will always have the simple form
+Tijλiλj= 0, (7)
+whereTijare the components of the surface stress energy tensor, which for a harmo-
+nious system of the kind under consideration here will be given [3] by
+Tij=κABΦA| | |iΦB| | |j+Lgij. (8)
+However for the internal perturbation modes, wihin the worldshee t, the form of the
+characteristic equation will depend on the specific details of the sys tem. In particular
+for the accoustic type modes of the harmonious system (6) the ch aracteristic equation
+will in general be rather complicated, reducing to a simple quadratic f orm like that of
+(7) only under special conditions, such as those of the simply harmo nious and perfectly
+harmonious categories that will be presented in the following section s.
+3 Harmoniously elastic models
+The category of harmonious models includes models of the perfect s olid type [16] which
+belong to the extensive harmoniously elastic subcategory that is characterised by the
+condition that the component matrix ˆwABshould have a well defined inverse ˆ γAB=
+ˆw−1
+AB, which, if it exists, will be interpretable as the tensorially well behave d metric that
+is locally induced on the target space by the section Φaccording to the specification
+ˆγACˆwCB=δB
+A, (9)
+This is something that will be possible only if the target-space dimensio nqdoes not
+exceed the dimension d=p+1 of the supporting base space, which if it is an embedded
+p-brane worldsheet can itself not exceed the dimension nof the background spacetime:
+q≤p+ 1≤n. The dimensionally maximal case q=p+ 1 includes various models
+of the recently investigated kind [17] referred to a hyperelastic, while perfect solids[16]
+and so, in particular, ordinary fluid models of the barotropic type, a re included in the
+caseq=p.
+Harmoniously elastic modelsareperfectly elastic inthe usual sense, meaning [18, 19]
+that they are governed by a Lagrangian that is determined by some prescribed intrinsic
+4structure as a function just of the induced metric ˆ γABon the relevant target space,
+but they are not elastic models of the most general kind: for non-h armoniously elastic
+models the prescribed intrinsic structure can include various other vectorial or tensorial
+fields (for example to allow for the anisotropic grain in wood) as well as the prescribed
+metric ˆgABwhich is all that is allowed in the harmonious case. In the usual approa ch
+[16, 18, 19, 20] to the treatment of elastic solid models, the locally ind uced metric ˆ γAB
+is what is used for lowering and raising of target space indices. It is th erefor important
+to remember that in the dimensionally unrestricted approach [1, 2, 3 ] followed here it is
+instead the globally prescribed target space metric ˆ gABthat it is used for this purpose.
+It will be convenient for the following discussion to introduce a new kin d of elasticity
+tensor that is defined, for any harmonious model, by
+EABCD=E(CD)(AB)= 2∂κAB
+∂ˆwCD. (10)
+In terms of this quantity the ordinary elasticity tensor of the usua l treatment [16, 18,
+19, 20] will be given by the expression
+EABCD=EABCD+2κA(CγD)B−κABγCD+2γA(CPD)B−γABPCD,(11)
+in which the pressure tensor is given in terms of the energy density ρ=−Lby
+PAB=κAB−ργAB. (12)
+A simple example [3] is that of the baby Skyrme model [21] for which the target space
+is a 2-sphere on which κAB=κ⋆gAB+α⋆(ˆwgAB−ˆwAB), whereκ⋆andα⋆are constants,
+so that one obtains EABCD= 2α⋆(gABgCD−gA(CgD)B).
+4 The sound cone in simply harmonious models
+Instead of working through the details of a complete perturbation analysis, the charac-
+teristic equation governing the propagation of infinitesimal discont inuities is obtainable
+efficiently by adaption, from the rather similar case of an elastic solid [2 0, 22], of a
+method due originally to Hadamard, of which the simplest illustration is p rovided by
+theDalembertianwaveequationforascalar ϕsay, namely ∇jϕj= 0whereϕj=gji∇iϕ.
+TheideaoftheHadamardmethodistousethefactthatthediscont inuityofthegradient
+of a continuous quantity will be aligned with the normal covector λiof the discontinuity
+surface. Applying this to the components ϕj, one sees that the discontinuity of their
+gradients will be given in terms of corresponding discontinuity amplitu de components
+/tildewideϕjby an expression of the form [ ∇iϕj] =λi/tildewideϕj. Moreover, taking the discontinuity of
+the integrability condition ∇[iϕj]= 0, one sees that the discontinuity amplitude will
+have to satisfy λ[i/tildewideϕj]= 0, and hence that it will be given in terms of some scalar am-
+plitude/tildewideϕby/tildewideϕi=/tildewideϕλi. Thus finally, taking the discontinuity of the the Dalembert
+5equation itself, one obtains the well known light-cone tangency con dition
+gijλiλj= 0. (13)
+Applying the same line of reasoning to the multiscalar field Φ, one sees that the
+discontinity of the gradient of its covariant derivative (2) will be give n in terms of some
+set of amplitude components /tildewideχAby
+[∇iΦA
+| | |j] =λiλj/tildewideχA. (14)
+It therefore follows from the definition (1) of the horizontally induc ed metric wABthat
+discontinuity of its gradient will be given by
+[∇iˆwAB] = 2λiλj/tildewideχ(AΦB)| | |j. (15)
+These quantities are needed for the evaluation of the discontinuity of the gradient of
+the current (5), which will be given by the formula
+[DiJAj] =κABgjk[∇iΦB
+| | |k]+∂κAB
+∂ˆwCDΦB| | |j[∇iˆwCD]. (16)
+in which the distinction between Diand∇idisappears, as the relevent [3] affine and
+gauge connection terms (but not their derivatives) will be continuo us. The discon-
+tinuity of the set of pseudo-conservation equations (6) thereby provides the required
+characteristic equation in the form
+λiλjQij
+AB/tildewideχB= 0, (17)
+in which, using the notation (10), we shall have
+Qij
+AB=gijκAB+EADBCΦC| | |iΦD| | |j. (18)
+It is to be remarked that this formula is simpler than the correspond ing expression
+using the usual elasticity tensor (11) of the traditional approach [16, 18, 19, 20].
+The eigenvalue equation ensuing from (17) may take a rather complic ated quartic or
+higher polynomial form when the target space dimension is two or mor e, with a generic
+equation of state involving dependence not just on the trace
+ˆw=ˆwAA= ˆgABˆwAB, (19)
+but also onhigher order invariants starting with ˆwABˆwBA. However it will conveniently
+separate into merely quadratic subsystems in what will be referred to as the simply
+harmonious case, namely that for which the Lagrangian depends onlyon the trace
+invariant ˆw, so that one obtains
+κAB=κˆgAB, (20)
+6with
+κ=−2dL
+dˆw. (21)
+In this simply harmonious case one obtains
+EABCD= 2dκ
+dˆwˆgABˆgCD, (22)
+which gives a characteristic equation of the form
+λiλi/tildewideχA+2
+κdκ
+dˆwλiΦA
+| | |iλjΦB
+| | |j/tildewideχB= 0. (23)
+It is evident that this will be trivially satisfied by a set of shake modes, propagating
+at the speed of light, with polarisation /tildewideχAthat is transverse to the current across the
+discontinuity, in the sense that
+/tildewideχAJAiλi= 0, (24)
+since for such a mode – regardless of the particular linear or non-line ar functional form
+of the equation of state – the characteristic equation will evidently reduce just to the
+same nullity condition as in the ordinary Dalembertian case (13), name ly
+λiλi= 0. (25)
+There will also be a less trivial set of sound type modes with polarisatio n that is
+longitudinal in the sense of being aligned with the current across the discontinuity, so
+that for such a mode the discontinuity amplitude vector /tildewideχAwill be given (modulo a
+multiplicative factor that can be absorbed into the normalisation of t he characteristic
+covectorλi) by the prescription
+/tildewideχA=ΦA
+| | |iλi. (26)
+This reduces the characteristic equation to a simple quadratic form – specifying what
+is describable as a sound cone – that will be given by
+/parenleftbigg
+gij+2
+κdκ
+dˆwwij/parenrightbigg
+λiλj= 0 (27)
+using the notation
+wij= ˆgABΦA
+| | |iΦB
+| | |j (28)
+for the gauge covariant pullback of the target space metric.
+Itwillbeshownbelowhownon-triviallynon-linearmodelsofthissimplyha rmonious
+type arise naturally in the treatment of string defects of multiscala r field theories of
+the common kinetically linear kind. However before doing that it is will be instructive,
+for the sake of comparison, to describe another noteworthy sub category, namely that
+ofperfectly harmonious models for which the characteristic equation will be similarly
+simplifiable
+75 The sound cone in perfectly harmonious models
+The subcategory of what are describable as perfectly harmonious models is physically
+important because it includes the generic (not necessarily irrotatio nal) case of an or-
+dinary barotropic perfect fluid. The perfectly harmonious subcat egory is defined by
+the requirement that the dependence of the Lagrangian on the ta rget space tensor ˆwAB
+should again involve only a single scalar invariant, but with the latter no w chosen to
+be the determinant det {ˆw}of the matrix with components
+ˆwAB= ˆgACˆwCB, (29)
+which will be admissible so long as the target space dimension qdoes not exceed the
+base space-time dimension d=p+ 1 (whereas for q > p+ 1 this determinant would
+vanish identically). Ordinary perfect fluids are of the particular kind for which the
+target space dimension is the same, q=p, as the space (as distinct from space-time)
+dimension of the supporting base, which is 3 in the usual terrestrial and astrophysical
+applications, but might be higher for exotic cosmological theories in w hich the space-
+time dimension is not 4 but 5 or more.
+In terms of the tensorial inverse matrix ˆ γAB=ˆw−1
+ABofˆwAB(which is interpretable
+as the metric locally induced on the target space by the section Φ) as defined by (9),
+the stipulation that Lshould depend only on det {ˆw}leads to the expression
+κAB=hˆγAB(30)
+with
+h=−2det{ˆw}dL
+d(det{ˆw}). (31)
+This quantity hwill be interpretable simply as the enthalpy density in the case of an
+ordinary perfect fluid, for which the pressure tensor (12) takes the formPAB=PˆγAB,
+in which the pressure is given in terms of the energy density ρ=−L, and the enthalpy
+densityhby the well known formula P=h−ρ.
+The ensuing formula
+∂κAB
+∂ˆwCD=det{ˆw}dh
+d(det{ˆw})ˆγABˆγCD−hˆγA(CˆγD)B, (32)
+can be used to reduce the characteristic matrix (18) to the form
+Qij
+AB= ˆγAB/parenleftbig
+hgij−κCDΦC| | |iΦD| | |j/parenrightbig
++/parenleftBig
+2det{ˆw}dh
+d(det{ˆw})−h/parenrightBig
+ˆγACΦC| | |iˆγBDΦD| | |j.(33)
+As before, this will be satisfied by trivial shake modes, with polarisat ion/tildewideχAthat is
+transverse to the current across the discontinuity in the sense s pecified by (24), since
+8for such a mode – regardless of the particular linear or non-linear fu nctional form of
+the equation of state – the characteristic equation will reduce to t he quadratic form
+/parenleftbig
+(h+L)gij−Tij/parenrightbig
+λiλj= 0, (34)
+withTijas given by (8).
+In the ordinary perfect fluid case this will take the degenerate for m (¯uiλi)2= 0,
+where ¯uiis the timelike (and physically well defined) unit fluid flow tangent vecto r
+that is characterised by the condition JA
+i¯ui= 0, meaning orthogonality to all the
+(separately unphysical, since target coordinate dependent) cur rents, and in terms of
+which the stress-energy tensor will take the familiar form Tij=h¯ui¯uj+Pgij.
+As before, there will also be a set of non-trivial sound type modes w ith polarisation
+that is longitudinal in the sense specified by (26), for which the char acteristic equation
+will reduce to the quadratic form
+/parenleftbigg
+hgij+2/parenleftBigdet{ˆw}dh
+hd(det{ˆw})−1/parenrightBig
+(Tij−Lgij)/parenrightbigg
+λiλj= 0, (35)
+which is what characterises the ordinary sound cone in the familiar pe rfect fluid case.
+6 Extended Witten models
+The physical relevance of the perfectly harmonious category pre sented in the imme-
+diately preceeding section is obvious, at least in the case of the ordin ary elastic solid
+and fluid applications for which the target space dimension is q= 3. However for
+the study of the simply harmonious category, as presented in the s ection before that,
+some physical motivation needs to be provided. In the case for whic h the target space
+is one dimensional (and for which simply harmonious means the same th ing as per-
+fectly harmonious) such a justification was provided by the demons tration [13, 14] that
+such models are what is appropriate for the macroscopic descriptio n of string defects
+in simple kinetically linear field models of a subcategory proposed by Witt en[12]. The
+purpose of the present section is to present a straightforward e xtension of Witten’s
+subcategory that can form string defects which will be shown in the following section
+to be macroscopically describable by simply harmonious models of a less trivial kind,
+with two – or higher– dimensional target spaces that are curved.
+Within the category characterised by a Lagrangian of the forced- harmonic type [3],
+the original Witten subcategory andthe extensions considered he re arecharacterised by
+two essential properties of which the first is that of having a (3+ q)-dimensional target
+spaceXof the ordinary flat kind, so that the symmetry group of the kinetic partLkin
+of the Lagrangian is 0 {3+q}. The second property is that the target space is however
+endowed withapotential function ˆVdepending onjust two scalar combinations, namely
+a squared “Higgs amplitude” Ψ2and a squared “carrier amplitude” X2. These are
+9obtained by decomposing the target space as a direct product of a 2-dimensional “Higgs
+field” space and a ( q+1)-dimensional “carrier field” space, so that the symmetry grou p
+of the whole Lagrangian,
+L=Lkin−ˆV, (36)
+will be generically broken down to the direct product, 0 {2}×0{q+1}, of a Higgs field
+symmetry group having the form 0 {2}with a “carrier” symmetry group having the
+form 0{q+1}. The idea is that the potential should be such that the 0 {2}symmetry of
+the Higgs part is spontaneously broken, so that the vacuum will adm it the occurrence
+of string type topological defects (which will be “local” if the symmet ry algebra of the
+“Higgs field” part is “gauged”) containing fields whose internal symm etries are just
+those of the carrier group 0 {q+1}.
+The “Higgs field” space can be taken to have flat coordinates Ψ1andΨ2say, which
+can be conveniently thought of as the real and imaginary parts of a complex field
+Ψ1+iΨ2=Ψeiψ, (37)
+in terms of which the 2-dimensional Higgs field part of the target spa ce metric will be
+dˆs2
+hig= dΨ12+dΨ22= dΨ2+Ψ2dψ2, (38)
+and corresponding Higgs field amplitude will be given by
+Ψ2=Ψ12+Ψ22. (39)
+Similarly the “carrier field” space can be taken to have flat target sp ace coordinates,
+Xasay,a= 1,...,q+1 in terms of which the carrier metric contribution will be
+dˆs2
+car=δabdXadXb, (40)
+whereδabis the unit matrix, and the carrier amplitude itself will be given by
+X2=δabXaXb. (41)
+These contributions combine to give the complete metric on the flat t arget space Xas
+dˆs2
+hig+dˆs2
+car, which means that the kinetic part of the Lagrangian will take the fo rm
+Lkin=Lhig+Lcar, (42)
+with
+Lhig=−1
+2/parenleftBig
+(DµΨ1)DµΨ1+(DµΨ2)DµΨ2/parenrightBig
+−FµνFµν
+16πe2, (43)
+and
+Lcar=−1
+2δab(∇µXa)∇µXb, (44)
+10where the internal gauge coupling of the Higgs field part has been inc orporated by the
+use of the covariant differentiation operation that is given by
+DµΨ1=∇µΨ1−AµΨ2,DµΨ2=∇µΨ2+AµΨ1,
+whereAµis a U{1}gauge form with curvature Fµν= 2∇[µAν], for which Gothic letters
+have beenusedtoindicatethat, althoughmathematically analagous , thisinternal gauge
+field is not meant to be physically interpretable as the ordinary electr omagnetic field.
+For a small but non-zero value of the coupling constant ethis gauge field enables the
+vortex defects of the model to be locally confined – without the loga rithmic energy
+divergence for which a long range “infra red” cut off would otherwise be needed.
+In his original formulation [12] Witten made the further postulate th at, as well as
+this internal gauge coupling of the Higgs field, there would also be an e xternal gauge
+coupling of the“carrier field” part to an analogous U {1}gauge form Aν, with curvature
+Fµν= 2∇[µAν], that was meant to be interpreted as that of an ordinary electrom agnetic
+field, with its own extra Lagrangian contribution FµνFνµ/16πe2. However – unless the
+corresponding coupling constant, esay, is set to zero – such a coupling engenders
+technical trouble by reintroducing the logarithmic “infra-red” dive rgence that had been
+removed by the other gauge coupling.
+The present treatment will be based on the supposition that gauge self-coupling of
+the “carrier field” part is weak enough to be neglected, so that the divergence problem
+is avoided, but this does not exclude allowance for passive coupling to an external
+background of electromagnetic or other conceivable radiation. It will nevertheless be
+supposed that such radiation is sufficiently weak to allow the gauge to be chosen so that
+the corresponding gauge form (namely Aνin the electromagnetic case) to be taken to
+be zero in a neigbourhood that is large compared with the internal dim ensions of the
+defect, so that within this neighbourhood there will be no further lo ss of generality in
+taking the kinetic part of the Lagrangian to have the simple form (43 ).
+The original Witten model was characterised by q= 1, so that the carrier target
+space coordinates could be considered as components of a complex fieldX1+iX2=
+Xeiχ. What I refer to as the minimally extended Witten model is character ised by
+q= 2, so that the carrier symmetry group will have the non-Abelian fo rm 0{3}, instead
+of the Abelian form 0 {2}that it had in the original Witten model.
+In a more elaborate – non-minimal – extension proposed for conside ration by Lilley
+et al.(private communication [23]) the carrier space dimension is taken to b e given by
+q+1 = 4. Instead of retaining the its full symmetry group 0 {4}, these authors took the
+carrier space to be endowed with a complex structure by grouping it s coordinates into
+a pair of complex fields X1+iX2andX3+iX4, so that the carrier symmetry group
+is reduced to the form U {2}. The latter has the structure of a direct product of an
+SU{2}group (which Lilley et al.took to be “gauged”) with a U {1}group (which they
+left as merely “global”, but which could just as well be taken to be cou pled to ordinary
+electromagnetism).
+11In all these cases the flat metric (40) of the ( q+1)-dimensional carrier part can be
+rewritten as
+dˆs2
+car= dX2+X2dˆΩ2(45)
+where dˆΩ2is the metric on the relevant symmetry-orbit space, X, which will be the
+unitq-sphere as given in terms of some system of coordinates χAby an expression of
+the form
+dˆΩ2= ˆgABdχAdχB. (46)
+More particularly, for the minimally extended model characterised b yq= 2, the
+standard choice χ1=ˆθandχ2= ˆϕwill beobtained by setting X1=Xsinˆθcosˆϕ,X2=
+Xsinˆθsinˆϕ, andX3=Xcosˆθ. In terms of these, the spherical metric components will
+be given by a prescription of the familiar form ˆ g11= 1, ˆg12= 0, ˆg22= sin2ˆθ.
+The basic idea behind cosmic string theory, as developed at first mos t notably
+by Kibble [24, 15] , was that short, effectively straight string segm ents in a locally
+uniform background neighbourhood could be approximated by Nielse n Olesen type
+vortex solutions of the underlying field model.The presence of longitu dinal currents in
+thevortexwasexcluded bytheratherstrongkindofcylindindrical symmetry postulated
+by anansatzof theNielsen Olsentype, but was admittedby a weaker kindof cylindrical
+symmetry ansatz that was subsequently introduced by Witten [12]. As shown by the
+recent work of Lilley et al.(private communication [23]) even the weaker kind of
+cylindical symmetry ansatz proposed by Witten is incompatible with th e simultaneous
+presence of several non-commuting longitudinal currents, whos e treatment will therefor
+require an ansatz of an even weaker kind that will be introduced in th e next section,
+whereby it is required that the cylindrical symmetry should hold only a pproximately
+in the relevant locally uniform background neigbourhood.
+Provided the external gauge coupling is sufficiently weak for its self c oupling to
+be neglected, it will be possible to choose the gauge so that the corr esponding (elec-
+tromagnetic) guage form Aνvanishes in the neighbourhood under consideration. The
+field equations for the remaining internal guage form Aνand for the flat target space
+components will then be expressible in kinetically decoupled form as a fi rst subsystem
+consisting of
+∇νFµν= 4πe2/parenleftbig
+Ψ2DµΨ1−Ψ1DµΨ2/parenrightbig
+, (47)
+and
+DµDµΨ1= 2∂ˆV
+∂(Ψ2)Ψ1,DµDµΨ2= 2∂ˆV
+∂(Ψ2)Ψ2, (48)
+for the gauge form and the Higgs field, and a second subsystem give n simply by
+∇µ∇µXa= 2∂ˆV
+∂(X2)Xa, (49)
+(in whicha= 1,...,q+1) for the carrier field.
+127 Witten’s weak symmetry ansatz
+Following Witten [12], attention will now be restricted to configuration s in which the
+Higgs subsystem is subject to the same symmetry conditions as in a s imple Nielsen-
+Olesen type vortex (for which the carrier subsystem is absent). T his means that in a
+flat spacetime, with respect to a cylindrical coordinates for which t he metric is
+ds2= d̺2+̺2dφ2+dz2−dt2, (50)
+the Higgs field and its gauge form are postulated to be longitudinally sy mmetric in
+the strong sense, to the effect that Ψ1andΨ1are independent of zandt, but they
+are required to be axially symmetric only in the weak (albeit strict [3]) se nse, meaning
+modulo an action of the primary symmetry group O {2}, so that the phase ψin (37) is
+allowed to have an angle dependence of the form
+ψ=nφ, (51)
+wherenis a fixed integer winding number, while the amplitude Ψcan depend only on
+̺. The corresponding ansatz for the internal gauge field is that it sh ould have the form
+Aµdxµ=Adφ, (52)
+in which the quantity Ais also a function only of ̺.
+Stillfollowing Witten[12], itwill bepostulatedthatthecarriersubsyst em isaxisym-
+metricinthestrongsense–meaningthatthefieldcomponents Xaareallindependent of
+φ–so thatusing a primefor differentiation withrespect to zandadot fordifferentiation
+with respect to t, their dynamical equations (49) will take the form
+1
+̺d
+d̺/parenleftbigg
+̺dXa
+d̺/parenrightbigg
+=¨Xa−X′′a+2∂ˆV
+∂(X2)Xa, (53)
+An ansatz of the restrictive Nielsen Olesen type postulated by Kibble [24] would
+also require staticity and cylindrical symmetry in the strong sense, meaning ˙Xa= 0
+andX′a= 0, so that the components Xashould depend only on ̺. The system of
+field equations would thereby be reduced from four to two dimension s, namely those of
+a flat cross section with fixed longitudinal coordinate values that ca n, without loss of
+generality, be taken to be t= 0 andz= 0. Macroscopic quantities such as the string
+energy per unit length will then be obtainable by integration over the cross section.
+Witten’s innovation [12] was to recognise that such a reduction to a t wo dimensional
+system on a flat cross section will still be obtainable from a less restr ictive ansatz
+whereby the longitudinal symmetry of the carrier field is required to be only of the
+weak type. The Witten ansatz can be decomposed into two success ive conditions, of
+which the first is that the amplitude Xcan depend only on ̺, so that
+˙X=X′= 0 (54)
+13but that subject, of course, to the ensuing restaints, namely
+δabXa˙Xb= 0, δabXaX′b= 0, (55)
+the longitudinal gradients ˙XbandX′bare allowed to have non vanishing values. The
+second condition of the Witten ansatz is that these gradient fields t hemselves should
+be longitudinally symmetric in the strong sense. It is this second cond ition that will
+have to be relaxed in the work that follows.
+There will be no obstacle to the implementation of such a Witten type s ymmetry
+ansatz provided the relevant symmetry-orbit space Xsay (meaning the quotient of
+the carrier field space by the action of its symmetry group) happen s to beflat– as
+in the single carrier component case originally considered by Witten, a s well as in
+multicomponent Abelian cases that have been considered more rece ntly [11]. In such
+cases a linear symmetry-preserving map fom the z,tplane to Xwill be available as a
+framework for parallel propagationof the field on the sample cross section att= 0, z=
+0, so as to construct a solution that remains valid for all values of tandz. Subject
+to the choice of a system of flatcoordinates χAon the symmetry-orbit space X, the
+Witten ansatz simply amounts to taking uniformly constant values fo rχ′Aand ˙χA. The
+supplementary requirement of invariance under the discrete symm etries of time and
+parity reversal imply the further restriction that the initial value o fχAatt= 0, z= 0
+be uniform over the cross section, meaning independent not just o fφbut also of ̺.
+Suchanidealprocedurewillunfortunatelybeavailableonlyforares trictedchoice[23]
+of the values of χ′Aand ˙χAif – as in the cases we are concerned with here – the rele-
+vantq-dimensional symmetry-orbit space Xiscurvedso that its symmetry algebra is
+non-Abelian, with the implication that Lie transport operations with r espect to differ-
+ent generators will fail to be mutually consistent. In order to obta in an effectively 2
+dimensional description on a sample cross section at t= 0,z= 0, in cases involving
+currents aligned with symmetry generators that do not commute, the Witten type weak
+symmetry ansatz will neeed to be replaced by something less restric tive.
+8 The local geodicity ansatz
+For the treatment of a generic current configuration, what I pro pose is something de-
+scribable as the local geodicity ansatz, whereby one is enabled to start from arbitrarily
+chosen uniform values of, χAand its derivatives χ′Aand ˙χAon an initial cross section at
+t= 0, z= 0. This ansatz retains the first condition of the Witten ansatz, as embodied
+in (54) and (55), but instead of a further symmetry requirement t he second condition
+of the local geodicity ansatz is that the value of χAthroughout a finite spacetime neigh-
+bourhood of the cross section should be obtained by the standard process of geodesic
+extrapolation with respect to the metric ˆ gABonX. According to this prescription, a
+coordinate pair {z,t}maps to a position specified by unit parameter value τ= 1 on
+14the affinely parametrized geodesic χA{τ}specified at τ= 0 by the tangent
+dχA
+dτ=t˙χA+zχ′A. (56)
+Solong asoneis concerned withderivatives ofatmost second order , which isall that
+is needed for the field equations in question, the application of this an satz is very easily
+achievable by choosing to work with local coordinates such that the relevant connection
+components vanish. In such a system the local geodicity ansatz sim ply means that all
+the second derivatives will also vanish:
+ˆΓB
+A C= 0⇒¨χA= ˙χ′A=χ′′A= 0. (57)
+When the symmetry orbit space Xis flat, this ansatz is evidently equivalent to
+a weak symmetry ansatz of the kind introduced by Witten. The adva ntage of the
+prescription (57) is that it is applicable even when Xis curved, and its disadvantage
+in that case is that it is not exactly applicable everywhere simultaneou sly, but only on
+the chosen cross section at t= 0, z= 0. It can however be adopted as a very good
+approximation so long as χAis restricted to a range that is small compared with the
+curvature scale of X.
+The concrete implementation of such an approximation procedure is conveniently
+achievable, for the extended Witten models introduced above, by t aking the local co-
+ordinates on the symmetry-orbit space Xto be specified by simply setting
+χA=Xa
+X,A=a−1,a= 2,...,q+1. (58)
+By substituting this in (40), one obtains the q-spherical metric (40) in the explicit form
+given by
+dˆΩ2=δABdχAdχB+(δABχAdχB)2
+1−χ2. (59)
+in which the deviation from flatness is attributable just to the last te rm, which will be
+negligible so long as the dimensionless quantity
+χ2=δABχAχB(60)
+is very small compared with unity.
+In the minimally extended case, for which q= 2, this metric will be that of an
+ordinary 2-sphere, with coordinates expressible as
+χ1= sinˆθsinˆϕ,χ2= cosˆθ. (61)
+15in terms of spherical coordinates of the usual kind, for which
+dˆΩ2= dˆθ2+sin2ˆθdˆϕ2. (62)
+Adoption of the convention that χA= 0, on the chosen initial cross section, as
+specified by t= 0, z= 0, is equivalent to requiring there that
+X1=X,Xa= 0∀a/ne}ationslash= 1, (63)
+which implies by (54) that the first derivatives of X1will vanish there,
+˙X1= 0,X′1= 0, (64)
+and by (55) that its second derivatives will be given there by
+¨X1
+X=−δAB˙χA˙χB,˙X′1
+X=−δAB˙χAχ′B,X′′1
+X=−δABχ′Aχ′B.(65)
+Since the metric (59) has the form required for the local geodicity a nsatz to take the
+form (57), whereby the second derivatives of χAall vanish where χA= 0, it follows that
+the second derivatives of the other components of Xawill also vanish there,
+¨Xa=˙X′a=X′′a= 0∀a/ne}ationslash= 1. (66)
+We thereby obtain
+¨X1−X′′1=ˆwX,¨Xa−X′′a= 0∀a/ne}ationslash= 1, (67)
+whereˆwis defined according to (19) as the the trace of the target-space tensor
+ˆwAB=χ′Aχ′B−˙χA˙χB,
+so that, where χA= 0, it will be given, independently of ̺, by
+ˆw=δAB(χ′Aχ′B−˙χA˙χB) =1
+X2δab(X′aX′b−˙Xa˙Xb).
+Under these circumstances, the dynamical equations of the carr ier subsystem (53)
+will be satisfied automatically for a/ne}ationslash= 1, so this subsystem reduces to a just a single
+non trivial equation, which will take the form
+1
+̺d
+d̺/parenleftbigg
+̺dX
+d̺/parenrightbigg
+=/parenleftBigg
+ˆw+2∂ˆV
+∂(X2)/parenrightBigg
+X. (68)
+This can be seen to be independent of the internal dimension q, and thus exactly the
+same as that of its analogue for the original unextended Witten mod el. It is evident
+16that the carrier contribution to the kinetic part (44) of the Lagra ngian density (44) will
+also reduce to the same form as for the original Witten model, namely
+Lcar=−1
+2ˆwX2. (69)
+Within the framework of the local geodicity ansatz, the preceeding work establishes
+that in terms of the constant parameter ˆwand the radially dependent field variable
+X2the carrier contribution in the extended model is indistinguishable fr om the carrier
+contribution in the original model. It follows that, in terms of the sca lar trace ˆw, the
+extended model will share, with the original Witten model, an equatio n of state of
+exactly the same simply harmonious form for the ensuing string mode l Lagrangian,
+L=/integraldisplay
+2π̺Ld̺, (70)
+as obtained [13, 14] by integration over the cross section using field values satisfying
+the dynamical equations consisting of (68) for the carrier amplitud e in conjunction with
+the Higgs subsystem consisting of (47) and (47).
+It is to be remarked that instead of interpreting the independent v ariableˆwin the
+simply harmonic equation of state as the trace of the target space tensorˆwAB, it can
+equivalently be characterised as the trace
+w=wi
+i, (71)
+of the worldsheet tensor defined by (28), which in the local neighbo urhood where AiA
+vanishes will take the simple form wij= ˆgABχA
+,iχB
+,j. As well as having the same trace,
+w=ˆw, (72)
+the target-space tensor ˆwABand base-space tensor wijshare their quadratic invariant
+wj
+iwi
+j=ˆwB
+AˆwA
+B, (73)
+which can thereby be seen to be positive definite, due to the Euclidea n signature of
+ˆgAB(whereas the signature of the 2-dimensional worldsheet metric gijis Lorentzian).
+9 Conclusions
+The upshot of the forgoing work is that the extension of Witten’s fie ld model will have
+vortices macroscopically describable by a conducting string model o f simply harmonic
+type, in the sense of being governed by a Lagrangian depending jus t on the trace wof
+the base space tensor wijdefined by (28) as the pullback of the prescribed target space
+17metric ˆgAB. It is instructive to present the properties of such models using a p referred
+worldsheet reference frame of orthonormal type, meaning
+g00=−1,g11= 1,g01= 0, (74)
+that is chosen so as to diagiagonalise the pullback wij, of which the non-vanishing
+components will be the eigenvalues that are given in terms of the inva riantswand
+wj
+iwi
+jby the expressions
+w00=/radicalbigg
+w2
+4−det{w}+w
+2,w11=/radicalbigg
+w2
+4−det{w} −w
+2,(75)
+in which the relevant invariant determinant, namely that of wj
+i, is defined as
+det{w}=w0
+0w1
+1−w1
+0w0
+1=1
+2(w2−wj
+iwi
+j). (76)
+It is to be noticed that positive definite character of the target me tric ˆgABentails the
+non-negativity of the eigenvalues,
+w00≥0,w11≥0, (77)
+and hence the non- positivity of the determinant:
+ˆwB
+AˆwA
+B−ˆw2=−2det{w} ≥0. (78)
+It is also to be observed that the determinant will necessarily vanish – meaning that
+the equality ˆwB
+AˆwA
+B=ˆw2will automatically be satisfied so that either w00orw11will
+be zero – in the special case [23] for which the currents happen to b e all aligned with the
+sameone-parameter subgroup generator in X, which is the only kind of configuration
+for which a stationary circular vorton type equilibrium state will be po ssible.
+In terms of the eigenvalues w00andw11, the corresponding non-vanishing compo-
+nents of the surface stress energy tensor (8) will be given by the formulae
+T00=κw00−L,T11=κw11+L, (79)
+in whichLandκdepend only on the trace, w=w11−w00.
+The original Witten field model gave rise to an ensuing conducting str ing model
+that was of ordinarily elastic type in the sense of having a target spa ce dimension q
+that was the same as the space dimension p(not the spacetime dimension p+1) of the
+supporting worldsheet, which in the string case is simply p= 1. It was found[13] that
+a good approximation for the equation of state of the ensuing strin g model could be
+provided in terms of a microscopic length scale δ⋆and a couple of mass scales mand
+m⋆– depending [14] on the particular functional form of the interactio n potential ˆV–
+by the formula
+L=−m2−1
+2m2
+⋆ln{1+δ2
+⋆w}, (80)
+18which gives
+κ=m2
+⋆δ2
+⋆
+1+δ2⋆w,1
+κdκ
+dw=−δ2
+⋆
+1+δ2⋆w. (81)
+The work in the preceeding sections implies that the minimally extended Witten model
+with the same interaction potential ˆVwill give rise to an ensuing conducting string
+model that will be governed by the sameequation of state, even though it will not be
+of the ordinary elastic type with q=p, but of the hyperelastic type [17] with q=p+1,
+that is to say with a target space having a dimension that is equal to t he space-time
+dimension of the supporting wordldsheet, namely p+ 1 = 2 in the string case under
+consideration. It can be seen from the characteristic equation (2 7) that, according to
+(81), the speed cLof longitudinal sound type perturbations relative to the preferre d
+reference system will be given in this hyperelastic case by
+c2
+L=1−δ2
+⋆(w00+w11)
+1+δ2⋆(w00+w11), (82)
+wherew00andw11are the – necessarily non-negative – eigenvalues given by (75), of
+which the sum in (82) will be expressible as w00+w11=/radicalBig
+2ˆwB
+AˆwA
+B−ˆw2.
+It is to be remarked that the dually related [25, 10] “electric” and “m agnetic”
+varieties of the ordinarily elastic case obtained from the original une xtended Witten
+model can be considered as degenerate limits of this hyperelastic ca se: the “electric”
+variety is characterised by w=−w00withw11= 0, and the “magnetic” variety is
+characterised by w=w11withw00= 0.
+As well as the longitudinal modes characterised by (81), there will o f course be
+extrinsic “wiggle” perturbation of the world sheet, with propagatio n speedcE, which
+according to (7) will be given in terms of the energy density U=T00and the string
+tensionT=−T11by the universally [10, 26] valid formula
+c2
+E=T
+U. (83)
+The novelty in the minimally extended Witten model, as contrasted with the original
+Witten model, is that the ensuing string model has a third kind of pert urbation mode.
+As well as the longitudinaly polarised sound type modes governed by ( 82) and the
+extrinsic “wiggle” modes governed by (83), there will also be intrinsic “shake” modes
+characterised (with respect to the curved target space X) by the transverse polarization
+condition (24) with propagation speed cTgiven according to (25) by
+c2
+T= 1, (84)
+which simply means that (like the perturbation modes of the underlyin g field model
+with the flat target space X) these shear type modes will travel at the speed of light.
+19Having completed the presentation of this simply harmonious hypere lastic string
+model, we still need to consider theextent towhich itsuse isjustifiab le asamacroscopic
+string description of vortex defects in the minimal extension of the Witten field model.
+The foregoing derivation relied on a local geodicity ansatz whose app licability depended
+on a local flatness approximation whose range of validity is limited by th e requirement
+that the dimensionless magnitude χ2defined by (60) should remain small.
+Subject to the conditions of (57), the local geodicity ansatz mean s that for small but
+nonzero values of tandzthe value of χAwill be given approximately by χA= ˙χAt+χ′Az,
+or more concisely, in terms of the worldsheet coordinates σ0=tandσ1=z, byχA=
+χA
+,iσi. It follows that the required magnitude χ2will be given in this approximation
+by the formula χ2=wijσiσj. Thus more particularly, with respect to the preferred
+longitudinal coordinate system characterised by (75), it will be give n by
+χ2=w00t2+w11z2. (85)
+For the approximate flatness approximation to be utilisable, the nec essary condition,
+χ2≪1, (86)
+must be satisfied in a range of tandzthat is large compared with the thickness δof
+the vortex deffect, which entails the requirements w00δ2≪1 andw11δ2≪1.
+This leads us to the final conclusion, which is that the condition for ne gligibility
+of the curvature term in (59) – and thus for the validity of the simply harmonious
+conducting stringmodel–will beexpressible simply asanupperbound onthequadratic
+field gradient invariant ˆwB
+AˆwA
+B(and hence by (78) also on ˆw2) in the form
+ˆwB
+AˆwA
+Bδ4≪1. (87)
+So long as the string thickness δis small enough, this condition will hold automat-
+ically by (78) as a consequence of the requirement for stability again st longitudinal
+perturbations, namely the condition c2
+L>0, which according to (82) will take the form
+(2ˆwB
+AˆwA
+B−ˆw2)δ4
+⋆<1. (88)
+Within this allowed range there will clearly be no danger of violation of (8 7) provided
+that (as is usually supposed in the cosmological context envisaged b y Witten) the
+length scales associated with the carrier field are relatively large, an d the mass scales
+correspondingly small, compared with those associated with the Higg s field:
+δ2
+⋆≫δ2,m2
+⋆≪m2. (89)
+There is never any possibility of instability of the shake modes charac terised by (84),
+and the usual extra requirement c2
+E>0, for stability against extrinsic wiggle pertur-
+bations – which is equivalent to the condition that the tension Tgiven by (79) should
+be positive – makes little difference in the circumstances of (89), as it can be seen that
+it will hold automatically if (88) is replaced by the only marginally stronge r condition
+(2ˆwB
+AˆwA
+B−ˆw2)δ4
+⋆<1−exp{−2m2/m2
+⋆}.
+20Acknowledgements
+The author is grateful to Marc Lilley, Jerome Martin, and Patrick Pe ter for stimu-
+lating conversations.
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+22
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+arXiv:1001.0913v1  [astro-ph.EP]  6 Jan 2010Version 2.5 — October 29, 2018
+Discovery of the Transiting Planet Kepler-5b
+David G. Koch,1William J. Borucki,1Jason F. Rowe,1,2Natalie M. Batalha,3
+Timothy M. Brown,4Douglas A. Caldwell,5John. Caldwell,6William D. Cochran,7
+Edna DeVore,5Edward W. Dunham,8Andrea K. Dupree,9Thomas N. Gautier III,10
+John C. Geary,9Ron L. Gilliland,11Steve B. Howell,12Jon M. Jenkins,5
+David W. Latham,9Jack J. Lissauer,1Geoff W. Marcy,13David Morrison,1Jill Tarter,5
+ABSTRACT
+We present 44 days of high duty cycle, ultra precise photometry of the 13th
+magnitude star Kepler-5 (KIC 8191672, Teff=6300 K, log g=4.1), which exhibits
+periodic transits with a depth of 0.7%. Detailed modeling of the transit is con-
+sistent with a planetary companion with an orbital period of 3 .548460±0.000032
+days and a radius of 1 .431+0.041
+−0.052RJ. Follow-up radial velocity measurements with
+the Keck HIRES spectrograph on 9 separate nights demonstrate that the planet
+is more than twice as massive as Jupiter with a mass of 2 .114+0.056
+−0.059MJand a
+mean density of 0 .894±0.079 g/cm3.
+1NASA Ames Research Center, Moffett Field, CA 94035
+2NASA Postdoctoral Program Fellow
+3San Jose State University, San Jose, CA 95192
+4Las Cumbres Observatory Global Telescope, Goleta, CA 93117
+5SETI Institute, Mountain View, CA 94043
+6York University, Toronto, Ontario, Canada
+7University of Texas, Austin, TX 78712
+8Lowell Observatory, Flagstaff, AZ 86001
+9Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02 138
+10Jet Propulsion Laboratory/California Institute of Technology, Pa sadena, CA 91109
+11Space Telescope Science Institute, Baltimore, MD 21218
+12National Optical Astronomy Observatory, Tucson, AZ 85719
+13University of California, Berkeley, Berkeley, CA 94720– 2 –
+Subject headings: planetarysystems —stars: individual (Kepler-5, KIC 8191672,
+2MASS 19573768+4402061) — techniques: spectroscopic — Facilitie s: The Ke-
+pler Mission.
+1. INTRODUCTION
+The launch of Kepleroffers a special opportunity to study the nature of transiting ex-
+trasolar planets through transit photometry. The Keplermission is designed to determine
+the frequency of terrestrial-size planets. The Keplermission also provides an excellent plat-
+form to study and understand the wide variety of extrasolar plane ts. From the spacecraft’s
+Earth-trailing orbit heliocentric photometric measurements are de void of artifacts usually
+associated with instruments in close proximity to the Earth, due to p roblems such as the
+day-night cycleandeffectsimposedbytheEarth’satmosphere ong roundbasedobservations,
+and orbital effects for satellites in low earth orbit.
+We report the discovery and confirmation of Kepler-5b, a strongly irradiated transiting
+hot Jupiter. We describe the Keplerphotometry and transit modeling, and the follow-up
+observations used to confirm that Kepler-5b is a planet, including an orbital solution using
+radial velocities obtained with HIRES on Keck 1.
+2.KEPLER PHOTOMETRY
+TheKeplerphotometer consists of a 0.95-m Schmidt telescope feeding an arra y of 42
+CCDs with a broad (430 - 890 nm) spectral response (Koch et al. 20 10). From its Earth-
+trailing heliocentric orbit and single field of view, Keplercan monitor more than 100,000
+stars nearly continuously for the lifetime of the mission.
+Photometry of Kepler-5b (KIC 8191672, α= 19h57m37s.68,δ= +44◦02′06′′.2, Kep-
+mag=13.369) was obtained during 1 May - 15 June 2009, at an integra tion time of 29.426
+minutes for each observation. This run includes 9.7 days of photome tric data taken while
+ground analysis was being performed in preparation for start of sc ience operations, and is
+referred to as the Q0 data set (Haas et al. 2010). Kelper science o perations began on 12 May
+2009, which compose the second data set of 33.5 days of photomet ry referred to as the Q1
+data set. There is a gap with a length of 30 hours between Q0 and Q1 t o facilitate contact
+between the spacecraft and groundstations for downlink of the Q 0 data stream.
+After pipeline data processing and photometry extraction, as des cribed in Jenkins et al.
+(2010) and Caldwell et al. (2010), the time series stream was detre nded with a running 1-day– 3 –
+mean. All observations that occurred during a transit were rejec ted from the evaluation of
+the mean. The detrended photometry is presented in Figure 1.
+2.1. Initial Transit Fits and False-Positive Detection
+From examination of Q0 light curves, Kepler-5 and a few dozen other candidates were
+initially flagged as KeplerObjects of Interest (KOI). To identify the best candidates for
+ground-based follow-up observations, a number of metrics are co mputed to help recognize
+andreject stellar binary systems thatcanmimic aplanetarytransit signal. Aquick summary
+of these steps, as it relates to Kepler-5 are listed below. For a full d escription of the steps
+taken to identify false positives see Batalha et al. (2010).
+The transit lightcurve is modeled using the analytic expressions of Ma ndel & Agol
+(2002) using V-band non-linear limb darkening parameters Claret (2000). The ste llar ra-
+dius (R⋆), effective temperature ( Teff) and surface gravity (log g) are fixed to values adopted
+fromKeplerInput Catalog (KIC). The stellar mass ( M⋆) is usually calculated from log g
+andR⋆. In the case of Kepler-5, a larger stellar radius of 1.9 R⊙was required to be con-
+sistent with the transit duration. With M⋆andR⋆fixed to their initial values, a transit fit
+is then computed to determine the orbital inclination, planetary rad ius, and depth of the
+occultation (passing behind the star) assuming a circular orbit. The best fit is found using
+a Levenberg-Marquardt minimization algorithm (Press et al. 1992).
+Theseinitialfitsareusedtodeterminewhetherthetransitisrepre sentitive ofaplanetary
+event. In the case of a stellar binary where the surface brightnes s of the two components
+differ, the observed depth of the odd and even numbered transits will differ. There was no
+measurable difference in the transit depths to report.
+For systems that show only primary transits, a search is made for a weak occultation
+assuming that the orbit is circular, namely at phase 0.5. In the cases where an occultation
+is found with significance greater than 2 σfor the depth, the dayside temperature of the
+planet can be estimated. From the depth of the occultation, the flu x ratio of the planet
+and star ( FP/F⋆) over the instrumental bandpass can be obtained. The depth of t he transit
+indicates the ratio of the planet and star radii. By assuming that the star and planet both
+behave as blackbodies and the flux ratio is bolometric, the dayside eff ective temperature can
+be estimated. For Kepler-5 we find FP/F⋆= 3.6×10−5. This gives a planetary effective
+temperature of Teff= 1720±214K, where an error of 30% is assumed for the input stellar
+luminosity andradius. This estimate isalower limit, asasignificant fract ionoftheplanetary
+flux is emitted at wavelengths longer than the red edge of the Keplerbandpass, but it is a– 4 –
+useful diagnostic in order to determine whether the depth of the o ccultation is consistent
+with a strongly irradiated planet. To make this comparison we can est imate the equilibrium
+temperature,
+Teq=T⋆(R⋆/2a)1/2[f(1−AB)]1/4, (1)
+for the companion, where R⋆andT⋆are the stellar radius and temperature, with the planet
+at distance awith a Bondalbedo of AB, andfisa proxy for atmospheric thermal circulation.
+We assume AB= 0.1 for highly irradiated planets (Rowe et al. 2006) and f= 1 for efficient
+heat distribution to the night side. Assuming stellar irradiationis the p rimary energy source,
+these parameter choices give a rough estimate for the dayside tem perature of the planet.
+Assuming a 30% error in the input stellar parameters and that the st ar and planet act as
+blackbodies we find Teq= 1810±289 for Kepler-5b. The consistency of TeffandTeqto first
+order suggests that the occultation is consistent with a strongly ir radiated planet. If the
+estimate of Teffwere found to be much larger than Teq, then the companion is likely to be
+self-luminous and is probably a star.
+2.2. Centroid Shifts
+TheKeplerpipeline provides measurements of the centroids of stellar images wit h a
+precision of about 0.1 millipixel for the brightness of Kepler-5 (Batalh a, 2010). Examination
+of the image centroids during transits revealed shifts with an amplitu de of 0.2 millipixels.
+Suchamotioncanresultifthephoto-apertureforKepler-5include slightfromanothersource.
+We examine and demonstrate with speckle and adaptive-optic (AO) im ages in Section 3.1
+that the centroid motion observed for Kepler-5 can be accounted for by the presence of
+nearby faint stars with constant brightness.
+3. FOLLOW-UP OBSERVATIONS
+After a KOI passes the above tests, additional ground-based fo llow-up observations are
+obtained as described in Gautier et al. (2010). These observations include high-resolution
+imaging to search for additional sources of flux within the Keplerphotometric aperture that
+would dilute the depth of the transit, and reconnaissance spectro scopy to confirm and refine
+the KIC stellar classification and to search for evidence of stellar co mpanions.– 5 –
+3.1. High-Resolution Imaging
+Ground-based visible-light speckle imaging from the WIYN Telescope a nd near-infrared
+AO imaging from the Mt. Palomar 5-m telescope show that Kepler-5 ha s two companions.
+The first companion is 0 .9′′away and 5.2 magnitudes fainter, as seen in the NIR-AO image,
+but is not visible in the speckle image. The other companion, KIC 81916 80, is 7.3′′away
+with aKeplermagnitude of 17.69. The difference in Keplermagnitude between Kepler-5
+and KIC 8191680 can be used to compute the expected centroid sh ift, depending on which
+object is assumed to be the source of the transit signal.
+If KIC 8191680is a background eclipsing binary, then the centroids are expected to shift
+by +14.0 and −0.8 millipixels in the column and row directions of the detector, respectiv ely.
+If KIC 8191680 shows no flux changes and Kepler-5 is indeed the sou rce of the transit
+signal, then the expected centroid shift is −0.3 and 0.0 millipixels, which is consistent with
+the centroid shift reported in Section 2.2. The transit depth is estim ated to be diluted by
+∼2±0.2%. We include the effects of dilution in our transitfits.
+The speckle image shows a single star within its 2′′square field. This rules out very
+close background eclipsing binaries that might simulate the observed transits, except for
+stars lying closer than about ∼0.1′′from the target star.
+3.2. Reconnaissance Spectroscopy
+The FIbre-fed Echelle Spectrograph (FIES) on the 2.5-m Nordic Op tical Telescope
+(NOT) was used on 4 June 2009 to obtain a spectrum of Kepler-5 for the purposes of
+stellar classification. Stellar parameters of log g= 4.0 andTeff= 6500K were estimated,
+which were used to refine the transit model fits described in §2.1. The spectrum shows no
+indication that the system includes an eclipsing binary or has a compos ite spectrum.
+3.3. HIRES Spectroscopy
+HIRES spectra were obtained from 3-6 June, 2-4 July, and on 6 Oct ober, 2009. Fig-
+ure 2 shows the radial velocities for Kepler-5 folded with the photom etric period of the
+planet. An analysis of the Keck/HIRES template spectrum by D. Fisc her using SME
+(Valenti & Piskunov 1996) measured the stellar parameters as liste d in Table 2. A fit to
+the velocities with the eccentricity fixed to zero measures a reflex v elocity semi-amplitude
+ofK= 228±8ms−1with residuals of 15ms−1. A bisector analysis of the HIRES velocities– 6 –
+does not show any significant variations correlated with the radial v elocities.
+4. ANALYSIS
+TheKeplerphotometricandHIRESradial-velocitymeasurementscanbesimulta neously
+modeled. The data is fit for the center of transit time, period, impac t parameter ( b), the
+scaled planetary radius (R p/R⋆), the amplitude of the radial velocity (K), photometric and
+velocity zeropoints and ζ/R⋆. The last term, ζ/R⋆is related to the transit duration ( Td=
+2(ζ/R⋆)−1) and the mean stellar density (P´ al et al. 2009). We also account fo r non-circular
+orbitsby modeling for ecos( ω), e sin(ω), where eis theorbital eccentricity and ωisargument
+of periastron.
+The modeled stellar density is strongly dependent on the impact para meter,b, of the
+planetary orbit. It was discovered that the adopted limb-darkenin g parameters were placing
+unrealistic constraints on b. Specifically, the best χ2would occur at b= 0 for a majority
+of the models of Keplerlight curves. The model fits were repeated for various choices of
+limb darkening, including values generated specifically for the Keplerbandpass with Atlas
+9 models by A. Prsa. Detailed examination of the fits revealed that th e limb darkening
+models were over-predicting the curvature of the base of the tra nsit shape. To alleviate
+this problem, limb darkening coefficients were computed for the thre e known exoplanet
+systems in the Keplerfield of view, TrES-2 (Daemgen et al. 2009), HAT-P-7 (P´ al et al.
+2008; Gilliland et al. 2010), and HAT-P-11 (Bakos et al. 2009), based on fits fixed to the
+published values of M⋆,R⋆, andi. We then linearly interpolate between these values for
+candidates with different temperatures. It is estimated this proce dure results in a ∼0.5◦
+systematic error on the quoted value of ireported in Table 2.
+The error distribution forthestellar densities thatfit the transit lig ht curve areobtained
+from a Markov-Chain-Monte-Carlo (MCMC) analysis. We adopt the a pproach outlined in
+§4.3 of Ford (2005). Convergence was tested by generating 10 Mar kov-chains each with 106
+steps and different initial conditions. We checked that each chain ha d statistically similar
+means and distributions. Given the stellar effective temperature fr om the Keck/HIRES tem-
+plate and the mean stellar density from transit photometry we emplo y theρ⋆method. This
+method uses stellar evolution Yale-Yonsei tracks to calculate the r ange of stellar parame-
+ters consistent with the observations. A full description of the ρ⋆method is described in
+citetbor10b.
+The output of the ρ⋆method is a set of Markov Chains giving a consistent set of 105
+allowedM⋆andR⋆pairs. The distribution of stellar mass and radius allows us to define– 7 –
+a transition probability to compute a MCMC analysis to determine the m ost likely model
+values. We fit for stellar mass and radius, planetary mass and radius , center time of transit,
+orbital period and inclination, depth of the occulation and allow for no n-circular orbits
+through the inclusion of e cos( ω) and e sin( ω). We generated 10 sets of Markov-Chains with
+106elements and use all 10 sets to generate distributions for the mode l fits.
+The errors in Table 2 are centered around the mode of each model p arameter which
+represents the value of highest relative probability. The mode is calc ulated by binning the
+bootstrappedparametersinto50equally-spaced bins. Wechoset ousethemodetodetermine
+the most likely system parameters as some distributions, such as th e stellar mass, are bi-
+modal. We report the bounds that enclose ±68% of the samples centred on the mode and
+report these values in Table 2.
+5. DISCUSSION
+Similar to Kepler-7 (Latham et al. 2010), the Kepler-5 host star is not much hotter
+than the Sun, Teff= 6297±60, but is much more massive and larger than the Sun, M⋆=
+1.374+0.040
+−0.059M⊙andR⋆= 1.793+0.043
+−0.062R⊙. With a mean stellar density of ρ⋆= 0.33±0.02
+gcm−3the star is on its way to becoming a shell-burning subgiant. Similar to Kepler-4
+(Borucki et al. 2010b) and Kepler-7 there are two distinct sets of model parameters that
+fit the observations equally well. There are two peaks in the mass dist ribution centered at
+M⋆= 1.21 and 1.38M⊙. While our analysis favours the larger mass, the lower mass choice
+can not be ruled out.
+The planet is found to have a mass of MP= 2.114+0.056
+−0.059MJand a radius of RP=
+1.431+0.041
+−0.052RJ, whereRJrefers to the equatorial radius of Jupiter. This makes the planet
+more massive and larger than Jupiter. The mean planetary density is 0.89±0.08 gcm−3,
+which is not uncommon amongst the known population of transiting Ju piter-sized planets.
+Figure 3 from Latham et al. (2010) compares the mass and radius of Kepler-5b to other
+known transiting extrasolar planets.
+Thephotometrypointstoaweak(2 σ)detection ofanoccultation. Ascorrelationsinthe
+timeseries may lead to a false detection of an occultation we compute d a Fourier Transform
+and autocorrelation of the dataset which did not show any significan t issues. There is no
+excess of power in the Fourier Transform from 1 to 12 hours has a 3 σdetection limit of ∼15
+ppm.
+Most of the observed light from the planet in the Keplerbandpass is due to thermal
+emission, but we can use the depth of the occultation to derive an up per limit on the albedo– 8 –
+of the planet. The geometric albedo,
+Ag=Fp
+F⋆a2
+R2
+p, (2)
+is defined as the ratio of the planet’s luminosity at full phase to the lum inosity from a
+Lambertian disk, where Rpis the planetary radius. This allows us to place boundaries on
+the dayside temperature of the planet as estimated by the equilibriu m temperature given
+by Equation 1. Figure 3 shows the boundaries for Kepler-4b, 5b, 6b and 7b where we
+have calculated Teqforf=1,2 and ABfrom zero to the 1- σupper limit. The Bond albedo
+is estimated from the geometric albedo by assuming a Lambertian sca tter as described in
+Rowe et al. (2006). Due to the relatively short periods of the first f our planets discovered by
+Kepler, it not surprising that all of them have temperatures greater tha n 1500 K. The upper
+limits show that Kepler-5b, 6b and 7b are less reflective than Jupiter and likely quiet dark,
+consistent with predictions and observations of other hot extras olar planets. Additional
+photometry of these planets will provide special opportunities to s tudy planets in extreme
+conditions.
+Funding for this Discovery mission is provided by NASA’s Science Mission Directorate.
+We would like to thank D. Fischer, A. Prsa and everyone that has con tributed to the Kepler
+mission.
+REFERENCES
+Bakos, G. ´A., et al. 2009, ApJ, in press (arXiv:0901.0282v1)
+Batalha, et al. 2010, ApJ, this issue
+Borucki, W. J., et al. 2010, Science, TBD
+Borucki, W. J., et al. 2010, ApJ, this issue
+Caldwell, D. A., et al. 2010, ApJ, this issue
+Claret, A. 2000 A&A, 363, 1081
+Daemgen, S., et al. 2009, A&A, 498, 567
+Dunham, E. W., et al. 2010, ApJ, this issue
+Ford, E.B. 2005, ApJ, 129, 1706– 9 –
+Fig. 1.— The phased light curve of Kepler-5 containing 12 transits obs erved by Kepler
+between 1 May and 15 June 2009. The upper panel shows the full 44 -day time series after
+detrending. The bottom panel shows the light curve folded with the orbital period. The
+lower curve shows the primary eclipse, with the fitted transit model overplotted in red and
+corresponding scaletotheleft. Theuppercurvecovers theexpe ctedtimeofoccultation, with
+the fitted model overplotted in green and corresponding scale fou nd to the right, expanded
+by a factor of 5 relative to the scale for the transit data.– 10 –
+Fig. 2.— The phased radial-velocity curve for Kepler-5 consisting of 8 epochs observed
+using the Keck/HIRES spectrometer spanning 126 days. The over plotted fit assumes a
+circular orbit, phased to match the transit photometry. A tabulat ion of the radial-velocity
+measurements used for this Figure may be found in the online materia ls.– 11 –
+Fig. 3.— The dayside temperatures and 1 σupper bounds on the Bond albedo for Kepler-4b
+(hatched), Kepler-5b (solid), Kepler-6b (outside) and Kepler-7b (cross hatched) based on
+the 1-σupper limits on the the depth of the occultation.– 12 –
+Gautier, T. N., et al. 2010, ApJ, this issue
+Gilliland, R., et al. 2010, PASP, submitted
+Haas, M. et al. 2010, ApJ, this issue
+Jenkins, J. J., et al. 2010, ApJ, this issue
+Koch, D. G., et al. 2010, ApJ, this issue
+Latham, D. W., et al. 2010, ApJ, this issue
+Mandel, K., & Agol, E. 2002, ApJ, 580, L171
+Morel, P. 1997, A&A, 124, 597
+P´ al, A., et al. 2008, ApJ, 680, 1450
+P´ al, A., et al. 2009, arXiv:0908.1705v2
+Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 19 92, Numerical Recipes
+in Fortran 77, Second Edition, Cambridge University Press, 678
+Rowe, J. F., et al. 2006, ApJ, 646, 1241
+Walker, G. A. H., et al. 2008, A&A, 482, 691
+Valenti, J. A., & Piskunov, N. 1996 A&AS, 118, 595
+Yi, S. K., Demarque, P., Kim, Y.-C., Lee, Y.-W., Ree, C. H., Lejeune, T., & Barnes, S.
+2001, ApJS, 136, 417
+This preprint was prepared with the AAS L ATEX macros v5.2.– 13 –
+Table 2. System Parameters for Kepler-5
+Parameter Value Notes
+Orbital period P(d) 3 .548460±0.000032 A
+Midtransit time E(HJD) 2454955 .90122±0.00021 A
+Scaled semimajor axis a/R⋆ 6.06±0.14 A
+Scaled planet radius RP/R⋆ 0.08195+0.00030
+−0.00047A
+Impact parameter b≡acosi/R⋆ 0.393+0.051
+−0.043A
+Orbital inclination i(deg) 86 .◦3±0.5 A
+Orbital semi-amplitude K(ms−1) 227 .5±2.8 A,B
+Orbital eccentricity e < 0.024 A,B,G
+Center-of-mass velocity γ(ms−1) 0 A,B
+Observed stellar parameters
+Effective temperature Teff(K) 6297 ±60 C
+Spectroscopic gravity log g(cgs) 3 .96±0.10 C
+Metallicity [Fe/H] +0 .04±0.06 C
+Projected rotation vsini(kms−1) 4 .8±1.0 C
+Mean radial velocity (kms−1) −46.7±4.1 B
+Derived stellar parameters
+MassM⋆(M⊙) 1 .374+0.040
+−0.059C,D
+RadiusR⋆(R⊙) 1 .793+0.043
+−0.062C,D
+Surface gravity log g⋆(cgs) 4 .07±0.02 C,D
+Luminosity L⋆(L⊙) 4 .67+0.63
+−0.59C,D
+Age (Gyr) 3.0±0.6 C,D
+Planetary parameters
+MassMP(MJ) 2 .114+0.056
+−0.059A,B,C,D
+RadiusRP(RJ, equatorial) 1 .431+0.041
+−0.052A,B,C,D
+Density ρP(gcm−3) 0 .894±0.079 A,B,C,D
+Surface gravity log gP(cgs) 3 .41±0.03 A,B,C,D
+Orbital semimajor axis a(AU) 0 .05064±0.00070 E
+Equilibrium temperature Teq(K) 1868 ±284 F
+Note. —
+A: Based on the photometry.
+B: Based on the radial velocities.
+C: Based on a MOOG analysis of the FIES spectra.
+D: Based on the Yale-Yonsei stellar evolution tracks.
+E: Based on Newton’s version of Kepler’s Third Law and total m ass.
+F: Assumes Bond albedo = 0.1 and complete redistribution.
+G: 1 sigma upper limit
\ No newline at end of file
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+arXiv:1001.0914v2  [cond-mat.mtrl-sci]  23 Jun 2010epl draft
+Superdiffusive, heterogeneous, and collective particle mo tion near
+the fluid-solid transition in athermal disordered material s
+Claus Heussinger1, Ludovic Berthier2andJean-Louis Barrat1
+1Universit´ e de Lyon; Univ. Lyon I, Laboratoire de Physique d e la Mati` ere Condens´ ee et Nanostructures; UMR CNRS
+5586, 69622 Villeurbanne, France
+2Laboratoire des Collo¨ ıdes, Verres et Nanomat´ eriaux, UMR CNRS 5587, Universit´ e Montpellier 2, 34095 Montpellier,
+France
+PACS05.20.Jj – Statistical mechanics of classical fluids
+PACS64.70.qd – Thermodynamics and statistical mechanics
+PACS83.80.Fg – Granular solids
+Abstract. - We use computer simulations to study the microscopic dynam ics of an athermal
+assembly of soft particles near the fluid-to-solid, jamming transition. Borrowing tools developed
+to study dynamic heterogeneity near glass transitions, we d iscover a number of original signatures
+of the jamming transition at the particle scale. We observe s uperdiffusive, spatially heterogeneous,
+andcollective particle motion over acharacteristic scale which displays asurprising non-monotonic
+behavior across the transition. In the solid phase, the dyna mics is an intermittent succession of
+elastic deformations and plastic relaxations, which are bo th characterized by scale-free spatial
+correlations and system size dependent dynamic susceptibi lities. Our results show that dynamic
+heterogeneities in dense athermal systems and glass-forme rs are very different, and shed light on
+recent experimental reports of ‘anomalous’ dynamical beha vior near the jamming transition of
+granular and colloidal assemblies.
+Many materials, from emulsions and suspensions to
+foams and granular materials are dense assemblies of non-
+Brownian particles [1]. Since thermal energy is irrelevant,
+the dynamics of these systems must be studied in driven
+non-equilibrium conditions. Together with the driving
+mechanism, a second important control parameter is the
+volume fraction of the particles, φ. These materials un-
+dergo a fluid-to-solid ‘jamming’ transition as φincreases
+beyond some critical density φc[2].
+The properties of this jamming transition have received
+considerable interest in recent years, and much progress
+was made through the analysis of idealized theoretical
+models, such as soft, frictionless, repulsive particles as
+studied below [3]. Structural and mechanical properties
+of systems on both sides of the jamming transition have
+been analyzed [4,5], and a number of remarkable features
+emerged, such as algebraic scaling of linear mechanical
+response [6], or the development of nontrivial timescales
+or lengthscales characterizing the macroscopic behavior of
+the system [7–9].
+With the developments of new experimental techniques
+(confocal microscopy, original light scattering techniques,etc.), thedynamicsofsystemsnearthejammingtransition
+can now be resolved at the particle scale. Very recently,
+a number of ‘anomalous’ or ‘unexpected’ dynamic behav-
+iors were reported in granular and colloidal assemblies:
+superdiffusive particle motion [10,11], nonmonotonic vari-
+ationsof characteristicscalesacrossthe transition [11–14],
+anomalous drop of dynamic correlations upon compres-
+sion [13,15]. Thus, it appears timely to investigate, at
+the fundamental level of single particle trajectories, the
+signatures of the jamming transition in idealized theoret-
+ical models. Our work reveals original signatures of the
+jamming transition in both fluid and solid phases which,
+we believe, shed light on recent experimental findings. We
+also establish a connection between single particle dynam-
+ics and collective particle motion, which allows us to de-
+velopan intuitive and appealing picture of jamming as the
+consequence of the diverging size of rigid particle clusters.
+We study a bidimensional 50:50 binary mixture of
+Nparticles of diameter ratio 1.4 with harmonic repul-
+sion [16], V(rij) =k(rij−σij)2, whererijis the distance
+between the centers of particle iandj,σij= (σi+σj)/2,
+andσiis the diameter of particle i. Particles only inter-
+p-1Claus Heussinger1 Ludovic Berthier 2 Jean-Louis Barrat1
+act when they overlap, V(rij> σij) = 0. We work in
+the zero temperature limit, so that all the dynamical pro-
+cesses in the system are induced by an external driving.
+We use a simple shear flow, and perform quasi-static shear
+simulations, as described before [17]. Very small shear
+strains are applied, followed by a minimization of the po-
+tential energy. At each step, particles are first affinely
+displaced along the y-axis by an amount δy=γ0xwith
+γ0= 5×10−5. AppropriateLees-Edwardsperiodic bound-
+ary conditions are used. During the subsequent energy
+minimization, additional nonaffine displacements occur in
+both directions. In the dilute limit, these nonaffine dis-
+placements are absent, and their presence directly reveals
+the influence of interactions and interparticle correlations.
+In the following, we focus on the purely nonaffinedisplace-
+ments occurring along the x-direction, i.e. transverse to
+the flow. We use system sizes N= 900, 1600 and 2500 to
+detect finite size effects. The unique control parameter is
+the volume fraction, φ. Below a critical value, φc≃0.843,
+which is well defined and sharp in the thermodynamic
+limit, the system flows with no resistance, while a yield
+stress grows continuously from 0, when φincreases above
+φc[18]. This corresponds to the jamming transition for
+the present system and driving conditions.
+The simplest quantity which is measured from parti-
+cle displacements is the root mean-squared displacement,
+∆(γ) =/angbracketleftBig
+1
+N/summationtextN
+i=1[xi(γ)−xi(0)]2/angbracketrightBig1/2
+,wherexi(γ)−xi(0)
+is the transverse displacement of particle iafter a total
+strainγis applied. In this expression, the brackets in-
+dicate averages taken over independent initial conditions
+all chosen in the driven stationary state at a particular
+density. We present the evolution of ∆( γ) at various vol-
+ume fractions across φcin Fig. 1. The displacements in
+the yield-stress flow regime above φcare diffusive at all
+strains, ∆ ∼√γ. This is a robust signal of the quasi-
+static dynamics in yield-stress materials. It can be traced
+back to the accumulation of plastic rearrangements span-
+ning the entire system [19,20]. The important observation
+here is that the diffusion constant depends only weakly on
+density and distance to φc.
+The regime below φcis more interesting. At any given
+volume fraction, particles move in a superdiffusive, ballis-
+tic manner at short strain, ∆ ∼γ, before crossing over
+to diffusive motion at large strain. Since we work at zero
+temperature, this ballistic regime is completely unrelated
+to the trivial short-time ballistic displacements in particle
+systems with classical Newtonian dynamics.
+Superdiffusive behavior can be interpreted from the ob-
+servation that below the jamming transition, the network
+of particle contacts is not sufficient to insure mechan-
+ical stability, and there exists a large number of zero-
+frequency modes allowing particle displacements at no en-
+ergy cost [21]. The actual displacement field is a superpo-
+sition ofthese modes, which can thus persist until the par-
+ticle configuration has evolved sufficiently to decorrelate
+the mode spectrum. Persistence of the modes, followed10-410-310-210-1100
+10-410-310-210-1100101∆(γ)
+γ∼ γ∼ γ1/2
+φ = 0.8500
+0.8550
+0.8600
+0.8400
+0.8380
+0.8340
+0.8250
+Fig. 1: Nonaffine displacement ∆( γ) for various volume frac-
+tionsφ. Forφ > φ c, diffusive motion is observed at all γ. In
+contrast, a superdiffusive regime develops for φ < φ c, with a
+particle ‘velocity’ ℓ∆(φ) = ∆(γ)/γwhich increases when ap-
+proaching φc.
+by their decorrelation explain superdiffusive and diffusive
+motion, respectively.
+Remarkably, Fig. 1 shows that particles move faster
+whenφis increased towards φcin both ballistic and diffu-
+sive regimes. In particular, the short-time particle ‘veloc-
+ities’ℓ∆(φ)≡limγ→0∆(γ)/γincrease with φ. This is a
+surprising finding because compression towards jamming
+usually yields slower dynamics [11,15]. Note that ℓ∆is a
+displacement per unit of strain and thus has the dimen-
+sion of length. Thus an increasing particle velocity in fact
+implies an increasing length scale. Indeed, we will argue
+below that superdiffusion reflects the concerted motion of
+solid-likeclustersof particlescorrelatedovera length scale
+comparable to ℓ∆(φ), and which grows towards φc.
+The displacements ∆ and the associated length scale
+ℓ∆(φ) measure how much particles have to move in addi-
+tion to the affine displacements, in order to accomodate
+the imposed shear flow. At φ= 0.840, for example, this
+additional displacement is about ten times larger than the
+affine contribution itself. This means, that close to φc
+the system is in a highly ‘fragile’ state and small changes
+in the boundary or loading conditions lead to large-scale
+motions [22].
+Interestingly, superdiffusion was recently used to iden-
+tify the location of the jamming transition in a bidimen-
+sional granularsystem [11]. Due to the ‘random agitation’
+driving mechanism in that experiment, superdiffusion is
+observed on a modest time window, and only very close to
+φc. This is consistent with our finding that superdiffusion
+is indeed most pronounced near the transition.
+Asecond, often studied, correlationfunction to quantify
+single particle dynamics is the ‘overlap’ function [11]
+Q(a,γ) =/angbracketleftBigg
+1
+NN/summationdisplay
+i=1exp/parenleftbigg
+−[xi(γ)−xi(0)]2
+2a2/parenrightbigg/angbracketrightBigg
+.(1)
+p-2Collective particle motion near fluid-solid transition
+ 0.2 0.4 0.6 0.8 1
+ 0.0002  0.0004  0.0006  0.0008Q
+γφ=0.834
+φ=0.846φ=0.900φ = 0.834
+0.838
+0.840
+0.846
+0.850
+0.900
+Fig. 2: The average overlap Q(a,γ) fora= 0.001 and different
+volume fractions across the jamming transition. The relax-
+ation becomes faster when φincreases towards φc, but slows
+down when φincreases above φc. The fluid and solid phases
+are thus readily distinguished by their distinct behavior u pon
+compression.
+Theoverlap Q(a,γ)goesfrom1to0astypicalparticledis-
+placementsget largerthan the probinglength scale a. It is
+thus very similar to a self-intermediate scattering function
+in liquids, with aplayingthe role ofan inversewavevector,
+andγthe role of time.
+The behavior of Q(a,γ) is presented in Fig. 2 for a fixed
+a= 0.001 (qualitatively similar results are obtained for
+different choices of a) and various volume fractions across
+φc. Forφ < φ c, we find that the overlap function de-
+cays faster when φincreases. This is consistent with the
+above observation that displacements get larger closer to
+φc. A qualitatively different behaviour is found above the
+transition, where the overlap decays more slowly when φ
+increases. This is in striking contrast with the behavior of
+the mean squared displacement in Fig. 1 which showed no
+such variations with φ.
+The data in Fig. 2 thus imply the existence of an un-
+expected nonmonotonic variation of the dynamical be-
+haviour, which is absent in the root mean-squared dis-
+placement, Fig. 1. To quantify these differences we define
+the analog ℓQ(φ) of the displacement scale, ℓ∆(φ), dis-
+cussed above. For a given strain γwe measure the value
+ofasuch that
+Q(a≡γℓQ(φ),γ) = 0.5. (2)
+The value 0.5 is arbitrary and we find similar results with
+0.3 and 0.7. For small strains within the ballistic regime
+this definition insures that ℓQis independent of strain γ
+and reflects a short-time velocity, as does its analog ℓ∆.
+For larger strains a crossover to diffusive behavior sets
+in similar as for the root mean-square displacement ∆
+(Fig. 1). In the following, we want to avoid this crossover
+and thus choose γas small as possible.
+In Fig. 3 we report the behaviour of ℓQ(φ) as defined 1 2 4 7
+ 0.83  0.84 0.85 0.86  0.87 0.88  0.89  0.9displacement scales l
+φ∼|δφ|−0.9∼δφ−0.25lQ
+lQel
+l∆
+Fig. 3: Relaxation length scale ℓQ(φ) quantifying the decay
+of the overlap Q(a,γ) for a single strain-step γ=γ0, from
+Eq. (2) ( ℓel
+Qis defined from purely elastic deformations above
+jamming). It is a nonmonotonic function, with a maximum at
+the jamming transition, mirroring the behavior of the overl ap
+in Fig. 2. It obeys power law behavior with different exponent s
+on both sides of the transition. In the fluid phase, ℓQbehaves
+similarly to the ‘velocity’ ℓ∆defined from superdiffusive be-
+haviour.
+from Eq. (2), which clearly reflects the nonmonotonic be-
+haviouroftheoverlapobservedinFig.2. Thisresultshows
+that a remarkably simple statistical analysis of single par-
+ticle displacements in athermal systems very easily reveals
+the existence of, and quite accurately locates, the fluid-to-
+solid jamming transition. This measurement requires no
+finite size scaling, or other involved or indirect statistical
+analysis [11,12,15].
+We have discussed two signatures of the jamming tran-
+sition from the behavior of ∆ (superdiffusion) and Q(a,γ)
+(nonmonotonic relaxation as a function of density). It
+should come as a third surprise that both dynamic quan-
+tities apparently provide distinct informations, in partic-
+ular for φ > φ c, although they both aim at quantifying
+dynamics at the particle scale. In fact, such discrepancies
+are well-known and well-studied in the field of the glass
+transition where several similar ‘decoupling’ phenomena
+have been studied [23]. Indeed, if the particle dynamics
+were a Gaussian process, the information content of ∆
+andQwould be mathematically equivalent. Their differ-
+ent behavior thus suggests that the distribution of single
+particle displacements is strongly non-Gaussian [25].
+To substantiate this claim we report the evo-
+lution of the van Hove distribution, P(x,γ) =/angbracketleftbig1
+N/summationtext
+iδ(x−[xi(γ)−xi(0)])/angbracketrightbig
+, as a function of volume
+fraction below φc(Fig. 4). Above φcthe shape of the van
+Hove function has a form well-known from recent work
+on sheared glasses [19,20]; therefore, we do not show our
+data here. Briefly, the distribution is ‘bimodal’, with a
+first component at small xresulting from displacements
+during reversible elastic branches, and a second compo-
+nent at larger xdue to irreversible plastic events. This
+p-3Claus Heussinger1 Ludovic Berthier 2 Jean-Louis Barrat1
+10-610-410-2100
+-40 -20  0  20  40∆⋅P(x)
+x/∆0.8400
+0.8380
+0.8340
+φ = 0.8250
+Fig. 4: Displacement distribution after a single strain ste p,γ=
+γ0, for different volume fractions φ < φ c, rescaled such that a
+Gaussian process (dashed line) has unit variance. Increasi ngly
+heterogeneous distributions of particle displacements ar e found
+when approaching φc, with the development of algebraic tails
+yielding a diverging kurtosis at φc.
+latter contribution completely dominates the second mo-
+ment of the distribution, and thus controls the diffusive
+behavior of ∆( γ) in Fig. 1.
+The decomposition between elastic and plastic branches
+aboveφcsuggests a separate analysis of these two types
+of dynamics. We have computed Q(a,γ) along elastic
+branches only, and defined ℓel
+Q(φ), the ‘elastic’ analog of
+the displacement scale ℓQ(φ) for the full overlap; it is
+shown in Fig. 3. Both ℓQandℓeq
+Qare very close, con-
+firming that Q(a,γ) is not very sensitive to plastic rear-
+rangements, in contrast to ∆( γ). The scaling of the typi-
+cal elastic response above φchas recently been discussed
+in great detail [24,26,27], and we simply quote the re-
+sult:ℓel
+Q∼(φc−φ)−1/4∼ℓQ, in good agreement with our
+numerical results.
+Belowφcthe distributions remain highly non-Gaussian
+(Fig. 4), even though plastic eventsare absent. Deviations
+from a Gaussian shape become stronger as φincreases to-
+wardsφc. The distribution develops polynomial tails, and
+it is thus not surprising that different averages taken over
+such distributions provide quantitatively distinct results.
+In Fig. 3, we show that ℓQ∼(φc−φ)−0.9, forφ < φ c.
+In contrast, we find that the scale ℓ∆, defined in Fig. 1,
+scales as ℓ∆∼(φc−φ)−1.1with an exponent slightly dis-
+tinct from the one of ℓQ. Hence, despite the appeal of the
+algebraic ‘scaling’ relations reported in Fig. 3, our parti-
+cledistributions P(x,γ)donotfollowsimplescalingforms,
+evenalongelasticbranches,inapparentcontradictionwith
+recent claims [27]. Single particle dynamics is not ‘univer-
+sal’ near the jamming transition and it is somewhat am-
+biguous to define ‘typical’ displacements from such broad
+distributions.
+It is tempting to speculate that the polynomial tails
+which develop near but below φcin Fig. 4 are in fact the
+‘precursors’ of plastic events taking place above φc. TheFig. 5: Grey-scale plot of the spatial fluctuations of the ove rlap
+in Eq. (1) (dark=‘immobile’, light=‘mobile’) for φ= 0.825<
+φc(left),φc= 0.840 (middle)and φ= 0.85> φc(right). Strain
+γ=γ0and probing length scale achosen such that Q≈0.5.
+Spatial correlations are increasing with φtowards φc, but stay
+comparable to system size above φcwhen the system is in the
+solid phase.
+tailsin factbecomesobroadthat thekurtosisofthe distri-
+bution (also called ‘non-Gaussian parameter’) actually di-
+verges at small γwhenφcis approached from below (data
+not shown). This implies that the kurtosis could be used
+as a second, simple statistical indicator of the underlying
+jamming transition. In glass-formers, the non-Gaussian
+parameter increases as the glass transition is approached,
+but muchmoremodestly [28], whilevan hovedistributions
+develop exponential rather than algebraic tails [25].
+Let us now turn to the discussion of particle correla-
+tions. This will allow us to make concrete the notion of
+‘rigid’ clusters that we have alluded to above. To this end,
+we first discuss the images shown in Fig. 5, where the spa-
+tialfluctuationsoftheoverlapareshown. Fromthisfigure,
+we can readily identify dynamically correlated clusters of
+particles, that clearly grow in size upon approaching φc
+frombelow. Theseimagesaredirectevidencethatdynam-
+ics becomes more collective as φcis approached. Above φc
+the system is now solid, and responds as an elastic body.
+Correspondingly, mobility fluctuations are correlated on a
+length scale comparable to the system size, the only visi-
+ble effect of the density being that the snapshot looks less
+‘disordered’ at the largest φ. Very similar observations
+were recently made experimentally in a soft granular ma-
+terial [14].
+To quantify these qualitative observations, we study the
+dynamical susceptibility χ4, defined as the variance of
+statistical fluctuations of the overlap Q[29],χ4(a,γ) =
+N/parenleftBig/angbracketleftBig
+ˆQ(a,γ)2/angbracketrightBig
+−Q(a,γ)2/parenrightBig
+, where ˆQrepresents the in-
+stantaneous contribution to the average Q. Our results
+are summarized in Fig. 6. In all cases, χ4(a,γ) displays
+a well-defined maximum when shown as a function of γ,
+which shifts towards lower strain when φincreases. This
+maximum identifies a strain for which fluctuations of the
+overlap are maximal, and is slaved to the typical strain
+overwhich Qdecaysin Fig. 2, as commonly found in stud-
+iesofdynamicheterogeneity[30,31]. As iswell-known,the
+height of this maximum is a good estimate of the number
+ofparticlesrelaxinginacorrelatedmanner[31], whichjus-
+tifies the fundamental importance of χ4. The increase of
+p-4Collective particle motion near fluid-solid transition
+ 0 10 20 30 40 50 60
+10−410−310−210−1χ4
+γa = 0.001
+a = 0.01
+a = 0.1φ = 0.8400
+0.8380
+0.8360
+0.8340
+ 20 40 60 80 100 120 140 160 180
+10-410-3χ4
+γN = 2500
+2500,elastic
+1600
+1600,elastic
+900
+900,elastic
+Fig. 6: Dynamical susceptibility χ4as function of strain γfor:
+(Top) different volume-fractions φ < φ cand probing length
+scalea. (Bottom) a single φ= 0.85> φcand different system
+sizes. The dynamic susceptibility increases when φincreases
+towards φcreflecting increasingly collective dynamics. Above
+φcdynamic correlations are system size dependent, even when
+only elastic branches are considered.
+the height of the peak of the dynamic susceptibility when
+φincreases towards φcin Fig. 6 thus quantitatively con-
+firms the visual impression given by the snapshots: the
+dynamics is increasingly collective when φincreases to-
+wardsφc. As shown in Fig. 6, this conclusion does not
+depend on our choice of a probing length scale a.
+To quantify the possible divergence of the spatial corre-
+lation of the dynamics, we analyze the data as above for
+the overlap. For a given strain γ, we look for asuch that
+χ4is maximum (it corresponds roughly to ℓQ(γ) defined
+in Eq. (2)) and measure its height. We report the evo-
+lution of this maximum as a function of volume fraction
+in Fig. 7. The data for three different system sizes super-
+impose below φc, and we can satisfactorily describe the
+φ-dependence as: χ4∼(φc−φ)−1.8. Comparing this ap-
+parent divergence with the much more modest increase of
+dynamic correlations in glass-forming liquids close to the
+glass transition [32], emphasizes once more the qualitative
+differences between both types of transitions. 10 100
+ 0.825  0.83 0.835  0.84 0.845  0.85 0.855  0.86χ4max
+φ∼δφ−1.8
+Fig. 7: Evolutionofthemaximumofthedynamicsusceptibili ty
+χ4across thejamming transition ( γ=γ0; symbols asinFig. 6).
+Itdiverges algebraically as φincreases towards φcandis system
+size independent. Above φc, the maximum is proportional to
+the system size, with an amplitude which decays slowly upon
+compression, both for plastic and elastic dynamics.
+Assuming that correlated domains have a compact ge-
+ometry, this finding immediately translates into a gen-
+uine dynamic correlation length scale, ξ4, which grows as
+ξ4=√χ4∼(φc−φ)−0.9. Strikingly, we found a similar
+power law divergence for both ℓQandℓ∆in Figs. 1 and 3.
+The exponents are furthermore close to the one obtained
+in previous work [17] using the spatial dependence of the
+displacement correlation function.
+These findings call for a simple relation between single-
+particle displacements, as characterized by ℓQorℓ∆, and
+particlecorrelations,asgivenby ξ4orbydisplacementcor-
+relations. The picture that we propose assumes particles
+to form ‘temporarily rigid’ clusters of the size of the corre-
+lation length ξ4. Driven by the shear-strain γthese clus-
+ters move or rotate in a solid-like manner over distances
+∆cluster=ξ4·γ. This motion shows up as pronounced bal-
+listic regime in the mean-squared displacement or in the
+decay of the overlap function Qand allows to make the
+identification, ℓQ≃ξ4. The cross-overtoparticlediffusion
+then corresponds to a typical cluster lifetime. On longer
+strains, clusters break up and lose their identity. This
+process is evident in the reduction of the dynamical cor-
+relation length as measured by the decreasing amplitude
+ofχ4withγin Fig. 6.
+Moving to φ > φcin Fig. 6, we observe that χ4still ex-
+hibits a maximum at a given strain γ, which again tracks
+the relaxation of Q. Remarkably, we now find that the
+height of this peak strongly depends on the system size,
+and increases roughly linearly with N, see Fig. 6. This
+suggests that spatial correlations saturate to the system
+size atφcand remain scale-free above φcin the entire
+solid phase [18]. This is in agreement with studies of the
+mechanicalresponseofdisorderedsolids,whichshowedev-
+idence of plastic rearrangements spanning the entire sys-
+p-5Claus Heussinger1 Ludovic Berthier 2 Jean-Louis Barrat1
+tem [20,33]. Additionally, we show in Figs. 6 and 7 that
+the dynamic susceptibility measured along purely elastic
+branches (discarding plastic events), is linearly growing
+withNas well. Indeed, the elastic response of a disor-
+dered solid is also scale-free, as shown theoretically [34,35]
+and numerically [36,37].
+For a given system size, χ4grows with φbelowφc, and
+becomes proportional to Naboveφc. However, the am-
+plitude of χ4decreases slowly with φaboveφc. The net
+result is that χ4, for a fixed system size N, has a striking
+nonmonotonic behaviour with φ, and presents an abso-
+lute maximum at φc, which adds to our list of microscopic
+signatures of the jamming transition.
+A similar maximum of χ4has recently been reported
+both for colloidal [13,15] and granular[11] assemblies, and
+given two distinct interpretations. It was associated to a
+nonmonotonic dynamic correlation length scale ξ4diverg-
+ing atφcin Ref. [11]. Alternatively, it was attributed to a
+nonmonotonicstrengthofessentiallyscale-freespatialcor-
+relations in Refs. [13,15]. Our results clearly favor the sec-
+ond interpretation to explain the behavior above φc. The
+dynamic susceptibility χ4aboveφcis system size depen-
+dent with scale-free spatial dynamic fluctuations (Fig. 7).
+This finding suggests that the length scale identified
+from the low-frequency part of the vibrational spectrum,
+andmuchstudiedrecently[7,27,37]doesnotdirectlyinflu-
+ence the dynamics at the particle scale beyond the linear
+regime where it belongs. We suggest, however, that it in-
+directly influences the behaviour of χ4, and leads to the
+decaying strength of the correlations as observed in Fig. 7
+aboveφc. The evolution of the displacement length scale
+ℓQaboveφcin Fig. 3 suggests a simple explanation for
+this decay. The overlap function Q(a,γ) relax when typi-
+cal displacements are of the order of a. Whenφincreases,
+typical displacements in response to strain increments be-
+comesmaller,andmorestepsareneededtodecorrelatethe
+overlap. If these steps are not perfectly correlated, more
+steps directly imply less dynamic fluctuations and thus a
+reducedχ4[31]. This physicalexplanationwasmademore
+quantitativeintheempiricalmodelofRef.[15], whichthen
+accounts for a nonmonotonic evolution of χ4acrossφc.
+To summarize, borrowing tools from studies of dynamic
+heterogeneity in glass-formers, we provided a detailed ac-
+count of the evolution of the microscopic dynamics of an
+idealized model system acrossthe jamming transition. We
+found a number of original signatures, that are strikingly
+different from the behaviour of viscous liquids, but might
+have been observed in recent experimental reports. Given
+the (relative) ease with which particle trajectories can
+be monitored in suspensions, foams, or granular assem-
+blies, we hope our study will motivate further experimen-
+tal studies along the lines of the present study.
+∗∗∗
+CH acknowledges financial support from the Humboldt
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+016001 (2004).
+[34] B. A. DiDonna and T. C. Lubensky, Phys. Rev. E 72,
+066619 (2005).
+[35] G. Picard, et al., Eur. Phys. J. E 15, 371 (2004).
+[36] C. E. Maloney. Phys. Rev. Lett. 97, 035503 (2006).
+[37] F. Leonforte, et al., Phys. Rev. B 70, 014203 (2004).
+p-6
\ No newline at end of file
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+arXiv:1001.0915v1  [astro-ph.EP]  6 Jan 2010Relativity in Fundamental Astronomy:
+Dynamics, Reference Frames, and Data Analysis
+Proceedings IAU Symposium No. 261, 2009
+S. Klioner, P. K. Seidelmann & M. Soffel, eds.c/circlecopyrt2009 International Astronomical Union
+DOI: 00.0000/X000000000000000X
+Relativistic aspects of rotational motion of
+celestial bodies
+S.A. Klioner, E. Gerlach, M.H. Soffel
+Lohrmann Observatory, Dresden Technical University, 0106 2 Dresden, Germany
+Abstract. Relativistic modelling of rotational motion of extended bo dies represents one of
+the most complicated problems of Applied Relativity. The re lativistic reference systems of IAU
+(2000) give a suitable theoretical framework for such a mode lling. Recent developments in the
+post-Newtonian theory of Earth rotation in the limit of rigi dly rotating multipoles are reported
+below. All components of the theory are summarized and the re sults are demonstrated. The
+experience with the relativistic Earth rotation theory can be directly applied to model the
+rotational motion of other celestial bodies. The high-prec ision theories of rotation of the Moon,
+Mars and Mercury can be expected to be of interest in the near f uture.
+1. Earth rotation and relativity
+Earth rotation is the only astronomical phenomenon which is observ ed with very high
+accuracy, but is traditionally modelled in a Newtonian way. Although a n umber of at-
+tempts to estimate and calculate the relativistic effects in Earth rot ation have been un-
+dertaken (e.g., Bizouard et al.(1992); Brumberg & Simon (2007) and reference therein)
+no consistent theory has appeared until now. As a result the calcu lations of different
+authors substantially differ from each other. Even the way geodet ic precession/nutation
+is usually taken into account is just a first-orderapproximationand is not fully consistent
+with relativity. On the other hand, the relativistic effects in Earth’s r otation are rela-
+tively large. For example, the geodetic precession (1.9′′per century) is about 3 ×10−4
+of general precession. The geodetic nutation (up to 200 µas) is 200 times larger than the
+goal accuracy of modern theories of Earth rotation. One more re ason to carefully inves-
+tigate relativistic effects in Earth rotation is the fact that the geod ynamical observations
+yield important tests of general relativity (e.g., the best estimate o f the PPN γusing
+large range of angular distances from the Sun comes from geodetic VLBI data) and it is
+dangerous to risk that these tests are biased because of a relativ istically flawed theory of
+Earth rotation.
+Early attempts to model rotational motion of the Earth in a relativis tic framework
+(see, for example, Brumberg 1972) made use of only one relativistic references system to
+describe both rotational and translational motions. That refere nce system was usually
+chosen to be quite similar to the BCRS. This resulted in a mathematically correct, but
+physically inadequate coordinate picture of rotational motion. For example, from that
+coordinate picture a prediction of seasonal LOD-variations with an amplitude of about
+75 microseconds has been put forward.
+At the end of the 1980s a better reference system for modelling of Earth rotation
+has been constructed, that after a number of modifications and im provements has been
+adopted as GCRS in the IAU 2000 Resolutions. The GCRS implements th e Einstein’s
+equivalence principle and represents a reference system in which th e gravitational influ-
+ence of external matter (the Moon, the Sun, planets, etc.) is red uced to tidal potentials.
+Thus, for physical phenomena occurring in the vicinity of the Earth the GCRS repre-
+sents a reference system, the coordinates of which are, in a sens e, as close as possible to
+12 S.A. Klioner, E. Gerlach, M.H. Soffel
+measurable quantities. This substantially simplifies the interpretatio n of the coordinate
+description of physical phenomena localized in the vicinity of the Eart h. One important
+application of the GCRS is modelling of Earth rotation. The price to pay when using
+GCRS is that one should deal not only with one relativistic reference s ystem, but with
+several reference systems, the most important of which are the BCRS and the GCRS.
+This makes it necessary to clearly and carefully distinguish between p arameters and
+quantities defined in the GCRS and those defined in the BCRS.
+2. Relativistic equations of Earth rotation
+Themodelwhichisusedinthisinvestigationwasdiscussedandpublishe dbyKlioner et al.
+(2001). Let us, however, repeat these equations once again not going into physical de-
+tails. The post-Newtonian equations of motion (omitting numerically n egligible terms as
+explained in Klioner et al.(2001)) read
+d
+dT/parenleftbig
+Cabωb/parenrightbig
+=∞/summationdisplay
+l=11
+l!εabcMbLGcL+La(C,ω,Ωiner), (2.1)
+whereT= TCG, C=Cabis the post-Newtonian tensor of inertia and ω=ωais
+the angular velocity of the post-Newtonian Tisserand axes (Klioner 1996),MLare the
+multipole moments of the Earth’s gravitational field defined in the GCR S,GLare the
+multipolemomentsoftheexternaltidalgravito-electricfieldin theG CRS.Inthe simplest
+situation (a number of mass monopoles) GLare explicitly given by Eqs. (19)–(23) of
+Klioneret al.(2001).
+The additional torque Ladepends on C,ω, as well as on the angular velocity Ωiner
+describingtherelativisticprecessions(geodetic,Lense-Thirringa ndThomasprecessions).
+The definition of Ωinercan be found, e.g., in Klioner et al.(2001). A detailed discussion
+ofLa, its structure and consequences will be published elsewhere (Klione ret al.2009).
+The model of rigidly rotatingmultipoles (Klioner et al.2001) representsa set of formal
+mathematical assumptions that make the general mathematical s tructure of Eqs. (2.1)
+similar to that of the Newtonian equations of rotation of a rigid body:
+Cab=PacPbdCcd,Ccd= const (2.2)
+Ma1a2...al=Pa1b1Pa2b2...PalblMb1b2...bl,Mb1b2...bl= const, l/greaterorequalslant2,(2.3)
+where the orthogonal matrix Pab(T) is assumed to be related to the angular velocity ωa
+used in (2.1) as
+ωa=1
+2εabcPdb(T)d
+dTPdc(T). (2.4)
+Themeaningoftheseassumptionsisthat both thetensorofinertia Cabandthe multipole
+moments of the Earth’s gravitational field MLare “rotating rigidly” and that their rigid
+rotationis described by the same angularvelocity ωathat appearsin the post-Newtonian
+equations of rotational motion. It means that in a reference syst em obtained from the
+GCRS by a time-dependent rotation of spatial axes both the tenso r of inertia and the
+multipole moments of the Earth’s gravitational field are constant.
+No acceptable definition of a physically rigid body exists in General Rela tivity. The
+model of rigidly rotating multipoles represent a minimal set of assump tions that allows
+one to develop the post-Newtonian theory of rotation in the same m anner as one usually
+does within Newtonian theory for rigid bodies. In the model of rigidly r otatingmultipolesRelativistic Aspects of Rotational Motion 3
+only those properties of Newtonian rigid bodies are saved which are in deed necessary for
+the theory of rotation. For example, no assumption on local physic al properties (“local
+rigidity”) is made. It has not been proved as a theorem, but it is rath er probable that
+no physical body can satisfy assumptions (2.2)–(2.4). The assump tions of the model of
+rigidly rotating multipoles will be relaxed in a later stage of the work whe n non-rigid
+effects are discussed.
+3. Post-Newtonian equations of rotational motions in numerical
+computations
+Looking at the post-Newtonian equations of motion (2.1)–(2.4) one can formulate
+several problems to be solved before the equations can be used in n umerical calculations:
+A. How to parametrize the matrix Pab?
+B. How to compute MLfrom the standard models of the Earth’s gravity field?
+C. How to compute GLfrom a solar system ephemeris?
+D. How to compute the torque εabcMbLGcLout ofMLandGL?
+E. How to deal with different time scales (TCG, TCB, TT, TDB) appear ing in the
+equations of motion, solar system ephemerides, used models of Ear th gravity, etc.?
+F. How to treat the relativistic scaling of various parameters when u sing TDB and/or
+TT instead of TCB and TCG?
+G. How to find relativistically meaningful numerical values for the initia l conditions
+and various parameters?
+These questions are discussed below.
+4. Relativistic definitions of the angles
+One of the tricky points is the definition of the angles describing the E arth orientation
+in the relativistic framework. Exactly as in Bretagnon et al.(1997, 1998) we first define
+the rotated BCRS coordinates( x,y,z) by two constant rotations of the BCRS as realized
+by the JPL’s DE403:
+
+x
+y
+z
+=Rx(23◦26′21.40928′′)Rz(−0.05294′′)
+x
+y
+z
+
+DE403. (4.1)
+Then the IAU 2000transformations between BCRS and GCRS are ap plied to the coordi-
+nates (t,x,y,z),tbeing TCB, to get the corresponding GCRS coordinates ( T,X,Y,Z ).
+The spatial coordinates ( X,Y,Z) are then rotated by the time-dependent matrix Pijto
+get the spatial coordinates of the terrestrial reference syste m (ξ,η,ζ). The matrix Pijis
+then represented as a product of three orthogonal matrices:
+
+ξ
+η
+ζ
+=Rz(φ)Rx(ω)Rz(ψ)
+X
+Y
+Z
+. (4.2)
+The angles φ,ψandωare used to parametrize the orthogonal matrix Paband therefore,
+to define the orientation of the Earth orientation in the GCRS. The m eaning of the
+terrestrial system ( ξ,η,ζ) here is the same as in Bretagnon et al.(1997): this is the
+reference system in which we define the harmonic expansion of the g ravitational field
+with the standard values of potential coefficients ClmandSlm.4 S.A. Klioner, E. Gerlach, M.H. Soffel
+5. STF model of the torque
+The relativistic torque requires computations with symmetric and tr ace-free cartesian
+(STF) tensors MLandGL. For this project special numerical algorithms for numerical
+calculations have been developed. The detailed algorithms and their d erivation will be
+published elsewhere. Let us give here only the most important formu las. For each lthe
+component Da=εabcMbL−1GcL−1of the torque in the right-hand side of Eq. (2.1) can
+be computed as ( Al= 4lπl!/(2l+1)!!,a+
+lm=/radicalbig
+l(l+1)−m(m+1) )
+D1=1
+Al/parenleftBiggl−1/summationdisplay
+m=0a+
+lm/parenleftbig
+−MR
+lmGI
+l,m+1+MI
+l,m+1GR
+lm/parenrightbig
++l−1/summationdisplay
+m=1a+
+lm/parenleftbig
+MI
+lmGR
+l,m+1−MR
+l,m+1GI
+lm/parenrightbig/parenrightBigg
+, (5.1)
+D2=1
+Al/parenleftBiggl−1/summationdisplay
+m=0a+
+lm/parenleftbig
+−MR
+lmGR
+l,m+1+MR
+l,m+1GR
+lm/parenrightbig
++l−1/summationdisplay
+m=1a+
+lm/parenleftbig
+−MI
+lmGI
+l,m+1+MI
+l,m+1GI
+lm/parenrightbig/parenrightBigg
+, (5.2)
+D3=2
+All/summationdisplay
+m=1m/parenleftbig
+MI
+lmGR
+lm−MR
+lmGI
+lm/parenrightbig
+. (5.3)
+The coefficients GR
+lmandGI
+lmcharacterizing the tidal field can be computed from Eqs.
+(19)–(23) of Klioner et al.(2001) as explicit functions of the parameters of the solar
+system bodies: their masses, positions, velocities and acceleration s. A Fortran code to
+compute GR
+lmandGI
+lmforl <7 and 0/lessorequalslantm/lessorequalslantlhas been generated automatically with
+a specially written software package for Mathematica . It is possible to develop a sort
+of recursive algorithm to compute GR
+lmandGI
+lmfor anylsimilar to the corresponding
+algorithms for, e.g., Legendre polynomials.
+The coefficients MR
+lmandMI
+lmcharacterizing the gravitational field of the Earth can
+be computed as
+MR
+l0=l!
+(2l−1)!!/parenleftbigg4π
+2l+1/parenrightbigg1/2
+MERl
+ECl0, (5.4)
+MR
+lm= (−1)m1
+2l!
+(2l−1)!!/parenleftbigg4π
+2l+1(l+m)!
+(l−m)!/parenrightbigg1/2
+MERl
+EClm,1/lessorequalslantm/lessorequalslantl,(5.5)
+MI
+lm= (−1)m+11
+2l!
+(2l−1)!!/parenleftbigg4π
+2l+1(l+m)!
+(l−m)!/parenrightbigg1/2
+MERl
+ESlm,1/lessorequalslantm/lessorequalslantl,(5.6)
+whereMEis the mass of the Earth, REits radius,ClmandSlmare usual potential coef-
+ficients of the Earth’s gravitational field. If only Newtonian terms a re considered in the
+torque this formulation with STF tensors is fully equivalent to the clas sical formulation
+with Legendre polynomials (e.g., Bretagnon et al.1997, 1998). If the relativistic terms
+are taken in account, the only known way to express the torque is t hat with STF tensors.
+6. Time transformations
+An important aspect of relativistic Earth rotation theory is the tre atment of different
+relativistic time scales. The transformation between TDB and TT at the geocenter (allRelativistic Aspects of Rotational Motion 5
+the transformations in this Section are meant to be “evaluated at t he geocenter”) are
+computed along the lines of Section 3 of Klioner (2008b). Namely,
+TT = TDB +∆TDB(TDB) , (6.1)
+TDB = TT −∆TT(TT), (6.2)
+TCG = TCB +∆TCB(TCB) , (6.3)
+TCB = TCG −∆TCG(TCG) , (6.4)
+so that
+d∆TDB
+dTDB=ATDB+BTDBd∆TCB
+dTCB, (6.5)
+ATDB=LB−LG
+1−LB, (6.6)
+BTDB=1−LG
+1−LB=ATDB+1, (6.7)
+d∆TT
+dTT=ATT+BTTd∆TCG
+dTCG, (6.8)
+ATT=LB−LG
+1−LG, (6.9)
+BTT=1−LB
+1−LG= 1−ATT, (6.10)
+d∆TCB
+dTCB=F(TCB) =1
+c2α(TCB)+1
+c4β(TCB), (6.11)
+d∆TCG
+dTCG=F(TCG−∆TCG)
+1+F(TCG−∆TCG), (6.12)
+where the functions αandβare given by Eqs. (3.3)–(3-4) of Klioner (2008b) and Eq.
+(6.12) represents a computational improvement of Eq. (3.8) of Klio ner (2008b). Clearly,
+the derivatives d∆TCB/dTCBandd∆TCG/dTCGmust be expressed as functions of
+TDB and TT, respectively, when used in (6.5)–(6.8).
+The differential equations for ∆TDB and ∆TT are first integrated nu merically for the
+wholerangeofthe used solarsystemephemeris(anyephemeriswith DE-likeinterfacecan
+beusedwiththecode).Theinitialconditionsfor∆TDBand∆TTarech osenaccordingto
+the IAU 2006 Resolution defining TDB: for JDTT= 2443144.5003725one has JDTDB=
+2443144.5003725−6.55×105/86400and vice versa.The results of the integrationsfor the
+pairs ∆TDB and d∆TCB/dTCB, and ∆TT and d∆TT/dTTare stored with a selected
+step in the corresponding time variable (TDB for ∆TDB and its derivat ive, and TT
+for ∆TT and its derivative). A cubic spline on the equidistant grid is the n constructed
+for each of these 4 quantities. The accuracy of the spline represe ntation is automatically
+estimated using additionaldatapoints computed duringthe numeric al integration.These
+additional data points lie between the grid points used for the spline a nd are only used
+to control the accuracy of the spline. The splines precomputed an d validated in this
+way are stored in files and read in by the main code upon request. The se splines are
+directly used fortime transformationduringthe numericalintegra tionsofEarthrotation.
+Althoughthissplinerepresentationrequiressignificantlymorestor edcoefficientsthan,for
+example,arepresentationwithChebyshevpolynomialswiththesame accuracy,thespline
+representation has been chosen because of its extremely high com putational efficiency.
+More sophisticated representations may be implemented in future v ersions of the code.6 S.A. Klioner, E. Gerlach, M.H. Soffel
+7. Relativistic scaling of parameters
+Obviously, there are two classes of quantities entering Eqs. (2.1)– (2.4) that are defined
+in the BCRS and GCRS and, therefore, naturally parametrized by TC B and TCG,
+respectively. It is important to realize that the post-Newtonian eq uations of motion are
+only valid if non-scaled time scales TCG and TCB are used. If TT and/or TDB are
+needed, the equations should be changed correspondingly.
+The relevant quantities defined in the GCRS and parametrized by TCG are: (1) the
+orthogonal matrix Paband quantities related to that matrix: angular velocity ωaand
+corresponding Euler angles ϕ,ψandω; (2) the tensor of inertia Cab; (3) the multipole
+moment of Earth’s gravitational field ML. In principle, (a) GLand (b) Ωa
+inerare also
+defined in the GCRS and parametrized by TCG, but these quantities a re computed
+using positions xA, velocities vAand accelerations aAof solarsystem bodies. The orbital
+motion of solar system bodies are modelled in BCRS and parametrized b y TCB or TDB.
+Thedefinition of GLisconceivedinsuchawaythat positions,velocitiesandaccelerations
+ofsolarsystembodies in theBCRS shouldbe takenatthe momentofT CBcorresponding
+to the requiredmoment ofTCG with spatiallocation taken at the geo center.Let us recall
+that the transformation between TCB and TCG is a 4-dimensional on e and require the
+spatial location of an event to be known.
+In all theoretical works aimed to derive and/or analyze the rotatio nal equations of
+motion in the GCRS one uses TCG as coordinate time scale parametrizin g the equa-
+tions. Although the natural time variable for the equations of Eart h rotation is TCG,
+in principle, using a corresponding re-parametrization any time scale (including TCG,
+TT, TCB and TDB) can be used as independent time variable. Thus, sim ple rescaling of
+the first and second derivatives of the angles entering the equatio ns of rotational motion
+should be applied to use TT instead of TCG:
+dθ
+dTCG= (1−LG)dθ
+dTT, (7.1)
+d2θ
+dTCG2= (1−LG)2d2θ
+dTT2, (7.2)
+whereθis any of the angles ϕ,ψandωused in the equations of motion to parametrize
+the orientation of the Earth. If TDB is used as independent variable the corresponding
+formulas are more complicated:
+dθ
+dTCG= (1−LG)/parenleftBigg
+dTT
+dTDB/vextendsingle/vextendsingle/vextendsingle/vextendsinglexE/parenrightBigg−1
+dθ
+dTDB, (7.3)
+d2θ
+dTCG2= (1−LG)2/parenleftBigg
+dTT
+dTDB/vextendsingle/vextendsingle/vextendsingle/vextendsinglexE/parenrightBigg−2
+d2θ
+dTDB2
+−(1−LG)2/parenleftBigg
+dTT
+dTDB/vextendsingle/vextendsingle/vextendsingle/vextendsinglexE/parenrightBigg−3
+d2TT
+dTDB2/vextendsingle/vextendsingle/vextendsingle/vextendsinglexEdθ
+dTDB, (7.4)
+where the derivatives of TT w.r.t. TDB should be evaluated at the geo center (i.e., for
+x=xE). These relations must be substituted into the equations of rotat ion motion to
+replace the derivatives of the angles ϕ,ψandωw.r.t. TCG as appear e.g., in Eqs. (7)–
+(9) of Bretagnon et al.(1998). It is clear that the parametrization with TDB makes the
+equations more complicated.
+The values of the parameters naturally entering the equations of r otational motion
+must be interpreted as unscaled (TCB-compatible or TCG-compatib le) values. If scaledRelativistic Aspects of Rotational Motion 7
+(TT-compatibleorTDB-compatible) valuesareused, the scalingmus tbe explicitly taken
+into account. The relativistic scaling of parameters read (see, e.g., Klioner 2008a):
+GMTT
+A= (1−LG)GMTCG
+A, GMTCG
+A=GMTCB
+A,
+GMTDB
+A= (1−LB)GMTCB
+A, (7.5)
+XTT= (1−LG)XTCG, xTDB= (1−LB)xTCB, (7.6)
+VTT=VTCG, vTDB=vTCB, (7.7)
+ATT= (1−LG)−1ATCG, aTDB= (1−LB)−1aTCB, (7.8)
+whereGMAis the mass parameter of a body, x,v, andaare parameters represents
+spatial coordinates (distances), velocities and accelerations in th e BCRS, respectively,
+whileX,V, andAare similar quantities in the GCRS.
+Now, considering the source of the numerical values of the parame ters used in the
+equations of Earth rotation we can see the following.
+a. The position xA, velocities vA, accelerations aAand mass parameters GMAof the
+massive solar system bodies are taken from standard JPL ephemer ides and are TDB-
+compatible.
+b. The radius of the Earth comes together with the potential coeffi cientsClmand
+Slmfrom a model of the Earth’s gravity field (e.g., GEMT3 was used in SMAR T). These
+valuescomefromSLRanddedicatedtechniqueslikeGRACE.GCRSwith TT-compatible
+quantities is used to process these data. Therefore, the values o f the radius of the Earth is
+TT-compatible. Obviously, ClmandSlmhave the same values when used with any time
+scale. The mass parameter GMEof the Earth coming with the Earth gravity models is
+also TT-compatible.
+c. From the definitions of MR
+lmandMI
+lmgiven above and formulas for GLgiven
+by Eqs. (19)–(23) of Klioner et al.(2001), it is easy to see that the TCG-compatible
+torqueFa=/summationtext∞
+l=11
+l!εabcMbLGcLcan be computed using TDB-compatible values of
+mass parameters GMTDB
+A, positions xTDB
+A, velocities vTDB
+Aand accelerations aTDB
+Aof
+all external bodies, TDB-compatible value of the mass parameter o f the Earth GMTDB
+E
+and the value of the Earth’s radius formally rescaled from TT to TDB a sRTDB
+E=
+(1−LB)(1−LG)−1RTT
+E. Denoting the resulting torque by Fa
+TDB, it can be seen that
+the TCG-compatible value is Fa
+TCG= (1−LB)−1Fa
+TDB.
+d. The values of the Earth’s moments of inertia Ai,i= 1,2,3 can be represented as
+GAi=GMER2
+Eki, wherekiis a factor characterizing the distribution of matter inside
+the Earth. Clearly, the factors kido not depend on the scaling. Therefore, the moments
+of inertia can be scaled as
+ATT
+i= (1−LG)3ATCG
+i. (7.9)
+The last question is how to interpret the values of the moments of ine rtiaAi=
+(A,B,C) and the initial conditions for the angles ϕ,ψandωand their derivatives given
+in Bretagnon et al.(1998). Obviously, the initial angles at J2000 are independent of th e
+scaling. Forthe other parametersin question it is not possible to clea rlyclaim if the given
+values are TDB-compatible or TT-compatible. Arguments in favor of both interpreta-
+tions can be given. A rigorous solution here is only possible when all calc ulations leading
+to these quantities are repeated in the framework of General Rela tivity. In this paper
+we prefer to interpret the SMART values of Ai, ˙ϕ,˙ψand ˙ωas being TT-compatible.
+Therefore, if TDB is used as independent variable, the values of the derivatives should8 S.A. Klioner, E. Gerlach, M.H. Soffel
+be changed accordingly. For any of these angles one has
+dα
+dTDB=/parenleftBigg
+dTT
+dTDB/vextendsingle/vextendsingle/vextendsingle/vextendsinglexE/parenrightBigg
+dα
+dTT. (7.10)
+Thus, we have all tools to treat correctlythe relativistic scaling of a ll relevant parameters
+of the Earth rotation theory as well as relativistic time scales.
+8. Geodetic precession and nutation
+In the frameworkof our model geodetic precession and nutation a re taken into account
+inanaturalwaybyincluding theadditionaltorquethat depends onΩa
+inerinthe equations
+of rotational motion:
+La=εabcCbdωdΩc
+iner−d
+dT/parenleftbig
+CabΩb
+iner/parenrightbig
+. (8.1)
+The first term of the additional torque reflects the fact that the GCRS of the IAU
+is defined to be kinematically non-rotating (see Soffel et al.2003). The second term
+has been usually hidden by the corresponding re-definition of the po st-Newtonian spin
+(Damour, Soffel & Xu 1993; Klioner & Soffel 2000). It can be demons trated that this
+second term must be explicitly taken into account to maintain the con sistency between
+dynamically and kinematically non-rotating solutions. Further details will be published
+elsewhere (Klioner et al.2009). Using the additional torque Lain Eq. (2.1) is a rigorous
+way to take geodetic precession/nutation into account.
+The standard way to account for geodetic precession/nutation t hat was used up to
+now by a number of authors can be described as follows: (1) solve th e purely Newto-
+nian equations of rotational motion and consider this solution as a re lativistic one in a
+dynamically non-rotating version of the GCRS and (2) add the preco mputed geodetic
+precession/nutation to it. The second step is fully correct since th e geodetic preces-
+sion/nutation is by definition the rotation between the kinematically a nd dynamically
+non-rotating versions of the GCRS and it can be precomputed since it is fully indepen-
+dent of the Earth rotation. The inconsistency of the first step co mes from the fact that
+in the computation of the Newtonian torque the coordinates of the solar system bod-
+ies are taken from an ephemeris constructed in the BCRS. However , the dynamically
+non-rotating version of the GCRS rotatesrelative to the BCRS with angular velocity
+Ωa
+iner(T). This means that the BCRS coordinates of solar system bodies sho uld be first
+rotated into “dynamically non-rotating coordinates” and only afte r that rotation those
+coordinatescan be used to compute the Newtonian torque. For th is reasonthis procedure
+does not lead to a correct solution in the kinematically non-rotating G CRS (see Fig. 1).
+We will call such solutions in this paper “SMART-like kinematical solution s”.
+On the other hand, there are two ways to obtain a correct kinemat ically non-rotating
+solution: (1) use the torque given by Eq. (8.1) in the equations of mo tion, (2) compute
+the geodetic precession/nutation matrix, apply the geodetic prec ession/nutation to the
+solar system ephemeris, integrate (2.1) without Lawith the obtained rotated ephemeris
+(the correct solution in a dynamically non-rotating version of the GC RS is obtained
+in this step), apply the geodetic precession/nutation matrix to the solution. We have
+implemented both ways in our code and checked explicitly that they giv e the same
+solution (to within about 0.001 µas over 150 years). It is interesting to note that the
+rotational matrix of geodetic precession/nutation (that is, the m atrix defining a rotation
+with the angularvelocity Ωa
+iner) cannot be parametrized by normal Euler angles. We have
+used therefore the quaternion representation for that matrix.Relativistic Aspects of Rotational Motion 9
+purely Newtonian 
+solution 
+dynamically  
+non-rotating solution kinematically  
+non-rotating solution SMART-like kinematical 
+solutions add  
+GP-corrections 
+add  
+GP-corrections GP-rotated 
+ephemeris GP-induced 
+torque 
+Figure 1. Scheme of the two ways to obtain a kinematically non-rotatin g solution from a purely
+Newtonian one,andan illustration oftherelation betweent hecorrect kinematically non-rotating
+solution and SMART-like kinematical solutions. “GP” stand s for geodetic precession/nutation.
+Each gray block represents a solution. A solid line means: ad d precomputed geodetic preces-
+sion/nutation into a solution to get a new one. A dashed line m eans: recompute a solution with
+indicated change in the torque model.
+9. Overview of the numerical code
+A code in Fortran 95 has been written to integrate the post-Newto nian equations of
+rotational motion numerically. The software is carefully coded to av oid numerical insta-
+bilities and excessive round-off errors. Two numerical integrators with dense output –
+ODEX and Adams-Bashforth-Moulton multistep integrator – can be used for numerical
+integrations.Thesetwointegratorscanbe used tocrosschecke achother.The integrations
+are automatically performed in two directions – forwards and backw ards – that allows
+one to directly estimate the accuracy of the integration. The code is able to use any type
+of arithmetic available with a given current hardware and compiler. Fo r a number of
+operations, which have been identified as precision-critical, one has the possibility to use
+either the library FMLIB Smith (2001) for arbitrary-precision arith metic or the pack-
+age DDFUN that uses two double-precision numbers to implement qua drupole-precision
+arithmetic (Bailey 2005). Our current baseline is to use ODEX with 80 b it arithmetic.
+The estimated errors of numerical integrations after 150 years o f integration are below
+0.001µas.
+Several relativistic features have been incorporated into the cod e: (1) the full post-
+Newtonian torque using the STF tensor machinery, (2) rigorous tr eatment of geodetic
+precession/nutation as an additional torque in the equations of mo tion, (3) rigorous
+treatmentoftimescales(anyofthefourtimescales–TT,TDB,TCB orTCG–evaluated
+at the geocenter can be used as the independent variable of the eq uations of motion
+(TCG beingphysicallypreferableforthisrole),(4) correctrelativis ticscalingofconstants
+and parameters. All these “sources of relativistic effects” can be switched on and off
+independently of each other.
+In order to test our code and the STF-tensor formulation of the t orque we have
+coded also the classical Newtonian torque with Legendre polynomials as described by
+Bretagnon et al.(1997, 1998) and integrated our equations for 150 years with the se two
+torque algorithms. Maximal deviations between these two integrat ions were 0.0004 µas
+forφ, 0.0001µas forψ, and 0.0002 µas forω. This demonstrates both the equivalence of
+the two formulations and the correctness of our code.
+We have also repeated the Newtonian dynamical solution of SMART97 using the New-
+toniantorque,theJPLephemerisDE403andthesameinitialvaluesa sinBretagnon et al.
+(1998). Jean-Louis Simon (2007) has provided us with the unpublish ed full version of
+SMART97 (involving about 70000 Poisson terms for each of the thre e angles). We have
+calculated the differences between that full SMART97 series and ou r numerical integra-10 S.A. Klioner, E. Gerlach, M.H. Soffel
+-2.00e+020.00e+002.00e+024.00e+026.00e+028.00e+021.00e+03
+-100 -80 -60 -40 -20  0  20  40[µas]∆ φ
+t - J2000 [yr]-2.00e+02-1.50e+02-1.00e+02-5.00e+010.00e+005.00e+011.00e+021.50e+022.00e+02
+-100 -80 -60 -40 -20  0  20  40[µas]∆ ψ
+t - J2000 [yr]-1.00e+02-8.00e+01-6.00e+01-4.00e+01-2.00e+010.00e+002.00e+014.00e+016.00e+018.00e+011.00e+02
+-100 -80 -60 -40 -20  0  20  40[µas]∆ ω
+t - J2000 [yr]
+Figure 2. Differences (in µas) between the published kinematical SMART97 solution and the
+correct kinematically non-rotating solution (with post-N ewtonian torques, relativistic scaling
+and time scaled neglected).
+tion over 150 years. Analysis of the results and a comparison to Bre tagnonet al.(1998)
+have demonstrated that our integrations reproduce SMART97 wit hin the full accuracy
+of the latter.
+10. Relativistic vs. Newtonian integrations
+We have performed a series of numerical calculations comparing pur ely Newtonian
+integration with integration where relativistic effects are taken into account. The same
+initial conditions and parameters that we used to reconstruct the SMART97 solution
+were used for all integrations (see below). The results are illustrat ed on Figs. 2–5. The
+difference between the kinematical SMART97 solution and the consis tent kinematically-
+non-rotating solution obtained as described in Section 8 is shown in Fig . 2. Fig. 3 shows
+the effects of the post-Newtonian torque. The effects of the rela tivistic scaling and time
+scales are depicted in Fig. 4. Finally, Fig. 5 demonstrates the differen ces between a
+SMART-like kinematical solution and our full post-Newtonian integra tion. A detailed
+analysis of these results will be done elsewhere.
+To complete the consistent post-Newtonian theory of Earth rota tion the parameters
+(first of all, the moments of inertia of the Earth) should be fitted to be consistent with
+the observed precession rate. This task will be discussed and trea ted in the near future.
+11. Relativistic effects in rotational bodies of other bodies
+The same numerical code can be applied to model the rotational mot ion of other
+bodies. Especially, high-accuracy models of rotational motion of th e Moon, Mercury and
+Mars are of interest because of the planned space missions to Merc ury and Mars, and
+the expected improvements of the accuracy of LLR. Most of the c hanges in the code are
+trivial and concern the numerical values of the constants. One imp ortant improvement
+of the code is necessary for the Moon: the figure-figure interact ion with the Earth must
+be taken into account. Using the STF approach to compute the tor que this task is not
+difficult.
+The relativistic effects in the rotation of Moon, Mars and Mercury ma y be significantlyRelativistic Aspects of Rotational Motion 11
+-5.00e+02 0.00e+00 5.00e+02 1.00e+03 1.50e+03 2.00e+03 2.50e+03 3.00e+03 
+-400 -300 -200 -100  0  100  200 [µas] ∆ φ
+t - J2000 [yr] -7.00e+02 -6.00e+02 -5.00e+02 -4.00e+02 -3.00e+02 -2.00e+02 -1.00e+02 0.00e+00 1.00e+02 2.00e+02 3.00e+02 4.00e+02 
+-400 -300 -200 -100  0  100  200 [µas] ∆ ψ
+t - J2000 [yr] -2.00e-01 0.00e+00 2.00e-01 4.00e-01 6.00e-01 8.00e-01 1.00e+00 1.20e+00 1.40e+00 1.60e+00 1.80e+00 2.00e+00 
+-400 -300 -200 -100  0  100  200 [µas] ∆ ω
+t - J2000 [yr] 
+Figure 3. The effect (in µas) of the post-Newtonian torque.
+-1.50e+02 -1.00e+02 -5.00e+01 0.00e+00 5.00e+01 1.00e+02 1.50e+02 2.00e+02 2.50e+02 3.00e+02 
+-400 -300 -200 -100  0  100  200 [µas] ∆ φ
+t - J2000 [yr] -3.00e+02 -2.50e+02 -2.00e+02 -1.50e+02 -1.00e+02 -5.00e+01 0.00e+00 5.00e+01 1.00e+02 1.50e+02 
+-400 -300 -200 -100  0  100  200 [µas] ∆ ψ
+t - J2000 [yr] -1.00e-01 -5.00e-02 0.00e+00 5.00e-02 1.00e-01 1.50e-01 2.00e-01 2.50e-01 3.00e-01 
+-400 -300 -200 -100  0  100  200 [µas] ∆ ω
+t - J2000 [yr] 
+Figure 4. The effect (in µas) of the relativistic scaling and time scales.
+larger than in the rotation of Earth. In Table 1 the amplitudes of geo detic precession
+and nutation are given for several solar system bodies. One can se e the large effects for
+Mercury and Mars. Besides an early investigation of Bois & Vokroulick y (1995) suggests
+that the effects of the relativistic torque for the Moon may attain 1 mas. Our approach
+allows one to investigate the rotational motion of the Moon, Mars an d Mercury in a
+rigorous relativistic framework.
+References
+Bailey, D.H., 2005, Computing in Science and Engineering, 7 (3), 54;see also
+http://crd.lbl.gov/ ∼dhbailey/mpdist/
+Bizouard C., Schastok, J., Soffel M.H., Souchay J., (1992) In :Journ´ ees 1992 , N. Capitaine (ed.),
+Observatoire de Paris, 76
+Bois, E., Vokrouhlicky, D., 1995, A&A, 300, 559
+Bretagnon, P., Francou, G., Rocher, P., Simon, J.L., 1997, A&A, 319, 30512 S.A. Klioner, E. Gerlach, M.H. Soffel
+-1.00e+03 -5.00e+02 0.00e+00 5.00e+02 1.00e+03 1.50e+03 2.00e+03 2.50e+03 3.00e+03 3.50e+03 
+-400 -300 -200 -100  0  100  200 [!as] ! "
+t - J2000 [yr] -2.00e+03 -1.50e+03 -1.00e+03 -5.00e+02 0.00e+00 5.00e+02 1.00e+03 
+-400 -300 -200 -100  0  100  200 [!as] ! #
+t - J2000 [yr] -3.00e+02 -2.00e+02 -1.00e+02 0.00e+00 1.00e+02 2.00e+02 3.00e+02 4.00e+02 5.00e+02 
+-400 -300 -200 -100  0  100  200 [!as] ! $
+t - J2000 [yr] 
+Figure 5. Difference between a purely Newtonian integration rotated f or geodetic preces-
+sion/nutation in a SMART-like way (see Section 8) and our sol ution that includes all relativistic
+effects discussed here.
+body geodetic precession geodetic nutation
+[′′per century] [ µas]
+Earth 1.92 153
+Moon 1.95 154
+Mercury 21.43 5080
+Venus 4.32 85
+Mars 0.68 567
+Table 1. Magnitude of geodetic precession/nutation for various bod ies
+Bretagnon, P., Francou, G., Rocher, P., Simon, J.L., 1998, A&A, 329, 329
+Brumberg, V.A., 1972, Relativistic Celestial Mechanics, N auka: Moscow, in Russian
+Brumberg, V.A., Simon, J.-L., 2007, Notes scientifique et te chniques de l’insitut de m´ echanique
+c´ eleste, S088
+Damour, T., Soffel, M., Xu, Ch., 1993, Phys. Rev. D 47, 3124
+Klioner, S.A., Voinov, A.V., 1993, Phys. Rev. D 48, 1451
+Klioner, S.A., 1996, In: “Dynamics, Ephemerides, and Astro metry of the Solar System”, Ferraz-
+Mello, S. and Morando, B., Arlot, J.-E. (eds.), Proc. of IAU S ymposium 172, Springer,
+New York, 309
+Klioner, S.A., Soffel, M., 2000, Phys. Rev. D 62, 024019
+Klioner, S.A., Soffel, M., Xu. C., Wu, X., 2001, In: Influence o f geophysics, time and space
+reference frames on Earth rotation studies (Proc. Journ´ ee s’2001), N. Capitaine (ed.), Paris
+Observatory, Paris, 232
+Klioner, S.A., 2008a, A&A, 478, 951
+Klioner, S.A., 2008b, In: A Giant Step: from Milli- to Micro- arcsecond Astrometry, W.Jin,
+I.Platais, M.Perryman(eds.)Proc.oftheIAUSymposium248 , CambridgeUniversityPress,
+Cambridge, 356
+Klioner, S.A., Gerlach, E., Soffel, M., 2009, “Rigorous trea tment of geodetic precession in the
+theory of Earth rotation”, in preparation
+Simon, J.-L., 2007, private communication
+Smith, D., 2001, “FMLIB”, http://myweb.lmu.edu/dmsmith/FMLIB.html
+Soffel, M., Klioner, S.A., Petit, G. et al., 2003,Astron.J., 126, 2687
\ No newline at end of file
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+arXiv:1001.0916v1  [math.GT]  6 Jan 2010Involutions on S6with 3-dimensional fixed point set
+Martin Olbermann
+October 31, 2018
+Abstract
+In this article, we classify all involutions on S6with 3-dimensional fixed point set. In
+particular, we discuss the relation between the classification of invo lutions with fixed
+point set a knotted 3-sphere and the classification of free involutio ns on homotopy CP3’s.
+1 Introduction
+Asageneralassumption, weareinterested insmoothinvolut ionsonconnected, closed, smooth
+manifolds.
+The study of group actions on very simple manifolds such as di sks, spheres or Euclidean
+spaces has been a very active subject. In his MathSciNet revi ew of [Paw02], Masuda notes:
+“Representations of groups provide examples of group actio ns on Euclidean spaces, disks
+or spheres, and an important natural question in transforma tion groups is to what extent
+arbitrary actions on those spaces resemble actions provide d by representations.” The first
+highlight theorems are due to P.A. Smith. For an involution o n a sphere, Smith proved that
+the fixed point set is a Z2-homology sphere. There are various converses to this theor em
+which can be found in the literature, e.g. [ DW99,Paw02] and references therein. However,
+the following theorem seems to be new:
+Theorem 1.1 [Olb10] EveryZ2-homology 3-sphere is the fixed point set of an involution on
+S6.
+Moreover, the method used in [ Olb10] generalizes to a classification of these involutions,
+which is the subject of the present paper. (A shorter proof of the above existence theorem
+can be given using Dovermann’s equivariant surgery approac h along the lines of [ Sch82] and
+[DMS86].)
+Recently, a class of involutions called conjugations was defined in [ HHP05] and various
+aspects of conjugations were studied in [ FP05,Olb07,HH08,HH09]. Conjugations τon
+topological spaces Xhave the property that the fixed point set has Z2-cohomology ring
+isomorphic to the Z2-cohomology ring of X, with the slight difference that all degrees are
+divided by two. In the case of smooth involutions on (positiv e-dimensional) spheres, the
+conjugations are exactly the involutions on even-dimensio nal spheres with half-dimensional
+fixedpointset. Indimension2, theSch¨ onfliestheoremgives aclassification: every conjugation
+is conjugate to the reflection of S2at the equator. In dimension 4, work of Gordon and
+1Sumners shows that there are infinitely many non-equivalent conjugations on S4. Hambleton
+and Hausmann recently reduced the study of such involutions to a non-equivariant four-
+dimensional knot theory question [ HH09]. Knot theory of k-spheres in S2kis easier for k >2,
+so that the study of conjugations gets a different flavor for k >2.
+The study of free involutions on simply-connected spin mani folds with the same homology
+groups as CP3was motivated by the question whether it is possible to define an “equivariant
+Montgomery-Yang correspondence”. After Haefliger [ Hae62] proved that the group C3
+3of
+knotted S3’s inS6(isotopy classes of smooth embeddings, or equivalently diffe omorphism
+classes relative to S3) is isomorphic to Z, Montgomery and Yang showed [ MY66] that there
+is a natural bijection between C3
+3and the set of diffeomorphism classes of homotopy CP3’s.
+Wall’s classification of all simply-connected spin manifol ds with the same homology groups
+asCP3[Wal66] also uses the bijection between diffeomorphism classes of su ch manifolds,
+together with a basis of H2, and isotopy classes of framed S3-knots in S6. In the equivariant
+setting, Li and L¨ u [ LL07] show that the existence of a free involution on a homotopy CP3
+implies the existence of an involution on S6with fixed point set the corresponding knotted
+S3. Similarly, in our approach, the same surgery arguments app ly in both cases. However, it
+is not possible to produce a nice bijection on the set of equiv ariant diffeomorphism classes of
+these, as claimed in [ Su09]. Our classification results are:
+Theorem 1.2 LetMbe a smooth closed simply-connected spin manifold with H2(M) =Z,
+andH3(M) = 0. Letx∈H2(M;Z)be a generator. We assume that M/\e}atio\slash∼=S2×S4.
+•If/angbracketleftBig
+p1(M)x−4x3
+24,[M]/angbracketrightBig
+is odd, there exists no free involution on M.
+•If/a\}bracketle{tx3,[M]/a\}bracketri}htis odd, and/angbracketleftBig
+p1(M)x−4x3
+24,[M]/angbracketrightBig
+is even, there exist up to diffeomorphism
+exactly two free involutions on M.
+•If/a\}bracketle{tx3,[M]/a\}bracketri}htis even, and/angbracketleftBig
+p1(M)x−4x3
+24,[M]/angbracketrightBig
+is even, there exist up to diffeomorphism
+exactly five free involutions on M.
+IfM∼=S2×S4, then the same classification holds for orientation-revers ing involutions which
+act by−1onH2(M).
+In the case of homotopy CP3’s the classification of free involutions was given by Petrie [Pet72]
+(whose result contains a mistake, corrected by Dovermann, M asuda and Schultz), and Su
+[Su09]. Li and Su (unpublished) give the answer to the existence qu estion in the case of odd
+/a\}bracketle{tx3,[M]/a\}bracketri}ht. Our method reproves all these results in a different way, exte nds to a larger class
+of manifolds, and gives classification results in all cases.
+Theorem 1.3 For every even element of C3
+3there are (up to equivariant diffeomorphism
+relativeS3) exactly four involutions (conjugations) on S6with the knot as fixed point set. For
+every odd element of C3
+3, there is no involution on S6with the knot as fixed point set.
+The new part of the theorem is the classification of these invo lutions. Li and L¨ u proved
+that the existence of an involution with fixed point set a give n knot is equivalent to the
+2existence of a free involution on the corresponding homotop yCP3under the Montgomery-
+Yangcorrespondence, sothattogether withSu’sworkmentio nedabove, theexistencequestion
+was answered.
+The case of involutions on S6with fixed point set S3is especially interesting since, given
+another 6-manifold with an involution that has a 3-dimensio nal fixed point set, (equivariant)
+connected sum gives a possibly different involution on the sam e 6-manifold with same fixed
+point set. (This is in analogy with the fact that connected su m with a homotopy sphere gives
+a possibly new smooth structure on the same underlying topol ogical manifold.) Similarly,
+connected sum with a conjugation on S6with fixed point set different from S3gives a new
+conjugation on the same 6-manifold, with different fixed point space.
+Our main result is a classification of smooth involutions on S6with arbitrary three-
+dimensional fixed point set, using a recent classification of embeddings of closed oriented
+connected 3-manifolds into S6by A. Skopenkov [ Sko08]. Skopenkov proves that for a 3-
+dimensional Z2-homology sphere Mthe set of isotopy classes of embeddings i:M→S6has
+a free action by C3
+3and the orbits are in canonical bijection with H1(M).
+Theorem 1.4 LetMbe aZ2-homology sphere of dimension 3. The set of isotopy classes
+of embeddings i:M→S6which are the fixed point sets of involutions (conjugations) is
+contained in the orbit corresponding to 0∈H1(M). Moreover, it is acted upon freely and
+transitively by 2Z⊆Z∼=C3
+3. There are up to equivariant diffeomorphism relative to iexactly
+four such conjugations for every such i.
+In our case, we can also classify involutions without additi onal identification of the fixed
+point set with a given 3-manifold. However, we consider invo lutions together with an ori-
+entation of their fixed point sets, and the equivalence relat ion is equivariant diffeomorphism
+which respects the orientations of both S6and the fixed point set. Equipping the involution
+with an orientation of the fixed point set is necessary in orde r to perform connected sums.
+Thus it is more natural to determine this more structured set of equivariant diffeomorphism
+classesInvM(S6).
+Since the action of the mapping class group of Mon the above set of equivariant diffeo-
+morphism classes of involutions relative to the fixed point s etMis trivial, we get the same
+classification as in theorem 1.4:
+Theorem 1.5 Conjugations up to conjugation (preserving orientations) w ith fixed point set
+of a fixed oriented diffeomorphism type of Z2-homology 3-spheres are in bijection with Z⊕Z4.
+Under connected sum, InvS3(S6)∼=Z⊕Z4is an isomorphism of groups, and InvS3(S6)acts
+freely and transitively on InvM(S6)for every Z2-homology 3-sphere M.
+Remark 1.6 In principle, using the machinery described in [ Kre85] it is also possible to
+prove classification results for conjugations on X6with fixed point set M3in other cases as
+X=S6. (However, the argument we use to show the surgery obstructi on is trivial does not
+extend to the case of free involutions on other manifolds.)
+3One would compute the set of equivariant diffeomorphism clas ses of the involution together
+with an identification of the fixed point set with a prescribed 3-manifold M, i.e. the set
+EmbZ2(M→X) ={f: (M,id)→(X,τ)}/∼
+wherefis an inclusion onto the fixed point set of τ, andf: (M,id)→(X,τ)is identified
+withf′: (M,id)→(X′,τ′)if there is an equivariant diffeomorphism φ: (X,τ)→(X′,τ′)
+making the obvious triangle commute. One of the difficulties t o overcome is “due” to Wall’s
+classification: in order to determine which of the resulting 6-manifolds are diffeomorphic to
+X, one would need a good understanding of the isomorphism clas ses of the algebraic invariants
+(trilinear forms), and this problem seems to be very hard in g eneral.
+Acknowledgements. I would like to thank Diarmuid Crowley, Jean-Claude Hausman n,
+Matthias Kreck and Arturo Prat-Waldron for many helpful dis cussions and remarks. Special
+thanks to Yang Su, whose article [ Su09] was the origin of this paper, and who shared with
+me the so far unpublished notes on a generalization of [ Su09] due to himself and Banghi Li.
+2 Preliminaries
+2.1 Modified surgery
+We will use Kreck’s modified surgery theory [ Kre99,Kre85] which is also suited to give
+classification results. By the equivariant tubular neighbo urhood theorem, we can write S6=
+M×D3∪∂V, whereVis a manifold with boundary M×S2equipped with a free involution,
+which restricts to ( id,−id) on the boundary. (It is not hard to see that the normal bundle of
+MinS6is trivial. See [ Olb07] for a proof.)
+A classification of manifolds Vwith free involutions τup to equivariant diffeomorphism
+is the same as a classification of the quotient manifolds W=V/τup to diffeomorphism. This
+is what modified surgery theory will give us.
+We firstdeterminethenormal 2-type ofthemanifolds Wunderconsideration. Thenormal
+2-typeBof a 6-manifold (see the precise definition 3.2) is a fibration B→BO. It carries
+roughly the information of a 3-skeleton of the manifold toge ther with the restriction of the
+normal bundle to this 3-skeleton. After computing the bordi sm group of manifolds with
+normalB-structure, we show that in every bordism class there exists a manifold (together
+with a map to B) which qualifies as the Wabove. Moreover, we show that given two normally
+B-bordant manifolds Was above, the obstruction, which a priori lies in the complic ated
+monoidl7(Z2,−1), for the existence of an s-cobordism (i.e. a diffeomorphism ) is zero. The
+diffeomorphism classification of the manifolds Wunder consideration is given by the set of
+orbits of the action of the group of fiber homotopy self-equiv alencesB→BO.
+2.2 Conjugations on manifolds
+This section explains what conjugation spaces are and shows that the smooth involutions on
+S2nwithn-dimensional fixed point set are exactly the smooth conjugat ions. The rest of the
+paper does not depend on the material in this section.
+4A conjugation on a topological space Xis an involution τ:X→X, which we consider as
+an action of the group Z2∼=C={id,τ}onX, and which satisfies the following cohomological
+pattern: We denote the Borel equivariant cohomology of XbyH∗
+C(X;Z2). It is a module
+overH∗
+C(pt;Z2) =Z2[u]. The restriction maps in equivariant cohomology are denot ed by
+ρ:H∗
+C(X;Z2)→H∗(X;Z2) andr:H∗
+C(X;Z2)→H∗
+C(Xτ;Z2)∼=H∗(Xτ;Z2)[u].
+Definition 2.1 [HHP05]Xis a conjugation space if
+•Hodd(X;Z2) = 0,
+•there exists a (ring) isomorphism κ:H2∗(X;Z2)→H∗(Xτ;Z2)
+•and a (multiplicative) section σ:H∗(X;Z2)→H∗
+C(X;Z2)ofρ
+•such that the so-called conjugation equation holds:
+rσ(x) =κ(x)uk+terms of lower degree in u.
+One does not need to require that κandσbe ring homomorphisms, it is a consequence of
+the definition. Moreover, the “structure maps” κandσare unique, and natural with respect
+to equivariant maps between conjugation spaces.
+There are many examples of such conjugations: complex conju gation on the projective
+spaceCPnand on complex Grassmannians, natural involutions on smoot h toric manifolds
+[DJ91] and on polygon spaces [ HK98]. Every cell complex with the property that each cell
+is a unit disk in Cnwith complex conjugation, and with equivariant attaching m aps, is a
+conjugation space. Coadjoint orbits of semi-simple Lie gro ups with the Chevalley involution
+are conjugation spaces. Moreover, there are various constr uctions of new conjugation spaces
+out of other conjugation spaces.
+A conjugation manifold is a conjugation space consisting of a smooth manifold Xwith
+a smooth involution τ. As a consequence, a closed conjugation manifold Xmust be even-
+dimensional, say of dimension 2 n, andMis of dimension n.
+In [Olb07] we proved that it is possible to give a definition of conjugat ion spaces without
+the non-geometric maps κandσ, which is moreover well-adapted to the case of conjugation
+manifolds, where the fixed point set has an equivariant tubul ar neighbourhood.
+Proposition 2.2 Every smooth involution on S2nwith (non-empty) n-dimensional fixed
+point set is a conjugation.
+Proof: Let pt∈S2nbe a fixed point of the involution. Then ptis a conjugation space and
+(S2n,pt) is a conjugation pair, by the same proof as in Example 3.5 of [ HHP05]. Then the
+extension property for triples, Prop. 4.1 in [ HHP05], shows that S2nis a conjugation space.
+(A slightly different proof is given in [ HH09].)q.e.d.
+3 Free involutions on certain 6-manifolds
+We are considering smooth involutions on smooth closed simp ly-connected spin manifolds M
+withH2(M) =Z, andH3(M) = 0. The classification by Wall in [ Wal66] also uses a generator
+5x∈H2(M) and an orientation of M. Then pairs ( M,x) up to diffeomorphism are classified
+by the bordism class of Mx→CP∞∈ΩSpin
+6(CP∞)∼=Z2, and (M,x) is mapped under the
+isomorphism to/parenleftbigg/angbracketleftbiggp1(M)x−4x3
+24,[M]/angbracketrightbigg
+,/a\}bracketle{tx3,[M]/a\}bracketri}ht/parenrightbigg
+.
+Both switching the sign of the generator of H2(M) and the orientation of Minduce multipli-
+cation with−1, so that diffeomorphism classes are in bijection with Z2/−1.
+As observed in [ LL07], the Lefschetz fixed point theorem implies:
+Lemma 3.1 If/a\}bracketle{tx3,[M]/a\}bracketri}htis non-zero, then a free involution on Mmust be orientation re-
+versing, and act by −1onH2(M).
+If/a\}bracketle{tx3,[M]/a\}bracketri}ht= 0, we consider only orientation reversing free involution s which are−1 on
+H2(M). From Li and Su we learned that except for the case M=S2×S4, these are all free
+involutions: If the involution acts by −1 onH4(M), the first Pontryagin class must be 0, and
+we use the classification. The remaining case is handled as ab ove by the Lefschetz fixed point
+theorem.
+Obviously, for M=S2×S4, our classification of orientation reversing free involuti ons
+which are−1 onH2(M) does not include all free involutions.
+3.1 The normal 2-type
+Definition 3.2 The normal 2-type of a compact manifold Nis a fibration B2(N)→BO
+which is obtained as a Postnikov factorization of the stable normal bundle map N→BO:
+There is a 3-connected map N→B2(N), the fibration B2(N)→BOis 3-coconnected (i.e.
+πi(BO,B 2(N)) = 0fori >3), and the composition is the stable normal bundle map. This
+determines B2(N)→BOup to fiber homotopy equivalence.
+Lemma 3.3 Letτbe an involution on Mas above, and let N=M/τbe the the quotient
+space of the involution. The second space in a Postnikov tower forNis either P= (CP∞×
+S∞)/(c,−1), wherecis complex conjugation, or Q= (CP∞×S∞)/(τ,−1), whereτis fiberwise
+the antipodal involution on S2→CP∞→HP∞.
+Proof: The first space in the Postnikov tower is a K(Z2,1), and the second space is a K(Z,2)
+fibration over it, with π1acting nontrivially on π2. Such fibrations are classified by their
+k-invariant in H3(K(Z2,1);Z−)∼=Z2. The spaces PandQhave the required properties,
+and they are not homotopy equivalent, as e.g. H2(P;Z2)∼=Z2
+2andH2(Q;Z2)∼=Z2. So
+they represent all isomorphism classes of fibrations. ( Phask-invariant 0, and Qhas nonzero
+k-invariant.) q.e.d.
+Lemma 3.4 TheZ2cohomology ring of NisZ2[q,t]//a\}bracketle{tt3,q2/a\}bracketri}ht, wheredeg(q) = 4,deg(t) = 1.
+Proof: (This was proved in [ Su09] in thecase of homotopy CP3’s, and we generalize his proof.)
+We consider the Serre spectral sequence of the fibration M→N→RP∞, withZ2(and also
+sometimes with integral) coefficients. The first case is that d3:E0,2
+3→E3,0
+3is non-trivial.
+Then by multiplicativity the E4-term has exactly one Z2inEp,0
+4andEp,4
+4for eachp= 0,1,2.
+6There are no further differentials, and we get the above cohomo logy ring. The second case
+is thatd3:E0,2
+3→E3,0
+3is trivial. We will show that this leads to a contradiction. B y
+multiplicativity, also d3:E0,6
+3→E3,4
+3is trivial. If we remember that we need the limit to
+have no cohomology in degrees >6, then we see in the sequence with integral coefficients that
+there must be a nontrivial d3-differential between the fourth and second line. Then the sam e
+must hold for the Z2coefficient sequence. And we get a d7-differential from the sixth to the
+zeroth line. The E∞-term has exactly one Z2inEp,0
+∞forp= 0,...,6 andEp,2
+∞forp= 0,1,2.
+This gives a cohomology ring with a generator t∈H1(N;Z2), and another generator
+x∈H2(N;Z2). Since in this case H2(N;Z2)∼=Z2
+2, the second Postnikov space must be P,
+and we can choose x∈H2(N;Z2) coming from P; we choose the class in H2(P;Z2) which
+maps to 0 under a section of P→RP∞. Note that xmaps nontrivially to H2(M;Z2).
+We have the relations t7=t3x=x2+at2x+t4= 0, where a∈Z2. (We have Sq1x=tx
+as this is true in P, thust3x=Sq1(t2x) = 0 since H5(N;Z) = 0. By Poincar´ e duality t2x2
+can’t be zero. This implies that in x2+at2x+bt4= 0 we have b= 1.)
+We see that Sq1(t5) =t6,Sq2(t4) = 0,Sq2(t2x) =t2x2/\e}atio\slash= 0. It follows that the first Wu
+class istand the second Wu class is x. But that implies that the second Stiefel-Whitney class
+ofNmaps to a non-trivial class in H2(M;Z2). But since Mis spin, this image must be zero,
+as it is the second Stiefel-Whitney class of M. Contradiction. q.e.d.
+Corollary 3.5 The second space in a Postnikov tower for NisQ.
+Proof: This follows from the fact that H2(N;Z2)∼=Z2.q.e.d.
+Proposition 3.6 If such a manifold Mwith/a\}bracketle{tx3,[M]/a\}bracketri}htodd has a free involution τ, then the
+quotient space Nhas normal 2-type B=BSpin×Q→BO×BO(1)⊕→BO.
+Proof: If/a\}bracketle{tx3,[M]/a\}bracketri}htis odd, then the map H4(Q;Z2)→H4(N;Z2) is a bijection, since then
+we have isomorphisms H4(Q;Z2)→H4(CP∞;Z2)→H4(M;Z2)←H4(N;Z2). Then Sq2:
+H4(N;Z2)→H6(N;Z2) is zero, so is the second Wu class of M/τ, and as a consequence the
+quotient space has a spin structure twisted by L→Q.q.e.d.
+Proposition 3.7 If such a manifold Mwith trivial square H2(M;Z2)→H4(M;Z2)has a
+free involution τ, then the quotient space Nhas one of the following normal 2-types:
+A=Q×BSpin→BO(1)×BO(1)×BO(1)×BO⊕→BO,
+B=Q×BSpin→BO(1)×BO⊕→BO,
+where the maps to all BO(1)’s are the projections p:Q→RP∞=BO(1).
+Proof: If/a\}bracketle{tx3,[M]/a\}bracketri}htis even, then there is a second possibility for the normal 2-t ype. If we fix
+the second space in a Postnikov tower to be Q, the second Stiefel-Whitney class of Ncan
+bet2or 0. Thus Neither admits spin structures twisted by Lor spin structures twisted by
+L⊕L⊕L= 3L.q.e.d.
+A normal B-structure on a manifold Ncan be defined in three equivalent ways.
+7•It is a vertical homotopy class of lifts of the normal bundle m apN→BOtoB(this
+is independent of the particular map N→BOcoming from an embedding of Ninto
+some Euclidean space).
+•It is a map f:N→Q(up to homotopy) together with a spin structure on the bundle
+νN−f∗(L), where Lis the nontrivial real line bundle on Q.
+•It is a map f:N→Qtogether with a homotopy η(and this up to homotopy) in the
+following square:
+Nν/d47/d47
+f
+/d15/d15BO
+w1×w2
+/d15/d15
+Qη/d51/d59/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110
+t×0/d47/d47K(Z2,1)×K(Z2,2)
+Here we fix maps corresponding to the classes w1∈H1(BO;Z2),w2∈H2(BO;Z2), the
+generator t∈H1(Q;Z2),0∈H2(Q;Z2).
+(For fixed f, the homotopy classes of homotopies ηhave a free and transitive action by
+π1((K(Z2,1)×K(Z2,2))N)∼=H1(N;Z2)×H0(N;Z2).)
+Similarly, normal A-structures are defined. (In the first definition, replace BbyA. In the
+second definition, replace Lby 3L. In the third definition, replace 0 by t2.)
+Remark 3.8 As a converse to proposition 3.7, a manifold with normal B-structure N6→B
+which is a 3-connected map is the quotient of an involution on a closed simply-connected spin
+manifold MwithH2(M) =Z, andH3(M) = 0if and only if H3(N;Λ) = 0. HereΛ =Z[Z2]
+is the group ring of the fundamental group. The same holds for n ormalA-structures.
+3.2 Computation of the bordism groups
+We compute bordismgroupsof manifolds with normal A-structures (resp. B-structures). The
+(co)homology of Qis described in [ Olb07]. To compute the bordism groups ΩA
+6and ΩB
+6we
+use the Atiyah-Hirzebruch spectral sequence (which comput es the group up to an extension
+problem) and an Adams spectral sequence (which can help solv e the extension problem). We
+get:
+Theorem 3.9 We have isomorphisms ΩA
+6∼=Z2⊕Z2andΩB
+6∼=Z2⊕Z4.
+Proof: The case of ΩB
+6was proven in [ Olb07]. For ΩA
+6, the Atiyah-Hirzebruch spectral se-
+quence is
+Hp(Q;ΩSpin
+q)⇒ΩA
+p+q,
+thed2-differential
+E2
+p,1∼=Hp(Q;Z2)→Hp−2(Q;Z2)∼=E2
+p−2,2
+is the dual of Sq2+tSq1+t2Sq0, and the d2-differential
+E2
+p,0∼=Hp(Q;Z−)→Hp−2(Q;Z2)∼=E2
+p−2,1
+8is reduction modulo 2 composed with the dual of Sq2+tSq1+t2Sq0. (We obtain this using
+the Thom isomorphism
+ΩSpin
+6(Q;3L)∼=ΩSpin
+9(D(3L),S(3L))
+and the Atiyah-Hirzebruch spectral sequence for the latter . See also [ Tei93].) From the
+calculations in [ Olb07] it immediately follows that the differentials d2:E2
+6,1→E2
+4,2and
+d2:E2
+6,0→E2
+4,1are non-trivial. This implies that the nonzero terms on the s ixth diagonal in
+theE∞-term are:
+E∞
+2,4∼=Z, E∞
+5,1∼=Z2, E∞
+6,0∼=2Z.
+Thus ΩA
+6∼=Z2⊕Z2or ΩA
+6∼=Z2.
+Now we consider the Adams spectral sequence
+Exts,t
+A(H∗(MSpin∧T(3L);Z2),Z2)⇒ΩA
+t−s−3,
+whereAis the mod 2 Steenrod algebra. We compute the left hand side fo rt−s−3≤6. We
+have
+H∗(MSpin∧T(3L);Z2)∼=H∗(MSpin;Z2)⊗˜H∗(T(3L);Z2),
+and˜H∗(T(3L);Z2) is a free H∗(Q;Z2)-module on one generator u3of degree 3 (the Thom
+class). We have
+Sq(u3) =w(3L)u3=u3+tu3+t2u3+t3u3.
+All of this allows to write down the A-module structure of H∗(MSpin∧T(3L);Z2) in degrees
+≤10. Then we compute a free A-resolution (in low degrees). From this we get the E2-term
+of the spectral sequence, which is displayed in the followin g diagram:
+01234567s
+2 3 4 5 6 7 8 9 10 t−s························
+••
+•
+•
+•
+•
+•
+•/d23/d23/d23
+••/d10/d10/d10/d10/d10/d10•/d127/d127/d127/d127/d127•
+••
+•
+•
+•
+••
+•
+•
+•
+•
+•
+••
+Since thereare no differentials starting or endingat ( t−s,s) = (9,1), we obtain ΩA
+6∼=Z2⊕Z2.
+q.e.d.
+93.3 Construction and classification up to normal B-bordism
+By Wall’s classification, every bordism class in ΩSpin
+6(K(Z,2))∼=Z2contains a unique normal
+2-smoothing with H3= 0 (up to diffeomorphism relative BSpin×K(Z,2)).
+Theorem 3.10 In every bordism class in ΩA
+6there is a unique manifold W→A(up to
+diffeomorphism relative to A) such that W→Ais 3-connected and H3(W;Λ) = 0. The same
+is true if one replaces AbyB.
+Proof: For the construction and classification of free invol utions on these manifolds, we use
+surgery theory. The existence proof is a simplified version o f the proof of the main theorem
+in [Olb10].
+We start with any 6-dimensional closed manifold with normal A-structure, and we do
+surgery to get manifolds Wsuch that the map W→Ais 3-connected and H3(W;Λ) = 0.
+Surgery below the middle dimension is always possible [ Kre99]. It allows to modify any
+closed 6-manifold Wwith normal A-structure into one with normal 2-type A. Let us denote
+this new manifold again by W. It remains to kill H3(W;Λ).
+By the Hurewicz theorem (its extended version) we have a surj ectionπ3(W)→H3(W;Λ),
+where Λ = Z[Z2] is the group ring of the fundamental group, and H3(W;Λ) can be identified
+with the homology of the universal cover of W.
+The map H3(W;Λ)→H3(W;Z2) factors through H3(W;Λ)⊗ΛZ2, more precisely the
+relation is given by a universal coefficient spectral sequenc e
+TorΛ
+p(Hq(W;Λ),Z2)⇒Hp+q(W;Z2).
+(This can also be interpreted as the Serre spectral sequence for the fibration ˜W→W→
+RP∞.) Herethezeroth andthesecond rowarerelated bynon-trivi al differentials: wecompare
+with the corresponding situation for the space Q. As a result, H3(W;Z2)∼=H3(W;Λ)⊗ΛZ2.
+By Poincar´ e duality, H3(W;Λ)∼=H3(W;Λ), and this is free over Z, as there is no Z-
+torsion in H2(W;Λ). Since H3(W;Λ) is free over Z, it is a sum of summands of the form
+Λ,Z+andZ−[CR62]. We also get that H3(W;Λ)∼=HomΛ(H3(W;Λ),Λ), for example
+again from a universal coefficient spectral sequence. The map H3(W;Λ)→H3(W;Λ)→
+HomΛ(H3(W;Λ),Λ) describes the Λ-valued intersection form on H3(W;Λ). The Z2-valued
+intersection form on H3(W;Z2) is given by tensoring with Z2.
+But this implies that for a class x∈H3(W;Λ) with Tx=±x, its image in H3(W;Z2)
+has intersection 0 with all other elements, hence it must be 0 : ifTx=±x, thenTλ(x,y) =
+λ(Tx,y) =λ(±x,y) =±λ(x,y), soλ(x,y) is a multiple of (1 ±T) and its reduction in Z2is
+zero.
+HenceH3(W;Λ) is a free Λ-module with non-degenerate intersection for m. Since we are
+in dimension 3, there is a quadratic refinement in Λ //a\}bracketle{tx+ ¯x,1/a\}bracketri}htwhich is uniquely determined
+by the intersection form. Hence we obtain an element in ˜L6(Λ,w=−) = 0, as L6∼=Z2is
+given by the Arf invariant. Thus it is possible (after stabil ization) to do surgery which makes
+H3(W;Λ) = 0.
+The argument shows that every class of ΩA
+6contains a manifold Wwith normal 2-type A
+andH3(W;Λ) = 0.
+10Fortheuniquenessresultwetaketwosuchmanifolds W0,W1, andassumethereisanormal
+A-bordism between them. By Kreck’s theory [ Kre99], p. 734, there is a surgery obstruction in
+˜l7(Λ,w=−1) for turning the normal A-bordism into an s-cobordism. By surgery below the
+middle dimension we may assume that the bordism Yis equipped with a 3-connected map to
+A. Now Kreck defines the surgery obstruction using a construct ion of a certain disjoint union
+Uofsubmanifoldsof Ydiffeomorphicto S3×D4, anddefinesthesurgeryobstructionto bethe
+kernel of H3(∂U;Λ)→H3(Y\int(U),W0;Λ). He also notes that the orthogonal complement
+ofthiskernelisthekernel of H3(∂U;Λ)→H3(Y\int(U),W1;Λ). Butinourcase H3(Wi;Λ) =
+0 so that both these kernels are equal to the kernel of H3(∂U;Λ)→H3(Y\int(U);Λ).
+This implies that the surgery obstruction just defined lies i n the group ˜L7(Λ,w=−1)
+which is trivial as computed by Wall. Hence we get as a result t hatA-bordant manifolds Wi
+with normal 2-type AandH3(W;Λ) = 0 are diffeomorphic (relative to A).
+Theprooffornormal 2-type Bisobtained byreplacingall occurrencesof AbyB.q.e.d.
+3.4 The transfer
+For the transfer (double cover) map ΩA
+6→Ω˜A
+6∼=ΩSpin
+6(CP∞) we compare the Atiyah-
+Hirzebruch spectral sequences. We fix the isomorphism
+ΩSpin
+6(CP∞)∼=Z2
+[f:M→CP∞]/mapsto→/parenleftbigg/angbracketleftbiggp1(M)f∗x−4f∗x3
+24,[M]/angbracketrightbigg
+,/a\}bracketle{tf∗x3,[M]/a\}bracketri}ht/parenrightbigg
+.
+One computes the homology transfers using the long exact seq uences coming from short exact
+coefficient sequences:
+H6(Q;Z−)∼=→H6(CP∞),
+H4(Q;Z2)0→H4(CP∞;Z2),
+H2(Q;Z−)∼=→H2(CP∞).
+We see that the transfer gives a map of short exact sequences
+F5∼=Z2⊕Z
+(a,b)/mapsto→2b
+/d15/d15(a,b)/mapsto→(a,b,0)/d47/d47ΩA
+6∼=Z2⊕Z2(a,b,c)/mapsto→c/d47/d47
+tr
+/d15/d15E∞
+6,0∼=Z
+c/mapsto→2c
+/d15/d15
+˜F5∼=Zb/mapsto→(b,0)/d47/d47Ω˜A
+6∼=Z2
+(b,c)/mapsto→c/d47/d47˜E∞
+6,0∼=Z
+which shows (using the snake lemma) that ΩA
+6→Ω˜A
+6has a cokernel of order 4. But the
+composition of projection and transfer: Ω˜A
+6→ΩA
+6→Ω˜A
+6is multiplication by 2. So the image
+of the transfer consists exactly of all classes divisible by 2.
+It also follows that one can find generators for the free summa nds in ΩA
+6as images of the
+generators of Ω˜A
+6.
+Theorem 3.11 [Olb10] The image of the transfer map ΩB
+6→Ω˜B
+6∼=ΩSpin
+6(CP∞)is2Z⊕Z.
+113.5 Generators for the bordism group ΩB
+6
+WeusetheThomisomorphismfortwistedspinbordism: ΩB
+6∼=ΩSpin
+6(Q;L)∼=ΩSpin
+7(DL,SL).
+Under this isomorphism, the boundary map ΩSpin
+7(DL,SL)→ΩSpin
+6corresponds to the
+transfer map ΩB
+6→Ω˜B
+6. It follows that the torsion elements in ΩB
+6∼=ΩSpin
+7(DL,SL) come
+from ΩSpin
+7(DL)∼=ΩSpin
+7(Q). Moreover, ΩSpin
+7(CP∞) = 0, as one sees easily from the Atiyah-
+Hirzebruchspectral sequence. ThusΩSpin
+7(Q)∼=Z4mustberesponsibleforthetorsion. More-
+over, astudyoftheAtiyah-Hirzebruch spectral sequencesh owsthat themapΩSpin
+7(CP3/τ)→
+ΩSpin
+7(Q) is an isomorphism. Now there is a bundle RP2→CP3/τ→S4. SoCP3/τ−RP2
+is anRP2-bundle over D4, so homotopy equivalent to RP2. Again the Atiyah-Hirzebruch
+spectral sequence shows that ΩSpin
+7(CP3/τ−RP2) = ΩSpin
+6(CP3/τ−RP2) = 0. Thus
+ΩSpin
+7(CP3/τ)∼=ΩSpin
+7(CP3/τ,CP3/τ−RP2), andthelatterisisomorphictoΩSpin
+3(RP2)∼=Z4
+by the Thom isomorphism. Again the Atiyah-Hirzebruch spect ral sequence shows that the
+transfer ΩSpin
+3(RP2)→ΩSpin
+3(S2)∼=Z2is surjective. Thus one can detect a generator of
+ΩSpin
+7(CP3/τ) by the composition
+ΩSpin
+7(CP3/τ)→ΩSpin
+7(CP3/τ,CP3/τ−RP2)→ΩSpin
+3(RP2)→ΩSpin
+3(S2).
+Now take the seven-dimensional manifold ( CP3×S1)/(τ,c), where cis complex conju-
+gation. The spin structure on CP3×S1which restricts to the non-bounding one on S1is
+preserved by theinvolution ( τ,c), sothat we obtain a spinstructureon thequotient. Themap
+toCP3/τis projection on the first coordinate. This intersects RP2=CP1/τtransversely,
+so that the Thom isomorphism maps it to the pullback ( CP1×S1)/(τ,c)→CP1/τwhose
+double cover is the projection CP1×S1→CP1, which is a generator for ΩSpin
+3(S2) since the
+Spin structure restricts to the non-boundant one on S1.
+We have to apply the Thom isomorphism ΩSpin
+7(DL|CP3/τ,SL|CP3/τ)→ΩSpin
+6(CP3/τ;L)
+to the map ( CP3×S1)/(τ,c)→CP3/τ→(CP3×D1)/(τ,−1). For this, we homotope the
+map to make it transversal to the zero section: take
+(CP3×S1)/(τ,c)→(CP3×D1)/(τ,−1)
+[x,y]/mapsto→[x,Im(y)]
+and intersect it with the zero section: we obtain two copies o fCP3/τwith different B-
+structures.
+It follows that the torsion Z4in ΩB
+6is generated by the sum (or the difference) of two
+copies of CP3/τwith different B-structures. Let us denote one of them by σand the other
+byσ′.
+Thus we obtain as generators for ΩB
+6∼=ΩSpin
+7(DL,SL):
+•X×D1f×id→CP∞×D1, whereXis asimply-connected spin6-manifold with H3(X) = 0,
+trivial cup product H2(X)×H2(X)→H4(X), andf:X→CP∞defines a generator
+ofH2(X) such that the first Pontrjagin class of Xis equal to 24 times the generator of
+H4(X) which is dual to f∈H2(X).
+•((CP3×D1)/(τ,−1),σ),
+12•((CP3×D1)/(τ,−1),σ)−((CP3×D1)/(τ,−1),σ′).
+Thetwo former generators generate free summands, the latte r generates the torsion summand
+Z4. In this basis, the map ΩB
+6∼=Z2⊕Z4→Z2∼=ΩSpin
+6(CP∞) is given as ( a,b,c)/mapsto→(2a,b).
+3.6 The automorphism groups of AandBand their action on the bordism
+groups
+Theautomorphismgroup Aut(B)BOoffiberhomotopyclassesoffiberhomotopyself-equivalen-
+ces ofBacts on ΩB
+6. We saw that ΩB
+6∼=ΩSpin
+7(DL,SL), where Lis the nontrivial real line
+bundle (CP∞×R)/(τ,−1).
+Lemma 3.12 The set of equivariant oriented diffeomorphism classes of fre e involutions on
+6-manifolds with H3= 0and whose quotient spaces have normal 2-type Bare given as the
+orbits. Again, Bcan be replaced by Ain the theorem.
+Proof: Equivariant diffeomorphism classes of free involutio ns are the same as diffeomorphism
+classes of the quotients. Now the theorem follows from the un iqueness of the Postnikov
+decomposition, i.e. for a given manifold Wwith normal 2-type B, the map W→Bis
+uniquely determined up to fiber homotopy self equivalences o fBoverBO. See also [ Kre85].
+q.e.d.
+The restriction of the first component of a fiber homotopy self -equivalence of Q×BSpin
+toQis a self-homotopy equivalence of Q. There is a unique free homotopy class of maps
+Q×BSpin→Qwhich is an isomorphism on π1, and thus (using obstruction theory to extend
+homotopiesfrom QtoQ×BSpin)auniquefreehomotopyclassofmaps Q×BSpin→Qwhich
+are an isomorphism on π1. The vertical homotopy classes of fiber homotopy equivalenc es thus
+correspond to the different choices (up to homotopy) of a homot opyηfromQ×BSpin→
+Q→K(Z2,1)×K(Z2,2) toQ×BSpin→BO→K(Z2,1)×K(Z2,2). So the group has
+four elements, and the action of the group on the set of normal B-structures on a manifold
+f:M→B(i.e. spin structures on νM−f∗L) is given by just changing the spin structure σ
+intoσ,−σ,σ+f∗t,−σ+f∗t, wheret∈H1(B;Z2) is the generator.
+On ΩSpin
+7(DL,SL), the group Aut(B)BOacts in the following way: the negative spin
+generator acts by -1, and the spin flip generator acts by
+((CP3×D1)/(τ,−1),σ)/mapsto→((CP3×D1)/(τ,−1),σ′),
+and is the identity on X. Thus in the decomposition ΩSpin
+7(DL,SL)∼=Z⊕Z⊕Z4given by
+the above generators, the negative spin generator acts by ( a,b,c)/mapsto→(−a,−b,−c), and the
+spin flip generator acts by ( a,b,c)/mapsto→(a,b,b−c). We obtain orbits of the form
+{(a,b,c),(a,b,b−c),(−a,−b,−c),(−a,−b,−b+c)}.
+The group Aut(A)BOalso has four elements which act on the set of normal A-structures
+on a manifold f:M→A(i.e. spin structures on νM−f∗(3L)) by just changing the spin
+structure σintoσ,−σ,σ+f∗t,−σ+f∗t, wheret∈H1(A;Z2) is the generator.
+13This is either the identity or minus the identity on the gener ators for the free summands
+in ΩA
+6as they are images of the generators of Ω˜A
+6unde projection. All group elements must
+act by the identity on the torsion generator. We obtain orbit s
+{(a,b,c),(−a,−b,−c)}.
+The group Aut(K(Z,2)×BSpin)BOalso has four elements, which act on the bordism
+group by reversing the spin structure and/or the class in H2. It follows that here the orbits
+are of the form
+{(a,b),(−a,−b)}.
+For theproofof theorem 1.2, it is now sufficient tocount preimages andorbits: Anelement
+of the form (2 a+1,b)∈ΩSpin
+6(CP∞) has no preimages in ΩA
+6or ΩB
+6. An element (2 a,2b+1)
+in ΩSpin
+6(CP∞) has four preimages ( a,2b+1,c) in ΩB
+6and no preimages in ΩA
+6. These four
+preimages, together with the four preimages of ( −2a,−2b−1), decompose into two orbits.
+For (2a,2b) in ΩSpin
+6(CP∞) we obtain four preimages ( a,2b,c) in ΩB
+6. These four preimages,
+together with the four preimages of ( −2a,−2b), decompose into three orbits. The element
+(2a,2b) in ΩSpin
+6(CP∞) has two preimages ( a,b,c) in ΩB
+6. These two preimages, together with
+the two preimages of ( −2a,−2b), decompose into two orbits.
+4 Involutions on S6with three-dimensional fixed point set
+4.1 Non-equivariant classification of embeddings in S6
+Before we classify involutions on S6, let us recall the non-equivariant results on embeddings
+of three-manifolds into S6.
+Naturally the most interesting case is the one of knotted 3-s pheres in the six-sphere. Here
+the results are due to Haefliger. One could consider various e quivalence relations on knotted
+S3’s inS6. In all cases there is an addition defined using connected sum s:
+First we can look at the group C3
+3of isotopy classes of embeddings of S3intoS6. Sec-
+ond, the group Θ of diffeomorphism classes of embeddings of S3into oriented manifolds X
+diffeomorphic to S6, relative to S3. (We require a diffeomorphism to be the identity on S3
+and to preserve orientations.) Third, the group Σ of orienta tion-preserving diffeomorphism
+classes of pairs ( X,M), wherethe oriented manifold Xis diffeomorphic to S6and the oriented
+submanifold Mis diffeomorphic to S3.
+There are obvious surjective group homomorphisms C3
+3→Θ→Σ. Haefliger showed that
+C3
+3and Σ are both isomorphic to Z, so that both of the above maps are isomorphisms.
+To prove that C3
+3→Θ is an isomorphism, one needs to show that a diffeomorphism
+S6→S6relative the embedded S3can be replaced by an isotopy. This is true since
+π0(Diff(Dn,∂))→π0(Diff(Sn)) is surjective. This means it is always possible to mod-
+ify the original diffeomorphism on a disk such that the resulti ng diffeomorphism is isotopic
+to the identity.
+One explanation of the isomorphism Θ →Σ is Cerf’s result that Diff+(S3) is connected.
+Thus every orientation-preserving diffeomorphism of S3is isotopic to the identity, and this
+isotopy extends to an ambient equivariant isotopy of S6.
+14The negative of an isotopy class is given by precomposing the embedding S3→S6with
+an orientation-reversing self-diffeomorphism of S3.
+Isotopy classes of framed embeddings S3×D3→S6are in bijection with Z2, the framing
+giving an additional integer invariant (the isomorphism to Z2depends on a choice). Note also
+that in the PL category, all non-framed knots are trivial, bu t the isotopy classes of framed
+knots are in bijection with smooth isotopy classes of smooth framed knots.
+Forembeddingsofgeneralclosedorientedconnected3-mani foldsM3intoS6, theargument
+that diffeomorphism relative to the submanifold implies isot opy of the embeddings still holds.
+Isotopy classes of embeddings Emb(M,S6) have been classified by Skopenkov in [ Sko08]. To
+an isotopy class of embeddings i:M→S6heassociates its Whitney invariant W(i)∈H1(N).
+(For the precise definition we refer to [ Sko08], we give a description in special cases in remark
+4.5.) The map W:Emb(M,S6)→H1(N) is surjective, and C3
+3∼=Zacts transitively on
+the fibers by connected sum. This action has non-trivial stab ilizers in general: There is a
+bijective map (the Kreck invariant) from the fiber over u∈H1(N) toZd(u), whered(u) is the
+divisibility of ¯ u∈H1(M)/{torsion}. In general, both the Whitney and the Kreck invariant
+depend on choices. For Z2-homology spheres M, the Whitney invariant does not depend on
+choices, and C3
+3acts freely on the fibers, so that Emb(M,S6) is in non-canonical bijection
+withZ×H1(M). Instead of a map to Z, the Kreck invariant describes an action of Zon
+Emb(M,S6) which leaves the Whitney invariant fixed.
+4.2 What should we classify in the equivariant case?
+In the equivariant case, there are again various equivalenc e relations one can put on the set of
+embeddings respectively involutions. However, in order to get a well-defined connected sum
+operation, it is necessary to orient both the 6-manifold and the fixed point set.
+Remark 4.1 The proof of the uniqueness of the non-equivariant connected sum construc-
+tion can be generalized to show that the equivariant diffeomo rphism type of the equivariant
+connected sum of two conjugation manifolds of dimension 2nonly depends on the connected
+component of the set of equivariant isomorphisms of a tangen t space at a fixed point with
+(R2n,(1n,−1n)). More generally, varying the chosen fixed point, we get a bund le of such iso-
+morphisms over the fixed point set, and the equivariant diffeo morphism type depends only on
+the connected component in the total space of this bundle. (Th e total space has two components
+if the fixed point set is orientable, and one component if it is not.) See also Definition 1.1
+and Lemma 1.2 of [ L¨ of81]. In particular, the connected sum is unique up to equivarian t dif-
+feomorphism if we are provided with orientations of the conj ugation manifolds and their fixed
+point sets, but in general depends also on an orientation of t he fixed point sets. This answers
+a question in [ HHP05], and it also explains why we are less interested in the class ification of
+conjugations without an orientation of the fixed point set.
+We fixM3, allow various involutions τonS6, and consider equivariant embeddings
+(M,id)→(S6,τ) such that the image of iis the fixed point set of τ. Again there are
+several equivalence relations which we can put on this set.
+15•Two embeddings i0: (M,id)→(S6,τ0) andi1: (M,id)→(S6,τ1) are equivalent if
+there is an equivariant, orientation-preserving diffeomorp hismφ: (S6,τ0)→(S6,τ1)
+relativeM, i.e. there is a commutative triangle
+(M,id)i0/d47/d47
+i1
+/d37/d37/d75/d75/d75/d75/d75/d75/d75/d75/d75(S6,τ0)
+φ
+/d15/d15
+(S6,τ1).
+This classsifies involutions together with an identificatio n of the fixed point set with the
+given 3-manifold M. We get a set EmbZ2(M,S6).
+•Two embeddings i0: (M,id)→(S6,τ0) andi1: (M,id)→(S6,τ1) are equivalent
+if there is an equivariant diffeomorphism φ: (S6,τ0)→(S6,τ1) which restricts to
+some self-diffeomorphism φMofM. We require that both φandφMare orientation-
+preserving. There is a commutative square
+(M,id)i0/d47/d47
+φM
+/d15/d15(S6,τ0)
+φ
+/d15/d15
+(M,id)i1/d47/d47(S6,τ1).
+This classifies (up to orientation-preserving diffeomorphis m) involutions plus an orien-
+tation of the fixed point set. One might call this the classific ation of conjugations up to
+conjugation. We get a set InvM(S6).
+Remark 4.2 One could also define an equivariant version of isotopy class es of embeddings:
+we say that two equivariant embeddings i0: (M,id)→(S6,τ0)andi1: (M,id)→(S6,τ1)
+are equivalent if there is an equivariant embedding i: (M×I,id)→(S6×I,τ)such that
+i(x,t) = (it(x),t)andτ(y,t) = (τt(y),t), hence in particular it restricts on both ends to i
+respectively i′. (We also require that the image of all the embeddings involv ed is the whole
+fixed point set.)
+4.3 Analysis of involutions on S6with three-dimensional fixed point set
+We recall the classical result:
+Theorem 4.3 (P.A. Smith [ Bre72])If an involution on Snhas fixed points, the fixed point
+set is aZ2-homology sphere.
+Comparing with Skopenkov’s classification of embeddings, o ur first result is the following.
+Proposition 4.4 Ifi:M3→S6is the embedding of a fixed point set of an involution, then
+the Whitney invariant of the embedding vanishes: W(i) = 0.
+16Proof: Let τbe such an involution, and let σbe the reflection of S6at the equator. In [ Sko08]
+it is proved that W(σ◦i) =−W(i). But we have a commutative square
+M
+=
+/d15/d15i/d47/d47S6
+σ◦τ
+/d15/d15
+Mσ◦i/d47/d47S6
+which shows that iandσ◦iare isotopic. Thus W(i) =−W(i), and since H1(M) consists of
+odd torsion only, we have W(i) = 0. q.e.d.
+Remark 4.5 Up to a multiplicative factor, which is invertible in the cas e ofZ2-homology
+spheres, the Whitney invariant can also be defined in the foll owing way: Let Ci=S6\i(M3).
+By Alexander duality, H2(Ci)∼=ZandH4(Ci)∼=H2(M)∼=H1(M). Then the invariant is
+given by the square of a generator of H2(Ci). In the case of a fixed point set of a conjugation,
+the involution acts by multiplication with −1on bothH2(Ci)andH4(Ci). It follows that the
+square of a generator of H2(Ci)must be 0.
+Recall that by the equivariant tubular neigborhood theorem , for every involution on S6
+with fixed point set Mwe can write S6=M×D3∪V, such that the involution is id×−id
+onM×D3, and free on V. The quotient W=V/τis a manifold with fundamental group Z2
+and boundary M×RP2. The normal 2-type of WisB. The inclusion of the boundary needs
+to induce an isomorphism π2(RP2)→π2(B).
+In order to define the bordism set Ω(B,M×RP2)
+6 , we have to fix a normal B-structure on
+M×RP2. Since we are interested in a classification, we first conside r all relevant normal
+B-structures.
+For any such structure, the homotopy class of maps M×RP2→Q→RP∞is the non-
+trivial class in H1(M×RP2;Z2). We have to lift this non-trivial map M×RP2→RP∞toQ.
+Any lift, together with a choice of a spin structure on νM×RP2−f∗(L), will then be a normal
+B-structure on M×RP2. It is easy to find a lift fonRP2, see [Olb07], p.47, and composing
+any lift with the projection M×RP2→RP2gives a lift S3×RP2→Q. Obstruction theory
+shows that pointed homotopy classes of lifts on RP2are classified by H2(RP2,Z−)∼=Z, and
+that every lift on RP2extends uniquely up to homotopy to a lift on M×RP2, see [Olb07],
+pp.50-52.
+But there is a further condition. Only two pointed homotopy c lasses of lifts induce an
+isomorphism π2(RP2)→π2(Q), andoneobtains onefromthe other by precomposingwith the
+nontrivialpointedhomotopyclassofmaps RP2→RP2. However, thismapisfreelyhomotopic
+to the identity of RP2. Thus we get up to (free) homotopy a unique map f:RP2→Q. There
+arefourspinstructureson νM×RP2−f∗(L), sincespinstructuresonunoriented(butorientable
+and spin) bundles over Nare in bijection with H0(N;Z2)×H1(N;Z2). Thus there are four
+distinguished normal B-structures on M×RP2which can be used in the construction.
+Theorem 4.6 For every distinguished normal B-structure on M×RP2, every element of
+the bordism set Ω(B,M×RP2)
+6 contains (up to diffeomorphism relative to Band the boundary)
+a unique manifold Wwhich produces a conjugation on S6.
+17Remark 4.7 Actually we also see that H3(W;Λ)consisting just of odd torsion is a necessary
+and sufficient condition for this.
+Proof: The proof of existence is a slight modification of the p roof of theorem 3.10. (For full
+details, see the proof of theorem 1.3 in [ Olb10].) Also the uniqueness extends from the proof
+of theorem 3.10: If we take two manifolds W0,W1with the same normal B-structure on the
+boundary, and which both produce conjugations on S6, and assume that there is a normal B-
+bordismbetween them, wehave tomodifytheargument from 3.3slightly to showthat Kreck’s
+surgery obstruction is 0: For both i= 0,1 the kernel of H3(∂U;Λ)→H3(Y\int(U),Wi;Λ)
+is equal to the kernel of H3(∂U;Λ)→H3(Y\int(U);Λ)/{torsion}. So again the surgery
+obstruction lies in ˜L7(Λ,w=−1) = 0. q.e.d.
+Corollary 4.8 Diffeomorphism classes of Wrelative to M×RP2and toB(whereM×RP2→
+Bis fixed) producing conjugations on S6are in bijection with Ω(B,M×RP2)
+6∼=Z2⊕Z4.
+Diffeomorphism classes of Wrelative to M×RP2and toB(whereM×RP2→Bis
+not fixed) producing conjugations on S6are in bijection with the disjoint union of these four
+bordism sets.
+Equivariant connected sum of two conjugations with fixed poi nt setsM1respectively
+M2corresponds to a map of bordism sets defined by gluing along pa rt of the boundary (or
+parametrized boundary connected sum): Choose disks D3inM1,M2centered at the points
+where the connected sum is performed. Then there is a map
+Ω(B,M1×RP2)
+6×Ω(B,M2×RP2)
+6→Ω(B,(M1#M2)×RP2)
+6
+(W1,W2)/mapsto→W1∪D3×RP2W2
+This is equivariant with respect to the action of the bordism group ΩB
+6. Hence it equips
+Ω(B,S3×RP2)
+6 with a group structure, and we obtain an action of Ω(B,S3×RP2)
+6 on Ω(B,M×RP2)
+6
+for anyM.
+To compare with the non-equivariant embedding results of Sk openkov, it suffices to con-
+sider the image under the transfer map: Comparing with [ Sko08], we see that for the embed-
+dings with Whitney invariant 0, the Kreck invariant describ es an action of ZonEmb(M,S6)
+which corresponds precisely to the action of
+ΩSpin
+6(K(Z,2),∂=S3×S2)/(0⊕Z⊂ΩSpin
+6(K(Z,2)))
+on the set
+ΩSpin
+6(K(Z,2),∂=M×S2)/(0⊕Z⊂ΩSpin
+6(K(Z,2))).
+We will see in the next section how the quotients arise also in the equivariant setting.
+184.4 Equivariant diffeomorphism classes as a quotient by grou p actions
+Now we have to relate the set of relative diffeomorphism classe s of the previous section to the
+setEmbZ2(M,S6) of equivariant diffeomorphism classes of six-spheres with i nvolution whose
+fixed point set is identified with M.
+Basically, we forget the B-structure, we construct the equivariant inclusion M×D3→X
+fromM×RP2→W, and we forget the tubular neighbourhood and the framing of t he normal
+bundle.
+More precisely, we have the following sets of equivalence cl asses:
+1. The set T1of diffeomorphism classes of manifolds Wrelative to M×RP2andBO.
+An element is represented by M×RP2→W→B, such that the first map is the
+inclusion of the boundary. Two representatives WandW′are equivalent if there is a
+diffeomorphism W→W′which commutes with the maps from M×RP2and toBO.
+2. The set T2of diffeomorphism classes of manifolds Wrelative to M×RP2. An element
+is represented by M×RP2→W, which is the inclusion of the boundary. Two represen-
+tativesWandW′are equivalent if there is a diffeomorphism W→W′which commutes
+with the maps from M×RP2.
+3. The set T3of equivariant diffeomorphism classes of manifolds Vrelative to M×S2.
+An element is represented by ( M×S2,(id,−id))→(V,τ), which is the inclusion of
+the boundary, and where τis a free involution. Two representatives VandV′are
+equivalent if there is a diffeomorphism ( V,τ)→(V′,τ′) which commutes with the maps
+fromM×S2.
+4. The set T4of equivariant diffeomorphism classes of closed manifolds Xrelative to M×
+D3. An element is represented by an equivariant embedding ( M×D3,(id,−id))→
+(X,τ), where τis free on the complement of the image. Two representatives XandX′
+are equivalent if there is a diffeomorphism ( X,τ)→(X′,τ′) which commutes with the
+maps from M×D3.
+5. The set EmbZ2(M,S6) of equivariant diffeomorphism classes of manifolds Xrelative to
+M. An element is represented by an equivariant embedding ( M,id)→(X,τ), where τ
+is free on the complement of the image. Two representatives XandX′are equivalent
+if there is a diffeomorphism ( X,τ)→(X′,τ′) which commutes with the maps from M.
+On the set T1, the automorphism group Aut(B)BOof fiber homotopy classes of fiber
+homotopy self-equivalences of Bacts by post-composing M×RP2→BwithB→B.
+We saw that the group Aut(B)BOhas four elements, and the action of the group on the
+set of normal B-structures on a manifold f:M→B(i.e. spin structures on νM−f∗L) is
+given by just changing the spin structure σintoσ,−σ,σ+f∗t,−σ+f∗t, wheret∈H1(B;Z2)
+is the generator.
+Thus the group Aut(B)BOacts freely and transitively on the set of distinguished nor mal
+B-structures on M×RP2. In particular each orbit of the action on the set in 1contains a
+unique element from each of the four bordism sets.
+19By uniqueness of the Postnikov decomposition, forgetting t he map to Bis a bijection from
+the quotient of T1byAut(B)BOto the set T2. See also [ Kre85]. Thus the set T2is in bijection
+with Ω(B,M×RP2)
+6∼=Z2⊕Z4.
+The sets T2,T3andT4are in bijective correspondence: One gets from M×D3→Xby
+restriction to M×S2→V=X\int(M×D3) andfrom M×S2→Vone takes the quotient by
+theinvolution toobtain M×RP2→W=V/τ. Vice-versa, from M×RP2→W,takeanynon-
+trivial double covering VofWand any map M×S2→Vinducing the given M×RP2→W.
+The involution on Vis the non-trivial deck transformation. Up to equivalence, this does not
+dependon the choices made. And from φ:M×S2→V, obtainM×D3→X=M×D3∪φV.
+The smooth structure on the latter depends on choices, but on ly up to equivalence.
+On the sets T2,T3andT4, we have an action of the group of bundle automorphisms
+Map(M,O(3)) ofM×D3: every bundle automorphism induces a self-diffeomorphism of the
+boundary M×S2which induces a self-diffeomorphism of M×RP2, and we precompose
+with these diffeomorphisms. A bundle automorphism f:M→O(3) corresponds to the
+diffeomorphism
+φf:M×RP2→M×RP2
+(x,±y)/mapsto→(x,±f(x)·y)
+fory∈S2. So the bundle automorphism which is minus the identity on ea ch fiber acts
+trivially on M×RP2, hence the action is trivial also on the other sets. (This cor responds
+inT4to the fact that the embeddings i:M×D3→XandM×D3id×−id→M×D3i→X
+are related by the equivariant diffeomorphism τ: (X,τ)→(X,τ).) Thus we may restrict to
+orientation preserving bundle automorphisms.
+The equivariant tubular neighbourhood of the fixed point set is unique up to isotopy
+and bundle automorphisms. But isotopies of embeddings can b e enlarged to isotopies of the
+ambient space, so that they act trivially on the set of equiva riant diffeomorphism classes.
+It follows that the action of the group of bundle automorphis ms descends to an action of
+[M,SO(3)], and that dividing out this action, we get the set EmbZ2(M,S6).
+The part which is a little more complicated is to see the actio n ofZ∼=π3(SO(3))∼=
+[M,SO(3)]∋fonabordismset Ω(B,M×RP2)
+6 . Wecanalso precomposewiththecorresponding
+self-diffeomorphism φfofM×RP2, but this changes the precise map M×RP2→B. Still
+the normal B-structure is preserved: Up to homotopy, we may assume that f:M→SO(3)
+is trivial on a whole disk D3, so that the normal B-structure on D3×RP2does not change.
+Spin structures on vector bundles over M×RP2are in bijection with spin structures on
+their restrictions to RP2(using the natural framing of the normal bundle), and the lat ter are
+invariant. Hence these bundle automorphisms preserves the normalB-structure on M×RP2,
+and we may consider the action of fon the bordism group as given by gluing a mapping
+cylinder of f.
+Lemma 4.9 LetB→BObe a fibration, let Nbe an(n−1)-manifold with normal B-
+structure, W1,W2,W3be normal B-nullbordisms of N, letφ:N→Nbe a diffeomorphism
+20preserving the normal B-structure, let Cφbe the mapping cylinder of φ, leti0,i1:N→Cφbe
+the two natural inclusions, and let Tφbe the mapping torus of φ. Then, in the bordism group
+ΩB
+n, we have
+(W1∪i0Cφ∪i1−W2)−(W1∪idN−W2) =Tφ=W3∪i0Cφ∪i1−W3.
+Proof:W×Ican be considered as a bordism between the manifolds with bou ndary∂W×I
+and−W∪W. The second equality in the statement follows from applying this toW=W3.
+Similarly, the first equality is obtained by applying this to W=W1andW=W2.q.e.d.
+It follows that the action of fon the bordism set Ω(B,S3×RP2)
+6 is the same as taking the
+disjoint sum with D4×RP2∪φfD4×RP2. The latter is an RP2-bundle on S4, and the
+corresponding double cover is a S2-bundle over S4which can be identified with S2→CP3→
+HP1. Thus the induced action on Ω(˜B,S3×S2)
+6∼=Z2is by a generator for the first summand of
+Ω˜B
+6∼=Z2.
+Itfollows thattheset EmbZ2(M,S6)oforbitsoftheaction on T4isinbijection with Z⊕Z4.
+In particular EmbZ2(M,S6) is a group which acts freely and transitively on EmbZ2(M,S6)
+for allM.
+Comparing with the non-equivariant classification of embed dings in S6, we see that for-
+getting the involution defines a map EmbZ2(M,S6)∼=Z⊕Z4→Emb(M,S6)∼=Z⊕H1(M)
+which is equivariant with respect to the group homomorphism EmbZ2(S3,S6)∼=Z⊕Z4→
+Emb(S3,S6)∼=Zgiven by ( a,b)/mapsto→2a. In particular the image of EmbZ2(M,S6) is acted
+upon freely by 2 Z⊆Z, and the map is 4-to-1.
+For embeddings of Z2-homology spheres we saw that the elements [ i:M→S6]∈
+Emb(M,S6) with vanishing Whitney invariant are acted freely and tran sitively upon by
+C3
+3∼=Z. The subset of isotopy classes of embeddings which are the fix ed point sets of con-
+jugations are acted freely and transitively upon by 2 Z⊆Z. There are up to equivariant
+diffeomorphism relative to iexactly four such conjugations for every i. This proves theorems
+1.3and1.4.
+4.5 From embeddings to submanifolds - proof of theorem 1.5
+The more natural thing is to classify involutions without th e additional identification of the
+fixed point set with a fixed 3-manifold M. The invariant of the involution should be its fixed
+point set, i.e. a submanifold of S6. In order to get from embeddings up to diffeomorphism
+to submanifolds up to diffeomorphism, it suffices to divide out t he action of the group of
+self-diffeomorphisms Diff(M). Since isotopies extend to ambient isotopies (this also ho lds
+in this equivariant case, since it suffices to extend a vector fi eld on the fixed point set to an
+equivariant vector field on the whole space), and these give e quivariant diffeomorphisms, the
+action of Diff(M) factors through the mapping class group of M.
+Since in our case, the map M×RP2→Qfactors through RP2, this map does not
+change, so that we get the same normal B-structure. Then the action of a self-diffeomorphism
+f:M→Mon the bordism set Ω(
+6B,M×RP2) is by disjoint union with the mapping torus
+Tf×idoff×id:M×RP2→RP2. NowTf×id=Tf×RP2, and the map to Qfactors again
+21through RP2. Spin structures twisted by LonTf×RP2are products of a spin structure on
+Mand a spin-structure twisted by LonRP2. Thus a spin-nullbordism of Tfgives a normal
+B-nullbordism of Tf×id. Recall that a four-dimensional spin manifold is zero borda nt iff its
+signature is zero, and that the signature of a mapping torus i s always zero. Hence the action
+of the mapping class group of Mon the bordism set is trivial. As a consequence, the mapping
+class group of Macts trivially on EmbZ2(M,S6). This proves theorem 1.5.
+References
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+Mathematics, Vol. 46. Academic Press, New York-London, 197 2.
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+Springer Lecture Notes, 53 (1968)
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+Algebras, Pure and Applied Mathematics XI, Wiley-Interscience Publish ers(1962)
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+on spheres. Comm. Pure Appl. Math. 52 (1999), no. 8, 935–947.
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+topy complex projective spaces. Japan. J. Math. (N.S.) 12 (1 986), no. 1, 1–35.
+[FP05] Franz, M. and Puppe, V. Steenrod squares on conjugati on spaces. C. R. Math.
+Acad. Sci. Paris 342 (2006), no. 3, 187–190.
+[Hae62] Haefliger, A. Knotted (4 k−1)-spheres in 6 k-spaces. Ann. of Math. 75 (1962), 452–
+466
+[HH08] Hausmann, J.-C. and Holm, T. Conjugation spaces and e dges of compatible torus
+actions.arXiv:0807.3289 (2008)
+[HH09] Hambleton, I. and Hausmann, J.-C. Conjugation space s and 4-manifolds.
+arXiv:0906.5057 (2009)
+[HHP05] Hausmann, J.-C. and Holm, T. and Puppe, V. Conjugati on spaces. Algebraic and
+Geometric Topology, Volume 5 (2005), 923–964.
+[HK98] Hausmann, J.-C. and Knutson, A. The cohomology ring o f polygon spaces. Annales
+de l’Institut Fourier (1998), 281–321.
+[Kre99] Kreck, M. Surgery and duality. Annals of Mathematics, 149 (1999), 707–754.
+[Kre85] Kreck, M. An extension of results of Browder, Noviko v and Wall about surgery on
+compact manifolds. Preprint, Mainz (1985)
+22[LL07] Li, B.-H. and L¨ u, Z. Smooth free involution of HCP3and Smith conjecture for
+imbeddings of S3inS6. Math. Ann. 339 (2007), no. 4, 879–889.
+[L¨ of81] L¨ offler, P. Homotopielineare Zp-Operationen auf Sph¨ aren. Topology 20 (1981), no.
+3, 291–312.
+[MY66] Montgomery, D. and Yang, C. T. Differentiable actions o n homotopy seven spheres.
+Trans. Amer. Math. Soc. 122 (1966), 480–498.
+[Olb07] Olbermann,M. Conjugationson6-Manifolds. PhDThe sis, Univ.Heidelberg(2007).
+Available online at http://www.ub.uni-heidelberg.de/archiv/7450/
+[Olb10] Olbermann, M. Conjugations on 6-manifolds with fre e integral cohomology.
+arxiv:1001.0911 (2010).
+[Paw02] Pawalowski, K. Manifolds as fixed point sets of smoot h compact Lie group actions.
+Current trends in transformation groups, 79–104, K-Monogr. Math., 7, Kluwer
+Acad. Publ., (2002).
+[Pet72] Petrie, T. Involutions on homotopy complex project ive spaces and related topics.
+Proceedings of the Second Conference on Compact Transforma tion Groups (Univ.
+Massachusetts, Amherst, Mass., 1971), Part I, pp.234–259. LectureNotes inMath.,
+Vol. 298, Springer, (1972).
+[Sch82] Schultz, R. Exotic spheres as stationary sets of hom otopy sphereinvolutions. Michi-
+gan Math. J., 29 (1982), 121–122.
+[Sko08] Skopenkov, A. A classification of smooth embeddings of 3-manifolds in 6-space.
+Math.Z. 260, (2008). 647–672.
+[Su09] Su, Y. A note on the Z2-equivariant Montgomery-Yang correspondence.
+arxiv:0908.3053 (2009)
+[Tei93] Teichner, P. On the signature of four-manifolds wit h universal covering spin. Math.
+Ann. 295, no. 4 (1993) 745–759
+[Wal66] Wall, C.T.C. Classification problems in differential topology. V. On certain 6-
+manifolds. Invent. math. 1 (1966) 355–374
+[Wal99] Wall, C.T.C. Surgery on compact manifolds, Second e dition. Edited and with a
+foreword by A. A. Ranicki. Mathematical Surveys and Monogra phs, 69. American
+Mathematical Society, Providence, RI (1999).
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+many
+E-mail:olber@mpim-bonn.mpg.de
+23
\ No newline at end of file
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+ 
+Thermally-Driven Atmospheric Escape 
+ 
+Robert E. Johnson     Engineering Physics, Thornton Hall B102, University of Virginia,  
+Charlottesville, VA 22902 
+    Physics Department, NYU, NY, NY 10003     rej@virginia.edu  
+Abstract 
+    Accurately determining escape rates fr om a planet's atmosphere is critical for 
+determining its evolution. Escape can be dr iven by upward thermal conduction of energy 
+deposited well below the exobase, as we ll as by non-thermal processes produced by 
+energy deposited in the exobase region. Recent applications of a model for escape driven 
+by upward thermal conduction, called the slow hydrodynamic escape model, have resulted in surprisingly large loss rates for th e thick atmosphere of Titan, Saturn's largest 
+moon. Based on a molecular kinetic simulation of the exobase region, these rates appear 
+to be orders of magnitude t oo large. Because of the large amount of Cassini data already 
+available for Titan's upper atmosphere and the wealth of data expected within the next 
+decade for the atmospheres of Pluto, Mars, and extrasolar planets, accurately determining 
+present escape rates is criti cal for understanding their e volution. Therefore, the slow 
+hydrodynamic model is evaluated here. It is shown that such a model cannot give a 
+reliable description of the at mospheric temperature profile unless  it is coupled to a 
+molecular kinetic description of the exobase region. Therefore, the present escape rates 
+for Titan and Pluto must be re-evaluated us ing atmospheric models described in this 
+paper.       
+Introduction 
+    There is considerable interest in de scriptions of atmospheric escape due to the 
+extensive Cassini measurements in Titan's thermosphere, the coming encounter of New 
+Horizons with Pluto, the MAVEN mission to study the thermosphere of Mars and the 
+evolution of extrasolar planet atmosphere s. For many solar system bodies atmospheric 
+escape is dominated by non-thermal processes induced by energy dir ectly deposited in 
+the exobase region, as discussed for Mars (e.g., Chaufray et al. 2007) and Titan (e.g., 
+Johnson 2009). Escape can also occur by ener gy deposited well belo w the exobase and 
+transported upward by thermal conduction, as  described for Pluto (e.g., Hunten and 
+Watson, 1982; McNutt 1989; Krasnopolsky 1999; Strobel 2008a). Surprisingly, analysis 
+of the extensive Cassini-Huygens data on the density profile in Titan's upper atmosphere has lead to huge differences in the estimates  of the present escape rates (e.g., Johnson et 
+al. 2009 for a review). Prior to Cassini's tour of the Saturnian system, escape of nitrogen 
+and methane from Titan had been assumed to occur by nonthermal processes (e.g., Shematovich et al. 2003; Michael  et al. 2005). However, the de nsity vs. altitude profiles 
+have been recently analyzed assuming  that escape is driven by upward thermal 
+conduction of energy deposited we ll below the exobase (Strobel  2008a, 2009; Yelle et al. 
+2008). A model called the `slow hydro-dynamic escap e' model, referred to here as the SHE model, was applied to Titan by Strobe l (2008a). This model was intended to 
+describe cases intermediate between Jean s escape and hydrodynamic escape (e.g., Hunten 
+1982). It is based on solving the 1D radial fluid dynamic equations when the net upward 
+flow velocity below the exobase is much less than the speed of sound, hence the word 
+`slow'.     The large escape rates derived from th e SHE model were suggest ed to be produced by 
+distortion of the molecular velocity distribut ion function in the exoba se region (Yelle et 
+al. 2008; Strobel 2009). However, recent kinetic Monte Carlo simulations  suggest that the 
+nitrogen and methane escape from Titan's atmosphere, obtained using the SHE model, 
+appear to be much too larg e (Tucker and Johnson 2009). Th erefore, the SHE model is 
+examined below in order to determine its ge neral applicability a nd, in particular, its 
+relevance to escape of nitrogen and methane from Titan.  
+Thermal Escape 
+    The Jeans parameter, λ, is typically used to determine the importance of thermal 
+models for escape. λ is the ratio between the gravitati onal binding energy of a molecule 
+in a planet's atmosphere, Φ
+g(r), to its thermal energy, kT : λ=Φg(r)/kT . Here k is the 
+Boltzmann constant, T the temperature, with Φg(r)=GMm /r, where G is the gravitational 
+constant, M the planet's mass, m the molecular mass, and r the distance from the center of 
+the planet. The Jeans parameter can also be written as  λ=(vesc/c)², where vesc is the escape 
+speed at r and c=(kT/m)1/2 is of the order of the speed of sound. When the net heating at 
+depth, after accounting for radi ative cooling, is high and ne ither downward nor horizontal 
+heat transport can remove energy fast e nough, then the temperature and pressure can 
+increase until the gas flows outward into spa ce removing energy. This process, referred to 
+as hydrodynamic escape, occurs when the valu e of the Jeans parameter at the exobase, λx, 
+becomes small (e.g., Hunten1982). It is often us ed to describe the evolution of young, hot 
+atmospheres having a large escaping hydrogen co mponent that can entrain and carry off 
+heavier species. On the other hand when λx is large, thermal escape occurs on a molecule 
+by molecule basis, in an evaporative pro cess called Jeans escape. The slow hydrodynamic 
+escape model (SHE), meant to be intermediate between these, is described in a number of 
+publications (e.g., Johnson et al. 2008 for a summary).     The mass loss rate for a single species atmosphere in the SHE model is obtained by 
+solving 1D radial fluid dynamic equations, typically using scaled variables. These 
+equations were initially used to describe the escape of solar wind ions assuming thermal 
+conduction is maintained by the electrons a nd occurs along the radi al field lines (e.g., 
+Parker 1964). The model was subsequently applied to atmospheres containing only neutrals. It was used to estimate the outfl ow from Pluto driven by thermal conduction 
+(Watson et al. 1981; McNutt 1989; Kras nopolsky, 1999; Strobel, 2008b) and was 
+recently applied to Titan (Strobel 2008a; 2009). 
+From the 1D continuity equation, atmospheric escape requires a constant net flow of 
+molecules from the lower boundary to the upper boundary: i.e., 4 πr²nv=φ, where v is the 
+radial flow velocity, r the radial position from the center of the body, n the density at r, 
+and φ the escape rate. The radial energy and mome ntum equations are solved in terms of 
+an escape rate, φ. The parameter φ is subsequently used to fit the available density data. 
+In this model 'slow' means that mv²/kT  is small below the exobase, so terms explicitly 
+containing v² are dropped while keeping φ fixed. Therefore, SHE is suggested to apply when λ = (v esc/c)² > ~10 and in regions where v <~0.3 c (e.g., Strobel 2008; Watson et al. 
+1981). The net heating rate, Q(r) (heating minus radiative coo ling), is used in the energy 
+conservation equation. For a thermal conductivity K, heat capacity per molecule Cp, and 
+gravitational energy Φg(r), the integrated energy equa tion gives the thermal flux: 
+ 
+                QdrrdrdTKr r TCr
+rr
+r g p2
+0024 )] ( 4))( ([ π π φ ∫= −Φ−                              (1) 
+ 
+    where the v² terms are dropped.                                    
+    For n and T specified at a lower boundary, Eq . 1 is solved along with the momentum 
+equation in which the terms v² are also dropped and viscosit y is typically ignored. The 
+heat flux through the lower boundary and φ are constrained in SHE by assuming that 
+T→0 as r→∞, and by the atmospheric density profile , as described below. Solutions for 
+the temperature profile and escape flux have be en obtained for Pluto and Titan with heat 
+supplied by upward conduction through the lower boundary ( Q(r)= 0 in Eq. 1) or with net 
+solar heating ( Q(r)≠0) in the integrated volume. 
+    Since φ is constant, v increases as the densit y decreases, eventually approaching the 
+speed of sound. Therefore, the energy equation is, typically, not solved to infinity (e.g., 
+McNutt 1989). Strobel (2008a, b), for instance,  solves these equations up to a region 
+above the exobase where v <~0.3 c and then finds solutions for which the asymptotic 
+behavior of n and T are consistent with the boundary conditions at infinity. 
+    Although ignoring the upward flow sp eed below the exobase is reasonable for λ > ~10, 
+continuing the 1D radial energy equation into the region above the e xobase is not correct. 
+That is, the probability of co llisions decreases exponentially with radial distance rapidly 
+becoming negligible, but the thermal conductivity used is independent of density. 
+However, it has been argued that distortions in  the tail of molecular velocity distribution 
+near the exobase act to power the heat tr ansport and produce escap e (Yelle et al. 2008; 
+Strobel 2009). These aspects are examined be low based on kinetic theory and simulations 
+of the Boltzmann equations, followed by a di scussion of the applicability of SHE. 
+ 
+Kinetic Theory 
+    The behavior of molecules in a planet ary atmosphere can be accurately described by 
+the Boltzmann equations (e.g., Chamberlain a nd Hunten 1987) in which the spatial and 
+temporal gradients in the phase space densit ies are determined by the collisions between 
+the molecules and by the forces acting on th e molecules. Taking the five principal 
+moments of these equations for each species (e.g., Chapman and Cowling 1970) results in 
+the fluid dynamic equations: the continuity , energy, and momentum equations for the 
+molecular density, energy density, and mean flow  velocity of the molecules. It is the 1D 
+radial versions of these equa tions that are used in SHE. These apply in regions of the 
+atmosphere in which the mean free path for collisions, ℓc, is much less than the scale for 
+significant changes in atmospheric propertie s, typically describe d by the local scale 
+height of atmosphere, H. The ratio, ℓc /H=Kn , is called the Knudsen number. Therefore, 
+for Kn <<1, the velocity distribution function ( VDF) for the molecules is well described 
+by a Maxwell-Boltzmann (MB) distributi on and the fluid dynamic equations can 
+accurately describe the behavior of the gas.     Gradients in these moments, which ar e locally averaged quan tities, result in the 
+thermal conduction, viscous and diffusion terms.  These account for the flow of molecules 
+between two adjacent volumes of gas, both a ssumed to have MB VDF, but with slightly 
+different temperatures, flow  speeds or compositions. The transport of energy and 
+momentum between neighboring volumes depe nds on the molecular speeds. Prior to 
+making a collision, more molecules will flow fr om the hotter region than from the colder 
+region, and those in the tail of the hotter region will be lost to the neighboring volume 
+faster. Therefore, in addition to the transf er of heat and momentum, the MB VDF is 
+distorted  by an amount that depends on the size of the gradients and on the mean free 
+path, and this distortion is largest in the ta il of the distribution where the velocities are 
+highest. When Kn <<1, the distortion of the VDF is typically ignored and considered a 
+second order effect, and the transport of en ergy, momentum and mass is well described 
+by the thermal conduction, viscous and diffu sion terms. But the presence of such 
+gradients implies the VDF have a sligh tly non-Maxwellian character even when Kn is 
+small.     The distortion of the VDF can become  significant near the exobase affecting the 
+transport properties and estimates of the escape  flux. The effect of the distortion can be 
+calculated using higher order moments of the Boltzmann equations, such as the 13 
+moment equations (e.g. Chap man and Cowling 1970, Hirsch felder et al. 1964; Schunk 
+and Nagy 2000). These are used in Cui et al . (2008) and Yelle et al. (2008) for H
+2 escape 
+from Titan's upper atmosphere. Since the firs t order perturbations to the VDF come from 
+gradients in temperature and flow speed , the thermal conductivity and viscosity 
+determine the size of the distortion. A simp le estimate of the change in the VDF, f(w), in 
+the z direction, with w the z (upward) component of the mol ecular velocity, due to a small 
+vertical temperature gradient is f(w) ~  f(0)(w) - ℓc(dT/dz) df(0)/dT.   Here  f(0) is the MD 
+VDF for a temperature T (e.g., Chapman and Cowling 1970) . Since the viscous effects 
+are typically assumed to be small in SHE, only the temperature gradient is considered 
+here. Writing df(0) /dT=(f(0) /2T)[mw²/kT-1],   it is seen that for large w, f can be 
+significantly affected, as discussed above.     If there is no external heat source for the exobase region, but escape is occurring, then 
+the heat removed by the exiting molecules leads to a negative temperature gradient in the 
+exobase region, a process referred to as ad iabatic cooling. In this case the exobase 
+temperature, T
+x, is lower than T below the exobase so that, in expression for f(w) above, 
+molecules with escape energies are replenishe d from below refilling and adding to the tail 
+of the VDF. Based on this, it is argued that  the presence of the temperature gradient 
+'enhances' Jeans escape for CH 4 and N 2 (Yelle et al. 2008; Str obel 2009). What is meant 
+is that the escape flux is enhanced over the Jeans rate calcula ted using the exobase 
+temperature, Tx. Since Tx is lower than T below the exobase due to escape, this is 
+somewhat circular. But a rough estimate of the enhancement is readily obtained. Since 
+the molecules which populate the hot tail in the exobase region come from below, their 
+contribution to escape can be estimated using the VDF below that altitude from which hot 
+molecules can directly escape to space, a few mean free paths below the exobase (see 
+e.g., Johnson et al. 2008). For λx >~10 the tail of the distributi on dominates escape, so that 
+the 'enhanced' escape rate is, roughly, the Jeans rate calculated using the T a few scale 
+heights below the exobase. The Jeans expressi on for the net escape in a 1D atmosphere 
+is: φJ=4πrx²(nxxv/4)(λx+1)exp(-λx) ; with xv=(8kT x/mπ)1/2 and the subscript x implies values at the nominal exobase. Repl acing their exobase temperatures, Tx, from Yelle et al. 
+(2008) and Strobel et al. (2008) by their values of T at a few scale heights below the 
+exobase, the `enhancement' in Jeans rate for escape of N2(λx~40) or for CH4 (λx~20) from 
+Titan is small and is orders of magnitude belo w the rates calculated by these authors. This 
+result is confirmed by direct simulations of the kinetics in the exoba se region as discussed 
+below.  
+Monte Carlo Simulations  
+    Monte Carlo simulations can b used to de scribe the kinetic beha vior of a gas. Since 
+such simulations are equivalent to solvi ng the Boltzmann equations (e.g., Bird 1994), 
+they reproduce the results of the fluid dyna mic equations well below the exobase where 
+Kn < ~0.1. In addition, such simulations are often more readily implemented than the 
+Boltzmann equations. In Monte Carlo simulati ons, the atmospheric density is described 
+by a large number of representative molecules. Collisions occurring between these 
+representative molecules are described by th e cross sections for real molecules and, 
+hence, can be calculated as accurately as necessary. Between collisions, the 
+representative molecules move subject to the gravitational force of th e planet/satellite and 
+any neighboring body. If ion motion is consider ed then the electromagnetic forces can be 
+included (Tseng et al. 2009). Since a finite  volume is simulated, appropriate boundary 
+conditions are applied: escape across so me upper boundary, absorption or supply of 
+molecules at a lower boundary, etc.     Starting with some initial distributi on of molecular positions and velocities, and 
+prescribed boundary conditions, the evolution of  an atmosphere is simulated. For fixed 
+external conditions, the simulations are run until the distribu tions of position and 
+velocities of the representative molecules yi eld steady-state densities and temperatures. 
+The the number of molecules entering and leav ing the simulated volume is also recorded. 
+From the position and velocities of the representative molecules, the steady state spatial 
+distributions and VDF are calculated. Becau se the cross sections, forces, and boundary 
+conditions can, in principal, be described as  realistically as one likes, the accuracy is 
+usually determined by the number of represen tative particles and the method for deciding 
+when a collision occurs. One such model is the so-called Direct Simulation Monte Carlo 
+(DSMC) model (Bird 1994) which has been applied extensively. 
+    Unlike the fluid equations, such simulati ons can, in principal, describe the behavior 
+and escape of molecules in the exobase region from Kn <<1 to Kn >>1. In practice, 
+consideration is given to those aspects that must be well represented. For escape from 
+Titan, the tail of the VDF above the exoba se must be accurately simulated requiring a 
+large number of representative molecules. A ccuracy can also be increased by the use of 
+adaptable grids and weights. Since the molecules well above the exobase move in large 
+ballistic trajectories, an upper boundary is typically chosen for which the collision 
+probability drops below some prescribed le vel. At this boundary the representative 
+molecules are tested to see if they would es cape or return to the atmosphere. Therefore, 
+the grid size, weights, boundaries, and nu mbers of representative molecules are 
+optimized. These procedures are well understood,  and have been applied, for example, to 
+cometary coma (e.g., Tenishev et al. 2008) and to escape from Titan induced by both the 
+incident plasma (e.g., Shematovich et al. 2003; Michael et al. 2005;  Michael and Johnson 
+2005) and thermal conduction (Tucker and Johnson 2009).     Simulations for a pure N2 atmosphere and an N2 plus CH4 atmosphere were carried out 
+to test the continuum models for escape from Titan. Values of temperature, density and 
+heat flux from Strobel (2008a) and Yelle et al. (2008) were  used at a lower boundary a 
+few scale heights below the exobase (Tucke r and Johnson 2009). These simulations were 
+carried out in both 1D rectangul ar and 3D spherical coordinate s to test escape driven by 
+heat transported from below, ignoring non-th ermal processes. For comparison with the 
+1D radial SHE model, the 3D spherical si mulations were performed for an isotropic 
+atmosphere in which the VDF, density and te mperature exhibit only radial variations. 
+Sensitivity tests were made on the cross se ction models, the upper and lower boundaries 
+positions, and the number of representative molecules. The simulations for the N ₂, as in 
+Strobel (2008b, 2009), and for the N ₂ + CH₄, as in Yelle et al. ( 2008), were reported. It 
+was found that, due to the large value of λx, the radial and rectangul ar simulations results 
+were not very different until well above Titan's exobase. More importantly, it was shown that the 1D continuum models, when exte nded above the exobase, overestimated the 
+escape rates by orders of magnitude, consistent  with the discussion above. Therefore, the 
+continuum estimates of escape at Titan appear  to be incorrect, so  that care should be 
+taken in applying SHE to other bod ies (e.g., Tucker et al. 2010). 
+ 
+Applicability of SHE 
+The SHE model can be understood using an anal ytic solution to Eq. 1. For an assumed 
+constant K with the heat deposited below the lower boundary, r0,  so that Q(r >r 0) = 0, 
+and applying the boundary condition ( T →0, r →∞), one obtains:   
+        
+       CpT(r )= [1-exp(-r φ /r)][-Φg (rφ)+ϕE/φ] + Φg(r)                               (2a) 
+ 
+Here rφ=[φCp/4πK] is a length scale indi cating when energy transport changes from 
+pure thermal conduction, φ=0, to heat transport and escape by molecular flow. The 
+quantity ϕE is the energy per unit time flowing through the system from the lower 
+boundary, so that the energy per escaping mol ecule transported upward from below is 
+ϕϕ/E . For transparency, the scaling of the vari ables typically carried out in SHE is not 
+used (appendix). Setting T = T 0 at the lower boundary r0, T(r)  is obtained from Eq. 2a: 
+ 
+         [Φg(r) - C pT(r)] / [Φg(r0) - C pT0] = [1 - exp(-r φ/r)] / [1 - exp(-r φ/r0)]         (2b) 
+     In published applications of SHE, 
+K is usually temperature dependent, but the change 
+in K is relatively small over the exobase region.  In addition, the solution is typically 
+terminated at some point above the exobase where v/c is still small, as discussed earlier, 
+but with the requirement that T(r) exhibits the asymptotic behavior ( T →0, r →∞). 
+However, the analytic expression in Eq. 2b is representative of  the applications of SHE to 
+planetary escape. 
+It is seen in Eq. 2b that the escape rate, φ, is a free parameter. It is not directly 
+determined by the energy transport unless there is direct measure of the temperature profile. As the viscosity is typically estimat ed to be small, the hydrostatic approximation 
+is used in SHE to determine the density, 
+n: d(nkT)/dr = n d( Φg(r))/dr . Inserting the T(r) 
+in Eq. 2b, and writing  x=rφ/r and x 0=rφ/r0, one finds:  
+       ∫−−− − =0]}))' exp(1('/[' )/( exp{]/[ )(0 0 0 0x
+xp x xx dx kC TTnrn β           (3) 
+ 
+    where β0 = [1 - (C p/k)/λ0] / [1-exp(-x 0)], with λ0 the value of the Jeans parameter at r0. 
+Since the atmospheric temperature is genera lly not directly meas ured, the hydrostatic 
+approximation is also used to extract the at mospheric temperatures. Therefore, the value 
+of φ can be determined by fitting n(r) to available density data below the exobase or to 
+the temperature profile extrac ted from that density data. 
+    Parameters at Titan's exobase vary  with solar conditions in Strobel (2009). 
+Representative exobase values are: rx ~ 4000 km; Kx ~ 1.4×10³ ergcm-1s-1K-1; λx ~ 40 for 
+N₂ or ~ 20 for CH ₄; Tx ~ 150 K; and nx ~ 3 ×10⁷N₂/cm³ with CH₄ less than 10% of the 
+total in the exobase re gion. Using the above K, the length scale in Eq. 2b is rφ ~ φ(3× 
+10-24 km s). Substituting the Jeans value for φ based on these parameters results in a 
+negligible length scale: rφ ~ (3×10⁻¹¹ -10⁻³) km, depending on whether one assumes 
+nitrogen or the methane is the predominant escaping molecule. For these very small 
+values of rφ, the temperature and density profiles  are determined only by the thermal 
+conductivity and the upp er boundary condition: T(r) ~ T 0 (r0/r) and n(r) ~ n 0[r0/r]λ o -1. 
+Such a profile is not representative of Titan’s atmosphere. Using instead K ∝Ts, T(r) 
+would decay somewhat more slowly,  T(r) ~ T 0(r0/r)-1/(1+s)  , but the resulting radial 
+profiles are still not characteristic.  
+    Therefore, unless one assumes that φ is many orders of magnitude larger than the Jeans 
+rate for these λx, the SHE model cannot  give realistic density profiles. For example, 
+Strobel (2008) finds a mass loss rate that is relatively large even for the case Q = 0 above 
+r0 = 3450 km: i.e., mφ ~2×1028amu/s . The recommended value in St robel (2009) is similar, 
+mφ ~ 3×1028amu/s , so that rφ ~ 200 km.These large rate are requi red to give a realistic 
+radial dependence for the atmosphere. The corresponding large upward flux of heat is 
+seen to cause T(r) to decay more slowly  over the exobase region than for the small φ 
+result above. This is the opposite to what is suggested in discussions  of the model. That 
+large φ are favored in the SHE model is also consistent with the scaling procedure 
+typically used, in which the ener gy equation (Eq.1) is divided by φ before solving 
+(appendix).  
+    As stated earlier, for λx >~10, and/ or in regions where v <~0.3c, Eq. 1 can describe the 
+temperature dependence of the atmosphere  a few scale heights below the exobase. 
+Therefore, rather than solv ing up to some region above the exobase, and forcing the 
+temperature to go to zero at very large r, the region above Kn ~ 0.1 should be described 
+by a molecular kinetic model. For this reas on we developed a hybrid model in which the 
+SHE equations are solved, using an estimate of the escape rate up to a radial distance ru at 
+which Kn ~ 0.1. These results are then used as a lower boundary c ondition for a DSMC 
+simulation describing the region from ru up to an altitude well above the exobase where 
+Kn >>1. The improved estimate of the escape rate  so obtained is then used to obtain a 
+new solution of the radial 1D SHE equations. This iterative procedure was applied to the 
+atmosphere of Pluto, a body much smaller th an Titan, for which the thermal escape flux 
+is expected to be significant. To avoid ques tions of the accuracy of the description of 
+Q(r),  the model was applied to the case in wh ich all the solar energy is deposited below 
+the lower boundary: i.e., Q = 0 in Eq. 1. The T(r) profile was found to be very different from that in Strobel (2008a ) and the atmospheric loss rate  was orders of magnitude 
+smaller (Tucker et al. 2010). Of  course, for very large hea ting rates, large escape rates 
+driven by upward thermal conduction are possible, as will be described in subsequent work.     One reason the SHE model disagrees with  the DSMC simulations is that the form for 
+the thermal conductivity, 
+K, is kept constant above the exobase. That is, in calculating K, 
+the product nℓc is typically independent of densit y in the fluid regime. The conductivity 
+term in the Eq. 1 requires that collisional ener gy transfer occurs over distances very small 
+compared to the scale of the density gradients: i.e., Kn <<1. This could be roughly 
+circumvented in an ad hoc manner by replacing nℓc by ~ nℓc /(1+Kn)  in the transition 
+region. Such a substitution would allow th e effective K to decrease with decreasing 
+density when the mean free path length for neutral-neutral collisions is longer than the 
+scale height.     The SHE model can also  be applicable if thermal conduction occurs by means other 
+than by neutral-neutral collisions. If, for exampl e, ions are dragged out of an atmosphere 
+along magnetic field lines, whic h is in fact the case at T itan, then collisional heat 
+transport can effectively pers ist well above the exobase de fined by the neutral-neutral 
+collision cross section. This is a process we have described as 'back  sputtering' (see e.g., 
+Johnson et al. 2008; Johnson 2009).     In SHE, the requirement that T goes to ze ro at infinity is also problematic as seen in 
+the analytic model above. This requires that  the thermal energy of the escaping molecules 
+is fully converted into flow energy. This assu mption is critical, since it is the flow energy 
+in SHE, and not the tail of the random therma l motion, that carries the molecules over the 
+gravitational barrier. However,  when significant escape occu rs, DSMC simulations show 
+that the radial and horizontal temperatures  can diverge significantly with increasing 
+altitude above the exobase  (Tucker et al. 2010). 
+    If the rate of heat de position in an atmosphere is very high, and downward thermal 
+conduction, radiation to space, and horizont al transport are inefficient, then the 
+atmospheric T will increase until the upward flow  is such that molecular escape can carry 
+off the excess energy (e.g., Hunten 1982). But th is does not imply that the SHE model is 
+required. In the absence of direct ener gy deposition in the exobase region, DSMC 
+simulations show that even for an artificially high 
+Tx at Titan ( λx ~10; Tucker and Johnson 
+2009), the relatively large escap e rate does not differ signifi cantly from the Jeans rate. 
+These results are consistent w ith the criterion suggested much  earlier (e.g., Hunten 1982): 
+if λ approaches ~2 above the exobase, then the escape rate is not t oo different from the 
+Jeans rate, but if λ approaches ~2 near or below th e exobase, then direct outflow can 
+occur and the atmosphere is highly extended approaching a cometary  description: i.e., ℓc~ 
+rx. The loss from Titan of the trace species, H2, with λx ~ 3 at the exobase, is borderline 
+(e.g., Cui et al. 2008). However, the resulting H2 flux is not sufficient to drive off the 
+heavy species, but acts to extract heat from the background molecules. For the heavy 
+species, N2 and CH4, with larger λx solutions to the radial 1D equations below the exobase 
+and should be attached to a DS MC model of escape, as discus sed above, or to a value of 
+the escape rate close to the Jeans rate.  
+Summary     Detailed in situ data on the struct ure of atmospheres in the exobase region from 
+spacecraft are now available for some solar system bodies, will soon be for a number of 
+others, and are of considerable interest  to modeling the evolution of exoplanet 
+atmospheres. In this region, also called the transition region, it is critical to use models 
+that can accurately describe the molecular kinetics. Although escap e driven by thermal 
+conduction can compete with non-thermal escap e processes (e.g., Johnson et al. 2008; 
+2009), it is shown that the 1D , slow hydrodynamic escape model, as presently applied, 
+can not be relied on to give accurate molecu lar escape rates. Alt hough the escape rates at 
+Titan could  be significantly different  from the Jeans rate due to nonthermal processes 
+(e.g., DeLaHaye et al. 2007; Johnson et al. 2009), extending conti nuum models, such as 
+the SHE model (Strobel 2008a, 2009) or a diffusi ve separation model (Yelle et al. 2009), 
+above the exobase can give escape rates for h eavy species that are orders of magnitude 
+too large. That such escape rates for Titan are incorrect is argued here based on kinetic 
+theory and the nature of the solutions to the SHE mode, and it has also been suggested 
+based on simulations of the gas kinetics (Tucke r and Johnson 2009). It is shown here that 
+the 1D SHE model, as applied, appears to fa vor large escape rates, and, therefore, it 
+cannot be used to readily e xplore the parameter space if th e loss rates are unknown. This 
+model can, in principal, be used in an ite rative procedure in wh ich the radial fluid 
+equations are solved below the exobase and are used with an estimate of the escape rate 
+based on Jeans escape or are attached to a molecular kinetic model, such as the DSMC model (Tucker et al. 20 10). In addition, if there is a si gnificant upward flux locally, a 3D 
+model allowing horizontal transport is likel y required. However, whether 1D or 3D 
+continuum models are used up to the exoba se, obtaining the thermal and nonthermal 
+escape rates requires the use of a kinetic description of the gas in the tran sition region. 
+ 
+Acknowledgement 
+    This work is supported by a grant from  NASA's Planetary Atmo sphere's Program and 
+by a NASA Cassini Data Analysis Grant.  
+Appendix 
+For clarity, a standard notation was used a bove. The notation typically used in the SHE 
+model is: F=φ/4π is used instead of the net escape rate φ and the variables in Eq. 1 are 
+replaced by scaled variables: λ = Φg/kT = GMm/rkT, ζ  = Fk/K 0r0λ0 , ψ = mv²/kT 0 , τ = 
+T/T 0, and c = (kT 0/m)1/2, where the subscript o implies the value at the lower boundary. 
+The thermal conductivity K, often written as κ, can have a simple temperature 
+dependence, K = K 0(T/T 0)s, and the symbol U =(2kT/m)1/2 is often used as the most 
+probable speed in a Maxwellian distribution. Scaling is then carried out by essentially dividing Eq. 1 by 
+φ, replacing r by λ and using the vari ables above in   
+     ∫−= −−r
+ror
+r p QdrrdrdTKr rUTC . 4 ]/) (4))( [(2 1 2
+0πφφ π  
+This results in the thermal conduc tion term being very large if φ approaches to the Jeans 
+rate and λo ~ 20 - 40, as it is at Titan for CH 4 and N 2. 
+      
+References 
+Bird, G.A. 1994, Molecular Gas Dynamics  a nd the Direct Simulation of Gas Flows, pp. 
+ 218--256. Clarendon Press, Oxford, England  Chamberlain, J. W., Hunten D. 1987, Theory of Planetary Atmosphere. Academic Press,  
+     New York  Chapman, S., and Cowling, T. G. 1970, Th e Mathematical Theory of Nonuniform    
+     Gases, 3rd ed., Cambridge Univ. Press, Cambridge, U.K  Chaufray, J. Y., Modolo, R., Leblanc, F., Ch anteur, G., Johnson, R. E., and  Luhmann,  
+     J. G. 2007, JGR, 112, E09009, doi:10.1029/2007JE002915  Cui, J., Yelle, R. V., and Volk, K. 2008, J. Geophys. Res., 113, E10004,       doi:10.1029/2007JE003032  De La Haye, V., Waite, J. H., Jr., Johns on, R. E., et al. 2007, JGR, 112, A07309,  
+     doi:10.1029/2006JA012222  Hirschfelder, J. O., Curtiss, C. F., Bird, R. B. 1964, Molecular Theory of Gases and  
+     Liquids. New York: W iley; 2nd corrected printing. 
+ Hunten, D. M. 1982, Planet Space Sci., 30. 773  Hunten, D. M.,Watson, A.J. 1982, Icarus, 51, 655  Johnson, R. E. 1990, Energetic Charged-Part icle Interactions w ith Atmospheres and  
+    Surfaces. Springer-Verlag, Berlin.  Johnson, R.E., Combi, M. R., Fox, J. L., Ip, W-H., Leblanc, F., McGrath, M. A.,  
+Shematovich, V.I., Strobel, D. F., Waite J. H., Jr. 2008, Atmospheric escape, chap. in 
+Aeronomy, ed.. A. Nagy et al., Spac e Sci. Rev., doi 10.1007/s11214-008-9415-3 
+ Johnson, R. E. 2009, Trans. R. Soc. A, 367, 753  Johnson, R. E. 2009, Mass Loss Processes in Titan's Upper Atmosphere, Chap. 15 in
+  
+    Titan from Cassini-Huygens  (eds. R.H. Brown et al.) pp373-391. 
+ Krasnopolsky, V.A. 1999, J. Geophys.Res., 104, 5955  McNutt, R. L. 1989, Geophys. Res.Lett., 16, 1225   Michael, M. and Johnson, R. E. 2005, Planetary & Space Sci., 53, 1510 
+ Michael, M., Johnson, R. E., Leblanc, F., Liu, M., Luhmann, J. G. & Shematovich, V. I.  
+     2005, Icarus, 175, 263 
+ Parker, E. N. 1964, Astrophys. J., 139, 93  Schunk, R. W., and Nagy, A. F. 2000, in I onospheres: Physics, Plasma Physics, and  
+     Chemistry, eds. A. J. Dressler, J. T. Houghton, and M. J. Rycroft, pp. 52-- 100,  
+     Cambridge Univ. Press, Cambridge, U.K.  Shematovich, V.I., Johnson, R. E., Michael, M. & Luhmann, J. G. 2003, J. Geophys.        Res., 108, No. E8, 5087, doi:10.1029/2003JE002094  Strobel, D. F. 2008a, Icarus, 193, 588  Strobel, D. F. 2008b, Icarus, 193, 612 
+ Strobel, D. F. 2009, Icarus, 202, 632  
+ Tenishev, V., Combi, M, Davi dsson, B. 2008, Ap. J., 685, 659 
+ Tseng, W-.L., Ip, W.-H., Johnson, R. E., Cassi dy, T. A., and Elrod, M. K. 2009, Icarus,  
+     in press Tucker, O.J. and Johnson, R.E. 2009,  Planet.SpaceSci., 57, 1889 
+Tucker, O.,J., Erwin, J., Volkov, A. N., Cassidy,  T. A., Johnson, R. E. 2010, Escape from  
+     Pluto's Atmosphere: Fluid/DSMC Hybrid Simulation, submitted. Yelle, R. V., Cui, J., Müller-Wodarg, I. C. F. 2008,  J.Geophys. Res.,113,       doi 10.1029/2007JE003031 Watson, A. J., Donahue, T. M., Walker, J. C. G. 1981, Icarus, 48,150 
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+arXiv:1001.0919v2  [cond-mat.mtrl-sci]  14 May 2010Sustained ferromagnetism induced by H-vacancies in grapha ne
+Julia Berashevich and Tapash Chakraborty
+Department of Physics and Astronomy, University of Manitob a, Winnipeg, Canada, R3T 2N2
+The electronic and magnetic properties of graphane with H-v acancies were investigated using the
+quantum-chemistry methods. The hybridization of the edges is found to be absolutely crucial in
+defining the size of the bandgap, which is increased from 3.04 eV to 7.51 eV when the hybridization
+is changed from the sp2to thesp3type. The H-vacancy defects also influence the size of the gap
+that depends on the number of defects and their distribution between the two sides of the graphane
+plane. Further, the H-vacancy defects induced on one side of the graphane plane and placed on the
+neighboring carbon atoms are found to be the source of ferrom agnetism which is distinguished by
+the high stability of the state with a large spin number in com parison to that of the singlet state
+and is expected to persist even at room temperatures. Howeve r, the ferromagnetic ordering of the
+spins is obtained to be limited by the concentration of H-vac ancy defects and ordering would be
+preserved if number of defects do not exceed eight.
+INTRODUCTION
+Graphene is the carbon-based wonder material which
+has gained wide attention due to its many unique elec-
+tronic and magnetic properties. Despite the high mo-
+bility of the charge carriers in graphene resulting from
+its zero effective mass [1–3], the absence of gap hin-
+ders its application in nanoelectronics. The magnetic
+properties of graphene arising from spin ordering of the
+localized states at the zigzag edges [4] or by the pres-
+ence of defects [6–10] might facilitate its application in
+carbon-based spintronics. If the localized states occupy
+the same sublattice then they can induce the sublattice
+imbalance and according to Lieb’s theorem [5] that can
+lead to the ground state being ferromagnetic [6, 7]. The
+room-temperature ferromagnetism in graphene has been
+obtained experimentally [11]. However, there are some
+issues involved in maintaining the ferromagnetic state
+whose stability depends on the concentration of the lo-
+calized states, distance between states and size of the
+graphene flakes through the size of the band gap [6–10].
+Therefore, disappearance of the gap in bulk graphene
+brings some inconsistencies in applying Lieb’s theorem
+[8].
+Recently discovered graphane [15–18] – hydrogenated
+graphene – has brought new impetus in the investiga-
+tion of carbon-based materials due to the predicted ad-
+vantages in its application in nanoelectronics and spin-
+tronics. Termination of the carbon atoms by hydrogens
+leads to the generation of sp3carbon network removing
+theπbands from its band structure thereby generating
+a gap. It was theoretically predicted that fully hydro-
+genated graphane is non-magnetic and a wide band gap
+semiconductor [15, 19]. However, H-vacancy defects in
+graphane generate localized states characterized by non-
+zero magnetic moments (each defect has µ= 1.0µB[20],
+whereµBis the Bohr magneton [Fig. ]). As graphane is
+characterized by a wide gap and the value of the charge
+transfer integral ( tσ∼-7.7 eV [14]) is higher than that
+in graphene ( tπ∼-2.4 eV [14]), according to the Hub-
+FIG. 1: The spin density for a single H-vacancy defect in
+graphane plotted with isovalues of ±0.001 e/˚A3.
+bard model [6, 7] these should stabilize ferromagnetism
+in graphane (for example through an increase of the crit-
+ical value of the on-site repulsion term [9]).
+We report here on our investigation of the electronic
+and magnetic properties of graphane and their modi-
+fication once the H-vacancy defects are introduced in
+the lattice. Our study is performed via the quantum-
+chemistry methods using the spin-polarized density func-
+tional theory with the semilocal gradient corrected func-
+tional (UB3LYP/6-31G) in the Jaguar 7.5 program [21].
+The H-vacancies are introduced in the originally opti-
+mized structure of defect-free graphane in the chair con-
+formation (for the size of the graphane flake see Fig. ),
+whose carbon atoms at the edges are terminated by two
+hydrogen atoms thereby preserving the sp3network over
+the whole lattice.
+SINGLE H-VACANCY DEFECT
+The size of the band gap of graphane nanoribbons de-
+creases exponentially with increasing nanoribbon width
+as a result of vanishing of the confinement effect [12] sim-
+ilarly to that in graphene [13]. Therefore, for graphane
+flakes confined by the edges from all sides of size
+10˚A×16˚A suggesting strong confinement effect, the large
+band gap in comparison to that obtained for nanorib-
+bons [12, 15, 19] is expected. Defect-free graphane flakes2
+of size 10 ˚A×16˚A with edges possessing sp3hybridization
+arefoundtobecharacterizedbydegeneratebandsandby
+a band gap of 7.51 eV (the highest and lowest molecular
+orbitals are HOMO=-6.09 eV and LUMO=1.42 eV, re-
+spectively),asshowninFig. (a). Forcomparisonwehave
+examined flakes of size 18 ˚A×16˚A and found a decrease
+in gap to 7.15 eV (HOMO=-5.87 eV and LUMO=1.27
+eV). The edges in the sp2hybridization, for which the
+edge carbon atoms are terminated by a single hydro-
+gen, possess the localized states. In this case the or-
+bital degeneracy is lifted and the gap decreases to 3.04
+eV (HOMO= −4.64 eV and LUMO= −1.58 eV). In the
+available experiment [16, 17], only a transformation of
+graphene from the conductor to an insulator due to its
+hydrogenation was reported, but the size of the gap and
+the type of edge hybridization were not indicated. Since
+we found that the gap is sensitive to edge hybridization
+and can be increased from 3.04 eV to a maximum of 7.51
+eV by its transformation from sp2tosp3type, this is-
+sue should be the top priority for further experimental
+investigations.
+A single H-vacancy defect in the graphane lattice gen-
+erates an unsaturated dangling bond on the carbon atom
+–πunpaired electron (perpendicular pzorbital). More-
+over, bonding of the carbon atom carrying the defect
+with its neighbors is changed from the sp3hybridization
+tosp2, thus providing a modification of the bond length
+from 1.55 ˚A to 1.52 ˚A. The perpendicular pzorbital and
+the C-C bonds possessing sp2hybridization participate
+in the formation of the localized state characterized by
+an unpaired spin (see the spin density in Fig. ). There-
+fore, this localized state is spin-polarized and generates
+a defect level in the band gap (see the bands in Fig. (b)).
+For theα-spin state, the defect level ( π) appears close
+to the valence band (HOMO= −4.55 eV) thereby sup-
+pressing the size of the gap to ∆α=6.06 eV, while for the
+β-spin state the π∗level occurs closer to the conduction
+band (LUMO= −0.84 eV) and the gap is ∆β=5.29 eV.
+DIFFERENT DISTRIBUTION OF THE
+H-VACANCY DEFECTS
+For several H-vacancy defects, ordering of spins of the
+localized states and the size of the bandgap are defined
+by the distance between the defects and the distribution
+of those defects between the sides of the graphane plane.
+The side dependence is related to the sublattice symme-
+try. For the chair conformation of graphane, the carbon
+atoms belonging to different sublattices are terminated
+by the hydrogen atoms from different sides of the plane.
+It was already known that in graphene [22], when the
+localized states occupy the same sublattice and if their
+spins have antiparallel alignment, then the contribution
+of theπstates to the total energy diminishes as a result
+of the destructive interference between the spin-up andspin-down tails. Therefore, according to Lieb’s theorem
+[5] for the localized states occupying the same sublattice
+a ferromagnetic ordering of their spins would be energet-
+ically favored, but for the states on different sublattices,
+one expects the antiferromagnetic ordering [6–9].
+For vacancies equally distributed between both sides
+of the graphane plane ( AB-distribution), the energet-
+ically favorable spin ordering is the antiparallel align-
+ment of spins between one side (A sublattice) and the
+other (B sublattice). Thus, for even number of vacan-
+cies in the AB-distribution, all spins are paired but the
+band degeneracy can be slightly lifted. In Fig. (c) we
+present the bands for two H-vacancy defects in the AB-
+distribution separated by a distance d= 3aC−C, where
+aC−Cis the length of the C-C bond. Therefore for the
+α- andβ-spin states, the obtained band gap of the size
+∆α,β= 5.41 eV is defined by the energy gap between the
+π(HOMO= −4.56 eV) and the π∗(LUMO= −0.85 eV)
+defect levels. For odd number of defects, one localized
+state would have unpaired spin which generates an extra
+levelπfor theα- andπ∗for theβ-spin states. How-
+ever, when several H-vacancy defects are located on the
+same side of the graphane plane the parallel alignment of
+their spins would be preferred because they belong to the
+same sublattice ( AA-distribution). In Fig. (d) we show
+the energetics of the bands for graphane with two H-
+vacancy defects separated by a distance of d= 4aC−Cin
+its triplet state. Each spin state induces a defect level in
+the valence band of the α-spin state ( πstates) and in the
+conduction band of the β-spin state ( π∗states). There-
+fore, the size of the bandgap for the α-state is defined by
+the conduction band and the defect level πin the valence
+band, while for the β-spin state by the valence band and
+the defect level π∗in the conduction band (∆α=6.08 eV,
+∆β=5.24 eV) that is similar to the case of the single H-
+vacancy (see Fig. (b)). However, the state characterized
+by antiparallel alignment of two spins (the singlet state),
+possesses the πandπ∗defect levels for both the α- and
+β-spin states and the size of the bandgap is defined by
+the energy gap between these defect levels, πandπ∗, i.e.,
+∆α≃∆β(for example see Fig. (c)).
+The destructive and constructive contributions of the
+spin tails of the localized states decrease with increasing
+distance between the defects [22]. Therefore, we have
+calculated the difference in the total energy between the
+triplet and singlet states ( E(2
+2;1
+2)) depending on the dis-
+tance between the two H-vacancy defects. If E(2
+2;1
+2)en-
+ergy is negative the ferromagnetic ordering of spins is
+energetically preferred, but an antiferromagnetic order-
+ing otherwise. We have calculated the two components:
+the energy E0
+(2
+2;1
+2)is considered before relaxation of the
+lattice induced by the presence of defects and the E(2
+2;1
+2)
+component after relaxation. These energies for the AA-
+andAB-distributions and splitting of the πlevels in the
+valence band ( ε1−ε2) are presented in Table I. For3
+FIG. 2: Energetics of the bands in graphane with sp3hybridized edges (solid lines) and the defect levels ( πandπ∗) induced
+by H-vacancies (dashed lines): (a) defect-free graphane; ( b) graphane containing a single H-vacancy; (c) graphane con taining
+two H-vacancies distributed between the two sides of the gra phane plane ( AB-distribution) and separated by a distance of
+d= 3aC−C(antiferromagnetic spin ordering); (d) graphane containi ng two H-vacancies located on one side of the graphane
+plane (AA-distribution) and separated by the distance of d= 4aC−C(ferromagnetic spin ordering). The spin density plotted
+with isovalues of ±0.001 e/˚A3is also presented.
+theAB-distribution the distance between the twodefects
+should be > aC−Cbecause for d=aC−Cthe spins of the
+two localized states are paired which leads to transfor-
+mation of a single bond in sp3hybridization between the
+nearest-neighbor carbon atoms to a double bond in sp2
+hybridization ( σandπbonds) of length 1.36 ˚A. Such a
+defect forms the πandπ∗defect levels which are ener-
+getically close to the edges of the conduction and valence
+bands of graphane. Therefore, the size of the band gap
+defined by the defect levels is 6.38eV, that is much larger
+than that for the levels formed by the non-bonded per-
+pendicular pzorbital (∆=3.71 eV in Fig. (c)).
+A significant energy difference between the triplet and
+the singlet states was found only for d≤2aC−Cin the
+AA-distribution for which the relaxation of the graphane
+lattice stabilizes the state with ferromagnetic spin order-
+ing, and for d <2aC−Cin theAB-distribution. As a
+result, for the nearest location of the defects, the order-
+ing of spins is according to Lieb’s theorem [5]. When the
+defects are spatially separated ( d >2aC−C) the energy
+difference between the state with a large spin number
+and the singlet state diminishes because of decoupling
+of the magnetic moments of the localized states and the
+random spin distribution with a minimum number of un-
+paired spins would be energetically favored. Therefore,
+for even number of defects the system would prefer to re-
+main in the singlet state independent of the distribution
+of defects over the sublattices, while for odd number of
+defects the triplet state is preferred.TABLE I: Difference in the total energy between the triplet
+and singlet states for the non-optimized plane of graphane
+with two defects E0
+(2
+2;1
+2)and after its full relaxation E(2
+2;1
+2).
+(ε1−ε2) is the energy splitting of the πorbitals for the relaxed
+lattice of graphane with two defects.
+distance E0
+(2
+2;1
+2)(eV)E(2
+2;1
+2)(eV)(ε1−ε2) (eV)
+AA-distribution (ferromagnetic ordering)
+d= 2aC−C−1.52 −1.23 2.82×10−1
+d= 4aC−C−0.26−1.32×10−21.52×10−1
+d= 6aC−C7.11×10−5−1.15×10−21.53×10−2
+d= 8aC−C1.76×10−5-7.93×10−36.52×10−3
+AB-distribution (antiferromagnetic ordering)
+d=aC−C1.34 2.98 -
+d= 3aC−C4.30×10−31.18×10−2-
+d= 5aC−C−1.08×10−46.13×10−3-
+We have calculated the fluctuation of the gap size with
+increasing number of defects for the system in its singlet
+state when the size of the gap is defined by the energy
+difference between the induced defect levels, πandπ∗,
+formed by the perpendicular pzorbitals. For the AB-
+distribution ( d > aC−C), the size of the gap can fluctu-
+ate from 1.2 to 3.7 eVdepending on the distance between
+the defects and their locations. The change in the gap
+size is related to the degree of the broken sublattice sym-4
+metry. Moreover, with increasing concentration of the
+H-vacancies ( N >8) we found that the number of local-
+ized states can be smaller than the number of defects.
+For theAA-distribution the size of the gap is found to
+gradually decrease from 3.75 to 2.72 eV with increasing
+number of defects from N= 2 toN= 18. Additionally,
+with a growing number of defects a significant distortion
+of the planarity of graphane, such as buckling of the lat-
+tice inherent for the AA-distribution, was observed.
+STABLE FERROMAGNETISM
+A prediction of the ordering of the AA-distributed
+localized states on the graphane surface is controver-
+sial. According to results in [24] for semihydrogenated
+graphene, i.e. graphene hydrogenated from one side
+(the so called graphone), the non-hydrogenated side
+of graphane is possessing the localized states and the
+spins all of them are ferromagnetically ordered. We be-
+lieve that this behavior is highly debatable because it
+is known for the carbon-like materials, such as diamond
+and graphitic structures, that the total magnetization
+is suppressed by increasing the vacancy concentrations,
+and particularly for graphitic structures this occurs more
+rapidly [10]. In contrast, authors of Ref. [20] did not con-
+sider the ferromagnetic ordering of the states in defected
+graphane at all because they found that for the vacancy
+defects located on the neighbors ( d= 2aC−C), the spins
+are paired, indicating a nonmagnetic state. When the
+distance exceeds2 aC−Cfor which pairingofthe spins can
+not occur, the interaction between the localized states
+vanish, thereby favoring the antiferromagnetic ordering
+of spins. However, we believe that pairing of the spins
+of the localized states located on the neighboring carbon
+atoms is unlikely, considering the significant distance be-
+tween the vacancies ( ∼2.55˚A). Moreover, pairing may
+occur only when the spins are antiferromagnetically or-
+dered (that for the states localized onthe samesublattice
+is against the Lieb’s theorem [5]) and it implies the for-
+mation of the bond, which again seems unlikely because
+of the large distance between the localized states (the
+typicalC−Cbond length in organic compounds is ∼1.4
+˚Aagainst ∼2.55˚Aforthe statesseparatedby d= 2aC−C
+in graphane). Therefore, the questionoforderingofspins
+of the localized states in their AAdistribution is unclear
+as yet which we set out to investigate here.
+We found that for the AA-distribution of the localized
+states formed by the H-vacancies, the ferromagnetic or-
+dering of their spins is possible when the vacancies are
+placed on the neighboring carbon atoms (see E0
+(2
+2;1
+2)for
+theAA-distribution in Table I), thereby generating a
+state characterized by a large spin number (see an ex-
+ample for graphene in Ref. [23]). Just as in [20] it was
+noticed that increasing the distance between the vacan-
+cies leads to vanishing of π-πinteraction between local-ized states resulting in the antiferromagnetic ordering of
+spins. Therefore,weinvestigatingherethestabilityofthe
+ferromagnetic ordering of the spins of the localized states
+placed on the neighboring carbon atoms ( d= 2aC−C)
+and formation of domains of defects depending on the
+concentration of the defects in domain and number of
+domains.
+We found that with increasing concentration of de-
+fects (N) the stability of the state with a large spin
+number, i.e., the difference in total energy between the
+state with a large spin number and the singlet states,
+can increase. Therefore, for two parallel lines of defects
+E0
+(4
+2;1
+2)=−3.32 eV for N= 4 (Fig. (a) for the spin
+density distribution of N= 4),E0
+(6
+2;1
+2)=−5.03 eV for
+N= 6 and E0
+(8
+2;1
+2)=−6.56 eV for N= 8. However, for
+N= 8 the states characterized by the lower spin num-
+ber are close in energy to the state with the larger spin
+number ( E0
+(8
+2;6
+2)=−0.035 eV,E0
+(8
+2;4
+2)=−0.065 eV and
+E0
+(8
+2;2
+2)=−0.12 eV) thereby destabilizing it and limit-
+ing the number of defects having ferromagnetically or-
+dered spins. A further increase of the defect concentra-
+tion(N >8)leadstosignificantsuppressionoftheenergy
+difference between the ferromagnetic and singlet states.
+Thus,E0
+(10
+2;1
+2)=−0.25 eV for N= 10,E0
+(12
+2;1
+2)=−0.17
+eV forN= 12 and E0
+(14
+2;1
+2)=−0.14 eV for N= 14.
+Therefore, just as for graphene [6–9], there is a critical
+value of defect concentration above which the ferromag-
+netic ordering of the spins of the localized states occupy-
+ingthesamesublatticenolongerexists. Anotherdestabi-
+lizing factor for ferromagnetism in graphene containing
+many H-vacancies in the AA-distribution is the lattice
+relaxation leading to the buckling of the graphane struc-
+ture (E(4
+2;1
+2)=−1.49 eV for N= 4 and E(8
+2;1
+2)= 1.29
+eV forN= 8). There is also a significant decrease in
+stability when the defects are divided in groups, because
+the state characterized by antiferromagnetic ordering of
+spins between the groups is close in energy to that of the
+statewithferromagneticorderingofallspins(seethespin
+density distribution for N= 4 in Fig. (b)). Finally, since
+graphane is a wide gap semiconductor and possess no
+localized states at the edges which could interact ( π−π
+interaction)with states formed by the H-vacancydefects,
+the increasing size of the graphane flakes leading to sup-
+pression of the graphane gap should not drastically alter
+the interaction of the localized states and their ordering,
+which was the case in graphene [8].
+To summarize, for the AA-distribution of the H-
+vacancies the size of the band gap can be tuned by
+the level of hydrogenation – the gap slowly decreases
+with increasing number of defects, while for the AB-
+distribution the size of the gap fluctuates in the range
+of 1.2 – 3.7 eV depending on the level of the broken sub-
+lattice symmetry. Moreover, formation of the H-vacancy
+defects redistributed over one side of the graphane plane5
+FIG. 3: Spin density for the AA-distribution of H-vacancies
+(a) 8 H-vacancies ( E0
+(8
+2;1
+2)=−6.36 eV), (b) 4 H-vacancies in
+the singlet state ( E0
+(4
+2;1
+2)= 4.81×10−5eV). Spin densities are
+plotted with isovalues of ±0.005 e/˚A3.
+(AA-distribution) and located on the neighboring car-
+bon atoms belonging to the same sublattice will gen-
+erate a stable state characterized by a large spin num-
+ber (ferromagnetic ordering). For a better stabilization
+of this state, the reorganization of the graphane lattice
+in response to the occurrence of defects should be mini-
+mized. Deformation of the graphane lattice can be mini-
+mized for free standing graphane in the low-temperature
+regime and through interaction of graphane with a sub-
+strate under the condition that the sp3hybridization of
+graphane lattice is preserved. If the rigidity of graphane
+on a substrate could be achieved, graphane containing
+H-vacancies on one side of the plane can even become a
+room-temperature ferromagnet, that obviously has enor-
+mous potentials for application in nanoelectronics and
+spintronics. Because each defect forms a perpendicu-
+larpzorbital possessing unpaired spin and, therefore, its
+contribution to the magnetization is 1 µB, the magnitude
+of magnetization of such room temperature ferromagnet
+can ideally be regulated by the number of H-vacancy de-
+fects. However, the number of defects to achieve the
+stable ferromagnetism at room temperature should be
+limited because above a critical value the ferromagnetic
+ordering of spins would be unstable.
+The work was supported by the Canada Research
+Chairs Program.
+References
+[1] Abergel D S L, Apalkov V, Berashevich J, Ziegler K,
+Chakraborty T Advances in Physics . in press.[2] Novoselov K S, Geim, A K, Morozov S V, Jiang D, Kat-
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+[3] Ando T in Nano-Physics & Bio-Electronics: A New
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+wald M L, Brus L E 2008 Nano Lett. 84597.
+[17] Elias D C, Nair R R, Mohiuddin T M G, Morozov S V,
+Blake P, Halsall M P, Ferrari A C, Boukhvalov D W,
+Katsnelson M I, Geim A K, Novoselov K S 2009 Science
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+2009Nanotechnology 20465704.
+[19] Leb` egue S, Klintenberg M, Eriksson O, Katsnelson M I
+2009Phys. Rev. B 79245117.
+[20] S ¸ahin H, Ataca C, Ciraci S 2009 Appl. Phys. Lett. 95
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+[23] Yazyev O V, Wang W L, Meng S, Kaxiras E 2008 Nano
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\ No newline at end of file
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+arXiv:1001.0920v2  [cs.DS]  3 Feb 2010Symposium on TheoreticalAspects of Computer Science 2010( Nancy, France), pp. 573-584
+www.stacs-conf.org
+ONLINE CORRELATION CLUSTERING
+CLAIRE MATHIEU1AND OCAN SANKUR1,2AND WARREN SCHUDY1
+1Department of Computer Science, Brown University, 115 Wate rman Street, Providence, RI 02912
+2Ecole Normale Sup´ erieure, 45, rue d’Ulm, 75005 Paris, Fran ce
+Abstract. We study the online clustering problem where data items arri ve in an online
+fashion. The algorithm maintains a clustering of data items into similarity classes. Upon
+arrival of v, the relation between v and previously arrived i tems is revealed, so that for
+each u we are told whether v is similar to u. The algorithm can c reate a new cluster for v
+and merge existing clusters.
+When the objective is to minimize disagreements between the clustering and the input,
+we prove that a natural greedy algorithm is O(n)-competitiv e, and this is optimal.
+When the objective is to maximize agreements between the clu stering and the input, we
+prove that the greedy algorithm is .5-competitive; that no o nline algorithm can be better
+than .834-competitive; we prove that it is possible to get be tter than 1/2, by exhibiting a
+randomized algorithm with competitive ratio .5+c for a smal l positive fixed constant c.
+1. Introduction
+We study online correlation clustering. In correlation clu stering [2, 15], the input is
+a complete graph whose edges are labeled either positive, meaning similar, or negative,
+meaning dissimilar. The goal is to produce a clustering that agrees as much as possible
+with the edge labels. More precisely, the output is a cluster ing that maximizes the number
+of agreements, i.e., the sum of positive edges within clusters and the negative e dges between
+clusters. Equivalently, this clustering minimizes the dis agreements. This has applications
+in information retrieval, e.g. [8, 10].
+In the online setting, vertices arrive one at a time and the to tal number of vertices is
+unknown to the algorithm a priori. Upon the arrival of a vertex, the labels of the edges that
+connect this new vertex to the previously discovered vertic es are revealed. The algorithm
+updates the clustering while preserving the clusters alrea dy identified (it is not permitted to
+split any pre-existing cluster). Motivated by information retrieval applications, this online
+model was proposed by Charikar, Chekuri, Feder and Motwani [ 5] (for another clustering
+problem). As in [5], our algorithms maintain Hierarchical Agglomerative Clusterings at all
+times; this is well suited for the applications of interest.
+1998 ACM Subject Classification: F.2.2 Nonnumerical Algorithms and Problems.
+Key words and phrases: correlation clustering, online algorithms.
+Part of this work was funded by NSF grant CCF 0728816.
+c/circlecopyrtC.Mathieu,O.Sankur,andW.Schudy
+CC/circlecopyrtCreativeCommonsAttribution-NoDerivsLicense574 C. MATHIEU, O. SANKUR, AND W. SCHUDY
+The problem of correlation clustering was introduced by Ben -Dor et al. [3] to cluster
+gene expression patterns. Unfortunately, it was shown that even the offline version of
+correlation clustering is NP-hard [15, 2]. The following ar e the two approximation problems
+that have been studied [2, 7, 1]: Given a complete graph whose edges are labeled positive
+or negative, find a clustering that minimizes the number of di sagreements, or maximizes
+the number of agreements. We will call these problems MinDisAgree andMaxAgree
+respectively. Bansal et al. [2] studied approximation algo rithms both for minimization
+and maximization problems, giving a constant factor algori thm for MinDisAgree , and
+aPolynomial Time Approximation Scheme (PTAS) forMaxAgree . Charikar et al. [7]
+proved that MinDisAgree is APX-hard and gave a factor 4 approximation. Ailon et al.
+[1] presented a randomized factor 2 .5 approximation for MinDisAgree , which is currently
+the best known factor. The problem has attracted significant attention, with further work
+on several variants [9, 6, 11, 13, 3, 12, 14].
+In this paper, we study online algorithms for MinDisAgree andMaxAgree . We
+prove that MinDisAgree is essentially hopeless in the online setting: the natural g reedy
+algorithm is O(n)-competitive, and this is optimal up to a constant factor, e ven with ran-
+domization (Theorem 3.4). The situation is better for MaxAgree : we prove that the
+greedy algorithm is a .5-competitive (Theorem 2.1), but that no algorithm can be be tter
+than 0.803 competitive (0 .834 for randomized algorithms, see Theorem 2.2). What is the
+optimal competitive ratio? We prove that it is better than .5 by exhibiting an algorithm
+with competitive ratio 0 .5+ǫ0whereǫ0is a small absolute constant (Theorem 2.6). Thus
+Greedy is not always the best choice!
+More formally, let v1,...,vndenote the sequence of vertices of the input graph, where
+nis not known in advance. Between any two vertices, viandvjfori/\e}atio\slash=j, there is an
+edge labeled positive or negative. In MinDisAgree (resp.MaxAgree ), the goal is to
+find a clustering C,i.e.a partition of the nodes, that minimizes the number of disagr ee-
+ments cost( C): the number of negative edges within clusters plus the numb er of positive
+edges between clusters (resp. maximizes the number of agree ments profit( C): the number
+of positive edges within clusters plus the number of negativ e edges between clusters). Al-
+though these problems are equivalent in terms of optimality , they differ from the point of
+view of approximation. Let OPT denote the optimum solution o fMinDisAgree and of
+MaxAgree .
+In the online setting, upon the arrival of a new vertex, the al gorithm updates the
+current clustering: it may either create a new singleton clu ster or add the new vertex to a
+pre-existing cluster, and may decide to merge some pre-exis ting clusters. It is not allowed
+to split pre-existing clusters.
+Ac-competitive algorithm for MinDisAgree outputs, onany input σ, aclustering C(σ)
+such that cost( C(σ))≤c·cost(OPT( σ)). ForMaxAgree , we must have profit( C(σ))≥
+c·profit(OPT( σ)). (When the algorithm is randomized, this must hold in expe ctation).
+2. Maximizing Agreements Online
+2.1. Competitiveness of Greedy
+For subsets of vertices SandTwe define Γ( S,T) as the set of edges between SandT.
+We write Γ+(S,T) (resp. Γ−(S,T)) for the set of positive (resp. negative) edges of Γ( S,T).ONLINE CORRELATION CLUSTERING 575
+We define the gainof merging SwithTas the change in the profit when clusters SandT
+are merged:
+gain(S,T) =|Γ+(S,T)|−|Γ−(S,T)|= 2|Γ+(S,T)|−|S||T|.
+We present the following greedy algorithm for online correl ation clustering.
+Algorithm 1 Algorithm Greedy
+1:Upon the arrival of vertexvdo
+2:Putvin a new cluster consisting of {v}.
+3:whilethere are two clusters C,C′such that gain( C,C′)>0do
+4:MergeCandC′
+5:end while
+6:end for
+Theorem 2.1. Let OPT denote the offline optimum.
+•For every instance, profit (Greedy )≥0.5profit(OPT).
+•There are instances with profit (Greedy )≤(0.5+o(1))profit(OPT).
+2.2. Bounding the optimal competitive ratio
+Theorem 2.2. The competitive ratio of any randomized online algorithm for MaxAgree is
+at most0.834. The competitive ratio of any deterministic online algorith m forMaxAgree
+is at most 0.803.
+The proof uses Yao’s Min-Max Theorem [4] (maximization vers ion).
+Theorem 2.3 (Yao’s Min-Max Theorem) .Fix a distribution Dover a set of inputs (Iσ)σ.
+The competitive ratio of any randomized online algorithm is a t most
+max{EI[profit(A(I))]
+EI[profit(OPT(I))]:Adeterministic online algorithm },
+where the expectations are over a random input Idrawn from distribution D.
+To prove Theorem 2.2, we first define two generic inputs that we will use to apply
+Theorem 2.3. The first input is a graph G1with 2mvertices and all positive edges between
+them The second input is a graph with 6 mvertices defined as follows. The first 2 mvertices
+have all positive edges between them, the next 2 mvertices have all positive edges between
+them, and the last 2 mvertices also have all positive edges between them. In each o f these
+three sets G1,G2,G3of 2mvertices, half are labelled “left side” vertices and the oth er half
+are labelled ”right side” vertices. All edges between left v ertices are positive, but edges
+between a vertex uon the left side of Giand a vertex von the right side of Gj,j/\e}atio\slash=i, are
+all negative.
+The online algorithm cannot distinguish between the two inp uts until time 2 m+1, so
+it must hedge against two very different possible optimal stru ctures.576 C. MATHIEU, O. SANKUR, AND W. SCHUDY
+2.3. Beating Greedy
+2.3.1.Designing the algorithm. Our algorithm is based on the observation that Algorithm
+Greedy always satisfies at least half of the edges. Thus, if profit(OP T) is less than (1 −
+α/2)|E|for some constant α, then the profit of Greedy is better than half of optimal.
+We design an algorithm called Dense, parameterized by constants αandτ, such that if
+profit(OPT) is greater than (1 −α/2)|E|, then the approximation factor is at least 0 .5+ηfor
+some positive constant η. We use both algorithms Greedy andDenseto define Algorithm
+2.
+Theorem 2.4. Letα∈(0,1),τ >1andη∈(0,1
+2)be such that
+η≤1.5−τ2−((2√
+3+9/2)α1/4+α1/4
+1−α1/4+α/2)22τ−1
+(τ−1). (2.1)
+Then, for every instance such that OPT ≥(1−α/2)E, Algorithm Dense α,τhas profit at
+least(1/2+η)OPT.
+Using Theorem 2.4 we can bound the competitive ratio of Algor ithm 2.
+Corollary 2.5. Letα,τandηbe as above, and let p=α/(2+2η(2−α)). Then Algorithm
+2 has competitive ratio at least1
+2+αη/2
+1+2η(1−α/2).
+Corollary 2.6. Forα= 10−12,τ= 1.0946,η= 0.0555andp= 4,5·10−13, Algorithm 2 is
+1
+2+2·10−14-competitive.
+Algorithm 2 A1
+2+ǫ0-competitive algorithm
+Givenp,α,τ,
+With probability 1 −p, runGreedy ,
+With probability p, runDense α,τ.
+Algorithm 3 Algorithm Dense α,τ
+1:LetC=/hatwideOPT1and for every cluster D∈ C, let repr1(D) :=D∈/hatwideOPT1.
+2:Upon the arrival of a vertex vat timetdo
+3:Putvin a new cluster {v}.
+4:ift=tifor some ithen
+5:forevery cluster Din/hatwiderOPTido
+6: Define a cluster D′′obtained by merging the restriction of Dto{ti−1,...,ti}
+with every cluster C∈ Cin{1,...,ti−1}such that repri−1(C) is defined and is
+half-contained in D.
+7: IfD′′is not empty, set repri(D′′) :=D∈/hatwiderOPTi.
+8:end for
+9:end if
+10:end for
+How do we define algorithm Dense? Using the PTAS of [2], one can compute offline a
+factor (1 −α/2) approximative solution OPT′of any instance of MaxAgree in polynomialONLINE CORRELATION CLUSTERING 577
+time. We will design algorithm Denseso that it guarantees an approximation factor of
+0.5+ηwhenever profit(OPT′)≥(1−α)|E|. Since profit(OPT) ≥(1−α/2)|E|implies that
+profit(OPT′)≥(1−α)|E|, Theorem 2.4 will follow.
+We say that OPT′
+tislargeif profit(OPT′
+t)≥(1−α)|E|. We define a sequence ( ti)i
+ofupdate times inductively as follows: By convention t0= 0. Time t1is the earliest time
+t≥100 such that OPT′
+tis large. Assume tiis already defined, and let jbe such that
+τj−1≤ti< τj. If OPT′
+τjis large, then ti+1=τj, elseti+1is the earliest time t≥τjsuch
+that OPT′
+tis large. Let t1,t2,...,tKbe the resulting sequence. We will note, with an abuse
+of notation, OPT′
+iinstead of OPT′
+tifor 1≤i≤K.
+We say that a cluster Aishalf-contained inBif|A∩B|>|A|/2. Letǫ=α1/4. For
+eachti, we inductively define a near optimal clustering of the nodes [1,ti]. For the base
+case, let /hatwideOPT1be the clustering obtained from OPT′
+1by keeping the 1 /ǫ2largest clusters
+and splitting the other clusters into singletons. For the ge neral case, to define /hatwiderOPTigiven
+/hatwiderOPTi−1, mark the clusters of OPT′
+ias follows. For any Din OPT′
+i, markDif either one
+of the 1/ǫ2−1/ǫlargest clusters of /hatwiderOPTi−1is half-contained in D, orDis one of the 1 /ǫ
+largest clusters OPT′
+i. Then/hatwiderOPTicontains all the marked clusters of OPT′
+iand the rest
+of the vertices in [1 ,ti] as singleton clusters. (Note that, by definition, any /hatwiderOPTicontains
+at most 1 /ǫ2non-singleton clusters; this will be useful in the analysis .)
+Note that Denseonly depends on parameters αandτindirectly via the definition of
+update times and of /hatwideOPT.
+2.3.2.Analysis: Proof of Theorem 2.4. The analysis is by induction on i, assuming that we
+start from clustering /hatwiderOPTiat timeti, then apply the above algorithm from time tito the
+final time t. Ifi= 1 this is exactly our algorithm, and if i=Kthen this is simply /hatwiderOPTK;
+in general it is a mixture of the two constructions.
+More formally, define a forest F(at time t) with one node for each ti≤tand cluster of
+/hatwiderOPTi. The node associated to a cluster Aof/hatwiderOPTi−1is a child of the node associated to a
+clusterBof/hatwiderOPTiif and only if Ais half-contained in B. With a slight abuse of notation,
+we define the following clustering Fassociated to the forest. There is one cluster Tfor each
+tree of the forest: for each node Aof the tree, if iis such that A∈/hatwiderOPTi, then cluster T
+contains A∩(ti−1,ti]. This defines T.
+One interpretation of Denseis that at all times t, there is an associated forest and
+clustering F; and our algorithm Densesimply maintains it. See Figure 1 for an example.
+Lemma 2.7. Algorithm 3 is an online algorithm that outputs clustering Fat timet.
+LetFibe the forest obtained from Fby erasing every node associated to clusters of
+/hatwiderOPTjfor every j < i. With a slight abuse of notation, we define the following clus tering
+Fiassociated to that forest: there is one cluster Cfor each tree of the forest defined as
+follows. For each node Aof the tree, let k≥ibe such that A∈/hatwiderOPTk: thenCcontains
+A∩(tk−1,tk] ifk > i, andCcontains Aifk=i. This defines a sequence of clusterings such
+thatF1=Fis the output of the algorithm, and FK=/hatwiderOPTK.
+Lemma 2.8 (Main lemma) .For any 2≤i≤K,
+cost(Fi−1)−cost(Fi)≤/parenleftbigg
+(4+2√
+3)ǫ+ǫ
+1−ǫ/parenrightbigg
+titK.578 C. MATHIEU, O. SANKUR, AND W. SCHUDY
+Figure 1: An example of a forest Fgiven in left, and the corresponding clustering given in
+right. Here, we have /hatwiderOPTi={B1,B2}and/hatwiderOPTi−1={A1,...,A 5}.
+We defer the proof of Lemma 2.8 to next section. Assuming Lemm a 2.8, we upper-
+bound the cost of clustering F.
+Lemma 2.9 (Lemma 14, [2]) .For any 0< c <1and clustering C, letC′be the clustering
+obtained from Cby splitting all clusters of Cof size less than cn, wherenis the number of
+vertices. Then cost (C′)≤cost(C)+cn2/2.
+Lemma 2.10. cost(F)≤((2√
+3+9/2)ǫ+ǫ
+1−ǫ+ǫ4/2)2τ−1
+τ−1t2
+K.
+Proof.We write: cost( F) = cost( /hatwiderOPTK) +/summationtextK
+i=2(cost(Fi−1)−cost(Fi)).By definition,
+/hatwiderOPTKcontains the 1 /ǫlargest clusters of OPT′
+K. Then the remaining clusters of OPT′
+K
+are of size at most ǫtK. By Lemma 2.9, the cost of /hatwiderOPTKis at most cost(OPT′
+K)+ǫt2
+K/2≤
+(α+ǫ)t2
+K/2. Applying Lemma 2.8, and summing over 2 ≤i≤K, we get
+cost(F)≤(α+ǫ)t2
+K/2+/parenleftbigg
+(4+2√
+3)ǫ+ǫ
+1−ǫ/parenrightbigg/summationdisplay
+ititK.
+By definition of the update times ( ti)i, for any j >0 there exists at most one tisuch that
+τj≤ti< τj+1. LetLbe such that τL≤tK< τL+1. Then
+/summationdisplay
+1≤i≤Kti≤/summationdisplay
+1≤i≤K−1ti+tK≤/summationdisplay
+1≤j≤Lτj+tK≤τL+1
+τ−1+tK≤2τ−1
+τ−1tK.
+Hence the desired bound on cost( F).ONLINE CORRELATION CLUSTERING 579
+Proof of Theorem 2.4. Fix an input graph of size n, such that profit(OPT) ≥(1−α/2)/parenleftbign
+2/parenrightbig
+.
+By Lemma 2.10, at time tK, Algorithm 3 has clustering Fwith cost( F)≤O(ǫ)2τ−1
+τ−1t2
+K.
+By definition of the update times, n < τt K. To guarantee a competitive ratio of 0 .5+η,
+for some η, the cost must not exceed (0 .5−η)/parenleftbign
+2/parenrightbig
+at timen, when all vertices tK+1,...,n
+are added as singleton clusters. Thenumber of new edges adde d to the graph between times
+tKandnis/parenleftbign−tK
+2/parenrightbig
++tK(n−tK). We must have
+2τ−1
+τ−1O(ǫ)t2
+K+/parenleftbiggn−tK
+2/parenrightbigg
++tK(n−tK)≤(0.5−η)/parenleftbiggn
+2/parenrightbigg
+, (2.2)
+for some 0 < η <0.5. Using the fact that n−tK≤(τ−1)tKandtK≤n−1, to satisfy
+(2.2), it suffices to have
+2τ−1
+τ−1O(ǫ)t2
+K+t2
+K(τ−1)2/2+(τ−1)t2
+K≤(0.5−η)t2
+K/2,
+whichisequivalentto(2.1). Moreover wehavethefollowing naturalconstraintsonconstants
+η,ǫandτ: 0< η <0.5, 0< ǫ <1, andτ >1. Then, for any set of values of constants η,ǫ,
+τverifying those constraints, Algorithm Denseis 0.5+η-competitive.
+2.3.3.The core of the analysis: proof of Lemma 2.8.
+Lemma 2.11. LetSibe the set of vertices of the non-singleton clusters that are not among
+the1/ǫ2−1/ǫlargest clusters of /hatwiderOPTi−1. Then|Si| ≤ǫ
+1−ǫti−1.
+Proof.LetCbe a cluster of /hatwiderOPTi−1, such that C⊆ Si. Then|C| ≤(1/ǫ2−1/ǫ)−1ti−1.
+Since there are at most 1 /ǫsuch clusters, the number of vertices of these are at most
+1/ǫ(1/ǫ2−1/ǫ)−1ti−1.
+Notation 2.12. For anyi/\e}atio\slash=j, and a cluster Bof OPT′
+i, we denote by γi,j
+Bthe square root
+of the number of edges of [1 ,tmin(i,j)]×[1,tmin(i,j)], adjacent to at least one node of B, and
+which are classified differently in OPT′
+iand in OPT′
+j.
+We refer to non singleton clusters as largeclusters.
+Lemma 2.13. LetTibe the set of vertices of those 1/ǫ2−1/ǫlargest clusters of /hatwiderOPTi−1
+that are not half-contained in any cluster of OPT′
+i. Then|Ti| ≤√
+6/summationtext
+largeC∈/hatwiderOPTi−1γi,i−1
+C.
+LetBbe a cluster of /hatwiderOPTi. For any j≤i, we define Cj(B) as the cluster associated
+with the tree of Fjthat contains B. For any B∈/hatwiderOPTi, we call Ci−1(B) theextension of
+Ci(B) toFi−1. By definition of Fi, the following lemma is easy.
+Lemma 2.14. For any B∈/hatwiderOPTi, the restriction of Ci−1(B)to(ti−1,tK]is equal to the
+restriction of Ci(B)to(ti−1,tK].
+Let (Aj)jdenote the clusters of /hatwiderOPTi−1that are half-contained in B. We define δi(B)
+as the symmetric difference of the restriction of Bto [1,ti−1] and∪jAj:
+δi(B) = (B∩[1,ti−1])∆∪jAj.580 C. MATHIEU, O. SANKUR, AND W. SCHUDY
+Lemma 2.15. For any cluster CiofFi, letC′
+idenote the extension of CitoFi−1. Then
+/uniondisplay
+Ci∈FiCi\C′
+i⊆ Si∪Ti∪/uniondisplay
+largeB∈/hatwiderOPTiδi(B)
+Proof.By Lemma 2.14, the partition of the vertices ( ti−1,tK] is the same for Cias forC′
+i.
+SoCiandC′
+ionly differ in the vertices of [1 ,ti−1):
+/uniondisplay
+Ci∈FiCi\C′
+i⊆/uniondisplay
+B∈/hatwiderOPTiδi(B).
+We will show that for a singleton cluster Bof/hatwiderOPTi,δi(B) is included in Si∪
+Ti/uniontext
+largeB∈/hatwiderOPTiδi(B), which yields the lemma.
+LetB={v}be a singleton cluster of /hatwiderOPTisuch that δi(B)/\e}atio\slash={}. A non-singleton
+cluster cannot be half-contained in a singleton cluster so w e conclude no clusters are half-
+contained in Band hence δi(B) ={v}. By definition of δi(B),v∈[1,ti−1]. So there exists
+a cluster Aof/hatwiderOPTi−1that contains v. Clearly Ais not a singleton since otherwise δi(B)
+would be {}. There are two cases.
+First, ifAis half-contained in a cluster B′/\e}atio\slash=Bof/hatwiderOPTithen cluster B′is necessarily
+large since it contains more than one vertex of A. Then we have v∈δi(B′).
+Second, if Ais not half-contained in any cluster of /hatwiderOPTithenA⊆ Si∪Ti. In fact, if
+Ais half-contained in a cluster of OPT′
+iwhich is split into singletons in /hatwiderOPTi, thenAis
+not one of the 1 /ǫ2−1/ǫlargest clusters of /hatwiderOPTi−1, andA⊆ Si. IfAis not half-contained
+in any cluster of OPT′
+i, thenA⊆ TiifAis one of the 1 /ǫ2−1/ǫlargest clusters of /hatwiderOPTi−1
+andA⊆ Siotherwise.
+Lemma 2.16. For any large cluster Bof/hatwiderOPTi,|δi(B)| ≤2√
+2γi,i−1
+B.
+Proof.LetB′denote the restriction of Bto [1,ti−1]. We first show that
+1/2(|∪jAj\B′|)2≤(γi,i−1
+B)2.
+Observe that ( γi,i−1
+B)2includes all edges uvsuch that one of the following two cases occurs.
+First, ifu∈Aj\Bandv∈Aj∩B: such edges are internal in the clustering OPT′
+i−1but
+external in the clustering OPT′
+i. The number of edges of this type is/summationtext
+j|Aj\B|·|Aj∩B|.
+SinceAjis half-contained in B, this is at least/summationtext
+j|Aj\B|2.
+Second, if u∈Aj∩Bandv∈Ak∩Bwithj/\e}atio\slash=k: such edges are external in the
+clustering OPT′
+i−1but internal in the clustering OPT′
+i. The number of edges of this type
+is/summationtext
+j<k|Aj∩B|·|Ak∩B| ≥/summationtext
+j<k|Aj\B|·|Ak\B|.
+Summing, it is easy to infer that ( γi,i−1
+B)2≥(1/2)/parenleftBig/summationtext
+j|Aj\B|/parenrightBig2
+= (1/2)|∪jAj\B′|2.
+Let (A′
+j)jdenote the clusters of /hatwiderOPTi−1that are not half-contained in B, but have non-
+empty intersections with B. We now show that
+1/2(|B′\∪jA′
+j|)2≤(γi,i−1
+B)2.
+We have B′\∪jAj=∪j(A′
+j∩B). Observe that any A′
+jis a large cluster of /hatwiderOPTi−1, thus a
+cluster of OPT′
+i−1. Then ( γi,i−1
+B)2includes all edges uvsuch that one of the following two
+cases occursONLINE CORRELATION CLUSTERING 581
+First, ifu∈A′
+j\Bandv∈A′
+j∩B: such edges are internal in the clustering OPT′
+i−1but
+external in the clustering OPT′
+i. The number of edges of this type is/summationtext
+j|A′
+j\B|·|A′
+j∩B|.
+SinceA′
+jis not half-contained in B, this is at least/summationtext
+j|A′
+j∩B|2.
+Second, if u∈A′
+j∩Bandv∈A′
+k∩Bwithj/\e}atio\slash=k: such edges are external in the
+clustering OPT′
+i−1but internal in the clustering OPT′
+i. The number of edges of this type
+is/summationtext
+j<k|A′
+j∩B|·|A′
+k∩B|.
+Summing, we get
+(γi,i−1
+B)2≥(1/2)
+/summationdisplay
+j|A′
+j∩B|
+2
+= (1/2)|B′\∪jA′
+j|2.
+Lemma 2.17. For any i≥1,/hatwiderOPTihas at most 1/ǫ2non singleton clusters, all of which
+are clusters of OPT′
+i
+Proof.By definition, /hatwiderOPT1has at most 1 /ǫ2non singleton clusters. For any i >1, a cluster
+of/hatwiderOPTi−1can only be half-contained in one cluster of OPT′
+i. Therefore given /hatwiderOPTi−1, at
+most 1/ǫ2clusters of OPT′
+iare marked. Thus /hatwiderOPTihas at most 1 /ǫ2clusters.
+We can now prove Lemma 2.8.
+Proof of Lemma 2.8. By Lemma 2.14, clusterings FiandFi−1only differ in their partition
+of [1,ti−1]. Then the set of the vertices that are classified differently i nFiandFi−1is
+∪iCi\Ci−1. Each of these vertices creates at most tKdisagreements:
+cost(Fi−1)−cost(Fi)≤/summationdisplay
+Ci∈Fi|Ci\Ci−1|tK (2.3)
+By Lemmas 2.15 and 2.16,
+/summationdisplay
+Ci∈Fi|Ci\Ci−1|tK≤
+2√
+2/parenleftBigg/summationdisplay
+largeB∈/hatwiderOPTiγi,i−1
+B/parenrightBigg
++|Si|+|Ti|
+tK.(2.4)
+By Lemmas 2.11 and 2.13,
+|Si| ≤ǫ
+1−ǫti−1and|Ti| ≤√
+6/summationdisplay
+largeB∈/hatwiderOPTi−1γi−1,i
+B(2.5)
+Theterm/summationtext
+largeB∈/hatwiderOPTi−1γi−1,i
+Bcanbeseenasthe ℓ1normofthevector( γi−1,i
+B)largeB. Since
+/hatwiderOPTi−1has at most 1 /ǫ2large clusters by Lemma 2.17, we can use H¨ older’s inequalit y:
+/summationdisplay
+largeB∈/hatwiderOPTi−1γi−1,i
+B=/bardbl(γi−1,i
+B)largeB/bardbl1≤1/ǫ/bardbl(γi−1,i
+B)largeB/bardbl2.
+By definition we have /bardbl(γi−1,i
+B)large B/bardbl2≤/radicalBig
+2(cost(OPT′
+i−1)+cost(OPT′
+i)). Thus
+/summationdisplay
+largeB∈/hatwiderOPTi−1γi−1,i
+B≤1/ǫ/radicalBig
+2(αt2
+i−1/2+αt2
+i/2)≤√
+2α
+ǫti. (2.6)582 C. MATHIEU, O. SANKUR, AND W. SCHUDY
+Similarly, we have
+/summationdisplay
+largeB∈/hatwiderOPTiγi,i−1
+B≤√
+2α
+ǫti. (2.7)
+Combining equations (2.3) through (2.7) and α=ǫ4yields
+cost(Fi−1)−cost(Fi)≤/parenleftbigg
+(4+2√
+3)ǫ+ǫ
+1−ǫ/parenrightbigg
+titK
+3. Minimizing Disagreements Online
+Theorem 3.1. Algorithm Greedy is(2n+1)-competitive for MinDisAgree .
+To prove Theorem 3.1, we need to compare the cost of the optima l clustering to the
+cost of the clustering constructed by the algorithm. The fol lowing lemma reduces this to,
+roughly, analyzing the number of vertices classified differen tly.
+Lemma 3.2. LetWandW′be two clusterings such that there is an injection W′
+i∈ W′→
+Wi∈ W. Then cost (W′)−cost(W)≤n/summationtext
+i|W′
+i\Wi|.
+For subsets of vertices S1,...,S m, we will write, with a slight abuse of notation,
+Γ+(S1,...,S m) for the set of edges in Γ+(Si,Sj) for any i/\e}atio\slash=j: Γ+(S1,...,S m) =
+∪i/negationslash=jΓ+(Si,Sj).
+Lemma 3.3. LetCbe a cluster created by Greedy , andW={W1,...,W K}denote
+the clusters of OPT. Then |C| ≤maxi|C∩Wi|+ 2|Γ+(C∩W1,...,C∩WK)|.. We call
+i0= argmax
+i|C∩Wi|theleaderofC.
+Proof of Theorem 3.1. LetCdenote the clustering given by Greedy . For every cluster
+Wiof OPT, merge all the clusters of Cthat have ias their leaders. Let C′= (W′
+i) be
+this new clustering. By definition of the greedy algorithm, t his operation can only in-
+crease the cost since every pair of clusters have a negative- majority cut at the end of the
+algorithm:cost( C)≤cost(C′).We apply Lemma 3.2 to W=OPT and W′=C′, and ob-
+tain: cost( C′)≤cost(OPT) + n/summationtext
+i|W′
+i\Wi|. By definition of C′we have |W′
+i\Wi|=/summationtext
+C∈C:leader(C)=i/summationtext
+j/negationslash=i|C∩Wj|,hence
+/summationdisplay
+i|W′
+i\Wi|=/summationdisplay
+C∈C/summationdisplay
+j/negationslash=leader( C)|C∩Wj|.
+By Lemma 3.3,/summationtext
+j/negationslash=leader( C)|C∩Wj| ≤2|Γ+(C∩W1,...,C∩WK)|. Finally, to boundOPT
+from below, we observe that, for any two clusterings CandW, it holds that the sum over
+C∈ Cof|Γ+(C∩W1,...,C∩WK)|is less than cost( W). Combining these inequalities
+yields the theorem.ONLINE CORRELATION CLUSTERING 583
+Theorem 3.4. Let ALG be a randomized algorithm for MinDisAgree . Then there exists
+an instance on which ALG has cost at least n−1−cost(OPT)where OPT is the offline
+optimum. If OPT is constant then cost (ALG) = Ω(n)cost(OPT).
+Proof.Consider two cliques AandB, each of size m, where all the internal edges of Aand
+Bare positive. Choose a vertex ainA, and a set of vertices b1,...,bkinB. Define the
+edge labels of abias positive, for all 1 ≤i≤kand the rest of the edges between AandB
+as negative. Define an input sequence starting with a,b1,...,bk, followed by the rest of the
+vertices in any order.
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+arXiv:1001.0922v1  [math.LO]  6 Jan 2010Understanding preservation theorems, II
+Chaz Schlindwein
+Department of Mathematics and Computing
+Lander University
+Greenwood, South Carolina 29649 USA
+cschlind@lander.edu
+November 18, 2018
+Abstract
+We present an exposition of much of Sections VI.3 and XVIII.3 from S helah’s
+bookProper and Improper Forcing . This covers numerous preservation theorems
+for countable support iterations of proper forcing, including pres ervation of the
+property “no new random reals over V,” the property “reals of the ground model
+form a non-meager set,” the property “every dense open set con tains a dense
+open set of the ground model,” and preservation theorems related to the weak
+bounding property, the weakωω-bounding property, and the property “the set
+of reals of the ground model has positive outer measure.”
+11 Introduction
+This is the fourth of a sequence of papers giving an exposition of por tions of
+Shelah’s book, Proper and Improper Forcing [9]. The earlier papers were [6], [7],
+and [8], which cover sections 2 through 8 of [9, Chapter XI], sections 2 and 3 of
+[9, Chapter XV], and sections 1 and 2 of [9, Chapter VI], respectively .
+In this paper, we give an exposition of much of [9, Sections VI.3 and XV III.3]
+dealing with preservation theorems. We include proofs of the prese rvation, under
+countable support iteration of proper forcing, of the property “ no new random
+reals,” the property “every open dense set contains an old open de nse set,” the
+property of non-meagerness of the reals of the ground model, an d preservation
+theorems related to weak bounding, weakωω-bounding, and “the set of reals of
+the ground model has positive outer measure.”
+Another treatment of preservation theorems, using different me thods, is given
+in [2], [3]. The results of [9, Section VI.3] included here as Theorem 2.5, T heorem
+3.5, and Theorem 4.13 may also be derived as corollaries of [1, Theorem 6.1.18];
+the proof there is essentially the same as the ones given by Shelah in [9 , Section
+VI.3].
+2 Preservation of weak bounding
+The most important tool in the study of preservation theorems fo r countable
+support forcing iterations is the Proper Iteration Lemma. Here, a nd throughout
+this paper, Pα,κis characterized by
+V[GPα]|= “Pα,κ={p[α,κ):p∈Pκandpα∈GPα}.”
+Theorem 2.1 (Proper Iteration Lemma, Shelah). Suppose /an}bracketle{tPη:η≤κ/an}bracketri}htisa
+countable support forcing iteration based on /an}bracketle{tQη:η < κ/an}bracketri}htand for every η < κwe
+have that 1/bardbl−Pη“Qηis proper.” Suppose also that α < κandλis a sufficiently
+large regular cardinal and Nis a countable elementary submodel of Hλand
+{Pκ,α} ∈Nandp∈PαisN-generic and p/bardbl−“q∈Pα,κ∩N[GPα].”Then there
+isr∈Pκsuch that risN-generic and rα=pandp/bardbl−“r[α,κ)≤q.”
+Proof: See (e.g.) [8, Theorem 2.1].
+We deal first with the weak bounding property.
+2Definition 2.2. Suppose AandBare sets of integers. We say A⊆∗Biff
+{n∈A:n /∈B}is finite.
+Definition 2.3. Suppose P ⊆[ω]ℵ0is a filter. We say Pis a P-filter iff P
+contains all co-finite subsets of ω, and(∀U ∈[P]ℵ0)(∃A∈ P)(∀B∈ U)(A⊆∗B).
+Definition 2.4. Suppose Pis a P-filter and Pis a forcing notion. We say that
+Pis weakly P-bounding iff 1/bardbl−P“(∀A∈[ω]ℵ0)(∃B∈ P)(A/ne}ationslash⊆∗B).”
+The following Theorem is [9, Conclusion VI.3.17(1)].
+Theorem 2.5. Suppose κis a limit ordinal and Pis a P-filter and /an}bracketle{tPη:η≤κ/an}bracketri}ht
+is a countable support forcing iteration based on /an}bracketle{tQη:η < κ/an}bracketri}ht. Suppose for every
+η < κwe have Pηis weakly P-bounding and 1/bardbl−Pη“Qηis proper.” ThenPκis
+weaklyP-bounding.
+Proof: This is clear if κhas uncountable cofinality, so assume cf( κ) =ω.
+Suppose p∈PκandAisaPκ-nameand p/bardbl−“A∈[ω]ℵ0.” Letλbeasufficiently
+largeregularcardinaland Nacountableelementarysubstructureof Hλsuchthat
+{Pκ,P,A,p} ∈N.
+Let/an}bracketle{tαk:k∈ω/an}bracketri}ht ∈Nbe an increasing sequence cofinal in κsuch that α0= 0.
+FixB∈ Psuch that ( ∀X∈ P∩N)(B⊆∗X). It suffices to show p/ne}ationslash/bardbl−“A⊆∗B.”
+Build/an}bracketle{tqk,pk,mk:k∈ω/an}bracketri}htsuch that q0=pand for every k∈ωwe have that
+each of the following holds:
+(1)pk∈PαkisN-generic, and
+(2)pk/bardbl−“qk+1∈Pαk,κ∩N[GPαk] andqk+1≤qk[αk,κ),” and
+(3)mkis aPαk-name for an integer and pk/bardbl−“ifk >0 thenmk> mk−1,” and
+(4)pk/bardbl−“qk+1/bardbl−‘mk/∈Bandmk∈A,’” and
+(5)pk+1αk=pk, and
+(6)pk/bardbl−“pk+1[αk,αk+1)≤qk+1αk+1.”
+Theconstructionproceedsasfollows. Given pk,qk, andmk−1, workin V[GPαk]
+withpk∈GPαk.
+BuildAk∈[ω]ℵ0∩N[GPαk] and/an}bracketle{tqi
+k:i∈ω/an}bracketri}ht ∈N[GPαk] such that q0
+k=
+qk[αk,κ), and for every i∈ωwe haveqi+1
+k≤qi
+k, andqi+1
+k/bardbl−“i∈Aiffi∈Ak.”
+Using the hypothesis on Pαkwe may choose Bk∈ Psuch that Ak/ne}ationslash⊆∗Bk. By
+elementarity we may assume Bk∈N[GPαk]. Because pkisN-generic, we have
+Bk∈V∩N[GPαk] =N.
+3Because Bk∈Nwe have B⊆∗Bk, and hence Ak/ne}ationslash⊆∗B. Therefore we may
+choosemk∈ωsuch that mk∈Akandmk/∈Band ifk >0 thenmk> mk−1.
+Letqk+1=qmk+1
+k. Clearly (2), (3), and (4) are satisfied.
+Using the Proper Iteration Lemma we may choose pk+1satisfying (1), (5), and
+(6).
+This completes the recursive construction.
+Letr∈Pκbe such that for every k∈ωwe have rαk=pk.
+Suppose, towards a contradiction, that r′≤randr′/bardbl−“A⊆∗B.” FixPκ-
+namesnandksuch that r′/bardbl−“A⊆n∪B, andmk> n.”
+By strengthening r′we may assume that kandmkare integers rather than
+merely names.
+Because r′≤(pk+1,qk+1) we have r′/bardbl−“mk∈A−n⊆B, andmk/∈B.” This
+is a contradiction.
+The Theorem is established.
+Lemma 2.8. Suppose Pis weakly P-bounding and 1/bardbl−P“Qis almost P-
+bounding.” ThenP∗Qis weakly P-bounding.
+Proof: Suppose ( p,q)/bardbl−P∗Q“A∈[ω]ℵ0.” Take q′andBinVPsuch that
+p/bardbl−“B∈ Pandq′≤qand
+(*)q′/bardbl−‘(∀Y∈[ω]ℵ0∩V[GP])(A∩Y/ne}ationslash⊆∗B).’”
+Takep′≤pandB′∈ Psuch that p′/bardbl−“B/ne}ationslash⊆∗B′.” By (*) we have ( p′,q′)/bardbl−
+“A∩(B−B′)/ne}ationslash⊆∗B.” Hence ( p′,q′)/bardbl−“A/ne}ationslash⊆∗B′.”
+The Lemma is established.
+Theorem 2.9. Suppose Pis a P-filter and /an}bracketle{tPη:η≤κ/an}bracketri}htis a countable support
+forcing iteration based on /an}bracketle{tQη:η < κ/an}bracketri}ht. Suppose for every η < κwe have 1/bardbl−Pη
+“Qηis proper and almost P-bounding.” ThenPκis weakly P-bounding.
+Proof: By Theorem 2.5 and Lemma 2.8.
+3 Preservation of weaklyωω-bounding
+In this section we give an exposition of a preservation theorem, due to Shelah,
+concerning the weakωω-bounding property.
+Definition 3.1. Suppose fandgare inωω. We say f≤∗giff(∃n∈ω)(∀k > n)
+(f(k)≤g(k)).
+4Definition 3.2. Suppose F⊆ωωandg∈ωω. We say that gboundsFiff
+(∀f∈F)(f≤∗g).
+Definition 3.3. Suppose Pis a forcing notion. We say that Pis weaklyωω-
+bounding iff 1/bardbl−“(∀f∈ωω)(∃g∈ωω∩V)(g/ne}ationslash≤∗f).”
+The following Theorem is [9, Conclusion VI.3.17(2)].
+Theorem 3.4. Suppose κis a limit ordinal and /an}bracketle{tPη:η≤κ/an}bracketri}htis a countable
+support forcing iteration based on /an}bracketle{tQη:η < κ/an}bracketri}ht. Suppose for every η < κwe
+havePηis weaklyωω-bounding and 1/bardbl−Pη“Qηis proper.” ThenPκis weakly
+ωω-bounding.
+Proof: UsetheproofofTheorem2.5with([ ω]ℵ0,P,⊇∗)replacedwith(ωω,ωω∩
+V,≤∗).
+The Theorem is established.
+The following definition is equivalent to [9, Definition VI.3.5(1)].
+Definition 3.5. Suppose Pis aforcing notion. We say Pis almostωω-bounding
+iff1/bardbl−“(∀f∈ωω)(∃g∈ωω∩V)(∀A∈[ω]ℵ0∩V)(∃∞n∈A)(f(n)< g(n)).”
+Lemma3.6. Suppose Pisalmostωω-bounding. Then Pisweaklyωω-bounding.
+Proof: Take A=ωin Definition 3.5.
+Lemma 3.7. Suppose Pis weaklyωω-bounding and 1/bardbl−P“Qis almostωω-
+bounding.” ThenP∗Qis weaklyωω-bounding.
+Proof: Like Lemma 2.7.
+The Lemma is established.
+Theorem 3.9. Suppose /an}bracketle{tPη:η≤κ/an}bracketri}htis a countable support forcing iteration
+based on /an}bracketle{tQη:η < κ/an}bracketri}ht. Suppose for every η < κwe have 1/bardbl−Pη“Qηis proper
+and almostωω-bounding.” ThenPκis weaklyωω-bounding.
+Proof: By Theorem 3.4 and Lemma 3.7.
+4 Preservation of no new random reals
+We now turn our attention to the preservation of the property “n o new random
+reals.”
+5Definition 4.1. Forτ∈<ω2, we letUτ={η∈ω2:ηextendsτ}.
+Recall that for A⊆ω2, the outer measure of Aisµ∗(A) = inf{Σ{2−lh(τ):τ∈
+C}:C⊆<ω2 andA⊆/uniontext{Uτ:τ∈C}}.Ais Lebesgue measurable iff ( ∀τ∈<ω2)
+(µ∗(A∩Uτ)+µ∗(Uτ−A) =µ∗(Uτ)), in which case we write µ(A) =µ∗(A).
+Definition 4.2. Suppose A⊆ω2. We say that Ais closed under rational
+translation iff (∀b∈A)(∀b∗=a.e.b)(b∗∈A).
+The following Lemma is known as “Kolmogorov’s zero-one Law.”
+Lemma 4.3. Suppose A⊆ω2is closed under rational translations and suppose
+thatAis Lebesgue measurable. Then µ(A) = 0orµ(A) = 1.
+Proof: Let γ=µ(A) and suppose, towards a contradiction, that 0 < γ <1.
+Claim 1. Whenever τ∈<ω2 andτ0andτ1are the immediate successors of τ,
+thenµ(A∩Uτ0) =µ(A∩Uτ1).
+Proof: We have µ(A∩Uτ0) = 2−lh(τ0)µ({b∈ω2:τ0ˆb∈A}) = 2−lh(τ1)µ({b∈
+ω2:τ1ˆb∈A}) =µ(A∩Uτ1).
+Claim 2: For all τ∈<ω2 we have µ(A∩Uτ) = 2−lh(τ)γ.
+Proof: By induction on τ, using Claim 1.
+Chooseδ > γsuch that δ2< γ. Choose C⊆<ω2 such that A⊆/uniontext{Uτ:τ∈C}
+and Σ{µ(Uτ):τ∈C}< δ.
+For each τ∈C, we may, using Claim 2, choose Cτ⊆<ω2 such that A∩Uτ⊆/uniontext{Uη:η∈Cτ}and Σ{µ(Uη):η∈Cτ}<2−lh(τ)δ.
+LetC∗=/uniontext{Cτ:τ∈C}.
+We have that A⊆/uniontext{Uη:η∈C∗}and Σ{µ(Uη):η∈C∗}< δ2< γ.
+This contradiction establishes the Lemma.
+Definition 4.4. Suppose Y⊆ω2. We define RT(Y), the “rational translates”
+ofY, to equal {b∈ω2:(∃b′∈Y)(b′=a.e.b)}.
+Definition 4.5. Suppose yandy′are perfect subsets ofω2of positive Lebesgue
+measure. We define y/precedesequaly′to mean y⊆RT(y′).
+Lemma 4.6. Suppose /an}bracketle{tyn:n∈ω/an}bracketri}htis a sequence of perfect subsets ofω2of pos-
+itive Lebesgue measure. Then there is a perfect set y⊆ω2of positive Lebesgue
+measure such that (∀n∈ω)(y/precedesequalyn).
+Proof: By Lemma 4.3 we have that µ(RT(yn)) = 1 for every n∈ω. For
+eachn∈ωletDn⊆ω2 be an open set such that µ(Dn)<2−n−1andDn∪
+6RT(yn) =ω2. LetC=ω2−/uniontext{Dn:n∈ω}. We have that Cis a closed set
+of positive measure. Let ybe the perfect kernel of C(see [5, page 66]). We
+have that yis a perfect set of positive measure, and for every n∈ωwe have
+y⊆C⊆ω2−Dn⊆RT(yn).
+The Lemma is established.
+Lemma 4.7. Suppose xandyare subsets ofω2. Then x∩RT(y) =∅iff
+RT(x)∩y=∅.
+Proof: Clear.
+Lemma 4.8. LetPbe any forcing. Then V[GP]|= “(∀x∈ω2)(xis random
+overViff (∀y∈V)(yis a perfect set of positive Lebesgue measure implies
+x∈RT(y))).”
+Proof: Work in V[GP]. Suppose x∈ω2 is not random over V. LetB∈V
+be a Borel set such that x∈Bandµ(B) = 0. Let D∈Vbe an open set such
+thatµ(D)<1 and RT( B)⊆D. Letybe the perfect kernel ofω2−D. Then
+y∈Vis a perfect set of positive measure, and because y∩RT(B) =∅, we have
+RT(y)∩B=∅, and therefore x /∈RT(y).
+In the other direction, suppose x∈ω2 andy∈Vis a perfect set of positive
+measure such that x /∈RT(y). We show that xis not random over V. Choose
+/an}bracketle{tDn:n∈ω/an}bracketri}ht ∈Va sequence of open sets such that for every n∈ωwe have
+µ(Dn)<1/nandω2−RT(y)⊆Dn. LetB=/intersectiontext{Dn:n∈ω}. We have that
+B∈Vis a Borel set of Lebesgue measure zero and x∈B. Therefore xis not
+random over V.
+The Lemma is established.
+The following is [9, Lemma VI.3.18]. Notice how the argument parallels the
+proof of Theorem 2.5.
+Theorem 4.9. Suppose κis a limit ordinal and /an}bracketle{tPη:η≤κ/an}bracketri}htis a countable
+support forcing iteration based on /an}bracketle{tQη:η < κ/an}bracketri}ht. Suppose for every η < κwe
+have1/bardbl−Pη“Qηis proper and there are no reals that are random over V.”Then
+1/bardbl−Pκ“there are no reals that are random over V.”
+Proof: For cf( κ)> ωthis is clear, so assume instead that cf( κ) =ω.
+Suppose p∈Pκandgis aPκ-name and p/bardbl−“g∈ω2.” Letλbe a sufficiently
+large regular cardinal and let Nbe a countable elementary substructure of Hλ
+7containing {Pκ,p,g}. Let/an}bracketle{tαn:n∈ω/an}bracketri}ht ∈Nbe an increasing sequence cofinal in
+κsuch that α0= 0.
+Using Lemma 4.6, fix y⊆ω2 a perfect set of positive Lebesgue measure such
+that for every perfect y′∈Nwithµ(y′)>0 we have y/precedesequaly′.
+Build/an}bracketle{tqk,pk,mk:k∈ω/an}bracketri}htsuch that q0=pand for each k∈ωwe have the
+following:
+(1)pk∈PαkisN-generic, and
+(2)pk+1αk=pk, and
+(3)pk/bardbl−“qk+1∈Pαk,κ∩N[GPαk] andqk+1≤qk[αk,κ),” and
+(4)pk/bardbl−“pk+1[αk,αk+1)≤qk+1αk+1,” and
+(5)pk/bardbl−“mk+1> mkandqk+1/bardbl−“(∀ρ∈mk2)(Uρˆg[mk,mk+1)∩y=∅).’”
+The construction proceeds as follows. Suppose we are given pkandqkandmk.
+Work in V[GPαk] withpk∈GPαk. Build /an}bracketle{tqi
+k:i∈ω/an}bracketri}ht ∈N[GPαk] a decreasing
+sequence of conditions in Pαk,κandfk∈ω2∩N[GPαk] such that q0
+k≤qkand
+for every i∈ωwe have qi
+k/bardbl−“fk(i) =g(i).” Using the hypothesis on Pαkand
+Lemma 4.8, we may choose a perfect set yk∈Vof positive measure such that
+fk/∈RT(yk). By elementarity we may assume yk∈N[GPαk]∩V=N.
+Because y/precedesequalykwe have RT( y)⊆RT(yk), and hence fk/∈RT(y). Hence by
+Lemma 4.7 we have RT( {fk})∩y=∅. Hence for each ρ∈mk2, we may let mρ
+k
+be an integer greater than mksuch that Uρˆfk[lh(ρ),mρ
+k)∩y=∅, using the fact
+thatyis closed.
+Letmk+1= max{mρ
+k:ρ∈mk2}. Letqk+1=qmk+1+1
+k. We have that qk+1
+satisfies (3) and (5). Using the Proper Iteration Lemma, we may ch oosepk+1
+satisfying (1), (2), and (4).
+This completes the recursive construction.
+Letr∈Pκbe chosen such that ( ∀k∈ω)(rαk=pk).
+We have r/bardbl−“RT({g})∩y=∅.” Hence by Lemmas 4.7 and 4.8 we have r/bardbl−“g
+is not random over V.”
+The Theorem is established.
+5 Preservation of “every new dense open set con-
+tains an old dense open set”
+In this section we prove preservation of the property “every new dense open set
+contains an old dense open set.” Shelah includes two very different pr oofs of this
+8fact in his book; we follow the proof given in [9, Section XVIII.3].
+Throughout this section we fix an enumeration /an}bracketle{tη∗
+n:n∈ω/an}bracketri}htof<ωωsuch that
+whenever η∗
+iis an initial segment of η∗
+jtheni≤j. Also, throughout this section
+we letBequal the set of functions from<ωωinto<ωω.
+Definition 5.1. Suppose fandgare inB. We say f≤Bgiff for every η∈<ωω
+there isν∈<ωωsuch that νˆf(ν)is an initial segment of ηˆg(η).
+WeremarkthatDefinition5.1differsfrom[9,ContextandDefinitionXV III.3.7A]
+because we have incorporated [9, Remark XVIII.3.7F(1)].
+Lemma 5.2. The relation ≤Bis a partial ordering of B.
+Proof: Immediate.
+Lemma 5.3. Suppose /an}bracketle{tfi:i∈ω/an}bracketri}htis a sequence of elements of B. Then there is
+g∈ Bsuch that for every i∈ωwe have fi≤Bg.
+Proof: For every η∈<ωωandk∈ωdefineg0(η) =ηandgk+1(η) =
+gk(η)ˆfk(ηˆgk(η)). Define g(η) to equal gn(η) whereη=η∗
+n.
+To see that fk≤Bgit suffices to note that whenever n > kthen
+(η∗
+n/hatwidegk(η∗
+n))/hatwidefk(η∗
+n/hatwidegk(η∗
+n)) =η∗
+n/hatwidegk+1(η∗
+n)⊆η∗
+n/hatwideg(η∗
+n).
+The Lemma is established.
+Lemma 5.4. Suppose Pis a forcing notion. Then every dense open subset of
+ωωinV[GP]contains a dense open subset ofωωinViffV[GP]|= “(∀f∈ B)
+(∃g∈ B ∩V)(f≤Bg).”
+Proof: We first establish the “if” direction. Work in V[GP]. Suppose Dis
+a dense open subset ofωω. Pickf∈ Bsuch that for every η∈<ωωwe have
+Uηˆf(η)⊆D. Fixg∈ B ∩Vsuch that f≤Bg. LetD′=/uniontext{Uηˆg(η):η∈<ωω}.
+We have D′is a dense open subset of DandD′∈V.
+For the “only if” direction, suppose f∈ B. Build/an}bracketle{tDn,ηn,xn:n∈ω/an}bracketri}htrecur-
+sively such that for every n∈ωwe have that either Uη∗
+n⊆/uniontext{Di:i < n}and
+Dn=Dn−1andxn=xn−1andηn=ηn−1, or all of the following::
+(1)ηnextendsη∗
+n, and
+(2)Dn=Uηnˆf(ηn)ˆ/angbracketleft0/angbracketright, and
+(3)Dnis disjoint from/uniontext{Di:i < n}∪{xi:i < n}, and
+(4)xn∈ωωextendsηnˆf(ηn)ˆ/an}bracketle{t1/an}bracketri}ht.
+9We may take D′∈Vopen dense such that D′⊆/uniontext{Dn:n∈ω}.
+Chooseg∈ B∩Vsuch that ( ∀η∈<ωω)(Uηˆg(η)⊆D′).
+Givenη∈<ωω, pickn∈ωsuch that Uηˆg(η)∩Uηnˆf(ηn)ˆ/angbracketleft0/angbracketright/ne}ationslash=∅. We have
+xn/∈D′, and so Uηˆg(η)⊆Uηnˆf(ηn). It follows that f≤Bg.
+The Lemma is established.
+The following is [9, Conclusion VI.2.15D] and [9, Claim XVIII.3.7D]; we follow
+the proof given in [9, Chapter XVIII].
+Theorem 5.5. Suppose /an}bracketle{tPη:η≤κ/an}bracketri}htis a countable support forcing iteration
+based on /an}bracketle{tQη:η < κ/an}bracketri}ht. Suppose for every η < κwe have 1/bardbl−Pη“Qηis proper and
+1/bardbl−Qη‘for every dense open D⊆ωωthere is a dense open D′⊆Dsuch that
+D′∈V[GPη].’”Then1/bardbl−Pκ“for every dense open D⊆ωωthere is a dense open
+D′⊆Dsuch that D′∈V.”
+Proof: By induction on κ. The induction step is clear for κa successor ordinal
+and, in light of Lemma 5.4, it is likewise clear for κof uncountable cofinality. So
+we assume cf( κ) =ω.
+Suppose p∈Pκandfis aPκ-name and p/bardbl−“f∈ B.” Choose λa sufficiently
+large regular cardinal and Na countable elementary substructure of Hλsuch
+that{Pκ,f,p} ∈N.
+Let/an}bracketle{tαk:k∈ω/an}bracketri}ht ∈Nbe an increasing sequence cofinal in κsuch that α0= 0.
+Using Lemma 5.3, fix g∈ Bsuch that ( ∀h∈ B∩N)(h≤Bg).
+Build/an}bracketle{tqk,pk,mk:k∈ω/an}bracketri}htsuch that q0=pand for every k∈ωwe have that
+each of the following holds:
+(1)pk∈PαkisN-generic, and
+(2)pk/bardbl−“qk+1∈Pαk,κ∩N[GPαk] andqk+1≤qk[αk,κ),” and
+(3)pk/bardbl−“qk+1/bardbl−‘mk∈ωandη∗
+kˆg(η∗
+k) extends η∗
+mkˆf(η∗
+mk),’” and
+(4)pk+1αk=pk, and
+(5)pk/bardbl−“pk+1[αk,αk+1)≤qk+1αk+1.”
+The construction proceeds as follows. Given pkandqk, work in V[GPαk] with
+pk∈GPαk.
+Build/an}bracketle{tqi
+k:i∈ω/an}bracketri}ht ∈N[GPαk] andfk∈ B ∩N[GPαk] such that q0
+k=qk[αk,κ),
+and for every i∈ωwe have the following:
+(1)qi+1
+k≤qi
+k, and
+(2)qi+1
+k/bardbl−“fk(η∗
+i) =f(η∗
+i).”
+10Using Lemma 5.4, choose gk∈ B ∩Vsuch that fk≤Bgk. We may assume
+gk∈N[GPαk]. Hence gk∈N. Hencegk≤Bg.
+By Lemma 5.2 we have fk≤Bg, so we may choose mksuch that η∗
+kˆg(η∗
+k)
+extendsη∗
+mkˆfk(η∗
+mk).
+Letqk+1=qmk+1
+k. We have that qk+1andmksatisfy (2) and (3).
+Using the Proper Iteration Lemma we may choose pk+1satisfying (1), (4), and
+(5).
+This completes the recursive construction.
+Letr∈Pκbe such that for every k∈ωwe have rαk=pk.
+Suppose, towards a contradiction, that r′≤randr′/bardbl−“f/ne}ationslash≤Bg.” Fix a
+Pκ-nameksuch that r′/bardbl−“η∗
+kˆg(η∗
+k) does not extend η∗
+mkˆf(η∗
+mk).”
+By strengthening r′we may assume that kandmkare integers rather than
+names.
+Because r′≤(pk+1,qk+1) we have r′/bardbl−“η∗
+kˆg(η∗
+k) extends η∗
+mkˆf(η∗
+mk).” This
+is a contradiction.
+The Theorem is established.
+6 On “the set of reals that are in the ground
+model has positive outer measure in the forc-
+ing extension”
+In this section we present a theorem of Shelah ([9, Claim XVIII.3.8B(3 )]) that
+gives a sufficient condition for a forcing iteration to satisfy µ∗(ω2∩V)>0. This
+notion has been investigated also by [4].
+Definition 6.1. We letB′be the set of functions ffromωinto<ω2such that
+Σ{µ(Uf(m)):m∈ω} ≤1.
+Lemma 6.2. Suppose g∈ω2andλis a sufficiently large regular cardinal and
+Nis a countable elementary substructure of Hλ. Thengis random over Niff
+(∀f∈ B′∩N)(∃m∈ω)(∀i≥m)(gdoes not extend f(i)).
+Proof: Wefirstestablishthe“onlyif”direction. Suppose g∈ω2andf∈ B′∩N
+and (∃∞m∈ω)(gextendsf(m)). LetB={h∈ω2:(∃∞m∈ω)(f(m) is an
+initial segment of h)}. ThenB⊆ω2 is a Borel set and g∈B∈N, andµ(B) = 0
+because for every n∈ωwe have that Bis covered by/uniontext{Uf(i):i≥n}, and
+limn→∞(µ(/uniontext{Uf(i):i≥n}) = 0. Therefore gis not random over N.
+11To prove the “if” direction, suppose that gis not random over N. We may
+chooseB∈Na Borel set of measure zero such that g∈B∈N. Let/an}bracketle{tDn:n∈
+ω/an}bracketri}ht ∈Nbe a sequence of open subsets ofω2 such that for every n∈ωwe have
+B⊆Dnandµ(Dn)<2−n. For each n∈ωchoosekn≤ωand/an}bracketle{tηn
+i:i < kn/an}bracketri}ht
+a sequence of pairwise incomparable elements of<ω2 such that Dn=/uniontext{Uηn
+i:
+i < kn}. Furthermore we may assume that /an}bracketle{t/an}bracketle{tηn
+i:i < kn/an}bracketri}ht:n∈ω/an}bracketri}htis an element
+ofN. Letf∈Nbe a one-to-one function mapping ωonto{ηn
+i:i < knand
+n∈ω}. Then we have that f∈ B′and (∃∞m∈ω)(g∈Uf(m)). The Lemma is
+established.
+Lemma 6.3. Suppose g∈ω2andλis a sufficiently large regular cardinal and
+Nis a countable elementary substructure of Hλ. Suppose gis random over N.
+Suppose Y∈Nis a subset of<ω2andΣ{µ(Uη):η∈Y}is finite. Then {η∈Y:
+g∈Uη}is finite.
+Proof: We may assume Yis infinite. Choose a finite integer mand infinite sets
+(not necessarily disjoint) Di⊆Yfori < msuch that each Diis inNand/uniontext{Di:
+i < m}=Yand for each i < mwe have Σ {µ(Uη):η∈Di} ≤1. For each i < m
+choosefi∈Nsuch that fimapsωontoDi. By Lemma 6.2, for every i < m
+there isβi∈ωsuch that ( ∀j≥βi)(gdoes not extend fi(j)). Hence {η∈Y:
+g∈Uη} ⊆/uniontext{{fi(j):j < βi}:i < m}, which is finite. The Lemma is established.
+Lemma 6.4. Suppose Pis a poset such that whenever λis a sufficiently large
+regular cardinal and Nis a countable elementary substructure of HλandP∈N
+andg∈ω2andgis random over N, thenV[GP]|= “gis random over N[GP].”
+ThenV[GP]|= “ω2∩Vhas positive outer measure.”
+Proof: Suppose, towards a contradiction, that in V[GP] we have that Bis a
+Borel subset ofω2 such thatω2∩V⊆Bandµ(B) = 0.
+InV, choose λa sufficiently large regular cardinal and Na countable elemen-
+tary substructure of Hλsuch that p∈Nand a name for Bis inN. Letg∈ω2 be
+random over N. By hypothesis, V[GP]|= “gis random over N[GP].” Therefore
+V[GP]|= “g /∈B.” This contradiction establishes the Lemma.
+Lemma 6.5. Suppose Pis a poset. Suppose χis a sufficiently large regular
+cardinal and λis a regular cardinal sufficiently larger than χ. Suppose Nis
+a countable elementary substructure of HλandN1andN2are countable ele-
+mentary substructures of Hχandχ∈NandP∈N1∈N2∈N. Suppose
+also
+12(1)G1⊆P∩N1is anN1-generic subset of P, and
+(2)p∈G1andG1∈N, and
+(3)/an}bracketle{tfl:l≤k/an}bracketri}ht ∈Nis a finite sequence of P-names such that p/bardbl−“fl∈ B′∩
+N1[GP]”for alll≤k, and
+(4)g∈ω2is random over N, and
+(5)/an}bracketle{tβl:l≤k/an}bracketri}htis a sequence of integers and for all l≤kwe have(∀j≥βl)(gdoes
+not extend fl[G1](j)).That is, for every j≥βlthere isp′∈G1andρ∈<ω2
+such that gdoes not extend ρandp′/bardbl−“ρ=fl(j).”
+Thenthere isG2⊆P∩N2anN2-generic subset of Psuch that p∈G2and
+G2∈Nand for all l≤kwe have (∀j≥βl)(gdoes not extend fl[G2](j)).
+Proof: Build /an}bracketle{tpn:n∈ω/an}bracketri}ht ∈Nand/an}bracketle{tmn:n∈ω/an}bracketri}ht ∈Nand/an}bracketle{tf∗
+l:l≤k/an}bracketri}ht ∈Nsuch
+thatp0=pand for each n∈ωwe have each of the following:
+(1)pn∈G1andpn+1≤pn, and
+(2)mnis an integer such that mn≥nandpn/bardbl−“Σ{µ(Ufl(i)):i≥mn}<2−n
+for each l≤k,” and
+(3) for every l≤kwe have f∗
+l∈Nmapsωinto<ω2, and
+(4)pn/bardbl−“flmn=f∗
+lmnfor each l≤k.”
+Claim 1. For l≤kwe have f∗
+l∈ B′.
+Proof. Suppose,towardsacontradiction,that l≤kandm∈ωandΣ{µ(Uf∗
+l(i)):
+i < m}>1. Because pm/bardbl−“flm=f∗
+lm,” it follows that pm/bardbl−“fl/∈ B′.” This
+contradiction establishes the Claim.
+Build/an}bracketle{tpn,m:m∈ω,n∈ω/an}bracketri}ht ∈Nand/an}bracketle{tf∗
+l,n:l≤k,n∈ω/an}bracketri}ht ∈Nsuch that each of
+the following holds:
+(1) for every n∈ωwe have that /an}bracketle{tpn,m:m∈ω/an}bracketri}htis anN2-generic sequence for
+Pandpn,0=pn, and
+(2) for every l≤kandn∈ωandm∈ωwe have pn,m/bardbl−“f∗
+l,nm=flm.”
+Claim 2. For l≤kandn∈ωwe have f∗
+l,n∈ B′.
+Proof: Similar to Claim 1.
+Claim 3. For every l≤kandn∈ωwe have Σ {µ(Uf∗
+l,n(i)):i≥mn} ≤2−n.
+Proof: Suppose landnconstitute a counterexample. Then we can choose
+an integer tso large that Σ {µ(Uf∗
+l,n(i)):mn≤i < t}>2−n. We have pn,t/bardbl−
+“Σ{µ(Uf∗
+l,n(i)):mn≤i < t}= Σ{µ(Uf∗
+l(i)):mn≤i < t}<Σ{µ(Uf∗
+l(i)):mn≤
+i < ω}<2−n.” This contradiction establishes the Claim.
+13For each l≤kandn∈ωletU∗
+l,n=/uniontext{Uf∗
+l,n(i):i∈ω}.
+Claim 4. For every l≤kandn∈ωwe have U∗
+l,n⊆/uniontext{Uf∗
+l(i):i∈ω} ∪/uniontext{Uf∗
+l,n(i):i≥mn}.
+Proof: The Claim is forced by the condition pn,n, hence it is true outright.
+Foreach l≤kletU∗
+l=/uniontext{U∗
+l,n:n∈ω}. ByClaims3and4wehavethat µ(U∗
+l)
+is finite for every l≤k. By Lemma 6.4 we have that {ρ∈<ω2:(∃l≤k)(∃n∈ω)
+(∃i∈ω)(ρ=f∗
+l,n(i) andgextendsρ)}is finite. Therefore, we may fix n∗so large
+that (∀l≤k)(∀n∈ω)(∀i∈ω)(gextendsf∗
+l,n(i) only if µ(Uf∗
+l,n(i))≥2−n∗).
+Claim 5. Suppose l≤kandi∈ωandn∈ωandµ(Uf∗
+l,n(i))≥2−n∗. Then
+i < mn∗.
+Proof: Suppose i≥mn∗. Thenpn,i+1/bardbl−“µ(Uf∗
+l,n(i)) =µ(Ufl(i))<Σ{µ(Ufl(j)):
+j≥mn∗}<2−n∗. This contradiction establishes the Claim.
+Fixt > m n∗such that t > β lfor every l≤k. For every l≤kwe have
+pn∗,t/bardbl−“f∗
+l,n∗t=f∗
+lt.” Thus, by Claim 5, we have that pn∗,t/bardbl−“(∀l≤k)
+(∀i≥βl)(gdoes not extend f∗
+l,n∗(i)).”
+LetG2={p′∈P∩N2:(∃m∈ω)(pn∗,m≤p′)}. We have that G2is as
+required.
+The Lemma is established.
+Definition 6.6. Suppose g∈ω2. We say that Pisg-good iff whenever
+(1)χis a sufficiently large regular cardinal and λis a regular cardinal sufficiently
+larger than χand
+(2)Nis a countable elementary substructure of Hλandχ∈NandN1is a
+countable elementary substructure of Hχand
+(3)P∈N1∈Nand
+(4)gis random over Nand
+(5)k∈ωand/an}bracketle{tfl:l < k/an}bracketri}ht ∈Nis a sequence of P-names and
+(6)p∈P∩N1and
+(7)p/bardbl−“(∀l < k)(fl∈ B′∩N1[GP]),”and
+(8)/an}bracketle{tf∗
+l:l < k/an}bracketri}htis a sequence of elements of B′and/an}bracketle{tβl:l < k/an}bracketri}htis a sequence of
+integers and for every l < kwe have (∀m≥βl)(gdoes not extend f∗
+l(m))and
+(9)G1⊆P∩N1andG1∈NandG1isN1-generic over Pandp∈G1and
+14(10) (∀l < k)(fl[G1] =f∗
+l),
+thenthere isq≤psuch that qisN-generic and q/bardbl−“gis random over N[GP]
+and (∀l < k)(∀m≥βl)(gdoes not extend fl(m)).”
+Lemma 6.7. Suppose we have that
+(1)g∈ω2and
+(2)1/bardbl−“Qisg-good,”and
+(3)χis a sufficiently large regular cardinal and λis a regular cardinal sufficiently
+larger than χ, and
+(4)Nis a countable elementary substructure of Hλand{P∗Q,χ} ∈N, and
+(5)p∈PisN-generic and qis aP-name and
+(6)p/bardbl−“N1is a countable elementary substructure of Hχ[GP] andN1∈N[GP]
+andgis random over N[GP] andq∈Q∩N1,”and
+(7)k∈ωandp/bardbl−“/an}bracketle{tfl:l < k/an}bracketri}ht ∈N[GP] is a sequence of Q-namesand q/bardbl−Q‘(∀l <
+k)(fl∈ B′∩N1[GQ]).,’”and
+(8)/an}bracketle{tf∗
+l:l < k/an}bracketri}htand/an}bracketle{tβl:l < k/an}bracketri}htare sequences of P-names and p/bardbl−“(∀l < k)
+(f∗
+l∈ B′∩N[GP] andβl∈ωand (∀i≥βl)(gdoes not extend f∗
+l(i))),”and
+(9)Gis aP-name and p/bardbl−“G⊆Q∩N1is generic over N1andq∈G∈N[GP]
+and (∀l < k)(f∗
+l=fl[G]).”
+Thenthere is a P-namersuch that p/bardbl−“r≤q”and(p,r)isN-generic and
+(p,r)/bardbl−“gis random over N[GP∗Q] and (∀l < k)(∀i≥βl)(gdoes not extend
+fl(i)).”
+Proof: Immediate.
+Theorem 6.8. Suppose g∈ω2and suppose /an}bracketle{tPη:η≤κ/an}bracketri}htis a countable support
+forcing iteration based on /an}bracketle{tQη:η < κ/an}bracketri}ht. Suppose for every η < κwe have 1/bardbl−Pη
+“Qηis proper and g-good.’” Suppose also
+(1)χis a sufficiently large regular cardinal and λis a regular cardinal sufficiently
+larger than χ, and
+(2)Nis a countable elementary substructure of Hλand{Pκ,χ} ∈Nand
+(3)α∈κ∩Nandp∈Pα∩Nand
+15(4)p/bardbl−“N′is a countable elementary substructure of Hχ[GPα] andPα,κ∈N′∈
+N[GPα] (so necessarily α∈N′),”and
+(5)pisN-generic and gis aPα-name and p/bardbl−“gis random over N[GPα] and
+q∈Pα,κ∩N′,”and
+(6)k∈ωandp/bardbl−“/an}bracketle{tfl:l < k/an}bracketri}ht ∈N[GPα] is a sequence of Pα,κ-names and
+q/bardbl−Pα,κ‘(∀l < k)(fl∈ B′∩N′[GPα,κ]),’”and
+(7)/an}bracketle{tf∗
+l:l < k/an}bracketri}htand/an}bracketle{tβl:l < k/an}bracketri}htare sequences of Pα-names and p/bardbl−“(∀l < k)
+(f∗
+l∈ B′∩N[GPα] andβl∈ωand (∀i≥βl)(gdoes not extend f∗
+l(i))),”and
+(8)GisaPα-nameand p/bardbl−“G⊆Pα,κ∩N′isgenericover N′andq∈G∈N[GPα]
+and (∀l < k)(f∗
+l=fl[G]).”
+Thenthere isr∈Pκsuch that rα=pandp/bardbl−“r[α,κ)≤q”andrisN-
+generic and r/bardbl−“gis random over N[GPκ] and (∀l < k)(∀i≥βl)(gdoes not
+extendfl(i)).”
+Proof: By induction on κ.
+Successor case: κ=γ+1.
+InV[GPα] letG1=GγandG2=G/G1. That is, G1={p′γ:p′∈G}and
+(∀p′∈Pα,γ)(∀r′)(p′/bardbl−“r′∈G2” iff (∀p∗≤p′)(∃q′∈G)(∃p#≤p∗)(p#≤q′γ
+andp#/bardbl−“r′=q′(γ)”)).
+ChooseN∗acountableelementarysubstructureof Hχ[GPα]suchthat N′[GPα]∈
+N∗∈N[GPα] andG∈N∗. Choose /an}bracketle{tf∗∗
+l:l < k/an}bracketri}htsuch that for all l < kwe have
+(p,qγ)/bardbl−Pγ“f∗∗
+l=fl[G2].” Because p/bardbl−“f∗∗
+l[G1] =f∗
+l” for all l < k, we have
+that (p,qγ)/bardbl−“(∀l < k)(∀j≥βl)(gdoes not extend f∗∗
+l(j)).”
+Use Lemma 6.5 to choose G′
+1such that p/bardbl−“G′
+1⊆Pα,γ∩N∗is generic over
+N∗andqγ∈G′
+1andG′
+1∈N[GPα] and (∀l < k)(∀j≥βl)(gdoes not extend
+f∗∗
+l[G′
+1](j)).”
+By the induction hypothesis, with G′
+1playing the role of Gand/an}bracketle{tf∗∗
+l:l < k/an}bracketri}ht
+playing the role of /an}bracketle{tfl:l < k/an}bracketri}ht, we can choose r′∈Pγsuch that r′α=pand
+p/bardbl−“r′[α,γ)≤qγ” andr′isN-generic and r′/bardbl−“gis random over N[GPγ]
+and (∀l < k)(∀i≥βl)(gdoes not extend f∗∗
+l(i)).”
+Using Lemma 6.7 with G2playing the role of GandN′[GPγ] playing the role
+ofN1, we may choose r∗such that r′/bardbl−“r∗∈Qγandr∗≤q(γ)” and (r′,r∗)/bardbl−“g
+is random over N[GPκ] and (∀l < k)(∀i≥βl)(gdoes not extend fl(i)).”
+Letr= (r′,r∗). This concludes the verification of the successor case.
+16Limit case: κis a limit ordinal.
+Let/an}bracketle{tαn:n∈ω/an}bracketri}htbe an increasing sequence from κ∩Ncofinal in sup( κ∩N)
+such that α0=α. Let/an}bracketle{tσn:n∈ω/an}bracketri}htlist allPκ-namesσsuch that σ∈Nand
+1/bardbl−Pκ“σis an ordinal.” Let /an}bracketle{tfl:l∈ω/an}bracketri}htbe a sequence that extends /an}bracketle{tfl:l < k/an}bracketri}ht,
+such that it lists the set of all Pκ-namesfinNsuch that ( p,q)/bardbl−“f∈ B′.”
+Build/an}bracketle{tpn,qn,βn,Gn,G∗
+n,G′
+n,Nn/an}bracketri}htsuch that p0=pandq0=qandG0=G
+andN0=N′and/an}bracketle{tβl:l∈ω/an}bracketri}htextends/an}bracketle{tβl:l < k/an}bracketri}ht, and for every n∈ωwe have
+that each of the following holds:
+(1)pn/bardbl−“G′
+n=Gnαn+1andGn+1=Gn/G′
+n(see the successorcase, above),”
+and
+(2)pn/bardbl−“Nn+1is a countable elementary substructure of Hχ[GPαn] and
+{Nn[GPαn−1,αn],Gn,fn,αn+1,σn} ∈Nn+1∈N[GPαn]” (ifn= 0 then replace
+N0[GPα−1,α0] withN0), and
+(3)βnis aPαn-name for an integer and pn/bardbl−“(∀j≥βn)(gdoes not extend
+fn[Gn](j)),” and
+(4)pn/bardbl−“G∗
+n⊆Pαn,αn+1∩Nn+1isNn+1-generic and qnαn+1∈G∗
+n+1∈
+N[GPαn] and (∀l <max(n+1,k))(∀j≥βl)(gdoes not extend fl[Gn+1][G∗
+n](j)),”
+and
+(5)pn+1∈Pαn+1isN-generic and pn+1/bardbl−“gis random over N[GPαn+1] and
+(∀i <max(n+1,k))(∀j≥βl)(gdoes not extend fl[Gn+1](j)),” and
+(6)pn/bardbl−“pn+1[αn,αn+1)≤qnαn+1,” and
+(7)pn+1/bardbl−“qn+1≤qn[αn+1,κ) andqn+1∈Gn+1andqn+1decides the value
+ofσnandqn+1decides the value of flnfor every l≤n.”
+The construction proceeds as follows. Given pnandqnandGn, construct G′
+n
+andGn+1as in (1) (see successor case, above). There is no problem in choos ing
+Nn+1as in (2). We have that pn/bardbl−“fn[Gn]∈ B′” by the reasoning of Claim 1
+in the proof of Lemma 6.5, hence we may choose βnas in (3) because of Lemma
+6.2. We may choose G∗
+nas in (4) by Lemma 6.5. We may choose pn+1satisfying
+(5) and (6) by using the induction hypothesis. There is no difficulty in c hoosing
+qn+1satisfying (7).
+Taker∈Pκsuch that for every n∈ωwe have rαn=pn.
+Claim.r/bardbl−“gis random over N[GPκ].
+Proof: Suppose not. By Lemma 6.2 we may take r′≤randl∈ωsuch that
+r′/bardbl−“(∃∞m∈ω)(gextendsfl(m)).” By strengthening r′further, wemayassume
+there is an integer β∗such that r′/bardbl−“βl=β∗.” By a further strengthening of r′
+17we may assume there is an integer j≥β∗such that r′/bardbl−“gextendsfl(j).” Let
+n= max(j+1,l+1). By (7) we havethat pn+1/bardbl−“qn+1/bardbl−‘fl[Gn+1](j) =fl(j).’”
+We have pn+1/bardbl−“gdoes not extend fl[Gn+1](j).” The Claim is established.
+We have that risN-generic by the usual argument on ordinal names in N,
+and it is clear that r/bardbl−“(∀l < k)(∀j≥βl)(gdoes not extend fl(j)).”
+The Theorem is established.
+The following Theorem is [9, Claim XVIII.3.8C(1)].
+Theorem 6.9. Suppose /an}bracketle{tPη:η≤κ/an}bracketri}htis a countable support iteration based on
+/an}bracketle{tQη:η < κ/an}bracketri}htand for every η < κwe have 1/bardbl−Pη“Qηis proper and for every
+g∈ω2 we have that Qηisg-good.” ThenV[GPκ]|= “ω2∩Vdoes not have
+measure zero.”
+Proof: By Theorem 6.8 with α=k= 0 and Lemma 6.2.
+7 Preservation of “the set of old reals is non-
+meager”
+LetB∗be the set of functions from<ω2 into<ω2.
+Definition 7.1. Suppose f∈ B∗andg∈ω2. We say fR†giff(∃∞m∈ω)
+(gmˆf(gm) is an initial segment of g).
+Lemma 7.2. Suppose X⊆ω2. ThenXis non-meager iff for every f∈ B∗there
+isg∈Xsuch that fR†g.
+Proof: Suppose Xis non-meager, and suppose f∈ B′.
+For every i∈ωletDi=/uniontext{Uτˆf(τ):(∃n > i)(τ∈n2)}. We have that each Di
+is an open dense set, so because Xis non-meager, we may fix g∈X∩/intersectiontext{Di:
+i∈ω}. Clearly fR†g.
+For the converse, suppose ( ∀f∈ B∗)(∃g∈X)(fR†g), and suppose /an}bracketle{tDi:i∈ω/an}bracketri}ht
+is a decreasing sequence of open dense subsets ofω2. We show X∩/intersectiontext{Di:i∈ω}
+is non-empty. It suffices to find g∈Xsuch that ( ∃∞j∈ω)(g∈Dj).
+Choosef∈ B∗such that for every η∈<ω2 we have Uηˆf(η)⊆Dlh(η). Fix
+g∈Xsuch that fR†g. Given i∈ωchoosej > isuch that gjˆf(gj) is an
+initial segment of g. Letη=gj. Theng∈Uηˆf(η)⊆Dj.
+The Lemma is established.
+18Lemma 7.3. Suppose λis a sufficiently large regular cardinal and Nis a count-
+able elementary substructure of Hλ, and suppose g∈ω2. The following are
+equivalent:
+(1) (∀f∈ B∗∩N)(fR†g).
+(2) (∀f∈ B∗∩N)(∃m∈ω)(gmˆf(gm) is an initial segment of g).
+(3)gis Cohen over N.
+Proof: It is obvious that (1) implies (2).
+Suppose (2) holds and D∈Nis an open dense subset ofω2. Choose f∈
+B∗∩Nsuch that ( ∀ν∈<ω2)(Uνˆf(ν)⊆D). Using (2), choose m∈ωsuch that
+gmˆf(gm) is an initial segment of g. We have g∈Ugmˆf(gm)⊆D. We
+conclude that g∈/intersectiontext{D∈N:Dis an open dense subset ofω2}, i.e.,gis Cohen
+overN.
+Finally, suppose (3) holds and f∈ B∗∩N. Suppose k∈ω. LetDk={h∈ω2:
+(∃m > k)(hmˆf(hm) is an initial segment of h)}. It is easyto see that for every
+k∈ωwe have Dkis an open dense subset ofω2. Because ( ∀k∈ω)(g∈Dk) we
+have that fR†g.
+The Lemma is established.
+The following Lemma, due to Goldstern and Shelah, is [9, Lemma XVIII.3 .11].
+Lemma 7.4. Suppose Pis a Suslin proper forcing (see [1, Section 7] ) and for
+every forcing Qwe have 1/bardbl−Q“Pis Suslin proper and 1/bardbl−P‘ω2∩V[GQ] is not
+meager.’” Suppose λis a sufficiently large regular cardinal and Nis a countable
+elementary submodel of HλandP∈Nandp∈P∩Nandg∈ω2is Cohen
+overN. Then there is q≤psuch that qisN-generic and q/bardbl−“gis Cohen over
+N[GP].”
+The proof presented in [9] is quite clear, so we do not repeat it here.
+Lemma 7.5. Suppose /an}bracketle{tPη:η≤κ/an}bracketri}htis a countable support forcing iteration based
+on/an}bracketle{tQη:η < κ/an}bracketri}ht. Suppose for every η < κwe have 1/bardbl−Pη“Qηis a Suslin proper
+forcing and for every forcing Qwe have 1/bardbl−Q‘Qηis Suslin proper and 1/bardbl−Qη
+“ω2∩V[GPη][GQ] is not meager.”’” Suppose λis a sufficiently large regular
+cardinal and Nis a countable elementary substructure of HλandPκ∈Nand
+α∈κ∩Nandp∈PαisN-generic and p/bardbl−“q∈Pα,κ∩N[GPα] andg∈ω2 is
+Cohen over N[GPα].”Then there is r∈Pκsuch that risN-generic and rα=p
+andp/bardbl−“r[α,κ)≤q” andr/bardbl−“gis Cohen over N[GPκ].”
+Proof: By induction on κ.
+19Case 1:κis a successor ordinal.
+Letβbe the immediate predecessor of κ. By the induction hypothesis we may
+taker′∈Pβsuch that r′isN-generic and r′α=pandp/bardbl−“r′≤qβ” and
+r′/bardbl−“gis Cohen over N[GPβ].” By Lemma 7.4 we may take r∗∈Qβsuch that
+r′/bardbl−“r∗≤q(β) andr∗isN[GPβ]-genericand r∗/bardbl−‘gisCohenover N[GPβ][Qβ].’”
+Letr∈Pκbe defined by rβ=r′andr(β) =r∗. We have that rsatisfies the
+requirements of the Lemma.
+Case 2:κis a limit ordinal.
+Let/an}bracketle{tαk:k∈ω/an}bracketri}htbe an increasing sequence from κ∩Ncofinal in sup( κ∩N)
+such that α0=α. Let/an}bracketle{tσk:k∈ω/an}bracketri}htlist allPκ-namesσinNsuch that 1/bardbl−Pκ“σ
+is an ordinal.”
+Let/an}bracketle{tfi:i∈ω/an}bracketri}htlist allPκ-namesfinNsuch that V[GPκ]|= “f∈ B∗,” and let
+/an}bracketle{tη′
+m:m∈ω/an}bracketri}htlist<ω2.
+Build/an}bracketle{tqk,pk,nk:k∈ω/an}bracketri}htsuch that p0=pandq0=qand for every k∈ωwe
+have that each of the following holds:
+(1)pk∈PαkisN-generic, and
+(2)pk/bardbl−“qk+1∈Pαk,κ∩N[GPαk] andqk+1≤qk[αk,κ),” and
+(3)pk/bardbl−“qk+1/bardbl−‘gis Cohen over N[GPαk] andσk∈Nandnk∈ωand
+gnkˆfk(gnk) is an initial segment of g,’” and
+(4)pk+1αk=pk, and
+(5)pk/bardbl−“pk+1[αk,αk+1)≤qk+1αk+1.”
+The construction proceeds as follows. Given pkandqk, work in V[GPαk] with
+pk∈GPαk.
+Build/an}bracketle{tqm
+k:m∈ω/an}bracketri}ht ∈N[GPαk] andf′
+k∈ B∗∩N[GPαk] such that /an}bracketle{tqm
+k:m∈ω/an}bracketri}ht
+is a decreasing sequence of elements of Pαk,κandq0
+k≤qk[αk,κ) and there
+is an ordinal τsuch that q0
+k/bardbl−“τ=σk,” and for every m∈ωwe have that
+qm
+k/bardbl−“f′
+k(η′
+m) =fk(η′
+m).” Necessarily τ∈N[GPαk] and therefore, because
+pk∈GPαkisN-generic, we have τ∈N. Because gis Cohen over N[GPαk] we
+may use Lemma 7.3 to take nksuch that gnkˆf′
+k(gnk) is an initial segment of
+g.
+Letqk+1=qnk+1
+k.
+Using the induction hypothesis, we may choose pk+1as required.
+This completes the recursive construction.
+Letr∈Pκbe such that for every k∈ωwe have rαk=pk.
+20We have that risN-generic, because for each k∈ωwe have pk+1/bardbl−“qk+1/bardbl−
+‘σk∈N.’”
+Suppose, towards a contradiction, that r′≤randr′/bardbl−“gis not Cohen over
+N[GPκ].” Choose r∗≤r′andk∈ωsuch that r/bardbl−“(∀m∈ω)(gmˆfk(gm) is
+not an initial segment of g).”
+Because r∗≤(pk+1,qk+1) we have r∗/bardbl−“gnkˆfk(gnk) is an initial segment of
+g.” This is a contradiction.
+The Lemma is established.
+The following Theorem is [9, Claim XVIII.Claim 3.10C].
+Theorem 7.6. Suppose /an}bracketle{tPη:η≤κ/an}bracketri}htis a countable support forcing iteration
+based on /an}bracketle{tQη:η < κ/an}bracketri}ht. Suppose for every η < κwe have 1/bardbl−Pη“Qηis Suslin
+proper forcing and for every forcing Qwe have 1/bardbl−Q‘Qηis Suslin proper and
+1/bardbl−Qη“ω2∩V[GPη][GQ] is not meager.”’” Then1/bardbl−Pκ“ω2∩Vis not meager.”
+Proof: Suppose, towards a contradiction, that q∈Pκandq/bardbl−“ω2∩Vis
+meager.” By Lemma 7.3 we may take faPκ-name for an element of B∗such
+thatq/bardbl−“(∀g∈ω2∩V)(fR†gfails).” Let λbe a sufficiently largeregularcardinal
+andletNbe acountableelementarysubstructureof Hλsuchthat {Pκ,q,f} ∈N.
+Letg∈ω2 be Cohen over N.
+By Lemma 7.5 with α= 0 we may take r≤qsuch that r/bardbl−“gis Cohen over
+N[GPκ].” By Lemma 7.3 we have r/bardbl−“fR†g.” This contradiction establishes
+the Theorem.
+8 References
+[1] Bartoszynski, T., and H. Judah, Set Theory: On the Structure of the Real
+Line, A K Peters, 1995.
+[2] Goldstern, M., Tools for Your Forcing Construction, Set theory of the re-
+als (Haim Judah, editor), Israel Mathematical Conference Proce edings, vol. 6,
+American Mathematical Society, pp. 305–360. (1993)
+[3] Goldstern, M., and J. Kellner, New reals: Can live with them, can live
+without them, Math. Log. Quart. 52, No. 2, pp. 115–124, 2006.
+[4] Kellner, J., and S. Shelah, Preserving preservation, Journal of Symbolic
+Logic, vol. 70, pp. 914–945, 2005.
+21[5] Moschovakis, Y., Descriptive Set Theory, Studies in Logic and the Founda-
+tions of Mathematics, v. 100, North-Holland, 1980.
+[6] Schlindwein, C., Shelah’s work on non-semi-proper iterations, I, A rchive for
+Mathematical Logic, vol. 47, pp. 579–606, 2008.
+[7] Schlindwein, C., Shelah’s work on non-semi-proper iterations, II, Journal
+of Symbolic Logic, vol. 66, pp. 1865–1883, 2001.
+[8] Schlindwein, C., Understanding preservation theorems: Chapte r VI of
+Proper and Improper Forcing ,” submitted.
+[9]Shelah, S., Proper and Improper Forcing , PerspectivesinMathematical
+Logic, Springer, Berlin, 1998.
+22
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+arXiv:1001.0923v1  [gr-qc]  6 Jan 2010Particle Swarm Optimization and gravitational wave data an alysis: Performance on a
+binary inspiral testbed
+Yan Wang
+Department of Astronomy, Nanjing University, Nanjing, 210 093, China∗
+Soumya D. Mohanty
+Center for Gravitational Wave Astronomy, Dept. of Physics a nd Astronomy,
+The University of Texas at Brownsville, 80 Fort Brown, Brown sville, TX 78520, U.S.A.†
+The detection and estimation of gravitational wave (GW) sig nals belonging to a parameterized
+family of waveforms requires, in general, the numerical max imization of a data-dependent func-
+tion of the signal parameters. Due to noise in the data, the fu nction to be maximized is often
+highly multi-modal with numerous local maxima. Searching f or the global maximum then becomes
+computationally expensive, which in turn can limit the scie ntific scope of the search. Stochastic op-
+timization is one possible approach to reducing computatio nal costs in such applications. We report
+results from a first investigation of the Particle Swarm Opti mization (PSO) method in this context.
+The method is applied to a testbed motivated by the problem of detection and estimation of a
+binary inspiral signal. Our results show that PSO works well in the presence of high multi-modality,
+making it a viable candidate method for further application s in GW data analysis.
+PACS numbers: 95.85.Sz,04.80.Nn, 07.05.Kf, 02.50.Tt, 02. 60.Pn
+I. INTRODUCTION
+The detection and estimation of a gravitational wave
+(GW) signal belonging to a parameterized family of
+waveforms requires, in general, the numerical maximiza-
+tion of some data-dependent function over the space of
+the signal parameters. For example, in the matched fil-
+tering[1, 2] method, which is the focus of this paper,
+the function to be maximized is a suitably defined inner
+productbetweenthedataandparameterizedsignalwave-
+forms. The global maximum of this function serves as a
+detection statistic. A point estimate of the signal param-
+eters is furnished by the location of the global maximum
+in parameter space.
+The presence of noise in the output of GW detectors
+leads to a large number of local maxima in this func-
+tion that are distributed randomly in parameter space.
+The search for the global maximum in this forest of lo-
+cal maxima then becomes a computationally expensive
+task. This can affect the sensitivity of a search by limit-
+ing either the volume that is searched in parameter space
+or the integration length of data required for accumulat-
+ing sufficient signal-to-noise ratio (SNR), or both. The
+computational efficiency ofthe searchforthe globalmax-
+imum is, thus, an important issue in GW data analysis.
+The various search strategies proposed in the GW liter-
+ature so far can be broadly divided into those based on
+sampling the function on predetermined grids of points
+in parameter space [3–5], and those that use stochastic
+optimization methods (e.g., [8–10]).
+∗Current address: Albert Einstein Institute, Callinstr. 38 , 30167
+Hannover, Germany
+†Corresponding author:mohanty@phys.utb.eduIn the class of grid-based methods, significant savings
+in computational costs have been demonstrated with a
+hierarchy of grids [4, 6, 7]. A nice feature of grid-based
+methods is that they are easy to characterizestatistically
+and, hence, design variables of the algorithm, such as the
+spacing of points, can be fixed systematically.
+Stochastic methods do not use pre-determined grids
+but employ some form of random walk through the pa-
+rameter space. The probabilistic rules of the random
+walkaretunedtomaximizethechancesofitsterminating
+closeto theglobalmaximum. Therearemanyalgorithms
+that fall under the class of stochastic methods, a hybrid
+of simulated annealing and Metropolis-Hastings Markov
+Chain Monte Carlo (MCMC) being the most widely ex-
+plored in GW data analysis [8–10].
+Since the number of points in a grid grows exponen-
+tially with the dimensionality of the parameter space,
+stochastic methods tend to outperform grid-based ones
+with an increase in the number of signal parameters.
+It is worth noting here that stochastic methods in GW
+data analysis incur the additional computational cost of
+generating signal waveforms on the fly. In grid-based
+methods, on the other hand, waveformscan be computed
+and stored in advance of processing the data. Stochas-
+tic methods can, therefore, lose their advantage if the
+computational cost of generating waveformsbecomes too
+high.
+The performance of a stochastic method may be sensi-
+tive to the values to which its design variables are tuned.
+Since the tuning is usually done on simulated data, it
+is not clear how robust current stochastic methods are
+against features of real data such as non-stationarity
+and non-Gaussianity. Additionally, the number of de-
+sign variables that require careful tuning is fairly large
+for some of the methods. In such cases, tuning becomes
+moreofanartthanawelldefinedprocedureandthismay2
+also affect robustness. In some methods, prior informa-
+tion is used about generic features of the function to be
+maximized. This may not be reliable if the assumptions
+behind the prior information, such as a particular noise
+model, become invalid. To properly address issues such
+as these it is important that a wide variety of stochastic
+methods be explored in GW data analysis .
+Particle Swarm Optimization (PSO) [11], first pro-
+posed by Kennedy and Eberhardt in 1995, is a stochas-
+tic method that has been garnering a lot of attention
+recently in many application areas [12]. An attractive
+feature of PSO is that, in its basic form, it has a small
+number of design variables. On standard testbeds, PSO
+has been found to have comparable or superior perfor-
+mance to other well known methods such as MCMC.
+ThispaperpresentsthefirstapplicationofPSOtoGW
+data analysis. We pose the following specific questions:
+1. Is PSO a viable method when applied to a function
+that is highly multi-modal and essentially stochas-
+tic in nature? This is the typical case in GW data
+analysis.
+2. How many design variables are there in PSO, and
+how many of them need to be tuned well?
+3. Can the tuning of these variables be done with-
+out requiring prior information about features of
+the function, thus increasing the robustness of the
+method?
+4. What is the computational cost of the method and
+what are the most important technical improve-
+ments required for the future?
+To answer these questions in the most direct and reliable
+manner, we construct a testbed based on the well un-
+derstood task of detecting and estimating binary inspiral
+signals in data from a single ground-based detector. This
+problem involves low-dimensionality but offers the more
+serious challenge of high multi-modality. To keep the fo-
+cus on the latter, a simplification is made regarding the
+shape of the search region such that it admits unphysical
+waveforms. Thus, the implementation of PSO presented
+here is not directly applicable to binary inspiral searches
+at present. The required technical refinements are dis-
+cussed in the paper. In addition, a novel and system-
+atic tuning procedure is introduced that is based on data
+containing only noise. This procedure may be useful for
+other stochastic methods also.
+The rest of the paper is organized as follows. Sec. II
+describes the testbed and Sec. III describes the PSO
+method. We explain our procedure for tuning the de-
+sign variables of PSO in Sec. IV. Sec. V then presents
+results from numerical simulations. Our conclusions and
+pointers to future work are presented in Sec VI.
+Notation
+x,y, etc: A time series with a finite number, N, ofsamples. The kthsample, 0 ≤k≤N−1,isdenoted
+byx[k].
+δs,T: The sampling interval and the duration of xre-
+spectively. The number of samples in xisN=
+[T/δs], where the square brackets denote trunca-
+tion to the nearest integer.
+Θ : The set of parameters describing a family of signals.
+s(Θ): The time series of the signal corresponding to
+parameter values Θ. The kthsample of s(Θ) is
+denoted by s[k;Θ]. In our case, signals have a
+well-defined start and stop time and the interval
+between them may be less than T. However, s(Θ)
+still consists of Nsamples with the samples outside
+the intervalenclosed bythe startand stop times set
+to 0 (zero-padding).
+/tildewidex: The Discrete Fourier Transform (DFT) of x.
+The DFT value at the frequency k/T,k=
+0,1,...,[N/2 + 1], is denoted as /tildewidex[k]. The DFT
+ofs(Θ) is denoted by /tildewides(Θ) and its value at the kth
+frequency by /tildewides[k;Θ].
+/an}bracketle{tx,y/an}bracketri}ht: The time series introduced above are elements
+ofRN, the vector space of real N-tuples. From
+the point of view of detection and estimation of
+a signal in data with additive stationary noise, a
+natural inner product can be introduced on this
+vector space,
+/an}bracketle{tx,y/an}bracketri}ht= 4R
+[N/2+1]/summationdisplay
+k=0/tildewidex∗[k]/tildewidey[k]
+Sn[k]T
+ (1)
+whereSn[k]istheone-sidedPowerSpectralDensity
+(PSD) of the noise.
+/bardbly/bardbl: the norm on RN,
+/bardbly/bardbl2=/an}bracketle{ty,y/an}bracketri}ht, (2)
+induced by the inner product defined above. The
+signal to noise ratio (SNR) of a signal s(Θ) is de-
+fined as/bardbls(Θ)/bardbl.
+II. TEST BED
+In this section, we describe the testbed to which PSO
+is applied. The testbed is constituted by the noise model,
+signal family and the function to be maximized.
+A. Noise model
+A GW signal incident on an interferometric ground-
+based detector produces a difference in the lengths of its
+two arms. After calibrating out the common arm length
+and the transfer function of the detector, the data, x,3
+contains the measured GW-induced strain added to in-
+strumental and environmental noise n. Thus, x=n
+when no GW signal is present, and x=s(Θ)+nwhen
+there is. In our simulations, nis a realization of a sta-
+tionary, Gaussian noise process with a PSD, Sn[k], that
+matches the initial LIGO [13] design sensitivity curve in
+shape [14].
+B. Signal waveforms
+We use the signal family associated with a non-
+spinning inspiraling binary system, computed up to the
+second post-Newtonian (2PN) order [15]. This sys-
+tem consists of two non-spinning compact stars (Neu-
+tron Stars or Black Holes) losing orbital binding energy
+through GW emission. Members of this signal family
+havechirpwaveforms with monotonically increasing in-
+stantaneous amplitude and frequency.
+For the case of a single detector, the parameters speci-
+fying the 2PN signal waveforms can be grouped into two
+sets. The first set is that of the chirp-time [16] param-
+eters,{τa},a= 0,1,1.5,2, that are constructed out ofthe masses of the two components of the binary. Expres-
+sions for the chirp-time parameters are provided in Ap-
+pendix A. The second set consists of the time-of-arrival ,
+ta, theinitial phase , Φaand the amplitude, A. Interfer-
+ometric ground-based detectors have a sharp rise in seis-
+mic noise below some frequency fa( = 40 Hz for inital
+LIGO). The chirp signal from a binary inspiral is essen-
+tially unobservable when its instantaneous frequency is
+belowfa. The time at which the signal becomes visible
+istaand the corresponding instantaneous phase of the
+signal is Φ a.
+Since all the four chirp-times depend on the masses of
+the twocompactstars, onlytwoofthem areindependent.
+We chooseτ0andτ1.5as the two independent chirp-time
+parameters. Thus, the set of signal parameters is Θ =
+{A,Φa,ta,θ={τ0,τ1.5}}.
+As discussed in Sec. I, the computational cost of gen-
+erating waveforms on the fly is important for stochastic
+methods like PSO. The 2PN signal family is amenable
+to a fast implementation because a sufficiently accurate
+analytical form exists for the Fourier transform of these
+waveforms [17],
+/tildewides[k;Θ] =
+
+0, k≤[faT],
+ANf−7/6exp/bracketleftbig
+−2πifta+iΦa−iψ(f;θ)+iπ
+4/bracketrightbig
+,[faT]<k≤[fcT],
+0, k>[fcT](3)
+where, the lower cutoff frequency fawas explained above
+and the upper cutoff frequency fcfollows from the termi-
+nation of the inspiral waveform when the binary reaches
+its last stable orbit. For our testbed, we set fc= 700 Hz
+although in a real search it depends on the mass of the
+binary system. The expression for ψ(f;θ) is given in Ap-
+pendix A. The normalization constant Nis defined such
+that,/bardbls({A= 1,Φa,ta,θ})/bardbl2= 1. It follows that Ais
+the SNR of the signal.
+Later on, we use the fact that although not all values
+ofθcorrespond to valid binary mass components, Eq. 3
+can still be used to generate perfectly normal waveforms.
+Thesewaveformsarealsochirpsbut theirphaseevolution
+does not correspond to any physical binary system.
+C. Fitness function
+The function to be maximized is,
+Λ(ta,θ|x) =/bracketleftbig
+/an}bracketle{tq0(ta,θ),x/an}bracketri}ht2+
+/an}bracketle{tqπ/2(ta,θ),x/an}bracketri}ht2/bracketrightbig1/2, (4)
+qφ(ta,θ) =s({A= 1,Φa=φ,ta,θ}).(5)This function is obtained by maximizing the log-
+likelihood, /an}bracketle{tx,s(Θ)/an}bracketri}ht −(1/2)/bardbls(Θ)/bardbl2analytically over A
+and Φ a.
+For givenθ, the evaluation of /an}bracketle{tqφ(ta,θ),x/an}bracketri}htoverta=
+mδs,m= 0,1,...,N−1, is a cross-correlationoperation
+that can be computed efficiently using the Fast Fourier
+Transform (FFT). Thus, the function that is maximized
+using PSO is,
+λ(θ|x) = max
+taΛ(ta,θ|x). (6)
+In the remainder of the paper, λ(θ|x) will be called the
+fitness function in keepingwith the standardterminology
+used in much of the literature on stochastic methods.
+The presence of noise in xmakes the fitness function
+highly multi-modal as shown in Fig 1. The large number
+of local maxima with random locations and sizes poses a
+strong challenge to stochastic methods. When the noise
+is stationary and Gaussian, and the signal present in the
+data is from the waveform family that one is searching
+for, certain characteristic features are present in the fit-
+ness function. For example, the shape of the peak in
+Fig. 1 is elongated on the average along a predictable di-
+rection. MCMC methods in the GW literature use this
+type of prior information about the fitness function in
+tuning the design variables [9].4
+15.81616.216.416.6
+0.60.650.70.750.84.555.566.57
+τ0 (sec)
+τ1.5 (sec)λ(θ| x)
+FIG. 1: A realization of the fitness function for the binary in -
+spiral testbed. The data contains a signal with an SNR=8.0.
+In the absence of noise, the fitness function has only one ex-
+tremum at the location identified by the chirp-times of the
+signal. The presence of noise leads to a forest of local max-
+ima.
+III. PARTICLE SWARM OPTIMIZATION
+The PSO algorithm is first described in terms of a
+general fitness function λ(θ), over some parameter set
+θ. Later, we specialize the discussion to the case of the
+binary inspiral testbed.
+A. The PSO Algorithm
+Letθ={θ1,θ2,...,θ D}denoteapointin RD, andλ(θ)
+be the fitness function. The essential idea behind PSO is
+to compute λ(θ) simultaneously at several locations and
+usethesesamplestoinfluencethelocationsforcomputing
+thenextsetofsamples. Thisprocesscontinuesiteratively
+until some stopping rule is satisfied. The process can be
+visualized by treating the sample locations as a swarm
+ofparticles that moves in RD, hence the name of the
+algorithm. A precise description now follows.
+Let thecoordinatesin RDoftheithparticlein aswarm
+ofNpparticles be Qi[k] at thekthstep in the search
+(k= 0,1,...). Associated with this particle is a velocity
+vectorVi[k] that determines Qi[k+1],
+Qi[k+1] =Qi[k]+Vi[k]. (7)
+ThePSOalgorithmisusuallystartedwith randomlycho-
+sen particle locations and velocities. In our implementa-
+tion, we position the particles initially on a regular grid
+while the initial velocities are kept random.
+Let the maximum value of λ(θ) found by the ithparti-
+cle overksteps beRi(k) and the location of Ri(k), called
+the particle’s best location pbest, bePi[k]. Thus,
+Ri(k) =λ(Pi[k])≥λ(Qi[j]);j≤k. (8)Let the maximum over {Ri(k)},i= 1,...,N p, beRg(k)
+and its location, called the global best location gbestbe
+Pg[k],
+Rg(k) =λ(Pg[k])≥λ(Pj[k]);∀j . (9)
+At any step, there is always one particle in the swarm
+whosepbestis also the gbest. We call this particle the
+best particle at stepk. Note that both pbestandgbest
+are locations found over the entire past history of the
+motionoftheparticles. Theyneednotnecessarilychange
+at every step.
+The velocity for the ithparticle at the next step, k+1,
+is determined by the dynamical equation,
+Vi[k+1] =wVi[k]+c1χ1(Pi[k]−Qi[k])+
+c2χ2(Pg[k]−Qi[k]), (10)
+wherew, which can depend on k, is called the inertia
+weight,c1andc2are called acceleration constants , and
+χ1,χ2are random numbers drawn independently at each
+step from the uniform distribution on [0 ,1].
+Finally, for any component Vi,m[p] of the particle ve-
+locityVi[p] = (Vi,1[p],...,V i,D[p]),
+Vi,m[p] =/braceleftbigg
+Vmax,m, Vi,m[p]>Vmax,m
+−Vmax,m, Vi,m[p]<−Vmax,m,(11)
+whereVmax,k>0,k= 1,...,D, andVmax=
+(Vmax,1,...,V max,D) is called the maximum velocity .
+Like all stochastic methods, PSO involves a compe-
+tition between wide ranging exploration of the fitness
+function and convergence to a best value. In order to
+avoid trapping by a local maximum, the method must be
+able to explore other parts of the parameter space, while
+to find the global maximum, the method must eventu-
+ally explore a progressively smaller region around some
+point. The way this competition is implemented in the
+PSO algorithm is seen clearly from Eqs. 10. The first
+term simply moves a particle along a straight line, while
+the remaining two terms are sources of acceleration, one
+pulling it towardsits pbestand anotherpulling it towards
+gbest. The last two effects are combined with random
+weightsχ1andχ2. The random deflections and inertial
+motion allow a particle to explore the fitness function,
+while the attractive pulls of pbestandgbestcounter this
+behavior. With a dynamic inertia weight that decreases
+in time, the attractive pull eventually wins over. A rudi-
+mentary emulation of real biological swarming behavior
+is built in through each particle being aware of gbest.
+The PSO algorithm has another interesting feature.
+The best particle, by definition, has its pbestPi[k] coin-
+cident with gbest,Pg[k], making the terms Pi[k]−Qi[k]
+andPg[k]−Qi[k] equal for it. This particle then acceler-
+ates towards gbestalone and only moves along a straight
+line through this location. This situation continues un-
+til a new gbestis found. In effect, one particle at any
+step shows a convergence behavior, exploring the neigh-
+borhood of the current gbest, while the other particles
+continue their exploration.5
+B. Termination criterion
+For stochastic methods, the probability of convergence
+to the global maximum is usually guaranteed only in the
+asymptoticlimit. Hence, anypracticalimplementationof
+a stochastic method must include a criterion for termi-
+natingthesearch. Thecriterionweadoptfortermination
+is specific to the fitness function for the binary inspiral
+testbed and, accordingly, θnow refers to the chirp time
+parameters.
+If the particles in PSO continue to move over several
+steps but do not find a significantly different gbest, it
+is likely that the current gbestlies close to the global
+maximum. A natural criterion for termination then is
+to check if gbeststays confined to a small region over a
+predetermined number of steps.
+When the data contains only a signal, x=
+s({A,Φa,ta,θ}), the fitness function is maximum at the
+locationθof the signal. ( s(θ)≡s({A,Φa,ta,θ}) for
+brevity in the following since the other parameters do
+not figure in the fitness function.) The fractional drop
+in the fitness function for a small displacement ∆ θ=
+(∆θ1= ∆τ0,∆θ2= ∆τ1.5) is given by,
+1−λ(θ+∆θ|s(θ))
+λ(θ|s(θ))≃−/summationtext2
+i,j=1Hij∆θi∆θj
+2λ(θ|s(θ)),(12)
+Hij=∂2λ(θ′|s(θ))
+∂θ′
+i∂θ′
+j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+θ′=θ,(13)
+whereθis the location of the signal. For a small frac-
+tionaldrop α, therefore,wegetanellipsoidalregion Sα(θ)
+centered at θsuch thatλ(θ′|s(θ))≥(1−α)λ(θ|s(θ)) if
+θ′∈ Sα(θ).
+Now, the neighborhood of the global maximum in the
+presence of noise is also Sα(θ) on the average for a frac-
+tional drop α. Therefore, it is natural to choose the re-
+gion of convergence to be Sα(θ) in general. This reduces
+the task of specifying the region to simply choosing a
+value forα. Following a convention widely used in the
+GW literature [3], we fix α= 0.03.
+Thus, we arrive at the following criterion for terminat-
+ing PSO. At each step k, (i) obtain the ellipsoid around
+gbest, that is, Sα(Pg[k]). (ii) If the best location Pg[k+1]
+falls outside Sα(Pg[k]), then reset the region of conver-
+gence to the new best location, i.e., use Sα(Pg[k+ 1]).
+(iii) If the region of convergence is not found to change
+overNtsuccessive steps, then terminate PSO.
+The termination criterion implies that if PSO termi-
+nates near the true global maximum, the fitness value
+found will have a fractional drop less than α. Conse-
+quently, it will have a performance comparable to a grid-
+based searchin which the templates arespaced according
+totheminimal match criterion[5]andtheminimalmatch
+is 1−α. This is important for situations where a grid-
+based search is infeasible as it guarantees that PSO will
+perform as well or better. The probabilityof convergence
+to the global maximum must be high, however, and this
+is the objective of the tuning process described later.C. Search Boundary
+Evenwiththeterminationcriterioninplace,thesearch
+region must be finite in order for PSO to terminate in a
+finite number of steps. Otherwise, the swarm may con-
+tinue to find a better gbestand the termination criterion
+may never be satisfied. This is especially relevant in the
+case when the data has only noise. Thus, the PSO dy-
+namics must be supplemented with appropriate bound-
+ary conditions. Many approaches to this problem have
+been proposed, with agoodsummaryprovidedin[18]. In
+this paper, we use the invisible wall boundary condition,
+but we also briefly describe some of the others below.
+1. Types of boundary conditions
+The boundary conditions proposed in the PSO litera-
+ture are as follows. (This list is taken from [18] and is by
+no means an exhaustive one.)
+Absorbing walls – When a particle crosses a rectangular
+boundary, the velocity component perpendicular to
+the boundary is zeroed. Eventually, this allows the
+particle to be pulled into the search domain.
+Reflecting walls – As with the absorbing walls condi-
+tion, the particle velocity is altered but instead of
+being zeroed, the velocity component perpendicu-
+lar to the wall is reversed in sign. This throws the
+particle back into the search domain.
+Invisible walls – No change is made to the dynamics of
+the boundary crossing particle but λ(θ|x) is set to
+zero and it is not evaluated as long as the particle
+stays outside the boundary.
+We have tried all three boundary conditions but, like the
+authors of [18], we find that the invisible wall condition
+tends to perform better than the other two.
+2. Search region for the testbed
+The simplest search region in θparameter space is a
+rectangleτ0,min≤τ0≤τ0,maxandτ1.5,min≤τ1.5≤
+τ1.5,max. A part of this region, however, admits wave-
+forms that do not correspond to a physically valid bi-
+nary system. This is due to the dependence of τ0,τ1.5on
+the symmetric combinations ofbinary component masses
+Mandµ, the total and reduced mass of the binary re-
+spectively, and the inequality M≥4µ. Nonetheless, as
+remarked in Sec. IIB, there is no technical problem in
+generating waveforms corresponding to the unphysical
+chirp times and nothing strange happens to the fitness
+function there. See Fig. 1, for example, where a part of
+the parameter region shown is unphysical. Since the pri-
+mary utility of the binary inspiral problem in this paper6
+is to provide a testbed for PSO, this physical constraint
+is ignored.
+The rectangular search region allows the coordinate
+transformation,
+x1= (τ0−τ0,min)/(τ0,max−τ0,min),(14)
+x2= (τ1.5−τ1.5,min)/(τ1.5,max−τ1.5,min),(15)
+such thatxi∈[0,1],i= 1,2. In our codes, all PSO
+equations use x1andx2and the corresponding velocity
+components.
+We choose τ0,min= 0.94 sec,τ0,max= 37.48 sec,
+τ1.5,min= 0.234 sec and τ1.5,max= 1.021 sec. With τ0
+andτ1.5along the horizontal and vertical axes respec-
+tively, the upper right hand corner corresponds to binary
+component masses (in M⊙)m1= 1.1 andm2= 1.1. The
+lower left hand corner corresponds to m1= 10.5 and
+m2= 9.7.
+D. PSO design variables
+One of the questions posed at the beginning was about
+the number of design variables in PSO. In our implemen-
+tation, there are a total of 9 that are listed below for
+reference.
+Np: Number of particles in the swarm.
+c1, c2: Acceleration constants.
+Vmax: Maximum velocity of a particle.
+α, Nt: The parameters used to specify the termination
+criterion for PSO. These parameters are not part
+of the standard PSO algorithm.
+Parameters governing the inertia decay law: The iner-
+tia weight is decreased in value as PSO progresses
+through a search. The PSO literature is full of
+different types of decay laws but, in general, it
+is known that a strictly linear decay law is not
+very useful. We have developed the following de-
+caylawthathaselementsofbothlinearityandnon-
+linearity. Let w[k] be the value ofthe inertiaweight
+at stepk,
+w[k] =w0−m(k−k0)/Nt, (16)
+wherew0>0 andm>0. The parameter k0starts
+with an initial value of k0= 0 and is kept fixed
+as long as gbeststays within the current region of
+convergence. If gbestexits the convergence region
+at some step k′without termination, k0is set equal
+tok′. Thus, the value of the inertia is reset to the
+starting value of w0every time termination fails
+and the linear decay of the inertia starts anew.Nrep: For given data x, independent runs of PSO yield
+different values of λ(θ|x) corresponding to the dif-
+ferent fitness values at termination. This is un-
+avoidable for any stochastic method. However, ter-
+minationnearthetrueglobalmaximuminindepen-
+dent runs of PSO on the same data should result
+in the clustering of the different values found and
+their locations. We can turn this argument around
+by runningPSO independently severaltimes on the
+samedataandusingtheformationofaclusterasan
+indicator ofsuccessful termination in the vicinity of
+the global maximum. The number of independent
+runsofPSOonthesamedata, Nrep, isalsoadesign
+variable.
+IV. TUNING THE DESIGN VARIABLES
+For any stochastic method, convergence to the global
+maximum can only be quantified as a probability. In
+some asymptotic limit, such as particle number Np→ ∞
+for PSO, this probability becomes unity. However, this
+also implies an infinitely large computing cost. Thus the
+design variables must be tuned to find the best trade-off
+between the probability of convergence and the associ-
+ated computational cost. We present here the procedure
+followed for tuning the design variables of PSO.
+In contrast to the tuning procedure used for most
+MCMC methods in the GW literature, our approach is
+not based on data containing a signal but data that is
+purely noise. The latter is the worst case scenario for
+any stochastic method. However, good performance in
+the pure noise case more or less guarantees success when
+a signal is present. Moreover, this approach to tuning
+avoids any bias due to the use of a particular set of sig-
+nals or SNRs.
+The tuning procedure presented here can be used, in
+principle, to tune all the nine design variables of PSO
+(c.f., Sec. IIID). However, applying the procedure to all
+of them is computationally too expensive, at least for
+the objectives of this paper. We focus instead on two
+of the most important variables for the performance of
+PSO,NpandNt. For the rest, we either choose values
+commonlyusedintheliteratureorsimplypickreasonable
+ones based on our experience with PSO. Thus, we set :
+c1=c2= 2,Vmax= (0.5,0.5),w0= 0.9,m= 0.4,
+α= 0.03 andNrep= 5.
+A. Criterion for optimal tuning
+Measuring the probability of convergence for the pure
+noise case presents a practical problem. In simulations
+where a large SNR signal is present, we know that the
+true maximum is most likely to be in close proximity to
+the location of the signal and it can be found reliably
+using, say, a small area grid-based search. For the pure
+noise case, however, the location is not known a priori,7
+even approximately, and the only reliable solution is a
+grid-based search over the entire search region. However,
+we avoid this solution because (i) the simulations become
+computationally very expensive, and more importantly,
+(ii) it would fail for higher dimensional problems where
+grid-based searches are infeasible.
+To circumvent this problem, we invoke the argument
+outlined in Sec. IIID for using Nrepwherein termination
+in the vicinity of the global maximum is indicated by the
+clustering of the fitness and parameter values over inde-
+pendent runs of PSO. One way to further confirm the
+association between a cluster and the global maximum is
+to increase the number of particles significantly and ver-
+ify that a cluster forms around the same location. This
+is similar to what is done, for example, in the numerical
+solution of differential equations. To check that a given
+solution is valid, the computational grid is made denser
+and the new solution is compared with the old one. The
+above ideas can be quantified as follows, allowing an ob-
+jective criterion for tuning to be developed.
+Let there be a number of independent trials, in each of
+which a new realization nof noise is obtained and PSO
+is runNreptimes on n. Thus, in each trial, Nrepvalues
+are obtained for each of the chirp-times τ0andτ1.5, and
+the corresponding fitness values λ(θ|n). We define a set
+ofNrep= 5 numbers to be clustered if at least 3 of them
+lie in a range that is less than 30% of the entire range
+of the 5 numbers. This definition of clustering is applied
+to each of the three sets of Nrepvalues above. We then
+define,
+Probability of clustering: Let Pτ0,Pτ1.5andPλbe the
+fraction of trials in which clustering occurs for τ0,
+τ1.5andλ(θ|n) respectively. The maximum among
+Pτ0,Pτ1.5andPλis defined as the probability of
+clustering .
+Consistency of clustering: If, for a given realization of
+noise, theNrepfitness values are found to be clus-
+tered, then the cluster is defined to be consistent
+if (i) the fitness values are also clustered for N′
+p
+sufficiently greater than Np, and (ii) the absolute
+difference between the maximum fitness values ρ
+andρ′, corresponding to NpandN′
+prespectively,
+is≤10% of their mean, ( ρ+ρ′)/2. We define the
+consistency of clustering as the fraction of trials in
+which the clusters are consistent.
+We deem a given combination of design variable values
+acceptable if both the probability and the consistency of
+clustering exceed 0.9 for that combination. Of all the
+combinations that are acceptable, the optimal is chosen
+to be the one that has the lowest computational cost in
+terms of the mean number of template evaluations.
+B. Simulations
+The tuning procedure described above is now applied
+to the two design variables NpandNt. The following setTABLEI:Computational costofPSOondatawithnosignals.
+For each combination of NpandNt, the mean number of
+fitness function evaluations is listed along with the maximu m
+(superscript) and minimum (subscript) over 50 trials. The
+mean values have been rounded off to the nearest integers.
+Np= 42 81 121
+Nt= 20 830912768
+5250 1628421465
+8910 2500639688
+13310
+40 1740124486
+9618 3169440824
+19521 4463261105
+25410
+80 2892037338
+22302 5266966825
+35559 7411595469
+53119
+120 3856751450
+32550 6998285293
+49410 101495143990
+75262
+160 4814768880
+38808 86759109755
+68040 126346161535
+98010
+of points is used to find the acceptable combinations:
+Np∈ {42,81,121}
+Nt∈ {20,40,80,120,160}
+This particular domain in the Np-Ntplane is chosen
+based on our empirical experience with PSO. ( Np= 42
+corresponds to a 7-by-6 grid of initial positions, 81 to a
+9-by-9and 121 to an 11-by-11one.) The number of trials
+is 50 and each realization of noise is 64 seconds long with
+sampling interval δs= 1/2048 sec.
+The tuning procedure proceeds as follows.
+a. Computational cost – For each point in the Np-
+Ntplane, we record the mean number of fitness function
+evaluations. The results are shown in Table I.
+b. Probability of clustering – Table II lists the prob-
+ability of clustering for each combination of NtandNp.
+Note that for the combination Np= 121 and Nt= 40,
+Pτ0= 94% is very different from Pλ= 76% and Pτ1.5=
+80%. This suggests that the abnormally high value of
+Pτ0here is most likely a statistical outlier. Therefore, we
+do not consider this combination as having a probability
+of clustering ≥90%.
+c. Consistency of clustering – Referring to Table II,
+we see that the consistency test is required only for
+Np≥81 andNt≥80 for which, as per the definition
+of acceptability above, the probability of clustering is
+≥90%. Further, for a given Nt, the computational cost
+is lower for Np= 81 thanNp= 121. Hence, we only tune
+overNt≥80 forNp= 81. No extra work is required for
+obtaining these results since for each trial, the same data
+realization was used for both Np= 81 andNp= 121 and
+the latter can be used to check if a cluster found by the
+former was consistent or not. In other words, Np= 81
+andN′
+p= 121 in the definition of the consistency of clus-
+tering given earlier.
+We obtain the following results for the consistency of
+clustering: 91%, 93% and 95% for Nt= 80, 120 and 160
+respectively. Thus, according to our final criterion, we
+pickNp= 81 andNt= 80 as this is the acceptable com-
+bination with the lowest computing cost (c.f., Table I).8
+TABLEII: Probabilityofclustering for differentcombinati ons
+ofNpandNt. For each combination, the fraction of trials
+(in %)Pλ,Pτ0andPτ1.5for which the fitness, τ0andτ1.5
+values respectively were found to be clustered are listed. T he
+probability of clustering, shown in bold, is the maximum ove r
+Pλ,Pτ0andPτ1.5. The number of trials for each combination
+is 50.
+Np= 42 81 121
+Nt= 20(Pλ)66
+(Pτ0)74
+(Pτ1.5)6860
+72
+7270
+82
+82
+4072
+82
+8676
+88
+7676
+94
+80
+8084
+84
+8884
+90
+8690
+92
+92
+12072
+78
+6888
+92
+8896
+92
+96
+16082
+88
+7886
+86
+8094
+94
+92
+C. Trials with no clustering
+So far, we have focussed on clustering as the main in-
+dicator of success in locating the true global maximum.
+Does this imply that in the trials in which there is no
+clustering, PSO fails to locate the global maximum? To
+address this, we carried out the following test. First, we
+retain the maximum among the Nrepfitness values from
+each trial. For each combination of NpandNt, we divide
+the set of maximum fitness values into two disjoint sub-
+sets: oneinwhichallparameters,thetwochirp-timesand
+the fitness, were clustered and the other in which at least
+one parameter did not show clustering. For the former
+set, clustering of all three parameters is a strong indica-
+tor of successful termination near the global maximum.
+A two-sample Kolmogorov-Smirnov (KS) test [19] is car-
+ried out to see if the two subsets were drawn from the
+same parent distribution. The results are summarized in
+Table III.
+As can be seen from the table, in all cases the test
+supports the hypothesis that the maximum fitness value
+is drawn from the same distribution irrespective of the
+clustering of the parameters. That this is a non-trivial
+result is further supported by the fact that if the same
+test is done with fitness values other than the maximum
+one, the null hypothesis is rejected strongly. Table III
+shows the results from the same test but using the set of
+minimum fitness values. In this case, it is seen that the
+values are drawn from different distributions, at least for
+Np= 81. Thus, we conclude that even in the absence ofTABLE III: Statistical difference in the distribution of max -
+imum fitness values. The table entries are the significance
+values from a two sample Kolmogorov-Smirnov test with the
+null hypothesis: the maximum fitness values from trials that
+show clustering of all parameters and trials that do not are
+drawn from the same parent distribution. The numbers in
+parentheses are the significance values for the test done wit h
+minimum fitness values.
+Np= 81 121
+Nt= 80 0.5(2×10−2) 0 .9(0.7)
+120 0.9(2×10−2) 0 .4(0.4)
+160 0.7(8×10−4) 0 .4(0.9)
+clustering, the PSO run that yields the maximum fitness
+value terminates, with high probability, in the vicinity
+of the global maximum for the Np= 81 andNt= 80
+combination.
+We have traced the lack of clustering to the presence
+of distant peaks in the fitness function that are similar in
+value. The probability of this happening in the presence
+of a sufficiently strong signal is very small, but this need
+not be so for noise-only data.
+D. Comments
+We have demonstrated a systematic tuning procedure
+for the design variables of PSO. It is important to note
+that no prior information about any special features of
+the fitness function was used. Hence, the procedure
+would stay the same if the testbed were changed.
+A larger number of trials or a finer spacing of grid
+points inNpandNtwill probably lead to a different end
+result. Instead of Np= 81, for example, Np= 121 may
+turn out to be the right choice. However, the main goal
+in this paper is to test the viability of PSO and, for this
+purpose, a coarse tuning such as the one presented here
+is adequate. Besides, a significant investment in refining
+the resultsofthe tuning procedurewouldbe renderedob-
+solete with future improvements in the implementation
+of PSO. Until a version of PSO is developed that is hard
+to improve upon, a strong focus on the results from tun-
+ing, as opposed to improvements to the tuning procedure
+itself, is not of much use.
+ForNp= 121, Table III shows that the minimum fit-
+ness values obtained with and without clustering are also
+mutually consistent. This observation suggests an alter-
+nativeapproachtotuningwhereoneofthemeasuresused
+for picking the optimal combination is this type of con-
+sistency. We leave this for future work to address.9
+0102030405060708090012345678910
+m1 (solar mass)m2 (solar mass)
+FIG. 2: The region in the m1,m2plane corresponding to
+the physically valid part of the τ0,τ1.5plane. The region is
+indicated by taking a regular grid of points in the τ0,τ1.5
+plane and mapping them to the corresponding values of m1,
+m2, where by convention m1≥m2. The⋆markers shows the
+signal locations used in the simulations.
+V. RESULTS WITH SIGNAL PRESENT
+In this section, we describe the results of simulations
+performed with signals added to data. We quantify the
+performanceofPSOatfourdifferentvaluesofsignalSNR
+and four different locations in the τ0,τ1.5plane,
+SNR∈ {9.0,8.0,7.0,6.0},
+(τ0,τ1.5)∈ {(5.0,0.6),(10.0,0.75),
+(16.0,0.762),(20.0,0.9)}.
+(The units for both τ0andτ1.5are in seconds).
+The corresponding masses (in M⊙) of the binary
+components are, respectively, {(m1= 7.78,m2=
+1.91),(4.71,1.35),(2.40,1.40),(2.61,1.03)}. Fig. 2 shows
+the physical part of the search region mapped into the
+m1,m2plane along with the signal locations.
+For each combination of signal location and SNR, 50
+independent data realizations are generated. The length
+of each realization is 64 sec, with δs= 1/2048 sec, and
+the signal is added at an offset of 10 sec from the start.
+A. Qualitative changes induced by a signal
+It is instructive to observe how a signal affects the be-
+havioroftheswarm. Ingeneral,thepresenceofthesignal
+leads to a broadening of the peak in the fitness function.
+As is well known, the broadening is more pronounced in
+one direction, due to the correlation between estimation
+errors, leading to the appearance of a thin ridge-like fea-
+ture (c.f., Fig 1).5 10 15 20 25 30 350.30.40.50.60.70.80.91
+τ0 (sec)τ1.5 (sec)
+FIG. 3: Evolution of a swarm in the presence of a signal.
+The ‘◦’ and ‘∗’ markers show the pbestlocations of Np= 81
+particles when 5% and 60% of the total number of steps were
+completed respectively. The lines show the paths followed b y
+thepbestlocations of 5 representative particles between these
+two steps. With time, the pbestlocations tend to congregate
+around the ridge-like feature produced by a signal.
+The particles begin by moving randomly in the param-
+eter space but each time a particle crosses the ridge, its
+pbesttends to fall closer to the flanks of the ridge. As
+time progresses, the pbestof all particles cluster around
+the ridge. This increasesits attractivepowerin the accel-
+eration of the particles, progressively drawing more par-
+ticles into exploration of the fitness function along the
+ridge.
+Fig.3showssnapshotsofPSOatdifferentstagesin the
+search and the progressive clustering of pbestlocations is
+seen clearly. A key point to note here is that no prior
+knowledge is built into PSO about the ridge-like feature.
+It is found by the particles as they explore the search
+region.
+B. Figures of merit
+In order to quantify the performance of PSO in the
+presence of a signal, we look at two figures of merit. The
+first is the probability of clustering defined in Sec. IVA.
+Since the tuning procedure requires a minimum value of
+90%, the probability of clustering in the presence of a
+strong signal should be significantly higher but it should
+be consistent with the pure noise case for weak signals.
+When a signal is added to the data, we do not need the
+consistency of clustering criterion of Sec. IVA in orderto
+confirm the association of a cluster with the global maxi-
+mum. Since we know the location of the signal and since
+the expectation of the fitness function must be maximum
+at that location, it suffices to check if the maximum fit-10
+TABLE IV: Probability of clustering for simulations with si g-
+nal present in the data. For each combination of signal SNR
+and location, the fraction of trials (in %) Pλ,Pτ0andPτ1.5for
+which the fitness, τ0andτ1.5values respectively were found to
+be clustered are listed. The probability of clustering, sho wn
+in bold, is the maximum over Pλ,Pτ0andPτ1.5. The number
+of trials for each combination is 50.
+(τ0,τ1.5)
+= (5.0,0.6)(10.0,0.75) (16.0,0.762) (20 .0,0.9)
+SNR=9.0(Pλ)98
+(Pτ0)94
+(Pτ1.5)9494
+92
+9494
+96
+9490
+98
+94
+8.096
+96
+9298
+96
+9698
+92
+9294
+92
+96
+7.096
+96
+9092
+90
+8898
+94
+9892
+88
+90
+6.092
+94
+8482
+86
+8688
+86
+8494
+96
+96
+ness in the cluster is larger than the value at the signal
+location. Our second figure of merit, therefore, is the
+fraction of trials in which this occurs. Ideally, this figure
+of merit should be unity.
+Table IV reports the first figure of merit for each com-
+bination ofsignalSNR and location. Asexpected, forthe
+case of strong signals (SNR ≥7) the probability of clus-
+tering is always, and often significantly, higher than 90%.
+For the weak signal SNR of 6 .0, the probability of clus-
+tering has an average value of 91% which is statistically
+consistent with the pure noise case of 90%.
+As far as the second figure of merit is concerned,
+we find that it is unity for all combinations of signal
+SNR and locations except for one, namely, SNR= 8.0,
+τ0= 10.0 sec andτ1.5= 0.75 sec, for which it was 0.98.
+Fig. 4 shows the scatterplot between the maximum fit-
+ness found by PSO and the value at this signal location
+for all signal SNR values. It is seen that in one trial the
+maximum fitness fell below the value at the signal loca-
+tion. However, the two values are so close that the figure
+of merit should be considered to be practically unity for
+this case too.
+Taken together, the figures of merit show that PSO
+almost always terminates near the true global maximum
+when a sufficiently strong signal is present. When the
+signal is weak, we recover the performance ensured by
+the tuning procedure for the pure noise case.4 5 6 7 8 9 10 114567891011
+Fitness at signal locationMax. Fitness found by PSO
+FIG. 4: Scatterplot of maximum fitness value found by PSO
+(Y axis) against the value at the known signal location, τ0=
+10.0 sec,τ1.5= 0.75 sec, for all signal SNR values. In one
+trial, the maximum fitness value (near 6.0 on the Y axis) dips
+below the line of equality (dashed).
+C. Signal detection and parameter estimation
+In order to cast the results obtained so far in terms of
+signal detection and parameter estimation performance,
+wechoosethemaximum fitnessvaluefound overthe Nrep
+runs as the detection statistic and the location corre-
+sponding to the maximum fitness value as the estimator
+for the chirp time parameters.
+1. Detection
+It was discussed in Sec IVC that, after tuning PSO,
+the distribution of the detection statistic in trials with
+and without clustering remains the same. As the simul-
+taneous clustering of the two chirp-times and the fitness
+values indicates termination in the vicinity of the global
+maximum, it follows that the probability distribution of
+the PSO detection statistic is about the same as that of
+the global maximum. Strictly speaking, the PSO detec-
+tion statistic will always have a value less than the global
+maximum but, given our termination criterion, the rel-
+ative difference between the two is less than 3%. Thus,
+the false alarm probabilities, for a givendetection thresh-
+old, correspondingto the PSO detection statistic and the
+true global maximum are also nearly the same, with the
+former being slightly smaller.
+In the presence of signals with an SNR of 8 or higher,
+which is the typical value sought in a real detection, it
+was shown that almost all trials exhibit clustering and
+that the detection statistic value was always higher than
+the fitness at the true signal location. Hence, the distri-
+bution of the detection statistic in the presence of a sig-
+nal also closely follows that of the true global maximum.11
+Strictly speaking, as with the false alarm probability, the
+detection probability will be slightly smaller for the PSO
+detection statistic, for a given threshold, as compared to
+that for the true global maximum.
+The above line of reasoning suggests that the Receiver
+Operating Characteristics (ROC) of the PSO detection
+statistic should nearly be the same as that of the true
+global maximum. The only way to rigorously verify this
+is to carry out simulations with a large number of tri-
+als in which both PSO and grid-based searches are per-
+formed. This is a computationally expensive task which
+we plan to undertake in the future. However, it is im-
+portant to note that such a comparison may not be pos-
+sible for searches that are too expensive for a grid-based
+search.
+2. Estimation
+Table V summarizes the errorsin the estimation of the
+chirp time parameters for signal SNR values ≥8 at the
+different signal locations used in the simulations. Each
+entry in the table is an estimate of the root mean-square
+error (rmse) defined as,
+rmse(θ) =/bracketleftbigg
+E/bracketleftbigg/parenleftBig
+/hatwideθ−θ/parenrightBig2/bracketrightbigg/bracketrightbigg1/2
+, (17)
+where/hatwideθis the estimator of θ. The rmse includes the
+effects of both estimator variance and bias.
+Since the search region in the current testbed includes
+unphysical chirp time parameters, the global maximum
+and, hence, the estimated chirp times fall there in some
+trials. Fig. 5 shows an example where the estimates from
+all the trials are shown for a signal SNR of 8. In an
+improved implementation of PSO, blocking the unphysi-
+cal region should improveparameter estimation accuracy
+significantly. As an indicator of this, we also show in Ta-
+ble V, the rmse obtainedby droppingthe trials wherethe
+estimate fell in the unphysical region. It is seen that the
+errors are reduced significantly, especially for the lower
+signal SNR value for which there is more scatter into the
+the unphysical region.
+A comparison of Table V with existing results [20]
+shows that the estimation errors due to PSO are con-
+sistent with grid-based searches.
+D. Computational cost
+The number of fitness function evaluations for each
+combination of signal SNR and location are shown in
+Table. VI. It is seen that for a signal SNR of 9.0, the
+maximumnumber ofevaluationsis aboutthe sameasthe
+meanin the purenoisecase(c.f., TableI). This reduction
+is consistent with the fact that a strong signal makes it
+easier for the swarm to find the global maximum.5 10 15 20 25 30 350.30.40.50.60.70.80.91
+τ0 (sec)τ1.5 (sec)
+(τ0=5.0,τ1.5=0.6)→ •
+(10.0,0.75) → +
+(16.2,0.762) → ×
+(20.0,0.9) → ∗
+FIG. 5: Estimation of parameter values for a signal SNR of
+8.0. The true locations of the signals are indicated by the ⋆
+marker and each of the markers, •, +,∗and×, indicates an
+estimated location corresponding to one of the true locatio ns.
+The association between the markers and the true signal loca -
+tions is indicated in the figure. For each true signal locatio n,
+the simulation consisted of 50 trials.
+TABLE V: Signal parameter estimation errors with PSO.
+Each entry in the table is of the form a(b), where aandbare
+the estimated root mean square errors (rmse) for τ0andτ1.5
+respectively (expressed as a percentage of the true paramet er
+value). Ineachrow, thetopandbottompairs ofnumbersrefer
+to mse obtained without and with the physical boundary cut
+respectively. All thenumbershavebeenroundedoff, giventh e
+expected precision from the 50 trials used per combination o f
+signal SNR and location.
+(τ0,τ1.5)
+= (5.0,0.6)(10.0,0.75) (16.2,0.762) (20 .0,0.9)
+SNR=9.02(13)
+2(12)1(11)
+0.5(6)0.3(6.0)
+0.2(4)1(11)
+0.2(4)
+8.044(13)
+10.5(10)32(16)
+1(10)12(16)
+0.3(5)11(17)
+12(10)
+TABLE VI: Computational cost of PSO on data containing a
+signal. For each combination of signal SNR and location, the
+mean number of fitness function evaluations, over 50 trials, is
+listed along with the maximum (superscript) and minimum
+(subscript). All numbers are in units of 104and rounded off
+to a single digit of precision.
+(τ0,τ1.5)
+= (5.0,0.6)(10.0,0.75) (16.0,0.762) (20 .0,0.9)
+SNR=9.0 4.45.2
+3.8 4.75.7
+4.1 4.85.6
+4.24.75.8
+4.2
+8.0 4.56.0
+3.8 4.76.5
+3.7 4.85.6
+4.14.85.9
+4.2
+7.0 4.76.4
+3.6 4.86.3
+3.9 4.96.2
+3.74.85.9
+3.8
+6.0 4.56.1
+3.3 4.87.0
+2.9 4.75.9
+3.24.76.0
+3.612
+For ground-based detectors, the dominant computa-
+tional cost comes from the pure noise case. Although our
+tuning procedure produced Np= 81 andNt= 80 as the
+optimal combination, there is statistical uncertainty in
+this result due to the finite and somewhat small number
+oftrialsused. Tomakeourestimateofthecomputational
+cost conservative, we use the combination Np= 121 and
+Nt= 80 instead for which all the performance measures
+aresignificantlybetter. FromTableI, thetypicalnumber
+of fitness evaluations required for the testbed considered
+here is∼7×104with a spread of about ±2×104. Of
+this, the termination criterion itself accounts for a fixed
+number, (Nt= 80)×(Np= 121) = 9680, of evaluations.
+A grid-based search provides a convenient perspective
+for evaluating the computational cost of PSO. According
+to [5], for 2PN waveform and initial LIGO noise PSD,
+the number of fitness function evaluations required in a
+single grid with a minimal match of 0.97 ( ⇒α= 0.03)
+is 1.1×104if the minimum mass used for constructing
+the template waveforms is 1 M⊙. In the current testbed,
+the search region in the mass plane (c.f., Fig. 2) is not
+the simple one considered in [5] although the minimum
+masses are similar. Additionally, [5] uses an analytic fit
+for the noise PSD that differs from the one used here.
+Ignoring these differences we find that the current imple-
+mentation of PSO requires about 7 times as many eval-
+uations, on the average, as a grid-based method.
+VI. CONCLUSIONS
+We applied PSO to the binary inspiral testbed where
+the main challenge was to locate the global maximum
+of a highly multi-modal fitness function. Such functions,
+with anunpredictable number ofextremahavingrandom
+locations and sizes, are typical in GW data analysis.
+In response to the questions posed at the beginning,
+the results obtained from simulations show clearly that:
+1. PSO is a viable method for signal detection and
+estimation in GW data analysis as it can success-
+fully handle the challenge of high multi-modality
+presented by such problems.
+2. Goodperformancewasachievedbytuningonlytwo
+out of the nine design variables involved in the
+method. Thus, PSO is a stochastic method that
+offers the possibility of having a small number of
+design variables in practice.
+3. The design variables were tuned using a systematic
+procedure that does not require any prior informa-
+tion about features of the fitness function. As such,
+the procedure should be widely applicable to other
+stochastic methods also.
+4. PSO is about 7 times more expensive than a grid-
+based search in the number of fitness function eval-
+uations required.The higher cost of PSO is not surprising since grid-based
+searches are usually more efficient than stochastic meth-
+ods in low-dimensional problems such as the one consid-
+ered here. The performance gain of stochastic methods
+appearsduetotheslowerriseintheircomputationalcost,
+with increase in dimensionality, compared to the expo-
+nential one of grid-based searches. Therefore, we expect
+PSO to be cheaper than grid-based searches in higher
+dimensional problems. However, a definitive answer re-
+quires an actual test on problems such as the inspiral
+of high mass spinning binary components or the LISA
+Galactic Binaryresolution problem [21]. The demonstra-
+tion in this paper that PSO can handle the more serious
+challenge of high multi-modality is the first step towards
+such future investigations.
+The computational cost of PSO may be significantly
+reduced by taking into account the physical boundary in
+parameter space (see Fig. 5). The current implementa-
+tion of PSO requires the search to extend over a large
+unphysical region. In fact, as far as the binary inspiral
+problem goes, we find this to be the most outstanding is-
+sue. We have tried the invisible walls condition with the
+curved physical boundary but find that the performance
+of PSO is negatively affected. Specifically, termination
+takes a much longer time and the probability of cluster-
+ing is significantly reduced. This behavior is attributable
+to the curved shape of the boundary allowing a signif-
+icantly larger number of particles to escape the search
+domain. Once outside, particles contribute nothing to
+the search and keep moving until they are pulled back.
+To solve this problem, it appears inevitable that the
+dynamical equations of PSO must be modified. For sig-
+nalsotherthanbinaryinspirals,suchasGalacticBinaries
+in the case of LISA, the nature of the boundary problem
+would be different and it may not be an issue in some
+applications.
+Finally, a comment about the use of Gaussian, station-
+ary noise in the testbed. We emphasize here that PSO
+is a method for finding the global maximum of a fitness
+function irrespective of what produces that peak, a gen-
+uine GW signal or an instrumental transient. Since, the
+implementation of PSO in this paper uses no prior infor-
+mation about features of the fitness function, it should
+find the peak regardless of its source. Thus, there should
+be no significant difference in the performance of PSO
+and a grid-based method even for non-stationary, non-
+Gaussian noise. In future work, we will verify this ex-
+plicitly by using non-GW signals in our simulations.
+Acknowledgments
+This work was supported by Research Corporation
+award CC6584. YW was supported by NSFC grant
+10773005 of China and NSF award HRD-0734800 to the
+Centre for Gravitational Wave Astronomy at the Univer-
+sity of Texas at Brownsville.13
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+[13] B P Abbott et al, Rep. Prog. Phys. 72, 076901 (2009).
+[14] The design sensitivity curve is available as a text file
+from the URL:
+http://www.ligo.caltech.edu/ ∼jzweizig/distribution/LSC Data/
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+nas and Propagation, 52, no. 2, 397 (2004).
+[19] R. R. Wilcox, Introduction to Robust Estimation and Hy-
+pothesis Testing , 2ndEdition (Academic Press, 2005).
+[20] R.Balasubramanian, B. S.SathyaprakashandS.V.Dhu-
+randhar, Phys. Rev. D 53, 3033 (1996).
+[21] S. Babak et al, Class. Quantum Grav. 25, 114037 (2008).
+Appendix A: Details of the signal waveform
+1. Chirp-time parameters
+The chirp-time parameters, {τa},a= 0,1,1.5,2, are
+given in terms of the masses, m1andm2≤m1, of the
+binary components (we use c=G= 1),
+τ0=5
+256M−5/3η−1(πfa)−8/3, (A1)
+τ1=5
+192µ(πfa)2/parenleftbigg743
+336+11
+4η/parenrightbigg
+, (A2)
+τ1.5=1
+8µ/parenleftbiggM
+π2fa5/parenrightbigg1/3
+, (A3)
+τ2=5
+128µ/parenleftbiggM
+π2fa2/parenrightbigg2/3/parenleftbigg3058673
+1016064+5429
+1008η+
+617
+144η2/parenrightbigg
+, (A4)whereM=m1+m2is the total mass of the compact
+binary,µ=m1m2/Mis the reduced mass and η=µ/M.
+Since all the chirp time parameters depend on m1and
+m2, only two of them are independent. It is convenient
+to chooseτ0andτ1.5asthe independent parameterssince
+Mandµcan be obtained algebraically from them,
+µ=1
+16f2a/parenleftbigg5
+4π4τ0τ2
+1.5/parenrightbigg1/3
+, (A5)
+M=5
+32faτ1.5
+π2τ0, (A6)
+allowingτ1andτ2to be obtained algebraically from τ0
+andτ1.5.
+2. The phase function
+In Eq. 3, the function ψ(f;θ) is given by,
+ψ(f;θ) =/summationdisplay
+i∈{0,1,1.5,2}αi(f)τi, (A7)
+α0(f) = 2πf−16πfa
+5+6πfa
+5/parenleftbiggf
+fa/parenrightbigg−5/3
+,(A8)
+α1(f) = 2πf−4πfa+2πfa/parenleftbiggf
+fa/parenrightbigg−1
+,(A9)
+α1.5(f) =−2πf+5πfa−3πfa/parenleftbiggf
+fa/parenrightbigg−2/3
+,(A10)
+α2(f) = 2πf−8πfa+6πfa/parenleftbiggf
+fa/parenrightbigg−1/3
+.(A11)
+The functions αa,a= 0,1,1.5,2 and thef−7/6factor in
+the amplitude of the signal (Eq. 3) can be pre-computed
+and stored, reducing the computational cost of generat-
+ing waveforms.
\ No newline at end of file
diff --git a/1001.0924.txt b/1001.0924.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d65cd6fda15c6a23db9f5440c717e11aa7a900b3
--- /dev/null
+++ b/1001.0924.txt
@@ -0,0 +1,1286 @@
+arXiv:1001.0924v1  [cond-mat.stat-mech]  6 Jan 2010,
+Bridging length and time scales in sheared demixing systems :
+from the Cahn-Hilliard to the Doi-Ohta model
+Asja Jeli´ c,∗Patrick Ilg, and Hans Christian ¨Ottinger
+Polymer Physics, Department of Materials,
+ETH Zurich, 8093 Zurich, Switzerland
+(Dated: October 30, 2018)
+Abstract
+We develop a systematic coarse-graining procedure which es tablishes the connection between
+models of mixtures of immiscible fluids at different length and time scales. We start from the
+Cahn-Hilliard model of spinodal decomposition in a binary fl uid mixture under flow from which we
+derive the coarse-grained description. The crucial step in this procedure is to identify the relevant
+coarse-grained variables and find the appropriate mapping w hich expresses them in terms of the
+more microscopic variables. In order to capture the physics of the Doi-Ohta level, we introduce the
+interfacial width as an additional variable at that level. I n this way, we account for the stretching
+of the interface under flow and derive analytically the conve ctive behavior of the relevant coarse-
+grained variables, which in the long wavelength limit recov ers the familiar phenomenological Doi-
+Ohta model. In addition, we obtain the expression for the int erfacial tension in terms of the
+Cahn-Hilliard parameters as a direct result of the develope d coarse-graining procedure. Finally,
+by analyzing the numerical results obtained from the simula tions on the Cahn-Hilliard level, we
+discuss that dissipative processes at the Doi-Ohta level ar e of the same origin as in the Cahn-
+Hilliard model. The way to estimate the interface relaxatio n times of the Doi-Ohta model from
+the underlying morphology dynamics simulated at the Cahn-H illiard level is established.
+PACS numbers: 05.70.Ln, 64.75.-g, 83.10.Bb, 47.50.Cd
+∗Present address: Universit´ e Pierre et Marie Curie - Paris VI, LPTH E UMR 7589, 4 Place Jussieu, 75252
+Paris Cedex 05, France; Electronic address: asja@lpthe.jussieu.f r
+1I. INTRODUCTION
+The problem of coarse graining in terms of bridging time and length sca les between
+microscopic and macroscopic levels of description is a crucial issue in t he physics of com-
+plex fluids, like polymer melts, colloids, liquid crystals and emulsions, out of equilibrium.
+The wide span of length and time scales is particularly evident in these s ystems, due to
+their internal structure leading to additional mesoscopic levels of d escription, intermediate
+between microscopic and macroscopic [1]. Many of the practical app lications of these flu-
+ids crucially depend on the evolution of their complex, multiphase morp hologies developed
+through equilibrium self-assembling or during non-equilibrium process ing of these systems.
+Some examples of such applications are food processing, membrane technology, encapsula-
+tion systems, drug delivery, coating, production of paint and cosm etics etc [1–4]. Therefore,
+the goal of connecting different levels of description of complex fluid s is of great importance.
+In the present work, this problem is approached by considering the phase separation of
+binary fluid mixtures subjected to a shear flow.
+When a binary (AB) fluid mixture is quenched from a high temperature homogeneous
+phase to an unstable region below the coexistence curve, it become s unstable with respect
+to long-wavelength fluctuations in the composition and starts to ph ase separate [5]. The
+interface between the A- and B-rich domains is initially very diffuse, bu t sharpens with time.
+In the later stages of phase separation, local equilibrium is achieved within each domain and
+the effects of interfacial energy become important. The domain pa ttern further coarsens in
+time, while the dynamics is governed by the minimization of the excess in terfacial energy of
+the system. The kinetics of this nonequilibrium phenomena is an area o f extensive research
+[5]. Especially the effects of a shear flow on the deformation and kinet ics of domain growth,
+and the corresponding rheological behavior, are challenging and te chnologically important
+topics [6]. The complexity of the problem lies in the presence of a comple x interface and
+its motions due to coagulation, rupture and deformation of domains , which significantly
+influences the macroscopic properties of a mixture. Although nume rous experimental [7–
+11], numerical [12–24] and analytical [14–17, 24–28] studies have b een done in order to
+understand the flow effects on the morphology and the dynamics in t his system, many
+questions still remain open, like the possibility of achieving a nonequilibr ium steady state
+[20–24].
+2Due to the complex morphology of this system, even in the case of tw o simple Newto-
+nian fluids, one can identify several length and time scales requiring d ifferent theoretical
+approaches and simulation methods. The standard mesoscopic mod el for understanding the
+dynamics of spinodal decomposition is based on the formulation of Ca hn and Hilliard [29].
+It describes the kinetics of the system in terms of the convective- diffusion equation for the
+order parameter - composition c, and the Navier-Stokes equation for the fluid velocity v
+(so-called ‘model H’, see e.g. [30]):
+∂c
+∂t+v·(∇c) =M∇2µc, (1.1a)
+ρ/parenleftbigg∂v
+∂t+(v·∇)v/parenrightbigg
+=η∇2v−∇p+µc∇c, (1.1b)
+whereMis the mobility coefficient, pthe pressure, ρthe mass density, and ηthe viscosity.
+The chemical potential µcis given as µc=δF[c]/δc, with the free energy functional of the
+Ginzburg-Landau form F[c] =/integraltext
+d3r/parenleftbig
+−(a/2)c2+(b/4)c4+(κ/2)|∇c|2/parenrightbig
+, (a,b >0), where
+the square brackets emphasize the occurence of functional inte grations. During the late
+stages of phase separation, the time evolution can be described by the change of the size
+andshapes ofthe domains. Dueto the dynamical scaling hypothesis , the system is described
+by a characteristic length scale L(t), which is related to the average domain size. By using
+dimensional arguments, one can estimate the size of the contribut ions from the specific
+terms in the equations (1.1), which leads to the distinction of the thr ee growth regimes
+[5, 31, 32]: diffusive L(t)∼(Mσt)1/3, viscous hydrodynamics L(t)∼σt/η, and inertial
+hydrodynamics L(t)∼(σt2/ρ)1/3, where σis the interfacial tension. Under shear flow
+the situation becomes more complex since the domains elongate along the flow direction
+and the system becomes anisotropic and can not be described by a s ingle length scale (see
+[14–17, 19, 20] and references therein).
+On the other hand, for understanding the rheological properties of multiphase systems,
+like polymer blends, under shear flow, Doi and Ohta suggested that the average information
+about the interface between the two phases is sufficient. Hence, t heir phenomenological
+model [28] focus on the interface, which is considered as a zero widt h surface embedded in
+the fluid. The presence of the interface is then represented thro ugh two state variables that
+describe the interfacial area per unit volume, Q, and its anisotropy, q, in a given flow field.
+They are defined in terms of the interface orientational distributio n function f(n), where n
+is a unit vector normal to the interface. This is a probability density o f finding the amount
+3of interface with the orientation n, so that the integral over all orientations gives the total
+amount of interface per unit volume. Therefore, for the state va riablesQandqone can
+write the following definitions
+Q=/integraldisplay
+f(n)d2n, (1.2a)
+q=/integraldisplay/parenleftbigg
+nn−1
+31/parenrightbigg
+f(n)d2n=nn−1
+3Q1, (1.2b)
+where1is the unit tensor, and/integraltext
+d2ndenotes an integration over the unit sphere. Introduc-
+tion of the distribution function and further averaging over all pos sible orientations imply
+the loss of information of the detailed morphology of the system. Th is coarse-grained model
+keeps only the information about the average amount of interfacia l area per unit volume and
+its orientation. However, the detailed information about the morph ology, i.e., the explicit
+interfacial position and orientation, is lost.
+The time evolution of the new configurational variables Qandqis determined by two
+factors: the external flow field which orients and enlarges the inte rface, and the interfacial
+tension which has the opposite effect and governs the relaxation of the interface. Therefore,
+the time evolution can be separated into two parts:
+∂Q
+∂t=∂Q
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+convection+∂Q
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+relaxation, (1.3a)
+∂q
+∂t=∂q
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+convection+∂q
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+relaxation. (1.3b)
+The convective part of the time evolution can be found by considerin g the convection be-
+havior of the unit vector nnormal to the interface under affine deformations and volume
+preservation assumptions. Then, the convection of the state va riables is derived using the
+above definitions (1.2)
+∂Q
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+convection=−∇·(Qv)−(∇v)T:q+2
+3Q1: (∇v)T, (1.4a)
+∂q
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+convection=−(∇v)·q−q·(∇v)T−1
+3Q˙γ−v·∇q+1
+31(∇v) :q
++/bracketleftbigg
+q+1
+9Q1/bracketrightbigg
+Tr(∇v)+nnnn: (∇v)T. (1.4b)
+where˙γ=∇v+(∇v)Tis the symmetrized velocity gradient tensor [33]. In the equation for
+theinterfacialshape(1.4b),thefourthmomentoftheinterface normalnappears, whichmust
+be expressed in terms of the state variables through an appropria te closure approximation
+4in order to obtain a self-contained set of time evolution equations. D oi and Ohta postulated
+the following closure approximation
+nnnn: (∇v)T=1
+Qnnnn: (∇v)T(1.5)
+and verified its accuracy. Using definition (1.2b), the convective tim e evolution of the
+anisotropy qtakes the form
+∂q
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+convection=−(∇v)·q−q·(∇v)T−1
+3Q˙γ−v·∇q+1
+31(∇v) :q
++/bracketleftbigg
+q+1
+9Q1/bracketrightbigg
+Tr(∇v)+1
+Q/parenleftbigg
+q+1
+3Q1/parenrightbigg/parenleftbigg
+q+1
+3Q1/parenrightbigg
+: (∇v).(1.6)
+As for the relaxation in the Doi-Ohta model, the relaxation of the sta te variables can be
+written in a generalized form
+∂Q
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+relaxation=−1
+τdo,1Q, (1.7a)
+∂q
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+relaxation=−1
+τdo,2q. (1.7b)
+In order to specify the empirical interface relaxation times τdo,1andτdo,2, assumptions on
+the appropriate interface relaxation mechanisms must be made. Fin ally, the full set of time
+evolutionequations fortheconfigurationalvariables Qandq, togetherwiththeequationsfor
+the hydrodynamic variables ρ,gandǫhas been formulated within the GENERIC formalism
+[33], which for the stress tensor gives
+Σ=−τ−σ/parenleftbigg
+q−2
+3Q1/parenrightbigg
+−p1, (1.8)
+whereτis the viscous stress tensor.
+Although very crude, the ability of the Doi-Ohta model to connect t he morphology of the
+system to its macroscopic behavior, through the above given set o f rheological constitutive
+equations, is very appealing for different theoretical and technolo gical purposes. The model
+and its variants have been used to analyze multiphase flow of polymer s, as well as simple
+fluids undergoing spinodal decomposition [7, 33–40]. However, the n eed for explicit assump-
+tions of the interfacial relaxation mechanisms and the appropriate relaxation times, which
+depend in an unknown manner on the underlying morphology, makes it difficult for wide
+and easy use. Therefore, in this paper we concentrate on relating the macroscopic Doi-Ohta
+model to the underlying morphology, described by the more detailed Cahn-Hilliard model.
+5The goal is to establish a systematic, thermodynamically guided meth od which determines
+the coarse-grained Doi-Ohta model of a phase separating binary fl uid from the more detailed
+Cahn-Hilliard model, in the region where the considered system can be described with both
+approaches. Toconnect thesetwodifferent levelsofdescription, theGeneralEquationforthe
+Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) framew ork is used [41–43].
+This approach allows to specify the mathematical structure which s hould be left invariant
+in the coarse-graining procedure and hence assures thermodyna mically consistent results.
+In the following, we first describe the GENERIC structure and form ulate the Cahn-
+Hilliardmodel within thisframework. The coarse-graining procedure which derives themore
+macroscopic Doi-Ohtamodel isthenpresented. Inparticular, wep resent some mathematical
+challenges which occur within the coarse-graining procedure, as we ll as the efficient way to
+extract the interface relaxation times of the Doi-Ohta level from t he underlying morphology.
+II. GENERIC FORMALISM
+The main points of the GENERIC framework of nonequilibrium thermod ynamics [41–43]
+can be summarized in the following way. In analogy to equilibrium thermo dynamics, one
+needs to choose a complete set of variables x, which describes the situation of interest to the
+desired detail. For a system out of equilibrium the time evolution of all t he relevant state
+variables can be divided into the reversible and theirreversible contributions
+˙x=˙x|rev+˙x|irrev. (2.1)
+Thesetwocontributionsareobtainedbyconsidering separatelyth eenergyEandtheentropy
+S— the two generators of reversible and irreversible dynamics, resp ectively. This can be
+explained by using the analogy with the classical Hamiltonian mechanics as following. Using
+the classical Hamiltonian mechanics, the reversible contribution of t he time evolution, ˙x|rev,
+is related to the energy gradient by way of a Poisson operator L, so that it is ˙x|rev=
+L·(δE/δx). The Poisson bracket associated to L, given by {A,B}= (δA/δx,L·δB/δx)
+with appropriate scalar product ( .,.), must be antisymmetric and satisfy Lebniz’ rule and
+the Jacobi identity, which are both features that capture the na ture of reversibility. The
+motivation for formulating the irreversible part of the dynamics com es from the reversible
+part. Forthereversibledynamics, theenergyplaysadistinctrole— itisaconservedquantity
+6for closed systems and it drives the reversible dynamics. Therefor e, for the irreversible
+dynamics the entropy presents an important quantity — it does not decrease for closed
+systems. FortheGENERICformulation, itisthenassumedthatthe irreversiblecontribution
+to the time evolution, ˙x|irrev, is driven by the entropy gradient, i.e., that it is of the form
+˙x|irrev=M·(δS/δx), withMa generalized friction matrix. This friction matrix contains
+transportcoefficientsandrelaxationtimesassociatedtothecorr espondingdissipative effects.
+It is required for Mto be (Onsager-Casimir) symmetric and the condition that it is positiv e
+semi-definite ensures that ˙S≥0 is fulfilled.
+From the above illustration, one concludes that the time evolution of xcan be expressed
+in terms of four “building blocks” E,S,LandMas
+˙x=L(x)·δE(x)
+δx+M(x)·δS(x)
+δx. (2.2)
+The two different contributions to the time evolution, reversible and irreversible, are not
+independent. They are related by the two complementary degener acy requirements
+L(x)·δS(x)
+δx=0,M(x)·δE(x)
+δx=0. (2.3)
+The first requirement expresses the reversible nature of the Lcontribution to the dynamics,
+demonstratingthefactthatthereversible dynamicscapturedin Ldoesnotaffecttheentropy
+functional. The second one expresses the conservation of the to tal energy of an isolated
+system by the irreversible contribution to the system dynamics cap tured in M.
+The presented GENERIC form of the time evolution equations for th e state variables x
+represents a mathematical structure which guarantees that th e chosen model is thermody-
+namically consistent. Moreover, by concentrating onthethermod ynamics building blocks E,
+S,LandMrather than on the time evolution equations in systematic coarse-g raining pro-
+cedures, the obtained coarse-grained equations are guarantee d to be also thermodynamically
+admissible.
+A. GENERIC formulation of the Cahn-Hilliard model
+Inordertodevelopasystematiccoarse-grainingprocedurewhich relatesthemoredetailed
+Cahn-Hilliard level to the more macroscopic Doi-Ohta level, we must fo rmulate the first one
+in the GENERIC structure. As a first step, we need to identify the lis t of independent
+7variables xwhich fully describe the considered thermodynamic system. In addit ion to the
+hydrodynamic fields, the mass density ρ, the momentum density g, and the internal energy
+densityǫ, we use the composition c– the mass fraction of one component – in order to
+describe the configuration of the system.
+Takinginto account theCahn-Hilliardfreeenergyofaphasesepara ting binarysystem, we
+assume the total energy and entropy of the system, in terms of t he variables x= (ρ,g,ǫ,c),
+to be
+E(1)(x) =/integraldisplay
+Vd3r/parenleftbigg1
+2g2
+ρ+ǫ+1
+2κE|∇c|2/parenrightbigg
+, (2.4a)
+S(1)(x) =/integraldisplay
+Vd3rs(ρ,ǫ,c), (2.4b)
+whereVis the total volume of the system. For simplicity, the gradient-squa red term, which
+describes the contribution due to the presence of the interface, is included only in the energy
+functional (2.4a), although it can be separated between both, en ergy and entropy [44, 45].
+The actual functional form of the entropy density s(ρ,ǫ,c) is not needed for the derivation
+of the Cahn-Hilliard time evolution equations in the GENERIC form. How ever, we use
+the assumption of local equilibrium, i.e., s(ρ,ǫ,c) has the same form as an equilibrium
+system at the corresponding state point. In the numerical simulat ions reported in this
+paper, we have used the popular c4-structure for the uniform part of the entropy density,
+i.e.,s(ρ,ǫ,c) =s0(ρ,ǫ)−ac2/2+bc4/4, which presents the entropic contribution to the bulk
+free energy density [14, 15, 19, 20, 22–24]. Other possible functio nal forms include the one
+for the van der Waals systems, for example Eq. 4 in [44]. The function al derivatives of the
+above energy and entropy functionals with respect to the indepen dent variables xare given
+with
+δE(1)
+δx=/parenleftbigg
+−1
+2v2,v,1,−κE∇2c/parenrightbiggT
+, (2.5a)
+δS(1)
+δx=/parenleftbigg∂s(ρ,ǫ,c)
+∂ρ,0,1
+T,∂s(ρ,ǫ,c)
+∂c/parenrightbiggT
+, (2.5b)
+where we omitted the position dependence for simplicity.
+The Poisson operator L(1)determines the reversible contributions to the full evolution
+equations of the variables x, and hence models the convective behavior. For our particular
+choice of variables and in view of the functional derivatives (2.5) one arrives at the following
+8expression for the Poisson operator
+L(1)(r,r′) =
+0ρ(r′)∂δ
+∂r′0 0
+−∂δ
+∂rρ(r)L(1)
+ggL(1)
+gǫ−(∇c(r))δ
+0 L(1)
+ǫg 0 0
+0−(∇c(r))δ0 0
+, (2.6a)
+with
+L(1)
+gg(r,r′) =g(r′)∂δ
+∂r′−∂δ
+∂rg(r), (2.6b)
+L(1)
+gǫ(r,r′) =−∂δ
+∂rǫ(r)−∂δ
+∂rp(r′), (2.6c)
+L(1)
+ǫg(r,r′) =ǫ(r′)∂δ
+∂r′+p(r)∂δ
+∂r′, (2.6d)
+whereδ=δ(r−r′) is Dirac’s δ-function. In addition to the usual hydrodynamic part
+of the Poisson matrix for the variables ρ,g,ǫ, element L(1)
+cgdictates the convection of
+the configurational variable c, which is a scalar, so the entry in the Poisson operator is
+as expected by [41]. The antisymmetry of the Poisson matrix (2.6) a nd the degeneracy
+requirement are satisfied by construction, and the Jacobi identit y holds. At this point we
+mention that, in general, the symbol “ ·” in (2.2) implies not only summation over discrete
+indices. If field variables are involved the operators LandMare written in terms of two
+space arguments ( r,r′), and an integration over r′must be performed when multiplied with
+a function of r′from the right. However, in the case of the field equations being loca l, one
+can express LandMin terms of a single variable ronly [41], and no integration is implied
+when these operators are multiplied from the right. Such single varia ble notation is used
+below for the friction matrix M(1).
+The irreversible effects in the Cahn-Hilliard model for a binary fluid und er flow arise due
+to viscosity andheat conduction, aswell as diffusion. Thus, to incor porate them throughthe
+friction matrix M(1), one can write it as the sum of two matrices M(1)=M(1),hyd+M(1),dif,
+whereM(1),hydis the usual hydrodynamic friction matrix, and M(1),difcontains all the
+transport coefficients related to diffusion of mass [41, 46]. The hydr odynamic part of the
+9friction matrix is given in the r-notation as
+M(1),hyd(r) =
+0 0 0 0
+0M(1),hyd
+ggM(1),hyd
+gǫ0
+0M(1),hyd
+ǫgM(1),hyd
+ǫǫ0
+0 0 0 0
+, (2.7a)
+with
+M(1),hyd
+gg(r) =−(∇ηT∇+1∇·ηT∇)T−∇ˆκT∇, (2.7b)
+M(1),hyd
+gǫ(r) =∇·ηT˙γ+∇ˆκT
+2tr˙γ, (2.7c)
+M(1),hyd
+ǫg(r) =−ηT˙γ·∇−ˆκT
+2tr˙γ∇, (2.7d)
+M(1),hyd
+ǫǫ(r) =ηT
+2˙γ:˙γ+ˆκT
+4(tr˙γ)2−∇·λqT2∇, (2.7e)
+with all derivative operators acting on everything to the right, i.e., a lso on functions multi-
+plied to the right side of the operator M, and with with the transport coefficient ˆ κ, being
+a combination of the dilatational viscosity κand the viscosity η, ˆκ=κ−2
+3η, andλqthe
+thermal conductivity [41].
+IfM(r) is a position dependent mobility coefficient, the diffusion part of the f riction
+matrix in the r-notation is
+M(1),dif(r) =
+0 0 0 0
+0 0 0 0
+0 0M(1),dif
+ǫǫM(1),dif
+ǫc
+0 0M(1),dif
+cǫM(1),dif
+cc
+, (2.8a)
+where the cc-element is describing the diffusion of c, so that
+M(1),dif
+cc(r) =−∇·MT∇, (2.8b)
+and the other three elements areobtained from the symmetry and the degeneracy conditions
+M(1),dif
+cǫ(r) =−∇·MT∇κE/parenleftbig
+∇2c/parenrightbig
+, (2.8c)
+M(1),dif
+ǫc(r) =−κE/parenleftbig
+∇2c/parenrightbig
+∇·MT∇, (2.8d)
+M(1),dif
+ǫǫ(r) =−κE/parenleftbig
+∇2c/parenrightbig
+∇·MT∇κE/parenleftbig
+∇2c/parenrightbig
+. (2.8e)
+Here again are all variables functions of r, and all derivative operators ∇ ≡∂
+∂racting on
+everything to the right, except when they are in brackets and bou nded to act on c. The
+10symmetry and the degeneracy requirement for the matrix M(1),difare satisfied by construc-
+tion.
+Finally, by inserting the above building blocks E,S,LandMinto the GENERIC
+equation (2.2), the full set of time evolution equations takes the fo rm
+∂ρ
+∂t=−∇·(ρv), (2.9a)
+∂g
+∂t=−∇·(vg)−∇·(Π+τ), (2.9b)
+∂ǫ
+∂t=−∇·(ǫv)−p∇·v−τ: (∇v)T−∇·jq
++κE/parenleftbig
+∇2c/parenrightbig
+∇·/bracketleftbigg
+MT∇/parenleftbigg
+−∂s(ρ,ǫ,c)
+∂c−κE
+T∇2c/parenrightbigg/bracketrightbigg
+, (2.9c)
+∂c
+∂t=−v·(∇c)+∇·/bracketleftbigg
+MT∇/parenleftbigg
+−∂s(ρ,ǫ,c)
+∂c−κE
+T∇2c/parenrightbigg/bracketrightbigg
+, (2.9d)
+whereΠ=/parenleftbig
+p−1
+2κE|∇c|2/parenrightbig
+1+κE(∇c)(∇c) is the pressure tensor, τ=η˙γ−ˆκ(∇·v)1is
+the viscous stress tensor, and jq=−λq∇Trepresents the conductive flow of internal energy.
+For an isothermal, incompressible flow, these time evolution equation s take the form of the
+standard Cahn-Hilliard model. Similar set of hydrodynamic equations f or phase separating
+fluid mixtures has been derived from an underlying microscopic dynam ics [46].
+III. COARSE GRAINING FROM CAHN-HILLIARD TO DOI-OHTA LEVEL
+A specific feature of the GENERIC formalism is that it is applicable at diff erent levels
+of description associated with different length and time scales. Each level of description is
+described by an appropriate set of state variables and building block s. The mathematical
+structure of GENERIC offers a possibility to perform coarse grainin g by focusing not on the
+time evolution equations, but on the building blocks E,S,LandM, see [47]. The transition
+between a more detailed (Cahn-Hilliard) level 1 to a less detailed (Doi-O hta) level 2, which
+involve the derivation of the building blocks from level 1 to level 2, can then be performed
+by systematic procedures [48]. The possibility to perform coarse-g raining by focusing on the
+basic building blocks guarantees that the thermodynamic structur e of the problem will be
+preserved.
+A systematic coarse-graining procedure is based on the idea of the separation of time
+scales. In this way, it is assumed that there exists a division of the “f ast” and “slow”
+11degrees of freedom at the more microscopic level 1 according to th eir relaxation times. By
+means of the projection-operator technique, one can then elimina te these “fast” degrees
+of freedom from the time evolution equations for the “slow” variable s. The latter are then
+associated to themacroscopic variables at level 2. The crucial ste ps inthis general procedure
+are to identify the proper mapping of the variables of one level to an other, Π( x) :x−→y,
+which average in a non-equilibrium ensemble ρy(x) is the new coarse-grained variable y
+y=/angbracketleftig
+Π(x)/angbracketrightig
+y=/integraldisplay
+Dxρy[x]Π(x). (3.1)
+A. Mappings and ensemble
+For coarse graining from Cahn-Hilliard level, the relevant set of varia bles at the coarse-
+grained level y= (ρ,g,ǫ,Q,q) is motivated by the phenomenological Doi-Ohta model
+[28, 33]. We assume that the hydrodynamic fields ρ,g, andǫare smooth on the more
+microscopic Cahn-Hilliard length scale. Then the mappings Π ρ,Πg, and Π ǫ, simply pick
+out the hydrodynamic fields, while we need to determine the relations hip of mappings Π Q
+andΠqtotheunderlying configuration. Wedo thisby following thephysical m eaning rather
+than theexact definitions (1.2) of the Doi-Ohtavariables. This is bec ause, first, the interface
+orientational distribution function f(n) is not given at the Cahn-Hilliard level, and, second,
+due to the difference in modeling of the interface. While in the Doi-Ohta model the interface
+us assumed to be sharp, in the diffuse-interface theories, like Cahn -Hilliard one, the interface
+is given through continuous variations of the composition c. Therefore, for determining the
+average interfacial area per unit volume Qand its orientation qone must use the appropri-
+ate combination of the gradients of the composition ∇c(r), and perform ensemble averaging
+which incorporates “smoothing” over a certain volume. We introduc e a smoothing function
+χ(r−r′), which averages the observable over a certain volume v(r) around position r, and
+satisfies the normalization condition/integraltext
+Vd3r′χ(r−r′) = 1. To obtain the variables which
+would correspond to the Doi-Ohta averaged interfacial area and it s orientation, volume v(r)
+must comprise many droplets for good statistics. That means that the length scale of the
+smoothing volume v(r) – smoothing length a– must satisfy
+ξ≪L(t)≪a≪Λ, (3.2)
+12whereξis a length of the size of the interfacial width, L(t) is the growing characteristic
+domain size, and Λ is the size of the system. The first part of the ineq uality (3.2), ξ≪L(t),
+denotes an important fact that we perform the coarse-graining p rocedure only during the
+late stages of phase separation, i.e., when the well defined domains a re formed and the
+coarsening can be described by the growth laws given in Section I.
+Although one can express the coarse-grained variables Qandqusing gradient of the com-
+position∇c(r) and average the expression as discussed above, there are still s ome questions
+which are not solved. As already mentioned, contrary to the pheno menological Doi-Ohta
+model, in the coarse-grained model, the interface not only has a finit e width, but it does
+not behave like a zero-width mathematical surface. Rather it defo rms under the applied
+flow and its stretching must be captured by the new variables, which is not the case in the
+phenomenological Doi-Ohta model. There are different ways to solve these problems. One
+way would be that instead of using the Cahn-Hilliard Poisson operator L(1), Eq.(2.6), one
+needs to use its modification with the appropriate constraint which w ould account for the
+stretching of the interface. Another possibility would be to make ce rtain modifications to
+the original Cahn-Hilliard model in such a way to make the interfacial w idth, as well as in-
+terfacial tension, fixed. However, maybe the most straightforw ard way to solve thepresented
+problems is to account for the finite interfacial width and its stretc hing by introducing an
+additional variable into our coarse-grained Doi-Ohta model. This add itional variable would
+be the average interfacial width l. This means that for the configurational variables of the
+coarse-grained Doi-Ohta model we use new variables {P,p,l}, which are obtained as ensem-
+ble averages of the suitable mappings Π P,Πpand Π l. The original Doi-Ohta model and
+the appropriate convective behavior of its variables are obtained b y transformation of the
+configurational variables {P(r),p(r),l(r)}to{Q(r),q(r),l(r)}as
+Q(r) =l(r)P(r),q(r) =l(r)p(r). (3.3)
+We choose the appropriate mappings for the new variables, having in mind that the
+variables Qandq, obtained from {P,p,l}as (3.3), should represent the average amount of
+interface per unit volume and its orientation, both of dimension m−1. Then, for suitable
+13mappings for the new variables we propose
+ΠP[c](r) =/integraldisplay
+Vd3r′|∇c(r′)|2χ(r−r′), (3.4a)
+Πp[c](r) =/integraldisplay
+Vd3r′/parenleftbigg
+(∇c(r′))(∇c(r′))−1
+3|∇c(r′)|21/parenrightbigg
+χ(r−r′),(3.4b)
+Πl[c](r) =/integraldisplay
+Vd3r′|∇c(r′)|χ(r−r′)
+/integraldisplay
+Vd3r′|∇c(r′)|2χ(r−r′). (3.4c)
+The next step in the coarse-graining procedure is the choice of the nonequilibrium en-
+semble. The natural choice is for it to be of the mixed ensemble due to the choice of the
+macroscopic variables [41]. Since the hydrodynamic fields are simply ma pped from the
+Cahn-Hilliard to the Doi-Ohta level, the appropriate probability distrib ution function is of
+the generalized microcanonical type. For the configurational var iables, on the other hand,
+we choose a generalized canonical ensemble with the corresponding Lagrange multipliers λQ,
+λqandλl. The total probability measure ρy[x] then takes the form
+ρy[x] =δ(Πρ−ρ)δ(Πg−g)δ(Πǫ−ǫ)ρ(Q,q)[c], (3.5a)
+ρ(Q,q)[c] =Ω1[x]
+N[y]exp/parenleftbigg
+−/integraldisplay
+Vd3rλ(r) :Π[c](r)/parenrightbigg
+, (3.5b)
+where the normalization factor N[y] is
+N[y] =/integraldisplay
+DcΩ1[x]exp/parenleftbigg
+−/integraldisplay
+Vd3rλ(r) :Π[c](r)/parenrightbigg
+, (3.5c)
+and Ω 1[x] = exp/parenleftbig
+S(1)(x)/kB/parenrightbig
+. The projection operators Π Q,Πq, and Π lare, for simplicity,
+denoted by Πin the above equations. The Lagrange multipliers λQ,λq, andλl, denoted by
+λ, are determined by the values of the slow variables y=/an}b∇acketle{tx/an}b∇acket∇i}htwhere the average is performed
+according to (3.1). Their interpretation and the identification of th e proper values for the
+situation of interest is difficult and requires dynamic material informa tion [48, 49]. However,
+for the results presented in this paper, the exact form of the non equilibrium ensemble will
+not be needed.
+14B. Energy
+The energy of the coarse-grained Doi-Ohta level is obtained by ave raging the energy of
+the more detailed Cahn-Hilliard level,
+E(2)(y) =/integraldisplay
+Vd3r/integraldisplay
+Dxρy[x]/integraldisplay
+Vd3r′χ(r−r′)/parenleftigg
+1
+2g(r′)2
+ρ(r′)+ǫ(r′)+1
+2κE|∇c(r′)|2/parenrightigg
+.(3.6)
+Sinceweassumethatthehydrodynamicfields ρ,g,andǫaresmoothonthemoremicroscopic
+Cahn-Hilliard length scale, the energy expression takes the form
+E(2)(y) =/integraldisplay
+Vd3r/parenleftigg
+1
+2g(r)2
+ρ(r)+ǫ(r)/parenrightigg
++/integraldisplay
+Vd3rκE
+2l(r)Q(r), (3.7)
+where in the last term on the right hand side of the equation, we have recognized the new
+variables Q(r) andl(r), determined by the mapping (3.4a) and (3.4c). The last term on
+the right hand side of the above equation describes the energy den sity due to the presence
+of the interface which is proportional to the interfacial tension σ. We therefore obtain the
+following well-known expression for the interfacial tension
+σ(r) =κE
+2l(r), (3.8)
+which here follows directly as the first result of the performed coar se-graining procedure.
+C. Entropy
+The coarse-grained entropy for the case of the generalized mixed ensemble ρy[x] is ob-
+tained from
+S(2)(y) =/integraldisplay
+Dcρ(Q,q)[c]/bracketleftbig
+S(1)(x)−kBlnρ(Q,q)[c]/bracketrightbig
+, (3.9)
+whereS(1)(x) is given by (2.4b). The additional entropy in the coarse-grained ex pression,
+beside a simple ensemble average of the entropy from the more deta iled level 1, is associated
+with the passage from the more microscopic configurational variab lec, to the more macro-
+scopic variables Q,q, andl, while the hydrodynamic variables are taken to the coarser level
+without affecting the entropy. The additional entropy takes into a ccount all the microstates
+with the composition c(r) consistent with the more coarse-grained state given with Q(r),
+q(r), andl(r). The functional derivative of the coarse-grained entropy is
+δS(2)(y)
+δy=/parenleftbigg∂s(ρ,ǫ,c)
+∂ρ,0,1
+T(r),λQ(r),λq(r),λl(r)/parenrightbiggT
+. (3.10)
+15As discussed above, there exists a systematic procedure to obta in the Lagrange multipliers
+from the thermodynamically guided simulations [48]. However, their de termination is a
+difficult task and for the calculation of the time friction matrix element s in section IIIE we
+will assume λ=0.
+D. Poisson operator
+The general expression for the coarse-grained Poisson operato r is given by
+L(2)(y) =/angbracketleftigg/parenleftbiggδΠ(x)
+δx/parenrightbiggT
+·L(1)(x)·/parenleftbiggδΠ(x)
+δx/parenrightbigg/angbracketrightigg
+y. (3.11)
+which for coarse graining from the Cahn-Hilliard to the Doi-Ohta level takes the form
+L(2)
+ij(r1,r2) =/integraldisplay
+Dxρy[x]/integraldisplay
+Vd3r′
+1/integraldisplay
+Vd3r′
+2χ(r1−r′
+1)χ(r2−r′
+2)
+×/integraldisplay
+Vd3r3/integraldisplay
+Vd3r4δΠi[x](r′
+1)
+δxk(r3)L(1)
+kl(r3,r4)δΠj[x](r′
+2)
+δxl(r4),(3.12)
+where the mappings Πare given by (3.4), and the elements of the Poisson operator L(1)
+by (2.6). To understand the above expression, note that the inte grations over r3andr4
+come from the contraction of the operator L(1)withδΠ/δx, which includes both matrix
+multiplication and integration over the position label. The obtained qua ntity must then be
+averaged in order to obtain the expression at the Doi-Ohta level. Th erefore, the integrations
+overr′
+1andr′
+2comefromthespatialsmoothing ofthisquantity, since theensemb le averaging
+also implies smoothing in space, as noted in the previous subsection.
+From the elements of L(1)in (2.6) and the mappings Πin (3.4), we conclude that the
+Poisson operator L(2)has the following form
+L(2)(r1,r2) =
+0L(2)
+ρg0 0 0 0
+L(2)
+gρL(2)
+ggL(2)
+gǫL(2)
+gPL(2)
+gpL(2)
+gl
+0L(2)
+ǫg0 0 0 0
+0L(2)
+Pg0 0 0 0
+0L(2)
+pg0 0 0 0
+0L(2)
+lg0 0 0 0
+. (3.13)
+16Intheanalyticalderivationofthecoarse-grainedPoissonoperat or,severalapproximations
+that are related to the difference in length scales (3.2) are used. Th e crossover in length scale
+from the more microscopic Cahn-Hilliard level to the more macroscop ic Doi-Ohta level is
+performed through the smoothing function χ(r1−r2). While the details of the derivation for
+some of the elements of L(2)are given in the Appendix A, here we give their final expressions
+L(2)
+ρg(r1,r2) =ρ(r2)∂χ
+∂r2. (3.14a)
+L(2)
+gg(r1,r2) =g(r2)∂χ
+∂r2−∂χ
+∂r1g(r1), (3.14b)
+L(2)
+ǫg(r1,r2) =ǫ(r2)∂χ
+∂r2+p(r1)∂χ
+∂r2, (3.14c)
+L(2)
+Pg,β(r1,r2) =∂
+∂r2α/bracketleftbigg/parenleftbigg
+2pαβ(r2)−1
+3P(r2)δαβ/parenrightbigg
+χ/bracketrightbigg
++P(r2)∂χ
+∂r2β,(3.14d)
+L(2)
+pg,αβγ(r1,r2) =∂
+∂r2α/bracketleftbigg/parenleftbigg
+pβγ(r2)+1
+3P(r2)δβγ/parenrightbigg
+χ/bracketrightbigg
++∂
+∂r2β/bracketleftbigg/parenleftbigg
+pαγ(r2)+1
+3P(r2)δαγ/parenrightbigg
+χ/bracketrightbigg
+−2
+3δαβ∂
+∂r2ν/bracketleftbigg/parenleftbigg
+pνγ(r2)+1
+3P(r2)δνγ/parenrightbigg
+χ/bracketrightbigg
+−/parenleftbigg∂
+∂r2γpαβ(r2)/parenrightbigg
+χ, (3.14e)
+L(2)
+lg,β(r1,r2) =−l(r1)
+P(r1)pαβ(r1)∂χ
+∂r2α−1
+3l(r1)∂χ
+∂r2β−∂l(r1)
+∂r1βχ. (3.14f)
+whereχ=χ(r1−r2). With this, we obtained all the elements of the coarse-grained Pois son
+operator L(2), since the elements L(2)
+gP,L(2)
+gp, andL(2)
+glare obtained from the antisymmetry
+condition for L(2). We note the occurence of the smoothing function χ(r1−r2) in the
+Poisson operator L(2). Indeed, when looking at the elements of L(2)which involve only
+the hydrodynamic fields ρ,g, andǫ, we see that they differ from the appropriate elements
+ofL(1)in (2.6) only in locality, i.e., instead of the Dirac delta function, we rathe r have a
+smoothing function χ(r1−r2). While the original Cahn-Hilliard model is local in space,
+the coarse-grained model, instead, takes into account the whole v olume element v(r1) of the
+smoothing function χ(r1−r2). By assumption of the length scales comparison (3.2), this
+smoothing function behaves simply as a delta function on the Doi-Oht a level due to the
+difference in length scales. Alternatively, the expressions for the e lements of the Poisson
+matrix (3.14d)-(3.14f) could be also obtained based only on the math ematical character of
+17the variables P,p, andl. Similar to the original Doi-Ohta derivation, one would consider
+the transformation properties of the vector ∇cand the mappings (3.4), in order to infer the
+convective behavior.
+The Poisson operator L(2)determines the convective behavior of the state variables,
+which can then be compared to the original Doi-Ohta model. The reve rsible time evolution
+equations for the variables {ρ,g,ǫ,P,p,l}are obtained as
+∂yi(r1)
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+rev=/integraldisplay
+Vd3r2L(2)
+ij(r1,r2)δE(2)(y)
+δyj(r2), (3.15)
+under thepreviously discussed assumptionthatthesmoothingfun ctionχ(r1−r2)isactingas
+aDiracdeltafunction δ(r1−r2)attheDoi-Ohtalevelofdescription. Then, bytransformation
+of the variables, the full set of the reversible time evolution equatio ns for the set of state
+variables {ρ,g,ǫ,Q,q,l}takes the form
+∂ρ
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+rev=−∇·(ρv), (3.16a)
+∂g
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+rev=−∇·(vg)−∇p−∇·Γ/parenleftbigg
+q−2
+3Q1/parenrightbigg
+−Q∇Γ, (3.16b)
+∂ǫ
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+rev=−∇·(ǫv)−p(∇·v), (3.16c)
+∂Q
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+rev=−∇·(Qv)−(∇v)T:q+2
+3Q(∇·v), (3.16d)
+∂q
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+rev=−(∇v)·q−q·(∇v)T−1
+3Q˙γ−v·∇q+1
+31(∇v) :q
++1
+3Q1(∇·v)+1
+Q/parenleftbigg
+q+1
+3Q1/parenrightbigg/parenleftbigg
+q+1
+3Q1/parenrightbigg
+: (∇v), (3.16e)
+∂l
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+rev=−∇·(lv)+l
+Qq: (∇v)T+4
+3l(∇·v). (3.16f)
+These equations correspond to the convective part of the time ev olution equations of the
+Doi-Ohta model expressed in the GENERIC formalism [33]. Indeed, w hen the closure ap-
+proximation (1.5) for the fourth moment of the interfacial normal vectornnnnis used, one
+obtains the exact equations as above (see Eqs. (1.4a) and (1.6)). Furthermore, since the Ja-
+cobi identity of the starting L(1)operator is fulfilled and the closure approximation we used
+corresponds to the one made by Doi and Ohta [28], and examined in [3 3], we assume that
+the Jacobi identity of the derived Poisson operator L(2)is valid. However, this assumption
+might be questioned due to the approximations used in its derivation a nd which take into
+account different length scales (see Appendix A) .
+18E. Irreversible dynamics
+In this section we turn to the dynamic material properties which are contained in a
+friction matrix. There are two contributions to the coarse-graine d friction operator, M(2)=
+M(2)′+M(2)′′. The first contribution, M(2)′, is obtained directly by averaging the elements
+of the friction matrix M(1), in the same way as for the Poisson operator in (3.11). This
+direct contribution describes dissipative effects that are carried o n from a more microscopic
+level of description. A second contribution to the coarse-grained friction operator, M(2)′′,
+results from the processes that are slower than the characteris tic time scale of the Cahn-
+Hilliard level (given by the diffusion time tD) but are fast compared to the processes at the
+Doi-Ohta level of description (with the time scale of the interface re laxation time τdo,1,2).
+This contribution presents an important feature of the coarse-g raining procedure, since it
+capturestheadditionaldissipationarisingfromtheadditionalproc esseswhichcanbetreated
+as fluctuations on the time scale of the Doi-Ohta level. It can be evalu ated from the Green-
+Kubo formula
+M(2)′′
+jk(y) =1
+kB/integraldisplayτs
+0dt/angbracketleftig
+˙Πf
+j(x(t))˙Πf
+k(x(0))/angbracketrightig
+y, (3.17)
+whereτsseparates the times scales between the fast and slow variables, ˙Πfis the rapidly
+fluctuating part of the time derivative of the microscopic expressio ns for the slow variables
+y, and the average is over the atomistic trajectories consistent wit h the coarse-grained state
+yatt= 0 and evolved according to the microscopic dynamics to the time t[41].
+In order to identify the main dissipative contribution at the coarse- grained level and its
+origin, one has to understand all the processes occurring in a phas e separation of a binary
+fluid under shear flow. In particular, through numerical simulations we tried to understand
+if there exist any processes in this system that can be considered a s fast compared to the
+time scale of the Doi-Ohta level, and to estimate their contribution to dissipation at the
+Doi-Ohta level through the Green-Kubo part of the friction matrix M(2)′′, Eq. (3.17). We
+performed simulations of the Cahn-Hilliard equation (2.9d) for a binar y mixture of volume
+fractionφ= 0.3 which was subjected to a shear flow at time t0= 200tDafter the start of a
+phase separating process (when the well-defined droplets were fo rmed). We have used the
+c4-structure for the entropy density s, as given in subsection IIA. We used a 1-dimensional
+equilibrium profilebetween thetwo coexisting bulkphases, withtheco mpositionfield c(x) =
+/radicalbig
+a/btanh(x/2ξ), in order to define the length and time scale using ξ=/radicalbig
+κE/aandtD=
+19FIG. 1: (Color online) Timeevolution ofthecoarse-grained Doi-Ohta variables forabinarymixture
+withφ= 0.3, subjected to the shear flow at time t0= 200tDafter the beginning of a phase
+separating process: interfacial width l(upper left), average amount of interface per unit volume Q
+(lower left), and shear viscosity, η=−qxy/˙γas a function of strain γ= ˙γt(right). The results for
+the shear rates ˙ γtD= 0.05 and ˙γtD= 0.06 are presented.
+ξ2/Ma. Hydrodynamic interaction was not included and, therefore, coar sening of domains
+proceeded governed by the diffusion process, i.e., through the eva poration-condensation
+mechanism and the growth law L(t)∼t1/3[5, 31, 32]. Once the well defined interfaces were
+formed, we looked for the fast processes permanently at work (lik e deformation of interface
+shapesthroughfastcoagulationandbreak-up, etc.), which would giverisetonewirreversible
+dynamics, going beyond the diffusion and hydrodynamic ones.
+The time evolution of the coarse-grained Doi-Ohta variables in Fig.1 is c omputed by a
+simple use of mappings (3.4). The shear viscosity, which arises from e xcess shear stress due
+to the domain interfaces, is related to the anisotropy element qxythroughη=−qxy/˙γ. From
+the time evolution of the average interfacial width l, one can see that this new structural
+variable relaxes very quickly and is afterwards only affected by the fl ow through the convec-
+tive behavior, Eq.(3.16f). This numerical analysis shows that the co arse-grained Doi-Ohta
+variables fluctuate in time (with the time scale of ∼1/˙γ) around their average values due
+to the competition between the flow field and the ordering mechanism s, see Fig.1. This can
+be also seen in the time evolution of the morphology. The snapshots o f the morphology,
+presented in Fig.2, show that at first domains are elongated due to t he shear flow, which
+tends to orient them towards the shear direction. The elongation is then followed by coagu-
+20FIG. 2: Configurations of a binary fluid with volume fraction φ= 0.3, simulated on a 128 ×128
+lattice during phase separation under shear flow at different v alues of strain γ= ˙γt. Shear flow
+was applied after initial time of t0= 200tD, with the shear rate ˙ γtD= 0.05. The snapshots were
+taken until the maximum strain γ= 12. The gray scale is used in order to capture the processes
+of interface deformation.
+lation, break-up, shape relaxation, as well as the evaporation-co ndensation of the droplets,
+which are exactly the mechanisms of interfacial relaxation propose d by Doi and Ohta [28],
+and Lee and Park [34]. Similar interfacial relaxation processes have been identified in other
+works which report simulations of the Cahn-Hilliard model with and with out hydrodynamic
+interaction [12, 14–18, 20–24]. The morphology time evolution shows no new processes that
+are fast compared to the Doi-Ohta level of description, and which w ould emerge as dissi-
+pation on the coarse-grained level through the Green-Kubo form ula (3.17). Moreover, the
+time evolution curves of the Doi-Ohta variables in Fig. 1 are smooth wit hout the presence
+of thermal fluctuations (which were not incorporated into our ana lysis). Since the above
+described mechanisms of interfaces relaxation are also the ones co nsidered for the analysis
+of different growth regimes in the late stage kinetics of phase separ ation (see Section I), we
+conclude that the main dissipative contribution at the coarse-grain ed level arises through
+the averaging of the already present dissipative terms at the Cahn -Hilliard level, M(2)′.
+The relaxation of the Doi-Ohta variable y={ρ,g,ǫ,P,p,l}could be expressed from the
+irreversible part of the GENERIC time evolution equation (2.2) as
+∂yi(r1)
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+irrev=/integraldisplay
+Vd3r2M(2)′
+ij(r1,r2)δS(2)(y)
+δyj(r2). (3.18)
+In order to derive the coarse-grained friction matrix M(2)′, we note that the Cahn-Hilliard
+friction matrix M(1)consists of a hydrodynamic and a diffusive part, M(1)=M(1),hyd+
+M(1),dif, given in subsection IIA. Therefore M(2)′will also comprise the hydrodynamic
+21part and a part arising from the diffusion. Since the hydrodynamic va riables are simply
+taken over from the Cahn-Hilliard level, the appropriate hydrodyna mic elements of the new
+friction matrix are the same as in (2.7). The diffusive part, on the oth er hand, is related to
+the configurational variable c, and the ensemble averaging must be performed in a similar
+way as for the Poisson operator in Section IIID. For the analytical derivation of the coarse-
+grained friction matrix M(2)′,dif, in analogy to Eq. (3.11), one starts from
+M(2)′,dif(y) =/angbracketleftigg/parenleftbiggδΠ(x)
+δx/parenrightbiggT
+·M(1),dif(x)·/parenleftbiggδΠ(x)
+δx/parenrightbigg/angbracketrightigg
+y, (3.19)
+and then employs similar procedure and approximations as in the deriv ation of the elements
+of the Poisson matrix L(2)(see Appendix A). For example, the relaxation of the Doi-Ohta
+variableQcould be expressed as an irreversible part of the GENERIC time evolu tion equa-
+tion (3.18) as
+∂Q(r1)
+∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+irrev=l(r1)/integraldisplay
+Vd3r2M(2)′,dif
+Pε(r1,r2)1
+T(r2), (3.20)
+where we have used the approximation ( ∂Q(r)/∂t)|irrev≈l(r) (∂P(r)/∂t)|irrev, since the
+interfacial width ldoes not change much, compared to the other two configurationa l vari-
+ables. Using the entropy functional derivative (3.10) with λ=0, results in the matrix
+elementM(2)′,dif
+Pεbeing the only relevant term for the variable Pwhich is left after the av-
+eraging (3.19). Then, putting the equation (3.20) in the Doi-Ohta eq uation (1.7a) for the
+relaxation of Qin a homogeneous case gives the following expression for the relaxat ion time
+1
+τdo,1=2κEMl
+VQ/integraldisplay
+Dcρ(Q,q)[c]/integraldisplay
+Vd3r/bracketleftbigg∂
+∂r/parenleftbigg∂2c(r)
+∂r2/parenrightbigg/bracketrightbigg2
+, (3.21)
+whereVis the volume of the system. Details of the calculation are given in Appe ndix B.
+Similar expression could be obtained for the relaxation time τdo,2. The above formula, gives
+the expression for the Doi-Ohta relaxation time τdo,1calculated from the transformation
+of the element of the friction matrix M(1),difcontaining the relaxation parameters of the
+Cahn-Hilliard level at any time t. This is the result of the presented systematic coarse-
+graining procedure developed within the GENERIC formalism which sho ws in which way
+the Doi-Ohta relaxation times can be obtained from the more detailed level of description.
+However, the above relation is difficult to express analytically in form o f the variables of
+the Doi-Ohta level, due to the diffusive terms coming from the Cahn-H illiard level. On the
+other hand, the above expression could, in principle, be calculated n umerically using short
+22time simulations at the Cahn-Hilliard level. However, the problems conc erning the third
+order derivatives of the composition field cmake this task rather complicated, as well as the
+need for determination of the Lagrange multipliers in order to employ the full GENERIC
+procedure developed in this paper.
+IV. CONCLUSIONS AND PERSPECTIVES
+In this paper, we have developed a systematic, thermodynamically g uided method which
+establishes the coarse-grained Doi-Ohta model, used for rheologic al behavior, from the more
+detailed Cahn-Hilliard model of a phase separating binary fluid. The ma in contributions
+of the present work can be summarized as: (1) the introduction of the average interfacial
+widthlas an additional structural variable; (2) the derivation of the coa rse-grained Poisson
+operator L(2), Eq.(3.11), from its finer level analogue; (3) no new dissipative proc esses arise
+during level jumping apart from the known hydrodynamic and diffusiv e dissipation; (4) the
+derivation of an expression (3.21) for the Doi-Ohta relaxation times as a function of the
+elements of the friction matrix describing relaxation at the finer Cah n-Hilliard level.
+The crucial step in the coarse-graining procedure was to identify t he relevant coarse-
+grained variables and find the appropriate mapping which expresses them in terms of the
+more microscopic variables. In order to capture the physics of the Doi-Ohta level, we in-
+troduced the interfacial width l(r,t) as an additional variable in the new model. In that
+way we could account for the stretching of the interface under flo w and derive analytically
+the reversible (convective) behavior of the variables Q(r,t) andq(r,t), which recovers the
+already established phenomenological Doi-Ohta model. Introductio n of the interfacial width
+land its time evolution equation, as an addition to Qandqof the original Doi-Ohta model
+with zero-width interface, offers new possibilities for modeling of the rheological behavior
+of multiphase flows. In addition, the expression for the interfacial tension (3.8) in terms
+of the Cahn-Hilliard parameters follows as the direct result of the de veloped systematic
+coarse-graining procedure.
+Considering the irreversible dynamics on the Doi-Ohta level, it has bee n shown that the
+dissipative processes at this coarse-grained level are carried on f rom the more microscopic
+Cahn-Hilliardlevel. Althoughtheiranalyticalderivationistoocomplexa ndrichinstructure,
+the way to connect the interface relaxation times of the Doi-Ohta m odel and the underlying
+23morphology dynamics simulated at the Cahn-Hilliard level is established . Analysis of the
+numerical investigation of phase separation under shear in the diffu sive regime revealed no
+new physical process that occur at the time scale which is slow compa red to the Cahn-
+Hilliard level and fast from the perspective of the Doi-Ohta level. The refore, there are no
+new dissipative effects arising in the performed coarse-graining ste p through the Green-
+Kubo type formula, i.e., no new physics appear. That leads us to the c onclusion that all
+the dissipation has been introduced into the model by coarse grainin g from the reversible
+atomistic level to the Cahn-Hilliard level. This coarse-graining step ha s been done in the
+derivation of the hydrodynamic equations of a phase separating flu id mixture from the
+underlying microscopic dynamics [46]. Furthermore, Espa´ nol and V ´ azquez showed in [50]
+that all dissipation is introduced already at the very first coarse-g raining step, going from
+the atomistic to the Fokker-Planck level. From there then follows th e conclusion that the
+Cahn-Hilliard level is not so fundamental considering that the same d issipation mechanisms
+occur even at the finer levels, and are afterwards transmitted to an even more coarse-grained
+Doi-Ohta level, with no new dissipative processes arising.
+The presented results could be used to deal with several interest ing open problems. First,
+possible extension of the procedure to more complicated phase sep arating systems with the
+addition of surfactants or block copolymers would be very interest ing for wider use. Second,
+the nature of the dissipative processes in a phase separating binar y mixture is still far
+from clear. While during the diffusive regime, in the absence of therma l noise, we do not
+recognize any further fast processes that could be considered a s fluctuations at the Doi-
+Ohta level, this becomes far from evident during the late stages of c oarsening when inertia
+becomes important. The nature of the domain coarsening in the iner tial hydrodynamics
+regime is still unknown, as well as the asymptotic behavior of a phase separating system
+under shear. Furthermore, recent studies showed that a noneq uilibrium steady state can
+be reached at high Reynolds numbers (low shear rates) only in the mix ed viscous-inertial
+regime, i.e., with any non-zero amount of inertia [21–24]. The fact tha t no steady state could
+be formed without presence of inertia, no matter how small, sugges ts that inertia plays the
+role of a singular perturbation in this problem [24]. The formed steady state, arising from a
+cyclic occurrence of elongation of domains followed by their break up , coagulation and shape
+relaxation, is characterized by irregular fluctuations around the a ttained steady state values.
+It would be interesting to understand the nonequilibrium steady sta te and the originof these
+24fluctuations, using the presented coarse-graining procedure. H owever, the phenomenological
+Doi-Ohtatheoryassumes lowReynolds number, i.e., neglects inertia. Deeper insight into the
+mixed viscous-inertial regime is therefore crucial in order to develo p the connection between
+the two levels and to understand the role of inertia in this problem.
+Appendix A
+We present a derivation of the elements L(2)
+Pg(r1,r2) andL(2)
+lg(r1,r2) of the coarse-grained
+Poisson operator L(2)in detail. For the element L(2)
+Pg(r1,r2) the general expression (3.12)
+gives
+L(2)
+Pg(r1,r2) =/integraldisplay
+Dxρy[x]/integraldisplay
+Vd3r′
+1/integraldisplay
+Vd3r′
+2χ(r1−r′
+1)χ(r2−r′
+2)
+×/integraldisplay
+Vd3r3/integraldisplay
+Vd3r4δΠP[c](r′
+1)
+δc(r3)L(1)
+cg(r3,r4)δΠg[g](r′
+2)
+δg(r4), (A1)
+which after substitution of the appropriate functional derivative s,L(1)
+cgelement, and the
+integration over r3andr4becomes
+L(2)
+Pg(r1,r2) =/integraldisplay
+Dcρ(Q,q)[c]/integraldisplay
+Vd3r′
+1/integraldisplay
+Vd3r′
+2χ(r1−r′
+1)χ(r2−r′
+2)
+×∂c
+∂r′
+2∂
+∂r′
+2·/parenleftbigg
+2∂c
+∂r′
+2χ(r′
+1−r′
+2)/parenrightbigg
+. (A2)
+For further calculations, we use the difference between the Cahn- Hilliard and Doi-Ohta
+length scales (3.2), so that/integraltext
+Vd3r′
+1χ(r1−r′
+1)χ(r′
+1−r′
+2)≈χ(r1−r′
+2) (see [45]). Then the last
+equation, after integration over r′
+1, becomes
+L(2)
+Pg(r1,r2) =/integraldisplay
+Dcρ(Q,q)[c]/integraldisplay
+Vd3r′
+2χ(r2−r′
+2)∂c
+∂r′
+2∂
+∂r′
+2·/parenleftbigg
+2∂c
+∂r′
+2χ(r1−r′
+2)/parenrightbigg
+.(A3)
+After integration by parts, in which surface terms vanish, using ap proximation χ(r1−
+r′
+2)χ(r2−r′
+2)≈χ(r1−r2)χ(r2−r′
+2) valid due to the fact that the smoothing length is much
+smaller than the Doi-Ohta length scale, equation (3.2), and the ident ity∂χ(r−r′)/∂r′=
+−∂χ(r−r′)/∂r, we come to the expression
+L(2)
+Pg,β(r1,r2) =/integraldisplay
+Dcρ(Q,q)[c]×/braceleftigg
+∂
+∂r2α/integraldisplay
+Vd3r′
+22∂c
+∂r′
+2α∂c
+∂r′
+2βχ(r1−r2)χ(r2−r′
+2)
+−∂
+∂r1β/integraldisplay
+Vd3r′
+2|∇c(r′
+2)|2χ(r1−r2)χ(r2−r′
+2)
+−∂
+∂r2β/integraldisplay
+Vd3r′
+2|∇c(r′
+2)|2χ(r1−r2)χ(r2−r′
+2)/bracerightbigg
+. (A4)
+25Finally, with the help of the mappings (3.4), we can recognize the appr opriate terms in the
+last equation, so that their ensemble average gives
+L(2)
+Pg,β(r1,r2) =∂
+∂r2α/bracketleftbigg/parenleftbigg
+2pαβ(r2)−1
+3P(r2)δαβ/parenrightbigg
+χ(r1−r2)/bracketrightbigg
++P(r2)∂χ(r1−r2)
+∂r2β.(A5)
+The Poisson operator element L(2)
+lg(r1,r2) is obtained under similar approximations as
+above, but also the following approximations
+/integraldisplay
+Vd3r′
+1Φ[c](r′
+1)χ(r1−r′
+1)χ(r′
+1−r′
+2)≈Φ[c](r1)χ(r1−r′
+2), (A6)
+which holdsunder theassumption ofthedifference inlengthscales (3 .2), andtheassumption
+/an}b∇acketle{t(ΠP[c](r))nΠl[c](r)/an}b∇acket∇i}ht ≈(/an}b∇acketle{tΠP[c](r)/an}b∇acket∇i}ht)n/an}b∇acketle{tΠl[c](r)/an}b∇acket∇i}ht, (A7)
+forn=±1. For the element L(2)
+lg(r1,r2), we then obtain
+L(2)
+lg,β(r1,r2) =1
+P(r1)/angbracketleftigg/integraldisplay
+Vd3r′∂c
+∂r′α∂c
+∂r′
+β|∇c(r′)|−1χ(r1−r′)/angbracketrightigg
+∂χ(r1−r2)
+∂r2α
+−∂l(r1)
+∂r1βχ(r1−r2)−2
+3l(r1)∂χ(r1−r2)
+∂r2β
+−2l(r1)
+P(r1)pαβ(r1)∂χ(r1−r2)
+∂r2α. (A8)
+In order to express the previous equation in terms of the coarse- grained Doi-Ohta variables
+{P,p,l}, we must use the following closure approximation for the first term o n the right
+hand side/angbracketleftigg/integraldisplay
+Vd3r′∂c
+∂r′α∂c
+∂r′
+β|∇c(r′)|−1χ(r1−r′)/angbracketrightigg
+≈
+C/angbracketleftigg/integraldisplay
+Vd3r′∂c
+∂r′α∂c
+∂r′
+βχ(r1−r′)/angbracketrightigg/angbracketleftbigg/integraldisplay
+Vd3r′|∇c(r′)|χ(r1−r′)/angbracketrightbigg
+, (A9)
+whereCis chosen in such a way that the previous expression is valid when its tr ace is taken.
+By putting α=βinto the above equation, we find
+C=/angbracketleftbigg/integraldisplay
+Vd3r′|∇c(r′)|2χ(r1−r′)/angbracketrightbigg−1
+. (A10)
+Using the above expressions, element L(2)
+lg(r1,r2) takes the final form
+L(2)
+lg,β(r1,r2) =−l(r1)
+P(r1)pαβ(r1)∂χ(r1−r2)
+∂r2α−1
+3l(r1)∂χ(r1−r2)
+∂r2β
+−∂l(r1)
+∂r1βχ(r1−r2). (A11)
+26Appendix B
+We present the derivation of the expression (3.21) for the time rela xationτdo,1. Substitu-
+tion of the equation (3.20) for the irreversible time evolution of the c onfigurational variable
+Qinto the Doi-Ohta relaxation equation (1.7a) gives
+1
+τdo,1=−l(r1)
+Q(r1)/integraldisplay
+Vd3r2M(2)′,dif
+Pε(r1,r2)1
+T(r2). (B1)
+Under the assumption of a homogeneous system, integration over the whole system size of
+the above expression gives
+1
+τdo,1=−l
+QVT/integraldisplay
+Vd3r1/integraldisplay
+Vd3r2M(2)′,dif
+Pε(r1,r2). (B2)
+The general expression (3.19) for the coarse-grained friction ma trixM(2)′,difgives
+M(2)′,dif
+Pε(r1,r2) =/integraldisplay
+Dxρy[x]/integraldisplay
+Vd3r′
+1/integraldisplay
+Vd3r′
+2χ(r1−r′
+1)χ(r2−r′
+2)
+×/integraldisplay
+Vd3r3/integraldisplay
+Vd3r4δΠP[c](r′
+1)
+δc(r3)M(1),dif
+cǫ(r3,r4)δΠǫ[ǫ](r′
+2)
+δǫ(r4),(B3)
+which after substitution of the appropriate functional derivative s,M(1),difelement, and the
+integration over r4,r′
+1, andr′
+2becomes
+M(2)′,dif
+Pε(r1,r2) =/integraldisplay
+Dcρ(Q,q)[c]/integraldisplay
+Vd3r′2κEMT∂
+∂r′α/parenleftbigg/parenleftbigg∂2c
+∂r′2/parenrightbigg
+χ(r2−r′)/parenrightbigg
+×∂2
+∂r′α∂r′
+β/parenleftigg
+∂c
+∂r′
+βχ(r1−r′)/parenrightigg
+, (B4)
+where we have used the assumption of a homogeneous ( M=const.) and isothermal system,
+as well as the approximations based on a difference between the Cah n-Hilliard and Doi-
+Ohta length scales, as in Appendix A. Substitution of the friction mat rix element (B4) into
+equation (B2), gives after using the normalization condition/integraltext
+Vd3r′χ(r−r′) = 1, the final
+formula for the Doi-Ohta time relaxation
+1
+τdo,1=2κEMl
+VQ/integraldisplay
+Dcρ(Q,q)[c]/integraldisplay
+Vd3r/bracketleftbigg∂
+∂r/parenleftbigg∂2c(r)
+∂r2/parenrightbigg/bracketrightbigg2
+. (B5)
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+arXiv:1001.0926v2  [math.GT]  29 Sep 2010TWISTED TORSION INVARIANTS AND LINK CONCORDANCE
+JAE CHOON CHA AND STEFAN FRIEDL
+Dedicated to the memory of Des Sheiham
+Abstract. The twisted torsion of a 3-manifold is well-known to be zero w hen-
+ever the corresponding twisted Alexander module is non-tor sion. Under mild
+extra assumptions we introduce a new twisted torsion invari ant which is always
+non-zero. We show how this torsion invariant relates to the t wisted intersec-
+tion form of a bounding 4-manifold, generalizing a theorem o f Milnor to the
+non-acyclic case. Using this result, we give new obstructio ns to 3-manifolds
+being homology cobordant and to links being concordant. The se obstructions
+are sufficiently strong to detect that the Bing double of the fig ure eight knot
+is not slice.
+1.Introduction
+In this paper we introduce new obstructions to links being concorda nt, which are
+obtained from twisted torsion invariants. Recall that an m–component (oriented)
+linkis an embedded ordered collection of mdisjoint (oriented) circles in S3. Given
+an oriented link Lwe denote by−Lthe same link with the orientation of each
+component reversed. We say that two m–component oriented links L0=L1
+0∪
+···∪Lm
+0andL1=L1
+1∪···∪Lm
+1areconcordant if there exists a collection of
+mdisjoint, locally flat, oriented annuli A1,...,AminS3×[0,1] such that ∂Ai=
+Li
+0×0∪−Li
+1×1,i= 1,...,m. We say that an m–component link is sliceif it is
+concordant to the trivial m–component link. Equivalently a link is slice if it bounds
+mdisjoint locally flat disks in D4.
+LetL⊂S3be an oriented m–component link. Throughout this paper we write
+EL=S3\νL, whereνLis a tubular neighborhood of L, andπ(L) =π1(EL).
+Letψ:H1(EL)→Hbe a non–trivial homomorphism to a free abelian group and
+letα:π(L)→GL(R,k) be a representation with Ra subring of C. We can then
+consider the twisted homology module H1(EL;R[H]k) and we define
+rank(L,ψ,α) = rankR[H]H1(EL;R[H]k),
+where rank R[H]Adenotes the dimension of A⊗R[H]Q(H) over the quotient field
+Q(H) of the integral domain R[H]. If rank(L,ψ,α) = 0, then it turns out that
+ELisQ(H)k–acyclic, i.e., H∗(EL;Q(H)k) = 0, and we can define the torsion
+τα⊗ψ(L)∈Q(H)×which is well–defined up to multiplication by an element of
+the form±dhwhered∈det(α(π(L))) andh∈H. This invariant generalizes the
+invariants introduced by Reidemeister, Milnor and Turaev and later b y Lin and
+Wada.
+In general though rank( L,ψ,α) will be non–zero, for example this is the case for
+any boundary link with at least two components (cf. Theorem 6.1) and the trivial
+representation. Suppose that R⊂Cis closed under complex conjugation. In
+that case the ring R[H] has a natural involution given by complex conjugation and
+h=h−1forh∈H, this involution furthermore extends naturally to an involution
+2000Mathematics Subject Classification. 57M25, 57M27, 57N70.
+Key words and phrases. Twisted Torsion, Homology Cobordism, Link Concordance.
+12 JAE CHOON CHA AND STEFAN FRIEDL
+onQ(H). Now suppose that α:π(L)→GL(R,k) is a unitary representation and
+suppose that ψis non–trivial on each meridian of L. Under these assumptions we
+define a torsion invariant
+τα⊗ψ(L)∈Q(H)×/N(Q(H))
+even if rank( L,ψ,α)/\e}atio\slash= 0. Here N(Q(H)) denotes the subgroup of norms of the
+multiplicative group Q(H)×, i.e.,N(Q(H)) ={±qq|q∈Q(H)×}. The torsion
+τα⊗ψ(L) viewed as an element in Q(H)×/N(Q(H)) is again well–defined up to
+multiplication by an element of the form ±dhwhered∈det(α(π(L))) andh∈H.
+The invariant τα⊗ψ(L) is the twisted version of an invariant first introduced by
+Turaev [Tu86, Section 5.1].
+Remark. IfKis a knot in S3,ψ:H1(EK)→Zis an isomorphism, and α:π(L)→
+GL(Z,1) is the trivial representation, then rank( K,ψ,α) = 0 and τα⊗ψ(K) =
+∆K(t)/(t−1) (cf. [Tu01]). In general, if rank( L,ψ,α) = 0, then τα⊗ψ(K) can
+be expressed in terms of twisted Alexander polynomials (cf. [ KL99a] and [FK06]).
+Ifαis the trivial representation and if rank( L,ψ,α)>0, then the torsion is re-
+lated to the Alexander polynomial defined using Tor Z[H](H1(EL;Z[H])) (cf. [Tu86,
+Theorem 5.1.1]). We expect that a similar result also holds in the twisted case.
+We can now state our sliceness obstruction. First, if Lis them–component
+unlink inS3with meridians µ1,...,µm, then given αandψas above we will show
+in Corollary 6.2that rank(L,ψ,α) =k(m−1) and
+τα⊗ψ(L) =±dh·m/productdisplay
+i=1det/parenleftbig
+id−ψ(µi)α(µi)/parenrightbig−1∈Q(H)×/N(Q(H))
+withd∈det(α(π(L))) andh∈H. The following theorem says that the torsion
+invariant of a slice link is given by the above expression if the represen tation factors
+through ap-group. This is a special case of our main result, Theorem 4.2, which is
+a more general result for link concordance.
+Theorem 1.1. LetLbe anm–component oriented slice link. Let R⊂Cbe a sub-
+ring closed under complex conjugation and let α:π(L)→GL(R,k)be a representa-
+tion which factors through a finite group of prime power order . Letψ:H1(EL)→H
+be an epimorphism onto a free abelian group which is non–triv ial on each meridian
+ofL. Then
+rank(L,ψ,α) =k(m−1)
+and
+τα⊗ψ(L) =±dh·m/productdisplay
+i=1det/parenleftbig
+id−ψ(µi)α(µi)/parenrightbig−1∈Q(H)×/N(Q(H))
+for somed∈det(α(π(L)))andh∈H, whereµ1,...,µmare meridians of L.
+Remark. LetK⊂S3be a knot. Fox and Milnor [ FM66] have shown that if K
+is slice, then the untwisted Alexander polynomial of Kfactors as±tlf(t)f(t−1)
+for some polynomial f(t) and some l∈Z. It is well–known that for a knot all
+representationswhichfactorthrougha p–grouparenecessarilyabelian,inparticular
+one can easily show that the sliceness obstruction of Theorem 1.1reduces to the
+Fox–Milnor obstruction. On the other hand for knots Kirk and Living ston [KL99a]
+usedtwistedtorsioncorrespondingto‘Casson–Gordon’–typerep resentationstogive
+sliceness obstructions which go beyond Fox–Milnor. We also refer to [KL99b],
+[Ta02],[HKL08], [Liv09]andalso[ FV09]formoreontwistedAlexanderpolynomials
+of knots and their relation to knot concordance.TWISTED TORSION INVARIANTS AND LINK CONCORDANCE 3
+Remark. (1)Ifαis the trivial representation, then Theorem 1.1was proved
+in the one–variable case by Murasugi [ Mu67] (cf. also [ Ka77]) and in the
+multi–variable case it was proved independently by Kawauchi [ Ka78, The-
+orem B] and Nakagawa [ Na78]. These results were reproved by Turaev
+[Tu86, Theorem 5.4.2] using torsion.
+(2)Theorem 1.1is related in spirit to [ Fr05, Theorem 2.2] where a sliceness
+obstruction for links is given using the Atiyah–Patodi–Singer eta inva riant
+and unitary representations factoring through p–groups. The metabelian
+case was considered earlier by the first author and Ko [ CKo99].
+(3)We expect the obstruction of Theorem 1.1to be closely related to the
+discriminant part of Hirzebruch-type invariants developed by the fi rst au-
+thor [Ch10].
+We prove a generalization of Milnor’s classical duality theorem for Reid emeister
+torsion [Mi62] and use it as a key ingredient in the proof of the above sliceness
+obstruction. For an even dimensional compact oriented manifold Wendowed with
+a homomorphism Z[π1(W)]→Qinto a field Qwith involution such that Wis
+Q–acyclic, Milnor proved that the Reidemeister torsion of M=∂WoverQis of
+the formqq, whereqis the Reidemeister torsion of W. In particular, up to norms,
+the torsion of Mis trivial. A generalization of this to the twisted acyclic case is
+given in [ KL99a]. We generalize these prior results to the non-acyclic case:the
+twisted torsion of an odd dimensional manifold Mis equal to the determinant of
+the twisted intersection pairing of a bounding even-dimens ional manifold W(up
+to norms) . For a precise statement, see Theorem 2.4. We hope that this is of
+independent interest and useful for further applications.
+The paper is organized as follows. In Section 2we introduce twisted torsion of
+3-manifolds and links with non-acyclic twisted homology and we show ho w these
+invariants relate to intersection forms of bounding 4-manifolds. In Section3we
+study the torsion invariant of homology cobordant 3-manifolds and in Section 4we
+apply these results to link concordance where we prove our main the orem (which
+implies Theorem 1.1). In Section 5we discuss and to a certain degree classify
+complex representations which factor through p–groups. In Section 6we show how
+to compute twisted torsion of boundary links using a boundary link Se ifert matrix
+and we discuss the behavior of our invariants for links which are boun dary slice.
+Finally in Section 7we prove a formula for the twisted invariants of satellite links
+and we use this formula to reprove the fact (first proved by the fir st author [ Ch10])
+that the Bing double of the Figure 8 knot is not slice.
+Conventions. The following are understood unless it says specifically otherwise:
+(i) groups are finitely generated, (ii) manifolds are compact, orient able and con-
+nected, (iii) rings are understood to be associative, commutative w ith unit element,
+and (iv) homology and cohomology groups are taken with integral co efficients.
+Acknowledgments. We would like to thank Baskar Balasubramanyam, David
+Cimasoni, Taehee Kim, Vladimir Turaev and Liam Watson for helpful com ments
+and conversations. We also thank an anonymousreferee for usef ul suggestions. The
+first author was supported by the National Research Foundation of Korea (NRF)
+grants funded by Korean government(MEST) (Grant No. 2009–0 094069 and R01–
+2007–000–11687–0).
+2.Twisted torsion of 3-manifolds and links
+In this section we define twisted torsion invariants of odd dimensiona l manifolds
+for finite dimensional unitary representations for which the corre sponding twisted
+homology groups are not necessarily acyclic. We furthermore inves tigate twisted4 JAE CHOON CHA AND STEFAN FRIEDL
+torsion of odd dimensional manifolds cobounding a manifold, along the way we
+generalize a theorem of Milnor to the nonacyclic case. In the last sec tion we then
+specialize to the case of 3-manifolds and link exteriors.
+2.1.Twisted homology and Poincar´ e duality. Let (X,Y) be a CW-pair with
+π=π1(X), andφ:π→GL(k,R) a finite dimensional representation over a ring R.
+Denote by p:/tildewideX→Xthe universal covering of Xand write/tildewideY=p−1(Y). Recall
+thatπacts on the left of /tildewideXas the deck transformation group. We consider the
+cellularchaincomplex C∗(/tildewideX,/tildewideY)asaright Zπ-modulevia σ·g:=g−1σforachainσ.
+TheRk–coefficient cellular chain complex of ( X,Y) is defined to be
+C∗(X,Y;Rk) =C∗(/tildewideX,/tildewideY)⊗ZπRk.
+whereRkis viewed as a ( Zπ,R)–bimodule via the representation φ. We define the
+ith twisted homology group to be theR-module
+Hφ
+i(X,Y;Rk) =Hi(C∗(X,Y;Rk)).
+If the representation is understood clearly, then we just write Hiinstead ofHφ
+i.
+Now assume that Ris a ring with a (possibly trivial) involution . Given an R–
+moduleAwedenoteby Atheopposite R–module, i.e. A=Aasabeliangroups,and
+the multiplication by r∈RonAis given by the multiplication by ronA. Finally
+recall that a representation φ:π→GL(k,R) is called unitaryifφ(g−1) =φ(g)tfor
+allg∈π.
+We now have the following twisted Poincar´ e duality theorem (cf. also [Le94, p.
+91], [FK06, Lemma 4.12], and [ KL99a]). Here, given a vector space Vover a field
+Q, we denote the dual vector space by V∗= HomQ(V,Q).
+Theorem 2.1. LetWbe ann–manifold with ∂W=M∪M′,MandM′sub-
+manifolds such that M∩M′=∂M=∂M′. Letφ:π1(W)→GL(k,Q)be a
+unitary representation over a field Qwith involution. Then there exists a natural
+isomorphism
+Hi(W,M;Qk)∼=Hn−i(W,M′;Qk)∗
+ofQ–vector spaces.
+Thetheoremcanbeprovedalongthesamelinesasthestandardpro ofofPoincar´ e
+duality. For the reader’s convenience we give an outline of the proof .
+Proof.Choose a cell structure of ( W,M) to define C∗(/tildewiderW,/tildewiderM) and letD∗(/tildewiderW,/tildewiderM′)
+be the chain complex of the dual cell structure. Then the equivaria nt intersection
+pairing (e.g., see [ Le94, p. 91]) gives an isomorphism of chain complexes
+C∗(W,M;Qk)∼=Dn−∗(W,M′;Qk)∗.
+Applying the universal coefficient theorem over Q, the conclusion follows. /square
+2.2.Basic definitions of twisted torsion.
+Torsion of a based chain complex. We begin by recalling the algebraic setup for
+torsion invariants. Suppose C={C∗}is a based chain complex over a field Qand
+B={B∗}is a basis for H∗(C), i.e.,Biis a basis of the Q–vector spaces Hi(C). The
+torsionτ(C,B)∈Q×:=Q\{0}is defined as in [ Tu01,Tu02]. (See also Milnor’s
+classic introduction to torsion [ Mi66], but note that we follow Turaev’s convention,
+which gives the reciprocal of the torsion in [ Mi66].) IfH∗(C) is identically zero,
+then we will just write τ(C)∈Q×for the torsion.
+In the following theorem, we collect well-known algebraic properties o f torsion
+which will be useful later. In the theorem, giventwo bases bandb′ofa vectorspace,
+we will denote the basis change matrix from btob′by (b|b′), as in Milnor [ Mi66]
+and Turaev [ Tu01,Tu02].TWISTED TORSION INVARIANTS AND LINK CONCORDANCE 5
+Theorem 2.2. SupposeCis a based chain complex over QandB={Bi}is a
+basis ofH∗(C).
+(1)SupposeB′={B′
+i}is another basis of H∗(C). Then
+τ(C,B) =τ(C,B′)·/productdisplay
+i(Bi|B′
+i)(−1)i+1.
+(2)LetC′be the dual based chain complex given by C′
+i= (Cn−i)∗andB′be the
+basis ofH∗(C′) =Hn−∗(C)∗dual toB. Thenτ(C,B) =τ(C′,B′)(−1)n+1.
+(3)SupposeC∗is acyclic. Choose a basis of the nth cycle submodule Zn=
+Ker{Cn→Cn−1}, and view
+C′={Zn−→Cn−→···−→ C0}
+C′′={···−→Cn+2−→Cn+1−→Zn}
+as acyclic based chain complexes (indexed so that C′
+0=C0,C′′
+0=Zn).
+Thenτ(C) =τ(C′)·τ(C′′)(−1)n.
+(4)Suppose 0→C′→C→C′′→0is a short exact sequence of based chain
+complexes andB′,B, andB′′are bases of H∗(C′),H∗(C)andH∗(C′′),
+respectively. We view the associated homology long exact se quence as an
+acyclic complex, say H, based byB,B′,B′′. Then
+τ(C,B) =τ(C′,B′)·τ(C′′,B′′)·τ(H).
+Twisted torsion of CW-complexes. Let (X,Y) be a finite CW-pair with π=π1(X)
+and letφ:π→GL(k,Q) be a representation over a field Q. LetB={B∗}be a
+basis ofH∗(X,Y;Qk). The universal cover ( ˜X,˜Y) has a natural cell structure, and
+the chain complex C∗(/tildewideX,/tildewideY) can be based over Zπby choosing a lift of each cell of
+(X,Y) and orienting it. This, together with the standard basis of Qk, gives rise to
+a basing of C∗(X,Y;Qk) overQ. We can then define the twisted torsion
+τφ(X,Y,B)∈Q×
+to be the torsion of C∗(X,Y;Qk) with respect to B. We will dropBfrom the
+notation if H∗(X,Y;Qk) = 0.
+Standard arguments (e.g., see [ Mi66,Tu86,Tu01,Tu02]) show that τφ(X,Y,B)
+is well-defined up to multiplication by an element in ±det(φ(π)), and is invariant
+undersimplehomotopypreserving B. ByChapman’stheorem[ Chp74]the invariant
+τφ(X,Y,B) only depends on the homeomorphism type of ( X,Y). In particular
+when (M,N) is a manifold pair, we can define τφ(M,N,B) by picking any finite
+CW–structure for ( M,N).
+Twisted torsion of odd dimensional manifolds. LetMbe an odd dimensional man-
+ifold withπ=π1(M) and letφ:π→GL(k,Q) be a representation over a field Q.
+In many interesting cases H∗(M;Qk) will be nontrivial. For example the untwisted
+multivariable Alexander module of a boundary link with at least two comp onents is
+always non-torsion. In order to define an invariant for the nonacy clic case without
+referring to a basis of homology, we now assume the following two con ditions hold:
+(1)Qis endowed with a (possibly trivial) involution, and φis unitary.
+(2)∂MisQk-acyclic, i.e., H∗(∂M;Qk) = 0.
+Sinceφis unitary, we have the Poincar´ e duality isomorphism
+Hi(M;Qk)∼=Hn−i(M,∂M;Qk)∗∼=Hn−i(M;Qk)∗
+ofQ–vector spaces, by Theorem 2.1and by Condition (2). Since Mis odd dimen-
+sional, one can pick a basis B={B∗}forH∗(M;Qk) with the following property:
+for eachi,Biis the dual basis of Bn−ivia the above Poincar´ e duality isomorphism.
+We call such a basis B={B∗}aself-dual basis forH∗(M;Qk).6 JAE CHOON CHA AND STEFAN FRIEDL
+Lemma 2.3. SupposeBandB′are self-dual bases for H∗(M;Qk). Then for some
+q∈Q×, up to the indeterminacy of the torsion,
+τφ(M,B′) =τφ(M,B)·qq.
+Proof.Let (Bi|B′
+i) be the determinant of the base change matrix from BitoB′
+i.
+Then (Bn−i|B′
+n−i) =(Bi|B′
+i)−1. The desired conclusion follows immediately from
+Theorem 2.2. /square
+We define the norm subgroup ofQ×to beN(Q) ={±qq|q∈Q×}, and we say
+f∈Q×is anormwhenf∈N(Q). We define
+τφ(M) =τφ(M,B) as an element in Q×/N(Q)
+whereBis a self-dual basis of H∗(M;Qk). By Lemma 2.3the invariant τφ(M) is
+well-defined up to multiplication by an element in ±det(φ(π1(M))). This invariant
+was first introduced by Turaev [ Tu86, Section 5.1] in the untwisted case.
+In the following, given f∈Q, we will sometimes write τφ(M).=f∈Q×/N(Q),
+to indicate that there exists a representative of τφ(M) which equals f.
+2.3.Twisted torsion and bounding manifolds. In this subsection we prove a
+non-acyclic generalization of a well-known theorem of Milnor [ Mi62, Theorem 2]
+and of its twisted analogue due to Kirk and Livingston [ KL99a, Theorem 5.1 and
+Corollary 5.3].
+LetWbe a 2r–dimensional manifold with (possibly disconnected) boundary M.
+Letφ:π1(W)→GL(k,Q) be a unitary representation. Now consider the map
+Hr(W;Qk)∼=− −→Hr(W,M′;Qk)∗ι∗
+− −→Hr(W;Qk)∗
+where the first map is the isomorphism given by Theorem 2.1and the second map
+is induced by ι: (W,∅)→(W,M). This map gives rise to a pairing
+λ:Hr(W;Qk)−→Hr(W;Qk)−→Q
+which is well-known to be ( −1)r-hermitian. This pairing is called the equivariant
+intersection form of (W,φ). This form is in general singular, in fact for any x∈
+Im{Hr(M;Qk)→Hr(W;Qk)}andy∈Hr(W;Qk) we haveλ(x,y) =λ(y,x) = 0.
+In particular λgives rise to a pairing on Hr(W;Qk)/i∗Hr(M;Qk) which turns out
+to be non-singular.
+We now pick a basis B={v1,...,vs}forHr(W;Qk)/i∗Hr(M;Qk) and we com-
+pute det(λ(vi,vj))∈Q×. Note that if we change the basis, then the determinant
+of the form changes by a norm. Put differently, we obtain a well-defin ed invariant
+Λ(W,φ) := det(λ(vi,vj))∈Q×/N(Q).
+Note that Λ( W,φ) =Λ(W,φ) sinceλis (−1)r-hermitian.
+Theorem 2.4. SupposeWis a2r–dimensional manifold with (possibly discon-
+nected) boundary M. Letφ:π1(W)→GL(k,Q)be a unitary representation. Then
+τφ(M) = Λ(W,φ)∈Q×/N(Q)
+up to the indeterminacy ±det(φ(π1(M))).
+In the following we will only apply the theorem to the case that Λ( W,φ) = 1,
+but we hope that the general case is of independent interest.
+To prove Theorem 2.4we need the following duality of torsion, which is essen-
+tially due to Milnor [ Mi62]. (See also Kirk and Livingston [ KL99a] for the twisted
+case.)TWISTED TORSION INVARIANTS AND LINK CONCORDANCE 7
+Lemma 2.5. SupposeWis ann–manifold and M,M′are submanifolds of ∂W
+such that∂W=M∪M′andM∩M′=∂M=∂M′. Supposeφ:π1(W)→GL(k,Q)
+is a unitary representation, B={B∗}is a basis for H∗(W,M;Qk), andB′is the
+dual basis for H∗(W,M′;Qk) =Hn−∗(W,M;Qk)∗. Then
+τφ(W,M,B) =τφ(W,M′,B′)(−1)n+1
+.
+Proof.The lemma follows immediately from the duality
+Ci(W,M;Qk)∼=Dn−i(W,M′;Qk)∗
+(see the proof of Theorem 2.1) and Theorem 2.2. /square
+Proof of Theorem 2.4. ChooseabasisBforH∗(W;Qk), andchooseaself-dualbasis
+B′forH∗(M;Qk). LetB′′be the basis of H∗(W,M;Qk) =Hn−∗(W;Qk)∗which
+is dual toB. From (the proof of) Theorem 2.1, Theorem 2.2, and Lemma 2.5, it
+follows that
+τφ(W,B) =τ0·τφ(M,B′)·τφ(W,B)(−1)n+1
+whereτ0is the torsion of the acyclic chain complex (= homology long exact se-
+quence)
+···−→Hi+1(W,M;Qk)−→Hi(M;Qk)−→Hi(W;Qk)−→Hi(W,M;Qk)−→···
+which is based by B,B′,B′′. We break the long exact sequence at i∗:Hr(W;Qk)→
+Hr(W,M;Qk) as in Theorem 2.2: letPbe the image of i∗, choose a basis bforP,
+and let
+C′={P−→Hr(W,M;Qk)−→···−→ H0(W,M;Qk)},
+C′′={···−→Hr(M;Qk)−→Hr(W;Qk)−→P}.
+Then by Theorem 2.2(3) we have τ0=τ(C′)τ(C′′)(−1)r. SinceB′is self-dual,
+C′′={C′′
+i}is canonically isomorphic, as a based chain complex, with the dual
+chain complex{C′
+3r+1−i∗}ofC′exceptC′′
+0=P. Therefore we have
+τ(C′′) =τ(C′)(−1)r
+·/parenleftbig
+b∗|b/parenrightbig
+whereb∗is the dual basis of bforP∗. It follows that
+τφ(M,B′) =τφ(W,B)·τφ(W,B)·τ(C′)−1·τ(C′)−1·/parenleftbig
+b|b∗/parenrightbig(−1)r
+.
+Note that by definition we have/parenleftbig
+b|b∗) = Λ(W,φ) =:D∈Q×/N(Q). Since the
+intersection form is ( −1)r-hermitian, D=±Dand soD−1≡DinQ×/N(Q). This
+completes the proof. /square
+Remark. IfWisQk–acyclic, then D= 1 automatically and τ(C′) = 1 in the above
+proof. Soτφ(M) =qqup to±det(φ(π1(M))), whereq=τφ(W). This shows our
+non-acyclicresult specializes to Milnor’s theorem [ Mi62, Theorem 2] and its twisted
+analogue due to Kirk and Livingston [ KL99a, Theorem 5.1 and Corollary 5.3].
+2.4.Twisted torsion of 3-manifolds and links. In this section we now spe-
+cialize to the case of 3-manifolds. Let Mbe a 3-manifold with empty or toroidal
+boundary (e.g. a link exterior). We write π=π1(M). Letψ:π→Hbe an
+epimorphism onto a nontrivial free abelian group H. LetRbe a domain with in-
+volution. We equip R[H] with the involution given by rh=rh−1forr∈Rand
+h∈H. This extends to an involution on Q(H), the quotient field of R[H]. Now
+letα:π→GL(k,R) be a unitary representation. Using αandψ, we define a left
+Z[π]-module structure on R[H]k:=Rk⊗RR[H] as follows:
+g·(v⊗p) := (α(g)·v)⊗(ψ(g)p)8 JAE CHOON CHA AND STEFAN FRIEDL
+whereg∈π,v∈Rkandp∈R[H]. This extends to a Z[π]-module structure on
+Q(H)k. It can be seen that α⊗ψ:π→GL(k,Q(H)) is unitary since αis unitary.
+We will several times make use of the following simple fact:
+det((α⊗ψ)(π))⊂det(α(π))·H.
+Wenowsaythat ψ:π→Hisadmissible ifψrestrictedtoanyboundarycomponent
+ofMisnontrivial. Notethat ψisalwaysadmissibleif Misclosed. If ψisadmissible
+then Condition (2) in Section 2.2is satisfied (e.g., see [ KL99a, Proposition 3.5] and
+[KL99a, Section 3.3]) and by the discussion of the previous section we thus o btain
+an invariant
+τα⊗ψ(M)∈Q(H)×/N(Q(H))
+well-defined up to multiplication by an element of the form ±dhwithd∈det(α(π))
+andh∈H.
+Remark. Note that if rank( H) is nonzero, then in the above setting we have
+Hi(M;Q(H)k) = 0 fori= 0,3 (see e.g. [ FK06,FK08]), so that it suffices to
+chooseB1in the definition of torsion.
+We now specialize even further, namely to the case of link exteriors. LetL⊂S3
+be an oriented m–componentlink. We denote the exteriorby ELand the (oriented)
+meridians by µ1,...,µm. Using the basis µ1,...,µmwe can now naturally identify
+H1(EL;Z) withZm. We say that a homomorphism ψ:Zm→Hisadmissible ifψis
+an epimorphism onto a nontrivial free abelian group Hsuch that the epimorphism
+is nontrivial on each subsummand of Zm=Z⊕···⊕Z. In this case the induced
+mapπ1(EL)→His admissible in the above sense, so that we obtain a well-defined
+torsion invariant
+τα⊗ψ(L) :=τα⊗ψ(EL)∈Q(H)×/N(Q(H)).
+3.Homology cobordism and twisted torsion
+In this section we investigate the behaviour of the twisted torsion u nder homol-
+ogy cobordism. Recall that two 3–manifolds MandM′(possibly with nonempty
+toroidalboundary)aresaidtobe homology cobordant ifthereexistsa4–manifold W
+containingMandM′as submanifolds such that ∂W=M∪−M′,∂M=M∩M′=
+∂M′, and the inclusions M→WandM′→Winduce isomorphisms on H∗(−;Z).
+From now on, R⊂Cis always assumed to be a subring closed under complex
+conjugation. We denote by Pk(R,π) the set of representations π→GL(k,R)
+factoring through a p-group. We will continuously make use of the well-known
+fact that complex representations factoring through finite grou ps are necessarily
+unitary.
+Forψ:π1(M)→Handα∈Pk(R,π), we define
+rank(M,α,ψ) := dimQ(H)Hα⊗ψ
+1(M;Q(H)k).
+Our main result is the following:
+Theorem 3.1. Suppose two 3–manifolds MandM′are homology cobordant and
+His a free abelian group. Then there are bijections
+Ψ: Hom(π1(M),H)−→Hom(π1(M′),H)
+Φ:Pk(R,π1(M))−→Pk(R,π1(M′))
+such that for any ψ:π1(M)→H,ψis admissible if and only if Ψ(ψ)is admissible,
+and in this case, for any α∈Pk(R,π1(M)), we have
+rank(M,α,ψ) = rank(M′,Φ(α),Ψ(ψ))TWISTED TORSION INVARIANTS AND LINK CONCORDANCE 9
+and
+τα⊗ψ(M) =±dhτΦ(α)⊗Ψ(ψ)(M′)inQ(H)×/N(Q(H))
+for somed∈det(α(π1(M))) = det((Ψ( α))(π1(M′)))andh∈H.
+Remark. In the proof of Theorem 3.1we will see that for a given homology cobor-
+dismWbetweenMandM′, the maps Ψ and Φ are both induced by π1(M)→
+π1(W)←π1(M′), in particular Ψ and Φ are ‘related’ bijections.
+3.1.Stallings Theorem and representations through p-groups. We first
+construct the bijections in Theorem 3.1using Stallings’ theorem [ Sta65].
+Proof of Theorem 3.1, Part I. SupposeWis a homology cobordism between M
+andM′. Obviously the induced isomorphisms on H1(−;Z) gives rise to bijections
+Hom(π1(M),H)≈Hom(π1(W),H)≈Hom(π1(M′),H).
+Their composition will be Ψ in the above statement. Since ∂M=∂M′it follows
+thatψ:π1(M)→His admissible if and only if so is Ψ( ψ).
+The bijection Φ is constructed as follows. For a group G, we denote the qth lower
+central subgroup by Gq, which is defined inductively via G1=G,Gq= [G,Gq−1].
+For anyα∈Pk(R,π1(M)),αfactors through π1(M)/π1(M)qfor someq, since any
+p-group is nilpotent. By Stallings’ theorem [ Sta65], we have
+π1(M)/π1(M)q∼=− −→π1(W)/π1(W)q.
+Thereforeαinduces a representation π1(W)→GL(k,R) which factors through
+π1(W)/π1(W)q. It is easily seen that this induces a bijection Pk(R,π1(M))≈← −
+Pk(R,π1(W)). Similarly for M′andW, and composing them, we obtain a bijection
+Φ:Pk(R,π1(M))→Pk(R,π1(M′)).
+3.2.Cohn local property and twisted coefficients. In our proof of the con-
+clusion on torsions in Theorem 3.1we will compute (the quotient of) the torsions of
+MandM′in terms of the Q(H)k-coefficient intersection pairing of a cobordism W,
+appealing to Theorem 2.4. The key point is that one can provethat the intersection
+pairing is trivial when Wis a homology cobordism. This is done following a stan-
+dard strategy, namely by controlling the size of the underlying Q(H)k-coefficient
+homology of ( W,M), appealing to a chain contraction argument originally due to
+Vogel.
+In Lemma 3.2below we denote by ǫ:Z[π]→Zthe augmentation map given by
+g/ma√sto→1,g∈π.
+Lemma 3.2. Supposeπis a group and Ris a domain with characteristic zero,
+α∈Pk(R,π), andψ:π→His an epimorphism onto a nontrivial free abelian
+groupH. IfAis a square matrix over Z[π]such thatǫ(A)is invertible over Z, then
+(α⊗ψ)(A)is invertible over Q(H).
+Remark. Another wayofphrasingLemma 3.2is to saythat the Z[π]-moduleQ(H)k
+is in fact a module over the Cohn localization ofZ[π].
+Our proof of Lemma 3.2depends heavily on a result originally due to Strebel
+[Str74] and Levine [ Le94].
+Proof.SinceRhas characteristic zero and since the quotient ring of Rgives the
+sameQ(H), we may assume that Rcontains Q. Supposeαfactors through f:π→10 JAE CHOON CHA AND STEFAN FRIEDL
+PwherePis ap-group. We have the following commutative diagram:
+EndZ[π](Z[π]n) EndR(R[H]nk) R[H]
+EndQ[P](Q[P]n) EndR(Rnk) R/d47/d47α⊗ψ
+/d15/d15f
+/d39/d39/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79/d79
+α/d47/d47det
+/d15/d15ǫR
+/d15/d15ǫR
+/d47/d47 /d47/d47det
+where the induced maps are denoted by the same symbol, as an abus e of notation.
+The augmentation R[H]→Ris denoted by ǫR.
+SupposeA∈EndZ[π](Z[π]n) is ann×nmatrix over Z[π] such that ǫ(A) is
+invertible. Due to Strebel [ Str74, Lemma 1.10] and Levine [ Le94, Lemma 4.3] this
+implies that f(A) is invertible over Q[P], i.e., a unit in End Q[P](Q[P]n). Since the
+maps in the above diagram are multiplicative, it follows that α(A) is invertible.
+Therefore det( α(A))/\e}atio\slash= 0 and det(( α⊗ψ)(A))/\e}atio\slash= 0. It follows that Ais injective
+overR[H] and hence invertible over the field Q(H). /square
+A standard well-known chain contraction argument originally due to V ogel (see
+also Levine [ Le94, Proposition 4.2] and Cochran-Orr-Teichner [ COT03]) shows the
+following statement as a consequence of Lemma 3.2:
+Lemma 3.3. Suppose (X,A)is a finite CW-pair, Ris of characteristic zero, α∈
+Pk(R,π1(X)), andψ:π1(X)→His an epimorphism onto a nontrivial free abelian
+group. IfH∗(X,A;Z) = 0, thenH∗(X,A;Q(H)k) = 0.
+Proof of Theorem 3.1, Part II. SupposeWis a homology cobordism between M
+andM′. Recall that we have constructed bijections
+Pk(R,π1(M))≈Pk(R,π1(W))≈Pk(R,π1(M′)),
+Hom(π1(M),H)≈Hom(π1(W),H)≈Hom(π1(M′),H).
+Fixα∈Pk(R,π1(M)) andψ:π1(M)→H, and as an abuse of notation, we denote
+byαandψthe corresponding representations and homomorphisms of π1(M′) and
+π1(W) as well.
+Forconveniencewrite Q=Q(H). SinceH∗(W,M) = 0,wehave H∗(W,M;Qk) =
+0 by Lemma 3.3, and soH∗(M;Qk)∼=H∗(W;Qk). Similarly for M′andW, and
+from this the conclusion on the ranks follows immediately.
+Now pick a self–dual basis BforH∗(M;Qk) and pick a self–dual basis B′for
+H∗(M′;Qk). Recall that H∗(∂M;Qk) =H∗(∂M′;Qk) = 0 since ψis admissible
+(see Section 2.4). In particularBgives rise to a basis for H∗(M,∂M;Qk) which we
+also denote byB. Note thatBandB′give rise to a self–dual basis for H∗(∂W;Qk)
+which we denote by B⊕B′.
+SinceH∗(W,M;Qk) = 0, we have Λ( W,φ) = 1. Therefore, by Theorem 2.4, we
+haveτα⊗ψ(∂W).=τα⊗ψ(∂W,B⊕B′).= 1. Applying Theorem 2.2to the excision
+short exact sequence
+0−→C∗(M′;Qk)−→C∗(∂W;Qk)−→C∗(M,∂M;Qk)−→0
+and then applying Lemma 2.5we obtain
+(∗)1 =τα⊗ψ(∂W,B⊕B′) =τα⊗ψ(M′,B′)·τα⊗ψ(M,∂M,B)·τ0
+=τα⊗ψ(M′,B′)·τα⊗ψ(M,B)·τ0,
+whereτ0is the torsion of the homology long exact sequence
+−→Hi(M′;Qk)−→Hi(∂W;Qk)−→Hi(M,∂M;Qk)−→Hi−1(M′;Qk)−→
+which is based by B′,B′⊕BandB. It follows easily that τ0= 1. Therefore,
+multiplying (∗) byτα⊗ψ(M,B), the conclusion follows. /squareTWISTED TORSION INVARIANTS AND LINK CONCORDANCE 11
+4.Link concordance and twisted torsion
+It is well known that if two links are concordant, then their exterior s are homol-
+ogycobordant. Moreprecisely, if m-componentlinks LandL′inS3areconcordant,
+then the exterior Wof a concordance has boundary EL∪m(S1×S1×[0,1])∪−EL′
+where we glue the i-th meridian (longitude) of Lto thei-th meridian (longitude)
+ofL′(Recall that links are assumed to be orderedcollections of circles). Rounding
+corners and extending collars outward, we can assume that ∂W=EL∪−EL′and
+∂EL=EL∩EL′=∂EL′. Using Alexander duality it is straightforward to verify
+thatWis indeed a homology cobordism.
+Therefore, we can apply the invariance of torsion (Theorem 3.1) immediately.
+Furthermore, in case of links, we can make further observations o n the correspon-
+dence between representations. The main aim of this section is to pr ovide a precise
+description of this correspondence.
+First, recall that given an oriented link L⊂S3withmcomponents, we can
+naturally identify H1(EL) withZmin such a way that the ith positive meridian
+µiofLrepresents the ith standard basis of Zm. Therefore Hom( π1(EL),H) is
+identified with Hom( Zm,H), whereHdenotes a nontrivial free abelian group as in
+the previous sections. Suppose two links LandL′are concordant, in particular the
+concordance exterior Wis a homology cobordism between ELandEL′. Recall that
+in Theorem 3.1, we defined a bijection Ψ: Hom( π1(EL),H)→Hom(π1(EL′),H)
+usingH1(EL)∼=H1(W)∼=H1(EL′). Since the meridians of LandL′are freely
+homotopic in W, it is easily seen that the bijection Ψ induces the identity on
+Hom(Zm,H), that is, a homomorphisms π1(EL)→Hand its image under Ψ are
+identified with the same homomorphism Zm→H.
+Recall that a homomorphism ψ:Zm→Hisadmissible ifψis an epimorphism
+which is nontrivial on each µi. In this case the corresponding map π1(EL)→H,
+which will also be denoted by φ, is admissible in the sense of Section 2. Thus the
+twistedtorsion τα⊗φ(EL)canbedefinedforanyunitaryrepresentation α:π1(EL)→
+GL(k,R).
+The correspondence between representations of π=π1(−) for concordant links
+is better described in terms of representations of the lower centr al quotients π/πq.
+It causes no loss of generality, since any representation of πfactoring through a
+p-group factors through π/πqfor someqsincep–groups are nilpotent. To proceed
+we will need some technicalities discussed in the following subsection.
+4.1.Concordance and lower central series. In this subsection we state some
+knownresultsandsomefolkloreresultsonlinkconcordanceandlowe rcentralseries.
+LetL⊂S3be an ordered, oriented m–componentlink. We denote the meridians
+byµ1,...,µmandbyabuseofnotationwewilldenotethecorrespondingelements in
+π1(EL) byµ1,...,µmas well. We now denote by Fthe free group on mgenerators
+x1,...,xm. Following ideas of Levine (see [ Le94, p. 101]) we define an F/Fq-
+structure forLto be a homomorphism ϕ:π→F/Fqsuch that for any i= 1,...,m
+the element ϕ(µi) is a conjugate of xi. (Here recall that given a group πwe denote
+byπqtheqth lower central subgroup.) Furthermore, we refer to a link Lequipped
+with anF/Fq–structure as an F/Fq–link.
+Aspecial automorphism of F/Fqis an automorphism hofF/Fqsuch thath(xi)
+is a conjugate of xifor eachi.
+Lemma 4.1. LetLbe an ordered oriented m-component link. We write π=
+π1(EL).
+(1)AnyF/Fq-structureϕ:π→F/Fqinduces an isomorphism π/πq→F/Fq.
+(2)Ifϕandϕ′areF/Fq–structures for L, then there exists a special automor-
+phismΘsuch thatϕ′= Θ◦ϕ.12 JAE CHOON CHA AND STEFAN FRIEDL
+Proof.We denote by γ:F→πthe map which sends xitoµi. We also pick a map
+ψ:F→Fsuch that the map Fψ− →F→F/Fqagrees with ϕ◦γ. Finally note
+thatγdescends to a map F/Fq→π/πqwhich we denote by γ. We now have the
+following commutative diagram, where the vertical maps are the obv ious projection
+maps:
+F F
+F/Fqπ/πqF/Fq./d47/d47ψ
+/d15/d15/d31/d31/d31/d31/d31/d31/d31/d31/d31
+/d31/d31/d63/d63/d63/d63/d63/d63/d63/d63/d63/d63/d63/d63
+γ
+/d15/d15/d31/d31/d31/d31/d31/d31/d31/d31/d31
+/d47/d47γ/d47/d47ϕ
+It follows from Stallings’ theorem [ Sta65] applied to ψand from the commutativity
+of the diagram that the map ϕ◦γis an isomorphism. On the other hand it follows
+from Milnor’s theorem [ Mi57, Theorem 4] that γis surjective. We now conclude
+thatϕis an isomorphism. This concludes the proof of (1).
+The second statement is an immediate consequence of (1). /square
+Note that it follows from Lemma 4.1(1) and [ Mi57] that a link admits an F/Fq-
+structure if and only if Milnor’s ¯ µ-invariants of the form ¯ µ(i1···iq−1) defined in
+[Mi57] vanish for L.
+4.2.Obstructions to links being concordant. Now we are ready to derive the
+following theorem:
+Theorem 4.2. Suppose two m–component ordered links LandL′are concordant,
+ψ:Zm→His an admissible homomorphism, and Ris a subring of Cclosed under
+complex conjugation. If ϕandϕ′are arbitrary F/Fq–structures for LandL′,
+respectively, then there exists a special automorphism ΘofF/Fqsuch that for any
+α∈Pk(R,F/Fq)we have
+rank(L,ψ,α◦ϕ) = rank(L′,ψ,α◦Θ◦ϕ′)
+and
+τ(α◦ϕ)⊗ψ(L) =±dh·τ(α◦Θ◦ϕ′)⊗ψ(L′)∈Q(H)×/N(Q(H))
+for somed∈det(α(F/Fq))andh∈H.
+Proof.LetWbe the exterior of a concordance, E=EL, andE′=EL′. Since
+H∗(E)∼=H∗(W)∼=H∗(E′) we can apply Stallings’ theorem [ Sta65] to conclude
+that the inclusion maps induce isomorphisms
+π1(E)/π1(E)q∼=− −→π1(W)/π1(W)q∼=← −π1(E′)/π1(E′)q.
+Observe that the composition sends meridians of Lto (conjugates of) meridians
+ofL′. It follows that the composition
+π1(E′)−→π1(E′)/π1(E′)q∼=π1(W)/π1(W)q∼=π1(E)/π1(E)q−→F/Fq
+gives anF/Fq–structure on L′which we denote by ϕ′′.
+Forα∈Pk(R,F/Fq),α◦ϕ∈Pk(R,π1(E)) correspondsto α◦ϕ′′∈Pk(R,π1(E′))
+under the bijection Φ in Theorem 3.1, by the definition of F/Fq–concordance and
+the definition of Φ (see Subsection 3.1). Now by Theorem 3.1, we have
+rank(L,ψ,α◦ϕ) = rank(L′,ψ,α◦ϕ′′)
+τ(α◦ϕ)⊗ψ(L) =τ(α◦ϕ′′)⊗ψ(L′).
+By Lemma 4.1there exists a special automorphism Θ such that ϕ′′= Θ◦ϕ′.
+The theorem now immediately follow from these observations. /square
+The following corollary is equivalent to Theorem 1.1given in the introduction.TWISTED TORSION INVARIANTS AND LINK CONCORDANCE 13
+Corollary 4.3. SupposeLis anm–component slice link, ψ:Zm→His an admis-
+sible homomorphism, and Ris a subring of Cclosed under complex conjugation.
+Then for any α∈Pk(R,π(L)), we have rank(L,ψ,α) =k(m−1)and
+τα⊗ψ(L) =±dh·m/productdisplay
+i=1det(id−α(µi)ti)−1∈Q(H)×/N(Q(H))
+for somed∈det(α(π(L)))andh∈H.
+Proof.Writeπ=π1(EL). Letα∈Pk(R,π(L)). By definition αfactors through
+ap-group. Since p-groups are nilpotent it follows that there exists a qsuch that
+αfactors through π/πq. SinceLis slice, by the proof of Theorem 4.2there exists
+anF/Fq–structureϕ:π→π/πq∼=− →F/FqonL. In particular there exists α0∈
+Pk(R,F/Fq) such that α=α0◦ϕ. SinceLis concordant to the trivial link,
+we may by Theorem 4.2assume without loss of generality that Lis already the
+trivial link. (Precisely speaking, this requires the change of α0toα0◦Θ for some
+special automorphism Θ of F/Fq, but this can be ignored since det(id −α(µi)ti) =
+det(id−α0(xi)ti) is left unchanged when xiis replaced with its conjugate.) The
+corollarynowfollows from the explicit calculation for the unlink, which w ill be done
+in Theorem 6.2. /square
+5.Representation theory of p–groups
+In this section we collect a few basic facts of the theory of represe ntations fac-
+toring through p-groups. This summary will be useful in the later discussion of
+examples.
+Givenkwe denote by P(k)⊂GL(C,k) the subgroup of permutation matrices,
+i.e. matrices with exactly one non–trivial entry in each row and in each column,
+and all the non–trivial entries are equal to one. Let R⊂Cbe a subring which
+is closed under complex conjugation. We then write D(R,k)⊂GL(R,k) for the
+subgroup of diagonal matrices.
+Proposition5.1. Letπbe a groupwith generators g1,...,gn. Letα:π→GL(C,k)
+be a representation. We write Xi=α(gi),i= 1,...,n. Letpbe a prime. Then α
+factors through a p–group if and only if there exists a matrix Q∈GL(C,k)such that
+fori= 1,...,nwe can write QXiQ−1=PiDiwithPi∈P(k)andDi∈D(C,k)
+such that
+(1)P1,...,Pngenerate a subgroup of p–power order, and
+(2)the diagonal entries of D1,...,Dnarep–power roots of unity.
+Proof.We will say that a matrix Disp–diagonal if Dis diagonal and if all the
+diagonal entries are p–power roots of unity. We will several times make use of the
+following two basic observations:
+(i)IfP,P′∈P(k) andD,D′∈D(R,k) withPD=P′D′, thenP=P′and
+D=D′.
+(ii)LetP,P′∈P(k) andD,D′∈D(R,k), thenPDP′D′=PP′·P′−1DP′D′
+andP′−1DP′D′is a diagonal matrix, furthermore, if D,D′arep–diagonal,
+thenP′−1DP′D′is alsop–diagonal.
+Now assume that we have a representation α:π→GL(C,k) with the following
+property: there exists a matrix Q∈GL(C,k) such that for i= 1,...,nwe can
+writeQα(gi)Q−1=PiDiwithPi∈P(k),Di∈D(C,k) which satisfy conditions
+(1) and (2).
+We claim that α(π) is ap–group. Without loss of generality we can assume
+thatQis the identity. Let Ybe any element in the group α(π), we can write
+Y= (Pi1Di1)ǫ1·····(PirDir)ǫrfor someij∈{1,...,n},ǫj∈{±1}. We have to14 JAE CHOON CHA AND STEFAN FRIEDL
+show thatYhasp–power order. Let X=Pǫ1
+i1·····Pǫr
+ir. By assumption Xps= id
+for somes. Using the above observations we can write
+Yps=/parenleftbig
+(Pi1Di1)ǫ1·····(PirDir)ǫr/parenrightbigps
+=Xps·D
+for somep–diagonal matrix D. It follows that Yps=D(and hence Y) hasp–power
+order. This shows that α(π) is ap–group.
+Now letα:π→GL(C,k) be a representation which factors through a p–group
+P. We write Xi=α(gi),i= 1,...,n. By [We03, Corollary 4.9] any representation
+α:P→GL(C,k) of thep-groupPis induced from a representation of degree 1.
+This means that there exists a finite index subgroup Q⊂Pand a one–dimensional
+representation Q→GL(C,1) such that αis given by the natural P–left action
+onC[P]⊗C[Q]C. Note that the one–dimensional representation is necessarily of
+p–power order. Now pick representatives p1,...,plforP/Q. These representatives
+defines a basis for C[P]⊗C[Q]C, and writing αwith respect to this basis we see that
+αis of the required type. /square
+6.Boundary links
+Boundary links form an important subclass of links which has been inte nsely
+studied over the years. In this section we will show how to calculate t wisted invari-
+ants for boundary links and we will study the twisted invariants of bo undary links
+that are ‘boundary slice’.
+6.1.Invariants of boundary links.
+Definition. Aboundary link is anm–component oriented link in S3which hasm
+disjointSeifertsurfaces, i.e. thereexist mdisjointorientedsurfaces V1,...,Vm⊂S3
+such that∂(Vi) =Li,i= 1,...,m. We refer to such a disjoint collection of Seifert
+surfaces as a boundary link Seifert surface .
+Remark. Not every link is a boundary link. For example it is clear that a bound-
+ary link has trivial linking numbers. Furthermore, Milnor’s [ Mi57] ¯µ–invariants are
+trivial for a boundary link, and in fact trivial for any link concordant to a boundary
+link. Cochran and Orr [ CO90,CO93] showed that there also exist links with van-
+ishing ¯µ–invariants which are not concordant to a boundary link (cf. also [ GL92]
+and [Le94]).
+Throughout this section let Fbe the free group on the generators x1,...,xm.
+AnF–linkis a pair (L,ϕ) whereLis an oriented m–component link in S3and
+ϕ:π(L)→Fis an epimorphism sending an ithmeridian to xi. Note that ϕgives
+rise to a map from ELto the wedge of mcircles, and by a standard transversality
+argument such a map then gives rise to a boundary link Seifert surfa ce. Con-
+versely, the existence of a boundary link Seifert surface VforLproduces such an
+epimorphism π(L)→Fby the Thom-Pontryagin construction.
+Given an oriented boundary link Lwith disjoint oriented surfaces V1,...,Vm⊂
+S3we can define the ‘boundary link Seifert matrix’ as follows: Let gibe the genus
+ofVi. Fori= 1,...,mpick a basis vi,1,...,vi,2giforH1(Vi). Fork,l∈{1,...,m}
+we defineAklto be the 2gk×2gl–matrix given by ( Aij)kl= lk(vi,k,(vj,l)+) where
+(vj,l)+denotes the positive push–off of vj,lfromVjintoS3\V. We now view
+(Aij)klas anm×m–matrix of matrices and we refer to it as a boundary link Seifert
+matrix. We refer to [ Lia77] and [Ko85,Ko87] for details and for more information
+on boundary link Seifert matrices.
+We now turn to the computation of twisted invariants of boundary lin ks. Let
+(L,ϕ) be anF-link andA= (Aij) a corresponding boundary link Seifert matrixTWISTED TORSION INVARIANTS AND LINK CONCORDANCE 15
+whereAijis anri×rj–matrix. Let r=/summationtextm
+i=1riand
+T:= diag(t1,...,t1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+r1,...,tm,...,tm/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
+rm).
+We viewAt−ATas anr×r–matrix over R[t±1
+1,...,t±1
+m]. Letψ:Zm→Hbe
+an admissible epimorphism to a nontrivial free abelian group Hand letα:F→
+GL(R,k) be a representation. Note that all entries of the matrix ( α⊗ψ)(At−AT)
+are sums of monomials in one variable, we can therefore unambiguous ly define the
+matrix (α⊗ψ)(At−AT)to be therk×rk–matrix over R[H] givenby replacing each
+summandatj
+iof an entry of At−ATby the matrix aα(xi)jψ(ti)∈GL(R[H],k).
+(Here and throughout this section we will identify the additive group Zmwith the
+multiplicative group generated by t1,...,tm.)
+Theorem 6.1. Let(L,ϕ)be anF–link. LetAbe a boundary link Seifert matrix
+corresponding to Seifert surfaces given by ϕ. Letψ:π(L)→Hbe an admissible
+epimorphism to a nontrivial free abelian group H. LetR⊂Cbe a subring closed
+under complex conjugation, and let α:π(L)→GL(R,k)be a unitary representation
+which factors through the epimorphism ϕ. Then the following hold:
+(1)We have rank(L,ψ,α)≥k(m−1)and equality holds if and only if det((α⊗
+ψ)(At−AT))/\e}atio\slash= 0,
+(2)Ifrank(L,ψ,α) =k(m−1), then
+τα⊗ψ(L).= det((α⊗ψ)(At−AT))·m/productdisplay
+i=1det(id−α(xi)ti)−1∈Q(H)×
+N(Q(H)).
+(3)Ifαfactors through a p–group, then rank(L,ψ,α) =k(m−1).
+Proof.LetV1∪···∪Vm⊂ELbe a properly embedded boundary link Seifert
+surface forLcorresponding to ϕtogether with bases vi,1,...,vi,2giforH1(Vi) such
+that (Aij) is the corresponding Seifert matrix, i.e. ( Aij)kl= lk(vi,k,(vj,l)+). Let
+C=S3\(νV1∪···∪νVm). Notethattheinducedmaps π1(Vi)→F,i= 1,...,mand
+π1(C)→Fare trivial, in particular αandψrestricted to π1(Vi),i= 1,...,mand
+π1(C) are trivial. We therefore get the following exact Mayer–Vietoris se quence:
+m/circleplusdisplay
+i=1H1(Vi)⊗ZR[H]k−→H1(C)⊗ZR[H]k−→H1(EL;R[H]k)
+−→m/circleplusdisplay
+i=1H0(Vi)⊗ZR[H]k−→H0(C)⊗ZR[H]k−→H0(EL;R[H]k)−→0
+(We refer to [ FK06, Section 3] for details.) Now let ci,1,...,ci,2gi,i= 1,...,mbe
+the basis for H1(C) dual to the basis vi,1,...,vi,2gi,i= 1,...,m, i.e. lk(ci,j,vk,l) =
+δik·δjl. Using these bases it is well–known that the map
+H1(Vi)⊗ZR[H]k− −→H1(C)⊗ZR[H]k
+is represented by ( α⊗ψ)(At−AT). We write n=/summationtextm
+i=12gik. Note that ( α⊗
+ψ)(At−AT) is ann×n-matrix. We now have
+rank(L,ψ,α) = dimQ(H)H1(EL;Q(H)k)
+= dim/parenleftiggm/circleplusdisplay
+i=1H0(Vi)⊗ZQ(H)k/parenrightigg
+−dim/parenleftbig
+H0(C)⊗ZQ(H)k/parenrightbig
++dim/parenleftbig
+H0(EL;Q(H)k/parenrightbig
++dimQ(H)(H1(C)⊗Q(H)k)
+−rank((α⊗ψ)(At−AT))
+=km−k+0+n−rank((α⊗ψ)(At−AT)).16 JAE CHOON CHA AND STEFAN FRIEDL
+In particular rank( L,ψ,α)≥km−k, and equality holds if and only if the rank of
+then×n-matrix (α⊗ψ)(At−AT) is equal to n. This proves (1).
+We now turn to the calculation of τα⊗ψ(L). Suppose that rank( L,ψ,α) =k(m−
+1). Fori= 1,...,mwepickapoint ui∈Vi. Notethat{ui,{vi,1,...,vi,2gi}}defines
+a basis for the free Z–moduleH∗(Vi) and
+τ(Vi;{ui,{vi
+1,...,vi
+2gi}}) =±1∈Z
+since±1 are the only units in the ring Z. We now denote by e1,...,ekthe standard
+basis ofQk. Fori= 1,...,mwe then let
+Vi0={ui⊗ej}j=1,...,k,
+Vi1={vi,j⊗el}j=1,...,2gi,l=1,...,k.
+Note that for i= 1,...,mthe set{Vi0,Vi1}defines a basis for the free Q(H)–
+moduleH∗(Vi)⊗ZQ(H)kand
+τ(Vi;{Vi0,Vi1}) =τ(Vi;{ui,{vi,1,...,vi,2gi}})k=±1∈Z.
+We now also pick a point b∈Cand fori= 1,...,m−1 we denote by Sithe result
+of gluingVi×−1 andVi×1 together along ∂Vi×[−1,1]. Note that S1,...,Sm−1
+are a basis for H2(C). We now let
+C0={p⊗ej}j=1,...,k,
+C1={ci,j⊗el}i=1,...,m,j=1,...,2gi,l=1,...,k,
+C2={Si⊗ej}i=1,...,m−1,j=1,...,k.
+Note that the set {C0,C1,C2}defines a basis for the free Q(H)–moduleH∗(C)⊗Z
+Q(H)kand a similar argument as above shows that
+τ(C;{C0,C1,C2}) =±1∈Z.
+Now note that rank( L,ψ,α) =k(m−1) implies by (1) that the map
+m/circleplusdisplay
+i=1H1(Vi)⊗ZQ(H)k−→H1(C)⊗ZQ(H)k
+is injective. This implies that the map
+H2(C)⊗Q(H)k−→H2(EL;Q(H)k)
+is an isomorphism, and we denote by B2be the image ofC2under this isomorphism.
+By a slight abuse of notation we also write C2:={Si⊗ej}i=1,...,m−1,j=1,...,k.
+Fori= 1,...,mwe now write Pi= id−α(xi)ti. Furthermore for i= 1,...,mwe
+letµibe a meridian of Liwhich is based at band which intersects Vigeometrically
+once in the positive direction and which is disjoint from Vj,j/\e}atio\slash=i.
+We now denote by p:/tildewiderEL→ELtheF–cover corresponding to ϕ. We pick a
+point˜boverband fori= 1,...,mwe denote by ˜ µithe component of p−1(µi) such
+that∂˜µi=˜b−xi·˜b.
+Claim.Fori∈{1,...,m−1}andj∈{1,...,k}we define
+b′
+ij:= ˜µi⊗P−1
+iej−˜µi+1⊗P−1
+i+1ej.
+Then the following hold:
+(1){b′
+ij}is a basis for H1(EL;Q(H)k),
+(2)for anyi,j,kandlwe have
+b′
+ij·(Sr⊗es) =/braceleftigg
+δjs,ifr=i,
+−δjs,ifr=i+1.TWISTED TORSION INVARIANTS AND LINK CONCORDANCE 17
+(3)for anyiandjwe have
+r(b′
+ij) =ui⊗P−1
+iej−ui+1⊗P−1
+i+1ej.
+In order to prove the claim, first note that
+∂(˜µi⊗ej) = (1−xi)˜b⊗ej
+=˜b⊗(α⊗ψ)(1−xi)ej
+=˜b⊗(id−α(xi)ti)ej=˜b⊗Piej.
+It therefore follows that
+∂(b′
+ij) =˜b⊗PiP−1
+iej−˜b⊗Pi+1P−1
+i+1ej= 0.
+The calculation of b′
+ij·(Sr⊗es) follows easily from the definitions. This shows
+in particular that the b′
+ijare linearly independent and hence form a basis for
+H1(EL;Q(H)k) Finally it is straightforward to verify that
+r(b′
+ij) =ui⊗P−1
+iej−ui+1⊗P−1
+i+1ej.
+This concludes the proof of the claim.
+Fori= 1,...,m−1 andj= 1,...,kwe now let
+bij=m−1/summationdisplay
+r=ib′
+rj.
+We writeC1:={bij}. It followsimmediately from the aboveclaimthat C1is abasis,
+and thatC1is dual toC2. We now consider the following short exact sequence of
+Q(H)–complexes:
+0−→m/circleplusdisplay
+i=1C∗(Vi;Q(H)k)−→C∗(C;Q(H)k)−→C∗(EL;Q(H)k)−→0
+together with the above bases. It follows from Theorem 2.2(4) together with the
+above calculations and from the definition of torsion that
+τα⊗ψ(L,{B1,B2}) = det((α⊗ψ)(At−AT))·τ(C),
+whereCis the following complex with the canonical bases:
+0−→Q(H)k(m−1)f− −→Q(H)kmg− −→Q(H)k−→0
+f=
+P−1
+10... 0
+−P−1
+2P−1
+2... 0
+0.........
+0...−P−1
+m−1P−1
+m−1
+0... 0−P−1
+m
+
+g=/parenleftbigP1... Pm/parenrightbig
+It now follows (cf. e.g. [ Tu01, Theorem 2.2]) that
+τ(C) =m/productdisplay
+i=1det(Pi)−1.
+This concludes the proof of the second statement of the theorem .
+Finallynotethatdet( A−At) =±1(see[Ko87]), inparticularif αfactorsthrough
+ap–group, then it follows immediately from Theorem 3.2that det((α⊗ψ)(At−
+AT))/\e}atio\slash= 0. This concludes the proof of part (3) of the theorem. /square
+The following corollary follows immediately from taking Ato be the trivial ma-
+trix.18 JAE CHOON CHA AND STEFAN FRIEDL
+Corollary 6.2. LetL⊂S3be them–component unlink in S3with meridians
+µ1,...,µm. Letψ:Zm→Hbe an admissible homomorphism. Let R⊂Cbe
+a subring closed under complex conjugation, and let α:π(L)→GL(R,k)be a
+unitary representation. Then
+rank(L,ψ,α) =k(m−1)
+and
+τα⊗ψ(L) =±dh·m/productdisplay
+i=1det/parenleftbig
+id−ψ(µi)α(µi)/parenrightbig−1∈Q(H)×/N(Q(H))
+withd∈det(α(π(L)))andh∈H.
+6.2.Boundary slice links. LetL⊂S3be anm–component boundary link. We
+sayLisboundary slice ifthereexistdisjointlyembedded3–manifolds W1,...,Wm∈
+D4such that for i= 1,...,m, the boundary ∂Wiis the union of a Seifert surface
+and a slice disk for Li. It is known that a boundary slice link is boundary slice with
+respect to anyboundary link Seifert surface. We note that boundary slice links are
+slice; it is a long-standing open question whether the converse holds for boundary
+links.
+Obstructions to being boundary slice has been studied by several a uthors, in-
+cluding Cappell and Shaneson [ CS80], Duval [ Du86], Ko [Ko87], Mio [Mi87], Shei-
+ham [Sh03,Sh06]. The following torsion obstruction to being boundary slice is a
+consequence of these works.
+Theorem 6.3. Let(L,ϕ)be anm-component F-link which is boundary slice. Let
+α:π(L)ϕ− →F→GL(R,k)be a unitary representation where R⊂Cis a subring
+closed under complex conjugation. Let ψ:Zm→Hbe an admissible homomor-
+phism. If rank(L,ψ,α) =k(m−1), then
+τα⊗ψ(L) =±dh·m/productdisplay
+i=1det/parenleftbig
+id−ψ(µi)α(µi)/parenrightbig−1∈Q(H)×/N(Q(H))
+for somed∈det(α(π(L)))andh∈H.
+Proof.LetV=V1∪···∪Vmbe a Seifert surface for Lcorresponding to ϕand
+let (Aij) be the corresponding boundary link Seifert matrix. We denote by githe
+genus ofVi. By the first remark of this subsection, there are disjointly embed ded
+3–manifolds WiinD4such that∂Wi=Vi∪(a slice disk for Li). By Ko [ Ko87,
+Lemma 3.3], it follows that ( Aij) is metabolic. This means that for i= 1,...,m
+there exists an invertible 2 gi×2gimatrixPisuch that each PiAijPt
+jis of the form
+/parenleftbigg0C
+D E/parenrightbigg
+where 0 is a gi×gj-matrix. (In fact Ko [ Ko87] has shown that in higher odd
+dimensions Abeing metabolic is in fact a sufficient condition for being boundary
+slice.) It follows easily that ( α⊗ψ)(At−AT) is equivalent to a matrix of the form
+±h/parenleftbigg0X
+XtY/parenrightbigg
+whereh∈HandXandYare matrices over R[H] of sizek/summationtext
+igi. It now follows
+that
+det((α⊗ψ)(At−AT)) =±udet(R)·det(Rt) =±udet(R)·det(R)
+for someu∈H. The theorem now follows immediately from Theorem 6.1and the
+assumption that rank( L,ψ,α) =k(m−1). /squareTWISTED TORSION INVARIANTS AND LINK CONCORDANCE 19
+Remark. LetLbe a boundary link. According to Theorem 4.2the twisted torsion
+corresponding to any representation factoring through a p–group gives a sliceness
+obstruction. On the other hand, according to Theorems 6.1and6.3the twisted
+torsions corresponding to many more representations give an obs truction to being
+boundary slice. This is very similar to the situation in [ Fr05] where the vanishing of
+the eta invariant for (boundary) slice links is investigated, as well as [CKo99,Sm89]
+(cf. [Fr05, Theorem 4.8] and [ Fr05, Theorem 4.3]). Levine [ Le07] used work ofVogel
+[Vo90] to resolve the apparent discrepancy in [ Fr05] between the vanishing of the
+eta invariants for boundary links which are slice and those which are b oundary
+slice.
+7.Computation for satellite links
+7.1.The satellite construction. LetL⊂S3be anm–component oriented link
+and letK⊂S3be a knot. Let A⊂ELbe a simple closed curve, unknotted in S3.
+ThenS3\νAis a solid torus. Let φ:∂(νA)→∂(νK) be a diffeomorphism which
+sends a meridian of Ato a longitude of K, and a longitude of Ato a meridian of
+K. The space/parenleftbig
+S3\νA/parenrightbig
+∪φ/parenleftbig
+S3\νK/parenrightbig
+is diffeomorphic to S3. The image of Lis denoted by S=S(L,K,A). We saySis
+thesatellite link with companion K, orbitLand axisA. Put differently, Sis the
+result of replacing a tubular neighborhood of Kby a link in a solid torus, namely
+byL⊂S3\νA. Note that Sinherits an orientation from L.
+An important example is given by letting Lbe the unlink with two components
+andA⊂ELas in Figure 1. The corresponding satellite knot S(L,K,A) is called
+K =
+LA
+B(K)
+Figure 1. The Bing double of the Figure 8 knot.
+theBing double of Kand referred to as B(K).
+We now return to the discussion of satellite links in general. Note that the
+abelianization map π1(S3\νK)→Zgives rise to a map of degree one S3\νKto
+νAwhich is a diffeomorphism on the boundary. In particular we get an indu ced
+map
+ES=/parenleftbig
+S3\νA\νL/parenrightbig
+∪φ/parenleftbig
+S3\νK/parenrightbig
+−→/parenleftbig
+S3\νA\νL/parenrightbig
+∪νA=EL
+which we denote by f. Note that fis a diffeomorphism on the boundary and that
+finduces an isomorphism of homology groups.
+Before we state the next lemma note that the curve Adefines an element in
+A∈π(L) which is well–defined up to conjugation.
+Lemma 7.1. LetL,K,A,S =S(L,K,A)andfas above. Let Qbe a subfield
+ofCand letα:π(L)→GL(Q,k)be a unitary representation. We denote the
+representation π(S)f− →π(L)α− →GL(Q,k)byαas well. Let ψ:Zm→Hbe an
+admissible homomorphism such that ψ(A) = 0. Denote by z1,...,zkthe eigenvalues20 JAE CHOON CHA AND STEFAN FRIEDL
+ofα(A)and let∆K∈Z[t±1]be a fixed representative of the Alexander polynomial
+ofK. Then the following hold:
+(1)rank(S,ψ,α) = rank(L,ψ,α)if and only if ∆K(zi)/\e}atio\slash= 0fori= 1,...,k,
+(2)if∆K(zi)/\e}atio\slash= 0fori= 1,...,k, then
+τα⊗ψ(S) =τα⊗ψ(L)·k/productdisplay
+i=1∆K(zi)∈Q(H)×/N(Q(H))
+up to multiplication by an element of the form ±dhwithd∈det(α(π(L)))
+andh∈H.
+Before we give the proof of Lemma 7.1we first state and prove the following
+result which is an immediate consequence of Lemma 7.1and Theorem 1.1.
+Corollary 7.2. LetLbe an oriented m–component slice link and let K,Aand
+S=S(L,K,A)as above. Let Qbe a subfield of Cclosed under complex conjuga-
+tion which has the unique factorization property and let α:π(L)→GL(R,k)be a
+representation which factors through a p–group. Let ψ:Zm→Hbe an admissible
+homomorphism. Denote by z1,...,zkthe eigenvalues of α(A)and let∆K∈Z[t±1]
+be a fixed representative of the Alexander polynomial of K. Assume that ψ(A) = 0.
+IfS=S(L,K,A)is slice, then ∆K(zi)/\e}atio\slash= 0fori= 1,...,kand
+k/productdisplay
+i=1∆K(zi) =±d·qq∈Q
+for somed∈det(α(π(L)))andq∈Q.
+Proof of Lemma 7.1. We writeT=νA. Consider the following commutative
+diagram of short exact sequences of cellular chain complexes (wher e we write
+V=Q(H)k):
+0C∗(∂(T);V)C∗(EL\νA;V)⊕C∗(S3\νC;V)C∗(ES;V)0
+0C∗(∂(T);V)C∗(EL\νA;V)⊕C∗(T;V)C∗(EL;V)0/d47/d47 /d47/d47
+/d15/d15id/d15/d15id⊕f/d47/d47 /d47/d47
+/d15/d15f
+/d47/d47 /d47/d47 /d47/d47 /d47/d47
+We assumed that ψ(A) = 0. It followsthat the map ψrestricted to Tand restricted
+toS3\νKis trivial. In particular Cα⊗ψ
+∗(EK;Q(H)k) =Cα
+∗(EK;Qk)⊗Q(H) and
+Cα⊗ψ
+∗(T;Q(H)k) =Cα
+∗(T;Qk)⊗Q(H). It is a consequence of [ Go78, Section 5]
+thatf∗:H1(EK;Qk)→H1(T;Qk) is always surjective and it is an isomorphism
+if and only if ∆ K(zi)/\e}atio\slash= 0 fori= 1,...,k. The first statement of the lemma
+now follows immediately from the commutative diagram of long exact ho mology
+sequences corresponding to the above diagram.
+Now suppose that ∆ K(zi)/\e}atio\slash= 0 fori= 1...,k.
+Claim.The mapf∗:Hj(EK;Qk)→Hj(T;Qk) is an isomorphism for any j.
+We already saw above that f∗:H1(EK;Qk)→H1(T;Qk) is an isomorphism.
+Note thatEKis homotopy equivalent to a 2–complex and that Tis homotopy
+equivalent to 1–complex. In particular H2(T;Qk) = 0 and it follows from the
+long exact sequence that H2(EK;Qk) = 0 as well. Note that π1(EK)→π1(T) is
+surjective, it follows that f∗:H0(EK;Qk)→H0(T;Qk) is surjective as well. On
+the other hand we have that χ(EK) = 0 =χ(T), hence
+b0(EK;Qk) =b1(EK;Qk)+kχ(EK) =b1(T;Qk)+kχ(T) =b0(T;Qk),
+i.e.f∗:H0(EK;Qk)→H0(T;Qk) is in fact an isomorphism. This concludes the
+proof of the claim.TWISTED TORSION INVARIANTS AND LINK CONCORDANCE 21
+LetDibeQ–bases for Hi(T;Qk),i= 0,1. We endow Hi(EK;Qk) with the
+corresponding basis f−1
+∗(Di),i= 0,1. It follows from the long exact sequence
+corresponding to f:EK→TthatH∗(f:EK→T;Qk) = 0. It follows from
+Theorem 2.2(4) that
+τα(T;{Di}i=0,1) =τα(EK,{f−1
+∗(Di)}i=0,1)·τα(f:EK−→T).
+(Here and in the following lines all equalities of torsion are up to multiplica tion by
+an element of the form ±dhwithd∈det(α(π(L))) andh∈H.) Note that the
+Diare alsoQ–bases forHi(T;Q(H)k) =Hi(T;Qk)⊗Q(H),i= 0,1. In particular
+the calculations in the previous lines also work for the torsions corre sponding to
+twisting by α⊗ψ.
+LetBbe a Seifert matrix for the knot K. It follows easily from [ Go78, Section 5]
+and [Tu01, Theorem 2.2] that
+τα(f:EK−→T)−1=k/productdisplay
+i=1det(Bt−Bzi) =k/productdisplay
+i=1∆K(zi).
+Now pick a Q(H)–basisBforH1(EL;V) and denote the dual basis for H2(EL;V)
+byB′. Finally pick bases Ei,i= 0,1,2 forHi(∂T;Q(H)k). Note that the bases
+Di,B,Eigive rise to bases Fi,i= 0,1,2 forH∗(EL\intT;Q(H)k). It now follows
+from Theorem 2.2(4) that
+τα⊗ψ(ES,{f−1(Bi)}) =τα⊗ψ(EL\intT;{Fi})·τα⊗ψ(EK;{f−1(Di)})
+·τα⊗ψ(∂T,{Ei})−1
+τα⊗ψ(EL,{Bi}) =τα⊗ψ(EL\intT;{Fi})·τα⊗ψ(T;{Di})·τα⊗ψ(∂T,{Ei})−1
+(Here and in the next line the equalities of torsion are up to multiplicatio n by an
+element of the form ±dhwithd∈det(α(π(L))) andh∈H.) Combining these
+results we now obtain that
+τα⊗ψ(ES,{f−1(Bi)}) =τα⊗ψ(EL,{Bi})·k/productdisplay
+i=1∆K(zi).
+This implies the desired equality. /square
+7.2.Bing doubles. We consider the Bing double B(K) of a knot K, in order to
+provide an example for a detailed computation. Note that if Kis a slice knot, then
+B(K) is in fact a slice link (cf. [ Ci06, Section 1]). The converse is a well–known
+folklore conjecture.
+Given any knot Kall the ‘classical’ sliceness obstructions (e.g. multivariable
+Alexander polynomial, Levine–Tristram signatures) of its Bing double are trivial
+(cf.[Ci06, Section1]forabeautiful survey). Itisthereforeaninterestin gquestionto
+determine which methods and invariants detect the non-sliceness o f Bing doubles.
+For interesting results, we refer to Cimasoni [ Ci06], Harvey [ Ha08], Cha [Ch10,
+Ch09], and Cha-Livingston-Ruberman [ CLR08]. In particular in [ CLR08] it was
+proved that if Kis not algebraically slice, B(K) is not slice. Also it is known that
+there are algebraically slice knots with non-slice Bing doubles [ CK08,CHL08].
+Now letK= 41be the Figure 8 knot. It is well–known that Kis not slice.
+For example this follows from [ FM66] since the Alexander polynomial ∆ 41(t) =
+t−1−3+tis not a norm, namely not of the form ±tkf(t)f(t−1). In what follows
+we show that an appropriate twisted torsion invariant of the Bing do ubleB(41) is
+not a norm, and consequently B(41) is not slice. (The fact that B(41) is not slice
+had first been shown in [ Ch10] using the discriminant invariant.)
+Letx,ybe the meridians of the two component unlink Land denote by Athe
+unknottedcurveinthedefinitionofBingdoubles. Itiswell-knowntha tArepresents22 JAE CHOON CHA AND STEFAN FRIEDL
+the commutator [ x,y]. We write ξ=e2πi/8. Note that Z[ξ] is a UFD (cf. [ Wa97,
+Theorem 11.1]). Consider the representation α:π(L) =/a\}bracketle{tx,y/a\}bracketri}ht→GL(Z[ξ],8) given
+by
+α(x) =
+0 0 0 ξ0 0 0 0
+1 0 0 0 0 0 0 0
+0 1 0 0 0 0 0 0
+0 0−i0 0 0 0 0
+0 0 0 0 0 0 0 ξ
+0 0 0 0−1 0 0 0
+0 0 0 0 0 1 0 0
+0 0 0 0 0 0 −ξ0
+
+and
+α(y) =
+0 0 0 0 ξ0 0 0
+0 0 0 0 0 0 0 −i
+0 0 0 0 0 0 ξ30
+0 0 0 0 0 ξ70 0
+i0 0 0 0 0 0 0
+0 0 0 ξ0 0 0 0
+0 0ξ30 0 0 0 0
+0−1 0 0 0 0 0 0
+.
+It follows easily from Proposition 5.1thatαfactors through a 2–group. We denote
+the representation π1(EB(K))f− →π(L)→GL(Z[ξ],8) byαas well, and we let ψbe
+the isomorphism H1(EB(K))∼=H1(EL)→Z2. (HereK= 41.) We compute the
+eigenvalues of α(A) =α([x,y]) to be
+{±1,±i,±e2πi/16,±e2π3i/16}.
+We then calculate
+∆K(1)·∆K(−1)·∆K(i)·∆K(−i)
+·∆K(e2πi/16)·∆K(−e2πi/16)·∆K(e2π3i/16)·∆K(−e2π3i/16)
+to equal 2115. If B(K) was slice, then by Corollary 7.2we would have
+2115 =±dq·q
+for somed∈det(α(π(L))) and for some qin the quotient field of Z[ξ]. Note that qq
+is a positive real number and note that det( α(π(L)))∩R=±1. It therefore follows
+that ifB(K) is slice, then 2115 = q·q. SinceZ[ξ] is a UFD we furthermore deduce
+thatqactually lies in Z[ξ]. Now note that 2115 = 47 ·45 = 47·(6+3i)(6−3i). Since
+Z[ξ] is a UFD we only have to show that 47 is not a norm. A direct calculation
+shows that any norm in Z[ξ] is of the form
+a2+b2+c2+d2+√
+2(a(b−d)+c(b+d))
+for somea,b,c,d∈Z. One can now easily deduce that 47 is not a norm in Z[ξ].
+This concludes the proof of the claim.
+Note that the process of Bing doubling can be iterated. We refer to the work
+of the first author [ Ch10], the first author and Kim [ CK08], Cochran, Harvey and
+Leidy [CHL08] and van Cott [ vCo09] for interesting results on iterated Bing dou-
+bling.
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+arXiv:1001.0928v2  [cond-mat.mes-hall]  7 Jan 2010Magnetic Nanostructures
+K. Bennemann
+Institute of Theoretical Physics FU-Berlin
+Arnimallee 14, 14195 Berlin
+keywords: Magnetism, Nanostructures
+Contents
+1 Introduction 2
+1.1 Magnetic Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
+1.2 Magnetic Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
+1.3 Tunnel Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
+2 Theory 10
+2.1 Magnetism : Electronic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
+2.2 Magnetism : Heisenberg type theory . . . . . . . . . . . . . . . . . . . . . . . . 14
+2.3 Electronic Structure of Mesoscopic Systems : Balian–Bl och type theory . . . . . 15
+2.4 Magnetooptics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
+2.5 Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
+2.6 Theory for Magnetic Films during Growth : Nonequilibriu m Magnetic Domain
+Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
+2.7 Tunnel Junctions: Spin Currents . . . . . . . . . . . . . . . . . . . . . . . . . 29
+2.8 Quantum dot Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
+3 Results 35
+3.1 Small Magnetic Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
+3.1.1 Single Magnetic Clusters . . . . . . . . . . . . . . . . . . . . . . . . .35
+3.1.2 Ensemble of Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
+3.2 Thin Magnetic Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
+3.2.1 Magnetic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
+3.2.2 Reorientation Transition of the Magnetization . . . . . . . . . . . . . . 45
+3.2.3 Magnetooptics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
+3.2.4 Nanostructured Thin Films with Magnetic Domains . . . . . . . . . . . 51
+3.3 Tunnel Junctions: Magnetic Effects . . . . . . . . . . . . . . . . . . . . . . . . 58
+4 Summary 64
+Abstract
+Characteristic results of magnetism in small particles and thin films ar e presented. As a conse-
+quence of the reduced atomic coordination in small clusters and thin films the electronic states
+and density of states modify. Thus magnetic moments and magnetiz ation are affected. In
+tunnel junctions interplay of magnetism, spin currents and super conductivity are of particular
+interest. Results are given for single transition metal clusters, clu ster ensembles, thin films and
+tunnel systems.
+11 Introduction
+Due to advances in preparing small particles, thin films, film multilayers and tunnel junctions
+the area of nanostructures in physics has received increased inte rest. Clearly engineering on
+an atomic scale condensed matter offers many possibilities regarding new physics and technical
+applications.
+Of particular interest is the occurrence of magnetism in nanostruc tures like single tran-
+sition metal clusters (Ni, Co, Fe etc.), ensembles of such clusters, for example in lattice like
+arrangements[1], ferromagnetic thin films, multilayers of such films [ 2] and tunnel junctions[3].
+The latter are of interest, for example, regarding switching of elec tric, charge and spin currents.
+Size and orientation of magnetic moments and magnetization depend in general sensitively on
+atomic environment, atomic coordination (surfaces, interfaces e tc.).
+Of course, one expects that magnetic fluctuations are significant due to the reduced di-
+mension of nanostructures. Regarding magnetic anisotropy, this will play an important role in
+general and also with respect to the role played effectively by the ma gnetic fluctuations. Due
+to magnetic anisotropy phase transitions occur in reduced dimensio ns and ultrathin films.
+Figure 1: Illustration of b.c.c.– and f.c.c. like clusters. Different atomic shells surrounding the
+center 1 are labelled by 2, 3 etc.
+In general the topology of the nanostructure affects strongly t he electronic structure and
+the orientation of the magnetization, its domain structure, and ma gnetic relaxation. The
+electron energy spectrum gets discrete and quantum well states (QWS) occur in thin films.
+2Note, in nanostructured thin films magnetic domain structure is fre quently present. The size
+of the domains and their relaxation depends on atomic structure of the film, anisotropy and
+temperature. The reversal of a domain magnetization may be relat ively slow, but may speed
+up at nonequilibrium for example due to hot electrons.
+In growing thin films interesting nonequilibrium behavior may result fro m the interde-
+pendence of atomic morphology of the nanostructure and its magn etic domain formation. This
+maybeseen observing timeresolved propertiesofgrowingnanostr uctures. Thus, itisofinterest
+to study, time resolved, the occurrence of uniform ferromagnet ism in thin films as a function
+of film thickness and domain density and size, in general on growth co nditions for producing
+nanostructures.
+For tunneling interplay of magnetism and superconductivity is of par ticular interest. Be-
+tween two magnets one expects as a result of the spin continuity eq uation and Landau–Lifshitz
+equation Josephson like spin currents.
+In the following magnetic nanostructures having different geometr y and tunneling are
+discussed:
+1.1 Magnetic Clusters
+While small clusters may have a complicated atomic structure, this will tend to become bulk
+like for larger ones. Hence, approximately one may assume first liquid like structures as the
+number of atoms Nin the cluster increases and then bulk like structures for larger clus ters.
+The cluster volume Vis given by V∼N3and the surface area by S∼N2/3[1].
+In Fig.1 f.c.c.- and b.c.c. like cluster structures are shown. These may approximate
+magnetic transition metal clusters. In such clusters DOS and magn etic moments are site
+dependent.
+In Fig.2 the discretization of the electron energy spectrum of such clusters is illustrated.
+LevelsareoccupieduptotheFermi–level εFandinteresting odd, eveneffectsoccurasafunction
+of cluster size (number Nof atoms).
+The magnetism of the cluster is characterized by its atomic shell ( i) dependent magnetic
+momentsµi(N), bythemagnetization M(T,N) andCurie–temperature Tc(N). Thesemagnetic
+properties may be calculated using an electronic theory and for exa mple a tight–binding Hub-
+bard like hamiltonian, or on a more phenomenological level a Heisenber g hamiltonian including
+magnetic anisotropy. Thus, one may estimate the Curie–temperat ure from
+Tc∼aJqeff(N), (1)
+whereaisafittingparameter, Jtheinteratomicexchangecouplingintegraland qefftheeffective
+coordination number [2].
+InFig.2 size effects for the electronic structure aresketched. No te, screening of Coulombic
+interactions and width of d–electron states varies with cluster size and affects thus magnetic
+activity. Also spacing of the n electronic states is
+δ≈εF/n≈(/planckover2pi1vF/R)(kFR)−2(N(0)V)−1. (2)
+Here,εFis the Fermi energy and vFandkFthe corresponding Fermi velocity and wave–
+vector.Vrefers to the volume and N(0) to the density of states (D.O.S.) at εF. One expects
+forT > δthat the discretization plays no great role. However, discretizatio n might affect
+3Figure 2: Sketch of size effects in small clusters having a diameter Randnelectrons with
+energyεn. The spacing of the electronic energy levels is δ. (ǫk→discrete spectrum as particle
+size decreases). The electronic spectrum exhibits shell structur e which reflects characteristi-
+cally the atomic structure. Many properties should reflect this, in p articular magnetism and
+superconductivity.
+many properties in grains and quantum dots, in particular magnetism and superconductivity
+(formation of magnetic moments, magnetization, size of Cooper pa irs, superfluid density etc.).
+The local electron density of states of atom iis given by Ni(ε,N) and generally spin
+dependent ( Niσ(ε,N)). This determines occurrence and size of local magnetic moments . Their
+direction is determined by magnetic anisotropy. Note, besides spin m agnetism also orbital one
+occurs and is typically enhanced compared with bulk one. Also genera lly the orbital magnetism
+is dependent on atomic site.
+Mie–scattering by spherical small particles is an interesting phenom ena. One expects that
+magnetism leaves a characteristic fingerprint on the Mie–scatterin g profile. The magnetic field
+of the incident light couples to the cluster magnetization. This affect s the Mie scattering in
+particular the backscattering intensity [4].
+This effect of magnetism on Mie scattering needs be studied more. It offers an interesting
+alternative to deflection experiments by a magnetic field regarding s tudy of small particle
+magnetism.
+In Fig.3 an ensemble of clusters (or quantum dots) arranged on a lat tice is shown. In
+general the lattice sites may be occupied by ferromagnetic (or par amagnetic) clusters or may
+be empty. Dependent on the cluster pattern one gets rich physica l behavior. For example,
+while the single clusters are ferromagnetic, for larger spacing betw een the clusters one might
+get (for higher temperature) no global magnetization of the whole cluster ensemble. For dense
+spacings ofthe grains, dotsandsufficient strong interaction amon gst them, local andalso global
+magnetization may occur as a function of temperature T. Note, typically Tci(N)>Tc, where
+irefers to the cluster iandTcto the Curie–temperature of the ensemble, cluster lattice. Of
+course, also antiferromagnetism may occur.
+In general interesting phase diagrams are expected for an ensem ble of magnetic grains
+(quantum dots). One gets for interactions J >0 andJ <0 ferromagnetism or antiferromag-
+netism for the grain ensemble. For a mixture of grains, for example a mixture of supercon-
+ducting and ferromagnetic ones, one may observe like for alloys inte resting behavior and one
+may get Josephson currents between the dots and due to electro nic charge transfers odd–even
+occupation effects, etc.
+4Figure 3: Illustration of an ensemble of small particles (grains) arra nged in a lattice. The
+lattice sites may be empty or occupied by ferromagnetic, paramagn etic, or superconducting
+clusters, for example. Removing irregularly clusters from the lattic e sites creates all sorts of
+nanostructures. Electrons move via hopping and tunneling betwee n the lattice sites.
+An interesting example of an important nanostructure is an ensemb le of quantum dots,
+an anti-dot lattice for example. In Fig.4 an anti–dot lattice immersed in a medium is sketched.
+Here, the anti–dots repel electrons moving for example between t he quantum dots. An external
+magnetic field B will change, deform the electron orbits spin–depend ently and this may yield
+interesting behavior of the electronic properties of the system. S ensitive quantum mechanical
+interferences appear in such nanostructures. As indicated in Fig.4 a few electronic orbits may
+determine mostly the electronic structure, see theory by Stampfl iet al.[5]. Quantum mechani-
+cal interference cause characteristic oscillations in the electronic density of states (DOS). These
+may be spin dependent.
+In summary, these are some cluster like nanostructures. Regard ing magnetism transition–
+metal and rare–earth atoms, metal–oxydes etc. are particularly interesting. The dependence
+of magnetic spin and orbital moments and spin correlations on partic le size and temperature
+are of interest. Magnetic moments, Curie temperature and magne tization control the magnetic
+behavior [1, 2, 6, 7, 8, 9, 10].
+1.2 Magnetic Films
+Forultrathinfilmsonemaygetmagneticbehaviorwhich coulddifferdra sticallyfromtheonefor
+thick ferromagnetic films. The latter are approaching as a function of film thickness dthe bulk
+behavior (b), for the Curie–temperature one finds Tc(d)−→Tc,bas film thickness dincreases.
+The increase of Tc(N) for increasing thickness of the ferromagnetic film is often describ ed by
+the scaling law ∆ Tc/Tcb∝N−λ, or more empirically by ∆ Tc/Tc(N)∝N−λ′
+with non–universal
+exponentλ′which fits better experiments, for details see Jensen et al.[2].
+Note, in ultrathin films, one or two atomic layers thick, ferromagnet ism results from mag-
+netic anisotropy (spin–orbit coupling etc.) suppressing two–dimens ional magnetic fluctuations.
+Various film structures are shown in Fig.5. Neighboring magnetic films m ay order parallel
+or antiparallel. Approximately, first for ultrathin films Tc(d)∼dtypically. As the thickness
+of the film increases the Curie–temperature approaches the bulk o ne. Generally one gets for
+multilayer structures a rich variety of magnetic phase diagrams. Fo r example, the change
+of the magnetization upon increasing the Cu–spacer thickness of a (Co/xCu/Ni) film system
+5Figure 4: Magnetic field effects on polygonal electron paths in an ant i–dot lattice. Here, a
+denotes the spacing of the anti–dots with radius d. The electrons m ove between the dots and
+are repelled by the anti–dot potential and thus selectively the polyg onal paths 1, 2, 3 etc. yield
+the most important contribution to the spin dependent D.O.S., magne toresistance, etc. Details
+of the theory determining the electronic structure of such mesos copic systems are given by
+Stampfli et al..
+Figure 5: Thin magnetic film structures are shown. Typical configur ations are given. (a) Thin
+ferromagnetic film of thickness N on a substrate, (b) two ferroma gnetic films (FM1, FM2)
+on a substrate and (c) multilayer film structure. The ferromagnet ic films are separated by
+a nonmagnetic one (NM). At the surface the magnetic film may be cov ered by nonmagnetic
+material (cap). Of particular importance is the interplay of magnet ism of magnetic thin films
+of a film multilayer system.
+6Figure6: Thereorientationtransition M⊥→M/bardblofthemagnetizationatthesurfaceorinterface
+of films is shown. This transition may be induced by temperature T, increasing film thickness
+Nand film morphology and external magnetic field B.
+Figure 7: Illustration of a magnetic stripe domain phase of a thin film. N eighboring domains
+are antiferromagnetically oriented and separated by Bloch type do main walls. The magnetic
+structure is generally controlled by the film morphology and magnetic anisotropy. Of course,
+dependent on this other domain structures may result, see Jense net al..
+reflects characteristically the coupling between the two ferromag nets Ni and Co. For a detailed
+discussion of theoretical and experimental results see Jensen et al.[2, 6, 7, 9, 10, 11].
+Magnetic anisotropy controls the orientation of the magnetization ,− →M(T,d,...), at the
+surface of the film. Dependent on temperature T, film thickness and structure and effective
+magneticfield− →Btheorientationofthesurfacemagnetization− →Mmaychangefromperpendicular
+M⊥to parallel one M/bardbl,M⊥−→M/bardbl. This reorientation transition, important for example for
+magnetic recording, is illustrated in Fig.6 [7].
+Obviously, the orientation of the magnetization at surfaces is relat ed to the magnetic
+domain structure occurring in thin films. This is also clear from the illust ration of domain
+structure in Fig.7.
+As expected on general physical grounds, dependent on film grow th conditions one gets
+for thin film growth on a substrate a variety of nanostructures. T his is illustrated in Fig.8. For
+a growing film the accompanied magnetism may not be at equilibrium, but changing as the
+film topology changes. One has a nonequilibrium situation and magnetic structure changes due
+to magnetic relaxation, of domains etc.. The latter may be relatively slow. Various pattern
+of magnetic domains occur in general. Then reversal of local domain magnetization occur on
+a ps (picosecond) to ns (nanosecond) time scale and change the glo bal film magnetization as
+7function of time, see for example studies by Brinzanik, Jensen, Ben nemann.
+Similarly as in the case of cluster ensembles on a lattice the magnetic do mains result-
+ing in nanostructured films may be dense or separated by larger dist ances. Correspondingly
+the domains are dominantly coupled via exchange or dipolar interactio ns, for example. This
+then will be reflected in the global magnetization, size of magnetic do mains and in particular
+magnetic relaxation during film growth.
+In Fig.9 the growth of a thin film resulting from atom by atom deposition at the surface is
+shown. For simplicity fast diffusion of chemisorbed atoms is assumed. Growing of a film occurs,
+since deposited atoms prefer to sit next to already deposited atom s thus gaining optimally
+cohesion. As time progresses islands of deposited atoms coalesce a nd nanostructured film is
+formed. Of course, via diffusion of the deposited atoms the film stru cture depends somewhat on
+the film structure of the substrate. Thus, one may get island form ation, striped structures and
+quasi uniform growth, see calculations by Jensen et al.. It is remarkable that already simple
+growth models yield most of the observed structures during film gro wth.
+Regarding magnetization this reflects the nanostructure and typ ically magnetic domains
+occur before global uniform film magnetization is present.
+In thin films one expects quantum–well states (Q.W.S.) which will chang e characteris-
+tically with film thickness. Such states were calculated for example by H¨ ubner, Luce et al..
+For details see Bennemann in Nonlinear Optics in Metals (Oxford Univer sity Press) [12]. The
+confinement in thin films causes the Q.W.S. In FM films these states are spin split reflecting
+characteristically magnetism. In Fig.10 QWS are sketched and their p arity is indicated by (+)
+and (-).
+Magnetooptics (S.H.G.: Second harmonic light generation) will reflect magnetism sensi-
+tively. Thus in particular for magnetic nano film structures with spin s plit Q.W.S. one will
+get interesting optical properties reflecting characteristically ma gnetism. This is illustrated in
+Fig.11. Optical interferences involving QWS cause oscillations in the MS HG as a function of
+film thickness.
+This summarizes then interesting nanostructures consisting of film s. Film multilayers
+offer particularly interesting magnetic pattern. Neighboring films int eract largely via exchange
+coupling and this controls the magnetic behavior of each film and the g lobal one. For example
+transportpropertieslike themagnetoresistance dependonther elativeorientationofthemagne-
+tization of neighboring films. During film growth magnetic relaxation co ntrolled by anisotropy
+plays an important role. This is reflected in the nonequilibrium magnetiz ation of the film and
+its relaxation towards the equilibrium one.
+1.3 Tunnel Junctions
+Tunnel junctions involving magnetism are interesting microstructu res, in particular regarding
+quantum mechanical behavior, switching devices and charge– and s pin currents and their in-
+terdependence. Coupling of the magnetic order parameter phase s, for both ferromagnets and
+antiferromagnets, on both sides of the tunnel medium yields Josep hson like (spin) currents,
+driven by the phase difference, for details see study by Nogueira et al.and others. Obviously,
+this will depend on the magnetic state of the medium through which tu nnelling occurs, on spin
+relaxation. The effective spin coupling between N1andN3depends on the spin susceptibility
+ofN2andJeff=Jeff(χ). Note, from the continuity equation follows
+− →jT∝Jeff− →S1×− →S3+... (3)
+8Figure 8: Nanostructures of ultrathin films and accompanying magn etic domain structures.
+The upper Fig.(a) shows various observed atomic nanostructures . Questions are listed regard-
+ing dependence of the resulting magnetic structures on film topolog y. Co/Cu(001) refers to
+0.9ML of Co (black:substrate, grey:first layer, white:2.adatom layer ), Fe/Cu(111)to 0.8ML of
+Fe (grey: chains of Fe atoms which will coalesce for increasing Fe cov erage). Fig.(b) The
+obtained magnetic domain structures are given. Only domain struct ure at surface is shown.
+Magnetic domains for Co/Cu (white: 2. layer, grey: 1. layer). The ir regularly shaped domains
+(white) for Co/Au have a perpendicular magnetization and lateral s ize of about 1um. Note,
+such structures are observed in various experiments. Typically ma gnetic domain size increases
+with increasing film thickness. Increasing for Co/Au the Co film thickn ess to 6ML turns their
+magnetization to in–plane (reorientation transition).
+Figure 9: Illustration of simple (Eden type) growth of a thin film (for a square lattice). One
+assumes fast surface diffusion and uses a Monte Carlo simulation of m olecular beam epitaxial
+(MBE) film growth. Surface atomic sites are indicated. Successively deposited atoms are given
+in black and due to cohesive energy prefer to cluster.
+9Figure 10: Sketch of electronic Q.W.S. in a thin film and resulting from th e confinement of the
+electrons. Such QWS occur for example in a Co/xML Cu/Co film system . Describing the film
+confinement by a square well potential one gets already quite prop erly such QWS and which
+parity is indicated by (+) and (-). Note, parity changes for increas ing quantum number.
+The tunneling is sketched in Fig. 12.
+Clearlytunnellingallowstostudymagneticeffects, ferromagneticve rsusantiferromagnetic
+configurations of the tunnel system ( ↑|T|↑), or (↑|T|↓) and interplay of magnetism and
+superconductivity ( T≡S.C.), as for example for junctions (FM/SC/FM) or (SC/FM/SC).
+Such junctions may serve as detectors for triplett superconduc tivity (TSC) or ferromagnetism
+(FM), antiferromagnetism(AF). Of course tunnelling is different fo r parallel or antiparallel
+magnetization of M1andM3. Also importantly, tunnel junctions may serve to study Onsager
+theory on a molecular–atomistic scale.
+Electron and Cooper pair transfer between two quantum dots exh ibits interesting behav-
+ior, for example v.St¨ uckelberg oscillations due to bouncing back and forth of electrons if an
+energy barrier is present. Then assistance of photons is needed t o overcome the energy barrier.
+The current (spin–, charge current) may depend on the pulse sha pe and duration, see Garcia,
+Grigorenko et al..
+Summary: For illustrating reasons various interesting nanostruct ures like clusters, films
+and tunnel junctions with important magnetic effects are discusse d. In the next chapter some
+theoretical methods useful for calculations are presented. The n results obtained this way are
+given.
+It is important to note that for illustrational purposes the physics has been simplified.
+Mostly, the analysis can be improved straightforwardly. However, likely this will not change
+the physical insights obtained from the simplified analysis. For more d etails see for clusters
+studies by Pastor et al.[1, 6] and by Stampfli et al.[5], and for films research by Jensen et
+al.[2, 7, 8], and for tunnel junctions studies by Nogueira, Morr et al.[3].
+2 Theory
+In general theory for nanostructures must allow for a local, atom ic like analysis of the electronic
+structure. Using Hubbard hamiltonian and tight–binding type theor y one may determine
+(a)Niσ(ǫ,...), the electron density of states at an atomic site iand for spin σ, dependent on
+10Figure11: IllustrationofmagneticS.H.G.(MSHG)resultingforthinfilms withQ.W.S.. Note, in
+magnetic films the Q.W.S. arespin polarizedand their energies depend o n film thickness. Thus,
+SHG involving QWS exhibits film thickness dependent effects. Oscillation s in the magnetoopti-
+cal signals occur. The interference of S.H.G. from the surface and the interface (film/substrate)
+yields a sensitive detection of magnetism. Characteristic difference s occur for (a) weak in-
+terference and (b) strong interference. This is observed for ex ample for xCu/Fe/Cu ( weak
+interference ) and xAu/Co/Au structures ( strong interferenc e ), respectively. Here, xrefers to
+the number of layers of the top film.
+11Figure 12: A tunnel junction ( N1|N2|N3), whereNirefers to the state, magnetic, super-
+conducting, normal. The current jTmay refer to spin– or charge transport and may be driven
+by the phase difference ∆ ϕ=ϕ3−ϕ1, for example. Both spin and charge current may couple
+in general. The continuity equation for the magnetization, ( ∂tMi+∂µjiµ,s= 0), indicates that
+magnetic dynamics M(t) may induce spin currents js. As a consequence interesting nonequi-
+librium behavior is expected. Note, applying a voltage Vto the sketched tunnel system may
+yield a two–level system (resembling a qubit, etc.)
+the local atomic configuration surrounding atom i,
+(b)µ, the atomic like magnetic moment, as a function of particle size (clust er size or film
+thickness),
+(c)M, the magnetization and the Curie–temperature ( Tc).
+Note, all properties result from the electronic Green’s function Giσ(ǫ,...). Including spin–
+orbit coupling ( Vso) yields magnetic anisotropy. Thus, also
+(d) orbital magnetism
+is obtained. Note, anisotropy and orbital magnetism get typically fo r nanostructures more
+important.
+Alternatively to an electronic theory one may use on a more phenome nological level the
+Heisenberg hamiltonianincluding magneticanisotropy toanalyzemagn etism innanostructures.
+2.1 Magnetism : Electronic theory
+To determine the size and structural dependence of the magnetic properties of small transi-
+tion metal clusters the Hubbard hamiltonian for d–electrons which are expected to contribute
+dominantly is used (see Pastor et al.[1, 6] and Moran–Lopez et al.[10])
+H=/summationdisplay
+i=jtijc+
+iσciσ+H′, (4)
+H′=/summationdisplay
+iǫiσniσ−Edc, (5)
+and effective on–site electron energies
+ǫiσ=ǫ0+U∆ni−σJ
+2µi. (6)
+Here,c+
+iσandciσarethe usual creationand annihilationoperator for electrons on s iteiand with
+spinσandtijdenotes the distance dependent hopping integral. Note, i,jrefer to atomic sites
+and includes orbital character ( di:eg,t2gorbitals,s,p-orbitals).H′describes interactions (in
+12the unrestricted Hatree–Fock approximation). The effective intr aatomic Coulomb interaction
+are denoted by Uand the exchange interaction by J(J=U↑↓−U↑↑,U= (U↑↓+U↑↑)/2).
+Here,Uσσ′, refers to electron spins σ,σ′.Edc= (1/2)/summationtext
+i,σ(ǫiσ−ǫ0)<niσ>corrects for double
+counting as usual. The charge transfer ∆ niis given by ∆ ni=ni−n0,n0= (1/n)/summationtext
+ini.
+The quasi local magnetic moment at site iis given by
+µi∝<ni↑−ni↓> , (7)
+with∝an}bracketle{tniσ∝an}bracketri}ht=/integraltextǫF
+−∞dǫNiσ(ǫ−ǫiσ)t. Here,Niσ(ǫ) is the density of states (DOS) at site ifor
+electrons with spin σ(and at time t).
+This theory applies also to thin films. Then, imay refer to film layer and one should keep
+µiandp+,−
+i.
+To calculate the magnetization M(T) one must take into account the orientation of the
+magnetic moments. Assuming a preferred magnetization axis one ge ts (Ising model, see Moran-
+Lopezet al., Liu) [10]
+M(T)≃/summationdisplay
+i{p+
+iµ+
+i(T)+p−
+iµ−
+i(T)}/µ+(0). (8)
+The probabilities p+,−
+irefer to finding moments µ+,−
+ipointing parallel or antiparallel to the
+preferred magnetization axis. For simplicity one may use the approx imationp+,−
+i=p+,−.
+Assuming spherical like clusters, then irefers to the atomic shell of the cluster and µ+,−
+ito
+the magnetic moment within shell ipointing in the direction of the magnetization (+) and in
+opposite direction ( −), respectively.
+Then one determines the order parameter ( ∝M)
+µi=p+
+i−p−
+i≃p+−p−(9)
+(µi≈µ) from minimizing the free–energy
+F= ∆E−TS . (10)
+Here, the entropy Sis given by
+S≃ −kN{p+lnp++p−lnp−}, (11)
+and∆E=E(µ)−E(0). The electronic energy is calculated using a hamiltonian H, for example
+Eq.(4). For calculating the Green’s functions the electronic energie s are determined from
+ǫ+,−
+iσ≃ǫ0
+iσ−σJ/summationdisplay
+jµ+,−
+iµ+,−
+j. (12)
+Note,ǫ+
+iσ≃ǫ0
+iσ−σµ+
+iJ/summationtext
+j{p+µ+
+j+p−µ−
+j}. The Curie–temperature is given by M(Tc) = 0.
+A similar analysis can be performed using functional–integral theory as developed by
+Hubbard et al., see Pastor et al.[6].
+Note, as already mentioned this theory can also be used for films. Th enimay refer to
+the film layer, etc..
+The magnetization can also be determined using the Bragg–Williams app roximation. As-
+suming for simplicity Ising type spins one finds for the magnetization ( see Jensen, Dreyss´ e et
+al.)
+Mi(T) = tanh{βJµi(z0Miµi+z1Mi+1µi+1+z−1Mi−1µi−1)−∆hi}. (13)
+13Here,β= 1/kT,z0,z1,z−1are nearest neighbor coordination numbers and ∆ hidenotes the
+Onsager reaction field. Referring to cluster shells (film layer) z0gives the neighboring atoms
+ofiwithin shell (layer) iandz−1andz1the nearest neighbor atoms in the shell (layer) below
+and above, respectively. It is
+∆hi= (βJµi)2Mi{z0µ2
+i(1−M2
+i)+z1µ2
+i+1(1−M2
+i+1)+z−1µ2
+i−1(1−M2
+i−1)}.(14)
+Applying these expressions to films one has z1= 0 (andµ1=o) ifirefers to the surface plane
+andz−1= 0,µ1= 0 ifirefers to the film layer on a nonmagnetic substrate.
+Note, forT≤Tcthe Eq.(13) can be linearized yielding a (tridiagonal) matrix equation
+which largest eigenvalue gives Tc(d). For the hamiltonian H one may use the Heisenberg one,
+for example.
+Orbitalmagnetism, anisotropy: Addingspin–orbitinteraction VsotoEq.(4),H→H+Vso,
+and using
+Vso=−v/summationdisplay
+α,β(− →Li·− →Si)α,βc+
+αcβ, (15)
+one may also determine magnetic anisotropy and also orbital magnet ic moments ( ∝an}bracketle{tLi∝an}bracketri}ht). (− →Li·− →Si)α,βrefers to intra–atomic matrix elements between orbitals α,β. Of course, the orbital
+moment− →Lidepends on cluster atom iand on orientation δof− →Swith respect to structural
+axis:− →Li→Li,δ(see Pastor et al.[6]). In case of films irefers to the film layer.
+Note, in clusters and thin films and at surfaces the spin–orbit couplin g and orbital mag-
+netism is typically enhanced. Orbital magnetism may play an interestin g role for nanostruc-
+tures.
+2.2 Magnetism : Heisenberg type theory
+One may also calculate the magnetism in nanostructures by using the Heisenberg type hamil-
+tonian, including magnetic anisotropy (see Jensen et al.[2, 7, 8, 9, 11]). As known magnetic
+anisotropy controls the magnetic structure like direction of the ma gnetization, domains, their
+size, shape, in thin films and at interfaces in particular. Then one det ermines the magnetic
+structure resulting from the quasi local magnetic moments (− →Si) using the hamiltonian
+H=−1
+2J/summationdisplay
+i,j− →Si·− →Sj+1
+2A/summationdisplay
+i,j/parenleftBigg− →Si− →Sj
+r3
+ij−3(− →Si·− →rij)(− →Sj·− →rij)
+r5
+ij/parenrightBigg
++Hanis.,(16)
+with exchange anisotropy hamiltonian
+Hanis.=−1
+4K/summationdisplay
+i,jSz
+iSz
+j−1
+4D/summationdisplay
+i,j(Sx
+iSx
+j+Sy
+iSy
+j). (17)
+Here,KandDare uniaxial and quartic in–plane (exchange) anisotropy constant s, respectively.
+Note, inHthe first term is the Heisenberg exchange interaction and the seco nd term refers
+to the magnetic dipole interaction, and A=µ0(gµB)2/a3
+0.− →Sidenotes Heisenberg spins (
+for Ising model spin1
+2) at sitei. The second term in Eq.(16), which results as interaction
+amongst magnetic moments in classical electrodynamics, reduces in particular at surfaces the
+magnetization normal to the surface. it gives the demagnetization field and is called the shape
+14anisotropy. Although this long–range magnetic dipole coupling is usua lly much smaller than
+the exchange one, it is on a mesoscopic length scale typically very impo rtant and determines
+the magnetic domain structure.
+Of course, the parameters, in particular KandDmay depend on i,j(shell, film layer).
+Typically anisotropy is larger at surfaces, interfaces. The interpla y ofKandD, which may
+depend on film thickness and temperature ( K→Ki, whereirefers to film layer), determines
+the normal and in–plane magnetization at surfaces and the reorien tation transition (see Jensen
+et al.). Note, one has also higher order anisotropy constants like K4due to noncollinearity
+of the spins (approximately K4∝J−1), see Jensen et al., which may play a role. It is also
+important to note that exchange and dipolar coupling have a differen t distance dependence.
+The parameters in particular the anisotropy ones depend in genera l on temperature. Of course,
+temperature induces changes of the atomic structure and this pla ys a role for thin films and for
+films during growth on a substrate.
+Applying standard methods of statistical mechanism one gets the f ree–energy F, the
+magnetization and phase–diagrams in terms of Jand the anisotropy forces for clusters and
+thin films and also for nanostructured films with magnetic domain stru cture. For a more
+detailed analysis see in particular P. Jensen et al.. The magnetic phase diagram (P.D.) follows
+from minimizing the free–energy.
+From Eq.(16) follows the magnetic reorientation transition, for exa mple as driven by
+temperature:
+M⊥→TRM/bardbl (18)
+occurring at temperature TR. Note, the temperature dependence of the effective anisotropy
+parameters and of the dipole coupling is mainly determined by the magn etization. Minimizing
+the free–energy (anisotropy contribution to free–energy) give s the transition M⊥−→M/bardblat
+temperature TR, see Jensen et al.[7].
+Note, the control parameters in Eq.(16) depend on atomic struct ure, morphology of the
+nannanostructures, film thickness for example. Thus, TRis affected.
+TheaboveHubbard-tight-bindingtypeelectronictheoryandtheH eisenbergtypeplenomeno-
+logical theory including magnetic anisotropy permit a calculation of th e magnetic properties of
+nanostructures. A basic understanding of the dependence of ma gnetism in nanostructures on
+atomic configuration and on more global morphology is obtained.
+Note, for alloys one may extend above theories using appropriate v ersions of C.P.A. like
+analysis (C.P.A: coherent potential approximation).
+2.3 Electronic Structure of Mesoscopic Systems : Balian–Bl och type
+theory
+The important electronic structure (shell structure) of mesosc opic systems like spherical clus-
+ters, discs, rings, (quantum) dots can be determined using a relat ively simple theory developed
+by Stampfli et al.extending original work by Balian–Bloch (Gutzwiller) [5]. One assumes a
+squarewell likegenerally spindependent potential ( Uσ). Thedominant contribution totheelec-
+tronic structure (near the Fermi–energy and for electronic wave vectork−1
+Flarger than atomic
+distance) results from (interfering) closed electronic orbital pat hs. Then the key quantity of the
+electronic structure of a quantum dot system, the density of sta tes (DOS), can be calculated
+from (n=number of atoms in nanostructure) from the Green’s function G. The Green’s func-
+tionG(− →r,− →r′) is derived by using multiple scattering theory, see Stampfli et al.. Interference
+15of different electron paths yields oscillations in the DOS et al.. This leaves a fingerprint on
+many properties.
+Thus, for example, oscillations in the electronic structure of mesos copic systems, in the
+magnetoresistance and other properties of quantum–dot syste ms can be calculated.
+From the electronic Green’s function Gone determines the (generally spin dependent)
+density of states (DOS)
+Nσ(E,n) =1
+2πi/integraldisplay
+Vddr{Gσ(− →r,− →r′,E+iǫ)−Gσ(− →r,− →r′,E−iǫ)}− →r′=− →r.(19)
+One gets (N= average DOS)
+Nσ(E,n) =Nσ(E,n)+∆Nσ(E,n), (20)
+where ∆Nσrefers to the oscillating part of the DOS due to interference of dom inating closed
+electron paths in clusters, thin films, and ensemble of repelling anti–d ots, see the theory by
+Stampfli et al.. Clearly the scattering can be spin–dependent (due to a potential Uσ) and can
+be manipulated by external magnetic fields B(s. cyclotron paths, Lorentz–force etc.) [5].
+Undercertainconditionsregardingthepotentialfeltbytheelectr onsinthenanostructures
+(square–well like potentials etc., and states with k−1> a, interatomic distance) one gets the
+result (see Stampfli et al.using extensions of the Balian–Bloch type theory)
+∆Nσ(E,n)≃/summationdisplay
+lAlσ(E,n)cos(kLl+φlσ). (21)
+Here,lrefers to closed orbits (polygons) of length L,k=/radicalbig
+|E|+iδ, andφlσdenotes the
+phase shift characterizing the scattering potential and the geom etry of the system. An external
+magnetic field affects ∆ Nσ(E,n) via path deformation and phase shifts resulting frommagnetic
+flux (see Aharonov–Bohm effect).
+Clearly, the DOS in particular ∆ Nσ(E,n) will be affected characteristically by the mag-
+netism in various nanostructures, in quantum dot systems. For de tails of the Balian–Bloch like
+analysis see the theory by Stampfli et al.[5].
+Note, the electronic structure due to the interferences of the d ominant electronic states
+near the Fermi–energy described approximately by Eq.(21) results in addition to the one due
+to the atomic symmetry of the nanostructure (spherical one for clusters, 2d–symmetry for thin
+films, etc.). Thus, for example, one may find for magnetic clusters a phase diagram temper-
+ature vs. cluster size which dependent on magnetization exhibits ph ases where atomic (shell)
+structure and then spin dependent electronic structure dominat es. Of course, for sufficiently
+large clusters these structures disappear.
+The interference of closed electron orbits in magnetic mesoscopic s ystems like spherical
+clusters, discs, rings, thin films and quantum dot lattices causes sp in dependently character-
+istic structure , oscillations in the DOS N(E,n) (yielding corresponding ones for example in
+the cohesive energy, occupation of d,selectron states, Slater–Pauling curve for magnetism,
+magnetoresistance, etc.).
+(a) Spherical Clusters: The properties of small spherical cluster s follow from the DOS
+given approximately by
+∆Nσ= ∆No+Σ∞
+t=1Σ∞
+p=2t+1At,p,σsinφt,p,σ, (22)
+16Figure 13: Examples of dominating electron paths characterized by (p,t) in (a) spheres, discs
+and (b) rings. Note, pandtrefer to the number of corners and t to the one of circulations
+aroundthecenter, respectively, ofthepolygonialpaths. In(c) theeffect ofanexternal magnetic
+fieldBis shown. Note, in particular for rings magnetic flux quantization may occur. Also the
+phase shift due to the Aharanov–Bohm effect may induce currents . For ferromagnets these are
+spin polarized.
+17where the orbits are characterized by the number of corners pandtdescribes how many times
+thecenter ofthesphereiscircled. Here, ∆ Noresultsforp= 2tandturnsouttobeunimportant.
+For a detailed derivation of the amplitudes At,p,σand phases φt,p,σsee analysis by Stampfli et
+al.[5]. One gets
+At,pσ= 2R2at,pσ/radicalBig
+sin3(πt/p)/pexp−(2pk2Rsin(πt/p)), (23)
+withat,p=/radicalbig
+k1R/π(−1)tcos(πt/p). Note, the amplitudes decrease rapidly for increasing num-
+ber p of corners. The most important contributions to the DOS res ult from triangular, square
+and lower polygon orbits. The structure in the DOS affects many pro perties (conductivity,
+magnetoresistance, spin susceptibility χ, spin dependent ionization potential, etc.).
+(b)CircularRings: Circularringswithouterradius Raandinnerradius Riareparticularly
+interesting, since Ri/Racontrols which orbits are most important. For example, one may
+eliminate DOS contributions due to triangular orbits. One gets, see F ig.13 for illustration,
+∆N= ∆N1+∆N2+∆N3, (24)
+where ∆N1results for orbits inside the ring (with πt/p <arccos(R′/R)) and which are also
+present for discs, ∆ N2due to orbits scattered by the outer and inner surface of the ring and
+which circle around the center, and ∆ N3results from ping–pong like orbits between RaandRi.
+In all cases one gets rapid oscillations (due to exponentials) and slow er ones due to cos and sin
+functions.
+Note, results for thin films follow for Ra,Ri−→ ∞.
+(c) Quantum Dots: The DOS of quantum dots is calculated similarly. Th us, one gets for
+example oscillations in the magnetoresistance rσ, sincerσ∝N2
+σ(E,n).
+Molecular fields and external magnetic field Bwill affect the electronic orbits, see Fig.13
+for illustration. For discs, rings and quantum dots one gets phase s hifts ∆φdue to path
+deformation and the flux through the polygons. Note, the Lorent z force deforms the orbits.
+Also the magnetic flux due to a vector potential− →Aperpendicular to the discs plane for example
+gives a phase shift
+∆φ′=/contintegraldisplay− →A·d− →r=− →B·− →S, (25)
+whereSis the area enclosed by the orbit. Note, structure in DOS due to Lan dau–levels may
+be included already in N(E,n).
+(d) Circular disc and quantum dots: For circular discs and quantum d ots one gets (for
+square well like potentials) in Eq.(22) that ∆ N0≈0 and for the second term that mainly orbits
+with p=2,3 are important, since At,p,σ∼1/p1/2. Note, now at,pσ= (αt,pσ/2)/radicalbig
+1/(πk1Rp) and
+At,p=αt,pR2[(1/πk1Rp)sin3(πt/p)]1/2exp−[2pk2Rsin(πt/p)], (26)
+withαt,p= 2 forp∝ne}ationslash= 2tandαt,p= 1 for p=2t and phases Φ t,p=p[2k1Rsin(pt/p)−π/2+δt,p]+
+3π/4asforspheres. Phases δt,presult from(possibly spin dependent) potential scattering at the
+surface of the system, see books on quantum mechanics [5], note δt,pσ=−πforU→ ∞. The
+phase shift δaffects level spacing (and vise versa) and appearance of electron ic shell structure.
+Magnetic field B: The magnetic field is taken to be perpendicular to the disc plane. Then
+one gets a change of the phase by
+∆Φ = ∆Φ 1+∆Φ2, (27)
+18where∆Φ 1=pkRcΦ0
+c, Φ0
+c= 2arcsin(Rsin(Φ0)/Rc), cyclotronorbit Rc=k/B(Rc≫R), is due
+to path deformation and ∆Φ 2= ∆Φ′, see previous Eq., due to the enclosed flux (Aharonov–
+Bohm). The angle Φ0
+crefers to the center of the cyclotron orbit, see Fig.13. Straightf orward
+analysis gives
+∆Φ′=±BS±, (28)
+with
+S+,−=S0±∆,∆ = (p/2)R2
+c(φ0
+c−sinφ0
+c). (29)
+Here, 2φ0= (2π/p)t,φ0
+cis the angle seen from the center of the disc, see Fig., and for the
+cyclotron orbit one assumes R≫R.±refers to clockwise and counterclockwise motion around
+the center of the orbit. ∆ denotes the area resulting from path de formation due to field B. It
+is then
+∆Nσ(E,B)∝Σt,pAt,pσcos(SB)(....) = Σt,pAt,pσcos(S0B)sinΦ t,pexp−(pk2(k1/B)Φ0
+c),(30)
+where Φ t,pdenote phase changes due to path deformation, see previous exp ression. Factor
+cos(S0B) describes Aharonov–Bohm effect plus interference effects and g ives oscillations peri-
+odic in B. Note, cos BScauses oscillations which are periodic in B, and oscillations change with
+(a-d). For large field B, 2Rc≤(a−d), see Fig.4, path 1 and 3 disappears and only 2 remains
+(see Landau level oscillations with periodicity (1/B), since S∝(1/B2)). ∆N∝cos(SB),
+φ2∝SB. The DOS oscillations are ∆ N∝cos(SB), (Φ2∝(SB)).
+The ratio (d/a) characterizes strength of antidot potential sca ttering. Of course, the
+oscillations in the DOS cause similar ones in the magnetoresistance, fo r example.
+Antidot lattice: One assumes a 2d–lattice of antidots scattering via a repulsive potential
+the electrons. The lattice of antidots is illustrated in Fig.4 and Fig.14. N ote, orbits 1, 2, 3 are
+most important. The scattering by the antidot lattice causes phas e shifts Φ t,pσ. Note, the area
+S enclosed by the orbits 1, 2 and 3 is nearly independent of magnetic fi eldB, such orbits occur
+for 2Rc>(a−d) andRc∝1/B. One assumes that orbits are dephasing for distances much
+larger than lattice distance a. The oscillations in the DOSaregiven by ( performing calculations
+similarly as for disc, see similar paths in Fig.13 and Fig.14)
+∆Nσ(E,B) = (√
+2(a−d)/k1π)Σ∞
+t=1(sinhϕ/sinh(4t+1)ϕ)1/2cos(BS)sinφt,p
+×exp(−4tk2(k1/B))ϕ1+∆Nr(E,B), (31)
+with phases φt,p= 4t[k1(k1/B)ϕ1+B/2(k1/B)2(ϕ1−sinϕ1)+δt,4t]+π/2,coshϕ= 2a/d−1,
+ϕ1= 2arcsin[( a−d)/(k1/B)/√
+2],S= 2t(a−d)2, etc., see Stampfli et al.[5]. The first term
+is due to path 1 and ∆ Nr(E,B) due to path 2, 3 etc..
+Note, Eq.32 takes into account that path reflections result from s mall antidots (rather
+than from spherical surfaces of discs with varying curvature). T he oscillations of ∆ Nσperiodic
+inBresult for orbits 1, 2, 3, 4 and change with (a-d). For very small fie ldB, noteRc∝(1/B),
+orbits become unstable and the large orbits get irregular due to dep hasing. For increasing
+magnetic field Bsuch that 2 Rc≤(a−d) first orbit 3 and then 1 disappears and only 2
+survives. This orbit causes then for increasing field BLandau level oscillations with period
+(1/B), since S∝R2
+c∝(1/B)2, flux∼SBand ∆N∝cos(SB). The oscillation periods are
+expected to decrease for decreasing lattice constant a and incre asing magnetic field B. [5]
+The oscillations in DOS ∆ Nσ(E,B) affect many properties, in particular the magnetore-
+sistance. A crossover from periodicity proportional in B to one in (1 /B) should occur for
+example for discs.
+19Figure 14: Effect of a magnetic field on various closed electron orbits , paths: (a) for a ring,
+(b) for a lattice of anti–dots, which scatter the electrons by a rep ulsive potential. Note, in
+particular for a narrow ring path 1 may dominate and magnetic flux qu antization occurs. If the
+mean free path of the electrons is relatively large, as compared to t he circumference of the ring,
+a magnetic field may induce a current driven by the phase shift due to the Aharonov–Bohm
+effect. On general grounds this current is expected to be spin pola rized for ferromagnets and
+to decrease proportional to R−2and for increasing temperature.
+The theory by Stampfli et al.can be used also for a system of quantum dots on a
+lattice, including then hopping between the dots. Furthermore, an ensemble of magnetic and
+superconductingquantumdotsmayexhibitinterestingbehaviorre garding(quantisationeffects)
+magnetic flux etc..
+Note, in general for lower dimensions (3 −→2) degrees of freedom are reduced and one
+expects stronger oscillations of the DOS, for example in a magnetic fi eld. The interference of
+the orbits yields generally rapid and slow oscillations. The spin depende nce is calculated from
+the spin dependent scattering potential Uσat the surfaces of the system.
+Note, a ring may be viewed as an antidot enclosed by the metallic ring. T he electrons are
+repelled by a strong repulsive potential from the inner core of the r ing. If a magnetic field B is
+present, due to the flux quantization the electronic DOS reflects t his and has fine structure as
+a result of the flux quantum φ0= 2π/planckover2pi1c/e. Also a beating pattern of the oscillations may reflect
+interference of several paths dominating the electronic structu re, see Figs.. These oscillations
+are also present in the electronic energy E=/integraltext
+dεεN(ε) +.... As also discussed later the
+Aharonov–Bohm effect may induce a current in rings and discs etc. w ith interesting structure.
+For further details of the analysis for rings and films see Stampfli et al.[5]. In particular
+these may exhibit interesting behavior at nonequilibrium ( due to hot e lectrons, for example ).
+It is important to compare the results by Stampfli et al.using the Balian– Bloch type
+theory [5] with quantum mechanical calculations of the DOS. Then
+N(E)σ= Σmδ(E−Emσ), (32)
+where the eigenvalues Emσof the state |mσ∝an}bracketri}htare determined by the Schr¨ odinger equation.
+2.4 Magnetooptics
+Interesting magnetooptical behavior is exhibited by magnetic films. Nonlinear optics, SHG is
+very surface sensitive and reflects the magnetic properties of th e film. SHG is generated at the
+surface and at the interface surface/substrate or at the inter face of two films.
+20Figure 15: Optical configuration for incoming ωlight and reflected 2 ωlight. Sketch of polar-
+ization of incoming and outgoing light characterized by the angle ϕand Φ. z is normal to the
+surface and the crystal axes x, y are in the surface plane. θdenotes the angle of incidence.
+In Fig.15 SHG at surfaces is shown (light ω→2ω). The 2ω–light is characterized by the
+response function χijl(− →M,ω). The Fig. illustrates the optical configuration. Clearly, since the
+response depends on− →M, noteM⊥andM/bardblyield different χijl(− →M), the reorientation transition
+of the magnetization can be studied optically.
+For thin films the nonlinear susceptibility, response function χ2may be split into the
+contribution χsfrom the surface and χifrom the interface. Then owing to the contribution
+χsχito the SHG intensity I∝ |χ|2the relative phase of χsandχiis important. Furthermore,
+the magnetic contrast
+∆I(2ω,M)∝I(2ω,M)−I(2ω,−M) (33)
+will reflect the film magnetism, since the susceptibility has contributio ns which are even and
+odd inM, see Bennemann et al.in Nonlinear Optics, Oxford University Press.[12]
+Note, high resolution interference studies are needed to detect a lso for example lateral
+magnetic domain structures of films. Also polarized light reflects mag netism and in particular
+the magnetic reorientation transition, see Fig.16 and following result s.
+In magnetooptics (MSHG) using different combinations of polarizatio n of the incoming
+and outgoing light, see Fig.15, one may analyze
+χijl(− →M). (34)
+Regarding the dependence on magnetization, for the susceptibility odd in magnetization one
+writes
+χo
+ijl=χijlmMm+... (35)
+(Note,χ=χe+χo). Thus, in particular χijl(Mc) changes characteristically for c=x,y, orz.
+For example, for incoming s–polarized light and outgoing p–polarized 2 ωlight, see Fig. 15 for
+illustration, the nonlinear susceptibility χijl(My) dominates. Similarly χijl(Mz) dominates in
+case of outgoing s–SHG polarization. For general analysis see H¨ ub neret al., Nonlinear Optics
+in Metals, Oxford Univ- Press.[12]
+Regarding susceptibility χeandχonote further interesting behavior could result from
+terms which are higher order in the magnetization M ( χe=χe
+0+aM2+..., etc.) and χijl
+reflects also the strength of the spin–orbit interaction. As discus sed by H¨ ubner et al.in case of
+(a)− →M∝bardbl− →x(longitudinal configuration), the tensor χijlinvolvesχyxx,χyyy,χyzz,χzxx,
+χzyy,χzzz,...and in case of
+21Figure 16: Polarization dependence of SHG. Note, the difference be tween magnetic Ni and
+nonmagnetic Cu. The angle ϕrefers to the incoming light polarization. Curves (a), (b), (c) for
+Cu result from using different input parameter for the diffraction ind exn(ω) and fork(ω), see
+calculations by H¨ ubner et al..
+(b)− →M∝bardbl− →z(polar configuration, optical plane x,z) elementsχzxx,χzzz,χxyzandχxzxof
+susceptibility χijloccur.
+For longitudinal configuration and polarization combination s→p χzyyand forp→s
+χyxx,χyzzare involved. Furthermore, in case of polar configuration and polar ization combina-
+tionss→pelementχzxxand forp→sthe element χxyzoccur. This demonstrates clearly
+that Nolimoke as well as Moke can observe the magnetic reorientatio n transition and other
+interesting magnetic properties of nanostructures.
+InFig.16resultsbyH¨ ubner et al.forthepolarizationdependence ofSHG–lightareshown.
+Note, agreement with experiments is very good. Obviously magnetis m is clearly reflected.
+Similar behavior regarding polarization dependence is expected for t he magnetic transition
+metals Cr, Fe, and Co. Due to ∝an}bracketle{td|z|d∝an}bracketri}ht ≈ ∝an}bracketle{td|x|d∝an}bracketri}htfor the dipole transition matrix elements
+one gets typically same results for s– and p–polarized light. To under stand the behavior of Cu,
+note∝an}bracketle{ts|z|s∝an}bracketri}ht ≈ ∝an}bracketle{ts|x|s∝an}bracketri}ht ≈0. Curves (a) and (b) refer to wavelenghts exciting and not
+exciting the Cu d–electrons, respectively. Generally as physically expected Is(p−SH)NM−→ω
+Is(p−SH)TM. Again for a detailed discussion of the interesting polarization depen dence see
+magnetooptics and discussion by H¨ ubner et al..
+In particular, interesting magnetooptical behavior of SHG and Mok e results also from
+the spin–dependent quantum–well states (Q.W.S.) occurring in thin fi lms. These states result
+from the square–well potential representing the confinement of the electrons in a thin film. In
+magnetic films the resulting electron states are of course spin–split . Characteristic magnetic
+properties follow. In contrast to band like–states for the electro ns in the film only Q.W.S.
+show a strong dependence on film thickness. Thus, to study film thic kness dependent (optical)
+behavior SHG involving Q.W.S. needs be studied.
+Oneexpects characteristic behavior oftheopticalresponse, its magnitudeanddependence
+on light frequency ωand− →M. Characteristic oscillations in the MSHG signal occur, since QWS
+involved in resonantly enhanced SHG occur periodically upon increasin g film thickness, see Fig.
+for illustration (note, SHG involving transitions i→j→land with one of these states being
+22Figure 17: Illustration of quantum well states (QWS) due to confine ment in a thin film of
+thickness d. Such states occur for example in a Co/10MLCu/Co film s ystem. Note, the DOS
+of the QWS is strongly located at the interface. In magnetic films one gets generally spin split
+QWS. The parity of the QWS changes for increasing energy.
+a QWS).
+Clearly, the Q.W.S. energies shift with varying film thickness. Then SHG light intensity
+(I2ω) involving these states may oscillate as a function of film thickness an d this in particular
+may reflect the magnetism of the film. Clearly, owing to the periodic ap pearance of Q.W.S.
+at certain energies for increasing film thickness, SHG involving these states may be resonantly
+enhanced and then oscillations occur as a function of film thickness.
+A detailed analysis, see calculation of QWS by Luce, Bennemann, show s that the SHG
+periods depend on the parity of the Q.W.S., the position of the Q.W.S. wit hin the Fermi–
+see or above, and on the interference of second harmonic light fro m the surface and interface
+film/substrate, etc. If this interference is important, then SHG r esponse is sensitive to the
+parity of Q.W.S., light phase shift at the interface and inversion symme try of the film. If this
+interference is not important, then SHG response and oscillations a re different, see Fig.11 for
+illustration of the interference.
+Thus, for an analysis of SHG involving QWS one may study:
+(a)χi≃χs, when interference is important. Different behavior of SHG may res ult then
+if 1. QWS is involved as final state, 2. as intermediate state, or 3. bo th QWS as intermediate
+and final state matter. Note, for χsχi→(−1), for example due to inversion symmetric films
+or a phase shift πat the interface, the destructive interference yields no SHG–sign al. Also if a
+final QWS near the Fermi–energy (and which may set the period of SH G–oscillation) has even
+parity, then in contrast to a QWS with odd parity no SHG signal occur s, since for the latter
+the product of the three dipole matrix elements is small.
+(b)χi∝ne}ationslash=χs, and interference is unimportant. Then different oscillation periods may
+occur. For example if a QWS above the Fermi–energy becomes availab le at film thickness
+d1for SHG then a first peak in SHG appears and then at film thickness 2 d1again at which
+previous situation is repeated, and so on. Of course, strength of signal depends on wavevector
+k in the BZ, DOS and frequency ωmust fit optical transition. In case of a FM film then the
+resonantly enhanced SHG transitions are spin dependent and an en hanced magnetic contrast
+∆Imay occur.
+If occupied QWS below the Fermi–energy are involved, then also oscilla tions occur, in
+particular due to DOS, see transition metals. If QWS below and above the Fermi–energy cause
+oscillations then the period of SHG may result from a superposition, a s an example see the
+23Figure 18: Illustration of electronic SHG transitions involving a QWS de termining the oscil-
+lation period. If the QWS has even parity then resulting SHG is small. On ly QWS with odd
+parity cause due to larger dipole matrix elements larger SHG and oscilla tions. In FM–films the
+QWS is spin split. Note, parity of QWS changes for increasing quantum number.
+behavior of xCu/Co/Cu (001) films.
+Characteristic properties of film SHG are listed in table 1. Various pro perties are listed
+for the case of no interference ( χi≫χs) and strong interference ( χi≈χs) of light from surface
+and interface. The first case is expected for a film system xCu/Fe/ Cu(001), for example, since
+the interface Cu/Fe dominates due to the QWS of the Cu film and the la rge DOS of Fe near
+Fermi energy εF(see SHG transitions: Cu→ωFe→ωQWS→2ωCu). Thus,χi≫χsand
+due to the QWS above εFone gets two and more SHG oscillation periods.
+The caseχi≈χsis expected for a film xAu/Co(0001)/Au(111), for example, since n o
+Cod–states (see band structure of Co) are available as intermediate s tates for SHG transitions
+and the QWS in Au just below the Fermi–energy controls the SHG cont ribution, see Fig.18
+for illustration. The parity of the QWS causes then oscillations Λ = 2Λ M(Mrefers to Moke),
+since clearly the 3 dipole (p) matrix elements ∝an}bracketle{td|p|QWS∝an}bracketri}ht∝an}bracketle{tQWS|p|d∝an}bracketri}ht∝an}bracketle{td|p|d∝an}bracketri}htare much
+smaller if QWS has even symmetry (note, pis odd and dis odd).
+The wave vector kdependence is controlled by the Brillouin zone structure (BZ). For a
+detailed discussion see Nonlinear Optics in Metals, Oxford University P ress. [12]
+The interesting S.H.G. (second harmonic light) interference resultin g from surface s and
+interfaces i and reflecting sensitively magnetic properties of the fi lm may be analyzed as follows.
+The SHG light intensity I(2ω) is approximately given by
+I(2ω)∼|χijl|2, (36)
+whereχdenotes the nonlinear susceptibility. Note, χmay be split as (s:surface, i:interface)
+χijl(2ω) =χs
+ijl+χi
+ijl. (37)
+One gets
+I(2ω)∼2|χs
+ijl|2+2χs
+ijlχi
+ijl+.... (38)
+Obviously the intensity I(2ω) depends on the resultant phase of the susceptibilities χsandχi.
+24Figure 19: Multifilm system with antiferromagnetically ordered neighb oring films. The shaded
+areas indicate interface regions important for SHG. Note, SHG res ults also at the surface.
+Typically approximately two atomic layers will contribute. The magnet ization is assumed to
+be parallel to the surface which normal is in z–direction.
+Then assuming, for example, that χsandχiare of nearly equal weight ( |χs|∼|χi|), it
+mayhappenthatthe2.terminEq.(38) cancels thefirst one, see for exampleinversion symmetry
+in films (χs
+ijl→ −χi
+ijl) or phase shift of the light by πat the interface.
+If the interference of light from the surface and interface is neglig ible then different oscil-
+lations of the outcoming SHG light as a function of film thickness occur .
+The weight of the optical transitions i→j→lchanges as the film thickness increases,
+see later discussion. Thus, Moke and Nolimoke oscillations occur.
+Regarding optical properties, the morphology of the thin film and its magnetic domain
+structure should play a role in general.
+Film multilayers: For magnetic film multilayers one expects interesting in terferences and
+magnetic optical behavior. Assuming for example the structure sh own in Fig.19, neglecting for
+simplicity QWS, one gets for SHG
+I(2ω)∼ |χs(− →M)+χint.1+χint.2+...|2. (39)
+. Writingχ=χe+χo,χo∼M, one has
+I(2ω)∼ |χe
+s+χo
+s(M)+Σiχe
+int.i+...|2(40)
+for the af structure. In the case of ferromagnetically aligned films it is
+I(2ω)∼ |χs+Σiχint.i(− →M)|2. (41)
+Note, to detect magnetic domain structure experiments must ach ieve high lateral reso-
+lution, via interferences for example, etc. Of course surfaces wit h a mixture of domains with
+magnetization M⊥andM∝bardblmight yield a behavior as observed for the magnetic reorientation
+transition.
+25Table 1: Characteristics of SHG response from thin films. Its depen dence on the film thickness
+involves QWS. The SHG oscillations reflect magnetic properties of the film.
+|χi|≫|χδ| |χi|=|χδ|
+kselectivity (xCu/Fe/Cu(001) for ex-
+ample)
+Strongmanetic signal due to
+strong (magnetic) interface
+contributionsWeakSHGsignal, fromonly
+fewkpoints and without
+strong interface contribu-
+tions
+Sharp SHG peaks due to
+few contributing kpoints
+resulting in strong reso-
+nancesDoubled period and addi-
+tional periods are frequency
+dependent
+Strong frequency depen-
+denceoftheSHGoscillation
+of the QWS in the k⊥direc-
+tionMOKE period absent; dou-
+bled and additional SHG
+period visible
+MOKE period and larger
+periods visible; no exact
+doubling of the MOKE pe-
+riod
+Nokselectivity (xAu/Co(0001)/Au(111)
+for example
+Strong magnetic signal,
+since strong interface con-
+tributionBroadSHGpeaks,
+since contributions come
+from many kpointsSmaller magnetic contribu-
+tion, since interface and
+(nonmagnetic) surface con-
+tributions are of the same
+magnitude
+Weakfrequencydependence
+of the oscillation periodBroad, smooth peaks, since
+interference effects do not
+change magnitude
+MOKE period and larger
+periods presentSHG oscillation periods
+rather independent of the
+frequency, since the SHG
+signal is caused by the
+QWS near EF
+MOKE period absent, dou-
+bled period present
+262.5 Magnetization Dynamics
+The time dependence of the magnetization is of great interest. Gen erally due to reduced
+dimension dynamics in nanostructures could be faster than in bulk. F or example switching
+of the magnetization in thin films is expected to be faster. Of course , this depends on film
+thickness, magneticanisotropyandmolecularfieldsofneighboringm agneticfilmsinamultilayer
+structure.
+To speed up a reversal of the magnetization (− →M↔ −− →M) within ps– times to fs times, one
+may use thin magnetic films for example with many excited electrons, h ot electrons, resulting
+fromlight irradiation. Inheterogeneous magneticfilmstructures t ransferofangular momentum
+may be fast so that a fairly fast switching of the topmost film magnet ization may occur.
+The magnetic dynamics is described by (see Landau–Lifshitz equatio n)
+d− →Mi/dt=−/parenleftBiggµB
+/planckover2pi1/parenrightBig− →Mi×− →Heff+G
+M2s− →Mi×(− →Mi×− →Heff). (42)
+Here,irefers to a film or a magnetic domain in a nano structured film and− →Heffto a molecular
+field acting on− →Miand resulting for example from neighboring films or magnetic domains. The
+first term describes precessional motion and the second one relax ation.Msis the saturation
+magnetization (Note, using− →Mi×− →Heff→∂− →Mi/∂tin the 2. term on the r.h.s. of Eq.(42) yields
+the Gilbert equation with damping parameter Gdescribing spin dissipation).
+One gets from Eq.(38) (and also from Boltzmann type theory) that the magnetization in
+thin films changes during times of the order of (controlled by angular –momentum conservation)
+1
+τM∝A(Tel)|V↑↓|2NN+... (43)
+Here, in case of excited electrons Telmay refer to the temperature of the hot electrons and
+V↑↓to the spin flip scattering potential causing changes in the magnetiz ation (for example:
+spin–orbit scattering or exchange interactions) and Nis the average DOS of the electrons. [13]
+Note, in case of transition metals with many hot electrons Eq.(35) ma y yield response
+timesτMof the order of 100 fs. Clearly, raising the temperature, T→Teldue to hot electrons,
+can speed up the magnetic dynamics.
+The Landau–Lifshitz (LL) equation is generally important for spin dy namics. One may
+use the spin continuity equation ( ∂tMi+∂µjiµ,σ= 0) to determine the spin currents− →jσ(t)
+(including spin Josephson one) induced by magnetization dynamics an dthen describe the latter
+by using the LL–equation [3].
+As will be discussed one gets for magnetic tunnel junctions (presu mably best if τs> τt)
+spin current driven by phase difference between magnets of the fo rm
+− →js∝∂− →M/∂t∼sin(∆Φ). (44)
+This follows from the LL–equation and the spin continuity equation (s ee nonequilibrium mag-
+netism).
+2.6 Theory for Magnetic Films during Growth : Nonequilibriu m
+Magnetic Domain Structure
+The magnetic structure of thin films can be controlled by film growth c onditions, since mag-
+netism depends on the atomic structure of the film. Obviously this is o f great interest for
+27engineering the magnetic properties of films [8, 11].
+During growth of the film one has a film structure changing in time. This changes the
+magnetization. Thus, also the film magnetization changes in time (not e, magnetic relaxation
+processes may occur somewhat time delayed). Then
+− →M(− →r,t,{atomic structure (t) }) (45)
+describes the nonequilibrium magnetization of growing films. Clearly, t he magnetic structure
+(domain structure) changes in time tas the atomic structure changes during growth of the film.
+For simplicity one may use a kinetic MC simulation (MC: Monte Carlo metho d) for molecular
+beam epitaxial film growth (MBE) to calculate the film structure and t he accompanying mag-
+netic one.. The nonequilibrium behavior reflects sensitively magnetic p roperties of the film,
+relaxation processes, range of magnetic forces and magnetic dom ain structure.
+The atomic structure of the growing film is determined from a particu lar film growth
+model. For example, in the Eden type growth model the atoms are ra ndomly deposited (to
+sitesi) with probability
+pi=∝e−Ei/kT, (46)
+whereEiis the atomic binding energy depending on the local coordination numb erz(Ei≈
+−A√zi). For illustration see Fig.9.
+Different film growth is obtained by using layer dependent parameter sA. One may also
+take into account diffusion of the deposited atoms. This yields then ir regular film structures
+during growth with varying clusters of deposited atoms (islands of d ifferent sizes and shapes).
+The resulting magnetic structure consists typically of an ensemble o f magnetic domains.
+The magnetic domains may not be at equilibrium and will respond to the in time changing
+atomic structure of the growing film. This may occur time delayed. Th us, successively atomic
+structure and magnetic structure change.
+Magnetic relaxation flipping the domain magnetizations is described by using a Markov
+equation and Arrhenius ansatz for the magnetic transition rates.
+In summary, for each obtained atomic structure of the growing film one calculates the
+corresponding magnetic (domain) structure. Using the Heisenber g type hamiltonian including
+anisotropy it is obvious how by changing the atomic coordination the e xchange interactions
+change and thus magnetism. For simplicity one uses first Eq.(16). Fo r details of the analysis
+see Brinzanik, Jensen et al.[11].
+Spin relaxation assumes coherent rotation of the spins of each ato mic cluster (deposited
+group of atoms). The magnetization (one may use for simplicity an Is ing model) of such atomic
+clusters is directed along the easy axis. Magnetic relaxation is (using Arrhenius type rates for
+transitions− →Mi→ −− →Mi) given by
+Γi= Γ0e−Eb
+i/kT. (47)
+Here,Eb
+igives the energy barrier against switching of the domain magnetizat ion. Note, if no
+energy barriers are present then one may use the Metropolis algor ithm. The magnetic barrier
+energy of a cluster controlling the domain relaxation is for example giv en by
+Ei=NiKi(T)cos2ϕ−gµBHiSicosϕ+... . (48)
+The first term is due to magnetic anisotropy and the second due to t he fieldHi(exchange
+interaction, external magnetic field, dipolar field) acting on spin Si.
+28Generally an energy barrier Eb
+iis present for the two magnetic orientations of magnetic
+clusteri. One applies then asmentioned already a kinetic MCsimulation during film growth to
+obtain the magnetic structure corresponding to a given atomic str ucture at time t. Correlated
+relaxation of neighboring magnetic domains must be taken into accou nt. Note, the time for
+each calculational step is set by the spin precession frequency Γ 0≈109−1012sec−1.
+Asuccessful analysis has been derived by Brinzanik andJensen et al.[11]. In their studies
+first the atomic structure of the film at time t is calculated using a Ede n type growth model.
+Then the corresponding magnetic structure is determined perfor ming a kinetic MC simulation
+and using Markov equation and for the system of magnetic domains ( interacting via exchange
+and dipolar coupling) the energy
+E≃ −(1/4)Σi,jγijLij− →Si·− →Sj−ΣiKiNi(Sz
+i)2+∆E. (49)
+Here,γijis the domain wall energy, Lijthe island surface area, and Nithe number of island
+atoms. The 2. term describes anisotropy. And
+∆E= Σi>j(µiµj/r3
+ij) [− →Si·− →Sj−3(− →rij·− →Sj)(− →rij·− →Sj)/r2
+ij]
+−Σiµi− →B·− →Si. (50)
+Here, forsimplicityislandswith Niatomsandalignedmagneticmomentsaretreatedasparticles
+with magnetic moments µi(T). The film magnetization results from averaging over the domain
+magnetization.
+Typically one gets first for the given nanostructured film magnetism which is not at
+equilibrium. Anon–saturatedmagnetization istypically obtained. Ast imeprogresses magnetic
+relaxation processes, magnetization reversals of domains, for ex ample, occur. These change the
+magnetization of the film. Thus, finally equilibrium magnetization is obta ined. For growing
+film thickness one might get an uniformly magnetized film, for details se e analysis by Jensen
+et al.[8, 11].
+2.7 Tunnel Junctions: Spin Currents
+Tunnel junctions ( N1|N2|N3) are interesting nanostructures regarding (ultrafast) switchin g
+effects, interplay of magnetism and superconductivity, and in gene ral quantum mechanical in-
+terference effects. Note, Nimay refer tomaterial which isferromagnetic orsuperconducting, for
+example. The situation is illustrated in Fig. 20. On general grounds on e may get spin currents
+driven by phase difference between two magnetic systems, ferrom agnets or antiferromagnets
+(which order parameter is also characterized by a phase) [3].
+Interesting tunnel junctions are shown in Fig.(20). In Fig. 20(a) is shown a two quantum
+dot system niσwith energy ǫiσcontrolling the tunnel current jiσ
+σ. The position of the energies
+ǫimay depend on occupation (see Hubbard like hamiltonian: εiσ=εo
+i+Uni¯σ+...). Note,niσ
+and thus the current jσcan be manipulated optically.
+ThetunnelsystemshowninFig.(20b)canbeusedasasensorfortr ipletsuperconductivity.
+Then the configuration of the angular momentum− →dof the triplet Cooper pairs and of the
+magnetization− →Mof the tunnel medium control characteristically the tunnel curre nt. The
+Josephson current exhibits interesting behavior, for example upo n rotating the magnetization
+relative to the angular momentum of the Cooper pairs.
+29In Fig.(20c) is illustrated how tunneling can be used (with the help of a b ias voltage) to
+control magnetoresistance and to determine the magnetization o f a ferromagnet (see Takahashi
+et al.) [3]. Note, upon changing ( ↑ |N2| ↑)→(↑ |N2| ↓) the magnetoresistance increases,
+see GMR effect, and accumulation of spin polarized electrons occurs inN2. This has been
+discussed by Takahashi et al.[3]. Furthermore, if N2becomes superconducting a competition
+between superconductivity and magnetization occurs for configu ration (↑ |N2| ↓) as a result of
+accumulating nonequilibrium spin density in N2. In Fig.20(b) and Fig.20(c) we assume τs>τt,
+whereτsandτtrefer to the spin diffusion and electron tunneling time, respectively. Electrons
+keep spin while tunneling.
+For tunnel configuration ( FM1|TSC|FM2) one expects similar interesting tunneling be-
+havior, for example regarding dependence on relative orientation o f− →Miand angular momentum− →dof the triplett Cooper pairs as for junctions ( TSC/FM/TSC ), see Fig.20(b).
+Furthermore, the current jJ
+sexpected for τs>τtdepends on relative magnetization of the
+two magnets, and on the state of N2(normal, superconducting singulet or triplet).
+Spin Currents : Using the continuity equation one can find the relatio nship between spin
+currents and magnetization dynamics for the magnetic tunnel jun ction illustrated in Fig.(12).
+One has
+∂tMi+∂µjiµ,σ= 0. (51)
+This may give under certain assumptions straightforwardly the con nection between magnetiza-
+tion and spin polarized electron currents (induced by hot electrons , temperature gradients or
+external fields). Note, the magnetic dynamics (characterized bydMi
+dt) may be described by the
+LL–equation. This yields generally a spin Josephson current betwee n magnets.
+Also according to Kirchhoff the emissivity (e) of the junction is relate d to its (time de-
+pendent) magnetization, magnetic resistance. It is
+∆e
+e≃a(GMR), (52)
+where(GMR)denotesthegiantmagnetoresistance resulting fort hejunctioniftheconfiguration
+(↑|N2|↑) changes to ( ↑|N2|↓). This changes the emissivity to ∆ e.
+Viewing the phase of the magnetic order parameter φi(magnetization Mi)
+M=|Mi|eiφi(53)
+similarly as the phase of the S.C. state, one gets for a junction ( FM|N|FM) a Josephson
+like current jJdriven by the phase difference of the spin polarizations on both sides of the
+tunnel junction. Using the continuity equation for the spins, integ rating using Gau integral,
+and the Landau–Lifshitz equation, one gets for the spin current
+jσ=j1
+σ(V)+jJ,jJ∝dM/dt∝− →ML×− →MR+...∝ |ML||MR|sin(φL−φR+...).(54)
+Here,LandRrefer to the left and right side of the tunnel junction, respective ly.j1
+σ(V) refers
+to the spin current due to the potential Vand may result from the spin dependent DOS. For
+details of the derivation of the spin Josephson current see Nogueir aet al.[3].
+Note, using the general formula j=∂F
+∂φ, whereFis the free–energy, one gets
+j=−(e//planckover2pi1)Σi∂Ei/∂φtanh(Ei/kT). (55)
+30Figure 20: Illustration of various tunnel junctions. (a) A two quan tum dot system with spin–
+dependent levels ǫiσ(niσ,t) is shown (for example εiσ=εo
+i+Uini¯σ). Two levels are separated
+by an energy barrier which can be overcome using an external elect ric field. (b) A triplet
+superconducting Josephson junction ( SC1|FM|SC3) is sketched. Here,− →drefers to the
+angular momentum of the Cooper pairs which order parameter has t he phaseφ. (c) A junction
+(FM1|SC|FM2) is illustrated. An external electrical field (potential V) shifts the electronic
+energy levels (bands). This may control the tunnel current (mag netoresistance), see Takahashi
+et al.. Of particular interest is the interplay of spin current and superco nductivity.
+31Then using Ei∝Jeff− →SL·− →SR+...,one gets also
+jJ
+σ∼Jeffsin(φL−φR)+.... (56)
+Here,Eigives theenergydifference between oppositedirectionsofthemag netization(molecular
+field).
+Of course, such a Josephson like spin current is expected on gener al grounds, since φand
+Szare canonical conjugate variables and ( approximately )
+[φ,Sz] =i. (57)
+Note, this holds for the Heisenberg hamiltonian as well as for itineran t magnetism described
+by the Hubbard hamiltonian, for example. The commutator relations hip suggests in analogy
+to the BCS–theory to derive the spin Josephson current from the hamiltonian
+H=−EJS2cos(φL−φR)+µ2
+B
+2Cs(Sz
+L−Sz
+R)2+... , (58)
+where again L,Rrefer to the left and right side of the junction and Csdenotes the spin
+capacitance. In general spin relaxation effects should be taken int o account (magnetization
+dissipation, see LL–equation). Using then the (classical) Hamiltonian equations of motio n
+(φ=∂H/∂Sz,Sz=−∂H/∂φ) one gets
+∆˙φ= 2µBVs, jJ
+S= (2EJS2/µB)sin∆φ . (59)
+Here, ∆φ=φ1−ϕ2andVs= (µB/Cs)(Sz
+L−Sz
+R). That ∆ ˙φ= 0 ifML∝bardblMRand ∆˙φ∝ne}ationslash= 0
+ifMLis antiferromagnetically aligned relative to MRcan be checked by experiment. It is
+jJ
+s∼sin(∆φ0+4Mt). Note, details of the analysis for the a.c. like effect are given by Nog ueira
+et al.
+Thus, interestingly the spin current in a FM|FMtunnel junction behaves in the same
+way as the superconductor Josephson current. Of course, as a lready mentioned magnetic re-
+laxation (see Landau–Lifshitz–Gilbert Eq.) affects jJ
+S. For further details see again Nogueira
+and Bennemann [3].
+Clearly, oneexpectsthatalsojunctionsinvolvingantiferromagnet s(af),(AF/F),(AF/AF)
+yield such Josephson currents, since using the order parameter f or an AF one gets also that
+Sz
+qandφare conjugate variables. (Treat af as consisting of two fm sublatt ices). Eq.57 should
+hold for both J >0 andJ <0, see Heisenberg hamiltonian.
+Note, alsoJeff=Jeff(χ) is a functional of the spin susceptibility χ, since the effective
+exchange coupling between the LandRside of a tunnel junction is mediated by the spin
+susceptibility of system N2, see Fig.12.
+The analysis may be easily extended if anexternal magnetic field− →Bispresent. Then from
+thecontinuity equationonegets js∼∂M
+∂tand∂− →M
+∂t=a− →ML×−→MR−gµB− →BL×− →SL+...(andsimilarly
+∂− − →MR
+∂t=a−→MR×− →ML+...). Alternatively one may use the Hamilton–Jacobi equations with the
+canonical conjugate variables Szandφ(˙Sz=∂H
+∂φ,˙φ=−∂H
+∂Sz) and changing the hamiltonian
+H−→H−gµB(− →B·− →SL+− →B·− →SR) to derive the currents jsandjJ
+s.
+Note, according to Maxwell Eqs. the spin current jSshould induce an electric field Ei
+given by
+∂xEy−∂yEx=−4πµB∂t∝an}bracketle{tSz∝an}bracketri}ht= 4πjs. (60)
+32Here, for simplicity we assume no voltage and ∂tB= 0 for an external magnetic field B.
+Tunnel JunctionswithSpinandChargeCurrent: Generallyonegets bothachargecurrent
+jC=−e˙NLand a spin current js=−µB(˙Sz
+L−˙Sz
+R) and these may interfere. This occurs
+for example for SCM|SCMjunctions, where SCMrefers to nonuniform superconductors
+coexisting with magnetic order (see Larkin–Ovchinikov state), and for a (SC/FM/SC) junction
+with a ferromagnet between two superconductors. Then one get s after some algebra for the
+Josephson currents (see Nogueira et al.)
+jJ
+c= (j1+j2cos∆ϕ)sin(∆φ+2πl
+φ0Hy) (61)
+and
+jJ
+s=jssin∆ϕcos(∆φ+2πlHy
+φ0). (62)
+Here,φ0istheelementaryfluxquantum, Hyanexternalmagneticfieldiny–direction, l= 2λ+d,
+withλbeing the penetration thickness, and dthe junction thickness. ∆ ϕandφrefer to the
+phase difference of magnetism and superconductivity, respective ly. The magnetic field Hyis
+perpendicular to the current direction.
+(TSC/FM/TSC) Junction: Regarding switching of tunnel current a nd analysis of triplet
+superconductivity the tunnel junction ( TSC|FM|TSC) is of interest. Here, TSCrefers
+to triplet superconductivity. The Josephson current flows then t hrough low–energy Andreev
+states. Relative orientation of the magnetization− →Mof the ferromagnet ( FM) andd–vectors
+of the triplet superconductors, see Fig.20(b), control the tunn el current. One gets, see Morr et
+al.[3],
+jJ
+c=−e
+/planckover2pi1/summationdisplay
+i∂Ei
+∂φtanh(Ei
+kT), (63)
+whereφis the phase difference between the two superconductors and Eiare the energies of
+the Andreev states and which are calculated using Bogoliubov–de Ge nnes analysis. Thus one
+derives an unusual temperature dependence of the Josephson c urrent on temperature and even
+thatjJ
+cmay change sign for certain directions of− →M, although ∆ φdid not change.
+(SC/FM/SC)Junction: Asdiscussed recentlybyKastening et al.[3]suchjunctionsreflect
+characteristically magnetism. Again the tunnel current is carried b y Andreev states. No net
+spin current flows fromleft to right, since the spin polarized curren t throughthe Andreev states
+iscompensated by thetunnel current throughcontinuum states asmust bedue to basic physics.
+In case of strong ferromagnetism single electrons tunnel, while for weak ferromagnetism Cooper
+pair tunneling occurs. The Josephson current may change sign for increasing temperature
+without a change in the relative phase of the two singlet supercondu ctors. Of course, dependent
+on coherence length, temperature and thickness of the FM and st rength of FM one may get
+jJ= 0 for the Josephson current.
+(FM/SC/FM) Junction: It is already obvious from Fig.20(c) that in th e presence of an
+applied voltage Vjunctions (FM1|SC2|FM3) carry currents which depend sensitively on the
+relative orientation of the magnetization of the two ferromagnets . If these are directed in oppo-
+site direction (AF configuration) one gets a maximal spin accumulatio n in the superconductor
+and thus one may suppress (at a critical voltage) superconductiv ity in singlet superconductors.
+As a consequence the magnetoresistance changes. Hence, such junctions exhibit currents
+jcσ=jcσ(− →M1,− →M3,SC2,T) (64)
+33which reflect sensitively superconductivity and ferromagnetism.
+Of course, the current jcσis affected by temperature gradients ∆ Tbetween the ferromag-
+nets and resulting for example at nonequilibrium from hot electrons in oneFM. This may
+cause interesting behaviour.
+Note, also a ring structure hollow at the center (and with magnetic fl ux Φ) and outer
+ring superconducting and a neighboring inner ring ferromagnetic an d which is also next to the
+hollow center may exhibit interesting behavior in the presence of an e xternal magnetic field B.
+Theequationforthecurrentdueto ∂E/∂φ, forthecurrentdrivenbythephasedependence
+of the electronic energy, is of general significance. From it one may derive the currents induced
+in rings, discs etc. due to the Aharonov–Bohm effect if a magnetic fie ld is present.
+2.8 Quantum dot Systems
+Interesting current pattern and interferences are also expect ed for the quantum dot grain struc-
+tureshowninFig.3. Assuming amixtureofferromagneticandsuperc onducting grains(clusters,
+quantum dots) one has the hamiltonian
+H=HT−(ze2/C)/summationdisplay
+i,j(Ni−Nj)2+H′. (65)
+Here,HTdenotes the tunneling hamiltonian. This includes also spin Josephson c urrents of
+the formEJ
+ijcosϕijand also Josephson Cooper pair current contributions resulting fr om the
+phase difference ϕij=ϕi−ϕjof the phases of the S.C. order parameter of grains iandj.
+The electrostatic effects due to different charges of grains i,jare given by the second term. H′
+denotes remaining affects, for example due to ferromagnetic grain s. One expects characteristic
+differences for singlet and triplet superconductivity and interplay o f spin currents jσbetween
+magnetic grains and Josephson–currents.
+Ingrainstheinterplay ofCooper pairsize anddistance between Coo perpairsmanipulated
+by confinement is expected to exhibit interesting behavior.
+For a lattice like array of quantum dots it is of interest to study phas e ordering of the
+order parameter of the various dots i, its dependence on distance between dots, etc. (see related
+situation for superconductors when Tc∝ρs, hereρsis the superfluid density).
+As speculated for two–band superconductors and as assumed fo rscq–bits (strong) phase
+coupling of two magnetic systems (quantum dots) may cause a cova lent splitting like process
+(mode–mode coupling like process). Thus, for example, one expect s for the combined, phase
+coupledsystem N1andN3, seeFig.20, thetwocovalentlylikesplitstates |N1N3∝an}bracketri}ht1and|N1N3∝an}bracketri}ht2.
+Light irradiation causes interesting responses, since occupation o f electronic states (by
+single electrons, Cooper pairs) can be manipulated.
+Also note, for a mixture of superconducting and magnetic quantum dots on a lattice
+(and presence of an external magnetic field B) one may expect interesting quantum mechanical
+effects.
+These examples may suffice already to demonstrate theinteresting behaviour displayed by
+nanostructures involving tunneling. This holds also for currents ac ross multilayers of magnetic
+films.
+34Figure 21: Typical electron density of states (D.O.S.) results for Fenclusters, n=15. The local
+D.O.S. refers to the atoms 1, 2, 3, see Fig.1. (a) refers to local DOS , (b) to average cluster
+DOS.
+3 Results
+Characteristic results are presented for clusters, films and tunn el junctions.
+3.1 Small Magnetic Particles
+3.1.1 Single Magnetic Clusters
+Insmall clusters thelocalD.O.S.exhibits thedependence oftheelec tronicstructure onthelocal
+atomic environment of a cluster atom. In Fig.21 the local D.O.S. is comp ared with the average
+D.O.S. Results were obtained by Pastor et al. using the tight–binding H ubbard hamiltonian.
+The D.O.S. is the most important property of the electronic structu re and determines the
+thermodynamical and magnetic behavior ( µn,Mn(T)).
+In table 2 results for the bond–length dn(dbrefers to bulk), cohesive energy Ecoh(n) and
+magnetic moments µnare given for Ninclusters.
+In Fig.22 the dependence of the magnetic properties on cluster size is given. The variation
+ofµnreflectstheatomicstructure(shellstructure). Theseresults wereobtainedbyS.Mukherjee
+et al.). In accordance with the electronic calculations by Pastor et al. a s imple model is used
+assuming that the moments depend on cluster shell ( i) and local atomic coordination, both
+changing with number of cluster atoms. Such behavior results also f or other clusters, for
+exampleNinclusters. Note, the discrepancy between theoretical and exper imental results for
+larger clusters. Of course, µi→µB, bulk behavior as clusters get larger.
+In Fig.23 results for µnof Fenare given. The results were obtained using the Hubbard
+hamiltonian and a tight–binding type calculation by Pastor et al.. For Fe one expects a particu-
+lar strong interdependence of magnetism and structure (see bcc vs fcc Fe). For each cluster one
+3536Figure 22: Average magnetic moment µ(N) ofFenclusters as a function of cluster size. The
+solid and dashed curves refer to calculations by Jensen et al.using different cluster shell models
+for the magentic moment µ(N).µ(N) is calculated by averaging over the shell (coordination)
+dependent moments µiand where these are given by the local atomic coordination (varying
+withN). Crosses refer to experimental results (de Heer et al.). Note, solid and dashed curves
+refer to calculations using different models for the dependence of t he local magnetic moments
+µion atomic coordination. Of course, for larger clusters µ→µb.
+assumes a structure which yields the largest cohesion. The variatio n ofµnas a function of n is
+due to the interplay of changes in coordination number and bond–len gths (note, approximately
+Wn∝d−5, where W is the band–width and d the bond–length). Interestingly, one gets for
+the small clusters µnlarger than µb, magnetic moments which are larger than the bulk one as
+expected. The sensitive dependence of the Fe magnetic moments a nd magnetism on structure
+is of general importance for nanostructures and of material invo lving Fe and Fe under pressure.
+One may also calculate using molecular field type methods the Curie–te mperature Tcas
+a function of cluster size. Results are shown in Fig.24. Typically one ge ts thatTcincreases for
+increasing cluster size. Note, the exchange interaction J, qeffandµimay change as the cluster
+grows.
+Orbital magnetism: Clusters as other low dimensional systems (sur faces, thin films) ex-
+hibitalsointeresting orbitalmagnetism. Forexample, forNicluster saremarkableenhancement
+of the average orbital moment as compared to bulk is observed. No te, one gets for the orbital
+moment of Ni ∝an}bracketle{tL(Ni7)∝an}bracketri}ht ≃0.5µBwhile for bulk ∝an}bracketle{tL(bulk)∝an}bracketri}ht ≃0.05µB.Using the tight–binding
+Hubbard hamiltonian including the spin–orbit interaction ( Hso=−a/summationtext(− →L·− →S)iα,iβa+a, here
+α,βrefer to orbitals), Pastor, Dorantes–Davila et al.obtain for the average orbital moment
+∝an}bracketle{tLδ∝an}bracketri}htthe results shown in Fig.25. Here, δrefers to the cluster crystal axis. The important DOS
+Nδ
+imσ(ε) is determined self consistently for each orientation δof the magnetization with respect
+to structure. The s–o coupling connects spin up and spin down stat es (dependent on relative
+orientation of− →Mand atomic structure). (− →L·− →S)...are intraatomic matrix elements.
+The results show that the orbital magnetic moment is an important c ontribution to the
+magnetic moment of clusters and quite generally to the one of nanos tructures of transition
+metals (TM). ∝an}bracketle{t− →L∝an}bracketri}htdepends sensitively on band filling. Also note, for example, ( Lz−Lx), etc.
+reflect magnetic anisotropy. For Ni the orbital moment aligns para llel to the spin moment.
+37Figure 23: Dependence of average magnetic moment of Fenon cluster size. Note, the de-
+pendence of the results on the relaxation of the atomic structure , see calculations using an
+electronic theory by Pastor et al.. For each n a structure with largest cohesion is assumed. For
+Fe–clusters one expects a sensitive dependence on structure an d possibly a particular strong
+interdependence of magnetism and structure. The vertical bars refer to experimental results for
+the depletion factor and which is assumed to be proportional to the average magnetic moment,
+at least approximately.
+Figure 24: The Curie–temperature TC(M(Tc) = 0) as a function of number of cluster atoms.
+The calculations by Jensen et al.assume two different cluster structures.
+38Figure 25: Orbital moment per atom ∝an}bracketle{tLδ∝an}bracketri}htfor a pentagonal bipyramid transition metal model
+cluster of 7 atoms as a function of d–state filling (band–filling) nd. Bulk n.n. neighbor atomic
+distancesaretaken. δreferstothemagnetizationdirectiontakenalong(principal Cnsymmetry)
+cluster axis ( δ=z: full curves, δ=x: dashed curves, for details see Pastor, Dorantes–Davila
+et al.).Note, full and open circles refer to calculations using different elec tron energies ǫiσ.
+Miescattering: Fordetermining magnetisminsmall clusters besides S tern–Gerlachdeflec-
+tion experiments (as performed by W. de Heer et al.) Mie–scattering could be used. Extending
+the usual Mie scattering theory to magnetic clusters one calculate s from Maxwell equations
+with spin orbit coupling, coupling the electromagnetic field and the clus ter magnetization, and
+with a dielectric function εof tensor form the Mie backscattering profile. Characteristic dif-
+ferences are obtained for Ni, Co, Fe. For example, magnetic Mie sca ttering is important for
+spherical clusters with radius approximately larger than 6 nm for Fe and 10 nm for Co.
+Generally− →Mhas a weak effect on the electromagnetic field. This is strongest whe n size
+of cluster is of same order as the wavelength of the electromagnet ic field (k−1). In view of
+the weak coupling one may treat the perturbation due to cluster ma gnetization− →Min lowest
+order (with respect to the off–diagonal element εxy=aλsoM+...of the dielectric tensor εij
+which is frequency dependent). The tensor εijmay be fitted using magnetooptics (Kerr–effect,
+ellipticity etc.).
+Results obtained by Tarento et al.are given in Fig.26. Note, the backward scattering
+and the angle dependent scattering profile reflect magnetism. Mag netic effects are maximal
+when magnetization is parallel to y, see Mie scattering configuration , Fig.26(a). Note, for
+angleθ=π/2 the backscattering intensity I(M=0) is very small and then the ch anges due
+to the cluster magnetization are relatively large. In Fig.26(b) ε0=−1.569+i5.58 andεxy=
+−0.059+i0.122isused. Note, inFig.26(c)weuseforthedielectric function εo=−4.592+i7.778
+andεxy=−0.1613−i0.0733.
+Mie scattering exhibits also characteristic differences regarding th e angular dependence
+of the backscattering intensity for Cr, Fe, Co, and Ni.
+39Figure 26: Angular dependence of Mie backscattering: (a) Mie–sca ttering configuration for
+magnetic clusters (Co clusters of radius 2600 ˚A). The incident plane wave propagates along
+z–direction and electric and magnetic fields are polarized along x– and y–direction. In Figs.
+(b),(c) different values for the dielectric tensor εijare used, see text. Results for the angle–
+dependent backscattering intensity are given, see analysis Taren toet al.. The solid curves refer
+to M=0, the dashed ones to magnetic clusters with− →Malong y–axis, and the angle to the
+polar one in the x–z plane. The axis units are arbitrary. In (c) the tw o dashed curves refer to
+magnetization taken along the x– and y–axis. (The field frequencies are such that /planckover2pi1ω=5eV in
+Fig.(b), and 3.5eV in Fig.(c))
+403.1.2 Ensemble of Clusters
+The behavior of an ensemble of magnetic clusters is expected to refl ect magnetic anisotropy of
+the cluster and cluster rotation. An external magnetic field Hwill align the magnetic clusters
+whichmagnetizationpointsinthedirectionoftheeasyclusteraxis. T hisissetbytheanisotropy
+axis of the cluster. Note, to align the magnetic clusters the pinning o f the cluster magnetization
+to the easy axis has to overcome. This should be reflected then in th e dependence of the cluster
+ensemble magnetization on an external magnetic field.
+Tblcharacterizes the energy which pins the cluster magnetization to it s easy axis. There-
+fore, different behavior is expected for the ensemble magnetizatio n< µZ>for temperature
+T >T bl(Langevin behavior) and T <T bl.
+In Figs. 27(a) and 27(b) results are given for the magnetization of a system of clusters.
+Note, in a magnetic field Hcluster anisotropy characterized by the blocking temperature Tbl
+andHinterfere. For temperatures T > T blthe internal anisotropy needs to be overcome to
+align the cluster magnetization via the external field.
+Results shown in Fig.27 give the magnetization observed in Stern–Ger lach experiments
+for an ensemble of clusters in a magnetic field H. Obviously magnetic anisotropy within the
+cluster determines the magnetic behavior. The temperature Tblcharacterizes the pinning of
+the magnetization to the easy axis of each cluster. Thermally activa ted depinning occurs for
+T >T bl(see calculations by Jensen et al.). Clearly, for temperatures larger than the blocking
+one Langevin type behavior is expected. For lower temperatures o ne needs a certain strength
+of the external magnetic field ( µH≥kT) for unpinning the cluster magnetization from its easy
+axis.
+Figure 27: Results for cluster ensemble: The component of the ave rage cluster magnetization
+in z–direction. (a) Comparison for T > T blandT < T bl, whereTblrefers to the blocking
+temperature. (b) Comparison with experimental results for Conclusters, see de Heer et al.and
+Bloomfield and others. Note, here Tbl<T.
+Quantumdot lattices: Typical results for a latticeof quantum dots oranti–dots areshown
+in the following figures. Of course, one expects that the density of statesN(E), generally if
+magnetism is involved spin dependent, reflects characteristically th e nanostructure.
+In Fig.28 results for the DOS oscillations are given which demonstrate their dependence
+on dimension and geometry of the nanostructure. For a sphere ma inly paths with t=1 and p=3
+and 4 contribute. Their interference yields a beating pattern. For circular rings path 3 is most
+important (see Fig.13) and note larger discrepancy between classic al and quantum–mechanical
+calculations. For circular discs note Ai∼k−1/2( in contrast to spheres with Ai∼k1/2).
+Approximately thespin upandspin downDOSaresplit by themolecular fi eldheffincase
+of ferromagnetic nanostructures. It is of interest to study the interplay of superconductivity
+41and magnetism and magnetic field B.
+In Fig.28 the DOS results obtained from the semiclassical Balian–Bloch –Stampfli the-
+ory are compared with quantum–mechanical calculations. The oscilla ting part of the DOS
+due to the interference of the dominant electronic paths is shown f or various nanostructures.
+The quantum–mechanical calculations were performed using Schr¨ odinger equation and Green’s
+function theory
+N(E,..) = (γ/π)/integraldisplay
+dE′N(E′)
+(E−E′)2+γ2. (66)
+Of course, the parameter γmust be choosen as usual such that it does not affect the structu re
+in the DOS.
+Note, in case of magnetism one has a spin dependent DOS Nσ(E) and to simplify one
+may assume that spin up and spin down states are split by the molecula r fieldheffwhich may
+include an external magnetic field B.
+Also hollow particles (see coated nanoparticles, for example) can be treated similarly as
+rings. This is of interest, for example, to study confined and bent 2 d electronic systems.
+In Fig.29 results are given for the oscillating part of the D.O.S. asa fun ction of energy and
+external magnetic field for a lattice of anti–dots which repel the ele ctrons. The magnetic field
+Bchanges the orbits and thus their interference. As a result DOS os cillations occur. Note,
+only a few closed electron orbits (1,2,3, ...) give the dominant contribution to the electronic
+structure, the DOS, see theory by Stampfli et al.and Eq.(19) and Eq.(21). The magnetic field
+causes paths deformation and phase shifts. The flux due to the ex ternal field Bisφ=SB
+and for orbits 1, 2, 3 Sis approximately independent of Bfor small field. The cyclotron orbit
+radius isRc=k
+Band we assume for simplicity 2 Rc>(a−d) andRc≫R. Then,Rc∼1
+Band
+the DOS depends on (d/a) and ∆ N∼cosSB. This controls also height of oscillations (note
+S∝R2
+c∝1
+B2).
+For increasing external magnetic field Bthe oscillations in the DOS change. Interestingly,
+the contributions of some orbits may be nearly eliminated. Oscillation p eriod inBdecrease for
+increasingBand decreasing lattice constant a.
+Note, an internal molecular field due to magnetism is expected to affe ct the DOS similarly
+as the external magnetic field. Note also, the above oscillations in th e DOS result from the
+confinement (closed electronic orbits) and additional structure m ay result from detailed atomic
+structure yielding the well known electronic shell structure in clust ers etc.
+Note, for orbit 4 (see Fig.) and similar ping pong orbits one has cos( SB)≈1. Of course,
+for increasing scattering potential U the amplitude and interferen ce changes ( Rc∼1/B). For
+decreasing field Bdephasing occurs and must be included.
+For increasing radius the results for thin rings may be similar to the on es for planar
+surfaces of thin films. For thin films one may perform the Balian–Bloch type calculations as
+sketched in the figure below.
+3.2 Thin Magnetic Films
+Thin films are of central importance for engineering magnetic mater ial on a nanoscale. Thin
+transition metal ( TM ) films, for example, exhibit magnetic propertie s of interest for many
+technical applications like magnetic recording. In the following typica l results are given which
+show the dependence of film magnetism on the atomic structure, to pology of the film and
+on film thickness. Magnetic anisotropy is very important. Magnetism is characterized by the
+42Figure 28: Dependence of the DOS oscillations on dimension and geome try of a mesoscopic
+system obtained by Stampfli et al.using an extension of the Balian–Bloch theory. Results
+for (a) sphere, (b) disc and (c) ring are given. The electrons are s cattered by the potential
+U. This potential may scatter spin dependently. The dashed curve s refer for comparison to
+quantum–mechanical calculations.
+43Figure 29: Oscillations of the D.O.S. as a function of magnetic field B for the orbits 1, 2, 3 due
+to scattering by repelling anti–dots, see calculations by Stampfli et al.. The inset shows the
+lattice of anti–dots. Note, the scattering may be spin–dependent and the density of states is
+spin split. Note 2 Rc≥(a−d) is assumed.
+Figure 30: Sketch of the important orbits for thin films. Here, one h as to include transmis-
+sion (T) and reflection ( R) coefficients at the corners of the paths. Magnetic films yield spin
+dependent DOS. Note T and R are spin dependent at interface Cu/C o.
+44magnetizationasafunctionoffilmthickness, anisotropy, bytheor ientationofthemagnetization
+at the film surface and by the Curie–temperature Tc. Nanostructured films exhibit depending
+on film growth conditions (controlling the film morphology) character istic magnetic domain
+structures. Thendependingonthedistancesbetweenthedomain svariousmagneticinteractions
+controlling the global film magnetization come into play.
+3.2.1 Magnetic Structure
+The magnetic properties of thin ferromagnetic films have been stud ied intensively. Due to
+anisotropylong–rangeferromagneticorderoccursevenforultr athin, quasi 2d–films. Competing
+anisotropic forces determine the orientation of the magnetization relative to the atomic lattice.
+We use the hamiltonian given in Eq.(16) with a local anisotropy
+Hanis=−Σi(K2cos2θi+K4cos4θi+Kssin4θi). (67)
+Note, forT >0 the in–plane anisotropy Ksgets important. After some standard algebra (see
+Jensenet al.) one finds the energy per spin. One gets for
+1. uniform magnetic phases:
+Eu(θ) =−(9/2)J−K2cos2θ−K4cos4θ−Kssin4θ+Eo(cos2θ−1/3),(68)
+whereθdenotes the angle between film normal and magnetization, and for
+2. stripe domains with wall width band domain periodicity 2 L:
+Edom(θ) =−J/2(q−π2/bL)−K2(cos2θ(1−b/L)+b/2L)
+−K4(cosθ4(1−b/L))+3b/8L−Ks(sin4θ(1−b/L)+3b/8L)+...(69)
+Here,θdenotes the canting angle of the domain magnetization along the z–axis (θ= 0).q= 4
+for a square monolayer. The domains have the periodicity 2L along th e y–direction and have
+an uniform magnetization along the x–direction, inside the domains th e magnetization is along
+±z–direction. This domain configuration is assumed for simplicity. One ge ts rich magnetic
+behavior for thin films. Magnetic anisotropy is important.
+Typical results for the magnetic structure are given in Fig.31. Gene rally one gets inter-
+esting phase diagrams, see Jensen et al.[2, 11].
+Multilayers of magnetic films reflect how the magnetization of neighbo ring films interfere,
+affect each other, act like a molecular field on the magnetization of a p articular film.
+In Fig.32 results are given for multilayer films Cu—Ni—Cu and Co—Cu—Ni. N ote,
+one finds for the Curie temperature Tc(Co)> Tc(Ni). Clearly, the Co film causes that the
+magnetization of the Ni film is present at higher temperatures. This results from interlayer
+exchange coupling.
+The results shown in Fig.33 demonstrate the dependence of Tcon film structure. Obvi-
+ously, interesting characterization of magnetic films is also given by t heir Curie–temperature.
+3.2.2 Reorientation Transition of the Magnetization
+The reorientation transition of the magnetization at surfaces (an d interfaces) is very important
+regarding applications, but also regarding understanding of magne tic anisotropy. Such a tran-
+sition may be driven by temperature, film thickness and film topology ( and at nonequilibrium
+45Figure 31: Magnetic structure of thin films obtained at T= 0 due to exchange coupling and
+magnetic anisotropy. Phases I,IIrefer to stripe domain structure with perpendicular and
+canted magnetization, respectively. Phases III,IVrefer to uniform magnetization with canted
+and in–plane magnetization, respectively. ( Eois the demagnetization energy). The anisotropy
+constantsK2,K4andKsare given in Eq.(49), their physical significance is clear. The dashed
+curves indicate what happens if only uniform phases are considered .
+Figure 32: Magnetization of a Co– and Ni–film separated by 3ML Cu–film , see inset Fig.. The
+experimental results (from XMCD) are indicated by squares and cir cles. The indicated shift of
+the Ni Curie–temperature results from interlayer exchange coup ling between Ni and Co film.
+46Figure 33: Curie–temperature Tcas a function of film thickness N and morphology. (a) Results
+obtained using MFA for f.c.c. films. (b) Tcdependence on film topology: (1) Layer by layer film
+growth. (2) Results refer to rough films and those obtained by ran dom deposition of atoms.
+The dashed curves refer to Bragg–Williams mean field–theory. The s tars refer to experimental
+results for f.c.c.–Co on Cu(111). (c) Homogeneous ferromagnetic films assuming different film
+structures, see Jensen et al..
+47Figure 34: The magnetic reorientation transition. (a) Transition M⊥−→M/bardbl. The solid
+curve gives the magnetization of an uniform (single domain) film. occu rring for increasing film
+thickness and for Fe |Cu films. (b) Free–energy change ∆ F=F(M⊥)−F(M∝bardbl) per spin and
+for increasing temperature. Magnetic anisotropy is temperature dependent and this causes the
+reorientation transition at TR. A fcc(100) film is assumed, see Jensen et al.. (c) Phase diagram
+for the orientation of the magnetization is controlled by the anisotr opy parameter K⊥
+2(T,N)
+andK⊥
+4(T,N), see Eq.(49). Note, in the coexistence phase the energy minima of the states
+with perpendicular ( ⊥) and parallel ( ∝bardbl) magnetization are separated by an energy barrier. The
+P.D. results from minimizing the free–energy ( K⊥
+4refers toM⊥).
+by hot electrons). Magnetooptically the reorientation transition is characterized by the linear
+responseχijor the nonvanishing elements of the nonlinear susceptibility χijl, for example.
+In the following typical results for the reorientation transition of t he magnetization are
+presented. In Fig.34 results are given (a) for thickness induced re orientation transition in
+b.c.c. Fe films,(b) for temperature induced transition at TR, and (c) for the dependence of the
+phase diagram (PD) on anisotropy constants, K4,K⊥
+4andK/bardbl
+4are included. The free–energy
+∆F=F(T,θ= 0)−F(T,θ=π/2) is calculated with F=−ΣiK2i(m(T))cos2θ+.... Here,i
+refers to the film layer and θto the angle between film normal and direction of magnetization,
+see Jensen et al.[2, 7]. Note, ∆ F <0 indicates a perpendicular magnetization and ∆ F >0 an
+in plane one.
+3.2.3 Magnetooptics
+Magnetism of thin films and multilayers thereof is well determined by th eir magnetooptical
+behavior. In particular SHG sensing sensitively structural symmet ry, surfaces and interfaces,
+48Figure 35: Magnetooptics of thin films (xCu/Fe(001) involving quant um well states (QWS)
+due to confinement. As a consequence of QWS presence the SHG is s trongly film thickness
+dependent. SHG involving QWS exhibits characteristic SHG oscillations . (a) QWS in thin
+films, the parity of the wave functions is indicated by (+) and (-). En ergies of QWS along
+Γ→Xdirection of BZ are given. Black dots refer to a 6ML film and black and g rey dots
+to a 12ML film. The parity of the states is indicated. (b) SHG involving a s intermediate
+state an occupied QWS below the Fermi–energy, the final state is a F e d–state. Transitions
+contributing dominantly to SHG are shown. One gets corresponding SHG oscillations as a
+function of film thickness. (c) Interference of SHG from surface and interface is negligible (
+for example (xCu/Fe(001)). The important SHG optical transition s involving QWS as final
+states are indicated. Transitions (i) are possible for Fe films with 6ML and 12ML, (ii) only for
+films with 12 ML Fe. Thus, one gets SHG oscillations having a 6ML period. For details see
+discussion by Luce and Bennemann ( Nonlinear Optics, Oxford ).
+49is very useful for studying magnetic properties of nanostructur es. Magnetic thin films cause
+characteristic SHG reflecting magnetism, since the nonlinear susce ptibilityχijldetermining the
+light intensity I(2ω) of SHG depends on magnetization− →M.
+As usually one splits the susceptibility in parts even and odd in Mand then the magnetic
+contrast
+∆I(2ω)∝I(2ω,M)−I(2ω,−M) (70)
+yields sensitively the magnetic behavior. Resonantly enhanced SHG t ransitions of the ferro-
+magnet which states are spin split are a sensitive measure of magnet ism. Note, the interference
+of SHG from the film surface and interface (film/substrate) is of int erest, in particular if mag-
+netism is different at surface and interface, see Fig.11.
+In table 1 characteristic features of SHG response from thin films a re given. We refer to
+weak interference of SHG when χi≫χsand to strong interference when χi≈χs. Note, the
+wave vector− →kselectivity is controlled by band–structure. For the film system Cu/ Fe/Cu the
+dipole matrix elements with d–states cause χi≫χs. For Au/Co/Au films no Co d–states are
+involved and therefore χi≈χs. The periodic appearance of QWS for increasing film thickness
+may cause oscillations in SHG.
+In thin films (which may be described with respect to their electronic s tructure by po-
+tential wells) one gets QWS (quantum well states) and correspond ing contributions to MSHG
+(magnetic second harmonic light). As supported by experiments QW S in thin films may con-
+tribute strongly to SHG (quasi resonantly enhance SHG). Since th e occurrence of QWS is
+dependent on film thickness, one gets QWS involving characteristic o scillations in SHG as a
+function of film thickness. These oscillations depend of course on ligh t frequency , magnetism,
+parity of the QWS, inversion symmetry of the film and interference o f SHG from surface and
+interface.
+If interference of nonlinear light from surface and interface is impo rtant, then SHG re-
+sponse depends on QWS parity. As a consequence only a period twice as large as for the
+linear Kerr effect optics appears. If this interference is not so impo rtant, then SHG exhibits
+typically two oscillation periods. For details of such behavior see Luce , Bennemann ([12]) and
+also discussion given previously, see Fig.18 etc..
+SHG involving QWS is shown in Fig.35. In Fig.35(a) thin film QWS and their par ity are
+sketched. The Figs. refer to xCu/Fe(001) system. For other sy stems the analysis is similar.
+Fig.35(a) shows the energies of the QWS. The parity is indicated by (+ ) and (-). Note, at
+the Fermi energy a parity change occurs. The spin polarized d–sta tes of Fe are also shown.
+In Fig.35(b) dominant SHG contribution resulting from an occupied QW S below the Fermi–
+energyisillustrated(finalstateofSHGtransitionisad–stateofFe ). InFig.35(c)SHGinvolving
+unoccupied QWS is shown. Note, states along Γ →Xdirection in the Brillouin zone (BZ)
+are most important. Interference of surface and interface SHG is negligible. Dominant SHG
+transitions are indicated: (i) are possible for 6ML and for 12ML of Cu . This gives oscillations
+with a period of 6ML. (ii) possible only for 12ML. Thus, one gets two diff erent oscillations
+(which are frequency dependent, at k1,k2in the BZ).
+Note, forχiχs−→ −1 one gets a cancellation of the thickness dependent contribution
+to SHG due to QWS. If χiandχsbecome comparable parity of QWS gets important. For
+example, for a dominant QWS close to the Fermi–energy one gets osc illations only for one with
+odd parity, see discussion by Luce et al.[12].
+This shows the interesting magnetooptical behavior of thin films.
+503.2.4 Nanostructured Thin Films with Magnetic Domains
+General remarks: In the following the magnetic properties of gene rally irregular nanostructures
+at equilibrium and nonequilibrium are discussed. Generally one observe s that the magnetic
+properties of nanostructured films depend on the film growth cond itions. During growth irreg-
+ular film structures with varying island sizes, shapes appear. Compe ting interactions between
+islands (exchange, dipole,..) occur. Magnetic anisotropy controls th e formation of domains,
+their size and shape and magnetic relaxation of the domains. The rela xations determine the
+transition from nonequilibrium to equilibrium. In view of the complex beh avior one may use
+Monte Carlo simulations. To describe the transition from isolated islan ds to connected ones
+correlations of magnetic relaxation (of neighboring islands) must be taken into account. For
+example cluster (island) spin flips will lead to a much faster magnetic re laxation than single
+spin flips.
+Main questions are (1) what is the domain structure, (2) what is the time dependent
+approach of the film magnetization towards equilibrium, (3) what are the controlling forces.
+If only short range interactions (exchange ones) are active then one expects that percolating
+atomic structures (coverage θ≥θp) are necessary for long range global magnetic ordering.
+Ordering for coverages below θpmight indicate dipolar interactions.
+The time–dependent structural and magnetic behavior of films dur ing growth reveals
+many interesting properties and in particular the important interpla y of atomic structure and
+formation of magnetic domains and global film magnetization. As the fi lm growth the size of
+the domains, its surface roughness and the orientation of the dom ain magnetization changes.
+Magnetic relaxation occurring typically on a ps to ns time scale causes in thin films a time
+dependent magnetizationwhich approaches only atrelatively long tim es (via thermal activation
+etc.) the equilibrium magnetization. Depending on the domain size and d istances between
+the domains exchange or magnetic dipole coupling may dominate and ca use corresponding,
+different relaxation. Thus, the approach of inhomogeneous ultrat hin films towards equilibrium
+magnetization and its dependence on film topology requires careful studies.
+Important parameters controlling film behavior are the blocking tem peratureTbl(Tbl∼
+KN/k Bln(texpΓo)∼5K), forN≃1000 atoms resulting from magnetic anisotropy and the
+percolation coverage θp. While magnetic ordering due to exchange coupling occurs for T <
+Tc∼J, long–range magnetic dipolar coupling causes ordering for T≤Tdip∼(Nµ)2/R3(which
+for N=1000 atoms gives about 5K). Rrefers to the distance between domains. As mentioned
+due to its complexity analysis requires generally Monte Carlo simulation s, see Jensen et al.
+[8, 11]. In Fig.36 magnetic domain relaxation is sketched and the typica l energy barriers
+against reversal of the direction of an island (domain) magnetizatio n are shown.
+The magnetic relaxation is calculated using a Monte Carlo method and a Markov master
+equation
+dP(x,t)/dt=−ΣΓ(x→x′)P(x,t)+ΣΓ(x′→x)P(x′,t),
+wherePgives the probability of a spin state x=S1,S2,...at timet. Γ(x→x′) denotes the
+transitionrate. ∝an}bracketle{tM(t)∝an}bracketri}ht ∝ΣiMi(t),i=timestepstakingintoaccountpossiblycoherent spinflips
+of connected domains, see Brinzanik, Jensen et al.[11]. The Markov equation in combination
+with the M.C. statistical methods yields the time dependent non-equ ilibrium magnetization
+and its relaxation towards the equilibrium one. Depending on the dens ity of the domains,
+their distance, correlated relaxation of neighboring domains occur s. This is demonstrated in
+Fig.36(c) where results obtained assuming cluster spin relaxation vs . single spin relaxation are
+51Figure36: Sketchof(a)coherentlyrelaxingconnecting magneticd omains. (b)relaxingdomains
+illustrating model assuming coherent relaxation and successively sin gle domain relaxation used
+by Jensen et al.(CSF: Coherent spin flip of neighboring domains, SSF: Single domain sp in flip).
+(c) energy barrier of a domain on which a molecular field heffacts for magnetization reversal.
+The barrier energy ( ǫi=Eb
+i/KN) controls the reversals of the domain magnetization and
+results from magnetic anisotropy ( ∼K) and (interdomain, interlayer) molecular field hi
+eff. (d)
+Test results for the relaxation of the magnetization using CSF appr oximation.
+52Figure 37: Magnetic relaxation in thin films for different film thickness a nd temperature. (a)
+Time dependent long range ordering in case of exchange and dipole co upling and for varying
+temperature. Note, θ < θ p. The relaxation starts from fully aligned domain magnetization.
+At low temperatures ( T∼1K) a stable magnetization due to dipole interactions is obtained.
+(b) Equilibrium magnetization as a function of temperature for θ < θp. Here,θpdenotes the
+percolationcoverage( θp= 0.9ML). (c)Equilibriummagnetizationasafunctionoftemperature
+for different anisotropy (K) and film thickness. Note, the differenc e between nonequilibrium
+and equilibrium magnetization.
+53Figure 38: Magnetic relaxation dependence on film structures, on g rowth mode, layer by layer
+vs. islandgrowth. Note, dependence onanisotropy Kandtemperature. (upperFig.) Forisland
+growth the magnetization depends strongly on magnetic anisotrop y (K). (lower Fig.) Mag-
+netization in case of layer by layer growth depends strongly on exch ange interactions between
+islands.
+54Figure 39: Dependence of Curie–temperature Tcand blocking temperature Tbon film thickness
+(coverageθ).θpdenotes the percolation coverage. Note, a semilogarithmic plot is us ed. The
+curve for the ordering temperature due to dipole coupling neglects for simplicity domain wall
+energy. As θ−→θp, the exchange interactions begin to dominate, for θ < θpdipole coupling
+plays a role and yields a small Tc. The nonequilibrium behavior is mainly due to magnetic
+anisotropy.
+shown.
+For a domain structured film the film magnetization is given by
+− →M∝ΣimiNi∝an}bracketle{t− →Si∝an}bracketri}ht, (71)
+where− →Mpointsalongtheeasyaxisand ireferstothe i−thislandwith mialignedmagneticmo-
+ments. Here, oneaveragesoverthemagnetizationdirectionsoft hedomains. (ThekineticMonte
+Carlo method gives after equilibration for the average magnetizatio n∝an}bracketle{tM∝an}bracketri}ht ≈(1/R)ΣR
+r=1Mr(t)).
+In Fig.37 magnetic relaxation of thin films is shown (for details see Jens enet al.) [11].
+Results obtained below and above percolation coverage θpare compared. Note, (for θ > θp)
+the strong influence of single island anisotropy Kon relaxation. For increasing connectivity
+of the islands faster relaxation results from exchange interaction s between islands. Note, the
+temperature and film thickness dependence of the relaxation.
+The MC results demonstrate that faster relaxation occurs for co rrelated spin flips (CSF),
+as expected of course. As also expected different behavior occur s below and above percolation
+coverage (θp).
+In Fig.38 it is shown how the relaxation of the magnetization depends o n the atomic film
+structure. The dependence of the remanent magnetization of a fi lm on its growth mode, layer
+by layer or island type one is given. Note, the dependence of the res ults by Jensen et al.on
+anisotropy and temperature.
+In Fig.39 ordering in thin films is characterized. Depending on the dens ity of domains
+exchange and dipolar coupling act differently. Clearly the percolation coverage is an impor-
+tant control–parameter for the magnetic behavior and range of m agnetic interactions. At low
+coverage dipole coupling may dominate and then ordering occurs at lo w temperatures.
+In Fig.40 the dependence of magnetic behavior, of domain structur e on film morphology
+is shown. Results were obtained by Jensen et al.using a kinetic M.C. simulation and Markov
+equationfortherelaxationofthemagneticdomainsaregiven. Filmst ructures resulting froman
+55Figure 40: Magnetic relaxation in domain structured thin films using Mo nte Carlo type cal-
+culations performed by Jensen et al.[11].λrefers to different spin clusters and θpto the
+percolation coverage. CSF refers to correlated cluster (island) s pin flips. For comparison also
+results are given assuming single spin flips (SSF). The atomic nanostr uctures corresponding
+to the magnetic domain structures are also shown and were obtaine d using an Eden growth
+model, for details see Jensen et al.. Regarding nanostructures white and grey areas refer to
+magnetic domains and black areas to uncovered surface of the thin film.
+56Figure41: (left) Filmnanostructures using Eden type growthmode l, (black: uncovered surface,
+grey (light) to first layer atoms, dark to first layer atoms); (right ) corresponding resulting
+magnetic domains and their dependence on film thickness. (black: un covered surf., grey: two
+oppositemagneticdirections). Onlydomainsatthesurfacearemar ked, seeresultsbyBrinzanik
+et al..
+57Figure 42: Average magnetic domain size as a function (a) of covera ge (film thickness) and
+(b) temperature. The film growth model is described in the text (se e Jensen et al.). The
+percolation threshold is θp∼0.9ML.
+Eden growth model and resulting magnetic domains are shown [8, 11 ]. Regarding calculations
+an area of 500 ×500 lattice constants is taken which contains about 1250 islands. λrefers to
+different spin cluster sizes.[2, 8, 11]
+Of course, magnetic ordering in thin films depends sensitively on topo logy and film thick-
+ness and the dominant magnetic coupling between domains. Results p resented demonstrate
+this.
+The results presented in Fig.41 for the domain size (and results on th eir roughness and
+surface) are very important regarding understanding of magnet ic behavior of irregular thin
+films and applications.
+In Fig.41 results are given for the magnetic domain structure in thin fi lms having different
+thickness. Of course, the magnetic domain structure changes an d the size of the domains
+increases as the film thickness increases.
+In Fig.42 results for magnetic domains and their size dependence on fi lm thickness and
+temperature are given. These were obtained using the previously d iscussed analysis by Jensen
+et al.. Note, as a function of temperature one may get a maximum in the do main size. The
+reason is that for low temperatures domain growth is hindered by en ergy barriers and at higher
+temperatures thermal activation disintegrates domains.
+Note, generally for increasing film thickness the domains which are fir st isolated begin
+to merge and form larger connected areas, still markedly affected by the atomic nanostructure
+of the film. The roughness of the domains is affected by the nanostr ucture and decreases for
+thicker films and for increasing temperature. Of course, this is exp ected on general physical
+grounds.
+3.3 Tunnel Junctions: Magnetic Effects
+Nanoscaled tunnel junctions offer interesting physics in particular regarding quantum mechan-
+ical effects and spin dependent currents, separation of charge a nd spin currents and their in-
+terdependence. Also one may use such nanostructures ( FM|M|FM),M= normal state,
+superconducting state, etc.) as interesting fast switches and ma gnetoresistance devices. Note,
+58Figure 43: Tunnel junction ( TSC|FM|TSC), where FM and TSC refer to a ferromagnet
+and triplett superconductor, respectively, φto the phase of the superconductor: (a) Josephson
+currentIJas a function of temperature (changing sign), (b) IJas a function of α, the angle
+between magnetization− →Mand direction normal to the current. Note, IJmay be switched
+between zero and a finite value.
+the Josephson tunnel current is carried by Andreev–states, se e Kastening, Morr et al.. In
+the following some examples are discussed. Note, different behavior may result dependent on
+timesτCP,τs,andτtreferring to Cooper pairs, spin diffusion and tunneling transport tim e,
+respectively.
+For a tunnel junction
+(TSC|FM|TS) (72)
+where TSC denotes a triplet superconductor and FM a ferromagne t, one gets interesting be-
+havior of the Josephson current jJas a function of temperature, αandθ. The angles αand
+θdetermine the direction of the magnetization of the ferromagnet a nd the direction of the
+TSC–order parameter, see Fig.20(b) for illustration. The Andreev states are determined from
+Hψj=Eψjusing Bogoliubov–de Gennes method.
+As shown by Fig.43 the Josephson current may change sign as a func tion of temperature
+and may be switched between zero and finite value upon varying the a nglesα,θ. The un-
+conventional change of sign of IJ(without change of phase) for increasing temperature results
+from the changing occupation of the Andreev states and from the factor∂Ei
+∂φin the expression
+for the tunnel current. Note, this sign change is different than th e one observed for singulet
+superconductors with a ferromagnetic barrier in between. The be havior ofIJfor rotating mag-
+netization, as a function of α, suggests to use such junctions as switching devices. Note, for
+increasingφthe current changes more drastically.
+Of particular interest is the behavior of IJas a function of phase φ(for T=0) shown
+in Fig.43(b). Then only the states Ei<0 are occupied. For (1) M∝bardbldL,R,α∝ne}ationslash=π/2, the
+ferromagnet (FM) couples Andreev states, and (2) for− →M⊥− →dL,Rthe Andreev states are not
+coupled by the FM and spins are well defined.
+Clearly the above junction would be a sensitive probe to detect triple t superconductivity,
+see results by Morr et al.[3].
+59Figure 44: Josephson current IJatT= 0 for a junction (SC/FM/SC) as a function of phase φ
+of the superconductor for several values of the scattering str engthgandzof the magnetic and
+nonmagnetic potential, respectively ( g= 0,1/3,2/3,1 from left to right, z= 0,1/3,2/3,1 from
+bottom to top), see results by Kastening, Morr et al.. Note, the ferromagnet is approximately
+represented by a potential barrier.
+Another tunnel system is one involving singlet superconductors,
+SC|FM|SC . (73)
+The FM is approximately represented by a barrier potential scatte ring spin dependently
+the electrons ( grefers to the magnetic scattering strength, zto the non–magnetic one). Again,
+depending on nonmagnetic and magnetic scattering of the tunnelling electrons the Josephson
+current may change sign as a function of temperature Tand furthermore as a function of the
+relativephase ofthetwo superconductors, seeFig.43andFig.44. ( Note,IJ=I↑+I↓,Is=I↑−I↓,
+the Josephson current IJis solely carried by the Andreev states, the total spin current Is=
+0, since spin current through Andreev states is canceled by the on e through the continuum
+states).
+Forz≫gone getsIJ= (e∆0//planckover2pi1)sinφ
+2Z2and non magnetic current is dominated by Cooper
+pairs. Forg≫zone hasIJ=−(e∆0//planckover2pi1)sinφ
+2g2and current is carried by single electrons. Note,
+then the phase shift by π. Results assume for the thickness of the tunnel junction dto be
+smaller than the coherence length, otherwise the situation is more c omplicated.
+Note, the sign change of the Josephson current as a function of t emperature shown in
+Fig.45 results not from a transition of the junction from a π–state at low temperature to a
+0–phase at high temperature, but from a change in the population o f the Andreev states (while
+the relative phase between the superconductors remains unchan ged).
+Also interestingly, the total spin polarization of the two supercond uctors ground state
+changes from ∝an}bracketle{tSz∝an}bracketri}ht= 0 to∝an}bracketle{tSz∝an}bracketri}ht= 1/2.
+Of course, as mentioned already interesting results are also expec ted for (FM1/SC/FM 2)
+junctions. The tunnel current depends on the superconducting state, singulet vs. triplet. Also
+a spin current is expected for τs> τt, times refer as before to times without spin flip and
+tunneling time. Singulet superconductivity may block tunneling. For a triplet superconductor
+the angle between Cooper pair angular momentum− →dand− →Mmay control the tunneling.
+60Figure 45: Josephson current IJof a tunnel junction (SC/FM/SC) as a function ofT
+Tcfor
+φ=π/2 and nonmagnetic scattering strength z= 0.5 and several values of g, the magnetic
+scattering strength.
+Transport between quantum dots: An interesting case of electro n transport (electron
+pump model) between two quantum dots involving possibly Coulomb– an d spin–blockade (see
+Hubbard hamiltonian with spin dependent on–site interaction) is illustr ated in Fig.20(a). Driv-
+ing the current with a pulsed (polarized) external field to overcome an energy barrier one gets
+spin–dependent charge transport with v. St¨ uckelberg oscillation s due to bouncing forth and
+back of the electrons between the two quantum dots.
+In Fig.46 results are given for the photon assisted tunneling betwee n two quantum dots.
+The applied electromagnetic field is given by V(t) =V0cos(ωt). Note the dependence of the
+charge transfer on the duration of the applied light pulse and the Ra bi oscillations due to the
+bouncing back and forth of the electrons. The Rabi oscillation freq uency is Ω = 2 ωJN(E=
+V0//planckover2pi1ω). HereJNis a Bessel function of order N, where N refers to number of photo ns absorbed
+to fulfill resonant condition N/planckover2pi1ω=√
+∆ε2+4ω2. Different frequencies are used which cause
+resonant absorption of one, two and three and possibly more phot ons. Of course this affects
+the charge transfer. The time resolved analysis of the occupation of the electronic states shows
+that system if connected to reservoirs acts as electron pump. Be fore action of the external
+electromagnetic field the initial state is n1= 1 andn2= 0. After the pulse is over oscillations
+disappear and one gets again the initial state via transferring one e lectron to the right quantum
+dot and the left reservoir donating one electron to the left quantu m dot, see Garcia et al.[3].
+Spin dependent quantum dot electron states cause spin currents .
+Of interest is also the coherent control of photon assisted tunne ling between quantum
+dots and its dependence on the shape of the light pulse, see Grigore nko, Garcia et al.[3]. The
+shape of the external electric (or magnetic pulse) may be optimized to get a maximal charge
+or spin current between the quantum dots connected to two meta llic contacts.
+Note, these results are also of interest for Fermion or Boson syst ems on optical lattices,
+its dynamics and interaction with external fields. Optical manipulatio n of molecular binding,
+in particular Boson formation induced by an electromagnetic field is an option. In intense
+fields nonlinear behavior may be particular interesting. Furthermor e, on optical lattices one
+may study in particular the interplay of magnetism and lattice struct ure, the transition from
+local, Heisenberg like to itinerant behavior of magnetism. Important parameter for the occur-
+rence of ferromagnetism (antiferromagnetism) are Coulomb inter actions and particle hopping
+(kinetic energy) between lattice sites, and possibly also spin–orbit c oupling. Note, as indicated
+61Figure 46: Charge transfer between two quantum dots exhibiting R abi (v.St¨ uckelberg) oscil-
+lations due to bouncing electrons between the quantum dots. (a) D ependence of transferred
+charge on pulse duration of external field and for different freque nciesω1,ω2,ω3which cause
+resonant tunneling by absorption of one, two and three photons, respectively; (b) Time depen-
+dent occupation of quantum dot states, see results by Garcia et al.. Note, for spin dependent
+quantum dot states (magnetic quantum dot), see Fig.20(a), the r esults depend on the light
+polarization and may involve spin dependent currents.
+by Hund’s rules ferromagnetism could occur already for a Fermi–liquid with strong enough re-
+pulsive Coulomb interactions, quasi irrespective of the lattice stru cture, since spin polarization
+(together with the Pauli principle) may minimize the repulsive Coulomb in teractions. In case
+where a.f. dimerization of neighboring lattice sites yields Boson forma tion and resulting BEC
+condensation optical influence, also of the magnetic excitations, is an important option.
+InFig.47thepossibilityofmanufactoringopticallyanultrafastswitch ingdeviceissketched.
+Note, in general changing in nanostructures the magnetization ge nerates in accordance with
+Maxwell–equations light.
+Via hot electrons fast changing temperature gradients may yield int eresting and novel
+behavior of tunnel junctions. Possibly in such way Onsager theory can be tested for nanostruc-
+tures.
+For further discussion of the interesting physics realized by tunne l systems, the interplay
+of spin and charge currents (see continuity equation for the spin d ensity) and of accompanying
+light (see Maxwell equations), one should study the literature, see Nogueira, Bennemann et al.
+[3] and Bennemann [13].
+Currents in Mesoscopic Structures: As mentioned already for mes oscopic structures like
+discs, quantum dots and rings, for example, one might get interest ing topological effects and
+persistent currents driven by phases resulting from the Aharono v–Bohm effect [14]. One may
+derive such currents induced by a magnetic field B from j=∂∆F
+∂φ, where F refers to the free–
+energy. From symmetry considerations one gets ∆ F∝cosBS, where BS is the flux enclosed
+by a ring for example, and for the current
+j∝sinBS. (74)
+Here, the phase shift ∆ φ=±BSresults from encircling the center ( of the ring, disc, etc.)
+62Figure 47: Illustration of a tunnel junction which may act as an ultra fast switching device due
+to optical excitation of (hot) electrons. Magnetization changes a s the electronic temperature
+changes.
+clockwise or counterclockwise. Note, as in the case of supercondu ctors flux quantization may
+occur. Dynamics due to B(t) or M(t) is of course reflected in the ind uced current. Also electron
+scattering causes dephasing etc. and affects the current.
+The induced currents due to the Aharonov–Bohm effect with details of their oscillations
+can be calculated from
+j=∂△F/∂φ=−e//planckover2pi1/summationdisplay
+∂Ei/∂φtanh(Ei/kT) (75)
+and using the Balian–Bloch type theory developed by Stampfli et al.for calculating Ei(φ) and
+taking into account clockwise and anticlockwise encircling the center . For narrow rings one
+expects that mainly orbits 1, 2, 3 are important. A magnetic field B ca uses the phase shift
+∆φ= ∆φ1+∆φ2due to path deformation and flux, respectively (∆ φ2=BS±, where S refers
+to the area enclosed by the orbit and ±to clockwise or conterclockwise encircling of B ), see
+Aharonov–Bohm effect. Thus, one gets for the current induced f or example by the field B into
+a narrow ring the expression
+j∝ΣtΣpatpsinφnsinφtp(...)+... (76)
+Here,φn=BS0denotes the quantized flux in units of hc/eof the orbit characterized by t
+andp(describes Aharonov–Bohm effect) and φtpresults from the path deformation due to the
+fieldB. Note, the factor (...) causes rapid oscillations described by expone ntials, see Stamfli
+it et al.. Interference of contributions from different paths may yie ld a beating pattern for
+the current oscillations. Generally the oscillations resemble those dis cussed for the DOS, see
+Figs. Since the coupling of the induced currents is given by terms jµAµ, whereAµrefers to
+the vector potential of B, one gets generally that the induced current decreases quadrat ically
+with the ring radius R ( j∝R−2). [14] The fine structure of the current oscillations reflects the
+quantization of the enclosed flux. In accordance with le Chatelier pr inciple the magnetic field
+associated with the induced current should counteract the exter nal fieldB.
+Of course, dephasing expected for strong scattering of the elec trons and large rings (and
+discs) weakens and possibly destroys the current. Also for ferro magnets (Ni, Fe, ...) oneexpects
+spin currents (due to DOS–effects etc.). For spin currents also sp in orbit coupling may play a
+role, see Aharonov–Casher effect. [14]
+As noted a ring shows similar behavior as a (repulsive) antidot covere d by a thin film.
+Of interest would be also to study a ring of graphene, due to the rem arkable properties of
+63graphene, or conducting nanotubes. To study interplay of electr ic and magnetic fields (Maxwell
+equationsatactioninnanostructures) double–ringsareofintere st. Generallythediscussion and
+results for superconducting cylinders, rings, squids, tunnel jun ctions, can be frequently applied
+somewhat to the topologically closed mesoscopic structures in case of relatively long free path
+of the electrons (see dominant paths in Balian–Bloch–Stampfli theo ry). This demonstrates the
+interesting properties of nanostructures.
+4 Summary
+Magnetic effects in nanostructures have been discussed. Typical properties of clusters, cluster
+ensembles, single films and film multilayers and microscopic tunnel syst ems are presented. Due
+to the reduced dimension and system size external fields may chang e sensitively the behav-
+ior. Spin dependent transport on a nanoscale, involving optical man ipulation, temperature
+gradients etc., see corresponding Onsager theory, offers intere sting possibilities. Also (strong)
+nonequilibrium behavior of nanostructure needs be studied.[13]
+Acknowledgements
+Essential general help in particular regarding organisation of the r eview, technical assistence
+and critical reading was given by Christof Bennemann. Furthermor e, discussions and many
+results by P. Jensen, M. Garcia, P. Stampfli, G. Pastor, W. H¨ ubne r, K. Baberschke, M. Kulic,
+and many others are gratefully acknowledged. I am particular grat eful to F.Nogueira for ideas,
+discussions and help. This review was supported by D.F.G through S.F.B .
+References
+[1] Pastor G.M., Bennemann K.H. (1999), Magnetism in Cluste rs in Metal Clusters, Editor
+Ekardt W. (Wiley, New York, 1999); Pastor G.M. in Atomic Clus ters and Nanopart-
+cles, p. 335 (EDP Sciences, Les Ulis/Springer, Berlin, 2001 ); Pastor G.M., Ph.D. thesis,
+F.U.Berlin (1989); Sugano S., Microcluster Physics, Sprin ger Series in Mat. Science 20
+(1991); Tomanek D., Mukherjee S., Bennemann K.H., Phys. Rev . B28, 665 (1983);
+Jellinek J., Theory of Atomic and Molecular Clusters, Sprin ger Series in Cluster Physics
+(1999), also Proceedings of ISSPIC–conferences
+[2] Jensen P.J., Bennemann K.H., Magnetic Structure of Film s, Surface Science reports 61,
+129 (2006); Brinzanik R., Jensen P.J., Bennemann K.H., Stud y of nonequilibrium mag-
+netic domain structure during growth of nanostructured ult rathin films, J. of Magnetism
+and Magnetic Materials 238, 258 (2002); Brinzanik R., Ph.D. thesis F.U. Berlin (2003),
+Monte Carlo study of Magnetic nanostructures during Growth ; Jensen P.J., Bennemann
+K.H., Magnetism of 2d–nanostructures, in book Magnetic Mat erial (Editor A.V.Narlikar,
+Springer, 2004)
+[3] Kastening B., Morr D.K., Alff L. and Bennemann K.H., Charg e Transport and Quantum
+Phase Transition in Tunnel Junctions, Phys. Rev.B 79, 144508 (2009); Kastening B.,
+Morr D.K., Manske D. and Bennemann K.H., Novel Josephson Effe ct in Tunnel Junc-
+tions, Phys. Rev. Lett. 96, 47009-1 (2006); Nogueira F. and Bennemann K.H., Spin
+Josephson effect in ferromagnetic/ferromagnetic tunnel ju nctions, Europhysics Lett. 67,
+64620 (2004); Takahashi S., Imamura H., Maekawa S., Spin Imbal ance and Magnetoresis-
+tance of Tunnel Junctions, Phys. Rev. Lett. 82, 3911 (1999); Garcia M., Habilitation
+thesis, FU Berlin (1999); Speer O., Garcia M., Bennemann K.H . Phys. Rev. 62, 2630
+(1996)
+[4] Tarento R.J., Bennemann K.H., Joyes P., van de Walle J., M ie Scattering of Magnetic
+Spheres, Phys. Rev. E 69, 166 (2004)
+[5] Stampfli P., Bennemann K.H., Electronic Shell Structure in Mesoscopic Systems: Balian–
+Bloch type theory, Z.Physik D 25, 87 (1992); Phys. Rev. Lett. 69, 3471 (1992); Tatievski
+B., Diploma thesis, Balian–Bloch type theory for Spheres, D iscs, Rings and Anti–Dot
+Lattices (F.U. Berlin, 1993); Tatievski B., Stampfli P. and B ennemann K.H., Analytical
+results for the electron shell structure in Mesoscopic Syst ems, Ann. der Physik 4, 202
+(1995)
+[6] Jensen P.J., Bennemann K.H., Magnetic Properties of Fer romagnetic Clusters in Stern–
+Gerlach Deflection Experiments, Comput. Mat. Science 2, 488 (1994); Jensen P.J.,
+Mukherjee S., Bennemann K.H., Theory for Spin Relaxation in Small Magnetic Clus-
+ters, Z. Phys. D bf 21, 349 (1991); Dorantes–Davila J., Guira do R.A., Pastor G.M.,
+Orbital Magnetism, in Physics of Low Dimensional Systems (E ditor J.L.Moran–Lopez,
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+mann K.H., Transition Metal Clusters, Physica B 149,22 (1988); Phys. Rev.B 40, 7642
+(1989); Dorantes–Davila J., Dreysse H., Pastor G.M., Physi cal Rev. Lett. 91, 197206
+(2003); Pastor G.M., Dorantes–Davila J., Bennemann K.H., S pin Fluctuations in T.M.
+Clusters, Phys. Rev. B 70, 1 (2004)
+[7] Jensen P.J., Morr D.K. and Bennemann K.H., Reorientatio n Transition in Thin fer-
+romagnetic films, Surface Science 307, 1109 (1994); Jensen P.J. and Bennemann K.H.,
+Temperature InducedReorientation of Magnetism at Surface s, Solid State Communic. 100,
+585 (1996)
+[8] Jensen P.J. and Bennemann K.H., Theory for Domain Wall Wi dth, Ann. Physik 2, 475
+(1993)
+[9] Jensen P.J., Dreysse H., and Bennemann K.H., Magnetism a nd Curie–temperature in thin
+transition metal films, Europhys. Lett. 18, 463 (1992), Surf. Sci. 269–270 , 627 (1992)
+[10] Moran–Lopez J.L., Bennemann K.H., Avignon M., Magneti sm in transition metals, Phys.
+Rev. B23, 5978 (1981); Liu S.H., Itinerant Magnetism, Phys. Rev. B, 17, 3629 (1978)
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+52, 16012 (1995) and Jensen P.J. in Ref.2; Brinzanik R., Jensen P.J., Bennemann
+K.H., Nonequilibrium Magnetic Domain Structure during Fil m Growth, J. Magn. Magn.
+Mat.238, 258 (2002)
+[12] Bennemann K.H., Nonlinear Optics in Metals (Oxford Uni versity Press, 1998)
+[13] Bennemann K.H., Ultrafast Dynamics in Solids, Ann. Phy sik18, 480 (2009)
+65[14] Butiker M., Imry I., Landauer R., Phys.Lett. 96A, 365 (1983); Riedel E.K., v.Oppen F.,
+Phys. Rev. B 47, 15449 (1993); Balatsky A.V., Altshuler B.L., Phys. Rev. Le tt.70, 1678
+(1993)
+66
\ No newline at end of file
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+Dusty Magnetohydr odynamics in S tar Forming Regions.
+S. VAN LOO1, S.A.E.G. F ALLE2, T.W.HAR TQUI ST1, O.HA VNES3,4,5, and 
+G.E.MO RFILL4
+1 School of Physics and Astronomy , Universi ty of Le eds, Leeds LS2 9JT , United  Kingdom
+2Departmen t of Applied Mathe matical Sciences,  Universi ty of Leeds,  Leeds LS2 9JT , United 
+Kingdom
+3Departmen t of Physics and Technology , University of Tromsö, 9037 Tromsö, Norway
+4Max-Planck-Institu t für extr aterrestr ische Physik, 85740 Garching,  Germany
+5University Cent er (UNI S) , 9071  Longye arbyen, Svalbard,  Norway.
+Abstract. Star formation  occurs  in dark  molecu lar regions  where the number  density  of 
+hydrogen  nucle i , nH, exceeds  104 cm-3 and the fraction al ionization is 10-7 or less.  Dust grains 
+with sizes  ranging  up to tenths of microns  and perhaps  down to tens of nanome ters conta in 
+just under  one percen t of the mass.   Reco mbinat ion on grains  is importan t for the remov al of 
+gas phase ions, which  are produced  by cosmic rays pene trating the dark reg ions.  Collisions of 
+neutrals  with charged grains  contribu te signifi cantly to the coupling  of the magnet ic field to 
+the neutr al gas.  Consequent ly, the dynami cs of the grains must  be included  in the 
+magnetohydrodyn amic mode ls of large scale collapse,  the evolut ion of waves  and the 
+structures of shocks import ant in  star format ion.
+11.  Introduction
+Trumpler  [1] in 1930  and Stebbins,  Huffer and Whitford  [2,3]  in 1934  and 1939   reported  the 
+results  of photome tric studies  that demonstrated  that dark “holes”  in the Galaxy , noted  by 
+Herschel close  to 150 years  before,  are due to obscurat ion by interstell ar dust.  In the first part 
+of his import ant May 1941  paper  on the charge carried by interste llar dust,  Spitzer [4] wrote 
+“The  presence  of dust particles implies that the physica l state of such  a medium may be 
+somewhat  different  than that discus sed in the classic al work  by Eddington,  in which  only 
+atoms  were  assumed  to be abundant”.   Despite Spitze r’s early  insight,  the first papers  on the 
+effects  of dust on the magnetohydrodyna mic behav ior of interstell ar matter appeared  only 
+about thre e decad es ago.
+In 1997  Hartquist,  Pilipp and Havnes  [5] published  an introduc tory review of work on dusty 
+plasma  in interst ellar clouds  and star forming  regions.   While  numerous  advances  have  since 
+occurred,  we refer  the reader  to that article for a more  detai led presentat ion of many  of the 
+basic  physica l processes  and equat ions import ant for the area.   Though  we ment ion later work 
+here,  as well as give a qualit ative  explanat ion of some  of the key conc epts,  we have  not 
+attempted  to be comprehensive.   Our curren t interest  in multi-fluid  shocks  in star forming 
+regions has guided  our choice  of emphasis, but we do men tion some other progress.
+Section  2 contains  a brief  descrip tion of the processes  establ ishing  the fraction al ionization in 
+dark regions  in dense  cores,  molecul ar structures  that are progeni tors  of stars;  recomb ination 
+on grains  is significan t.  Section  3 provides  a descrip tion of the role that dust plays  in 
+ambipo lar diffusion,  the motion  of ions relative to neutra ls resul ting from  magnet ic field 
+gradients,  in dense  cores  and protostell ar format ion.  Section 4 contains a brief  summary  of 
+the results  of an early  study  of the effects of dust on the damping  of Alfvén  waves  in dense 
+cores.   Section  5 is a selec tive short  review  of work  on the effects  of dust on the structure  of 
+shocks   assumed  to propagate  perpend icularly to the magne tic field in star forming  regions. 
+Section  6 contains  a simil ar review  of work published  prior  to 2009  on shocks   propagat ing 
+2oblique ly to the upstrea m magnet ic field.   Section  7 concerns  more  recent  work  on such 
+shocks.  Sect ion 8 conc ludes the  paper . 
+2. The Fractional  Ionizat ion in Dark S tar Forming Regions.  
+Stars  form  in dense  cores,  which are molecu lar condensa tions  embedd ed in more diffuse 
+molecu lar gas that is many  tens or more  times less dense  than the dense  cores.   Most,  if not 
+all, of the more  diffuse gas surrounding  dense  cores  is in transluc ent clumps  making  up Giant 
+Molecular  Clouds  (GMCs)  or GMC complex es (see,   e.g., section 2 of [5] for a review). An 
+nH (i.e. the  number  density of hydrogen nucl ei) range of 104 – 105 cm-3 is typi cal of dense cores 
+in which  solar -like stars form,  but an nH range  of 106 – 107 cm-3 is typical of regions  in which 
+high-mass  stars, those  that conta in at least  8 solar masses, are born  [6].  Of course,  star 
+format ion leads  to values  of nH greatly  exceed ing these,  but most  of this paper  conc erns 
+phenomena  occurring  in the  nH range of 104 – 107 cm-3.
+Oppenhe imer and Dalgarno  [7], Gail and Sedlmayr  [8], Elmegre en [9], Draine  and Sutin  [10] 
+and Umebayashi  and Nakano  [11] are amongst  those  who  have  contr ibuted  to the 
+developm ent of our understanding  of the ioniz ation  structur e and the calculation  of the grain 
+charge in dark star forming  regions.  Many  of the considera tions  have  much  in common  with 
+those addressed by Shukla  and  Mamun[12].
+Cosmic ray ioniza tion of H2 at a rate of the order  of 10-17 – 10-16 s-1 leads to H2+ , which  reacts 
+with H2 to form  H3+.  H3+ in turn reacts with neutral  species including  O, H2O and CO (the 
+most  abundant  gas phase  mole cule other  than H2 in most  circumstan ces) to form   molecu lar 
+ions.  Dissociat ive recomb ination  is a major  mech anism  for removing  mole cular  ions in dense 
+cores,  but its effectiv eness  is moderat ed if metals,  like magnesium  and sodium,  are not too 
+deplet ed onto  grains.   Char ge transfer  between  molecul ar ions and such  metals produces 
+meta llic ions.   The gas phase  process  removing  them is radiative recombin ation,  which  is 
+3many  orders  of magni tude slower than dissocia tive recombinat ion.  The prim ary mechan ism 
+removing  meta llic ions is recomb ination on grains.
+At the typical tempera tures  of dense  cores  of the order  of 10 K and typica l core densities, 
+most  grains  will carry  one negativ e charge if nearly  all of the grain material is in grains of 
+sizes  of around  0.1 μm. Considerab le uncert ainty  concern ing  the grain  size distribution 
+exists.   Certa inly in more  diffuse clouds  many  grains  of much  smaller  size exist.   However , in 
+dense,  dark regions  the format ion of ice  mantles by the depletion  of gas phase  species  may 
+lead to almost  all grains having  sizes of about 0.1 μ m.
+The grain  charge probabil ity distribu tion function  and the fractional  ionization must  be 
+calcu lated self-consisten tly.  Some of the authors  cited  above  and many  others  have  done  so. 
+Fig.7  of [5] shows results  for the fraction al abundan ces of electrons,  gas phase  ions,  grains 
+carrying  a single  negat ive charge, grains  carrying  a single  positive   charge and neutral  grains 
+as funct ions of  nH ranging  from  104 to 1014  cm-3.  All grains were assumed to be  spherica l with 
+radii of 0.1 μm and to have  a fraction al abundan ce relative to nH of 4x10-12.  The cosmic  ray 
+induced  ioniza tion ra te and  the te mperatur e were tak en to be  10-17 s-1  and 10 K , respect ively.
+For nH  up to about  109 cm-3, the fractional  ionization  drops  roughly  as nH-½  from  3x10-8.  The 
+fraction al abundances  of ions and electrons  are nearly  equal  and almost  all grains  are 
+negativ ely charged.  At nH above  1010 cm-3 and below  1012 cm-3  most  grains  are neutral and 
+the fractiona l abundanc es of ions and of negat ively  charged grains  are nearly  equal,  while the 
+fraction al abundanc e of electrons  is lower .  At nH above   1012 cm-3, the fractional  abundanc es 
+of nega tively charged and posit ively  charged grains  are nearly  equa l, while the fraction al 
+abundance  of ions is lower .
+3. Ambipolar  Diffusion in De nse Cor e Collapse.
+There  has been  considerab le deba te abou t the relative roles,  in dense  core formation  and 
+evolut ion, of magne tically regul ated,  gravi tation ally driven  collapse  and non-linear 
+4magnetohydrodyn amic processes  that occur  even  in the absence   of gravi ty [13].   Ambipolar 
+diffusion  is import ant for the relev ant non-linear  magnetohydrodyn amic processes  [14] and 
+grains  must  be included  in the treatment  of its effect on the those  processes.   Howeve r, the 
+bulk of the studies  on the importan ce of grains  for ambipol ar diffusion in star format ion have 
+been  performed  for a picture in which  a dense  core is supported  by a combina tion of thermal 
+pressure and magne tic forces.   In this picture collapse  occurs  as magneti c force  drives  the 
+charged particles to move  relative to the neutr als, thus reduc ing the magnet ic contr ibution  to 
+the support.
+The timescale  for the decrease  in the magn etic support  depends  upon  the force  per unit 
+volume  due to collisions  between  neutra ls and charged species.  Baker  [15],  Elmegr een [9], 
+and Nakano  and Umebay ashi [16] were  the first to recognize  the importan ce of the 
+contribu tion of collisions  between  neutra ls and charged grains  to this force  per unit volume 
+and to ca lculate its ef fect on the ion  slip veloc ity and ambipo lar dif fusion/ collapse tim e scale.
+We summar ize the condi tions  [5] under  which  grain-neutr al collisions  signifi cantly affect the 
+veloci ty of electrons  relative to the neutra ls in a region under going  gravitational  induced 
+collapse  as a consequen ce of ambipolar  diffusion.   We assume that the charges on all grains 
+are equal  and do not fluctuate.  Ω is the magn itude  of the grain  gyrofrequency , νjk  is the 
+frequency  at which  a single  particle of species  j  under goes   collisions  with  partic les of 
+species k.  The subscripts  can be n, g  or i indica ting neutrals,  grains and ions,  respect ively .  If 
+νng << Ω , the condi tion is
+                   νng >> νni 
+If  νng >> Ω ,  the condi tion is
+                νni2
+νng2νgn2
+2<<1
+5In a series  of papers  Ciolek  and Mou schovi as,  e.g. [17],  and Tassis and Mouschovias, 
+e.g.[18],  have  presented the results  of the most  thorough  studies  of ambipol ar-diffusion 
+regulat ed, gravi tationally  driven  collapse.   They have  considered the collapse  of thin disks. 
+They  performed  self-consisten t calcu lations of the fractional  ionization  and probabi lity 
+distribut ion function  for the charges on grains.   Though  all grains  were  taken  to  have  the 
+same  size,  fluid  equa tions  were  included  for grains  of each charge.  For the low tempera tures 
+involved  only three  grain char ges needed to  be considered.
+4.  The Effects of D ust on Alfvén Waves.
+Of course,  the most  famous  paper  on the effects  of dust on waves  is that of Rao,  Shukla  and 
+Yu [19]  on dust  acousti c waves.   By now the literature on waves  in dusty plasmas  is immense. 
+Here,  we consider  results  of an early  study  [20] of the propagation  of Alfvén  waves  in dusty 
+star forming  regions  in order  to draw  atten tion to some  of the relevan t physics.   Waves are 
+import ant as they contribu te to the widths  of the observed  line profiles  in star forming  regions 
+and their non-line ar evolution  may contribute  to the product ion of dense  cores.   Kulsrud  and 
+Pearce  [21] considered the damping  of Alfvén  waves  in a weakly  ionized dustless plasma  due 
+to ion-neutra l friction.   An analysis restricted  to linear waves  propaga ting paral lel to the large 
+scale  magnet ic fie ld leads to an approxi mate dispersion rel ation  of
+           k2=4πρi
+B02ω21νin
+νni−iω
+where k is the compl ex wavenumber , ω is the real angul ar frequency , B0 is the strength  of the 
+large-scal e magn etic field   and ρi is the  mass density of ions.
+6As grain- neutral  collisions  in star form ing regions  had been  recogni zed to be import ant for 
+ambipo lar diffusion,  the initiation of studies  of  their role in wave  propagation  was a natural 
+step.
+One set of resul ts from  a linear  analysis  [20] of Alfvén  wave s propagat ing parall el to the 
+large-scal e  magn etic field in a dense  core were  for nH = 2x104 cm-3, T=20 K and B0=10-4 G. 
+The dust grains  were  assumed  to contain one percent  of the mass  and have  uniform  radii of 
+0.1 μm and mass  density  1 g cm-3.  The number  densit ies of ions and electrons  were  taken  to 
+be 10-3 cm-3.  Results  for the interesting  range  of ω of 10-4 to 10-5 y-1 show  a reduct ion in the 
+damping  rate of a factor  of 2 or 3 for cases  in which  the dynami cs of neutral  and negat ively 
+charged grains  are included  comp ared to those  in which  they are excluded.   The reduct ion 
+would have  been  much  larger for differen t reasonable  assumptions  for the ion and electron 
+number densit ies.
+Van Loo  et al. [14] have   found  that the nonline ar develop ment  of waves  with ω in about  this 
+range  is sensitiv e to the degree  of collisiona l coupling  between  charged and neutra l fluids.   In 
+the near future,  the inclusion  of at least  two grain  fluids  in treatments  of the nonlinear 
+developm ent of w aves in  star forming  regions will  be expect ed.
+The effect of the charge fluctuat ions of grains  on the dissipation  of the waves  was reveal ed in 
+the initial study  of Alfvén waves  in dusty media [20].  The influenc e of charge fluctuations on 
+wave damping  has been  studied   sub sequen tly in  many cont exts.
+5.  Perpendicular  shocks.
+Once stars form,  their outflows  affect the environ ment.   The interac tion of the outflows  with 
+ambien t molecular gas leads  to the forma tion of shocks  that may lead to the compression  of 
+inhomogene ities to trigg er more  star form ation  or possibly  to the disruption  of them  result ing 
+7in the termin ation  of stellar birth.   The shocks  also alter the chemi cal and dust contents  of the 
+star forming  regions, in part  by sputtering  dust grains.
+Two key papers  on the natur e of shocks  in star form ing regions  are those  by Draine  [22] and 
+Draine, Rober ge and Dalgarno [23]. 
+Shock s propagat ing perpendicu lar to the large-scal e magnet ic field are fast-mode  shocks.   We 
+will assume  that the magne tic pressure  is signifi cantly greater  in the  preshock  medium than 
+the thermal pressure.   Thus,  the fast-mode  and Alfvén  speeds  are about  equal.   Two upstream 
+Alfvén  speeds  are relevan t.  One is the Alfvén  speed  of low frequency  waves,  VAl, which  is 
+determ ined by the upstream  magne tic field strength,  B0 , and the sum of the mass  densit ies of 
+all species.  The other  is the Alfvén  speed  of high frequency  wave s, VAh , which depends  on B0 
+and the mass density of char ged species that are well-coupled  to the magne tic field.   If VS , the 
+shock  speed,  exceeds  VAh the shock  will have   jumps  in all fluid  parameters.   However , for a 
+range  of VS < VAh but >VAl, the flow variab les in all fluids  are continuous.   We consider  only 
+such C-type  shocks.   In a frame  comoving  with  the C-type  shock,  the charged particles 
+decel erate  from  the shock  veloci ty further  upstream   than the neutra ls do.  This creat es a 
+precursor .  Requir ing the gradien t in the magneti c pressure to be comparab le to the drag force 
+per unit volume  on ions due to collisions  with  neutra ls, one can find that the precursor 
+thickness is roughly  
+             Δ=VAl2
+2νniVS
+Δ  is the lengthsca le over which dissipation occurs,  and the smaller Δ is the hotter the gas 
+becomes.   If grains  were  present  and well- coupled  to the magne tic field, the expression  for Δ 
+would contain the sum of νni and νng , rather than  νni alone.   However , collisions  tend to 
+decouple  the grains from  the magne tic field, an effect considered  by Draine [22] and Draine, 
+Rober ge and Dalgarno  [23] whose models  of perpendicul ar shocks  are reliab le for cases  in 
+which the  preshock value  of nH  is ≤  106 cm-3.
+8At higher  densit ies a self consisten t calcul ation  of the average  charge on the grains  and the 
+fraction al ioniz ation  is required  to obtain  accurat e results for  νni and for the effect that grains 
+have  on the shock  structure.   Pilipp,  Hartquist  and Havnes  [24] includ ed such a calculation  in 
+each  of their  models of perpendi cular  shocks.   They  also includ ed   fluid  equations  for the 
+grains  rather  than adop t a simpl er approxi mation to calculate the effects of dust [22, 23]. This 
+led them  to discover  a run-away  proces s operat ing in perpend icular shocks  for which  the 
+preshock  value of nH is 107 cm-3 or highe r.  At such  densities,  the ratio  of n(e)/|Zg|ng drops 
+below  unity  within  the precursors  of sufficiently fast shocks.  n(e)  is the electron number 
+density , |Zg|e is the magni tude of the average  charge carried  by grains  and ng is the number 
+density  of grains.   Once this ratio drops  below  unity , |Zg| begins  to drop  as there  are 
+insuf ficien t electrons  to continue  charging the grains.   Assume  that the shock  propagates  in 
+the x-direc tion and the magnet ic field  is in the y-direct ion and that Ω/νgn , the grain  Hall 
+parame ter, is small.   Then  the grains  separate  from  the other  charged particles sufficien tly to 
+generat e an x-compon ent of the ele ctric field with a  magnitud e given  approxima tely by
+         
+               Ex=mgνgn∣vnx−vgx∣
+∣Zg∣e
+Here  mg is the mass  of a grain.   This creat es an ion drift veloc ity componen t in the z-direc tion 
+with a magn itude  of cEx/By .  As |Zg| drops, this component  of the drift velocity increases. 
+This causes  an incre ase in the rate at which a grain  exper iences  collisions  with ions,  which 
+leads to  a further drop in  |Zg|.  Hence, a runaway  occurs.
+The recent  work  of Guill et, Jones and Pineau  des Forêts  [25, 26] represents  a signif icant 
+developm ent in the modeling  of perpendicu lar shocks  in dusty  star forming  regions.   In their 
+work on C-typ e shocks  [25],  they adopt ed a hybrid  approach.   The gaseous  species were 
+described  as fluids, whereas the trajectories  and charges of many  individua l dust particles 
+were  followed.   The treatmen t gave  self-consistent  results  for the grain charges, gas phase  ion 
+9and electron  abundan ces and dynam ics.  Though  the fluid  approach  of Pilipp  et al. [24] is 
+valid  for a wide  range  of parame ter space,  the Guille t et al.  [25] method  is required  in cases 
+in which  dust gyroradi i are comparab le to or larger than the scales  on which  variations  of 
+parame ters vary in  shocks.
+6. Steady-S tate Models of Oblique Shock s.
+Pilipp  and Hartquist  [27] adopt ed a fluid description  of grain  dynami cs in studies  of steady 
+shocks  propagat ing obliquely  to the upstrea m magnet ic field  in dusty  star forming  regions. 
+We will assume  that a shock  propagates  in the x-dire ction  and that the upstream magn etic 
+field  has x and y componen ts but its z component  is zero.   They  found  that grain-neutr al 
+collisions  lead to a rotation of the magneti c field in a C-type  shock  precursor  around  the x-
+direct ion.
+The following  considera tions  show why such  rotation occurs.   In the shock  frame  the z-
+component  of the electr ic field,  Ez , is VSBy0/c , where  By0 is the y-componen t of the upstream 
+magnet ic field.   Thus,  there is a componen t of ExB drift in the x-dire ction.   In the absenc e of 
+collisions,  the x-compon ents of the  ExB  drift velocities of all species are equal.   However , 
+grain-neutr al collisions  are signific ant leading  to a non-zero  x-component  of the Hall current. 
+For a steady  shock,  the equa tion of charge conserva tion requires  that the x-component  of the 
+total current  is zero.   Thus,  a current  parallel to the magneti c field having  a x-componen t that 
+cance ls the Hall  current  component  in the  x direc tion must  exist.  Use of Ampere ’s Law sho ws 
+that the curren t along  the magnet ic field gener ates a component  of the magne tic field in the z 
+direct ion.
+By integrat ing from  an upstrea m point  in the downstrea m direct ion, Pilipp  and Hartquist  [27] 
+succeeded  in finding  only  intermedi ate-mode  shock  solutions.   Such solutions  are 
+inadm issible [28].
+10Wardle [29] showed  that integra tion in the downstrea m direc tion will not yield  steady  fast-
+mode  solut ions because  the down stream state corresponds  to a saddle  point.  He found  fast-
+mode solut ions by integra ting upstream from  the downstream  state.
+Chapman  and Wardle [30] have  extended  this work and shown  that the inclusion  of PAHs 
+leads  to a drop  in the gas phase  electron abundance  and enhanc ed rotation   of the magn etic 
+field.  PAHs are nano-part icles  thought  to be abundant  in clouds  more  diffuse than star 
+forming  regions.   As mentioned  earlier, in such  regions  desorpt ion of material from  the gas 
+phase  may  result in a ll part icles gro wing  to sizes close  to 0.1 μm.
+Integrat ion in the upstream direc tion is not appropri ate if condi tions  anywhere  in a shock 
+deviat e from  equilibr ium.   After  shocked  gas has cooled,  the abundanc e of H2O, an impor tant 
+coolant  in shocked  dense  core mater ial, remains far from  its equil ibrium  value for many  times 
+the flow t ime through a shock.   Other che mical species also have abundances that are far from 
+their  equilibr ium values  for considerab le periods.   Consequen tly, the calcula tion of shock 
+structures  by integrat ion in the upstream  direc tion is inappropr iate, and the use of a time-
+dependent,  rather than a steady-state,  approa ch is necessary  to overcome  the difficulties found 
+by Pilipp  and Hartquist  [27] and explain ed by Wardle [29].   Falle [31] has deve loped  an 
+appropriat e time-depend ent approach.
+7.  Time-Depen dent Mode ls of Oblique Shock s in Dusty R egions.
+Van Loo et al. [32] have  used the techn ique deve loped  by Falle [31] to model  oblique  shocks 
+in uniform  density  medi a.  Figures  1 and 2 show  results  obtain ed for a shock  that has evolved 
+to a steady- state  structur e.  As shown  by Wardle  [29] when  field rotation is signific ant the 
+trajectory  in By , Bz phase  space  corresponds  to a spiral node  in the vicini ty of the upstream 
+fast-mode  state.   As seen from  Figures  1 and 2, the veloc ity structure  is compl icated where 
+magnet ic field rot ation  is significan t.
+11The availability of a time-dependent  code  opens  the possibility of studying  shocks  in 
+inhomogeneous  media.   Star forming  regions  conta in density  structures  on a variety  of scales 
+and, as ment ioned  in Section  4, the respon ses of such structures  to shocks  determ ine whether 
+star formation  induc es further  stellar birth  or causes  it to cease.  Ashmore  et al [33] used the 
+Falle- Van Loo et al. code  to study  obliqu e shocks  in inhomogeneous  star forming  regions. 
+Results simi lar to  those that  they present ed are  displayed in  Figure 3.
+8. Conclusion
+The major challenge  in star formation  theory  will be the incorpora tion of the effects  of dust in 
+multidimensional,  time- dependent  magne tohydrodynam ic simul ations.   The work  reported  in 
+many  of the papers  cited in this brief review  demonstrates  the exist ence of a community  that 
+appreci ates the role that dust plays  in the phenomena  invest igated   in simula tions  of the 
+dynamics  of star forma tion.   However , so far the assumption  of non-ideal 
+magnetohydrodyn amics has been  relaxed  in only a handful  of the multidimensiona l numeric al 
+studies, e.g. [14].  Hard but  interest ing work remains.
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+14Figure  1. The veloc ity structure,  in the shock  frame,  along  the shock  normal  for a steady- state 
+C-type  shock  propaga ting at 25 km/s  through  a homogeneous  mediu m with  nH =105 cm-3. 
+Each  grain  has a radius  of 0.4 μm and a mass  of 8x10-13 g and one percent of the mass is in 
+grains.  The upstrea m magn etic field  strength  is 10-4 G. The shock  veloci ty is at an angle  of 
+45owith respect  to the upstream magnet ic field.  The  blue line represents  the neutr al fluid, the 
+red line  the ion and ele ctron  fluids and the   black  line th e grain  fluid.
+15
+Figure 2. The ratio of Bz to By in the  shock  for w hich results are  shown in Fig.1.
+16
+Figure  3.  The veloci ty structure,  in the initial shock  frame,  along the shock  normal  5000  yr 
+after the steady  shock  for which results  are shown  in Fig.1  propaga tes into a region  of lower 
+density  (from  nH  = 105 cm-3  to 104 cm-3).  The blue line represents  the neutr al fluid, the red 
+line the ion  and elec tron fluids  and the bl ack lin e the  grain flu id.
+17
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